Convergence of matrix continued fractions

The aim of this work is to give some criteria on the convergence of matrix continued fractions. We begin by presenting some new results which generalize the links between the convergent elements of real continued fractions. Secondly, we give necessary and sufficient conditions for the convergence of continued fractions of matrix arguments. This paper will be completed by illustrating the theoretical results with some examples.     

基于Samelson逆的向量值连分式的收敛准则

The aim of this work is to give some criteria on the convergence of vector valued continued fractions defined by Samelson inverse. We give a new approach to prove the convergence theory of continued fractions. First, by means of the modified classical backward recurrence relation, we obtain a formula between the m-th and n-th convergence of vector valued continued fractions. Second, using this formula, we give necessary and sufficient conditions for the convergence of vector valued continued fractions.     

Matrix-valued continued fractions

We discuss the properties of matrix-valued continued fractions based on Samelson inverse. We begin to establish a recurrence relation for the approximants of matrix-valued continued fractions. Using this recurrence relation, we obtain a formula for the difference between mth and nth approximants of matrix-valued continued fractions. Based on this formula, we give some necessary and sufficient conditions for the convergence of matrix-valued continued fractions, and at the same time, we give the estimate of the rate of convergence. This paper shows that some famous results in the scalar case can be generalized to the matrix case, even some of them are exact generalizations of the scalar results.     

The doubling property of conformal measures of infinite iterated function systems

We provide sufficient conditions for the conformal measures induced by regular conformal infinite iterated function systems to satisfy the doubling property. We apply these conditions to iterated function systems derived from the continued fraction algorithm—continued fractions with restricted entries. For these systems our conditions are expressed in terms of the asymptotic density properties of the allowed entries. As examples, we give some relatively large classes of sets of continued fractions with restricted entries for which the corresponding conformal measures have the doubling property. Similarly, we give some other classes for which the conformal measure does not have the doubling property.     

二元矩阵有理插值函数的构造

一般构造矩阵值有理函数的方法是利用连分式给出的,其算法的可行性不易预知,且计算量大.本文对于二元矩阵值有理插值的计算,通过引入多个参数,定义一对二元多项式:代数多项式和矩阵多项式,利用两多项式相等的充分必要条件通过求解线性方程组确定参数,并由此给出了矩阵值有理插值公式.该公式简单,具有广阔的应用前景.     

On the transcendence and algebraic independence of certain continued fractions

The transcendence of continued fractions =[a ₀;a ₁,a ₂,...] is proved under growth conditions involving the denominatorsq _n of the convergents and shifted partial quotientsa _n+k . Extending this idea, conditions for the algebraic independence of several continued fractions are given. The proofs use the approximation properties of continued fractions in combination with the Thue-Siegel-Roth Theorem or a criterion for algebraic independence of Bundschuh.     

Simple continued fraction solutions for diophantine equations

We present elementary necessary and sufficient conditions for the solvability of the Diophantine equation x² − Dy² = n for any n and any nonsquare integer D > 0, using only simple continued fraction expansions. This includes a simple device for finding fundamental solutions of such equations.     

The area of empty axis-parallel boxes amidst 2-dimensional lattice points

The dispersion of a point set in the unit square is the area of the largest empty axis-parallel box. In this paper we are interested in the dispersion of lattices in the plane, that is, the supremum of the area of the empty axis-parallel boxes amidst the lattice points. We introduce a framework with which to study this based on the continued fractions expansions of the lattice generators. We give necessary and sufficient conditions under which a lattice has finite dispersion. We obtain an exact formula for the dispersion of the lattices associated to subrings of the ring of integers of quadratic fields. We have tight bounds for the dispersion of a lattice based on the largest continued fraction coefficient of the generators, accurate to within one half. We provide an equivalent formulation of Zaremba's conjecture. Using this framework we are able to give very short proofs of previous results.     

Ordered OLPS and CF Nth numerators

Definitions, theorems and examples are established for a general model of Laurent polynomial spaces and ordered orthogonal Laurent polynomial sequences, ordered with respect to ordered bases and orthogonal with respect to inner products ·=L°⊙ decomposed into transition functional ⊙ and strong moment functional, or, more generally, sample functional L couplings. Under this formulation that is shown to subsume those in the existing literature, new fundamental results are produced, including necessary and sufficient conditions for ordered OLPS to be sequences of nth numerators of continued fractions, in contrast to the classical result concerning nth denominators which is shown to hold only in special cases.     

