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Journal articles on the topic 'Linear metric space'

Author: Grafiati

Published: 24 April 2022

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1

KRÖN, BERNHARD, JÖRG LEHNERT, NORBERT SEIFTER, and ELMAR TEUFL. "LINEAR AND PROJECTIVE BOUNDARY OF NILPOTENT GROUPS." Glasgow Mathematical Journal 57, no. 3 (December 22, 2014): 591–632. http://dx.doi.org/10.1017/s0017089514000512.

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AbstractWe define a pseudometric on the set of all unbounded subsets of a metric space. The Kolmogorov quotient of this pseudometric space is a complete metric space. The definition of the pseudometric is guided by the principle that two unbounded subsets have distance 0 whenever they stay sublinearly close. Based on this pseudometric we introduce and study a general concept of boundaries of metric spaces. Such a boundary is the closure of a subset in the Kolmogorov quotient determined by an arbitrarily chosen family of unbounded subsets. Our interest lies in those boundaries which we get by choosing unbounded cyclic sub(semi)groups of a finitely generated group (or more general of a compactly generated, locally compact Hausdorff group). We show that these boundaries are quasi-isometric invariants and determine them in the case of nilpotent groups as a disjoint union of certain spheres (or projective spaces). In addition we apply this concept to vertex-transitive graphs with polynomial growth and to random walks on nilpotent groups.

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2

Michael, George. "A metric linear space is an open cone." Kyoto Journal of Mathematics 52, no. 4 (2012): 833–38. http://dx.doi.org/10.1215/21562261-1728893.

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3

Pluzhnikova, Elena Aleksandrovna, Tatyana Vladimirovna Zhukovskaya, and Yuriy Anatol’evich Moiseev. "ON SETS OF METRIC REGULARITY OF MAPPINGS IN SPACES WITH VECTOR-VALUED METRIC." Tambov University Reports. Series: Natural and Technical Sciences, no. 123 (2018): 547–54. http://dx.doi.org/10.20310/1810-0198-2018-23-123-547-554.

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Spaces with vector-valued metric are considered. The values of a vectorvalued metric are elements of a cone in some linear normed space. The concept of the set of metric regularity for mapping in spaces with vector-valued metric is formulated. A statement on the stability of the set of metric regularity of a given mapping for its Lipschitz perturbations in spaces with vector-valued metric is obtained.

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4

Rohen, Yumnam, Tatjana Dosenovic, and Stojan Radenovic. "A note on the paper "A fixed point theorems in Sb-metric spaces"." Filomat 31, no. 11 (2017): 3335–46. http://dx.doi.org/10.2298/fil1711335r.

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Very recently, N. Souayan and N. Mlaiki [Nazir Souayan and Nabil Mlaiki, A fixed point theorem in Sb-metric spaces, J. Math. Comput. Sci. 16 (2016), 131-139] and S. Sedghi et al. [S. Sedghi, A. Gholidahneb, T. Dosenovic, J. Esfahani, S. Radenovic, Common fixed point of four maps in Sb-metric spaces, to appear in J. Linear Topol. Algebra], introduced the concept of Sb-metric space as a generalization of S-metric space. In this paper, we modified the definition of Sb-metric introduced by Souayan and Mlaiki and prove some coupled common fixed point theorems in Sb-metric space. We also present an example to confirm our theoretical results.

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5

Holá, Ľubica. "The Attouch-Wets topology and a characterisation of normable linear spaces." Bulletin of the Australian Mathematical Society 44, no. 1 (August 1991): 11–18. http://dx.doi.org/10.1017/s0004972700029415.

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Let X and Y be metric spaces and C(X, Y) be the space of all continuous functions from X to Y. If X is a locally connected space, the compact-open topology on C(X, Y) is weaker than the Attouch-Wets topology on C(X, Y). The result is applied on the space of continuous linear functions. Let X be a locally convex topological linear space metrisable with an invariant metric and X* be a continuous dual. X is normable if and only if the strong topology on X* and the Attouch-Wets topology coincide.

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6

Roy, Kushal, Hossein Alaeidizaji, Mantu Saha, Babak Mohammadi, and Vahid Parvaneh. "Some Fixed-Point Theorems over a Generalized F -Metric Space." Advances in Mathematical Physics 2021 (May 5, 2021): 1–7. http://dx.doi.org/10.1155/2021/5570653.

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In this article, the concept of sequential F -metric spaces has been introduced as a generalization of usual metric spaces, b -metric spaces, J S -metric spaces, and mainly F -metric spaces. Some topological properties of such spaces have been discussed here. By considering this notion, we prove fixed-point theorems for some classes of contractive mappings over such spaces. Examples have been given in order to examine the validity of the underlying space and in support of our fixed-point theorems. Moreover, our fixed-point theorem is applied to obtain solution of a system of linear algebraic equations.

