Dynamics of difference equation x n + 1 = f ( x n − l , x n − k ) $x_{n+1}=f( x_{n-l},x_{n-k})$

Dynamics of higher order rational difference equation $x_{n+1}=(alpha+beta x_{n})/(A + Bx_{n}+ Cx_{n-k})$

The main goal of this paper is to investigate the periodic character, invariant intervals, oscillation and global stability and other new results of all positive solutions of the equation$$x_{n+1}=frac{alpha+beta x_{n}}{A + Bx_{n}+ Cx_{n-k}},~~ n=0,1,2,ldots,$$where the parameters $alpha$, $beta$, $A$, $B$ and $C$ are positive, and the initial conditions $x_{-k},x_{-k+1},ldots,x_{-1},x_{0}$ are...

Polarization constant $mathcal{K}(n,X)=1$ for entire functions of exponential type

In this paper we will prove that if $L$ is a continuous symmetric n-linear form on a Hilbert space and $widehat{L}$ is the associated continuous n-homogeneous polynomial, then $||L||=||widehat{L}||$. For the proof we are using a classical generalized  inequality due to S. Bernstein for entire functions of exponential type. Furthermore we study the case that if X is a Banach space then we have t...

FROM K(n +1)∗(X) TO K(n)∗(X) NORIHIKO MINAMI

Let X be a space of finite type. Set q = 2(p − 1) as usual, and define the mod q support of K(n)∗(X) by S(X,K(n)) = {m ∈ Z/qZ | ⊕ d≡mmod q K(n) d 6= 0} for n > 0. Call K(n)∗(X) sparse if there is no m ∈ Z/qZ with m,m+ 1 ∈ S(X,K(n)). Then we show the relation S(X,K(n)) j S(X,K(n+1)) for any finite type space X with K(n+ 1)∗(X) being sparse. As a special case, we have K(n+ 1)odd(X) = 0 =⇒ K(n)odd...

On the Rational Recursive Sequence X_{N+1}=ƔX_{N-K}+(AX_N+BX_{N-K})⁄(CX_N-DX_{N-K})

Characterization of a new photorefractive material: K(l-y)L(y)T(1-x)N(x).

We report the growth and characterization of a new photorefractive material, potassium lithium tantalate niobate (KLTN). A KLTN crystal doped with copper is demonstrated to yield high diffraction efficiency of photorefractive gratings in the paraelectric phase. Voltage-controllable index gratings with n(1) = 8.5 x 10(-5) were achieved, which yielded diffraction efficiencies of 75% in a 2.9-mm-t...