Some approximation theorems for modified Bernstein-Durrmeyer operators

The modified Bernstein-Durrmeyer operators discussed in this paper are given byM_nf≡M_n(f,x)=(n+2)P_(n,k)∫_0~1p_n+1.k(t)f(t)dt,whereWe will show,for 0<α<1 and 1≤p≤∞     

On a system of second order differential equations with periodic impulse coefficients

A thorough investigation of the systemd~2y(x):dx~2 p(x)y(x)=0with periodic impulse coefficientsp(x)={1,0≤xx_0>0) -η, x_0≤x<2π(η>0)p(x)=p(x 2π),-∞相似文献   

The problem of the solvability of nonlinear two-point boundary-value problems

We establish sufficient conditions for the solvability of boundary-value problems of the form $$\begin{gathered} u'' = f(t,u,u'); \hfill \\ \begin{array}{*{20}c} {(u(0),} & {u'(0)) \in S_0 ,} & {(u(1),} & {u'(1)) \in S_1 .} \\ \end{array} \hfill \\ \end{gathered} $$      

Optimal trajectories for quadratic variational processes via invariant imbedding

Consider minimizing the integral $$I = \int_0^T {[\dot w^2 + g(y)w^2 ] dy}$$ where $$w = w(y), \dot w = dw/dy, w(T) = 1, w(0) = free$$ ForT sufficiently small, it is shown that $$w_{opt} = x(t,T), 0 \leqslant t \leqslant T$$ where the functionx, viewed as a function ofT, is a solution of the Cauchy problem $$\begin{gathered} x_T (t,T) = r(T)x(t,T), T \geqslant t \hfill \\ x(t,t) = 1 \hfill \\\end{gathered}$$ and the auxiliary functionr satisfies the Riccati system $$\begin{gathered} r_T = ---g(T) + r^2 , T \geqslant 0 \hfill \\ r(0) = 0 \hfill \\\end{gathered}$$ In the derivation of the Cauchy problem, no use is made of Euler equations, dynamic programming, or Pontryagin's maximum principle. Only ordinary differential equations are employed. The Cauchy problem provides a one-sweep integration procedure; it is intimately connected with the theory of the second variation.     

Stability of the Faedo-Galerkin approximation of nonlinear wave-equations

Present investigation analyses the Ljapunov stability of the systems of ordinary differential equations arising in then-th step of the Faedo-Galerkin approximation for the nonlinear wave-equation $$\begin{gathered} u_{tt} - u_{xx} + M(u) = 0 \hfill \\ u(0,t) = u(1,t) = 0 \hfill \\ u(x,0) = \Phi (x); u_t (x,0) = \Psi (x). \hfill \\ \end{gathered}$$ For the nonlinearities of the classM (u)=u ^{2 p+1} ,p ∈N, ann-independent stability result is given. Thus also the stability of the original equation is shown.     

Asymptotic analysis of a singular Sturm-Liouville boundary value problem

Asymptotic expansions are given for the eigenvalues λ_n and eigenfunctions u_n of the following singular Sturm-Liouville problem with indefinite weight: $$\begin{gathered} - ((1 - x^2 )u'(x))' = \lambda xu(x) on ( - 1,1), \hfill \\ lim_{| x | \to 1} u(x) finite \hfill \\ \end{gathered} $$ This eigenvalue problem arises if one separates variables in a partial differential equation which describes electron scattering in a one-dimensional slab configuration. Asymptotic expansions of the normalization constants of the eigenfunctions are also given. The constants in these asymptotic expansions involve complete elliptic integrals. The asymptotic results are compared with the results of numerical calculations.     

Perturbazioni per processi di controllo con parametri distribuiti

We consider the control processes $$\begin{gathered} (E) z_{xy} + A(x,y)z_x + B(x,y)z_y + C(x,y)z = F(x,y)U(x,y) \hfill \\ q.o. in R = [0,\alpha [ \times [0,\beta [, \hfill \\ \end{gathered} $$ $$\begin{gathered} (\tilde E) z_{xy} + \bar A(x,y)z_x + \bar B(x,y)z_y + \bar C(x,y)z = \bar F(x,y)U(x,y) \hfill \\ q.o. in R \hfill \\ \end{gathered} $$ We show that under appropriate assumptions on the dataA, B, C, F, if the process (E) is completely controllable, then the perturbed process (ē) is completely controllable too. The result is obteined proving for the evolution matrixV, a continuous dependence on the coefficientsA, B, C.     

