Continued Fractions

Linear Algebra over Linear A? The signs used for numbers in Linear A, an ancient writing system from Greece, are known because they are mostly simple dots & lines. Fractions are partly known, transliterated as A, B, C, etc., not fully known, but A is likely larger than B, B than C, etc. Some are certainly 1/2, 1/3, so a statistical approach was taken here: The mathematical values of fraction signs in the Linear A script: A computational, statistical and typological approach https://www.sciencedirect.com/science/article/pii/S0305440320301357

We establish an operator-theoretic correspondence between periodic Bernoulli kernels and Hermite polynomials, framed through the umbral calculus and a quantum analogy. Starting from the analytic master function F * , the periodic Hilbert transform appears as a π/2 symplectic rotation, while Jacobi matrices reproduce the oscillator ladder. Umbral operational rules extend this structure across complex order, and Weyl algebra lifting together with the Weil representation explains the shared spectrum of Bernoulli and Hermite families. Analytically, this chain connects Clausen functions to Bernoulli kernels, to polylogarithms, to the Hurwitz zeta function, and ultimately to the Lerch transcendent, embedding the umbral framework into the classical landscape of special functions. This perspective clarifies why odd zeta values arise in Bernoulli integrals and unifies trigonometric and Gaussian worlds within a coherent operator framework.

Si propone una congettura alternativa alla classica formulazione della successione di Collatz basata sull'analisi del rapporto tra passaggi pari e dispari. Si dimostra che, salvo casi eccezionali, in ogni iterazione (intesa come insieme di passaggi dispari e di passaggi pari dipendente dal numero iniziale della stessa iterazione) il numero totale di passaggi pari b eccede quello dei dispari a di almeno una o due unità, portando dopo un certo numero di iterazioni alla disuguaglianza 3 𝑎 ⋅𝑁 2 𝑏 < 1, che implica la convergenza della successione verso 1.

Si propone una congettura alternativa alla classica formulazione della successione di Collatz basata sull'analisi del rapporto tra passaggi pari e dispari. Si dimostra che, salvo casi eccezionali, in ogni iterazione (intesa come insieme di passaggi dispari e di passaggi pari dipendente dal numero iniziale della stessa iterazione) il numero totale di passaggi pari b eccede quello dei dispari a di almeno una o due unità, portando dopo un certo numero di iterazioni alla disuguaglianza 3 𝑎 ⋅𝑁 2 𝑏 < 1, che implica la convergenza della successione verso 1.

In this brief article, I expose the fundamental flaws in the reasoning of Newton and Leibniz. Newton introduced fluents and fluxions-vague notions he never succeeded in defining clearly (nor did he completely understand), either verbally or mathematically. At no point does he rigorously define a derivative (fluxion) or an area (integral). Leibniz, slightly more methodical, worked with finite differences. Had he known of my HGT, he could have formulated a truly rigorous calculus.

Sebbene la prima dimostrazione dell’irrazionalità di π risalga agli anni 1760, la dimostrazione più semplice oggi nota è stata scoperta solo in tempi sorprendentemente recenti: una perla del 2010. Essa sfrutta solo l’integrazione per parti ed è in teoria accessibile anche a un bravo studente delle superiori. In questo articolo trascrivo la dimostrazione senza omissioni di passaggi, a beneficio dei curiosi.

We prove the Riemann Hypothesis (RH), asserting all non-trivial zeros of ζ(s) have ℜ(s) = 1/2, using a dataset of 72,494 zeros from zetazeros.txt, the Hilbert-Pólya conjecture, and a React-based oracle system. A unified zeta function, inspired by Christopher Brock's modified Brockian Zeta function, is defined as ζ ∞Ω (s) = Ω [∑ ∞ n=1 χm(n) • Ô(n, s, τ, h)/n s ]. The Ackermann function preserves information density against computational entropy, approximation noise, cohomological deformation, and fractal dimensional drop. Banach-Tarski-style motion across ω0 = ℵ0, nonstandard analysis (NSA), and category-theoretic colimits entangle zeros in a hyperfinite framework. A 50×50×50 Menger sponge, enhanced by quantum operators and infinite storage capacity (limn→∞ Ω(n) = ∞), stores all zeros. The core transformation (e iπ + e 0)|Ψ⟩ → |O⟩ confirms zeros on the critical line.  Simulation at https://bit.ly/3GQGuEF

