Algorithms for nonlinear piecewise polynomial approximation: Theoretical aspects
Borislav
Karaivanov;
Pencho
Petrushev;
Robert
C.
Sharpley
2585-2631
Abstract: In this article algorithms are developed for nonlinear $n$-term Courant element approximation of functions in $L_p$ ( $0 < p \le \infty$) on bounded polygonal domains in $\mathbb{R} ^2$. Redundant collections of Courant elements, which are generated by multilevel nested triangulations allowing arbitrarily sharp angles, are investigated. Scalable algorithms are derived for nonlinear approximation which both capture the rate of the best approximation and provide the basis for numerical implementation. Simple thresholding criteria enable approximation of a target function $f$ to optimally high asymptotic rates which are determined and automatically achieved by the inherent smoothness of $f$. The algorithms provide direct approximation estimates and permit utilization of the general Jackson-Bernstein machinery to characterize $n$-term Courant element approximation in terms of a scale of smoothness spaces ($B$-spaces) which govern the approximation rates.
The almost-disjointness number may have countable cofinality
Jörg
Brendle
2633-2649
Abstract: We show that it is consistent for the almost-disjointness number $\mathfrak{a}$ to have countable cofinality. For example, it may be equal to $\aleph_\omega$.
Cyclicity of CM elliptic curves modulo $p$
Alina
Carmen
Cojocaru
2651-2662
Abstract: Let $E$ be an elliptic curve defined over $\mathbb{Q}$ and with complex multiplication. For a prime $p$ of good reduction, let $\overline{E}$ be the reduction of $E$ modulo $p.$ We find the density of the primes $p \leq x$ for which $\overline{E}(\mathbb{F} _p)$ is a cyclic group. An asymptotic formula for these primes had been obtained conditionally by J.-P. Serre in 1976, and unconditionally by Ram Murty in 1979. The aim of this paper is to give a new simpler unconditional proof of this asymptotic formula and also to provide explicit error terms in the formula.
Taylor expansion of an Eisenstein series
Tonghai
Yang
2663-2674
Abstract: In this paper, we give an explicit formula for the first two terms of the Taylor expansion of a classical Eisenstein series of weight $2k+1$ for $\Gamma_{0}(q)$. Both the first term and the second term have interesting arithmetic interpretations. We apply the result to compute the central derivative of some Hecke $L$-functions.
Systems of diagonal Diophantine inequalities
Eric
Freeman
2675-2713
Abstract: We treat systems of real diagonal forms $F_1({\mathbf x}), F_2({\mathbf x}), \ldots, F_R({\mathbf x})$ of degree $k$, in $s$ variables. We give a lower bound $s_0(R,k)$, which depends only on $R$ and $k$, such that if $s \geq s_0(R,k)$ holds, then, under certain conditions on the forms, and for any positive real number $\epsilon$, there is a nonzero integral simultaneous solution $\displaystyle{{\mathbf x}\in {\mathbb Z}^s}$ of the system of Diophantine inequalities $\vert F_i({\mathbf x})\vert < \epsilon$ for $1 \leq i \leq R$. In particular, our result is one of the first to treat systems of inequalities of even degree. The result is an extension of earlier work by the author on quadratic forms. Also, a restriction in that work is removed, which enables us to now treat combined systems of Diophantine equations and inequalities.
On the canonical rings of covers of surfaces of minimal degree
Francisco
Javier
Gallego;
Bangere
P.
Purnaprajna
2715-2732
Abstract: In one of the main results of this paper, we find the degrees of the generators of the canonical ring of a regular algebraic surface $X$ of general type defined over a field of characteristic $0$, under the hypothesis that the canonical divisor of $X$ determines a morphism $\varphi$ from $X$ to a surface of minimal degree $Y$. As a corollary of our results and results of Ciliberto and Green, we obtain a necessary and sufficient condition for the canonical ring of $X$ to be generated in degree less than or equal to $2$. We construct new examples of surfaces satisfying the hypothesis of our theorem and prove results which show that many a priori plausible examples cannot exist. Our methods are to exploit the $\mathcal{O}_{Y}$-algebra structure on $\varphi_{*}\mathcal{O}_{X}$. These methods have other applications, including those on Calabi-Yau threefolds. We prove new results on homogeneous rings associated to a polarized Calabi-Yau threefold and also prove some existence theorems for Calabi-Yau covers of threefolds of minimal degree. These have consequences towards constructing new examples of Calabi-Yau threefolds.
A classification and examples of rank one chain domains
H.
H.
Brungs;
N.
I.
