On the existence of multiple steady-state solutions in the theory of electrodiffusion. Part I: the nonelectroneutral case. Part II: a constructive method for the electroneutral case
Fatiha
Alabau
4709-4756
Abstract: We give a constructive method for giving examples of doping functions and geometry of the device for which the nonelectroneutral voltage driven equations have multiple solutions. We show in particular, by performing a singular perturbation analysis of the current driven equations that if the electroneutral voltage driven equations have multiple solutions then the nonelectroneutral voltage driven equations have multiple solutions for sufficiently small normed Debye length. We then give a constructive method for giving examples of data for which the electroneutral voltage driven equations have multiple solutions.
Convergence of Polynomial Level Sets
J.
Ferrera
4757-4773
Abstract: In this paper we give a characterization of pointwise and uniform convergence of sequences of homogeneous polynomials on a Banach space by means of the convergence of their level sets. Results are obtained both in the real and the complex cases, as well as some generalizations to the nonhomogeneous case and to holomorphic functions in the complex case. Kuratowski convergence of closed sets is used in order to characterize pointwise convergence. We require uniform convergence of the distance function to get uniform convergence of the sequence of polynomials.
The Stable Homotopy Types of Stunted Lens Spaces mod 4
Huajian
Yang
4775-4798
Abstract: Let $L^{n+k}_n$ be the mod $4$ stunted lens space $L^{n+k}/L^{n-1}$. Let $\nu(m)$ denote the exponent of $2$ in $m$, and $\phi (k)$ the number of integers $j$ satisfying $j\equiv 0,1, 2, 4 (\operatorname{mod}8)$, and $0< j\leq k$. In this paper we complete the classification of the stable homotopy types of mod $4$ stunted lens spaces. The main result (Theorem 1.3 (i)) is that, under some appropriate conditions, $L^{n+k}_n$ and $L^{m+k}_m$ are stably equivalent iff $\nu(n-m)\geq \phi(k)+\delta$, where $\delta=-1, 0$ or $1$.
Symmetric functional differential equations and neural networks with memory
Jianhong
Wu
4799-4838
Abstract: We establish an analytic local Hopf bifurcation theorem and a topological global Hopf bifurcation theorem to detect the existence and to describe the spatial-temporal pattern, the asymptotic form and the global continuation of bifurcations of periodic wave solutions for functional differential equations in the presence of symmetry. We apply these general results to obtain the coexistence of multiple large-amplitude wave solutions for the delayed Hopfield-Cohen-Grossberg model of neural networks with a symmetric circulant connection matrix.
Baire and $\sigma$-Borel characterizations of weakly compact sets in $M(T)$
T.
V.
Panchapagesan
4839-4847
Abstract: Let $T$ be a locally compact Hausdorff space and let $M(T)$ be the Banach space of all bounded complex Radon measures on $T$. Let $\mathcal{B}_o(T)$ and $\mathcal{B}_c(T)$ be the $\sigma$-rings generated by the compact $G_\delta$ subsets and by the compact subsets of $T$, respectively. The members of $\mathcal{B}_o(T)$ are called Baire sets of $T$ and those of $\mathcal{B}_c(T)$ are called $\sigma$-Borel sets of $T$ (since they are precisely the $\sigma$-bounded Borel sets of $T$). Identifying $M(T)$ with the Banach space of all Borel regular complex measures on $T$, in this note we characterize weakly compact subsets $A$ of $M(T)$ in terms of the Baire and $\sigma$-Borel restrictions of the members of $A$. These characterizations permit us to give a generalization of a theorem of Dieudonné which is stronger and more natural than that given by Grothendieck.
Characterizations of weakly compact operators on $C_o(T)$
T.
V.