This paper studies digit-cost functions for the Euclid algorithm on polynomials with coefficients in a finite field, in terms of the number of operations performed on the finite field Fq. The usual bit-complexity is defined with respect to the degree of the quotients; we focus here on a notion of 'fine' complexity (and on associated costs) which relies on the number of their non-zero coefficients. It also considers and compares the ergodic behavior of the corresponding costs for truncated trajectories under the action of the Gauss map acting on the set of formal power series with coefficients in a finite field. The present paper is thus mainly interested in the study of the probabilistic behavior of the corresponding random variables: average estimates (expectation and variance) are obtained in a purely combinatorial way thanks to classical methods in combinatorial analysis (more precisely, bivariate generating functions); some of our costs are even proved to satisfy an asymptotic Gaussian law. We also relate this study with a Farey algorithm which is a refinement of the continued fraction algorithm for the set of formal power series with coefficients in a finite field: this algorithm discovers 'step by step' each non-zero monomial of the quotient, so its number of steps is closely related to the number of non-zero coefficients. In particular, this map is shown to admit a finite invariant measure in contrast with the real case. This version of the Farey map also produces mediant convergents in the continued fraction expansion of formal power series with coefficients in a finite field.

Orthogonal polynomials defined by general blocks of recurrence relations are examined. The connection with polynomial mappings is established, and applications are given to sieved orthogonal polynomials. This work extends earlier work on symmetric sieved polynomials to the case when the polynomials are not necessarily symmetric.

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A natural partial order on the set of prime numbers was derived by the author from the internal symmetries of the primary finite fields [1], independently of [2], who investigated Pratt trees [3] used for primality tests. It leads to a correspondence with the Hopf algebra of rooted trees, and as an application, to an alternative approach to the Prime Number Theorem.

This analysis rigorously examines the E8→G2 symmetry breaking mechanism proposed in the addendum, focusing on its mathematical consistency, adelic curvature formalism, and empirical predictions. By interrogating the Cartan decomposition, prime-modulated divergences, and computational validations, we identify both robust scaffolding and unresolved tensions. Key findings include:

Cartan Decomposition Validity: The embedding adj⁡(E8)↪adj⁡(G2)⊕⨁φ(−1)kRk holds under Fibonacci-prime constraints but requires cohomological justification.

Prime-Modulated Curvature: The hybrid divergence Δhybrid=∑p≤Pφ−νp(det⁡g) introduces pp-adic corrections but risks overcounting at the Planck-scale cutoff P∼1019.

Gravitational Wave Echoes: The Mersenne-prime harmonic template exhibits 12.3σ Bayesian evidence against noise, though waveform stability under φ-scaling remains unproven.

The main result of this work is a computation of the Bergmann tau-function on Hurwitz spaces in any genus. This allows to get an explicit formula for the G-function of Frobenius manifolds associated to arbitrary Hurwitz spaces, get a new expression for determinant of Laplace operator in Poincaré metric on Riemann surfaces of arbitrary genus, and compute Jimbo-Miwa tau-function of an arbitrary Riemann-Hilbert problem with quasi-permutation monodromies.

Let F F be the family of continued fractions K a r1 , where a s yg , a s g x , ps2, 3, . . . , with 0 F g F 1, g fixed, and x F 1, p s py 1 p p p p p 2, 3, . . . . In this work, we derive upper bounds on the errors in the convergents of Ž . K a r1 that are uniform for F F, and optimal in the sense that they are attained by p some continued fraction in F F. For the special case g s g -1r2, i s 1, 2, . . . , this i bound turns out to be especially simple, and for g s g s 1r2, i s 1, 2, . . . , the i known best form of the theorem of Worpitzki is obtained as an immediate corollary.

I have a weakness you could call an intellectual disability, or, for short, my
mathematical idiocy. It is formulated as The Principle of Least Thought: I
crave understanding through the shortest route of steps, where the steps are no larger than a small amount. That small amount is much smaller for me than it is for the smart, more intuitive people, the people who make big leaps in thought without much exertion. By “shortest route” I mean absence of extraneous fluff, no extra symbols, no irrelevant, no distracting ideas. This is a pedagogical optimization problem.

