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.     

Fractional integration associated with second order divergence operators on R<Superscript>n</Superscript>

The classical Hardy-Littlewood-Sobolev theorems for Riesz potentials (−Δ)^−α/2 are extended to the generalised fractional integrals L ^–α/2 for 0 < α < n, where L=−div A∇ is a uniformly complex elliptic operator with bounded measurable coefficients in ℝⁿ.     

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.     

Empirical multifractal moment measures of self-similar measure for <Emphasis Type="Italic">q</Emphasis> < 0<Superscript>*</Superscript>

For q ≥ 0, Olsen [1] has attained the exact rate of convergence of the L ^q -spectrum of a self-similar measure and showed that the so-called empirical multifractal moment measures converges weakly to the normalized multifractal measures. Unfortunately, nothing is known for q < 0. Indeed, the problem of analysing the L ^q - spectrum for q < 0 is generally considered significantly more difficult since the L ^q -spectrum is extremely sensitive to small variations of μ for q < 0. In [2] we showed that self-similar measures satisfying the Open Set Condition (OSC) are Ahlfors regular and, using this fact, we obtained the exact rate of convergence of the L ^q -spectrum of a self-similar measure satisfying the OSC for q < 0. In this paper, we apply the results from [2] to show the empirical multifractal q’th moment measures of self-similar measures satisfying the OSC converges weakly to the normalized multifractal Hausdorff measures for q < 0. Authors’ addresses: Jiaqing Xiao, School of Science, Wuhan University of Technology, Wuhan 430070, China; Wu Min, School of Mathematical Sciences, South China University of Technology, Guangzhou, 510640, China     

Traces of functions with a dominating mixed derivative in ℝ<Superscript>3</Superscript>

We investigate traces of functions, belonging to a class of functions with dominating mixed smoothness in ℝ³, with respect to planes in oblique position. In comparison with the classical theory for isotropic spaces a few new phenomenona occur. We shall present two different approaches. One is based on the use of the Fourier transform and restricted to p = 2. The other one is applicable in the general case of Besov-Lizorkin-Triebel spaces and based on atomic decompositions.     

Sharp Estimates in Bergman and Besov Spaces on Bounded Symmetric Domains

Sharp estimates of the point-evaluation functional in weighted Bergman spaces L ^p _a (Ω, dν _α) and for the point-evaluation derivalive functional in Besov spaces B ^p (Ω) are obtained for bounded symmetric domains Ω in ℂ ⁿ . Received October 25, 1999, Accepted December 6, 2000     

Quasi-<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> norm orthogonal Galerkin expansions in sums of Jacobi polynomials

In the study of differential equations on [ − 1,1] subject to linear homogeneous boundary conditions of finite order, it is often expedient to represent the solution in a Galerkin expansion, that is, as a sum of basis functions, each of which satisfies the given boundary conditions. In order that the functions be maximally distinct, one can use the Gram-Schmidt method to generate a set orthogonal with respect to a particular weight function. Here we consider all such sets associated with the Jacobi weight function, w(x) = (1 − x) ^α (1 + x) ^β . However, this procedure is not only cumbersome for sets of large degree, but does not provide any intrinsic means to characterize the functions that result. We show here that each basis function can be written as the sum of a small number of Jacobi polynomials, whose coefficients are found by imposing the boundary conditions and orthogonality to the first few basis functions only. That orthogonality of the entire set follows—a property we term “auto-orthogonality”—is remarkable. Additionally, these basis functions are shown to behave asymptotically like individual Jacobi polynomials and share many of the latter’s useful properties. Of particular note is that these basis sets retain the exponential convergence characteristic of Jacobi expansions for expansion of an arbitrary function satisfying the boundary conditions imposed. Further, the associated error is asymptotically minimized in an L ^p(α) norm given the appropriate choice of α = β. The rich algebraic structure underlying these properties remains partially obscured by the rather difficult form of the non-standard weighted integrals of Jacobi polynomials upon which our analysis rests. Nevertheless, we are able to prove most of these results in specific cases and certain of the results in the general case. However a proof that such expansions can satisfy linear boundary conditions of arbitrary order and form appears extremely difficult.     

