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In this article, we present two resilient algorithms, the improved alternative hard c‐means (IAHCM) and the improved alternative fuzzy c‐means (IAFCM). We implement the Gaussian distance‐dependent function proposed by Zhang and Chen (D.‐Q. Zhang and Chen, 2004). In some cases, Zhang and Chen's metric distance does not account for the clustering centroid effect predicted by the large value. R* is employed in IAHCM and IAFCM to discover robust results while minimizing its sensitivity. Experiments are conducted using two‐and three‐dimensional data, including Diamond and Iris real‐world data. The results are based on demonstrating the robust simplicity and applicability of the offered algorithms. Similarly, computational complexity is assessed. [ABSTRACT FROM AUTHOR]

Topology is the most important branch of modern mathematics, which plays an important role in applications. In this paper, we use the concept of the topology, based on the concept of multiset to solve an important problems in life (DNA and RNA mutation) to detect diseases and help biologists in the treatment of diseases. Also, we introduce a new theory that explains if there is an existence of a mutation or not, and we have a set of metric functions through which we examine the congruence and similarity and dissimilarity between “types,” which may be a strings of bits, vectors, DNA or RNA sequences, …, etc. Finally, we will introduce a theory that which can be used to know the existence and place of mutation. [ABSTRACT FROM AUTHOR]

ABSTRACT Morphospaces are spatial depictions of morphological variation among biological forms that have become an integral part of the analytical toolkit of evolutionary biologists and palaeobiologists. Nevertheless, the term morphospace brings together a great variety of spaces with different geometries. In particular, many morphospaces lack the metric properties underlying the notions of distance and direction, which are, however, central to the analysis of morphological differences and evolutionary transitions. The problem is illustrated here with the iconic morphospace of coiled shells implemented by Raup 50 years ago. The model, which allows the description of shell coiling geometry of various invertebrate taxa, is a seminal reference in theoretical morphology and morphospace theory, but also a morphometric framework frequently used in empirical studies, particularly of ammonoids. Because of the definition of its underlying parameters, Raup's morphospace does not possess a Euclidean structure and a meaningful interpretation of the spread and spacing of taxa within it is not guaranteed. Focusing on the region of the morphospace occupied by most ammonoids, I detail a landmark-based morphospace circumventing this problem and built from the same input measurements required for the calculation of Raup's parameters. From simulations and a reanalysis of Palaeozoic ammonoid shell disparity, the properties of these morphospaces are compared and their algebraic and geometric relationships highlighted. While Raup's model remains a valuable tool for describing ammonoid shells and relating their shapes to the coiling process, it is demonstrated that quantitative analyses of morphological patterns should be carried out within the landmark-based framework. Beyond this specific case, the increasing use and diversity of morphospaces in evolutionary morphology call for caution when interpreting patterns and comparing results drawn from different types of morphospaces. [ABSTRACT FROM AUTHOR]

We present a form of the Mean Value Theorem (MVT) for a continuous function f between metric spaces, connecting it with the possibility to choose the [ABSTRACT FROM AUTHOR]

A form of Kripke's schema turns out to be equivalent to each of the following two statements from metric topology: every open subspace of a separable metric space is separable; every open subset of a separable metric space is a countable union of open balls. Thus Kripke's schema serves as a point of reference for classifying theorems of classical mathematics within Bishop-style constructive reverse mathematics. [ABSTRACT FROM AUTHOR]

Manifolds are curved spaces that admit local coordinate systems which look like Euclidean space. Much of the classical geometry of Euclidean space extends and generalizes to manifolds. A generalization of Euclidean geometry is Riemannian geometry, where distances are defined which, in small regions, look like Euclidean geometry. This article describes how familiar concepts of Euclidean geometry generalize to the context of Riemannian manifolds, as well as more general structures. WIREs Comput Stat 2012 doi: 10.1002/wics.1220 [ABSTRACT FROM AUTHOR]

A test set of 10 molecules (open and ring forms of ozone and sulfur dioxide as well as water and hydrogen sulfide and their respective fluoro- and chloro-substituted analogs) of specific atmospheric interest has been formed as to assess the performance of various density functional theory methods in (hyper)polarizability calculations against well-established ab initio methods. The choice of these molecules was further based on (i) the profound change in the physics between isomeric systems, e.g., open (C2v) and ring (D3h) forms of ozone, (ii) the relation between isomeric forms, e.g., open and ring form of sulfur dioxide (both of C2v symmetry), and (iii) the effect of the substitution, e.g., in fluoro- and chloro-substituted water analogs. The analysis is aided by arguments chosen from the information theory, graph theory, and pattern recognition fields of Mathematics: In brief, a multidimensional space is formed by the methods which are playing the role of vectors with the independent components of the electric properties to act as the coordinates of these vectors, hence the relation between different vectors (e.g., methods) can be quantified by a proximity measure. Results are in agreement with previous studies revealing the acceptable and consistent behavior of the mPW1PW91, B3P86, and PBE0 methods. It is worth noting the remarkable good performance of the double hybrid functionals (namely: B2PLYP and mPW2PLYP) which are for the first time used in calculations of electric response properties. © 2009 Wiley Periodicals, Inc. J Comput Chem 2010 [ABSTRACT FROM AUTHOR]

In a recent Forum paper, it is argued that, in most studies, ordinal data such as the Braun-Blanquet abundance/dominance scale are not properly treated by multivariate methods. This is because conventional multivariate methods are generally adequate for ratio-scale variables only, while for ordinal variables differences between states and their ratios are not interpreted. Conversely, in this paper it is shown that using conventional multivariate procedures for evaluating ordinal data should imply a shift from a metric space to a topological data space; as such the use of ordinal data does not represent a serious methodological error, provided that results are interpreted accordingly. [ABSTRACT FROM AUTHOR]

We study within the framework of Zermelo-Fraenkel set theory ZF the role that the axiom of choice plays in the theory of Lindelöf metric spaces. We show that in ZF the weak choice principles: (i) Every Lindelöf metric space is separable and (ii) Every Lindelöf metric space is second countable (Forms 340 and 341, respectively, in [10]) are equivalent. We also prove that a Lindelöf metric space is hereditarily separable iff it is hereditarily Lindelöf iff it hold as well the axiom of choice restricted to countable sets and to topologies of Lindelöf metric spaces as the countable union theorem restricted to Lindelöf metric spaces. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

We study conditions for a topological space to be metrizable, properties of metrizable spaces, and the role the axiom of choice plays in these matters. [ABSTRACT FROM AUTHOR]

In the realm of metric spaces the role of choice principles is investigated. [ABSTRACT FROM AUTHOR]