A question of Gol’dberg concerning entire functions with prescribed zeros

Let (z_j) be a sequence of complex numbers satisfying |z_j|→ ∞ asj → ∞ and denote by n(r) the number of z_j satisfying |z_j|≤ r. Suppose that lim inf_{r → ⇈} log n(r)/ logr > 0. Let ϕ be a positive, non-decreasing function satisfying ∫^∞ (ϕ(t)t logt)⁻¹ dt < ∞. It is proved that there exists an entire functionf whose zeros are the z_j such that log log M(r,f) = o((log n(r))²ϕ(log n(r))) asr → ∞ outside some exceptional set of finite logarithmic measure, and that the integral condition on ϕ is best possible here. These results answer a question by A. A. Gol’dberg.     

Limit theorems for large deviations probabilities of certain quadratic forms

Let (ξ _k ,F _k ) be a martingale difference sequence. The paper concerns the tail behavior of the quadratic formS _n = ∑ _k=1 ⁿ ∑ _j=1 ^k−1 β _n ^k−j χ _k χ _j , where β _n asn→∞. The main conclusions aboutP}n ⁻¹ S _n >x _n }, wherex _n →∞, asn→∞, are obtained using the tail behavior of a martingale with values in a certain Hilbert space. Vilnius University, Naugarduko 24; Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 37, No. 4, pp. 532–549, October–December, 1997.     

Norms of powers of absolutely convergent fourier series

Letf(t) = ∑a _k e ^ikt be infinitely differentiable on R, |f(t)|<1. It is known that under these assumptions ‖ⁿ‖ converges to a finite limitl asn → ∞ (l ² = sec(arga),a = (f′(0))² -f″(0)). We obtain here more precise results: (i) an asymptotic series (in powers ofn ^-1/2) for the Fourier coefficientsa _nk off ⁿ , which holds uniformly ink asn → ∞; (ii) an asymptotic series (this time only powers ofn ^-1 are present!) for ‖f ⁿ ‖; (iii) the fact that ifi ^j f ^(j)(0) is real forj = 1,2,..., 2h + 2 then ‖f ⁿ ‖ = l + o(n ^-h ),n → ∞. More generally, we obtain analogous finite asymptotic expansions whenf is assumed to be differentiable only finitely many times.     

The Bieberbach-Rademacher problem for the Monge-Ampère operator

Assume that Ω is a bounded, strictly convex, smooth domain in ℝ^N withN≥2. We consider the problem det ((∂ _iju(x)))=f(x,u(x)),u(x)→∞ asx→∂Ω, where (∂ _iju(x)) denotes the Hessian ofu(x) andf meets some natural regularity and growth conditions. We prove that there exists a unique smooth, strictly convex solution of this problem. The boundary-blow-up rate ofu(x) is characterized in terms of the distance ofx from ∂Ω. Partially supported by the Royal Swedish Academy of Sciences, Gustaf Sigurd Magnuson's fund.     

Decrease rate of the probabilities ofε-deviations for the means of stationary processes

The asymptotic behavior asn → ∞ of the normed sumsσn =n ⁻¹ Σ _k ⁼⁰ⁿ⁻¹ Xk for a stationary processX = (X _n ,n ∈ ℤ) is studied. For a fixedε > 0, upper estimates for P(sup _k≥n |σ _k | ≥ε) asn → ∞ are obtained. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 366–372, September, 1998.     

Local linear operators and multicontour solutions of homogeneous coupled map lattices

Theorems are proved establishing a relationship between the spectra of the linear operators of the formA+Ωg _iB_ig_i ⁻¹ andA+B _i, whereg _i∈G, andG is a group acting by linear isometric operators. It is assumed that the closed operatorsA andB _i possess the following property: ‖B _iA⁻¹gB_jA⁻¹‖→0 asd(e,g)→∞. Hered is a left-invariant metric onG ande is the unit ofG. Moreover, the operatorA is invariant with respect to the action of the groupG. These theorems are applied to the proof of the existence of multicontour solutions of dynamical systems on lattices. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 37–47, January, 1999.     

Characteristic features of the dynamics of the Ginzburg-Landau equation in a plane domain

We study the boundary value problem w_t=ℵ₀Δw+ℵ₁w-ℵ₂w|w|²,w|∂Ω₀=0 in the domain Ω₀={(x,y):0 ≤ x ≤ l₁,0 ≤ y ≤ l₂}. Here, w is a complex-valued function, Δ is the laplace operator, and ℵ_j, j=0,1,2, are complex constants withRe ℵ_j > 0. We show that under a rather general choice of the parameters l₁ and l₂, the number of stable invariant tori in the problem, as well as their dimensions, grows infinitely asRe ℵ₀ → 0 andRe ℵ₀ → 0. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 2, pp. 205–220, November, 2000.     