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     

Saturation of the linear methods of summation of Fourier series in the spaces <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript><Subscript>φ</Subscript>

We consider the problem of saturation of the linear methods of summation of Fourier series in the spaces S ^p _φ specified by arbitrary sequences of functions defined in a certain subset of the space ℂ. Sufficient conditions for the saturation of the indicated methods in these spaces are established. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 6, pp. 815–828, June, 2008.     

On the parity of the class number of an imaginary abelian field of conductor 2<Superscript><Emphasis Type="Italic">a</Emphasis></Superscript><Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">b</Emphasis></Superscript>

Let p be an odd prime number, and p^n₀{p^{n_0}} the highest power of p dividing 2 ^p−1 − 1. Let K_n=Q(z_pⁿ⁺¹){K_n={\bf Q}(\zeta_{p^{n+1}})} and L_n,j=K_n⁺(z_2^j+2){L_{n,j}=K_n^+(\zeta_{2^{j+2}})} for j ≥ 0. Let h_n^*{h_n^*} be the relative class number of K _n , and h _n,j the class number of L _n,j , respectively. Let n be an integer with n ≥ n ₀. We prove that if the ratio h_n^*/h_n-1^*{h_n^*/h_{n-1}^*} is odd, then h _n,j /h _n−1,j is odd for any j ≥ 0.     

Weighted Integrals and Bloch Spaces of n-Harmonic Functions on the Polydisc

We study anisotropic mixed norm spaces h(p,q,α) consisting of n-harmonic functions on the unit polydisc of by means of fractional integro-differentiation including small 0 < p < 1 and multi-indices α = (α ₁,...,α _n ) with non-positive α _j  ≤ 0. As an application, two different Bloch spaces of n-harmonic functions are characterized.      

Approximation by K-finite functions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

Let Γ ⊂ ℝⁿ, n ≥ 2, be the boundary of a bounded domain. We prove that the translates by elements of Γ of functions which transform according to a fixed irreducible representation of the orthogonal group form a dense class in L ^p (ℝⁿ) for . A similar problem for noncompact symmetric spaces of rank one is also considered. We also study the connection of the above problem with the injectivity sets for weighted spherical mean operators. The first author was supported in part by a grant from UGC via DSA-SAP Phase IV.     

ON LAGRANGE INTERPOLATION TO ｜x｜^α（1 〈α 〈 2） WITH EQUALLY SPACED NODES

S.M. Lozinskii proved the exact convergence rate at the zero of Lagrange interpolation polynomials to |x| based on equidistant nodes in [−1,1], In 2000, M. Rever generalized S.M. Lozinskii’s result to |x|^α(0≤α≤1). In this paper we will present the exact rate of convergence at the point zero for the interpolants of |x|^α(1<α<2).     

Star product algebras of test functions

We prove that the Gelfand-Shilov spaces S _α ^β are topological algebras under the Moyal *-product if and only if α ≥ β. These spaces of test functions can be used to construct a noncommutative field theory. The star product depends on the noncommutativity parameter continuously in their topology. We also prove that the series expansion of the Moyal product converges absolutely in S _α ^β if and only if β < 1/2. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 1, pp. 3–17, October, 2007.     

The convolution structure for Jacobi function expansions

The product ϕ _λ ^(α,β) (t₁)ϕ _λ ^(α,β) (t₂) of two Jacobi functions is expressed as an integral in terms of ϕ _λ ^(α,β) (t₃) with explicit non-negative kernel, when α≧β≧−1/2. The resulting convolution structure for Jacobi function expansions is studied. For special values of α and β the results are known from the theory of symmetric spaces.     

Orthogonality criteria for compactly supported refinable functions and refinable function vectors

A refinable function φ(x):ℝⁿ→ℝ or, more generally, a refinable function vector Φ(x)=[φ₁(x),...,φ_r(x)]^T is an L¹ solution of a system of (vector-valued) refinement equations involving expansion by a dilation matrix A, which is an expanding integer matrix. A refinable function vector is called orthogonal if {φ_j(x−α):α∈ℤⁿ, 1≤j≤r form an orthogonal set of functions in L²(ℝⁿ). Compactly supported orthogonal refinable functions and function vectors can be used to construct orthonormal wavelet and multiwavelet bases of L²(ℝⁿ). In this paper we give a comprehensive set of necessary and sufficient conditions for the orthogonality of compactly supported refinable functions and refinable function vectors.     

