Irrational Research Articles

Infinite temperature spin dc conductivity of the spin-1/2 XXZ chain

Abstract Using the Bethe ansatz method and the thermodynamic Bethe ansatz equations for the higher-spin integrable XXZ chain, the regular zero frequency contribution to the spin current correlation (spin dc conductivity) is analyzed for the spin-1/2 XXZ chain with the anisotropy 0 ⩽ Δ < 1 . In the high temperature limit, we write down the dressed scattering kernels by one quasi-particle bare energies and exactly evaluate the spin dc conductivity at zero magnetic field, which is proportional to the inverse temperature β at the β → 0 limit. We find that the high temperature proportionality constant σ 0 reg of the spin dc conductivity is discontinuous at all rational numbers of the anisotropy parameter p 0 = π / cos − 1 ⁡ Δ ⩾ 2 with the gap increasing larger than the second power of growing magnetization on one quasi-particle. The isotropic Δ = 1 point is exceptional. Close to this point, σ 0 reg slowly increases in proportion to the first power of the one quasi-particle magnetization. On the other hand, σ 0 reg is proportional to the second power of the same when p 0 approaches irrational numbers.

Cardioprotective effect of Perakine against myocardial ischemia-reperfusion injury of type-2 diabetic rat in Langendorff-Perfused Rat Hearts: Role of TLR4/NF-kB signalling pathway.

Myocardial ischemia-reperfusion (I/R) injury is a grave life-threatening situation, if not treated well in time. The development of natural remedies for myocardial I/R injury has witnessed dramatic growth in the last decade. Prompted by the above, in the present study, we have elucidated the pharmacological effect of Perakine (PER), an indole alkaloid in myocardial ischemia-reperfusion injury in type 2 diabetic rats. The model was established by inducing diabetes in experimental rats followed by the development of myocardial I/R injury model of isolated rat heart by the use of improved Langendorff retrograde perfusion technology. Results of the study suggest that PER significantly lowered the infarct size and infarct volume, and improvement in cardiac ability (LVSP, ±dP/dtmax, and heart rate). It also showed a significant lowering of cardiac biomarkers (CK, CK(MB), ALT, AST, and LDH) in a dose-dependent manner as compared to unprotected IR rats. The level of oxidative stress (MDA, SOD, and GSH) was also found lowered in IR rats together with a reduction in the production of pro-inflammatory cytokines (IL-1β, IL-6, IL-17, and TNF-α). The anti-inflammatory acting of PER on IR rats is believed to be linked with the reduction of TNF-α, NF-κB, and TLR4 in both RT-qPCR and western blot analysis. Our research showed that Perakine could potentially alleviate cardiac ischemia-reperfusion injury in rats by blocking the TLR4/NF-κB signaling cascade.

Diophantine approximation by rational numbers of certain parity types

AbstractFor a given irrational number, we consider the properties of best rational approximations of given parities. There are three different kinds of rational numbers according to the parity of the numerator and denominator, say odd/odd, even/odd, and odd/even rational numbers. We study algorithms to find best approximations by rational numbers of given parities and compare these algorithms with continued fraction expansions.

The tasty world of irrational numbers – explorative mathematical visualizations generated by AI image models

In this paper, images containing mathematical ideas that were generated by different image-generating AI models (DALL-E 2, Leonardo.Ai, and Midjourney) are comparatively interpreted from a mathematics aesthetical perspective. Our exploratory study aimed to scrutinize how AI image models visualize abstract mathematical ideas and whether this imagination holds any potential for use in the context of learning mathematics. Two different mathematics-containing prompts, representing different aspects of mathematics, were used as input. Our results show that based on our examples, currently, Midjourney generates the most detailed outcome that can be used as an initial point for discussing mathematical ideas with students in the classroom.

Closure property and cognitive biases

In this note, we discuss two specific cognitive biases towards closure property as a learning obstacle in school mathematics in the context of number sets. These biases are related to rational and irrational numbers, two important and challenging issues from an educational point of view. The first bias concerns incorrect application of rules and properties of closure property in rational numbers to irrational numbers, and the second deals with the improper generalisation of the closure property in finite states to infinite states. These biases indicate the importance of teaching and learning the closure property.

