Resolvable 4-cycle group divisible designs with two associate classes: Part size even

Let ${\lambda }_1{K}_a$ denote the graph on a vertices with ${\lambda }_1$ edges between every pair of vertices. Take p copies of this graph ${\lambda }_1{K}_a$, and join each pair of vertices in different copies with ${\lambda }_2$ edges. The resulting graph is denoted by $K(a,p;{\lambda }_1,{\lambda }_2)$, a graph that was of particular interest to Bose and Shimamoto in their study of group divisible designs with two associate classes. The existence of z-cycle decompositions of this graph have been found when $z\in \{3,4\}$. In this paper we consider resolvable decompositions, finding necessary and sufficient conditions for a 4-cycle factorization of $K(a,p;{\lambda }_1,{\lambda }_2)$ (when ${\lambda }_1$ is even) or of $K(a,p;{\lambda }_1,{\lambda }_2)$ minus a 1-factor (when ${\lambda }_1$ is odd) whenever a is even.     

On maximizing the fundamental frequency of the complement of an obstacle

Let $\mathrm{\Omega }?{\mathbb{R}}^n$ be a bounded domain satisfying a Hayman-type asymmetry condition, and let D be an arbitrary bounded domain referred to as an “obstacle”. We are interested in the behavior of the first Dirichlet eigenvalue ${\lambda }_1(\mathrm{\Omega }?(x+D))$.First, we prove an upper bound on ${\lambda }_1(\mathrm{\Omega }?(x+D))$ in terms of the distance of the set $x+D$ to the set of maximum points $x_0$ of the first Dirichlet ground state ${?}_{{\lambda }_1}>0$ of Ω. In short, a direct corollary is that if

(1)${\mu }_{\mathrm{\Omega }}:=\underset{x}{\max }?{\lambda }_1(\mathrm{\Omega }?(x+D))$

is large enough in terms of ${\lambda }_1(\mathrm{\Omega })$, then all maximizer sets $x+D$ of ${\mu }_{\mathrm{\Omega }}$ are close to each maximum point $x_0$ of ${?}_{{\lambda }_1}$.Second, we discuss the distribution of ${?}_{{\lambda }_1(\mathrm{\Omega })}$ and the possibility to inscribe wavelength balls at a given point in Ω.Finally, we specify our observations to convex obstacles D and show that if ${\mu }_{\mathrm{\Omega }}$ is sufficiently large with respect to ${\lambda }_1(\mathrm{\Omega })$, then all maximizers $x+D$ of ${\mu }_{\mathrm{\Omega }}$ contain all maximum points $x_0$ of ${?}_{{\lambda }_1(\mathrm{\Omega })}$.     

An exact upper bound for sums of element orders in non-cyclic finite groups

Denote the sum of element orders in a finite group G by $\psi (G)$ and let $C_n$ denote the cyclic group of order n. Suppose that G is a non-cyclic finite group of order n and q is the least prime divisor of n. We proved that $\psi (G)\le \frac{7}{11}\psi (C_n)$ and $\psi (G)<\frac{1}{q?1}\psi (C_n)$. The first result is best possible, since for each $n=4k$, k odd, there exists a group G of order n satisfying $\psi (G)=\frac{7}{11}\psi (C_n)$ and the second result implies that if G is of odd order, then $\psi (G)<\frac{1}{2}\psi (C_n)$. Our results improve the inequality $\psi (G)<\psi (C_n)$ obtained by H. Amiri, S.M. Jafarian Amiri and I.M. Isaacs in 2009, as well as other results obtained by S.M. Jafarian Amiri and M. Amiri in 2014 and by R. Shen, G. Chen and C. Wu in 2015. Furthermore, we obtained some $\psi (G)$-based sufficient conditions for the solvability of G.     

