Mean-square approximation by polynomials on the unit disk
Thomas L.
Kriete;
Barbara D.
MacCluer
1-34
Abstract: We investigate ${P^2}(\mu )$, the closure in ${L^2}(\mu )$ of the complex polynomials, for certain measures $\mu$ on the closed unit disk in the complex plane. Specifically, we present a condition on $\mu$ which guarantees that ${P^2}(\mu )$ decomposes into a natural direct sum.
The fixed point theorem in equivariant cohomology
J. D. S.
Jones;
S. B.
Petrack
35-49
Abstract: In this paper we study the ${S^1}$-equivariant de Rham cohomology of infinite dimensional ${S^1}$-manifolds. Our main example is the free loop space $LX$ where $X$ is a finite dimensional manifold with the circle acting by rotating loops. We construct a new form of equivariant cohomology $h_T^*$ which agrees with the usual periodic equivariant cohomology in finite dimensions and we prove a suitable analogue of the classical fixed point theorem which is valid for loop spaces $LX$. This gives a cohomological framework for studying differential forms on loop spaces and we apply these methods to various questions which arise from the work of Witten [16], Atiyah [2], and Bismut [5]. In particular we show, following Atiyah in [2], that the $\hat A$-polynomial of $X$ arises as an equivariant characteristic class, in the theory $h_T^*$, of the normal bundle to $X$, considered as the space of constant loops, in $LX$.
Generalized second derivatives of convex functions and saddle functions
R. T.
Rockafellar
51-77
Abstract: The theory of second-order epi-derivatives of extended-real-valued functions is applied to convex functions on ${\mathbb{R}^n}$ and shown to be closely tied to proto-differentiation of the corresponding subgradient multifunctions, as well as to second-order epi-differentiation of conjugate functions. An extension is then made to saddle functions, which by definition are convex in one argument and concave in another. For this case a concept of epi-hypo-differentiability is introduced. The saddle function results provide a foundation for the sensitivity analysis of primal and dual optimal solutions to general finite-dimensional problems in convex optimization, since such solutions are characterized as saddlepoints of a convex-concave Lagrangian function, or equivalently as subgradients of the saddle function conjugate to the Lagrangian.
The crank of partitions mod $8,\;9$ and $10$
Frank G.
Garvan
79-94
Abstract: Recently new combinatorial interpretations of Ramanujan's partition congruences modulo $5$, $7$ and $11$ were found. These were in terms of the crank. A refinement of the congruence modulo $ 5$ is proved. The number of partitions of $5n + 4$ with even crank is congruent to 0 modulo $ 5$. The residue of the even crank modulo $10$ divides these partitions into five equal classes. Other relations for the crank modulo $ 8$, $9$ and $10$ are also proved. The dissections of certain generating functions associated with these results are calculated. All of the results are proved by elementary methods.
Homotopy theory of minimal simplicial spaces
Mark D.
Pritt
95-119
Abstract: Various aspects of homotopy theory in the category of minimal simplicial spaces are studied. It is shown that the usual results of homotopy theory hold in this category, and necessary and sufficient conditions are given under which a simplicial space has the homotopy type of a minimal simplicial space. Continuous cohomology in this category is also studied.
Existence and uniqueness of the Riemann problem for a nonlinear system of conservation laws of mixed type
L.
Hsiao;
P.
de Mottoni
121-158
Abstract: We study the system of conservation laws given by $\displaystyle \left\{ {_{{\upsilon _t} + {{[\upsilon (a + u)]}_x} = 0\quad (a >... ...{\text{a}}{\text{constant}}),}^{{u_t} + {{[u(1 - \upsilon )]}_x} = 0,}} \right.$ with any Riemann initial data $({u_ \mp },{\upsilon _ \mp })$. The system is elliptic in the domain where $ {(\upsilon - u + a - 1)^2} + 4(a - 1)u < 0$ and strictly hyperbolic when $ {(\upsilon - u + a - 1)^2} + 4(a - 1)u > 0$. We combine and generalize Lax criterion and Oleinik-Liu criterion to introduce the generalized entropy condition (G.E.C.) by which we can show that the Riemann problem always has a weak solution (any discontinuity satisfies the G.E.C.) for any initial data, however not necessarily unique. We introduce the minimum principle then in the definition of an admissible weak solution for the Riemann problem and the existence and uniqueness of the solution for any Riemann data.
The diffeotopy group of the twisted $2$-sphere bundle over the circle
M. Ho
Kim;
Frank
Raymond
159-168
Abstract: The diffeotopy group of the nontrivial $2$-sphere bundle over the circle is shown to be isomorphic to ${\mathbb{Z}_2} \oplus {\mathbb{Z}_2}$. The first generator is induced by a reflection across the base circle, while a second generator comes from rotating the $2$-sphere fiber as one travels around the base circle. The technique employed also shows that homotopic diffeomorphisms are diffeotopic.
Rotation and winding numbers for planar polygons and curves
Branko
Grünbaum;
G. C.
Shephard
169-187
Abstract: The winding and rotation numbers for closed plane polygons and curves appear in various contexts. Here alternative definitions are presented, and relations between these characteristics and several other integer-valued functions are investigated. In particular, a point-dependent "tangent number" is defined, and it is shown that the sum of the winding and tangent numbers is independent of the point with respect to which they are taken, and equals the rotation number.
Basic dual homotopy invariants of Riemannian foliations
Peter Y. H.
Pang
189-199
Abstract: In this paper, we use the Sullivan minimal model construction to produce invariants for Riemannian foliations. Existence and vanishing results are proved for these invariants.
