Witten-Helffer-Sjöstrand theory for $S^1$-equivariant cohomology
Hon-kit
Wai
2141-2182
Abstract: Given an $S^1$-invariant Morse function $f$ and an $S^1$-invariant Riemannian metric $g$, a family of finite dimensional subcomplexes $(\widetilde \Omega^*_{inv,sm}(M,t), D(t))$, $t\in [0,\infty)$, of the Witten deformation of the $S^1$-equivariant de Rham complex is constructed, by studying the asymptotic behavior of the spectrum of the corresponding Laplacian $\widetilde \Delta ^k(t)=D^*_k(t)D_k(t)+D_{k-1}(t)D^*_{k-1}(t)$ as $t\to \infty$. In fact the spectrum of $\widetilde \Delta^k(t)$ can be separated into the small eigenvalues, finite eigenvalues and the large eigenvalues. Then one obtains $( \widetilde \Omega^*_{inv,sm}(M,t),D(t))$ as the complex of eigenforms corresponding to the small eigenvalues of $\widetilde \Delta(t)$. This permits us to verify the $S^1$-equivariant Morse inequalities. Moreover suppose $f$ is self-indexing and $(f,g)$ satisfies the Morse-Smale condition, then it is shown that this family of subcomplexes converges as $t\to \infty$ to a geometric complex which is induced by $(f,g)$ and calculates the $S^1$-equivariant cohomology of $M$.
Compact Composition Operators on BMOA
P.
S.
Bourdon;
J.
A.
Cima;
A.
L.
Matheson
2183-2196
Abstract: We characterize the compact composition operators on BMOA, the space consisting of those holomorphic functions on the open unit disk $U$ that are Poisson integrals of functions on $\partial U$, that have bounded mean oscillation. We then use our characterization to show that compactness of a composition operator on BMOA implies its compactness on the Hardy spaces (a simple example shows the converse does not hold). We also explore how compactness of the composition operator $C_\phi: \operatorname{BMOA}\rightarrow\operatorname{BMOA}$ relates to the shape of $\phi(U)$ near $\partial U$, introducing the notion of mean order of contact. Finally, we discuss the relationships among compactness conditions for composition operators on BMOA, VMOA, and the big and little Bloch spaces.
Towards a Halphen theory of linear series on curves
L.
Chiantini;
C.
Ciliberto
2197-2212
Abstract: A linear series $g^{N}_{\delta }$ on a curve $C\subset \mathbf{P}^{3}$ is primary when it does not contain the series cut by planes. For such series, we provide a lower bound for the degree $\delta$, in terms of deg($C$), g($C$) and of the number $s=\min \{i:h^{0}\mathcal{I}_{C}(i)\neq 0\}$. Examples show that the bound is sharp. Extensions to the case of general linear series and to the case of curves in higher projective spaces are considered.
The diagonal subring and the Cohen-Macaulay property of a multigraded ring
Eero
Hyry
2213-2232
Abstract: Let $T$ be a multigraded ring defined over a local ring $(A,\mathfrak{m})$. This paper deals with the question how the Cohen-Macaulay property of $T$ is related to that of its diagonal subring $T^\Delta$. In the bigraded case we are able to give necessary and sufficient conditions for the Cohen-Macaulayness of $T$. If $I_1,\dotsc,I_r\subset A$ are ideals of positive height, we can then compare the Cohen-Macaulay property of the multi-Rees algebra $R_A(I_1,\dotsc,I_r)$ with the Cohen-Macaulay property of the usual Rees algebra $R_A(I_1\cdots I_r)$. We also obtain a bound for the joint reduction numbers of two $\mathfrak{m}$-primary ideals in the case the corresponding multi-Rees algebra is Cohen-Macaulay.
