Indiana University Mathematics Journal: Table of contents for volume 56, issue 1, 2007
Annotated Table of Contents, Indiana Univ. Math. J. 56, Number 2 (2007)
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- A determinantal formula for the exterior powers of the polynomial ring
By Dan Laksov, Anders Thorup: We present a very general, conceptually natural, explicit and computationally efficient \emph{Schubert calculus}. It consists of two strongly interrelated parts, a structure theorem for an exterior power of the polynomial ring in one variable as a module over the ring of symmetric polynomials and a determinantal formula. Both parts are similar to corresponding results in algebra, combinatorics and geometry. To emphasize the connections with these fields we give several proofs of our main results, each proof illuminating the theory from a different algebraic, combinatorial, or geometric angle. The main application of our theory is to the Schubert calculus of Grassmann schemes, where it gives a natural homology and cohomology theory for the Grassmannians in a very general setting.
Wed, 25 Apr 2007 15:18:26 EST - A priori estimates of stationary solutions of an activator-inhibitor system
By Huiqiang Jiang, Wei-ming Ni: We consider positive solutions of the stationary Gierer-Meinhardt system \begin{eqnarray*} {}&&d_{1}\Delta u-u+\frac{u^{p}}{v^{q}}+\sigma=0\quad\mbox{\ in }\Omega,\\[2pt] {}&&d_{2}\Delta v-v+\frac{u^{r}}{v^{s}}=0hphantom{\ =\ 0}\quad\mbox{in }\Omega,\\[2pt] {}&&\frac{\partial u}{\partial\nu}=\frac{\partial v}{\partial\nu}=0\hphantom{\ = \ 0 \ =\ 0}\quad\mbox{on }\partial\Omega, \end{eqnarray*} where $\Delta$ is the Laplace operator, $\Omega$ is a bounded smooth domain in $\mathbb{R}^{n}$, $n\geq1$, and $\nu$ is the unit outer normal to $\partial\Omega$. Under suitable conditions on the exponents $p$, $q$, $r$, and $s$, different types of \textit{a priori} estimates are obtained, existence and non-existence results of nontrivial solutions are derived, for both $\sigma>0$ and $\sigma=0$ cases.
Wed, 25 Apr 2007 15:18:26 EST - Approximate and pseudo-amenability of the Fourier algebra
By Fereidoun Ghahramani, Ross Stokke: Let $G$ be a locally compact group and let $A(G)$ be its Fourier algebra. We find sufficient conditions for $A(G)$ to be approximately/pseudo-amenable without being amenable. We also study $A(G)$ for its operator approximate/pseudo-amenability.
Wed, 25 Apr 2007 15:18:26 EST - Continuity estimates for $n$-harmonic equations
By Tadeusz Iwaniec, Jani Onninen: We investigate the nonhomogeneous $n$-harmonic equation $$\mbox{div}\, |\nabla u|^{n-2}\nabla u =f$$ for $u$ in the Sobolev space $\mathscr{W}^{1,n}(\Omega)$, where $f$ is a given function in the Zygmund class $\mathscr{L}\log^\alpha \mathscr{L}(\Omega)$. In dimension $n=2$ the solutions are continuous whenever $f$ lies in the Hardy space $\mathscr{H}^1(\Omega)$, so in particular, if $f\in \mathscr{L}\log \mathscr{L}(\Omega)$. We show in higher dimensions that within the Zygmund classes the condition $\alpha > n-1$ is both necessary and sufficient for the solutions to be continuous. We also investigate continuity of the map $f \rightarrow \nabla u$, from $\mathscr{L}\log^\alpha \mathscr{L}(\Omega)$ into $\mathscr{L}^n\log^\beta \mathscr{L}(\Omega)$, for \[ -1 < \beta < \frac{n\alpha}{n-1}-1. \] These and other results of the present paper, though anticipated by simple examples, are in fact far from routine. Certainly, they are central in the $p$-harmonic theory.
Wed, 25 Apr 2007 15:18:26 EST - Doubling Properties of Self-similar Measures
By Po-Lam Yung: Let $\{F_i\}_{i=1}^N$ be a system of similitudes in $\mathbb{R}^n$. We study necessary and sufficient conditions for their associated self-similar measures to be doubling on its su pport. An equivalent condition is obtained when $\{F_i\}$ satisfies the open set condition . The condition allows us to construct many examples of interest. In the case where the op en set condition is not satisfied, we study an infinitely convoluted Bernoulli measure (as sociated with the golden ratio $\rho=(\sqrt{5}-1)/2$) and give a necessary and sufficient condition for it to be doubling on its support $[0,1]$.
