Shadows and traces in bicategories

Traces in symmetric monoidal categories are well-known and have many applications; for instance, their functoriality directly implies the Lefschetz fixed point theorem. However, for some applications, such as generalizations of the Lefschetz theorem, one needs "noncommutative" traces, such as the Hattori-Stallings trace for modules over noncommutative rings. In this paper we study a generalization of the symmetric monoidal trace which applies to noncommutative situations; its context is a bicategory equipped with an extra structure called a "shadow." In particular, we prove its functoriality and 2-functoriality, which are essential to its applications in fixed-point theory. Throughout we make use of an appropriate "cylindrical" type of string diagram, which we justify formally in an appendix.Comment: 46 pages; v2: reorganized and shortened, added proof for cylindrical string diagrams; v3: final version, to appear in JHR

Classical Duals, Legendre Transforms and the Vainshtein Mechanism

We show how to generalize the classical duals found by Gabadadze {\it et al} to a very large class of self-interacting theories. This enables one to adopt a perturbative description beyond the scale at which classical perturbation theory breaks down in the original theory. This is particularly relevant if we want to test modified gravity scenarios that exhibit Vainshtein screening on solar system scales. We recognise the duals as being related to the Legendre transform of the original Lagrangian, and present a practical method for finding the dual in general; our methods can also be applied to self-interacting theories with a hierarchy of strong coupling scales, and with multiple fields. We find the classical dual of the full quintic galileon theory as an example.Comment: 16 page

Axiomatic Foundations for Metrics of Distributive Justice Shown by the Example of Needs-Based Justice

Distributive justice deals with allocations of goods and bads within a group. Different principles and results of distributions are seen as possible ideals. Often those normative approaches are solely framed verbally, which complicates the application to different concrete distribution situations that are supposed to be evaluated in regard to justice. One possibility in order to frame this precisely and to allow for a fine-grained evaluation of justice lies in formal modelling of these ideals by metrics. Choosing a metric that is supposed to map a certain ideal has to be justified. Such justification might be given by demanding specific substantiated axioms, which have to be met by a metric. This paper introduces such axioms for metrics of distributive justice shown by the example of needs-based justice. Furthermore, some exemplary metrics of needs-based justice and a three dimensional method for visualisation of non-comparative justice axioms or evaluations are presented. Therewith, a base worth discussing for the evaluation and modelling of metrics of distributive justice is given

Accelerated Backpressure Algorithm

We develop an Accelerated Back Pressure (ABP) algorithm using Accelerated Dual Descent (ADD), a distributed approximate Newton-like algorithm that only uses local information. Our construction is based on writing the backpressure algorithm as the solution to a network feasibility problem solved via stochastic dual subgradient descent. We apply stochastic ADD in place of the stochastic gradient descent algorithm. We prove that the ABP algorithm guarantees stable queues. Our numerical experiments demonstrate a significant improvement in convergence rate, especially when the packet arrival statistics vary over time.Comment: 9 pages, 4 figures. A version of this work with significantly extended proofs is being submitted for journal publicatio

Structure theorems for linear and non-linear differential operators admitting invariant polynomial subspaces

In this paper we derive structure theorems that characterize the spaces of linear and non-linear differential operators that preserve finite dimensional subspaces generated by polynomials in one or several variables. By means of the useful concept of deficiency, we can write explicit basis for these spaces of differential operators. In the case of linear operators, these results apply to the theory of quasi-exact solvability in quantum mechanics, specially in the multivariate case where the Lie algebraic approach is harder to apply. In the case of non-linear operators, the structure theorems in this paper can be applied to the method of finding special solutions of non-linear evolution equations by nonlinear separation of variables.Comment: 23 pages, typed in AMS-LaTe