Continued Fraction Research Articles

Pegmatite zonation and the use of muscovite as a geochemical indicator for tin-tantalum-tungsten mineralization: Case studies from the Kalehe and Idjwi areas, Democratic Republic of Congo

The central part of the Karagwe-Ankole Belt (KAB) in the Great Lakes area of Central Africa is rich in various rare and strategic elements. The area north of Kalehe and Idjwi island within the Sud-Kivu Province of eastern Democratic Republic of the Congo (DRC) contains Nb, Ta, Sn and W mineralization hosted in pegmatites and quartz veins that are related to leucogranite intrusions. In this work, two granite-pegmatite fields were studied: the Bugarula field in the northern part of Idjwi island and the Kalehe field, along the west coast of Lake Kivu, with a view to determine the co-genetic link between different pegmatites and leucogranites in these regions. Based on mineralogical assemblages, five pegmatite zones were identified, occurring with some spatial overlap, but with an overall increasing distance from the inferred parental granite: 1) albite pegmatites, 2) two-mica pegmatites, 3) muscovite pegmatites, 4) beryl-bearing Nb–Ta–Sn mineralized pegmatites and 5) tourmaline-bearing Nb–Ta–Sn mineralized pegmatites. Muscovite compositions were analyzed from each zone and used to calculate pegmatite fractionation trends by using Rayleigh fractionation modelling. The model uses published partition coefficients (Kd) and measured contents of K, Rb and Cs in muscovite. Muscovites from the consecutive pegmatite zones in both the Bugarula and Kalehe pegmatite fields record a continuous fractionation trend, indicating a co-genetic magmatic evolution. A parental leucogranitic composition of 3.6 wt% K2O, 8 ppm Cs and 300 ppm Rb was iteratively determined to yield a best fit fractionation trend with the data. This composition is within the range of exposed leucogranites sampled in this study, which average at 4.2 wt% K2O, 7 ppm Cs and 400 ppm Rb (n = 12). It should be noted that individually, none of the sampled leucogranites gave satisfactory fractionation trajectories that would suggest a direct co-genetic link between them and the pegmatites. In the Idjwi granite-pegmatite field, the degree of fractionation with respect to the inferred parental leucogranite ranges from 20% to 99% for the least fractionated and most evolved Nb–Ta–Sn mineralized pegmatites, respectively. In the Kalehe area, fractionation values also range from 22% up to 99% for the least fractionated and most evolved (mineralized) pegmatites, respectively. Muscovites from mineralized pegmatites of zones 4 and 5 show high concentrations of Nb, Ta and Sn. As such, we show that muscovite compositions in these pegmatites provide a powerful tool to track Nb–Ta, Sn (and W) enrichment and potential mineralization. These high degrees of fractionation associated with the enrichment in economically interesting elements are similar to reported values for pegmatite fields elsewhere in the Karagwe-Ankole Belt.

Continued fractions related to a group of linear fractional transformations

Abstract There are strong relations between the theory of continued fractions and groups of linear fractional transformations. We consider the group G 3 , 3 {G}_{3,3} generated by the linear fractional transformations a = 1 − 1 ∕ z a=1-1/z and b = z + 2 b=z+2 . This group is the unique subgroup of the modular group PSL ( 2 , Z ) {\rm{PSL}}(2,{\mathbb{Z}}) with index 2. We calculate the cusp point of an element given as a word in generators. Conversely, we use the continued fraction expansion of a given rational number p ∕ q p/q , to obtain an element in G 3 , 3 {G}_{3,3} with cusp point p ∕ q p/q . As a result, we say that the action of G 3 , 3 {G}_{3,3} on rational numbers is transitive.

On the asymptotic behaviour of Sudler products for badly approximable numbers

Given a badly approximable number α, we study the asymptotic behaviour of the Sudler product defined by PN(α)=∏r=1N2|sin⁡πrα|. We show that liminfN→∞PN(α)=0 and limsupN→∞PN(α)/N=∞ whenever the sequence of partial quotients in the continued fraction expansion of α exceeds 7 infinitely often. This improves results obtained by Lubinsky for the general case, and by Grepstad, Neumüller and Zafeiropoulos for the special case of quadratic irrationals. Furthermore, we prove that this threshold value 7 is optimal, even when restricting α to be a quadratic irrational, which gives a negative answer to a question of the latter authors.

