On ∗-minimal ∗-ideals and ∗-biideals in involution rings

For semiprime involution rings, we determine some ∗-minimal ∗-ideals using idempotent elements. Nevertheless, ∗-minimal ∗-biideals are characterized by idempotent elements. Moreover, the involutive version of a theorem due to Steinfeld, which investigates a semiprime involution ring A if A=SocA, is given. Finally, semiprime involution rings having no proper nonzero ∗-biideals are characterized.     

A conditionally almost linear filtering problem with non-gaussian initial condition ∗

We derive a formula for computing explicitly the optimal non-linear filter for a class of problems which admit finite dimensional filters. The result includes all known results of this kind as special cases     

Numerical ranges and dilations ∗

Denote by W(A) the numerical range of a bounded linear operator A. For two operators A and B (which may act on different Hilbert spaces), we study the relation between the inclusion relation W(A)?W(B) and the condition that A can be dilated to an operator of the form B?I. We also investigate the possibilities of dilating an operator A to operators with simple structure under the assumption that W(A) is included in a special region.     

Tilting modules and ∗-modules

The relation between ?-modules-studied in [MO], [D], [C], [DH], [CM], [Z] and [T]-and Tiltng modules over an arbitrary ring is analyzed. In particular we prove that Tilting modules are exactly the faithful and finendo ?-modules. This answers a question of Trlifaj [T, Problem 1.5], showing that for any ring R the class of ?-modules generating the injectives and that one of Tiltings coincides. As a first application, we give an easy proof of the fact that every faithful ?-module over a finite-dimensional K-algebra is a classical Tilting module (see [DH, Theorem 1]). As a second application, we characterise the Tiltings as those modules which induce an equivalence between two categories with suitable dual properties.     

Beck–Chevalley condition and Goursat categories

We characterise regular Goursat categories through a specific stability property of regular epimorphisms with respect to pullbacks. Under the assumption of the existence of some pushouts this property can be also expressed as a restricted Beck–Chevalley condition, with respect to the fibration of points, for a special class of commutative squares. In the case of varieties of universal algebras these results give, in particular, a structural explanation of the existence of the ternary operations characterising 3-permutable varieties of universal algebras.     

Semialgebraic sets and the Łojasiewicz-Siciak condition

The paper contains a full geometric characterization of compact semialgebraic sets in C satisfying the ?ojasiewicz-Siciak condition. The ?ojasiewicz-Siciak condition is a certain estimate for the Siciak extremal function. In a previous paper, we gave a sufficient criterion for a compact, connected, and semialgebraic set in C to satisfy this condition. In the present paper, we remove completely the connectedness assumption and prove that the aforementioned sufficient condition is also necessary. Moreover, we obtain some new results concerning the ?ojasiewicz-Siciak condition in C^N. For example, we prove that if K₁,...,K_p are compact, nonpluripolar, and pairwise disjoint subsets of C^N, each satisfying the ?ojasiewicz-Siciak condition, and K:= K₁?· · ·?K_p is polynomially convex, then K satisfies this condition as well.     

On discrete solvgroups and Poénaru's condition

We prove that a class of discrete cocompact solvgroups do not fulfill Poénaru's condition P(2) (see [3, 4], improving the result of [2]. This corresponds to the high concavity of balls in the Cayley graphs of such groups.     

Products of symmetric and skew-symmetric Matrices ∗

The class of matrices which can be represented as products of two matrices, each of which is either symmetric or skew-symmetric, is identified. Possible ranks of the factors in such representations of a given matrix are identified as well.     

Group inverse and group involutory Matrices ∗

In this work we deal with group involutory matrices, i.e.A ^#=A. We give necessary and sufficient conditions to characterize these matrices in terms of different representations of the group inverse. First, we give different expressions of the group inverse of a square matrix A. In addition, the special case of integer matrices is considered.     

Piecewise affine functions and polyhedral sets ∗

In this paper we present a number of characterizations of piecewise affine and piecewise linear functions defined on finite dimesional normed vector spaces. In particular we prove that a real-valued function is piecewise affine [resp. piecewise linear] if both its epigraph and its hypograph are (nonconvex) polyhedral sets[resp..Polyhedral cones]. Also,We show that the collection of all piecewise affine[resp.piecewise linear] functions. Furthermore, we prove that a function is piecewise affine[resp.piecewise linear] if it can be represented as a difference of two convex [resp.,sublinear] polyhedral fucntions.     

Schreier theory for singular extensions of categorical groups and homotopy classification∗ ∗

If G is a categorical group, a G-module is defined to be a braided categorical group (A c) together with an action of G on (A,c). In this work we define the notions of singular extension of G by the G-module (A,c) and of 1-cocycle of G with coefficients in (A,c) and we obtain, first, a bijection between the set of equivalence classes of singular extensions of G by (Ac) and the set of equivalence classes of 1-cocycles. Next, we associate to any G-module (Ac) a Kan fibration of simplicial sets ?: Ner(GAc)) → Ner(G)and then we show that there is a bijection between the set of equivalence classes of singular extensions of G by (A,c) and Γ[Ner(G,A,c/Ner(G)]the set of fibre homotopy classes of cross-sections of the fibration ?.     

Lie-admissible algebras and generalized witt algebras ∗

The symmetric group 𝔖_n+1 of degree n + 1 admits an n-dimensional irreducible Q𝔖_n-module V corresponding to the hook partition (2, 1^n?1). By the work of Craig and Plesken, we know that there are σ(n + 1) many isomorphism classes of Z𝔖_n+1-lattices which are rationally equivalent to V, where σ denotes the divisor counting function. In the present article, we explicitly compute the Solomon zeta function of these lattices. As an application we obtain the Solomon zeta function of the Z𝔖_n+1-lattice defined by the Specht basis.     

Hermite and ps-rings of hurwitz series ∗

Let R be a commutative ring and H R the ring of Hurwitz series over R. In this note, we consider some properties of rings, which are shared by R and HR. In particular, we show that for the rings R and H R, if either ring is (i) a Hermite ring, or (ii) a PF-ring in the sense that every finitely generated projective R-module is free, then so is another. We also show that if R is a PS-ring in the sense that the socle Soc( _RR) is projective and char(R) = 0, then H R is also a PS-ring.     

Operational Research Education for Practitioners and Managers∗

The paper reviews the history of education in operational research and indicates the needs both of a practitioner and of a manager. It describes how these can be met by standard course teaching and suggests some areas of development.

     

Weak Graph Map Homotopy and Its Applications∗

The authors introduce a notion of a weak graph map homotopy (they call it M-homotopy), discuss its properties and applications. They prove that the weak graph map homotopy equivalence between graphs coincides with the graph homotopy equivalence defined by Yau et al in 2001. The difference between them is that the weak graph map homotopy transformation is defined in terms of maps, while the graph homotopy transformation is defined by means of combinatorial operations. They discuss its advantages over the graph homotopy transformation. As its applications, they investigate the mapping class group of a graph and the 1-order MP-homotopy group of a pointed simple graph. Moreover, they show that the 1-order MP-homotopy group of a pointed simple graph is invariant up to the weak graph map homotopy equivalence.