bluuebunny/crossref_metadata_2025 huggingface dataset:
Well Ramified Extensions of Complete Discrete Valuation Fields with Applications to the Kato Conductor
Authors: Spriano Luca
Published: December 2010

We study extensions L/K of complete discrete valuation fields K with residue field of characteristic p > 0, which we do not assume to be perfect. Our work concerns ramification theory for such extensions, in particular we show that all classical properties which are true under the hypothesis “the residue field extensionis separable” are still valid under the more general hypothesis that the valuation ring extension is monogenic. We also show that conversely, if classical ramification properties hold true for an extension L/K, then the extension of valuation rings is monogenic. These are the “well ramified” extensions. We show that there are only three possible types of well ramified extensions and we give examples. In the last part of the paper we consider, for the three types, Kato’s generalization of the conductor, which we show how to bound in certain cases.
On the Tempered Spectrum of Quasi-Split Classical Groups II
Authors: Goldberg David , Shahidi Freydoon
Published: December 2010

We determine the poles of the standard intertwining operators for a maximal parabolic subgroup of the quasi-split unitary group defined by a quadratic extension E/F of p-adic fields of characteristic zero. We study the case where the Levi component M ≃ GLn(E) × Um(F), with n ≡ m (mod 2). This, along with earlier work, determines the poles of the local Rankin-Selberg product L-function L(s, t′ × τ), with t′ an irreducible unitary supercuspidal representation of GLn(E) and τ a generic irreducible unitary supercuspidal representation of Um(F). The results are interpreted using the theory of twisted endoscopy.
Inequivalent Transitive Factorizations into Transpositions
Authors: Goulden I. P. , Jackson D. M. , Latour F. G.
Published: December 2010

The question of counting minimal factorizations of permutations into transpositions that act transitively on a set has been studied extensively in the geometrical setting of ramified coverings of the sphere and in the algebraic setting of symmetric functions.

It is natural, however, from a combinatorial point of view to ask how such results are affected by counting up to equivalence of factorizations, where two factorizations are equivalent if they differ only by the interchange of adjacent factors that commute. We obtain an explicit and elegant result for the number of such factorizations of permutations with precisely two factors. The approach used is a combinatorial one that rests on two constructions.

We believe that this approach, and the combinatorial primitives that have been developed for the “cut and join” analysis, will also assist with the general case.
The Symplectic Geometry of Polygons in the 3-Sphere
Authors: Treloar Thomas
Published: December 2010

We study the symplectic geometry of the moduli spaces Mr = Mr() of closed n-gons with fixed side-lengths in the 3-sphere. We prove that these moduli spaces have symplectic structures obtained by reduction of the fusion product of n conjugacy classes in SU(2) by the diagonal conjugation action of SU(2). Here the fusion product of n conjugacy classes is a Hamiltonian quasi-Poisson SU(2)-manifold in the sense of [AKSM]. An integrable Hamiltonian system is constructed on Mr in which the Hamiltonian flows are given by bending polygons along a maximal collection of nonintersecting diagonals. Finally, we show the symplectic structure on Mr relates to the symplectic structure obtained from gauge-theoretic description of Mr. The results of this paper are analogues for the 3-sphere of results obtained for Mr(), the moduli space of n-gons with fixed side-lengths in hyperbolic 3-space [KMT], and for Mr(), the moduli space of n-gons with fixed side-lengths in [KM1].
On the Classification of Simple Stably Projectionless C*-Algebras
Authors: Razak Shaloub
Published: December 2010

It is shown that simple stably projectionless C*-algebras which are inductive limits of certain specified building blocks with trivial K-theory are classified by their cone of positive traces with distinguished subset. This is the first example of an isomorphism theorem verifying the conjecture of Elliott for a subclass of the stably projectionless algebras.
Lie Algebras of Pro-Affine Algebraic Groups
Authors: Nahlus Nazih
Published: December 2010

We extend the basic theory of Lie algebras of affine algebraic groups to the case of pro-affine algebraic groups over an algebraically closed fieldKof characteristic 0. However, some modifications are needed in some extensions. So we introduce the pro-discrete topology on the Lie algebra ℒ(G) of the pro-affine algebraic groupGoverK, which is discrete in the finite-dimensional case and linearly compact in general. As an example, ifLis any sub Lie algebra of ℒ(G), we show that the closure of [L,L] in ℒ(G) is algebraic in ℒ(G).

We also discuss the Hopf algebra of representative functions H(L) of a residually finite dimensional Lie algebraL. As an example, we show that ifLis a sub Lie algebra of ℒ(G) andGis connected, then the canonical Hopf algebra morphism fromK[G] intoH(L) is injective if and only ifLis algebraically dense in ℒ(G).
q-Integral and Moment Representations for q-Orthogonal Polynomials
Authors: Ismail Mourad E. H. , Stanton Dennis
Published: December 2010

We develop a method for deriving integral representations of certain orthogonal polynomials as moments. These moment representations are applied to find linear and multilinear generating functions for q-orthogonal polynomials. As a byproduct we establish new transformation formulas for combinations of basic hypergeometric functions, including a new representation of the q-exponential function εq.
Admissible Majorants for Model Subspaces of H2, Part I: Slow Winding of the Generating Inner Function
Authors: Havin Victor , Mashreghi Javad
Published: December 2010

