Society for Industrial and Applied Mathematics: SIAM Journal on Applied Algebra and Geometry: Table of Contents
Table of Contents for SIAM Journal on Applied Algebra and Geometry. List of articles from both the latest and ahead of print issues.
https://epubs.siam.org/loi/sjaabq?af=R
Society for Industrial and Applied Mathematics: SIAM Journal on Applied Algebra and Geometry: Table of Contents
Society for Industrial and Applied Mathematics
enUS
SIAM Journal on Applied Algebra and Geometry
SIAM Journal on Applied Algebra and Geometry
https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaabq/cover.jpg
https://epubs.siam.org/loi/sjaabq?af=R

Differential Equations for Gaussian Statistical Models with Rational Maximum Likelihood Estimator
https://epubs.siam.org/doi/abs/10.1137/23M1569228?af=R
SIAM Journal on Applied Algebra and Geometry, <a href="https://epubs.siam.org/toc/sjaabq/8/3">Volume 8, Issue 3</a>, Page 465492, September 2024. <br/> Abstract.We study multivariate Gaussian statistical models whose maximum likelihood estimator (MLE) is a rational function of the observed data. We establish a onetoone correspondence between such models and the solutions to a nonlinear firstorder partial differential equation (PDE). Using our correspondence, we reinterpret familiar classes of models with rational MLE, such as directed (and decomposable undirected) Gaussian graphical models. We also find new models with rational MLE. For linear concentration models with rational MLE, we show that homaloidal polynomials from birational geometry lead to solutions to the PDE. We thus shed light on the problem of classifying Gaussian models with rational MLE by relating it to the open problem in birational geometry of classifying homaloidal polynomials.
SIAM Journal on Applied Algebra and Geometry, Volume 8, Issue 3, Page 465492, September 2024. <br/> Abstract.We study multivariate Gaussian statistical models whose maximum likelihood estimator (MLE) is a rational function of the observed data. We establish a onetoone correspondence between such models and the solutions to a nonlinear firstorder partial differential equation (PDE). Using our correspondence, we reinterpret familiar classes of models with rational MLE, such as directed (and decomposable undirected) Gaussian graphical models. We also find new models with rational MLE. For linear concentration models with rational MLE, we show that homaloidal polynomials from birational geometry lead to solutions to the PDE. We thus shed light on the problem of classifying Gaussian models with rational MLE by relating it to the open problem in birational geometry of classifying homaloidal polynomials. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaabq/cover.jpg" alttext="cover image"/></p>
Differential Equations for Gaussian Statistical Models with Rational Maximum Likelihood Estimator
10.1137/23M1569228
SIAM Journal on Applied Algebra and Geometry
20240731T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Carlos Améndola
Lukas Gustafsson
Kathlén Kohn
Orlando Marigliano
Anna Seigal
Differential Equations for Gaussian Statistical Models with Rational Maximum Likelihood Estimator
8
3
465
492
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/23M1569228
https://epubs.siam.org/doi/abs/10.1137/23M1569228?af=R
© 2024 Society for Industrial and Applied Mathematics

Clubs and Their Applications
https://epubs.siam.org/doi/abs/10.1137/22M1523534?af=R
SIAM Journal on Applied Algebra and Geometry, <a href="https://epubs.siam.org/toc/sjaabq/8/3">Volume 8, Issue 3</a>, Page 493518, September 2024. <br/> Abstract.Clubs of rank [math] are wellcelebrated objects in finite geometries introduced by Fancsali and Sziklai in 2006. After the connection with a special type of arc known as a KMarc, they renewed their interest. This paper aims to study clubs of rank [math] in [math]. We provide a classification result for [math]clubs of rank [math], and we analyze the [math]equivalence of the known subspaces defining clubs; for some of them the problem is then translated into determining whether or not certain scattered spaces are equivalent. Then we find a polynomial description of the known families of clubs via some linearized polynomials. Then we apply our results to the theory of blocking sets, KMarcs, polynomials, and rank metric codes, obtaining new constructions and classification results.
SIAM Journal on Applied Algebra and Geometry, Volume 8, Issue 3, Page 493518, September 2024. <br/> Abstract.Clubs of rank [math] are wellcelebrated objects in finite geometries introduced by Fancsali and Sziklai in 2006. After the connection with a special type of arc known as a KMarc, they renewed their interest. This paper aims to study clubs of rank [math] in [math]. We provide a classification result for [math]clubs of rank [math], and we analyze the [math]equivalence of the known subspaces defining clubs; for some of them the problem is then translated into determining whether or not certain scattered spaces are equivalent. Then we find a polynomial description of the known families of clubs via some linearized polynomials. Then we apply our results to the theory of blocking sets, KMarcs, polynomials, and rank metric codes, obtaining new constructions and classification results. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaabq/cover.jpg" alttext="cover image"/></p>
Clubs and Their Applications
10.1137/22M1523534
SIAM Journal on Applied Algebra and Geometry
20240812T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Vito Napolitano
Olga Polverino
Paolo Santonastaso
Ferdinando Zullo
Clubs and Their Applications
8
3
493
518
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/22M1523534
https://epubs.siam.org/doi/abs/10.1137/22M1523534?af=R
© 2024 Society for Industrial and Applied Mathematics

