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]

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

Subgross and macroscopic investigation of the coeliac artery in the chinchilla (chinchilla lanigera)

The knowledge of branching and variations of the coeliac artery is clinicallyimportant, especially in the surgical operations and non-surgical treatments.Moreover, the chinchillas abdominal region have been used as a model in somesurgical experimental researches. In this frame, we have aimed to explain thebranching of this artery in the chinchillas detailedly. A total of 10 adult, healthy,male chinchillas (chinchilla lanigera) were used to investigate the origin and thecourse of the coeliac artery and its branches. Coloured latex was injected intothe carotid arteries, following conventional anatomical applications. The resultsindicated that the coeliac artery was divided into 4 branches such as left gastricartery, hepatic artery, splenic artery and gastrolienal artery. The left gastric arterywas a continuity of the coeliac artery and the main vessel of the stomach. Thehepatic artery was divided into the left lateral branch, the left medial branch andthe right branch. The splenic artery was covered by the pancreas tissue and sentbranches to the pancreas. The gastrolienal artery was supplying the fundus ofthe stomach and the dorsal extremity of the spleen. We believe that the findingswill be of help to the researchers interested in the anatomical area, surgeons andexperimental researches

A review of the ecology, palaeontology and distribution of atlantid heteropods (Caenogastropoda: Pterotracheoidea: Atlantidae)

Fewer than 1% of marine gastropod species live a holoplanktic life. Of these, the shelled heteropods of the family Atlantidae are among the most poorly understood. The atlantids potentially make up an important part of the ocean zooplankton, composing up to 69% of shelled holoplanktic gastropods in the Late Pleistocene to Recent fossil record. They are also likely to be at high risk from current and future global changes, including anthropogenic ocean acidification. However, due to their small size (<12 mm), difficulty of sampling and complicated morphology, we still lack key information about atlantid taxonomy and ecology. This makes it difficult to understand how important they are in the ocean foodweb and how they will be affected by environmental change. Although many studies have been carried out on the atlantids, these have generally been broad and unconnected. Here, we draw together this previous research, summarizing what is currently known about atlantid taxonomy, palaeontology, ecology and biogeography, and aiming to provide a foundation for future research on this group. The data indicate complex behaviours involving seasonal and vertical migration, and demonstrate extended geographical ranges, with implications for understanding the role of atlantids in the ocean foodweb and their sensitivity to environmental changes. This review highlights the urgent need for further taxonomic research on the atlantids, including molecular analysis, and for improved sampling techniques

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

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