Relevant bibliographies by topics / Fractal measure

Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Fractal measure'

Author: Grafiati
Published: 4 June 2021
Last updated: 1 February 2022

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Journal articles on the topic "Fractal measure"

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Mörters, Peter, and David Preiss. "Tangent measure distributions of fractal measures." Mathematische Annalen 312, no. 1 (September 1, 1998): 53–93. http://dx.doi.org/10.1007/s002080050212.

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Chen, Yanguang. "Fractal Modeling and Fractal Dimension Description of Urban Morphology." Entropy 22, no. 9 (August 30, 2020): 961. http://dx.doi.org/10.3390/e22090961.

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The conventional mathematical methods are based on characteristic length, while urban form has no characteristic length in many aspects. Urban area is a scale-dependence measure, which indicates the scale-free distribution of urban patterns. Thus, the urban description based on characteristic lengths should be replaced by urban characterization based on scaling. Fractal geometry is one powerful tool for the scaling analysis of cities. Fractal parameters can be defined by entropy and correlation functions. However, the question of how to understand city fractals is still pending. By means of logic deduction and ideas from fractal theory, this paper is devoted to discussing fractals and fractal dimensions of urban landscape. The main points of this work are as follows. Firstly, urban form can be treated as pre-fractals rather than real fractals, and fractal properties of cities are only valid within certain scaling ranges. Secondly, the topological dimension of city fractals based on the urban area is 0; thus, the minimum fractal dimension value of fractal cities is equal to or greater than 0. Thirdly, the fractal dimension of urban form is used to substitute the urban area, and it is better to define city fractals in a two-dimensional embedding space; thus, the maximum fractal dimension value of urban form is 2. A conclusion can be reached that urban form can be explored as fractals within certain ranges of scales and fractal geometry can be applied to the spatial analysis of the scale-free aspects of urban morphology.
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Kadanoff, Leo P. "Fractal Singularities in a Measure and How to Measure Singularities on a Fractal." Progress of Theoretical Physics Supplement 86 (1986): 383–86. http://dx.doi.org/10.1143/ptps.86.383.

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SENGUPTA, KAUSHIK, and K. J. VINOY. "A NEW MEASURE OF LACUNARITY FOR GENERALIZED FRACTALS AND ITS IMPACT IN THE ELECTROMAGNETIC BEHAVIOR OF KOCH DIPOLE ANTENNAS." Fractals 14, no. 04 (December 2006): 271–82. http://dx.doi.org/10.1142/s0218348x06003313.

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In recent years, fractal geometries have been explored in various branches of science and engineering. In antenna engineering several of these geometries have been studied due to their purported potential of realizing multi-resonant antennas. Although due to the complex nature of fractals most of these previous studies were experimental, there have been some analytical investigations on the performance of the antennas using them. One such analytical attempt was aimed at quantitatively relating fractal dimension with antenna characteristics within a single fractal set. It is however desirable to have all fractal geometries covered under one framework for antenna design and other similar applications. With this objective as the final goal, we strive in this paper to extend an earlier approach to more generalized situations, by incorporating the lacunarity of fractal geometries as a measure of its spatial distribution. Since the available measure of lacunarity was found to be inconsistent, in this paper we propose to use a new measure to quantize the fractal lacunarity. We also demonstrate the use of this new measure in uniquely explaining the behavior of dipole antennas made of generalized Koch curves and go on to show how fundamental lacunarity is in influencing electromagnetic behavior of fractal antennas. It is expected that this averaged measure of lacunarity may find applications in areas beyond antennas.
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Falconer, Kenneth J., and Gerald A. Edgar. "Measure, Topology and Fractal Geometry." Mathematical Gazette 75, no. 472 (June 1991): 237. http://dx.doi.org/10.2307/3620293.

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Norton, Alec, and Gerald A. Edgar. "Measure, Topology, and Fractal Geometry." American Mathematical Monthly 99, no. 4 (April 1992): 378. http://dx.doi.org/10.2307/2324919.

