A Topological String The Rasetti-Regge Lagrangian, Topological Quantum Field Theory, and Vo

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rbgm包用户手册:Atlantis生态系统模型的盒子几何模型(BGM)文件和拓扑工具说明书

rbgm包用户手册:Atlantis生态系统模型的盒子几何模型(BGM)文件和拓扑工具说明书

Package‘rbgm’October14,2022Type PackageTitle Tools for'Box Geometry Model'(BGM)Files and Topology for theAtlantis Ecosystem ModelVersion0.1.0Depends R(>=3.2.2),raster,spImports dplyr,geosphere,rlang,reproj,sfheadersSuggests bgmfiles,covr,knitr,rmarkdown,roxygen2,testthatDescription Facilities for working with Atlantis box-geometry model(BGM)files.Atlantis is a deterministic,biogeochemical,whole-of-ecosystem model.Functions are provided to read from BGMfiles directly,preserving theirinternal topology,as well as helper functions to generate spatial data from thesemesh forms.This functionality aims to simplify the creation and modification of box and geometry as well as the ability to integrate with other data sources. NeedsCompilation noByteCompile yesLicense GPL-3RoxygenNote7.1.0Encoding UTF-8URL https://research.csiro.au/atlantis/BugReports https:///AustralianAntarcticDivision/rbgm/issues/ VignetteBuilder knitrAuthor Michael D.Sumner[aut,cre]Maintainer Michael D.Sumner<******************>Repository CRANDate/Publication2020-04-1205:30:04UTC12rbgm-package R topics documented:rbgm-package (2)bgmfile (3)boxSpatial (4)build_dz (5)nodeSpatial (6)Index8 rbgm-package Utilities for BGMfiles for AtlantisDescriptionTools for handling network data for Atlantis from box-geometry model(BGM)filesrbgm features•read.bgmfiles and faithfully store all information so it can be round-tripped•conversion from.bgm forms to Spatial classes(lines and polygons)•(not yet implemented:write to.bgm)I.Importbgmfile read directly from a.bgmfileII.ConversionboxSpatial convert boxes to a SpatialPolygonsDataFramefaceSpatial convert faces to a SpatialLinesDataFrameboundarySpatial convert boundary to a single-row SpatialPolygonsDataFramenodeSpatial obtain all vertices as pointspointSpatial obtain all instances of vertices as pointsIII.Miscellaneousbuild_dz Build Atlantis dz Valuesbgmfile3bgmfile Read BGMDescriptionRead geometry and full topology from BGMfiles.Usagebgmfile(x,...)read_bgm(x,...)Argumentsx path to a bgmfile...ignored for nowDetailsBGM is afile format used for the’Box Geometry Model’in the Atlantis Ecosystem Model.This function reads everything from the.bgmfile and returns it as linked tables.See AlsoSee helper functions to convert the bgm tables to‘Spatial‘objects,boxSpatial,faceSpatial, nodeSpatial,boundarySpatial,pointSpatialExampleslibrary(bgmfiles)bfile<-sample(bgmfiles(),1L)bgm<-bgmfile(bfile)str(bgm)4boxSpatial boxSpatial Convert to spatial formatDescriptionTake the output of bgmfile and return a Spatial object or a sf object.UsageboxSpatial(bgm)box_sp(bgm)box_sf(bgm)boundarySpatial(bgm)boundary_sp(bgm)boundary_sf(bgm)node_sp(bgm)point_sp(bgm)faceSpatial(bgm)face_sp(bgm)face_sf(bgm)Argumentsbgm output of a BGMfile,as returned by bgmfileDetailsNote that the‘_sp‘forms are aliased to original functions called‘*Spatial‘,and now have‘_sf‘counterparts to return that format.ValueSpatial*object or sf object•box_sp SpatialPolygonsDataFrame•face_sp SpatialLinesDataFrame•boundary_sp SpatialPolygonsDataFramebuild_dz5•node_sp SpatialPointsDataFrame•point_sp SpatialPointsDataFrame•box_sf sf with sfc_POLYGON column•face_sf sf with sfc_LINESTRING column•boundary_sf sf with sfc_POLYGON column•node_sf sf with sfc_POINT column•point_sf sf with sfc_POINT columnWarningThe sf objects created by‘box_sf()‘,‘node_sf()‘,‘face_sf()‘,‘boundary_sf()‘and‘point_sf()‘were not created by the sf package.They were created with reference to the sf format prior to November 2019.If you have problems it may be necessary to recreate the’crs’part of the of the object with code like‘x<-box_sf(bgm);library(sf);st_crs(x)<-st_crs(attr(x$geometry,"crs")$proj)‘.Get in touch([create an issue](https:///AustralianAntarcticDivision/rbgm/issues))if you have any troubles.Examplesfname<-bgmfiles::bgmfiles(pattern="antarctica_28")bgm<-bgmfile(fname)spdf<-box_sp(bgm)sfdf<-box_sf(bgm)sldf<-face_sp(bgm)plot(spdf,col=grey(seq(0,1,length=nrow(bgm$boxes))))plot(sldf,col=rainbow(nrow(bgm$faces)),lwd=2,add=TRUE)build_dz Build Atlantis dz ValuesDescriptionBuild dz layer values for Atlantis from a bottom value,up through successive intervals.Each value is the positive offset required to rise to the top of the current interval.Usagebuild_dz(z,zlayers=c(-Inf,-2000,-1000,-750,-400,-300,-200,-100,-50,-20,0) )Argumentsz lowermost valuezlayers intervals of layer valuesDetailsOffset values are returned to move from z against the intervals in zlayers.The intervals are assumed to be sorted and increasing in value from-Inf inity.Once the maximum layer is reached the result is padded by that top value.Valuenumeric vector of offset valuesExamples##sanity testsbuild_dz(-5000)build_dz(-1500)##build_dz(300)##errorbuild_dz(0)##ok##datadd<-c(-4396.49,-2100.84,-4448.81,-411.96,-2703.56,-5232.96,-4176.25,-2862.37,-3795.6,-1024.64,-897.93,-1695.82,-4949.76, -5264.24,-2886.81)##all values in a matrix for checking##[zlayers,dd]dzvals<-sapply(dd,build_dz)##process into textf1<-function(x)sprintf("somelabel,%i,%s",x,paste(build_dz(dd[x]),collapse=",")) tex1<-sapply(seq(length(dd)),f1)##for examplef2<-function(x){sprintf("morelabel,%i,%s",x,paste(as.integer(build_dz(dd[x])),collapse=","))}tex2<-sapply(seq(length(dd)),f2)nodeSpatial Vertices as Spatial points.DescriptionObtain all vertices as a SpatialPointsDataFrame or a sf dataframe.UsagenodeSpatial(bgm)node_sf(bgm)pointSpatial(bgm)point_sf(bgm)Argumentsbgm BGM object from bgmfileDetailsNodes are the unique coordinates(or vertices),points are the instances of those coordinates that exist in the model.point_sp or point_sf return all instances of the vertices with information about which boxes they belong to.node_sp and node_sf return all vertices.ValueSpatialPointsDataFrame or sf data frameWarningThe sf objects created by‘box_sf()‘,‘node_sf()‘,‘face_sf()‘,‘boundary_sf()‘and‘point_sf()‘were not created by the sf package.They were created with reference to the sf format prior to November 2019.If you have problems it may be necessary to recreate the’crs’part of the of the object with code like‘x<-node_sf(bgm);library(sf);st_crs(x)<-st_crs(attr(x$geometry,"crs")$proj)‘.Get in touch([create an issue](https:///AustralianAntarcticDivision/rbgm/issues))if you have any troubles.Examplesfname<-bgmfiles::bgmfiles(pattern="antarctica_28")bgm<-bgmfile(fname)spnode<-node_sp(bgm)names(spnode)nrow(spnode)##only unique verticesnrow(bgm$vertices)sppoints<-point_sp(bgm)names(sppoints)nrow(sppoints)names(point_sf(bgm))Indexbgmfile,2,3,4,7boundary_sf(boxSpatial),4boundary_sp(boxSpatial),4 boundarySpatial,2,3boundarySpatial(boxSpatial),4box_sf(boxSpatial),4box_sp(boxSpatial),4boxSpatial,2,3,4build_dz,2,5face_sf(boxSpatial),4face_sp(boxSpatial),4faceSpatial,2,3faceSpatial(boxSpatial),4node_sf,7node_sf(nodeSpatial),6node_sp,7node_sp(boxSpatial),4nodeSpatial,2,3,6point_sf,7point_sf(nodeSpatial),6point_sp,7point_sp(boxSpatial),4pointSpatial,2,3pointSpatial(nodeSpatial),6rbgm-package,2read_bgm(bgmfile),3Spatial,4SpatialLinesDataFrame,2,4 SpatialPointsDataFrame,5–7 SpatialPolygonsDataFrame,2,48。

ASAP学习教程 脚本语言篇

ASAP学习教程 脚本语言篇

Opening an ASAP Editor Window
Open a new editor window
Open an existing script. Choose files of type *.inr.
Editing Script Files
• ASAP command scripts are text-only files, so any text editor can be used. • The ASAP editor is a full-featured editor that includes Find/Replace, block copying, and drag-and-drop editing. • The ASAP editor also allows us to “step” through a program, has color-coded commands, and other features of modern programming environments.
IN MM YD FT MI M KM MIL UM CM UIN A NM UM MM CM M INCHES MILLIMETE • System units and YARDS wavelength units apply FEET to the entire project. MILES METERS KILOMETERS •System units must be MILS defined before MICRONS wavelength units. CENTIMETERS MICROINCHES ANGSTROMS NANOMETERS MICRONS MILLIMETERS CENTIMETERS METERS Names go in single quotes where •

Conformal collider physics Energy and charge correlations

Conformal collider physics Energy and charge correlations
µ electromagnetic current jem . From the point of view of QCD this current is simply a global
symmetry. In this case the theory is not conformal, but at high enough energies we can approximate the process as a conformal one to the extent that we can ignore the running of the coupling and the details of the hadronization process. In this paper we will analyze similar processes but in conformal field theories.
Fig. 1: A localized excitation is produced in a conformal field theory and its decay products are measured by calorimeters sitting far away.
1
For early work on the applications of scale invariance to strong interactions and, in particular, collisions, see [1].
arXiv:0803.1467v3 [hep-th] 4 May 2008
Conformal collider physics: Energy and charge correlations

正则表达式(Regularexpressions)使用笔记

正则表达式(Regularexpressions)使用笔记

正则表达式(Regularexpressions)使⽤笔记Regular expressions are a powerful language for matching text patterns. This page gives a basic introduction to regularexpressions themselves sufficient for our Python exercises and shows how regular expressions work in Python. ThePython "re" module provides regular expression support.In Python a regular expression search is typically written as:match = re.search(pat, str)The re.search() method takes a regular expression pattern and a string and searches for that pattern within the string. Ifthe search is successful, search() returns a match object or None otherwise. Therefore, the search is usually immediatelyfollowed by an if-statement to test if the search succeeded, as shown in the following example which searches for thepattern 'word:' followed by a 3 letter word (details below):str = 'an example word:cat!!'match = re.search(r'word:\w\w\w', str)# If-statement after search() tests if it succeededif match:print 'found', match.group() ## 'found word:cat'else:print 'did not find'The code match = re.search(pat, str) stores the search result in a variable named "match". Then the if-statement tests thematch -- if true the search succeeded and match.group() is the matching text (e.g. 'word:cat'). Otherwise if the match isfalse (None to be more specific), then the search did not succeed, and there is no matching text.The 'r' at the start of the pattern string designates a python "raw" string which passes through backslashes withoutchange which is very handy for regular expressions (Java needs this feature badly!). I recommend that you always writepattern strings with the 'r' just as a habit.Note: match.group() returns a string of matched expression(type:str)Basic PatternsThe power of regular expressions is that they can specify patterns, not just fixed characters. Here are the most basicpatterns which match single chars:a, X, 9, < -- ordinary characters just match themselves exactly. The meta-characters which do not match themselves because they have special meanings are: . ^ $ * + ? { [ ] \ | (). (a period) -- matches any single character except newline '\n'\w -- (lowercase w) matches a "word" character: a letter or digit or underbar [a-zA-Z0-9_]. Note that although "word" is the mnemonic for this, it only matches a single word char, not a whole word. \W (upper case W) matches any non-word character.\b -- boundary between word and non-word\s -- (lowercase s) matches a single whitespace character -- space, newline, return, tab, form [ \n\r\t\f]. \S (upper case S) matches any non-whitespace character.\t, \n, \r -- tab, newline, return\d -- decimal digit [0-9]^ = start, $ = end -- match the start or end of the string\ -- inhibit the "specialness" of a character. So, for example, use \. to match a period or \\ to match a slash. If you are unsure if a character has special meaning, such as '@', you can put a slash in front of it, @, to make sure it is treated just as a character.Basic FeaturesThe basic rules of regular expression search for a pattern within a string are:The search proceeds through the string from start to end, stopping at the first match foundAll of the pattern must be matched, but not all of the stringIf match = re.search(pat, str) is successful, match is not None and in particular match.group() is the matching text RepetitionThings get more interesting when you use + and * to specify repetition in the pattern+ -- 1 or more occurrences of the pattern to its left, e.g. 'i+' = one or more i's'*' -- 0 or more occurrences of the pattern to its left? -- match 0 or 1 occurrences of the pattern to its leftLeftmost & LargestFirst the search finds the leftmost match for the pattern, and second it tries to use up as much of the string as possible --i.e. + and * go as far as possible (the + and * are said to be "greedy").## i+ = one or more i's, as many as possible.match = re.search(r'pi+', 'piiig') => found, match.group() == "piii"## Finds the first/leftmost solution, and within it drives the +## as far as possible (aka 'leftmost and largest').## In this example, note that it does not get to the second set of i's.match = re.search(r'i+', 'piigiiii') => found, match.group() == "ii"## \s* = zero or more whitespace chars## Here look for 3 digits, possibly separated by whitespace.match = re.search(r'\d\s*\d\s*\d', 'xx1 2 3xx') => found, match.group() == "1 2 3"match = re.search(r'\d\s*\d\s*\d', 'xx12 3xx') => found, match.group() == "12 3"match = re.search(r'\d\s*\d\s*\d', 'xx123xx') => found, match.group() == "123"## ^ = matches the start of string, so this fails:match = re.search(r'^b\w+', 'foobar') => not found, match == None## but without the ^ it succeeds:match = re.search(r'b\w+', 'foobar') => found, match.group() == "bar"Emails ExampleSuppose you want to find the email address inside the string 'xyz alice-b@ purple monkey'. We'll use this as a running example to demonstrate more regular expression features. Here's an attempt using the pattern r'\w+@\w+':str = 'purple alice-b@ monkey dishwasher'match = re.search(r'\w+@\w+', str)if match:print match.group() ## 'b@google'The search does not get the whole email address in this case because the \w does not match the '-' or '.' in the address. We'll fix this using the regular expression features below.Square BracketsSquare brackets can be used to indicate a set of chars, so [abc] matches 'a' or 'b' or 'c'. The codes \w, \s etc. work insidesquare brackets too with the one exception that dot (.) just means a literal dot. For the emails problem, the square brackets are an easy way to add '.' and '-' to the set of chars which can appear around the @ with the pattern r'[\w.-]+@[\w.-]+' to get the whole email address:match = re.search(r'[\w.-]+@[\w.-]+', str)if match:print match.group() ## 'alice-b@'You can also use a dash to indicate a range, so1. [a-z] matches all lowercase letters.2. To use a dash without indicating a range, put the dash last, e.g. [abc-].3. An up-hat (^) at the start of a square-bracket set inverts it, so [^ab] means any char except 'a' or 'b'.Group ExtractionThe "group" feature of a regular expression allows you to pick out parts of the matching text. Suppose for the emails problem that we want to extract the username and host separately. To do this, add parenthesis ( ) around the username and host in the pattern, like this: r'([\w.-]+)@([\w.-]+)'. In this case, the parenthesis do not change what the pattern will match, instead they establish logical "groups" inside of the match text. On a successful search, match.group(1) is the match text corresponding to the 1st left parenthesis, and match.group(2) is the text corresponding to the 2nd left parenthesis. The plain match.group() is still the whole match text as usual.str = 'purple alice-b@ monkey dishwasher'match = re.search('([\w.-]+)@([\w.-]+)', str)if match:print match.group() ## 'alice-b@' (the whole match)print match.group(1) ## 'alice-b' (the username, group 1)print match.group(2) ## '' (the host, group 2)A common workflow(⼯作流程) with regular expressions is that you write a pattern for the thing you are looking for, adding parenthesis groups to extract the parts you want.Note: match.group(1) is the match text corresponding to the 1st left parenthesis, and match.group(2) is the text corresponding to the 2nd left parenthesisfindallfindall() is probably the single most powerful function in the re module. Above we used re.search() to find the first match for a pattern. findall() finds all the matches and returns them as a list of strings(list), with each string representing one match.## Suppose we have a text with many email addressesstr = 'purple alice@, blah monkey bob@ blah dishwasher'## Here re.findall() returns a list of all the found email stringsemails = re.findall(r'[\w\.-]+@[\w\.-]+', str) ## ['alice@', 'bob@']for email in emails:# do something with each found email stringprint emailfindall With FilesFor files, you may be in the habit of writing a loop to iterate over the lines of the file, and you could then call findall() on each line. Instead, let findall() do the iteration for you -- much better! Just feed the whole file text into findall() and let it return a list of all the matches in a single step (recall that f.read() returns the whole text of a file in a single string):# Open filef = open('test.txt', 'r')# Feed the file text into findall(); it returns a list of all the found stringsstrings = re.findall(r'some pattern', f.read())findall and GroupsThe parenthesis ( ) group mechanism can be combined with findall(). If the pattern includes 2 or more parenthesis groups, then instead of returning a list of strings, findall() returns a list of tuples. Each tuple represents one match of the pattern, and inside the tuple is the group(1), group(2) .. data. So if 2 parenthesis groups are added to the email pattern, then findall() returns a list of tuples, each length 2 containing the username and host, e.g. ('alice', '').str = 'purple alice@, blah monkey bob@ blah dishwasher'tuples = re.findall(r'([\w\.-]+)@([\w\.-]+)', str)print tuples ## [('alice', ''), ('bob', '')]for tuple in tuples:print tuple[0] ## usernameprint tuple[1] ## hostOnce you have the list of tuples, you can loop over it to do some computation for each tuple. If the pattern includes no parenthesis, then findall() returns a list of found strings as in earlier examples. If the pattern includes a single set of parenthesis, then findall() returns a list of strings corresponding to that single group.Obscure optional feature:Sometimes you have paren ( ) groupings in the pattern, but which you do not want to extract. In that case, write the parens with a ?: at the start, e.g. (?: ) and that left paren will not count as a group result.Reference:1.2.Thanks!<完>。

