Asymptotics for the nonlinear dissipative wave equation
Tokio
Matsuyama
865-899
Abstract: We are interested in the asymptotic behaviour of global classical solutions to the initial-boundary value problem for the nonlinear dissipative wave equation in the whole space or the exterior domain outside a star-shaped obstacle. We shall treat the nonlinear dissipative term like $a_1 (1+\vert x \vert)^{-\delta} \vert u_t \vert^{\beta} u_t$ $(a_1$, $\beta$, $\delta>0)$ and prove that the energy does not in general decay. Further, we can deduce that the classical solution is asymptotically free and the local energy decays at a certain rate as the time goes to infinity.
Hölder regularity for a Kolmogorov equation
Andrea
Pascucci
901-924
Abstract: We study the interior regularity properties of the solutions to the degenerate parabolic equation, \begin{displaymath}\Delta_{x}u+b\partial_{y}u-\partial_{t}u=f, \qquad (x,y,t)\in \mathbb{R} ^{N}\times \mathbb{R}\times\mathbb{R} ,\end{displaymath} which arises in mathematical finance and in the theory of diffusion processes.
Some properties of the Schouten tensor and applications to conformal geometry
Pengfei
Guan;
Jeff
Viaclovsky;
Guofang
Wang
925-933
Abstract: The Riemannian curvature tensor decomposes into a conformally invariant part, the Weyl tensor, and a non-conformally invariant part, the Schouten tensor. A study of the $k$th elementary symmetric function of the eigenvalues of the Schouten tensor was initiated in an earlier paper by the second author, and a natural condition to impose is that the eigenvalues of the Schouten tensor are in a certain cone, $\Gamma_k^+$. We prove that this eigenvalue condition for $k \geq n/2$ implies that the Ricci curvature is positive. We then consider some applications to the locally conformally flat case, in particular, to extremal metrics of $\sigma_k$-curvature functionals and conformal quermassintegral inequalities, using the results of the first and third authors.
The double of a hyperbolic manifold and non-positively curved exotic $PL$ structures
Pedro
Ontaneda
935-965
Abstract: We give examples of non-compact finite volume real hyperbolic manifolds of dimension greater than five, such that their doubles admit at least three non-equivalent smoothable $PL$ structures, two of which admit a Riemannian metric of non-positive curvature while the third does not. We also prove that the doubles of non-compact finite volume real hyperbolic manifolds of dimension greater than four are differentiably rigid.
Non-independence of excursions of the Brownian sheet and of additive Brownian motion
Robert
C.
Dalang;
T.
Mountford
967-985
Abstract: A classical and important property of Brownian motion is that given its zero set, distinct excursions away from zero are independent. In this paper, we examine the analogous question for the Brownian sheet, and also for additive Brownian motion. Our main result is that given the level set of the Brownian sheet at level zero, distinct excursions of the sheet away from zero are not independent. In fact, given the zero set of the Brownian sheet in the entire non-negative quadrant, and the sign of all but a finite number of excursions away from zero, the signs of the remaining excursions are determined. For additive Brownian motion, we prove the following definitive result: given the zero set of additive Brownian motion and the sign of a single excursion, the signs of all other excursions are determined. In an appendix by John B. Walsh, it is shown that given the absolute value of the sheet in the entire quadrant and, in addition, the sign of the sheet at a fixed, non-random time point, then the whole sheet can be recovered.
Supercongruences between truncated $_{2}F_{1}$ hypergeometric functions and their Gaussian analogs
Eric
Mortenson
987-1007
Abstract: Fernando Rodriguez-Villegas has conjectured a number of supercongruences for hypergeometric Calabi-Yau manifolds of dimension $d\le 3$. For manifolds of dimension $d=1$, he observed four potential supercongruences. Later the author proved one of the four. Motivated by Rodriguez-Villegas's work, in the present paper we prove a general result on supercongruences between values of truncated $_{2}F_{1}$hypergeometric functions and Gaussian hypergeometric functions. As a corollary to that result, we prove the three remaining supercongruences.
Cyclic covers of rings with rational singularities
Anurag
K.
