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.     

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|.     

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 the permutability of Backlund transformations

Let(?)=B_ηu:2(q-(?))+(⊿((?)-2q))+(2q_x+(?)_x))η=0,2(r-(?)+(⊿(2(?)-r)+(r_x+2(?)_x))η=0,u=(q,r)~Tbe the Backlund transformation (BT) of the hierarchy of AKNS equations,where η is a parameterand Δ=integral from -∞ to x (qr-(?))dx′.It is shown in this paper the infinitesimal BT B_(η+ε)B_η~(-1) admits thefollowing expansionB_(η+ε)B_η~(-1)u=u+εsum from n=0 to ∞ β_n(JL~(n+1)u)η~n,β_n=1+(-1)~n2~(-n-1),where L is the recurrence operator of the hierarchy and ε is an infinitesimal parameter.Thisexpansion implies the equivalence between the permutabiliy of BTs and the involution in pairs ofconserved densities.     

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.     

H convergence for quasi-linear elliptic equations with quadratic growth

We consider in this paper the limit behavior of the solutionsu ^? of the problem $$\begin{gathered} - div(a^\varepsilon Du^\varepsilon ) + \gamma u^\varepsilon = H^\varepsilon (x, u^\varepsilon , Du^\varepsilon ), \hfill \\ u^\varepsilon \in H_0^1 (\Omega ) \cap L^\infty (\Omega ), \hfill \\ \end{gathered}$$ whereH ^? has quadratic growth inDu ^? anda ^? (x) is a family of matrices satisfying the general assumptions of abstract homogenization. We also consider the problem $$\begin{gathered} - div(a^\varepsilon Du^\varepsilon ) + G^\varepsilon (x, u^\varepsilon , Du^\varepsilon ) = f \in H^{ - 1} (\Omega ), \hfill \\ u^\varepsilon \in H_0^1 (\Omega ), G^\varepsilon (x, u^\varepsilon , Du^\varepsilon ) \in L^1 (\Omega ), u^\varepsilon G^\varepsilon (x, u^\varepsilon , Du^\varepsilon ) \in L^1 (\Omega ) \hfill \\ \end{gathered}$$ whereG ^? has quadratic growth inDu ^? and satisfiesG ^? (x, s, ξ)s ≥ 0. Note that in this last modelu ^? is in general unbounded, which gives extra difficulties for the homogenization process. In both cases we pass to the limit and obtain an homogenized equation having the same structure.     

On a Żołądek theorem

We prove the existence of cubic systems of the form $$ \begin{gathered} \dot x = y[1 - 2r(5 + 3r^2 )x + \gamma \lambda ^2 x^2 ] + a_0 x + a_1 x^2 + a_2 xy + a_3 y^2 + a_4 x^3 + a_5 x^2 y + a_6 xy^2 , \hfill \\ \dot y = - x(1 - 8rx)(1 - 3r\gamma x) - 2x[2(1 - 3r^2 ) - r\gamma (7 - 15r^2 )x]y \hfill \\ - [r(11 + r^2 ) + \gamma (1 - 22r^2 - 3r^4 )x]y^2 \hfill \\ - 2r\gamma \delta y^3 + a_0 y + a_7 x^2 + a_8 xy + a_9 y^2 + a_{10} x^3 + a_{11} x^2 y, \hfill \\ \end{gathered} $$ where α = 3r ² + 17, γ = r ² + 3, δ = 1 ? r ², and λ = 3r ² + 1, that have at least eleven limit cycles in a neighborhood of the point O(0, 0).     

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 $$\sum\limits_{m< x} {g(m)<< x^{1 + \varepsilon } .} $$      

A note on periodic solutions of a differential equation with two parameters

In [1] (p. 215), the authors Andronov, Leontovich-Andronova, Gordon, and Maier, consider the following equation: $$\left\{ \begin{gathered} \tfrac{{dx}}{{dt}} = y, \hfill \\ \tfrac{{dy}}{{dt}} = x + x^2 - \left( {\varepsilon _1 + \varepsilon _2 x} \right)y, \hfill \\ \end{gathered} \right.$$ whereε ₁ andε ₂ are real constants andε ₁ andε ₂ are not both zero. They proved that there are no non-trivial periodic solutions except possibly for the case $0< \tfrac{{\varepsilon _1 }}{{\varepsilon _2 }}< \tfrac{3}{2}$ . They left that case as an open problem. In this note we prove that there are indeed no non-trivial periodic solutions in the case $0< \tfrac{{\varepsilon _1 }}{{\varepsilon _2 }}< \tfrac{3}{2}$ either. Our method of proof consists essentially of constructing a Dulac function (see [6] and [9]) and using the conception of Duff's rotated vector field (see [4], [7], [8], [10], and [11]).     

