On convergence of types and processes in Euclidean space

Let be an Euclidean space; Y _n , Z, U random vectors in ; h _n , g _n affine transformations and let þ be a subgroup of the group G of all the in vertible affine transformations, closed relative to G. Suppose that g_n and where Z is nonsingular. The behaviour of _n = h _n g _n ^–1 as n is discussed first. The results are used then to prove that if for all t(0, ), where h _n þ and Z ₁ is nonsingular and nonsymmetric with respect to þ then H, for all t(0,) and is a continuous homomorphism of the multiplicative group of (0, ) into þ. The explicit forms of the possible are shown.     

On the number of nowhere zero points in linear mappings

LetA be a nonsingularn byn matrix over the finite fieldGF _q ,k=n/2,q=p ^a ,a1, wherep is prime. LetP(A,q) denote the number of vectorsx in (GF _q ) ⁿ such that bothx andAx have no zero component. We prove that forn2, and ,P(A,q)[(q–1)(q–3)] ^k (q–2) ^n–2k and describe all matricesA for which the equality holds. We also prove that the result conjectured in [1], namely thatP(A,q)1, is true for allqn+23 orqn+14.     

Admissibility of linear estimators of regression coefficients under quadratic loss

For the general fixed effects linear model:Y=X+, N(0,V),V0, we obtain the necessary and sufficient conditions forLY+a to be admissible for a linear estimable functionS in the class of all estimators under the loss function (d -S)D(d -S), whereD0 is known. For the general random effects linear model: =XV ₁₁ X+XV ₁₂+V ₂₁ X+V ₂₂0, we also get the necessary and sufficient conditions forLY+a to be admissible for a linear estimable functionS+Q in the class of all estimators under the loss function (d -S -Q)D(d -S -Q), whereD0 is known.     

Weak Laws with Random Indices for Arrays of Random Elements in Rademacher Type p Banach Spaces

For a sequence of constants {a _n,n1}, an array of rowwise independent and stochastically dominated random elements { V _nj, j1, n1} in a real separable Rademacher type p (1p2) Banach space, and a sequence of positive integer-valued random variables {T _n, n1}, a general weak law of large numbers of the form is established where {c _nj, j1, n1}, _n , b _n are suitable sequences. Some related results are also presented. No assumption is made concerning the existence of expected values or absolute moments of the {V _nj, j1, n1}. Illustrative examples include one wherein the strong law of large numbers fails.     

Zur Interpolation und Integration differenzierbarer periodischer Funktionen

Summary Letx ₀<x ₁<...<x _n–1<x ₀+2 be nodes having multiplicitiesv ₀,...,v _n–1, 1v _k r (0k<n). We approximate the evaluation functional ,x fixed, and the integral respectively by linear functionals of the form and determine optimal weights for the Favard classesW _r C ₂. In the even case of optimal interpolation these weights are unique except forr=1,x(x _k +x _k–1)/2 mod 2. Moreover we get periodic polynomial splinesw _{k, j} (0k<n, 0j<v _k ) of orderr such that are the optimal weights. Certain optimal quadrature formulas are shown to be of interpolatory type with respect to these splines. For the odd case of optimal interpolation we merely have obtained a partial solution.

Bojanov hat in [4, 5] ähnliche Resultate wie wir erzielt. Um Wiederholungen zu vermeiden, werden Resultate, deren Beweise man bereits in [4, 5] findet, nur zitiert     

Convergence of series of translations

For a mean zero norm one sequence (f _n )L ₂[0, 1], the sequence (f _n {nx+y}) is an orthonormal sequence inL ₂([0, 1]²); so if , then converges for a.e. (x, y)[0, 1]² and has a maximal function inL ₂([0, 1]²). But for a mean zerofL ₂[0, 1], it is harder to give necessary and sufficient conditions for theL ₂-norm convergence or a.e. convergence of . Ifc _n 0 and , then this series will not converge inL ₂-norm on a denseG subset of the mean zero functions inL ₂[0, 1]. Also, there are mean zerofL[0, 1] such that never converges and there is a mean zero continuous functionf with a.e. However, iff is mean zero and of bounded variation or in some Lip() with 1/2<1, and if |c _n | = 0(n ^–) for >1/2, then converges a.e. and unconditionally inL ₂[0, 1]. In addition, for any mean zerof of bounded variation, the series has its maximal function in allL _p[0, 1] with 1p<. Finally, if (f _n )L [0, 1] is a uniformly bounded mean zero sequence, then is a necessary and sufficient condition for to converge for a.e.y and a.e. (x _n )[0, 1]. Moreover, iffL [0, 1] is mean zero and , then for a.e. (x _n )[0, 1], converges for a.e.y and in allL _p [0, 1] with 1p<. Some of these theorems can be generalized simply to other compact groups besides [0, 1] under addition modulo one.     

