Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture
Michel L.
Lapidus
465-529
Abstract: Let $\Omega$ be a bounded open set of $ {\mathbb{R}^n}\;(n \geq 1)$ with "fractal" boundary $\Gamma$. We extend Hermann Weyl's classical theorem by establishing a precise remainder estimate for the asymptotics of the eigenvalues of positive elliptic operators of order $ 2m\;(m \geq 1)$ on $ \Omega$. We consider both Dirichlet and Neumann boundary conditions. Our estimate--which is expressed in terms of the Minkowski rather than the Hausdorff dimension of $\Gamma $--specifies and partially solves the Weyl-Berry conjecture for the eigenvalues of the Laplacian. Berry's conjecture--which extends to "fractals" Weyl's conjecture--is closely related to Kac's question "Can one hear the shape of a drum?"; further, it has significant physical applications, for example to the scattering of waves by "fractal" surfaces or the study of porous media. We also deduce from our results new remainder estimates for the asymptotics of the associated "partition function" (or trace of the heat semigroup). In addition, we provide examples showing that our remainder estimates are sharp in every possible "fractal" (i.e., Minkowski) dimension. The techniques used in this paper belong to the theory of partial differential equations, the calculus of variations, approximation theory and--to a lesser extent--geometric measure theory. An interesting aspect of this work is that it establishes new connections between spectral and "fractal" geometry.
Partially acyclic manifold decompositions yielding generalized manifolds
David F.
Snyder
531-571
Abstract: Let $G$ be an upper semicontinuous decomposition (used) of the $(n + k)$-manifold $M$ into subcontinua having the shape of closed orientable $n$-manifolds $(2 < n,k)$. We define $G$ to be $j$-acyclic if for every element $g$ of $G$ the reduced Čech homology of $g$ vanishes up through dimension $ j$. The primary objective of this investigation is to determine the local connectivity properties of the decomposition space $ B = M/G$ if $G$ is $(k - 2)$-acyclic and $B$ is finite dimensional. The Leray-Grothendieck spectral sequence of the decomposition map $ p$ is analyzed, which relegates the principal part of the investigation to studying the structure of the Leray sheaf of $p$ and its relation to the local cohomology of $B$. Let $E$ denote the subset of $B$ over which the Leray sheaf is not locally constant, $K$ the subset of $E$ over which the Leray sheaf is not locally Hausdorff, and $D = E - K$. Then we get as our main result, which extends work of R. J. Daverman and J. J. Walsh, and generalizes a result of D. S. Coram and P. Duvall as well, Theorem. Let $G$ be a $(k - 2)$-acyclic decomposition of the $ (n + k)$-manifold $ M$ such that $ k < n + 2$, $ B = M/G$ is finite dimensional, and the set $E$ does not locally separate $B$. Then $B$ is a generalized $ k$-manifold, if either $k = n + 1$, or $k < n + 1$ and $M$ is orientable.
On surfaces and Heegaard surfaces
Klaus
Johannson
573-591
Abstract: This paper is concerned with the intersection of surfaces and Heegaard surfaces in closed orientable $3$-manifolds $M$. Given a Heegaard decomposition $(M,{V_1},{V_2})$ it will be shown that any surface (orientable or not) in $M$ is equivalent to a surface which intersects $ {V_1}$ in discs whose total number is limited from above by some function in the genus of $ \partial {V_1}$ alone. The equivalence relation in question is generated by disc- and annulus-compressions.
Recursive functionals and quantifiers of finite types revisited. V
S. C.
Kleene
593-630
Abstract: This is the last in a sequence of papers that redoes the theory of recursion in finite types. A key feature of the theory is that a computation can succeed (or finish) even if some of its subcomputations do not, if these turn out to be irrelevant to the total computation. I give a detailed description of computations involving oracles for type $3$ functionals. The computation may be viewed formally as a transfinite sequence of symbolic expressions, but I also describe a semantics in which each expression is given a concrete realization.
