The topological structure of complete noncompact submanifolds in spheres

In this paper, we first show that δ-super stable complete noncompact minimal submanifolds in $S^{m+n}$ or $R^{m+n}$ with $\delta >{(\frac{m?1}{m})}^2$ admit no nontrivial $L^2$ harmonic 1-forms and have only one nonparabolic end, which generalizes Cao–Shen–Zhu's result in [2] on stable minimal hypersurface in $R^{m+1}$ and Lin's result in [13] on $\frac{m?1}{m}$-super stable minimal submanifolds in $R^{m+n}$. Second, we prove that the dimension of the space of $L^2$ harmonic p-forms on $M^m$ is zero or finite and there is only one nonparabolic end or finitely many nonparabolic ends of M under the assumptions on the Schrödinger operators involving the squared norm of the traceless second fundamental form.     

A note on conservative galaxies,Skolem systems,cyclic cycle decompositions,and Heffter arrays

The conservative number of a graph $G$ is the minimum positive integer $M$, such that $G$ admits an orientation and a labeling of its edges by distinct integers in $\{1,2,\dots ,M\}$, such that at each vertex of degree at least three, the sum of the labels on the in-coming edges is equal to the sum of the labels on the out-going edges. A graph is conservative if $M=\left|E\left(G\right)\right|$. It is worth noting that determining whether certain biregular graphs are conservative is equivalent to find integer Heffter arrays.In this work we show that the conservative number of a galaxy (a disjoint union of stars) of size $M$ is $M$ for $M\equiv 0$, $3(mod4)$, and $M+1$ otherwise. Consequently, given positive integers $m_1$, $m_2$, …, $m_n$ with $m_i\ge 3$ for $1\le i\le n$, we construct a cyclic $(m_1,m_2,\dots ,m_n)$-cycle system of infinitely many circulant graphs, generalizing a result of Bryant, Gavlas and Ling (2003). In particular, it allows us to construct a cyclic $(m_1,m_2,\dots ,m_n)$-cycle system of the complete graph $K_{2M+1}$, where $M={\sum }_{i=1}^n{m}_i$. Also, we prove necessary and sufficient conditions for the existence of a cyclic $(m_1,m_2,\dots ,m_n)$-cycle system of $K_{2M+2}?F$, where $F$ is a 1-factor. Furthermore, we give a sufficient condition for a subset of ${\mathbb{Z}}_v?\left\{0\right\}$ to be sequenceable.     

The de Rham Witt complex,cohomological kernels and pm-extensions in characteristic p

In earlier work the authors determined the Brauer kernel of extensions of degree p in characteristic $p>2$ where the Galois group is a semidirect product of order ps for $s|(p?1)$. This result is extended here and tools are developed to compute the cohomological kernels $H_{p^m}^{n+1}(E_m/F)$ for all $n\ge 0$ where $[E_m:F]=p^m$ and the Galois closure is a semidirect product of cyclic groups order $p^m$ and s where $s|(p?1)$. A six-term exact sequence describing the K-theory and cohomology of the extension is obtained. As an application it is shown that any F-division p-algebra of index $p^m$ split in $E_m$ is cyclic; a characteristic p analogue of a result of Vishne. The proofs use the de Rham Witt complex and Izhboldin groups, extending techniques developed earlier for the study of degree 4 extensions in characteristic two. The paper also provides background on the de Rham Witt Complex and Izhboldin groups difficult to track down in the literature.     

New concepts of regular and (highly) irregular vague graphs with applications

In this paper, some types of vague graphs are introdaced such as $d_m$-regular, $t{d}_m$-regular, $m$-highly irregular and $m$-highly totally irregular vague graphs are introduced and some properties of them are discussed. Comparative study between $d_m$-regular ($m$-highly irregular) vague graph and $t{d}_m$-regular ($m$-highly totally irregular) vague graph are done. In addition, $d_m$-regularity and $m$-highly irregularity on some vague graphs, which underlying crisp graphs are a cycle or a path is also studied. Finally, some applications of regular vague graphs are given for demonstration of fullerene molecules, road transport network and wireless multihop networks.     

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.     

Bijections between generalized Catalan families of types A and C

Motivated by the relation ${\mathsf{N}}^m\left(C_n\right)=(mn+1){\mathsf{N}}^m\left(A_{n?1}\right)$, holding for the $m$-generalized Catalan numbers of type $A$ and $C$, the connection between dominant regions of the $m$-Shi arrangement of type $A_{n?1}$ and $C_n$ is investigated. More precisely, it is explicitly shown how $mn+1$ copies of the set of dominant regions of the $m$-Shi arrangement of type $A_{n?1}$, biject onto the set of type $C_n$ such regions. This is achieved by exploiting two different viewpoints of the representative alcove of each region: the Shi tableau and the abacus diagram. In the same line of thought, a bijection between $mn+1$ copies of the set of $m$-Dyck paths of height $n$ and the set of $N?E$ lattice paths inside an $n\times mn$ rectangle is provided.     

Approximately locating an invisible agent in a graph with relative distance queries

In a pursuit evasion game on a finite, simple, undirected, and connected graph $G$, a first player visits vertices $m_1,m_2,\dots$ of $G$, where $m_{i+1}$ is in the closed neighborhood of $m_i$ for every $i$, and a second player probes arbitrary vertices $c_1,c_2,\dots$ of $G$, and learns whether or not the distance between $c_{i+1}$ and $m_{i+1}$ is at most the distance between $c_i$ and $m_i$. Up to what distance $d$ can the second player determine the position of the first? For trees of bounded maximum degree and grids, we show that $d$ is bounded by a constant. We conjecture that $d=O(logn)$ for every graph $G$ of order $n$, and show that $d=0$ if $m_{i+1}$ may differ from $m_i$ only if $i$ is a multiple of some sufficiently large integer.     

