25 results on '"Closed set"'
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2. Some nil-ai-semiring varieties.
- Author
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Wu, Y. N. and Zhao, X. Z.
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VOCABULARY - Abstract
We study some nil-ai-semiring varieties. We establish a model for the free object in the variety FC generated by all commutative flat semirings. Also, we provide two sufficient conditions under which a finite ai-semiring is nonfinitely based. As a consequence, we show that the power semiring P S ˙ c (W) of the finite nil-semigroup S ˙ c (W) is nonfinitely based, where W is a finite set of words in the free commutative semigroup X c + over an alphabet X, whenever the maximum of lengths of words in W is k ≥ 3 and W does not contain the kth power of a letter. This partially answers a problem raised by Jackson et al. (J Algebr 611: 211–245, 2022). [ABSTRACT FROM AUTHOR]
- Published
- 2024
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3. Structural Similarity of Objects Represented by Ordinary Graphs.
- Author
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Gusakova, S. M.
- Abstract
The operations of structural similarity on the objects with a factual basis presented as ordinary graphs are considered. Conditions are given under which such objects are similar, and a characterization of the set of graphs with isomorphic results of the similarity operation is presented. The proofs use the apparatus of the Galois correspondence associated with the graph. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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4. From Open Set to Closed Set: Supervised Spatial Divide-and-Conquer for Object Counting.
- Author
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Xiong, Haipeng, Lu, Hao, Liu, Chengxin, Liu, Liang, Shen, Chunhua, and Cao, Zhiguo
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MATHEMATICAL analysis , *COUNTING , *ACQUISITION of data - Abstract
Visual counting, a task that aims to estimate the number of objects from an image/video, is an open-set problem by nature as the number of population can vary in [ 0 , + ∞) in theory. However, collected data are limited in reality, which means that only a closed set is observed. Existing methods typically model this task through regression, while they are prone to suffer from unseen scenes with counts out of the scope of the closed set. In fact, counting has an interesting and exclusive property—spatially decomposable. A dense region can always be divided until sub-region counts are within the previously observed closed set. We therefore introduce the idea of spatial divide-and-conquer (S-DC) that transforms open-set counting into a closed set problem. This idea is implemented by a novel Supervised Spatial Divide-and-Conquer Network (SS-DCNet). It can learn from a closed set but generalize to open-set scenarios via S-DC. We provide mathematical analyses and a controlled experiment on synthetic data, demonstrating why closed-set modeling works well. Experiments show that SS-DCNet achieves state-of-the-art performance in crowd counting, vehicle counting and plant counting. SS-DCNet also demonstrates superior transferablity under the cross-dataset setting. Code and models are available at: https://git.io/SS-DCNet. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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5. A Proof of Fatou’s Interpolation Theorem.
- Author
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Danielyan, Arthur A.
- Abstract
For Fatou’s interpolation theorem of 1906 we suggest a new elementary proof. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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6. Topological properties of closed weakly m-semiconvex sets.
- Author
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Osipchuk, Tetiana M.
- Subjects
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TOPOLOGICAL property , *CONVEX sets , *POINT set theory - Abstract
The present work considers properties of generally convex sets in the n-dimensional real Euclidean space ℝn, n > 1, known as weakly m-semiconvex, m = 1, 2, ... , n − 1. For all that, the subclass of not m-semiconvex sets is distinguished from the class of weakly m-semiconvex sets. A set of the space ℝn is called m-semiconvex if, for any point of the complement of the set to the whole space, there is an m-dimensional half-plane passing through this point and not intersecting the set. An open set of ℝn is called weaklym-semiconvex if, for any point of the boundary of the set, there exists an m-dimensional half-plane passing through this point and not intersecting the given set. A closed set of ℝn is called weaklym-semiconvex if it is approximated from the outside by a family of open weakly m-semiconvex sets. An example of a closed set with three connected components of the subclass of weakly 1-semiconvex but not 1-semiconvex sets in the plane is constructed. It is proved that this number of components is minimal for any closed set of the subclass. An example of a closed set of the subclass with a smooth boundary and four components is constructed. It is proved that this number of components is minimal for any closed, bounded set of the subclass having a smooth boundary and a not 1-semiconvex interior. It is also proved that the interior of a closed, weakly 1-semiconvex set with a finite number of components in the plane is weakly 1-semiconvex. Weakly m-semiconvex but not m-semiconvex domains and closed connected sets in ℝn are constructed for any n ≥ 3 and any m = 1, 2, ... , n − 2. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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7. Distance Functions Between Sets in (q1, q2)-Quasimetric Spaces.
