Open Problems in Algebraic Statistics
从打好基础开始作文

从打好基础开始作文英文回答:To start with, building a strong foundation is crucial in any endeavor. Just like constructing a house, a solid foundation is necessary to ensure stability and longevity. In the context of education, laying a strong foundation means acquiring a solid understanding of fundamental concepts and skills. This can be achieved through a combination of studying, practice, and hands-on experience.For example, in language learning, it is essential to first master the basics of grammar, vocabulary, and pronunciation. Without a strong foundation in these areas, it becomes difficult to communicate effectively and comprehend more complex language structures. By dedicating time and effort to learning and practicing the basics, one can build a strong linguistic foundation that will serve as a solid base for further language development.Similarly, in subjects like mathematics and science, a strong foundation is necessary to comprehend advanced concepts and solve complex problems. For instance, in mathematics, understanding basic arithmetic operations, algebraic equations, and geometric principles is essential before moving on to more advanced topics like calculus or statistics. Without a strong foundation in these fundamental concepts, it becomes challenging to grasp and apply more complex mathematical principles.In conclusion, building a strong foundation is crucial in any endeavor, including education. By focusing on mastering the basics and acquiring a solid understanding of fundamental concepts and skills, individuals can lay the groundwork for further growth and development. Just as a house needs a solid foundation to stand strong, a strong educational foundation is essential for success in any field.中文回答:首先,打好基础对于任何事情都至关重要。
小学数学六年级教材培训讲稿

小学数学六年级教材培训讲稿小学数学六年级教材培训讲稿(上册)研读教材,把握课堂教学的落脚点,研读学生,重视数学思考的过程性》亲爱的老师朋友们:在这里,我不是专家学者,我们是同行是朋友。
今天非常荣幸由我来和大家一起研究材法。
通过一段时间的准备并结合自己的教学实践和思考,我想对六年级两册教材的教学作出自己的理解和建议,并希望能为大家教学好全年册教材作好参谋。
教学是一门艺术”,我对这门艺术的研究,暂定为两个研读,即:“研读教材、研读学生”。
因此本次交流的题目定位为“研读教材,把握课堂教学的落脚点;研读学生,重视数学思考的过程性。
”想法是:教材是理念的文本体现,是实施教学的一个载体,需要研读;学生是从师生共同的活动中去活化教材,赢得自主发展,这是目的,需要研读。
在讲解之前,我想给大家几点建议:1.认真研究教师教学用书,因为“教师用书”将教材的编写意图、教学方式的建议、课时的划分等都写得非常清楚。
2.认真研究教材,并结合教师用书和自己对教材的理解,才能科学地预设好适合本地学生研究的教学策略。
3.我们教学的对象是小学高年级的学生,绝对不是没有任何数学经验的生命体,因此动手操作、自主探索、合作交流、迁移类推、猜测验证等研究方式应该成为他们研究数学的重要方式。
引言新一轮课改以来,部分深入课堂观察的教育理论工作者发现:“课堂教学改革要求教师具备解读文本、解读学生的智慧……”怎样解读文本?遵循从整体到局部的思路。
首先从整体上把握教材:研究“课标”;了解教材编排体系、编写特点;了解教学内容的承前启后。
然后从科学性(知识的、编写的)、思想性(数学的、人文的)、趣味性等方面进行单元分析。
在此基础上进行课时设计。
一、这一册教材包括下面一些内容:1.数与代数2.分数乘法3.分数除法4.圆5.百分数分数乘法和除法、圆、百分数等是本册教材的重点教学内容。
二、教材的编写特点由于教学内容的不同,本册教材还具有下面几个明显的特点:1.空间与图形2.位置3.统计与概率4.数学思想方法5.数学综合应用6.统计—扇形7.数学广角---鸡确定起跑线统计图、兔同笼已删除。
spss大学考试题及答案

spss大学考试题及答案一、选择题(每题2分,共20分)1. 在SPSS中,以下哪项不是数据视图(Data View)中的数据属性?A. 数字B. 日期C. 图片D. 标签答案:C2. SPSS中,用于描述性统计分析的命令是:A. AnalyzeB. TransformC. GraphD. File答案:A3. 在SPSS中,要进行t检验,应该选择以下哪个菜单?A. Analyze > Compare MeansB. Analyze > RegressionC. Analyze > Descriptive StatisticsD. Analyze > Nonparametric Tests答案:A4. 在SPSS中,如果需要计算一个变量的总和,应该使用以下哪个功能?A. ComputeB. AggregateC. AlgebraicD. Recode答案:B5. 在SPSS中,以下哪个命令用于因子分析?A. FactorB. ClusterC. Reliability AnalysisD. Canonical Correlation答案:A6. 要在SPSS中创建一个频率分布表,应该选择以下哪个命令?A. Analyze > Descriptive Statistics > FrequenciesB. Analyze > Descriptive Statistics > DescriptivesC. Analyze > Descriptive Statistics > ExploreD. Analyze > Descriptive Statistics > Crosstabs答案:A7. 在SPSS中,如果需要对数据进行排序,应该使用以下哪个命令?A. Sort CasesB. Rank CasesC. Order CasesD. Arrange Cases答案:A8. 在SPSS中,要进行卡方检验,应该选择以下哪个菜单?A. Analyze > Descriptive Statistics > CrosstabsB. Analyze > Compare Means > Independent-Samples T TestC. Analyze > Nonparametric Tests > Chi-SquareD. Analyze > Regression > Binary Logistic答案:C9. 在SPSS中,以下哪项不是数据录入时的变量属性?A. 变量类型B. 变量标签C. 缺失值D. 数据格式答案:D10. 在SPSS中,要进行相关性分析,应该选择以下哪个命令?A. Analyze > CorrelationB. Analyze > RegressionC. Analyze > FactorD. Analyze > Cluster答案:A二、简答题(每题5分,共30分)1. 描述SPSS中的数据录入过程。
对内存溢出comsol官方的解释

