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In this paper, an extension of first order logic is introduced. In such logics atomic formulas may have infinite lengths. An Omitting Types Theorem is proved. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

In the realm of evolutionary game theory, the majority of scenarios involve players with incomplete knowledge, specially regarding their opponents' actions and payoffs compounded by the ever‐shifting landscape of players' interactions. These dynamics present formidable challenges in both the analysis and optimization of game evolution. To address this, a novel model named the networked evolutionary game (NEG) is proposed based on incomplete information with switched topologies. This model captures situations where players possess limited insight into their opponents' benefits, yet make decisions based on their own payoffs while adapting to different networks and new players. To bridge the gap between incomplete and complete information games, R. Selten's transformation method is leveraged, a renowned approach that converts an incomplete information game into an interim agent game, thereby establishing the equivalence of pure Nash equilibria (NE) in both scenarios. Employing the semi‐tensor product (STP) of matrices, a powerful tool in logistic system, the evolution of the model is articulated through algebraic relationships. This enables to unravel the patterns of game evolution and identify the corresponding pure Nash equilibria. By introducing control players, strategically positioned within the game, optimized control is facilitated over the evolutionary trajectory, ultimately leading to convergence towards an optimal outcome. Finally, these concepts are illustrated with a practical example within the paper. [ABSTRACT FROM AUTHOR]

The study of how children learn numbers has yielded one of the most productive research programs in cognitive development, spanning empirical and computational methods, as well as nativist and empiricist philosophies. This paper provides a tutorial on how to think computationally about learning models in a domain like number, where learners take finite data and go far beyond what they directly observe or perceive. To illustrate, this paper then outlines a model which acquires a counting procedure using observations of sets and words, extending the proposal of Piantadosi et al. (2012). This new version of the model responds to several critiques of the original work and outlines an approach which is likely appropriate for acquiring further aspects of mathematics. [ABSTRACT FROM AUTHOR]

There are different definitions of axioms, but the one that seems to have general approval is that axioms are statements whose truths are universally accepted but cannot be proven; they are the foundation from which further propositional truths are derived. Previous attempts, led by David Hilbert, to show that all of mathematics can be built into an axiomatic system that is complete and consistent failed when Kurt Gödel proved that there will always be statements which are known to be true but can never be proven within the same axiomatic system. But Gödel and his followers took no account of brain mechanisms that generate and mediate logic. In this largely theoretical paper, but backed by previous experiments and our new ones reported below, we show that in the case of so‐called 'optical illusions', there exists a significant and irreconcilable difference between their visual perception and their description according to Euclidean geometry; when participants are asked to adjust, from an initial randomised state, the perceptual geometric axioms to conform to the Euclidean description, the two never match, although the degree of mismatch varies between individuals. These results provide evidence that perceptual axioms, or statements known to be perceptually true, cannot be described mathematically. Thus, the logic of the visual perceptual system is irreconcilable with the cognitive (mathematical) system and cannot be updated even when knowledge of the difference between the two is available. Hence, no one brain reality is more 'objective' than any other. [ABSTRACT FROM AUTHOR]

One of the basic requirements of our modern day society is personal authentication. Biometric recognition should make a human‐like identity determination by identifying its physiological and/or behavioral characteristics. In comparison to traditional knowledge‐based approaches, biometric identification systems have the potential to bring benefits. However, because of the difficulties in extracting non‐class discriminative features, the lack of protection during storage of extracted features, and poor recognition accuracy, most frequently used biometric systems lack model protection and robustness. This research proposed a Mimus multimodal biometric system focused on the combination of multiple modalities and optimal level fusion of features to resolve these problems. Initially, the novel Blob‐funk method extracts the complementary non‐class discriminatory information among different modalities, which accomplishes the biometric data enrollment. Thus, it extracts the different properties by comparing surrounding regions based on finding the local maxima and minima of the function. After extracting the features, they need to be stored in a secure manner in a database. Therefore, the paper incorporates the new code block protection strategy to achieve an effectual protection of continuous monitoring via the generation of non‐invertible features, which is used to create the templates, thus storing them in a database. Finally, the novel Lucynomial logistic regression system incorporates user authentication and thus achieves greater recognition accuracy through estimation of threshold value with confrontation of spoof attacks. Hence, compared to the existing techniques such as SVM, PCA, and DBN, the outcome of the proposed work attains 97.53% accuracy, 0.020% FAR, 96.44% recall, and 97.85% precision, thus exemplifying the competence of the novel system. [ABSTRACT FROM AUTHOR]