Irrational numbers of journal editors and of editorial positions: a threat to society

Irrational numbers of journal editors and of editorial positions: a threat to society

Isoliquiritigenin attenuates myocardial ischemia reperfusion through autophagy activation mediated by AMPK/mTOR/ULK1 signaling

BackgroundIschemia reperfusion (IR) causes impaired myocardial function, and autophagy activation ameliorates myocardial IR injury. Isoliquiritigenin (ISO) has been found to protect myocardial tissues via AMPK, with exerting anti-tumor property through autophagy activation. This study aims to investigate ISO capacity to attenuate myocardial IR through autophagy activation mediated by AMPK/mTOR/ULK1 signaling.MethodsISO effects were explored by SD rats and H9c2 cells. IR rats and IR-induced H9c2 cell models were established by ligating left anterior descending (LAD) coronary artery and hypoxia/re-oxygenation, respectively, followed by low, medium and high dosages of ISO intervention (Rats: 10, 20, and 40 mg/kg; H9c2 cells: 1, 10, and 100 μmol/L). Myocardial tissue injury in rats was assessed by myocardial function-related index, HE staining, Masson trichrome staining, TTC staining, and ELISA. Autophagy of H9c2 cells was detected by transmission electron microscopy (TEM) and immunofluorescence. Autophagy-related and AMPK/mTOR/ULK1 pathway-related protein expressions were detected with western blot.ResultsISO treatment caused myocardial function improvement, and inhibition of myocardial inflammatory infiltration, fibrosis, infarct area, oxidative stress, CK-MB, cTnI, and cTnT expression in IR rats. In IR-modeled H9c2 cells, ISO treatment lowered apoptosis rate and activated autophagy and LC3 fluorescence expression. In vivo and in vitro, ISO intervention exhibited enhanced Beclin1, LC3II/LC3I, and p-AMPK/AMPK levels, whereas inhibited P62, p-mTOR/mTOR and p-ULK1(S757)/ULK1 protein expression, activating autophagy and protecting myocardial tissues from IR injury.ConclusionISO treatment may induce autophagy by regulating AMPK/mTOR/ULK1 signaling, thereby improving myocardial IR injury, as a potential candidate for treatment of myocardial IR injury.

Topology Meets Number Theory

Liouville proved the existence of a set L of transcendental real numbers now known as Liouville numbers. Erdős proved that while L is a small set in that its Lebesgue measure is zero, and even its s-dimensional Hausdorff measure, for each s > 0, equals zero, it has the Erdős property, that is, every real number is the sum of two numbers in L . He proved L is a dense G δ -subset of R and every dense G δ -subset of R has the Erdős property. While being a dense G δ -subset of R is a purely topological property, all such sets contain c Liouville numbers. Each dense G δ -subset of R , including L , is homeomorphic to the product N ℵ 0 of copies of the discrete space N of all natural numbers. Also this product space is homeomorphic to the space P of all irrational real numbers and the space T of all transcendental real numbers. Hence every dense G δ -subset of R has cardinality c . Indeed, any dense G δ -subset of R has a chain Xm , m ∈ ( 0 , ∞ ) of homeomorphic dense G δ -subsets such that X m ⊂ X n , for n < m, and X n ∖ X m has cardinality c . Finally, every real number r ≠ 1 is equal to ab , for some a , b ∈ L .

The high type quadratic Siegel disks are Jordan domains

Let \alpha be an irrational number of sufficiently high type and suppose P_{\alpha}(z)=e^{2\pi\textup{i}\alpha}z+z^{2} has a Siegel disk \Delta_{\alpha} centered at the origin. We prove that the boundary of \Delta_{\alpha} is a Jordan curve, and that it contains the critical point -e^{2\pi\textup{i}\alpha}/2 if and only if \alpha is a Herman number.

MACDONALDIZATION AND DISNEYIZATION AS GLOBALIZATION TRENDS

The article explores the dimensions of the social phenomenon of McDonaldization, introduced into scientific circulation by sociologist George Ritzer. Using the principles established by the McDonald's fast food corporation as a metaphor, Ritzer delves into the consequences of rationalization and standardization observed in modern society. McDonaldization serves as a catalyst for larger-scale social transformations, spreading the process of globalization and generating the phenomenon of Disneyization in all areas of human activity. It is determined that the McDonaldization drive for efficiency and control leads to the «irrationality of rationality» in various aspects of modern human life. Benjamin Barber's concept illustrates strategies for the development of a globalized culture in which McWorld and Jihad coexist and interact, impeding the development of democratic processes, but each in its own way. The article reveals the nature of Disneyization and its implementation in the globalization narratives of our time. The emergence of the McDonald's and Disneyized worlds emphasize the spread of the logic of consumerism, which plays a key role in the formation of a new world order with an emphasis on universalization, predictability and control of all social processes that fall into the project. Disneyization is defined as a transformative force that embellishes reality, emphasizes compliance with consumer norms, and distorts historical narratives by making them simplistic. The trivialization and sanitization of media products are the next stages in the evolution of the McDonaldized and Disneyized world, which gives rise to hybrid consumption and the desire to expand the territory of influence. The findings of this study contribute to a comprehensive understanding of the impact of McDonaldization, which is not limited to the fast food industry but extends to social, educational, artistic, political, economic, and other spheres. The interconnection of McDonald's with related phenomena, such as Disneyization, allows us to take a fresh look at the development of modern society.

