Some properties of c-covers of a pair of Lie algebras

In [H. Safa and H. Arabyani, On c-nilpotent multiplier and c-covers of a pair of Lie algebras, Commun. Algebra 45(10) (2017), 4429–4434], we characterized the structure of the c-nilpotent multiplier of a pair of Lie algebras in terms of its c-covering pairs and discussed some results on the existence of c-covers of a pair of Lie algebras. In the present paper, it is shown under some conditions that a relative c-central extension of a pair of Lie algebras is a homomorphic image of a c-covering pair. Moreover, we prove that a c-cover of a pair of finite dimensional Lie algebras, under some assumptions, has a unique domain up to isomorphism and also that every perfect pair of Lie algebras admits at least one c-cover. Finally, we discuss a result concerning the isoclinism of c-covering pairs.     

Extensions of Isometric Dual Representations of Semigroups

Let T be a dual representation of a suitable subsemigroup Sof a locally compact abelian group G by isometries on a dualBanach space X=(X_*)^*. It is shown that (X, T) can be extendedto a dual representation of G on a dual Banach space Y containingX, and that this extension can be done in a canonical way. Inthe case of a representation by *-monomorphisms of a von Neumannalgebra, the extension is a representation of G by *-automorphismsof a von Neumann algebra.     

A characterization of p-bases of rings of constants

We obtain two equivalent conditions for m polynomials in n variables to form a p-basis of a ring of constants of some polynomial K-derivation, where K is a unique factorization domain of characteristic p > 0. One of these conditions involves Jacobians while the other some properties of factors. In the case m = n this extends the known theorem of Nousiainen, and we obtain a new formulation of the Jacobian conjecture in positive characteristic.     

On divisors of semigroups of order-preserving mappings of a finite chain

We show that if a semigroup T divides a semigroup of full order preserving transformations of a finite chain, then so does any semidirect product S⋊T where S is a finite semilattice whose natural order makes S a chain.     

Characterizations of optimal scalings of matrices

A scaling of a non-negative, square matrixA ≠ 0 is a matrix of the formDAD ^?1, whereD is a non-negative, non-singular, diagonal, square matrix. For a non-negative, rectangular matrixB ≠ 0 we define a scaling to be a matrixCBE ^?1 whereC andE are non-negative, non-singular, diagonal, square matrices of the corresponding dimension. (For square matrices the latter definition allows more scalings.) A measure of the goodness of a scalingX is the maximal ratio of non-zero elements ofX. We characterize the minimal value of this measure over the set of all scalings of a given matrix. This is obtained in terms of cyclic products associated with a graph corresponding to the matrix. Our analysis is based on converting the scaling problem into a linear program. We then characterize the extreme points of the polytope which occurs in the linear program.     

Contractions of infrainvariant systems of subgroups

We create a method which allows an arbitrary group G with an infrainvariant system ℒ(G) of subgroups to be embedded in a group G* with an infrainvariant system ℒ(G*) of subgroups, so that G _α^* ∩G ∈ ℒ(G) for every subgroup G _α^* ∩G ∈ ℒ(G*) and each factor B/A of a jump of subgroups in ℒ(G*) is isomorphic to a factor of a jump in ℒ(G), or to any specified group H. Using this method, we state new results on right-ordered groups. In particular, it is proved that every Conrad right-ordered group is embedded with preservation of order in a Conrad right-ordered group of Hahn type (i.e., a right-ordered group whose factors of jumps of convex subgroups are order isomorphic to the additive group ℝ); every right-ordered Smirnov group is embedded in a right-ordered Smirnov group of Hahn type; a new proof is given for the Holland–McCleary theorem on embedding every linearly ordered group in a linearly ordered group of Hahn type.     

Dichotomies of the set of test measures of a Haar-null set

We prove that ifX is a Polish space andF a face ofP(X) with the Baire property, thenF is either a meager or a co-meager subset ofP(X). As a consequence we show that for every abelian Polish groupX and every analytic Haar-null set Λ⊆X, the set of test measuresT(Λ) of Λ is either meager or co-meager. We characterize the non-locally-compact groups as the ones for which there exists a closed Haar-null setF⊆X withT(F) meager, Moreover, we answer negatively a question of J. Mycielski by showing that for every non-locally-compact abelian Polish group and every σ-compact subgroupG ofX there exists aG-invariantF _σ subset ofX which is neither prevalent nor Haar-null. Research supported by a grant of EPEAEK program “Pythagoras”.     