Asymptotic solution for a new class of forbiddenr-graphs

We consider the problem of finding ex (n; G), defined as the maximal number of edges anr-graph onn vertices can have that contains no subgraph isomorphic toG. We construct certainr-graphsG for which we find the coefficient(G) of the asymptotic expansion ex(n;G)== asn.     

Boundary Asymptotics for Orthogonal Rational Functions on the Unit Circle

Let w() be a positive weight function on the unit circle of the complex plane. For a sequence of points { _k } _{k = 1} included in a compact subset of the unit disk, we consider the orthogonal rational functions _n that are obtained by orthogonalization of the sequence { 1, z / ₁, z ² / ₂, ... } where , with respect to the inner product In this paper we discuss the behaviour of _n (t) for t = 1 and n under certain conditions. The main condition on the weight is that it satisfies a Lipschitz–Dini condition and that it is bounded away from zero. This generalizes a theorem given by Szeg in the polynomial case, that is when all _k = 0.     

The maximum number of balancing sets

Leta ₁, ...,a _n be a sequence of nonzero real numbers with sum zero.A subsetB of {1, 2,...,n} is called a balancing set if a _b = 0 (b B). S. Nabeya showed that the number of balancing sets is bounded above by and this bound achieved forn even witha _j =(–1) ^j . Here his conjecture is verified, showing a tight upper bound whenn = 2k + 1. The essentially unique extremal configuration is:a ₁ = 2,a ₂ = ... =a _k = 1,a _k+1 = ... =a _2k+1 = -1.     

On Kolmogorov-Type Inequalities Taking into Account the Number of Changes in the Sign of Derivatives

For 2-periodic functions and arbitrary q [1, ] and p (0, ], we obtain the new exact Kolmogorov-type inequality which takes into account the number of changes in the sign of the derivatives (x ^(k)) over the period. Here, = (r – k + 1/q)/(r + 1/p), _r is the Euler perfect spline of degree r, and . The inequality indicated turns into the equality for functions of the form x(t) = a _r (nt + b), a, b R, n N. We also obtain an analog of this inequality in the case where k = 0 and q = and prove new exact Bernstein-type inequalities for trigonometric polynomials and splines.     

Joint convergence of conditional expectations

Let P_n, nIN{0}, be probability measures on a-fieldA; f_n, nIN{0}, be a family of uniformly boundedA-measurable functions andA _n, nIN, be a sequence of sub--fields ofA, increasing or decreasing to the-fieldA _o. It is shown in this paper that the conditional expectations converge in P_o-measure to with k, n, m , if P_n|A, nIN, converges uniformly to P_n|A and f_n, nIN, converges in P_o-measure to f_o.     

On exponents in arithmetical semigroups

Let be the factoring of a natural numbern into primes. In 1969Niven found the average order of the functionsh(n)= min {₁,... _r } andH(n)= max {₁,... _r }. In this note a generalization ofNiven's results is discussed, namely the average order of the corresponding functions in certain arithmetical semigroups.     

Non-symmetric association schemes of symmetric matrices

Let X_n be the set of n×n symmetric matrices over a finite field F_q,where q is a power of an odd prime.For S_1,S_2 ∈ X_n,we define (S_1,S_2)∈ R_0 iff S_1=S_2;(S_1,S_2)∈R_(r,ε)iff S_1-S_2 is congruent to where?=1 or z,z being afixed non-square element of F_q.Then X_n=(X_n,{R_0,R_(r,ε)|1≤r≤n,?=1 or z}) is a non-symmetric association scheme of class 2n on X_n.The parameters of X_n have been computed.And we also prove that X_n is commutative.     

