Nearly flat almost monotone measures are big pieces of lipschitz graphs

A concentrated (ξ, m) almost monotone measure inR ⁿ is a Radon measure Φ satisfying the two following conditions: (1) Θ ^m (Φ,x)≥1 for every x ∈spt (Φ) and (2) for everyx ∈R ⁿ the ratioexp [ξ(r)]r^−mΦ(B(x,r)) is increasing as a function of r>0. Here ξ is an increasing function such thatlim _r→0-ξ(r)=0. We prove that there is a relatively open dense setReg (Φ) ∋spt (Φ) such that at each x∈Reg(Φ) the support of Φ has the following regularity property: given ε>0 and λ>0 there is an m dimensional spaceW ⊂R ⁿ and a λ-Lipschitz function f from x+W into x+W^‖ so that (100-ε)% ofspt(Φ) ∩B (x, r) coincides with the graph of f, at some scale r>0 depending on x, ε, and λ.     

High-Dimensional Centrally Symmetric Polytopes with Neighborliness Proportional to Dimension

Let A be a d by n matrix, d < n. Let C be the regular cross polytope (octahedron) in Rⁿ. It has recently been shown that properties of the centrosymmetric polytope P = AC are of interest for finding sparse solutions to the underdetermined system of equations y = Ax [9]. In particular, it is valuable to know that P is centrally k-neighborly. We study the face numbers of randomly projected cross polytopes in the proportional-dimensional case where d ∼ δn, where the projector A is chosen uniformly at random from the Grassmann manifold of d-dimensional orthoprojectors of Rⁿ. We derive ρ_N(δ) > 0 with the property that, for any ρ < ρ_N(δ), with overwhelming probability for large d, the number of k-dimensional faces of P = AC is the same as for C, for 0 ≤ k ≤ ρd. This implies that P is centrally ⌊ ρ d ⌋-neighborly, and its skeleton Skel_{⌊ ρ d ⌋}(P) is combinatorially equivalent to Skel_{⌊ ρ d⌋}(C). We display graphs of ρ_N. Two weaker notions of neighborliness are also important for understanding sparse solutions of linear equations: weak neighborliness and sectional neighborliness [9]; we study both. Weak (k,ε)-neighborliness asks if the k-faces are all simplicial and if the number of k-dimensional faces f_k(P) ≥ f_k(C)(1 – ε). We characterize and compute the critical proportion ρ_W(δ) > 0 such that weak (k,ε) neighborliness holds at k significantly smaller than ρ_W · d and fails for k significantly larger than ρ_W · d. Sectional (k,ε)-neighborliness asks whether all, except for a small fraction ε, of the k-dimensional intrinsic sections of P are k-dimensional cross polytopes. (Intrinsic sections intersect P with k-dimensional subspaces spanned by vertices of P.) We characterize and compute a proportion ρ_S(δ) > 0 guaranteeing this property for k/d ∼ ρ < ρ_S(δ). We display graphs of ρ_S and ρ_W.     

Extensions of the measurable choice theorem by means of forcing

Using the method of forcing of set theory, we prove the following two theorems on the existence of measurable choice functions: LetT be the closed unit interval [0,1] and letm be the usual Lebesgue measure defined on the Borel subsets ofT. Theorem1. LetS⊂T×T be a Borel set such that for alltεT,S _t ^def={x|(t,x)εS} is countable and non-empty. Then there exists a countable series of Lebesgue-measurable functionsf _n: T→T such thatS _t={f_n(t)|nεω} for alltε[0,1],W _x={y|(x,y)εW} is uncountable. Then there exists a functionh:[0,1]×[0,1]→W with the following properties: (a) for each xε[0,1], the functionh(x,·) is one-one and ontoW _x and is Borel measurable; (b) for eachy, h(·, y) is Lebesgue measurable; (c) the functionh is Lebesgue measurable.     

On the stability of the quadratic functional equation on bounded domains

A result of Skof and Terracini will be generalized; More precisely, we will prove that if a functionf : [-t, t]ⁿ →E satisfies the inequality (1) for some δ > 0 and for allx, y ∈ [-t, t]ⁿ withx + y, x - y ∈ [-t, t]ⁿ, then there exists a quadratic functionq: ℝⁿ →E such that ∥f(x) -q(x)∥ < (2912n² + 1872n + 334)δ for anyx ∈ [-t, t] ⁿ .     

Range Minima Queries with Respect to a Random Permutation,and Approximate Range Counting

In approximate halfspace range counting, one is given a set P of n points in ℝ ^d , and an ε>0, and the goal is to preprocess P into a data structure which can answer efficiently queries of the form: Given a halfspace h, compute an estimate N such that (1−ε)|P∩h|≤N≤(1+ε)|P∩h|.     

