1-Homogeneous Graphs with Cocktail Party μ-Graphs

Let be a graph with diameter d 2. Recall is 1-homogeneous (in the sense of Nomura) whenever for every edge xy of the distance partition{{z V() | (z, y) = i, (x, z) = j} | 0 i, j d}is equitable and its parameters do not depend on the edge xy. Let be 1-homogeneous. Then is distance-regular and also locally strongly regular with parameters (v,k,,), where v = k, k = a ₁, (v – k – 1) = k(k – 1 – ) and c ₂ + 1, since a -graph is a regular graph with valency . If c ₂ = + 1 and c ₂ 1, then is a Terwilliger graph, i.e., all the -graphs of are complete. In [11] we classified the Terwilliger 1-homogeneous graphs with c ₂ 2 and obtained that there are only three such examples. In this article we consider the case c ₂ = + 2 3, i.e., the case when the -graphs of are the Cocktail Party graphs, and obtain that either = 0, = 2 or is one of the following graphs: (i) a Johnson graph J(2m, m) with m 2, (ii) a folded Johnson graph J¯(4m, 2m) with m 3, (iii) a halved m-cube with m 4, (iv) a folded halved (2m)-cube with m 5, (v) a Cocktail Party graph K _{m × 2} with m 3, (vi) the Schläfli graph, (vii) the Gosset graph.     

A graded scale of parametric families of distributions,and parameter estimates based on the sample mean

Summary Denote by _k a class of familiesP={P} of distributions on the line R¹ depending on a general scalar parameter , being an interval of R¹, and such that the moments µ₁()=xdP ,...,µ_2k ()=x ^2k dP are finite, ₁ (), ..., _k (), _k+1 () ..., _k () exist and are continuous, with ₁ () 0, and _j +1 ()= ₁ () _j () +[₂() -₁()²] _j ()/ ₁ (), J=2, ..., k. Let ₁=¯x=x ₁ + ... +x _n/n, ₂=x ₁ ² + ... +x _n ²/n, ..., _k =(x ₁ ^k + ... +x _n ^k/n denote the sample moments constructed for a sample x₁, ..., x_n from a population with distribution Pg. We prove that the estimator of the parameter by the method of moments determined from the equation ₁= ₁() and depending on the observations x₁, ..., x_n only via the sample mean ¯x is asymptotically admissible (and optimal) in the class _k of the estimators determined by the estimator equations of the form ₀ () + ₁ () ₁ + ... + _k () _k =0 if and only ifP _k .The asymptotic admissibility (respectively, optimality) means that the variance of the limit, as n (normal) distribution of an estimator normalized in a standard way is less than the same characteristic for any estimator in the class under consideration for at least one 9 (respectively, for every ).The scales arise of classes _{1 2}... of parametric families and of classes _{1 2} ... of estimators related so that the asymptotic admissibility of an estimator by the method of moments in the class _k is equivalent to the membership of the familyP in the class _k .The intersection consists only of the families of distributions with densities of the form h(x) exp {C₀() + C₁() x } when for the latter the problem of moments is definite, that is, there is no other family with the same moments ₁ (), ₂ (), ...Such scales in the problem of estimating the location parameter were predicted by Linnik about 20 years ago and were constructed by the author in [1] (see also [2, 3]) in exact, not asymptotic, formulation.Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei, pp. 41–47, 1981.     

Complete hyperbolicity of Hartogs domain

Let X be a complex space and an upper semicontinuous function on X. Consider the Hartogs domain (X) given by (X)={(z, w)X×C: |w| < e ^–(z) }. In this article, some necessary and sufficient conditions on the complete hyperbolicity of (X) are established. Mathematics Subject Classification (2000):32A10, 32C10, 32H20, 32A17     

Mathematical expectations of functions of sums of a random number of independent terms

Conditions are found which must be imposed on a function g(x) in order that M g(₁+₂+ + _v < if M g(_i) < and M g(v) < ,, ₁, ₂, , _n, ... being non-negative and independent, being integral, and {_i} being identically distributed. The result is applied to the theory of branching processes.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 387–394, April, 1968.     

Canonical Semigroups of States and Cocycles for the Group of Automorphisms of a Homogeneous Tree

Let G=A ut(T) be the group of automorphisms of a homogeneous tree and let d(v,gv) denote the natural tree distance. Fix a base vertex e in T. The function (g)=exp(–d(e,ge)), being positive definte on G, gives rise to a semigroup of states on G whose infinitesimal generator d/d|₌₀=log() is conditionally positive definite but not positive definite. Hence, log() corresponds to a nontrivial cocycle (g): GH in some representation space H . In contrast with the case of PGL(2,), the representation is not irreducible.Let _o (g) be the derivative of the spherical function corresponding to the complementary series of A ut(T). We show that –d(e,ge) and _o (g) come from cohomologous cocycles. Moreover, _o is associated to one of the two (irreducible) special representations of A ut(T).     

