共查询到10条相似文献,搜索用时 109 毫秒
1.
B. B. Tsegaw 《Differential Equations》2015,51(12):1607-1619
We study the nonexistence of weak solutions of higher-order elliptic and parabolic inequalities of the following types: \(\sum {_{i = 1}^N\sum\nolimits_{{e_i} \leqslant {\alpha _i} \leqslant {m_i}} {D_{{x_i}}^{{\alpha _i}}\left( {{A_{{\alpha _i}}}\left( {x,u} \right)} \right)} \geqslant f\left( {x,u} \right),} x \in {\mathbb{R}^N}\), and \({u_t} + \sum {_{i = 1}^N\sum\nolimits_{{k_i} \leqslant {\beta _i} \leqslant {n_i}} {D_{{x_i}}^{{\beta _i}}\left( {{B_{{\beta _i}}}\left( {x,t,u} \right)} \right)} > g\left( {x,t,u} \right),\left( {x,t} \right)} \in {\mathbb{R}^N} \times {\mathbb{R}_ + }\), where l i , m i , k i , n i ∈ N satisfy the condition l i , k i > 1 for all i = 1,..., N, and A αi (x, u), B βi (x, t, u), f(x, u), and g(x, t, u) are some given Carathéodory functions. Under appropriate conditions on the functions A αi , B βi , f, and g, we prove theorems on the nonexistence of solutions of these inequalities. 相似文献
2.
M. A. Korolev 《Proceedings of the Steklov Institute of Mathematics》2017,296(1):154-171
The work is devoted to generalized Kloosterman sums modulo a prime, i.e., trigonometric sums of the form \(\sum\nolimits_{p \leqslant x} {\exp \left\{ {2\pi i\left( {a\bar p + {F_k}\left( p \right)} \right)/q} \right\}} \) and \(\sum\nolimits_{n \leqslant x} {\mu \left( n \right)\exp \left\{ {2\pi i\left( {a\bar n + {F_k}\left( n \right)} \right)/q} \right\}} \), where q is a prime number, \(\left( {a,q} \right) = 1,m\bar m \equiv 1\left( {\bmod {\kern 1pt} q} \right)\), F k (u) is a polynomial of degree k ≥ 2 with integer coefficients, and p runs over prime numbers. An upper estimate with a power saving is obtained for the absolute values of such sums for x ≥ q1/2+ε. 相似文献
3.
The uncertain system is considered, where the coefficients a ij (n) of the m×m matrix A n are functionals of any nature subject to the constraints
Such systems include, in particular, switched-type systems, whose matrix A can take values in a given finite set.By using a special Lyapunov function, a bound δ ≤ δ(α0,α*) ensuring the global asymptotic stability of the system is found. In particular, the system is stable if the last inequality is replaced by a i,j (n) = 0 for j < i.It is shown that pulse-width modulated systems reduce to the uncertain systems under consideration; moreover, in the case of a pulse-width modulation of the first kind, the coefficients of the matrix A are functions of x(n), and in the case of a modulation of the second kind, they are functionals. 相似文献
$x_{n + 1} = A_n x_n , n = 0,1,2, \ldots ,$
$\begin{array}{*{20}c} {\left| {a_{i,i} (n)} \right| \leqslant \alpha _ * < 1,} \\ {\left| {a_{i,j} (n)} \right| \leqslant \alpha _0 for j \geqslant i + 1,} \\ {\left| {a_{i,j} (n)} \right| \leqslant \delta for j < i.} \\ \end{array} $
4.
Let \({\{ {f_{\lambda ;j}}\} _{\lambda \in V;1 \leqslant j \leqslant k}}\) be families of holomorphic functions in the open unit disk \({\text{D}} \subset {\Bbb C}\) ? ? depending holomorphically on a parameter λ ∈ V ? ? n . We establish a Rolle type theorem for the generalized multiplicity (called cyclicity) of zeros of the family of univariate holomorphic functions \({\left\{ {\sum\nolimits_{j = 1}^k {{f_{\lambda ;j}}} } \right\}_{\lambda \in V}}\) at 0 ∈ D. As a corollary, we estimate the cyclicity of the family of generalized exponential polynomials, that is, the family of entire functions of the form \(\sum\nolimits_{k = 1}^m {{P_k}(z){e^{{Q_k}(z)}}} \), z ∈ ?, where P k and Q k are holomorphic polynomials of degrees p and q, respectively, parameterized by vectors of coefficients of P k and Q k . 相似文献
5.
