MAXIMUM TREES OF FINITE SEQUENCES

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On the fourier-stieltjes coefficients of cantor-type distributions

For a givenn-tuple of non-negative numbers (p ₀,p ₁,...,p _n?1) whose sum is equal to unity let μ(t) denote the probability that Σ _j ^{= 1/∞} X _j /n ^j ≦t, where the independent random variablesX _j assume the values 0,1,...,n?1 with probabilitiesp ₀,p ₁,...,p _n?1 respectively. For mostn-tuples we obtain upper and lower bounds on |û(m)|; these estimates involve then-ary representation ofm, or in some cases of 2m, so that a very simple and explicit characterization of the sequences on whichû(m) approaches zero can be given. In particular, for the Cantor middle-third measure, corresponding to the triple (1/2, 0, 1/2), the following criterion is obtained.û(m) approaches zero on a sequenceT of integers if and only if Ω(2m) approaches infinity onT, where Ω(k) is the sum of the following three quantities associated with the ternary representation ofk: the number of runs of zeros, the number of runs of twos and the number of ones. The results obtained are easily extended to the case when then-tuple varies withj (subject to certain mild restrictions).     

A theorem on non-associative algebras and its application to differential equations

For a given field F, the set of F-algebras (resp. commutative F-algebras) of arity n≥2 and F-dimension m can be identified with the mⁿ⁺¹ (resp. m(^m+n?1 _n)) dimensional F-affine space S of structure coefficients. We show: If F is algebraically closed, then there exists an affine subvariety A of S with A≠S, which is defined over the prime field of F, such that all F-algebras corresponding to the points of S-A posses precisely n^m?1 idempotent elements ≠0 and fail to have nil potent elements ≠0. This implies for a system of ordinary differential equations $$\left( * \right)\dot X_i = D_i \left( {X_l ,..,X_m } \right),i = l,..,m,$$ with D_i(X_i,...,X_m)∈?[X₁,...,X_m] homogeneous polynomials of degree n: If the coefficients of the polynomials D_i, i=1,...,m, are algebraically independent over the field of rationals, then (*) possesses precisely n^m?1 ray solutions and fails to have a critical point other than the origin.     

On the number of permutatively inequivalent basic sequences in a Banach space

Let X be a Banach space with a Schauder basis (en)_n∈N. The relation E₀ is Borel reducible to permutative equivalence between normalized block-sequences of (en)_n∈N or X is c₀ or ?p saturated for some 1?p<+∞. If (en)_n∈N is shrinking unconditional then either it is equivalent to the canonical basis of c₀ or ?p, 1<p<+∞, or the relation E₀ is Borel reducible to permutative equivalence between sequences of normalized disjoint blocks of X or of X^∗. If (en)_n∈N is unconditional, then either X is isomorphic to ?₂, or X contains ω² subspaces or ω² quotients which are spanned by pairwise permutatively inequivalent normalized unconditional bases.     

A note on the probabilistic approach to Turán's problem

The Turán number T(n, l, k) is the smallest possible number of edges in a k-graph on n vertices such that every l-set of vertices contains an edge. Given a k-graph H = (V(H), E(H)), we let X_s(S) equal the number of edges contained in S, for any s-set S?V(H). Turán's problem is equivalent to estimating the expectation E(X_l), given that min(X_l) ≥ 1. The following lower bound on the variance of X_s is proved:

$\text{Var(X}{}_{\mathrm{s}}\text{)?mm}\text{n?2k}\text{s?k}\text{n}\text{s}{}^{\mathrm{?1}}\text{n}\text{k}{}^1$

, where m = |E(H)| and $\text{m}\text{= (}{}_{\mathrm{k}}{}^{\mathrm{n}}\text{) ? m}$. This implies the following: putting t(k, l) = lim_n→∞T(n, l, k)(_kⁿ)^?1 then t(k, l) ≥ T(s, l, k)((_k^s) ? 1)^?1, whenever s ≥ l > k ≥ 2. A connection of these results with the existence of certain t-designs is mentioned.     

Characterizations of symmetrized polydisc

Let Γ_n, n ≥ 2, denote the symmetrized polydisc in ?ⁿ, and Γ₁ be the closed unit disc in ?. We provide some characterizations of elements in Γ_n. In particular, an element (s₁,..., s_n?1, p) ∈ ?ⁿ is in Γ_n if and only if \({s_j} = {\beta _j} + \overline {{\beta _{n - j}}}p\), j = 1,..., n ? 1, for some (β₁,..., β_n?1) ∈ Γ_n?1, and |p| ≤ 1.     

On the normal approximation of a binomial random sum

We present upper bounds of the L _s norms of the normal approximation for random sums of independent identically distributed random variables X ₁ , X ₂ , . . . with finite absolute moments of order 2 + δ, 0 < δ ≤ 1, where the number of summands N is a binomial random variable independent of the summands X ₁ , X ₂ , . . . . The upper bounds obtained are of order (E N) ^?δ/2 for all 1 ≤ s ≤ ∞.     

