Isoperimetric inequalities for the least harmonic majorant of $\vert x\vert \sp p$
Makoto
Sakai
431-472
Abstract: Let $D$ be an open set in the $d$-dimensional Euclidean space ${{\mathbf{R}}^d}$ containing the origin 0 and let $ {h^{(p)}}(x,D)$ be the least harmonic majorant of $\vert x{\vert^p}$ in $D$, where $ 0 < p < \infty$ if $d \geqslant 2$ and $1 \leqslant p < \infty $ if $d = 1$. We shall be concerned with the following isoperimetric inequalities $ {h^{(p)}}{(0,D)^{1/p}} \leqslant cr(D)$, where $r(D)$ denotes the volume radius of $D$, namely, a ball with radius $ r(D)$ has the same volume as $D$ has and $c$ is a constant dependent on $d$ and $p$ but independent of $D$. We fix $d$ and denote by $c(p)$ the infimum of such constants $ c$. As a function of $ p$, $c(p)$ is nondecreasing and satisfies $c(p) \geqslant 1$. We shall show (1) there are positive constants ${C_1}$ and ${C_2}$ such that $ {C_1}{p^{(d - 1)/d}} \leqslant c(p) \leqslant {C_2}{p^{(d - 1)/d}}$ for $p \geqslant 1$, (2) $c(p) = 1$ if $p \leqslant d + {2^{1 - d}}$. Many estimations of $ {h^{(p)}}(0,D)$ and their applications are also given.
Some results on locally finitely presentable categories
M.
Makkai;
A. M.
Pitts
473-496
Abstract: We prove that any full subcategory of a locally finitely presentable (l.f.p.) category having small limits and filtered colimits preserved by the inclusion functor is itself l.f.p. Here "full" may be weakened to "full with respect to isomorphisms." Further, we characterize those left exact functors $I:{\mathbf{C}} \to {\mathbf{D}}$ between small categories with finite limits for which the functor $ {I^{\ast}}:{\mathbf{LEX}}({\mathbf{D}},{\text{Set)}} \to {\mathbf{LEX}}{\text{(}}{\mathbf{C}}{\text{,Set)}}$ induced by composition is full and faithful. As an application, we prove a theorem on sheaf representations, a consequence of which is that, for any site $\mathcal{C} = ({\mathbf{C}},J)$ on a category $ {\mathbf{C}}$ with finite limits, defined by a subcanonical Grothendieck topology $J$, the closure in $ {\mathbf{LEX}}({\mathbf{C}},{\text{Set)}}$ under small limits and filtered colimits of the models of $ \mathcal{C}$ is the whole of $ {\mathbf{LEX}}({\mathbf{C}},{\text{Set)}}$.
On the theory of fundamental norming bounded biorthogonal systems in Banach spaces
Paolo
Terenzi
497-511
Abstract: Let $X$ and $Y$ be quasi complementary subspaces of a separable Banach space $B$ and let $({z_n})$ be a sequence complete in $X$. Then (a) there exists a uniformly minimal norming $M$-basis $({x_n})$ of $X$ with ${x_m} \in \operatorname{span} {({z_n})_{n \geqslant {q_m}}}$ for every $m$, $ {q_m} \to \infty$; (b) if $({x_n})$ is a uniformly minimal norming $ M$-basis of $X$, there exists a uniformly minimal norming $M$-basis of $B$ which is an extension of $({x_n})$; (c) there exists a uniformly minimal norming $M$-basis $ ({x_n}) \cup ({y_n})$ of $ B$ with $({x_n}) \subset X$ and $({y_n}) \subset Y$.
