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

A non-smooth two points boundary value problem on Riemannian manifolds

Let (?,〈,〉_R) be a Riemannian manifold, x₀, x₁ ∈ ? and V: ?→? a locally Lipschitz continuous potential function. In this paper we look for the solutions x:[0, 1]→ →? of the differential inclusion (0.1) $$D_t \dot x(t) \in \partial V\left( {x(t)} \right)$$ with boundary conditions (0.2) $$x(0) = x_0 ,x(1) = x_1 $$ where D_tx(t) denotes the covariant derivative of x(t) along the direction of x(t) and ?V(x(t)) the generalized gradient of V in x(t). Using a variant of the Lusternik-Schnirelman critical point theory, we state the existence of infinitely many solutions of problem (0.1)-(0.2) when ? is not contractible in itself.     

On positive solutions of a reciprocal difference equation with minimum

In this paper we consider positive solutions of the following difference equation $$x_{n + 1} = \min \left\{ {\frac{A}{{x_n }},\frac{B}{{x_{n - 2} }}} \right\}, A, B > 0.$$ We prove that every positive solution is eventually periodic. Also, we present here some results concerning positive solutions of the difference equation $$x_{n + 1} = \min \left\{ {\frac{A}{{x_n x_{n - 1} ...x_{n - k} }},\frac{B}{{x_{n - (k + 2)} ...x_{n - (2k + 2)} }}} \right\}, A, B > 0.$$      

On the convergence to infinity of Fourier series along dense subsequences of numbers

В статье доказываетс я Теорема.Какова бы ни была возрастающая последовательность натуральных чисел {H _k } _{k = 1} ^∞ c $$\mathop {\lim }\limits_{k \to \infty } \frac{{H_k }}{k} = + \infty$$ , существует функцияf∈L(0, 2π) такая, что для почт и всех x∈(0, 2π) можно найти возраст ающую последовательность номеров {n_k(x)} _k=1 ^∞ ,удовлетворяющую усл овиям 1) $$n_k (x) \leqq H_k , k = 1,2, ...,$$ 2) $$\mathop {\lim }\limits_{t \to \infty } S_{n_{2t} (x)} (x,f) = + \infty ,$$ 3) $$\mathop {\lim }\limits_{t \to \infty } S_{n_{2t - 1} (x)} (x,f) = - \infty$$ .     

О стРУктУРЕ цЕлОИ ФУН кцИИ, ОБРАЩАУЩЕИсь В НУль Н А жАДАННОИ пОслЕДОВАтЕльНОстИ тОЧЕк

The following result is proved. Theorem.Let λ _n ,0<λ _n ↑∞, be a sequence of positive numbers with finite density $$\sigma = \mathop {\lim }\limits_{n \to \infty } \frac{n}{{\lambda _n }}$$ and let a compact set K has the following property: it intersects the real axis along the interval [a, b], where a is the very left point of K, B is the very right point of K; furthermore, K intersects every vertical straight line Re z=α, a≤α≤b, along an interval. If 1) $$F(z) \in [1,S_{ - \pi \sigma }^{\pi \sigma } \cup K(\alpha + i\pi \sigma ) \cup K(\alpha - i\pi \sigma )], \alpha \in R;$$ 2) 2) $$F( \pm \lambda _n ) = 0, n = 1,2,...,$$ then $$F(z) = A(z)e^{\alpha z} \alpha (z),$$ where $$A(z) \in [1,K], \alpha (z) = \prod\limits_1^\pi {\left( {1 - \frac{{z^2 }}{{\lambda _n^2 }}} \right)}$$ . This result generalizes the theorem of Kaz'min [3]. Three corollaries are also proved, which generalize the theorems ofBoas [1] andPólya [6]. In the theorems of Boas and Pólya, we haveF(n)=0, ?n ε Z. In our case $$F( \pm \lambda _n ) = 0, 0< \lambda _n \uparrow \infty , \sigma = \mathop {\lim }\limits_{n \to \infty } \frac{n}{{\lambda _n }}$$ .     