A model subspace Kϴ of the Hardy space H2 = H2(ℂ+) for the upper half plane ℂ+ is H2(ℂ+) ϴ ϴH2(ℂ+) where ϴ is an inner function in ℂ+. A function ω: ⟼ [0,∞) is called an admissible majorant for Kϴ if there exists an f ∈ Kϴ, f ≢ 0, |f(x)| ≤ ω(x) almost everywhere on ℝ. For some (mainly meromorphic) ϴ's some parts of Adm ϴ (the set of all admissible majorants for Kϴ) are explicitly described. These descriptions depend on the rate of growth of argϴ along ℝ. This paper is about slowly growing arguments (slower than x). Our results exhibit the dependence of Adm B on the geometry of the zeros of the Blaschke product B. A complete description of Adm B is obtained for B's with purely imaginary (“vertical”) zeros. We show that in this case a unique minimal admissible majorant exists.
Ternary Diophantine Equations via Galois Representations and Modular Forms
Authors: Bennett Michael A. , Skinner Chris M.
Published: December 2010

In this paper, we develop techniques for solving ternary Diophantine equations of the shape Axn + Byn = Cz2, based upon the theory of Galois representations and modular forms. We subsequently utilize these methods to completely solve such equations for various choices of the parameters A, B andC. We conclude with an application of our results to certain classical polynomial-exponential equations, such as those of Ramanujan–Nagell type.
Éléments unipotents réguliers des sous-groupes de Levi
Authors: Bonnafé Cédric
Published: December 2010

Nous étudions la structure du centralisateur d'un élément unipotent régulier d'un sousgroupe de Levi d'un groupe réductif, ainsi que la structure du groupe des composantes de ce centralisateur en relation avec la notion de système local cuspidal définie par Lusztig. Nous déterminons son radical unipotent, montrons l'existence d'un complément de Levi et étudions la structure de son groupe de Weyl. Comme application, nous démontrons des résultats qui étaient annoncés dans un précédent article de l'auteur sur les éléments unipotents réguliers.
Non-Abelian Generalizations of the Erdős-Kac Theorem
Authors: Murty M. Ram , Saidak Filip
Published: December 2010

Let a be a natural number greater than 1. Let fa(n) be the order of a mod n. Denote by ω(n) the number of distinct prime factors of n. Assuming a weak form of the generalised Riemann hypothesis, we prove the following conjecture of Erdös and Pomerance:

The number of n ≤ x coprime to a satisfying

is asymptotic to as x tends to infinity.
On the Neumann Problem for the Schrödinger Equations with Singular Potentials in Lipschitz Domains
Authors: Tao Xiangxing , Wang Henggeng
Published: December 2010

We consider the Neumann problem for the Schrödinger equations –Δu + Vu = 0, with singular nonnegative potentials V belonging to the reverse Hölder class ℬn, in a connected Lipschitz domain Ω Rn. Given boundary data g in Hp or Lp for 1 – ɛ < p ≤ 2, where 0 < ɛ < , it is shown that there is a unique solution, u, that solves the Neumann problem for the given data and such that the nontangential maximal function of ▽u is in Lp(∂Ω). Moreover, the uniform estimates are found.
Variations of Mixed Hodge Structures of Multiple Polylogarithms
Authors: Zhao Jianqiang
Published: December 2010

It is well known that multiple polylogarithms give rise to good unipotent variations of mixed Hodge-Tate structures. In this paper we shall explicitly determine these structures related to multiple logarithms and some other multiple polylogarithms of lower weights. The purpose of this explicit construction is to give some important applications. First we study the limit of mixed Hodge-Tate structures and make a conjecture relating the variations of mixed Hodge-Tate structures of multiple logarithms to those of general multiple polylogarithms. Then following Deligne and Beilinson we describe an approach to defining the single-valued real analytic version of the multiple polylogarithms which generalizes the well-known result of Zagier on classical polylogarithms. In the process we find some interesting identities relating single-valued multiple polylogarithms of the same weight k when k = 2 and 3. At the end of this paper, motivated by Zagier's conjecture we pose a problem which relates the special values of multiple Dedekind zeta functions of a number field to the single-valued version of multiple polylogarithms.
Extensions by Simple C*-Algebras: Quasidiagonal Extensions
Authors: Lin Huaxin
Published: December 2010

Let A be an amenable separable C*-algebra and B be a non-unital but σ-unital simple C*- algebra with continuous scale. We show that two essential extensions τ1 and τ2 of A by B are approximately unitarily equivalent if and only if

If A is assumed to satisfy the Universal Coefficient Theorem, there is a bijection fromapproximate unitary equivalence classes of the abovementioned extensions to KL(A,M(B)/B). Using KL(A,M(B)/B), we compute exactly when an essential extension is quasidiagonal. We show that quasidiagonal extensions may not be approximately trivial. We also study the approximately trivial extensions.
Convex Polynomial Approximation in the Uniform Norm: Conclusion
Authors: Kopotun K. A. , Leviatan D. , Shevchuk I. A.
Published: December 2010

Estimating the degree of approximation in the uniform norm, of a convex function on a finite interval, by convex algebraic polynomials, has received wide attention over the last twenty years. However, while much progress has been made especially in recent years by, among others, the authors of this article, separately and jointly, there have been left some interesting open questions. In this paper we give final answers to all those open problems. We are able to say, for each r-th differentiable convex function, whether or not its degree of convex polynomial approximation in the uniform norm may be estimated by a Jackson-type estimate involving the weighted Ditzian–Totik kth modulus of smoothness, and how the constants in this estimate behave. It turns out that for some pairs (k, r) we have such estimate with constants depending only on these parameters. For other pairs the estimate is valid, but only with constants that depend on the function being approximated, while there are pairs for which the Jackson-type estimate is, in general, invalid.
Extremal Metric for the First Eigenvalue on a Klein Bottle
Authors: Jakobson Dmitry , Nadirashvili Nikolai , Polterovich Iosif
Published: December 2010

