Finite-difference method for solving the first boundary-value problem for a second-order nonlinear ordinary differential equation with a divergent principal part

The approximation is studied of the first boundary-value problem for the equation (1) $$- \frac{d}{{dx}}K(x,\frac{{du}}{{dx}}) + f(x,u) = 0,0< x< 1,$$ with boundary conditions (2) $$u(0) = u(1) = 0$$ by difference boundary-value problems of form (3) $$- \left[ {a(x,w_{\bar x} )} \right]_x + \varphi (x,w) = 0,x \in w_r ,$$ (4) $$w(0) = w(1) = 0.$$ Theorems are established on the solvability of problem (3), (4). Theorems are proved on uniform convergence and on the order of uniform convergence. Here, as usual, boundedness is not assumed, but just the summability of the corresponding derivatives of the solutions of problem (1), (2). Also considered are singular boundary-value problems of form (1), (2), where uniform convergence with order h is proved under assumption of piecewise absolute continuity of the functionf(x,u(x)).     

Sufficient conditions for absolute asymptotic stability of linear equations in a Banach space

For a linear differential equation of the type (1) $$\frac{{dx}}{{dt}} = A_0 x(t) + A_1 x(t - \Delta _1 ) + ... + A_n x(t - \Delta _n )$$ we establish the followingTHEOREM. If $$\overline {\left| {z_1 } \right| = ...\underline{\underline \cup } \left| z \right|_n = 1\sigma \left( {A_0 + \sum\nolimits_{k = 1}^n {z_k A_k } } \right)} \subset \left\{ {\lambda :\operatorname{Re} \lambda< 0} \right\}$$ then system (1) is absolutely asymptotically stable.     

On the order of the error function of the square-full integers

LetL(x) denote the number of square full integers ≤x. By a square-full integer, we mean a positive integer all of whose prime factors have multiplicity at least two. It is well known that $$\left. {L(x)} \right| \sim \frac{{\zeta ({3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2})}}{{\zeta (3)}}x^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + \frac{{\zeta ({2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3})}}{{\zeta (2)}}x^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} ,$$ where ζ(s) denotes the Riemann Zeta function. Let Δ(x) denote the error function in the asymptotic formula forL(x). On the basis of the Riemann hypothesis (R.H.), it is known that \(\Delta (x) = O(x^{\tfrac{{13}}{{81}} + \varepsilon } )\) for every ε>0. In this paper, we prove the following results on the assumption of R.H.: (1) $$\frac{1}{x}\int\limits_1^x {\Delta (t)dt} = O(x^{\tfrac{1}{{12}} + \varepsilon } ),$$ (2) $$\int\limits_1^x {\frac{{\Delta (t)}}{t}\log } ^{v - 1} \left( {\frac{x}{t}} \right) = O(x^{\tfrac{1}{{12}} + \varepsilon } )$$ for any integer ν≥1. In fact, we prove some general results and deduce the above from them. On the basis of (1) and (2) above, we conjecture that \(\Delta (x) = O(x^{{1 \mathord{\left/ {\vphantom {1 {12}}} \right. \kern-0em} {12}} + \varepsilon } )\) under the assumption of R.H.     

A maximum principle for time-lag control problems with bounded state

A maximum principle is obtained for control problems involving a constant time lag τ in both the control and state variables. The problem considered is that of minimizing $$I(x) = \int_{t^0 }^{t^1 } {L (t,x(t), x(t - \tau ), u(t), u(t - \tau )) dt} $$ subject to the constraints 1 $$\begin{gathered} \dot x(t) = f(t,x(t),x(t - \tau ),u(t),u(t - \tau )), \hfill \\ x(t) = \phi (t), u(t) = \eta (t), t^0 - \tau \leqslant t \leqslant t^0 , \hfill \\ \end{gathered} $$ 1 $$\psi _\alpha (t,x(t),x(t - \tau )) \leqslant 0,\alpha = 1, \ldots ,m,$$ 1 $$x^i (t^1 ) = X^i ,i = 1, \ldots ,n$$ . The results are obtained using the method of Hestenes.     

