Characterization theorem of generalized polynomial of best approximation having bounded coefficients

Let the set of generalized polynomials having bounded coefficients beK={p= _jg_j. _{j j j},j=1, 2, ...,n}, whereg ₁,g ₂, ...,g _n are linearly independent continuous functions defined on the interval [a, b], _j, _j are extended real numbers satisfying _j<+, _j>-, and _{j j}. Assume thatf is a continuous function defined on a compact setX [a, b]. This paper gives the characterization theorem forp being the best uniform approximation tof fromK, and points out that the characterization theorem can be applied in calculating the approximate solution of best approximation tof fromK.     

Weak convergence using higher-order cumulants

Denote byc _j (F) thejth cumulant (or semi-invariant) of the distribution functionF. We say thatF is specified by its higher-order cumulants if it is the unique distribution functionG having the following property: there exists a positive integerJ such thatc _j (G)=c _j (F) forj=1,2 andjJ. Let (F _n n1) be a sequence of distribution functions, and suppose that there existsJ such thatc _j (F _n )c _j (F) asn, forj=1,2 andjJ. It is proved thatF _n F so long asF is specified by its higher-order cumulants. It is an open problem to characterize the family of distributions which are specified by their higher-order cumulants.     

Wave operators in a multistratified strip

One investigates the scattering theory for the positive self-adjoint operatorH=–· acting in with = × and a bounded open set in ^n–1,n2. The real-valued function belongs toL (), is bounded from below byc>0 and there exist real-valued functions ₁ and ₂ inL () such that – _j ,j=1,2 is a short range perturbation of _j when (–1) ^j x _n +. One assumes _j = ^(j) 1_R,j=1,2, with ^(j) L bounded from below byc>0. One proves the existence and completeness of the generalized wave operators _j ^± =s – _j e ^–itHj ,j=1,2, withH _j =–· _j and _j : equal to 1 if (–1) ^j x _n >0 and to 0 if (–1) ^j x _n <0. The ranges ofW _j ^± :=( _j ^± )^* are characterized so that W ₁ ^± =Ran and . The scattering operator can then be defined.     

Independence of Likelihood Ratio Criteria for Homogeneity of Several Populations

Let _i be an i-tb population with a probability density function f(· | _i ) with one dimensional unknown parameter _i = 1, 2, ... , k. Let n _i sample be drawn from each _i . The likelihood ratio criteria _j|(j–1) for testing hypothesis that the first j parameters are equal against alternative hypothesis that the first (j – 1) parameters are equal and the j-th parameter is different with the previous ones are defined, j = 2, 3, ... , k. The paper shows the asymptotic independence of _j|(j–1)'s up to the order 1/n under a hypothesis of equality of k parameters, where n is a number of total samples.     

Summability of Hermite Series

Let fL ^p (R), 1pt8, and c _j be the inner product of f and the Hermite function h _j . Assume that c _j 's satisfy If r=5/4, then the Hermite series c _j h _j conerges to f almost everywhere. If r=9/4-1/p, the c _j h _j converges to f in L ^p (R).     

Arithmetic of the Modular Function j1,8

Since the genus of the modular curve X_1 (8) = _1 (8) * is zero, we find a field generator j _1,8(z) = ₃(2z)/₃(4z) (₃(z) := _n ein ^2z ) such that the function field over X ₁(8) is (j _1,8). We apply this modular function j _1,8 to the construction of some class fields over an imaginary quadratic field K, and compute the minimal polynomial of the singular value of the Hauptmodul N(j _1,8) of (j _1,8).     

On the Heyde Theorem for Finite Abelian Groups

It is well-known Heyde's characterization theorem for the Gaussian distribution on the real line: if _j are independent random variables, _j , _j are nonzero constants such that _i ± _j ^–1 _j 0 for all i j and the conditional distribution of L ₂=_{1 1} + ··· + _{n n} given L ₁=_{1 1} + ··· + _{n n} is symmetric, then all random variables _j are Gaussian. We prove some analogs of this theorem, assuming that independent random variables take on values in a finite Abelian group X and the coefficients _j , _j are automorphisms of X.     

A Remark on Bernstein,Prokhorov, Bennett,Hoeffding, and Talagrand Inequalities

Let X ₁,..., X_n be independent random variables such that {X_j 1}=1 and E X _j=0 for all j. We prove an upper bound for the tail probabilities of the sum M _n=X₁+...+ X_n. Namely, we prove the inequality {M _nx} 3.7 {S_n x}, where S _n=₁+...+ _n is a sum of centered independent identically distributed Bernoulli random variables such that E S _n ² =ME M _n ² and {_k=1}=E S _n ² /(n+E S _n ² ) for all k (we call a random variable Bernoulli if it assumes at most two values). The inequality holds for x at which the survival function x{S _nx} has a jump down. For remaining x, the inequality still holds provided that we interpolate the function between the adjacent jump points linearly or log-linearly. If necessary, in order to estimate {S _nx} one can use special bounds for binomial probabilities. Up to the factor at most 2.375, the inequality is final. The inequality improves the classical Bernstein, Prokhorov, Bennett, Hoeffding, Talagrand, and other bounds.     