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7

HYTÖNEN, TUOMAS, DACHUN YANG, and DONGYONG YANG. "The Hardy space H1 on non-homogeneous metric spaces." Mathematical Proceedings of the Cambridge Philosophical Society 153, no. 1 (December 8, 2011): 9–31. http://dx.doi.org/10.1017/s0305004111000776.

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AbstractLet (, d, μ) be a metric measure space and satisfy the so-called upper doubling condition and the geometrical doubling condition. We introduce the atomic Hardy space H1(μ) and prove that its dual space is the known space RBMO(μ) in this context. Using this duality, we establish a criterion for the boundedness of linear operators from H1(μ) to any Banach space. As an application of this criterion, we obtain the boundedness of Calderón–Zygmund operators from H1(μ) to L1(μ).

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8

Larotonda, Gabriel. "Metric geometry of infinite-dimensional Lie groups and their homogeneous spaces." Forum Mathematicum 31, no. 6 (November 1, 2019): 1567–605. http://dx.doi.org/10.1515/forum-2019-0127.

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AbstractWe study the geometry of Lie groups G with a continuous Finsler metric, in presence of a subgroup K such that the metric is right-invariant for the action of K. We present a systematic study of the metric and geodesic structure of homogeneous spaces M obtained by the quotient {M\simeq G/K}. Of particular interest are left-invariant metrics of G which are then bi-invariant for the action of K. We then focus on the geodesic structure of groups K that admit bi-invariant metrics, proving that one-parameter groups are short paths for those metrics, and characterizing all other short paths. We provide applications of the results obtained, in two settings: manifolds of Banach space linear operators, and groups of maps from compact manifolds.

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9

Constantini, Camillo, and Wieslaw Kubís. "Paths in hyperspaces." Applied General Topology 4, no. 2 (October 1, 2003): 377. http://dx.doi.org/10.4995/agt.2003.2040.

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<p>We prove that the hyperspace of closed bounded sets with the Hausdor_ topology, over an almost convex metric space, is an absolute retract. Dense subspaces of normed linear spaces are examples of, not necessarily connected, almost convex metric spaces. We give some necessary conditions for the path-wise connectedness of the Hausdorff metric topology on closed bounded sets. Finally, we describe properties of a separable metric space, under which its hyperspace with the Wijsman topology is path-wise connected.</p>

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10

Zhang, Zihou, and Chunyan Liu. "The Representations and Continuity of the Metric Projections on Two Classes of Half-Spaces in Banach Spaces." Abstract and Applied Analysis 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/908676.

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We show a necessary and sufficient condition for the existence of metric projection on a class of half-spaceKx0*,c={x∈X:x*(x)≤c}in Banach space. Two representations of metric projectionsPKx0*,candPKx0,care given, respectively, whereKx0,cstands for dual half-space ofKx0*,cin dual spaceX*. By these representations, a series of continuity results of the metric projectionsPKx0*,candPKx0,care given. We also provide the characterization that a metric projection is a linear bounded operator.

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11

Zhukovskaia, Tatiana, and Elena Pluzhnikova. "The set of regularity of a multivalued mapping in a space with a vector-valued metric." Tambov University Reports. Series: Natural and Technical Sciences, no. 125 (2019): 39–46. http://dx.doi.org/10.20310/1810-0198-2019-24-125-39-46.

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We consider multivalued mappings acting in spaces with a vector-valued metric. A vector-valued metric is understood as a mapping satisfying the axioms “of an ordinary metric” with values in the cone of a linear normed space. The concept of the regularity set of a multivalued mapping is defined. A set of regularity is used in the study of inclusions in spaces with a vector-valued metric.

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12

Wang, Hui, and Yuwen Wang. "Metric Generalized Inverse of Linear Operator in Banach Space." Chinese Annals of Mathematics 24, no. 04 (October 2003): 509–20. http://dx.doi.org/10.1142/s0252959903000517.

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13

KNIEPER, GERHARD. "A second derivative formula of the Liouville entropy at spaces of constant negative curvature." Ergodic Theory and Dynamical Systems 17, no. 5 (October 1997): 1131–35. http://dx.doi.org/10.1017/s0143385797086446.

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In this paper we study a new functional on the space of metrics with negative curvature on a compact manifold. It is a linear combination of Liouville entropy and total scalar curvature. Locally symmetric spaces are critical points of this functional. We provide an explicit formula for its second derivative at metrics of constant negative curvature. In particular, this shows that a metric of constant curvature is a local maximum.

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14

Koh, T. Y., and F. Wan. "Theory of the norm-induced metric in atmospheric dynamics." Atmospheric Chemistry and Physics 15, no. 5 (March 9, 2015): 2571–94. http://dx.doi.org/10.5194/acp-15-2571-2015.

