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Abstract: Abstract In Ciaramella et al. (2020) we defined a new partition of unity for the Bank–Jimack domain decomposition method in 1D and proved that with the new partition of unity, the Bank–Jimack method is an optimal Schwarz method in 1D and thus converges in two iterations for two subdomains: it becomes a direct solver, and this independently of the outer coarse mesh one uses! In this paper, we show that the Bank–Jimack method in 2D is an optimized Schwarz method and its convergence behavior depends on the structure of the outer coarse mesh each subdomain is using. For an equally spaced coarse mesh its convergence behavior is not as good as the convergence behavior of optimized Schwarz. However, if a stretched coarse mesh is used, then the Bank–Jimack method becomes faster then optimized Schwarz with Robin or Ventcell transmission conditions. Our analysis leads to a conjecture stating that the convergence factor of the Bank–Jimack method with overlap L and m geometrically stretched outer coarse mesh cells is \(1-O(L^{\frac {1}{2 m}})\) . PubDate: 2022-05-24

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Abstract: Abstract Our work firstly investigates the unique existence and the continuous dependence (on the singular kernel and initial data) of solutions to nonlocal evolution equations on Hilbert spaces. Secondly, we prove the well-definedness of a related semi-dynamical system consisting of Lipschitz continuous mappings in the space of continuous functions by constructing a metric utilizing the kernel of nonlocal derivative. Our results extend and generalize the existing results on Caputo fractional differential equations, namely the stability and structural stability results in Diethelm and Ford (J. Math. Anal. Appl. 265, 229–248, 2002), the related semi-dynamical systems in Son and Kloeden (Vietnam J. Math. 49, 1305–1315, 2021), to the case of nonlocal differential equations. PubDate: 2022-05-23

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Abstract: Abstract In this paper we give an overview of some recent and older results concerning free boundary problems governed by elliptic operators. PubDate: 2022-05-18

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Abstract: Abstract Denote by Σn and Qn the set of all n × n symmetric and skew-symmetric matrices over a field \(\mathbb {F}\) , respectively, where \(\text {char}(\mathbb {F})\neq 2\) and \( \mathbb {F} \geq n^{2}+1\) . A characterization of \(\phi ,\psi :{\varSigma }_{n} \rightarrow {\varSigma }_{n}\) , for which at least one of them is surjective, satisfying $ \det (\phi (x)+\psi (y))=\det (x+y)\qquad (x,y\in {\varSigma }_{n}) $ is given. Furthermore, if n is even and \(\phi ,\psi :Q_{n} \rightarrow Q_{n}\) , for which ψ is surjective and ψ(0) = 0, satisfy $\det (\phi (x)+\psi (y))=\det (x+y)\qquad (x,y\in Q_{n}), $ then ϕ = ψ and ψ must be a bijective linear map preserving the determinant. PubDate: 2022-05-13

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Abstract: Abstract The Internodes method is a general purpose method to deal with non-conforming discretizations of partial differential equations on 2D and 3D regions partitioned into disjoint subdomains. In this paper we are interested in measuring how much the Internodes method is conservative across the interface. If hp-fem discretizations are employed, we prove that both the total force and total work generated by the numerical solution at the interface of the decomposition vanish in an optimal way when the mesh size tends to zero, i.e., like \(\mathcal {O}(h^{p})\) , where p is the local polynomial degree and h the mesh-size. This is the same as the error decay in the H1-broken norm. We observe that the conservation properties of a method are intrinsic to the method itself because they depend on the way the interface conditions are enforced rather then on the problem we are called to approximate. For this reason, in this paper, we focus on second-order elliptic PDEs, although we use the terminology (of forces and works) proper of linear elasticity. Two and three dimensional numerical experiments corroborate the theoretical findings, also by comparing Internodes with Mortar and WACA methods. PubDate: 2022-05-10

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Abstract: Abstract Results of Koebe (Ber. Sächs. Akad. Wiss. Leipzig, Math.-phys. Kl. 88, 141–164, 1936), Schramm (Invent. Math. 107(3), 543560, 1992), and Springborn (Math. Z. 249, 513–517, 2005) yield realizations of 3-polytopes with edges tangent to the unit sphere. Here we study the algebraic degrees of such realizations. This initiates the research on constrained realization spaces of polytopes. PubDate: 2022-05-06

