Dissertationen zum Thema „Generalized Metric Spaces“

On retrouve le raisonnement algébrique partout en mathématique et en informatique, et il a déjà été généralisé à pleins de contextes différents. En 2016, Mardare, Panangaden et Plotkin ont introduit les algèbres quantitatives, c'est-à-dire, des éspaces métriques équippés d'opérations $1$-lipschitzienne relativement à la métrique. Ils ont prouvées des homologues à des résultats importants en algèbre universelle, et en particulier ils ont donné un système de deduction correct et complet qui génralise la logique équationelle de Birkhoff en remplçant l'égalité par l'égalité à $\varepsilon$ près. Ça leur a permis de donné une axiomatisation algébrique pour quelques métriques importantes comme la distance de Hausdorff et celle de Kantorovich.Dans cette thèse, on modifie deux aspects du cadre de Mardare et al. Premièrement, on remplace les métriques par une notion plus générale qui englobe les pseudométriques, les ordres partiels, les métriques probabilistes, entre autres. Deuxièmement, on n'exige pas que les operations de nos algèbres quantitatives soient lipshitzienne. On donne un système de deduction correct et complet, on construit les algèbres quantitatives libres, et on démontre la valeur de notre généralisation en prouvant que toute monade sur les éspaces métriques généralisés qui est le relèvement d'une monade finitaire sur les ensembles peut être présentées par une théorique algébrique quantitative. On applique ce dernier résultat pour obtenir une axiomatization de la distance de \L ukaszyk--Karmowski
Algebraic reasoning is ubiquitous in mathematics and computer science, and it has been generalized to many different settings. In 2016, Mardare, Panangaden, and Plotkin introduced quantitative algebras, that is, metric spaces equipped with operations that are nonexpansive relative to the metric. They proved counterparts to important results in universal algebra, and in particular they provided a sound and complete deduction system generalizing Birkhoff's equational logic by replacing equality with equality up to $\varepsilon$. This allowed them to give algebraic axiomatizations for several important metrics like the Hausdorff and Kantorovich distances.In this thesis, we make two modifications to Mardare et al.'s framework. First, we replace metrics with a more general notion that captures pseudometrics, partial orders, probabilistic metrics, and more. Second, we do not require the operations in a quantitative algebra to be nonexpansive. We provide a sound and complete deduction system, we construct free quantitative algebras, and we demonstrate the value of our generalization by proving that any monad on generalized metric spaces that lifts a monad on sets can be presented with a quantitative algebraic theory. We apply this last result to obtain an axiomatization for the \L ukaszyk--Karmowski distance

