Structural properties of the one-dimensional drift-diffusion models for semiconductors
Fatiha
Alabau
823-871
Abstract: This paper is devoted to the analysis of the one-dimensional current and voltage drift-diffusion models for arbitrary types of semiconductor devices and under the assumption of vanishing generation recombination. We show in the course of this paper, that these models satisfy structural properties, which are due to the particular form of the coupling of the involved systems. These structural properties allow us to prove an existence and uniqueness result for the solutions of the current driven model together with monotonicity properties with respect to the total current $I$, of the electron and hole current densities and of the electric field at the contacts. We also prove analytic dependence of the solutions on $I$. These results allow us to establish several qualitative properties of the current voltage characteristic. In particular, we give the nature of the (possible) bifurcation points of this curve, we show that the voltage function is an analytic function of the total current and we characterize the asymptotic behavior of the currents for large voltages. As a consequence, we show that the currents never saturate as the voltage goes to $\pm \infty$, contrary to what was predicted by numerical simulations by M. S. Mock (Compel. 1 (1982), pp. 165--174). We also analyze the drift-diffusion models under the assumption of quasi-neutral approximation. We show, in particular, that the reduced current driven model has at most one solution, but that it does not always have a solution. Then, we compare the full and the reduced voltage driven models and we show that, in general, the quasi-neutral approximation is not accurate for large voltages, even if no saturation phenomenon occurs. Finally, we prove a local existence and uniqueness result for the current driven model in the case of small generation recombination terms.
Connection coefficients, matchings, maps and combinatorial conjectures for Jack symmetric functions
I.
P.
Goulden;
D.
M.
Jackson
873-892
Abstract: A power series is introduced that is an extension to three sets of variables of the Cauchy sum for Jack symmetric functions in the Jack parameter $\alpha.$ We conjecture that the coefficients of this series with respect to the power sum basis are nonnegative integer polynomials in $b$, the Jack parameter shifted by $1$. More strongly, we make the Matchings-Jack Conjecture, that the coefficients are counting series in $b$ for matchings with respect to a parameter of nonbipartiteness. Evidence is presented for these conjectures and they are proved for two infinite families. The coefficients of a second series, essentially the logarithm of the first, specialize at values $1$ and $2$ of the Jack parameter to the numbers of hypermaps in orientable and locally orientable surfaces, respectively. We conjecture that these coefficients are also nonnegative integer polynomials in $b$, and we make the Hypermap-Jack Conjecture, that the coefficients are counting series in $b$ for hypermaps in locally orientable surfaces with respect to a parameter of nonorientability.
Iterated Spectra of Numbers---Elementary, Dynamical, and Algebraic Approaches
Vitaly
Bergelson;
Neil
Hindman;
Bryna
Kra
893-912
Abstract: $IP^*$ sets and central sets are subsets of $\mathbb N$ which arise out of applications of topological dynamics to number theory and are known to have rich combinatorial structure. Spectra of numbers are often studied sets of the form $\{[n\alpha+\gamma]\colon n\in\mathbb N\}$. Iterated spectra are similarly defined with $n$ coming from another spectrum. Using elementary, dynamical, and algebraic approaches we show that iterated spectra have significantly richer combinatorial structure than was previously known. For example we show that if $\alpha>0$ and $0<\gamma<1$, then $\{[n\alpha+\gamma]\colon n\in\mathbb N\}$ is an $IP^*$ set and consequently contains an infinite sequence together with all finite sums and products of terms from that sequence without repetition.
Distinguished representations and quadratic base change for $GL(3)$
Herve
Jacquet;
Yangbo
Ye
913-939
Abstract: Let $E/F$ be a quadratic extension of number fields. Suppose that every real place of $F$ splits in $E$ and let $H$ be the unitary group in 3 variables. Suppose that $\Pi$ is an automorphic cuspidal representation of $GL(3,E_{\mathbb{A}})$. We prove that there is a form $\phi$ in the space of $\Pi$ such that the integral of $\phi$ over $H(F)\setminus H(F_{\mathbb{A}})$ is non zero. Our proof is based on earlier results and the notion, discussed in this paper, of Shalika germs for certain Kloosterman integrals.
Geometrizing Infinite Dimensional Locally Compact Groups
Conrad
Plaut
941-962
Abstract: We study groups having invariant metrics of curvature bounded below in the sense of Alexandrov. Such groups are a generalization of Lie groups with invariant Riemannian metrics, but form a much larger class. We prove that every locally compact, arcwise connected, first countable group has such a metric. These groups may not be (even infinite dimensional) manifolds. We show a number of relationships between the algebraic and geometric structures of groups equipped with such metrics. Many results do not require local compactness.
