Corner diffusion is shown to play a crucial role in determining the asymptotic mound coarsening exponent n in the case of unstable epitaxial growth on (001) and (111) surfaces. For the case of island-relaxation without corner diffusion the asymptotic exponent is found to satisfy n.1/4. However, when rapid cornerdiffusion is allowed, the coarsening exponent is found to approach 1/3. An explanati...

We consider the nonlinear half-Laplacian heat equation $$\begin{aligned} u_t+(-\Delta )^{\frac{1}{2}} u-|u|^{p-1}u=0,\quad {\mathbb {R}}^n\times (0,T). \end{aligned}$$We prove that all blows-up are type I, provided \(n \le 4\) and \( 1<p<p_{*} (n)\) where p_{*} is an explicit exponent which below \(\frac{n+1}{n-1}\), critical Sobolev exponent. Central to our proof a Giga-Kohn monotonicity formu...

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we prove that for $p>7$ there are‎ ‎[‎ ‎p^{4}+2p^{3}+20p^{2}+147p+(3p+29)gcd (p-1,3)+5gcd (p-1,4)+1246‎ ‎] ‎groups of order $p^{8}$ with exponent $p$‎. ‎if $p$ is a group of order $p^{8}$‎ ‎and exponent $p$‎, ‎and if $p$ has class $c>1$ then $p$ is a descendant of $‎p/gamma _{c}(p)$‎. ‎for each group of exponent $p$ with order less than $‎p^{8} $ we calculate the number of descendants of o...

Let X1, . . . , Xm denote smooth projective curves of genus gi ≥ 2 over an algebraically closed field of characteristic 0 and let n denote any integer at least equal to 1 + max i=1 gi. We show that the product JX1 × · · · × JXm of the corresponding Jacobian varieties admits the structure of a Prym-Tyurin variety of exponent n. This exponent is considerably smaller than the exponent of the struc...

Given knowledge of one or more of the primes in a multiprime RSA modulus we show that the private exponent can be recovered provided it is sufficiently small. In particular, we present a simple and efficient method that given v of the u primes dividing the modulus N recovers any private exponent d satisfying d < Nv/u− . When only one prime is known, this bound can be increased to approximately ...

We study the asymptotic behaviour of resistance scaling and fluctuation of resistance that give rise to flicker noise in an n-simplex lattice. We propose a simple method to calculate the resistance scaling and give a closed-form formula to calculate the exponent, βL, associated with resistance scaling, for any n. Using current cumulant method we calculate the exact noise exponent for n-simplex ...