Minimal sets and varieties
Keith
A.
Kearnes;
Emil
W.
Kiss;
Matthew
A.
Valeriote
1-41
Abstract: The aim of this paper is twofold. First some machinery is established to reveal the structure of abelian congruences. Then we describe all minimal, locally finite, locally solvable varieties. For locally solvable varieties, this solves problems 9 and 10 of Hobby and McKenzie. We generalize part of this result by proving that all locally finite varieties generated by nilpotent algebras that have a trivial locally strongly solvable subvariety are congruence permutable.
Decomposition theorems and approximation by a ``floating" system of exponentials
E.
S.
Belinskii
43-53
Abstract: The main problem considered in this paper is the approximation of a trigonometric polynomial by a trigonometric polynomial with a prescribed number of harmonics. The method proposed here gives an opportunity to consider approximation in different spaces, among them the space of continuous functions, the space of functions with uniformly convergent Fourier series, and the space of continuous analytic functions. Applications are given to approximation of the Sobolev classes by trigonometric polynomials with prescribed number of harmonics, and to the widths of the Sobolev classes. This work supplements investigations by Maiorov, Makovoz and the author where similar results were given in the integral metric.
Rumely's local global principle for algebraic ${\mathrm P}{\mathcal S}{\mathrm C}$ fields over rings
Moshe
Jarden;
Aharon
Razon
55-85
Abstract: Let $\mathcal{S}$ be a finite set of rational primes. We denote the maximal Galois extension of $\mathbb{Q}$ in which all $p\in \mathcal{S}$ totally decompose by $N$. We also denote the fixed field in $N$ of $e$ elements $\sigma _{1},\ldots , \sigma _{e}$ in the absolute Galois group $G( \mathbb{Q})$ of $\mathbb{Q}$ by $N( {\boldsymbol \sigma })$. We denote the ring of integers of a given algebraic extension $M$ of $\mathbb{Q}$ by $\mathbb{Z}_{M}$. We also denote the set of all valuations of $M$ (resp., which lie over $S$) by $\mathcal{V}_{M}$ (resp., $\mathcal{S}_{M}$). If $v\in \mathcal{V}_{M}$, then $O_{M,v}$ denotes the ring of integers of a Henselization of $M$ with respect to $v$. We prove that for almost all ${\boldsymbol \sigma }\in G( \mathbb{Q})^{e}$, the field $M=N( {\boldsymbol \sigma })$ satisfies the following local global principle: Let $V$ be an affine absolutely irreducible variety defined over $M$. Suppose that $V(O_{M,v})\not =\varnothing$ for each $v\in \mathcal{V}_{M}\backslash \mathcal{S}_{M}$ and $V_{\mathrm{sim}}(O_{M,v})\not =\varnothing$ for each $v\in \mathcal{S}_{M}$. Then $V(O_{M})\not =\varnothing$. We also prove two approximation theorems for $M$.
Realizing homology boundary links with arbitrary patterns
Paul
Bellis
87-100
Abstract: Homology boundary links have become an increasingly important class of links, largely due to their significance in the ongoing concordance classification of links. Tim Cochran and Jerome Levine defined an algebraic object called a pattern associated to a homology boundary link which can be used to study the deviance of a homology boundary link from being a boundary link. Since a pattern is a set of $m$ elements which normally generates the free group of rank $m$, any invariants which detect non-trivial patterns can be applied to the purely algebraic question of when such a set is a set of conjugates of a generating set for the free group. We will give a constructive geometric proof that all patterns are realized by some homology boundary link $L^n$ in $S^{n+2}$. We shall also prove an analogous existence theorem for calibrations of $\mathbb{E}$-links, a more general and less understood class of links tha homology boundary links.
Weighted ergodic theorems for mean ergodic $L_1$-contractions
Dogan
Çömez;
Michael
Lin;
James
Olsen
101-117
Abstract: It is shown that any bounded weight sequence which is good for all probability preserving transformations (a universally good weight) is also a good weight for any $L_{1}$-contraction with mean ergodic (ME) modulus, and for any positive contraction of $L_{p}$ with $1 < p <\infty$. We extend the return times theorem by proving that if $S$ is a Dunford-Schwartz operator (not necessarily positive) on a Lebesgue space, then for any $g$ bounded measurable $\{S^{n} g(\omega )\}$ is a universally good weight for a.e. $\omega .$ We prove that if a bounded sequence has "Fourier coefficents", then its weighted averages for any $L_{1}$-contraction with mean ergodic modulus converge in $L_{1}$-norm. In order to produce weights, good for weighted ergodic theorems for $L_{1}$-contractions with quasi-ME modulus (i.e., so that the modulus has a positive fixed point supported on its conservative part), we show that the modulus of the tensor product of $L_{1}$-contractions is the product of their moduli, and that the tensor product of positive quasi-ME $L_{1}$-contractions is quasi-ME.
