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In this work we give a bound for computing the least element with m≥2 factorizations of same length in a numerical semigroup with embedding dimension p, being p≥3 and we present an algorith...

Let S=&lang, a1,&hellip, ap&rang, be a numerical semigroup, let s&isin, S and let Z(s) be its set of factorizations. The set of lengths is denoted by L(s)={L(x1,⋯,xp)∣(x1,⋯,xp)&isin, Z(s)}, where L(x1,⋯,xp)=x1+⋯+xp. The following sets can then be defined: W(n)={s&isin, S∣&exist, x&isin, Z(s)suchthatL(x)=n}, &nu, (n)=⋃s&isin, W(n)L(s)={l1&lt, l2&lt, ⋯&lt, lr} and &Delta, &nu, (n)={l2&minus, l1,&hellip, lr&minus, 1}. In this paper, we prove that the function &Delta, N&rarr, P(N) is almost periodic with period lcm(a1,ap).

In this paper we show that a $k$-shellable simplicial complex is the expansion of a shellable complex. We prove that the face ring of a pure $k$-shellable simplicial complex satisfies the Stanley conjecture. In this way, by applying expansion functor to the face ring of a given pure shellable complex, we construct a large class of rings satisfying the Stanley conjecture. Also, by presenting some characterizations of $k$-shellable graphs, we extend some results due to Castrill\'{o}n-Cruz, Cruz-Estrada and Van Tuyl-Villareal., Comment: To appear in Math. Scand

Bijective maps are commonly used in many computer graphics and scientific computing applications, including texture, displacement, and bump mapping. However, their computation is numerically challenging due to the global nature of the problem, which makes standard smooth optimization techniques prohibitively expensive. We propose to use a scaffold structure to reduce this challenging and global problem to a local injectivity condition. This construction allows us to benefit from the recent advancements in locally injective maps optimization to efficiently compute large scale bijective maps (both in 2D and 3D), sidestepping the need to explicitly detect and avoid collisions. Our algorithm is guaranteed to robustly compute a globally bijective map, both in 2D and 3D. To demonstrate the practical applicability, we use it to compute globally bijective single patch parametrizations, to pack multiple charts into a single UV domain, to remove self-intersections from existing models, and to deform 3D objects while preventing self-intersections. Our approach is simple to implement, efficient (two orders of magnitude faster than competing methods), and robust, as we demonstrate in a stress test on a parametrization dataset with over a hundred meshes.

金沢大学理工学域 数物科学系 名誉教授, In this paper, we propose an extension of quantum searches on graphs driven by quantum walks to simplicial complexes. To this end, we define a new quantum walk on simplicial complex which is an alternative of preceding studies by authors. We show that the quantum search on the specific simplicial complex corresponding to the triangulation of n-dimensional unit square driven by this new simplicial quantum walk works well, namely, a marked simplex can be found with probability 1+o(1) 1+o(1) within a time O(N − − √ ) O(N), where N is the number of simplices with the dimension of marked simplex., Embargo Period 12 months

This work extends the results known for the Delta sets of non-symmetric numerical semigroups with embedding dimension three to the symmetric case. Thus, we have a fast algorithm to compute the Delta set of any embedding dimension three numerical semigroup. Also, as a consequence of these resutls, the sets that can be realized as Delta sets of numerical semigroups of embedding dimension three are fully characterized.

We get a classification of 1-truncated simplicial algebras under certain conditions by using GAP.

This work extends the results known for the Delta sets of non-symmetric numerical semigroups with embedding dimension three to the symmetric case. Thus, we have a fast algorithm to compute the Delta set of any embedding dimension three numerical semigroup. Also, as a consequence of these results, the sets that can be realized as Delta sets of numerical semigroups of embedding dimension three are fully characterized.

