Year-2024

On logical structures

2025-10-07T12:18:57Z

On logical structures Roy, Sayantan; Basu, Sankha S. (Advisor); Chakraborty, Mihir K. (Advisor) Universal logic, as described by Béziau, is a structural approach to logic which aims to unify the large kaleidoscopic variety of logics, while preserving their diversity. In order to do this, Béziau’s original proposal was to consider logics irrespective of how they are generated −beyond, e.g., the dichotomy of syntax/semantics − via logical structures (which, according to his initial proposal, are pairs of the form (L , |−) where L is a set and |− ⊆ P(L ) × L ).In this thesis, we investigate the fruitfulness of this idea in detail. The results we obtain can be divided into three classes. The first two chapters (after the introductory one) are concerned with the properties of some particular types of logical structures. The fourth one hovers around Suszko’s Thesis and many-valued logics. In this chapter, we provide a definition of many-valued logical structures taking into account the language/metalanguage hierarchy; a generalized version of Suszko’s Thesis is outlined as well. The fifth chapter deals with the question: is it possible to describe paraconsistency independently of the language or connectives? We show that the answer to this question is affirmative, and propose several syntax independent defnitions of paraconsistency. The sixth and the final one Is a contribution to proof theory. In this chapter, we investigate cut-elimination theorems in some detail. We provide two non-algorithmic proofs of the same for the propositional fragment of LK, followed by an addressal of the more general question of elimination of an arbitrary rule. This forces us to consider the notions of ‘rules’, ‘sequent systems’ and related notions in more detail. We generalise these concepts and manage to prove such‘ RULE-elimination theorems’ in these generalised settings. It is important to notice that in spite of the diversity of the areas to which these results are generally assumed to belong, the underlying unity rests entirely upon the very simple definition of logical structures. The establishment of this simple yet nontrivial fact, is perhaps one of the most important contributions of this thesis

Splitting subspaces, Krylov subspaces and polynomial matrices over finite fields

2024-10-10T22:00:15Z

Splitting subspaces, Krylov subspaces and polynomial matrices over finite fields Aggarwal, Divya; Ram, Samrith (Advisor) Let V be a vector space of dimension n over the finite field Fq and T be a linear operator on V . Given an integer m that divides n, an m-dimensional subspace W of V is T-splitting if V = W ⊕ TW ⊕ · · · ⊕ Td−1W where d = n/m. Let σ(m, d; T) denote the number of m-dimensional T-splitting subspaces. Determining σ(m, d; T) for an arbitrary operator T is an interesting problem. We prove that σ(m, d; T) depends only on the similarity class type of T and give an explicit formula in the special case where T is cyclic and nilpotent. Denote by σ(m, d; τ ) the number of m-dimensional splitting subspaces for a linear operator of similarity class type τ over an Fq-vector space of dimension md. For fixed values of m, d and τ , we show that σ(m, d; τ ) is a polynomial in q. This problem is closely related to another open problem on Krylov spaces. We discuss this connection and give explicit formulae for σ(m, d; T) in the case where the invariant factors of T satisfy certain degree conditions. A connection with another enumeration problem on polynomial matrices is also discussed. We finally present a brief review of some recent developments in the splitting subspaces problem. In particular, the connection of this problem to the theory of symmetric functions is highlighted. Lastly, we conclude with some future directions for research.

On the distribution and applications of Ramanujan sums

2024-09-02T22:00:23Z

On the distribution and applications of Ramanujan sums Goel, Shivani; Chaubey, Sneha (Advisor) While studying the trigonometric series expansion of certain arithmetic functions, Ramanujan, in 1918, defined a sum of the nth power of the primitive qth roots of unity and denoted it as cq(n). These sums are now known as Ramanujan sums. Since then, Ramanujan sums have been widely used and studied in mathematics and other areas. Most importantly, it is used in the proof of Vinogradov’s theorem that every sufficiently large odd number is the sum of three primes. It is also used to simplify the computations of Arithmetic Fourier Transform (AFT), Discrete Fourier Transform (DFT), and Discrete Cosine Transform (DCT) coefficients for a special type of signal. We study Ramanujan sums in the context of the k-tuple prime conjecture. A twin prime is a prime number that is either two less or two more than another prime number. It is conjectured that there are infinitely many twin primes. Hardy and Littlewood generalized the twin prime conjecture and gave the k-tuple conjecture. Let d1, · · · , dk be distinct integers, and b(p) is the number of distinct residue classes (mod p) represented by di . If b(p) < p for every prime p, the k-tuple conjecture gives an asymptotic formula for the number of n ≤ x such that all the k numbers n + di are primes. We study a heuristic proof of the k-tuple conjecture using the convolution of Ramanujan sums. Additionally, we study questions on the distribution of Ramanujan sums. One way to study distribution is via moments of averages. Chan and Kumchev studied the first and second moments of Ramanujan sums. In this thesis, we estimate the higher moment of their averages using the theory of functions of several variables initiated by Vaidyanathaswamy. Ramanujan sums can also be generalized over number fields. A number field is an extension field K of the field of rational numbers Q such that the field extension K/Q has a finite degree. Nowak first studied the first moment for Ramanujan sums over quadratic number fields, and later, it was estimated for the higher degree number fields as well. For a general number field, assuming generalized Lindelöf Hypothesis, we improve the first moment result and also study the second moment. Furthermore, unconditionally, we estimate asymptotic formulas for the second moment for quadratic, cubic, and cyclotomic number fields. Our primary tool for these results is a Perron-type formula. Finally, we obtain the second moment result for certain integral domains called Prüfer domains.

