Graham Higman has defined coset diagrams for PSL(2,ℤ). These diagrams are composed of fragments, and the fragments are further composed of two or more circuits. Q. Mushtaq has proved in 1983 that existence of a certain fragment γ of a coset diagram in a coset diagram is a polynomial f in ℤ[z]. Higman has conjectured that, the polynomials related to the fragments are monic and for a fixed degree...

In this paper we find all solutions of four kinds of the Diophantine equations begin{equation*} ~x^{2}pm V_{t}xy-y^{2}pm x=0text{ and}~x^{2}pm V_{t}xy-y^{2}pm y=0, end{equation*}% for an odd number $t$, and, begin{equation*} ~x^{2}pm V_{t}xy+y^{2}-x=0text{ and}text{ }x^{2}pm V_{t}xy+y^{2}-y=0, end{equation*}% for an even number $t$, where $V_{n}$ is a generalized Lucas number. This pape...

We consider the 2D J 1 −J 2 classical XY model on a square lattice. In the frustrated phase corresponding to J 2 > J 1 /2, an Ising order parameter emerges by an " order due to disorder " effect. This leads to a discrete symmetry plus the O(2) global one. We formulate the problem in a Coulomb gas language and show by a renormalization group analysis that only two phases are still possible: a lo...

Equation Basepoint Parameter Dimension Restrictions of Isotropy #1 W = XY + Z (0, 0, 0, 0) 4 #2 W 2 = XY + Z + 1 (1, 0, 0, 0) 3 #3 W = XY + Z +X (0, 0, 0, 0) 2 #4 W = XY + Z +XZ + αX (0, 0, 0, 0) α arbitrary 1 #5 W = XY + Z +XZ (0, 0, 0, 0) 1 #6 W 2 = XY +XY +XZ (1, 1, 0, 1) 1 #7 W = XY + Z (1, 0, 0, 1) α 6= 0, 1, 2 1 #8 W = XY + e (1, 0, 0, 0) 1 #9 W = XY + logZ (0, 0, 0, 1) 1 #10 W = XY + Z l...

graham higman has defined coset diagrams for psl(2,ℤ). these diagrams are composed of fragments, and the fragments are further composed of two or more circuits. q. mushtaq has proved in 1983 that existence of a certain fragment γ of a coset diagram in a coset diagram is a polynomial f in ℤ[z]. higman has conjectured that, the polynomials related to the fragments are monic and for a fixed degree...

This study aimed to evaluate the potential of commercial enzymes cellulase (CE), xylanase (XY) and the combination of CE + XY to improve rumen fermentation of Guinea grass ecotype ‘A’ (Panicum maximum). The experiment was carried out in a randomized complete block design. In vitro incubations were performed with four doses of individual enzymes and their combinations (1:1 from each enzyme) as 5...

‎Let $R$ be a ring with involution $*$‎. ‎An additive mapping $T:Rto R$ is called a left(respectively right) centralizer if $T(xy)=T(x)y$ (respectively $T(xy)=xT(y)$) for all $x,yin R$‎. ‎The purpose of this paper is to examine the commutativity of prime rings with involution satisfying certain identities involving left centralizers.

In [9], the author introduced a digital version of this theorem where the exponents appearing in (1) are viewed as sums of digits. To illustrate this, consider the binomial theorem for N = 2: (x + y) = xy + xy + xy + xy. (2) It is easy to verify that (2) is equivalent to (x+ y) = xy + xy + xy + xy, (3) where s(k) denotes the sum of digits of k expressed in binary. For example, s(3) = s(1 · 2 + ...