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Showing 1 to 13 of 13 for “"roots of unity"”.
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Explicit Factorization of Generalized Cyclotomic Polynomials of Order $2^m 3$ Over a Finite Field $F_q$
We give explicit factorizations of $a$-cyclotomic polynomials of order $2^m 3$, $Q_{2^m3,a}(x)$, over a finite field $F_q$ with $q$ elements where $q$ is a prime power, $m$ is a nonnegative integer and $a$ is a nonnegative element of $F_q$. We use the relation between usual cyclotomic polynomials …
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On the Convergence and Divergence of Q-Continued Fractions on and Off the Unit Circle
… in the general sense at an uncountable set of points on the unit circle. We also show that the Rogers-Ramanujan continued fraction converges generally at all roots of unity (in contrast to classical convergence) and that it does not converge generally at any point outside the unit circle."
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Quadratic Reciprocity: Proofs and Applications
The law of quadratic reciprocity is an important result in number theory. The purpose of this thesis is to present several proofs as well as applications of the law of quadratic reciprocity. I will present three proofs of the quadratic reciprocity. We begin with a proof that depends on Gauss's …
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Semisimple filtrations of tilting modules for algebraic groups
… group over an algebraically closed field $k$ of characteristic $p > 0$. The indecomposable tilting modules $\{T(\lambda)\}$ for $G$, which are labeled by highest weight, form an important class of self-dual representations over $k$. In this thesis we investigate semisimple filtrations of …
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Infinite Series Identities in the Theory of Elliptic Functions and Q-Series
… In Chapter 2, We extend C. L. Siegel's method of proving the Dedekind-eta function transformation by integrating some selected functions over a positively oriented polygon, generalizing Siegel's integration over a parallelogram. As consequences, we obtain a generalization of the Dedekind-eta …
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Steinitz Classes of Tamely Ramified Nonabelian Extensions of Algebraic Number Fields of Degree P(3)
Let L/k be a finite extension of algebraic number fields of degree n with rings of integers ${\cal D}\sb{L}$ and ${\cal D}\sb{k}$. As an ${\cal D}\sb{k}$-module ${\cal D}\sb{L}$ is finitely generated and torsion free and thus can be written as a direct sum of $n - 1$ copies of ${\cal D}\sb{k}$ and …
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Constructing and Tabulating Dihedral Function Fields
… odd prime degree l dihedral extensions of a rational function field k_0(x), where k_0 is a perfect field with characteristic not dividing 2l. We begin with a class field theoretic construction algorithm when k0 is a finite field. We also describe modifications to this algorithm to …
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Product Identities for Theta Functions
… that if Zn can be taken as the disjoint union of a lattice generated by n linearly independent vectors in Z n and a finite number of its translates, certain products of theta functions can be written as linear combinations of other products of theta functions. Many known identities, including …
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Error-Correcting Codes Associated With Generalized Hadamard Matrices Over Groups
… these Hadamard matrices. Indeed, the success of early Mars deep-space probes was strongly dependent upon this communication technology.</p> <p>The concept of Hadamard matrices with elements drawn from an Abelian group is a natural generalization of the concept. For the case in which the …
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On Multiplication Groups of Quasigroups
… approach. We first construct a family of quasigroups which behave in a group-like fashion. We then focus on the multiplication groups of quasigroups, which have first appeared in the work of A. A. Albert. These permutation groups allow us to study quasigroups using group theory. We also …
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On the main conjectures of Iwasawa theory for certain elliptic curves with complex multiplication
The conjecture of Birch and Swinnerton-Dyer is unquestionably one of the most important open problems in number theory today. Let $E$ be an elliptic curve defined over an imaginary quadratic field $K$ contained in $\mathbb{C}$, and suppose that $E$ has complex multiplication by the ring of integers …
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Computational analysis of thermo-acoustic instabilities in combustion chambers and afterburners
… effects. Gas turbine designs typically consist of a periodic assembly of N identical units; as evidenced by documented studies, the coupling across sectors may give rise to unstable modes, which are the highlight of this study. In the proposed model, the governing equations are linearized in the …
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ΠΡΩΤΟΙ ΑΡΙΘΜΟΙ ΚΑΙ ΚΥΚΛΟΤΟΜΙΑ-ΠΡΩΤΟΙ ΑΡΙΘΜΟΙ ΤΗΣ ΜΟΡΦΗΣ Χ + (Χ+1)
THE PRESENT DOCTORAL THESIS CONSISTS OF THREE CHAPTERS, NAMELY: CHAPTER 1: CRITERIA AND FORMULAE FOR PRIME NUMBERS. CHAPTER 2: PRIMES AND CYCLOTOMY. CHAPTER3: PRIME AND COMPOSITE NUMBERS OF THE FORM X2+(X+1)2. IN CHAPTER 1 NEW PRIMALITY CRITERIA ARE OBTAINED, ALONG WITH NEW FORMULAE FOR THE …