ORCID Profile
0000-0002-9170-8295
Current Organisation
NC State University College of Sciences
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Publisher: Springer Science and Business Media LLC
Date: 10-12-2022
DOI: 10.1007/S00220-022-04569-6
Abstract: The connection between Feynman integrals and GKZ A -hypergeometric systems has been a topic of recent interest with advances in mathematical techniques and computational tools opening new possibilities in this paper we continue to explore this connection. To each such hypergeometric system there is an associated toric ideal, we prove that the latter has the Cohen-Macaulay property for two large families of Feynman integrals. This implies, for ex le, that both the number of independent solutions and dynamical singularities are independent of space-time dimension and generalized propagator powers. Furthermore, in particular, it means that the process of finding a series representation of these integrals is fully algorithmic.
Publisher: AIP Publishing
Date: 05-2019
DOI: 10.1063/1.5030475
Abstract: We derive a general formula for the Euler characteristic of a fibration of projective hypersurfaces in terms of invariants of an arbitrary base variety. When the general fiber is an elliptic curve, such formulas have appeared in the physics literature in the context of calculating D-brane charge for M-/F-theory and type-IIB compactifications of string vacua. While there are various methods for computing Euler characteristics of algebraic varieties, we prove a base-independent pushforward formula which reduces the computation of the Euler characteristic of relative hypersurfaces to simple algebraic manipulations of rational expressions determined by its isor class in a projective bundle. We illustrate our methods by applying them to an explicit family of relative hypersurfaces whose fibers are of arbitrary dimension and degree.
Publisher: arXiv
Date: 2020
Publisher: Elsevier BV
Date: 06-2017
Publisher: ACM
Date: 03-11-2011
Publisher: Elsevier BV
Date: 09-2019
Publisher: American Mathematical Society (AMS)
Date: 24-05-2019
DOI: 10.1090/MCOM/3448
Publisher: Springer Science and Business Media LLC
Date: 18-12-2018
DOI: 10.1007/S11538-018-00555-Z
Abstract: Bistability and multistationarity are properties of reaction networks linked to switch-like responses and connected to cell memory and cell decision making. Determining whether and when a network exhibits bistability is a hard and open mathematical problem. One successful strategy consists of analyzing small networks and deducing that some of the properties are preserved upon passage to the full network. Motivated by this, we study chemical reaction networks with few chemical complexes. Under mass action kinetics, the steady states of these networks are described by fewnomial systems, that is polynomial systems having few distinct monomials. Such systems of polynomials are often studied in real algebraic geometry by the use of Gale dual systems. Using this Gale duality, we give precise conditions in terms of the reaction rate constants for the number and stability of the steady states of families of reaction networks with one non-flow reaction.
Publisher: Association for Computing Machinery (ACM)
Date: 14-08-2015
Publisher: Elsevier BV
Date: 05-2019
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 08-03-2023
DOI: 10.1137/22M1475533
Publisher: World Scientific Pub Co Pte Lt
Date: 05-2019
DOI: 10.1142/S0219498819500956
Abstract: Suppose that [Formula: see text] is a toric variety of codimension two defined by an [Formula: see text] integer matrix [Formula: see text], and let [Formula: see text] be a Gale dual of [Formula: see text]. In this paper, we compute the Euclidean distance degree and polar degrees of [Formula: see text] (along with other associated invariants) combinatorially working from the matrix [Formula: see text]. Our approach allows for the consideration of ex les that would be impractical using algebraic or geometric methods. It also yields considerably simpler computational formulas for these invariants, allowing much larger ex les to be computed much more quickly than the analogous combinatorial methods using the matrix [Formula: see text] in the codimension two case.
