Irrational Research Articles

Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point

Abstract A singular nonlinear differential equation z σ d w d z = a w + z w f ( z , w ) , {z}^{\sigma }\frac{{\rm{d}}w}{{\rm{d}}z}=aw+zwf\left(z,w), where σ > 1 \sigma \gt 1 , is considered in a neighbourhood of the point z = 0 z=0 located either in the complex plane C {\mathbb{C}} if σ \sigma is a natural number, in a Riemann surface of a rational function if σ \sigma is a rational number, or in the Riemann surface of logarithmic function if σ \sigma is an irrational number. It is assumed that w = w ( z ) w=w\left(z) , a ∈ C ⧹ { 0 } a\in {\mathbb{C}}\setminus \left\{0\right\} , and that the function f f is analytic in a neighbourhood of the origin in C × C {\mathbb{C}}\times {\mathbb{C}} . Considering σ \sigma to be an integer, a rational, or an irrational number, for each of the above-mentioned cases, the existence is proved of analytic solutions w = w ( z ) w=w\left(z) in a domain that is part of a neighbourhood of the point z = 0 z=0 in C {\mathbb{C}} or in the Riemann surface of either a rational or a logarithmic function. Within this domain, the property lim z → 0 w ( z ) = 0 {\mathrm{lim}}_{z\to 0}w\left(z)=0 is proved and an asymptotic behaviour of w ( z ) w\left(z) is established. Several examples and figures illustrate the results derived. The blow-up phenomenon is discussed as well.

A threshold for the best two-term underapproximation by Egyptian fractions

Let G be the greedy algorithm that, for each θ∈(0,1], produces an infinite sequence of positive integers (an)n=1∞ satisfying ∑n=1∞1/an=θ. For natural numbers p<q, let Υ(p,q) denote the smallest positive integer j such that p divides q+j. Continuing Nathanson’s study of two-term underapproximations, we show that whenever Υ(p,q)⩽3, G gives the (unique) best two-term underapproximation of p/q; i.e., if 1/x1+1/x2<p/q for some x1,x2∈N, then 1/x1+1/x2⩽1/a1+1/a2. However, the same conclusion fails for every Υ(p,q)⩾4. Next, we study stepwise underapproximation by G. Let em=θ−∑n=1m1/an be the mth error term. We compare 1/am to a superior underapproximation of em−1, denoted by N/bm (N∈N⩾2), and characterize when 1/am=N/bm. One characterization is am+1⩾Nam2−am+1. Hence, for rational θ, we only have 1/am=N/bm for finitely many m. However, there are irrational numbers such that 1/am=N/bm for all m. Along the way, various auxiliary results are encountered.

Irrationality exponent and convergence exponent in continued fraction expansions

Let x∈(0,1) be an irrational number with continued fraction expansion [a1(x),a2(x),⋯,an(x),⋯] . We give the multifractal spectrum of the irrationality exponent and the convergence exponent of x defined by v(x):=sup{v>0:|x−pq|<1qv for infinitely many (q,p)∈ℕ×ℤ} and τx:=infs⩾0:∑n⩾1an−sx<∞ respectively. To be precise, we completely determine the Hausdorff dimension of Eα,v=x∈0,1: τx=α, vx=v for any α⩾0, v⩾2 .

Metrical properties of the large products of partial quotients in continued fractions

The study of products of consecutive partial quotients in the continued fraction arises naturally out of the improvements to Dirichlet’s theorem. We study the distribution of the two large products of partial quotients among the first n terms. More precisely, writing the continued fraction expansion of an irrational number , for a non-decreasing function , we completely determine the size of the set F2(φ)=x∈[0,1):∃1⩽k≠l⩽n, ak(x)ak+1(x)⩾φ(n),al(x)al+1(x)⩾φ(n) for infinitely many n∈N in terms of Lebesgue measure and Hausdorff dimension.

