On sets that are uniquely determined by a restricted set of integrals
J. H. B.
Kemperman
417-458
Abstract: In many applied areas, such as tomography and crystallography, one is confronted by an unknown subset $S$ of a measure space $ (X,\lambda )$ such as ${{\mathbf{R}}^n}$, or an unknown function $0 \leqslant \phi \leqslant 1$ on $ X$, having known moments (integrals) relative to a specified class $ F$ of functions $f:X \to {\mathbf{R}}$. Usually, these $ F$-moments do not fully determine the object $S$ or function $\phi$. We will say that $S$ is a set of uniqueness if no other function $ 0 \leqslant \psi \leqslant 1$ has the same $F$-moments as $S$ in so far as the latter moments exist. Here, $ S$ is identified with its indicator function. Within this general setting and with no further assumptions, we develop a powerful sufficient condition for uniqueness, called generalized additivity. When $F$ is a finite class, this condition of generalized additivity is shown to be also necessary for uniqueness. For each $\phi$, which is not the indicator function of a set of uniqueness, there exist infinitely many sets having the same $F$-moments as $\phi$, provided $ (X,\lambda ,F)$ is nonatomic or regular and, moreover, `strongly rich', a condition which is satisfied in many applications. Using such general results, we also study the uniqueness problem for measures with given marginals relative to a given system of projections ${\pi _j}:X \to {Y_j}(j \in J)$. Here, one likes to know, for instance, what subsets $S$ of $X$ are uniquely determined by the corresponding set of projections (X-ray pictures). It is allowed that $ \lambda (S) = \infty$. Our results are also relevant to a wide class of optimization problems.
Epicomplete Archimedean $l$-groups and vector lattices
Richard N.
Ball;
Anthony W.
Hager
459-478
Abstract: An object $ G$ in a category is epicomplete provided that the only morphisms out of $ G$ which are simultaneously epi and mono are the isomorphisms. We characterize the epicomplete objects in the category ${\mathbf{Arch}}$, whose objects are the archimedean lattice-ordered groups (archimedean $\ell $-groups) and whose morphisms are the maps preserving both group and lattice structure ($\ell$-homomorphisms). Recall that a space is basically disconnected if the closure of each cozero subset is open. Theorem. The following are equivalent for $G \in {\mathbf{Arch}}$. (a) $G$ is ${\mathbf{Arch}}$ epicomplete. (b) $ G$ is an ${\mathbf{Arch}}$ extremal suboject of $ D(Y)$ for some basically disconnected compact Hausdorff space $ Y$. Here $ D(Y)$ denotes the continuous extended real-valued functions on $ Y$ which are finite on a dense subset. (c) $G$ is conditionally and laterally $ \sigma$-complete (meaning each countable subset of positive elements of $G$ which is either bounded or pairwise disjoint has a supremum), and $G$ is divisible. The analysis of $ {\mathbf{Arch}}$ rests on an analysis of the closely related category ${\mathbf{W}}$, whose objects are of the form $ (G,u)$, where $G \in {\mathbf{Arch}}$ and $u$ is a weak unit (meaning $g \wedge u = 0$ implies $g = 0$ for all $g \in G$), and whose morphisms are the $ \ell$-homomorphism preserving the weak unit. Theorem. The following are equivalent for $(G,u) \in {\mathbf{W}}$. (a) $(G,u)$ is ${\mathbf{W}}$ epicomplete. (b) $ (G,u)$ is ${\mathbf{W}}$ isomorphic to $ (D(Y),1)$. (c) $ (G,u)$ is conditionally and laterally $\sigma$-complete, and $G$ is divisible.
