Hiroshima Math. J. 36 (2006), 289–329 Tilings from non-Pisot unimodular matrices

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历届菲尔兹奖获得者1936 美国（芬兰裔）阿尔斯·阿尔福斯(Ahlfors，Lars Valerian) 邓若瓦猜想、覆盖理论1936 美国杰西·道格拉斯(Douglas，Jesse) 普拉托极小曲面问题、变分问题的反问题1950 法国坎布里奇罗朗·施瓦尔茨(Schwartz，Laurent) 广义函数论1950 美国（挪威裔）阿特尔·赛尔伯格(Selberg，Atle) 素数定理的初等证明、调和分析等1954 日本小平邦彦(Kodaira Kunihiko) 推广黎曼-罗赫定理、小平邦彦消解定理1954 法国让-皮埃尔·塞尔(Serre，Jean-Pierre) 一般纤空间概念、同伦的局部化方法、同伦论的一些重要结果1958 英国（德国裔）克劳斯·费里德里希·罗斯(Roth，Klaus Friedrich) 代数数有理逼近的瑟厄-西格尔-罗斯定理1958 法国雷内·托姆(Thom，Rene) 拓扑学配边理论、奇点理论、拓扑流形理论1962 瑞典拉尔斯·赫尔曼德尔(Hormander，Lars Valter) 线性偏微分算子理论、伪微分算子理论1962 美国约翰·米尔诺(Milnor，John Willard) 7维球面的微分结构、否定庞加莱主猜想、代数k 理论1966 英国迈克尔·法兰西斯·阿蒂亚(Atiyah，Michael Francis) 阿提雅-辛格指标定理、拓扑k 理论1966 美国鲍尔·约瑟夫·科恩(Cohen，Paul Joseph) 力迫法、连续统假设与zf系统的独立性1966 法国亚力山大·格罗腾迪克(Grothendieck，Alexandre) 代数几何体系、泛函分析中的核空间、张量积1966 美国斯蒂芬·斯梅尔(Smale，Stephen) 广义庞加莱猜想、微分动力系统理论1970 尼斯阿兰·贝克(Baker，Alan) 数论中的一些问题、二次域的类数问题1970 日本广中平祐(Hironaka Heisuke) 代数簇的奇点消解问题1970 苏联谢尔盖·彼得洛维奇·诺维科夫(Новиков,Сергей петрович) 微分拓扑学配边理论、微分流形理论庞特里雅金示性类的拓扑不变性1970 美国约翰·格里格·汤普森(Thompson，John Grggs) 有限单群的伯恩德赛猜想和弗洛贝纽斯猜想1974 美国（英国裔）大卫·布赖恩特·曼福德 (Mumford，David Bryart) 代数几何学参模理论、代数曲面的分类1974 意大利恩里科·庞比里(Bombieri，Enrico) 有限单群分类问题、哥德巴赫猜想的（1，3）命题1978 美国查里斯·费弗曼(Fefferman，Charles) 奇异积分算子、偏微分方程1978 比利时皮埃尔·德林(Deligne，Pierre) 代数几何中的部分韦伊猜想1978 美国丹尼尔·奎伦(Quillen，Daniel) 代数k理论的亚当斯猜想、塞尔猜想1978 苏联格·阿·玛古利斯(Margulis，G.A.) 关于李群的离散子群的塞尔伯格猜想1983 法国阿兰·孔耐(Connes，Alan) 算子代数、代数分类问题1983 美国威廉·瑟斯顿(Thurston，William) 3维流形的叶状结构及其分类1983 美国（中国裔）丘成桐(Yan Sheng-Tung) 卡拉比猜想、正质量猜想1986 美国 M.弗里德曼(Freedman，Michael) 4维流形的庞加莱猜想1986 英国 S.唐纳森(Donaldson，Simon) 4维流形的拓扑学1986 德国 G.法尔廷斯(Faltings，Gerd) 莫德尔猜想1990 苏联 V.德里费尔德(Drifel’d，Vladimir) 模理论、与量子群有关的hopf代数1990 新西兰 F.R.J. 沃恩(Vaughan， F.R.Jones) 扭结理论1990 日本森重文(Shigffumi Mori) 3维代数簇的分类1990 美国 E. 威滕(Witten，Edward) 弦理论、对超弦理论作了统一的数学处理1994 比利时 J.布尔盖恩(Bourgain，Jean) 无限维的偏微分方程1994 法国 P.L. 利翁(Lions，Pierre-Louis) 非线性偏微分方程、玻尔兹曼方程1994 法国 J.C. 约克兹(Yoccoz，Jean Christophe) 一般复动力系统的性状和分类1994 俄罗斯 E. 泽尔曼诺夫(Zelmanov，Efim) 群论的弱伯恩赛得猜想1998 英国 R.E. 博切尔兹(Borcherds，Richard E.) 魔群月光猜想、卡茨-穆迪代数1998 英国 W.T. 高尔斯(Gowers，W.Timothy) 巴拿赫空间理论、超平面猜想1998 俄罗斯 M. 孔采维奇(Kontsevich，Maxin) 线理论、扭结分类猜想1998 美国 C.T. 麦克马兰(Mcmullen，Curtis T) 混沌理论、复动力系统的主猜想1998 英国安德鲁·怀尔斯(Wiles，Andrew) 费尔马猜想2002 法国洛朗·拉佛阁证明了与函数域相应的整体朗兰兹纲领，从而在数论与分析两大领域之间建立了新的联系2002 俄罗斯符拉基米尔·弗沃特斯基发展了新的代数簇上同调理论而获奖。

最大叶子生成树(MLST问题)

Spanning directed trees with many leaves Noga Alon1,Fedor V.Fomin2,Gregory Gutin3,Michael Krivelevich1,andSaket Saurabh21Department of Mathematics,Tel Aviv UniversityTel Aviv69978,Israel{nogaa,krivelev}@post.tau.ac.il2Department of Informatics,University of BergenPOB7803,5020Bergen,Norway{fedor.fomin,saket}@ii.uib.no3Department of Computer ScienceRoyal Holloway,University of LondonEgham,Surrey TW200EX,UKgutin@Abstract.The Directed Maximum Leaf Out-Branching problemis toﬁnd an out-branching(i.e.a rooted oriented spanning tree)in a givendigraph with the maximum number of leaves.In this paper,we obtaintwo combinatorial results on the number of leaves in out-branchings.Weshow that–every strongly connected n-vertex digraph D with minimum in-degree at least3has an out-branching with at least(n/4)1/3−1leaves;–if a strongly connected digraph D does not contain an out-branching with k leaves,then the pathwidth of its underlying graph UG(D)isO(k log k).Moreover,if the digraph is acyclic,the pathwidth is atmost4k.The last result implies that it can be decided in time2O(k log2k)·n O(1)whether a strongly connected digraph on n vertices has an out-branchingwith at least k leaves.On acyclic digraphs the running time of our algo-rithm is2O(k log k)·n O(1).1IntroductionIn this paper,we initiate the combinatorial and algorithmic study of a natural generalization of the well studied Maximum Leaf Spanning Tree problem on connected undirected graphs[11,16,19–21,24,25,31,33].Given a digraph D,a subdigraph T of D is an out-tree if T is an oriented tree with only one vertex s of in-degree zero(called the root).If T is a spanning out-tree,i.e.V(T)=V(D), then T is called an out-branching of D.The vertices of T of out-degree zero are called leaves.The Directed Maximum Leaf Out-Branching(DMLOB) Preliminary extended abstracts of this paper have been presented at FSTTCS2007[5]and ICALP2007[4]2N.Alon,F.V.Fomin,G.Gutin,M.Krivelevich,and S.Saurabhproblem is to ﬁnd an out-branching in a given digraph with the maximum num-ber of leaves.The combinatorial study of spanning trees with maximum number of leaves in undirected graphs has an extensive history.Linial conjectured around 1987that every connected graph on n vertices with minimum vertex degree δhas a spanning tree with at least n (δ−2)/(δ+1)+c δleaves,where c δdepends on δ.This is indeed the case for all δ≤5.Kleitman and West [27]and Linial and Sturtevant [30]showed that every connected undirected graph G on n vertices with minimum degree at least 3has a spanning tree with at least n/4+2leaves.Griggs and Wu [25]proved that the maximum number of leaves in a spanning tree is at least n/2+2when δ=5and at least 2n/5+8/5when δ=4.All these results are tight.The situation is less clear for δ≥6;the ﬁrst author observed that Linial’s conjecture is false for all large values of δ.Indeed,the results in [2]imply that there are undirected graphs with n vertices and minimum degree δinwhich no tree has more than (1−(1+o (1))ln(δ+1)δ+1)n leaves,where the o (1)-termtends to zero an δtends to inﬁnity,and this is essentially tight.See also [3],pp.4-5and [13]for more information.In this paper we prove an analogue of the Kleitman-West result for directed graphs:every strongly connected digraph D of order n with minimum in-degree at least 3has an out-branching with at least (n/4)1/3−1leaves.We do not know whether this bound is tight,however we show that there are strongly connected digraphs with minimum in-degree 3in which every out-branching has at most O (√n )leaves.Unlike its undirected counterpart which has attracted a lot of attention in all algorithmic paradigms like approximation algorithms [24,31,33],parameterized algorithms [11,19,21],exact exponential time algorithms [20]and also combina-torial studies [16,25,27,30],the Directed Maximum Leaf Out-Branching problem has largely been neglected until recently.The only paper we are aware of is the very recent paper [18]that describes an O (√opt )-approximation algo-rithms for DMLOB.Our second combinatorial result relates the number of leaves in a DMLOB of a directed graph D with the pathwidth of its underlying graph UG(D ).(We postpone the deﬁnition of pathwidth till the next section.)If an undirected graph G contains a star K 1,k as a minor,then it is possible to construct a spanning tree with at least k leaves from this minor.Otherwise,there is no K 1,k minor in G ,and it is possible to prove that the pathwidth of G is O (k ).(See,e.g.[8].)Actually,a much more general result due to Bienstock et al.[10])is that any undirected graph of pathwidth at least k ,contains all trees on k vertices as a minor.We prove a result that can be viewed as a generalization of known bounds on the number of leaves in a spanning tree of an undirected graph in terms of its pathwidth,to strongly connected digraphs.We show that either a strongly connected digraph D has a DMLOB with at least k leaves or the pathwidth of UG(D )is O (k log k ).For an acyclic digraph with a DMLOB having k leaves,we prove that the pathwidth is at most 4k .This almost matches the boundSpanning directed trees with many leaves3 for undirected graphs.These combinatorial results are useful in the design of parameterized algorithms.In parameterized algorithms,for decision problems with input size n,and a parameter k,the goal is to design an algorithm with runtime f(k)n O(1),where f is a function of k alone.(For DMLOB such a parameter is the number of leaves in the out-tree.)Problems having such an algorithm are said to beﬁxed parameter tractable(FPT).The book by Downey and Fellows[17]provides an introduction to the topic of parameterized complexity.For recent developments see the books by Flum and Grohe[23]and by Niedermeier[32].The parameterized version of DMLOB is deﬁned as follows:Given a digraph D and a positive integral parameter k,does D contain an out-branching with at least k leaves?We denote the parameterized versions of DMLOB by k-DMLOB. If in the above deﬁnition we do not insist on an out-branching and ask whether there exists an out-tree with at least k leaves,we get the parameterized Di-rected Maximum Leaf Out-Tree problem(denoted k-DMLOT).Our combinatorial bounds,combined with dynamic programming on graphs of bounded pathwidth imply theﬁrst parameterized algorithms for k-DMLOB on strongly connected digraphs and acyclic digraphs.We remark that the algorith-mic results presented here also hold for all digraphs if we consider k-DMLOT rather than k-DMLOB.This answers an open question of Mike Fellows[14, 22,26].However,we mainly restrict ourselves to k-DMLOB for clarity and the harder challenges it poses,and we brieﬂy consider k-DMLOT only in the last section.Very recently,using a modiﬁcation of our approach,Bonsma and Dorn[12] proved that either an arbitrary digraph D has an out-branching with at most k leaves or the pathwidth of UG(D )is O(k3),where D is the digraph obtained from D by deleting all arcs not contained in any out-branching of D.The bound O(k3)is much larger than our bounds for strongly connected and acyclic di-graphs,but it suﬃces to allow Bonsma and Dorn to show that k-DMLOB is FPT,settling another open question of Fellows[22,26].This paper is organized as follows.In Section2we provide additional ter-minology and notation as well as some well-known results.We introduce locally optimal out-branchings in Section3.Bounds on the number of leaves in maxi-mum leaf out-branchings of strongly connected and acyclic digraphs are obtained in Section4.In Section5we prove upper bounds on the pathwidth of the un-derlying graph of strongly connected and acyclic digraphs that do not contain out-branchings with at least k leaves.In Section6we conclude with discussions and open problems.2PreliminariesLet D be a digraph.By V(D)and A(D)we represent the vertex set and arc set of D,respectively.An oriented graph is a digraph with no directed2-cycle. Given a subset V ⊆V(D)of a digraph D,let D[V ]denote the digraph induced by V .The underlying graph UG(D)of D is obtained from D by omitting all4N.Alon,F.V.Fomin,G.Gutin,M.Krivelevich,and S.Saurabhorientations of arcs and by deleting one edge from each resulting pair of parallel edges.The connectivity components of D are the subdigraphs of D induced by the vertices of components of UG(D ).A digraph D is strongly connected if,for every pair x,y of vertices there are directed paths from x to y and from y to x.A maximal strongly connected subdigraph of D is called a strong component .A vertex u of D is an in-neighbor (out-neighbor )of a vertex v if uv ∈A (D )(vu ∈A (D ),respectively).The in-degree d −(v )(out-degree d +(v ))of a vertex v is the number of its in-neighbors (out-neighbors).We denote by (D )the maximum number of leaves in an out-tree of a digraph D and by s (D )we denote the maximum possible number of leaves in an out-branching of a digraph D .When D has no out-branching,we write s (D )=0.The following simple result gives necessary and suﬃcient conditions for a digraph to have an out-branching.This assertion allows us to check whether s (D )>0in time O (|V (D )|+|A (D )|).Proposition 1([7]).A digraph D has an out-branching if and only if D has a unique strong component with no incoming arcs.Let P =u 1u 2...u q be a directed path in a digraph D .An arc u i u j of D is a forward (backward )arc for P if i ≤j −2(j <i ,respectively).Every backward arc of the type v i +1v i is called double .For a natural number n ,[n ]denotes the set {1,2,...,n }.A tree decomposition of an (undirected)graph G is a pair (X,U )where U is a tree whose vertices we will call nodes and X =({X i |i ∈V (U )})is a collection of subsets of V (G )such that 1. i ∈V (U )X i =V (G ),2.for each edge {v,w }∈E (G ),there is an i ∈V (U )such that v,w ∈X i ,and3.for each v ∈V (G )the set of nodes {i |v ∈X i }forms a subtree of U .The width of a tree decomposition ({X i |i ∈V (U )},U )equals max i ∈V (U ){|X i |−1}.The treewidth of a graph G is the minimum width over all tree decompositions of G .If in the deﬁnitions of a tree decomposition and treewidth we restrict U to be a path,then we have the deﬁnitions of path decomposition and pathwidth.We use the notation tw (G )and pw (G )to denote the treewidth and the pathwidth of a graph G .We also need an equivalent deﬁnition of pathwidth in terms of vertex sepa-rators with respect to a linear ordering of the vertices.Let G be a graph and let σ=(v 1,v 2,...,v n )be an ordering of V (G ).For j ∈[n ]put V j ={v i :i ∈[j ]}and denote by ∂V j all vertices of V j that have neighbors in V \V j .Setting vs (G,σ)=max i ∈[n ]|∂V i |,we deﬁne the vertex separation of G asvs (G )=min {vs (G,σ):σis an ordering of V (G )}.The following assertion is well-known.It follows directly from the results of Kirousis and Papadimitriou [29]on interval width of a graph,see also [28].Proposition 2([28,29]).For any graph G ,vs (G )=pw (G ).Spanning directed trees with many leaves53Locally Optimal Out-BranchingsOur bounds are based onﬁnding locally optimal out-branchings.Given a di-graph,D and an out-branching T,we call a vertex leaf,link and branch if its out-degree in T is0,1and≥2respectively.Let S+≥2(T)be the set of branch vertices,S+1(T)the set of link vertices and L(T)the set of leaves in the tree T. Let P2(T)be the set of maximal paths consisting of link vertices.By p(v)we denote the parent of a vertex v in T;p(v)is the unique in-neighbor of v.We calla pair of vertices u and v siblings if they do not belong to the same path from the root r in T.We start with the following well known and easy to observe facts.Fact1|S+≥2(T)|≤|L(T)|−1.Fact2|P2(T)|≤2|L(T)|−1.Now we deﬁne the notion of local exchange which is intensively used in our proofs.Deﬁnition3 -Arc Exchange( -AE)optimal out-branching:An out-branching T of a directed graph D with k leaves is -AE optimal if for all arc subsets F⊆A(T)and X⊆A(D)−A(T)of size ,(A(T)\F)∪X is either notan out-branching,or an out-branching with at most k leaves.In other words,Tis -AE optimal if it can’t be turned into an out-branching with more leaves by exchanging arcs.Let us remark,that for everyﬁxed ,an -AE optimal out-branching can be obtained in polynomial time.In our proofs we use only1-AE optimal out-branchings.We need the following simple properties of1-AE optimal out-branchings. Lemma1.Let T be an1-AE optimal out-branching rooted at r in a digraph D. Then the following holds:(a)For every pair of siblings u,v∈V(T)\L with d+T (p(v))=1,there is no arce=(u,v)∈A(D)\A(T);(b)For every pair of vertices u,v/∈L,d+T (p(v))=1,which are on the samepath from the root with dist(r,u)<dist(r,v)there is no arc e=(u,v)∈A(D)\A(T)(here dist(r,u)is the distance to u in T from the root r); (c)There is no arc(v,r),v/∈L such that the directed cycle formed by the(r,v)-path and the arc(v,r)contains a vertex x such that d+T(p(x))=1. Proof.The proof easily follows from the fact that the existence of any of these arcs contradicts the local optimality of T with respect to1-AE. 4Combinatorial BoundsWe start with a lemma that allows us to obtain lower bounds on s(D).6N.Alon,F.V.Fomin,G.Gutin,M.Krivelevich,and S.SaurabhLemma 2.Let D be a oriented graph of order n in which every vertex is of in-degree 2and let D have an out-branching.If D has no out-tree with k leaves,then n ≤4k 3.Proof.Let us assume that D has no out-tree with k leaves.Consider an out-branching T of D with p <k leaves which is 1-AE optimal.Let r be the root of T .We will bound the number n of vertices in T as follows.Every vertex of T is either a leaf,or a branch vertex,or a link vertex.By Facts 1and 2we already have bounds on the number of leaf and branch vertices as well as the number of maximal paths consisting of link vertices.So to get an upper bound on n in terms of k ,it suﬃces to bound the length of each maximal path consisting of link vertices.Let us consider such a path P and let x,y be the ﬁrst and last vertices of P ,respectively.The vertices of V (T )\V (P )can be partitioned into four classes as follows:(a )ancestor vertices :the vertices which appear before x on the (r,x )-path of T ;(b )descendant vertices :the vertices appearing after the vertices of P on pathsof T starting at r and passing through y ;(c )sink vertices :the vertices which are leaves but not descendant vertices;(d )special vertices :none-of-the-above vertices.Let P =P −x ,let z be the out-neighbor of y on T and let T z be the subtree of T rooted at z .By Lemma 1,there are no arcs from special or ancestor vertices to the path P .Let uv be an arc of A (D )\A (P )such that v ∈V (P ).There are two possibilities for u :(i)u ∈V (P ),(ii)u ∈V (P )and uv is backward for P (there are no forward arcs for P since T is 1-AE optimal).Note that every vertex of type (i)is either a descendant vertex or a sink.Observe also that the backward arcs for P form a vertex-disjoint collection of out-trees with roots at vertices that are not terminal vertices of backward arcs for P .These roots are terminal vertices of arcs in which ﬁrst vertices are descendant vertices or sinks.We denote by {u 1,u 2,...,u s }and {v 1,v 2,...,v t }the sets of vertices on P which have in-neighbors that are descendant vertices and sinks,respectively.Let the out-tree formed by backward arcs for P rooted at w ∈{u 1,...,u s ,v 1,...,v t }be denoted by T (w )and let l (w )denote the number of leaves in T (w ).Observe that the following is an out-tree rooted at z :T z ∪{(in (u 1),u 1),...,(in (u s ),u s )}∪si =1T (u i ),where {in (u 1),...,in (u s )}are the in-neighbors of {u 1,...,u s }on T z .This out-tree has at least s i =1l (u i )leaves and,thus, s i =1l (u i )≤k −1.Let us denote the subtree of T rooted at x by T x and let {in (v 1),...,in (v t )}be the in-neighbors of {v 1,...,v t }on T −V (T x ).Then we have the following out-tree:(T −V (T x ))∪{(in (v 1),v 1),...,(in (v t ),v t )}∪ti =1T (v i )Spanning directed trees with many leaves 7with at least t i =1l (v i )leaves.Thus, t i =1l (v i )≤k −1.Consider a path R =v 0v 1...v r formed by backward arcs.Observe that the arcs {v i v i +1:0≤i ≤r −1}∪{v j v +j :1≤j ≤r }form an out-tree withr leaves,where v +j is the out-neighbor of v j on P.Thus,there is no path ofbackward arcs of length more than k −1.Every out-tree T (w ),w ∈{u 1,...,u s }has l (w )leaves and,thus,its arcs can be decomposed into l (w )paths,each of length at most k −1.Now we can bound the number of arcs in all the trees T (w ),w ∈{u 1,...,u s },as follows: s i =1l (u i )(k −1)≤(k −1)2.We can similarly bound the number of arcs in all the trees T (w ),w ∈{v 1,...,v s }by (k −1)2.Recall that the vertices of P can be either terminal vertices of backward arcsfor P or vertices in {u 1,...,u s ,v 1,...,v t }.Observe that s +t ≤2(k −1)since s i =1l (u i )≤k −1and ti =1l (v i )≤k −1.Thus,the number of vertices in P is bounded from above by 1+2(k −1)+2(k −1)2.Therefore,n =|L (T )|+|S +≥2(T )|+|S +1(T )|=|L (T )|+|S +≥2(T )|+ P ∈P 2(T )|V (P )|≤(k −1)+(k −2)+(2k −3)(2k 2−2k +1)<4k 3.Thus,we conclude that n ≤4k 3.Theorem 4.Let D be a strongly connected digraph with n vertices.(a)If D is an oriented graph with minimum in-degree at least 2,then s (D )≥(n/4)1/3−1.(b)If D is a digraph with minimum in-degree at least 3,then s (D )≥(n/4)1/3−1.Proof.Since D is strongly connected,we have (D )= s (D )>0.Let T be an 1-AE optimal out-branching of D with maximum number of leaves.(a)Delete some arcs from A (D )\A (T ),if needed,such that the in-degree of each vertex of D becomes 2.Now the inequality s (D )≥(n/4)1/3−1follows from Lemma 2and the fact that (D )= s (D ).(b)Let P be the path formed in the proof of Lemma 2.(Note that A (P )⊆A (T ).)Delete every double arc of P ,in case there are any,and delete some more arcs from A (D )\A (T ),if needed,to ensure that the in-degree of each vertex of D becomes 2.It is not diﬃcult to see that the proof of Lemma 2remains valid for the new digraph D .Now the inequality s (D )≥(n/4)1/3−1follows from Lemma 2and the fact that (D )= s (D ). Remark 5It is easy to see that Theorem 4holds also for acyclic digraphs D with s (D )>0.While we do not know whether the bounds of Theorem 4are tight,we can show that no linear bounds are possible.The following result is formulated for Part (b)of Theorem 4,but a similar result holds for Part (a)as well.8N.Alon,F.V.Fomin,G.Gutin,M.Krivelevich,and S.SaurabhTheorem6.For each t≥6there is a strongly connected digraph H t of order n=t2+1with minimum in-degree3such that0< s(H t)=O(t).Proof.Let V(H t)={r}∪{u i1,u i2,...,u i t|i∈[t]}andA(H t)=u i j u i j+1,u i j+1u i j|i∈[t],j∈{0,1,...,t−3}u i j u i j−2|i∈[t],j∈{3,4,...,t−2}u i j u i q|i∈[t],t−3≤j=q≤t,where u i0=r for every i∈[t].It is easy to check that0< s(H t)=O(t). 5Pathwidth of underlying graphs and parameterized algorithmsBy Proposition1,an acyclic digraph D has an out-branching if and only if D possesses a single vertex of in-degree zero.Theorem7.Let D be an acyclic digraph with a single vertex of in-degree zero. Then either s(D)≥k or the underlying undirected graph of D is of pathwidth at most4k and we can obtain this path decomposition in polynomial time. Proof.Assume that s(D)≤k−1.Consider a1-AE optimal out-branching T of D.Notice that|L(T)|≤k−1.Now remove all the leaves and branch vertices from the tree T.The remaining vertices form maximal directed paths consisting of link vertices.Delete theﬁrst vertices of all paths.As a result we obtain a collection Q of directed paths.Let H=∪P∈Q P.We will show that every arc uv with u,v∈V(H)is in H.Let P ∈Q.As in the proof of Lemma2,we see that there are no forward arcs for P .Since D is acyclic,there are no backward arcs for P .Suppose uv is an arc of D such that u∈R and v∈P ,where R and P are distinct paths from Q.As in the proof of Lemma2,we see that u is either a sink or a descendent vertex for P in T.Since R contains no sinks of T,u is a descendent vertex, which is impossible as D is acyclic.Thus,we have proved that pw(UG(H))=1.Consider a path decomposition of H of width1.We can obtain a path de-composition of UG(D)by adding all the vertices of L(T)∪S+≥2(T)∪F(T),whereF(T)is the set ofﬁrst vertices of maximal directed paths consisting of link ver-tices of T,to each of the bags of a path decomposition of H of width1.Observe that the pathwidth of this decomposition is bounded from above by|L(T)|+|S+≥2(T)|+|F(T)|+1≤(k−1)+(k−2)+(2k−3)+1≤4k−5. The bounds on the various sets in the inequality above follows from Facts1and 2.This proves the theorem. Corollary1.For acyclic digraphs,the problem k-DMLOB can solved in time 2O(k log k)·n O(1).Spanning directed trees with many leaves 9Proof.The proof of Theorem 7can be easily turned into a polynomial time algorithm to either build an out-branching of D with at least k leaves or to show that pw (UG(D ))≤4k and provide the corresponding path decomposition.A standard dynamic programming over the path (tree)decomposition (see e.g.[6])gives us an algorithm of running time 2O (k log k )·n O (1).The following simple lemma is well known,see,e.g.,[15].Lemma 3.Let T =(V,E )be an undirected tree and let w :V →R +∪{0}be a weight function on its vertices.There exists a vertex v ∈T such that the weight of every subtree T of T −v is at most w (T )/2,where w (T )= v ∈V w (v ).Let D be a strongly connected digraph with s (D )=λand let T be an out-branching of D with λleaves.Consider the following decomposition of T (called a β-decomposition )which will be useful in the proof of Theorem 8.Assign weight 1to all leaves of T and weight 0to all non-leaves of T .By Lemma 3,T has a vertex v such that each component of T −v has at most λ/2+1leaves (if v is not the root and its in-neighbor v −in T is a link vertex,then v −becomes a new leaf).Let T 1,T 2,...,T s be the components of T −v and let l 1,l 2,...,l s be the numbers of leaves in the components.Notice that λ≤ s i =1l i ≤λ+1(we may get a new leaf).We may assume that l s ≤l s −1≤···≤l 1≤λ/2+1.Let j be the ﬁrst index such that j i =1l i ≥λ2+1.Considertwo cases:(a)l j ≤(λ+2)/4and (b)l j >(λ+2)/4.In Case (a),we haveλ+22≤j i =1l i ≤3(λ+2)4and λ−64≤s i =j +1l i ≤λ2.In Case (b),we have j =2andλ+2≤l 1≤λ+2and λ−2≤si =2l i ≤3λ+2.Let p =j in Case (a)and p =1in Case (b).Add to D and T a copy v of v (with the same in-and out-neighbors).Then the number of leaves in each of the out-treesT =T [{v }∪(∪pi =1V (T i ))]and T =T [{v }∪(∪s i =p +1V (T i ))]is between λ(1+o (1))/4and 3λ(1+o (1))/4.Observe that the vertices of T have at most λ+1out-neighbors in T and the vertices of T have at most λ+1out-neighbors in T (we add 1to λdue to the fact that v ‘belongs’to both T and T ).Similarly to deriving T and T from T ,we can obtain two out-trees from T and two out-trees from T in which the numbers of leaves are approximately between a quarter and three quarters of the number of leaves in T and T ,respectively.Observe that after O (log λ)‘dividing’steps,we will end up with O (λ)out-trees with just one leaf,i.e.,directed paths.These paths contain O (λ)copies of vertices of D (such as v above).After deleting the copies,we obtain a collection of O (λ)disjoint directed paths covering V (D ).10N.Alon,F.V.Fomin,G.Gutin,M.Krivelevich,and S.SaurabhTheorem8.Let D be a strongly connected digraph.Then either s(D)≥k or the underlying undirected graph of D is of pathwidth O(k log k).Proof.We may assume that s(D)<k.Let T be be a1-AE optimal out-branching.Consider aβ-decomposition of T.The decomposition process can be viewed as a tree T rooted in a node(associated with)T.The children of T in T are nodes(associated with)T and T ;the leaves of T are the directed paths of the decomposition.Theﬁrst layer of T is the node T,the second layer are T and T ,the third layer are the children of T and T ,etc.In what follows, we do not distinguish between a node Q of T and the tree associated with the node.Assume that T has t layers.Notice that the last layer consists of(some) leaves of T and that t=O(log k),which was proved above(k≤λ−1).Let Q be a node of T at layer j.We will prove thatpw(UG(D[V(Q)]))<2(t−j+2.5)k(1) Since t=O(log k),(1)for j=1implies that the underlying undirected graph of D is of pathwidth O(k log k).Weﬁrst prove(1)for j=t when Q is a path from the decomposition.LetW=(L(T)∪S+≥2(T)∪F(T))∩V(Q),where F(T)is the set ofﬁrst vertices ofmaximal paths of T consisting of link vertices.As in the proof of Theorem7,it follows from Facts1and2that|W|<4k.Obtain a digraph R by deleting from D[V(Q)]all arcs in which at least one end-vertex is in W and which are not arcs of Q.As in the proof of Theorem7,it follows from Lemma1and1-AE opti-mality of T that there are no forward arcs for Q in R.Let Q=v1v2...v q.For every j∈[q],let V j={v i:i∈[j]}.If for some j the set V j contained k vertices,say{v 1,v 2,···,vk },having in-neighbors in the set{v j+1,v j+2,...,v q},then Dwould contain an out-tree with k leaves formed by the path v j+1v j+2...v q to-gether with a backward arc terminating at v i from a vertex on the path for each 1≤i≤k,a contradiction.Thus vs(UG(D2[P]))≤k.By Proposition2,the pathwidth of UG(R)is at most k.Let(X1,X2,...,X s)be a path decomposition of UG(R)of width at most k.Then(X1∪W,X2∪W,...,X s∪W)is a path decomposition of UG(D[V(Q)])of width less than k+4k.Thus,pw(UG(D[V(Q)]))<5k(2) Now assume that we have proved(1)for j=i and show it for j=i−1. Let Q be a node of layer i−1.If Q is a leaf of T,we are done by(2).So,we may assume that Q has children Q and Q which are nodes of layer i.In the β-decomposition of T given before this theorem,we saw that the vertices of T have at mostλ+1out-neighbors in T and the vertices of T have at mostλ+1 out-neighbors in T .Similarly,we can see that(in theβ-decomposition of this proof)the vertices of Q have at most k out-neighbors in Q and the vertices of Q have at most k out-neighbors in Q (since k≤λ−1).Let Y denote the set of the above-mentioned out-neighbors on Q and Q ;|Y|≤2k.Delete from D[V(Q )∪V(Q )]all arcs in which at least one end-vertex is in Y and which do not belong to Q ∪QSpanning directed trees with many leaves11 Let G denote the obtained digraph.Observe that G is disconnected and G[V(Q )]and G[V(Q )]are components of G.Thus,pw(UG(G))≤b,where b=max{pw(UG(G[V(Q )])),pw(UG(G[V(Q )]))}<2(t−i+4.5)k(3) Let(Z1,Z2,...,Z r)be a path decomposition of G of width at most b.Then (Z1∪Y,Z2∪Y,...,Z r∪Y)is a path decomposition of UG(D[V(Q )∪V(Q )]) of width at most b+2k<2(t−i+2.5)k. Similar to the proof of Corollary1,we obtain the following:Corollary2.For a strongly connected digraph D,the problem k-DMLOB can be solved in time2O(k log2k)·n O(1).6Discussion and Open ProblemsIn this paper,we initiated the algorithmic and combinatorial study of the Di-rected Maximum Leaf Out-Branching problem.In particular,we showed that for every strongly connected digraph D of order n and with minimum in-degree at least3, s(D)=Ω(n1/3).An interesting open combinatorial question here is whether this bound is tight.If it is not,it would be interesting toﬁnd the maximum number r such that s(D)=Ω(n r)for every strongly connected digraph D of order n and with minimum in-degree at least3.It follows from ourresults that13≤r≤12.We also provided an algorithm of time complexity2O(k log2k)·n O(1)which solves the k-DMLOB problem for a strongly connected digraph D.The algo-rithm is based on a combinatorial bound on the pathwidth of the underlying graph of D.Instead of using results from Section5,one can use Bodlaender’s algorithm[9]computing(forﬁxed k)tree decomposition of width k(if such a decomposition exists)in linear bined with our combinatorial bounds this yields a linear time algorithm for k-DMLOB(for a strongly connected di-graphs).However,the exponential dependence of k in Bodlaender’s algorithm is c k3for some large constant c.Finally,let us observe that while our results are for strongly connected di-graphs,they can be extended to a larger class of digraphs.Notice that (D)≥ s(D)for each digraph D.Let L be the family of digraphs D for which either s(D)=0or s(D)= (D).The following assertion shows that L includes a large number digraphs including all strongly connected digraphs and acyclic di-graphs(and,also,the well-studied classes of semicomplete multipartite digraphs and quasi-transitive digraphs,see[7]for the deﬁnitions).Proposition3([5]).Suppose that a digraph D satisﬁes the following property: for every pair R and Q of distinct strong components of D,if there is an arc from R to Q then each vertex of Q has an in-neighbor in R.Then D∈L.。