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Abstract. We suggest that some metrics for quantifying distances in phase space are based on linearized flows about unrealistic reference states and hence may not be applicable to atmospheric flows. A new approach of defining a norm-induced metric based on the total energy norm is proposed. The approach is based on the rigorous mathematics of normed vector spaces and the law of energy conservation in physics. It involves the innovative construction of the phase space so that energy (or a certain physical invariant) takes the form of a Euclidean norm. The metric can be applied to both linear and nonlinear flows and for small and large separations in phase space. The new metric is derived for models of various levels of sophistication: the 2-D barotropic model, the shallow-water model and the 3-D dry, compressible atmosphere in different vertical coordinates. Numerical calculations of the new metric are illustrated with analytic dynamical systems as well as with global reanalysis data. The differences from a commonly used metric and the potential for application in ensemble prediction, error growth analysis and predictability studies are discussed.

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15

Duchoň, Miloslav, and Peter Maličký. "A Helly theorem for functions with values in metric spaces." Tatra Mountains Mathematical Publications 44, no. 1 (December 1, 2009): 159–68. http://dx.doi.org/10.2478/v10127-009-0056-z.

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Abstract We present a Helly type theorem for sequences of functions with values in metric spaces and apply it to representations of some mappings on the space of continuous functions. A generalization of the Riesz theorem is formulated and proved. More concretely, a representation of certain majored linear operators on the space of continuous functions into a complete metric space.

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16

Bridges, Douglas, and Ray Mines. "Sequentially continuous linear mappings in constructive analysis." Journal of Symbolic Logic 63, no. 2 (June 1998): 579–83. http://dx.doi.org/10.2307/2586851.

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A mapping u: X → Y between metric spaces is sequentially continuous if for each sequence (xn) converging to x ∈ X, (u(xn)) converges to u(x). It is well known in classical mathematics that a sequentially continuous mapping between metric spaces is continuous; but, as all proofs of this result involve the law of excluded middle, there appears to be a constructive distinction between sequential continuity and continuity. Although this distinction is worth exploring in its own right, there is another reason why sequential continuity is interesting to the constructive mathematician: Ishihara [8] has a version of Banach's inverse mapping theorem in functional analysis that involves the sequential continuity, rather than continuity, of the linear mappings; if this result could be upgraded by deleting the word “sequential”, then we could prove constructively the standard versions of the inverse mapping theorem and the closed graph theorem.Troelstra [9] showed that in Brouwer's intuitionistic mathematics (INT) a sequentially continuous mapping on a separable metric space is continuous. On the other hand, Ishihara [6, 7] proved constructively that the continuity of sequentially continuous mappings on a separable metric space is equivalent to a certain boundedness principle for subsets of ℕ; in the same paper, he showed that the latter principle holds within the recursive constructive mathematics (RUSS) of the Markov School. Since it is not known whether that principle holds within Bishop's constructive mathematics (BISH), of which INT and RUSS are models and which can be regarded as the constructive core of mathematics, the exploration of sequential continuity within BISH holds some interest.

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17

Das, Ruchi, and Tarun Das. "Asymptotic Properties of -Expansive Homeomorphisms on a Metric -Space." Abstract and Applied Analysis 2012 (2012): 1–10. http://dx.doi.org/10.1155/2012/237820.

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We define and study the notions of positively and negatively -asymptotic points for a homeomorphism on a metric -space. We obtain necessary and sufficient conditions for two points to be positively/negatively -asymptotic. Also, we show that the problem of studying -expansive homeomorphisms on a bounded subset of a normed linear -space is equivalent to the problem of studying linear -expansive homeomorphisms on a bounded subset of another normed linear -space.

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18

Panek, L., and N. M. P. Panek. "Group of Isometries of Niederreiter-Rosenbloom-Tsfasman Block Space." TEMA (São Carlos) 21, no. 2 (July 22, 2020): 271. http://dx.doi.org/10.5540/tema.2020.021.02.271.

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Let P = ({1, 2, ..., n}, ≤) be a poset that is an union of disjoint chains of the same length and V = F^N_q be the space of N-tuples over the ﬁnite ﬁeld Fq. Let Vi = F^{k_i}_q , with 1 ≤ i ≤ n, be a family of ﬁnite-dimensional linear spaces such that k_1 + k_2 + ... + k_n = N and let V = V_1×V_2×...×V_n endow with the poset block metric d_(P,π) induced by the poset P and the partition π = (k_1, k_2, ..., k_n), encompassing both Niederreiter-Rosenbloom-Tsfasman metric and error-block metric. In this paper, we give a complete description of group of isometries of the metric space (V, d_(P,π)), also called the Niederreiter-Rosenbloom-Tsfasman block space. In particular, we reobtain the group of isometries of the Niederreiter-Rosenbloom-Tsfasman space and obtain the group of isometries of the error-block metric space.

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19

Hammad, Hasanen A., Hassen Aydi, and Yaé Ulrich Gaba. "Exciting Fixed Point Results on a Novel Space with Supportive Applications." Journal of Function Spaces 2021 (January 27, 2021): 1–12. http://dx.doi.org/10.1155/2021/6613774.

