Distribution functions of binary solutions (exact analytic solution)

We show that the general solution of the Ornstein-Zernike system of equations for multicomponent solutions has the form h_αβ=∑A _αβ ^j exp(-λ_jr)/r, where λ_j are the roots of the transcendental equation 1-ρΔ(λ_j)=0 and the amplitudes A_αβ ^j can be calculated if the direct correlation functions are given. We investigate the properties of this solution including the behavior of the roots A _αβ ^j and amplitudes A_αβ ^j in both the low-density limit and the vicinity of the critical point. Several relations on A_αβ ^j and C_αβ are found. In the vicinity of the critical point, we find the state equation for a liquid, which confirms the Van der Waals similarity hypothesis. The expansion under consideration is asymptotic because we expand functions in series in eigenfunctions of the asymptotic Ornstein-Zernike equation valid at r→∞. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 3, pp. 500–515, June, 2000.     

Thermodynamic formalism of transcendental entire maps of finite singular type

We build a version of a thermodynamic formalism for maps of the form f(z) = ∑ _{j = 0} ^{p + q} a _j e ^(j−p)z where p, q > 0 and . We show in particular the existence and uniqueness of (t,α)-conformal measures and that the Hausdorff dimension HD(J _f ^r ) = h is the unique zero of the pressure function t ↦ P(t) for t > 1, where the set J _f ^r is the radial Julia set. Partially supported by NSF Grant DMS 0100078. Partially supported by Warsaw University of Technology Grant No. 504G11200023000, Polish KBN Grant No. 2PO3A03425 and Chilean FONDECYT Grant No. 11060280.     

Systems of equations driven by stable processes

Let Z^j_t, j = 1, . . . , d, be independent one-dimensional symmetric stable processes of index α ∈ (0,2). We consider the system of stochastic differential equations where the matrix A(x)=(A_ij(x))1≤ i, j ≤ d is continuous and bounded in x and nondegenerate for each x. We prove existence and uniqueness of a weak solution to this system. The approach of this paper uses the martingale problem method. For this, we establish some estimates for pseudodifferential operators with singular state-dependent symbols. Let λ₂ > λ₁ > 0. We show that for any two vectors a, b∈ ℝ_d with |a|, |b| ∈ (λ₁, λ₂) and p sufficiently large, the L^p-norm of the operator whose Fourier multiplier is (|u · a|^α - |u · b|^α) / ∑_j=1^d |u_i|^α is bounded by a constant multiple of |a−b|^θ for some θ > 0, where u=(u₁ , . . . , u_d) ∈ ℝ^d. We deduce from this the L^p-boundedness of pseudodifferential operators with symbols of the form ψ(x,u)=|u · a(x)|^α / ∑_j=1^d |u_i|^α, where u=(u₁,...,u_d) and a is a continuous function on ℝ^d with |a(x)|∈ (λ₁, λ₂) for all x∈ ℝ^d. Research partially supported by NSF grant DMS-0244737. Research partially supported by NSF grant DMS-0303310.     

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.     

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.     

The Banach-Mazur distance to the cube and the Dvoretzky-Rogers factorization

We show the existence of a sequence (λ _n ) of scalars withλ _n =o(n) such that, for any symmetric compact convex bodyB ⊂R ⁿ , there is an affine transformationT satisfyingQ ⊂T(B) ⊂λ _n Q, whereQ is then-dimensional cube. This complements results of the second-named author regarding the lower bound on suchλ _n . We also show that ifX is ann-dimensional Banach space andm=[n/2], then there are operatorsα:l ₂ ^m →X andβ:X→l _∞ ^m with ‖α‖·‖β‖≦C, whereC is a universal constant; this may be called “the proportional Dvoretzky-Rogers factorization”. These facts and their corollaries reveal new features of the structure of the Banach-Mazur compactum. Research performed while this author was visiting IHES. Supported in part by the NSF Grant DMS-8702058 and the Sloan Research Fellowship.     

Gersgorin variations II: On themes of Fan and Gudkov

Assume F={f₁,. . .,f_n} is a family of nonnegative functions of n−1 nonnegative variables such that, for every matrix A of order n, |a_ii|>f_i (moduli of off-diagonal entries in row i of A) for all i implies A nonsingular. We show that there is a positive vector x, depending only on F, such that for all A=(a_ij), and all i, f_i≥∑_j|a_ij|{x_j}/x_i. This improves a theorem of Ky Fan, and yields a generalization of a nonsingularity criterion of Gudkov. Dedicated to Charles A. Micchelli, in celebration of his 60th birthday and our 30 years of friendship     

Arithmetical properties of the number of t-core partitions

Let Λ={λ ₁≥⋅⋅⋅≥λ _s ≥1} be a partition of an integer n. Then the Ferrers-Young diagram of Λ is an array of nodes with λ _i nodes in the ith row. Let λ _j ′ denote the number of nodes in column j in the Ferrers-Young diagram of Λ. The hook number of the (i,j) node in the Ferrers-Young diagram of Λ is denoted by H(i,j):=λ _i +λ _j ′−i−j+1. A partition of n is called a t-core partition of n if none of the hook numbers is a multiple of t. The number of t-core partitions of n is denoted by a(t;n). In the present paper, some congruences and distribution properties of the number of 2 ^t -core partitions of n are obtained. A simple convolution identity for t-cores is also given.      