小学上册J卷英语第4单元暑期作业

小学上册英语第4单元暑期作业英语试题一、综合题(本题有100小题，每小题1分，共100分.每小题不选、错误，均不给分)1.What do we call the act of giving someone a gift?A. PresentingB. BestowingC. OfferingD. All of the AboveD2. A reduction reaction involves the ______ of electrons.3.My _____ (遥控车) can go very fast.4.When a solid dissolves in a liquid, it is called _______. (溶解)5.What is the coldest season?A. SpringB. SummerC. WinterD. FallC6.In autumn, we celebrate ______ (感恩节).7.What is the main ingredient in a Caesar salad?A. LettuceB. TomatoC. CucumberD. Spinach8.What is 5 x 2?A. 8B. 9C. 10D. 119.What do we call the place where we can watch performances?A. StadiumB. TheaterC. LibraryD. Office10.What is the name of the famous explorer who discovered America?A. ColumbusB. MagellanC. Vasco da GamaD. Marco Polo11. A chemical change is often indicated by a change in _______.12.What is a baby dog called?A. KittenB. PuppyC. CalfD. Chick13.The ________ (basketball) game was exciting.14.We visit the ______ (动物园) to see new animals.15.What is the name of the famous wizarding school in Harry Potter?A. HogwartsB. NarniaC. Middle-earthD. WonderlandA Hogwarts16.What do you call a person who writes stories?A. ScientistB. JournalistC. AuthorD. Poet17.Elements in the same group of the periodic table have similar ______.18.The ant builds a ________________ (巢) underground.19.She is ___ (reading/writing) a letter.20.The state of matter that has a definite volume but no definite shape is ______.21.My grandma has a lovely _____ at her house.22. A wave can be represented graphically as a ______.23.What is the capital of Italy?A. RomeB. VeniceC. FlorenceD. Milan24.In my neighborhood, there is a __________. I often go there to play with my friends. We like to __________ and sometimes have picnics on the grass.25.The anaconda is one of the largest ________________ (蛇).26.I wish for a warm ______ (冬天).27. A ____ is a playful creature that loves to jump around.28.My birthday is in ___ (June).29.She is climbing the ___. (tree)30.ssance artist Michelangelo painted the ceiling of the _______. (西斯廷教堂) The Rena31.The fastest land animal is the __________.32.What is the name of the action taken to protect the environment?A. PollutionB. ConservationC. WasteD. DisposalB33.Which instrument has strings and is played with a bow?A. GuitarB. ViolinC. DrumsD. FluteB34.Wildflowers grow without ______ (照顾).35.I have a ______ for math. (test)36.I want to learn how to ________.37.What is the capital of the United Kingdom?A. LondonB. EdinburghC. DublinD. CardiffA38. A saturated solution can be created by adding solute until no more can be ______.39.What is the main purpose of a fridge?A. CookingB. CoolingC. CleaningD. Heating40.The __________ (历史的传播) relies on multiple mediums.41.The mountain is very _______ (high).42.The _____ (植物研究) helps improve agricultural practices.43.What is the name of the famous wizarding school in Harry Potter?A. HogwartsB. NarniaC. OzD. CamelotA44.What do we call the process by which animals adapt to their environment?A. EvolutionB. MigrationC. HibernationD. AdaptationA45.What do you call a baby rabbit?A. KitB. PupC. CalfD. Cub46.The process of creating energy in cells is called ______.47.The _____ (library) has many books.48.What is the primary color of a stoplight for "go"?A. YellowB. GreenC. RedD. Blue49.The flowers are ________ in the garden.50.The _____ (grass/flower) is green.51.What do you call a baby ferret?A. KitB. PupC. CalfD. Cub52.The ice cream is ______ (melting) in the heat.53.What is the opposite of tall?A. ShortB. LongC. HighD. Wide54.My sister, ______ (我妹妹), is interested in fashion design.55.The __________ is the largest ocean on earth.56.My uncle is my funny _______ who tells hilarious stories.57.The _____ (flower) is pretty.58.What do you call the main character in a story?A. ProtagonistB. AntagonistC. NarratorD. AuthorA59.The flowers are _______ (opening) in the spring.60.The capital of Togo is ________ (洛美).61.What do we call the force that pulls objects toward the Earth?A. MagnetismB. FrictionC. GravityD. InertiaC Gravity62.The Earth's surface is covered by a variety of ______, including forests and grasslands.63.What is the main ingredient in sushi?A. FishB. RiceC. SeaweedD. Vegetables64.My ______ loves to participate in competitions.65.The process of changing from a solid to a gas without becoming liquid is called _______.66.What is the opposite of ‘hard’?A. SolidB. RoughC. SoftD. Tough67.The cat is very _________. (可爱的)68.The ancient civilizations of Mesopotamia contributed to the development of ________.69.What is the capital city of Japan?A. TokyoB. OsakaC. KyotoD. HiroshimaA70.trial Revolution began in _____. The Indu71.The ______ is known for its long neck and spots.72.What shape has three sides?A. SquareB. TriangleC. CircleD. RectangleB73.We have ______ (English) class at noon.74. A squirrel collects ______ (食物) for the winter months.75.We have a ________ (party) for Halloween.76.Every substance has unique _____.77.The dress is very ___. (expensive)78.Reactivity is the ability of a substance to undergo a _____.79.The process of ______ can contribute to the nutrient cycle.80.I love to ________ (聚会) with friends.81.The first person to climb Mount Everest was ______ (希拉里).82.The ancient Greeks engaged in ________ (体育) during the Olympics.83.The element with the symbol Zr is __________.84.What do you call a story that explains how something came to be?A. LegendB. MythC. FableD. Tale85.The _______ (青蛙) croaks at night.86.What do we use to write?A. BrushB. PencilC. HammerD. SpoonB87.The skunk can spray a _______ (臭味).88.The __________ is the largest desert in the world. (撒哈拉沙漠)89.Which instrument has keys and can be played with both hands?A. GuitarB. ViolinC. PianoD. DrumsC90.What is the term for a baby cow?A. CalfB. KidC. LambD. FoalA91.He is learning to ________ a guitar.92.I saw a ________ in the garden.93.The chemical symbol for selenium is __________.94. A ________ (气候变化) impacts weather patterns.95.The chemical formula for lutetium oxide is _____.96.We can _______ a picnic by the lake.97.I love the sound of ______ (雨) falling on the roof. It makes me feel cozy inside.98.What is the capital city of France?A. LondonB. ParisC. RomeD. BerlinB99.I think it's fun to swap ________ (玩具名) with my classmates.100.The __________ was so bright that I had to wear sunglasses. (阳光)。

用文体学分析Hiroshima

An Analysis of Language Features inHiroshima -the “Liveliest” CityAbstract: Hiroshima is a feature story or simply a feature, a type of journalistic writing .Feature stories may appear in newspapers, magazines, on TV or radio. The one we are studying was presented on an American radio program. This paper tries to analyze the features of speech such as sarcasm, alliteration, metaphor, and synecdoche, etc to research the effect of the narrative skill and reflect the theme and expressing art value of this novel.Key words: Hiroshima narrative liveliest feature stories1.IntroductionA feature story covers a selected issue in depth .The purpose of a feature story is to hold the readers‟ attention and make them read the story from beginning to end .For such an aim ,the reporter, first of all ,chooses something interesting to write about and a unique angle to present the story. The subject of a feature story can be a person, or a group of people, a place, a thing, an event, an accident ect. The subject of the feature story “Hiroshima-the …Liveliest‟ City in Japan” is about a place –Hiroshima .The story is mainly divided into three parts: the writer‟s arrival at Hiroshima, the reception by the city mayor, and his visit to the atomic ward in the hospital. The writer uses the first-person narrative voice .The story begins with the writer‟s arrival at the railway station. The writer does not try to conceal his emotions about the city or his attitude toward the atomic bomb. In the very first paragraph he says, “I had a lump in my throat and a lot of sad thoughts on my mind.”He asked, “Was I not at the scene of the crime?” On his way to his destination he observed the crowds of Japanese.” At the reception, the writer expected the mayor to talk about the atomic bomb and its tragic impact. To his great surprise, the mayor referred to Hiroshima as the “liveliest city in Japan.”The puzzled writer was told by an elder Japanese man that there were two schools of thought in Hiroshima about the bomb. With many prepared questions, thewriter visited the atomic ward in the hospital .he interviewed atomic bomb victims and came to his conclusion about Hiroshima.Ⅰ.Analysis of the feature of speechSarcasm is a way of using words that are the opposite of what you mean in order to be unpleasant to somebody or to make fun of them. “Hiroshima-the …Liveliest‟ City in Japan .(Title) If you write about this city, do not forget to say that it is the gayest city in Japan, even if many of the town‟s people still bear hidden wounds, and burns.（L3, Para.27）Both the “Liveliest” City and the “gayest city” used sarcasm. Alliteration is the use of the same so words that are close together. …..as the fastest train in the world slipped to a stop in Hiroshima Station. (L.2, Para.1) “Slipped” and “stop” begin with /s/. I felt sick, and ever since then they have been testing and treating me. Both “testing”and “treating” begin with /t/. Metaphor is a figure of speech that describes something by referring to it as something else, in order to show that the two things have the same qualities and to make the description more powerful. (1)And secondly, because I had a lump in my throat and a lot of sad thoughts on my mind that had little to do with anything a Nippon railways official might say. (L5, Para.1) (2) The usher bowed deeply and heaved a long, almost musical sigh, when I show him the invitation which the mayor had sent me in response to my request for an interview.(L.2,Para.5) The first sentence means that at secondly, because I was choked with emotions and occupied with sorrow and there was no connection between my sad thoughts and the words of a Japanese railways official.Synecdoche is a figure of speech in which a part of something is used to represent a whole, or a whole is used to represent a part of something. The rather arresting spectacle of little old Japan adrift amid beige concrete skyscrapers is the very symbol of the incessant struggle between the kimono and the miniskirt. (L5.Pare .7) “Kimono” and“miniskirt” represent the tradition and modern .Onomatopoeia is a device that uses words which imitate the sounds made by an object（animate or inanimate）,or which are associated with or suggestive of some action or movement. Just as I was beginning to find the ride long, the taxi screechedto a halt, and the driver got out and went over to policemen to ask the way. It means that just when I was starting to find the journey long, the taxi stopped suddenly with a harsh piercing sound, and the driver got out of his car and went over to a policeman to ask for direction. Anti-climax refers to the sudden appearance of a ridiculous or trivial idea following one or more significant or elevated ideas in a descending order of significance or intensity, from strong to weak, from weighty to light or frivolous. Anti-climax is usually comic in effect. Seldom has a city gained such world renown, and I am proud and happy to welcome you to Hiroshima, a town known throughout the word for its –oysters.（L, 1, Para.17）It expressed the writer‟s surprise.ConclusionIn response to the author‟s request for an interview, the mayor of Hiroshima invited him to have dinner with other foreigners on the restaurant boat .Upon stepping on the soil, his mind was full of sad thoughts. However, the following experience was out of his expectation. The life in Hiroshima seemed much the same as the same as in other Japanese cities: traditional kimonos and Western dress, constant bows and taxi drivers. In the reception, he thought the mayor would say something about the atomic bomb. But the mayor made a speech of oysters. A small Japanese man told him that people didn‟t want to talk about it any more and even wanted to demolish the monument, for it hurt everybody .Then the author went to another place-the atomic ward in a hospital where many patients suffering from radiation were treated .A former fisherman as a victim told him that it was humiliating to survive in this city. Even their children would encounter prejudice. But every day he folded a paper bird to commemorate .The author previously wondered whether Hiroshima was the liveliest city in Japan, but now he could read the answer in every eye.Bibliography[1] 张汉熙. 高级英语[M]. 北京: 外语教学与研究出版社，2010[2] 王军高卫红. 高级英语大学教材全解[M]. 北京: 中国海洋大学出版社，2010。

小学上册第二次英语第1单元测验卷

小学上册英语第1单元测验卷英语试题一、综合题(本题有100小题，每小题1分，共100分.每小题不选、错误，均不给分)1.What do you call the sport of riding on water using a board?A. SurfingB. SkatingC. DivingD. SwimmingA2. A saturated solution cannot dissolve any more ______.3.My coach is very . (我的教练很。

)4.The _____ (照片记录) of plants shows their changes over time.5.The jellyfish has tentacles that can sting its ________________ (猎物).6.My brother is a ______. He enjoys programming.7.What is the main gas found in the air we breathe?A. OxygenB. HydrogenC. NitrogenD. Carbon DioxideC8.My brother is a ______. He dreams of playing in the NBA.9.What do you call a person who studies insects?A. EntomologistB. BiologistC. ZoologistD. BotanistA10. A beaker is a common piece of ______ glassware.11.What is the capital of Japan?A. KyotoB. TokyoC. OsakaD. Hiroshima12.What do you call a group of stars?A. GalaxyB. Solar SystemC. ConstellationD. Nebula13. A bird builds a ______ (巢) in trees.14.Which of these is a winter sport?A. SwimmingB. SkiingC. SoccerD. TennisB15.What instrument has strings and is played with a bow?A. ViolinB. DrumsC. TrumpetD. GuitarA16.This ____ is very colorful and fun. (玩具名称)17.Having a variety of ________ (玩具名) allows me to be more ________ (形容词) in play.18.The _______ (青蛙) can leap great distances.19.The ________ was a significant event during the American Revolution.20.The ________ (生态系统保护) ensures biodiversity.21. A ______ is a symbol of freedom.22.I like to explore the ______ (森林).23.What is the term for a person who travels to space?A. AstronautB. CosmonautC. PilotD. ScientistA24.Which color do you get by mixing red and white?A. PinkB. PurpleC. OrangeD. BrownA25. A ____ is a friendly animal that enjoys human companionship.26.The _____ (爸爸) is cooking dinner.27.I enjoy _____ (reading/writing) stories.28.The Great Depression started in the year ________.29. A shadow is formed when an object blocks _______.30.Plants can create a ______ (宁静的环境) in our lives.31.Many animals rely on __________ for their survival.32. A __________ is a mixture that can be separated by filtration.33.I can ______ very fast.34.They are going to _____ (商店).35.What is the main purpose of a library?A. Borrow booksB. Watch moviesC. Play gamesD. Study scienceA36.The _____ (飞机) takes off smoothly.37.What do you call the center of an atom?A. ElectronB. NeutronC. ProtonD. NucleusD38.I enjoy _______ (散步) in the evening.39.The _____ (cupcake) is decorated.40.What do we call the act of moving quickly on foot?A. WalkingB. RunningC. JoggingD. SprintingB41.My best friend is very __________. (体贴)42. A _____ (灌木) can provide privacy in a garden.43.The ______ (鸭子) quacks when it swims.44.The teacher is _____ the lesson. (starting)45.The parrot has a colorful ______ (羽毛).46.He is eating a ___. (sandwich)47. A ________ (猴子) is very clever and enjoys swinging from trees.48. A __________ is a large area of grassland in North America.49.What is the main purpose of a refrigerator?A. To cook foodB. To freeze foodC. To keep food coldD. To wash foodC To keep food cold50.The _______ of sound can be increased by amplifying it.51.How many days are in February during a leap year?A. 28B. 29C. 30D. 31答案:B52.What is 50 - 25?A. 20B. 25C. 30D. 3553.What is a common color for grass?A. RedB. BlueC. GreenD. YellowC54.What is the name of the famous American musician known for "Someone Like You"?A. AdeleB. Taylor SwiftC. Kelly ClarksonD. BeyoncéA55.The ______ has a strong family bond.56.The ____ has bright colors and loves to fly.57.This boy, ______ (这个男孩), is good at puzzles.58.I built a ________ (城堡) out of my building blocks.59.I like to ___ (listen/sing) to songs.60.Mahatma Gandhi led India to independence through ______ (非暴力) resistance.61.What is 8 divided by 2?A. 2B. 4C. 6D. 8B62.The ______ (水分) in soil is crucial.63.What is the main gas in the air we breathe?A. OxygenB. NitrogenC. Carbon dioxideD. Helium64.The kids are _____ in the park. (running)65.Which ocean is located between Africa and Australia?A. AtlanticB. IndianC. ArcticD. PacificBbustion is a reaction that produces ________ and water.67.What do we call the science of numbers?A. ChemistryB. PhysicsC. MathematicsD. BiologyC68.Many plants have _____ (香味) that attract insects.69.What is the capital city of Kyrgyzstan?A. BishkekB. OshC. Jalal-AbadD. Tokmok70.The ant works hard to find ________________ (食物).71.We will _____ (travel/stay) at home.72.The ________ (rocket) launches into space.73.The _____ shows the positions of planets in the sky.74.The French Revolution began in the year _______.75.In my culture, we often greet each other with "Good morning, ." (在我的文化中，我们常常用“早上好，”互相问候。

