Fundamental Gaps in Numerical Semigroups with Respect to Their Multiplicity

For a numerical semigroup, we introduce the concept of a fundamental gap with respect tothe multiplicity of the semigroup.The semigroup is fully determined by its multiplicity and these gaps.We study the case when a set of non-negative integers is the set of fundamental gaps with respect to themultiplicity of a numerical semigroup.Numerical semigroups with maximum and mininmm number ofthis kind of gaps are described.     

Making Sense of Abstract Algebra: Exploring Secondary Teachers’ Understandings of Inverse Functions in Relation to Its Group Structure

This article draws on semi-structured, task-based interviews to explore secondary teachers’ (N = 7) understandings of inverse functions in relation to abstract algebra. In particular, a concept map task is used to understand the degree to which participants, having recently taken an abstract algebra course, situated inverse functions within its group structure (i.e., the set of invertible functions under composition). In addition, their particular conceptions of functions and function composition throughout the interviews were then also considered as a means to explore further their responses during the interviews. Findings indicate that only two participants showed evidence of the desirable mathematically powerful understandings from abstract algebra in relation to inverse functions, and further analysis suggests a variety of challenges in terms of developing meaningful connections, which were more related to conceptions about secondary content than to the abstract algebra content. Implications for the mathematical preparation of secondary teachers are discussed.     

Hölder–Lipschitz Norms and Their Duals on Spaces with Semigroups,with Applications to Earth Mover’s Distance

We introduce a family of bounded, multiscale distances on any space equipped with an operator semigroup. In many examples, these distances are equivalent to a snowflake of the natural distance on the space. Under weak regularity assumptions on the kernels defining the semigroup, we derive simple characterizations of the Hölder–Lipschitz norm and its dual with respect to these distances. As the dual norm of the difference of two probability measures is the Earth Mover’s Distance (EMD) between these measures, our characterizations give simple formulas for a metric equivalent to EMD. We extend these results to the mixed Hölder–Lipschitz norm and its dual on the product of spaces, each of which is equipped with its own semigroup. Additionally, we derive an approximation theorem for mixed Lipschitz functions in this setting.     

Barbashin and krasovskii’s asymptotic stability theorem in application to control systems on smooth manifolds

We introduce the notion of so-called standard control system, whose phase space is a finite-dimensional smooth manifold satisfying a number of conditions; in particular, it is supposed to be connected, orientable, and having a countable atlas. For a given standard control system, we consider a set of time translations and construct the closure of this set in the topology of uniform convergence on compact sets. In these terms, we study the conditions of uniform local reachability of a given trajectory. The main result is formulated in terms of a modified Lyapunov function. A simple example is considered.     

Remark on the Regularities of Kato’s Solutions to Navier-Stokes Equations with Initial Data in L^d(R^d)

Motivated by the results of J. Y. Chemin in ＂J. Anal. Math., 77, 1999, 27- 50＂ and G. Furioli et al in ＂Revista Mat. Iberoamer., 16, 2002, 605-667＂, the author considers further regularities of the mild solutions to Navier-Stokes equation with initial data uo ∈ L^d（R^d）. In particular, it is proved that if u C ∈（[0, T^＊）; L^d（R^d）） is a mild solution of （NSv）, then u（t,x）- e^vt△uo ∈ L^∞（（0, T）;B2/4^1,∞）~∩L^1 （（0, T）; B2/4^3 ,∞） for any T 〈 T^＊.     

Simon’s team’s contributions to scientific progress in mathematics education: A commentary on the Learning Through Activity (LTA) research program

I discuss two ways in which the Learning Through Activity (LTA) research program contributes to scientific progress in mathematics education: (a) providing general and content-specific constructs to explain conceptual learning and instructional design that corroborate and/or elaborate on previous work and (b) raising new questions/issues. The general constructs include using instructional design as testable models of learning and using theoretical constructs to guide real-time, instructional adaptations. In this sense, the general constructs promote understanding of linkages between conceptual learning and instruction in mathematics. The concept-specific constructs consist of empirically-grounded, hypothetical learning trajectories (HLTs) for fractional and multiplicative reasoning. Each HLT consists of specific, intended conceptual changes and tasks that can bring them forth. Questions raised for me by the LTA work involve inconsistencies between the stance on learning and reported teaching-learning interactions that effectively led to students’ abstraction of the intended mathematical concepts.     

From design-based research to re-sourcing ‘in the wild’: reflections on studies of the co-evolution of mathematics teaching resources and practices

This commentary paper looks across the studies of the design and use of mathematics teaching resources included in this issue. It analyses everyday and educational notions of resource; particularly how usage of that term varies across the papers. Key characteristics of each study are identified and the studies are organized into five broad groups representing different lines of investigation of resource design and use. Across the studies, the broad notion of appropriation is influential in conceptualizing user/tool relations, with the instrumental/documentational approach particularly prominent, although close inspection shows that researchers draw on this in different ways. A majority of the studies relate to professional development interventions in a context of large-scale reform efforts, offering insights into a range of approaches to providing practical support for teacher appropriation of resources. One study investigates the resource systems established by ordinary teachers: alongside other research, this highlights the challenges of a re-sourcing approach in which teachers collaborate to curate a localized resource system, and points to conditions which are conducive to the success of such efforts.     

Generalizations of Korn’s inequality based on gradient estimates in Orlicz spaces and applications to variational problems in 2D involving the trace free part of the symmetric gradient

We prove variants of Korn’s inequality involving the deviatoric part of the symmetric gradient of fields u:\mathbbR² é W? \mathbbR² u:{\mathbb{R}^2} \supset \Omega \to {\mathbb{R}^2} belonging to Orlicz–Sobolev classes. These inequalities are derived with the help of gradient estimates for the Poisson equation in Orlicz spaces. We apply these Korn type inequalities to variational integrals of the form

Comment on ‘Correcting for Simulation Bias in Monte Carlo Methods to Value Exotic Options in Models Driven by Lévy Processes’ by C. Ribeiro and N. Webber

Abstract

This paper proposes a pricing method for path-dependent derivatives with discrete monitoring when an underlying asset price is driven by a time-changed Lévy process. The key to our method is to derive a backward recurrence relation for computing the multivariate characteristic function of the intertemporal joint distribution of the time-changed Lévy process. Using the derived representation of the characteristic function, we obtain semi-analytical pricing formulas for geometric Asian, forward start, barrier, fader and lookback options, all of which are discretely monitored.