On the *q*-Laplace Transform and Related Special Functions*Axioms* **2016**, *5*(3), 24; doi:10.3390/axioms5030024 - 6 September 2016**Abstract **

Motivated by statistical mechanics contexts, we study the properties of the *q*-Laplace transform, which is an extension of the well-known Laplace transform. In many circumstances, the kernel function to evaluate certain integral forms has been studied. In this article, we establish
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Motivated by statistical mechanics contexts, we study the properties of the *q*-Laplace transform, which is an extension of the well-known Laplace transform. In many circumstances, the kernel function to evaluate certain integral forms has been studied. In this article, we establish relationships between *q*-exponential and other well-known functional forms, such as Mittag–Leffler functions, hypergeometric and *H*-function, by means of the kernel function of the integral. Traditionally, we have been applying the Laplace transform method to solve differential equations and boundary value problems. Here, we propose an alternative, the *q*-Laplace transform method, to solve differential equations, such as as the fractional space-time diffusion equation, the generalized kinetic equation and the time fractional heat equation.
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The Universe in Leśniewski’s Mereology: Some Comments on Sobociński’s Reflections*Axioms* **2016**, *5*(3), 23; doi:10.3390/axioms5030023 - 6 September 2016**Abstract **

Stanisław Leśniewski’s mereology was originally conceived as a theory of foundations of mathematics and it is also for this reason that it has philosophical connotations. The ‘philosophical significance’ of mereology was upheld by Bolesław Sobociński who expressed the view in his correspondence
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Stanisław Leśniewski’s mereology was originally conceived as a theory of foundations of mathematics and it is also for this reason that it has philosophical connotations. The ‘philosophical significance’ of mereology was upheld by Bolesław Sobociński who expressed the view in his correspondence with J.M. Bocheński. As he wrote to Bocheński in 1948: “[...] it is interesting that, being such a simple deductive theory, mereology may prove a number of very general theses reminiscent of metaphysical ontology”. The theses which Sobociński had in mind were related to the mereological notion of “the Universe”. Sobociński listed them in the letter adding his philosophical commentary but he did not give proofs for them and did not specify precisely the theory lying behind them. This is what we want to supply in the first part of our paper. We indicate some connections between the notion of the universe and other specific mereological notions. Motivated by Sobociński’s informal suggestions showing his preference for mereology over the axiomatic set theory in application to philosophy we propose to consider Sobociński’s formalism in a new frame which is the ZFM theory—an extension of Zermelo-Fraenkel set theory by mereological axioms, developed by A. Pietruszczak. In this systematic part we investigate reasons of ’philosophical hopes’ mentioned by Sobociński, pinned on the mereological concept of “the Universe”.
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A Method for Ordering of LR-Type Fuzzy Numbers: An Important Decision Criteria*Axioms* **2016**, *5*(3), 22; doi:10.3390/axioms5030022 - 31 August 2016**Abstract **

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Methods for ordering fuzzy numbers play an important role as decision criteria, with applications in areas such as optimization and data mining, among others. Although there are several proposals for ordering methods in the fuzzy literature, many of them are difficult to
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Methods for ordering fuzzy numbers play an important role as decision criteria, with applications in areas such as optimization and data mining, among others. Although there are several proposals for ordering methods in the fuzzy literature, many of them are difficult to apply and present some problems with ranking computation. For that reason, this work proposes an ordering method for fuzzy numbers based on a simple application of a polynomial function. We study some properties of our new method, comparing our results with those generated by other methods previously discussed in literature.
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Is Kazimierz Ajdukiewicz’s Concept of a Real Definition Still Important?*Axioms* **2016**, *5*(3), 21; doi:10.3390/axioms5030021 - 17 August 2016**Abstract **

The concept of a real definition worked out by Kazimierz Ajdukiewicz is still important in the theory of definition and can be developed by applying Hilary Putnam’s theory of reference of natural kind terms and Karl Popper’s fallibilism. On the one hand,
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The concept of a real definition worked out by Kazimierz Ajdukiewicz is still important in the theory of definition and can be developed by applying Hilary Putnam’s theory of reference of natural kind terms and Karl Popper’s fallibilism. On the one hand, the *definiendum* of a real definition refers to a natural kind of things and, on the other hand, the *definiens* of such a definition expresses actual, empirical, fallible knowledge which can be revised and changed.
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Approach of Complexity in Nature: Entropic Nonuniqueness*Axioms* **2016**, *5*(3), 20; doi:10.3390/axioms5030020 - 12 August 2016**Abstract **