Panchapagesan
4849-4867
Abstract: Let $T$ be a locally compact Hausdorff space and let $C_o(T)= \{f\,: T \rightarrow \mathbb{C}$, $f$ is continuous and vanishes at infinity} be provided with the supremum norm. Let $\mathcal{B}_c(T)$ and $\mathcal{B}_o(T)$ be the $\sigma$-rings generated by the compact subsets and by the compact $G_\delta$ subsets of $T$, respectively. The members of $\mathcal{B}_c(T)$ are called $\sigma$-Borel sets of $T$ since they are precisely the $\sigma$-bounded Borel sets of $T$. The members of $\mathcal{B}_o(T)$ are called the Baire sets of $T$. $M(T)$ denotes the dual of $C_o(T)$. Let $X$ be a quasicomplete locally convex Hausdorff space. Suppose $u: C_o(T) \rightarrow X$ is a continuous linear operator. Using the Baire and $\sigma$-Borel characterizations of weakly compact sets in $M(T)$ as given in a previous paper of the author's and combining the integration technique of Bartle, Dunford and Schwartz, we obtain 35 characterizations for the operator $u$ to be weakly compact, several of which are new. The independent results on the regularity and on the regular Borel extendability of $\sigma$-additive $X$-valued Baire measures are deduced as an immediate consequence of these characterizations. Some other applications are also included.
Homology of the configuration spaces of quasi-equilateral polygon linkages
Yasuhiko
Kamiyama;
Michishige
Tezuka;
Tsuguyoshi
Toma
4869-4896
Abstract: We consider the configuration space $M_{n,r}$ of quasi-equilateral polygon linkages with $n$ vertices each edge having length $1$ except for one fixed edge having length $r \; (r \geq 0)$ in the Euclidean plane $\mathbf{R}^{2}.$ In this paper, we determine $H_{\ast }(M_{n,r}; \mathrm{\bf Z})$.
Two dimensional elliptic equation with critical nonlinear growth
Takayoshi
Ogawa;
Takashi
Suzuki
4897-4918
Abstract: We study the asymptotic behavior of solutions to a semilinear elliptic equation associated with the critical nonlinear growth in two dimensions. \begin{equation*}\left\{ \begin{array}{cc} -\Delta u= \lambda ue^{u^2}, u>0 & \text{in} \ \Omega , u=0 & \text{on} \ \partial \Omega , \end{array} \right. \tag{1.1} \end{equation*} where $\Omega$ is a unit disk in $\mathbb{R}^2$ and $\lambda$ denotes a positive parameter. We show that for a radially symmetric solution of (1.1) satisfies \begin{equation*}\int _{D}\left\vert\nabla u\right\vert^{2}dx\rightarrow 4\pi,\quad\lambda \searrow 0. \end{equation*} Moreover, by using the Pohozaev identity to the rescaled equation, we show that for any finite energy radially symmetric solutions to (1.1), there is a rescaled asymptotics such as \begin{equation*}u_m^2(\gamma _m x)-u_m^2 (\gamma _m)\to 2\log\frac{2}{1+|x|^2} \quad\text{as }\lambda _m\searrow 0 \end{equation*} locally uniformly in $x\in\mathbb R^2$. We also show some extensions of the above results for general two dimensional domains.
Kähler Differentials, the $T$-functor, and a Theorem of Steinberg
W.
G.
Dwyer;
C.
W.
Wilkerson
4919-4930
Abstract: Let $T$ be the functor on the category of unstable algebras over the Steenrod algebra constructed by Lannes. We use an argument involving Kähler differentials to show that $T$ preserves polynomial algebras. This leads to new and relatively simple proofs of some topological and algebraic theorems.
The singular limit of a vector-valued reaction-diffusion process
Lia
Bronsard;
Barbara
Stoth
4931-4953
Abstract: We study the asymptotic behaviour of the solution to the vector-valued reaction-diffusion equation \begin{equation*}\varepsilon {\partial _{t}}\varphi -\varepsilon \triangle \varphi + {\frac{1}{\varepsilon }} \tilde W_{,\varphi } (\varphi ) = 0 \quad \text{ in } \Omega _{T}, \end{equation*} where $\varphi _{\varepsilon }=\varphi :\Omega _{T}:=(0,T)\times \Omega \longrightarrow \mathbf{R}^{2}$. We assume that the the potential $\tilde W$ depends only on the modulus of $\varphi$ and vanishes along two concentric circles. We present a priori estimates for the solution $\varphi$, and, in the spatially radially symmetric case, we show rigorously that in the singular limit as $\varepsilon \to 0$, two phases are created. The interface separating the bulk phases evolves by its mean curvature, while $\varphi$ evolves according to a harmonic map flow on the respective circles, coupled across the interfaces by a jump condition in the gradient.
