Bürgisser, Peter 1962
Overview
Works:  41 works in 121 publications in 3 languages and 1,783 library holdings 

Genres:  Academic theses 
Roles:  Author, Other, dgs, Contributor, htt 
Classifications:  QA267.7, 511.3 
Publication Timeline
.
Most widely held works by
Peter Bürgisser
Condition : the geometry of numerical algorithms by
Peter Bürgisser(
)
16 editions published in 2013 in 3 languages and held by 583 WorldCat member libraries worldwide
This book gathers threads that have evolved across different mathematical disciplines into seamless narrative. It deals with condition as a main aspect in the understanding of the performance regarding both stability and complexity of numerical algorithms. While the role of condition was shaped in the last halfcentury, so far there has not been a monograph treating this subject in a uniform and systematic way. The book puts special emphasis on the probabilistic analysis of numerical algorithms via the analysis of the corresponding condition. The exposition's level increases along the book, starting in the context of linear algebra at an undergraduate level and reaching in its third part the recent developments and partial solutions for Smale's 17th problem which can be explained within a graduate course. Its middle part contains a conditionbased course on linear programming that fills a gap between the current elementary expositions of the subject based on the simplex method and those focusing on convex programming
16 editions published in 2013 in 3 languages and held by 583 WorldCat member libraries worldwide
This book gathers threads that have evolved across different mathematical disciplines into seamless narrative. It deals with condition as a main aspect in the understanding of the performance regarding both stability and complexity of numerical algorithms. While the role of condition was shaped in the last halfcentury, so far there has not been a monograph treating this subject in a uniform and systematic way. The book puts special emphasis on the probabilistic analysis of numerical algorithms via the analysis of the corresponding condition. The exposition's level increases along the book, starting in the context of linear algebra at an undergraduate level and reaching in its third part the recent developments and partial solutions for Smale's 17th problem which can be explained within a graduate course. Its middle part contains a conditionbased course on linear programming that fills a gap between the current elementary expositions of the subject based on the simplex method and those focusing on convex programming
Algebraic complexity theory by
Peter Bürgisser(
Book
)
19 editions published between 1997 and 2011 in English and held by 417 WorldCat member libraries worldwide
This is the first book to present an uptodate and selfcontained account of Algebraic Complexity Theory that is both comprehensive and unified. Requiring of the reader only some basic algebra and offering over 350 exercises, it is wellsuited as a textbook for beginners at graduate level. With its extensive bibliography covering about 500 research papers, this text is also an ideal reference book for the professional researcher. The subdivision of the contents into 21 more or less independent chapters enables readers to familiarize themselves quickly with a specific topic, and facilitates the use of this book as a basis for complementary courses in other areas such as computer algebra
19 editions published between 1997 and 2011 in English and held by 417 WorldCat member libraries worldwide
This is the first book to present an uptodate and selfcontained account of Algebraic Complexity Theory that is both comprehensive and unified. Requiring of the reader only some basic algebra and offering over 350 exercises, it is wellsuited as a textbook for beginners at graduate level. With its extensive bibliography covering about 500 research papers, this text is also an ideal reference book for the professional researcher. The subdivision of the contents into 21 more or less independent chapters enables readers to familiarize themselves quickly with a specific topic, and facilitates the use of this book as a basis for complementary courses in other areas such as computer algebra
Completeness and reduction in algebraic complexity theory by
Peter Bürgisser(
Book
)
19 editions published between 1998 and 2011 in English and German and held by 315 WorldCat member libraries worldwide
"The theory of NPcompleteness is a cornerstone of computational complexity. This monograph provides a thorough and comprehensive treatment of this concept in the framework of algebraic complexity theory. Many of the results presented are new and published for the first time. Topics include: complete treatment of Valiant's algebraic theory of NPcompleteness, interrelations with the classical theory as well as the BlumShubSmale model of computation, questions of structural complexity, fast evaluation of representations of general linear groups, and complexity of immanants. The book can be used at the advanced undergraduate or at the beginning graduate level in either mathematics or computer science."Jacket
