This thesis develops ways of localizing the Ritz values of non-Hermitian matrices. The restarted Arnoldi method with exact shifts, useful for determining a few desired eigenvalues of a matrix, employs Ritz values to refine eigenvalue estimates. In the Hermitian case, using selected Ritz values produces convergence due to interlacing. No generalization of interlacing exists for non-Hermitian matrices, and as a consequence no satisfactory general convergence theory exists. To study Ritz values, I propose the inverse field of values problem for k Ritz values, which asks if a set of k complex numbers can be Ritz values of a matrix. This problem is always solvable for k = 1 for any complex number in the field of values; I provide an improved algorithm for finding a Ritz vector in this case. I show that majorization can be used to characterize, as well as localize, Ritz values. To illustrate the difficulties of characterizing Ritz values, this work provides a complete analysis of the Ritz values of two 3 × 3 matrices: a Jordan block and a normal matrix. By constructing conditions for localizing the Ritz values of a matrix with one simple, normal, sought-after eigenvalue, this work develops sufficient conditions that guarantee convergence of the restarted Arnoldi method with exact shifts. For general matrices, the conditions provide insight into the subspace dimensions that ensure that shifts do not cluster near the wanted eigenvalue. As Ritz values form the basis for many iterative methods for determining eigenvalues and solving linear systems, an understanding of Ritz value behavior for non-Hermitian matrices has the potential to inform a broad range of analysis.

Neste trabalho se apresenta o desenvolvimento de algoritmos hermitianos de integração das equações diferenciais do filtro digital de Butterworth mediante operadores de integração numérica de ordem elevada com passo único. A teoria do filtro de Butterworth é apresentada mediante o emprego da transformada de Fourier. Exemplos de aplicação apresentados através destes algoritmos mostram que os resultados são, como esperado, mais precisos que os resultantes dos métodos usuais presentes na literatura especializada / In this work is presented the development of hermitian algorithm for integration of the Butterworth digital filter’s differential equations by means of high order numerical one step operators. The Butterworth filter’s theory is presented based on the Fourier transform. Numerical examples show that the results of the developed hermitian algorithm are more accurate than the usual methods present in the specialized literature, as expected

Orientador: Fernando Eduardo Torres Orihuela / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-10T07:01:07Z (GMT). No. of bitstreams: 1 SepulvedaCastellanos_Alonso_D.pdf: 783003 bytes, checksum: 2af4bba938cd5b7d31fcd02a5c79ac85 (MD5) Previous issue date: 2008 / Resumo: Estudamos os códigos de Goppa (códigos GH) sobre certos corpos de funções algébricas com muitos lugares racionais. Estes códigos generalizam os bem conhecidos códigos Hermitianos; portanto podemos esperar que estes códigos tenham bons parâmetros. Bulygin (IEEE Trans. Inform. Theory 52 (10), 4664¿4669 (2006)) inicia o estudo dos códigos GH; enquanto Bulygin considerou somente característica par, nosso trabalho 'e feito em qualquer característica. Em qualquer caso, nosso trabalho é fortemente influenciado pelo de Bulygin. A seguir, listamos alguns dos nossos resultados com respeito aos códigos GH. ¿ Calculamos ¿distâncias mínimas exatas¿, em particular, melhoramos os resultados de Bulygin; ¿ Encontramos cotas para os pesos generalizados de Hamming, al'em disso, mostramos um algoritmo para aplicar estes cálculos na criptografia; ¿ Calculamos um subgrupo de Automorfismos; ¿ Consideramos códigos em determinados subcorpos dos corpos usados para construir os códigos GH / Abstract: We study Goppa codes (GH codes) based on certain algebraic function fields whose number of rational places is large. These codes generalize the well-known Hermitian codes; thus we might expect that they have good parameters. Bulygin (IEEE Trans. Inform. Theory 52 (10), 4664¿4669 (2006)) initiate the study of GH-codes; while he considered only the even characteristic, our work is done regardless the characteristic. In any case our work was strongly influenced by Bulygin¿s. Next we list some of the results of our work with respect to GH-codes. ¿ We calculate ¿true minimum distances¿, in particular, we improve Bulygin¿s results; ¿ We find bounds on the generalized Hamming weights, moreover, we show an algorithm to apply these computations to the cryptography; ¿ We calculate an Automorphism subgroup; ¿ We consider codes on certain subfields of the fields used for to construct GH-codes / Doutorado / Algebra (Geometria Algebrica) / Doutor em Matemática