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The goal of this paper is to present a new space, a complex valued controlled rectangular b -metric space (for short, υ ℂ -metric space). Some examples and topological properties of υ ℂ -metric spaces are given. Also, some related common fixed point results are discussed. Our results generalize a lot of works in this direction. Moreover, we apply the theoretical results to find a unique solution of a complex valued Atangana-Baleanu fractional integral operator and a system of complex linear equations. Finally, a numerical example to find the current that passes through the RLC circuit is illustrated.

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20

Ehrlacher, Virginie, Damiano Lombardi, Olga Mula, and François-Xavier Vialard. "Nonlinear model reduction on metric spaces. Application to one-dimensional conservative PDEs in Wasserstein spaces." ESAIM: Mathematical Modelling and Numerical Analysis 54, no. 6 (November 2020): 2159–97. http://dx.doi.org/10.1051/m2an/2020013.

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We consider the problem of model reduction of parametrized PDEs where the goal is to approximate any function belonging to the set of solutions at a reduced computational cost. For this, the bottom line of most strategies has so far been based on the approximation of the solution set by linear spaces on Hilbert or Banach spaces. This approach can be expected to be successful only when the Kolmogorov width of the set decays fast. While this is the case on certain parabolic or elliptic problems, most transport-dominated problems are expected to present a slow decaying width and require to study nonlinear approximation methods. In this work, we propose to address the reduction problem from the perspective of general metric spaces with a suitably defined notion of distance. We develop and compare two different approaches, one based on barycenters and another one using tangent spaces when the metric space has an additional Riemannian structure. Since the notion of linear vectorial spaces does not exist in general metric spaces, both approaches result in nonlinear approximation methods. We give theoretical and numerical evidence of their efficiency to reduce complexity for one-dimensional conservative PDEs where the underlying metric space can be chosen to be the L2-Wasserstein space.

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21

Smrz, P. K. "Construction of Real Space{Time from Complex Linear Metric Connections." Australian Journal of Physics 50, no. 4 (1997): 793. http://dx.doi.org/10.1071/p96100.

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A construction of real space-time based on metric linear connections in a complex manifold is described. The construction works only in two or four dimensions. The four-dimensional case based on a connection reducible to group U(2, 2) can generate Riemann-Cartan geometry on the real submanifold of the original complex manifold. The possibility of connecting the appearance of Dirac fields with anholonomic complex frames is discussed.

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Hirasawa, Go. "A Metric for Unbounded Linear Operators in a Hilbert Space." Integral Equations and Operator Theory 70, no. 3 (December 4, 2010): 363–78. http://dx.doi.org/10.1007/s00020-010-1851-2.

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23

Gnanaprakasam, Arul Joseph, Salah Mahmoud Boulaaras, Gunaseelan Mani, Bahri Cherif, and Sahar Ahmed Idris. "Solving system of linear equations via bicomplex valued metric space." Demonstratio Mathematica 54, no. 1 (January 1, 2021): 474–87. http://dx.doi.org/10.1515/dema-2021-0046.

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Abstract In this paper, we prove some common fixed point theorems on bicomplex metric space. Our results generalize and expand some of the literature’s well-known results. We also explore some of the applications of our key results.

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24

Suantai, Suthep, Yeol Je Cho, and Jukrapong Tiammee. "Common ﬁxed points for generalized ψ-contractions in weak non-Archimedean fuzzy metric spaces." Applied General Topology 20, no. 1 (April 1, 2019): 1. http://dx.doi.org/10.4995/agt.2019.7638.

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<p>Fixed point theory in fuzzy metric spaces plays very important role in theory of nonlinear problems in applied science. In this paper, we prove an existence result of common fixed point of four nonlinear mappings satisfying a new type of contractive condition in a generalized fuzzy metric space, called weak non-Archimedean fuzzy metric space. Our main results can be applied to solve the existence of solutions of non-linear equations in fuzzy metric spaces. Some examples supporting our main theorem are also given. Our results improve and generalize some recent results contained in Vetro (2011)[16]to generalized contractive conditions under some suitable conditions and many known results in the literature.</p>

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25

Chen, Lili, Chaobo Li, Radoslaw Kaczmarek, and Yanfeng Zhao. "Several Fixed Point Theorems in Convex b-Metric Spaces and Applications." Mathematics 8, no. 2 (February 14, 2020): 242. http://dx.doi.org/10.3390/math8020242.

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Our paper is devoted to indicating a way of generalizing Mann’s iteration algorithm and a series of fixed point results in the framework of b-metric spaces. First, the concept of a convex b-metric space by means of a convex structure is introduced and Mann’s iteration algorithm is extended to this space. Next, by the help of Mann’s iteration scheme, strong convergence theorems for two types of contraction mappings in convex b-metric spaces are obtained. Some examples supporting our main results are also presented. Moreover, the problem of the T-stability of Mann’s iteration procedure for the above mappings in complete convex b-metric spaces is considered. As an application, we apply our main result to approximating the solution of the Fredholm linear integral equation.

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Hejmej, Beata. "Stability of Functional Equations in Dislocated Quasi-Metric Spaces." Annales Mathematicae Silesianae 32, no. 1 (September 1, 2018): 215–25. http://dx.doi.org/10.2478/amsil-2018-0005.