The first eigenvalue of the Laplacian on a surface can be viewed as a functional on the space of Riemannian metrics of a given area. Critical points of this functional are called extremal metrics. The only known extremal metrics are a round sphere, a standard projective plane, a Clifford torus and an equilateral torus. We construct an extremal metric on a Klein bottle. It is a metric of revolution, admitting a minimal isometric embedding into a sphere by the first eigenfunctions. Also, this Klein bottle is a bipolar surface for Lawson's -torus. We conjecture that an extremal metric for the first eigenvalue on a Klein bottle is unique, and hence it provides a sharp upper bound for λ1 on a Klein bottle of a given area. We present numerical evidence and prove the first results towards this conjecture.
Smooth Values of the Iterates of the Euler Phi-Function
Authors: Lamzouri Youness
Published: December 2010

Let ϕ(n) be the Euler phi-function, define ϕ0(n) = n and ϕk+1(n) = ϕ(ϕk(n)) for all k ≥ 0. We will determine an asymptotic formula for the set of integers n less than x for which ϕk(n) is y-smooth, conditionally on a weak form of the Elliott–Halberstam conjecture.
Eigenvalues,K-theory and Minimal Flows
Authors: Itzá-Ortiz Benjamín A.
Published: December 2010

Let (Y,T) be a minimal suspension flow built over a dynamical system (X,S) and with (strictly positive, continuous) ceiling functionf:X→ ℝ. We show that the eigenvalues of (Y,T) are contained in the range of a trace on theK0-group of (X,S). Moreover, a trace gives an order isomorphism of a subgroup ofK0(C(X) ⋊Sℤ) with the group of eigenvalues of (Y,T). Using this result, we relate the values oftfor which the time-tmap on the minimal suspension flow is minimal with theK-theory of the base of this suspension.
Hecke L-Functions and the Distribution of Totally Positive Integers
Authors: Ash Avner , Friedberg Solomon
Published: December 2010

Let K be a totally real number field of degree n. We show that the number of totally positive integers (or more generally the number of totally positive elements of a given fractional ideal) of given trace is evenly distributed around its expected value, which is obtained fromgeometric considerations. This result depends on unfolding an integral over a compact torus.
Large Sieve Inequalities via Subharmonic Methods and the Mahler Measure of the Fekete Polynomials
Authors: Erdélyi T. , Lubinsky D. S.
Published: December 2010

We investigate large sieve inequalities such as where is convex and increasing, P is a polynomial or an exponential of a potential, and the constant C depends on the degree of P, and the distribution of the points 0 ≤ Ƭ1 < Ƭ2 < … < Ƭm ≤ 2π. The method allows greater generality and is in some ways simpler than earlier ones. We apply our results to estimate the Mahler measure of Fekete polynomials.
Non-Backtracking Random Walks and Cogrowth of Graphs
Authors: Ortner Ronald , Woess Wolfgang
Published: December 2010

LetXbe a locally finite, connected graph without vertices of degree 1. Non-backtracking random walk moves at each step with equal probability to one of the “forward” neighbours of the actual state,i.e.,it does not go back along the preceding edge to the preceding state. This is not a Markov chain, but can be turned into a Markov chain whose state space is the set of oriented edges ofX. Thus we obtain for infiniteXthat then-step non-backtracking transition probabilities tend to zero, and we can also compute their limit whenXis finite. This provides a short proof of an old result concerning cogrowth of groups, and makes the extension of that result to arbitrary regular graphs rigorous. Even whenXis non-regular, butsmall cycles are dense in X, we show that the graphXis non-amenable if and only if the non-backtrackingn-step transition probabilities decay exponentially fast. This is a partial generalization of the cogrowth criterion for regular graphs which comprises the original cogrowth criterion for finitely generated groups of Grigorchuk and Cohen.
The Geometry of L0
Authors: Kalton N. J. , Koldobsky A. , Yaskin V. , Yaskina M.
Published: December 2010

Suppose that we have the unit Euclidean ball in ℝn and construct new bodies using three operations — linear transformations, closure in the radial metric, and multiplicative summation defined by We prove that in dimension 3 this procedure gives all origin-symmetric convex bodies, while this is no longer true in dimensions 4 and higher. We introduce the concept of embedding of a normed space in L0 that naturally extends the corresponding properties of Lp-spaces with p ≠ 0, and show that the procedure described above gives exactly the unit balls of subspaces of L0 in every dimension. We provide Fourier analytic and geometric characterizations of spaces embedding in L0, and prove several facts confirming the place of L0 in the scale of Lp-spaces.
Convexity of the Field of a Linear Transformation
Authors: Goldman A. J. , Marcus M.
Published: December 2010

Let Unbe an n-dimensional unitary space with inner productIn Unlet Sn-1denote the unit sphere:

Let A be an arbitrary linear transformation of Un.
Some Properties of Beatty Sequences II
Authors: Connell Ian G.
Published: December 2010

In a previous paper [l] we discussed a property of the complementary sequences

1

where square brackets denote the greatest integer function and α is any positive irrational. We called {un} and {vn} Beatty sequences of argument α.
Some Examples of Complemented Modular Lattices
Authors: Grätzer G. , Wonenburger Maria J.
Published: December 2010

Let L be a complemented, χ-complete modular lattice. A theorem of Amemiya
and Halperin (see [l], Theorem 4.3) asserts that if the intervals [O, a] and
[O, b], a, bεL, are upper χ-continuous then [O, a∪b] is also upper
χ-continuous. Roughly speaking, in L upper χ-continuity is additive. The
following question arises naturally: is χ-completeness an additive property
of complemented modular lattices? It follows from Corollary 1 to Theorem 1
below that the answer to this question is in the negative.