This paper extends previous work in the development of a methodology for ill-structured policy problems. It argues that recent advances in the structure and logic of argumentation allow one to develop a new basis for treating policy issues. In brief, policies may be viewed as the outcome of a process of reasoning (argumentation) whose purpose is to establish the basis (rationale) for the policy. This paper shows how the tools of symbolic logic may be applied to handle arguments containing strong inconsistencies. The tools presented allow treatment of one of the most fundamental issues concerned with ill-structured problems, the divergence between opposing formulations or perceptions of a problem. [ABSTRACT FROM AUTHOR]

We investigate Galvin's property, a striking feature of the filter of closed unbounded subsets of an infinite cardinal. In particular, we continue the work of Abraham and Shelah (J. Symbolic Logic 51 (1986), no. 1, 180–189) by developing new methods to handle singular cardinals. In addition, the paper explores some new strengthenings of Galvin's property and analyzes their connections with other classical properties of filters. [ABSTRACT FROM AUTHOR]

This paper investigates a chemotaxis‐Navier–Stokes system with logistic source and a fractional diffusion of order α∈(0,1)$$ \alpha \in \left(0,1\right) $$nt+u·∇n=−(−Δ)αn−χ∇·(n∇c)+an−bn2,ct+u·∇c=Δc+n−c,ut+(u·∇)u=Δu−∇P+n∇ϕ+f,∇·u=0,$$ \left\{\begin{array}{l}{n}_t+u\cdotp \nabla n=-{\left(-\Delta \right)}^{\alpha }n-\chi \nabla \cdotp \left(n\nabla c\right)+ an-b{n}^2,\\ {}{c}_t+u\cdotp \nabla c=\Delta c+n-c,\\ {}{u}_t+\left(u\cdotp \nabla \right)u=\Delta u-\nabla P+n\nabla \phi +f,\nabla \cdotp u=0,\end{array}\right. $$on the two‐dimensional periodic torus 핋2. It is proved that if α∈12,1,a,χ≥0,b>0,f∈C1핋2×[0,∞)∩L∞핋2×(0,∞)∩L2(0,∞);L2핋2 and the initial data (n0,c0,u0)$$ \left({n}_0,{c}_0,{u}_0\right) $$ satisfy some regular conditions, the considered problem admits a global classical solution. Through constructing a new Lyapunov functional different from the previous literature, we can obtain the following asymptotic stabilization: If a=0$$ a=0 $$, the solution components n$$ n $$ and c$$ c $$ satisfy n→0andc→0inL∞핋2ast→∞.If a>0$$ a>0 $$ and b>B04$$ b>\frac{B_0}{4} $$, where B0$$ {B}_0 $$ is a constant determined in (6.24), the solution components n$$ n $$ and c$$ c $$ satisfy n→abandc→abinL∞핋2ast→∞.In either case, the solution component u$$ u $$ satisfies u→0inL∞핋2ast→∞. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a parabolic–elliptic Keller–Segel system with a logistic term on ℝN$$ {\mathrm{\mathbb{R}}}^N $$, involving mixed local and nonlocal operators −Δ+(−Δ)s$$ -\Delta +{\left(-\Delta \right)}^s $$ with s∈(0,1)$$ s\in \left(0,1\right) $$. At first, we prove that the semigroup {T(x)}t>0$$ {\left\{T(x)\right\}}_{t>0} $$ generated by Δ−(−Δ)s−I$$ \Delta -{\left(-\Delta \right)}^s-I $$ is an analytical semigroup and uses blow‐up arguments in combination with the classical Liouville‐type theorem to demonstrate the regularity of weak solutions of the parabolic equation with the mixed operators. Next, the local existence and uniqueness of classical solutions are established by applying the semigroup theory and regularity results in case of s∈N2N+4,1$$ \mathbf{s}\in \left(\frac{N}{2N+4},1\right) $$. Moreover, we obtain the global existence and boundedness of classical solutions under some conditions on given initial values and the asymptotic behavior of the global solutions with strictly positive initial conditions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a parabolic–elliptic Keller–Segel system with a logistic term on ℝN$$ {\mathrm{\mathbb{R}}}^N $$, involving mixed local and nonlocal operators −Δ+(−Δ)s$$ -\Delta +{\left(-\Delta \right)}^s $$ with s∈(0,1)$$ s\in \left(0,1\right) $$. At first, we prove that the semigroup {T(x)}t>0$$ {\left\{T(x)\right\}}_{t>0} $$ generated by Δ−(−Δ)s−I$$ \Delta -{\left(-\Delta \right)}^s-I $$ is an analytical semigroup and uses blow‐up arguments in combination with the classical Liouville‐type theorem to demonstrate the regularity of weak solutions of the parabolic equation with the mixed operators. Next, the local existence and uniqueness of classical solutions are established by applying the semigroup theory and regularity results in case of s∈N2N+4,1$$ \mathbf{s}\in \left(\frac{N}{2N+4},1\right) $$. Moreover, we obtain the global existence and boundedness of classical solutions under some conditions on given initial values and the asymptotic behavior of the global solutions with strictly positive initial conditions. [ABSTRACT FROM AUTHOR]