Singh
1009-1024
Abstract: We examine some recent work of Phillip Griffith on étale covers and fibered products from the point of view of tight closure theory. While it is known that cyclic covers of Gorenstein rings with rational singularities are Cohen-Macaulay, we show this is not true in general in the absence of the Gorenstein hypothesis. Specifically, we show that the canonical cover of a $\mathbb Q$-Gorenstein ring with rational singularities need not be Cohen-Macaulay.
Homological properties of balanced Cohen-Macaulay algebras
Izuru
Mori
1025-1042
Abstract: A balanced Cohen-Macaulay algebra is a connected algebra $A$ having a balanced dualizing complex $\omega_A[d]$ in the sense of Yekutieli (1992) for some integer $d$ and some graded $A$-$A$ bimodule $\omega_A$. We study some homological properties of a balanced Cohen-Macaulay algebra. In particular, we will prove the following theorem: \begin{thm0} Let$A$\space be a Noetherian balanced Cohen-Macaulay algebra, and ... ...s^{r_0}_{j=1} \omega_A(-l_{0j})\to M\to 0. \end{align*}\end{enumerate}\end{thm0} As a corollary, we will have the following characterizations of AS Gorenstein algebras and AS regular algebras: \begin{cor0} Let$A$\space be a Noetherian balanced Cohen-Macaulay algebra. \beg... ...ximal Cohen-Macaulay graded left$A$-module is free. \end{enumerate}\end{cor0}
Noetherian PI Hopf algebras are Gorenstein
Q.-S.
Wu;
J.
J.
Zhang
1043-1066
Abstract: We prove that every noetherian affine PI Hopf algebra has finite injective dimension, which answers a question of Brown (1998).
Expanding maps on infra-nilmanifolds of homogeneous type
Karel
Dekimpe;
Kyung
Bai
Lee
1067-1077
Abstract: In this paper we investigate expanding maps on infra-nilmanifolds. Such manifolds are obtained as a quotient $E\backslash L$, where $L$ is a connected and simply connected nilpotent Lie group and $E$is a torsion-free uniform discrete subgroup of $L {\mathbb o} C$, with $C$ a compact subgroup of $\operatorname{Aut}(L)$. We show that if the Lie algebra of $L$ is homogeneous (i.e., graded and generated by elements of degree 1), then the corresponding infra-nilmanifolds admit an expanding map. This is a generalization of the result of H. Lee and K. B. Lee, who treated the 2-step nilpotent case.
Derivations and invariant forms of Jordan and alternative tori
Erhard
Neher;
Yoji
Yoshii
1079-1108
Abstract: Jordan and alternative tori are the coordinate algebras of extended affine Lie algebras of types ${A}_1$ and ${A}_2$. In this paper we show that the derivation algebra of a Jordan torus is a semidirect product of the ideal of inner derivations and the subalgebra of central derivations. In the course of proving this result, we investigate derivations of the more general class of division graded Jordan and alternative algebras. We also describe invariant forms of these algebras.
Limits of interpolatory processes
W.
R.
Madych
1109-1133
Abstract: Given $N$ distinct real numbers $\nu_1, \ldots, \nu_N$ and a positive approximation of the identity $\phi_{\epsilon}$, which converges weakly to the Dirac delta measure as $\epsilon$goes to zero, we investigate the polynomials $P_{\epsilon}(x)= \sum c_{\epsilon , j} e^{-i \nu_j x}$ which solve the interpolation problem \begin{displaymath}\int P_{\epsilon}(x) e^{i \nu_k x} \phi_{\epsilon}(x)dx=f_{\epsilon,k}, \quad k=1, \ldots, N,\end{displaymath} with prescribed data $f_{\epsilon,1}, \dots, f_{\epsilon,N}$. More specifically, we are interested in the behavior of $P_{\epsilon}(x)$ when the data is of the form $f_{\epsilon, k}=\int f(x) e^{i \nu_k x} \phi_{\epsilon}(x)dx$ for some prescribed function $f$. One of our results asserts that if $f$ is sufficiently nice and $\phi_{\epsilon}$ has sufficiently well-behaved moments, then $P_{\epsilon}$ converges to a limit $P$ which can be completely characterized. As an application we identify the limits of certain fundamental interpolatory splines whose knot set is $\mathbb{Z} \setminus \mathcal{N}$, where $\mathcal{N}$ is an arbitrary finite subset of the integer lattice $\mathbb{Z}$, as their degree goes to infinity.
