Companion <Emphasis Type="Italic">d</Emphasis>-algebras

In this paper we develop a theory of companion d-algebras in sufficient detail to demonstrate considerable parallelism with the theory of BCK-algebras as well as obtaining a collection of results of a novel type. Included among the latter are results on certain natural posets associated with companion d-algebras as well as constructions on Bin(X), the collection of binary operations on the set X, which permit construction of new companion d-algebras from companion d-algebras X also in natural ways. Supported by Korea Research Foundation Grant (KRF-2002-041-C00003).     

Zn[i] 的平方映射图

本文研究了模n 高斯整数环Z_n[i] 的平方映射图Γ(n). 利用数论、图论与群论等方法, 获得了Γ(n) 中顶点0 及1 的入度, 并研究了Γ(n) 的零因子子图的半正则性. 同时, 获得了Γ(n) 中顶点的高度公式.推广了Somer 等人给出的模n 剩余类环平方映射图的相关结论.     

Controlling the space with preassigned responses

It is shown that large classes of control systems, which include certain systems of the typex+A(t)x=B(t)u, can be handled in such a way that the control functionsu(t) are actually associated with responsesx(t) that belong to known families of functions. In particular, it is possible, for a variety of perturbationsB(t)u and operatorsA(t) with convex domains, to have responses that are line segments joining the origin to the reachable states.The present approach establishes the fact that a vast number of results from functional analysis concerning ranges of operators can be effectively applied to the general theory of control. It is also rather significant that the present theory does not necessarily require the solvability of the associated Cauchy problem.The operatorsB(t)u do not have to be invertible inu. However, it is shown that continuous controlsu(t) can be obtained for a variety of problems whenB ^–1(t)u exists and is continuous int.     

<Emphasis Type="Italic">ω</Emphasis>-Convergence theory of filters in <Emphasis Type="Italic">Lω</Emphasis> -spaces

In this paper, an ω-convergence theory of filters in an Lω-space is established. By means of the ω-convergence theory, some important characterizations with respective to the ω-closed sets, ω T ₂ separation and (ω ₁, ω ₂)-continuous mappings are obtained. Moreover, the mutual relationships among ω-convergence of molecular nets, ω-convergence of ideals and ω-convergence of filters are given.     

A unified theory of R-continuity

We introduce and investigate R-M-continuous functions defined between sets satisfying some minimal conditions. The functions enable us to formulate a unified theory of modifications of R-continuity [22]: R-irresoluteness [6], R-preirresoluteness [7].      

Subcategories of Filter Tower Spaces

The concept of a convergence tower space, or equivalently, a convergence approach space is formulated here in the context of a Cauchy setting in order to include a completion theory. Subcategories of filter tower spaces are defined in terms of axioms involving a general t-norm, T, in order to include a broad range of spaces. A T-regular sequence for a filter tower space is defined and, moreover, it is shown that the category of T-regular objects is a bireflective subcategory of all filter tower spaces. A completion theory for subcategories of filter tower spaces is given.     

A remark on congruences for coefficients of Faber polynomials

Faber polynomials appear when weight zero Hecke operators act on the modular j-invariant. They are polynomials in j with rational integral coefficents. Using the theory of p-adic modular forms we establish some congruences and divisibilities for these coefficients. 2000 Mathematics Subject Classification Primary—11F33 Supported by NSF grant DMS-0501225.     

Pseudo-differential operators with operator-valued symbols in the Mellin-edge-approach

The pseudo-differential Mellin-edge-approach is a tool for studying differential and pseudo-differential operators on manifolds with corners. The Mellin transform, acting on the corner axis ₊, is a substitute for the Fourier transform along edge variables in the calculus of wedge pseudo-differential operators. The basic elements of that theory (cf. Schulze [6,8]) are extended to edges like ₊ t with a control of symbols and smoothing operators near the vertext=0. The authors study the weighted Mellin wedge Sobolev spaces, the operator-valued Mellin convention translating Fourier symbols into Mellin ones under preserved smoothness up tot=0, and develop an operator calculus with its characterization on the level of symbols. Throughout the theory, there are involved one-parameter groups of isomorphisms acting on the Banach spaces that are the abstract analogues of the weighted cone Sobolev spaces.     

On the computation of the projective surgery obstruction groups

The computation of the projective surgery obstruction groupsL _n ^p (ZG), forG a hyperelementary finite group, is reduced to standard calculations in number theory, mostly involving class groups. Both the exponent of the torsion subgroup and the precise divisibility of the signatures are determined. ForG a 2-hyperelementary group, theL _n ^p (ZG) are detected by restriction to certain subquotients ofG, and a complete set of invariants is given for oriented surgery obstructions.Partially supported by NSERC grant A4000.Partially supported by NSF grant DMS-8610730(1) and the Danish Research Council.     

Semigroups of linear operators on p-Fréchet spaces, 0 < p < 1

We develop the beginning of a theory of semigroups of linear operators on p-Fréchet spaces, 0 < p < 1 (which are non-locally convex F-spaces), and give some applications.     

The Algebraic K-Theory Spectrum of a 2-Adic Local Field

We explicitly determine the homotopy type of the 2-completed algebraic K-theory spectrum KF, where F is an arbitrary finite extension of the 2-adic rational numbers. The answer is formulated in terms of topological complex K-theory and the K-theory of suitable finite fields, suspended copies of which are glued together by connecting maps that depend on the Iwasawa theory of F.     