Dynamics of Ostrowski skew-product: 1. Limit laws and Hausdorff dimensions

We present a dynamical study of Ostrowski’s map based on the use of transfer operators. The Ostrowski dynamical system is obtained as a skew-product of the Gauss map (it has the Gauss map as a base and intervals as fibers) and produces expansions of real numbers with respect to an irrational base given by continued fractions. By studying spectral properties of the associated transfer operators, we show that the absolutely continuous invariant measure of the Ostrowski dynamical system has exponential mixing properties. We deduce a central limit theorem for random variables of an arithmetic nature, and motivated by applications in inhomogeneous Diophantine approximation, we also get Bowen–Ruelle type implicit estimates in terms of spectral elements for the Hausdorff dimension of a bounded digit set.

ON THE FINITENESS OF THE SET OF GENERALIZED JACOBIANS WITH NONTRIVIAL TORSION POINTS OVER ALGEBRAIC NUMBER FIELDS

For a smooth projective curve \(\mathcal{C}\) defined over algebraic number field k, we investigate the question of finiteness of the set of generalized Jacobians \({{J}_{\mathfrak{m}}}\) of a curve \(\mathcal{C}\) associated with modules \(\mathfrak{m}\) defined over k such that a fixed divisor representing a class of finite order in the Jacobian J of the curve \(\mathcal{C}\) provides the torsion class in the generalized Jacobian \({{J}_{\mathfrak{m}}}\). Various results on the finiteness and infiniteness of the set of generalized Jacobians with the above property are obtained depending on the geometric conditions on the support of \(\mathfrak{m}\), as well as on the conditions on the field \(k\). These results were applied to the problem of the periodicity of a continuous fraction decomposition constructed in the field of formal power series \(k((1{\text{/}}x))\), for the special elements of the field of functions \(k(\tilde {\mathcal{C}})\) of the hyperelliptic curve \(\tilde {\mathcal{C}}:{{y}^{2}} = f(x)\).

Counting derangements with signed right-to-left minima and excedances

Recently Alexandersson and Getachew proved some multivariate generalizations of a formula for enumerating signed excedances in derangements. In this paper we first relate their work to a recent continued fraction for permutations and confirm some of their observations. Our second main result is two refinements of their multivariate identities, which clearly explain the meaning of each term in their main formulas. We also explore some similar formulas for permutations of type B.

Convergence conditions for p-adic continued fractions

Continued fractions have been introduced in the field of p-adic numbers Qp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}_p$$\\end{document} by several authors. However, a standard definition is still missing since all the proposed algorithms are not able to replicate all the properties of continued fractions in R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}$$\\end{document}. In particular, an analogue of the Lagrange’s Theorem is not yet proved for any attempt of generalizing continued fractions in Qp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}_p$$\\end{document}. Thus, it is worth to study the definition of new algorithms for p-adic continued fractions. The main condition that a new method needs to fulfill is the convergence in Qp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}_p$$\\end{document} of the continued fractions. In this paper we study some convergence conditions for continued fractions in Qp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}_p$$\\end{document}. These results allow to define many new families of continued fractions whose convergence is guaranteed. Then we provide some new algorithms exploiting the new convergence condition and we prove that one of them terminates in a finite number of steps when the input is rational, as it happens for real continued fractions.

2-charge circular fuzz-balls and their perturbations

We perform a detailed study of perturbations around 2-charge circular fuzz-balls and compare the results with the ones obtained in the case of ‘small’ BHs. In addition to the photon-sphere modes that govern the prompt ring-down, we also find a branch of long-lived QNMs localised inside the photon-sphere at the (meta)stable minimum of the radial effective potential. The latter are expected to dominate late time signals in the form of ‘echoes’. Moreover, contrary to ‘small’ BHs, some ‘static’ tidal Love numbers are non-zero and independent of the mass, charges and angular momentum of the fuzz-ball. We rely on the recently established connection between BH or fuzz-ball perturbation theory and quantum Seiberg-Witten curves for mathcal{N} = 2 SYM theories, which in turn are related to Liouville CFT via the AGT correspondence. We test our results against numerical results obtained with Leaver’s method of continuous fractions or Breit-Wigner resonance method for direct integration and with the WKB approximation based on geodesic motion. In the probe approximation, we also exclude rotational super-radiance, due to the absence of an ergo-region, and absorption, due to the absence of a horizon.

Quasinormal modes and stability of higher dimensional rotating black holes under massive scalar perturbations

We consider the stability of six-dimensional singly rotating Myers-Perry black holes under massive scalar perturbations. Using Leaver's continued fraction method, we compute the quasinormal modes of the massive scalar fields. All modes found are damped under the quasinormal boundary conditions. It is also found that long-living modes called quasiresonances exist for large scalar masses as in the four-dimensional Kerr black hole case. Our numerical results provide a direct and complement evidence for the stability of six-dimensional MP black holes under massive scalar perturbation.