Derivatives of Wronskians with applications to families of special Weierstrass points
Letterio
Gatto;
Fabrizio
Ponza
2233-2255
Abstract: Let $\pi :\mathfrak{X}\longrightarrow S$ be a flat proper family of smooth connected projective curves parametrized by some smooth scheme of finite type over $\mathbb{C}$. On every such a family, suitable derivatives along the fibers" (in the sense of Lax) of the relative wronskian, as defined by Laksov and Thorup, are constructed. They are sections of suitable jets extensions of the $g(g+1)/2$-th tensor power of the relative canonical bundle of the family itself. The geometrical meaning of such sections is discussed: the zero schemes of the $(k-1)$-th derivative ($k\geq 1$) of a relative wronskian correspond to families of Weierstrass Points (WP's) having weight at least $k$. The locus in $M_{g}$, the coarse moduli space of smooth projective curves of genus $g$, of curves possessing a WP of weight at least $k$, is denoted by $wt(k)$. The fact that $wt(2)$ has the expected dimension for all $g\geq 2$ was implicitly known in the literature. The main result of this paper hence consists in showing that $wt(3)$ has the expected dimension for all $g\geq 4$. As an application we compute the codimension $2$ Chow ($Q$-)class of $wt(3)$ for all $g\geq 4$, the main ingredient being the definition of the $k$-th derivative of a relative wronskian, which is the crucial tool which the paper is built on. In the concluding remarks we show how this result may be used to get relations among some codimension $2$ Chow ($Q$-)classes in $M_{4}$ ($g\geq 4$), corresponding to varieties of curves having a point $P$ with a suitable prescribed Weierstrass Gap Sequence, relating to previous work of Lax.
A Combinatorial Proof of Bass's Evaluations of the Ihara-Selberg Zeta Function for Graphs
Dominique
Foata;
Doron
Zeilberger
2257-2274
Abstract: We derive combinatorial proofs of the main two evaluations of the Ihara-Selberg zeta function associated with a graph. We give three proofs of the first evaluation all based on the algebra of Lyndon words. In the third proof it is shown that the first evaluation is an immediate consequence of Amitsur's identity on the characteristic polynomial of a sum of matrices. The second evaluation of the Ihara-Selberg zeta function is first derived by means of a sign-changing involution technique. Our second approach makes use of a short matrix-algebra argument.
Dehn surgery on arborescent links
Ying-Qing
Wu
2275-2294
Abstract: This paper studies Dehn surgery on a large class of links, called arborescent links. It will be shown that if an arborescent link $L$ is sufficiently complicated, in the sense that it is composed of at least $4$ rational tangles $T(p_{i}/q_{i})$ with all $q_{i} > 2$, and none of its length 2 tangles are of the form $T(1/2q_{1}, 1/2q_{2})$, then all complete surgeries on $L$ produce Haken manifolds. The proof needs some result on surgery on knots in tangle spaces. Let $T(r/2s, p/2q) = (B, t_{1}\cup t_{2}\cup K)$ be a tangle with $K$ a closed circle, and let $M = B - \operatorname{Int} N(t_{1}\cup t_{2})$. We will show that if $s>1$ and $p \not \equiv \pm 1$ mod $2q$, then $\partial M$ remains incompressible after all nontrivial surgeries on $K$. Two bridge links are a subclass of arborescent links. For such a link $L(p/q)$, most Dehn surgeries on it are non-Haken. However, it will be shown that all complete surgeries yield manifolds containing essential laminations, unless $p/q$ has a partial fraction decomposition of the form $1/(r-1/s)$, in which case it does admit non-laminar surgeries.
Farey polytopes and continued fractions associated with discrete hyperbolic groups
L.
Ya.
Vulakh
2295-2323
Abstract: The known definitions of Farey polytopes and continued fractions are generalized and applied to diophantine approximation in $n$-dimensional euclidean spaces. A generalized Remak-Rogers isolation theorem is proved and applied to show that certain Hurwitz constants for discrete groups acting in a hyperbolic space are isolated. The approximation constant for the imaginary quadratic field of discriminant $-15$ is found.
The iterated transfer analogue of the new doomsday conjecture
Norihiko
Minami
2325-2351
Abstract: A strong general restriction is given on the stable Hurewicz image of the classifying spaces of elementary abelian $p$-groups. In particular, this implies the iterated transfer analogue of the new doomsday conjecture.
