Estimates for k-Hessian operator and some applications

The k-convex functions are the viscosity subsolutions to the fully nonlinear elliptic equations F _k [u] = 0, where F _k [u] is the elementary symmetric function of order k, 1 ? ? 6 n, of the eigenvalues of the Hessian matrix D ² u. For example, F ₁[u] is the Laplacian Δu and F _n [u] is the real Monge-Ampère operator detD ² u, while 1-convex functions and n-convex functions are subharmonic and convex in the classical sense, respectively. In this paper, we establish an approximation theorem for negative k-convex functions, and give several estimates for the mixed k-Hessian operator. Applications of these estimates to the k-Green functions are also established.     

Collinear subsets of lattice point sequences—An analog of Szemerédi's theorem

Szemerédi's theorem states that given any positive number B and natural number k, there is a number n(k, B) such that if n ? n(k, B) and 0 < a₁ < … < a_n is a sequence of integers with a_n ? Bn, then some k of the a_i form an arithmetic progression. We prove that given any B and k, there is a number m(k, B) such that if m ? m(k, B) and u₀, u₁, …, u_m is a sequence of plane lattice points with ∑_i=1^m…u_i ? u_i?1… ? Bm, then some k of the u_i are collinear. Our result, while similar to Szemerédi's theorem, does not appear to imply it, nor does Szemerédi's theorem appear to imply our result.     

A note on the theorem of Lakshmikantham and Leela

We study the initial value problem u′ = f(t, u), u(0) = u₀, in a real Banach space. Using the fixed point principle of Daher and the theorem of Mönch and von Harten, we obtain a theorem which is an extension of that established by Lakshmikantham and Leela in [1].     

Cyclic quartic fields and genus theory of their subfields

Let k = $\text{Q}$(√u) (u ≠ 1 squarefree), K any possible cyclic quartic field containing k. A close relation is established between K and the genus group of k. In particular: (1) Each K can be written uniquely as K = $\text{Q}$(√vwη), where η is fixed in k and satisfies η ? 1, (η) = $\text{U}$²√u, |$\text{U}$²| = |(√u)|, (v, u) = 1, v ∈ $\text{Z}$ is squarefree, w|u, 0 < w < √u. Thus if u ≠ a² + b², there is no K ? k. If u = a² + b² then for each fixed v there are 2^{g ? 1}K ? k, where g is the number of prime divisors of u. (2) $\text{K}\text{k}$ has a relative integral basis (RIB) (i.e., O_K is free over O_k) iff N(ε₀) = ?1 and w = 1, where ε₀ is the fundamental unit of k, (or, equivalently, iff K = $\text{Q}$(√vε₀√u), (v, u) = 1). (3) A RIB is constructed explicitly whenever it exists. (4) disc(K) is given. In particular, the following results are special cases of (2): (i) Narkiewicz showed in 1974 that $\text{K}\text{k}$ has a RIB if u is a prime; (ii) Edgar and Peterson (J. Number Theory12 (1980), 77–83) showed that for u composite there is at least one K ? k having no RIB. Besides, it follows from (4) that the classification and integral basis of K given by Albert (Ann. of Math.31 (1930), 381–418) are wrong.     

LELONG-DEMAILLY NUMBERS IN TERMS OF CAPACITY AND WEAK CONVERGENCE FOR CLOSED POSITIVE CURRENTS

In this paper we give a new definition of the Lelong-Demailly number in terms of the CT-capacity, where T is a closed positive current and CT is the capacity associated to T. This derived from some esimate on the growth of the CT-capacity of the sublevel sets of a weighted plurisubharmonic （psh for short） function. These estimates enable us to give another proof of the Demailly＇s comparaison theorem as well as a generalization of some results due to Xing concerning the characterization of bounded psh functions. Another problem that we consider here is related to the existence of a psh function v that satisfies the equality CT（K） ： fK T ∧ （dd^cu）^p, where K is a compact subset. Finally, we give some conditions on the capacity CT that guarantees the weak convergence ukTk → uT, for positive closed currents T, Tk and psh functions uk, u.     

Uniqueness of solutions to an Abel type nonlinear integral equation on the half line

We consider a convolution-type integral equation u = k ? g(u) on the half line (???; a), a ?? ?, with kernel k(x) = x ^???1, 0 < ??, and function g(u), continuous and nondecreasing, such that g(0) = 0 and 0 < g(u) for 0 < u. We concentrate on the uniqueness problem for this equation, and we prove that if ?? ?? (1, 4), then for any two nontrivial solutions u ₁, u ₂ there exists a constant c ?? ? such that u ₂(x) = u ₁(x +c), ??? < x. The results are obtained by applying Hilbert projective metrics.     