A complemented modular lattice is called a Von Neumann geometry or
continuous geometry if it is complete and continuous. In particular a
complete Boolean algebra is a Von Neumann geometry. In any case in a Von
Neumann geometry the set of elements which possess a unique complement form
a complete Boolean algebra. This Boolean algebra is called the centre of the
Von Neumann geometry. Theorem 2 shows that any complete Boolean algebra can
be the centre of a Von Neumann geometry with a homogeneous basis of order n
(see [3] Part II, definition 3.2 for the definition of a homogeneous basis),
n being any fixed natural integer.
The Non-Biplanar Character of the Complete 9-Graph
Authors: Tutte W. T.
Published: December 2010

Let us define a planar partition of a graph G as a pair {H1, H2} of subgraphs of G with the following properties

(i)

Each of H1 and H2 includes all the vertices of G.

(ii)

Each edge of G belongs to just one of H1 and H2.

(iii)

H1 and H2 are planar graphs.

It is not required that H1 and H2 are connected. Moreover either of these graphs may have isolated vertices, incident with none of its edges.

We describe a graph having a planar partition as biplanar.
On Edge-Disjoint Cycles in a Graph
Authors: Moon J. W.
Published: December 2010

Let g(k) denote the least integer such that every graph, with n vertices and n+g(k) edges, contains at least k edge-disjoint cycles; let h(k) be similarly defined for planar graphs. Loops and multiple edges (i.e., cycles of length one and two) are permitted in both cases.
A Dual form of Kuratowski's Theorem
Authors: Harary Frank , Tutte William T.
Published: December 2010

The celebrated criterion of Kuratowski [2] for the planarity of a graph G involves the determination of whether G contains a subgraph homeomorphic to K5 or K3, 3 shown in Figure 1.
Unitary Perfect Numbers
Authors: Subbarao M.V. , Warren L. J.
Published: December 2010

Let σ*(N) denote the sum of the unitary divisors of N, that is,

It is easily seen that σ*(N) is multiplicative. In fact σ*(l) = 1 if N > 1 has the prime decomposition . Let us define a positive integer to be unitary perfect whenever σ*(N) = 2N. The first four such numbers are 6, 60, 90 and 87, 360. In a recent abstract [ l ] published by one of us, the last of these numbers was overlooked. No other unitary perfect numbers are known to the authors.
On Very Large One Sided Ideals of a Ring
Authors: Koh Kwangil
Published: December 2010

If R is a ring, a right (left) ideal of R is said to be large if it has non-zero intersection with each non-zero right (left) ideal of R [8]. If S is a set, let |S| be the cardinal number of S. We say a right (left) ideal I of a ring R is very large if |R/I| < < No. If a is an element of a ring R such that (a)r = {r ∊ R|ar = 0} is very large then we say a is very singular. The set of all very singular elements of a ring R is a two sided ideal of R. If R is a prime ring, then 0 is the only very singular element of R and a very large right (left) ideal of R is indeed large provided that R is not finite.
On Fundamental Constructions and Adjoint Functors
Authors: Maranda J. -M.
Published: December 2010

A fundamental construction of a category ζ((2), Appendice) is a triple (S, p, k), where S is a functor from ζ to itself and 2 where p:S2→S and k:1ζ→S are natural transformations such that
Some Remarks on a Paper of D. W. Kahn
Authors: Hoo C.S.
Published: December 2010

Suppose X is a simply-connected CW- complex whose homotopy groups are finite. For each n, let be the class of torsion groups whose p - primary components are zero for all primes p which do not figure in the homotopy groups πi(X) for i ≤ n.
A Non-Trivial Ring with Non-Rational Injective Hull
Authors: Berthiaume P.
Published: December 2010

Several authors have investigated "rings of quotients" of a given ring R . (See, for example, Johnson [7], Johnson and Wong [8], Utumi [11], Findlay and Lambek [5], Lambek [9], and Bourbaki [2].)
Ring Theoretic Properties of Matrix Rings
Authors: Kaye S.M.
Published: December 2010

K. Morita has shown that, given two rings R and S, there is an isomorphism between the category of left R-modules and the category of left S-modules if and only if there exists an R-S bimodule U such that

(1) U is a progenerator in the category of left R-modules, and

(2) S ≅ (EndR U)opp as rings.
A Theorem on Henselian Rings
Authors: Sankaran N.
Published: December 2010

It is known that if K is a field, then the ring of formal power series in one or more variables, with coefficients in K, is Henselian at its maximal ideal. In this note we show that if R is a ring (commutative and with identity element) which is Henselian at the maximal ideals M1, M2, … then R[[x]] - the ring of formal power series in x with coefficients from R - is also Henselian at the maximal ideals M1 ⋅ R[[x]] + x⋅ R[[x]], etc.
Transformation of Life-Test Data
Authors: Draper Norman R. , Guttman Irwin
Published: December 2010

In a recent paper Box and Cox (1964) considered the problem of transforming dependent variables in regression and analysis of variance problems, in order to achieve the usual assumptions of Normality, constant variance and additivity of effects. Here we adopt the same approach to investigate transformations of data which allow the transformed observations to follow a Gamma distribution. A special case of this is the exponential distribution, valuable in life-testing, for which examples are given.
An Edge but not Vertex Transitive Cubic Graph*
Authors: Bouwer I. Z.
Published: December 2010