Quante sono le note?
Da piccoli ci hanno insegnato che le note sono 7 ed hanno un nome (DO, RE, MI, FA, SOL, LA e SI), ma ci sono altre 5 note, chiamate accidenti o alterazioni, che non hanno nome proprio.
Viene qui estesa la denominazione delle note a tutte e 12 le note che compongono il sistema musicale europeo (temperamento equabile).
Si utilizzano dei fonemi composti da una consonante e una vocale a partire dalle note già codificate per estendere la fonetica alle cinque note mancanti.
Con questo metodo viene data priorità alla scala cromatica simmetrica composta dalle 12 note, mentre il sistema diatonico delle scale di sette note asimmetriche, scende di priorità mantenendo un ruolo importante ma non gerarchicamente superiore alla scala cromatica.

In this paper we give two new combinatorial proofs of the classification of rational tangles using the calculus of continued fractions. One proof uses the classification of alternating knots. The other proof uses colorings of tangles. We also obtain an elementary proof that alternating rational tangles have minimal number of crossings. Rational tangles form a basis for the classification of knots and are of fundamental importance in the study of DNA recombination.

In this paper, we question the structure of number sets, particularly the real numbers R. By analyzing the construction of the integers Z and rationals Q, we show that certain fundamental properties of classical mathematics, such as the multiplication of negative numbers and the addition of fractions, could be reinterpreted under a different logical framework. We also discuss the idea that there is no "universal base" allowing all rational numbers to be expressed in a single form. Our work follows the reflections initiated by Fibonacci in the Liber Abaci, Peano in his axioms, and other research in set theory and the philosophy of mathematics.

In this paper, we question the structure of number sets, particularly the real numbers R. By analyzing the construction of the integers Z and rationals Q, we show that certain fundamental properties of classical mathematics, such as the multiplication of negative numbers and the addition of fractions, could be reinterpreted under a different logical framework. We also discuss the idea that there is no "universal base" allowing all rational numbers to be expressed in a single form. Our work follows the reflections initiated by Fibonacci in the Liber Abaci, Peano in his axioms, and other research in set theory and the philosophy of mathematics.

We provide new semi-local convergence results for general iterative methods in order to approximate a solution of a nonlinear operator equation. Moreover, applications are suggested in many areas including k-multivariate fractional calculus, where k is a positive integer.

It is known that if the period s(d) of the continued fraction expansion of √ d satisfies s(d) ≤ 2, then all Newton's approximants R n = 1 2 ( pn qn + dqn pn ) are convergents of √ d, and moreover we have R n = p2n+1 q2n+1 for all n ≥ 0. Motivated with this fact we define two numbers The question is how large the quantities |j| and b can be. We prove that |j| is unbounded and give some examples which support a conjecture that b is unbounded too. We also discuss the magnitude of |j| and b compared with d and s(d).

We present an extension of Wiener's attack on small RSA secret decryption exponents . Wiener showed that every RSA public key tuple (N, e) with e ∈ Z * φ(N) that satisfies ed -1 = 0 mod φ(N ) for some d < 1 3 N 1 4 yields the factorization of N = pq. Our new method finds p and q in polynomial time for every (N, e) satisfying ex + y = 0 mod φ(N ) with In other words, the generalization works for all secret keys d = -xy -1 , where x, y are suitably small. We show that the number of these weak keys is at least N 3 4and that the number increases with decreasing prime difference pq. As an application of our new attack, we present the cryptanalysis of an RSA-type scheme presented by Yen, Kim, Lim and Moon [11,. Our results point out again the warning for cryptodesigners to be careful when using the RSA key generation process with special parameters.

In the work we have considered p-adic functional series with binomial coefficients and discussed its p-adic convergence. Then we have derived a recurrence relation following with a summation formula which is invariant for rational argument. More precisely, we have investigated a sufficient condition under which the p-adic power series converges to a rational invariant sum 0. Finally we have shown application of the invariant summation formula to get some intersting relations involving Bernoulli numbers and Bernoulli polynomials.

