Pure Mathematics Research Articles

Coevolution between institutions and scientific organizations: The case of IMPA

AbstractIn this study, the properties of coevolution were analysed and presented as a promising avenue to investigate the interaction between institutions and scientific organizations, which has implications for the development of scientific fields and public scientific policies. These properties were used to analyse the interaction between the Institute for Pure and Applied Mathematics (IMPA) and national and international institutions from 1951 to 2019. The findings reveal that if an area of knowledge is weak in a country, it can develop through different institutional actions by interested parties such as researchers. The institutional agency of IMPA researchers has functioned as an element that feeds coevolutionary dynamics. This dynamic has caused changes in the IMPA (micro-level), scientific community (meso-level), and governmental institutions and civil society (macro-level). This study’s main contribution is the cross-fertilization of coevolution with the agency approach of institutional theory.

Homotopical Algebra and Higher Structures

Homotopical algebra and higher category theory play an increasingly important role in pure mathematics, and higher methods have seen tremendous development in the last couple of decades. The talks delivered at the workshop described some of the latest progress in this area and applications to various problems of algebra, geometry, and combinatorics.

The Brownian Castle

AbstractWe introduce a 1+1‐dimensional temperature‐dependent model such that the classical ballistic deposition model is recovered as its zero‐temperature limit. Its ∞‐temperature version, which we refer to as the 0‐Ballistic Deposition (0‐BD) model, is a randomly evolving interface which, surprisingly enough, does not belong to either the Edwards–Wilkinson (EW) or the Kardar–Parisi–Zhang (KPZ) universality class. We show that 0‐BD has a scaling limit, a new stochastic process that we call Brownian Castle (BC) which, although it is “free”, is distinct from EW and, like any other renormalisation fixed point, is scale‐invariant, in this case under the 1:1:2 scaling (as opposed to 1:2:3 for KPZ and 1:2:4 for EW). In the present article, we not only derive its finite‐dimensional distributions, but also provide a “global” construction of the Brownian Castle which has the advantage of highlighting the fact that it admits backward characteristics given by the (backward) Brownian Web (see [37, 16]). Among others, this characterisation enables us to establish fine pathwise properties of BC and to relate these to special points of the Web. We prove that the Brownian Castle is a (strong) Markov and Feller process on a suitable space of càdlàg functions and determine its long‐time behaviour. Finally, we give a glimpse to its universality by proving the convergence of 0‐BD to BC in a rather strong sense. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

On the structure of the solutions to the matrix equation G⁎JG = J

We study the mathematical structure of the solution set (and its tangent space) to the matrix equation G⁎JG=J for a given square matrix J. In the language of pure mathematics, this is a Lie group which is the isometry group for a bilinear (or a sesquilinear) form. Generally these groups are described as intersections of a few special groups.The tangent space to {G:G⁎JG=J} consists of solutions to the linear matrix equation X⁎J+JX=0. For the complex case, the solution set of this linear equation was computed by De Terán and Dopico.We found that on its own, the equation X⁎J+JX=0 is hard to solve. By throwing into the mix the complementary linear equation X⁎J−JX=0, we find that the direct sum of the two solution sets is an easier to compute linear space. Thus, we obtain the two solution sets from projection maps. Not only is it possible to now solve the original problem, but we can approach the broader algebraic and geometric structure. One implication is that the two equations form an h and m pair familiar in the study of pseudo-Riemannian symmetric spaces.We explicitly demonstrate the computation of the solutions to the equation X⁎J±XJ=0 for real and complex matrices. However, real, complex or quaternionic case with an arbitrary involution (e.g., transpose, conjugate transpose, and the various quaternion transposes) can be effectively solved with the same strategy. We provide numerical examples and visualizations.

Ballistic Transport for One‐Dimensional Quasiperiodic Schrödinger Operators

AbstractIn this paper, we show that one‐dimensional discrete multifrequency quasiperiodic Schrödinger operators with smooth potentials demonstrate ballistic motion on the set of energies on which the corresponding Schrödinger cocycles are smoothly reducible to constant rotations. The proof is performed by establishing a local version of strong ballistic transport on an exhausting sequence of subsets on which reducibility can be achieved by a conjugation uniformly bounded in the Cℓ‐norm. We also establish global strong ballistic transport under an additional integral condition on the norms of conjugation matrices. The latter condition is quite mild and is satisfied in many known examples. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

Some Advances in Pure Mathematics Made in the 19th Century [Around the Globe

This article is reprinted with permission of the James Clerk Maxwell Foundation (JCMF), which is dedicated to the life and history of Clerk Maxwell. A wealth of information is available at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">http://www.clerkmaxwellfoundation.org/</uri> . In addition, the JCMF owns and maintains an extensive collection of Maxwell material at his birthplace, 14 India Street, Edinburgh, Scotland. Visitors are most welcome once the travel situation returns to normal. This article was arranged by David Forfar and James Rautio, both trustees of the JCMF.

