Infinitesimal obstructions to weakly mixing

Let ^t be the flow (parametrized with respect to arc length) of a smooth unit vector field v on a closed Riemannian manifold M ⁿ , whose orbits are geodesics. Then the (n-1)-plane field normal to v, v, is invariant under d ^t and, for each x M, we define a smooth real function _x (t) : (1 + _i (t)), where the _i(t) are the eigenvalues of AA ^T, A being the matrix (with respect to orthonormal bases) of the non-singular linear map d^2t , restricted to v at the point _x ^-t M ⁿ.Among other things, we prove the Theorem (Theorem II, below). Assume v is also volume preserving and that _x ^' (t) 0 for all x M and real t; then, if _x ^t : M M is weakly missng for some t, it is necessary that vx 0 at all x M.     

A statistical estimate of the structure of multi-extremal problems

The solution of ak-extremal problem is defined as the set of pairs (x _i ^* , _i),i = 1, ,k, where x _t ^* isi ^th local minimum and _i is the volume of the set of attraction of this minimum. A Bayesian estimate ofk and ( ₁ , , _k ) is constructed.This paper has been written while the author was a CNR visiting professor at the Institute of Mathematics of the Milano University.     

An Extension of the Roots Separation Theorem

Let T _n be an n×n unreduced symmetric tridiagonal matrix with eigenvalues ₁<₂<< _n and W _k is an (n–1)×(n–1) submatrix by deleting the kth row and the kth column from T _n , k=1,2,...,n. Let _{12 n–1} be the eigenvalues of W _k . It is proved that if W _k has no multiple eigenvalue, then ₁<₁<₂<₂<< _n–1< _n–1< _n ; otherwise if _i = _i+1 is a multiple eigenvalue of W _k , then the above relationship still holds except that the inequality _i < _i+1< _i+1 is replaced by _i = _i+1= _i+1.     

Approximation of Distributions by Parametric Sets

Let P be a probability distribution on a separable metric space (S,d). We study the following problem of approximation of a distribution P by a set from a given class A2 ^S : W(A,P) _S (d(x,A))P(dx)min _AA , where is a nondecreasing function. A special case where A is a parametric class A={A():T} is considered in detail. Our main interest is to obtain convergence results for sequences {A ^* _n }, where A ^* _n is an optimal set for a measure P _n satisfying P _n P, as n.     

The Minimality Properties of Chebyshev Polynomials and Their Lacunary Series

By considering four kinds of Chebyshev polynomials, an extended set of (real) results are given for Chebyshev polynomial minimality in suitably weighted Hölder norms on [–1,1], as well as (L ) minimax properties, and best L ₁ sufficiency requirements based on Chebyshev interpolation. Finally we establish best L _p , L and L ₁ approximation by partial sums of lacunary Chebyshev series of the form _i=0 a ⁱ _b ⁱ(x) where _n (x) is a Chebyshev polynomial and b is an odd integer 3. A complete set of proofs is provided.     

The Eta Invariant and the Real Connective K-Theory of the Classifying Space for Quaternion Groups

We express the real connective K-theory groups o_4k–1(B Q ) ofthe quaternion group Q of order = 2 ^j 8 in terms of therepresentation theory of Q by showing o_4k–1(B Q ) = Sp(S ^4k+3/Q )where is any fixed point free representation of Q in U(2k + 2).     

Automorphisms on a Lie group contracting modulo a compact subgroup and applications to semistable convolution semigroups

Let ( _t ) _t0 be a -semistable convolution semigroup of probability measures on a Lie groupG whose idempotent ₀ is the Haar measure on some compact subgroupK. Then all the measures ₁ are supported by theK-contraction groupC _K() of the topological automorphism ofG. We prove here the structure theoremC _K()=C()K, whereC() is the contraction group of . Then it turns out that it is sufficient to study semistable convolution semigroups on simply connected nilpotent Lie groups that have Lie algebras with a positive graduation.     