19 editions published between 1998 and 2011 in English and German and held by 315 WorldCat member libraries worldwide
"The theory of NPcompleteness is a cornerstone of computational complexity. This monograph provides a thorough and comprehensive treatment of this concept in the framework of algebraic complexity theory. Many of the results presented are new and published for the first time. Topics include: complete treatment of Valiant's algebraic theory of NPcompleteness, interrelations with the classical theory as well as the BlumShubSmale model of computation, questions of structural complexity, fast evaluation of representations of general linear groups, and complexity of immanants. The book can be used at the advanced undergraduate or at the beginning graduate level in either mathematics or computer science."Jacket
La double illusion de l'or et de l'amour chez Villiers de l'IsleAdam by
Peter Bürgisser(
Book
)
18 editions published between 1967 and 1969 in 3 languages and held by 136 WorldCat member libraries worldwide
18 editions published between 1967 and 1969 in 3 languages and held by 136 WorldCat member libraries worldwide
Algebraic Complexity Theory : With the Collaboration of Thomas Lickteig by
Peter Bürgisser(
)
3 editions published in 1997 in English and German and held by 84 WorldCat member libraries worldwide
This is the first book to present an uptodate and selfcontained account of Algebraic Complexity Theory that is both comprehensive and unified. Requiring of the reader only some basic algebra and offering over 350 exercises, it is wellsuited as a textbook for beginners at graduate level. With its extensive bibliography covering about 500 research papers, this text is also an ideal reference book for the professional researcher. The subdivision of the contents into 21 more or less independent chapters enables readers to familiarize themselves quickly with a specific topic, and facilitates the use of this book as a basis for complementary courses in other areas such as computer algebra
3 editions published in 1997 in English and German and held by 84 WorldCat member libraries worldwide
This is the first book to present an uptodate and selfcontained account of Algebraic Complexity Theory that is both comprehensive and unified. Requiring of the reader only some basic algebra and offering over 350 exercises, it is wellsuited as a textbook for beginners at graduate level. With its extensive bibliography covering about 500 research papers, this text is also an ideal reference book for the professional researcher. The subdivision of the contents into 21 more or less independent chapters enables readers to familiarize themselves quickly with a specific topic, and facilitates the use of this book as a basis for complementary courses in other areas such as computer algebra
Conjunctive queries, arithmetic circuits and counting complexity by
Stefan Mengel(
)
1 edition published in 2013 in English and held by 29 WorldCat member libraries worldwide
This thesis deals with several subjects from counting complexity and arithmetic circuit complexity.The first part explores the complexity of counting solutions to conjunctive queries, which are a basic class of queries from database theory. We introduce a parameter, called the quantified star size of a query phi, which measures how the free variables are spread in phi. As usual in database theory, we associate a hypergraph to a query phi. We show that for classes of queries for which these associated hypergraphs have bounded generalized hypertree width, bounded quantified star size exactly characterizes the subclasses of queries for which counting the number of solutions is tractable. In the case of bounded arity, this allows us to fully characterize the classes of conjunctive queries for which counting the solutions is tractable. Finally, we also analyze the complexity of computing the quantified star size of a conjunctive query.In the second part we characterize different classes from arithmetic circuit complexity by different means, including conjunctive queries and constraint satisfaction problems, graph polynomials on bounded treewidth graphs, and an extension of the classical arithmetic branching program model by stack memory.In particular, this yields new characterizations of the arithmetic circuit class VP, a class that is central to the area but arguably not well understood.Finally, the third part studies the complexity of two questions on polynomials given by arithmetic circuits: testing whether a monomial is present and counting the number of its monomials. We show that these problems are complete for different levels of the counting hierarchy, which had few or no known natural complete problems before. ; eng
1 edition published in 2013 in English and held by 29 WorldCat member libraries worldwide