On super 2-restricted and 3-restricted edge-connected vertex transitive graphs

Let $G=(V\left(G\right),E\left(G\right))$ be a simple connected graph and $F?E\left(G\right)$. An edge set $F$ is an $m$-restricted edge cut if $G?F$ is disconnected and each component of $G?F$ contains at least $m$ vertices. Let ${\lambda }^{\left(m\right)}\left(G\right)$ be the minimum size of all $m$-restricted edge cuts and ${\xi }_m\left(G\right)=min\{\left|\omega \left(U\right)\right|:\left|U\right|=m\text{ and }G\left[U\right]\text{ is connected}\}$, where $\omega \left(U\right)$ is the set of edges with exactly one end vertex in $U$ and $G\left[U\right]$ is the subgraph of $G$ induced by $U$. A graph $G$ is optimal-${\lambda }^{\left(m\right)}$ if ${\lambda }^{\left(m\right)}\left(G\right)={\xi }_m\left(G\right)$. An optimal-${\lambda }^{\left(m\right)}$ graph is called super $m$-restricted edge-connected if every minimum $m$-restricted edge cut is $\omega \left(U\right)$ for some vertex set $U$ with $\left|U\right|=m$ and $G\left[U\right]$ being connected. In this note, we give a characterization of super 2-restricted edge-connected vertex transitive graphs and obtain a sharp sufficient condition for an optimal-${\lambda }^{\left(3\right)}$ vertex transitive graph to be super 3-restricted edge-connected. In particular, a complete characterization for an optimal-${\lambda }^{\left(2\right)}$ minimal Cayley graph to be super 2-restricted edge-connected is obtained.     

Explicit factorizations of generalized Dickson polynomials of order 2m via generalized cyclotomic polynomials over finite fields

We derive explicit factorizations of generalized cyclotomic polynomials of order $2^m$ and generalized Dickson polynomials of the first kind of order $2^m$ over finite field $F_q$.     

Rate of decay of s-numbers

For an operator $T\in B(X,Y)$, we denote by $a_m\left(T\right)$, $c_m\left(T\right)$, $d_m\left(T\right)$, and $t_m\left(T\right)$ its approximation, Gelfand, Kolmogorov, and absolute numbers, respectively. We show that, for any infinite-dimensional Banach spaces $X$ and $Y$, and any sequence ${\alpha }_m\searrow 0$, there exists $T\in B(X,Y)$ for which the inequality $3{\alpha }_{?m/6?}?a_m\left(T\right)?max\{c_m\left(t\right),d_m\left(T\right)\}?min\{c_m\left(t\right),d_m\left(T\right)\}?t_m\left(T\right)?{\alpha }_m/9$ holds for every $m\in \mathbb{N}$. Similar results are obtained for other $s$-scales.     

On the additive cyclic structure of quasi-cyclic codes

An index $?$, length $m?$ quasi-cyclic code can be viewed as a cyclic code of length $m$ over the field ${\mathbb{F}}_{q^{?}}$ via a basis of the extension ${\mathbb{F}}_{q^{?}}∕{\mathbb{F}}_q$. However, this cyclic code is only linear over ${\mathbb{F}}_q$, making it an additive cyclic code, or an ${\mathbb{F}}_q$-linear cyclic code, over the alphabet ${\mathbb{F}}_{q^{?}}$. This approach was recently used in Shi et al. (2017) [16] to study a class of quasi-cyclic codes, and more importantly in Shi et al. (2017) [17] to settle a long-standing question on the asymptotic performance of cyclic codes. Here, we answer one of the problems posed in these two articles, and characterize those quasi-cyclic codes which have ${\mathbb{F}}_{q^{?}}$-linear cyclic images under a basis of the extension ${\mathbb{F}}_{q^{?}}∕{\mathbb{F}}_q$. Our characterizations are based on the module structure of quasi-cyclic codes, as well as on their CRT decompositions into constituents. In the case of a polynomial basis, we characterize the constituents by using the theory of invariant subspaces of operators. We also observe that analogous results extend to the case of quasi-twisted codes.     

Sums of powers of Catalan triangle numbers

In this paper, we consider combinatorial numbers ${\left(C_{m,k}\right)}_{m\ge 1,k\ge 0}$, mentioned as Catalan triangle numbers where $C_{m,k}?\left(\genfrac{}{}{0ex}{}{m?1}{k}\right)?\left(\genfrac{}{}{0ex}{}{m?1}{k?1}\right)$. These numbers unify the entries of the Catalan triangles $B_{n,k}$ and $A_{n,k}$ for appropriate values of parameters $m$ and $k$, i.e., $B_{n,k}=C_{2n,n?k}$ and $A_{n,k}=C_{2n+1,n+1?k}$. In fact, these numbers are suitable rearrangements of the known ballot numbers and some of these numbers are the well-known Catalan numbers $C_n$ that is $C_{2n,n?1}=C_{2n+1,n}=C_n$.We present identities for sums (and alternating sums) of $C_{m,k}$, squares and cubes of $C_{m,k}$ and, consequently, for $B_{n,k}$ and $A_{n,k}$. In particular, one of these identities solves an open problem posed in Gutiérrez et al. (2008). We also give some identities between ${\left(C_{m,k}\right)}_{m\ge 1,k\ge 0}$ and harmonic numbers ${\left(H_n\right)}_{n\ge 1}$. Finally, in the last section, new open problems and identities involving ${\left(C_n\right)}_{n\ge 0}$ are conjectured.