On the Elliptic Genera of Manifolds of Spin(7) Holonomy

Superstring compactification on a manifold of Spin(7) holonomy gives rise to a 2d worldsheet conformal field theory with an extended supersymmetry algebra. The \({\mathcal{N} = 1}\) superconformal algebra is extended by additional generators of spins 2 and 5/2, and instead of just superconformal symmetry one has a c = 12 realization of the symmetry group \({\mathcal{S}W(3/2,2)}\). In this paper, we compute the characters of this supergroup and decompose the elliptic genus of a general Spin(7) compactification in terms of these characters. We find suggestive relations to various sporadic groups, which are made more precise in a companion paper.     

To what extent are students expected to participate in specialised mathematical discourse? Change over time in school mathematics in England

ABSTRACT

From a discursive perspective, differences in the language in which mathematics questions are posed change the nature of the mathematics with which students are expected to engage. The project The Evolution of the Discourse of School Mathematics (EDSM) analysed the discourse of mathematics examination papers set in the UK between 1980 and 2011. In this article we address the issue of how students over this period have been expected to engage with the specialised discourse of school mathematics. We explain our analytic methods and present some outcomes of the analysis. We identify changes in engagement with algebraic manipulation, proving, relating mathematics to non-mathematical contexts and making connections between specialised mathematical objects. These changes are discussed in the light of public and policy domain debates about ‘standards’ of examinations.     

Conjugacy languages in groups

We study the regularity of several languages derived from conjugacy classes in a finitely generated group G for a variety of examples including word hyperbolic, virtually abelian, Artin, and Garside groups. We also determine the rationality of the growth series of the shortlex conjugacy language in virtually cyclic groups, proving one direction of a conjecture of Rivin.     

A new approach to discretization applied to a continuous, nonlinear, sliding-mode-like control using nonsmooth analysis

This paper introduces a new approach to discretization of nonlinearcontrol laws with a Lipschitz property. The sampling time isdefined as a parameter, which must be selected sufficientlysmall so that the closed-loop system is stable. In contrastto similar results, the stabilizing effect of the control istaken into account. This can result in less conservative constraintson the minimum sampling frequency. The discretization techniquesare explained on a general nonlinear model and applied to thediscretization of a novel nonlinear, robust sliding-mode-likecontrol law. Similar robustness features as for continuous controlare demonstrated. Nonsmooth Lyapunov functions are used forthe discretized sliding-mode-like control introducing cone shapedregions of the state space. One of these cone shaped regionscoincides with a cone shaped layer around the sliding mode definedby the continuous sliding-mode-like control. A stability theoremusing nonsmooth Lyapunov functions is provided.     

Bases for Cooperations in K-Theory

Gaussian polynomials are used to define bases with good multiplicative properties for the algebra K _*(K) of cooperations in K-theory and for the invariants under conjugation.     

Decomposition behavior in aggregated data sets

One way to aggregate data is to combine several sets with the same structure, but no overlap in their ranges of values — for instance, aggregating prices before and after a period of hyperinflation. Looking at nonparametric tests on three ‘items’, we compute the relation of the decomposition of the underlying voting profiles of such aggregated sets to those for the original data. We focus on the Basic components, including examples of ‘pure Basic’ sets, computed using Sage. This yields several interesting results about consistency of nonparametric tests with respect to this kind of aggregation, and suggests types of non-uniformity which are not detected by standard tests.     

Discrete particle swarm optimization for the team orienteering problem

Orienteering problem is a well researched routing problem which is a generalization of the traveling salesman problem. Team orienteering problem (TOP) is the extended version of the orienteering problem with more than one member in the team. In this paper the first known discrete particle swarm optimization (DPSO) algorithm has been developed for 2, 3 and 4-member TOP. In the DPSO meta-heuristic novel methods have been introduced for the initial particle generation process. Reduced variable neighborhood search and 2-opt were applied as the local search tools. The efficacy of the algorithm was tested using seven commonly used benchmark problem sets ranging in size from 21 to 102 nodes. The results of the DPSO algorithm were compared against seven other heuristic algorithms that have been developed for TOP. It was concluded that the developed DPSO algorithm for the TOP is competitive and robust across the benchmark problem sets.     

Dynamic Evasion-Interrogation Games with Uncertainty in the Context of Electromagnetics

We consider two player electromagnetic evasion-pursuit games where each player must incorporate significant uncertainty into their design strategies to disguise their intension and confuse their opponent. In this paper, the evader is allowed to make dynamic changes to his strategies in response to the dynamic input with uncertainty from the interrogator. The problem is formulated in two different ways. One is based on the evolution of the probability density function of the intensity of reflected signal and leads to a controlled forward Kolmogorov or Fokker-Planck equation. The other formulation is based on the evolution of expected value of the intensity of reflected signal and leads to controlled backward Kolmogorov equations. In addition, a number of numerical results are presented to illustrate the usefulness of the proposed approach in exploring problems of control in a general dynamic game setting.