下面是comsol 官方的解释:原因有来自硬件,但也有来自软件的设置,请参考It is a delicate task to give a general rule how much memory is needed for a specific problem. It depends on Number of nodes in the geometry mesh (base mesh). Type of shape function, (for example 2nd order Lagrange elements). The shape function type together with the geometry mesh determines the size of computational mesh (extended mesh). Number of dependent and independent variables. Sparsity of system matrices. This is in turn governed by the shape of the geometry and mesh. An extended ellipsoid gives sparser matrices than a sphere. Spheres and cubes are the toughest shapes with regards to memory demand. Further, the degree of coupling between different equations affect sparsity. From the last item above we understand that the amount of memory required to solve a problem depends on the number of non-zero entries in the Jacobian matrix, not the number of degrees of freedom. For example, for a non-isothermal flow problem coupled to the conduction-convection equation and ideal gas law equation of state, all 5 dependent variables (u, v, w, p, T) appear in all 5 equations (in 3D). Hence the Jacobian matrix will be much fuller than a simple heat conduction problem. This is also the reason why solving a thermal radiation problems often uses up a lot of memory. The contribution to the Jacobian matrix from any surface elements creates a block in the matrix which is full. Read more about the definition of Degrees of Freedom (DOF) in COMSOL Multiphysics. Selecting Proper Solver Type The solvers in COMSOL Multiphysics always break down each problem into the solution of one or several linear systems of equations. Thus, in all the solvers the choice between direct and iterative solvers for linear systems affects the solution time and memory requirements. The direct solvers solve a linear system by Gaussian elimination. This stable and reliable process is well suited even for for ill-conditioned systems. These solvers require less tuning and are often faster than the iterative solvers in 1D and 2D, where they are the default setting. The elimination process sometimes requires large memory resources and long computing times, an effect particularly noticeable for 3D models. COMSOL Multiphysics 3.3 support automatic detection for symmetric matrices system, by default the option "Automatic" is selected. However if your linear solver does not take advantage of symmetry, such as UMFPACK, you won't see any improvement. (Note: Symmetric in this case does not mean a symmetric geometry. Rather, it refers to the Jacobian matrix being symmetric along the diagonal, see further COMSOL Multiphysics User's Guide, section Solving the Model). The default setting for 3D calls on iterative solvers. They generally use less memory than direct solvers and are often faster in 3D. For optimal performance, though, the iterative solvers need the careful selection of a preconditioner. Some examples of preconditioners are Incomplete LU Algebraic and Geometric Multigrid SSOR Vector et.c. For large problems, it is imperative to choose a preconditioner that is suitable for the problem type at hand. Please refer to the COMSOL Multiphysics 3.3 User's guide, section The Linear System Solver for more information. Efficient Geometry Modeling In general, the first step is to try to reduce the model geometry as much as possible. Often you can find symmetry planes and therefore reduce the model to 1/2 or 1/4 or even 1/8 of the original size. The memory usage does not scale linearly, rather polynomially (A*n^k, k>1), where A is a constant, n the number of degrees of freedom, and k a real number describing the polynomial order. The value of k may vary depending on the geometry dimension (1D, 2D, or 3D) and solver settings, but is always greater than one. This means that you save more than half the memory if you find a symmetryplane and cut the geometry size by half. Note that symmetry should be for both the geometry and the physics. Avoid small geometry objects where they are not needed (for curved segments, use one Bézier-curve instead many small line segments, for example) Use interactive meshing and assembly in order to generate the mesh. This will give you more flexibility as the nodes of the element between subdomains does not need to be connected anymore. This allow you to drop significantly the number of elements. Also, the quality of the mesh is important if an iterative solver is used. Convergence will be faster and more robust if the element quality is high. If an angle in a mesh element corner approaches 180 degrees, the quality will be too low. This is often the case for triangular or tetrahedron elements in thin layer regions. Quad and hexahedron elements can also have low quality in some cases. If you select Mesh/Statistics, you will see the Minimum element quality. It should not be lower that 0.01. The problem can be that the triangles get very thin when they approach the sharp corners of the geometry, leading to poor element quality there. Hardware and software settings Benefit of a 64-bit architecture A 32-bit processor system (e.g., Windows platform) can potentially allocate up to 4 GB of memory, but in most practical cases only 1.5 GB due to operating system footprint. This is regardless of how big you set the swap space. Some operating systems require special settings to be able to use high amounts of memory. COMSOL Multiphysics comes in an optional 64-bit version. Using a 64-bit processor with a 64-bit operating system enables you to address substantially more RAM memory than 4 GB. Using a 64-bit platform will not prevent the out-of-memory message all the time, but at least you can be sure that all memory is used by COMSOL Multiphysics (unlike a 32-bit architecture where this is not always possible). If using a 64-bit machine is not an option, see the paragraph below about virtual memory and swap space allocation on 32-bit architecture. Software Settings There are several ways of increasing the available memory for the solvers. Reducing the Java heap size will make more memory available for the solvers. When running COMSOL Multiphysics, you can decrease the maximum Java heap size. On Windows, modify the file COMSOL33/bin/comsol.opts and on UNIX/Linux/Mac OSX, modify the file COMSOL33/bin/comsol. Decrease the value of parameter MAXHEAP=256m from 256MB to a lower value, say 128, by changing to MAXHEAP=128m. By default, you get more memory available in a COMSOL Multiphysics server. Connect to the COMSOL Multiphysics server from a COMSOL Multiphysics client. You can also modify the Java heap size on a COMSOL Multiphysics server: On Windows, modify the file COMSOL33/bin/comsol.opts and on UNIX/Linux/Mac OSX, modify the file COMSOL33/bin/comsol. Decrease the value of parameter MAXHEAPSERVER=128m from 128MB to a lower value, say 64, by changing to MAXHEAPSERVER=64m. A corollary of this is that many users report that they get more free memory by running a COMSOL Multiphysics server and a COMSOL Multiphysics client on the same machine. If you run out of memory during postprocessing, you can increase the maximum Java heap size. On Windows, modify the file COMSOL33/bin/comsol.opts and on UNIX/Linux/Mac OSX, modify the file COMSOL33/bin/comsol. Increase the value of parameter MAXHEAP=256m from 256MB to a higher value, say 512, by changing to MAXHEAP=512m. Run client server mode To run a COMSOL Multiphysics server and a COMSOL Multiphysics client on different computers, you need a floating network license. All COMSOL Multiphysics licenses allow you to run a COMSOL Multiphysics server and a COMSOL Multiphysics client on the same computer. If you run out of memory when saving an mph-file from the COMSOL Multiphysics graphical user interface when you are running COMSOL Multiphysics with MATLAB you might have to increase the Javaheap size within MATLAB. When you are running COMSOL Multiphysics with MATLAB the Java heap within MATLAB will function like the Java heap within the COMSOL Multiphysics server. To set the Java heap size in MATLAB, create a java.opts file with the contents -Xmx256m for example, to set the maximum Java heap to 256MB. Create java.opts either in the current directory or in the directory MATLAB/bin/ARCH, where MATLAB is the MATLAB installation directory, e. g., C:\MATLAB71, and ARCH is the architecture, win32, win64, glnx86, glnxa64, sol2, or mac. See also how to use parallel processors in Knowledgebase solution 1001. Virtual Memory and Swap Space for 32-bit architecture Increasing the swap file size on the system is only interesting if you can live with the notable slowness in operation that occurs as soon as the hard disk swapping starts. In many cases, hard disk swapping is not an option. In Windows XP, you only have 2 GB of virtual address space available initially for a user process. This memory space is also sometimes fragmented by shared libraries. On Linux, UNIX, and Mac, you typically get between 3-4 GB of virtual address space available for your process. This address space is usually not fragmented. COMSOL Multiphysics supports the option in Windows XP Professional and Windows 2003 Server to have 3 GB of virtual address space available user processes. This reduces the available address space for the operating system to 1 GB. To access it you must boot the system using an additional boot parameter. The following example shows how to enable application memory tuning by adding the /3GB parameter in the boot.ini file: [boot loader] timeout=30 default=multi(0)disk(0)rdisk(0)partition(2)\WINNT[operating systems] multi(0)disk(0)rdisk(0)partition(2)\WINNT="????" /3GB The "????" above can be the name of your operating system for example "Microsoft Windows XP Professional". Please note that the file boot.ini is critical to booting your system. To access the boot.ini file make sure that the Windows system file are not hidden. Use Notepad to edit the file, other text editor may use different character mark which can cause the computer fail to boot. Note about memory monitoring Especially in 3D modelling, extensive memory usage urges some extra precautions. The "Out of Memory"-problem occurs whenever COMSOL Multiphysics tries to allocate an array which does not fit sequentially in memory. It is common that the amount of available memory seems large enough for the array - but there may exist no continuous block of that size, due to fragmentation. If you for example look in the task manager of Windows, this may have the effect that you can't see any high memory allocation, and get "out of memory" anyway. The "Memory Usage" column in the Windows task manager can also be misleading, because it only shows the currently swapped portion. A better indicator of total usage can be found like this (Win2000, WinNT, Win XP): Press Ctrl-Alt-Delete to bring up the Task Manager. Go to the "processes" tab. From the View menu, choose Select Columns. Select the Virtual Memory Size check box. Now find the comsol.exe process that corresponds to the COMSOL Multiphysics process in the process list and look in the "VM Size" column. This shows a better value for the actual memory used.。
ap课程数学教材

ap课程数学教材根据我的了解,AP数学课程主要有以下几个方向,分别是AP Calculus AB(微积分AB)、AP Calculus BC(微积分BC)、AP Statistics(统计学)和AP Computer Science A(计算机科学A)。
关于这些课程的教材,我可以为你提供一些建议。
1. AP Calculus AB(微积分AB):常用的教材包括《Calculus: Graphical, Numerical, Algebraic》(Thomas, Finney, Demana, Waits),《Calculus》(James Stewart),以及《Barron's AP Calculus》等。
2. AP Calculus BC(微积分BC):对于这门课程,你可以考虑使用《Calculus: Early Transcendentals》(James Stewart),《Barron's AP Calculus》或者《Calculus: Concepts and Contexts》(James Stewart)等教材。
3. AP Statistics(统计学):推荐的教材有《The Practice of Statistics》(Daren S. Starnes, Dan Yates, David S. Moore),《Barron's AP Statistics》以及《Introduction to the Practice of Statistics》(David S. Moore, George P. McCabe)等。
4. AP Computer Science A(计算机科学A):常用的教材包括《Big Java: Early Objects》(Cay S. Horstmann),《Barron's AP Computer Science A》以及《Java: How to Program》(Paul Deitel, Harvey Deitel)等。
数学天地的英文

数学天地的英文Mathematical WorldMathematics, often referred to as the language of the universe, encompasses a vast and fascinating world. The study of numbers, shapes, patterns, and relationships not only helps us understand the world around us but also forms the foundation of many scientific and technological advancements. In this article, we will delve into the English vocabulary related to mathematics and explore the various branches and applications of this intriguing field.1. Numbers and OperationsNumbers are the building blocks of mathematics. From the basic integers to the complex realm of imaginary numbers, each numerical concept holds its own significance. Addition, subtraction, multiplication, and division are the fundamental operations that manipulate numbers and create mathematical expressions. Furthermore, concepts such as fractions, decimals, and percentages expand our understanding of numerical values and their representation in everyday life.2. Geometry and ShapesGeometry deals with the study of shapes and their properties. Euclidean geometry, named after the ancient Greek mathematician Euclid, is the most widely recognized branch. It explores the properties of lines, angles, curves, polygons, and three-dimensional figures. Geometric concepts, like symmetry and congruence, play essential roles in architectural design, art, and navigation systems.3. Algebra and EquationsAlgebra introduces variables and symbols into mathematical expressions and equations. By using algebraic techniques, we can solve equations, simplify expressions, and analyze patterns and relationships between variables. Algebraic concepts find applications in fields such as physics, engineering, and economics, providing tools to model and solve real-world problems.4. Calculus and AnalysisCalculus is a branch of mathematics that studies change and motion. It comprises differential calculus, which examines the rate of change, and integral calculus, which analyzes accumulation. Calculus has revolutionized the fields of physics and engineering, enabling the understanding of complex phenomena such as motion, acceleration, and rates of growth.5. Statistics and ProbabilityStatistics deals with data collection, analysis, interpretation, and presentation. It provides methods to make inferences, draw conclusions, and make informed decisions based on available information. Probability, a fundamental concept in statistics, quantifies the likelihood of events occurring. Statistics and probability are widely used in scientific research, social sciences, and business analytics.6. Applied MathematicsApplied mathematics bridges the gap between theory and real-world applications. This interdisciplinary field combines mathematical techniques with other areas like physics, computer science, and finance to solvepractical problems. Examples include mathematical modeling of climate patterns, optimization algorithms, cryptography, and financial forecasting.ConclusionMathematics is a universal language that transcends cultural and linguistic boundaries. Its concepts, vocabulary, and applications are valuable tools in many aspects of life. From counting and measuring to predicting and analyzing, mathematics permeates our everyday experiences. Embracing the beauty and power of mathematics opens doors to endless possibilities and deepens our understanding of the world we live in.Note: The word count of this article is 618 words. If you require additional content, please let me know.。
AlgebraicGraphTheory