On the interplay between measurable and topological dynamics

On the interplay between measurable and topological dynamics

a rX iv:mat h /48328v1[mat h.DS]24Aug24ON THE INTERPLAY BETWEEN MEASURABLE AND TOPOLOGICAL DYNAMICS E.GLASNER AND B.WEISS Contents Introduction 2Part 1.Analogies 31.Poincar´e recurrence vs.Birkhoff’s recurrence 31.1.Poincar´e recurrence theorem and topological recurrence 31.2.The existence of Borel cross-sections 41.3.Recurrence sequences and Poincar´e sequences 52.The equivalence of weak mixing and continuous spectrum 73.Disjointness:measure vs.topological d mixing:measure vs.topological 125.Distal systems:topological vs.measure 196.Furstenberg-Zimmer structure theorem vs.its topological PI version 217.Entropy:measure and topological 227.1.The classical variational principle 227.2.Entropy pairs and UPE systems 227.3.A measure attaining the topological entropy of an open cover 237.4.The variational principle for open covers 287.5.Further results connecting topological and measure entropy 307.6.Topological determinism and zero entropy 31Part 2.Meeting grounds 338.Unique ergodicity 339.The relative Jewett-Krieger theorem3410.Models for other commutative diagrams3911.The Furstenberg-Weiss almost 1-1extension theorem4012.Cantor minimal representations4013.Other related theorems41References442 E.GLASNER AND B.WEISSIntroductionRecurrent-wandering,conservative-dissipative,contracting-expanding,deter-ministic-chaotic,isometric-mixing,periodic-turbulent,distal-proximal,the list can go on and on.These(pairs of)words—all of which can be found in the dictio-nary—convey dynamical images and were therefore adopted by mathematicians to denote one or another mathematical aspect of a dynamical system.The two sister branches of the theory of dynamical systems called ergodic theory(or measurable dynamics)and topological dynamics use these words to describe different but parallel notions in their respective theories and the surprising fact is that many of the corresponding results are rather similar.In the following article we have tried to demonstrate both the parallelism and the discord between ergodic theory and topo-logical dynamics.We hope that the subjects we chose to deal with will successfully demonstrate this duality.The table of contents gives a detailed listing of the topics covered.In thefirst part we have detailed the strong analogies between ergodic theory and topological dynamics as shown in the treatment of recurrence phenomena,equicontinuity and weak mixing,distality and entropy.In the case of distality the topological version camefirst and the theory of measurable distality was strongly influenced by the topo-logical results.For entropy theory the influence clearly was in the opposite direction. The prototypical result of the second part is the statement that any abstract mea-sure probability preserving system can be represented as a continuous transformation of a compact space,and thus in some sense ergodic theory embeds into topological dynamics.We have not attempted in any way to be either systematic or comprehensive. Rather our choice of subjects was motivated by taste,interest and knowledge and to great extent is random.We did try to make the survey accessible to non-specialists, and for this reason we deal throughout with the simplest case of actions of Z.Most of the discussion carries over to noninvertible mappings and to R actions.Indeed much of what we describe can be carried over to general amenable groups.Similarly, we have for the most part given rather complete definitions.Nonetheless,we did take advantage of the fact that this article is part of a handbook and for some of the definitions,basic notions and well known results we refer the reader to the earlier introductory chapters of volume I.Finally,we should acknowledge the fact that we made use of parts of our previous expositions[86]and[35].We made the writing of this survey more pleasurable for us by the introduction of a few original results.In particular the following results are entirely or partially new.Theorem1.2(the equivalence of the existence of a Borel cross-section with the coincidence of recurrence and periodicity),most of the material in Section4 (on topological mild-mixing),all of subsection7.4(the converse side of the local variational principle)and subsection7.6(on topological determinism).MEASURABLE AND TOPOLOGICAL DYNAMICS3 Part1.Analogies1.Poincar´e recurrence vs.Birkhoff’s recurrence1.1.Poincar´e recurrence theorem and topological recurrence.The simplest dynamical systems are the periodic ones.In the absence of periodicity the crudest approximation to this is approximate periodicity where instead of some iterate T n x returning exactly to x it returns to a neighborhood of x.Thefirst theorem in abstract measure dynamics is Poincar´e’s recurrence theorem which asserts that for afinite measure preserving system(X,B,µ,T)and any measurable set A,µ-a.e.point of A returns to A(see[46,Theorem4.3.1]).The proof of this basic fact is rather simple and depends on identifying the set of points W⊂A that never return to A.These are called the wandering points and their measurability follows from the formulaW=A∩ ∞ k=1T−k(X\A) .Now for n≥0,the sets T−n W are pairwise disjoint since x∈T−n W means that the forward orbit of x visits A for the last time at moment n.Sinceµ(T−n W)=µ(W) it follows thatµ(W)=0which is the assertion of Poincar´e’s theorem.Noting that A∩T−n W describes the points of A which visit A for the last time at moment n, and thatµ(∪∞n=0T−n W)=0we have established the following stronger formulation of Poincar´e’s theorem.1.1.Theorem.For afinite measure preserving system(X,B,µ,T)and any measur-able set A,µ-a.e.point of A returns to A infinitely often.Note that only sets of the form T−n B appeared in the above discussion so that the invertibility of T is not needed for this result.In the situation of classical dynam-ics,which was Poincar´e’s main interest,X is also equipped with a separable metric topology.In such a situation we can apply the theorem to a refining sequence of partitions P m,where each P m is a countable partition into sets of diameter at most1m ofitself,and since the intersection of a sequence of sets of full measure has full measure, we deduce the corollary thatµ-a.e.point of X is recurrent.This is the measure theoretical path to the recurrence phenomenon which depends on the presence of afinite invariant measure.The necessity of such measure is clear from considering translation by one on the integers.The system is dissipative,in the sense that no recurrence takes place even though there is an infinite invariant measure.There is also a topological path to recurrence which was developed in an abstract setting by G.D.Birkhoff.Here the above example is eliminated by requiring that the topological space X,on which our continuous transformation T acts,be compact.It is possible to show that in this setting afinite T-invariant measure always exists,and so we can retrieve the measure theoretical picture,but a purely topological discussion will give us better insight.4 E.GLASNER AND B.WEISSA key notion here is that of minimality.A nonempty closed,T-invariant set E⊂X, is said to be minimal if F⊂E,closed and T-invariant implies F=∅or F=E.If X itself is a minimal set we say that the system(X,T)is a minimal system. Fix now a point x0∈X and consider∞ n=1ω(x0)=MEASURABLE AND TOPOLOGICAL DYNAMICS5 that the converse is also valid—namely if there are no conservative quasi-invariant measures then there is a Borel cross-section.Note that the periodic points of(X,T)form a Borel subset for which a cross-section always exists,so that we can conclude from the above discussion the following statement in which no explicit mention is made of measures.1.2.Theorem.For a system(X,T),with X a completely metrizable separable space, there exists a Borel cross-section if and only if the only recurrent points are the peri-odic ones.1.3.Remark.Already in[42]as well as in[21]onefinds many equivalent conditions for the existence of a Borel section for a system(X,T).However one doesn’tfind there explicit mention of conditions in terms of recurrence.Silvestrov and Tomiyama [76]established the theorem in this formulation for X compact(using C∗-algebra methods).We thank zar for drawing our attention to their paper.1.3.Recurrence sequences and Poincar´e sequences.We will conclude this sec-tion with a discussion of recurrence sequences and Poincar´e sequences.First for some definitions.Let us say that D is a recurrence set if for any dynamical system(Y,T) with compatible metricρand anyǫ>0there is a point y0and a d∈D withρ(T d y0,y0)<ǫ.Since any system contains minimal sets it suffices to restrict attention here to minimal systems.For minimal systems the set of such y’s for afixedǫis a dense open set. To see this fact,let U be an open set.By the minimality there is some N such that for any y∈Y,and some0≤n≤N,we have T n y∈ing the uniform continuity of T n,wefind now aδ>0such that ifρ(u,v)<δthen for all0≤n≤Nρ(T n u,T n v)<ǫ.Now let z0be a point in Y and d0∈D such that(1)p(T d0z0,z0)<δ.For some0≤n0≤N we have T n0z0=y0∈U and from(1)we getρ(T d0y0,y0)<ǫ. Thus points thatǫreturn form an open dense set.Intersecting overǫ→0gives a dense Gδin Y of points y for whichρ(T d y,y)=0.infd∈DThus there are points which actually recur along times drawn from the given recur-rence set.A nice example of a recurrence set is the set of squares.To see this it is easier to prove a stronger property which is the analogue in ergodic theory of recurrence sets.1.4.Definition.A sequence{s j}is said to be a Poincar´e sequence if for anyfinite measure preserving system(X,B,µ,T)and any B∈B with positive measure we haveµ(T s j B∩B)>0for some s j in the sequence.6 E.GLASNER AND B.WEISSSince any minimal topological system(Y,T)hasfinite invariant measures with global support,µany Poincar´e sequence is recurrence sequence.Indeed for any presumptive constant b>0which would witness the non-recurrence of{s j}for(Y,T), there would have to be an open set B with diameter less than b and having positiveµ-measure such that T s j B∩B is empty for all{s j}.Here is a sufficient condition for a sequence to be a Poincar´e sequence:1.5.Lemma.If for everyα∈(0,2π)limn→∞1nn1U s k(1B−f0) L2−→0or1nn1µ(B∩T−s k B)= f0 2>0which clearly implies that{s k}is a Poincar´e sequence. The proof we have just given is in fact von-Neumann’s original proof for the mean ergodic theorem.He used the fact that N satisfies the assumptions of the proposition, which is Weyl’s famous theorem on the equidistribution of{nα}.Returning to the squares Weyl also showed that{n2α}is equidistributed for all irrationalα.For rationalαthe exponential sum in the lemma needn’t vanish,however the recurrence along squares for the rational part of the spectrum is easily verified directly so that we can conclude that indeed the squares are a Poincar´e sequence and hence a recurrence sequence.The converse is not always true,i.e.there are recurrence sequences that are not Poincar´e sequences.This wasfirst shown by I.Kriz[60]in a beautiful example(see also[86,Chapter5]).Finally here is a simple problem.MEASURABLE AND TOPOLOGICAL DYNAMICS7 Problem:If D is a recurrence sequence for all circle rotations is it a recurrence set?A little bit of evidence for a positive answer to that problem comes from looking at a slightly different characterization of recurrence sets.Let N denote the collection of sets of the formN(U,U)={n:T−n U∩U=∅},(U open and nonempty),where T is a minimal transformation.Denote by N∗the subsets of N that have a non-empty intersection with every element of N.Then N∗is exactly the class of recurrence sets.For minimal transformations,another description of N(U,U)is obtained by fixing some y0and denotingN(y0,U)={n:T n y0∈U}Then N(U,U)=N(y0,U)−N(y0,U).Notice that the minimality of T implies that N(y0,U)is a syndetic set(a set with bounded gaps)and so any N(U,U)is the set of differences of a syndetic set.Thus N consists essentially of all sets of the form S−S where S is a syndetic set.Given afinite set of real numbers{λ1,λ2,...,λk}andǫ>0setV(λ1,λ2,...,λk;ǫ)={n∈Z:maxj{ nλj <ǫ}},where · denotes the distance to the closest integer.The collection of such sets forms a basis of neighborhoods at zero for a topology on Z which makes it a topological group.This topology is called the Bohr topology.(The corresponding uniform structure is totally bounded and the completion of Z with respect to it is a compact topological group called the Bohr compactification of Z.)Veech proved in[78]that any set of the form S−S with S⊂Z syndetic contains a neighborhood of zero in the Bohr topology up to a set of zero density.It is not known if in that statement the zero density set can be omitted.If it could then a positive answer to the above problem would follow(see also[32]).2.The equivalence of weak mixing and continuous spectrumIn order to analyze the structure of a dynamical system X there are,a priori,two possible approaches.In thefirst approach one considers the collection of subsystems Y⊂X(i.e.closed T-invariant subsets)and tries to understand how X is built up by these subsystems.In the other approach one is interested in the collection of factors Xπ→Y of the system X.In the measure theoretical case thefirst approach leads to the ergodic decomposition and thereby to the study of the“indecomposable”or ergodic components of the system.In the topological setup there is,unfortunately,no such convenient decomposition describing the system in terms of its indecomposable parts and one has to use some less satisfactory substitutes.Natural candidates for in-decomposable components of a topological dynamical system are the“orbit closures”(i.e.the topologically transitive subsystems)or the“prolongation”cells(which often coincide with the orbit closures),see[4].The minimal subsystems are of particular importance here.Although we can not say,in any reasonable sense,that the study of the general system can be reduced to that of its minimal components,the analysis of8 E.GLASNER AND B.WEISSthe minimal systems is nevertheless an important step towards a better understanding of the general system.This reasoning leads us to the study of the collection of indecomposable systems (ergodic systems in the measure category and transitive or minimal systems in thetopological case)and their factors.The simplest and best understood indecomposable dynamical systems are the ergodic translations of a compact monothetic group(a cyclic permutation on Z p for a prime number p,the“adding machine”on ∞n=0Z2, an irrational rotation z→e2πiαz on S1={z∈C:|z|=1}etc.).It is not hard toshow that this class of ergodic actions is characterized as those dynamical systems which admit a model(X,X,µ,T)where X is a compact metric space,T:X→X a surjective isometry andµis T-ergodic.We call these systems Kronecker or isometric systems.Thus ourfirst question concerning the existence of factors should be:given an ergodic dynamical system X which are its Kronecker factors?Recall that a measure dynamical system X=(X,X,µ,T)is called weakly mixing if the product system(X×X,X⊗X,µ×µ,T×T)is ergodic.The following classical theorem is due to von Neumann.The short and elegant proof we give was suggested by Y.Katznelson.2.1.Theorem.An ergodic system X is weakly mixing iffit admits no nontrivial Kronecker factor.Proof.Suppose X is weakly mixing and admits an isometric factor.Now a factor of a weakly mixing system is also weakly mixing and the only system which is both isometric and weakly mixing is the trivial system(an easy exercise).Thus a weakly mixing system does not admit a nontrivial Kronecker factor.For the other direction,if X is non-weakly mixing then in the product space X×X there exists a T-invariant measurable subset W such that0<(µ×µ)(W)<1.For every x∈X let W(x)={x′∈X:(x,x′)∈W}and let f x=1W(x),a function in L∞(µ).It is easy to check that U T f x=f T−1x so that the mapπ:X→L2(µ)defined byπ(x)=f x,x∈X is a Borel factor map.Denotingπ(X)=Y⊂L2(µ),andν=π∗(µ),we now have a factor mapπ:X→(Y,ν).Now the function π(x) is clearly measurable and invariant and by ergodicity it is a constantµ-a.e.;say π(x) =1. The dynamical system(Y,ν)is thus a subsystem of the compact dynamical system (B,U T),where B is the unit ball of the Hilbert space L2(µ)and U T is the Koopman unitary operator induced by T on L2(µ).Now it is well known(see e.g.[35])that a compact topologically transitive subsystem which carries an invariant probability measure must be a Kronecker system and our proof is complete.Concerning the terminology we used in the proof of Theorem2.1,B.O.Koopman, a student of G.D.Birkhoffand a co-author of both Birkhoffand von Neumann introduced the crucial idea of associating with a measure dynamical system X= (X,X,µ,T)the unitary operator U T on the Hilbert space L2(µ).It is now an easy matter to see that Theorem2.1can be re-formulated as saying that the system X is weakly mixing iffthe point spectrum of the Koopman operator U T comprises the single complex number1with multiplicity1.Or,put otherwise,that the one dimensional space of constant functions is the eigenspace corresponding to the eigenvalue1(thisMEASURABLE AND TOPOLOGICAL DYNAMICS9 fact alone is equivalent to the ergodicity of the dynamical system)and that the restriction of U T to the orthogonal complement of the space of constant functions has a continuous spectrum.We now consider a topological analogue of this theorem.Recall that a topo-logical system(X,T)is topologically weakly mixing when the product system (X×X,T×T)is topologically transitive.It is equicontinuous when the family {T n:n∈Z}is an equicontinuous family of maps.Again an equivalent condition is the existence of a compatible metric with respect to which T is an isometry.And,moreover,a minimal system is equicontinuous iffit is a minimal translation on a compact monothetic group.We will need the following lemma.2.2.Lemma.Let(X,T)be a minimal system and f:X→R a T-invariant function with at least one point of continuity(for example this is the case when f is lower or upper semi-continuous or more generally when it is the pointwise limit of a sequence of continuous functions),then f is a constant.Proof.Let x0be a continuity point and x an arbitrary point in X.Since{T n x: n∈Z}is dense and as the value f(T n x)does not depend on n it follows that f(x)=f(x0).2.3.Theorem.Let(X,T)be a minimal system then(X,T)is topologically weakly mixing iffit has no non-trivial equicontinuous factor.Proof.Suppose(X,T)is minimal and topologically weakly mixing and letπ:(X,T)→(Y,T)be an equicontinuous factor.If(x,x′)is a point whose T×T orbit is dense in X×X then(y,y′)=(π(x),π(x′))has a dense orbit in Y×Y.However,if(Y,T) is equicontinuous then Y admits a compatible metric with respect to which T is an isometry and the existence of a transitive point in Y×Y implies that Y is a trivial one point space.Conversely,assuming that(X×X,T×T)is not transitive we will construct an equicontinuous factor(Z,T)of(X,T).As(X,T)is a minimal system,there exists a T-invariant probability measureµon X with full support.By assumption there exists an open T-invariant subset U of X×X,such that cls U:=M X×X.By minimality the projections of M to both X coordinates are onto.For every y∈X let M(y)={x∈X:(x,y)∈M},and let f y=1M(y)be the indicator function of the set M(y),considered as an element of L1(X,µ).Denote byπ:X→L1(X,µ)the map y→f y.We will show thatπis a continuous homomorphism,where we consider L1(X,µ)as a dynamical system with the isometric action of the group{U n T:n∈Z}and U T f(x)=f(T x).Fix y0∈X andǫ>0.There exists an open neighborhood V of the closed set M(y0)withµ(V\M(y0))<ǫ.Since M is closed the set map y→M(y),X→2X is upper semi-continuous and we can find a neighborhood W of y0such that M(y)⊂V for every y∈W.Thus for every y∈W we haveµ(M(y)\M(y0))<ǫ.In particular,µ(M(y))≤µ(M(y0))+ǫand it follows that the map y→µ(M(y))is upper semi-continuous.A simple computation shows that it is T-invariant,hence,by Lemma2.2,a constant.10 E.GLASNER AND B.WEISSWith y0,ǫand V,W as above,for every y∈W,µ(M(y)\M(y0))<ǫandµ(M(y))=µ(M(y0)),thusµ(M(y)∆M(y0))<2ǫ,i.e., f y−f y0 1<2ǫ.This proves the claim thatπis continuous.Let Z=π(X)be the image of X in L1(µ).Sinceπis continuous,Z is compact. It is easy to see that the T-invariance of M implies that for every n∈Z and y∈X, f T−n y=f y◦T n so that Z is U T-invariant andπ:(Y,T)→(Z,U T)is a homomorphism. Clearly(Z,U T)is minimal and equicontinuous(in fact isometric).Theorem2.3is due to Keynes and Robertson[57]who developed an idea of Fursten-berg,[22];and independently to K.Petersen[70]who utilized a previous work of W.A.Veech,[78].The proof we presented is an elaboration of a work of McMahon[66]due to Blanchard,Host and Maass,[13].We take this opportunity to point outa curious phenomenon which recurs again and again.Some problems in topological dynamics—like the one we just discussed—whose formulation is purely topological, can be solved using the fact that a Z dynamical system always carries an invariant probability measure,and then employing a machinery provided by ergodic theory.In several cases this approach is the only one presently known for solving the problem. In the present case however purely topological proofs exist,e.g.the Petersen-Veech proof is one such.3.Disjointness:measure vs.topologicalIn the ring of integers Z two integers m and n have no common factor if whenever k|m and k|n then k=±1.They are disjoint if m·n is the least common multiple of m and n.Of course in Z these two notions coincide.In his seminal paper of 1967[23],H.Furstenberg introduced the same notions in the context of dynamical systems,both measure-preserving transformations and homeomorphisms of compact spaces,and asked whether in these categories as well the two are equivalent.The notion of a factor in,say the measure category,is the natural one:the dynamical system Y=(Y,Y,ν,T)is a factor of the dynamical system X=(X,X,µ,T)if there exists a measurable mapπ:X→Y withπ(µ)=νthat T◦π=π◦T.A common factor of two systems X and Y is thus a third system Z which is a factor of both.A joining of the two systems X and Y is any system W which admits both as factors and is in turn spanned by them.According to Furstenberg’s definition the systems X and Y are disjoint if the product system X×Y is the only joining they admit.In the topological category,a joining of(X,T)and(Y,S)is any subsystem W⊂X×Y of the product system(X×Y,T×S)whose projections on both coordinates are full;i.e.πX(W)=X andπY(W)=Y.(X,T)and(Y,S)are disjoint if X×Y is the unique joining of these two systems.It is easy to verify that if(X,T)and(Y,S)are disjoint then at least one of them is minimal.Also,if both systems are minimal then they are disjoint iffthe product system(X×Y,T×S)is minimal.In1979,D.Rudolph,using joining techniques,provided thefirst example of a pair of ergodic measure preserving transformations with no common factor which are not disjoint[72].In this work Rudolph laid the foundation of joining theory.He introduced the class of dynamical systems having“minimal self-joinings”(MSJ),and constructed a rank one mixing dynamical system having minimal self-joinings of all orders.MEASURABLE AND TOPOLOGICAL DYNAMICS11 Given a dynamical system X=(X,X,µ,T)a probability measureλon the product of k copies of X denoted X1,X2,...,X k,invariant under the product transformation and projecting ontoµin each coordinate is a k-fold self-joining.It is called an off-diagonal if it is a“graph”measure of the formλ=gr(µ,T n1,...,T n k),i.e.λis the image ofµunder the map x→ T n1x,T n2x,...,T n k x of X into k i=1X i.The joiningλis a product of off-diagonals if there exists a partition(J1,...,J m)of {1,...,k}such that(i)For each l,the projection ofλon i∈J l X i is an off-diagonal,(ii) The systems i∈J l X i,1≤l≤m,are independent.An ergodic system X has minimal self-joinings of order k if every k-fold ergodic self-joining of X is a product of off-diagonals.In[72]Rudolph shows how any dynamical system with MSJ can be used to con-struct a counter example to Furstenberg’s question as well as a wealth of other counter examples to various questions in ergodic theory.In[52]del Junco,Rahe and Swanson were able to show that the classical example of Chac´o n[16]has MSJ,answering a question of Rudolph whether a weakly but not strongly mixing system with MSJ exists.In[38]Glasner and Weiss provide a topological counterexample,which also serves as a natural counterexample in the measure category.The example consists of two horocycleflows which have no nontrivial common factor but are nevertheless not disjoint.It is based on deep results of Ratner[71]which provide a complete description of the self joinings of a horocycleflow.More recently an even more strik-ing example was given in the topological category by E.Lindenstrauss,where two minimal dynamical systems with no nontrivial factor share a common almost1-1 extension,[63].Beginning with the pioneering works of Furstenberg and Rudolph,the notion of joinings was exploited by many authors;Furstenberg1977[24],Rudolph1979[72], Veech1982[81],Ratner1983[71],del Junco and Rudolph1987[53],Host1991 [47],King1992[58],Glasner,Host and Rudolph1992[36],Thouvenot1993[77], Ryzhikov1994[73],Kammeyer and Rudolph1995(2002)[55],del Junco,Lema´n czyk and Mentzen1995[51],and Lema´n czyk,Parreau and Thouvenot2000[62],to men-tion a few.The negative answer to Furstenberg’s question and the consequent works on joinings and disjointness show that in order to study the relationship between two dynamical systems it is necessary to know all the possible joinings of the two systems and to understand the nature of these joinings.Some of the best known disjointness relations among families of dynamical systems are the following:•id⊥ergodic,•distal⊥weakly mixing([23]),•rigid⊥mild mixing([27]),•zero entropy⊥K-systems([23]),in the measure category and•F-systems⊥minimal([23]),•minimal distal⊥weakly mixing,•minimal zero entropy⊥minimal UPE-systems([9]),12 E.GLASNER AND B.WEISSin the topological category.d mixing:measure vs.topological4.1.Definition.Let X=(X,X,µ,T)be a measure dynamical system.1.The system X is rigid if there exists a sequence n kր∞such thatlimµ(T n k A∩A)=µ(A)for every measurable subset A of X.We say that X is{n k}-rigid.2.An ergodic system is mildly mixing if it has no non-trivial rigid factor. These notions were introduced in[27].The authors show that the mild mixing property is equivalent to the following multiplier property.4.2.Theorem.An ergodic system X=(X,X,µ,T)is mildly mixing ifffor every ergodic(finite or infinite)measure preserving system(Y,Y,ν,T),the product system(X×Y,µ×ν,T×T),is ergodic.Since every Kronecker system is rigid it follows from Theorem2.1that mild mixing implies weak mixing.Clearly strong mixing implies mild mixing.It is not hard to construct rigid weakly mixing systems,so that the class of mildly mixing systems is properly contained in the class of weakly mixing systems.Finally there are mildly but not strongly mixing systems;e.g.Chac´o n’s system is an example(see Aaronson and Weiss[1]).We also have the following analytic characterization of mild mixing.4.3.Proposition.An ergodic system X is mildly mixing iffφf(n)<1,lim supn→∞for every matrix coefficientφf,where for f∈L2(X,µ), f =1,φf(n):= U T n f,f . Proof.If X→Y is a rigid factor,then there exists a sequence n i→∞such that U T n i→id strongly on L2(Y,ν).For any function f∈L20(Y,ν)with f =1, we have lim i→∞φf(n i)=1.Conversely,if lim i→∞φf(n i)=1for some n iր∞and f∈L20(X,µ), f =1,then lim i→∞U T n i f=f.Clearly f can be replaced by a bounded function and we let A be the sub-algebra of L∞(X,µ)generated by {U T n f:n∈Z}.The algebra A defines a non-trivial factor X→Y such that U T n i→id strongly on L2(Y,ν). We say that a collection F of nonempty subsets of Z is a family if it is hereditary upward and proper(i.e.A⊂B and A∈F implies B∈F,and F is neither empty nor all of2Z).With a family F of nonempty subsets of Z we associate the dual familyF∗={E:E∩F=∅,∀F∈F}.It is easily verified that F∗is indeed a family.Also,for families,F1⊂F2⇒F∗1⊃F∗2, and F∗∗=F.。

