On the sectionwise connectedness of a contingent

Let X be a real normed space and let f: ? → X be a continuous mapping. Let T _f (t ₀) be the contingent of the graph G(f) at a point (t ₀, f(t ₀)) and let S ⁺ ? (0,∞) × X be the “right” unit hemisphere centered at (0, 0 _X ). We show that

1. 1.
   If dimX < ∞ and the dilation D(f, t ₀) of f at t ₀ is finite then T _f (t ₀) ∩ S ⁺ is compact and connected. The result holds for \(T_f (t_0 ) \cap \overline {S^ + } \) even with infinite dilation in the case f: [0,∞) → X.
    
2. 2.
   If dimX = ∞, then, given any compact set F ? S ⁺, there exists a Lipschitz mapping f: ? → X such that T _f (t ₀) ∩ S ⁺ = F.
    
3. 3.
   But if a closed set F ? S ⁺ has cardinality greater than that of the continuum then the relation T _f (t ₀) ∩ S ⁺ = F does not hold for any Lipschitz f: ? → X.
    

     

Connecting orbits of Lagrangian systems in a nonstationary force field

We study connecting orbits of a natural Lagrangian system defined on a complete Riemannian manifold subjected to the action of a nonstationary force field with potential U(q, t) = f(t)V(q). It is assumed that the factor f(t) tends to ∞ as t→±∞ and vanishes at a unique point t ₀ ∈ ?. Let X ₊, X _? denote the sets of isolated critical points of V (x) at which U(x, t) as a function of x distinguishes its maximum for any fixed t > t ₀ and t < t ₀, respectively. Under nondegeneracy conditions on points of X _± we prove the existence of infinitely many doubly asymptotic trajectories connecting X _? and X ₊.     

Probabilities of high extremes for a Gaussian stationary process in a random environment

Let ξ(t) be a zero-mean stationary Gaussian process with the covariance function r(t) of Pickands type, i.e., r(t) = 1 ? |t| ^α + o(|t| ^α ), t → 0, 0 < α ≤ 2, and η(t), ζ(t) be periodic random processes. The exact asymptotic behavior of the probabilities P(max _t∈[0,T] η(t)ξ(t) > u), P(max _t∈[0,T] (ξ(t) + η(t)) > u) and P(max _t∈[0,T] (η(t)ξ(t) + ζ(t)) > u) is obtained for u → ∞ for any T > 0 and independent ξ(t), η(t), ζ(t).     

Singularly perturbed two-dimensional parabolic problem in the case of intersecting roots of the reduced equation

The singularly perturbed parabolic equation ?u _t + ε²Δu ? f(u, x, ε) = 0, x ∈ D ? ?², t > 0 with Robin conditions on the boundary of D is considered. The asymptotic stability as t → ∞ and the global domain of attraction are analyzed for the stationary solution whose limit as ε → 0 is a nonsmooth solution to the reduced equation f(u, x, 0) = 0 that consists of two intersecting roots of this equation.     

Extremes of Gaussian processes with smooth random expectation and smooth random variance

Let ξ(t), t ∈ [0, T],T > 0, be a Gaussian stationary process with expectation 0 and variance 1, and let η(t) and μ(t) be other sufficiently smooth random processes independent of ξ(t). In this paper, we obtain an asymptotic exact result for P(sup _t∈[0,T](η(t)ξ(t) + μ(t)) > u) as u→∞.     

Existence of Semilinear Differential Equations with Nonlocal Initial Conditions

In this paper we give the existence of mild solutions for semilinear Cauchy problems u′（t） = Au（t） ＋f（t, u（t））, t ∈ I, a.e. with nonlocal initial condition u（O） = g（u） ＋uo when the map g loses compactness in Banach spaces.     

Two-Sided Estimates of Fourier Sums Lebesgue Functions with Respect to Polynomials Orthogonal on Nonuniform Grids

Let Ω = {t₀, t₁, …, t_N} and Ω_N = {x₀, x₁, …, x_N–1}, where x_j = (t_j + t_{j + 1})/2, j = 0, 1, …, N–1 be arbitrary systems of distinct points of the segment [–1, 1]. For each function f(x) continuous on the segment [–1, 1], we construct discrete Fourier sums S_{n, N}( f, x) with respect to the system of polynomials {p?_k,N(x)} _k=0 ^N–1 , forming an orthonormal system on nonuniform point systems Ω_N consisting of finite number N of points from the segment [–1, 1] with weight Δt_j = t_{j + 1}–t_j. We find the growth order for the Lebesgue function L_n,N (x) of the considered partial discrete Fourier sums S_n,N ( f, x) as n = O(δ _N ^?2/7 ), δ_N = max_{0≤ j≤N?1} Δt_j More exactly, we have a two-sided pointwise estimate for the Lebesgue function L_{n, N}(x), depending on n and the position of the point x from [–1, 1].     

