A probabilistic approach to positive harmonic functions in a slab with alternating Dirichlet and Neumann boundary conditions
Ross
G.
Pinsky
2445-2477
Abstract: Let $\Omega = R^{d}\times (-1,1)$, $d\ge 2$, be a $d+1$ dimensional slab. Denote points $z\in R^{d+1}$ by $z=(r,\theta ,y)$, where $(r,\theta )\in [0,\infty )\times S^{d-1}$ and $y\in R$. Denoting the boundary of the slab by $\Gamma =\partial \Omega$, let \begin{displaymath}\Gamma _{D}=\{z=(r,\theta ,y)\in \Gamma : r\in \bigcup _{n=1}^{\infty }(a_{n},b_{n})\},\end{displaymath} where $\{(a_{n},b_{n})\}_{n=1}^{\infty }$is an ordered sequence of intervals on the right half line (that is, $a_{n+1}>b_{n}$). Assume that the lengths of the intervals are bounded and that the spaces between consecutive intervals are bounded and bounded away from zero. Let $\Gamma _{N}=\Gamma -\bar \Gamma _{D}$. Let $C_{B}(\Omega ;\Gamma _{D}, \Gamma _{N})$ and $C_{P}(\Omega ; \Gamma _{D}, \Gamma _{N})$denote respectively the cone of bounded, positive harmonic functions in $\Omega$ and the cone of positive harmonic functions in $\Omega$ which satisfy the Dirichlet boundary condition on $\Gamma _{D}$ and the Neumann boundary condition on $\Gamma _{N}$. Letting $\rho _{n}\equiv b_{n}-a_{n}$, the main result of this paper, under a modest assumption on the sequence $\{\rho _{n}\}$, may be summarized as follows when $d\ge 3$: 1. If $\sum _{n=1}^{\infty }\frac{n}{\vert\log \rho _{n}\vert} <\infty$, then $\mathcal C_B(\Omega,\Gamma_D,\Gamma _N)$ and $\mathcal C_P(\Omega ,\Gamma _D,\Gamma _N)$ are both one-dimensional (as in the case of the Neumann boundary condition on the entire boundary). In particular, this occurs if $\rho _{n}=\exp (-n^{l})$ with $l>2$. 2. If $\sum _{n=1}^{\infty }\frac{n}{\vert\log \rho _{n}\vert} =\infty$and $\sum _{n=1}^{\infty }\frac{\vert\log \rho _{n}\vert^{\frac{1}{2}}}{n^{2}}=\infty$, then $\mathcal C_B(\Omega ,\Gamma _D,\Gamma _N) =\varnothing$ and $\mathcal C_P(\Omega ,\Gamma _D,\Gamma _N)$ is one-dimensional. In particular, this occurs if $\rho _{n}=\exp (-n^{2})$. 3. If $\sum _{n=1}^{\infty }\frac{\vert\log \rho _{n}\vert^{\frac{1}{2}}}{n^{2}}<\infty$, then $\mathcal C_B(\Omega ,\Gamma _D,\Gamma _N)=\varnothing$ and the set of minimal elements generating $\mathcal C_P(\Omega ,\Gamma _D,\Gamma _N)$ is isomorphic to $S^{d-1}$ (as in the case of the Dirichlet boundary condition on the entire boundary). In particular, this occurs if $\rho _{n}=\exp (-n^{l})$with $0\le l<2$. When $d=2$, $\mathcal C_B(\Omega ,\Gamma _D,\Gamma _N)=\varnothing$ as soon as there is at least one interval of Dirichlet boundary condition. The dichotomy for $\mathcal C_P(\Omega ,\Gamma _D,\Gamma _N)$ is as above.
Avoidable algebraic subsets of Euclidean space
James
H.
