MULTIFRACTAL ANALYSIS OF CONVERGENCE EXPONENTS FOR PRODUCTS OF CONSECUTIVE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS

For each real number x∈(0,1),let[a_(1)(x),a_(2)(x),…,a_n(x),…]denote its continued fraction expansion.We study the convergence exponent defined byτ(x)=inf{s≥0:∞∑n=1(a_(n)(x)a_(n+1)(x))^(-s)<∞},which reflects the growth rate of the product of two consecutive partial quotients.As a main result,the Hausdorff dimensions of the level sets ofτ(x)are determined.

ARBITRARILY LONG ARITHMETIC PROGRESSIONS FOR CONTINUED FRACTIONS OF LAURENT SERIES

A famous theorem of Szemer＇edi asserts that any subset of integers with posi- tive upper density contains arbitrarily arithmetic progressions. Let Fq be a finite field with q elements and Fq（（X^-1）） be the power field of formal series with coefficients lying in Fq. In this paper, we concern with the analogous Szemeredi problem for continued fractions of Laurent series： we will show that the set of points x ∈ Fq（（X-1）） of whose sequence of degrees of partial quotients is strictly increasing and contain arbitrarily long arithmetic progressions is of Hausdorff dimension 1/2.

A SUFFICIENT CONDITION OF CONVERGENCE FOR CLIFFORD CONTINUED FRACTIONS

In this article, a sufficient condition for a Clifford continued fraction to be convergent is established, and some applications are given.

MULTIFRACTAL ANALYSIS OF THE CONVERGENCE EXPONENT IN CONTINUED FRACTIONS

Let x∈(0,1)be a real number with continued fraction expansion[a_(1)(x),a_(2)(x),a_(3)(x),⋯].This paper is concerned with the multifractal spectrum of the convergence exponent of{a_(n)(x)}_(n≥1) defined by τ(x):=inf{s≥0:∑n≥1an^(-s)(x)<∞}.

Two Dual Expansions for Generalized Bivariate Thiele-Type Matrix Valued Interpolating Continued Fractions

A new method for the construction of bivariate matrix valued rational interpolants on a rectangulargrid is introduced. The rational interpolants are expressed in the continued fraction form with scalardenominator. Tile matrix quotients are based oil the generalized inverse for a matrix, Which is found to beeffective in continued fraction interpolation. In this paper, tWo dual expansions for bivariate matrix valuedThiele-type interpolating continued fractions are presented, then, tWo dual rational interpolants are definedout of them.

On Continued Fractions and Their Applications

Continued fractions constitute a very important subject in mathematics. Their importance lies in the fact that they have very interesting and beautiful applications in many fields in pure and applied sciences. This review article will reveal some of these applications and will reflect the beauty behind their uses in calculating roots of real numbers, getting solutions of algebraic Equations of the second degree, and their uses in solving special ordinary differential Equations such as Legendre, Hermite, and Laguerre Equations;moreover and most important, their use in physics in solving Schrodinger Equation for a certain potential. A comparison will also be given between the results obtained via continued fractions and those obtained through the use of well-known numerical methods. Advances in the subject will be discussed at the end of this review article.

Pacman Renormalization in Siegel Parameters of Bounded Type

A novel method of renormalization called Pacman renormalization allows us to study (unicritical) Siegel functions through Pacman-type functions. It has been used to investigate the Siegel parameters with combinatorially periodic rotation number in the main cardioid of the Mandelbrot set. It is already known that it can be defined a Pacman renormalization operator such that for Siegel pacmen, with combinatorially periodic rotation numbers, the operator is compact, analytic and has a unique fixed point, at which it is hyperbolic with one-dimensional unstable manifold. In this paper we observe that this Pacman renormalization operator is compact and analytic at any Siegel Pacman or Siegel map with combinatorially bounded rotation number. This allows us to define a renormalization operator on the hybrid classes of the standard Siegel pacmen to which we built its horseshoe where the operator is topologically semiconjugated to the left shift on the space of bi-infinite sequences of natural numbers bounded by some constant.

On the Family of Thue Equation |x^3+mx^2y-(m+3)xy^2+y^3|=k

The family of cubic Thue equation which depend on two parameters ｜ x^3 ＋ mx^2 y-（m＋3） xy^2＋y^3｜=k is studied. Using rational approximation, we give a smaller upper bound of the solution of the equation, that is quite better than the present result. Moreover, we study two inequalities ｜ x^3 ＋ mx^2y-（m ＋ 3） xy^2＋y^3 ｜ =k≤2m＋3 and ｜x^3 ＋mx^2y- （m＋3）xy^2 ＋ y^3｜ = k≤ （2m＋3）^2 separately. Our result of upper bound make it easy to solve those inequalities by simple method of continuous fraction expansion.