Dubrovin
2733-2753
Abstract: A chain order of a skew field $D$ is a subring $R$ of $D$ so that $d\in D\backslash R$ implies $d^{-1}\in R.$ Such a ring $R$ has rank one if $J(R)$, the Jacobson radical of $R,$ is its only nonzero completely prime ideal. We show that a rank one chain order of $D$ is either invariant, in which case $R$ corresponds to a real-valued valuation of $D,$ or $R$ is nearly simple, in which case $R,$ $J(R)$ and $(0)$ are the only ideals of $R,$ or $R$ is exceptional in which case $R$ contains a prime ideal $Q$that is not completely prime. We use the group $\mathcal{M}(R)$ of divisorial $R{\text{-}ideals}$ of $D$ with the subgroup $\mathcal{H}(R)$ of principal $R{\text{-}ideals}$ to characterize these cases. The exceptional case subdivides further into infinitely many cases depending on the index $k$of $\mathcal{H}(R)$ in $\mathcal{M}(R).$Using the covering group $\mathbb{G}$ of $\operatorname{SL}(2,\mathbb{R} )$ and the result that the group ring $T\mathbb{G}$ is embeddable into a skew field for $T$ a skew field, examples of rank one chain orders are constructed for each possible exceptional case.
On the spectral sequence constructors of Guichardet and Stefan
Donald
W.
Barnes
2755-2769
Abstract: The concept of a spectral sequence constructor is generalised to Hopf Galois extensions. The spectral sequence constructions that are given by Guichardet for crossed product algebras are also generalised and shown to provide examples. It is shown that all spectral sequence constructors for Hopf Galois extensions construct the same spectral sequence.
Formality in an equivariant setting
Steven
Lillywhite
2771-2793
Abstract: We define and discuss $G$-formality for certain spaces endowed with an action by a compact Lie group. This concept is essentially formality of the Borel construction of the space in a category of commutative differential graded algebras over $R=H^\bullet(BG)$. These results may be applied in computing the equivariant cohomology of their loop spaces.
Large rectangular semigroups in Stone-Cech compactifications
Neil
Hindman;
Dona
Strauss;
Yevhen
Zelenyuk
2795-2812
Abstract: We show that large rectangular semigroups can be found in certain Stone-Cech compactifications. In particular, there are copies of the $2^{\mathfrak{c}}\times 2^{\mathfrak{c}}$rectangular semigroup in the smallest ideal of $(\beta\mathbb{N},+)$, and so, a semigroup consisting of idempotents can be embedded in the smallest ideal of $(\beta\mathbb{N},+)$ if and only if it is a subsemigroup of the $2^{\mathfrak{c}}\times 2^{\mathfrak{c}}$ rectangular semigroup. In fact, we show that for any ordinal $\lambda$ with cardinality at most $\mathfrak{c}$, $\beta{\mathbb{N}}$ contains a semigroup of idempotents whose rectangular components are all copies of the $2^{\mathfrak{c}}\times 2^{\mathfrak{c}}$ rectangular semigroup and form a decreasing chain indexed by $\lambda+1$, with the minimum component contained in the smallest ideal of $\beta\mathbb{N}$. As a fortuitous corollary we obtain the fact that there are $\leq_{L}$-chains of idempotents of length $\mathfrak{c}$ in $\beta \mathbb{N}$. We show also that there are copies of the direct product of the $2^{\mathfrak{c}}\times 2^{\mathfrak{c}}$ rectangular semigroup with the free group on $2^{\mathfrak{c}}$ generators contained in the smallest ideal of $\beta\mathbb{N}$.
Galois groups of quantum group actions and regularity of fixed-point algebras
Takehiko
Yamanouchi
2813-2828
Abstract: It is shown that, for a minimal and integrable action of a locally compact quantum group on a factor, the group of automorphisms of the factor leaving the fixed-point algebra pointwise invariant is identified with the intrinsic group of the dual quantum group. It is proven also that, for such an action, the regularity of the fixed-point algebra is equivalent to the cocommutativity of the quantum group.
Composition operators acting on holomorphic Sobolev spaces
Boo
Rim
Choe;
Hyungwoon
Koo;
Wayne
Smith
2829-2855
Abstract: We study the action of composition operators on Sobolev spaces of analytic functions having fractional derivatives in some weighted Bergman space or Hardy space on the unit disk. Criteria for when such operators are bounded or compact are given. In particular, we find the precise range of orders of fractional derivatives for which all composition operators are bounded on such spaces. Sharp results about boundedness and compactness of a composition operator are also given when the inducing map is polygonal.
Distributions of corank 1 and their characteristic vector fields
B.
Jakubczyk;
M.
Zhitomirskii
2857-2883
Abstract: We prove that any 1-parameter family of corank 1 distributions (or Pfaff equations) on a compact manifold $M^{n}$ is trivializable, i.e., transformable to a constant family by a family of diffeomorphisms, if all distributions of the family have the same characteristic line field. The characteristic line field is a field of tangent lines which is invariantly assigned to a corank one distribution. It is defined on $M^{n}$, if $n=2k$, or on a subset of $M^{n}$ called the Martinet hypersurface, if $n=2k+1$. Our second main result states that if two corank one distributions have the same characteristic line field and are close to each other, then they are equivalent via a diffeomorphism. This holds under a weak assumption on the singularities of the distributions. The second result implies that the abnormal curves of a distribution determine the equivalence class of the distribution, among distributions close to a given one.
