Syzygies of unimodular Lawrence ideals (2009)
Dave Bayer, Sorin Popescu, Bernd Sturmfels
Abstract: Infinite hyperplane arrangements whose vertices form a lattice are studied from the point of view of commutative algebra. The quotient of such an arrangement modulo the lattice action...
Geometry of the restricted Boltzmann machine (2009)
Cueto, Maria Angelica, Morton, Jason, Sturmfels, Bernd
The restricted Boltzmann machine is a graphical model for binary random variables. Based on a complete bipartite graph separating hidden and observed variables, it is the binary analog to the factor...
On the toric algebra of graphical models (2009)
Dan Geiger, Christopher Meek, Bernd Sturmfels
We formulate necessary and sufficient conditions for an arbitrary discrete probability distribution to factor according to an undirected graphical model, or a log-linear model, or other more general...
Blow-ups of $\mathbb{P}^{n-3}$ at $n$ points and spinor varieties (2009)
Sturmfels, Bernd, Velasco, Mauricio
Work of Dolgachev and Castravet-Tevelev establishes a bijection between the $2^{n-1}$ weights of the half-spin representations of $\mathfrak{so}_{2n}$ and the generators of the Cox ring of the...
Multivariate Gaussians, Semidefinite Matrix Completion, and Convex Algebraic Geometry (2009)
Sturmfels, Bernd, Uhler, Caroline
We study multivariate normal models that are described by linear constraints on the inverse of the covariance matrix. Maximum likelihood estimation for such models leads to the problem of maximizing...
COMMUTING BIRTH-AND-DEATH PROCESSES (2009)
Steven N. Evans, Bernd Sturmfels, Caroline Uhler
Abstract. We use methods from combinatorics and algebraic statistics to study analogues of birth-and-death processes that have as their state space a finite subset of the m-dimensional lattice and...
Siphons in chemical reaction networks (2009)
Siphons in a chemical reaction system are subsets of the species that have the potential of being absent in a steady state. We present a characterization of minimal siphons in terms of primary...
Classification of Six-Point Metrics (2009)
Abstract There are 339 combinatorial types of generic metrics on six points. They correspond to the 339 regular triangulations of the second hypersimplex \Delta (6; 2), which also has 14 non-regular...
Reconstructing Spatiotemporal Gene Expression Data from Partial Observations (2009)
Cartwright, Dustin A., Brady, Siobhan M., Orlando, David A., Sturmfels, Bernd, Benfey, Philip N.
Developmental transcriptional networks in plants and animals operate in both space and time. To understand these transcriptional networks it is essential to obtain whole-genome expression data at...
John Maynard Smith, Lior Pachter, Bernd Sturmfels
www.cambridge.org Information on this title: www.cambridge.org/9780521857000 c ○ Cambridge University Press 2005 This book is in copyright. Subject to statutory exception and to the provisions of...
The Hilbert scheme of the diagonal in a product of projective spaces (2009)
Cartwright, Dustin, Sturmfels, Bernd
The diagonal in a product of projective spaces is cut out by the ideal of 2x2-minors of a matrix of unknowns. The multigraded Hilbert scheme which classifies its degenerations has a unique...
Reconstructing spatiotemporal gene expression data from partial observations (2009)
Cartwright, Dustin A., Brady, Siobhan M., Orlando, David A., Sturmfels, Bernd, Benfey, Philip N.
Motivation: Developmental transcriptional networks in plants and animals operate in both space and time. To understand these transcriptional networks it is essential to obtain whole-genome expression...
Commuting birth-and-death processes (2008)
Evans, Steven N., Sturmfels, Bernd, Uhler, Caroline
We use methods from combinatorics and algebraic statistics to study analogues of birth-and-death processes that have as their state space a finite subset of the $m$-dimensional lattice and for which...
Polynomial relations among principal minors of a 4x4-matrix (2008)
Lin, Shaowei, Sturmfels, Bernd
The image of the principal minor map for n x n-matrices is shown to be closed. In the 19th century, Nansen and Muir studied the implicitization problem of finding all relations among principal minors...
Graphical models for correlated defaults (2008)
Filiz, I. Onur, Guo, Xin, Morton, Jason, Sturmfels, Bernd
A simple graphical model for correlated defaults is proposed, with explicit formulas for the loss distribution. Algebraic geometry techniques are employed to show that this model is well posed for...
How to Draw Tropical Planes (2008)
Herrmann, Sven, Jensen, Anders, Joswig, Michael, Sturmfels, Bernd
The tropical Grassmannian parameterizes tropicalizations of linear spaces, while the Dressian parameterizes all planes in $\TP^{n-1}$. We study these parameter spaces and we compute them explicitly...
References Constrained Model–Based Clustering (2008)
Gunter Ritter, Hans Achatz, Peter Kleinschmidt, Bernd Sturmfels, Ravindra K. Ahuja, ...
algorithms for bipartite network flows. SIAM J. Computing, 23:906–933, 1994. [3] María Teresa Gallegos and Gunter Ritter. A robust method for cluster analysis. Annals
Marginal Likelihood Integrals for Mixtures of Independence Models (2008)
Lin, Shaowei, Sturmfels, Bernd, Xu, Zhiqiang
Inference in Bayesian statistics involves the evaluation of marginal likelihood integrals. We present algebraic algorithms for computing such integrals exactly for discrete data of small sample size....
Matrix Cubes Parametrized by Eigenvalues (2008)
Nie, Jiawang, Sturmfels, Bernd
An elimination problem in semidefinite programming is solved by means of tensor algebra. It concerns families of matrix cube problems whose constraints are the minimum and maximum eigenvalue function...
Can Biology Lead to New Theorems? CAN BIOLOGY LEAD TO NEW THEOREMS? (2008)
Abstract. This article argues for an affirmative answer to the question in the title. In future interactions between mathematics and biology, both fields will contribute to each other, and, in...
Profile of Bow Street Staff (2008)
James A. Carlson, James Arthur Archive, Raoul Bott Library, Maria Chudnovsky, Bernd Sturmfels, ...
2 3 4 6 9 10 13 14 15 17 18 20 27 28
SOME CHAPTERS FROM “COMBINATORIAL COMMUTATIVE ALGEBRA” (2008)
Of the 18 chapters in the book, we will only cover the 5 listed below and thus allow for a more leisurely pace. There is also some supplementary material in the bibliography to expand upon if needed....
Ideals, Varieties and Macaulay 2 (2008)
Bernd Sturmfels, O Polynomialring
This chapter introduces Macaulay 2 commands for some elementary computations in algebraic geometry. Familiarity with Gröbner bases is assumed. Many students and researchers alike have their first...
