Indecomposability and the number of links

Let (?, ?) be a linear matrix problem induced from a finite dimensional algebra ∧. Then an? ×? matrix M in R(?, ?) is indecomposable if and only if the number of links in the canonical formM (∞) of M is equal to. ?-dim? ? 1. On the other hand, the dimension of the endomorphism ring of M is equal to ?-dim? ? σ(M).

CYCCYCLIC VECTORS AND CELLULAR INDECOMPOSABLE OPERATORS ON Q_p SPACES

We identify the functions whose polynomial multiples are weak＊ dense in Qp spaces and prove that if ｜f（z）｜ ≥ ｜g（z）｜ and g is cyclic in Qp, then f is cyclic in Qp. We also show that the multiplication operator Mx on Qp spaces is cellular indecomposable.

SOME RESULTS OF MODULAR LIE SUPERALGEBRAS

In the present article, the authors give some properties on subinvariant subalgebras of modular Lie superalgebras and obtain the derivation tower theorem of modular Lie superalgebras, which is analogous to the automorphism tower theorem of finite groups. Moreover, they announce and prove some results of modular complete Lie superalgebras.

Indecomposable Representations of the Square—Root Lie Algebras of Vector Type and Their Boson Realizations

The explicit expressions for indecomposable representations of nine square-root Lie algebras of vector type, , are obtained on the space of universal enveloping algebra of two-state Heisenberg–Weyl algebra, the invariant subspaces and the quotient spaces. From Fock representations corresponding to these indecomposable representations, the inhomogeneous boson realizations of are given. The expectation values of in the angular momentum coherent states are calculated as well as the corresponding classical limits.

Almost split sequences for symmetric non-semisimple Hopf algebras

We first prove that for a finite dimensional non-semisimple Hopfalgebra H, the trivial H-module is not projective and so the almost split sequence ended with k exists. By this exact sequence, for all indecomposable H-module X, we can construct a special kind of exact sequence ending with it. The main aim of this paper is to determine when this special exact sequence is an almost split one. For this aim, we restrict H to be tmimodular and the square of its antipode to be an inner automorphism. As a special case, we give an application to the quantum double D（H）=（H^op）^＊∞ H） of any non-semisimple Hopf algebra.

EXPLICIT DESCRIPTION OF A CLASS OF INDECOMPOSABLE INJECTIVE MODULES

Let R be a commutative Noetherian ring and p be a prime ideal of R such that the ideal pRp is principal and ht(p)≠0. In this note, the anthors describe the explicit structure of the injective envelope of the R-module R/p.

Two Dimensional Indecomposable Modules over Infinite Dimensional Hereditary Path Algebras

Non-isomorphic two dimensional indecomposable modules over infinite dimensional hereditary path algebras are described. We infer that none of them can be determined by their dimension vectors.

PARTIAL FERRERS MATRICES

We introduce a class of n×n matrices of 0's and l's which can be regarded as generalizations of the classical Ferrets boards of rook theory. Assuming the matrices are fully indecomposable, we determine the minimum permanent and the minimum numberof l's as a function of n. We also characterize these matrices in terms of weighted,top-rooted trees.

THE STRUCTURE OF INVERTIBLE SUBSTITUTIONS ON A THREE-LETTER ALPHABET

We study the structure of invertible substitutions on three-letter alphabet. We show that there exists a finite set S of invertible substitutions such that any invertible substitution can be written as I wσ 1σ 2…σ k,where I w is the inner automorphism associated with w, and σ j∈ S for 1≤j≤k. As a consequence,M is the matrix of an invertible substitution if and only if it is a finite product of non-negative elementary matrices.

The Spectrum of Linear Operators on Quotient Indecomposable Banach Spaces

This paper discusses the special properties of the spectrum of linear operators （in particular, bounded linear operators） on quotient indecomposable Banach spaces; shows that in such spaces generators of C0-groups are always bounded linear operators, and that generators of C0-semigroups satisfy the spectral mapping theorem; and gives an example to show that the generators of C0-semigroups in quotient indecomposable spaces are not necessarily bounded.

RETRACTED: Disintegration of Group Representations on Direct Integrals of Banach Spaces

Short Retraction Notice? The paper does not meet the standards of "Advances in Pure Mathematics".? This article has been retracted to straighten the academic record. In making this decision the Editorial Board follows COPE's Retraction Guidelines. The aim is to promote the circulation of scientific research by offering an ideal research publication platform with due consideration of internationally accepted standards on publication ethics. The Editorial Board would like to extend its sincere apologies for any inconvenience this retraction may have caused.? Editor guiding this retraction: Editorial Board (EIC of APM).? Please see the article page for more details. The full retraction notice in PDF is preceding the original paper which is marked "RETRACTED".

