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About Space Time
, M.Sc. Thesis Sharif University of Technology ; Rastegar, Arash (Supervisor)
Abstract
Einstein's general theory of relativity is an admirable successful and unifier geometrical modeling among concepts of space, time and gravity. This theory along with special theory of relativity made a massive change in our physical view and this, especially in its historical context, has deep and interesting consequences in man's philosophical viewpoint which can be studied. Some parts of this thesis relates to these consequences. Some of these interesting consequences may be hidden by computional or mathematical viewpoint and often these equations do not contain enough intuition. in one part, we provide a formulizatin of Einstein's equaion that is intuitional which can be translated in...
Tropical Algebraic Geometry
, M.Sc. Thesis Sharif University of Technology ; Rastegar, Arash (Supervisor)
Abstract
Tropical algebraic geometry is a fairly new branch in geometry which is called so in honor of Brazilian mathematician Imre Simon who was pioneer of this field.The set equipped with the addition and multiplication is a semifield.Algebraic geometry objects like algebraic varieties can be defined over.Polynomials and rational functions are defined over.The functions that they define are piecewise linear and concave functions and the set of points where they are nonlinear is a tropical variety which is a concave polyhedral. Thus, tropical algebraic geometry is a piecewise linear version of algebraic geometry.Another approach to tropical algebraic geometry comes back to the works of Russian...
A^1-Homotopy Theory
, M.Sc. Thesis Sharif University of Technology ; Rastegar, Arash (Supervisor)
Abstract
A^1-homotopy theory or the motivic homotopy theory is a homotopy theory for smooth schemes of finite type over a Noetherian base scheme of finite dimension. A^1-homotopy was introduced by Vladimir Voevodsky and Fabien Morel in the 90s. The fundamental idea of this theory is that for schemes the affine line A^1 should play the role of the interval I = [0,1]. A^1-homotopy has been proved to be an important theory. Some familiar cohomology theories like motivic cohomology and algebraic K-theory are representable in this theory. Motivic cohomology and A^1-homotopy theory also appear in Voevodsky’s proof of Milnor’s conjecture and Block-Kato conjecture. In this thesis,we will give an introduction...
A Tropical Approach to Arithmetic Geometry
, M.Sc. Thesis Sharif University of Technology ; Rastegar, Arash (Supervisor)
Abstract
In this thesis we introduce Tropical Geometry and It’s Application in Arithmetic Geometry
An Introduction to Rigid Analytic Geometry
, M.Sc. Thesis Sharif University of Technology ; Rastegar, Arash (Supervisor)
Abstract
Rigid analytic geometry was developed by John Tate in 1971. Although it seems that a rigid analytic spaces on non-Archimedean field F, is like something called F-analytic manifold; however in general these two are distinct concepts.
In this thesis; we introduce rigid analytic spaces. It begins in chapter one by non-Archimedean fields. In this chapter also contains theorem 1-3-7; which there is no proof for it without usage of methods of this chapter; up to the present. Topics discussed in this chapter are interesting subjects of other applica¬tions of non-Archimedean fields. There is a visualization of the field Q_p; and an appendix includes some definitions and consequences of ordered...
In this thesis; we introduce rigid analytic spaces. It begins in chapter one by non-Archimedean fields. In this chapter also contains theorem 1-3-7; which there is no proof for it without usage of methods of this chapter; up to the present. Topics discussed in this chapter are interesting subjects of other applica¬tions of non-Archimedean fields. There is a visualization of the field Q_p; and an appendix includes some definitions and consequences of ordered...
Kolyvagin’s Theorem
, M.Sc. Thesis Sharif University of Technology ; Rastegar, Arash (Supervisor)
Abstract
Kolyvagin’s theorem, stated in the late 1980s, is a very important theorem in the theory of elliptic curves, which, together with Gross-Zagier’s work, results in a special case of the Birch and Swinerton-Dyer conjecture, which relates algebraic and analytic properties of the rational elliptic curves to each other. Although the correctness of this conjecture, in general, is an open problem, Kolyvagin’s idea and the concepts he introduced while proving his theorem are very powerful and can be applied to more general situations. The purpose of writing this dissertation is to collect the equivalent forms of Kolyvagin’s theorem with proofs of the main results. We will also give a brief overview...
Average Rank of Elliptic Curves Over Q
, M.Sc. Thesis Sharif University of Technology ; Rastegar, Arash (Supervisor) ; Rajaei, Ali (Co-Advisor)
Abstract
In elliptic curves equation, if the number of solution is infinite, complication of calculation increase quickly. Even the simplest solutioncan be large. y2 + y = x35115523309x − 140826120488927 The x-coordinate number of smallest solution has 5454 digits. The theorem of Mordell and Weil states that “The set E(Q) of rational solutions has the structure of an infinitely generated abelian group.” E(Q) = (Z)rankE(Q) ⊕T Can the rank be arbitrary large? The current record is rank(E)=28.Manjul Behargava has recently made progress on the study of the average rank for all elliptic curves with rational coefficient. Every such curve has a unique equation of the form y2 = x3 +Ax+B, where A and B are...