- Author
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Greshnov, A. V.
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SET functions , *COMPLETENESS theorem , *DISTANCES , *SPACE - Abstract
We prove completeness theorems for the set of all d-closed d-bounded sets in a (q1, q2)-quasimetric space (X, d) equipped with suitable analogs of the Hausdorff distance. [ABSTRACT FROM AUTHOR]
- Published
- 2020
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8. On almost everywhere differentiability of the metric projection on closed sets in lp(ℝn), 2 < p < ∞.
- Author
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Sjödin, Tord
- Abstract
Let F be a closed subset of ℝ
n and let P(x) denote the metric projection (closest point mapping) of x ∈ ℝn onto F in lp -norm. A classical result of Asplund states that P is (Fréchet) differentiable almost everywhere (a.e.) in ℝn in the Euclidean case p = 2. We consider the case 2 < p < ∞ and prove that the ith component Pi (x) of P(x) is differentiable a.e. if Pi (x) 6= xi and satisfies Hölder condition of order 1/(p−1) if Pi (x) = xi . [ABSTRACT FROM AUTHOR]- Published
- 2018
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9. Analysis of Spaces of Similarity Generated by a Fact Base in JSM Problems.
- Author
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Gusakova, S. M.
- Abstract
In this paper, we investigate spaces of similarity generated by fact bases of intelligent JSM systems and present a classification for the set of potential hypotheses. Conditions on similarity spaces are imposed to reduce the number of classes in this classification. The results obtained in this work make it possible to estimate the set of hypotheses (including minimal ones) found by inductive reasoning and can be used to solve the problems of opinion analysis and formation of a social structure. [ABSTRACT FROM AUTHOR]
- Published
- 2018
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10. Application of IF set oscillation in the field of face recognition.
- Author
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Bhattacharya (Halder), Sharmistha and Roy, Srijita
- Abstract
The situation when uncertainty rules the circumstances some inefficiency of crisp set can be observed. This was overcome by application of fuzzy set for solving different real life problems. Intuitionistic fuzzy set was introduced to handle complex circumstances and to provide better result. Fuzzy set proves its efficiency in field data processing. Minimal structure fuzzy oscillation can be efficiently applied for image processing, especially for face recognition. In this paper we introduce the concept of Intuitionistic fuzzy set or IF Set based minimal structure oscillation. A face database which may be considered as training set, is used to form IF set minimal structure. Again non membership pixel values are also calculated with those pixel values which are used to form IF set minimal structure accordingly. The pixels from membership and nonmembership images are applied to compute two new oscillatory operators. We introduce four new conditions according to the values of the oscillatory operators, which is used for recognition of the face. This new concept can be applied in different types of real life data analysis such as data mining. Along with the theoretical explanation we also presented some experimental results which proves the precision of the face recognition algorithm. In this paper proposed new face recognition algorithm executed and tested using MATLAB 7.9 software and experiments are performed on Face fix Database and ORL database. Accuracy of the results describes the application of IF set minimal structure oscillation in the field of face recognition. [ABSTRACT FROM AUTHOR]
- Published
- 2017
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11. (H, F)-Closed set and coupled coincidence point theorems for a generalized compatible in partially G-metric spaces.
