Carleson measures and the generalized Campanato spaces of vector-valued martingales

In this paper, the so-called(p, φ)-Carleson measure is introduced and the relationship between vector-valued martingales in the general Campanato spaces Lp,φ(X) and the(p, φ)-Carleson measures is investigated. Specifically, it is proved that for q ∈ [2, ∞), the measure dμ := ||dfk||~qdP ? dm is a(q, φ)-Carleson measure on ? × N for every f ∈ L_q,φ(X)if and only if X has an equivalent norm which is q-uniformly convex; while for p ∈(1, 2], the measure dμ :=||dfk||~pdP ? dm is a(p, φ)-Carleson measure on ? × N implies that f ∈ L_p,φ(X)if and only if X admits an equivalent norm which is p-uniformly smooth. This result extends an earlier result in the literature from BMO spaces to general Campanato spaces.     

Criteria for embedding of spaces constructed by the method of means with arbitrary quasi-concave functional parameters

We consider a generalization ?(X₀,X₁)_p0,p1 of the method of means to arbitrary non-degenerate functional parameter. In this case non-trivial embedding ?(X₀,X₁)_p0,p1⊂ψ(X₀,X₁)_q0,q1 take place. We find necessary and sufficient condition for such embedding if 1?q₀?p₀?∞ and 1?q₁?p₁?∞ or 1?p₀?q₀?∞ and 1?p₁?q₁?∞.     

Path-fan Ramsey numbers

For two given graphs F and H, the Ramsey number R(F,H) is the smallest positive integer p such that for every graph G on p vertices the following holds: either G contains F as a subgraph or the complement of G contains H as a subgraph. In this paper, we study the Ramsey numbers R(P_n,F_m), where P_n is a path on n vertices and F_m is the graph obtained from m disjoint triangles by identifying precisely one vertex of every triangle (F_m is the join of K₁ and mK₂). We determine the exact values of R(P_n,F_m) for the following values of n and m: 1?n?5 and m?2; n?6 and 2?m?(n+1)/2; 6?n?7 and m?n-1; n?8 and n-1?m?n or ((q·n-2q+1)/2?m?(q·n-q+2)/2 with 3?q?n-5) or m?(n-3)²/2; odd n?9 and ((q·n-3q+1)/2?m?(q·n-2q)/2 with 3?q?(n-3)/2) or ((q·n-q-n+4)/2?m?(q·n-2q)/2 with (n-1)/2?q?n-5). Moreover, we give nontrivial lower bounds and upper bounds for R(P_n,F_m) for the other values of m and n.     

A practical error formula for multivariate rational interpolation and approximation

We consider exact and approximate multivariate interpolation of a function f(x ₁?,?.?.?.?,?x _d ) by a rational function p _n,m /q _n,m (x ₁?,?.?.?.?,?x _d ) and develop an error formula for the difference f???p _n,m /q _n,m . The similarity with a well-known univariate formula for the error in rational interpolation is striking. Exact interpolation is through point values for f and approximate interpolation is through intervals bounding f. The latter allows for some measurement error on the function values, which is controlled and limited by the nature of the interval data. To achieve this result we make use of an error formula obtained for multivariate polynomial interpolation, which we first present in a more general form. The practical usefulness of the error formula in multivariate rational interpolation is illustrated by means of a 4-dimensional example, which is only one of the several problems we tested it on.     

Regularity of patterns in the factorization of n!

Consider the multiplicities ep₁(n),ep₂(n),…,epk(n) in which the primes p₁,p₂,…,pk appear in the factorization of n!. We show that these multiplicities are jointly uniformly distributed modulo (m₁,m₂,…,mk) for any fixed integers m₁,m₂,…,mk, thus improving a result of Luca and St?nic? [F. Luca, P. St?nic?, On the prime power factorization of n!, J. Number Theory 102 (2003) 298-305]. To prove the theorem, we obtain a result regarding the joint distribution of several completely q-additive functions, which seems to be of independent interest.     

Partial integrability of Hamiltonian systems with homogeneous potential

In this paper we consider systems with n degrees of freedom given by the natural Hamiltonian function of the form $$ H = \frac{1} {2}p^T Mp + V(q), $$ where q = (q ₁, …, q _n ) ∈ ? ⁿ , p = (p ₁, …, p _n ) ∈ ? ⁿ , are the canonical coordinates and momenta, M is a symmetric non-singular matrix, and V (q) is a homogeneous function of degree k ∈ ?*. We assume that the system admits 1 ? m < n independent and commuting first integrals F ₁, … F _m . Our main results give easily computable and effective necessary conditions for the existence of one more additional first integral F _m+1 such that all integrals F ₁, … F _m+1 are independent and pairwise commute. These conditions are derived from an analysis of the differential Galois group of variational equations along a particular solution of the system. We apply our result analysing the partial integrability of a certain n body problem on a line and the planar three body problem.     

