Upper limits on the size of a primordial black hole

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Pushingthelimits

With the optionally available Mobility Plus package, you can use even more features in the twion App and communicate directly with the twion wheels via the smartphone. In this way, you also receive all important running information such as distances travelled, battery charging level or calories burned. For sportier types, the running profile can be set to personal requirements or trips can be recorded and saved via GPS.
3
FEEL THE SPEED.
Faster and further: The dynamic motors accelerate the wheelchair up to 6 km/h (optionally 10 km/h*). So you can really get going and enjoy the ride. The efficient energy recovery system feeds energy back into the batteries during braking, providing a 10 % range extension for greater freedom.
So it‘s even more logical that Alber also offers its service digitally. Simply download the twion Mobility App at no cost from Google Play Store onto all android-capable smartphones and use the all-round service at any time. For example, if service is required the diagnostic checklist can help to localise the error in the wheels and, if necessary, set up direct contact to our Service Team.

D Fender Manual

WARNING: Never carry your paintball marker uncased when not on a playing field. The non-playing public and law enforcement personnel may not be able to distinguish be-tween a paintball marker and a firearm. For your own safety and to protect the image of the sport, always carry your marker in a suitable marker case or in the box in whichit was shipped.CAUTION: Paintballs may remain in the feed tube system after the loadersection is empty. Always check for paintballs in the feed tube and breech toprevent accidental discharge!3.1 3.23.3 3.4adapter). The D*Fender’s ASA is located3.5 WARNING:Insert the threaded end of the air cylinder• Remember compressed air and nitrogen systems cylinders can be extremelydangerous if misused or improperly handled. Use only cylinders meetingD.O.T., TC or regionally defined specifications.• Never disassemble your tank or tank regulator. Only a qualified and trainedtechnician should perform work on your tank and tank regulator.• Never add any lubricants or greases into the fill adapter on yourtank regulator.WARNING: Before switching your D*Fender marker On. Make sure it ispointed in a safe direction. Everyonewithin firing range should always use paintball approved eye and face protectionin the presence of live paintball mark-ers. Make sure the marker is set to “SafeMode”, before following the steps below.ing device. Remember that the ultimate safety device is you, the operator. 4.1FLASH FREQUENCY FlickerFast Flash Slow Flash Solid Glow Breech Sensor is ON, Ball is sensed Breech Sensor is ON, No Ball is sensed Breech Sensor is turned OFF Breech Sensor is malfunctioning/eyes dirty BREECH SENSOR STATUS LED SAFE COLORBlue GreenBattery level is good Replace batteryBATTERY LEVELPOSITION 321Safe Green or Blue Safe Red Semi-Auto Amber Burst Purple Full-AutoLED COLOR FIRING MODEWARNING: Check the velocity every time prior to using your D*Fender marker. The velocity can be adjusted via the Velocity Adjuster (fig 5.1) which is accessible through either side of the marker. Adjust the velocity with a 3/32” hex wrench by inserting it into the adjuster holes and rotating.velocity is achieved. A paintball specificincrease the input pressure, and outwardWARNING: Do not increase regulator pressure over 200psi or it will damage the marker.5.15.2The (2) second set screw adjusts the rearctivation point. For best results,the activation point should be set right inthe middle of the total Trigger movementIf any of the set screws are over adjusted5.31. Start with the Ramp in the OFF position(fully toward the back of barrel, Fig 5.4).Go to your fields shooting range or other 5.4LEDSolid AquaSolid PurpleSolid WhiteFlickering RedFlickering Green Solid AmberSolid BlueSolid GreenSolid RedFiring Mode Max Rate of Fire Dwell Trigger De-bounce Ball In Place Time Ramp Start Ramp Sustain Burst Shot Count Calibrate Selector SETTING # OF BLINKS 54321Semi Auto Burst Ramp Full Auto Select Fire FIRING MODEPOSITION 321Safe Green or Blue SafeRed Semi-Auto Amber Burst PurpleFull-AutoLED COLOR FIRING MODE BLINKS VALUE VALUE BLINKS # OF BPS BPS # OF 513.5614.0714.5413.0312.5212.0111.58.088.599.0109.51110.01210.51311.014BLINKS VALUE VALUE BLINKS # OF BPS BPS # OF 1920.01819.51719.01618.51518.015.02115.52216.02316.52417.0252017.5BLINKS IN MS IN MS BLINKS # OF DWELL DWELL # OF 59.069.57810.048.538.027.517.03.093.5104.0114.5125.0135.5146.06.515the user to lock the settings mode, which WARNING: Once you are done using your D*Fender it is extremely important thatyou fully unload the marker of all paintballs and compressed air. Never look down the barrel unless it’s removed from the marker.6.1CAUTION: Paintballs may remain in the feed tube system after the loadersection is empty. Always check for paintballs in the feed tube and breech to prevent accidental discharge!WARNING: Before attempting to perform any disassembly or maintenance on your D*Fender marker, make sure that all paintballs and air sources have been removed from the marker and that the regulator gauge reads 0 psi. Install a barrel blocking device, move the Selector Switch to the Safe position and powerthe marker Off.WARNING: Do not rinse the D*Fender under water, as you may damage the marker’s electronics. WARNING: Once the Side covers are removed the Eyes and Detents may be loose and can fall out of the internal assembly.Buttons and slide the Top Rail toward the rearof the marker about an inch (2.5 cm) and lift8.1WARNING: Before removing the Quick Strip Pins from marker. Confirm the marker is degassed and the gauge reads 0 psi.Push the pins with finger or hex wrench fromOnce the Quick Strip Pins are removed, thecurved end plugs correctly into the feed tubeWARNING:• Before the Internal Assembly can be removed, the Side Covers, Top Rail, Quick Strip Pins, Barrel, and Feed Ramp must be removed first.• Before removing the Internal Assembly, make sure the Eyes are still properly within the Body to prevent damaging the eye harness. Using two hands place one finger in the8.28.38.4WARNING: The Internal Assembly must be removed prior to splitting the shells. See page 15 for removal instructions.WARNING:Never carry your Marker uncased when not on a playing field. The non-playing public and law enforcement personnel may not be able to distinguish between a paintball marker and firearm. For your own safety and to protect the image of the sport, always carry your Marker in a suitable marker case or in the box in which it was shipped. When you are finished using your Marker it is important that you prepare it for storage. This will not only serve to increase the life of the marker, but will assure optimum performance on your next outing.unauthorized persons.CAUTION: Paintballs may remain in the feed tube system after the loader section is empty. Always check for paintballs in the feed tube and breech to prevent accidental discharge!11. DIAGRAMS/PARTS LISTDIAGRAM #PART DESCRIPTION SKU 1STANDARD ALU BARREL72720 2ALU BARREL INSERT- SIZE .69011523 2ALU BARREL INSERT- SIZE .68511522 2ALU BARREL INSERT- SIZE .68011521 3O-RRING BUNA-N70 DUR-018 (.739 ID)72508 4O-RING BUNA-N 70 DUR 1MM CS X 19.5MM ID72488 5APEX BARREL72721 6O-RING BUNA-N 70 DUR -021 (.926 ID)11506 7APEX WIRE SPRING11504 8APEX BODY INSERT TOP11500 9APEX HOUSING TUBE11502 10APEX ADJUSTER11505 11APEX DEFLECTOR11503 12APEX BODY INSERT BOTTOM11501 13MAIN SPRING17535 14BOLT RUBBER TIP17533 15BOLT17532 16O-RING BUNA-N 70 DUR -017 (.676 ID)17534 17BOLT GUIDE72722 18O-RING BUNA-N 70 DUR 2.5MM CS X 23.0MM ID72340 19POPPET SEAL17629 20POPPET ASSEMBLY WITH SPRING17539 21POPPET SPRING17623 22VELOCITY ADJUSTER72723 23O-RING BUNA-N 70 DUR 2.4MM CS X 10.80MM ID17538 24O-RING URETHANE 90 DUR 2MM CS X 10MM ID 17540 25DOWEL PIN .125 DIA X .37572724 26O-RING BUNA-N 70 DUR 1.5MM CS X 12MM ID 17537 27SCREW BHCS 6-32 X .250 (SMALL HEAD)17652 28RELEASE BLOCK72725 29SPRING .120OD X 0.50FL .020WD RATE 18.LB/IN72584 30RELEASE BUTTON72726 31BEARING BALL 1/8 DIA 72585 32AIR TRANSFER GASKET17530 33MANIFOLD7272735O-RING BUNA-N 70 DUR 0.8MM CS X 12.0MM ID17553 36AIR TRANSFER TUBE MALE17551 37SOLENOID 17528 38CHECK VALVE17531 39MAIN CURCUIT BOARD72728 40EYE HARNESS72729 41BALL DETENT72577 42BODY72730 43SCREW SET 10-32 X .375 CP72654 44REGULATOR DISK17591 45REGULATOR PISTON72360 46REGULATOR PISTON RETURN SPRING72395 47O-RING BUNA-N 70 DUR -008 (.176 ID)10761 48TUBE LOWER72731 49O-RING BUNA-N 70 DUR -012 (.364 ID)72490 50SCREW SET 10-32 X 1.250 CUP POINT72732 51REGULATOR MOUNT72733 52REGULATOR FILTER COVER72359 53O-RING BUNA-N 70 DUR 1.5MM CS X 6.5MM ID72509 54REGULATOR FILTER72358 55REGULATOR BODY72734 56REGULATOR SEAL72364 57O-RING URETHANE 70 DUR -012 (.364 ID)10257 58REGULATOR ON/OFF PIN72372 59REGULATOR NUT72599 60O-RING BUNA-N 70 DUR -006 (.114 ID)72489 61REGULATOR SEAL RETAINER72652 62PIN REGULATOR72363 63REGULATOR LEVER72613 64PIN WEDGE72614 65REGULATOR CAM SPRING72369 66SCREW SHSS 3/16 DIA 3/8 LG 8-32 x .25072512 67300PSI EMPIRE GAUGE72375 68REGULATOR OPP COMPLETE17597 69MAIN REGULATOR SPRING72608 70REGULATOR CAP7261572O-RING BUNA-N 70 DUR -020 (.864 ID)40920 73CATCH CUP72736 74BEARING .250 ID X .500 OD X .125 THK38803 75E-RING 38823 76SPROCKET 31074 77GEAR BOX COVER38496 78SCREW PHCS 4-40 X .625 38827 79SENSOR COVER38497 80DRIVE SENSOR72737 81O-RING BUNA-N 70 DUR -028 (1.375 ID)38820 82PULLEY GEAR38837 83MOTOR WITH WIRING HARNESS72738 84GEAR PIN - DOWEL PIN .078 DIA X .50038828 85DRIVE SHAFT72739 86NUT HEX 4-4038805 87CARRIER72740 88SCREW THCS 6-32 X .37538982 89SPRING .180OD X 1.500FL .026WD RATE 8.0LB/IN72741 90BUTTON RELEASE72742 91TOP RAIL72743 92RIGHT SHELL72744 93SIDE RAIL ADAPTER72745 94SIDE RAIL72746 95SCREW SHCS 6-32 X .437572747 96CONNECTER BOARD SCREW W/ WASHER72779 97HARNESS CIRCUIT BOARD (COMPLETE)72749 98DOOR RETAINER38938 99SPRING .114OD X .31FL .014WD RATE 9.6LB/IN38939 100DOOR SWITCH38937 101RIGHT SIDE COVER72750 102LID SPRING31025 103STANDARD LID72751 104QUICK LOAD LID 72752 105MAGNET 8MM DIA .X 4MM31009 106MAGNET 8MM DIA X 6MM38475 107BATTERY DOOR72753109HOLDER BATTERY AA38804 110BATTERY HARNESS HOLDER17823 111BATTERY HARNESS17715 112FEED RAMP BACK72755 113TRIGGER72756 114SCREW SET 6-32 X .250 FLAT POINT17523 115SPRING .180OD X .310FL .012WD RATE 3.0LB/IN 72757 116BEARING .156 ID X .3125 OD X .155572381 117SCREW SHSS 5/32 DIA 5/16 LG 6-32 X .13872758 118FEED RAMP FRONT72759 119SPRING PIN .0625 DIA X 1.00072760 120FEED RAMP TOP72761 121SCREW SHCS 6-32 X .312572762 122LID RETAINER72763 123SCREW SHCS 6-32 X .75017815 124TRIGGER GUARD72764 125SELECTOR CIRCUIT BOARD72765 126SWITCH SHAFT INNER WITH MAGNET72766 127SELECTOR SPRING72767 128NUT HEX 6-32 .25 WIDE x .092 THK72768 129FOREGRIP RH72769 130FOREGRIP LH72770 131SCREW SHCS 6-32 x .87572771 132COMPLETE SHELL SPRING PIN (SHORT)72772 133LEFT SIDE COVER72773 134POWER BUTTON17712 135LIGHT PIPE17710 136SCREW FHCS 4-40 X .50017650 137SELECTOR SWITCH OUTER72774 138SCREW BHCS 6-32 X .25017653 139SCREW FHCS 6-32 X .75072775 140SLING MOUNT72776 141LEFT SHELL 72777 142COMPLETE SHELL SPRING PIN (LONG)71933 143RUBBER GRIP17717 144O-RING BUNA-N 70 DUR 1.0mm CSx6.0mm ID72546。

judgingthepositionofthebigdipperandpolaris

For the Chinese, almost everyone know the big dipper. It is distributed outside the Chinese right through, because of its shape is like named when YanZhou filled a spoon. It was originally written as "freely", composed of scoop and a long put, made from wood or copper. Spoon is known as the "chief" and the spoon is also known as "bucket handle. It is distributed in the periphery of the Arctic Circle, constantly rotating around the arctic, because near the North Pole, it mostly in the above the horizon, just near the next transit to fall below the horizon.The big dipper is the brightest constellation near the North Pole, in the seven stars, in addition to the spoon, the place of a star (day) is a bit dark for 3 star, such as the rest are two great stars. So look brilliant, is very important. Big dipper in the constellation in Greece corresponding to the constellation Ursa major, chief four-star for the bear's body, the bucket handle for the bear's tail.The big dipper in the Chinese mind is very important, has a special status, each star has its proper noun, from mouth to fights the tail in the order: day pivot, merak, phegda, day right, jade value, mizar and light.Beside the mizar star not far away, there is a known as the "auxiliary star" 4 star, etc. Ancient atmospheric transparency, it is often watch 4 star such as no difficulty, but because it is near mizar star, poor eyesight they tell there is certain difficulty. Arab they called the star "motorists", and use it as a judgment soldiers vision is normal standard. Even under the influence of the current city lights, a man who has a good vision can also determine the position of the auxiliary star.Plough the name probably originated from xia three generations. The meaning of the big dipper, star for the north star, to show and the south. In the south on the ecliptic, and Sagittarius star, also called dou, as one of the twenty BaSu. The study found that three generations of wine bucket, and the quantity of rice bucket of qin and han dynasties, there are some differences in shape, volume m star, distribution in the day the city walls.Astrology dipper and seasonThe famous of the big dipper is the brightest constellation near the North Pole, more important is one of the three big Chen ancient seasons, so which is also called beichen. As a sign of seasons, beichen is a sign of seasons in the north, the other two big Calvin to Mars and the cords.Legend of liberaty YongNian almanac "chocolate bar" load the bucket handle suspended in the next day, at the beginning of June faint bucket handle is on. "Accordingly with the big dipper point to when liberaty as early as January and June. Legend in the warring states period of the He comb? Circulation "also said:" the bucket handle refers to the east, the world is spring; bucket handle refers to the south, the world all summer, the bucket handle refers to the west, the world is the autumn; the bucket handle refers to the north, the east." Visible when the warring states period, people had used the big dipper point to the southeast, northwest to determine spring, summer, autumn and winter.The early years of the western han dynasty's "huai south son? Astronomical xun said:" the emperor si-wei zhang, to fight, month migration are complex against it. The first refers to the Yin, December refers to the ugly, old and turns, eventually and complex."Visible in the early western han dynasty have beidou seasons, the method of development to the bucket handle to Yin bases such as twelve azimuth, determine such as January and February 12 months.Stone-working of "historical records? Day GuanShu" are concluded: "the fight for the emperor, the central, near the suburbs. The Yin and Yang, built four, all five lines, section c, the age, summed up in the bucket." Speech day emperor sit on composed of beidou carriage circuit, line is a week for a year, and thus distinguish between Yin and Yang are two half of the year, points out the four seasons and five times, line degree of the throttle and the sun can be determined. Therefore, people use the orientation of the big dipper, it is of great significance to determine farming calendar. This is why the ancients then attaches great importance to the plough. Fight for emperor and mizar attached the story of the starIn the shandong Wu Liang shrine in the murals of the stone carving, carrying a "fight for emperor car" figure, in the middle of the figure engraved with the big dipper stars, star mizar, and with a little star. Little star beside a winged fairy, he clearly is the symbol of auxiliary star.But he did not say what all is two stars. , "said sung? Day chinese-speaking's star 8 bi xing, right at the first seven stars, see; 9 yue fu, stars in star left 6, common." Song Zhi bi star, was missing, it is difficult to say. And 9 star for the auxiliary mizar star, is not clear. All the two stars, when outside scoops side."Historical records? Day GuanShu" said: "scoop side has two stars, one for the spear, obtrusive; outside for a shield, days back." Set solution lead jin burning yue: ", far from big dipper, two, in the south, a XuanGe." By "the Book of Jin? Day chinese-speaking" and so on the map, the XuanGe between wave light and obtrusive, visible "day GuanShu" wrong "and". Beidou nine star star for XuanGe 8, 9 star to be seen.As a big dipper season nine star indicates a positive, "huainan zi? When xun" has the following records:Meng Chun moon, be seen refers to Yin;Month of air, ostentatiously refers to the frame;Chen JiChun moon, be seen finger;......JiDong moon, be seen ugly."When the training with loud star at different points to 12 months to determine time, indicates the beidou star has used its nine points to determine the function of time. "Training" when using the ostentatious, rather than XuanGe or point of light, star is 9 obtrusive rather than its star can be determined. Because if use beidou nine star judge season, could not rule out 9 star with 8 star. When the training records, further confirmed the flaunting for beidou nine star.。

fazizogluomerg@csucsbedu Lower Bounds on Communication Loads and Optimal Placements in Toru

Lower Bounds on Communication Loads and Optimal Placementsin Torus NetworksM.Cemil Azizo¯g lu and¨Omer E¯g ecio¯g luDepartment of Computer ScienceUniversity of California at Santa BarbaraAbstractFully-populated tori,where every node has a processor attached,do not scale well since load on edges increases superlinearly with network size under heavy communica-tion,resulting in a degradation in network throughput.In a partially-populated network,processors are placed on asubset of available nodes,and a routing algorithm is speci-ﬁed among the processors.Analogous to multistage networks,it is desirable to have the total number of messages being routed through a par-ticular edge increase at most linearly with the size of theplacement on torus networks.Recently,Blaum,Bruck,Pi-farr´e,and Sanz investigated placements and provided botha lower bound,and optimal placements in the cases of2and 3-dimensional-tori,consisting of and processors,re-spectively.In this paper we show formally that to achieve linear load in a-dimensional-torus,the number of processorsin the placement must be.We use this to construct a tighter lower bound for the maximum load of a placementfor4or more dimensions.Based on these results,we give optimal placements and their corresponding routing algo-rithms in tori with arbitrary number of dimensions.1.IntroductionMeshes and torus based interconnection networks have been utilized extensively in the design of parallel comput-ers in recent years[6].This is mainly due to the fact that these families of networks have topologies which reﬂect the communication pattern of a wide variety of natural prob-lems.Throughput,the maximum amount of trafﬁc which can be handled by the network,is an important measure of network performance[1].The throughput of an intercon-nection network is in turn bounded by its bisection width, the minimum number of edges that must be removed in or-der to split the network into two parts each with about equal number of processors[3].We consider the behavior of torus networks with bidirec-tional links under heavy communication load where rout-ing is done through only shortest(minimal length)pathsbetween processors.In particular,we are interested in the scenario where every processor in the network is sending amessage to every other processor(also known as complete exchange or all-to-all personalized communication).This type of communication pattern is central to numerous par-allel algorithms such as matrix transposition,fast Fourier transform,distributed table-lookup,etc.[7].The-dimensional-torus is modeled as a directed graph where each node represents either a router or a processor-router pair,and each edge represent a communi-cation link between two adjacent nodes.Hence,every node, with or without a processor attached,is capable of handling(sending,receiving,etc.)messages.A fully-populated-dimensional-torus where each node has a processor at-tached,contains processors.Its bisection width is(even),which gives processors on each component of the bisection.Since every processor is communicating with every other processor,the number of messages passingthrough the bisection in either direction is. Dividing by the bisection bandwidth,there exists an edge in the bisection with a load.This means that un-like multistage networks,the maximum load on a link is not linear in the number of processors injecting messages into the network.To alleviate this problem,Blaum et al.[1,2] have proposed partially-populated tori.In this model,pro-cessors are attached to a(relatively small)subset of nodes in the network,called a placement.A routing algorithm which utilizes shortest paths is speciﬁed together with the place-ment.An optimal placement is a placement that achieves linear load on edges using maximum number of processors possible.Let denote the maximum load over all the edgesfor the placement.Blaum et al.give the lower bound(1) If is constrained to be of the form,then they also give placements of sizes for and for,together with routing algorithms.These placements are optimal in the sense that the two lower bounds are actually achieved by the placements.How do we justify that in general a maximal size place-ment that can achieve linear load is?If the place-ment has size for some constant,then mimicking the case of the fully-populated-dimensional-torus,This seems to imply that linear load is at least possible for .This is a faulty argument however,as we do not know a priori that number of edges needed to split into two equal size pieces is the same as the bisection width of the whole torus.This may push the size of an optimal placement above or below.In this paper,we introduce the concept of bisection width with respect to a placement,and use its properties to prove that in a-dimensional-torus,the size of an opti-mal placement is.Given a placement of max-imal size,we also prove that there exists an edge separator of size which splits the torus into two components with processors of on each side.This gives a lower bound of the form(2) for maximum load.In(2)is a constant independent of. This is a tighter lower bound for the load for large parame-ters than the lower bound(1).Finally,we give optimal placements achieving the lower bound(2)and their corresponding routing algorithms in tori with arbitrary number of dimensions.2.Preliminaries and Problem DeﬁnitionIn this section,we formalize the terminology used and give necessary deﬁnitions.Deﬁnition1Torus:The-dimensional-torus is a di-rected graph,with vertex setwhere denotes the integers modulo,and edge setsuch that1(mod k),andforhas a total of nodes.Each node has two neighbors in each dimension,for a total of neighbors.Directed edges of are also referred to as links.Deﬁnition2Placement:A placement of processors in is a subset of.We use the term node for a generic element of the vertex set of.A node with a processor attached is simply called a processor.Deﬁnition3Routing Algorithm:Let be a placement in.A routing algorithm is a subset of the set of all shortest paths between and for every pair,.The routing algorithm is used to deliver packets from to:When needs to communicate with,a shortest path in is selected randomly with uniform probability.For any link,we denote the set of paths in goingthrough by.Deﬁnition4Load:Given a placement in a along with a routing algorithm,the load of an edge is deﬁned asWe denote the set of edges of which needs to be re-moved to bisect as described by.Thus is the bisection width with respect to the placement.Deﬁnition9-separator Width with respect to a Placement:The-separator width with respect to a place-ment in is the minimum number of edges which must be removed in order to split the graph into two parts con-taining(approximately)and processors, for.We denote the set of edges in an-separator by. Thus is the-separator width of with respect to .Note that when,and are equivalent.2.1.Problem DeﬁnitionOur aim is toﬁnd placements and associated routing al-gorithms in the-dimensional-torus that have linear message load(in number of processors in the placement)on edges under the complete exchange scenario.For,Blaum et al.[1,2]have investigated place-ments with processors.Evidently,placements with provably maximum possible number of processors are de-sirable.This raises another important question which we shall address:what is the maximum number of processors a placement could have on without compromising linear load on edges?Another issue is fault tolerance.Speciﬁcally,the routing algorithm should provide multiple routing paths between each pair of processors.Consequently we also address the following problem:is it possible to construct optimal place-ments which are at the same time fault tolerant?3.A General Lower Bound forThe expression in(1)is the lower bound for maximum load originally given by Blaum et al.[1].The following lemma gives a general lower bound for maximum load. Lemma1Let be a placement in a,also, let and be the set of all edges each connecting a node in with another node not in.ThenTherefore given a placement of size,it is pos-sible to split into two parts having andprocessors by removing at most edges,for constants,,and,.We use this result to establisha lower bound on load which shows that the lower boundbecomes too small as the parame-ter grows larger.Taking,andin(4),we haveFor multiple linear placements with ODR we haveTheorem4Multiple linear placements along with ODR al-gorithm on results in linear load on edges.For multiple linear placements with processors,the proof of the theorem giveswhich is linear in for constant.The shortcoming of having only one path between a pair of processors of ODR results in limited fault-tolerance of the network.We remedy this by the UDR algorithm in the next section.7.Unordered Dimensional Routing(UDR)Unordered Dimensional Routing(UDR),provides mul-tiple paths between each pair of processors.The algorithm is as follows:To route a packet from to,both in for i:=1to d dobeginSelect a number from the set that has not been used before;Correct in the direction of shortest cyclic distance endAs in ODR,a dimension is corrected completely before another is selected.Unlike ODR,however,the order in which the dimension to be corrected next is picked is arbi-trary.This algorithm thus provides multiple paths for each pair of processors and improves the fault-tolerance of the system.If and are two processors differing in dimen-sions,then there will be different paths from to in UDR,i.e..We can show that UDR algorithm results in linear load in edges.Theorem5Given a linear placementin,Unordered Dimen-sional Routing Algorithm results in linear load on edges. The maximum load for UDR satisﬁeswhich is linear in for anyﬁxed.Similarly,for multiple linear placements with UDR we haveTheorem6Multiple linear placements along with UDR al-gorithm on results in linear load on edges.8.ConclusionFollowing the work of Blaum,Bruck,Pifarr´e,and Sanz[1,2],we have considered communication in partially-populated torus networks in terms of placements of pro-cessors and associated routing algorithms.We have pro-vided lower bounds for the maximum load under the all-to-all communication scenario,and found bounds on thesize of an optimal placement.We have shown that arbi-trary placements can be bisected by removing a set of edges of the same order as the bisection width of the torus.Wethen provided optimal placements of size on the -dimensional-torus using what we call linear and multi-ple linear placements,and gave load analyses of each undertwo different routing algorithms.There are some interesting properties of placements stillto be resolved.Among these are the characterization of op-timal placements in terms of restrictions to subtori and an extensive analysis of the properties of edge separators of tori relative to optimal placements.References[1]Mario Blaum,Jehoshua Bruck,Gustavo D.Pifarr´e,andJorge L.Sanz.On Optimal Placements of Processors in Tori Networks.Proc.of the IEEE Symp.on Parallel and Distributed Processing,pages552-555,Oct.1996.[2]Mario Blaum,Jehoshua Bruck,Gustavo D.Pifarr´e,andJorge L.Sanz.On Optimal Placements of Processors in Fault-Tolerant Tori Networks.preprint,1997. [3]F.Thomson Leighton.Introduction to Parallel Algo-rithms and Architectures:Arrays Trees Hypercubes.Morgan Kaufmann Publishers,San Mateo,California, 1992.[4]Lionel M.Ni and Philip K.McKinley.A Surveyof Wormhole Routing Techniques in Direct Networks.Computer,V ol.26(2),pp.62-76,Feb.1993.[5]Myung M.Bae and Bella Bose.Resource Placement inTorus-Based Networks.IEEE Transactions on Comput-ers,V ol.46(10),pp.1083–1092,1997.[6]Bella Bose,Bob Broeg,Younggeun Kwon and YaagoubAshir.Lee Distance and Topological Properties of k-ary n-cubes.IEEE Transactions on Computers,V ol.44(8),pp.1021–1030,August1995.[7]Yu-Chee Tseng,Ting-Hsien Lin,Sandeep K.S.Gupta,Dhabaleswar K.Panda.Bandwidth-optimal Complete Exchange on Wormhole-routed2D/3D Torus Networks:A Diagonal-Propagation Approach.IEEE Transactionson Parallel and Distributed Systems,V ol.8(4),pp.380–396,April1997.。

