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We report on a study focused on identifying and describing the representations of mathematicians contained in Mexican textbooks of lower secondary level. We considered representations not only in the text but also in drawings, photographs, and illustrations in general. The term mathematician was understood in a broad sense: any person (or group of people) that in the textbook either (1) was explicitly referred to as a mathematician, (2) was credited with the development of a mathematical concept or tool, or (3) was displayed performing some sort of mathematical activity (such as counting, modelling, etc.). The results show that the representations that most frequently appear in the textbooks are white male mathematicians (mainly Europeans), who lived in ancient times; the representations of female mathematicians are almost nil. At the end of the paper the implications of these results are discussed, and some directions for future research are suggested.

Given the increase in the number of fictional women mathematicians and scientists on television and in the movies, educators who wish to incorporate pop culture into their classrooms need tools with which to evaluate these portrayals. In this article we summarize studies related to the impact of Hollywood representations on girls and then we provide a case study example of the character Fred Burkle from the television series Angel. In addition we provide a theoretical foundation and popular culture role model checklist that can be used to analyze other representations.

Here we explore the educational implications of classroom activities related to the backgrounds and motivations of talented Hollywood comedy writers and the mathematical moments they created for the Emmy Award-winning animated sitcom Futurama.

A bstract The quiver Yangian, an infinite-dimensional algebra introduced recently in [ 1 ], is the algebra underlying BPS state counting problems for toric Calabi-Yau three-folds. We introduce trigonometric and elliptic analogues of quiver Yangians, which we call toroidal quiver algebras and elliptic quiver algebras, respectively. We construct the representations of the shifted toroidal and elliptic algebras in terms of the statistical model of crystal melting. We also derive the algebras and their representations from equivariant localization of three-dimensional N = 2 supersymmetric quiver gauge theories, and their dimensionally-reduced counterparts. The analysis of supersymmetric gauge theories suggests that there exist even richer classes of algebras associated with higher-genus Riemann surfaces and generalized cohomology theories.

In this paper, we prove results about solutions of the Diophantine equation $x^p+y^p=z^3$ over various number fields using the modular method. First, by assuming some standard modularity conjecture, we prove an asymptotic result for general number fields of narrow class number one satisfying some technical conditions. Second, we show that there is an explicit bound such that the equation $x^p+y^p=z^3$ does not have a particular type of solution over $K=\\mathbb {Q}(\\sqrt {-d})$ , where $d=1,7,19,43,67$ whenever p is bigger than this bound. During the course of the proof, we prove various results about the irreducibility of Galois representations, image of inertia groups, and Bianchi newforms.

The dynamics and existence results of generalized Caputo fractional derivatives have been studied by several authors. Uniform stability and equilibrium in fractional-order neural networks with generalized Caputo derivatives in real-valued settings, however, have not been extensively studied. In contrast to earlier studies, we first investigate the uniform stability and equilibrium results for complex-valued neural networks within the framework of a generalized Caputo fractional derivative. We investigate the intermittent behavior of complex-valued neural networks in generalized Caputo fractional-order contexts. Numerical results are supplied to demonstrate the viability and accuracy of the presented results. At the end of the article, a few open questions are posed.

A bstract We construct an integrable physical model of a single particle scattering with impurities spread on a circle. The S -matrices of the scattering with the impurities are such that the quantized energies of this system, coming from the Bethe Ansatz equations, correspond to the imaginary parts of the non-trivial zeros of the the Riemann ζ ( s ) function along the axis of the complex s -plane. A simple and natural generalization of the original scattering problem leads instead to Bethe Ansatz equations whose solutions are the non-trivial zeros of the Dirichlet L -functions again along the axis . The conjecture that all the non-trivial zeros of these functions are aligned along this axis of the complex s -plane is known as the Generalised Riemann Hypothesis (GRH). In the language of the scattering problem analysed in this paper the validity of the GRH is equivalent to the completeness of the Bethe Ansatz equations. Moreover the idea that the validity of the GRH requires both the duality equation (i.e. the mapping s → 1 – s ) and the Euler product representation of the Dirichlet L -functions finds additional and novel support from the physical scattering model analysed in this paper. This is further illustrated by an explicit counterexample provided by the solutions of the Bethe Ansatz equations which employ the Davenport-Heilbronn function , i.e. a function whose completion satisfies the duality equation χ ( s ) = χ (1 – s ) but that does not have an Euler product representation. In this case, even though there are infinitely many solutions of the Bethe Ansatz equations along the axis , there are also infinitely many pairs of solutions away from this axis and symmetrically placed with respect to it.

Motivated by the importance of diffusion equations in many physical situations in general and in plasma physics in particular, therefore, in this study, we try to find some novel solutions to fractional-order diffusion equations to explain many of the ambiguities about the phenomena in plasma physics and many other fields. In this article, we implement two well-known analytical methods for the solution of diffusion equations. We suggest the modified form of homotopy perturbation method and Adomian decomposition methods using Jafari-Yang transform. Furthermore, illustrative examples are introduced to show the accuracy of the proposed methods. It is observed that the proposed method solution has the desire rate of convergence toward the exact solution. The suggested method’s main advantage is less number of calculations. The proposed methods give series form solution which converges quickly towards the exact solution. To show the reliability of the proposed method, we present some graphical representations of the exact and analytical results, which are in strong agreement with each other. The results we showed through graphs and tables for different fractional-order confirm that the results converge towards exact solution as the fractional-order tends towards integer-order. Moreover, it can solve physical problems having fractional order in different areas of applied sciences. Also, the proposed method helps many plasma physicists in modeling several nonlinear structures such as solitons, shocks, and rogue waves in different plasma systems.

This article discusses the quantization of linear Hamiltonian systems, a historically rich but under explored line of research. The key idea is that a classical linear Hamiltonian system induces on its phase space a compatible complex structure and scalar product, giving rise to a complex Hilbert space where classical dynamics becomes a one-parameter unitary group. Boson Fock quantization of this group then recovers, up to unitary equivalence, the results of canonical quantization. This expository overview traces the development of this framework from foundational works to modern symplectic perspectives, offering a case study in the dialogue between analysis, geometry, and physics.

Spinors can be expressed as Lie algebra of infinitesimal rotations. Spinors are also defined as elements of a vector space which carries a linear representation of the Clifford algebra typically. The motivation for this study is to define a new and particular sequence. An essential feature of this sequence is that while a generalization is being made, spinors, which have a lot of use in physics, are used. This new sequence defined using spinor representations is called the Horadam spinor sequence; formulas such as the Binet formula, generating function formula, and Cassini formula are given. The Horadam spinors given in this study are a generalization of the spinor representations of Horadam quaternion sequences.