Some topological properties of ω-covering sets

We prove the following theorems:1. There exists an -covering with the property s ₀.2. Under cov there exists X such that is not an -covering orX \ B is not an -covering].3. Also we characterize the property of being an -covering.     

A Note on π-Engel Conditions

In this paper we show that the weakly -Engel conditions are closely related to the existance of normal -complements; while the -Engel conditions are closely related to the -nilpotent groups.AMS Subject Classification (2000): 20D20     

(φ,ξ,η,λ)‐Structures of Parabolic Type

We consider (,,,)structures of parabolic type on hypersurfaces of dual spaces and study the rank of the affinor . We consider almost contact metric structures of parabolic type of the first kind on hypersurfaces of 4dimensional dual metric space. We study the properties of these structures and give examples of normal, integrable, and Sasakian parabolic structures.     

On Fraenkel’s <Emphasis Type="Italic">N</Emphasis>-Heap Wythoff’s Conjectures

The N-heap Wythoffs game is a two-player impartial game with N piles of tokens of sizes Players take turns removing any number of tokens from a single pile, or removing (a₁,..., a_N) from all piles - a_i tokens from the i-th pile, providing that where is the nim addition. The first player that cannot make a move loses. Denote all the P-positions (i.e., losing positions) by Two conjectures were proposed on the game by Fraenkel [7]. When are fixed, i) there exists an integer N₁ such that when . ii) there exist integers N₂ and _2 such that when , the golden section.In this paper, we provide a sufficient condition for the conjectures to hold, and subsequently prove them for the three-heap Wythoffs game with the first piles having up to 10 tokens.AMS Subject Classification: 91A46, 68R05.     

On <Emphasis Type="Italic">L</Emphasis><Subscript>α,ω</Subscript> complete extensions of complete theories of Boolean algebras

For a complete first order theory of Boolean algebras T which has nonisomorphic countable models, we determine the first limit ordinal = (T) such that We show that for some and for all other Ts, A nonprincipal ideal I of B is almost principal, if a is a principal ideal of B} is a maximal ideal of B. We show that the theory of Boolean algebras with an almost principal ideal has complete extensions and characterize them by invariants similar to the Tarskis invariants.Mathematics Subject Classification (2000): Primary 03C15, Secondary 03C35, 06E05Revised version: 2 February 2004     

On the Operator Semirings of a Γ-Semiring

In this paper we introduce the notion of operator semirings of a -semiring to study -semirings. It is shown that the lattices of all left (right) ideals (two-sided ideals) of a -semiring and its right (respectively left) operator semiring are isomorphic. This has many applications to characterize various -semirings.AMS Subject Classification (2000): 16Y60, 16Y99     

On convergence of the method of fundamental solutions for solving the Dirichlet problem of Poisson’s equation

In this paper the convergence of using the method of fundamental solutions for solving the boundary value problem of Laplaces equation in R² is established, where the boundaries of the domain and fictitious domain are assumed to be concentric circles. Fourier series is then used to find the particular solutions of Poissons equation, which the derivatives of particular solutions are estimated under the L² norm. The convergent order of solving the Dirichlet problem of Poissons equation by the method of particular solution and method of fundamental solution is derived. Dedicated to Charles A. Micchelli with esteem on the occasion of his 60th birthdayAMS subject classification 35J05, 31A99     

On the Irreducibility of Commuting Varieties Associated with Involutions of Simple Lie Algebras

Let be a reductive Lie algebra over an algebraically closed field of characteristic zero and an arbitrary -grading. We consider the variety , which is called the commuting variety associated with the -grading. Earlier it was proved by the author that is irreducible, if the -grading is of maximal rank. Now we show that is irreducible for and (E₆,F₄). In the case of symmetric pairs of rank one, we show that the number of irreducible components of is equal to that of nonzero non--regular nilpotent G ₀-orbits in . We also discuss a general problem of the irreducibility of commuting varieties.     

Metric and Complete Metric σ-Frames

A -frame is a lattice in which countable joins exist and binary meets distribute over countable joins. In this paper, the category MFrm, of metric -frames, is introduced, and it is shown to be equivalent to the category MLFrm _u, of metric Lindelöf frames.Finally, it is shown that the complete metric -frames are exactly the cozero parts of complete metric Lindelöf frames.     

A Question in the Theory of Totally Local Formations of Finite Groups

It is proved that all proper totally local subformations of a non one-generated totally local formation of finite groups are one-generated iff coincides with a formation of all soluble -groups, where ||=2.     

