scholarly journals Fractal tube formulas for compact sets and relative fractal drums: oscillations, complex dimensions and fractality

2018 ◽  
Vol 5 (1) ◽  
pp. 1-119
Author(s):  
Michel Lapidus ◽  
Goran Radunović ◽  
Darko Žubrinić
Fractals ◽  
2010 ◽  
Vol 18 (03) ◽  
pp. 349-361 ◽  
Author(s):  
BÜNYAMIN DEMÍR ◽  
ALI DENÍZ ◽  
ŞAHIN KOÇAK ◽  
A. ERSIN ÜREYEN

Lapidus and Pearse proved recently an interesting formula about the volume of tubular neighborhoods of fractal sprays, including the self-similar fractals. We consider the graph-directed fractals in the sense of graph self-similarity of Mauldin-Williams within this framework of Lapidus-Pearse. Extending the notion of complex dimensions to the graph-directed fractals we compute the volumes of tubular neighborhoods of their associated tilings and give a simplified and pointwise proof of a version of Lapidus-Pearse formula, which can be applied to both self-similar and graph-directed fractals.


2018 ◽  
Vol 2 (4) ◽  
pp. 26 ◽  
Author(s):  
Michel Lapidus ◽  
Hùng Lũ’ ◽  
Machiel Frankenhuijsen

The theory of complex dimensions describes the oscillations in the geometry (spectra and dynamics) of fractal strings. Such geometric oscillations can be seen most clearly in the explicit volume formula for the tubular neighborhoods of a p-adic fractal string L p , expressed in terms of the underlying complex dimensions. The general fractal tube formula obtained in this paper is illustrated by several examples, including the nonarchimedean Cantor and Euler strings. Moreover, we show that the Minkowski dimension of a p-adic fractal string coincides with the abscissa of convergence of the geometric zeta function associated with the string, as well as with the asymptotic growth rate of the corresponding geometric counting function. The proof of this new result can be applied to both real and p-adic fractal strings and hence, yields a unifying explanation of a key result in the theory of complex dimensions for fractal strings, even in the archimedean (or real) case.


2010 ◽  
Vol 112 (1) ◽  
pp. 91-136 ◽  
Author(s):  
Michel L. Lapidus ◽  
Erin P. J. Pearse

2018 ◽  
Vol 10 (3) ◽  
pp. 304
Author(s):  
Familia Novita Simanjuntak

ABSTRACTSustainable development urges to merge the three complex dimensions: global economy, global society and physically earth environment. Sachs (2015) states that sustainable development is the expert effort to comprehend the world and the method to solve the crowded earth issues by the global population growth that nine times increase than the first industry era. Education is one of fatal element for sustainable development phase especially for the human (society) development. Human development becomes the main core of invesment for economy development because it is prepared for the youngst as the next generation to develop economic improvement individually for their family and also for their community (include for the State’s development interest). Marshall, Hine and East (2017) studied about the education which develop the autonomous motivation to support individu execute the pro-environmental behaviors (PEBs). This autonomous motivation will establish the environmental attitude and personality in decision making and action of sustainable environment protection and preservation.Keywords: sustainable development, education, pro-environmental behaviors ABSTRAKPembangunan berkelanjutan berupaya untuk mengkaitkan tiga sistem yang rumit yaitu sistem ekonomi dunia, sistem sosial dunia dan lingkungan fisik bumi. Sachs (2015) menyatakan bahwa pembangunan berkelanjutan menjadi cara para pakar untuk memahami dunia dan sebuah metode untuk menyelesaikan permasalahan dunia yang berawal dari sesaknya bumi akibat pertumbuhan penduduk dunia yang sudah mencapai sembilan kali lebih banyak dari populasi manusia yang hidup pada jaman dimulainya revolusi industri. Pendidikan adalah salah satu komponen yang penting dalam proses pembangunan berkelanjutan terutama pembangunan yang terkait manusia (sosial). Pembangunan manusia menjadi bagian vital dari investasi yang dibutuhkan dalam pembangunan ekonomi karena merupakan jalur investasi yang disiapkan untuk anak-anak sebagai generasi penerus yang akan melanjutkan perbaikan ekonomi baik secara individu bagi keluarganya maupun secara berkelompok bagi komunitasnya (termasuk kepentingan pembangunan di Negaranya). Penelitian Marshall, Hine and East (2017) menyatakan bahwa pendidikan dapat membentuk dorongan dari dalam setiap individu untuk melakukan perilaku pro lingkungan hidup (pro-environmental behaviours). Dorongan dari dalam individu ini secara otonomi membentuk watak dan karakter yang ramah lingkungan untuk membuat keputusan dan bertindak yang melindungi dan menjaga keberlanjutan lingkungan hidup.Kata kunci: pembangunan berkelanjutan, pendidikan, pro-environmental beharviors