Picard Groups of Rings of Coinvariants

Let k be a field, H a Hopf k-algebra with bijective antipode, A a right H-comodule algebra and C a Hopf algebra with bijective antipode which is also a right H-module coalgebra. Under some appropriate assumptions, and assuming that the set of grouplike elements G(A ⊗ C) of the coring A ⊗ C is a group, we show how to calculate, via an exact sequence, the Picard group of the subring of coinvariants in terms of the Picard group of A and various subgroups of G(A ⊗ C). Presented by: Claus Ringel.     

Quasi-Permutation Representations of p-Groups of Class 2

If G is a finite linear group of degree n, that is, a finitegroup of automorphisms of an n-dimensional complex vector space(or, equivalently, a finite group of non-singular matrices oforder n with complex coefficients), we shall say that G is aquasi-permutation group if the trace of every element of G isa non-negative rational integer. The reason for this terminologyis that, if G is a permutation group of degree n, its elements,considered as acting on the elements of a basis of an n-dimensionalcomplex vector space V, induce automorphisms of V forming agroup isomorphic to G. The trace of the automorphism correspondingto an element x of G is equal to the number of letters leftfixed by x, and so is a non-negative integer. Thus, a permutationgroup of degree n has a representation as a quasi-permutationgroup of degree n. See [8].     

Periodicity of the Spectrum of a Finite Union of Intervals

A set Ω, of Lebesgue measure 1, in the real line is called spectral if there is a set Λ of real numbers such that the exponential functions e _λ (x)=exp (2πiλx), λ∈Λ, form a complete orthonormal system on L ²(Ω). Such a set Λ is called a spectrum of Ω. In this note we present a simplified proof of the fact that any spectrum Λ of a set Ω which is finite union of intervals must be periodic. The original proof is due to Bose and Madan.     

Derived Equivalences of Actions of a Category

Let $\mathbb{k}$ be a commutative ring and I a category. As a generalization of a $\mathbb{k}$ -category with a (pseudo) action of a group we consider a family of $\mathbb{k}$ -categories with a (pseudo, lax, or oplax) action of I, namely an oplax functor from I to the 2-category of small $\mathbb{k}$ -categories. We investigate derived equivalences of those oplax functors, and establish a Morita type theorem for them. This gives a base of investigations of derived equivalences of Grothendieck constructions of those oplax functors.     

Products of operators of multidimensional structured model of systems

The author has previously defined the concept of a general system in terms of operators and operands. An operand is a mapping defined on a subset of an m-fold Cartesian product instead of the usual set and collection of k-ary relations on it. An operator is a kind of mapping between two collections of operands. Here subsystems, extensions, and the notion of P-semiexactness is studied. In particular we derive conditions such that P-semiexactness of a composition of operators, and of one factor, implies P-semiexactness of the other factor.     

On the stability of invariant sets of leaves of three-dimensional periodic systems

The small C ¹ perturbations of differential equations are studied. The concepts of a weakly hyperbolic set K and a leaf ? are introduced for a system of ordinary differential equations. The Lipschitz condition is not supposed. It is shown that, if the perturbation is small enough, then there exists a continuous mapping h: ? → ? ^Y , where ? ^Y is a leaf of the perturbed system.     

