91 research outputs found

    Laplacian spectra of complex networks and random walks on them: Are scale-free architectures really important?

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    We study the Laplacian operator of an uncorrelated random network and, as an application, consider hopping processes (diffusion, random walks, signal propagation, etc.) on networks. We develop a strict approach to these problems. We derive an exact closed set of integral equations, which provide the averages of the Laplacian operator's resolvent. This enables us to describe the propagation of a signal and random walks on the network. We show that the determining parameter in this problem is the minimum degree qmq_m of vertices in the network and that the high-degree part of the degree distribution is not that essential. The position of the lower edge of the Laplacian spectrum λc\lambda_c appears to be the same as in the regular Bethe lattice with the coordination number qmq_m. Namely, λc>0\lambda_c>0 if qm>2q_m>2, and λc=0\lambda_c=0 if qm2q_m\leq2. In both these cases the density of eigenvalues ρ(λ)0\rho(\lambda)\to0 as λλc+0\lambda\to\lambda_c+0, but the limiting behaviors near λc\lambda_c are very different. In terms of a distance from a starting vertex, the hopping propagator is a steady moving Gaussian, broadening with time. This picture qualitatively coincides with that for a regular Bethe lattice. Our analytical results include the spectral density ρ(λ)\rho(\lambda) near λc\lambda_c and the long-time asymptotics of the autocorrelator and the propagator.Comment: 25 pages, 4 figure

    Boundedness and Stability of Impulsively Perturbed Systems in a Banach Space

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    Consider a linear impulsive equation in a Banach space x˙(t)+A(t)x(t)=f(t), t0,\dot{x}(t)+A(t)x(t) = f(t), ~t \geq 0, x(τi+0)=Bix(τi0)+αi,x(\tau_i +0)= B_i x(\tau_i -0) + \alpha_i, with limiτi=\lim_{i \rightarrow \infty} \tau_i = \infty . Suppose each solution of the corresponding semi-homogeneous equation x˙(t)+A(t)x(t)=0,\dot{x}(t)+A(t)x(t) = 0, (2) is bounded for any bounded sequence {αi}\{ \alpha_i \}. The conditions are determined ensuring (a) the solution of the corresponding homogeneous equation has an exponential estimate; (b) each solution of (1),(2) is bounded on the half-line for any bounded ff and bounded sequence {αi}\{ \alpha_i \} ; (c) limtx(t)=0\lim_{t \rightarrow \infty}x(t)=0 for any f,αif, \alpha_i tending to zero; (d) exponential estimate of ff implies a similar estimate for xx.Comment: 19 pages, LaTex-fil

    СХОДИМОСТЬ ПОСЛЕДОВАТЕЛЬНЫХ ПРИБЛИЖЕНИЙ ДЛЯ УРАВНЕНИЙ С НОРМАЛЬНЫМИ ОПЕРАТОРАМИ

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    The article deals with normal linear operators B with a unit spectral radius in Hilbert spaces, for which the successive approximations xn+1 = Bxn + f with an arbitrarily initial approximation x0 converge to a solution of the equation x = Bx + f (under condition that these solutions exist). Sufficient conditions for the convergence of successive approximations on subspaces of source-wise represented functions and in weakened norms are established. The behavior of residuals and corrections of these approximations is studied, too. Moreover, the behavior of “approximate” successive approximations is also investigated.В сообщении изучаются действующие в гильбертовом пространстве X нормальные линейные операторы B с единичным спектральным радиусом, для которых, однако, последовательные приближения xn+1 = Bxn + f сходятся при любом начальном приближении x0 к одному из решений уравнения x = Bx + f при условии, что такие решения существуют. Получены достаточные условия сходимости последовательных приближений на подпространствах истокообразно представимых функций и сходимость приближений в более слабой, чем исходная, норме гильбертова пространства. Исследовано поведение невязок и поправок. Изучено также поведение последовательных приближений при вычислениях с малыми ошибками

    О решении задачи Коши с неограниченной правой частью для уравнений дробного порядка

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    In this article, we study the question of the solvability of an analogue of the Cauchy problem for ordinary differential equations with fractional Riemann-Liouville derivatives on the unbounded right-hand side in certain function spaces. The solvability conditions of the problem under consideration in given function spaces, as well as the existence conditions of a unique solution are presented. The study uses the method of reducing the problem to the second-kind Volterra equation, the Schauder principle of a fixed point in a Banach space, and the Banach-Cachoppoli principle of a fixed point in a complete metric space.Изучается вопрос о разрешимости аналога задачи Коши для обыкновенных дифференциальных уравнений с дробными производными Римана-Лиувилля с неограниченной правой частью в определенных пространствах функций. Приводятся условия разрешимости рассматриваемой задачи в данных функциональных пространствах, а также условия существования единственного решения. При исследовании используются метод сведения задачи к уравнению Вольтерра второго рода, принцип Шаудера неподвижной точки в банаховом пространстве и принцип Банаха-Каччиопполи неподвижной точки в полном метрическом пространстве
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