A Comparison Framework for Interleaved Persistence Modules

We present a generalization of the induced matching theorem and use it to prove a generalization of the algebraic stability theorem for $\mathbb{R}$-indexed pointwise finite-dimensional persistence modules. Via numerous examples, we show how the generalized algebraic stability theorem enables the computation of rigorous error bounds in the space of persistence diagrams that go beyond the typical formulation in terms of bottleneck (or log bottleneck) distance

Parameterized Verification of Graph Transformation Systems with Whole Neighbourhood Operations

We introduce a new class of graph transformation systems in which rewrite rules can be guarded by universally quantified conditions on the neighbourhood of nodes. These conditions are defined via special graph patterns which may be transformed by the rule as well. For the new class for graph rewrite rules, we provide a symbolic procedure working on minimal representations of upward closed sets of configurations. We prove correctness and effectiveness of the procedure by a categorical presentation of rewrite rules as well as the involved order, and using results for well-structured transition systems. We apply the resulting procedure to the analysis of the Distributed Dining Philosophers protocol on an arbitrary network structure.Comment: Extended version of a submittion accepted at RP'14 Worksho

Spectrum of the non-commutative spherical well

We give precise meaning to piecewise constant potentials in non-commutative quantum mechanics. In particular we discuss the infinite and finite non-commutative spherical well in two dimensions. Using this, bound-states and scattering can be discussed unambiguously. Here we focus on the infinite well and solve for the eigenvalues and eigenfunctions. We find that time reversal symmetry is broken by the non-commutativity. We show that in the commutative and thermodynamic limits the eigenstates and eigenfunctions of the commutative spherical well are recovered and time reversal symmetry is restored

Scattering in three-dimensional fuzzy space

We develop scattering theory in a non-commutative space defined by a $su(2)$ coordinate algebra. By introducing a positive operator valued measure as a replacement for strong position measurements, we are able to derive explicit expressions for the probability current, differential and total cross-sections. We show that at low incident energies the kinematics of these expressions is identical to that of commutative scattering theory. The consequences of spacial non-commutativity are found to be more pronounced at the dynamical level where, even at low incident energies, the phase shifts of the partial waves can deviate strongly from commutative results. This is demonstrated for scattering from a spherical well. The impact of non-commutativity on the well's spectrum and on the properties of its bound and scattering states are considered in detail. It is found that for sufficiently large well-depths the potential effectively becomes repulsive and that the cross-section tends towards that of hard sphere scattering. This can occur even at low incident energies when the particle's wave-length inside the well becomes comparable to the non-commutative length-scale.Comment: 12 pages, 6 figure

Quantum like modelling of decision making: quantifying uncertainty with the aid of the Heisenberg-Robertson inequality

This paper contributes to quantum-like modeling of decision making (DM) under uncertainty through application of Heisenberg’s uncertainty principle (in the form of the Robertson inequality). In this paper we apply this instrument to quantify uncertainty in DM performed by quantum-like agents. As an example, we apply the Heisenberg uncertainty principle to the determination of mutual interrelation of uncertainties for “incompatible questions” used to be asked in political opinion pools. We also consider the problem of representation of decision problems, e.g., in the form of questions, by Hermitian operators, commuting and noncommuting, corresponding to compatible and incompatible questions respectively. Our construction unifies the two different situations (compatible versus incompatible mental observables), by means of a single Hilbert space and of a deformation parameter which can be tuned to describe these opposite cases. One of the main foundational consequences of this paper for cognitive psychology is formalization of the mutual uncertainty about incompatible questions with the aid of Heisenberg’s uncertainty principle implying the mental state dependence of (in)compatibility of questions