Gelfand-Kirillov Dimension and Local Finiteness of Symmetrizations of Associative Superalgebras

In this work we extend to superalgebras a result of Skosyrskii [Algebra and Logic, 18 (1) (1979) 49–57, Lemma 2] relating associative and Jordan structures. As an application, we show that the Gelfand-Kirillov dimension of an associative superalgebra coincides with that of its symmetrization, and that local finiteness is equivalent in associative superalgebras and in their symmetrizations. In this situation we obtain that having zero Gelfand-Kirillov dimension is equivalent to being locally finite.     

Hurwitz and Tasoev Continued Fractions

The continued fractions studied by Tasoev are not widely known although their characteristics are very similar to those of Hurwitz continued fractions. Recently, the author found several general forms of Tasoev continued fractions, and by applying this method he also obtained some more general forms of Hurwitz continued fractions belonging to so called tanh-type and tan-type. In this paper, we constitute a new class of general forms of Hurwitz continued fractions of e-type. The known continued fraction expansions of e^1/a (a 1), ae^1/a and (1/a)e^1/a are included as special cases. The corresponding Tasoev continued fractions are also derived.     

On the Packing Densities and the Covering Densities of the Cartesian Products of Convex Bodies

In this article we study the following problem: Is the covering (packing) density of a Cartesian product of two convex bodies always equal to the product of their corresponding covering (packing) densities? For the covering case we get a negative answer. For the packing case we get a combinatorial version which seems to be important for its own interest.     

The Boothby-Wang Fibration of the Iwasawa Manifold as a Critical Point of the Energy

In this paper, we show that the Boothby-Wang fibration of the Iwasawa manifold is an unstable critical point for the energy of a distribution. The work of the first author is partially supported by TBAG-?G/2.     

Typical <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>-Dimensions of Measures

For a probability measure μ on a subset of , the lower and upper L^q-dimensions of order are defined by We study the typical behaviour (in the sense of Baire’s category) of the L^q-dimensions and . We prove that a typical measure μ is as irregular as possible: for all q ≥ 1, the lower L^q-dimension attains the smallest possible value and the upper L^q-dimension attains the largest possible value.     

Complete Minimal Hypersurfaces in a Sphere

In this paper we investigate complete minimal hypersurfaces with at most two principal curvatures. We prove that if the squared norm S of the second fundamental form satisfies S ≥ n, then S = n and f(Mⁿ) is a minimal Clifford torus.     

Characterizations of Some Product Manifolds by Their Spectrum

Let Sⁿ(c) denote the n-dimensional Euclidean sphere of constant sectional curvature c and denote by CPⁿ(c) the complex projective space of complex dimension n and of holomorphic sectional curvature c. In this paper, we obtain some characterizations of the manifolds S²(c) × S²(c′), S⁴(c) × S⁴(c′), CP²(c) × CP²(c′) by their spectrum.     

Finding the Best Face on a Voronoi Polyhedron – The Strong Dodecahedral Conjecture Revisited

In this paper we prove the following theorem. The surface area density of a unit ball in any face cone of a Voronoi cell in an arbitrary packing of unit balls of Euclidean 3-space is at most and so the surface area of any Voronoi cell in a packing with unit balls in Euclidean 3-space is at least This result and the ideas of its proof support the Strong Dodecahedral Conjecture according to which the surface area of any Voronoi cell in a packing with unit balls in Euclidean 3-space is at least as large as 16.6508..., the surface area of a regular dodecahedron of inradius 1.The authors were partially supported by the Hung. Nat. Sci. Found (OTKA), grant no. T043556.Supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant.     

Charakterisierung der <Emphasis Type="Italic">q</Emphasis>-Orthogonalpolynome in <Emphasis Type="Italic">x</Emphasis>

Characterization of q-Orthogonal Polynomials. Im Anschluß an die Arbeit Orthogonalpolynome in x und q^–x als Lösungen von reellen q-Operatorgleichungen zweiter Ordnung (Monatsh. Math. 132, 123–140 (2001); im folgenden als [4] zitiert) werden alle Möglichkeiten für q-Orthogonalpolynome in x als Lösungen von q-Operatorgleichungen zweiter Ordnung angegeben (Orthogonalität im positiv definiten Sinne). Dabei erfolgt die Numerierung der Abschnitte und die Angabe der Formel-nummern unter Einbeziehung von [4].     

Analytic Multifunctions, Holomorphic Motions and Hausdorff Dimension in IFSs

We study iterated function systems of contractions which depend holomorphically on a complex parameter λ. We first restrict our attention to systems which consist of similarities that satisfy the OSC. In this setting, we prove that the Hausdorff dimension of the limit set J(λ) is a continuous, subharmonic function of λ. In the remainder of the paper, systems consisting of conformal contractions are considered. We give conditions under which J(λ) and A(λ) = describe a holomorphic motion, and construct an example that shows that this is not the case in general. We finally show that A(λ) is best described as an analytic multifunction of λ, a notion that generalizes that of holomorphic motion. This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and by the Fonds Québécois de Recherche sur la Nature et les Technologies (FQRNT). This research was supported by the FQRNT.     