Weakly metric spaces

The paper contains some initial results of the theory of weakly metric spaces. The weak triangle axiom: for any ε > 0 there exists δ > 0 such that for any points x, y, and z with d(y, z) ≤ δ, the inequality d(x, z) ≤ d(x, y) + ε holds. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 352, 2008, pp. 94–105.     

On the zeros of a class of generalised Dirichlet series-XIV

We prove a general theorem on the zeros of a class of generalised Dirichlet series. We quote the following results as samples. Theorem A.Let 0<θ<1/2and let {a _n }be a sequence of complex numbers satisfying the inequality for N = 1,2,3,…,also for n = 1,2,3,…let α _n be real and |α_n| ≤ C(θ)where C(θ) > 0is a certain (small)constant depending only on θ. Then the number of zeros of the function in the rectangle (1/2-δ⩽σ⩽1/2+δ,T⩽t⩽2T) (where 0<δ<1/2)is ≥C(θ,δ)T logT where C(θ,δ)is a positive constant independent of T provided T ≥T ₀(θ,δ)a large positive constant. Theorem B.In the above theorem we can relax the condition on a _n to and |a_N| ≤ (1/2-θ)^-1.Then the lower bound for the number of zeros in (σ⩾1/3−δ,T⩽t⩽2T)is > C(θ,δ) Tlog T(log logT)^-1.The upper bound for the number of zeros in σ⩾1/3+δ,T⩽t⩽2T) isO(T)provided for every ε > 0. Dedicated to the memory of Professor K G Ramanathan     

NON-EXISTENCE OF POSITIVE ENTIRE SOLUTIONS FOR ELLIPTIC INEQUALITIES OF P-LAPLACIAN

In this paper,the nonexistence of positive entire solutions for div(|Du|^1-2Du)≥q(x)f(u),x∈R^N,is establisbed,where p＞1,DU=(D,u……dnu),qsR^N→(o,∞)and f2(0,∞)→(o,∞)are continuous functions.     

SOLVING SYMMETRIC MONOTONE LINEAR VARIATIONAL INEQUALITIES BY SOME MODIFIED LEVITIN-POLYAK PROJECTION METHODS

Some modified Levitin-Polyak projection methods are proposed in this paper for solving monotone linear variational inequality x∈Ω，（x′-x)^T(Hx c)≤0，for any x′∈Ω.It is pointed out that there are similar methods for solving a general linear variational inequality.     

A note on intersections of free submonoids of a free monoid

According to a theorem of Tilson [6] any intersection of free submonoids of a free monoid is free. Here we consider intersections of the form {x, y}^* ∩ {u, v}^*, where x, y, u and v are words in a finitely generated free monoid Σ^*, and show that if both the monoids {x, y}^* and {u, v}^* are of the rank two, then the intersection is a free monoid generated either by (at most) two words or by a regular language of the form β₀ + β((γ(1+ δ + ... δ^t))^*ε for some words β⁰, β, γ, δ and ε, and some integer t≥0. An example is given showing that the latter possibility may occur for each t≥0 with nonempty values of the words.     

The central connection problem at turning points of linear differential equations

A system of linear differential equations of the vectorial form εdy/dx=A (x, ε) y is considered, where ε is a positive parameter, and the matrixA (x, ε) is holomorphic in |x|⩽x ₀, 0 < ε ⩽ ε₀ , with an asymptotic expansionsA (x, ε) ∼ ∑ _r=0 ^∞ A _r (x) ε ^r , as ε→0. The eigenvalues ofA ₀(x) are supposed to coalesce atx=0 so as to make this point a simple turning point. With the help of refinements of the representations for the inner and outer asymptotic solutions, as ε→0, that were introduced in the articles [9] and [10] by the author (see the references at the end of the paper), explicit connection formulas between these solutions are calculated. As part of this derivation it is shown that only the diagonal entries of the connection matrix are asymptotically relevant.     

A collocation by spline in tension

The collocation method by spline in tension for the problem: −εy"+p(x)y=f(x), y(0)=α₀,y(1)=α₁, p(x)>0, 0<ε<<1, is derived. The method has the second order of the global uniform convergence. For the corresponding difference scheme the optimal estimate: O (h_imin(h_i, ε) is obtained. This research was supported partly by NSF and SIZ for Science of SAP Vojvodina through funds made available to the U.S.—Yugoalav Joint Board on Scientific and Tchnological Cooperation (grants JF554, JF799).     

Tiling Transitive Tournaments and Their Blow-ups

Let TT _k denote the transitive tournament on k vertices. Let TT(h,k) denote the graph obtained from TT _k by replacing each vertex with an independent set of size h≥1. The following result is proved: Let c ₂=1/2, c ₃=5/6 and c _k =1−2^{−k−log k} for k≥4. For every ∈>0 there exists N=N(∈,h,k) such that for every undirected graph G with n>N vertices and with δ(G)≥c _k n, every orientation of G contains vertex disjoint copies of TT(h,k) that cover all but at most ∈n vertices. In the cases k=2 and k=3 the result is asymptotically tight. For k≥4, c _k cannot be improved to less than 1−2^{−0.5k(1+o(1))}. This revised version was published online in June 2006 with corrections to the Cover Date.     

