Some transplantation theorems for the generalized Mehler transform and related asymptotic expansions
Susan
Schindler
257-291
Abstract: Let $P_{ - 1/2 + ix}^m(z)$ be the associated Legendre function of order $m$ and degree $- 1/2 + ix$. We give, here, two integral transforms ${G^m}$ and ${H^m}$, arising naturally from the generalized Mehler transform, which is induced by $P_{ - 1/2 + ix}^m(\cosh y)$, such thatb ${H^m}{G^m}$ = Identity (formally). We show that if $1 < p < \infty , - 1/p < \alpha < 1 - 1/p,m \leqq 1/2$ or $ m = 1,2, \ldots ,$ then $ \vert\vert{G^m}f\vert{\vert _{p,\alpha }} \leqq A_{p,\alpha }^m\vert\vert\hat f\vert{\vert _{p,\alpha }}$ and $ \vert\vert{H^m}f\vert{\vert _{p,\alpha }} \leqq A_{p,\alpha }^m\vert\vert\hat f\vert{\vert _{p,\alpha }}$, where $^ \wedge$ denotes the Fourier cosine transform. We also prove that ${G^m}f,{H^m}f$ exist as limits in ${L^{p,\alpha }}$ of partial integrals, and we prove inequalities equivalent to the above pair: $\vert\vert{G^m}\hat f\vert{\vert _{p,\alpha }} \leqq A_{p,\alpha }^m\vert\vert f\vert{\vert _{p,\alpha }}$ and $ \vert\vert{H^m}\hat f\vert{\vert _{p,\alpha }} \leqq A_{p,\alpha }^m\vert\vert f\vert{\vert _{p,\alpha }}$. These we dualize to $ \vert\vert{({H^m}f)^ \wedge }\vert{\vert _{p,\alpha }} \leqq A_{p,\alpha }^m\vert\vert f\vert{\vert _{p,\alpha }}$, and $\vert\vert{({G^m}f)^ \wedge }\vert{\vert _{p,\alpha }} \leqq A_{p,\alpha }^m\vert\vert f\vert{\vert _{p,\alpha }}$. ${G^m}$ and ${H^m}$ are given by ${G^m}(f;y) = \int_0^\infty {f(x){K^m}(x,y)dx}$ and ${H^m}(f;x) = \int_0^\infty {f(y){K^m}(x,y)dy\;} (0 \leqq y < \infty )$, where $\displaystyle {K^m}(x,y) = \vert\Gamma (1/2 - m - ix)/\Gamma ( - ix)/{(\sinh y)^{1/2}}P_{ - 1/2 + ix}^m(\cosh y).$ The principal method of proving the inequalities involves getting asymptotic expansions for ${K^m}(x,y)$; these are in terms of sines and cosines for large $y$, and in terms of Bessel functions for $ y$ small. Then we can use Fourier and Hankel multiplier theorems. The main consequences of our results are the typical ones for transplantation theorems: mean convergence and multiplier theorems. They can easily be restated in terms of the more usual Mehler transform pair $\displaystyle g(y) = \int_0^\infty {f(x){P_{ - 1/2 + ix}}(y)dx}$ and $ f(x) = {\pi ^{ - 1}}x\sinh \pi x \cdot \Gamma (1/2 - m + ix)\Gamma (1/2 - m - ix)\int_0^\infty {g(y){P_{ - 1/2 + ix}}(y)dy.}$
Bifunctors and adjoint pairs
J. Fisher
Palmquist;
David C.
Newell
293-303
Abstract: We use a definition of tensor products of functors to generalize some theorems of homological algebra. We show that adjoint pairs of functors between additive functor categories correspond to bifunctors and that composition of such adjoint pairs corresponds to the tensor product of the bifunctors. We also generalize some homological characterizations of finitely generated projective modules to characterizations of small projectives in a functor category. We apply our results to adjoint pairs arising from satellites and from a functor on the domain categories.
Some examples in topology
S. P.
Franklin;
M.
Rajagopalan
305-314
Abstract: §1 is concerned with variations on the theme of an ordinal compactification of the integers. Several applications are found, yielding, for instance, an example previously known only modulo the continuum hypothesis, and a counter-example to a published assertion. §2 is concerned with zero-one sequences and §3 with spaces built from sequential fans. Of two old problems of Čech, one is solved and one partly solved. Since the sections are more or less independent, each will have its own introduction. Sequential spaces form the connecting thread, although not all the examples are concerned with them.
Contractions on $L\sb{1}$-spaces
M. A.
Akcoglu;
A.
