Boundedness of Reset P/T Nets

We investigate the decidability of termination, reachability, coverability and deadlock-freeness of Petri nets endowed with a hierarchy of places, and with inhibitor arcs, reset arcs and transfer arcs that respect this hierarchy. We also investigate what happens when we have a mix of these special arcs, some of which respect the hierarchy, while others do not. We settle the decidability status of the above four problems for all combinations of hierarchy, inhibitor, reset and transfer arcs, except the termination problem for two combinations. For both these combinations, we show that deciding termination is as hard as deciding the positivity problem on linear recurrence sequences -- a long-standing open problem.

The study is continued investigation of mathematical and functional/physical interpretation of image analysis and processing operations used as sets of operations (ring elements) in descriptive image algebras (DIA) with one ring. The main result is the determination and characterization of interpretation domains of DIA operations: image algebras that make it possible to operate with both the main image models and main models of transformation procedures that ensure effective synthesis and realization of the basic procedures involved in the formal description, processing, analysis, and recognition of images. The applicability of DIAs in practice is determined by the realizability—the possibility of interpretation—of its operations. The interpretation is considered as a transition from a meaningful description of the operation to its mathematical or algorithmic implementation. The main types of interpretability are defined, and examples of interpretability of operations of the descriptive image algebras with one ring, are given.

We analyze the two possible ways of simulating complex systems with cellular automata: by using the familiar timewise updating or by using the complementary spacewise updating. Both updating algorithms operate on identical sets of initial conditions defining the state of the automaton. While timewise growth generally probes just vanishingly small sets of initial conditions producing statistical samples of the asymptotic attractors, spacewise growth operates with much restricted sets which allow one to simulate them all, exhaustively. Our main result is the derivation of an exact analytical formula to quantify precisely one of the two sources of algorithmic complexity of spacewise detection of the complete set of attractors for elementary 1D cellular automata with generic non-periodic architectures of any arbitrary size. The formula gives the total number of initial conditions that need to be investigated to locate rigorously all possible patterns for any given rule. As simple applications, we illustrate how this knowledge may be used (i) to uncover missing patterns in previous classifications in the literature and (ii) to obtain surprisingly novel patterns that are totally unreachable with the time-honored technique of artificially imposing spatially periodic boundary conditions.

In this paper, we provide a complete study of the spectrum of a constant coefficients differential operator on a scale of localized Sobolev spaces, $$H^{s}_{\mathrm{loc}}(I),$$ which are Fréchet spaces. This is quite different from what we find in the literature, where all the relevant results are concerned with spectrum on Banach spaces. Our aim is to understand the behavior of all the three types of spectrum (point, residual, and continuous) and the relation between them and those of the dual operator. The main result we present shows that there is no complex number in the resolvent set of such operators, which suggest a new way to define spectrum if we want to reproduce the classical theorems of the Spectral Theory in Fréchet spaces.

For each irreducible module of the symmetric group $\\mathcal{S}_{N}$ there is\na set of parametrized nonsymmetric Jack polynomials in $N$ variables taking\nvalues in the module. These polynomials are simultaneous eigenfunctions of a\ncommutative set of operators, self-adjoint with respect to two Hermitian forms,\none called the contravariant form and the other is with respect to a\nmatrix-valued measure on the $N$-torus. The latter is valid for the parameter\nlying in an interval about zero which depends on the module. The author in a\nprevious paper [SIGMA 12 (2016), 033, 27 pages, arXiv:1511.06721] proved the\nexistence of the measure and that its absolutely continuous part satisfies a\nsystem of linear differential equations. In this paper the system is analyzed\nin detail. The $N$-torus is divided into $(N-1)!$ connected components by the\nhyperplanes $x_{i}=x_{j}$, $i<j$, which are the singularities of the system.\nThe main result is that the orthogonality measure has no singular part with\nrespect to Haar measure, and thus is given by a matrix function times Haar\nmeasure. This function is analytic on each of the connected components.\n

According to the Gottesman-Knill theorem, quantum algorithms which utilise only the operations belonging to a certain restricted set are efficiently simulable classically. Since some of the operations in this set generate entangled states, it is commonly concluded that entanglement is insufficient to enable quantum computers to outperform classical computers. I argue in this paper that this conclusion is misleading. First, the statement of the theorem (that the particular set of quantum operations in question can be simulated using a classical computer) is, on reflection, already evident when we consider Bell's and related inequalities in the context of a discussion of computational machines. This, in turn, helps us to understand that the appropriate conclusion to draw from the Gottesman-Knill theorem is not that entanglement is insufficient to enable a quantum performance advantage, but rather that if we limit ourselves to the operations referred to in the Gottesman-Knill theorem, we will not have used the resources provided by an entangled quantum system to their full potential.

