Quasi-metric geometry

2014 ◽  
Author(s):  
◽  
Dan Brigham
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Linear Structure ◽  
Abstract Objects ◽  
Metric Geometry ◽  
Natural Structure ◽  
University Of Missouri ◽  
The University

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] Every time one sees |x-y|, one is looking at a specific metric acting on x and y, whatever they may happen to be, usually numbers or vectors. The notion of the distance between two objects is one of the most fundamental and ubiquitous in many branches of mathematics. A quasi-metric is a generalization of the familiar notion of metric. This dissertation examines what happens in this new setting of quasi-metrics. In particular, in the first chapter we introduce quasi-metrics, provide examples of them, then, given an arbitrary quasi-metric, develop a procedure which allows us to construct a better quasi-metric. Then we look at topological matters, such as openness and continuity. After that, we look at functions on abstract objects called groupoids, which is yet another step toward generality, since the objects we consider here contain the class of quasi-metrics. Dealing with groupoids is useful because it provides a natural structure into which quasi-metrics and quasi-norms fit. After these preliminary chapters, we then introduce linear structure, meaning the quasi-metrics studied are defined on sets in which one can add two points together, and multiply points by numbers, as this is not possible in an abstract set. Next we quantify smoothness of quasi-metric spaces, and throw in measures. For the first six chapters, we worked within a given quasi-metric space, assigning to points the distance between them. The seventh and final chapter deals with the "distance'' between two distinct quasi-metrics.

The use of metaheuristics for feature selection and the derivation of diagnostic rules

2019 ◽  
Author(s):  
◽  
Jordan Stevens
Keyword(s):  
Genetic Algorithm ◽  
Feature Selection ◽  
Alcohol Use Disorder ◽  
Learning Context ◽  
University Of Missouri ◽  
Search Spaces ◽  
History Of ◽  
The University ◽  
New Algorithms

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] This dissertation includes four chapters that discuss 1) the history of metaheuristics, 2) the development of a genetic algorithm for feature selection, 3) the development of a genetic algorithm for deriving psychiatric diagnoses and 4) a demonstration of deriving shortened diagnostic rules for alcohol use disorder that optimally agree with the DSM-5. The first chapter offers an overview of novel developments in the metaheuristics literature, along with suggestions for future developments. The second and third chapters of this dissertation 1) propose new algorithms that can handle search spaces that are not accessible by current algorithms and 2) examine each component of the proposed algorithms to identify subordinate heuristics that are essential for the success of the algorithm. The final chapter utilizes information obtained from the previous two chapters to assess the performance of an algorithm for deriving diagnostic rules in a supervised learning context.

Metadrama in the chorus : the first choral ode of Seneca's Oedipus

2016 ◽  
Author(s):  
◽  
Maria Silvia Sarais
Keyword(s):  
Literary Criticism ◽  
Main Topic ◽  
Apparent Discrepancy ◽  
University Of Missouri ◽  
Level Of Significance ◽  
The University ◽  
Verbal Analysis ◽  
Artistic Skill ◽  
Insight Into

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] The main topic of this dissertation is the first choral ode of Seneca's Oedipus. Seneca's handling of the choral parts has often suffered from negative criticism, mainly due to the fact that often, at first glance, the interventions of the chorus seem to not be an integral part of the play. For this reason, Seneca's choral odes have often been considered a mere display of the poets' rhetorical abilities, frequently dismissed as lyrical evasions, and occasionally even charged with being at odds with the context in which they are inserted. These types of assessments have been conducive to negative evaluations of Seneca's handling of the chorus, and of his dramatic competence, and have often been used to support the argument that Seneca's tragedies were not intended to be staged. My analysis shows that Seneca's variation of the poetic sources is never meaningless for the dramatic context, and argues that the chorus is an integral part of the play, one that is essential to fully understand and appreciate Seneca's dramatic art. By combining an intertextual approach and verbal analysis, I show that omissions, additions, and variations lead the audience to see the first ode as a symbolic account of Oedipus' destiny, one that prefigures the ways in which the tragic narrative is about to develop, and that prepares the audience to recognize the ways in which Seneca's poetic rewriting of Oedipus' myth is going to be original. The ode, in fact, displays the presence of a language that is highly ambiguous and multireferential, and that draws on the technical language of literary criticism and of programmatic statements of poetics. This language permits the audience to detect a meta-dramatic level of communication in the ode, one where Oedipus is characterized as a surrogate of the tragic poet. The final chapter provides evidence of the fact that all of the remaining odes display the same polysemous language that continues to sustain a meta-dramatic level of significance. My study shows that a recognition of metadrama in the odes is important in several ways. It increases the tragic irony. It complicates traditional notions of tragic fate, thereby providing an explanation for the apparent discrepancy between Seneca the Philosopher and Seneca the Tragedian that does not see the tragedies as the result of Seneca's retraction of his Stoic ideas. It offers an insight into Seneca's tragic poetics, while pointing to a possible reconciliation between Callimachean artistic skill and Bacchic inspiration. Finally, it points to a particular type of expected audience, one that is rational, literarily well-educated, and hence able to recognize all of the subtleties of the poets' sophisticated poetic enterprise.

