Harish-Chandra modules with the unique embedding property
David H.
Collingwood
1-48
Abstract: Let $G$ be a connected semisimple real matrix group. In view of Casselman's subrepresentation theorem, every irreducible admissible representation of $ G$ may be realized as a submodule of some principal series representation. We give a classification of representations with a unique embedding into principal series, in the case of regular infinitesimal character. Our basic philosophy is to link the theory of asymptotic behavior of matrix coefficients with the theory of coherent continuation of characters. This is accomplished by using the "Jacquet functor" and the Kazhdan-Lusztig conjectures.
Zero distribution for pairs of holomorphic functions with applications to eigenvalue distribution
A. A.
Shkalikov
49-63
Abstract: Let $f$ and $g$ be holomorphic in an angle $\Lambda$. Theorem 1 shows that the zero-distributions of $f$ and $g$ are comparable if, near $\partial \Lambda$, $f$ and $g$ grow similarly. This result is applied to analyse the asymptotic behavior of eigenvalues of certain perturbed normal operators.
Sets of essentially unitary operators
Ridgley
Lange
65-75
Abstract: Let ${U_e}$ be the set of essentially unitary operators on a separable Hilbert space $H$; for $1 \leqslant p \leqslant \infty$, let $ {U_p}$ be the set of operators $T$ such that $I - T^{\ast}T$ lies in the Schatten $p$-ideal and the spectrum of $T$ does not fill the unit disc; and let $U_e^n$ be the set of operators in $ {U_e}$ of Fredholm index $ n$. The author proves that each $U_e^n$ is closed and path connected, that $ {U_p}$ is dense in $ {U_e}^0$ and $ {U_p}$ is path connected for each $p$, and that all these sets are invariant under Cayley transform. It is proved that the spectrum is continuous on ${U_\infty }$ but not on ${U_e}$, while the spectral radius is continuous on ${U_e}$. Sufficient conditions that an operator in ${U_e}$ have a nontrivial hyperinvariant subspace are given, and it is proved that the general hyperinvariant subspace problem can be reduced to that problem for perturbations of the bilateral shift. The product of commuting operators in ${U_p}$ is ${U_p}$, but this result is false in general. Quasisimilarity in ${U_e}$ is also studied; quasisimilar operators in $ {U_e}\backslash {U_\infty }$ are unitarily equivalent modulo the ideal of compacts, and this result also holds in ${U_\infty }$ if the spectrum is also preserved.
Weighted norm inequalities for strongly singular convolution operators
Sagun
Chanillo
77-107
Abstract: We derive sharp function estimates for convolution operators whose kernels are more singular than Calderon-Zygmund kernels. This leads to weighted norm inequalities. Weighted weak $(1,1)$ results are also proved. All the results obtained are in the context of ${A_p}$ weights.
An algebraic decomposition of the recursively enumerable degrees and the coincidence of several degree classes with the promptly simple degrees
Klaus
Ambos-Spies;
Carl G.
Jockusch;
Richard A.
Shore;
Robert I.
Soare
109-128
Abstract: We specify a definable decomposition of the upper semilattice of recursively enumerable (r.e.) degrees $\mathbf{R}$ as the disjoint union of an ideal $\mathbf{M}$ and a strong filter $\mathbf{NC}$. The ideal $ \mathbf{M}$ consists of $\mathbf{0}$ together with all degrees which are parts of r.e. minimal pairs, and thus the degrees in $\mathbf{NC}$ are called noncappable degrees. Furthermore, $ \mathbf{NC}$ coincides with five other apparently unrelated subclasses of $\mathbf{R: ENC}$, the effectively noncappable degrees; $\mathbf{PS}$, the degrees of promptly simple sets; $ \mathbf{LC}$, the r.e. degrees cuppable to $ {\mathbf{0}}^{\prime}$ by a low r.e. degree; $ {\mathbf{SP\bar H}}$, the degrees of non-$hh$-simple r.e. sets with the splitting property; and $\mathbf{G}$, the degrees in the orbit of an r.e. generic set under automorphisms of the lattice of r.e. sets.