A condition on the value function both necessary and sufficient for full regularity of minimizers of one-dimensional variational problems
M.
A.
Sychev;
V.
J.
Mizel
119-133
Abstract: We study two-point Lagrange problems for integrands $L= L(t,u,v)$: \begin{equation}\begin{split} F[u]=\int _a^b L(t,u(t),\dot u(t))&\,dt \to \inf, & u\in\mathcal A=\{v\in W^{1,1} ([a,b];\mathbb R^n)|v(a)=A,v(b)=B\}. \end{split}\tag{P}\label{tagp} \end{equation} Under very weak regularity hypotheses [$L$ is Hölder continuous and locally elliptic on each compact subset of $\mathbb R\times\mathbb R^n\times\mathbb R^n$] we obtain, when $L$ is of superlinear growth in $v$, a characterization of problems in which the minimizers of (P) are $C^1$-regular for all boundary data. This characterization involves the behavior of the value function $S$: $\mathbb R\times\mathbb R^n\times\mathbb R\times\mathbb R^n\to\mathbb R$ defined by $S(a,A,b,B)=\inf _{\mathcal A} F$. Namely, all minimizers for (P) are $C^1$-regular in neighborhoods of $a$ and $b$ if and only if $S$ is Lipschitz continuous at $(a,A,b,B)$. Consequently problems (P) possessing no singular minimizers are characterized in cases where not even a weak form of the Euler-Lagrange equations is available for guidance. Full regularity results for problems where $L$ is nearly autonomous, nearly independent of $u$, or jointly convex in $(u,v)$ are presented.
Divisor spaces on punctured Riemann surfaces
Sadok
Kallel
135-164
Abstract: In this paper, we study the topology of spaces of $n$-tuples of positive divisors on (punctured) Riemann surfaces which have no points in common (the divisor spaces). These spaces arise in connection with spaces of based holomorphic maps from Riemann surfaces to complex projective spaces. We find that there are Eilenberg-Moore type spectral sequences converging to their homology. These spectral sequences collapse at the $E^2$ term, and we essentially obtain complete homology calculations. We recover for instance results of F. Cohen, R. Cohen, B. Mann and J. Milgram, The topology of rational functions and divisors of surfaces, Acta Math. 166 (1991), 163-221. We also study the homotopy type of certain mapping spaces obtained as a suitable direct limit of the divisor spaces. These mapping spaces, first considered by G. Segal, were studied in a special case by F. Cohen, R. Cohen, B. Mann and J. Milgram, who conjectured that they split. In this paper, we show that the splitting does occur provided we invert the prime two.
Factorisation in nest algebras. II
M.
Anoussis;
E.
G.
Katsoulis
165-183
Abstract: The main result of this paper is Theorem 5, which provides a necessary and sufficient condition on a positive operator $A$ for the existence of an operator $B$ in the nest algebra $AlgN$ of a nest $N$ satisfying $A=BB^{*}$ (resp. $A=B^{*}B)$. In Section 3 we give a new proof of a result of Power concerning outer factorisation of operators. We also show that a positive operator $A$ has the property that there exists for every nest $N$ an operator $B_N$ in $AlgN$ satisfying $A=B_NB_N^{*}$ (resp. $A=B_N^{*}B_N$) if and only if $A$ is a Fredholm operator. In Section 4 we show that for a given operator $A$ in $B(H)$ there exists an operator $B$ in $AlgN$ satisfying $AA^{*}=BB^{*}$ if and only if the range $r(A)$ of $A$ is equal to the range of some operator in $AlgN$. We also determine the algebraic structure of the set of ranges of operators in $AlgN$. Let $F_r(N)$ be the set of positive operators $A$ for which there exists an operator $B$ in $AlgN$ satisfying $A=BB^{*}$. In Section 5 we obtain information about this set. In particular we discuss the following question: Assume $A$ and $B$ are positive operators such that $A\leq B$ and $A$ belongs to $F_r(N)$. Which further conditions permit us to conclude that $B$ belongs to $F_r(N)$?
New subfactors from braid group representations
Juliana
Erlijman
185-211
Abstract: This paper is about the construction of new examples of pairs of subfactors of the hyperfinite II$_{1}$ factor, and the computation of their indices and relative commutants. The construction is done in general by considering unitary braid representations with certain properties that are satisfied in natural examples. We compute the indices explicitly for the particular cases in which the braid representations are obtained in connection with representation theory of Lie algebras of types A,B,C,D.
