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.     

A Symmetric Function Resolution of the Number of Permutations With Respect to Block-Stable Elements

We consider a question raised by Suhov and Voice from quantum information theory and quantum computing. An element of a partition of {1, ..., n} is said to be block-stable for if it is not moved to another block under the action of π. The problem concerns the determination of the generating series for elements of with respect to the number of block-stable elements of a canonical partition of a finite n-set, with block sizes k₁, ..., k_r, in terms of the moment (power) sums p_q(k₁, ..., k_r). We also consider the limit subject to the condition that exists for q = 1, 2,.... Received January 31, 2006     

C*-Modular Vector States

Let be a C*-algebra and X a Hilbert C* -module. If is a projection, let be the p-sphere of X. For φ a state of with support p in and consider the modular vector state φ_x of given by The spheres provide fibrations

The dimension of the kernel in finite and infinite intersections of starshaped sets

Summary. We establish the following Helly-type result for infinite families of starshaped sets in Define the function f on {1, 2} by f(1) = 4, f(2) = 3. Let be a fixed positive number, and let be a uniformly bounded family of compact sets in the plane. For k = 1, 2, if every f(k) (not necessarily distinct) members of intersect in a starshaped set whose kernel contains a k-dimensional neighborhood of radius , then is a starshaped set whose kernel is at least k-dimensional. The number f(k) is best in each case. In addition, we present a few results concerning the dimension of the kernel in an intersection of starshaped sets in Some of these involve finite families of sets, while others involve infinite families and make use of the Hausdorff metric.     

The Maximum Size Of Graphs With A Unique<Emphasis Type="Italic">k</Emphasis>- Factor

For suitable positive integers n and k let m(n, k) denote the maximum number of edges in a graph of order n which has a unique k-factor. In 1964, Hetyei and in 1984, Hendry proved for even n and , respectively. Recently, Johann confirmed the following conjectures of Hendry: for and kn even and for n = 2kq, where q is a positive integer. In this paper we prove for and kn even, and we determine m(n, 3).     

A classification of martingale Hardy spaces associated with rearrangement-invariant function spaces

Let X be a rearrangement-invariant Banach function space over a complete probability space , and denote by the Hardy space consisting of all martingales such that . We prove that implies for any filtration if and only if Doobs inequality holds in X, where denotes the martingale defined by , n = 0, 1, 2, ..., and a.s.Received: 1 August 2000     

Harmonic mean curvature lines on surfaces immersed in $ \mathbb{R}^{3} $ \mathbb{R}^{3}

Consider oriented surfaces immersed in . Associated to them, here are studied pairs of transversal foliations with singularities, defined on the Elliptic region, where the Gaussian curvature , given by the product of the principal curvatures k ₁, k ₂ is positive. The leaves of the foliations are the lines of harmonic mean curvature, also called characteristic or diagonal lines, along which the normal curvature of the immersion is given by , where is the arithmetic mean curvature. That is, is the harmonic mean of the principal curvatures k ₁, k ₂ of the immersion. The singularities of the foliations are the umbilic points and parabolic curves, where k ₁ = k ₂ and , respectively.Here are determined the structurally stable patterns of harmonic mean curvature lines near the umbilic points, parabolic curves and harmonic mean curvature cycles, the periodic leaves of the foliations. The genericity of these patterns is established.This provides the three essential local ingredients to establish sufficient conditions, likely to be also necessary, for Harmonic Mean Curvature Structural Stability of immersed surfaces. This study, outlined towards the end of the paper, is a natural analog and complement for that carried out previously by the authors for the Arithmetic Mean Curvature and the Asymptotic Structural Stability of immersed surfaces, [13, 14, 17], and also extended recently to the case of the Geometric Mean Curvature Configuration [15].The first author was partially supported by FUNAPE/UFG. Both authors are fellows of CNPq. This work was done under the project PRONEX/FINEP/MCT - Conv. 76.97.1080.00 - Teoria Qualitativa das Equações Diferenciais Ordinárias and CNPq - Grant 476886/2001-5.     

A Helly-type theorem for countable intersections of starshaped sets

Let k and d be fixed integers, 0kd, and let be a collection of sets in If every countable subfamily of has a starshaped intersection, then is (nonempty and) starshaped as well. Moreover, if every countable subfamily of has as its intersection a starshaped set whose kernel is at least k-dimensional, then the kernel of is at least k-dimensional, too. Finally, dual statements hold for unions of sets.Received: 3 April 2004     

Systèmes dynamiques non archimédiens et corps des norms (Non-Archimedean Dynamic Systems and Fields of Norms)

Let _k be the ring of integers of a finite extension k of the field _p of p-adic numbers. The endomorphisms of a formal group law defined over _k provide nontrivial examples of commuting formal series with coefficients in _k . This article deals with the inverse problem formulated by Jonathan Lubin within the context of non-Archimedean dynamical systems. We present a large family of series, with coefficients in _p , which satisfy Lubin's conjecture. These series are constructed with the help of Lubin–Tate formal group laws over _p . We introduce the notion of minimally ramified series which turn out to be modulo p reductions of some series of this family. The commutant monoids of these minimally ramified series are determined by using the Fontaine–Wintenberger theory of the field of norms which allows an interpretation of them as automorphisms of _p -extensions of local fields of characteristic zero. A particularly effective example illustrating the paper is given by a family of series generalizing ebyev polynomials     

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.     

On a conditional Gołab-Schinzel equation

Let We show that for every function satisfying the conditional equation

Transformations of Grassmannians preserving the class of base subsets

Let be a finite-dimensional projective space and be the Grassmannian consisting of all k-dimensional subspaces of . In the paper we show that transformations of sending base subsets to base subsets are induced by collineations of to itself or to the dual projective space . This statement generalizes the main result of the authors paper [19].     

Beckman-Quarles type theorems for mappings from $$ \mathbb{R}^n $$ to $$ \mathbb{C}^n $$

Summary. Let We say that preserves the distance d 0 if for each implies Let A _n denote the set of all positive numbers d such that any map that preserves unit distance preserves also distance d. Let D _n denote the set of all positive numbers d with the property: if and then there exists a finite set S _xy with such that any map that preserves unit distance preserves also the distance between x and y. Obviously, We prove: (1) (2) for n 2 D _n is a dense subset of (2) implies that each mapping f from to (n 2) preserving unit distance preserves all distances, if f is continuous with respect to the product topologies on and      

A Sharp Bound for the Number of Sets that Pairwise Intersect at <Emphasis Type="Italic">k</Emphasis> Positive Values

In this paper we prove that if is a set of k positive integers and {A ₁, ..., A _m } is a family of subsets of an n-element set satisfying , for all 1 i < j m, then . The case k = 1 was proven 50 years ago by Majumdar.     

Remarks on ample vector bundles of curve genus two

Let be an ample vector bundle of rank n – 1 on a smooth complex projective variety X of dimension n≥ 3 such that X is a -bundle over and that for any fiber F of the bundle projection . The pairs with = 2 are classified, where is the curve genus of . This allows us to improve some previous results. Received: 13 June 2006     

An Application of U(g)-bimodules to Representation Theory of Affine Lie Algebras

Let be the affine Lie algebra associated to the simple finite-dimensional Lie algebra . We consider the tensor product of the loop -module associated to the irreducible finite-dimensional -module V() and the irreducible highest weight -module L _k,. Then L _k, can be viewed as an irreducible module for the vertex operator algebra M _k,0. Let A(L _k,) be the corresponding -bimodule. We prove that if the -module is zero, then the -module is irreducible. As an example, we apply this result on integrable representations for affine Lie algebras.