TEM study of boron additions to cobalt aluminide

1988 ◽  
Vol 46 ◽  
pp. 546-547
Author(s):  
Gerald B. Feldewerth
Keyword(s):  
High Temperature ◽  
Turbine Engines ◽  
Major Drawback ◽  
University Of Missouri ◽  
Specific Strength ◽  
Aerospace Applications ◽  
Wide Range ◽  
Ordered Alloys ◽  
The University ◽  
Very High

In recent years an increasing emphasis has been placed on the study of high temperature intermetallic compounds for possible aerospace applications. One group of interest is the B2 aiuminides. This group of intermetaliics has a very high melting temperature, good high temperature, and excellent specific strength. These qualities make it a candidate for applications such as turbine engines. The B2 aiuminides exist over a wide range of compositions and also have a large solubility for third element substitutional additions, which may allow alloying additions to overcome their major drawback, their brittle nature.One B2 aluminide currently being studied is cobalt aluminide. Optical microscopy of CoAl alloys produced at the University of Missouri-Rolla showed a dramatic decrease in the grain size which affects the yield strength and flow stress of long range ordered alloys, and a change in the grain shape with the addition of 0.5 % boron.

MARS — Its Tenth Anniversary of Operation and its Future

1980 ◽  
Vol 19 (03) ◽  
pp. 125-132
Author(s):  
G. S. Lodwick ◽  
C. R. Wickizer ◽  
E. Dickhaus
Keyword(s):  
Conceptual Development ◽  
Administrative Support ◽  
Final Section ◽  
Economic Benefits ◽  
Hospital Environment ◽  
Computer Scientist ◽  
University Of Missouri ◽  
Complete Bibliography ◽  
The University ◽  
Economic Considerations

The Missouri Automated Radiology System recently passed its tenth year of clinical operation at the University of Missouri. This article presents the views of a radiologist who has been instrumental in the conceptual development and administrative support of MARS for most of this period, an economist who evaluated MARS from 1972 to 1974 as part of her doctoral dissertation, and a computer scientist who has worked for two years in the development of a Standard MUMPS version of MARS. The first section provides a historical perspective. The second deals with economic considerations of the present MARS system, and suggests those improvements which offer the greatest economic benefits. The final section discusses the new approaches employed in the latest version of MARS, as well as areas for further application in the overall radiology and hospital environment. A complete bibliography on MARS is provided for further reading.

On Asymmetric Distances

2013 ◽  
Vol 1 ◽  
pp. 200-231 ◽  
Author(s):  
Andrea C.G. Mennucci
Keyword(s):  
Total Variation ◽  
Calculus Of Variations ◽  
Distance Function ◽  
Metric Space ◽  
Metric Spaces ◽  
Geodesic Segment ◽  
Intrinsic Metric ◽  
Typical Application ◽  
Variation Formula

Abstract In this paper we discuss asymmetric length structures and asymmetric metric spaces. A length structure induces a (semi)distance function; by using the total variation formula, a (semi)distance function induces a length. In the first part we identify a topology in the set of paths that best describes when the above operations are idempotent. As a typical application, we consider the length of paths defined by a Finslerian functional in Calculus of Variations. In the second part we generalize the setting of General metric spaces of Busemann, and discuss the newly found aspects of the theory: we identify three interesting classes of paths, and compare them; we note that a geodesic segment (as defined by Busemann) is not necessarily continuous in our setting; hence we present three different notions of intrinsic metric space.