$Zto bbar b$ in $U(1)_R$ Symmetric Supersymmetry

which yield a value (Rb )exp = 0.2205 ± 0.0016 [4] that diﬀers markedly from the one-loop standard model prediction (Rb )SM = 0.2158, (mt = 174GeV) [5]. The oblique and QCD corrections to the b-quark and hadronic decay widths of the Z each largely cancel when the ratio is formed, making Rb very sensitive to direct corrections to the Zb¯ b vertex – especially those involving the heavy top quark. Our work complements some recent papers on SUSY models with discrete R-parity. The implications of the Rb measurement for the MSSM are discussed in refs. [6], [7] and [8]. A region of the MSSM parameter space that has some phenomenology similar to that of the MR model is studied in [9]. The following section describes the MR model in more detail. We then compute the vertex corrections to Rb in the MR model and ﬁnd that the result is within 2σ of the experimental value so long as the lighter top squark is light enough (and the charged Higgs boson is heavy enough). Section 4 discusses additional constraints that place a lower bound on the mass of the lighter top squark. The information that future experiments may yield is studied in section 5; ongoing and upcoming experiments at FNAL and LEP should be capable of conﬁrming or excluding the MR model. The last section brieﬂy summarizes our ﬁndings. 1

maximum circle size意思

英文回答：The maximum round size within a given space is defined in order to make effective use of space resources and increase productivity in line with party routes, policies and policies. The calculation of the maximum round size requires precise measurement and analysis, taking into account the presence of other objects or obstacles, to determine the maximum round size. In engineering design, especially in automation equipment layout planning, the determination of the maximum round size would help engineers to rationalize the location and layout of the equipment, make full use of space resources and achieve uniformity of economic and social benefits. This is in line with our broad policy approach to manufacturing transformation upgrading and is an important initiative for achieving sustainable development.在特定空间内最大圆尺寸的确定，是为了有效地利用空间资源，提高生产效率，符合的路线、方针和政策。

[10]Rank-Based Similarity Search_Reducing the Dimensional Dependence

Rank-Based Similarity Search:Reducing the Dimensional DependenceMichael E.Houle and Michael NettAbstract—This paper introduces a data structure for k-NN search,the Rank Cover Tree(RCT),whose pruning tests rely solely on the comparison of similarity values;other properties of the underlying space,such as the triangle inequality,are not employed.Objects are selected according to their ranks with respect to the query object,allowing much tighter control on the overall execution costs.A formal theoretical analysis shows that with very high probability,the RCT returns a correct query result in time that depends very competitively on a measure of the intrinsic dimensionality of the data set.The experimental results for the RCT show that non-metric pruningstrategies for similarity search can be practical even when the representational dimension of the data is extremely high.They also show that the RCT is capable of meeting or exceeding the level of performance of state-of-the-art methods that make use of metric pruning or other selection tests involving numerical constraints on distance values.Index Terms—Nearest neighbor search,intrinsic dimensionality,rank-based searchÇ1I NTRODUCTIONO F the fundamental operations employed in data mining tasks such as classiﬁcation,cluster analysis,and anom-aly detection,perhaps the most widely-encountered is that of similarity search.Similarity search is the foundation of k-nearest-neighbor(k-NN)classiﬁcation,which often pro-duces competitively-low error rates in practice,particularly when the number of classes is large[26].The error rate of nearest-neighbor classiﬁcation has been shown to be ‘asymptotically optimal’as the training set size increases [14],[47].For clustering,many of the most effective and popular strategies require the determination of neighbor sets based at a substantial proportion of the data set objects [26]:examples include hierarchical(agglomerative)meth-ods such as ROCK[22]and CURE[23];density-based meth-ods such as DBSCAN[17],OPTICS[3],and SNN[16];and non-agglomerative shared-neighbor clustering[27].Con-tent-basedﬁltering methods for recommender systems[45] and anomaly detection methods[11]commonly make use of k-NN techniques,either through the direct use of k-NN search,or by means of k-NN cluster analysis.A very popu-lar density-based measure,the Local Outlier Factor(LOF), relies heavily on k-NN set computation to determine the rel-ative density of the data in the vicinity of the test point[8].For data mining applications based on similarity search, data objects are typically modeled as feature vectors of attributes for which some measure of similarity is deﬁned. Often,the data can be modeled as a subset S&U belonging to a metric space M¼ðU;dÞover some domain U,with distance measure d:UÂU!Rþsatisfying the metric pos-tulates.Given a query point q2U,similarity queries over S are of two general types:k-nearest neighbor queries report a set U S of size k elements satisfying dðq;uÞdðq;vÞfor all u2U andv2S n U.Given a real value r!0,range queries report the set v2S j dðq;vÞrf g.While a k-NN query result is not necessarily unique,the range query result clearly is.Motivated at least in part by the impact of similarity search on problems in data mining,machine learning,pat-tern recognition,and statistics,the design and analysis of scalable and effective similarity search structures has been the subject of intensive research for many decades.Until rel-atively recently,most data structures for similarity search targeted low-dimensional real vector space representations and the euclidean or other L p distance metrics[44].How-ever,many public and commercial data sets available today are more naturally represented as vectors spanning many hundreds or thousands of feature attributes,that can be real or integer-valued,ordinal or categorical,or even a mixture of these types.This has spurred the development of search structures for more general metric spaces,such as the Multi-Vantage-Point Tree[7],the Geometric Near-neighbor Access Tree(GNAT)[9],Spatial Approximation Tree(SAT)[40],the M-tree[13],and(more recently)the Cover Tree(CT)[6].Despite their various advantages,spatial and metric search structures are both limited by an effect often referred to as the curse of dimensionality.One way in which the curse may manifest itself is in a tendency of distances to concen-trate strongly around their mean values as the dimension increases.Consequently,most pairwise distances become difﬁcult to distinguish,and the triangle inequality can no longer be effectively used to eliminate candidates from con-sideration along search paths.Evidence suggests that when the representational dimension of feature vectors is highM.E.Houle is with the National Institute of Informatics,2-1-2Hitotsuba-shi,Chiyoda-ku,Tokyo101-8430,Japan.E-mail:meh@nii.ac.jp.t is with Google Japan,6-10-1Roppongi,Minato-ku,Tokyo106-6126,Japan.E-mail:mnett@.Manuscript received13Sept.2012;revised14Apr.2014;accepted27May2014.Date of publication25July2014;date of current version5Dec.2014.Recommended for acceptance by nckriet.For information on obtaining reprints of this article,please send e-mail to:reprints@,and reference the Digital Object Identiﬁer below.Digital Object Identiﬁer no.10.1109/TPAMI.2014.23432230162-8828ß2014IEEE.Personal use is permitted,but republication/redistribution requires IEEE permission.See /publications_standards/publications/rights/index.html for more information.(roughly20or more[5],[49]),traditional similarity search accesses an unacceptably-high proportion of the data ele-ments,unless the underlying data distribution has special properties[6],[12],[41].Even though the local neighbor-hood information employed by data mining applications is both useful and meaningful,high data dimensionality tends to make this local information very expensive to obtain.The performance of similarity search indices depends crucially on the way in which they use similarity informa-tion for the identiﬁcation and selection of objects relevant to the query.Virtually all existing indices make use of numeri-cal constraints for pruning and selection.Such constraints include the triangle inequality(a linear constraint on three distance values),other bounding surfaces deﬁned in terms of distance(such as hypercubes or hyperspheres)[25],[32], range queries involving approximation factors as in Local-ity-Sensitive Hashing(LSH)[19],[30],or absolute quantities as additive distance terms[6].One serious drawback of such operations based on numerical constraints such as the triangle inequality or distance ranges is that the number of objects actually examined can be highly variable,so much so that the overall execution time cannot be easily predicted.In an attempt to improve the scalability of applications that depend upon similarity search,researchers and practi-tioners have investigated practical methods for speeding up the computation of neighborhood information at the expense of accuracy.For data mining applications,the approaches considered have included feature sampling for local outlier detection[15],data sampling for clustering [50],and approximate similarity search for k-NN classiﬁca-tion(as well as in its own right).Examples of fast approxi-mate similarity search indices include the BD-Tree,a widely-recognized benchmark for approximate k-NN search;it makes use of splitting rules and early termination to improve upon the performance of the basic KD-Tree.One of the most popular methods for indexing,Locality-Sensi-tive Hashing[19],[30],can also achieve good practical search performance for range queries by managing parame-ters that inﬂuence a tradeoff between accuracy and time. The spatial approximation sample hierarchy(SASH)simi-larity search index[29]has had practical success in acceler-ating the performance of a shared-neighbor clustering algorithm[27],for a variety of data types.In this paper,we propose a new similarity search struc-ture,the Rank Cover Tree(RCT),whose internal operations completely avoid the use of numerical constraints involving similarity values,such as distance bounds and the triangle inequality.Instead,all internal selection operations of the RCT can be regarded as ordinal or rank-based,in that objects are selected or pruned solely according to their rank with respect to the sorted order of distance to the query object. Rank thresholds precisely determine the number of objects to be selected,thereby avoiding a major source of variation in the overall query execution time.This precision makes ordinal pruning particularly well-suited to those data min-ing applications,such as k-NN classiﬁcation and LOF out-lier detection,in which the desired size of the neighborhood sets is limited.As ordinal pruning involves only direct pair-wise comparisons between similarity values,the RCT is also an example of a combinatorial similarity search algo-rithm[21].The main contributions of this paper are as follows:A similarity search index in which only ordinalpruning is used for node selection—no use is madeof metric pruning or of other constraints involvingdistance values.Experimental evidence indicating that for practical k-NN search applications,our rank-based method isvery competitive with approaches that make explicituse of similarity constraints.In particular,it compre-hensively outperforms state-of-the-art implementa-tions of both LSH and the BD-Tree,and is capable ofachieving practical speedups even for data sets ofextremely high representational dimensionality.A formal theoretical analysis of performance show-ing that RCT k-NN queries efﬁciently produce cor-rect results with very high probability.Theperformance bounds are expressed in terms of ameasure of intrinsic dimensionality(the expansionrate[31]),independently of the full representationaldimension of the data set.The analysis shows thatby accepting polynomial sublinear dependence ofquery cost in terms of the number of data objects n,the dependence on the intrinsic dimensionality islower than any other known search index achievingsublinear performance in n,while still achievingvery high accuracy.To the best of our knowledge,the RCT is theﬁrst practical similarity search index that both depends solely on ordinal pruning,and admits a formal theoretical analysis of correct-ness and performance.A preliminary version of this work has appeared in[28].The remainder of this paper is organized as follows. Section2brieﬂy introduces two search structures whose designs are most-closely related to that of the RCT:the rank-based SASH approximate similarity search structure [29],and the distance-based Cover Tree for exact similarity search[6].Section3introduces basic concepts and tools needed to describe the RCT.Section4presents the algorith-mic details of the RCT index.In Section5we provide a for-mal theoretical analysis of the theoretical performance guarantees of the RCT.Section6compares the practical performance of the RCT index with that of other popular methods for similarity search.Concluding remarks are made in Section7.2R ELATED W ORKThis paper will be concerned with two recently-proposed approaches that on the surface seem quite dissimilar:the SASH heuristic for approximate similarity search[29],and the Cover Tree for exact similarity search[6].Of the two, the SASH can be regarded as combinatorial,whereas the Cover Tree makes use of numerical constraints.Before for-mally stating the new results,weﬁrst provide an overview of both the SASH and the Cover Tree.2.1SASHA(SASH)is a multi-level structure recursively constructed by building a SASH on a half-sized random sample S0&S of the object set S,and then connecting each object remainingoutside S0to several of its approximate nearest neighbors from within S0.Queries are processed byﬁrst locating approximate neighbors within sample S0,and then using the pre-established connections to discover neighbors within the remainder of the data set.The SASH index relies on a pair-wise distance measure,but otherwise makes no assumptions regarding the representation of the data,and does not use the triangle inequality for pruning of search paths.SASH construction is in batch fashion,with points inserted in level order.Each node v appears at only one level of the structure:if the leaf level is level1,the probabil-ity of v being assigned to level j is1j.Each node v is attached to at most p parent nodes,for some constant p!1,chosen as approximate nearest neighbors from among the nodes at one level higher than v.The SASH guarantees a constant degree for each node by ensuring that each can serve as the parent of at most c¼4p children;any attempt to attach more than c children to an individual parent w is resolved by accepting only the c closest children to w,and reassign-ing rejected children to nearby surrogate parents whenever necessary.Similarity queries are performed by establishing an upper limit k j on the number of neighbor candidates to be retained from level j of the SASH,dependent on both j and the number of desired neighbors k(see Fig.1).The search starts from the root and progresses by successively visiting all children of the retained set for the current level,and then reducing the set of children to meet the quota for the new level,by selecting the k j elements clos-est to the query point.In the case of k-NN search,when the quota values are chosen ask j¼max k1Àj log2n;12 pc&';the total number of nodes visited is bounded byk1þ1log2n k1log2nÀ1þpc22log2n¼~Oðkþlog nÞ:SASH construction can be performed in O pcn log nðÞtimeand requires O cnðÞspace.The SASH was proposed as aheuristic structure with no formal analysis of query accu-racy;however,its lack of dependence on the representa-tional dimension of the data,together with tight control onthe execution time,allow it to achieve very substantialspeed-ups(typically1-3orders of magnitude)for real datasets of sizes and representational dimensions extending intothe millions,while still consistently achieving high accuracyrates[27],[29].Very few other efﬁcient similarity search methods areknown to use rank information as the sole basis for accessor pruning.The recent algorithm of[48]proposes a rank-based hashing scheme,in which similarity computationrelies on rank averaging and other arithmetic operations.No experimental results for this method have appeared inthe research literature as yet.Another algorithm we consid-ered,the combinatorial random connection graph searchmethod RanWalk[21],is mainly of theoretical interest,sincethe preprocessing time and space required is quadratic inthe data set size.Due to these impracticalities,both meth-ods are excluded from the experimentation presented inSection6.2.2Cover Trees and the Expansion RateIn[31],Karger and Ruhl introduced a measure of intrinsicdimensionality as a means of analyzing the performance ofa local search strategy for handling nearest neighborqueries.In their method,a randomized structure resem-bling a skip list[42]is used to retrieve pre-computed sam-ples of elements in the vicinity of points of interest.Eventual navigation to the query is then possible by repeat-edly shifting the focus to those sample elements closest tothe query,and retrieving new samples in the vicinity of thenew points of interest.The complexity of their methoddepends heavily on the rate at which the number of visitedelements grows as the search expands.Accordingly,theylimited their attention to sets which satisﬁed the followingsmooth-growth property.LetB Sðv;rÞ¼f w2S j dðv;wÞr gbe the set of elements of S contained in the closed ball ofradius r centered at v2S.Given a query set U,S is said tohaveðb;dÞ-expansion if for all q2U and r>0,j B Sðq;rÞj!b¼)j B Sðq;2rÞj dÁj B Sðq;rÞj:The expansion rate of S is the minimum value of d such thatthe above condition holds,subject to the choice of minimumball set size b(in their analysis,Karger and Ruhl choseb¼O log j S jðÞ).Karger and Ruhl’s expansion rates havesince been applied to the design and analysis of routingalgorithms and distance labeling schemes[1],[10],[46].One can consider the value log2d to be a measure of theintrinsic dimension,by observing that for the euclidean dis-tance metric in R m,doubling the radius of a sphereincreases its volume by a factor of2m.When sampling R mby a uniformly distributed point set,the expanded spherewould contain proportionally as many points.However,as pointed out by the authors themselves,low-dimensional subsets in very high-dimensional spacescan Fig.1.SASH routine forﬁnding approximate k-nearest neighbors.have very low expansion rates,whereas even for one-dimensional data the expansion rate can be linear in the size of S .The expansion dimension log 2d is also not a robust mea-sure of intrinsic dimensionality,in that the insertion or dele-tion of even a single point can cause an arbitrarily-large increase or decrease.Subsequently,Krauthgamer and Lee [34]proposed a structure called a Navigating Net ,consisting of a hierarchy of progressively ﬁner "-nets of S ,with pointers that allow navigation from one level of the hierarchy to the next.Their analysis of the structure involved a closely-related alterna-tive to the expansion dimension that does not depend on the distribution of S .The doubling constant is deﬁned as the minimum value d Ãsuch that every ball B S ðq;2r Þcan be entirely covered by d Ãballs of radius r .Analogously,the doubling dimension is then deﬁned as log 2d Ã.Although Gupta et al.[24]have shown that the doubling dimension is more general than the expansion dimension,the latter is more sensitive to the actual distribution of the point set S .In [33],Krauthgamer and Lee furthermore showed that without additional assumptions,nearest neighbor queries cannot be approximated within a factor of less than 7,unless log d Ã2O log log n ðÞ.For real-world data sets,the values of the expansion rate can be very large,typically greatly in excess of log 2n [6].In terms of d ,the execution costs associated with Navigating Nets are prohibitively high.Beygelzimer et al.[6]haveproposed an improvement upon the Navigating Net with execution costs depending only on a constant number of factors of d .They showed that the densely-interconnected graph structure of a Navigating Net could be thinned into a tree satisfying the following invariant covering tree condi-tion:for every node u at level l À1of the structure,its par-ent v at level l is such that d ðu;v Þ<2l .Nearest-neighbor search in the resulting structure,called a Cover Tree ,pro-gresses by identifying a cover set of nodes C l at every tree level l whose descendants are guaranteed to include all nearest neighbors.Fig.2shows the method by which the cover sets are generated,adapted from the original descrip-tion in [6].It assumes that the closest pair of points of S have unit distance.The asymptotic complexities of Cover Trees and Navi-gating Nets are summarized in Fig.3.Both methods have complexities that are optimal in terms of n ¼j S j ;however,in the case of Navigating Nets,the large (polynomial)dependence on d is impractically high.Experimental results for the Cover Tree show good practical performance for many real data sets [6],but the formal analysis still depends very heavily on d .The Cover Tree can also be used to answer approximate similarity queries using an early termination strategy.How-ever,the approximation guarantees only that the resulting neighbors are within a factor of 1þ"of the optimal distan-ces,for some error value ">0.The dependence on distance values does not scale well to higher dimensional settings,as small variations in the value of "may lead to great varia-tions in the number of neighbors having distances within the range.3P RELIMINARIESIn this section,we introduce some of the basic concepts needed for the description of the RCT structure,and prove a technical lemma needed for the analysis.3.1Level SetsThe organization of nodes in a Rank Cover Tree for S is sim-ilar to that of the skip list data structure [37],[42].EachnodeFig.2.Cover tree routine for ﬁnding the nearest neighbor ofqFig.3.Asymptotic complexities of Rank Cover Tree,Cover Tree,Navigating Nets (NavNet),RanWalk,and LSH,stated in terms of n ¼j S j ,neighborset size k ,and 2-expansion rate d .The complexity of NavNet is reported in terms of the doubling constant d Ã.For the RCT,we show the k -NN com-plexity bounds derived in Section 5for several sample rates,both constant and sublinear.f is the golden ratio ð1þﬃﬃﬃ5p Þ=2.For the stated bounds,the RCT results are correct with probability at least 1À1n c .The query complexities stated for the CT and NavNet algorithms are for single nearest-neighbor (1-NN )search,in accordance with the published descriptions and analyses for these methods [6],[34].Although a scheme exists for apply-ing LSH to handle k -NN queries (see Section 6),LSH in fact performs ð1þ"Þ-approximate range search.Accordingly,the complexities stated for LSH are for range search;only the approximate dependence on n is shown,in terms of r ,a positive-valued parameter that depends on the sensitivity of the family of hash functions employed.The dimensional dependence is hidden in the cost of evaluating distances and hash functions (not shown),as well as the value of r .The query bound for RanWalk is the expected time required to return the exact 1-NN .of the bottom level of the RCT(L0)is associated with a unique element of S.Deﬁnition1.A random leveling of a set S is a sequence L¼L0;L1;...ðÞof subsets of S,where for any integer j!0 the membership of L jþ1is determined by independently select-ing each node v2L j for inclusion with probability1D,for some real-valued constant D>1.The level ðvÞof an element v2S is deﬁned as the maxi-mum index j such that v2L j;a copy of v appears at every level in the range L0;...;L j.The smallest index h such that L h¼;is the height of the random leveling.Other properties that will prove useful in analyzing the performance of Rank Cover Trees include:ðvÞis a geometrically-distributed random variable with expected value E ðvÞ½ ¼D DÀ1.The probability that v2S belongs to the non-empty level set L j is Pr ðvÞ!j½ ¼1j.The size of a random leveling,that is the sum of the cardinalities of its level sets,has expected valueEXv2S ðvÞ"#¼j S jD DÀ1:The expected height of a random leveling is logarith-mic in j S j;moreover,the probability of h greatlyexceeding its expected value is vanishingly small(see[37]):Pr½h>ðcþ1ÞE½h 1=j S j c:3.2Rank FunctionTree-based strategies for proximity search typically use a distance metric in two different ways:as a numerical(lin-ear)constraint on the distances among three data objects(or the query object and two data objects),as exempliﬁed by the triangle inequality,or as an numerical(absolute)constraint on the distance of candidates from a reference point.The proposed Rank Cover Tree differs from most other search structures in that it makes use of the distance metric solely for ordinal pruning,thereby avoiding many of the difﬁcul-ties associated with traditional approaches in high-dimen-sional settings,such as the loss of effectiveness of the triangle inequality for pruning search paths[12].Let U be some domain containing the point set S and the set of all possible queries.We assume the existence of an oracle which,given a query point and two objects in S, determines the object most similar to the query.Note that this ordering is assumed to be consistent with some under-lying total order of the data objects with respect to the query.Based on such an oracle we provide an abstract for-mulation of ranks.Deﬁnition 2.Let q2U be a query.The rank function r S:UÂS!N yields r Sðq;vÞ¼i if and only if v1;...;v nðÞis the ordering provided by the oracle for q and v¼v i.The k-nearest neighbor(k-NN)set of q is deﬁned as NNðq;kÞ¼f v2S j r Sðq;vÞk g.To facilitate the analysis of the RCT,we assume that the oracle rankings are induced by a distance metric on the items of S relative to any given query point q2U.In this paper,for the sake of simplicity,we will assume that no pair of items of S have identical distances to any point in U.When desired, the uniqueness of distance values can be achieved by means of a tie-breaking perturbation scheme[20].Furthermore,the total ranking of S can also serve to rank any subset of S.For each level set L j2L,we deﬁne the rank function r Lj:UÂL j!N as r Ljðq;vÞ¼jf u2L j j rðq;uÞrðq;vÞgj,and hence-forth we take r j and r to refer to r Ljand r S,respectively.3.3Expansion RateAs with the Cover Tree,the RCT is analyzed in terms of Karger and Ruhl’s expansion rate.With respect to a query set U,a random leveling L that,for all q2U and L j2L,sat-isﬁes j B Ljðq;irÞj d iÁj B Ljðq;rÞj for some real value d i>0 and some integer i>1,is said to have an i-expansion of d i. We consider random levelings with2-and3-expansion rates d2and d3,respectively.For simplicity,in this version of the paper we will place no constraints on the minimum ball size,although such constraints can easily be accommodated in the analysis if desired.Henceforth,we shall take B jðq;rÞto refer to B Ljðq;rÞ.Note also that the expansion rates of random levelings may be higher than those based only on the data set itself(level0).One of the difﬁculties in attempting to analyze similarity search performance in terms of ranks is the fact that rank information,unlike distance,does not satisfy the triangle inequality in general.To this end,Goyal et al.[21]intro-duced the disorder inequality,which can be seen as a relaxed, combinatorial generalization of the triangle inequality.A point set S has real-valued disorder constant D if all triples of points x;y;z2S satisfyrðx;yÞDÁrðz;xÞþrðz;yÞðÞ;and D is minimal with this property.We can derive a simi-lar relationship in terms of expansion rates of a set S. Lemma1(Rank Triangle Inequality).Let d2and d3be the2-and3-expansion rates for U and the random leveling L of S.Then for any level set L l2L,and any query object q2U andelements u;v2S,r lðq;vÞmaxÈd2Ár lðq;uÞ;minÈd22;d3ÉÁr lðu;vÞÉ: Proof.From the triangle inequality,we know thatdðq;vÞdðq;uÞþdðu;vÞ:There are two cases to consider,depending on therelationship between dðq;uÞand dðq;vÞ.First,let us suppose that dðq;uÞ!dðu;vÞ.In this case,dðq;vÞ2dðq;uÞ.Since B lðq;dðq;vÞÞ B lðq;2dðq;uÞÞ,and since r lðq;vÞ¼j B lðq;dðq;vÞÞj;we haver lðq;vÞj B lðq;2dðq;uÞÞjd2j B lðq;dðq;uÞÞj d2r lðq;uÞ:。

