Relatively coarse sequential convergence

We generalize the notion of a coarse sequential convergence compatible with an algebraic structure to a coarse one in a given class of convergences. In particular, we investigate coarseness in the class of all compatible convergences (with unique limits) the restriction of which to a given subset is fixed. We characterize such convergences and study relative coarseness in connection with extensions and completions of groups and rings. E.g., we show that: (i) each relatively coarse dense group precompletion of the group of rational numbers (equipped with the usual metric convergence) is complete; (ii) there are exactly exp exp such completions; (iii) the real line is the only one of them the convergence of which is Fréchet. Analogous results hold for the relatively coarse dense field precompletions of the subfield of all complex numbers both coordinates of which are rational numbers.     

Classes of abelian groups related to sequential convergence

We study classes of abelian groups related to sequential com¬pactness and its generalizations (completeness, coarseness and sequential pre-compactness) in convergence groups. In particular, we describe the algebraic structure of the abelian groups on which every coarse convergence is complete and we prove that: i) every abelian group admits a sequentially precompact convergence; ii) every algebraically compact abelian group admits a sequen¬tially compact convergence.     

Symbolic dynamics in investigation of quaternionic Julia sets

We focus our attention on the dynamics of the simplest quaternionic quadratic function f_Q(X) = X² + Q. The discussion can be reduced to a complex parameter Q and a three dimensional subspace. The images of quaternionic Julia sets suggest a natural decomposition. We find that it can be derived from a certain symbolic dynamics giving rise to fractal fibrations. The starting point are the equators and their preimages. If the parameter Q is real, fibrations are trivial, obtained by rotation of the complex Julia set. Repeating itineraries, on the other hand, define curves connecting periodic points.     

Sequential convergences on cyclically ordered groups without Urysohn’s axiom

In this paper we investigate sequential convergences on a cyclically ordered group G which are compatible with the group operation and with the relation of cyclic order; we do not assume the validity of the Urysohn’s axiom. The system convG of convergences under consideration is partially ordered by means of the set-theoretical inclusion. We prove that convG is a Brouwerian lattice. This work has been partially supported by the Slovak Academy of Sciences via the project Center of Excellence-Physics of Information, grant I/2/2005.     

Coloring abelian groups

A sufficient (and necessary, if n=2) condition for the existence of a particular kind of n-coloring of an abelian group is given, and applied to show that (a) the real line is colorable with two colors so that the distance 1 is forbidden for one color, and the distance s>0 for the other, or so that both 1 and s are forbidden for both colors, if and only if s is not the ratio of an odd and an even integer; (b) the chromatic number of Q² and Q³ is 2, but that of Qⁿ is greater than 2 for n>3.     

Bounds on a class of partial partitions of a vector space over gf(2):a graph theoretical approach

A collection Q of linearly independent w-suhicfs of the n-dimensional vector space V(n) over GF(2) is a w-quilt if whenever X and Y are distinct elements of Q, then X is disjoint from the linear span of Y. The main problem is to determine the maximum possibility cardinality of a w-quilt in V(n) for fixed w and n. Here a graph T(Q) is associated with each quilt Q. The connected components of T(Q) are shown to be complete graphs and the structure of the subquilts corresponding to these components is completely determined. By use of Ramsey type arguments these results are shown to lead to new upper bounds on the cardinality of a w-quilt in V(n) where n = w + 2, a case of particular interest.     

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.     

Manin’s conjecture for a class of singular cubic hypersurfaces

Let l > 2 be a fixed positive integer and Q(y) be a positive definite quadratic form in l variables with integral coefficients. The aim of this paper is to count rational points of bounded height on the cubic hypersurface defined by u³ = Q(y)z. We can get a power-saving result for a class of special quadratic forms and improve on some previous work.     

On the multisearching problem for hypercubes

In this paper we give improved bounds for the multisearch problem on a hypercube. This is a parallel search problem where the elements in the structure S to be searched are totally ordered, but where it is not possible to compare in constant time any two given queries q and q′. More precisely, we are given on a n-processor hypercube a sorted n-element sequence S, and a set Q of n queries, and we need to find for each query q Q its location in the sorted S. We present an improved algorithm for the multisearch problem, one that takes O(log n(log log n)³) time on a n-processor hypercube. This problem is fundamental in computational geometry, for example it models planar point location in a slab. We give as application a trapezoidal decomposition algorithm with the same time complexity on a n log n-processor hypercube. The hypercube model for which we claim our bounds is the standard one, SIMD, with O(1) memory registers per processor, and with one-port communication. Each register can store O(log n) bits, so that a processor knows its ID.     

Rates of convergence for continued fractions of irrational numbers

The dynamics of the Gauss Map suggests a way to compare the convergence to a real number ζ ε(0,l) of a continued fraction and the divergence of the orbit of ζ Of particular interest is the comparison of the rate of convergence to ζ of its simple continued fraction and the rate of divergence by the Gauss Map of the orbit of ζ for all irrational numbers in (0,l). We state and prove sharp inequalities for the convergence of the sequence of rational convergents of an irrational number ζ. We show that the product of the rate of convergence of the continued fraction of ζ and the rate of divergence by the Gauss Map of the orbit of ζ equals 1.     

Zero-term rank preservers

We obtain characterizations of those linear operators that preserve zero-term rank on the m×n matrices over antinegative semirings. That is, a linear operator T preserves zero-term rank if and only if it has the form T(X)=P(B∘X)Q, where P, Q are permutation matrices and B∘X is the Schur product with B whose entries are all nonzero and not zero-divisors.     