Let G be an undirected graph, without loops or multiple edges. An automorphism of G is a permutation of the vertices of G that preserves adjacency. G is vertex transitive if, given any two vertices of G, there is an automorphism of the graph that maps one to the other. Similarly, G is edge transitive if for any two edges (a, b) and (c, d) of G there exists an automorphism f of G such that {c, d} = {f(a), f(b)}. A graph is regular of degree d if each vertex belongs to exactly d edges.
A General Selection Principle, with Applications in Analysis and Algebra
Authors: Rice Norman M.
Published: December 2010

The General Selection Principle referred to in the title is really Tychonoff's theorem on products of compact spaces, but in a somewhat disguised form (c.f. Theorem 2.1, below). It is believed that this form is one which lends itself very well to many applications. More specifically, one of the immediate corollaries of the main theorem is a theorem due to Rado (c.f. Cor 2.1.1.), which has been used by Erdos and de Bruijn [2], Luxemburg [7], and the author [9] to give simple proofs of a variety of results.
A Combinatorial Proof of a Conjecture of Goldberg and Moon
Authors: Alspach Brian
Published: December 2010

Let Tn denote a tournament of order n, let G(Tn) denote the automorphism group of Tn, let |G| denote the order of the group G, and let g(n) denote the maximum of |G(Tn)| taken over all tournaments Tn of order n. Goldberg and Moon conjectured [2] that for all n≥1 with equality holding if and only if n is a power of 3. In an addendum to [2] it was pointed out that their conjecture is equivalent to the conjecture that if G is any odd order subgroup of Sn, the symmetric group of degree n, then with equality possible if and only if n is a power of 3. The latter conjecture was proved in [1] by John D. Dixon who made use of the Feit-Thompson theorem in his proof. In this paper we avoid use of the Feit-Thompson result and give a combinatorial proof of the Goldberg-Moon conjecture.
Non-Smooth Geodesic Flows and Classical Mechanics
Authors: Marsden J. E.
Published: December 2010

As is well known, there is an intimate connection between geodesic flows and Hamiltonian systems. In fact, if g is a Riemannian, or pseudo-Riemannian metric on a manifold M (we think of M as q-space or the configuration space), we may define a smooth function Tg on the cotangent bundle T*M (q-p-space, or the phase space). This function is the kinetic energy of q, and locally is given by
A Characterization of the Normal and
Weibull Distributions
Authors: Seshadri V.
Published: December 2010

Let X and Y be two independent normal variates each distributed with zero mean and a common variance. Then the quotient X/Y has the Cauchy distribution symmetrical about the origin. Of particular interest in recent years has been the converse problem and examples of non-normal distributions with a Cauchy distribution for the quotient have been illustrated by Mauldon [9], Laha [2; 3; 4] and Steck [10].
Pure Simple and Indecomposable Rings
Authors: Fieldhouse David J.
Published: December 2010

P. M. Cohn [7] calls a submodule P of the left A-module M pure iff 0 → E ⊗ P → E ⊗ M is exact for all right modules E. This concept has been studied in [11] and [12]. We will call a non-zero module pure simple iff its only pure submodules are 0 and itself, and the ring A left pure simple iff it is pure simple as a left A-module. We relate these concepts to the PP and PF rings of Hattori [13], and give several new characterizations of these rings. In order to establish these, we use the following known result: the Jacobson radical of any module is the sum of all its small submodules.

Parts of this paper are contained in the author's doctoral thesis [10] at McGill University.
A Simple Solution to the Word Problem for
Lattices
Authors: Day Alan
Published: December 2010

Whitman [2] solved the word problem for lattices by giving an explicit
construction of the free lattice, FL(X),
on a given set of generators X.

The solution is the following:

For x, y ∊ X, and
a, b, c,
d ∊ FL(X),

(W1)

(W2)

(W3)

(W4)

where [p, q] = {x;
p ≤ x ≤ q}.

The purpose of this note is to give a simple nonconstructive proof that the
condition (W4) must hold in every projective (hence every free) lattice.
Jonsson [1] has shown that in every equational class of lattices (Wl), (W2),
and (W3) hold. Therefore the combination of these results gives a complete
nonconstructive solution to the word problem for lattices.
Left Cauchy Integral Bases in Linear Topological Spaces
Authors: Dyer James A.
Published: December 2010

The purpose of this paper is to consider a representation for the elements of a linear topological space in the form of a σ-integral over a linearly ordered subset of V; this ordered subset is what will be called an L basis. The formal definition of an L basis is essentially an abstraction from ideas used, often tacitly, in proofs of many of the theorems concerning integral representations for continuous linear functionals on function spaces.

The L basis constructed in this paper differs in several basic ways from the integral basis considered by Edwards in [5]. Since the integrals used here are of Hellinger type rather than Radon type one has in the approximating sums for the integral an immediate and natural analogue to the partial sum operators of summation basis theory.
The Toeplitz-Hausdorff Theorem Explained
Authors: Davis Chandler
Published: December 2010

Paul Halmos expressed [3, p. 110] the general dissatisfaction with the usual proofs of this famous and important theorem. They all make it seem like an accidental product of a computation. A more conceptual proof was devised by N. P. Dekker. In spite of the elegance of his proof, the one offered below may have some claim to be regarded as "the reason the theorem is true".
Generalized de la Vallée Poussin Disconjugacy Tests for Linear Differential Equations(1)
Authors: Willett D.
Published: December 2010

In this paper, we study the oscillatory behavior of the solutions of the linear differential equation

(1.1)

where

(1.2)

and all functions are assumed to be continuous on a bounded interval [a, b). An «th-order linear equation is said to be disconjugate on an interval I provided it has no nontrivial solution with more than n — 1 zeros, counting multiplicities, in I.
An Analytic Proof of a Theorem of Felix Klein
Authors: Griffith Gareth J.
Published: December 2010