[ES] En 1965 L. Zadeh introdujo el concepto de conjunto fuzzy, estableciendo una nueva línea de investigación, conocida como matemática fuzzy. Desde entonces, varios autores han estado investigando la construcción de una definición consistente de espacio métrico fuzzy. En 1994, George y Veeramani introdujeron y estudiaron un concepto de espacio métrico fuzzy, que era una adecuada modificación del concepto dado por Kramosil y Michalek. Estos conceptos han sido estudiados y desarrollados en diversas líneas durante los últimos 25 años. Con la intención de contribuir a este desarrollo de la teoría fuzzy, en esta tesis hemos introducido y estudiado los siguientes ítems: 1. Hemos introducido el concepto de espacio métrico fuzzy extendido M0, que es una extensión adecuada de una GV-métrica fuzzy donde el parámetro t puede tomar el valor 0. Además, hemos estudiado conceptos relacionados con la convergencia y las sucesiones de Cauchy en este contexto, así como teoremas sobre contractividad y punto fijo. 2. Hemos probado la existencia de sucesiones contractivas en el sentido de D.Mihet en un espacio métrico fuzzy en el sentido de George y Veeramani que no son de Cauchy. En consecuencia, hemos introducido y estudiado un concepto adecuado de sucesión estrictamente contractiva y hemos corregido el Lema 3.2 de [V. Gregori and J.J. Miñana, On fuzzy psi-contractive sequences and fixed point theorems, Fuzzy Sets and Systems 300 (2016), 93-101]. 3. Hemos introducido y estudiado una noción de (GV-)espacio métrico parcial fuzzy (X,P,*) sin ninguna condición adicional sobre la t-norma *. Después, hemos definido una topologia T_P sobre X deducida de P y hemos demostrado que (X,T_P) es un espacio T0. 4. Hemos relacionado el mencionado concepto de GV-espacio métrico parcial fuzzy con la noción de GV-espacio casi-métrico fuzzy definido por Gregori y Romaguera en [V. Gregori and S. Romaguera, Fuzzy quasi-metric spaces, Applied General Topology 5 (2004), 129-136]. Se ha estudiado una dualidad entre estos espacios, imitando las técnicas utilizadas por Matthews en [S.G.Matthews, Partial metric topology, Annals of the New York Academy of Sciences 728 (1994), 183-197].
[CAT] En 1965, L. Zadeh va introduir el concepte de conjunt fuzzy, establint una nova línia d'investigació, coneguda com matemàtica fuzzy. Des d'aquell moment, molts autors han investigat la construcció d'una definició consistent d'espai mètric fuzzy. En 1994, George i Veeramani van introduir i estudiar una noció d'espai mètric fuzzy, realitzant una modificació adequada del concepte donat per Kramosil i Michalek. Aquests conceptes han estat estudiats i desenvolupats en diversos sentits durant els últims 25 anys. Amb la intenció de contribuir a aquest desenvolupament de la teoria fuzzy, en aquesta tesi hem introduït i estudiat els següents continguts: 1. Hem introduït el concepte d'espai mètric extés M0, que és una extensió adequada d'una GV -mètrica fuzzy M on el paràmetre t pot prendre el valor 0. A més, hem estudiat conceptes relacionats amb la convergència i les successions de Cauchy en aquest context, així com teoremes sobre contractivitat i punt fixe. 2. Hem provat l'existència de successions contractives en el sentit de D. Mihet en un GV -espai mètric fuzzy que no són Cauchy. Conseqüentment, hem aportat i estudiat un concepte apropiat de successió estrictament contractiva i hem corregit el Lema 3.2 de [V. Gregori and J.J. Miñana, On fuzzy psi-contractive sequences and fixed point theorems, Fuzzy Sets and Systems 300 (2016), 93-101]. 3. Hem introduït i estudiat una noció de (GV -)espai mètric parcial fuzzy (X,P,*) sense cap tipus de condició addicional sobre la t-norma contínua *. A continuació, hem definit una topologia T_P sobre X deduïda de P i hem demostrat que (X, T_P) es un espai T0. 4. Hem relacionat el ja mencionat concepte de GV -espai mètric parcial fuzzy amb la noció de GV -espai quasi-mètric fuzzy definit per Gregori i Romaguera en [V. Gregori and S. Romaguera, Fuzzy quasi-metric spaces, Applied General Topology 5 (2004), 129-136]. S'ha estudiat una dualitat entre ambdós espais, imitant les tècniques utilitzades per Matthews en [S.G.Matthews, Partial metric topology, Annals of the New York Academy of Sciences 728 (1994), 183-197].
[EN] In 1965, L. Zadeh introduced the concept of fuzzy set, and thus established a new topic of research, known as fuzzy mathematics. Since then, several authors have been investigating the approach of a consistent fuzzy metric space theory. In 1994, George and Veeramani introduced and studied a concept of fuzzy metric space which was a proper modification of the concept given by Kramosil and Michalek. These notions have been studied and developed in several ways during the last 25 years. With the purpose of contributing to the development of the study of the fuzzy theory, in this thesis we have introduced and studied the following items: 1. We have introduced the concept of extended fuzzy metric M0 which is an appropriate extension of a GV -fuzzy metric M where the parameter t can take the value 0. Furthermore, we have studied convergence and Cauchyness concepts in this context, as well as contractivity and fixed point theorems. 2. We have proved the existence of contractive sequences in the sense of D. Mihet in a GV -fuzzy metric space which are not Cauchy. Then we have given and studied an appropriate concept of strictly contractive sequence and we have corrected Lemma 3.2 of [V. Gregori and J.J. Miñana, On fuzzy psi-contractive sequences and fixed point theorems, Fuzzy Sets and Systems 300 (2016), 93-101]. 3. We have introduced and studied a concept of (GV -)fuzzy partial metric space (X,P,*) without any extra conditions on the continuous t-norm *. Then we have defined a topology T_P on X deduced from P and we have proved that (X, T_P) is a T0 space. 4. We have related the aforementioned notion of GV -fuzzy partial metric space with the concept of GV -fuzzy quasi-metric space given by Gregori and Romaguera in [V. Gregori and S. Romaguera, Fuzzy quasi-metric spaces, Applied General Topology 5 (2004), 129-136]. A duality is studied by mimicking the techniques used in [S.G.Matthews, Partial metric topology, Annals of the New York Academy of Sciences 728 (1994), 183-197] by Matthews.
Miravet Fortuño, D. (2019). GENERALIZED FUZZY METRIC SPACES DEFINED BY MEANS OF T-NORMS [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/124816
TESIS

The aim of the thesis is to work towards a many-valued logic over a commutative unital quantale and, at the same time, towards a generalisation of coalgebraic logic enriched over a commutative unital quantale Ω. This is done by noticing that the contravariant powerset adjunction can be generalised to categories enriched over a commutative unital quantale. From here we define categorical algebras for the monad generated by this adjunction. We finish by showing that these categorical algebras are algebras over Set with operations and equations, and show that in some cases we can restrict the arity of those operations to be finite.