The Arf and Sato link concordance invariants
Rachel
Sturm Beiss
479-491
Abstract: The Kervaire-Arf invariant is a $Z/2$ valued concordance invariant of knots and proper links. The $\beta$ invariant (or Sato's invariant) is a $ Z$ valued concordance invariant of two component links of linking number zero discovered by J. Levine and studied by Sato, Cochran, and Daniel Ruberman. Cochran has found a sequence of invariants $\{ {\beta _i}\} $ associated with a two component link of linking number zero where each ${\beta _i}$ is a $Z$ valued concordance invariant and ${\beta _0} = \beta$. In this paper we demonstrate a formula for the Arf invariant of a two component link $L = X \cup Y$ of linking number zero in terms of the $\beta$ invariant of the link: $\displaystyle \operatorname{arf} (X \cup Y) = \operatorname{arf} (X) + \operatorname{arf} (Y) + \beta (X \cup Y)\quad (\bmod 2).$ This leads to the result that the Arf invariant of a link of linking number zero is independent of the orientation of the link's components. We then establish a formula for $\vert\beta \vert$ in terms of the link's Alexander polynomial $\Delta (x,y) = (x - 1)(y - 1)f(x,y)$: $\displaystyle \vert\beta (L)\vert = \vert f(1,1)\vert.$ Finally we find a relationship between the ${\beta _i}$ invariants and linking numbers of lifts of $X$ and $Y$ in a $Z/2$ cover of the compliment of $X \cup Y$.
Multipliers, linear functionals and the Fr\'echet envelope of the Smirnov class $N\sb *({\bf U}\sp n)$
Marek
Nawrocki
493-506
Abstract: Linear topological properties of the Smirnov class $ {N_{\ast}}({\mathbb{U}^n})$ of the unit polydisk $ {\mathbb{U}^n}$ in ${\mathbb{C}^n}$ are studied. All multipliers of $ {N_{\ast}}({\mathbb{U}^n})$ into the Hardy spaces $ {H_p}({\mathbb{U}^n}),\;0 < p \leqslant \infty$, are described. A representation of the continuous linear functionals on $ {N_{\ast}}({\mathbb{U}^n})$ is obtained. The Fréchet envelope of $ {N_{\ast}}({\mathbb{U}^n})$ is constructed. It is proved that if $ n > 1$, then $ {N_{\ast}}({\mathbb{U}^n})$ is not isomorphic to $ {N_{\ast}}(\mathbb{U}{^1})$.
Varieties of group representations and Casson's invariant for rational homology $3$-spheres
S.
Boyer;
A.
Nicas
507-522
Abstract: Andrew Casson's ${\mathbf{Z}}$-valued invariant for ${\mathbf{Z}}$-homology $3$-spheres is shown to extend to a ${\mathbf{Q}}$-valued invariant for ${\mathbf{Q}}$-homology $3$-spheres which is additive with respect to connected sums. We analyze conditions under which the set of abelian ${\operatorname{SL} _2}({\mathbf{C}})$ and $ \operatorname{SU} (2)$ representations of a finitely generated group is isolated. A formula for the dimension of the Zariski tangent space to an abelian ${\operatorname{SL} _2}({\mathbf{C}})$ or $ \operatorname{SU} (2)$ representation is obtained. We also derive a sum theorem for Casson's invariant with respect to toroidal splittings of a $ {\mathbf{Z}}$-homology $ 3$-sphere.
The connection matrix in Morse-Smale flows
James F.
Reineck
523-545
Abstract: In a Morse-Smale flow with no periodic orbits, it is shown that the connection matrix is unique. In the case of periodic orbits, nonuniqueness can occur. We show that on $2$-manifolds, with some technical assumptions, given a connection matrix for the flow, one can replace the periodic orbits with doubly-connected rest points and obtain a new flow with no periodic orbits having the given connection matrix.
Baer modules over domains
Paul C.
Eklof;
László
Fuchs;
Saharon
Shelah
547-560
Abstract: For a commutative domain $R$ with $1$, an $R$-module $B$ is called a Baer module if $ \operatorname{Ext} _R^1(B,T) = 0$ for all torsion $R$-modules $T$. The structure of Baer modules over arbitrary domains is investigated, and the problem is reduced to the case of countably generated Baer modules. This requires a general version of the singular compactness theorem. As an application we show that over $ h$-local Prüfer domains, Baer modules are necessarily projective. In addition, we establish an independence result for a weaker version of Baer modules.
