Multiplier Hopf group coalgebras from algebraic and analytical point of views

The Multiplier Hopf Group Coalgebra was introduced by Hegazi in 2002 [7] as a generalization of Hope group caolgebra, introduced by Turaev in 2000 [5], in the non-unital case. We prove that the concepts introduced by A.Van Daele in constructing multiplier Hopf algebra \cite{4} can be adapted to serve again in our construction. A multiplier Hopf group coalgebra is a family of algebras $A=\{A_{\alpha}\}_{\alpha \in \pi}$, ($\pi$ is a discrete group) equipped with a family of homomorphisms $\Delta=\{\Delta_{\alpha,\beta}:A_{\alpha\beta}\longrightarrow M(A_{\alpha}\otimes A_{\beta})\}_{\alpha,\beta \in \pi}$ which is called a comultiplication under some conditions, where $M(A_{\alpha}\otimes A_{\beta})$ is the multiplier algebra of $A_{\alpha}\otimes A_{\beta}$. In 2003 A. Van Daele suggest a new approach to study the same structure by consider the direct sum of the algebras $A_p$'s which will be a multiplier Hopf algebra called later group cograded multiplier Hope algebra \cite{11}. And hence there exist a one to one correspondence between multiplier Hopf Group Coalgebra and group cograded multiplier Hopf algebra. By using this one-one correspondence we studied multiplier Hopf Group Coalgebra \

Construction of Nilpotent Jordan Algebras Over any Arbitrary Fields

We give a computional method to construct and classify nilpotent Jordan algebras over any arbitrary fields by the second cohomolgy of nilpotent Jordan algebras of low dimension "analogue of Skjelbred-Sund method", we see that every nilpotent Jordan algebras can be constructed by the second cohomolgy of nilpotent Jordan algebras of low dimension. We use this method to classify nilpotent Jordan algebras up to dimension three over any field and nilpotent Jordan algebras of dimension four over an algebraic closed field of characteristic not 2 and over the real field R. Also commutative nilpotent associative algebras are classified, we show that there are up to isomorphism 13 nilpotent Jordan algebras of dimension 4 over an algebraic closed field of characteristic not 2, four of those are not associative, yielding 9 commutative nilpotent associative algebras. Also up to isomorphism there are 17 nilpotent Jordan algebras of dimension 4 over the real field R, five of those are not associative, yielding 12 commutative nilpotent associative algebras

Survival May Not be for the Fittest (Lessons from some TV games)

In this paper we argue that biological fitness is a multi-objective concept hence the statement "fittest" is inappropriate. The following statement is proposed "Survival is mostly for those with non-dominated fitness". Also we use some TV games to show that under the following conditions: i) There are no dominant players. ii) At each time step successful players may eliminate some of their less successful competitors, Then the ultimate winner may not be the fittest (but close).Comment: non

Nilpotent evolution algebras over arbitrary fields

The paper is devoted to the study of annihilator extensions of evolution algebras and suggests an approach to classify finite-dimensional nilpotent evolution algebras. Subsequently nilpotent evolution algebras of dimension up to four are classfied.Comment: arXiv admin note: text overlap with arXiv:1312.4685 by other author

Five-dimensional nilpotent evolution algebras

The paper is devoted to give a complete classification of five-dimension nilpotent evolution algebras over an algebraically closed field. We obtained a list of 27 isolated non-isomorphic nilpotent evolution algebras and 2 families of non-isomorphic algebras depending on one parameter.Comment: arXiv admin note: text overlap with arXiv:1508.0686

An Overview of Complex Adaptive Systems

Almost every biological, economic and social system is a complex adaptive system (CAS). Mathematical and computer models are relevant to CAS. Some approaches to modeling CAS are given. Applications in vaccination and the immune system are studied. Mathematical topics motivated by CAS are discussed.Comment: 26 pages, accepted in Mansoura J. Mat

On Persistence and Stability of some Biological Systems with Cross Diffusion

The concept of cross diffusion is applied to some biological systems. The conditions for persistence and Turing instability in the presence of cross diffusion are derived. Many examples including: predator-prey, epidemics (with and without delay), hawk-dove-retaliate and prisoner's dilemma games are given.Comment: 15 pages, Accepted in Adv. Complex Sys

Differential calculus on Hopf Group Coalgebra

In this paper we construct the Differential calculus on the Hopf Group Coalgebra introduced by Turaev [10]. We proved that the concepts introduced by S.L.Woronowicz in constructing Differential calculus on Hopf Compact Matrix Pseudogroups (Quantum Groups)[7] can be adapted to serve again in our construction

Geometry and reducibility of induction of Hopf group co-algebras

In this work we study the induction (induced and coinduced)theory for Hopf group coalgebra. We define a substructure B of a Hopf group coalgebra $H$, called subHopf group coalgebra. Also, we have introduced the definition of Hopf group suboalgebra and group coisotropic quantum subgroup of $H$. The properties of the algebraic structure of the induced and coinduced are given. Moreover, a framework of the geometric interperation and simplicity theory of such representation strructure are stuided

Induced and Coinduced Representations of Hopf Group Coalgebra

In this work we study the induction theory for Hopf group coalgebra. To reach this goal we define a substructure B of a Hopf group coalgebra $H$, called subHopf group coalgebra. Also, we introduced the definition of Hopf group suboalgebra and group coisotropic quantum subgroup of $H$