The Barycenter in Free Nilpotent Lie Groups and Its Application to IteratedIntegrals Signatures
https://epubs.siam.org/doi/abs/10.1137/23M159024X?af=R
SIAM Journal on Applied Algebra and Geometry, <a href="https://epubs.siam.org/toc/sjaabq/8/3">Volume 8, Issue 3</a>, Page 519552, September 2024. <br/> Abstract.We establish the welldefinedness of the barycenter (in the sense of Buser and Karcher) for every integrable measure on the free nilpotent Lie group of step [math] (over [math]). We provide two algorithms for computing it, using methods from Lie theory (namely, the Baker–Campbell–Hausdorff formula) and from the theory of Gröbner bases of modules. Our main motivation stems from measures induced by iteratedintegrals signatures, and we calculate the barycenter for the signature of the Brownian motion.
SIAM Journal on Applied Algebra and Geometry, Volume 8, Issue 3, Page 519552, September 2024. <br/> Abstract.We establish the welldefinedness of the barycenter (in the sense of Buser and Karcher) for every integrable measure on the free nilpotent Lie group of step [math] (over [math]). We provide two algorithms for computing it, using methods from Lie theory (namely, the Baker–Campbell–Hausdorff formula) and from the theory of Gröbner bases of modules. Our main motivation stems from measures induced by iteratedintegrals signatures, and we calculate the barycenter for the signature of the Brownian motion. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaabq/cover.jpg" alttext="cover image"/></p>
The Barycenter in Free Nilpotent Lie Groups and Its Application to IteratedIntegrals Signatures
10.1137/23M159024X
SIAM Journal on Applied Algebra and Geometry
20240813T07:00:00Z
© 2024 M. Clausel, J. Diehl, R. Mignot, L. Schmitz, N. Sugiura, and K. Usevich
Marianne Clausel
Joscha Diehl
Raphael Mignot
Leonard Schmitz
Nozomi Sugiura
Konstantin Usevich
The Barycenter in Free Nilpotent Lie Groups and Its Application to IteratedIntegrals Signatures
8
3
519
552
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/23M159024X
https://epubs.siam.org/doi/abs/10.1137/23M159024X?af=R
© 2024 M. Clausel, J. Diehl, R. Mignot, L. Schmitz, N. Sugiura, and K. Usevich

The Service Rate Region Polytope
https://epubs.siam.org/doi/abs/10.1137/23M1557829?af=R
SIAM Journal on Applied Algebra and Geometry, <a href="https://epubs.siam.org/toc/sjaabq/8/3">Volume 8, Issue 3</a>, Page 553582, September 2024. <br/> Abstract.We investigate the properties of a family of polytopes that naturally arise in connection with a problem in distributed data storage, namely service rate region polytopes. The service rate region of a distributed coded system describes the data access requests that the underlying system can support. In this paper, we study the polytope structure of the service rate region with the primary goal of describing its geometric shape and properties. We achieve this by introducing various structural parameters of the service rate region and establishing upper and lower bounds for them. The techniques we apply in this paper range from coding theory to optimization. One of our main results shows that every rational point of the service rate region has a socalled rational allocation, answering an open question in the research area.
SIAM Journal on Applied Algebra and Geometry, Volume 8, Issue 3, Page 553582, September 2024. <br/> Abstract.We investigate the properties of a family of polytopes that naturally arise in connection with a problem in distributed data storage, namely service rate region polytopes. The service rate region of a distributed coded system describes the data access requests that the underlying system can support. In this paper, we study the polytope structure of the service rate region with the primary goal of describing its geometric shape and properties. We achieve this by introducing various structural parameters of the service rate region and establishing upper and lower bounds for them. The techniques we apply in this paper range from coding theory to optimization. One of our main results shows that every rational point of the service rate region has a socalled rational allocation, answering an open question in the research area. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaabq/cover.jpg" alttext="cover image"/></p>
The Service Rate Region Polytope
10.1137/23M1557829
SIAM Journal on Applied Algebra and Geometry
20240813T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Gianira N. Alfarano
Altan B. Kılıç
Alberto Ravagnani
Emina Soljanin
The Service Rate Region Polytope
8
3
553
582
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/23M1557829
https://epubs.siam.org/doi/abs/10.1137/23M1557829?af=R
© 2024 Society for Industrial and Applied Mathematics