This thesis deals with several subjects from counting complexity and arithmetic circuit complexity.The first part explores the complexity of counting solutions to conjunctive queries, which are a basic class of queries from database theory. We introduce a parameter, called the quantified star size of a query phi, which measures how the free variables are spread in phi. As usual in database theory, we associate a hypergraph to a query phi. We show that for classes of queries for which these associated hypergraphs have bounded generalized hypertree width, bounded quantified star size exactly characterizes the subclasses of queries for which counting the number of solutions is tractable. In the case of bounded arity, this allows us to fully characterize the classes of conjunctive queries for which counting the solutions is tractable. Finally, we also analyze the complexity of computing the quantified star size of a conjunctive query.In the second part we characterize different classes from arithmetic circuit complexity by different means, including conjunctive queries and constraint satisfaction problems, graph polynomials on bounded treewidth graphs, and an extension of the classical arithmetic branching program model by stack memory.In particular, this yields new characterizations of the arithmetic circuit class VP, a class that is central to the area but arguably not well understood.Finally, the third part studies the complexity of two questions on polynomials given by arithmetic circuits: testing whether a monomial is present and counting the number of its monomials. We show that these problems are complete for different levels of the counting hierarchy, which had few or no known natural complete problems before. ; eng
Degenerationsordnung und Trägerfunktional bilinearer Abbildungen by
Peter Bürgisser(
Book
)
4 editions published between 1990 and 2006 in German and held by 29 WorldCat member libraries worldwide
4 editions published between 1990 and 2006 in German and held by 29 WorldCat member libraries worldwide
Condition : the geometry of numerical algorithms by
Peter Bürgisser(
)
1 edition published in 2013 in English and held by 22 WorldCat member libraries worldwide
This book gathers threads that have evolved across different mathematical disciplines into seamless narrative. It deals with condition as a main aspect in the understanding of the performance regarding both stability and complexity of numerical algorithms. While the role of condition was shaped in the last halfcentury, so far there has not been a monograph treating this subject in a uniform and systematic way. The book puts special emphasis on the probabilistic analysis of numerical algorithms via the analysis of the corresponding condition. The exposition's level increases along the book, starting in the context of linear algebra at an undergraduate level and reaching in its third part the recent developments and partial solutions for Smale's 17th problem which can be explained within a graduate course. Its middle part contains a conditionbased course on linear programming that fills a gap between the current elementary expositions of the subject based on the simplex method and those focusing on convex programming
1 edition published in 2013 in English and held by 22 WorldCat member libraries worldwide
This book gathers threads that have evolved across different mathematical disciplines into seamless narrative. It deals with condition as a main aspect in the understanding of the performance regarding both stability and complexity of numerical algorithms. While the role of condition was shaped in the last halfcentury, so far there has not been a monograph treating this subject in a uniform and systematic way. The book puts special emphasis on the probabilistic analysis of numerical algorithms via the analysis of the corresponding condition. The exposition's level increases along the book, starting in the context of linear algebra at an undergraduate level and reaching in its third part the recent developments and partial solutions for Smale's 17th problem which can be explained within a graduate course. Its middle part contains a conditionbased course on linear programming that fills a gap between the current elementary expositions of the subject based on the simplex method and those focusing on convex programming
Geometric analysis of the condition of the convex feasibility problem by
Dennis Amelunxen(
)
1 edition published in 2011 in English and held by 18 WorldCat member libraries worldwide
The focus of this paper is the homogeneous convex feasibility problem, which is the following question: Given an mdimensional subspace of R n, does this subspace intersect a fixed convex cone solely in the origin or are there further intersection points? This problem includes as special cases the linear, the second order, and the semidefinite feasibility problems, where one simply chooses the positive orthant, a product of Lorentz cones, or the cone of positive semidefinite matrices, respectively. An important role for the running time of algorithms solving the convex feasibility problem is played by Renegar's condition number. The (inverse of the) condition of an input is basically the magnitude of the smallest perturbation, which changes the status of the input, i.e., which makes a feasible input infeasible, or the other way round. We will give an average analysis of this condition for several classes of convex cones, and one that is independent of the underlying convex cone. We will also describe a way of deriving smoothed analyses from our approach. We will achieve these results by adopting a purely geometric viewpoint leading to computations in the Grassmann manifold. Besides these main results about the random behavior of the condition of the convex feasibility problem, we will obtain a couple of byproduct results in the domain of spherical convex geometry. ; eng
