Complete linear Weingarten surfaces of Bryant type. A Plateau problem at infinity
José
Antonio
Gálvez;
Antonio
Martínez;
Francisco
Milán
3405-3428
Abstract: In this paper we study a large class of Weingarten surfaces which includes the constant mean curvature one surfaces and flat surfaces in the hyperbolic 3-space. We show that these surfaces can be parametrized by holomorphic data like minimal surfaces in the Euclidean 3-space and we use it to study their completeness. We also establish some existence and uniqueness theorems by studing the Plateau problem at infinity: when is a given curve on the ideal boundary the asymptotic boundary of a complete surface in our family? and, how many embedded solutions are there?
$L^p\to L^q$ regularity of Fourier integral operators with caustics
Andrew
Comech
3429-3454
Abstract: The caustics of Fourier integral operators are defined as caustics of the corresponding Schwartz kernels (Lagrangian distributions on $X\times Y$). The caustic set $\Sigma(\matheurb{ C})$of the canonical relation is characterized as the set of points where the rank of the projection $\pi:\matheurb{ C}\to X\times Y$is smaller than its maximal value, $\dim(X\times Y)-1$. We derive the $L^ p(Y)\to L^ q(X)$ estimates on Fourier integral operators with caustics of corank $1$(such as caustics of type $A_{m+1}$, $m\in{\mathbb N}$). For the values of $p$ and $q$outside of a certain neighborhood of the line of duality, $q=p'$, the $L^ p\to L^ q$ estimates are proved to be caustics-insensitive. We apply our results to the analysis of the blow-up of the estimates on the half-wave operator just before the geodesic flow forms caustics.
Jack polynomials and some identities for partitions
Michel
Lassalle
3455-3476
Abstract: We prove an identity about partitions involving new combinatorial coefficients. The proof given is using a generating function. As an application we obtain the explicit expression of two shifted symmetric functions, related with Jack polynomials. These quantities are the moments of the ``$\alpha$-content'' random variable with respect to some transition probability distributions.
The best constant of the Moser-Trudinger inequality on $\textbf{S}^2$
Yuji
Sano
3477-3482
Abstract: We consider the best constant of the Moser-Trudinger inequality on $\textbf{S}^2$ under a certain orthogonality condition. Applying Moser's calculation, we construct a counterexample to the sharper inequality with the condition.
Definability in the lattice of equational theories of commutative semigroups
Andrzej
Kisielewicz
3483-3504
Abstract: In this paper we study first-order definability in the lattice of equational theories of commutative semigroups. In a series of papers, J. Jezek, solving problems posed by A. Tarski and R. McKenzie, has proved, in particular, that each equational theory is first-order definable in the lattice of equational theories of a given type, up to automorphism, and that such lattices have no automorphisms besides the obvious syntactically defined ones (with exceptions for special unary types). He has proved also that the most important classes of theories of a given type are so definable. In a later paper, Jezek and McKenzie have ``almost proved" the same facts for the lattice of equational theories of semigroups. There were good reasons to believe that the same can be proved for the lattice of equational theories of commutative semigroups. In this paper, however, we show that the case of commutative semigroups is different.
An extended urn model with application to approximation
Fengxin
Chen
3505-3515
Abstract: Pólya's urn model from probability theory is extended to obtain a class of approximation operators for which the Weierstrass Approximation Theorem holds.
Parabolic evolution equations with asymptotically autonomous delay
Roland
Schnaubelt
3517-3543
Abstract: We study retarded parabolic non-autonomous evolution equations whose coefficients converge as $t\to\infty$, such that the autonomous problem in the limit has an exponential dichotomy. Then the non-autonomous problem inherits the exponential dichotomy, and the solution of the inhomogeneous equation tends to the stationary solution at infinity. We use a generalized characteristic equation to deduce the exponential dichotomy and new representation formulas for the solution of the inhomogeneous equation.
Nonlinearizable actions of dihedral groups on affine space
Kayo
Masuda
3545-3556
Abstract: Let $G$ be a reductive, non-abelian, algebraic group defined over $\mathbb{C}$. We investigate algebraic $G$-actions on the total spaces of non-trivial algebraic $G$-vector bundles over $G$-modules with great interest in the case that $G$ is a dihedral group. We construct a map classifying such actions of a dihedral group in some cases and describe the spaces of those non-linearizable actions in some examples.
On the structure of a sofic shift space
Klaus
Thomsen
3557-3619
Abstract: The structure of a sofic shift space is investigated, and Krieger's embedding theorem and Boyle's factor theorem are generalized to a large class of sofic shifts.
Realizability of modules over Tate cohomology
David
Benson;
Henning
Krause;
Stefan
Schwede
3621-3668
Abstract: Let $k$ be a field and let $G$ be a finite group. There is a canonical element in the Hochschild cohomology of the Tate cohomology $\gamma_G\in HH^{3,-1}\hat H^*(G,k)$ with the following property. Given a graded $\hat H^*(G,k)$-module $X$, the image of $\gamma_G$in ${\text{\rm Ext}}^{3,-1}_{\hat H^*(G,k)}(X,X)$ vanishes if and only if $X$ is isomorphic to a direct summand of $\hat H^*(G,M)$ for some $kG$-module $M$. The description of the realizability obstruction works in any triangulated category with direct sums. We show that in the case of the derived category of a differential graded algebra $A$, there is also a canonical element of Hochschild cohomology $HH^{3,-1}H^*(A)$ which is a predecessor for these obstructions.