Algebraic Graph TheoryChris Godsil(University of Waterloo),Mike Newman(University of Ottawa)April25–291Overview of the FieldAlgebraic graph theory comprises both the study of algebraic objects arising in connection with graphs, for example,automorphism groups of graphs along with the use of algebraic tools to establish interesting properties of combinatorial objects.One of the oldest themes in the area is the investigation of the relation between properties of a graph and the spectrum of its adjacency matrix.A central topic and important source of tools is the theory of association schemes.An association scheme is,roughly speaking,a collection of graphs on a common vertex set whichfit together in a highly regular fashion.These arise regularly in connection with extremal structures:such structures often have an unex-pected degree of regularity and,because of this,often give rise to an association scheme.This in turn leads to a semisimple commutative algebra and the representation theory of this algebra provides useful restrictions on the underlying combinatorial object.Thus in coding theory we look for codes that are as large as possible, since such codes are most effective in transmitting information over noisy channels.The theory of association schemes provides the most effective means for determining just how large is actually possible;this theory rests on Delsarte’s thesis[4],which showed how to use schemes to translate the problem into a question that be solved by linear programming.2Recent Developments and Open ProblemsBrouwer,Haemers and Cioabˇa have recently shown how information on the spectrum of a graph can be used to proved that certain classes of graphs must contain perfect matchings.Brouwer and others have also investigated the connectivity of strongly-regular and distance-regular graphs.This is an old question,but much remains to be done.Recently Brouwer and Koolen[2]proved that the vertex connectivity of a distance-regular graph is equal to its valency.Haemers and Van Dam have worked on extensively on the question of which graphs are characterized by the spectrum of their adjacency matrix.They consider both general graphs and special classes,such as distance-regular graphs.One very significant and unexpected outcome of this work was the construction,by Koolen and Van Dam[10],of a new family of distance-regular graphs with the same parameters as the Grassmann graphs.(The vertices of these graphs are the k-dimensional subspaces of a vector space of dimension v over thefinitefield GF(q);two vertices are adjacent if their intersection has dimension k1.The graphs are q-analog of the Johnson graphs,which play a role in design theory.)These graphs showed that the widely held belief that we knew all distance-regular graphs of“large diameter”was false,and they indicate that the classification of distance-regular graphs will be more complex(and more interesting?)than we expected.1Association schemes have long been applied to problems in extremal set theory and coding theory.In his(very)recent thesis,Vanhove[14]has demonstrated that they can also provide many interesting results in finite geometry.Recent work by Schrijver and others[13]showed how schemes could used in combination with semidef-inite programming to provide significant improvements to the best known bounds.However these methods are difficult to use,we do not yet have a feel for we might most usefully apply them and their underlying theory is imperfectly understood.Work in Quantum Information theory is leading to a wide range of questions which can be successfully studied using ideas and tools from Algebraic Graph Theory.Methods fromfinite geometry provide the most effective means of constructing mutually unbiased bases,which play a role in quantum information theory and in certain cryptographic protocols.One important question is to determine the maximum size of a set of mutually unbiased bases in d-dimensional complex space.If d is a prime power the geometric methods just mentioned provide sets of size d+1,which is the largest possible.But if d is twice an odd integer then in most cases no set larger than three has been found.Whether larger sets exist is an important open problem. 3Presentation HighlightsThe talks mostlyfitted into one of four areas,which we discuss separately.3.1SpectraWillem Haemers spoke on universal adjacency matrices with only two distinct eigenvalues.Such matrices are linear combinations of I,J,D and A(where D is the diagonal matrix of vertex degrees and A the usual adjacency matrix).Any matrix usually considered in spectral graph theory has this form,but Willem is considering these matrices in general.His talk focussed on the graphs for which some universal adjacency matrix has only two eigenvalues.With Omidi he has proved that such a graph must either be strong(its Seidel matrix has only two eigenvalues)or it has exactly two different vertex degrees and the subgraph induced by the vertices of a given degree must be regular.Brouwer formulated a conjecture on the minimum size of a subset S of the vertices of a strongly-regular graph X such that no component of X\S was a single vertex.Cioabˇa spoke on his recent work with Jack Koolen on this conjecture.They proved that it is false,and there are four infinite families of counterexamples.3.2PhysicsAs noted above,algebraic graph theory has many applications and potential applications to problems in quantum computing,although the connection has become apparent only very recently.A number of talks were related to this connection.One important problem in quantum computing is whether there is a quantum algorithm for the graph isomorphism problem that would be faster than the classical approaches.Currently the situation is quite open.Martin Roetteler’s talk described recent work[1]on this problem.For our workshop’s viewpoint,one surprising feature is that the work made use of the Bose-Mesner algebra of a related association scheme; this connection had not been made before.Severini discussed quantum applications of what is known as the Lov´a sz theta-function of a graph.This function can be viewed as an eigenvalue bound and is closely related to both the LP bound of Delsarte and the Delsarte-Hoffman bound on the size of an independent set in a regular graph.Severini’s work shows that Lov´a sz’s theta-function provides a bound on the capacity of a certain channel arising in quantum communication theoryWork in quantum information theory has lead to interest in complex Hadamard matrices—these are d×d complex matrices H such that all entries of H have the same absolute value and HH∗=dI.Both Chan and Sz¨o ll˝o si dealt with these in their talks.Aidan Roy spoke on complex spherical designs.Real spherical designs were much studied by Seidel and his coworkers,because of their many applications in combinatorics and other areas.The complex case languished because there were no apparent applications,but now we have learnt that these manifest them-selves in quantum information theory under acronyms such as MUBs and SIC-POVMs.Roy’s talk focussedon a recent 45page paper with Suda [12],where (among other things)they showed that extremal complex designs gave rise to association schemes.One feature of this work is that the matrices in their schemes are not symmetric,which is surprising because we have very few interesting examples of non-symmetric schemes that do not arise as conjugacy class schemes of finite groups.3.3Extremal Set TheoryCoherent configurations are a non-commutative extension of association schemes.They have played a sig-nificant role in work on the graph isomorphism problem but,in comparison with association schemes,they have provided much less information about interesting extremal structures.The work presented by Hobart and Williford may improve matters,since they have been able to extend and use some of the standard bounds from the theory of schemes.Delsarte [4]showed how association schemes could be used to derive linear programs,whose values provided strong upper bounds on the size of codes.Association schemes have both a combinatorial structure and an algebraic structure and these two structures are in some sense dual to one another.In Delsarte’s work,both the combinatorial and the algebraic structure had a natural linear ordering (the schemes are both metric and cometric)and this played an important role in his work.Martin explained how this linearity constraint could be relaxed.This work is important since it could lead to new bounds,and also provide a better understanding of duality.One of Rick Wilson’s many important contributions to combinatorics was his use of association schemes to prove a sharp form of the Erd˝o s-Ko-Rado theorem [15].The Erd˝o s-Ko-Rado theorem itself ([5])can certainly be called a seminal result,and by now there are many analogs and extensions of it which have been derived by a range of methods.More recently it has been realized that most of these extensions can be derived in a very natural way using the theory of association schemes.Karen Meagher presented recent joint work (with Godsil,and with Spiga,[8,11])on the case where the subsets in the Erd˝o s-Ko-Rado theorem are replaced by permutations.It has long been known that there is an interesting association scheme on permutations,but this scheme is much less manageable than the schemes used by Delsarte and,prior to the work presented by Meagher,no useful combinatorial information had been obtained from it.Chowdhury presented her recent work on a conjecture of Frankl and F¨u redi.This concerns families F of m -subsets of a set X such that any two distinct elements of have exactly λelements in common.Frankl and F¨u redi conjectured that the m -sets in any such family contain at least m 2 pairs of elements of X .Chowdhury verified this conjecture in a number of cases;she used classical combinatorial techniques and it remains to see whether algebraic methods can yield any leverage in problems of this type.3.4Finite GeometryEric Moorhouse spoke on questions concerning automorphism groups of projective planes,focussing on connections between the finite and infinite case.Thus for a group acting on a finite plane,the number of orbits on points must be equal to the number of orbits on lines.It is not known if this must be true for planes of infinite order.Is there an infinite plane such that for each positive integer k ,the automorphism group has only finitely many orbits on k -tuples?This question is open even for k =4.Simeon Ball considered the structure of subsets S of a k -dimensional vector space over a field of order q such that each d -subset of S is a basis.The canonical examples arise by adding a point at infinity to the point set of a rational normal curve.These sets arise in coding theory as maximum distance separable codes and in matroid theory,in the study of the representability of uniform matroids (to mention just two applications).It is conjectured that,if k ≤q −1then |S |≤q +1unless q is even and k =3or k =q −1,in which case |S |≤q +2.Simeon presented a proof of this theorem when q is a prime and commented on the general case.He developed a connection to Segre’s classical characterization of conics in planes of odd order,as sets of q +1points such that no three are collinear.There are many analogs between finite geometry and extremal set theory;questions about the geometry of subspaces can often be viewed as q -analogs of questions in extremal set theory.So the EKR-problem,which concerns characterizations of intersecting families of k -subsets of a fixed set,leads naturally to a study of intersecting families of k -subspaces of a finite vector space.In terms of association schemes this means we move from the Johnson scheme to the Grassmann scheme.This is fairly well understood,with thebasic results obtained by Frankl and Wilson[6].But infinite geometry,polar spaces form an important topic. Roughly speaking the object here is to study the families of subspaces that are isotropic relative to some form, for example the subspaces that lie on a smooth quadric.In group theoretic terms we are now dealing with symplectic,orthogonal and unitary groups.There are related association schemes on the isotropic subspaces of maximum dimension.Vanhove spoke on important work from his Ph.D.thesis,where he investigated the appropriate versions of the EKR problem in these schemes.4Outcome of the MeetingIt is too early to offer much in the way of concrete evidence of impact.Matt DeV os observed that a conjecture of Brouwer on the vertex connectivity of graphs in an association scheme was wrong,in a quite simple way. This indicates that the question is more complex than expected,and quite possibly more interesting.That this observation was made testifies to the scope of the meeting.On a broader level,one of the successes of the meeting was the wide variety of seemingly disparate topics that were able to come together;the ideas of algebraic graph theory touch a number of things that would at first glance seem neither algebraic nor graph theoretical.There was a lively interaction between researchers from different domains.The proportion of post-docs and graduate students was relatively high.This had a positive impact on the level of excitement and interaction at the meeting.The combination of expert and beginning researchers created a lively atmosphere for mathematical discussion.References[1]A.Ambainis,L.Magnin,M.Roetteler,J.Roland.Symmetry-assisted adversaries for quantum state gen-eration,arXiv1012.2112,35pp.[2]A.E.Brouwer,J.H.Koolen.The vertex connectivity of a distance-regular graph.European bina-torics30(2009),668–673.[3]A.E.Brouwer,D.M.Mesner.The connectivity of strongly regular graphs.European binatorics,6(1985),215–216.[4]P.Delsarte.An algebraic approach to the association schemes of coding theory.Philips Res.Rep.Suppl.,(10):vi+97,1973.[5]P.Erd˝o s,C.Ko,R.Rado.Intersection theorems for systems offinite sets.Quart.J.Math.Oxford Ser.(2),12(1961),313–320.[6]P.Frankl,R.M.Wilson.The Erd˝o s-Ko-Rado theorem for vector binatorial Theory,SeriesA,43(1986),228–236.[7]D.Gijswijt,A.Schrijver,H.Tanaka.New upper bounds for nonbinary codes based on the Terwilligeralgebra and semidefinite binatorial Theory,Series A,113(2006),1719–1731. [8]C.D.Godsil,K.Meagher.A new proof of the Erd˝o s-Ko-Rado theorem for intersecting families of per-mutations.arXiv0710.2109,18pp.[9]C.D.Godsil,G.F.Royle.Algebraic Graph Theory,Springer-Verlag,(New York),2001.[10]J.H.Koolen,E.R.van Dam.A new family of distance-regular graphs with unbounded diameter.Inven-tiones Mathematicae,162(2005),189-193.[11]K.Meagher,P.Spiga.An Erdos-Ko-Rado theorem for the derangement graph of PGL(2,q)acting onthe projective line.arXiv0910.3193,17pp.[12]A.P.Roy,plex spherical Codes and designs,(2011),arXiv1104.4692,45pp.[13]A.Schrijver.New code upper bounds from the Terwilliger algebra and semidefinite programming.IEEETransactions on Information Theory51(2005),2859–2866.[14]F.Vanhove.Incidence geometry from an algebraic graph theory point of view.Ph.D.Thesis,Gent2011.[15]R.M.Wilson.The exact bound in the Erds-Ko-Rado binatorica,4(1984),247–257.。
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。
OpenProblemsList

