Finite Extensions of Topogenities

Let X = Y Z, Y Z = Ø, < be a topogenity on Y, a topology on X. A (<, )-extension is a topogenity < on X such that < ¦Y = <, (<) = . We establish some properties of (<, )-extensions and construct all of them in the case of a finite Z.     

Simultaneous Extensions Revisited

After an introduction presenting the problem of simultaneous extensions, some concrete examples show the difficulties in examining this problem.     

Special Simultaneous Quasi-uniform Extensions

A necessary and sufficient condition is presented for the existence of a quasi-uniformity for a prescribed topology, such that the quasi-uniformity is transitive, doubly uniformly strict and uniformly regular, respectively.     

Separation Properties of Simultaneous Topological Extensions

Let X be a set and s i( x ) be a (possibly zero) filter in X_i X for i I and x X . A topology pi on X is said to be a simultaneous extension compatible with...     

Simultaneous Surface Resolution in Cyclic Galois Extensions

We show that simultaneous surface resolution is not always possible in a cyclic extension whose degree is greater than three and is not divisible by the characteristic. This answers a recent question of Ted Chinburg.

     

Simultaneous Extensions of Topologies Through Traces of Neighbourhood Filters

Let X be a set, X _i X for i I and, for x X, a filter in X _i. The paper gives necessary and sufficient conditions for the existence of a topology on X compatible with , i.e. such that is the trace on X _i of the -neighbourhood filter of x. It is shown that, among these compatible topologies,there are always a coarsest one and a finest one. Some applications are also given.     

Field Extensions and Extensions of Linear Differential Operators

Let K R P be a tower of fields, N be a P-module, and : R N be a K-linear differential operator. The aim of this paper is to investigate whether the operator has an extension to P, i.e. if these exists a differential operator : P N such that |_R = . The results of this paper were published in Russian in Mat. Zametki 30(2) (1981), 237–248.     

Double Ore Extensions Versus Iterated Ore Extensions

Motivated by the construction of new examples of Artin–Schelter regular algebras of global dimension four, Zhang and Zhang [6 Zhang , J. J. , Zhang , J. ( 2008 ). Double Ore extensions . J. Pure Appl. Algebra 212 ( 12 ): 2668 – 2690 .[Crossref], [Web of Science ®] , [Google Scholar]] introduced an algebra extension A _P [y ₁, y ₂; σ, δ, τ] of A, which they called a double Ore extension. This construction seems to be similar to that of a two-step iterated Ore extension over A. The aim of this article is to describe those double Ore extensions which can be presented as iterated Ore extensions of the form A[y ₁; σ₁, δ₁][y ₂; σ₂, δ₂]. We also give partial answers to some questions posed in Zhang and Zhang [6 Zhang , J. J. , Zhang , J. ( 2008 ). Double Ore extensions . J. Pure Appl. Algebra 212 ( 12 ): 2668 – 2690 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Clifford Extensions

Formulating the construction of Clifford algebras, we introduce the notion of Clifford extensions and show that Clifford extensions are Frobenius extensions. Consequently, Clifford extensions of Auslander–Gorenstein rings are Auslander–Gorenstein rings.     

Convex-Concave Extensions

In this paper we introduce a new notion which we call convex-concave extensions. Convex-concave extensions provide for given nonlinear functions convex lower bound functions and concave upper bound functions, and can be viewed as a generalization of interval extensions. Convex-concave extensions can approximate the shape of a given function in a better way than interval extensions which deliver only constant lower and upper bounds for the range. Therefore, convex-concave extensions can be applied in a more flexible manner. For example, they can be used to construct convex relaxations. Moreover, it is demonstrated that in many cases the overestimation which is due to interval extensions can be drastically reduced. Applications and some numerical examples, including constrained global optimization problems of large scale, are presented.     

Zero-Dimensional Principal Extensions

We prove that every topological dynamical system (X,T) has a zero-dimensional principal extension, i.e. a zero-dimensional extension (Y,S) such that for every S-invariant measure ν on Y the conditional entropy h(ν|X) is zero. This reduces the discussion of many entropy-related properties to the zero-dimensional case which gives access to the various useful tools of symbolic dynamics.     

t-Pebbling and Extensions

Graph pebbling is the study of moving discrete pebbles from certain initial distributions on the vertices of a graph to various target distributions via pebbling moves. A pebbling move removes two pebbles from a vertex and places one pebble on one of its neighbors (losing the other as a toll). For t ≥ 1 the t-pebbling number of a graph is the minimum number of pebbles necessary so that from any initial distribution of them it is possible to move t pebbles to any vertex. We provide the best possible upper bound on the t-pebbling number of a diameter two graph, proving a conjecture of Curtis et al., in the process. We also give a linear time (in the number of edges) algorithm to t-pebble such graphs, as well as a quartic time (in the number of vertices) algorithm to compute the pebbling number of such graphs, improving the best known result of Bekmetjev and Cusack. Furthermore, we show that, for complete graphs, cycles, trees, and cubes, we can allow the target to be any distribution of t pebbles without increasing the corresponding t-pebbling numbers; we conjecture that this behavior holds for all graphs. Finally, we explore fractional and optimal fractional versions of pebbling, proving the fractional pebbling number conjecture of Hurlbert and using linear optimization to reveal results on the optimal fractional pebbling number of vertex-transitive graphs.     

Covariant Projective Extensions

The theory of crossed products of C~*-algebras by groups of automorphisms is a well-developed area of the theory of operator algebras. Given the importance and the success ofthat theory, it is natural to attempt to extend it to a more general situation by, for example,developing a theory of crossed products of C~*-algebras by semigroups of automorphisms, or evenof endomorphisms. Indeed, in recent years a number of papers have appeared that are concernedwith such non-classical theories of covariance algebras, see, for instance [1-3].     

Projective Extensions of Fields

Every field K admits proper projective extensions, that is,Galois extensions where the Galois group is a non-trivial projectivegroup, unless K is separably closed or K is a pythagorean formallyreal field without cyclic extensions of odd degree. As a consequence,it turns out that almost all absolute Galois groups decomposeas proper semidirect products. We show that each local field has a unique maximal projectiveextension, and that the same holds for each global field ofpositive characteristic. In characteristic 0, we prove thatLeopoldt's conjecture for all totally real number fields isequivalent to the statement that, for all totally real numberfields, all projective extensions are cyclotomic. So the realizabilityof any non-procyclic projective group as Galois group over Qproduces counterexamples to the Leopoldt conjecture.