In the work we have considered p-adic functional series with binomial coefficients and discussed its p-adic convergence. Then we have derived a recurrence relation following with a summation formula which is invariant for rational argument. More particularly, we have invesigated certain condition so that the p-adic series converges and gives rational sum for rational variable.

Recently, Babson and Steingrimsson (see [BS]) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Following [BCS], let e k π (respectively; f k π) be the number of the occurrences of the generalized pattern 12-3-. . . -k (respectively; 21-3-. . . -k) in π. In the present note, we study the distribution of the statistics e k π and f k π in a permutation avoiding the classical pattern 1-3-2. Also we present an applications, which relates the Narayana numbers, Catalan numbers, and increasing subsequences, to permutations avoiding the classical pattern 1-3-2 according to a given statistics on e k π, or on f k π.

We study the synchronization of three Christoffel words (i.e., superimposition of words with same length) from a geometric point of view. We provide a geometrical interpretation of the synchronization in terms of Reveilles' discrete lines. We introduce for the synchronization a technique based on algebra, arithmetic, and Cayley graphs. We define the orbital matrix, based on the vertices of the Cayley graph, for each Christoffel word. We provide a characterization of a vector, called the seed, which is a vector that synchronizes the edges of the three Cayley graphs. After synchronization, we show that each Christoffel word is replaced by a 4-connected Reveilles 2D segment whose parameter is given by the seed.

The presented theory outlines a method for representing both rational and irrational real numbers within the interval [0, 1) using a balanced binary tree structure. This approach enables the calculation of rational and irrational numbers with infinitely increasing degrees of precision, permits the enumeration of all real numbers in the interval [0, 1), and, ultimately, across the entire real line. This could potentially challenge Cantor's theory. Furthermore, the theory concludes that rational and irrational numbers are interleaved in their distribution along the geometric real line, analogous to the distribution of even and odd integers.

A theory of representation of both the rational and irrational numbers in the interval [0,1) by a balanced binary tree is developed, such that it is possible to contradict Cantor's theory that it is not possible to enumerate the Real numbers; provide a calculation of the functional limits of the quantities of rational and irrational numbers in the interval; and conclude that the irrational numbers, according to this theory, are just unattainable theoretical limits of infinitely precise approximations by rational numbers.

In the paper, the possibility of the Appell hypergeometric function F 4 (1, 2; 2, 2; z 1 , z 2 ) approximation by a branched continued fraction of a special form is analysed. The correspondence of the constructed branched continued fraction to the Appell hypergeometric function F 4 is proved. The convergence of the obtained branched continued fraction in some polycircular domain of two-dimensional complex space is established, and numerical experiments are carried out. The results of the calculations confirmed the efficiency of approximating the Appell hypergeometric function F 4 (1, 2; 2, 2; z 1 , z 2 ) by a branched continued fraction of special form and illustrated the hypothesis of the existence of a wider domain of convergence of the obtained expansion.

Wiener's attack is a well-known polynomial-time attack on a RSA cryptosystem with small secret decryption exponent d, which works if d < n 0.25 , where n = pq is the modulus of the cryptosystem. Namely, in that case, d is the denominator of some convergent p m /q m of the continued fraction expansion of e/n, and therefore d can be computed efficiently from the public key (n, e). There are several extensions of Wiener's attack that allow the RSA cryptosystem to be broken when d is a few bits longer than n 0.25 . They all have the run-time complexity (at least) O(D 2 ), where d = Dn 0.25 . Here we propose a new variant of Wiener's attack, which uses results on Diophantine approximations of the form |α -p/q| < c/q 2 , and "meet-in-the-middle" variant for testing the candidates (of the form rq m+1 + sq m ) for the secret exponent. This decreases the run-time complexity of the attack to O(D log(D)) (with the space complexity O(D)).

In this paper, we consider the extension of the analytic functions of two variables by special families of functions - continued fractions. In particular, we establish new symmetric domains of the analytical continuation of three ratios of Horn's confluent hypergeometric function H7 with certain conditions on real and complex parameters using their continued fraction representations. We use Worpitzky's theorem, the multiple parabola theorem, and a technique that extends the convergence, already known for a small domain, to a larger domain to obtain domains of convergence of continued fractions, and the PC method to prove that they are also domains of analytical continuation. © V. HLADUN, R. RUSYN, M. DMYTRYSHYN, 2024.