In all areas of scientific study, definitions are used to describe the meaning of terms. Thus, a good set of definitions aids the scientific process by enabling researchers to communicate in a common language. In this regard, the Systems Engineering community has allocated significant effort to understanding the nature and scope of common definitions of a system. However, less attention has been given to formally examining whether these common definitions of a system are adequate. In this paper, we argue that the common definitions of a system are limited in their ability to adequately define a system's boundary. Furthermore, we argue that the common definitions of a system rely on context and prior understanding to communicate the boundary of a system. Finally, by using concepts from philosophy and mathematical logic, we show that the common definitions of a system are nominal in their ability to define a system's boundary. [ABSTRACT FROM AUTHOR]

In this paper, we study the spatial propagation dynamics of a parabolic–elliptic chemotaxis system with logistic source which reduces to the well‐known Fisher‐KPP equation without chemotaxis. It is known that for fast decaying initial functions, this system has a finite spreading speed. For slowly decaying initial functions, we show that the accelerating propagation will occur and chemotaxis does not affect the propagation mode determined by slowly decaying initial functions if the logistic damping is strong, that is, the system has the same upper and lower bounds of the accelerating propagation as for the classical Fisher‐KPP equation. The main new idea of proving our results is the construction of auxiliary equations to overcome the lack of comparison principle due to chemotaxis. [ABSTRACT FROM AUTHOR]

Recent evidence demonstrates that the perceived, not the actual, level of income inequality influences the redistribution policy. The perception of inequality, as conceptualized in this paper, is closely related to both objective inequality and prospect equality. An axiomatic system of individual preferences is suggested and demonstrated to characterize an index of perceived inequality. Prospect equality reflects the individual ideal level of equality, and it serves as a reference point for perception. I adopt the proposed notion to study voting on redistribution. I theoretically identify the conditions under which a more equal society will demand redistribution while a less equal society blocks redistribution. These insights help explain the redistribution puzzle observed across nations. [ABSTRACT FROM AUTHOR]

Optimal coordination is essential for multi‐automated guided vehicle (AGV) systems, particularly in logistic transportation cases, where the system task completion time needs to be minimized, with the guarantee of safe operation. This is because an optimal coordination strategy (OCS), if achieved, can significantly improve the transportation system's efficiency. In this paper, to deal with the dynamic interaction process among AGVs, and sensing and communication range limits, we formulate the optimal coordination problem into a distributed differential game (DDG) framework, where individual AGVs only use information communicated from nearby AGVs to design their optimal operation trajectories. This helps to significantly reduce the computational and communication requirements for the multi‐AGV logistic transportation systems. Targeting operation safety and working efficiency requirements, we incorporate collision avoidance and trajectory optimization objectives into the proposed framework. It is shown that local OCS, obtained by solving the DDG problem for each AGV, will converge to the global Nash equilibrium, which represents the most efficient operating condition for the entire logistic transportation system. Finally, the efficacy of the proposed method is demonstrated, based on simulations and experiments, benchmarked with existing logistic warehousing planning and differential game methods. Compared with conventional methods, the proposed framework successfully helps reduce the task completion time by up to 16%. [ABSTRACT FROM AUTHOR]