Introducing acoustics as music in the interdisciplinary and inclusive classroom through ski-hill graph pedagogy to teach the meter fundamentals as sound (psychoacoustics)

Although music is sound (acoustics) and listeners experience the physics of acoustics as music, the meter of most music textbooks is the notated meter signature and group of beats notated in ‘measures’ on the page with vertical bar lines. Notation-based representation of the meter is antiquated music theory because new research in acoustics, cognitive neuroscience, and critically music theory (Cohn, 2020) informs otherwise. Cohn’s modern meter theory, a distillation of contemporary meter theory from North America, is the first comprehensive meter theory that acknowledges the meter's psychoacoustic experience and augments notation-based understandings of the meter through the meter the fundamental mathematics of the Ski-hill graph. This paper is a music teacher’s response to the availability of modern meter theory’s Ski-hill graph to represent the meter's psychoacoustic (mind and body) experience. The paper illustrates how listeners articulate their quantification of low meter frequencies such as the duple meter in sets of pulses in ratio 2:1 initiated by 1.2 hz and 2.4 hz experienced concurrently. Also, the triple meter sets of pulses in ratios 3:1 0.8 hz and 2.4 hz (Nozaradan et al., 2011).

Like a Sparrow, Small but Complete

Summary By investigating an interesting number theory problem and its generalization, a reader will experience the process of research in pure mathematics. The content is completely elementary and accessible to any university students. We hope that the article will serve as an invitation to mathematical research.

𝒩-Structures Applied to Commutative Ideals of BCI-Algebras

The study of symmetry is one of the most important and beautiful themes uniting various areas of contemporary arithmetic. Algebraic structures are useful structures in pure mathematics for learning a geometrical object’s symmetries. In order to provide a mathematical tool for dealing with negative information, a negative-valued function came into existence along with N-structures. In the present analysis, the notion of N-structures is applied to the ideals, especially the commutative ideals of BCI-algebras. Firstly, several properties of N-subalgebras and N-ideals in BCI-algebras are investigated. Furthermore, the notion of a commutative N-ideal is defined, and related properties are investigated. In addition, useful characterizations of commutative N-ideals are established. A condition for a closed N-ideal to be a commutative N-ideal is provided. Finally, it is proved that in a commutative BCI-algebra, every closed N-ideal is a commutative N-ideal.

Understanding the paradox between tragedies of the commons and the anticommons: From a cognitive psychology perspective.

The tragedy of the commons refers to the overuse of resources which are rival in consumption but lack excludability and it also refers to rent dissipation. While the tragedy of the anticommons is a tragedy closely connected with underuse of resources that are rival in consumption and with too strong excludability. The prior studies proved that the tragedy of the commons and the tragedy of the anticommons are symmetric from the perspective of pure mathematics, especially the game theory, which was later refuted by behavioral economics experiments. According to them, the tragedy of the anticommons is severer than the tragedy of the commons. The asymmetry of the tragedy of the commons and the tragedy of the anticommons is a paradox by these different research methods. This paradox shows that there are imperfections in the completely rational economic man hypothesis set up by neoclassical economics. As a fundamental theory, the tragedy of the commons is quite influential in many disciplines, such as microeconomics, public sector economics, ecological economics, environmental economics, management, sociology, property law, and political science. And the tragedy of the anticommons theory has also opened its door of both theoretical research and practical implications since its acceptance by Nobel laureate Buchanan, the main founder of public choice school. Only when theoretical issues are thoroughly discussed and made clear enough, can people avoid misunderstanding or misusing the commons theory. Thus, it is necessary to elucidate the paradox between them. Based on Simon’s bounded rationality, Kahneman and Tversky’s prospect theory, value function, Thaler’s mental accounting, endowment effect, and other cognitive psychological tools, this study clearly shows that agents’ decision-making process is not just based on the long-believed marginal benefit and marginal cost analysis advocated by traditional neoclassical economists. Agents’ decision-making is a process in which agents selectively absorb, code the objective marginal revenue and marginal cost, and feed relevant information to their brain. Therefore, what plays a directly decisive role is not the objective marginal revenue and marginal cost per se, but the mentally perceived subjective utility of marginal revenue and marginal cost by the human brain. Followed by this research clue, the paradox between the tragedy of the commons and the tragedy of the anticommons is elucidated from the perspective of cognitive psychology.

