scholarly journals Polish Metric Spaces: Their Classification and Isometry Groups

2001 ◽  
Vol 7 (3) ◽  
pp. 361-375 ◽  
Author(s):  
John D. Clemens ◽  
Su Gao ◽  
Alexander S. Kechris
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Classification Problem ◽  
Classification Problems ◽  
Isometry Groups ◽  
Polish Spaces ◽  
Basic Ideas ◽  
In The Beginning ◽  
Theory Of Complexity

§ 1. Introduction. In this communication we present some recent results on the classification of Polish metric spaces up to isometry and on the isometry groups of Polish metric spaces. A Polish metric space is a complete separable metric space (X, d).Our first goal is to determine the exact complexity of the classification problem of general Polish metric spaces up to isometry. This work was motivated by a paper of Vershik [1998], where he remarks (in the beginning of Section 2): “The classification of Polish spaces up to isometry is an enormous task. More precisely, this classification is not ‘smooth’ in the modern terminology.” Our Theorem 2.1 below quantifies precisely the enormity of this task.After doing this, we turn to special classes of Polish metric spaces and investigate the classification problems associated with them. Note that these classification problems are in principle no more complicated than the general one above. However, the determination of their exact complexity is not necessarily easier.The investigation of the classification problems naturally leads to some interesting results on the groups of isometries of Polish metric spaces. We shall also present these results below.The rest of this section is devoted to an introduction of some basic ideas of a theory of complexity for classification problems, which will help to put our results in perspective. Detailed expositions of this general theory can be found, e.g., in Hjorth [2000], Kechris [1999], [2001].

scholarly journals Research on Complex Classification Algorithm of Breast Cancer Chip Based on SVM-RFE Gene Feature Screening

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Guobin Chen ◽  
Xianzhong Xie ◽  
Shijin Li
Keyword(s):  
Gene Expression ◽  
Classification Problem ◽  
Classification Algorithm ◽  
Gene Chip ◽  
Classification Problems ◽  
Gene Chips ◽  
Gene Expression Matrix ◽  
Performance Indexes ◽  
Characteristic Sequences

Screening and classification of characteristic genes is a complex classification problem, and the characteristic sequences of gene expression show high-dimensional characteristics. How to select an effective gene screening algorithm is the main problem to be solved by analyzing gene chips. The combination of KNN, SVM, and SVM-RFE is selected to screen complex classification problems, and a new method to solve complex classification problems is provided. In the process of gene chip pretreatment, LogFC and P value equivalents in the gene expression matrix are screened, and different gene features are screened, and then SVM-RFE algorithm is used to sort and screen genes. Firstly, the characteristics of gene chips are analyzed and the number between probes and genes is counted. Clustering analysis among each sample and PCA classification analysis of different samples are carried out. Secondly, the basic algorithms of SVM and KNN are tested, and the important indexes such as error rate and accuracy rate of the algorithms are tested to obtain the optimal parameters. Finally, the performance indexes of accuracy, precision, recall, and F1 of several complex classification algorithms are compared through the complex classification of SVM, KNN, KNN-PCA, SVM-PCA, SVM-RFE-SVM, and SVM-RFE-KNN at P=0. 01,0.05,0.001. SVM-RFE-SVM has the best classification effect and can be used as a gene chip classification algorithm to analyze the characteristics of genes.

scholarly journals Absoluteness theorems for arbitrary Polish spaces

2019 ◽  
Vol 53 (2) ◽  
pp. 109-123
Author(s):  
Diego Alejandro Mejía ◽  
Ismael E. Rivera-Madrid
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Classical Theorem ◽  
General Version ◽  
Polish Spaces ◽  
Analytic Sets

By coding Polish metric spaces with metrics on countable sets, we propose an interpretation of Polish metric spaces in models of ZFC and extend Mostowski's classical theorem of absoluteness of analytic sets for any Polish metric space in general. In addition, we prove a general version of Shoenfield's absoluteness theorem.

scholarly journals Restricted Boltzmann Machines for Classification of Hepatocellular Carcinoma

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
James A. Koziol ◽  
Eng M. Tan ◽  
Liping Dai ◽  
Pengfei Ren ◽  
Jian-Ying Zhang
Keyword(s):  
Hepatocellular Carcinoma ◽  
Cancer Detection ◽  
Classification Problem ◽  
Classification Problems ◽  
Operating Characteristics ◽  
Restricted Boltzmann Machines ◽  
Boltzmann Machines ◽  
Multiple Antigen ◽  
Detection And Diagnosis

