Accelerating the convergence of the method of alternating projections
Heinz
H.
Bauschke;
Frank
Deutsch;
Hein
Hundal;
Sung-Ho
Park
3433-3461
Abstract: The powerful von Neumann-Halperin method of alternating projections (MAP) is an algorithm for determining the best approximation to any given point in a Hilbert space from the intersection of a finite number of subspaces. It achieves this by reducing the problem to an iterative scheme which involves only computing best approximations from the individual subspaces which make up the intersection. The main practical drawback of this algorithm, at least for some applications, is that the method is slowly convergent. In this paper, we consider a general class of iterative methods which includes the MAP as a special case. For such methods, we study an ``accelerated'' version of this algorithm that was considered earlier by Gubin, Polyak, and Raik (1967) and by Gearhart and Koshy (1989). We show that the accelerated algorithm converges faster than the MAP in the case of two subspaces, but is, in general, not faster than the MAP for more than two subspaces! However, for a ``symmetric'' version of the MAP, the accelerated algorithm always converges faster for any number of subspaces. Our proof seems to require the use of the Spectral Theorem for selfadjoint mappings.
Anderson's double complex and gamma monomials for rational function fields
Sunghan
Bae;
Ernst-Ulrich
Gekeler;
Pyung-Lyun
Kang;
Linsheng
Yin
3463-3474
Abstract: We investigate algebraic $\Gamma$-monomials of Thakur's positive characteristic $\Gamma$-function, by using Anderson and Das' double complex method of computing the sign cohomology of the universal ordinary distribution. We prove that the $\Gamma$-monomial associated to an element of the second sign cohomology of the universal ordinary distribution of $\mathbb{F} _{q}(T)$generates a Kummer extension of some Carlitz cyclotomic function field, which is also a Galois extension of the base field $\mathbb{F} _{q}(T)$. These results are characteristic-$p$ analogues of those of Deligne on classical $\Gamma$-monomials, proofs of which were given by Das using the double complex method. In this paper, we also obtain some results on $e$-monomials of Carlitz's exponential function.
Remarks about uniform boundedness of rational points over function fields
Lucia
Caporaso
3475-3484
Abstract: We prove certain uniform versions of the Mordell Conjecture and of the Shafarevich Conjecture for curves over function fields and their rational points.
Irreducibility of equisingular families of curves
Thomas
Keilen
3485-3512
Abstract: In 1985 Joe Harris proved the long-standing claim of Severi that equisingular families of plane nodal curves are irreducible whenever they are nonempty. For families with more complicated singularities this is no longer true. Given a divisor $D$ on a smooth projective surface $\Sigma$ it thus makes sense to look for conditions which ensure that the family $V_{\vert D\vert}^{irr}\big(\mathcal{S}_1,\ldots,\mathcal{S}_r\big)$ of irreducible curves in the linear system $\vert D\vert _l$ with precisely $r$ singular points of types $\mathcal{S}_1,\ldots,\mathcal{S}_r$ is irreducible. Considering different surfaces, including general surfaces in $\mathbb P_{\mathbb C}^3$ and products of curves, we produce a sufficient condition of the type \begin{displaymath}\sum\limits_{i=1}^r\deg\big(X(\mathcal{S}_i)\big)^2 < \gamma\cdot (D- K_\Sigma)^2, \end{displaymath} where $\gamma$ is some constant and $X(\mathcal{S}_i)$ some zero-dimensional scheme associated to the singularity type. Our results carry the same asymptotics as the best known results in this direction in the plane case, even though the coefficient is worse. For most of the surfaces considered these are the only known results in that direction.
Planar convex bodies, Fourier transform, lattice points, and irregularities of distribution
L.
Brandolini;
A.
Iosevich;
G.
Travaglini
3513-3535
Abstract: Let $B$ be a convex body in the plane. The purpose of this paper is a systematic study of the geometric properties of the boundary of $B$, and the consequences of these properties for the distribution of lattice points in rotated and translated copies of $\rho B$ ($\rho$ being a large positive number), irregularities of distribution, and the spherical average decay of the Fourier transform of the characteristic function of $B$. The analysis makes use of two notions of ``dimension'' of a convex set. The first notion is defined in terms of the number of sides required to approximate a convex set by a polygon up to a certain degree of accuracy. The second is the fractal dimension of the image of the Gauss map of $B$. The results stated in terms of these quantities are essentially sharp and lead to a nearly complete description of the problems in question.