Zhishan Yang 《Frontiers of Mathematics in China》2017,12(4):981-992
For a cubic algebraic extension K of ?, the behavior of the ideal counting function is considered in this paper. More precisely, let a K (n) be the number of integral ideals of the field K with norm n, we prove an asymptotic formula for the sum \(\sum\nolimits_{n_1^2 + n_2^2 \leqslant x} {a_K \left( {n_1^2 + n_2^2 } \right)} \). 相似文献
6.
Let f and g be multiplicative functions of modulus 1. Assume that \( {\lim_{x \to \infty }}\frac{1}{x}\left| {\sum\nolimits_{n \leqslant x} {f(n)} } \right| = A > 0 \) and \( {\lim_{x \to \infty }}\frac{1}{x}\left| {\sum\nolimits_{n \leqslant x} {g(n)} } \right| = 0 \). We prove that, under these conditions,Concerning the Liouville function λ, we find an upper estimate for \( \frac{1}{x}\left| {\sum\limits_{n \leqslant x} {\lambda (n)\lambda (n + 1)} } \right| \) under the unproved hypothesis that L(s, χ) have Siegel zeros for an infinite sequence of L-functions.
相似文献
$ \mathop {\lim }\limits_{x \to \infty } \frac{1}{x}\sum\limits_{n \leqslant x} {f(n)g(n + 1) = 0.}$
7.
The problem of finding the number and the most likely shape of solutions of the system \(\sum\nolimits_{j = 0}^\infty {{\lambda _j}{n_j}} \leqslant M,\;\sum\nolimits_{j = 1}^\infty {{n_j}} = N\), where λj,M,N > 0 and N is an integer, as M,N →∞, can naturally be interpreted as a problem of analytic number theory. We solve this problem for the case in which the counting function of λj is of the order of λd/2, where d, the number of degrees of freedom, is less than two. 相似文献
8.
L. P. Kuptsov 《Mathematical Notes》1970,8(5):820-826
A class of nonlinear second-order equations of divergent form is distinguished, whose solutions have properties recalling the properties of solutions of ordinary elliptic equations. In the linear case these are equations of the form $$\sum\nolimits_{j = 1}^k \lambda _j (x)A_j^2 u + \sum\nolimits_{j = 1}^k {\mu _j (x)A_j u + c(x)u + f(x) = 0,} $$ where the \(A_j = \sum\nolimits_{\alpha = 1}^n {\alpha _j^\alpha (x)\frac{\partial }{{\partial x^\alpha }}(1 \leqslant j \leqslant k)} \) are linearly independent first-order differential operators whose Lie algebra is of rank n, 2 ? k ? n, and theλj(x) ? 0 are functions which can become0 zero or increase in a definite way. Harnack's inequality is proved for nonnegative solutions of these equations. 相似文献
9.
Driss Drissi 《Complex Analysis and Operator Theory》2010,4(4):813-819
Let {x n } be a sequence of complex numbers and let \({\Delta^nx_j = \sum\nolimits_{k=0}^{n} (-1)^k\break\left(\begin{array}{l}n\\ k\\\end{array} \right)x_{n-k+j}}\) . In this paper, we will show that if \({ |x_n| = O(n^k)}\) , as n → ∞ for some positive integer k, and \({n|\Delta^n x_j|^{\frac{1}{n}} \to 0}\) as n→ ∞, then \({\Delta^{k+1} x_j = 0}\) . More importantly, applications to the orbits of operators and invariant subspace problem are also given; this helps to improve former results obtained by Gelfand–Hille, Mbekhta–Zemánek and others. 相似文献
10.
The system ? i = ? i (?) + x i+2, \(i \in \overline {1,n - 2} \), ? n?1 = ? n?1(?) + u 1, ? n = ? n (?) + u 2,where ? i (·) are nonanticipating functionals of an arbitrary nature with the following properties—\(\left| {{\varphi _i}\left( \cdot \right)} \right| \leqslant c\sum\nolimits_{k = 1}^i {\left| {{x_k}\left( t \right)} \right|} \), \(i \in \overline {1,n} \), c = const—and u 1 and u 2 are the controls is considered. It is assumed that only the outputs x 1 and x 2 are measurable. The problem of synthesis of both continuous and impulsive controls u1 and u2, which make the system globally asymptotically stable, is solved. The solution of the problem is based on the construction of the observer-based equations, the quadratic Lyapunov function, and the averaging method. 相似文献