Filling by holomorphic curves in symplectic 4-manifolds
Rugang
Ye
213-250
Abstract: We develop a general framework for embedded (immersed) $J$-holomorphic curves and a systematic treatment of the theory of filling by holomorphic curves in 4-dimensional symplectic manifolds. In particular, a deformation theory and an intersection theory for $J$-holomorphic curves with boundary are developed. Bishop's local filling theorem is extended to almost complex manifolds. Existence and uniqueness of global fillings are given complete proofs. Then they are extended to the situation with nontrivial $J$-holomorphic spheres, culminating in the construction of singular fillings.
Geometry of families of nodal curves on the blown-up projective plane
Gert-Martin
Greuel;
Christoph
Lossen;
Eugenii
Shustin
251-274
Abstract: Let $\mathbb P^2_r\,$ be the projective plane blown up at $r$ generic points. Denote by $E_0,E_1,\ldots ,E_r$ the strict transform of a generic straight line on $\mathbb P^2$ and the exceptional divisors of the blown-up points on $\mathbb P^2_r$ respectively. We consider the variety $V_{irr}(d;\,d_1,\ldots,d_r;\,k)$ of all irreducible curves $C$ in $|dE_0-\sum _{i=1}^{r} d_iE_i|$ with $k$ nodes as the only singularities and give asymptotically nearly optimal sufficient conditions for its smoothness, irreducibility and non-emptiness. Moreover, we extend our conditions for the smoothness and the irreducibility to families of reducible curves. For $r\leq 9$ we give the complete answer concerning the existence of nodal curves in $V_{irr}(d;\,d_1,\ldots,d_r;\,k)$.
On the classification of irregular surfaces of general type with nonbirational bicanonical map
Fabrizio
Catanese;
Ciro
Ciliberto;
Margarida
Mendes
Lopes
275-308
Abstract: The present paper is devoted to the classification of irregular surfaces of general type with $p_{g}\geq 3$ and nonbirational bicanonical map. Our main result is that, if $S$ is such a surface and if $S$ is minimal with no pencil of curves of genus $2$, then $S$ is the symmetric product of a curve of genus $3$, and therefore $p_{g}=q=3$ and $K^{2}=6$. Furthermore we obtain some results towards the classification of minimal surfaces with $p_{g}=q=3$. Such surfaces have $6\leq K^{2}\leq 9$, and we show that $K^{2}=6$ if and only if $S$ is the symmetric product of a curve of genus $3$. We also classify the minimal surfaces with $p_{g}=q=3$ with a pencil of curves of genus $2$, proving in particular that for those one has $K^{2}=8$.
Wandering vectors for irrational rotation unitary systems
Deguang
Han
309-320
Abstract: An abstract characterization for those irrational rotation unitary systems with complete wandering subspaces is given. We prove that an irrational rotation unitary system has a complete wandering vector if and only if the von Neumann algebra generated by the unitary system is finite and shares a cyclic vector with its commutant. We solve a factorization problem of Dai and Larson negatively for wandering vector multipliers, and strengthen this by showing that for an irrational rotation unitary system $\mathcal{U}$, every unitary operator in $w^{*}(\mathcal{U})$ is a wandering vector multiplier. Moreover, we show that there is a class of wandering vector multipliers, induced in a natural way by pairs of characters of the integer group $\mathbb{Z}$, which fail to factor even as the product of a unitary in $\mathcal{U}'$ and a unitary in $w^{*}(\mathcal{U})$. Incomplete maximal wandering subspaces are also considered, and some questions are raised.
Widths of Subgroups
Rita
Gitik;
Mahan
Mitra;
Eliyahu
Rips;
Michah
Sageev
321-329
Abstract: We say that the width of an infinite subgroup $H$ in $G$ is $n$ if there exists a collection of $n$ essentially distinct conjugates of $H$ such that the intersection of any two elements of the collection is infinite and $n$ is maximal possible. We define the width of a finite subgroup to be $0$. We prove that a quasiconvex subgroup of a negatively curved group has finite width. It follows that geometrically finite surfaces in closed hyperbolic $3$-manifolds satisfy the $k$-plane property for some $k$.
