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

Abstract: Abstract We construct a family of self-similar and bounded weak solutions for all positive time to the two-dimensional shallow water equations with the bottom topography a quadratic function with respect to the reciprocal of radius under axisymmetry assumptions. The source term generated from the bottom topography leads to a different structure in solutions with that obtained in hyperbolic conservative system. The key point is that the steady-state solution is used to deal with the singularity near the origin in the reduced system under axisymmetric transformation. The obtained results may apply to other systems with similar structures. PubDate: 2022-02-22

Abstract: Abstract Recently, Gekeler proved that the group of invertible analytic functions modulo constant functions on Drinfeld’s upper half space is isomorphic to the dual of an integral generalized Steinberg representation. In this note, we show that the group of invertible functions is the dual of a universal extension of that Steinberg representation. As an application, we show that lifting obstructions of rigid analytic theta cocycles of Hilbert modular forms in the sense of Darmon–Vonk can be computed in terms of \(\mathcal {L}\) -invariants of the associated Galois representation. The same argument applies to theta cocycles for definite unitary groups. PubDate: 2022-02-10

Abstract: Abstract In this work, we construct, for any \(d \ge 2\) , a new foliation on \(\mathbb {CP}^2\) of degree d with a unique singular point of multiplicity \(d-1\) without invariant algebraic curves that contain all its separatrices. We also prove that if X is a foliation on \(\mathbb {CP}^2\) with a unique nilpotent singular point, then X has no algebraic leaves. Finally, we characterize logarithmic foliations on \(\mathbb {CP}^2\) with a unique singular point. And we give some new examples of this kind of foliations. PubDate: 2022-02-09

Abstract: Abstract We develop a theory of multiplicities and mixed multiplicities of filtrations, extending the theory for filtrations of m-primary ideals to arbitrary (not necessarily Noetherian) filtrations. The mixed multiplicities of r filtrations on an analytically unramified local ring R come from the coefficients of a suitable homogeneous polynomial in r variables of degree equal to the dimension of the ring, analogously to the classical case of the mixed multiplicities of m-primary ideals in a local ring. We prove that the Minkowski inequalities hold for arbitrary filtrations. The characterization of equality in the Minkowski inequality for m-primary ideals in a local ring by Teissier, Rees and Sharp and Katz does not extend to arbitrary filtrations, but we show that they are true in a large and important subcategory of filtrations. We define divisorial and bounded filtrations. The filtration of powers of a fixed ideal is a bounded filtration, as is a divisorial filtration. We show that in an excellent local domain, the characterization of equality in the Minkowski equality is characterized by the condition that the integral closures of suitable Rees like algebras are the same, strictly generalizing the theorem of Teissier, Rees and Sharp and Katz. We also prove that a theorem of Rees characterizing the inclusion of ideals with the same multiplicity generalizes to bounded filtrations in excellent local domains. We give a number of other applications, extending classical theorems for ideals. PubDate: 2022-01-27

Abstract: Abstract The aims of this work are to study Rees algebras of filtrations of monomial ideals associated with covering polyhedra of rational matrices with nonnegative entries and nonzero columns using combinatorial optimization and integer programming and to study powers of monomial ideals and their integral closures using irreducible decompositions and polyhedral geometry. We study the Waldschmidt constant and the ic-resurgence of the filtration associated with a covering polyhedron and show how to compute these constants using linear programming. Then, we show a lower bound for the ic-resurgence of the ideal of covers of a graph and prove that the lower bound is attained when the graph is perfect. We also show lower bounds for the ic-resurgence of the edge ideal of a graph and give an algorithm to compute the asymptotic resurgence of squarefree monomial ideals. A classification of when Newton’s polyhedron is the irreducible polyhedron is presented using integral closure. PubDate: 2022-01-25