This paper proposes a novel image compression‐encryption scheme based on compressive sensing and bit‐level XOR. In the proposed scheme, the encrypted sequences are generated from a 4‐D hyper‐chaotic Lorenz system and a Logistic system. First, an original image is sampled by a secure block parallel compressive sensing (PCS) scheme. In the PCS phase, a key‐controlled discrete cosine transform sparse basis matrix and a key‐controlled complex Hadamard measurement matrix are employed. Next, the real part and imaginary part of the resulting complex‐valued data are quantized and transformed into bit streams, respectively. After that, the two generated bit streams are combined into one bit stream. Then, the resulting bit stream is further encrypted by a bit‐level XOR operation, where the encrypted key is generated from a chaotic system. The experiment results show the effectiveness and reliability of the proposed joint compression‐encryption scheme. The proposed scheme can not only enhance the security of the compressed image but also improve the reconstructed quality of the compressed image. [ABSTRACT FROM AUTHOR]

Let M be the basic set theory that consists of the axioms of extensionality, emptyset, pair, union, powerset, infinity, transitive containment, Δ0‐separation and set foundation. This paper studies the relative strength of set theories obtained by adding fragments of the set‐theoretic collection scheme to M. We focus on two common parameterisations of the collection: Πn‐collection, which is the usual collection scheme restricted to Πn‐formulae, and strongΠn‐collection, which is equivalent to Πn‐collection plus Σn+1‐separation. The main result of this paper shows that for all n≥1, M+Πn+1− collection +Σn+2− inductionon ω proves that there exists a transitive model of Zermelo Set Theory plus Πn‐collection,the theory M+Πn+1− collection is Πn+3‐conservative over the theory M+ strong Πn− collection. It is also shown that (2) holds for n=0 when the Axiom of Choice is included in the base theory. The final section indicates how the proofs of (1) and (2) can be modified to obtain analogues of these results for theories obtained by adding fragments of collection to a base theory (Kripke‐Platek Set Theory with Infinity plus V=L) that does not include the powerset axiom. [ABSTRACT FROM AUTHOR]

Of course, Dummett saw Wittgenstein as talking about proofs in a modern formalism, and he might concede the change introduced by paradigmatic proofs as again a banal point, that is what makes them paradigmatic. Ramsey, a presumed bridge between Peirce and Wittgenstein, anticipated some ideas of epistemic ("human") logic in his papers written around 1929, when he worked with Wittgenstein at Cambridge. In his middle and late periods, Wittgenstein reached conclusions that sound paradoxical, and at variance with mathematical practice as commonly understood. Wittgenstein and his interpreters largely treated all proofs as being of a kind, but C. S. Peirce, who pondered the puzzle of deduction a century earlier, distinguished between corollarial (routine) and theorematic (creative) proofs. [Extracted from the article]

Unlike Martin-Löf randomness and Schnorr randomness, computable randomness has not been defined, except for a few ad hoc cases, outside of Cantor space. This paper offers such a definition (actually, several equivalent definitions), and further, provides a general method for abstracting 'bit-wise' definitions of randomness from Cantor space to arbitrary computable probability spaces. This same method is also applied to give machine characterizations of computable and Schnorr randomness for computable probability spaces, extending the previously known results. The paper contains a new type of randomness-endomorphism randomness-which the author hopes will shed light on the open question of whether Kolmogorov-Loveland randomness is equivalent to Martin-Löf randomness. The last section contains ideas for future research. [ABSTRACT FROM AUTHOR]

In this paper, the positive fragment of the logic [ABSTRACT FROM AUTHOR]

When Professor Katarzyna Szymanska-Debowska invited me to write a few lines about my friend Jean Mawhin, I agreed without hesitation. At that time Professor Roseau was a well-known mathematician, and Jean Mawhin was not 28 yet. In his talk entitled I A tribute to Juliusz Schauder i (see I Antiquitates Mathematicae i . 2018; 12(1):220-257 for an expanded version), Jean Mawhin said: I restrain myself to a somewhat impressionist tribute, and have mostly the words of some of the greatest mathematicians of the XXth century to show the high rank they all give to SCHAUDER's contributions. I First thought: about the title under which these thoughts are collected. i I oscillated between two possible titles: (1) I The First 80 Years of Nonlinear Analysis of Jean Mawhin i and (2) I An exercise of admiration: Jean Mawhin at the 80th anniversary i . [Extracted from the article]

