$F$-regularity, test elements, and smooth base change
Melvin
Hochster;
Craig
Huneke
1-62
Abstract: This paper deals with tight closure theory in positive characteristic. After a good deal of preliminary work in the first five sections, including a treatment of $ F$-rationality and a treatment of $F$-regularity for Gorenstein rings, a very widely applicable theory of test elements for tight closure is developed in $\S6$ and is then applied in $\S7$ to prove that both tight closure and $ F$-regularity commute with smooth base change under many circumstances (where "smooth" is used to mean flat with geometrically regular fibers). For example, it is shown in $\S6$ that for a reduced ring $ R$ essentially of finite type over an excellent local ring of characteristic $ p$, if $c$ is not in any minimal prime of $ R$ and ${R_c}$ is regular, then $c$ has a power that is a test element. It is shown in $\S7$ that if $S$ is a flat $R$-algebra with regular fibers and $R$ is $F$-regular then $S$ is $F$-regular. The general problem of showing that tight closure commutes with smooth base change remains open, but is reduced here to showing that tight closure commutes with localization.
$q$-Hermite polynomials, biorthogonal rational functions, and $q$-beta integrals
M. E. H.
Ismail;
D. R.
Masson
63-116
Abstract: We characterize the solutions of the indeterminate moment problem associated with the continuous $q$-Hermite polynomials when $ q > 1$ in terms of their Stieltjes transforms. The extremal measures are found explicitly. An analog of the Askey-Wilson integral is evaluated. It involves integrating a kernel, similar to the Askey-Wilson kernel, against any solution of the $q$-Hermite moment problem, provided that certain integrability conditions hold. This led to direct evaluation of several $q$-beta integrals and their discrete analogs as well as a generalization of Bailey's ${}_6{\psi _6}$, sum containing infinitely many parameters. A system of biorthogonal rational functions is also introduced.
The profile near blowup time for solution of the heat equation with a nonlinear boundary condition
Bei
Hu;
Hong-Ming
Yin
117-135
Abstract: This paper studies the blowup profile near the blowup time for the heat equation ${u_t} = \Delta u$ with the nonlinear boundary condition $ {u_n} = {u^p}$ on $\partial \Omega \times [0,T)$. Under certain assumptions, the exact rate of the blowup is established. It is also proved that the blowup will not occur in the interior of the domain. The asymptotic behavior near the blowup point is also studied.
Univalent functions and the Pompeiu problem
Nicola
Garofalo;
Fausto
Segàla
137-146
Abstract: In this paper we prove a result on the Pompeiu problem. If the Schwarz function $\Phi$ of the boundary of a simply-connected domain $ \Omega \subset {\mathbb{R}^2}$ extends meromorphically into a certain portion $ E$ of $\Omega$ with a pole at some point ${z_0} \in E$, then $\Omega$ has the Pompeiu property unless $ \Phi$ is a Möbius transformation, in which case $\Omega$ is a disk.
On the fundamental periods of Hilbert modular forms
Ze-Li
Dou
147-158
Abstract: The main purpose of this paper is to establish the existence of fundamental periods of primitive cusp forms of Hilbert modular type of several variables, as well as the relationship between those fundamental periods and the special values of the associated $L$-functions. These results, together with some recent results of Shimura, give us the means of translating with ease results concerning periods of automorphic forms derived from various points of view. We also verify several conjectures of Shimura on the properties of such fundamental periods.
Optimal drift on $[0,1]$
Susan
Lee
159-175
Abstract: Consider one-dimensional diffusions on the interval $[0,1]$ of the form $d{X_t} = d{B_t} + b({X_t})dt$, with 0 a reflecting boundary, $ b(x) \geqslant 0$, and $\int_0^1 {b(x)dx = 1}$. In this paper, we show that there is a unique drift which minimizes the expected time for ${X_t}$ to hit $1$, starting from ${X_0} = 0$. In the deterministic case $d{X_t} = b({X_t})dt$, the optimal drift is the function which is identically equal to $1$. By contrast, if $d{X_t} = d{B_t} + b({X_t})dt$, then the optimal drift is the step function which is $ 2$ on the interval $ [1/4,3/4]$ and is 0 otherwise. We also solve this problem for arbitrary starting point ${X_0} = {x_0}$ and find that the unique optimal drift depends on the starting point, $ {x_0}$, in a curious manner.
