Let $q\ge 3$ be an integer, let $\chi $ denote a Dirichlet character modulo $q.$ For any real number $a\ge 0$ we define the generalized Dirichlet $L$-functions $$ L(s,\chi ,a)=\sum _{n=1}^{\infty }\frac {\chi (n)}{(n+a)^s}, $$ where $s=\sigma +{\rm i} t$ with $\sigma >1$ and $t$ both real. They can be extended to all $s$ by analytic continuation. In this paper we study the mean value properties of the generalized Dirichlet $L$-functions especially for $s=1$ and $s=\frac 12+{\rm i} t$, and obtain two sharp asymptotic formulas by using the analytic method and the theory of van der Corput.
The main purpose of the paper is to study, using the analytic method and the property of the Ramanujan's sum, the computational problem of the mean value of the mixed exponential sums with Dirichlet characters and general Gauss sum. For integers $m$, $ n$, $ k$, $ q$, with $k\geq {1}$ and $q\geq {3}$, and Dirichlet characters $\chi $, $\bar {\chi }$ modulo $q$ we define a mixed exponential sum $$ C(m,n;k;\chi ;\bar {\chi };q)= \sum \limits _{a=1}^{q}{\mkern -4mu\vrule width0pt height1em}' \chi (a)G_{k}(a,\bar {\chi })e \Big (\frac {ma^{k}+n\overline {a^{k}}}{q}\Big ), $$ with Dirichlet character $\chi $ and general Gauss sum $G_{k}(a,\bar {\chi })$ as coefficient, where $\sum \nolimits '$ denotes the summation over all $a$ such that $(a,q)=1$, $a\bar {a}\equiv {1}\mod {q}$ and $e(y)={\rm e}^{2\pi {\rm i} y}$. We mean value of $$ \sum _{m}\sum _{\chi }\sum _{\bar {\chi }}|C(m,n;k;\chi ;\bar {\chi };q)|^{4}, $$ and give an exact computational formula for it.
An optimization problem for the unilateral contact between a pseudoplate and a rigid obstacle is considered. The variable thickness of the pseudoplate plays the role of a control variable. The cost functional is a regular functional only in the smooth case. The existence of an optimal thickness is verified. The penalized optimal control problem is considered in the general case.
Let $G = (V,E)$ be a simple graph. A $3$-valued function $f\:V(G)\rightarrow \lbrace -1,0,1\rbrace $ is said to be a minus dominating function if for every vertex $v\in V$, $f(N[v]) = \sum _{u\in N[v]}f(u)\ge 1$, where $N[v]$ is the closed neighborhood of $v$. The weight of a minus dominating function $f$ on $G$ is $f(V) = \sum _{v\in V}f(v)$. The minus domination number of a graph $G$, denoted by $\gamma ^-(G)$, equals the minimum weight of a minus dominating function on $G$. In this paper, the following two results are obtained. (1) If $G$ is a bipartite graph of order $n$, then
\[ \gamma ^-(G)\ge 4\bigl (\sqrt{n + 1}-1\bigr )-n. \] (2) For any negative integer $k$ and any positive integer $m\ge 3$, there exists a graph $G$ with girth $m$ such that $\gamma ^-(G)\le k$. Therefore, two open problems about minus domination number are solved.
We consider the mixed problem for the hyperbolic partial differential-functional equation of the first order \[ D_xz(x,y) = f(x,y,z_{(x,y)}, D_yz(x,y)), \] where $z_{(x,y)} \: [-\tau ,0] \times [0,h] \rightarrow \mathbb{R}$ is a function defined by $z_{(x,y)}(t,s) = z(x+t, y+s)$, $(t,s) \in [-\tau ,0] \times [0,h]$. Using the method of bicharacteristics and the method of successive approximations for a certain integral-functional system we prove, under suitable assumptions, a theorem of the local existence of generalized solutions of this problem.
Blood's non-Newtonian behaviour is investigated in an idealized coronary 3D bypass model, which includes both the proximal and distal parts of the occluded native artery and the connected end-to- side bypass graft. Considering the blood to be a generalized Newtonian fluid, the shear-dependent viscosity is given by two well-known macroscopic non-Newtonian models (the Carrea-Yasuda model and the modified Cross model). Both non-Newtonian steady flow fields are analyzed with regard to the bypass geometry and are compared with the case of the Newtonian fluid. In order to perform all numerical simulations, we developed an incompressible Navier.Stokes solver based on the pseudo-compressibility approach and on the cell-centred finite volume formulation of the central explicit fourth-stage Runge-Kutta time stepping scheme defined on unstructured hexahedral computational grid. and Obsahuje seznam literatury
The study of the bulla from 18 lemaeopodid copepod species collected on 15 marine fish species and one freshwater fish species taken mainly from the Gulf of Lions in the Mediterranean Sea reveals a great morphological and structural variability. It is however possible to bring forth three general remarks: - the bulla of Lernaeopodidae parasites of Selachii have a remarkably constant structure probably due to the tegument nature of the attachment substratum; - the bulla of Lernaeopodidae parasites of Teleostei has a morphology influenced by the nature of the attachment tissue; - when species of a same genus (i.e. Clavellotis) are attached on a same organ, the shape of the bulla can constitute a taxonomic characteristic.
In this paper we investigate the effect on the multiplicity of Laplacian eigenvalues of two disjoint connected graphs when adding an edge between them. As an application of the result, the multiplicity of 1 as a Laplacian eigenvalue of trees is also considered.