Precise evaluation of a polynomial at a point given in staggered correction format

The problem of the evaluation in floating-point arithmetic of a polynomial with floating-point coefficients at a point which is a finite sum of floating-point numbers is studied. The solution is obtained as an infinite convergent series of floating-point numbers. The algorithm requires a precise scalar product, but this can always be implemented by software in a high-level language without assembly language routines as we indicate. A convergence result is proved under a very weak restriction on the size of the degree of the polynomial in terms of the unit roundoff u; roughly speaking, the degree should not be larger than the square root of (1 + u)(2u). Even in the particular case when the point at which to evaluate the polynomial reduces to one floating-point number, we find a new simplified algorithm among the whole family that the preceding convergence result allows.

This problem occurs, among others, in the convergence of the Newton method to some real root of the given polynomial p. If we simply use the Horner scheme to evaluate the polynomial p in a neighbourhood of the root, in some cases the evaluation will contain no correct digits and will prevent us from getting convergence even to machine accuracy. The convergence of iterative methods, among which the Newton method, with added perturbations was the central theme of my talk given at the ICCAM'92. The second part will appear in a forthcoming paper. These added perturbations can represent for example forward or backward errors occurring in finite-precision computations.

The problem discussed here appears in validating some hypotheses of these general convergence results (see the forthcoming paper).     

并行求解约束优化问题的QP-free型算法

针对约束块可分的最优化问题,引入序列线性方程组方法和有效集策略,提出了一个求解约束块可分优化问题的QP-free型并行变量分配（PVD）算法.算法中用三个系数具有对称结构的线性方程组来代替PVD算法中的二次规划问题以求解线搜索方向,避免了约束不相容,减小了计算量.并且算法不要求约束是凸的.最后证明了QP-free型PVD算法的全局收敛性.     

More examples on ordered set reconstruction

We show that there are nonisomorphic ordered sets P and Q that have the same maximal and minimal decks and a rank k such that there is a map B from the elements of rank k in P to the elements of rank k in Q such that P{x} is isomorphic to Q{B(x)} for all x of rank k in P. The examples are preceded by a criterion as to when two nonisomorphic ordered sets will have a rank k and a map B as above.     

Some properties of the level convergence topology on fuzzy number space E

In this paper, we study topological structure of level convergence on fuzzy number space Eⁿ and give a characterization of compact subsets in (Eⁿ,τ(l)), where τ(l) is the level convergence topology on Eⁿ.     

拟Fuchs群的收敛指数

根据H2上的双曲距离在拟共形变换下的拟不变性,给出了K-拟共形抛物循环Fuchs群的收敛指数的估计．     

Smoothness of the gradient of weak solutions of degenerate linear equations

Let Q(x) be a nonnegative definite, symmetric matrix such that (Q(x))(1/2) is Lipschitz continuous. Given a real-valued function b(x) and a weak solution u(x) of div(Q▽u) = b, we find sufficient conditions in order that Q(1/2)▽u has some first order smoothness. Specifically, if Ω is a bounded open set in R~n, we study when the components of Q(1/2)▽u belong to the first order Sobolev space W_Q~(1,2)(Ω)defined by Sawyer and Wheeden. Alternately, we study when each of n first order Lipschitz vector field derivatives X_iu has some first order smoothness if u is a weak solution in Ω of ∑_(i=1)~n X′_iX_(iu) + b = 0.We do not assume that {X_i} is a Hormander collection of vector fields in Ω. The results signal ones for more general equations.     

On the orders of directly indecomposable groups

We investigate the set of those integers n for which directly indecomposable groups of order n exist. For even n such groups are easily constructed. In contrast, we show that the density of the set of odd numbers with this property is zero. For each n we define a graph whose connected components describe uniform direct decompositions of all groups of order n. We prove that for almost all odd numbers (i.e., with the exception of a set of density zero) this graph has a single ‘big’ connected component and all other vertices are isolated. We also give an asymptotic formula for the number of isolated vertices of the graph, i.e., for the number of prime divisors q of n such that every group of order n has a cyclic direct factor of order q.     

Siegmund Duality for Continuous Time Markov Chains on ℤ<Stack><Subscript>+</Subscript><Superscript>d</Superscript></Stack>

For the continuous time Markov chain with transition function P(t) on Z₊^d, we give the necessary and sufficient conditions for the existence of its Siegmund dual with transition function P(t). If Q, the q-matrix of P(t), is uniformly bounded, we show that the Siegmund dual relation can be expressed directly in terms of q-matrices, and a sufficient condition under which the Q-function is the Siegmund dual of some Q-function is also given.     

Jacobi-Sobolev-type orthogonal polynomials: Second-order differential equation and zeros

We obtain an explicit expression for the Sobolev-type orthogonal polynomials {Q_n} associated with the inner product

, where p(x) = (1 − x)(1 + x)^β is the Jacobi weight function, ,β> − 1, A₁,B₁,A₂,B₂0 and p, q P, the linear space of polynomials with real coefficients. The hypergeometric representation (₆F₅) and the second-order linear differential equation that such polynomials satisfy are also obtained. The asymptotic behaviour of such polynomials in [−1, 1] is studied. Furthermore, we obtain some estimates for the largest zero of Q_n(x). Such a zero is located outside the interval [−1, 1]. We deduce his dependence of the masses. Finally, the WKB analysis for the distribution of zeros is presented.