Accurate phylogenetic reconstruction methods are inherently computationally heavy and therefore are limited to relatively small numbers of taxa. Supertree construction is the task of amalgamating small trees over partial sets into a big tree over the complete taxa set. The need for fast and accurate supertree methods has become crucial due to the enormous number of new genomic sequences generat...

We study the uniqueness and expansion properties of the positive solution of the logistic equation ∆u + au = b(x)f(u) in a smooth bounded domain Ω, subject to the singular boundary condition u = +∞ on ∂Ω. The absorption term f is a positive function satisfying the Keller–Osserman condition and such that the mapping f(u)/u is increasing on (0,+∞). We assume that b is non-negative, while the valu...

Let $ A be a closed linear operator defined on complex Banach space X $. We show novel representation, using strongly continuous families of bounded operators \mathbb{N}_0 $, for the unique solution following time-stepping scheme \begin{document}$ \begin{eqnarray*} (*)\left\{ \begin{array}{rcl} \,_C \nabla^{\alpha} u^n& = &Au^{n}+f^{n}, \quad n\geq 2;\\ u^0& &u_0; \\ u^1& &a...

In this paper, we shall be concerned with the existence result of Unilateral problem associated to the equations of the form, Au+ g(x, u,∇u) = f, where A is a Leray-Lions operator from its domain D(A) ⊂ W 1 0 LM (Ω) into WEM (Ω). On the nonlinear lower order term g(x, u,∇u), we assume that it is a Carathéodory function having natural growth with respect to |∇u|, and satisfies the sign condition...

is considered. Stationary traveling waves of above equation are proved to exist when f(u) is symmetric and g(u) is antisymmetric about u = 0. Solutions of initial value problems tends to almost piecewise constant functions within O(1) time. The almost constant pieces are separated by sharp interior layers, called fronts. The motion of these fronts are studied by asymptotic expansion. The equati...

We consider the non-linear impulsive evolution equation u′(t) = Au(t) + f(t, u(t), Tu(t), Su(t)), 0 < t < T0, t 6= ti, u(0) = u0, ∆u(ti) = Ii(u(ti)), i = 1, 2, 3, . . . , p. in a Banach space X, where A is the infinitesimal generator of a C0 semigroup. We study the existence and uniqueness of the mild solutions of the evolution equation by using semigroup theory and then show that the mild solu...

We present QNet, a method for constructing split networks from weighted quartet trees. QNet can be viewed as a quartet analogue of the distance-based Neighbor-Net (NNet) method for network construction. Just as NNet, QNet works by agglomeratively computing a collection of circular weighted splits of the taxa set which is subsequently represented by a planar split network. To illustrate the appl...

The system under consideration is −∆u+ auu = u 3 − βuv, u = u(x), −∆v + avv = v 3 − βuv, v = v(x), x ∈ R, u||x|→∞ = v||x|→∞ = 0, where au, av and β are positive constants. We prove the existence of a componentwise positive smooth radially symmetric solution of this system. This result is a part of the results presented in the recent paper [1]; in our opinion, our method allows one to treat the ...

Under general hypotheses, we prove the existence of a nontrivial solution for the equation Lu = N(u), where u belongs to a Hilbert space H , L is an invertible continuous selfadjoint operator, and N is superlinear. We are particularly interested in the case where L is strongly indefinite and N is not compact. We apply the result to the Choquard-Pekar equation -Au(x)+p(x)u(x) = u(x) f U <'y\ dy,...

Assume that A is a bounded self-adjoint operator in a Hilbert space H. Then, the variational principle max v |(Au, v)|2 (Av, v) = (Au, u) (*) holds if and only if A ≥ 0, that is, if (Av, v) ≥ 0 for all v ∈ H. We define the quotient on the left-hand side in (*) to be zero if (Av, v) = 0. As an application of this principle a variational principle for the electrical capacitance of a conductor of ...