Irrational Research Articles

On radiuses of convergence of q-metallic numbers and related q-rational numbers

The q-rational numbers and the q-irrational numbers are introduced by S. Morier-Genoud and V. Ovsienko. In this paper, we focus on q-real quadratic irrational numbers, especially q-metallic numbers and q-rational sequences which converge to q-metallic numbers, and consider the radiuses of convergence of them when we assume that q is a complex number. We construct two sequences given by recurrence formula as a generalization of the q-deformation of Fibonacci numbers and Pell numbers which are introduced by S. Morier-Genoud and V. Ovsienko. We give an estimation of radiuses of convergence of them, and we solve on conjecture of the lower bound expected which is introduced by L. Leclere, S. Morier-Genoud, V. Ovsienko, and A. Veselov for the metallic numbers and its convergence rational sequence. In addition, we obtain a relationship between the radius of convergence of the [n,n,...,n,...]_q and [n,n,...,n]_q in the case of n=3 and n=4.

The golden number seen in a mechanical oscillator

A seemingly ubiquitous irrational number often appearing in nature and in man-made things like structures, paintings, and physical systems, is the golden number. Here, we show that this astonishing number appears in the periodic solutions of an underactuated mass-spring oscillator driven by a nonlinear self-excitation. Specifically, by using the two-time scale perturbation method, it is analytically demonstrated that the golden number appears in the ratio of amplitudes, as well as in the oscillation frequency of the periodic solution, which is referred to as golden solution and, by applying the Poincaré method, it is demonstrated that this solution is asymptotically stable. Additionally, the analytic results are illustrated by means of numerical simulations and also, an experimental study is conducted.

Computational ghost imaging encryption with a pattern compression from 3D to 0D

The principle of computational ghost imaging (GI) offers a potential application in optical encryption. Nevertheless, large numbers of keys composed of random or specific patterns set an obstacle to its application. Here, we propose a series of pattern compression methods based on computational GI, in which thousands of patterns are replaced by a single standard image (i.e., two-dimensional data), a sequence of numbers (i.e., one-dimensional data) or the fractional part of an irrational number (i.e., zero-dimensional data). Different pattern compression methods are tested in both simulations and experiments, and their error tolerances in encryption are further discussed. Our proposed methods can greatly reduce the pattern amount and enhance encryption security, which pushes forward the application of computational GI, especially in optical encryption.

Partitions into Beatty sequences

Let $$\alpha >1$$ be an irrational number. We establish asymptotic formulas for the number of partitions of n into summands and distinct summands, chosen from the Beatty sequence $$(\lfloor \alpha m\rfloor )$$ . This improves some results of Erdös and Richmond established in 1977.

An Alternative Definition of Irrational Numbers

An algebraic real number is irrational when the index of at least one of its prime factors is incongruent with respect to the desired ischolar_main of the number. In this article the author has discussed about an alternative definition of irrational numbers.

Odd-odd continued fraction algorithm

By using a jump transformation associated to the Romik map, we define a new continued fraction algorithm called odd-odd continued fraction, whose principal convergents are rational numbers of odd denominators and odd numerators. Among others, it is proved that all the best approximating rationals of odd denominators and odd numerators of an irrational number are given by the principal convergents of the odd-odd continued fraction algorithm and vice versa.

Mobius Group Generated by Two Elements of Order 2, 4, and Reduced Quadratic Irrational Numbers

The construction of circuits for the evolution of orbits and reduced quadratic irrational numbers under the action of Mobius groups have many applications like in construction of substitution box (s-box), strong-substitution box (s.s-box), image processing, data encryption, in interest for security experts, and other fields of sciences. In this paper, we investigate the behavior of reduced quadratic irrational numbers (RQINs) in the coset diagrams of the set Q ′ ′ m = η / s : η ∈ Q ∗ m , s = 1 , 2 under the action of group H = < x ′ , y ′ : x ′ 2 = y ′ 4 = 1 > , where m is square free integer and Q ∗ m = a ′ + m / c ′ , a ′ , a ′ 2 − m / c ′ c ′ = 1 , c ′ ≠ 0 . We discuss the type and reduced cardinality of the orbit Q ′ ′ p . By using the notion of congruence, we give the general form of reduced numbers (RNs) in particular orbits under certain conditions on prime p . Further, we classify that for a reduced number r whether − r , r ¯ , − r ¯ lying in orbit or not. AMS Mathematics subject classification (2010): 05C25, 20G401.

