bluuebunny/crossref_metadata_2025_split huggingface dataset:
Generalized Reductive Lie Algebras: Connections With Extended Affine Lie Algebras and Lie Tori
Authors: Azam Saeid 
Published: December 2010

AbstractWe investigate a class of Lie algebras which we call generalized reductive Lie algebras. These are generalizations of semi-simple, reductive, and affine Kac–Moody Lie algebras. A generalized reductive Lie algebra which has an irreducible root system is said to be irreducible and we note that this class of algebras has been under intensive investigation in recent years. They have also been called extended affine Lie algebras. The larger class of generalized reductive Lie algebras has not been so intensively investigated. We study themin this paper and note that one way they arise is as fixed point subalgebras of finite order automorphisms. We show that the core modulo the center of a generalized reductive Lie algebra is a direct sum of centerless Lie tori. Therefore one can use the results known about the classification of centerless Lie tori to classify the cores modulo centers of generalized reductive Lie algebras.
Extension Theorems on Weighted Sobolev Spaces and Some Applications
Authors: Chua Seng-Kee 
Published: December 2010

AbstractWe extend the extension theorems to weighted Sobolev spaces on (ε, δ) domains with doubling weight w that satisfies a Poincaré inequality and such that w–1/p is locally Lp′. We also make use of the main theorem to improve weighted Sobolev interpolation inequalities.
Hypoelliptic Bi-Invariant Laplacians on Infinite Dimensional Compact Groups
Authors: Bendikov A. , Saloff-Coste L. 
Published: December 2010

AbstractOn a compact connected group G, consider the infinitesimal generator –L of a central symmetric Gaussian convolution semigroup (μt)t>0. Using appropriate notions of distribution and smooth function spaces, we prove that L is hypoelliptic if and only if (μt)t>0 is absolutely continuous with respect to Haar measure and admits a continuous density x ⟼ μt (x), t > 0, such that limt!0t log μt(e) = 0. In particular, this condition holds if and only if any Borel measure u which is solution of Lu = 0 in an open set Ω can be represented by a continuous function in Ω. Examples are discussed.
Decomposability of von Neumann Algebras and the Mazur Property of Higher Level
Authors: Hu Zhiguo , Neufang Matthias 
Published: December 2010

AbstractThe decomposability number of a von Neumann algebra ℳ (denoted by dec(ℳ)) is the greatest cardinality of a family of pairwise orthogonal non-zero projections in ℳ. In this paper, we explore the close connection between dec(ℳ) and the cardinal level of the Mazur property for the predual ℳ* of ℳ, the study of which was initiated by the second author. Here, our main focus is on those von Neumann algebras whose preduals constitute such important Banach algebras on a locally compact group G as the group algebra L1(G), the Fourier algebra A(G), the measure algebra M(G), the algebra LUC(G)*, etc. We show that for any of these von Neumann algebras, say ℳ, the cardinal number dec(ℳ) and a certain cardinal level of the Mazur property of ℳ* are completely encoded in the underlying group structure. In fact, they can be expressed precisely by two dual cardinal invariants of G: the compact covering number κ(G) of G and the least cardinality ᙭(G) of an open basis at the identity of G. We also present an application of the Mazur property of higher level to the topological centre problem for the Banach algebra A(G)**.
On the Cohomology of Moduli of Vector Bundles and the Tamagawa Number of SLn
Authors: Dhillon Ajneet 
Published: December 2010

AbstractWe compute some Hodge and Betti numbers of the moduli space of stable rank r, degree d vector bundles on a smooth projective curve. We do not assume r and d are coprime. In the process we equip the cohomology of an arbitrary algebraic stack with a functorial mixed Hodge structure. This Hodge structure is computed in the case of the moduli stack of rank r, degree d vector bundles on a curve. Our methods also yield a formula for the Poincaré polynomial of the moduli stack that is valid over any ground field. In the last section we use the previous sections to give a proof that the Tamagawa number of SLn is one.
Partial *-Automorphisms, Normalizers, and Submodules in Monotone Complete C*-Algebras
Authors: Hamana Masamichi 
Published: December 2010

