Matrix summability and a generalized Gibbs phenomenon

Letf be a real-valued function sequence {f _k } that converges to on a deleted neighborhoodD of . If there is a subsequence {f _k(j) } and a number sequencex such that lim _j x _j = and either lim _j f _k(j) (x _j )>lim sup _x (x) or lim _j f _k(j) (x _j ) x (x), thenf is said to display theGibbs phenomenon at . IfA is a (real) summability matrix, thenAf is a function sequence given by(Af) _n (x)= _k=0 a _n,k f _k (x). IfAf displays the Gibbs phenomenon wheneverf does, thenA is said to beGP-preserving. By replacingf _k (x) withf _k (x _j )F _k,j , the Gibbs phenomenon is viewed as a property of the matrixF, andGP-preserving matrices are determined by properties of the matrix productAF. The general results give explicit conditions on the entries {a _n,k } that are necessary and/or sufficient forA to beGP-preserving. For example: if(x)0 thenF displaysGP iff lim _k,j F _k,j 0, and ifA isGP-preserving then lim _n,k A _n,k 0. IfA is a triangular matrix that is stronger than convergence, thenA is notGP-preserving. The general results are used to study the preservation of the Gibbs phenomenon by matrix methods of Nörlund, Hausdorff, and others.     

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.     

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 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.     

K-theory of reduced C*-algebras and rapidly decreasing functions on groups

We associate to any length function L on a group a space of rapidly decreasing functions on (in the l ² sense), denoted by H _L (). When H _L () is contained in the reduced C*-algebra C _r ^* () of (), then it is a dense *-subalgebra of C _r ^* () and we prove a theorem of A. Connes which asserts that under this hypothesis H _L () has the same K-theory as C _r ^* (). We introduce another space of rapidly decreasing functions on (in the l ¹ sense), denoted by H _L ^1, (), which is always a dense *-subalgebra of the Banach algebra l ¹(), and we show that H _L ^1, () has the same K-theory as l ¹().     

Rudiments of an average case complexity theory for piecewise-linear path following algorithms

We examine the efficiency of PL path following algorithms in followingF _T ^-1 (0), whereF _T is the PL approximation, induced by the simplicial triangulationT, to a mapf: ^{n n-1}. In particular, we consider the problem of determining an upper bound on the expected number of pivots made per unit length off ^–1(0) that is approximated. We show that if the sizes of the simplices ofT are sufficiently small, where sufficiently small is an explicitly given quantity dependent on measurements of how nicef is, then the average directional density ofT, as introduced by Todd, really does give a good approximation to the expected number of pivots made, confirming what researchers have believed on intuitive grounds for a decade. Because what constitutes sufficiently small is a precisely given quantity, i.e., non-asymptotic, we are able to provide some rigorous justification for the claim that the expected number of pivots grows only polynomially inn, the number of variables.Several other issues are also examined.Research supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship. This research was performed while the author was a member of the Mathematical Sciences Research Institute, Berkeley, California.     

Some Conditions for a C0-Semigroup to Be Asymptotically Finite-Dimensional

We study the class of bounded C ₀-semigroups T=(T _t ) _t0 on a Banach space X satisfying the asymptotic finite dimensionality condition: codim X ₀(T)<, where X ₀(T):={x X:lim_t T _t x=0}. We prove a theorem which provides some necessary and sufficient conditions for asymptotic finite dimensionality.     

Multi-Faithful Spanning Trees of Infinite Graphs

For an end and a tree T of a graph G we denote respectively by m() and m _T () the maximum numbers of pairwise disjoint rays of G and T belonging to , and we define tm() := min{m _T(): T is a spanning tree of G}. In this paper we give partial answers — affirmative and negative ones — to the general problem of determining if, for a function f mapping every end of G to a cardinal f() such that tm() f() m(), there exists a spanning tree T of G such that m _T () = f() for every end of G.     

A structure theorem for sum form functional equations

Summary It is proved that, iff _ij:]0, 1[ C (i = 1, ,k;j = 1, ,l) are measurable, satisfy the equation (1) (with some functionsg _it, h_jt:]0, 1[ C), then eachf _ij is in a linear space (called Euler space) spanned by the functionsx x ^j(logx) ^k (x ]0, 1[;j = 1, ,M;k = 0, ,m _j – 1), where ₁, , _M are distinct complex numbers andm ₁, , m_M natural numbers. The dimension of this linear space is bounded by a linear function ofN.     

Asymptotically gaussian distribution for random perturbations of rotations of the circle

Let T _, be a transformation of the two-dimensional torus T ₂ given by the formula T _, (x, y)(2x, y+ +x) mod 1.A version of the functional central limit theorem is formulated for variables of the form ,where is an irrational number and f belongs to a class of real- valued functions on T ₂ described in terms of . The proof will be published elsewhere.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 240, 1997, pp. 78–81.This work was performed as part of the Russian-German project DFG-RFBR, grant 96-01-00096. The second author was also supported by the Russian Foundation for Basic Research, grant 96-01-00672.     