Existence of multiple positive solutions for one-dimensional p-Laplacian operator

In this paper, we consider the multipoint boundary value problem for one-dimensional p-Laplacian $$(\phi_{p}(u'))'+f(t,u,u')=0,\quad t\in [0,1],$$ subject to the boundary value conditions: $$u'(0)=\sum_{i=1}^{n-2}\alpha_{i}u'(\xi_{i}),\qquad u(1)=\sum_{i=1}^{n-2}\beta_{i}u(\xi_{i}),$$ where φ _p (s)=|s| ^p?2?s,p>1;ξ _i ∈(0,1) with 0<ξ ₁<ξ ₂<???<ξ _n?2<1 and α _i ,β _i satisfy α _i ,β _i ∈[0,∞),0≤∑ _i=1 ^n?2 α _i <1 and 0≤∑ _i=1 ^n?2 β _i <1. Using a fixed point theorem for operators in a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem.     

The exact density function of the ratio of two dependent linear combinations of chi-square variables

A computable expression is derived for the raw moments of the random variableZ=N/D whereN= ₁ ⁿ m _iX_i+ _n ^+1s m _iX_i,D= _n ^+1s l _iX_i+ _s ^+1r n _iX_i, and theX _i's are independently distributed central chi-square variables. The first four moments are required for approximating the distribution ofZ by means of Pearson curves. The exact density function ofZ is obtained in terms of sums of generalized hypergeometric functions by taking the inverse Mellin transform of theh-th moment of the ratioN/D whereh is a complex number. The casen=1,s=2 andr=3 is discussed in detail and a general technique which applies to any ratio having the structure ofZ is also described. A theoretical example shows that the inverse Mellin transform technique yields the exact density function of a ratio whose density can be obtained by means of the transformation of variables technique. In the second example, the exact density function of a ratio of dependent quardratic forms is evaluated at various points and then compared with simulated values.     

Conditional limit theorems for exponential families and finite versions of de Finetti's theorem

Consider an exponential familyP _λ which is maximal, smooth, and has uniformly bounded standardized fourth moments. Consider a sequenceX ₁,X ₂,... of i.i.d. random variables with parameter λ. LetQ _nsk be the law ofX ₁,...,X _k given thatS _n=X ₁+...+X _n=s. Choose λ so thatE _λ(X ₁)=s/n. Ifk andn→∞ butk/n→0, then $$\parallel Q_{nsk} - P_\lambda ^k \parallel = \gamma \frac{k}{n} + o\left( {\frac{k}{n}} \right)$$ where γ=1/2E{|1?Z ²|} andZ isN(0,1). The error term is uniform ins, the value ofS _n. Similar results are given fork/n→θ and for mixtures of theP _λ ^k . Versions of de Finetti's theorem follow.     

Reducibility of Punctual Hilbert Schemes of Cone Varieties

In this short note we show that for any pair of positive integers (d, n) with n > 2 and d > 1 or n = 2 and d > 4, there always exist projective varieties X ? ? ^N of dimension n and degree d and an integer s ₀ such that Hilb ^s (X) is reducible for all s ≥ s ₀. X will be a projective cone in ? ^N over an arbitrary projective variety Y ? ? ^N?1. In particular, we show that, opposite to the case of smooth surfaces, there exist projective surfaces with a single isolated singularity which have reducible Hilbert scheme of points.     

On covering a product of sets with products of their subsets

Suppose that S₁,…,S_N are collections of subsets of X₁,…,X_N, respectively, such that n_i subsets belonging to S_i, and no fewer, cover X_i for all i. the main result of this paper is that to cover X₁ x…x X_N requires no fewer than σ^N_i=1 (n_i–1) + 1 and no more than Π^N_i=1n_i subsets of the form A₁ x…x A_N, where A_i∈S₁foralli. Moreo ver, these bounds cannot be improved. Identical bounds for the spanning number of a normal product of graphs are also obtained.     

Classifying Hilbert bundles. II

A Hilbert bundle (p, B, X) is a type of fibre space p: B → X such that each fibre p^?1(x) is a Hilbert space. However, p^?1(x) may vary in dimension as x varies in X, even when X is connected. We give two “homotopy” type classification theorems for Hilbert bundles having primarily finite dimensional fibres. An (m, n)-bundle over the pair (X, A) is a Hilbert bundle over (p, B, X) such that the dimension of p^?1(x) is m for x in A and n otherwise. As a special case, we show that if X is a compact metric space, C₊X the upper cone of the suspension SX, then the isomorphism classes of (m, n)-bundles over (SX, C₊X) are in one-to-one correspondence with the members of [X, V_m($\text{C}$ⁿ)] where V_m($\text{C}$ⁿ) is the Stiefel manifold. The results are all applicable to the classification of separable, continuous trace C¹-algebras, with specific results given to illustrate.     