斯普林格数学研究生教材丛书

《斯普林格数学研究生教材丛书》（Graduate Texts in Mathematics）GTM001《Introduction to Axiomatic Set Theory》Gaisi Takeuti, Wilson M.Zaring GTM002《Measure and Category》John C.Oxtoby（测度和范畴）（2ed.）GTM003《Topological Vector Spaces》H.H.Schaefer, M.P.Wolff（2ed.）GTM004《A Course in Homological Algebra》P.J.Hilton, U.Stammbach（2ed.）（同调代数教程）GTM005《Categories for the Working Mathematician》Saunders Mac Lane（2ed.）GTM006《Projective Planes》Daniel R.Hughes, Fred C.Piper（投射平面）GTM007《A Course in Arithmetic》Jean-Pierre Serre（数论教程）GTM008《Axiomatic set theory》Gaisi Takeuti, Wilson M.Zaring（2ed.）GTM009《Introduction to Lie Algebras and Representation Theory》James E.Humphreys（李代数和表示论导论）GTM010《A Course in Simple-Homotopy Theory》M.M CohenGTM011《Functions of One Complex VariableⅠ》John B.ConwayGTM012《Advanced Mathematical Analysis》Richard BealsGTM013《Rings and Categories of Modules》Frank W.Anderson, Kent R.Fuller（环和模的范畴）（2ed.）GTM014《Stable Mappings and Their Singularities》Martin Golubitsky, Victor Guillemin （稳定映射及其奇点）GTM015《Lectures in Functional Analysis and Operator Theory》Sterling K.Berberian GTM016《The Structure of Fields》David J.Winter（域结构）GTM017《Random Processes》Murray RosenblattGTM018《Measure Theory》Paul R.Halmos（测度论）GTM019《A Hilbert Space Problem Book》Paul R.Halmos（希尔伯特问题集）GTM020《Fibre Bundles》Dale Husemoller（纤维丛）GTM021《Linear Algebraic Groups》James E.Humphreys（线性代数群）GTM022《An Algebraic Introduction to Mathematical Logic》Donald W.Barnes, John M.MackGTM023《Linear Algebra》Werner H.Greub（线性代数）GTM024《Geometric Functional Analysis and Its Applications》Paul R.HolmesGTM025《Real and Abstract Analysis》Edwin Hewitt, Karl StrombergGTM026《Algebraic Theories》Ernest G.ManesGTM027《General Topology》John L.Kelley（一般拓扑学）GTM028《Commutative Algebra》VolumeⅠOscar Zariski, Pierre Samuel（交换代数）GTM029《Commutative Algebra》VolumeⅡOscar Zariski, Pierre Samuel（交换代数）GTM030《Lectures in Abstract AlgebraⅠ.Basic Concepts》Nathan Jacobson（抽象代数讲义Ⅰ基本概念分册）GTM031《Lectures in Abstract AlgebraⅡ.Linear Algabra》Nathan.Jacobson（抽象代数讲义Ⅱ线性代数分册）GTM032《Lectures in Abstract AlgebraⅢ.Theory of Fields and Galois Theory》Nathan.Jacobson（抽象代数讲义Ⅲ域和伽罗瓦理论）GTM033《Differential Topology》Morris W.Hirsch（微分拓扑）GTM034《Principles of Random Walk》Frank Spitzer（2ed.）（随机游动原理）GTM035《Several Complex Variables and Banach Algebras》Herbert Alexander, John Wermer（多复变和Banach代数）GTM036《Linear Topological Spaces》John L.Kelley, Isaac Namioka（线性拓扑空间）GTM037《Mathematical Logic》J.Donald Monk（数理逻辑）GTM038《Several Complex Variables》H.Grauert, K.FritzsheGTM039《An Invitation to C*-Algebras》William Arveson（C*-代数引论）GTM040《Denumerable Markov Chains》John G.Kemeny, urie Snell, Anthony W.KnappGTM041《Modular Functions and Dirichlet Series in Number Theory》Tom M.Apostol （数论中的模函数和Dirichlet序列）GTM042《Linear Representations of Finite Groups》Jean-Pierre Serre（有限群的线性表示）GTM043《Rings of Continuous Functions》Leonard Gillman, Meyer JerisonGTM044《Elementary Algebraic Geometry》Keith KendigGTM045《Probability TheoryⅠ》M.Loève（概率论Ⅰ）（4ed.）GTM046《Probability TheoryⅡ》M.Loève（概率论Ⅱ）（4ed.）GTM047《Geometric Topology in Dimensions 2 and 3》Edwin E.MoiseGTM048《General Relativity for Mathematicians》Rainer.K.Sachs, H.Wu伍鸿熙（为数学家写的广义相对论）GTM049《Linear Geometry》K.W.Gruenberg, A.J.Weir（2ed.）GTM050《Fermat's Last Theorem》Harold M.EdwardsGTM051《A Course in Differential Geometry》Wilhelm Klingenberg（微分几何教程）GTM052《Algebraic Geometry》Robin Hartshorne（代数几何）GTM053《A Course in Mathematical Logic for Mathematicians》Yu.I.Manin（2ed.）GTM054《Combinatorics with Emphasis on the Theory of Graphs》Jack E.Graver, Mark E.WatkinsGTM055《Introduction to Operator TheoryⅠ》Arlen Brown, Carl PearcyGTM056《Algebraic Topology：An Introduction》W.S.MasseyGTM057《Introduction to Knot Theory》Richard.H.Crowell, Ralph.H.FoxGTM058《p-adic Numbers, p-adic Analysis, and Zeta-Functions》Neal Koblitz（p-adic 数、p-adic分析和Z函数）GTM059《Cyclotomic Fields》Serge LangGTM060《Mathematical Methods of Classical Mechanics》V.I.Arnold（经典力学的数学方法）（2ed.）GTM061《Elements of Homotopy Theory》George W.Whitehead（同论论基础）GTM062《Fundamentals of the Theory of Groups》M.I.Kargapolov, Ju.I.Merzljakov GTM063《Modern Graph Theory》Béla BollobásGTM064《Fourier Series：A Modern Introduction》VolumeⅠ（2ed.）R.E.Edwards（傅里叶级数）GTM065《Differential Analysis on Complex Manifolds》Raymond O.Wells, Jr.（3ed.）GTM066《Introduction to Affine Group Schemes》William C.Waterhouse（仿射群概型引论）GTM067《Local Fields》Jean-Pierre Serre（局部域）GTM069《Cyclotomic FieldsⅠandⅡ》Serge LangGTM070《Singular Homology Theory》William S.MasseyGTM071《Riemann Surfaces》Herschel M.Farkas, Irwin Kra（黎曼曲面）GTM072《Classical Topology and Combinatorial Group Theory》John Stillwell（经典拓扑和组合群论）GTM073《Algebra》Thomas W.Hungerford（代数）GTM074《Multiplicative Number Theory》Harold Davenport（乘法数论）（3ed.）GTM075《Basic Theory of Algebraic Groups and Lie Algebras》G.P.HochschildGTM076《Algebraic Geometry：An Introduction to Birational Geometry of Algebraic Varieties》Shigeru IitakaGTM077《Lectures on the Theory of Algebraic Numbers》Erich HeckeGTM078《A Course in Universal Algebra》Stanley Burris, H.P.Sankappanavar（泛代数教程）GTM079《An Introduction to Ergodic Theory》Peter Walters（遍历性理论引论）GTM080《A Course in_the Theory of Groups》Derek J.S.RobinsonGTM081《Lectures on Riemann Surfaces》Otto ForsterGTM082《Differential Forms in Algebraic Topology》Raoul Bott, Loring W.Tu（代数拓扑中的微分形式）GTM083《Introduction to Cyclotomic Fields》Lawrence C.Washington（割圆域引论）GTM084《A Classical Introduction to Modern Number Theory》Kenneth Ireland, Michael Rosen（现代数论经典引论）GTM085《Fourier Series A Modern Introduction》Volume 1（2ed.）R.E.Edwards GTM086《Introduction to Coding Theory》J.H.van Lint（3ed .）GTM087《Cohomology of Groups》Kenneth S.Brown（上同调群）GTM088《Associative Algebras》Richard S.PierceGTM089《Introduction to Algebraic and Abelian Functions》Serge Lang（代数和交换函数引论）GTM090《An Introduction to Convex Polytopes》Ame BrondstedGTM091《The Geometry of Discrete Groups》Alan F.BeardonGTM092《Sequences and Series in BanachSpaces》Joseph DiestelGTM093《Modern Geometry-Methods and Applications》（PartⅠ.The of geometry Surfaces Transformation Groups and Fields）B.A.Dubrovin, A.T.Fomenko, S.P.Novikov （现代几何学方法和应用）GTM094《Foundations of Differentiable Manifolds and Lie Groups》Frank W.Warner（可微流形和李群基础）GTM095《Probability》A.N.Shiryaev（2ed.）GTM096《A Course in Functional Analysis》John B.Conway（泛函分析教程）GTM097《Introduction to Elliptic Curves and Modular Forms》Neal Koblitz（椭圆曲线和模形式引论）GTM098《Representations of Compact Lie Groups》Theodor Breöcker, Tammo tom DieckGTM099《Finite Reflection Groups》L.C.Grove, C.T.Benson（2ed.）GTM100《Harmonic Analysis on Semigroups》Christensen Berg, Jens Peter Reus Christensen, Paul ResselGTM101《Galois Theory》Harold M.Edwards（伽罗瓦理论）GTM102《Lie Groups, Lie Algebras, and Their Representation》V.S.Varadarajan（李群、李代数及其表示）GTM103《Complex Analysis》Serge LangGTM104《Modern Geometry-Methods and Applications》（PartⅡ.Geometry and Topology of Manifolds）B.A.Dubrovin, A.T.Fomenko, S.P.Novikov（现代几何学方法和应用）GTM105《SL₂ (R)》Serge Lang（SL₂ (R)群）GTM106《The Arithmetic of Elliptic Curves》Joseph H.Silverman（椭圆曲线的算术理论）GTM107《Applications of Lie Groups to Differential Equations》Peter J.Olver（李群在微分方程中的应用）GTM108《Holomorphic Functions and Integral Representations in Several Complex Variables》R.Michael RangeGTM109《Univalent Functions and Teichmueller Spaces》Lehto OlliGTM110《Algebraic Number Theory》Serge Lang（代数数论）GTM111《Elliptic Curves》Dale Husemoeller（椭圆曲线）GTM112《Elliptic Functions》Serge Lang（椭圆函数）GTM113《Brownian Motion and Stochastic Calculus》Ioannis Karatzas, Steven E.Shreve （布朗运动和随机计算）GTM114《A Course in Number Theory and Cryptography》Neal Koblitz（数论和密码学教程）GTM115《Differential Geometry：Manifolds, Curves, and Surfaces》M.Berger, B.Gostiaux GTM116《Measure and Integral》Volume1 John L.Kelley, T.P.SrinivasanGTM117《Algebraic Groups and Class Fields》Jean-Pierre Serre（代数群和类域）GTM118《Analysis Now》Gert K.Pedersen（现代分析）GTM119《An introduction to Algebraic Topology》Jossph J.Rotman（代数拓扑导论）GTM120《Weakly Differentiable Functions》William P.Ziemer（弱可微函数）GTM121《Cyclotomic Fields》Serge LangGTM122《Theory of Complex Functions》Reinhold RemmertGTM123《Numbers》H.-D.Ebbinghaus, H.Hermes, F.Hirzebruch, M.Koecher, K.Mainzer, J.Neukirch, A.Prestel, R.Remmert（2ed.）GTM124《Modern Geometry-Methods and Applications》（PartⅢ.Introduction to Homology Theory）B.A.Dubrovin, A.T.Fomenko, S.P.Novikov（现代几何学方法和应用）GTM125《Complex Variables：An introduction》Garlos A.Berenstein, Roger Gay GTM126《Linear Algebraic Groups》Armand Borel（线性代数群）GTM127《A Basic Course in Algebraic Topology》William S.Massey（代数拓扑基础教程）GTM128《Partial Differential Equations》Jeffrey RauchGTM129《Representation Theory：A First Course》William Fulton, Joe HarrisGTM130《Tensor Geometry》C.T.J.Dodson, T.Poston（张量几何）GTM131《A First Course in Noncommutative Rings》m（非交换环初级教程）GTM132《Iteration of Rational Functions：Complex Analytic Dynamical Systems》AlanF.Beardon（有理函数的迭代：复解析动力系统）GTM133《Algebraic Geometry：A First Course》Joe Harris（代数几何）GTM134《Coding and Information Theory》Steven RomanGTM135《Advanced Linear Algebra》Steven RomanGTM136《Algebra：An Approach via Module Theory》William A.Adkins, Steven H.WeintraubGTM137《Harmonic Function Theory》Sheldon Axler, Paul Bourdon, Wade Ramey（调和函数理论）GTM138《A Course in Computational Algebraic Number Theory》Henri Cohen（计算代数数论教程）GTM139《Topology and Geometry》Glen E.BredonGTM140《Optima and Equilibria：An Introduction to Nonlinear Analysis》Jean-Pierre AubinGTM141《A Computational Approach to Commutative Algebra》Gröbner Bases, Thomas Becker, Volker Weispfenning, Heinz KredelGTM142《Real and Functional Analysis》Serge Lang（3ed.）GTM143《Measure Theory》J.L.DoobGTM144《Noncommutative Algebra》Benson Farb, R.Keith DennisGTM145《Homology Theory：An Introduction to Algebraic Topology》James W.Vick（同调论：代数拓扑简介）GTM146《Computability：A Mathematical Sketchbook》Douglas S.BridgesGTM147《Algebraic K-Theory and Its Applications》Jonathan Rosenberg（代数K理论及其应用）GTM148《An Introduction to the Theory of Groups》Joseph J.Rotman（群论入门）GTM149《Foundations of Hyperbolic Manifolds》John G.Ratcliffe（双曲流形基础）GTM150《Commutative Algebra with a view toward Algebraic Geometry》David EisenbudGTM151《Advanced Topics in the Arithmetic of Elliptic Curves》Joseph H.Silverman（椭圆曲线的算术高级选题）GTM152《Lectures on Polytopes》Günter M.ZieglerGTM153《Algebraic Topology：A First Course》William Fulton（代数拓扑）GTM154《An introduction to Analysis》Arlen Brown, Carl PearcyGTM155《Quantum Groups》Christian Kassel（量子群）GTM156《Classical Descriptive Set Theory》Alexander S.KechrisGTM157《Integration and Probability》Paul MalliavinGTM158《Field theory》Steven Roman（2ed.）GTM159《Functions of One Complex Variable VolⅡ》John B.ConwayGTM160《Differential and Riemannian Manifolds》Serge Lang（微分流形和黎曼流形）GTM161《Polynomials and Polynomial Inequalities》Peter Borwein, Tamás Erdélyi（多项式和多项式不等式）GTM162《Groups and Representations》J.L.Alperin, Rowen B.Bell（群及其表示）GTM163《Permutation Groups》John D.Dixon, Brian Mortime rGTM164《Additive Number Theory：The Classical Bases》Melvyn B.NathansonGTM165《Additive Number Theory：Inverse Problems and the Geometry of Sumsets》Melvyn B.NathansonGTM166《Differential Geometry：Cartan's Generalization of Klein's Erlangen Program》R.W.SharpeGTM167《Field and Galois Theory》Patrick MorandiGTM168《Combinatorial Convexity and Algebraic Geometry》Günter Ewald（组合凸面体和代数几何）GTM169《Matrix Analysis》Rajendra BhatiaGTM170《Sheaf Theory》Glen E.Bredon（2ed.）GTM171《Riemannian Geometry》Peter Petersen（黎曼几何）GTM172《Classical Topics in Complex Function Theory》Reinhold RemmertGTM173《Graph Theory》Reinhard Diestel（图论）（3ed.）GTM174《Foundations of Real and Abstract Analysis》Douglas S.Bridges（实分析和抽象分析基础）GTM175《An Introduction to Knot Theory》W.B.Raymond LickorishGTM176《Riemannian Manifolds：An Introduction to Curvature》John M.LeeGTM177《Analytic Number Theory》Donald J.Newman（解析数论）GTM178《Nonsmooth Analysis and Control Theory》F.H.clarke, Yu.S.Ledyaev, R.J.Stern, P.R.Wolenski（非光滑分析和控制论）GTM179《Banach Algebra Techniques in Operator Theory》Ronald G.Douglas（2ed.）GTM180《A Course on Borel Sets》S.M.Srivastava（Borel 集教程）GTM181《Numerical Analysis》Rainer KressGTM182《Ordinary Differential Equations》Wolfgang WalterGTM183《An introduction to Banach Spaces》Robert E.MegginsonGTM184《Modern Graph Theory》Béla Bollobás（现代图论）GTM185《Using Algebraic Geomety》David A.Cox, John Little, Donal O’Shea（应用代数几何）GTM186《Fourier Analysis on Number Fields》Dinakar Ramakrishnan, Robert J.Valenza GTM187《Moduli of Curves》Joe Harris, Ian Morrison（曲线模）GTM188《Lectures on the Hyperreals：An Introduction to Nonstandard Analysis》Robert GoldblattGTM189《Lectures on Modules and Rings》m（模和环讲义）GTM190《Problems in Algebraic Number Theory》M.Ram Murty, Jody Esmonde（代数数论中的问题）GTM191《Fundamentals of Differential Geometry》Serge Lang（微分几何基础）GTM192《Elements of Functional Analysis》Francis Hirsch, Gilles LacombeGTM193《Advanced Topics in Computational Number Theory》Henri CohenGTM194《One-Parameter Semigroups for Linear Evolution Equations》Klaus-Jochen Engel, Rainer Nagel（线性发展方程的单参数半群）GTM195《Elementary Methods in Number Theory》Melvyn B.Nathanson（数论中的基本方法）GTM196《Basic Homological Algebra》M.Scott OsborneGTM197《The Geometry of Schemes》David Eisenbud, Joe HarrisGTM198《A Course in p-adic Analysis》Alain M.RobertGTM199《Theory of Bergman Spaces》Hakan Hedenmalm, Boris Korenblum, Kehe Zhu（Bergman空间理论）GTM200《An Introduction to Riemann-Finsler Geometry》D.Bao, S.-S.Chern, Z.Shen GTM201《Diophantine Geometry An Introduction》Marc Hindry, Joseph H.Silverman GTM202《Introduction to Topological Manifolds》John M.LeeGTM203《The Symmetric Group》Bruce E.SaganGTM204《Galois Theory》Jean-Pierre EscofierGTM205《Rational Homotopy Theory》Yves Félix, Stephen Halperin, Jean-Claude Thomas（有理同伦论）GTM206《Problems in Analytic Number Theory》M.Ram MurtyGTM207《Algebraic Graph Theory》Chris Godsil, Gordon Royle（代数图论）GTM208《Analysis for Applied Mathematics》Ward CheneyGTM209《A Short Course on Spectral Theory》William Arveson（谱理论简明教程）GTM210《Number Theory in Function Fields》Michael RosenGTM211《Algebra》Serge Lang（代数）GTM212《Lectures on Discrete Geometry》Jiri Matousek（离散几何讲义）GTM213《From Holomorphic Functions to Complex Manifolds》Klaus Fritzsche, Hans Grauert（从正则函数到复流形）GTM214《Partial Differential Equations》Jüergen Jost（偏微分方程）GTM215《Algebraic Functions and Projective Curves》David M.Goldschmidt（代数函数和投影曲线）GTM216《Matrices：Theory and Applications》Denis Serre（矩阵：理论及应用）GTM217《Model Theory An Introduction》David Marker（模型论引论）GTM218《Introduction to Smooth Manifolds》John M.Lee（光滑流形引论）GTM219《The Arithmetic of Hyperbolic 3-Manifolds》Colin Maclachlan, Alan W.Reid GTM220《Smooth Manifolds and Observables》Jet Nestruev（光滑流形和直观）GTM221《Convex Polytopes》Branko GrüenbaumGTM222《Lie Groups, Lie Algebras, and Representations》Brian C.Hall（李群、李代数和表示）GTM223《Fourier Analysis and its Applications》Anders Vretblad（傅立叶分析及其应用）GTM224《Metric Structures in Differential Geometry》Gerard Walschap（微分几何中的度量结构）GTM225《Lie Groups》Daniel Bump（李群）GTM226《Spaces of Holomorphic Functions in the Unit Ball》Kehe Zhu（单位球内的全纯函数空间）GTM227《Combinatorial Commutative Algebra》Ezra Miller, Bernd Sturmfels（组合交换代数）GTM228《A First Course in Modular Forms》Fred Diamond, Jerry Shurman（模形式初级教程）GTM229《The Geometry of Syzygies》David Eisenbud（合冲几何）GTM230《An Introduction to Markov Processes》Daniel W.Stroock（马尔可夫过程引论）GTM231《Combinatorics of Coxeter Groups》Anders Bjröner, Francesco Brenti（Coxeter 群的组合学）GTM232《An Introduction to Number Theory》Graham Everest, Thomas Ward（数论入门）GTM233《Topics in Banach Space Theory》Fenando Albiac, Nigel J.Kalton（Banach空间理论选题）GTM234《Analysis and Probability：Wavelets, Signals, Fractals》Palle E.T.Jorgensen（分析与概率）GTM235《Compact Lie Groups》Mark R.Sepanski（紧致李群）GTM236《Bounded Analytic Functions》John B.Garnett（有界解析函数）GTM237《An Introduction to Operators on the Hardy-Hilbert Space》Rubén A.Martínez-Avendano, Peter Rosenthal（哈代-希尔伯特空间算子引论）GTM238《A Course in Enumeration》Martin Aigner（枚举教程）GTM239《Number Theory：VolumeⅠTools and Diophantine Equations》Henri Cohen GTM240《Number Theory：VolumeⅡAnalytic and Modern Tools》Henri Cohen GTM241《The Arithmetic of Dynamical Systems》Joseph H.SilvermanGTM242《Abstract Algebra》Pierre Antoine Grillet（抽象代数）GTM243《Topological Methods in Group Theory》Ross GeogheganGTM244《Graph Theory》J.A.Bondy, U.S.R.MurtyGTM245《Complex Analysis：In the Spirit of Lipman Bers》Jane P.Gilman, Irwin Kra, Rubi E.RodriguezGTM246《A Course in Commutative Banach Algebras》Eberhard KaniuthGTM247《Braid Groups》Christian Kassel, Vladimir TuraevGTM248《Buildings Theory and Applications》Peter Abramenko, Kenneth S.Brown GTM249《Classical Fourier Analysis》Loukas Grafakos（经典傅里叶分析）GTM250《Modern Fourier Analysis》Loukas Grafakos（现代傅里叶分析）GTM251《The Finite Simple Groups》Robert A.WilsonGTM252《Distributions and Operators》Gerd GrubbGTM253《Elementary Functional Analysis》Barbara D.MacCluerGTM254《Algebraic Function Fields and Codes》Henning StichtenothGTM255《Symmetry Representations and Invariants》Roe Goodman, Nolan R.Wallach GTM256《A Course in Commutative Algebra》Kemper GregorGTM257《Deformation Theory》Robin HartshorneGTM258《Foundation of Optimization》Osman GülerGTM259《Ergodic Theory：with a view towards Number Theory》Manfred Einsiedler, Thomas WardGTM260《Monomial Ideals》Jurgen Herzog, Takayuki HibiGTM261《Probability and Stochastics》Erhan CinlarGTM262《Essentials of Integration Theory for Analysis》Daniel W.StroockGTM263《Analysis on Fock Spaces》Kehe ZhuGTM264《Functional Analysis, Calculus of Variations and Optimal Control》Francis ClarkeGTM265《Unbounded Self-adjoint Operatorson Hilbert Space》Konrad Schmüdgen GTM266《Calculus Without Derivatives》Jean-Paul PenotGTM267《Quantum Theory for Mathematicians》Brian C.HallGTM268《Geometric Analysis of the Bergman Kernel and Metric》Steven G.Krantz GTM269《Locally Convex Spaces》M.Scott Osborne。