The lattice point discrepancy of a torus in ℝ3

This article provides an asymptotic formula for the number of integer points in the three-dimensional body $$ \left( \begin{gathered} x \hfill \\ y \hfill \\ z \hfill \\ \end{gathered} \right) = t\left( \begin{gathered} (a + r\cos \alpha )\cos \beta \hfill \\ (a + r\cos \alpha )\sin \beta \hfill \\ r\sin \alpha \hfill \\ \end{gathered} \right),0 \leqq \alpha ,\beta < 2\pi ,0 \leqq r \leqq b, $$ for fixed a > b > 0 and large t.     

A Noetherian and confluent rewrite system for idempotent semigroups

Let B be a semigroup with the additional relation $$\begin{gathered} xx \Rightarrow x \hfill \\ xyz \Rightarrow xz if x \mathop {CI}\limits_ = z and xy\mathop {CI}\limits_ = z \hfill \\ \end{gathered} $$ B is called aband or anidempotent semigroup [3]. It is shown in this paper that the replacement rules (rewrites) resulting from the axiom of idempotence: $$\forall w \in B.ww = w$$ can be replaced by theNoetherian, confluent, conditional rewrites (i. e. a terminating replacement system having the Church-Rosser-Property): $$\begin{gathered} xx \Rightarrow x \hfill \\ x \Rightarrow xx \hfill \\ \end{gathered} $$ These rewrites are used to obtain a unique normal form for words in B and hence are the basis for a decision procedure for word equality in B. The proof techniques are based uponterm rewriting systems [7] rather than the usual algebraic approach. Alternative and simpler proofs of a result reported earlier by Green and Rees [4] and Gerhardt [6] have been obtained.     

Bang-bang property for Bolza problems in two dimensions

Consider the following Bolza problem: $$\begin{gathered} \min \int {h(x,u) dt,} \hfill \\ \dot x = F(x) + uG(x), \hfill \\ \left| u \right| \leqslant 1, x \in \Omega \subset \mathbb{R}^2 , \hfill \\ x(0) = x_0 , x(1) = x_1 . \hfill \\ \end{gathered} $$ We show that, under suitable assumptions onF, G, h, all optimal trajectories are bang-bang. The proof relies on a geometrical approach that works for every smooth two-dimensional manifold. As a corollary, we obtain existence results for nonconvex optimization problems.     

On the convergence rate of the partial sums of positive entire Dirichlet series

Пусть Λ=(λ_n) — возрастаю щая к+∞ последователь ность неотрицательных чис ел, λ₀=0, а S₊(Λ) — класс абсолют но сходящихся в С рядо в Дирихле вида $$F\left( z \right) = \mathop \sum \limits_{k = 0}^\infty a_k \exp \left\{ {z\lambda _k } \right\},$$ где a₀=1 и a_k>0 (k∈N). Положим $$\begin{gathered} S_n \left( z \right) = \mathop \sum \limits_{k = 1}^\infty a_k \exp \left\{ {z\lambda _k } \right\}, \hfill \\ \sigma _n \left( F \right) = \max \left\{ {\frac{1}{{S_n \left( x \right)}} - \frac{1}{{F\left( x \right)}}:x \in R} \right\}. \hfill \\ \end{gathered} $$ Доказано, что для того, чтобы для любой функц ии F∈S₊(Λ) выполнялось равенст во $$\mathop {\lim \sup }\limits_{n \to \infty } \frac{1}{{\ln n}}\ln \frac{1}{{\sigma _n \left( F \right)}} = + \infty ,$$ необходимо и достато чно, чтобы $$\mathop \sum \limits_{n = 1}^\infty \frac{1}{{n\lambda _n }}< + \infty .$$ Аналогичные результ ы получены для различ ных подклассов классаS ₊ (Λ), определяемых условиями на убывани е коэффициентова _n.     