A Random Copositive Matrix Is Completely Positive with Positive Probability
https://epubs.siam.org/doi/abs/10.1137/23M1583491?af=R
SIAM Journal on Applied Algebra and Geometry, <a href="https://epubs.siam.org/toc/sjaabq/8/3">Volume 8, Issue 3</a>, Page 583611, September 2024. <br/> Abstract.An [math] symmetric matrix [math] is copositive if the quadratic form [math] is nonnegative on the nonnegative orthant [math]. The cone of copositive matrices strictly contains the cone of completely positive matrices, i.e., all matrices of the form [math] for some [math] matrix [math] with nonnegative entries. The main result, proved using Blekherman’s real algebraic geometry inspired techniques and tools of convex geometry, shows that asymptotically, as [math] goes to infinity, the ratio of volume radii of the two cones is strictly positive. Consequently, the same holds true for the ratio of volume radii of any two cones sandwiched between them, e.g., the cones of positive semidefinite matrices, matrices with nonnegative entries, their intersection, and their Minkowski sum.
SIAM Journal on Applied Algebra and Geometry, Volume 8, Issue 3, Page 583611, September 2024. <br/> Abstract.An [math] symmetric matrix [math] is copositive if the quadratic form [math] is nonnegative on the nonnegative orthant [math]. The cone of copositive matrices strictly contains the cone of completely positive matrices, i.e., all matrices of the form [math] for some [math] matrix [math] with nonnegative entries. The main result, proved using Blekherman’s real algebraic geometry inspired techniques and tools of convex geometry, shows that asymptotically, as [math] goes to infinity, the ratio of volume radii of the two cones is strictly positive. Consequently, the same holds true for the ratio of volume radii of any two cones sandwiched between them, e.g., the cones of positive semidefinite matrices, matrices with nonnegative entries, their intersection, and their Minkowski sum. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaabq/cover.jpg" alttext="cover image"/></p>
A Random Copositive Matrix Is Completely Positive with Positive Probability
10.1137/23M1583491
SIAM Journal on Applied Algebra and Geometry
20240819T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Igor Klep
Tea Štrekelj
Aljaž Zalar
A Random Copositive Matrix Is Completely Positive with Positive Probability
8
3
583
611
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/23M1583491
https://epubs.siam.org/doi/abs/10.1137/23M1583491?af=R
© 2024 Society for Industrial and Applied Mathematics

Orbit Spaces of Weyl Groups Acting on Compact Tori: A Unified and Explicit Polynomial Description
https://epubs.siam.org/doi/abs/10.1137/23M158173X?af=R
SIAM Journal on Applied Algebra and Geometry, <a href="https://epubs.siam.org/toc/sjaabq/8/3">Volume 8, Issue 3</a>, Page 612649, September 2024. <br/> Abstract.The Weyl group of a crystallographic root system has a multiplicative action on the compact torus. The orbit space of this action is a compact basic semialgebraic set. We present a polynomial description of this set for the Weyl groups associated to root systems of types [math], [math], [math], [math], and [math]. Our description is given through a polynomial matrix inequality. The novelty lies in an approach via Hermite quadratic forms and a closed form formula for the matrix entries. The orbit space of the multiplicative Weyl group action is the orthogonality region of generalized Chebyshev polynomials. In this polynomial basis, we show that the matrices obtained for the five types follow the same, surprisingly simple pattern. This is applied to the optimization of trigonometric polynomials with crystallographic symmetries.
SIAM Journal on Applied Algebra and Geometry, Volume 8, Issue 3, Page 612649, September 2024. <br/> Abstract.The Weyl group of a crystallographic root system has a multiplicative action on the compact torus. The orbit space of this action is a compact basic semialgebraic set. We present a polynomial description of this set for the Weyl groups associated to root systems of types [math], [math], [math], [math], and [math]. Our description is given through a polynomial matrix inequality. The novelty lies in an approach via Hermite quadratic forms and a closed form formula for the matrix entries. The orbit space of the multiplicative Weyl group action is the orthogonality region of generalized Chebyshev polynomials. In this polynomial basis, we show that the matrices obtained for the five types follow the same, surprisingly simple pattern. This is applied to the optimization of trigonometric polynomials with crystallographic symmetries. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaabq/cover.jpg" alttext="cover image"/></p>
Orbit Spaces of Weyl Groups Acting on Compact Tori: A Unified and Explicit Polynomial Description
10.1137/23M158173X
SIAM Journal on Applied Algebra and Geometry
20240819T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Evelyne Hubert
Tobias Metzlaff
Cordian Riener
Orbit Spaces of Weyl Groups Acting on Compact Tori: A Unified and Explicit Polynomial Description
8
3
612
649
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/23M158173X
https://epubs.siam.org/doi/abs/10.1137/23M158173X?af=R
© 2024 Society for Industrial and Applied Mathematics

