Abstract: Dyson’s rank function and the Andrews–Garvan crank function famously give combinatorial witnesses for Ramanujan’s partition function congruences modulo 5, 7, and 11. While these functions can be used to show that the corresponding sets of partitions split into 5, 7, or 11 equally sized sets, one may ask how to make the resulting bijections between partitions organized by rank or crank combinatorially explicit. Stanton recently made conjectures which aim to uncover a deeper combinatorial structure along these lines, where it turns out that minor modifications of the rank and crank are required. Here, we prove two of these conjectures. We also provide abstract criteria for quotients of polynomials by certain cyclotomic polynomials to have non-negative coefficients based on unimodality and symmetry. Furthermore, we extend Stanton’s conjecture to an infinite family of cranks. This suggests further applications to other combinatorial objects. We also discuss numerical evidence for our conjectures, connections with other analytic conjectures such as the distribution of partition ranks. PubDate: 2022-06-06
Abstract: Abstract Following Bauschke and Combettes (Convex analysis and monotone operator theory in Hilbert spaces, Springer, Cham, 2017), we introduce ProxNet, a collection of deep neural networks with ReLU activation which emulate numerical solution operators of variational inequalities (VIs). We analyze the expression rates of ProxNets in emulating solution operators for variational inequality problems posed on closed, convex cones in real, separable Hilbert spaces, covering the classical contact problems in mechanics, and early exercise problems as arise, e.g., in valuation of American-style contracts in Black–Scholes financial market models. In the finite-dimensional setting, the VIs reduce to matrix VIs in Euclidean space, and ProxNets emulate classical projected matrix iterations, such as projected Jacobi and projected SOR methods. PubDate: 2022-06-04
Abstract: Abstract We study monomial ideals with linear presentation or partially linear resolution. We give combinatorial characterizations of linear presentation for square-free ideals of degree 3, and for primary ideals whose resolutions are linear except for the last step (the “almost linear” case). We also give sharp bounds on Castelnuovo–Mumford regularity and numbers of generators in some cases. It is a basic observation that linearity properties are inherited by the restriction of an ideal to a subset of variables, and we study when the converse holds. We construct fractal examples of almost linear primary ideals with relatively few generators related to the Sierpiński triangle. Our results also lead to classes of highly connected simplicial complexes \(\Delta \) that cannot be extended to the complete \(\mathrm{dim}\Delta \) -skeleton of the simplex on the same variables by shelling. PubDate: 2022-05-29
Abstract: Abstract An exact transformation, which we call the master identity, is obtained for the first time for the series \(\sum _{n=1}^{\infty }\sigma _{a}(n)e^{-ny}\) for \(a\in {\mathbb {C}}\) and Re \((y)>0\) . New modular-type transformations when a is a nonzero even integer are obtained as its special cases. The precise obstruction to modularity is explicitly seen in these transformations. These include a novel companion to Ramanujan’s famous formula for \(\zeta (2m+1)\) . The Wigert–Bellman identity arising from the \(a=0\) case of the master identity is derived too. When a is an odd integer, the well-known modular transformations of the Eisenstein series on \(SL _{2}\left( {\mathbb {Z}}\right) \) , that of the Dedekind eta function as well as Ramanujan’s formula for \(\zeta (2m+1)\) are derived from the master identity. The latter identity itself is derived using Guinand’s version of the Voronoï summation formula and an integral evaluation of N. S. Koshliakov involving a generalization of the modified Bessel function \(K_{\nu }(z)\) . Koshliakov’s integral evaluation is proved for the first time. It is then generalized using a well-known kernel of Watson to obtain an interesting two-variable generalization of the modified Bessel function. This generalization allows us to obtain a new modular-type transformation involving the sums-of-squares function \(r_k(n)\) . Some results on functions self-reciprocal in the Watson kernel are also obtained. PubDate: 2022-05-19
Abstract: Abstract Deep generative neural networks have enabled modeling complex distributions, but incorporating physics knowledge into the neural networks is still challenging and is at the core of current physics-based machine learning research. To this end, we propose a physics generative neural network (PhysGNN), a new class of generative neural networks for learning unknown distributions in a physical system described by partial differential equations (PDE). PhysGNN couples PDE systems with generative neural networks. It is a fully differentiable model that allows back-propagation of gradients through both numerical PDE solvers and generative neural networks, and is trained by minimizing the discrete Wasserstein distance between generated and observed probability distributions of the PDE outputs using the stochastic gradient descent method. Moreover, PhysGNN does not require adversarial training like standard generative neural networks, which offers better stability than adversarial training. We show that PhysGNN can learn complex distributions in stochastic inverse problems, where conventional methods such as maximum likelihood estimation and momentum matching methods may be inapplicable when little knowledge is known about the form of unknown distributions or the physical model is too complex. Our method allows physics-based generative neural network training for learning complex distributions in the context of differential equations. PubDate: 2022-05-17