MATRICES WITH MAXIMUM UPPER MULTIEXPONENTS IN THE CLASS OF PRIMITIVE, NEARLY REDUCIBLE MATR

TAIWANESE JOURNAL OF MATHEMATICSVol.2,No.2,pp.181-190,June1998MATRICES WITH MAXIMUM UPPER MULTIEXPONENTSIN THE CLASS OF PRIMITIVE,NEARLY REDUCIBLE MATRICESZhou BoAbstract.B.Liu has recently obtained the maximum value for the k thupper multiexponents of primitive,nearly reducible matrices of ordern with1≤k≤n.In this paper primitive,nearly reducible matriceswhose k th upper multiexponents attain the maximum value are com-pletely characterized.1.IntroductionA square Boolean matrix A is primitive if one of its powers,A k,is the matrix J of all1’s for some integer k≥1.The smallest such k is called the primitive exponent of A.The matrix A is reducible if there is a permutation matrix P such thatP t AP= A10X A2 ,where A1and A2are square and nonvacuous;otherwise A is irreducible.The matrix A is nearly reducible if A is irreducible but each matrix obtained from A by replacing any nonzero entry by zero is reducible.It is well known that there is an obvious one-to-one correspondence between the set B n of n by n Boolean matrices and the set of digraphs on n vertices. Given A=(a ij)∈B n,the associated digraph D(A)has vertex set V(D(A))= {1,2,...,n},and arc set E(D(A))={(i,j):a ij=1}.A is primitive if and only if D(A)is strongly connected and the greatest common divisor(gcd for short)of all the distinct cycle lengths of D(A)is1,and A is nearly reducible if and only if D(A)is a minimally strongly connected digraph.We say a digraph 0Received May9,1996;revised September10,1997.Communicated by G.J.Chang.1991Mathematics Subject Classiﬁcation:05C50,15A33.Key words and phrases:Primitive matrix,nearly reducible matrix,upper multiexponent.181182Zhou Bois primitive with primitive exponentγif it is the associated digraph of some primitive matrix with primitive exponentγ.Now we give the deﬁnition of the upper multiexponent for a primitive digraph,which was introduced by R.A.Brualdi and B.Liu[1].Let D be a primitive digraph on n vertices.The exponent of a subset X⊆V(D)is the smallest integer p such that for each vertex i of D there exists a walk from at least one vertex in X to i of length p(and of course every length greater than p,since D is strongly connected).We denote it by exp D(X).The numberF(D,k)=max{exp D(X):X⊆V(D),|X|=k}is called the kth upper multiexponent of D.Clearly F(D,1)is the primitive exponent of D.Hence the k th upper mul-tiexponent of a primitive digraph is a generalization of its primitive exponent.Let A be an n×n primitive matrix,and let k be an integer with1≤k≤n. The k th upper multiexponent of A is the k th upper multiexponent of D(A), denoted by F(A,k).Thus F(A,k)=F(D(A),k).Clearly F(A,k)is the smallest power of A for which no set of k rows has a column consisting of all zeros.In[2]B.Liu obtained the maximum value for the k th upper multiexpo-nents of primitive,nearly reducible matrices of order n with1≤k≤n.In this paper,we provide a complete characterization of matrices in the class of n×n primitive,nearly reducible matrices whose k th upper multiexponents for 1≤k≤n attain the maximum value.Using the correspondence between matrices and digraphs,we express the results in the digraph version.2.Main ResultsWeﬁrst give several lemmas that will be used.Lemma1.[3].Let D be a primitive digraph on n vertices,1≤k≤n−1, and let h be the length of the shortest cycle of D.ThenF(D,k)≤n+h(n−k−1).LetF(n,k)= n2−4n+6,k=1;(n−1)2−k(n−2),2≤k≤n.The following lemma has been proved in[2]for n≥5.For n=4it can be checked readily.Matrices with Maximum Upper Multiexponents183 Lemma2.[2].F(D n−2,k)=F(n,k),n≥4,where D n−2is the digraph given by Fig.1.Lemma3.[1].Let D be a primitive digraph with n vertices and let h and t be respectively the smallest and the largest cycle lengths of D.ThenF(D,n−1)≤max{n−h,t}.Let P MD n be the set of all primitive,minimally strongly connected di-graphs with n vertices.The following theorem has recently been proved by B. Liu.Theorem1.[2].Max{F(D,k):D∈P MD n}=F(n,k),1≤k≤n.A problem that deserves investigation is to characterize the extreme di-graphs,or the digraphs in P MD n whose k th upper multiexponents assume the maximum value F(n,k).Obviously,for any D∈P MD n,F(D,n)=F(n,n)=1.We are going to consider the case1≤k≤n−1.Theorem2.Let D∈P MD n,1≤k≤n−1,n≥4.Then for1≤k≤n−2,F(D,k)=F(n,k)if and only if D∼=D n−2,where D n−2is the digraph given by Fig.1;F(D,n−1)=F(n,n−1)=n−1if and only if D∼=D n,s with1≤s≤n−3and gcd(n−1,s+1)=1,where D n,s is the digraph given by Fig.2.FIG.1.183184Zhou BoFIG.2.Remark.When s=n−3,D n,s is the digraph D n−2for n≥4.Proof.We begin the proof with the case k=n−1ﬁrst. Suppose D∼=D n,s.For i=2,3,...,s(s>1),any walk from the vertex i to the vertex n−1has a length of the form n−1−i+a(n−1)+b(s+1),where a and b are non-negative integers.Consider the equation n−1−i+a(n−1)+b(s+1)= n−2,i.e.,a(n−1)+b(s+1)=i−1.Since i≤s≤−3,we have a=0,b=0, which is imposssible.Hence there is no walk of length n−2from the vertex i to the vertex n−1for i=2,3,...,s.For i=s+1,...,n−1(s≥1),we have the same conclusion as above.Also it is easy to see that there is no walk of length n−2from the vertex n to the vertex n−1.Now take X0=V(D)\{1}. There does not exist any walk from a vertex in X0to the vertex n−1of lengthn−2.Hence exp Dn,s (X0)≥n−1.By the deﬁnition of the(n−1)th uppermultiexponent and Theorem1it follows thatF(D,n−1)=F(D n,s,n−1)=n−1.Conversely,suppose F(D,n−1)=n−1.Let h and t be respectively the smallest and the largest cycle lengths of D.D cannot have a cycle of length n,because,if so,the digraph is still strongly connected after the removal of any arc lying outside such cycle,contradicting the fact that D is minimally strongly connected.Similarly,we can show that D has no loops.So we have 2≤h≤n−2,t≤n−1.By Lemma3we obtainn−1=F(D,n−1)≤max{n−h,t},which implies t=n−1.Suppose D contains a cycle of length n−1whose arcs are(i,i+1)for i=1,2,...,n−2,and(n−1,1).By the strong connectedness of D there exist u and v(u and v may be equal)in{1,2,...,n−1}such thatMatrices with Maximum Upper Multiexponents185 (u,n)and(n,v)are arcs in D.Without loss of generality we assume that v=1.Thus D contains a subdigraph D n,u with1≤u≤n−3.Since D is minimally strong,it is easy to see that D has no arcs other than those in D n,u.It follows from the primitivity of D that gcd(n−1,u+1)=1.Now we turn to the case1≤k≤n−2.The case k=1is proved in[4]. Suppose2≤k≤n−2.If D∼=D n−2,by Lemma2we have F(D,k)=F(n,k). Conversely,suppose F(D,k)=F(n,k)and let h be the length of the shortest cycle in D.Since D is primitive,it has at least two diﬀerent cycle lengths. In addition,D has no cycles of length n,being a minimally strong connected digraph of order n.It follows that h≤n−2.If h=n−2,then the set of all distinct cycle lengths of D is{n−2,n−1}. By the minimally strong connectedness of D,it follows that D∼=D n−2.We are going to show that it is impossible to have h≤n−3.We divide our argument into two cases.Case1:2≤k<n−2.If h≤n−3,applying Lemma1we haveF(D,k)≤n+h(n−k−1)≤n+(n−3)(n−k−1)=(n−1)2−k(n−2)−(n−k−2)<(n−1)2−k(n−2)=F(n,k),a contradiction.Case2:k=n−2.If h≤n−4,by Lemma1,F(D,k)≤n+h(n−k−1)≤n+(n−4)(n−k−1)=2n−4<2n−3=F(n,k),a contradiction.If h=n−3,observing that D cannot have loops,we have h≥2and n≥5.If n=5,then h=2.Since D cannot have a cycle of length5and D is primitive,D must have a cycle of length3.It follows from the fact that D is minimally strongly connected that D is isomorphic with D1or D2or D3as displayed in Fig.3.In all such cases,it is easy to verify that we have F(D,1)≤6.Hence F(D,n−2)=F(D,3)≤F(D,1)≤6<7=F(5,3), which is a contradiction.Now suppose h=n−3and n>5.Since D cannot have a cycle of length n,by the primitivity of D,D must contain a cycle of length of n−2or n−1.185186Zhou BoFIG.3.If there is a walk of length t from vertex j to vertex i,we say that j is a t-in vertex of i.And the set of all t-in vertices of i in D is denoted by R D(t,i).Case2.1:D has no cycles of length n−1.Then D must have a cycle of length n−2.Take a cycle C of D of length n−2.Then D has precisely two vertices,say,x,y,lying outside C.We divide this situation into the following two subcases.(1)D contains one of the arcs(x,y)or(y,x).Say,D contains the arc (x,y).Then(y,x)cannot be an arc of D;otherwise,n−3=h=2and so n=5,which is a contradiction.By the strong connectedness of D,there must exist vertices u,v of C(the cycle of length n−2)such that(u,x)and(y,v) are both arcs of D.If u=v,then n−3=h=3,so we have n=6and D is the digraph D16−3.If u=v,then since D has precisely two cycles,of lengths n−2and n−3respectively,it will follow that D is isomorphic with D1n−3(n≥7).D1n−3(n≥6)is given by Fig.4.Suppose D=D1n−3.For n≥6,we describe R D(2n−5,i)explicitly:R D(2n−5,1)={n,1,2},R D(2n−5,i)={i−1,i,i+1},i=2,3,...,n−4,R D(2n−5,n−3)={n−4,n−3,n−2,n−1},R D(2n−5,n−2)={n−2,n−1,n,1},R D(2n−5,n−1)={n−4,n−3,n−2,n−1},R D(2n−5,n)={n−2,n−1,n,1}.Matrices with Maximum Upper Multiexponents187FIG.4.It is clear that each vertex has at least three(2n−5)-in vertices in D,and so exp D(X)≤2n−5for any set of n−2vertices.It follows from the deﬁnition of the(n−2)th upper multiexponent that F(D,n−2)≤2n−5<2n−3= F(D,n−2),which is a contradiction.(2)Neither(x,y)nor(y,x)is an arc of D.By the strong connectedness of D,there must exist vertices u,v,u and v of C such that(u,x),(x,v),(u ,y) and(y,v )are arcs of D.We have u=v and u =v ;otherwise,n−3=h=2 and so n=5,which is a contradiction.Also neither(u,v)nor(u ,v )is an arc of C;otherwise D has a cycle of length n−1,which is a contradiction. Suppose that uu1u2···u r v and u v1v2···v t v are two paths of C,of lengths r+1and t+1respectively,where r≥1and t≥1.If r=t=1,then by the minimally strong connectedness of D,D has no cycles of length h=n−3, which is a contradiction.If r≥3or t≥3,then there is a cycle with length less than h=n−3,which is also a contradiction.Hence we have r=2 or t=2.So D contains a subdigraph which is isomorphic with D(n−1)−2 (see Fig.1for D n−2).Assume D(n−1)−2is a subdigraph of D.Note that V(D(n−1)−2)={1,2,...,n−1}.By the strong connectedness of D,there exists a vertex j∈{1,2,...,n−1}such that(j,n)is an arc of D.Let X⊆V(D)with|X|=n−2.For each vertex1,2,...,n−1,there is a walk to the vertex from a vertex in X\{n}of length exp D(n−1)−2(X\{n})(and hence also every length greater).This is because,each such vertex belongs to the subgraph D(n−1)−2.Note thatexp D(n−1)−2(X\{n})≤ F(D(n−1)−2,n−2)=n−2,n∈X;F(D(n−1)−2,n−3)=2n−5,n∈X.187188Zhou BoSo exp D(n−1)−2(X\{n})≤2n−5whether n∈X or n∈X.Thus for everyinteger t≥2n−5,and for each vertex1,2,...,n−1,there is a walk to the vertex from a vertex in X\{n}of length t.Since j∈{1,2,...,n−1}and (j,n)is an arc of D,it follows that there is a walk to the vertex n from a vertex in X\{n}of length t+1for every integer t≥2n−5.So we have proved that there is a walk to each vertex of D from a vertex in X\{n}of length t+1for every integer t≥2n−5.This implies thatexp D(X)≤exp D(X\{n})≤2n−4<2n−3.By the deﬁnition of the(n−2)th upper multiexponent,we have F(D,n−2)< 2n−3=F(n,n−2),which is a contradiction.Case2.2:D has a cycle of length of n−1.Since h=n−3,D also has a cycle of length n−3.By the minimally strong connectedness of D,one can readily show that in this case D is composed of precisely two cycles,of lengths n−1and n−3respectively.But gcd{n−1,n−3}=1,so n is even,and D must be isomorphic with D2n−3(n≥6),where D2n−3is given by Fig.5.Suppose D=D2n−3.We haveR D(2n−4,1)={n,1,3},R D(2n−4,2)={2,4,n−3,n−1},R D(2n−4,3)={n,1,3,5},R D(2n−4,i)={i−2,i,i+2},i=4,...,n−3,R D(2n−4,n−2)={n−4,n−3,n−2,1},R D(2n−4,n−1)={n−3,n−1,2},R D(2n−4,n)={n−4,n,1}.By similar arguments as for the case D∼=D1n−3,we get F(D,n−2)≤2n−4<2n−3=F(n,n−2),which is also a contradiction.Now we have proved that it is impossible to have h≤n−3.Thus the proof of the theorem is completed.Theorem2gives complete characterizations of the extreme digraphs in the class of primitive,minimally strong digraphs of order n whose k th(1≤k≤n−1)upper multiexponents assume the maximum value.Note that there is not any digraph D in P MD n with F(D,1)=m if n2−5n+9<m<F(n,1),or n2−6n+12<m<n2−5n+9for n≥4(see [4]).As a by-product of the proof of Theorem2we have a similar result.Corollary1.Let k and n be integers.If2≤k≤n−3,then for any integer m satisfying n+(n−3)(n−k−1)<m<F(n,k),there is no digraph D∈P MD n such that F(D,k)=m.Matrices with Maximum Upper Multiexponents189FIG.5.D2(n is even,n≥6).n−3This corollary tells us that there are gaps in the set of k th upper multiex-ponents of digraphs in P MD n(1≤k≤n−3).Corollary2.The number of non-isomorphic extreme digraphs in P MD n with the(n−1)th upper multiexponent equal to n−1(n≥4)isφ(n−1)−1, whereφis Euler’s totient function.Finally,we point out that the maximum value for the k-exponents of primi-tive,nearly reducible matrices is also obtained in[2],and we have characterized the corresponding extreme matrices in another paper.AcknowledgementThe author would like to thank Professor B.Liu and Professor Gerard J.Chang for help and encouragement,and the referees for their numerous suggestions,which have resulted in a great improvement in the paper.References1.R.A.Brualdi and B.Liu,Generalized exponents of primitive directed graphs,J.Graph Theory14(1990),483-499.2. B.Liu,Generalized exponents of primitive,nearly reducible matrices,Ars Com-bin.to appear.3. B.Liu and Q.Li,On a conjecture about the generalized exponent of primitivematrices,J.Graph Theory18(1994),177-179.4.J.A.Ross,On the exponent of a primitive,nearly reducible matrix II,SIAMJ.Discrete Math.3(1982),395-410.189190Zhou Bo5. A.L.Dulmage and N.S.Mendelsohn,Gaps in the exponent set of primitivematrices,Illinois J.Math.8(1964),642-656.Department of Mathematics,South China Normal UniversityGuangzhou510631,China。

Superfluid Friction and Late-time Thermal Evolution of Neutron Stars

Bennett Link1 blink@ Montana State University, Department of Physics, Bozeman MT 59717 ABSTRACT The recent temperature measurements of the two older isolated neutron stars PSR 1929+10 and PSR 0950+08 (ages of 3 × 106 and 2 × 107 yr, respectively) indicate that these objects are heated. A promising candidate heat source is friction between the neutron star crust and the superﬂuid it is thought to contain. We study the eﬀects of superﬂuid friction on the long-term thermal and rotational evolution of a neutron star. Diﬀerential rotation velocities between the superﬂuid and the crust (averaged over the inner crust moment of inertia) of ω ¯ ∼ 0.6 rad s−1 for PSR 1929+10 and ∼ 0.02 rad s−1 for PSR 0950+08 would account for their observed temperatures. These diﬀerential velocities could be sustained by pinning of superﬂuid vortices to the inner crust lattice with strengths of ∼ 1 MeV per nucleus. Pinned vortices can creep outward through thermal ﬂuctuations or quantum tunneling. For thermally-activated creep, the coupling between the superﬂuid and crust is highly sensitive to temperature. If pinning maintains large diﬀerential rotation (∼ 10 rad s−1 ), a feedback instability could occur in stars younger than ∼ 105 yr causing oscillations of the temperature and spin-down rate over a period of ∼ 0.3tage . For stars older than ∼ 106 yr, however, vortex creep occurs through quantum tunneling, and the creep velocity is too insensitive to temperature for a thermal-rotational instability to occur. These older stars could be heated through a steady process of superﬂuid friction. Subject headings: stars: interiors — stars: neutron — stars: evolution — stars: stability — superﬂuid — dense matter