Embeddings of open manifolds
Nancy
Cardim
2353-2373
Abstract: Let $TOP(M)$ be the simplicial group of homeomorphisms of $M$. The following theorems are proved. Theorem A. Let $M$ be a topological manifold of dim $\geq$ 5 with a finite number of tame ends $\varepsilon _{i}$, $1\leq i\leq k$. Let $TOP^{ep}(M)$ be the simplicial group of end preserving homeomorphisms of $M$. Let $W_{i}$ be a periodic neighborhood of each end in $M$, and let $p_{i}: W_{i} \to \mathbb{R}$ be manifold approximate fibrations. Then there exists a map $f: TOP^{ep}(M) \to \prod _{i} TOP^{ep}(W_{i})$ such that the homotopy fiber of $f$ is equivalent to $TOP_{cs}(M)$, the simplicial group of homeomorphisms of $M$ which have compact support. Theorem B. Let $M$ be a compact topological manifold of dim $\geq$ 5, with connected boundary $\partial M$, and denote the interior of $M$ by $Int M$. Let $f: TOP(M)\to TOP(Int M)$ be the restriction map and let $\mathcal{G}$ be the homotopy fiber of $f$ over $id_{Int M}$. Then $\pi _{i} \mathcal{G}$ is isomorphic to $\pi _{i} \mathcal{C} (\partial M)$ for $i > 0$, where $\mathcal{C} (\partial M)$ is the concordance space of $\partial M$. Theorem C. Let $q_{0}: W \to \mathbb{R}$ be a manifold approximate fibration with dim $W \geq$ 5. Then there exist maps $\alpha : \pi _{i} TOP^{ep}(W) \to \pi _{i} TOP(\hat W)$ and $\beta : \pi _{i} TOP(\hat W) \to \pi _{i} TOP^{ep}(W)$ for $i >1$, such that $\beta \circ \alpha \simeq id$, where $\hat W$ is a compact and connected manifold and $W$ is the infinite cyclic cover of $\hat W$.
The viscous Cahn-Hilliard equation: Morse decomposition and structure of the global attractor
M.
Grinfeld;
A.
Novick-Cohen
2375-2406
Abstract: In this paper we establish a Morse decomposition of the stationary solutions of the one-dimensional viscous Cahn-Hilliard equation by explicit energy calculations. Strong non-degeneracy of the stationary solutions is proven away from turning points and points of bifurcation from the homogeneous state and the dimension of the unstable manifold is calculated for all stationary states. In the unstable case, the flow on the global attractor is shown to be semi-conjugate to the flow on the global attractor of the Chaffee-Infante equation, and in the metastable case close to the nonlocal reaction-diffusion limit, a partial description of the structure of the global attractor is obtained by connection matrix arguments, employing a partial energy ordering and the existence of a weak lap number principle.
Multivariate matrix refinable functions with arbitrary matrix dilation
Qingtang
Jiang
2407-2438
Abstract: Characterizations of the stability and orthonormality of a multivariate matrix refinable function $\Phi$ with arbitrary matrix dilation $M$ are provided in terms of the eigenvalue and $1$-eigenvector properties of the restricted transition operator. Under mild conditions, it is shown that the approximation order of $\Phi$ is equivalent to the order of the vanishing moment conditions of the matrix refinement mask $\{\mathbf{P}_{\alpha}\}$. The restricted transition operator associated with the matrix refinement mask $\{\mathbf{P}_{\alpha}\}$ is represented by a finite matrix $({\mathcal A} _{Mi-j})_{i, j}$, with ${\mathcal A} _j=|\hbox{det$(M)$}|^{-1}\sum _{\kappa }\mathbf{P}_{\kappa -j}\otimes \mathbf{P}_{\kappa }$ and $\mathbf{P}_{\kappa -j}\otimes \mathbf{P}_{\kappa }$ being the Kronecker product of matrices $\mathbf{P}_{\kappa -j}$ and $\mathbf{P}_{\kappa }$. The spectral properties of the transition operator are studied. The Sobolev regularity estimate of a matrix refinable function $\Phi$ is given in terms of the spectral radius of the restricted transition operator to an invariant subspace. This estimate is analyzed in an example.