Linking numbers in rational homology $3$-spheres, cyclic branched covers and infinite cyclic covers
Józef
H.
Przytycki;
Akira
Yasuhara
3669-3685
Abstract: We study the linking numbers in a rational homology $3$-sphere and in the infinite cyclic cover of the complement of a knot. They take values in $\mathbb{Q}$ and in ${Q}(\mathbb{Z}[t,t^{-1}])$, respectively, where ${Q}(\mathbb{Z}[t,t^{-1}])$ denotes the quotient field of $\mathbb{Z}[t,t^{-1}]$. It is known that the modulo- $\mathbb{Z}$ linking number in the rational homology $3$-sphere is determined by the linking matrix of the framed link and that the modulo- $\mathbb{Z}[t,t^{-1}]$ linking number in the infinite cyclic cover of the complement of a knot is determined by the Seifert matrix of the knot. We eliminate `modulo $\mathbb{Z}$' and `modulo $\mathbb{Z}[t,t^{-1}]$'. When the finite cyclic cover of the $3$-sphere branched over a knot is a rational homology $3$-sphere, the linking number of a pair in the preimage of a link in the $3$-sphere is determined by the Goeritz/Seifert matrix of the knot.
Ideals in a perfect closure, linear growth of primary decompositions, and tight closure
Rodney
Y.
Sharp;
Nicole
Nossem
3687-3720
Abstract: This paper is concerned with tight closure in a commutative Noetherian ring $R$ of prime characteristic $p$, and is motivated by an argument of K. E. Smith and I. Swanson that shows that, if the sequence of Frobenius powers of a proper ideal ${\mathfrak{a}}$ of $R$has linear growth of primary decompositions, then tight closure (of ${\mathfrak{a}}$) `commutes with localization at the powers of a single element'. It is shown in this paper that, provided $R$ has a weak test element, linear growth of primary decompositions for other sequences of ideals of $R$ that approximate, in a certain sense, the sequence of Frobenius powers of ${\mathfrak{a}}$ would not only be just as good in this context, but, in the presence of a certain additional finiteness property, would actually imply that tight closure (of ${\mathfrak{a}}$) commutes with localization at an arbitrary multiplicatively closed subset of $R$. Work of M. Katzman on the localization problem for tight closure raised the question as to whether the union of the associated primes of the tight closures of the Frobenius powers of ${\mathfrak{a}}$has only finitely many maximal members. This paper develops, through a careful analysis of the ideal theory of the perfect closure of $R$, strategies for showing that tight closure (of a specified ideal ${\mathfrak{a}}$ of $R$) commutes with localization at an arbitrary multiplicatively closed subset of $R$and for showing that the union of the associated primes of the tight closures of the Frobenius powers of ${\mathfrak{a}}$is actually a finite set. Several applications of the strategies are presented; in most of them it was already known that tight closure commutes with localization, but the resulting affirmative answers to Katzman's question in the various situations considered are believed to be new.
A positivstellensatz for non-commutative polynomials
J.
William
Helton;
Scott
A.
McCullough
3721-3737
Abstract: A non-commutative polynomial which is positive on a bounded semi-algebraic set of operators has a weighted sum of squares representation. This Positivstellensatz parallels similar results in the commutative case. A broader issue is, to what extent does real semi-algebraic geometry extend to non-commutative polynomials? Our ``strict" Positivstellensatz is positive news, on the opposite extreme from strict positivity would be a Real Nullstellensatz. We give an example which shows that there is no non-commutative Real Nullstellensatz along certain lines. However, we include a successful type of non-commutative Nullstellensatz proved by George Bergman.
Non-isotopic symplectic tori in the same homology class
Tolga
Etgü;
B.
Doug
Park
3739-3750
Abstract: For any pair of integers $n\geq 1$ and $q\geq 2$, we construct an infinite family of mutually non-isotopic symplectic tori representing the homology class $q[F]$ of an elliptic surface $E(n)$, where $[F]$ is the homology class of the fiber. We also show how such families can be non-isotopically and symplectically embedded into a more general class of symplectic $4$-manifolds.
A class of processes on the path space over a compact Riemannian manifold with unbounded diffusion
Jörg-Uwe
Löbus
3751-3767
Abstract: A class of diffusion processes on the path space over a compact Riemannian manifold is constructed. The diffusion of such a process is governed by an unbounded operator. A representation of the associated generator is derived and the existence of a certain local second moment is shown.
The double bubble problem on the flat two-torus
Joseph
Corneli;
Paul
Holt;
George
Lee;
Nicholas
Leger;
Eric
Schoenfeld;
Benjamin
Steinhurst
3769-3820
Abstract: We characterize the perimeter-minimizing double bubbles on all flat two-tori and, as corollaries, on the flat infinite cylinder and the flat infinite strip with free boundary. Specifically, we show that there are five distinct types of minimizers on flat two-tori, depending on the areas to be enclosed.
Addendum to ``Symmetrization, symmetric stable processes, and Riesz capacities''
Dimitrios
Betsakos
3821-3821