Fractional parts of generalized polynomials at prime arguments

Let f(X) be a monic polynomial of degree d≥2 with real coefficients. In this paper, we obtain an upper bound for the discrepancy of the sequence ([f(p)α]β) generated by the generalized polynomial [f(X)α]β, where p runs through the set of all primes and α, β are irrational numbers satisfying certain conditions. We also study the small fractional parts of the sequence ([f(p)α]β) and prove a result analogous to that of Madritsch and Tichy (Theorem 2.3, [13]).

Go(Φ)d is Number: Plotting the Divided Line &amp; the Problem of the Irrational

Plato believed that behind everything in the universe lie mathematical principles. Plato was inspired by Pythagoras (571 BCE), who developed a school of mathematics at Crotona that studied sacred geometry as a form of religion. The school’s motto was “God is number,” or “All is Number”. Pythagoras believed that numbers represented God in pattern, symmetry, and infinity. When one of its students, Hippasus told the world the secret of the existence of irrational numbers, Greek geometry was born and Pythagoras’ idea of divinity in numbers died because how could God not be perfect and symmetrical? In Plato’s Republic he discusses something called The Divided Line, which is a map, of sorts, for reaching what he calls the highest Good, which is the ultimate truth where one realizes the true state of the universe and can see the world for what it really is. Many mathematicians have attempted to plot Plato’s Divided Line only to come across a litany of problems and conundrums. Some have said that it the Divided Line cannot be plotted and is merely an allegory not meant to be plotted. This paper discusses some of the conundrums preventing the plotting of Plato’s Divided Line (not an exhaustive list), including Whole ‘vs’ Separate, Equality ‘vs’ Ontological Dissimilarity, Linear ‘vs’ Non-linear, and Infinity ‘vs’ Finite. This paper also explores a new understanding of the Allegory of the Cave in light of ‘the problem of the irrational.’ In exploring the link between the Divided Line and the ‘the problem of the irrational,’ I was able to plot it. It was found that the Divided Line is not a line in the linear sense, but a spiral, the Golden Ratio! This paper is an example of a new category of scholarly inquiry I call “Math Theory” based on scholarly mathematical axioms in theory, rather than including actual maths. In my papers I use existing mathematical equations and place them in an encompassing theory, rather than finding new formulae to fit an existing theory. Keywords: Pythagoras, divided line, math theory, highest good, all is number

Square Root Expressions and Irrational Numbers in Middle School Mathematics Textbooks Throughout the History of the Republic

Irrational numbers, a fundamental set that extends rational numbers to real numbers, have been a subject of limited study in the context of textbook analyses. This study, therefore, examines how square root expressions and irrational numbers have been addressed in middle school mathematics textbooks throughout the history of the Republic of Türkiye. The research was conducted using a comprehensive approach, with qualitative research methods and content analysis was applied to 37 mathematics textbooks (seven 5th grade, twelve 6th grade, nine 7th grade, and nine 8th grade) from 1932 to 2022. Data were collected from mathematics textbooks available at the Ferit Ragıp Tuncor Archive and Documentation Library affiliated with the Ministry of National Education. The results of this study indicate that changes in mathematics curricula are reflected in textbooks, and this reflection is directly influenced by the learning theory adopted in the curriculum. The study also provides insights into how irrational numbers are introduced in textbooks based on the sequence of topics, and recommendations are made based on the findings.

On the Hausdorff dimension of invariant measures of piecewise smooth circle homeomorphisms

Abstract We show that, generically, the unique invariant measure of a sufficiently regular piecewise smooth circle homeomorphism with irrational rotation number and zero mean nonlinearity (e.g. piecewise linear) has zero Hausdorff dimension. To encode this generic condition, we consider piecewise smooth homeomorphisms as generalized interval exchange transformations (GIETs) of the interval and rely on the notion of combinatorial rotation number for GIETs, which can be seen as an extension of the classical notion of rotation number for circle homeomorphisms to the GIET setting.

Uniqueness of Denjoy minimal sets for twist maps with zero entropy *

For monotone twist maps with zero topological entropy, we show that the set of recurrent points with irrational rotation number α can be described by a single orientation preserving circle homeomorphism and hence there is either an invariant circle, or a unique Denjoy minimal set, of rotation number α.

Evolving Patterns in Irrational Numbers Using Waiting Times between Digits

There is an increasing interest in determining if there exist observable patterns or structures within the digits of irrational numbers. We extend this search by investigating the interval in position between two consecutive occurrences of the same digit, a kind of waiting time statistics. We characterise these by the burstiness measure which distinguishes if the inter-event times are periodic, bursty, or Poisson processes. Furthermore, the complexity–entropy plane was used to determine if the intervals are stochastic or chaotic. We analyse sequences of the first 1 million digits of the numbers π, e, 2, and ϕ. We find that the intervals between single, double, and triple digits are Poisson processes with a burstiness measure in the range −0.05≤B≤0.05 for the four numbers studied. This result is supported by a complexity–entropy plane analysis, which shows that the time intervals have the same characteristics as Gaussian noise. The four irrational numbers have identical degrees of complexity and burstiness in their inter-event analysis.