University of Bialystok

University of Bialystok

JOURNAL OF FORMALIZED MATHEMATICSVolume12,Released2000,Published2001Inst.of Computer Science,University of Bia l ystokGauges and Cages.Part I1Artur Korni l owicz University of Bia l ystokRobert Milewski University of Bia l ystokAdam Naumowicz University of Bia l ystokAndrzej Trybulec University of Bia l ystokMML Identifier:JORDAN1A.WWW:/JFM/Vol12/jordan1a.htmlThe articles[32],[38],[37],[12],[35],[1],[34],[29],[3],[4],[2],[36],[26],[19],[28],[39],[27], [8],[18],[13],[30],[17],[15],[9],[16],[10],[11],[6],[21],[31],[23],[33],[22],[20],[24],[5], [7],[25],and[14]provide the notation and terminology for this paper.1.PreliminariesFor simplicity,we adopt the following convention:i,i1,i2,j,j1,j2,k,m,n,t are naturalnumbers,D is a non empty subset of E2T ,E is a compact non vertical non horizontal subsetof E2T ,C is a compact connected non vertical non horizontal subset of E2T,G is a Go-board,p,q,x are points of E2T ,and r,s are real numbers.We now state several propositions:(1)For all real numbers s1,s3,s4,l such that s1≤s3and s1≤s4and0≤l and l≤1holds s1≤(1−l)·s3+l·s4.(2)For all real numbers s1,s3,s4,l such that s3≤s1and s4≤s1and0≤l and l≤1holds(1−l)·s3+l·s4≤s1.(3)If n>0,then m n mod m=0.(4)If j>0and i mod j=0,then i÷j=ij.(5)If n>0,then i n÷i=i n.(6)If0<n and1<r,then1<r n.(7)If r>1and m>n,then r m>r n.(8)Let T be a non empty topological space,A be a subset of T,and B,C be subsetsof the carrier of T.If A is connected and C is a component of B and A∩C=∅and A⊆B,then A⊆C.Let f be afinite sequence.The functor Center f yielding a natural number is defined as follows:(Def.1)Center f=(len f÷2)+1.1This work has been partially supported by CALCULEMUS grant HPRN-CT-2000-00102.1c Association of Mizar UsersWe now state two propositions:(9)For everyfinite sequence f such that len f is odd holds len f=2·Center f−1.(10)For everyfinite sequence f such that len f is even holds len f=2·Center f−2.2.Some Subsets of the PlaneOne can verify the following observations:∗there exists a subset of E2T which is compact,non vertical,non horizontal,and non empty and satisfies conditions of simple closed curve,∗there exists a subset of E2T which is compact,non empty,and horizontal,and∗there exists a subset of E2T which is compact,non empty,and vertical.We now state a number of propositions:(11)If p∈N-most D,then p2=N-bound D.(12)If p∈E-most D,then p1=E-bound D.(13)If p∈S-most D,then p2=S-bound D.(14)If p∈W-most D,then p1=W-bound D.(15)BDD D misses D.(16)For every compact non empty subset S of E2T satisfying conditions of simple closedcurve holds LowerArc S⊆S and UpperArc S⊆S.(17)p∈VerticalLine p1.(18)[r,s]∈VerticalLine r.(19)For every subset A of E2Tsuch that A⊆VerticalLine s holds A is vertical.(20)(proj2)([r,s])=s and(proj1)([r,s])=r.(21)If p1=q1and r∈[(proj2)(p),(proj2)(q)],then[p1,r]∈L(p,q).(22)If p2=q2and r∈[(proj1)(p),(proj1)(q)],then[r,p2]∈L(p,q).(23)If p∈VerticalLine s and q∈VerticalLine s,then L(p,q)⊆VerticalLine s.Let S be a non empty subset of E2T satisfying conditions of simple closed curve.Onecan verify that LowerArc S is non empty and compact and UpperArc S is non empty and compact.One can prove the following propositions:(24)For all subsets A,B of E2Tsuch that A meets B holds(proj2)◦A meets(proj2)◦B.(25)For all subsets A,B of E2T such that A misses B and A⊆VerticalLine s andB⊆VerticalLine s holds(proj2)◦A misses(proj2)◦B.(26)For every closed subset S of E2Tsuch that S is Bounded holds(proj2)◦S is closed.(27)For every subset S of E2Tsuch that S is Bounded holds(proj2)◦S is bounded.(28)For every compact subset S of E2Tholds(proj2)◦S is compact.In this article we present several logical schemes.The scheme TRSubsetEx deals with a natural number A and a unary predicate P,and states that:There exists a subset A of E AT such that for every point p of E ATholds p∈AiffP[p]for all values of the parameters.The scheme TRSubsetUniq deals with a natural number A and a unary predicate P, and states that:Let A,B be subsets of E AT .Suppose for every point p of E ATholds p∈A iffP[p]and for every point p of E A T holds p∈B iffP[p].Then A=B for all values of the parameters.Let p be a point of E2T .The functor NorthHalfline p yielding a subset of E2Tis definedby:(Def.2)For every point x of E2Tholds x∈NorthHalfline p iffx1=p1and x2≥p2.The functor EastHalfline p yielding a subset of E2Tis defined by:(Def.3)For every point x of E2Tholds x∈EastHalfline p iffx1≥p1and x2=p2.The functor SouthHalfline p yielding a subset of E2Tis defined by:(Def.4)For every point x of E2Tholds x∈SouthHalfline p iffx1=p1and x2≤p2.The functor WestHalfline p yields a subset of E2Tand is defined as follows:(Def.5)For every point x of E2Tholds x∈WestHalfline p iffx1≤p1and x2=p2.The following propositions are true:(29)NorthHalfline p={q;q ranges over points of E2T:q1=p1∧q2≥p2}.(30)NorthHalfline p={[p1,r];r ranges over elements of R:r≥p2}.(31)EastHalfline p={q;q ranges over points of E2T:q1≥p1∧q2=p2}.(32)EastHalfline p={[r,p2];r ranges over elements of R:r≥p1}.(33)SouthHalfline p={q;q ranges over points of E2T:q1=p1∧q2≤p2}.(34)SouthHalfline p={[p1,r];r ranges over elements of R:r≤p2}.(35)WestHalfline p={q;q ranges over points of E2T:q1≤p1∧q2=p2}.(36)WestHalfline p={[r,p2];r ranges over elements of R:r≤p1}.Let p be a point of E2T.One can check the following observations:∗NorthHalfline p is non empty and convex,∗EastHalfline p is non empty and convex,∗SouthHalfline p is non empty and convex,and∗WestHalfline p is non empty and convex.3.GoboardsWe now state a number of propositions:(37)If1≤i and i≤len G and1≤j and j≤width G,then G◦(i,j)∈L(G◦(i,1),G◦(i,width G)).(38)If1≤i and i≤len G and1≤j and j≤width G,then G◦(i,j)∈L(G◦(1,j),G◦(len G,j)).(39)If1≤j1and j1≤width G and1≤j2and j2≤width G and1≤i1and i1≤i2and i2≤len G,then(G◦(i1,j1))1≤(G◦(i2,j2))1.(40)If1≤i1and i1≤len G and1≤i2and i2≤len G and1≤j1and j1≤j2andj2≤width G,then(G◦(i1,j1))2≤(G◦(i2,j2))2.(41)Let f be a non constant standard special circular sequence.Suppose f is a sequencewhich elements belong to G and1≤t and t≤len G.Then(G◦(t,width G))2≥N-bound L(f).(42)Let f be a non constant standard special circular sequence.Suppose f is a se-quence which elements belong to G and1≤t and t≤width G.Then(G◦(1,t))1≤W-bound L(f).(43)Let f be a non constant standard special circular sequence.Suppose f is a se-quence which elements belong to G and1≤t and t≤len G.Then(G◦(t,1))2≤S-bound L(f).(44)Let f be a non constant standard special circular sequence.Suppose f is a sequencewhich elements belong to G and1≤t and t≤width G.Then(G◦(len G,t))1≥E-bound L(f).(45)If i≤len G and j≤width G,then cell(G,i,j)is non empty.(46)If i≤len G and j≤width G,then cell(G,i,j)is connected.(47)If i≤len G,then cell(G,i,0)is not Bounded.(48)If i≤len G,then cell(G,i,width G)is not Bounded.4.GaugesNext we state a number of propositions:(49)width Gauge(D,n)=2n+3.(50)If i<j,then len Gauge(D,i)<len Gauge(D,j).(51)If i≤j,then len Gauge(D,i)≤len Gauge(D,j).(52)If m≤n and1<i and i<len Gauge(D,m),then1<2n− m·(i−2)+2and2n− m·(i−2)+2<len Gauge(D,n).(53)If m≤n and1<i and i<width Gauge(D,m),then1<2n− m·(i−2)+2and2n− m·(i−2)+2<width Gauge(D,n).(54)Suppose m≤n and1<i and i<len Gauge(D,m)and1<j and j<width Gauge(D,m).Let i1,j1be natural numbers.If i1=2n− m·(i−2)+2and j1=2n− m·(j−2)+2,then Gauge(D,m)◦(i,j)=Gauge(D,n)◦(i1,j1).(55)If m≤n and1<i and i+1<len Gauge(D,m),then1<2n− m·(i−1)+2and2n− m·(i−1)+2≤len Gauge(D,n).(56)If m≤n and1<i and i+1<width Gauge(D,m),then1<2n− m·(i−1)+2and2n− m·(i−1)+2≤width Gauge(D,n).(57)If1≤i and i≤len Gauge(D,n)and1≤j and j≤len Gauge(D,m)and n>0and m>0or n=0and m=0,then(Gauge(D,n)◦(Center Gauge(D,n),i))1=(Gauge(D,m)◦(Center Gauge(D,m),j))1.(58)If1≤i and i≤len Gauge(D,n)and1≤j and j≤len Gauge(D,m)and n>0and m>0or n=0and m=0,then(Gauge(D,n)◦(i,Center Gauge(D,n)))2=(Gauge(D,m)◦(j,Center Gauge(D,m)))2.(59)If1≤i and i≤len Gauge(C,1),then(Gauge(C,1)◦(Center Gauge(C,1),i))1=W-bound C+E-bound C.2(60)If1≤i and i≤len Gauge(C,1),then(Gauge(C,1)◦(i,Center Gauge(C,1)))2=S-bound C+N-bound C.2(61)If1≤i and i≤len Gauge(E,n)and1≤j and j≤len Gauge(E,m)and m≤n,then(Gauge(E,n)◦(i,len Gauge(E,n)))2≤(Gauge(E,m)◦(j,len Gauge(E,m)))2.(62)If1≤i and i≤len Gauge(E,n)and1≤j and j≤len Gauge(E,m)and m≤n,then(Gauge(E,n)◦(len Gauge(E,n),i))1≤(Gauge(E,m)◦(len Gauge(E,m),j))1.(63)If1≤i and i≤len Gauge(E,n)and1≤j and j≤len Gauge(E,m)and m≤n,then(Gauge(E,m)◦(1,j))1≤(Gauge(E,n)◦(1,i))1.(64)If1≤i and i≤len Gauge(E,n)and1≤j and j≤len Gauge(E,m)and m≤n,then(Gauge(E,m)◦(j,1))2≤(Gauge(E,n)◦(i,1))2.(65)If1≤m and m≤n,then L(Gauge(E,n)◦(Center Gauge(E,n),1),Gauge(E,n)◦(Center Gauge(E,n),len Gauge(E,n)))⊆L(Gauge(E,m)◦(Center Gauge(E,m),1),Gauge(E,m)◦(Center Gauge(E,m),len Gauge(E,m))).(66)If1≤m and m≤n and1≤j and j≤width Gauge(E,n),then L(Gauge(E,n)◦(Center Gauge(E,n),1),Gauge(E,n)◦(Center Gauge(E,n),j))⊆L(Gauge(E,m)◦(Center Gauge(E,m),1),Gauge(E,n)◦(Center Gauge(E,n),j)).(67)If1≤m and m≤n and1≤j and j≤width Gauge(E,n),then L(Gauge(E,m)◦(Center Gauge(E,m),1),Gauge(E,n)◦(Center Gauge(E,n),j))⊆L(Gauge(E,m)◦(Center Gauge(E,m),1),Gauge(E,m)◦(Center Gauge(E,m),len Gauge(E,m))).(68)Suppose m≤n and1<i and i+1<len Gauge(E,m)and1<j and j+1<width Gauge(E,m).Let i1,j1be natural numbers.Suppose2n− m·(i−2)+2≤i1and i1<2n− m·(i−1)+2and2n− m·(j−2)+2≤j1and j1<2n− m·(j−1)+2.Then cell(Gauge(E,n),i1,j1)⊆cell(Gauge(E,m),i,j).(69)Suppose m≤n and3≤i and i<len Gauge(E,m)and1<j and j+1<width Gauge(E,m).Let i1,j1be natural numbers.If i1=2n− m·(i−2)+2andj1=2n− m·(j−2)+2,then cell(Gauge(E,n),i1− 1,j1)⊆cell(Gauge(E,m),i− 1,j).(70)If i≤len Gauge(C,n),then cell(Gauge(C,n),i,0)⊆UBD C.(71)If i≤len Gauge(E,n),then cell(Gauge(E,n),i,width Gauge(E,n))⊆UBD E.5.CagesThe following propositions are true:(72)If p∈C,then NorthHalfline p meets L(Cage(C,n)).(73)If p∈C,then EastHalfline p meets L(Cage(C,n)).(74)If p∈C,then SouthHalfline p meets L(Cage(C,n)).(75)If p∈C,then WestHalfline p meets L(Cage(C,n)).(76)There exist k,t such that1≤k and k<len Cage(C,n)and1≤t and t≤width Gauge(C,n)and(Cage(C,n))k=Gauge(C,n)◦(1,t).(77)There exist k,t such that1≤k and k<len Cage(C,n)and1≤t and t≤len Gauge(C,n)and(Cage(C,n))k=Gauge(C,n)◦(t,1).(78)There exist k,t such that1≤k and k<len Cage(C,n)and1≤t and t≤width Gauge(C,n)and(Cage(C,n))k=Gauge(C,n)◦(len Gauge(C,n),t).(79)If1≤k and k≤len Cage(C,n)and1≤t and t≤len Gauge(C,n)and(Cage(C,n))k=Gauge(C,n)◦(t,width Gauge(C,n)),then(Cage(C,n))k∈N-most L(Cage(C,n)).(80)If1≤k and k≤len Cage(C,n)and1≤t and t≤width Gauge(C,n)and (Cage(C,n))k=Gauge(C,n)◦(1,t),then(Cage(C,n))k∈W-most L(Cage(C,n)).(81)If1≤k and k≤len Cage(C,n)and1≤t and t≤len Gauge(C,n)and (Cage(C,n))k=Gauge(C,n)◦(t,1),then(Cage(C,n))k∈S-most L(Cage(C,n)).(82)If1≤k and k≤len Cage(C,n)and1≤t and t≤width Gauge(C,n)and(Cage(C,n))k=Gauge(C,n)◦(len Gauge(C,n),t),then(Cage(C,n))k∈E-most L(Cage(C,n)).(83)W-bound L(Cage(C,n))=W-bound C−E-bound C−W-bound C.n.(84)S-bound L(Cage(C,n))=S-bound C−N-bound C−S-bound C2n.(85)E-bound L(Cage(C,n))=E-bound C+E-bound C−W-bound C2n(86)N-bound L(Cage(C,n))+S-bound L(Cage(C,n))=N-bound L(Cage(C,m))+ S-bound L(Cage(C,m)).(87)E-bound L(Cage(C,n))+W-bound L(Cage(C,n))=E-bound L(Cage(C,m))+ W-bound L(Cage(C,m)).(88)If i<j,then E-bound L(Cage(C,j))<E-bound L(Cage(C,i)).(89)If i<j,then W-bound L(Cage(C,i))<W-bound L(Cage(C,j)).(90)If i<j,then S-bound L(Cage(C,i))<S-bound L(Cage(C,j)).(91)If1≤i and i≤len Gauge(C,n),then N-bound L(Cage(C,n))=(Gauge(C,n)◦(i,len Gauge(C,n)))2.(92)If1≤i and i≤len Gauge(C,n),then E-bound L(Cage(C,n))=(Gauge(C,n)◦(len Gauge(C,n),i))1.(93)If1≤i and i≤len Gauge(C,n),then S-bound L(Cage(C,n))=(Gauge(C,n)◦(i,1))2.(94)If1≤i and i≤len Gauge(C,n),then W-bound L(Cage(C,n))=(Gauge(C,n)◦(1,i))1.(95)If x∈C and p∈NorthHalfline x∩ L(Cage(C,n)),then p2>x2.(96)If x∈C and p∈EastHalfline x∩ L(Cage(C,n)),then p1>x1.(97)If x∈C and p∈SouthHalfline x∩ L(Cage(C,n)),then p2<x2.(98)If x∈C and p∈WestHalfline x∩ L(Cage(C,n)),then p1<x1.(99)If x∈N-most C and p∈NorthHalfline x and1≤i and i<len Cage(C,n)andp∈L(Cage(C,n),i),then L(Cage(C,n),i)is horizontal.(100)If x∈E-most C and p∈EastHalfline x and1≤i and i<len Cage(C,n)and p∈L(Cage(C,n),i),then L(Cage(C,n),i)is vertical.(101)If x∈S-most C and p∈SouthHalfline x and1≤i and i<len Cage(C,n)and p∈L(Cage(C,n),i),then L(Cage(C,n),i)is horizontal.(102)If x∈W-most C and p∈WestHalfline x and1≤i and i<len Cage(C,n)and p∈L(Cage(C,n),i),then L(Cage(C,n),i)is vertical.(103)If x∈N-most C and p∈NorthHalfline x∩ L(Cage(C,n)),then p2= N-bound L(Cage(C,n)).(104)If x∈E-most C and p∈EastHalfline x∩ L(Cage(C,n)),then p1= E-bound L(Cage(C,n)).(105)If x∈S-most C and p∈SouthHalfline x∩ L(Cage(C,n)),then p2= S-bound L(Cage(C,n)).(106)If x∈W-most C and p∈WestHalfline x∩ L(Cage(C,n)),then p1=W-bound L(Cage(C,n)).(107)If x∈N-most C,then there exists a point p of E2T such that NorthHalfline x∩L(Cage(C,n))={p}.(108)If x∈E-most C,then there exists a point p of E2T such that EastHalfline x∩L(Cage(C,n))={p}.(109)If x∈S-most C,then there exists a point p of E2T such that SouthHalfline x∩L(Cage(C,n))={p}.(110)If x∈W-most C,then there exists a point p of E2T such that WestHalfline x∩L(Cage(C,n))={p}.References[1]Grzegorz Bancerek.The fundamental properties of natural numbers.Journal of Formalized Mathematics,1,1989./JFM/Vol1/nat_1.html.[2]Grzegorz Bancerek and Krzysztof Hryniewiecki.Segments of natural numbers andfinite sequences.Journalof Formalized Mathematics,1,1989./JFM/Vol1/finseq_1.html.[3]Czes l aw Byli´n ski.Functions and their basic properties.Journal of Formalized Mathematics,1,1989.http:///JFM/Vol1/funct_1.html.[4]Czes l aw Byli´n ski.Functions from a set to a set.Journal of Formalized Mathematics,1,1989.http:///JFM/Vol1/funct_2.html.[5]Czes l aw Byli´n ski.Gauges.Journal of Formalized Mathematics,11,1999./JFM/Vol11/jordan8.html.[6]Czes l aw Byli´n ski and Piotr Rudnicki.Bounding boxes for compact sets in E2.Journal of FormalizedMathematics,9,1997./JFM/Vol9/pscomp_1.html.[7]Czes l aw Byli´n ski and Mariusz˙Zynel.Cages,external aproximation of jordan’s curve.Journal of 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Single superpartner production at Tevatron Run II