Parabolic Implosion and Julia-Lavaurs Sets in the Exponential Family

We deal with all the maps from the exponential family f _ε(z) = (e ⁻¹ + ε)exp(z), with ε ≥ 0. Let h _ε = HD(J _r,ε) be the Hausdorff dimension of the radial Julia sets J _r,ε. Observing the phenomenon of parabolic implosion, it is shown that the function ε ↦ h _ε is not continuous from the right.     

Global existence and time decay of small solutions to the landau-ginzburg type equations

We study the Cauchy problem for the nonlinear dissipative equations (0.1) uo∂u-αδu + Β|u|^2/n u = 0,x ∃ Rⁿ,t } 0,u(0,x) = u0(x),x ∃ Rⁿ, where α,Β ∃ C, ℜα 0. We are interested in the dissipative case ℜα 0, and ℜδ(α,Β)≥ 0, θ = |∫ u0(x)dx| ⊋ 0, where δ(α, Β) = ##|α|^n-1n^n/2 / ((n + 1)|α|² + α^{2 n/2}. Furthermore, we assume that the initial data u₀ ∃ L^p are such that (1 + |x|)^αu0 ∃ L¹, with sufficiently small norm ∃ = (1 + |x|)^α u₀ 1 + u₀ p, wherep 1, α ∃ (0,1). Then there exists a unique solution of the Cauchy problem (0.1)u(t, x) ∃ C ((0, ∞); L^∞) ∩ C ([0, ∞); L¹ ∩ L^p) satisfying the time decay estimates for allt0 u(t)||_∞ Cɛt^-n/2(1 + η log 〈t〉)^-n/2, if hg = θ^2/n 2_π ℜδ(α, Β) 0; u(t)||_∞ Cɛt^-n/2(1 + Μ log 〈t〉)^-n/4, if η = 0 and Μ = θ^4/n 4π)² (ℑδ(α, Β))² ℜ((1 + 1/n) υ1-1 υ2) 0; and u(t)||_∞ Cɛt^-n/2(1 + κ log 〈t〉)^-n/6, if η = 0, Μ = 0, κ 0, where υl,l = 1,2 are defined in (1.2), κ is a positive constant defined in (2.31).     

Decomposition of the conjugacy representation of the symmetric groups

Consider the two natural representations of the symmetric groupS _n on the group algebra ℂ[S _n ]: the regular representation and the conjugacy representation (acting on the basis by conjugation). Letm(λ) be the multiplicity of the irreducible representationS ^λ in the conjugacy representation and letf ^λ be the multiplicity ofS ^λ in the regular representation. By the character estimates of [R1] and [Wa] we prove

N(2,2,0)代数的E-反演半群

本文研究了N(2,2,0)代数(S,*,△,0)的E-反演半群.利用N(2,2,0)代数的幂等元,弱逆元,中间单位元的性质和同宇关系,得到了N(2,2,0)代数的半群(S,*)构成E-反演半群的条件及元素α的右伴随非零零因子唯一,且为α的弱逆元等结论,这些结果进一步刻画了N(2,2,0)代数的结构.     

Existence of solutions with turning points for nonlinear singularly perturbed boundary-value problems

We consider the following singularly perturbed boundary-value problem:

On the range of the derivative of Gâteaux-smooth functions on separable banach spaces

We prove that there exists a Lipschitz function froml ¹ into ℝ² which is Gateaux-differentiable at every point and such that for everyx, y εl ¹, the norm off′(x) −f′(y) is bigger than 1. On the other hand, for every Lipschitz and Gateaux-differentiable function from an arbitrary Banach spaceX into ℝ and for everyε > 0, there always exist two pointsx, y εX such that ‖f′(x) −f′(y)‖ is less thanε. We also construct, in every infinite dimensional separable Banach space, a real valued functionf onX, which is Gateaux-differentiable at every point, has bounded non-empty support, and with the properties thatf′ is norm to weak* continuous andf′(X) has an isolated pointa, and that necessarilya ε 0. This work has been initiated while the second-named author was visiting the University of Bordeaux. The second-named author is supported by grant AV 1019003, A1 019 205, GA CR 201 01 1198.     

Parabolic Implosion and Julia-Lavaurs Sets in the Exponential Family

We deal with all the maps from the exponential family f _ε(z) = (e ⁻¹ + ε)exp(z), with ε ≥ 0. Let h _ε = HD(J _r,ε) be the Hausdorff dimension of the radial Julia sets J _r,ε. Observing the phenomenon of parabolic implosion, it is shown that the function ε ↦ h _ε is not continuous from the right. The research of the first author was supported in part by the NSF Grant DMS 0100078.