Asymptotics of degrees of someS n-sub regular representations

The numbers % MathType!End!2!1!, λ ⊢n appear in the enumeration of various objects, as well as coefficients inS _nrepresentations associated with products of higher commutators. We study their asymptotics asn→∞ and show that if (λ₁, λ₂, …)≈(α ₁,α ₂, …)n, if (λ′₁, λ′₂, …)≈(β ₁,β ₂, …)n and ifγ=1− Σ _k⩽1(α _k⩽1+β _k⩽1), then % MathType!End!2!1!. Work partially supported by N.S.F. Grant No. DMS 94-01197.     

Multiplicative isometries on the Smirnov class

We show that T is a surjective multiplicative (but not necessarily linear) isometry from the Smirnov class on the open unit disk, the ball, or the polydisk onto itself, if and only if there exists a holomorphic automorphism Φ such that T(f)=f ○ Φ for every class element f or T(f) = [`(f<sup>°</sup> [`(j)] )]\overline {f^\circ \bar \varphi } for every class element f, where the automorphism Φ is a unitary transformation in the case of the ball and Φ(z ₁, ..., z _n ) = (l₁ z_i1 ,...,l_n z_in )(\lambda _1 z_{i_1 } ,...,\lambda _n z_{i_n } ) for |λ _j | = 1, 1 ≤ j ≤ n, and (i ₁; ..., i _n )is some permutation of the integers from 1through n in the case of the n-dimensional polydisk.     

The cotype constant and an almost euclidean decomposition for finite-dimensional normed spaces

Every 2n-dimensional normed spaceE contains twon-dimensional subspacesE ₁ andE ₂ which are orthogonal with respect to the inner product induced by the John ellipsoid ofE and which satisfyd(E _i, l ₂ ⁿ )≦f(K ₂(E)), wheref(K ₂(E)) is some number that depends only on the cotype constant ofE, denotedK ₂(E). Supported in part by NSF grant DMS 8401906.     

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     

A Reflexivity Result Concerning Banach Space Operators with a Multiply Connected Spectrum

We consider a multiply connected domain Ω which is obtained by removing n closed disks which are centered at λ _j with radius r _j for j = 1, . . . , n from the unit disk. We assume that T is a bounded linear operator on a separable reflexive Banach space whose spectrum contains ∂Ω and does not contain the points λ₁, λ₂, . . . , λ _n , and the operators T and r _j (T − λ _j I)⁻¹ are polynomially bounded. Then either T has a nontrivial hyperinvariant subspace or the WOT-closure of the algebra {f(T) : f is a rational function with poles off [`(W)]{\overline\Omega}} is reflexive.     

L p − Lq estimates for convolution operators with n-Dimensional singular measures

Let S ⊂ ℜⁿ⁺¹ be the graph of the function ϕ :[−1, 1] ⁿ → ℜ defined by ϕ (x ₁ , …, x_n) = ∑ _j=1 ⁿ |x_j|^αj, with1<α ₁ ≤ … ≤ α_n, let σ the Euclidean area measure on S. In this article we study the L^p − L^q boundedness of convolution operators with the singular Borel measure on Rⁿ⁺¹ given by μ (E)=σ (E ∩ S)     

On double sine and cosine transforms,Lipschitz and Zygmund classes

We consider complex-valued functions f ∈ L 1 (R+2),where R +:= [0,∞),and prove sufficient conditions under which the double sine Fourier transform f ss and the double cosine Fourier transform f cc belong to one of the two-dimensional Lipschitz classes Lip(α,β) for some 0 α,β≤ 1;or to one of the Zygmund classes Zyg(α,β) for some 0 α,β≤ 2.These sufficient conditions are best possible in the sense that they are also necessary for nonnegative-valued functions f ∈ L 1 (R+2).     

Comparison of exact constants in Kolmogorov-type inequalities for periodic and nonperiodic functions of many variables

We investigate the correlation between the constants K(ℝⁿ) and , where

Convergence of mock Fourier series

For certain Cantor measures μ on ℝⁿ, it was shown by Jorgensen and Pedersen that there exists an orthonormal basis of exponentialse ^2πiγ·x for λεΛ. a discrete subset of ℝⁿ called aspectrum for μ. For anyL ¹ functionf, we define coefficientsc _γ(f)=∝f(y)e ^−2πiγiy dμ(y) and form the Mock Fourier series ∑_λ∈Λ^cλ(f)e ^2πiλ·x . There is a natural sequence of finite subsets Λ_n increasing to Λ asn→∞, and we define the partial sums of the Mock Fourier series by We prove, under mild technical assumptions on μ and Λ, thats _n(f) converges uniformly tof for any continuous functionf and obtain the rate of convergence in terms of the modulus of continuity off. We also show, under somewhat stronger hypotheses, almost everywhere convergence forfεL ¹. Research supported in part by the National Science Foundation, Grant DMS-0140194.     

Inequalities and sphere-packing inl p

If 1<p<∞, there is a constantr _p <1/2 so that ifr>r _p only a bounded number of balls inl _p of radiusr can be packed into the unit ball ofl _p . We obtain the exact value of this bound for eachp andr as a consequence of several new inequalities relating the expressions Σλ _i λ _j ‖x _i –x _j ‖ ^p , Σλ _i ‖x _i ‖ ^p and Σλ _i ^/2 for sequences (x _i ) ₁ ⁿ ⊂l _p and (λ _i ) ₁ ⁿ ⊂R.     

Admissible functions with multiple discontinuities

LetT be the unit circle, α irrational andF: T → R a step function. A necessary and sufficient condition for the skew of the α-rotation byf (considered as taking values mod 1) to be minimal is given. Also, the boundedness of Σ _i=1 ⁿ f(x+iα asn → α is resolved.