In this paper we extend three classical and fundamental results in polyhedral geometry, namely, Carath'{e}odory's theorem, the Minkowski-Weyl theorem, and Gordan's lemma to infinite dimensional spaces, in which considered cones and monoids are invariant under actions of symmetric groups.
{"title":"Theorems of Carathéodory, Minkowski-Weyl, and Gordan up to Symmetry","authors":"D. Le, Tim Römer","doi":"10.1137/22M148865X","DOIUrl":"https://doi.org/10.1137/22M148865X","url":null,"abstract":"In this paper we extend three classical and fundamental results in polyhedral geometry, namely, Carath'{e}odory's theorem, the Minkowski-Weyl theorem, and Gordan's lemma to infinite dimensional spaces, in which considered cones and monoids are invariant under actions of symmetric groups.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"22 1","pages":"291-310"},"PeriodicalIF":1.2,"publicationDate":"2021-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83903251","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Javaloyes, Enrique Pend'as-Recondo, Miguel Sánchez Caja
A geometric model for the computation of the firefront of a forest wildfire which takes into account several effects (possibly time-dependent wind, anisotropies and slope of the ground) is introduced. It relies on a general theoretical framework, which reduces the hyperbolic PDE system of any wave to an ODE in a Lorentz-Finsler framework. The wind induces a sort of double semi-elliptical fire growth, while the influence of the slope is modeled by means of a term which comes from the Matsumoto metric (i.e., the standard non-reversible Finsler metric that measures the time when going up and down a hill). These contributions make a significant difference from previous models because, now, the infinitesimal wavefronts are not restricted to be elliptical. Even though this is a technical complication, the wavefronts remain computable in real time. Some simulations of evolution are shown, paying special attention to possible crossovers of the fire.
{"title":"A General Model for Wildfire Propagation with Wind and Slope","authors":"M. Javaloyes, Enrique Pend'as-Recondo, Miguel Sánchez Caja","doi":"10.1137/22M1477866","DOIUrl":"https://doi.org/10.1137/22M1477866","url":null,"abstract":"A geometric model for the computation of the firefront of a forest wildfire which takes into account several effects (possibly time-dependent wind, anisotropies and slope of the ground) is introduced. It relies on a general theoretical framework, which reduces the hyperbolic PDE system of any wave to an ODE in a Lorentz-Finsler framework. The wind induces a sort of double semi-elliptical fire growth, while the influence of the slope is modeled by means of a term which comes from the Matsumoto metric (i.e., the standard non-reversible Finsler metric that measures the time when going up and down a hill). These contributions make a significant difference from previous models because, now, the infinitesimal wavefronts are not restricted to be elliptical. Even though this is a technical complication, the wavefronts remain computable in real time. Some simulations of evolution are shown, paying special attention to possible crossovers of the fire.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"84 1","pages":"414-439"},"PeriodicalIF":1.2,"publicationDate":"2021-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87393140","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce invariants for compact $C^1$-orientable surfaces (with boundary) in $mathbb{R}^3$ up to rigid transformations. Our invariants are certain degree four polynomials in the moments of the delta function of the surface. We give an effective and numerically stable inversion algorithm for retrieving the surface from the invariants, which works on a comeagre subset of $C^3$-surfaces.
{"title":"Complete ({boldsymbol{SE(3)}}) Invariants for a Comeagre Set of ({boldsymbol{C^3}}) Compact Orientable Surfaces in (mathbb{R}^{boldsymbol{3}})","authors":"Yair Hayut, D. Lehavi","doi":"10.1137/21M1445776","DOIUrl":"https://doi.org/10.1137/21M1445776","url":null,"abstract":"We introduce invariants for compact $C^1$-orientable surfaces (with boundary) in $mathbb{R}^3$ up to rigid transformations. Our invariants are certain degree four polynomials in the moments of the delta function of the surface. We give an effective and numerically stable inversion algorithm for retrieving the surface from the invariants, which works on a comeagre subset of $C^3$-surfaces.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"11 1","pages":"311-344"},"PeriodicalIF":1.2,"publicationDate":"2021-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75971680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study a family of convex polytopes, called SIM-bodies, which were introduced by Giannakopoulos and Koutsoupias (2018) to analyze so-called Straight-Jacket Auctions. First, we show that the SIM-bodies belong to the class of generalized permutahedra. Second, we prove an optimality result for the Straight-Jacket Auctions among certain deterministic auctions. Third, we employ computer algebra methods and mathematical software to explicitly determine optimal prices and revenues.