Maximal functions with polynomial densities in lacunary directions
Kathryn
Hare;
Fulvio
Ricci
1135-1144
Abstract: Given a real polynomial $p(t)$ in one variable such that $p(0)=0$, we consider the maximal operator in $\mathbb{R}^{2}$, \begin{displaymath}M_{p}f(x_{1},x_{2})=\sup _{h>0\,,\,i,j\in \mathbb{Z}}\frac{1... ...t f\big (x_{1}-2^{i}p(t),x_{2}-2^{j}p(t)\big )\big \vert\,dt . \end{displaymath} We prove that $M_{p}$ is bounded on $L^{q}(\mathbb{R}^{2})$ for $q>1$ with bounds that only depend on the degree of $p$.
Singular integrals with rough kernels along real-analytic submanifolds in ${\mathbf{R}}^3$
Dashan
Fan;
Kanghui
Guo;
Yibiao
Pan
1145-1165
Abstract: $L^p$ mapping properties will be established in this paper for singular Radon transforms with rough kernels defined by translates of a real-analytic submanifold in $\mathbf{R}^3$.
Spherical maximal operator on symmetric spaces of constant curvature
Amos
Nevo;
P.
K.
Ratnakumar
1167-1182
Abstract: We prove an endpoint weak-type maximal inequality for the spherical maximal operator applied to radial funcions on symmetric spaces of constant curvature and dimension $n\ge 2$. More explicitly, in the Lorentz space associated with the natural isometry-invariant measure, we show that, for every radial function $f$, \begin{displaymath}\Vert{\mathcal M}f\Vert _{\,n^{\prime},\infty}\leq C_n \Vert f \Vert _{n^{\prime},1},\,\,\,\, n^\prime=\frac{n}{n-1}.\end{displaymath} The proof uses only geometric arguments and volume estimates, and applies uniformly in every dimension.
The Mori cones of moduli spaces of pointed curves of small genus
Gavril
Farkas;
Angela
Gibney
1183-1199
Abstract: We compute the Mori cones of the moduli spaces $\overline M_{g,n}$ of $n$pointed stable curves of genus $g$, when $g$ and $n$ are relatively small. For instance we show that for $g<14$ every curve in $\overline M_g$ is equivalent to an effective combination of the components of the locus of curves with $3g-4$ nodes. We completely describe the cone of nef divisors for the space $\overline M_{0,6}$, thus verifying Fulton's conjecture for this space. Using this description we obtain a classification of all the fibrations of $\overline M_{0,6}$.
On the inversion of the convolution and Laplace transform
Boris
Baeumer
1201-1212
Abstract: We present a new inversion formula for the classical, finite, and asymptotic Laplace transform $\hat f$ of continuous or generalized functions $f$. The inversion is given as a limit of a sequence of finite linear combinations of exponential functions whose construction requires only the values of $\hat f$ evaluated on a Müntz set of real numbers. The inversion sequence converges in the strongest possible sense. The limit is uniform if $f$is continuous, it is in $L^{1}$ if $f\in L^{1}$, and converges in an appropriate norm or Fréchet topology for generalized functions $f$. As a corollary we obtain a new constructive inversion procedure for the convolution transform ${\mathcal K}:f\mapsto k\star f$; i.e., for given $g$ and $k$ we construct a sequence of continuous functions $f_{n}$ such that $k\star f_{n}\to g$.
Infinite partition regular matrices: solutions in central sets
Neil
Hindman;
Imre
Leader;
Dona
Strauss
1213-1235
Abstract: A finite or infinite matrix $A$ is image partition regular provided that whenever ${\mathbb N}$ is finitely colored, there must be some $\vec {x}$with entries from ${\mathbb N}$ such that all entries of $A\vec {x}$ are in the same color class. In contrast to the finite case, infinite image partition regular matrices seem very hard to analyze: they do not enjoy the closure and consistency properties of the finite case, and it is difficult to construct new ones from old. In this paper we introduce the stronger notion of central image partition regularity, meaning that $A$ must have images in every central subset of ${\mathbb N}$. We describe some classes of centrally image partition regular matrices and investigate the extent to which they are better behaved than ordinary image partition regular matrices. It turns out that the centrally image partition regular matrices are closed under some natural operations, and this allows us to give new examples of image partition regular matrices. In particular, we are able to solve a vexing open problem by showing that whenever ${\mathbb N}$ is finitely colored, there must exist injective sequences $\langle x_n\rangle_{n=0}^\infty$ and $\langle z_n\rangle_{n=0}^\infty$ in ${\mathbb N}$ with all sums of the forms $x_n+x_m$ and $z_n+2z_m$ with $n<m$ in the same color class. This is the first example of an image partition regular system whose regularity is not guaranteed by the Milliken-Taylor Theorem, or variants thereof.