AbstractFor monotone complete C*-algebras A ⊂ B with A contained in B as a monotone closed C*-subalgebra, the relation X = AsA gives a bijection between the set of all monotone closed linear subspaces X of B such that AX + XA ⊂ X and XX* + X*X ⊂ A and a set of certain partial isometries s in the “normalizer” of A in B, and similarly for the map s ⟼ Ad s between the latter set and a set of certain “partial *-automorphisms” of A. We introduce natural inverse semigroup structures in the set of such X's and the set of partial *-automorphisms of A, modulo a certain relation, so that the composition of these maps induces an inverse semigroup homomorphism between them. For a large enough B the homomorphism becomes surjective and all the partial *-automorphisms of A are realized via partial isometries in B. In particular, the inverse semigroup associated with a type II1 von Neumann factor, modulo the outer automorphism group, can be viewed as the fundamental group of the factor. We also consider the C*-algebra version of these results.
Growth and Zeros of the Zeta Function for Hyperbolic Rational Maps
Authors: Christianson Hans 
Published: December 2010

AbstractThis paper describes new results on the growth and zeros of the Ruelle zeta function for the Julia set of a hyperbolic rational map. It is shown that the zeta function is bounded by exp(CK|s|δ) in strips | Res| ≤K, whereδis the dimension of the Julia set. This leads to bounds on the number of zeros in strips (interpreted as the Pollicott–Ruelle resonances of this dynamical system). An upper bound on the number of zeros in polynomial regions {| Res| ≤ | Ims|α} is given, followed by weaker lower bound estimates in strips {Res> –C, | Ims| ≤r}, and logarithmic neighbourhoods {| Res| ≤ ρlog | Ims|}. Recent numerical work of Strain–Zworski suggests the upper bounds in strips are optimal.
Searching for Absolute ᘓℛ–Epic Spaces
Authors: Barr Michael , Kennison John F. , Raphael R. 
Published: December 2010

AbstractIn previous papers, Barr and Raphael investigated the situation of a topological space Y and a subspace X such that the induced map C(Y ) → C(X) is an epimorphism in the category ᘓℛ of commutative rings (with units). We call such an embedding a ᘓℛ-epic embedding and we say that X is absolute ᘓℛ-epic if every embedding of X is ᘓℛ-epic. We continue this investigation. Our most notable result shows that a Lindelöf space X is absolute ᘓℛ-epic if a countable intersection of βX-neighbourhoods of X is a βX-neighbourhood of X. This condition is stable under countable sums, the formation of closed subspaces, cozero-subspaces, and being the domain or codomain of a perfect map. A strengthening of the Lindelöf property leads to a new class with the same closure properties that is also closed under finite products. Moreover, all σ-compact spaces and all Lindelöf P-spaces satisfy this stronger condition. We get some results in the non-Lindelöf case that are sufficient to show that the Dieudonné plank and some closely related spaces are absolute ᘓℛ-epic.
Osculating Varieties of Veronese Varieties and Their Higher Secant Varieties
Authors: Bernardi A. , Catalisano M. V. , Gimigliano A. , Idà M. 
Published: December 2010

AbstractWe consider the k-osculating varieties Ok,n.d to the (Veronese) d-uple embeddings of ℙn. We study the dimension of their higher secant varieties via inverse systems (apolarity). By associating certain 0-dimensional schemes Y ⊂ ℙn to and by studying their Hilbert functions, we are able, in several cases, to determine whether those secant varieties are defective or not.
Cyclic Groups and the Three Distance Theorem
Authors: Chevallier Nicolas 
Published: December 2010

AbstractWe give a two dimensional extension of the three distance theorem. Let θ be in R2 and let q be in N. There exists a triangulation of R2 invariant by Z2-translations, whose set of vertices is Z2 + ﹛0, θ, … , qθ﹜, and whose number of different triangles, up to translations, is bounded above by a constant which does not depend on θ and q.
Preduals and Nuclear Operators Associated with Bounded, p-Convex, p-Concave and Positive p-Summing Operators
Authors: Labuschagne C. C. A. 
Published: December 2010

AbstractWe use Krivine's form of the Grothendieck inequality to renorm the space of bounded linear maps acting between Banach lattices. We construct preduals and describe the nuclear operators associated with these preduals for this renormed space of bounded operators as well as for the spaces of p-convex, p-concave and positive p-summing operators acting between Banach lattices and Banach spaces. The nuclear operators obtained are described in terms of factorizations through classical Banach spaces via positive operators.
Geometry and Spectra of Closed Extensions of Elliptic Cone Operators
Authors: Gil Juan B. , Krainer Thomas , Mendoza Gerardo A. 
Published: December 2010

AbstractWe study the geometry of the set of closed extensions of index 0 of an elliptic differential cone operator and its model operator in connection with the spectra of the extensions, and we give a necessary and sufficient condition for the existence of rays of minimal growth for such operators.
The Choquet–Deny Equation in a Banach Space
Authors: Jaworski Wojciech , Neufang Matthias 
Published: December 2010

