A Fixed Point Theorem for Stronger Association Rules and Its Computational Aspects

Discrete systems such as sets, monoids, groups are familiar categories. The internal structure of the latter two is defined by an algebraic operator. In this paper we concentrate on discrete systems that are characterized by unary operators; these include choice operators σ, encountered in economics and social theory, and closure operators φ, encountered in discrete geometry and data mining. Because, for many arbitrary operators α, it is easy to induce a closure structure on the base set, closure operators play a central role in discrete systems.

A computational algorithm for optimally covering a plane region

In this paper, we prove an analogous version of fixed point theorem due to Kannan [Bull. Calcutta Math. Soc., 60, (1968), 71–76] via conditional contractive mappings and some other relational metrical notions. In the proof of present results, we utilize the notion of comparable mappings and some other well known classical fixed point theorems in the settings of relational and ordered metric spaces. We also, highlight the close connection of $$\alpha $$ -admissible mappings with the binary relation and partial order relation as well. Radically, these investigations open another new direction of metric fixed point theory for contractive type mappings. Moreover, non-trivial examples are given to demonstrate the importance and usefulness of such findings.

Recently Berinde and Păcurar [Approximating fixed points of enriched contractions in Banach spaces. {\em J. Fixed Point Theory Appl.} {\bf 22} (2020), no. 2., 1--10], first introduced the idea of enriched contraction mappings and proved the existence of a fixed point of an enriched contraction mapping using the well-known fact that any fixed point of {the averaged mapping $T_\lambda$, where $\lambda\in (0,1]$, is also a fixed point of the initial mapping $T$}. In this work, we introduce the idea of weak enriched contraction mappings, and a new generalization of an averaged mapping called double averaged mapping. The first attempt is to prove the existence and uniqueness of the fixed point of a double averaged mapping associated with a weak enriched contraction mapping. Based on this result on Banach spaces, we give some sufficient conditions for the equality of all fixed points of a double averaged mapping and the set {of all fixed points of a weak enriched contraction mapping.} Moreover, our results show that an appropriate Kirk's iterative algorithm can be used to approximate a fixed point of a weak enriched contraction mapping. An illustrative example for showing the efficiency of our results is given.

By a metric fixed point theorem we mean an existence result for a fixed point of a mapping f under conditions which depend on a metric d, and which are not invariant when we replace d by an equivalent metric. The best known metric fixed point theorem is the Banach theorem, also called the contractive mapping principle: “every contraction from a complete metric space into itself has a (unique) fixed point”. It is clear that a contractive mapping can lose this property if d is replaced by an equivalent metric. The Banach theorem is a basic tool in functional analysis, nonlinear analysis and differential equations. If we relax the contractive condition, requiring only that the mapping be nonexpansive, that is, d(f (x), f (y)) ≤ d(x, y), then trivial examples show that the Banach theorem need no longer hold. This failure may have been the reason why no significant result about the existence of fixed points for nonexpansive mappings was obtained for many years. However, in 1965, Browder [Br1 and Br2], Göhde [Go] and Kirk [Ki1] proved the following results: “let X be a Banach space, C a closed, convex and bounded subset of X and T : C → C a nonexpansive mapping. If X is either a Hilbert space, or a uniformly convex Banach space or a reflexive Banach space with normal structure, then T has a fixed point”. This result is, in some sense, surprising because it uses convexity hypotheses (more usual in topological fixed point theory) and geometric properties of the Banach spaces (commonly used in linear functional analysis, but rarely considered in nonlinear analysis prior to this time). The above results were the starting point for a new mathematical field: the application of the geometric theory of Banach spaces to fixed point theory. The texts [GK1], [AK], [KZ] and [Z] constitute excellent surveys of this theory.KeywordsBanach SpaceFixed Point TheoremNonexpansive MappingFixed Point TheoryConvex Banach SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction Program schemata (also called schemes, or abstract programs) are widely known as devices for treating properties of computer programs which are invariant across a range of interpretatio ns. Since the intent of schemata is to formalize a specific aspect of effective computation, the properties studied are usually those which have been prescribed in other models of computation: e.g., totality, equivalence, decision power. It is perhaps for this reason that the methodology for dealing with schemata has evolved around constructive mathematics and direct demonstration rather than continuous mathematics, transfinite mathematics and indirect proof. Recently, however, it has been pointed out (Kfoury [3], DeMillo [2]) that useful properties of schemata can be established using arguments which rest on nonconstructi ve foundations; in this case the model theory of first order logic. The outcome of this activity has been to recast some deep questions regarding, for instance, what can and cannot be proved about computer programs into a form which is similar to the corresponding logical questions, where some of the answers are known. In this note,* we present three such applications of model theory. We allow a free wheeling notion of schemata (allowing, for example, first order oracles of any sort desired) and show that direct limitations exist in the ability to draw inferences about schemata by examining finite entailment and axiomatizability. The third application shows that even the expansive concept of effectiveness allowed here does not admit more relative decision power than the standard models of computation. 2 Notation Logical notation follows such standard sources as Bell and Slomson [1]. In particular, such notions as first order language, similarity type, relational structure, and interpretation are assumed in this presentation. A given first order language always talks about

