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≤∞     

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.     

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.     

Über die kleinste natürliche Zahl maximaler Ordnung modm

modm. Ifm is natural,a an integer with (a, m)=1 put $$\begin{gathered} {}^om(a): = min\{ h\left| {h \in \mathbb{N},} \right.a^h \equiv 1(modm)\} , \hfill \\ \psi (m): = \max \{ o_m (a)\left| a \right. \in \mathbb{Z},(a,m) = 1\} , \hfill \\ g(m): = \min \{ a\left| {a \in \mathbb{N},(a,m) = 1,o_m (a) = } \right.\psi (m)\} . \hfill \\ \end{gathered} $$ Form prime,g(m) is the least natural primitive root modm. We establish the estimation $$\sum\limits_{m< x} {g(m)<< x^{1 + \varepsilon } .} $$      

Some variations on Kac's formula for Brownian motion

We show that under suitable conditions $$\begin{gathered} E_x f\left\{ {a + \int_0^t \beta \left[ {b + \int_0^s {\alpha \left( {X_r } \right)dr, c + s, X_s } } \right]ds, b + \int_0^t {\alpha \left( {X_s } \right)ds, c + t, X_t } } \right\} \hfill \\ = e^{tG} f\left[ {a, b, c, x} \right] \hfill \\ \end{gathered} $$ whereX _t is a Brownian motion andG is the generator of a (C ₀) contraction semigroupe ^tG.     

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.     

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.     

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.     

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|.     

Fractal relaxed Dirichlet problems

Given aself similar fractal K ? ? ⁿ of Hausdorff dimension α>n?2, andc ₁>0, we give an easy and explicit construction, using the self similarity properties ofK, of a sequence of closed sets? _h such that for every bounded open setΩ?? ⁿ and for everyf ∈ L²(Ω) the solutions to $$\left\{ \begin{gathered} - \Delta u_h = f in \Omega \backslash \varepsilon _h \hfill \\ u_h = 0 on \partial (\Omega \backslash \varepsilon _h ) \hfill \\ \end{gathered} \right.$$ converge to the solution of the relaxed Dirichlet boundary value problem $$\left\{ \begin{gathered} - \Delta u + uc_1 \mathcal{H}_{\left| K \right.}^\alpha = f in \Omega \hfill \\ u = 0 on \partial \Omega \hfill \\ \end{gathered} \right.$$ (H _∣ ^α denotes the restriction of the α-dimensional Hausdorff measure toK). The condition α>n?2 is strict.     

The number of strictly increasing mappings of fences

Let X and Y be fences of size n and m, respectively and n, m be either both even or both odd integers (i.e., |m-n| is an even integer). Let \(r = \left\lfloor {{{(n - 1)} \mathord{\left/ {\vphantom {{(n - 1)} 2}} \right. \kern-0em} 2}} \right\rfloor\) . If 1<n<-m then there are \(a_{n,m} = (m + 1)2^{n - 2} - 2(n - 1)(\begin{array}{*{20}c} {n - 2} \\ r \\ \end{array} )\) of strictly increasing mappings of X to Y. If 1<-m<-n<-2m and s=1/2(n?m) then there are a _n,m+b _n,m+c _n of such mappings, where $$\begin{gathered} b_{n,m} = 8\sum\limits_{i = 0}^{s - 2} {\left( {\begin{array}{*{20}c} {m + 2i + 1} \\ l \\ \end{array} } \right)4^{s - 2 - 1} } \hfill \\ {\text{ }}c_n = \left\{ \begin{gathered} \left( {\begin{array}{*{20}c} {n - 1} \\ {s - 1} \\ \end{array} } \right){\text{ if both }}n,m{\text{ are even;}} \hfill \\ {\text{ 0 if both }}n,m{\text{ are odd}}{\text{.}} \hfill \\ \end{gathered} \right. \hfill \\ \end{gathered} $$      

Regularity results for a class of obstacle problems in Heisenberg groups

We study regularity results for solutions u ∈ HW ^1,p (Ω) to the obstacle problem $$\int_\Omega \mathcal{A} \left( {x,\nabla _{\mathbb{H}^u } } \right)\nabla _\mathbb{H} \left( {v - u} \right)dx \geqslant 0 \forall v \in \mathcal{K}_{\psi ,u} \left( \Omega \right)$$ such that u ? ψ a.e. in Ω, where $xxx$ , in Heisenberg groups ? ⁿ . In particular, we obtain weak differentiability in the T-direction and horizontal estimates of Calderon-Zygmund type, i.e. $$\begin{gathered} T\psi \in HW_{loc}^{1,p} \left( \Omega \right) \Rightarrow Tu \in L_{loc}^p \left( \Omega \right), \hfill \\ \left| {\nabla _{\mathbb{H}\psi } } \right|^p \in L_{loc}^q \left( \Omega \right) \Rightarrow \left| {\nabla _{\mathbb{H}^u } } \right|^p \in L_{loc}^q \left( \Omega \right), \hfill \\ \end{gathered}$$ where 2 < p < 4, q > 1.     