<p>Open quantum systems have become a rapidly developing sector for research. Such systems present novel physical phenomena, such as topological chirality, enhanced sensitivity, and unidirectional invisibility resulting from both their non-equilibrium dynamics and the presence of exceptional points.</p> <p><br></p> <p>We begin by introducing the core features of open systems governed by non-Hermitian Hamiltonians, providing the PT -dimer as an illustrative example. Proceeding, we introduce the Lindblad master equation which provides a working description of decoherence in quantum systems, and investigate its properties through the Decohering Dimer and periodic potentials. We then detail our preferred experimental apparatus governed by the Lindbladian. Finally, we introduce the Liouvillian, its relation to non-Hermitian Hamiltonians and Lindbladians, and through it investigate multiple properties of open quantum systems.</p>

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Dans cette thèse nous nous intéressons aux flots de Monge-Ampère complexes, à leurs généralisations et à leurs applications géométriques sur les variétés hermitiennes compactes. Dans les deux premiers chapitres, nous prouvons qu'un flot de Monge-Ampère complexe sur une variété hermitienne compacte peut être exécuté à partir d'une condition initiale arbitraire avec un nombre Lelong nul en tous points. En utilisant cette propriété, nous con- firmons une conjecture de Tosatti-Weinkove: le flot de Chern-Ricci effectue une contraction chirurgicale canonique. Enfin, nous étudions une généralisation du flot de Chern-Ricci sur des variétés hermitiennes compactes, le flot de Chern-Ricci tordu. Cette partie a donné lieu à deux publications indépendantes. Dans le troisième chapitre, une notion de C -sous-solution parabolique est introduite pour les équations paraboliques, étendant la théorie des C -sous-solutions développée récem- ment par B. Guan et plus spécifiquement G. Székelyhidi pour les équations elliptiques. La théorie parabolique qui en résulte fournit une approche unifiée et pratique pour l'étude de nombreux flots géométriques. Il s'agit ici d'une collaboration avec Duong H. Phong (Université Columbia ) Dans le quatrième chapitre, une approche de viscosité est introduite pour le problème de Dirichlet associé aux équations complexes de type hessienne sur les domaines de Cn. Les arguments sont modélisés sur la théorie des solutions de viscosité pour les équations réelles de type hessienne développées par Trudinger. En conséquence, nous résolvons le problème de Dirichlet pour les équations de quotient de hessiennes et lagrangiennes spéciales. Nous établissons également des résultats de régularité de base pour les solutions. Il s'agit ici d'une collaboration avec Sl-awomir Dinew (Université Jagellonne) et Hoang-Son Do (Institut de Mathématiques de Hanoi). / In this thesis we study the complex Monge-Ampère flows, and their generalizations and geometric applications on compact Hermitian manifods. In the first two chapters, we prove that a general complex Monge-Ampère flow on a compact Hermitian manifold can be run from an arbitrary initial condition with zero Lelong number at all points. Using this property, we confirm a conjecture of Tosatti- Weinkove: the Chern-Ricci flow performs a canonical surgical contraction. Finally, we study a generalization of the Chern-Ricci flow on compact Hermitian manifolds, namely the twisted Chern-Ricci flow. This part gave rise to two independent publications. In the third chapter, a notion of parabolic C -subsolution is introduced for parabolic non-linear equations, extending the theory of C -subsolutions recently developed by B. Guan and more specifically G. Székelyhidi for elliptic equations. The resulting parabolic theory provides a convenient unified approach for the study of many geometric flows. This part is a joint work with Duong H. Phong (Columbia University) In the fourth chapter, a viscosity approach is introduced for the Dirichlet problem associated to complex Hessian type equations on domains in Cn. The arguments are modelled on the theory of viscosity solutions for real Hessian type equations developed by Trudinger. As consequence we solve the Dirichlet problem for the Hessian quotient and special Lagrangian equations. We also establish basic regularity results for the solutions. This part is a joint work with Sl-awomir Dinew (Jagiellonian University) and Hoang-Son Do (Hanoi Institute of Mathematics).