Odd‐Stoichiometry and Secondary Phase Relations in Higher Manganese Silicides

AbstractHigher manganese silicides (denoted as MnSiγ) are a prominent class of intermetallic compound for thermoelectric applications that crystallizes into a Nowotny Chimney Ladder (NCL) phase and constitutes a higher concentration of silicon with odd stoichiometry, typically represented by an irrational number. In this work, the genesis of Si odd stoichiometry and secondary phase relations in silicon‐rich and silicon‐deficient MnSix is presented to clarify the significance of labile (i.e., easily alterable) Si‐subsystem in inheriting the modulated lattice structure with characteristic anharmonicity. The transport properties show an interrelation to the presence of a secondary phase, which is driven by the odd stoichiometry, where the thermoelectric figure of merit (zT) is found to be optimal for MnSi1.74. The structural characterization of rapidly solidified melt‐spun ribbons and spark plasma sintered bulk specimens reveal a characteristic presence of MnSi and Si as secondary phases in Si‐deficient (x ≈ 1.5) and Si‐excess (x ≈ 2) compositions of nominal MnSix, respectively, that coexists with the major MnSiγ based NCL phase. This study provides the fundamental basis of phase evolution and the relevance of odd stoichiometry in silicon‐rich manganese silicides which exhibit fascinating phonon dynamics in optimal MnSi1.74 stoichiometry.

Democracy Incentivizes Bullshit

Democracies face an epistemic crisis: incentivizing bullshit. Here “bullshit”—coined by philosopher Harry Frankfurt—means convincing truth-insensitive statements or claims. This paper focuses on several democratic factors that incentivize bullshit: deliberative transparency, epistemic spillover effects, and rational irrationality. These factors pollute the epistemic commons, decrease institutional trust, and enact epistemic injustice. Unfortunately, it is difficult to separate democratic governance from incentivizing bullshit.

The Literary Genre of Tales in Teaching Irrational Numbers

The Literary Genre of Tales in Teaching Irrational Numbers

On the convergence of periodic continued fractions via linearly recurring sequences

In this paper, we give a new process to study the convergence of periodic continued fractions, making use of the approach on the linear difference equations with periodic coefficients, based on the analytic formula of the linearly recurring sequences. Some new explicit expressions of nth numerators and nth denominators of the convergents of periodic continued fractions lead to discuss their convergence. Moreover, we provide a new discussion on the closed link between quadratic irrational numbers and simple periodic continued fractions. Some applications of the Pell’s equations are given and some numerical examples which illustrate the theory are supplied.

Identification of Algebraic and Transcendental Numbers Using Some Elementary Functions

In this paper, to identify irrational numbers while appropriately emphasizing basic functions. All rational numbers can be expressed algebraically, but not all irrational numbers. Rational, irrational, algebraic numbers and transcendental numbers are mostly concerned with irrational numbers using some elementary functions.

Golden Ratio: Construction, Geometry, Beauty, and Diversity

Nature is attractive due to the proper ratios of various components in them and in between others. In nature, leaves, flowers, fruits, Himalayas, hills, valleys, springs, rivers, etc. show a rhythmic balance and seem harmonic and aesthetically pleasing. The golden ratio or the golden (divine) proportion is considered the most pleasing to human visual sensation and is not limited only to aesthetic beauty but also found prominently in the natural world. It is an irrational number denoted by φ and is approximately 1.618034. It is a fascinating topic for mathematicians, artists, natural scientists, philosophers, biologists, artists, architects, musicians, psychologists, designers, etc.&#x0D; This research work presents a panoramic view of the golden ratio; their construction algorithms, and structures; mathematical presentations, geometry, patterns, and properties from mathematical perspectives and opens the broad horizon for further research. Besides these, the paper also presents their existences, diversities, their relationships with nature, the universe, arts, design, architecture, and engineering in ancient and modern mathematics and sciences.