( de Gruyter 2003 Alexander duality in subdivisions of Lawrence polytopes (2008)
Francisco Santos, Bernd Sturmfels
(Communicated by G. Ziegler) Abstract. The class of simplicial complexes representing triangulations and subdivisions of Lawrence polytopes is closed under Alexander duality. This gives a new...
Extension Spaces of Oriented Matriods Extension Spaces of Oriented Matroids (2008)
Bernd Sturmfels, Günter M. Ziegler, Bernd Sturmfels, Günter M. Ziegler
Abstract. We study the space of all extensions of a real hyperplane arrangement by a new pseudo-hyperplane, and, more generally, of an oriented matroid by a new element. The question whether this...
Sagbi Bases of Cox-Nagata Rings (2008)
Sturmfels, Bernd, Xu, Zhiqiang
We degenerate Cox-Nagata rings to toric algebras by means of sagbi bases induced by configurations over the rational function field. For del Pezzo surfaces, this degeneration implies the...
A Gröbner basis is a set of multivariate polynomials that has desirable algorithmic properties. Every set of polynomials can be transformed into a Gröbner basis. This process generalizes three...
1. Introduction. MONOMIAL RESOLUTIONS (2008)
Dave Bayer, Irena Peeva, Bernd Sturmfels
Let M be a monomial ideal in the polynomial ring S = k[x1,...,xn] over a field k. We are interested in the problem, first posed by Kaplansky in the early 1960’s, of finding a minimal
Normal Toric Ideals of Low Codimension (2008)
Dueck, Pierre, Hosten, Serkan, Sturmfels, Bernd
Every normal toric ideal of codimension two is minimally generated by a Grobner basis with squarefree initial monomials. A polynomial time algorithm is presented for checking whether a toric ideal of...
Research Contributions Overview Shmuel Onn (2008)
Imre Bárány, Yael Berstein, Jesus De Loera, Antoine Deza, Raymond Hemmecke, Peter Kleinschmidt, ...
My research aims at the development of efficient algebraic and geometric methods for discrete optimization, the investigation of the underlying mathematical structures, and the employment of such...
Graphical Models for Correlated Defaults (2008)
I. Onur Filiz, Xin Guo, Jason Morton, Bernd Sturmfels
A simple graphical model for correlated defaults is proposed, with explicit formulas for the loss distribution. Algebraic geometry techniques are employed to show that this model is well posed for...
Computer algebra in systems biology (2007)
Laubenbacher, Reinhard, Sturmfels, Bernd
Systems biology focuses on the study of entire biological systems rather than on their individual components. With the emergence of high-throughput data generation technologies for molecular biology...
Bernd Sturmfels, Robert Weismantel, Gunter M. Ziegler
Abstract. There are very close connections between the arithmetic of integer lattices, algebraic properties of the associated ideals, and the geometry and the combinatorics of corresponding...
Syzygies of Unimodular Lawrence Ideals (2007)
Dave Bayer, Sorin Popescu, Bernd Sturmfels
: Innite hyperplane arrangements whose vertices form a lattice are studied from the point of view of commutative algebra. The quotient of such an arrangement modulo the lattice action represents the...
A Toric Ring With Irrational Poincaré-Betti Series (2007)
Jan-Erik Roos, Bernd Sturmfels
.-- We show that there exists a toric curve in P 8 , whose homogeneous coordinate ring has a presentation with 12 quadratic relations and whose Poincar'e-Betti series is irrational. The example...
Supernormal vector configurations (2007)
Serkan Hosten, Diane Maclagan, Bernd Sturmfels, Let B
Abstract. A configuration of lattice vectors is supernormal if it contains a Hilbert basis for every cone spanned by a subset. We study such configurations from various perspectives, including...
Supernormal vector configurations (2007)
Diane Maclagan, Bernd Sturmfels
Abstract. A conguration of lattice vectors is supernormal if it contains a Hilbert basis for every cone spanned by a subset. We study such congurations from various perspectives, including...
Serkan Hosten, Francisco Santos, Bernd Sturmfels, Communicated Gunter, M. Ziegler
Abstract. We study the convex hull PA of the 0-1 incidence vectors of all triangulations of a point configuration A. This was called the universal polytope in [4]. The affine span of PA is described...
Serkan Hosten, Bernd Sturmfels
Integer programming (IP) is concerned with solving linear equations over the non-negative integers, subject to the requirement that a linear cost function is minimized. We consider families of...
Frank Sottile, Bernd Sturmfels
Dedicated to the memory of Gian-Carlo Rota Abstract. The maximal minors of a p (m + p)-matrix of univariate polynomials of degree n with indeterminate coecients are themselves polynomials of degree...
Classification of Six-Point Metrics (2007)
There are 339 combinatorial types of generic metrics on six points. They correspond to the 339 regular triangulations of the second hypersimplex ∆(6, 2), which also has 14 non-regular...
Contemporary Mathematics First Steps in Tropical Geometry (2007)
Jürgen Richter-gebert, Bernd Sturmfels, Thorsten Theobald
Tropical algebraic geometry is the study of piecewise-linear objects which behave like algebraic varieties. We give an introduction to this theory, with an emphasis on tropical varieties in...
Toric dynamical systems (2007)
Craciun, Gheorghe, Dickenstein, Alicia, Shiu, Anne, Sturmfels, Bernd
Toric dynamical systems are known as complex balancing mass action systems in the mathematical chemistry literature, where many of their remarkable properties have been established. They include as...
The Cyclohedron Test for Finding Periodic Genes in Time Course Expression Studies (2007)
Morton, Jason, Pachter, Lior, Shiu, Anne, Sturmfels, Bernd
The problem of finding periodically expressed genes from time course microarray experiments is at the center of numerous efforts to identify the molecular components of biological clocks. We present...
The Cyclohedron Test for Finding Periodic Genes in Time Course Expression Studies (2007)
Morton, Jason, Pachter, Lior, Shiu, Anne, Sturmfels, Bernd
The problem of finding periodically expressed genes from time course microarray experiments is at the center of numerous efforts to identify the molecular components of biological clocks. We present...
The Cyclohedron Test for Finding Periodic Genes in Time Course Expression Studies (2007)
Morton, Jason, Pachter, Lior, Shiu, Anne, Sturmfels, Bernd
The problem of finding periodically expressed genes from time course microarray experiments is at the center of numerous efforts to identify the molecular components of biological clocks. We present...
The Cyclohedron Test for Finding Periodic Genes in Time Course Expression Studies (2007)
Morton, Jason, Pachter, Lior, Shiu, Anne, Sturmfels, Bernd
The problem of finding periodically expressed genes from time course microarray experiments is at the center of numerous efforts to identify the molecular components of biological clocks. We present...