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Boltzmann introduced in the 1870s a logarithmic measure for the connection between the thermodynamical entropy and the probabilities of the microscopic configurations of the system. His celebrated entropic functional for classical systems was then extended by Gibbs to the entire phase space
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Boltzmann introduced in the 1870s a logarithmic measure for the connection between the thermodynamical entropy and the probabilities of the microscopic configurations of the system. His celebrated entropic functional for classical systems was then extended by Gibbs to the entire phase space of a many-body system and by von Neumann in order to cover quantum systems, as well. Finally, it was used by Shannon within the theory of information. The simplest expression of this functional corresponds to a discrete set of *W* microscopic possibilities and is given by ${S}_{BG}=-k{\sum}_{i=1}^{W}{p}_{i}ln{p}_{i}$ (*k* is a positive universal constant; BG stands for Boltzmann–Gibbs). This relation enables the construction of BGstatistical mechanics, which, together with the Maxwell equations and classical, quantum and relativistic mechanics, constitutes one of the pillars of contemporary physics. The BG theory has provided uncountable important applications in physics, chemistry, computational sciences, economics, biology, networks and others. As argued in the textbooks, its application in physical systems is legitimate whenever the hypothesis of ergodicity is satisfied, i.e., when ensemble and time averages coincide. However, what can we do when ergodicity and similar simple hypotheses are violated, which indeed happens in very many natural, artificial and social complex systems. The possibility of generalizing BG statistical mechanics through a family of non-additive entropies was advanced in 1988, namely ${S}_{q}=k\frac{1-{\sum}_{i=1}^{W}{p}_{i}^{q}}{q-1}$ , which recovers the additive ${S}_{BG}$ entropy in the *q*→ 1 limit. The index *q* is to be determined from mechanical first principles, corresponding to complexity universality classes. Along three decades, this idea intensively evolved world-wide (see the Bibliography in http://tsallis.cat.cbpf.br/biblio.htm ) and led to a plethora of predictions, verifications and applications in physical systems and elsewhere. As expected, whenever a paradigm shift is explored, some controversy naturally emerged, as well, in the community. The present status of the general picture is here described, starting from its dynamical and thermodynamical foundations and ending with its most recent physical applications.
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A Logical Analysis of Existential Dependence and Some Other Ontological Concepts—A Comment to Some Ideas of Eugenia Ginsberg-Blaustein*Axioms* **2016**, *5*(3), 19; doi:10.3390/axioms5030019 - 15 July 2016**Abstract **

This paper deals with several problems concerning notion of existential dependence and ontological notions of existence, necessity and fusion. Following some ideas of Eugenia Ginsberg-Blaustein, the notions are treated in reference to objects, in relation to the concepts of state of affairs
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This paper deals with several problems concerning notion of existential dependence and ontological notions of existence, necessity and fusion. Following some ideas of Eugenia Ginsberg-Blaustein, the notions are treated in reference to objects, in relation to the concepts of state of affairs and subject of state of affairs. It provides an axiomatic characterization of these concepts within the framework of a multi-modal propositional logic and then presents a semantic analysis of these concepts. The semantics are a slight modification to the standard relational semantics for normal modal propositional logic.
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Potential Infinity, Abstraction Principles and Arithmetic (Leśniewski Style)*Axioms* **2016**, *5*(2), 18; doi:10.3390/axioms5020018 - 15 June 2016**Abstract **

This paper starts with an explanation of how the logicist research program can be approached within the framework of Leśniewski’s systems. One nice feature of the system is that Hume’s Principle is derivable in it from an explicit definition of natural numbers.
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This paper starts with an explanation of how the logicist research program can be approached within the framework of Leśniewski’s systems. One nice feature of the system is that Hume’s Principle is derivable in it from an explicit definition of natural numbers. I generalize this result to show that all predicative abstraction principles corresponding to second-level relations, which are provably equivalence relations, are provable. However, the system fails, despite being much neater than the construction of *Principia Mathematica* (PM). One of the key reasons is that, just as in the case of the system of PM, without the assumption that infinitely many objects exist, (renderings of) most of the standard axioms of Peano Arithmetic are not derivable in the system. I prove that introducing modal quantifiers meant to capture the intuitions behind potential infinity results in the (renderings of) axioms of Peano Arithmetic (PA) being valid in all relational models (i.e. Kripke-style models, to be defined later on) of the extended language. The second, historical part of the paper contains a user-friendly description of Leśniewski’s own arithmetic and a brief investigation into its properties.
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On the Mutual Definability of the Notions of Entailment, Rejection, and Inconsistency*Axioms* **2016**, *5*(2), 15; doi:10.3390/axioms5020015 - 7 June 2016**Abstract **