Exaⅿpⅼes of Topoⅼogyⅽaⅼ Operaⅾs anⅾ Grassⅿⅿan Operaⅾs
, M.Sc. Thesis Sharif University of Technology ; Jafari, Aⅿir (Supervisor) ; Rastegar, Arash (Co-Supervisor)
Abstract
The projeⅽt first explores the ⅽonⅽept of operaⅾs anⅾ the basiⅽ ⅾefinitions of the theory suⅽh as the ⅾifferent ⅾefini− tions of operaⅾs ، anⅾ then expⅼores the types of exaⅿpⅼes of topoⅼogiⅽaⅼ operaⅾs anⅾ their properties anⅾ exaⅿines speⅽifiⅽaⅼⅼy the ⅽonstruⅽtion of the Grassⅿann props by us− ing the aⅼgebraiⅽ geoⅿetry tooⅼ anⅾ finally we ⅽoⅿpute the ⅽohoⅿoⅼogy group of Grassⅿann props by using the aⅼgebraiⅽ geoⅿetry tooⅼ anⅾ aⅼgebraiⅽ topology tool
Galois Representation and Modular Forms
, M.Sc. Thesis Sharif University of Technology ; Rastegar, Arash (Supervisor) ; Rajaei, Ali (Co-Supervisor)
Abstract
In this thesis, we studied the representation of Galois groups and modular forms.Also, we presented the Serre’s conjecture which has been proven in two papers published in 2005 and 2008. Moreover, the main aim of this text is presenting the Fontaine-Mazur and Serre’s conjecture
Various Versions of the Sato-Tate Conjecture
, M.Sc. Thesis Sharif University of Technology ; Rastegar, Arash (Supervisor) ; Gholamzadeh Mahmoudi, Mohammad (Supervisor)
Abstract
The Sato-Tate conjecture is an important conjecture regarding the distribution of the Frobenius traces of a family of elliptic curves over finite fields obtained from the reductions of an elliptic curve without CM over a number field modulo the prime ideals of its ring of integers. The statement is that the sequence of normalized Frobenius traces should follow a semicircle distribution. It was discovered by Mikio Sato and reformulated by John Tate in terms of L-functions around 1960. A complete proof of the conjecture for elliptic curves over totally real fields was published in 2008 by R. Taylor et al. under some mild technical assumptions. In addition to the original Sato-Tate conjecture,...
Existence of Arithmetic Progressions in Subsets of Natural Numbers
, M.Sc. Thesis Sharif University of Technology ; Rastegar, Arash (Supervisor) ; Hatami Varzaneh, Omid (Supervisor)
Abstract
Szemeredi's theorem is one of the significant theorems in additive combinatorics which was started by Van Der Waerden's theorem in 1927. Erdos and Turan conjectured generalized versions of Van Der Waerden's theorem in several ways included Szemeredi's theorem. In 1975 Szemeredi proved the conjecture using complicated combinatorial methods. In 1977 H. Furstenberg proved Szemeredi's theorem via the Ergodic theory approach which led to prove polynomial Szemeredi's theorem and multi-dimensional Szemeredi's theorem. The Ergodic approach is the only known approach so far to these generalizations of this theorem which is named Ergodic Ramsey theory and led to some other problems in Ergodic theory...
Hegel's ontology of power : the structure of social domination in capitalism
, Book
Cambridge University Press
2020
Deformation of outer representations of galois group II
, Article Iranian Journal of Mathematical Sciences and Informatics ; Volume 6, Issue 2 , 2011 , Pages 33-41 ; 17354463 (ISSN) ; Sharif University of Technology
2011
Abstract
This paper is devoted to deformation theory of "anabelian" representations of the absolute Galois group landing in outer automorphism group of the algebraic fundamental group of a hyperbolic smooth curve defined over a number-field. In the first part of this paper, we obtained several universal deformations for Lie-algebra versions of the above representation using the Schlessinger criteria for functors on Artin local rings. In the second part, we use a version of Schlessinger criteria for functors on the Artinian category of nilpotent Lie algebras which is formulated by Pridham, and explore arithmetic applications
Deformation of outer representations of Galois group
, Article Iranian Journal of Mathematical Sciences and Informatics ; Volume 6, Issue 1 , 2011 , Pages 35-52 ; 17354463 (ISSN) ; Sharif University of Technology
2011
Abstract
To a hyperbolic smooth curve defined over a number-field one naturally associates "ananabelian" representation of the absolute Galois group of the base field landing in outer automorphism group of the algebraic fundamental group. In this paper, we introduce several deformation problems for Lie-algebra versions of the above representation and show that, this way we get a richer structure than those coming from deformations of "abelian" Galois representations induced by the Tate module of associated Jacobian variety. We develop an arithmetic deformation theory of graded Lie algebras with finite dimensional graded components to serve our purpose
Arithmetic Teichmuller theory
, Article Iranian Journal of Mathematical Sciences and Informatics ; Volume 14, Issue 2 , 2019 , Pages 157-171 ; 17354463 (ISSN) ; Sharif University of Technology
Iranian Academic Center for Education, Culture and Research
2019
Abstract