Solution to the boundary blowup problem for k-curvature equation

We consider the boundary blowup problem for k-curvature equation, i.e., H _k [u] = f(u) g(|Du|) in an n-dimensional domain Ω, with the boundary condition u(x) → ∞ as dist (x,∂Ω) → 0. We prove the existence result under some hypotheses. We also establish the asymptotic behavior of a solution near the boundary ∂Ω. Mathematics Subject Classification (2000) 35J65, 35B40, 53C21     

On the average hyperoscillations of planted plane trees

Assume that the leaves of a planted plane tree are enumerated from left to right by 1, 2, .... Thej-ths-turn of the tree is defined to be the root of the (unique) subtree of minimal height with leavesj, j+1, ...,j+s−1. If all trees withn nodes are regarded equally likely, the average level number of thej-ths-turn tends to a finite limitα _s (j), which is of orderj ^1/2. Thej-th ”s-hyperoscillation”α ₁(j)−α _s+1(j) is given by 1/2α ₁(s)+O(j ^−1/2) and therefore tends (forj → ∞) to a constant behaving like √8/π·s ^1/2 fors → ∞. These results are obtained by setting up appropriate generating functions, which are expanded about their (algebraic) singularities nearest to the origin, so that the asymptotic formulas are consequences of the so-called Darboux-Pólyamethod.     

Transient phenomena for random walks in the absence of the expected value of jumps

Let ξ, ξ₁, ξ₂, ... be independent identically distributed random variables, and S _n :=Σ _j=1 ⁿ ,ξ _j , $ \bar S $ \bar S := sup _n≥0 S _n . If Eξ = −a < 0 then we call transient those phenomena that happen to the distribution $ \bar S $ \bar S as a → 0 and $ \bar S $ \bar S tends to infinity in probability. We consider the case when Eξ fails to exist and study transient phenomena as a → 0 for the following two random walk models:

Expectation of random polytopes

A random polytopeP _n in a convex bodyC is the convex hull ofn identically and independently distributed points inC. Its expectation is a convex body in the interior ofC. We study the deviation of the expectation ofP _n fromC asn→∞: while forC of classC ^k+1,k≥1, precise asymptotic expansions for the deviation exist, the behaviour of the deviation is extremely irregular for most convex bodiesC of classC ¹. Dedicated to my teacher and friend Professor Edmund Hlawka on the occasion of his 80^th birthday     

On the bestL 1-approximations of functional classes by splines under restrictions imposed on their derivatives

We find the exact asymptotics (asn→∞) of the bestL ₁-approximations of classesW ₁ ^r of periodic functions by spliness∈S _{2n, r∼-1} (S _{2n, r∼-1} is a set of 2π-periodic polynomial splines of orderr−1, defect one, and with nodes at the pointskπ/n,k∈ℤ) such that V ₀ ^2π s( ^r-1)≤1+ɛ _n , where {ɛ _n } _n=1 ^∞ is a decreasing sequence of positive numbers such that ɛ _n n ²→∞ and ɛ _n →0 asn→∞. Dnepropetrovsk University, Dnepropetrovsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 4, pp. 435–444, April, 1999.     

Large solutions to the p-Laplacian for large p

In this work we consider the behaviour for large values of p of the unique positive weak solution u _p to Δ _p u = u ^q in Ω, u = +∞ on , where q > p − 1. We take q = q(p) and analyze the limit of u _p as p → ∞. We find that when q(p)/p → Q the behaviour strongly depends on Q. If 1 < Q < ∞ then solutions converge uniformly in compacts to a viscosity solution of with u = +∞ on . If Q = 1 then solutions go to ∞ in the whole Ω and when Q = ∞ solutions converge to 1 uniformly in compact subsets of Ω, hence the boundary blow-up is lost in the limit.     

Estimation of the variance for strongly mixing sequences

Abstract. Let {Xn,n≥1} be a stationary strongly mixing random sequence satisfying EX1=u,     

On some nonlinear boundary-value problems in cylindrical domains

The paper deals with a linear elliptic second-order equation in divergent form in a semi-infinite cylinder Ω=Ω×[0,∞), where Ω is a bounded domain, Ω⊂ℝⁿ⁻¹, 0≤x_n<∞. The solution under consideration satisfies the nonlinear boundary condition, ∂u/∂v+a₀∣u∣^q−1u=0, where ϖ/ϖν is the differentiation along the external conormal a₀=const>0, q=const. The asymptotic behavior of the solution for x_n→+∞ is studied. Bibliography: 15 titles. Dedicated to O. A. Oleinik The investigations described in this paper were supported in part by the Russian Foundation for Basic Research, project No. 93-011-16035, and by the International Science Foundation, grant MIE000. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 19, pp. 000-000, 0000.     