Schmerl
2479-2489
Abstract: Fix an integer $n\ge 1$ and consider real $n$-dimensional $\mathbb{R}^n$. A partition of $\mathbb{R}^n$ avoids the polynomial $p(x_0,x_1,\dotsc,x_{k-1})\in\mathbb R[x_0,x_1,\dotsc,x_{k-1}]$, where each $x_i$ is an $n$-tuple of variables, if there is no set of the partition which contains distinct $a_0,a_1,\dotsc,a_{k-1}$ such that $p(a_0,a_1,\dotsc,a_{k-1})=0$. The polynomial is avoidable if some countable partition avoids it. The avoidable polynomials are studied here. The polynomial $\|x-y\|^2-\|y-z\|^2$ is an especially interesting example of an avoidable one. We find (1) a countable partition which avoids every avoidable polynomial over $Q$, and (2) a characterization of the avoidable polynomials. An important feature is that both the ``master'' partition in (1) and the characterization in (2) depend on the cardinality of $\mathbb R$.
On Better-Quasi-Ordering Countable Series-Parallel Orders
Stéphan
Thomassé
2491-2505
Abstract: We prove that any infinite sequence of countable series-parallel orders contains an increasing (with respect to embedding) infinite subsequence. This result generalizes Laver's and Corominas' theorems concerning better-quasi-order of the classes of countable chains and trees.
On reflection of stationary sets in $\mathcal{P}_\kappa\lambda$
Thomas
Jech;
Saharon
Shelah
2507-2515
Abstract: Let $\kappa$ be an inaccessible cardinal, and let $E_{0} = \{x \in \mathcal{P}_{\kappa }\kappa ^{+} : \text{cf} \; \lambda _{x} = \text{cf} \; \kappa _{x}\}$ and $E_{1} = \{x \in \mathcal{P}_{\kappa }\kappa ^{+} : \kappa _{x}$ is regular and $\lambda _{x} = \kappa _{x}^{+}\}$. It is consistent that the set $E_{1}$ is stationary and that every stationary subset of $E_{0}$ reflects at almost every $a \in E_{1}$.
On Macaulayfication of Noetherian schemes
Takesi
Kawasaki
2517-2552
Abstract: The Macaulayfication of a Noetherian scheme $X$ is a birational proper morphism from a Cohen-Macaulay scheme to $X$. In 1978 Faltings gave a Macaulayfication of a quasi-projective scheme if its non-Cohen-Macaulay locus is of dimension $0$ or $1$. In the present article, we construct a Macaulayfication of Noetherian schemes without any assumption on the non-Cohen-Macaulay locus. Of course, a desingularization is a Macaulayfication and, in 1964, Hironaka already gave a desingularization of an algebraic variety over a field of characteristic $0$. Our method, however, to construct a Macaulayfication is independent of the characteristic.
Theta line bundles and the determinant of the Hodge bundle
Alexis
Kouvidakis
2553-2568
Abstract: We give an expression of the determinant of the push forward of a symmetric line bundle on a complex abelian fibration, in terms of the pull back of the relative dualizing sheaf via the zero section.
On syzygies of abelian varieties
Elena
Rubei
2569-2579
Abstract: In this paper we prove the following result: Let $X$ be a complex torus and $M$ a normally generated line bundle on $X$; then, for every $p \geq 0$, the line bundle $M^{p+1}$ satisfies Property $N_{p}$ of Green-Lazarsfeld.
Partitions into Primes
Yifan
Yang
2581-2600
Abstract: We investigate the asymptotic behavior of the partition function $p_{\Lambda} (n)$ defined by $\sum ^{\infty }_{n=0}p_{\Lambda} (n)x^{n} =\prod ^{\infty }_{m=1}(1-x^{m})^{-\Lambda (m)}$, where $\Lambda (n)$ denotes the von Mangoldt function. Improving a result of Richmond, we show that $\log p_{\Lambda} (n)=2\sqrt {\zeta (2)n}+O(\sqrt n\exp \{-c(\log n) (\log _{2} n)^{-2/3}(\log _{3} n)^{-1/3}\})$, where $c$ is a positive constant and $\log _{k}$ denotes the $k$ times iterated logarithm. We also show that the error term can be improved to $O(n^{1/4})$ if and only if the Riemann Hypothesis holds.