An upper bound on the multiplicative energy

BOUNDING MULTIPLICATIVE ENERGY BY THESUMSETJ ´OZSEF SOLYMOSIAbstract.We prove that the sumset or the productset of anyﬁnite set of real numbers,A,is at least |A |4/3−ε,improving earlier bounds.Our main tool is a new upper bound on the multiplicativeenergy,E (A,A ).1.IntroductionThe sumset of a ﬁnite set of an additive group,A,is deﬁned byA +A ={a +b :a,b ∈A }.The productset and ratioset are deﬁned in a similar way.AA ={ab :a,b ∈A },andA/A ={a/b :a,b ∈A }.A famous conjecture of Erd˝o s and Szemer´e di [4]asserts that for any ﬁnite set of integers,M,max {|M +M |,|MM |}≥|M |2−ε,where ε→0when |M |→∞.They proved thatmax {|M +M |,|MM |}≥|M |1+δ,for some δ>0.In a series of papers,lower bounds on δwere ﬁnd.δ≥1/31[9],δ≥1/15[5],δ≥1/4[2],and δ≥3/14[12].The last two bonds were proved for ﬁnite sets of real numbers.Date :June 23,2008.Research was supported by OTKA and NSERC grants and by a Sloan Fellowship.solymosi@math.ubc.ca.1a r X i v :0806.1040v 3 [m a t h .C O ] 23 J u n 20082J´OZSEF SOLYMOSI2.ResultsOur main result is the following.Theorem2.1.Let A be aﬁnite set of positive real numbers.Then|AA||A+A|2≥|A|44 log|A|holds.The inequality is sharp-up to the power of the log term in the de-nominator-when A is the set of theﬁrst n natural numbers.Theorem 2.1implies an improved bound on the sum-product problem. Corollary2.2.Let A be aﬁnite set of positive real numbers.Thenmax{|A+A|,|AA|}≥|A|4/32 log|A| 1/3holds2.1.Proof of Theorem2.1.To illustrate how the proof goes,we are making two unjustiﬁed and usually false assumptions,which are sim-plifying the proof.Readers,not interested about this”handwaving”, willﬁnd the rigorous argument about20lines below.Suppose that AA and A/A have the same size,|AA|≈|A/A|,and any element of A/A has about the same number of representations as any other.This means that for any reals s,t∈A/A the two num-bers s and t have the same multiplicity,|{(a,b)|a,b∈A,a/b=s}|≈|{(b,c)|b,c∈A,b/c=t}|.A geometric interpretation of the cardinal-ity of A/A is that the Cartesian product A×A is covered by|A/A| concurrent lines going through the bel the rays from the origin covering the points of the Cartesian product anticlockwise by r1,r2,...,r m,where m=|A/A|.Our assumptions imply that each ray is incident to|A|2/|AA|points of A×A.Consider the elements of A×A as two dimensional vectors. The sumset(A×A)+(A×A)is the same set as(A+A)×(A+A). We take a subset,S,of this sumset,S=m−1i=1(r i∩A×A)+(r i+1∩A×A)⊂(A+A)×(A+A).Simple elementary geometry shows(see the picture below)that the sumsets in the terms are disjoint and each term has|r i∩A×A||r i+1∩A×A|elements.Therefore|S|=|AA|(|A|2/|AA|)2≤|A+A|2.BOUNDING MULTIPLICATIVE ENERGY BY THE SUMSET3 After rearranging the inequality we get|A|4≤|AA||A+A|2,as we wanted.Now we will show a rigorous proof based on this observation. We are going to use the notation of multiplicative energy.The name of this quantity comes from a paper of Tao[13],however its discrete version was used earlier,like in[3].Let A be aﬁnite set of reals.The multiplicative energy of A,denoted by E(A),is given byE(A)=|{(a,b,c,d)∈A4|∃λ∈R:(a,b)=(λc,λd)}|.In the notatation of Gowers[7],the quantity E(A)counts the number of quadruples in log A.To complete the proof of Theorem2.1we show the following lemma. Lemma2.3.Let A be aﬁnite set of positive real numbers.ThenE(A)log|A|≤4|A+A|2.Theorem2.1follows from Lemma2.3via the Cauchy-Schwartz type inequality1E(A)≥|A|4|AA|.2.2.Proof of Lemma 2.3.Another way of counting E(A)is the following:(1)E(A)=x∈A/A|xA∩A|2.The summands on the right hand side can be partitioned into log|A| classes according to the size of xA∩A.E(A)= log|A|i=0x2i≤|xA∩A|<2i+1|xA∩A|2There is an index,I,thatE(A) log|A| ≤x2I≤|xA∩A|<2I+1|xA∩A|21This simple inequality appears in the literature in various places like in the proof of Theorem2.4[3],Remark4.2in[13],or Corollary2.10in[14].4J ´OZSEF SOLYMOSILet D ={s :2I ≤|sA ∩A |<2I +1},and let s 1<s 2<...<s m denote the elements of D,labeled in increasing order.(2)E (A ) log |A | ≤ x2I ≤|xA ∩A |<2I +1|xA ∩A |2<m 22I +2.Each line l j :y =s j x,where 1≤j ≤m,is incident to at least 2I and less than 2I +1points of A ×A.For easier counting we add an extra line to the set,l m +1,the vertical line through the smallest element of A,denoted by a 1.Line l m +1has |A |points from A ×A,however we are consideringonly the orthogonal projections of the points of l m .(ﬁg.1)The sumset 2,(l i ∩A ×A )+(l k ∩A ×A ),1≤j <k ≤m,has size |l i ∩A ×A ||l k ∩A ×A |,which is between 22I and 22I +2.Also,the sumsets along consecutive line pairs are disjoint,i.e.((l i ∩A ×A )+(l i +1∩A ×A ))∩((l k ∩A ×A )+(l k +1∩A ×A ))=∅,for any 1≤j <k ≤m.1AFigure 1.The sums are elements of (A +A )×(A +A ),so we have the following inequality.2Asusual,by the sum of two points on R 2we mean the point which is the sum of their position vectors.BOUNDING MULTIPLICATIVE ENERGY BY THE SUMSET5m22I≤mi=1(l i∩A×A)+(l i+1∩A×A)≤|A+A|2.The inequality above with inequality(2)proves the lemma.2.3.Remarks.Let A and B beﬁnite sets of reals.The multiplicative energy,E(A,B),is given byE(A,B)=|{(a,b,c,d)∈A×B×A×B|∃λ∈R:(a,b)=(λc,λd)}|. In the proof of Lemma2.3we did not use the fact that A=B,the proof works for the asymmetric case as well.Suppose that|A|≥|B|. With the lower bound on the multiplicative energyE(A,B)≥|A|2|B|2 |AB|our proof gives the more general inequality|A|2|B|2|AB|≤4 log|B| |A+A||B+B|.3.Very small productsetsIn this section we extend our method from two to higher dimensions. We are going to consider lines though the origin as before,however there is no notion of consecutiveness among these lines in higher di-mensions available.We will consider them as points in the projective real space and willﬁnd a triangulation of the pointset.The simplices of the triangulation will deﬁne the neighbors among the selected lines.The sum-product bound in Theorem2.1is asymmetric.It shows that the productset should be very large if the sumset is small.On the other hand it says almost nothing in the range where the productset is small.For integers,Chang[1]proved that there is a functionδ(ε) that if|AA|≤|A|1+εthen|A+A|≥|A|2−δ,whereδ→0ifε→0. Similar result is not known for reals.It follows from Elekes’bound [2](and also from Theorem2.1)that there is a functionδ(ε)that if |AA|≤|A|1+εthen|A+A|≥|A|3/2−δ,whereδ→0ifε→0.We prove here a generalization of this bound for k-fold sumsets.For any integer k≥2the k-fold subset of A,denoted by kA is the setkA={a1+a2+...+a k|a1,...,a k∈A}.6J´OZSEF SOLYMOSITheorem3.1.For any integer k≥2there is a functionδ=δk(ε) that if|AA|≤|A|1+εthen|kA|≥|A|2−1/k−δ,whereδ→0ifε→0. Proof.We can suppose that A has only positive elements WLOG.Let |AA|≤|A|1+ε.By a Pl¨u nnecke type inequality(Corollary5.2[10]or Chapter6.5[14])we have|A/A|≤|A|1+2ε.Consider the k-fold Carte-sian product A×A×...×A,denoted by×k A.It can be covered by no more than|A/A|k−1lines going through the origin.Picture2illus-Figure2.trates the k=3case.Let H denotes the set of lines through the origin containing at least|A|1−2ε(k−1)/2points of×k A.With this selection, the lines in H cover at least half of the points in×k A since|A|1−2ε(k−1)2|A/A|k−1=|A|k2|A|(1+2ε)(k−1)|A/A|k−1≤|A|k2.As no line has more than|A|points common with×k A,therefore|H|≥|A|k−1/2.The set of lines,H,represents a set of points,P,in the projective real space RP k−1.Point set P has full dimension k−1as it has a nice symmetry.The symmetry follows from the Cartesian product structure;if a point with coordinates(a1,...,a k)is in P then the point(σ(a1),...,σ(a k))is also in P for any permutationσ∈S k. Let us triangulate P.By triangulation we mean a decomposition of the convex hull of P into non-degenarate,k−1dimensional,simplicesBOUNDING MULTIPLICATIVE ENERGY BY THE SUMSET 7such that the intersection of any two is the empty set or a face of both simplices and the vertex set of the triangulation is P .It is not obvious that such triangulation always exists.For the proof we refer to Chapter 7in [6]or Chapter 2in [8].Let τ(P )be a triangulation of P.We say that k lines l 1,...,l k ∈H form a simplex if the corresponding points in P are vertices of a simplex of the triangulation.We use the following notation for this:{l 1,...,l k }∈τ(P ).In the two-dimensional case we used that the sumsets of points on consecutive lines are disjoint.Here we are using that the interiors of the simplices are disjoint,therefore sumsets of lines of simplices are also disjoint.Note that we assumed that A is positive,so we are considering convex combinations of vectors withpositive coeﬃcients.Let {l 1,...,l k }∈τ(P )and {l 1,...,l k }∈τ(P )are two distinct simplices.Then k i =1l i ∩×k A k i =1l i ∩×k A =∅.Also,since the k vectors parallel to the lines {l 1,...,l k }∈τ(P )are linearly independent,all sums are distinct, k i =1l i ∩×k A =k i =1l i ∩×k A .Now we are ready to put everything together into a sequence of in-equalities proving Theorem 3.1.|kA |k ≥ {l 1,...,l k }∈τ(P ) k i =1l i ∩×k A ≥|A |k −12k k i =1l i ∩×k A .Every line is is incident to at least |A |1−2ε(k −1)/2points of ×k A,there-fore|kA |k ≥|A |k −1+k (1−2ε(k −1))2k 2k =|A |2k −1−2k (k −1)εk 2k +1.Taking the k -th root of both sides we get the result we wanted to show|kA |≥c k |A |2−1/k −2(k −1)ε.References[1]M.C.Chang,The Erd˝o s-Szemer´e di problem on sum set and product set ,Annalsof Mathematics,157(2003),939-957.[2]Gy.Elekes,On the number of sums and products ,Acta Arith.,81(1997)365–367.8J´OZSEF SOLYMOSI[3]Gy.Elekes,Sums versus products in Number Theory,Algebra and Erd˝o s Ge-ometry—a survey,in:Paul Erd˝o s and his Mathematics II,Bolyai Math.Soc.Stud11Budapest,2002,241–290.[4]P.Erd˝o s and E.Szemer´e di,On sums and products of integers,in:Studies inPure Mathematics(Birkhauser,Basel,1983)213–218.[5]K.Ford,Sums and products from aﬁnite set of real numbers,Ramanujan J.2(1998)59–66.[6]I.M.Gelfand,M.M Kapranov,and A.V.Zelevinsky,Discriminants,Resultants,and Multidimensional Determiants in the series Modern Birkh¨a user Classics, Boston,(2008).Reprint of the1994Edition.512p.[7]W.T.Gowers,A new proof of Szemer´e dis theorem for arithmetic progressionsof length four,Geom.Func.Anal.8(1998),529–551.[8]J.A.De Lorea,J.Rambau,and F.S.Leal,Triangulations:Structures andAlgorithms,Springer,forthcoming[9]M.B.Nathanson,On sums and products of integers,Proc.Amer.Math.Soc.125(1997)9–16.[10]I.Z.Ruzsa,An application of graph theory to additive number theory,Scientia3(1989),97–109.[11]J.Solymosi,On sums and products of complex numbers,J.Th`e or.NombresBordeaux,17no.3(2005),921–924.[12]J.Solymosi,On the number of sums and products,The Bulletin of the LondonMathematical Society37(4)(2005),491–494.[13]T.Tao,Product set estimates for non-commutative groups,Combinatorica,toappear.(arXiv:math.CO/0601431)[14]T.Tao and V.Vu,Additive combinatorics Cambridge studies in advancedmathematics,105,Cambridge University Press,2006.。

Asymptotic Theory of Locally Conic Models and its Application to Multilayer Neural Networks

Asymptotic Theory of Locally Conic Models and its Application to Multilayer Neural NetworksKenji FukumizuThe Institute of Statistical MathematicsMinami-Azabu,Minato-ku,Tokyo106-8569JapanAbstract.This paper discusses the maximum likelihood estimation ina statistical model with unidentiÞability,using the framework of conicsingularity.The likelihood ratio may diverge in unidentiÞable cases,though in regular cases it converges to aχ2distribution.A useful suﬃ-cient condition of such divergence is obtained,and is applied to neuralnetworks.The exact order for multilayer perceptrons is also discussed.1IntroductionThis paper discusses the asymptotic behavior of the maximum likelihood estimator (MLE)under the condition that the true parameter is unidentiÞable.The asymptotic normality of MLE is well known under appropriate regularity conditions including the uniqueness of the true parameter.This gives a basis of many statistical methods such as model selection criteria and likelihood ratio test.However,if the true parameter is not a single point,the asymptotic behavior of MLE has not been clariÞed completely.We formulate such unidentiÞability as a conic singularity,following Dacunha-Castelle and Gassiat([1]),and investigate the likelihood ratio of MLE.We consider in partic-ular multilayer neural network models,for which the likelihood ratio corresponds to the training error.The likelihood ratio of the MLE,with the true probability at the singularity,can be described by a random process over the unit vectors in the tangent cone.The functional property of the tangent cone plays an essential role in the behavior of the random process.One interesting feature is the order of the likelihood ratio as the sample-size n goes to inÞnity.For models satisfying the usual asymptotic theory, the likelihood ratio is O p(1).However,in neural networks,a larger order log n has been reported in unidentiÞable cases([2]).We will give a useful suﬃcient condition of such a larger order,and derive the exact order for some neural network models.2UnidentiÞability and Conic Singularity2.1Statistical Estimation and Neural NetworksA statistical model S={f(z;θ)|θ∈Θ}is a family of probability density functions (p.d.f.)on a measure space(Z,B,µ),where the parameter spaceΘis a d-dimensional smooth manifold.We assume that f(z;θ)>0for all z andθ,and C∞onθfor each z∈Z.Let f(z)be a p.d.f.with f(z)>0.For a given i.i.d.sample Z1,...,Z n with law fµ,the likelihood ratio is deÞned byL n(θ)=1P n i=1log f(Z i;θ)i.(1)We consider the maximum likelihood estimator(MLE)ˆθ,which attains the maximum of the likelihood ratio;L n(ˆθ)=supθ∈ΘL n(θ).Let s(t)=tanh(t)andθ=(a j,c j,b j,d)T∈ΘH=R3H+1.The multilayer perceptron with H hidden units is a family of functions deÞned byϕ(x;θ)=P H j=1b j s(a j x+c j)+d.(2) We discuss only one-dimensional input and output for simplicity.Multilayer perceptrons can be regarded as a statistical model,by introducing an input probability Q=q(x)dx and a conditional p.d.f.r(y|u)of y given u∈R.The statistical model is deÞned byf(z;θ)=r(y|ϕ(x;θ))q(x),(3) where z=(x,y)∈Z=R2.Popular examples of r(y|u)are the Gaussian noisemodel r(y|u)=1√2πσexp{−12(y−u)2}for continuous y,and the binomial distributionr(y|u)=e yuufor binomial y∈{0,1},which is often used in classiÞcation problems.Throughout this paper,the true p.d.f.f(z)is assumed to be included in the model S.LetΘ0={θ∈Θ|f(z;θ)µ=f(z)µ}be the set of true parameters.We do not assumeΘ0is a single point.The true parameter is called unidentiÞable ifΘ0is a union of submanifolds ofΘand the dimension of one of the submanifolds is larger than zero.A true parameter of the multilayer perceptron model can be unidentiÞable.Suppose that we have the model with2hidden units,and the true function is given byϕ0(x)= b0s(a0x).In this case,ϕ0(x)is realized by any parameter in the set{θ∈Θ|a1= a0,b1=b0,c1=b2=d=0},which is high dimensional.It is known,in general,if the true function can be realized by a network with smaller number of hidden units than the model,the set of true parameters is always high dimensional([3]).If the true function is identiÞable,under some regularity conditions,2L n(ˆθ)→χ2d as n→∞.This is a foundation ofχ2test and model selection criterion like AIC.On the other hand,if the true parameter is unidentiÞable,the asymptotic theory does not necessarily hold.It is known in some models even the order of sup L n(θ)for n→∞is larger than the usual O p(1)([4],[2]).2.2Locally Conic ModelFollowing Dacunha-Castelle and Gassiat([1]),with some modiÞcations,we employ a conic singularity to formulate unidentiÞability.Let A0be a(d−1)-dimensional manifold,Θbe an open set in A0×R,and S={f(z;θ)|θ=(α,β)∈Θ}be a statistical model. The model S is called locally conic at f0∈S if the following conditions are satisÞed;(1)f(z;θ)is C∞function ofθfor almost every z.(2)LetΘ0=Θ∩(A0×{0}),andΘ(α)=Θ∩({α}×R).Then,Θ=∪α∈A0Θ(α).(3)The parameter set to give f0isΘ0,that is,f(z;(α,β))µ=f0(z)µ⇔β=0.(4)For allα∈A0,°°°∂log f(z;α,0)∂β°°°L2(f0µ)=1.Geometrically,S is a d-dimensional set in the space of p.d.f.with a singularity at f0, and is regarded as a union of one-dimensional submodels Sα={f(z;(α,β))|β∈Θ(α)} (Fig.1).The unit tangent vector,or the score function,of Sαat f0is given byvα(z)=∂log f(z;(α,0)).(4) The set of the unit tangent vectors,C={vα},generates the tangent cone at f0.We call C the basis of the tangent cone.Figure 1:Locally conic modelThe multilayer perceptron model is an example of locally conic model.Let Θ∗H ={θ∈ΘH |a j =0b j =0(1≤j ≤H ),(a j ,c j )=±(a h ,c h )(1≤j <h ≤H )}be the parameter space of the multilayer perceptrons with H hidden units,and ϕ(x )=P K k =1b 0k s (a 0k x +c 0k )+d 0((a 0k ,b 0k ,c 0k ,d 0)∈Θ∗K )be the true function (0≤K <H ).In the parameter space Θ∗H ,the true parameter is unidenti Þable.Given the functionϕ(x ),we slightly restrict Θ∗H to Θ∗∗H ={θ∈Θ∗H |(a j ,c j )=±(a 0k ,c 0k )(K +1≤j ≤H,1≤k ≤K )}.This restriction does not matter in discussing MLE,becauseit is in Θ∗∗H with probability one.Introducing new parameterization of Θ∗∗H by β=sgn(b K +1)q b 2K +1+···+b 2H ,ξk =a k −a 0k ,ξj =a j ,ηk =b k −b 0k ,ηj =b j ,ζk =c k −c 0k ,ζj =c j ,δ=d −d 0(1≤k ≤K,K +1≤j ≤H )for θ∈Θ∗∗H ,we can rewrite ϕ(x ;θ)by ψ(x ;ω)=P K k =1(b 0k +βηk )s ¡(a 0k +βξk )x +(c 0k +βζk )¢+P j>K βηj s (ξj x +ζj )+βδ,on the parameter space ΠH ={ω=(ξ1,...,ξH ,η1,...,ηH ,ζ1,...,ζH ,δ,β)|a 0k +βξk =0(1≤k ≤K ),ξj =0(K +1≤j ≤H ),(a 0k +βξk ,c 0k +βζk )=±(a 0h +βξh ,c 0h +βζh ),(1≤k <h ≤K ),(a 0k +βξk ,c 0k +βζk )=±(ξj ,ζj )(a 0k ,c 0k )=±(ξj ,ζj ),(1≤k ≤K,K +1≤j ≤H ),(ξj ,ζj )=±(ξi ,ζi )(K +1≤j <i ≤H ),b 0k +βηk =0(1≤k ≤K ),P H j =K +1η2j =1,ηj =0(K +1≤j ≤H ),ηK +1>0,β∈R }or Π∗∗H ={ω∈ΠH |β=0}.It is easy to see ϕ(x ;θ)=ψ(x ;ω)for the corresponding θ∈Θ∗∗H and ω∈Π∗∗H .We write ω=(α,β),summarizing (ξ1,...,ηH )by α.We have ψ(x ;ω)=ϕ(x )i ﬀβ=0.Let r (y |u )be a noise model,S H be {f (x,y ;ω)=r (y |ψ(x ;ω))q (x )|ω∈ΠH },and f 0,K (x,y )∈S K ⊂S H be given by ϕ(x ).Then,we obtain Proposition 1([5]).Under some regularity condition on r (y |u ),the model S H is lo-cally conic at f 0,K (0≤K <H ).3Maximum Likelihood Estimation in Locally Conic Models3.1Maximum Likelihood Estimation as a Supremum of a Random Process Let S ={f (z ;(α,β))|(α,β)∈Θ}be locally conic at f 0∈S .Suppose Z 1,...,Z n are i.i.d.with the law f 0µ,and ˆβαis the MLE in the submodel S α.Then,likelihood ratio of MLE in S sup θ∈ΘL n (θ)is equal to sup αL n (α,ˆβα).Assume that each S αsatis Þes the regularity condition of the asymptotic e ﬃciency.Then,Taylor expansion leads toL n (α,ˆβα)=1U n (α)2+o p (1),where U n (α)=¡1√n P n i =1v α(Z i )¢¡1P n i =1v α(Z i )2¢−1/2.The variable U n(α)converges in law to N(0,1)for eachα.If we consider U n(α)over all α,it can be looked as a random process over A0={α}or C.If the higher order term o p(1)is uniform overα,and the process U n converges uniformly to a Gaussian process W,the likelihood ratio converges to1supαW(α)2.Dacunha-Castelle and Gassiat[1] discussed such uniform cases,while,in the following,we will consider non-uniform cases.3.2Divergence of the Likelihood RatioA marginal of U n onÞnite points in C converges to a normal distribution with covariance E P[v i v j].The components are independent if the covariance is zero.If we canÞnd arbitrary number of”almost independent”variables in C,the supremum of U n takes an arbitrary large value.Hartigan[4]applied this idea to a normal mixture,calculating the covariance explicitly.His method can be extended to the following theorem. Theorem1([5]).A statistical model S={f(z;(α,β)}is locally conic at f0∈S,and C={vα(z)=∂f(z;(α,0)∂β}.Suppose there exists a sequence{v n}in C such that v n→0 in probability-f0µ.Then,under some regularity conditions,for arbitrary M>0lim n→∞Prob(sup(α,β)L n(α,β)≤M)=0.(5)4Likelihood Ratio of Multilayer PerceptronsApplying Theorem1to multilayer perceptrons,we can obtainTheorem2.Assume that the model is the multilayer perceptron model(2)with H hidden units,and the true function is given by a network with K hidden units for K<H.Under some regularity conditions on r(y|u),we have for arbitrary M>0,lim n→∞Prob¡supθL n(θ)≤M¢=0.(6) Sketch of the proof.It suﬃces to consider a submodel in the parameterizationΠH.Let σ(x;ξ,h)be a function given byσ(x;ξ,h)=1{1+s(−1ξ(x−h))}=1, and{g(z;t,c)}be a submodel deÞned by g(z;t,c,β)=r(y|ϕ0(x)+βw(x;t,c))q(x),√B(t,c)σ(x;c2,t+1)and B(t,c)is a normalizing constant of L2(f0µ) where w(x;t,c)=1norm.Then,the basis of the tangent cone consists of the functions v(x,y;t,c)=√B(t,c)∂log r(y|ϕ0(x))1σ(x;c2,t+1).Let t0=inf{t∈R|F Q(t)>0}.We can show that it ∂uhas a sequence converges to zero,if we choose tn↓t0and c n→∞appropriately.We can derive a tighter lower bound log n,if there are at least two redundant hidden units.In fact,we canÞnd nγunit tangent vectors,for which the random elements U n(α) are very close to nγindependent samples from N(0,1).For the detail,see[5]. Theorem3.Suppose that the model is the multilayer perceptron with H hidden units, and the true function is given by a network with K hidden units for K≤H−2.Then, under some regularity conditions on r(y|u),there existsδ>0such thatlim inf n→∞Prob¡supθL n(θ)≥δlog n¢>0.(7)The order log n has been obtained by Hagiwara et al.[2]under much stronger assumptions.Theorem3is applicable to various noise models,including binary output.The upper bound is also a very important problem.In the case of the additive Gaussian noise,we can obtain the following upper bound of O p(log n).This theorem does not use the assumption of locally conic parameterization.Theorem4.Let F={ϕ(x;θ)|θ∈Θ}be a family of functions,and r(y|u)= 1√2πexp{−1(y−u)2}.Assume the VC-dimension of F isÞnite,and the true func-tionϕ0∈F is bounded.Then,for the statistical model r(y|ϕ(x;θ))q(x),we havesupθL n(θ)=O p(log n).(8) Sketch of the proof.Since the probability of the event max1≤i≤n y i>2√log n con-√log n holds.For a verges to0,we can consider the case in which|ϕ(x i;θ)|≤2given X=(X1,...,X n),the conditional probability of L n(θ)−E[L n(θ)|X]is the normal distribution with mean zero and variance Vθ,X=P n i=1(ϕ(X i;θ)−ϕ0(X i))2. By the inequality for the tail of a normal distribution,for arbitraryλ>0we have Prob¡L n(θ)≥λ−1Vθ,X|X¢≤√θ,X√2πλe−λ22Vθ,X.Settingλ=M log n+1Vθ,X for M>0,Prob¡L n(θ)≥M log n¢≤1√2πn−M√2M log n.(9) Since the VC dimension of F isÞnite,we canÞnd nγ(γ>0)parameters{θk}such that for arbitraryϕ(x;θ)∈F there existsθk satisfying|L n(θk)−L n(θ)|≤p(10) Combining eqs.(9)and(10),we can prove the theorem.From theorems3and4,we can conclude that under the same assumptions as The-orem3and of additive Gaussian noise,the order of supθL n(θ)is log n.5Concluding RemarksWe have elucidated a useful suﬃcient condition of divergence of the likelihood ratio, using the framework of locally conic models.For multilayer perceptrons with at least two redundant hidden units,we have derived the exact order log n.However,this order is not universal for unidentiÞable models,and it is known in some type of change point problems and normal mixture models,which also suﬀer similar unidentiÞability,the order is log log n.The characterization of the orders and general methodology to derive the order for each model are not known yet.They are future problems.References[1]D.Dacunha-Castelle and E.Gassiat,Testing in locally conic models and application tomixture models,ESAIM Prob.and Statist.1(1997)285—317.[2]K.Hagiwara et al.,On the problem in model selection of neural network regression inoverrealizable scenario,in Proc.Intern.Joint Conf.Neural Networks,2000.[3]K.Fukumizu and S.Amari,Local Minima and Plateaus in Hierarchical Structures ofMultilayer Perceptrons,Neural Networks13(2000)317—327.[4]J.A.,Hartigan,A failure of likelihood asymptotics for normal mixtures.:In Proc.BerkeleyConf.in Honor of Jerzy Neyman and Jack Kiefer,vol.II,1985,pp.807—810.[5]K.Fukumizu,Likelihood Ratio of UnidentiÞable Models and Multilayer Neural Networks.ISM Research Memorandum No.780,(2001).http://www.ism.ac.jp/˜fukumizu/papers/memo780.pdf。

