The asymptotic distributions of incomplete U-statistics

Summary An incomplete U-statistic is obtained by sampling the terms of an U-statistic. This paper derives the asymptotic distribution (if the variance is finite). Depending on the number of sampled terms, the resulting distribution is either the same as for the U-statistic, a normal distribution, or something intermediate. Also the case of a non-random sampling of the terms is treated. As an example, a non-parametric test of the independence of two circular random variables is studied. The results are generalized to generalized U-statistics.     

Continuity of generalized derivations on JB*-algebras

We prove that every generalized Jordan derivation D from a JB?-algebra 𝒜 into itself or into its dual space is automatically continuous. In particular, we establish that every generalized Jordan derivation from a C?-algebra to a Jordan Banach module is continuous. As a consequence, every generalized derivation from a C?-algebra to a Banach bimodule is continuous.     

Generalized Jordan left derivations on semiprime algebras

Let A{\mathcal{A}} be a semiprime algebra of characteristic not 2. Then any generalized Jordan left derivation on A{\mathcal{A}} is a generalized left derivation and is also a generalized derivation. This gives an affirmative answer to a question in Ashraf and Ali (Bull Korean Math Soc 45:253–261, 2008). Moreover, we prove that there are no nonzero generalized Jordan left derivations that take only nilpotent values on A{\mathcal{A}} .     

M-structure in dual Banach spaces

A result previously known only for certain ordered Banach spaces is generalized to arbitrary real Banach spaces. Let ℒ be the Banach algebra of operators generated by theL-projections of a real Banach spaceU, and let ℳ (U ^* be the bounded operators on the dual spaceU ^* with adjoint in ℒ(U ^**. Then the adjoint operation maps ℒ (U) onto ℳ (U ^*). In particular, anyM-projection ofU ^* is weak^* continuous. Supported in part by the National Science Foundation.     

Jordan triple derivations on J-subspace lattice algebras

Let L be a J-subspace lattice on a Banach space X and Alg L the associated J-subspace lattice algebra. Let A be a standard operator subalgebra (i.e., it contains all finite rank operators in AlgL) of AlgL and M■B(X) the Alg L-bimodule. It is shown that every linear Jordan triple derivation from A into M is a derivation, and that every generalized Jordan (triple) derivation from A into M is a generalized derivation.     

On generalized Jordan derivations of Lie triple systems

Under some conditions we prove that every generalized Jordan triple derivation on a Lie triple system is a generalized derivation. Specially, we conclude that every Jordan triple θ-derivation on a Lie triple system is a θ-derivation.     

Modified generalized Hadamard matrices and constructions for transversal designs

It is well known that there exists a transversal design TD_λ[k; u] admitting a class regular automorphism group U if and only if there exists a generalized Hadamard matrix GH(u, λ) over U. Note that in this case the resulting transversal design is symmetric by Jungnickel’s result. In this article we define a modified generalized Hadamard matrix and show that transversal designs which are not necessarily symmetric can be constructed from these under a modified condition similar to class regularity even if it admits no class regular automorphism group.     

Struttura delle fibrazioni di una classe di piani di André generalizzati finiti di ordineq 1+1

Every finite generalized André plane is associated with a spreadF′ of the projective spacePG(2t+1,q), which is obtained from a regular preadF replacing in a switching setU some of the subspaces ofF. The construction ofU is realized by an opportune setA of non-identical automorphisms of the fieldGF(q ^t+1). In this paper we characterize the irreducible components ofU, whenU is obtained by a setA consisting of two automorphisms. In the second paragraph we prove that such switching sets are only of two types. In the third paragraph we provide a constructive rule which is a necessary and sufficient condition for the existence of both the types. In such a way we describe the structure of the spreadF′ associated with any finite generalized André plane such that |A|=2.      