Single superpartner production at Tevatron Run II

a r X i v :h e p -p h /0007288v 2 25 J u l 2000Single superpartner production at Tevatron Run IIF.D´e liot 1,G.Moreau 2,C.Royon 1,3,41:Service de Physique des Particules,DAPNIA CEA/Saclay 91191Gif-sur-Yvette Cedex,France2:Service de Physique Th´e oriqueCEA/Saclay 91191Gif-sur-Yvette Cedex,France3:Brookhaven National Laboratory,Upton,New York,11973;4:University of Texas,Arlington,Texas,76019February 1,2008AbstractWe study the single productions of supersymmetric particles at Tevatron Run II which occur inthe 2→2−body processes involving R-parity violating couplings of type λ′ijk L i Q j D ck .We focus on the single gaugino productions which receive contributions from the resonant slepton productions.We first calculate the amplitudes of the single gaugino productions.Then we perform analyses of the single gaugino productions based on the three charged leptons and like sign dilepton signatures.These analyses allow to probe supersymmetric particles masses beyond the present experimental limits,and many of the λ′ijk coupling constants down to values smaller than the low-energy bounds.Finally,we show that the studies of the single gaugino productions offer the opportunity to reconstruct the ˜χ01,˜χ±1,˜νL and ˜l ±L masses with a good accuracy in a model independent way.1IntroductionIn the Minimal Supersymmetric Standard Model (MSSM),the supersymmetric (SUSY)particles must be produced in pairs.The phase space is largely suppressed in pair produc-tion of SUSY particles due to the important masses of the superpartners.The R-parity violating (R p )extension of the MSSM contains the following additional terms in the superpotential,which are trilinear in the quarks and leptons superfields,W R p=i,j,k12λ′′ijk U c i D c j D ck,(1)where i,j,k are flavour indices.These R p couplings offer the opportunity to produce thescalar supersymmetric particles as resonances [1,2].Although the R p coupling constants are severely constrained by the low-energy experimental bounds [3,4,5],the resonant superpartner production reaches high cross sections both at leptonic [6]and hadronic [7]colliders.The resonant production of SUSY particle has another interest:since its cross section is proportional to a power 2of the relevant R p coupling,this reaction would allow an eas-ier determination of the R p couplings than the pair production provided the R p coupling is large enough.As a matter of fact in the pair production study,the sensitivity on the R p couplings is mainly provided by the displaced vertex analysis of the Lightest Super-symmetric Particle (LSP)decay which is difficult experimentally,especially at hadronic colliders.Neither the Grand Unified Theories(GUT),the string theories nor the study of the discrete gauge symmetries give a strong theoretical argument in favor of the R-parity violating or R-parity conserving scenarios[3].Hence,the resonant production of SUSY particle through R p couplings is an attractive possibility which must be considered in the phenomenology of supersymmetry.The hadronic colliders have an advantage in detecting new particles resonance.Indeed, due to the wide energy distribution of the colliding partons,the resonance can be probed in a wide range of the new particle mass.This is in contrast with the leptonic colliders for which the center of mass energy must befine-tuned in order to discover new narrow width resonances.At hadronic colliders,either a slepton or a squark can be produced at the resonance respectively through aλ′or aλ′′coupling constant.In the hypothesis of a single dominant R p coupling constant,the resonant scalar particle can decay through the same R p coupling as in the production,leading to a two quarkfinal state for the hard process[8,9,10,11,12]. In the case where bothλ′andλcouplings are non-vanishing,the slepton produced viaλ′can decay throughλgiving rise to the samefinal state as in Drell-Yan process,namely two leptons[11,13,15].However,for reasonable values of the R p coupling constants, the decays of the resonant scalar particle via gauge interactions are typically dominant if kinematically allowed[6,16].The main decay of the resonant scalar particle through gauge interactions is the decay into its Standard Model partner plus a gaugino.Indeed,in the case where the resonant scalar particle is a squark,it is produced throughλ′′interactions so that it must be a Right squark˜q R and thus it cannot decay into the W±-boson,which is the only other possible decay channel via gauge interactions.Besides,in the case where the resonant scalar particle is a slepton,it is a Left slepton produced via aλ′coupling but it cannot generally decay as˜l±L→W±˜νL or as˜νL→W±˜l∓L.The reason is that in most of the SUSY models,as for example the supergravity or the gauge mediated models,the mass difference between the Left charged slepton and the Left sneutrino is due to the D-termsso that it isfixed by the relation m2˜l±L−m2˜νL=cos2βM2W[17]and thus it does not exceedthe W±-boson mass.Nevertheless,we note that in the large tanβscenario,a resonant scalar particle of the third generation can generally decay into the W±-boson due to the large mixing in the third family sfermion sector.For instance,in the SUGRA model with a large tanβa tau-sneutrino produced at the resonance can decay as˜ντ→W±˜τ∓1,˜τ∓1 being the lightest stau.The resonant scalar particle production at hadronic colliders leads thus mainly to the single gaugino production,in case where the decay of the relevant scalar particle into gaugino is kinematically allowed.In this paper,we study the single gaugino productions at Tevatron Run II.The single gaugino productions at hadronic colliders werefirst studied in[2,7].Later,studies on the single neutralino[18]and single chargino[19]productions at Tevatron have been performed.The single neutralino[20]and single chargino[21] productions have also been considered in the context of physics at LHC.In the present article,we also study the single superpartner productions at Tevatron Run II which occur via2→2−body processes and do not receive contributions from resonant SUSY particle productions.The singly produced superpartner initiates a cascade decay ended typically by the R p decay of the LSP.In case of a single dominantλ′′coupling constant,the LSP decays into quarks so that this cascade decay leads to multijetfinal states having a large QCD background[7,8].Nevertheless,if some leptonic decays,as for instance˜χ±→l±ν˜χ0,˜χ±being the chargino and˜χ0the neutralino,enter the chain reaction,clearer leptonic signatures can be investigated[22].In contrast,in the hypothesis of a single dominantλ′coupling constant,the LSP decay into charged leptons naturally favors leptonic signatures [2].We will thus study the single superpartner production reaction at Tevatron Run II within the scenario of a single dominantλ′ijk coupling constant.In section2,we define our theoretical framework.In section3,we present the values of the cross sections for the various single superpartner productions viaλ′ijk at Tevatron Run II and we discuss the interesting multileptonic signatures that these processes can generate.In section4,we analyse the three lepton signature induced by the single chargino production.In section5,we study the like sign dileptonfinal state generated by the single neutralino and chargino productions.2Theoretical frameworkOur framework throughout this paper will be the so-called minimal supergravity model (mSUGRA)which assumes the existence of a grand unified gauge theory and family universal boundary conditions on the supersymmetry breaking parameters.We choose the5following parameters:m0the universal scalars mass at the unification scale M X, m1/2the universal gauginos mass at M X,A=A t=A b=Aτthe trilinear Yukawa coupling at M X,sign(µ)the sign of theµ(t)parameter(t=log(M2X/Q2),Q denoting the running scale)and tanβ=<H u>/<H d>where<H u>and<H d>denote the vacuum expectation values of the Higgsfields.In this model,the higgsino mixing parameter |µ|is determined by the radiative electroweak symmetry breaking condition.Note also that the parameters m1/2and M2(t)(˜W wino mass)are related by the solution of the one loop renormalization group equations m1/2=(1−βa t)M a(t)withβa=g2X b a/(4π)2, whereβa are the beta functions,g X is the coupling constant at M X and b a=[3,−1,−11], a=[3,2,1]corresponding to the gauge group factors SU(3)c,SU(2)L,SU(1)Y.We shall set the unification scale at M X=21016GeV and the running scale at the Z0-boson mass: Q=m Z0.We also assume the infraredfixed point hypothesis for the top quark Yukawa coupling [23]that provides a natural explanation of a large top quark mass m top.In the infrared fixed point approach,tanβisfixed up to the ambiguity associated with large or low tanβsolutions.The low solution of tanβisfixed by the equation m top=C sinβ,where C≈190−210GeV forαs(m Z0)=0.11−0.13.For instance,with a top quark mass of m top=174.2GeV[24],the low solution is given by tanβ≈1.5.The second important effect of the infraredfixed point hypothesis is that the dependence of the electroweak symmetry breaking constraint on the A parameter becomes weak so that|µ|is a known function of the m0,m1/2and tanβparameters[23].Finally,we consider the R p extension of the mSUGRA model characterised by a single dominant R p coupling constant of typeλ′ijk.3Single superpartner productions viaλ′ijk at Tevatron Run II 3.1Resonant superpartner productionAt hadronic colliders,either a sneutrino(˜ν)or a charged slepton(˜l)can be produced at the resonance via theλ′ijk coupling.As explained in Section1,for most of the SUSY models, the slepton produced at the resonance has two possible gauge decays,namely a decayjd d dd jil~χ+χ~-d~-ilkliχkuu cj~dddk~ci νiiν~χ+~χ+u jχj+νuku jdd jddνi d kνdkd iνicd jjχ~χ~χ~0d dkdjujiχ0~χ0il~χ0u ~lkj i lcd u~(d)(c)(b)(a)Figure 1:Feynman diagrams for the 4single production reactions involving λ′ijk at hadronic colliders which receive acontribution from a resonant supersymmetric particle production.The λ′ijk coupling constant is symbolised by a smallcircle and the arrows denote the flow of the particle momentum.into either a chargino or a neutralino.Therefore,in the scenario of a single dominant λ′ijk coupling and for most of the SUSY models,either a chargino or a neutralino is singly produced together with either a charged lepton or a neutrino,through the resonant superpartner production at hadronic colliders.There are thus four main possible types of single superpartner production reaction involving λ′ijk at hadronic colliders which receive a contribution from resonant SUSY particle production.The diagrams associated to these four reactions are drawn in Fig.1.As can be seen in this figure,these single superpartner productions receive also some contributions from both the t and u channels.Note that all the single superpartner production processes drawn in Fig.1have charge conjugated processes.We have calculated the amplitudes of the processes shown in Fig.1and the results are given in Appendix A.3.1.1Cross sectionsIn this section,we discuss the dependence of the single gaugino production cross sections on the various supersymmetric parameters.We will not assume here the radiative elec-σ(p p –→χ~1+ µ-)µ = -200 GeV µ = -500 GeVtan βσ (p b )M 2 = 200 GeVm 0 = 200 GeVλ’ 211=0.0900.20.40.60.810102030405060Figure 2:Cross sections (in pb )of the single chargino production p ¯p →˜χ+1µ−at a center of mass energy of 2T eV as a function of the tan βparameter for λ′211=0.09,M 2=200GeV ,m 0=200GeV and two values of the µparameter:µ=−200GeV,−500GeV .troweak symmetry breaking condition in order to study the variations of the cross sections with the higgsino mixing parameter µ.First,we study the cross section of the single chargino production p ¯p →˜χ+l −i which occurs through the λ′ijk coupling (see Fig.1(a)).The differences between the ˜χ+e −,˜χ+µ−and ˜χ+τ−production (occuring respectively through the λ′1jk ,λ′2jk and λ′3jk couplings with identical j and k indices)cross sections involve m l i lepton mass terms (see AppendixA)and are thus negligible.The p ¯p →˜χ+l −i reaction receives contributions from the s channel sneutrino exchange and the t and u channels squark exchanges as shown in Fig.1.However,the t and u channels represent small contributions to the whole single chargino production cross section when the sneutrino exchanged in the s channel is real,namely for m ˜νiL >m ˜χ±.The t and u channels cross sections will be relevant only when the produced sneutrino is virtual since the s channel contribution is small.In this situation the single chargino production rate is greatly reduced compared to the case where the exchanged sneutrino is produced as a resonance.Hence,The t and u channels do notrepresent important contributions to the ˜χ+l −i production rate.The dependence of the ˜χ+l −i production rate on the A coupling is weak.Indeed,theσ(p p –→χ~+µ-)σ(χ~1+ µ-)σ(χ~2+ µ-)µ (GeV)σ (p b )M 2 = 200 GeV m 0 = 200 GeVtan β = 1.5λ’ 211=0.090.20.40.60.81-800-600-400-2000200400600800Figure 3:Cross sections (in pb )of the single chargino productions p ¯p →˜χ+1,2µ−as a function of the µparameter (in GeV )for λ′211=0.09,M 2=200GeV ,tan β=1.5and m 0=200GeV at a center of mass energy of 2T eV .rate depends on the A parameter only through the masses of the third generation squarkseventually exchanged in the t and u channels (see Fig.1).Similarly,the dependences on the A coupling of the rates of the other single gaugino productions shown in Fig.1are weak.Therefore,in this article we present the results for A =ter,we will discuss the effects of large A couplings on the cascade decays which are similar to the effects of large tan βvalues.tan βdependence:The dependence of the ˜χ+l −i production rate on tan βis also weak,except for tan β<10.This can be seen in Fig.2where the cross section of the p ¯p →˜χ+1µ−reaction occuring through the λ′211coupling is shown as a function of the tan βparameter.The choice of the λ′211coupling is motivated by the fact that the analysis in Sections 4and 5are explicitly made for this R p coupling.In Fig.2,we have taken the λ′211valueequal to its low-energy experimental bound for m ˜d R =100GeV which is λ′211<0.09[4].At this stage,some remarks on the values of the cross sections presented in this section must be done.First,the single gaugino production rates must be multiplied by a factor 2in order to take into account the charge conjugated process,which is for example in the present case p ¯p →˜χ−µ+.Furthermore,the values of the cross sections for all the single gaugino productions are obtained using the CTEQ4L structure function [25].σ(p p –→ χ~1+ µ-)M 2 (G e V )m 0 (Ge V )σ (p b )λ’ 211 =0.09 µ = -200 GeV tan β = 1.510015020025030035012515017520022525027530032535037512345Figure 4:Cross section (in pb )of the single chargino production p ¯p →˜χ+1µ−as a function of the m 0(in GeV )and M 2(in GeV )parameters.The center of mass energy is√σ(p p –→ χ~+µ-)σ(χ~1+ µ-)σ(χ~2+ µ-)m 0 (GeV)σ (p b )λ’ 211=0.09 λ’ 231=0.22λ’ 211=0.09λ’ 221=0.18λ’ 212=0.09M 2 = 200 GeV µ = -200 GeV tan β = 1.500.10.20.30.40.50.60.70.80.9100200300400500600Figure 5:Cross sections (in pb )of the single chargino productions p ¯p →˜χ+1,2µ−as a function of the m 0parameter (in GeV ).The center of mass energy is taken at√through various R p couplings of type λ′2jk are presented.In this figure,we only consider the R p couplings giving the highest cross sections.The values of the considered λ′2jk couplings have been taken at their low-energy limit [4]for a squark mass of 100GeV .Therate of the ˜χ+2µ−production through λ′211is also shown in this figure.We already notice that the cross section is significant for many R p couplings and we will come back on this important statement in the following.The σ(p ¯p →˜χ+µ−)rates decrease as m 0increases for the same reason as in Fig.4.A decrease of the rates also occurs at small values of m 0.The reason is the following.When m 0decreases,the ˜νmass is getting closer to the ˜χ±masses so that the phase space factor associated to the decay ˜νµ→˜χ±µ∓decreases.We also observe that the single ˜χ+2production rate is much smaller than the single ˜χ+1production rate,as in Fig.3.The differences between the ˜χ+1µ−production rates occuring via the various λ′2jk couplings are explained by the different parton densities.Indeed,as shown in Fig.1the hard processassociated to the ˜χ+1µ−production occuring through the λ′2jk coupling constant has a partonic initial state ¯q j q k .The ˜χ+1µ−production via the λ′211coupling has first generation quarks in the initial state which provide the maximum parton density.We now discuss the rate behaviours for the reactions p ¯p →˜χ−νµ,p ¯p →˜χ0µ−and p ¯p →˜χ0νµwhich occur via λ′211,in the SUSY parameter space.The dependences of these rates on the A ,tan β,µand M 2parameters are typically the same as for the ˜χ+µ−production rate.The variations of the ˜χ−1νµ,˜χ01,2µ−and ˜χ01νµproductions cross sectionswith the m 0parameter are shown in Fig.6.The ˜χ−2νµ,˜χ03,4µ−and ˜χ03,4νµproduction rates are comparatively negligible and thus have not been represented.We observe in this figure that the cross sections decrease at large m 0values like the ˜χ+µ−production rate.However,while the single ˜χ±1productions rates decrease at small m 0values (see Fig.5and Fig.6),this is not true for the single ˜χ01productions (see Fig.6).The reasonis that in mSUGRA the ˜χ01and ˜l iL (l i =l ±i ,νi )masses are never close enough to inducea significant decrease of the cross section associated to the reaction p ¯p →˜l iL →˜χ01l i ,where l i =l ±i ,νi (see Fig.1(c)(d)),caused by a phase space factor reduction.Therefore,the resonant slepton contribution to the single ˜χ01production is not reduced at small m 0values like the resonant slepton contribution to the single ˜χ±1production.For the samereason,the single ˜χ01productions have much higher cross sections than the single ˜χ±1productions in most of the mSUGRA parameter space,as illustrate Fig.5and Fig.6.We note that in the particular case of a single dominant λ′3jk coupling constant and of largetan βvalues,the rate of the reaction p ¯p →˜τ±1→˜χ01τ±(see Fig.1(d)),where ˜τ±1is the lightest tau-slepton,can be reduced at low m 0values since then m ˜τ±1can be closed to m ˜χ01due to the large mixing occuring in the staus sector.By analysing Fig.5and Fig.6,we also remark that the ˜χ−νµ(˜χ0µ−)production rate is larger than the ˜χ+µ−(˜χ0νµ)one.The explanation is that in p ¯p collisions the initial states of the resonant charged sleptonproduction u j ¯d k ,¯u j d k have higher partonic densities than the initial states of the resonantsneutrino production d j ¯d k ,¯d j d k .This phenomenon also increases the difference between the rates of the ˜χ01µ−and ˜χ+1µ−productions at Tevatron.Although the single ˜χ±1production cross sections are smaller than the ˜χ01ones,it is interesting to study both of them since they have quite high values.σ(χ~10 µ-)σ(χ~20 µ-)σ(χ~10 νµ)σ(χ~1- νµ)m 0 (GeV)σ (p b )λ’ 211=0.09M 2 = 200 GeVµ = -200 GeV tan β = 1.5510152025100150200250300350400450500Figure 6:Cross sections (in pb )of the reactions p ¯p →˜χ−1ν,p ¯p →˜χ01,2µ−and p ¯p →˜χ01νas a function of the m 0parameter (in GeV ).The center of mass energy is taken at√•The squark production d k g→˜u jL l i via the exchange of a˜u jL squark(d k quark)in the t(s)channel.•The sneutrino production¯d j d k→Z˜νiL via the exchange of a d k or d j quark(˜νiL sneutrino)in the t(s)channel.•The charged slepton production¯u j d k→Z˜l iL via the exchange of a d k or u j quark (˜l iL slepton)in the t(s)channel.•The sneutrino production¯u j d k→W−˜νiL via the exchange of a d j quark(˜l iL sneu-trino)in the t(s)channel.•The charged slepton production¯d j d k→W+˜l iL via the exchange of a u j quark(˜νiL sneutrino)in the t(s)channel.The single gluino production cannot reach high cross sections due to the strong ex-perimental limits on the squarks and gluinos masses which are typically about m˜q,m˜g>∼200GeV[30].Indeed,the single gluino production occurs through the exchange of squarks in the t and u channels,as described above,so that the cross section of this productiondecreases as the squarks and gluinos masses increase.For the value m˜q=m˜g=250GeV which is close to the experimental limits,wefind the single gluino production rateσ(p¯p→˜gµ)≈10−2pb which is consistent with the results of[7].The cross sections√given in this section are computed at a center of mass energy ofFor m ˜ν,m ˜l ,m ˜q ,m ˜χ02>m ˜χ±1,the branching ratio B (˜χ±1→˜χ01l ±ν)is typically of order 30%and is smaller than for the other possible decay ˜χ±1→˜χ01¯q p q ′p because of the color factor.Since in our framework the ˜χ01is the LSP,it can only decay via λ′ijk ,either as ˜χ01→l i u j d k or as ˜χ01→νi d j d k ,with a branching ratio B (˜χ01→l i u j d k )ranging between ∼40%and ∼70%.The three lepton signature is particularly attractive at hadronic colliders because of the possibility to reduce the associated Standard Model background.In Section 4.2we describe this Standard Model background and in Section 4.4we show how it can be reduced.4.2Standard Model background of the 3lepton signature at TevatronThe first source of Standard Model background for the three leptons final state is the topquark pair production q ¯q →t ¯tor gg →t ¯t .Since the top quark life time is smaller than its hadronisation time,the top decays and its main channel is the decay into a W gauge bosonand a bottom quark as t →bW .The t ¯tproduction can thus give rise to a 3l final state if the W bosons and one of the b-quarks undergo leptonic decays simultaneously.The cross section,calculated at leading order with PYTHIA [32]using the CTEQ2L structurefunction,times the branching fraction is σ(p ¯p →t ¯t )×B 2(W →l p νp )≈863fb (704fb )with p =1,2,3at√s =2T eV .The W ±Z 0production gives also a small contribution to the 3leptons background through the decays:W →bu p and Z →b ¯b ,W →lνand Z →b ¯b or W →bu p and Z →l ¯l ,if a lepton is produced in each of the b jets.Similarly,the Z 0Z 0production followed by the decays Z →l ¯l (l =e,µ),Z →τ¯τ,where one of the τdecays into lepton while the other decays into jet,leads to three leptons in the final state.Within the same framework as above,the cross section is of order σ(p ¯p →ZZ →3l )≈2fb .The Z 0Z 0production can also contribute weakly to the 3leptons background via the decays:Z →l ¯l and Z →b ¯b or Z →b ¯b and Z →b ¯b ,since a lepton can be produced in a b jet.It has been pointed out recently that the W Z ∗(throughout this paper a star indicates a virtual particle)and the W γ∗productions could represent important contributions to the trilepton background [33,34].The complete list of contributions to the 3leptons final state from the W Z ,W γ∗and ZZ productions,including cases where either one or both of the gauge bosons can be virtual,has been calculated in [35].The authors of [35]have found that the W Z ,W γ∗and ZZ backgrounds (including virtual boson(s))at the upgraded Tevatron have together a cross section of order 0.5fb after the following cuts have been implemented:P t (l 1)>20GeV ,P t (l 2)>15GeV ,P t (l 3)>10GeV ;|η(l 1,l 2,3)|<1.0,2.0;ISO δR =0.4<2GeV ;E /T >25GeV ;81GeV <M inv (l ¯l )<101GeV ;12GeV <M inv (l ¯l );60GeV <m T (l,E /T )<85GeV .We note that there is at most one hard jet in the 3leptons backgrounds generated by the W Z ,W γ∗and ZZ productions (including virtual boson(s)).Since the number of hard jets is equal to 2in our signal (see Section 4.1),a jet veto can thus reduce this Standardm1/2\m0200GeV400GeV100GeV3.3762.974200GeV0.1220.130300GeV1.310−21.310−2s=2T eV.These rates have been calculated with HERWIG[44]using the EHLQ2 structure function.Model background with respect to the signal.Other small sources of Standard Model background have been estimated in[36]:The productions like Zb,W t or W t¯t.After applying cuts on the geometrical acceptance,thetransverse momentum and the isolation,these backgrounds are expected to be at most√of order10−4pb in p¯p collisions with a center of mass energy ofs=2T eV.The authors of[38]have also estimated the background from the three-jet events faking trilepton signals to be around10−3fb.Hence for the study of the Standard Model background associated to the3lepton signature at Tevatron Run II,we consider the W±Z0production and both the physics and non-physics contributions generated by the Z0Z0and t¯t productions.4.3Supersymmetric background of the3lepton signature at TevatronIf an excess of events is observed in the three lepton channel at Tevatron,one would wonder what is the origin of those anomalous events.One would thus have to consider all of the supersymmetric productions leading to the three lepton signature.In the present context of R-parity violation,multileptonicfinal states can be generated by the single chargino production involving R p couplings,but also by the supersymmetric particle pair production which involves only gauge couplings[39,40].In R p models,the superpartner pair production can even lead to the trilepton signature[41,42,43].As a matter of fact, both of the produced supersymmetric particles decay,either directly or through cascade decays,into the LSP which is the neutralino in our framework.In the hypothesis of a dominantλ′coupling constant,each of the2produced neutralinos can decay into a charged lepton and two quarks:at least two charged leptons and four jets in thefinal state are produced.The third charged lepton can be generated in the cascade decays as for example at the level of the chargino decay˜χ±→˜χ0l±ν.In Table1,we show for different mSUGRA points the cross section of the sum ofall superpartner pair productions,namely the R p conserving SUSY background of the3lepton signature generated by the single chargino production.As can be seen in this table, the summed superpartner pair production rate decreases as m0and m1/2increase.This is due to the increase of the superpartner masses as the m0or m1/2parameter increases. The SUSY background will be important only for low values of m0and m1/2as we will see in the following.4.4CutsIn order to simulate the single chargino production p¯p→˜χ±1l∓at Tevatron,the matrix elements(see Appendix A)of this process have been implemented in a version of the SUSYGEN event generator[45]allowing the generation of p¯p reactions[46].The Stan-dard Model background(W±Z0,Z0Z0and t¯t productions)has been simulated using the PYTHIA event generator[32]and the SUSY background(all SUSY particles pair pro-ductions)using the HERWIG event generator[44].SUSYGEN,PYTHIA and HERWIG have been interfaced with the SHW detector simulation package[37],which mimics an average of the CDF and D0Run II detector performance.We have developped a series of cuts in order to enhance the signal-to-background ratio. First,we have selected the events with at least three leptons where the leptons are either an electron,a muon or a tau reconstructed from a jet,namely N l≥3[l=e,µ,τ].We have also considered the case where the selected leptons are only electrons and muons, namely N l≥3[l=e,µ].The selection criteria on the jets was to have a number of jets greater or equal to two, where the jets have a transverse momentum higher than10GeV,namely N j≥2with P t(j)>10GeV.This jet veto reduces the3lepton backgrounds coming from the W±Z0 and Z0Z0productions.Indeed,the W±Z0production generates no hard jets and the Z0Z0production generates at most one hard jet.Moreover,the hard jet produced in the Z0Z0background is generated by a tau decay(see Section4.2)and can thus be identified as a tau.Besides,some effective cuts concerning the energies of the produced leptons have been applied.In Fig.7,we show the distributions of the third leading lepton energy in the3 lepton events produced by the Standard Model background(W±Z0,Z0Z0and t¯t)and the SUSY signal.Based on those kinds of distributions,we have chosen the following cut on the third leading lepton energy:E(l3)>10GeV.Similarly,we have required that the energies of the2leading leptons verify E(l2)>20GeV and E(l1)>20GeV.We will refer to all the selection criteria described above,namely N l≥3[l=e,µ,τ] with E(l1)>20GeV,E(l2)>20GeV,E(l3)>10GeV,and N j≥2with P t(j)>10GeV, as cut1.Finally,since the leptons originating from the hadron decays(as in the t¯t production)are not well isolated,we have applied some cuts on the lepton isolation.We have imposed√the isolation cut∆R=。