R-torsion and zeta functions for analytic Anosov flows on 3-manifolds
Héctor
Sánchez-Morgado
963-973
Abstract: We improve previous results relating R-torsion, for an acyclic representation of the fundamental group, with a special value of the torsion zeta function of an analytic Anosov flow on a 3-manifold. By using the new techniques of Rugh and Fried we get rid of the unpleasent assumptions about the regularity of the invariant foliations.
An Extension of Lomonosov's Techniques to Non-compact Operators
Aleksander
Simonic
975-995
Abstract: The aim of this work is to generalize Lomonosov's techniques in order to apply them to a wider class of not necessarily compact operators. We start by establishing a connection between the existence of invariant subspaces and density of what we define as the associated Lomonosov space in a certain function space. On a Hilbert space, approximation with Lomonosov functions results in an extended version of Burnside's Theorem. An application of this theorem to the algebra generated by an essentially self-adjoint operator $A$ yields the existence of vector states on the space of all polynomials restricted to the essential spectrum of $A$. Finally, the invariant subspace problem for compact perturbations of self-adjoint operators acting on a real Hilbert space is translated into an extreme problem and the solution is obtained upon differentiating certain real-valued functions at their extreme.
Partially hyperbolic fixed points with constraints
Patrick
Bonckaert
997-1011
Abstract: We investigate the local conjugacy, at a partially hyperbolic fixed point, of a diffeomorphism (vector field) to its normally linear part in the presence of constraints, where the change of variables also must satisfy the constraints. The main result is applied to vector fields respecting a singular foliation, encountered, by F. Dumortier and R. Roussarie, in the desingularization of families of vector fields.
Quasiadditivity and measure property of capacity and the tangential boundary behavior of harmonic functions
H.
Aikawa;
A.
A.
Borichev
1013-1030
Abstract: We show that if a set $E$ is dispersely decomposed into subsets, then the capacity of $E$ is comparable to the summation of the capacities of the subsets. From this fact it is derived that the Lebesgue measure of a certain expanded set is estimated by the capacity of $E$. These properties hold for classical capacities, $L^{p}$-capacities and energy capacities of general kernels. The estimation is applied to the boundary behavior of harmonic functions. We introduce a boundary thin set and show a fine limit type boundary behavior of harmonic functions. We show that a thin set does not meet essentially Nagel-Stein and Nagel-Rudin-Shapiro type approaching regions at almost all bounary points.
The behavior of Fourier transforms for nilpotent Lie groups
Ronald
L.
Lipsman;
Jonathan
Rosenberg
1031-1050
Abstract: We study weak analogues of the Paley-Wiener Theorem for both the scalar-valued and the operator-valued Fourier transforms on a nilpotent Lie group $G$. Such theorems should assert that the appropriate Fourier transform of a function or distribution of compact support on $G$ extends to be ``holomorphic'' on an appropriate complexification of (a part of) $\hat G$. We prove the weak scalar-valued Paley-Wiener Theorem for some nilpotent Lie groups but show that it is false in general. We also prove a weak operator-valued Paley-Wiener Theorem for arbitrary nilpotent Lie groups, which in turn establishes the truth of a conjecture of Moss. Finally, we prove a conjecture about Dixmier-Douady invariants of continuous-trace subquotients of $C^{*}(G)$ when $G$ is two-step nilpotent.
Operator Semigroup Compactifications
H.
D.
Junghenn
1051-1073
Abstract: A weakly continuous, equicontinuous representation of a semitopological semigroup $S$ on a locally convex topological vector space $X$ gives rise to a family of operator semigroup compactifications of $S$, one for each invariant subspace of $X$. We consider those invariant subspaces which are maximal with respect to the associated compactification possessing a given property of semigroup compactifications and show that under suitable hypotheses this maximality is preserved under the formation of projective limits, strict inductive limits and tensor products.
Real analysis related to the Monge-Ampère equation
Luis
A.
Caffarelli;
Cristian
E.
Gutiérrez
1075-1092
Abstract: In this paper we consider a family of convex sets in $\mathbf{R}^{n}$, $\mathcal{F}= \{S(x,t)\}$, $x\in \mathbf{R}^{n}$, $t>0$, satisfying certain axioms of affine invariance, and a Borel measure $\mu$ satisfying a doubling condition with respect to the family $\mathcal{F}.$ The axioms are modelled on the properties of the solutions of the real Monge-Ampère equation. The purpose of the paper is to show a variant of the Calderón-Zygmund decomposition in terms of the members of $\mathcal{F}.$ This is achieved by showing first a Besicovitch-type covering lemma for the family $\mathcal{F}$ and then using the doubling property of the measure $\mu .$ The decomposition is motivated by the study of the properties of the linearized Monge-Ampère equation. We show certain applications to maximal functions, and we prove a John and Nirenberg-type inequality for functions with bounded mean oscillation with respect to $\mathcal{F}.$
Affine Dupin Surfaces
Ross
Niebergall;
Patrick
J.
