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.     

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.     

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.     

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.     

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.     

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.     

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.     

Embeddings of quasicells of iterative algebras

We study the relation between the projective and totally restricted extensions of preiterative algebras. We prove that each degree 1 projective extension of a quasicell of the algebra P _k ^* is a maximal subalgebra of a degree 1 totally restricted extension of the same quasicell. We show also that a projective extension of a quasicell can always be distinguished from its totally restricted extension in the same algebra by hyperidentities.     

Properties of Standard n-Ideals of a Lattice

An n-ideal of a lattice L is a convex sublattice containing a fixed element n L and it is called standard if it is a standard element of the lattice of n-ideals I_n(L). In this paper we have shown that, for a neutral element n of a lattice L, the principal n-ideal a_n of a lattice L is a standard n-ideal if and only if a n is standard and a n is dual standard. We have also shown that if n is a neutral element and (n] and [n) are relatively complemented, then every homomorphism n-kernels of L is a standard n-ideal and every standard n-ideal is the n-kernel of precisely one congruence relation. Finally, we have shown that, for a relatively complemented lattice L with 0 and 1, C(L) is a Boolean algebra if and only if every standard n-ideal of L is a principal n-ideal.AMS Subject Classification (2000) 06B10 06B99 06C15     

Rigidity of harmonic maps of maximum rank

Thehomotopical rank of a mapf:M →N is, by definition, min{dimg(M) ¦g homotopic tof}. We give upper bounds for this invariant whenM is compact Kähler andN is a compact discrete quotient of a classical symmetric space, e.g., the space of positive definite matrices. In many cases the upper bound is sharp and is attained by geodesic immersions of locally hermitian symmetric spaces. An example is constructed (Section 9) to show that there do, in addition, exist harmonic maps of quite a different character. A byproduct is construction of an algebraic surface with large and interesting fundamental group. Finally, a criterion for lifting harmonic maps to holomorphic ones is given, as is a factorization theorem for representations of the fundamental group of a compact Kähler manifold. The technique for the main result is a combination of harmonic map theory, algebra, and combinatorics; it follows the path pioneered by Siu in his ridigity theorem and later extended by Sampson.     

Enumeration of perfect matchings of a type of Cartesian products of graphs

Let G be a graph and let Pm(G) denote the number of perfect matchings of G.We denote the path with m vertices by P_m and the Cartesian product of graphs G and H by G×H. In this paper, as the continuance of our paper [W. Yan, F. Zhang, Enumeration of perfect matchings of graphs with reflective symmetry by Pfaffians, Adv. Appl. Math. 32 (2004) 175-188], we enumerate perfect matchings in a type of Cartesian products of graphs by the Pfaffian method, which was discovered by Kasteleyn. Here are some of our results:1. Let T be a tree and let C_n denote the cycle with n vertices. Then Pm(C₄×T)=∏(2+α²), where the product ranges over all eigenvalues α of T. Moreover, we prove that Pm(C₄×T) is always a square or double a square.2. Let T be a tree. Then Pm(P₄×T)=∏(1+3α²+α⁴), where the product ranges over all non-negative eigenvalues α of T.3. Let T be a tree with a perfect matching. Then Pm(P₃×T)=∏(2+α²), where the product ranges over all positive eigenvalues α of T. Moreover, we prove that Pm(C₄×T)=[Pm(P₃×T)]².     

Depths of Powers of the Edge Ideal of a Tree

Lower bounds are given for the depths of R/I ^t for t ≥ 1 when I is the edge ideal of a tree or forest. The bounds are given in terms of the diameter of the tree, or in case of a forest, the largest diameter of a connected component and the number of connected components. These lower bounds provide a lower bound on the power for which the depths stabilize.     

