Infinite-dimensional linear systems with unbounded control and observation: a functional analytic approach
Dietmar
Salamon
383-431
Abstract: The object of this paper is to develop a unifying framework for the functional analytic representation of infinite dimensional linear systems with unbounded input and output operators. On the basis of the general approach new results are derived on the wellposedness of feedback systems and on the linear quadratic control problem. The implications of the theory for large classes of functional and partial differential equations are discussed in detail.
Sufficiency conditions for $L\sp p$-multipliers with power weights
Benjamin
Muckenhoupt;
Richard L.
Wheeden;
Wo-Sang
Young
433-461
Abstract: Weighted norm inequalities in ${R^1}$ are proved for multiplier operators with the multiplier function of Hörmander type. The operators are initially defined on the space ${\mathcal{S}_{0,0}}$ of Schwartz functions whose Fourier transforms have compact support not including 0. This restriction on the domain of definition makes it possible to use weight functions of the form ${\left\vert x \right\vert^\alpha }$ for $ \alpha$ larger than usually considered. For these weight functions, if $(\alpha + 1)/p$ is not an integer, a strict inequality on $\alpha$ is shown to be sufficient for a norm inequality to hold. A sequel to this paper shows that the weak version of this inequality is necessary.
Sufficiency conditions for $L\sp p$ multipliers with general weights
Benjamin
Muckenhoupt;
Richard L.
Wheeden;
Wo-Sang
Young
463-502
Abstract: Weighted norm inequalities in ${R^1}$ are proved for multiplier operators with the multiplier function satisfying Hörmander type conditions. The operators are initially defined on the space $ {\mathcal{S}_{0,0}}$ of Schwartz functions whose Fourier transforms have compact support not including 0. This restriction on the domain of definition makes it possible to use a larger class of weight functions than usually considered; weight functions used here are of the form ${\left\vert {g(x)} \right\vert^p}V(x)$ where $ g(x)$) is a polynomial of arbitrarily high degree and $V(x)$ is in ${A_p}$. For weight functions in ${A_p}$, the results hold for all Schwartz functions. The periodic case is also considered.
Necessity conditions for $L\sp p$ multipliers with power weights
Benjamin
Muckenhoupt
503-520
Abstract: It is shown that if multiplier operators are bounded on with weight $ {\left\vert x \right\vert^\alpha }$ for all functions in the space ${\mathcal{S}_{0,0}}$ of Schwartz functions whose Fourier transforms have compact support not including 0 and all multiplier functions in a standard Hörmander type multiplier class, then $\alpha$ must satisfy certain inequalities. This is a sequel to a previous paper in which conditions on $ \alpha$ that were almost the same were shown to be sufficient for the norm inequality to hold.
Some weighted norm inequalities for the Fourier transform of functions with vanishing moments
Cora
Sadosky;
Richard L.
Wheeden
521-533
Abstract: Weighted $ L^p$ norm inequalities are derived between a function and its Fourier transform in case the function has vanishing moments up to some order. For weights of the form ${\left\vert x \right\vert^\gamma }$, the results concern values of $\gamma$ which are outside the range which is normally considered.
Polynomial algebras have polynomial growth
David R.
Finston
535-556
Abstract: The definitions and basic properties of Gelfand-Kirillov dimension are extended to algebras over a field which are not necessarily associative. The results are applied to the algebra of polynomial functions on an arbitrary finite dimensional algebra to obtain polynomial growth (i.e. integral G-K dimension) for these algebras. The G-K dimension of the polynomial algebra in one indeterminate is shown to be constant on the category of all finite dimensional nomial extensions of an associative algebra.
Near coherence of filters. II. Applications to operator ideals, the Stone-\v Cech remainder of a half-line, order ideals of sequences, and slenderness of groups
Andreas
Blass
557-581
Abstract: The set-theoretic principle of near coherence of filters (NCF) is known to be neither provable nor refutable from the usual axioms of set theory. We show that NCF is equivalent to the following statements, among others: (1) The ideal of compact operators on Hilbert space is not the sum of two smaller ideals. (2) The Stone-Čech remainder of a half-line has only one composant. (This was first proved by J. Mioduszewski.) (3) The partial ordering of slenderness classes of abelian groups, minus its top element, is directed upward (and in fact has a top element). Thus, all these statements are also consistent and independent.
The MacRae invariant and the first local Chern character
Paul
Roberts
583-591
Abstract: The first local Chern character of a bounded complex of locally free sheaves on a scheme $Y$ is given by intersection with a Cartier divisor. In the case of the resolution of a module of finite projective dimension, this is the invariant defined by MacRae.