1 edition published in 2011 in English and held by 18 WorldCat member libraries worldwide
The focus of this paper is the homogeneous convex feasibility problem, which is the following question: Given an mdimensional subspace of R n, does this subspace intersect a fixed convex cone solely in the origin or are there further intersection points? This problem includes as special cases the linear, the second order, and the semidefinite feasibility problems, where one simply chooses the positive orthant, a product of Lorentz cones, or the cone of positive semidefinite matrices, respectively. An important role for the running time of algorithms solving the convex feasibility problem is played by Renegar's condition number. The (inverse of the) condition of an input is basically the magnitude of the smallest perturbation, which changes the status of the input, i.e., which makes a feasible input infeasible, or the other way round. We will give an average analysis of this condition for several classes of convex cones, and one that is independent of the underlying convex cone. We will also describe a way of deriving smoothed analyses from our approach. We will achieve these results by adopting a purely geometric viewpoint leading to computations in the Grassmann manifold. Besides these main results about the random behavior of the condition of the convex feasibility problem, we will obtain a couple of byproduct results in the domain of spherical convex geometry. ; eng
Isotropic and coisotropic subvarieties of Grassmannians by
Kathlén Kohn(
)
1 edition published in 2018 in English and held by 18 WorldCat member libraries worldwide
1 edition published in 2018 in English and held by 18 WorldCat member libraries worldwide
Condition : the Geometry of Numerical Algorithms by
Peter Bürgisser(
)
1 edition published in 2013 in English and held by 17 WorldCat member libraries worldwide
1 edition published in 2013 in English and held by 17 WorldCat member libraries worldwide
Numerical and statistical aspects of tensor decompositions by
Paul Breiding(
)
1 edition published in 2017 in English and held by 15 WorldCat member libraries worldwide
1 edition published in 2017 in English and held by 15 WorldCat member libraries worldwide
On the complexity of numerical analysis(
)
1 edition published in 2006 in English and held by 14 WorldCat member libraries worldwide
1 edition published in 2006 in English and held by 14 WorldCat member libraries worldwide
Geometric complexity theory and orbit closures of homogeneous forms by
Jesko Hüttenhain(
)
1 edition published in 2017 in English and held by 13 WorldCat member libraries worldwide
1 edition published in 2017 in English and held by 13 WorldCat member libraries worldwide
On randomized algebraic test complexity by
Peter Bürgisser(
Book
)
2 editions published in 1992 in German and English and held by 7 WorldCat member libraries worldwide
We prove a general lower bound on the average decision complexity for testing membership in an irreducible algebraic subset X [union] [real numbers][superscript m] and apply it to kgeneric complete intersection of polynomials of the same degree, extending results in [4, 6]. We also give applications to nongeneric cases, such as graphs of elementary symmetric functions, SL(m, [real numbers]), and determinant varieties, extending results in [Li 90]."
2 editions published in 1992 in German and English and held by 7 WorldCat member libraries worldwide
We prove a general lower bound on the average decision complexity for testing membership in an irreducible algebraic subset X [union] [real numbers][superscript m] and apply it to kgeneric complete intersection of polynomials of the same degree, extending results in [4, 6]. We also give applications to nongeneric cases, such as graphs of elementary symmetric functions, SL(m, [real numbers]), and determinant varieties, extending results in [Li 90]."
Completeness and Reduction in Algebraic Complexity Theory by
Peter Bürgisser(
)
in English and held by 7 WorldCat member libraries worldwide
in English and held by 7 WorldCat member libraries worldwide
Algebraic Complexity Theory : With the Collaboration of Thomas Lickteig by
Peter Bürgisser(
)
in English and held by 7 WorldCat member libraries worldwide
in English and held by 7 WorldCat member libraries worldwide
Some computational problems in linear algebra as hard as matrix multiplication by
Peter Bürgisser(
Book
)
4 editions published in 1991 in German and English and held by 6 WorldCat member libraries worldwide
Abstract: "We define the complexity of a computational problem given by a relation using the model of computation trees together with the Ostrowski complexity measure. Natural examples from linear algebra are: KER[subscript n]: Compute a basis of the kernel for a given n x nmatrix, OGB [subscript n]: Find an invertible matrix that transforms a given symmetric n x nmatrix (quadratic form) to diagonal form, SPR[subscript n]: Find a sparse representation of a given n x nmatrix. To such a sequence of problems we assign an exponent similar as for matrix multiplication