<p>This thesis deals with different aspects of string and /p/-brane theories. One of the motivations for string theory is to unify the forces in nature and produce a quantum theory of gravity. /p/-branes and related objects arise in string theory and are related to a non-perturbative definition of the theory. The results of this thesis might help in understanding string theory better. The first part of the thesis introduces and discusses relevant topics for the second part of the thesis which consists of five papers.</p><p>In the three first papers we develop and treat a perturbative approach to relativistic /p/-branes around stretched geometries. The unperturbed theory is described by a string- or particle-like theory. The theory is solved, within perturbation theory, by constructing successive canonical transformations which map the theory to the unperturbed one order by order. The result is used to define a quantum theory which requires for consistency d = 25 + p dimensions for the bosonic /p/-branes and d = 11 for the supermembrane. This is one of the first quantum results for extended objects beyond string theory and is a confirmation of the expectation of an eleven-dimensional quantum membrane.</p><p>The two last papers deal with a gauged WZNW-approach to strings moving on non-trivial space-times. The groups used in the formulation of these models are connected to Hermitian symmetric spaces of non-compact type. We have found that the GKO-construction does not yield a unitary spectrum. We will show that there exists, however, a different approach, the BRST approach, which gives unitarity under certain conditions. This is the first example of a difference between the GKO- and BRST construction. This is one of the first proofs of unitarity of a string theory in a non-trivial non-compact space-time. Furthermore, new critical string theories in dimensions less then 26 or 10 is found for the bosonic and supersymmetric string, respectively.</p>

Probabilistic simultaneous polynomial reconstruction algorithm of Bleichenbacher-Kiayias-Yung is extended to the polynomials whose degrees are allowed to be distinct. Furthermore, it is observed that probability of the algorithm can be increased. Specifically, for a finite field $F$, we present a probabilistic algorithm which can recover polynomials $p_1,ldots, p_r in F[x]$ of degree less than $k_1,k_2,ldots,k_r$, respectively with given field evaluations $p_l(z_i) = y_{i,l}$ for all $i in I$, $|I|=t$ and $l in [r]$ with probability at least $1 - (n - t)/|F|$ and with time complexity at most $O((nr)^3)$. Next, by using this algorithm, we present a probabilistic decoder for interleaved Reed-Solomon codes. It is observed that interleaved Reed-Solomon codes over $F$ with rate $R$ can be decoded up to burst error rate $frac{r}{r+1}(1 - R)$ probabilistically for an interleaving parameter $r$. It is proved that a Reed-Solomon code RS$(n / k)$ can be decoded up to error rate $frac{r}{r+1}(1 - R&#039 / )$ for $R&#039 / = frac{(k-1)(r+1)+2}{2n}$ when probabilistic interleaved Reed-Solomon decoders are applied. Similarly, for a finite field $F_{q^2}$, it is proved that $q$-folded Hermitian codes over $F_{q^{2q}}$ with rate $R$ can be decoded up to error rate $frac{q}{q+1}(1 - R)$ probabilistically. On the other hand, it is observed that interleaved codes whose subcodes would have different minimum distances can be list decodable up to radius of minimum of list decoding radiuses of subcodes. Specifically, we present a list decoding algorithm for $C$, which is interleaving of $C_1,ldots, C_b$ whose minimum distances would be different, decoding up to radius of minimum of list decoding radiuses of $C_1,ldots, C_b$ with list size polynomial in the maximum of list sizes of $C_1,ldots, C_b$ and with time complexity polynomial in list size of $C$ and $b$. Next, by using this list decoding algorithm for interleaved codes, we obtained new list decoding algorithm for $qh$-folded Hermitian codes for $q$ standing for field size the code defined and $h$ is any positive integer. The decoding algorithm list decodes $qh$-folded Hermitian codes for radius that is generally better than radius of Guruswami-Sudan algorithm, with time complexity and list size polynomial in list size of $h$-folded Reed-Solomon codes defined over $F_{q^2}$.