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Tan, Teewoon, and Hong Yan. "The fractal neighbor distance measure." Pattern Recognition 35, no. 6 (June 2002): 1371–87. http://dx.doi.org/10.1016/s0031-3203(01)00125-x.

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Kolb, M. "Harmonic measure for fractal objects." Nuclear Physics B - Proceedings Supplements 5, no. 1 (September 1988): 129–34. http://dx.doi.org/10.1016/0920-5632(88)90027-8.

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Chen, Yanguang. "Characterizing Growth and Form of Fractal Cities with Allometric Scaling Exponents." Discrete Dynamics in Nature and Society 2010 (2010): 1–22. http://dx.doi.org/10.1155/2010/194715.

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Fractal growth is a kind of allometric growth, and the allometric scaling exponents can be employed to describe growing fractal phenomena such as cities. The spatial features of the regular fractals can be characterized by fractal dimension. However, for the real systems with statistical fractality, it is incomplete to measure the structure of scaling invariance only by fractal dimension. Sometimes, we need to know the ratio of different dimensions rather than the fractal dimensions themselves. A fractal-dimension ratio can make an allometric scaling exponent (ASE). As compared with fractal dimension, ASEs have three advantages. First, the values of ASEs are easy to be estimated in practice; second, ASEs can reflect the dynamical characters of system's evolution; third, the analysis of ASEs can be made through prefractal structure with limited scale. Therefore, the ASEs based on fractal dimensions are more functional than fractal dimensions for real fractal systems. In this paper, the definition and calculation method of ASEs are illustrated by starting from mathematical fractals, and, then, China's cities are taken as examples to show how to apply ASEs to depiction of growth and form of fractal cities.
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SIMMONS, DAVID. "On interpreting Patterson–Sullivan measures of geometrically finite groups as Hausdorff and packing measures." Ergodic Theory and Dynamical Systems 36, no. 8 (July 21, 2015): 2675–86. http://dx.doi.org/10.1017/etds.2015.27.

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We provide a new proof of a theorem whose proof was sketched by Sullivan [Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics. Acta Math.149(3–4) (1982), 215–237], namely that if the Poincaré exponent of a geometrically finite Kleinian group $G$ is strictly between its minimal and maximal cusp ranks, then the Patterson–Sullivan measure of $G$ is not proportional to the Hausdorff or packing measure of any gauge function. This disproves a conjecture of Stratmann [Multiple fractal aspects of conformal measures; a survey. Workshop on Fractals and Dynamics (Mathematica Gottingensis, 5). Eds. M. Denker, S.-M. Heinemann and B. Stratmann. Springer, Berlin, 1997, pp. 65–71; Fractal geometry on hyperbolic manifolds. Non-Euclidean Geometries (Mathematical Applications (N.Y.), 581). Springer, New York, 2006, pp. 227–247].
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Dissertations / Theses on the topic "Fractal measure"

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Törnblom, Arvid. "Measure theory, fractal geometry and their applications on digital sundials." Thesis, Uppsala universitet, Analys och sannolikhetsteori, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-435354.

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Childress, Scot Paul. "Quantum measures, arithmetic coils, and generalized fractal strings." Diss., UC access only, 2009. http://proquest.umi.com/pqdweb?index=128&did=1871850181&SrchMode=1&sid=1&Fmt=7&retrieveGroup=0&VType=PQD&VInst=PROD&RQT=309&VName=PQD&TS=1270491013&clientId=48051.

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Thesis (Ph. D.)--University of California, Riverside, 2009.
Includes abstract. Includes bibliographical references (leaves 202-204) and index. Issued in print and online. Available via ProQuest Digital Dissertations.
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Siebert, Kitzeln B. "A modern presentation of "dimension and outer measure"." Columbus, Ohio : Ohio State University, 2008. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1211395297.