The $b{\rm o}$-Adams spectral sequence
Wolfgang
Lellmann;
Mark
Mahowald
593-623
Abstract: Due to its relation to the image of the $J$-homomorphism and first order periodicity (Bott periodicity), connective real $K$-theory is well suited for problems in $ 2$-local stable homotopy that arise geometrically. On the other hand the use of generalized homology theories in the construction of Adams type spectral sequences has proved to be quite fruitful provided one is able to get a hold on the respective $ {E_2}$-terms. In this paper we make a first attempt to construct an algebraic and computational theory of the ${E_2}$-term of the bo-Adams spectral sequence. This allows for some concrete computations which are then used to give a proof of the bounded torsion theorem of [8] as used in the geometric application of [2]. The final table of the $ {E_2}$-term for $\pi _ \ast ^S$ in ${\operatorname{dim}}\, \leq 20$ shows that the statement of this theorem cannot be improved. No higher differentials appear in this range of the bo-Adams spectral sequence. We observe, however, that such a differential has to exist in dim 30.
A bilinear form for Spin manifolds
Peter S.
Landweber;
Robert E.
Stong
625-640
Abstract: This paper studies the bilinear form on $ {H^j}(M;{Z_2})$ defined by $\left[ {x,\,y} \right] = x\,{\text{S}}{{\text{q}}^2}y[M]$ when $M$ is a closed Spin manifold of dimension $ 2j + 2$. In analogy with the work of Lusztig, Milnor, and Peterson for oriented manifolds, the rank of this form on integral classes gives rise to a cobordism invariant.
Modules and stability theory
Anand
Pillay;
Mike
Prest
641-662
Abstract: Modules are now widely recognized as important examples of stable structures. In fact, in the light of results and conjectures of Zilber [Zi] ( $ {\aleph _1}$-categorical structures are ``field-like'', ``module-like'' or ``trivial''), we may consider modules as one of the typical examples of stable structures. Our aim here is both to prove some new results in the model theory of modules and to highlight the particularly clear form of, and the algebraic content of, the concepts of stability theory when applied to modules. One of the main themes of this paper is the connection between stability-theoretic notions, such as ranks, and algebraic decomposition of models. We will usually work with $T$, a complete theory of $R$-modules, for some ring $R$. In $\S2$ we show that the various stability-theoretic ranks, when defined, are the same. In $\S3$ we show that $T$ (not necessarily superstable) is nonmultidimensional (in the sence of Shelah [Sh1]). In $\S4$ we consider the algebraic content of saturation and we show, for example, that if $ M$ is a superstable module then $M$ is $ F_{{\aleph _0}}^a$-saturated just if $M$ is pure-injective and realizes all types in finitely many free variables over $\phi$. In $\S5$ we use our methods to reprove Ziegler's theorem on the possible spectrum functions. In $\S6$ we show the profusion (in a variety of senses) of regular types. In $\S7$ we give a structure theorem for the models of $ T$ in the case where $ T$ has $U$-rank 1.
On nonbinary $3$-connected matroids
James G.
Oxley
663-679
Abstract: It is well known that a matroid is binary if and only if it has no minor isomorphic to ${U_{2,4}}$, the $4$-point line. Extending this result, Bixby proved that every element in a nonbinary connected matroid is in a ${U_{2,4}}$-minor. The result was further extended by Seymour who showed that every pair of elements in a nonbinary $3$-connected matroid is in a $ {U_{2,4}}$-minor. This paper extends Seymour's theorem by proving that if $\left\{ {x,\,y,\,z} \right\}$ is contained in a nonbinary $3$-connected matroid $M$, then either $M$ has a ${U_{2,4}}$-minor using $\left\{ {x,\,y,\,z} \right\}$, or $ M$ has a minor isomorphic to the rank-$3$ whirl that uses $\left\{ {x,\,y,\,z} \right\}$ as its rim or its spokes.
Analytic functions with prescribed cluster sets
L. W.
Brinn
681-693
Abstract: Suppose that $0 < R \leq + \infty$. A monotonic boundary path (mb-path) in $\left\vert z \right\vert < R$ is a simple continuous curve $z = z(s)$, $0 \leq s < 1$, in $\left\vert z \right\vert < R$ such that $ \left\vert {z(s)} \right\vert \to R$ strictly monotonically as $s \to 1$. Suppose that $ f$ is a complex valued function, defined in $ \left\vert z \right\vert < R$, and that $t$ is any mb-path in $\left\vert z \right\vert < R$. The cluster set of $f$ on $t$ is the set of those points $w$ on the Riemann sphere for which there exists a sequence $ \{ {z_n}\}$ of points of $ t$ with ${\operatorname{lim}_{n \to \infty }}\left\vert {{z_n}} \right\vert = R$ and $ {\operatorname{lim}_{n \to \infty }}f({z_n}) = w$. The cluster set is denoted by $ {C_t}(f)$. If the cluster set is a single point set, that point is called the asymptotic value of $f$ on $t$. If the function $f$ is continuous, then ${C_t}(f)$ is a continuum on the Riemann sphere. It is a conjecture of F. Bagemihl and W. Seidel that if $\mathcal{T}$ is a family of mb-paths in $\left\vert z \right\vert < R$ satisfying certain conditions, and if $\mathcal{K}$ is an analytic set of continua on the Riemann sphere, then there exists a function $ f$, analytic in $\left\vert z \right\vert < R$, such that $\left\{ {{C_t}(f)\vert t \in \mathcal{T}} \right\} = \mathcal{K}$. A restricted form of the conjecture is mentioned in [3, p. 100]. Our principal results show the correctness of the conjecture in the case that $\mathcal{K}$ is the collection of all continua on the Riemann sphere and $\mathcal{T}$ is a tress of a certain type. The results are generalized in several directions. In particular, our technique for constructing the analytic function $f$ extends immediately to the case in which $\mathcal{K}$ is any closed set of continua on the sphere. Specific examples of closed sets lead to several corollaries.
