Distance Integral Complete Multipartite Graphs with s=5, 6

Let D(G) = (dij )n×n denote the distance matrix of a connected graph G with order n, where dij is equal to the distance between vertices vi and vj in G. A graph is called distance integral if all eigenvalues of its distance matrix are integers. In 2014, Yang and Wang gave a su?cient and necessary condition for complete r-partite graphs Kp1,p2,··· ,pr =Ka1·p1,a2·p2,··· ,as···ps to be distance integral and obtained such distance integral graphs with s = 1, 2, 3, 4. However distance integral complete multipartite graphs Ka1·p1,a2·p2,··· ,as·ps with s>4 have not been found. In this paper, we find and construct some infinite classes of these distance integral graphs Ka1·p1,a2·p2,··· ,as·ps with s = 5, 6. The problem of the existence of such distance integral graphs Ka1·p1,a2·p2,··· ,as·ps with arbitrarily large number s remains open.     

Normal pairs of noncommutative rings

This paper extends the concept of a normal pair from commutative ring theory to the context of a pair of (associative unital) rings. This is done by using the notion of integrality introduced by Atterton. It is shown that if $$R \subseteq S$$ are rings and $$D=(d_{ij})$$ is an $$n\times n$$ matrix with entries in S, then D is integral (in the sense of Atterton) over the full ring of $$n\times n$$ matrices with entries in R if and only if each $$d_{ij}$$ is integral over R. If $$R \subseteq S$$ are rings with corresponding full rings of $$n\times n$$ matrices $$R_n$$ and $$S_n$$, then $$(R_n,S_n)$$ is a normal pair if and only if (R, S) is a normal pair. Examples are given of a pair $$(\Lambda , \Gamma )$$ of noncommutative (in fact, full matrix) rings such that $$\Lambda \subset \Gamma $$ is (resp., is not) a minimal ring extension; it can be further arranged that $$(\Lambda , \Gamma )$$ is a normal pair or that $$\Lambda \subset \Gamma $$ is an integral extension.     

Distance spectra and distance energy of integral circulant graphs

The distance energy of a graph G is a recently developed energy-type invariant, defined as the sum of absolute values of the eigenvalues of the distance matrix of G. There was a vast research for the pairs and families of non-cospectral graphs having equal distance energy, and most of these constructions were based on the join of graphs. A graph is called circulant if it is Cayley graph on the circulant group, i.e. its adjacency matrix is circulant. A graph is called integral if all eigenvalues of its adjacency matrix are integers. Integral circulant graphs play an important role in modeling quantum spin networks supporting the perfect state transfer. In this paper, we characterize the distance spectra of integral circulant graphs and prove that these graphs have integral eigenvalues of distance matrix D. Furthermore, we calculate the distance spectra and distance energy of unitary Cayley graphs. In conclusion, we present two families of pairs (G₁,G₂) of integral circulant graphs with equal distance energy - in the first family G₁ is subgraph of G₂, while in the second family the diameter of both graphs is three.     

On realization of fusion rings from generalized Cartan matrices

The Casimir element of a fusion ring (R, B) gives rise to the so called Casimir matrix C of (R, B). This enables us to construct a generalized Cartan matrix D-C in the sense of Kac for a suitable diagonal matrix D. In this paper, we study some elementary properties of the Casimir matrix C and use them to realize certain fusion rings from the generalized Cartan matrix D-C of finite (resp. affine) type. It turns out that there exists a fusion ring with D-C being of finite (resp. affine) type if and only if D-C has only the form A₂ (resp. A₁⁽¹⁾. We also realize all fusion rings with D-C being a particular generalized Cartan matrix of indefinite type.     

CONSTRUCTION OF INDECOMPOSABLEDEFINITE HERMITIAN FORMS

ＣＯＮＳＴＲＵＣＴＩＯＮＯＦＩＮＤＥＣＯＭＰＯＳＡＢＬＥＤＥＦＩＮＩＴＥＨＥＲＭＩＴＩＡＮＦＯＲＭＳ￥ＺＨＵＦＵＺＵ（ＤｅｐａｒｔｍｅｌｌｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＥａｓｔＣｈｉｎａＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙｔＳｈａｎｇｈａｉ２０００６２，...     

A characterization of subclasses of semi-selfdecomposable distributions by stochastic integral representations

Characterizations of the classes of selfdecomposable (semi-selfdecomposable, resp.) by a stochastic integral with respect to Lévy process (semi-Lévy process, resp.) are known. A similar characterization for the Urbanik–Sato nested subclasses of the class of selfdecomposable distributions is also known. In this paper, a characterization of the nested subclasses of the class of semi-selfdecomposable distributions is given in terms of stochastic integral with respect to semi-Lévy process.     

Spectra of Cayley graphs of complex reflection groups

Renteln proved that the eigenvalues of the distance matrix of a Cayley graph of a real reflection group with respect to the set of all reflections are integral and provided a combinatorial formula for some such spectra. We prove the eigenvalues of the distance, adjacency, and codimension matrices of Cayley graphs of complex reflection groups with connection sets consisting of all reflections are integral and provide a combinatorial formula for the codimension spectra for a family of monomial complex reflection groups.     

GCD-Sets in Integral Domains. II

Abstract

Let D be an integral domain and S ≠ U(D) a saturated multiplicative subset of D. We say that S is a GCD-set (resp., factorial-set) if S is a GCD-monoid (resp., factorial-monoid) under the product of D and that S is a Marot-set if every integral ideal of D intersecting S is generated by a set of elements in S. In this paper, we study Marot GCD-sets and Marot factorial-sets.     