In 1876, Klein published the following result: “If a crunode of a real, irreducible, plane, algebraic curve changes into an acnode via the intermediary stage of a real cusp, two real inflexions are introduced in a neighborhood of the double points” [2].
A Lebesgue Decomposition Theorem for C* Algebras
Authors: Henle Michael
Published: December 2010

This paper, by generalizing von Neumann's proof of the Radon-Nikodym and Lebesgue decomposition theorems [3], obtains analogous results for positive linear functional on a C* algebra. The concept of "absolute continuity" used and the Radon-Nikodym portion of the resulting theorem are due to Dye [2].
Some Theorems on Convex Polygons
Authors: Altman E.
Published: December 2010

In this paper diagonals of various orders in a (strict) convex polygon Pn are considered. The sums of lengths of diagonals of the same order are studied. A relationship between the number of consecutive diagonals which do not intersect a given maximal diagonal and lie on one side of it and the order of the smallest diagonal among them is established. Finally a new proof of a conjecture of P. Erdos, considered already in [1], is given.
Functions with a Finite Number of Negative Squares
Authors: Stewart James
Published: December 2010

Let f be a complex-valued function defined on the real line R with the property that for every x∊R. If k is a nonnegative integer,f is said to have k negative squares, or to be indefinite of order k, if the Hermitian form
A Cure for the Telephone Disease
Authors: Hajnal A. , Milner E. C. , Szemerédi E.
Published: December 2010

The following problem due to A. Boyd, has enjoyed a certain popularity in recent months with several mathematicians. A different solution to the one given here has been given independently by R. T. Bumby and J. Spencer.

The Problem, There are n ladies, and each one of them knows an item of scandal which is not known to any of the others. They communicate by telephone, and whenever two ladies make a call, they pass on to each other, as much scandal as they know at that time. How many calls are needed before all the ladies know all the scandal?
Pullbacks in Regular Categories
Authors: Meisen Jeanne
Published: December 2010

Given a pair of maps

in a category, we would like to know whether they form part of a pullback diagram as follows:

I am indebted to Basil Rattray for mentioning the solution of this problem for the category of sets. Here we shall solve it for any regular category in the sense of Barr [1].

It will be useful to make the following definition.
Frieze Patterns in the Hyperbolic Plane
Authors: Garner C. W. L.
Published: December 2010

It is well known that in the Euclidean plane there are seven distinct frieze patterns, i.e. seven ways to generate an infinite design bounded by two parallel lines. In the hyperbolic plane, this can be generalized to two types of frieze patterns, those bounded by concentric horocycles and those bounded by concentric equidistant curves. There are nine such frieze patterns; as in the Euclidean case, their symmetry groups are and
Etats Accessibles Dans Un Processus De Galton-Watson
Authors: Dubuc Serge
Published: December 2010

Si S est un ensemble dénombrable sur lequel porte une chaîne de Markov, si pn(x, y) est la probabilité de transition de x à y au ne instant, on posera An(x) la totalité des y de S tels que pn(x, y)>0, ce sont les états accessibles à partir de x à l’instant n. Dans le cas de deux types de chaînes de Markov, nous recherchons des informations sur les états accessibles; ce sera premièrement les marches aléatoires sur Z et deuxièmement les processus de Galton-Watson sur ℕ. Nous recherchons en particulier les obstructions arithmétiques à ce qu’un état soit accessible.
A Note on Dualizing Goldie Dimension
Authors: Fleury Patrick
Published: December 2010

The purpose of this note is to offer a equalization of the concept of Goldie dimension and to prove a structure theorem (Theorem 3.1) for modules satisfying the conditions of this dualization. In this paper, all rings considered are associative with unit and all modules are unital.
The Existence of Injective Effacements
Authors: Barr Michael
Published: December 2010

The main result of this paper is little more than a juxtaposition of a remark in [Leroux] (§5, part A), with a theorem of [Banaschewski]. However, the results are individually little known and their juxtaposition not at all. More explicitly, Leroux remarks, in somewhat different form,

Let the category X be complete and cocomplete, and have a set of cogenerators. Then a necessary and sufficient condition that there exist infective effacements (with respect to the class of regular monomorphisms) is that X be coregular and satisfy, in addition, condition COEX 4*.
The Fifth Unitary Perfect Number
Authors: Wall Charles R.
Published: December 2010

A divisor d of a positive integer n is a unitary divisor if d and n/d are relatively prime. An integer is said to be unitary perfect if it equals the sum of its proper unitary divisors. Subbarao and Warren [2] gave the first four unitary perfect numbers: 6, 60, 90 and 87360. In 1969,1 reported [3] that

is also unitary perfect. The purpose of this paper is to show that this last number, which for brevity we denote by W, is indeed the next unitary perfect number after 87360.
On the Existence and Uniqueness of Solutions of the Equation
Authors: Cleménts John C.
Published: December 2010

The existence and uniqueness of strong global solutions of initial-boundary value problems for the quasilinear equation utt—∂σi(uxi)/∂xi—ΔNut= f is established for functions σi(ξ), i = 1, …, N, satisfying: σi,(ξ) ∊ C1(-∞, ∞), σi(0) = 0 and
for some constant K0.
Independence in Combinatorial Geometries of Rank Three
Authors: Hall Japheth
Published: December 2010

The class of allcombinatorial geometries of rank threeshall coincide with the class of all pairs (V, S) such thatVis a set andSis a collection of non-empty subsets ofVsuch that each pair of distinct elements ofVbelong to exactly one member ofS. (See [3].)

