scholarly journals Fractal Complex Dimensions, Riemann Hypothesis and Invertibility of the Spectral Operator

Author(s):  
Hafedh Herichi ◽  
Michel Lapidus
Author(s):  
Michel L. Lapidus

This research expository article not only contains a survey of earlier work but also contains a main new result, which we first describe. Given c ≥0, the spectral operator can be thought of intuitively as the operator which sends the geometry onto the spectrum of a fractal string of dimension not exceeding c . Rigorously, it turns out to coincide with a suitable quantization of the Riemann zeta function ζ = ζ ( s ): , where ∂=∂ c is the infinitesimal shift of the real line acting on the weighted Hilbert space . In this paper, we establish a new asymmetric criterion for the Riemann hypothesis (RH), expressed in terms of the invertibility of the spectral operator for all values of the dimension parameter (i.e. for all c in the left half of the critical interval (0,1)). This corresponds (conditionally) to a mathematical (and perhaps also, physical) ‘phase transition’ occurring in the midfractal case when . Both the universality and the non-universality of ζ = ζ ( s ) in the right (resp., left) critical strip (resp., ) play a key role in this context. These new results are presented here. We also briefly discuss earlier joint work on the complex dimensions of fractal strings, and we survey earlier related work of the author with Maier and with Herichi, respectively, in which were established symmetric criteria for the RH, expressed, respectively, in terms of a family of natural inverse spectral problems for fractal strings of Minkowski dimension D ∈(0,1), with , and of the quasi-invertibility of the family of spectral operators (with ).


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


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 ...


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.


Mathematika ◽  
2016 ◽  
Vol 63 (1) ◽  
pp. 29-33 ◽  
Author(s):  
Sandro Bettin ◽  
Steven M. Gonek
Keyword(s):  

Author(s):  
CARLO SANNA

Abstract Let $g \geq 2$ be an integer. A natural number is said to be a base-g Niven number if it is divisible by the sum of its base-g digits. Assuming Hooley’s Riemann hypothesis, we prove that the set of base-g Niven numbers is an additive basis, that is, there exists a positive integer $C_g$ such that every natural number is the sum of at most $C_g$ base-g Niven numbers.


Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1254
Author(s):  
Xue Han ◽  
Xiaofei Yan ◽  
Deyu Zhang

Let Pc(x)={p≤x|p,[pc]areprimes},c∈R+∖N and λsym2f(n) be the n-th Fourier coefficient associated with the symmetric square L-function L(s,sym2f). For any A>0, we prove that the mean value of λsym2f(n) over Pc(x) is ≪xlog−A−2x for almost all c∈ε,(5+3)/8−ε in the sense of Lebesgue measure. Furthermore, it holds for all c∈(0,1) under the Riemann Hypothesis. Furthermore, we obtain that asymptotic formula for λf2(n) over Pc(x) is ∑p,qprimep≤x,q=[pc]λf2(p)=xclog2x(1+o(1)), for almost all c∈ε,(5+3)/8−ε, where λf(n) is the normalized n-th Fourier coefficient associated with a holomorphic cusp form f for the full modular group.


Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 289 ◽  
Author(s):  
Matt Visser

The gap between what we can explicitly prove regarding the distribution of primes and what we suspect regarding the distribution of primes is enormous. It is (reasonably) well-known that the Riemann hypothesis is not sufficient to prove Andrica’s conjecture: ∀n≥1, is p n + 1 - p n ≤ 1 ? However, can one at least get tolerably close? I shall first show that with a logarithmic modification, provided one assumes the Riemann hypothesis, one has p n + 1 /ln p n + 1 - p n /ln p n < 11/25; (n ≥ 1). Then, by considering more general mth roots, again assuming the Riemann hypothesis, I show that p n + 1 m - p n m < 44/(25 e[m < 2]); (n ≥ 3; m > 2). In counterpoint, if we limit ourselves to what we can currently prove unconditionally, then the only explicit Andrica-like results seem to be variants on the relatively weak results below: ln2 pn + 1 - ln2 pn < 9; ln3 pn + 1 - ln3 pn < 52; ln4 pn + 1 - ln4 pn < 991; (n ≥ 1). I shall also update the region on which Andrica’s conjecture is unconditionally verified.


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