Bonds for a functional on a class of functions with positive real parts

We establish sharp upper and lower bounds for the functionalJ (p)- Re{p (z)-zp'(z)/p(z), 1. where ¦z¦=r is fixed, 0 <r <1, p(z).P_n (A, B), –1 B < a 1, a certain class of regular functions in the disk with values in the right halfplane.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 686–689, May, 1990.     

Asymptotics of a Boundary Crossing Probability of a Brownian Bridge with General Trend

Let us consider a signal-plus-noise model h(z)+B ₀(z), z [0,1], where > 0, h: [0,1] , and B ₀ is a Brownian bridge. We establish the asymptotics for the boundary crossing probability of the weighted signal-plus-noise model for , that is P (sup _{z [0,1]} w(z)( h(z)+B ₀(z))>c), for , (1) where w: [0,1] [0, is a weight function and c>0 is arbitrary. By the large deviation principle one gets a result with a constant which is the solution of a minimizing problem. In this paper we show an asymptotic expansion that is stronger than large deviation. As byproduct of our result we obtain the solution of the minimizing problem occurring in the large deviation expression. It is worth mentioning that the probability considered in (1) appears as power of the weighted Kolmogorov test applied to the test problem H ₀: h 0 against the alternative K: h>0 in the signal-plus-noise model.     

Minimization of the Conformal Radius under Circular Cutting of a Domain

Let D be a simply connected domain on the complex plane such that 0 D. For r > 0 , let D _r be the connected component of D {z : |z| < r} containing the origin. For fixed r, we solve the problem on minimization of the conformal radius R(D _r, 0) among all domains D with given conformal radius R(D, 0). This also leads to the solution of the problem on maximization of the logarithmic capacity of the local -extension E (a) of E among all continua E with given logarithmic capacity. Here, E (a) = E {z : |z – a| }, a E, > 0. Bibliography: 12 titles.     

Capacities of Julia sets of rational functions

For any polynomialp(z)=a ₀ z ⁿ +a|z ^n–1++a _n , a₀0, n2,F is the Julia set and ^* is the equilibrium distribution onF. Hans Brolin^[1] proved that (F)>0, andS ^* =F. Up to now, we know nothing about rational functions. The aim of this paper is to discuss the case of rational functions.Project supported by the Science Fund of the Chinese Academy of Sciences.     

Compact interval spaces in which all closed subsets are homeomorphic to clopen ones,I

A topological space X whose topology is the order topology of some linear ordering on X, is called an interval space. A space in which every closed subspace is homeomorphic to a clopen subspace, is called a CO space. We regard linear orderings as topological spaces, by equipping them with their order topology. If L and K are linear orderings, then L ^*, L+K, L·K denote respectively the reverse orderings of L, the ordered sum of L and K and the lexicographic order on L×K (so ·2=+ and 2·=). Ordinals are considered as linear orderings, and cardinals are initial ordinals. For cardinals , 0, let L(, )= + 1 + ^* . Main theorem. Let X be a compact interval space. Then X is a CO space if and only if X is homeomorphic to a space of the form + 1 + _i L( _i , _i ), where is any ordinal, n, for every ii, _i are regular cardinals and _{i i}, and if n>0, then max({ _i: i}) · . This first part is devoted to show the following result. Theorem: If X is a compact interval CO space, then X is a scattered space (that means that every subspace of X has an isolated point).Supported by the Université Claude-Bernard (Lyon-1), the Ben Gurion University of the Negev, and the C.N.R.S.: UPR 9016Supported by the City of Lyon     

Diophantine complexity

It is well known that any recursively enumerable set S of natural numbers can be represented by formulas of the following types a S x yx, z₁...Z_n (D (a,x,y,z₁,...,z_n)=0)¦(Davis normal form); aS z₁...z_n (E (a,z₁..., z_n)=E₂(a,z₁,...,z_n)) (exponential diophantine representation); and aS z₁... z_n, (D (a-, z₁),...)Z_n)=0) (diophantine representation). Each of these three representations yields a different measure for the complexity of S as a whole and the complexity of accepting individual members of S. A survey is presented of various results concerning such complexity measures, due to different authors.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeliniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 174, pp. 122–131, 1988.     

On the concentration of stresses due to a small elliptic inclusion on the neutral axis of a deep beam under constant bending moment

Zusammenfassung Es wird die Spannungsverteilung untersucht, die sich in einem breiten Balken mit konstanter Höhe unter einem konstanten Biegemoment ausbildet, wenn er eine kleine elliptische Einschliessung mit Zentrum auf der Neutralachse enthält. Insbesondere werden die Fälle eines sehr starren Einschlusses sowie eines elliptischen Loches im Detail diskutiert.

