Some Decision Problems for Extended Modular Groups

In this paper we investigate solvability of the word problem for Extended Modular groups, Extended Hecke groups and Picard groups in terms of complete rewriting systems. At the final part of the paper we examine the other important decision problem (conjugacy problem) for only Extended Modular groups

Conjugacy for Free Groups under Split Extensions

At the present paper we show that conjugacy is preserved and reflected by the natural homomorphism defined from a semigroup S to a group G, where G defines split extensions of some free groups. The main idea in the proofs is based on a geometrical structure as applied in the paper [8]

On the first Zagreb index and multiplicative Zagreb coindices of graphs

For a (molecular) graph G with vertex set V (G) and edge set E(G), the first Zagreb index of G is defined as M-1(G) = Sigma v(i is an element of V(G))d(C)(v(i))(2), where d(G) (v(i)) is the degree of vertex v(i), in G. Recently Xu et al. introduced two graphical invariants (Pi) over bar (1) (G) = Pi v(i)v(j is an element of E(G)) (dG (v(i))+dG (v(j))) and (Pi) over bar (2)(G) = Pi(vivj is an element of E(G)) (dG (v(i))+dG (v(j))) named as first multiplicative Zagreb coindex and second multiplicative Zagreb coindex, respectively. The Narumi-Katayama index of a graph G, denoted by NK(G), is equal to the product of the degrees of the vertices of G, that is, NK(G) = Pi(n)(i=1) d(G) (v(i)). The irregularity index t(G) of G is defined as the num=1 ber of distinct terms in the degree sequence of G. In this paper, we give some lower and upper bounds on the first Zagreb index M-1(G) of graphs and trees in terms of number of vertices, irregularity index, maximum degree, and characterize the extremal graphs. Moreover, we obtain some lower and upper bounds on the (first and second) multiplicative Zagreb coindices of graphs and characterize the extremal graphs. Finally, we present some relations between first Zagreb index and NarumiKatayama index, and (first and second) multiplicative Zagreb index and coindices of graphs.Korean Government - 2013R1A1A2009341Necmettin Erbakan ÜniversitesiSelçuk Üniversites

The next step of the word problem over monoids

It is known that a group presentation P can be regarded as a 2-complex with a single 0-cell. Thus we can consider a 3-complex with a single 0-cell which is known as a 3-presentation. Similarly, we can also consider 3-presentations for monoids. In this paper, by using spher- ical monoid pictures, we show that there exists a finite 3-monoid-presentation which has unsolvable ‘‘generalized identity problem’’ that can be thought as the next step (or one- dimension higher) of the word problem for monoids. We note that the method used in this paper has chemical and physical applications

Finite derivation type for graph products of monoids

Bu çalışma, 08-11, Ağustos 2014 tarihlerinde Gyeongju[Güney Kore]’de düzenlenen 22. International Conference on Finite and Infinite Dimensional Complex Analysis and Applications (ICFIDCAA) Kongresi‘nde bildiri olarak sunulmuştur.The aim of this paper is to show that the class of monoids of finite derivation type is closed under graph products.Balıkesir Üniversitesi - 2014/95, 2015/4

Archaeogenetic analysis of Neolithic sheep from Anatolia suggests a complex demographic history since domestication

Yurtman, ozer, Yuncu et al. provide an ancient DNA data set to demonstrate the impact of human activity on the demographic history of domestic sheep. The authors demonstrate that there may have been multiple domestication events with notable changes to the gene pool of European and Anatolian sheep since the Neolithic. Sheep were among the first domesticated animals, but their demographic history is little understood. Here we analyzed nuclear polymorphism and mitochondrial data (mtDNA) from ancient central and west Anatolian sheep dating from Epipaleolithic to late Neolithic, comparatively with modern-day breeds and central Asian Neolithic/Bronze Age sheep (OBI). Analyzing ancient nuclear data, we found that Anatolian Neolithic sheep (ANS) are genetically closest to present-day European breeds relative to Asian breeds, a conclusion supported by mtDNA haplogroup frequencies. In contrast, OBI showed higher genetic affinity to present-day Asian breeds. These results suggest that the east-west genetic structure observed in present-day breeds had already emerged by 6000 BCE, hinting at multiple sheep domestication episodes or early wild introgression in southwest Asia. Furthermore, we found that ANS are genetically distinct from all modern breeds. Our results suggest that European and Anatolian domestic sheep gene pools have been strongly remolded since the Neolithic

One dimension higher of the word problem

Just as a group presentation ${\mathcal P}$ can be regarded as a $2$-complex with a single $0$-cell, so we can consider a $3$-complex with a single $0$-cell, known as a $3$-presentaton. In this paper, by using a geometric way, called spherical pictures, we show that there exist a finite $3$-presentation which has unsolvable generalised identity problem which can be thought as one-dimension higher of the word problem