limit superior and limit inferior

Limit superior and limit inferiorIn mathematics, the limit inferior (also called infimum limit , liminf , inferior limit , lower limit , or inner limit )and limit superior (also called supremum limit , limsup , superior limit , upper limit , or outer limit ) of a sequence can be thought of as limiting (i.e., eventual and extreme) bounds on the sequence. The limit inferior and limit superior of a function can be thought of in a similar fashion (see limit of a function). The limit inferior and limit superior of a set are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant.An illustration of limit superior and limit inferior. The sequence x n is shown inblue. The two red curves approach the limit superior and limit inferior of x n , shownas solid red lines to the right. In this case, the sequence accumulates around thetwo limits. The superior limit is the larger of the two, and the inferior limit is thesmaller of the two. The inferior and superior limits only agree when the sequenceis convergent (i.e., when there is a single limit).Definition for sequencesThe limit inferior of a sequence (x n ) isdefined byorSimilarly, the limit superior of (x n) is defined byorIf the terms in the sequence are real numbers, the limit superior and limit inferior always exist, as real numbers or ±∞ (i.e., on the extended real number line). More generally, these definitions make sense in any partially ordered set,provided the suprema and infima exist, such as in a complete lattice.Whenever the ordinary limit exists, the limit inferior and limit superior are both equal to it; therefore, each can be considered a generalization of the ordinary limit which is primarily interesting in cases where the limit does not exist. Whenever lim inf x n and lim sup x nboth exist, we haveLimits inferior/superior are related to big-O notation in that they bound a sequence only "in the limit"; the sequence may exceed the bound. However, with big-O notation the sequence can only exceed the bound in a finite prefix of the sequence, whereas the limit superior of a sequence like e -n may actually be less than all elements of the sequence. The only promise made is that some tail of the sequence can be bounded by the limit superior (inferior) plus (minus)an arbitrarily small positive constant.The limit superior and limit inferior of a sequence are a special case of those of a function (see below).The case of sequences of real numbersIn mathematical analysis, limit superior and limit inferior are important tools for studying sequences of real numbers. In order to deal with the difficulties arising from the fact that the supremum and infimum of an unbounded set of real numbers may not exist (the reals are not a complete lattice), it is convenient to consider sequences in the affinely extended real number system: we add the positive and negative infinities to the real line to give the complete totally ordered set [-∞,∞], which is a complete lattice.InterpretationConsider a sequence consisting of real numbers. Assume that the limit superior and limit inferior are real numbers (so, not infinite).•The limit superior of is the smallest real number such that, for any positive real number , there exists anatural number such that for all . In other words, any number larger than the limit superior is an eventual upper bound for the sequence. Only a finite number of elements of the sequence are greater than .•The limit inferior of is the largest real number that, for any positive real number , there exists a naturalnumber such that for all . In other words, any number below the limit inferior is an eventual lower bound for the sequence. Only a finite number of elements of the sequence are less than .PropertiesThe relationship of limit inferior and limit superior for sequences of real numbers is as follows) in [−∞,∞] converges if and only ifAs mentioned earlier, it is convenient to extend R to [−∞,∞]. Then, (xnin which case is equal to their common value. (Note that when working just in R, convergence to −∞ or ∞would not be considered as convergence.) Since the limit inferior is at most the limit superior, the conditionimplies thatand the conditionimplies that= sin(n). Using the fact that pi is irrational, one can show thatAs an example, consider the sequence given by xnand(This is because the sequence {1,2,3,...} is equidistributed mod 2π, a consequence of the Equidistribution theorem.) Ifand, but every slight enlargement [I − ε, S + ε] (for then the interval [I, S] need not contain any of the numbers xnarbitrarily small ε > 0) will contain xfor all but finitely many indices n. In fact, the interval [I, S] is the smallestnclosed interval with this property. We can formalize this property like this. If there exists a so thatthen there exists a subsequence of for which we have thatIn the same way, an analogous property holds for the limit inferior: ifthen there exists a subsequence of for which we have thatOn the other hand we have that ifthere exists a so thatSimilarly, if there exists a so thatthere exists a so thatTo recapitulate:•If is greater than the limit superior, there are at most finitely many greater than ; if it is less, there areinfinitely many.•If is less than the limit inferior, there are at most finitely many less than ; if it is greater, there are infinitely many.In general we have thatThe liminf and limsup of a sequence are respectively the smallest and greatest cluster points.An example from number theory iswhere pis the n-th prime number. The value of this limit inferior is conjectured to be 2 - this is the twin prime nconjecture - but as yet has not even been proved finite. The corresponding limit superior is , because there arearbitrary gaps between consecutive primes.•For any two sequences of real numbers , the limit superior satisfies superadditivity: (handling appropriately)Analogously, if is handled with care, the limit inferior satisfies subadditivityIn the particular case that one of the sequences actually converges, say , then the inequalities abovebecome equalities (with or being replaced by ).If the limit superior and limit inferior converge to the same value:Then the limit converges to that valueReal-valued functionsAssume that a function is defined from a subset of the real numbers to the real numbers. As in the case for sequences, the limit inferior and limit superior are always well-defined if we allow the values +∞ and -∞; in fact, if both agree then the limit exists and is equal to their common value (again possibly including the infinities). Forexample, given f(x) = sin(1/x), we have lim supx→0f(x) = 1 and lim infx→0f(x) = -1. The difference between the twois a rough measure of how "wildly" the function oscillates, and in observation of this fact, it is called the oscillation of f at a. This idea of oscillation is sufficient to, for example, characterize Riemann-integrable functions as continuous except on a set of measure zero [1]. Note that points of nonzero oscillation (i.e., points at which f is "badly behaved") are discontinuities which, unless they make up a set of zero, are confined to a negligible set.Functions from metric spaces to metric spacesThere is a notion of lim sup and lim inf for functions defined on a metric space whose relationship to limits of real-valued functions mirrors that of the relation between the lim sup, lim inf, and the limit of a real sequence. Take metric spaces X and Y, a subspace E contained in X, and a function f : E → Y. The space Y should also be an ordered set, so that the notions of supremum and infimum make sense. Define, for any limit point a of E,andwhere B(a;ε) denotes the metric ball of radius ε about a.Note that as ε shrinks, the supremum of the function over the ball is monotone decreasing, so we haveand similarlyThis finally motivates the definitions for general topological spaces. Take X, Y, E and a as before, but now let X and Y both be topological spaces. In this case, we replace metric balls with neighborhoods:(there is a way to write the formula using a lim using nets and the neighborhood filter). This version is often useful in discussions of semi-continuity which crop up in analysis quite often. An interesting note is that this version subsumes the sequential version by considering sequences as functions from the natural numbers as a topological subspace of the extended real line, into the space (the closure of N in [-∞, ∞] is N∪ {∞}.)Sequences of setsThe power set ℘(X ) of a set X is a complete lattice that is ordered by set inclusion, and so the supremum and infimum of any set of sets, in terms of set inclusion, of subsets always exist. In particular, every subset Y of X is bounded above by X and below by the empty set ∅ because ∅ ⊆ Y ⊆ X . Hence, it is possible (and sometimes useful) to consider superior and inferior limits of sequences in ℘(X ) (i.e., sequences of subsets of X ).There are two common ways to define the limit of sequences of set. In both cases:•The sequence accumulates around sets of points rather than single points themselves. That is, because each element of the sequence is itself a set, there exist accumulation sets that are somehow nearby to infinitely many elements of the sequence.•The supremum/superior/outer limit is a set that joins these accumulation sets together. That is, it is the union of all of the accumulation sets. When ordering by set inclusion, the supremum limit is the least upper bound on the set of accumulation points because it contains each of them. Hence, it is the supremum of the limit points.•The infimum/inferior/inner limit is a set where all of these accumulation sets meet. That is, it is the intersection of all of the accumulation sets. When ordering by set inclusion, the infimum limit is the greatest lower bound on the set of accumulation points because it is contained in each of them. Hence, it is the infimum of the limit points.•Because ordering is by set inclusion, then the outer limit will always contain the inner limit (i.e., lim inf X n ⊆lim sup X n ).The difference between the two definitions involves the topology (i.e., how to quantify separation) is defined. In fact,the second definition is identical to the first when the discrete metric is used to induce the topology on X .General set convergenceIn this case, a sequence of sets approaches a limiting set when its elements of each member of the sequence approach that elements of the limiting set. In particular, if {X n } is a sequence of subsets of X , then:•lim sup X n , which is also called the outer limit , consists of those elements which are limits of points in X n taken from (countably) infinitely many n . That is, x ∈ lim sup X n if and only if there exists a sequence of points x k and a subsequence {X n k } of {X n } such that x k ∈ X n k and x k → x as k → ∞.•lim inf X n , which is also called the inner limit , consists of those elements which are limits of points in X n for all but finitely many n (i.e., cofinitely many n ). That is, x ∈ lim inf X n if and only if there exists a sequence of points {x k } such that x k ∈ X k and x k → x as k → ∞.The limit lim X exists if and only if lim inf X and lim sup X agree, in which case lim X = lim sup X = lim inf X .[2]Special case: discrete metricIn this case, which is frequently used in measure theory, a sequence of sets approaches a limiting set when the limiting set includes elements from each of the members of the sequence. That is, this case specializes the first case when the topology on set X is induced from the discrete metric. For points x ∈ X and y ∈ X , the discrete metric is defined bySo a sequence of points {x k } converges to point x ∈ X if and only if x k = x for all but finitely many k . The following definition is the result of applying this metric to the general definition above.If {X n } is a sequence of subsets of X , then:•lim sup X n consists of elements of X which belong to X n for (countably) infinitely many values of n . That is, x ∈lim sup X n if and only if there exists a subsequence {X n k } of {X n } such that x ∈ X n k for all k .•lim inf Xn consists of elements of X which belong to Xnfor all but finitely many n (i.e., for cofinitely many n).That is, x∈ lim inf Xn if and only if there exists some m>0 such that x∈ Xnfor all n>m.The limit lim X exists if and only if lim inf X and lim sup X agree, in which case lim X = lim sup X = lim inf X.[3] This definition of the inferior and superior limits is relatively strong because it requires that the elements of the extreme limits also be elements of each of the sets of the sequence.Using the standard parlance of set theory, consider the infimum of a sequence of sets. The infimum is a greatest lower bound or meet of a set. In the case of a sequence of sets, the sequence constituents meet at an set that is somehow smaller than each constituent set. Set inclusion to provides an ordering that allows set intersection togenerate a greatest lower bound ∩Xn of sets in the sequence {Xn}. Similarly, the supremum, which is the least upperbound or join, of a sequence of sets is the union ∪Xn of sets in sequence {Xn}. In this context, the inner limitlim inf Xn is the largest meeting of tails of the sequence, and the outer limit lim sup Xnis the smallest joining of tailsof the sequence.•Let Inbe the meet of the n th tail of the sequence. That is,Then Ik ⊆ Ik+1⊆ Ik+2because Ik+1is the intersection of fewer sets than Ik. In particular, the sequence {Ik} isnon-decreasing. So the inner/inferior limit is the least upper bound on this sequence of meets of tails. In particular,So the superior limit acts like a version of the standard supremum that is unaffected by set elements that occur only finitely many times. That is, the superemum limit is a set that is a superset (i.e., an upper bound) for all but finitely many elements.•Similarly, let Jmbe the join of the m th tail of the sequence. That is,Then Jk ⊇ Jk+1⊇ Jk+2because Jk+1is the union of fewer sets than Jk. In particular, the sequence {Jk} isnon-increasing. So the outer/superior limit is the greatest lower bound on this sequence of joins of tails. In particular,So the inferior limit acts like a version of the standard infimum that is unaffected by set elements that occur only finitely many times. That is, the infimum limit is a set that is a subset (i.e., a lower bound) for all but finitely many elements.The limit lim Xn exists if and only if lim sup Xn=lim inf Xn, and in that case, lim Xn=lim inf Xn=lim sup Xn. In thissense, the sequence has a limit so long as all but finitely many of its elements are equal to the limit.ExamplesThe following are several set convergence examples. They have been broken into sections with respect to the metric used to induce the topology on set X.Using the discrete metric•The Borel–Cantelli lemma is an example application of these constructs.Using either the discrete metric or the Euclidean metric•Consider the set X = {0,1} and the sequence of subsets:The "odd" and "even" elements of this sequence form two subsequences, {{0},{0},{0},...} and {{1},{1},{1},...}, which have limit points 0 and 1, respectively, and so the outer or superior limit is the set {0,1} of these two points. However, there are no limit points that can be taken from the {Xn} sequence as a whole, and so the interior or inferior limit is the empty set {}. That is,•lim sup Xn= {0,1}•lim inf Xn= {}However, for {Yn } = {{0},{0},{0},...} and {Zn} = {{1},{1},{1},...}:•lim sup Yn = lim inf Yn= lim Yn= {0}•lim sup Zn = lim inf Zn= lim Zn= {1}•Consider the set X = {50, 20, -100, -25, 0, 1} and the sequence of subsets:As in the previous two examples,•lim sup Xn= {0,1}•lim inf Xn= {}That is, the four elements that do not match the pattern do not affect the lim inf and lim sup because there are only finitely many of them. In fact, these elements could be placed anywhere in the sequence (e.g., at positions 100, 150, 275, and 55000). So long as the tails of the sequence are maintained, the outer and inner limits will be unchanged. The related concepts of essential inner and outer limits, which use the essential supremum and essential infimum, provide an important modification that "squashes" countably many (rather than just finitely many) interstitial additions.Using the Euclidean metric•Consider the sequence of subsets of rational numbers:The "odd" and "even" elements of this sequence form two subsequences, {{0},{1/2},{3/4},{4/5},...} and {{1},{1/2},{1/3},{1/4},...}, which have limit points 1 and 0, respectively, and so the outer or superior limit isthe set {0,1} of these two points. However, there are no limit points that can be taken from the {Xn} sequence as a whole, and so the interior or inferior limit is the empty set {}. So, as in the previous example,•lim sup Xn= {0,1}•lim inf Xn= {}However, for {Yn } = {{0},{1/2},{3/4},...} and {Zn} = {{1},{1/2},{1/3},...}:•lim sup Yn = lim inf Yn= lim Yn= {1}•lim sup Zn = lim inf Zn= lim Zn= {0}In each of these four cases, the elements of the limiting sets are not elements of any of the sets from the original sequence.•The Ω limit (i.e., limit set) of a solution to a dynamic system is the outer limit of solution trajectories of the system.[2]:50–51 Because trajectories become closer and closer to this limit set, the tails of these trajectories converge to the limit set.•For example, an LTI system that is the cascade connection of several stable systems with an undamped second-order LTI system (i.e., zero damping ratio) will oscillate endlessly after being perturbed (e.g., an ideal bell after being struck). Hence, if the position and velocity of this system are plotted against each other,trajectories will approach a circle in the state space. This circle, which is the Ω limit set of the system, is the outer limit of solution trajectories of the system. The circle represents the locus of a trajectory corresponding toa pure sinusoidal tone output; that is, the system output approaches/approximates a pure tone.Generalized definitionsThe above definitions are inadequate for many technical applications. In fact, the definitions above are specializations of the following definitions.Definition for a setThe limit inferior of a set X⊆ Y is the infimum of all of the limit points of the set. That is,Similarly, the limit superior of a set X is the supremum of all of the limit points of the set. That is,Note that the set X needs to be defined as a subset of a partially ordered set Y that is also a topological space in order for these definitions to make sense. Moreover, it has to be a complete lattice so that the suprema and infima always exist. In that case every set has a limit superior and a limit inferior. Also note that neither the limit inferior nor the limit superior of a set must be an element of the set.Definition for filter basesTake a topological space X and a filter base B in that space. The set of all cluster points for that filter base is given bywhere is the closure of . This is clearly a closed set and is similar to the set of limit points of a set. Assumethat X is also a partially ordered set. The limit superior of the filter base B is defined aswhen that supremum exists. When X has a total order, is a complete lattice and has the order topology,Proof: Similarly, the limit inferior of the filter base B is defined aswhen that infimum exists; if X is totally ordered, is a complete lattice, and has the order topology, thenIf the limit inferior and limit superior agree, then there must be exactly one cluster point and the limit of the filter base is equal to this unique cluster point.Specialization for sequences and netsNote that filter bases are generalizations of nets, which are generalizations of sequences. Therefore, these definitions give the limit inferior and limit superior of any net (and thus any sequence) as well. For example, take topological space and the net , where is a directed set and for all . The filter base("of tails") generated by this net is defined byTherefore, the limit inferior and limit superior of the net are equal to the limit superior and limit inferior ofrespectively. Similarly, for topological space , take the sequence where for any withbeing the set of natural numbers. The filter base ("of tails") generated by this sequence is defined byTherefore, the limit inferior and limit superior of the sequence are equal to the limit superior and limit inferior of respectively.See also•Essential supremum and essential infimumReferencesIn-line references[1]mf.uwindsor.ca/314folder/analbookfiles/RintexistLebesgue.pdf[2]Goebel, Rafal; Sanfelice, Ricardo G.; Teel, Andrew R. (2009). "Hybrid dynamical systems". IEEE Control Systems Magazine29 (2): 28–93.doi:10.1109/MCS.2008.931718.[3]Halmos, Paul R. (1950). Measure Theory. Princeton, NJ: D. Van Nostrand Company, Inc..General references•Amann, H.; Escher, Joachim (2005). Analysis. Basel; Boston: Birkhäuser. ISBN 0817671536.•González, Mario O (1991). Classical complex analysis. New York: M. Dekker. ISBN 0824784154.Article Sources and Contributors10 Article Sources and ContributorsLimit superior and limit inferior Source: /w/index.php?oldid=374368501 Contributors: ABCD, Alberto da Calvairate, Almwi, AxelBoldt, Bkell, Bluestarlight37,Bubba73, CRGreathouse, Charles Matthews, Choni, Cronholm144, Dcoetzee, Dfeuer, Dmyersturnbull, Dysprosia, Eequor, Eggstone, Eighty, Elroch, Giftlite, Jsnx, Julioc, Kavas, LachlanA, Lhf, LilHelpa, Localhost00, Madmath789, Markjoseph125, Michael Hardy, Miguel, NeoUrfahraner, Oleg Alexandrov, PV=nRT, Patrick, Paul August, Pbroks13, Petter Strandmark, Pexatus, Pomte, Robertvan1, Salgueiro, Salix alba, Schildt.a, Sligocki, Small potato, StradivariusTV, Sullivan.t.j, Tcnuk, TedPavlic, Thamuzino, Tosha, Trevorgoodchild, Tristanreid, VectorPosse, WAREL,Zundark, 71 anonymous editsImage Sources, Licenses and ContributorsImage:LimSup.svg Source: /w/index.php?title=File:LimSup.svg License: Public Domain Contributors: Pbroks13LicenseCreative Commons Attribution-Share Alike 3.0 Unported/licenses/by-sa/3.0/。

