Sharif Digital Repository

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 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 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... 

In this thesis we introduce Tropical Geometry and It’s Application in Arithmetic Geometry  

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... 

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... 

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... 

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  

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  

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,... 

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... 

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,...