Systems of sets of lengths of Puiseux monoids

In this paper we study the system of sets of lengths of non-finitely generated atomic Puiseux monoids (a Puiseux monoid is an additive submonoid of ${\mathbb{Q}}_{\ge 0}$). We begin by presenting a BF-monoid M with full system of sets of lengths, which means that for each subset S of ${\mathbb{Z}}_{\ge 2}$ there exists an element $x\in M$ whose set of lengths $\mathsf{L}(x)$ is S. It is well known that systems of sets of lengths do not characterize numerical monoids. Here, we prove that systems of sets of lengths do not characterize non-finitely generated atomic Puiseux monoids. In a recent paper, Geroldinger and Schmid found the intersection of systems of sets of lengths of numerical monoids. Motivated by this, we extend their result to the setting of atomic Puiseux monoids. Finally, we relate the sets of lengths of the Puiseux monoid $P=〈1/p|p\text{is prime}〉$ with the Goldbach's conjecture; in particular, we show that $\mathsf{L}(2)$ is precisely the set of Goldbach's numbers.     

Spectra and Laplacian spectra of arbitrary powers of lexicographic products of graphs

Consider two graphs $G$ and $H$. Let $H^k\left[G\right]$ be the lexicographic product of $H^k$ and $G$, where $H^k$ is the lexicographic product of the graph $H$ by itself $k$ times. In this paper, we determine the spectrum of $H^k\left[G\right]$ and $H^k$ when $G$ and $H$ are regular and the Laplacian spectrum of $H^k\left[G\right]$ and $H^k$ for $G$ and $H$ arbitrary. Particular emphasis is given to the least eigenvalue of the adjacency matrix in the case of lexicographic powers of regular graphs, and to the algebraic connectivity and the largest Laplacian eigenvalues in the case of lexicographic powers of arbitrary graphs. This approach allows the determination of the spectrum (in case of regular graphs) and Laplacian spectrum (for arbitrary graphs) of huge graphs. As an example, the spectrum of the lexicographic power of the Petersen graph with the googol number (that is, 10¹⁰⁰ ) of vertices is determined. The paper finishes with the extension of some well known spectral and combinatorial invariant properties of graphs to its lexicographic powers.     

Behavior of zeros of polynomials of near best approximation

The purpose of this paper is to study the asymptotic behavior of the zeros of polynomials of near best approximation to continuous functions f on a compact set E in the case when f is analytic on the interior of E but not everywhere on the boundary. For example, suppose E is a finite union of compact intervals of the real line and f is a continuous function on E, but is not analytic on E; then we show (cf. Corollary 2.2) that every point of E is a limit point of zeros of the polynomials of best uniform approximation to f on E. This fact answers a question posed by P. Borwein who showed that, for the case when E is a single interval and f is real-valued, then the above hypotheses on f imply that at least one point of E is the limit point of zeros of such polynomials.     

Independence number of products of Kneser graphs

We study the independence number of a product of Kneser graph $K(n,k)$ with itself, where we consider all four standard graph products. The cases of the direct, the lexicographic and the strong product of Kneser graphs are not difficult (the formula for $\alpha (K(n,k)⊠K(n,k))$ is presented in this paper), while the case of the Cartesian product of Kneser graphs is much more involved. We establish a lower bound and an upper bound for the independence number of $K(n,2)\square K(n,2)$, which are asymptotically tending to $n^3∕3$ and $3n^3∕8$, respectively. The former is obtained by a construction, which differs from the standard diagonalization procedure, while for the upper bound the $\ell$-independence number of Kneser graphs can be applied. We also establish some constructions in odd graphs $K(2k+1,k)$, which give a lower bound for the 2-independence number of these graphs, and prove that two such constructions give the same lower bound as a previously known one. Finally, we consider the $s$-stable Kneser graphs $K{(ks+1,k)}_{s-stab}$, derive a formula for their $\ell$-independence number, and give the exact value of the independence number of the Cartesian square of $K{(ks+1,k)}_{s-stab}$.     