- Author
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Narawadee Na Nan and Phakdi Charoensawan
- Subjects
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MATHEMATICS theorems , *MONOTONE operators , *OPERATOR theory , *FUNCTIONAL analysis , *INTEGRAL operators - Abstract
In this work, we present a notion of an (H, F)-closed set and prove the existence of a coupled coincidence point theorem for a pair {F,H} of mappings F,H : X × X → X with ϕ-contraction mappings in partially ordered metric spaces without H-increasing property of F and mixed monotone property of H. We give some examples of a nonlinear contraction mapping, which is not applied to the existence of coupled coincidence point by H using the mixed monotone property and H-increasing property of F. We also show the uniqueness of a coupled coincidence point of the given mappings. Further, we apply our results to the existence and uniqueness of a coupled coincidence point of the given mappings in partially ordered G-metric spaces with H-increasing property of F and mixed monotone property of H. These results generalize some recent results in the literature. [ABSTRACT FROM AUTHOR]
- Published
- 2014
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12. A mathematical model of dynamic social networks.
- Author
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Pfaltz, John L.
- Abstract
A mathematical model for dynamic networks is developed that is based on closed, rather than open, sets. For a social network, it seems appropriate to use a neighborhood concept to establish these sets. We then define a rigorous concept of continuous change, and show that it shares some of the properties associated with the continuity of the calculus. We demonstrate that continuity is local in nature, in that if the network change is discontinuous, it will be so at a single point and the discontinuity will be apparent in that point’s immediate neighborhood. Necessary and sufficient criteria for continuity are provided when the change involves only the addition, or deletion, of individual nodes or connections (edges). To illustrate large scale continuous change, we choose a practical process which reduces a complex network to its fundamental cycles, in the course of which most triadically closed subportions are removed. Finally, we explore several variants of the neighborhood concept, and prove that a rigorous notion of fuzzy closure can be defined. [ABSTRACT FROM AUTHOR]
- Published
- 2013
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13. Closed products of sets and the axiom of choice.
- Author
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Jesus, Joao and Silva, Samuel
- Subjects
- *
AXIOM of choice , *SET theory , *TOPOLOGY , *MATHEMATICAL analysis , *MATHEMATICS , *ALGEBRA , *AXIOMATIC set theory - Abstract
With a slight modification of a previous argument due to Schechter, we show that the Axiom of Choice is equivalent to the following topological statement: 'If a product of a non-empty family of sets is closed in a topological (Tychonoff) product, then at least one of the factors is closed'. We also discuss the case on which one adds the hypothesis that the closed product of sets is a non-empty set. [ABSTRACT FROM AUTHOR]
- Published
- 2011
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14. Closed classes of functions, generalized constraints, and clusters.
- Author
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Lehtonen, Erkko
- Subjects
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FUNCTION algebras , *SET theory , *CLUSTER analysis (Statistics) , *PERMUTATIONS , *MATHEMATICAL variables , *GALOIS theory - Abstract
Classes of functions of several variables on arbitrary nonempty domains that are closed under permutation of variables and addition of dummy variables are characterized by generalized constraints, and hereby Hellerstein's Galois theory of functions and generalized constraints is extended to infinite domains. Furthermore, classes of operations on arbitrary nonempty domains that are closed under permutation of variables, addition of dummy variables, and composition are characterized by clusters, and a Galois connection is established between operations and clusters. [ABSTRACT FROM AUTHOR]
- Published
- 2010
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15. Solutions of the average cost optimality equation for finite Markov decision chains: risk-sensitive and risk-neutral criteria.
- Author
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Cavazos-Cadena, Rolando
- Subjects
MARKOV processes ,EQUATIONS ,STOCHASTIC analysis ,MATRICES (Mathematics) ,PROBABILITY theory ,TOPOLOGY - Abstract
This work is concerned with controlled Markov chains with finite state and action spaces. It is assumed that the decision maker has an arbitrary but constant risk sensitivity coefficient, and that the performance of a control policy is measured by the long-run average cost criterion. Within this framework, the existence of solutions of the corresponding risk-sensitive optimality equation for arbitrary cost function is characterized in terms of communication properties of the transition law. [ABSTRACT FROM AUTHOR]
- Published
- 2009
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16. The poset scheduling problem.