A new Ф-generalized quasi metric space with some fixed point results and applications

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3157-3172
Author(s):  
Mujahid Abbas ◽  
Bahru Leyew ◽  
Safeer Khan
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fixed Points ◽  
Periodic Solutions ◽  
Metric Space ◽  
Delay Differential Equations ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Existence Of Periodic Solutions ◽  
Delay Differential

In this paper, the concept of a new ?-generalized quasi metric space is introduced. A number of well-known quasi metric spaces are retrieved from ?-generalized quasi metric space. Some general fixed point theorems in a ?-generalized quasi metric spaces are proved, which generalize, modify and unify some existing fixed point theorems in the literature. We also give applications of our results to obtain fixed points for contraction mappings in the domain of words and to prove the existence of periodic solutions of delay differential equations.

scholarly journals On Some New Contractive Conditions in Complete Metric Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 118
Author(s):  
Jelena Vujaković ◽  
Eugen Ljajko ◽  
Mirjana Pavlović ◽  
Stojan Radenović
Keyword(s):  
Current Literature ◽  
Metric Space ◽  
Metric Spaces ◽  
Complete Metric Spaces ◽  
Nonlinear Anal ◽  
Increasing Mapping ◽  
Strictly Increasing Mapping

One of the main goals of this paper is to obtain new contractive conditions using the method of a strictly increasing mapping F:(0,+∞)→(−∞,+∞). According to the recently obtained results, this was possible (Wardowski’s method) only if two more properties (F2) and (F3) were used instead of the aforementioned strictly increasing (F1). Using only the fact that the function F is strictly increasing, we came to new families of contractive conditions that have not been found in the existing literature so far. Assuming that α(u,v)=1 for every u and v from metric space Ξ, we obtain some contractive conditions that can be found in the research of Rhoades (Trans. Amer. Math. Soc. 1977, 222) and Collaco and Silva (Nonlinear Anal. TMA 1997). Results of the paper significantly improve, complement, unify, generalize and enrich several results known in the current literature. In addition, we give examples with results in line with the ones we obtained.

scholarly journals An Intrinsic Characterization of Five Points in a CAT(0) Space

2020 ◽  
Vol 8 (1) ◽  
pp. 114-165
Author(s):  
Tetsu Toyoda
Keyword(s):  
Metric Space ◽  
Isometric Embedding ◽  
Metric Spaces ◽  
Necessary Conditions ◽  
New Approach ◽  
Intrinsic Characterization ◽  
New Family

AbstractGromov (2001) and Sturm (2003) proved that any four points in a CAT(0) space satisfy a certain family of inequalities. We call those inequalities the ⊠-inequalities, following the notation used by Gromov. In this paper, we prove that a metric space X containing at most five points admits an isometric embedding into a CAT(0) space if and only if any four points in X satisfy the ⊠-inequalities. To prove this, we introduce a new family of necessary conditions for a metric space to admit an isometric embedding into a CAT(0) space by modifying and generalizing Gromov’s cycle conditions. Furthermore, we prove that if a metric space satisfies all those necessary conditions, then it admits an isometric embedding into a CAT(0) space. This work presents a new approach to characterizing those metric spaces that admit an isometric embedding into a CAT(0) space.

Fuzzy double controlled metric spaces and related results

2021 ◽  
Vol 40 (5) ◽  
pp. 9977-9985
Author(s):  
Naeem Saleem ◽  
Hüseyin Işık ◽  
Salman Furqan ◽  
Choonkil Park
Keyword(s):  
Integral Equation ◽  
Metric Space ◽  
Existence And Uniqueness ◽  
Metric Spaces ◽  
Banach Contraction Principle ◽  
Uniqueness Of Solution ◽  
Contraction Principle ◽  
Triangular Inequality ◽  
Banach Contraction ◽  
The Banach Contraction Principle

In this paper, we introduce the concept of fuzzy double controlled metric space that can be regarded as the generalization of fuzzy b-metric space, extended fuzzy b-metric space and controlled fuzzy metric space. We use two non-comparable functions α and β in the triangular inequality as: M q ( x , z , t α ( x , y ) + s β ( y , z ) ) ≥ M q ( x , y , t ) ∗ M q ( y , z , s ) . We prove Banach contraction principle in fuzzy double controlled metric space and generalize the Banach contraction principle in aforementioned spaces. We give some examples to support our main results. An application to existence and uniqueness of solution for an integral equation is also presented in this work.

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