Strong Approximation Property Research Articles

On the density of S-adic integers near some projective G-varieties

We provide some general conditions which ensure that a system of inequalities involving homogeneous polynomials with coefficients in a \(S\)-adic field has nontrivial \(S\)-integral solutions. The proofs are based on the strong approximation property for Zariski-dense subgroups and adelic geometry of numbers. We give some examples of applications for systems involving quadratic and linear forms.

Arithmetic purity of the Hardy-Littlewood property and geometric sieve for affine quadrics

We establish the Hardy-Littlewood property (à la Borovoi-Rudnick) for Zariski open subsets in affine quadrics of the form q(x1,⋯,xn)=m, where q is a non-degenerate integral quadratic form in n⩾3 variables and m is a non-zero integer. This gives asymptotic formulas for the density of integral points taking coprime polynomial values, which is a quantitative version of the arithmetic purity of strong approximation property off infinity for affine quadrics.

On the Compact Operators Case of the Bishop–Phelps–Bollobás Property for Numerical Radius

We study the Bishop–Phelps–Bollobás property for numerical radius restricted to the case of compact operators (BPBp-nu for compact operators in short). We show that $$C_0(L)$$ spaces have the BPBp-nu for compact operators for every Hausdorff topological locally compact space L. To this end, on the one hand, we provide some techniques allowing to pass the BPBp-nu for compact operators from subspaces to the whole space and, on the other hand, we prove some strong approximation property of $$C_0(L)$$ spaces and their duals. Besides, we also show that real Hilbert spaces and isometric preduals of $$\ell _1$$ have the BPBp-nu for compact operators.

Exactness and SOAP of crossed products via Herz–Schur multipliers

Given a C⁎-dynamical system (A,G,α), with G a discrete group, Schur A-multipliers and Herz–Schur (A,G,α)-multipliers are used to implement approximation properties, namely exactness and the strong operator approximation property (SOAP), of A⋊α,rG. The resulting characterisations of exactness and SOAP of A⋊α,rG generalise the corresponding statements for the reduced group C⁎-algebra.

The approximation property and exactness of locally compact groups

We extend a theorem of Haagerup and Kraus in the C*-algebra context: for a locally compact group with the approximation property (AP), the reduced C*-crossed product construction preserves the strong operator approximation property (SOAP). In particular their reduced group C*-algebras have the SOAP. Our method also solves another open problem: the AP implies exactness for general locally compact groups.

E-approximation properties

In this paper, we introduce the E-approximation property and the $$E^u$$ -approximation property which generalize Sinha and Karn’s p-approximation property. Characterizations of these properties are given parallel to Lima and Oja’s results on the strong approximation property and the weak bounded approximation property. Representation theorems for the dual of $${\mathcal {L}} (X, Y)$$ under the topology of uniform convergence on E-compact sets and $$E^u$$ -compact sets are also provided. As an application, the representations are used to generalize the main theorem of Choi and Kim in [1]. These results build up fundamental bases for future investigations of these properties.

History, Developments and Open Problems on Approximation Properties

In this paper, we give a comprehensive review of the classical approximation property. Then, we present some important results on modern variants, such as the weak bounded approximation property, the strong approximation property and p-approximation property. Most recent progress on E-approximation property and open problems are discussed at the end.

The Calderón problem for the fractional Schrödinger equation

We show global uniqueness in an inverse problem for the fractional Schr\"odinger equation: an unknown potential in a bounded domain is uniquely determined by exterior measurements of solutions. We also show global uniqueness in the partial data problem where the measurements are taken in arbitrary open, possibly disjoint, subsets of the exterior. The results apply in any dimension $\geq 2$ and are based on a strong approximation property of the fractional equation that extends earlier work. This special feature of the nonlocal equation renders the analysis of related inverse problems radically different from the traditional Calder\'on problem.

Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data

In this note we discuss the conditional stability issue for the finite dimensional Calderon problem for the fractional Schrodinger equation with a finite number of measurements. More precisely, we assume that the unknown potential \begin{document}$ q \in L^{\infty}(\Omega) $\end{document} in the equation \begin{document}$ ((- \Delta)^s+ q)u = 0 \mbox{ in } \Omega\subset \mathbb{R}^n $\end{document} satisfies the a priori assumption that it is contained in a finite dimensional subspace of \begin{document}$ L^{\infty}(\Omega) $\end{document} . Under this condition we prove Lipschitz stability estimates for the fractional Calderon problem by means of finitely many Cauchy data depending on \begin{document}$ q $\end{document} . We allow for the possibility of zero being a Dirichlet eigenvalue of the associated fractional Schrodinger equation. Our result relies on the strong Runge approximation property of the fractional Schrodinger equation.

Nested Artin strong approximation property

We study the Artin Approximation property with constraints in a different frame. As a consequence we give a nested Artin Strong Approximation property for algebraic power series rings over a field.