The group of homotopy equivalences of products of spheres and of Lie groups

We investigate the group of self homotopy equivalences of a space X which induce the identity homomorphism on all homotopy groups. We obtain results on the structure of provided the p-localization of X has the homotopy type of a p-local product of odd-dimensional spheres. In particular, we show that is a semidirect product of certain homotopy groups . We also show that has a central series whose successive quotients are , which are direct sums of homotopy groups of p-local spheres. This leads to a determination of the order of the p-torsion subgroup of and an upper bound for its p-exponent. These results apply to any Lie group G at a regular prime p. We derive some general properties of and give numerous explicit calculations. Received: 14 April 2001; in final form: 10 September 2001 / Published online: 17 June 2002     

On Morrey’s estimate of the Sobolev norms of solutions of elliptic equations

We give a complete proof of Morrey’s estimate for the W ^1,p -norm of a solution of a second-order elliptic equation on a domain in terms of the L ₁-norm of this solution. The dependence of the constant in this estimate on the coefficients of the equation is investigated.     

Prime Lie Rings of Derivations of Commutative Rings

Let R be a 2-torsion free commutative ring with identity, and δ a nonzero derivation of R such that R is δ-prime. Then Rδ is a prime Lie ring and any nonzero ideal of Rδ contains an ideal of the form Jδ where J is a nonzero δ-ideal of R.     

Another type of dimensions of modules and rings

Abstract

We say that a class Q of left R-modules is a monic class if a nonzero submodule of a module in Q is also a module in Q. For a monic class Q, we define a Q-dimension of modules that measures how far modules are from the modules in Q. For a monic class Q of indecomposable modules we characterize rings whose modules have Q-dimension. We prove that for an artinian principal ideal ring the Q-dimension coincides with the uniserial dimension. We also characterize when every module has Q-dimension.     

Sub-Riemannian geometry of the coefficients of univalent functions

We consider coefficient bodies Mn for univalent functions. Based on the Löwner-Kufarev parametric representation we get a partially integrable Hamiltonian system in which the first integrals are Kirillov's operators for a representation of the Virasoro algebra. Then Mn are defined as sub-Riemannian manifolds. Given a Lie-Poisson bracket they form a grading of subspaces with the first subspace as a bracket-generating distribution of complex dimension two. With this sub-Riemannian structure we construct a new Hamiltonian system to calculate regular geodesics which turn to be horizontal. Lagrangian formulation is also given in the particular case M₃.     

Weighted scattering matrices of regular coverings of graphs

We give a decomposition formula for the determinant det(I ? U(λ)) of the weighted bond scattering matrix U(λ) of a regular covering of G. Furthermore, we define an L-function of G, and give a determinant expression of it. As a corollary, we express some determinant of the weighted bond scattering matrix of a regular covering of G by means of its L-functions.     

Universal Overconvergence of Polynomial Expansions of Harmonic Functions

For each compact subset K of ^N let (K) denote the space of functions that are harmonic on some neighbourhood of K. The space (K) is equipped with the topology of uniform convergence on K. Let Ω be an open subset of ^N such that 0Ω and ^N\Ω is connected. It is shown that there exists a series ∑H_n, where H_n is a homogeneous harmonic polynomial of degree n on ^N, such that (i) ∑H_n converges on some ball of centre 0 to a function that is continuous on Ω and harmonic on Ω, (ii) the partial sums of ∑H_n are dense in (K) for every compact subset K of ^N\Ω with connected complement. Some refinements are given and our results are compared with an analogous theorem concerning overconvergence of power series.     

Properties of Hamilton cycles of circuit graphs of matroids

Let G be a circuit graph of a connected matroid. P. Li and G. Liu [Comput. Math. Appl., 2008, 55: 654–659] proved that G has a Hamilton cycle including e and another Hamilton cycle excluding e for any edge e of G if G has at least four vertices. This paper proves that G has a Hamilton cycle including e and excluding e′ for any two edges e and e′ of G if G has at least five vertices. This result is best possible in some sense. An open problem is proposed in the end of this paper.