An extension of Wilf’s conjecture to affine semigroups

Wilf Conjecture on numerical semigroups is a question posed by Wilf in 1978 and is an inequality connecting the Frobenius number, embedding dimension and the genus of the semigroup. The conjecture is still open in general. We prove that this Wilf inequality is preserved under gluing of numerical semigroups. If the numerical semigroups minimally generated by [Formula: see text] and [Formula: see text] satisfy the Wilf inequality, then so does their gluing which is minimally generated by [Formula: see text]. We discuss the extended Wilf’s Conjecture in higher dimensions for certain affine semigroups and prove an analogous result.

A numerical semigroups is a submonoid of non-negative integers whose complement is finite. Generalizing this concept a generalized numerical semigroup is a submonoid of $\mathbb{N}^d$ whose the complement is finite. In the context of numerical semigroups, the study about Wilf's Conjecture brought new ways of thinking about monoids and remain open problems. In this work we will show a generalization to the Wilf's Conjecture and we proving it to several families of generalized numerical semigroups and show their relationship with the natural generalization proposed by García-García, Marín-Aragón and Vigneron-Tenorio.

In this paper we introduce a particular semigroup transform [Formula: see text] that fixes the invariants involved in Wilf's conjecture, except the embedding dimension. It also allows one to arrange the set of non-ordinary and non-irreducible numerical semigroups in a family of rooted trees. In addition, we study another transform, having similar features, that has been introduced by Bras-Amorós, and we make a comparison of them. In particular, we study the behavior of the embedding dimension under the action of such transforms, providing some consequences concerning Wilf's conjecture.

An affine semigroup is a finitely generated subsemigroup of $(\mathbb Z_{\ge 0}^d, +)$, and a numerical semigroup is an affine semigroup with $d = 1$. A growing body of recent work examines shifted families of numerical semigroups, that is, families of numerical semigroups of the form $M_n = \langle n + r_1, \ldots, n + r_k \rangle$ for fixed $r_1, \ldots, r_k$, with one semigroup for each value of the shift parameter $n$. It has been shown that within any shifted family of numerical semigroups, the size of any minimal presentation is bounded (in fact, this size is eventually periodic in $n$). In this paper, we consider shifted families of affine semigroups, and demonstrate that some, but not all, shifted families of 4-generated affine semigroups have arbitrarily large minimal presentations.

Topology optimization (TO) aims at defining the ideal layout of structures in order to minimize the use of material within a given design space. Such goal is pursued by applying an optimization strategy, where the possible solutions are the feasible geometries, and the functional to minimize is the mass structure. Obtaining high performance in terms of mass reduction depends on many factors. The most well-known and widespread topological optimization approaches can be divided into two main categories: microscopic and macroscopic. In microscopic approaches, such as SIMP and BESO, the feasible domain available for the solid material is divided in a finite number of discrete elements (FEM). The density of the material of each element represents the design variables, which may vary with continuity in SIMP method (density approach) or being binary in BESO Method (evolutionary approach). In macroscopic approaches, such as Level Set, the parameters describing the evolution of the boundaries of the solid domain of the structure are the design variables. Again, a discretization of the design domain is required, even if the implementation of this strategy in the optimization algorithm doesn’t intrinsically need the decomposition in a finite number of elements.

We provide algorithms for performing computations in generalized numerical semigroups, that is, submonoids of [Formula: see text] with finite complement in [Formula: see text]. These semigroups are affine semigroups, which in particular implies that they are finitely generated. For a given finite set of elements in [Formula: see text] we show how to deduce if the monoid spanned by this set is a generalized numerical semigroup and, if so, we calculate its set of gaps. Also, given a finite set of elements in [Formula: see text] we can determine if it is the set of gaps of a generalized numerical semigroup and, if so, compute the minimal generators of this monoid. We provide a new algorithm to compute the set of all generalized numerical semigroups with a prescribed genus (the cardinality of their sets of gaps). Its implementation allowed us to compute (for various dimensions) the number of numerical semigroups of higher genus than has previously been computed.