地理信息系统导论chap07

地理信息系统导论chap07

7.6 Other Editing Operations 7.6.1 Edgematching 7.6.2 Line Simplification and Smoothing Key Concepts and Terms Review Questions Applications: Spatial Data Editing Task 1: Edit a Shapefile Task 2: Use Cluster Tolerance to Fix Digitizing Errors Between Two Shapefiles Task 3: Use Topology Rule to Fix Dangles Task 4: Use Topology Rule to Ensure That Two Polygon Layers Cover Each Other Challenge Task References
In the United States, the development of spatial data accuracy standards has gone through three phases.
1. U.S. National Map Accuracy Standard, revised and adopted in 1947 2. Accuracy standards for large-scale maps proposed by the American Society for Photogrammetry and Remote Sensing in 1990 3. National Standard for Spatial Data Accuracy established by the Federal Geographic Data Committee in 1998

编译原理龙书课后部分答案(英文版)

编译原理龙书课后部分答案(英文版)

1) What is the difference between a compiler and an interpreter?∙ A compiler is a program that can read a program in one language - the source language - and translate it into an equivalent program in another language – the target language and report any errors in the source program that it detects during the translation process.∙ Interpreter directly executes the operations specified in the source program on inputs supplied by the user.2) What are the advantages of:(a) a compiler over an interpretera. The machine-language target program produced by a compiler is usually much faster than an interpreter at mapping inputs to outputs.(b) an interpreter over a compiler?b. An interpreter can usually give better error diagnostics than a compiler, because it executes the source program statement by statement.3) What advantages are there to a language-processing system in which the compiler produces assembly language rather than machine language?The compiler may produce an assembly-language program as its output, becauseassembly language is easier to produce as output and is easier to debug.4.2.3 Design grammars for the following languages:a) The set of all strings of 0s and 1s such that every 0 is immediately followed by at least 1.S -> SS | 1 | 01 | ε4.3.1 The following is a grammar for the regular expressions over symbols a and b only, using + in place of | for unions, to avoid conflict with the use of vertical bar as meta-symbol in grammars:rexpr -> rexpr + rterm | rtermrterm -> rterm rfactor | rfactorrfactor -> rfactor * | rprimaryrprimary -> a | ba) Left factor this grammar.rexpr -> rexpr + rterm | rtermrterm -> rterm rfactor | rfactorrfactor -> rfactor * | rprimaryrprimary -> a | bb) Does left factoring make the grammar suitable for top-down parsing?No, left recursion is still in the grammar.c) In addition to left factoring, eliminate left recursion from the original grammar.rexpr -> rterm rexpr’rexpr’ -> + rterm rexpr | εrterm -> rfactor rterm’rterm’ -> rfactor rterm | εrfactor -> rprimary rfactor’rfactor’ -> * rfactor’ | εrprimary -> a | bd) Is the resulting grammar suitable for top-down parsing?Yes.Exercise 4.4.1 For each of the following grammars, derive predictive parsers and show the parsing tables. You may left-factor and/or eliminate left-recursion from your grammars first.A predictive parser may be derived by recursive decent or by the table driven approach. Either way you must also show the predictive parse table.a) The grammar of exercise4.2.2 a) S -> 0S1 | 01This grammar has no left recursion. It could possibly benefit from left factoring. Here is the recursive decent PP code.s() {ma tch(‘0’);if (lookahead == ‘0’)s();match(‘1’);}OrLeft factoring the grammar first:S -> 0S’S’ -> S1 | 1s() {match(‘0’); s’();}s’() {if (lookahead == ‘0’)s(); match(‘1’);elsematch(‘1’);}Now we will build the PP table S -> 0S’S’ -> S1 | 1First(S) = {0}First(S’) = {0, 1}Follow(S) = {1, $}non-recursive predictive parsing.b) The grammar of exercise4.2.2 b) S -> +SS | *SS | a with string +*aaa.Left factoring does not apply and there is no left recursion to remove.s() {if(lookahead == ‘+’)match(‘+’); s(); s();else if(lookahead == ‘*’)match(‘*’); s(); s();else if(lookahead == ‘a’)match(‘a’);elsereport(“syntax error”);}First(S) = {+, *, a}non-recursive predictive parsing.5.1.1 a, b, c: Investigating GraphViz as a solution to presenting trees Extend the SDD of Fig. 5.4 to handle expressions as in Fig. 5.1:1.L -> E N1.L.val = E.syn2. E -> F E'1. E.syn = E'.syn2.E'.inh = F.val3.E' -> + T Esubone'1.Esubone'.inh = E'.inh + T.syn2.E'.syn = Esubone'.syn4.T -> F T'1.T'.inh = F.val2.T.syn = T'.syn5.T' -> * F Tsubone'1.Tsubone'.inh = T'.inh * F.val2.T'.syn = Tsubone'.syn6.T' -> epsilon1.T'.syn = T'.inh7.E' -> epsilon1.E'.syn = E'.inh8. F -> digit1. F.val = digit.lexval9. F -> ( E )1. F.val = E.syn10.E -> T1. E.syn = T.syn5.1.3 a, b, c: Investigating GraphViz as a solution to presenting treesWhat are all the topological sorts for the dependency graph of Fig. 5.7?1.1, 2, 3, 4, 5, 6, 7, 8, 92.1, 2, 3, 5, 4, 6, 7, 8, 93.1, 2, 4, 3, 5, 6, 7, 8, 94.1, 3, 2, 4, 5, 6, 7, 8, 95.1, 3, 2, 5, 4, 6, 7, 8, 96.1, 3, 5, 2, 4, 6, 7, 8, 97.2, 1, 3, 4, 5, 6, 7, 8, 98.2, 1, 3, 5, 4, 6, 7, 8, 99.2, 1, 4, 3, 5, 6, 7, 8, 910.2, 4, 1, 3, 5, 6, 7, 8, 95.2.2 a, b: Investigating GraphViz as a solution to presenting treesSuppose that we have a production A -> BCD. Each of the four nonterminals A, B, C, and D have two attributes: s is a synthesized attribute, and i is an inherited attribute. For each of the sets of rules below, tell whether (1) the rules are consistent with anS-attributed definition (2) the rules are consistent with an L-attributed definition, and (3) whether the rules are consistent with any evaluation order at all?a) A.s = B.i + C.s1.No--contains inherited attribute2.Yes--"From above or from the left"3.Yes--L-attributed so no cyclesb) A.s = B.i + C.s and D.i = A.i + B.s1.No--contains inherited attributes2.Yes--"From above or from the left"3.Yes--L-attributed so no cyclesc) A.s = B.s + D.s1.Yes--all attributes synthesized2.Yes--all attributes synthesized3.Yes--S- and L-attributed, so no cyclesd)∙ A.s = D.i∙ B.i = A.s + C.s∙ C.i = B.s∙ D.i = B.i + C.i1.No--contains inherited attributes2.No--B.i uses A.s, which depends on D.i, which depends on B.i (cycle)3.No--Cycle implies no topological sorts (evaluation orders) using the rules Below is a grammar for expressions involving operator + and integer orfloating-point operands. Floating-point numbers are distinguished by having a decimal point.1. E -> E + T | T2.T -> num . num | numa) Give an SDD to determine the type of each term T and expression E.1. E -> Esubone + T1. E.type = if (E.type == float || T.type == float) { E.type = float } else{ E.type = integer }2. E -> T1. E.type = T.type3.T -> numsubone . numsubtwo1.T.type = float4.T -> num1.T.type = integerb) Extend your SDD of (a) to translate expressions into postfix notation. Use the binary operator intToFloat to turn an integer into an equivalent float.Note: I use character ',' to separate floating point numbers in the resulting postfix notation. Also, the symbol "||" implies concatenation.1. E -> Esubone + T1. E.val = Esubone.val || ',' || T.val || '+'2. E -> T1. E.val = T.val3.T -> numsubone . numsubtwo1.T.val = numsubone.val || '.' || numsubtwo.val4.T -> num1.T.val = intToFloat(num.val)5.3.2 Give an SDD to translate infix expressions with + and * into equivalent expressions without redundant parenthesis. For example, since both operators associate from the left, and * takes precedence over +, ((a*(b+c))*(d)) translates into a*(b+c)*d. Note: symbol "||" implies concatenation.1.S -> E1. E.iop = nil2.S.equation = E.equation2. E -> Esubone + T1.Esubone.iop = E.iop2.T.iop = E.iop3. E.equation = Esubone.equation || '+' || T.equation4. E.sop = '+'3. E -> T1.T.iop = E.iop2. E.equation = T.equation3. E.sop = T.sop4.T -> Tsubone * F1.Tsubone.iop = '*'2. F.iop = '*'3.T.equation = Tsubone.equation || '*' || F.equation4.T.sop = '*'5.T -> F1. F.iop = T.iop2.T.equation = F.equation3.T.sop = F.sop6. F -> char1. F.equation = char.lexval2. F.sop = nil7. F -> ( E )1.if (F.iop == '*' && E.sop == '+') { F.equation = '(' || E.equation || ')' }else { F.equation = E.equation }2. F.sop = nilGive an SDD to differentiate expressions such as x * (3*x + x * x) involving the operators + and *, the variable x, and constants. Assume that no simplification occurs, so that, for example, 3*x will be translated into 3*1 + 0*x. Note: symbol "||" implies concatenation. Also, differentiation(x*y) = (x * differentiation(y) + differentiation(x) * y) and differentiation(x+y) = differentiation(x) + differentiation(y).1.S -> E1.S.d = E.d2. E -> T1. E.d = T.d2. E.val = T.val3.T -> F1.T.d = F.d2.T.val = F.val4.T -> Tsubone * F1.T.d = '(' || Tsubone.val || ") * (" || F.d || ") + (" || Tsubone.d || ") * (" ||F.val || ')'2.T.val = Tsubone.val || '*' || F.val5. E -> Esubone + T1. E.d = '(' || Esubone.d || ") + (" || T.d || ')'2. E.val = Esubone.val || '+' || T.val6. F -> ( E )1. F.d = E.d2. F.val = '(' || E.val || ')'7. F -> char1. F.d = 12. F.val = char.lexval8. F -> constant1. F.d = 02. F.val = constant.lexval3.。