$C\sp \infty$ loop algebras and noncommutative Bott periodicity
N. Christopher
Phillips
631-659
Abstract: We construct the noncommutative analogs ${\Omega _\infty }A$ and ${\Omega _{{\text{lip}}}}A$ of the ${C^\infty }$ and Lipschitz loop spaces for a pro-${C^\ast}$-algebra $A$ equipped with a suitable dense subalgebra. With $ {U_{{\text{nc}}}}$ and $ P$ being the classifying algebras for $K$-theory earlier introduced by the author, we then prove that there are homotopy equivalences ${\Omega _\infty }{U_{{\text{nc}}}} \simeq P$ and ${\Omega _\infty }P \simeq {U_{{\text{nc}}}}$. This result is a noncommutative analog of Bott periodicity in the form $\Omega U \simeq {\mathbf{Z}} \times BU$ and $\Omega ({\mathbf{Z}} \times BU) \simeq U$.
Absolute continuity results for superprocesses with some applications
Steven N.
Evans;
Edwin
Perkins
661-681
Abstract: Let ${X^1}$ and ${X^2}$ be instances of a measure-valued Dawson-Watanabe $\xi$-super process where the underlying spatial motions are given by a Borel right process, $ \xi$, and where the branching mechanism has finite variance. A necessary and sufficient condition on $X_0^1$ and $X_0^2$ is found for the law of $X_s^1$ to be absolutely continuous with respect to the law of $X_t^2$. The conditions are the natural absolute continuity conditions on $\xi$, but some care must be taken with the set of times $s$, $t$ being considered. The result is used to study the closed support of super-Brownian motion and give sufficient conditions for the existence of a nontrivial "collision measure" for a pair of independent super-Lévy processes or, more generally, for a super-Lévy process and a fixed measure. The collision measure gauges the extent of overlap of the two measures. As a final application, we give an elementary proof of the instantaneous propagation of a super-Lévy process to all points to which the underlying Lévy process can jump. This result is then extended to a much larger class of superprocesses using different techniques.
Second-order elliptic operators and heat kernels on Lie groups
Ola
Bratteli;
Derek W.
Robinson
683-713
Abstract: Arendt, Batty, and Robinson proved that each second-order strongly elliptic operator $C$ associated with left translations on the $ {L_p}$-spaces of a Lie group $G$ generates an interpolating family of semigroups $T$, whenever the coefficients of $ C$ are sufficiently smooth. We establish that $T$ has an integral kernel $K$ satisfying the bounds $ {{\mathbf{R}}^d}$.
Quasi-projective and relative cohomological dimension of groups
Juan M.
Alonso
715-739
Abstract: We give a geometric interpretation of the quasi-projective dimension of groups, a notion introduced by Howie and Schneebeli [H-S1] as a generalization of the Identity Property.
A spectral commutant lifting theorem
Hari
Bercovici;
Ciprian
Foias;
Allen
Tannenbaum
741-763
Abstract: The commutant lifting theorem of [24] may be regarded as a very general interpolation theorem from which a number of classical interpolation results may be deduced. In this paper we prove a spectral version of the commutant lifting theorem in which one bounds the spectral radius of the interpolant and not the norm. We relate this to a spectral analogue of classical matricial Nevanlinna-Pick interpolation.
Alternating sequences and induced operators
M. A.
Akcoglu;
R. E.
Bradley
765-791
Abstract: We show that when a positive ${L_p}$ contraction is equipped with a norming function having full support, then it is related in a natural way to an operator on any other ${L_p}$ space, $1 < p < \infty$. This construction is used to generalize a theorem of Rota concerning the convergence of alternating sequences.
$K\sb 1$-groups, quasidiagonality, and interpolation by multiplier projections
Shuang
Zhang
793-818
Abstract: We relate the following conditions on a $\sigma$-unital ${C^\ast}$-algebra $A$ with the " $ {\text{FS}}$ property": (a) ${K_1}(A) = 0$; (b) every projection in $M(A)/A$ lifts; (c) the general Weyl-von Neumann theorem holds in $M(A)$: Any selfadjoint element $h$ in $M(A)$ can be written as $ h = \sum\nolimits_{i = 1}^\infty {{\lambda _i}{p_i} + a} $ for some selfadjoint element $a$ in $A$, some bounded real sequence $\{ {\lambda _i}\}$, and some mutually orthogonal projections $\{ {p_i}\}$ in $A$ with $\sum\nolimits_{i = 1}^\infty {{p_i} = 1}$; (d) $M(A)$ has $ {\text{FS}}$; and (e) interpolation by multiplier projections holds: For any closed projections $p$ and $q$ in $ {A^{\ast \ast}}$ with $ pq = 0$, there is a projection $r$ in $M(A)$ such that $p \leq r \leq 1 - q$. We prove various equivalent versions of (a)-(e), and show that (e) $\Leftrightarrow$ (d) $\Leftrightarrow$ (c) $ \Rightarrow$ (b) $\Leftarrow$ (a), and that (a) $\Leftrightarrow$ (b) if, in addition, $ A$ is stable. Combining the above results, we obtain counterexamples to the conjecture of G. K. Pedersen "$A$ has $ FS \Rightarrow M(A)$ has ${\text{FS}}$" (for example the stabilized Bunce-Deddens algebras). Hence the generalized Weyl-von Neumann theorem does not generally hold in $ L({H_A})$ for $ \sigma$-unital $ {C^\ast}$-algebras with ${\text{FS}}$.