Tangent Space and Dimension Estimation with the Wasserstein Distance
https://epubs.siam.org/doi/abs/10.1137/22M1522711?af=R
SIAM Journal on Applied Algebra and Geometry, <a href="https://epubs.siam.org/toc/sjaabq/8/3">Volume 8, Issue 3</a>, Page 650685, September 2024. <br/> Abstract.Consider a set of points sampled independently near a smooth compact submanifold of Euclidean space. We provide mathematically rigorous bounds on the number of sample points required to estimate both the dimension and the tangent spaces of that manifold with high confidence. The algorithm for this estimation is Local PCA, a local version of principal component analysis. Our results accommodate for noisy nonuniform data distribution with the noise that may vary across the manifold, and allow simultaneous estimation at multiple points. Crucially, all of the constants appearing in our bound are explicitly described. The proof uses a matrix concentration inequality to estimate covariance matrices and a Wasserstein distance bound for quantifying nonlinearity of the underlying manifold and nonuniformity of the probability measure.
SIAM Journal on Applied Algebra and Geometry, Volume 8, Issue 3, Page 650685, September 2024. <br/> Abstract.Consider a set of points sampled independently near a smooth compact submanifold of Euclidean space. We provide mathematically rigorous bounds on the number of sample points required to estimate both the dimension and the tangent spaces of that manifold with high confidence. The algorithm for this estimation is Local PCA, a local version of principal component analysis. Our results accommodate for noisy nonuniform data distribution with the noise that may vary across the manifold, and allow simultaneous estimation at multiple points. Crucially, all of the constants appearing in our bound are explicitly described. The proof uses a matrix concentration inequality to estimate covariance matrices and a Wasserstein distance bound for quantifying nonlinearity of the underlying manifold and nonuniformity of the probability measure. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaabq/cover.jpg" alttext="cover image"/></p>
Tangent Space and Dimension Estimation with the Wasserstein Distance
10.1137/22M1522711
SIAM Journal on Applied Algebra and Geometry
20240823T07:00:00Z
© 2024 Uzu Lim
Uzu Lim
Harald Oberhauser
Vidit Nanda
Tangent Space and Dimension Estimation with the Wasserstein Distance
8
3
650
685
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/22M1522711
https://epubs.siam.org/doi/abs/10.1137/22M1522711?af=R
© 2024 Uzu Lim

Bivariate Splines on a Triangulation with a Single Totally Interior Edge
https://epubs.siam.org/doi/abs/10.1137/23M158317X?af=R
SIAM Journal on Applied Algebra and Geometry, <a href="https://epubs.siam.org/toc/sjaabq/8/3">Volume 8, Issue 3</a>, Page 686712, September 2024. <br/> Abstract.We derive an explicit formula, valid for all integers [math], for the dimension of the vector space [math] of piecewise polynomial functions continuously differentiable to order [math] and whose constituents have degree at most [math], where [math] is a planar triangulation that has a single totally interior edge. This extends previous results of Tohǎneanu, Mináč, and Sorokina. Our result is a natural successor of Schumaker’s 1979 dimension formula for splines on a planar vertex star. Indeed, there has not been a dimension formula in this level of generality (valid for all integers [math] and any vertex coordinates) since Schumaker’s result. We derive our results using commutative algebra.
SIAM Journal on Applied Algebra and Geometry, Volume 8, Issue 3, Page 686712, September 2024. <br/> Abstract.We derive an explicit formula, valid for all integers [math], for the dimension of the vector space [math] of piecewise polynomial functions continuously differentiable to order [math] and whose constituents have degree at most [math], where [math] is a planar triangulation that has a single totally interior edge. This extends previous results of Tohǎneanu, Mináč, and Sorokina. Our result is a natural successor of Schumaker’s 1979 dimension formula for splines on a planar vertex star. Indeed, there has not been a dimension formula in this level of generality (valid for all integers [math] and any vertex coordinates) since Schumaker’s result. We derive our results using commutative algebra. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaabq/cover.jpg" alttext="cover image"/></p>
Bivariate Splines on a Triangulation with a Single Totally Interior Edge
10.1137/23M158317X
SIAM Journal on Applied Algebra and Geometry
20240823T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Michael DiPasquale
Beihui Yuan
Bivariate Splines on a Triangulation with a Single Totally Interior Edge
8
3
686
712
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/23M158317X
https://epubs.siam.org/doi/abs/10.1137/23M158317X?af=R
© 2024 Society for Industrial and Applied Mathematics

