On a conjecture of Pomerance

We say that k is a P-integer if the first phi(k) primes coprime to k form a reduced residue system modulo k. In 1980 Pomerance proved the finiteness of the set of P-integers and conjectured that 30 is the largest P-integer. We prove the conjecture assuming the Riemann Hypothesis. We further prove that there is no P-integer between 30 and 10^11 and none above 10^3500.Comment: 10 pages. Submitted to Acta Arithmetic

Coefficient Bounds and Fekete-Szeg¨o inequality for a Certain Families of Bi-Prestarlike Functions Defined by (M,N)-Lucas Polynomials

In the current work, we use the (M,N)-Lucas Polynomials to introduce a new families of holomorphic and bi-Prestarlike functions defined in the unit disk O and establish upper bounds for the second and third coefficients of the Taylor-Maclaurin series expansions of functions belonging to these families. Also, we debate Fekete-Szeg¨o problem for thesefamilies. Further, we point out several certain special cases for our results

Geometrical Theory of Analytic Functions

The book contains papers published in the Mathematics Special Issue, entitled "Geometrical Theory of Analytic Functions". Fifteen papers devoted to the study concerning complex-valued functions of one variable present new outcomes related to special classes of univalent functions, differential equations in view of geometric function theory, quantum calculus and its applications in geometric function theory, operators and special functions associated with differential subordination and superordination theories and starlikeness, and convexity criteria

Some topics in set theory

This thesis is divided into two parts. In the first of these we consider Ackermann-type set theories and many of our results concern natural models. We prove a number of results about the existence of natural models of Ackermann's set theory, A, and applications of this work are shown to answer several questions raised by Reinhardt in [56]. A+ (introduced in [56]) is another Ackermann-type set theory and we show that its set theoretic part is precisely ZF. Then we introduce the notion of natural models of A + and show how our results on natural models of A extend to these models. There are a number of results about other Ackermann-type set theories and some of the work which was already known for ZF is extended to A. This includes permutation models, which are shown to answer another of Reinhardt's questions. In the second part we consider the different approaches to set theory; dealing mainly with the more philosophical aspects. We reconsider Cantor's work, suggest that it has frequently been misunderstood and indicate how quasi-constructive set theories seem to use a definite part of Cantor's earlier ideas. Other approaches to set theory are also considered and criticised. The section on NF includes some more technical observations on ordered pairs. There is also an appendix, in which we outline some results on extended ordinal arithmetic.<p

A barrier to be broken - change and continuity in the transition between Bronze and Iron Age Aegean. From the observation of burial contexts and grave goods

This work discusses change and continuities taking place in the transition from the Late Bronze to the Early Iron Ages in Greece. The geographical range will cover the eastern mainland (including Euboea), Naxos and Knossos, in a period starting from the final palatial culture of LH IIIB2/13th century and ending with the Proto-Geometric/10th century burial evidence. In order to collect and observe the archaeological evidence several tombs assemblages have been researched from both original reports and visits to relevant Greek museums. Finds have been tabulated, and continuities, innovations and losses have been identified. The major categories of material evidence analysed included pottery, metalwork and jewellery but also the form of the tombs and the manner of the burials were considered. The final analysis of these categories of evidence refutes theories of major and or abrupt change, whether caused by invasion or natural phenomena. It rather indicates social modifications following the loss of the palatial centres and their administration and culminating in their gradual replacement by new forms of social structure. Although not directly demonstrable from the existing evidence, a possible scenario is proposed to explain the frequent indications of influence from SE and Central Europe during this transition