Applications of simplicial $M$-sets to proper and strong shape theories
L. J.
Hernández Paricio
363-409
Abstract: In this paper we have tried to reduce the classical classification problems for spaces and maps of the proper category and of the strong shape category to similar problems in the homotopy category of simplicial sets or in the homotopy category of simplicial $M$-sets, which $M$ is the monoid of proper selfmaps of the discrete space $ \mathbb{N}$ of nonnegative integers. Given a prospace (prosimplicial set) $ Y$, we have constructed a simplicial set ${\overline {\mathcal{P}} ^R}Y$ such that the Hurewicz homotopy groups of ${\overline {\mathcal{P}} ^R}Y$ are the Grossman homotopy groups of $Y$. For the case of the end prospace $Y = \varepsilon X$ of a space $X$, we obtain Brown's proper homotopy groups; and for the Vietoris prospace $Y = VX$ (introduced by Porter) of a compact metrisable space $X$, we have Quigley's inward groups. The simplicial subset ${\overline {\mathcal{P}} ^R}Y$ of a tower $ Y$ contains, as a simplicial subset, the homotopy limit ${\lim ^R}Y$. The inclusion $ {\lim ^R}Y \to {\overline {\mathcal{P}} ^R}Y$ induces many relations between the homotopy and (co)homology invariants of the prospace $ Y$. Using the functor $ {\overline {\mathcal{P}} ^R}$ we prove Whitehead theorems for proper homotopy, prohomotopy, and strong shape theories as a particular case of the standard Whitehead theorem. The algebraic condition is given in terms of Brown's proper groups, Grossman's homotopy groups and Quigley's inward groups, respectively. In all these cases an equivalent cohomological condition can be given by taking twisted coefficients. The "singular" homology groups of ${\overline {\mathcal{P}} ^R}Y$ provide homology theories for the Brown, Grossman and Quigley homotopy groups that satisfy Hurewicz theorems in the corresponding settings. However, there are other homology theories for the homotopy groups above satisfying other Hurewicz theorems. We also analyse the notion of $\overline {\mathcal{P}} $-movable prospace. For a $ \overline {\mathcal{P}}$-movable tower we prove easily (without $ {\lim ^1}$ functors) that the strong homotopy groups agree with the Čech homotopy groups and the Grossman homotopy groups are determined by the Čech (or strong) groups by the formula $^G{\pi_q} = \overline{\mathcal{P}} \check{\pi}_q$. This implies that the algebraic condition of the Whitehead theorem can be given in terms of strong (Čech) groups when the condition of $\overline {\mathcal{P}} $-movability is included. We also study homology theories for the strong (Steenrod) homotopy groups which satisfy Hurewicz theorems but in general do not agree with the corresponding Steenrod-Sitnikov homology theories.
A free-boundary problem for the heat equation arising in flame propagation
Luis A.
Caffarelli;
Juan L.
Vázquez
411-441
Abstract: We introduce a new free-boundary problem for the heat equation, of interest in combustion theory. It is obtained in the description of laminar flames as an asymptotic limit for high activation energy. The problem asks for the determination of a domain in space-time, $\Omega \subset {{\mathbf{R}}^n} \times (0,T)$, and a function $u(x,t) \geqslant 0$ defined in $\Omega$, such that ${u_t} = \Delta u$ in $ \Omega ,\;u$ takes certain initial conditions, $u(x,0) = {u_0}(x)$ for $x \in {\Omega _0} = \partial \Omega \cap \{ t = 0\}$, and two conditions are satisfied on the free boundary $\Gamma = \partial \Omega \cap \{ t > 0\} :u = 0$ and ${u_\nu } = - 1$, where ${u_\nu }$ denotes the derivative of $ u$ along the spatial exterior normal to $\Gamma$. We approximate this problem by means of a certain regularization on the boundary and prove the existence of a weak solution under suitable assumptions on the initial data.
Epi-derivatives of integral functionals with applications
Philip D.
Loewen;
Harry H.
Zheng
443-459
Abstract: We study first- and second-order epi-differentiability for integral functionals defined on ${L^2}[0,T]$, and apply the results to obtain first- and second-order necessary conditions for optimality in free endpoint control problems.