A Langford sequence of order m and defect d can be identified with a labeling of the vertices of a path of order 2m in which each label from d up to d + m − 1 appears twice and in which the vertices that have been labeled with k are at distance k. In this paper, we introduce two generalizations of this labeling that are related to distances. The basic idea is to assign nonnegative integers to vertices in such a way that if n vertices (n > 1) have been labeled with k then they are mutually at distance k. We study these labelings for some well known families of graphs. We also study the existence of these labelings in general. Finally, given a sequence or a set of nonnegative integers, we study the existence of graphs that can be labeled according to this sequence or set. The research conducted in this document by the first author has been supported by the Spanish Research Council under project MTM2011-28800-C02-01 and symbolically by the Catalan Research Council under grant 2014SGR1147.

We generalize the geometric sequence $\{a^p, a^{p-1}b, a^{p-2}b^2,...,b^p\}$ to allow the $p$ copies of $a$ (resp. $b$) to all be different. We call the sequence $\{a_1a_2a_3\cdots a_p, b_1a_2a_3\cdots a_p, b_1b_2a_3\cdots a_p,\ldots, b_1b_2b_3\cdots b_p\}$ a \emph{compound sequence}. We consider numerical semigroups whose minimal set of generators form a compound sequence, and compute various semigroup and arithmetical invariants, including the Frobenius number, Ap\'ery sets, Betti elements, and catenary degree. We compute bounds on the delta set and the tame degree.

We give a formula for the simplicial tree-numbers of the independent set complex of a finite matroid M as a product of eigenvalues of the total combinatorial Laplacians on this complex. Two matroid invariants emerge naturally in describing the multiplicities of these eigenvalues in the formula: one is the unsigned reduced Euler characteristic of the independent set complex and the other is the β -invariant of a matroid. We will demonstrate various applications of this formula including a "matroid theoretic" derivation of Kalai's simplicial tree-numbers of a standard simplex.

A numerical semigroup is a submonoid of , the set of nonnegative integers, under addition and with finite complement in . If the numerical semigroup is the form with an integer not divisible by tree, then is a pseudo symmetric numerical semigroup with embedding dimension and multiplicity three. We present procedures to calculate the delta of pseudo- symmetric numerical semigroups as given above. Also, we will give a relation between the betti numbers and the delta sets of these semigroups.

For a square-free monomial ideal I ⊂ S = k[x 1, x 2,…, x n ], we introduce the notion of quasi-linear quotients. By using the quasi-linear quotients, we give a new algebraic criterion for the shellability of a pure simplicial complex Δ over [n]. Also, we provide a new criterion for the Cohen–Macaulayness of the face ring of a pure simplicial complex Δ. Moreover, we show that the face ring of the spanning simplicial complex (defined in [2]) of an r-cyclic graph is Cohen–Macaulay.

Let $$\{a_1,\ldots ,a_p\}$$ be the minimal generating set of a numerical monoid S. For any $$s\in S$$ , its Delta set is defined by $$\Delta (s)=\{l_{i}-l_{i-1}\mid i=2,\ldots ,k\}$$ where $$\{l_1

The delta set of a numerical semigroup $S$, denoted $\Delta(S)$, is a factorization invariant that measures the complexity of the sets of lengths of elements in $S$. We study the following problem: Which finite sets occur as the delta set of a numerical semigroup $S$? It is known that $\min \Delta(S) = \gcd \Delta(S)$ is a necessary condition. For any two-element set $\{d,td\}$ we produce a semigroup $S$ with this delta set. We then show that for $t\ge 2$, the set $\{d,td\}$ occurs as the delta set of some numerical semigroup of embedding dimension three if and only if $t=2$., Comment: 13 pages

We study the commutation of the stabilizer generators embedded in the q-representation of higher dimensional simplicial complex. The specific geometric structure and topological characteristics of 1-simplex connectivity are generalized to higher dimensional structure of spin networks encoded in ordered complex via combinatorial optimization of a closed compact space. Obtained results of a consistent homology-chain basis are used to define connectivity and dynamical self organization of spin network system via continuous sequences of simplicial maps.

