Large solutions of coupled sublinear/superlinear elliptic equations

We show that large positive solutions exist for the semilinear elliptic equation Δu = p(x)u ^α + q(x)v ^β on bounded domains in R ⁿ , n ≥ 3, for the superlinear case 0 < α ≤ β, β > 1, but not the sublinear case 0 < α ≤ β ≤ 1. We also show that entire large positive solutions exist for both the superlinear and sublinear cases provided the nonnegative continuous functions p and q satisfy certain decay conditions at infinity. Existence and nonexistence of entire bounded solutions are established as well.     

Large Monotone Paths in Graphs with Bounded Degree

 We prove that for every ε>0 and positive integer r, there exists Δ₀=Δ₀(ε) such that if Δ>Δ₀ and n>n(Δ,ε,r) then there exists a packing of K _n with ⌊(n−1)/Δ⌋ graphs, each having maximum degree at most Δ and girth at least r, where at most εn ² edges are unpacked. This result is used to prove the following: Let f be an assignment of real numbers to the edges of a graph G. Let α(G,f) denote the maximum length of a monotone simple path of G with respect to f. Let α(G) be the minimum of α(G,f), ranging over all possible assignments. Now let α_Δ be the maximum of α(G) ranging over all graphs with maximum degree at most Δ. We prove that Δ+1≥α_Δ≥Δ(1−o(1)). This extends some results of Graham and Kleitman [6] and of Calderbank et al. [4] who considered α(K _n ). Received: March 15, 1999?Final version received: October 22, 1999     

The behavior of bounded solutions to the equation Δu−c(x)u=0 on riemannian manifolds of special type

In the paper we consider solutions of the equation Δu−c(x)u=0,c(x)≥0, on complete Riemannian manifolds constituted as follows: the exterior of some compact set is isometric to the direct product of the semiaxis by some compact manifold with the metricds ²=h ²(r)dr ²+g ²(r)dθ². Necessary and sufficient conditions under which bounded solutions of the equation have a limit independent of θ asr→∞ are obtained and also conditions under which the two-sided Liouville theorem is valid. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 215–221, February, 1999.     

Asymptotic separation for independent trajectories of Markov processes

We say that n independent trajectories ξ₁(t),…,ξ _n (t) of a stochastic process ξ(t)on a metric space are asymptotically separated if, for some ɛ > 0, the distance between ξ _i (t _i ) and ξ _j (t _j ) is at least ɛ, for some indices i, j and for all large enough t ₁,…,t _n , with probability 1. We prove sufficient conitions for asymptotic separationin terms of the Green function and the transition function, for a wide class of Markov processes. In particular,if ξ is the diffusion on a Riemannian manifold generated by the Laplace operator Δ, and the heat kernel p(t, x, y) satisfies the inequality p(t, x, x) ≤ Ct ^−ν/2 then n trajectories of ξ are asymptotically separated provided . Moreover, if for some α∈(0, 2)then n trajectories of ξ^(α) are asymptotically separated, where ξ^(α) is the α-process generated by −(−Δ)^α/2. Received: 10 June 1999 / Revised version: 20 April 2000 / Published online: 14 December 2000 RID="*" ID="*" Supported by the EPSRC Research Fellowship B/94/AF/1782 RID="**" ID="**" Partially supported by the EPSRC Visiting Fellowship GR/M61573     

On the order of growth of solutions of algebraic differential equations

Assume thatf is an integer transcendental solution of the differential equationP _n (z, f, f′)=P _n−1(z, f, f′, ... f ^(p)), whereP _n andP _n−1 are polynomials in all variables, the degree ofP _n with respect tof andf′ is equal ton, and the degree ofP _n−1 with respect tof, f′, ... f ^(p) is at mostn−1. We prove that the order ρ of growth off satisfies the relation 1/2≤ρ<∞. We also prove that if ρ=1/2, then, for a certain real ν, in the domain {z: ν<argz<ν+2π}/E _*, whereE _* is a certain set of disks with finite sum of radii, the estimate lnf(z)=z ^1/2 (β+o(1)), β∈C, holds forz=re ^iϕ,r≥r(ϕ)≥0. Furthermore, on the ray {z: argz=ν}, the following relation is true: ln‖f(re ^iν)‖=o(r ^1/2),r→+∞,r>0, , where Δ is a certain set on the semiaxisr>0 with mes Δ<∞. “L'vivs'ka Politekhnika” University, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 1, pp. 69–77, January, 1999.     