Author(s):  
Ehud Hrushovski ◽  
François Loeser

This chapter introduces the concept of stable completion and provides a concrete representation of unit vector Mathematical Double-Struck Capital A superscript n in terms of spaces of semi-lattices, with particular emphasis on the frontier between the definable and the topological categories. It begins by constructing a topological embedding of unit vector Mathematical Double-Struck Capital A superscript n into the inverse limit of a system of spaces of semi-lattices L(Hsubscript d) endowed with the linear topology, where Hsubscript d are finite-dimensional vector spaces. The description is extended to the projective setting. The linear topology is then related to the one induced by the finite level morphism L(Hsubscript d). The chapter also considers the condition that if a definable set in L(Hsubscript d) is an intersection of relatively compact sets, then it is itself relatively compact.


2006 ◽  
Vol 23 (3) ◽  
pp. 98-100
Author(s):  
Muhamad Ali

Studies of Islam in Southeast Asia have sought to better understand its multifacetedand complex dimensions, although one may make a generalizedcategorization of Muslim beliefs and practices based on a fundamental differencein ideologies and strategies, such as cultural and political Islam.Anna M. Gade’s Perfection Makes Practice stresses the cultural aspect ofIndonesian Muslim practices by analyzing the practices of reciting andmemorizing the Qur’an, as well as the annual competition.Muslim engagement with the Qur’an has tended to emphasize the cognitiveover the psychological dimension. Perfection Makes Practice analyzesthe role of emotion in these undertakings through a combination ofapproaches, particularly the history of religions, ethnography, psychology,and anthropology. By investigating Qur’anic practitioners in Makassar,South Sulawesi, during the 1990s, Gade argues that the perfection of theQur’an as a perceived, learned, and performed text has made and remade thepractitioners, as well as other members of the Muslim community, to renewor increase their engagement with the holy text. In this process, she suggests,moods and motivation are crucial to preserving the recited Qur’an and revitalizingthe Muslim community.In chapter 1, Gade begins with a theoretical consideration for her casestudy. Drawing from concepts that emphasize the importance of feeling andemotion in ritual and religious experience, she develops a conceptualizationof this engagement. In chapter 2, Gade explains memorization within thecontext of the self and social relations. She argues that Qur’anic memorizershave a special relationship with its style and structure, as well as with thesocial milieu. Although Qur’anic memorization is a normal practice for mostMuslims, its practitioners have learned how to memorize and recite beautifullysome or all of the Qur’an’s verses, a process that requires emotion ...


1982 ◽  
Vol 8 (2) ◽  
pp. 455
Author(s):  
Akemann ◽  
Bruckner

2017 ◽  
Vol 4 (1) ◽  
pp. 43-72 ◽  
Author(s):  
Martin de Borbon

Abstract The goal of this article is to provide a construction and classification, in the case of two complex dimensions, of the possible tangent cones at points of limit spaces of non-collapsed sequences of Kähler-Einstein metrics with cone singularities. The proofs and constructions are completely elementary, nevertheless they have an intrinsic beauty. In a few words; tangent cones correspond to spherical metrics with cone singularities in the projective line by means of the Kähler quotient construction with respect to the S1-action generated by the Reeb vector field, except in the irregular case ℂβ₁×ℂβ₂ with β₂/ β₁ ∉ Q.


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