Translational absolute continuity and Fourier frames on a sum of singular measures

A finite Borel measure μ in ${\mathbb{R}}^d$ is called a frame-spectral measure if it admits an exponential frame (or Fourier frame) for $L^2(\mu )$. It has been conjectured that a frame-spectral measure must be translationally absolutely continuous, which is a criterion describing the local uniformity of a measure on its support. In this paper, we show that if any measures ν and λ without atoms whose supports form a packing pair, then $\nu ?\lambda +{\delta }_t?\nu$ is translationally singular and it does not admit any Fourier frame. In particular, we show that the sum of one-fourth and one-sixteenth Cantor measure ${\mu }_4+{\mu }_{16}$ does not admit any Fourier frame. We also interpolate the mixed-type frame-spectral measures studied by Lev and the measure we studied. In doing so, we demonstrate a discontinuity behavior: For any anticlockwise rotation mapping $R_{\theta }$ with $\theta \ne \pm \pi /2$, the two-dimensional measure ${\rho }_{\theta }(?):=({\mu }_4\times {\delta }_0)(?)+({\delta }_0\times {\mu }_{16})(R_{\theta }^{?1}?)$, supported on the union of x-axis and $y=(\cot ?\theta )x$, always admit a Fourier frame. Furthermore, we can find ${\{e^{2\pi i〈\lambda ,x〉}\}}_{\lambda \in {\mathrm{\Lambda }}_{\theta }}$ such that it forms a Fourier frame for ${\rho }_{\theta }$ with frame bounds independent of θ. Nonetheless, ${\rho }_{\pm \pi /2}$ does not admit any Fourier frame.     

Upper bounds for the sum of Laplacian eigenvalues of graphs

Let G be a graph with n vertices and $e(G)$ edges, and let ${\mu }_1(G)?{\mu }_2(G)???{\mu }_n(G)=0$ be the Laplacian eigenvalues of G. Let $S_k(G)={\sum }_{i=1}^k{\mu }_i(G)$, where $1?k?n$. Brouwer conjectured that $S_k(G)?e(G)+\left(\genfrac{}{}{0ex}{}{k+1}{2}\right)$ for $1?k?n$. It has been shown in Haemers et al. [7] that the conjecture is true for trees. We give upper bounds for $S_k(G)$, and in particular, we show that the conjecture is true for unicyclic and bicyclic graphs.     

Classification of some 3-subgroups of the finite groups of Lie type E6

We consider the finite exceptional group of Lie type $G=E_6^{\epsilon }(q)$ (universal version) with $3|q?\epsilon$, where $E_6^{+1}(q)=E_6(q)$ and $E_6^{?1}(q){=}^2{E}_6(q)$. We classify, up to conjugacy, all maximal-proper 3-local subgroups of G, that is, all 3-local $M<G$ which are maximal with respect to inclusion among all proper subgroups of G which are 3-local. To this end, we also determine, up to conjugacy, all elementary-abelian 3-subgroups containing $Z(G)$, all extraspecial subgroups containing $Z(G)$, and all cyclic groups of order 9 containing $Z(G)$. These classifications are an important first step towards a classification of the 3-radical subgroups of G, which play a crucial role in many open conjectures in modular representation theory.     

Proof of a conjecture on monomial graphs

Let e be a positive integer, p be an odd prime, $q=p^e$, and ${\mathbb{F}}_q$ be the finite field of q elements. Let $f,g\in {\mathbb{F}}_q[X,Y]$. The graph $G_q(f,g)$ is a bipartite graph with vertex partitions $P={\mathbb{F}}_q^3$ and $L={\mathbb{F}}_q^3$, and edges defined as follows: a vertex $(p)=(p_1,p_2,p_3)\in P$ is adjacent to a vertex $[l]=[l_1,l_2,l_3]\in L$ if and only if $p_2+l_2=f(p_1,l_1)$ and $p_3+l_3=g(p_1,l_1)$. If $f=XY$ and $g=X{Y}^2$, the graph $G_q(XY,X{Y}^2)$ contains no cycles of length less than eight and is edge-transitive. Motivated by certain questions in extremal graph theory and finite geometry, people search for examples of graphs $G_q(f,g)$ containing no cycles of length less than eight and not isomorphic to the graph $G_q(XY,X{Y}^2)$, even without requiring them to be edge-transitive. So far, no such graphs $G_q(f,g)$ have been found. It was conjectured that if both f and g are monomials, then no such graphs exist. In this paper we prove the conjecture.