Problem of the extremal decomposition of a closed plane

One considers the problem of the maximum of the product of powers of conformal radii of nonoverlapping domains in the following formulation. Let A=a₁, ..., a_n and B=(b₁, ..., b_m be systems of distinguished points in ¯C and let ={₁,..., _m} be a system of positive numbers. ByU(D,b ) we denote the reduced modulus of the simply connected domain D relative to the pointb D. Find the maximum of the sum in the familyD of all systems of nonoverlapping simply connected domains D_j, j=1, ..., m, satisfying the following condition: the domain D_j does not contain points b_i B, different from b_j, and some collection A_j, for each domain, of points from A, _j=1 ^m A _j =A. The solution of this problem is obtained by the simultaneous use of the method of variation and of the method of the moduli of families of curves and is given by Theorem 1 of the present paper. As consequences of Theorem 1 one obtains Theorems 2 and 3, strengthening the corresponding results of a previous paper of the author.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 149–154, 1985.     

On the convergence of the LJ search method

The convergence of the Luus-Jaakola search method for unconstrained optimization problems is established.Notation E _n Euclideann-space - f Gradient off(x) - ² f Hessian matrix - (·) ^T Transpose of (·) - I Index set {1, 2, ...,n} - [x _i1 ^*(j) ] Point around which search is made in the (j + 1)th iteration, i.e., [x _1l ^*(j) ,x _2l ^*(j) ,...,x _n1 ^*(j) ] - r _i ⁽ⁱ⁾ Range ofx _il ^*(i) in the (j + 1)th iteration - l ₁ min_i {r _i ⁽⁰⁾ } - l ₂ min_i {r _i ⁽⁰⁾ } - A _j Region of search in thejth iteration, i.e., {x E _n:x _il ^*(j-1) –0.5r _i ^(j-1) x _ix _il ^*(j-1) +0.5r _i ^(j-1) ,i I} - S _j Closed sphere with center origin and radius _j - Reduction factor in each iteration - 1– - (·) Gamma function Many discussions with Dr. S. N. Iyer, Professor of Electrical Engineering, College of Engineering, Trivandrum, India, are gratefully acknowledged. The author has great pleasure to thank Dr. K. Surendran, Professor, Department of Electrical Engineering, P.S.G. College of Technology, Coimbatore, India, for suggesting this work.     

Estimating the Diffusion Coefficient for Diffusions Driven by fBm

When the Hurst coefficient of a fBm B _t ^H is greater than 1/2, it is possible to define a stochastic integral with respect to B _t ^H as the pathwise limit of Riemann sums. In this article we consider diffusion equations of the type X_t = x₀ + ₀ ^T (X_s) dB_s ^H. We then construct a simple-to-use estimator of the diffusion coefficient (x), based on the number of crossings of level x of the process X _t. We then study consistency in probability of this estimator and calculate convergence rates in probability.     

Quasiclassical asymptotic behavior of the scattering cross section for asymptotically homogeneous potentials

One considers the total scattering cross section on the potential gV(x), x^m, m3, for large values of the coupling constant g and of the wave number k. One assumes that V(x)(x/|1x|)|x|^–, 2>m+1, as ¦x¦. It is shown that for gk^–1 , g^3–ak^2(a–2) the scattering cross section is equal asymptotically to _a(gk^–1), x=(m–1)(–1)^–1. Here the coefficient _a is determined only by the function and the number . Under the additional conditions >0, V>0, the indicated asymptotic behavior holds in the large domain gk^–1 , gk^a–z c(gk^–1), >0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 152, pp. 105–136, 1986.     

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.     

Mouvement moyen et système dynamique gaussien

Summary In the situation of the classical mean motion, we haven planets moving in the plane, planetk+1 being a satellite of planetk. A classcal result then states that planetn has a mean motion,i.e. its mean angular speed between time 0 and timet has a limit whent. We show in this article that any real gaussian dynamical system can be interpreted as the limit of this situation, whenn. From a given nonatomic probability measure on [0,], we construct a transformationT of the complex brownian path (B _u)_0u1 which preserves Wiener measure.T is defined as the limit of a sequenceT _n, whereT _n acts as the motion of 2ⁿ planets. In this way we get a real gaussian dynamical system, whose spectral measure is the symetric probability on [-,] obtained from . The transformationT can be inserted in a flow (T ^t) _t, and the orbitstZ _t=B ₁T ^t still have almost surely a mean motion, which is the mean of .     

On TDP permutations

A permutation :i| _i , 0i<n is called a TDP permutation ifi–a _i j–a _j (modn) fori j. This paper finds all TDP permutations forn15, discusses the method for generating TDP permutations, and finally by applying MLE method obtains a formula for estimating the number of TDP permutations forn> 15.Project supported by National Natural Science Foundation of China.     

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.     

A note on "More Operator Versions of the Schwarz Inequality"

It is shown that for any (n + 1)-positive (possibly non-linear) map and any bounded linear operators A _i ,i = 1,¨,n we have [(A _i ^* A _j )] _{i,j = 1} ^*[(A _i )^*(A _j )] _{i,j = 1} ^*, and that the statement is false if "(n + 1)-positive" is replaced by "n-positive". This resolves an issue raised by Bhatia and Davis in relation to a Schwartz inequality which can be regarded as a non-commutative variance-covariance inequality [2]     

Complex whitham deformations in problems with “integrable instability”

The focusing nonlinear Schrödinger equation with finite-density boundary conditions as |x| is considered. The asymptotic behavior of the solution ast is investigated by means of the complex theory of deformations of Whitham.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 3, pp. 393–419, September, 1995.