Group analysis of the one-dimensional boltzmann equation: II. Equivalence groups and symmetry groups in the special case

We obtain relations that define the equivalence algebra of the family of one-dimensional Boltzmann equations f _t + cf _x + F(t, x, c)f _c = 0 and show that all equations of that form are locally equivalent. We carry out the group classification of the equation with respect to the function F in the special case where the function F and the transformations of the variables t and x are assumed to be independent of c. We show that, under such constraints for the transformation and the family of equations, the maximum possible symmetry algebra is eight-dimensional, which corresponds to an equation with a linear function F.     

Proof of a tournament partition conjecture and an application to 1-factors with prescribed cycle lengths

In 1982 Thomassen asked whether there exists an integer f(k,t) such that every strongly f(k,t)-connected tournament T admits a partition of its vertex set into t vertex classes V ₁,…V _t such that for all i the subtournament T[V _i] induced on T by V _i is strongly k-connected. Our main result implies an affirmative answer to this question. In particular we show that f(k, t)=O(k ⁷ t ⁴) suffices. As another application of our main result we give an affirmative answer to a question of Song as to whether, for any integer t, there exists aninteger h(t) such that every strongly h(t)-connected tournament has a 1-factor consisting of t vertex-disjoint cycles of prescribed lengths. We show that h(t)=O(t ⁵) suffices.     

On vanishing coefficients of algebraic power series over fields of positive characteristic

Let K be a field of characteristic p>0 and let f(t ₁,…,t _d ) be a power series in d variables with coefficients in K that is algebraic over the field of multivariate rational functions K(t ₁,…,t _d ). We prove a generalization of both Derksen’s recent analogue of the Skolem–Mahler–Lech theorem in positive characteristic and a classical theorem of Christol, by showing that the set of indices (n ₁,…,n _d )∈? ^d for which the coefficient of \(t_{1}^{n_{1}}\cdots t_{d}^{n_{d}}\) in f(t ₁,…,t _d ) is zero is a p-automatic set. Applying this result to multivariate rational functions leads to interesting effective results concerning some Diophantine equations related to S-unit equations and more generally to the Mordell–Lang Theorem over fields of positive characteristic.     

On the metric type of measurable functions and convergence in distribution

In the present paper, sequences of real measurable functions defined on a measure space ([0, 1], µ), where µ is the Lebesgue measure, are studied. It is proved that for every sequence f_n that converges to f in distribution, there exists a sequence of automorphisms S_n of ([0, 1], µ) such that f_n(S_n(t)) converges to f(t) in measure. Connection with some known results is also discussed.     

Factorization of the reaction-diffusion equation,the wave equation,and other equations

We investigate equations of the form D _t u = Δu + ξ? _u for an unknown function u(t, x), t ∈ ?, x ∈ X, where D _t u = a ₀(u, t) + Σ _k=1 ^r a _k (t, u)? _t ^k u, Δ is the Laplace-Beltrami operator on a Riemannian manifold X, and ξ is a smooth vector field on X. More exactly, we study morphisms from this equation within the category PDE of partial differential equations, which was introduced by the author earlier. We restrict ourselves to morphisms of a special form—the so-called geometric morphisms, which are given by maps of X to other smooth manifolds (of the same or smaller dimension). It is shown that a map f: X → Y defines a morphism from the equation D _t u = Δu + ξ? _u if and only if, for some vector field Ξ and a metric on Y, the equality (Δ + ξ?)f*v = f*(Δ + Ξ?)v holds for any smooth function v: Y → ?. In this case, the quotient equation is D _t v = Δv + Ξ?v for an unknown function v(t, y), y ∈ Y. It is also shown that, if a map f: X → Y is a locally trivial bundle, then f defines a morphism from the equation D _t u = Δu if and only if fibers of f are parallel and, for any path γ on Y, the expansion factor of a fiber translated along the horizontal lift γ to X depends on γ only.     

Continuous wavelets on compact manifolds

Let M be a smooth compact oriented Riemannian manifold, and let Δ _M be the Laplace–Beltrami operator on M. Say \({0 \neq f \in \mathcal{S}(\mathbb {R}^+)}\) , and that f (0)  =  0. For t  >  0, let K _t (x, y) denote the kernel of f (t ² Δ _M ). We show that K _t is well-localized near the diagonal, in the sense that it satisfies estimates akin to those satisfied by the kernel of the convolution operator f (t ²Δ) on \({\mathbb {R}^n}\) . We define continuous \({\mathcal {S}}\)-wavelets on M, in such a manner that K _t (x, y) satisfies this definition, because of its localization near the diagonal. Continuous \({\mathcal {S}}\)-wavelets on M are analogous to continuous wavelets on \({\mathbb {R}^n}\) in \({\mathcal {S}}\) (\({\mathbb {R}^n}\)). In particular, we are able to characterize the Hölder continuous functions on M by the size of their continuous \({\mathcal {S}}\)-wavelet transforms, for Hölder exponents strictly between 0 and 1. If M is the torus \({\mathbb T^2}\) or the sphere S ², and f (s)  =  se ^?s (the “Mexican hat” situation), we obtain two explicit approximate formulas for K _t , one to be used when t is large, and one to be used when t is small.     