Sur la multiplicité de la première valeur propre de l'opérateur de Schrödinger avec champ magnétique sur la sphère
Gérard
Besson;
Bruno
Colbois;
Gilles
Courtois
331-345
Abstract: L'objet de cet article est d'étudier la multiplicité de la première valeur propre de l'opérateur de Schrödinger avec champ magnétique sur la sphère $S^{2}$, et, répondant en cela à une question posée par Y. Colin de Verdière, de montrer d'une part que cette multiplicité peut être arbitrairement grande, mais que, d'autre part, elle est toujours bornée en fonction de la courbure de la connexion associée.ABSTRACT. The purpose of this text is to study the first eigenvalue of the Schrödinger operator with magnetic field on the 2-sphere and to show that its multiplicity can be arbitrarily high. We also show that this multiplicity is bounded in terms of the curvature of the corresponding connection. This answers a question asked by Y. Colin de Verdière.
On differential equations for Sobolev-type Laguerre polynomials
J.
Koekoek;
R.
Koekoek;
H.
Bavinck
347-393
Abstract: The Sobolev-type Laguerre polynomials $\{L_n^{\alpha,M,N}(x)\}_{n=0}^{\infty}$ are orthogonal with respect to the inner product \begin{displaymath}\langle f,g\rangle\;=\frac{1}{\Gamma(\alpha+1)}\int _0^{\infty}x^{\alpha}e^{-x}f(x)g(x)dx+Mf(0)g(0)+ Nf'(0)g'(0),\end{displaymath} where $\alpha>-1$, $M\ge 0$ and $N\ge 0$. In 1990 the first and second author showed that in the case $M>0$ and $N=0$ the polynomials are eigenfunctions of a unique differential operator of the form \begin{displaymath}M\sum _{i=1}^{\infty}a_i(x)D^i+xD^2+(\alpha+1-x)D,\end{displaymath} where $\left\{a_i(x)\right\}_{i=1}^{\infty}$ are independent of $n$. This differential operator is of order $2\alpha+4$ if $\alpha$ is a nonnegative integer, and of infinite order otherwise. In this paper we construct all differential equations of the form \begin{align}&M\sum _{i=0}^{\infty}a_i(x)y^{(i)}(x)+ N\sum _{i=0}^{\infty}b_i(x)y^{(i)}(x)\nonumber &\hspace{1cm}{}+MN\sum _{i=0}^{\infty}c_i(x)y^{(i)}(x)+ xy''(x)+(\alpha +1-x)y'(x)+ny(x)=0,\nonumber \end{align} where the coefficients $\left\{a_i(x)\right\}_{i=1}^{\infty}$, $\left\{b_i(x)\right\}_{i=1}^{\infty}$ and $\left\{c_i(x)\right\}_{i=1}^{\infty}$ are independent of $n$ and the coefficients $a_0(x)$, $b_0(x)$ and $c_0(x)$ are independent of $x$, satisfied by the Sobolev-type Laguerre polynomials $\{L_n^{\alpha,M,N}(x)\}_{n=0}^{\infty}$. Further, we show that in the case $M=0$ and $N>0$ the polynomials are eigenfunctions of a linear differential operator, which is of order $2\alpha+8$ if $\alpha$ is a nonnegative integer and of infinite order otherwise. Finally, we show that in the case $M>0$ and $N>0$ the polynomials are eigenfunctions of a linear differential operator, which is of order $4\alpha+10$ if $\alpha$ is a nonnegative integer and of infinite order otherwise.
Double Walsh series with coefficients of bounded variation of higher order
Chang-Pao
Chen;
Ching-Tang
Wu
395-417
Abstract: Let $D_{j}^{k}(x)$ denote the Cesàro sums of order $k$ of the Walsh functions. The estimates of $D_{j}^{k}(x)$ given by Fine back in 1949 are extended to the case $k>2$. As a corollary, the following properties are established for the rectangular partial sums of those double Walsh series whose coefficients satisfy conditions of bounded variation of order $(p,0), (0,p)$, and $(p,p)$ for some $p\ge 1$: (a) regular convergence; (b) uniform convergence; (c) $L^{r}$-integrability and $L^{r}$-metric convergence for $0<r<1/p$; and (d) Parseval's formula. Extensions to those with coefficients of generalized bounded variation are also derived.
Local Boundary Regularity of the Szego Projection and Biholomorphic Mappings of Non-Pseudoconvex Domains
Peiming
Ma
419-428
Abstract: It is shown that the Szeg\H{o} projection $S$ of a smoothly bounded domain $\Omega$, not necessarily pseudoconvex, satisfies local regularity estimates at certain boundary points, provided that condition $R$ holds for $\Omega$. It is also shown that any biholomorphic mapping $f:\Omega \rightarrow D$ between smoothly bounded domains extends smoothly near such points, provided that a weak regularity assumption holds for $D$.