Wed, 25 Apr 2007 15:18:26 EST - Erratum: "On Marcinkiewicz integral with variable kernels", Vol. 53 (2004), 805-822
By Chin-cheng Lin, Yong Ding, Ying-Chieh Lin:
Wed, 25 Apr 2007 15:18:26 EST - Gevrey regularity of solutions to the 3-D Navier-Stokes equations with weighted $l_p$ initial data
By Animikh Biswas, David Swanson:
Wed, 25 Apr 2007 15:18:26 EST - Hamiltonian stability and index of minimal Lagrangian surfaces of the complex projective plane
By Francisco Urbano: We show that the Clifford torus and the totally geodesic real projective plane $\mathbb{R}\mathbb{P}^2$ in the complex projective plane $\mathbb{C}\mathbb{P}^2$ are the unique Hamiltonian stable minimal Lagrangian compact surfaces of $\mathbb{C}\mathbb{P}^2$ with genus $g\leq4$, when the surface is orientable, and with Euler characteristic $\chi\geq-1$, when the surface is nonorientable. Also we characterize $\mathbb{R}\mathbb{P}^2$ in $\mathbb{C}\mathbb{P}^2$ as the least possible index minimal Lagrangian compact nonorientable surface of $\mathbb{C}\mathbb{P}^2$.
Wed, 25 Apr 2007 15:18:26 EST - How to get common universal vectors
By F. Bayart, E. Matheron: We prove the existence of common universal vectors for various uncountable families of universal sequence of linear operators. In particular, we give a criterion for a one-parameter family of operators on a Banach space to have a common hypercyclic vector. This criterion relies on some tools from Probability Theory and depends on the geometry of the underlying Banach space. We also study several specific examples, such as shift operators or translation-dilation operators.
Wed, 25 Apr 2007 15:18:26 EST - Irreducible noncommutative defining polynomials for convex sets have degree 4 or less
By Harry Dym, J. William Helton, Scott McCullough:
Wed, 25 Apr 2007 15:18:26 EST - Navier-Stokes equations in thin 3D domains with Navier boundary conditions
By Dragos Iftimie, Genevieve Raugel, George R. Sell:
Wed, 25 Apr 2007 15:18:26 EST - On global existence for the spherically symmetric Einstein-Vlasov system in Schwarzschild coordinates
By Hakan Andreasson: The spherically symmetric Einstein-Vlasov system in Schwarzschild coordinates (i.e., polar slicing and areal radial coordinate) is considered. An improved continuation criterion for global existence of classical solutions is given. Two other types of criteria which prevent finite time blow-up are also given.
Wed, 25 Apr 2007 15:18:26 EST - Orbifold $eta$-invariants
By C. Farsi: In this paper we generalize the classical Atiayh-Patodi-Singer Theorem to orbifolds. From this result, it follows that orbifold $\eta$-invariants can be classically defined.
Wed, 25 Apr 2007 15:18:26 EST - Periodic pulses of coupled nonlinear Schr\"odinger equations in optics
By Jaime Angulo Pava, Felipe Linares: A system of coupled nonlinear Schr\"odinger equations arising in nonlinear optics is considered. The existence of periodic pulses as well as the stability and instability of such solutions are studied. It is shown the existence of a smooth curve of periodic pulses that are of cnoidal type. The Grillakis, Shatah and Strauss theory is set forward to prove the stability results. Regarding instability a general criteria introduced by Grillakis and Jones is used. The well-posedness of the periodic boundary value problem is also studied. Results in the same spirit of the ones obtained for single quadratic semilinear Schr\"odinger equation by Kenig, Ponce and Vega are established.
Wed, 25 Apr 2007 15:18:26 EST - Representations of additive categories and direct-sum decompositions of objects
By Alberto Facchini: The aim of this paper is to embed additive categories in which direct-sum decompositions into indecomposables are not unique but have a regular geometric behavior into categories in which the Krull-Schmidt Theorem holds, that is, to give a representation of additive categories into categories with unique direct-sum decompositions into indecomposables. Cf. Theorems 4.8, 6.1, 6.2, 7.2, and 8.2.
Wed, 25 Apr 2007 15:18:26 EST