Open Problems ListArising from MathsCSP Workshop,Oxford,March2006Version0.3,April25,20061Complexity and Tractability of CSPQuestion1.0(The Dichotomy Conjecture)Let B be a relational structure.The problem of deciding whether a given relational structure has a homomorphism to B is denoted CSP(B).For which(finite)structures is CSP(B)decidable in polynomial time?Is it true that for anyfinite structure B the problem CSP(B)is either decidable in polynomial time or NP-complete?Communicated by:Tomas Feder&Moshe Vardi(1993) Question1.1A relational structure B is called hereditarily tractable if CSP(B )is tractable for all substructures B of B.Which structures B are hereditarily tractable?Communicated by:Pavol Hell Question1.2A weak near-unanimity term is defined to be one that satisfies the following identities:f(x,...,x)=x and f(x,y,....y)=f(y,x,y,....y)=...=f(y,...,y,x).Is CSP(B)tractable for any(finite)structure B which is preserved by a weak near-unanimity term?Communicated by:Benoit Larose,Matt Valeriote Question1.3A constraint language1S is called globally tractable for a problem P,if P(S)is tractable,and it is called(locally)tractable if for everyfinite L⊆S,P(L)is tractable.These two notions of tractability do not coincide in the Abduction problem(see talk by Nadia Creignou).•For which computational problems related to the CSP do these two notions of tractability coincide?•In particular,do they coincide for the standard CSP decision problem?Communicated by:Nadia Creignou 1That is,a(possibly infinite)set of relations over somefixed set.1Question1.4(see also Question3.5)It has been shown that when a structure B has bounded pathwidth duality the corresponding problem CSP(B)is in the complexity class NL (see talk by Victor Dalmau).Is the converse also true(modulo some natural complexity-theoretic assumptions)?Communicated by:Victor Dalmau Question1.5Is there a good(numerical)parameterization for constraint satisfaction problems that makes themfixed-parameter tractable?Question1.6Further develop techniques based on delta-matroids to complete the com-plexity classification of the Boolean CSP(with constants)with at most two occurrences per variable(see talk by Tomas Feder).Communicated by:Tomas Feder Question1.7Classify the complexity of uniform Boolean CSPs(where both structure and constraint relations are specified in the input).Communicated by:Heribert Vollmer Question1.8The microstructure graph of a binary CSP has vertices for each variable/value pair,and edges that join all pairs of vertices that are compatible with the constraints.What properties of this graph are sufficient to ensure tractability?Are there properties that do not rely on the constraint language or the constraint graph individually?2Approximability and Soft ConstraintsQuestion2.1Is it true that Max CSP(L)is APX-complete whenever Max CSP(L)is NP-hard?Communicated by:Peter Jonsson Question2.2Prove or disprove that Max CSP(L)is in PO if the core of L is super-modular on some lattice,and otherwise this problem is APX-complete.The above has been proved for languages with domain size3,and for languages contain-ing all constants by a computer-assisted case analysis(see talk by Peter Jonsson).Develop techniques that allow one to prove such results without computer-assisted analysis.Communicated by:Peter Jonsson Question2.3For some constraint languages L,the problem Max CSP(L)is hard to approximate better than the random mindless algorithm on satisfiable or almost satisfiable instances.Such problems are called approximation resistant(see talk by Johan Hastad).Is a single random predicate over Boolean variables with large arity approximation resistant?What properties of predicates make a CSP approximation resistant?What transformations of predicates preserve approximation resistance?Communicated by:Johan Hastad2Question2.4Many optimisation problems involving constraints(such as Max-Sat,Max CSP,Min-Ones SAT)can be represented using soft constraints where each constraint is specified by a cost function assigning some measure of cost to each tuple of values in its scope.Are all tractable classes of soft constraints characterized by their multimorphisms?(see talk by Peter Jeavons)Communicated by:Peter Jeavons 3AlgebraQuestion3.1The Galois connection between sets of relations and sets of operations that preserve them has been used to analyse several different computational problems such as the satisfiability of the CSP,and counting the number of solutions.How can we characterise the computational goals for which we can use this Galois connection?Communicated by:Nadia Creignou Question3.2For any relational structure B=(B,R1,...,R k),let co-CSP(B)denote the class of structures which do not have a homomorphism to B.It has been shown that the question of whether co-CSP(B)is definable in Datalog is determined by P ol(B),the polymorphisms of the relations of B(see talk by Andrei Bulatov).Let B be a core,F the set of all idempotent polymorphisms of B and V the variety generated by the algebra(B,F).Is it true that co-CSP(B)is definable in Datalog if and only if V omits types1and2(that is,the local structure of anyfinite algebra in V does not contain a G-set or an affine algebra)?Communicated by:Andrei Bulatov Question3.3Does every tractable clone of polynomials over a group contain a Mal’tsev operation?Communicated by:Pascal Tesson Question3.4Classify(w.r.t.tractability of corresponding CSPs)clones of polynomials of semigroups.Communicated by:Pascal Tesson Question3.5Is it true that for any structure B which is invariant under a near-unanimity operation the problem CSP(B)is in the complexity class NL?Does every such structure have bounded pathwidth duality?(see also Question1.4)Both results are known to hold for a2-element domain(Dalmau)and for majority operations(Dalmau,Krokhin).Communicated by:Victor Dalmau,Benoit Larose3Question3.6Is it decidable whether a given structure is invariant under a near-unanimity function(of some arity)?Communicated by:Benoit Larose Question3.7Let L be afixedfinite lattice.Given an integer-valued supermodular func-tion f on L n,is there an algorithm that maximizes f in polynomial time in n if the function f is given by an oracle?The answer is yes if L is a distributive lattice(see“Supermodular Functions and the Complexity of Max-CSP”,Cohen,Cooper,Jeavons,Krokhin,Discrete Applied Mathemat-ics,2005).More generally,the answer is yes if L is obtained fromfinite distributive lattices via Mal’tsev products(Krokhin,Larose–see talk by Peter Jonsson).The smallest lattice for which the answer is not known is the3-diamond.Communicated by:Andrei Krokhin Question3.8Find the exact relationship between width and relational width.(It is known that one is bounded if and and only if the other is bounded.)Also,what types of width are preserved under natural algebraic constructions?Communicated by:Victor Dalmau 4LogicQuestion4.1The(basic)Propositional Circumscription problem is defined as fol-lows:Input:a propositional formulaφwith atomic relations from a set S,and a clause c.Question:is c satisfied in every minimal model ofφ?It is conjectured(Kirousis,Kolaitis)that there is a trichotomy for this problem,that it iseither in P,coNP-complete or inΠP2,depending on the choice of S.Does this conjecturehold?Communicated by:Nadia Creignou Question4.2The Inverse Satisfiability problem is defined as follows: Input:afinite set of relations S and a relation R.Question:is R expressible by a CNF(S)-formula without existential variables?A dichotomy theorem was obtained by Kavvadias and Sideri for the complexity of this problem with constants.Does a dichotomy hold without the constants?Are the Schaefer cases still tractable?Communicated by:Nadia Creignou4Question4.3Let LFP denote classes of structures definable infirst-order logic with a least-fixed-point operator,let HOM denote classes of structures which are closed under homomorphisms,and let co-CSP denote classes of structures defined by not having a homomorphism to somefixed target structure.•Is LFP∩HOM⊆Datalog?•Is LFP∩co-CSP⊆Datalog?(forfinite target structures)•Is LFP∩co-CSP⊆Datalog?(forω-categorical target structures)Communicated by:Albert Atserias,Manuel BodirskyQuestion4.4(see also Question3.2)Definability of co-CSP(B)in k-Datalog is a sufficient condition for tractability of CSP(B),which is sometimes referred to as having width k. There is a game-theoretic characterisation of definability in k-Datalog in terms of(∃,k)-pebble games(see talk by Phokion Kolaitis).•Is there an algorithm to decide for a given structure B whether co-CSP(B)is definable in k-Datalog(for afixed k)?•Is the width hierarchy strict?The same question when B isω-categorical,but not necessarilyfinite?Communicated by:Phokion Kolaitis,Manuel BodirskyQuestion4.5Find a good logic to capture CSP with“nice”(e.g.,ω-categorical)infinite templates.Communicated by:Iain Stewart 5Graph TheoryQuestion5.1The list homomorphism problem for a(directed)graph H is equivalent to the problem CSP(H∗)where H∗equals H together with all unary relations.•It is conjectured that the list homomorphism problem for a reflexive digraph is tractable if H has the X-underbar property(which is the same as having the bi-nary polymorphism min w.r.t.some total ordering on the set of vertices),and NP-complete otherwise.•It is conjectured that the list homomorphism problem for an irreflexive digraph is tractable if H is preserved by a majority operation,and NP-complete otherwise. Do these conjectures hold?Communicated by:Tomas Feder&Pavol Hell5Question5.2“An island of tractability?”Let A m be the class of all relational structures of the form(A,E1,...,E m)where each E i is an irreflexive symmetric binary relation and the relations E i together satisfy the following‘fullness’condition:any two distinct elements x,y are related in exactly one of the relations E i.Let B m be the single relational structure({1,...,m},E1,...,E m)where each E i is the symmetric binary relation containing all pairs xy except the pair ii.(Note that the relations E i are not irreflexive.)The problem CSP(A m,B m)is defined as:Given A∈A m,is there a homomorphism from A to B m?When m=2,this problem is solvable in polynomial time-it is the recognition problem for split graphs(see“Algorithmic Graph Theory and Perfect Graphs”,M.C.Golumbic, Academic Press,New York,1980)When m>3,this problem is NP-complete(see“Full constraint satisfaction problems”,T.Feder and P.Hell,to appear in SIAM Journal on Computing).What happens when m=3?Is this an“island of tractability”?Quasi-polynomial algorithms are known for this problem(see“Full constraint satisfaction problems”,T. Feder and P.Hell,,to appear in SIAM Journal on Computing,and“Two algorithms for list matrix partitions”,T.Feder,P.Hell,D.Kral,and J.Sgall,SODA2005).Note that a similar problem for m=3was investigated in“The list partition problem for graphs”, K.Cameron,E.E.Eschen,C.T.Hoang and R.Sritharan,SODA2004.Communicated by:Tomas Feder&Pavol Hell Question5.3Finding the generalized hypertree-width,w(H)of a hypergraph H is known to be NP-complete.However it is possible to compute a hypertree-decomposition of H in polynomial time,and the hypertree-width of H is at most3w(H)+1(see talk by Georg Gottlob).Are there other decompositions giving better approximations of the generalized hypertree-width that can be found in polynomial time?Communicated by:Georg Gottlob Question5.4It is known that a CSP whose constraint hypergraph has bounded fractional hypertree width is tractable(see talk by Daniel Marx).Is there a hypergraph property more general than bounded fractional hypertree width that makes the associated CSP polynomial-time solvable?Are there classes of CSP that are tractable due to structural restrictions and have unbounded fractional hypertree width?Communicated by:Georg Gottlob,Daniel Marx Question5.5Prove that there exist two functions f1(w),f2(w)such that,for every w, there is an algorithm that constructs in time n f1(w)a fractional hypertree decomposition of width at most f2(w)for any hypergraph of fractional hypertree width at most w(See talk by Daniel Marx).Communicated by:Daniel Marx6Question5.6Turn the connection between the Robber and Army game and fractional hypertree width into an algorithm for approximating fractional hypertree width.Communicated by:Daniel Marx Question5.7Close the complexity gap between(H,C,K)-colouring and (H,C,K)-colouring (see talk by Dimitrios Thilikos)Find a tight characterization for thefixed-parameter tractable(H,C,K)-colouring problems.•For the(H,C,K)-colouring problems,find nice properties for the non-parameterisedpart(H−C)that guaranteefixed-parameter tractability.•Clarify the role of loops in the parameterised part C forfixed-parameter hardnessresults.Communicated by:Dimitrios Thilikos6Constraint Programming and ModellingQuestion6.1In a constraint programming system there is usually a search procedure that assigns values to particular variables in some order,interspersed with a constraint propagation process which modifies the constraints in the light of these assignments.Is it possible to choose an ordering for the variables and values assigned which changes each problem instance as soon as possible into a new instance which is in a tractable class? Can this be done efficiently?Are there useful heuristics?Question6.2The time taken by a constraint programming system tofind a solution toa given instance can be dramatically altered by modelling the problem differently.Can the efficiency of different constraint models be objectively compared,or does it depend entirely on the solution algorithm?Question6.3For practical constraint solving it is important to eliminate symmetry,in order to avoid wasted search effort.Under what conditions is it tractable to detect the symmetry in a given problem in-stance?7Notes•Representations of constraints-implicit representation-effect on complexity•Unique games conjecture-structural restrictions that make it false-connectionsbetween definability and approximation•MMSNP-characterise tractable problems apart from CSP7•Migrate theoretical results to tools•What restrictions do practical problems actually satisfy?•Practical parallel algorithms-does this align with tractable classes?•Practically relevant constraint languages(”global constraints”)•For what kinds of problems do constraint algorithms/heuristics give good results?8。
INSEARCHOFEXCELLENCE