Heat equation for an arbitrary doubly-connected region inR 2 with mixed boundary conditions

The basic problem in this paper is that of determining the geometry of an arbitrary doubly-connected region inR ² with mixed boundary conditions, from the complete knowledge of the eigenvalues { _j } _j=1 for the Laplace operator, using the asymptotic expansion of the spectral function (t)= _j=1 exp(–t _j ) ast0.     

A family of noetherian rings with their finite length modules under control

We investigate the category mod of finite length modules over the ring =A _k , where is a V-ring, i.e. a ring for which every simple module is injective, k a subfield of its centre and A an elementary k-algebra. Each simple module E _j gives rise to a quasiprogenerator P _j = A E _j . By a result of K. Fuller, P _j induces a category equivalence from which we deduce that mod _j mod EndP _j . As a consequence we can(1) construct for each elementary k-algebra A over a finite field k a nonartinian noetherian ring such that modA mod(2) find twisted versions of algebras of wild representation type such that itself is of finite or tame representation type (in mod)(3) describe for certain rings the minimal almost split morphisms in mod and observe that almost all of these maps are not almost split in Mod.     

Fast Summation of Power Series with Coefficients Analytic at Infinity

We propose a fast summation algorithm for slowly convergent power series of the form _{j=j 0} z ^j _j j _i=1 ^s (j+ _i )^– _i , where R, _i 0 and _i C, 1is, are known parameters, and _j =(j), being a given real or complex function, analytic at infinity. Such series embody many cases treated by specific methods in the recent literature on acceleration. Our approach rests on explicit asymptotic summation, started from the efficient numerical computation of the Laurent coefficients of . The effectiveness of the resulting method, termed ASM (Asymptotic Summation Method), is shown by several numerical tests.     

Strong uniqueness of generalized polynomial of best approximation having bounded coefficients

Let the set of generalized polynomials haviug bounded coefficients be whereg ₁,g ₂, ...,g _n are linearly independent continuous functions defined on the interval [a, b], _j , _j are extended real numbers satisfying _j <+, _j >– and _{j j} . Assume thatf is a continuous function defined on a compact setX [a, b]. In the paper, we first give the sufficient conditions for the polynomial of best uniform approximation tof fromK being unique and strongly unique. Furthermore, we give two forms of necessary and sufficient conditions for the best approximation to be strongly unique.     

On solvability of linear difference equations in smooth and real analytic vector functions of several variables

We investigate the multidimensional equations _j=1 ^q A_j(x)y(x+e _j )=f(x),e _j ⁿ wherex ⁿ andA _j : ⁿ Hom( ^p , ^m ),f : ^{n m} are given maps. Sufficient conditions for smooth and analytic solvability for anyf C ^k ,k are found.Research partially supported by the Israel Ministry of ScienceAMS classification 39B Functional equations     

ALTERNATION THEORY IN APPROXIMATION BY POLYNOMIALS HAVING BOUNDED COEFFICIENTS

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Un théorème sur les séries de Dirichlet

We considerk Dirichlet series a _j (n)n ^–s (1jk),k2. We suppose that for eachj the series a _j (n)n ^–s converges fors=s _j =j+it _j , and that Max j<1/(k–1). We prove that the (Dirichlet) product of these series converges uniformly on every bounded segment of the line es = (₁+...+ _k )/k+1–1/k and we estimate the rate of convergence. The number 1–1/k cannot be replaced by a smaller one.     

Stress Intensity Factors near the Cracks on the Edges of Openings in Composite Plates

Based on the photoelasticity method, the behavior of stress intensity factors (SIFs) near cracks propagating from the edges of openings in plates made of elastic and linearly viscoelastic fibrous composite materials is studied. It is found that the relative value of the SIF, K ₁/K ₁ ⁰ (K ₁ ⁰=₀c), near the crack tips on the edges of openings in composite plates is a function of the ratio c/R, whose numerical values depend on the mechanical properties of materials of the plates. Using the quasi-elastic method for solving the viscoelastic problems, the effect of viscoelastic properties of the plate material on the value of K ₁/K ₁ ⁰ is estimated. It is shown that the values of the function K ₁(t)/>/K ₁ ⁰ near the cracks on the opening edges in plates made of linearly viscoelastic fibrous composites grow under creep.     

Transforming binary sequences using priority queues

A priority queue transforms an input sequence into an output sequence which is a re-ordering of the sequence . The setR of all such related pairs is studied in the case that is a binary sequence. It is proved thatR is a partial order and that ¦R¦=c _n+1, the (n+1)th Catalan number. An efficient (O(n ²)) algorithm is given for computing the number of outputs achievable from a given input.     

A0-contractivity of the variable-step BDF's of order less than four

We construct a polyhedral norm in which the fixed-step backward differentiation formulas of order one to three are contractive when applied to any differential equation of the formy=(t)y, (t) 0 (i.e. A₀-contractive). The result also holds for non-uniform step-sequences if combined with certain restrictions on the stepwise order-selection.