Secondary School Math Teachers' Views on Irrational Numbers: A Qualitative Study

Bu araştırma ortaokul matematik öğretmenlerinin irrasyonel sayılara ilişkin görüşlerini belirlemeye yönelik nitel bir araştırmadır. Araştırma, ortaokulda görev yapmakta olan matematik öğretmenlerinin irrasyonel sayılara ilişkin algılarını belirleme amacı taşımaktadır. Yapılan bu araştırmada fenomonolojik desen kullanılmıştır. Araştırmaya halen ortaokulda matematik öğretmenliği görevini ifa etmekte olan 31 öğretmen dâhil olmuştur. Veri toplama aracı olarak on sorudan oluşan yarı yapılandırılmış görüşme formu kullanılmış, görüşmeler yüz yüze veya çevrimiçi görüşmeler şeklinde yürütülmüştür. Verilerin çözümlenmesinde içerik analizine başvurulmuştur. Araştırma sonucunda katılımcıların irrasyonel sayılara ilişkin istenen düzeyde bir deneyime sahip olmadıkları, bunun sonucunda irrasyonel sayılara ilişkin yeterli bir farkındalığa sahip olmadıkları görülmüştür. Özellikle gündelik yaşam sürecinde irrasyonel sayılarla yeterli bir düzeyde karşılaşmayan öğretmenlerin bu sayılara ilişkin algıları ve kavramsal farkındalıklarının yetersiz olduğu görülmüştür. Buna göre günlük yaşamda daha hassas hesaplamalar için anahtar bir rol oynayan irrasyonel sayıların öneminin yeterli düzeyde kavranmadığı kanısı oluşmuştur. Bu durum öğretmenlerin hizmet öncesi eğitimlerinde olduğu gibi hizmet sürecinde de irrasyonel sayılara ilişkin beklenen kazanımlarının gözden geçirilmesi gerektiği kanaatine yol açmıştır.

The Amalgamation Formulae and Iterations of Mono-digit Pedigree

In this paper we present equations to generate the concatenation of natural numbers A∈N. The main ingredient of our study is the innocuous repunit of a number R_A ; and in this study we introduce three amalgamation operators: ↕ for the constant, ↑ for the incremental and ↓ for the detrimental cases of concatenation according to a variable arithmetic progression. Our method of amalgamation demonstrates the generation of common number sequences, as well as how any two different natural numbers can be joined and alternated, depending on which of the numbers should be displayed first. We also introduce an operator Ω(A ) based on repunits and whose dynamics yields two constants H_(o )and H_(e )which we calculate. Amongst the few corollaries from this conjecture is an identity stating that the product of the repunits of those irrational numbers R_(H_o ) R_(H_e )is equal to unity.

On Transcendental Regular Continued Fractions

ABSTRACT We establish explicit constructions of real transcendental numbers that are not U-numbers with respect to Mahler’s classification by using regular continued fraction expansions of irrational real algebraic numbers.

Game of strokes: Optimal & conversion strategy algorithms with simulations & application

Strategic decision-making for sequential move games requires rationality and continuity of rationality to guarantee maximum payoffs at all nodes/stages/levels. Rationality and continuity of rationality in a player's behaviour are not often observed and/or maintained thus, leading to less optimal outcomes. More so, the belief in an opponent's rationality, on the other hand, co-determines the level of effort a player employs while making strategic decisions. Given irrationality and discontinuity of rationality in a sequential move game with mover advantages, there are strategic steps (algorithms) to convert and/or maintain the mover advantages of an irrational player. In this paper, the conversion strategy algorithms, as well as the optimal strategy algorithms, are developed using the Beta Limit Sum (BLS) strategy model and the game of strokes. The simulation exercises confirm that the BLS strategy model is an optimal solution for the finite sequential game of strokes. One of the key applications of these strategies is that of resource economics like environmental resources (clean water, air & land). These are public goods, as such, the optimal strategy entails that the community cooperates (as one entity) and takes the same actions or strategy to maintain a healthy and clean state of the communal environmental resources.

