Centralizers of generalized derivations on multilinear polynomials in prime rings

Let R be a prime ring of characteristic different from 2, with Utumi quotient ring U and extended centroid C, δ a nonzero derivation of R, G a nonzero generalized derivation of R, and f(x ₁, …, x _n ) a noncentral multilinear polynomial over C. If δ(G(f(r ₁, …, r _n ))f(r ₁, …, r _n )) = 0 for all r ₁, …, r _n ∈ R, then f(x ₁, …, x _n )² is central-valued on R. Moreover there exists a ∈ U such that G(x) = ax for all x ∈ R and δ is an inner derivation of R such that δ(a) = 0.     

Almost sure central limit theorem for partial sums and maxima

Let X, X₁, X₂, … be i.i.d. random variables with nondegenerate common distribution function F, satisfying EX = 0, EX² = 1. Let X_i and M_n = max{X_i, 1 ≤ i ≤ n }. Suppose there exists constants a_n > 0, b_n ∈ R and a nondegenrate distribution G (y) such that Then, we have almost surely, where f (x, y) denotes the bounded Lipschitz 1 function and Φ(x) is the standard normal distribution function (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A conjecture of Kippenhahn about the characteristic polynomial of a pencil generated by two Hermitian matrices. II

Let A be an n × n matrix; write A = H+iK, where i²=—1 and H and K are Hermitian. Let f(x,y,z) = det(zI?xH?yK). We first show that a pair of matrices over an algebraically closed field, which satisfy quadratic polynomials, can be put into block, upper triangular form, with diagonal blocks of size 1×1 or 2×2, via a simultaneous similarity. This is used to prove that if $\text{f(x,y,z) = [g(x,y,z)]}{}^{\mathrm{<mtext>n</mtext><mtext>2</mtext>}}$, where g has degree 2, then for some unitary matrix U, the matrix U¹AU is the direct sum of $\text{n}\text{2}$ copies of a 2×2 matrix A₁, where A₁ is determined, up to unitary similarity, by the polynomial g(x,y,z). We use the connection between f(x,y,z) and the numerical range of A to investigate the case where f(x,y,z) has the form (z?αax? βy)^r[g(x,y,z)]^s, where g(x,y,z) is irreducible of degree 2.     

On a problem of Katona on minimal separating systems

Let S be an n-element set. In this paper, we determine the smallest number f(n) for which there exists a family of subsets of S{A₁,A₂,…,A_f(n)} with the following property: Given any two elements x, y ∈ S (x ≠ y), there exist k, l such that A_k ∩ A_l= ?, and x ∈ A_k, y ∈ A_l. In particular it is shown that f(n)= 3 log₃n when n is a power of 3.     

On maximal antichains consisting of sets and their complements

Let 1 ? k₁ ? k₂ ? … ? k_n be integers and let S denote the set of all vectors x = (x₁, x₂, …, x_n) with integral coordinates satisfying 0 ? x_i ? k_i, i = 1, 2, …, n. The complement of x is (k₁ ? x₁, k₂ ? x₂, …, k_n ? x_n) and a subset X of S is an antichain provided that for any two distinct elements x, y of X, the inequalities x_i ? y_i, i = 1, 2, …, n, do not all hold. We determine an LYM inequality and the maximal cardinality of an antichain consisting of vectors and its complements. Also a generalization of the Erdös-Ko-Rado theorem is given.     

Comonotone approximation of periodic functions

Suppose that a continuous 2π-periodic function f on the real axis ? changes its monotonicity at different ordered fixed points y _i ∈ [?π,π), i = 1, …, 2s, s ∈ ?. In other words, there is a set Y: = {y _i } _i∈? of points y _i = y _i+2s + 2π on ? such that f is nondecreasing on [y _i ,y _i?1] if i is odd and not increasing if i is even. For each n ≥ N(Y), we construct a trigonometric polynomial P _n of order ≤ n changing its monotonicity at the same points y _i ∈ Y as f and such that $$ \parallel f - P_n \parallel \leqslant c(s) \omega _2 \left( {f,\frac{\pi } {n}} \right), $$ where N(Y) is a constant depending only on Y, c(s) is a constant depending only on s, ω₂(f,·) is the modulus of continuity of second order of the function f, and ∥ · ∥ is the max-norm.     