Traveling Wave Solutions to the Free Boundary Incompressible Navier‐Stokes Equations

AbstractIn this paper we study a finite‐depth layer of viscous incompressible fluid in dimension , modeled by the Navier‐Stokes equations. The fluid is assumed to be bounded below by a flat rigid surface and above by a free, moving interface. A uniform gravitational field acts perpendicularly to the flat surface, and we consider the cases with and without surface tension acting on the free interface. In addition to these gravity‐capillary effects, we allow for a second force field in the bulk and an external stress tensor on the free interface, both of which are posited to be in traveling wave form, i.e., time‐independent when viewed in a coordinate system moving at a constant velocity parallel to the rigid lower boundary. We prove that, with surface tension in dimension and without surface tension in dimension , for every nontrivial traveling velocity there exists a nonempty open set of force and stress data that give rise to traveling wave solutions. While the existence of inviscid traveling waves is well‐known, to the best of our knowledge this is the first construction of viscous traveling wave solutions.Our proof involves a number of novel analytic ingredients, including: the study of an overdetermined Stokes problem and its underdetermined adjoint problem, a delicate asymptotic development of the symbol for a normal‐stress to normal‐Dirichlet map defined via the Stokes operator, a new scale of specialized anisotropic Sobolev spaces, and the study of a pseudodifferential operator that synthesizes the various operators acting on the free surface functions. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

The Poincaré Index on Singular Varieties

In this paper, we discuss a few simple methods for computing the local topological index and its various analogs for vector fields and differential forms given on complex varieties with singularities of different types. They are based on properties of regular meromorphic and logarithmic differential forms, of the dualizing (canonical) module and related constructions. In particular, we show how to compute the index on Cohen–Macaulay, Gorenstein and monomial curves, on normal and non-normal surfaces and some others. In contrast with known traditional approaches, we use neither computers, nor integration, perturbations, deformations, resolution of singularities, spectral sequences or other related standard tools of pure mathematics.

Topological Spaces on Fuzzy Structures

Apart from their applications to almost all sectors of the human activity, fuzzy mathematics is also importantly developed on a theoretical basis providing useful links even to classical branches of pure mathematics, like Algebra, Analysis, Geometry, Topology, etc. The present paper comes across the steps that enabled the extension of the concept of topological space, the most general category of mathematical spaces, to fuzzy structures. Fuzzy and soft topological spaces are introduced in particular, the fundamental concepts of limits, continuity, compactness and Hausdorff space are defined on them and examples are provided illustrating them.

Design and Computational Flow Analysis of Different Rocket Nozzle Profile

In recent days the launching value of increasing the industrial business starts with lowering prices related to the launch vehicle operation. Reducing the price begins with rising the propulsion systems, creating these vehicles a lot of economical with a restricted fuel amount. The Linear Aerospike engine has incontestable bigger thrust potency over this engines used on launch vehicles. However, unresolved problems with warming plague this standard. This study proposes an approach amendment to the rocket motor that explores a way to increase the performance of the nozzle body, and lower the chance of failure because of the acute pressure environments. The results show that however the most exhaust flow of an Aerospike is full of couture of the spike surface. Victimization in Ansys CFD software system and acting a straight forward experiment with whoosh rockets, the study explores however exhaust flow changes the performance characteristics, in the main speed close to the wall. The results with some improvement is created by CFD nozzle flow studies, to make sure the bottom pure mathematics isn't a contributory issue to the film-cooling results.

Video Pembelajaran Youtube untuk Media Pembelajaran Daring Mahasiswa

This act of community dedication is aimed to provide video lessons of First Fundamental Mathematics/ First Calculus for the undergraduate students. The videos are free lesson source, accessible anytime and anywhere from Youtube. During the activity, there are a lot of positive responses from the students who are taking the same lessons as provided by the video materials. It can be concluded that the activity went very well and the result is as good as expected. This can be seen from the viewers’ enthusiasm who contribute in discussion on the comment section if they got stuck, appreciating the content, and also the high traffic. The fact that the most viewers are undergraduate students can be seen on Youtube analytic. As for the suggestion, if there is another team who wants to get the better result, it is strongly advised to promote and to share related information regarding the videos massively.

A Novel Intelligent Ground Moving Target Indication Using Meta-Heuristic-Based Simplified Fractional Fourier Transform

One of the most significant subfields in “Synthetic Aperture Radar (SAR)” research is considered to be target detection. Numerous studies have been conducted on target identification, with the majority of them favoring filter-oriented methods. The fundamental goal of radar systems is to “detect moving targets on the ground.” Decomposing a complex matrix into a structured sparse matrix and a low-rank matrix is a fundamental mathematics issue. Surveillance and reconnaissance rely heavily on “Ground Moving Target Indication (GMTI),” but it's not a simple task. The SAR ATI was first developed for calculating the radial velocity of ground-moving objects. Yet, overlapping stationary clutter can corrupt the recorded differential phase, resulting in mistakes in position and velocity calculations. The main concept of this paper is to propose a novel “Adaptive Simplified Fractional Fourier Transform (A-SFrFT)” using the intelligent meta-heuristic improvement. This adaptive SFrFT efficiently estimates the “Doppler parameters of the moving targets.” The improved “Harris Hawks Optimization (HHO)” termed Trio Updating HHO (TU-HHO) is used as the meta-heuristic algorithm that enhances the performance of the SFrFT-based target estimation. The mathematical analysis and simulation findings show that the suggested methods recommended strategy is successful.