Strong convergence of the modified hybrid steepest-descent methods for general variational inequalities

In this paper, we consider the general variational inequality GVI(F, g, C), whereF andg are mappings from a Hilbert space into itself andC is the fixed point set of a nonexpansive mapping. We suggest and analyze a new modified hybrid steepest-descent method of type methodu _n+1=(1?α+θ _n+1)Tu _n +αu _n ?θ _n+1 g (Tu _n )?λ _n+1 μF(Tu _n ),n≥0. for solving the general variational inequalities. The sequencex _n is shown to converge in norm to the solutions of the general variational inequality GVI(F, g, C) under some mild conditions. Application to constrained generalized pseudo-inverse is included. Results proved in the paper can be viewed as an refinement and improvement of previously known results.     

A function with prescribed initial derivatives in different Banach spaces

Let X₀ ? X₁ ? ··· ? X_p be Banach spaces with continuous injection of X_k into X_{k + 1} for 0 ? k ? p ? 1, and with X₀ dense in X_p. We seek a function u: [0, 1] → X₀ such that its kth derivative u^(k), k = 0, 1,…, p, is continuous from [0, 1] into x_k, and satisfies the initial condition u^(k)(0) = a_k?X_k. It is shown that such a function exists if and only if the initial values a₀, a₁, …, a_p satisfy a certain condition reminiscent of interpolation theory. This condition always holds when p = 1; when p ? 2, the spaces X_k (k = 0, 1, …, p) may or may not be such that the desired function exists for any given initial values a_k?X_k.     

Binomial coefficients, Catalan numbers and Lucas quotients

Let p be an odd prime and let a,m ∈ Z with a 0 and p ︱ m.In this paper we determinep ∑k=0 pa-1(2k k=d)/mk mod p2 for d=0,1;for example,where(-) is the Jacobi symbol and {un}n≥0 is the Lucas sequence given by u0 = 0,u1 = 1 and un+1 =(m-2)un-un-1(n = 1,2,3,...).As an application,we determine ∑0kpa,k≡r(mod p-1) Ck modulo p2 for any integer r,where Ck denotes the Catalan number 2kk /(k + 1).We also pose some related conjectures.     

Almost sure limit theorem for the maximum of a classof quasi-stationary sequences简

This paper investigates the problem of almost sure limit theorem for the maximum of quasi-stationary sequence based on the result of Turkman and Walker. We prove an almost sure limit theorem for the maximum of a class of quasi-stationary sequence under weak dependence conditions of D(uk,un) and αtn,ln = O(log log n).(1+ε).     

Lucas's criterion for the primality of numbers of the form N = h2n−1

The following theorem is proved. Let N = h2ⁿ-1, where n ≥ 2, h is odd, 1 <-h < 2ⁿ, and suppose that v is a positive integer, v ≥ 3,α is a root of the equation $$(v^2 - 4,N) = 1,\left( {\frac{{v - 2}}{N}} \right) = 1,\left( {\frac{{v + 2}}{N}} \right) = - 1$$ . Then for N to be prime, it is necessary and sufficient that s_n?2≡0(modN), where S_k+1=S _k ² ? 2 (k = 0, 1...), s_o=a^h+ a^?h. For given N, an algorithm is described for the construction of the smallest v satisfying the conditions of this theorem.     

Simple continued fractions for some irrational numbers,II

In this paper we prove a theorem allowing us to determine the continued fraction expansion for Σ_k=0^∞u^?c(k), where c(k) is any sequence of positive integers that grows sufficiently quickly. As an application, we determine the continued fraction expansion for Liouville's famous transcendental number Σ_k=0^∞m^{?(k + 1)!}.     

Investigation of solvability of the multidimensional two-phase Stefan and the nonstationary filtration Florin problems for second order parabolic equations in weighted Hölder spaces of functions

Multidimensional two-phase Stefan (k=1) and nonstationary filtration Florin (k=0) problems for second order parabolic equations in the case when the free boundary is a graph of a functionx _n =Ψ _k (x′t),x′∈ ^n?1 ,n≥2,t∈(0,T) are studied. A unique solvability theorem in weighted Hölder spaces of functions with time power weight is proved, coercive estimates for solutions are obtained. Bibliography: 30 titles.     