학교수학에서 무리수 도입으로 인한 담론 변화 분석

This study examined the method of discovering incommensurable quantities, the origin of irrational numbers, and the changes in Greek mathematics resulting from the discovery of these quantities from Sfard's commognitive perspective. In addition, in light of the history of mathematics, this study looked into changes in the discourse of school mathematics due to the introduction of irrational numbers. As a result of the analysis, the discovery of incommensurable quantities in the history of mathematics was an opportunity to change from empirical mathematics to formal and theoretical mathematics, and was an event that brought about changes in various mathematical concepts, proportion theory and roles of mathematical tools. In other words, the discovery of incommensurable magnitude transformed Greek mathematics into an incommensurable discourse. These changes in the history of mathematics appear similarly in the changes in the discourse of school mathematics due to the introduction of irrational numbers. The introduction of irrational numbers into school mathematics resulted in the application of formal definitions of numbers, changed the concepts of number and area, and led to the emergence of narratives that seemed to contradict previous discourses. It is understood that the introduction of irrational numbers has changed the discourse of school mathematics into a discourse that is incommensurable with the previous one. The analysis of this study could be basic data for the design of teaching and learning methods for students facing the task of transitioning to the incommensurable discourse of irrational number learning.

Derivative of the Minkowski function for numbers with bounded partial quotients

It is well known that the derivative of the Minkowski function ? ( x ) ?(x) (whenever exists) may take only two values: 0 0 and + ∞ +\infty . Let E n \textbf {E}_n be the set of irrational numbers on the interval [ 0 ; 1 ] [0; 1] whose partial quotients (related to the continued fraction expansion) do not exceed n n . It is also known that the quantity ? ′ ( x ) ?’(x) at a point x = [ 0 ; a 1 , a 2 , … , a t , … ] x=[0;a_1,a_2,\dots ,a_t,\dots ] is linked with the limit behavior of the arithmetic means ( a 1 + a 2 + ⋯ + a t ) / t (a_1+a_2+\dots +a_t)/t . In particular, A. Dushistova, I. Kan, and N. Moshchevitin showed that if x ∈ E n x\in \textbf {E}_n satisfies a 1 + a 2 + ⋯ + a t &gt; ( κ 1 ( n ) − ε ) t a_1+a_2+\dots +a_t&gt;(\kappa ^{(n)}_1-\varepsilon ) t , where ε &gt; 0 \varepsilon &gt;0 and κ 1 ( n ) \kappa ^{(n)}_1 is a certain explicit constant, then ? ′ ( x ) = + ∞ ?’(x)=+\infty . They also showed that the quantity κ 1 ( n ) \kappa ^{(n)}_1 cannot be increased. In the present paper, a dual problem is treated: how small may the quantity a 1 + a 2 + ⋯ + a t − κ 1 ( n ) t a_1+a_2+\dots +a_t-\kappa ^{(n)}_1 t be if ? ′ ( x ) = 0 ?’(x)=0 ? Optimal estimates in this problem are found.

The Accuracy of Computational Results from Wolfram Mathematica in the Context of Summation in Trigonometry

This article explores the accessibility of symbolic computations, such as using the Wolfram Mathematica environment, in promoting the shift from informal experimentation to formal mathematical justifications. We investigate the accuracy of computational results from mathematical software in the context of a certain summation in trigonometry. In particular, the key issue addressed here is the calculated sum ∑n=044tan⁡1+4n°. This paper utilizes Wolfram Mathematica to handle the irrational numbers in the sum more accurately, which it achieves by representing them symbolically rather than using numerical approximations. Can we rely on the calculated result from Wolfram, especially if almost all the addends are irrational, or must the students eventually prove it mathematically? It is clear that the problem can be solved using software; however, the nature of the result raises questions about its correctness, and this inherent informality can encourage a few students to seek viable mathematical proofs. In this way, a balance is reached between formal and informal mathematics.