On the divergence of Lagrange interpolation processes on a set of second category

В статье даны полные д оказательства следу ющих утверждений. Пустьω — непрерывная неубывающая полуадд итивная функций на [0, ∞),ω(0)=0 и пусть M?[0, 1] — матрица узл ов интерполирования. Если $$\mathop {\lim sup}\limits_{n \to \infty } \omega \left( {\frac{1}{n}} \right)\log n > 0$$ то существует точкаx ₀∈[0,1] и функцияf ∈ С[0,1] таки е, чтоω(f, δ)=О(ω(δ)), для которой $$\mathop {\lim sup}\limits_{n \to \infty } |L_n (\mathfrak{M},f,x_0 ) - f(x_0 )| > 0$$ Если же $$\mathop {\lim sup}\limits_{n \to \infty } \omega \left( {\frac{1}{n}} \right)\log n = \infty$$ , то существуют множес твоE второй категори и и функцияf ∈ С[0,1],ω(f, δ)=o(ω(δ)) та кие, что для всехx∈E $$\mathop {\lim sup}\limits_{n \to \infty } |L_n (\mathfrak{M},f,x)| = \infty$$ . Исправлена погрешно сть, допущенная автор ом в [5], и отмеченная в работе П. Вертеши [9].     

Partial regularity for quasilinear nonuniformly elliptic systems of the general type

We establish the partial C^1,α-regularity of weak solutions of nonhomogeneous nonuniformly elliptic systems of the type $$ - \frac{\partial }{{\partial x_\alpha }}A_\alpha ^i (x,u,u_x ) = B^i (x,u,u_x ),{\text{ }}i = 1,...,n$$ . The system of Euler equations of the variational problem of finding a minimum of the integral $\int\limits_\Omega {\mathcal{F}(u_x )dx} $ with an integrand of the type $$\mathcal{F}(p) = a|p|^2 + b|p|^m + \sqrt {1 + \det ^2 p,} {\text{ }}a > 0,{\text{ }}b > 0$$ , for b large enough, is a typical example of systems under consideration. Bibliography: 11 titles.     

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   

Conditionally oscillatory half-linear differential equations

We consider a nonoscillatory half-linear second order differential equation (*) $$ (r(t)\Phi (x'))' + c(t)\Phi (x) = 0,\Phi (x) = \left| x \right|^{p - 2} x,p > 1, $$ and suppose that we know its solution h. Using this solution we construct a function d such that the equation (**) $$ (r(t)\Phi (x'))' + [c(t) + \lambda d(t)]\Phi (x) = 0 $$ is conditionally oscillatory. Then we study oscillations of the perturbed equation (**). The obtained (non)oscillation criteria extend existing results for perturbed half-linear Euler and Euler-Weber equations.     

On the interrelation of almost sure invariance principles for certain stochastic adaptive algorithms and for partial sums of random variables

Since the novel work of Berkes and Philipp⁽³⁾ much effort has been focused on establishing almost sure invariance principles of the form (1) $$\left| {\sum\limits_{i = 1}^{|\_t\_|} {x_1 - X_t } } \right| \ll t^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} - \gamma } $$ where {x _i ,i=1,2,3,...} is a sequence of random vectors and {X _t ,t>-0} is a Brownian motion. In this note, we show that if {A _k ,k=1,2,3,...} and {b _k ,k=1,2,3,...} are processes satisfying almost-sure bounds analogous to Eq. (1), (where {X _t ,t≥0} could be a more general Gauss-Markov process) then {h _k ,k=1,2,3...}, the solution of the stochastic approximation or adaptive filtering algorithm (2) $$h_{k + 1} = h_k + \frac{1}{k}(b_k - A_k h_k )for{\text{ }}k{\text{ = 1,2,3}}...$$ also satisfies and almost sure invariance principle of the same type.     

On minor forms of Desargues and Pappus

Let Π be a projective plane coordinatized by a ternary ring (R, F). In addition to the two operations + and ·, defined bya+b =F(a,1,b and \(a \cdot b = F(a,b,0)\) , a third operation * is defined by \(a * b = F(1,a,b),\forall a,b \in R\) Several minor forms of the propositions of Desargues and Pappus are introduced in Π and their geometrical properties are developed. Several algebraic results are obtained in connection with these minor forms. For example, the first minor form of DesarguesD ₁ is proved to be equivalent to each of the following algebraic identities in every (R, F): (1) $$a \cdot c = c \cdot a \Rightarrow F(a,c,b) = F(c,a,b),$$ (2) $$a \cdot (1 + b) = a + a \cdot b,$$ (3) $$a * b = a + b$$ (4) $$F(x,m,k) = (x \cdot m) * k,\forall a,b,c,k,m,x \in R.$$ Some more algebraic identities are characterized byD ₂ andD ₃.     

