Local subgroups of the Monster and odd code loops
Thomas M.
Richardson
1453-1531
Abstract: The main result of this work is an explicit construction of $ p$-local subgroups of the Monster, the largest sporadic simple group. The groups constructed are the normalizers in the Monster of certain subgroups of order $3^{2}$, $5^{2}$, and $7^{2}$ and have shapes $\displaystyle {3^{2 + 5 + 10}}\cdot ({M_{11}} \times GL(2,3)),\quad {5^{2 + 2 + 4}}\cdot {S_3} \times GL(2,5)),\quad {\text{and}}{7^{2 + 1 + 2}}\cdot GL(2,7)$ . These groups result from a general construction which proceeds in three steps. We start with a self-orthogonal code $C$ of length $n$ over the field $ {\mathbb{F}_p}$, where $ p$ is an odd prime. The first step is to define a code loop $L$ whose structure is based on $ C$. The second step is to define a group $N$ of permutations of functions from $\mathbb{F}_p^2$ to $L$. The final step is to show that $N$ has a normal subgroup $ K$ of order ${p^2}$. The result of this construction is the quotient group $N/K$ of shape ${p^{2 + m + 2m}}(S \times GL(2,p))$, where $m + 1 = \dim (C)$ and $S$ is the group of permutations of Aut$(C)$. To show that the groups we construct are contained in the Monster, we make use of certain lattices $\Lambda (C)$, defined in terms of the code $ C$. One step in demonstrating this is to show that the centralizer of an element of order $p$ in $N/K$ is contained in the centralizer of an element of order $p$ in the Monster. The lattices are useful in this regard since a quotient of the automorphism group of the lattice is a composition factor of the appropriate centralizer in the Monster. This work was inspired by a similar construction using code loops based on binary codes that John Conway used to construct a subgroup of the Monster of shape $ {2^{2 + 11 + 22}}\cdot ({M_{24}} \times GL(2,2))$.
Geometrical evolution of developed interfaces
Piero
de Mottoni;
Michelle
Schatzman
1533-1589
Abstract: Consider the reaction-diffusion equation in ${\mathbb{R}^N} \times {\mathbb{R}^ + }:{u_t} - {h^2}\Delta u + \varphi (u) = 0;\varphi$ is the derivative of a bistable potential with wells of equal depth and $h$ is a small parameter. If the initial data has an interface, we give an asymptotic expansion of arbitrarily high order and error estimates valid up to time $O({h^{ - 2}})$. At lowest order, the interface evolves normally, with a velocity proportional to the mean curvature. Soit l'équation de réaction-diffusion dans ${\mathbb{R}^N} \times {\mathbb{R}^ + },\quad {u_t} - {h^2}\Delta u + \varphi (u) = 0$, avec $ \varphi$ la dérivée d'un potentiel bistable à puits également profonds et $h$ un petit paramètre. Pour une condition initiale possédant une interface, on donne un développement asymptotique d'ordre arbitrairement élevé, ainsi que des estimations d'erreur valides jusqu'à un temps en $O({h^{ - 2}})$. A l'ordre le plus bas, l'interface évolue normalement, à une vitesse proportionnelle à la courbure moyenne.
Schur's partition theorem, companions, refinements and generalizations
Krishnaswami
Alladi;
Basil
Gordon
1591-1608
Abstract: Schur's partition theorem asserts the equality $S(n) = {S_1}(n)$, where $S(n)$ is the number of partitions of $n$ into distinct parts $\equiv 1,2(\mod 3)$ and ${S_1}(n)$ is the number of partitions of $ n$ into parts with minimal difference $3$ and no consecutive multiples of $3$. Using a computer search Andrews found a companion result $ S(n) = {S_2}(n)$, where $ {S_2}(n)$ is the number of partitions of $n$ whose parts ${e_i}$ satisfy ${e_i} - {e_{i + 1}} \geqslant 3,2or5$ according as ${e_i} \equiv 1,2$ or $(\bmod 3)$. By means of a new technique called the method of weighted words, a combinatorial as well as a generating function proof of both these theorems are given simultaneously. It is shown that ${S_1}(n)$ and ${S_2}(n)$ are only two of six companion partition functions ${S_j}(n),j = 1,2, \ldots 6$, all equal to $ S(n)$. A three parameter refinement and generalization of these results is obtained.