小学上册第十二次英语第三单元期中试卷

小学上册英语第三单元期中试卷英语试题一、综合题(本题有100小题，每小题1分，共100分.每小题不选、错误，均不给分)1.The girl loves to ________.2.What do you call a story with magic and fantasy elements?A. Realistic FictionB. Science FictionC. FantasyD. Non-fictionC3.The ________ (向日葵) turns towards the sun and is very bright.4.The firefly lights up the ______ (夜晚).5.What do you call a large body of saltwater?A. RiverB. OceanC. LakeD. Pond6.How many colors are there in a rainbow?A. 5B. 6C. 7D. 87.ts produce _____ (果汁) that is tasty. Some pla8.My favorite ice cream flavor is ________.9.The _______ serves as a habitat for many creatures.10.The ________ was a famous explorer who discovered new lands.11.Planting bulbs can lead to beautiful ______ (花朵) in spring.12.What is the name of the popular movie genre that involves superheroes?A. DramaB. ComedyC. ActionD. AdventureC13.What do we call the largest organ in the human body?A. HeartB. SkinC. LiverD. BrainB Skin14.What is the capital of Palau?A. NgerulmudB. KororC. MelekeokD. AiraiA15.The ________ (train) goes to the city.16.The weather is _____ outside today. (nice)17.rt in community gardening projects can foster ______ and cooperation. (参与社区园艺项目可以促进团结与合作。

具有反射群对称性的球面图案自动生成

具有反射群对称性的球面图案自动生成王新长;刘满凤;欧阳培昌【摘要】Equivariant mapping method is not only difficult to be implemented, but also constrained by the order of symmetry group. Drawn on the experience of the invariant theory of finite reflection group, this paper proposes an invariant mapping method to yield aesthetical spherical patterns and establishes a method to create infinite spherical patterns automatically. This method not only is easy to be implemented, but also can be extended to deal with the cases in the higher dimensional spaces.%等变映射方法在生成艺术图案中具有构造困难，受对称群阶数瓶颈限制等缺点。

借鉴有限反射群不变论的结论，提出不变映射方法生成具有正多面体反射群对称性的球面艺术图案，建立了一种可生成无穷无尽球面图案的自动化方法。

该方法不仅实施容易，且可类似地推广到高维空间中。

【期刊名称】《计算机工程与应用》【年(卷),期】2013(000)023【总页数】4页(P27-30)【关键词】有限反射群;正多面体;不变论;不变映射【作者】王新长;刘满凤;欧阳培昌【作者单位】江西财经大学信息管理学院，南昌 330013; 井冈山大学数理学院，江西吉安 343009;江西财经大学信息管理学院，南昌 330013;井冈山大学数理学院，江西吉安 343009【正文语种】中文【中图分类】TP391利用计算机技术自动生成艺术图案是一个实用的新兴课题，借助迅猛发展精工技艺（如激光喷墨、3D打印等），其研究结果可以广泛地应用到壁纸、瓷砖、包装材料、纺织等与装饰领域有关的行业，生成美观的工艺品，不仅可以满足人们对于美的追求与赏析，而且具有可观的经济价值。

福建省8种蝴蝶新纪录

第14卷第1期漳州师范学院学报（自然科学版）Vol.14 No.1 2001年2月 Journal of Zhangzhou Teachers College (Nat.Sci.) Feb.2001福建省８种蝴蝶新纪录　徐奇涵江凡福建　漳州　３６３０００福建农林大学植保系　摘要本文报道福建省蝴蝶新纪录８种地理分布及采集时间它们的形态特征１粉蝶科　Ｐｉｅｒｉｄａｅ　(1) 红腋斑粉蝶 Delias acalis (Godart)形态特征翅面淡黑色中室端脉白斑呈飞鸟形翅反面色斑同正面臀室斑白黄色缅甸泰国不丹马来西亚云南福建采于漳州市郊采集时间２蛺蝶科　Ｎｙｍｐｈａｌｉｄａｅ　(2) 渡带翠蛺蝶 Euthalia duda Staudinger形态特征翅正面黑褐色中带斑列白色后面3个向外扩散后翅白色横带的外侧有蓝绿色的过渡带区地理分布四川福建采于南靖武夷山1980年6月1998年7月(3)波纹眼蛺蝶 Junonia atlites (Linnaeus)形态特征翅面灰褐色亚外缘有褐色波状线2条前翅中室内有横线4条其中前翅r5室cu 1室内的斑和后翅m 1室外半部黑色地理分布泰国斯里兰卡我国的广东云南四川福建采于漳州市郊2000年11月翅展约28mm但前翅反面外侧带细后翅cu 1室内眼状斑显著地理分布印度尼西亚等福建采于南靖），　男，　福建龙海人，　讲师　漳州师范学院学报(自然科学版) 2001年区的北限2000年7月形态特征与霓纱燕灰蝶 R. nissa相似后翅外横带在M3脉处错位,2aÄÚ²àÒàÏâ×Å°×É«Ï¸Ïß¸£½¨²ÉÓÚÕÄÖÝÊÐ½¼2000年11月翅展约47mmÇ°³á°×É«Ð±ÖÐ´øÓÉ3个斑纹连接而成后翅无斑点地理分布泰国印度尼西亚等福建采于华安采集时间(7) 锷弄蝶属一种 Aeromachus sp.形态特征翅正面黑色有金属光泽雄蝶前翅反面有1列黄白色小斑组成的弧形外横线止于Cu 2脉福建采于南靖2000年10月形态特征近似盒纹孔弄蝶 P. theca (Evans)½üÔ²ÐÎ³Ê»ÒÂÌÉ«ÅÅ³É1行福建采于南靖2000年10月）　翅型(反面). 图 2: Aeromachus sp. () 翅型(左边为反面,右边为正面).参考文献[1]周尧主编. 中国蝶类志[M]. 郑州:河南科学技术出版社 ,1999.[2]周尧. 中国蝴蝶分类与鉴定[M]. 郑州:河南科学技术出版社,1998.[3]顾茂彬, 陈佩珍. 海南岛蝴蝶[M]. 北京:中国林业出版社,1997.[4]李传隆, 朱宝云. 中国蝶类图谱[M]. 上海:上海远东出版社,1992.[5]李传隆主编. 云南蝴蝶[M]. 北京: 中国林业出版社,1995.[6]童雪松主编. 浙江蝶类志[M]. 杭州:浙江科学技术出版社,1993. (下转第17页)第1期邓立虎17¹ØÓÚ¾ßÓÐÆ«²îÂÏÔªµÄË«Çú·½³Ì½âµÄÇ¿ÆÈÕñ¶¯ÐÔ[J]In this paper, by means of averaging technique combing with Green’s formula and Jesen’s inequality to study the oscillatory of Solution of hyperbolic equations with deviating arguments under Dirichlet boundary condition. The sufficient conditions of the oscillatory of Solution are obtained.Key words: Hyperbolic differential equation with deviating arguments; Dirichlet boundary condition; Oscillation.(上接第78页)[7]白水隆. 原色台湾蝶类大图鉴[M]. 东京:保育社,1960.[8]张保信. 台湾蝴蝶鉴定指南[M].台北:渡假出版社有限公司,1982.[9]滨野荣茨. 台湾蝶类生态大图鉴[M]. 台北:牛顿出版社,1986 .[10]赵力,王效岳. 四川省蝴蝶[M].台北:台湾省立博物馆,1996.[11]丁冬荪. 江西省武夷山自然保护区昆虫名录[J]. 江西植保,1997,20 Suppl.:8-27.[12]黄邦侃主编. 福建昆虫志鳞翅目: 蝶类[M].福州:福建科学技术出版社,2001.[13] 赵修复. 福建昆虫名录[M].福州:福建科学技术出版社,1981.[14]王之珉. 福建省蝴蝶新纪录[J]. 武夷科学Zhangzhou Education College, Fujian 363000。