Oscillatory hyper Hilbert transforms along general curves

We consider the oscillatory hyper Hilbert transform H _γ,α,β f(x) = ∫ ₀ ^∞ f(x - Γ(t))e^it-β t^-(1+α)dt; where Γ(t) = (t, γ(t)) in ?² is a general curve. When γ is convex, we give a simple condition on γ such that H _γ,α,β is bounded on L ² when β > 3α, β > 0: As a corollary, under this condition, we obtain the L ^p -boundedness of H _γ,α,β when 2β/(2β - 3α) < p < 2β/(3α). When Γ is a general nonconvex curve, we give some more complicated conditions on γ such that H _γ,α,β is bounded on L ²: As an application, we construct a class of strictly convex curves along which H _γ,α,β is bounded on L ² only if β > 2α > 0.     

Solvability of Mixed Problems for the Klein–Gordon–Fock Equation in the Class <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> for <Emphasis Type="Italic">p</Emphasis> ≥ 1

We prove that the mixed problem for the Klein–Gordon–Fock equation u _tt (x, t) ? u _xx (x, t) + au(x, t) = 0, where a ≥ 0, in the rectangle Q _T = [0 ≤ x ≤ l] × [0 ≤ t ≤ T] with zero initial conditions and with the boundary conditions u(0, t) = μ(t) ∈ L _p [0, T ], u(l, t) = 0, has a unique generalized solution u(x, t) in the class L _p (Q _T ) for p ≥ 1. We construct the solution in explicit analytic form.     

Meromorphic Solutions for a Class of Differential Equations and Their Applications

In this note, we study the admissible meromorphic solutions for algebraic differential equation fⁿf' + P_n?1(f) = R(z)e^α(z), where P_n?1(f) is a differential polynomial in f of degree ≤ n ? 1 with small function coefficients, R is a non-vanishing small function of f, and α is an entire function. We show that this equation does not possess any meromorphic solution f(z) satisfying N(r, f) = S(r, f) unless P_n?1(f) ≡ 0. Using this result, we generalize a well-known result by Hayman.     

Inverse problem with nonlocal observation of finding the coefficient multiplying <Emphasis Type="Italic">u</Emphasis> <Subscript> <Emphasis Type="Italic">t</Emphasis> </Subscript> in the parabolic equation

We study the inverse problem of the reconstruction of the coefficient ?(x, t) = ?⁰(x, t) + r(x) multiplying u_t in a nonstationary parabolic equation. Here ?⁰(x, t) ≥ ?⁰ > 0 is a given function, and r(x) ≥ 0 is an unknown function of the class L_∞(Ω). In addition to the initial and boundary conditions (the data of the direct problem), we pose the problem of nonlocal observation in the form ∫₀^Tu(x, t) dμ(t) = χ(x) with a known measure dμ(t) and a function χ(x). We separately consider the case dμ(t) = ω(t)dt of integral observation with a smooth function ω(t). We obtain sufficient conditions for the existence and uniqueness of the solution of the inverse problem, which have the form of ready-to-verify inequalities. We suggest an iterative procedure for finding the solution and prove its convergence. Examples of particular inverse problems for which the assumptions of our theorems hold are presented.     

Systems that generate solutions with a small period

Let (j₁,..., j_n) be a permutation of the n-tuple (1, ..., n). A system of differential equations \(\dot x = {f_i}\left( {{x_{{j_i}}}} \right),i = 1, \ldots ,n\) in which each function f_i is continuous on ? is considered. This system is said to have the property of generation of solutions with a small period if, for any number M > 0, there exists a number ω₀ = ω₀(M) > 0 such that if 0 < ω ≤ ω₀ and h_i(t, x₁, ..., x_n) are continuous functions on ? × ?ⁿ ω-periodic in t that satisfy the inequalities |h_i| ≤ M the system \(\dot x = {f_i}\left( {{x_{{j_i}}}} \right),i = 1, \ldots ,n\) has an ω-periodic solution. It is shown that a system has the property of generation of solutions with a small period if and only if f_i(?) = ? for i = 1,..., n. It is also shown that the smallness condition on the period is essential.     

A tracing of the fractional temperature field

This paper is devoted to a study of L~q-tracing of the fractional temperature field u(t, x)—the weak solution of the fractional heat equation(?_t +(-?_x)~α)u(t, x) = g(t, x) in L~p(R_+~(1+n)) subject to the initial temperature u(0, x) = f(x) in L~p(R~n).