On the decomposition of Langlands subrepresentations for a group in the Harish-Chandra class
Eugenio
Garnica-Vigil
1609-1648
Abstract: When a group $ G$ is in the Harish-Chandra class, the goal of classifying its tempered representations and the goal of decomposing the Langlands subrepresentation for any of its standard representations are equivalent. The main result of this work is given in Theorem (5.3.5) that consists of a formula for decomposing any Langlands subrepresentation for the group $G$. The classification of tempered representations is a consequence of this theorem (Corollary (5.3.6)).
On the period-two-property of the majority operator in infinite graphs
Gadi
Moran
1649-1667
Abstract: A self-mapping $ M:X \to X$ of a nonempty set $X$ has the Period-Two-Property (p2p) if ${M^2}x = x$ holds for every $M$-periodic point $x \in X$. Let $X$ be the set of all $\{ 0,1\}$-labelings $x:V \to \{ 0,1\}$ of the set of vertices $ V$ of a locally finite connected graph $G$. For $x \in X$ let $Mx \in X$ label $v \in V$ by the majority bit that $x$ applies to its neighbors, retaining $ \upsilon$'s $ x$-label in case of a tie. We show that $M$ has the p2p if there is a finite bound on the degrees in $G$ and $\frac{1} {n}\log {b_n} \to 0$, where ${b_n}$ is the number of $\upsilon \in V$ at a distance at most $ n$ from a fixed vertex ${\upsilon _0} \in V$.
Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions
Konstantin
Mischaikow;
Hal
Smith;
Horst R.
Thieme
1669-1685
Abstract: From the work of C. Conley, it is known that the omega limit set of a precompact orbit of an autonomous semiflow is a chain recurrent set. Here, we improve a result of L. Markus by showing that the omega limit set of a solution of an asymptotically autonomous semiflow is a chain recurrent set relative to the limiting autonomous semiflow. In the special case that there is a Lyapunov function for the limiting semiflow, sufficient conditions are given for an omega limit set of the asymptotically autonomous semiflow to be contained in a level set of the Lyapunov function.
Singular limit of solutions of $u\sb t=\Delta u\sp m-A\cdot \nabla (u\sp q/q)$ as $q\to\infty$
Kin Ming
Hui
1687-1712
Abstract: We will show that the solutions of ${u_t} = \Delta {u^m} - A\nabla ({u^q}/q)$ in $ {R^n} \times (0,T),T > 0,m > 1,u(x,0) = f(x) \in {L^1}({R^n}) \cap {L^\infty }({R^n})$ converge weakly in ${({L^\infty }(G))^ * }$ for any compact subset $ G$ of ${R^n} \times (0,T)$ as $ q \to \infty$ to the solution of the porous medium equation ${\upsilon _t} = \Delta {\upsilon ^m}$ in ${R^n} \times (0,T)$ with $\upsilon (x,0) = g(x)$ where $ g \in {L^1}({R^n}),0 \leqslant g \leqslant 1$, satisfies $\tilde{g}(x) \in {L^1}({R^n}),\quad \tilde{g}(x) \geqslant 0$ such that $g(x) = f(x),\quad \tilde{g}(x) = 0$ whenever $ g(x) < 1$ a.e. $x \in {R^n}$. The convergence is uniform on compact subsets of $ {R^n} \times (0,T)\quad {\text{if}}\quad f \in {C_0}({R^n})$.
Inequalities for zero-balanced hypergeometric functions
G. D.
Anderson;
R. W.
Barnard;
K. C.
Richards;
M. K.
Vamanamurthy;
M.