Mathematics students’ conceptions and reactions to questions concerning the nature of rational and irrational numbers

In this paper, we present parts of an online qualitative research project which lasted a whole semester about the understanding of real numbers by sixty first-year Greek mathematics undergraduate students during the COVID-19 pandemic. We focus on the distinction between rational and irrational numbers and their location on the number line using straightedge and compass. The results, came of questionnaires, focused interviews and tasks, disclosed some gaps regarding real numbers that the students had from school and revealed confusion between irrational numbers and their decimal approximation. The results also led us to group students’ conceptions of rational and irrational numbers into five categories related to the number and pattern of decimal digits. Teaching practices and perspectives as well as the, negative, impact of the pandemic on teaching and learning mathematics are discussed.

Hofstetter Kurt: Ich Schaue in den Himmel um Mich zu Erden (I Look up to the Sky to Ground Myself)

Hofstetter Kurt: Ich Schaue in den Himmel um Mich zu Erden (I Look up to the Sky to Ground Myself)

Irrational self-similar sets

Let $K\subset\mathbb{R}$ be a self-similar set defined on $\mathbb{R}$. It is easy to prove that if the Lebesgue measure of $K$ is zero, then for Lebesgue almost every $t$, $$K+t=\{x+t:x\in K\}$$ only consists of irrational or transcendental numbers. In this note, we shall consider some classes of self-similar sets, and explicitly construct such $t$'s. Our main idea is from the $q$-expansions.

A CONTINUOUS HOMOMORPHISM OF A THIN SET ONTO A FAT SET

AbstractA thin set is defined to be an uncountable dense zero-dimensional subset of measure zero and Hausdorff measure zero of an Euclidean space. A fat set is defined to be an uncountable dense path-connected subset of an Euclidean space which has full measure, that is, its complement has measure zero. While there are well-known pathological maps of a set of measure zero, such as the Cantor set, onto an interval, we show that the standard addition on $\mathbb {R}$ maps a thin set onto a fat set; in fact the fat set is all of $\mathbb {R}$ . Our argument depends on the theorem of Paul Erdős that every real number is a sum of two Liouville numbers. Our thin set is the set $\mathcal {L}^{2}$ , where $\mathcal {L}$ is the set of all Liouville numbers, and the fat set is $\mathbb {R}$ itself. Finally, it is shown that $\mathcal {L}$ and $\mathcal {L}^{2}$ are both homeomorphic to $\mathbb {P}$ , the space of all irrational numbers.

Rational approximations to two irrational numbers

For real $\xi$ we consider the irrationality measure function $\psi_\xi(t) = \min_{1\leqslant q \leqslant t, q\in\mathbb{Z}} || q\xi ||$, where $||\cdot||$ - distance to the nearest integer. We prove that in the case $\alpha\pm\beta\notin\mathbb{Z}$ there exist arbitrary large values of $t$ with $$\Bigl | \frac{1}{\psi_\alpha(t)} - \frac{1}{\psi_\beta(t)} \Bigl | \geqslant \sqrt5\left(1-\sqrt{\frac{\sqrt5-1}{2}}\right)t.$$ The constant on the right-hand side is optimal.