Recently, quality improvement of housing and the living environment has become a priority issue. The quality improvement of the acoustic field in a direct voice transmission as a fundamental method of information transmission is no exception. It is important to realize an acoustical environment in which the audience can concentrate on hearing sound without distraction due to external noise. The relationship between the voice signal and the external noise has been studied extensively by a number of researchers, thus significant achievements have been reported. Most of the results, however have more or less focused on the voice signals from the viewpoint of perception and understanding of the voice with respect to the masking effect and articulation. This paper differs from these studies in the following two aspects. First, the intent is to evaluate the external noise instead of the voice signal. Second, the sensory evaluation is based on the subjective judgment of human beings. The fuzziness necessarily associated with the subjective quantity is taken into account from the outset. Specifically, by using the fuzzy theory, this paper studies the problem of estimating or predicting the psychological judgment of the audience when meaningless nonwhite noise with an arbitrary power spectrum eaten externally.The validity of the method proposed here is verified by means of the measured data from an indoor psychological experiment. [ABSTRACT FROM AUTHOR]

Logic criteria demand inputs that guarantee detection of certain faults. One such criterion, MUMCUT, is composed of three criteria, where each constituent criterion ensures the detection of specific faults. In practice, the criteria may overlap in terms of faults detected, leading to redundant tests, but due to the fact that infeasible requirements do not result in tests, all the constituent criteria are needed. The key insight of this paper is that analysis of the feasibility of the constituent criteria can reduce test set size without sacrificing fault detection for specific faults. This paper introduces a new logic criterion, Minimal-MUMCUT, and shows how it can apply to minimal DNF, minimal CNF, and general form Boolean expressions. With Minimal-MUMCUT, a determination is made of which constituent criteria are feasible, and hence necessary, at the level of individual literals and terms. An empirical study found that Minimal-MUMCUT reduces the test set size, without sacrificing fault detection, regardless of the predicate format. Copyright © 2011 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Model‐Based Systems Design (MBSD) formalizes the application of system modeling. However, it is still difficult for designers to implement MBSD in practice. One of the challenges is to make the field of design more unified, as different MBSD methodologies focus on specific areas, and various designers from different areas use ambiguous terms. Moreover, the failure of some products originates in undiscovered couplings of functions in the design stage. Axiomatic Design (AD) provides a scientific foundation for system design. In order to reduce ambiguity and alleviate functional couplings, this paper integrates AD with the MBSD and proposes a unified systems design approach, called Axiomatic Model‐Based Systems Design (A‐MBSD). A‐MBSD uses four pillars: (a) the fundamental framework of AD; (b) the Independence Axiom; (c) a new behavior domain (BD); and (d) an A‐MBSD modeling profile. Finally, the design of a Forest Fire Satellite Monitoring and Control System demonstrates that the proposed approach can improve unity and alleviate the coupling of functions in a system design. [ABSTRACT FROM AUTHOR]

Technologically advanced aircraft rely on robust and responsive logistics systems to ensure a high state of operational readiness. This paper fills a critical gap in the literature for combat models by closely relating effectiveness of the logistics system to determinants of success in combat. We present a stochastic diffusion model of an aerial battle between Blue and Red forces. The number of aircraft of Blue forces aloft and ready to be aloft on combat missions is limited by the maximum number of assigned aircraft, the reliability of aircraft subsystems, and the logistic system's ability to repair and replenish those subsystems. Our parsimonious model can illustrate important trade‐offs between logistics decision variables and operational success. [ABSTRACT FROM AUTHOR]

This paper is a contribution to the development of model theory of fuzzy logic in narrow sense. We consider a formal system EvŁ of fuzzy logic that has evaluated syntax, i. e. axioms need not be fully convincing and so, they form a fuzzy set only. Consequently, formulas are provable in some general degree. A generalization of Gödel's completeness theorem does hold in EvŁ. The truth values form an MV-algebra that is either finite or Łukasiewicz algebra on [0, 1]. The classical omitting types theorem states that given a formal theory T and a set Σ(x1, ... , xn ) of formulas with the same free variables, we can construct a model of T which omits Σ, i. e. there is always a formula from Σ not true in it. In this paper, we generalize this theorem for EvŁ, that is, we prove that if T is a fuzzy theory and Σ(x1, ... , xn ) forms a fuzzy set , then a model omitting Σ also exists. We will prove this theorem for two essential cases of EvŁ: either EvŁ has logical (truth) constants for all truth values, or it has these constants for truth values from [0, 1] ∩ ℚ only. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