In this paper, two axiomatic theories *T*^{−} and *T*′ are constructed, which are dual to Tarski’s theory *T*^{+} (1930) of deductive systems based on classical propositional calculus. While in Tarski’s theory *T*^{+} the primitive notion is the classical
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In this paper, two axiomatic theories *T*^{−} and *T*′ are constructed, which are dual to Tarski’s theory *T*^{+} (1930) of deductive systems based on classical propositional calculus. While in Tarski’s theory *T*^{+} the primitive notion is the classical consequence function (entailment) *Cn*^{+}, in the dual theory *T*^{−} it is replaced by the notion of Słupecki’s rejection consequence *Cn*^{−} and in the dual theory *T*′ it is replaced by the notion of the family *Incons* of inconsistent sets. The author has proved that the theories *T*^{+}, *T*^{−}, and *T*′ are equivalent.
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An Overview of the Fuzzy Axiomatic Systems and Characterizations Proposed at Ghent University*Axioms* **2016**, *5*(2), 17; doi:10.3390/axioms5020017 - 7 June 2016**Abstract **

During the past 40 years of fuzzy research at the Fuzziness and Uncertainty Modeling research unit of Ghent University several axiomatic systems and characterizations have been introduced. In this paper we highlight some of them. The main purpose of this paper consists
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During the past 40 years of fuzzy research at the Fuzziness and Uncertainty Modeling research unit of Ghent University several axiomatic systems and characterizations have been introduced. In this paper we highlight some of them. The main purpose of this paper consists of an invitation to continue research on these first attempts to axiomatize important concepts and systems in fuzzy set theory. Currently, these attempts are spread over many journals; with this paper they are now collected in a neat overview. In the literature, many axiom systems have been introduced, but as far as we know the axiomatic system of Huntington concerning a Boolean algebra has been the only one where the axioms have been proven independent. Another line of further research could be with respect to the simplification of these systems, in discovering redundancies between the axioms.
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Contribution of Warsaw Logicians to Computational Logic*Axioms* **2016**, *5*(2), 16; doi:10.3390/axioms5020016 - 3 June 2016**Abstract **

The newly emerging branch of research of Computer Science received encouragement from the successors of the Warsaw mathematical school: Kuratowski, Mazur, Mostowski, Grzegorczyk, and Rasiowa. Rasiowa realized very early that the spectrum of computer programs should be incorporated into the realm of
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The newly emerging branch of research of Computer Science received encouragement from the successors of the Warsaw mathematical school: Kuratowski, Mazur, Mostowski, Grzegorczyk, and Rasiowa. Rasiowa realized very early that the spectrum of computer programs should be incorporated into the realm of mathematical logic in order to make a rigorous treatment of program correctness. This gave rise to the concept of algorithmic logic developed since the 1970s by Rasiowa, Salwicki, Mirkowska, and their followers. Together with Pratt’s dynamic logic, algorithmic logic evolved into a mainstream branch of research: logic of programs. In the late 1980s, Warsaw logicians Tiuryn and Urzyczyn categorized various logics of programs, depending on the class of programs involved. Quite unexpectedly, they discovered that some persistent open questions about the expressive power of logics are equivalent to famous open problems in complexity theory. This, along with parallel discoveries by Harel, Immerman and Vardi, contributed to the creation of an important area of theoretical computer science: descriptive complexity. By that time, the modal *μ*-calculus was recognized as a sort of a universal logic of programs. The mid 1990s saw a landmark result by Walukiewicz, who showed completeness of a natural axiomatization for the *μ*-calculus proposed by Kozen. The difficult proof of this result, based on automata theory, opened a path to further investigations. Later, Bojanczyk opened a new chapter by introducing an unboundedness quantifier, which allowed for expressing some quantitative properties of programs. Yet another topic, linking the past with the future, is the subject of automata founded in the Fraenkel-Mostowski set theory. The studies on intuitionism found their continuation in the studies of Curry-Howard isomorphism. ukasiewicz’s landmark idea of many-valued logic found its continuation in various approaches to incompleteness and uncertainty.
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An Axiomatic Account of Question Evocation: The Propositional Case*Axioms* **2016**, *5*(2), 14; doi:10.3390/axioms5020014 - 26 May 2016**Abstract **
An axiomatic system for question evocation in Classical Propositional Logic is proposed. Soundness and completeness of the system are proven.
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Fundamental Results for Pseudo-Differential Operators of Type 1, 1*Axioms* **2016**, *5*(2), 13; doi:10.3390/axioms5020013 - 19 May 2016**Abstract **

