Let G=(V,E), $V={v_1,v_2,ldots,v_n}$, be a simple connected graph with $%n$ vertices, $m$ edges and a sequence of vertex degrees $d_1geqd_2geqcdotsgeq d_n>0$, $d_i=d(v_i)$. Let ${A}=(a_{ij})_{ntimes n}$ and ${%D}=mathrm{diag }(d_1,d_2,ldots , d_n)$ be the adjacency and the diagonaldegree matrix of $G$, respectively. Denote by ${mathcal{L}^+}(G)={D}^{-1/2}(D+A) {D}^{-1/2}$ the normalized signles...

Let $$\mu _{\infty }\subseteq \mathbb {C}$$ be the collection of roots unity and $$\mathcal {C}_{n}:=\{(s_{1},\ldots ,s_{n})\in \mu }^{n}:s_{i}\ne s_{j}\text { for any }1\le i<j\le n\}$$ . Two elements $$(s_{1},\ldots ,s_{n})$$ $$(t_{1},\ldots ,t_{n})$$ {C}_{n}$$ are said to projectively equivalent if there exists $$\gamma \in PGL (2,\mathbb {C})$$ such that (s_{i})=t_{i}$$ $$1\le i\le n$$ In t...

Abstract Let $\mathrm{AP}_k=\{a,a+d,\ldots,a+(k-1)d\}$ be an arithmetic progression. For $\varepsilon&gt;0$ we call a set $\mathrm{AP}_k(\varepsilon)=\{x_0,\ldots,x_{k-1}\}$ $\varepsilon$ -approximate progression if for some and d , $|x_i-(a+id)|&lt;\varepsilon d$ holds all $i\in\{0,1\ldots,k-1\}$ . Complementing earlier results of Dumitrescu (2011, J. Comput. Geom. 2 (1) 16–29), in this paper ...

A birth-death chain is a discrete-time Markov on the integers whose transition probabilities $p_{i,j}$ are non-zero if and only $|i-j|=1$. We consider chains birth $p_{i,i+1}$ form periodic sequence, so that $p_{i,i+1}=p_{i \mod m}$ for some $m$ $p_0,\ldots,p_{m-1}$. The trajectory $(X_n)_{n=0,1,\ldots}$ of such satisfies strong law large numbers central limit theorem. study effect reordering $...

A ring $R$ with an automorphism $sigma$ and a $sigma$-derivation $delta$ is called $delta$-quasi-Baer (resp., $sigma$-invariant quasi-Baer) if the right annihilator of every $delta$-ideal (resp., $sigma$-invariant ideal) of $R$ is generated by an idempotent, as a right ideal. In this paper, we study Baer and quasi-Baer properties of skew PBW extensions. More exactly, let $A=sigma(R)leftlangle x...

We construct infinite families of abstract regular polytopes Schläfli type $\{4,p_1,\ldots,p_{n-1}\}$ from extensions centrally symmetric spherical $n$-polytopes. In addition, by applying the halving operation, we obtain both locally and toroidal hypertopes $\left\{\genfrac{}{}{0pt}{}{p_1}{p_1},\ldots,p_{n-1}\right\}$.

Let $(X_j, d_j, \mu _j)$, $j=0,1,\ldots , m$, be metric measure spaces. Given $0 \lt p^\kappa \le \infty $ for $\kappa = 1, \ldots and an analytic family of multilinear operators $$ T_z: L^{p^1}(X_1)\times \cdots \times L^{p^m}(X_m) \to L^1_{\rm loc

Let $\mathbb {N}=\{0,1,2,\ldots \}$ and $A\subset \mathbb {N}$. $h\geq 2$ let $r_h(A,n)=\sharp \{ (a_1,\ldots ,a_h) \in A^{h}: a_1+\cdots +a_h=n\}.$ The set $A$ is called an asymptotic basis of order $h$ if $r_h(A,n)\geq 1$ for all sufficiently la

Let $$n\ge 3$$ be an integer and p a prime with $$p\equiv 1\pmod {n}$$ . In this paper, we show that $$\begin{aligned} {}_nF_{n-1}\bigg [\begin{array}{llll} \frac{n-1}{n}&{}\frac{n-1}{n}&{}\ldots &{}\frac{n-1}{n}\\ &{}1&{}\ldots &{}1\end{array}\bigg | \, 1\bigg ]_{p-1}\equiv -\Gamma _p\bigg (\frac{1}{n}\bigg )^n\pmod {p^3}, \end{aligned}$$ where the truncated hypergeometric series x_1&{}x_2&{}\...

We study some properties of the Rovelli-Smolin-DePietri volume operator in loop quantum gravity, which significantly simplify diagonalization problem and shed light on pattern degeneracy eigenstates. The is defined by its action spaces tensor products $\mathcal{H}_{j_1}\otimes \ldots \otimes \mathcal{H}_{j_N}$ irreducible SU(2) representation $\mathcal{H}_{j_i}, i=1,\ldots,N$, labelled with spi...