On some special classes of additive codes over finite fields

2024-06-10T22:00:21Z

On some special classes of additive codes over finite fields Sharma, Sandeep; Sharma, Anuradha (Advisor) In this thesis, we define and study a new class of additive codes over finite fields, viz. multi-twisted (MT) additive codes, which is a generalization of constacyclic additive codes and an extension of MT (linear) codes introduced by Aydin and Halilović [5]. We study their algebraic structures by writing a canonical form decomposition of these codes using the Chinese Remainder Theorem and provide an enumeration formula for these codes. With the help of their canonical form decomposition, we also provide a trace description for all MT additive codes over finite fields. We further apply probabilistic methods to study the asymptotic properties of the rates and relative Hamming distances of a special subclass of 1-generator MT additive codes. We show that there exists an asymptotically good infinite sequence of MT additive codes of length p↵` and block length p↵ ! 1 over Fqt with rate v p⌘`t and relative Hamming distance at least , where ` 1 and t 2 are integers, q is a prime power, Fqt is the finite field of order qt , p is an odd prime satisfying gcd(p, q)=1, v = ordp(q) is the multiplicative order of q modulo p, ⌘ is the largest positive integer such that p⌘ | (qv 1) and is a positive real number satisfying hqt () < 1 1 `t, (here hqt (·) denotes the qt -ary entropy function). This shows that the family of MT additive codes over finite fields is asymptotically good. As special cases, we deduce that the families of constacyclic and cyclic additive codes over finite fields are asymptotically good. By placing ordinary, Hermitian and ⇤ trace bilinear forms, we study the dual codes of MT additive codes over finite fields and derive necessary and sufficient conditions under which an MT additive code is (i) self-orthogonal, (ii) self-dual and (iii) an additive code with complementary dual (or an ACD code). We also derive a necessary and sufficient condition for the existence of a self-dual MT additive code over a finite field and provide enumeration formulae for all self-orthogonal, self-dual and ACD MT additive codes over finite fields with respect to the aforementioned trace bilinear forms. We further employ probabilistic methods and results from groups and geometry to study the asymptotic behavior of the rates and relative Hamming distances of self-orthogonal, self-dual and ACD MT additive codes over finite fields with respect to the aforementioned trace bilinear forms. We establish the existence of asymptotically good infinite sequences of self-orthogonal and ACD MT additive codes of length p↵` and block length p↵ ! 1 over Fqt with relative Hamming distance at least and rates v p⌘`t and 2v p⌘`t with respect to the aforementioned trace bilinear forms, where is a positive real number satisfying hqt () < 1 2 1 2`t, p is an odd prime coprime to q, v = ordp(q) is the multiplicative order of q modulo p, and ⌘ is the largest positive integer satisfying p⌘ | (qv 1). This shows that self-orthogonal and ACD MT additive codes over finite fields are asymptotically good. As special cases, we deduce that self-orthogonal and ACD constacyclic additive codes over finite fields are asymptotically good. We also establish the existence of asymptotically good infinite sequences of self-dual MT additive codes of length p↵` and block length p↵ ! 1 over Fqt with relative Hamming distance at least and rate 1 2 with respect to the aforementioned trace bilinear forms, where is a positive real number satisfying hqt () < 1 2`t, p is an odd prime coprime to q such that v = ordp(q) is even. As special cases, we deduce that self-dual cyclic and negacyclic additive codes over finite fields are asymptotically good. We also define and study a special class of self-dual MT additive codes over Fqt with respect to the ordinary trace bilinear form, viz. doubly-even self-dual MT additive codes and characterize a special class of these codes in terms of their constituents with the help of their trace description, where q is an even prime power. With the help of this characterization and using probabilistic methods and results from groups and geometry, we further study the asymptotic behaviour of their relative Hamming distances and show that doublyeven self-dual MT additive codes over F2t are asymptotically good. As a special case, we also deduce that doubly-even self-dual cyclic additive codes over F2t are asymptotically good.We next define and study a new class of additive codes over finite fields, viz. additive quasi-Abelian (QA) codes, which is a generalization of a special class of MT additive codes over finite fields and an extension of linear QA codes over finite fields. We study the algebraic structures of these codes and their dual codes with respect to ordinary, Hermitian and ⇤ trace bilinear forms. We further express these codes as direct sums of additive codes over finite fields and derive necessary and sufficient conditions under which an additive QA code is (i) self-orthogonal, (ii) self-dual and (iii) ACD. We also derive necessary and sufficient conditions for the existence of a self-dual additive QA code over finite fields. Besides this, we obtain explicit enumeration formulae for all self-orthogonal, self-dual and ACD additive QA codes over finite fields. We also list several MDS and almost MDS codes belonging to the family of additive QA codes over finite fields, which shows that additive QA codes over finite fields is a promising class of codes to find codes with good and optimal parameters.