“Do we teach subjects or students?” Analyzing science and mathematics teacher conversations about issues of equity in the classroom

Teachers involved in a Master's level course in diversity participated in virtual, synchronous, anonymized discussions around issues of ethnic and racial diversity, gender, and stereotypes that could impact their students’ participation in fields related to science, technology, engineering, and mathematics (STEM). Guided by theoretical frameworks from Social Cognitive Career Theory (SCCT) and Critical Race Theory (CRT), a convenience sample of 14 science and mathematics teachers participated in a series of virtual chats using open‐ended questioning and facilitated by two university instructors. Using conversation and critical discourse analyses, three primary themes emerged: understanding of issues related to stereotypes, encouragement of females and minorities to pursue careers in STEM, and the place for diversity discussions in science and mathematics classrooms. The teachers felt burdened by curricular and administrative constraints that inhibit their ability to participate in thought‐provoking critical conversations. The paper concludes with a discussion of ways teachers can assist in the STEM career identity development of their underrepresented females and students of color and calls for research that combines the key findings in SCCT and CRT to build confidence and capacity for teachers to effectively confront issues of racism, sexism, and stereotyping in science and mathematics classrooms.     

What Sense Do Children Make of Three‐Dimensional,Life‐Sized “Representations” of Animals?

One way of investigating what children learn about animals is to examine, the menial models they reveal through their talk when they come face to face with animal representations. In this study, representations were provided by (a) robotic models in a museum, (b) preserved animals in a museum, and (c) preserved animals borrowed from a museum and presented in a school setting. By recording pupil talk in these three settings and then analyzing the content of the talk for comments about the anatomy of animals, their behaviour, habitat features, and so on, both similarities and significant differences are revealed by this study, depending on the setting. Based on these findings, recommendations are made for science education in both classroom and museum settings.     

Comparison Theorems for the Eigenvalue Gap of Schr?dinger Operators on the Real Line

We establish several comparison results on the eigenvalue gap for Schr?dinger operators on the real line. The potentials we consider here include symmetric single-well, double-well, U _c -class, M _c -class as well as their perturbations. Some related results on the eigenvalue ratio are also discussed.     

Optimal Interpolation on Some Classes of Smooth Functions Defined on the Whole Real Axis

In this paper a new research direction in the optimal interpolation theory is introduced, where the classes of functions to be considered are defined on the whole real axis, and a set of permissible interpolating knots is a denumerable set satisfying certain regular conditions.Some basic problems are formulated, and the exact results obtained in recent years are surveyed.Moreover, many unsolved problems have been put forward.     

Instability and violence in Imperial Rome: A “laboratory” for studying social contagion?

Imperial Rome with its >50% assassination rate of emperors, many of whom are depicted in history as ‘deranged’, initially appears a chaotic period of history beyond the purview of science. But time series analysis indicates this violence occurred non‐randomly: reign length was autocorrelated and demonstrated ‘memory persistence,’ and short reigns occurred in clusters. Additionally, deviations from average reign‐length occurred in patterns matching the Empire's rise and decline. A model is proposed for how army‐backed usurpation and post‐coup instability likely generated the observed cycles. The five‐century span of Imperial Rome likely makes it the longest‐lived regime with fair documentation, and potentially provides a ‘laboratory’ with ongoing relevance for studying transmission of violence and instability. © 2016 Wiley Periodicals, Inc. Complexity 21: 613–622, 2016     

Vector Interpretation of the Matrix Orthogonality on the Real Line

In this paper we study sequences of vector orthogonal polynomials. The vector orthogonality presented here provides a reinterpretation of what is known in the literature as matrix orthogonality. These systems of orthogonal polynomials satisfy three-term recurrence relations with matrix coefficients that do not obey to any type of symmetry. In this sense the vectorial reinterpretation allows us to study a non-symmetric case of the matrix orthogonality. We also prove that our systems of polynomials are indeed orthonormal with respect to a complex measure of orthogonality. Approximation problems of Hermite-Padé type are also discussed. Finally, a Markov’s type theorem is presented.     

Gelfand-Kirillov dimensions of the ℤ-graded oscillator representations of 𝔬(n,ℂ) and 𝔰𝔭(2n,ℂ)

In this paper, we give a method to compute the Gelfand–Kirillov dimensions of some polynomial–type weight modules. These modules are infinite-dimensional irreducible 𝔬(n,?)-modules and 𝔰𝔭(2n,?)-modules that appeared in the ?-graded oscillator generalizations of the classical theorem on harmonic polynomials established by Luo and Xu. We also found that some of these modules have the secondly minimal GK-dimension, and some of them have the larger GK-dimension than the maximal GK-dimension apearing in unitary highest-weight modules.     

On 𝔬𝔰𝔭(1 | 2)-Relative Cohomology on S 1|1

We compute the second 𝔬𝔰𝔭(1 | 2) ?relative cohomology space of 𝒦(1) with coefficients in the module of λ-densities 𝔉_λ on S ^1|1. This result allows us to compute the second 𝔬𝔰𝔭(1 | 2) ?relative cohomology space of 𝒦(1) with coefficients in the Poisson superalgebra 𝒮𝒫. We explicitly give 2-cocycles spanning these cohomology spaces.     