Sequential reactive systems include hardware devices and software programs which operate in continuous interaction with the external environment, from which they receive streams of input signals (data, commands) and in response to them form streams of output signals. Systems of this type include controllers, network switches, program interpreters, system drivers. The behavior of some reactive systems is determined not only by the sequence of values of input signals, but also by the time of their arrival at the inputs of the system and the delays in computing the output signals. These aspects of reactive system computations are taken into account by real-time models of computation which include, in particular, realtime finite state machines (TFSMs). However, in most works where this class of real-time automata is studied a simple variant of TFSM semantics is used: the transduction relation computed by a TFSM is defined so that the elements of an output stream, regardless oftheir timestamps, follow in the same order as the corresponding elements ofthe input stream. This straightforward approach makes the model easier to analyze and manipulate, but it misses many important features of real-time computation. In this paper we study a more realistic semantics of TFSMs and show how to represent it by means of Labeled Transition Systems. The use of the new TFSM model also requires new approaches to the solution of verification problems in the framework of this model. For this purpose, we propose an alternative definition of TFSM computations by means of Labeled Transition Systems and show that the two definitions of semantics for the considered class of real-time finite state machines are in good agreement with each other. The use of TFSM semantics based on Labeled Transition Systems opens up the possibility of adapting well known real-time model checking techniques to the verification ofsequential reactive systems.

In this paper, we discuss some properties of generalized modular metric spaces. This space was first introduced by Turkoglu and Manav in 2018. Later, we extend the concepts about contraction mappings in generalized modular metric spaces. We also develop a convexity structure and defined the N normal structure in this space.The results show us that if X is a complete modular metric space and TN is a contraction mapping then T has a unique fixed point.Moreover, we need the constant of generalized modular metric axiom should be less than or equal to 1 to guarantee the ball is closed. The closeness property of the ball is necessary to generate normal structure in generalized modular metric space. This normal structure property assures the existence of the fixed point of a nonexpansive mapping.

<abstract><p>We introduce a new class of generalized graphic fuzzy $ F $- contractive mappings on metric spaces and establish the existence of common fuzzy coincidence and fixed point results for such contractions. It is significant to note that we do not use any form of continuity of mappings to prove these results. Some examples are provided to verify our proven results. Various developments in the existing literature are generalized and extended by our results. It is aimed that the initiated concepts in this work will encourage new research aspects in fixed point theory and related hybrid models in the literature of fuzzy mathematics.</p></abstract>

The classical Schrödinger bridge seeks the most likely probability law for a diffusion process, in path space, that matches marginals at two end points in time; the likelihood is quantified by the relative entropy between the sought law and a prior. Jamison proved that the new law is obtained through a multiplicative functional transformation of the prior. This transformation is characterised by an automorphism on the space of endpoints probability measures, which has been studied by Fortet, Beurling, and others. A similar question can be raised for processes evolving in a discrete time and space as well as for processes defined over non-commutative probability spaces. The present paper builds on earlier work by Pavon and Ticozzi and begins by establishing solutions to Schrödinger systems for Markov chains. Our approach is based on the Hilbert metric and shows that the solution to the Schrödinger bridge is provided by the fixed point of a contractive map. We approach, in a similar manner, the steering of a quantum system across a quantum channel. We are able to establish existence of quantum transitions that are multiplicative functional transformations of a given Kraus map for the cases where the marginals are either uniform or pure states. As in the Markov chain case, and for uniform density matrices, the solution of the quantum bridge can be constructed from the fixed point of a certain contractive map. For arbitrary marginal densities, extensive numerical simulations indicate that iteration of a similar map leads to fixed points from which we can construct a quantum bridge. For this general case, however, a proof of convergence remains elusive.