Abstract: Abstract We give a necessary and sufficient condition for the existence of infinitely many complete intersections in the shifted family of a numerical semigroup. PubDate: 2022-05-11
Abstract: Abstract This paper is concerned with the question of whether geometric structures such as cell complexes can be used to simultaneously describe the minimal free resolutions of all powers of a monomial ideal. We provide a full answer in the case of square-free monomial ideals of projective dimension one by introducing a combinatorial construction of a family of (cubical) cell complexes whose 1-skeletons are powers of a graph that supports the resolution of the ideal. PubDate: 2022-05-09
Abstract: Abstract In the present paper, we study algebraic properties of edge ideals associated with plane curve arrangements via their Levi graphs. Using combinatorial properties of such Levi graphs, we are able to describe those monomial algebras being Cohen–Macaulay, Buchsbaum, and sequentially Cohen–Macaulay. We also consider the projective dimension and the Castelnuovo-Mumford regularity for these edge ideals. We provide effective lower and upper bounds on them. As a by-product of our study, we connect, in general, various Buchsbaum properties of squarefree modules. PubDate: 2022-05-07
Abstract: Abstract A very well-covered graph is a well-covered graph without isolated vertices such that the height of its edge ideal is half of the number of vertices. In this survey article, we gather together most of the old and new results on the edge and cover ideals of these graphs. PubDate: 2022-05-06
Abstract: Abstract We present a conjecture about the reduced Hilbert series of the coordinate ring of a simple polyomino in terms of particular arrangements of non-attacking rooks that can be placed on the polyomino. By using a computational approach, we prove that the above conjecture holds for all simple polyominoes up to rank 11. In addition, we prove that the conjecture holds true for the class of parallelogram polyominoes, by looking at those as simple planar distributive lattices. Finally, we give a combinatorial interpretation of the Gorensteinness of parallelogram polyominoes. PubDate: 2022-04-28
Abstract: Abstract We explore the dependence of the Betti numbers of monomial ideals on the characteristic of the field. A first observation is that for a fixed prime p either the i-th Betti number of all high enough powers of a monomial ideal differs in characteristic 0 and in characteristic p or it is the same for all high enough powers. In our main results, we provide constructions and explicit examples of monomial ideals all of whose powers have some characteristic-dependent Betti numbers or whose asymptotic regularity depends on the field. We prove that, adding a monomial on new variables to a monomial ideal allows to spread the characteristic dependence to all powers. For any given prime p, this produces an edge ideal such that all its powers have some Betti numbers that are different over \(\mathbb {Q}\) and over \(\mathbb {Z}_p\) . Moreover, we show that, for every \(r \ge 0\) and \(i \ge 3\) there is a monomial ideal I such that some coefficient in a degree \(\ge r\) of the Kodiyalam polynomials \({\mathfrak {P}}_3(I),\ldots ,{\mathfrak {P}}_{i+r}(I)\) depends on the characteristic. We also provide a summary of related results and speculate about the behavior of other combinatorially defined ideals. PubDate: 2022-04-18
Abstract: Abstract We introduce a framework for generating, organizing, and reasoning with computational knowledge. It is motivated by the observation that most problems in Computational Sciences and Engineering (CSE) can be described as that of completing (from data) a computational graph (or hypergraph) representing dependencies between functions and variables. In that setting nodes represent variables and edges (or hyperedges) represent functions (or functionals). Functions and variables may be known, unknown, or random. Data come in the form of observations of distinct values of a finite number of subsets of the variables of the graph (satisfying its functional dependencies). The underlying problem combines a regression problem (approximating unknown functions) with a matrix completion problem (recovering unobserved variables in the data). Replacing unknown functions by Gaussian processes and conditioning on observed data provides a simple but efficient approach to completing such graphs. Since the proposed framework is highly expressive, it has a vast potential application scope. Since the completion process can be automatized, as one solves \(\sqrt{\sqrt{2}+\sqrt{3}}\) on a pocket calculator without thinking about it, one could, with the proposed framework, solve a complex CSE problem by drawing a diagram. Compared to traditional regression/kriging, the proposed framework can be used to recover unknown functions with much scarcer data by exploiting interdependencies between multiple functions and variables. The computational graph completion (CGC) problem addressed by the proposed framework could therefore also be interpreted as a generalization of that of solving linear systems of equations to that of approximating unknown variables and functions with noisy, incomplete, and nonlinear dependencies. Numerous examples illustrate the flexibility, scope, efficacy, and robustness of the CGC framework and show how it can be used as a pathway to identifying simple solutions to classical CSE problems. These examples include the seamless CGC representation of known methods (for solving/learning PDEs, surrogate/multiscale modeling, mode decomposition, deep learning) and the discovery of new ones (digital twin modeling, dimension reduction). PubDate: 2022-04-18