Brunel
315-325
Abstract: It is shown that a linear contraction on a complex ${L_1}$-space can be represented in terms of its linear modulus. This result is then used to give a direct proof of Chacon's general ratio ergodic theorem.
On embeddings with locally nice cross-sections
J. L.
Bryant
327-332
Abstract: A $k$-dimensional compactum ${X^k}$ in euclidean space ${E^n}(n - k \geqq 3)$ is said to be locally nice in $ {E^n}$ if ${E^n} - {X^k}$ is $1$-ULC. In this paper we prove a general theorem which implies, in particular, that ${X^k}$ is locally nice in $ {E^n}$ if the intersection of ${X^k}$ with each horizontal hyperplane of $ {E^n}$ is locally nice in the hyperplane. From known results we obtain immediately that a $k$-dimensional polyhedron $P$ in ${E^n}$ ( $n - k \geqq 3$ and $n \geqq 5$) is tame in ${E^n}$ if each $({E^{n - 1}} \times \{ w\} ) - P(w \in {E^1})$ is $1$-ULC. However, by strengthening our general theorem in the case $n = 4$, we are able to prove this result for $ n = 4$ as well. For example, an arc $A$ in ${E^4}$ is tame if each horizontal cross-section of $ A$ is tame in the cross-sectional hyperplane (that is, lies in an arc that is tame in the hyperplane).
Certain dense embeddings of regular semigroups
Mario
Petrich
333-343
Abstract: In a previous paper, the author has introduced a number of homomorphisms of an arbitrary semigroup into the translational hull of certain Rees matrix semigroups or orthogonal sums thereof. For regular semigroups, it is proved here that all of these homomorphisms have the property that the image is a densely embedded subsemigroup, i.e., is a densely embedded ideal of its idealizer, and that the corresponding Rees matrix semigroups are regular. Several of these homomorphisms are 1-1, in each case they furnish a different dense embedding of an arbitrary regular semigroup into the translational hull of a regular Rees matrix semigroup or orthogonal sums thereof. A new representation for regular semigroups is introduced.
Maximal orders over regular local rings
Mark
Ramras
345-352
Abstract: In this paper various sufficient conditions are given for the maximality of an $R$-order in a finite-dimensional central simple $K$-algebra, where $R$ is a regular local ring whose quotient field is $ K$. Stronger results are obtained when we assume the dimension of $R$ to be three. This work depends upon earlier results of this author [5] for regular local rings of dimension two, and the fundamental work of Auslander and Goldman [1] for dimension one.
Partial orders on the types in $\beta N$
Mary Ellen
Rudin
353-362
Abstract: Three partial orders on the types of points in $\beta N$ are defined and studied in this paper. Their relation to the types of points in $\beta N - N$ is also described.
Regularity conditions in nonnoetherian rings
T.
Kabele
363-374
Abstract: We show that properties of $R$-sequences and the Koszul complex which hold for noetherian local rings do not hold for nonnoetherian local rings. For example, we construct a local ring with finitely generated maximal ideal such that ${\text{hd} _R}M < \infty $ but $M$ is not generated by an $ R$-sequence. In fact, every element of $M - {M^2}$ is a zero divisor. Generalizing a result of Dieudonné, we show that even in local (nonnoetherian) integral domains a permutation of an $R$-sequence is not necessarily an $ R$-sequence.
Partitions with a restriction on the multiplicity of the summands
Peter
Hagis
375-384
Abstract: Using the circle dissection method, a convergent series and several asymptotic formulae are obtained for $p(n,t)$, the number of partitions of the positive integer $n$ in which no part may be repeated more than $ t$ times.
On a generalization of alternative and Lie rings
Erwin
Kleinfeld
385-395
Abstract: Alternative as well as Lie rings satisfy all of the following four identities: (i) $({x^2},y,z) = x(x,y,z) + (x,y,z)x$, (ii) $ (x,{y^2},z) = y(x,y,z) + (x,y,z)y$, (iii) $(x,y,{z^2}) = z(x,y,z) + (x,y,z)z$, (iv) $(x,x,x) = 0$, where the associator $(a,b,c)$ is defined by $(a,b,c) = (ab)c - a(bc)$. If $R$ is a ring of characteristic different from two and satisfies (iv) and any two of the first three identities, then it is shown that a necessary and sufficient condition for $R$ to be alternative is that whenever $a,b,c$ are contained in a subring $ S$ of $R$ which can be generated by two elements and whenever $ {(a,b,c)^2} = 0$, then $(a,b,c) = 0$.
A construction of Lie algebras from a class of ternary algebras
John R.