This paper was prepared for presentation at the 47th Annual Fall Meeting of the Society of Petroleum Engineers held in San Antonio, Tex., Oct. 8–11, 1972. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgment of where and by who the paper is presented. Publication elsewhere after publication in the JOURNAL paper is presented. Publication elsewhere after publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL is usually granted upon request to the Editor of the appropriate journal provided agreement to give proper credit is made. provided agreement to give proper credit is made. Discussion of this paper is invited. Three copies of any discussion should be sent to the Society of Petroleum Engineers office. Such discussion may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines. Abstract A natural gas pipeline system is herein defined as an arrangement of pipeline and compressor stations connected in some arbitrary manner so as to permit the steady-state flow of gas from one or more points of supply to on or more points of delivery. A network is code in nodal notation to provide the description of the existing facilities (pipelines and compressors). The network thus defined must be of the form of a tree, i.e., the flow rate through each facility is known. Pipelines operating in parallel with each other, having common endpoint pressures are permitted. At each source of supply, the available flow rate and a maximum available pressure are known. At each point of delivery, except the terminal, the flow rate and the minimum delivery pressure are known. The pressure at the pressure are known. The pressure at the terminal is constant, and the flow rate at the terminal is calculated from material balance and fuel considerations. The objective of optimization is to minimize the annual cost of owning and operating the facilities required to transport gas through the system. If the existing facilities are adequate, the sum of the annual costs to operate the existing facilities is minimized, i.e., previous sunk, fixed costs are constant and ignored. If the existing facilities are inadequate, new facilities are added to the system in such a manner that the sum of the annual costs of owning and operating the new facilities plus the operating cost of the existing facilities is minimal. A form of dynamic programming is used as the optimizer. Introduction The design of natural gas pipelines systems requires that a set of pipeline facilities be selected that will permit the flow of gas from points of supply to points of demand. In points of supply to points of demand. In general, there are many solutions (sets of facilities), each of which will satisfy or solve the design problem of interest. The engineer, having defined the design problem, usually employs the following procedure in his effort to develop a solution: assume a set of independent variables and assign these values, solve the appropriate set of equations for values of the dependent variables, determine if the resultant solution is feasible, i.e., all restraints are satisfied, and perhaps estimate the cost of All required facilities, either accept the solution or revalue one or more of the independent variables and repeat Steps 2 through 4. Computer programs have been written that mechanize the computational and data processing aspects of the problem.