Localization of Frames

Motivated by Balan, Casazza, Heil, and Landau's results on localization of frames in a separable Hilbert space ℋ, we investigate the transitivity of localization relation of the frame ℱ = {f i } i∈I and sequence ℰ = {e j } j∈G in ℋ with respect to the associated map a: I → G, where G is a discrete abelian group and I is a index set. We also study some properties of ℓ p -column decay, ℓ p -row decay, weak homogeneous approximation property, and strong homogeneous approximation property of frame ℱ = {f i } i∈I , and sequence ℰ = {e j } j∈G , with respect to the associated map a.

Boltzmann Machine and Hyperbolic Activation Function in Higher Order Network

For higher-order programming, higher-order network architecture is necessary to provide faster convergence rate, greater storage capacity, stronger approximation property, and higher fault tolerance than lower-order neural networks. Thus, the higher-order clauses for logic programming in Hopfield Networks have been considered in this paper. The goal is to perform logic programming based on the energy minimization scheme is to achieve the best global minimum. However, there is no guarantee to find the best minimum in the network. However, by using Boltzmann Machines and hyperbolic tangent activation function this problem can be overcome.

The strong approximation property and the weak bounded approximation property

We show that the strong approximation property (strong AP) (respectively, strong CAP) and the weak bounded approximation property (respectively, weak BCAP) are equivalent for every Banach space. This gives a negative answer to Oja's conjecture. As a consequence, we show that each of the spaces c0 and ℓ1 has a subspace which has the AP but fails to have the strong AP.

THE STABILITY PROPERTIES OF STRONG INVARIANT APPROXIMATION PROPERTY

Let G be a countable exact discrete group. G has the strong invariant approximation property(SIAP) if and only if C � u(G,S) G = C � � (G) ⊗ S for any Hilbert space H and closed subspace S ⊆ H. We shall use results of Haagerup and Kraus on the approximation property (AP) to investigate some permanence properties of the SIAP for discrete groups. This can be done most efficiently for exact groups. In this paper we describe that the stability properties of the SIAP property pass to semi direct products, and extensions for discrete exact groups.

Elementary geometric local–global principles for fields

We define and investigate a family of local–global principles for fields involving both orderings and p-valuations. This family contains the PAC, PRC and PpC fields and exhausts the class of pseudo classically closed fields. We show that the fields satisfying such a local–global principle form an elementary class, admit diophantine definitions of holomorphy domains, and their orderings satisfy the strong approximation property.

Manin obstruction to strong approximation for homogeneous spaces

For a homogeneous space X (not necessarily principal) of a connected algebraic group G (not necessarily linear) over a number field k , we prove a theorem of strong approximation for the adelic points of X in the Brauer–Manin set. Namely, for an adelic point x of X orthogonal to a certain subgroup (which may contain transcendental elements) of the Brauer group \operatorname{Br}(X) of X with respect to the Manin pairing, we prove a strong approximation property for x away from a finite set S of places of k . Our result extends a result of Harari for torsors of semiabelian varieties and a result of Colliot-Thélène and Xu for homogeneous spaces of simply connected semisimple groups, and our proof uses those results.

A unified approach to the strong approximation property and the weak bounded approximation property of Banach spaces

A unified approach to the strong approximation property and the weak bounded approximation property of Banach spaces

COARSE SPACES BY ALGEBRAIC MULTIGRID: MULTIGRID CONVERGENCE AND UPSCALING ERROR ESTIMATES

We give an overview of a number of algebraic multigrid methods targeting finite element discretization problems. The focus is on the properties of the constructed hierarchy of coarse spaces that guarantee (two-grid) convergence. In particular, a necessary condition known as "weak approximation property," and a sufficient one, referred to as "strong approximation property," are discussed. Their role in proving convergence of the TG method (as iterative method) and also on the approximation properties of the algebraic mottigrid (AMG) coarse spaces if used as discretization tool is pointed out. Some preliminary numerical results illustrating the latter aspect are also reported.

Equilibria for set-valued maps on nonsmooth domains

A set-valued map defined on a compact lipschitzian retract of a normed space with nontrivial Euler characteristic and satisfying (i) a strong graph approximation property and (ii) a tangency condition expressed in terms of Clarke’s tangent cone, admits an equilibrium. This result extends in a simple way known solvability theorems to a large class of nonconvex set-valued maps defined on nonsmooth domains.

$R$-equivalence in adjoint classical groups over fields of virtual cohomological dimension $2$

Let $F$ be a field of characteristic not $2$ whose virtual cohomological dimension is at most $2$. Let $G$ be a semisimple group of adjoint type defined over $F$. Let $RG(F)$ denote the normal subgroup of $G(F)$ consisting of elements $R$-equivalent to identity. We show that if $G$ is of classical type not containing a factor of type $D_n$, $G(F)/RG(F) = 0$. If $G$ is a simple classical adjoint group of type $D_n$, we show that if $F$ and its multi-quadratic extensions satisfy strong approximation property, then $G(F)/RG(F) = 0$. This leads to a new proof of the $R$-triviality of $F$-rational points of adjoint classical groups defined over number fields.