A free boundary problem for a singular system of differential equations: An application to a model of tumor growth
Shangbin
Cui;
Avner
Friedman
3537-3590
Abstract: In this paper we consider a free boundary problem for a nonlinear system of two ordinary differential equations, one of which is singular at some points, including the initial point $r=0$. Because of the singularity at $r=0$, the initial value problem has a one-parameter family of solutions. We prove that there exists a unique solution to the free boundary problem. The proof of existence employs two ``shooting'' parameters. Analysis of the profiles of solutions of the initial value problem and tools such as comparison theorems and weak limits of solutions play an important role in the proof. The system considered here is motivated by a model in tumor growth, but the methods developed should be applicable to more general systems.
Sharp Fourier type and cotype with respect to compact semisimple Lie groups
José
García-Cuerva;
José
Manuel
Marco;
Javier
Parcet
3591-3609
Abstract: Sharp Fourier type and cotype of Lebesgue spaces and Schatten classes with respect to an arbitrary compact semisimple Lie group are investigated. In the process, a local variant of the Hausdorff-Young inequality on such groups is given.
Left-determined model categories and universal homotopy theories
J.
Rosicky;
W.
Tholen
3611-3623
Abstract: We say that a model category is left-determined if the weak equivalences are generated (in a sense specified below) by the cofibrations. While the model category of simplicial sets is not left-determined, we show that its non-oriented variant, the category of symmetric simplicial sets (in the sense of Lawvere and Grandis) carries a natural left-determined model category structure. This is used to give another and, as we believe simpler, proof of a recent result of D. Dugger about universal homotopy theories.
The combinatorial rigidity conjecture is false for cubic polynomials
Christian
Henriksen
3625-3639
Abstract: We show that there exist two cubic polynomials with connected Julia sets which are combinatorially equivalent but not topologically conjugate on their Julia sets. This disproves a conjecture by McMullen from 1995.
Zero entropy, non-integrable geodesic flows and a non-commutative rotation vector
Leo
T.
Butler
3641-3650
Abstract: Let $\mathfrak g$ be a $2$-step nilpotent Lie algebra; we say $\mathfrak g$ is non-integrable if, for a generic pair of points $\mathfrak g ={\mathrm {Lie}}(G)$ is non-integrable, $D < G$ is a cocompact subgroup, and ${\mathbf g}$ is a left-invariant Riemannian metric, then the geodesic flow of ${\mathbf g}$ on $T^*(D \backslash G)$ is neither Liouville nor non-commutatively integrable with $C^0$ first integrals. The proof uses a generalization of the rotation vector pioneered by Benardete and Mitchell.
Complete homogeneous varieties: Structure and classification
Carlos
Sancho de Salas
3651-3667
Abstract: Homogeneous varieties are those whose group of automorphisms acts transitively on them. In this paper we prove that any complete homogeneous variety splits in a unique way as a product of an abelian variety and a parabolic variety. This is obtained by proving a rigidity theorem for the parabolic subgroups of a linear group. Finally, using the results of Wenzel on the classification of parabolic subgroups of a linear group and the results of Demazure on the automorphisms of a flag variety, we obtain the classification of the parabolic varieties (in characteristic different from $2,3$). This, together with the moduli of abelian varieties, concludes the classification of the complete homogeneous varieties.
A path-transformation for random walks and the Robinson-Schensted correspondence
Neil
O'Connell
3669-3697
Abstract: The author and Marc Yor recently introduced a path-transformation $G^{(k)}$ with the property that, for $X$ belonging to a certain class of random walks on $\mathbb{Z}_+^k$, the transformed walk $G^{(k)}(X)$has the same law as the original walk conditioned never to exit the Weyl chamber $\{x: x_1\le\cdots\le x_k\}$. In this paper, we show that $G^{(k)}$ is closely related to the Robinson-Schensted algorithm, and use this connection to give a new proof of the above representation theorem. The new proof is valid for a larger class of random walks and yields additional information about the joint law of $X$ and $G^{(k)}(X)$. The corresponding results for the Brownian model are recovered by Donsker's theorem. These are connected with Hermitian Brownian motion and the Gaussian Unitary Ensemble of random matrix theory. The connection we make between the path-transformation $G^{(k)}$ and the Robinson-Schensted algorithm also provides a new formula and interpretation for the latter. This can be used to study properties of the Robinson-Schensted algorithm and, moreover, extends easily to a continuous setting.
