For given graphs G and F, the Turán number ex(G,F) is defined to be maximum of edges in an F-free subgraph G. Foucaud, Krivelevich Perarnau later independently Briggs Cox introduced a dual version this problem wherein for k, one maximizes host graph which ex(G,H)<k. Addressing Cox, we determine asymptotic value inverse paths length 4 5 provide improved lower bound all even length. Moreover, obt...

Let m, d and k be positive integers such that k ≤ me , where e = (m, d). Let p be an prime number and π a primitive element of Fpm . To each ~a = (a0, · · · , ak−1) ∈ F k pm , we associate the linearized polynomial f~a(x) = k−1 ∑ j=0 ajx p . And, to each f~a(x), we associated the sequence c~a = (f~a(1), f~a(π), · · · , f~a(π pm−2)). Let C = {c~a | ~a ∈ F k pm} be the cyclic code formed by the s...

Let k be an algebraically closed field of characteristic p > 0. The purpose of this paper is to describe the Hopf algebra of a finite commutative infinitesimal unipotent k-group scheme which is uniserial, i.e., which has a unique composition series. As there is only one simple finite commutative infinitesimal unipotent group scheme (namely αp := ker {F : Ga → Ga} , with Ga being the additive gr...

If g is a complex simple Lie algebra, and k does not exceed the dual Coxeter number of g, then the k coefficient of the dim g power of the Euler product may be given by the dimension of a subspace of ∧g defined by all abelian subalgebras of g of dimension k. This has implications for all the coefficients of all the powers of the Euler product. Involved in the main results are Dale Peterson’s 2 ...

In this work we examine noncommutativity of position coordinates in classical symplectic mechanics and its quantisation. In coordinates {q i , p k } the canonical symplectic two-form is ω 0 = dq i ∧ dp i. It is well known in symplectic mechanics [5, 6, 9] that the interaction of a charged particle with a magnetic field can be described in a Hamil-tonian formalism without a choice of a potential...

Let C be a geometrically irreducible curve over a field k, let k be an algebraic closure of k, and let m be any positive integer not divisible by the characteristic of k. The Jacobian variety J of C comes equipped with a principal p~ar ization A, which is in particular an isomorphism from J to its dual variety J . The polarization A gives us an isomorphism between the m-torsion J m of J and its...