Ryan
1093-1115
Abstract: In this paper we study nondegenerate affine surfaces in ${\mathbb R} ^{3}$ whose affine principal curvatures are constant along their lines of curvature. We give a complete local classification of these surfaces assuming that the lines of curvature are planar, and there are no umbilics.
A Note on Bernoulli Numbers and Shintani Generalized Bernoulli Polynomials
Minking
Eie
1117-1136
Abstract: Generalized Bernoulli polynomials were introduced by Shintani in 1976 in order to express the special values at non-positive integers of Dedekind zeta functions for totally real numbers. The coefficients of such polynomials are finite combinations of products of Bernoulli numbers which are difficult to get hold of. On the other hand, Zagier was able to get the explicit formula for the special values in cases of real quadratic number fields. In this paper, we shall improve Shintani's formula by proving that the special values can be determined by a finite set of polynomials. This provides a convenient way to evaluate the special values of various types of Dedekind functions. Indeed, a much broader class of zeta functions considered by the author [4] admits a similar formula for its special values. As a consequence, we are able to find infinitely many identities among Bernoulli numbers through identities among zeta functions. All these identities are difficult to prove otherwise.
Even Linkage Classes
Scott
Nollet
1137-1162
Abstract: In this paper we generalize the $\mathcal{E}$ and $\mathcal{N}$-type resolutions used by Martin-Deschamps and Perrin for curves in $\mathbb{P}^{3}$ to subschemes of pure codimension in projective space, and shows that these resolutions are interchanged by the mapping cone procedure under a simple linkage. Via these resolutions, Rao's correspondence is extended to give a bijection between even linkage classes of subschemes of pure codimension two and stable equivalence classes of reflexive sheaves $\mathcal{E}$ satisfying $H^{1}_{*}( \mathcal{E})=0$ and $\mathop{\mathcal{E}xt}^{1}( \mathcal{E}^{\vee }, \mathcal{O})=0$. Further, these resolutions are used to extend the work of Martin-Deschamps and Perrin for Cohen-Macaulay curves in $\mathbb{P}^{3}$ to subschemes of pure codimension two in $\mathbb{P}^{n}$. In particular, even linkage classes of such subschemes satisfy the Lazarsfeld-Rao property and any minimal subscheme for an even linkage class links directly to a minimal subscheme for the dual class.
Geometric Isomorphisms between Infinite Dimensional Teichmüller Spaces
Clifford
J.
Earle;
Frederick
P.
Gardiner
1163-1190
Abstract: Let $X$ and $Y$ be the interiors of bordered Riemann surfaces with finitely generated fundamental groups and nonempty borders. We prove that every holomorphic isomorphism of the Teichmüller space of $X$ onto the Teichmüller space of $Y$ is induced by a quasiconformal homeomorphism of $X$ onto $Y$. These Teichmüller spaces are not finite dimensional and their groups of holomorphic automorphisms do not act properly discontinuously, so the proof presents difficulties not present in the classical case. To overcome them we study weak continuity properties of isometries of the tangent spaces to Teichmüller space and special properties of Teichmüller disks.
Topological centers of certain dual algebras
Anthony
To-Ming
Lau;
Ali
Ülger
1191-1212
Abstract: Let $A$ be a Banach algebra with a bounded approximate identity. Let $Z_1$ and $\widetilde Z_2$ be, respectively, the topological centers of the algebras $A^{**}$ and $(A^*A)^*$. In this paper, for weakly sequentially complete Banach algebras, in particular for the group and Fourier algebras $L^1(G)$ and $A(G)$, we study the sets $Z_1$, $\widetilde Z_2$, the relations between them and with several other subspaces of $A^{**}$ or $A^*$.
On the Poles of Rankin-Selberg Convolutions of Modular Forms
Xian-jin
Li
1213-1234
Abstract: The Rankin-Selberg convolution is usually normalized by the multiplication of a zeta factor. One naturally expects that the non-normalized convolution will have poles where the zeta factor has zeros, and that these poles will have the same order as the zeros of the zeta factor. However, this will only happen if the normalized convolution does not vanish at the zeros of the zeta factor. In this paper, we prove that given any point inside the critical strip, which is not equal to $\frac{1}{2}$ and is not a zero of the Riemann zeta function, there exist infinitely many cusp forms whose normalized convolutions do not vanish at that point.