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The question of when a vertex algebra is a quantization of the arc space of its associated scheme has recently received a lot of attention in both the mathematics and physics literature. This property was first studied by Tomoyuki Arakawa and Anne Moreau (see their paper in the references), and was given the name \lq\lq classical freeness" by Jethro van Ekeren and Reimundo Heluani [Comm. Math. Phys. 386 (2021), no. 1, pp. 495-550] in their work on chiral homology. Later, it was extended to vertex superalgebras by Hao Li [Eur. J. Math. 7 (2021), pp. 1689–1728]. In this note, we prove the classical freeness of the simple affine vertex superalgebra L_n(\mathfrak {o}\mathfrak {s}\mathfrak {p}_{m|2r}) for all positive integers m,n,r satisfying -\frac {m}{2} + r +n+1 > 0. In particular, it holds for the rational vertex superalgebras L_n(\mathfrak {o}\mathfrak {s}\mathfrak {p}_{1|2r}) for all positive integers r,n. [ABSTRACT FROM AUTHOR]

This paper details a literature review of mathematics vocabulary intervention studies for students with mathematics difficulty. The primary aim was to identify instructional practices that support mathematics vocabulary development. We conducted a database search to identify mathematics intervention studies either focused exclusively on vocabulary or with an embedded vocabulary component. Ultimately, 13 studies with participants from kindergarten to Grade 8 were included in the review. The majority of included studies had dual foci on vocabulary and other mathematics content, including whole number computation, word problem solving, fractions, algebra, or geometry. All studies that measured mathematics vocabulary performance indicated positive student outcomes. In addition, multiple studies indicated positive effects on measures of other mathematics content beyond mathematics vocabulary knowledge. We noted six instructional practices across studies that bolstered the mathematics vocabulary performance of students with mathematics difficulty. From most to least common, these practices included formal vocabulary use, explicit instruction, use of representations, repeated exposures, pre‐teaching, and graphic organizers. Implications are addressed for both researchers and practitioners. [ABSTRACT FROM AUTHOR]

Parity is an important and easy to recognise property for meromorphic functions. On the parity of meromorphic functions, Liu, Liu and Korhonen [J Math Anal Appl 512(2022):126129] obtained some meaningful results. In this paper, we investigate the parity of a meromorphic function y(z) under the hypothesis that y(z)2n-2y(z)n is even. In addition, we discuss the relationship on the parity of a meromorphic function with its q-difference polynomials and differential expressions. For instance, we consider the parity of a meromorphic function y(z) under the assumption that y'(z)/ y(z)n and y(qz)/y(z)n are odd or even functions, where n is a positive integer. [ABSTRACT FROM AUTHOR]

For $k\geq 2$ and a nonzero integer n , a generalised Diophantine m -tuple with property $D_k(n)$ is a set of m positive integers $S = \{a_1,a_2,\ldots , a_m\}$ such that $a_ia_j + n$ is a k th power for $1\leq i. Define $M_k(n):= \text {sup}\{|S| : S$ having property $D_k(n)\}$. Dixit et al. ['Generalised Diophantine m -tuples', Proc. Amer. Math. Soc. 150 (4) (2022), 1455–1465] proved that $M_k(n)=O(\log n)$ , for a fixed k , as n varies. In this paper, we obtain effective upper bounds on $M_k(n)$. In particular, we show that for $k\geq 2$ , $M_k(n) \leq 3\,\phi (k) \log n$ if n is sufficiently large compared to k. [ABSTRACT FROM AUTHOR]

In this paper we resolve a conjecture of Kresch and Tamvakis [Duke Math. J. 110 (2001), pp. 359–376]. Our result is the following. Theorem : For any positive integer D and any integers i,j (0\leq i,j \leq D), \; the absolute value of the following hypergeometric series is at most 1: \begin{equation*} {_4F_3} \left [ \begin {array}{c} -i, \; i+1, \; -j, \; j+1 \\ 1, \; D+2, \; -D \end{array} ; 1 \right ]. \end{equation*} To prove this theorem, we use the Biedenharn-Elliott identity, the theory of Leonard pairs, and the Perron-Frobenius theorem. [ABSTRACT FROM AUTHOR]

The number of non-negative integer matrices with given row and column sums features in a variety of problems in mathematics and statistics but no closed-form expression for it is known, so we rely on approximations. In this paper, we describe a new such approximation, motivated by consideration of the statistics of matrices with non-integer numbers of columns. This estimate can be evaluated in time linear in the size of the matrix and returns results of accuracy as good as or better than existing linear-time approximations across a wide range of settings. We show that the estimate is asymptotically exact in the regime of sparse tables, while empirically performing at least as well as other linear-time estimates in the regime of dense tables. We also use the new estimate as the starting point for an improved numerical method for either counting or sampling matrices with given margins using sequential importance sampling. Code implementing our methods is available. [ABSTRACT FROM AUTHOR]