The relative Burnside module and the stable maps between classifying spaces of compact Lie groups
Norihiko
Minami
461-498
Abstract: Tom Dieck's Burnside ring of compact Lie groups is generalized to the relative case: For any $G \triangleright N$, a compact Lie group and its normal subgroup $A(G \triangleright N)$ is defined to be an appropriate set of the equivalence classes of compact $ G$-ENR's with free $ N$-action, in such a way that $\psi :A(G \triangleright N) \simeq \pi _{G/N}^0({S^0};B{(N,G)_ + })$, where $B(N,G)$ is the classifying space of principal $(N,G)$-bundle. Under the "product" situation, i.e. $G = F \times K,\;N = K,\;A(F \times K \triangleright K)$ is also denoted by $A(F,K)$, as it turns out to be the usual $ A(F,K)$ when both $ F$ and $K$ are finite. Then a couple of applications are given to the study of stable maps between classifying spaces of compact Lie groups: a conceptual proof of Feshbach's double coset formula, and a density theorem on the map $\alpha _p^ \wedge :A(L,H)_p^ \wedge \to \{ B{L_{ + ,}}B{H_ + }\} _p^ \wedge$ for any compact Lie groups $L,\;K$ when $p$ is odd. (Some restriction is applied to $ L$ when $p = 2$.) This latter result may be regarded as the pushout of Feshbach's density theorem and the theorem of May-Snaith-Zelewski, over the celebrated Carlsson solution of Segal's Burnside ring conjecture.
Functional rotation numbers for one-dimensional maps
A. M.
Blokh
499-513
Abstract: We introduce functional rotation numbers and sets for one-dimensional maps (we call them $f$-rotation numbers and sets) and deduce some of their properties (density of ${\text{f}}$-rotation numbers of periodic points in the $ {\text{f}}$-rotation set, conditions for the connectedness of the ${\text{f}}$-rotation set) from the spectral decomposition theorem for one-dimensional maps.
An optimal condition for the LIL for trigonometric series
I.
Berkes
515-530
Abstract: By a classical theorem (Salem-Zygmund [6], Erdős-Gàl [3]), if $ ({n_k})$ is a sequence of positive integers satisfying $ {n_{k + 1}}/{n_k} \geqslant q > 1\;(k = 1,2, \ldots )$ then $(\cos {n_k}x)$ obeys the law of the iterated logarithm, i.e., (1) $\displaystyle \mathop {\lim \sup }\limits_{N \to \infty } {(N\log \log N)^{ - 1... ...\limits_{k \leqslant N} {\cos {n_k}x = 1\quad {\text{a}}{\text{.e}}{\text{.}}}$ It is also known (Takahashi [7, 8]) that the Hadamard gap condition ${n_{k + 1}}/{n_k} \geqslant q > 1$ can be essentially weakened here but the problem of finding the precise gap condition for the LIL (1) has remained open. In this paper we find, using combinatorial methods, an optimal gap condition for the upper half of the LIL, i.e., the inequality $\leqslant 1$ in (1).
The vanishing viscosity method in one-dimensional thermoelasticity
Gui Qiang
Chen;
Constantine M.
Dafermos
531-541
Abstract: The vanishing viscosity method is applied to the system of conservation laws of mass, momentum, and energy for a special class of one-dimensional thermoelastic media that do not conduct heat. Two types of vanishing "viscosity" are considered: Newtonian and artificial, in both cases accompanied by vanishing heat conductivity. It is shown that in either case one can pass to the zero viscosity limit by the method of compensated compactness, provided that velocity and pressure are uniformly bounded. Oscillations in the entropy field may propagate along the linearly degenerate characteristic field but do not affect the compactness of the velocity field or the pressure field. A priori bounds on velocity and pressure are established, albeit only for the case of artificial viscosity.