IN SEARCH OF EXCELLENCEExcellence is a journey and not a destination. In science itimplies perpetual efforts to advance the frontiers of knowledge.This often leads to progressively increasing specialization andemergence of newer disciplines. A brief summary of salientcontributions of Indian scientists in various disciplines isintroduced in this section.92P U R S U I T A N D P R O M O T I O N O F S C I E N C EThe modern period of mathematics research in India started with Srinivasa Ramanujan whose work on analytic number theory and modular forms ishighly relevant even today. In the pre-Independence period mathematicians like S.S. Pillai,Vaidyanathaswamy, Ananda Rau and others contributed a lot.Particular mention should be made of universities in Allahabad, Varanasi, Kolkata,Chennai and Waltair and later at Chandigarh,Hyderabad, Bangalore and Delhi (JNU). The Department of Atomic Energy came in a big way to boost mathematical research by starting and nurturing the Tata Institute of Fundamental Research (TIFR), which, under the leadership of Chandrasekharan, blossomed into a great school of learning of international standard. The Indian Statistical Institute, started by P.C. Mahalanobis,made its mark in an international scene and continues to flourish. Applied mathematics community owes a great deal to the services of three giants Ñ N.R. Sen, B.R. Seth and P .L. Khastgir. Some of the areas in which significant contributions have been made are briefly described here.A LGEBRAOne might say that the work on modern algebra in India started with the beautiful piece of work in 1958 on the proof of SerreÕs conjecture for n =2. A particular case of the conjecture is to imply that a unimodular vector with polynomial entries in n vari-ables can be completed to a matrix of determinantone. Another important school from India was start-ed in Panjab University whose work centres around Zassanhaus conjecture on groupings.A LGEBRAIC G EOMETRYThe study of algebraic geometry began with a seminal paper in 1964 on vector bundles. With further study on vector bundles that led to the mod-uli of parabolic bundles, principle bundles, algebraic differential equations (and more recently the rela-tionship with string theory and physics), TIFR has become a leading school in algebraic geometry. Of the later generation, two pieces of work need special mention: the work on characterization of affine plane purely topologically as a smooth affine surface, sim-ply connected at infinity and the work on Kodaira vanishing. There is also some work giving purely algebraic geometry description of the topologically invariants of algebraic varieties. In particular this can be used to study the Galois Module Structure of these invariants.L IE T HEORYThe inspiration of a work in Lie theory in India came from the monumental work on infinite dimensional representation theory by Harish Chandra, who has, in some sense, brought the sub-ject from the periphery of mathematics to centre stage. In India, the initial study was on the discrete subgroups of Lie groups from number theoretic angle. The subject received an impetus after an inter-national conference in 1960 in TIFR, where the lead-ing lights on the subject, including A. Selberg partic-M ATHEMATICAL S CIENCESC H A P T E R V I Iipated. Then work on rigidity questions was initiat-ed. The question is whether the lattices in arithmetic groups can have interesting deformations except for the well-known classical cases. Many important cases in this question were settled.D IFFERENTIALE QUATIONA fter the study of L-functions were found to beuseful in number theory and arithmetic geome-try, it became natural to study the L-functions arising out of the eigenvalues of discrete spectrum of the dif-ferential equations. MinakshisundaramÕs result on the corresponding result for the differential equation leading to the Epstein Zeta function and his paper with A. Pleijel on the same for the connected com-pact Riemanian manifold are works of great impor-tance. The idea of the paper (namely using the heat equation) lead to further improvement in the hands of Patodi. The results on regularity of weak solution is an important piece of work. In the later 1970s a good school on non-linear partial differential equa-tions that was set up as a joint venture between TIFR and IISc, has come up very well and an impressive lists of results to its credit.For differential equations in applied mathematics, the result of P.L. Bhatnagar, BGK model (by Bhatnagar, Gross, Krook) in collision process in gas and an explanation of Ramdas Paradox (that the temperature minimum happens about 30 cm above the surface) will stand out as good mathematical models. Further significant contributions have been made to the area of group theoretic methods for the exact solutions of non-liner partial differential equations of physical and engineering systems.E RGODIC T HEORYE arliest important contribution to the Ergodic the-ory in India came from the Indian Statistical Institute. Around 1970, there was work on spectra of unitary operators associated to non-singular trans-formation of flows and their twisted version, involv-ing a cocycle.Two results in the subjects from 1980s and 1990s are quoted. If G is lattice in SL(2,R) and {uÐt} a unipotent one parameter subgroup of G, then all non-periodic orbits of {uÐt} on GÐ1 are uniformly distributed. If Q is non-generate in definite quadratic form in n=variables, which is not a multiple of rational form, then the number of lattice points xÐwith a< ½Q(x)½< b, ½½x½½< r, is at least comparable to the volume of the corresponding region.N UMBER T HEORYT he tradition on number theory started with Ramanujan. His work on the cusp form for the full modular group was a breakthrough in the study of modular form. His conjectures on the coefficient of this cusp form (called RamanujanÕs tau function) and the connection of these conjectures with conjectures of A. Weil in algebraic geometry opened new research areas in mathematics. RamanujanÕs work (with Hardy) on an asymptotic formula for the parti-tion of n, led a new approach (in the hands of Hardy-Littlewood) to attack such problems called circle method. This idea was further refined and S.S. Pillai settled WaringÕs Conjecture for the 6th power by this method. Later the only remaining case namely 4th powers was settled in mid-1980s. After Independence, the major work in number theory was in analytic number theory, by the school in TIFR and in geometry of numbers by the school in Panjab University. The work on elliptic units and the con-struction of ray class fields over imaginary quadratic fields of elliptic units are some of the important achievements of Indian number theory school. Pioneering work in BakerÕs Theory of linear forms in logarithms and work on geometry of numbers and in particular the MinkowskiÕs theorem for n = 5 are worth mentioning.P ROBABILITY T HEORYS ome of the landmarks in research in probability theory at the Indian Statistical Institute are the following:93 P U R S U I T A N D P R O M O T I O N O F S C I E N C Eq A comprehensive study of the topology of weak convergence in the space of probability measures on topological spaces, particularly, metric spaces. This includes central limit theorems in locally compact abelian groups and Milhert spaces, arithmetic of probability distributions under convolution in topological groups, Levy-khichini representations for characteristic functions of probability distributions on group and vector spaces.q Characterization problems of mathematical statistics with emphasis on the derivation of probability laws under natural constraints on statistics evaluated from independent observations.q Development of quantum stochastic calculus based on a quantum version of ItoÕs formula for non-commutative stochastic processes in Fock spaces. This includes the study of quantum stochastic integrals and differential equations leading to the construction of operator Markov processes describing the evolution of irreversible quantum processes.q Martingale methods in the study of diffusion processes in infinite dimensional spaces.q Stochastic processes in financial mathematics.C OMBINATORICST hough the work in combinatorics had been ini-tiated in India purely through the efforts of R.C.Bose at the Indian Statistical Institute in late thirties, it reached its peak in late fifties at the University of North Carolina, USA, where he was joined by his former student S.S.Shrikhande. They provided the first counter-example to the celebrat-ed conjecture of Euler (1782) and jointly with Parker further improved it. The last result is regarded a classic.In the absence of these giants there was practically no research activity in this area in India. However, with the return of Shrikhande to India in 1960 activities in the area flourished and many notable results in the areas of embedding of residual designs in symmetric designs, A-design conjecture and t-designs and codes were reported.T HEORY OF R ELATIVITYI n a strict sense the subject falls well within the purview of physics but due to the overwhelming response by workers with strong foundation in applied mathematics the activity could blossom in some of the departments of mathematics of certain universities/institutes. Groups in BHU, Gujarat University, Ahmedabad, Calcutta University, and IIT, Kharagpur, have contributed generously to the area of exact solutions of Einstein equations of gen-eral relativity, unified field theory and others. However, one exact solution which has come to be known as Vaidya metric and seems to have wide application in high-energy astrophysics deserves a special mention.N UMERICAL A NALYSIST he work in this area commenced with an attempt to solve non-linear partial differential equations governing many a physical and engineering system with special reference to the study of Navier-Stabes equations and cross-viscous forces in non-Newtonian fluids. The work on N-S equation has turned out to be a basic paper in the sense that it reappeared in the volume, Selected Papers on Numerical Solution of Equations of Fluid Dynamics, Applied Mathematics, through the Physical Society of Japan. The work on non-Newtonian fluid has found a place in the most prestigious volume on Principles of Classical Mechanics & Field Theory by Truesdell and Toupin. The other works which deserve mention are the development of extremal point collocation method and stiffy stable method.A PPLIED M ATHEMATICST ill 1950, except for a group of research enthusi-asts working under the guidance of N.R.Sen at Calcutta University there was practically no output in applied mathematics. However, with directives from the centre to emphasize on research in basic94P U R S U I T A N D P R O M O T I O N O F S C I E N C Eand applied sciences and liberal central fundings through central and state sponsored laboratories, the activity did receive an impetus. The department of mathematics at IIT, Kharagpur, established at the very inception of the institute of national importance in 1951, under the dynamic leadership of B.R.Seth took the lead role in developing a group of excellence in certain areas of mathematical sciences. In fact, the research carried out there in various disciplines of applied mathematics such as elasticity-plasticity, non-linear mechanics, rheological fluid mechanics, hydroelasticity, thermoelasticity, numerical analysis, theory of relativity, cosmology, magneto hydrody-namics and high-temperature gasdynamics turned out to be a trend setting one for other IITs, RECs, other Technical Institutes and Universities that were in the formative stages. B.R. SethÕs own researches on the study of Saint-VenamtÕs problem and transi-tion theory to unify elastic-plastic behaviour of mate-rials earned him the prestigious EulerÕs bronze medal of the Soviet Academy of Sciences in 1957. The other areas in which applied mathematicians con-tributed generously are biomechanics, CFD, chaotic dynamics, theory of turbulence, bifurcation analysis, porous media, magnetics fluids and mathematicalphysiology.95 P U R S U I T A N D P R O M O T I O N O F S C I E N C E。
数学英语知识点归纳总结

