Given an integer g $g$ and also some given integers m $m$ (sufficiently large) c 1 , ? $c_1,\dots c_m$ we show that the number of all non-negative n ? M $n\leqslant M$ with property there exist k $k_1,\dots k_m$ such 2 = ? i \begin{equation*} n^2=\sum _{i=1}^m c_i g^{k_i} \end{equation*} is o ( log ) ? / $o((\log M)^{m-1/2})$ . We obtain a similar bound when dealing more general inequalities Q ...

Given integers r ⩾ 2 $r\geqslant 2$ and n , t 1 $n,t\geqslant 1$ we call families F ⋯ ⊆ P ( [ ] ) $\mathcal {F}_1,\dots ,\mathcal {F}_r\subseteq \mathcal {P}([n])$ $r$ -cross $t$ -intersecting if for all i ∈ $F_i\in {F}_i$ $i\in [r]$ have | ∩ $\vert \bigcap _{i\in [r]}F_i\vert \geqslant t$ . We obtain a strong generalisation of the classic Hilton–Milner theorem on cross-intersecting families. I...

A graph $G$ is $k$-$weighted-list-antimagic$ if for any vertex weighting $\omega\colon V(G)\to\mathbb{R}$ and list assignment $L\colon E(G)\to2^{\mathbb{R}}$ with $|L(e)|\geq |E(G)|+k$ there exists an edge labeling $f$ such that $f(e)\in L(e)$ all $e\in E(G)$, labels of edges are pairwise distinct, the sum on incident to a plus weight distinct from at every other vertex. In this paper we prove ...

In this paper, we study the $$M_n/M_n/c/(K_1+K_2)+M_n$$ system with two finite-size queues where underlying exponential random variables have state-dependent rates. When all servers are busy, upon arrival customers may join online or offline/callback queue simply balk. Customers waiting in impatient and if their patience expires, they choose to callback instead of abandoning for good. assumed b...

The Cohn-Umans group-theoretic approach to matrix multiplication suggests embedding matrix multiplication into group algebra multiplication, and bounding $\omega$ in terms of the representation theory of the host group. This framework is general enough to capture the best known upper bounds on $\omega$ and is conjectured to be powerful enough to prove $\omega = 2$, although finding a suitable g...

We calculate the masses of vector and axial-vector mesons as well nucleon delta resonance in chiral symmetry restored vacuum. This is accomplished by separating quark operators appearing QCD sum rules for these hadrons into symmetric breaking parts depending on contributions fermion zero modes. then extract vacuum expectation values all separated using rule relations with their widths. By takin...

Let $$C_{k_1}, \ldots , C_{k_n}$$ be cycles with $$k_i\ge 2$$ vertices ( $$1\le i\le n$$ ). By attaching these n together in a linear order, we obtain graph called polygon chain. cyclic graph, which is ring if it can embedded on the plane; and twisted Möbius band. It known that sandpile group of chain always cyclic. Furthermore, there exist edge generators. In this paper, not only show any (twi...

A graph $G$ is a non-separating planar if there drawing $D$ of on the plane such that (1) no two edges cross each other in and (2) for any cycle $C$ $D$, vertices not are same side $D$.
 Non-separating graphs closed under taking minors subclass superclass outerplanar graphs.
 In this paper, we show only it does contain $K_1 \cup K_4$ or K_{2,3}$ $K_{1,1,3}$ as minor.
 Furthermore...

Quantum code construction from two classical codes $D_1[n,k_1,d_1]$ and $D_2[n,k_2,d_2]$ over the field $\mathbb {F}_{p^m}$ ( $p$ is prime $m$ an integer) satisfying dual containing criteria $D_1^{\perp } \subset D_2$ using Calderbank–Shor–Steane (CSS) framework well-studied. We show that generalization of CSS for qubits to qudits yields different classes codes, namely, {F}_{p}$ -linear well-kn...

We show that the spectral action, when perturbed by a gauge potential, can be written as series of Chern–Simons actions and Yang–Mills all orders. In odd orders, generalized forms are integrated against an $(b,B)$-cocycle, whereas, in even powers curvature $(b,B)$-cocycles Hochschild cocycles well. both cases, cochains derived from Taylor expansion action $\\operatorname{Tr}(f(D+V))$ $V=\\pi_D(...