This dissertation presents results about computations of Graev metrics on free groups and characterizes isometry groups of countable noncompact Heine-Borel Polish ultrametric spaces. In Chapter 2, computations of Graev metrics are performed on free groups. One of the related results answers an open question of Van Den Dries and Gao. In Chapter 3, isometry groups of countable noncompact Heine-Borel Polish ultrametric spaces are characterized. The notion of generalized tree is defined and a correspondence between the isomorphism group of a generalized tree and the isometry group of a Heine-Borel Polish ultrametric space is established. The concept of a weak inverse limit is introduced to capture the characterization of isomorphism groups of generalized trees. In Chapter 4, partial results of isometry groups of uncountable compact ultrametric spaces are given. It turns out that every compact ultrametric space has a unique countable orbital decomposition. An orbital space consists of disjoint orbits. An orbit subspace of an orbital space is actually a compact homogeneous ultrametric subspace.

U ovoj tezi je definisana uopštena konvolucija koja pripada domenu pseudo-analize i ima veliku primenu u mnogim matematičkim teorijama, npr. u proba-bilističkim metričkim prostorima, PDJ, teorijama odlučivanja, sistema, kontrole i fazi brojeva. Dokazane su bitne osobine ove operacije sa funkcijama. Dokazana je veza izmedju pseudo-konvolucija baziranih na poluprstenima različitih klasa Definisana je (5, C/)-konvolucija bazirana na uslovno distributivnom poluprstenu ([0,1], S, U)).Dat je još jedan vid uopštenja konvolucije baziran na uopštenim pseudo-operacijama.
In this thesis the generalized convolution have been defined. This operation with functions has applications in different mathematical theo­ ries, for example in Probabilistic Metric Spaces, PDE, System and Control Theory, Fuzzy numbers. Some basic properties of this operation has been proved, as well as connection between generalized convolutions based on dif­ferent classes of semirings. (5, U)-convolution has been defined, as well as convolution based on generalized pseudo-operations.

The research topic of this PhD thesis is a comparative analysis of classical specic geometric frameworks and of their anisotropic extensions; the construction of three different types of Finsler frameworks, which are suitable for the analysis of the cancer cells population dynamical system; the development of the anisotropic Beltrami framework theory with the derivation of the evolution ow equations corresponding to different classes of anisotropic metrics, and tentative applications in image processing.
Predmet istraživanja doktorske disertacije je uporedna analiza klasičnih i specifičnih geometrijskih radnih okruženja i njihovih anizotropnih proširenja; konstrukcija  tri Finslerova radna okruženja različitog tipa koja su pogodna za analizu dinamičkog  sistema populacije kanceroznih ćelija; razvoj teorije anizotropnog Beltramijevog radnog okruženja i formiranje jednačina evolutivnog toka za različite klase anizotropnih metrika, kao i mogućnost primene dobijenih teorijskih rezultata u digitalnoj obradi slika.

Mathematical models have extensively been used in problems related to engineering, computer sciences, economics, social, natural and medical sciences etc. It has become very common to use mathematical tools to solve, study the behavior and different aspects of a system and its different subsystems. Because of various uncertainties arising in real world situations, methods of classical mathematics may not be successfully applied to solve them. Thus, new mathematical theories such as probability theory and fuzzy set theory have been introduced by mathematicians and computer scientists to handle the problems associated with the uncertainties of a model. But there are certain deficiencies pertaining to the parametrization in fuzzy set theory. Soft set theory aims to provide enough tools in the form of parameters to deal with the uncertainty in a data and to represent it in a useful way. The distinguishing attribute of soft set theory is that unlike probability theory and fuzzy set theory, it does not uphold a precise quantity. This attribute has facilitated applications in decision making, demand analysis, forecasting, information sciences, mathematics and other disciplines. In this thesis we will discuss several algebraic and topological properties of soft sets and fuzzy soft sets. Since soft sets can be considered as setvalued maps, the study of fixed point theory for multivalued maps on soft topological spaces and on other related structures will be also explored. The contributions of the study carried out in this thesis can be summarized as follows: i) Revisit of basic operations in soft set theory and proving some new results based on these modifications which would certainly set a new dimension to explore this theory further and would help to extend its limits further in different directions. Our findings can be applied to develop and modify the existing literature on soft topological spaces ii) Defining some new classes of mappings and then proving the existence and uniqueness of such mappings which can be viewed as a positive contribution towards an advancement of metric fixed point theory iii) Initiative of soft fixed point theory in framework of soft metric spaces and proving the results lying at the intersection of soft set theory and fixed point theory which would help in establishing a bridge between these two flourishing areas of research. iv) This study is also a starting point for the future research in the area of fuzzy soft fixed point theory.
Abbas, M. (2014). Soft Set Theory: Generalizations, Fixed Point Theorems, and Applications [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/48470
TESIS