Are Hamiltonian flows geodesic flows?
Christopher
McCord;
Kenneth
R.
Meyer;
Daniel
Offin
1237-1250
Abstract: When a Hamiltonian system has a ``Kinetic + Potential'' structure, the resulting flow is locally a geodesic flow. But there may be singularities of the geodesic structure; so the local structure does not always imply that the flow is globally a geodesic flow. In order for a flow to be a geodesic flow, the underlying manifold must have the structure of a unit tangent bundle. We develop homological conditions for a manifold to have such a structure. We apply these criteria to several classical examples: a particle in a potential well, the double spherical pendulum, the Kovalevskaya top, and the $N$-body problem. We show that the flow of the reduced planar $N$-body problem and the reduced spatial 3-body are never geodesic flows except when the angular momentum is zero and the energy is positive.
An extension theorem for separately holomorphic functions with pluripolar singularities
Marek
Jarnicki;
Peter
Pflug
1251-1267
Abstract: Let $D_j\subset\mathbb{C} ^{n_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluriregular set, $j=1,\dots,N$. Put \begin{displaymath}X:=\bigcup_{j=1}^N A_1\times\dots\times A_{j-1}\times D_j\tim... ...thbb{C} ^{n_1}\times\dots\times\mathbb{C} ^{n_N}=\mathbb{C} ^n.\end{displaymath} Let $U\subset\mathbb{C} ^n$ be an open neighborhood of $X$ and let $M\subset U$ be a relatively closed subset of $U$. For $j\in\{1,\dots,N\}$ let $\Sigma_j$ be the set of all $M_{(z',\cdot,z'')}:=\{z_j\in\mathbb{C} ^{n_j}: (z',z_j,z'')\in M\}$ is not pluripolar. Assume that $\Sigma_1,\dots,\Sigma_N$ are pluripolar. Put \begin{multline*}X':=\bigcup_{j=1}^N\{(z',z_j,z'')\in(A_1\times\dots\times A_{j-... ...imes(A_{j+1}\times\dots\times A_N): (z',z'')\notin\Sigma_j\}. \end{multline*} Then there exists a relatively closed pluripolar subset $\widehat{M}\subset\widehat X$ of the ``envelope of holomorphy'' $\widehat{X}\subset\mathbb{C} ^n$ of $X$ such that: $\bullet$ $X\setminus M$ there exists exactly one function $\widehat f$ holomorphic on $\widehat X\setminus\widehat M$ with $\widehat f=f$ on $\widehat M$ is singular with respect to the family of all functions $\widehat f$.
Hyperbolic mean growth of bounded holomorphic functions in the ball
E.
G.
Kwon
1269-1294
Abstract: We consider the hyperbolic Hardy class $\varrho H^{p}(B)$, $0<p<\infty$. It consists of $\phi$ holomorphic in the unit complex ball $B$ for which $\vert \phi \vert < 1$ and \begin{displaymath}\sup _{0<r<1} \, \int _{\partial B} \left \{ \varrho (\phi (r\zeta ), 0)\right \}^{p} \, d\sigma (\zeta ) ~<~ \infty ,\end{displaymath} where $\varrho$denotes the hyperbolic distance of the unit disc. The hyperbolic version of the Littlewood-Paley type $g$-function and the area function are defined in terms of the invariant gradient of $B$, and membership of $\varrho H^{p}(B)$ is expressed by the $L^{p}$ property of the functions. As an application, we can characterize the boundedness and the compactness of the composition operator $\mathcal{C}_{\phi }$, defined by $\mathcal{C}_{\phi }f = f\circ \phi$, from the Bloch space into the Hardy space $H^{p}(B)$.
Erratum to ``Subgroup properties of fully residually free groups''
Ilya
Kapovich
1295-1296