Let R be a commutative ring with nonzero identity and J(R) the Jacobson radical of R. The Jacobson graph of R, denoted by JR, is defined as the graph with vertex set RJ(R) such that two distinct vertices x and y are adjacent if and only if 1 − xy is not a unit of R. The genus of a simple graph G is the smallest nonnegative integer n such that G can be embedded into an orientable surface Sn. In this paper, we investigate the genus number of the compact Riemann surface in which JR can be embedded and explicitly determine all finite commutative rings R (up to isomorphism) such that JR is toroidal., Krishnan Selvakumar, Manoharan Subajini., and Obsahuje seznam literatury
For a finitely generated group, we study the relations between its rank, the maximal rank of its free quotient, called co-rank (inner rank, cut number), and the maximal rank of its free abelian quotient, called the Betti number. We show that any combination of the group’s rank, co-rank, and Betti number within obvious constraints is realized for some finitely presented group (for Betti number equal to rank, the group can be chosen torsion-free). In addition, we show that the Betti number is additive with respect to the free product and the direct product of groups. Our results are important for the theory of foliations and for manifold topology, where the corresponding notions are related with the cut-number (or genus) and the isotropy index of the manifold, as well as with the operations of connected sum and direct product of manifolds., Irina Gelbukh., and Obsahuje seznam literatury
The nullity of a graph G is the multiplicity of zero as an eigenvalue in the spectrum of its adjacency matrix. From the interlacing theorem, derived from Cauchy’s inequalities for matrices, a vertex of a graph can be a core vertex if, on deleting the vertex, the nullity decreases, or a Fiedler vertex, otherwise. We adopt a graph theoretical approach to determine conditions required for the identification of a pair of prescribed types of root vertices of two graphs to form a cut-vertex of unique type in the coalescence. Moreover, the nullity of subgraphs obtained by perturbations of the coalescence G is determined relative to the nullity of G. This has direct applications in spectral graph theory as well as in the construction of certain ipso-connected nano-molecular insulators., Didar A. Ali, John Baptist Gauci, Irene Sciriha, Khidir R. Sharaf., and Obsahuje seznam literatury
Let $(R,\mathfrak m)$ be a commutative Noetherian regular local ring of dimension $d$ and $I$ be a proper ideal of $R$ such that ${\rm mAss}_R(R/I)={\rm Assh}_R(I)$. It is shown that the $R$-module $H^{{\rm ht}(I)}_I(R)$ is $I$-cofinite if and only if
${\rm cd}(I,R)={\rm ht}(I)$. Also we present a sufficient condition under which this condition the $R$-module $H^i_I(R)$ is finitely generated if and only if it vanishes., Jafar A'zami, Naser Pourreza., and Obsahuje bibliografické odkazy
Let \Omega \in L^{s}\left ( S^{n-1} \right ) for s\geqslant 1 be a homogeneous function of degree zero and b a BMO function. The commutator generated by the Marcinkiewicz integral μΩ and b is defined by \left[ {b,{\mu _\Omega }} \right](f)(x) = {\left( {\int_0^\infty {{{\left| {\int_{\left| {x - y} \right| \leqslant t} {\frac{{\Omega (x - y)}}{{{{\left| {x - y} \right|}^{n - 1}}}}\left[ {b(x) - b(y)} \right]f(y){\text{d}}y} } \right|}^2}\frac{{{\text{d}}t}}{{{t^3}}}} } \right)^{1/2}}. In this paper, the author proves the \left (L^{p\left ( \cdot \right )}\left ( \mathbb{R}^{n} \right ),L^{p\left ( \cdot \right )}\left ( \mathbb{R}^{n} \right ) \right )-boundedness of the Marcinkiewicz integral operator μΩ and its commutator [b, μΩ ] when p(·) satisfies some conditions. Moreover, the author obtains the corresponding result about μΩ and [b, μΩ ] on Herz spaces with variable exponent., Hongbin Wang., and Obsahuje seznam literatury
The numerical range of an n × n matrix is determined by an n degree hyperbolic ternary form. Helton-Vinnikov confirmed conversely that an n degree hyperbolic ternary form admits a symmetric determinantal representation. We determine the types of Riemann theta functions appearing in the Helton-Vinnikov formula for the real symmetric determinantal representation of hyperbolic forms for the genus g = 1. We reformulate the Fiedler-Helton-Vinnikov formulae for the genus g = 0, 1, and present an elementary computation of the reformulation. Several examples are provided for computing the real symmetric matrices using the reformulation., Mao-Ting Chien, Hiroshi Nakazato., and Obsahuje seznam literatury
Let Ω \subset \mathbb{R}^{n} be a domain and let α < n − 1. We prove the Concentration-Compactness Principle for the embedding of the space W_{0}^{1}L^{n} log^{\alpha } L(Ω) into an Orlicz space corresponding to a Young function which behaves like (t^{n/n-1-\alpha }) for large t. We also give the result for the embedding into multiple exponential spaces. Our main result is Theorem 1.6 where we show that if one passes to unbounded domains, then, after the usual modification of the integrand in the Moser functional, the statement of the Concentration-Compactnes Principle is very similar to the statement in the case of a bounded domain. In particular, in the case of a nontrivial weak limit the borderline exponent is still given by the formula P: = \left( {1 - \left\| {\Phi (|\nabla u|)} \right\|_{L^1 (\mathbb{R}^n )} } \right)^{ - 1/(n - 1)} ., Robert Černý., and Obsahuje seznam literatury
The eigenvalues of graphs are related to many of its combinatorial properties. In his fundamental work, Fiedler showed the close connections between the Laplacian eigenvalues and eigenvectors of a graph and its vertex-connectivity and edge-connectivity. We present some new results describing the connections between the spectrum of a regular graph and other combinatorial parameters such as its generalized connectivity, toughness, and the existence of spanning trees with bounded degree., Sebastian M. Cioabă, Xiaofeng Gu., and Obsahuje seznam literatury
We investigate how one can detect the dualizing property for a chain complex over a commutative local Noetherian ring R. Our focus is on homological properties of contracting endomorphisms of R, e.g., the Frobenius endomorphism when R contains a field of positive characteristic. For instance, in this case, when R is F-finite and C is a semidualizing R-complex, we prove that the following conditions are equivalent: (i) C is a dualizing R-complex; (ii) C\sim RHom_{R}(^{n}R,C) for some n > 0; (iii) G_{C}-dim^{n}R < ∞ and C is derived RHom_{R}(^{n}R,C)-reflexive for some n > 0; and (iv) G_{C}-dim^{n}R < ∞ for infinitely many n > 0., Saeed Nasseh, Sean Sather-Wagstaff., and Obsahuje seznam literatury
We extend Rump’s verified method (S.Oishi, K.Tanabe, T.Ogita, S.M.Rump (2007)) for computing the inverse of extremely ill-conditioned square matrices to computing the Moore-Penrose inverse of extremely ill-conditioned rectangular matrices with full column (row) rank. We establish the convergence of our numerical verified method for computing the Moore-Penrose inverse. We also discuss the rank-deficient case and test some ill-conditioned examples. We provide our Matlab codes for computing the Moore-Penrose inverse., Yunkun Chen, Xinghua Shi, Yimin Wei., and Obsahuje seznam literatury