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Nomenclature x, y Cartesian coordinates - , elliptic coordinates - u, v (u ,u )=components of displacements - , unit elongations in -and -directions - shearing strain - , normal stress components in elliptical coordinates - shearing stress in elliptic coordinates - _x , _y normal stress in Cartesian coordinates - _xy shearing stress in Cartesian coordinates - E Young's modulus for the beam - v Poisson's ratio for the beam - 1/h ₁, 1/h ₂ stretch ratios - e _x + _y dilatation - 2 rotation - M bending moment     

Ergodic properties of Poisson processes with almost periodic intensity

Summary We discuss in this paper a non-homogeneous Poisson process A driven by an almost periodic intensity function. We give the stationary version A ^* and the Palm version A ⁰ corresponding to A ^*. Let (T _i ,i) be the inter-point distance sequence in A and (T _i ⁰ ,i) in A ⁰. We prove that forj, the sequence (T _i+j,i) converges in distribution to (T _i ⁰ ,i). If the intensity function is periodic then the convergence is in variation.     

A spectral condition for strong convergence of Markov operators

Summary If P is a conservative Markov operator on L ₁(m) with no unimodular spectral points except 1, then u(I–P)Pⁿ₁0 for every uL ₁.     

On feasible sets defined through Chebyshev approximation

LetZ be a compact set of the real space with at leastn + 2 points;f,h1,h2:Z continuous functions,h1,h2 strictly positive andP(x,z),x(x ₀,...,x _n ) ⁿ⁺¹,z , a polynomial of degree at mostn. Consider a feasible setM {x ⁿ⁺¹z Z, –h ₂(z) P(x, z)–f(z)h ₁(z)}. Here it is proved the null vector 0 of ⁿ⁺¹ belongs to the compact convex hull of the gradients ± (1,z,...,z ⁿ ), wherez Z are the index points in which the constraint functions are active for a givenx* M, if and only ifM is a singleton.This work was partially supported by CONACYT-MEXICO.     

Weighted Composition Operators on Bergman and Dirichlet Spaces

Let H() denote a functional Hilbert space of analytic functions on a domain . Let w : C and : be such that w f is in H() for every f in H(). The operator wC given by f w f is called a weighted composition operator on H(). In this paper we characterize such operators and those for which (wC )* is a composition operator. Compact weighted composition operators on some functional Hilbert spaces are also characterized. We give sufficient conditions for the compactness of such operators on weighted Dirichlet spaces.     

Convolution Operators with Fractional Measures Associated to Holomorphic Functions

Let be an open set in the complex plane and let be a holomorphic function on . Let K be a compact subset of with nonempty interior such that 0 K. Let be the Borel measure of R ⁴ C ² given by(E = _K E(z, (z))|z|^–2 d(z)where 0 < 2 and d(x ₁ + ix ₂) = dx ₁ dx ₂ denotes the Lebesgue measure on C. Let T be the convolution operator T f = * f. In this paper we characterize the type set E associated to T .     

Estimation of the Intensity of the Flow of Nonmonotone Refusals in the Queuing System (≤ λ)/G/m

We consider a queuing system ()/G/m, where the symbol () means that, independently of prehistory, the probability of arrival of a call during the time interval dtdoes not exceed dt. The case where the queue length first attains the level r m+ 1 during a busy period is called the refusal of the system. We determine a bound for the intensity ₁(t) of the flow of homogeneous events associated with the monotone refusals of the system, namely, ₁(t) = O( ^{r+ 1}₁ ^{m– 1} _{r– m+ 1}), where _k is the kth moment of the service-time distribution.     

Fractional integrodifferentiation in Hölder classes of arbitrary order

Hölder classes of variable order (x) are introduced and it is shown that the fractional integralI ₀₊ has Hölder order (x)+ (0 < , ₊, +₊ < 1, ₊ = sup (x)).     

An approach to the constructivization of Cantor's set theory

A new approach is proposed for the construction of constructive analogs of set theory in hyperarithmetic languages , where is a scale of constructive ordinals. For every ordinal in the language , a special relation of equality = is defined for codes of one-parameter formulas (conditions) of the level in a constructive hyperarithmetic hierarchy corresponding to the scale . The membership relation, (also expressible in the language ), is defined by the conditionx y=z(z= x&z y), where the relation is obtained by suitable refinement of the traditional representations of the constructive relation of membership. This results in a hierarchy of constructive analogsM of the theory of sets (in which the sets are represented by codes of conditions of level , identified modulo the relation =, and is taken as the relation of membership). Some properties of this hierarchy are introduced which show that for the limits ,M is sufficiently rich from the traditional set theoretic standpoint.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 68, pp. 38–49, 1977.