This paper develops some deeper consequences of an extended definition, proposed previously by the author, of pseudo-differential operators that are of type $1,1$ in Hörmander’s sense. Thus, it contributes to the long-standing problem of creating a systematic theory of such
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This paper develops some deeper consequences of an extended definition, proposed previously by the author, of pseudo-differential operators that are of type $1,1$ in Hörmander’s sense. Thus, it contributes to the long-standing problem of creating a systematic theory of such operators. It is shown that type $1,1$ -operators are defined and continuous on the full space of temperate distributions, if they fulfil Hörmander’s twisted diagonal condition, or more generally if they belong to the self-adjoint subclass; and that they are always defined on the temperate smooth functions. As a main tool the paradifferential decomposition is derived for type $1,1$ -operators, and to confirm a natural hypothesis the symmetric term is shown to cause the domain restrictions; whereas the other terms are shown to define nice type $1,1$ -operators fulfilling the twisted diagonal condition. The decomposition is analysed in the type $1,1$ -context by combining the Spectral Support Rule and the factorisation inequality, which gives pointwise estimates of pseudo-differential operators in terms of maximal functions.
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Infinite-dimensional Lie Algebras, Representations, Hermitian Duality and the Operators of Stochastic Calculus*Axioms* **2016**, *5*(2), 12; doi:10.3390/axioms5020012 - 17 May 2016**Abstract **

We study densely defined unbounded operators acting between different Hilbert spaces. For these, we introduce a notion of symmetric (closable) pairs of operators. The purpose of our paper is to give applications to selected themes at the cross road of operator commutation
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We study densely defined unbounded operators acting between different Hilbert spaces. For these, we introduce a notion of symmetric (closable) pairs of operators. The purpose of our paper is to give applications to selected themes at the cross road of operator commutation relations and stochastic calculus. We study a family of representations of the canonical commutation relations (CCR)-algebra (an infinite number of degrees of freedom), which we call admissible. The family of admissible representations includes the Fock-vacuum representation. We show that, to every admissible representation, there is an associated Gaussian stochastic calculus, and we point out that the case of the Fock-vacuum CCR-representation in a natural way yields the operators of Malliavin calculus. We thus get the operators of Malliavin’s calculus of variation from a more algebraic approach than is common. We further obtain explicit and natural formulas, and rules, for the operators of stochastic calculus. Our approach makes use of a notion of symmetric (closable) pairs of operators. The Fock-vacuum representation yields a maximal symmetric pair. This duality viewpoint has the further advantage that issues with unbounded operators and dense domains can be resolved much easier than what is possible with alternative tools. With the use of CCR representation theory, we also obtain, as a byproduct, a number of new results in multi-variable operator theory which we feel are of independent interest.
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An Overview of Topological Groups: Yesterday, Today, Tomorrow*Axioms* **2016**, *5*(2), 11; doi:10.3390/axioms5020011 - 5 May 2016**Abstract **
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Applications of Skew Models Using Generalized Logistic Distribution*Axioms* **2016**, *5*(2), 10; doi:10.3390/axioms5020010 - 15 April 2016**Abstract **

We use the skew distribution generation procedure proposed by Azzalini [*Scand. J. Stat.*, 1985, *12*, 171–178] to create three new probability distribution functions. These models make use of normal, student-*t* and generalized logistic distribution, see Rathie and Swamee
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We use the skew distribution generation procedure proposed by Azzalini [*Scand. J. Stat.*, 1985, *12*, 171–178] to create three new probability distribution functions. These models make use of normal, student-*t* and generalized logistic distribution, see Rathie and Swamee [Technical Research Report No. 07/2006. Department of Statistics, University of Brasilia: Brasilia, Brazil, 2006]. Expressions for the moments about origin are derived. Graphical illustrations are also provided. The distributions derived in this paper can be seen as generalizations of the distributions given by Nadarajah and Kotz [*Acta Appl. Math.*, 2006, *91*, 1–37]. Applications with unimodal and bimodal data are given to illustrate the applicability of the results derived in this paper. The applications include the analysis of the following data sets: (a) spending on public education in various countries in 2003; (b) total expenditure on health in 2009 in various countries and (c) waiting time between eruptions of the Old Faithful Geyser in the Yellow Stone National Park, Wyoming, USA. We compare the fit of the distributions introduced in this paper with the distributions given by Nadarajah and Kotz [*Acta Appl. Math.*, 2006, *91*, 1–37]. The results show that our distributions, in general, fit better the data sets. The general *R* codes for fitting the distributions introduced in this paper are given in Appendix A .
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The Lvov-Warsaw School and Its Future*Axioms* **2016**, *5*(2), 9; doi:10.3390/axioms5020009 - 11 April 2016**Abstract **
The Lvov-Warsaw School (L-WS) was the most important movement in the history of Polish philosophy, and certainly prominent in the general history of philosophy, and 20th century logics and mathematics in particular.[...]
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Summary of Data Farming*Axioms* **2016**, *5*(1), 8; doi:10.3390/axioms5010008 - 1 March 2016**Abstract **