Multiple antigen miniarrays can provide accurate tools for cancer detection and diagnosis. These miniarrays can be validated by examining their operating characteristics in classifying individuals as either cancer patients or normal (non-cancer) subjects. We describe the use of restricted Boltzmann machines for this classification problem, relative to diagnosis of hepatocellular carcinoma. In this setting, we find that its operating characteristics are similar to a logistic regression standard and suggest that restricted Boltzmann machines merit further consideration for classification problems.

scholarly journals Isometry groups of separable metric spaces

2009 ◽  
Vol 146 (1) ◽  
pp. 67-81 ◽  
Author(s):  
MACIEJ MALICKI ◽  
SŁAWOMIR SOLECKI
Keyword(s):  
Metric Space ◽  
Isometry Group ◽  
Metric Spaces ◽  
Structure Theorem ◽  
Ultrametric Space ◽  
Isometry Groups ◽  
Locally Compact ◽  
Natural Action ◽  
Polish Group ◽  
Separable Metric Space

AbstractWe show that every locally compact Polish group is isomorphic to the isometry group of a proper separable metric space. This answers a question of Gao and Kechris. We also analyze the natural action of the isometry group of a separable ultrametric space on the space. This leads us to a structure theorem representing an arbitrary separable ultrametric space as a bundle with an ultrametric base and with ultrahomogeneous fibers which are invariant under the action of the isometry group.

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 Assessment of health/functioning of older adults who consume psychoactive substances

2018 ◽  
Vol 71 (3) ◽  
pp. 942-950
Author(s):  
Vania Dias Cruz ◽  
Silvana Sidney Costa Santos ◽  
Jamila Geri Tomaschewski-Barlem ◽  
Bárbara Tarouco da Silva ◽  
Celmira Lange ◽  
...  
Keyword(s):  
Older Adults ◽  
Qualitative Case Study ◽  
International Classification Of Functioning ◽  
Psychoactive Substances ◽  
Systematic Observation ◽  
Health Functioning ◽  
Final Consideration ◽  
Theory Of Complexity

ABSTRACT Objective: To assess the health/functioning of the older adult who consumes psychoactive substances through the International Classification of Functioning, Disability and Health, considering the theory of complexity. Method: Qualitative case study, with 11 older adults, held between December 2015 and February 2016 in the state of Rio Grande do Sul, using interviews, documents and non-systematic observation. It was approved by the ethics committee. The analysis followed the propositions of the case study, using the complexity of Morin as theoretical basis. Results: We identified older adults who consider themselves healthy and show alterations - the alterations can be exacerbated by the use of psychoactive substances - of health/functioning expected according to the natural course of aging such as: systemic arterial hypertension; depressive symptoms; dizziness; tinnitus; harmed sleep/rest; and inadequate food and water consumption. Final consideration: The assessment of health/functioning of older adults who use psychoactive substances, guided by complex thinking, exceeds the accuracy limits to risk the understanding of the phenomena in its complexity.

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 aggregate method for thorax diseases classification

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Bayu Adhi Nugroho
Keyword(s):  
Network Architecture ◽  
Classification Problem ◽  
Calculation Algorithm ◽  
Training Pattern ◽  
Deep Network ◽  
Network Training ◽  
Chest X Ray ◽  
Medical Image Classification ◽  
Positive Pattern

AbstractA common problem found in real-word medical image classification is the inherent imbalance of the positive and negative patterns in the dataset where positive patterns are usually rare. Moreover, in the classification of multiple classes with neural network, a training pattern is treated as a positive pattern in one output node and negative in all the remaining output nodes. In this paper, the weights of a training pattern in the loss function are designed based not only on the number of the training patterns in the class but also on the different nodes where one of them treats this training pattern as positive and the others treat it as negative. We propose a combined approach of weights calculation algorithm for deep network training and the training optimization from the state-of-the-art deep network architecture for thorax diseases classification problem. Experimental results on the Chest X-Ray image dataset demonstrate that this new weighting scheme improves classification performances, also the training optimization from the EfficientNet improves the performance furthermore. We compare the aggregate method with several performances from the previous study of thorax diseases classifications to provide the fair comparisons against the proposed method.

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.

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