On Ramanujan’s continued fractions of order twenty-four

Two continued fractions U(q) and V(q) of order twenty-four are obtained from a general continued fraction identity of Ramanujan. Some theta-function and modular identities for U(q) and V(q) are established to prove general theorems for the explicit evaluations of U(±q) and V(±q). From the theta-function identities of U(q) and V(q), three colour partition identities are derived as application to partition theory of integer. Further, 2-, 4- and 8-dissection formulas are established for the continued fractions U∗(q)≔q−5/2U(q) and V∗(q)≔q−1/2V(q), and their reciprocals.

New Equivalents of Kurepa’s Hypothesis for Left Factorial

Kurepa’s hypothesis for the left factorial has been an unsolved problem for more than 50 years. In this paper, we have proposed new equivalents for Kurepa’s hypothesis for the left factorial. The connection between the left factorial and the continued fractions is given. The new equivalent based on the properties of the integer part of real numbers is proven. Moreover, a new equivalent based on the properties of two well-known sequences is given. A new representation of the left factorial is listed. Since derangement numbers are closely related to Kurepa’s hypothesis, we made some notes about the derangement numbers and defined a new sequence of natural numbers based on the derangement numbers. In this paper, we indicate a possible direction for further research through solving quadratic equations.

Combinatorial properties of multidimensional continued fractions

The study of combinatorial properties of mathematical objects is a very important research field and continued fractions have been deeply studied in this sense. However, multidimensional continued fractions, which are a generalization arising from an algorithm due to Jacobi, have been poorly investigated in this sense, up to now. In this paper, we propose a combinatorial interpretation of the convergents of multidimensional continued fractions in terms of counting some particular tilings, generalizing some results that hold for classical continued fractions.

Positive bidiagonal factorization of tetradiagonal Hessenberg matrices

Recently, a spectral Favard theorem was presented for bounded banded lower Hessenberg matrices that possess a positive bidiagonal factorization. The paper establishes conditions, expressed in terms of continued fractions, under which an oscillatory tetradiagonal Hessenberg matrix can have such a positive bidiagonal factorization. Oscillatory tetradiagonal Toeplitz matrices are examined as a case study of matrices that admit a positive bidiagonal factorization. Furthermore, the paper proves that oscillatory banded Hessenberg matrices are organized in rays, where the origin of the ray does not have a positive bidiagonal factorization, but all the interior points of the ray do have such a positive bidiagonal factorization.

Learning to Extrapolate Using Continued Fractions: Predicting the Critical Temperature of Superconductor Materials

In the field of Artificial Intelligence (AI) and Machine Learning (ML), a common objective is the approximation of unknown target functions y=f(x) using limited instances S=(x(i),y(i)), where x(i)∈D and D represents the domain of interest. We refer to S as the training set and aim to identify a low-complexity mathematical model that can effectively approximate this target function for new instances x. Consequently, the model’s generalization ability is evaluated on a separate set T={x(j)}⊂D, where T≠S, frequently with T∩S=∅, to assess its performance beyond the training set. However, certain applications require accurate approximation not only within the original domain D but in an extended domain D′ that encompasses D as well. This becomes particularly relevant in scenarios involving the design of new structures, where minimizing errors in approximations is crucial. For example, when developing new materials through data-driven approaches, the AI/ML system can provide valuable insights to guide the design process by serving as a surrogate function. Consequently, the learned model can be employed to facilitate the design of new laboratory experiments. In this paper, we propose a method for multivariate regression based on iterative fitting of a continued fraction, incorporating additive spline models. We compare the performance of our method with established techniques, including AdaBoost, Kernel Ridge, Linear Regression, Lasso Lars, Linear Support Vector Regression, Multi-Layer Perceptrons, Random Forest, Stochastic Gradient Descent, and XGBoost. To evaluate these methods, we focus on an important problem in the field, namely, predicting the critical temperature of superconductors based on their physical–chemical characteristics.

Large algebraic integers

An algebraic integer is said large if all its real or complex embeddings have absolute value larger than [Formula: see text]. An integral ideal is said large if it admits a large generator. We investigate the notion of largeness, relating it to some arithmetic invariants of the field involved, such as the regulator and the covering radius of the lattice of units. We also study its connection with the Weil height and the Bogomolov property. We provide an algorithm for testing largeness and give some applications to the construction of floor functions arising in the theory of continued fractions.