{"title":"Generalized permutahedra and optimal auctions","authors":"M. Joswig, Max Klimm, Sylvain Spitz","doi":"10.1137/21m1441286","DOIUrl":"https://doi.org/10.1137/21m1441286","url":null,"abstract":"We study a family of convex polytopes, called SIM-bodies, which were introduced by Giannakopoulos and Koutsoupias (2018) to analyze so-called Straight-Jacket Auctions. First, we show that the SIM-bodies belong to the class of generalized permutahedra. Second, we prove an optimality result for the Straight-Jacket Auctions among certain deterministic auctions. Third, we employ computer algebra methods and mathematical software to explicitly determine optimal prices and revenues.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"6 1","pages":"711-739"},"PeriodicalIF":1.2,"publicationDate":"2021-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81588775","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study a new invariant of circular planar electrical networks, well known to phylogeneticists: the circular split system. We use our invariant to answer some open questions about levels of complexity of networks and their related Kalmanson metrics. The key to our analysis is the realization that certain matrices arising from weighted split systems are studied in another guise: the Kron reductions of Laplacian matrices of planar electrical networks. Specifically we show that a response matrix of a circular planar electrical network corresponds to a unique resistance metric obeying the Kalmanson condition, and thus a unique weighted circular split system. Our results allow interchange of methods: phylogenetic reconstruction using theorems about electrical networks, and circuit reconstruction using phylogenetic techniques.
{"title":"Circular Planar Electrical Networks, Split Systems, and Phylogenetic Networks","authors":"S. Forcey","doi":"10.1137/22m1473844","DOIUrl":"https://doi.org/10.1137/22m1473844","url":null,"abstract":"We study a new invariant of circular planar electrical networks, well known to phylogeneticists: the circular split system. We use our invariant to answer some open questions about levels of complexity of networks and their related Kalmanson metrics. The key to our analysis is the realization that certain matrices arising from weighted split systems are studied in another guise: the Kron reductions of Laplacian matrices of planar electrical networks. Specifically we show that a response matrix of a circular planar electrical network corresponds to a unique resistance metric obeying the Kalmanson condition, and thus a unique weighted circular split system. Our results allow interchange of methods: phylogenetic reconstruction using theorems about electrical networks, and circuit reconstruction using phylogenetic techniques.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"220 1","pages":"49-76"},"PeriodicalIF":1.2,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73773826","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give algorithms to compute decompositions of a given polynomial, or more generally mixed tensor, as sum of rank one tensors, and to establish whether such a decomposition is unique. In particular, we present methods to compute the decomposition of a general plane quintic in seven powers, and of a general space cubic in five powers; the two decompositions of a general plane sextic of rank nine, and the five decompositions of a general plane septic. Furthermore, we give Magma implementations of all our algorithms.
{"title":"Decomposition algorithms for tensors and polynomials","authors":"A. Laface, Alex Massarenti, Rick Rischter","doi":"10.1137/21m1453712","DOIUrl":"https://doi.org/10.1137/21m1453712","url":null,"abstract":"We give algorithms to compute decompositions of a given polynomial, or more generally mixed tensor, as sum of rank one tensors, and to establish whether such a decomposition is unique. In particular, we present methods to compute the decomposition of a general plane quintic in seven powers, and of a general space cubic in five powers; the two decompositions of a general plane sextic of rank nine, and the five decompositions of a general plane septic. Furthermore, we give Magma implementations of all our algorithms.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"77 1","pages":"264-290"},"PeriodicalIF":1.2,"publicationDate":"2021-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86929303","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Signomials are obtained by generalizing polynomials to allow for arbitrary real exponents. This generalization offers great expressive power, but has historically sacrificed the organizing principle of ``degree'' that is central to polynomial optimization theory. We reclaim that principle here through the concept of signomial rings, which we use to derive complete convex relaxation hierarchies of upper and lower bounds for signomial optimization via sums of arithmetic-geometric exponentials (SAGE) nonnegativity certificates. The Positivstellensatz underlying the lower bounds relies on the concept of conditional SAGE and removes regularity conditions required by earlier works, such as convexity and Archimedeanity of the feasible set. Through worked examples we illustrate the practicality of this hierarchy in areas such as chemical reaction network theory and chemical engineering. These examples include comparisons to direct global solvers (e.g., BARON and ANTIGONE) and the Lasserre hierarchy (where appropriate). The completeness of our hierarchy of upper bounds follows from a generic construction whereby a Positivstellensatz for signomial nonnegativity over a compact set provides for arbitrarily strong outer approximations of the corresponding cone of nonnegative signomials. While working toward that result, we prove basic facts on the existence and uniqueness of solutions to signomial moment problems.