Interpolation by polyhedral functions

A polyhedral functionlp(Δ_n) (f). interpolating a function f, defined on a polygon Φ, is defined by a set of interpolating nodes Δ_n ?Φ and a partition P(Δ_n) of the polygon Φ into triangles with vertices at the points of Δ_n. In this article we will compute for convex moduli of continuity the quatities $$\begin{gathered} E (H_\Phi ^\omega ; P (\Delta _n )) = sup || f - l_{p(\Delta _n )} (f)||, \hfill \\ f \in H_\Phi ^\omega \hfill \\ \end{gathered} $$ and also give an asymptotic estimate of the quantities $$\begin{gathered} E_n (H_\Phi ^\omega ) = infinf E (H_\Phi ^\omega ; P (\Delta _n )). \hfill \\ \Delta _n P(\Delta _n ) \hfill \\ \end{gathered} $$      

Boundary parametrization of planar self-affine tiles with collinear digit set

We consider a class of planar self-affine tiles T = M-1 a∈D(T + a) generated by an expanding integral matrix M and a collinear digit set D as follows:M =(0-B 1-A),D = {(00),...,(|B|0-1)}.We give a parametrization S1 →T of the boundary of T with the following standard properties.It is H¨older continuous and associated with a sequence of simple closed polygonal approximations whose vertices lie on T and have algebraic preimages.We derive a new proof that T is homeomorphic to a disk if and only if 2|A| |B + 2|.     

Approximation of the function sign x in the uniform and integral metrics by means of rational functions

Estimates are obtained for the nonsymmetric deviations R_n [sign x] and R_n [sign x]_L of the function sign x from rational functions of degree ≤n, respectively, in the metric $$c([ - 1, - \delta ] \cup [\delta ,1]), 0< \delta< exp( - \alpha \surd \overline n ), \alpha > 0,$$ and in the metric L[?1, 1]: $$\begin{gathered} R_n [sign x] _{\frown }^\smile exp \{ - \pi ^2 n/(2 ln 1/\delta )\} , n \to \infty , \hfill \\ 10^{ - 3} n^{ - 2} \exp ( - 2\pi \surd \overline n )< R_n [sign x_{|L}< \exp ( - \pi \surd \overline {n/2} + 150). \hfill \\ \end{gathered} $$ Let 0 < δ < 1, Δ (δ)=[?1, ? δ] ∪ [δ, 1]; $$\begin{gathered} R_n [f;\Delta (\delta )] = R_n [f] = inf max |f(x) - R(x)|, \hfill \\ R_n [f;[ - 1,1] ]_L = R_n [f]_L = \mathop {inf}\limits_{R(x)} \smallint _{ - 1}^1 |f(x) - R(x)|dx, \hfill \\ \end{gathered} $$ where R(x) is a rational function of order at most n. Bulanov [1] proved that for δ ε [e^?n, e^?1] the inequality $$\exp \left( {\frac{{\pi ^2 n}}{{2\ln (1/\delta }}} \right) \leqslant R_n [sign x] \leqslant 30 exp\left( {\frac{{\pi ^2 n}}{{2\ln (1/\delta + 4 ln ln (e/\delta ) + 4}}} \right)$$ is valid. The lower estimate in this inequality was previously obtained by Gonchar ([2], cf. also [1]).     

Necessary conditions for optimality in the identification of elliptic systems with parameter constraints

We consider the problems of dientifying the parametersa _ij (x), b _i (x), c(x) in a 2nd order, linear, uniformly elliptic equation, $$\begin{gathered} - \partial _i (a_{ij} (x)\partial _j u) + b_i (x)\partial _i u + c(x)u = f(x),in\Omega , \hfill \\ \partial _v u|_{\partial \Omega } = \phi (s),s \in \partial \Omega , \hfill \\ \end{gathered} $$ on the basis of measurement data $$u(s) = z(s),s \in B \subset \partial \Omega ,$$ with an equality constraint and inequality constraints on the parameters. The cost functionals are one-sided Gâteaux differentiable with respect to the state variables and the parameters. Using the Duboviskii-Milyutin lemma, we get maximum principles for the identification problems, which are necessary conditions for the existence of optimal parameters.     

Boundedness results for impulsive functional differential equations with infinite delays

In this paper, boundedness criteria are established for solutions of a class of impulsive functional differential equations with infinite delays of the form $$\begin{gathered} x'(t) = F(t,x( \cdot )), t > t^ * \hfill \\ \Delta x(t_k ) = I(t_k ,x(t_k^ - )), k = 1,2,... \hfill \\ \end{gathered} $$ By using Lyapunov functions and Razumikhin technique, some new Bazumikhin-type theorems on boundedness are obtained.     