Boundedness of Multilinear Operators in Herz-type Hardy Space

Let κ∈ℕ. We prove that the multilinear operators of finite sums of products of singular integrals on ℝⁿ are bounded from HK ^α1,p1 _q1 (ℝⁿ) ×···×HK ^αk,pk _qk (ℝⁿ) into HK ^α,p _q (ℝⁿ) if they have vanishing moments up to a certain order dictated by the target spaces. These conditions on vanishing moments satisfied by the multilinear operators are also necessary when α_j≥ 0 and the singular integrals considered here include the Calderón-Zygmund singular integrals and the fractional integrals of any orders. Received September 6, 1999, Revised November 17, 1999, Accepted December 9, 1999     

Optimal Domains for <Emphasis Type="Italic">L</Emphasis><Superscript>0</Superscript>-valued Operators Via Stochastic Measures

We study extension of operators T: E→ L⁰([0, 1]), where E is an F–function space and L⁰([0, 1]) the space of measurable functions with the topology of convergence in measure, to domains larger than E, and we study the properties of such domains. The main tool is the integration of scalar functions with respect to stochastic measures and the corresponding spaces of integrable functions. Partially supported by D.G.I. #MTM2006-13000-C03-01 (Spain).     

A theorem on cyclic polytopes

LetC(ν, d) represent a cyclic polytope withν vertices ind dimensions. A criterion is given for deciding whether a given subset of the vertices ofC(ν, d) is the set of vertices of some face ofC(ν, d). This enables us to determine, in a simple manner, the number ofj-faces ofC(ν, d) for each value ofj (1≦j≦d−1).     

On a Sharper Lower Bound for a Percentile of a Student’s <Emphasis Type="Italic">t</Emphasis> Distribution with an Application

In this communication, we first compare z _α and t _ν,α , the upper 100α% points of a standard normal and a Student’s t _ν distributions respectively. We begin with a proof of a well-known result, namely, for every fixed 0 < a < \frac120<\alpha <\frac{1}{2} and the degree of freedom ν, one has t _ν,α  > z _α . Next, Theorem 3.1 provides a new and explicit expression b _ν ( > 1) such that for every fixed 0 < a < \frac120<\alpha < \frac{1}{2} and ν, we can conclude t _ν,α  > b _ν z _α . This is clearly a significant improvement over the result that is customarily quoted in nearly every textbook and elsewhere. A proof of Theorem 3.1 is surprisingly simple and pretty. We also extend Theorem 3.1 in the case of a non-central Student’s t distribution (Section 3.3). In the context of Stein’s (Ann Math Stat 16:243–258, 1945; Econometrica 17:77–78, 1949) 100(1 − α)% fixed-width confidence intervals for the mean of a normal distribution having an unknown variance, we have examined the oversampling rate on an average for a variety of choices of m, the pilot sample size. We ran simulations to investigate this issue. We have found that the oversampling rates are approximated well by t_n,a/2²z_a/2^-2t_{\nu ,\alpha /2}^{2}z_{\alpha /2}^{-2} for small and moderate values of m( ≤ 50) all across Table 2 where ν = m − 1. However, when m is chosen large (≥ 100), we find from Table 3 that the oversampling rates are not approximated by t_n,a/2²z_a/2^-2t_{\nu ,\alpha /2}^{2}z_{\alpha /2}^{-2} very well anymore in some cases, and in those cases the oversampling rates either exceed the new lower bound of t_n,a/2²z_a/2^-2,t_{\nu ,\alpha /2}^{2}z_{\alpha /2}^{-2}, namely b_n²,b_{\nu }^{2}, or comes incredibly close to b_n²b_{\nu }^{2} where ν = m − 1. That is, the new lower bound for a percentile of a Student’s t distribution may play an important role in order to come up with diagnostics in our understanding of simulated output under Stein’s fixed-width confidence interval method.