A finiteness condition on regular local overrings of a local domain
Bernard
Johnston
513-524
Abstract: The local factorization theorem of Zariski and Abhyankar implies that between a given pair of $2$-dimensional regular local rings, $S \supseteq R$, having the same quotient field, every chain of regular local rings must be finite. It is shown in this paper that this property extends to every such pair of regular local rings, regardless of dimension. An example is given to show that this does not hold if "regular" is replaced by "Cohen-Macaulay," by "normal," or by "rational singularity." More generally, it is shown that the set $\mathcal{R}(R)$ of $n$dimensional regular local rings birationally dominating a given $n$-dimensional local domain, $R$, and ordered by containment, satisfies the descending chain condition. An example is given to show that if $R$ is regular the two examples of minimal elements of $ \mathcal{R}(R)$ given by J. Sally do not exhaust the set of minimal elements of $ \mathcal{R}(R)$.
Porous sets and quasisymmetric maps
Jussi
Väisälä
525-533
Abstract: A set $ A$ in ${R^n}$ is called porous if there is $\alpha > 0$ such that every ball $\overline B (x,r)$ contains a point whose distance from $A$ is at least $\alpha r$. We show that porosity is preserved by quasisymmetric maps, in particular, by bilipschitz maps. Local versions are also given.
The variation of the de Rham zeta function
Steven
Rosenberg
535-557
Abstract: Special values of the zeta function $\zeta (s)$ for the Laplacian on forms $ \Delta$ on a compact Riemannian manifold are known to have geometric significance. We compute the variation of these special values with respect to the variation of the metric and write down the Euler-Lagrange equation for conformal variations. The invariant metric on a locally symmetric space is shown to be critical for every local Lagrangian. We also compute the variation of
Parallel translation of curvature along geodesics
James J.
Hebda
559-572
Abstract: According to the Cartan-Ambrose-Hicks Theorem, two simply-connected, complete Riemannian manifolds are isometric if, given a certain correspondence between all the broken geodesics emanating from a point in one manifold, and all those emanating from a point in the other, the parallel translates of the curvature tensor agree along corresponding broken geodesics. For generic metrics on a surface, the hypothesis can be refined so that it is enough to compare curvature along corresponding unbroken geodesics in order to obtain the isometry.
Norms of Hankel operators and uniform algebras
Takahiko
Nakazi
573-580
Abstract: Two generalizations of the classical Hankel operators are defined on an abstract Hardy space that is associated with a uniform algebra. In this paper the norms of Hankel operators are studied. This has applications to weighted norm inequalities for conjugation operators, and invertible Topelitz operators. The results in this paper have applications to concrete uniform algebras, for example, a polydisc algebra and a uniform algebra which consists of rational functions.
Application of a theorem of M. G. Kre\u\i n to singular integrals
Rainer
Wittmann
581-599
Abstract: We give Hölder and ${L^2}$ estimates for singular integrals on homogeneous spaces in the sense of Coifman and Weiss. The fundamental tool which allows us to pass from Hölder to $ {L^2}$ estimates, is a theorem of M. G. Krein.
Polynomial invariants of graphs
Seiya
Negami
601-622
Abstract: We define two polynomials $f(G)$ and $ {f^{\ast}}(G)$ for a graph $ G$ by a recursive formula with respect to deformation of graphs. Analyzing their various properties, we shall discuss when two graphs have the same polynomials.
Endomorphisms of right ideals of the Weyl algebra
J. T.
Stafford
623-639
Abstract: Let $A = A(k)$ be the first Weyl algebra over an infinite field $k$, let $P$ be any noncyclic, projective right ideal of $ A$ and set $S = \operatorname{End} (P)$. We prove that, as $ k$-algebras, $S\not \cong A$. In contrast, there exists a noncyclic, projective right ideal $Q$ of $S$ such that $S \cong \operatorname{End} (Q)$. Thus, despite the fact that they are Morita equivalent, $S$ and $A$ have surprisingly different properties. For example, under the canonical maps, $ {\operatorname{Aut} _k}(A) \cong {\operatorname{Pic} _k}(A) \cong {\operatorname{Pic} _k}(S)$. In contrast, ${\operatorname{Aut} _k}(S)$ has infinite index in ${\operatorname{Pic} _k}(S)$.