Spot Self-Replication and Dynamics for the Schnakenburg Model in a Two-Dimensional Domain

The dynamical behavior of multi-spot solutions in a two-dimensional domain Ω is analyzed for the two-component Schnakenburg reaction–diffusion model in the singularly perturbed limit of small diffusivity ε for one of the two components. In the limit ε→0, a quasi-equilibrium spot pattern in the region away from the spots is constructed by representing each localized spot as a logarithmic singularity of unknown strength S _j for j=1,…,K at unknown spot locations x _j ∈Ω for j=1,…,K. A formal asymptotic analysis, which has the effect of summing infinite logarithmic series in powers of −1/log ε, is then used to derive an ODE differential algebraic system (DAE) for the collective coordinates S _j and x _j for j=1,…,K, which characterizes the slow dynamics of a spot pattern. This DAE system involves the Neumann Green’s function for the Laplacian. By numerically examining the stability thresholds for a single spot solution, a specific criterion in terms of the source strengths S _j , for j=1,…,K, is then formulated to theoretically predict the initiation of a spot-splitting event. The analytical theory is illustrated for spot patterns in the unit disk and the unit square, and is compared with full numerical results computed directly from the Schnakenburg model.      

L 1-convergence of double cosine- and Walsh-Fourier series

We prove the convergence inL ¹([−gp, π)²)-norm of the double Fourier series of an integrable functionf(x, y) which is periodic and even with respect tox andy, with coefficientsa _jk satisfying certain conditions of Hardy-Karamata kind, and such thata _jk logj logk→0 asj, k→∞. These sufficient conditions become quite natural in particular cases. Then we extend these results to the convergence of double Walsh-Fourier series inL ¹ (0, 1)²)- norm. As a by-product, we obtain Tauberian conditions ensuring the convergence of a double numerical series provided it is Cesàro summable. This research was partially supported by the Hungarian National Foundation for Scientific Research under Grant # 234.     

Asymptotic behavior of solutions for a class of retarded difference equations

Consider the retarded difference equationx _n −x _n−1 =F(−f(x _n )+g(x _n−k )), wherek is a positive integer,F,f,g:R→R are continuous,F andf are increasing onR, anduF(u)>0 for allu≠0. We show that whenf(y)≥g(y) (resp. f(y)≤g(y)) fory∈R, every solution of (*) tends to either a constant or −∞ (resp. ∞) asn→∞. Furthermore, iff(y)≡g(y) fory∈R, then every solution of (*) tends to a constant asn→∞. Project supported by NNSF (19601016) of China and NSF (97-37-42) of Hunan     

Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type

LetX be a Banach space,K a nonempty, bounded, closed and convex subset ofX, and supposeT:K→K satisfies: for eachx∈K, lim sup _i→∞{sup _y∈K ‖t ⁱx−Tⁱy∼−‖x−y‖}≦0. IfT ^N is continuous for some positive integerN, and if either (a)X is uniformly convex, or (b)K is compact, thenT has a fixed point inK. The former generalizes a theorem of Goebel and Kirk for asymptotically nonexpansive mappings. These are mappingsT:K→K satisfying, fori sufficiently large, ‖Tⁱx−Tⁱy‖≦k _i‖x−y∼,x,y∈K, wherek _i→1 asi→∞. The precise assumption in (a) is somewhat weaker than uniform convexity, requiring only that Goebel’s characteristic of convexity, ɛ₀ (X), be less than one. Research supported by National Science Foundation Grant GP 18045.     

The limiting distribution of the trace of a random plane partition

We study the asymptotic behaviour of the trace (the sum of the diagonal parts) τ _n = τ _n (ω) of a plane partition ω of the positive integer n, assuming that ω is chosen uniformly at random from the set of all such partitions. We prove that (τ _n − c ₀ n ^2/3)/c ₁ n ^1/3 log^1/2 n converges weakly, as n → ∞, to the standard normal distribution, where c ₀ = ζ(2)/ [2ζ(3)]^2/3, c ₁ = √(1/3/) [2ζ(3)]^1/3 and ζ(s) = Σ _j=1^∞ j ^−s . Partial support given by the National Science Fund of the Bulgarian Ministry of Education and Science, grant No. VU-MI-105/2005.