Relevant bibliographies by topics / Hausdorff measures. Measure theory. Topological groups

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Academic literature on the topic 'Hausdorff measures. Measure theory. Topological groups'

Author: Grafiati
Published: 4 June 2021
Last updated: 16 February 2022

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Journal articles on the topic "Hausdorff measures. Measure theory. Topological groups"

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BERLANGA, RICARDO. "A TOPOLOGISED MEASURE HOMOLOGY." Glasgow Mathematical Journal 50, no. 3 (September 2008): 359–69. http://dx.doi.org/10.1017/s0017089508004266.

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AbstractA homology theory based on measures, first mentioned by Thurston, is naturally defined here as a functor into the category of locally convex topological vector spaces. It is proved that the first homology space is Hausdorff.
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STRATMANN, B. O., and M. URBAŃSKI. "The geometry of conformal measures for parabolic rational maps." Mathematical Proceedings of the Cambridge Philosophical Society 128, no. 1 (January 2000): 141–56. http://dx.doi.org/10.1017/s0305004199003837.

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We study the h-conformal measure for parabolic rational maps, where h denotes the Hausdorff dimension of the associated Julia sets. We derive a formula which describes in a uniform way the scaling of this measure at arbitrary elements of the Julia set. Furthermore, we establish the Khintchine Limit Law for parabolic rational maps (the analogue of the ‘logarithmic law for geodesics’ in the theory of Kleinian groups) and show that this law provides some efficient control for the fluctuation of the h-conformal measure. We then show that these results lead to some refinements of the description of this measure in terms of Hausdorff and packing measures with respect to some gauge functions. Also, we derive a simple proof of the fact that the Julia set of a parabolic rational map is uniformly perfect. Finally, we obtain that the conformal measure is a regular doubling measure, we show that its Renyi dimension and its information dimension are equal to h and we compute its logarithmic index.
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Li, Qiongling, Xinwei Li, Xuetong Wang, Yuxia Li, Kuncheng Li, Yang Yu, Changhao Yin, Shuyu Li, and Ying Han. "Topological Properties of Large-Scale Cortical Networks Based on Multiple Morphological Features in Amnestic Mild Cognitive Impairment." Neural Plasticity 2016 (2016): 1–14. http://dx.doi.org/10.1155/2016/3462309.

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Previous studies have demonstrated that amnestic mild cognitive impairment (aMCI) has disrupted properties of large-scale cortical networks based on cortical thickness and gray matter volume. However, it is largely unknown whether the topological properties of cortical networks based on geometric measures (i.e., sulcal depth, curvature, and metric distortion) change in aMCI patients compared with normal controls because these geometric features of cerebral cortex may be related to its intrinsic connectivity. Here, we compare properties in cortical networks constructed by six different morphological features in 36 aMCI participants and 36 normal controls. Six cortical features (3 volumetric and 3 geometric features) were extracted for each participant, and brain abnormities in aMCI were identified by cortical network based on graph theory method. All the cortical networks showed small-world properties. Regions showing significant differences mainly located in the medial temporal lobe and supramarginal and right inferior parietal lobe. In addition, we also found that the cortical networks constructed by cortical thickness and sulcal depth showed significant differences between the two groups. Our results indicated that geometric measure (i.e., sulcal depth) can be used to construct network to discriminate individuals with aMCI from controls besides volumetric measures.
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Sawicki, Adam, Michał Oszmaniec, and Marek Kuś. "Convexity of momentum map, Morse index, and quantum entanglement." Reviews in Mathematical Physics 26, no. 03 (April 2014): 1450004. http://dx.doi.org/10.1142/s0129055x14500044.