Uniform Diophantine approximation with restricted denominators

Let b≥2 be an integer and A=(an)n=1∞ be a strictly increasing subsequence of positive integers with η:=lim supn→∞an+1an<+∞. For each irrational real number ξ, we denote by vˆb,A(ξ) the supremum of the real numbers vˆ for which, for every sufficiently large integer N, the equation ‖banξ‖<(baN)−vˆ has a solution n with 1≤n≤N. For every vˆ∈[0,η], let Vˆb,A(vˆ) (Vˆb,A⁎(vˆ)) be the set of all real numbers ξ such that vˆb,A(ξ)≥vˆ (vˆb,A(ξ)=vˆ) respectively. In this paper, we give some results of the Hausdorfff dimensions of Vˆb,A(vˆ) and Vˆb,A⁎(vˆ). When η=1, we prove that the Hausdorfff dimensions of Vˆb,A(vˆ) and Vˆb,A⁎(vˆ) are equal to (1−vˆ1+vˆ)2 for any vˆ∈[0,1]. When η>1 and limn→∞⁡an+1an exists, we show that the Hausdorfff dimension of Vˆb,A(vˆ) is strictly less than (η−vˆη+vˆ)2 for some vˆ, which is different with the case η=1, and we give a lower bound of the Hausdorfff dimensions of Vˆb,A(vˆ) and Vˆb,A⁎(vˆ) for any vˆ∈[0,η]. Furthermore, we show that this lower bound can be reached for some vˆ.

Interplay between insertion of zeros and the complexity of Dedekind cuts

The topic of this paper is the study of the possibility of effective conversions between different representations of irrational numbers. We are interested in subrecursive computations, in which we disallow the use of unbounded search. It is well-known that the base expansion of an irrational number can be subrecursively computed from its Dedekind cut, but in the reverse direction this is not subrecursively possible. Thus it would be natural to try to understand how additional properties of the base expansion, such as the presence of specific patterns of zero digits, correlate with the complexity of the Dedekind cut. Our first result roughly states that if large parts of zero digits dominate over smaller parts of arbitrary digits and the base expansion has elementary complexity, then the Dedekind cut also has elementary complexity. Much more interesting is the case when one allows large parts of zero digits to alternate with large parts of arbitrary digits. We compare the complexity of the Dedekind cuts of an arbitrary irrational number and another irrational number, obtained by inserting long sequences of zero digits in its base expansion. We provide examples in which (1) both numbers have elementary Dedekind cuts, (2) the first number has complex Dedekind cut and elementary base expansion, but the second number has elementary Dedekind cut and (3) both numbers have complex Dedekind cuts and elementary base expansions. An open question remains, whether it is possible for the first number to have elementary Dedekind cut, while the second number has complex Dedekind cut.

직관주의 수학 및 고전 수학 관점에 따른 교과서의 귀류법 증명에서 무리수 √2의 존재성 문제

Intuitionistic mathematics philosophy, which distanced itself from classical mathematics, takes the view that mathematical objects are constructed by the human mind, like constructivism, which has been discussed in mathematics education. This paper aims to analyze the content of school mathematics and obtain educational implications based on a discussion of central issues in intuitionistic mathematics. Concerning the proof by contradiction, which can reveal the difference between intuitionistic mathematics and classical mathematics, among the expressions of the proposition presented as an example of the proof by contradiction in textbooks, in ‘√2 is not a rational number’, contrast to ‘√2 is an irrational number’, avoided a mention about its existence as an irrational number. With this background, we examined the issues of intuitionistic mathematical philosophy, focusing on the existence of mathematical objects, and based on this, we analyzed the textbook's related proofs by contradiction in terms of the existence of √2 as an irrational number. This analysis found that the perspective of intuitionistic mathematics permeates the expression and the proof which do not mention and avoid the existence of √2 as an irrational number. Based on the above, we discussed the reflection on the existence of mathematical objects in school mathematics and the educational value of mathematics from a humanistic aspect.

An Effective Refutation of General Relativity and Quantum Theories

It is briefly shown that all physical theories that rely on a mathematical application of the circle number as an irrational dimensionless number are irrational and flawed in their explanatory content, since they place the physical dimensions of space (L) [length] and time (T) in an inaccurate and irrational relationship. Albert Einstein's most important postulate (speed of light in a vacuum = constant) is thus refuted and shown that there π are no natural constants in nature – apart from the circle number π