Rigidity of harmonic maps of maximum rank

Thehomotopical rank of a mapf:M →N is, by definition, min{dimg(M) ¦g homotopic tof}. We give upper bounds for this invariant whenM is compact Kähler andN is a compact discrete quotient of a classical symmetric space, e.g., the space of positive definite matrices. In many cases the upper bound is sharp and is attained by geodesic immersions of locally hermitian symmetric spaces. An example is constructed (Section 9) to show that there do, in addition, exist harmonic maps of quite a different character. A byproduct is construction of an algebraic surface with large and interesting fundamental group. Finally, a criterion for lifting harmonic maps to holomorphic ones is given, as is a factorization theorem for representations of the fundamental group of a compact Kähler manifold. The technique for the main result is a combination of harmonic map theory, algebra, and combinatorics; it follows the path pioneered by Siu in his ridigity theorem and later extended by Sampson.     

Unimodality of independence polynomials of the incidence product of graphs

Given two graphs $G$ and $H$, assume that $V\left(G\right)=\{v_1,v_2,\dots ,v_n\}$ and $U$ is a subset of $V\left(H\right)$. We introduce a new graph operation called the incidence product, denoted by $G\odot H^U$, as follows: insert a new vertex into each edge of $G$, then join with edges those pairs of new vertices on adjacent edges of $G$. Finally, for every vertex $v_i\in V\left(G\right)$, replace it by a copy of the graph $H$ and join every new vertex being adjacent to $v_i$ to every vertex of $U$. It generalizes the line graph operation. We prove that the independence polynomial

$I\left(G\odot H^U;x\right)=I^n(H;x)M\left(G;\frac{x{I}^2(H?U;x)}{I^2(H;x)}\right),$

where $M(G;x)$ is its matching polynomial. Based on this formula, we show that the incidence product of some graphs preserves symmetry, unimodality, reality of zeros of independence polynomials. As applications, we obtain some graphs so-formed having symmetric and unimodal independence polynomials. In particular, the graph $Q\left(G\right)$ introduced by Cvetkovi?, Doob and Sachs has a symmetric and unimodal independence polynomial.     

On sums of squares of k-nomials

In 2005, Boman et al. introduced the concept of factor width for a real symmetric positive semidefinite matrix. This is the smallest positive integer k for which the matrix A can be written as $A=V{V}^T$ with each column of V containing at most k non-zeros. The cones of matrices of bounded factor width give a hierarchy of inner approximations to the PSD cone. In the polynomial optimization context, a Gram matrix of a polynomial having factor width k corresponds to the polynomial being a sum of squares of polynomials of support at most k. Recently, Ahmadi and Majumdar [1], explored this connection for case $k=2$ and proposed to relax the reliance on polynomials that are sums of squares in semidefinite programming to polynomials that are sums of binomial squares In this paper, we prove some results on the geometry of the cones of matrices with bounded factor widths and their duals, and use them to derive new results on the limitations of certificates of nonnegativity of quadratic forms by sums of k-nomial squares using standard multipliers. In particular we show that they never help for symmetric quadratics, for any quadratic if $k=2$, and any quaternary quadratic if $k=3$. Furthermore we give some evidence that those are a complete list of such cases.     

Super-multiplicativity and a lower bound for the decay of the signature of a path of finite length

For a path of length $L>0$, if for all $n\ge 1$, we multiply the n-th term of the signature by $n!L^{?n}$, we say that the resulting signature is ‘normalised’. It has been established (T. J. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths, Springer, 2007) that the norm of the n-th term of the normalised signature of a bounded-variation path is bounded above by 1. In this article, we discuss the super-multiplicativity of the norm of the signature of a path with finite length, and prove by Fekete's lemma the existence of a non-zero limit of the n-th root of the norm of the n-th term in the normalised signature as n approaches infinity.     

The structure of DGA resolutions of monomial ideals

Let $I?\mathbb{k}[x_1,\dots ,x_n]$ be a squarefree monomial ideal in a polynomial ring. In this paper we study multiplications on the minimal free resolution $\mathbb{F}$ of $\mathbb{k}[x_1,\dots ,x_n]/I$. In particular, we characterize the possible vectors of total Betti numbers for such ideals which admit a differential graded algebra (DGA) structure on $\mathbb{F}$. We also show that under these assumptions the maximal shifts of the graded Betti numbers are subadditive.On the other hand, we present an example of a strongly generic monomial ideal which does not admit a DGA structure on its minimal free resolution. In particular, this demonstrates that the Hull resolution and the Lyubeznik resolution do not admit DGA structures in general.Finally, we show that it is enough to modify the last map of $\mathbb{F}$ to ensure that it admits the structure of a DG algebra.     