Multiple criteria group site selection problems involve a group of individuals evaluating a set of alternative sites on the basis of multiple criteria. This paper presents an application of a new fuzzy algorithm for finding and exploring potential solutions to these problems in a raster Geographical Information System (GIS) environment. Linguistic assessments from decision-makers are represented as triangular fuzzy numbers (TFN's), which are adjusted for uncertainty in the source data and their relationship to suitability by using an approach based on type-2 fuzzy sets. The first aggregation of inputs is a compensatory one based on fuzzy multiattribute decision-making (MADM) theory. An adjusted aggregation then factors in conflicts, risks and uncertainties to enable a variety of compensatory and non-compensatory outcomes to be generated based on decision-maker preferences. The algorithm was implemented in ArcView GIS as part of an ongoing collaborative project with Brisbane Airport. This paper outlines the fuzzy algorithm and its use in site selection for a recycling facility on the Brisbane Airport site. Copyright © 2005 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

To help guide certain problem‐solving efforts in a timely and relevant manner, public agencies should consider the use of simple multiplicative component models. Such models involve basic algebra, arithmetic, practical logic, and straightforward cross‐tabulations and tables. Two actual case examples are presented. Additional diverse applications are briefly discussed to illustrate the potential value of this tool to guide a number of evidence‐informed efforts. [ABSTRACT FROM AUTHOR]

Using the analysis developed in our earlier paper , we show that every uncountable cardinal in Gitik's model of in which all uncountable cardinals are singular is almost Ramsey and is also a Rowbottom cardinal carrying a Rowbottom filter. We assume that the model of is constructed from a proper class of strongly compact cardinals, each of which is a limit of measurable cardinals. Our work consequently reduces the best previously known upper bound in consistency strength for the theory [ABSTRACT FROM AUTHOR]

A structure is called weakly oligomorphic if its endomorphism monoid has only finitely many invariant relations of every arity. The goal of this paper is to show that the notions of homomorphism-homogeneity, and weak oligomorphy are not only completely analogous to the classical notions of homogeneity and oligomorphy, but are actually closely related. We first prove a Fraïssé-type theorem for homomorphism-homogeneous relational structures. We then show that the countable models of the theories of countable weakly oligomorphic structures are mutually homomorphism-equivalent (we call first order theories with this property weakly ω-categorical). Furthermore we show that every weakly oligomorphic homomorphism-homogeneous structure contains (up to isomorphism) a unique homogeneous, homomorphism-homogeneous core, to which it is homomorphism-equivalent. As a consequence we obtain that every countable weakly oligomorphic structure is homomorphism-equivalent to a finite or ω-categorical structure. As a corollary we obtain a characterization of positive existential theories of weakly oligomorphic structures as the positive existential parts of ω-categorical theories. [ABSTRACT FROM AUTHOR]

This paper is concerned with the possible values of the cofinality of the least Berkeley cardinal. Berkeley cardinals are very large cardinal axioms incompatible with the Axiom of Choice, and the interest in the cofinality of the least Berkeley arises from a result in [1], showing it is connected with the failure of AC. In fact, by a theorem of Bagaria, Koellner and Woodin, if γ is the cofinality of the least Berkeley cardinal then γ‐DC fails. We shall prove that this result is optimal for γ=ω or γ=ω1. In particular, it will follow that the cofinality of the least Berkeley is independent of ZF. [ABSTRACT FROM AUTHOR]

The main result of this paper is Theorem [1.1], which shows that it is possible for derived models to satisfy [ABSTRACT FROM AUTHOR]

In this paper we prove some cases of the gap‐2 cardinal transfer theorem for uncountable languages assuming the axiom of constructibility. Consider uncountable cardinals κ,λ, with λ regular, a first order language L with at least one unary predicate symbol U, with |L|

The study of Martin-Löf randomness on a computable metric space with a computable measure has seen much progress recently. In this paper we study Martin-Löf randomness on a more general space, that is, a computable topological space with a computable measure. On such a space, Martin-Löf randomness may not be a natural notion because there is no universal test, and Martin-Löf randomness and complexity randomness (defined in this paper) do not coincide in general. We show that SCT3 is a sufficient condition for the existence and coincidence, and study how much we can weaken this condition. [ABSTRACT FROM AUTHOR]