Vuorinen
1713-1723
Abstract: The authors study certain monotoneity and convexity properties of the Gaussian hypergeometric function and those of the Euler gamma function.
Some recurrence formulas for spherical polynomials on tube domains
Gen Kai
Zhang
1725-1734
Abstract: For a tube domain $ G/K$ we study the tensor products of two spherical representations of the maximal compact group $K$ and the product of the corresponding spherical polynomials. When one of these is a fundamental representation, we prove that the spherical representations appear with multiplicity at most one and we then find all the coefficients in the recurrence formula for the product of the spherical polynomials. This generalizes the previous result of L. Vretare and proves for certain cases a conjecture of R. Stanley on Jack symmetric polynomials.
Spectral and Fredholm properties of operators in elementary nest algebras
Bruce A.
Barnes;
Jon M.
Clauss
1735-1741
Abstract: Some spectral and Fredholm properties are proved for linear operators which leave invariant certain nests of closed subspaces.
On closed minimal submanifolds in pinched Riemannian manifolds
Hong Wei
Xu
1743-1751
Abstract: In this paper, we first prove a generalized Simons integral inequality for closed minimal submanifolds in a Riemannian manifold. Second, we prove a pinching theorem for closed minimal submanifolds in a complete simply connected pinched Riemannian manifold, which generalizes the results obtained by S. S. Chern, M. do Carmo, and S. Kobayashi and A. M. Li and J. M. Li respectively. Finally, we obtain a distribution theorem for the square norm of the second fundamental form of $M$ under the assumption that $M$ is a minimal submanifold with parallel second fundamental form in a Riemannian manifold.
On the $L\sp 2$ inequalities involving trigonometric polynomials and their derivatives
Weiyu
Chen
1753-1761
Abstract: In this note we study the upper bound of the integral $\displaystyle \int_0^\pi {{{({t^{(k)}}(x))}^2}w(x)} dx$ where $t(x)$ is a trigonometric polynomial with real coefficients such that $\left\Vert t \right\Vert\infty \leqslant 1$ and $w(x)$ is a nonnegative function defined on $ [0,\pi ]$. When $w(x) = \sin ^jx$, where $j$ is a positive integer, we obtain the exact upper bound for the above integral.
Hausdorff measure and level sets of typical continuous mappings in Euclidean spaces
Bernd
Kirchheim
1763-1777
Abstract: We determine the Hausdorff dimension of level sets and of sets of points of multiplicity for mappings in a residual subset of the space of all continuous mappings from ${\mathbb{R}^n}$ to $ {\mathbb{R}^m}$.
Groupoids associated with endomorphisms
Valentin
Deaconu
1779-1786
Abstract: To a compact Hausdorff space which covers itself, we associate an $ r$-discrete locally compact Hausdorff groupoid. Its ${{\mathbf{C}}^ * }$-algebra carries an action of the circle allowing it to be regarded as a crossed product by an endomorphism and as a generalization of the Cuntz algebra ${O_p}$. We consider examples related to coverings of the circle and of a Heisenberg $ 3$-manifold.
Holomorphic martingales and interpolation between Hardy spaces: the complex method
P. F. X.
Müller
1787-1792
Abstract: A probabilistic proof is given to identify the complex interpolation space of $ {H^1}(\mathbb{T})$ and $ {H^\infty }(\mathbb{T})$ as $ {H^p}(\mathbb{T})$.
Hypersurfaces in space forms satisfying the condition $\Delta x=Ax+B$
Luis J.
Alías;
Angel
Ferrández;
Pascual
Lucas
1793-1801
Abstract: In this work we study and classify pseudo-Riemannian hypersurfaces in pseudo-Riemannian space forms which satisfy the condition $ \Delta x = Ax + B$, where $ A$ is an endomorphism, $ B$ is a constant vector, and $x$ stands for the isometric immersion. We prove that the family of such hypersurfaces consists of open pieces of minimal hypersurfaces, totally umbilical hypersurfaces, products of two nonflat totally umbilical submanifolds, and a special class of quadratic hypersurfaces.