Nonparametric estimation of mixed partial derivatives of a multivariate density

On the basis of a random sample of size n on an m-dimensional random vector X, this note proposes a class of estimators f_n^(p) of f^(p), where f is a density of X w.r.t. a σ-finite measure dominated by the Lebesgue measure on R^m, p = (p₁,…,p_m), p_j ≥ 0, fixed integers, and for x = (x₁,…,x_m) in R^m, f^(p)(x) = ?^p₁+…+p_m f(x)/(?^p₁x₁ … ?^p_mx_m). Asymptotic unbiasedness as well as both almost sure and mean square consistencies of f_n^(p) are examined. Further, a necessary and sufficient condition for uniform asymptotic unbisedness or for uniform mean square consistency of f_n^(p) is given. Finally, applications of estimators of this note to certain statistical problems are pointed out.     

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.     

On the order of an automorphism of a smooth hypersurface

In this paper we give an effective criterion as to when a positive integer q is the order of an automorphism of a smooth hypersurface of dimension n and degree d, for every d ≥ 3, n ≥ 2, (n, d) ≠ (2, 4), and gcd(q, d) = gcd(q, d ? 1) = 1. This allows us to give a complete criterion in the case where q = p is a prime number. In particular, we show the following result: If X is a smooth hypersurface of dimension n and degree d admitting an automorphism of prime order p then p < (d ? 1) ⁿ⁺¹; and if p > (d ? 1) ⁿ then X is isomorphic to the Klein hypersurface, n = 2 or n + 2 is prime, and p = Φ _n+2(1 ? d) where Φ _n+2 is the (n+2)-th cyclotomic polynomial. Finally, we provide some applications to intermediate jacobians of Klein hypersurfaces.     

Path-kipas Ramsey numbers

For two given graphs F and H, the Ramsey number R(F,H) is the smallest positive integer p such that for every graph G on p vertices the following holds: either G contains F as a subgraph or the complement of G contains H as a subgraph. In this paper, we study the Ramsey numbers , where P_n is a path on n vertices and is the graph obtained from the join of K₁ and P_m. We determine the exact values of for the following values of n and m: 1?n?5 and m?3; n?6 and (m is odd, 3?m?2n-1) or (m is even, 4?m?n+1); 6?n≤7 and m=2n-2 or m?2n; n?8 and m=2n-2 or m=2n or (q·n-2q+1?m?q·n-q+2 with 3?q?n-5) or m?(n-3)²; odd n?9 and (q·n-3q+1?m?q·n-2q with 3?q?(n-3)/2) or (q·n-q-n+4?m?q·n-2q with (n-1)/2?q?n-4). Moreover, we give lower bounds and upper bounds for for the other values of m and n.     

On the spectrum of the hierarchical Laplacian

Let (X, d) be a locally compact separable ultrametric space. We assume that (X, d) is proper, that is, any closed ball B?X is a compact set. Given a measure m on X and a function C(B) defined on the set of balls (the choice function) we define the hierarchical Laplacian L _C which is closely related to the concept of the hierarchical lattice of F.J. Dyson. L _C is a non-negative definite self-adjoint operator in L ²(X, m). In this paper we address the following question: How general can be the spectrum \(\mathsf {Spec}(L_{C})\subseteq \mathbb {R}_{+}?\) When (X, d) is compact, S p e c(L _C ) is an increasing sequence of eigenvalues of finite multiplicity which contains 0. Assuming that (X, d) is not compact we show that under some natural conditions concerning the structure of the hierarchical lattice (≡ the tree of d-balls) any given closed subset S ? ?₊, which contains 0 as an accumulation point and is unbounded if X is non-discrete, may appear as S p e c(L _C ) for some appropriately chosen function C(B). The operator ?L _C extends to L ^q (X, m), 1 ≦ q < ∞, as Markov generator and its spectrum does not depend on q. As an example, we consider the operator ?? ^α of fractional derivative defined on the field ? _p of p-adic numbers.     

Forbidden (0,1)-Vectors in Hyperplanes of ℝ n : The Restricted Case

In this paper we continue our investigation on “Extremal problems under dimension constraint” introduced in [2]. Let E(n, k) be the set of (0,1)-vectors in ? ⁿ with k one's. Given 1 ≤ m, w ≤ n let X ? E(n, m) satisfy span (X) ∩ E(n, w) = ?. How big can |X| be? This is the main problem studied in this paper. We solve this problem for all parameters 1 ≤ m, w ≤ n and n > n ₀(m, w).     

On complexity of three-dimensional hyperbolic manifolds with geodesic boundary

The nonintersecting classes ? _p,q are defined, with p, q ?? ? and p ?? q ?? 1, of orientable hyperbolic 3-manifolds with geodesic boundary. If M ?? ? _p,q , then the complexity c(M) and the Euler characteristic ??(M) of M are related by the formula c(M) = p???(M). The classes ? _q,q , q ?? 1, and ?_2,1 are known to contain infinite series of manifolds for each of which the exact values of complexity were found. There is given an infinite series of manifolds from ?_3,1 and obtained exact values of complexity for these manifolds. The method of proof is based on calculating the ?-invariants of manifolds.     