Open Problems in Algebraic Statistics (2007)
Algebraic statistics is concerned with the study of probabilistic models and techniques for statistical inference using methods from algebra and geometry. This article presents a list of open...
Affine Buildings and Tropical Convexity (2007)
Joswig, Michael, Sturmfels, Bernd, Yu, Josephine
The notion of convexity in tropical geometry is closely related to notions of convexity in the theory of affine buildings. We explore this relationship from a combinatorial and computational...
Tropical Implicitization and Mixed Fiber Polytopes (2007)
Sturmfels, Bernd, Yu, Josephine
The software TrIm offers implementations of tropical implicitization and tropical elimination, as developed by Tevelev and the authors. Given a polynomial map with generic coefficients, TrIm computes...
Elimination Theory for Tropical Varieties (2007)
Sturmfels, Bernd, Tevelev, Jenia
Tropical algebraic geometry offers new tools for elimination theory and implicitization. We determine the tropicalization of the image of a subvariety of an algebraic torus under any homomorphism...
Analysis of epistatic interactions and fitness landscapes using a new geometric approach (2007)
Beerenwinkel, Niko, Pachter, Lior, Sturmfels, Bernd, Elena, Santiago F, Lenski, Richard E
Abstract Background Understanding interactions between mutations and how they affect fitness is a central problem in evolutionary biology that bears on such fundamental issues as the structure of...
The Cyclohedron Test for Finding Periodic Genes in Time Course Expression Studies (2007)
Morton, Jason, Pachter, Lior, Shiu, Anne, Sturmfels, Bernd
The problem of finding periodically expressed genes from time course microarray experiments is at the center of numerous efforts to identify the molecular components of biological clocks. We present...
Convex Rank Tests and Semigraphoids (2007)
Morton, Jason, Pachter, Lior, Shiu, Anne, Sturmfels, Bernd, Wienand, Oliver
Convex rank tests are partitions of the symmetric group which have desirable geometric properties. The statistical tests defined by such partitions involve counting all permutations in the...
Semidefinite Representation of the $k$-Ellipse (2007)
Nie, Jiawang, Parrilo, Pablo A., Sturmfels, Bernd
The $k$-ellipse is the plane algebraic curve consisting of all points whose sum of distances from $k$ given points is a fixed number. The polynomial equation defining the $k$-ellipse has degree $2^k$...
DOI 10.1007/s11538-007-9244-7 (2007)
Peter Huggins, Lior Pachter, Bernd Sturmfels
Abstract The human genotope is the convex hull of all allele frequency vectors that can be obtained from the genotypes present in the human population. In this paper, we take a few initial steps...
The Algebraic Degree of Semidefinite Programming (2006)
Nie, Jiawang, Ranestad, Kristian, Sturmfels, Bernd
Given a generic semidefinite program, specified by matrices with rational entries, each coordinate of its optimal solution is an algebraic number. We study the degree of the minimal polynomials of...
Towards the Human Genotope (2006)
Huggins, Peter, Pachter, Lior, Sturmfels, Bernd
The human genotope is the convex hull of all allele frequency vectors that can be obtained from the genotypes present in the human population. In this paper we take a few initial steps towards a...
Three Counterexamples on Semigraphoids (2006)
Hemmecke, Raymond, Morton, Jason, Shiu, Anne, Sturmfels, Bernd, Wienand, Oliver
Semigraphoids are combinatorial structures that arise in statistical learning theory. They are equivalent to convex rank tests and to polyhedral fans that coarsen the reflection arrangement of the...
Conjunctive Bayesian networks (2006)
Beerenwinkel, Niko, Eriksson, Nicholas, Sturmfels, Bernd
Conjunctive Bayesian networks (CBNs) are graphical models that describe the accumulation of events which are constrained in the order of their occurrence. A CBN is given by a partial order on a...
On the toric algebra of graphical models (2006)
Geiger, Dan, Meek, Christopher, Sturmfels, Bernd
We formulate necessary and sufficient conditions for an arbitrary discrete probability distribution to factor according to an undirected graphical model, or a log-linear model, or other more general...
The Newton Polytope of the Implicit Equation (2006)
Sturmfels, Bernd, Tevelev, Jenia, Yu, Josephine
We apply tropical geometry to study the image of a map defined by Laurent polynomials with generic coefficients. If this image is a hypersurface then our approach gives a construction of its Newton...
Toric geometry of cuts and splits (2006)
Sturmfels, Bernd, Sullivant, Seth
Associated to any graph is a toric ideal whose generators record relations among the cuts of the graph. We study these ideals and the geometry of the corresponding toric varieties. Our theorems and...
On the toric algebra of graphical models (2006)
Geiger, Dan, Meek, Christopher, Sturmfels, Bernd
We formulate necessary and sufficient conditions for an arbitrary discrete probability distribution to factor according to an undirected graphical model, or a log-linear model, or other more general...
Parametric Alignment of Drosophila Genomes (2006)
Colin N. Dewey, Peter M. Huggins, Kevin Woods, Bernd Sturmfels, Lior Pachter
The classic algorithms of Needleman–Wunsch and Smith–Waterman find a maximum a posteriori probability alignment for a pair hidden Markov model (PHMM). To process large genomes that have undergone...
Morton, Jason, Pachter, Lior, Shiu, Anne, Sturmfels, Bernd, Wienand, Oliver
We study partitions of the symmetric group which have desirable geometric properties. The statistical tests defined by such partitions involve counting all permutations in the equivalence classes....
Parametric alignment of Drosophila genomes (2006)
Colin Dewey, Peter Huggins, Kevin Woods, Bernd Sturmfels, Lior Pachter
The classic algorithms of Needleman--Wunsch and Smith--Waterman find a maximum a posteriori probability alignment for a pair hidden Markov model (PHMM). In order to process large genomes that have...
Hyperdeterminantal relations among symmetric principal minors (2006)
The principal minors of a symmetric $n{\times}n$-matrix form a vector of length $2^n$. We characterize these vectors in terms of algebraic equations derived from the $...
Epistasis and Shapes of Fitness Landscapes (2006)
Beerenwinkel, Niko, Pachter, Lior, Sturmfels, Bernd
The relationship between the shape of a fitness landscape and the underlying gene interactions, or epistasis, has been extensively studied in the two-locus case. Gene interactions among multiple loci...
The Hyperdeterminant and Triangulations of the 4-Cube (2006)
Huggins, Peter, Sturmfels, Bernd, Yu, Josephine, Yuster, Debbie
The hyperdeterminant of format 2 x 2 x 2 x 2 is a polynomial of degree 24 in 16 unknowns which has 2894276 terms. We compute the Newton polytope of this polynomial and the secondary polytope of the...