Abstract: Nowadays outcome‐based methods are adapted in e‐learning system to meet the need of the novel development of e‐learning systems for improved web‐based retrieval results. Typically, knowledge retrieval process is denoted by the production rule, frame, semantic network, fuzzy logic, predicate logic, and group of skill concept. To get the optimized result in proposed skill‐based e‐learning, the fuzzy knowledge model is applied. In knowledge retrieval, the fuzzy membership value of the knowledge and the combinations of framed rules are used to acquire the knowledge. The fuzzy techniques are adapted for analyzing the retrieved knowledge concept of individual skills like Inadequate (Ia), Adequate (A), Value added adequate (Vaa), and Integrated skill (I) and in fuzzy inference system in skill‐based education provides a decision about the learner community skill set and it promotes the excellence skill through the delivery of the suitable courses to the learners instead of supplying common courses to different skilled persons. In the existing knowledge modeling methods known as the Knowledge Capturing and Modeling, concept map‐based knowledge modeling confirm the learners to have known or unknown domain concept. Further, many of the researchers present on the analysis of the learner performances, behavior, learning environment, etc. The proposed paper investigate the individual skill abilities and it is suggested a set of courses in adaptive curriculum and syllabus to the learner and also it adapts andragogy in skill‐based education, to model the fuzzy knowledge, the fuzzy membership function and rules are used. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of recursive equilibria in a dynamic stochastic model with infinitely lived heterogeneous agents, several commodities, and general inter- and intratemporal production. We illustrate the usefulness of our result by providing sufficient conditions for the existence of recursive equilibria in heterogeneous agent versions of both the Lucas asset pricing model and the neoclassical stochastic growth model. [ABSTRACT FROM AUTHOR]

This paper provides a finer analysis of the well-known form of the Local Deduction Theorem in contraction-free logics (understood as Tarski-style finitary structural consequence relations). An infinite hierarchy of its natural strengthenings is introduced and studied. The main results are the separation of its initial four members and the subsequent collapse of the hierarchy. [ABSTRACT FROM AUTHOR]

In this paper, we show that Ogasawa's theorem has a proof in Bishop style constructive mathematics (BISH). In , we introduced the elementary constructive theory of C*-algebras in BISH, but we did not discuss the powers of positive elements there. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim [ABSTRACT FROM AUTHOR]

In this paper a first order theory for the logics defined through literal paraconsistent-paracomplete matrices is developed. These logics are intended to model situations in which the ground level information may be contradictory or incomplete, but it is treated within a classical framework. This means that literal formulas, i.e. atomic formulas and their iterated negations, may behave poorly specially regarding their negations, but more complex formulas, i.e. formulas that include a binary connective are well behaved. This situation may and does appear for instance in data bases (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

The purpose of this paper is to survey the correspondence between bounded arithmetic and propositional proof systems. In addition, it also contains some new results which have appeared as an extended abstract in the proceedings of the conference TAMC 2008 [11]. Bounded arithmetic is closely related to propositional proof systems; this relation has found many fruitful applications. The aim of this paper is to explain and develop the general correspondence between propositional proof systems and arithmetic theories, as introduced by Krajíček and Pudlák [46]. Instead of focusing on the relation between particular proof systems and theories, we favour a general axiomatic approach to this correspondence. In the course of the development we particularly highlight the role played by logical closure properties of propositional proof systems, thereby obtaining a characterization of extensions of EF in terms of a simple combination of these closure properties (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

The notion of an ℐ -matrix as a model of a given π -institution ℐ is introduced. The main difference from the approach followed so far in Categorical Abstract Algebraic Logic (CAAL) and the one adopted here is that an ℐ -matrix is considered modulo the entire class of morphisms from the underlying N -algebraic system of ℐ into its own underlying algebraic system, rather than modulo a single fixed (N,N ′)-logical morphism. The motivation for introducing ℐ -matrices comes from a desire to formulate a correspondence property for N -protoalgebraic π -institutions closer in spirit to the one for sentential logics than that considered in CAAL before. As a result, in the previously established hierarchy of syntactically protoalgebraic π -institutions, i. e., those with an implication system, and of protoalgebraic π -institutions, i. e., those with a monotone Leibniz operator, the present paper interjects the class of those π -institutions with the correspondence property, as applied to ℐ -matrices. Moreover, this work on ℐ -matrices enables us to prove many results pertaining to the local deduction-detachment theorems, paralleling classical results in Abstract Algebraic Logic formulated, first, by Czelakowski and Blok and Pigozzi. Those results will appear in a sequel to this paper. (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