Some complete metrics on spaces of fuzzy subsets

The classes of the L_p,∞- and L_p-metrics play an important role to develop a probability theory in fuzzy sample spaces. All of these metrics are known to be separable, but not complete. The classes are closely related as for each L_p,∞-metric there exists some L_p-metric which induces the same topology. This paper deals with the completion of the L_p,∞- and L_p-metrics. We can also show that the relationship between the classes of L_p,∞- and L_p-metrics still holds for the obtained respective classes of their completions.     

Nonlinear Dynamics Equation in p-Adic String Theory

We investigate nonlinear pseudodifferential equations with infinitely many derivatives. These are equations of a new class, and they originally appeared in p-adic string theory. Their investigation is of interest in mathematical physics and its applications, in particular, in string theory and cosmology. We undertake a systematic mathematical investigation of the properties of these equations and prove the main uniqueness theorem for the solution in an algebra of generalized functions. We discuss boundary problems for bounded solutions and prove the existence theorem for spatially homogeneous solutions for odd p. For even p, we prove the absence of a continuous nonnegative solution interpolating between two vacuums and indicate the possible existence of discontinuous solutions. We also consider the multidimensional equation and discuss soliton and q-brane solutions.     

Lp-Approximable Sequences of Vectors and Limit Distribution of Quadratic Forms of Random Variables

The properties of L₂-approximable sequences established here form a complete toolkit for statistical results concerning weighted sums of random variables, where the weights are nonstochastic sequences approximated in some sense by square-integrable functions and the random variables are “two-wing” averages of martingale differences. The results constitute the first significant advancement in the theory of L₂-approximable sequences since 1976 when Moussatat introduced a narrower notion of L₂-generated sequences. The method relies on a study of certain linear operators in the spaces L_p and l_p. A criterion of L_p-approximability is given. The results are new even when the weight generating function is identically 1. A central limit theorem for quadratic forms of random variables illustrates the method.     

A remark on complete positivity of elementary operators

We show that many of the features of the theory of hypercyclic and supercyclic operators extend to that of finitely hypercyclic/supercyclic operators. In particular, subnormal operators, Banach space isometries, and thereforeC ₁ contractions are not finitely supercyclic.     

Homotopy Theory of A∞ Ring Spectra and Applications to MU-Modules

We give a definition of a derivation of an A ring spectrum and relate this notion to topological Hochschild cohomology. Strict multiplicative structure is introduced into Postnikov towers and generalized Adams towers of A -ring spectra. An obstruction theory for lifting multiplicative maps is constructed. The developed techniques are then applied to show that a broad class of complex-oriented spectra admit structures of MU-algebras where MU is the complex cobordism spectrum. Various computations of topological derivations and topological Hochschild cohomology are made.     

Resonant States in High-Temperature Superconductors with Impurities

A microscopic theory of resonant states for the Zn-doped CuO₂ plane in the superconducting phase is formulated in the effective t–J model. In the model derived from the original p–d model, Zn impurities are considered as vacancies for the d states at Cu sites. In the superconducting phase, in addition to the local static perturbation induced by the vacancy, a dynamical perturbation appears that results in a frequency-dependent perturbation matrix. Using the T-matrix formalism for the Green's functions in terms of the Hubbard operators, we calculate the local density of electronic states with d, p, and s symmetries.     

两个正规p-超可解子群的积

本文研究了两个正规的p-超可解子群的积构成的极小非P-超可解群的结构的问题.利用有限群论和群类论的一些基本方法,获得了两个正规的p-超可解子群的积仍为P-超可解群的一些充分条件,并推广了文[1]中关于超可解群的情况.     

Dense trees: a new look at degenerate graphs

Dense trees are undirected graphs defined as natural extensions of trees. They are already known in the realm of graph coloring under the name of k-degenerate graphs. For a given integer k1, a k-dense cycle is a connected graph, where the degree of each vertex is greater than k. A k-dense forest F=(V,E) is a graph without k-dense cycles as subgraphs. If F is connected, then is a k-dense tree. 1-dense trees are standard trees. We have |E|k|V|−k(k+1)/2. If equality holds F is connected and is called a maximal k-dense tree. k-trees (a subfamily of triangulated graphs) are special cases of maximal k-dense trees.We review the basic theory of dense trees in the family of graphs and show their relation with k-trees. Vertex and edge connectivity is thoroughly investigated, and the role of maximal k-dense trees as “reinforced” spanning trees of arbitrary graphs is presented. Then it is shown how a k-dense forest or tree can be decomposed into a set of standard spanning trees connected through a common “root” of k vertices. All sections include efficient construction algorithms. Applications of k-dense trees in the fields of distributed systems and data structures are finally indicated.     

<Emphasis Type="Italic">∂</Emphasis>-operations and <Emphasis Type="Italic">H</Emphasis><Subscript><Emphasis Type="Italic">V</Emphasis></Subscript>-fields

The hyperoperations, called theta-operations （δ）, are motivated from the usual property, which the derivative has on the derivation of a product of functions. Using any map on a set, one can define δ-operations. In this paper, we continue our study on the δ-operations on groupoids, rings, fields and vector spaces or on the corresponding hyperstructures. Using δ-operations one obtains, mainly, Hwstructures, which form the largest class of the hyperstructures. For representation theory of hyperstructures, by hypermatrices, one needs special Hv-rings or Hy-fields, so these hyperstructures can be used. Moreover, we study the relation of these δ-structures with other classes of hyperstructures, especially with the Hv-structures.