Thesis (Ph. D.)--University of Wisconsin--Madison, 1988.
Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 53-55).

碩士
國立高雄師範大學
數學系
94
In this paper , we first establish some new fixed point theorems for multivalued maps. Using these theorems, we can prove generalized Kannan's type and generalized Chatterjea's type fixed point theorems for multivalued maps. The primitive Kannan's type and Chatterjea's type fixed point theorems for single-valued maps are special cases of our theorems.

碩士
國立新竹教育大學
數學教育學系碩士班
95
In this paper, we use the property of hyperconvex metric space to establish an intersection property about a family of admissible sets. Applying this intersection property we get a generalized theorem, a matching theorem and a coincidence theorem. As the application, we use this generalized theorem to establish some existence theorems about variational inequalities and minimax inequalities.

碩士
國立新竹教育大學
應用數學系碩士班
95
In this paper, we prove a generalized 2-gKKM theorem in hyperconvex metric space. By using this theorem we get a matching theorem, a coincidence theorem, and some fixed point theorems under some assumptions. As applications, we use this generalized 2-gKKM theorem to establish some existence theorems about variational inequalities.

碩士
國立新竹教育大學
人資處數學教育碩士班
95
In this paper, we first define 2-KKM mapping and generalized 2-KKM mapping. Then we apply the property of hyperconvex metric space to get a KKM theorem and a fixed point theorem without compact assumption. By using this KKM theorem we get some theorems about variational inequalities and minimax inequalities.

碩士
國立高雄師範大學
數學系
105
Let A and B be nonempty subsets of a metric space (X,d) and T:A∪B → A∪B be a cyclic mapping. In this paper, we establish some new convergence theorems satisfying the following condition: (G) there exists an MT-funtion ϕ:[0,∞) → [0,1) such that d(Tx,Ty)≤ϕ(d(x,y))max{(1/4)[d(x,Ty)+2d(Tx,Ty)+d(y,Tx)], (1/8)[d(x,Ty)+3d(x,Tx)+3d(y,Ty)+d(y,Tx)]}+[1-ϕ(d(x,y))]dist(A,B) for all x∈A and y∈B.

碩士
國立新竹教育大學
人資處數學教育碩士班
95
In this paper, we use the property of hyperconvex metric space to establish an intersection property about a family of admissible sets. Applying this intersection property we establish some generalized variational inequality theorems. We also establish some minimax inequality theorems concerning four real-valued mappings under some assumptions.

碩士
國立新竹教育大學
應用數學系碩士班
100
We introduce the notion of weaker (Φ,ϕ)-contractive mapping in completemetric spaces and prove the periodic points and fixed points for this type of contraction. Our results generalize or improve many recent fixed point theorems in the literature

The aim of this thesis is to build the foundations of generalized ordinary differ- ential equation theory in metric spaces. While differential equations in metric spaces have been studied before, the chosen approach cannot be extended to in- clude more general types of integral equations. We introduce a definition which combines the added generality of metric spaces with the strength of Kurzweil's generalized ordinary differential equations. Additionally, we present existence and uniqueness theorems which offer new results even in the context of Euclidean spaces.

碩士
國立清華大學
應用數學系所
105
The paper is concerned with the result of the Korovkin type approximation theorem related to functions of single-valued and multi-valued. For the study of Korovkin type approximation theorem of single-valued functions, we discuss the uniform convergence in the space of functions define on the space of partial metric and metric-like. As for the Korovkin type approximation theorem of multi-valued functions, we discuss the uniform convergence in the space of functions from partial metric space to closed and bounded sets of real number and the functions from metric-like space to closed and bounded sets of real number.