General Structures of Block Based Interpolational Function

We construct general structures of one and two variable interpolation function, without depending on the existence of divided difference or inverse differences, and we also discuss the block based osculatory interpolation in one variable case. Clearly, our method offers marly flexible interpolation schemes for choices. Error terms for the interpolation are determined and numerical examples are given to show the effectlveness of the results.

Werner-Type Matrix Valued Rational Interpolation and Its Recurrence Algorithms

In this paper, a practical Werner-type continued fraction method for solving matrix valued rational interpolation problem is provided by using a generalized inverse of matrices. In order to reduce the continued fraction form to rational function form of the interpolants, an efficient forward recurrence algorithm is obtained.

The Limiting Case of Blending Differences for Bivariate Blending Continued Fraction Expansions

For a univariate function given by its Taylor series expansion, a continued fraction expansion can be obtained with the Viscovatov＇s algorithm, as the limiting value of a Thiele interpolating continued fraction or by means of the determinantal formulas for inverse and reciprocal differences with coincident data points. In this paper, both Viscovatov-like algorithms and Taylor-like expansions are incorporated to yield bivariate blending continued expansions which are computed as the limiting value of bivariate blending rational interpolants, which are constructed based on symmetric blending differences. Numerical examples are given to show the effectiveness of our methods.

Completing Einstein’s Spacetime

The four-dimensional character of Einstein’s spacetime is generally accepted in mainstream physics as beyond reasonable doubt correct. However the real problem is when we require scale invariance and that this spacetime be four-dimensional on all scales. It is true that on our classical scale, the 4D decouples into 3D plus one time dimension and that on very large scale only the curvature of spacetime becomes noticeable. However the critical problem is that such spacetime must remain 4D no matter how small the scale we are probing is. This is something of crucial importance for quantum physics. The present work addresses this basic, natural and logical requirement and shows how many contradictory results and shortcomings of relativity and quantum gravity could be eliminated when we “complete” Einstein’s spacetime in such a geometrical gauge invariant way. Concurrently the work serves also as a review of the vast Literature on E-Infinity theory used here.

THE THEORY OF STATIC DECAY IN COMPUTATIONAL MECHANICS

In this paper, a new mathematical form, matrix, continued fraction (MCF) is introduced to describe the decay of effects of an equilibrant system of forces acting on a sphere of an elastic body. By this way, the famous Saint-Venant's principle is proved often but not always valid in computational mechanics.

CONSIDER SAINT-VENANT'S PRINCIPLE BY MEANS OF CHAIN MODEL

A precise background theory of computational mechanics is formed. Saint_Venant's principle is discussed in chain model by means of this precise theory. The classical continued fraction is developed into operator continued fraction to be the constrictive formulation of the chain model. The decay of effect of a self_equilibrated system of forces in chain model is decided by the convergence of operator continued fraction, so the reasonable part of Saint_Venant's principle is described as the convergence of operator continued fraction. In case of divergence the effect of a self_equilibrated system of forces may be non_zero at even infinite distant sections, so Saint_Venant's principle is not a common principle.

Gravity Field Imaging by Continued Fraction Downward Continuation: A Case Study of the Nechako Basin(Canada)

Interpretation of gravity data plays an important role in the study of geologic structure and resource exploration in the deep part of the earth,like the lower crust,the upper mantle(Lüet al.,2013,2019).The gravity anomaly reflects the lateral resolution of the underground mass distribution.

ON THETA-TYPE FUNCTIONS IN THE FORM(x;q)∞

As in our previous work [14], a function is said to be of theta-type when its asymptotic behavior near any root of unity is similar to what happened for Jacobi theta functions. It is shown that only four Euler infinite products have this property. That this is the case is obtained by investigating the analyticity obstacle of a Laplace-type integral of the exponential generating function of Bernoulli numbers.

Three-Dimensional Generalized Inverse Matrix Rational Interpolation

In this paper, a three dimensional matrix valued rational interpolant (TGMRI) is first constructed by making use of the generalized inverse of matrices. The interpolants are of the Thiele type branched continued fraction form, with matrix numerator and scalar denominator. Some properties of TGMRI are given. An efficient recursive algorithm is proposed. The results in the paper can be extend to n variable.

Continued Fraction Algorithm for Matrix Exponentials

A recursive rational algorithm for matrix exponentials was obtained by making use of the generalized inverse of a matrix in this paper. On the basis of the n th convergence of Thiele type continued fraction expansion, a new type of the generalized inverse matrix valued Padé approximant (GMPA) for matrix exponentials was defined and its remainder formula was proved. The results of this paper were illustrated by some examples.

p-adic Continued Fractions Ⅲ

1.Introduction In order to discuss the irrationality, the transcendence and the algebraic independence for p-adic numbers, the first author introduced in two previous papers [1, 2] a simple form for p-adic continued fraction which is called p-adic simple continued fraction by making use of the algebraic theory of continued fraction in the real field mentioned by Schmidt, and gave a sufficient condition for certain p-adic integers which and whose sum, defference, product and quotient are all p-adic transcendental numbers.