Data Farming is a process that has been developed to support decision-makers by answering questions that are not currently addressed. Data farming uses an inter-disciplinary approach that includes modeling and simulation, high performance computing, and statistical analysis to examine questions of interest
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Data Farming is a process that has been developed to support decision-makers by answering questions that are not currently addressed. Data farming uses an inter-disciplinary approach that includes modeling and simulation, high performance computing, and statistical analysis to examine questions of interest with a large number of alternatives. Data farming allows for the examination of uncertain events with numerous possible outcomes and provides the capability of executing enough experiments so that both overall and unexpected results may be captured and examined for insights. Harnessing the power of data farming to apply it to our questions is essential to providing support not currently available to decision-makers. This support is critically needed in answering questions inherent in the scenarios we expect to confront in the future as the challenges our forces face become more complex and uncertain. This article was created on the basis of work conducted by Task Group MSG-088 “Data Farming in Support of NATO”, which is being applied in MSG-124 “Developing Actionable Data Farming Decision Support for NATO” of the Science and Technology Organization, North Atlantic Treaty Organization (STO NATO).
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Tactical Size Unit as Distribution in a Data Farming Environment*Axioms* **2016**, *5*(1), 7; doi:10.3390/axioms5010007 - 22 February 2016**Abstract **

In agent based models, the agents are usually platforms (individual soldiers, tanks, helicopters, *etc.*), not military units. In the Sandis software, the agents can be platoon size units. As there are about 30 soldiers in a platoon, there is a need
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In agent based models, the agents are usually platforms (individual soldiers, tanks, helicopters, *etc.*), not military units. In the Sandis software, the agents can be platoon size units. As there are about 30 soldiers in a platoon, there is a need for strength distribution in simulations. The contribution of this paper is a conceptual model of the platoon level agent, the needed mathematical models and concepts, and references earlier studies of how simulations have been conducted in a data farming environment with platoon/squad size unit agents with strength distribution.
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Entropy Production Rate of a One-Dimensional Alpha-Fractional Diffusion Process*Axioms* **2016**, *5*(1), 6; doi:10.3390/axioms5010006 - 5 February 2016**Abstract **

In this paper, the one-dimensional *α*-fractional diffusion equation is revisited. This equation is a particular case of the time- and space-fractional diffusion equation with the quotient of the orders of the time- and space-fractional derivatives equal to one-half. First, some integral
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In this paper, the one-dimensional *α*-fractional diffusion equation is revisited. This equation is a particular case of the time- and space-fractional diffusion equation with the quotient of the orders of the time- and space-fractional derivatives equal to one-half. First, some integral representations of its fundamental solution including the Mellin-Barnes integral representation are derived. Then a series representation and asymptotics of the fundamental solution are discussed. The fundamental solution is interpreted as a probability density function and its entropy in the Shannon sense is calculated. The entropy production rate of the stochastic process governed by the *α*-fractional diffusion equation is shown to be equal to one of the conventional diffusion equation.
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Modular Nuclearity: A Generally Covariant Perspective*Axioms* **2016**, *5*(1), 5; doi:10.3390/axioms5010005 - 29 January 2016**Abstract **

A quantum field theory in its algebraic description may admit many irregular states. So far, selection criteria to distinguish physically reasonable states have been restricted to free fields (Hadamard condition) or to flat spacetimes (e.g., Buchholz-Wichmann nuclearity). We propose instead to use
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A quantum field theory in its algebraic description may admit many irregular states. So far, selection criteria to distinguish physically reasonable states have been restricted to free fields (Hadamard condition) or to flat spacetimes (e.g., Buchholz-Wichmann nuclearity). We propose instead to use a modular ℓp -condition, which is an extension of a strengthened modular nuclearity condition to generally covariant theories. The modular nuclearity condition was previously introduced in Minkowski space, where it played an important role in constructive two dimensional algebraic QFT’s. We show that our generally covariant extension of this condition makes sense for a vast range of theories, and that it behaves well under causal propagation and taking mixtures. In addition we show that our modular ℓp -condition holds for every quasi-free Hadamard state of a free scalar quantum field (regardless of mass or scalar curvature coupling). However, our condition is not equivalent to the Hadamard condition.
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