A closed separable subspace of $\beta{\bf N}$ which is not a retract
Petr
Simon
641-655
Abstract: We shall exhibit a countable subset, $X$, of $ {{\mathbf{N}}^{\ast}}$ whose closure is not a retract of $\beta {\mathbf{N}}$. The points of $X$ are constructed in $ c$ steps with the aid of an independent matrix of subsets of $\omega$.
Seifert matrices and boundary link cobordisms
Ki Hyoung
Ko
657-681
Abstract: To an $ m$-component boundary link of odd dimension, a matrix is associated by taking the Seifert pairing on a Seifert surface of the link. An algebraic description of the set of boundary link cobordism classes of boundary links is obtained by using this matrix invariant.
A classification of simple Lie modules having a $1$-dimensional weight space
D. J.
Britten;
F. W.
Lemire
683-697
Abstract: Let $L$ denote a simple Lie algebra over the complex numbers. In this paper, we classify and construct all simple $L$ modules which may be infinite dimensional but have at least one $1$-dimensional weight space. This completes the study begun earlier by the authors for the case of $ L = {A_n}$. The approach presented here relies heavily on the results of Suren Fernando whose dissertation dealt with simple weight modules and their weight systems.
Some moduli spaces for rank $2$ stable reflexive sheaves on ${\bf P}\sp 3$
Rosa M.
Miró-Roig
699-717
Abstract: In [Ma], Maruyama proved that the set $M({c_1},{c_2},{c_3})$ of isomorphism classes of rank $ 2$ stable reflexive sheaves on $ {{\mathbf{P}}^3}$ with Chern classes $ ({c_1},{c_2},{c_3})$ has a natural structure as an algebraic scheme. Until now, there are no general results about these schemes concerning dimension, irreducibility, rationality, etc. and only in a few cases a precise description of them is known. This paper is devoted to the following cases: (i) $M( - 1,{c_2},c_2^2 - 2r{c_2} + 2r(r + 1))$ with ${c_2} \geqslant 4$, $ 1 \leqslant r \leqslant ( - 1 + \sqrt {4{c_2} - 7} )/2$; and (ii) $ M( - 1,{c_2},c_2^2 - 2(r - 1){c_2})$ with $ {c_2} \geqslant 8$, $2 \leqslant r \leqslant ( - 1 + \sqrt {4{c_2} - 7} )/2$.
The adelic zeta function associated with the space of binary cubic forms with coefficients in a function field
Boris A.
Datskovsky
719-745
Abstract: In this paper we study the adelic zeta function associated with the prehomogeneous vector space of binary cubic forms, defined over a function field. We establish its rationality, find its poles and residues and a simple functional equation that this zeta function satisfies.
Representations of crossed products by coactions and principal bundles
M. B.
Landstad;
J.
Phillips;
I.
Raeburn;
C. E.
Sutherland
747-784
Abstract: The main purpose of this paper is to establish a covariant representation theory for coactions of locally compact groups on ${C^{\ast}}$-algebras (including a notion of exterior equivalence), to show how these results extend the usual notions for actions of groups on ${C^{\ast}}$-algebras, and to apply these ideas to classes of coactions termed pointwise unitary and locally unitary to obtain a complete realization of the isomorphism theory of locally trivial principal $ G$-bundles in this context. We are also able to obtain all (Cartan) principal $ G$-bundles in this context, but their isomorphism theory remains elusive.
Counting cycles in permutations by group characters, with an application to a topological problem
D. M.
Jackson
785-801
Abstract: The character theory of the symmetric group is used to derive properties of the number of permutations, with $k$ cycles, which are expressible as the product of a full cycle with an element of an arbitrary, but fixed, conjugacy class. For the conjugacy class of fixed point free involutions, this problem has application to the analysis of singularities in surfaces.
On the canonical element conjecture
Sankar P.
Dutta
803-811
Abstract: The canonical element conjecture is proved in the following two cases: (i) depth $A = \dim A - 1$, $H_m^{n - 1}(A)$ is decomposable; (ii) depth $A = \dim A - 1$, $H_m^{n - 1}{(A)^ \vee }$ is cyclic. The equivalence of the C.E.C. and the improved new intersection theorem is also established.