Une Formule Approximative de Dimension pour Certains Produits de Riesz

Consider the Riesz product $\mu _a = \mathop \prod \limits_{n = 1}^\infty (1 + r\cos (q^n t + \varphi _n ))$ . We prove the following approximative formula for the dimension ofμ _a. $$\dim \mu _a = 1 - \frac{1}{{\log q}}\int_0^{2\pi } {(1 + r\cos x)\log (1 + r\cos x)\frac{{dx}}{{2\pi }} + 0\left( {\frac{r}{{q^2 \log q}}} \right).}$$      

Regularized sums of half-integer powers of a Sturm-Liouville operator

We investigate the question of the regularized sums of part of the eigenvalues z_n (lying along a direction) of a Sturm-Liouville operator. The first regularized sum is $$\sum\nolimits_{n = 1}^\infty {(z_n - n - \frac{{c_1 }}{n} + \frac{2}{\pi } \cdot z_n arctg \frac{1}{{z_n }} - \frac{2}{\pi }) = \frac{{B_2 }}{2} - c_1 \cdot \gamma + \int_1^\infty {\left[ {R(z) - \frac{{l_0 }}{{\sqrt z }} - \frac{{l_1 }}{z} - \frac{{l_2 }}{{z\sqrt z }}} \right]} } \sqrt z dz,$$ where the z_n are eigenvalues lying along the positive semi-axis, z _n ² =λ_n, $$l_0 = \frac{\pi }{2}, l_1 = - \frac{1}{2}, l_2 = - \frac{1}{4}\int_0^\pi {q(x) dx,} c_1 = - \frac{2}{\pi }l_2 ,$$ , B₂ is a Bernoulli number, γ is Euler's constant, and \(R(z)\) is the trace of the resolvent of a Sturm-Liouville operator.     

On strong summahility of Fourier series and differentiability of functions

ПустьS _n (f, x) — суммы Фурье периодической сумми руемой функцииf(x). Доказано, что если фун кцияФ(u), определенная, непрерывная и выпукл ая вверх для u≧0 (Ф(0)=0), удовлетворяет ус ловию (1) $$\int\limits_{ + 0} {\frac{{du}}{{\Phi (u)}}< \infty ,} $$ то имеет место следую щее вложение классов функций (2) $$S(\Phi ) = \left\{ {f:\mathop {\max }\limits_x \sum\limits_{n = o}^\infty \Phi (\left| {f(x) - S_n (fx)} \right|)< \infty } \right\} \subset Lip1,$$ и, более того, при услов ии (1) все функции из кла ссаS(Ф) непрерывно дифферен цируемы, а их производные имеют равномерно сходящие ся ряды Фурье. Установлено также, чт о если функция Ф удовл етворяет условию lim supФ(u/2)/Ф(u)<1, то условие (1) является н е только достаточным, но и необходимым для влож ения (2).     

Convergent approximations in parabolic variational inequalities II: Hamilton-jacobi inequalities

In this paper we consider two-sided parabolic inequalities of the form (li) $$\psi _1 \leqslant u \leqslant \psi _2 , in{\mathbf{ }}Q;$$ (lii) $$\left[ { - \frac{{\partial u}}{{\partial t}} + A(t)u + H(x,t,u,Du)} \right]e \geqslant 0, in{\mathbf{ }}Q,$$ for alle in the convex support cone of the solution given by $$K(u) = \left\{ {\lambda (\upsilon - u):\psi _1 \leqslant \upsilon \leqslant \psi _2 ,\lambda > 0} \right\}{\mathbf{ }};$$ (liii) $$\left. {\frac{{\partial u}}{{\partial v}}} \right|_\Sigma = 0, u( \cdot ,T) = \bar u$$ where $$Q = \Omega \times (0,T), \sum = \partial \Omega \times (0,T).$$ Such inequalities arise in the characterization of saddle-point payoffsu in two person differential games with stopping times as strategies. In this case,H is the Hamiltonian in the formulation. A numerical scheme for approximatingu is obtained by the continuous time, piecewise linear, Galerkin approximation of a so-called penalized equation. A rate of convergence tou of orderO(h ^1/2) is demonstrated in theL ²(0,T; H ¹(Ω)) norm, whereh is the maximum diameter of a given triangulation.     