A semi-analytic method for the computation of the effective properties of composites of two isotropic constituent materials

We develop a computational framework for the semi-analytic representation of the effective transport properties of two-component composite materials, in the spirit of Bergman’s spectral representation. The components of the effective permittivity (or conductivity) tensor are expanded as power series in a contrast parameter between the two phases. The coefficients (the moments of a spectral measure) of these expansions are determined recursively through the numerical evaluations of integrals defined solely on the boundary between the constituent phases, and discretized by high-order (possibly exponentially converging) quadrature schemes. The high precision reached by these coefficients allows the series representation to be recast into Padé approximants (or continued fractions) to provide analytical expressions valid over large (possibly infinite) regions of the parameter complex plane. Examples, thus far limited to two-dimensional rhombic unit cells, demonstrate that the method is reliable for microgeometries and material parameters for which other methods are far less dependable. The analytical representations are particularly useful to study the optical or electrostatic resonant behaviour of some composite materials.

Approximation of Differential Operators with Boundary Conditions

The use of spectral methods for solution of boundary value problems is very effective but involves great technical difficulties associated with the implementation of the boundary conditions. There exist several methods of such an implementation, but they are either very cumbersome or require a preliminary analysis of the problem and its reduction to an integral form. We propose a universal means of implementation of the boundary conditions for linear differential operators on a finite interval, which is very simple in its realization. The use of the rational arithmetic allows to assess the effectiveness of this method without interference of the round-off errors. We apply this approach for computation of rational approximations for some fundamental constants. We obtained approximations that in a number of cases are better than those that are given by convergents of regular continued fractions of these constants.

Some results related to Hurwitz stability of combinatorial polynomials

Many important problems are closely related to the zeros of certain polynomials derived from combinatorial objects. The aim of this paper is to observe some results and applications for the Hurwitz stability of polynomials in combinatorics and study other related problems.We first present a criterion for the Hurwitz stability of the Turán expressions of recursive polynomials. In particular, it implies the q-log-convexity or q-log-concavity of the original polynomials. We also give a criterion for the Hurwitz stability of recursive polynomials and prove that the Hurwitz stability of any palindromic polynomial implies its semi-γ-positivity, which illustrates that the original polynomial with odd degree is unimodal. In particular, we get that the semi-γ-positivity of polynomials implies their parity-unimodality and the Hurwitz stability of polynomials implies their parity-log-concavity. Those results generalize the connections between real-rootedness, γ-positivity, log-concavity and unimodality to Hurwitz stability, semi-γ-positivity, parity-log-concavity and parity-unimodality (unimodality). As applications of these criteria, we derive some Hurwitz stability results occurred in the literature in a unified manner. In addition, we obtain the Hurwitz stability of Turán expressions for alternating run polynomials of types A and B and the Hurwitz stability for alternating run polynomials defined on a dual set of Stirling permutations.Finally, we study a class of recursive palindromic polynomials and derive many nice properties including Hurwitz stability, semi-γ-positivity, non-γ-positivity, unimodality, strong q-log-convexity, the Jacobi continued fraction expansion and the relation with derivative polynomials. In particular, these properties of the alternating descents polynomials of types A and B can be implied in a unified approach.

Representation of Some Ratios of Horn’s Hypergeometric Functions H7 by Continued Fractions

The paper deals with the problem of representation of Horn’s hypergeometric functions via continued fractions and branched continued fractions. We construct the formal continued fraction expansions for three ratios of Horn’s hypergeometric functions H7. The method employed is a two-dimensional generalization of the classical method of constructing a Gaussian continued fraction. It is proved that the continued fraction, which is an expansion of each ratio, uniformly converges to a holomorphic function of two variables on every compact subset of some domain of C2, and that this function is an analytic continuation of such a ratio in this domain. To illustrate this, we provide some numerical experiments at the end.

A new algorithm for 𝑝-adic continued fractions

Continued fractions in the field of p p -adic numbers have been recently studied by several authors. It is known that the real continued fraction of a positive quadratic irrational is eventually periodic (Lagrange’s Theorem). It is still not known if a p p -adic continued fraction algorithm exists that shares a similar property. In this paper we modify and improve one of Browkin’s algorithms. This algorithm is considered one of the best at the present time. Our new algorithm shows better properties of periodicity. We show for the square root of integers that if our algorithm produces a periodic expansion, then this periodic expansion will have pre-period one. It appears experimentally that our algorithm produces more periodic continued fractions for quadratic irrationals than Browkin’s algorithm. Hence, it is closer to an algorithm to which an analogue of Lagrange’s Theorem would apply.