The symbolic representation of billiards without boundary condition
Takehiko
Morita
819-828
Abstract: We consider a dynamical system with elastic reflections in the whole plane and show that such a dynamical system can be represented as a symbolic flow over a mixing subshift of finite type. This fact enables us to prove an analogue of the prime number theorem for the closed orbits of such a dynamical system.
Support algebras of $\sigma$-unital $C\sp *$-algebras and their quasi-multipliers
Hua Xin
Lin
829-854
Abstract: We study certain dense hereditary $^\ast$-subalgebras of $\sigma$-unital ${C^\ast}$-algebras and their relations with the Pedersen ideals. The quasi-multipliers of the dense hereditary $^\ast$-subalgebras are also studied.
Groups of prime power order as Frobenius-Wielandt complements
Carlo M.
Scoppola
855-874
Abstract: It is known that the Sylow subgroups of a Frobenius complement are cyclic or generalized quaternion. In this paper it is shown that there are no restrictions at all on the structure of the Sylow subgroups of the Frobenius-Wielandt complements that appear in the well-known Wielandt's generalization of Frobenius' Theorem. Some examples of explicit constructions are also given.
Partitions, irreducible characters, and inequalities for generalized matrix functions
Thomas H.
Pate
875-894
Abstract: Given a partition $\alpha = \{ {\alpha _1},{\alpha _2}, \ldots ,{\alpha _s}\}$, ${\alpha _1} \geq {\alpha _2} \geq \cdots \geq {\alpha _s}$, of $n$ we let $ {X_\alpha }$ denote the derived irreducible character of ${S_n}$, and we associate with $\alpha$ a derived partition $\displaystyle \alpha\prime = \{ {\alpha _1} - 1,{\alpha _2} - 1, \ldots ,{\alpha _t} - 1,{\alpha _{t + 1}}, \ldots ,{\alpha _s},{1^t}\}$ where $ t$ denotes the smallest positive integer such that $ {\alpha _t} > {\alpha _{t + 1}}\;({\alpha _{s + 1}} = 0)$. We show that if $ Y$ is a decomposable $\mathbb{C}$-valued $n$-linear function on ${\mathbb{C}^m} \times {\mathbb{C}^m} \times \cdots \times {\mathbb{C}^m}$ ($n$-copies) then $\left\langle {{X_\alpha }Y,Y} \right\rangle \geq \left\langle {{X_\alpha },Y,Y} \right\rangle$. Translating into the notation of matrix theory we obtain an inequality involving the generalized matrix functions ${d_{{X_\alpha }}}$ and $ {d_{{X_{\alpha\prime}}}}$, namely that $\displaystyle {({X_\alpha }(e))^{ - 1}}{d_{{X_\alpha }}}(B) \geq {({X_{\alpha\prime}}(e))^{ - 1}}{d_{{X_{\alpha\prime}}}}(B)$ for each $ n \times n$ positive semidefinite Hermitian matrix $B$. This result generalizes a classical result of I. Schur and includes many other known inequalities as special cases.
A Picard theorem with an application to minimal surfaces. II
Peter
Hall
895-902
Abstract: Let $ f:{\mathbf{C}} \to {{\mathbf{R}}^n}$ be a parabolic minimal surface such that the normals to $f$ omit $n + k$ directions in general position, $k \geq 0$. We obtain sharp bounds on the dimension of the affine span of $ f$ and of the linear span of the Gauss map of $f$.
Covering moves
R.
Piergallini
903-920
Abstract: In this paper we give a positive answer to a long standing question posed by Montesinos, by introducing new covering moves, in order to relate any two colored diagrams representing the same $3$-manifold as simple branched $3$-covering of ${S^3}$.