Diophantine approximation with prime restriction in function fields

In the thirties of the last century, I. M. Vinogradov established uniform distribution modulo 1 of the sequence pα when α is a fixed irrational real number and p runs over the primes. In particular, he showed that the inequality ||pα||≤p−1/5+ε has infinitely prime solutions p, where ||.|| denotes the distance to a nearest integer. This result has subsequently been improved by many authors. The current record is due to Matomäki (2009) who showed the infinitude of prime solutions of the inequality ||pα||≤p−1/3+ε. This exponent 1/3 is considered the limit of the current technology. We prove function field analogues of this result for the fields k=Fq(T) and imaginary quadratic extensions K of k. Essential in our method is the Dirichlet approximation theorem for function fields which is established in general form in the appendix authored by Arijit Ganguly.

On Inhomogeneous Extension of Thue–Roth’s Type Inequality with Moving Targets

AbstractLet $\Gamma \subset \overline{\mathbb Q}^{\times }$ be a finitely generated multiplicative group of algebraic numbers. Let $\delta ,\linebreak\beta \in \overline{\mathbb Q}^\times $ be algebraic numbers with $\beta $ irrational. In this paper, we prove that there exist only finitely many triples $(u, q, p)\in \Gamma \times \mathbb{Z}^2$ with $d = [\mathbb{Q}(u):\mathbb{Q}]$ such that $|\delta qu|>1$ and $$\begin{align*} & 0<|\delta qu+\beta-p|<\frac{1}{H(u)^\varepsilon q^{d+\varepsilon}}, \end{align*}$$where $H(u)$ denotes the absolute Weil height. This is an inhomogeneous analogue of the main theorem in [ 2]. As an application of our result, we also prove a transcendence result, which states as follows: let $\alpha>1$ be a real number. Let $\beta $ be an algebraic irrational number and $\lambda $ be a non-zero real algebraic number. For a given real number $\varepsilon>0$, if there are infinitely many natural numbers $n$ for which $||\lambda \alpha ^n+\beta || < 2^{- \varepsilon n}$ holds true, then $\alpha $ is transcendental, where $||x||$ denotes the distance from its nearest integer. When $\alpha $ and $\beta $ both are algebraic numbers satisfying same conditions, then a particular result of Kulkarni et al. [ 3] asserts that $\alpha ^d$ is a Pisot number. When $\beta $ is an algebraic irrational, our result implies that no algebraic number $\alpha $ satisfies the inequality for infinitely many natural numbers $n$. Also, our result strengthens a result of Wagner and Ziegler [ 7].

The Topological Dimension of Radial Julia Sets

We prove that the meandering set for $$f_a(z)=e^z+a$$ is homeomorphic to the space of irrational numbers whenever a belongs to the Fatou set of $$f_a$$ . This extends recent results by Vasiliki Evdoridou and Lasse Rempe. It implies that the radial Julia set of $$f_a$$ has topological dimension zero for all attracting and parabolic parameters, including all $$a\in (-\infty ,-1]$$ . Similar results are obtained for Fatou’s function $$f(z)=z+1+e^{-z}$$ .

Analytic maps of parabolic and elliptic type with trivial centralisers

We prove that for a dense set of irrational numbers \alpha , the analytic centraliser of the map e^{2\pi i \alpha} z+ z^2 near 0 is trivial. We also prove that some analytic circle diffeomorphisms in the Arnol’d family, with irrational rotation numbers, have trivial centralisers. These provide the first examples of such maps with trivial centralisers.

Renormalization of Bicritical Circle Maps

A general ansatz in Renormalization Theory, already established in many important situations, states that exponential convergence of renormalization orbits implies that topological conjugacies are actually smooth (when restricted to the attractors of the original systems). In this paper we establish this principle for a large class of bicritical circle maps, which are $C^3$ circle homeomorphisms with irrational rotation number and exactly two (non-flat) critical points. The proof presented here is an adaptation, to the bicritical setting, of the one given by de Faria and de Melo for the case of a single critical point. When combined with some recent papers, our main theorem implies $C^{1+\alpha}$ rigidity for real-analytic bicritical circle maps with rotation number of bounded type.