On the E- and MV-optimality of block designs having k≥v

In this paper we consider the problem of determining and constructing E- and MV-optimal block designs to use in experimental settings where treatments are applied to experimental units occurring in b blocks of size k, k. It is shown that some of the well-known methods for constructing E- and MV-optimal unequally replicated designs having k fail to yield optimal designs in the case where . Some sufficient conditions are derived for the E- and MV-optimality of block designs having and methods for constructing designs satisfying these sufficient conditions are given.     

Algebraic systems of quadratic forms of number fields and function fields

In this paper we examine for which Witt classes ,..., _n over a number field or a function fieldF there exist a finite extensionL/F and ₂,..., _n L^* such thatT _L/F ()=1 andTr _L/F (_i)=_i fori=2,...n.     

Приближение непреры вных функций средним и Эйлера

Let _n(1,f,x)=1/2 ⁿ _k=1 ⁿ C _k ⁿ S_k(x,f) denote the Euler means of the Fourier series of the 2-periodic functionf(x). For a function the main term of deviationf(x)– _n (1,f, x) is calculated. Asymptotically exact order of decrease of the upper bound of such deviations over the classH ⁽⁾ is also obtained.     

The Higher K-Theory of Real Curves

If X is a smooth curve defined over the real numbers , we show that K _n (X) is the sum of a divisible group and a finite elementary Abelian 2-group when n 2. We determine the torsion subgroup of K _n (X), which is a finite sum of copies of and 2, only depending on the topological invariants of X() and X(), and show that (for n 2) these torsion subgroups are periodic of order 8.     

The Lattice of Interpretability Types of Cantor Varieties

For integers 1 m < n, a Cantor variety with m basic n-ary operations _i and n basic m-ary operations _k is a variety of algebras defined by identities _k(₁( ), ... , _m( )) = _k and i(₁( ), ... ,_n( )) = y _i, where = (x ₁., ... , x _n) and = (y ₁, ... , y _m). We prove that interpretability types of Cantor varieties form a distributive lattice, , which is dual to the direct product ₁ × ₂ of a lattice, ₁, of positive integers respecting the natural linear ordering and a lattice, ₂, of positive integers with divisibility. The lattice is an upper subsemilattice of the lattice of all interpretability types of varieties of algebras.     

Some greedyt-intersecting families of finite sequences

Letn, s ₁,s ₂, ... ands _n be positive integers. Assume is an integer for eachi}. For , , and , denotes _p (a)={j|1jn,a _j p}, , and . is called anI _t ^p -intersecting family if, for any a,b ,a _i b _i =min(a _i ,b _i )p for at leastt i's. is called a greedyI _t ^P -intersecting family if is anI _t ^p -intersecting family andW _p (A)W _p (B+A ^c ) for anyAS _p ( ) and any with |B|=t–1.In this paper, we obtain a sharp upper bound of | | for greedyI _t ^p -intersecting families in for the case 2ps _i (1in) ands ₁>s ₂>...>s _n .This project is partially supported by the National Natural Science Foundation of China (No.19401008) and by Postdoctoral Science Foundation of China.     

Differential geometry of tensor product immersions II

In the first part of this series, we prove that the tensor product immersionf ₁ f _2k of2k isometric spherical immersions of a Riemannian manifoldM in Euclidean space is of-type with k and classify tensor product immersionsf ₁ f _2k which are ofk-type. In this article we investigate the tensor product immersionsf ₁ f _2k which are of (k+1)-type. Several classification theorems are obtained.     

