Feynman’s Operational Calculi: Decomposing Disentanglings

Let X be a Banach space and suppose that A ₁,…,A _n are noncommuting (that is, not necessarily commuting) elements in ℒ(X), the space of bounded linear operators on X. Further, for each i∈{1,…,n}, let μ _i be a continuous probability measure on ℬ([0,1]), the Borel class of [0,1]. Each such n-tuple of operator-measure pairs (A _i ,μ _i ), i=1,…,n, determines an operational calculus or disentangling map T_m1,...,mn{mathcal{T}}_{mu_{1},dots,mu_{n}} from a commutative Banach algebra mathbbD(A₁,...,A_n){mathbb{D}}(A_{1},dots,A_{n}) of analytic functions, called the disentangling algebra , into the noncommutative Banach algebra ℒ(X). The disentanglings are the central processes of Feynman’s operational calculi.     

Stability Properties for Feynman’s Operational Calculus in the Combined Continuous/Discrete Setting

We establish some stability theorems for Feynman’s operational calculus in the setting where the time-ordering measures are allowed to have both continuous and discrete parts. In particular, we investigate stability in a number of special cases of this blended approach to the operational calculus.Mathematics Subject Classifications (2000) Primary: 46J15, 47A56, 47A60, 60B10; secondary: 46N50, 47N50.     

Towards a Comprehensive Stability Theory for Feynman’s Operational Calculus: The Time-Dependent Setting

We obtain lower bounds on blow-up of solutions for the 3D magneto-micropolar equations. More precisely, we establish some estimates for the solution \((\mathbf{u},\mathbf{w},\mathbf{b}) (t)\) in its maximal interval \([0,T^{*})\) provided that \(T^{*}<\infty\), which show for \(\delta\in(0,1)\) that \(\|(\mathbf{u},\mathbf{w},\mathbf{b})(t)\|_{\dot{H}^{s}}\) is at least of the order \((T^{*}-t)^{-(\delta s)/(1+2\delta)}\) for \(s\geq1/2+\delta\). In particular, by choosing a suitable \(\delta\), one concludes that \(\|(\mathbf{u},\mathbf{w},\mathbf{b})(t)\|_{\dot{H}^{s}}\) is at least of the order \((T^{*}-t)^{-s/4}\), and \((T^{*}-t)^{1/4-s/2}\) for \(s\geq1\), and \(1/2< s<3/2\), respectively. We also show that \((T^{*}-t)^{-s/3}\) is a lower rate for \(\|(\mathbf{u},\mathbf{w},\mathbf{b})(t)\|_{\dot{H}^{s}}\) if \(s>3/2\).     

Feynman’s operational calculi for noncommuting operators: The monogenic calculus

In recent papers the authors presented their approach to Feynman’s operational calculi for a system of not necessarily commuting bounded linear operators acting on a Banach space. The central objects of the theory are the disentangling algebra, a commutative Banach algebra, and the disentangling map which carries this commutative structure into the noncommutative algebra of operators. Under assumptions concerning the growth of disentangled exponential expressions, the associated functional calculus for the system of operators is a distribution with compact support which we view as the joint spectrum of the operators with respect to the disentangling map. In this paper, the functional calculus is represented in terms of a higher-dimensional analogue of the Riesz-Dunford calculus using Clifford analysis.     

Analyticity and the Deviant Logician: Williamson’s Argument from Disagreement

One way to discredit the suggestion that a statement is true just in virtue of its meaning is to observe that its truth is the subject of genuine disagreement. By appealing to the case of the unorthodox philosopher, Timothy Williamson has recast this response as an argument foreclosing any appeal to analyticity. Reconciling Quine’s epistemological holism with his treatment of the ‘deviant logician’, I show that we may discharge the demands of charitable interpretation even while attributing trivial semantic error to Williamson’s philosophers. Williamson’s effort to generalize the argument from disagreement therefore fails.     

Borch’s Theorem from the perspective of comonotonicity

This short note revisits the classical Theorem of Borch on the characterization of Pareto optimal risk exchange treaties under the expected utility paradigm. Our objective is to approach the optimal risk exchange problem by a new method, which is based on a Breeden–Litzenberger type integral representation formula for increasing convex functions and the theory of comonotonicity. Our method allows us to derive Borch’s characterization without using Kuhn–Tucker theory, and also without the need of assuming that all utility functions are continuously differentiable everywhere. We demonstrate that our approach can be used effectively to solve the Pareto optimal risk-sharing problem with a positivity constraint being imposed on the admissible allocations when the aggregate risk is positive.     

Analysis of the hyper-chaos generated from Chen’s system

This letter does some further researches on a newly proposed hyper-chaos found in [Gao TG, Chen ZQ, Chen G. A hyper-chaos Generated from Chen’s system, Int J Mod Phys C 2006;17(4):471–8], and gets some new results with some parameters of the system are definite and the other one varies. The basic dynamical behavior of the system under various conditions is analyzed, and the new findings is not only demonstrated by computer simulations but also verified with bifurcation analysis.     