The distribution of 4-full numbers

Let Q(x) denote the number of 4-full numbers not exceeding x. It is well known that $$Q(x) = \sum\limits_{j = 4}^7 {r_j x^{1/j} + R(x)}$$ where $$r_j = \mathop {res}\limits_{s = 1/j} (F(s)/s), F(s) = \mathop \prod \limits_P \left( {1 + \frac{{p^{ - 4s} }}{{1 - p^{ - s} }}} \right)$$ and R(x) is the remainder. This paper proves that $$R(x) \ll x^{3626/35461 + \varepsilon }$$ where ε is any positive number.     

On the modulus of continuity in connection with a problem of J. Szabados concerning strong approximation

Пустьf(x) — интегрируемая 2π-периодическая функция, aω(f,δ) иs _n(x)=s_n(f, x). соответственно, модуль непрерывности иn-ая сумма Фурье этой функции. В настоящей работе, продолжающей исследования Г. Фрейда, Л. Лейндлера—E. M. Никищина, И. Сабадоша и К. И. Осколкова, доказывается следующая теорема.Если Ω(u) — выпуклая или вогнутая непрерывная функция и если (1) 1 $$\left\| {\left. {\sum\limits_{k = 1}^\infty \Omega (|S_k (x) - f(x)|)} \right\|_C } \right.$$ то 1 $$\omega (f;\delta ) = O\left( {\delta \int\limits_\delta ^1 {\frac{{\bar \Omega (v)}}{{v^2 }}dv} } \right),$$ где ¯Ω(v) —функция, обратная к Ω(и). При этом существует функция f₀(х), удовлетворяющая условию (1), для которой $$\omega (f;\delta ) = c\delta \int\limits_\delta ^1 {\frac{{\bar \Omega (v)}}{{v^2 }}dv} (c > 0).$$ ЕслиΩ(u)— вогнутая функция, то интеграл \(\int\limits_\delta ^1 {\frac{{\bar \Omega (v)}}{{v^2 }}dv} \) можно заменить на \(\int\limits_{\bar \Omega (\delta )}^1 {\frac{{du}}{{\Omega (u)}}.} \) . Отсюда вытекает, что еслиΩ(u) — функция типа модуля непрерывности, то для того, чтобы (1) всегда влекло принадлежность f(x) классу Lip 1, необходимо и достаточно условие \(\int\limits_0^1 {\frac{{du}}{{\Omega (u)}}}< \infty .\)      

Schrödinger equation and oscillatory Hilbert transforms of second degree

Let $h(t,x): = p.v. \sum\limits_{n \in Z\backslash \left| 0 \right|} {\frac{{e^{\pi i(tn^2 + 2xn)} }}{{2\pi in}}} = \mathop {\lim }\limits_{N \to \infty } \sum\limits_{0< \left| n \right| \leqslant N} {\frac{{e^{\pi i(tn^2 + 2xn)} }}{{2\pi in}}} $ ( $(i = \sqrt { - 1;} t,x$ -real variables). It is proved that in the rectangle $D: = \left\{ {(t,x):0< t< 1,\left| x \right| \leqslant \frac{1}{2}} \right\}$ , the function h satisfies the followingfunctional inequality: $\left| {h(t,x)} \right| \leqslant \sqrt t \left| {h\left( {\frac{1}{t},\frac{x}{t}} \right)} \right| + c,$ where c is an absolute positive constant. Iterations of this relation provide another, more elementary, proof of the known global boundedness result $\left\| {h; L^\infty (E^2 )} \right\| : = ess sup \left| {h(t,x)} \right|< \infty .$ The above functional inequality is derived from a general duality relation, of theta-function type, for solutions of the Cauchy initial value problem for Schrödinger equation of a free particle. Variation and complexity of solutions of Schrödinger equation are discussed.     