AbstractLet G be a locally compact group and π a representation of G by weakly* continuous isometries acting in a dual Banach space E. Given a probability measure μ on G, we study the Choquet–Deny equation π(μ)x = x, x ∈ E. We prove that the solutions of this equation form the range of a projection of norm 1 and can be represented by means of a “Poisson formula” on the same boundary space that is used to represent the bounded harmonic functions of the random walk of law μ. The relation between the space of solutions of the Choquet–Deny equation in E and the space of bounded harmonic functions can be understood in terms of a construction resembling theW*-crossed product and coinciding precisely with the crossed product in the special case of the Choquet–Deny equation in the space E = B(L2(G)) of bounded linear operators on L2(G). Other general properties of the Choquet–Deny equation in a Banach space are also discussed.
Ideas from Zariski Topology in the Study of Cubical Homology
Authors: Kaczynski Tomasz , Mrozek Marian , Trahan Anik 
Published: December 2010

AbstractCubical sets and their homology have been used in dynamical systems as well as in digital imaging. We take a fresh look at this topic, following Zariski ideas from algebraic geometry. The cubical topology is defined to be a topology in ℝd in which a set is closed if and only if it is cubical. This concept is a convenient frame for describing a variety of important features of cubical sets. Separation axioms which, in general, are not satisfied here, characterize exactly those pairs of points which we want to distinguish. The noetherian property guarantees the correctness of the algorithms. Moreover, maps between cubical sets which are continuous and closed with respect to the cubical topology are precisely those for whom the homology map can be defined and computed without grid subdivisions. A combinatorial version of the Vietoris–Begle theorem is derived. This theorem plays the central role in an algorithm computing homology of maps which are continuous with respect to the Euclidean topology.
Sobolev Extensions of Hölder Continuous and Characteristic Functions on Metric Spaces
Authors: Björn Anders , Björn Jana , Shanmugalingam Nageswari 
Published: December 2010

AbstractWe study when characteristic and Hölder continuous functions are traces of Sobolev functions on doubling metric measure spaces. We provide analytic and geometric conditions sufficient for extending characteristic and Hölder continuous functions into globally defined Sobolev functions.
On Effective Witt Decomposition and the Cartan–Dieudonné Theorem
Authors: Fukshansky Lenny 
Published: December 2010

AbstractLetKbe a number field, and letFbe a symmetric bilinear form in 2Nvariables overK. LetZbe a subspace ofKN. A classical theorem of Witt states that the bilinear space (Z,F) can be decomposed into an orthogonal sum of hyperbolic planes and singular and anisotropic components. We prove the existence of such a decomposition of small height, where all bounds on height are explicit in terms of heights ofFandZ. We also prove a special version of Siegel's lemma for a bilinear space, which provides a small-height orthogonal decomposition into one-dimensional subspaces. Finally, we prove an effective version of the Cartan–Dieudonné theorem. Namely, we show that every isometry σ of a regular bilinear space (Z,F) can be represented as a product of reflections of bounded heights with an explicit bound on heights in terms of heights ofF,Z, and σ.
Strichartz Inequalities for the Wave Equation with the Full Laplacian on the Heisenberg Group
Authors: Furioli Giulia , Melzi Camillo , Veneruso Alessandro 
Published: December 2010

AbstractWe prove dispersive and Strichartz inequalities for the solution of the wave equation related to the full Laplacian on the Heisenberg group, by means of Besov spaces defined by a Littlewood–Paley decomposition related to the spectral resolution of the full Laplacian. This requires a careful analysis due also to the non-homogeneous nature of the full Laplacian. This result has to be compared to a previous one by Bahouri, Gérard and Xu concerning the solution of the wave equation related to the Kohn Laplacian.
On the Convergence of Successive Approximations in the Theory of Ordinary Differential Equations
Authors: Luxemburg W. A. J. 
Published: December 2010

Let R denote the rectangle: |t-t0| ≤ a, | x-x0| ≤ b (a,b > 0) in the (t,x) plane and let f(t, x) be a function of two real variables t and x, defined and continuous on R. If I is the interval |t—t0| ≤ d with d = min(a,b/M), where M = max(|f(t, x)|, (t, x) ϵ R), then every solution x = x (t) of the differential equation x' = f(t, x) defined on I and which satisfies the initial condition x(t0) = x0, satisfies the integral equation1.1and conversely. In some cases, in order to prove the existence and uniqueness of the solutions of (1.1) on I, one forms the successive approximations1.2
On Some Properties of Functions Regular in the Unit Circle
Authors: Rooney P.G. 
Published: December 2010