Entire functions, in the classification of differentiable germs tangent to the identity, in one or two variables
Patrick
Ahern;
Jean-Pierre
Rosay
543-572
Abstract: This paper presents a survey and some (hopefully) new facts on germs of maps tangent to the identity (in $\mathbb{R},\mathbb{C},$ or ${\mathbb{R}^2}$), (maps $f$ defined near 0, such that $f(0) = 0$, and $f'(0)$ is the identity). Proofs are mostly original. The paper is mostly oriented towards precise examples and the questions of descriptions of members in the conjugacy class, flows, $k$th root. It happened that entire functions provide clear and easy examples. However they should be considered just as a tool, not as the main topic. For example in Proposition $2$ the function $z \mapsto z + {z^2}$ should be better thought of as the map $(x,y) \to (x + {x^2} - {y^2},y + 2xy)$.
Diophantine approximation in ${\bf R}\sp n$
L. Ya.
Vulakh
573-585
Abstract: A modification of the Ford geometric approach to the problem of approximation of irrational real numbers by rational fractions is developed. It is applied to find an upper bound for the Hurwitz constant for a discrete group acting in a hyperbolic space. A generalized Khinchine's approximation theorem is also given.
Weighted boundary limits of the generalized Kobayashi-Royden metrics on weakly pseudoconvex domains
Ji Ye
Yu
587-614
Abstract: The purpose of this paper is to study the existence of weighted boundary limits of the generalized Kobayashi-Royden metrics on weakly pseudoconvex domains in ${\mathbb{C}^n}$ and to explore the connections between the limits and the Levi invariants. The main result extends Graham's result on strongly pseudoconvex domains to a large class of weakly pseudoconvex domains.
Contiguity relations for generalized hypergeometric functions
Alan
Adolphson;
Bernard
Dwork
615-625
Abstract: It is well known that the hypergeometric functions $\displaystyle _2{F_1}(\alpha \pm 1,\beta ,\gamma ;t),{\quad _2}{F_1}(\alpha ,\beta \pm 1,\gamma ;t),{\quad _2}{F_1}(\alpha ,\beta ,\gamma \pm 1;t),$ which are contiguous to $_2{F_1}(\alpha ,\beta ,\gamma ;t)$, can be expressed in terms of $\displaystyle _2{F_1}(\alpha ,\beta ,\gamma ;t)\quad {\text{and}}{\quad _2}F_1^\prime (\alpha ,\beta ,\gamma ;t).$ We explain how to derive analogous formulas for generalized hypergeometric functions. Our main point is that such relations can be deduced from the geometry of the cone associated in a recent paper by B. Dwork and F. Loeser to a generalized hypergeometric series.
Assessing prediction error in autoregressive models
Ping
Zhang;
Paul
Shaman
627-637
Abstract: Assessing prediction error is a problem which arises in time series analysis. The distinction between the conditional prediction error $e$ and the unconditional prediction error $ E(e)$ has not received much attention in the literature. Although one can argue that the conditional version is more practical, we show in this article that assessing $e$ is nearly impossible. In particular, we use the correlation coefficient $ \operatorname{corr} (\hat e,e)$, where $\hat e$ is an estimate of $e$, as a measure of performance and show that ${\lim _{T \to \infty }}\sqrt T \operatorname{corr} (\hat e,e) = C$ where $T$ is the sample size and $C > 0$ is some constant. Furthermore, the value of $C$ is large only when the process is extremely non-Gaussian or nearly nonstationary.
Prime ideals in polynomial rings over one-dimensional domains
William
Heinzer;
Sylvia
Wiegand
639-650
Abstract: Let $R$ be a one-dimensional integral domain with only finitely many maximal ideals and let $ x$ be an indeterminate over $R$. We study the prime spectrum of the polynomial ring $R[x]$ as a partially ordered set. In the case where $R$ is countable we classify $\operatorname{Spec} (R[x])$ in terms of splitting properties of the maximal ideals ${\mathbf{m}}$ of $R$ and the valuative dimension of ${R_{\mathbf{m}}}_{}$.
On the existence of global Tchebychev nets
Sandra L.
Samelson;
W. P.
Dayawansa
651-660
Abstract: Let $S$ be a complete, open simply connected surface. Suppose that the integral of the Gauss curvature over arbitrary measurable sets is less than $ \pi /2$ in magnitude. We show that the surface admits a global Tchebychev net.