{"title":"Algebraic Perspectives on Signomial Optimization","authors":"Mareike Dressler, Riley Murray","doi":"10.1137/21m1462568","DOIUrl":"https://doi.org/10.1137/21m1462568","url":null,"abstract":"Signomials are obtained by generalizing polynomials to allow for arbitrary real exponents. This generalization offers great expressive power, but has historically sacrificed the organizing principle of ``degree'' that is central to polynomial optimization theory. We reclaim that principle here through the concept of signomial rings, which we use to derive complete convex relaxation hierarchies of upper and lower bounds for signomial optimization via sums of arithmetic-geometric exponentials (SAGE) nonnegativity certificates. The Positivstellensatz underlying the lower bounds relies on the concept of conditional SAGE and removes regularity conditions required by earlier works, such as convexity and Archimedeanity of the feasible set. Through worked examples we illustrate the practicality of this hierarchy in areas such as chemical reaction network theory and chemical engineering. These examples include comparisons to direct global solvers (e.g., BARON and ANTIGONE) and the Lasserre hierarchy (where appropriate). The completeness of our hierarchy of upper bounds follows from a generic construction whereby a Positivstellensatz for signomial nonnegativity over a compact set provides for arbitrarily strong outer approximations of the corresponding cone of nonnegative signomials. While working toward that result, we prove basic facts on the existence and uniqueness of solutions to signomial moment problems.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"16 1","pages":"650-684"},"PeriodicalIF":1.2,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83201654","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For a given lattice polytope P ⊂ R, consider the space LP of trivariate polynomials over a finite field Fq, whose Newton polytopes are contained in P . We give upper bounds for the maximum number of Fq-zeros of polynomials in LP in terms of the Minkowski length of P and q, the size of the field. Consequently, this produces lower bounds for the minimum distance of toric codes defined by evaluating elements of LP at the points of the algebraic torus (Fq). Our approach is based on understanding factorizations of polynomials in LP with the largest possible number of non-unit factors. The related combinatorial result that we obtain is a description of Minkowski sums of lattice polytopes contained in P with the largest possible number of non-trivial summands.
{"title":"$mathbb{F}_q$-Zeros of Sparse Trivariate Polynomials and Toric 3-Fold Codes","authors":"Kyle P. Meyer, Ivan Soprunov, Jenya Soprunova","doi":"10.1137/21m1436890","DOIUrl":"https://doi.org/10.1137/21m1436890","url":null,"abstract":"For a given lattice polytope P ⊂ R, consider the space LP of trivariate polynomials over a finite field Fq, whose Newton polytopes are contained in P . We give upper bounds for the maximum number of Fq-zeros of polynomials in LP in terms of the Minkowski length of P and q, the size of the field. Consequently, this produces lower bounds for the minimum distance of toric codes defined by evaluating elements of LP at the points of the algebraic torus (Fq). Our approach is based on understanding factorizations of polynomials in LP with the largest possible number of non-unit factors. The related combinatorial result that we obtain is a description of Minkowski sums of lattice polytopes contained in P with the largest possible number of non-trivial summands.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"25 1","pages":"432-467"},"PeriodicalIF":1.2,"publicationDate":"2021-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83529874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Nature of Mathematics","authors":"M. Lawson","doi":"10.1201/9781003098072-2","DOIUrl":"https://doi.org/10.1201/9781003098072-2","url":null,"abstract":"","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"43 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89661415","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. Lindberg, Nathan Nicholson, J. Rodriguez, Zinan Wang
We consider statistical models arising from the common set of solutions to a sparse polynomial system with general coefficients. The maximum likelihood degree counts the number of critical points of the likelihood function restricted to the model. We prove the maximum likelihood degree of a sparse polynomial system is determined by its Newton polytopes and equals the mixed volume of a related Lagrange system of equations.
{"title":"The Maximum Likelihood Degree of Sparse Polynomial Systems","authors":"J. Lindberg, Nathan Nicholson, J. Rodriguez, Zinan Wang","doi":"10.1137/21m1422550","DOIUrl":"https://doi.org/10.1137/21m1422550","url":null,"abstract":"We consider statistical models arising from the common set of solutions to a sparse polynomial system with general coefficients. The maximum likelihood degree counts the number of critical points of the likelihood function restricted to the model. We prove the maximum likelihood degree of a sparse polynomial system is determined by its Newton polytopes and equals the mixed volume of a related Lagrange system of equations.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"1 1","pages":"159-171"},"PeriodicalIF":1.2,"publicationDate":"2021-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77090473","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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