The Dirichlet problem for Monge-Ampère equations in non-convex domains and spacelike hypersurfaces of constant Gauss curvature
Bo
Guan
4955-4971
Abstract: In this paper we extend the well known results on the existence and regularity of solutions of the Dirichlet problem for Monge-Ampère equations in a strictly convex domain to an arbitrary smooth bounded domain in $\mathbb R^n$ as well as in a general Riemannian manifold. We prove for the nondegenerate case that a sufficient (and necessary) condition for the classical solvability is the existence of a subsolution. For the totally degenerate case we show that the solution is in $C^{1,1} (\overline {\Omega})$ if the given boundary data extends to a locally strictly convex $C^2$ function on $\overline {\Omega}$. As an application we prove some existence results for spacelike hypersurfaces of constant Gauss-Kronecker curvature in Minkowski space spanning a prescribed boundary.
An index for periodic orbits of local semidynamical systems
Christian
C.
Fenske
4973-4991
Abstract: We define an index of Fuller type counting the number of periodic orbits of a semiflow on an ANR by a suitable approximation process.
Global analytic regularity for sums of squares of vector fields
Paulo
D.
Cordaro;
A.
Alexandrou
Himonas
4993-5001
Abstract: We consider a class of operators in the form of a sum of squares of vector fields with real analytic coefficients on the torus and we show that the zero order term may influence their global analytic hypoellipticity. Also we extend a result of Cordaro-Himonas.
The lifting of an exponential sum to a cyclic algebraic number field of prime degree
Yangbo
Ye
5003-5015
Abstract: Let $E$ be a cyclic algebraic number field of prime degree. We prove an identity which lifts an exponential sum similar to the Kloosterman sum to an exponential sum taken over certain algebraic integers in $E$.
Degenerate principal series and local theta correspondence
Soo
Teck
Lee;
Chen-bo
Zhu
5017-5046
Abstract: In this paper we determine the structure of the natural $\widetilde{U}(n,n)$ module ${\Omega^{p,q}(l)}$ which is the Howe quotient corresponding to the determinant character $\det^l$ of $U(p,q)$. We first give a description of the tempered distributions on $M_{p+q,n}(\mathbb C)$ which transform according to the character $\det^{-l}$ under the linear action of $U(p,q)$. We then show that after tensoring with a character, ${\Omega^{p,q}(l)}$ can be embedded into one of the degenerate series representations of $U(n,n)$. This allows us to determine the module structure of ${\Omega^{p,q}(l)}$. Moreover we show that certain irreducible constituents in the degenerate series can be identified with some of these representations ${\Omega^{p,q}(l)}$ or their irreducible quotients. We also compute the Gelfand-Kirillov dimensions of the irreducible constituents of the degenerate series.
Character sums associated to finite Coxeter groups
Jan
Denef;
François
Loeser
5047-5066
Abstract: The main result of this paper is a character sum identity for Coxeter arrangements over finite fields which is an analogue of Macdonald's conjecture proved by Opdam.
On Shintani zeta functions for $\mathrm{GL}(2)$
Akihiko
Yukie
5067-5094
Abstract: In this paper we consider an analogue of the zeta function for not necessarily prehomogeneous representations of $\text{GL}(2)$ and compute some of the poles.
A generalized Dedekind-Mertens lemma and its converse
Alberto
Corso;
William
Heinzer;
Craig
Huneke
5095-5109
Abstract: We study content ideals of polynomials and their behavior under multiplication. We give a generalization of the Lemma of Dedekind-Mertens and prove the converse under suitable dimensionality restrictions.
On transversality with deficiency and a conjecture of Sard
Carlos
Biasi;
Osamu
Saeki
5111-5122
Abstract: Let $f : M \to N$ be a $C^{r}$ map between $C^{r}$ manifolds $(r \geq 1)$ and $K$ a $C^{r}$ manifold. In this paper, by using the Sard theorem, we study the topological properties of the space of $C^{r}$ maps $g : K \to N$ which satisfy a certain transversality condition with respect to $f$ in a weak sense. As an application, by considering the case where $K$ is a point, we obtain some new results about the topological properties of $f(R_{q}(f))$, where $R_{q}(f)$ is the set of points of $M$ where the rank of the differential of $f$ is less than or equal to $q$. In particular, we show a result about the topological dimension of $f(R_{q}(f))$, which is closely related to a conjecture of Sard concerning the Hausdorff measure of $f(R_{q}(f))$.