Publisher: Informa UK Limited
Date: 26-03-2020
Publisher: Det Kgl. Bibliotek/Royal Danish Library
Date: 08-04-2018
DOI: 10.7146/MATH.SCAND.A-101478
Abstract: We determine the Euclidean distance degree of a projective toric variety. This extends the formula of Matsui and Takeuchi for the degree of the $A$-discriminant in terms of Euler obstructions. Our primary goal is the development of reliable algorithmic tools for computing the points on a real toric variety that are closest to a given data point.
Publisher: Springer Science and Business Media LLC
Date: 14-06-2022
DOI: 10.1007/S10208-022-09574-8
Abstract: We describe a new algorithm for computing Whitney stratifications of complex projective varieties. The main ingredients are (a) an algebraic criterion, due to Lê and Teissier, which reformulates Whitney regularity in terms of conormal spaces and maps, and (b) a new interpretation of this conormal criterion via ideal saturations, which can be practically implemented on a computer. We show that this algorithm improves upon the existing state of the art by several orders of magnitude, even for relatively small input varieties. En route, we introduce related algorithms for efficiently stratifying affine varieties, flags on a given variety, and algebraic maps.
Publisher: Springer Science and Business Media LLC
Date: 04-05-2021
DOI: 10.1007/S40315-021-00373-W
Abstract: We study questions of existence and uniqueness of quadrature domains using computational tools from real algebraic geometry. These problems are transformed into questions about the number of solutions to an associated real semi-algebraic system, which is analyzed using the method of real comprehensive triangular decomposition.
Publisher: Springer Science and Business Media LLC
Date: 25-10-2023
Publisher: Elsevier BV
Date: 04-2017
Publisher: Cambridge University Press (CUP)
Date: 15-07-2009
DOI: 10.1017/S0305004109990120
Abstract: Let G be a finite group acting on vector spaces V and W and consider a smooth G -equivariant mapping f : V → W . This paper addresses the question of the zero set of f near a zero x with isotropy subgroup G . It is known from results of Bierstone and Field on G -transversality theory that the zero set in a neighbourhood of x is a stratified set. The purpose of this paper is to partially determine the structure of the stratified set near x using only information from the representations V and W . We define an index s (Σ) for isotropy subgroups Σ of G which is the difference of the dimension of the fixed point subspace of Σ in V and W . Our main result states that if V contains a subspace G -isomorphic to W , then for every maximal isotropy subgroup Σ satisfying s (Σ) s ( G ), the zero set of f near x contains a smooth manifold of zeros with isotropy subgroup Σ of dimension s (Σ). We also present partial results in the case of group representations V and W which do not satisfy the conditions of our main theorem. The paper contains many ex les and raises several questions concerning the computation of zero sets of equivariant maps. These results have application to the bifurcation theory of G -reversible equivariant vector fields.
Publisher: Elsevier BV
Date: 05-2024
Publisher: Springer Science and Business Media LLC
Date: 24-08-2020
DOI: 10.1007/S10910-020-01162-X
Abstract: We study families of chemical reaction networks whose positive steady states are toric, and therefore can be parameterized by monomials. Families are constructed algorithmically from a core network we show that if a family member is multistationary, then so are all subsequent networks in the family. Further, we address the questions of model selection and experimental design for families by investigating the algebraic dependencies of the chemical concentrations using matroids. Given a family with toric steady states and a constant number of conservation relations, we construct a matroid that encodes important information regarding the steady state behaviour of the entire family. Among other things, this gives necessary conditions for the distinguishability of families of reaction networks with respect to a data set of measured chemical concentrations. We illustrate our results using multi-site phosphorylation networks.
Publisher: Springer International Publishing
Date: 2017
Publisher: Elsevier BV
Date: 03-2015
Publisher: ACM
Date: 28-07-2014
Location: United States of America
Location: United States of America
Start Date: 2016
End Date: 2017
Funder: Natural Sciences and Engineering Research Council
View Funded ActivityStart Date: 2015
End Date: 2016
Funder: Natural Sciences and Engineering Research Council
View Funded ActivityStart Date: 2022
End Date: 2027
Funder: United States Air Force
View Funded Activity