数学英语知识点归纳总结1. Basic arithmeticArithmetic is the foundation of mathematics and covers the basic operations of addition, subtraction, multiplication, and division. These operations are used to solve everyday problems and form the basis for more advanced mathematical concepts. Understanding basic arithmetic is essential for performing more complex mathematical calculations and solving problems in various fields.2. AlgebraAlgebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It includes topics such as equations, inequalities, polynomials, and functions. Algebraic concepts are used to solve problems in areas such as physics, engineering, and computer science, making it an important topic for students to learn and understand.3. GeometryGeometry is the study of shapes, sizes, and properties of space. It includes topics such as lines, angles, triangles, circles, and polygons. Understanding geometry is essential for understanding the structure of objects and solving problems related to spatial relationships and measurements. Geometry also plays a crucial role in fields such as architecture, art, and design.4. TrigonometryTrigonometry is the study of the relationships between the angles and sides of triangles. It includes topics such as sine, cosine, tangent, and their inverses. Trigonometric concepts are used in fields such as physics, engineering, and navigation to solve problems related to angles and distances. Understanding trigonometry is essential for students who want to pursue careers in these fields.5. CalculusCalculus is a branch of mathematics that deals with the study of change and motion. It includes topics such as derivatives, integrals, limits, and infinite series. Calculus is used to solve problems in areas such as physics, engineering, and economics, making it an important topic for students to learn and understand. Understanding calculus is essential for students who want to pursue careers in these fields.6. StatisticsStatistics is the study of data collection, analysis, interpretation, and presentation. It includes topics such as probability, distribution, hypothesis testing, and regression analysis. Statistics is used in fields such as business, economics, and social sciences to make informeddecisions and draw meaningful conclusions from data. Understanding statistics is essential for students who want to pursue careers in these fields.7. Discrete mathematicsDiscrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. It includes topics such as set theory, logic, graph theory, and combinatorics. Discrete mathematics is used in fields such as computer science, cryptography, and information technology to solve problems related to discrete structures and algorithms. Understanding discrete mathematics is essential for students who want to pursue careers in these fields.In conclusion, mathematics is a vast and diverse subject that covers a wide range of topics and concepts. From basic arithmetic to advanced calculus, there is always something new to learn and explore. Understanding the key mathematical concepts and principles is essential for students who want to pursue careers in fields such as science, engineering, and technology. By mastering these concepts, students can develop the skills and knowledge they need to succeed in their chosen fields and make meaningful contributions to the world of mathematics.。
Openproblem

TA (x)i =
i2 ,··· ,im =1
k Then we may see that TA : n → n for any positive integer k . The nonlinear operator TA has appeared in [2] and some other papers on eigenvalues of nonnegative tensors.
2.2.
The Norm of a Nonlinear Operator
n
Let F :
→
n
. Define F = sup{ F (x) : x = 1}.
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Here · is a certain norm in n . If F is a continuous operator, then definitely k F is finite. We may see that TA is a continuous operator for any positive 1 k k integer k . Thus, ||TA || is well-defined. Hence, the qnear operator on
1
n
:
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TA = (Axm−1 ) m−1 . That is, for x ∈
n
, TA (x) ∈
n
and its ith component is
n
Title Page
m1 −1 aii2 ···im xi2 · · · xim .
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博士必读数学