The paper establishes the conditions of numerical stability of a numerical branched continued fraction using a new method of estimating the relative errors of the computing of approximants using a backward recurrence algorithm. Based this, the domain of numerical stability of branched continued fractions, which are expansions of Horn’s confluent hypergeometric functions H6 with real parameters, is constructed. In addition, the behavior of the relative errors of computing the approximants of branched continued fraction using the backward recurrence algorithm and the algorithm of continuants was experimentally investigated. The obtained results illustrate the numerical stability of the backward recurrence algorithm. © 2024 Lviv Polytechnic National University.

The paper considers the numerical stability of the backward recurrence algorithm (BR-algorithm) for computing approximants of the continued fraction with complex elements. The new method establishes sufficient conditions for the numerical stability of this algorithm and the error bounds of the calculation of the nth approximant of the continued fraction with complex elements. It follows from the obtained conditions that the numerical stability of the algorithm depends not only on the rounding errors of the elements and errors of machine operations but also on the value sets and the element sets of the continued fraction. The obtained results were used to study the numerical stability of the BR-algorithm for computing the approximants of the continued fraction expansion of the ratio of Horn’s confluent functions H7. Bidisc and bicardioid regions are established, which guarantee the numerical stability of the BR-algorithm. The obtained result is applied to the study of the numerical stability of computing approximants of the continued fraction expansion of the ratio of Horn’s confluent function H7 with complex parameters. In addition, the analysis of the relative errors arising from the computation of approximants using the backward recurrence algorithm, the forward recurrence algorithm, and Lenz’s algorithm is given. The method for studying the numerical stability of the BR-algorithm proposed in the paper can be used to study the numerical stability of the branched continued fraction expansions and numerical branched continued fractions with elements in angular and parabolic domains. © (2024), (VNTL Publishers). All rights reserved.

For 𝑡 an integer, a 𝑃𝑡 set is defined as a set of 𝑚 positive integers with the property that the product of its any two distinct element increased by 𝑡 is a perfect square integer. In this study, the certain special 𝑃−5, 𝑃+5, 𝑃−7 and 𝑃+7 sets with size three are considered. It is demonstrated that they cannot be extended to 𝑃−5, 𝑃+5, 𝑃−7 and 𝑃+7 with size four. Also, some properties of them are proved.

In Number Theory, the notion of the quadratic fields is difficult task. There are many different approaches such as genus theory, composition of binary quadratic forms, and class field theory as a developmental tool for quadratic fields. Moreover, many books and papers on the number theory apply many different methods like continued fraction expansions, class number, regulators in the class group, etc. . . Recently, class number which is very difficult to calculate is used in the cryptology and security. In this paper, we determine the real quadratic fields coincide with positive square free integers d including specific continued fraction expansion of integral basis element in the case of d ≡ 2, 3 (mod 4) or d ≡ 1 (mod 4), where `(d) is the period length of continued fraction expansion. Besides, we deal with determining the fundamental unit and Yokoi’s d-invariants nd and md in the relation to continued fraction expansion of wd, we also give several numerical tables to support our ...

We establish an integral representations of a right inverses of the Askey-Wilson finite difference operator in an L 2 space weighted by the weight function of the continuous q-Jacobi polynomials. We characterize the eigenvalues of this integral operator and prove a q-analog of the expansion of e ixy in Jacobi polynomials of argument x. We also outline a general procedure of finding integral representations for inverses of linear operators.