We present some general results concerning the topological space of cuts of a countable model of arithmetic given by a particular indicator Y. The notion of “indicator” is de.ned in a novel way, without initially specifying what property is indicated and is used to de.ne a topological space of cuts of the model. Various familiar properties of cuts (strength, regularity, saturation, coding properties) are investigated in this sense, and several results are given stating whether or not the set of cuts having the property is comeagre. A new notion of “generic cut” is introduced and investigated and it is shown in the case of countable arithmetically saturated models M ⊧ PA that generic cuts exist, indeed the set of generic cuts is comeagre in the sense of Baire, and furthermore that two generic cuts within the same “small interval” of the model are conjugate by an automorphism of the model.The paper concludes by outlining some applications to constructions of cuts satisfying properties incompatible with genericity, and discussing in model-theoretic terms those properties for which there is an indicator Y. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

In a previous paper we constructed a full and faithful functor ℳ from the category of locally compact metric spaces to the category of formal topologies (representations of locales). Here we show that for a real-valued continuous function f, ℳ(f) factors through the localic positive reals if, and only if, f has a uniform positive lower bound on each ball in the locally compact space. We work within the framework of Bishop constructive mathematics, where the latter notion is strictly stronger than point-wise positivity. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

This paper provides a constructive topological semantics for non-deducibility of a first order intuitionistic formula. Formal topology theory, in particular the recently introduced notion of a binary positivity predicate, and co-induction are two needful tools. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

We examine the concept of almost everywhere domination from the viewpoint of mass problems. Let AED and MLR be the sets of reals which are almost everywhere dominating and Martin-Löf random, respectively. Let b1, b2, and b3 be the degrees of unsolvability of the mass problems associated with AED, MLR × AED, and MLR ∩ AED, respectively. Let &Pscr;w be the lattice of degrees of unsolvability of mass problems associated with nonempty Π01 subsets of 2ω. Let 1 and 0 be the top and bottom elements of &Pscr;w. We show that inf(b1, 1), inf(b2, 1), and inf(b3, 1) belong to &Pscr;w and 0 < inf(b1, 1) < inf(b2, 1) < inf(b3, 1) < 1. Under the natural embedding of the recursively enumerable Turing degrees into &Pscr;w, we show that inf(b1, 1) and inf(b3, 1) but not inf(b2, 1) are comparable with some recursively enumerable Turing degrees other than 0 and 0′. In order to make this paper more self-contained, we exposit the proofs of some recent theorems due to Hirschfeldt, Miller, Nies, and Stephan. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

This paper proposes an extension of Chaitin's halting probability Ω to a measurement operator in an infinite dimensional quantum system. Chaitin's Ω is defined as the probability that the universal self-delimiting Turing machine U halts, and plays a central role in the development of algorithmic information theory. In the theory, there are two equivalent ways to define the program-size complexity H (s) of a given finite binary string s. In the standard way, H (s) is defined as the length of the shortest input string for U to output s. In the other way, the so-called universal probability m is introduced first, and then H (s) is defined as –log2m (s) without reference to the concept of program-size. Mathematically, the statistics of outcomes in a quantum measurement are described by a positive operator-valued measure (POVM) in the most general setting. Based on the theory of computability structures on a Banach space developed by Pour-El and Richards, we extend the universal probability to an analogue of POVM in an infinite dimensional quantum system, called a universal semi-POVM. We also give another characterization of Chaitin's Ω numbers by universal probabilities. Then, based on this characterization, we propose to define an extension of Ω as a sum of the POVM elements of a universal semi-POVM. The validity of this definition is discussed. In what follows, we introduce an operator version $ \hat H $(s) of H (s) in a Hilbert space of infinite dimension using a universal semi-POVM, and study its properties. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

A set of propositional connectives is said to be functionally complete if all propositional formulae can be expressed using only connectives from that set. In this paper we give sufficient and necessary conditions for a one-element set of propositional connectives to be functionally complete. These conditions provide a simple and elegant characterization of functionally complete one-element sets of propositional connectives (of arbitrary arity). (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

The paper deals with a generalization of Helmholtz' conditions for the existence of a first-order kinetic potential related to a given set of 2nd order ordinary differential equations. The extension affects the consideration of such ODE systems which are not self-adjoint due to dissipation. Necessary and sufficient conditions are given for the simultaneous existence of two state functions – a Lagrangian and a dissipation function – both of the first-order such that the given set of 2nd order ODE results from the well-known Lagrangean approach. [ABSTRACT FROM AUTHOR]