Arithmetic Properties of Generalized Hypergeometric <Emphasis Type="Italic">F</Emphasis>-Series

A generalization of the Siegel–Shidlovskii method in the theory of transcendental numbers is used to prove the infinite algebraic independence of elements (generated by generalized hypergeometric series) of direct products of fields \(\mathbb{K}_v\), which are completions of an algebraic number field \(\mathbb{K}\) of finite degree over the field of rational numbers with respect to valuations v of \(\mathbb{K}\) extending p-adic valuations of the field ? over all primes p, except for a finite number of them.     

Thoughts on Quotient-fine Nearness Frames

We characterize nearness frames whose completions are fine (we call them quotient-fine), and show that the subcategory QfNFrm they form is reflective in the category of strong nearness frames. The resulting functor commutes with the completion functor. QfNFrm is isomorphic to the subcategory of the functor category (RegFrm) ² given by the dense onto \(h\colon M\to L\), where 2 denotes the category with only two objects and exactly one morphism between them.     

The elliptic functions and integrals of the “{1 \mathord{\left/ {\vphantom {1 9}} \right. \kern-\nulldelimiterspace} 9}” problem

We describe the functions needed in the determination of the rate of convergence of best $L^\infty $ rational approximation to $\exp ( - x)$ on [0,∞) when the degree n of the approximation tends to ∞ (“1/9” problem).     

On three problems for sequences

If ξ∈ (0,1) and A=a_n, n?? is a sequence of real numbers define S_n(ξ,A)∶=Σ{a_k∶:k=[nξ]+1 to n}, n??, where [x] is the greatest integer less than or equal to x. In the theory of regularly varying sequences the problem arose to conclude from the convergence of the sequence S_n (ξ,A), n??, for all ξ in an appropriate set K of real numbers, that the sequence a_n, n??, converges to zero. It was shown that such a conclusion is possible if K={ξ,1?ξ} with ξ∈ (0,1) irrational. Then the following three questions were posed and will be answered in this paper:

1. does the convergence of S_n (ξ,A), n??, for a single irrational number ξ imply a_n→0.
2. does the convergence of S_n(ξ,A), n??, for finitely many rational numbers ξ∈ (0, 1) imply a_n→0.
3. does the convergence of S_n (ξ,A), n??, for all rational numbers ξ∈ (0,1) imply a_n→0?

     

Standard completions for quasiordered sets

A standard completion for a quasiordered set Q is a closure system whose point closures are the principal ideals of Q. We characterize the following types of standard completions by means of their closure operators:

1. V-distributive completions,
2. Completely distributive completions,
3. A-completions (i.e. standard completions which are completely distributive algebraic lattices),
4. Boolean completions.

Moreover, completely distributive completions are described by certain idempotent relations, and the A-completions are shown to be in one-to-one correspondence with the join-dense subsets of Q. If a pseudocomplemented meet-semilattice Q has a Boolean completion ?, then Q must be a Boolean lattice and ? its MacNeille completion.     

Convergence analysis of the direct extension of ADMM for multiple-block separable convex minimization

Recently, the alternating direction method of multipliers (ADMM) has found many efficient applications in various areas; and it has been shown that the convergence is not guaranteed when it is directly extended to the multiple-block case of separable convex minimization problems where there are m ≥ 3 functions without coupled variables in the objective. This fact has given great impetus to investigate various conditions on both the model and the algorithm’s parameter that can ensure the convergence of the direct extension of ADMM (abbreviated as “e-ADMM”). Despite some results under very strong conditions (e.g., at least (m ? 1) functions should be strongly convex) that are applicable to the generic case with a general m, some others concentrate on the special case of m = 3 under the relatively milder condition that only one function is assumed to be strongly convex. We focus on extending the convergence analysis from the case of m = 3 to the more general case of m ≥ 3. That is, we show the convergence of e-ADMM for the case of m ≥ 3 with the assumption of only (m ? 2) functions being strongly convex; and establish its convergence rates in different scenarios such as the worst-case convergence rates measured by iteration complexity and the globally linear convergence rate under stronger assumptions. Thus the convergence of e-ADMM for the general case of m ≥ 4 is proved; this result seems to be still unknown even though it is intuitive given the known result of the case of m = 3. Even for the special case of m = 3, our convergence results turn out to be more general than the existing results that are derived specifically for the case of m = 3.     