Likelihood ratio orderings of spacings of heterogeneous exponential random variables

Let X₁,X₂,…,X_n be independent exponential random variables such that X_i has failure rate λ for i=1,…,p and X_j has failure rate λ^* for j=p+1,…,n, where p≥1 and q=n-p≥1. Denote by D_i:n(p,q)=X_i:n-X_i-1:n the ith spacing of the order statistics , where X_0:n≡0. It is shown that D_i:n(p,q)?_lrD_i+1:n(p,q) for i=1,…,n-1, and that if λ?λ^* then , and for i=1,…,n, where ?_lr denotes the likelihood ratio order. The main results are used to establish the dispersive orderings between spacings.     

The generic division rings

LetA=k (X ₁, X₂..., X_m) be the division ring generated by genericn×n matrices over a fieldk; thenA is not a crossed product in the following cases: (i) there exists a primeq such thatq ³ℛn;(ii)[k:Q]=m, whereQ is the field of rationals, then if eitherq ³ℛn for someq for whichq-1ℛm, orq ²/nn for some other prime; (iii)k=Z _p ^r a finite field ofp ^r elements and eitherq ³ℛn for sameqℛp ^r-1 orq ²ℛn for some other primes. Other cases are also considered.     

Conditional ordering of order statistics

For any positive integers m and n, let X₁,X₂,…,X_m∨n be independent random variables with possibly nonidentical distributions. Let X_1:n≤X_2:n≤?≤X_n:n be order statistics of random variables X₁,X₂,…,X_n, and let X_1:m≤X_2:m≤?≤X_m:m be order statistics of random variables X₁,X₂,…,X_m. It is shown that (X_j:n,X_j+1:n,…,X_n:n) given X_i:m>y for j−i≥max{n−m,0}, and (X_1:n,X_2:n,…,X_j:n) given X_i:m≤y for j−i≤min{n−m,0} are all increasing in y with respect to the usual multivariate stochastic order. We thus extend the main results in Dubhashi and Häggström (2008) [1] and Hu and Chen (2008) [2].     

Multivariate model with a Kronecker product covariance structure: S. N. Roy method of estimation

In this paper we consider a (p × q)-matrix X = (X ₁, ..., X _q ), where a pq-vector vec (X) = (X ₁ ^T , ...,X _q ^T ) ^T is assumed to be distributed normally with mean vector vec (M) = (M ₁ ^T , ...,M _q ^T ) ^T and a positive definite covariance matrix Λ. Suppose that Λ follows a Kronecker product covariance structure, that is Λ = Φ?Σ, where Φ = (? _ij ) is a (q × q)-matrix and Σ = (σ _ij ) is a (p × p)-matrix and the matrices Φ, Σ are positive definite. Such a model is considered in [4], where the maximum likelihood estimates of the parameters M, Φ, Σ are obtained. Using S. N. Roy’s technique (see, e.g., [3]) of the multivariate statistical analysis, we obtain consistent and unbiased estimates of M, Φ, Σ as in [4], but with less calculations.     

Atomic decomposition characterizations of weighted multiparameter Hardy spaces

Let w ?? A _??. In this paper, we introduce weighted-(p, q) atomic Hardy spaces H _w ^p,q (? ⁿ ×? ^m ) for 0 < p ? 1, q >q _w and show that the weighted Hardy space H _w ^p (? ⁿ × ? ^m ) defined via Littlewood-Paley square functions coincides with H _w ^p,q (? ⁿ × ? ^m ) for 0 < p ? 1, q > q _w . As applications, we get a general principle on the H _w ^p (? ⁿ × ? ^m ) to L _w ^p (? ⁿ ×? ^m ) boundedness and a boundedness criterion for two parameter singular integrals on the weighted Hardy spaces.     

Classifying Hilbert bundles

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. We generalize the classical homotopy classification theory of vector bundles to a “homotopy” classification of certain Hilbert bundles. An (m, n)-bundle over the pair (X, A) is a Hilbert bundle (p, B, X) such that the dimension of p^?1(x) is m for x in A and n otherwise. The main result here is that if A is a compact set lying in the “edge” of the metric space X (e.g. if X is a topological manifold and A is a compact subset of the boundary of X), then the problem of classifying (m, n)-bundles over (X, A) reduces to a problem in the classical theory of vector bundles. In particular, we show there is a one-to-one correspondence between the members of the orbit set, [A, G_m($\text{C}$ⁿ)]/[X, U(n)] ¦ A, and the isomorphism classes of (m, n)-bundles over (X, A) which are trivial over X, A.     

On additive polynomials and certain maximal curves

We show that a maximal curve over F_q² given by an equation A(X)=F(Y), where A(X)∈F_q²[X] is additive and separable and where F(Y)∈F_q²[Y] has degree m prime to the characteristic p, is such that all roots of A(X) belong to F_q². In the particular case where F(Y)=Y^m, we show that the degree m is a divisor of q+1.