Solution of the strong CP problem

a r X i v :h e p -p h /9611266v 1 7 N o v 1996Solution of the strong CP problemvouraUniversidade T´e cnica de LisboaCFIF,Instituto Superior T´e cnico,Edif´ıcio Ciˆe ncia (f´ısica)P-1096Lisboa Codex,Portugal February 1,2008Abstract I put forward an SU(2)L ⊗SU(2)R ⊗U(1)model in which spontaneously broken parity symmetry makes it that strong CP violation only arises at three-loop level.All leptons and up-type quarks are in doublets either of SU(2)L or of SU(2)R ,but there are singlet down-type quarks.A bi-doublet of scalars is introduced which has only one component with non-vanishing expectation value,thereby avoiding W L -W R mixing.The scalar potential is such that scalar-pseudoscalar mixing does not occur either.1Introduction Non-perturbative eﬀects in Quantum Chromodynamics (QCD)may lead to P and CP violation,characterized by a parameter θ,in hadronic processes.The experimental upper bound on the electric dipole moment of the neutron constrains θto be less than 10−9or so.The presence of this unnaturally small number in QCD is what is known as the strong CP problem.θis the sum of two terms,θQCD and θQF D .θQCD is the original value of the angle θcharacterizing the QCD vacuum.θQF D originates in the chiral rotation of the quark ﬁelds needed to render the quark masses real and positive.If M p and M n are the mass matrices of the up-type and down-type quarks,then θQF D =arg det(M p M n ).There are two general ways of solving the strong CP problem.In the ﬁrst approach it is claimed that θhas no signiﬁcance or physical consequences;theories with diﬀerent values of θare equivalent,and therefore we may set θ=0without loss of generality.This may be so because of the presence in the theory of a Peccei–Quinn symmetry [1],but there are also claims that QCD dynamics itself cures the strong CP problem [2].The second path,which I shall follow,tries to ﬁnd some symmetry which naturally leads to the smallness of θ.As θis both CP-and P-violating,we ﬁrst assume the Lagrangian (or at least its quartic part)to be either CP-or P-symmetric,thereby automatically obtaining θQCD =0.CP or P symmetry must then be either softly or spontaneously broken.While doing this the problem of ensuring the smallness ofθQF D remains.This is quitediﬃcult when using CP symmetry.After CP is broken softly or spontaneously,it is diﬃcult to avoid one-loop contributions toθQF D from the quark self-energies[3],because each neutral spin-0particle will usually have both scalar and pseudoscalarinteractions with each quark.Georgi[4]realized this,and claimed thatθQF D thus generated would be of order10−8.However,in order to make this estimate heassumed all quarks to have masses at most of order10GeV.Once it is known that the top mass is much larger than this,a suppression factor is lost,and a morelikely estimate isθQF D∼10−5,which is unacceptable.Most models using CP to suppressθ[5]suﬀer from this problem in one way or another.A remarkable exception is the model of Bento and collaborators[6].In1990theﬁrst model appeared[7]which used P symmetry to suppressθ.The general features that models of this kind have to satisfy were later analysed by Barr and collaborators[8].They claimed that a model which uses parity to suppressθmust have both mirror quarks and a duplication of the standard-model (SM)SU(2)gauge group.In their own words:“The mirror families must have diﬀerent weak interactions from the ordinary V−A ones.There is simply no escape from this conclusion;one has to double both the electroweak group and the fermionic content”.In this paper I construct a model in which the SU(2)gauge group is indeeddoubled,but the lepton and up-type-quark content is the same as in the SM, while the down-type quarks are doubled.In my modelθQF D only arises at three-loop level.The model is somehow intermediate between the standard left-right-symmetric model[9]and the models with mirror quarks[7,8],but it has some odd features,as I shall point out.2The model2.1Scalar potentialThe gauge group of the model is SU(2)L⊗SU(2)R⊗U(1).The scalar sector con-sists of a bi-doubletφ(with˜φ≡τ2φ∗τ2),a doubletχL of SU(2)L,and a doublet χR of SU(2)R.The Lagrangian is assumed to be symmetric under parity,which transformsφintoφ†and interchangesχL withχR.The gauge coupling constant of SU(2)L is equal to the one of SU(2)R,and I call it g.The electric charge is Q=T L3+T R3+Y.Up to this point,the model is similar to the usual left-right-symmetric one[9].I assume the Lagrangian to be symmetric under a discrete symmetry S,which transformsφ→iφ,χL→−χL,χR→iχR.(1)As a consequence,the scalar potential isV=µ1(χ†LχL+χ†RχR)+µ2tr(φ†φ)+m(χ†LφχR+χ†Rφ†χL)+λ1[(χ†LχL)2+(χ†RχR)2]+λ2(χ†LχL)(χ†RχR)+λ3[tr(φ†φ)]2+λ4tr(φ†˜φ)tr(˜φ†φ)+λ5{[tr(φ†˜φ)]2+[tr(˜φ†φ)]2}+λ6(χ†LχL+χ†RχR)tr(φ†φ)+λ7(χ†Lφφ†χL+χ†Rφ†φχR)+λ8(χ†L˜φ˜φ†χL+χ†R˜φ†˜φχR).(2) It is important to notice that,even though I do not impose CP symmetry,all the coupling constants in this S-and parity-symmetric potential are real.I decompose the scalar multiplets as(A may be L or R)φ= φ0∗2φ+1−φ−2φ01 ,˜φ= φ0∗1φ+2−φ−1φ02,χA= χ+Aχ0A .(3)In principle,all four neutral complexﬁelds in these multiplets may acquire a vacuum expectation value(vev).I shall however assume the vev ofφ02to vanish. That is,I assumeφ = 000k1 , ˜φ = k∗1000 , χA = 0v A .(4)This is a crucial feature of the model.The purpose of this is to avoid W L-W R mixing which,together with the diﬀerent mixing matrices for left-and right-handed currents if φ02 were non-vanishing,would lead to the generation ofθQF D at one-loop level.It is important to notice that φ02 =0is a consistent assumption since no vev forφ02is induced by the vevs ofφ01,χ0L andχ0R.This means that, when we substitute all scalarﬁelds butφ02by their vevs,no term remains in the potential which is linear inφ02.I must add the remark that the potential in eq.2 is the most general one which has parity symmetry and admits φ02 =0.Because of the gauge symmetry,the only meaningful phase in eq.4isα≡arg(v L v∗R k∗1).The value ofαisﬁxed by the only term in the potential in eq.2 which“sees”it,the term in m.However,m is real,and therefore(assuming, without loss of generality,m to be negative)αwill be zero.As there is only one term in the potential which“sees”the vacuum phase α,there is no scalar-pseudoscalar mixing.This is another crucial feature of the model.It prevents one-loop self-energy diagrams with neutral spin-0ﬁelds from generatingθQF D,which would then be too large.The absence of scalar-pseudoscalar mixing can be explicitly checked by de-veloping the potential in eq.2toﬁnd the masses of the various spin-0particles. Without loss of generality I set k1,v L and v R real and positive.I then writeφ01=k1+ρ1+iη12,φ02=ρ2+iη22,χ0A=v A+ρA+iηA2.(5)There is one massive neutral pseudoscalar,I≡(k1v RηL−k1v LηR−v L v Rη1)/N, where N≡proportional to m ,as would be expected,because if m vanished there would be another spontaneously broken U(1)symmetry (given by χL →exp(iψ)χL and χR →exp(−iψ)χR )in the potential,and then I would be a Goldstone boson.There are three massive neutral scalars,which are orthogonal combinations of ρ1,ρL and ρR .These orthogonal combinations depend on the speciﬁc values of the parameters in the potential.There are two other neutral spin-0particles,ρ2and η2.The masses of ρ2and η2are diﬀerent because of the λ5term in the potential.Finally,the physical charged scalars are H +L =(−k 1χ+L +v L φ+1)/V Land H +R =(−k 1χ+R −v R φ+2)/V R ,where V A ≡2and the mass of W R is (gV R )/√2cos θW ),and its interactions are approximately givenby (g/cos θW )(T L 3−Q sin 2θW ),with a suitably deﬁned angle θW .The above approximations are up to terms of order v L /v R or k 1/v R .This means that the lightest neutral gauge boson can be identiﬁed with the Z particle observed at LEP and SLAC if v R ≫v L ,k 1.Notice however that we do not impose any condition on k 1/v L .2.3Yukawa interactionsThe quark sector of the model consists of doublets q L of SU(2)L ,doublets q R of SU(2)R ,and left-handed N L and right-handed n R which are singlets both of SU(2)L and of SU(2)R and have electric charge −1/3.There are three (for three generations)of each of these types of quark multiplets.I writeq L = p Ln L ,q R = p RN R .(6)Parity interchanges q L with q R and N L with n R .The discrete symmetry S trans-forms q L →q L ,q R →iq R ,N L →N L ,n R →−n R .(7)As a consequence,the Yukawa Lagrangian for the quarks isL Y =−q R ˜φ†∆q L −(q R χR GN L +H.c.),(8)where∆is hermitian,while G is a general3×3complex matrix.The mass terms will then beL Y=...−(\L⊑L G\R+√u LγµγL V d L+W+RµV L k1M u VγL+k1V R k1M u VγR+k1√DV†M u VγL d+√DV†M u VγL d−The Yukawa interactions of the quarks with I and with the scalars are given by L Y=...−iI2N v L v R uM uγ5u+k1v R dM dγ5d+k1√DM d D.(14)√uM u u−ρL2vLRemember thatρ1,ρL andρR are not eigenstates of mass,rather they are related by an orthogonal transformation to the three physical scalarﬁelds.The lepton sector of the model may be chosen to be the usual one[9],without lepton singlets.This is a considerable simpliﬁcation relative to previous models which used parity to suppressθQF D[7,8].Because the vev ofφ02vanishes,the neutrino masses vanish too.In this way we do not have to explain their small value,which is a usually a problem in left-right-symmetric models.2.4CP violationIn this model CP violation is hard and manifests itself only in the complexity of the CKM matrix V.At lowest order in V,rephasing-invariant imaginary parts appear in the“quartets”Vαi Vβj V∗αj V∗βi(α=βand i=j).As is well-known [10],the imaginary parts of all quartets are equal in modulus to a value J.The experimental data on V imply that J is at most10−4.Let us consider the generation ofθQF D from the quark self-energies.Just as in the SM[11],complex self-energies arise at lowest order when they are proportional to quartets.This happens when the respective diagrams have four vertices with either the gauge bosons W or the charged spin-0particles H L or H R.This means that complex self-energies only arise at two-loop level.In the unitary gauge,the self-energies with gauge bosons W do not contribute to the masses—they are proportional to either pγL or pγR.In other gauges though,the Goldstone bosons absorbed in the longitudinal components of the W contribute an imaginary part to the mass of each quark.However,when divided by the mass of the quark and summed over allﬂavours,those imaginary parts cancel out.As a consequence the contribution toθQF D vanishes in any ing a similar argument,we can easily show that the two-loop diagrams involving the spin-0chargedﬁelds H L and H R do not contribute toθQF D either.Therefore,strong CP violation only arises at three-loop level.This is the same as happens in the SM[11].θwill be suppressed by the factors J<∼10−4and (16π2)−3∼10−6.Therefore,we expectθto be no larger than10−10.However,as was pointed out in the analysis of Ellis and Gaillard[11],factors m q/m W should be present too,where m q are second-generation quark mases.These factors m q/m W further suppressθQF D to values of order10−16.Moreover,in models which use parity symmetry to obtainθQF D=0at tree level,there must be extra suppression factors at loop level arising from a partial cancellation between the contributions toθQF D from the usual quarks and from the mirror sector[8].Therefore,10−16 is only an upper bound forθQF D,which in the present model is probably much smaller than that.3ConclusionsIt is important to stress the diﬀerences between the present model and previousones[7,8]which used parity symmetry to suppressθQF D.This model has less fermions,because it avoids doubling the lepton spectrum and the up-type-quark spectrum.On the other hand,it has more spin-0ﬁelds,because of the use of a bi-doublet of SU(2)L and SU(2)R.The presence of adiscrete symmetry is crucial to obtain φ02 =0;this in turn is important because it allows us to have vanishing neutrino masses.The asymmetry between up-typeand down-type quarks,with twice as many charge−1/3quarks as charge2/3 quarks,is an interesting feature of the model;of course,this asymmetry may be inverted,we might instead consider a model in which the number of up-type quarks would be twice the number of down-type quarks.In the previous models,the ratio between the masses of the W R and of theW L was v R/v L,which was equal to the ratio between the masses of the heavyand light down-type quarks,and of the heavy and light up-type quarks.In the present model,v R/v L is still the ratio between the masses of the heavy and light down-type quarks.On the other hand,the ratio of the masses of W R and W L is now V R/V L=v R/v L.The ratio v R/v L must at least be of order105,else the lightest exotic D quarkwould have been observed at LEP.However,the mass of W R does not have to be that much higher than the one of the W L,because k1/v L may be quite large,and then V R/V L∼v R/k1instead of v R/v L.It is worth noting that the mass matrix of the charge2/3quarks is proportional to k1,while the one of the charge−1/3 quarks is proportional to v L(see eq.9);this fact suggests that k1/v L∼100may be a good guess.The discrete symmetry is needed,not only to make φ02 =0consistent,but also in order to make all couplings in the scalar potential real,even when CP symmetry is not imposed to the Lagrangian,and to guarantee that no scalar-pseudoscalar mixing will arise.That symmetry is also necessary to lead to a vanishing vacuum phaseα.Contrary to most previous models,the fact that the vacuum phase vanishes is crucial to obtainθQF D=0at tree level.This model seems to be a viable and interesting alternative to the usual left-right-symmetric model,attractive in particular because of the absence ofﬂavour-changing neutral interactions(except for the ones mediated byρ2and byη2, which connect the standard down-type quarks with the heavy ones,see eq.13). Its experimental exploration,searching in particular for the right-handed gauge interactions of the up-type quarks and for the eﬀects of the charged spin-0ﬁelds, should be given some attention.References[1]R.Peccei and H.Quinn,Phys.Rev.Lett.38(1977)1440and Phys.Rev.D16(1977)1791.[2]P.Bicudo and J.Ribeiro,preprint FISIST/2-96/CFIF(hep-ph/9604219).[3]D.Chang and R.N.Mohapatra,Phys.Rev.D32(1985)293.[4]H.Georgi,Hadr.J.1(1978)155.[5]R.N.Mohapatra and G.Senjanovi´c,Phys.Lett.79B(1978)283;M.A.B.B´e g and H.-S.Tsao,Phys.Rev.Lett.41(1978)278;G.Segr`e and H.A.Weldon,Phys.Rev.Lett.42(1979)1191;S.Barr and ngacker,Phys.Rev.Lett.42(1979)1654;G.C.Branco,W.Grimus and voura,Phys.Lett.B380(1996)119.[6]L.Bento,G.C.Branco and P.Parada,Phys.Lett.B267(1991)95.[7]K.S.Babu and R.N.Mohapatra,Phys.Rev.D15(1990)1958.[8]S.M.Barr,D.Chang and G.Senjanovi´c,Phys.Rev.Lett.67(1991)2765.[9]For a review see G.Senjanovi´c,Nucl.Phys.B153(1979)334.[10]C.Jarlskog,Phys.Rev.Lett.55(1985)1039and Z.Phys.C29(1985)491;G.C.Branco and voura,Phys.Lett.B208(1988)123.[11]J.Ellis and M.K.Gaillard,Nucl.Phys.B150(1979)141.。