Search-extension method for finding multiple solutions of nonlinear problem

Based on the eigensystem {λj,φj}of -Δ, the multiple solutions for nonlinear problem Δu f(u) = 0 in Ω, u = 0 on (?)Ωare approximated. A new search-extension method (SEM) is proposed, which consists of three algorithms in three level subspaces. Numerical experiments for f(u) = u3 in a square and L-shape domain are presented. The results show that there exist at least 3k - 1 distinct nonzero solutions corresponding to each κ-ple eigenvalue of -Δ(Conjecture 1).     

On the proof of continued fraction expansions for irrationals

We use matrices to prove two theorems in regard to the continued fraction expansion for Σ_{k = 0}^∞u^−2k and for Σ_{k = 0}^∞u^−c(k), where c(k) is any sequence of positive integers that increase quickly.     

Regularized semigroups,iterated Cauchy problems and equipartition of energy

The iterated Cauchy problem under consideration is $$\Pi _{k = 1}^n (d/dt - A_k )u(t) = 0(t \geqslant 0).(*)$$ Here {A ₁,..., A_n} are unbounded linear operators on a Banach space. The initial value problem for (*) is governed by a semigroup of some sort. When eachA _k is a (C ₀) semigroup generator, this semigroup is of class (C ₀) and was studied by J. T. Sandefur [26]. This result is extended to the case when eachA _k generates aC-regularized semigroup (withC independent ofk). This means one can solveu′=Au, u(0)=f∈C (Dom (A)) and getu(t)→0 wheneverC ^?1f→0; hereC is bounded and injective. When theA _k are commuting generator withA _k-A_j injective fork≠j, then the Goldstein-Sandefur d'Alembert formula [19] is extended, viz. solutions of (*) (with suitable restrictions on the initial data) are of the form \(u = \sum\nolimits_{i = 1}^n {u_i } \) whereu _i is a solution ofu′ _i=A_iu_i. Examples and applications are given. Included among the examples is the establishment of a form of equipartition of energy for the Laplace equation; equipartition of energy is wellknown for the wave equation. A final section of the paper deals with the absence of necessary conditions for equipartition of energy.     

On uniqueness properties of solutions of the k-generalized KdV equations

In this paper we study uniqueness properties of solutions of the so-called k-generalized Korteweg-de Vries equations. Our goal is to obtain sufficient conditions on the behavior of the difference u₁−u₂ of two solutions u₁,u₂ of (1.1) at two different times t₀=0 and t₁=1 which guarantee that u₁≡u₂.     

A priori estimates and continuous dependence of solutions of mixed problems for parabolic equations as nonlocal boundary conditions pass into local ones

In the rectangle G = (0, 1) × (0, T), we consider the family of problems

$$\begin{gathered} \frac{1}{{a(x,t)}}\frac{{\partial u_\alpha }}{{\partial t}} - \frac{{\partial ^2 u_\alpha }}{{\partial x^2 }} = f(x,t), u_\alpha (x,0) = \phi _\alpha (x), u_\alpha (0,t) = 0, 0 \leqslant \alpha \leqslant 1, \hfill \\ u_0 (1,t) = h(t), \frac{{\partial u_1 (1,t)}}{{\partial x}} = h(t), \frac{{u_\alpha (1,t) - u_\alpha (\alpha ,t)}}{{1 - \alpha }} = h(t), 0 < \alpha < 1, \hfill \\ a_1 \geqslant a(x,t) \geqslant a_0 > 0, h \in W_2^1 (0,T), \phi _\alpha \in W_2^1 (0,T), \phi _\alpha (0) = 0, 0 \leqslant \alpha \leqslant 1, \hfill \\ \phi _0 (1) = h(0), \phi '_1 (1) = h(0), \frac{{\phi _\alpha (1) - \phi _\alpha (0)}}{{1 - \alpha }} = h(0), 0 < \alpha < 1, f \in L_2 (G) \hfill \\ \end{gathered} $$

. It is well known that, for α = 0 and α = 1, the corresponding problems with local conditions are solvable, and the solutions are unique and belong to W ₂ ^2,1 (G).

We prove the existence and uniqueness of solutions of the family of problems with nonlocal conditions for each α ∈ (0, 1). For the differences u _α ? u ₀ and u _α ? u ₁ (0 < α < 1), we establish a priori estimates and use them to prove that if ? _α → ? ₀ as α → 0, then u _α → u ₀ and if ? _α → ? ₁ as α → 1, then u _α → u ₁.