On the existence of convex classical solutions for multilayer free boundary problems with general nonlinear joining conditions
Andrew
Acker
2981-3020
Abstract: We prove the existence of convex classical solutions for a general multidimensional, multilayer free-boundary problem. The geometric context of this problem is a nested family of closed, convex surfaces. Except for the innermost and outermost surfaces, which are given, these surfaces are interpreted as unknown layer-interfaces, where the layers are the bounded annular domains between them. Each unknown interface is characterized by a quite general nonlinear equation, called a joining condition, which relates the first derivatives (along the interface) of the capacitary potentials in the two adjoining layers, as well as the spatial variables. A well-known special case of this problem involves several stationary, immiscible, two-dimensional flows of ideal fluid, related along their interfaces by Bernoulli's law.
Small subalgebras of Steenrod and Morava stabilizer algebras
N.
Yagita
3021-3041
Abstract: Let $P(j)$ (resp. $S(n)_{(j)})$ be the subalgebra of the Steenrod algebra $\mathcal{A}_p$ (resp. $n$th Morava stabilizer algebra) generated by reduced powers $\mathcal{P}^{p^i}$, $0\le i\le j$ (resp. $t_i$, $1\le i\le j)$. In this paper we identify the dual $P(j-1)^*$ of $P(j-1)$ (resp. $S(n)_{(j)}$, for $j\le n)$ with some Frobenius kernel (resp. $F_{p^n}$-points) of a unipotent subgroup $G(j+1)$ of the general linear algebraic group $GL_{j+1}$. Using these facts, we get the additive structure of $H^*(P(1))=\operatorname{Ext}_{P(1)}(Z/p,Z/p)$ for odd primes.
Morita equivalence for crossed products by Hilbert $C^\ast$-bimodules
Beatriz
Abadie;
Søren
Eilers;
Ruy
Exel
3043-3054
Abstract: We introduce the notion of the crossed product $A \rtimes _X{\Bbb{Z}}$ of a $C^*$-algebra $A$ by a Hilbert $C^*$-bimodule $X$. It is shown that given a $C^*$-algebra $B$ which carries a semi-saturated action of the circle group (in the sense that $B$ is generated by the spectral subspaces $B_0$ and $B_1$), then $B$ is isomorphic to the crossed product $B_0 \rtimes _{B_1}{\Bbb{Z}}$. We then present our main result, in which we show that the crossed products $A \rtimes _X{\Bbb{Z}}$ and $B \rtimes _Y{\Bbb{Z}}$ are strongly Morita equivalent to each other, provided that $A$ and $B$ are strongly Morita equivalent under an imprimitivity bimodule $M$ satisfying $X\otimes _A M \simeq M\otimes _B Y$ as $A-B$ Hilbert $C^*$-bimodules. We also present a six-term exact sequence for $K$-groups of crossed products by Hilbert $C^*$-bimodules.
Homogeneity in powers of subspaces of the real line
L.
Brian
Lawrence
3055-3064
Abstract: Working in ZFC, we prove that for every zero-dimensional subspace $S$ of the real line, the Tychonoff power ${}^\omega S$ is homogeneous ($\omega$ denotes the nonnegative integers). It then follows as a corollary that ${}^\omega S$ is homogeneous whenever $S$ is a separable zero-dimensional metrizable space. The question of homogeneity in powers of this type was first raised by Ben Fitzpatrick, and was subsequently popularized by Gary Gruenhage and Hao-xuan Zhou.
Liouvillian integration and Bernoulli foliations
D.
Cerveau;
P.
Sad
3065-3081
Abstract: Analytic foliations in the 2-dimensional complex projective space with algebraic invariant curves are studied when the holonomy groups of these curves are solvable. It is shown that such a condition leads to the existence of a Liouville type first integral, and, under ``generic'' extra conditions, it is proven that these foliations can be defined by Bernoulli equations.
Equations for the Jacobian of a hyperelliptic curve
Paul
van Wamelen
3083-3106
Abstract: We give an explicit embedding of the Jacobian of a hyperelliptic curve, $y^2 = f(x)$, into projective space such that the image is isomorphic to the Jacobian over the splitting field of $f$. The embedding is a modification of the usual embedding by theta functions with half integer characteristics.
Cusp forms for congruence subgroups of $Sp_n(\mathbb{Z})$ and theta functions
Yaacov
Kopeliovich
3107-3118
Abstract: In this paper we use theta functions with rational characteristic to construct cusp forms for congruence subgroups $\Gamma _g(p)$ of $Sp(g,\mathbb Z)$.The action of the quotient group $Sp(g,\mathbb Z_p)$ on these forms is conjugate to the linear action of $Sp(g,\mathbb Z_p)$ on $(\mathbb Z_p)^{2g}$. We show that these forms are higher-dimensional analogues of the Fricke functions.
Complicated dynamics of parabolic equations with simple gradient dependence
Martino
Prizzi;
Krzysztof
P.