D λ -cycles in 3-cyclable graphs

Letk and be positive integers, andG a 2-connected graph of ordern with minimum degree and independence number. A cycleC ofG is called aD -cycle if every component ofG – V(C) has order smaller than. The graphG isk-cyclable if anyk vertices ofG lie on a common cycle. A previous result of the author is that if k 2, G isk-connected and every connected subgraphH ofG of order has at leastn +k ² + 1/k + 1 – vertices outsideH adjacent to at least one vertex ofH, thenG contains aD -cycle. Here it is conjectured that k-connected can be replaced by k-cyclable, and this is proved fork = 3. As a consequence it is shown that ifn 4 – 6, or ifG is triangle-free andn 8 – 10, thenG contains aD ₃-cycle orG , where denotes a well-known class of nonhamiltonian graphs of connectivity 2. As an analogue of a result of Nash-Williams it follows that ifn 4 – 6 and – 1, thenG is hamiltonian orG . The results are all best possible and compare favorably with recent results on hamiltonicity of graphs which are close to claw-free.     

p-Frames in Separable Banach Spaces

Let X be a separable Banach space with dual X ^*. A countable family of elements {g _i }X ^* is a p-frame (1 p ) if the norm _X is equivalent to the ^p -norm of the sequence {g _i ()}. Without further assumptions, we prove that a p-frame allows every gX ^* to be represented as an unconditionally convergent series g=d _i g _i for coefficients {d _i } ^q , where 1/p+1/q=1. A p-frame {g _i } is not necessarily linear independent, so {g _i } is some kind of overcomplete basis for X ^*. We prove that a q-Riesz basis for X ^* is a p-frame for X and that the associated coefficient functionals {f _i } constitutes a p-Riesz basis allowing us to expand every fX (respectively gX ^*) as f=g _i (f)f _i (respectively g=g(f _i )g _i ). In the general case of a p-frame such expansions are only possible under extra assumptions.     

Asymptotically minimax tests of composite hypotheses

Summary X ₁,,X> _n are independent, identically distributed random variables with common density function f(x¦ ₁ ,, _k , _k+1 ), assumed to satisfy certain standard regularity conditions. The k+1 parameters are unknown, and the problem is to test the hypothesis that _k+1 =b against the alternative that _k+1 =b+cn ^–1/2 . ₁ ,, _k are nuisance parameters. For this problem, the following artificial problem is temporarily substituted. It is known that ¦ ₁ -a _i ¦n ^–1/2 M(n) for i=1,,k, where a ₁ , ,a _k are known, and M(n) approaches infinity as n increases but n ^–1/2 M(n) approaches zero as n increases. A Bayes decision rule is constructed for this artificial problem, relative to the a priori distribution which assigns weight A to _k+1 =b, and weight 1-A to _k+1 =b+cn ^–1/2 , in each case the weight being spread uniformly over the possible values of ₁ ,, _k in the artificial problem. An analysis of the structure of the Bayes rule shows that if estimates of ₁ ,..., _k are substituted for a ₁ ..., a _k respectively, the resulting rule is a solution to the original problem, and this rule has the same asymptotic properties as a solution to the artificial problem as the Bayes rule for the artificial problem, no matter what the values a ₁ ..., a _k are.Research supported by the U.S. Air Force under Grant AF-AFOSR-68-1472.     

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.     

Spline polynomials with a prescribed sequence of extrema

In the present note a theorem about strong suitability of the space of algebraic polynomials of degree n in C[a,b] (Theorem A in [1]) is generalized to the space of spline polynomials _{[a, b} ^{]n, k} (n2, 0) in C[a, b]. Namely, it is shown that the following theorem is valid: for arbitrary numbers ₀, ₁, ..., _n+k, satisfying the conditions (_i–_i–1) (_{i+1{ i}< 0(i=1, ..., n +k–1), there is a unique polynomials _n,k (t) _{[a, b} ^]/n,k and pointsa=₀,<₁<...< _n+k– 1< _n+k = b (₁1 <_n, ..., _kk<_n+k–1), such that s_n,k(_i) = _i(i=0, ..., n + k), s_n,k(_i)=0 (i=1, ..., n + k–1).Translated from Matematicheskii Zametki, Vol. 11, No. 3, pp. 251–258, March, 1972.