Words and Quadratic Numbers

In this paper we study the balance matrix that gives the order of balance of any binary word. In addition, we define for Christoffel words a new matrix called second order balance matrix. This matrix gives more information on the balance property of a word that codes the number of occurrences of the letter 1 in successive blocks of the same length for the studied Christoffel word. By taking the maximum of the Second order balance matrix we define the second order of balance and we are able to order the Christoffel words according to these values. Our construction uses extensively the continued fraction associated with the slope of each Christoffel word, and we prove a recursive formula based on fine properties of the Stern-Brocot tree to construct second order matrices.

We study the symbolic sequences associated with a two-dimensional nonlinear map describing a digital filter with 2's complement arithmetic, and the role of the filter parameter a = 2cos θ in determining admissibility of these sequences. We solve an appropriately specialized form of Chua and Lin's inequalities governing admissibility, and obtain a complete identification of the infinite class of periodic symbolic sequences admissible in an open interval having parameter value θ = 0 as left end-point. A key step in the argument, easily established by parity for sequences having odd period, requires a more penetrating analysis in the even case. Computer searches for sequences with periods up to 200 have revealed that this class accounts for the greater number of admissible periodic sequences associated with the map. Our solution provides an efficient construction for the periodic symbolic sequences admissible adjacent to zero, parametrized by the period and the number of +− digit pairs present. We also show that the precise interval of admissibility is (0, π/N), where N is the least period, which permits us to relate the sequences to the points of state space that give rise to them. We thereby obtain a bound on the measure of the subset of state space consisting of the points associated with periodic symbolic sequences, improving previously published bounds. A similar identification of the periodic sequences admissible at the far end of the parameter range, adjacent to θ = π, is deduced.

Периодичность и квазипериодичность функциональных непрерывных дробей в гиперэллиптическом поле $$L = \mathbb{Q}(x)(\sqrt{f})$$ имеет более сложную природу, чем периодичность числовых непрерывных дробей элементов квадратичных полей. Известно, что периодичность непрерывной дроби элемента $$\sqrt{f}/h^{g+1}$$, построенной по нормированию, связанному с многочленом h первой степени, эквивалентна наличию нетривиальных S-единиц в поле L рода g и эквивалентна наличию нетривиального кручения в группе классов дивизоров. В данной статье найден точный промежуток значений $$s \in \mathbb{Z}$$ таких, что элементы $$\sqrt{f}/h^s$$ имеют периодическое разложение в непрерывную дробь, где $$f \in \mathbb{Q}[x]$$ - свободный от квадратов многочлен четной степени. Для многочленов f нечетной степени проблема периодичности непрерывных дробей элементов вида $$\sqrt{f}/h^s$$ рассмотрена в статье [5], причем там доказано, что длина квазипериода не превосходит степени фундаментальной S-единицы поля L. Проблема периодичности непрерывных дробей элементов вида $$\sqrt{f}/h^s$$ для многочленов f четной степени является более сложной. Это подчеркивается найденным нами примером многочлена f степени 4, для которого соответствующие непрерывные дроби имеют аномально большую длину периода. Ранее в статье [5] также были найдены примеры непрерывных дробей элементов гиперэллиптического поля L с длиной квазипериода значительно превосходившей степень фундаментальной S-единицы поля L.

To date, the only non-group structure that has been suitably employed as the key space for Diffie-Hellman-type cryptographic key exchange is the infrastructure of a real quadratic (number or function) field. We present an implementation of a Diffie-Hellman-type protocol based on real quadratic number field arithmetic that provides a significant improvement in performance over previous versions of this scheme. This dramatic speed-up is achieved by replacing the ordinary multiplication and reduction procedures for reduced ideals by a new version of the NUCOMP algorithm due to Shanks.

Any sequence claimed to be random, real numbers or fuzzy numbers, must be tested for randomness. We first test our sequence of fuzzy random numbers, obtained from Sobol quasi-random numbers, for randomness using a run test and then a frequency test. We identified two types of triangular shaped fuzzy numbers from Chapter 4: (1) quadratic fuzzy numbers generated from 7-tuples; and (2) quadratic Bezier generated fuzzy numbers (QBGFNs). For reasons given there we direct our attention to QBGFNs. A run test depends on what definition of ≤ between fuzzy numbers we are using. So we do the run test three times on the Bezier fuzzy numbers; first using Buckley’s Method of ≤ (Section 2.6.1) next using Kerre’s Method of ≤ (Section 2.6.2) and lastly using Chen’s Method of ≤ (Section 2.6.3). We must also test our sequence of random fuzzy vectors for randomness. We have seen that sequences of random numbers can pass randomness tests but when they are used to build vectors the resulting sequence of vectors can fail randomness tests (Chapter 3). We will test our sequences of random vectors, whose components are all TFNs, for randomness using a chi-square test.