Sturmfels: The mathematics of phylogenomics (2006)
“The lack of real contact between mathematics and biology is either a tragedy, a scandal or a challenge, it is hard to decide which. ” – Gian-Carlo Rota, [26, p. 2] 1
Parametric alignment of Drosophila genomes (2006)
Colin Dewey, Peter Huggins, Kevin Woods, Bernd Sturmfels, Lior Pachter
The classic algorithms of Needleman–Wunsch and Smith–Waterman find a maximum a posteriori probability alignment for a pair hidden Markov model (PHMM). In order to process large genomes that have...
Jason Morton, Lior Pachter, Anne Shiu, Bernd Sturmfels, Oliver Wien
We study partitions of the symmetric group which have desirable geometric properties. The statistical tests defined by such partitions involve counting all permutations in the equivalence classes....
Parametric Alignment of Drosophila Genomes (2005)
Dewey, Colin, Huggins, Peter, Woods, Kevin, Sturmfels, Bernd, Pachter, Lior
The classic algorithms of Needleman--Wunsch and Smith--Waterman find a maximum a posteriori probability alignment for a pair hidden Markov model (PHMM). In order to process large genomes that have...
Evolution on distributive lattices (2005)
Beerenwinkel, Niko, Eriksson, Nicholas, Sturmfels, Bernd
We consider the directed evolution of a population after an intervention that has significantly altered the underlying fitness landscape. We model the space of genotypes as a distributive lattice;...
Dickenstein, Alicia, Feichtner, Eva Maria, Sturmfels, Bernd
Tropical geometry is used to develop a new approach to the theory of discriminants and resultants in the sense of Gel'fand, Kapranov and Zelevinsky. The tropical A-discriminant, which is the...
Algebraic Factor Analysis: Tetrads, Pentads and Beyond (2005)
Drton, Mathias, Sturmfels, Bernd, Sullivant, Seth
Factor analysis refers to a statistical model in which observed variables are conditionally independent given fewer hidden variables, known as factors, and all the random variables follow a...
An algebraic geometry approach to nonlinear parametric optimization in control (2005)
Fotiou, Ioannis A., Rostalski, Philipp, Sturmfels, Bernd, Morari, Manfred
We present a method for nonlinear parametric optimization based on algebraic geometry. The problem to be studied, which arises in optimal control, is to minimize a polynomial function with parameters...
Computing Tropical Varieties (2005)
Bogart, Tristram, Jensen, Anders, Speyer, David, Sturmfels, Bernd, Thomas, Rekha
The tropical variety of a $d$-dimensional prime ideal in a polynomial ring with complex coefficients is a pure $d$-dimensional polyhedral fan. This fan is shown to be connected in codimension one. We...
Combinatorial secant varieties (2005)
Sturmfels, Bernd, Sullivant, Seth
The construction of joins and secant varieties is studied in the combinatorial context of monomial ideals. For ideals generated by quadratic monomials, the generators of the secant ideals are...
Can biology lead to new theorems? (2005)
This article argues for an affirmative answer to the question in the title. In future interactions between mathematics and biology, both fields will contribute to each other, and, in particular,...
Toric ideals of phylogenetic invariants (2005)
Bernd Sturmfels, Seth Sullivant
Statistical models of evolution are algebraic varieties in the space of joint probability distributions on the leaf colorations of a phylogenetic tree. The phylogenetic invariants of a model are the...
Alicia Dickenstein, Eva Maria Feichtner, Bernd Sturmfels, Dedicated Pilar, Pisón Casares
Let A be an integer d × n-matrix such that (1, 1,...,1) is in the row span of A. This defines a projective toric variety XA in CPn−1.Itsdual variety X ∗ A is the closure in the projective space...
Minimizing Polynomials via Sum of Squares over the Gradient Ideal (2004)
Nie, Jiawang, Demmel, James W., Sturmfels, Bernd
This paper has been withdrawn by the authors due to its publication
Matroid polytopes, nested sets and Bergman fans (2004)
Feichtner, Eva Maria, Sturmfels, Bernd
The tropical variety defined by linear equations with constant coefficients is the Bergman fan of the corresponding matroid. Building on a self-contained introduction to matroid polytopes, we present...
The Mathematics of Phylogenomics (2004)
Pachter, Lior, Sturmfels, Bernd
The grand challenges in biology today are being shaped by powerful high-throughput technologies that have revealed the genomes of many organisms, global expression patterns of genes and detailed...
Solving the Likelihood Equations (2004)
Hosten, Serkan, Khetan, Amit, Sturmfels, Bernd
Given a model in algebraic statistics and some data, the likelihood function is a rational function on a projective variety. Algebraic algorithms are presented for computing all critical points of...
Speyer, David, Sturmfels, Bernd
These are the notes for the Clay Mathematics Institute Senior Scholar Lecture which was delivered by Bernd Sturmfels in Park City, Utah, on July 22, 2004. The topic of this lecture is the ``tropical...
Phylogenetic Algebraic Geometry (2004)
Eriksson, Nicholas, Ranestad, Kristian, Sturmfels, Bernd, Sullivant, Seth
Phylogenetic algebraic geometry is concerned with certain complex projective algebraic varieties derived from finite trees. Real positive points on these varieties represent probabilistic models of...
The Maximum Likelihood Degree (2004)
Catanese, Fabrizio, Hosten, Serkan, Khetan, Amit, Sturmfels, Bernd
Maximum likelihood estimation in statistics leads to the problem of maximizing a product of powers of polynomials. We study the algebraic degree of the critical equations of this optimization...
Resultants in Genetic Linkage Analysis (2004)
Hallgrimsdottir, Ingileif B., Sturmfels, Bernd
Statistical models for genetic linkage analysis of k-locus diseases are k-dimensional subvarieties of a (3^k-1)-dimensional probability simplex. We determine the algebraic invariants of these models...
Classification of Six-Point Metrics (2004)
Sturmfels, Bernd, Yu, Josephine
There are 339 combinatorial types of generic metrics on six points. They correspond to the 339 regular triangulations of the second hypersimplex \Delta(6,2), which also has 14 non-regular...
Toric ideals of phylogenetic invariants (2004)
Sturmfels, Bernd, Sullivant, Seth
Statistical models of evolution are algebraic varieties in the space of joint probability distributions on the leaf colorations of a phylogenetic tree. The phylogenetic invariants of a model are the...
Parametric Inference for Biological Sequence Analysis (2004)
Pachter, Lior, Sturmfels, Bernd
One of the major successes in computational biology has been the unification, using the graphical model formalism, of a multitude of algorithms for annotating and comparing biological sequences....