The space Hp, 1 ≤ p ≤ ∞ consists of those analytic functions f(z) regular in the unit circle, for which Mp (f;r) is bounded for O ≤ r ≤ 1, whereThese spaces have been extensively studied.One well known result concerning these spaces is that if f(z) = Σ ∞n=0 anzn and {an} ɛ lp for some p, 1 ≤ p ≤ 2, then f ɛ Hq, where p-1+q-1 = 1, and conversely if f ɛ Hp, 1 ≤ p ≤ 2, then {an} ɛ lq. We propose to generalize this result to deal with functions f(z) = Σ ∞n=0 anzn with {n-λ an; n = 1, 2,...} ɛ lp, where λ ≥ 0. The resulting generalization is contained in the theorems below.However, in order to make these generalizations we must first generalize the spaces Hp. To this end we make the following definition.
On a Normal Form of the Orthogonal Transformation I
Authors: Zassenhaus Hans 
Published: December 2010

At the Edmonton Meeting of the Canadian Mathematical Congress E. Wigner asked me whether one knew something about the distribution of the characteristic roots of the linear transformations that leave invariant the quadratic form t2+x2-y2-z2, just as one knows that a Lorentz transformation has two complex conjugate characteristic roots and two real characteristic roots that are either inverse to one another or the numbers 1 and -1.In this paper an answer to E. Wigner’s question will be obtained.We are concerned with the pairs of matrices (X,A) with coefficients in a field of reference F such that the condition0.1is satisfied, where XT = (ξki) is the transpose of the matrix X = (ξki). It follows that both matrices are quadratic of the same degree d.
A Regular Quaternion Polygon
Authors: Crowe Donald W. 
Published: December 2010

Repeated application of the unitary reflectionsto the point (1,0) and line x + iy = 1 yields 24 points and 24 lines. These are the vertices and edges of the regular complex polygon 4 {3} 4 whose group has the abstract definition R4 = I, RSR = SRS [l]. The purpose of this note is to introduce the notion of regular quaternion polygon and give an example analogous to the preceding one.
Minkowski's Theorem with Curvature Limitations (I)
Authors: Melzak Z. A. 
Published: December 2010

The well-known theorem of Minkowski, [1], [2], states that;(M) a plane convex region, symmetric about the origin O, includes a lattice point other than O if its area is greater than 4.By a lattice point we shall understand a point in the plane, both of whose coordinates are rational integers. In connection with (M) a critical region is defined to be a convex symmetric region of area 4, which includes no lattice point other than O. One such region is the open square S={(x, y)| |x| < 1, |y|<1} an infinite set of critical regions is formed by the parallelograms bounded by the lines y = x+1, y = x-1, y-1 = k (x-l), y+1 = k(x+l), 0≤-k<∞. Finally, there is a critical hexagon H, bounded by the following six lines: y-1 = -(x-l), y+1 = -(x+1), y-1 = mx, y+1 = mx, y = (1/m)(x-1), y = (l/m)(x+l), where m = tan π/12. All the vertex angles of H are equal to 2π/3.
A Generalization of Wythoff's Game*
Authors: Connell Ian G. 
Published: December 2010

W.A. Wythoff [1] in 1907 defined a modification of the game of Nim by the following rules:(i)there are two players who play alternately;(ii)initially there are two piles of matches, an arbitrary number in each pile;(iii)a player may take an arbitrary number of matches from one pile or an equal number from both piles but he must take at least one match;(iv)the player who takes the last match wins the game.If, after his move, a player leaves one match in one pile and two in the other he can force a win; for if his opponent takes one match from the pile containing two he can take both remaining matches; and similarly for the other possibilities.
Sur la Structure d'Une Classe d'Anneaux
Authors: Thierrin Gabriel 
Published: December 2010

L'objet de ce travail est de déterminer la structure d'une classe d'anneaux, celle des anneaux bipotents à droite.1. Anneaux bipotents à droite. Un anneau A est dit bipotent à droite, si l'on apour tout a ϵ A, On voit facilement qu'un anneau A est bipotent à droite, si et seulement si pour tout couple a, b ϵ A, il existe x ϵ A tel que l'on ait ab = a2x.
A Note on Hilbert's Tenth Problem
Authors: Melzak Z. A. 
Published: December 2010