Faulkner
397-408
Abstract: A class of algebras with a ternary composition and alternating bilinear form is defined. The construction of a Lie algebra from a member of this class is given, and the Lie algebra is shown to be simple if the form is nondegenerate. A characterization of the Lie algebras so constructed in terms of their structure as modules for the three-dimensional simple Lie algebra is obtained in the case the base ring contains 1/2. Finally, some of the Lie algebras are identified; in particular, Lie algebras of type ${E_8}$ are obtained.
Nonlinear evolution equations and product stable operators on Banach spaces
G. F.
Webb
409-426
Abstract: The method of product integration is used to obtain solutions to the time dependent Banach space differential equation $ [0,\infty )$ to the set of nonlinear operators from the Banach space $ X$ to itself and $ u$ is a function from $[0,\infty )$ to $X$. The main requirements placed on $A$ are that $A$ is $m$-dissipative and product stable on its domain. Applications are given to a linear partial differential equation, to nonlinear dissipative operators in Hilbert space, and to continuous, $ m$-dissipative, everywhere defined operators in Banach spaces.
On conformal maps of infinitely connected Dirichlet regions
V. C.
Williams
427-453
Abstract: Let $D$ be a plane region of arbitrary connectivity $( > 1)$ for which the Dirichlet problem is solvable. There exists a conformal map of $D$ onto a region bounded by two level loci of $H$, a nontrivial harmonic measure. $H$ is essentially the difference of two logarithmic potentials. The two measures involved are mutually singular probability measures. Further properties of these measures, and of $ H$, are derived. The special case in which $D$ is of connectivity 2 is the classical theorem which states that an annular region is conformally equivalent to a region bounded by two circles. The case in which $D$ is of finite connectivity was treated by J. L. Walsh in 1956. A similar generalization of the Riemann mapping theorem is also established. Finally, converses of the above results are also valid.
Regular representations of Dirichlet spaces
Masatoshi
Fukushima
455-473
Abstract: We construct a regular and a strongly regular Dirichlet space which are equivalent to a given Dirichlet space in the sense that their associated function algebras are isomorphic and isometric. There is an appropriate strong Markov process called a Ray process on the underlying space of each strongly regular Dirichlet space.
Common partial transversals and integral matrices
R. A.
Brualdi
475-492
Abstract: Certain packing and covering problems associated with the common partial transversals of two families $\mathfrak{A}$ and $ \mathfrak{B}$ of subsets of a set $E$ are investigated. Under suitable finitary restrictions, necessary and sufficient conditions are obtained for there to exist pairwise disjoint sets ${F_1}, \ldots ,{F_t}$ where each $ {F_i}$ is a partial transversal of $ \mathfrak{A}$ with defect at most $p$ and a partial transversal of $\mathfrak{B}$ with defect at most $q$. We also prove that (i) $E = \cup _{i = 1}^t{T_i}$ where each $ {T_i}$ is a common partial transversal of $ \mathfrak{A}$ and $\mathfrak{B}$ if and only if (ii)
Systems of division problems for distributions
B.
Roth
493-504
Abstract: Suppose $ {({f_{ij}})_{1 \leqq i,j \leqq p}}$ is a $p \times p$ matrix of real-valued infinitely (respectively $m$-times continuously) differentiable functions on an open subset $\Omega$ of ${R^n}$. Then ${({f_{ij}})_{1 \leqq i,j \leqq p}}$ maps the space of $p$-tuples of distributions on $ \Omega$ (respectively distributions of order $\leqq m$ on $\Omega$) into itself. In the present paper, the $p \times p$ matrices ${({f_{ij}})_{1 \leqq i,j \leqq p}}$ for which this mapping is onto are characterized in terms of the zeros of the determinant of ${({f_{ij}})_{1 \leqq i,j \leqq p}}$ when the $ {f_{ij}}$ are infinitely differentiable on $ \Omega \subset {R^1}$ and when the ${f_{ij}}$ are $m$-times continuously differentiable on $\Omega \subset {R^n}$. Finally, partial results are obtained when the ${f_{ij}}$ are infinitely differentiable on $\Omega \subset {R^n}$ and extensions are made to $p \times q$ systems of division problems for distributions.
Zero divisors in Noetherian-like rings
E. Graham
Evans
505-512
Abstract: The zero divisors of $R/I$ for every ideal $I$ of a Noetherian ring is a finite union of primes. We take this property as a definition and study the class of rings so defined. Such rings are stable under localization and quotients. They are not stable under integral closure and are highly unstable under polynomial adjunction. The length of maximal $R$ sequences is well defined on them. In this paper all rings are commutative with unit and all modules are unitary.