A result on hypothesis testing for a multivariate normal distribution when some observations are missing

Let U_i = (X_i, Y_i), i = 1, 2,…, n, be a random sample from a bivariate normal distribution with mean μ = (μ_x, μ_y) and covariance matrix

. Let X_i, i = n + 1,…, N represent additional independent observations on the X population. Consider the hypothesis testing problem H₀ : μ = 0 vs. H₁ : μ ≠ 0. We prove that Hotelling's T² test, which uses (X_i, Y_i), i = 1, 2,…, n (and discards X_i, i = n + 1,…, N) is an admissible test. In addition, and from a practical point of view, the proof will enable us to identify the region of the parameter space where the T²-test cannot be beaten. A similar result is also proved for the problem of testing μ_x ? μ_y = 0. A Bayes test and other competitors which are similar tests are discussed.     

Adjacency preserving functions on symmetric elements and linear preservers of non-zero decomposable symmetric tensors

Let A be a non-empty set and m be a positive integer. Let ≡ be the equivalence relation defined on A ^m such that (x ₁, …, x _m ) ≡ (y ₁, …, y _m ) if there exists a permutation σ on {1, …, m} such that y _σ(i) = x _i for all i. Let A ^(m) denote the set of all equivalence classes determined by ≡. Two elements X and Y in A ^(m) are said to be adjacent if (x ₁, …, x _m?1, a) ∈ X and (x ₁, …, x _m?1, b) ∈ Y for some x ₁, …, x _m?1 ∈ A and some distinct elements a, b ∈ A. We study the structure of functions from A ^(m) to B ⁽ⁿ⁾ that send adjacent elements to adjacent elements when A has at least n + 2 elements and its application to linear preservers of non-zero decomposable symmetric tensors.     

Fractal functions and interpolation

Let a data set {(x _i,y _i) ∈I×R;i=0,1,?,N} be given, whereI=[x ₀,x _N]?R. We introduce iterated function systems whose attractorsG are graphs of continuous functionsf∶I→R, which interpolate the data according tof(x _i)=y _i fori ε {0,1,?,N}. Results are presented on the existence, coding theory, functional equations and moment theory for such fractal interpolation functions. Applications to the approximation of naturally wiggly functions, which may show some kind of geometrical self-similarity under magnification, such as profiles of cloud tops and mountain ranges, are envisaged.     

Comonotone approximation of periodic functions

Suppose that a continuous 2π-periodic function f on the real axis ? changes its monotonicity at different ordered fixed points y _i ∈ [? π, π), i = 1, …, 2s, s ∈ ?. In other words, there is a set Y:= {y _i } _i∈? of points y _i = y _i+2s + 2π on ? such that, on [y _i , y _i?1], f is nondecreasing if i is odd and nonincreasing if i is even. For each n ≥ N(Y), we construct a trigonometric polynomial P _n of order ≤ n changing its monotonicity at the same points y _i ∈ Y as f and such that $$ \left\| {f - P_n } \right\| \leqslant c\left( s \right)\omega _2 \left( {f,\frac{\pi } {n}} \right), $$ where N(Y) is a constant depending only on Y, c(s) is a constant depending only on s, ω ₂(f, ·) is the modulus of continuity of second order of the function f, and ∥ · ∥ is the max-norm.     

Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions

A function f(x) defined on $\text{X}$ = $\text{X}$₁ × $\text{X}$₂ × … × $\text{X}$_n where each $\text{X}$_i is totally ordered satisfying f(x ∨ y) f(x ∧ y) ≥ f(x) f(y), where the lattice operations ∨ and ∧ refer to the usual ordering on $\text{X}$, is said to be multivariate totally positive of order 2 (MTP₂). A random vector Z = (Z₁, Z₂,…, Z_n) of n-real components is MTP₂ if its density is MTP₂. Classes of examples include independent random variables, absolute value multinormal whose covariance matrix Σ satisfies ?DΣ^?1D with nonnegative off-diagonal elements for some diagonal matrix D, characteristic roots of random Wishart matrices, multivariate logistic, gamma and F distributions, and others. Composition and marginal operations preserve the MTP₂ properties. The MTP₂ property facilitate the characterization of bounds for confidence sets, the calculation of coverage probabilities, securing estimates of multivariate ranking, in establishing a hierarchy of correlation inequalities, and in studying monotone Markov processes. Extensions on the theory of MTP₂ kernels are presented and amplified by a wide variety of applications.     