Criterion of polynomial increase of a function and its derivatives

ДОкАжАНО, ЧтО Дль тОгО, ЧтОБы Дльr РАж ДИФФЕРЕНцИРУЕМОИ НА пРОМЕжУткЕ [А, + ∞) ФУНкцИИf сУЩЕстВОВА л тАкОИ МНОгОЧлЕН (1) $$P(x) = \mathop \Sigma \limits_{\kappa = 0}^{r - 1} a_k x^k ,$$ , ЧтО (2) $$\mathop {\lim }\limits_{x \to + \infty } (f(x) - P(x))^{(k)} = 0,k = 0,1,...,r - 1,$$ , НЕОБхОДИМО И ДОстАтО ЧНО, ЧтОБы схОДИлсь ИН тЕгРАл (3) $$\int\limits_a^{ + \infty } {dt_1 } \int\limits_{t_1 }^{ + \infty } {dt_2 ...} \int\limits_{t_{r - 1} }^{ + \infty } {f^{(r)} (t)dt.}$$ ЕслИ ЁтОт ИНтЕгРАл сх ОДИтсь, тО Дль кОЁФФИц ИЕНтОВ МНОгОЧлЕНА (1) ИМЕУт МЕс тО ФОРМУлы $$\begin{gathered} a_{r - m} = \frac{1}{{(r - m)!}}\left( {\mathop \Sigma \limits_{j = 1}^m \frac{{( - 1)^{m - j} f^{(r - j)} (x_0 )}}{{(m - j)!}}} \right.x_0^{m - j} + \hfill \\ + ( - 1)^{m - 1} \left. {\mathop \Sigma \limits_{l = 0}^{m - 1} \frac{{x_0^l }}{{l!}}\int\limits_a^{ + \infty } {dt_1 } \int\limits_{t_1 }^{ + \infty } {dt_2 ...} \int\limits_{t_{m - l - 1} }^{ + \infty } {f^{(r)} (t_{m - 1} )dt_{m - 1} } } \right),m = 1,2,...,r. \hfill \\ \end{gathered}$$ ДОстАтОЧНыМ, НО НЕ НЕОБхОДИМыМ Усл ОВИЕМ схОДИМОстИ кРА тНОгО ИНтЕгРАлА (3) ьВльЕтсь схОДИМОсть ИНтЕгРАл А \(\int\limits_a^{ + \infty } {x^{r - 1} f^{(r)} (x)dx}\)      

Asymptotic expansions of solutions of the Cauchy problem for nonlinear parabolic equations

This paper is concerned with the Cauchy problem for the nonlinear parabolic equation $${\partial _t}u| = \vartriangle u + F(x,t,u,\nabla u){\text{ in }}{{\text{R}}^N} \times (0,\infty ),{\text{ }}u(x,0) = \varphi (x){\text{ in }}{{\text{R}}^N},$$ , where $$\begin{gathered} N \geqslant 1, \hfill \\ F \in C(R^N \times (0,\infty ) \times R \times R^N ), \hfill \\ \phi \in L^\infty (R^N ) \cap L^1 (R^N ,(1 + |x|^K )dx)forsomeK \geqslant 0 \hfill \\ \end{gathered} $$ . We give a sufficient condition for the solution to behave like a multiple of the Gauss kernel as t → ∞ and obtain the higher order asymptotic expansions of the solution in W ^1,q (R ^N ) with 1 ≤ q ≤ ∞.     

Remarks on eigenvalues and eigenfunctions of a special elliptic system

LetΛ ₁(Ω) be the first eigenvalue of the vector-valued problem $$\begin{gathered} \Delta u + \alpha grad div u + \Delta u = 0 in \Omega , \hfill \\ u = 0 in \partial \Omega , \hfill \\ \end{gathered} $$ , withα>0. Letλ ₁(Ω) be the first eigenvalue of the scalar problem $$\begin{gathered} \Delta u + \lambda u = 0 in \Omega , \hfill \\ u = 0 on \partial \Omega . \hfill \\ \end{gathered} $$ . The paper contains a proof of the inequality $$\left( {1 + \frac{\alpha }{n}} \right)\lambda _1 \left( \Omega \right) > \Lambda _1 \left( \Omega \right) > \left( \Omega \right)$$ and improves recent estimates of Sprössig [15] and Levine and Protter [11]. Moreover we show, ifΩ is a ball, that an eigensolution u₁, associated withΛ ₁(Ω) is not unique and that the eigensolutions for this and higher eigenvalues are never rotationally invariant. Finally we calculate some eigensolutions explicitly.