Dualizing complexes of affine semigroup rings
Uwe
Schäfer;
Peter
Schenzel
561-582
Abstract: For an affine semigroup ring we construct the dualizing complex in terms of the semigroup and the homology of the face lattice of the polyhedral cone spanned by the semigroup. As a consequence there are characterizations of locally Cohen-Macaulay rings, Buchsbaum rings, and Cohen-Macaulay rings as well as Serre's condition ${\mathcal{S}_l}$.
Weak Chebyshev subspaces and $A$-subspaces of $C[a,b]$
Wu
Li
583-591
Abstract: In this paper we show some very interesting properties of weak Chebyshev subspaces and use them to simplify Pinkus's characterization of $A$subspaces of $C[a,b]$. As a consequence we obtain that if the metric projection ${P_G}$ from $C[a,b]$ onto a finite-dimensional subspace $ G$ has a continuous selection and elements of $G$ have no common zeros on $(a,b)$, then $G$ is an $A$-subspace.
$N$-body Schr\"odinger operators with finitely many bound states
W. D.
Evans;
Roger T.
Lewis
593-626
Abstract: In this paper we consider a class of second-order elliptic operators which includes atomic-type $N$-body operators for $N > 2$. Our concern is the problem of predicting the existence of only a finite number of bound states corresponding to eigenvalues below the essential spectrum. We obtain a criterion which is natural for the problem and easy to apply as is demonstrated with various examples. While the criterion applies to general second-order elliptic operators, sharp results are obtained when the Hamiltonian of an atom with an infinitely heavy nucleus of charge $Z$ and $N$ electrons of charge $1$ and mass $ \tfrac{1} {2}$ is considered.
Global families of limit cycles of planar analytic systems
L. M.
Perko
627-656
Abstract: The global behavior of any one-parameter family of limit cycles of a planar analytic system $\dot x = f(x,\lambda )$ depending on a parameter $\lambda \in R$ is determined. It is shown that any one-parameter family of limit cycles belongs to a maximal one-parameter family which is either open or cyclic. If the family is open, then it terminates as the parameter or the orbits become unbounded, or it terminates at a critical point or on a (compound) separatrix cycle of the system. This implies that the periods in a one-parameter family of limit cycles can become unbounded only if the orbits become unbounded or if they approach a degenerate critical point or (compound) separatrix cycle of the system. This is a more specific result for planar analytic systems than Wintner's principle of natural termination for $n$-dimensional systems where the periods can become unbounded in strange ways. This work generalizes Duffs results for one-parameter families of limit cycles generated by a one-parameter family of rotated vector fields. In particular, it is shown that the behavior at a nonsingular, multiple limit cycle of any one-parameter family of limit cycles is exactly the same as the behavior at a multiple limit cycle of a one-parameter family of limit cycles generated by a one-parameter family of rotated vector fields.
Solving Beltrami equations by circle packing
Zheng-Xu
He
657-670
Abstract: We use Andreev-Thurston's theorem on the existence of circle packings to construct approximating solutions to the Beltrami equations on Riemann surfaces. The convergence of the approximating solutions on compact subsets will be shown. This gives a constructive proof of the existence theorem for Beltrami equations.
Mean summability methods for Laguerre series
Krzysztof
Stempak
671-690
Abstract: We apply a construction of generalized convolution in $\displaystyle {L^1}({\mathbb{R}_ + } \times \mathbb{R},{x^{2\alpha - 1}}dxdt),\qquad \alpha \geqslant 1,$ cf. [8], to investigate the mean convergence of expansions in Laguerre series. Following ideas of [4, 5] we construct a functional calculus for the operator $\displaystyle L = - \left( {\frac{{{\partial ^2}}} {{\partial {x^2}}} + \frac{{... ... {t^2}}}} \right),\qquad x > 0,\quad t \in \mathbb{R},\quad \alpha \geqslant 1.$ Then, arguing as in [3], we prove results concerning the mean convergence of some summability methods for Laguerre series. In particular, the classical Abel-Poisson and Bochner-Riesz summability methods are included.
On the smoothness of convex envelopes
A.