Parametric inference for biological sequence analysis (2004)
One of the major successes in computational biology has been the unification, using the graphical model formalism, of a multitude of algorithms for annotating and comparing biological sequences....
Bernd Sturmfels, Lucile Packard Fellowship
rship: Journal of the American Mathematical Society, Duke Mathematical Journal, Collecteana Mathematica, Beitrage zur Geometrie und Algebra, Order, Discrete and Computational Geometry, Applicable...
Documenta Math. 205 Erratum for “Tropical Convexity” (2004)
Mike Develin, Bernd Sturmfels, Communicated Günter Ziegler, Mike Develin, Bernd Sturmfels
Abstract. Theorem 29 and Corollary 30 of [1] are incorrect. This only concerns the application of tropical convexity to phylogenetic trees. None of the results on tropical convexity itself is...
Tropical Geometry of Statistical Models (2003)
Pachter, Lior, Sturmfels, Bernd
This paper presents a unified mathematical framework for inference in graphical models, building on the observation that graphical models are algebraic varieties. From this geometric viewpoint,...
Algebraic Recipes for Integer Programming (2003)
Integer programming is concerned with solving linear systems of equations over the non-negative integers. The basic question is to find a solution which minimizes a given linear objective function...
Develin, Mike, Sturmfels, Bernd
The notions of convexity and convex polytopes are introduced in the setting of tropical geometry. Combinatorial types of tropical polytopes are shown to be in bijection with regular triangulations of...
Short Rational Functions for Toric Algebra and Applications (2003)
De Loera, Jesus, Haws, David, Hemmecke, Raymond, Huggins, Peter, Sturmfels, Bernd, Yoshida, Ruriko
We encode the binomials belonging to the toric ideal $I_A$ associated with an integral $d \times n$ matrix $A$ using a short sum of rational functions as introduced by Barvinok \cite{bar,newbar}....
First steps in tropical geometry (2003)
Richter-Gebert, Jürgen, Sturmfels, Bernd, Theobald, Thorsten
Tropical algebraic geometry is the geometry of the tropical semiring $(\mathbb{R},\min,+)$. Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We give an...
The Tropical Grassmannian (2003)
Speyer, David, Sturmfels, Bernd
In tropical algebraic geometry, the solution sets of polynomial equations are piecewise-linear. We introduce the tropical variety of a polynomial ideal, and we identify it with a polyhedral...
Computing the Integer Programming Gap (2003)
Hosten, Serkan, Sturmfels, Bernd
We determine the maximal gap between the optimal values of an integer program and its linear programming relaxation, where the matrix and cost function are fixed but the right hand side is...
Algebraic Geometry of Bayesian Networks (2003)
Garcia, Luis David, Stillman, Michael, Sturmfels, Bernd
We study the algebraic varieties defined by the conditional independence statements of Bayesian Networks. A complete algebraic classification is given for Bayesian Networks on at most five random...
Algebraic unimodular counting (2003)
Abstract We study algebraic algorithms for expressing the number of non-negative integer solutions to a unimodular system of linear equations as a function of the right hand side. Our methods include...
Documenta Math. 1 Tropical Convexity (2003)
Mike Develin, Bernd Sturmfels, Communicated Günter Ziegler
Abstract. The notions of convexity and convex polytopes are introduced in the setting of tropical geometry. Combinatorial types of tropical polytopes are shown to be in bijection with regular...
Higher Lawrence configurations (2002)
Santos, Francisco, Sturmfels, Bernd
Any configuration of lattice vectors gives rise to a hierarchy of higher-dimensional configurations which generalize the Lawrence construction in geometric combinatorics. We prove finiteness results...
The Graph of Monomial Ideals (2002)
Altmann, Klaus, Sturmfels, Bernd
There is a natural infinite graph whose vertices are the monomial ideals in a polynomial ring. The definition involves Gr\"obner bases or the action of an algebraic torus. We present algorithms for...
Toric Hyperkahler Varieties (2002)
Hausel, Tamas, Sturmfels, Bernd
Extending work of Bielawski-Dancer and Konno, we develop a theory of toric hyperkahler varieties, which involves toric geometry, matroid theory and convex polyhedra. The framework is a detailed study...
Alexander duality in subdivisions of Lawrence polytopes (2002)
Santos, Francisco, Sturmfels, Bernd
The class of simplicial complexes representing triangulations and subdivisions of Lawrence polytopes is closed under Alexander duality. This gives a new geometric model for oriented matroid duality.
Syzygies of oriented matroids (2002)
Novik, Isabella, Postnikov, Alexander, Sturmfels, Bernd
We construct minimal cellular resolutions of squarefree monomial ideals arising from hyperplane arrangements, matroids, and oriented matroids. These are Stanley-Reisner ideals of complexes of...
Multigraded Hilbert Schemes (2002)
Haiman, Mark, Sturmfels, Bernd
We introduce the multigraded Hilbert scheme, which parametrizes all homogeneous ideals with fixed Hilbert function in a polynomial ring that is graded by any abelian group. Our construction is widely...
Documenta Math. 495 Toric Hyperkähler Varieties (2002)
Tamás Hausel, Bernd Sturmfels, Communicated Günter, M. Ziegler
Abstract. Extending work of Bielawski-Dancer [3] and Konno [14], we develop a theory of toric hyperkähler varieties, which involves toric geometry, matroid theory and convex polyhedra. The framework...
SOLVING SYSTEMS OF POLYNOMIAL EQUATIONS (2002)
Howdy Readers, Bernd Sturmfels
These are the lecture notes for ten lectures to be given at the CBMS
Supernormal Vector Configurations (2001)
Hosten, Serkan, Maclagan, Diane, Sturmfels, Bernd
A configuration of lattice vectors is supernormal if it contains a Hilbert basis for every cone spanned by a subset. We study such configurations from various perspectives, including triangulations,...
Algebraic Unimodular Counting (2001)
De Loera, Jesus A., Sturmfels, Bernd
We study algebraic algorithms for expressing the number of non-negative integer solutions to a unimodular system of linear equations as a function of the right hand side. Our methods include Todd...
Minimizing Polynomial Functions (2001)
Parrilo, Pablo A., Sturmfels, Bernd
We compare algorithms for global optimization of polynomial functions in many variables. It is demonstrated that existing algebraic methods (Gr\"obner bases, resultants, homotopy methods) are...
Elimination Theory in Codimension Two (2001)
Dickenstein, Alicia, Sturmfels, Bernd
New formulas are given for Chow forms, discriminants and resultants arising from (not necessarily normal) toric varieties of codimension 2. Exact descriptions are also given for the secondary polygon...