Fourier coefficients of Siegel cusp forms of genus n

Let F(Z) be a cusp form of integral weight k relative to the Siegel modular group Sp_n(Z) and let f(N) be its Fourier coefficient with index N. Making use of Rankin's convolution, one proves the estimate (1) $$f(\mathcal{N}) = O(\left| \mathcal{N} \right|^{\tfrac{k}{2} - \tfrac{1}{2}\delta (n)} ),$$ where $$\delta (n) = \frac{{n + 1}}{{\left( {n + 1} \right)\left( {2n + \tfrac{{1 + ( - 1)^n }}{2}} \right) + 1}}.$$ Previously, for n ≥ 2 one has known Raghavan's estimate $$f(\mathcal{N}) = O(\left| \mathcal{N} \right|^{\tfrac{k}{2}} )$$ In the case n=2, Kitaoka has obtained a result, sharper than (1), namely: (2) $$f(\mathcal{N}) = O(\left| \mathcal{N} \right|^{\tfrac{k}{2} - \tfrac{1}{4} + \varepsilon } ).$$ At the end of the paper one investigates specially the case n=2. It is shown that in some cases the result (2) can be improved to, apparently, unimprovable estimates if one assumes some analogues of the Petersson conjecture. These results lead to a conjecture regarding the optimal estimates of f(N), n=2.     

A stable Nyström interpolant for some Mellin convolution equations

We consider the numerical solution of second kind integral equations of the form $$u(y) - \int\limits_0^1 {k(y/x)\frac{{u(x)}}{x}dx = f(y), 0 \le y \le 1,} $$ for some given kernelk(t). These equations, usually indicated as of Mellin type, arise in a variety of applications. In particular, we examine a Nyström interpolant based on the following product quadrature rule: $$\int\limits_0^1 {k(y/x)\frac{{u(x)}}{x}dx \approx \sum\limits_{i = 0}^n {w_{ni} (y)u(x_{mi} ).} } $$ This rule is obtained by interpolatingu(x) by the Lagrange polynomial associated with the set of Gauss-Radau nodes {x _ni}. Under certain assumptions on the kernelk(t), we are able to prove the stability of our interpolant and derive convergence estimates.     

Existence of a solution of a difference scheme for one variational problem

The problem of minimizing the functional (A) $${}_a\smallint ^b \varphi (x,y,y',y'')dx$$ under the conditions (B) $$y(a) = a_0 ,y'(a) = a_1 ,y(b) = b_0 ,y'(b) = b_1$$ is replaced by the problem of finding the vector (y₁,y₂,...,y_n?1) on which the sum (C) $$\sum\limits_{\kappa = 0}^n {C_\kappa \varphi (x_\kappa ,y_\kappa ,\left. {\frac{{y_{\kappa + 1} - y_\kappa }}{h},\frac{{y_{\kappa + 1} - 2y_\kappa + y_{\kappa + 1} )}}{{h^2 }}} \right)}$$ takes a minimal value. Under certain conditions on ? andC _k it is proved that a solution exists for the difference scheme constructed. The method of differentiation with respect to a parameter is used for the proof.     

Bases formed of successive primitives

Necessary and sufficient conditions are found in order for the system of successive primitives $$\left\{ {F_n (z) = \sum\nolimits_{k = 0}^\infty {\frac{{a_{k - n} }}{{k!}}z^k } } \right\}, n = 0,1,2, ...,$$ generated by the integer-valued function \(F_n (z) = \sum\nolimits_{k = 0}^\infty {\frac{{a_k }}{{k!}}zk} \) of growth no higher than first order of the normal typeσ(F₀(z) ε [1;σ] to form a quasi-power basis in the class [1; σ].     