On Shimura, Shintani and Eichler-Zagier correspondences
M.
Manickam;
B.
Ramakrishnan
2601-2617
Abstract: In this paper, we set up Shimura and Shintani correspondences between Jacobi forms and modular forms of integral weight for arbitrary level and character, and generalize the Eichler-Zagier isomorphism between Jacobi forms and modular forms of half-integral weight to higher levels. Using this together with the known results, we get a strong multiplicity 1 theorem in certain cases for both Jacobi cusp newforms and half-integral weight cusp newforms. As a consequence, we get, among other results, the explicit Waldspurger theorem.
Rigidity of Coxeter groups
Stratos
Prassidis;
Barry
Spieler
2619-2642
Abstract: Let $W$ be a Coxeter group acting properly discontinuously and cocompactly on manifolds $N$ and $M ({\partial}M = {\emptyset})$ such that the fixed point sets of finite subgroups are contractible. Let $f: (N, {\partial}N) \to (M{\times}D^k, M{\times}S^{k-1})$ be a $W$-homotopy equivalence which restricts to a $W$-homeomorphism on the boundary. Under an assumption on the three dimensional fixed point sets, we show that then $f$ is $W$-homotopic to a $W$-homeomorphism.
Homology decompositions for classifying spaces of compact Lie groups
Alexei
Strounine
2643-2657
Abstract: Let $p$ be a prime number and $G$ be a compact Lie group. A homology decomposition for the classifying space $BG$ is a way of building $BG$ up to mod $p$ homology as a homotopy colimit of classifying spaces of subgroups of $G$. In this paper we develop techniques for constructing such homology decompositions. Jackowski, McClure and Oliver (Homotopy classification of self-maps of BG via $G$-actions, Ann. of Math. 135 (1992), 183-270) construct a homology decomposition of $BG$ by classifying spaces of $p$-stubborn subgroups of $G$. Their decomposition is based on the existence of a finite-dimensional mod $p$ acyclic $G$-$CW$-complex with restricted set of orbit types. We apply our techniques to give a parallel proof of the $p$-stubborn decomposition of $BG$which does not use this geometric construction.
Cohomology of uniformly powerful $p$-groups
William
Browder;
Jonathan
Pakianathan
2659-2688
Abstract: In this paper we will study the cohomology of a family of $p$-groups associated to $\mathbb{F}_p$-Lie algebras. More precisely, we study a category $\mathbf{BGrp}$ of $p$-groups which will be equivalent to the category of $\mathbb{F}_p$-bracket algebras (Lie algebras minus the Jacobi identity). We then show that for a group $G$ in this category, its $\mathbb{F}_p$-cohomology is that of an elementary abelian $p$-group if and only if it is associated to a Lie algebra. We then proceed to study the exponent of $H^*(G ;\mathbb{Z})$ in the case that $G$ is associated to a Lie algebra $\mathfrak{L}$. To do this, we use the Bockstein spectral sequence and derive a formula that gives $B_2^*$ in terms of the Lie algebra cohomologies of $\mathfrak{L}$. We then expand some of these results to a wider category of $p$-groups. In particular, we calculate the cohomology of the $p$-groups $\Gamma _{n,k}$ which are defined to be the kernel of the mod $p$ reduction $GL_n(\mathbb{Z}/p^{k+1}\mathbb{Z}) \overset{mod}{\longrightarrow} GL_n(\mathbb{F}_p).$
Group actions and group extensions
Ergün
Yalçin
2689-2700
Abstract: In this paper we study finite group extensions represented by special cohomology classes. As an application, we obtain some restrictions on finite groups which can act freely on a product of spheres or on a product of real projective spaces. In particular, we prove that if $(Z/p)^r$ acts freely on $(S^1)^k$, then $r \leq k$.