A characterization of nonhomogeneous dual and weak dual wavelet superframes for Walsh‐reducing subspace of L2(ℝ+,ℂL)$$ {L}&amp;amp;amp;amp;amp;#x0005E;2\left({\mathbb{R}}_{&amp;amp;amp;amp;amp;#x0002B;},{\mathbb{C}}&amp;amp;amp;amp;amp;#x0005E;L\right) $$

The concept of superframe was first introduced by Balan in the context of signal multiplexing and by Han and Larson from the perspective of pure mathematics. Since then, the theory of super wavelet (Gabor) frames has been well developed. In recent years, due to enjoying the same fast algorithm as affine wavelet frames and having more freedom to design wavelet masks, nonhomogeneous wavelet frames have attracted much attention of many mathematicians. This paper addresses (weak) nonhomogeneous dual wavelet superframes in the setting of Walsh‐reducing subspaces of . We present a characterization of nonhomogeneous (weak) dual wavelet superframe pairs in a Walsh‐reducing subspace of . Some examples are also provided.

Statistics education at a time of global disruption and crises: a growing challenge for the curriculum, classroom and beyond

COVID-19 provided the world at large, including the world of mathematics education, with a challenge demanding attention and understanding. If met, this could potentially provide society with many of the skills needed to tackle the challenges of hyperobjects (Morton, 2013) such as climate change, which are potentially more threatening in the long-term. The COVID-19 context and the massive amount of data it has produced are the most recent examples of the growing recognition that the school mathematics curriculum has a role to play outside of the pure mathematics classroom. This paper considers COVID-19 as a stimulus for increasing the importance of statistical literacy and data literacy in preparing society for coping with world crises. Topics considered include the importance of acknowledging statistics as a significant component of mathematical ways of knowing, the contextual motivation provided by the COVID-19 crisis, the importance of statistics and statistical literacy, the place of statistics in the wider school curriculum, and finally, its place in the classroom. These topics need to be taken into account by both policy makers and teachers.

The Principle of Mathematical Induction: Applications in Physical Optics

The bare rudiments of the principle of mathematical induction as a method of proof date back to ancient times. In the contemporary university milieu, the demonstrative scheme is taught as part of a course in discrete mathematics, set theory, number theory, graph theory, group theory, game theory, linear algebra, logic, and combinatorics. In theoretical computer science, it bears the pivotal role of developing the appropriate cognitive skills necessary for the effective design and implementation of algorithms, assessing for both their correctness and complexity. Pure mathematics and computer science aside, the scope of its utility in the physical sciences remains limited. Following an outline of some elementary concepts from vector algebra and phasor analysis, the proofs by induction of a couple of salient results in multiple-slit interferometry are presented, viz., the fringe intensity distribution formula and the upper bound of the total fringe count. These specific optical instantiations serve to illustrate the versatility and power of the principle at tackling real-world problems. It thereby makes a welcome departure from the popular view of induction as a mere last resort for proving abstract mathematical statements.

Subdifferential set of an operator

Let X, Y be complex Banach spaces. Let {mathcal L}(X,Y) be the space of bounded operators. An important aspect of understanding differentiability is to study the subdifferential of the norm at a point, say x in X, this is the set, {f in X^*:Vert fVert =1~,f(x)=Vert xVert }. See page 7 in Deville et al. (Pitman Monographs and Surveys in Pure and Applied Mathematics. 64. Harlow: Longman Scientific and Technical. New York: John Wiley and Sons, Inc. 1993). Motivated by recent results of Singla (Singla in Linear Alg. Appl. 629:208–218, 2021) in the context of Hilbert spaces, for T in {{mathcal {L}}}(X,Y), we determine the subdifferential of the operator norm at T, partial _T = {Lambda in {{mathcal {L}}}(X,Y)^*: Lambda (T) = Vert TVert ~,~Vert Lambda Vert =1}. Our approach is based on the ‘position’ of the space of compact operators and the Calkin norm of T. Our ideas give a unified approach and extend several results from Singla (Linear Alg. Appl. 629:208–218, 2021) to the case of ell ^p-spaces for 1<p<infty . We also investigate the converse, using the structure of the subdifferential set to decide when the Calkin norm is a strict contraction. As an application of these ideas, we partially solve the open problem of relating the subdifferential of the operator norm at a compact operator T to that of T(x_0), where x_0 is a unit vector where T attains its norm.