VC-Dimension of Sets of Permutations

VC-dimension of a set of permutations to be the maximal k such that there exist distinct that appear in A in all possible linear orders, that is, every linear order of is equivalent to the standard order of for at least one permutation . In other words, the VC-dimension of A is the maximal k such that for some the restriction of A to contains all possible linear orders. This is analogous to the VC-dimension of a set of strings. Our main result is that there exists a universal constant C such that any set of permutations with VC-dimension 2 is of size . This is analogous to Sauer's lemma for the case of VC-dimension 2. One corollary of our main result is that any acyclic set of linear orders of is of size , (a set A of linear orders on is called acyclic if no 3 elements appear in A in all 3 orders (i,j,k), (k,i,j) and (j,k,i)). The size of the largest acyclic set of linear orders has interested researchers for many years because it is the largest number of linear orders of n alternatives such that the following is always satisfied: if each one of a set of voters chooses one of these orders as his preference then the majority relation between each two alternatives is transitive. Received August 6, 1998     

On estimating of the number of constituents of a finite mixture of continuous distributions

Summary Suppose thatH is a mixture of distributions for a given familyF A necessary and sufficient condition is obtained under whichH is, in fact, a finite mixture. An estimator of the number of distributions constituting the mixture is proposed assuming that the mixture is finite and its asymptotic properties are investigated.     

Denseness of volatile and nonvolatile sequences of functions

In a recent paper by Jonasson and Steif, definitions to describe the volatility of sequences of Boolean functions, $f_n:{\{?1,1\}}^n\to \{?1,1\}$ were introduced. We continue their study of how these definitions relate to noise stability and noise sensitivity. Our main results are that the set of volatile sequences of Boolean functions is a natural way “dense” in the set of all sequences of Boolean functions, and that the set of non-volatile Boolean sequences is not “dense” in the set of noise stable sequences of Boolean functions.     

Classification of bm-Nambu structures of top degree

In this paper, we classify $b^m$-Nambu structures via $b^m$-cohomology. The complex of $b^m$-forms is an extension of the De Rham complex, which allows us to consider singular forms. $b^m$-Cohomology is well understood thanks to Scott (2016) [12], and it can be expressed in terms of the De Rham cohomology of the manifold and of the critical hypersurface using a Mazzeo–Melrose-type formula. Each of the terms in $b^m$-Mazzeo–Melrose formula acquires a geometrical interpretation in this classification. We also give equivariant versions of this classification scheme.     

On the ordinariness of coverings of stable curves

In the present paper, we study the ordinariness of coverings of stable curves. Let $f:Y\to X$ be a morphism of stable curves over a discrete valuation ring R with algebraically closed residue field of characteristic $p>0$. Write S for Spec R and η (resp. s) for the generic point (resp. closed point) of S. Suppose that the generic fiber $X_{\eta }$ of X is smooth over η, that the morphism $f_{\eta }:Y_{\eta }\to X_{\eta }$ over η on the generic fiber induced by f is a Galois étale covering (hence $Y_{\eta }$ is smooth over η too) whose Galois group is a solvable group G, that the genus of the normalization of each irreducible component of the special fiber $X_s$ is ≥2, and that $Y_s$ is ordinary. Then we have that the morphism $f_s:Y_s\to X_s$ over s induced by f is an admissible covering. This result extends a result of M. Raynaud concerning the ordinariness of coverings to the case where $X_s$ is a stable curve. If, moreover, we suppose that G is a p-group, and that the p-rank of the normalization of each irreducible component of $X_s$ is ≥2, we can give a numerical criterion for the admissibility of $f_s$.     

Well-posedness of phase-lock equations of superconductivity

In this article we give an affirmative answer to an open question raised in [M. Zhan, Phase-lock equations and its connections to Ginzburg–Landau equations of superconductivity, J. Nonlinear Anal. 42 (2000) 1063–1075]. There, the author introduced the phase-lock equations to model superconductivity phenomena. We showed that the phase-lock equations have unique solutions with $L^2$ initial data for space dimensions $n=2,3$.