Commutator theory without join-distributivity
Paolo
Lipparini
177-202
Abstract: We develop Commutator Theory for congruences of general algebraic systems (henceforth called algebras) assuming only the existence of a ternary term $d$ such that $d(a,b,b)[\alpha ,\alpha ]a[\alpha ,\alpha ]d(b,b,a)$, whenever $\alpha$ is a congruence and $a\alpha b$. Our results apply in particular to congruence modular and $n$-permutable varieties, to most locally finite varieties, and to inverse semigroups. We obtain results concerning permutability of congruences, abelian and solvable congruences, connections between congruence identities and commutator identities. We show that many lattices cannot be embedded in the congruence lattice of algebras satisfying our hypothesis. For other lattices, some intervals are forced to be abelian, and others are forced to be nonabelian. We give simplified proofs of some results about the commutator in modular varieties, and generalize some of them to single algebras having a modular congruence lattice.
Geometric consequences of extremal behavior in a theorem of Macaulay
Anna
Bigatti;
Anthony V.
Geramita;
Juan C.
Migliore
203-235
Abstract: F. S. Macaulay gave necessary and sufficient conditions on the growth of a nonnegative integer-valued function which determine when such a function can be the Hilbert function of a standard graded $k$-algebra. We investigate some algebraic and geometric consequences which arise from the extremal cases of Macaulay's theorem. Our work also builds on the fundamental work of G. Gotzmann. Our principal applications are to the study of Hilbert functions of zero-schemes with uniformity conditions. As a consequence, we have new strong limitations on the possible Hilbert functions of the points which arise as a general hyperplane section of an irreducible curve.
Singular polynomials for finite reflection groups
C. F.
Dunkl;
M. F. E.
de Jeu;
E. M.
Opdam
237-256
Abstract: The Dunkl operators involve a multiplicity function as parameter. For generic values of this function the simultaneous kernel of these operators, acting on polynomials, is equal to the constants. For special values, however, this kernel is larger. We determine these singular values completely and give partial results on the representations of $G$ that occur in this kernel.
On the oblique derivative problem for diffusion processes and diffusion equations with H\"older continuous coefficients
Masaaki
Tsuchiya
257-281
Abstract: On a ${C^2}$-domain in a Euclidean space, we consider the oblique derivative problem for a diffusion equation and assume the coefficients of the diffusion and boundary operators are Hölder continuous. We then prove the uniqueness of diffusion processes and fundamental solutions corresponding to the problem. For the purpose, obtaining a stochastic representation of some solutions to the problem plays a key role; in our situation, a difficulty arises from the absence of a fundamental solution with ${C^2}$-smoothness up to the boundary. It is overcome by showing some stability of a fundamental solution and a diffusion process, respectively, under approximation of the domain. In particular, the stability of the fundamental solution is verified through construction: it is done by applying the parametrix method twice to a parametrix with explicit expression.
Distribution of partial sums of the Taylor development of rational functions
V.
Nestoridis
283-295
Abstract: Let $f$ be a rational function regular at 0, which is not a polynomial; let $ {S_N}(z),\;N = 0,1,2, \ldots ,z \in \mathbb{C}$, denote the partial sums of the Taylor development of $f$. We investigate the angular distribution of the sequence ${S_N}(z),\;N = 0,1,2, \ldots$, around $ f(z)$. We show that for all $z$ in the plane, except a denumerable union of straight lines passing through 0, this angular distribution exists and is uniform.