Further results on convergence acceleration for continued fractions $K(a\sb{n}/1)$
Lisa
Jacobsen
129-146
Abstract: If $K(a_n^{\prime}/1)$ is a convergent continued fraction with known tails, it can be used to construct modified approximants $ f_n^{\ast}$ for other continued fractions $ K({a_n}/1)$ with unknown values. These modified approximants may converge faster to the value $f$ of $ K({a_n}/1)$ than the ordinary approximants ${f_n}$ do. In particular, if ${a_n} - a_n^{\prime} \to 0$ fast enough, we obtain $\vert f - f_n^{\ast}\vert/\vert f - {f_n}\vert \to 0$; i.e. convergence acceleration. the present paper also gives bounds for this ratio of the two truncation errors, in many cases.
Semidirect products and reduction in mechanics
Jerrold E.
Marsden;
Tudor
Raţiu;
Alan
Weinstein
147-177
Abstract: This paper shows how to reduce a Hamiltonian system on the cotangent bundle of a Lie group to a Hamiltonian system in the dual of the Lie algebra of a semidirect product. The procedure simplifies, unifies, and extends work of Greene, Guillemin, Holm, Holmes, Kupershmidt, Marsden, Morrison, Ratiu, Sternberg and others. The heavy top, compressible fluids, magnetohydrodynamics, elasticity, the Maxwell-Vlasov equations and multifluid plasmas are presented as examples. Starting with Lagrangian variables, our method explains in a direct way why semidirect products occur so frequently in examples. It also provides a framework for the systematic introduction of Clebsch, or canonical, variables.
A linear homogenization problem with time dependent coefficient
Maria Luisa
Mascarenhas
179-195
Abstract: We consider: the homogenization problem $\displaystyle \left\{ {\begin{array}{*{20}{c}} {(\partial u\varepsilon /\partia... ...0,} {{u_\varepsilon }(x,0) = \phi (x), } & {} \end{array} } \right.$ where $ \beta$ is a strictly positive bounded real function, periodic of period $ 1$, and $ {\beta_\varepsilon }(x) = \beta (x/\varepsilon )$; the equivalent integral equation $\displaystyle {u_\varepsilon }(x,t) + \int_0^t {{\beta_\varepsilon }(x)\,{u_\varepsilon }(x,s)\;ds = \phi (x)};$ and the homogenized equation $\displaystyle {u_0}(x,t) + \int_0^t {K(t - s)\,{u_0}(s)\,ds = \phi (x)},$ where $ K$ is a unique, well-defined function depending on $\beta$. We study this problem for a time dependent $ \beta$, and characterize a two-variable function $K(s,t)$ satisfying $\displaystyle {u_0}(x,t) + \int_0^t {K(s,t - s)\,{u_0}(x,s)\;ds = \phi (x)}$ and study its uniqueness.
Homologically homogeneous rings
K. A.
Brown;
C. R.
Hajarnavis
197-208
Abstract: In this paper we study the structure of a right Noetherian ring $ R$ of finite right global dimesion integral over a central subring $ C$ and satisfying the following condition: if $V,W$ are irreducible right $R$-modules with ${r_C}(V) = {r_C}(W)$ then $\operatorname{pr}\, \dim (V) = \operatorname{pr}\, \dim (W)$.
Additivity of measure implies additivity of category
Tomek
Bartoszyński
209-213
Abstract: In this paper it is proved that $ {2^\omega }$-additivity of category follows from $ {2^\omega }$-additivity of measure, and a combinatorial characterization of additivity of measure is found.
Nonlinear stability of asymptotic suction
Milan
Miklavčič
215-231
Abstract: The semigroup approach to the Navier-Stokes equation in halfspace is used to prove that the stability of the asymptotic suction velocity profile is determined by the eigenvalues of the classical Orr-Sommerfeld equation. The usual obstacle, namely, that the corresponding linear operator contains 0 in the spectrum is removed with the use of weighted spaces.