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.     

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.     

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.     

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 $$\begin{gathered} b_{n,m} = 8\sum\limits_{i = 0}^{s - 2} {\left( {\begin{array}{*{20}c} {m + 2i + 1} \\ l \\ \end{array} } \right)4^{s - 2 - 1} } \hfill \\ {\text{ }}c_n = \left\{ \begin{gathered} \left( {\begin{array}{*{20}c} {n - 1} \\ {s - 1} \\ \end{array} } \right){\text{ if both }}n,m{\text{ are even;}} \hfill \\ {\text{ 0 if both }}n,m{\text{ are odd}}{\text{.}} \hfill \\ \end{gathered} \right. \hfill \\ \end{gathered} $$      

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: $$\begin{gathered} \left\| {y - u} \right\|_{W_2^2 \left( \omega \right)} = 0\left( {h^{\min \left( {k - 2;1,5} \right)} } \right),u \in W_\infty ^k \left( \Omega \right),\left\| {y - u} \right\|_{W_2^2 \left( \omega \right)} = 0\left( {h^{k - 2,5} } \right), \hfill \\ u \in W_2^k \left( \Omega \right),k \in \left[ {3,4} \right] \hfill \\ \end{gathered} $$ .     

On the null-controllability in function space of nonlinear systems of neutral functional differential equations with limited controls

Consider the following functional equations of neutral type: $$\begin{gathered} (i) (d/dt)D(t,x_t ) = L(t,x_t ), \hfill \\ (ii) (d/dt)D(t,x_t ) = L(t,x_t ) + B(t)u(t), \hfill \\ (iii) (d/dt)D(t,x_t ) = L(t,x_t ) + B(t)u(t) + f(t,x(t),u(t)), \hfill \\ \end{gathered} $$ whereD, L are bounded linear operators fromC([?h, 0],E ⁿ) intoE ⁿ for eacht?(σ, ∞) =J, B is ann ×m continuous matrix function,u:J →C ^m is square integrable with values in the unitm-dimensional cubeC ^m, andf(t, 0, 0)=0. We prove that, if the system (i) is uniformly asymptotically stable and if the controlled system (ii) is controllable, then the system (iii) is null-controllable with constraints, provided that $$f = f_1 + f_2 $$ , where $$\begin{gathered} |f_1 (t,\phi ,0)| \leqslant \varepsilon \parallel \phi \parallel , |f_2 (t,\phi ,0)| \leqslant \pi (t)\parallel \phi \parallel , t \geqslant \sigma , \hfill \\ \Pi = \int_0^\infty {\pi (t)dt< \infty .} \hfill \\ \end{gathered} $$      

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   

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.     

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.     

Multiple rational trigonometric sums and multiple integrals

We obtain an estimate of the modulus of a complete multiple rational trigonometric sum: $$\left| {\sum {_{x_{1, \ldots ,} x_r = 1^{\exp \left( {{{2\pi if\left( {x_{1, \ldots ,} x_r } \right)} \mathord{\left/ {\vphantom {{2\pi if\left( {x_{1, \ldots ,} x_r } \right)} q}} \right. \kern-\nulldelimiterspace} q}} \right)} }^q } } \right| \ll q^{{{r - 1} \mathord{\left/ {\vphantom {{r - 1} {n + \varepsilon }}} \right. \kern-\nulldelimiterspace} {n + \varepsilon }}} ,$$ where $$\begin{gathered} f\left( {x_{1, \ldots ,} x_r } \right) = \sum {_{0 \leqslant t_1 , \ldots ,t_r \leqslant n^a t_1 , \ldots ,t_r x_1^{t_1 } \ldots x_r^{t_r } ,} } \hfill \\ a_{0, \ldots ,0} = 0,\left( {a_{0, \ldots ,0,1} , \ldots ,a_{n, \ldots ,n,} q} \right) = 1 \hfill \\ \end{gathered} $$ , and an estimate of the modulus of a multiple trigonometric integral.     

BOUNDARY VALUE PROBLEMS OF SINGULARLY PERTURBED INTEGRO－DIFFERENTIAL EQUATIONS

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