Zeros of Non-Baxter Paraorthogonal Polynomials on?the Unit Circle

We provide leading-order asymptotics for the size of the gap in the zeros around 1 of paraorthogonal polynomials on the unit circle whose Verblunsky coefficients satisfy a slow decay condition and are inside the interval (−1,0). We also include related results that impose less restrictive conditions on the Verblunsky coefficients.     

Are the eigenvalues of the B‐spline isogeometric analysis approximation of −Δu = λu known in almost closed form?

We consider the B‐spline isogeometric analysis approximation of the Laplacian eigenvalue problem −Δu = λu over the d‐dimensional hypercube (0,1)^d. By using tensor‐product arguments, we show that the eigenvalue–eigenvector structure of the resulting discretization matrix is completely determined by the eigenvalue–eigenvector structure of the matrix arising from the isogeometric analysis approximation based on B‐splines of degree p of the unidimensional problem . Here, n is the mesh fineness parameter, and the size of is N(n,p) = n + p − 2. In previous works, it was established that the normalized sequence enjoys an asymptotic spectral distribution described by a function e_p(θ), the so‐called spectral symbol. The contributions of this paper can be summarized as follows:

1. the eigenvalues of are arranged in ascending order, ;
2. is a sequence of functions from [0,π] to , which depends only on p;
3. h = 1/n and θ_j,n = jπh for j = 1,…,n; and
4. is the remainder, which satisfies for some constant depending only on α and p. We also provide a proof of this expansion for α = 0 and j = 1,…,N(n,p) −(4p − 2), where 4p − 2 represents a theoretical estimate of the number of outliers .
5. We show through numerical experiments that, for p ≥ 3 and k ≥ 1, there exists a point θ( p,k) ∈ (0,π) such that vanishes on [0,θ( p,k)]. Moreover, as it is suggested by the numerics of this paper, the infimum of θ(p,k) over all k ≥ 1, say y_p, is strictly positive, and the equation holds numerically whenever θ_j,n < θ( p), where θ( p) is a point in (0,y_p] which grows with p.
6. For p ≥ 3, based on the asymptotic expansion in the above item 3, we propose a parallel interpolation–extrapolation algorithm for computing the eigenvalues of , excluding the outliers. The performance of the algorithm is illustrated through numerical experiments. Note that, by the previous item 4, the algorithm is actually not necessary for computing the eigenvalues corresponding to points θ_j,n < θ( p).

     

Bojanov–Naidenov Problem for Differentiable Functions on the Real Line and the Inequalities of Various Metrics

Ukrainian Mathematical Journal - For given r ∈ N, p, λ &gt; 0 and any fixed interval [a, b] ? R, we solve the extreme problem $$ underset{a}{overset{b}{int...     

Coherent Continuation on the Category of (𝔤, K)-Modules

Abstract

Let 𝔤 be a complex semisimple Lie algebra. Let K be an algebraic group acting on the flag variety of 𝔤 with finitely many orbits. We give a geometric interpretation of the coherent continuation on the category of finitely generated (𝔤, K )-modules in terms of the intertwining functors on the category of K-equivariant 𝒟-modules.     

The Nikol'skii—Timan—Dzjadyk Theorem for Functions on Compact Sets of the Real Line

The Nikol'skii—Timan—Dzjadyk theorem concerning the polynomial approximation of functions on the interval [-1,1] is generalized to the case of the approximation of functions given on a compact set on the real line which can consist of an infinite number of intervals. December 8, 1999. Date revised: August 15, 2000. Date accepted: October 19, 2000.     

Functoriality of the BGG Category 𝒪††

This article aims to contribute to the study of algebras with triangular decomposition over a Hopf algebra, as well as the BGG Category 𝒪. We study functorial properties of 𝒪 across various setups. The first setup is over a skew group ring, involving a finite group Γ acting on a regular triangular algebra A. We develop Clifford theory for A?Γ, and obtain results on block decomposition, complete reducibility, and enough projectives. 𝒪 is shown to be a highest weight category when A satisfies one of the “Conditions (S);” the BGG Reciprocity formula is slightly different because the duality functor need not preserve each simple module.

Next, we turn to tensor products of such skew group rings; such a product is also a skew group ring. We are thus able to relate four different types of Categories 𝒪; more precisely, we list several conditions, each of which is equivalent in any one setup, to any other setup, and which yield information about 𝒪.     

On Completeness and Minimality of Random Exponential System in a Weighted Banach Space of Functions Continuous on the Real Line

In this paper, the completeness and minimality properties of some random exponential system in a weighted Banach space of complex functions continuous on the real line for convex nonnegative weight are studied. The results may be viewed as a probabilistic version of Malliavin's classical results.     

Paley–Wiener Theorems of Generalized Fourier Transform Associated with a Cherednik Type Operator on the Real Line

We consider a singular differential-difference operator \(\Lambda \) on the real line which generalizes the Cherednik operator associated with the reflection group \(\mathbb {Z}_2\) on \(\mathbb {R}\). We establish the Paley–Wiener theorems for the generalized Fourier transform on \(\mathbb {R}\) tied to \(\Lambda \).