Arithmetic calculus of Fourier transforms by Igusa local zeta functions
Tatsuo
Kimura
297-306
Abstract: We show the possibility of explicit calculation of the Fourier transforms of complex powers of relative invariants of some prehomogeneous vector spaces over $ \mathbb{R}$ by using the explicit form of $p$-adic Igusa local zeta functions.
Asymptotics for orthogonal rational functions
A.
Bultheel;
P.
González-Vera;
E.
Hendriksen;
O.
Njåstad
307-329
Abstract: Let $\{ {\alpha _n}\}$ be a sequence of (not necessarily distinct) points in the open unit disk, and let $\displaystyle {B_0} = 1,\quad {B_n}(z) = \prod\limits_{m = 1}^n {\frac{{\overli... ...alpha _m} - z)}} {{(1 - \overline {{\alpha _m}} z}}),\qquad n = 1,2, \ldots ,}$ ( $\frac{{\overline {{\alpha _n}} }} {{\vert{\alpha _n}\vert}} = - 1$ when $ {\alpha _n} = 0$). Let $ \mu$ be a finite (positive) Borel measure on the unit circle, and let $\{ {\varphi _n}(z)\}$ be orthonormal functions obtained by orthogonalization of $\{ {B_n}:n = 0,1,2, \ldots \}$ with respect to $ \mu$. Boundedness and convergence properties of the reciprocal orthogonal functions $\varphi _n^*(z) = {B_n}(z)\overline {{\varphi _n}(1/\overline z )}$ and the reproducing kernels $ {k_n}(z,w) = \sum\nolimits_{m = 0}^n {{\varphi _m}(z)\overline {{\varphi _m}(w)} }$ are discussed in the situation $\vert{\alpha _n}\vert \leqslant R < 1$ for all $n$, in particular their relationship to the Szegö condition ${L_2}(\mu )$ of the system $ \{ {\varphi _n}(z):n = 0,1,2, \ldots \}$. Limit functions of $\varphi _n^{\ast}(z)$ and ${k_n}(z,w)$ are obtained. In particular, if a subsequence $ \{ {\alpha _{n(s)}}\}$ converge to $\alpha$, then the subsequence $\{ \varphi _{n(s)}^{\ast}(z)\}$ converges to $\displaystyle {e^{i\lambda }}\frac{{\sqrt {1 - \vert\alpha {\vert^2}} }} {{1 - ... ...ine \alpha z}}\frac{1} {{{\sigma _{\mu (z)}}}},\qquad \lambda \in {\mathbf{R}},$ where $\{ {k_n}(z,w)\}$ converge to $1/(1 - z\overline w ){\sigma _\mu }(z)\overline {{\sigma _\mu }(w)}$. The results generalize corresponding results from the classical Szegö theory (concerned with the polynomial situation ${\alpha _n} = 0$ for all $n$).
On orthogonal polynomials with respect to varying measures on the unit circle
K.
Pan
331-340
Abstract: Let $\{ {\phi _n}(d\mu )\}$ be a system of orthonormal polynomials on the unit circle with respect to $ d\mu$ and $\{ {\psi _{n,m}}(d\mu )\} $ be a system of orthonormal polynomials on the unit circle with respect to the varying measures $d\mu /\vert{w_n}(z){\vert^2},\;z = {e^{i\theta }}$, where $\{ {w_n}(z)\}$ is a sequence of polynomials, $\deg {w_n} = n$, whose zeros ${w_{n,1}}, \ldots ,{w_{n,n}}$ lie in $ \vert z\vert < 1$ The asymptotic behavior of the ratio of the two systems on and outside the unit circle is obtained.
Conical limit points and groups of divergence type
Sungbok
Hong
341-357
Abstract: We use the Patterson-Sullivan measure to generalize Agard's theorem to all groups of divergence type. As a consequence, we prove that for a nonelementary group $ \Gamma$ of divergence type, the conical limit set has positive Patterson-Sullivan measure.