Unprovability and phase transitions in Ramsey theory

The first mathematically interesting, first-order arithmetical example of incompleteness was given in the late seventies and is know as the Paris-Harrington principle. It is a strengthened form of the finite Ramsey theorem which can not be proved, nor refuted in Peano Arithmetic. In this dissertation we investigate several other unprovable statements of Ramseyan nature and determine the threshold functions for the related phase transitions. Chapter 1 sketches out the historical development of unprovability and phase transitions, and offers a little information on Ramsey theory. In addition, it introduces the necessary mathematical background by giving definitions and some useful lemmas. Chapter 2 deals with the pigeonhole principle, presumably the most well-known, finite instance of the Ramsey theorem. Although straightforward in itself, the principle gives rise to unprovable statements. We investigate the related phase transitions and determine the threshold functions. Chapter 3 explores a phase transition related to the so-called infinite subsequence principle, which is another instance of Ramsey’s theorem. Chapter 4 considers the Ramsey theorem without restrictions on the dimensions and colours. First, generalisations of results on partitioning α-large sets are proved, as they are needed later. Second, we show that an iteration of a finite version of the Ramsey theorem leads to unprovability. Chapter 5 investigates the template “thin implies Ramsey”, of which one of the theorems of Nash-Williams is an example. After proving a more universal instance, we study the strength of the original Nash-Williams theorem. We conclude this chapter by presenting an unprovable statement related to Schreier families. Chapter 6 is intended as a vast introduction to the Atlas of prefixed polynomial equations. We begin with the necessary definitions, present some specific members of the Atlas, discuss several issues and give technical details

Factors Influencing Customer Satisfaction towards E-shopping in Malaysia

Online shopping or e-shopping has changed the world of business and quite a few people have decided to work with these features. What their primary concerns precisely and the responses from the globalisation are the competency of incorporation while doing their businesses. E-shopping has also increased substantially in Malaysia in recent years. The rapid increase in the e-commerce industry in Malaysia has created the demand to emphasize on how to increase customer satisfaction while operating in the e-retailing environment. It is very important that customers are satisfied with the website, or else, they would not return. Therefore, a crucial fact to look into is that companies must ensure that their customers are satisfied with their purchases that are really essential from the ecommerce’s point of view. With is in mind, this study aimed at investigating customer satisfaction towards e-shopping in Malaysia. A total of 400 questionnaires were distributed among students randomly selected from various public and private universities located within Klang valley area. Total 369 questionnaires were returned, out of which 341 questionnaires were found usable for further analysis. Finally, SEM was employed to test the hypotheses. This study found that customer satisfaction towards e-shopping in Malaysia is to a great extent influenced by ease of use, trust, design of the website, online security and e-service quality. Finally, recommendations and future study direction is provided. Keywords: E-shopping, Customer satisfaction, Trust, Online security, E-service quality, Malaysia