Solvability of the Dirichlet problem for degenerate quasilinear elliptic equations

In a bounded domain of the n -dimensional (n?2) space one considers a class of degenerate quasilinear elliptic equations, whose model is the equation $$\sum\limits_{i = 1}^n {\frac{{\partial F}}{{\partial x_i }}} (a^{\ell _i } (u)\left| {u_{x_i } } \right|^{m_i - 2} u_{x_i } ) = f(x),$$ where x =(x₁,..., x_r), l_i?0, m_i>1, the function f is summable with some power, the nonnegative continuous function a(u) vanishes at a finite number of points and satisfies \(\frac{{lim}}{{\left| u \right| \to \infty }}a(u) > 0\) . One proves the existence of bounded generalized solutions with a finite integral $$\int\limits_\Omega {\sum\limits_{i = 1}^n {a^{\ell _i } (u)\left| {u_{x_i } } \right|^{m_i } dx} }$$ of the Dirichlet problem with zero boundary conditions.     

On the normalizing multiplier of the generalized Jackson kernel

We consider the question of evaluating the normalizing multiplier $$\gamma _{n,k} = \frac{1}{\pi }\int_{ - \pi }^\pi {\left( {\frac{{sin\tfrac{{nt}}{2}}}{{sin\tfrac{t}{2}}}} \right)^{2k} dt} $$ for the generalized Jackson kernel J _n,k (t). We obtain the explicit formula $$\gamma _{n,k} = 2\sum\limits_{p = 0}^{\left[ {k - \tfrac{k}{n}} \right]} {( - 1)\left( {\begin{array}{*{20}c} {2k} \\ p \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {k(n + 1) - np - 1} \\ {k(n - 1) - np} \\ \end{array} } \right)} $$ and the representation $$\gamma _{n,k} = \sqrt {\frac{{24}}{\pi }} \cdot \frac{{(n - 1)^{2k - 1} }}{{\sqrt {2k - 1} }}\left[ {1\frac{1}{8} \cdot \frac{1}{{2k - 1}} + \omega (n,k)} \right],$$ , where $$\left| {\omega (n,k)} \right| < \frac{4}{{(2k - 1)\sqrt {ln(2k - 1)} }} + \sqrt {12\pi } \cdot \frac{{k^{\tfrac{3}{2}} }}{{n - 1}}\left( {1 + \frac{1}{{n - 1}}} \right)^{2k - 2} .$$ .     

Derivation of Freund - Witten adelic formula for four-point Veneziano amplitudes

On the basis of an analysis on the adelic group (Tate's formula) a regularization is proposed for the divergent infinite product ofp-adic Γ functions: $$\Gamma _p (\alpha ) = \frac{{1 - p^{\alpha - 1} }}{{1 - p^{ - \alpha } }}, p = 2,3,5...,$$ and the adelic formula $$reg\prod\limits_{p = 2}^\infty {\Gamma p(\alpha ) = \frac{{(\zeta \alpha )}}{{\zeta (1 - \alpha )}},} $$ where ζ(α) is the Riemann ζ function, is proved.     

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 \(\int\limits_a^{ + \infty } {x^{r - 1} f^{(r)} (x)dx}\)      

Well-posedness of a semilinear heat equation with weak initial data

This article mainly consists of two parts. In the first part the initial value problem (IVP) of the semilinear heat equation $$\begin{gathered} \partial _t u - \Delta u = \left| u \right|^{k - 1} u, on \mathbb{R}^n x(0,\infty ), k \geqslant 2 \hfill \\ u(x,0) = u_0 (x), x \in \mathbb{R}^n \hfill \\ \end{gathered} $$ with initial data in $\dot L_{r,p} $ is studied. We prove the well-posedness when $$1< p< \infty , \frac{2}{{k(k - 1)}}< \frac{n}{p} \leqslant \frac{2}{{k - 1}}, and r =< \frac{n}{p} - \frac{2}{{k - 1}}( \leqslant 0)$$ and construct non-unique solutions for $$1< p< \frac{{n(k - 1)}}{2}< k + 1, and r< \frac{n}{p} - \frac{2}{{k - 1}}.$$ In the second part the well-posedness of the avove IVP for k=2 with μ₀?H ^s (? ⁿ ) is proved if $$ - 1< s, for n = 1, \frac{n}{2} - 2< s, for n \geqslant 2.$$ and this result is then extended for more general nonlinear terms and initial data. By taking special values of r, p, s, and u₀, these well-posedness results reduce to some of those previously obtained by other authors [4, 14].