- Author
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Chang, Gerard and Edmonds, Jack
- Abstract
Let P and Q be two finite posets and for each p∈P and q∈Q let c(p, q) be a specified (real-valued) cost. The poset scheduling problem is to find a function s: P→Q such that Σ c( p, s( p)) is minimized, subject to the constraints that p
- Published
- 1985
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17. Closed set of the uniqueness conditions and bifurcation criteria in generalized coupled thermoplasticity for small deformations
- Author
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Zdzisław Śloderbach
- Subjects
Mathematical problem ,Basis (linear algebra) ,Closed set ,Thermodynamic equilibrium ,Mathematical analysis ,Boundary problem ,General Physics and Astronomy ,02 engineering and technology ,Physics and Astronomy(all) ,021001 nanoscience & nanotechnology ,020303 mechanical engineering & transports ,0203 mechanical engineering ,Materials Science(all) ,Mechanics of Materials ,General Materials Science ,Uniqueness ,Boundary value problem ,0210 nano-technology ,Bifurcation ,Mathematics - Abstract
This paper reports the results of a study into global and local conditions of uniqueness and the criteria excluding the possibility of bifurcation of the equilibrium state for small strains. The conditions and criteria are derived on the basis of an analysis of the problem of uniqueness of a solution involving the basic incremental boundary problem of coupled generalized thermo-elasto-plasticity. This work forms a follow-up of previous research (Śloderbach in Bifurcations criteria for equilibrium states in generalized thermoplasticity, IFTR Reports, 1980, Arch Mech 3(35):337–349, 351–367, 1983), but contains a new derivation of global and local criteria excluding a possibility of bifurcation of an equilibrium state regarding a comparison body dependent on the admissible fields of stress rate. The thermal elasto-plastic coupling effects, non-associated laws of plastic flow and influence of plastic strains on thermoplastic properties of a body were taken into account in this work. Thus, the mathematical problem considered here is not a self-conjugated problem.
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18. Deterministic epidemiological models at the individual level
- Author
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Kieran J. Sharkey
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Mathematical optimization ,Closed set ,Population ,Population Dynamics ,Systems Theory ,Context (language use) ,Communicable Diseases ,Disease Outbreaks ,Modelling and Simulation ,Stochastic simulation ,Master equation ,Deterministic simulation ,Disease Transmission, Infectious ,State space ,Cluster Analysis ,Humans ,Quantitative Biology::Populations and Evolution ,education ,Ecosystem ,Mathematics ,education.field_of_study ,Stochastic Processes ,Models, Statistical ,Applied Mathematics ,Agricultural and Biological Sciences (miscellaneous) ,Systems Integration ,Modeling and Simulation ,Carrier State ,Neural Networks, Computer ,Epidemiologic Methods ,Deterministic system - Abstract
In many fields of science including population dynamics, the vast state spaces inhabited by all but the very simplest of systems can preclude a deterministic analysis. Here, a class of approximate deterministic models is introduced into the field of epidemiology that reduces this state space to one that is numerically feasible. However, these reduced state space master equations do not in general form a closed set. To resolve this, the equations are approximated using closure approximations. This process results in a method for constructing deterministic differential equation models with a potentially large scope of application including dynamic directed contact networks and heterogeneous systems using time dependent parameters. The method is exemplified in the case of an SIR (susceptible-infectious-removed) epidemiological model and is numerically evaluated on a range of networks from spatially local to random. In the context of epidemics propagated on contact networks, this work assists in clarifying the link between stochastic simulation and traditional population level deterministic models.