Identification of irrational numbers through elementary functions

The paper explores on the identification of irrational numbers properly focusing on elementary function. Irrational numbers are the primary concern for rational, irrational and algebraic and can be interpreted that all rational are algebraic numbers but all irrational numbers are not algebraic. Reviewing on various books, articles, criticisms, and research on irrational numbers, what I have found that there is no proper research irrational numbers through elementary functions. So, the paper purposes to examine on how irrational number are identified through elementary functions. The paper applies descriptive method prioritizing on qualitative research design adopting data from secondary sources. The significance of the study is for determining irrational numbers via elementary function.

An Irrational Lagrangian Density of a Single Hypergraph

The Turán number of an $r$-uniform graph $F$, denoted by $ex(n,F)$, is the maximum number of edges in an $F$-free $r$-uniform graph on $n$ vertices. The Turán density of $F$ is defined as $\pi(F)=\underset{{n\rightarrow\infty}}{\lim}{ex(n,F) \over {n \choose r }}.$ Denote $\Pi_{\infty}^{(r)}={ \pi(\cal F): \cal F is a family of r{-uniform graphs}},$ $\Pi_{fin}^{(r)}=\{ \pi(\cal F): \cal F {is a \ finite \ family of} r{{-}uniform graphs}\}$, and $\Pi_{t}^{(r)}=\{\pi(\cal F): \cal F is a family of r{-uniform graphs, and}|\cal F|\le t}.$ For graphs, Erdös and Simonovits [Studia Sci. Mat. Hungar. 1 (1966), pp. 51--57] and Erdös and Stone [Bull. Amer. Math. Soc., 52 (1946), pp. 1087--1091] showed that $\Pi_{\infty}^{(2)}=\Pi_{fin}^{(2)}=\Pi_{1}^{(2)}={0, {1 \over 2}, {2 \over 3}, ...,{l-1 \over l}, ...}.$ We know quite little about the Turán density of an $r$-uniform graph for $r\ge 3$. Baber and Talbot [Electron. J. Combin., 19 (2011)] and Pikhurko [Israel J. Math., 20 (2014), pp. 415--454] showed that there is an irrational number in $\Pi_{3}^{(3)}$ and $\Pi_{fin}^{(3)}$, respectively, disproving a conjecture of Chung and Graham [Erdös on Graphs: His Legacy of Unsolved Problems, A. K. Peters, Natick, MA, 1999]. Baber and Talbot [Electron. J. Combin., 19 (2011)] asked whether $\Pi_{1}^{(r)}$ contains an irrational number. The Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems. The Lagrangian density of an $r$-uniform graph $F$ is $\pi_{\lambda}(F)=\sup \{r! \lambda(G):G\;is;F-free\}$, where $\lambda(G)$ is the Lagrangian of an $r$-uniform graph $G$. Sidorenko [Combinatorica, 9 (1989), pp. 207--215] showed that the Lagrangian density of an $r$-uniform hypergraph $F$ is the same as the Turán density of the extension of $F$. In this paper, we show that the Lagrangian density of $F={123, 124, 134, 234, 567}$ (the disjoint union of $K_4^3$ and an edge) is ${\sqrt 3\over 3}$, and consequently, the Turán density of the extension of $F$ is an irrational number, answering the question of Baber and Talbot.

Tunable spatiotemporal resolution photoacoustic microscopy by combining quasi-periodic scanning and register-fusion algorithm

Quasi-periodic scanning combined with a register-fusion algorithm is proposed to realize tunable spatiotemporal resolution photoacoustic microscopy. Quasi-periodic scanning involves an irrational number ratio for the periods of scanning signals in two directions. It can provide sub-pixel spatial sampling for each frame. The proposed method can adjust the temporal and spatial resolutions by changing the data length for image reconstruction. For moving targets, the method can obtain a series of low-resolution images with a high imaging frame rate. A high-spatial-resolution image can be fused from these images using the register-fusion algorithm. The proposed method can acquire both motion and structural details of moving targets.