The 5th Dimension and its Implications for the String Theory, Conservation of Energy and Heisenberg Uncertainty Principle

In 1919, the mathematician Theodor Kaluza began to play around with Albert Einstein’s formulas for gravity. Out of curiosity, he reworked the equations to see how they‘d look in five, rather than four, dimensions. The exercise yielded an extra set of equations, which were the same as James Clerk Maxwell’s equations for the electromagnetic field. His idea of a fifth dimension had produced a mathematical unification of gravity and electromagnetism. Since Kaluza’s calculations yielded an extra set of equations with Einstein’s formulas for gravity, it’s logical to assume that extending the number of known dimensions to five might also extend E = mc2. This article also visits conservation of mass and energy / mass-energy equivalence, Electromagnetism, Dark Matter, Dark Energy, the Matrix, Antigravitons, Incompleteness Theorems, New Irrationals, Calculus, Celluloid Motion, and Vector-Tensor-Scalar Geometry. And as the text accompanying Figure 5 says, “The counterclockwise rotation does not have to encompass 360 degrees. It can be divided into single degrees – and even arcseconds which are 3,600 times smaller than a degree. Each arcsecond (or tiny part of it) could correspond to a separate dimension and the total dimensions might make up a temporal multiverse that’s reminiscent of string theory, the cosmological framework that says particles are composed of one-dimensional objects called strings (a group of binary digits is called a single-dimensional bit string).” Finally, Heisenberg’s Principle of Indeterminacy - in which the position and the velocity of a particle cannot both be measured exactly, at the same time, even in theory - is revised. Quantum uncertainty is changed into quantum certainty by software containing a new type of irrational number – actually, irrational equations - that’s based on mathematics’ Matrix and topology’s Mobius strip, as well as being interdimensional. Ludwig Wittgenstein, an Austrian-British philosopher, once stated, "The limits of my language mean the limits of my world." Wittgenstein proposed that our understanding of the world is shaped by the language we use to describe it. This concept extends to mathematics, which can be seen as a specialized language. So if a person has no doubt that 1 + 1 = 2, he or she can never arrive at a truly Unified Theory in which only one thing exists. The maths accepted in 2023 is indispensable in enabling advancements in technology and science that have transformed our lives but is incapable of fully describing Unification. Scientific history has shown that it can go partway. Scientists consequently assume it will eventually go all the way.

Exact Short Products From Truncated Multipliers

Abstract We sometimes need to compute the most significant digits of the product of small integers with a multiplier requiring much storage, e.g. a large integer (e.g. $5^{100}$) or an irrational number ($\pi $). We only need to access the most significant digits of the multiplier—as long as the integers are sufficiently small. We provide an efficient algorithm to compute the range of integers given a truncated multiplier and a desired number of digits.

Quantum graphs and microwave networks as narrow-band filters for quantum and microwave devices.

We investigate properties of the transmission amplitude of quantum graphs and microwave networks composed of regular polygons such as triangles and squares. We show that for the graphs composed of regular polygons, with the edges of the length l, the transmission amplitude displays a band of transmission suppression with some narrow peaks of full transmission. The peaks are distributed symmetrically with respect to the symmetry axis kl=π, where k is the wave vector. For microwave networks the transmission peak amplitudes are reduced and their symmetry is broken due to the influence of internal absorption. We demonstrate that for the graphs composed of the same polygons but separated by the edges of length l^{'}<l, the transmission spectrum is generally not symmetric according to the axis kl^{'}=π. We also show that graphs composed of regular polygons of different size with the edges being irrational numbers are not fully chaotic and their level spacing distribution and the spectral rigidity are well described by the Berry-Robnik distributions. Moreover, the transmission spectrum of such a graph displays peaks which are very close to one. Furthermore, the microwave networks are investigated in the time-domain using short Gaussian pulses. In this case the delay-time distributions, though very sensitive to the internal structure of the networks, show the sequences of transmitted peaks with the amplitudes much smaller than the input one. The analyzed properties of the graphs and networks suggest that they can be effectively used to manipulate quantum and wave transport.