Griewank;
P. J.
Rabier
691-709
Abstract: We examine differentiability properties of the convex envelope $\operatorname{conv} E$ of a given function $ E:{{\mathbf{R}}^n} \to ( - \infty ,\infty ]$ in terms of properties of $ E$. It is shown that $ {C^1}$ as well as optimal ${C^{1,\alpha }}$ regularity results, $0 < \alpha \leqslant 1$, can be obtained under general conditions.
An asymptotic formula for hypo-analytic pseudodifferential operators
S.
Berhanu
711-729
Abstract: An asymptotic expansion formula for hypo-analytic pseudodifferential operators is proved and applications are given.
Random Blaschke products
W. George
Cochran
731-755
Abstract: Let $\{ {\theta _n}(\omega )\} $ be a sequence of independent random variables uniformly distributed on $ [0,2\pi ]$, and let ${z_n}(\omega ) = {r_n}{e^{i{\theta _n}(\omega )}}$ for a fixed but arbitrary sequence of radii $ {r_n}$ satisfying the Blaschke condition $\sum {(1 - {r_n}) < \infty }$. We show that the random Blaschke product with zeros ${z_n}(\omega )$ is almost surely not in the little Bloch space, and we describe necessary conditions and sufficient conditions on the radii ${r_n}$ so that $\{ {z_n}(\omega )\}$ is almost surely an interpolating sequence.
Lie algebra modules with finite-dimensional weight spaces. I
S. L.
Fernando
757-781
Abstract: Let $\mathfrak{g}$ denote a reductive Lie algebra over an algebraically closed field of characteristic zero, and let $\mathfrak{h}$ denote a Cartan subalgebra of $\mathfrak{g}$. In this paper we study finitely generated $ \mathfrak{g}$-modules that decompose into direct sums of finite dimensional $\mathfrak{h}$-weight spaces. We show that the classification of irreducible modules in this category can be reduced to the classification of a certain class of irreducible modules, those we call torsion free modules. We also show that if $\mathfrak{g}$ is a simple Lie algebra that admits a torsion free module, then $\mathfrak{g}$ is of type $A$ or $C$.
Tauberian theorems for the Laplace-Stieltjes transform
C. J. K.
Batty
783-804
Abstract: Let $ \alpha :[0,\infty ) \to {\mathbf{C}}$ be a function of locally bounded variation, with $\alpha (0) = 0$, whose Laplace-Stieltjes transform $g(z) = \int_0^\infty {{e^{ - zt}}d\alpha (t)}$ is absolutely convergent for $\operatorname{Re} z > 0$. Let $E$ be the singular set of $ g$ in $i{\mathbf{R}}$, and suppose that $0 \notin E$. Various estimates for $ \lim {\sup _{t \to \infty }}\vert\alpha (t) - g(0)\vert$ are obtained. In particular, $ \alpha (t) \to g(0)$ as $t \to \infty$ if \begin{displaymath}\begin{gathered}({\text{i)}}\quad E\,{\text{is null,}} ... ...vert\alpha (s) - \alpha (t)\vert = 0. \end{gathered} \end{displaymath} This result contains Tauberian theorems for Laplace transforms, power series, and Dirichlet series.
Stability of individual elements under one-parameter semigroups
Charles J. K.
Batty;
Vù Quôc Phóng
805-818
Abstract: Let $\{ T(t):t \geqslant 0\}$ be a $ {C_0}$-semigroup on a Banach space $X$ with generator $A$, and let $x \in X$. If $\sigma (A) \cap i{\mathbf{R}}$ is empty and $t \mapsto T(t)x$ is uniformly continuous, then $ \vert\vert T(t)x\vert\vert \to 0$ as $t \to \infty $. If the semigroup is sun-reflexive, $\sigma (A) \cap i{\mathbf{R}}$ is countable, $ P\sigma (A) \cap i{\mathbf{R}}$ is empty, and $ t \mapsto T(t)x$ is uniformly weakly continuous, then $ T(t)x \to 0$ weakly as $t \to \infty$. Questions of almost periodicity and of stabilization of contraction semigroups on Hilbert space are also discussed.