Problème de Dirichlet pour une équation de Monge-Ampère réelle elliptique dégénérée en dimension $n$
Amel
Atallah
2701-2721
Abstract: RÉSUMÉ. On considère dans un ouvert borné $\Omega$ de $\mathbb{R}^n$, à bord régulier, le problème de Dirichlet \begin{equation*}\left\{ \begin{split} & \det u_{ij}=f(x)\text{ dans }\Omega, & u\vert _{\partial \Omega}=\varphi, \end{split}\right.\tag{1} \end{equation*} où $f\in C^{s_*}(\overline\Omega), \varphi\in C^{s_*+2,\alpha}(\Omega)$, $f$est positive et s'annule sur $\Sigma$ un ensemble fini de points de $\Omega$. On démontre alors sous certaines hypothèses sur $\varphi$ et si $\vert\det \varphi_{ij}-f\vert _{C^{s_*}}$ est assez petit, que le problème (1) possède une solution convexe unique $u\in C^{[s_*-3-n/2]}(\overline\Omega)$. ABSTRACT. We consider in a bounded open set $\Omega$ of $\mathbb{R}^n$, with regular boundary, the Dirichlet problem \begin{equation*}\left\{ \begin{split} & \det u_{ij}=f(x)\text{ in }\Omega, & u\vert _{\partial \Omega}=\varphi, \end{split}\right.\tag{1} \end{equation*} where $f\in C^{s_*}(\overline\Omega), \varphi\in C^{s_*+2,\alpha}(\Omega)$, $f$is positive and vanishes on $\Sigma$, a finite set of points in $\Omega$. We prove, under some hypothesis on $\varphi$ and if $\vert\det \varphi_{ij}-f\vert _{C^{s_*}}$ is sufficiently small, that the problem (1) has a unique convex solution $u\in C^{[s_*-3-n/2]}(\overline\Omega)$.
A new result on the Pompeiu problem
R.
Dalmasso
2723-2736
Abstract: A nonempty bounded open set $\Omega \subset {\mathbb{R}}^{n}$ ($n \geq 2$) is said to have the Pompeiu property if and only if the only continuous function $f$ on ${\mathbb{R}}^{n}$ for which the integral of $f$ over $\sigma (\Omega )$ is zero for all rigid motions $\sigma$ of ${\mathbb{R}}^{n}$ is $f \equiv 0$. We consider a nonempty bounded open set $\Omega \subset {\mathbb{R}}^{n}$ $(n \geq 2)$ with Lipschitz boundary and we assume that the complement of $\overline{\Omega }$ is connected. We show that the failure of the Pompeiu property for $\Omega$ implies some geometric conditions. Using these conditions we prove that a special kind of solid tori in ${\mathbb{R}}^{n}$, $n \geq 3$, has the Pompeiu property. So far the result was proved only for solid tori in ${\mathbb{R}}^{4}$. We also examine the case of planar domains. Finally we extend the example of solid tori to domains in ${\mathbb{R}}^{n}$ bounded by hypersurfaces of revolution.
Quadratic integral games and causal synthesis
Yuncheng
You
2737-2764
Abstract: The game problem for an input-output system governed by a Volterra integral equation with respect to a quadratic performance functional is an untouched open problem. In this paper, it is studied by a new approach called projection causality. The main result is the causal synthesis which provides a causal feedback implementation of the optimal strategies in the saddle point sense. The linear feedback operator is determined by the solution of a Fredholm integral operator equation, which is independent of data functions and control functions. Two application examples are included. The first one is quadratic differential games of a linear system with arbitrary finite delays in the state variable and control variables. The second is the standard linear-quadratic differential games, for which it is proved that the causal synthesis can be reduced to a known result where the feedback operator is determined by the solution of a differential Riccati operator equation.
Inverse spectral analysis with partial information on the potential, II. The case of discrete spectrum
Fritz
Gesztesy;
Barry
Simon
2765 - 2787
Abstract: We discuss results where the discrete spectrum (or partial information on the discrete spectrum) and partial information on the potential $q$ of a one-dimensional Schrödinger operator $H=-\frac{d^{2}}{dx^{2}}+q$ determine the potential completely. Included are theorems for finite intervals and for the whole line. In particular, we pose and solve a new type of inverse spectral problem involving fractions of the eigenvalues of $H$ on a finite interval and knowledge of $q$ over a corresponding fraction of the interval. The methods employed rest on Weyl $m$-function techniques and densities of zeros of a class of entire functions.