数学期刊名称缩写

Abbreviations of Names of SerialsThis list gives the form of references used in Mathematical Reviews(MR).The abbreviation is followed by the complete title,the place of publication and other pertinent information.∗not previously listed E available electronically §journal reviewed cover-to-cover V videocassette series †monographic series¶bibliographic journal∗Abh.Braunschw.Wiss.Ges.Abhandlungen derBraunschweigischen Wissenschaftlichen Gesellschaft.J.Cramer Verlag,Braunschweig.(Formerly Abh.Braunschweig.Wiss.Ges.)Abh.Braunschweig.Wiss.Ges.Abhandlungen derBraunschweigischen Wissenschaftlichen Gesellschaft.Goltze,G¨o ttingen.(Continued as Abh.Braunschw.Wiss.Ges.)§Abh.Math.Sem.Univ.Hamburg Abhandlungen aus dem Mathematischen Seminar der Universit¨a t Hamburg.Vandenhoeck&Ruprecht,G¨o ttingen.ISSN0025-5858.†Abh.Math.-Naturwiss.Kl.Akad.Wiss.Lit.Mainz Abhandlungen der Mathematisch-NaturwissenschaftlichenKlasse.Akademie der Wissenschaften und der Literaturin Mainz.[Transactions of the Mathematical-ScientiﬁcSection.Academy of Sciences and Literature in Mainz]Steiner,Stuttgart.ISSN0002-2993.§Abstr.Appl.Anal.Abstract and Applied Analysis.Mancorp,Tampa,FL.ISSN1085-3375.¶Abstracts Amer.Math.Soc.Abstracts of Papers Presented to the American Mathematical Society.Amer.Math.Soc.,Providence,RI.ISSN0192-5857.Acad.Roy.Belg.Bull.Cl.Sci.(6)Acad´e mie Royale deBelgique.Bulletin de la Classe des Sciences.6e S´e rie.Acad.Roy.Belgique,Brussels.ISSN0001-4141.Acad.Roy.Belg.Cl.Sci.M´e m.Collect.8o(3)Acad´e mieRoyale de Belgique.Classe des Sciences.M´e moires.Collection in-8o.3e S´e rie.Acad.Roy.Belgique,Brussels.ISSN0365-0936.Acad.Serbe Sci.Arts Glas Acad´e mie Serbe des Scienceset des Arts.Glas.Classe des Sciences Naturelles etMath´e matiques.Srpska Akad.Nauk.i Umetnost.,Belgrade.ISSN0374-7956.†Acc`e s Sci.Acc`e s Sciences.[Access to Sciences]De Boeck Univ.,Brussels.§E ACM J.Exp.Algorithmics The ACM Journal ofExperimental Algorithmics.ACM,New York.ISSN1084-6654.E ACM Trans.Math.Software Association for ComputingMachinery.Transactions on Mathematical Software.ACM,New York.ISSN0098-3500.∗§Acta Acad.Paedagog.Agriensis Sect.Mat.(N.S.)Acta Academiae Paedagogicae Agriensis.Nova Series.SectioMatematicae.Eszterh´a zy K´a roly Coll.,Eger.∗§Acta Anal.Funct.Appl.Acta Analysis Functionalis Applicata.AAFA.Yingyong Fanhanfenxi Xuebao.SciencePress,Beijing.ISSN1009-1327.§E Acta Appl.Math.Acta Applicandae Mathematicae.An International Survey Journal on Applying Mathematics andMathematical Applications.Kluwer Acad.Publ.,Dordrecht.ISSN0167-8019.§Acta Arith.Acta Arithmetica.Polish Acad.Sci.,Warsaw.ISSN0065-1036.Acta Astronom.Sinica Acta Astronomica Sinica.TianwenXuebao.Kexue Chubanshe(Science Press),Beijing.(Translated in Chinese Astronom.Astrophys.)ISSN0001-5245.Acta Astrophys.Sinica Acta Astrophysica Sinica.TiantiWuli Xuebao.Kexue Chubanshe(Science Press),Beijing.(Translated in Chinese Astronom.Astrophys.)ISSN0253-2379.Acta Automat.Sinica Acta Automatica Sinica.ZidonghuaXuebao.Kexue Chubanshe(Science Press),Beijing.ISSN0254-4156.Acta Cienc.Indica Math.Acta Ciencia Indica.Mathematics.Pragati Prakashan,Meerut.ISSN0970-0455.Acta Cient.Venezolana Acta Cient´ıﬁca Venezolana.Asociaci´o n Venezolana para el Avance de la Ciencia.Asoc.Venezolana Avance Cien.,Caracas.ISSN0001-5504.Acta Comment.Univ.Tartu.Math.Acta etCommentationes Universitatis Tartuensis de Mathematica.Univ.Tartu,Fac.Math.,Tartu.ISSN1406-2283.E Acta Cryst.Sect.A Acta Crystallographica.Section A:Foundations of Crystallography.Munksgaard,Copenhagen.ISSN0108-7673.§Acta Cybernet.Acta Cybernetica.J´o zsef Attila Univ.Szeged,Szeged.ISSN0324-721X.Acta Hist.Leopold.Acta Historica Leopoldina.DeutscheAkad.Naturforscher Leopoldina,Halle an der Saale.ISSN0001-5857.§E Acta Inform.Acta Informatica.Springer,Heidelberg.ISSN0001-5903.§Acta Math.Acta Mathematica.Inst.Mittag-Lefﬂer, Djursholm.ISSN0001-5962.§E Acta Math.Acad.Paedagog.Nyh´a zi.(N.S.)Acta Mathematica.Academiae Paedagogicae Ny´ıregyh´a ziensis.New Series.Bessenyei Gy¨o rgy Coll.,Ny´ıregyh´a za.ISSN0866-0182.§Acta Math.Appl.Sinica Acta Mathematicae Applicatae Sinica.Yingyong Shuxue Xuebao.Kexue Chubanshe(Science Press),Beijing.ISSN0254-3079.§Acta Math.Appl.Sinica(English Ser.)Acta Mathematicae Applicatae Sinica.English Series.Yingyong ShuxueXuebao.Science Press,Beijing.ISSN0168-9673.§E Acta Math.Hungar.Acta Mathematica Hungarica.Akad.Kiad´o,Budapest.ISSN0236-5294.§Acta rm.Univ.Ostraviensis Acta Mathematica et Informatica Universitatis Ostraviensis.Univ.Ostrava,Ostrava.ISSN1211-4774.§Acta Math.Sci.(Chinese)Acta Mathematica Scientia.Series A.Shuxue Wuli Xuebao.Chinese Edition.KexueChubanshe(Science Press),Beijing.(See also Acta Math.Sci.(English Ed.))ISSN1003-3998.§Acta Math.Sci.(English Ed.)Acta Mathematica Scientia.Series B.English Edition.Shuxue Wuli Xuebao.SciencePress,Beijing.(See also Acta Math.Sci.(Chinese))ISSN0252-9602.§E Acta Math.Sin.(Engl.Ser.)Acta Mathematica Sinica.English Series.Springer,Heidelberg.ISSN1000-9574.§Acta Math.Sinica Acta Mathematica Sinica.Chinese Math.Soc.,Acta Math.Sinica m.,Beijing.ISSN0583-1431.§E Acta enian.(N.S.)Acta Mathematica Universitatis Comenianae.New enius Univ.Press,Bratislava.ISSN0862-9544.§Acta Math.Vietnam.Acta Mathematica Vietnamica.Nat.Center Natur.Sci.Tech.,Hanoi.ISSN0251-4184.∗Acta Mech.Sin.Engl.Ser.Acta Mechanica Sinica.English Series.The Chinese Society of Theoretical and AppliedMechanics.Chinese J.Mech.Press,Beijing.(FormerlyActa Mech.Sinica(English Ed.))ISSN0567-7718.Acta Mech.Sinica(Beijing)Acta Mechanica Sinica.LixueXuebao.Chinese J.Mech.Press,Beijing.(See also ActaMech.Sinica(English Ed.))ISSN0459-1879.Acta Mech.Sinica(English Ed.)Acta Mechanica Sinica.English Edition.Lixue Xuebao.Kexue Chubanshe(SciencePress),Beijing.(Continued as Acta Mech.Sin.Engl.Ser.)(See also Acta Mech.Sinica(Beijing))ISSN0567-7718.∗Acta Mech.Solida Sin.Acta Mechanica Solida Sinica.Chinese Journal of Solid Mechanics.Huazhong Univ.Sci.Tech.,Wuhan.ISSN0894-9166.†Acta Numer.Acta Numerica.Cambridge Univ.Press, Cambridge.ISSN0962-4929.Acta Phys.Polon.B Jagellonian University.Institute ofPhysics and Polish Physical Society.Acta Physica PolonicaB.Jagellonian Univ.,Krak´o w.ISSN0587-4254.Acta Phys.Sinica Acta Physica Sinica.Wuli Xuebao.Chinese Phys.Soc.,Beijing.ISSN1000-3290.Acta put.Manage.Eng.Ser.Acta Polytechnica Scandinavica.Mathematics,Computingand Management in Engineering Series.Finn.Acad.Tech.,Espoo.ISSN1238-9803.§Acta Sci.Math.(Szeged)Acta Universitatis Szegediensis.Acta Scientiarum Mathematicarum.Univ.Szeged,Szeged.ISSN0001-6969.Acta Sci.Natur.Univ.Jilin.Acta Scientiarum NaturaliumUniversitatis Jilinensis.Jilin Daxue.Ziran Kexue Xuebao.Jilin University.Natural Sciences Journal.Jilin Univ.Nat.Sci.J.,Editor.Dept.,Changchun.ISSN0529-0279.Acta Sci.Natur.Univ.Norm.Hunan.Acta ScientiarumNaturalium Universitatis Normalis Hunanensis.HunanShifan Daxue Ziran Kexue Xuebao.J.Hunan Norm.Univ.,Editor.Dept.,Changsha.ISSN1000-2537.Acta Sci.Natur.Univ.Pekinensis See Beijing DaxueXuebao Ziran Kexue BanActa Sci.Natur.Univ.Sunyatseni Acta ScientiarumNaturalium Universitatis Sunyatseni.Zhongshan DaxueXuebao.Ziran Kexue Ban.Journal of Sun Yatsen University.Natural Sciences.J.Zhongshan Univ.,Editor.Dept.,Guangzhou.ISSN0529-6579.Acta Tech.CSA V Acta Technica CSA V.Acad.Sci.CzechRepub.,Prague.ISSN0001-7043.§Acta Univ.Carolin.Math.Phys.Acta Universitatis Carolinae.Mathematica et Physica.Karolinum,Prague.ISSN0001-7140.§Acta Univ.Lodz.Folia Math.Acta UniversitatisLodziensis.Folia Mathematica.Wydawn.Uniw.Ł´o dzkiego,Ł´o d´z.ISSN0208-6204.Acta Univ.Lodz.Folia Philos.Acta UniversitatisLodziensis.Folia Philosophica.Wydawn.Uniw.Ł´o dzkiego,Ł´o d´z.ISSN0208-6107.∗§Acta Univ.M.Belii Ser.Math.Acta Universitatis Matthiae Belii.Natural Science Series.Series Mathematics.MatejBel Univ.,Bansk´a Bystrica.(Formerly Acta Univ.MathaeiBelii Nat.Sci.Ser.Ser.Math.)§Acta Univ.Mathaei Belii Nat.Sci.Ser.Ser.Math.Matej Bel University.Acta.Natural Science Series.Series Mathematics.Matej Bel Univ.,Bansk´a Bystrica.(Continued as Acta Univ.M.Belii Ser.Math.)Acta Univ.Oulu.Ser.A Sci.Rerum Natur.ActaUniversitatis Ouluensis.Series A.Scientiae RerumNaturalium.Univ.Oulu,Oulu.ISSN0355-3191.§Acta Univ.Palack.Olomuc.Fac.Rerum Natur.Math.Acta Universitatis Palackianae Olomucensis.Facultas Rerum Naturalium.Mathematica.ISSN0231-9721.†Acta Univ.Ups.Stud.Philos.Ups.Acta Universitatis Upsaliensis.Studia Philosophica Upsaliensia.Uppsala Univ., Uppsala.ISSN0585-5497.†Acta Univ.Upsaliensis Skr.Uppsala Univ.C Organ.Hist.Acta Universitatis Upsaliensis.Skrifter r¨o randeUppsala anisation och Historia.[ActaUniversitatis Upsaliensis.Publications concerning Uppsalaanization and History]Uppsala Univ.,Uppsala.ISSN0502-7454.†Actualit´e s Math.Actualit´e s Math´e matiques.[Current Mathematical Topics]Hermann,Paris.†Actualit´e s Sci.Indust.Actualit´e s Scientiﬁques etIndustrielles.[Current Scientiﬁc and Industrial Topics]Hermann,Paris.†Adapt.Learn.Syst.Signal mun.Control Adaptive and Learning Systems for Signal Processing,Communications,and Control.Wiley,New York.Adv.Appl.Clifford Algebras Advances in Applied Clifford Algebras.Univ.Nac.Aut´o noma M´e xico,M´e xico.ISSN0188-7009.†Adv.Appl.Mech.Advances in Applied Mechanics.Academic Press,Boston,MA.ISSN0065-2165.∗†Adv.Astron.Astrophys.Advances in Astronomy andAstrophysics.Gordon and Breach,Amsterdam.ISSN1025-8206.†Adv.Book Class.Advanced Book Classics.Perseus, Reading,MA.†Adv.Bound.Elem.Ser.Advances in Boundary Elements put.Mech.,Southampton.(Continued as Int.Ser.Adv.Bound.Elem.)ISSN1368-258X.†Adv.Chem.Phys.Advances in Chemical Physics.Wiley, New York.†put.Econom.Advances in Computational Economics.Kluwer Acad.Publ.,Dordrecht.§E put.Math.Advances in ComputationalMathematics.Baltzer,Bussum.ISSN1019-7168.†put.Sci.Advances in Computing Science.Springer,Vienna.ISSN1433-0113.∗†Adv.Des.Control Advances in Design and Control.SIAM, Philadelphia,PA.§Adv.Differential Equations Advances in Differential Equations.Khayyam,Athens,OH.ISSN1079-9389.†Adv.Discrete Math.Appl.Advances in DiscreteMathematics and Applications.Gordon and Breach,Amsterdam.ISSN1028-3129.†Adv.Fluid Mech.Advances in Fluid put.Mech.,Southampton.ISSN1353-808X.†Adv.Fuzzy Systems Appl.Theory Advances in Fuzzy Systems—Applications and Theory.World Sci.Publishing,River Edge,NJ.§E Adv.in Appl.Math.Advances in Applied Mathematics.Academic Press,Orlando,FL.ISSN0196-8858.§Adv.in Appl.Probab.Advances in Applied Probability.Appl.Probab.Trust,Shefﬁeld.ISSN0001-8678.∗†Adv.Ind.Control Advances in Industrial Control.Springer, London.†Adv.Lectures Math.Advanced Lectures in Mathematics.Vieweg,Braunschweig.ISSN0932-7134.§E Adv.Math.Advances in Mathematics.Academic Press, Orlando,FL.ISSN0001-8708.§Adv.Math.(China)Advances in Mathematics(China).Shuxue Jinzhan.Peking Univ.Press,Beijing.ISSN1000-0917.†Adv.Math.Econ.Advances in Mathematical Economics.Springer,Tokyo.†Adv.Math.Sci.Advances in the Mathematical Sciences.Amer.Math.Soc.,Providence,RI.§Adv.Math.Sci.Appl.Advances in Mathematical Sciences and Applications.An International Journal.Gakk¯o tosho,Tokyo.ISSN1343-4373.§Adv.Nonlinear Var.Inequal.Advances in Nonlinear Variational Inequalities.An International Journal.Internat.Publ.,Orlando,FL.ISSN1092-910X.†Adv.Numer.Math.Advances in Numerical Mathematics.Teubner,Stuttgart.†Adv.Partial Differ.Equ.Advances in Partial Differential Equations.Wiley-VCH,Berlin.†Adv.Partial Differential Equations Advances in Partial Differential Equations.Akademie Verlag,Berlin.†Adv.Ser.Dynam.Systems Advanced Series in Dynamical Systems.World Sci.Publishing,River Edge,NJ.†Adv.Ser.Math.Phys.Advanced Series in Mathematical Physics.World Sci.Publishing,River Edge,NJ.†Adv.Ser.Math.Sci.Eng.Advanced Series in Mathematical Science and Engineering.World Fed.Publ.,Atlanta,GA.†Adv.Ser.Neurosci.Advanced Series in Neuroscience.World Sci.Publishing,River Edge,NJ.†Adv.Ser.Nonlinear Dynam.Advanced Series in Nonlinear Dynamics.World Sci.Publishing,River Edge,NJ.†Adv.Ser.Stat.Sci.Appl.Probab.Advanced Series on Statistical Science&Applied Probability.World Sci.Publishing,River Edge,NJ.†Adv.Ser.Theoret.Phys.Sci.Advanced Series onTheoretical Physical Science.World Sci.Publishing,RiverEdge,NJ.∗†Adv.Soft Comput.Advances in Soft Computing.Physica, Heidelberg.†Adv.Spat.Sci.Advances in Spatial Science.Springer, Berlin.†Adv.Stud.Contemp.Math.Advanced Studies inContemporary Mathematics.Gordon and Breach,New York.∗§Adv.Stud.Contemp.Math.(Pusan)Advanced Studies in Contemporary Mathematics(Pusan).Adv.Stud.Contemp.Math.,m.,Saga.ISSN1229-3067.†Adv.Stud.Pure Math.Advanced Studies in PureMathematics.Kinokuniya,Tokyo.†Adv.Textb.Control Signal Process.Advanced Textbooks in Control and Signal Processing.Springer,London.∗†Adv.Texts Phys.Advanced Texts in Physics.Springer,Berlin.ISSN1439-2674.§E Adv.Theor.Math.Phys.Advances in Theoretical and Mathematical Physics.Internat.Press,Cambridge,MA.ISSN1095-0761.†Adv.Theory put.Math.Advances in the Theory of Computation and Computational Mathematics.Nova Sci.Publ.,Commack,NY.†Adv.Top.Math.Advanced Topics in Mathematics.PWN, Warsaw.§E Aequationes Math.Aequationes Mathematicae.Birkh¨a user,Basel.ISSN0001-9054.§Afrika Mat.(3)Afrika Matematika.Journal of the African Mathematical Union.Journal de l’Union Math´e matiqueAfricaine.S´e rie3.Union Math.Africaine,Caluire.†Agr´e g.Math.Agr´e gation de Math´e matiques.Masson,Paris.AI Commun.AI Communications.The European Journal onArtiﬁcial Intelligence.IOS,Amsterdam.ISSN0921-7126.†AIAA Ed.Ser.AIAA Education Series.AIAA,Washington, DC.†AIP Conf.Proc.AIP Conference Proceedings.Amer.Inst.Phys.,New York.ISSN0094-243X.†AIP Ser.Modern Acoust.Signal Process.AIP Series in Modern Acoustics and Signal Processing.Amer.Inst.Phys.,New York.†AKP Class.AKP Classics.A K Peters,Wellesley,MA.∗†Al-Furq¯a n Islam.Herit.Found.Publ.Al-Furq¯a n Islamic Heritage Foundation Publication.Al-Furq¯a n Islam.Herit.Found.,London.†Albion Math.Appl.Ser.Albion Mathematics&Applications Series.Albion,Chichester.§E Algebr.Represent.Theory Algebras and Representation Theory.Kluwer Acad.Publ.,Dordrecht.ISSN1386-923X.Algebra and Logic Algebra and Logic.Consultants Bureau,New York.(Translation of Algebra Log.and Algebra iLogika)ISSN0002-5232.†Algebra Ber.Algebra Berichte.[Algebra Reports]Fischer, Munich.ISSN0942-1270.§E Algebra Colloq.Algebra Colloquium.Springer,Singapore.ISSN1005-3867.§Algebra i Analiz Rossi˘ıskaya Akademiya Nauk.Algebrai Analiz.“Nauka”S.-Peterburg.Otdel.,St.Petersburg.(Translated in St.Petersburg Math.J.)ISSN0234-0852.§Algebra i Logika Sibirski˘ıFond Algebry i Logiki.Algebrai Logika.Izdat.NII Mat.-Inform.Osnov Obuch.NGU,Novosibirsk.(Continued as Algebra Log.)(Translatedin Algebra and Logic)ISSN0373-9252.∗§Algebra Log.Algebra i Logika.Institut Diskretno˘ıMatematiki i Informatiki.Sib.Fond Algebry Log.,Novosibirsk.(Formerly Algebra i Logika)(Translatedin Algebra and Logic)ISSN0373-9252.†Algebra Logic Appl.Algebra,Logic and Applications.Gordon and Breach,Amsterdam.ISSN1041-5394.§E Algebra Universalis Algebra Universalis.Univ.Manitoba, Winnipeg,MB.ISSN0002-5240.§Algebras Groups Geom.Algebras,Groups and Geometries.Hadronic Press,Palm Harbor,FL.ISSN0741-9937.§E Algorithmica Algorithmica.An International Journal in Computer Science.Springer,New York.ISSN0178-4617.†Algorithms Combin.Algorithms and Combinatorics.Springer,Berlin.ISSN0937-5511.†Algorithms Comput.Math.Algorithms and Computation in Mathematics.Springer,Berlin.ISSN1431-1550.§Aligarh Bull.Math.The Aligarh Bulletin of Mathematics.Aligarh Muslim Univ.,Aligarh.Aligarh J.Statist.The Aligarh Journal of Statistics.AligarhMuslim Univ.,Aligarh.ISSN0971-0388.§pok Alkalmazott Matematikai Lapok.Magyar Tudom´a nyos Akad.,Budapest.ISSN0133-3399.∗Allg.Stat.Arch.Allgemeines Statistisches Archiv.AStA.Journal of the German Statistical Society.Physica,Heidelberg.ISSN0002-6018.†´Alxebra´Alxebra.[Algebra]Univ.Santiago deCompostela,Santiago de Compostela.†Am.Univ.Stud.Ser.IX Hist.American University Studies.Series IX:ng,New York.ISSN0740-0462.∗§E AMA Algebra Montp.Announc.AMA.AlgebraMontpellier Announcements.AMA Algebra Montp.Announc.,Montpellier.§E Amer.J.Math.American Journal of Mathematics.Johns Hopkins Univ.Press,Baltimore,MD.ISSN0002-9327.Amer.J.Math.Management Sci.American Journal ofMathematical and Management Sciences.Amer.Sci.Press,Syracuse,NY.ISSN0196-6324.E Amer.J.Phys.American Journal of Physics.Amer.Assoc.Phys.Teach.,College Park,MD.ISSN0002-9505.Amer.Math.Monthly The American MathematicalMonthly.Math.Assoc.America,Washington,DC.ISSN0002-9890.†Amer.Math.Soc.Colloq.Publ.American Mathematical Society Colloquium Publications.Amer.Math.Soc.,Providence,RI.ISSN0065-9258.†Amer.Math.Soc.Transl.Ser.2American Mathematical Society Translations,Series2.Amer.Math.Soc.,Providence,RI.(Selected translations of Russian language publications Tr.St.-Peterbg.Mat.Obshch.)ISSN0065-9290.E Amer.Statist.The American Statistician.Amer.Statist.Assoc.,Alexandria,V A.ISSN0003-1305.†Amer.Univ.Stud.Ser.V Philos.American University Studies.Series V:ng,New York.ISSN0739-6392.†AMS Progr.Math.Lecture Ser.AMS Progress in Mathematics Lecture Series.Amer.Math.Soc.,Providence,RI.†AMS Short Course Lecture Notes AMS Short Course Lecture Notes.Amer.Math.Soc.,Providence,RI.†AMS-MAA Joint Lecture Ser.AMS-MAA Joint Lecture Series.Amer.Math.Soc.,Providence,RI.†AMS/IP Stud.Adv.Math.AMS/IP Studies in Advanced Mathematics.Amer.Math.Soc.,Providence,RI.ISSN1089-3288.E An.Acad.Brasil.Ciˆe nc.Anais da Academia Brasileira deCiˆe ncias.Acad.Brasil.Ciˆe nc.,Rio de Janeiro.ISSN0001-3765.†An.F´ıs.Monogr.Anales de F´ısica.Monograf´ıas.[Annals of Physics.Monographs]CIEMAT,Madrid.§An.S¸tiint¸.Univ.Al.I.Cuza Ias¸i Inform.(N.S.)Analele S¸tiint¸iﬁce ale Universit˘a t¸ii“Al.I.Cuza”din Ias¸i.Informatic˘a.Serie Nou˘a.Ed.Univ.“Al.I.Cuza”,Ias¸i.ISSN1224-2268.§An.S¸tiint¸.Univ.Al.I.Cuza Ias¸i.Mat.(N.S.)Analele S¸tiint¸iﬁce ale Universit˘a tii“Al.I.Cuza”din Ias¸i.SerieNou˘a.Matematic˘a.Univ.Al.I.Cuza,Ias¸i.ISSN1221-8421.§An.S¸tiint¸.Univ.Ovidius Constant¸a Ser.Mat.Universit˘a t¸ii “Ovidius”Constant¸a.Analele S¸tiint¸iﬁce.Seria Matematic˘a.“Ovidius”Univ.Press,Constant¸a.ISSN1223-723X.§An.Univ.Bucures¸ti Mat.Analele Universit˘a t¸ii Bucures¸ti.Matematic˘a.Univ.Bucharest,Bucharest.ISSN1013-4123.An.Univ.Craiova rm.Analele Universitˇa t¸iidin Craiova.Seria Matematic˘a-Informatic˘a.Univ.Craiova,Craiova.ISSN1223-6934.∗§An.Univ.Oradea Fasc.Mat.Analele Universit˘a t¸ii din Oradea.Fascicola Matematica.Univ.Oradea,Oradea.ISSN1221-1265.§An.Univ.Timis¸oara Ser.Mat.-Inform.Universit˘a t¸ii din Timis¸oara.Analele.Seria Matematic˘a-Informatic˘a.Univ.Timis¸oara,Timis¸oara.ISSN1224-970X.An.Univ.Timis¸oara Ser.S¸tiint¸.Fiz.Analele Universit˘a t¸iidin Timis¸oara.Seria S¸tiint¸e Fizice.Univ.Vest Timis¸oara,Timis¸oara.†Anal.Appl.Analysis and its Applications.IOS,Amsterdam.ISSN1345-4240.§E Anal.Math.Analysis Mathematica.Akad.Kiad´o,Budapest.ISSN0133-3852.†Anal.Methods Spec.Funct.Analytical Methods and Special Functions.Gordon and Breach,Amsterdam.ISSN1027-0264.†Anal.Modern.Apl.Analiz˘a Modern˘a s¸i Aplicat¸ii.[Modern Analysis and Applications]Ed.Acad.Romˆa ne,Bucharest.§Analysis(Munich)Analysis.International Mathematical Journal of Analysis and its Applications.Oldenbourg,Munich.ISSN0174-4747.E Analysis(Oxford)Analysis.Blackwell,Oxford.ISSN0003-2638.†Angew.Statist.¨Okonom.Angewandte Statistik und¨Okonometrie.[Applied Statistics and Econometrics]Vandenhoeck&Ruprecht,G¨o ttingen.§E Ann.Acad.Sci.Fenn.Math.Annales AcademiæScientiarium Fennicæ.Mathematica.Acad.Sci.Fennica,Helsinki.ISSN1239-629X.§Ann.Acad.Sci.Fenn.Math.Diss.AcademiæScientiarum Fennicæ.Annales.Mathematica.Dissertationes.Acad.Sci.Fennica,Helsinki.ISSN1239-6303.§E Ann.Appl.Probab.The Annals of Applied Probability.Inst.Math.Statist.,Hayward,CA.ISSN1050-5164.§E b.Annals of Combinatorics.Springer,Singapore.(Continued as b.)ISSN0218-0006.∗§E b.Annals of Combinatorics.Birkh¨a user,Basel.(Formerly b.)ISSN0218-0006.§Ann.Differential Equations Annals of DifferentialEquations.Weifen Fangcheng Niankan.Fuzhou Univ.,Fuzhou.ISSN1002-0942.†Ann.Discrete Math.Annals of Discrete Mathematics.North-Holland,Amsterdam.Ann.´Econom.Statist.Annales d’´Economie et deStatistique.Inst.Nat.Statist.´Etud.´Econom.,Amiens.ISSN0769-489X.§Ann.Fac.Sci.Toulouse Math.(6)Annales de la Facult´e des Sciences de Toulouse.Math´e matiques.S´e rie6.Univ.Paul Sabatier,Toulouse.ISSN0240-2963.†Ann.Fac.Sci.Univ.Kinshasa Annales de la Facult´e des Sciences.Universit´e de Kinshasa.[Annals of the Facultyof Science.University of Kinshasa]Presses Univ.Kinshasa,Kinshasa.Ann.Fond.Louis de Broglie Fondation Louis de Broglie.Annales.Fond.Louis de Broglie,Paris.ISSN0182-4295.§E Ann.Global Anal.Geom.Annals of Global Analysis and Geometry.Kluwer Acad.Publ.,Dordrecht.ISSN0232-704X.∗E Ann.Henri Poincar´e Annales Henri Poincar´e.A Journal of Theoretical and Mathematical Physics.Birkh¨a user,Basel.(Merged from Ann.Inst.H.Poincar´e Phys.Th´e or.and Helv.Phys.Acta)ISSN1424-0637.∗Ann.I.S.U.P.Annales de l’I.S.U.P..Univ.Paris,Inst.Stat., Paris.§E Ann.Inst.Fourier(Grenoble)Universit´e de Grenoble.Annales de l’Institut Fourier.Univ.Grenoble I,Saint-Martin-d’H`e res.ISSN0373-0956.§E Ann.Inst.H.Poincar´e Anal.Non Lin´e aire Annales de l’Institut Henri Poincar´e.Analyse Non Lin´e aire.Gauthier-Villars,´Ed.Sci.M´e d.Elsevier,Paris.ISSN0294-1449.Ann.Inst.H.Poincar´e Phys.Th´e or.Annales de l’InstitutHenri Poincar´e.Physique Th´e orique.Gauthier-Villars,´Ed.Sci.M´e d.Elsevier,Paris.(Merged into Ann.HenriPoincar´e)ISSN0246-0211.§E Ann.Inst.H.Poincar´e Probab.Statist.Annales de l’Institut Henri Poincar´e.Probabilit´e s et Statistiques.Gauthier-Villars,´Ed.Sci.M´e d.Elsevier,Paris.ISSN0246-0203.E Ann.Inst.Statist.Math.Annals of the Institute ofStatistical Mathematics.Kluwer Acad.Publ.,Norwell,MA.ISSN0020-3157.†Ann.Internat.Soc.Dynam.Games Annals of the International Society of Dynamic Games.Birkh¨a userBoston,Boston,MA.†Ann.Israel Phys.Soc.Annals of the Israel Physical Society.IOP,Bristol.ISSN0309-8710.Ann.Japan Assoc.Philos.Sci.Annals of the JapanAssociation for Philosophy of Science.Japan Assoc.Philos.Sci.,Tokyo.ISSN0453-0691.§Ann.Mat.Pura Appl.(4)Annali di Matematica Pura ed Applicata.Serie Quarta.Zanichelli,Bologna.ISSN0003-4622.E Ann.Math.Artiﬁcial Intelligence Annals of Mathematicsand Artiﬁcial Intelligence.Baltzer,Bussum.ISSN1012-2443.§Ann.Math.Blaise Pascal Annales Math´e matiques Blaise Pascal.Univ.Blaise Pascal,Lab.Math.Pures Appl.,Aubi`e re.ISSN1259-1734.§Ann.Math.Sil.Annales Mathematicae Silesianae.Wydawn.Uniw.´Sl‘askiego,Katowice.(See also Pr.Nauk.Uniw.´Sl.Katow.)ISSN0860-2107.†Ann.New York Acad.Sci.Annals of the New York Academy of Sciences.New York Acad.Sci.,New York.ISSN0077-8923.§E Ann.of Math.(2)Annals of Mathematics.Second Series.Princeton Univ.Press,Princeton,NJ.ISSN0003-486X.†Ann.of Math.Stud.Annals of Mathematics Studies.Princeton Univ.Press,Princeton,NJ.E Ann.of Sci.Annals of Science.Taylor&Francis,London.ISSN0003-3790.E Ann.Oper.Res.Annals of Operations Research.Baltzer,Bussum.ISSN0254-5330.E Ann.Phys.(8)Annalen der Physik(8).Wiley-VCH,Berlin.ISSN0003-3804.E Ann.Physics Annals of Physics.Academic Press,Orlando,FL.ISSN0003-4916.§Ann.Polon.Math.Annales Polonici Mathematici.Polish Acad.Sci.,Warsaw.ISSN0066-2216.§Ann.Probab.The Annals of Probability.Inst.Math.Statist., Bethesda,MD.ISSN0091-1798.§E Ann.Pure Appl.Logic Annals of Pure and Applied Logic.North-Holland,Amsterdam.ISSN0168-0072.§E Ann.Sci.´Ecole Norm.Sup.(4)Annales Scientiﬁques de l’´Ecole Normale Sup´e rieure.Quatri`e me S´e rie.Gauthier-Villars,´Ed.Sci.M´e d.Elsevier,Paris.ISSN0012-9593.§Ann.Sci.Math.Qu´e bec Annales des SciencesMath´e matiques du Qu´e bec.Groupe Cherch.Sci.Math.,Montreal,QC.ISSN0707-9109.§Ann.Scuola Norm.Sup.Pisa Cl.Sci.(4)Annali della Scuola Normale Superiore di Pisa.Classe di Scienze.SerieIV.Scuola Norm.Sup.,Pisa.§Ann.Statist.The Annals of Statistics.Inst.Math.Statist., Hayward,CA.ISSN0090-5364.§Ann.Univ.Ferrara Sez.VII(N.S.)Annali dell’Universit`a di Ferrara.Nuova Serie.Sezione VII.Scienze Matematiche.Univ.Ferrara,Ferrara.§Ann.Univ.Mariae Curie-Skłodowska Sect.A Annales Universitatis Mariae Curie-Skłodowska.Sectio A.Mathematica.Uniw.Marii Curie-Skłodowskiej,Lublin.ISSN0365-1029.Ann.Univ.Sarav.Ser.Math.Annales UniversitatisSaraviensis.Series Mathematicae.Univ.Saarlandes,Saarbr¨u cken.ISSN0933-8268.§Ann.Univ.Sci.Budapest.E¨o tv¨o s Sect.Math.Annales Universitatis Scientiarum Budapestinensis de RolandoE¨o tv¨o s Nominatae.Sectio Mathematica.E¨o tv¨o s Lor´a ndUniv.,Budapest.ISSN0524-9007.§put.AnnalesUniversitatis Scientiarum Budapestinensis de RolandoE¨o tv¨o s Nominatae.Sectio Computatorica.E¨o tv¨o s Lor´a ndUniv.,Budapest.ISSN0138-9491.†Annu.Rev.Fluid Mech.Annual Review of FluidMechanics.Annual Reviews,Palo Alto,CA.ISSN0066-4189.Annuaire Univ.Soﬁa rm.Godishnik naSoﬁ˘ıskiya Universitet“Sv.Kliment Okhridski”.Fakultetpo Matematika i Informatika.Annuaire de l’Universit´e deSoﬁa“St.Kliment Ohridski”.Facult´e de Math´e matiques etInformatique.Presses Univ.“St.Kliment Ohridski”,Soﬁa.ISSN0205-0808.Annuaire Univ.Soﬁa Fac.Phys.Godishnik na Soﬁ˘ıskiyaUniversitet“Sv.Kliment Okhridski”.Fizicheski Fakultet.Annuaire de l’Universit´e de Soﬁa“St.Kliment Ohridski”.Facult´e de Physique.Presses Univ.“St.Kliment Ohridski”,Soﬁa.ISSN0584-0279.†Anu.Filol.Univ.Barc.Anuari de Filologia(Universitat de Barcelona).[Philology Yearbook(University ofBarcelona)]Univ.Barcelona,Barcelona.†Anwend.orientier.Stat.Anwendungsorientierte Statistik.[Applications-Oriented Statistics]Lang,Frankfurt am Main.ISSN1431-7982.Anz.¨Osterreich.Akad.Wiss.Math.-Natur.Kl.¨Osterreichische Akademie der Wissenschaften.Mathematisch-Naturwissenschaftliche Klasse.Anzeiger.¨Osterreich.Akad.Wissensch.,Vienna.∗§E ANZIAM J.The ANZIAM Journal.The Australian& New Zealand Industrial and Applied Mathematics Journal.Austral.Math.Soc.,Canberra.(Formerly J.Austral.Math.Soc.Ser.B)ISSN0334-2700.†Aportaciones Mat.Aportaciones Matem´a ticas.[Mathematical Contributions]Soc.Mat.Mexicana,M´e xico.†Aportaciones un.Aportaciones Matem´a ticas: Comunicaciones.[Mathematical Contributions:Communications]Soc.Mat.Mexicana,M´e xico.∗†Aportaciones Mat.Invest.Aportaciones Matem´a ticas: Investigaci´o n.[Mathematical Contributions:Research]Soc.Mat.Mexicana,M´e xico.(Formerly Aportaciones Mat.Notas Investigaci´o n)†Aportaciones Mat.Notas Investigaci´o n Aportaciones Matem´a ticas:Notas de Investigaci´o n.[MathematicalContributions:Research Notes]Soc.Mat.Mexicana,M´e xico.(Continued as Aportaciones Mat.Invest.)†Aportaciones Mat.Textos Aportaciones Matem´a ticas: Textos.[Mathematical Contributions:Texts]Soc.Mat.Mexicana,M´e xico.§E Appl.Algebra put.Applicable Algebra in Engineering,Communication and Computing.Springer,Heidelberg.ISSN0938-1279.§E Appl.Anal.Applicable Analysis.An International Journal.Gordon and Breach,Yverdon.ISSN0003-6811.§E Appl.Categ.Structures Applied Categorical Structures.A Journal Devoted to Applications of Categorical Methods inAlgebra,Analysis,Order,Topology and Computer Science.Kluwer Acad.Publ.,Dordrecht.ISSN0927-2852.†put.Control Signals Circuits Applied and Computational Control,Signals,and Circuits.Birkh¨a userBoston,Boston,MA.ISSN1522-8363.§E put.Harmon.Anal.Applied and Computational Harmonic Analysis.Time-Frequency and Time-Scale Analysis,Wavelets,Numerical Algorithms,andApplications.Academic Press,Orlando,FL.ISSN1063-5203.†Appl.Log.Ser.Applied Logic Series.Kluwer Acad.Publ., Dordrecht.†Appl.Math.Applications of Mathematics.Springer,New York.ISSN0172-4568.§Appl.Math.Applications of Mathematics.Acad.Sci.Czech Repub.,Prague.ISSN0862-7940.∗†Appl.Math.Applied Mathematics.Chapman&Hall/CRC, Boca Raton,FL.(Formerly put.)§E put.Applied Mathematics andComputation.North-Holland,New York.ISSN0096-3003.†Appl.Math.Engrg.Sci.Texts Applied Mathematics and Engineering Science Texts.CRC,Boca Raton,FL.∗rm.Applied Mathematics and Informatics.Tbilisi Univ.Press,Tbilisi.ISSN1512-0074.∗§Appl.Math.J.Chinese Univ.Ser.A Applied Mathematics.A Journal of Chinese Universities.Series A.Appl.Math.J.Chinese Univ.,m.,Hangzhou.(FormerlyGaoxiao Yingyong Shuxue Xuebao Ser.A)(See also Appl.Math.J.Chinese Univ.Ser.B)ISSN1000-4424.§Appl.Math.J.Chinese Univ.Ser.B Applied Mathematics.A Journal of Chinese Universities.Ser.B.Appl.Math.J.Chinese Univ.,m.,Hangzhou.(See also Appl.Math.J.Chinese Univ.Ser.A and Gaoxiao Yingyong Shuxue。