$M$-structure in the Banach algebra of operators on $C\sb{0}(\Omega )$
P. H.
Flinn;
R. R.
Smith
233-242
Abstract: The $M$-ideals in $B({C_0}(\Omega ))$, the space of continuous linear operators on $ {C_0}(\Omega )$, are determined where $\Omega$ is a locally compact Hausdorff countably paracompact space. A one-to-one correspondence between $M$-ideals in $ B({C_0}(\Omega ))$, open subsets of the Stone-Čech compactification of $\Omega$, and lower semicontinuous Hermitian projections in $B{({C_0}(\Omega ))^{\ast\ast}}$ is established.
James maps, Segal maps, and the Kahn-Priddy theorem
J.
Caruso;
F. R.
Cohen;
J. P.
May;
L. R.
Taylor
243-283
Abstract: The standard combinatorial approximation $ C({R^n},X)$ to ${\Omega ^n}{\Sigma ^n}X$ is a filtered space with easily understood filtration quotients ${D_q}({R^n},X)$. Stably, $ C({R^n},X)$ splits as the wedge of the $ {D_q}({R^n},X)$. We here analyze the multiplicative properties of the James maps which give rise to the splitting and of various related combinatorially derived maps between iterated loop spaces. The target of the total James map $\displaystyle j = ({j_q}):{\Omega ^n}{\Sigma ^n}X \to \mathop \times \limits_{q \geqslant 0} \;{\Omega ^{2nq}}{\Sigma ^{2nq}}{D_q}({R^n},X)$ is a ring space, and $j$ is an exponential $H$-map. There is a total Segal map $\displaystyle s = \mathop \times \limits_{q \geqslant 0} \;{s_{q}}:\mathop \tim... ... \mathop \times \limits_{q \geqslant 0} \;\Omega ^{3nq}\,\Sigma ^{3nq}{X^{[q]}}$ which is a ring map between ring spaces. There is a total partial power map $\displaystyle k = ({k_q}): {\Omega ^{n}}\,{\Sigma ^{n}}X \to \mathop \times \limits_{q \geqslant 0} \;{\Omega ^{n\,q}}\,{\Sigma ^{n\,q}}{X^{[q]}}$ which is an exponential $H$-map. There is a noncommutative binomial theorem for the computation of the smash power ${\Omega ^n}{\Sigma ^n}X \to {\Omega ^{nq}}{\Sigma ^{nq}}{X^{[q]}}$ in terms of the ${k_m}$ for $m \leqslant q$. The composite of $s$ and $j$ agrees with the composite of $k$ and the natural inclusion $\displaystyle \mathop \times \limits_{q \geqslant 0} \;{\Omega ^{n\,q}}\,{\Sigm... ...es \limits_{q \geqslant 0} \,{\Omega ^{3\,n\,q}}\,{\Sigma ^{3\,n\,q}}{X^{[q]}}.$ This analysis applies to essentially arbitrary spaces $ X$. When specialized to $ X = {S^0}$, it implies an unstable version of the Kahn-Priddy theorem. The exponential property of the James maps leads to an analysis of the behavior of loop addition with respect to the stable splitting of ${\Omega ^n}{\Sigma ^n}X$ when $X$ is connected, and there is an analogous analysis relating loop addition to the stable splitting of $ Q({X^ + })$.
James maps and $E\sb{n}$ ring spaces
F. R.
Cohen;
J. P.
May;
L. R.
Taylor
285-295
Abstract: We parametrize by operad actions the multiplicative analysis of the total James map given by Caruso and ourselves. The target of the total James map $\displaystyle j = \sum {{j_q}} :C({R^n},X) \to \prod\limits_{q \geqslant 0} {Q{D_q}({R^n},X)}$ is an ${E_n}$ ring space and $j$ is a $ {\mathcal{C}_n}$-map, where $ {\mathcal{C}_n}$ is the little $n$-cubes operad. This implies that $j$ has an $n$-fold delooping with domain ${\Sigma^n}X$. It also implies an algorithm for the calculation of $ {j_{\ast}}$ and thus of each $ {({j_q})_{\ast}}$ on $ \bmod\, p$ homology. When $n = \infty$ and $p = 2$, this algorithm is the essential starting point for Kuhn's proof of the Whitehead conjecture.
