Counting Number Fields by Discriminant

The central topic of this dissertation is counting number fields ordered by discriminant. We fix a base field k and let Nd(k,G;X) be the number of extensions N/k up to isomorphism with Nk/Q(dN/k) ≤ X, [N : k] = d and the Galois closure of N/k is equal to G. We establish two main results in this work. In the first result we establish upper bounds for N|G| (k,G;X) in the case that G is a finite group with an abelian normal subgroup. Further, we establish upper bounds for the case N |F| (k,G;X) where G is a Frobenius group with an abelian Frobenius kernel F. In the second result we establish is an asymptotic expression for N6(Q;A4;X). We show that N6(Q,A4;X) = CX1/2 + O(X0.426...) and indicate what is expecedted under the `-torsion conjecture and the Lindelöf Hypothesis. We begin this work by stating the results that are established here precisely, and giving a historical overview of the problem of counting number fields. In Chapter 2, we establish background material in the areas of ramification of prime numbers and analytic number theory. In Chapter 3, we establish the asymptotic result for N6(Q,A4;X). In Chapter 4, we establish upper bounds for Nd(k,G;X) for groups with a normal abelian subgroup and for Frobenius groups. Finally we conclude with Chapter 5 with certain extensions of the method. In particular, we indicate how to count extensions of different degrees and discuss how to use tools about average results on the size of the torsion of the class group on almost all extensions in a certain family

A Survey of Word Reordering Model in Statistical Machine Translation

Machine translation is the process of translating one natural language in to another natural language by computers. In statistical machine translation word reordering is a big challenge between distant language pair. It is important factor for its quality and efficiency. Word reordering is major challenge For Indian languages who have big structural difference like English and Hindi language. This paper present description about statistical machine translation, reordering model and reordering types

Convexifying Transformers: Improving optimization and understanding of transformer networks

Understanding the fundamental mechanism behind the success of transformer networks is still an open problem in the deep learning literature. Although their remarkable performance has been mostly attributed to the self-attention mechanism, the literature still lacks a solid analysis of these networks and interpretation of the functions learned by them. To this end, we study the training problem of attention/transformer networks and introduce a novel convex analytic approach to improve the understanding and optimization of these networks. Particularly, we first introduce a convex alternative to the self-attention mechanism and reformulate the regularized training problem of transformer networks with our alternative convex attention. Then, we cast the reformulation as a convex optimization problem that is interpretable and easier to optimize. Moreover, as a byproduct of our convex analysis, we reveal an implicit regularization mechanism, which promotes sparsity across tokens. Therefore, we not only improve the optimization of attention/transformer networks but also provide a solid theoretical understanding of the functions learned by them. We also demonstrate the effectiveness of our theory through several numerical experiments

Mechanic: A Learning Rate Tuner

We introduce a technique for tuning the learning rate scale factor of any base optimization algorithm and schedule automatically, which we call \textsc{mechanic}. Our method provides a practical realization of recent theoretical reductions for accomplishing a similar goal in online convex optimization. We rigorously evaluate \textsc{mechanic} on a range of large scale deep learning tasks with varying batch sizes, schedules, and base optimization algorithms. These experiments demonstrate that depending on the problem, \textsc{mechanic} either comes very close to, matches or even improves upon manual tuning of learning rates