In this work we study the framework of mathematical morphology on simplicial complex spaces. Simplicial complexes are widely used to represent multidimensional data, such as meshes, that are two dimensional complexes, or graphs, that can be interpreted as one dimensional complexes. Mathematical morphology is one of the most powerful frameworks for image processing, including the processing of digital structures, and is heavily used for many applications. However, mathematical morphology operators on simplicial complex spaces is not a concept fully developed in the literature. Specifically, we explore properties of the dimensional operators, small, versatile operators that can be used to define new operators on simplicial complexes, while maintaining properties from mathematical morphology. These operators can also be used to recover many morphological operators from the literature. Matlab code and additional material, including the proofs of the original properties, are freely available at http://code.google.com/p/math-morpho-simplicial-complexes.

Let S be a numerical monoid with minimal generating set 〈n1, …, nt〉. For m ∈ S, if [Formula: see text], then [Formula: see text] is called a factorization length of m. We denote by ℒ(m) = {m1, …, mk} (where mi < mi+1 for each 1 ≤ i < k) the set of all possible factorization lengths of m. The Delta set of m is defined by Δ(m) = {mi+1 - mi | 1 ≤ i < k} and the Delta set of S by Δ(S) = ⋃m∈SΔ(m). In this paper, we expand on the study of Δ(S) begun in [C. Bowles, S. T. Chapman, N. Kaplan and D. Reiser, On delta sets of numerical monoids, J. Algebra Appl. 5 (2006) 1–24] in the following manner. Let r1, r2, …, rt be an increasing sequence of positive integers and Mn = 〈n, n + r1, n + r2, …, n + rt〉 a numerical monoid where n is some positive integer. We prove that there exists a positive integer N such that if n > N then |Δ(Mn)| = 1. If t = 2 and r1 and r2 are relatively prime, then we determine a value for N which is sharp.

International audience; This paper introduces a data structure, called simplex tree, to represent abstract simplicial complexes of any dimension. All faces of the simplicial complex are explicitly stored in a trie whose nodes are in bijection with the faces of the complex. This data structure allows to efficiently implement a large range of basic operations on simplicial complexes. We provide theoretical complexity analysis as well as detailed experimental results. We more specifically study Rips and witness complexes.; Nous définissons dans cet article une nouvelle structure de données, appelée ''simplex tree'', pour représenter les complexes simpliciaux abstraits de toutes dimensions. Le complexe simplicial est représenté par un arbre préfixe dont les nœuds sont en bijection avec les faces du complexe. Cette structure de données permet de calculer efficacement un grand nombre d'opérations de bases sur les complexes simpliciaux. Nous développons dans cet article une analyse théorique de la complexité de ces algorithmes, ainsi qu'une analyse expérimentale détaillée. Nous étudions plus particulièrement la construction des complexes de Rips et des witness complexes.

Simplicial Dirac structures as finite analogues of the canonical Stokes Dirac structure, capturing the topological laws of the system, are defined on simplicial manifolds in terms of primal and dual cochains related by the coboundary operators. These finite-dimensional Dirac structures offer a framework for the formulation of standard input output finite-dimensional port-Hamiltonian systems that emulate the behavior of distributed-parameter port-Hamiltonian systems. This paper elaborates on the matrix representations of simplicial Dirac structures and the resulting port-Hamiltonian systems on simplicial manifolds. Employing these representations, we consider the existence of structural invariants and demonstrate how they pertain to the energy shaping of port-Hamiltonian systems on simplicial manifolds. (C) 2013 Elsevier Ltd. All rights reserved.

We present several new algorithms for computing factorization invariant values over affine semigroups. In particular, we give (i) the first known algorithm to compute the delta set of any affine semigroup, (ii) an improved method of computing the tame degree of an affine semigroup, and (iii) a dynamic algorithm to compute catenary degrees of affine semigroup elements. Our algorithms rely on theoretical results from combinatorial commutative algebra involving Gröbner bases, Hilbert bases, and other standard techniques. Implementation in the computer algebra system GAP is discussed.