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Abstract We present a result on the generalized Hyers-Ulam stability of a functional equation in a single variable for functions that have values in a complete dislocated quasi-metric space. Next, we show how to apply it to prove stability of the Cauchy functional equation and the linear functional equation in two variables, also for functions taking values in a complete dislocated quasimetric space. In this way we generalize some earlier results proved for classical complete metric spaces.

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27

Amirbostaghi, Gholamhossein, Mehdi Asadi, and Mohammad Mardanbeigi. "M-convex structure on b-metric spaces." Filomat 35, no. 14 (2021): 4765–76. http://dx.doi.org/10.2298/fil2114765a.

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We apply the concept of a m-convex b-metric space by introducing of m-convex structure on b-metric spaces. We obtain fixed point theorems in this structure. Recent results are concluded in our targets, as well. Some illustrated examples are presented to confirm our main results. As an application, we apply our main result to finding existence and uniqueness the solution of the Fredholm non-linear integral equation.

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Ambrosio, Luigi, and Daniele Puglisi. "Linear extension operators between spaces of Lipschitz maps and optimal transport." Journal für die reine und angewandte Mathematik (Crelles Journal) 2020, no. 764 (July 1, 2020): 1–21. http://dx.doi.org/10.1515/crelle-2018-0037.

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AbstractMotivated by the notion of {K\hskip-0.284528pt}-gentle partition of unity introduced in [J. R. Lee and A. Naor, Extending Lipschitz functions via random metric partitions, Invent. Math. 160 (2005), no. 1, 59–95] and the notion of {K\kern-0.284528pt}-Lipschitz retract studied in [S. I. Ohta, Extending Lipschitz and Hölder maps between metric spaces, Positivity 13 (2009), no. 2, 407–425], we study a weaker notion related to the Kantorovich–Rubinstein transport distance that we call {K\kern-0.284528pt}-random projection. We show that {K\kern-0.284528pt}-random projections can still be used to provide linear extension operators for Lipschitz maps. We also prove that the existence of these random projections is necessary and sufficient for the existence of weak{{}^{*}} continuous operators. Finally, we use this notion to characterize the metric spaces {(X,d)} such that the free space {\mathcal{F}(X)} has the bounded approximation propriety.

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29

Preda, Ciprian. "On the uniform exponential stability of linear skew-product semiflows." Journal of Function Spaces and Applications 4, no. 2 (2006): 145–61. http://dx.doi.org/10.1155/2006/703620.

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The problem of uniform exponential stability of linear skew-product semiflows on locally compact metric space with Banach fibers, is discussed. It is established a connection between the uniform exponential stability of linear skewproduct semiflows and some admissibility-type condition. This approach is based on the method of “test functions”, using a very large class of function spaces, the so-called Orlicz spaces.

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30

Nožička, František. "About a special metric in the linear space and Kepler motions." Applications of Mathematics 33, no. 1 (1988): 49–67. http://dx.doi.org/10.21136/am.1988.104286.

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31

Galley, Chad R. "Gravitational Self-force from Quantized Linear Metric Perturbations in Curved Space." Foundations of Physics 37, no. 4-5 (March 13, 2007): 460–79. http://dx.doi.org/10.1007/s10701-007-9111-2.

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REICH, SIMEON, and ALEXANDER J. ZASLAVSKI. "AN EXAMPLE CONCERNING BOUNDED LINEAR REGULARITY OF SUBSPACES IN HILBERT SPACE." Bulletin of the Australian Mathematical Society 89, no. 2 (September 12, 2013): 217–26. http://dx.doi.org/10.1017/s0004972713000749.

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AbstractWe study bounded linear regularity of finite sets of closed subspaces in a Hilbert space. In particular, we construct for each natural number $n\geq 3$ a set of $n$ closed subspaces of ${\ell }^{2} $ which has the bounded linear regularity property, while the bounded linear regularity property does not hold for each one of its nonempty, proper nonsingleton subsets. We also establish a related theorem regarding the bounded regularity property in metric spaces.

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33

Dobrowolski, Tadeusz. "Revisiting Cauty's proof of the Schauder conjecture." Abstract and Applied Analysis 2003, no. 7 (2003): 407–33. http://dx.doi.org/10.1155/s1085337503211015.

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The Schauder conjecture that every compact convex subset of a metric linear space has the fixed-point property was recently established by Cauty (2001). This paper elaborates on Cauty's proof in order to make it more detailed, and therefore more accessible. Such a detailed analysis allows us to show that the convex compacta in metric linear spaces possess the simplicial approximation property introduced by Kalton, Peck, and Roberts. The latter demonstrates that the original Schauder approach to solve the conjecture is in some sense correctable.

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34

Drozdowski, Robert. "On the generalized variation and generalized weak variation of maps with values in metric linear spaces." Tatra Mountains Mathematical Publications 42, no. 1 (December 1, 2009): 131–48. http://dx.doi.org/10.2478/v10127-009-0013-x.

