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 .

It is an open question whether a non-trivial convex combination of triangular norms (resp. conorms) can be a triangular norm (resp. conorm). We investigate the analogous question for S-implications and R-implications.

An (n, q, r, s) book is a collection of r-subspaces in PG(n, q) called pages, which cover the whole projective space and intersect in a common s-subspace called the spine such that any point outside the spine is in exactly one page. An (n, q, r, s) book t-spread is a t-spread in PG(n, q) for which there exists an (n, q, r, s) book, such that the points of each page of this book and hence the po...

We develop the inverse scattering transform for the KdV equation with real singular initial data q (x) of the form q (x) = r′ (x) + r (x), where r ∈ Lloc, r|R+ = 0. As a consequence we show that the solution q (x, t) is a meromorphic function with no real poles for any t > 0.

The aim of the paper is to find efficient conditions for the unique solvability of the problem u′(t) = `(u)(t) + q(t), u(a) = h(u) + c, where ` : C([a, b];R) → L([a, b];R) and h : C([a, b];R) → R are linear bounded operators, q ∈ L([a, b];R), and c ∈ R.

An $(n, r)$-arc is a set of $n$ points of a projective plane such that some $r$, but no $r+1$ of them, are collinear. The maximum size of an $(n, r)$-arc in  $PG(2, q)$ is denoted by $m_r(2,q)$.  In this paper we present  a new $(184,12)$-arc in PG$(2,17),$  a new $(244,14)$-arc and a new $(267,15$)-arc in $PG(2,19).$