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Abstract: Abstract Let A be a commutative Noetherian ring containing a field of characteristic zero. Let R = A[X1,…,Xm] be a polynomial ring and Am(A) = A〈X1,…,Xm, ∂1,…,∂m〉 be the m th Weyl algebra over A, where ∂i = ∂/∂Xi. Consider standard gradings on R and Am(A) by setting \(\deg z=0\) for all z ∈ A, \(\deg X_{i}=1\) , and \(\deg \partial _{i} =-1\) for i = 1,…,m. We present a few results about the behavior of the graded components of local cohomology modules \({H_{I}^{i}}(R)\) , where I is an arbitrary homogeneous ideal in R. We mostly restrict our attention to the vanishing, tameness, and rigidity properties. To obtain this, we use the theory of D-modules and show that generalized Eulerian Am(A)-modules exhibit these properties. As a corollary, we further get that components of graded local cohomology modules with respect to a pair of ideals display similar behavior. PubDate: 2022-04-27

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Abstract: Abstract In this paper, we study the multiple-sets split common null point problem (MSCNPP) in Banach spaces. We introduce a new self-adaptive algorithm for solving this problem. Under suitable conditions, we prove a strong convergence theorem of the proposed algorithm. An application of the main theorem to the multiple-sets split feasibility problem in Banach spaces is also presented. Finally, we provide the numerical experiments which show the efficiency and implementation of the proposed method. PubDate: 2022-04-27

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Abstract: Abstract We explore four approaches to the question of defectivity for a complex projective toric variety XA associated with an integral configuration A. The explicit tropicalization of the dual variety \(X_{A}^{\vee }\) due to Dickenstein, Feichtner, and Sturmfels allows for the computation of the defect in terms of an affine combinatorial invariant ρ(A). We express ρ(A) in terms of affine invariants ι(A) associated to Esterov’s iterated circuits and λ(A), an invariant defined by Curran and Cattani in terms of a Gale dual of A. Thus we obtain formulae for the dual defect in terms of iterated circuits and Gale duals. An alternative expression for the dual defect of XA is given by Furukawa–Ito in terms of Cayley decompositions of A. We give a Gale dual interpretation of these decompositions and apply it to the study of defective configurations. PubDate: 2022-04-27

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Abstract: Abstract In this paper, we survey the theory of Cartwright–Sturmfels ideals. These are \(\mathbb {Z}^{n}\) -graded ideals, whose multigraded generic initial ideal is radical. Cartwright–Sturmfels ideals have surprising properties, mostly stemming from the fact that their Hilbert scheme only contains one Borel-fixed point. This has consequences, e.g., on their universal Gröbner bases and on the family of their initial ideals. In this paper, we discuss several known classes of Cartwright–Sturmfels ideals and we find a new one. Among determinantal ideals of same-size minors of a matrix of variables and Schubert determinantal ideals, we are able to characterize those that are Cartwright–Sturmfels. PubDate: 2022-04-20

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Abstract: Abstract In this paper we study the set of tensors that admit a special type of decomposition called an orthogonal tensor train decomposition. Finding equations defining varieties of low-rank tensors is generally a hard problem, however, the set of orthogonally decomposable tensors is defined by appealing quadratic equations. The tensors we consider are an extension of orthogonally decomposable tensors. We show that they are defined by similar quadratic equations, as well as an interesting higher-degree additional equation. PubDate: 2022-04-08

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Abstract: Abstract Neural codes are lists of subsets of neurons that fire together. Of particular interest are neurons called place cells, which fire when an animal is in specific, usually convex regions in space. A fundamental question, therefore, is to determine which neural codes arise from the regions of some collection of open convex sets or closed convex sets in Euclidean space. This work focuses on how these two classes of codes – open convex and closed convex codes – are related. As a starting point, open convex codes have a desirable monotonicity property, namely, adding non-maximal codewords preserves open convexity; but here we show that this property fails to hold for closed convex codes. Additionally, while adding non-maximal codewords can only increase the open embedding dimension by 1, here we demonstrate that adding a single such codeword can increase the closed embedding dimension by an arbitrarily large amount. Finally, we disprove a conjecture of Goldrup and Phillipson, and also present an example of a code that is neither open convex nor closed convex. PubDate: 2022-04-01