On the Iwasawa $\lambda$-invariants of real abelian fields
Takae
Tsuji
3699-3714
Abstract: For a prime number $p$ and a number field $k$, let $A_\infty$ denote the projective limit of the $p$-parts of the ideal class groups of the intermediate fields of the cyclotomic $\mathbb{Z} _p$-extension over $k$. It is conjectured that $A_\infty$ is finite if $k$ is totally real. When $p$ is an odd prime and $k$ is a real abelian field, we give a criterion for the conjecture, which is a generalization of results of Ichimura and Sumida. Furthermore, in a special case where $p$ divides the degree of $k$, we also obtain a rather simple criterion.
Pseudo-holomorphic curves in complex Grassmann manifolds
Xiaoxiang
Jiao;
Jiagui
Peng
3715-3726
Abstract: It is proved that the Kähler angle of the pseudo-holomorphic sphere of constant curvature in complex Grassmannians is constant. At the same time we also prove several pinching theorems for the curvature and the Kähler angle of the pseudo-holomorphic spheres in complex Grassmannians with non-degenerate associated harmonic sequence.
The periodic Euler-Bernoulli equation
Vassilis
G.
Papanicolaou
3727-3759
Abstract: We continue the study of the Floquet (spectral) theory of the beam equation, namely the fourth-order eigenvalue problem \begin{displaymath}\left[ a(x)u^{\prime \prime }(x)\right] ^{\prime \prime }=\lambda \rho (x)u(x),\qquad -\infty <x<\infty , \end{displaymath} where the functions $a$ and $\rho$ are periodic and strictly positive. This equation models the transverse vibrations of a thin straight (periodic) beam whose physical characteristics are described by $a$ and $\rho$. Here we develop a theory analogous to the theory of the Hill operator $-(d/dx)^2+q(x)$. We first review some facts and notions from our previous works, including the concept of the pseudospectrum, or $\psi$-spectrum. Our new analysis begins with a detailed study of the zeros of the function $F(\lambda ;k)$, for any given ``quasimomentum'' $k\in \mathbb{C}$, where $F(\lambda ;k)=0$ is the Floquet-Bloch variety of the beam equation (the Hill quantity corresponding to $F(\lambda ;k)$ is $\Delta (\lambda )-2\cos (kb)$, where $\Delta (\lambda )$ is the discriminant and $b$ the period of $q$). We show that the multiplicity $m(\lambda ^{\ast })$ of any zero $\lambda ^{\ast }$ of $F(\lambda ;k)$ can be one or two and $m(\lambda ^{\ast })=2$ (for some $k$) if and only if $\lambda ^{\ast }$ is also a zero of another entire function $D(\lambda )$, independent of $k$. Furthermore, we show that $D(\lambda )$ has exactly one zero in each gap of the spectrum and two zeros (counting multiplicities) in each $\psi$-gap. If $\lambda ^{\ast }$ is a double zero of $F(\lambda ;k)$, it may happen that there is only one Floquet solution with quasimomentum $k$; thus, there are exceptional cases where the algebraic and geometric multiplicities do not agree. Next we show that if $(\alpha ,\beta )$ is an open $\psi$-gap of the pseudospectrum (i.e., $\alpha <\beta$), then the Floquet matrix $T(\lambda )$has a specific Jordan anomaly at $\lambda =\alpha$ and $\lambda =\beta$. We then introduce a multipoint (Dirichlet-type) eigenvalue problem which is the analogue of the Dirichlet problem for the Hill equation. We denote by $\{\mu _n\}_{n\in \mathbb{Z}}$ the eigenvalues of this multipoint problem and show that $\{\mu _n\}_{n\in \mathbb{Z}}$ is also characterized as the set of values of $\lambda$ for which there is a proper Floquet solution $f(x;\lambda )$ such that $f(0;\lambda )=0$. We also show (Theorem 7) that each gap of the $L^{2}(\mathbb{R})$-spectrum contains exactly one $\mu _{n}$ and each $\psi$-gap of the pseudospectrum contains exactly two $\mu _{n}$'s, counting multiplicities. Here when we say ``gap'' or ``$\psi$-gap'' we also include the endpoints (so that when two consecutive bands or $\psi$-bands touch, the in-between collapsed gap, or $\psi$-gap, is a point). We believe that $\{\mu _{n}\}_{n\in \mathbb{Z}}$ can be used to formulate the associated inverse spectral problem. As an application of Theorem 7, we show that if $\nu ^{*}$ is a collapsed (``closed'') $\psi$-gap, then the Floquet matrix $T(\nu ^{*})$ is diagonalizable. Some of the above results were conjectured in our previous works. However, our conjecture that if all the $\psi$-gaps are closed, then the beam operator is the square of a second-order (Hill-type) operator, is still open.