Projection orthogonale sur le graphe d'une relation linéaire fermé
Yahya
Mezroui
2789-2800
Abstract: Let ${LR(H)}$ denote the set of all closed linear relations on a Hilbert space $H$ (which contains all closed linear operators on $H$). In this paper, for every $E \in {\mathcal LR(H)}$ we define and study two associated linear operators on $H$, $\cos(E)$ and $\sin(E)$, which play an important role in the study of linear relations. These operators satisfy conditions quite analogous to trigonometric identities (whence their names) and appear, in particular, in the formula that gives the orthogonal projection on the graph of $E$, a formula first established for linear operators by M. H. Stone and extended to linear relations by H. De Snoo. We prove here a slightly modified version of the De Snoo formula. Several other applications of the $\cos(E)$ and $\sin(E)$ operators to operator theory will be given in a forthcoming paper.
Geometry of Banach spaces having shrinking approximations of the identity
Eve
Oja
2801-2823
Abstract: Let $a,c\geq 0$ and let $B$ be a compact set of scalars. We introduce property $M^{\ast }(a,B,c)$ of Banach spaces $X$ by the requirement that \begin{equation*}\limsup _{\nu }\Vert ax_{\nu }^{\ast } +bx^{\ast }+cy^{\ast }\V... ...q \limsup _{\nu }\Vert x_{\nu }^{\ast }\Vert\quad \forall b\in B \end{equation*}whenever $(x_{\nu }^{\ast })$ is a bounded net converging weak$^{\ast }$ to $x^{\ast }$ in $X^{\ast }$ and $\Vert y^{\ast }\Vert\leq \Vert x^{\ast }\Vert$. Using $M^{\ast }(a,B,c)$ with $\max \vert B\vert+c>1$, we characterize the existence of certain shrinking approximations of the identity (in particular, those related to $M$-, $u$-, and $h$-ideals of compact or approximable operators). We also show that the existence of these approximations of the identity is separably determined.
The truncated complex $K$-moment problem
Raúl
Curto;
Lawrence
A.
Fialkow
2825-2855
Abstract: Let $\gamma \equiv \gamma^{\left( 2n\right) }$ denote a sequence of complex numbers $\gamma _{00}, \gamma _{01}, \gamma _{10}, \dots , \gamma _{0,2n}, \dots , \gamma _{2n,0}$ ( $\gamma _{00}>0, \gamma _{ij}=\bar{\gamma}_{ji}$), and let $K$ denote a closed subset of the complex plane $\mathbb{C}$. The Truncated Complex $K$-Moment Problem for $\gamma$ entails determining whether there exists a positive Borel measure $\mu$ on $\mathbb{C}$ such that $\gamma _{ij}=\int \bar{z}^{i}z^{j}\,d\mu$ ( $0\leq i+j\leq 2n$) and $\operatorname{supp}\mu \subseteq K$. For $K\equiv K_{\mathcal{P}}$ a semi-algebraic set determined by a collection of complex polynomials $\mathcal{P} =\left\{ p_{i}\left( z,\bar{z}\right) \right\} _{i=1}^{m}$, we characterize the existence of a finitely atomic representing measure with the fewest possible atoms in terms of positivity and extension properties of the moment matrix $M\left( n\right) \left( \gamma \right)$and the localizing matrices $M_{p_{i}}$. We prove that there exists a $\operatorname{rank}M\left( n\right)$-atomic representing measure for $\gamma ^{\left( 2n\right) }$supported in $K_{\mathcal{P}}$if and only if $M\left( n\right) \geq 0$and there is some rank-preserving extension $M\left( n+1\right)$for which $M_{p_{i}}\left( n+k_{i}\right) \geq 0$, where $\deg p_{i}=2k_{i}$ or $2k_{i}-1$ $(1\leq i\leq m)$.