Distribution of Increasing ℓ-sequences in a Random Permutation

This paper examines the distribution of the number, k, of increasing -sequences in a random permutation of . A new solution is determined based on the compositions of n which requires, at most, summands. This solution easily yields existing results for the special case and provides an alternate form for the case . The expected number of increasing -sequences in a random permutation is determined and it is shown that the limiting distribution is degenerate about 0 for 2$$ " align="middle" border="0"> . An alternate algorithm to determine the exact distribution is presented, based on the partitions of n, which is easy to implement and efficient for small n. Applications in non-parametric statistics and graph theory are discussed.     

On the Representation of Approximate Subdifferentials for a Class of Generalized Convex Functions

Given a family of real-valued functions defined in a normed vector space X, we study a class of -convex functions having a simpler representation for the --subdifferential. The case =X^* with X being a Banach space (the Fenchel case) is particularly analysed, and we find that the sublinear lower semicontinuous functions satisfy the simpler representation with respect to X^*. As a side result, we provide various new subdifferential-type charaterizations of positively homogeneous functions among those which are lower semicontinuous and convex. In addition, we also discuss that family related to the the so-called prox-bounded functions. In this more general framework our simpler representation may give rise to a new notion of enlargement of the subdifferential.Mathematics Subject Classifications (2000) 47H05, 46B99, 47H17.This work is based on research material supported in part by CONICYT-Chile through FONDECYT 101-0116 and FONDAP-Matemáticas Aplicadas II.     

On Non-Negative Measurable Solutions of a Difference Functional Equation

Following M. Laczkovich [5] we consider the iterative functional equation

On the Movement of Transitive Permutation p-Groups

Let G be a transitive permutation group on a set and m a positive integer. If | – | m for every subset of and all g G, then || 2mp/(p – 1) where p is the least odd prime dividing |G|. It was shown by Mann and Praeger [13] that, for p = 3, the 3-groups G which attain this bound have exponent p. In this paper we will show a generalization of this result for any odd primes.AMS Subject Classification (2000), 20BXX     

Complete stability of large order statistics

Fix an integerr1. For eachnr, letM _nr be the rth largest ofX ₁,...,X _n, where {X _n,n1} is a sequence of i.i.d. random variables. Necessary and sufficient conditions are given for the convergence of _n=r n P[|M _nr /a _n –1|<] for every >0, where {a _n} is a real sequence and –1. Moreover, it is shown that if this series converges for somer1 and some >–1, then it converges for everyr1 and every >–1.     

On the Ordered K0‐Group of Universal, Free Product C*‐Algebras

The ordered K₀-group of the universal, unital free product C^*-algebra M_k()*M_l()is calculated in the case where k is prime and not a divisor in l. It is shown that the positive cone of K₀(M_k()*M_l())is as small as possible in this case. The article also contains results (full and partial) on the ordered K₀0-group of more general universal, unital free product C^* algebras.     

Generalizations of Perfect, Semiperfect, and Semiregular Rings

For a ring R and a right R-module M, a submodule N of M is said to be -small in M if, whenever N + X = M with M/X singular, we have X = M. If there exists an epimorphism p: P M such that P is projective and Ker(p) is -small in P, then we say that P is a projective -cover of M. A ring R is called -perfect (resp., -semiperfect, -semiregular) if every R-module (resp., simple R-module, cyclically presented R-module) has a projective -cover. The class of all -perfect (resp., -semiperfect, -semiregular) rings contains properly the class of all right perfect (resp., semiperfect, semiregular) rings. This paper is devoted to various properties and characterizations of -perfect, -semiperfect, and -semiregular rings. We define (R) by (R)/Soc(R_R) = Jac(R/Soc(R_R)) and show, among others, the following results:

Controllability and extremality in nonconvex differential inclusions

LetF:[0, T]×R ⁿ 2 ^{R n} be a set-valued map with compact values; let :R ⁿ R ^m be a locally Lipschitzian map,z(t) a given trajectory, andR the reachable set atT of the differential inclusion . We prove sufficient conditions for (z(T))intR and establish necessary conditions in maximum principle form for (z(T))(R). As a consequence of these results, we show that every boundary trajectory is simultaneously a Pontryagin extremal, Lagrangian extremal, and relaxed Lagrangian extremal.The author is grateful to an anonymous referee for his valuable remarks and comments which have helped to improve the paper.The paper was written while the author was visiting the laboratory of Prof. S. Suzuki, Department of Mechanical Engineering, Sophia University, Tokyo, Japan.