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Abstract In this paper, the class of maps (with values in a metric linear space) of a bounded generalized variation (bounded generalized weak variation) is described. Connections between those kinds of spaces are investigated.

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35

Sumati Kumari, Panda, Obaid Alqahtani, and Erdal Karapınar. "Some Fixed-Point Theorems in b-Dislocated Metric Space and Applications." Symmetry 10, no. 12 (December 2, 2018): 691. http://dx.doi.org/10.3390/sym10120691.

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In this article, we prove some fixed-point theorems in b-dislocated metric space. Thereafter, we propose a simple and efficient solution for a non-linear integral equation and non-linear fractional differential equations of Caputo type by using the technique of fixed point.

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36

Kausar, Samina, and Andre O. Falcao. "Analysis and Comparison of Vector Space and Metric Space Representations in QSAR Modeling." Molecules 24, no. 9 (April 30, 2019): 1698. http://dx.doi.org/10.3390/molecules24091698.

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The performance of quantitative structure–activity relationship (QSAR) models largely depends on the relevance of the selected molecular representation used as input data matrices. This work presents a thorough comparative analysis of two main categories of molecular representations (vector space and metric space) for fitting robust machine learning models in QSAR problems. For the assessment of these methods, seven different molecular representations that included RDKit descriptors, five different fingerprints types (MACCS, PubChem, FP2-based, Atom Pair, and ECFP4), and a graph matching approach (non-contiguous atom matching structure similarity; NAMS) in both vector space and metric space, were subjected to state-of-art machine learning methods that included different dimensionality reduction methods (feature selection and linear dimensionality reduction). Five distinct QSAR data sets were used for direct assessment and analysis. Results show that, in general, metric-space and vector-space representations are able to produce equivalent models, but there are significant differences between individual approaches. The NAMS-based similarity approach consistently outperformed most fingerprint representations in model quality, closely followed by Atom Pair fingerprints. To further verify these findings, the metric space-based models were fitted to the same data sets with the closest neighbors removed. These latter results further strengthened the above conclusions. The metric space graph-based approach appeared significantly superior to the other representations, albeit at a significant computational cost.

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37

Newton, Nigel J. "A class of non-parametric statistical manifolds modelled on Sobolev space." Information Geometry 2, no. 2 (November 25, 2019): 283–312. http://dx.doi.org/10.1007/s41884-019-00024-z.

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AbstractWe construct a family of non-parametric (infinite-dimensional) manifolds of finite measures on $${\mathbb {R}}^d$$Rd, each containing a smoothly embedded submanifold of probability measures. The manifolds are modelled on a variety of weighted Sobolev spaces, including Hilbert–Sobolev spaces and mixed-norm spaces, and support the Fisher–Rao metric as a weak Riemannian metric. Densities are expressed in terms of a deformed exponential function having linear growth. Unusually for the Sobolev context, and as a consequence of its linear growth, this “lifts” to a nonlinear superposition (Nemytskii) operator that acts continuously on a particular class of mixed-norm model spaces, and on the fixed norm space $$W^{2,1}$$W2,1; i.e. it maps each of these spaces continuously into itself. In contrast with non-parametric exponential manifolds, the density itself belongs to the model space, and the range of the chart is the whole of this space. Some of the results make essential use of a log-Sobolev embedding theorem, which also sharpens existing results concerning the regularity of statistical divergences on the manifolds. Applications to the stochastic partial differential equations of nonlinear filtering (and hence to the Fokker–Planck equation) are outlined.

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38

D’ALOTTO, LOUIS, and CHARLES GIARDINA. "THE KOLMOGOROV METRIC AND A GENERALIZATION ON A CLASSIFICATION OF CELLULAR AUTOMATA." International Journal on Artificial Intelligence Tools 03, no. 03 (September 1994): 311–26. http://dx.doi.org/10.1142/s0218213094000157.

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This paper introduces and applies a new metric, on the space of bi-infinite strings (sequences), to the linear cellular automata classification approach of R. Gilman. The metric presented herein is a generalization of the metric used in the classification work of Gilman and it is shown that those original classification results also hold with this generalized metric.

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Martínez-Giménez, Félix, Alfred Peris, and Francisco Rodenas. "Chaos on Fuzzy Dynamical Systems." Mathematics 9, no. 20 (October 18, 2021): 2629. http://dx.doi.org/10.3390/math9202629.

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Given a continuous map f:X→X on a metric space, it induces the maps f¯:K(X)→K(X), on the hyperspace of nonempty compact subspaces of X, and f^:F(X)→F(X), on the space of normal fuzzy sets, consisting of the upper semicontinuous functions u:X→[0,1] with compact support. Each of these spaces can be endowed with a respective metric. In this work, we studied the relationships among the dynamical systems (X,f), (K(X),f¯), and (F(X),f^). In particular, we considered several dynamical properties related to chaos: Devaney chaos, A-transitivity, Li–Yorke chaos, and distributional chaos, extending some results in work by Jardón, Sánchez and Sanchis (Mathematics 2020, 8, 1862) and work by Bernardes, Peris and Rodenas (Integr. Equ. Oper. Theory 2017, 88, 451–463). Especial attention is given to the dynamics of (continuous and linear) operators on metrizable topological vector spaces (linear dynamics).

