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In the paper (J Math Soc Jpn 72(1):303–331, 2020), Tse-Chung Yang and the first two current authors computed explicitly the number | SSp 2 (F q) | of isomorphism classes of superspecial abelian surfaces over an arbitrary finite field F q of even degree over the prime field F p . There it was assumed that certain commutative Z p -orders satisfy an étale condition that excludes the primes p = 2 , 3 , 5 . We treat these remaining primes in the present paper, where the computations are more involved because of the ramification. This completes the calculation of | SSp 2 (F q) | in the even degree case. The odd degree case was previous treated by Tse-Chung Yang and the first two current authors in (Doc Math 21:1607–1643, 2016). To complete the proof of our main theorem, we give a classification of lattices over local quaternion Bass orders, which is a new input to our previous works. [ABSTRACT FROM AUTHOR]

Learning to calculate with natural numbers by structuring seems promising but how this can be taught in a sustainable manner remains an open question. An eight-month-long intervention based on the idea of using a structural approach to addition and subtraction, and particularly bridging through ten, was implemented in four Swedish first-grade classes. One goal was that by the end of first grade, students would be able to solve tasks such as subtracting 8 from 15 by using part-whole number relations. In this paper, we report on learning outcomes from task-based interviews with intervention and control groups before, immediately after, and one year after the intervention, in order to investigate long-term effects and whether students used a structural approach when solving tasks in a higher number range in the second grade. In comparison to controls, students in the intervention group showed higher increases in their learning outcomes. Moreover, the intervention group used a structural approach to a larger extent when solving tasks in higher number ranges, whereas students in the control group more commonly used single-unit counting to solve such tasks. These findings have implications both for teaching and for research on students’ development of arithmetic skills. [ABSTRACT FROM AUTHOR]

In recent years philosophers have used results from cognitive science to formulate epistemologies of arithmetic (e.g. Giaquinto in J Philos 98(1):5–18, 2001). Such epistemologies have, however, been criticised, e.g. by Azzouni (Talking about nothing: numbers, hallucinations and fictions, Oxford University Press, 2010), for interpreting the capacities found by cognitive science in an overly numerical way. I offer an alternative framework for the way these psychological processes can be combined, forming the basis for an epistemology for arithmetic. The resulting framework avoids assigning numerical content to the Approximate Number System and Object Tracking System, two systems that have so far been the basis of epistemologies of arithmetic informed by cognitive science. The resulting account is, however, only a framework for an epistemology: in the final part of the paper I argue that it is compatible with both platonist and nominalist views of numbers by fitting it into an epistemology for ante rem structuralism and one for fictionalism. Unsurprisingly, cognitive science does not settle the debate between these positions in the philosophy of mathematics, but I it can be used to refine existing epistemologies and restrict our focus to the capacities that cognitive science has found to underly our mathematical knowledge. [ABSTRACT FROM AUTHOR]

Existing research suggests that young children can develop a partial understanding of the equal sign as an operator rather than as a relational symbol of equivalence. This partial understanding can be the result of overemphasis on canonical equation syntaxes of the type a + b = c in elementary school mathematics. This paper presents an examination of context and syntax nuances of relevant sections from the grade 1 Greek series of textbooks and workbooks. Using a conceptual framework of context variation, the analysis shows qualitative differences between equations of similar syntax and provides a nuanced determination of contextual and structural aspects of 'variation' in how the equal sign is presented in elementary mathematics. The paper proposes that since equations have context-specific meanings, context variations should constitute a separate element of analysis when investigating how the equal sign is presented. The implication for practice and future research is that nuanced considerations of equation syntax within varied contexts are needed for elaborating analyses of the equal sign presentation that move beyond dichotomized categorizations of canonical/non-canonical syntaxes. [ABSTRACT FROM AUTHOR]

Tile low-rank general matrix multiplication (TLR GEMM) is a novel method of matrix multiplication on large data-sparse matrices, which can significantly reduce storage footprint and arithmetic complexity under given accuracy. To implement high-performance TLR GEMM on Sunway many-core processor, the following challenges remain to be addressed: 1) design an efficient parallel scheme; 2) provide an efficient kernel library of math functions commonly used in TLR GEMM. This paper proposes swTLR GEMM, an efficient implementation of TLR GEMM. We assign LR GEMM computation to a single computing processing element (CPE) and use grouped task queue to process different data tiles of the TLR matrix. Moreover, we implement an efficient kernel library (swLR Kernels) for low-rank matrix operations. To scale to massive (CGs), we organize the CGs into the CG grid and partition the matrices into blocks accordingly. We also apply Cannon's algorithm to enable efficient communication when processing the matrix blocks across CGs simultaneously. The experiment results show that the DGEMM kernel in swLR Kernels achieves 102 × speedup on average. In terms of overall performance, swTLR GEMM-LLD and swTLR GEMM-LLL achieve 91 × and 20.1 × speedup on average, respectively. In addition, our implementation of swTLR GEMM exhibits good scalability when running on 1,024 CGs of Sunway processors (66,560 cores in total). [ABSTRACT FROM AUTHOR]

The arithmetic version of Anderson localization (AL), i.e., AL with explicit arithmetic description on both the localization frequency and the localization phase, was first given by Jitomirskaya (Ann Math 150:1159–1175, 1999) for the almost Mathieu operators (AMO). Later, the result was generalized by Bourgain and Jitomirskaya (Invent Math 148:453–463, 2002) to a class of one dimensional quasi-periodic long-range operators. In this paper, we propose a novel approach based on an arithmetic version of Aubry duality and quantitative reducibility. Our method enables us to prove the same result for the class of quasi-periodic long-range operators in all dimensions, which includes Jitomirskaya (Ann Math 150:1159–1175, 1999) and Bourgain and Jitomirskaya (Invent Math 148:453–463, 2002) as special cases. [ABSTRACT FROM AUTHOR]

In this paper we study the theory Q. We prove a basic result that says that, in a sense explained in the paper, Q can be split into two parts. We prove some consequences of this result. (i) Q is not a poly-pair theory. This means that, in a strong sense, pairing cannot be defined in Q. (ii) Q does not have the Pudlák Property. This means that there two interpretations of $$\mathsf{S}^1_2$$ in Q which do not have a definably isomorphic cut. (iii) Q is not sententially equivalent with $$\mathsf{PA}^-$$ . This tells us that we cannot do much better than mutual faithful interpretability as a measure of sameness of Q and $$\mathsf{PA}^-$$ . We briefly consider the idea of characterizing Q as the minimal-in-some-sense theory of some kind modulo some equivalence relation. We show that at least one possible road towards this aim is closed. [ABSTRACT FROM AUTHOR]

In this paper we investigate continuity properties of functions $${f : \mathbb {R}_+ \to \mathbb {R}_+}$$ that satisfy the ( p, q)-Jensen convexity inequality where H stands for the pth power (or Hölder) mean. One of the main results shows that there exist discontinuous multiplicative functions that are ( p, p)-Jensen convex for all positive rational numbers p. A counterpart of this result states that if f is ( p, p)-Jensen convex for all $${p \in P \subseteq \mathbb {R}_+}$$ , where P is a set of positive Lebesgue measure, then f must be continuous. [ABSTRACT FROM AUTHOR]