GAME DATA DISPLAY DEVICE

专利名称：GAME DATA DISPLAY DEVICE发明人：KATASE HIROYUKI,片瀬宏之,HASE SATORU,長谷哲,MUKOYAMA KOJI,向山幸治申请号：JP2004108931申请日：20040401公开号：JP2005287885A公开日：20051020专利内容由知识产权出版社提供专利附图：摘要：PROBLEM TO BE SOLVED: To provide a game data display device capable of informing a player of the number of increased/reduced stocks without causing operation loweringof a specific game machine.SOLUTION: A main control part 401 of a slot machine 4 controls the machine to stock bonuses obtained by internal winning in a RAM 413 and to generate the stocked bonuses when a prescribed stock release condition is satisfied. The CPU 311 of an each-stand information display terminal 3 calculates the number of increased/reduced stocks in a period from bonus occurrence of last time to that of this time, and the display of an inter-bonus game number history/increased and reduced stock number display area 31b is switched to increased/reduced stock number display to display "the number of increased/reduced stocks".COPYRIGHT: (C)2006,JPO&NCIPI申请人：DAIKOKU DENKI CO LTD,ダイコク電機株式会社地址：愛知県名古屋市中村区那古野１丁目４７番１号名古屋国際センタービル２階国籍：JP代理人：菅原正倫更多信息请下载全文后查看。

(完整word版)Hiroshima课文

Hiroshima —--- The ”Liveliest" City in Japan（excerpts)by Jacques Danvoir “Hiroshima！Everybody off!” That must be what the man in the Japanese stationmaster's uniform shouted, as the fastest train in the world slipped to a stop in Hiroshima Station。

I did not understand what he was saying. First of all，because he was shouting in Japanese。

And secondly，because I had a lump in my throat and a lot of sad thoughts on my mind that had little to do with anything a Nippon railways official might say. The very act of stepping on this soil, in breathing this air of Hiroshima，was for me a far greater adventure than any trip or any reportorial assignment I'd previously taken。

Was I not at the scene of the crime？The Japanese crowd did not appear to have the same preoccupations that I had. From the sidewalk outside the station, things seemed much the same as in other Japanese cities。

小学上册第二次英语第三单元暑期作业

小学上册英语第三单元暑期作业英语试题一、综合题(本题有100小题，每小题1分，共100分.每小题不选、错误，均不给分)1.The invention of ________ has reshaped modern transportation.2.She is _____ (跳舞) happily.3. A vulture plays an important role in ______ (生态系统).4.What is the name of the famous princess who lost her glass slipper?A. RapunzelB. BelleC. CinderellaD. Ariel5.The ________ has long arms and loves bananas.6.Which country is known for pyramids?A. MexicoB. ChinaC. EgyptD. GreeceC Egypt7.What do you call the middle of the Earth?A. CrustB. MantleC. CoreD. ShellC8.The sun _____ (rises/sets) in the east.9.The _____ (户外探索) can reveal many plant species.10.The _____ (灯光) is bright.11.I enjoy _____ (observing) nature in my garden.12.What is the capital city of Hungary?A. BudapestB. PragueC. ViennaD. BratislavaA13.The __________ (生态研究) informs public policy.14.The raccoon is often found rummaging through ______ (垃圾).15.What is the main purpose of a museum?A. To entertainB. To educateC. To sell goodsD. To provide shelterB16.The _______ (老虎) is a symbol of strength.17.How many points is a touchdown worth in American football?A. 3B. 5C. 6D. 7C18.In _____ (西班牙), flamenco dancing is popular.19.My sister is very ________.20.The tree has _____ (leaves/branches).21.The _____ (elephant) has big ears.22.What is the largest planet in our solar system?A. EarthB. MarsC. JupiterD. VenusC23.Acids can change the color of indicators to ______.24.Which instrument has keys and is played by pressing them?A. GuitarB. ViolinC. PianoD. DrumC25.What is the name of the closest galaxy to our Milky Way?A. AndromedaB. TriangulumC. WhirlpoolD. Sombrero26.The __________ is the largest organ in the human body.27.What is the name of the ocean on the east coast of the United States?A. AtlanticB. PacificC. IndianD. Arctic28.I like to ___ (listen/sing) to songs.29.The _______ (The Ottoman Empire) was a powerful empire that lasted for centuries.30.What is the name of the ocean located to the west of the United States?A. Atlantic OceanB. Indian OceanC. Arctic OceanD. Pacific OceanD31.The computer is very ___ (slow/fast).32.What is the term for an animal that eats both plants and meat?A. HerbivoreB. CarnivoreC. OmnivoreD. InsectivoreC33.The _____ (浣熊) is known for its masked face.34. A ____ has a strong beak and is often seen in parks.35.At the beach, I found a _______ (小螃蟹) hiding in the sand.36.Which bird is known for its colorful feathers?A. CrowB. PigeonC. PeacockD. Sparrow37.What do we call the feeling of being worried?A. JoyB. AnxietyC. HappinessD. AngerB Anxiety38.What is the longest river in the world?A. AmazonB. NileC. MississippiD. YangtzeB39.We enjoy ______ (看) the sunrise together.40.The chemical formula for ethanol is ______.41.Hydrogen is the lightest ______.42.Certain plants can adapt to ______ environments over time. (某些植物可以随着时间的推移适应极端环境。

小学下册Y卷英语第6单元暑期作业

小学下册英语第6单元暑期作业英语试题一、综合题(本题有100小题，每小题1分，共100分.每小题不选、错误，均不给分)1.What is 6 x 7?A. 42B. 36C. 48D. 54A2.What is the main language spoken in Spain?A. SpanishB. FrenchC. ItalianD. Portuguese3.I enjoy ______ (画) pictures in art class.4.The __________ is a major river in North America. (密西西比河)5.What do you call a place where you can borrow books?A. LibraryB. BookstoreC. SchoolD. Office6.The squirrel collects acorns in ________________ (秋天).7.What is the name of the famous beach in Rio de Janeiro?A. CopacabanaB. IpanemaC. BondiD. Waikiki8.The ______ is a part of the plant that produces leaves.9.I can engage my senses with my ________ (玩具名称).10.My favorite fruit is _____ (banana/apple).11.The element that is a gas at room temperature and essential for life is _______.12.I love my teddy _______ with a red bow tie.13.The __________ is often very pleasant in spring. (气候)14.The children are ______ in the playground. (laughing)15.What do you call the person who teaches you at school?A. DoctorB. TeacherC. ChefD. ArtistB16.The _______ (Space Race) was a competition between the US and USSR for space exploration.17.I love to try new ______ with my family.18.Which animal has a long neck?A. ElephantB. GiraffeC. ZebraD. LionB19.The _____ (nectar) attracts pollinators.20.The ________ was a major event in the history of England.21.What is the capital of Japan?A. TokyoB. KyotoC. OsakaD. HiroshimaA22.In _____ (南非), you can see the Big Five animals.23.What do you call a person who helps you learn?A. DriverB. TeacherC. ChefD. ArtistB24.________ (热带雨林) has a rich variety of plants.25.I like to ___ (play) chess with my dad.26.We have ___ (sport/music) practice today.27.What do we call a person who studies history?A. BiologistB. HistorianC. GeologistD. Chemist28.My favorite subject is ______ (science).29.The forest is very _______ (神秘的).30.What is the term for a baby tiger?A. CubB. PupC. KitD. FawnA31. Wall of China was built to __________ (保护) the country from invasions. The Grea32.The __________ of an atom is determined by its protons.33.I need to clean my ________.34.I enjoy solving puzzles. My favorite puzzle type is __________.35.What do we call the feeling of being afraid?A. HappinessB. FearC. ExcitementD. AngerB36.What is 18 + 2?A. 19B. 20C. 21D. 22B37.The tree is home to many ______.38.What do we call the event that occurs every four years when athletes compete internationally?A. OlympicsB. World CupC. Commonwealth GamesD. Pan American Games39.Elements in the same column of the periodic table have similar __________.40.What is the capital of Tanzania?A. Dar es SalaamB. DodomaC. ZanzibarD. ArushaB Dodoma41.The chemical symbol for phosphorus is __________.42.The _____ (种子发芽) is an exciting moment for gardeners.43.What do you put on a sandwich?A. BreadB. WaterC. PaperD. Cloth44.The _____ (小羊) bleats softly in the field. It is very fluffy. 小羊在田野里轻声叫唤。

小学上册第16次英语第3单元综合卷

小学上册英语第3单元综合卷英语试题一、综合题(本题有100小题，每小题1分，共100分.每小题不选、错误，均不给分)1.She is learning to ________ the guitar.2.The ancient Egyptians built ________ to showcase their power.3.What is the name of the famous explorer who sailed around the world?A. Ferdinand MagellanB. Christopher ColumbusC. Vasco da GamaD. Captain CookA4.What is the capital of the United Kingdom?A. LondonB. EdinburghC. CardiffD. Belfast5.What do we call a person who studies wildlife?A. Wildlife BiologistB. EcologistC. ConservationistD. All of the above6.Which of these shapes has three sides?A. SquareB. TriangleC. RectangleD. Circle7.What is the name of the famous American musician known for "Counting Stars"?A. OneRepublicB. Maroon 5C. ColdplayD. Imagine DragonsA8.My mom loves to _______ (动词) new recipes. 她的厨艺非常 _______ (形容词).9.What is the name of the famous American national park known for its giant redwoods?A. SequoiaB. YosemiteC. Grand CanyonD. YellowstoneA10.What is the value of 5 + 5 × 0?A. 0B. 5C. 10D. 15C11.I share my lunch with my ____.12.The __________ (历史的交响) celebrates diversity.13.The turtle swims slowly in the _________. (水)14.What do you call the person who studies rocks?A. BiologistB. GeologistC. ChemistD. Physicist15.I have a pet ___ . (dog)16. A reaction that absorbs energy involves breaking ______.17.What is the color of a typical blackboard?A. GreenB. YellowC. BlackD. White18.What is the capital of Kenya?A. NairobiB. KampalaC. Addis AbabaD. Dar es SalaamA19.The ______ (小鸭) loves splashing in the water.20.Which shape has four equal sides?A. RectangleB. CircleC. SquareD. TriangleC21. A reaction that occurs when a solid dissolves in a liquid is called a ______ reaction.22.What is the name of the famous castle in Scotland associated with legends of monsters?A. Edinburgh CastleB. Windsor CastleC. Loch Ness CastleD. Stirling CastleC23.What is the primary ingredient in a traditional bagel?A. Rye flourB. Wheat flourC. CornmealD. Sourdough24.The girl loves to ________.25.I like to draw characters from my favorite ____. (玩具名称)26.What type of animal is a dolphin?A. FishB. MammalC. ReptileD. AmphibianB27. A ______ (猫) likes to sit in sunny spots.28. A __________ is a famous historical landmark.29.We have a ______ (快乐的) celebration for birthdays.30.What do we call the process of freezing a liquid to make it solid?A. MeltingB. SolidifyingC. CoolingD. Hardening31.What do you need to ride a bike?A. HelmetB. ShoesC. SunglassesD. GlovesA32.What do you call a place where you can borrow books?A. SchoolB. LibraryC. StoreD. ParkB33.I love _______ (制作)手工艺品.34.My sister is my best _______ who loves to share secrets.35.The __________ (自然奇观) attract visitors from afar.36.What is the capital of New Zealand?A. AucklandB. WellingtonC. ChristchurchD. HamiltonB37.The concert was _______ (令人兴奋的).38.I love to listen to ______ (古典音乐) while I read.39.My pet _____ loves to cuddle and play.40.What do we call the king of the jungle?A. LeopardB. TigerC. LionD. BearC41.The _______ of sound can be influenced by obstacles in its path.42.The ostrich cannot ______ (飞).43.What do you call a story that is made up?A. FictionB. Non-fictionC. BiographyD. AutobiographyA44.Many _______ have medicinal uses in traditional practices.45.How many players are on a rugby team?A. FiveB. SixC. SevenD. Fifteen46. A ____ is a tiny insect that helps plants grow.47.My favorite subject is ___ (science/math).48.I share my secrets with my ____.49.What is the main color of a ripe banana?A. GreenB. YellowC. BrownD. Red50.It is hot in the ___. (summer)51.What do you call a group of ants?A. ColonyB. SwarmC. FlockD. PodA52.The earliest known human artifacts date back to the ________ Age.53.What do you call a sweet, baked good made with flour?A. CookieB. BrownieC. CakeD. All of the aboveD54.The porcupine raises its _________. (刺)55.What do you call a person who sells flowers?A. FloristB. GardenerC. FarmerD. BotanistA56.What is the process of a seed developing into a plant called?A. GerminationB. PropagationC. CultivationD. GrowthA57.What is the capital of Japan?A. OsakaB. KyotoC. TokyoD. Hiroshima58.My favorite sport is _______ (basketball).ets have distinctive tails made of gas and ______.60.Which fruit is typically red and round?A. BananaB. AppleC. OrangeD. Peach61.The sun rises in the ______. (east)62.The __________ (历史的纪录片) provide visual representations.63.Which of these animals can fly?A. ElephantB. DogC. ParrotD. Frog64.What do you call a person who performs magic tricks?A. IllusionistB. MagicianC. PerformerD. EntertainerB65.What is the hardest natural substance on Earth?A. GoldB. IronC. DiamondD. SilverC66.What is the capital of Azerbaijan?A. BakuB. GanjaC. SumqayitD. MingachevirA67.What do we call the act of saving money over time?A. SpendingB. InvestingC. SavingD. BudgetingC68.I love to watch the ______ (太阳) set in the evening. It paints the sky a beautiful ______ (颜色).69.What is the largest continent?A. AfricaB. AsiaC. EuropeD. Australia70.My sister is very __________ (敏感).71.I have a pet ______ (猫), and she is very ______ (可爱). She loves to play with ______ (球) in the garden.72.The _____ (豆荚) is green and crunchy.73.The ________ has colorful scales and swims fast.74.What is the capital of Finland?A. HelsinkiB. OsloC. StockholmD. TallinnA75.I want to learn how to ________ (编织).76.The ________ is a small creature that hops around.77.What is the soft, fluffy material used for stuffing pillows and jackets?A. CottonB. WoolC. DownD. PolyesterC78. A polar molecule has regions of _______ charge.79.The _____ (土壤) should be rich and full of nutrients.80.What do bees make?A. HoneyB. MilkC. ButterD. CheeseA81.The grass is very ___. (green)82.I like to help my dad ________ (修理) things around the house.83.My ________ (玩具名称) is my favorite holiday gift.84.How many wheels does a car typically have?A. 2B. 3C. 4D. 5C85.I think that kindness can make the world a better __________.86.The ________ was a famous figure in the American Revolution.87. A suspension is a mixture where particles settle ______.88.The _____ (狮子) is powerful and majestic.89.The ____ is a playful animal that enjoys chasing after balls.90. A flamingo is pink because of its _______ (饮食).91.What do we call a story that is made up?A. Non-fictionB. BiographyC. FictionD. HistoryC92.We built a ________ out of blocks.93.The seal barks excitedly on the ______ (沙滩).94.The __________ (历史的符号) convey meanings.95.Which one is a vegetable?A. AppleB. CarrotC. BananaD. GrapeB96. A geese migrates to warmer ______ during winter.97.The __________ is the area where most geological activity occurs.98.I want to _____ (see/watch) a movie tonight.99.The __________ is a large city in China known for its skyline. (上海)100.The ________ was a key event in the struggle for civil rights.。

hiroshima

hiroshimaHiroshimaHiroshima(広島市Hiroshima-shi) is the capital of Hiroshima Prefecture, and the largest city in the Chūgoku region of western Honshu, the largest island of Japan. It became best known as the first city in history to be destroyed by a nuclear weapon when the United States Army Air Forces (USAAF) dropped an atomic bomb on it at 8:15 A.M. on August 6, 1945, near the end of World War II.Hiroshima gained city status on April 1, 1889. On April 1, 1980, Hiroshima became a designated city. The city's current mayor since April 2011 is Kazumi Matsui.HistorySengoku periodHiroshima was founded on the river delta coastline of the Seto Inland Sea in 1589 by the powerful warlord Mōri Terumoto, who made it his capital after leaving Koriyama Castle in Aki Province. Hiroshima Castle was quickly built, and Terumoto moved in in 1593. Terumoto was on the losing side at the Battle of Sekigahara. The winner, Tokugawa Ieyasu, deprived Mori Terumoto of most of his fiefs including Hiroshima and gave Aki province to Masanori Fukushima, a daimyo who had supported Tokugawa.Tokugawa periodThe castle passed to Asano Nagaakira in 1619, and Asano was appointed the daimyo of this area. Under Asano rule, the city prospered, developed, and expanded, with few military conflicts or disturbances. Asano's descendants continued to rule until the Meiji Restoration in 1868. Hiroshima served as the capital ofHiroshima Domain during the Tokugawa period.Imperial periodAfter the han was abolished in 1871, the city became the capital of Hiroshima prefecture. Hiroshima became a major urban center during the imperial period as the Japanese economy shifted from primarily rural to urban industries.During 1870s, one of the seven government-sponsored English language schools was established in Hiroshima. Ujina Harbor was constructed through the efforts of Hiroshima Governor Sadaaki Senda in the 1880s, allowing Hiroshima to become an important port city.The Sanyo Railway was extended to Hiroshima in 1894, and a rail line from the main station to the harbor was constructed for military transportation during the First Sino-Japanese War. During that war, the Japanese government moved temporarily to Hiroshima, and the Emperor Mutsuhito maintained his headquarters at Hiroshima Castle from September 15, 1894 to April 27, 1895. The significance of Hiroshima for the Japanese government can be discerned from the fact that the first round of talks between Chinese and Japanese representatives to end the Sino-Japanese War was held in Hiroshima from February 1 to February 4, 1895. New industrial plants, including cotton mills, were established in Hiroshima in the late 19th century. Further industrialization in Hiroshima was stimulated during the Russo-Japanese War in 1904, which required development and production of military supplies. The Hiroshima Prefectural Commercial Exhibition Hall was constructed in 1915 as a center for trade and exhibition of new products. Later, its name was changed to Hiroshima Prefectural Product Exhibition Hall, and again to Hiroshima Prefectural Industrial Promotion Hall.During the First World War, Hiroshima became a focal point of military activity, as the Japanese government entered the war on the Allied side. About 500 German prisoners of war were held in Ninoshima Island in Hiroshima Bay.The growth of Hiroshima as a city continued after the First World War, as the city now attracted the attention of the Catholic Church, and on May 4, 1923, an Apostolic Vicar was appointed for that city.World War II and atomic bombingDuring World War II, the Second Army and Chugoku Regional Army were headquartered in Hiroshima, and the Army Marine Headquarters was located at Ujina port. The city also had large depots of military supplies, and was a key center for shipping.The bombing of Tokyo and other cities in Japan during World War II caused widespread destruction and hundreds of thousands of deaths, nearly all civilians, predominantly women and children. For example, Toyama, an urban area of 128,000, was nearly fully destroyed, and incendiary attacks on T okyo are believed to have claimed 90,000 lives. There were no such air raids in Hiroshima. However, the threat was certainly there and toprotect against potential firebombings in Hiroshima, students (between 11–14 years) were mobilized to demolish houses and create firebreaks.On Monday, August 6, 1945, at 8:15 AM, the Atomic Bomb "Little Boy" was dropped on Hiroshima by an American B-29 bomber, the Enola Gay, directly killing an estimated 80,000 people. By the end of the year, injury and radiation brought total casualties to 90,000–140,000. Approximately 69% of the city'sbuildings were completely destroyed, and about 7% severely damaged.Research about the effects of the attack was restricted during the occupation of Japan, and information censored until the signing of the San Francisco Peace Treaty in 1951, restoring control to the Japanese.The oleander is the official flower of the city of Hiroshima because it was the first to bloom again after the explosion of the atomic bomb in 1945.Postwar periodOn September 17, 1945, Hiroshima was struck by the Makurazaki Typhoon (Typhoon Ida). Hiroshima prefecture suffered more than 3,000 deaths and injuries, about half the national total. More than half the bridges in the city were destroyed, along with heavy damage to roads and railroads, further devastating the city.Hiroshima was rebuilt after the war, with the help from the national government through the Hiroshima Peace Memorial City Construction Law passed in 1949. It provided financial assistance for reconstruction, along with land donated that was previously owned by the national government and used for military purposes.In 1949, a design was selected for the Hiroshima Peace Memorial Park. Hiroshima Prefectural Industrial Promotion Hall, the closest surviving building to the location of the bomb's detonation, was designated the Genbaku Dome (原爆ドーム) or "Atomic Dome", a part of the Hiroshima Peace Memorial Park. The Hiroshima Peace Memorial Museum was opened in 1955 in the Peace Park.Hiroshima was proclaimed a City of Peace by the Japaneseparliament in 1949, at the initiative of its mayor, Shinzo Hamai (1905–1968). As a result, the city of Hiroshima received more international attention as a desirable location for holding international conferences on peace as well as social issues. As part of that effort, the Hiroshima Interpreters'and Guide's Association (HIGA) was established in 1992 in order to facilitate interpretation for conferences, and the Hiroshima Peace Institute was established in 1998 within the Hiroshima University. The city government continues to advocate the abolition of all nuclear weapons and the Mayor of Hiroshima is the president of Mayors for Peace, an international mayoral organization mobilizing cities and citizens worldwide to abolish and eliminate nuclear weapons by the year 2020 Mayors for Peace2020 Vision Campaign.EconomyHiroshima is the center of industry for the Chūgoku-Shikoku region, and is by and large centered along the coastal areas. Hiroshima has long been a port city and Hiroshima port or Hiroshima International Airport can be used for the transportation of goods.Its largest industry is the manufacturing industry with core industries being the production of Mazda cars, car parts and industrial equipment. Mazda Motor Corporation is by far Hiroshima's dominant company. Mazda accounts for 32% of Hiroshima's GDP. Mazda makes many models in Hiroshima for worldwide export, including the popular MX-5/Miata, Mazda Demio (Mazda2), Mazda CX-9 and Mazda RX-8. The Mazda CX-7 has been built there since early 2006. Other Mazda factories are in Hofu and Flat Rock, Michigan.General machinery and equipment also account for a largeportion of exports. Because these industries require research and design capabilities, it has also had the offshoot that Hiroshima has many innovative companies actively engaged in new growth fields (for example, Hiroshima Vehicle Engineering Company (HIVEC). Many of these companies hold the top market shares in Japan and the world, or are alone in their particular field. Tertiary industries in the wholesale and retail areas are also very developed.Another result of the concentration of industry is an accumulation of skilled personnel and fundamental technologies. This is considered by business to be a major reason for location in Hiroshima. Business setup costs are also much lower than other large cities in the country and there is a comprehensive system of tax breaks, etc. on offer for businesses which locate in Hiroshima. This is especially true of two projects: the Hiroshima Station Urban Development District and the Seifu Shinto area which offer capital installments (up to 501 million yen over 5 years), tax breaks and employee subsidies. Seifu Shinto, which translates as WestWind, New Town is the largest construction project in the region and is an attempt to build "a city within a city." It is attempting to design from the ground up a place to work, play, relax and live.One important industry in Hiroshima is the steel industry. The Japan Steel Works (formerly Nihon Seiko, established in 1907) has one of its plants in Hiroshima out of total of three plants (the other two are at Muroran and Yokohama).Hiroshima recently made it onto Lonely Planet's list of the top cities in the world. Commuting times rank amongst the shortest in Japan and the cost of living is lower than other largecities in Japan such as Tokyo, Osaka, Kyoto, or Fukuoka.CultureHiroshima has a professional symphony orchestra, which has performed at Wel City Hiroshima since 1963. There are also many museums in Hiroshima, including the Hiroshima Peace Memorial Museum, along with several art museums. The Hiroshima Museum of Art, which has a large collection of French renaissance art, opened in 1978. The Hiroshima Prefectural Art Museum opened in 1968, and is located near Shukkei-en gardens. The Hiroshima City Museum of Contemporary Art, which opened in 1989, is located near Hijiyama Park. Festivals include Hiroshima Flower Festival and Hiroshima International Animation Festival.Hiroshima Peace Memorial Park, which includes the Hiroshima Peace Memorial, draws many visitors from around the world, especially for the Hiroshima Peace Memorial Ceremony, an annual commemoration held on the date of the atomic bombing. The park also contains a large collection of monuments, including the Children's Peace Monument, the Hiroshima National Peace Memorial Hall for the Atomic Bomb Victims and many others.Hiroshima's rebuil t castle (nicknamed Rijō, meaning Koi Castle) houses a museum of life in the Edo period. Hiroshima Gokoku Shrine is within the walls of the castle. Other attractions in Hiroshima include Shukkei-en, Fudōin, Mitaki-dera, and Hijiyama Park.EducationHiroshima University was established in 1949, as part of a national restructuring of the education system. One national university was set up in each prefecture, including Hiroshima University, which combined eight existing institutions (HiroshimaUniversity of Literature and Science,Hiroshima School of Secondary Education, Hiroshima School of Education, Hiroshima Women's School of Secondary Education, Hiroshima School of Education for Youth, Hiroshima Higher School, Hiroshima Higher Technical School, and Hiroshima Municipal Higher Technical School), with the Hiroshima Prefectural Medical College added in 1953.TransportationLocal public transportation in Hiroshima is provided by a Tram system, operated by Hiroshima Electric Railway called "Hiroden" (広電?) for short. Hiroden also operates buses in and around Hiroshima Prefecture. Hiroshima Electric Railway was established on June 18, 1910, in Hiroshima. While many other Japanese cities abandoned the streetcar system by the 1980s, Hiroshima retained it because the construction of a subway system was too expensive for the city to afford, as it is located on a delta. During the 1960s, Hiroshima Electric Railway, or Hiroden, bought extra trams from other Japanese cities. Although trams in Hiroshima are now being replaced by newer models, most retain their original appearance. Thus, the tram system is sometimes called a "Moving Museum" by railroad buffs. Of the four trams that survived the war, two are still in operation as of July 2006 (Hiroden Numbers 651 and 652). There are seven tram lines, many of which terminate at Hiroshima Station.The Astram Line opened for the 1994 Asian Games in Hiroshima, with one line from central Hiroshima to Seifu Shinto and Hiroshima Big Arch, the main stadium of the Asian Games. Astram uses rubber-tyred metro cars, and provides service to areas towards the suburbs that are not served by Hiroden streetcars. The Skyrail Midorizaka Line is a monorail that operatesbetween Midoriguchi and Midori-Chūō, serving three stops.The JR West Hiroshima Station offers inter-city rail service, including Sanyō Shinkansen which provides high speed service between Shin-ōsaka and Fukuoka. Sanyō Shinkansen began providing service to Hiroshima in 1975, when the Osaka-Hakata extension opened. Other rail service includes the Sanyō Main Line, Kabe Line, Geibi Line, and Kure Line.Ferries are operated by JR Miyajima Ferry and Miyajima Matsudai Kisen to Miyajima. Hiroden provides service to Miyajimaguchi Station, which is located near the ferry terminal for service to Miyajima. Hiroshima Port is the main passenger ferry terminal for Hiroshima, with service to Etajima, Matsuyama, and other destinations. There is also an international ferry terminal which has service to Busan and Ulsan in South Korea, Shanghai, Dalian, Qingdao and Ningbo in China, Keelung and Kaohsiung in Taiwan, as well as Hong Kong. There is also a boat taxi service that runs along the ota-gawa channels into the city center.Hiroshima Airport, located nearby in the city of Mihara, provides air service within Japan to Tokyo, Sapporo, Okinawa, and Sendai. International air service is provided to Seoul, Guam, Bangkok, Taipei, Shanghai, Beijing, and Dalian. Commuter air service is also available at Hiroshima-Nishi Airport.Hiroshima during World War IIAt the time of its bombing, Hiroshima was a city of some industrial and military significance. A number of military camps were located nearby, including the headquarters of the Fifth Division and Field Marshal Shunroku Hata's 2nd General Army Headquarters, which commanded the defense of all of southern Japan.[22] Hiroshima was a minor supply and logistics base forthe Japanese military. The city was a communications center, a storage point, and an assembly area for troops. It was one of several Japanese cities left deliberately untouched by American bombing, allowing a pristine environment to measure the damage caused by the atomic bomb.[23][24]The center of the city contained several reinforced concrete buildings and lighter structures. Outside the center, the area was congested by a dense collection of small wooden workshops set among Japanese houses. A few larger industrial plants lay near the outskirts of the city. The houses were constructed of wood with tile roofs, and many of the industrial buildings were also built around wood frames. The city as a whole was highly susceptible to fire damage.The population of Hiroshima had reached a peak of over 381,000 earlier in the war, but prior to the atomic bombing the population had steadily decreased because of a systematic evacuation ordered by the Japanese government. At the time of the attack, the population was approximately 340,000–350,000.[1] Because official documents were burned, the exact population is uncertain.。