Diophantine properties of IETs and general systems: quantitative proximality and connectivity

Three properties of dynamical systems (recurrence, connectivity and proximality) are quantified by introducing and studying the gauges (measurable functions) corresponding to each of these properties. The properties of the proximality gauge are related to the results in the active field of shrinking targets. The emphasis in the present paper is on the IETs (interval exchange transformations) $( \mathcal {I},T)$ , $\mathcal {I}=[0,1)$ . In particular, we prove that if an IET T is ergodic (relative to the Lebesgue measure λ), then the equality A1 $$ \liminf_{n\to\infty} \, n\, \bigl|T^n(x)-y \bigr|=0 $$ holds for λ×λ-a.a. $(x,y)\in \mathcal {I}^{2}$ . The ergodicity assumption is essential: the result does not extend to all minimal IETs. Also, the factor? n? in (A1) is optimal (e.g., it cannot be replaced by n?ln(ln(lnn))). On the other hand, for Lebesgue almost all 3-IETs $( \mathcal {I},T)$ we prove that for all ?>0 A2 $$ \liminf_{n\to\infty} \, n^ \epsilon \bigl |T^n(x)-T^n(y)\bigr| = \infty,\quad\text{for Lebesgue a.a.} \ (x,y)\in \mathcal {I}^2. $$ This should be contrasted with the equality lim?inf _n→∞?|T ⁿ (x)?T ⁿ (y)|=0, for a.a. $(x,y)\in \mathcal {I}^{2}$ , which holds since $( \mathcal {I}^{2}, T\times T)$ is ergodic (because generic 3-IETs $( \mathcal {I},T)$ are weakly mixing). We introduce the notion of τ-entropy of an IET which is related to obtaining estimates of type (A2). We also prove that no 3-IET is strongly topologically mixing.     

Nonautonomous locally Lipschitz continuous functional differential equations in spaces of continuous functions

A local existence and uniqueness result for the functional differential equation in a Banach space X (FDE) $$\begin{gathered} x\prime (t) \in f(t)x(t) + g(t)x_t ,0 \leqslant t \leqslant T, \hfill \\ x_0 = \phi \in C( - R,0;X) \hfill \\ \end{gathered} $$ is obtained, for the case where the operatorsf(t) satisfy only a local dissipativity condition and the operatorsg(t) are only locally Lipschitz continuous. The conditions include equations defined on cones.     

On symmetric and nonsymmetric blowup for a weakly quasilinear heat equation

We construct blow-up patterns for the quasilinear heat equation (QHE) $$u_t = \nabla \cdot (k(u)\nabla u) + Q(u)$$ in Ω×(0,T), Ω being a bounded open convex set in ? ^N with smooth boundary, with zero Dirichet boundary condition and nonnegative initial data. The nonlinear coefficients of the equation are assumed to be smooth and positive functions and moreoverk(u) andQ(u)/u ^p with a fixedp>1 are of slow variation asu→∞, so that (QHE) can be treated as a quasilinear perturbation of the well-known semilinear heat equation (SHE) $$u_t = \nabla u) + u^p .$$ We prove that the blow-up patterns for the (QHE) and the (SHE) coincide in a structural sense under the extra assumption $$\smallint ^\infty k(f(e^s ))ds = \infty ,$$ wheref(v) is a monotone solution of the ODEf′(v)=Q(f(v))/v ^p defined for allv?1. If the integral is finite then the (QHE) is shown to admit an infinite number of different blow-up patterns.     

Inequalities for sine sums with more variables

A result of Vietoris states that if the real numbers \(a_1,\ldots ,a_n\) satisfy

$$\begin{aligned} \text{(*) } \qquad a_1\ge \frac{a_2}{2} \ge \cdots \ge \frac{a_n}{n}>0 \quad \text{ and } \quad a_{2k-1}\ge a_{2k} \quad (1\le k\le n/2), \end{aligned}$$

then, for \(x_1,\ldots ,x_m>0\) with \(x_1+\cdots +x_m <\pi \),

$$\begin{aligned} \begin{aligned} \text{(**) } \qquad \sum _{k=1}^n a_k \frac{\sin (k x_1) \cdots \sin (k x_m)}{k^m}>0. \end{aligned} \end{aligned}$$

We prove that \((**)\) (with “\(\ge \)” instead of “>”) holds under weaker conditions. It suffices to assume, instead of \((*)\), that

$$\begin{aligned} \sum _{k=1}^N a_k \frac{\sin (kt)}{k}>0 \quad (N=1,\ldots ,n; \, 0<t<\pi ), \end{aligned}$$

and, moreover, \((**)\) is valid for a larger region, namely, \(x_1,\ldots ,x_m\in (0,\pi )\).