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Poosapadi, Arjunan Sridhar, and sridhar arjunan@rmit edu au. "Fractal features of Surface Electromyogram: A new measure for low level muscle activation." RMIT University. Electrical and Computer Engineering, 2009. http://adt.lib.rmit.edu.au/adt/public/adt-VIT20090629.095851.

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Identifying finger and wrist flexion based actions using single channel surface electromyogram have a number of rehabilitation, defence and human computer interface applications. These applications are currently infeasible because of unreliability in classification of sEMG when the level of muscle contraction is low and when there are multiple active muscles. The presence of noise and cross-talk from closely located and simultaneously active muscles is exaggerated when muscles are weakly active such as during maintained wrist and finger flexion. It has been established in literature that surface electromyogram (sEMG) and other such biosignals are fractal signals. Some researchers have determined that fractal dimension (FD) is related to strength of muscle contraction. On careful analysis of fractal properties of sEMG, this research work has established that FD is related to the muscle size and complexity and not to the strength of muscle contraction. The work has also identified a novel feature, maximum fractal length (MFL) of the signal, as a good measure of strength of contraction of the muscle. From the analysis, it is observed that while at high level of contraction, root mean square (RMS) is an indicator of strength of contraction of the muscle, this relationship is not very strong when the muscle contraction is less than 50% maximum voluntary contraction. This work has established that MFL is a more reliable measure of strength of contraction compared to RMS, especially at low levels of contraction. This research work reports the use of fractal properties of sEMG to identify the small changes in strength of muscle contraction and the location of the active muscles. It is observed that fractal dimension (FD) of the signal is related with the properties of the muscle while maximum fractal length (MFL) is related to the strength of contraction of the associated muscle. The results show that classifying MFL and FD of a single channel sEMG from the forearm it is possible to accurately identify a set of finger and wrist flexion based actions even when the muscle activity is very weak. It is proposed that such a system could be used to control a prosthetic hand or for human computer interface.
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Leifsson, Patrik. "Fractal sets and dimensions." Thesis, Linköping University, Department of Mathematics, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-7320.

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Fractal analysis is an important tool when we need to study geometrical objects less regular than ordinary ones, e.g. a set with a non-integer dimension value. It has developed intensively over the last 30 years which gives a hint to its young age as a branch within mathematics.

In this thesis we take a look at some basic measure theory needed to introduce certain definitions of fractal dimensions, which can be used to measure a set's fractal degree. Comparisons of these definitions are done and we investigate when they coincide. With these tools different fractals are studied and compared.

A key idea in this thesis has been to sum up different names and definitions referring to similar concepts.

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Simonini, Marina. "Fractal sets and their applications in medicine." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2015. http://amslaurea.unibo.it/8763/.

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La geometria euclidea risulta spesso inadeguata a descrivere le forme della natura. I Frattali, oggetti interrotti e irregolari, come indica il nome stesso, sono più adatti a rappresentare la forma frastagliata delle linee costiere o altri elementi naturali. Lo strumento necessario per studiare rigorosamente i frattali sono i teoremi riguardanti la misura di Hausdorff, con i quali possono definirsi gli s-sets, dove s è la dimensione di Hausdorff. Se s non è intero, l'insieme in gioco può riconoscersi come frattale e non presenta tangenti e densità in quasi nessun punto. I frattali più classici, come gli insiemi di Cantor, Koch e Sierpinski, presentano anche la proprietà di auto-similarità e la dimensione di similitudine viene a coincidere con quella di Hausdorff. Una tecnica basata sulla dimensione frattale, detta box-counting, interviene in applicazioni bio-mediche e risulta utile per studiare le placche senili di varie specie di mammiferi tra cui l'uomo o anche per distinguere un melanoma maligno da una diversa lesione della cute.
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Inui, Kanji. "Study of the fractals generated by contractive mappings and their dimensions." Kyoto University, 2020. http://hdl.handle.net/2433/253370.