DUKE MATHEMATICAL JOURNAL Vol. 111, No. 2, c ○ 2002 SYZYGIES OF ORIENTED MATROIDS (2001)
Isabella Novik, Alexander Postnikov, Bernd Sturmfels
We construct minimal cellular resolutions of squarefree monomial ideals arising from hyperplane arrangements, matroids, and oriented matroids. These are Stanley-Reisner ideals of complexes of...
Algorithms for the Toric Hilbert Scheme (2000)
Stillman, Michael, Sturmfels, Bernd, Thomas, Rekha R.
The toric Hilbert scheme parametrizes all algebras isomorphic to a given semigroup algebra as a multigraded vectorspace. All components of the scheme are toric varieties, and among them, there is a...
Cattani, Eduardo, Dickenstein, Alicia, Sturmfels, Bernd
A binomial residue is a rational function defined by a hypergeometric integral whose kernel is singular along binomial divisors. Binomial residues provide an integral representation for rational...
Macaulay 2 and the geometry of schemes (2000)
Smith, Gregory G., Sturmfels, Bernd
This tutorial illustrates how to use Grayson and Stillman's computer algebra system, Macaulay2, to study schemes. The examples are taken from the homework for an algebraic geometry class given at the...
as follows. The Nn-graded k[x]-module of i-chains is (2000)
Isabella Novik, Alexander Postnikov, Bernd Sturmfels
Abstract. We construct minimal cellular resolutions of squarefree monomial ideals arising from hyperplane arrangements, matroids and oriented matroids. These are Stanley-Reisner ideals of complexes...
Syzygies of Oriented Matroids (2000)
Isabella Novik, Alexander Postnikov, Bernd Sturmfels
. We construct minimal cellular resolutions of squarefree monomial ideals arising from hyperplane arrangements, matroids and oriented matroids. These are Stanley-Reisner ideals of complexes of...
Syzygies of Unimodular Lawrence Ideals (1999)
Bayer, Dave, Popescu, Sorin, Sturmfels, Bernd
Infinite hyperplane arrangements whose vertices form a lattice are studied from the point of view of commutative algebra. The quotient of such an arrangement modulo the lattice action represents the...
Rational Hypergeometric Functions (1999)
Cattani, Eduardo, Dickenstein, Alicia, Sturmfels, Bernd
Multivariate hypergeometric functions associated with toric varieties were introduced by Gel'fand, Kapranov and Zelevinsky. Singularities of such functions are discriminants, that is, divisors...
A sagbi basis for the quantum Grassmannian (1999)
Sottile, Frank, Sturmfels, Bernd
The maximal minors of a p by (m + p) matrix of univariate polynomials of degree n with indeterminate coefficients are themselves polynomials of degree np. The subalgebra generated by their...
CUTTING CORNERS Shmuel Onn (1999)
Bernd Sturmfe Ls, Shmuel Onn, Bernd Sturmfels
this paper is the corner cut polyhedron, which we define as follows: P
CUTTING CORNERS Shmuel Onn (1999)
Bernd Sturmfe Ls, Shmuel Onn, Bernd Sturmfels
this paper is the corner cut polyhedron, which we define as follows: P
Generic and Cogeneric Monomial Ideals (1999)
Ezra Miller, Bernd Sturmfels, Kohji Yanagawa
This paper is a study of genericity properties of monomial ideals, initiated by Bayer, Peeva, and Sturmfels (1998). We will often use results from prior papers on this subject, although we have tried...
Monomial Ideals and Planar Graphs (1999)
. Grobner basis theory reduces questions about systems of polynomial equations to the combinatorial study of monomial ideals, or staircases. This article gives an elementary introduction to current...
A Sagbi Basis for the Quantum Grassmannian (1999)
Frank Sottile, Bernd Sturmfels
. The maximal minors of a p\Theta(m+p)-matrix of univariate polynomials of degree n with indeterminate coefficients are themselves polynomials of degree np. The subalgebra generated by their...
The LCM-lattice in monomial resolutions (1999)
Dave Bayer, Irena Peeva, Bernd Sturmfels
this paper is that there exists a genericity condition, which ensures simply structured homological behavior. The same idea is developed further for toric varieties in [PS]. We call a monomial ideal...
this paper is the corner cut polyhedron, which we define as follows: P
Multigraded Hilbert schemes (1999)
Michael Stillman, Bernd Sturmfels, Rekha Thomas
The toric Hilbert scheme parametrizes all algebras isomorphic to a given semigroup algebra as a multigraded vector space. All components of the scheme are toric varieties, and among them, there is a...
The object of this study in this paper is the corner cut polyhedron, which we define as follows: d 1 n 1 n d d Pn � conv � � �� � � � : �,..., � are n distinct vectors in N 4 �...
Generic and Cogeneric Monomial Ideals (1998)
Miller, Ezra, Sturmfels, Bernd, Yanagawa, Kohji
Monomial ideals which are generic with respect to either their generators or irreducible components have minimal free resolutions derived from simplicial complexes. For a generic monomial ideal, the...
Algebraic algorithms for sampling from conditional distributions (1998)
Diaconis, Persi, Sturmfels, Bernd
We construct Markov chain algorithms for sampling from discrete exponential families conditional on a sufficient statistic. Examples include contingency tables, logistic regression, and spectral...
Let S = k[x1,...,xn] be a polynomial ring over a field k and I a homogeneous ideal in S. A basic problem in commutative algebra is to construct the minimal free resolution FI of S/I over S. The...
Irena Peeva, Victor Reiner, Bernd Sturmfels
Abstract. We study monoid algebras which possess an initial ideal that is the Stanley-Reisner ideal of a poset. We construct a quadratic non-commutative Grobner basis which induces non-pure shellings...
Syzygies of codimension 2 lattice ideals (1998)
The study of semigroup algebras has a long tradition in commutative algebra. Presentation ideals of semigroup algebras are called toric ideals, inreference to their prominent role in geometry. In...
Residues and resultants (1998)
Eduardo Cattani, Alicia Dickenstein, Bernd Sturmfels
Abstract. Resultants, Jacobians and residues are basic invariants of multivariate polynomial systems. We examine their interrelations in the context of toric geometry. The global residue in the...
Irena Peeva, Victor Reiner, Bernd Sturmfels
, we consider minimal free resolutions of a field k as a module over the monoid algebra k[]. Using a result of Laudal and Sletsje which inteprets the ranks of the free modules in the resolution as...
Let S = k[x1,..., xn] be a polynomial ring over a field k and I a homogeneous ideal in S. A basic problem in commutative algebra is to construct the minimal free resolution FI of S/I over S. The...
Non-commutative Grobner bases for commutative algebras (1998)
David Eisenbud, Irena Peeva, Bernd Sturmfels
Abstract. We show that an ideal I in the free associative algebra k〈X1,..., Xn〉 over a field k has a finite Gröbner basis if the algebra defined by I is commutative; in characteristic 0 and...