Summability of Fourier orthogonal series for Jacobi weight on a ball in $\mathbb{R}^d$
Yuan
Xu
2439-2458
Abstract: Fourier orthogonal series with respect to the weight function $(1-|\mathbf x|^{2})^{\mu - 1/2}$ on the unit ball in $\mathbb{R}^{d}$ are studied. Compact formulae for the sum of the product of orthonormal polynomials in several variables and for the reproducing kernel are derived and used to study the summability of the Fourier orthogonal series. The main result states that the expansion of a continuous function in the Fourier orthogonal series with respect to $(1-|\mathbf x|^{2})^{\mu -1/2}$ is uniformly $(C, \delta )$ summable on the ball if and only if $\delta > \mu + (d-1)/2$.
Hardy inequalities in Orlicz spaces
Andrea
Cianchi
2459-2478
Abstract: We establish a sharp extension, in the framework of Orlicz spaces, of the ($n$-dimensional) Hardy inequality, involving functions defined on a domain $G$, their gradients and the distance function from the boundary of $G$.
Embedded singular continuous spectrum for one-dimensional Schrödinger operators
Christian
Remling
2479-2497
Abstract: We investigate one-dimensional Schrödinger operators with sparse potentials (i.e. the potential consists of a sequence of bumps with rapidly growing barrier separations). These examples illuminate various phenomena related to embedded singular continuous spectrum.
Newton's method on the complex exponential function
Mako
E.
Haruta
2499-2513
Abstract: We show that when Newton's method is applied to the product of a polynomial and the exponential function in the complex plane, the basins of attraction of roots have finite area.
The ideal structure of some analytic crossed products
Miron
Shpigel
2515-2538
Abstract: We study the ideal structure of a class of some analytic crossed products. For an $r$-discrete, principal, minimal groupoid $G$, we consider the analytic crossed product $C^*(G,\sigma)\times _\alpha \mathbb{Z}_+$, where $\alpha$ is given by a cocycle $c$. We show that the maximal ideal space $\mathcal{M}$ of $C^*(G,\sigma)\times _\alpha \mathbb{Z}_+$ depends on the asymptotic range of $c$, $R_\infty(c)$; that is, $\mathcal{M}$ is homeomorphic to $\overline{\mathbb{D}}\mid R_\infty(c)$ for $R_\infty(c)$ finite, and $\cal M$ consists of the unique maximal ideal for $R_\infty(c)=\mathbb{T}$. We also prove that $C^*(G,\sigma)\times _\alpha \mathbb{Z}_+$ is semisimple in both cases, and that $R_\infty(c)$ is invariant under isometric isomorphism.
Representation Theory of Reductive Normal Algebraic Monoids
Stephen
Doty
2539-2551
Abstract: New results in the representation theory of ``semisimple'' algebraic monoids are obtained, based on Renner's monoid version of Chevalley's big cell. (The semisimple algebraic monoids have been classified by Renner.) The rational representations of such a monoid are the same thing as ``polynomial'' representations of the associated reductive group of units in the monoid, and this representation category splits into a direct sum of subcategories by ``homogeneous'' degree. We show that each of these homogeneous subcategories is a highest weight category, in the sense of Cline, Parshall, and Scott, and so equivalent with the module category of a certain finite-dimensional quasihereditary algebra, which we show is a generalized Schur algebra in S. Donkin's sense.
Thermal capacity estimates on the Allen-Cahn equation
Richard
B.
Sowers;
Jang-Mei
Wu
2553-2567
Abstract: We consider the Allen-Cahn equation in a well-known scaling regime which gives motion by mean curvature. A well-known transformation of this PDE, using its standing wave, yields a PDE the solution of which is approximately the distance function to an interface moving by mean curvature. We give bounds on this last fact in terms of thermal capacity. Our techniques hinge upon the analysis of a certain semimartingale associated with a certain PDE (the PDE for the approximate distance function) and an analogue of some results by Bañuelos and Øksendal relating lifetimes of diffusions to exterior capacities.