Discretization of Compact Riemannian Manifolds Applied to the Spectrum of Laplacian

For κ ⩾ 0 and r₀ > 0 let ℳ(n, κ, r₀) be the set of all connected, compact n-dimensional Riemannian manifolds (Mⁿ, g) with Ricci (M, g) ⩾ −(n−1) κ g and Inj (M) ⩾ r₀. We study the relation between the kth eigenvalue λ_k(M) of the Laplacian associated to (Mⁿ,g), Δ = −div(grad), and the kth eigenvalue λ_k(X) of a combinatorial Laplacian associated to a discretization X of M. We show that there exist constants c, C > 0 (depending only on n, κ and r₀) such that for all M ∈ ℳ(n, κ, r₀) and X a discretization of for all k < |X|. Then, we obtain the same kind of result for two compact manifolds M and N ∈ ℳ(n, κ, r₀) such that the Gromov–Hausdorff distance between M and N is smaller than some η > 0. We show that there exist constants c, C > 0 depending on η, n, κ and r₀ such that for all . Mathematics Subject Classification (2000): 58J50, 53C20 Supported by Swiss National Science Foundation, grant No. 20-101 469     

Oscillatory of unstable type second-order neutral difference equations

We consider the problem of oscillation and nonoscillation solutions for unstable type second-order neutral difference equation: Δ²(x(n) − p(n)x(n −τ)) =q(n)x(g(n)). (1) In this paper, we obtain some conditions for the bounded solutions of Eq(1) to be oscillatory and for the existence of the nonoscillatory solutions.     

Sequence spaces and inverse of an infinite matrix

We give here some properties of the sets _α(uΔ) generalizing the space of generalized difference sequencesl ^∞(uΔ). Then we study spaces related to the sets of sequences that are strongly convergent or strongly bounded. Next we define from the sets of spaces that are (N,q) summable or bounded the sets of spaces that are (N,q)α-bounded orr-bounded. Then we give some properties of these spaces using Banach space of the forms _α.     

Necessary and Sufficient Conditions for a Fourth Order Functional Differential Equation to be Oscillatory

Necessary and sufficient conditions for a fourth order functional differential equation of the form (1) [r(t)y″(t)]″+f(t,y(h₁(t)), y(h₂(t)), …, y(h_n(t)))=0 to be oscillatory are given when f is strongly superlinear or strongly sublinear.     

Random matrix products and applications to cellular automata

Let λ be the upper Lyapunov exponent corresponding to a product of i.i.d. randomm×m matrices (X _i) _i ^0/∞ over ℂ. Assume that theX _i's are chosen from a finite set {D ₀,D ₁...,D _t-1(ℂ), withP(X _i=D_j)>0, and that the monoid generated byD ₀, D₁,…, D_q−1 contains a matrix of rank 1. We obtain an explicit formula for λ as a sum of a convergent series. We also consider the case where theX _i's are chosen according to a Markov process and thus generalize a result of Lima and Rahibe [22]. Our results on λ enable us to provide an approximation for the numberN _≠0(F(x)ⁿ,r) of nonzero coefficients inF(x) ⁿ.(modr), whereF(x) ∈ ℤ[x] andr≥2. We prove the existence of and supply a formula for a constant α (<1) such thatN _≠0(F(x)ⁿ,r) ≈n ^α for “almost” everyn. Supported in part by FWF Project P16004-N05     

The size-Ramsey number of trees

IfG andH are graphs, let us writeG→(H)₂ ifG contains a monochromatic copy ofH in any 2-colouring of the edges ofG. Thesize-Ramsey number r _e(H) of a graphH is the smallest possible number of edges a graphG may have ifG→(H)₂. SupposeT is a tree of order |T|≥2, and lett ₀,t ₁ be the cardinalities of the vertex classes ofT as a bipartite graph, and let Δ(T) be the maximal degree ofT. Moreover, let Δ₀, Δ₁ be the maxima of the degrees of the vertices in the respective vertex classes, and letβ(T)=T ₀Δ₀+t ₁Δ₁. Beck [7] proved thatβ(T)/4≤r _e(T)=O{β(T)(log|T|)¹²}, improving on a previous result of his [6] stating thatr _e(T)≤Δ(T)|T|(log|T|)¹². In [6], Beck conjectures thatr _e(T)=O{Δ(T)|T|}, and in [7] he puts forward the stronger conjecture thatr _e(T)=O{β(T)}. Here, we prove the first of these conjectures, and come quite close to proving the second by showing thatr _e(T)=O{β(T)logΔ(T)}.     

Asymptotic behavior of solutions to viscous conservation laws with slowly varying external forces

This paper considers the existence and large time behavior of solutions to the convection-diffusion equation u _t −Δu+b(x)·∇(u|u| ^{q −1})=f(x, t) in ℝ ⁿ ×[0,∞), where f(x, t) is slowly decaying and q≥1+1/n (or in some particular cases q≥1). The initial condition u ₀ is supposed to be in an appropriate L ^p space. Uniform and nonuniform decay of the solutions will be established depending on the data and the forcing term.This work is partially supported by an AMO Grant     

The smallest degree sum that yields potentially Kr,r-graphic sequences

We consider a variation of a classical Turán-type extremal problem as follows: Determine the smallest even integer σ(K_r,r,n) such that every n-term graphic sequence π = (d₁,d₂,...,d_n) with term sum σ(π) = d₁ + d₂ + ... + d_n ≥ σ(K_r,r,n) is potentially K_r,r-graphic, where K_r,r is an r × r complete bipartite graph, i.e. π has a realization G containing K_r,r as its subgraph. In this paper, the values σ(K_r,r,n) for even r and n ≥ 4r² - r - 6 and for odd r and n ≥ 4r² + 3r - 8 are determined.     