Nonlinear analysis of RR interval in euthymic bipolar disorder

The method of minimization of complex systems attractor representation through sequences of distances along phase trajectory by means of orientation sequence introduction is proposed. Orientation sequence is defined from support hyperplane determination in multi-dimensional phase space. This method is suitable for arbitrary construction of phase trajectories. It has been shown that, in such a way, phase trajectories can be represented by means of a set possibly containing a minimal quantity of elements. The proposed method is developed for an attractor reconstructed by the Takens method. In this paper, we develop the localized topological analysis of attractor trajectories by means of introducing asymptotic measures. As the exact topological stabilization proved to be impossible for complex systems attractors, we can consider only asymptotic convergence to it. Numerical experiments with time series calculated from differential delay equations show good convergence and accuracy with proposed methods. On the basis of proposed methods, some statistical characteristics of the investigated attractor (minimal embedding dimension and normalized deviation from exact topological stabilization) have been defined.

Estimating the number of reaction intermediates for formaldehyde oxidation by analysis of the chaotic behavior

Local exponential divergence plot and optimal embedding of a chaotic time series

The theoretical substantiation of a locally topological method for defining a minimum attractor embedding dimension on the basis of state-space method of a dynamic system description is supposed. The numerical modeling of various types of discrete dynamic systems on the computer is carried out with the purpose of verification of theoretical principles, which are the underlying principles for a minimum dimension computation method of such system attractor embedding.

Questions concerning the lengths of factorizations into irreducible elements in numerical monoids have gained much attention in the recent literature. In this note, we show that a numerical monoid has an element with two different irreducible factorizations of the same length if and only if its embedding dimension is greater than two. We find formulas in embedding dimension three for the smallest element with two different irreducible factorizations of the same length and the largest element whose different irreducible factorizations all have distinct lengths. We show that these formulas do not naturally extend to higher embedding dimensions.

A numerical semigroup is perfect if it does not have isolated gaps. In this paper we will order the perfect numerical semigroups with a fixed multiplicity. This ordering allows us to give an algorithm procedure to obtain them. We also study the perfect monoid, which is a subset of $\N$ that can be expressed as an intersection of perfect numerical semigroups, and we present the perfect monoid generated by a subset of $\N$. We give an algorithm to calculate it. We study the perfect closure of a numerical semigroup, as well as the perfect numerical semigroup with maximal embedding dimension, in particular Arf and saturated numerical semigroups.

The purpose of this paper is to give a new proof of A. H. Stone's Theorem that every metric space is paracompact.' We feel that the combinatorics involved are of some interest in their own right and that our method may be useful in other contexts. The statement that a space X is paracompact means that every open cover of X has a locally finite open refinement that covers X. (An open refinement of a collection of open sets is a new collection of open sets such that each set in the new collection is contained in some set of the original collection. A cover is said to be locally finite if each point has a neighborhood intersecting only a finite number of elements in the cover.) PROOF. (A) We shall first get a refinement such that each point is covered only a finite number of times. To do this we will order the elements of the given cover. Call the cover { C,} . Let R(x, m) be the open circle of radius 1/2m about x. A chosen circle (with respect to C,) is a circle R(x, n.+1) such that (i) R(x, n) CC,C,, (ii) n. is the smallest integer for which (i) holds and (iii) R(x, n,)CCC for some f3 m then y<?. Hence the C, that contain x form a descending sequence of ordinals and therefore only a finite number are distinct.

In this paper we describe an algorithm visiting all numerical semigroups up to a given genus using a well-suited representation. The interest of this algorithm is that it fits particularly well the architecture of modern computers allowing very large optimizations: we obtain the number of numerical semigroups of genus g ⩽ 67 g\leqslant 67 and we confirm the Wilf conjecture for g ⩽ 60 g\leqslant 60 .