     

A problem of Carlitz and its generalizations

Let ${\mathbb{F}_q}$ be the finite field of characteristic p > 2 with q elements. Carlitz proposed the problem of finding an explicit formula for the number of solutions to the equation $$(x_1+ x_2+\cdots+x_n)^2=a\, x_1x_2\cdots x_n,$$ where ${a\in \mathbb{F}_q^*}$ and n ≥ 3. By using the augmented degree matrix and Gauss sums, we consider the generalizations of the above equation and partially solve Carlitz’s problem. Moreover, the technique developed in this paper may be applied to other equations of the form ${h_1^\lambda=h_2}$ with ${h_1, h_2 \in \mathbb{F}_q[x_1,\ldots,x_n]}$ and ${\lambda \in \mathbb{N}}$ .     

On minimum norm solutions

This note investigates the problem $$\min x_p^p /p,s.t.Ax \geqslant b,$$ where 1<p<∞. It is proved that the dual of this problem has the form $$\max b^T y - A^T y_q^q /q,s.t.y \geqslant 0,$$ whereq=p/(p?1). The main contribution is an explicit rule for retrieving a primal solution from a dual one. If an inequality is replaced by an equality, then the corresponding dual variable is not restricted to stay nonnegative. A similar modification exists for interval constraints. Partially regularized problems are also discussed. Finally, we extend an observation of Luenberger, showing that the dual of $$\min x_p ,s.t.Ax \geqslant b,$$ is $$\max b^T y,s.t.y \geqslant 0,A^T y_q \leqslant 1,$$ and sharpening the relation between a primal solution and a dual solution.     

On multiplicative congruences

Let ${\varepsilon}$ be a fixed positive quantity, m be a large integer, x _j denote integer variables. We prove that for any positive integers N ₁, N ₂, N ₃ with ${N_1N_2N_3 > m^{1+\varepsilon}, }$ the set $$\{x_1x_2x_3 \quad ({\rm mod}\,m): \quad x_j\in [1,N_j]\}$$ contains almost all the residue classes modulo m (i.e., its cardinality is equal to m + o(m)). We further show that if m is cubefree, then for any positive integers N ₁, N ₂, N ₃, N ₄ with ${ N_1N_2N_3N_4 > m^{1+\varepsilon}, }$ the set $$\{x_1x_2x_3x_4 \quad ({\rm mod}\,m): \quad x_j\in [1,N_j]\}$$ also contains almost all the residue classes modulo m. Let p be a large prime parameter and let ${p > N > p^{63/76+\varepsilon}.}$ We prove that for any nonzero integer constant k and any integer ${\lambda\not\equiv 0 \,\, ({\rm mod}\,p)}$ the congruence $$p_1p_2(p_3+k)\equiv \lambda \quad ({\rm mod}\, p) $$ admits (1 + o(1))π(N)³/p solutions in prime numbers p ₁, p ₂, p ₃ ≤ N.     

On Waring's Problem for Six Cubes and Higher Degrees

It is proved that the equation $$n = x_{\,1}^3 + x_{\,2}^3 + x_{\,3}^3 + x_{\,4}^3 + x_{\,5}^3 + x_{\,6}^3 + u^4 + v^9$$ has nonnegative integral solutions if $n \equiv 1\left( {\bmod 5} \right)$ is even and sufficiently large. Bibliography: 8 titles.     

Existence and uniqueness of a regular solution of the Cauchy-Dirichlet problem for a class of doubly nonlinear parabolic equations

The class of equations of the type (1) $$\partial u/\partial t - div\overrightarrow a (u,\nabla u) = f,$$ such that (2) $$\begin{array}{l} \overrightarrow a (u,p) \cdot p \ge v_0 |u|^l |p|^m - \Phi _0 (u), \\ |\overrightarrow a (u,p)| \le \mu _1 |u|^l |p|^{m - 1} + \Phi _1 (u) \\ \end{array}$$ with some m ∈ (1,2), l≥0, and Φ _i (u)≥0 is studied. Similar equations arise in the study of turbulent filtration of gas or liquid through porous media. Existence and uniqueness in some class of Hölder continuous generalized solutions of the Cauchy-Dirichlet problem for equations of the type (1), (2), is proved. Bibliography: 9 titles.