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Kyoto University (京都大学)
0048
新制・課程博士
博士(人間・環境学)
甲第22534号
人博第937号
新制||人||223(附属図書館)
2019||人博||937(吉田南総合図書館)
京都大学大学院人間・環境学研究科共生人間学専攻
(主査)教授 角 大輝, 教授 上木 直昌, 准教授 木坂 正史
学位規則第4条第1項該当
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Wang, Nancy. "Fractal Sets: Dynamical, Dimensional and Topological Properties." Thesis, KTH, Skolan för teknikvetenskap (SCI), 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-233147.

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Fractals is a relatively new mathematical topic which received thorough treatment only starting with 1960's. Fractals can be observed everywhere in nature and in day-to-day life. To give a few examples, common fractals are the spiral cactus, the romanesco broccoli, human brain and the outline of the Swedish map. Fractal dimension is a dimension which need not take integer values. In fractal geometry, a fractal dimension is a ratio providing an index of the complexity of fractal pattern with regard to how the local geometry changes with the scale at which it is measured. In recent years, fractal analysis is used increasingly in many areas of engineering and technology. Among others, fractal analysis is used in signal and image compression, computer and video design, neuroscience and fractal based cancer modelling and diagnosing.   This study consists of two main parts. The first part of the study aims to understand the appearance of an irregular Cantor set generated by the chaotic dynamical system generated by the logistic function on the unit interval [0,1]. In order to understand this irregular Cantor set, we studied the topological properties of the Cantor Middle-thirds set and the generalised Cantor sets, all of which have zero length. The necessity to compare these sets with regard to their size led us to the second part of this paper, namely the dimension studies of fractals. More complex fractals were presented in the second part, three definitions of dimension were introduced. The fractal dimension of the irregular Cantor set generated by the logistic mapping was estimated and we found that the Hausdorff dimension has the widest scope and greatest flexibility in the fractal studies.
Fraktaler är ett relativt nytt ämne inom matematik som fick sitt stora genomslag först efter 60-talet.  En fraktal är ett självliknande mönster med struktur i alla skalor. Några vardagliga exempel på fraktaler är spiralkaktus, romanescobroccoli, mänskliga hjärnan, blodkärlen och Sveriges fastlandskust. Bråktalsdimension är en typ av dimension där dimensionsindexet tillåts att anta alla icke-negativa reella tal. Inom fraktalgeometri kan dimensionsindexet betraktas som ett komplexitetsindex av mönstret med avseende på hur den lokala geometrin förändras beroende på vilken skala mönstret betraktas i. Under det senaste decenniet har fraktalanalysen använts alltmer flitigt inom tekniska och vetenskapliga tillämpningar. Bland annat har fraktalanalysen använts i signal- och bildkompression, dator- och videoformgivning, neurovetenskap och fraktalbaserad cancerdiagnos.   Denna studie består av två huvuddelar. Den första delen fokuserar på att förstår hur en fraktal kan uppstå i ett kaotiskt dynamiskt system. För att vara mer specifik studerades den logistiska funktionen och hur denna ickelinjära avbildning genererar en oregelbunden Cantormängd på intervalet [0,1]. Vidare, för att förstå den oregelbundna Cantormängden studerades Cantormängden (eng. the Cantor Middle-Thirds set) och de generaliserade Cantormängderna, vilka alla har noll längd. För att kunna jämföra de olika Cantormängderna med avseende på storlek, leds denna studie vidare till dimensionsanalys av fraktaler som är huvudämnet i den andra delen av denna studie. Olika topologiska fraktaler presenterades, tre olika definitioner av dimension introducerades, bland annat lådräkningsdimensionen och Hausdorffdimensionen. Slutligen approximerades dimensionen av den oregelbundna Cantormängden med hjälp av Hausdorffdimensionen. Denna studie demonstrerar att Hausdorffdimensionen har större omfattning och mer flexibilitet för fraktalstudier.
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Farkas, Ábel. "Dimension and measure theory of self-similar structures with no separation condition." Thesis, University of St Andrews, 2015. http://hdl.handle.net/10023/7854.