空调表冷器几何参数对换热性能的影响

空调表冷器几何参数对换热性能的影响

第21卷第5期凋卒窒词2021年5月REFRIGERATION AND AIR-CONDITIONING10-14+技术研究f {本栏目投稿邮箱:} +zldt@chinajourn |空调表冷器几何参数对换热性能的影响郭月姣顾鑫鑫顾忱徐彤苏梦雨冯国增(江苏科技大学能源与动力学院)摘要表冷器的换热效率受几何结构设计参数影响较大。

为此,本文以某表冷器为研究对象,针对影响表冷器换热性能的主要结构尺寸参数,如管间距,肋片厚度,肋片间距以及管径进行研究。

首先,根据理论分析的方法研究各结构参数对换热性能的影响规律,得出各结构参数的优化范围;然后利用田口正交试验法设计了25组混合正交试验表以研究结构参数对换热效率的耦合影响;最后通过进行参数分析,得出影响表冷器换热性能因素从高到低的顺序为:肋片间距、管径、管间距、肋片厚度。

结果表明:当管间距20.5mm,肋片厚度0.31mm,肋片间距2.65mm,管径9.0mm时,表冷器单位体积换热量最高为7939.22kW/m30研究为表冷器结构参数设计提供了一定的科学依据。

关键词空调表冷器;换热效率;正交试验设计;结构优化Influence of air conditioner cooler geometric parameters on heat exchangeperformanceGuo Yuejiao Gu Xinxin Gu Chen Xu Tong Su Mengyu Feng Guozeng(School of Energy and Power Jiangsu University of Science and TechnologyABSTRACT The heat exchange efficiency of the surface air cooler is greatly affected bythe geometric design parameters.Therefore,this article takes a surface air cooler as theresearch object,and studies the main structural size parameters that affect the heat ex­change performance of the surface air cooler?such as tube spacing,fin thickness,fin spac­ing and tube diameter.First,according to the method of theoretical analysis,the influenceof each structural parameter on the heat transfer performance was studied,and the opti­mization range of each structural parameter was obtained.Then,using the Taguchi or­thogonal test method,25sets of mixed orthogonal test tables were designed to study thestructural parameters on heat transfer.Coupling effect of efficiency.Finally?through pa­rameter analysis,the order of factors affecting the heat exchange performance of the sur­face cooler from high to low is:fin spacing,tube diameter,tube spacing,fin thickness.The results show that when the tube spacing is20.5mm,the fin thickness is0.31mm,the fin spacing is2.65mm,and the tube diameter is9.0mm,the maximum heat transferper unit volume of the surface air cooler is7793.22kW/m3.The research provides a cer­tain scientific basis for the design of the structural parameters of the surface air cooler.KEY WORDS air-conditioning surface air cooler;heat exchange efficiency;orthogonalexperiment design;structure optimization基于住房与城乡建设部报告中数据显示,我不断上升的趋势①,这与全球低碳节能减排的大趋国建筑能耗占全国能耗的比例约为33%,并具有势相悖。

which option is a repeating pattern -回复

which option is a repeating pattern -回复

which option is a repeating pattern -回复Which Option is a Repeating Pattern?Introduction:Repeating patterns are a fundamental concept in mathematics, art, and design. They are sequences that repeat in a predictable manner. These patterns can be found in nature, architecture, music, and many other aspects of our daily lives. In this article, we will explore the concept of repeating patterns and analyze various options to recognize and identify them accurately.Understanding Repeating Patterns:To understand repeating patterns, we need to grasp the concept of repetition. Repetition refers to the act of doing or saying something more than once. When a sequence of elements or shapes repeats in a predictable manner, it forms a repeating pattern. These patterns can be simple or complex, consisting of one or more elements that reoccur in a specific order.Options to Identify Repeating Patterns:Option 1: Analyzing the Elements:The first option to identify a repeating pattern is to analyze the elements present in the sequence. Look for any visible patterns in shapes, colors, or symbols that repeat in a consistent manner. For example, in a sequence of circles, if the sizes or colors repeat in a specific order, it indicates a repeating pattern.Option 2: Observing the Order:Another option to recognize a repeating pattern is to observe the order in which the elements occur. If the sequence follows a specific order in terms of placement or arrangement, then it is likely to be a repeating pattern. For instance, in a sequence of numbers, if the numbers increase or decrease by a constant value each time, it forms a repeating pattern.Option 3: Extending the Sequence:Extending the sequence is another way to identify repeating patterns. By continuing the pattern beyond the given elements orshapes, you can determine if it repeats indefinitely. If the sequence follows a consistent pattern beyond the given elements, it is a repeating pattern. However, it is important to note that not all extended sequences form repeating patterns.Option 4: Using Algebraic Expressions:Algebraic expressions can also help identify repeating patterns. By representing the elements in the pattern using variables and constants, you can establish a general rule for the sequence. This rule will allow you to determine the next elements in the sequence and verify if it forms a repeating pattern.Option 5: Using Mathematical Notation:In mathematics, repeating patterns can be described using mathematical notation. For example, a repeating pattern can be represented by expressing the elements in terms of a formula or using a recursive sequence. Thus, using mathematical notation can be an effective way to identify and describe repeating patterns.Conclusion:Recognizing repeating patterns is an essential skill across various disciplines. Whether in mathematics, art, or design, understanding and identifying repeating patterns enhance our problem-solving abilities and creativity. By analyzing the elements, observing the order, extending the sequence, using algebraic expressions, and mathematical notation, we can accurately discern and describe repeating patterns. The ability to perceive patterns enables us to appreciate the beauty and symmetry present in the world around us.。

String Gyratons in Supergravity

String Gyratons in Supergravity


1 1 Hµνλ H µνλ ] + 12 2
dD x
|g |Bµν J µν + Sm .
(1)
Here G is the D−dimensional gravitational (Newtonian) coupling constant, and Sm is the action for the string matter source. The string coupling constant gs is determined by the vacuum expectation value of the dilaton field φ0 , gs = exp(φ0 ). The 3-form flux Hµνλ = ∂µ Bνλ + ∂ν Bλν + ∂λ Bµν (2)
common sector. The fields in the common sector are the metric gµν , the Kalb-Ramond antisymmetric field Bµν and the dilaton field φ. The corresponding action, which is also the low-energy superstring effective action, is S= 1 16πG dD x |g |e−2φ [R − 4(∇φ)2
II.
BASIC EQUATIONS
We consider the massless bosonic sector of supergravity. We restrict ourselves by discussing what is called the
∗ Electronic † Electronic

marshal.stringtohglobalansi java -回复

marshal.stringtohglobalansi java -回复

marshal.stringtohglobalansi java -回复Marshal.StringToHGlobalAnsi in Java: A Comprehensive GuideIntroductionIn the world of programming, it is not uncommon to encounter the need to interface with native code in order to enhance the functionality and performance of applications. One such scenario is when working with Java, where the Marshal.StringToHGlobalAnsi method comes into play. In this article, we will explore the purpose, usage, and step-by-step implementation ofMarshal.StringToHGlobalAnsi in Java.What is Marshal.StringToHGlobalAnsi?Marshal.StringToHGlobalAnsi is a method used for marshaling a managed string into a native representation in ANSI format. This method is primarily used when calling unmanaged code from managed code. Marshaling refers to the process of converting data between different formats to ensure compatibility.Purpose of Marshal.StringToHGlobalAnsiThe main purpose of Marshal.StringToHGlobalAnsi is to convert a Java string into a C-style (null-terminated) Unicode stringrepresented in ANSI format. This conversion enables the string to be passed to unmanaged code that expects a null-terminated ANSI string.Usage of Marshal.StringToHGlobalAnsiTo use Marshal.StringToHGlobalAnsi, the Java Native Interface (JNI) is required. JNI provides a framework for integrating Java code with native code, allowing for seamless communication between the two. Here is how Marshal.StringToHGlobalAnsi is used:Step 1: Setting up the Native MethodFirst, the native method declaration must be added to a Java class. This declaration signifies the method that will be called from native code. For example:javapublic class NativeLibrary {public native void myMethod(String str);}Step 2: Generating the C++ HeaderNext, generate the C++ header file for the native method using the Java Native Interface's javah utility. This utility scans the class files and extracts the native method signatures. In the command line, navigate to the class file location and execute the following command:javah NativeLibraryThis command generates a C++ header file named "NativeLibrary.h".Step 3: Implementing the Native MethodOpen the generated NativeLibrary.h file and implement the native method. Include the required libraries, such as "jni.h", and use the Marshal.StringToHGlobalAnsi method to transform the Java string into a null-terminated ANSI string. Here is an example implementation:cpp#include "NativeLibrary.h"#include <jni.h>JNIEXPORT void JNICALL Java_NativeLibrary_myMethod(JNIEnv *env, jobject obj, jstring str) {const char *nativeString = env->GetStringUTFChars(str, 0);Use the nativeString in the unmanaged codeenv->ReleaseStringUTFChars(str, nativeString);}Step 4: Compiling the Native CodeCompile the C++ source code, including the generated NativeLibrary.h header file, into a shared library (DLL in Windows) using a C++ compiler. The compilation process may vary depending on the operating system and compiler being used.Step 5: Loading the Native LibraryIn the Java code, load the compiled native library using the System.loadLibrary method. This step must be performed before calling any native methods. Here is an example:javapublic class Main {static {System.loadLibrary("NativeLibrary");}public static void main(String[] args) {NativeLibrary lib = new NativeLibrary();lib.myMethod("Hello World");}}Step 6: Running the Java CodeFinally, compile and run the Java code as usual. The Java runtime environment will load the native library at runtime, allowing the native code to be executed.ConclusionMarshal.StringToHGlobalAnsi is a powerful tool in Java for interacting with unmanaged code. Its purpose is to convert a Javastring into a null-terminated ANSI string, making it suitable for passing to unmanaged code. By following the step-by-step guide outlined in this article, you can effectively utilizeMarshal.StringToHGlobalAnsi in your Java applications and unlock the potential of integrating with native code.。

基于BIM建筑要素拓扑关系的室内封闭空间划分

基于BIM建筑要素拓扑关系的室内封闭空间划分

第38卷第4期计算机仿真2021年4月文章编号:1006 - 9348(2021)04 - 0381 -04基于B IM建筑要素拓扑关系的室内封闭空间划分李扬,刘平(齐齐哈尔大学建筑与土木工程学院,黑龙江齐齐哈尔161006)摘要:针对传统方法未能考虑室内封闭空间拓扑关系提取问题,导致室内封闭空间划分准确度下降,划分费用上升的问题,提出基于BIM建筑要素拓扑关系的室内封闭空间划分方法。

利用BIM模型对封闭室内各空间的连通和隔离对应进行信息分析,并提取出封闭室内各空间的连通关系的拓扑表示,以实现室内封闭空间拓扑关系提取。

通过建筑信息模型组建RCARG模型,将一组对比分量定量作为模型的固有特征。

利用建筑信息模型的领域属性,获取一组描述建筑模型内部空间结构的特征,完成室内封闭空间划分。

仿真结果表明,所提方法不仅能够有效提升室内封闭空间划分准确度,同时还能够降 低室内封闭空间划分费用。

关键词:建筑要素;拓扑关系;室内封闭空间;空间划分中图分类号:TP393 文献标识码:BDivision of Indoor Enclosed Space Based on the topologicalRelationship of BIM Architectural ElementsL I Yang,LIU Ping(S c h o o l of A rchitecture and Civil E n g in e erin g,Q iqihar U n iv ersity,Q iq ih ar H eilongjiang 161006,C h in a)A B S T R A C T:A im ing at the problem that the traditional m ethod fails to co nsider the extraction of the topological re­lationship of the indoor enclosed sp a c e,w h ic h leads to the decrease of the accuracy of the indoor enclosed space divi­sion and the in crease of the division c o s t,a m ethod of dividing the indoor enclosed space based on the topological re-lationship of BIM building elem ents is proposed.U se the BIM m odel to analyze the connection and isolation of each space in the enclosed ro o m,an d extract the topological representation of the connection relationship of each space in the enclosed room to realize the extraction of the topological relationship of the indoor enclosed sp a ce.T he RCARG m odel is constructed through the building inform ation m o d el,and a set of contrast com ponents are q uantified as th e in­herent ch aracteristics of th e m e the dom ain attrib u tes of the building inform ation m odel to obtain a set of characteristics d escribing the internal spatial structure of the building m o d el,and com plete the indoor enclosed space division.T he sim ulation results show that the proposed m ethod can not only effectively im prove the accuracy of indoor enclosed space d iv isio n,but also red u ce the cost of indoor enclosed space division.K E Y W O R D S:B IM;architectural e le m e n ts;topological relatio n sh ip;indoor enclosed s p a c e;space divisioni引言B IM数据具有包含范围广、表达能力强等特点,在建筑基金项目:黑龙江省省属本科高校基本科研业务费青年创新人才项目(135409231 )、全域旅游视阈下历史建筑保护开发与评价标准研究(135409257);黑龙江省教育科学规划基于BIM技术的高校城乡规划专业课程改革研究(GJD1319028);齐齐哈大学创新创业校级立项齐齐哈尔龙沙公园古建筑数字化档案建设(201910232244)收稿日期:2020 - 03 -12工程领域取得了十分广泛的应用相对于传统的建筑模型,BIM数据::_能够帮助人们更好地挖掘以及理解建筑空间的功能奠定坚实的基础。