On the D-stability and additive D-stability of matrices and Svicobians

We improve the available necessary conditions and sufficient conditions for the Dstability and additive D-stability of matrices. We define these dynamical properties with respect to a finite parallelepiped and propose an algorithm for checking them. For a particular class of matrices called Svicobians, we obtain some constructive necessary and sufficient conditions for Dstabilizability, D-stability, and additive D-stability.     

The structure of transitive ordered permutation groups

We give some necessary and sufficient conditions for transitive l-permutation groups to be 2-transitive. We also discuss primitive components and give necessary and sufficient conditions for transitive l-permutation groups to be normal-valued.     

Pullbacks of Arithmetical Rings

We give necessary and sufficient conditions that the pullback of a conductor square be a chain ring (i.e., a ring whose ideals are totally ordered by inclusion). We also give necessary and sufficient conditions that the pullback of a conductor square be an arithmetical ring (i.e., a ring which is locally a chain ring at every maximal ideal). For any integral domain D with field of fractions K, we characterize all Prüfer domains R between D[X] and K[X] such that the conductor C of K[X] into R is nonzero. As an application, we show that for n ≥ 2, such a ring R has the n-generator property (every finitely generated ideal can be generated by n elements) if and only if R/C has the same property.     

Hurwitz and Tasoev Continued Fractions

The continued fractions studied by Tasoev are not widely known although their characteristics are very similar to those of Hurwitz continued fractions. Recently, the author found several general forms of Tasoev continued fractions, and by applying this method he also obtained some more general forms of Hurwitz continued fractions belonging to so called tanh-type and tan-type. In this paper, we constitute a new class of general forms of Hurwitz continued fractions of e-type. The known continued fraction expansions of e^1/a (a 1), ae^1/a and (1/a)e^1/a are included as special cases. The corresponding Tasoev continued fractions are also derived.     

Hurwitz and Tasoev Continued Fractions

The continued fractions studied by Tasoev are not widely known although their characteristics are very similar to those of Hurwitz continued fractions. Recently, the author found several general forms of Tasoev continued fractions, and by applying this method he also obtained some more general forms of Hurwitz continued fractions belonging to so called tanh-type and tan-type. In this paper, we constitute a new class of general forms of Hurwitz continued fractions of e-type. The known continued fraction expansions of e^1/a (a 1), ae^1/a and (1/a)e^1/a are included as special cases. The corresponding Tasoev continued fractions are also derived.     

On Koch's convergence criterion for branching continued fractions

Extension of the necessary condition for convergence of branching continued fractions with real positive elements to the case when the remainders are complex numbers leads to a multidimensional analog of Koch's theorem. However the part of the theorem that is known in the literature as the Stern-Stolz convergence criterion does not hold in this case.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 36, 1992, pp. 10–13.     

Solvability of a periodic boundary-value problem for systems of functional differential equations with cyclic matrices

We consider first-order systems of linear functional differential equations with regular operators. For families of systems of two equations we obtain the general necessary and sufficient conditions for the unique solvability of a periodic boundary-value problem. For families of systems of n linear functional differential equations with cyclic matrices we obtain effective necessary and sufficient conditions for the unique solvability of a periodic boundary-value problem.     

Generalized Semi-Infinite Programming: Optimality Conditions Involving Reverse Convex Problems

This article deals with a generalized semi-infinite programming problem (S). Under appropriate assumptions, for such a problem we give necessary and sufficient optimality conditions via reverse convex problems. In particular, a necessary and sufficient optimality condition reduces the problem (S) to a min-max problem constrained with compact convex linked constraints.     

Some twin regions of convergence for branched continued fractions

We study twin regions of convergence for branched continued fractions and establish an estimate of the rate of convergence; we construct a counterexample showing that the natural formulation of Thron's convergence criterion for continued fractions does not extend to branched continued fractions. Translated fromMatematichni Metodi ta Fiziko-Makhanichni Polya, Vol. 39, No. 2, 1996, pp. 62–64.     

Two Results on Brzeziński π-crossed Product

We give the necessary and sufficient conditions for a family of Brzezínski crossed product algebras with suitable comultiplication and counit to be a Hopf π-coalgebra. On the other hand, necessary and sufficient conditions for the Brzeziński π-crossed product A?H to be a coquasitriangular Hopf π-coalgebra are derived, then the category ^A?H ? of the left π-comodules over A?H is braided.     

Nota sulle frazioni continuep-adiche

Summary In this note it is proved that for p-adic continued fractions a result analogous to Galois theorem for ordinary continued fractions holds. Moreover some results concerning the p- adic continued fraction expansions of the square roots of the integers are obtained.