Consider a combinatorial geometry (V, S) of rank three.
A Fractional Differentiation Theorem for the Laplace Transform
Authors: Conlan J. , Koh E. L.
Published: December 2010

In certain systems analysis ([1], [2], [3]), it is essential to invert the n-dimensional Laplace transform and specify the inverse image at a single variable t.
On the Global Dimensions of D+M
Authors: Dobbs David E.
Published: December 2010

This note answers affirmatively a question of the author [4, p. 456], by producing an example of an integrally closed quasi-local non valuation domain of global dimension 3, each of whose overrings is a goingdown ring. Although [4, Proposition 4.5] shows that such an example cannot be constructed by means of restrained power series, an approach via the more generalD+Mconstruction succeeds.
Support Points of the Class of Close-to-Convex Functions
Authors: Grassmann E. , Hengartner W. , Schober G.
Published: December 2010

Let H(U) be the linear space of holomorphic functions on U = {z:|z|<1} endowed with the topology of compact convergence, and denote by H′(U) its topological dual space. Let be a compact subset of H(U) and ƒ∈F. We say ƒ is a support point of if there exists an L∈H'(U), non-constant on , such that On the other hand, ƒ is an extreme point of if ƒ is not a proper convex combination of two other points of .
Solution of the ‘Cube’ Functional Equation in Terms of ‘Trilinear Coefficients’
Authors: Hemdan H. T.
Published: December 2010

We consider the following three functional equations

1

2

3

where f:R3→R.

Considering their geometric meaning, equations (1) and (2) are known as ‘Cube’ and ‘Octahedron’ functional equations, respectively. Under the assumption of continuity, Haruki [2] has proved that (1) and (2) are equivalent. Etigson [3], has proved the equivalence of (1) and (2) under no regularity assumption. We will give here another proof. Also, under the assumption of continuity, Haruki has solved the ‘Cube’ functional equation. He gave the solution as a certain polynomial of fifth degree in x, y, z individually whose terms are the partial derivatives of a given polynomial.
Amenability and Substantial Semigroups
Authors: Wong James C. S.
Published: December 2010

In this paper, we introduce the concept of topological left substantial subsemigroup of a locally compact semigroup S and prove that if T is such a subsemigroup in S and M(T) is the measure algebra of T, then M(T)* is topological left amenable iff M(S)* is topological left amenable, an extension of a similar result for discrete semigroups.
Convex Sets, Cantor Sets and a Midpoint Property
Authors: Reiter Harold
Published: December 2010

It is well known that every point of the closed unit interval I can be expressed as the midpoint of two points of the Cantor ternary set D. See [2, p. 549] and [3, p. 105]. Regarding J as a one dimensional compact convex set, it seems natural to try to generalize the above result to higher dimensional convex sets. We prove in section 3 that every convex polytope K in Euclidean space Rd contains a topological copy C of D such that each point of K is expressible as a midpoint of two points of C. Also, we give necessary and sufficient conditions on a planar compact convex set for it to contain a copy of D with the midpoint property above. In the final section we prove a result on minimal midpoint sets.
A Kneser Theorem for Higher order Elliptic Equations
Authors: Allegretto W.
Published: December 2010

The problem of establishing oscillation and non-oscillation criteria for elliptic equations has recently been considered by several authors. Extensive bibliographies may be found in the books by C. A. Swanson, [7], and by K. Kreith, [3].

Most of the interest has so far centered on equations of second order with some results also established for fourth order equations. Non-oscillation theorems for higher order equations have recently been established by the author, [1], and by Noussair and Yoshida, [5]. In particular, both in [1] and [5], Kneser-type theorems were established for classes of higher order elliptic equations.
Direct Product Decompositions of Elation Groups
Authors: Brown Julia M. Nowlin
Published: December 2010

Let G be a collineation group of a projective plane π. Let E be the subgroup generated by all elations in G. In the case that π is finite and G fixes no point or line, F. Piper [6; 7] has proved that if G contains certain combinations of perspectivities, then E is isomorphic to for some finite field g.
Intersection Theorems for Systems of Sets
Authors: Spencer Joel
Published: December 2010

Let n and k be positive integers, k≥3. Denote by ϕ(n, k) the least positive integer such that if F is any family of more than ϕ(n, k) sets, each set with n elements, then some k members of F have pairwise the same intersection. In this paper we obtain a new asymptotic upper bound for ϕ(n, k), k fixed, n approaching infinity.
Instabilité Des Corps Formellement Réels
Authors: Duret Jean-Louis
Published: December 2010

‖ A ‖ désignera le cardinal de A. Soient T une théorie complète du premier ordre dans un langage dénombrable, M un modèle de T, X un sous-ensemble de M; soit T(X) l'ensemble des formules closes à paramètres dans X satisfaites par M; on appelle n-type de T sur X un ensemble consistant avec T(X) et maximal de formules à paramètres dans X à n variables libres.
A Note on Lipschitzian Mappings in Convex Metric Spaces
Authors: Kirk W. A. , Ray WIlliam O.
Published: December 2010

Local criteria are given which imply the lipschitzian character of mappings defined in complete and convex metric spaces.
Pointwise Contraction Criteria for the Existence of Fixed Points
Authors: Clarke Frank H.
Published: December 2010

We show that, in a complete metric space, every selfmap that is a “weak directional contraction” admits a fixed point.
On a Theorem of Sullivan
Authors: Penna Michael A.
Published: December 2010

The purpose of this note is to give an elementary geometric proof of the following result stated by Sullivan (see (4)).