On Closed Weingarten Surfaces

We investigate closed surfaces in Euclidean 3-space satisfying certain functional relations κ = F(λ) between the principal curvatures κ, λ. In particular we find analytic closed surfaces of genus zero where F is a quadratic polynomial or F(λ) = cλ²ⁿ⁺¹. This generalizes results by H. Hopf on the case where F is linear and the case of ellipsoids of revolution where F(λ) = cλ³.     

Minkowski Arrangements of Spheres

Let be a non-negative number not greater than 1. Consider an arrangement of (not necessarily congruent) spheres with positive homogenity in the n-dimensional Euclidean space, i.e., in which the infimum of the radii of the spheres divided by the supremum of the radii of the spheres is a positive number. With each sphere S of associate a concentric sphere of radius times the radius of S. We call this sphere the -kernel of S. The arrangement is said to be a Minkowski arrangement of order if no sphere of overlaps the -kernel of another sphere. The problem is to find the greatest possible density of n-dimensional Minkowski sphere arrangements of order . In this paper we give upper bounds on for .     

Intersections and Unions of Orthogonal Polygons Starshaped Via Staircase n-Paths

For n ≥ 1, define p (n) to be the smallest natural number r for which the following is true: For any finite family of simply connected orthogonal polygons in the plane and points x and y in , if every r (not necessarily distinct) members of contain a common staircase n-path from x to y, then contains such a path. We show that p(1) = 1 and p(n) = 2 (n − 1) for n ≥ 2. The numbers p(n) yield an improved Helly theorem for intersections of sets starshaped via staircase n-paths. Moreover, we establish the following dual result for unions of these sets: Let be any finite family of orthogonal polygons in the plane, with simply connected. If every three (not necessarily distinct) members of have a union which is starshaped via staircase n-paths, then T is starshaped via staircase (n + 1)-paths. The number n + 1 in the theorem is best for every n ≥ 2.     

Hausdorff Dimension of Operator Stable Sample Paths

The Hausdorff dimension of the sample paths of a stochastic process with stationary independent operator stable increments is computed. With probability one, every sample path has the same dimension, depending on the real parts of the eigenvalues of the operator stable exponent.Received May 28, 2002; in revised form October 2, 2002 Published online May 15, 2003     

On an Extremal Problem About the Arc-Length of Algebraic Polynomials

In this work we investigate polynomials of maximal (minimal) arc-length in the interval [−1, 1] amongst all monic polynomials of fixed degree n with n real zeros in [−1, 1].     

Local Well-posedness of Kinetic Chemotaxis Models

We present a general functional analytic setting in which the Cauchy problem for mild solutions of kinetic chemotaxis models is well-posed, locally in time, in general physical dimensions. The models consist of a hyperbolic transport equation that is non-linearly and non-locally coupled to a reaction-diffusion system through kernel operators. Three examples are elaborated throughout the paper in which the latter system is (1) a single linear equation, (2) a FitzHugh-Nagumo system and (3) a piecewise linear approximation thereof. Finally we present a limit argument to obtain results on solutions in L ¹ ∩ L ^∞. Sander C. Hille: This work is supported by PIONIER grant 600-61410 of the Netherlands Organisation for Scientific Research, NWO.     

An Exceptional Set in the Ergodic Theory of Expanding Maps on Manifolds

Under a general hypothesis an expanding map T of a Riemannian manifold M is known to preserve a measure equivalent to the Liouville measure on that manifold. As a consequence of this and Birkhoff’s pointwise ergodic theorem, the orbits of almost all points on the manifold are asymptotically distributed with regard to this Liouville measure. Let T be Lipschitz of class τ for some τ in (0,1], let Ω(x) denote the forward orbit closure of x and for a positive real number δ and let E(x₀, δ) denote the set of points x in M such that the distance from x₀ to Ω is at least δ. Let dim A denote the Hausdorff dimension of the set A. In this paper we prove a result which implies that there is a constant C(T) > 0 such that if τ = 1 and if τ < 1. This gives a quantitative converse to the above asymptotic distribution phenomenon. The result we prove is of sufficient generality that a similar result for expanding hyperbolic rational maps of degree not less than two follows as a special case.     

Real Hypersurfaces of Type B in Complex Two-Plane Grassmannians

In this paper we give a characterization of real hypersurfaces of type B, that is, a tube over a totally real totally geodesic in complex two-plane Grassmannians with the shape operator A satisfying Aφ + φA = kφ, k is non-zero constant, for the structure tensor φ.     

Book Reviews

For , let E(λ_*, λ^*) be the set It has been proved in [1] and [3] that E(λ_*, λ^*) is an uncountable set. In the present paper, we strengthen this result by showing that where dim denotes the Hausdorff dimension.