Cosmic strings in dilaton gravity

Cosmic strings in dilaton gravity

a rXiv:g r-qc/97114v26J un1997DTP/97/1gr-qc/9701014COSMIC STRINGS IN DILATON GRA VITY Ruth Gregory &Caroline Santos ∗Centre for Particle Theory,University of Durham,Durham,DH13LE,U.K.ABSTRACT We examine the metric of an isolated self-gravitating abelian-Higgs vortex in dilatonic gravity for arbitrary coupling of the vortex fields to the dilaton.We look for solutions in both massless and massive dilaton gravity.We compare our results to existing metrics for strings in Einstein and Jordan-Brans-Dicke theory.We explore the generalization of Bogomolnyi arguments for our vortices and comment on the effects on test particles.PACS numbers:04.40.-b,11.27.+d Keywords:gravity,topological defects1.Introduction.Topological defects and other soliton structures have a wide application to many areas of physics.Cosmologists are interested in defects as possible sources for the density perturbations which seeded galaxy formation.String theorists are interested in defects not only as solutions of the low energy effective action,but as true solitons in the full non-perturbative theory which are required for consistency and inter-relation of the full spectrum of string theories.A topological defect is a discontinuity in the vacuum,and in conventionalfield theory can be classified according to the topology of the vacuum manifold of the particularfield theory being used to model the physical set up:disconnected vacuum manifolds give domain walls,non-simply connected manifolds,strings,and manifolds with non-trivial2-and3-spheres give monopoles and textures respectively.In this paper,we are concerned with defects associated with non-simply connected vacuum manifolds:cosmic strings[1]. The gravity of cosmic strings within the context of Einstein theory has been well explored, both in the case of‘model’strings,where the core of the string is modelled by a simplified energy-momentum tensor[2],and in the case of the fully coupled Einstein-Abelian-Higgs system[3,4];with the result that the spacetime of a self-gravitating local string is found to be generically conical,with the angular deficit given by8πGµ,µbeing the energy per unit length of the vortex.It seems likely however,that gravity is not given by the Einstein action,at least at sufficiently high energy scales,and the most promising alternative seems to be that offered by string theory,where the gravity becomes scalar-tensor in nature[5].Scalar-tensor gravity is not new,it was pioneered by Jordan,Brans and Dicke[6],who sought to incorporate Mach’s principle into gravity.The implications of such actions on general Friedmann-Robertson-Walker cosmological models have been well explored[7,8],however, the implications for theories of structure formation have not been so well studied.Broadly speaking,there are two views on explaining structure formation–inflation or defects,the latter consisting of two subsets:cosmic string or texture[9]induced perturbations.While there is little to choose between these from the particle physics or large scale structure point of view,the implications of each of these theories for the perturbations of the microwavebackground are distinct.However,calculations on the microwave background multipole moments do assume Einstein gravity[10],therefore it is interesting to question whether these conclusions are still valid in the context of scalar-tensor gravity.Even if the dilaton acquires a mass at a fairly high energy scale(with respect to the recombination temperature of the universe that is),at the core of a defect symmetry is restored and the physics is determined by the GUT scale,at which the dilaton might have rather different properties, impacting back on the cosmic microwave background.Calculations involving radiation from a cosmic string network generally make use of a “worldsheet-approximation”in which the string is treated as an infinitesimally thin source which moves according to,and has an energy momentum tensor appropriate for a two-dimensional worldsheet governed by the Nambu action.That this action is appropriate for the local string has been convincingly argued in the absence of gravity[11,12],but as yet no proof exists in the presence of gravity.This is generally believed to be related to the problems of using distributional sources of codimension greater than one in general relativity[13].Nonetheless,the fact that the self-gravitating infinite local vortex has a relatively small effect on spacetime lends credence to the worldsheet approximation for the string.In the presence of a dilaton,the worldsheet approximation may no longer be appro-priate.If the dilaton is massless,there is no reason to expect that the string will not have a long range effect on the dilaton,and even if the dilaton is massive,it introduces an additional length scale which may still have significant impact.In this paper,we take a modest step towards resolving this issue by examining the gravi-dilatonfield of a self-gravitating cosmic string in dilaton gravity.We consider a reasonably general form for the interaction with the dilaton,assuming that the abelian-Higgs lagrangian couples to the dilaton via an arbitrary coupling,e2aφL,in the stringframe.We consider both massive and massless dilatons.Our results for the massless dilaton are very similar to those of Gundlach,Ortiz and others[14],who considered cosmic strings in JBD theory.For the massive dilaton wefind that,apart from an intermediate annular region,the long-range structure of the string is as for Einstein gravity,as might be expected.The main exception to this qualitative and expected picture is that for aspecial value(a=−1)of the coupling of the dilaton to thefields which constitute the vortex the dilaton effectively decouples from the string,showing little or no reaction to its presence.This occurs independent of whether the dilaton is massive,and independent of the specifics of the U(1)model,i.e.whether it is type I,II,or supersymmetric.The layout of the paper is as follows:In the next section we review the Nielsen-Olesen vortex in the abelian-Higgs model.In section three we derive the main results of this paper,namely the gravitational and dilatonfields for the self-gravitating vortex in both massless and massive dilatonic gravity.In section four we consider Bogomolnyi bounds for the string,and show that these can only be saturated in the special case a=−1.In this case,the dilaton effectively decouples from the string.In sectionfive we consider the motion of test particles in the background of the string,and in section six we conclude.2.The Abelian Higgs Vortex.We start by briefly reviewing the U(1)vortex in order to establish notation and conventions.We take the abelian Higgs lagrangianL[Φ,A a]=D aΦ†D aΦ−14(Φ†Φ−η2)2(2.1) whereΦis a complex scalarfield,D a=∇a+ieA a is the usual gauge covariant derivative, and˜F ab thefield strength associated with A a.We use units in which¯h=c=1and a mostly minus signature.For cosmic strings associated with galaxy formationη∼1015GeV.We rewrite thefields in a way which makes manifest the physical degrees of freedom of the model:Φ(xα)=ηX(xα)e iχ(xα)(2.2a)1A a(xα)=F ab F ab−λη44e2⊔⊓X−P a P a X+λη2e[P0(R)−1]∇aφ(2.5) where R=√R +P20X02X0(X20−1)=0(2.6a)−P′′0+P′0dR,andβ=λ/2e2=m2X/m2P is the Bogomolnyi parameter[15] (β=1corresponds to the vortex being supersymmetrizable).Note that in these rescaled coordinates,the string has width of order unity.This string has winding number one; for winding number N,we replaceχby Nχ,and hence P by NP.Figure1shows the Nielsen-Olesen solutions for X and P for aβ=1winding number one string.It is also useful to briefly review the self-gravitating NO vortex in Einstein gravity, as much of the formalism can be used directly in the next section.To include the self-gravity of the string,we require a metric which exhibits the symmetries of the source,02468100.20.40.60.81X[r]P[r]FIGURE (1):X and P for a β=N =1vortex.namely,translational invariance along its length and rotational invariance around the core,i.e.cylindrical symmetry.The general cylindrically symmetric metric was given by Thorne[16]ds 2=e 2(γ−ψ)(dt 2−dr 2)−e 2ψdz 2−˜α2e −2ψdφ2(2.7)(where γ,ψ,˜αare independent of z,φ).The string couples to this metric via its energy-momentum tensorG ab =8πGT ab =8πG 2η2∇a X ∇b X +2η2X 2P a P b −2βληr as before,α=√α2+βP ′2α2−βP ′2ˆTθ=−Pθ=e−γX′2−eγX2P2α2+(X2−1)2/4(2.9c)θˆT z=−P z=ˆT00.(2.9d) zThe Einstein equations can then be read offas[16]α′′=−ǫαeγ(E−P R)(2.10a)(αγ′)′=ǫαeγ(P R+Pθ)(2.10b)αγ′2α′γ′=γ′α−only case in which these stresses do vanish is whenβ=1.In this case thefield equations reduce toX′=XP/αP′=1ληr)2e−C dθ2(2.17)=dˆt2−dˆr2−dˆz2−ˆr2(1−A)2e−2C dθ2whereˆt=e C/2t,ˆz=e C/2z,ˆr=e C/2(r+B/(1−A))(2.18) This is seen to be conical with a deficit angle∆=2π(A+C)=2πǫ R E0dR=16π2G rT00dr=8πGµ(2.19) whereµis the energy per unit length of the string.Notice that the deficit angle is inde-pendent of the radial stresses,but that there is a red/blue-shift of time between infinity and the core of the string if they do not vanish.Now let us examine the behaviour of the string with a dilaton present.3.Cosmic strings in dilaton gravity.We are interested in the behaviour of the isolated string metric(2.7)when the gravita-tional interactions take a form typical of low energy string theory[5].In its most minimal form,string gravity replaces the gravitational constant,G by a scalarfield,the dilaton in a rather analogous fashion to that of Jordan,Brans and Dicke who were motivated by Mach’s principle.We take an empirical approach to cosmic strings in this background theory,not concerning ourselves with the origin of thefields that form the vortex,but inputting‘by hand’the abelian-Higgs lagrangian(2.1).To take account of the(unknown) coupling of the cosmic string to the dilaton,we chooseˆS= d4x−g −R+2(∇φ)2−V(φ)+e2(a+2)φL{X,P,e2φg} (3.3) where V(φ)=e2φˆV(φ).Note however that this complicates the matter part of the la-grangian–a factor of e−2φbeing picked up each timeˆg ab is used:T ab=2δL[X,P,e2φg]λe−4φF ac F c b−L g ab(3.4)The“Einstein”equations are nowG ab=12V(φ)g ab−(∇φ)2g ab(3.6)represents the energy-momentum of the dilaton,which has as its equation of motion ⊔⊓φ=1∂φ+(a+1)2λF2e−4φ−λη2ληr,α=√α2 +e−4φβP′2α2 +e−4φβP′2α2 +e−4φβP′22αeγ˜V(φ)+αγ′24∂˜V2αeγE−1α(αX′)′=−2(a+1)X′φ′+XP22X(X2−1)eγ+2φ(3.10e)α P′2ǫαγ′eγ[P R−Pθ−2E]−α′φ′2−(αφ′2)′+1∂φ(3.11)We start by examining the case V(φ)≡0,i.e.a massless dilaton,as this ought to be qualitatively the same as a cosmic string in Brans-Dicke gravity.3.1Massless dilatonic gravity.In the case that the dilaton is massless the equations(3.10)are rather reminiscent of the pure Einstein gravity vortex(2.10),however,there is one crucial difference-the con-straint equation(3.10c)now contains anαφ′2term,and unless a=−1,αφ′will definitely be nonzero.In order to explore this solution,let usfirst consider the“wire approximation”, namelyαeγE(R)=ˆµδ(R);P R=Pθ=0(3.12) whereˆµ=µ/4πǫrepresents the energy per unit length of the cosmic string in units natural to the vortex,and is of order unity.(Recall thatǫsets the gravitational strength of the string.)In this case,eqns.(3.10)are readily integrated to giveα(R)=(1−ǫˆµ)R(3.13a)γ(R)=0(3.13b)ǫˆµ(a+1)φ(R)=ln(dR+b)(3.14b)dfφ=φ0+4dc−c2from(3.10c).This gives a Levi-Civita[17]solution for the metric. (Note that ifφis constant,we have c=0or4d,corresponding to the vacuum general relativistic solutions.)The constants b,c,d,f are given by integrating(3.10)and to order ǫare(a+1)ǫˆµ(3.15) d=1−A,b=B,c=0,f=12where A,B,C are given in(2.16).We can therefore see that c cannot remain zero,and to orderǫ2,c=1λη)(a+1)2ǫ2ˆµ2/4(≃ǫ−ǫ2)≃1.This metric agrees with Gundlach and Ortiz[14],who derived the metric for a Jordan-Brans-Dicke cosmic string.In the string frame,dˆs2=e2φds2=ˆr(a+1)ǫˆµ4 dˆt2−dˆr2−dˆz2−(1−ǫˆµ)2ˆr2−2(a+1)2ǫ2ˆµ2/4dθ2 (3.17) which is almost,but not quite,a conformally rescaled cone.Note[14]that the radius at which non-conical effects become important is when R≃e4ληe4Note that the back-reaction of the linearized solution(3.16)on the vortexfields is to alter the long-range fall-offof the X and Pfields:1−X≃exp{−R1+(a+1)ǫˆµ/4}P≃exp{−R1+(a+1)ǫˆµ/4/ληis the ratio of the dilaton mass to Higgs mass.Of course,wedo not expect that this will be the exact form of the dilaton potential,however,a quadratic approximation will be valid providedφremains close to the minimum of the potential.For a GUT string we expect10−11≤M≤1,representing a range for the unknown dilatonmass of1TeV-1015GeV.The dilaton equation(3.10d),then becomes(αφ′)′=αeγM2φ+ǫ(a+1)2ǫαeγ(P R+Pθ)(3.20)Once again,we begin by considering the wire model for the string which again gives α(R)andγ(R)as in(3.13a,b).However,the presence of the mass term in(3.20)now alters the form of the dilaton;integrating(3.20)for the wire model givesφw=−12(a+1)ǫˆµln M=O(ǫ),hence the quadratic approximation for the potential appears to be justified.For an extended source,we may solve(3.20)implicitly using its Green’s function:φ=−12ǫI0(MR) ∞R K0(MR′)R′[(a+1)E(R′)−(P R(R′)+Pθ(R′))]dR′≃−1102030405060r -5-4-3-2-10φThe dilaton field produced by the vortexM=1M=0.1M=0.01FIGURE (2):A plot of the dilaton field generated by aβ=N =1vortex for various values of the dilaton mass.The factor (a +1)ǫˆµhas been scaledout of the dilaton.Note the reciprocal dependence of the dilaton fall-offon themass,compared to the logarithmic dependence of the amplitude.Thus the spacetime is asymptotically conical in both string and Einstein frames.Now consider a =−1.In this case the dilaton is very strongly damped to zero outside the core:φ≃ǫMsimplify–they becomefirst order–and the vortex saturates an energy bound determined by the winding number of the vortex[21].For the dilatonic vortex,this delicate balance appears to be destroyed,except in the special case a=−1.In this section we would like to formalise this by presenting an energetic argument that a topological bound can be saturated if and only ifβ=−a=1.Since the cosmic string is cylindrically symmetric,and we do not a priori wish to make any assumptions about the global behaviour of the spacetime,we use an energy tailored to the system at hand–the C-energy introduced by Thorne[16]:E c=4π γ−ln∂˜α2παeγ∂E cαeγ γ′−α′′α′ ǫE+12˜V(φ) (4.3)Clearly every term inˆP0is positive semi-definite,and all vanish only inflat space,the latter three vanishing ifφ=γ=0.Now consider E,we may rewrite this asE=e2(a+1)φ e−γ X′−eγXPαe−φ−1α2e−2φ+1φ=γ=0and(2.15)implies that all terms inˆP0vanish except for the last expression in equation(4.4)for E.We thus obtainE c= 2ǫWe begin by commenting on the massive dilaton.Here the metric is given by(2.17) outside the Compton radius of the dilaton,and is therefore conical.Geodesics are therefore the same as for the Einstein cosmic string,and indeed,since the corrections within the Compton radius of the dilaton are extremely small(O(ǫ2)),the geodesics throughout the whole spacetime in the Einstein frame are essentially the same as for the Einstein self-gravitating string.Now consider the massless dilaton.In the string frame the metric is given by eq.(3.17) and the radial motion of a test particle in a plane transverse to the string,dˆz=0,is given by:˙ˆr2+h2ˆrν(1+ν)=E22,with an effective potential given by:Veff =h22(5.3)which is an identical effective potential to that of a particle moving inflat space.(The presence of theǫˆµterms in the definition of h shows that the spacetime is not globally flat,but conical.)All non-static trajectories therefore escape to infinity,and satisfyˆr≥hE2−k(5.4)In addition,there exist static trajectories for massive particles:ˆr=ˆr0,E=1.Now consider a=−1.For comparison with the effective potential(5.3),it is useful to redefine the radial coordinateˆr viaρ=ˆrν2+ν+12,with an effective potential given by:Ueff =h2ν2+ν+1+kν2+ν+1(5.6)Sinceν=O(ǫ),to leading order this isU eff≃h22ρν.(5.7)First considerν>0,i.e.a>−1.For massless particles,Ueff≃V eff,and thus photons escape to infinity,however,note that asˆr→∞,g tt=ˆrν(ν+1)→∞,hence these photons will be infinitely redshifted.(Note that this will happen in either frame,although thered-shifting in the Einstein frame occurs at a rate proportional toǫ2rather thanǫ.)For massive particles,Ueffis now a potential well,(seefigure3)hence all trajectoriesof massive particles are bounded,however,forρ≪e1/ν,Ueff≃V eff hence trajectories approaching‘close’to the cosmic string(i.e.on all scales of cosmological interest)behave as if in a conical spacetime.Such orbits will be highly eccentric,and have an outer bound ofˆr=O(E1/ν).Note that there are no static geodesics in this case,all particles initially at rest will be attracted to the string by an acceleration of orderǫ/r0.Ifν<0,i.e.a<−1,then Ueffis once more a scattering potential and all particles escape to infinity.Since g tt→0in this case,photons will now be infinitely blue-shifted. Once again there are no static solutions to the massive particle geodesic equation,this time the particles are repelled from the string.In the Einstein frame,the photon trajectories are identical to that of the string frame, but now all the massive particle trajectories are bound,as can be seen by removing all the terms involvingνfrom(5.6).0246810ln r00.20.40.60.811.21.4VUFIGURE (3):A comparison of the effective potential for a massive particlein a conical (dashed)and the massless dilaton (solid)background for ν=0.1,h =1.6.DiscussionIn this paper,we have derived the metric for U(1)local cosmic strings in dilaton gravity both with and without a potential for the dilaton.The (unknown)coupling of the abelian-Higgs model to the dilaton is accounted for by coupling the Lagrangian to the gravitational sector by an arbitrary e 2aφfactor.For a massless dilaton,the results are qualitatively the same as those of Gundlach and Ortiz [14],who considered cosmic strings in JBD theory.Essentially,the metric is the same as the usual cosmic string,i.e.conical,in the Einstein frame,and conformally conical in the string frame on scales of cosmological interest.However,on the very large scale,(r ∼√(a +1)2ǫ2ˆµ2),there is additional curvature,and the spacetime is not asymptotically locally flat in either frame.The exception is the special case a =−1,in which the metric is conical in either frame,and the dilaton is shifted in the core relative to infinity,the direction of the shift depending on whether the cosmic string is type I or II,no alteration in the dilaton occurring for the boundary between types I and II:β=1.For a massive dilaton,as expected,the metric asymptotes a conical metric,in bothstring and Einstein frames,however,the string does generate a dilaton‘cloud’,approx-imately of width m H/mφ,which is schematically depicted infigure3,for a=−1.For a=−1the dilaton is only perturbed away from its vacuum value in the core of the string,all.and forβ=1,it is not affected atstring for a=−1.Although it is beyond the scope of this paper to derive the effective action of the cosmic string,the results do support a Nambu approximation for the string,since they show that the metric is little affected on cosmological length scales,and remains approximatelyflat locally(unlike the global string[18]).Damour and Vilenkin[23]have recently explored the impact of a massive dilaton on string networks using a model for the interactions which modifies the Nambu approximation by making the mass per unit length interact with the (massive)dilaton.In other words,the worldsheets act as sources for the dilaton which has a mass mφ.They concluded that a TeV mass dilaton was incompatible with a GUT string network.Our results largely back up this calculation,but with one important caveat:The model used by Damour and Vilenkin makes no reference to the details of the dilaton coupling to the particle physics model producing the strings,the abelian-Higgs lagrangian, i.e.their coupling is independent of our variable a.Therefore,one should renormalize their calculations by factors of(a+1).This means that the conclusion that a TeV mass dilaton is incompatible with string theories of structure formation is only valid if a is not close to −1.For a=−1,such as might be the case if thefields composing the string are derivedfrom heterotic string theory or the NS-NS sector of type II string theory for example,there will be little dilatonic radiation from the cosmic string network,and hence a much weaker constraint.To sum up:the gravitationalfield of a cosmic string in dilaton gravity is surprisingly close to that of an Einstein cosmic string on cosmological distance scales.However,it is the microwave background rather than cosmological observations,that provides the tightest constraint on the cosmic string theory of structure formation.If the strings couple to the dilaton directly(a=−1),then such constraints are identical to those derived in Einstein gravity.However,if the 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A String Quartet Performance

A String Quartet Performance

**A String Quartet Performance**In the realm of musical experiences, a string quartet performance stands as an intimate and profound encounter that touches the very core of our souls.The prelude to a string quartet performance is an atmosphere steeped in hushed anticipation. The concert venue, often small and intimate, awaits the arrival of four masterful musicians and the delicate strains of their instruments. As the audience settles in, a sense of quiet reverence pervades the space.The opening notes of the quartet rise like a gentle breeze, weaving a sonic tapestry that instantly captivates. The initial moments are a delicate invitation into a world of refined beauty and emotional depth.The middle of the performance is a symphony of harmony and conversation among the strings. The violins soar with lyricism, the viola adds a rich and soulful undertone, and the cello grounds the ensemble with its resonant warmth. I recall a particular performance where the quartet played Beethoven's String Quartet No. 14. The way the instruments intertwined and responded to each other, creating a complex web of emotions and musical ideas, was truly transcendent.One of the most enchanting aspects of a string quartet performance is the intense communication and connection among the musicians. Their eyes meet, their breathing synchronizes, and every gesture and nuance conveys a shared understanding and passion for the music. It is a display of unity and collaboration at its finest.String quartet music has a rich history that spans centuries, carrying within it the voices of countless composers and the stories of human experience.As Arthur Schopenhauer said, "Music is the language of the spirit. It opens the secret of life bringing peace, abolishing strife." A string quartet performance embodies this wisdom, offering a refuge from the chaos of the world and a connection to something greater than ourselves.In conclusion, a string quartet performance is not just a musical event; it is a journey of the heart and mind, a communion of spirits through the language of strings.It is a moment where time stands still, and we are transported to a realm of pure beauty and emotion. The magic of a string quartet performance lies in its ability to touch the innermost recesses of our being and leave an indelible mark on our souls.。