Analog Category and Complexity
https://epubs.siam.org/doi/abs/10.1137/24M1635004?af=R
SIAM Journal on Applied Algebra and Geometry, <a href="https://epubs.siam.org/toc/sjaabq/8/3">Volume 8, Issue 3</a>, Page 713732, September 2024. <br/> Abstract.We study probabilistic variants of the Lusternik–Schnirelmann category and topological complexity, which bound the classical invariants from below. We present a number of computations illustrating both wide agreement and wide disagreement with the classical notions. In the aspherical case, where our invariants are group invariants, we establish a counterpart of the Eilenberg–Ganea theorem in the torsionfree case, as well as a contrasting universal upper bound in the finite case.
SIAM Journal on Applied Algebra and Geometry, Volume 8, Issue 3, Page 713732, September 2024. <br/> Abstract.We study probabilistic variants of the Lusternik–Schnirelmann category and topological complexity, which bound the classical invariants from below. We present a number of computations illustrating both wide agreement and wide disagreement with the classical notions. In the aspherical case, where our invariants are group invariants, we establish a counterpart of the Eilenberg–Ganea theorem in the torsionfree case, as well as a contrasting universal upper bound in the finite case. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaabq/cover.jpg" alttext="cover image"/></p>
Analog Category and Complexity
10.1137/24M1635004
SIAM Journal on Applied Algebra and Geometry
20240826T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Ben Knudsen
Shmuel Weinberger
Analog Category and Complexity
8
3
713
732
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/24M1635004
https://epubs.siam.org/doi/abs/10.1137/24M1635004?af=R
© 2024 Society for Industrial and Applied Mathematics

Orbit Recovery for BandLimited Functions
https://epubs.siam.org/doi/abs/10.1137/23M1577808?af=R
SIAM Journal on Applied Algebra and Geometry, <a href="https://epubs.siam.org/toc/sjaabq/8/3">Volume 8, Issue 3</a>, Page 733755, September 2024. <br/> Abstract.We study the third moment for functions on arbitrary compact Lie groups. We use techniques of representation theory to generalize the notion of bandlimited functions in classical Fourier theory to functions on the compact groups [math]. We then prove that for generic bandlimited functions the third moment or its Fourier equivalent, the bispectrum, determines the function up to translation by a single unitary matrix. Moreover, if [math] or [math], we prove that the third moment determines the [math]orbit of a bandlimited function. As a corollary, we obtain a large class of finitedimensional representations of these groups for which the third moment determines the orbit of a generic vector. When [math] this gives a result relevant to cryoEM, which was our original motivation for studying this problem.
SIAM Journal on Applied Algebra and Geometry, Volume 8, Issue 3, Page 733755, September 2024. <br/> Abstract.We study the third moment for functions on arbitrary compact Lie groups. We use techniques of representation theory to generalize the notion of bandlimited functions in classical Fourier theory to functions on the compact groups [math]. We then prove that for generic bandlimited functions the third moment or its Fourier equivalent, the bispectrum, determines the function up to translation by a single unitary matrix. Moreover, if [math] or [math], we prove that the third moment determines the [math]orbit of a bandlimited function. As a corollary, we obtain a large class of finitedimensional representations of these groups for which the third moment determines the orbit of a generic vector. When [math] this gives a result relevant to cryoEM, which was our original motivation for studying this problem. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaabq/cover.jpg" alttext="cover image"/></p>
Orbit Recovery for BandLimited Functions
10.1137/23M1577808
SIAM Journal on Applied Algebra and Geometry
20240827T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Dan Edidin
Matthew Satriano
Orbit Recovery for BandLimited Functions
8
3
733
755
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/23M1577808
https://epubs.siam.org/doi/abs/10.1137/23M1577808?af=R
© 2024 Society for Industrial and Applied Mathematics