Singularities of the hypergeometric system associated with a monomial curve
Francisco
Jesús
Castro-Jiménez;
Nobuki
Takayama
3761-3775
Abstract: We compute, using $\mathcal{D}$-module restrictions, the slopes of the irregular hypergeometric system associated with a monomial curve. We also study rational solutions and reducibility of such systems.
Asymptotics for logical limit laws: When the growth of the components is in an RT class
Jason
P.
Bell;
Stanley
N.
Burris
3777-3794
Abstract: Compton's method of proving monadic second-order limit laws is based on analyzing the generating function of a class of finite structures. For applications of his deeper results we previously relied on asymptotics obtained using Cauchy's integral formula. In this paper we develop elementary techniques, based on a Tauberian theorem of Schur, that significantly extend the classes of structures for which we know that Compton's theory can be applied.
Combinatorics of rooted trees and Hopf algebras
Michael
E.
Hoffman
3795-3811
Abstract: We begin by considering the graded vector space with a basis consisting of rooted trees, with grading given by the count of non-root vertices. We define two linear operators on this vector space, the growth and pruning operators, which respectively raise and lower grading; their commutator is the operator that multiplies a rooted tree by its number of vertices, and each operator naturally associates a multiplicity to each pair of rooted trees. By using symmetry groups of trees we define an inner product with respect to which the growth and pruning operators are adjoint, and obtain several results about the associated multiplicities. Now the symmetric algebra on the vector space of rooted trees (after a degree shift) can be endowed with a coproduct to make a Hopf algebra; this was defined by Kreimer in connection with renormalization. We extend the growth and pruning operators, as well as the inner product mentioned above, to Kreimer's Hopf algebra. On the other hand, the vector space of rooted trees itself can be given a noncommutative multiplication: with an appropriate coproduct, this leads to the Hopf algebra of Grossman and Larson. We show that the inner product on rooted trees leads to an isomorphism of the Grossman-Larson Hopf algebra with the graded dual of Kreimer's Hopf algebra, correcting an earlier result of Panaite.
Connections with prescribed first Pontrjagin form
Mahuya
Datta
3813-3824
Abstract: Let $P$ be a principal $O(n)$ bundle over a $C^\infty$manifold $M$ of dimension $m$. If $n\geq 5m+4+4\binom{m+1}{4}$, then we prove that every differential 4-form representing the first Pontrjagin class of $P$ is the Pontrjagin form of some connection on $P$.
Self-intersection class for singularities and its application to fold maps
Toru
Ohmoto;
Osamu
Saeki;
Kazuhiro
Sakuma
3825-3838
Abstract: Let $f :M \to N$ be a generic smooth map with corank one singularities between manifolds, and let $S(f)$ be the singular point set of $f$. We define the self-intersection class $I(S(f)) \in H^*(M; \mathbf{Z})$ of $S(f)$using an incident class introduced by Rimányi but with twisted coefficients, and give a formula for $I(S(f))$ in terms of characteristic classes of the manifolds. We then apply the formula to the existence problem of fold maps.
Erratum to ``Arens regularity of the algebra $A \hat{\otimes} B$''
A.
Ülger
3839-3839
Erratum to ``Spherical classes and the algebraic transfer''
Nguyên
H. V.
Hung
3841-3842