4 editions published in 1991 in German and English and held by 6 WorldCat member libraries worldwide
Abstract: "We define the complexity of a computational problem given by a relation using the model of computation trees together with the Ostrowski complexity measure. Natural examples from linear algebra are: KER[subscript n]: Compute a basis of the kernel for a given n x nmatrix, OGB [subscript n]: Find an invertible matrix that transforms a given symmetric n x nmatrix (quadratic form) to diagonal form, SPR[subscript n]: Find a sparse representation of a given n x nmatrix. To such a sequence of problems we assign an exponent similar as for matrix multiplication
Decision complexity of generic complete intersections by
Peter Bürgisser(
Book
)
1 edition published in 1992 in German and held by 5 WorldCat member libraries worldwide
1 edition published in 1992 in German and held by 5 WorldCat member libraries worldwide
Geometric complexity theory, tensor rank, and LittlewoodRichardson coefficients by Christian Ikenmeyer(
)
2 editions published in 2012 in English and held by 4 WorldCat member libraries worldwide
We provide a thorough introduction to Geometric Complexity Theory, an approach towards computational complexity lower bounds via methods from algebraic geometry and representation theory. Then we focus on the relevant representation theoretic multiplicities, namely plethysm coefficients, Kronecker coefficients, and LittlewoodRichardson coefficients. These multiplicities can be described as dimensions of highest weight vector spaces for which explicit bases are known only in the LittlewoodRichardson case.By explicit construction of highest weight vectors we can show that the border rank of m x m matrix multiplication is a least 3 m 2  2 and the border rank of 2 x 2 matrix multiplication is exactly seven. The latter gives a new proof of a result by Landsberg (J. Amer. Math. Soc., 19:447459, 2005).Moreover, we obtain new nonvanishing results for rectangular Kronecker coefficients and we prove a conjecture by Weintraub (J. Algebra, 129 (1): 103114, 1990) about the nonvanishing of plethysm coefficients of even partitions.Our indepth study of LittlewoodRichardson coefficients c_{\lambda,\mu} \nu yields a polynomial time algorithm for deciding c_{\lambda,\mu} \nu >= t in time polynomial in n and quadratic in t, where n denotes the number of parts of \nu. For t = 1, i.e., for checking positivity of c_{\lambda,\mu} \nu, we even obtain a running time of n 3 log \nu_1.Moreover, our insights lead to a proof of a conjecture by King, Tollu, and Toumazet (CRM Proc. Lecture Notes, 34, Symmetry in Physics: 99112), stating that c_{\lambda,\mu} \nu = 2 implies c_{M\lambda,M\mu} {M\nu} = M + 1 for all M \in \N. ; eng
2 editions published in 2012 in English and held by 4 WorldCat member libraries worldwide
We provide a thorough introduction to Geometric Complexity Theory, an approach towards computational complexity lower bounds via methods from algebraic geometry and representation theory. Then we focus on the relevant representation theoretic multiplicities, namely plethysm coefficients, Kronecker coefficients, and LittlewoodRichardson coefficients. These multiplicities can be described as dimensions of highest weight vector spaces for which explicit bases are known only in the LittlewoodRichardson case.By explicit construction of highest weight vectors we can show that the border rank of m x m matrix multiplication is a least 3 m 2  2 and the border rank of 2 x 2 matrix multiplication is exactly seven. The latter gives a new proof of a result by Landsberg (J. Amer. Math. Soc., 19:447459, 2005).Moreover, we obtain new nonvanishing results for rectangular Kronecker coefficients and we prove a conjecture by Weintraub (J. Algebra, 129 (1): 103114, 1990) about the nonvanishing of plethysm coefficients of even partitions.Our indepth study of LittlewoodRichardson coefficients c_{\lambda,\mu} \nu yields a polynomial time algorithm for deciding c_{\lambda,\mu} \nu >= t in time polynomial in n and quadratic in t, where n denotes the number of parts of \nu. For t = 1, i.e., for checking positivity of c_{\lambda,\mu} \nu, we even obtain a running time of n 3 log \nu_1.Moreover, our insights lead to a proof of a conjecture by King, Tollu, and Toumazet (CRM Proc. Lecture Notes, 34, Symmetry in Physics: 99112), stating that c_{\lambda,\mu} \nu = 2 implies c_{M\lambda,M\mu} {M\nu} = M + 1 for all M \in \N. ; eng
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Algebra Algorithms Combinatorial analysis Computational complexity Computers Computer science Computer scienceMathematics Computer software Distribution (Probability theory) Geometry, Algebraic Grassmann manifolds Group theory Information theory Linear programming Logic, Symbolic and mathematical Mathematical optimization Mathematics Matrices Numerical analysis Villiers de L'IsleAdam, Auguste,comte de,