博士必读数学前面几篇谈了一些对数学的粗浅看法。
其实,如果对某门数学有兴趣,最好的方法就是走进那个世界去学习和体验。
这里说说几本我看过后觉得不错的数学教科书。
1. 线性代数(Linear Algebra):我想国内的大学生都会学过这门课程,但是,未必每一位老师都能贯彻它的精要。
这门学科对于Learning 是必备的基础,对它的透彻掌握是必不可少的。
我在科大一年级的时候就学习了这门课,后来到了香港后,又重新把线性代数读了一遍,所读的是Introduction to Linear Algebra (3rd Ed.) by Gilbert Strang.这本书是MIT的线性代数课使用的教材,也是被很多其它大学选用的经典教材。
它的难度适中,讲解清晰,重要的是对许多核心的概念讨论得比较透彻。
我个人觉得,学习线性代数,最重要的不是去熟练矩阵运算和解方程的方法——这些在实际工作中MATLAB可以代劳,关键的是要深入理解几个基础而又重要的概念:子空间(Subspace),正交(Orthogonality),特征值和特征向量(Eigenvalues and eigenvectors),和线性变换(Linear transform)。
(如果你能理解傅立叶变化究竟做了一件什么事情,你才能说你知道了子空间!学线性代数一定要理解MATLAB能为你做的事情之外其他的东西,这才是精髓。
而很遗憾,很多高校的线性代数考试只测试学生的计算能力。
有几个数学老师能告诉学生:我们为什么要计算特征值?)从我的角度看来,一本线代教科书的质量,就在于它能否给这些根本概念以足够的重视,能否把它们的联系讲清楚。
Strang 的这本书在这方面是做得很好的。
而且,这本书有个得天独厚的优势。
书的作者长期在MIT讲授线性代数课(18.06),课程的video在MIT 的Open courseware网站上有提供。
有时间的朋友可以一边看着名师授课的录像,一边对照课本学习或者复习。
普林斯顿数学指南(英文版)

普林斯顿数学指南(英文版)The Princeton Companion to Mathematics is a comprehensive guide that explores the vast and intricate world of mathematics. This encyclopedic reference work is an indispensable resource for students, educators, and anyonewith an interest in the subject. It covers a wide range of topics, from the fundamentals of arithmetic and geometry tothe more advanced concepts of calculus, number theory, and topology.The book is organized into several sections, eachfocusing on a specific area of mathematics. The first section, "The Foundations of Mathematics," provides an overview of the basic principles and concepts that underlie all mathematical disciplines. This includes an introduction to the history of mathematics, its major branches, and the key figures who have shaped its development over the centuries.The second section, "Algebra and Number Theory," delves into the study of abstract structures, such as groups, rings, and fields, as well as the properties and relationships of numbers. This section also covers the foundations of algebraic geometry, which is concerned with the study of geometric objects defined by polynomial equations.The third section, "Analysis and Calculus," explores the study of continuity, change, and limits, as well as the techniques used to solve problems involving rates of change, such as differentiation and integration. This section also covers the theory of complex analysis, which extends the ideas of real analysis to complex numbers.The fourth section, "Geometry and Topology," focuses on the study of shape, size, and spatial relationships, as well as the properties of abstract spaces that are not necessarily Euclidean. This section includes discussions of classical geometry, such as Euclidean and non-Euclidean geometries, aswell as more modern areas like topology and differential geometry.The fifth section, "Probability and Statistics," deals with the study of randomness and uncertainty, as well as the collection, analysis, and interpretation of data. This section covers the basic principles of probability theory, statistical inference, and statistical modeling.The final section, "Applied Mathematics," highlights the many ways in which mathematics is used to solve real-world problems in fields such as physics, engineering, economics, and computer science. This section includes discussions of optimization, game theory, cryptography, and other areas where mathematical techniques are essential for solving practical problems.Throughout the book, readers will find numerous examples, exercises, and applications that illustrate the concepts and techniques discussed in each section. These include bothhistorical examples, such as the development of calculus or the proof of Fermat's Last Theorem, and contemporary applications, such as the use of chaos theory in meteorology or the role of Fourier analysis in image processing.In addition to its comprehensive coverage of mathematical topics, The Princeton Companion to Mathematics also features biographical sketches of many of the most influential mathematicians in history, from ancient Greek geometers like Euclid and Archimedes to modern giants like Isaac Newton, Carl Friedrich Gauss, and Emmy Noether. These profiles provide insights into the lives and achievements of these remarkable individuals, as well as their contributions to the development of mathematics.Furthermore, the book includes a detailed glossary of mathematical terms and symbols, which can be especially helpful for readers who are new to the subject or encounter unfamiliar concepts. The glossary defines key terms andprovides examples that illustrate their meanings and uses in various contexts.As a reference work, The Princeton Companion to Mathematics is designed to be accessible to a wide range of readers, from beginners to advanced students and professionals. Its clear explanations, extensive examples, and engaging historical narratives make it an ideal resource for anyone seeking to deepen their understanding of mathematics or explore new areas of the subject.In conclusion, The Princeton Companion to Mathematics is a valuable resource for anyone interested in learning more about the fascinating world of mathematics. Its comprehensive coverage of topics, engaging historical narratives, and clear explanations make it an invaluable tool for anyone seeking to enhance their knowledge and appreciation of this vital field of study. Whether you are a student, educator, or simply acurious mind looking to explore the beauty and elegance of mathematics, this guide is an excellent place to start.。
山东大学基础数学与应用数学专业本科生培养方案

数学与应用数学培养方案(070101)(Mathematics and Applied Mathematics 070101)一、专业简介(Ⅰ、Major Introduction)数学与应用数学专业的专业方向有:基础数学和应用数学。
本专业十分重视学生数学基础知识和专业基础知识的学习,注重对他们的创造性和创新能力的培养,为培养高级数学专业人才打好基础。
经过四年学习,使学生初步具备在基础数学或应用数学某个方向从事当代学术前沿问题研究的能力。
毕业后能从事数学及相关学科的教学和科学研究工作,并可继续深造,到高等学校或科研机构的基础数学、应用数学及其他交叉学科做研究生。
The major (Mathematics and Applied Mathematics) have two branches: Pure Mathematics and Applied Mathematics. The major focus teaching on both basic and professional theory of mathematics, and is committed to cultivating the high-level mathematical talents with the innovative and creative ability. After four-years-study, the students should have researching ability for academic open problems in some directions of pure or applied mathematics. When the students in the major graduate, they can teach or study mathematics and related subjects, or they can be postgraduate students of universities or institutes in Pure Mathematics or Applied Mathematics or some other related branches.二、培养目标(Ⅱ、Cultivation Objective)培养掌握数学科学的基本理论与基本方法,具有运用数学知识或使用计算机解决实际问题的能力,受到科学研究训练的高级专门人才,能在科技、教育、经济和企事业等部门从事研究、教学工作或在生产经营及管理部门从事实际应用、开发研究和管理工作,或能继续攻读研究生学位。
数学代入法解题方法

数学代入法解题方法Mathematical substitution method is a powerful tool in solving various problems in mathematics. It involves replacing variables in an equation with specific values in order to simplify the problem andfind a solution. This method is particularly useful in algebraic equations, where it can help eliminate complex terms and make calculations more manageable. By carefully selecting appropriate values to substitute, the problem can be transformed into a simpler form that is easier to solve.数学代入法是解决数学问题的强大工具。
它涉及用特定值替换方程中的变量,以简化问题并找到解决方案。
这种方法在代数方程中特别有用,它可以帮助消除复杂的项并使计算更容易处理。
通过仔细选择适当的替代值,问题可以转化为更简单的形式,更容易解决。
One of the key benefits of using the mathematical substitution method is that it allows us to break down complex problems into smaller, more manageable parts. By systematically replacing variables with known values, we can focus on solving simpler equations before bringing everything back together to find the finalsolution. This method helps to reduce the risk of errors and makes the problem-solving process more organized and structured.使用数学代入法的一个关键好处是它使我们能够将复杂问题分解为更小,更易处理的部分。
数学课都学什么英语作文

数学课都学什么英语作文In math class, we learn a variety of topics and concepts that are crucial for understanding the language of numbers and the principles of mathematics. Some of the main areaswe cover in math class include arithmetic, algebra, geometry, statistics, and calculus.Arithmetic is the branch of mathematics that deals with the basic operations of addition, subtraction, multiplication, and division. We learn techniques for performing these operations with whole numbers, fractions, decimals, and percentages. This lays the foundation for more advanced mathematical concepts.Algebra introduces the use of variables and symbols to represent unknown quantities and to solve equations and inequalities. We learn about linear equations, quadratic equations, and polynomial functions, and how to manipulate algebraic expressions. This helps us develop problem-solving skills and logical reasoning.Geometry focuses on the study of shapes, sizes, and properties of space. We learn about angles, lines, polygons,circles, and three-dimensional figures. We also explore concepts such as congruence, similarity, and transformations. Understanding geometry is essential for fields such as architecture, engineering, and design.Statistics involves the collection, analysis, interpretation, and presentation of data. We learn about different types of data, measures of central tendency, measures of dispersion, and probability. We also study various methods for making inferences and drawing conclusions from data.Calculus is the branch of mathematics that deals with rates of change and accumulation. We learn about limits, derivatives, integrals, and applications of calculus in science, engineering, and economics. This subject is particularly important for students pursuing careers in STEM fields.Overall, math class provides us with the essential knowledge and skills needed to solve problems, think critically, and make informed decisions in various areas of life. It helps us develop a deeper understanding of theworld around us and prepares us for further studies in higher education and future careers.在数学课上,我们学习各种重要的主题和概念,这些对理解数字语言和数学原理至关重要。
因式分解英语