The existence of soliton metrics for nilpotent Lie groups

We show that a left-invariant metric g on a nilpotent Lie group N is a soliton metric if and only if a matrix U and vector v associated the manifold (N, g) satisfy the matrix equation U v = [1], where [1] is a vector with every entry a one. We associate a generalized Cartan matrix to the matrix U and use the theory of Kac–Moody algebras to analyze the solution spaces for such linear systems. An application to the existence of soliton metrics on certain filiform Lie groups is given.     

Higher derivations on Lie ideals of triangular algebras

Let T be a triangular algebra and let U be an admissible Lie ideal of T. We mainly consider the question whether each Jordan higher derivation of U into T is a higher derivation of U into T. We also give some characterizations for the Jordan triple higher derivations of U.     

Sonin Type Expansions for Solutions to a Generalized Bessel Equation of Order m and Their Application

We prove theorems which express a solution to a generalized Bessel equation of order m as a Neumann type series and then apply them to derivation of more complicated relations for products of such solutions, the so-called Bateman type multiplication theorems.     

Special tight frames

Special vector systems, in which every element is the preceding element multiplied by a unitary matrix U, are introduced. Necessary and sufficient conditions for such a system to be a tight frame are obtained (Theorem 1). Examples illustrating the necessity of every condition are given. The theorem is applied to the Mercedes-Benz frame. Let P denote the matrix composed of orthonormal eigenvectors of U. A new system of vectors in which every element equals the corresponding element of the initial system multiplied by P* is considered. It is proved that this system is a generalized harmonic frame if and only if the assumptions of Theorem 1 hold. This result is applied to show how to transform the Mercedes-Benz frame into a generalized harmonic frame.     

IDENTITIES WITH GENERALIZED DERIVATIONS

Let R be a prime ring with extended centroid C. By a generalized derivation of R we mean an additive map g: R → R such that (xy) ^g = x^gy + xy ^δ for all x ∈ R, where δ is a derivation of R. In this paper we prove a version of Kharchenko's theorem for generalized derivations and present some results concerning certain identities with generalized derivations.     

Tanaka Formula for Multidimensional Brownian Motions

We study the Tanaka formula for multidimensional Brownian motions in the framework of generalized Wiener functionals. More precisely, we show that the submartingale U(B _t –x) is decomposed in the sence of generalized Wiener functionals into the sum of a martingale and the Brownian local time, U being twice of the kernel of Newtonian potential and B _t the multidimensional Brownian motion. We also discuss on an aspect of the Tanaka formula for multidimensional Brownian motions as the Doob–Meyer decomposition.     

The Bar-Radical of a Class of Almost Alternative Baric Algebras

In this article we prove that, if (U, ω) is a finite dimensional baric algebra of (γ, δ) type over a field F of characteristic ≠ 2,3,5 such that γ² ? δ² + δ = 1 and δ ≠ 0,1, then rad(U) = R(U) ∩ (bar(U))², where R(U) is the nilradical (maximal nil ideal) of U.     

Centers of generalized quantum groups

This paper treats the generalized quantum group $U=U(\chi ,\pi )$ with a bi-homomorphism χ for which the corresponding generalized root system is a finite set. We establish a Harish-Chandra type theorem describing the (skew) centers of U.     

Classroom note: Using Platonic bodies to generate a random sample from a population having two characteristics linked with a predetermined coefficient of correlation

This note describes a simple method of generating a random sample of N pairs (U _i ,?W _i ) from a population whose elements have two characteristics U  and W associated with a known coefficient of correlation. Although the method described is extremely advantageous when Platonic bodies are used, it can be generalized to include any discrete uniform distribution.     

On Lie ideals with generalized derivations

Let R be a prime ring with characteristic different from 2, let U be a nonzero Lie ideal of R, and let f be a generalized derivation associated with d. We prove the following results: (i) If a ∈ R and [a, f(U)] = 0 then a ∈ Z or d(a) = 0 or U ? Z; (ii) If f ²(U) = 0 then U ? Z; (iii) If u ² ∈ U for all u ∈ U and f acts as a homomorphism or antihomomorphism on U then either d = 0 or U ? Z.