MATHEMATICSEDUCATIONJOURNALS：数学教育学报

MATHEMATICS EDUCATION JOURNALSRESEARCH JOURNALS• Adults Learning Mathematics - An International Journal• Canadian Journal of Science, Mathematics and Technology Education• CASTME (Commonwealth Association for Science, Technology & Mathematics Education) Journal• Educational Studies in Mathematics• Electronic Journal of Mathematics and Technology• Eurasia Journal of Mathematics, Science & Technology Education• Focus on Learning Problems in Mathematics (RCML)• For the Learning of Mathematics• Hiroshima Journal of Mathematics Education• International Electronic Journal of Mathematics Education• International Journal for the History of Mathematics Education• International Journal for Mathematics Teaching and Learning• International Journal of Mathematical Education: Policy and Practice (IJME)• International Journal of Technology in Mathematics Education• International Journal of Computers for Mathematical Learning• International Journal of Mathematical Education in Science and Technology • International Journal of Science and Mathematics Education• International Online Journal of Science and Mathematics Education• Issues in the Undergraduate Mathematics Preparation of School Teachers: The Journal (IUMPST)• Journal for Research in Mathematics Education• Journal of Computers in Mathematics and Science Teaching• Journal of Mathematical Behavior• Journal of Mathematics Teacher Education• Journal of STEM (Science, Technology, Engineering, and Mathematics) Education: Innovations and Research• Literacy and Numeracy Studies: An International Journal in the Education and Training of Adults• Mathematical Thinking and Learning• Mathematics Education Research Journal• Mathematics Education Review• Mathematics Educator• Mathematics Educator• Mediterranean Journal for Research in Mathematics Education• The Montana Mathematics Enthusiast• Nordisk Matematikk Didaktikk/Nordic Studies in Mathematics Education • Philosophy of Mathematics Education journal• PRIMUS (Problems, Resources, Issues in Undergraduate Mathematics Studies)• Proceedings of the British Society for Research into Learning Mathematics • Récherches en Didactiques des Mathématiques• Research in Collegiate Mathematics Education (RCME)• Research in Mathematics Education• School Science and Mathematics• Spreadsheets in Education• Statistics Education Research Journal• Teaching Mathematics and its Applications• ZDM: The International Journal on Mathematics EducationPROFESSIONAL JOURNALS• Australian Mathematics Teacher• Australian Primary Mathematics Classroom• Australian Senior Mathematics Journal• Equals: Mathematics and Special Educational Needs• Learning and Teaching Mathematics• Mathematics in School• Mathematics Teacher• Mathematics Teaching• Mathematics Teaching in the Middle School• Primary Mathematics• Teaching Children Mathematics• The Teaching of Mathematics• Teaching Statistics。

Hiroshima英语简介

Hiroshima英语简介Hiroshima is the capital of Hiroshima Prefecture and the largest city in the Chūgoku region, JapanHiroshima may also refer to:Hiroshima Prefecture, JapanSanfrecce Hiroshima, Hiroshima's professional football (soccer) club Hiroshima Toyo Carp, Hiroshima's professional baseball club, though it is more often referred to as ToyoHiroshima (film), a 1995 Japanese–Canadian film about the decision process behind the dropping of the nuclear bomb on Hiroshima Hiroshima (band), an American jazz band formed in 1974 Kitahiroshima, Hokkaidō, a city located in Ishikari, Hokkaidō, Japan, formerly named "Hiroshima"the Hiroshima meteorite of 2003, which fell in Chūgoku, Japan (see meteorite falls)Hiroshima (book), a 1946 book written by John HerseyHiroshima (documentary), a 2005 television documentary "Hiroshima" (song), a 1990 single by Sandra Cretu"Hiroshima (B-B-B-Benny Hit His Head)", a 2008 single by Ben Folds Hiroshima Mon Amour, a 1959 filmHotel class submarine, Soviet submarine K-19 unofficially gained onfleet its nickname "Hiroshima"During World War II, the Second Army and Chugoku Regional Army were headquartered in Hiroshima, and the Army Marine Headquarters was located at Ujina port. The city also had large depots of military supplies, and was a key center for shipping.[13]The bombing of Tokyo and other cities in Japan during World War II caused widespread destruction and hundreds of thousands of deaths, nearly all civilians.[14] For example, Toyama, an urban area of 128,000, was nearly fully destroyed, and incendiary attacks on Tokyo are credited with claiming 90,000 lives. There were no such air raids in Hiroshima. However, the threat was certainly there and to protect against potential firebombings in Hiroshima, students (between 11–14 years) were mobilized to demolish houses and create firebreaks.On Monday, August 6, 1945, at 8:15 AM, the nuclear bomb "Little Boy" was dropped on Hiroshima by an American B-29 bomber, the Enola Gay,[16] directly killing an estimated 80,000 people. By the end of the year, injury and radiation brought total casualties to 90,000–140,000.[17] Approximately 69% of the city's buildings were completely destroyed, and about 7% severely damaged.Research about the effects of the attack was restricted during the occupation of Japan, and information censored until the signing of the San Francisco Peace Treaty in 1951, restoring control to the Japanese.[18]On September 17, 1945, Hiroshima was struck by the Makurazaki Typhoon (Typhoon Ida). Hiroshima prefecture suffered more than 3,000 deaths and injuries, about half the national total.[19] More than half the bridges in the city were destroyed, along with heavy damage to roads and railroads, further devastating the city.[20]Hiroshima was rebuilt after the war, with the help from the national government through the Hiroshima Peace Memorial City Construction Law passed in 1949. It provided financial assistance for reconstruction, along with land donated that was previously owned by the national government and used for military purposes.[21]Atomic Bomb Dome and modern HiroshimaIn 1949, a design was selected for the Hiroshima Peace Memorial Park. Hiroshima Prefectural Industrial Promotion Hall, the closest surviving building to the location of the bomb's detonation, was designated the Genbaku Dome (原爆ドーム) or "Atomic Dome", a part of the Hiroshima Peace Memorial Park. The Hiroshima Peace Memorial Museum was opened in 1955 in the Peace Park.[22]Hiroshima was proclaimed a City of Peace by the Japanese parliament in 1949, at the initiative of its mayor, Shinzo Hamai (1905–1968). As a result, the city of Hiroshima received more international attention as a desirable location for holding international conferences on peace as well as social issues. As part of that effort, the Hiroshima Interpreters' andGuide's Association (HIGA) was established in 1992 in order to facilitate interpretation for conferences, and the Hiroshima Peace Institute was established in 1998 within the Hiroshima University. The city government continues to advocate the abolition of all nuclear weapons and the Mayor of Hiroshima is the president of Mayors for Peace, an international mayoral organization mobilizing cities and citizens worldwide to abolish and eliminate nuclear weapons by the year 2020 Mayors for Peace 2020 Vision Campaign.。

多线性Hardy型算子在变指数Herz-Morrey乘积空间上的有界性

多线性Hardy型算子在变指数Herz-Morrey乘积空间上的有界性武江龙; 张璞【期刊名称】《《高校应用数学学报A辑》》【年(卷),期】2013(000)002【总页数】11页(P154-164)【关键词】Hardy算子; 多线性算子; 变指数Herz-Morrey空间; 乘积空间【作者】武江龙; 张璞【作者单位】牡丹江师范学院数学系，黑龙江牡丹江157011【正文语种】中文【中图分类】O174.2自1975年Coifman和Meyer[1]研究了双线性奇异积分的有界性之后,多线性算子的研究得到了广泛关注(见[2-3]及相关文献).1920年Hardy[4]证明了Hardy积分不等式,此后众多学者对Hardy积分不等式和Hardy算子进行了深入的研究(如文[5-8]等).1995年,Christ和Grafakos[9]研究了如下定义的n维Hardy算子并建立了它们的交换子在Lebesgue空间和齐次Herz空间中有界的刻画.最近,武江龙等人[11-12]讨论了分数次Hardy算子交换子在Herz-Morrey空间的有界性.另一方面,由于变指数函数空间在流体动力学及具有非标准增长条件的微分方程等领域有着广泛的应用,近二十年来逐步受到人们的重视,调和分析中的许多经典算子在变指数函数空间中的有界性问题也得到了广泛研究(见[13-23]及相关参考文献). 近来,Izuki[15-16]在变指数Herz-Morrey空间中分别研究了向量值次线性算子和分数次积分的有界性.受上述工作的启发,本文的主要目的是研究多线性Hardy型算子在变指数Herz-Morrey空间的有界性.下面先给出多线性分数次Hardy算子的定义.定义1.1设m,n是正整数且m≥1,n≥2,向量值函数=(f1,f2,...,fm),其中fi(i=1,2,...,m)是Rn上的局部可积函数,0≤β＜mn.m-阶多线性分数次Hardy算子定义为显然,当m=1时,另外,当β=0时,分别用Hm和表示相应于Hardy算子H的多线性算子H0,m和相应于共轭算子H∗:=的多线性算子.在叙述主要结果之前,先回顾几个相关概念.设E是Rn的Lebesgue可测子集且|E|＞0.先给出变指数Lebesgue空间的定义. 定义1.2设q(·):E→[1,∞)是可测函数.变指数Lebesgue空间Lq(·)(E)定义为局部可积的变指数Lebesgue空间(E)定义为(E)='f是可测函数:对所有的紧子集K⊂E,有f∈Lq(·)(K)“.当赋予如下的范数时,Lq(·)(E)成为Banach空间,用P(Rn)表示Rn上满足以下条件的所有可测函数q(·):Rn→[1,∞)构成的集合,并用q′(·)表示q(·)的共轭指数,即:1/q′(x)+1/q(x)=1.设M为Hardy-Littlewood极大算子.用B(Rn)表示P(Rn)中所有使M在Lq(·)(Rn)上有界的函数q(·)构成的集合.下面给出变指数Herz-Morrey空间的定义.对任意的k∈Z,令Bk=B(0,2k)={x∈Rn: |x|≤2k},Ak=Bk\Bk-1.用χk=χAk表示Ak的特征函数.在下文中,对于Rn的可测子集S,用|S|表示S的Lebesgue测度,χS表示S的特征函数.用C表示与主要参数无关的常数,且其取值在不同的位置可以不尽相同.本节给出与主要结果相关的几个命题和引理.命题2.1[18]设q(·)∈P(Rn)且满足下列条件则有q(·)∈B(Rn).上述结论分别被Nekvinda[18]和Cruz-Uribe等[19]独立证得.下面的结论是属于Diening[20]的,实际上,Diening证明了Musielak-Orlicz空间中更一般的结果,这里只转述了本文需要的部分结论(见文[16]或[21]).命题2.2[20]若q(·)∈P(Rn),则q(·)∈B(Rn)当且仅当q′(·)∈B(Rn).设0＜β＜n,分数次积分算子Iβ定义为2004年和2007年,Diening[13]和Capone等[14]分别研究了Iβ在变指数Lebesgue空间中的有界性,下面的结果属于Capone等[14].命题2.3设q1(·)∈P(Rn)满足(1)和(2).如果0＜β＜n/(q1)+且q2(·)由下式确定则存在常数C＞0,对任意的f∈Lq1(·)(Rn),有为证明主要结果,还需要以下几个引理.由Hβ,2的定义及引理2.1中的(3),得令1/u(x)=1/q1(x)+1/q2(x),则1/q(x)=1/u(x)-β/n.注意到χBj(x)≤C2-jβIβ(χBj)(x) (见[16],p350),使用命题2.3及引理2.1中的式(4),得再利用(8)式,引理2.3和(5)式,可得从而定理得证.对相应于分数次Hardy算子的共轭算子H∗β的多线性算子,有下面对应的结论成立.定理3.2设i=1,2,...,m(m∈Z+,m≥1);qi(·)∈P(Rn)且满足(1)和(2),变指数q(·)由下式确定证不失一般性,仅考虑m=2的情形.当m∈Z+(m≥1)时可类似证明.当m=2时,有类似于(9)式,使用式(8),引理2.3和式(5),可得于是,由式(14)和(15),类似于(10)式有令1/v=1/p1+1/p2,则1/p=1/v-β/n,从而p＞v.利用式(16),式(11)和序列形式的H¨older不等式,类似于(12)式,可得由式(13)和条件αi＞λi+β/2-nδi2,得从而定理得证.对于n维Hardy算子H及其共轭算子H∗:=的多线性算子,也有类似于定理3.1和定理3.2的结论成立.定理4.1设i=1,2,...,m(m∈Z+,m≥1);qi(·)∈P(Rn)且满足(1)和(2),变指数q(·)由下式确定证只需对定理3.1和定理3.2的证明稍作修改.下面只给出(i)的证明思路,(ii)的证明与定理3.2的证明类似.不失一般性,仍考虑m=2的情形.由Hm的定义并使用引理2.1,类似于(7)式,得到其中最后一步用到了(5)中的第一个式子.接下来与定理3.1的证明完全类似,便可得到所要证的结果.略去证明的细节.注3另外,作者们在[25]中已经考虑了Hardy算子在m=1时的情形,而本文建立了Hardy型算子在m≥1时结论,所以本文推广了[25]的结果.【相关文献】[1]Coifman R,Meyer Y.On commutators of singular integrals and bilinear singular integrals[J]. Trans Amer Math Soc,1975,212:315-331.[2]Kenig C,Stein E M.Multilinear estimates and fractional integration[J].Math Res Lett, 1999,6:1-15.[3]Grafakos L,Torres R H.Multilinear Calder´on-Zygmund theory[J].Adv Math,2002,165: 124-164.[4]Hardy G.Note on a theorem of Hilbert[J].Math Z,1920,6(3-4):314-317.[5]Hardy G,Littlewood J,Polya G.Inequalities,2nd ed[M].London:Cambridge University Press,1952.[6]Anderson K,Muckenhoupt B.Weighted weak type Hardy inequalities with application to Hilbert transforms and maximal functions[J].Studia Math,1982,72:9–26.[7]Sawyer E.Weighted Lebesgue and Lorentz norm inequalities for the Hardyoperator[J].Trans Amer Math Soc,1984,281(1):329–337.[8]Golubov B.Boundedness of the Hardy and the Hardy-Littlewood operators in the spaces ReH1and BMO[J].Sb Math,1997,188(7):1041-1054.[9]Chirst M,Grafakos L.Best constants for two non-convolution inequalities[J].Proc Amer Math Soc,1995,123:1687-1693.[10]Fu Zunwei,Liu Zongguang,Lu Shanzhen,et al.Characterization for commutators of N-dimensional fractional Hardy operators[J].Sci China Ser A,2007,50(10):1418-1426.[11]武江龙,王婧敏.多线性分数次Hardy算子交换子的有界性[J].高校应用数学学报,2010,25(1): 115-121.[12]武江龙.分数次Hardy算子多线性交换子的有界性[J].数学物理学学报,2011,31A(4):1055-1062.[13]Diening L.Riesz potential and Sobolev embeddings on generalized Lebesgue spaces and Sobolev spaces Lp(·)and Wk,p(·)[J].Math Nachr,2004,268:31-43.[14]Capone C,Cruz-Uribe D,Fiorenza A.The fractional maximal operator and fractional integrals on variable Lpspaces[J].Rev Mat Iberoamericanna,2007,23:743-770.[15]Izuki M.Boundedness of vector-valued sublinear operators on Herz-Morrey spaces with variable exponent[J].Math Sci Res J,2009,13:243-253.[16]Izuki M.Fractional integrals on Herz-Morrey spaces with variableexponent[J].Hiroshima Math J,2010,40:343-355.[17]Kovacik O,Rakosnik J.On spaces Lp(x)and Wk,p(x)[J].Czechoslovak Math J,1991,41(4): 592-618.[18]Nekvinda A.Hardy-Littlewood maximal operator On Lp(x)(Rn)[J].Math Inequal Appl, 2004,7:255-265.[19]Cruz-Uribe D,Fiorenza A,Neugebauer C.The maximal function on variable Lpspaces[J]. Ann Acad Sci Fenn Math,2003,28:223-238.[20]Diening L.Maximal functions on Musielak-Orlicz spaces and generalized Lebesgue spaces[J]. Bull Sci Math,2005,129:657-700.[21]Cruz-Uribe D,Fiorenza A,Martell J M,et al.The boundedness of classical operators on variable Lpspaces[J].Ann Acad Sci Fenn Math,2006,31:239-264.[22]Harjulehto P,H¨ast¨o P,Le U V,et al.Overview of differential equations with non-standard growth[J].Nonlinear Anal:Theory,Methods&Appl,2010,72(12):4551-4574.[23]Diening L,Harjulehto P,H¨ast¨o P,et al.Lebesgue and Sobolev Spaces with Variable Exponents[M].Lecture Notes in Mathematics Vol.2017,Berlin:Springer-Verlag,2011.[24]Huang Aiwu,Xu Jingshi.Multilinear singular integrals and commutators in variable exponent Lebesgue spaces[J].Appl Math J Chinese Univ,2010,25(1):69-77.[25]张璞,武江龙.分数次Hardy算子在变指数Herz-Morrey空间中的有界性[J].数学的实践与认识,2013,43(7):247-254.。