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We analyze from the topological perspective the space of all SLOCC (Stochastic Local Operations with Classical Communication) classes of pure states for composite quantum systems. We do it for both distinguishable and indistinguishable particles. In general, the topology of this space is rather complicated as it is a non-Hausdorff space. Using geometric invariant theory (GIT) and momentum map geometry, we propose a way to divide the space of all SLOCC classes into mathematically and physically meaningful families. Each family consists of possibly many "asymptotically" equivalent SLOCC classes. Moreover, each contains exactly one distinguished SLOCC class on which the total variance (a well-defined measure of entanglement) of the state Var [v] attains maximum. We provide an algorithm for finding critical sets of Var [v], which makes use of the convexity of the momentum map and allows classification of such defined families of SLOCC classes. The number of families is in general infinite. We introduce an additional refinement into finitely many groups of families using some developments in the momentum map geometry known as the Kirwan–Ness stratification. We also discuss how to define it equivalently using the convexity of the momentum map applied to SLOCC classes. Moreover, we note that the Morse index at the critical set of the total variance of state has an interpretation of number of non-SLOCC directions in which entanglement increases and calculate it for several exemplary systems. Finally, we introduce the SLOCC-invariant measure of entanglement as a square root of the total variance of state at the critical point and explain its geometric meaning.
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Yeo, Ronald A., Sephira G. Ryman, Martijn P. van den Heuvel, Marcel A. de Reus, Rex E. Jung, Jessica Pommy, Andrew R. Mayer, et al. "Graph Metrics of Structural Brain Networks in Individuals with Schizophrenia and Healthy Controls: Group Differences, Relationships with Intelligence, and Genetics." Journal of the International Neuropsychological Society 22, no. 2 (February 2016): 240–49. http://dx.doi.org/10.1017/s1355617715000867.

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AbstractObjectives: One of the most prominent features of schizophrenia is relatively lower general cognitive ability (GCA). An emerging approach to understanding the roots of variation in GCA relies on network properties of the brain. In this multi-center study, we determined global characteristics of brain networks using graph theory and related these to GCA in healthy controls and individuals with schizophrenia. Methods: Participants (N=116 controls, 80 patients with schizophrenia) were recruited from four sites. GCA was represented by the first principal component of a large battery of neurocognitive tests. Graph metrics were derived from diffusion-weighted imaging. Results: The global metrics of longer characteristic path length and reduced overall connectivity predicted lower GCA across groups, and group differences were noted for both variables. Measures of clustering, efficiency, and modularity did not differ across groups or predict GCA. Follow-up analyses investigated three topological types of connectivity—connections among high degree “rich club” nodes, “feeder” connections to these rich club nodes, and “local” connections not involving the rich club. Rich club and local connectivity predicted performance across groups. In a subsample (N=101 controls, 56 patients), a genetic measure reflecting mutation load, based on rare copy number deletions, was associated with longer characteristic path length. Conclusions: Results highlight the importance of characteristic path lengths and rich club connectivity for GCA and provide no evidence for group differences in the relationships between graph metrics and GCA. (JINS, 2016, 22, 240–249)
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Dissertations / Theses on the topic "Hausdorff measures. Measure theory. Topological groups"

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Akhvlediani, Andrei. "Hausdorff and Gromov distances in quantale-enriched categories /." 2008. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:MR45921.

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Thesis (M.A.)--York University, 2008. Graduate Programme in Mathematics and Statistics.
Typescript. Includes bibliographical references (leaves 166-167). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:MR45921
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Books on the topic "Hausdorff measures. Measure theory. Topological groups"

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1943-, Mauldin R. Daniel, and Williams S. C. 1952-, eds. The exact Hausdorff dimension in random recursive constructions. Providence, R.I., USA: American Mathematical Society, 1988.

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Graf, Siegfried. The exact Hausdorff dimension in random recursive constructions. Providence, R.I: American Mathematical Society, 1988.

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3

Koli︠a︡da, S. F. Dynamics and numbers: A special program, June 1-July 31, 2014, Max Planck Institute for Mathematics, Bonn, Germany : international conference, July 21-25, 2014, Max Planck Institute for Mathematics, Bonn, Germany. Edited by Max-Planck-Institut für Mathematik. Providence, Rhode Island: American Mathematical Society, 2016.

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