No Arabic abstract
Note that the family of closed curves C_N={(x,y)in R^2;x^(2N)+y^(2N)=1} for N=1,2,3,... approaches the boundary of [-1,1]^2 as N to infty. In this paper we exhibit a natural parameterization of these curves and generalize to a larger class of equations.
The polynomial $f_{2n}(x)=1+x+cdots+x^{2n}$ and its minimizer on the real line $x_{2n}=operatorname{arg,inf} f_{2n}(x)$ for $ninBbb N$ are studied. Results show that $x_{2n}$ exists, is unique, corresponds to $partial_x f_{2n}(x)=0$, and resides on the interval $[-1,-1/2]$ for all $n$. It is further shown that $inf f_{2n}(x)=(1+2n)/(1+2n(1-x_{2n}))$ and $inf f_{2n}(x)in[1/2,3/4]$ for all $n$ with an exact solution for $x_{2n}$ given in the form of a finite sum of hypergeometric functions of unity argument. Perturbation theory is applied to generate rapidly converging and asymptotically exact approximations to $x_{2n}$. Numerical studies are carried out to show how many terms of the perturbation expansion for $x_{2n}$ are needed to obtain suitably accurate approximations to the exact value.
The treated matrix equation $(1+ae^{-frac{|X|}{b}})X=Y$ in this short note has its origin in a modelling approach to describe the nonlinear time-dependent mechanical behaviour of rubber. We classify the solvability of $(1+ae^{-frac{|X|}{b}})X=Y$ in general normed spaces $(E,|cdot|)$ w.r.t. the parameters $a,binmathbb{R}$, $b eq 0$, and give an algorithm to numerically compute its solutions in $E=mathbb{R}^{mtimes n}$, $m,ninmathbb{N}$, $m,ngeq 2$, equipped with the Frobenius norm.
We study ternary sequences associated with a multidimensional continued fraction algorithm introduced by the first author. The algorithm is defined by two matrices and we show that it is measurably isomorphic to the shift on the set ${1,2}^mathbb{N}$ of directive sequences. For a given set $mathcal{C}$ of two substitutions, we show that there exists a $mathcal{C}$-adic sequence for every vector of letter frequencies or, equivalently, for every directive sequence. We show that their factor complexity is at most $2n+1$ and is $2n+1$ if and only if the letter frequencies are rationally independent if and only if the $mathcal{C}$-adic representation is primitive. It turns out that in this case, the sequences are dendric. We also prove that $mu$-almost every $mathcal{C}$-adic sequence is balanced, where $mu$ is any shift-invariant ergodic Borel probability measure on ${1,2}^mathbb{N}$ giving a positive measure to the cylinder $[12121212]$. We also prove that the second Lyapunov exponent of the matrix cocycle associated with the measure $mu$ is negative.
A ${00,01,10,11}$-valued function on the vertices of the $n$-cube is called a $t$-resilient $(n,2)$-function if it has the same number of $00$s, $01$s, $10$s and $11$s among the vertices of every subcube of dimension $t$. The Friedman and Fon-Der-Flaass bounds on the correlation immunity order say that such a function must satisfy $tle 2n/3-1$; moreover, the $(2n/3-1)$-resilient $(n,2)$-functions correspond to the equitable partitions of the $n$-cube with the quotient matrix $[[0,r,r,r],[r,0,r,r],[r,r,0,r],[r,r,r,0]]$, $r=n/3$. We suggest constructions of such functions and corresponding partitions, show connections with Latin hypercubes and binary $1$-perfect codes, characterize the non-full-rank and the reducible functions from the considered class, and discuss the possibility to make a complete characterization of the class.
We show that Bose-Einstein correlations of identical particles in hadron and nucleus high energy collisions, lead to long range rapidity correlations in the azimuthal angle. These correlations are inherent features of the CGC/saturation approach, however, their origin is more general than this approach. In framework of the proposed technique both even and odd $v_n$ occur naturally, independent of the type of target and projectile. We are of the opinion that it is premature to conclude that the appearance of azimuthal correlations are due to the hydrodynamical behaviour of the quark-gluon plasma.