Notes on APN functions, semibiplanes and dimensional dual hyperovals

Two geometric objects, incidence graphs of semibiplanes and dimensional dual hyperovals, are respectively associated with APN and quadratic APN functions. From Proposition 2 (resp. Proposition 5), two APN (resp. quadratic APN) functions are CCZ (resp. extended affine) equivalent if and only if the associated graphs (resp. dimensional dual hyperovals) are isomorphic. The former graphs for almost bent functions are distance regular graphs by Proposition 4. The structures of automorphism groups of these geometric objects are investigated in Proposition 3 and Lemma 7. In particular, (Edel and Pott, Adv Math Commun 3:59–81 (2009), Question 2) is negatively answered.     

On Striking Identities About the Exponential Functionals of the Brownian Bridge and Brownian Motion

The negative moment of order 1, resp. of order 1/2, for the integral on (0, 1) of the exponential of α times the Brownian bridge, resp. the Brownian motion, does not depend on α. We give a simple explanation and a reinforcement of this property in the case of the Brownian bridge. We then discuss how different the case of the Brownian motion is.     

Pointwise fourier inversion and related eigenfunction expansions

Necessary and sufficient conditions are found for the convergence at a pre-assigned point of the spherical partial sums (resp. integrals) of the Fourier series (resp. integral) in the class of piecewise smooth functions on Euclidean space. These results carry over unchanged to spherical harmonic expansions, Fourier transforms on hyperbolic space, and Dirichlet eigenfunction expansions with respect to the Laplace operator on a class of Riemannian manifolds. © 1994 John Wiley & Sons, Inc.     

Convex sets of schur stable and stable matrices

We derive characterizations for the Schur stability and the stability of all convex combinations of kk≥2, given real square matrices. Moreover, we characterize these properties for the set r(A, B) (c(A,B), resp.) of square matrices whose rows (columns, resp.) are independent convex combinations of the rows (columns, resp.) of two real matrices A and B. Our results can be viewed as contributions to the problem of robustness of matrix properties. This paper continues our paper [4].     

Factorization in Integral Domains,IV

Let D be an integral domain such that every nonzero nonunit of D is a finite product of irreducible elements. In this article, we introduce and study several unifying concepts for the theory of nonunique factorization in D. They give a new way to measure, in some sense, how far an half-factorial domain (resp., bounded factorization domain, atomic domain) D is from being a UFD (resp., finite factorization domain, Cohen–Kaplansky domain) based on equivalence relations on the set of irreducible elements of D.     

两类六角系统的$A\alpha$-特征多项式和$A\alpha$-谱

2017年, Nikiforov首次提出研究图$G$的$A\alpha$-矩阵, 其定义为:$A\alpha(G)=\alpha D(G)+(1-\alpha)A(G) (\alpha\in [0,1])$, 其中$A(G)$和$D(G)$分别为图$G$的邻接矩阵和度对角矩阵. 设$F_n$和$M_n$分别为圈状六角系统和M\"{o}bius带状六角系统图. 根据循环矩阵的行列式和特征值, 本文首先给出图$F_n$和$M_n$的$A\alph$-特征多项式和$A\alpha$-谱, 进一步得到图$F_n$和$M_n$的$A\alpha$-能量的上界.     

Backward stochastic dynamics with a subdifferential operator and non-local parabolic variational inequalities

In this article, we provide the first systematic study on the unique existence of the solution of backward stochastic dynamical variational inequalities on a general complete filtered probability space. We also build up a comprehensive analysis of the correspondence between these stochastic variational inequalities (resp. backward stochastic dynamics) and the weak solutions (instead of viscosity ones due to the intrinsic non-local nature of the integral of the gradient involved) of a class of non-local parabolic variational inequalities (resp. parabolic partial differential equations), which is barely touched in the existing literature due to its unconventional setting.     

Six notes on chains of prime ideals

Summary The following topics are considered: saturated chains of prime ideals in quadratic (resp., simple, local, and level) integral extension domains of a local domain R; the heights of a prime ideal and its contraction in integral extension domains of R; and, the nonexistence of intermediate rings between R and finite integral extension domains of R that are minimal with spect to having certain properties.Research on this paper was supported in part by the National Science Foundation, Grant MCS 8001597.     

Extension of DFM and MORPH in representative directions to viscous effects via Prony series approach

In this paper we present an extension of the material models DFM and MORPH formulated in representative directions via an approach for viscohyperelasticity based on a Prony series resp. integral approach. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some remarks on one-sided regularity

A ring is said to be right (resp., left) regular-duo if every right (resp., left) regular element is regular. The structure of one-sided regular elements is studied in various kinds of rings, especially, upper triangular matrix rings over one-sided Ore domains. We study the structure of (one-sided) regular-duo rings, and the relations between one-sided regular-duo rings and related ring theoretic properties.     

广义Fitting模的扩张

In this paper, we study the closeness of strongly （∞）-hopfian properties under some constructions such as the ring of Morita context, direct products, triangular matrix, fraction ring etc. Also, we prove that if M[X] is strongly hopfian （resp. strongly co-hopfian） in R[X]-Mod, then M is strongly hopfian （resp. strongly co-hopfian） in R-Mod.