On some subalgebras of a von Neumann algebra crossed product
Baruch
Solel
297-308
Abstract: We study conditions for a nonselfadjoint subalgebra of a von Neumann crossed product $ \mathcal{L}$ to be an algebra of analytic operators with respect to a flow on $\mathcal{L}$. We restrict ourselves to the case where $\mathcal{L}$ is constructed from a finite von Neumann algebra $M$ with a trace preserving $ ^{\ast}$-automorphism $ \alpha$ that acts ergodically on the center of $M$.
Extensions of algebraic systems
Awad A.
Iskander
309-327
Abstract: There are several generalizations to universal algebras of the notion "The group $ \mathfrak{A}$ is an extension of the group $ \mathfrak{B}$ by the group $ \mathfrak{C}$". In this paper we study three such generalizations and the corresponding products of classes of algebraic systems. Various results are presented. One such theorem characterizes the weakly congruence regular varieties admitting extensions of a particular sort. Another result gives, under a weak congruence permutability condition, an equational basis for the variety obtained by applying one such product to two other varieties.
Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator
Eric
Sawyer
329-337
Abstract: Characterizations are obtained for those pairs of weight functions $w,\upsilon$ for which the Hardy operator $ Tf(x) = \int_0^x {f(s)\;ds}$ is bounded from the Lorentz space $ {L^{r,s}}((0,\infty ),\upsilon \,dx)$ to $ {L^{p,q}}((0,\infty ),w\,dx),0 < p,q,r,s \leqslant \infty$. The modified Hardy operators ${T_\eta }f(x) = {x^{ - \eta }}Tf(x)$ for $ \eta$ real are also treated.
A two weight weak type inequality for fractional integrals
Eric
Sawyer
339-345
Abstract: For $1 < p \leqslant q < \infty ,0 < \alpha < n$ and $w(x),\upsilon (x)$ nonnegative weight functions on ${R^n}$ we show that the weak type inequality $\displaystyle \int_{\{ {T_\alpha }f > \lambda \} }\,w(x)\;dx \leqslant A{\lambda ^{ - q}}{\left( \int \vert f(x){\vert^p}\;\upsilon (x)\;dx \right)^{q/p}}$ holds for all $f \geqslant 0$ if and only if $\displaystyle \int_Q\,[{T_\alpha }({\chi_Q}w)\,(x)]^{p'}\upsilon (x)^{1 - p'}\,dx \leqslant B\left( \int_Qw \right)^{p^{\prime}/q^{\prime}} < \infty$ for all cubes $ Q$ in ${R^n}$. Here $ {T_\alpha }$ denotes the fractional integral of order $\alpha ,{T_\alpha }f(x) = \int \vert x - y{\vert^{\alpha - n}}f(y)\,dy$. More generally we can replace ${T_\alpha }$ by any suitable convolution operator with radial kernel decreasing in $\vert x\vert$.
An improved stability result for resonances
Mark S.
Ashbaugh;
Carl
Sundberg
347-360
Abstract: We prove stability of shape resonances for the sequence of Schrödinger equations $( - {d^2}/d{x^2} + U(x) + {W_n}(x))\psi (x) = E\psi (x),0 \leqslant x\, < \infty$, in the limit $n \to \infty$ where the barrier potentials ${W_n}(x)$ are integrable, nonnegative, supported in the interval $[1,a]\;(1 < a < \infty )$, and approach infinity pointwise a.e. for $x \in [1,a]$ as $n \to \infty$. In the course of our investigation we prove that for suitable complex initial conditions the solution to the Riccati equation $ S^{\prime}(x) = 1 - ({W_n}(x) - E){[S(x)]^2}$ goes to 0 as $n \to \infty$ uniformly on compact subsets of $ [1,a]$. Our approach is via ordinary differential equations using outgoing wave boundary conditions to define resonances. Our stability result extends a similar result of Ashbaugh and Harrell, who use an argument based on asymptotics and the implicit function theorem to study the above problem with $ \lambda V(x)$ replacing $ {W_n}(x)$. Our approach is to use the Riccati equation analysis mentioned above and an application of Hurwitz's Theorem from complex variable theory.
