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We study homogeneous nucleation in the two-dimensional $q-$state Potts model for $q=3,5,10,20$ and ferromagnetic couplings $J_{ij} \propto \Theta (R - |i-j|)$, by means of Monte Carlo simulations employing heat bath dynamics. Metastability is induced low temperature phase through an instantaneous quench magnetic field coupled to one $q$ spin states. The depth adjusted, depending on value $T$, i...

Proof. Argue by contradiction that T is not a tree, then there must exist a node v 2 T such that v has more than one parent in T . This means that there exist at least two paths R, p1, . . . , v and R, q1, . . . , v that connect the root of S(T ), which we denote by R, and v. Let t be the last node in R, p1, . . . , v and R, q1, . . . , v such that R, . . . , t are common prefix of these two pa...

For simplicity, we adopt the following convention: n is a natural number, p, q, u, w are points of En T, S is a subset of En T, A, B are convex subsets of En T, and r is a real number. Next we state several propositions: (1) (1− r) · p+ r · q = p+ r · (q − p). (2) If u, w ∈ halfline(p, q) and |u− p| = |w − p|, then u = w. (3) Let given S. Suppose p ∈ S and p 6= q and S ∩ halfline(p, q) is Bound...

By using averaging functions, several new oscillation criteria are established for the half-linear damped differential equation-2-2-2-2 r(t) (x) +p(t) |x| x,r(t) (x) +q(t)|x| x=0, dx dx dx dx dt dt dt dt α α α α ψ ϕ ψ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ and the more general equation-2-2 r(t) (x) +p(t) g(x),r(t) (x) +q(t)g(x)=0, d dx dx dx dx dt dt dt dt dt α α ψ ϕ ψ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ...

We study the decay of I (t) = R u(:; t) where u is a nonnegative solution to u t ? u + jruj q = 0 in R n R + with n 1. If 1 q n+2 n+1 then I (t) ! 0 if I (0) < 1 and inf t>0 I (t) > 0 if q > n+2 n+1 .