In connection with the publication of the catalogue B. Grünbaum, A catalogue of simplicial arrangements in the real projective plane, Ars Mathematica Contemporanea 2 (2009), 1-25 of known simplicial arrangements of lines in the real projective plane, and the note B. Grünbaum, Small unstretchable simplicial arrangements of pseudolines, Geombinatorics 18 (2009), 153-160 about small simplicial arrangements of pseudolines, several developments of these topics deserve to be mentioned. The present paper puts these results in perspective, and provides appropriate illustrations. Članek podaja nadaljni razvoj in ilustracije rezultatov iz kataloga znanih trikotnih ureditev premic v realni projektivni ravnini B. Grünbaum, A catalogue of simplicial arrangements in the real projective plane, Ars Mathematica Contemporanea 2 (2009), 1-25 in majhnih trikotnih ureditev psevdo premic B. Grünbaum, Small unstretchable simplicial arrangements of pseudolines, Geombinatorics 18 (2009), 153-160.

Topological spaces—such as classifying spaces, configuration spaces and spacetimes—often admit directionality. Qualitative invariants on such directed spaces often are more informative, yet more difficult, to calculate than classical homotopy invariants because directed spaces rarely decompose as homotopy colimits of simpler directed spaces. Directed spaces often arise as geometric realizations of simplicial sets and cubical sets equipped with order-theoretic structure encoding the orientations of simplices and 1-cubes. We prove dual simplicial and cubical approximation theorems appropriate for the directed setting and give criteria for two different homotopy relations on directed maps in the literature to coincide.

We first develop a general framework for Laplace operators defined in terms of the combinatorial structure of a simplicial complex. This includes, among others, the graph Laplacian, the combinatorial Laplacian on simplicial complexes, the weighted Laplacian, and the normalized graph Laplacian. This framework then allows us to define the normalized Laplace operator $\Delta_{i}^{up}$ on simplicial complexes which we then systematically investigate. We study the effects of a wedge sum, a join and a duplication of a motif on the spectrum of the normalized Laplace operator, and identify some of the combinatorial features of a simplicial complex that are encoded in its spectrum., Comment: 46 pages, 5 figures

For δ ≥ 1 and n ≥ 1, consider the simplicial complex of graphs on n vertices in which each vertex has degree at most δ; we identify a given graph with its edge set and admit one loop at each vertex. This complex is of some importance in the theory of semigroup algebras. When δ = 1, we obtain the matching complex, for which it is known that there is 3-torsion in degree d of the homology whenever (n − 4)/3 ≤ d ≤ (n − 6)/2. This paper establishes similar bounds for δ ≥ 2. Specifically, there is 3-torsion in degree d wheneverThe procedure for detecting torsion is to construct an explicit cycle z that is easily seen to have the property that 3zis a boundary. Defining a homomorphism that sends z to a non-boundary element in the chain complex of a certain matching complex, we obtain that z itself is a non-boundary. In particular, the homology class of z has order 3.

Let $M$ be a commutative cancellative monoid. The set $\Delta(M)$, which consists of all positive integers which are distances between consecutive factorization lengths of elements in $M$, is a widely studied object in the theory of nonunique factorizations. If $M$ is a Krull monoid with cyclic class group of order $n \ge 3$, then it is well-known that $\Delta(M) \subseteq \{1, \dots, n-2\}$. Moreover, equality holds for this containment when each class contains a prime divisor from $M$. In this note, we consider the question of determining which subsets of $\{1, \dots, n-2\}$ occur as the delta set of an individual element from $M$. We first prove for $x \in M$ that if $n - 2 \in \Delta(x)$, then $\Delta(x) = \{n-2\}$ (i.e., not all subsets of $\{1,\dots, n-2\}$ can be realized as delta sets of individual elements). We close by proving an Archimedean-type property for delta sets from Krull monoids with finite cyclic class group: for every natural number m, there exist a Krull monoid $M$ with finite cyclic class group such that $M$ has an element $x$ with $|\Delta(x)| \ge m$., Comment: 10 pages

We study the Cox ring and monoid of effective divisor classes of M¯¯¯¯0,n≅BlPn−3⁠, over a ring R. We provide a bijection between elements of the Cox ring, not divisible by any exceptional divisor section, and pure-dimensional singular simplicial complexes on {1,…,n−1} with weights in R∖{0} satisfying a zero-tension condition. This leads to a combinatorial criterion, satisfied by many triangulations of closed manifolds, for a divisor class to be among the minimal generators for the effective monoid. For classes obtained as the strict transform of quadrics, we present a complete classification of minimal generators, generalizing to all n the well-known Keel–Vermeire classes for n = 6. We use this classification to construct new divisors with interesting properties for all n≥7⁠.