Some Characterizations of Hereditarily Indecomposable Banach Spaces

In this paper we first discuss the relations between some G-M-type spaces, and the previous eight kinds of G-M-type Banach spaces are merged into four different kinds. Then we build a Generalized Operator Extension Theorem, and introduce the concept of complete minimal sequences. Some sufficient and necessary conditions under which a Banach space is a hereditarily indecomposable space are given. Finally, we give some characterizations of hereditarily indecomposable Banach Spaces.

The Cartan Invariant Matrix for the Finite Group G₂(13) of Type G₂

The Cartan invariant matrixfor the finite group G2(13) of type G2 over the finite field with 13 elements is determined in this paper.

Annihilator Ideals of Indecomposable Modules of Finite-Dimensional Pointed Hopf Algebras of Rank One

Let H be a finite-dimensional pointed Hopf algebra of rank one over an algebraically closed feld of characteristic zero.We describe all annihilator ideals of indecomposable H-modules by generators.In particular,we give the classification of all ideals of the finite-dimensional pointed rank one Hopf algebra of nilpotent type over the Klein 4-group.

PBW-Basis of Reduced Drinfeld Double Hall Algebra of Type An via Grobner-Shirshov Basis

In the Ringel-Hall algebra of Dynkin type,the set of all commutator relations between the isoclasses of indecomposable representations forms a minimal Grobner-Shirshov basis and the set of the corresponding irreducible elements forms a PBW-type basis of the Ringel-Hall algebra.We aim to generalize this result to the reduced Drinfeld double Hall algebra of type A_(n).First,we compute a minimal Grobner-Shirshov basis for the reduced Drinfeld double Hall algebra of type An by proving that all possible compositions between the commutator relations are trivial.Then,by taking the corresponding irreducible monomials,we construct a PBW-type basis for the reduced Drinfeld double Hall algebra of type A_(n).

Further study on indecomposable cryptographic functions

1 Introduction Boolean functions have important applications in stream ciphers and block ciphers.Over the last decades,the constructions of cryptographic Boolean functions have paid a lot of attention[1,2].Direct sum is a well-known secondary construction of cryptographic functions[3].By using the direct sum,a lot of functions with high nonlinearities can be obtained[4,5].However,the direct sum of two functions are decomposable functions,which have numerous null secondorder derivatives(which represents a potential weakness with respect to the higher order differential attack)[6].(In)decomposable functions were also studied in[7]by Zheng and Zhang under the name(non)separable functions.They provided some sufficient conditions that the functions are indecomposable[7].

Several Transitive Properties and Devaney's Chaos

The relation among transitivity, indecomposability and Z-transitivity is discussed. It is shown that for a non-wandering system （each point is non-wandering）, indecomposability is equivalent to transitivity, and for the dynamical systems without isolated points, Z-transitivity and transitivity are equivalent. Besides, a new transitive level as weak transitivity is introduced and some equivalent conditions of Devaney＇s chaos are given by weak transitivity. Moreover, it is proved that both d- shadowing property and d-shadowing property imply weak transitivity.

Gowers-Maurey's space and its conjugate space

RECENTLY Gowers and Maurey constructed the first example of Banach space containing no unconditional basic sequence. We denote this space by X_G, in this note. Using the results in ref. [1], some further studies and reconstructions of this space result in some satisfactory answers of a series of open questions in the Banach spaces theory. There is a general description about this remarkable development. Just as indicated in ref. [1], the most important characteristic of the Banach space X_G

Grbner-Shirshov Basis of Quantum Group of Type D_4

The authors take all isomorphism classes of indecomposable representations as new generators, and obtain all skew-commutators between these generators by using the Ringel-Hall algebra method. Then they prove that the set of these skew-commutators is a GrSbner-Shirshov basis for quantum group of type D4.

Skew-commutator relations and Grobner- Shirshov basis of quantum group of type F4

We give a Grobner-Shirshov basis of quantum group of type F4 by using the Ringel-Hall algebra approach. We compute all skew-commutator relations between the isoclasses of indecomposable representations of Ringel- Hall algebras of type F4 by using an ＇inductive＇ method. Precisely, we do not use the traditional way of computing the skew-commutative relations, that is first compute all Hall polynomials then compute the corresponding skew- commutator relations; instead, we compute the ＇easier＇ skew-commutator relations which correspond to those exact sequences with middle term indecomposable or the split exact sequences first, then ＇deduce＇ others from these ＇easier＇ ones and this in turn gives Hall polynomials as a byproduct. Then using the composition-diamond lemma prove that the set of these relations constitute a minimal CrSbner-Shirshov basis of the positive part of the quantum group of type F4. Dually, we get a Grobner-Shirshov basis of the negative part of the quantum group of type F4. And finally, we give a Gr6bner-Shirshov basis for the whole quantum group of type F4.