The analysis of similarity of matrices over fields, as well as integral domains which are not fields, is a classical problem in Linear Algebra and has received considerable attention. A related problem is that of simultaneous similarity of matrices. Many interesting algebraic questions that arise in such problems are discussed by Shmuel Friedland[1]. A special case of this problem is that of Simultaneous Unitary Similarity of hermitian matrices, which we describe as follows: Given a collection of m ordered pairs of similar n×n hermitian matrices denoted by {(Hl,Dl)}ml=1, 1. determine if there exists a unitary matrix U such that UHl U∗ = Dl for all l, 2. and in the case where a U exists, find such a U, (where U∗is the transpose conjugate of U ).The problem is easy for m =1. The problem is challenging for m > 1.The problem stated above is the algorithmic version of the problem of classifying hermitian matrices upto unitary similarity. Any problem involving classification of matrices up to similarity is considered to be “wild”[2]. The difficulty in solving the problem of classifying matrices up to unitary similarity is a indicator of, the toughness of problems involving matrices in unitary spaces [3](pg, 44-46 ).Suppose in the statement of the problem we replace the collection {(Hl,Dl)}ml=1, by a collection of m ordered pairs of complex square matrices denoted by {(Al,Bl) ml=1, then we get the Simultaneous Unitary Similarity problem for square matrices. Suppose we consider k ordered pairs of complex rectangular m ×n matrices denoted by {(Yl,Zl)}kl=1, then the Simultaneous Unitary Equivalence problem for rectangular matrices is the problem of finding whether there exists a m×m unitary matrix U and a n×n unitary matrix V such that UYlV ∗= Zl for all l and in the case they exist find them. In this thesis we describe algorithms to solve these problems. The Simultaneous Unitary Similarity problem for square matrices is challenging for even a single pair (m = 1) if the matrices involved i,e A1,B1 are not normal. In an expository article, Shapiro[4]describes the methods available to solve this problem by arriving at a canonical form. That is A1 or B1 is used to arrive at a canonical form and the matrices are unitarily similar if and only if the other matrix also leads to the same canonical form. In this thesis, in the second chapter we propose an iterative algorithm to solve the Simultaneous Unitary Similarity problem for hermitian matrices. In each iteration we either get a step closer to “the simple case” or end up solving the problem. The simple case which we describe in detail in the first chapter corresponds to finding whether there exists a diagonal unitary matrix U such that UHlU∗= Dl for all l. Solving this case involves defining “paths” made up of non-zero entries of Hl (or Dl). We use these paths to define an equivalence relation that partitions L = {1,…n}. Using these paths we associate scalars with each Hl(i,j) and Dl(i,j)denoted by pr(Hl(i,j)) and pr(Dl(i,j)) (pr is used to indicate that these scalars are obtained by considering products of non-zero elements along the paths from i,j to their class representative). Suppose i (I Є L)belongs to the class[d(i)](d(i) Є L) we denote by uisol a modulus one scalar expressed in terms of ud(i) using the path from i to d( i). The free variable ud(i) can be chosen to be any modulus one scalar. Let U sol be a diagonal unitary matrix given by U sol = diag(u1 sol , u2 sol , unsol ). We show that a diagonal U such that U HlU∗ = Dl exists if and only if pr(Hl(i, j)) = pr(Dl(i, j))for all l, i, j and UsolHlUsol∗= Dl. Solving the simple case sets the trend for solving the general case. In the general case in an iteration we are looking for a unitary U such that U = blk −diag(U1,…, Ur) where each Ui is a pi ×p (i, j Є L = {1,… , r}) unitary matrix such that U HlU ∗= Dl. Our aim in each iteration is to get at least a step closer to the simple case. Based on pi we partition the rows and columns of Hl and Dl to obtain pi ×pj sub-matrices denoted by Flij in Hl and Glij in D1. The aim is to diagonalize either Flij∗Flij Flij∗ and a get a step closer to the simple case. If square sub-matrices are multiples of unitary and rectangular sub-matrices are zeros we say that the collection is in Non-reductive-form and in this case we cannot get a step closer to the simple case. In Non- reductive-form just as in the simple case we define a relation on L using paths made up of these non-zero (multiples of unitary) sub-matrices. We have a partition of L. Using these paths we associate with Flij and (G1ij ) matrices denoted by pr(F1ij) and pr(G1ij) respectively where pr(F1ij) and