A convergence theory of multilevel additive Schwarz methods on unstructured meshes

We develop a convergence theory for two level and multilevel additive Schwarz domain decomposition methods for elliptic and parabolic problems on general unstructured meshes in two and three dimensions. The coarse and fine grids are assumed only to be shape regular, and the domains formed by the coarse and fine grids need not be identical. In this general setting, our convergence theory leads to completely local bounds for the condition numbers of two level additive Schwarz methods, which imply that these condition numbers are optimal, or independent of fine and coarse mesh sizes and subdomain sizes if the overlap amount of a subdomain with its neighbors varies proportionally to the subdomain size. In particular, we will show that additive Schwarz algorithms are still very efficient for nonselfadjoint parabolic problems with only symmetric, positive definite solvers both for local subproblems and for the global coarse problem. These conclusions for elliptic and parabolic problems improve our earlier results in [12, 15, 16]. Finally, the convergence theory is applied to multilevel additive Schwarz algorithms. Under some very weak assumptions on the fine mesh and coarser meshes, e.g., no requirements on the relation between neighboring coarse level meshes, we are able to derive a condition number bound of the orderO(² L ²), where = max₁lL(h _l +_l– 1)/ _l,h _l is the element size of thelth level mesh, _l the overlap of subdomains on thelth level mesh, andL the number of mesh levels.The work was partially supported by the NSF under contract ASC 92-01266, and ONR under contract ONR-N00014-92-J-1890. The second author was also partially supported by HKRGC grants no. CUHK 316/94E and the Direct Grant of CUHK.     

A Completion of \mathbb{Z} is a Field

We define various ring sequential convergences on and . We describe their properties and properties of their convergence completions. In particular, we define a convergence on by means of a nonprincipal ultrafilter on the positive prime numbers such that the underlying set of the completion is the ultraproduct of the prime finite fields Further, we show that is sequentially precompact but fails to be strongly sequentially precompact; this solves a problem posed by D. Dikranjan.     

Convergence of some finite element iterative methods related to different Reynolds numbers for the 2D/3D stationary incompressible magnetohydrodynamics

Based on finite element method (FEM), some iterative methods related to different Reynolds numbers are designed and analyzed for solving the 2D/3D stationary incompressible magnetohydrodynamics (MHD) numerically. Two-level finite element iterative methods, consisting of the classical m-iteration methods on a coarse grid and corrections on a fine grid, are designed to solve the system at low Reynolds numbers under the strong uniqueness condition. One-level Oseen-type iterative method is investigated on a fine mesh at high Reynolds numbers under the weak uniqueness condition. Furthermore, the uniform stability and convergence of these methods with respect to equation parameters R_e,R_m, S_c, mesh sizes h,H and iterative step m are provided. Finally, the efficiency of the proposed methods is confirmed by numerical investigations.     

Continuous completions

The paper generalizes some classical results about algebraic completions. Given a posetP with an auxiliary relation , it is shown how to construct all those completions ofP which are continuous lattices and whose way-below relation agrees with . The largest and the smallest such completions are investigated.Presented by R. W. Quackenbush.     

On the convergence rate of grid search for polynomial optimization over the simplex

We consider the approximate minimization of a given polynomial on the standard simplex, obtained by taking the minimum value over all rational grid points with given denominator \({r} \in \mathbb {N}\). It was shown in De Klerk et al. (SIAM J Optim 25(3):1498–1514, 2015) that the accuracy of this approximation depends on r as \(O(1/r^2)\) if there exists a rational global minimizer. In this note we show that the rational minimizer condition is not necessary to obtain the \(O(1/r^2)\) bound.     

Optimal Frame Completions with Prescribed Norms for Majorization

Given a finite sequence of vectors \(\mathcal F_0\) in a \(d\) -dimensional complex Hilbert space \({{\mathcal {H}}}\) we characterize in a complete and explicit way the optimal completions of \(\mathcal F_0\) obtained by appending a finite sequence of vectors with prescribed norms, where optimality is measured with respect to majorization (of the eigenvalues of the frame operators of the completed sequences). Indeed, we construct (in terms of a fast algorithm) a vector—that depends on the eigenvalues of the frame operator of the initial sequence \({\mathcal {F}}_0\) and the sequence of prescribed norms—that is a minimum for majorization among all eigenvalues of frame operators of completions with prescribed norms. Then, using the eigenspaces of the frame operator of the initial sequence \({\mathcal {F}}_0\) we describe the frame operators of all optimal completions for majorization. Hence, the concrete optimal completions with prescribed norms can be obtained using recent algorithmic constructions related with the Schur-Horn theorem.     