Abstract: Abstract This works concerns cohomological support varieties of modules over commutative local rings. The main result is that the support of a derived tensor product of a pair of differential graded modules over a Koszul complex is the join of the supports of the modules. This generalizes, and gives another proof of, a result of Dao and the third author dealing with Tor-independent modules over complete intersection rings. The result for Koszul complexes has a broader applicability, including to exterior algebras over local rings. PubDate: 2022-04-15
Abstract: Abstract This paper deals with the issue of stability in determining the absorption and the diffusion coefficients in quantitative photoacoustic imaging. We establish a global conditional Hölder stability inequality from the knowledge of two internal data obtained from optical waves, generated by two point sources in a region where the optical coefficients are known. PubDate: 2022-04-12
Abstract: Abstract In 1999, Herzog and Hibi introduced componentwise linear ideals. A homogeneous ideal I is componentwise linear if for all nonnegative integers d, the ideal generated by the homogeneous elements of degree d in I has a linear resolution. For square-free monomial ideals, componentwise linearity is related via Alexander duality to the property of being sequentially Cohen–Macaulay for the corresponding simplicial complexes. In general, the property of being componentwise linear is not preserved by taking powers. In 2011, Herzog, Hibi, and Ohsugi conjectured that if I is the cover ideal of a chordal graph, then \(I^s\) is componentwise linear for all \(s \ge 1\) . We survey some of the basic properties of componentwise linear ideals and then specialize to the progress on the Herzog–Hibi–Ohsugi conjecture during the last decade. We also survey the related problem of determining when the symbolic powers of a cover ideal are componentwise linear PubDate: 2022-04-07
Abstract: Abstract A semigroup \(\langle C\rangle \) in \(\mathbb {N}^n\) is a gluing of \(\langle A\rangle \) and \(\langle B\rangle \) if its finite set of generators C splits into two parts, \(C=k_1A\sqcup k_2B\) with \(k_1,k_2\ge 1\) , and the defining ideals of the corresponding semigroup rings satisfy that \(I_C\) is generated by \(I_A+I_B\) and one extra element. Two semigroups \(\langle A\rangle \) and \(\langle B\rangle \) can be glued if there exist positive integers \(k_1,k_2\) such that for \(C=k_1A\sqcup k_2B\) , \(\langle C\rangle \) is a gluing of \(\langle A\rangle \) and \(\langle B\rangle \) . Although any two numerical semigroups, namely semigroups in dimension \(n=1\) , can always be glued, it is no longer the case in higher dimensions. In this paper, we give necessary and sufficient conditions on A and B for the existence of a gluing of \(\langle A\rangle \) and \(\langle B\rangle \) , and give examples to illustrate why they are necessary. These generalize and explain the previous known results on existence of gluing. We also prove that the glued semigroup \(\langle C\rangle \) inherits the properties like Gorenstein or Cohen–Macaulay from the two parts \(\langle A\rangle \) and \(\langle B\rangle \) . PubDate: 2022-04-07
Abstract: Abstract In this article, Eisenstein cohomology of the arithmetic group \(G_2(\mathbb {Z})\) with coefficients in any finite dimensional highest weight irreducible representation has been determined. This is accomplished by studying the cohomology of the boundary of the Borel–Serre compactification. PubDate: 2022-04-05
Abstract: Abstract We discuss the computation of automorphism groups and normal forms of cones and polyhedra in Normaliz and indicate its implementation via nauty. The types of automorphisms include integral, rational, Euclidean and combinatorial, as well as algebraic for polytopes defined over real algebraic number fields. Examples treated in detail are the icosahedron and linear-ordering polytopes whose Euclidean automorphism groups are determined. PubDate: 2022-04-01
Abstract: Abstract We show that the Hunter–Saxton equation \(u_t+uu_x=\frac{1}{4}\big (\int _{-\infty }^x \hbox {d}\mu (t,z)- \int ^{\infty }_x \hbox {d}\mu (t,z)\big )\) and \(\mu _t+(u\mu )_x=0\) has a unique, global, weak, and conservative solution \((u,\mu )\) of the Cauchy problem on the line. PubDate: 2022-03-19
Abstract: Abstract We study the two-player zero-sum game with mixed strategies. For a class of commonly used regularizers and a class of metrics, we show the existence of a Lyapunov function of the gradient ascent descent dynamics. We also propose for a new particle method for a specific combination of regularizers and metrics. PubDate: 2022-03-03
Open Access journal
[228 journals]