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Rashed Alfuraidan, Monther. "The Contraction Principle for Multivalued Mappings on a Modular Metric Space with a Graph." Canadian Mathematical Bulletin 59, no. 01 (March 2016): 3–12. http://dx.doi.org/10.4153/cmb-2015-029-x.

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Abstract We study the existence of fixed points for contraction multivalued mappings in modular metric spaces endowed with a graph. The notion of a modular metric on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces like Orlicz spaces, were recently introduced. This paper can be seen as a generalization of Nadler and Edelstein’s fixed point theorems to modular metric spaces endowed with a graph.

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41

Gusev, Alexander. "Nonsingular Metric Elastic Universe." Symposium - International Astronomical Union 168 (1996): 571–72. http://dx.doi.org/10.1017/s0074180900110721.

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In the RTFD(Gusev (1986)) the conception of a Sakharov - Wheeler Metric Elasticity(SWME)(Sakharov (1967), Wheeler (1970)) had been worked out. On the basis of the exact solutions of Einstein equations and qualitative analysis RTFD the global evolution have been studied and the phase portraits of the early Universe is being constructed. An analysis of phase portraits show on the possibility description of spontaneous creation of Universe from an initial Minkowskian's vacuum to an inflationary de Sitter space-time in the frame of phenomenological non-quantum theory (Guth (1991)). During the past decade, a radically new picture of cosmology has emerged. The present homogeneous expanding Universe would have stated out with a de Sitter phase. The purpose of this paper is to shown that the geometry-dynamical approach to the Einstein's gravitation theory in the frame RTFD also is leaded to the nonsingular cosmological models (Brandenberger (1993)). Let us to propose that before the some moment of time the Universe is at the vacuum state and is described the geometry of Minkowskian's space. Deformations of vacuum state, identifying with empty Mikowskian's space are described by the deformations tensor, An arising of deformation ∊αβis leaded to appearance of the stress tensor ∊αβand the energy-momentumTαβ(∊γδ) which is connected with “creating” particles in the Universe. Here we are considered the deformations of Minkowskian's space (the initial vacuum state with∞αβ = 0) at the linear theory (~ ∊) of finite deformations. The final deformation stategαβare searched in the metric class of Friedmann's cosmological spaces. In the comoving reference systemUα(0, 0, 0, 1) the Friedmann's equations have form (Narlikar & Padmanabhan (1983), and Gusev (1989)):where R(t) is so called the expansion factor at the Robertson - Walker line element, k is the curvature parameter with the possible values −1, 0, + 1, P is pressure,k1,k2are the some combination from a Lame coefficients,l02is a “initial radius” Universe, a free parameter model. The phase space of this model is the two-dimensional (R,Ṙ) plane. We note that there is only two singular points (Ṙ= 0,Ṙ= 0) in the phase plane. The one of those points isR=l0,Ṙ= 0 and corresponds to Minkowski space - time. There are two classes trajectories which are asymptotically de Sitter. Those starting at large positive values ofṘgo off toṘ= + ∞, reaching their asymptotic value of H from above. Those starting with large negative values ofṘtend toR= + ∞ withṘ> 0. For small values ofṘand R we can see that there are periodic solutions about Minkowski space. The corresponding solutions oscillate with frequency given byH0(which is possible equal planck scale) about Minkowski space. Based on the preceding discussion of asymptotic solutions we see that there is a separatrix (Gusev, (1989)) in phase space dividing solutions which tend toR= + ∞ from those which oscillate or tend toR=l0. The above analyses of the phase portraits is an indication that in our theory Minkowski space may be unstable toward homogeneous deformations. We stress that all the general features of the phase portrait analyses are true for quadratic deformations of gravitational vacuum. Our model incorporates a very important feature: in the asymptotic de Sitter region, the quadratic deformations and temperature effects does not have an important effect on the geometry. The effective gravitational constant of coupling goes to zero as space - time approaches de Sitter space. In this sense the model is asymptotically free (gravitational confinement Linde, (1990)). At the late times the solutions are described a evolution of the de Sitter UniverseR~expHt(Hoyle et al. (1993)).

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Lee, Kwankyu. "The automorphism group of a linear space with the Rosenbloom–Tsfasman metric." European Journal of Combinatorics 24, no. 6 (August 2003): 607–12. http://dx.doi.org/10.1016/s0195-6698(03)00077-5.

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Singh, S. L., Charu Bhatnagar, and S. N. Mishra. "Stability of Jungck-type iterative procedures." International Journal of Mathematics and Mathematical Sciences 2005, no. 19 (2005): 3035–43. http://dx.doi.org/10.1155/ijmms.2005.3035.