Cellular resolutions of monomial modules (1998)
Abstract: We construct a canonical free resolution for arbitrary monomial modules and lattice ideals. This includes monomial ideals and defining ideals of toric varieties, and it generalizes our...
Algebraic algorithms for sampling from conditional distributions (1998)
Persi Diaconis, Bernd Sturmfels
We construct Markov chain algorithms for sampling from discrete exponential families conditional on a sufficient statistic. Examples include generating tables with fixed row and column sums and...
Residues and Resultants (1998)
Eduardo Cattani, Alicia Dickenstein, Bernd Sturmfels
Resultants, Jacobians and residues are basic invariants of multivariate polynomial systems. We examine their interrelations in the context of toric geometry. The global residue in the torus, studied...
Cellular Resolutions of Monomial Modules (1998)
: We construct a canonical free resolution for arbitrary monomial modules and lattice ideals. This includes monomial ideals and defining ideals of toric varieties, and it generalizes our joint...
Dave Bayer, Irena Peeva, Bernd Sturmfels
this paper we prove that the minimal free resolution for any generic monomial ideal M
Four Counterexamples In Combinatorial Algebraic Geometry (1998)
this article. 1 1. Regularity of powers of ideals
Gröbner Bases and Hypergeometric Functions (1998)
Bernd Sturmfels, Nobuki Takayama
The purpose of this tutorial 1 is to illustrate the use of Grobner bases and Buchberger's algorithm in the algebraic study of linear partial differential equations. Our reference example is the...
Grobner Bases and Hypergeometric Functions (1998)
Bernd Sturmfels And, Bernd Sturmfels, Nobuki Takayama
Introduction The purpose of this tutorial 1 is to illustrate the use of Grobner bases and Buchberger's algorithm in the algebraic study of linear partial differential equations. Our reference...
GENERIC LATTICE IDEALS Irena Peeva Bernd Sturmfels (1998)
this paper are:
Syzygies Of Codimension 2 Lattice Ideals (1998)
this paper we consider the more general class of lattice ideals. Fix a polynomial ring S = k[x 1 ; : : : ; x n ] over a field k and identify monomials x
Non-Commutative Gröbner Bases For Commutative Algebras (1998)
David Eisenbud, Irena Peeva, Bernd Sturmfels
this paper. Typeset by A M S-T E X 2 DAVID EISENBUD, IRENA PEEVA AND BERND STURMFELS
Dave Bayer, Irena Peeva, Bernd Sturmfels
this paper is that there exists a genericity condition, which ensures simply structured homological behavior. The same idea is developed further for toric varieties in [PS]. We call a monomial ideal...
Numerical Schubert Calculus (1998)
Birkett Huber, Frank Sottile, Bernd Sturmfels
We develop numerical homotopy algorithms for solving systems of polynomial equations arising from the classical Schubert calculus. These homotopies are optimal in that...
Irena Peeva, Victor Reiner, Bernd Sturmfels
Abstract. For a finitely generated submonoid Λ of N d, we consider the minimal free resolution of a field k as a module over the monoid algebra k[Λ]. Interpreting the ranks of the free modules in...
Cellular Resolutions of Monomial Modules (1997)
We construct a canonical free resolution for arbitrary monomial modules and lattice ideals. This includes monomial ideals and defining ideals of toric varieties, and it generalizes our joint results...
Numerical Schubert calculus (1997)
Huber, Birkett, Sottile, Frank, Sturmfels, Bernd
We develop numerical homotopy algorithms for solving systems of polynomial equations arising from the classical Schubert calculus. These homotopies are optimal in that generically no paths diverge....
Residues and Resultants (1997)
Cattani, Eduardo, Dickenstein, Alicia, Sturmfels, Bernd
Resultants, Jacobians and residues are basic invariants of multivariate polynomial systems. We examine their interrelations in the context of toric geometry. The global residue in the torus, studied...
Equations defining toric varieties (1997)
defining ideals of subvarieties of affine or projective space which are parametrized by monomials. With one exception there will be no proofs given in this paper. For most assertions which are stated...
Hypergeometric Polynomials and Integer Programming (1997)
Mutsumi Saito, Bernd Sturmfels, Nobuki Takayama
this paper we examine connections between hypergeometric differential equations and the theory of integer programming. Let A = (a ij ) be a non-negative integer d \Theta n-matrix
Hypergeometric Polynomials and Integer Programming (1997)
Mutsumi Saito, Bernd Sturmfels, Nobuki Takayama
. We examine connections between A-hypergeometric differential equations and the theory of integer programming. In the first part, we develop a "hypergeometric sensitivity analysis" for...
Constructive Invariant Theory (1997)
Bernd Sturmfels, Instructor Bernd Sturmfels, Harrison Tsai
this paper is to present an algorithm due to Derksen that computes such a nite set of generators for the invariant ring of an arbitrary representation of a linearly reductive group (3). Along the way...
Numerical Schubert Calculus (1997)
Birkett Huber, Frank Sottile, Bernd Sturmfels
. We develop numerical homotopy algorithms for solving systems of polynomial equations arising from the classical Schubert calculus. These homotopies are optimal in that generically no paths diverge....
Irena Peeva, Victor Reiner, Bernd Sturmfels
. For a finitely generated submonoid of N d , we consider the minimal free resolution of a field k as a module over the monoid algebra k[]. Interpreting the ranks of the free modules in the...
Gröbner Bases and Hypergeometric Functions (1997)
Bernd Sturmfels, Nobuki Takayama
Introduction The purpose of this tutorial is to illustrate the use of Grobner bases and Buchberger's algorithm in the algebraic study of linear partial differential equations. Our reference...
Equations Defining Toric Varieties (1996)
This article will appear in the proceedings of the AMS Summer Institute in Algebraic Geometry at Santa Cruz, July 1995. The topic is toric ideals, by which I mean the defining ideals of subvarieties...
Bayer, Dave, Peeva, Irena, Sturmfels, Bernd
Call a monomial ideal M "generic" if no variable appears with the same nonzero exponent in two distinct monomial generators. Using a convex polytope first studied by Scarf, we obtain a minimal free...
Grobner bases and convex polytopes / Bernd Sturmfels (1996)
Incluye bibliografía e índice
Solving Algebraic Equations in Terms of A-Hypergeometric Series (1996)
The roots of the general equation of degree n satisfy an A-hypergeometric system of differential equations in the sense of Gel'fand, Kapranov and Zelevinsky. We construct the n distinct...