Examples of $B(D,\lambda)$-refinable and weak $\overline\theta$-refinable spaces
Stephen H.
Fast;
J. C.
Smith
1803-1809
Abstract: In 1980, J. C. Smith asked for examples which would demonstrate the relationships between the properties $B(D,\lambda )$-refinability, $B(D,{\omega _0})$-refinability, and weak $\bar \theta $-refinability. This paper gives such examples in the class of ${T_4}$ spaces.
A measure theoretical subsequence characterization of statistical convergence
Harry I.
Miller
1811-1819
Abstract: The concept of statistical convergence of a sequence was first introduced by H. Fast. Statistical convergence was generalized by R. C. Buck, and studied by other authors, using a regular nonnegative summability matrix $ A$ in place of $ {C_1}$. The main result in this paper is a theorem that gives meaning to the statement: $S = \{ {s_n}\} $ converges to $ L$ statistically $ (T)$ if and only if "most" of the subsequences of $S$ converge, in the ordinary sense, to $ L$. Here $T$ is a regular, nonnegative and triangular matrix. Corresponding results for lacunary statistical convergence, recently defined and studied by J. A. Fridy and C. Orhan, are also presented.
Standard Lyndon bases of Lie algebras and enveloping algebras
Pierre
Lalonde;
Arun
Ram
1821-1830
Abstract: It is well known that the standard bracketings of Lyndon words in an alphabet $A$ form a basis for the free Lie algebra ${\text{Lie}}(A)$ generated by $A$. Suppose that $\mathfrak{g} \cong {\text{Lie}}(A)/J$ is a Lie algebra given by a generating set $ A$ and a Lie ideal $ J$ of relations. Using a Gröbner basis type approach we define a set of "standard" Lyndon words, a subset of the set Lyndon words, such that the standard bracketings of these words form a basis of the Lie algebra $\mathfrak{g}$. We show that a similar approach to the universal enveloping algebra $\mathfrak{g}$ naturally leads to a Poincaré-Birkhoff-Witt type basis of the enveloping algebra of $\mathfrak{g}$. We prove that the standard words satisfy the property that any factor of a standard word is again standard. Given root tables, this property is nearly sufficient to determine the standard Lyndon words for the complex finite-dimensional simple Lie algebras. We give an inductive procedure for computing the standard Lyndon words and give a complete list of the standard Lyndon words for the complex finite-dimensional simple Lie algebras. These results were announced in [LR].
On the oscillation of differential equations with an oscillatory coefficient
B. J.
Harris;
Q.
Kong
1831-1839
Abstract: We derive lower bounds for the distance between consecutive zeros of solutions of
Tate cohomology of periodic $K$-theory with reality is trivial
Lisbeth
Fajstrup
1841-1846
Abstract: We calculate the $ RO(\mathbb{Z}/2)$-graded spectrum for Atiyah's periodic $K$-theory with reality and the Tate cohomology associated to it. The latter is shown to be trivial.
Uniformly ergodic multioperators
M.
Mbekhta;
F.-H.
Vasilescu
1847-1854
Abstract: A version of the uniform ergodic theorem valid for commuting multioperators is given.
Statistical inference based on the possibility and belief measures
Yuan Yan
Chen
1855-1863
Abstract: In statistical inference, we infer the population parameter based on the realization of sample statistics. This can be considered in the framework of inductive inference. We showed, in Chen (1993), that if we measure a parameter by the possibility (or belief) measure, we can have an inductive inference similar to the Bayesian inference in belief update. In this article we apply this inference to statistical estimation and hypotheses evaluation (testing) for some parametric models, and compare them to the classical statistical inferences for both one-sample and two-sample problems.