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19. On Closures of Preimages of Metric Projection Mappings in Hilbert Spaces
- Author
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Dariusz Zagrodny
- Subjects
Discrete mathematics ,Statistics and Probability ,Sequence ,Numerical Analysis ,Closed set ,Applied Mathematics ,Hilbert space ,Boundary (topology) ,Mathematics::General Topology ,Convex metric space ,symbols.namesake ,symbols ,Countable set ,Geometry and Topology ,Dykstra's projection algorithm ,Analysis ,Mathematics ,Complement (set theory) - Abstract
The closure of preimages (inverse images) of metric projection mappings to a given set in a Hilbert space are investigated. In particular, some properties of fibers over singletons (level sets or preimages of singletons) of the metric projection are provided. One of them, a sufficient condition for the convergence of minimizing sequence for a giving point, ensures the convergence of a subsequence of minimizing points, thus the limit of the subsequence belongs to the image of the metric projection. Several examples preserving this sufficient condition are provided. It is also shown that the set of points for which the sufficient condition can be applied is dense in the boundary of the preimage of each set from a large class of subsets of the Hilbert space. As an application of obtained properties of preimages we show that if the complement of a nonconvex set is a countable union of preimages of convex closed sets then there is a point such that the value of the metric projection mapping is not a singleton. It is also shown that the Klee result, stating that only convex closed sets can be weakly closed Chebyshev sets, can be obtained for locally weakly closed sets.
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20. Equality of averaged and quenched large deviations for random walks in random environments in dimensions four and higher
- Author
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Atilla Yilmaz
- Subjects
Statistics and Probability ,Closed set ,Mathematical finance ,Probability (math.PR) ,Mathematical analysis ,Random walk ,Probability theory ,Dimension (vector space) ,Mathematics::Probability ,FOS: Mathematics ,Large deviations theory ,Renewal theorem ,60K37, 60F10, 82C41 ,Statistics, Probability and Uncertainty ,Rate function ,Mathematics - Probability ,Analysis ,Mathematics - Abstract
We consider large deviations for nearest-neighbor random walk in a uniformly elliptic i.i.d. environment. It is easy to see that the quenched and the averaged rate functions are not identically equal. When the dimension is at least four and Sznitman's transience condition (T) is satisfied, we prove that these rate functions are finite and equal on a closed set whose interior contains every nonzero velocity at which the rate functions vanish., 17 pages. Minor revision. In particular, note the change in the title of the paper. To appear in Probability Theory and Related Fields.
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21. F-closed sets and coupled fixed point theorems without the mixed monotone property
- Author
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Marwan Amin Kutbi, A. Roldán, Juan Martínez-Moreno, Wutiphol Sintunavarat, and C. Roldán
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Partially ordered set ,Closed set ,Applied Mathematics ,F-invariant set ,Preorder ,Fixed-point theorem ,Fixed point ,Fixed-point property ,Least fixed point ,Combinatorics ,Monotone polygon ,Mixed monotone property ,Geometry and Topology ,Contractive mapping ,Mathematics - Abstract
In this paper we present the notion of F-closed set (which is weaker than the concept of F-invariant set introduced in Samet and Vetro (Ann. Funct. Anal. 1:46-56, 2010), and we prove some coupled fixed point theorems without the condition of mixed monotone property. Furthermore, we interpret the transitive property as a partial preorder and, then, some results in that paper and in Sintunavarat et al. (Fixed Point Theory Appl. 2012:170, 2012) can be reduced to the unidimensional case. MSC:46T99, 47H10, 47H09, 54H25.
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22. Torricellian points in normed linear spaces
- Author
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Dan Comǎnescu, Sever S Dragomir, and Eder Kikianty
- Subjects
Discrete mathematics ,Closed set ,Applied Mathematics ,Perfect set ,Combinatorics ,Strictly convex space ,Isolated point ,Inner product space ,Set function ,Discrete Mathematics and Combinatorics ,Extreme point ,Analysis ,Mathematics ,Normed vector space - Abstract
Given a set of n (distinct) points A in a normed space, we consider the set of Torricellian points, that is, the set of points which minimises the sum of distances to the points in A . We introduce the Torricellian functional associated to a set of distinct points A , which calculates the sum of distances of a point x to the points in A . The Torricellian point is defined as the infimum (over all vectors) of this functional. We discuss the existence of Torricellian points in reflexive normed spaces, non-expansive subspaces and evidently, inner product spaces. A case for collinear points is given and is utilised to characterise strict convexity. For a non-collinear case, it is shown that the set of Torricellian points contains a unique point when the space is strictly convex. However, we show that the uniqueness of Torricellian point of a non-collinear set does not characterise strict convexity. We consider a particular example of the Torricellian problem in a space endowed with the Taxicab geometry. MSC:46B20, 49J27.