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   

Exact Non-Self-Similar Solutions of the Equation $$u_t = \Delta \ln u$$

In this paper, we obtain new exact non-self-similar solutions of the nonlinear diffusion equation $$\begin{gathered} {\text{ }}u_t = \Delta \ln u, \hfill \\ u \triangleq u\left( {x,t} \right):\Omega \times \mathbb{R}^ + \to \mathbb{R},{\text{ }} x \in \mathbb{R}^n , \hfill \\ \end{gathered} $$ where $\Omega \subset \mathbb{R}^n $ is the domain and $\mathbb{R}^ + = \left\{ {t:0 \leqslant t < + \infty } \right\},{\text{ }}u\left( {x,t} \right) \geqslant 0$ is the temperature of the medium.     

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.     

Difference schemes for one class of variational inequalities and fourth-order mixed boundary-value problem

A difference scheme is constructed for the solution of the variational equation $$\begin{gathered} a\left( {u, v} \right)---u \geqslant \left( {f, v---u} \right)\forall v \varepsilon K,K \{ vv \varepsilon W_2^2 \left( \Omega \right) \cap \mathop {W_2^1 \left( \Omega \right)}\limits^0 ,\frac{{\partial v}}{{\partial u}} \geqslant 0 a.e. on \Gamma \} ; \hfill \\ \Omega = \{ x = (x_1 ,x_2 ):0 \leqslant x_\alpha< l_\alpha ,\alpha = 1, 2\} \Gamma = \bar \Omega - \Omega ,a(u, v) = \hfill \\ = \int\limits_\Omega {\Delta u\Delta } vdx \equiv (\Delta u,\Delta v, \hfill \\ \end{gathered} $$ The following bound is obtained for this scheme: $$\left\| {y - u} \right\|_{W_2 \left( \omega \right)}^2 = 0(h^{(2k - 5)/4} )u \in W_2^k \left( \Omega \right),\left\| {y - u} \right\|_{W_2^2 \left( \omega \right)} = 0(h^{\min (k - 2;1,5)/2} ),u \in W_\infty ^k \left( \Omega \right) \cap W_2^3 \left( \Omega \right)$$ The following bounds are obtained for the mixed boundary-value problem: $$\begin{gathered} \left\| {y - u} \right\|_{W_2^2 \left( \omega \right)} = 0\left( {h^{\min \left( {k - 2;1,5} \right)} } \right),u \in W_\infty ^k \left( \Omega \right),\left\| {y - u} \right\|_{W_2^2 \left( \omega \right)} = 0\left( {h^{k - 2,5} } \right), \hfill \\ u \in W_2^k \left( \Omega \right),k \in \left[ {3,4} \right] \hfill \\ \end{gathered} $$ .     

Existence and iteration of positive solution for a three-point boundary value problem with ap-Laplacian operator

In the paper, we obtain the existence of positive solutions and establish a corresponding iterative scheme for BVPs $$\left\{ \begin{gathered} (\phi _p (u\prime ))\prime + q(t)f(t, u) = 0,0< t< 1, \hfill \\ u(0) - B(u\prime (\eta )) = 0, u\prime (1) = 0 \hfill \\ \end{gathered} \right.$$ and $$\left\{ \begin{gathered} (\phi _p (u\prime ))\prime + q(t)f(t, u) = 0,0< t< 1, \hfill \\ u\prime (0) = 0, u(1) + B(u\prime (\eta )) = 0 \hfill \\ \end{gathered} \right.$$ The main tool is the monotone iterative technique. Here, the coefficientq(t) may be singular att = 0,1.     

Some dichotomy results for the quenching problem

It is shown that any solution to the semilinear problem{ ut = uxx + δ(1-u)-p , (x, t) ∈ (-1 , 1) × (0 , T ), u( ±1 , t) = 0, t ∈ (0 , T ), u(x, 0) = u0(x) 1, x ∈ [ 1 , 1] either touches 1 in finite time or converges smoothly to a steady state as t →∞. Some extensions of this result to higher dimensions are also discussed.     

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 ≤ ∞.