CoBarS: Fast Reweighted Sampling for Polygon Spaces in Any Dimension
https://epubs.siam.org/doi/abs/10.1137/23M1620740?af=R
SIAM Journal on Applied Algebra and Geometry, <a href="https://epubs.siam.org/toc/sjaabq/8/3">Volume 8, Issue 3</a>, Page 756781, September 2024. <br/> Abstract. We present the first algorithm for sampling random configurations of closed [math]gons with any fixed edgelengths [math] in any dimension [math] which is proved to sample correctly from standard probability measures on these spaces. We generate open [math]gons as weighted sets of edge vectors on the unit sphere and close them by taking a Möbius transformation of the sphere which moves the center of mass of the edges to the origin. Using previous results of the authors, such a Möbius transformation can be found in [math] time. The resulting closed polygons are distributed according to a pushforward measure. The main contribution of the present paper is the explicit calculation of reweighting factors which transform this pushforward measure to any one of a family of standard measures on closed polygon space, including the symplectic volume for polygons in [math]. For fixed dimension, these reweighting factors may be computed in [math] time. Experimental results show that our algorithm is efficient and accurate in practice, and an opensource reference implementation is provided.
SIAM Journal on Applied Algebra and Geometry, Volume 8, Issue 3, Page 756781, September 2024. <br/> Abstract. We present the first algorithm for sampling random configurations of closed [math]gons with any fixed edgelengths [math] in any dimension [math] which is proved to sample correctly from standard probability measures on these spaces. We generate open [math]gons as weighted sets of edge vectors on the unit sphere and close them by taking a Möbius transformation of the sphere which moves the center of mass of the edges to the origin. Using previous results of the authors, such a Möbius transformation can be found in [math] time. The resulting closed polygons are distributed according to a pushforward measure. The main contribution of the present paper is the explicit calculation of reweighting factors which transform this pushforward measure to any one of a family of standard measures on closed polygon space, including the symplectic volume for polygons in [math]. For fixed dimension, these reweighting factors may be computed in [math] time. Experimental results show that our algorithm is efficient and accurate in practice, and an opensource reference implementation is provided. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaabq/cover.jpg" alttext="cover image"/></p>
CoBarS: Fast Reweighted Sampling for Polygon Spaces in Any Dimension
10.1137/23M1620740
SIAM Journal on Applied Algebra and Geometry
20240923T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Jason Cantarella
Henrik Schumacher
CoBarS: Fast Reweighted Sampling for Polygon Spaces in Any Dimension
8
3
756
781
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/23M1620740
https://epubs.siam.org/doi/abs/10.1137/23M1620740?af=R
© 2024 Society for Industrial and Applied Mathematics

Area Formula for Spherical Polygons via Prequantization
https://epubs.siam.org/doi/abs/10.1137/23M1565255?af=R
SIAM Journal on Applied Algebra and Geometry, <a href="https://epubs.siam.org/toc/sjaabq/8/3">Volume 8, Issue 3</a>, Page 782796, September 2024. <br/> Abstract. We present a formula for the signed area of a spherical polygon via prequantization. In contrast to the traditional formula based on the Gauss–Bonnet theorem that requires measuring angles, the new formula mimics Green’s theorem and is applicable to a wider range of degenerate spherical curves and polygons.
SIAM Journal on Applied Algebra and Geometry, Volume 8, Issue 3, Page 782796, September 2024. <br/> Abstract. We present a formula for the signed area of a spherical polygon via prequantization. In contrast to the traditional formula based on the Gauss–Bonnet theorem that requires measuring angles, the new formula mimics Green’s theorem and is applicable to a wider range of degenerate spherical curves and polygons. <p><img src="https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjaabq/cover.jpg" alttext="cover image"/></p>
Area Formula for Spherical Polygons via Prequantization
10.1137/23M1565255
SIAM Journal on Applied Algebra and Geometry
20240923T07:00:00Z
© 2024 Society for Industrial and Applied Mathematics
Albert Chern
Sadashige Ishida
Area Formula for Spherical Polygons via Prequantization
8
3
782
796
20240930T07:00:00Z
20240930T07:00:00Z
10.1137/23M1565255
https://epubs.siam.org/doi/abs/10.1137/23M1565255?af=R
© 2024 Society for Industrial and Applied Mathematics