因式分解英语
"因式分解"(Factorization)是将一个数或代数表达式表示为两个或更多数或代数表达式的乘积的过程。
在数学中,因式分解被广泛运用,可以应用于多项式、整数、甚至质数分解。
其他数学知识点的英文包括:
1."代数方程"(Algebraic Equations):用字母表示数,探索未知数值的方程。
2."几何形状"(Geometric Shapes):描述平面和立体图形的属性和特征。
3."微积分"(Calculus):研究变化和运动中的极限、微分和积分。
4."统计学"(Statistics):收集、分析和解释数据的方法和技术。
5."三角学"(Trigonometry):研究三角函数和三角形的性质和关系。
6."概率论"(Probability Theory):探讨事件发生的可能性和概率。
7."线性代数"(Linear Algebra):研究向量、线性方程组和线性映射。
8."数论"(Number Theory):研究整数性质和结构。
9."离散数学"(Discrete Mathematics):研究离散结构和对象,如图论和逻辑。
10."向量分析"(Vector Analysis):运用向量和张量研究的数学领域。
数学学习计划英文

数学学习计划英文IntroductionStudying mathematics can be a challenging yet rewarding experience. It requires dedication, discipline, and a strong commitment to learning. With the right study plan and strategies, anyone can improve their mathematical skills and achieve success in this subject. In this study plan, we will discuss the best ways to approach mathematics learning, set goals, and create a detailed plan to achieve those goals.Setting GoalsBefore starting a mathematics study plan, it is essential to set clear and achievable goals. These goals should be specific, measurable, and time-bound. Whether you want to improve your grades, pass a standardized test, or simply become more confident in your mathematical abilities, it's crucial to identify what you hope to accomplish. Once you have a clear understanding of your goals, you can create a plan to reach them.Study PlanNow that we have established the importance of setting clear goals, let's discuss the key components of a successful mathematics study plan.1. Assess Your Current LevelThe first step in creating a study plan is to assess your current mathematical abilities. This might involve taking a diagnostic test, reviewing your past performance in math classes, or simply reflecting on your strengths and weaknesses in the subject. By understanding where you stand, you can better tailor your study plan to address areas of improvement.2. Set a Realistic ScheduleMathematics requires consistent practice and repetition. Therefore, it's important to set aside regular, dedicated time for studying. Consider your other commitments, such as school, work, and activities, and determine how much time you can realistically dedicate to math. Be sure to schedule regular study sessions to stay on track with your goals.3. Identify ResourcesThere are countless resources available for mathematics learning, including textbooks, online tutorials, practice problems, and educational websites. Identify the resources that best suit your learning style and needs. It's also helpful to seek out additional support, such as a tutor or study group, to aid in your understanding of the material.4. Focus on Conceptual UnderstandingMathematics is not just about memorizing formulas and procedures; it's about understanding the underlying concepts. As you study, focus on developing a deepunderstanding of the foundational principles of mathematics. This will not only help you solve problems more effectively but also build a strong foundation for more advanced concepts.5. Practice, Practice, PracticePractice is essential for mastering mathematics. Be sure to work through a variety of problems in different areas of mathematics, such as algebra, geometry, calculus, and statistics. Repetition and exposure to different types of problems will help solidify your understanding and improve your problem-solving abilities.6. Review and ReflectRegularly review the material you have learned and reflect on your progress. Take the time to identify areas of strength and areas that need improvement. Additionally, consider seeking feedback from teachers, tutors, or peers to gain insight into your performance and areas for growth.7. Test Your KnowledgeIncorporate practice tests and quizzes into your study plan. This will help identify areas of weakness and provide a measure of your progress. Again, seek feedback on your performance to understand where you need to focus your efforts.8. Adjust and AdaptAs you progress through your study plan, be prepared to adjust and adapt your approach. If you find that certain methods or resources are not effective, don't be afraid to pivot and try something new. The key is to be flexible and open to different ways of learning.Sample Study PlanTo illustrate the above components, let's create a sample study plan for a high school student aiming to improve their mathematics grades.Goal: Improve overall mathematics grade from a C to a B+ by the end of the semester. Assessment: Review past exams and homework assignments to identify specific areas of weakness (e.g., algebraic concepts, word problems, geometry).Schedule:- Monday, Wednesday, and Friday from 4-5:30 PM: Complete practice problems and review textbook material.- Tuesday and Thursday from 6-7 PM: Attend math tutoring sessions at the school library. - Saturday from 10 AM-12 PM: Work on additional practice problems and review past exams.- Sunday: Rest and review material from the week.Resources:- Textbook: "Algebra and Geometry Fundamentals"- Online tutorials: Khan Academy- Math tutoring at school libraryFocus:- Develop a deep understanding of algebraic concepts, including factoring, solving equations, and graphing functions.- Improve problem-solving skills in word problems through practice and application.- Gain confidence in geometric concepts, such as angles, congruence, and area calculations. Reflect:- After each study session, reflect on the material covered and identify areas of improvement.- Seek feedback from the math tutor during tutoring sessions.Test:- Complete practice tests and quizzes to gauge progress.- Review and analyze exam results to identify areas of improvement.Adjust and Adapt:- If a certain study method is not effective, try a new approach.- Seek additional resources or support if needed.ConclusionCreating a successful mathematics study plan requires careful planning, dedication, and a willingness to adapt. By setting clear goals, assessing your current level, and following a structured plan, anyone can improve their mathematical skills. Remember, consistent practice, conceptual understanding, and a focus on problem-solving are key to success in mathematics. With the right study plan and strategies, you can achieve your mathematics goals and excel in this challenging yet rewarding subject.。
AP Statistics Syllabus

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We are interested in statistical models for discrete data which can be represented by polynomial constraints. As is customary in algebraic geometry, we consider varieties over the field of complex numbers, with the tacit understanding that statisticians mostly care about points whose coordinates are real and non-negative. The model referred to in the Specific Problem lives in the 64-dimensional space C4 ⊗ C4 ⊗ C4 of 4×4×4-tables (pijk ), where i, j, k ∈ {A, C, G, T}. It has the parametric representation pijk = ρAi · σAj · θAk + ρCi · σCj · θCk + ρGi · σGj · θGk + ρTi · σTj · θTk . (2.1)
Our problem is to compute the homogeneous prime ideal I of all polynomials which vanish on this model. The desired ideal I lives in the polynomial ring Q pAAA , pAAC , pAAT , . . . , pTTG , pTTT with 64 unknowns. In principle, one can compute generators of I by applying Gr¨ obner bases methods to the parametrization (2.1). However, our problem has 64 probabilities and 48 parameters, and it is simply too big for the kind of computations which were performed in [24, §3.2] using the software package Singular [13]. Given that Gr¨ obner basis methods appear to be too slow for any problem size which is actually relevant for real data, skeptics may wonder why a statistician should bother learning the language of ideals and varieties. One possible response to the practitioner’s legitimate question “Why (pure) mathematics?” is offered by the following quote due to Henri Poincar´ e: “Mathematics is the Art of Giving the Same Name to Different Things”. Indeed, our prime ideal I gives the same name to the following things: • the set of 4×4×4-tables of tensor rank ≤ 4, • the mixture of four models for three independent random variables, • the naive Bayes model with four classes, • the conditional independence model [X1 ⊥ ⊥ X2 ⊥ ⊥ X3 | Y ], • the fourth secant variety of the Segre variety P3 ×P3 ×P3 , • the general Markov model for the phylogenetic tree K1,3 , • superposition of four pure states in a quantum system [4, 14]. These different terms have been used in the literature for the geometric object represented by (2.1). The concise language of commutative algebra and algebraic geometry can be an effective channel of communication for the different communities of statisticians, computer scientists, physicists, engineers and biologists, all of whom have encountered formulas like (2.1). The generators of lowest degree in our ideal I have degree five, and the known generators of highest degree have degree nine. The analysis of Landsberg and Manivel in [20, Proposition 6.3] on 3×3×4-tables of tensor rank four implies the existence of additional ideal generators of degree six in I . This analysis had been overlooked by the authors of [24] when they formulated their Conjecture 3.24. Readers of [24, Chapter 3] are herewith kindly asked to replace “of degree 5 and 9” by “of degree at most 9”.
1. Introduction. This article is based on a lecture given in March 2007 at the workshop on Statistics, Biology and Dynamics held at the Institute for Mathematics and its Applications (IMA) in Minneapolis as part of the 2006/07 program on Applications of Algebraic Geometry. In four sections we present mathematical problems whose solutions would likely become important contributions to the emerging interactions between algebraic geometry and computational statistics. Each of the four sections starts out with a “specific problem” which plays the role of representing the broader research agenda. The latter is summarized in a “general problem”. Algebraic statistics is concerned with the study of probabilistic models and techniques for statistical inference using methods from algebra and geometry. The term was coined in the book of Pistone, Riccomagno and Wynn [25] and subsequently developed for biological applications in [24]. Readers from statistics will enjoy the introduction and review recently given by Drton and Sullivant [8], while readers from algebra will find various points of entry cited in our discussion and listed among our references. 2. Graphical Models with Hidden Variables. Our first question concerns three-dimensional contingency tables (pijk ) whose indices i, j, k range over a set of four elements, such as the set {A, C, G, T} of DNA bases. Specific Problem: Consider the variety of 4×4×4-tables of tensor rank at most 4. There are certain known polynomials of degree at most nine which vanish on this variety. Do they suffice to cut out the variety? This particular open problem appears in [24, Conjecture 3.24], and it here serves as a placeholder for the following broader direction of inquiry. General Problem: Study the geometry and commutative algebra of graphical models with hidden random variables. Construct these varieties by gluing familiar secant varieties, and by applying representation theory.