Μια πολλαπλότητα Riemann (M, g) ονομάζεται Einstein αν έχει σταθερή καμπυλότητα Ricci. Είναι γνωστό ότι αν (M=G/K, g) είναι μια συμπαγής ομογενής πολλαπλότητα Riemann, τότε οι G-αναλλοίωτες μετρικές Einstein μοναδιαίου όγκου, είναι τα κρίσιμα σημεία του συναρτησοειδούς ολικής βαθμωτής καμπυλότητας περιορισμένο στο χώρο των G-αναλλοίωτων μετρικών με όγκο 1. Για μια G-αναλλοίωτη μετρική Riemann η εξίσωση Einstein ανάγεται σε ένα σύστημα αλγεβρικών εξισώσεων. Οι θετικές πραγματικές λύσεις του συστήματος αυτού είναι ακριβώς οι G-αναλλοίωτες μετρικές Einstein που δέχεται η πολλαπλότητα Μ. Μια σημαντική οικογένεια συμπαγών ομογενών χώρων αποτελείται από τις γενικευμένες πολλαπλότητες σημαιών. Κάθε τέτοιος χώρος είναι μια τροχιά της συζυγούς αναπαράστασης μιας συμπαγούς, συνεκτικής, ημι-απλής ομάδας Lie G. Πρόκειται για ομογενείς πολλαπλότητες της μορφής G/C(S), όπου C(S) είναι ο κεντροποιητής ενός δακτυλίου S στην G. Κάθε τέτοιος χώρος δέχεται ένα πεπερασμένο πλήθος από G-αναλλοίωτες μετρικές Kahler-EInstein. Στην παρούσα διατριβή ταξινομούμε όλες τις πολλαπλότητες σημαιών G/K που αντιστοιχούν σε μια απλή ομάδα Lie G, των οποίων η ισοτροπική αναπαράσταση διασπάται σε 2 ή 4 μη αναγώγιμους και μη ισοδύναμους Ad(K)-αναλλοίωτους προσθετέους. Για κάθε τέτοιο χώρο λύνουμε την αναλλοίωτη εξίσωση Εinstein, και παρουσιάζουμε την αναλυτική μορφή νέων G-αναλλοίωτων μετρικών Einstein. Στις περισσότερες περιπτώσεις παρουσιάζουμε την πλήρη ταξινόμηση των αναλλοίωτων μετρικών Einstein. Επίσης εξετάζουμε το ισομετρικό πρόβλημα. Για την κατασκευή της εξίσωσης Einstein σε κάποιες πολλαπλότητες σημαιών με 4 ισοτροπικούς προσθετέους χρησιμοποιούμε την νηματοποίηση συστροφής που δέχεται κάθε πολλαπλότητα σημαιών επί ενός ισοτροπικά μη αναγώγιμου συμμετρικού χώρου συμπαγούς τύπου. Αυτή η μέθοδος είναι καινούργια και μπορεί να εφαρμοστεί και σε άλλες πολλαπλότητες σημαιών.
A Riemannian manifold (M, g) is called Einstein, if it has constant Ricci curvature. It is well known that if (M=G/K, g) is a compact homogeneous Riemannian manifold, then the G-invariant \tl{Einstein} metrics of unit volume, are the critical points of the scalar curvature function restricted to the space of all G-invariant metrics with volume 1. For a G-invariant Riemannian metric the Einstein equation reduces to a system of algebraic equations. The positive real solutions of this system are the $G$-invariant Einstein metrics on M. An important family of compact homogeneous spaces consists of the generalized flag manifolds. These are adjoint orbits of a compact semisimple Lie group. Flag manifolds of a compact connected semisimple Lie group exhaust all compact and simply connected homogeneous Kahler manifolds and are of the form G/C(S), where C(S) is the centralizer (in G) of a torus S in G. Such homogeneous spaces admit a finite number of G-invariant complex structures, and for any such complex structure there is a unique compatible G-invariant Kahler-Einstein metric. In this thesis we classify all flag manifolds M=G/K of a compact simple Lie group G, whose isotropy representation decomposes into 2 or 4, isotropy summands. For these spaces we solve the (homogeneous) Einstein equation, and we obtain the explicit form of new G-invariant Einstein metrics. For most cases we give the classification of homogeneous Einstein metrics. We also examine the isometric problem. For the construction of the Einstein equation on certain flag manifolds with four isotropy summands, we apply for first time the twistor fibration of a flag manifold over an isotropy irreducible symmetric space of compact type. This method is new and it can be used also for other flag manifolds. For flag manifolds with two isotropy summands, we use the restricted Hessian and we characterize the new Einstein metrics as local minimum points of the scalar curvature function restricted to the space of G-invariant Riemannian metrics of volume 1. We mention that the classification of flag manifolds with two isotropy summands gives us new examples of homogeneous spaces, for which the motion of a charged particle under the electromagnetic field, and the geodesics curves, are completely determined.