Vertex-disjoint quadrilaterals containing specified edges in a bipartite graph

The theory of vertex-disjoint cycles and 2-factors of graphs is the extension and generation of the well-known Hamiltonian cycles theory and it has important applications in network communication. In this paper we first prove the following result. Let G=(V ₁,V ₂;E) be a bipartite graph with |V ₁|=|V ₂|=n such that n≥2k+1, where k≥1 is an integer. If d(x)+d(y)≥?(4n+2k?1)/3? for each pair of nonadjacent vertices x and y of G with x∈V ₁ and y∈V ₂, then, for any k independent edges e ₁,…,e _k of G, G contains k vertex-disjoint quadrilaterals C ₁,…,C _k such that e _i ∈E(C _i ) for each i∈{1,…,k}. We further show that the degree condition above is sharp. If |V ₁|=|V ₂|=2k, we give degree conditions that G has a 2-factor with k vertex-disjoint quadrilaterals C ₁,…,C _k containing specified edges of G.     

On a conjecture of Kippenhahn about the characteristic polynomial of a pencil generated by two Hermitian matrices. I

Let A be an n × n complex matrix, and write A = H + iK, where i² = ?1 and H and K are Hermitian matrices. The characteristic polynomial of the pencil xH + yK is f(x, y, z) = det(zI ? xH ? yK). Suppose f(x, y, z) is factored into a product of irreducible polynomials. Kippenhahn [5, p. 212] conjectured that if there is a repeated factor, then there is a unitary matrix U such that U^?1AU is block diagonal. We prove that if f(x, y, z) has a linear factor of multiplicity greater than n?3, then H and K have a common eigenvector. This may be viewed as a special case of Kippenhahn’s conjecture.     

Solvability of a Multi-Point Boundary Value Problem at Resonance

Let f: [0, 1] × R² → R be a function satisfying Caratheodory’s conditions and e(t) ∈ L¹[0, 1]. Let η_i ∈ (0, 1), i = 1, …, k, with 0 s< η₁ < … < η_k < 1, be given. This paper is concerned with the problem of existence of a solution for the multi-point boundary value problem . Conditions for the existence of a solution for the above boundary value problem are given using Leray Schauder Continuation theorem.     

A Characterization of the free n-generated MV-algebra

An MV-algebra A=(A,0,¬,⊕) is an abelian monoid (A,0,⊕) equipped with a unary operation ¬ such that ¬¬x=x,x⊕¬0=¬0, and y⊕¬(y⊕¬x)=x⊕¬(x⊕¬y). Chang proved that the equational class of MV-algebras is generated by the real unit interval [0,1] equipped with the operations ¬x=1?x and x⊕y=min(1,x+y). Therefore, the free n-generated MV-algebra Free _n is the algebra of [0,1]-valued functions over the n-cube [0,1] ⁿ generated by the coordinate functions ξ _i ,i=1, . . . ,n, with pointwise operations. Any such function f is a McNaughton function, i.e., f is continuous, piecewise linear, and each piece has integer coefficients. Conversely, McNaughton proved that all McNaughton functions f: [0,1] ⁿ →[0,1] are in Free _n . The elements of Free _n are logical equivalence classes of n-variable formulas in the infinite-valued calculus of ?ukasiewicz. The aim of this paper is to provide an alternative, representation-free, characterization of Free _n .     