Theorem 1 (Sullivan). Let K be a finite simplicial complex with vertices v1, …, vN and corresponding barycentric coordinates b1, …, bN. Then the algebra of rational PL forms on K
Content Algebras
Authors: Rush David E.
Published: December 2010

Let R be a commutative ring with identity and let X be an indeterminate. In [5] and [16] it was shown that if f, g∈R[X], then for some integer n≥l, c(f)n+1c(g) = c(g)nc(fg), where c(h) denotes the additive subgroup of JR generated by the coefficients of h ∈ R[X]. Actually the statements in [5J and [16] are not so general as this; however, the proofs are. Specifically, in [16] Mertens considers only the case that R is a polynomial ring over the integers, but this gives the result for any ring by specializing the coefficients of f and g. In [5] Dedekind considers only the case that JR is a ring of algebraic integers, but his proof is completely general. Further, the above formula then holds if one lets c(h) denote the S-submodule of R generated by the coefficients of h ∈ R[X], S a subring of R, and it is in this form that it usually appears, especially the case S = R. Dedekind′s very elegant proof is reproduced in [15, p. 9, Lemma 6.1].
Hardy Inequalities with Mixed Norms
Authors: Bradley J. Scott
Published: December 2010

We give a necessary and sufficient condition on weight functions u and v such that for l≤p≤q≤∞ there exists a constant C for which

A corresponding dual result is also given. This extends a result of B. Muckenhoupt which appeared in Studia Math., 34 (1972).
Limit Distributions for the Extreme Order Statistics
Authors: Mejzler D.
Published: December 2010

Let Xl, …, Xn be independent random variables with the same distribution function (df) F(x) and let Xln≤X2n≤…≤nr be the corresponding order statistics. The (df) of Xkn will be denoted always by Fkn(x). Many authors have investigated the asymptotic behaviour of the maximal term Xnn as n → ∞. Gnedenko [3] proved the following
Tournaments whose Subtournaments are Irreducible or Transitive
Authors: Moon J. W.
Published: December 2010

Beineke and Harary gave an example of a family of tournaments Tn such that every subtournament of Tn is irreducible or transitive. We characterize all tournaments with this property.
On the Positive Definiteness of a Functional
Authors: Pandey J. N.
Published: December 2010

In this note, we have shown that the set of conditions given by Gel'fand and Vilenkin for the positive definiteness of a functional L(ϕ) [1, Theorem 7, p. 285] though sufficient is not necessary, thereby providing an answer to an open problem posed by them [1, Theorem 7, p. 285].
Brownian Motion—Wiener Process
Authors: Csörgő Miklós
Published: December 2010

Bachelier (1900), Einstein (1905) and Smoluchowski (1915) provided a theory of the peculiar erratic motion of small particles suspended in a liquid, first described in 1826 by the English botanist Brown. In a series of papers beginning in 1920 Wiener undertook a mathematical analysis of Brownian motion.
Central Limit Theorem for Absolute Deviations from the Sample Mean and Applications
Authors: McLeish D. L.
Published: December 2010

The following type of argument is rendered almost believable by its frequent occurrence in elementary courses in statistics. Let ξi be a sequence of independent identically distributed random variables with means μ variances σ2.
A Note on Derivations II
Authors: Herstein I. N.
Published: December 2010

In a previous note on derivations [1] we determined the structure of a prime ring R which has a derivation d≠0 such that the values of d commute, that is, for which d(x) d(y) = d(y) d(x) for all x, y∈R. Perhaps even more natural might be the question: what elements in a prime ring commute with all the values of a non-zero derivation? We address ourselves to this question here, and settle it.
Note on Automorphisms of a Free Abelian Group
Authors: Macedońska-Nosalska Olga
Published: December 2010

Let F be a free group. Denote by the quotient group by the commutator subgroup which is a free abelian group. The fact that the natural map from Aut(F) into Aut() is an epimorphism, in case when F is finitely generated, was known as a consequence of the theory of Nielsen transformations ([2]) Proposition 4.4 and [3] Corollary 3.5.1).
Stable Rings
Authors: Page S. S.
Published: December 2010

Let R be an associative ring with identity. If R is von- Neumann regular of a left v-ring, then for each left ideal, I, we have I2 = I. In this note we study rings such that for each left ideal L there exists an integer n = n(L)>0 such that Ln = Ln+1. We call such rings stable rings. We completely describe the stable commutative rings. These descriptions give rise to concepts related to, but more general than, finite Goldie dimension and T-nilpotence, and a notion of power pure.
Retracts and the Fixed Point Problem for Finite Partially Ordered Sets
Authors: Duffus Dwight , Poguntke Werner , Rival Ivan
Published: December 2010

A partially ordered set P has the fixed point property if every orderpreserving mapping f of P to P has a fixed point, that is, f(a) = a for some aϵP; call P fixed point free if P does not have the fixed point property.
Compact Derivations on Von Neumann Algebras
Authors: Tsui Sze-Kai
Published: December 2010

If is a von Neumann algebra that thas no nonzero finite discrete central projection, then there is no nontrivial compact derivation of into itself.
Commutators in Free Groups
Authors: Lyndon R. C. , Wicks M. J.
Published: December 2010

We study the representations of an element of a free group as a commutator. For a given element g of a free group F, we are interested in the set of all pairs (x, y) of elements of F such that

(1)

where [x, y] = xyx-1 y-1. If g = 1, the problem is trivial. We assume henceforth that g ≠ 1.
When are Immersions Diffeomorphisms?
Authors: Ho Chung-Wu
Published: December 2010

It is shown in this paper that an immersion of a connected, closed n-manifold into another connected n-manifold is a diffeomorphism if and only if the induced homomorphism between the fundamental groups is surjective at some point. This is proved as a consequence of a more general assertion about topological spaces.