On the structure of weight modules
Ivan
Dimitrov;
Olivier
Mathieu;
Ivan
Penkov
2857-2869
Abstract: Given any simple Lie superalgebra ${\mathfrak{g}}$, we investigate the structure of an arbitrary simple weight ${\mathfrak{g}}$-module. We introduce two invariants of simple weight modules: the shadow and the small Weyl group. Generalizing results of Fernando and Futorny we show that any simple module is obtained by parabolic induction from a cuspidal module of a Levi subsuperalgebra. Then we classify the cuspidal Levi subsuperalgebras of all simple classical Lie superalgebras and of the Lie superalgebra W$(n)$. Most of them are simply Levi subalgebras of ${\mathfrak{g}}_{0}$, in which case the classification of all finite cuspidal representations has recently been carried out by one of us (Mathieu). Our results reduce the classification of the finite simple weight modules over all classical simple Lie superalgebras to classifying the finite cuspidal modules over certain Lie superalgebras which we list explicitly.
Bivariate factorizations connecting Dickson polynomials and Galois theory
Shreeram
S.
Abhyankar;
Stephen
D.
Cohen;
Michael
E.
Zieve
2871-2887
Abstract: In his Ph.D. Thesis of 1897, Dickson introduced certain permutation polynomials whose Galois groups are essentially the dihedral groups. These are now called Dickson polynomials of the first kind, to distinguish them from their variations introduced by Schur in 1923, which are now called Dickson polynomials of the second kind. In the last few decades there have been extensive investigations of both of these types, which are related to the classical Chebyshev polynomials. We give new bivariate factorizations involving both types of Dickson polynomials. These factorizations demonstrate certain isomorphisms between dihedral groups and orthogonal groups, and lead to the construction of explicit equations with orthogonal groups as Galois groups.
The structure of conjugacy closed loops
Kenneth
Kunen
2889-2911
Abstract: We study structure theorems for the conjugacy closed (CC-) loops, a specific variety of G-loops (loops isomorphic to all their loop isotopes). These theorems give a description all such loops of small order. For example, if $p$ and $q$ are primes, $p < q$, and $q-1$ is not divisible by $p$, then the only CC-loop of order $pq$ is the cyclic group of order $pq$. For any prime $q > 2$, there is exactly one non-group CC-loop in order $2q$, and there are exactly three in order $q^2$. We also derive a number of equations valid in all CC-loops. By contrast, every equation valid in all G-loops is valid in all loops.
Infinite convolution products and refinable distributions on Lie groups
Wayne
Lawton
2913-2936
Abstract: Sufficient conditions for the convergence in distribution of an infinite convolution product $\mu_1*\mu_2*\ldots$ of measures on a connected Lie group $\mathcal G$ with respect to left invariant Haar measure are derived. These conditions are used to construct distributions $\phi$ that satisfy $T\phi = \phi$where $T$ is a refinement operator constructed from a measure $\mu$and a dilation automorphism $A$. The existence of $A$ implies $\mathcal G$ is nilpotent and simply connected and the exponential map is an analytic homeomorphism. Furthermore, there exists a unique minimal compact subset $\mathcal K \subset \mathcal G$such that for any open set $\mathcal U$ containing $\mathcal K,$ and for any distribution $f$ on $\mathcal G$ with compact support, there exists an integer $n(\mathcal U,f)$ such that $n \geq n(\mathcal U,f)$implies $\hbox{supp}(T^{n}f) \subset\mathcal U.$If $\mu$ is supported on an $A$-invariant uniform subgroup $\Gamma,$ then $T$ is related, by an intertwining operator, to a transition operator $W$ on $\mathbb C(\Gamma).$ Necessary and sufficient conditions for $T^{n}f$ to converge to $\phi \in L^{2}$, and for the $\Gamma$-translates of $\phi$ to be orthogonal or to form a Riesz basis, are characterized in terms of the spectrum of the restriction of $W$ to functions supported on $\Omega := \mathcal K \mathcal K^{-1} \cap \Gamma.$