Rybakowski
3119-3130
Abstract: Let $\Omega \subset \mathbb R^{2}$ be a smooth bounded domain. Given positive integers $n$, $k$ and $q_{l}~\le ~l$, $l=1$, ..., $k$, consider the semilinear parabolic equation \begin{alignat*}{2} u_{t}&=u_{xx}+u_{yy}+a(x,y)u+ \smash{\sum _{l=1}^{k}}a_{l}(x,y) u^{l-q_{l}}(u_{y})^{q_{l}},&\quad &t>0, (x,y)\in \Omega,\tag{E} u&=0,&\quad& t>0, (x,y)\in \partial \Omega . \end{alignat*} where $a(x,y)$ and $a_{l}(x,y)$ are smooth functions. By refining and extending previous results of Polácik we show that arbitrary $k$-jets of vector fields in $\mathbb R^{n}$ can be realized in equations of the form (E). In particular, taking $q_{l}\equiv 1$ we see that very complicated (chaotic) behavior is possible for reaction-diffusion-convection equations with linear dependence on $\nabla u$.
Convergence of Madelung-like lattice sums
David
Borwein;
Jonathan
M.
Borwein;
Christopher
Pinner
3131-3167
Abstract: We make a general study of the convergence properties of lattice sums, involving potentials, of the form occurring in mathematical chemistry and physics. Many specific examples are studied in detail. The prototype is Madelung's constant for NaCl: \begin{equation*}\sum _{-\infty}^{\infty} \frac{(-1)^{n+m+p}} {\sqrt{n^2+m^2+p^2}} = -1.74756459 \cdots, \end{equation*} presuming that one appropriately interprets the summation proccess.
Similarity to a contraction, for power-bounded operators with finite peripheral spectrum
Ralph
deLaubenfels
3169-3191
Abstract: Suppose $T$ is a power-bounded linear opertor on a Hilbert space with finite peripheral spectrum (spectrum on the unit circle). Several sufficient conditions are given for $T$ to be similar to a contraction. A natural growth condition on the resolvent in half-planes tangent to the unit circle at the peripheral spectrum is shown to be equivalent to $T$ having an $H^\infty(\mathcal P)\cap C(\overline{\mathcal P})$ functional calculus, for some open polygon $\mathcal P$ contained in the unit disc, which, in turn, is equivalent to $T$ being similar to a contraction with numerical range contained in a closed polygon in the closed unit disc. Having certain orbits of $T$ be square summable also implies that $T$ is similar to a contraction.
On hyper Kähler manifolds associated to Lagrangian Kähler submanifolds of $T^\ast\mathbb{C}^n$
Vicente
Cortés
3193-3205
Abstract: For any Lagrangian Kähler submanifold $M \subset T^*\mathbb{C}^n$, there exists a canonical hyper Kähler metric on $T^*M$. A Kähler potential for this metric is given by the generalized Calabi Ansatz of the theoretical physicists Cecotti, Ferrara and Girardello. This correspondence provides a method for the construction of (pseudo) hyper Kähler manifolds with large automorphism group. Using it, an interesting class of pseudo hyper Kähler manifolds of complex signature $(2,2n)$ is constructed. For any manifold $N$ in this class a group of automorphisms with a codimension one orbit on $N$ is specified. Finally, it is shown that the bundle of intermediate Jacobians over the moduli space of gauged Calabi Yau 3-folds admits a natural pseudo hyper Kähler metric of complex signature $(2,2n)$.
Based algebras and standard bases for quasi-hereditary algebras
Jie
Du;
Hebing
Rui
3207-3235
Abstract: Quasi-hereditary algebras can be viewed as a Lie theory approach to the theory of finite dimensional algebras. Motivated by the existence of certain nice bases for representations of semisimple Lie algebras and algebraic groups, we will construct in this paper nice bases for (split) quasi-hereditary algebras and characterize them using these bases. We first introduce the notion of a standardly based algebra, which is a generalized version of a cellular algebra introduced by Graham and Lehrer, and discuss their representation theory. The main result is that an algebra over a commutative local noetherian ring with finite rank is split quasi-hereditary if and only if it is standardly full-based. As an application, we will give an elementary proof of the fact that split symmetric algebras are not quasi-hereditary unless they are semisimple. Finally, some relations between standardly based algebras and cellular algebras are also discussed.
On the best constant for Hardy's inequality in $\mathbb{R}^n$
Moshe
Marcus;
Victor
J.