Extensions and generalizations of a theorem of Widder and of the theory of symmetric local semigroups

The theory of symmetric local semigroups due to A. Klein and L. Landau (J. Funct. Anal.44 (1981), 121–136) is generalized to semigroups indexed by subsets of $\text{R}$ⁿ for n > 1. The result implies a similar result of A. E. Nussbaum (J. Funct. Anal.48 (1982), 213–223). It is further generalized to semigroups that are symmetric local in some directions and unitary in others. The results are used to give a simple proof of A. Devinatz's (Duke Math. J.22 (1955), 185–192) and N. I. Akhiezer's (“the Classical Moment Problem and Some Related Questions,” Hafner, New York, 1965) generalization of a theorem of Widder concerning the representation of functions as Laplace integrals. This result is extended to the representation as a Laplace integral of a function taking values in $\text{B}$($\text{R}$), the set of bounded linear operators on a Hilbert space $\text{R}$. Also, a theorem is proved encompassing both the result of Devinatz and Akhiezer, and Bochner's theorem on the representation of positive definite functions as Fourier integrals.     

Existence of normal Hall subgroups by means of orders of products

Let G be a finite group, let π be a set of primes and let p be a prime. We characterize the existence of a normal Hall π‐subgroup in G in terms of the order of products of certain elements of G. This theorem generalizes a characterization of A. Moretó and the second author by using the orders of products of elements for those groups having a normal Sylow p‐subgroup 6 . As a consequence, we also give a π‐decomposability criterion for a finite group also by means of the orders of products.     

On the generation of bicliques of a graph

An independent set of a graph is a subset of pairwise non-adjacent vertices. A complete bipartite set B is a subset of vertices admitting a bipartition B=X∪Y, such that both X and Y are independent sets, and all vertices of X are adjacent to those of Y. If both X,Y≠∅, then B is called proper. A biclique is a maximal proper complete bipartite set of a graph. When the requirement that X and Y are independent sets of G is dropped, we have a non-induced biclique. We show that it is NP-complete to test whether a subset of the vertices of a graph is part of a biclique. We propose an algorithm that generates all non-induced bicliques of a graph. In addition, we propose specialized efficient algorithms for generating the bicliques of special classes of graphs.     

Filtration of components of processes of random evolution

The problem of estimation of a nonobservable component θ_t for a two-dimensional process (θ_t, ξ_t) of random evolution (θ _t,ξ_t);x_t, 0≤t≤T, is investigated on the basis of observations of ξ_s. s≤t, where x _t is a homogeneous Markov process with infinitesimal operator Q. Applications to stochastic models of a (B,S)-market of securities is described under conditions of incomplete market. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1701–1705, December, 1998.     

Submodules of Direct Sums of Compressible Modules

Let R be a ring. A right R-module M is called “essentially compressible” if it embeds in each of its essential submodules. Also a module X _R is called “completely essentially compressible” if every submodule of X _R is an essentially compressible R-module. In this aricle, it is shown that a right R-module M embeds in a direct sum of compressible right R-modules if and only if M _R is essentially compressible and every nonzero essentially compressible submodule of M _R contains a compressible submodule. Every essentially compressible R-module is shown to be retractable. Moreover, if either R _R has Krull dimension, or R is Morita equivalent to a right duo ring, then a right R-module embeds in a direct sum of compressible right R-modules if and only if it is completely essentially compressible.     

Eigenvector p-Bases of Rings of Constants of Derivations

Let A be a UFD of characteristic p > 0, let 𝒵 be a set of some eigenvectors of a derivation of A. We prove, under some additional assumptions, a necessary and sufficient condition for 𝒵 to be a p-basis of the minimal ring of constants containing 𝒵. The main preparatory result is the unique decomposition theorem with respect to a factor from a given subalgebra containing A^p.     

Nilpotent Ranks of Semigroups of Partial Transformations

A subset U of a semigroup S is a generating set for S if every element of S may be written as a finite product of elements of U. The rank of a finite semigroup S is the size of a minimal generating set of S, and the nilpotent rank of S is the size of a minimal generating set of S consisting of nilpotents in S. A partition of a q-element subset of the set X_n = {1,2,..., n} is said to be of type τ if the sizes of its classes form the partition τ of the positive integer q ≤ n. A non-trivial partition τ of q consists of k < q elements. For a non-trivial partition τ of q < n, the semigroup S(τ), generated by all the transformations with kernels of type τ, is nilpotent-generated. We prove that if τ is a non-trivial partition of q < n, then the rank and the nilpotent rank of S(τ) are both equal to the number of partitions X_n of type τ.