The Polytope of All Triangulations of a Point Configuration (1996)
Serkan Hosten, Francisco Santos, Bernd Sturmfels
We study the convex hull PA of the 0-1 incidence vectors of all triangulations of a point configuration A. This was called the universal polytope in [4]. The affine span of PA is described in terms...
Structural Gröbner Basis Detection (1996)
Bernd Sturmfels, Markus Wiegelmann
We determine the computational complexity of deciding whether m polynomials in n variables have relatively prime leading terms with respect to some term order. This problem is NP-complete in general,...
Solving Algebraic Equations in Terms of A-Hypergeometric Series (1996)
The roots of the general equation of degree n satisfy an A-hypergeometric system of differential equations in the sense of Gel'fand, Kapranov and Zelevinsky. We construct the n distinct...
The geometry of A-graded algebras (1994)
We study algebras k[x_1,...,x_n]/I which admit a grading by a subsemigroup of N^d such that every graded component is a one-dimensional k-vector space. V.I.~Arnold and coworkers proved that for d = 1...
Computing Multidimensional Residues (1994)
Cattani, Eduardo, Dickenstein, Alicia, Sturmfels, Bernd
Given n polynomials in n variables with a finite number of complex roots, for any of their roots there is a local residue operator assigning a complex number to any polynomial. This is an algebraic,...
Intersection Theory on Toric Varieties (1994)
Fulton, William, Sturmfels, Bernd
The operational Chow cohomology classes of a complete toric variety are identified with certain functions, called Minkowski weights, on the corresponding fan. The natural product of Chow cohomology...
Eisenbud, David, Sturmfels, Bernd
We investigate the structure of ideals generated by binomials (polynomials with at most two terms) and the schemes and varieties associated to them. The class of binomial ideals contains many...
A Note On Lattice Simplices And Toric Varieties (1994)
such that den(A \Gamma1 ) = jdet(A)j = n and den(e d \Delta A \Gamma1 ) = 1. Implication. The simplicial cones defined by the matrices in part (b) of this Theorem define a family of affine toric...
Variation of Cost Functions in Integer Programming (1994)
Bernd Sturmfels, Rekha R. Thomas
We study the problem of minimizing c \Delta x subject to A \Delta x = b, x 0 and x integral, for a fixed matrix A. Two cost functions c and c 0 are considered equivalent if they give the same optimal...
Gröbner Bases And Triangulations Of The Second Hypersimplex (1994)
Bernd Sturmfels, Rekha R. Thomas
The algebraic technique of Grobner bases is applied to study triangulations of the second hypersimplex \Delta(2; n). We present a quadratic Grobner basis for the associated toric ideal I(Kn ). The...
A Quantitative Steinitz' Theorem (1994)
. Any 3-dimensional convex polytope with n vertices can be realized in Euclidean 3-space with all coordinates of all vertices being integers of absolute value not exceeding n 169n 3 . 1. Introduction...
A Quantitative Steinitz' Theorem (1994)
. Any 3-dimensional convex polytope with n vertices can be realized in Euclidean 3-space with all coordinates of all vertices being integers of absolute value not exceeding n 169n 3 . 1. Introduction...
Finding Sparse Systems of Parameters (1993)
Eisenbud, David, Sturmfels, Bernd
For several computational procedures such as finding radicals and Noether normalizations, it is important to choose as sparse as possible a system of parameters in a polynomial ideal or modulo a...
Computing Hopf Bifurcations I (1993)
John Guckenheimer, Mark Myers, Bernd Sturmfels
. This paper addresses the problems of detecting Hopf bifurcations in systems of ordinary differential equations and following curves of Hopf points in two parameter families of vector fields. The...
Computational synthetic geometry / Jürgen Bokowski, Bernd Sturmfels (1989)
Bokowski, Jürgen, Sturmfels, Bernd
Incluye bibliografía
Boundary complexes of convex polytopes cannot be characterized locally (1987)
It is well known that there is no local criterion to decide the linear readability of matroids or oriented matroids. We use the set-up of chirotopes or oriented matroids to derive a similar result in...
Tropical geometry of statistical models
Pachter, Lior, Sturmfels, Bernd
This article presents a unified mathematical framework for inference in graphical models, building on the observation that graphical models are algebraic varieties. From this geometric viewpoint,...
Parametric inference for biological sequence analysis
Pachter, Lior, Sturmfels, Bernd
One of the major successes in computational biology has been the unification, by using the graphical model formalism, of a multitude of algorithms for annotating and comparing biological sequences....
Parametric Alignment of Drosophila Genomes
Dewey, Colin N, Huggins, Peter M, Woods, Kevin, Sturmfels, Bernd, Pachter, Lior
The classic algorithms of Needleman–Wunsch and Smith–Waterman find a maximum a posteriori probability alignment for a pair hidden Markov model (PHMM). To process large genomes that have undergone...
Tropical geometry of statistical models
Pachter, Lior, Sturmfels, Bernd
This article presents a unified mathematical framework for inference in graphical models, building on the observation that graphical models are algebraic varieties. From this geometric viewpoint,...
Parametric inference for biological sequence analysis
Pachter, Lior, Sturmfels, Bernd
One of the major successes in computational biology has been the unification, by using the graphical model formalism, of a multitude of algorithms for annotating and comparing biological sequences....
Parametric Alignment of Drosophila Genomes
Dewey, Colin N, Huggins, Peter M, Woods, Kevin, Sturmfels, Bernd, Pachter, Lior
The classic algorithms of Needleman–Wunsch and Smith–Waterman find a maximum a posteriori probability alignment for a pair hidden Markov model (PHMM). To process large genomes that have undergone...
Analysis of epistatic interactions and fitness landscapes using a new geometric approach
Beerenwinkel, Niko, Pachter, Lior, Sturmfels, Bernd, Elena, Santiago F, Lenski, Richard E
The Cyclohedron Test for Finding Periodic Genes in Time Course Expression Studies
Jason Morton, Lior Pachter, Anne Shiu, Bernd Sturmfels
The problem of finding periodically expressed genes from time course microarray experiments is at the center of numerous efforts to identify the molecular components of biological clocks. We present...
Comparison of Pattern Detection Methods in Microarray Time Series of the Segmentation Clock
Dequéant, Mary-Lee, Ahnert, Sebastian, Edelsbrunner, Herbert, Fink, Thomas M. A., Glynn, Earl F., Hattem, Gaye, ...
While genome-wide gene expression data are generated at an increasing rate, the repertoire of approaches for pattern discovery in these data is still limited. Identifying subtle patterns of interest...
Graphical models for correlated defaults
I. Onur Filiz, Xin Guo, Jason Morton, Bernd Sturmfels
A simple graphical model for correlated defaults is proposed, with explicit formulas for the loss distribution. Algebraic geometry techniques are employed to show that this model is well posed for...