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BACHIR, MOHAMMED. "AN EXTENSION OF THE BANACH–STONE THEOREM." Journal of the Australian Mathematical Society 105, no. 1 (November 8, 2017): 1–23. http://dx.doi.org/10.1017/s1446788717000271.

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We establish an extension of the Banach–Stone theorem to a class of isomorphisms more general than isometries in a noncompact framework. Some applications are given. In particular, we give a canonical representation of some (not necessarily linear) operators between products of function spaces. Our results are established for an abstract class of function spaces included in the space of all continuous and bounded functions defined on a complete metric space.

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Bahamonde, Sebastian, David Benisty, and Eduardo Guendelman. "Linear Potentials in Galaxy Halos by Asymmetric Wormholes." Universe 4, no. 11 (October 29, 2018): 112. http://dx.doi.org/10.3390/universe4110112.

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A spherically symmetric space-time solution for a diffusive two measures theory is studied. An asymmetric wormhole geometry is obtained where the metric coefficients has a linear term for galactic distances and the analysis of Mannheim and collaborators, can then be used to describe the galactic rotation curves. For cosmological distances a de-Sitter space-time is realized. Center of gravity coordinates for the wormhole are introduced which are the most suitable for the collective motion of a wormhole. The wormholes connect universes with different vacuum energy densities which may represent different universes in a “landscape scenario”. The metric coefficients depend on the asymmetric wormhole parameters. The coefficient of the linear potential is proportional to both the mass of the wormhole and the cosmological constant of the observed universe. Similar results are also expected in other theories like k-essence theories, that may support wormholes.

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Eveson, Simon P., and Roger D. Nussbaum. "An elementary proof of the Birkhoff-Hopf theorem." Mathematical Proceedings of the Cambridge Philosophical Society 117, no. 1 (January 1995): 31–55. http://dx.doi.org/10.1017/s0305004100072911.

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In important work some thirty years ago, G. Birkhoff[2, 3] and E. Hopf [16, 17] showed that large classes of positive linear operators behave like contraction mappings with respect to certain ‘almost’ metrics. Hopf worked in a space of measurable functions and took as his ‘almost’ metric the oscillation ω(y/x) of functions y and x with x(t) > 0 almost everywhere, defined by

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Miesch, Benjamin, and Maël Pavón. "Weakly externally hyperconvex subsets and hyperconvex gluings." Journal of Topology and Analysis 09, no. 03 (May 18, 2016): 379–407. http://dx.doi.org/10.1142/s1793525317500145.

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We give a necessary and sufficient condition under which gluings of hyperconvex metric spaces along weakly externally hyperconvex subsets are hyperconvex. This leads to a full characterization of hyperconvex gluings of two isometric copies of the same hyperconvex space. Furthermore, we investigate the case of gluings of finite dimensional hyperconvex linear spaces along linear subspaces. For this purpose, we characterize the weakly externally hyperconvex subsets of [Formula: see text] endowed with the maximum norm.

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KLEINBOCK, DMITRY, GREGORY MARGULIS, and JUNBO WANG. "METRIC DIOPHANTINE APPROXIMATION FOR SYSTEMS OF LINEAR FORMS VIA DYNAMICS." International Journal of Number Theory 06, no. 05 (August 2010): 1139–68. http://dx.doi.org/10.1142/s1793042110003423.

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The goal of this paper is to generalize the main results of [21] and subsequent papers on metric with dependent quantities to the set-up of systems of linear forms. In particular, we establish "joint strong extremality" of arbitrary finite collection of smooth non-degenerate submanifolds of ℝn. The proofs are based on generalized quantitative non-divergence estimates for translates of measures on the space of lattices.

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Szász-Friedl, Annamária. "Deformation of complex Finsler metrics." Analele Universitatii "Ovidius" Constanta - Seria Matematica 26, no. 3 (December 1, 2018): 229–44. http://dx.doi.org/10.2478/auom-2018-0043.

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AbstractThe aim of this paper is to describe the infinitesimal deformation (M, V) of a complex Finsler space family {(M, Lt)}t∈ℝ and to study some of its geometrical objects (metric tensor, non-linear connection, etc). In this circumstances the induced non-linear connection on (M, V) is defined. Moreover we have elaborate the inverse problem, the problem of the first order deformation of the metric. A special part is devoted to the study of particular cases of the perturbed metric.

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Baziv, N. M., and O. B. Hrybel. "On the algebraic dimension of Riesz spaces." Matematychni Studii 56, no. 1 (October 23, 2021): 67–71. http://dx.doi.org/10.30970/ms.56.1.67-71.

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We prove that the algebraic dimension of an infinite dimensional $C$-$\sigma$-complete Riesz space (in particular, of a Dedekind $\sigma$-complete and a laterally $\sigma$-complete Riesz space) with the principal projection property which either has a weak order unit or is not purely atomic, is at least continuum. A similar (incomparable to ours) result for complete metric linear spaces is well known.

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