Decomposability of symmetric multilinear functions

Let V denote a finite dimensional vector space over a field K of characteristic 0, let T_n(V) denote the vector space whose elements are the K-valued n-linear functions on V, and let S_n(V) denote the subspace of T_n(V) whose members are the fully symmetric members of T_n(V). If $\text{L}$_n denotes the symmetric group on {1,2,…,n} then we define the projection $\text{P}{}_{\mathrm{<mtext>L</mtext>}}\text{: T}{}_{\mathrm{n}}\text{(V) → S}{}_{\mathrm{n}}\text{(V)}$ by the formula , where P_σ : T_n(V) → T_n(V) is defined so that P_σ(A)(y₁,y₂,…,y_n = A(y_σ(1),y_σ(2),…,y_σ(n)) for each A?T_n(V) and y_i?V, 1 ? i ? n. If $\text{x}{}_{\mathrm{i}}\text{? V}{}^1\text{, 1 ? i ? n}$, then x₁?x₂? … ?x_n denotes the member of T_n(V) such that $\text{(x}{}_1\text{?x}{}_2\text{· ? ? ?x}{}_{\mathrm{n}}\text{)(y}{}_1\text{,y}{}_2\text{,…,y}{}_{\mathrm{n}}\text{) = П}{}^{\mathrm{n}}{}_{\mathrm{i=1}}\text{x}{}_{\mathrm{i}}\text{(y}{}_{\mathrm{i}}\text{)}$ for each y₁ ,₂,…,y_n in V, and x₁·x₂… x_n denotes $\text{P}{}_{\mathrm{<mtext>L</mtext>}}\text{(x}{}_1\text{?x}{}_2\text{? … ?x}{}_{\mathrm{n}}\text{)}$. If B? S_n(V) and there exists $\text{x}{}_{\mathrm{i}}\text{? V}{}^1\text{, 1 ? i ? n}$, such that B = x₁·x₂…x_n, then B is said to be decomposable. We present two sets of necessary and sufficient conditions for a member B of S_n(V) to be decomposable. One of these sets is valid for an arbitrary field of characteristic zero, while the other requires that K = R or C.     

An application of L-functions of imaginary quadratic fields

Let F₁(x, y),…, F_2h+1(x, y) be the representatives of equivalent classes of positive definite binary quadratic forms of discriminant ?q (q is a prime such that q ≡ 3 mod 4) with integer coefficients, then the number of integer solutions of F_i(x, y) = n (i = 1,…, 2h + 1) can be calculated for each natural number n using L-functions of imaginary quadratic field $\text{Q}$((?q)^1/2).     

A new inequality for symmetric functions

Let x_i ≥ 0, y_i ≥ 0 for i = 1,…, n; and let a_j(x) be the elementary symmetric function of n variables given by a_j(x) = ∑_{1 ≤ ii < … <ij ≤ n}x_ii … x_ij. Define the partical ordering x <y if a_j(x) ≤ a_j(y), j = 1,… n. We show that $\text{x}\text{\$}\text{?}\text{y ? x}{}^{\mathrm{\alpha }}\text{\$}\text{?}\text{y}{}^{\mathrm{\alpha }}\text{, 0}\text{\$}\text{?}\text{α ≤ 1}$, where {x^α}_i = x^α_i. We also give a necessary and sufficient condition on a function f(t) such that x <y ? f(x) <f(y). Both results depend crucially on the following: If x <y there exists a piecewise differentiable path z(t), with z_i(t) ≥ 0, such that z(0) = x, z(1) = y, and z(s) <z(t) if 0 ≤ s ≤ t ≤ 1.     

On the form of homomorphisms into the differential group Ls 1 and their extensibility

Let L_s ¹ (s ∈ ?) be the s-th differential group, that is the set {(x1,…,x_s): x₁ ≠ 0, x_n ∈ K, n =1,2,…,s} (K ∈ {?,?}) together with the group operation which describes the chain rules (up to order s) for C^s-functions with fixed point 0. We consider homomorphisms Φ_s, Φ_s = (f₁,…,f_s) from an abelian group (G,+) into L_s ¹ such that f₁ = 1, f₂ = … = f_p+2 = 0, 0_p+2 ≠ 0 for a fixed, but arbitrary p ≥ 0 such that p + 2 ≤ s (then f_p+2 is necessarily a homomorphism from (G, +) to (K, +). Let l ∈ ? or l = ∞. We present a criterion for the extensibility of Φ_s to a homomorphism Φ_s+l from (G, +) to L_s+1 ¹ (L_∞ ¹, if l = ∞), by proving that such an extension (continuation) exists iff the component functions f_n of Φ_s with s - p ≤ n ≤ min(s - p + l - 1,s) are certain polynomials in f_P+2 (see Theorem 1). We also formulate the problem in the language of truncated formal power series in one indeterminate X over K. The somewhat easier situation f ₁ ≠ 1 will be studied in a separate paper.