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We introduce methods to cope with self-similar sets when we do not assume any separation condition. For a self-similar set K ⊆ ℝᵈ we establish a similarity dimension-like formula for Hausdorff dimension regardless of any separation condition. By the application of this result we deduce that the Hausdorff measure and Hausdorff content of K are equal, which implies that K is Ahlfors regular if and only if Hᵗ (K) > 0 where t = dim[sub]H K. We further show that if t = dim[sub]H K < 1 then Hᵗ (K) > 0 is also equivalent to the weak separation property. Regarding Hausdorff dimension, we give a dimension approximation method that provides a tool to generalise results on non-overlapping self-similar sets to overlapping self-similar sets. We investigate how the Hausdorff dimension and measure of a self-similar set K ⊆ ℝᵈ behave under linear mappings. This depends on the nature of the group T generated by the orthogonal parts of the defining maps of K. We show that if T is finite then every linear image of K is a graph directed attractor and there exists at least one projection of K such that the dimension drops under projection. In general, with no restrictions on T we establish that Hᵗ (L ∘ O(K)) = Hᵗ (L(K)) for every element O of the closure of T , where L is a linear map and t = dim[sub]H K. We also prove that for disjoint subsets A and B of K we have that Hᵗ (L(A) ∩ L(B)) = 0. Hochman and Shmerkin showed that if T is dense in SO(d; ℝ) and the strong separation condition is satisfied then dim[sub]H (g(K)) = min {dim[sub]H K; l} for every continuously differentiable map g of rank l. We deduce the same result without any separation condition and we generalize a result of Eroğlu by obtaining that Hᵗ (g(K)) = 0. We show that for the attractor (K1, … ,Kq) of a graph directed iterated function system, for each 1 ≤ j ≤ q and ε > 0 there exists a self-similar set K ⊆ Kj that satisfies the strong separation condition and dim[sub]H Kj - ε < dim[sub]H K. We show that we can further assume convenient conditions on the orthogonal parts and similarity ratios of the defining similarities of K. Using this property we obtain results on a range of topics including on dimensions of projections, intersections, distance sets and sums and products of sets. We study the situations where the Hausdorff measure and Hausdorff content of a set are equal in the critical dimension. Our main result here shows that this equality holds for any subset of a set corresponding to a nontrivial cylinder of an irreducible subshift of finite type, and thus also for any self-similar or graph directed self-similar set, regardless of separation conditions. The main tool in the proof is an exhaustion lemma for Hausdorff measure based on the Vitali's Covering Theorem. We also give several examples showing that one cannot hope for the equality to hold in general if one moves in a number of the natural directions away from `self-similar'. Finally we consider an analogous version of the problem for packing measure. In this case we need the strong separation condition and can only prove that the packing measure and δ-approximate packing pre-measure coincide for sufficiently small δ > 0.
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Boore, Graeme C. "Directed graph iterated function systems." Thesis, University of St Andrews, 2011. http://hdl.handle.net/10023/2109.