A function assigning a composition to space-time points is called a compositional or simplicial field. These fields can be analyzed using the compositional analysis tools. In order to study compositions depending on space and/or time, reformulation and interpretation of traditional partial differential operators is required. These operators such as: partial derivatives, compositional gradient, directional derivative and divergence are of primary importance to state alternative models of processes as diffusion, advection and waves, from the compositional perspective. This kind of models, usually based on continuity of mass, circulation of a vector field along a curve and flux through surfaces, should be analyzed when compositional operators are used instead of the traditional gradient or divergence. This study is aimed at setting up the definitions, mathematical basis and interpretation of such operators.

In this paper we propose a model of random simplicial complexes with randomness in all dimensions. We start with a set of n vertices and retain each of them with probability p0; on the next step we connect every pair of retained vertices by an edge with probability p1, and then fill in every triangle in the obtained random graph with probability p2, and so on. As the result we obtain a random simplicial complex depending on the set of probability parameters (\(p_{0},p_{1},\ldots,p_{r}\)), 0 ≤ p i ≤ 1. The multi-parameter random simplicial complex includes both Linial-Meshulam and random clique complexes as special cases. Topological and geometric properties of this random simplicial complex depend on the whole set of parameters and their thresholds can be understood as convex subsets and not as single numbers as in all the previously studied models. We mainly focus on foundations and on containment properties of our multi-parameter random simplicial complexes. One may associate to any finite simplicial complex S a reduced density domain \(\tilde{\mu }(S) \subset \mathbf{R}^{r}\) (a convex domain) which fully controls information about the values of the multi-parameter for which the random complex contains S as a simplicial subcomplex. We also analyse balanced simplicial complexes and give positive and negative examples. We apply these results to describe dimension of a random simplicial complex.

In Duursma and Park (2010) [7] , the authors formulate new coset bounds for algebraic geometric codes. The bounds give improved lower bounds for the minimum distance of algebraic geometric codes as well as improved thresholds for algebraic geometric linear secret sharing schemes. The bounds depend on the delta set of a coset and on the choice of a sequence of divisors inside the delta set. In this paper we give general properties of delta sets and we analyze sequences of divisors supported in two points on Hermitian and Suzuki curves.

Higher Dimensional Automata (HDA) are topological models for the study of concurrency phenomena. The state space for an HDA is given as a pre-cubical complex in which a set of directed paths (d-paths) is singled out. The aim of this paper is to describe a general method that determines the space of directed paths with given end points in a pre-cubical complex as the nerve of a particular category. ¶ The paper generalizes the results from Raussen [Algebr. Geom. Topol. 10 (2010) 1683–1714; Appl. Algebra Engrg. Comm. Comput. 23 (2012) 59–84] in which we had to assume that the HDA in question arises from a semaphore model. In particular, important for applications, it allows for models in which directed loops occur in the processes involved.

We examine the Delta set of a cancellative and reduced atomic monoid S where every set of lengths of the factorizations of each element in S is bounded. In particular, we show the connection between the elements of �( S) and the Betti elements of S. We prove how the minimum and maximum element of �( S) can be determined using the Betti elements of S. This leads to a determination of when �( S) is a singleton. We then apply these results to the particular case where S is a numerical monoid that requires three generators. Conclusions are drawn in the cases where S has a unique minimal presentation, or has multiplicity three.

Generalizing the notion of the girth of a graph, a sequence of simplicial girths is assigned to each simplicial complex. Given a simplicial girth, lower bounds on higher simplicial girths are proven. When a simplicial girth is given and the Stanley---Reisner ring has a pure resolution, upper bounds on the number of vertices are proven.