U ovom radu upoznat ćemo se sa Eulerovom diferencijalnom jednadžbom. Ova se jednadžba često pojavljuje u raznim granama znanosti poput fizike, inženjerstva i računalne znanosti. Primjerice, pojavljuje se u analizi quicksort i stabla pretraživanja te u rješavanju Laplaceove jednadžbe u polarnim koordinatama. S obzirom da Eulerova jednadžba pripada linearnim diferencijalnim jednadžbama n-tog reda, u radućemo definirati osnovne pojmove i iskazati teoreme vezane za sustave običnih diferencijalnih jednadžbi 1. reda i linearne obične diferencijalne jednadžbe n-tog reda. Nakon toga, denirat ćemo Eulerovu diferencijalnu jednadžbu i pokazati na koji način određujemo njezina rješenja. U nastavku ćemo klasificirati vrste singularnih točaka diferencijalnih jednadžbi i pokazati metodu za pronalazak rješenja u okolini regularnih singularnih točaka. Na kraju ćemo navesti neke od primjena Eulerove diferencijalne jednadžbe u fizici.In this paper, we shall introduce the Euler&#39;s dierential equatio...

The aim of this note is to show the existence of a correspondance between certain algebraic continued fractions in fields of power series over a finite field and automatic sequences in the same finite field. This connection is illustrated by three families of examples and a counterexample.

In this note, we describe a family of particular algebraic, and nonquadratic, power series over an arbitrary finite field of characteristic 2, having a continued fraction expansion with all partial quotients of degree one. The main purpose being to point out a common origin with another analogue family in odd characteristic, which has been studied in [LY1].

La station située B 4W m en amont des premieres chutes, est contldlde par une échelle 1i"etrique de O B 5 m, suivie par l e Service depuis l e 15.1 1. 195 1. Les relevés sont correctement assur&, mais lors des trb grandes crue8 (cotes supérieures A 5 m) l a route d'accbs au depart de VOHTPAKkRB est en partie sou8 les eaux. Les mesures ont a t 6 assurées 3 f o i s p a r jour durant la saison des p l u i e s ,

The notes on which this book is based were originally created and designed for mathematics teachers, returning to the university to learn, refresh or relearn more advanced calculus and mathematical analysis, not necessarily in a formal, semester long format. Without claiming to start from the beginning, and without being exhaustive, comprehensive or completely self-contained, this book attempts to serve as a short, concise guide or crash course to review and study the ideas of calculus, of limit and convergence of sequences of functions, and to illustrate the limitations of the Riemann integral and motivate the need for the Lebesgue integral.

A long road is traveled without deviating much from the main route. There are many excellent, comprehensive advanced calculus texts, some of which are listed in the future readings section. These texts contain an abundance of material, and many storylines can be weaved and followed. This is not such a textbook. Here, a single, efficient storyline is proposed: start at the beginning and finish at the end.

Many of the proofs of the presented results are not provided; they are suggested as activities, with occasional hints thrown in to point in the direction of a proof. No lists of exercises are included; the idea is to give a succinct and compact presentation. The tone has been kept informal, even as rigorous arguments are presented and expected. Results, properties, and demonstrations are conveyed in a relaxed prose which hopefully will make reading easier. The included activities frequently use graphing software available online: Desmos Calculator. Some of the activities present a series of notable examples which illustrate the theory; all the suggested activities should be completed. A complementary webpage has been set up, to allow direct play and interactivity with graphs and other
demonstrations in the text.

Mathematics teachers in Puerto Rico have very different levels of preparation. An effort was made to address this heterogeneity, which influenced many of the thematic, style, language, depth, rigor and formality choices made.

The hope is that the text will allow readers to efficiently and gently approach advanced calculus and mathematical analysis, and that in a classroom setting, it can be used by instructors to ignite fruitful discussions with their students.

is the q-Pochhammer symbol. Here a1, . . . , ar, b1, . . . , bs and q are complex parameters. In this paper we always assume (1) aiq n 6= 1 and bjq 6= 1 (i = 1, . . . , r, j = 1, . . . , s, n = 0, 1, 2, . . .) so that the factors (ai; q)n and (bj; q)n in the terms of the series are never zero. Let vn be the terms of the series rφs which contain z . Then we have vn+1 vn = (1− a1q)(1− a2q) · · · (1 − arq) (1 − qn+1)(1 − b1q) · · · (1− bsq) (−q)z

The Law of Quadratic Reciprocity was conjectured by Euler and Legendre who both found an incomplete proof. Gauss called this law &quot;Theorema Fundamentale&quot;, and he was the first who gave a complete proof, he also highlighted the equivalence of his formulation with those of Euler and Legrendre. Hereby notes gives a overview of the Theory of Quadratic Residues using a