The tenth problem on Hilbert's well known list [1] is the following.(H 10) For an arbitrary polynomial P = P(x1,x2,…,xn) with integer coefficients to determine whether or not the equation P = 0 has a solution in integers.By 'integers' we always mean 'rational integers'. The problem (H 10) is still unsolved but it appears likely that no decision procedure exists; in this connection see [2]. It will be shown here that (H 10) is equivalent to deciding whether or not every member of a certain given countable sec of rational functions of a single variable x is absolutely monotonie. We recall that f(x) is absolutely monotonie in I if f(x) possesses non-negative derivatives of all orders at every x ∊ I.
On an Algorithm of G. Birkhoff Concerning Doubly Stochastic Matrices
Authors: Johnson Diane M. , Dulmage A. L. , Mendelsohn N. S. 
Published: December 2010

In [1] G. Birkhoff stated an algorithm for expressing a doubly stochastic matrix as an average of permutation matrices. In this note we prove two graphical lemmas and use these to find an upper bound for the number of permutation matrices which the Birkhoff algorithm may use.A doubly stochastic matrix is a matrix of non-negative elements with row and column sums equal to unity and is there - fore a square matrix. A permutation matrix is an n × n doubly stochastic matrix which has n2-n zeros and consequently has n ones, one in each row and one in each column. It has been shown by Birkhoff [1],Hoffman and Wielandt [5] and von Neumann [7] that the set of all doubly stochastic matrices, considered as a set of points in a space of n2 dimensions constitute the convex hull of permutation matrices.
On Certain Pairs of Matrices which do not Generate a Free Group
Authors: Ree Rimhak 
Published: December 2010

A complex number λ will be said to be free if the multiplicative group Fλ generated by the two matricesis a free group, and non-free, otherwise. Very little is known about the distribution of free and non-free numbers [1]. It is, for instance, unknown whether the domaincontains an open set which consists of only free points.
On Continuous Regular Rings
Authors: Utumi Y. 
Published: December 2010

A subset K of a lattice is said to be directed if for any a, b∊K there is c∊K with c ≥ a, b. A complete lattice L is called upper continuous if for every directed subset (aα) and every element b.The following is a slight improvement of [4; Anmerkung 1. 11, p. 11].
Definition of Affine Geometry by a Group of Transformations
Authors: Lipman Joe 
Published: December 2010

In his Geometric Algebra (New York, 1957) E. Artin poses the problem of co-ordinatizing an affine plane in the following terms.How little do we have to assume, from a geometric point of view, about an affine plane, in order to be able to describe its points by pairs of elements of a field, and its lines by linear equations?
A Family of Regular Maps of Type {6, 6}
Authors: Sherk F.A. 
Published: December 2010

In (4), pp. 25−27, Coxeter divided the regular maps on a surface of genus 1
into three infinite families. They are:(i)Maps of type {4, 4}.(ii)Maps of type {6, 3}.(iii)Maps of type {3, 6} (the duals of (ii)).We consider the family (iii). By adjoining an element to the group of any
map in (iii) we shall derive the group of a regular map of type {6, 6}. Thus
we produce a 1−1 correspondence between the members of the family (iii) and
of the new family. Corresponding members in the two families have certain
properties in common, the most interesting of which is the property of
reflexibility. Our results are summarized in Theorems 1 and 2.
A Note on a Theorem of Moser and Whitney
Authors: Narayana T. V. , Pettigrew H. M. 
Published: December 2010

In a recent paper [1], L. Moser and E.L. Whitney have proved the
following.Theorem: The number of compositions of n into parts=1, 2, 4 or 5 (mod 6) and
involving an even number of parts=4 or 5 (mod 6) exceeds by n the number of
compositions of n into parts=1, 2, 4 or 5 (mod 6) and involving an odd
number of parts=4 or 5 (mod 6).Their method of proof utilizes the notion of weighted compositions and the
method of generating series. They remark that they have not been able to
find a direct combinatorial proof. The purpose of this note is to give a
direct proof of a more general result.
A Pair of Generators for the Unimodular Group
Authors: Trott Stanton M. 
Published: December 2010

It is well known that ℳn, the multiplicative group n consisting of n-rowed square matrices with integer entries and determinant equal to ±1 can be generated by:It is also known that U4 can be generated by U1, U2, and U3 (cf.[2] p.85).However, by a construction which is much simpler than the one just mentioned for U4, it is possible to generate U3 by just U2 and U4. Since U2 and U4 affect only the first