When are the tangent sphere bundles of a Riemannian manifold reducible?
E.
Boeckx
2885-2903
Abstract: We determine all Riemannian manifolds for which the tangent sphere bundles, equipped with the Sasaki metric, are local or global Riemannian product manifolds.
Criteria for large deviations
Henri
Comman
2905-2923
Abstract: We give the general variational form of \begin{displaymath}\limsup(\int_X e^{h(x)/t_{\alpha}}\mu_{\alpha}(dx))^{t_{\alpha}}\end{displaymath} for any bounded above Borel measurable function $h$ on a topological space $X$, where $(\mu_{\alpha})$ is a net of Borel probability measures on $X$, and $(t_{\alpha})$ a net in $]0,\infty[$ converging to $0$. When $X$ is normal, we obtain a criterion in order to have a limit in the above expression for all $h$ continuous bounded, and deduce new criteria of a large deviation principle with not necessarily tight rate function; this allows us to remove the tightness hypothesis in various classical theorems.
Integration by parts formulas involving generalized Fourier-Feynman transforms on function space
Seung
Jun
Chang;
Jae
Gil
Choi;
David
Skoug
2925-2948
Abstract: In an upcoming paper, Chang and Skoug used a generalized Brownian motion process to define a generalized analytic Feynman integral and a generalized analytic Fourier-Feynman transform. In this paper we establish several integration by parts formulas involving generalized Feynman integrals, generalized Fourier-Feynman transforms, and the first variation of functionals of the form $F(x)=f(\langle {\alpha _{1} , x}\rangle, \dots , \langle {\alpha _{n} , x}\rangle )$ where $\langle {\alpha ,x}\rangle$ denotes the Paley-Wiener-Zygmund stochastic integral $\int _{0}^{T} \alpha (t) d x(t)$.
Thermodynamic formalism for countable to one Markov systems
Michiko
Yuri
2949-2971
Abstract: For countable to one transitive Markov systems we establish thermodynamic formalism for non-Hölder potentials in nonhyperbolic situations. We present a new method for the construction of conformal measures that satisfy the weak Gibbs property for potentials of weak bounded variation and show the existence of equilibrium states equivalent to the weak Gibbs measures. We see that certain periodic orbits cause a phase transition, non-Gibbsianness and force the decay of correlations to be slow. We apply our results to higher-dimensional maps with indifferent periodic points.
Strongly indefinite functionals and multiple solutions of elliptic systems
D.
G.
De Figueiredo;
Y.
H.
Ding
2973-2989
Abstract: We study existence and multiplicity of solutions of the elliptic system \begin{displaymath}\begin{cases}-\Delta u =H_u(x,u,v) & \text{in} \Omega, -\D... ...\quad u(x) =v(x)=0 \quad\text{on} \partial \Omega ,\end{cases}\end{displaymath} where $\Omega\subset\mathbb{R}^N, N\geq 3$, is a smooth bounded domain and $H\in \mathcal{C}^1(\bar{\Omega}\times\mathbb{R}^2, \mathbb{R})$. We assume that the nonlinear term \begin{displaymath}H(x,u,v)\sim \vert u\vert^p + \vert v\vert^q + R(x,u,v) \te... ...ert\to\infty}\frac{R(x,u,v)}{\vert u\vert^p+\vert v\vert^q}=0, \end{displaymath} where $p\in (1, 2^*)$, $2^*:=2N/(N-2)$, and $q\in (1, \infty)$. So some supercritical systems are included. Nontrivial solutions are obtained. When $H(x,u,v)$ is even in $(u,v)$, we show that the system possesses a sequence of solutions associated with a sequence of positive energies (resp. negative energies) going toward infinity (resp. zero) if $p>2$ (resp. $p<2$). All results are proved using variational methods. Some new critical point theorems for strongly indefinite functionals are proved.
Stability of small amplitude boundary layers for mixed hyperbolic-parabolic systems
F.
Rousset
2991-3008
Abstract: We consider an initial boundary value problem for a symmetrizable mixed hyperbolic-parabolic system of conservation laws with a small viscosity $\varepsilon$, $u^\varepsilon_t+F(u^\varepsilon)_x =\varepsilon(B(u^\varepsilon) u^\varepsilon_x )_x .$ When the boundary is noncharacteristic for both the viscous and the inviscid system, and the boundary condition dissipative, we show that $u^\varepsilon$ converges to a solution of the inviscid system before the formation of shocks if the amplitude of the boundary layer is sufficiently small. This generalizes previous results obtained for $B$ invertible and the linear study of Serre and Zumbrun obtained for a pure Dirichlet's boundary condition.