In the computational geometry field, simplicial complexes have been used to describe an underlying geometric shape knowing a point cloud sampled on it. In this article, an adequate statistical framework is first proposed for the choice of a simplicial complex among a parametrized family. A least-squares penalized criterion is introduced to choose a complex, and a model selection theorem states how to select the "best" model, from a statistical point of view. This result gives the shape of the penalty, and then the "slope heuristics method" is used to calibrate the penalty from the data. Some experimental studies on simulated and real datasets illustrate the method for the selection of graphs and simplicial complexes of dimension two.

A finite poset $P$ is called "simplicial", if it has the smallest element $\hat{0}$, and every interval $[\hat{0}, x]$ is a boolean algebra. The face poset of a simplicial complex is a typical example. Generalizing the Stanley-Reisner ring of a simplicial complex, Stanley assigned the graded ring $A_P$ to $P$. This ring has been studied from both combinatorial and topological perspective. In this paper, we will give a concise description of a dualizing complex of $A_P$, which has many applications., Comment: 16 pages. Simplified the proof of the main theorem. Added the remark that a theorem of Murai-Terai also holds for simplicial posets.

It is proved that, for n ⩾ 4, the function u = u n (z), z = (z 1, …, z n ) ∈ ℂ n , defined by the equation u n + z 1 u n−1 + … + z n = 0 cannot be a branch of an entire algebraic function g on ℂ n that is a composition of entire algebraic functions depending on fewer than n − 1 variables and has the same discriminant set as u n . A key role is played by a description of holomorphic maps between configuration spaces of ℂ and ℂℙ1, which, in turn, involves Teichmuller spaces and new holomorphically combinatorial invariants of complex spaces.

Several recent articles have studied the structure of the delta set of a numerical monoid. We continue this work with the assumption that the generating set S chosen for the numerical monoid M is not necessarily minimal. We show that for certain choices of S, the resulting delta set can be made (in terms of cardinality) arbitrarily large or small. We close with a close analysis of the case where M =⟨n 1, n 2, in 1 + jn 2⟩for non-negative i and j.

In this article we treat a notion of continuity for a multi-valued function F and we compute the descriptive set-theoretic complexity of the set of all x for which F is continuous at x. We give conditions under which the latter set is either a G_\delta set or the countable union of G_\delta sets. Also we provide a counterexample which shows that the latter result is optimum under the same conditions. Moreover we prove that those conditions are necessary in order to obtain that the set of points of continuity of F is Borel i.e., we show that if we drop some of the previous conditions then there is a multi-valued function F whose graph is a Borel set and the set of points of continuity of F is not a Borel set. Finally we give some analogue results regarding a stronger notion of continuity for a multi-valued function. This article is motivated by a question of M. Ziegler in "Real Computation with Least Discrete Advice: A Complexity Theory of Nonuniform Computability with Applications to Linear Algebra", (submitted).

In this paper, an algorithm computing the terms E 1 and E 2 of the Bousfield-Kan spectral sequence of a 1-reduced simplicial set X is defined. In order to compute the ordinary description of the first level E 1, some elementary operations of Homological Algebra are sufficient. On the contrary, to compute the stage E 2 it is necessary to know more information about the previous groups, in particular with respect to the generators. This additional information can be reached by computing the effective homology of RX, RX being the free simplicial Abelian group generated by X. The algorithm to get the effective homology of RX from the effective homology of X can be considered the main result in our paper. Moreover, we include a combinatorial proof of the convergence of the Bousfield-Kan spectral sequence, based on the tapered nature of the stage E 1.

Path categories are defined, and their basic properties are described, for simplicial and cubical sets. A calculational method for describing the path category P(K) of a finite oriented simplicial complex K is introduced, which involves a finite 2-category which can be specified by generators and relations. This method specializes to higher dimensional automata via the triangulation functor from cubical to simplicial sets, and leads to calculations of their associated execution paths.