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Abstract: Abstract In this paper, we consider the Poisson equation on a “long” domain which is the Cartesian product of a one-dimensional long interval with a (d − 1)-dimensional domain. The right-hand side is assumed to have a rank-1 tensor structure. We will present and compare methods to construct approximations of the solution which have tensor structure and the computational effort is governed by only solving elliptic problems on lower-dimensional domains. A zero-th order tensor approximation is derived by using tools from asymptotic analysis (method 1). The resulting approximation is an elementary tensor and, hence has a fixed error which turns out to be very close to the best possible approximation of zero-th order. This approximation can be used as a starting guess for the derivation of higher-order tensor approximations by a greedy-type method (method 2). Numerical experiments show that this method is converging towards the exact solution. Method 3 is based on the derivation of a tensor approximation via exponential sums applied to discretized differential operators and their inverses. It can be proved that this method converges exponentially with respect to the tensor rank. We present numerical experiments which compare the performance and sensitivity of these three methods. PubDate: 2022-04-01

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Abstract: Abstract The study of Chow varieties of decomposable forms lies at the confluence of algebraic geometry, commutative algebra, representation theory and combinatorics. There are many open questions about homological properties of Chow varieties and interesting classes of modules supported on them. The goal of this note is to survey some fundamental constructions and properties of these objects, and to propose some new directions of research. Our main focus will be on the study of certain maximal Cohen–Macaulay modules of covariants supported on Chow varieties, and on defining equations and syzygies. We also explain how to assemble Tor groups over Veronese subalgebras into modules over a Chow variety, leading to a result on the polynomial growth of these groups. PubDate: 2022-04-01

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Abstract: Abstract This paper studies how to learn parameters in diagonal Gaussian mixture models. The problem can be formulated as computing incomplete symmetric tensor decompositions. We use generating polynomials to compute incomplete symmetric tensor decompositions and approximations. Then the tensor approximation method is used to learn diagonal Gaussian mixture models. We also do the stability analysis. When the first and third order moments are sufficiently accurate, we show that the obtained parameters for the Gaussian mixture models are also highly accurate. Numerical experiments are also provided. PubDate: 2022-04-01

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Abstract: Abstract We provide an algorithm that takes as an input a given parametric family of homogeneous polynomials, which is invariant under the action of the general linear group, and an integer d. It outputs the ideal of that family intersected with the space of homogeneous polynomials of degree d. Our motivation comes from Question 7 in Ranestad and Sturmfels (Le Math. 75, 411–424, 2020) and Problem 13 in Sturmfels (2014), which ask to find equations for varieties of cubic and quartic symmetroids. The algorithm relies on a database of specific Young tableaux and highest weight polynomials. We provide the database and the implementation of the database construction algorithm. Moreover, we provide a Julia implementation to run the algorithm using the database, so that more varieties of homogeneous polynomials can easily be treated in the future. PubDate: 2022-02-24 DOI: 10.1007/s10013-022-00549-4

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Abstract: Abstract We study the univariate moment problem of piecewise-constant density functions on the interval [0,1] and its consequences for an inference problem in population genetics. We show that, up to closure, any collection of n moments is achieved by a step function with at most n − 1 breakpoints and that this bound is tight. We use this to show that any point in the n th coalescence manifold in population genetics can be attained by a piecewise constant population history with at most n − 2 changes. Both the moment cones and the coalescence manifold are projected spectrahedra and we describe the problem of finding a nearest point on them as a semidefinite program. PubDate: 2022-02-22 DOI: 10.1007/s10013-022-00548-5

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Abstract: Abstract We work out details of the extrinsic geometry for two Hilbert schemes of some contemporary interest: the Hilbert scheme \(\text {Hilb}^{2} \mathbb {P}^{2}\) of two points on \(\mathbb {P}^{2}\) and the dense open set parametrizing non-planar clusters in the punctual Hilbert scheme \(\text {Hilb}^{4}_{0}(\mathbb {A}^{3})\) of clusters of length four on \(\mathbb {A}^{3}\) with support at the origin. We find explicit equations in projective, respectively affine, embeddings for these spaces. In particular, we answer a question of Bernd Sturmfels who asked for a description of the latter space that is amenable to further computations. While the explicit equations we find are controlled in a precise way by the representation theory of SL3, our arguments also rely on computer algebra. PubDate: 2022-01-22 DOI: 10.1007/s10013-021-00545-0

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Abstract: Abstract Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths. PubDate: 2022-01-13 DOI: 10.1007/s10013-021-00526-3

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Abstract: Abstract Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths. PubDate: 2022-01-13 DOI: 10.1007/s10013-021-00526-3