Abstract A progenitor developed by Robert T. Curtis is a type of infinite groups formed by the semi-direct product of a free group m∗n and a transitive permutation group of degree n. To produce finite homomorphic images we had to add relations to the progenitor of the form 2∗n : N. In this thesis we have investigated several permutations progenitors and monomials, 2∗12 : S4, 2∗12 : S4 × 2, 2∗13 : (13 : 4), 2∗30 : ((2• : 3) : 5), 2∗13 :13,2∗13 :(13:2),2∗13 :(13:S3),53∗2 :m (13:4),7∗8 :m (32 :8),and 53∗4 :m (13 : 4). We have discovered that the permutations progenitors produced the following finite homomorphic images, we have found P GL(2, 13), U3 (4) : 2, 2 × Sz (8), PSL(2,7), PGL(2,27), PSL(2,8), PSL(3,3), 4•S4(5), PSL2(53), and 13 : PGL2(53) as homomorphic images of this progenitors. We will construct double coset enumeration for the homomorphic images, 2 × Sz (8) over (13 : 4) Suzuki twisted group, P GL(2, 13) over S4,and PSL(2,7) over S4 and Maximal subgroups of 2×PGL(2,27) over 2•(13 : 2), P SL(2, 8) over (9 : 2), and P SL(3, 3) over (13 : 3). We will also give our techniques of finding finite homomorphic images and their isomorphism images.

In cryptography, periodic sequences with terms in F2 are used almost everywhere. These sequences should have large linear complexity to be cryptographically strong. In fact, the linear complexity of a sequence should be close to its period. In this thesis, we study the expected value for N-periodic sequences with terms in the finite field Fq. This study is entirely devoted to W. Meidl and Harald Niederreiter&rsquo / s paper which is &ldquo / On the Expected Value of the Linear Complexity and the k-Error Linear Complexity of Periodic Sequences&rdquo / We only expand this paper, there is no improvement. In this paper there are important theorems and results about the expected value of linear complexity of periodic sequences.

Dans la première partie de cette thèse, on présente des bornes supérieures fines pour le nombre de sous-variétés irréductibles de torsion maximales dans une sous-variété du tore complexe algébrique $(\mathbb{C}^{\times})^n$ et d'une variété abélienne. Dans les deux cas, on donne une borne explicite en termes du degré des polynômes définissants et la variété ambiante. De plus, la dépendance en le degré des polynômes est optimale. Dans le cas du tore complexe, on donne aussi une borne explicite en termes du degré torique de la sous-variété. En conséquence de ce dernier résultat, on démontre les conjectures de Ruppert, et Aliev et Smyth pour le nombre de points de torsion isolés dans une hypersurface. Ces conjectures bornent ce nombre en terme, respectivement, du multi-degré et du volume du polytope de Newton d'un polynôme définissant l'hypersurface.Dans la deuxième partie de cette thèse, on présente une borne supérieure pour la hauteur des zéros isolés, dans le tore, d'un système de polynômes de Laurent sur un corps adélique qui satisfait la formule du produit. Cette borne s'exprime en termes des intégrales mixtes des fonctions toit locales associées à la hauteur choisie et le système des polynômes de Laurent. On montre aussi que cette borne est presque optimale dans quelques familles d'exemples. Ce résultat est un analogue arithmétique du théorème de Bern\v{s}tein-Ku\v{s}nirenko. / In the first part of this thesis we present sharp bounds on the number of maximal torsion cosets in a subvariety of a complex algebraic torus $(\mathbb{C}^{\times})^n$ and of an Abelian variety. In both cases, we give an explicit bound in terms of the degree of the defining polynomials and the ambient variety. Moreover, the dependence on the degree of the polynomials is sharp. In the case of the complex torus, we also give an effective bound in terms of the toric degree of the subvariety. As a consequence of the latter result, we prove the conjectures of Ruppert, and Aliev and Smyth on the number of isolated torsion points of a hypersurface. These conjectures bound this number in terms of the multidegree and the volume of the Newton polytope of a polynomial defining the hypersurface, respectively.In the second part of the thesis, we present an upper bound for the height of isolated zeros, in the torus, of a system of Laurent polynomials over an adelic field satisfying the product formula. This upper bound is expressed in terms of the mixed integrals of the local roof functions associated to the chosen height function and to the system of Laurent polynomials. We also show that this bound is close to optimal in some families of examples. This result is an arithmetic analogue of the classical Bern\v{s}tein-Ku\v{s}nirenko theorem.

Made available in DSpace on 2014-07-29T16:02:23Z (GMT). No. of bitstreams: 1 silvio sandro.pdf: 761623 bytes, checksum: 55f280c9ca185766a1ed91423c5edfad (MD5) Previous issue date: 2010-06-28 / The group of weak commutativity for bijection G(H;K;σ) = {H;K|[h;hσ] = 1, for all h H} belongs is defined as the quotient of the free product H * K the normal closure of {[h;hσ] : h belongs to all H} in H * K. In this dissertation, we studied the results obtained in 2009 by Sidka and Oliveira [7] that support the following conjecture: If H,K ~= Zp X...X Zp, then G(H,K,σ)is a p-group. / O grupo de comutatividade fraca por bijeção G(H;K;σ) = {H;K|[h;hσ] = 1, para todo h pertence H} é definido como sendo o quociente do produto livre H * K pelo fecho normal de {[h;hσ] : para todo h pertence H} emH * K. Nessa dissertação, estudamos os resultados obtidos em 2009 por Oliveira e Sidki [7] que suportam a seguinte conjectura: Se H,K ~= Zp X...X Zp, então G(H,K,σ) é um p-grupo.

Employing an error control code is one of the techniques to reduce the Peak-to-Average Power Ratio (PAPR) in an Orthogonal Frequency Division Multiplexing system; a well known class of such codes being the cosets of Reed-Muller codes. In this thesis, classes of such coset-codes of arbitrary linear codes are considered. It has been proved that the size of such a code can be doubled with marginal/no increase in the PAPR. Conditions for employing this method iteratively have been enunciated. In fact this method has enabled to get the optimal coset-codes. The PAPR of the coset-codes of the extended codes is obtained from the PAPR of the corresponding coset-codes of the parent code. Utility of a special type of lengthening is established in PAPR studies