We investigate the limitations that emerge in thermodynamic tasks as a result of having local control only over the components of a thermal machine. These limitations are particularly relevant for devices composed of interacting many-body systems. Specifically, we study protocols of work extraction that employ a many-body system as a working medium whose evolution can be driven by tuning the on-site Hamiltonian terms. This provides a restricted set of thermodynamic operations, giving rise to alternative bounds for the performance of engines. Our findings show that those limitations in control render it, in general, impossible to reach Carnot efficiency; in its extreme ramification it can even forbid to reach a finite efficiency or finite work per particle. We focus on the one-dimensional Ising model in the thermodynamic limit as a case study. We show that in the limit of strong interactions the ferromagnetic case becomes useless for work extraction, while the antiferromagnetic case improves its performance with the strength of the couplings, reaching Carnot in the limit of arbitrary strong interactions. Our results provide a promising connection between the study of quantum control and thermodynamics and introduce a more realistic set of physical operations well suited to capture current experimental scenarios.

This paper proposes a set of specific crossover and mutation operators for the delineation of functional regions through evolutionary computation. We consider a problem of dividing a given territory into local labor market areas based on spatial interaction data. Such areas are defined so that a high degree of inter-regional separation and intra-regional integration - in both cases in terms of commuting flows - exist. A genetic algorithm has been designed based on the maximization of a fitness function that measures aggregate intra-region interaction under constraints of inter-region separation and minimum size. Additional requirements, typical of any functional regionalization, include the absence of overlapping between delineated regions and an exhaustive coverage of the whole territory (so all basic spatial units must be allocated to one and only one-region). The complex set of restrictions results in conventional operators often generating invalid solutions, impeding or delaying the evolutionary process. This is the reason why an extensive set of operators has been designed that incorporates knowledge about the problem, allowing the evolution of the set of solutions towards the final result.

We compute the anomalous dimensions of a set of composite operators which involve derivatives at four loops in \(\bar{\rm MS}\) in \(\phi^4\) theory as a function of the operator moment \(n\). These operators are similar to the twist-2 operators which arise in QCD in the operator product expansion in deep inelastic scattering. By regarding their inverse Mellin transform as being equivalent to the DGLAP splitting functions we explore to what extent taking a restricted set of operator moments can give a good approximation to the exact four loop result.

Conditional strategies in iterated quantum games

Some of the expressive power of certain programming languages depends on the provision of convenient means of operating on collections, or “aggregates,“ of data values. This paper argues that certain aggregate operations could be implemented hy applying transformations to fragments of a program that access the elements of an aggregate. The techniques provide a means of isolating the mechanics of a restricted set of aggregate operators from the details of the programming mechanism needed to access individual members of the aggregate.

We introduce a notion of generalized Triebel-Lizorkin spaces associated with sectorial operators in Banach function spaces. Our approach is based on holomorphic functional calculus techniques. Using the concept of \(\mathcal {R}_{s}\)-sectorial operators, which in turn is based on the notion of \(\mathcal {R}_{s}\)-bounded sets of operators introduced by Lutz Weis, we obtain a neat theory including equivalence of various norms and a precise description of real and complex interpolation spaces. Another main result of this article is that an \(\mathcal {R}_{s}\)-sectorial operator always has a bounded H∞-functional calculus in its associated generalized Triebel-Lizorkin spaces.

Quantum resource theory is perhaps the most revolutionary framework that quantum physics has ever experienced. It plays vigorous roles in unifying the quantification methods of a requisite quantum effect as wells as in identifying protocols that optimize its usefulness in a given application in areas ranging from quantum information to computation. Moreover, the resource theories have transmuted radical quantum phenomena like coherence, nonclassicality and entanglement from being just intriguing to being helpful in executing realistic thoughts. A general quantum resource theoretical framework relies on the method of categorization of all possible quantum states into two sets, namely, the free set and the resource set. Associated with the set of free states there is a number of free quantum operations emerging from the natural constraints attributed to the corresponding physical system. Then, the task of quantum resource theory is to discover possible aspects arising from the restricted set of operations as resources. Along with the rapid growth of various resource theories corresponding to standard harmonic oscillator quantum optical states, significant advancement has been expedited along the same direction for generalized quantum optical states. Generalized quantum optical framework strives to bring in several prosperous contemporary ideas including nonlinearity, -symmetric non-Hermitian theories, q-deformed bosonic systems, etc., to accomplish similar but elevated objectives of the standard quantum optics and information theories. In this article, we review the developments of nonclassical resource theories of different generalized quantum optical states and their usefulness in the context of quantum information theories.