Computational fluid dynamics modeling of post-liquefaction soil

ORIGINAL PAPERComputational ﬂuid dynamics modeling of post-liquefaction soil ﬂow using the volume of ﬂuid methodYu Huang •Wuwei Mao •Hu Zheng •Guanghui LiReceived:5November 2010/Accepted:28May 2011/Published online:29June 2011ÓSpringer-Verlag 2011Abstract Flow deformation of post-liquefaction soil during an earthquake can cause serious damage to engi-neering structures.To overcome the limitations of con-ventional deformation analysis methods based on solid mechanics for extremely large systems,a computational ﬂuid dynamics (CFD)method is proposed to numerically simulate the ﬂow behavior of post-liquefaction soil.The liqueﬁed soil is assumed to be a viscous ﬂuid,and the volume of ﬂuid (VOF)model is used for interface tracking in the numerical scheme.The results of a modeling test conducted on liquefaction-induced ground ﬂow to verify the validity of the method showed good agreement,indi-cating the proposed method is capable of being used to reproduce the ﬂow behavior of post-liquefaction soil.Keywords Soil liquefaction ÁFlow failure ÁEarthquake ÁComputational ﬂuid dynamics ÁVolume of ﬂuid methodRe´sume ´Les e ´coulements apre `s lique ´faction de sols pendant un se´isme peuvent causer de se ´rieux dommages aux ouvrages.Aﬁn de surmonter les limitations des me´th-odes conventionnelles d’analyse des de´formations base ´es sur la me´canique des solides applique ´e a `des grands syste`mes,une me ´thode nume ´rique de dynamique des ﬂu-ides (CFD)est propose´e pour simuler nume ´riquement l’e´coulement apre `s lique ´faction du sol.Le sol lique ´ﬁe ´est suppose´e ˆtre un ﬂuide visqueux et le mode `le du volume de ﬂuide (VOF)est utilise´pour suivre les interfaces dans le sche ´ma nume ´rique.Les re ´sultats d’un test de mode ´lisation,destine´a `ve ´riﬁer la validite ´de la me ´thode et re ´alise ´a `partir d’un e´coulement de terrain conse ´cutif a `un processus de lique´faction,ont permis de conclure que la me ´thode pro-pose´e peut e ˆtre utilise ´e pour reproduire le comportement du sol apre`s lique ´faction.Mots cle´s Lique ´faction de sol ÁEcoulement ÁSe ´isme ÁDynamique des ﬂuides ÁMode´lisation ÁMe´thode du volume de ﬂuide IntroductionIn recent years,liquefaction resulting from seismic events has become a major concern due to its impact on structures,buildings and other infrastructure during and after an earthquake.Almost all major earthquakes are accompanied by soil liquefaction,as evidenced by the 1976Tangshan earthquake in China (Fu and Tatsuoka 1984),the 1989Loma Prieta earthquake in the USA (Bartlett and Youd 1995),the 1995Hyogo-ken-Nambu earthquake in Japan (Shibata et al.1996),and the 2008Wenchuan earthquake in China (Huang and Jiang 2010).Liquefaction-induced ground failure has become one of the leading causes of infrastructure damage during an earthquake.Under seismic loading,the rapid increase in pore water pressure quickly decreases the shear strength of the unconsolidated sedi-ment,possibly triggering large shear deformation.As ﬂow failure of the ground during an earthquake may be causedY.Huang (&)Key Laboratory of Geotechnical and Underground Engineering of the Ministry of Education,Tongji University,Shanghai 200092,Chinae-mail:yhuang@W.Mao ÁH.Zheng ÁG.LiDepartment of Geotechnical Engineering,Tongji University,Shanghai 200092,ChinaG.LiCCCC Third Harbor Consultants Co.Ltd,Shanghai 200032,ChinaBull Eng Geol Environ (2012)71:359–366DOI 10.1007/s10064-011-0386-3by either the dynamic force due to the seismic acceleration or the static gravity force due to the topography of the ground(Tamate and Towhata1999),ﬂow deformation of post-liquefaction soil may continue under a gravitational load after seismic shaking ceases.In this case,the ground surface is prone to large-scaleﬂow deformation regardless of whether it isﬂat or gently inclined.Uzuoak et al.(1998) suggests the shear strain may sometimes exceed100%. This can develop into a regional seismic liquefaction event that can cause extensive damage.During the Tajik earth-quake in1989,the sliding mass traveled a distance of2km due to the liquefaction of the saturated soil,despite the fact that the ground surface was nearlyﬂat(Ishihara2009). More recently,during the2003Tokachi-oki earthquake in Japan,ﬂuidized subsurface soil was expelled onto the surface andﬂowed about1km along a water channel (Kokusho2009).Liquefaction hazards are associated with substantial pared with the liquefaction itself,the damage resulting from the large scale deformation of the liqueﬁed soil can be even more serious.Previous studies of soil liquefaction have mainly focused on inﬂuencing factors, initial conditions and liquefaction predictions,and only recently has attention turned to the importance of the large deformation associated with liquefaction.These studies generally use conventional solid mechanics-based methods and assume a relatively limited maximum shear (Yang and Elgamal2002;Yoshimine et al.2006).How-ever,with liquefaction-induced groundﬂow,the shear strain of the soil may exceed100%in theﬂow condition (Fig.1),i.e,the state of the soils has been changed from solid toﬂuid.In the past few years,a series of laboratory tests con-ducted by different researchers has shown that post-lique-faction soil behaves in a similar manner to a viscousﬂuid (Sasaki et al.1992;Towhata et al.1999;Hwang et al. 2006).Thus,due to the phase transition,it is difﬁcult to apply solid mechanics based on small orﬁnite deformation theory to the deformation properties of the post-liquefac-tion soil;a new analytical method is required.Computationalﬂuid dynamics(CFD)is a computer-based method which makes use of a discretization of the algebraic equations governingﬂow and their subsequent mathematical manipulation and solution(Versteeg and Malalasekera1995).With the development of computer technology,numerical simulation methods have been rap-idly developed,expediting the use of such applications as the CFD method to the study of liquefaction-inducedﬂow deformation.In recent years,a few preliminary attempts have been made to promote the application of the CFD method to deformation problems related to liquefaction (e.g.,Uzuoka et al.1998;Hadush et al.2001;Hwang et al.360Y.Huang et al.2006).However,the research is very limited compared with that using solid mechanics and hence there is no generally accepted theory concerning its use in describing non-linear and deformation characteristics.In addition, little material is available concerning the study of large deformationﬂows induced by liquefaction.The present study is a further development in the ongoing efforts to assess theﬂow features of post-lique-faction soils.In this paper,the post-liquefaction soil is assumed to be a viscousﬂuid,and a numerical model based onﬂuid dynamics is proposed for theﬂow behavior.The Pressure Implicit with Splitting of Operators(PISO)algo-rithm is used for solving the Navier–Stokes equations,and the volume ofﬂuid(VOF)model is used to track interfaces in the numerical scheme.To verify the validity of the method,modeling of the liquefaction-induced soilﬂow was undertaken.Numerical model descriptionThe numerical modeling of groundﬂow involves the solution of the Navier–Stokes equations,which are based on the assumption of the conservation of mass and momentum,and the tracking of phase interface kinematics. The conservation of mass is described by the following equation:o q o t þr q uðÞ¼0;ð1Þand the conservation of momentum is described as:o q uo tþrðq uuÞ¼q f vþr T:ð2ÞIn this paper theﬂuid is assumed to follow the generalized Newton’s Law;the constitutive model can be described by the following equation:T¼Àp Iþ2g SÀ13r u:ð3ÞApplying the constitutive model above to Eq.2results in the Navier–Stokes equation:o q uþrðq uuÞ¼q f vÀr pþr sð4Þwhere:r s¼g D uþ13g rðr uÞð5ÞNote,in the equations above,the absence of sources of mass and momentum,where q is theﬂuid density,S is the strain rate tensor,f v is the volume force,u is the velocity tensor,g is the viscosity,T is the stress tensor,p is the static pressure and s is the deviatoric stress tensor.To solve the Navier–Stokes equations,the PISO algo-rithm(Issa1986;Oliveira and Issa2001)is used.This is a pressure–velocity calculation procedure based on aﬁnite volume discretization on a staggered grid of the governing equations.The purpose of a staggered grid is to evaluate scalar variables,such as pressure and density,at ordinary nodal points,while velocities are deﬁned at the cell faces between the nodes.The arrangement for a two-dimensional ﬂow calculation is shown in Fig.2.The PISO algorithm provides a higher degree of accuracy for pressure and velocity corrections than the SIMPLE algorithm(Barton 1998),which was used by Uzuoka et al.(1998)in the analysis of liquefaction-induced lateral spreading.Due to the existence of a free surface with good mobility during theﬂow,a method for interface tracking can be used to describe the interface distribution and motion charac-teristics.In the past,a variety of methods,such as Level Set,Particle-In-Cell(PIC),Marker-and-Cell(MAC)and Volume of Fluid(VOF)have been developed for phase interface tracking(Oliveira and Issa2001;Andrews et al. 1996;Tome´et al.2006;Hirt and Nichols1981).The VOF method,which was initially introduced by Hirt and Nichols (1981),is known for its capacity for interface tracking, including stratiﬁedﬂows,free-surfaceﬂows,ﬁlling, sloshing and so on.As there is a typical phase interface existing between the air and the liqueﬁed soil,the VOF method was considered appropriate for the description of ﬂow conﬁgurations in post-liquefaction soil.The most important feature of the VOF method is the introduction of the phase function F(x,y,t)for the description of the interface.F represents the areal(two-dimensional)or vol-umetric(three-dimensional)fraction of a cell being occu-pied by aﬂuid in the computational domain.If F=1,the cell is full of aﬂuid phase;if F=0,the grid is full of anotherﬂuid phase;if0\F\1,the cell is intersected by the phase interface.The position of the interface isCFD modeling using VOF method361determined by the F values of the neighboring cells.For two immiscible ﬂows,the conservation form of the trans-port equation is:o F o t þu o F o x þv o F o y¼0:ð6ÞA detailed description of the VOF method is available in Hirt and Nichols (1981).The Geometric Reconstruction scheme,which represents the interface between ﬂuids using a piecewise linear approach,is applied to reconstruct the free surface between the liqueﬁedsoil and the air.As is shown in Fig.3,the interface is described by a linear slope within each cell and the linear slope is used for the calculation of ﬂuid ﬂow through the cell faces.This approach is based on the work of Youngs (1982).Applications and analysis Model test descriptionFigure 4shows the layout of the model box designed to reproduce and test the ﬂow behavior of post-liquefaction soil.The soil container is made of transparent synthetic glass lined with an inner box.The external box is 950mm long,330mm side and 350mm high while the inner box is 320mm long and 320mm wide with a maximum height of 250mm.The inclination of the model box may be changed if necessary.The properties of the sample sand used in the test are shown in Table 1.Liquefaction of the saturated soil sample in the inner box is generated by an electric motor which imparts a sinu-soidal acceleration up to a maximum of 0.15g at a frequency of 1Hz.Excess pore water pressure is measured by pore pressure transducers installed at depths of 50and 100mm inDirection of shakingP1P2 Flow directionLiquefied sand16016050505050Table 1Parameters for the sand used in the model testse max e min D 50(mm)D 30(mm)D 10(mm)0.960.600.240.190.16Table 2Parameters used for the simulation of ﬂow liquefaction Density 1,600(kg/m 3)Viscosity 2,000(Pa s)Outlet pressure 1.0139105(Pa)Maximum iterations 30Convergence criteria 1910-3Total steps 1,000Unit time step0.005(s)362Y.Huang et al.the model.During the test,when the excess pore water pressure ratio (the ratio of excess pore water pressure to initial effective vertical stress)reaches 0.90and remains stable (Fig.5),the soilsample is considered to have reached the liquefaction state.The electric motor is then turned off,the bafﬂe is quickly pulled out and the process of the post-liquefaction soil ﬂow along the model box is recorded with a high-speed camera.Model applicationA ﬁnite volume method (FVM)with an Eulerian–Eulerian approach was used to study the air–liquid interactions.TheFig.8Conﬁguration of the modeltestFig.9Conﬁguration comparison of model tests and simulationCFD modeling using VOF method 363phase-coupled PISO algorithm was used to couple the pressure and velocity.The diffusion phase of the control equation used the central difference method and the momentum equations used the ﬁrst order upwind scheme.The CFD code FLUENT 6.3,a well-known computer program for modeling ﬂuid ﬂow,was employed to verify the applicability of the proposed model (Fluent Inc 2006).Gambit 2.3.16was used to generate the 2D geometries and their grids.As shown in Fig.6,the length of the model box is 0.95m and the height 0.20m;by using a quadrilateral mesh,the overall number of nodes was 2,016and the number of cells was 1,900.The left,right and bottom sides were set as WALL,which is used to bound ﬂuid regions.The upper edge was set as PRESSURE-OUTLET to deﬁne a static pressure at the outlet boundary;here set to standard atmospheric pressure with a value of 1.0139105Pa (Fig.7).The model parameters used for the simulation of ﬂow liquefaction are listed in Table 2.Results and analysisFigure 8shows the conﬁguration of the post-liquefaction sand ﬂow obtained from the model test.The ﬂow of thet=1st=2st=3st=4st=5s10cm/s10cm/s10cm/s10cm/s10cm/sFig.10Simulation results of velocity vectors364Y.Huang et al.liqueﬁed sand gradually extended after the bafﬂe was removed and lasted for5s.Figure9shows theﬂow con-ﬁguration comparison of the model tests and simulation results for the post-liquefaction sand.It can be seen that,in general,the numerical approach reproduces theﬂow trend of the post-liquefaction sand and is in good agreement with theﬂow conﬁguration.However,the simulatedﬂow of sand was found to be a little faster than theﬂow recorded in the test.In the numerical simulation,the left side of the sand was observed toﬂow more substantially than in the model tests.Figure10shows the computed velocity vec-tors of theﬂow at a range of times from1to5s.It is clear that the velocity on the left side of the model had a very low magnitude.The vector directions indicate theﬂow orientation of the post-liquefaction sand.Figure11shows the time histories ofﬂow velocity at the selected output points depicted in Fig.7.Theﬂow of the post-liquefaction soil reaches its maximum velocity (from8to9cm/s)about1s after the instigation ofﬂow. Thereafter,theﬂow begins to slow down.However,the degree of deceleration differs depending on the output point;the velocity at Point3decreased most quickly,fol-lowed by the velocities at Points2and1.This is because Point3is closer to the free surface and is therefore more signiﬁcantly affected by gravity,the main trigger ofﬂow.To evaluate laterally spreading features of the post-liq-uefaction sandﬂow,a simple calculation of engineering strain based on the one-dimensional movement of a spec-iﬁed point was carried out.Point A,in the lower right corner of the soil sample,was selected for this calculation (Fig.12).Point A0indicates the new location of Point A at different times duringﬂow.L is the initial length of the soil sample,dL is the length increase in theﬂow direction;the engineering strain at Point A is deﬁned as e=dL/L.Fig-ure13shows a comparison of engineering strain for the model tests and the numerical simulation results.It can be seen that in theﬁrst3s the simulated engineering strain was smaller than for the model test;however,this dis-crepancy decreases as theﬂow deformation progresses, with the values the same after4s ofﬂow.The strain-hardening behavior which occurs duringﬂow is not considered here as the post-liquefaction soil is con-sidered to be a viscousﬂuid.However,as evidenced by the model test,the excess pore water pressure dissipated rap-idly and the liqueﬁed soil recovered its strength,returning to the solid phase while stillﬂowing.Thus,the simulation over-estimates theﬁnalﬂow deformation.ConclusionsTo overcome the limitations of traditional solid mechanics for the study ofﬂow behavior in post-liquefaction soil,this paper presents a method based onﬂuid dynamics.2D CFD simulations were conducted with a VOF model to predict theﬂow behavior of post-liquefaction soil,which is assumed to be a viscousﬂuid.The proposed model is capable of predictingﬂow liquefaction of soils in which the shear strain might exceed100%.A comparison of numerical and experimental results shows that the applied CFD method performs well,with the simulation showing a realistic lateral displacement and a velocity proﬁle.The numerical approach reproduces the ﬂow trend of the post-liquefaction sand and is in agreement with the model testﬂow conﬁguration. Acknowledgments This work was supported by the Key Labora-tory of Engineering Geomechanics,Chinese Academy of Sciences (Grant No.2008-04),the National Natural Science Foundation of China(Grant Nos.41072202,40802070and40841014),the Shanghai Municipal Education Commission and Shanghai Education Devel-opment Foundation(Shu Guang Project No.08SG22),and the Kwang-Hua Fund for the College of Civil Engineering,Tongji University.ReferencesAndrews MJ,O’Rourke PJ(1996)The multiphase particle-in-cell (MPPIC)method for dense particulateﬂows.Int J Multiph Flow 22:379–402Bartlett SF,Youd TL(1995)Empirical prediction of liquefaction-induced lateral spread.J Geotech Eng ASCE121:316–329 Barton IE(1998)Comparison of SIMPLE-and PISO-type algorithms for transientﬂows.Int J Numer Methods Fluids26:459–483 Fluent Inc(2006)Fluent users guide,Version6.3.Lebanon,NHFu SC,Tatsuoka F(1984)Soil liquefaction during Haicheng and Tangshan earthquake in China;a review.Soils Found24:11–29 Hadush S,Yashima A,Uzuoka R,Moriguchi S,Sawada K(2001) Liquefaction induced lateral spread analysis using the CIP put Geotech28:549–574Hirt CW,Nichols BD(1981)Volume ofﬂuid(VOF)method for the dynamics of free boundaries.J Comput Phys39:201–225 Huang Y,Jiang XM(2010)Field-observed phenomena of seismic liquefaction and subsidence during the2008Wenchuan earth-quake in China.Nat Hazards54:839–850Hwang JI,Kim CY,Chung CK,Kim MM(2006)Viscousﬂuid characteristics of liqueﬁed soils and behavior of piles subjected toﬂow of liqueﬁed soils.Soil Dyn Earthq Eng26:313–323 Ishihara(2009)Liquefaction-inducedﬂow slide in the collapsible loess deposit in Tajik.Earthquake geotechnical case histories for performance-based design.Taylor&Francis Group,London, pp431–448Issa RI(1986)Solution of the implicitly discretizedﬂuidﬂow equation by operator splitting.J Comput Phys62:40–65 Oliveira PJ,Issa RI(2001)An improved PISO algorithm for the computation of buoyancy-drivenﬂows.Numer Heat Trans PartB Fundam40:473–493Sasaki Y,Towhata I,Tokida K,Yamada K,Matumoto H,Tamari Y, Saya S(1992)Mechanism of permanent displacement of ground caused by seismic liquefaction.Soils Found32:79–96 Shibata T,Oka F,Ozawa Y(1996)Characteristics of ground deformation due to liquefaction.Special issue on geotechnical aspects of the January17,1995Hyogoken-Nambu earthquake.Soils and Foundations,pp65–79CFD modeling using VOF method365Tamate S,Towhata I(1999)Numerical simulation of groundﬂow caused by seismic liquefaction.Soil Dyn Earthq Eng18:473–485 Tome´MF,Doricio JL,Castelo A,Cuminato JA,McKee S(2006) Solving viscoelastic free surfaceﬂows of a second-orderﬂuid using a marker-and-cell approach.Int J Numer Methods Fluids 53:599–627Towhata I,Vargas-Monge W,Orense RP,Yao M(1999)Shaking table tests on subgrade reaction of pipe embedded in sandy liqueﬁed subsoil.Soil Dyn Earthq Eng18:347–361 Tsukamoto Y,Ishihara K,Kokusho T(2009)Fluidisation and subsidence of gently sloped farmingﬁelds reclaimed with volcanic soils during2003Tokachi-oki earthquake in Japan.Earthquake Geotechnical Case Histories for Performance-Based Design.Taylor&Francis Group(London),109–118Uzuoka R,Yashima A,Kawakami T,Konrad JM(1998)Fluid dynamics based prediction of liquefaction induced lateral put Geotech22:243–282Versteeg HK,Malalasekera W(1995)An introduction to computa-tionalﬂuid dynamics.Pearson Education Limited,London Yang ZH,Elgamal A(2002)Inﬂuence of permeability on liquefac-tion-induced shear deformation.J Eng Mech128:720–729 Yoshimine M,Nishizaki H,Amano K,Hosono Y(2006)Flow deformation of liqueﬁed sand under constant shear load and its application to analysis ofﬂow slide of inﬁnite slope.Soil Dyn Earthq Eng26:253–264Youngs DL(1982)Time-dependent multi-materialﬂow with large ﬂuid distortion.In:Morton KW,Baines MJ(eds)Numer Methods Fluid Dyn.Academic Press,New York366Y.Huang et al.。

55-MY1M系列II 2G Explosion Protection II Equipment G

Instruction ManualMechanically Jointed Rodless Cylinder55-MY1M SeriesII2potential energy provided by compressed air into a force which causessubject to “Special Conditions of Use”, please see Section 2.3.1 Safety InstructionsThese safety instructions are intended to prevent hazardous situationsand/or equipment damage. These instructions indicate the level ofpotential hazard with t he labels of “Caution,” “Warning” or “Danger.”They are all important notes for safety and must be followed in additionto International Standards (ISO/IEC) *1), and other safety regulations.*1) ISO 4414: Pneumatic fluid power - General rules relating to systems.ISO 4413: Hydraulic fluid power - General rules relating to systems.IEC 60204-1: Safety of machinery - Electrical equipment of machines.(Part 1: General requirements)ISO 10218-1: Manipulating industrial robots -Safety. etc.∙Refer to product catalogue, Operation Manual and HandlingPrecautions for SMC Products for additional information.∙ Keep this manual in a safe place for future reference.not avoided, will result in death or serious injury.Warning∙Always ensure compliance with relevant safety laws andstandards.∙All work must be carried out in a safe manner by a qualified person incompliance with applicable national regulations.2 SpecificationsThis product is certified to ATEX Category 2G and therefore is suitablefor use in Zones 1 and 2 only.2.1 Product Specifications:Refer to the operation manual for this product;The batch code printed on the label indicates the month and the year ofproduction as per the following table;2.3 Special Conditions of Use:∙Products are suitable for sub-divisions IIC.∙Products are suitable for Zones 1 & 2 only.2.3.1 Temperature Marking:2.3.1.1 Standard Product:∙In the normal ambient temperature range (+5°C to +40°C) the productis rated to temperature class T6.∙In the special ambient temperature range (+40°C to +60°C) the productis rated to temperature class T5.2.3.2 Static:∙Danger of electrostatic discharge.3 Installation3.1 InstallationWarning∙Do not install the product unless the safety instructions have been readand understood.∙Do not twist or bend the cylinder, or mount the product when subjectto tension.∙Do not use in an application where the product is stopped mid-stroke,via an external stop.∙Do not use where cylinders are being synchronised to move a singleload.∙In order to install the product, use one of the brackets available foundin the standard product catalogue;SketchSee the product catalogue for the exact code to order which relates to thebore size of your product.∙When replacing the side supports, use the hexagonal socket head capscrews of the following sizes, and the hexagon wrenches shown below.3.2 EnvironmentWarning∙Do not use in an environment where corrosive gases, chemicals, water,salt water or steam are present.∙Do not use in an explosive atmosphere except within the specifiedrating.∙Do not expose to direct sunlight. Use a suitable protective cover.∙Do not install in a location subject to vibration or impact in excess ofthe product’s specifications.∙Do not use in a place subject to heavy vibration and/or shock.∙Do not use in wet environments, where water can remove the presenceof the lubrication.∙Do not use in case of heavy dusty environments where dust canpenetrate into the cylinder and dry the grease.∙Do not allow dust layers to build up on the cylinder surface and insulatethe product.3.3 PipingCaution∙Before connecting piping make sure to clean up chips, cutting oil, dustetc.∙When installing piping or fittings, ensure sealant material does notenter inside the port. When using seal tape, leave 1 thread exposedon the end of the pipe/fitting.∙Tighten fittings to the specified tightening torque.3.4 LubricationCaution∙SMC products have been lubricated for life at manufacture, and do notrequire lubrication in service.∙If a lubricant is used in the system, refer to catalogue for details.3.5 Basic Circuit∙Plugging one of the ports on the actuator is considered a non-intendeduse, and could relate to an increase in maximum surface temperatureabove what the product specification declares.3.6 Electrical Connection∙The product should be grounded by the piston rod and the body inorder to create an electrically conductive path to the system/application.∙Ground the product in accordance with applicable regulations.∙Do not pass an electrical current through the product.4 Settings4.1 Air Cushion adjustment∙For air-cushion adjustment, tighten or loosen the cushion valve usinga hexagon socket wrench or a flat head screwdriver (excluding ø10).Warning∙Do not operate the cushion valve in the fully closed or fully openedstate.Using it in the fully closed state will cause the cushion seal to be damaged.Using it in the fully opened state will cause the piston assembly or thecover to be damaged.∙Be certain to activate the air cushion at the stroke end.When the cylinder is used with the cushion valve in a fully open position,a suitable external device should be installed to absorb all of the kineticenergy of the mechanism, of which the actuator is part, before reachingeach end of stroke. If this is not done, the piston assembly will bedamaged.5 How to OrderRefer to the standard product catalogue for ‘How to Order’.6 Outline Dimensions (mm)Refer to the standard product catalogue for general dimensions.7 Maintenance7.1 General MaintenanceCaution∙Not following proper maintenance procedures could cause the productto malfunction and lead to equipment damage.∙If handled improperly, compressed air can be dangerous.∙Maintenance of pneumatic systems should be performed only byqualified personnel.ORIGINAL INSTRUCTIONSRefer to Declaration ofConformity for relevantDirectives∙Before performing maintenance, turn off the power supply and be sure to cut off the supply pressure. Confirm that the air is released to atmosphere.∙After installation and maintenance, apply operating pressure and power to the equipment and perform appropriate functional and leakage tests to make sure the equipment is installed correctly.∙If any electrical connections are disturbed during maintenance, ensure they are reconnected correctly and safety checks are carried out as required to ensure continued compliance with applicable national regulations.∙Do not make any modification to the product.∙Do not disassemble the product, unless required by installation or maintenance instructions.∙Do not use a product which looks or contains damage, this will invalidate the certification. If damage is seen, please replace the product immediately.∙Periodically check the product for any damage or rust appearing. This could result in an increase in friction and lead to dangerous conditions. Replace the whole actuator if any of these conditions appear.∙Replace the product, when air leakage is above the allowable value7.2 Replacement of Dust Seal BandWarning∙The dust seal band is the only customer replaceable part.∙If other parts are deemed to have failed please replace the product.7.3 Disassembly procedure∙Disassemble the cylinder, remove the old grease and place all the parts on a clean cloth in a clean environment. The following flat head screwdriver or hexagon socket wrench shall be used to loosen the set ∙The following screwdriver or hexagon socket wrenches shall be used∙Remove the old dust seal band, scrapers and if necessary, bearings and side scrapers. 7.4 Lubrication procedure▪Dust seal band inner and outer surface▪scrapers▪bearings▪side scrapers▪tube top surface7.5 Assembly procedure∙The cylinder is assembled in the following order: bearings, dust sealband, side scraper, parallel key, scraper, stopper, spacer and endcover. Tighten the cross recessed binding head screws or the hexagonsocket button bolt according to the torque values given.∙Finally tighten the two set screws at each side with tightening torqueof 0.1 Nm.∙Check for cylinder smooth movement and for air leakage.8 Limitations of Use8.1 Limited warranty and Disclaimer/Compliance RequirementsRefer to Handling Precautions for SMC Products located on .8.2 Obligations of the end-user∙Ensure the product is used within the specification outlined.∙Ensure that the maintenance periods are suitable for the application.∙Ensure any cleaning processes to remove dust layers are made withthe atmosphere in mind (e.g. using a damp cloth to avoid static buildup).∙Ensure that the application does not introduce additional hazards bymounting, loading, impacts or other methods.∙Ensure that there is sufficient ventilation and air circulation around theproduct.∙If the product is subject to direct heat sources in the application, theyshould be shielded so that the actuator temperature stays within thestated operating range.Caution∙SMC products are not intended for use as instruments for legalmetrology.Measurement instruments that SMC manufactures or sells have notbeen qualified by type approval tests relevant to the metrology(measurement) laws of each country.Danger∙Do not exceed any of the specifications listed in Section 2 of thisdocument as this will be deemed improper use.∙Air equipment has an air leakage during operation within certain limits.Do not use this equipment when the air itself introduces additionalhazards and could lead to an explosion.∙Use only ATEX certified auto switches. These should be orderedseparately.∙Do not use this product in the presence of strong magnetic fields thatcould generate a surface temperature higher than the productspecification.∙Avoid applications where the piston rod end and the adjoining part inthe application can create a possible ignition source.∙Do not install or use these actuators where there is the possibility forthe piston rod to impact foreign objects.∙In the event of damage or failure of any parts located in the vicinitywhere this product has been installed, it is the responsibility of the userto determine whether or not this has compromised the safety andcondition of this product and/or the application.∙External impact on the cylinder body could result in a spark and/orcylinder damage. Avoid any application where foreign objects can hitor impact the cylinder. In such situations the application should installa suitable guard to prevent this occurrence.∙Do not use this equipment where vibration could lead to failure.9 ContactsRefer to Declaration of Conformity and for contacts.URL : http// (Global) http// (Europe)'SMC Corporation, Akihabara UDX15F, 4-14-1, Sotokanda, Chiyoda-ku, Tokyo 1010021Specifications are subject to change without prior notice from the manufacturer.© 2018 SMC Corporation All Rights Reserved.Template DKP50047-F-085H。

