A Recipe for State Dependent Distributed Delay Differential Equations

We use the McKendrick equation with variable ageing rate and randomly distributed maturation time to derive a state dependent distributed delay differential equation. We show that the resulting delay differential equation preserves non-negativity of initial conditions and we characterise local stability of equilibria. By specifying the distribution of maturation age, we recover state dependent discrete, uniform and gamma distributed delay differential equations. We show how to reduce the uniform case to a system of state dependent discrete delay equations and the gamma distributed case to a system of ordinary differential equations. To illustrate the benefits of these reductions, we convert previously published transit compartment models into equivalent distributed delay differential equations.Comment: 28 page

Massive Orbifold

We study some aspects of 2d supersymmetric sigma models on orbifolds. It turns out that independently of whether the 2d QFT is conformal the operator products of twist operators are non-singular, suggesting that massive (non-conformal) orbifolds also `resolve singularities' just as in the conformal case. Moreover we recover the OPE of twist operators for conformal theories by considering the UV limit of the massive orbifold correlation functions. Alternatively, we can use the OPE of twist fields at the conformal point to derive conditions for the existence of non-singular solutions to special non-linear differential equations (such as Painleve III).Comment: 12 page

Semiclassical regime of Regge calculus and spin foams

Recent attempts to recover the graviton propagator from spin foam models involve the use of a boundary quantum state peaked on a classical geometry. The question arises whether beyond the case of a single simplex this suffices for peaking the interior geometry in a semiclassical configuration. In this paper we explore this issue in the context of quantum Regge calculus with a general triangulation. Via a stationary phase approximation, we show that the boundary state succeeds in peaking the interior in the appropriate configuration, and that boundary correlations can be computed order by order in an asymptotic expansion. Further, we show that if we replace at each simplex the exponential of the Regge action by its cosine -- as expected from the semiclassical limit of spin foam models -- then the contribution from the sign-reversed terms is suppressed in the semiclassical regime and the results match those of conventional Regge calculus.Comment: 30 pages, no figures. Updated version with minor corrections, one reference adde

Presumptive Reasoning in a Paraconsistent Setting

We explore presumptive reasoning in the paraconsistent case. Specifically, we provide semantics for non-trivial reasoning with presumptive arguments with contradictory assumptions or conclusions. We adapt the case models proposed by Verheij and define the paraconsistent analogues of the three types of validity defined therein: coherent, presumptively valid, and conclusive ones. To formalise the reasoning, we define case models that use $\mathsf{BD}\triangle$, an expansion of the Belnap--Dunn logic with the Baaz Delta operator. We also show how to recover presumptive reasoning in the original, classical context from our paraconsistent version of case models. Finally, we construct a~two-layered logic over $\mathsf{BD}\triangle$ and $\mathsf{biG}$ (an expansion of G\"{o}del logic with a coimplication or $\triangle$) and obtain a faithful translation of presumptive arguments into formulas