Mizel;
Yehuda
Pinchover
3237-3255
Abstract: Let $\Omega$ be a domain in $\mathbb R^n$ and $p\in (1,\infty)$. We consider the (generalized) Hardy inequality $\int _\Omega |\nabla u|^p\geq K\int _\Omega |u/\delta |^p$, where $\delta (x)=\operatorname{dist}\,(x,\partial \Omega )$. The inequality is valid for a large family of domains, including all bounded domains with Lipschitz boundary. We here explore the connection between the value of the Hardy constant $\mu _p(\Omega )=\inf _{\stackrel{\circ}{W}_{1,p}(\Omega )}\left (\int _\Omega |\nabla u|^p\,/\,\int _\Omega |u/\delta |^p \right )$ and the existence of a minimizer for this Rayleigh quotient. It is shown that for all smooth $n$-dimensional domains, $\mu _p(\Omega )\leq c_p$, where $c_p=(1-{1\over p})^p$ is the one-dimensional Hardy constant. Moreover it is shown that $\mu _p(\Omega )=c_p$ for all those domains not possessing a minimizer for the above Rayleigh quotient. Finally, for $p=2$, it is proved that $\mu _2(\Omega )<c_2=1/4$ if and only if the Rayleigh quotient possesses a minimizer. Examples show that strict inequality may occur even for bounded smooth domains, but $\mu _p=c_p$ for convex domains.
Criteria for $\bar{d}$-continuity
Zaqueu
Coelho;
Anthony
N.
Quas
3257-3268
Abstract: Bernoullicity is the strongest mixing property that a measure-theoretic dynamical system can have. This is known to be intimately connected to the so-called $\bar d$ metric on processes, introduced by Ornstein. In this paper, we consider families of measures arising in a number of contexts and give conditions under which the measures depend $\bar d$-continuously on the parameters. At points where there is $\bar d$-continuity, it is often straightforward to establish that the measures have the Bernoulli property.
A family of quantum projective spaces and related $q$-hypergeometric orthogonal polynomials
Mathijs
S.
Dijkhuizen;
Masatoshi
Noumi
3269-3296
Abstract: A one-parameter family of two-sided coideals in $\mathcal{U}_{q} (\mathfrak{g}\mathfrak{l}(n))$ is defined and the corresponding algebras of infinitesimally right invariant functions on the quantum unitary group $U_{q}(n)$ are studied. The Plancherel decomposition of these algebras with respect to the natural transitive $U_{q}(n)$-action is shown to be the same as in the case of a complex projective space. By computing the radial part of a suitable Casimir operator, we identify the zonal spherical functions (i.e. infinitesimally bi-invariant matrix coefficients of finite-dimensional irreducible representations) as Askey-Wilson polynomials containing two continuous and one discrete parameter. In certain limit cases, the zonal spherical functions are expressed as big and little $q$-Jacobi polynomials depending on one discrete parameter.
Quantized enveloping algebras for Borcherds superalgebras
Georgia
Benkart;
Seok-Jin
Kang;
Duncan
Melville
3297-3319
Abstract: We construct quantum deformations of enveloping algebras of Borcherds superalgebras, their Verma modules, and their irreducible highest weight modules.
Lie ideals in triangular operator algebras
T.
D.
Hudson;
L.
W.
Marcoux;
A.
R.
Sourour
3321-3339
Abstract: We study Lie ideals in two classes of triangular operator algebras: nest algebras and triangular UHF algebras. Our main results show that if ${\mathfrak L}$ is a closed Lie ideal of the triangular operator algebra ${\mathcal A}$, then there exist a closed associative ideal ${\mathcal K}$ and a closed subalgebra ${\mathfrak D}_{\mathcal K}$ of the diagonal ${\mathcal A}\cap {\mathcal A}^*$ so that ${\mathcal K}\subseteq {\mathfrak L}\subseteq {\mathcal K}+ {\mathfrak D}_{\mathcal K}$.
Discrete series characters and two-structures
Rebecca
A.
Herb
3341-3369
Abstract: Let $G$ be a connected semisimple real Lie group with compact Cartan subgroup. Harish-Chandra gave formulas for discrete series characters which are completely explicit except for certain interger constants appearing in the numerators. The main result of this paper is a new formula for these constants using two-structures. The new formula avoids endoscopy and stable discrete series entirely, expressing (unaveraged) discrete series constants directly in terms of (unaveraged) discrete series constants corresponding to two-structures of noncompact type.
Weighted norm inequalities for integral operators
Igor
E.
Verbitsky;
Richard
L.
Wheeden
3371-3391
Abstract: We consider a large class of positive integral operators acting on functions which are defined on a space of homogeneous type with a group structure. We show that any such operator has a discrete (dyadic) version which is always essentially equivalent in norm to the original operator. As an application, we study conditions of ``testing type,'' like those initially introduced by E. Sawyer in relation to the Hardy-Littlewood maximal function, which determine when a positive integral operator satisfies two-weight weak-type or strong-type $(L^{p}, L^{q})$ estimates. We show that in such a space it is possible to characterize these estimates by testing them only over ``cubes''. We also study some pointwise conditions which are sufficient for strong-type estimates and have applications to solvability of certain nonlinear equations.
Filling-invariants at infinity for manifolds of nonpositive curvature
Noel
Brady;
Benson
Farb
3393-3405
Abstract: In this paper we construct and study isoperimetric functions at infinity for Hadamard manifolds. These quasi-isometry invariants give a measure of the spread of geodesics in such a manifold.
Correction to ``Fox calculus, symplectic forms, and moduli spaces''
Valentino
Zocca
3407