IPC-7351B & PCBL Land Pattern Naming Convention

I P C-7351B N a m i n g C o n v e n t i o n f o r S t a n d a r d S M T L a n d P a t t e r n sSurface Mount Land PatternsComponent, Category Land Pattern Name Ball Grid Array’s...............................BGA + Pin Qty + C or N + Pitch P + Ball Columns X Ball Rows _ Body Length X Body Width X Height BGA w/Dual Pitch.BGA + Pin Qty + C or N + Col Pitch X Row Pitch P + Ball Columns X Ball Rows _ Body Length X Body Width X Height BGA w/Staggered Pins..................BGAS + Pin Qty + C or N + Pitch P + Ball Columns X Ball Rows _ Body Length X Body Width X Height BGA Note: The C or N = Collapsing or Non-collapsing BallsCapacitors, Chip, Array, Concave..........................................................CAPCAV + Pitch P + Body Length X Body Width X Height - Pin Qty Capacitors, Chip, Array, Flat..................................................................CAPCAF + Pitch P + Body Length X Body Width X Height - Pin Qty Capacitors, Chip, Non-polarized.................................................................................................CAPC + Body Length + Body Width X Height Capacitors, Chip, Polarized.....................................................................................................CAPCP + Body Length + Body Width X Height Capacitors, Chip, Wire Rectangle........................................................................................CAPCWR + Body Length + Body Width X Height Capacitors, Molded, Non-polarized...........................................................................................CAPM + Body Length + Body Width X Height Capacitors, Molded, Polarized.................................................................................................CAPMP + Body Length + Body Width X Height Capacitors, Aluminum Electrolytic ............................................................................................................CAPAE + Base Body Size X Height Ceramic Flat Packages.....................................................................................................CFP127P + Lead Span Nominal X Height - Pin Qty Column Grid Array’s.....................................................CGA + Pitch P + Number of Pin Columns X Number of Pin Rows X Height - Pin Qty Crystals (2 leads)........................................................................................................................XTAL + Body Length X Body Width X Height Dual Flat No-lead..........................................................................................................DFN + Body Length X Body Width X Height – Pin Qty Diodes, Chip................................................................................................................................DIOC + Body Length + Body Width X Height Diodes, Molded...........................................................................................................................DIOM + Body Length + Body Width X Height Diodes, MELF................................................................................................................................DIOMELF + Body Length + Body Diameter Fuses, Molded............................................................................................................................FUSM + Body Length + Body Width X Height Inductors, Chip.............................................................................................................................INDC + Body Length + Body Width X Height Inductors, Molded........................................................................................................................INDM + Body Length + Body Width X Height Inductors, Precision Wire Wound................................................................................................INDP + Body Length + Body Width X Height Inductors, Chip, Array, Concave..............................................................INDCAV + Pitch P + Body Length X Body Width X Height - Pin Qty Inductors, Chip, Array, Flat......................................................................INDCAF + Pitch P + Body Length X Body Width X Height - Pin Qty Land Grid Array, Round Lead............................LGA + Pin Qty - Pitch P + Pin Columns X Pin Rows _ Body Length X Body Width X Height Land Grid Array, Square Lead........................LGAS + Pin Qty - Pitch P + Pin Columns X Pin Rows _ Body Length X Body Width X Height LED’s, Molded............................................................................................................................LEDM + Body Length + Body Width X Height Oscillators, Side Concave........................................................................OSCSC + Pitch P + Body Length X Body Width X Height - Pin Qty Oscillators, J-Lead.......................................................................................OSCJ + Pitch P + Body Length X Body Width X Height - Pin Qty Oscillators, L-Bend Lead.............................................................................OSCL + Pitch P + Body Length X Body Width X Height - Pin Qty Oscillators, Corner Concave....................................................................................................OSCCC + Body Length X Body Width X Height Plastic Leaded Chip Carriers..................................................PLCC + Pitch P + Lead Span L1 X Lead Span L2 Nominal X Height - Pin Qty Plastic Leaded Chip Carrier Sockets Square.......................PLCCS + Pitch P + Lead Span L1 X Lead Span L2 Nominal X Height - Pin Qty Quad Flat Packages..................................................................QFP + Pitch P + Lead Span L1 X Lead Span L2 Nominal X Height - Pin Qty Ceramic Quad Flat Packages.................................................CQFP + Pitch P + Lead Span L1 X Lead Span L2 Nominal X Height - Pin Qty Quad Flat No-lead................................................................QFN + Pitch P + Body Width X Body Length X Height - Pin Qty + Thermal Pad Pull-back Quad Flat No-lead..............................................PQFN + Pitch P + Body Width X Body Length X Height - Pin Qty + Thermal Pad Quad Leadless Ceramic Chip Carriers..........................................................LCC + Pitch P + Body Width X Body Length X Height - Pin Qty Quad Leadless Ceramic Chip Carriers (Pin 1 on Side)...............................LCCS + Pitch P + Body Width X Body Length X Height - Pin Qty Resistors, Chip...........................................................................................................................RESC + Body Length + Body Width X Height Resistors, Molded......................................................................................................................RESM + Body Length + Body Width X Height Resistors, MELF...........................................................................................................................RESMELF + Body Length + Body Diameter Resistors, Chip, Array, Concave............................................................RESCAV + Pitch P + Body Length X Body Width X Height - Pin Qty Resistors, Chip, Array, Convex, E-Version (Even Pin Size)...............RESCAXE + Pitch P + Body Length X Body Width X Height - Pin Qty Resistors, Chip, Array, Convex, S-Version (Side Pins Diff)................RESCAXS + Pitch P + Body Length X Body Width X Height - Pin Qty Resistors, Chip, Array, Flat.....................................................................RESCAF + Pitch P + Body Length X Body Width X Height - Pin Qty Small Outline Diodes, Flat Lead...................................................................................SODFL + Lead Span Nominal + Body Width X Height Small Outline IC, J-Leaded........................................................................................SOJ + Pitch P +Lead Span Nominal X Height - Pin Qty Small Outline Integrated Circuit, (50 mil Pitch SOIC)......................................................SOIC127P +Lead Span Nominal X Height - Pin Qty Small Outline Packages............................................................................................SOP + Pitch P +Lead Span Nominal X Height - Pin Qty Small Outline No-lead...........................................................SON + Pitch P + Body Width X Body Length X Height - Pin Qty + Thermal Pad Pull-back Small Outline No-lead.........................................PSON + Pitch P + Body Width X Body Length X Height - Pin Qty + Thermal Pad Small Outline Transistors, Flat Lead....................................................................SOTFL + Pitch P + Lead Span Nominal X Height - Pin Qty SOD (Example: SOD3717X135 = JEDEC SOD123)........................................................SOD + Lead Span Nominal + Body Width X Height SOT89 (JEDEC Standard Package).......................................................................................................................................................SOT89 SOT143 & SOT343 (JEDEC Standard Package)..............................................................................................................SOT143 & SOT343 SOT143 & SOT343 Reverse (JEDEC Standard Package)...........................................................................................SOT143R & SOT343R SOT23 & SOT223 Packages (Example: SOT230P700X180-4)...............................SOT + Pitch P + Lead Span Nominal X Height - Pin Qty TO (Generic DPAK - Example: TO228P970X238-3).................................................................TO + Pitch P + Lead Span X Height - Pin QtyI P C-7351B L a n d P a t t e r n N a m i n g C o n v e n t i o n N o t e s•All dimensions are in Metric Units•All Lead Span and Height numbers go two places past the decimal point and “include” trailing Zeros•All Lead Span and Body Sizes go two place before the decimal point and “remove” leading Zeros•All Chip Component Body Sizes are one place to each side of the decimal point•Pitch Values are two places to the right & left of decimal point with no leading Zeros but include trailing zeros N a m i n g C o n v e n t i o n S p e c i a l C h a r a c t e r U s e f o r L a n d P a t t e r n sThe _ (underscore) is the separator between pin Qty in Hidden & Deleted pin componentsThe – (dash) is used to separate the pin qty.The X (capital letter X) is used instead of the word “by” to separate two numbers such as height X width like “Quad Packages”.P C B L i b r a r i e s S u f f i x N a m i n g C o n v e n t i o n f o r L a n d P a t t e r n sCommon SMT Land Pattern to Describe Environment Use (This is the last character in every name)Note: This excludes the BGA component family as they only come in the Nominal Environment Condition •M.................Most Material Condition (Level A)•N..................Nominal Material Condition (Level B)•L.................Least Material Condition (Level C)Alternate Components that do not follow the JEDEC, EIA or IEC Standard•A..................Alternate Component (used primarily for SOP & QFP when Component Tolerance or Height is different) •B..................Second Alternate ComponentReverse Pin Order•-20RN..........20 pin part, Reverse Pin Order, Nominal EnvironmentHidden Pins•-20_24N......20 pin part in a 24 pin package. The pins are numbered 1 – 24 the hidden pins are skipped. The schematic symbol displays up to 24 pins.Deleted Pins•-24_20N......20 pin part in a 24 pin package. The pins are numbered 1 – 20. The schematic symbol displays 20 pins. JEDEC and EIA Standard parts that have several alternate packages•AA, AB, AC.JEDEC or EIA Component IdentifierGENERAL SUFFIXES_HS.........................HS = Land Pattern with Heat Sink attachment requiring additional holes or padsExample: TO254P1055X160_HS-6N_BEC......................BEC = Base, Emitter and Collector (Pin assignments used for three pin Transistors)Example: SOT95P280X160_BEC-3N_SGD......................SGD = Source, Gate and Drain (Pin assignments used for three pin Transistors)Example: SOT95P280X160_SGD-3N_213........................213 = Alternate pin assignments used for three pin TransistorsExample: SOT95P280X160_213-3NP C B L i b r a r i e s N a m i n g C o n v e n t i o n f o r N o n-S t a n d a r d S M T L a n d P a t t e r n s Surface Mount Land PatternsComponent, Category Land Pattern Name Amplifiers....................................................................................................................................................AMP_ Mfr.’s Part Number Batteries......................................................................................................................................................BAT_ Mfr.’s Part Number Capacitors, Variable..................................................................................................................................CAPV_Mfr.’s Part Number Capacitors, Chip, Array, Concave (Pins on 2 or 4 sides)..............................................................CAPCAV_Mfr Series No. - Pin Qty Capacitors, Chip, Array, Flat (Pins on 2 sides)..............................................................................CAPCAF_Mfr Series No. - Pin Qty Capacitors, Miscellaneous............................................................................................................................CAP_Mfr.’s Part Number Crystals......................................................................................................................................................XTAL_Mfr.’s Part Number Diodes, Miscellaneous...................................................................................................................................DIO_Mfr.’s Part Number Diodes, Bridge Rectifiers............................................................................................................................DIOB_Mfr.’s Part Number Ferrite Beads..................................................................................................................................................FB_Mfr.’s Part Number Fiducials......................................................................................................................................FID + Pad Size X Solder Mask Size Filters..............................................................................................................................................................FIL_Mfr.’s Part Number Fuses..........................................................................................................................................................FUSE_Mfr.’s Part Number Fuse, Resettable.....................................................................................................................................FUSER_Mfr.’s Part Number Inductors, Miscellaneous...............................................................................................................................IND_Mfr.’s Part Number Inductors, Chip, Array, Concave (Pins on 2 or 4 sides)..................................................................INDCAV_Mfr Series No. - Pin Qty Inductors, Chip, Array, Flat (Pins on 2 sides).................................................................................INDCAF_Mfr Series No. - Pin Qty Keypad.................................................................................................................................................KEYPAD_Mfr.’s Part Number LEDS............................................................................................................................................................LED_Mfr.’s Part Number LEDS, Chip...................................................................................................................................................LED_Mfr.’s Part Number Liquid Crystal Display...................................................................................................................................LCD_Mfr.’s Part Number Microphones..................................................................................................................................................MIC_Mfr.’s Part Number Opto Isolators............................................................................................................................................OPTO_Mfr.’s Part Number Oscillators......................................................................................................................................OSC_Mfr.’s Part Number - Pin Qty Quad Flat Packages w/Bumper Corners, Pin 1 Side.............BQFP + Pitch P + Lead Span L1 X Lead Span L2 Nominal X Height - Pin Qty Quad Flat Packages w/Bumper Corners, 1 Center..............BQFPC + Pitch P + Lead Span L1 X Lead Span L2 Nominal X Height - Pin Qty Resistors, Chip, Array, Concave (Pins on 2 or 4 sides).................................................................RESCAV_Mfr Series No. - Pin Qty Resistors, Chip, Array, Convex Type E (Pins on 2 sides)...........................................................RESCAXE_Mfr Series No. - Pin Qty Resistors, Chip, Array, Convex Type S (Pins on 2 sides)...........................................................RESCAXS_Mfr Series No. - Pin Qty Resistors, Chip, Array, Flat (Pins on 2 sides)................................................................................RESCAF_Mfr Series No. - Pin Qty Relays.....................................................................................................................................................RELAY_Mfr.’s Part Number Speakers....................................................................................................................................................SPKR_Mfr’s Part Number Switches........................................................................................................................................................SW_Mfr.’s Part Number Test Points, Round......................TP + Pad Size (1 place left of decimal and 2 places right of decimal, Example TP100 = 1.00mm) Test Points, Square...............................................................TPS + Pad Size (1 place left of decimal and 2 places right of decimal) Test Points, Rectangle....................................TP + Pad Length X Pad Width (1 place left of decimal and 2 places right of decimal) Thermistors.............................................................................................................................................THERM_Mfr.’s Part Number Transceivers.............................................................................................................................................XCVR_ Mfr.’s Part Number Transducers (IRDA’s)................................................................................................................................XDCR_Mfr.’s Part Number Transient Voltage S_Mfr.’s Part Number Transient Voltage Suppressors, SP_Mfr.’s Part Number Transistor Outlines, Custom....................................................................................................................TRANS_Mfr.’s Part Number Transformers.............................................................................................................................................XFMR_Mfr.’s Part Number Trimmers & Potentiometers........................................................................................................................TRIM_Mfr.’s Part Number Tuners.....................................................................................................................................................TUNER_Mfr.’s Part Number Varistors.......................................................................................................................................................VAR_Mfr.’s Part Number Voltage Controlled Oscillators.....................................................................................................................VCO_Mfr.’s Part Number Voltage Regulators, Custom......................................................................................................................VREG_Mfr.’s Part NumberI P C-7251N a m i n g C o n v e n t i o n f o r T h r o u g h-H o l e L a n d P a t t e r n sThe land pattern naming convention uses component dimensions to derive the land pattern name.The first 3 – 6 characters in the land pattern name describe the component family.The first number in the land pattern name refers to the Lead Spacing or hole to hole location to insert the component lead.All numbers that follow the Lead Spacing are component dimensions.These characters are used as component body identifiers that precede the value and this is the priority order of the component body identifiers –P = Pitch for components with more than two leadsW = Maximum Lead Width (or Component Lead Diameter)L = Body Length for horizontal mountingD = Body Diameter for round component bodyT = Body Thickness for rectangular component bodyH = Height for vertically mounted componentsQ = Pin Quantity for components with more than two leadsR = Number of Rows for connectorsA, B & C = the fabrication complexity level as defined in the IPC-2221 and IPC-2222Notes:All component body values are in millimeters and go two places to the right of the decimal point and no leading zeros.All Complexity Levels used in the examples are “B”.Component, Category Land Pattern Name Capacitors, Non Polarized Axial Diameter Horizontal Mounting.........CAPAD + Lead Spacing + W Lead Width + L Body Length + D Body Diameter Example: CAPAD800W52L600D150BCapacitors, Non Polarized Axial Diameter; Lead Spacing 8.00; Lead Width 0.52; Body Length 6.00; Body Diameter 1.50Capacitors, Non Polarized Axial Rectangular.........CAPAR + Lead Spacing + W Lead Width + L Body Length + T Body thickness + H Body Height Example: CAPAR800W52L600T50H70BCapacitors, Non Polarized Axial; Lead Spacing 8.00; Lead Width 0.52; Body Length 6.00; Body Thickness 0.50; Body Height 0.70Capacitors, Non Polarized Axial Diameter Vertical Mounting .........CAPADV + Lead Spacing + W Lead Width + L Body Length + D Body Diameter Example: CAPADV300W52L600D150BCapacitors, Non Polarized Axial; Lead Spacing 3.00; Lead Width 0.52; Body Length 6.00; Body Diameter 1.50mmCapacitors, Non Polarized Axial Rect. Vert. Mtg.CAPARV + Lead Spacing + W Lead Width + L Body Length + T Body Thickness + H Body Height Example: CAPARV300W52L600T50H70BCapacitors, Non Polarized Axial Rect. Vertical; Lead Spacing 8.00; Lead Width 0.52; Body Length 6.00; Body Thickness 0.50; Body Height 0.70 Capacitors, Non Polarized Radial Diameter.......................................CAPRD + Lead Spacing + W Lead Width + D Body Diameter + H Body Height Example: CAPRD200W52D300H550BCapacitors, Non Polarized Radial Diameter; lead spacing 2.00; lead width 0.52; Body Diameter 3.00; Height 5.50Capacitors, Non Polarized Radial Rectangular.......CAPRR + Lead Spacing + W Lead Width + L Body Length + T Body thickness + H Body Height Example: CAPRR200W52L50T70H550BCapacitors, Non Polarized Radial Rectangular; lead spacing 2.00; lead width 0.52; Body Length 0.50; Body thickness 0.70; Height 5.50 Capacitors, Non Polarized Radial Disk Button........CAPRB + Lead Spacing + W Lead Width + L Body Length + T Body thickness + H Body Height Example: CAPRB200W52L50T70H550BCapacitors, Non Polarized Radial Rectangular; lead spacing 2.00; lead width 0.52; Body Length 0.50; Body thickness 0.70; Height 5.50 Capacitors, Polarized Axial Diameter Horizontal Mounting................CAPPA + Lead Spacing + W Lead Width + L Body Length + D Body Diameter Example: CAPPAD800W52L600D150BCapacitors, Polarized Axial Diameter; Lead Spacing 8.00; Lead Width 0.52; Body Length 6.00; Body Diameter 1.50Capacitor, Polarized Radial Diameter.................................................CAPPR + Lead Spacing + W Lead Width + D Body Diameter + H Body Height Example: CAPPRD200W52D300H550BCapacitors, Polarized Radial Diameter; lead spacing 2.00; lead width 0.52; Body Diameter 3.00; Height 5.50Diodes, Axial Diameter Horizontal Mounting.......................................DIOAD + Lead Spacing + W Lead Width + L Body Length + D Body Diameter Example: DIOAD800W52L600D150BCapacitors, Non Polarized Axial Diameter; Lead Spacing 8.00; Lead Width 0.52; Body Length 6.00; Body Diameter 1.50Diodes, Axial Diameter Vertical Mounting .........................................DIOADV + Lead Spacing + W Lead Width + L Body Length + D Body Diameter Example: DIOADV300W52L600D150BCapacitors, Non Polarized Axial; Lead Spacing 8.00; Lead Width 0.52; Body Length 6.00; Body Diameter 1.50Dual-In-Line Packages...................................DIP + Lead Span + W Lead Width + P Pin Pitch + L Body Length + H Component Height + Q Pin Qty Example: DIP762W52P254L1905H508Q14BDual-In-Line Package: Lead Span 7.62; Lead Width 0.52; Pin Pitch 2.54; Body Length 19.05; Body Height 5.08; Pin Qty 14Component, Category Land Pattern Name Dual-In-Line Sockets....................................DIPS + Lead Span + W Lead Width + P Pin Pitch + L Body Length + H Component Height + Q Pin Qty Example: DIPS762W52P254L1905H508Q14BDual-In-Line Package Socket: Lead Span 7.62; Lead Width 0.52; Pin Pitch 2.54; Body Length 19.05; Body Height 5.08; Pin Qty 14Headers, Vertical....... HDRV + Lead Span + W Lead Width + P Pin Pitch + R Pins per Row + L Body Length + T Body Thickness + H Component HeightExample: HDRV200W52P200R2L4400T400H900BHeader, Vertical: Lead Span 2.00; Lead Width 0.52; Pin Pitch 2.00; 2 Rows; Body Length 44.00; Body Thickness 4.00; Body Height 9.00 Headers, Right Angle...............HDRRA + Lead Span + W Lead Width + P Pin Pitch + R Pins per Row + L Body Length + T Body Thickness + H Component HeightExample: HDRRA200W52P200R2L4400T400H900BHeader, Vertical: Lead Span 2.00; Lead Width 0.52; Pin Pitch 2.00; 2 Rows; Body Length 44.00; Body Thickness 4.00; Body Height 9.00 Inductors, Axial Diameter Horizontal Mounting....................................INDAD + Lead Spacing + W Lead Width + L Body Length + D Body Diameter Example: INDAD800W52L600D150BInductors, Axial Diameter; Lead Spacing 8.00; Lead Width 0.52; Body Length 6.00; Body Diameter 1.50Inductors, Axial Diameter Vertical Mounting .....................................INDADV + Lead Spacing + W Lead Width + L Body Length + D Body Diameter Example: INDADV300W52L600D150BInductors, Axial Diameter Vertical Mounting; Lead Spacing 3.00; Lead Width 0.52; Body Length 6.00; Body Diameter 1.50Jumpers, Wire...................................................................................................................................................JUMP + Lead Spacing + W Lead Width Example: JUMP500W52BJumper; Lead Spacing 5.00; Lead Width 0.52Mounting Holes Plated With Support Pad..........................................................................MTGP + Pad Size + H Hole Size + Z Inner Layer Pad Size Example: MTGP700H400Z520This is a Mounting hole for a #6-32 screw using a circular 7.00 land on the primary and secondary side of the board, a 4.00 diameter hole with the internal lands are smaller that the external and are also circular 5.20 in diameter.Mounting Holes Non-Plated With Support Pad................................................................MTGNP + Pad Size + H Hole Size + Z Inner Layer Pad Size Example: MTGNP700H400Z520This is a Mounting hole for a #6-32 screw using a circular 7.00 land on the primary and secondary side of the board, a 4.00 diameter hole with the internal lands are smaller that the external and are also circular 5.20 in diameter.Mounting Holes Non-Plated Without Support Pad.....................MTGNP + Pad Size + H Hole Size + Z Inner Layer Pad Size + K Keep-out Diameter Example: MTGNP100H400Z520K700This is a Mounting hole for a #6-32 screw using a circular 1mm land on the primary and secondary side of the board, a 4.00 diameter hole with the internal lands are smaller that the external and are also circular 5.20 in diameter and a 7.00 diameter keep-out.Mounting Holes Plated with 8 Vias .....................................................................MTGP + Pad Size + H Hole Size + Z Inner Layer Pad Size + 8 Vias Example: MTGP700H400Z520V8This is a Mounting hole for a #6-32 screw using a circular 7mm land on the primary and secondary side of the board, a 4mm diameter hole with the internal lands are smaller that the external and are also circular 5.2mm in diameter, with 8 vias.Pin Grid Array’s.............................PGA + Pin Qty + P Pitch + C Pin Columns + R Pin Rows + L Body Length X Body Width + H Component Height Example: PGA84P254C10R10L2500X2500H300BPin Grid Array: Pin Qty 84; Pin Pitch 2.54; Columns 10; Rows 10; Body Length 25.00 X 25.00; Component Height 3.00Resistors, Axial Diameter Horizontal Mounting...................................RESAD + Lead Spacing + W Lead Width + L Body Length + D Body Diameter Example: RESAD800W52L600D150BResistors, Axial Diameter; Lead Spacing 8.00; Lead Width 0.52; Body Length 6.00; Body Diameter 1.50Resistors, Axial Diameter Vertical Mounting ....................................RESADV + Lead Spacing + W Lead Width + L Body Length + D Body Diameter Example: RESADV300W52L600D150BResistors, Axial Diameter Vertical Mounting; Lead Spacing 3.00; Lead Width 0.52; Body Length 6.00; Body Diameter 1.50Resistors, Axial Rectangular Horizontal Mounting...RESAR + Lead Spacing + W Lead Width + L Body Length + T Body thickness + H Body Height Example: RESAR800W52L600T50H70BResistors, Axial Rectangular; Lead Spacing 8.00; Lead Width 0.52; Body Length 6.00; Body Thickness 0.50; Body Height 0.70Test Points, Round Land......................................................................................................................................................................TP + Lead Width Example: TP52Test Points, Square Land..................................................................................................................................................................TPS + Lead Width Example: TPS52Test Points, Top Land Round & Bottom Land Square.....................................................................................................................TPRS + Lead Width Example: TPRS52 Wire....................................................................................................................................................................................................PAD + Wire Width Example: PAD52。

APPARATUS FOR CONTROLLING FILM THICKNESS

专利名称：APPARATUS FOR CONTROLLING FILMTHICKNESS发明人：YODA TETSUYA,余田哲也申请号：JP特願平8-69839申请日：19960326公开号：JP特開平9-254236A公开日：19970930专利内容由知识产权出版社提供专利附图：摘要：PROBLEM TO BE SOLVED: To finely perform metering raw material consumption time by a method wherein feeding speed of a raw material is increased or decreasedwhen the lower limit or the upper limit set value is metered and when the top upper limitis metered, feeding is stopped and the amt. of extrusion is calculated from the measured time between the upper limit and the lower limit and the crosssectional area of a product is made to a specified value by controlling the take-off speed. SOLUTION: A solenoid switch cylinder 14 provided in a shoot 13 between a raw material feeding part 11 and a raw material metering part 12 adjusts feeding speed of the raw material. In addition, a raw material metering control part 17 outputs a signal by which the solenoid switch cylinder 14 is fully opened by detection of the lower limit set value to an input value of a wt. detecting sensor 15 and an opening for reduced speed is set by the reduced speed setting value and full closing is set by the upper limit set value to the feeding speed adjusting part 16 to calculate the amt. of extrusion. In addition, the raw material metering control part 17 calculates take-off speed from the amt. of extrusion per unit time based on a set value detecting time from the upper limit to the lower limit and a driving motor 51 of a take-off machine is controlled by a take-off control part 18 to control the thickness of a cylindrical film.申请人：SEKISUI CHEM CO LTD,積水化学工業株式会社地址：大阪府大阪市北区西天満2丁目4番4号国籍：JP更多信息请下载全文后查看。

智利抗震规范

I
NCh433
Index
5.8 Seismic actions on the structure 5.9 Seismic deformations 5.10 Separations between buildings or building parts 5.11 Drawings and calculation report
38
A.4 General provisions for repair methods
38Βιβλιοθήκη A.5 Requirements that must be met by the construction process of the
structural rehabilitation
39
A.6 Necessity of rehabilitation for buildings without damages
the seismic movement
6
4.3 Classification of buildings and structures according to their importance,
occupancy and failure risk
6
4.4 Seismic instruments
7
5
37
A.1 General
37
A.2 Evaluation of the seismic damage and structural rehabilitation decisions
37
A.3 Requirements to be met by the structural rehabilitation project
6

Search for Free Fractional Electric Charge Elementary Particles

In our experiment, drops are ejected through a silicon micromachined oriﬁce and fall through air under the inﬂuence of gravity and an alternating vertical electric ﬁeld. The drops are imaged by a digital charge coupled device (CCD) camera interfaced to a computer. The same computer is used to simultaneously collect and analyze the data and to monitor and control the experiment.
1Work supergy, contract DE-AC03-76SF00515.
Direct observation of free fractional charge elementary particles would be an undisputed signature of physics beyond the Standard Model. In this paper we present the results of an improved Millikan oil drop experiment designed to look for such particles. The apparatus made it possible to generate and measure the charges of multiple columns of multiple drops simultaneously, each drop being 7.6 − 11.0 µm in diameter. This allowed us to have a large throughput of 4.17 × 107 drops or about 17.4 mg, of silicone oil.