By Grothendieck’s anabelian conjectures, Galois representations landing in outer automorphism group of the algebraic fundamental group which are associated to hyperbolic smooth curves defined over number-fields encode all the arithmetic information of these curves. The Goal of this paper is to develop an arithmetic Teichmuller theory, by which we mean, introducing arithmetic objects summarizing the arithmetic information coming from all curves of the same topological type defined over number-fields. We also introduce Hecke-Teichmuller Lie algebra which plays the role of Hecke algebra in the anabelian framework. © 2019 Academic Center for Education, Culture and Research TMU
Arithmetic teichmuller theory
, Article Iranian Journal of Mathematical Sciences and Informatics ; Volume 14, Issue 2 , 2019 , Pages 157-171 ; 17354463 (ISSN) ; Sharif University of Technology
Iranian Academic Center for Education, Culture and Research
2019
Abstract
By Grothendieck’s anabelian conjectures, Galois representations landing in outer automorphism group of the algebraic fundamental group which are associated to hyperbolic smooth curves defined over number-fields encode all the arithmetic information of these curves. The Goal of this paper is to develop an arithmetic Teichmuller theory, by which we mean, introducing arithmetic objects summarizing the arithmetic information coming from all curves of the same topological type defined over number-fields. We also introduce Hecke-Teichmuller Lie algebra which plays the role of Hecke algebra in the anabelian framework. © 2019 Academic Center for Education, Culture and Research TMU
Ihara-Type results for siegel modular forms
, Article Bulletin of the Iranian Mathematical Society ; Volume 46, Issue 3 , 2020 , Pages 693-716 ; Sharif University of Technology
Springer
2020
Abstract
Let p be a prime not dividing the integer n. By an Ihara result, we mean existence of a cokernel torsion-free injection from a full lattice in the space of p-old modular forms into a full lattice in the space of all modular forms of level pn. In this paper, we will prove an Ihara result in the number field case, for Siegel modular forms. The case of elliptic modular forms is discussed in Ihara (Discrete subgroups of Lie groups and applications to moduli, Oxford University Press, Bombay, 1975). We will use a geometric formulation for the notion of p-old Siegel modular forms (Rastegar in BIMS 43(7):1–23, 2017). Then, we apply an argument by Pappas, and prove the Ihara result using density of...
On a theorem of Ihara
, Article Scientia Iranica ; Volume 12, Issue 1 , 2005 , Pages 1-9 ; 10263098 (ISSN) ; Sharif University of Technology
Sharif University of Technology
2005
Abstract
Let p be a prime number and let n be a positive integer prime to p. By an Ihara-result, one means the existence of an injection with torsion-free cokernel, from a full lattice, in the space of p-old modular forms, into a full lattice, in the space of all modular forms of level np. In this paper, Ihara-results are proven for genus two Siegel modular forms, Siegel-Jacobi forms and Hilbert modular forms. Ihara did the genus one case of elliptic modular forms [1]. A geometric formulation is proposed for the notion of p-old Siegel modular forms of genus two, using clarifying comments by R. Schmidt [2] and, then, following suggestions in an earlier paper [3] on how to prove Ihara results. The main...
Investigating the impacts of plug-in hybrid electric vehicles on power distribution systems
, Article IEEE Transactions on Smart Grid ; Volume 4, Issue 3 , 2013 , Pages 1351-1360 ; 19493053 (ISSN) ; Fotuhi Firuzabad, M ; Rastegar, M
2013
Abstract
Despite the economic and environmental advantages of plug-in hybrid electric vehicles (PHEVs), the increased utilization of PHEVs brings up new concerns for power distribution system decision makers. Impacts of PHEVs on distribution networks, although have been proven to be noticeable, have not been thoroughly investigated for future years. In this paper, a comprehensive model is proposed to study the PHEV impacts on residential distribution systems. In so doing, PHEV fundamental characteristics, i.e., PHEV battery capacity, PHEV state of charge (SOC), and PHEV energy consumption in daily trips, are accurately modeled. As some of these effective characteristics depend on vehicle owner's...
Synthesis and Characterization of Electrospun Ceramic Nanofibers
, M.Sc. Thesis Sharif University of Technology ; Bagheri, Habib (Supervisor)
Abstract
Electrospinning is a simple and versatile technique for producing polymeric and ceramic nanofibers. The conventional procedure for fabrication of ceramic nanofibers is a combination of sol-gel techniques and electrospinning. The main challenge is the spin polymer nanofiber in fabricate of fibers with diameter from 1 to 100 nm, while their standard deviation are as low as possible. Uniform beads free polyamidenanofibers with lower diameter were fabricated.A 18% w/w of polyamide in formic acid was chosen and the effect of magnetic field, adding magnetic ionic liquid surfactant, auxiliary electrode on the main fiber were investigated. They termsfiber 1 to fiber 5. The mean diameters of 489,...