Weights andL Φ-boundedness of the Poisson integral operator

In this paper, we get a necessary and sufficient condition on the weights (μ,v) for the Poisson integral operator to be bounded fromL _Φ(R ⁿ, v(x)dx) to weak-L _Φ(R ₊ ⁿ⁺¹ ,dμ), where Φ is anN-function satisfying the Δ₂-condition. We also find a necessary and sufficient condition on the weights (μ,v) for the Poisson integral operator to be bounded fromL _Φ(R ⁿ,v(x)dx) toL _Φ(R ₊ ⁿ⁺¹ ,dμ) under some additional condition. Partially supported by NNSF of P.R. China     

Structure of augmentation quotients of finite homocyclic abelian groups

Let G be a finite abelian group and its Sylow p-subgroup a direct product of copies of a cyclic group of order p~r,i.e.,a finite homocyclic abelian group.LetΔ~n (G) denote the n-th power of the augmentation idealΔ(G) of the integral group ring ZG.The paper gives an explicit structure of the consecutive quotient group Q_n(G)=Δ~n(G)/Δ~(n 1)(G) for any natural number n and as a consequence settles a problem of Karpilovsky for this particular class of finite abelian groups.     

Difference estimates for continuous and discrete operator semigroups

For suitable bounded operator semigroups (e ^tA ) _t≥0 in a Banach space, we characterize the estimate ‖Ae ^tA ‖≤c/F(t) for large t, where F is a function satisfying a sublinear growth condition. The characterizations are by holomorphy estimates on the semigroup, and by estimates on powers of the resolvent. We give similar characterizations of the difference estimate ‖T ⁿ −T ⁿ⁺¹‖≤c/F(n) for a power-bounded linear operator T, when F(n) grows faster than n ^1/2 for large n.     

Minimax goodness-of-fit testing in multivariate nonparametric regression

We consider an unknown response function f defined on Δ = [0, 1] ^d , 1 ≤ d ≤ ∞, taken at n random uniform design points and observed with Gaussian noise of known variance. Given a positive sequence r _n → 0 as n → ∞ and a known function f ₀ ∈ L ₂(Δ), we propose, under general conditions, a unified framework for goodness-of-fit testing the null hypothesis H ₀: f = f ₀ against the alternative H ₁: f ∈ $ \mathcal{F} $ \mathcal{F} , ∥f − f ₀∥ ≥ r _n , where $ \mathcal{F} $ \mathcal{F} is an ellipsoid in the Hilbert space L ₂(Δ) with respect to the tensor product Fourier basis and ∥ · ∥ is the norm in L ₂(Δ). We obtain both rate and sharp asymptotics for the error probabilities in the minimax setup. The derived tests are inherently non-adaptive. Several illustrative examples are presented. In particular, we consider functions belonging to ellipsoids arising from the well-known multidimensional Sobolev and tensor product Sobolev norms as well as from the less-known Sloan-Woźniakowski norm and a norm constructed from multivariable analytic functions on the complex strip.     

Cyclic elements of the shift operator in weighted anisotropic spaces of holomorphic functions in the polydisk

Let ϕ(r) = (ϕ₁(r₁), …, ϕ_n(r_n)) be a vector-valued function on R ₊ ⁿ . A necessary and sufficient condition is obtained under which any function f ∈, H^∞ (D ⁿ ), f(z) ≠ 0, z ∈, D ⁿ , is cyclic in the corresponding weighted space L^p(ϕ), where D ⁿ is the unit polydisk in C ⁿ. Bibliography: 13 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 327, 2005, pp. 226–234.     

Are the degrees of best (co)convex and unconstrained polynomial approximation the same?

Let ℂ[−1,1] be the space of continuous functions on [−,1], and denote by Δ² the set of convex functions f ∈ ℂ[−,1]. Also, let E _n (f) and E _n ⁽²⁾ (f) denote the degrees of best unconstrained and convex approximation of f ∈ Δ² by algebraic polynomials of degree < n, respectively. Clearly, En (f) ≦ E _n ⁽²⁾ (f), and Lorentz and Zeller proved that the inverse inequality E _n ⁽²⁾ (f) ≦ cE _n (f) is invalid even with the constant c = c(f) which depends on the function f ∈ Δ². In this paper we prove, for every α > 0 and function f ∈ Δ², that

Negative result in pointwise 3-convex polynomial approximation

Let Δ³ be the set of functions three times continuously differentiable on [−1, 1] and such that f″′(x) ≥ 0, x ∈ [−1, 1]. We prove that, for any n ∈ ℕ and r ≥ 5, there exists a function f ∈ C ^r [−1, 1] ⋂ Δ³ [−1, 1] such that ∥f ^(r)∥ _{C[−1, 1]} ≤ 1 and, for an arbitrary algebraic polynomial P ∈ Δ³ [−1, 1], there exists x such that