The null space and the range of a convolution operator in a fading memory space
Olof J.
Staffans
361-388
Abstract: We study the convolution equation $(\ast)$ $\displaystyle \mu \; \ast \;x^{\prime}(t) + v\; \ast \;x(t) = f(t) \quad ( - \infty < t\, < \infty )$ , as well as a perturbed version of $(\ast)$, namely $ (\ast\ast)$ $\displaystyle \mu \;\ast\;x^{\prime}(t) + v\,\ast\;x(t) = \,F(x)\,(t)\quad ( - \infty < t < \infty ).$ Here $x$ is a $ {{\mathbf{R}}^n}$-valued function on $( - \infty ,\infty ),x^{\prime}(t) = dx(t)/dt$, and $\mu$ and $\nu$ are matrix-valued measures. If $\mu$ and $\nu$ are supported on $ [0,\infty )$, with $ \mu$ atomic at zero, then $ (\ast)$ can be regarded as a linear, autonomous, neutral functional differential equation with infinite delay. However, most of the time we do not consider the ordinary Cauchy problem for the neutral equation, i.e. we do not suppose that $ \mu$ and $\nu$ are supported on $[0,\infty )$, prescribe an initial condition of the type $ x(t) = \xi (t)\,(t \leqslant 0)$, and require $(\ast)$ and $ (\ast\ast)$ to hold only for $t \geqslant 0$. Instead we permit $ (\ast)$ and $(\ast\ast)$ to be of "Fredholm" type, i.e. $ \mu$ and $\nu$ need not vanish on $( - \infty ,0)$, we restrict the growth rate of $ x$ and $f$ at plus and minus infinity, and we look at the problem of the existence and uniqueness of solutions of $(\ast)$ and $ (\ast\ast)$ on the whole real line, satisfying conditions like $\vert x(t)\vert \leqslant C\eta (t)\;( - \infty < t < \infty )$, where $C$ is a constant, depending on $x$, and $\eta$ is a predefined function. Some authors use the word "admissible" when discussing problems of this type. In the case when the homogeneous version of $ (\ast)$ has nonzero solutions, we decompose the solutions into components with different exponential growth rates, and give a priori bounds on the growth rates of the solutions. As an application of the basic theory, we look at the Cauchy problem for a neutral functional differential equation, and prove the existence of stable and unstable manifolds.
Approximate subdifferentials and applications. I. The finite-dimensional theory
A. D.
Ioffe
389-416
Abstract: We introduce and study a new class of subdifferentials associated with arbitrary functions. Among the questions considered are: connection with other derivative-like objects (e.g. derivatives, convex subdifferentials, generalized gradients of Clarke and derivate containers of Warga), calculus of approximate subdifferentials and applications to analysis of set-valued maps and to optimization. It turns out that approximate subdifferentials are minimal (as sets) among other conceivable subdifferentials satisfying some natural requirements. This shows that certain results involving approximate subdifferentials are the best possible and, at the same time, marks certain limitations of nonsmooth analysis. Another important property of approximate subdifferentials is that, being essentially nonconvex, they admit a rich calculus that covers the calculus of convex subdifferentials and leads to more precise and sometimes new results for generalized gradients of Clarke.
Generalized Hua-operators and parabolic subgroups. The cases of ${\rm SL}(n,\,{\bf C})$ and ${\rm SL}(n,\,{\bf R})$
Kenneth D.
Johnson
417-429
Abstract: Suppose $G = {\text{SL}}(n,{\mathbf{C}})$ or $ {\text{SL}}(n,{\mathbf{R}})$ and $K$ is a maximal compact subgroup of $G$. If $P$ is any parabolic subgroup of $G$, we determine a system of differential equations on $G/K$ with the property that any function on $ G/K$ satisfies these differential equations if and only if it is the Poisson integral of a hyperfunction on $G/P$.