Compactness in Categories of S-Net Spaces

The $S$ -net spaces studied are convergence structures whose convergences are expressed by using generalized nets, the so called $S$ -nets, which are obtained from the usual nets by replacing the category of directed sets and cofinal maps with an arbitrary construct $S$ . We investigate compactness in categories of $S$ -net spaces defined by introducing continuous maps in a natural way and imposing some usual convergence axioms.     

A convergence on Boolean algebras generalizing the convergence on the Aleksandrov cube

We compare the forcing-related properties of a complete Boolean algebra B with the properties of the convergences λ_s (the algebraic convergence) and λ_ls on B generalizing the convergence on the Cantor and Aleksandrov cube, respectively. In particular, we show that λ_ls is a topological convergence iff forcing by B does not produce new reals and that λ_ls is weakly topological if B satisfies condition (?) (implied by the t-cc). On the other hand, if λ_ls is a weakly topological convergence, then B is a 2^h-cc algebra or in some generic extension the distributivity number of the ground model is greater than or equal to the tower number of the extension. So, the statement “The convergence λ_ls on the collapsing algebra \(B = {\text{ro}}{(^{ < \omega }}{\omega _2})\) is weakly topological” is independent of ZFC.     

Ein Approximationssatz für holomorphe Funktionen auf nicht-kompakten Riemannschen Flächen

Let R be a non-compact Riemann surface andO(R) the algebra of all holomorphic functions on R. A subalgebraA ?O(R) is calledfull (“voll”), if (F1) for every point ??R there is a function f∈A with a simple zero at ? and no other zeros; (F2) if f, g∈A and f/g has no poles, then f/∈A. In 1971 Ian RICHARDS set the problem whether full subalgebras are dense inO(R), with respect to the topology of compact convergence. We answer this question in the positive, using a lemma of I. RICHARDS and theorems of R. ARENS and the author. Does this approximation theorem remain true for Stein manifolds of dimension n>1, if one modifies condition (F1) in a natural way? A counterexample is provided by a domain of holomorphy G??² and a full, but not dense subalgebraA ?O(G).     

Cauchy completions of MV-algebras

Abstract. An MV-convergence is a convergence on an MV-algebra which renders the operations continuous. We show that such convergences on a given MV-algebra A are exactly the restrictions of the bounded -convergences on the abelian -group in which A appears as the unit interval. Thus the theory of -convergence and Cauchy structures transfers to MV-algebras.?We outline the general theory, and then apply it to three particular MV-convergences and their corresponding Cauchy completions. The Cauchy completion arising from order convergence coincides with the Dedekind-MacNeille completion of an MV-algebra. The Cauchy completion arising from polar convergence allows a tidy proof of the existence and uniqueness of the lateral completion of an MV-algebra. And the Cauchy completion arising from α-convergence gives rise to the cut completion of an MV-algebra. Received August 8, 2001; accepted in final form October 18, 2001.     

On a class of Wiener-Hopf equations of the first kind in a Sobolev space

This paper deals with the study of a class of Wiener-Hopf equations of the first kind in the Sobolev space H ₊ ^–2,1 () of Bessel potentials with the right-hand side in L ₁ ⁺ (). It is shown that the associated integral operator is a Fredholm operator and its nullity and defect numbers are obtained. Explicit formulas for the solutions are given.     

Generalized power series on a non-Archimedean field

Power series with rational exponents on the real numbers field and the Levi-Civita field are studied. We derive a radius of convergence for power series with rational exponents over the field of real numbers that depends on the coefficients and on the density of the exponents in the series. Then we generalize that result and study power series with rational exponents on the Levi-Civita field. A radius of convergence is established that asserts convergence under a weak topology and reduces to the conventional radius of convergence for real power series. It also asserts strong (order) convergence for points whose distance from the center is infinitely smaller than the radius of convergence. Then we study a class of functions that are given locally by power series with rational exponents, which are shown to form a commutative algebra over the Levi-Civita field; and we study the differentiability properties of such functions within their domain of convergence.