pr(G1ij) are multiples of unitary. If there exist pr(Flij) which are not multiples of identity then we diagonalize these matrices and move a step closer to the simple case and the given collection is said to be in Reduction-form. If not, the collection is in Solution-form. In Solution-form we identify a unitary matrix U sol = blk −diag(U1sol , U2 sol , …, Ur sol )where U isol is a pi ×pi unitary matrix that is expressed in terms of Ud(i) by using the path from i to[d(i)]( i Є [d(i)], d(i) Є L, Ud(i) is free). We show that there exists U such that U HlU∗ = Dl if and only if pr((Flij) = pr(G1ij) and U solHlU sol∗ = Dl. Thus in a maximum of n steps the algorithm solves the Simultaneous Unitary Similarity problem for hermitian matrices. In the second chapter we also relate the Simultaneous Unitary Similarity problem for hermitian matrices to the simultaneous closed system evolution problem for quantum states. In the third chapter we describe algorithms to solve the Unitary Similarity problem for square matrices (single ordered pair) and the Simultaneous Unitary Equivalence problem for rectangular matrices. These problems are related to the Simultaneous Unitary Similarity problem for hermitian matrices. The algorithms described in this chapter are similar in flow to the algorithm described in the second chapter. This shows that it is the fact that we are looking for unitary similarity that makes these forms possible. The hermitian (or normal)nature of the matrices is of secondary importance. Non-reductive-form is the same as in the hermitian case. The definition of the paths changes a little. But once the paths are defined and the set L is partitioned the definitions of Reduction-form and Solution-form are similar to their counterparts in the hermitian case. In the fourth chapter we analyze the worst case complexity of the proposed algorithms. The main computation in all these algorithms is that of diagonalizing normal matrices, partitioning L and calculating the products pr((Flij) = pr(G1ij). Finding the partition of L is like partitioning an undirected graph in the square case and partitioning a bi-graph in the rectangular case. Also, in this chapter we demonstrate the working of the proposed algorithms by running through the steps of the algorithms for three examples. In the fifth and the final chapter we show that finding if a given collection of ordered pairs of normal matrices is Simultaneously Similar is same as finding if the collection is Simultaneously Unitarily Similar. We also discuss why an algorithm to solve the Simultaneous Similarity problem, along the lines of the algorithms we have discussed in this thesis, may not exist. (For equations pl refer the pdf file)

This thesis deals with different aspects of string and /p/-brane theories. One of the motivations for string theory is to unify the forces in nature and produce a quantum theory of gravity. /p/-branes and related objects arise in string theory and are related to a non-perturbative definition of the theory. The results of this thesis might help in understanding string theory better. The first part of the thesis introduces and discusses relevant topics for the second part of the thesis which consists of five papers. In the three first papers we develop and treat a perturbative approach to relativistic /p/-branes around stretched geometries. The unperturbed theory is described by a string- or particle-like theory. The theory is solved, within perturbation theory, by constructing successive canonical transformations which map the theory to the unperturbed one order by order. The result is used to define a quantum theory which requires for consistency d = 25 + p dimensions for the bosonic /p/-branes and d = 11 for the supermembrane. This is one of the first quantum results for extended objects beyond string theory and is a confirmation of the expectation of an eleven-dimensional quantum membrane. The two last papers deal with a gauged WZNW-approach to strings moving on non-trivial space-times. The groups used in the formulation of these models are connected to Hermitian symmetric spaces of non-compact type. We have found that the GKO-construction does not yield a unitary spectrum. We will show that there exists, however, a different approach, the BRST approach, which gives unitarity under certain conditions. This is the first example of a difference between the GKO- and BRST construction. This is one of the first proofs of unitarity of a string theory in a non-trivial non-compact space-time. Furthermore, new critical string theories in dimensions less then 26 or 10 is found for the bosonic and supersymmetric string, respectively.