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23. Results on proximal and generalized weak proximal contractions including the case of iteration-dependent range sets
- Author
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Manuel De la Sen, Asier Ibeas, and Ravi P. Agarwal
- Subjects
Discrete mathematics ,Pure mathematics ,Closed set ,GEOMETRY AND TOPOLOGY ,Applied Mathematics ,metric-spaces ,mappings ,proximal contraction ,best proximity point ,weak proximal contraction ,set-theoretic limit ,Domain (mathematical analysis) ,Moore-Penrose pseudo-inverse ,points ,adaptive-control ,Metric space ,MATHEMATICS, APPLIED ,Differential geometry ,Set-theoretic limit ,Countable set ,Uniqueness ,Geometry and Topology ,Algebraic number ,Mathematics - Abstract
This paper presents some further results on proximal and asymptotic proximal contractions and on a class of generalized weak proximal contractions in metric spaces. The generalizations are stated for non-self-mappings of the forms T n : A n → B n for n ∈ Z 0 + and T : ⋃ j ∈ Z 0 + A 0 j → ⋃ j ∈ Z 0 + B 0 j , or T : A → ( ⋃ B n ) , subject to T ( A 0 n ) ⊆ B 0 n and T n ( A n ) ⊆ B n , such that T n converges uniformly to T, and the distances D n = d ( A n , B n ) are iteration-dependent, where A 0 n , A n , B 0 n and B n are non-empty subsets of X, for n ∈ Z 0 + , where ( X , d ) is a metric space, provided that the set-theoretic limit of the sequences of closed sets { A n } and { B n } exist as n → ∞ and that the countable infinite unions of the closed sets are closed. The convergence of the sequences in the domain and the image sets of the non-self-mapping, as well as the existence and uniqueness of the best proximity points, are also investigated if the metric space is complete. Two application examples are also given, being concerned, respectively, with the solutions through pseudo-inverses of both compatible and incompatible linear algebraic systems and with the parametrical identification of dynamic systems.
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24. (G,F)-Closed set and tripled point of coincidence theorems for generalized compatibility in partially metric spaces
- Author
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Chaiporn Thangthong and Phakdi Charoensawan
- Subjects
Combinatorics ,Metric space ,Monotone polygon ,Closed set ,Applied Mathematics ,Compatibility (mechanics) ,Discrete Mathematics and Combinatorics ,Uniqueness ,Nonlinear contraction ,Coincidence point ,Coincidence ,Analysis ,Mathematics - Abstract
In this work, we prove the existence of a tripled point of coincidence theorem for a pair of mappings with φ-contraction mappings in partially ordered metric spaces without G-increasing property of F and mixed monotone property of G, using the concept of a -closed set. We give some examples of a nonlinear contraction mapping, which is not applied to the existence of tripled coincidence point by G using the mixed monotone property. We also show the uniqueness of a tripled point of coincidence of the given mapping. Further, we apply our results to the existence and uniqueness of a tripled point of coincidence of the given mapping with G-increasing property of F and mixed monotone property of G in partially ordered metric spaces.
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25. Measures of Noncircularity and Fixed Points of Contractive Multifunctions
- Author
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Isabel Marrero
- Subjects
Discrete mathematics ,T57-57.97 ,QA299.6-433 ,Applied mathematics. Quantitative methods ,Closed set ,Applied Mathematics ,Banach space ,Fixed-point theorem ,Type (model theory) ,Fixed point ,Measure (mathematics) ,Differential geometry ,Hausdorff measure ,Geometry and Topology ,Analysis ,Mathematics - Abstract
In analogy to the Eisenfeld-Lakshmikantham measure of nonconvexity and the Hausdorff measure of noncompactness, we introduce two mutually equivalent measures of noncircularity for Banach spaces satisfying a Cantor type property, and apply them to establish a fixed point theorem of Darbo type for multifunctions. Namely, we prove that every multifunction with closed values, defined on a closed set and contractive with respect to any one of these measures, has the origin as a fixed point.
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