The Dirac Sea Contribution To The Energy Of An Electroweak String

The Dirac Sea Contribution To The Energy Of An Electroweak String

arise in extensions of the minimal electroweak which admit greater CP violation [9]. The stability analysis discussed thus far has focussed on the pure Z-string solutions. It is however possible that the formation of bound states of other particles on the string may have a non-trivial effect on the stability of electroweak strings. Of particular importance for a string like solution is the effect of a finite density of bound states moving along the string. In [10] the authors considered a toy model in which they extended the electroweak theory by adding an extra global U(1) scalar to the bosonic sector of the electroweak theory. They constructed solutions with a finite U(1) charge per unit length of the string and then investigated the stability of the resulting solutions. They found that it was pos
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a r X i v :c o n d -m a t /0208561v 1 [c o n d -m a t .m e s -h a l l ] 28 A u g 2002A Topological String:The Rasetti-Regge Lagrangian,Topological Quantum Field Theory,and Vortices in Quantum Fluids A.D.Speliotopoulos ∗National Research Council,Board on Physics and Astronomy,2101Constitution Avenue,NW,Washington,DC 20418†(Dated:July 29,2002)Abstract The kinetic part of the Rasetti-Regge action I RR for vortex lines is studied and links to string theory are made.It is shown that both I RR and the Polyakov string action I P ol can be constructed with the same field X µ.Unlike I NG ,however,I RR describes a Schwarz-type topological quantum field ing generators of classical Lie algebras,I RR is generalized to higher dimensions.In all dimensions,the momentum 1-form P constructed from the canonical momentum for the vortex belongs to the first cohomology class H 1(M,R m )of the worldsheet M swept-out by the vortex line.The dynamics of the vortex line thus depend directly on the topology of M .For a vortex ring,the equations of motion reduce to the Serret-Frenet equations in R 3,and in higher dimensions they reduce to the Maurer-Cartan equations for so (m ).INTRODUCTIONThat vortex lines play an important role in many physical systems is well-known[1,2]. Following the original ideas of Onsager[3]and Feynman[4],researchers have even used vortex rings in an attempt to explain the underlying cause of the the lambda transition for superfluid He4[5,6].More recently,the discovery of Bose-Einstein condensates(BEC)[7,8] has renewed interest in the study of vortex lines,and within the last year vortex excitations have been seen experimentally in BEC[9,10,11,12].There has been a corresponding theoretical interest in the formation and stability of vortex lines in BEC(see,for example, [13]and[14]).While experimental studies of vortex lines in quantumfluids have been remarkable,the-oretical understanding of the dynamics and interactions of vortex lines on a quantum level has proceeded at a much slower pace.Recent theoretical work on vortices in BEC has mostly been focused only on the formation and stability of vortices in the condensate and not on the properties and dynamics of the vortices once they are formed.Much of the efforts in developing a deeper understanding of the dynamics of vortex lines on a quantum level have been based on the work of M.Rasetti and ing arguments from classicalfluid dynamics of idealfluids,they[15]proposed a lagrangian for studying the quantum theory of vortex lines in quantumfluids in three dimensions.Vortex lines are treated as extended objects and like a string in string theory,a vortex line sweeps out a two-dimensional worldsheet M as it propagates in time.Current algebra methods leading to the study of Sdiff(R3),the diffeomorphic group in R3,were then used by them and subsequent researchers[16,17,18,19,20,21,22,23]in an effort to quantize thefield.In this paper we propose a different approach to understanding the dynamics of vortex lines.Focusing on a single vortex,we start with thefield Xµand make use of the so(3)Lie algebra to rewrite the kinetic part of the full Rasetti-Regge action I RR in terms of differential forms,and demonstrate how I RR is related to the Polyakov form[24]of the Nambu-Goto action I P ol[25,26](see also[27]for a different approach).It is then straightforward to see that while I P ol defines a propagating string,I RR defines,when quantized,a Schwarz-type topological quantumfield theory(TQFT)[28,29,30].Indeed,I RR is very similar in form to the Chern-Simons lagrangian.Making use of other classical Lie algebras,we then extend this construction to higher dimensions;the linkage between I P ol and I RR still hold(see also[31]).However,unlike I P ol,which can be constructed in any dimension,I RR exists only ina discrete number of dimension corresponding to the dimensions of the Lie algebra used in its construction.Using this approach,it becomes clear that the understanding of the quantum-and thus statistical behavior-of vortex lines will be thefirst real world application of TQFT.Con-versely,the vortex system provides a means of studying experimentally a TQFT for thefirst time.The purpose of this paper is thus to make the connection between I RR and TQFT, and our approach is strictly classical,and our analysis formal.Nonetheless,a great deal can immediately be discerned about the properties of vortex lines from the fact that it is a TQFT and this classical analysis.As is well known,a TQFT does not define a dynamical system in a traditional sense;a single vortex line does not,strictly speaking,have dynamical variables that evolve with time.TQFT’s are interesting nonetheless,[28,29].While our approach is strictly classical, even at this level wefind deep connections between topology and the dynamics of vortices. Indeed,we show that the momentum1-form P constructed from the canonical momentum of the vortex line belongs to thefirst cohomology class of M;the dynamics of vortices depend directly on the topology of M.Going further,we show formally that the solution to the equations of motion in three-dimensions reduces to the Serret-Frenet equations for arbitrary curves in R3.These equations are themselves equivalent to the equation of motion of a charged particle constrained to move on a unit sphere in the the presence of a dyon located at the center.In higher dimensions the equations of motion reduce to the Maurer-Cartan equations for so(m).The Maurer-Cartan1-forms can be interpreted as a“pure gauge”non-abelian vector potential,and as is the case for TQFT,we are working withflat vector bundles.With this analogy,explicit solutions of the equations of motion can be found using Wilson path ordering.Connections between I RR and string theory goes beyond the construction of I P ol,how-ever.A term of the form BµνdXµ∧dXν,where Bµνis a antisymmetric tensor functional of the stringfield,was added to I pol by Callan,Friedan,Martinec,and Perry[32]in their backgroundfield treatment of string.Bµνgenerates an all-pervasive magneticfield in space-time.While similar in form to I RR,in their treatment the specific functional dependence of Bµνon Xµwas determined by requiring that the trace anomally of the total string action vanish.This resulted in a Bµνthat is dramatically different from what is considered here.Along similar lines,Giveon,Rabinovici,and Veneziano[33]also considered a Bµνterm in the string lagrangian,but relaxed the trace anomally condition and considered the effect of constant Bµνon the string.GENERAL CONSTRUCTION OF I RRWe begin with a classical,real Lie algebra g with generators T a such that[T a,T b]=f c ab T c,where f cabare the structure constants for g,and indices run from1to m,thedimension of the g.T a is represented by matrices and following the convention in[34],the Killing form h ab≡T r{T a T b}=−δab is used to raise and lower indices:A a=h ab A b=−A b.With this orthonormality condition,we can use the set{T a}as a natural basis for R m,with V∈R m given by V=V a T a.The inner product on R m is then V,U ≡−T r{VU}for V,U∈R m.Furthermore,using the identity matrix I of g we can extend this construction to the m+1-dimensional Minkowski space Min by taking V=V0I/√m+X a T a, the usual Nambu-Goto string action is obtained throughI P ol≡−Tr dX∧∗dX= √3Tr XdX∧dX=−13 f abc X a∂0X b∂1X c dx0dx1.(2) Xµin this case describes a vortex line.Note,however,that g AB does not explicitly appear;I RR is a topological invariant and describes a Schwarz-type TQFT similar to Chern-Simons theory.(This corresponds to an antisymmetric-field langrangian in background-field string theory with B ab∼ǫabc X c in three dimensions.)Note also that I RR is translationally invari-ant;the lagrangian changes by a total derivative,XdX∧dX→XdX∧dX+Ξd(XdX)under the uniform translation X→X+Ξ.Indeed,I RR is the only translationally invariant topological action that can be constructed directly from Xµ.In the special case of g=so(3), I RR is proportional to the lagrangian in[15],but without the coupling due to self-interaction.Notice that I RR does not depend on X0,the time component of Xµ.This is expected: topological lagrangians describe systems with no dynamical degrees of freedom.We will thus work solely with X from this point on.This X is a section of a the vector bundle R m over M, and is at the same time an element of g,a vector on R m,and a0-form(and thus a function) on M.Seen thusly,the structure constants f abc form a rank-3,totally antisymmetric tensor on R m.The1-form F=F A dx A=F a A dx A T a is then a vector valued or,equivalently,a Lie algebra valued1-form on M,meaning that each of its two components F A are both vectors in R m and members of g.The equations of motion,d X∧d X=0,from eq.(2)can be integrated once to giveP≡[X,d X],(3) where P is a closed Lie algebra valued1-form on the worldsheet:d P=0.The two compo-nents of P are P0≡P c0T c=f c ab X a∂0X b T c and P1≡P c1T c=f c ab X a∂1X b T c.P is related to the canonical momentum for X through the dual formΠ≡∗P:ΠA a=1−gδI RRvortex lines:The dynamics of an open vortex line,which sweeps-out a2-d open sheet in R m,differ dramatically from that of a closed vortex line(vortex ring),which sweeps out a closed surface.For the open vortex line,H1(M,R m)=0and we can always make a translation to a frame in which the momentum vanishes,P=0so that0=[X,d X].The solution for X in this case is particularly simple.For g=so(3),su(2),sp(2),X=a(x0,x1)H,where a is an arbitrary function and H is a constant vector.The vortex line is constrained to move along one direction:H.For other Lie algebras,X∈c,the Cartan subalgebra for g,so that X=X i H i where{H i}form the bases for c[34],and X propagates within a linear subspaceof R m.For the vortex ring,on the other hand,H1(M,R m)=Z m,the integers,and P need not vanish.The dynamics of vortex rings are thus much more interesting,and we shall focus on them for the rest of the paper.We begin with g=so(3),su(2)or sp(2).The vortex is propagating in R3and it’s dynamics are especially contrained.VORTEX RINGS IN R3When g=so(3),f abc=ǫabc,and we are dealing with a vortex line propagating in R3. Although we can revert to the usual vector notation in this case,doing so will add notational complexity.Instead,we introduce a slight abuse of notation and write the cross product of two vectors V,U∈R3as VxU≡[V,U].It is straightforward to show that[P0,P1]=0;the two vectors are proportional to one another.Consequently,we can write P=ˆb p where p is a scalar1-form on M andˆb∈R3. (The hat denotes a unit vector:|ˆb|2= ˆb,ˆb =1.)Because d P=0,dˆb∧p+ˆb dp=0;each term must vanish separately.Consequently,dp=0,and p is a closed1-form.For the other term,dˆb∧p=0,and from Cartan’s lemma[36],dˆb must be proportional to p:dˆb=−τˆn p, whereτis an arbitrary function andˆn is a unit vector in R3orthogonal toˆb.Doing this trick once again and noting that ddˆb=0,dˆn=(−κˆt+τˆb)p whereκis another arbitrary function on M andˆt=ˆn xˆb.Once again ˆn,ˆt =0.It is then straightforward to show that dˆt=κˆn,and no more terms need to be introduced.To complete the solution for X,we note thatˆt,ˆn,ˆb form a moving orthogonal coordinate system on R3.Taking X=|X|(αˆt+βˆn+γˆb)for constantsα,β,γ,we require that this Xsolves eq.(3).Thenα=1and X lies alongˆt,while|X|2=1/κ.Solution of equations of motion therefore reduces tofinding solutions forˆt,ˆn,andˆb for givenκ,τand p.From ˆb,ˆn =0and dˆt=κˆn,we see that dτand dκare both proportional to the1-form p;the functionsκ,τthat determineˆt,ˆn,ˆb all depend upon p.Consequently,there is a function s(x0,x1)such that locally ds=p andˆt′=κ(s)ˆn,ˆn′=−κ(s)ˆt+τ(s)ˆb,ˆb′=−τ(s)ˆn,(5)where the prime denotes derivative wrt to s.These are the Serret-Frenet equations[36]for a curve c:[a,b]→R3parameterized by its arclength s.κ=1/|X|2is the local curvature of c and is positive definite,as required,whileτis the local torsion.Because P is a closed 1-form that is not exact,c is a closed loop in R3[36].The existence of solutions to the Serret-Frenet equations is guaranteed[36].It is neverthe-less instructive to look further into their explicit form for two special cases.Letκ=̟cos u,τ=̟sin u where u=u(s)and−π/2≤u≤π/2becauseκ≥0.Working with the coor-dinates dt=̟ds,eqs.(5)can be combined into¨ˆn=−ˆn+˙uˆn x˙ˆn,where the dot denotes derivative wrt t.This is similar to the equation of motion for a particle constrained to move on a sphere in the presence of a electric and magnetic dipole(a dyon)at the center of it,but in this case the ratio of the magnetic to electric“charge”of the dyon is˙u and can depend on time.Takingˆl=ˆn x˙ˆn,the torque˙ˆl=−˙u˙ˆn is opposite of the velocity of the particle˙ˆn and has strength˙u.Consequently the total volumeˆn·˙ˆn x¨ˆn swept out byˆn is just˙u.If this volume is a constant,then takingω=√ωsin u cos(ωt) T1−cos u cos(ωt)+˙uωT3ˆb=− sin u sin(ωt)+˙uωcos u sin(ωt) T2+cos uω{cos(ωt)T1+sin(ωt)T2+˙u T3}.(6) Furthermore,if˙u=0,thenˆb=T3and c is confined to the1−2plane.Periodicity ofˆtand ˆn for a closed curve c gives 2πn =b a ̟ds = b a ̟p .This is a well-known result [36]for closed curves and is the fundamental reason why p (and consequently P )is a close but not exact 1-form.For general u ,0=...ˆn −¨u ¨ˆn /˙u +(1+˙u 2)˙ˆn −¨u ˆn /˙u with the boundary conditions|ˆn |=1,|˙ˆn |=1,and ˆn (0)=T 1.VORTEX RINGS IN R m To solve eq.(3)for general g we follow an approach similar to that in the previous section and introduce a set of linearly independent vectors {ˆt r }∈R m on M where ˆt r =R a r T a such that ˆt r ,ˆt s =δr,s .The set {ˆt r }forms a moving frame on R m for points on M (letters in the second have of the alphabet denote coordinates in the moving frame).ThenR ra R sa =δrs and R ∈so (m );similarly,R ra R rb =δab .In addition,[ˆt r ,ˆt s ]=f t rs ˆt t ,but now f t rs =R a r R b s R t c f c ab are the “structure constants”in the moving frame.Because they depend on R a r ,in this frame f t rs need not be constant.Since {ˆt r }are orthonormal and span R m ,d ˆt r =−κrs ˆt s ,where κrs =−κsr are 1-formson M .Moreover,from dd ˆt r =0,0=dκrs +κrt ∧κts .(7)These are the Maurer-Cartan equations [36]and κrs are the Maurer-Cartan 1-forms forso (m ).Indeed,let S ˜a be the generators of so (m ),the symmetry group of R m ,such that[S ˜a ,S ˜b ]=k ˜a˜b ˜c S ˜c ,Tr {S ˜a S ˜b }=−δ˜a ˜b ,and ˜a runs from 1to m (m −1)/2.For a fixed ˜a ,S ˜a are m ×m antisymmetric matrices with elements (S ˜a )rs (we are not working in the adjointrepresentation for so (m )).We then introduce the 1-forms A =A ˜a S ˜a with values in the Liealgebra so (m )such that κrs ≡(A ˜a S ˜a )rs .Then dA +A ∧A =0;A can be seen as a non-abelian“vector potential”for the group so (m ).The field strength for A vanishes,however,and A is a “pure gauge”vector potential.As expected,A does not contain any physical degrees of freedom.Indeed,written in terms of matrices of so (m ),R t dR =−A .With this interpretation of the Maurier-Cartan equations it is straightforward to see thatˆt r =P exp s0A a rT a ,(8)where P denote Wilson path ordering.Solution to eq.(3)now follows straightforwardly.Given a set ofκrs,we constructˆt r using eq.(8).We then choose X=|X|ˆt1so that P=|X|2f1rsκ1rˆt s.Because d P=0,0={d log|X|2f1r′s′−κ1t′f tr′s′}∧κ1r′,(9)where r′,s′,t′>1and we have used d f rst=−κrn f nst−κsn f rnt−κtn f rsn.In addition, the choice ofκrs must satisfy the constraint0=f1r′s′κ1r′∧κ1s′;{κ1r′}therefore can not linearly independent.One solution of this constraint equation isκ1r′=κr′π,whereκa are functions on M andπis a1-form on M.This choice ofκ1r′does not restrictκr′s′and 0=dκr′s′+κr′t′∧κt′s′still.Integration of eq.(9)then gives|X|2=exp{ απ}for any functionαon M,and we are done.X is determined by the arbitrary functionα,and Maurer-Cartan1-formsκr′π,andκr′s′.Except for so(3),su(2),and sp(2),this choice ofκ1r′is not the most general one that satisfies the constraint equation.Indeed,with this choice,P0∝P1and like the R3case,the two components of P are proportional to one another.It is expected that when the general solution to the constraint equation is used,this relationship between the components of P will no longer hold.CONCLUDING REMARKSWe have shown in this paper the deep connection between I RR,on the one hand,and string theory and TQFT on the other.Indeed,the topological nature of the theory,and the fundamental role it plays in determining vortex dynamics,is manifest in our approach in analyzing the system.Moreover,with this approach generalization of vortex dynamics to higher dimensions becomes straightforward.With the goal being to establish the links between TQFT,string theory,and the study of vortex lines,the approach we have taken in this paper has been purposefully formal. We have focused on establishing mathematical structures,and using these structures in understanding the general physical properties of vortices propagating in superfluids that is due solely to the kinetic part of the full Rasette-Regge lagrangian.We have focused on only the kinetic part of the lagrangian for two reasons.First,the traditional interaction term between vortex lines found in[15]is extremely nonlinear.Some degree of perturbative analysis,based on the kinetic term,would most likely be needed.To this end,a thorough understanding of the“free”kinetic term is needed.Second,the interaction term has a1/r type of divergence singularity at the classical level,which has traditionally been regulated by introducing afinite vortex core.However,it is expected that the degree of divergence will be weakened in the full quantumfield theoretic treatment of the system,and a complete treatment of this divergence will most fruitfully be delayed until then.Thefirst step in the quantumfield theoretic approach is to the quantization of the “free”(kinetic)part of the Rasette-Regge lagrangian I RR.How to treat the many-vortex system is still an open question.Once more than one vortex line is introduced,a whole host of questions come to the fore.One particular issue is the question of how interactions between them should be incorporated into the approach outlined here.One can certainly choose to use the classical interaction term found in[15]. Another approach could be to follow the approach of string theory where the interaction of strings are represented by the merging and breaking of strings(which for closed strings fundamentally changes the genus,and thus topology,of the surface it sweeps-out).Much of the techniques developed for string theory could then conceivably be applied to the analysis of interacting vortex lines.Which of these two approaches will be more fruitful is unclear, especially in light of the two points listed above.The question of how to include interactions between vortices goes beyond a discussion of field-theoretic techniques and methodology,however.As we have mentioned in the intro-duction,a TQFT has no dynamics in the traditional sense;since the lagrangian does not depend on the metric,there is no notion of time.Will the inclusion of the interaction terms necessitate the introduction of the metric?While it is possible to use de Rham’s method of generalized forms[37]to rewrite and generalize the interaction term found in[15]in terms of differential forms(which will thus automatically be independent of the metric),it is unclear if such an approach is physically meaningful.Moreover,making sense of this interaction term will require the introduction of a high energy cut-off(the vortex core size),which may bring along its own particular set of problems.∗Electronic address:adspelio@†Present address:Department of Physics,University of California at Berkeley,Berkeley,CA94720-7300.[1]R.J.Donnelly,Quantized Vortices in Helium II(Cambridge University Press,New York,1991).[2]H.Nielsen and P.Olesen,Nucl.Phys.B61,45(1973).[3]L.Onsager,Nuovo Cimento Suppl.6,249(1949).[4]R.P.Feynman,in Progress in Low Temperature Physics,edited by C.Gorter(Horth-Holland,Amsterdam,1955),vol.1,p.17.[5]G.A.Williams,Phys.Rev.Lett.59,1926(1987).[6] F.Lund,A.Reisenegger,and C.Utreras,Phys.Rev.B41,155(1990).[7]M.Anderson,J.Ensher,M.Matthews,C.Wieman,and E.Cornell,Science269,198(1995).[8]W.Ketterle and N.van Druten,Phys.Rev.A54,656(1996).[9]M.R.Matthews,B.P.Anderson,P.C.Haljan,D.S.Hall,C.E.Wieman,and E.A.Cornell,Phys.Rev.Lett.83,2498(1999).[10]K.W.Madison,F.Chevy,W.Wohlleben,and J.Dalibard,Phys.Rev.Lett.84,806(2000).[11] F.Chevy,K.W.Madison,and J.Dalibard,Phys.Rev.Lett.85,2223(2000).[12] B.P.Anderson,P.C.Haljan,C.E.Wieman,and E.A.Cornell,Phys.Rev.Lett.85,2857(2000).[13] D.S.Rokshar,Phys.Rev.Lett.79,2164(1997).[14]J.Garc´ia-Ripoll,G.Molina-Terriza,V.P´e rez-G´a rcia,and et al.,Phys.Rev.Lett.87,140403(2001).[15]M.Rasetti and T.Regge,Physica A80,217(1975).[16]M.Rasetti and T.Regge,Lecture Notes in Physics,vol.201(Springer-Verlag,New York,1984).[17]G.A.Goldin,R.Menikoff,and D.H.Sharp,Phys.Rev.Lett58,2162(1987).[18]V.Penna and M.Spera,J.Math.Phys30,2778(1989).[19]U.K.Albertin and H.L.Morrison,Physica A159,188(1989).[20]U.K.Albertin and H.L.Morrison,J.Math.Phys.31,1535(1990).[21]G.A.Goldin,R.Menikoff,and D.H.Sharp,Phys.Rev.Lett.67,3499(1991).[22]G.A.Goldin and U.Moschella,J.Phys.A28,L475(1995).[23]V.Penna and M.Spera,Phys.Rev.B62,14547(2000).[24] A.M.Polyakov,Phys.Lett.B103,207(1981).[25]Y.Nambu,Lectures at the Copenhagen Summer Symposium(1970).[26]T.Goto,Prog.Theor.Phys.46,1560(1971).[27] F.Lund and T.Regge,Phys.Rev.D14,1524(1976).[28] E.Witten,Commun.Math.Phys.117,353(1988).[29] E.Witten,Commun.Math.Phys.117,351(1989).[30] A.S.Schwarz,Lett.Math.Phys.2,247(1978).[31]R.Ricca,Phys.Rev.A43,4281(1991).[32] C.G.Callan,D.Friedan,E.J.Martinec,and M.J.Perry,Nucl.Phys.p.593(1985).[33] A.Giveon,E.Rabinovici,and G.Veneziano,Nucl.Phys.B322,167(1989).[34] F.Jurgen and C.Schweigert,Symmetries,Lie Algebras and Representations:A GraduateCourse for Physicists(Cambride University Press,New York,1997).[35]R.Bott and L.W.Tu,Differential Forms in Algebraic Geometry(Springer-Verlag,New York,1982).[36]M.Spivak,A Comprehensive Introducetion to Differential Geometry,vol.2(Publish or Perish,Wilmington,DE,1970).[37]G.de Rham,Differentiable Manifolds:Forms,Currents,Harmonic Forms(Springer-Verlag,New York,1984).。

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