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This thesis concerns an active research area within fractal geometry. In the first part, in Chapters 2 and 3, for directed graph iterated function systems (IFSs) defined on ℝ, we prove that a class of 2-vertex directed graph IFSs have attractors that cannot be the attractors of standard (1-vertex directed graph) IFSs, with or without separation conditions. We also calculate their exact Hausdorff measure. Thus we are able to identify a new class of attractors for which the exact Hausdorff measure is known. We give a constructive algorithm for calculating the set of gap lengths of any attractor as a finite union of cosets of finitely generated semigroups of positive real numbers. The generators of these semigroups are contracting similarity ratios of simple cycles in the directed graph. The algorithm works for any IFS defined on ℝ with no limit on the number of vertices in the directed graph, provided a separation condition holds. The second part, in Chapter 4, applies to directed graph IFSs defined on ℝⁿ . We obtain an explicit calculable value for the power law behaviour as r → 0⁺ , of the qth packing moment of μ[subscript(u)], the self-similar measure at a vertex u, for the non-lattice case, with a corresponding limit for the lattice case. We do this (i) for any q ∈ ℝ if the strong separation condition holds, (ii) for q ≥ 0 if the weaker open set condition holds and a specified non-negative matrix associated with the system is irreducible. In the non-lattice case this enables the rate of convergence of the packing L[superscript(q)]-spectrum of μ[subscript(u)] to be determined. We also show, for (ii) but allowing q ∈ ℝ, that the upper multifractal q box-dimension with respect to μ[subscript(u)], of the set consisting of all the intersections of the components of F[subscript(u)], is strictly less than the multifractal q Hausdorff dimension with respect to μ[subscript(u)] of F[subscript(u)].
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Books on the topic "Fractal measure"

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Edgar, Gerald A. Measure, topology, and fractal geometry. New York: Springer-Verlag, 1990.

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Edgar, Gerald A. Measure, topology and fractal geometry. New York: Springer-Verlag, 1990.

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Edgar, Gerald A. Measure, topology, and fractal geometry. 2nd ed. New York: Springer-Verlag, 2008.

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Edgar, Gerald A. Measure, Topology, and Fractal Geometry. New York, NY: Springer New York, 1990. http://dx.doi.org/10.1007/978-1-4757-4134-6.

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Gavriluţ, Alina, Ioan Mercheş, and Maricel Agop. Atomicity through Fractal Measure Theory. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-29593-6.

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Edgar, Gerald, ed. Measure, Topology, and Fractal Geometry. New York, NY: Springer New York, 2008. http://dx.doi.org/10.1007/978-0-387-74749-1.

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Measure, topology, and fractal geometry. 2nd ed. New York: Springer-Verlag, 2008.

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Integral, probability, and fractal measures. New York: Springer, 1998.

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Edgar, Gerald A. Integral, Probability, and Fractal Measures. New York, NY: Springer New York, 1998.

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Falconer, K. J. The geometry of fractal sets. Cambridge [Cambridgeshire]: Cambridge University Press, 1985.

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Book chapters on the topic "Fractal measure"

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Barnsley, Michael F., Brendan Harding, and Miroslav Rypka. "Measure Preserving Fractal Homeomorphisms." In Fractals, Wavelets, and their Applications, 79–102. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-08105-2_5.

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Tricot, Claude. "Perfect Sets and Their Measure." In Curves and Fractal Dimension, 1–12. New York, NY: Springer New York, 1995. http://dx.doi.org/10.1007/978-1-4612-4170-6_1.

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Blath, Jochen. "Measure-valued Processes, Self-similarity and Flickering Random Measures." In Fractal Geometry and Stochastics IV, 175–96. Basel: Birkhäuser Basel, 2009. http://dx.doi.org/10.1007/978-3-0346-0030-9_6.

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Tricot, Claude. "Parameterized Curves, Support of a Measure." In Curves and Fractal Dimension, 59–70. New York, NY: Springer New York, 1995. http://dx.doi.org/10.1007/978-1-4612-4170-6_6.

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Gavriluţ, Alina, Ioan Mercheş, and Maricel Agop. "Several hypertopologies: A short overview." In Atomicity through Fractal Measure Theory, 1–7. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-29593-6_1.

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Gavriluţ, Alina, Ioan Mercheş, and Maricel Agop. "Extended atomicity through non-differentiability and its physical implications." In Atomicity through Fractal Measure Theory, 133–62. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-29593-6_10.

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Gavriluţ, Alina, Ioan Mercheş, and Maricel Agop. "On a multifractal theory of motion in a non-differentiable space: Toward a possible multifractal theory of measure." In Atomicity through Fractal Measure Theory, 163–79. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-29593-6_11.