Aufgrund fallender Preise zur Speicherung von Daten kann man derzeit eine explosionsartige Zunahme in der Menge der verfügbaren Daten beobachten. Diese Entwicklung gibt Unternehmen und wissenschaftliche Institutionen die Möglichkeit empirische Daten in ungekannter Größenordnung zu analysieren. Für viele Firmen ist die Analyse der gesammelten Daten aus ihrem operationalen Geschäft längst zu einem zentralen strategischen Aspekt geworden. Im Gegensatz zu der seit längerem schon betriebenen Business Intelligence, bestehen diese Analysen nicht mehr nur aus traditionellen relationalen Anfragen. In zunehmendem Anteil kommen komplexe Algorithmen aus den Bereichen Data Mining und Maschinelles Lernen hinzu, um versteckte Muster in den Daten zu erkennen, oder Vorhersagemodelle zu trainieren. Mit zunehmender Datenmenge und Komplexität der Analysen wird jedoch eine neue Generation von Systemen benötigt, die diese Kombination aus Anfragekomplexität und Datenvolumen gewachsen sind. Relationale Datenbanken waren lange Zeit das Zugpferd der Datenanalyse im großen Stil. Grund dafür war zum großen Teil ihre deklarativen Anfragesprache, welche es ermöglichte die logischen und physischen Aspekte der Datenspeicherung und Verarbeitung zu trennen, und Anfragen automatisch zu optimieren. Das starres Datenmodell und ihre beschränkte Menge von möglichen Operationen schränken jedoch die Anwendbarkeit von relationalen Datenbanken für viele der neueren analytischen Probleme stark ein. Diese Erkenntnis hat die Entwicklung einer neuen Generation von Systemen und Architekturen eingeläutet, die sich durch sehr generische Abstraktionen für parallelisierbare analytische Programme auszeichnen; MapReduce kann hier beispielhaft genannt werden, als der zweifelsohne prominenteste Vertreter dieser Systeme. Zwar vereinfachte und erschloss diese neue Generation von Systemen die Datenanalyse in diversen neuen Anwendungsfeldern, sie ist jedoch nicht in der Lage komplexe Anwendungen aus den Bereichen Data Mining und Maschinelles Lernen effizient abzubilden, ohne sich dabei extrem auf spezifische Anwendungen zu spezialisieren. Verglichen mit den relationalen Datenbanken haben MapReduce und vergleichbare Systeme außerdem die deklarative Abstraktion aufgegeben und zwingen den Anwender dazu systemnahe Programme zu schreiben und diese manuell zu optimieren. In dieser Dissertation werden verschiedene Techniken vorgestellt, die es ermöglichen etliche der zentralen Eigenschaften von relationalen Datenbanken im Kontext dieser neuen Generation von daten-parallelen Analysesystemen zu realisieren. Mithilfe dieser Techniken ist es möglich ein Analysesystem zu beschreiben, dessen Programme gleichzeitig sowohl generische und ausdrucksstark, als auch prägnant und deklarativ sind. Im einzelnen stellen wir folgende Techniken vor: Erstens, eine Programmierabstraktion die generisch ist und mit komplexen Datenmodellen umgehen kann, aber gleichzeitig viele der deklarativen Eigenschaften der relationalen Algebra erhält. Programme, die gegen dies Abstraktion entwickelt werden können ähnlich optimiert werden wie relationale Anfragen. Zweitens stellen wir eine Abstraktion für iterative daten-parallele Algorithmen vor. Die Abstraktion unterstützt inkrementelle (delta-basierte) Berechnungen und geht mit zustandsbehafteteten Berechnungen transparent um. Wir beschreiben wie man einen relationalen Anfrageoptimierer erweitern kann so dass dieser iterative Anfragen effektiv optimiert. Wir zeigen dabei dass der Optimierer dadurch in die Lage versetzt wird automatisch Ausführungspläne zu erzeugen, die wohlbekannten, manuell erstellten Programmen entsprechen. Die Abstraktion subsumiert dadurch spezialisierte Systeme (wie Pregel) und bietet vergleichbare Performanz. Drittens stellen wir Methoden vor, um die Programmierabstraktion in eine funktionale Sprachen einzubetten. Diese Integration ermögliche es prägnante Programme zu schreiben und einfach wiederzuverwendenden Komponenten und Bibliotheken, sowie Domänenspezifische Sprachen, zu erstellen. Wir legen dar wie man die Übersetzung und Optimierung des daten-parallelen Programms mit dem Sprachübersetzer der funktionalen Sprache so integriert, dass maximales Optimierungspotenzial besteht.

American Journal of Agricultural EconomicsVolume 71, Issue 3 p. 817-819 Comments and Replie Stochastic Technology and Prices: Reply Quirino Paris, Quirino Paris professor in the Department of Agricultural Economics University of California, DavisSearch for more papers by this author Quirino Paris, Quirino Paris professor in the Department of Agricultural Economics University of California, DavisSearch for more papers by this author First published: 01 August 1989 https://doi.org/10.2307/1242042Citations: 1AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL No abstract is available for this article.Citing Literature Volume71, Issue3August 1989Pages 817-819 RelatedInformation

Coincidence and fixed points for multi-valued mappings and its application to nonconvex integral inclusions

In a wide spectrum of mathematical issues, the presence of a fixed point (FP) is equal to the presence of a appropriate map solution. Thus in several fields of math and science, the presence of a fixed point is important. Furthermore, an interesting field of mathematics has been the study of the existence and uniqueness of common fixed point (CFP) and coincidence points of mappings fulfilling the contractive conditions. Therefore, the existence of a FP is of significant importance in several fields of mathematics and science. Results of the FP, coincidence point (CP) contribute conditions under which maps have solutions. The aim of this paper is to explore these conditions (mappings) used to obtain the FP, CP and CFP of a neutrosophic soft set. We study some of these mappings (conditions) such as contraction map, L-lipschitz map, non-expansive map, compatible map, commuting map, weakly commuting map, increasing map, dominating map, dominated map of a neutrosophic soft set. Moreover we introduce some new points like a coincidence point, common fixed point and periodic point of neutrosophic soft mapping. We establish some basic results, particular examples on these mappings and points. In these results we show the link between FP and CP. Moreover we show the importance of mappings for obtaining the FP, CP and CFP of neutrosophic soft mapping.