Analytic Fourier-Feynman transforms and convolution
Timothy
Huffman;
Chull
Park;
David
Skoug
661-673
Abstract: In this paper we develop an ${L_p}$ Fourier-Feynman theory for a class of functionals on Wiener space of the form $F(x) = f(\int_0^T {{\alpha _1}dx, \ldots ,\int_0^T {{\alpha _n}dx)} }$. We then define a convolution product for functionals on Wiener space and show that the Fourier-Feynman transform of the convolution product is a product of Fourier-Feynman transforms.
On the tangential interpolation problem for $H\sb 2$ functions
Daniel
Alpay;
Vladimir
Bolotnikov;
Yossi
Peretz
675-686
Abstract: The aim of this paper is to solve a matrix-valued version of the Nevanlinna-Pick interpolation problem for $ {H_2}$ functions. We reduce this problem to a Nevanlinna-Pick interpolation problem for Schur functions and obtain a linear fractional transformation which describes the set of all solutions.
Convexity of the ideal boundary for complete open surfaces
Jin-Whan
Yim
687-700
Abstract: For complete open surfaces admitting total curvature, we define several kinds of convexity for the ideal boundary, and provide examples of each of them. We also prove that a surface with most strongly convex ideal boundary is in fact a generalization of a Hadamard manifold in the sense that the ideal boundary consists entirely of Busemann functions.
On the embedded primary components of ideals. IV
William
Heinzer;
L. J.
Ratliff;
Kishor
Shah
701-708
Abstract: The results in this paper expand the fundamental decomposition theory of ideals pioneered by Emmy Noether. Specifically, let $I$ be an ideal in a local ring $(R,M)$ that has $M$ as an embedded prime divisor, and for a prime divisor $P$ of $I$ let $I{C_P}(I)$ be the set of irreducible components $ q$ of $I$ that are $P$-primary (so there exists a decomposition of $ I$ as an irredundant finite intersection of irreducible ideals that has $ q$ as a factor). Then the main results show: (a) $I{C_M}(I) = \cup \{ I{C_M}(Q);Q\;{\text{is a }}\operatorname{MEC} {\text{ of }}I\}$ ($Q$ is a MEC of $I$ in case $Q$ is maximal in the set of $M$-primary components of $I$); (b) if $I = \cap \{ {q_i};i = 1, \ldots ,n\}$ is an irredundant irreducible decomposition of $ I$ such that $ {q_i}$ is $M$-primary if and only if $i = 1, \ldots ,k < n$, then $\cap \{ {q_i};i = 1, \ldots ,k\}$ is an irredundant irreducible decomposition of a MEC of $ I$, and, conversely, if $ Q$ is a MEC of $ I$ and if $\cap \{ {Q_j};j = 1, \ldots ,m\}$ (resp., $ \cap \{ {q_i};i = 1, \ldots ,n\}$) is an irredundant irreducible decomposition of $ Q$ (resp., $I$) such that ${q_1}, \ldots ,{q_k}$ are the $M$-primary ideals in $\{ {q_1}, \ldots ,{q_n}\} $, then $m = k$ and $( \cap \{ {q_i};i = k + 1, \ldots ,n\} ) \cap ( \cap \{ {Q_j};j = 1, \ldots ,m\} )$ is an irredundant irreducible decomposition of $I$; (c) $I{C_M}(I) = \{ q,q\;{\text{is maximal in the set of ideals that contain }}I\;{\text{and do not contain }}I:M\}$; (d) if $Q$ is a MEC of $I$, then $I{C_M}(Q) = \{ q;Q \subseteq q \in I{C_M}(I)\}$; (e) if $J$ is an ideal that lies between $I$ and an ideal $Q \in I{C_M}(I)$, then $J = \cap \{ q;J \subseteq q \in I{C_M}(I)\}$; and, (f) there are no containment relations among the ideals in $ \cup \{ I{C_P}(I)$; $ P$ is a prime divisor of $ I$}.
Random quadratic forms
John
Gregory;
H. R.
Hughes
709-717
Abstract: The results of Boyce for random Sturm-Liouville problems are generalized to random quadratic forms. Order relationships are proved between the means of eigenvalues of a random quadratic form and the eigenvalues of an associated mean quadratic form. Finite-dimensional and infinite-dimensional examples that show these are the best possible results are given. Also included are some results for a general approximation theory for random quadratic forms.