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     

Inequality of Petrović and Giaccardi for Convex Function of Higher Order

In this paper we use Lidstone polynomials to prove further generalization of Giaccardi generalization of the well-known Petrovis inequality.     

Some Properties of Isologism of Groups and Baer-Invariants

Let be a variety of groups defined by the set of laws V. In this paper we study the concept of -isologism of groups in terms of -extensions and their connections with the Baer-invariant of groups are also discussed.AMS Subject Classification (2000): primary 20F14, 20F19, secondary 20E10     

Infinite independent systems of the identities of the associative algebra over an infinite field of characteristic <Emphasis Type="Italic">p</Emphasis> > 0

In this paper some infinitely based varieties of groups are constructed and these results are transferred to the associative algebras (or Lie algebras) over an infinite field of an arbitrary positive characteristic.     

From Pappus’ Theorem to the Twisted Cubic

We discuss a classical result in planar projective geometry known as Steiners theorem involving 12 interlocking applications of Pappus theorem. We prove this result using three dimensional projective geometry then uncover the dynamics of this construction and relate them to the geometry of the twisted cubic.Mathematics Subject Classification (2000). Primary 51N15.     

Explicit Formulas for Green’s Functions on the Annulus and on the Möbius Strip

Given a fixed point free antianalytic involution k of a domain G in thecomplex plane, bounded by a finite number of analytic curves, k-invariant Greensfunctions are defined on G. The Lindelöfs principle is extended to k-invariantGreens functions. When G is the annulus, k-invariant Greens functions areobtained in the explicit form. Since the factorization of the annulus by the group kgenerated by k produces a Möbius strip, the respective result helped us to obtain explicitforms for Greens functions on the Möbius strip.     

Hyperbolic Geometry with Clifford Algebra

The Clifford algebra in D. Hestenes formulation is used to study hyperbolic geometry and some interesting theorems are obtained. The computational power of this formulation is fully revealed by the ease of extending old results and discovering new ones. An important new result is the formulas on the area and perimeter of a convex polygon, based on extending Gauss equalities.     

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 Some Generalizations and Refinements of a Ky Fan Inequality

In this paper, a Ky Fan inequality and an inequality by Wu and Wang [10] will be generalized. Some new and improved refinements of the Ky Fan inequality will be put forward.AMS Subject Classification (2000) 26D15     

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:

Theta Pairs and the Structure of Finite Groups

We introduce the notion of -pair of a proper subgroup of a finite group and give necessary and sufficient conditions for supersolvability and nilpotence of a finite group. As application, we give a proof to a recent open problem of Skiba in the unsolved problems in group theory.     

On the Total Domination Number of Cartesian Products of Graphs

The most famous open problem involving domination in graphs is Vizings conjecture which states the domination number of the Cartesian product of any two graphs is at least as large as the product of their domination numbers. In this paper, we investigate a similar problem for total domination. In particular, we prove that the product of the total domination numbers of any nontrivial tree and any graph without isolated vertices is at most twice the total domination number of their Cartesian product, and we characterize the extremal graphs.Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal     

A polynomial of degree four not satisfying Rolle’s theorem in the unit ball of <Emphasis Type="Italic">l</Emphasis><Subscript>2</Subscript>

We give an example of a fourth degree polynomial which does not satisfy Rolles Theorem in the unit ball of l ₂.The author has been partially supported by MCyT and FEDER Project BFM2002-01423.     

A Note on the Interpolation of Entire Functions

In this note, we obtain the interpolation theorems of entire function with limited growth conditions.AMS Subject Classifications (1991): 30D15The project supported by the National Natural Science Foundation of China (No. 10071005).     

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     

Connections between Interval and Unit Circle for Sobolev Orthogonal Polynomials. Strong Asymptotics on the Real Line

In this paper we show the connection between Sobolev orthogonal Laurent polynomials on the unit circle and Sobolev orthogonal polynomials on a bounded interval of the real line. As a consequence we deduce the strong outer asymptotics for Sobolev orthogonal polynomials with respect to the inner product

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     

Matrix Units for Signed Brauer’s Algebras

Brauers centerlizer algebras have a basis consisting of undirected graphs. Signed Brauers algebras have a basis consisting of directed graphs, which generalizes Brauers algebras. In this paper, we compute a complete set of matrix units of Signed Brauers algebras.AMS Subject Classification: 99c:16028, 16S99(16K20).     

Metric σ-Frames versus Metric Lindelof Spaces

The adjoint relation between the category RegFrm, of regular -frames, Alex, of Alexandroff spaces, are studied in [9]. Here, we introduce the category MFrm, of metric -frames and give the adjoint relation between this category and the category MLSp, of metric Lindelof spaces, and show that MLSp is dually equivalent to the category of Alexandroff metric -frames.AMS Subject Classification: 06D99-54B30     

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.