Diversity of firm’s life cycle adapted from the firm’s technology investment decision

The stylized model presented is an optimal control model of technology investment decision of a single product firm. The firm’s technology investment does not have only a long-run positive effect but also a short-run adverse effect on its sales volume. We examine the case of high adverse investment effects where the firm finally leaves the market but we have observed different life cycles till this happens. Depending on the firm’s initial technology stock and sales volume, we compute different firm’s life cycles, which are driven by a trade-off between two strategies: technology versus sales focus strategy. Indifference curves, where managers are indifferent to apply initially technology or sales focus strategies, separate founding conditions of the firm to various classes distinguishable because of the firm’s life cycle.     

Operational aspect of the policy coordination for financial stability: role of Jeffreys–Lindley’s paradox in operations research

This study analyses the implications of Jeffery–Lindley’s paradox and Global Financial Crisis (GFC) for the operational aspect of macroeconomic policy coordination for financial stability. Using a Bayesian Vector Auto-regressive model and data from Jan 1985 to June 2016, our key findings suggest that the claim of macroeconomic policy interaction, interdependence and significance of coordinated policy operations for the financial stability holds its ground. The argument in the support for policy coordination for financial stability was found to be robust against the Jeffreys–Lindley’s paradox and in the Post-GFC era. A profound practical, operational and philosophical implication of this study is the positive aspects of Jeffreys–Lindley’s paradox and the possibility of employing the Frequentist and Bayesian estimation techniques as complementing rather competing frameworks.

     

Unifying voting theory from Nakamura’s to Greenberg’s theorems

Cycles, empty cores, intransitivities, and other complexities affect group decision and voting rules. Approaches that prevent these difficulties include the Nakamura number, Greenberg’s theorem, and single peaked preferences. The results derived here subsume and significantly extend these assertions while providing a common explanation for these seemingly dissimilar conclusions.     

Expansive learning: lessons from one teacher’s learning journey

The provision of quality learning experiences for teachers is critical to mathematics reform agendas aimed at equitable and culturally responsive teaching. In this paper we use an activity theory framework to explore one teacher’s learning journey. Drawing on the teacher’s self-report of his journey 1 year after his participation in an intervention designed to support the introduction of mathematical inquiry practices we examine those factors that supported expansive learning. In seeking to understand our pedagogical stance within the intervention we gained new insights into the provision of research based tools to support learning, the provision of space for individual and collective learning, and the provision of a safe learning environment both within the programme, the class, and the wider professional community. These factors are important in understanding transformational changes associated with ambitious pedagogy.     

Regular 4-polytopes from the Livingstone graph of Janko’s first group

The Janko group J ₁ has, up to duality, exactly two regular rank four polytopes, of respective Schl?fli types {5,3,5} and {5,6,5}. The aim of this paper is to give geometric constructions of these two polytopes, starting from the Livingstone graph.     

Characterizing exemplary mathematics instruction in Japanese classrooms from the learner’s perspective

This paper aims to examine key characteristics of exemplary mathematics instruction in Japanese classrooms. The selected findings of large-scale international studies of classroom practices in mathematics are reviewed for discussing the uniqueness of how Japanese teachers structure and deliver their lessons and what Japanese teachers value in their instruction from a teacher’s perspective. Then an analysis of post-lesson video-stimulated interviews with 60 students in three “well-taught” eighth-grade mathematics classrooms in Tokyo is reported to explore the learners’ views on what constitutes a “good” mathematics lesson. The co-constructed nature of quality mathematics instruction that focus on the role of students’ thinking in the classroom is discussed by recasting the characteristics of how lessons are structured and delivered and what experienced teachers tend to value in their instruction from the learner’s perspective. Valuing students’ thinking as necessary elements to be incorporated into the development of a lesson is the key to the approach taken by Japanese teachers to develop and maintain quality mathematics instruction.     

Primary care network development: the regulator’s perspective

Our research is motivated by the proliferation of primary care models in Ontario, Canada. Currently, primary care is mainly provided by facilities belonging to six models of care. These models are remunerated by various schemes—a mixture of fee-for-service, capitation and salary. In addition, they provide different levels of care and several are better adjusted than others to treat complex health needs. The proposed mixed integer programming model allows the regulator to test the outcomes of locating different types of primary care facilities on the overall cost, accessibility and appropriateness of provided care. The network design is fitted to the heterogeneity of the population residing in a defined geographical area, directly using an index (deprivation index) that was found to correlate with increased health needs and barriers to care. The model capabilities are illustrated on the geographical area of Kingston, Ontario.     

Kac’s conjecture from Nakajima quiver varieties

We prove a generating function formula for the Betti numbers of Nakajima quiver varieties. We prove that it is a q-deformation of the Weyl-Kac character formula. In particular this implies that the constant term of the polynomial counting the number of absolutely indecomposable representations of a quiver equals the multiplicity of a certain weight in the corresponding Kac-Moody algebra, which was conjectured by Kac in 1982.