Nonlinear trigonometric approximations of multivariate function classes

Order-sharp estimates are established for the best N-term approximations of functions from Nikol’skii–Besov type classes B_pq^sm(T^k) with respect to the multiple trigonometric system T^(k) in the metric of L_r(T^k) for a number of relations between the parameters s, p, q, r, and m (s = (s₁,..., s_n) ∈ R₊ⁿ, 1 ≤ p, q, r ≤ ∞, m = (m₁,..., m_n) ∈ Nⁿ, k = m₁ +... + m_n). Constructive methods of nonlinear trigonometric approximation—variants of the so-called greedy algorithms—are used in the proofs of upper estimates.     

Asymptotics for transportation cost in high dimensions

LetX ₁,...,X _n ,Y ₁,...,Y _n be i.i.d. with the law μ on the cube [0, 1] ^d ,d?3. LetL _n (μ)=inf_πΣ _i=1 ⁿ ||X _i ?Y _π(i)|| denote the optimal bipartite matching of theX andY points, where π ranges over all permutations of the integers 1, 2,...,n, and where ‖·‖ is a norm on ? ^d . If μ is Lebesgue measure it is shown that $$\mathop {\lim }\limits_{n \to \infty } L_n (\mu )/n^{(d - 1)/d} = \alpha {\text{a}}{\text{.s}}{\text{.}}$$ where α is a finite constant depending on ‖ ‖ andd only. More generally, for arbitrary μ it is shown that $$\mathop {\lim }\limits_{n \to \infty } L_n (\mu )/n^{(d - 1)/d} = \alpha \int {(f{\text{(}}x{\text{)}})^{(d - 1)/d} dxa.s.} $$ wheref is the density of the absolutely continuous part of μ. We also find the rate of convergence.     

Banach spaces embedding intoL 0

Our main result in this paper is that a Banach spaceX embeds intoL ₁ if and only ifl ₁(X) embeds intoL ₀; more generally if 1≦p<2,X embeds intoL _p if and only ifl _p(X) embeds intoL ₀. Research supported by NSF grant MCS-8301099.     

The mixing advantage is less than 2

Corresponding to n independent non-negative random variables X ₁,...,X _n , are values M ₁,...,M _n , where each M _i is the expected value of the maximum of n independent copies of X _i . We obtain an upper bound for the expected value of the maximum of X ₁,...,X _n in terms of M ₁,...,M _n . This inequality is sharp in the sense that the random variables can be chosen so that the bound is approached arbitrarily closely. We also present related comparison results.      

Complete convergence of weighted sums of martingale differences

LetF _oF ₁ ... be an increasing family of -algebras. For eachn1,X _n isF _n-measurable, andE(X _n|F_n–1) is zero almost surely, andE(|E_n|^p|F_n–1) is bounded by a finite constant almost surely for somep2. Leta _n1,...,a _nn be constants. Conditions are given to establish the complete convergence of (a _n1 X ₁+...+a _nnX_n)/n ^1/p , thereby obtaining an extension of Chow's (1966) result for the case of independent and identically distributed random variables. Whenp>2, the conditions are an improvement on existing results for the case of independence and identical distribution.     

Synthesis of easy tested diagrams in the basis {&;, ∨, <Superscript>−</Superscript>} for systems of functions from some classes

A new method for synthesis of easy tested diagrams of functional elements in the basis {&;, ∨, ^?} is proposed for systems of m Boolean functions differing from constants and representable as disjunctive normal forms containing l variables x ₁,...,x _l , l > 0, without negations and the other variables x _l+1,...,x _n with negations only. All faults are assumed to be stack-at-1 faults at the outputs of elements. It is proved that the length of a complete verification test is not greater than min {m, l} for such diagrams.     

Multiple ergodic theorems

Given measure preserving transformationsT ₁,T ₂,...,T _s of a probability space (X,B, μ) we are interested in the asymptotic behaviour of ergodic averages of the form $$\frac{1}{N}\sum\limits_{n = 0}^{N - 1} {T_1^n f_1 \cdot T_2^n f_2 } \cdot \cdots \cdot T_s^n f_s $$ wheref ₁,f ₂,...,f _s ?L ^∞(X,B,μ). In the general case we study, mainly for commuting transformations, conditions under which the limit of (1) inL ²-norm is ∫ _x f ₁ dμ·∫ _x f ₂ dμ...∫ _x f _s dμ for anyf ₁,f ₂...,f _s ?L ^∞(X,B,μ). If the transformations are commuting epimorphisms of a compact abelian group, then this limit exists almost everywhere. A few results are also obtained for some classes of non-commuting epimorphisms of compact abelian groups, and for commuting epimorphisms of arbitrary compact groups.