Bounded universal functions for sequences of holomorphic self-maps of the disk

We give several characterizations of those sequences of holomorphic self-maps {φ _n } _n≥1 of the unit disk for which there exists a function F in the unit ball of H ^∞ such that the orbit {F∘φ _n :n∈ℕ} is locally uniformly dense in . Such a function F is said to be a -universal function. One of our conditions is stated in terms of the hyperbolic derivatives of the functions φ _n . As a consequence we will see that if φ _n is the nth iterate of a map φ of into , then {φ _n } _n≥1 admits a -universal function if and only if φ is a parabolic or hyperbolic automorphism of . We show that whenever there exists a -universal function, then this function can be chosen to be a Blaschke product. Further, if there is a -universal function, we show that there exist uniformly closed subspaces consisting entirely of universal functions.     

A representation theorem for weak automorphisms of a universal algebra

Given a group G and a descending chainG ₀,G ₁,...,G _n, of normal subgroups ofG, we prove that there exists a universal algebra , such that the chain ...W_n( )...W₁( }) W₀( )W( ) is isomorphic to the chain ...G _n ...G ₁G ₀G, where W( ) is the group of weak automorphisms of , and W_n( ) is the group of weak automorphisms of that leaves alln-ary operations fixed.We also prove that there are an infinite number of non-isomorphic algebras that satisfy the above.These results are a generalization of those proved by J. Sichler, in the special case when G=G₀, and G₁=G₂=...=G_n=....Presented by J. Mycielski.This paper comprises part of the author's doctoral dissertation at the University of Notre Dame in 1983. The author wishes to express her deep gratitude to Professor Abraham Goetz for suggesting this problem, for being extremely generous with his time and experience, and for giving her his constant encouragement. The author also thanks the reviewer for his helpful comments.     

On computing the number of subgroups of a finite Abelian group

We consider the numberN _A (r) of subgroups of orderp ^r ofA, whereA is a finite Abelianp-group of type =₁,₂,..., _l ()), i.e. the direct sum of cyclic groups of order ⁱⁱ. Formulas for computingN _A (r) are well known. Here we derive a recurrence relation forN _A (r), which enables us to prove a conjecture of P. E. Dyubyuk about congruences betweenN _A (r) and the Gaussian binomial coefficient .     

Hyponormality of Toeplitz Operators with Polynomial and Symmetric-Type Symbols

Suppose that is a trigonometric polynomial of the form (z) = ^N_n=-N a_n zⁿ. It is well-known that T is normal if and only if | a_{– N}| =  | a_N| and the Fourier coefficients of satisfy the following symmetry condition:

Die Verteilung der schlichten Funktionen in einem Funktionenraum

We consider the set of regular functions . We construct a Borel measure and a class of outer measures _h onH. With these and _h we show that: (HS)=0 and _h (HS)=0, (S is the set of normed univalent functions). From _h (HS)=0 follows—forh=t —that the Hausdorff—Billingsley-dimension ofHS is zero.     

A Remarkable q,t-Catalan Sequence and q-Lagrange Inversion

We introduce a rational function C _n(q, t) and conjecture that it always evaluates to a polynomial in q, t with non-negative integer coefficients summing to the familiar Catalan number . We give supporting evidence by computing the specializations and C _n (q) = C _n(q,1) = C _n(1,q). We show that, in fact, D _n(q) q-counts Dyck words by the major index and C _n(q) q-counts Dyck paths by area. We also show that C _n(q, t) is the coefficient of the elementary symmetric function e _nin a symmetric polynomial DH_n(x; q, t) which is the conjectured Frobenius characteristic of the module of diagonal harmonic polynomials. On the validity of certain conjectures this yields that C _n(q, t) is the Hilbert series of the diagonal harmonic alternants. It develops that the specialization DH_n(x; q, 1) yields a novel and combinatorial way of expressing the solution of the q-Lagrange inversion problem studied by Andrews [2], Garsia [5] and Gessel [11]. Our proofs involve manipulations with the Macdonald basis {P (x; q, t)} which are best dealt with in -ring notation. In particular we derive here the -ring version of several symmetric function identities.Work carried out under NSF grant support.