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Gavriluţ, Alina, Ioan Mercheş, and Maricel Agop. "A mathematical-physical approach on regularity in hit-and-miss hypertopologies for fuzzy set multifunctions." In Atomicity through Fractal Measure Theory, 9–19. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-29593-6_2.

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Gavriluţ, Alina, Ioan Mercheş, and Maricel Agop. "Non-atomic set multifunctions." In Atomicity through Fractal Measure Theory, 21–35. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-29593-6_3.

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Gavriluţ, Alina, Ioan Mercheş, and Maricel Agop. "Non-atomicity and the Darboux property for fuzzy and non-fuzzy Borel/Baire multivalued set functions." In Atomicity through Fractal Measure Theory, 37–50. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-29593-6_4.

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Conference papers on the topic "Fractal measure"

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Abiyev, Rahib, and Kemal Ihsan Kilic. "An efficient fractal measure for image texture recognition." In 2009 Fifth International Conference on Soft Computing, Computing with Words and Perceptions in System Analysis, Decision and Control (ICSCCW). IEEE, 2009. http://dx.doi.org/10.1109/icsccw.2009.5379454.

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Angeline, Peter J. "Results on a fractal measure for evolutionary optimization." In AeroSense 2002, edited by Kevin L. Priddy, Paul E. Keller, and Peter J. Angeline. SPIE, 2002. http://dx.doi.org/10.1117/12.458701.

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NEKKA, FAHIMA, and JUN LI. "CHARACTERIZATION OF FRACTAL STRUCTURES THROUGH A HAUSDORFF MEASURE BASED METHOD." In Fractals and Related Phenomena in Nature. WORLD SCIENTIFIC, 2004. http://dx.doi.org/10.1142/9789812702746_0017.

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Podgorelec, V., M. Hericko, and M. B. Juric. "Assessing software complexity from UML using fractal complexity measure." In Second IEEE International Conference on Computational Cybernetics. IEEE, 2004. http://dx.doi.org/10.1109/icccyb.2004.1437717.

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Kinsner, W., and R. Dansereau. "A Relative Fractal Dimension Spectrum as a Complexity Measure." In 2006 5th IEEE International Conference on Cognitive Informatics. IEEE, 2006. http://dx.doi.org/10.1109/coginf.2006.365697.

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de Melo, R. H. C., and A. Conci. "Succolarity: Defining a method to calculate this fractal measure." In 2008 International Conference on Systems, Signals and Image Processing (IWSSIP). IEEE, 2008. http://dx.doi.org/10.1109/iwssip.2008.4604424.

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7

Liang, Ying, Qingyang Xiu, and Yunfeng Bi. "Geomagnetic Anomaly Mapping by Multi-fractal Measure Kriging Interpolation Method." In the 2nd International Conference. New York, New York, USA: ACM Press, 2018. http://dx.doi.org/10.1145/3291801.3291824.

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Torre, Davide La, and Edward R. Vrscay. "Fractal-based measure approximation with entropy maximization and sparsity constraints." In BAYESIAN INFERENCE AND MAXIMUM ENTROPY METHODS IN SCIENCE AND ENGINEERING: 31st International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering. AIP, 2012. http://dx.doi.org/10.1063/1.3703621.

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Dhok, S. B., R. B. Deshmukh, and A. G. Keskar. "Fast Fractal Encoding through FFT using Modified Crosscorrelation based Similarity Measure." In Annual International Conference on Advances in Distributed and Parallel Computing ADPC 2010. Global Science and Technology Forum, 2010. http://dx.doi.org/10.5176/978-981-08-7656-2atai2010-60.

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Jayasuriya, Surani Anuradha, and Alan Wee-Chung Liew. "Fractal dimension as a symmetry measure in 3D brain MRI analysis." In 2012 International Conference on Machine Learning and Cybernetics (ICMLC). IEEE, 2012. http://dx.doi.org/10.1109/icmlc.2012.6359511.

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