On Certain Algebraic Aspects of a Family of Orthogonal Polynomials

[Abstract not available

COMMUTATIVE ALGEBRA

The main aim of this project is to learn a branch of Mathematics that studies commutative rings with unity. The central notion in commutative algebra is that of prime ideal. This provides common generalization of primes of airthmetics and points of geometry. The geometric notion of concentrating attention near a point has as its algebraic analogue the important process localizing a ring at prime ideal, therefore result about lacalization can be thought in term of geometry

Poet: Product-oriented Video Captioner for E-commerce

In e-commerce, a growing number of user-generated videos are used for product promotion. How to generate video descriptions that narrate the user-preferred product characteristics depicted in the video is vital for successful promoting. Traditional video captioning methods, which focus on routinely describing what exists and happens in a video, are not amenable for product-oriented video captioning. To address this problem, we propose a product-oriented video captioner framework, abbreviated as Poet. Poet firstly represents the videos as product-oriented spatial-temporal graphs. Then, based on the aspects of the video-associated product, we perform knowledge-enhanced spatial-temporal inference on those graphs for capturing the dynamic change of fine-grained product-part characteristics. The knowledge leveraging module in Poet differs from the traditional design by performing knowledge filtering and dynamic memory modeling. We show that Poet achieves consistent performance improvement over previous methods concerning generation quality, product aspects capturing, and lexical diversity. Experiments are performed on two product-oriented video captioning datasets, buyer-generated fashion video dataset (BFVD) and fan-generated fashion video dataset (FFVD), collected from Mobile Taobao. We will release the desensitized datasets to promote further investigations on both video captioning and general video analysis problems.Comment: 10 pages, 3 figures, to appear in ACM MM 2020 proceeding

On A Conjecture of Pal Turan and Investigations Into Galois Groups of Generalized Laguerre Polynomials

In this dissertation we consider two problems. The first problem concerns a conjecture of Pal Turan on distance of a polynomial with integer coefficients from irreducible polynomial. Th problem remains open for polynomials with degree greater than 35. A. Schinzel, in 1970, reformulated Turan\u27s conjecture and subsequently proved the same. In the first part of this dissertation we give a refinement of Schinzel\u27s result. In the second half we investigate the Galois groups associated with the generalized Laguerre polynomials. We are able to classify Laguerre polynomials with the alternating group as the Galois group. We further compute the Galois groups in certain particular cases

On Galois groups of a one-parameter orthogonal family of polynomials

For a fixed integer t>1, we show that if t is not equal to 2, a square ≥4, or three times a square, then the discriminant of the generalized Laguerre polynomial L(s/t)n(x) is a nonzero square for at most finitely many pairs (n,s). Otherwise, the discriminant of L(s/t)n(x) is a nonzero square if and only if (n,s) belongs to one of seven explicitly describable infinite sets or to an additional finite set. This extends the results obtained for t=1 by P. Banerjee, M. Filaseta, C. Finch and J. Leidy. As a consequence, if α is a fixed rational number not equal to 1, 3, 5, or a negative integer, then for all but finitely many n, L(α)n(x) has Galois group Sn, thereby refining a previous result of M. Filaseta – T. Y. Lam and F. Hajir. As an illustration, we give for t=2 infinitely many integer specializations (n,s(n)) such that L(s(n)/2)n(x) has Galois group An. For n≤5, the set of rational numbers α for which the discriminant of L(α)n(x) is a nonzero square is explicitly computed by solving certain generalized Pell-like equations

On Galois groups of generalized Laguerre polynomials whose discriminants are squares

In this paper, we compute Galois groups over the rationals associated with generalized Laguerre polynomials Ln(∝)(x) whose discriminants are rational squares, where n and α are integers. An explicit description of the integer pairs ((n,∝)for which the discriminant of Ln(∝)(x) is a rational square was recently obtained by the author in a joint work with Filaseta, Finch and Leidy. Among these pairs ((n,∝), we show that for 2≤n≤5, the associated Galois group of Ln(∝)(x) is always

Classifying families of orthogonal polynomials having Galois group the alternating group

We give a classification of the generalized Laguerre polynomials Ln(α)(x), where α≠n is an integer, having the alternating group An as the Galois group over the rationals for up to a finitely many exceptions. © 2022 Elsevier Inc

Representation Theory of Finite Groups

When we have a finite group we can identify the elements of the group as matrices over some field, so we can study the properties of the groups by studying the matrices only.Representation theory is very useful in number theory and to solve many group theorytical problems. In this thesis I have tried to show some useful applications representation theory of finite groups

Cryptography

The methodology of concealing the content of messsages ,comes from the greeek root word Kryptos ,meaning hidden and graphikos, meaning writing . It is the practice and study of techniques for secure communication in the presence of third parties called adversaries(enemy) who prevent the users of cryptosystem from achieving their goal.The modern scienti�c study of cryptography is sometimes referred to as cryptology. SymmetricCipher A symmetric key cipher (also called a secret-key cipher, or a one-key cipher, or a private-key cipher, or a shared-key cipher) one that uses the same (necessarily secret) key to encrypt messages as it does to decrypt messages. This is the simplest kind of encryption that involves only one secret key to cipher and decipher information. Symmetrical encryption is an old and best-known technique. A�ne Cipher, Hill cipher,Substitution Cipher are examples of symmetric encryption. The main disadvantage of the symmetric key encryption is that all parties involved have to exchange the key used to encrypt the data before they can decrypt it. Asymmetric Cipher Asymmetric cryptography, also known as public key cryptography (PKC), uses public and private keys to encrypt and decrypt data respectively. The keys are simply large numbers that have been paired together but are not identical (asymmetric). One key in the pair can be shared with everyone; it is called the public key. The other key in the pair is kept secret; it is called the private key. Either of the keys can be used to encrypt a message; the opposite key from the one used to encrypt the message is used for decryption. In contrast to the symmetric ciphers, there are only three known asymetric cipher techniques- Di�e-Hellman Key exchange method, RSA Exchange Method and Digital Signatures