Unusually large gaps between consecutive primes
Helmut
Maier;
Carl
Pomerance
201-237
Abstract: Let $G(x)$ denote the largest gap between consecutive primes below $x$. In a series of papers from 1935 to 1963, Erdàs, Rankin, and Schànhage showed that $\displaystyle G(x) \geq (c + o(1)){\operatorname{log}}x{\operatorname{loglog}}x{\operatorname{loglogloglog}}x{({\operatorname{logloglog}}x)^{ - 2}}$ , where $c = {e^\gamma }$ and $\gamma$ is Euler's constant. Here, this result is shown with $c = {c_0}{e^\gamma }$ where ${c_0} = 1.31256 \ldots$ is the solution of the equation $ 4/{c_0} - {e^{ - 4/{c_0}}} = 3$. The principal new tool used is a result of independent interest, namely, a mean value theorem for generalized twin primes lying in a residue class with a large modulus.
$L\sp 1$-approximation with constraints
Allan
Pinkus;
Hans
Strauss
239-261
Abstract: In this paper we study problems of best ${L^1}$-approximation to continuous functions from finite-dimensional subspaces under a variety of constraints. Included are problems of bounded coefficient approximation, approximation with interpolation, restricted range approximation, and restricted range and derivative approximation. Emphasis is placed on problems of uniqueness.
Boundary behavior of the fast diffusion equation
Y. C.
Kwong
263-283
Abstract: The fast diffusion equation $\Delta {\upsilon ^m} = {\upsilon _t}$, $0 < m < 1$, is a degenerate nonlinear parabolic equation of which the existence of a unique continuous weak solution has been established. In this paper we are going to obtain a Lipschitz growth rate of the solution at the boundary of $ \Omega$ and estimate that in terms of the various data.
A class of algebras similar to the enveloping algebra of ${\rm sl}(2)$
S. P.
Smith
285-314
Abstract: Fix $f \in {\mathbf{C}}[X]$. Define $R = {\mathbf{C}}[A,B,H]$ subject to the relations $\displaystyle HA - AH = A,\quad HB - BH = - B,\quad AB - BA = f(H)$ . We study these algebras (for different $ f$) and in particular show how they are similar to (and different from) $U({\text{sl}}(2))$, the enveloping algebra of $ {\text{sl}}(2,{\mathbf{C}})$. There is a notion of highest weight modules and a category $ \mathcal{O}$ for such $ R$. For each $ n > 0$, if $ f(x) = {(x + 1)^{n + 1}} - {x^{n + 1}}$, then $R$ has precisely $n$ simple modules in each finite dimension, and every finite-dimensional $R$-module is semisimple.
Interpretations of Euclidean geometry
S.
Świerczkowski
315-328
Abstract: Following Tarski, we view $n$-dimensional Euclidean geometry as a first-order theory ${E_n}$ with an infinite set of axioms about the relations of betweenness (among points on a line) and equidistance (among pairs of points). We show that for $k < n$, ${E_n}$ does not admit a $k$-dimensional interpretation in the theory RCF of real closed fields, and we deduce that $ {E_n}$ cannot be interpreted $r$-dimensionally in ${E_s}$, when $r \cdot s < n$.
Tiled orders of finite global dimension
Hisaaki
Fujita
329-341
Abstract: We define a projective link between maximal ideals, with respect to which an idealizer preserves being of finite global dimension. Let $D$ be a local Dedekind domain with the quotient ring $K$. We show that for $2 \leq n \leq 5$, every tiled $D$-order of finite global dimension in ${(K)_n}$ is obtained by iterating idealizers w.r.t. projective links from a hereditary order. For $n \geq 6$, we give a tiled $ D$-order in ${(K)_n}$ without this property, which is also a counterexample to Tarsy's conjecture, saying that the maximum finite global dimension of such an order is $n - 1$.
A character-theoretic approach to embeddings of rooted maps in an orientable surface of given genus
D. M.
Jackson;
T. I.
Visentin
343-363
Abstract: The group algebra of the symmetric group and properties of the irreducible characters are used to derive combinatorial properties of embeddings of rooted maps in orientable surfaces of arbitrary genus. In particular, we show that there exists, for each genus, a correspondence between the set of rooted quadrangulations and a set of rooted maps of all lower genera with a distinguished subset of vertices.
Character theory and rooted maps in an orientable surface of given genus: face-colored maps
D. M.
Jackson;
T. I.
Visentin
365-376
Abstract: The character theoretic approach [5] to the enumeration of rooted maps in an orientable surface of arbitrary genus is extended to $2$-face-colorable rooted maps. In particular, we show that there exists, for each genus, a correspondence between the set of $2$-colored triangulations and a set of $2$-colored rooted maps of all lower genera with a distinguished subset of vertices.
On a multiplier conjecture for univalent functions
V.
Gruenberg;
F.
Rønning;
St.
Ruscheweyh
377-393
Abstract: Let $\mathcal{S}$ be the set of normalized univalent functions, and let $ \mathcal{D}$ be the subset of $\mathcal{S}$ containing functions with the property:
Time-delay operators in semiclassical limit. II. Short-range potentials
Xue Ping
Wang
395-416
Abstract: This work is a continuation of [27]. We prove that quantum time-delay operator localized in a nontrapping energy interval is in fact an $h$-pseudodifferential operator with the $ h$-principal symbol given by the classical time-delay function composed with the incoming wave operator in classical mechanics. The classical limit of time-delay operator is also given.