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Hiroshima Math.J.36(2006),289–329Tilings from non-Pisot unimodular matricesMaki Furukado(Received July5,2005)(Revised November8,2005)A ing the unimodular Pisot substitution of the free monoid on d letters,the existence of graph-directed self-similar sets f X i g i¼1;2;...;d satisfying the set equation(0.0.1)with the positive measure on the A-invariant contracting plane P is well-known,where A is the incidence matrix of the substitution.Moreover,under some conditions,the set f X i g i¼1;2;...;d is the prototile of the quasi-periodic tiling of P(see Figure1).Inthis paper,even in the case of non-Pisot matrix A,the generating method of graph-directed self-similar sets and quasi-periodic tilings is proposed under the‘‘blockingcondition’’.0.IntroductionThe following fact is well-known:using the unimodular Pisot substitution s of the free monoid on d letters,we obtain the prototiles f X i g i¼1;2;...;d with fractal boundary of the A-invariant contracting plane P,satisfying the set equation:l iAÀ1X i¼6ðvðiÞjþX jÞðnon-overlappingÞð0:0:1Þj¼1where the transformation A is the incidence matrix of the substitution s and vectors vðiÞj A P,1a j a l i are some translations.Moreover,under the super coincidence condition in[14],we see that the prototiles f X i g i¼1;2;...;d give us a graph directed self-similar tiling of P(see Figure1).The prototiles from the substitution have been studiedﬁrst by Rauzy in[20].Since Rauzy(see Figure 1),several properties of prototiles have been studied by many authors.For example,basic properties of f X i g i¼1;2;...;d have been studied in[16],[4],[10],[21] and[2],the estimation of the Hausdor¤dimention of q X i in[10],topological properties of X i in[22],[1],the relation with the Markov partition generated by f X i g i¼1;2;...;d in[4],[18],the relation with the algebraic b-expansion in[15], [14],Diophantine approximation in[13],quasi-periodic tiling in[14],[17],etc. In fact,we know that to study the structure of f X i g i¼1;2;...;d is useful and2000Mathematics Subject Classﬁcation.52C23,37A45,28A80.Key words and phrases.non-Pisot unimodular substitutions,graph-directed self-a‰ne tilings.important to research of fractal geometry,number theory,tiling theory,ergodic theory and dynamical systems.However,the study of the case that the matrix A is unimodular and non-Pisot is very few (see [11]).The purpose of this paper is to give a su‰cient condition of the existence of prototiles f X i g i ¼1;2;...;K which satisﬁes the set equation (0.0.1)and generates a quasi-periodic tiling of the contracting eigenspace P ,starting from the 4Â4non-Pisot unimodular hyperbolic integer matrix A .1.Deﬁnition and notations1.1.non-Pisot matrixIn this paper,we consider that the integer matrix A satisﬁes the following conditions.Assumption 1.1.Let us assume that(1)the eigenvalues l i ,i ¼1;2;3;4of the matrix A satisfyj l 1j b j l 2j >1>j l 3j b j l 4jðhyperbolic non -Pisot condition Þ;(2)det A ¼G 1(unimodular condition);(3)the matrix A has the standard position property which is mentioned later.For eigenvalues l i ,i ¼1;2;3;4of the matrix A ,let f v 1;v 2;v 3;v 4g be the corresponding basis of R 4generated by eigenvectors,that is,we consider that the 2-dimensional contracting eigenspace P c of the linear transformation A is spanned by f v 3;v 4g and that the 2-dimesional expanding eigenspace P e is spanned by f v 1;v 2g .And let p c ðx v 1þy v 2þz v 3þw v 4Þ¼z v 3þw v 4andFig.1.Rauzy fractal tiling.290Maki Furukadop e ðx v 1þy v 2þz v 3þw v 4Þ¼x v 1þy v 2be the projections to P c and P e respec-tively.Then,following relations P c A ¼A P c and P e A ¼A P e hold.Using the representation by ½e 1;e 2;e 3;e 4 ¼½v 1;v 2;v 3;v 4 x 11x 12x 13x 14x 21x 22x 23x 24x 31x 32x 33x 34x 41x 42x 43x 442666437775;the projected vectors p c e i A P c and p e e i A P e of the canonical basis f e i j i ¼1;2;3;4g are given byp c e i ¼x 3i v 3þx 4i v 4F ½x 3i ;x 4i t ;p e e i ¼x 1i v 1þx 2i v 2F ½x 1i ;x 2i trespectively.We say that p c e i and p c e j are in standard position for i ,j ði 0j Þif p c e i is not parallel to p c e j .And we say that the matrix A has the standard position property if any pair of p c e i ,i ¼1;2;3;4are in standard position.For easy undrstanding of several deﬁnitions and properties,we introduce an example at the end of each section.Example .Let us consider the following matrix A ¼10010011011010112666437775.The characteristic polynomial of A is given by F A ðx Þ¼x 4À3x 3þx 2þx þ1and it is irreducible,moreover l i ,1a i a 4of A satisfy l 1>l 2>1>j l 3j ¼j l 4j .Thebasis f v 1;v 2;v 3;v 4g is chosen as v 1:¼u 1,v 2:¼u 2,v 3:¼u 3þu 42,v 4:¼u 3Àu 42i where u j is the eigenvector of l j ,1a j a 4.In this example,vectors p c e i ,i ¼1;2;3;4are represented by the following ﬁgure (see Figure 2):It is clear that the matrix A in Example satisﬁes Assumption 1.1(1)(2)(3).Fig.2.Vectors p c e i ði ¼1;2;3;4Þin Example.291Tilings from non-Pisot matrices1.2.Parallelograms and segmentsFrom now on,we denote p instead of the projection p c and P instead of the plane P c for simplicity.For i;j A f1;2;3;4g;let i5j be the symbolic parallelogram generated by vectors p e i and p e j where i5jði0jÞis chosen if the counterclockwise angle a between p e i and p e j satisﬁes0<a<p.We write the set of symbolic par-allelograms asV2:¼i5ji;j A f1;2;3;4g;i0j;the angle a between p e i and p e j is chosen by0<a<p&':It is clear that the cardinality of V2is equal to6ð¼4C2Þfrom the standard position property.For i A f1;2;3;4g;let i be the symbolic segment generated by p e i.We write the set of symbolic segments asV1:¼f i j i A f1;2;3;4gg:A pairðx;i5jÞA p Z4ÂV2means geometrically the positive oriented parallelogram i5j with the base-point x of P,that is,ðx;i5jÞ:¼f xþmp e iþnp e j j0a m;n a1g(see Figure3).Let us deﬁne the set of all of theﬁnite formal sum of the parallelogram with the base-point l A p Z4ÂV2as follows:G2:¼Xl A p Z4ÂV2m l l j m l A Z;a f l j m l00g<þy8 <:9 =;:We call an element of G2a patch.For patches g and d such that g¼Pl A p Z4ÂV2m l l,d¼Pl A p Z4ÂV2n l l,we deﬁne the sumbyFig.3.ðx;i5jÞ.292Maki Furukadog þd ¼Xl A p Z 4ÂV 2m l l þX l A p Z 4ÂV 2n l l :¼X l A p Z 4ÂV 2ðm l þn l Þl :Then we know that G 2is a free Z -module.On the notation L ðþÞd ¼f n l l j n l 00;n l A N g and L ðÀÞd ¼f n l l j n l 00;n l A ÀN g for the pach d ¼P l A Z 4ÂV 2n l l ,we say that d is the subpatch of g if L ðþÞg I L ðþÞd ,L ðÀÞg I L ðÀÞd ,and denote g 1d .(see Figure 4).We continue to deﬁne the symbolic segment and the set of all of the ﬁnite formal sum of the segments analogously.A pair ðx ;i ÞA p Z 4ÂV 1means geometrically the positive segment i with the base-point x of P ,that is,ðx ;i Þ:¼f x þmp e i j 0a m a 1g :Let us deﬁne the set of all of the ﬁnite formal sum of the segment with the base-point l A p Z 4ÂV 1as follows:G 1:¼X l A p Z 4ÂV 1m l l j m l A Z ;a f l j m l 00g <y 8<:9=;:Then,G 1is the Z -module analogously.In Example,the set of symbolic parallelograms V 2is chosen asV 2¼f 251;153;154;253;452;354g(see Figure 2).1.3.The maps E 2ðy Þand E 1ðy Þgenerated by an automorphism y of thefree group F h 1;2;3;4iIn this section,we consider the covering of P by parallelograms with base-points.The map E i ðy Þð0a i a d Þis deﬁned in [7]and has led many results in the Pisot case.From this fact,we introduce maps E 2ðy Þ:G 2!G 2and E 1ðy Þ:G 1!G 1for making the covering rule of P by parallelograms with base-points in the non-Pisotcase.Fig.4.Patch d is the subpach of g .293Tilings from non-Pisot matricesLet F h1;2;3;4i be the free group on the alphabet f1;2;3;4g and let s be an automorphism of F h1;2;3;4i.For an automorphism s,we deﬁne the automorphism y as the mirror image of sÀ1and we denoteyðiÞ:¼wðiÞ1wðiÞ2...wðiÞkÀðiÞkþ1...wðiÞl i¼PðiÞk wðiÞkSðiÞkwhere l i is the length of yðiÞ,PðiÞ1¼e(the empty word)and SðiÞl i¼e for anyi A f1;2;3;4g.We call PðiÞk the k-preﬁx and SðiÞkthe k-su‰x of the element yðiÞof the free group F h1;2;3;4i respectively(see[7]).Remark.y is the mirror image of sÀ1,that is,yðiÞ¼wðiÞ1wðiÞ2...wðiÞl iÀ1wðiÞl iis given by sÀ1ðiÞ¼wðiÞl iwðiÞl iÀ1...wðiÞ2wðiÞ1:The natural homomorphism f:F h1;2;3;4i!Z4is deﬁned by fði aÞ¼a e i for a A Z and fðw1w2Þ¼fðw1Þþfðw1Þ.For an automorphism s,the corresponding linear representation(or in-cidence matrix)of s is given byL s:¼½fðsð1ÞÞ;fðsð2ÞÞ;fðsð3ÞÞ;fðsð4ÞÞ :Then,the commutative relationL s f¼f sholds.From now on,we assume that the incidence matrix L s of s coincides with A.Remark.For any unimodular matrix A,there exists an automorphism s of the free group F h1;2;3;4i such that L s¼A by Theorem7.3.4in[12].Let us deﬁne the map E2ðyÞon G2as follows:E2ðyÞð0;i5jÞ:¼ð0;yðiÞ5yðjÞÞ¼X1a k a l i1a l a l j ðfðPðiÞkÞþfðPðjÞlÞ;wðiÞk5wðjÞlÞðmod V2ÞE2ðyÞðx;i5jÞ:¼AÀ1xþE2ðyÞð0;i5jÞE2ðyÞXl A p Z4ÂV2m l l:¼Xl A p Z4ÂV2m l E2ðyÞl:ð1:3:2Þ294Maki FurukadoRemark .For two oriented parallelograms ðx ;p 5q Þand ðy ;r 5s Þ,p ;q ;r ;s A f 1G 1;2G 1;3G 1;4G 1g ,we sayðx ;p 5q Þ1ðy ;r 5s Þðmod V 2Þif two oriented parallelograms coincide each other including the orientation.The formula (1.3.2)means that the elements of ð0;y ði Þ5y ðj ÞÞare rewritten by G ðx ;k 5l Þ,k 5l A V 2.For example,ð0;i 5j Þ1ð0;i 5j Þðmod V 2Þð0;i À15j Þ1ÀðÀe i ;i 5j Þðmod V 2Þð0;i 5j À1Þ1ÀðÀe j ;i 5j Þðmod V 2Þð0;i À15j À1Þ1ðÀe i Àe j ;i 5j Þðmod V 2Þ(see Figure 5).By the way,for the positive oriented parallelogram ð0;i 5j Þ,in most cases,the patch E 2ðy Þð0;i 5j Þincludes the negative oriented parallelograms.To clarify this fact,we introduce the concept of the matrix A Ãasfollows.Fig.5.295Tilings from non-Pisot matricesLet us denote AÀ1¼½a ij 1a i;j a4and deﬁne the matrix AÃ:¼½aÃi5j;k5l i5j;k5l A V2byaÃi5j;k5l :¼deta ik a ila jk a jl!:¼“the number of the positive parallelogram i5j”À“the number of the negative parallelogram j5i”in the patch E2ðyÞð0;k5lÞ(see[11]).Then we know that AÃis not always AÃb O and that AÃmust be positive if all elements of E2ðyÞð0;i5jÞare positive.Let us deﬁne the map E1ðyÞon G1analogously,E1ðyÞð0;iÞ:¼ð0;yðiÞÞ¼X1a k a l i ðfðPðiÞkÞ;wðiÞkÞðmod V1ÞE1ðyÞðx;iÞ:¼AÀ1p xþE1ðyÞð0;iÞE1ðyÞXl A p Z4ÂV1m l l0 @1A:¼Xl A p Z4ÂV1m l E1ðyÞl:For two oriented segmentsðx;pÞandðy;rÞ,p;r A f1G1;2G1;3G1;4G1g,we sayðx;pÞ1ðy;rÞðmod V1Þanalogously.Let us deﬁne the boundary map q:G2!G1qð0;i5jÞ:¼ð0;iÞþðe i;jÞÀðe j;iÞÀð0;jÞqðx;i5jÞ:¼p xþqð0;i5jÞ:Then,we know the following lemma.Lemma 1.1([7]).The commutative diagram holds:q G2 !E2ðyÞG2 ???y???yG1 !E1ðyÞG1q:296Maki FurukadoIn Example,from the matrix A ¼10010011011010112666437775let us choose the automorphism s :1!142!33!4234!1428>>><>>>:,then the mirror image y of s À1is determined by s À1:1!423À12!1À143!24!32À14À118>>><>>>:;y :1!3À1242!41À13!24!14À12À138>>><>>>::Then,the map E 2ðy Þis given byE 2ðy Þð0;251Þ¼ð0;y ð2Þ5y ð1ÞÞ¼ð0;41À153À124Þ¼ð0;453À1Þþðf ð3À1Þ;452Þþðf ð3À12Þ;454Þþðf ð4Þ;1À153À1Þþðf ð4Þþf ð3À1Þ;1À152Þþðf ð4Þþf ð3À12Þ;1À154Þðmod V 2Þ¼ðÀe 3;354ÞþðÀe 3;452Þþðe 4Àe 1Àe 3;153Þþðe 4Àe 3Àe 1;251ÞÀðe 4Àe 3þe 2Àe 1;154Þ:Analogously,we obtainE 2ðy Þð0;153Þ¼ðÀe 3;253Þþðe 2Àe 3;452ÞE 2ðy Þð0;154Þ¼ðÀe 3;153ÞþðÀe 3;251ÞÀðe 2Àe 3;154ÞE 2ðy Þð0;253Þ¼ð0;452ÞþðÀe 1þe 4;251ÞE 2ðy Þð0;452Þ¼ðe 1Àe 2Àe 4;452ÞþðÀe 2;251Þþðe 1Àe 2Àe 4;354ÞþðÀe 2;153ÞE 2ðy Þð0;354Þ¼ð0;251Þþðe 1Àe 4;452Þþðe 1Àe 2Àe 4;253Þ(see Figure 6).The colors of the positive and negative oriented parallelograms are gray and black respectively in this paper.From the previous calculation for E 2ðy Þð0;i 5j Þ,i 5j A V 2,A Ãis given by297Tilings from non-Pisot matricesFig.6.ð0;i 5j Þand E 2ðy Þð0;i 5j Þ,i 5j A V 2in Example.298Maki FurukadoAÃ¼101111101010À10À100001000111011110001026666666643777777775:Moreover,the map E1ðyÞis given byE1ðyÞð0;1Þ¼ð0;yð1ÞÞ¼ð0;3À124Þ¼ð0;3À1Þþðfð3À1Þ;2Þþðfð3À12Þ;4Þðmod V1Þ¼ÀðÀe3;3ÞþðÀe3;2ÞþðÀe3þe2;4Þ: Analogously,we obtainE1ðyÞð0;2Þ¼ð0;4ÞÀðe4Àe1;1ÞE1ðyÞð0;3Þ¼ð0;2ÞE1ðyÞð0;4Þ¼ð0;1ÞÀðe1Àe4;4ÞÀðe1Àe4Àe2;2Þþðe1Àe4Àe2;3Þ(see Figure7).2.The seed U of E2ðyÞand the covering substitutionThe patch g¼P Lk¼1m kðx k;d kÞA G2,m k00,that is,the formalﬁnitesum of oriented parallelograms,can be considered as the compact set g¼6Lk¼1ðx k;d kÞH P.We should take care thatthe boundary of the patch g0the boundary of the compact set g (see Figure8).In this section,we consider the topological property of compact setsX i5j:¼limn!y A n E n2ðyÞð0;i5jÞ;i5j A V2:Definition 2.1.If we canﬁnd the patch U:¼P Mk¼1m kðx k;d kÞA G2,m k¼G1which satisﬁes the following conditions,we say that U is the seed of E2ðyÞand that E2ðyÞis a covering substitution of P:(a)U C0,where U means the interior of the compact set U¼6Mk¼1m kðx k;d kÞ;(b)there exists N A N such that E N2ðyÞU1U;(c)dðqðE n2ðyÞUÞ;0Þ!yðn!yÞ.Remark .(1)Even if there exists a seed U of E 2ðy Þsuch that E N2ðy ÞU 1U ,we cannot say that E nN2ðy ÞU 1E ðn À1ÞN 2ðy ÞU for all n ,because of the inﬂuence that E 2ðy ÞnN U has often negative parallelograms which bring about cancellations.(2)Our conjecture is that there exists a ﬁxed point ^F,which is the inﬁnite sum of parallelograms,that is,b ^F:^F ¼E 2ðy Þ^F and ^F might be givenby Fig.7.ð0;i Þand E 1ðy Þð0;i Þ,i A V 1in Example.lim n !y E n2ðy ÞU .But we leave it at the moment and we claim that we can set up the fractal tiling in this paper.Lemma 2.1.The limit set X i 5j :¼lim n !y A n E n2ðy Þð0;i 5j Þ,i 5j A V 2exists in the sense of the Hausdor¤metric on P.Proof .We put d :¼maxi 5j A V 2max g :g 0E 2ðy Þð0;i 5j Þf D ðA À1ð0;i 5j Þ;g Þg8<:9=;where D is the Hausdor¤metric on P .On the notation E n2ðy Þð0;i 5j Þ¼P L ðn Þk ¼1ðx ðn Þk ;d ðn Þk Þand the fundamental property of the Hausdor¤metric such that D ðA 1U A 2;B 1U B 2Þa max f D ðA 1;B 1Þ;D ðA 2;B 2Þg (see [5]),we see that D ðA À1E n 2ðy Þð0;i 5j Þ;E n þ12ðy Þð0;i 5j ÞÞ¼D A À16L ðn Þk ¼1ðx ðn Þk ;d ðn Þk Þ;E 2ðy ÞX L ðn Þk ¼1ðx ðn Þk ;d ðn Þk Þ! !¼D 6L ðn Þk ¼1A À1ðx ðn Þk ;d ðn Þk Þ;6L ðn Þk ¼1ðE 2ðy Þðx ðn Þk ;d ðn Þk ÞV E n þ12ð0;i 5j ÞÞ!amax 1a k a L ðn Þf D ðA À1ðx ðn Þk ;d ðn Þk Þ;E 2ðy Þðx ðn Þk ;d ðn Þk ÞV E n þ12ðy Þð0;i 5j ÞÞg a dwhere g V d ¼fðx ;i 5j Þj ðx ;i 5j ÞA g and ðx ;i 5j ÞA d g .By the operation ofA n þ1,wehaveFig.8.DðA n E n2ðyÞð0;i5jÞ;A nþ1E nþ12ðyÞð0;i5jÞÞa l nþ1dwhere l¼max fj l3j;j l4jg<1.Therefore,the limit set X i5j exists.rCorollary 2.1.Let U¼P Mk¼1m kðx k;d kÞbe a seed of E2ðyÞ,then thelimit set X asX:¼limn!y A n E n2ðyÞUexists andX¼6Mk¼1m kðx kþX dkÞwhere X dk¼lim n!y A n E n2ðyÞð0;d kÞ.The proof is obtained analogously with Lemma2.1.Lemma 2.2.The compact set X satisﬁes X 0q.Proof.We put C i the‘fractal’curve generated by E1ðyÞfrom the segmentð0;iÞ,r i the Hausdor¤metric D between C i andð0;iÞ,and E i the r i-neighbors of the segment ofð0;iÞ,that is,C i:¼limn!y A n E n1ðyÞð0;iÞ;r i:¼DðC i;ð0;iÞÞ;E i:¼f z j dðz;ð0;iÞÞa r i g;and moreover we put r:¼max1a i a4f r i g.Then,it is clear that C i H E i.Let C:¼lim n!y A n E n1ðyÞq U,then from the fact that A is continuous,we knowAÀm C¼limn!y A nÀm E nÀm1ðyÞE m1ðyÞq U for any m:ð2:0:3ÞOn the notationE m 1ðyÞq U¼X RðmÞk¼1ðxðmÞk;iðmÞkÞ;we can write the relation(2.0.3)byAÀm C H6RðmÞk¼1limn!yA nÀmðE nÀm1ðyÞðxðmÞk;iðmÞkÞÞ:Therefore,we haveA Àm C H6ðx ðm Þk ;i ðm Þk ÞA E m1ðy Þq Uðx ðm Þk þE i ðm ÞkÞandd ðA Àm C ;0Þb d 6ðx ðm Þk ;i ðm Þk ÞA E m 1ðy Þq Uðx ðm Þk þE i ðm ÞkÞ;00@1A b d ðE m1ðy Þq U ;0ÞÀr :From the assumption of Deﬁnition 2.1(c),if we take m satisfyingd ðE m1ðy Þq U ;0Þb 2r ;thend ðA Àm C ;0Þb r :Therefore,there exists N such that n b N impliesd ðA Àm þn E n1ðy Þq U ;0Þ>r 2:This means that the compact set A Àm A n E n 2ðy ÞU satisﬁes A Àm A n E n2ðy ÞU I B 0;r 2ÀÁfor all n b N ,where B 0;r 2ÀÁis a ball of the center 0and the radius r2.Therefore,we have A Àm X I B 0;r 2ÀÁand X I A m B 0;r2ÀÁ.rIn Example,let us choose the patch U on the plane P by U :¼ðe 1Àe 2À2e 3þe 4;251Þþðe 1Àe 2À2e 3þe 4;153ÞÀðe 1Àe 2Àe 3;154Þþð2e 1Àe 2À2e 3;253Þþð2e 1Àe 2À2e 3;452Þþð2e 1Àe 2À2e 3;354Þþðe 1À2e 2Àe 3;251Þ:Then,we can see that the patch U satisﬁes the seed condition of E 2ðy Þin Deﬁnition 2.1(a),(b)as N ¼1,(c)(see Figure 9).3.Blocking 3.1.Blocking patchTo treat the map E 2ðy Þwhich generates not only positive orientated parallelograms but also negative ones from the positive parallelogram,we use the new idea ‘‘blocking’’in this section.Definition 3.1.Let B be the family of the ﬁnite number of patchesg p ,B :¼f g p j 1a p a K ;g p A G 2g :If B satisﬁes the following conditions,we call B a family of blocking patchesassociated with s and we say s satisﬁes the blocking condition.(1)For each p,there exist a translation vectors x ðp Þk A p Z 4and the patchg V ðp ÞkA B such thatE 2ðy Þg p ¼X L p k ¼1ðx ðp Þk þg V ðp ÞkÞH P;Fig.9.q ðE n 2ðy ÞU Þand E n2ðy ÞU ,n ¼0;1;2;3in Example.that is,the patch E 2ðy Þg p can be decomposed by the translation of patches f g V ðp Þk g 1a k a L p ,g V ðp ÞkA B;(2)Let ^M¼½^m pq 1a p ;q a K be a K ÂK non-negative integer matrix given by ^mpq :¼a k j g V ðq Þk¼g p ;E 2ðy Þg q ¼X L qk ¼1ðx ðq Þk þg V ðq ÞkÞ():We assume that ^Mis primitive,i.e.,b M 1:^M M 1>O and that the maximal eigenvalue of ^Mcoincides with j l 1j Áj l 2j where j l 1j and j l 2j are the absolute values of the eingevalues of A satisfying j l 1j b j l 2j >1>j l 3j b j l 4j .Wecall ^Mthe incidence matrix of E 2ðy Þin the sense of blocking;(3)There exists a seed U of E 2ðy Þgiven by Deﬁnition 2.1such that(a)U is decomposed in the sense of blocking,i.e.,b i 1;i 2;...;i L Af 1;2;...;Kg :g i k A B,U ¼P Lk ¼1ðx i k þg i k Þ.To distinguish U ¼P Mk ¼1m k ðx k ;d k Þwhich is constructed by the parallelogram,we denote U ¼P Lk ¼1ðx i k þg i k Þ;(b)there exist g p A B,N 0A N and z A p Z 4such thatE N2ðy Þg p 1z þU ðin the sense of blocking Þ:Remark .About the condition (2),if the seed U is constructed by 6pieces parallelograms ðx i 5j ;i 5j Þ,i 5j A V 2and moreover all the elements ofE 2ðy Þð0;i 5j Þ,i 5j A V 2are positive,then the matrix ^Mcoincides with A Ãand the maximal eigenvalue of ^Mis equal to j l 1j Áj l 2j .In Example,let us introduce the family of blocking patches B ¼f g i g 1a i a 6associated with s byg 1:¼ð0;251Þg 2:¼ð0;153Þg 3:¼Àð0;154ÞþðÀe 2;251Þg 4:¼ð0;253Þg 5:¼ð0;452Þg 6:¼ð0;354Þ8>>>>>>>><>>>>>>>>::Then,the covering substitution E 2ðy Þfor g i ,1a i a 6is represented by the following in the sense of blocking:E 2ðy Þg 1:¼ðÀe 3þg 6ÞþðÀe 3þg 5Þþðe 4Àe 1Àe 3þg 2Þþðe 4Àe 3þe 2Àe 1þg 3ÞE 2ðy Þg 2:¼ðÀe 3þg 4Þþðe 2Àe 3þg 5ÞE 2ðy Þg 3:¼ðe 1Àe 3Àe 4þg 6Þþðe 1Àe 3Àe 4þg 5Þð3:1:4ÞE2ðyÞg4:¼g5þðÀe1þe4þg1ÞE2ðyÞg5:¼ðe1Àe2Àe4þg5ÞþðÀe2þg1Þþðe1Àe2Àe4þg6ÞþðÀe2þg2ÞE2ðyÞg6:¼g1þðe1Àe4þg5Þþðe1Àe2Àe4þg4Þ(see Figure10).Therefore,the incidence matrix^M of E2ðyÞin the sense of blocking is given by^M¼2666666664g1000111 g2100010 g3100000 g4010001 g5111111 g61010103777777775and we know the maximal eigenvalue l^M of^M coincides with l^M¼j l1jÁj l2j¼3:18....Let us consider the family of patches U instead of U byU:¼ðe1Àe2À2e3þe4þg1Þ;ðe1Àe2À2e3þe4þg2Þ;ðe1Àe2Àe3þg3Þ;ð2e1Àe2À2e3þg4Þ;ð2e1Àe2À2e3þg5Þ;ð2e1Àe2À2e3þg6Þ():Then,we can see that B satisﬁes Deﬁnition3.1(1),(2),(3)(a)(b)as N0¼1 (see Figure11).3.2.Graph of the blockingFrom the formula E2ðyÞg p¼P L pk¼1ðxðpÞkþgVðpÞkÞ,we put twoﬁnite sets Vcalled vertices and E called edges,and two functions i:E!V and t:E!V as follows;V:¼f g1;g2;...;g K g F f1;2;...;K gE:¼pkj1a p a K;VðpÞkA f1;2;...;K g;1a k a L p &' ipk:¼p;tpk:¼VðpÞk:Then,we obtain the directed graph G:¼f V;E;i;t g and the set of its ad-missible sequence as。