This is a bibliographical review that provides a brief glimpse into the beauty of continued fractions and how they can be very useful, both in obtaining good rational approximations of a given real number and in their representation as a continued fraction. Furthermore, it presents important properties, such as the relationship between quadratic irrationals and periodic continued fractions. On the other hand, with a view to a potential introduction of this topic in elementary education, a method for obtaining square root approximations through continued fractions is presented. Finally, using more advanced tools, we present an infinite continued fraction representation of the Euler number, which consequently implies the irrationality of e.

A Sturmian sequence related to the uniqueness conjecture for Markoff numbers

Given a polynomial f of odd degree, the nontrivial S-units can be effectively related to the continued fraction expansions of the elements associated with \(\sqrt f \) only in the case where S contains an infinite valuation and a finite valuation determined by first-degree polynomial. A quasi-periodicity criterion for any element of the field of formal power series in a first-degree polynomial is obtained. For key elements, a more accurate criterion is found. The criterion is used to show that, for S specified above, in the presence of a nontrivial S-unit, the expansion of \(\sqrt f \) can be both nonperiodic and periodic. Estimates relating the quasi-period to the degree of the fundamental S-unit are obtained. Examples in which the bounds of these estimates are attained are given.

Hankel continued fraction and its applications

defined over the field of rational numbers Q is normal. They do this by noting that, over a field of characteristic 3, F (X) equals 1/ √ 1 + 1/X. The latter has a periodic continued fraction and, as a consequence, they are able to explicitly verify that F (X) is normal. They then use an “automatic” proof to show that the Laurent series of F (X), defined over Q, is normal. Here we shall give a different proof of the latter result (Theorem 4, below) and show, more generally, that if a formal Laurent series Γ (X) defined over Q is defined and normal (mod p) for any prime p, then it is also normal over Q. We shall also consider the continued fraction of the more general surd 1/ √ 1 + u/X + v/X2 over any field not of characteristic 2. Put δ = u/4−v and note that √ 1 + u/X + v/X2 = 1 + u/(2X) when δ = 0. Hence, when δ = 0, the continued fraction expansion is trivial and is not normal. Define

In the classical case, the connection between the periodicity of the continued fraction of $$\sqrt f $$ and the existence of a fundamental unit of the corresponding hyperelliptic field $$\mathcal{L} = K(x)(\sqrt f )$$ , where K is a field of characteristic different from 2, has long been known. For the element $$\sqrt f $$ , the period length of the continued fraction in $$K((1{\text{/}}x))$$ can be trivially estimated from above by the doubled degree of the fundamental unit. Much more complicated and interesting is the problem of estimating (from above) the period length of other elements of $$\mathcal{L}$$ that have a periodic continued fraction. Among these elements, those of the form $$\sqrt f {\text{/}}{{x}^{s}}$$ , $$s \in \mathbb{Z}$$ , play a key role. For such elements, the period length can be many times greater than the double degree of the fundamental unit. In this article, we find upper bounds for the period length of key elements of hyperelliptic fields $$\mathcal{L}$$ over number fields K. An example is found that demonstrates the sharpness of the proven upper bounds.

Equidistribution properties of quadratic congruential pseudorandom numbers


 
 
 A method based on circle logarithm to prove Goldbach’s conjecture and Pebonacci sequence is proposed. Its essence is to deal with the real infinite series, each of the finite three elements (prime numbers, number series) has asymmetry problems, forming a basic even function one-variable quadratic equation and odd function one-variable three-dimensional number sequence; it is converted to "The irrelevant mathematical model expands latently in a closed interval of 0 to 1," forming a five-dimensional vortex space structure.
 
 

Coset diagrams for the orbit of the groupG=〈x,y∶x2=y4=1〉 acting on real quadratic fields give some interesting information. By using these coset diagrams, we show that in the orbitpG, where\({{p = (a + \sqrt n )} \mathord{\left/ {\vphantom {{p = (a + \sqrt n )} c}} \right. \kern-\nulldelimiterspace} c}\), the non-square positive integern does not change its value and the real quadratic irrational numbers of the formp, wherep and its algebraic conjugate\({{(a - \sqrt n )} \mathord{\left/ {\vphantom {{(a - \sqrt n )} c}} \right. \kern-\nulldelimiterspace} c}\) have different signs, are finite in number and that part of the coset diagram containing such numbers forms a single closed path, which is the only closed path in the orbit ofp.

A high-performance iterative quadratic convergence fixed-point, real and complex number, divider circuit using a previously developed redundant binary inner-product processor core is presented. The intermediate quotient coefficients are kept in redundant binary (RB) form without converting back to normal binary numbers. A unified multiplier using redundant binary representation for both signed and unsigned numbers is investigated. Goldschmidt and Newton-Raphson division methods are compared and the mathematical equivalence of these two methods is provided.