A characterization of the kernel of the Poincar\'e series operator
Makoto
Masumoto
695-704
Abstract: Let $\Gamma$ be a finitely generated Fuchsian group of the first kind acting on the unit disk $ \Delta$. The kernel of the Poincaré series operator of the Hardy space ${H^p},\,1 < p < \infty$, onto the Bers space $ {A_q}(\Gamma )$ of integrable holomorphic automorphic forms of weight $- 2q,\,q \in {\mathbf{Z}},\,q \geq 2$, on $\Delta$ for $\Gamma$ is characterized in terms of Eichler integrals of order $1 - q$ on $\Delta$ for $\Gamma$.
A generalized Fatou theorem
B. A.
Mair;
David
Singman
705-719
Abstract: In this paper, a general Fatou theorem is obtained for functions which are integrals of kernels against measures on ${{\mathbf{R}}^n}$. These include solutions of Laplace's equation on an upper half-space, parabolic equations on an infinite slab and the heat equation on a right half-space. Lebesgue almost everywhere boundary limits are obtained within regions which contain sequences approaching the boundary with any prescribed degree of tangency.
On bounded analytic functions in finitely connected domains
Zbigniew
Slodkowski
721-736
Abstract: A new proof of the corona theorem for finitely connected domains is given. It is based on a result on the existence of a meromorphic selection from an analytic set-valued function. The latter fact is also applied to the study of finitely generated ideals of $ {H^\infty }$ over multiply connected domains.
Eichler-Shimura homology, intersection numbers and rational structures on spaces of modular forms
Svetlana
Katok;
John J.
Millson
737-757
Abstract: In this paper we reinterpret the main results of [8] using the intersection theory of cycles with coefficients. To this end we give a functorial interpretation of Eichler-Schimura periods.
A renewal theorem for random walks in multidimensional time
J.
Galambos;
K.-H.
Indlekofer;
I.
Kátai
759-769
Abstract: Let $X,\,{X_1},\,{X_2}, \ldots $ be a family of integer valued, independent and identically distributed random variables with positive mean and finite (positive) variance. Let ${S_n} = {X_1} + \,{X_2} + \cdots + {X_n}$. The asymptotic behavior of the weighted sum $R(k) = \sum {a_n}P({S_n} = k)$, with summation over $n \geq 1$, is investigated as $k \to + \infty$. In the special case ${a_n} = {d_r}(n)$, the number of solutions of the equation $n = {n_1}{n_2} \cdots {n_r}$ in positive integers $ {n_j},\,1 \leq j \leq r,\,R(k)$ becomes the renewal function $Q(k)$ for a random walk in $ r$-dimensional time whose terms are distributed as $X$. Under some assumptions on the magnitude of ${a_n}$ and of $A(x) = \sum\nolimits_{n \leq x} {{a_n}}$, (i) it is shown that $R(k)$ is asymptotically distribution free as $k \to + \infty$, (ii) the proper order of magnitude of $R(k)$ is determined, and under some further restrictions on $A(x)$, (iii) a simple asymptotic formula is given for $R(k)$. From (i), the known asymptotic formula for $ Q(k)$ with $r = 2$ or 3 is deduced under the sole assumption of finite variance. The relaxation of previous moment assumptions requires a new inequality for the sum of the divisor function ${d_r}(n),\,1 \leq n \leq x$, which by itself is of interest.
A general theory of canonical forms
Richard S.
Palais;
Chuu-Lian
Terng
771-789
Abstract: If $G$ is a compact Lie group and $ M$ a Riemannian $ G$-manifold with principal orbits of codimension $k$ then a section or canonical form for $ M$ is a closed, smooth $ k$-dimensional submanifold of $M$ which meets all orbits of $M$ orthogonally. We discuss some of the remarkable properties of $G$-manifolds that admit sections, develop methods for constructing sections, and consider several applications.
${\bf Z}/p{\bf Z}$ actions on $(S\sp n)\sp k$
Alejandro
Adem
791-809
Abstract: Let ${\mathbf{Z}}/p$ act on a finitistic space $ X$ with integral cohomology isomorphic to that of $ {({S^n})^k}$ as a ring. We show a direct relationship between the ${\mathbf{Z}}/p$-module structure of ${H^n}(X;{\mathbf{Z}})$ and the nature of the fixed-point set. In particular, we obtain a significant restriction on $ {H^n}(X;{\mathbf{Z}})$ for free actions.
Addendum to: ``Group-graded rings, smash products, and group actions'' [Trans. Amer. Math. Soc. {\bf 282} (1984), no. 1, 237--258; MR0728711 (85i:16002)]
M.
Cohen;
S.
Montgomery
810-811