Keywords: Sturm–Liouville Spectral parameter Taylor monomials Time scale a b s t r a c t We will derive formulas for finding two linearly independent solutions of the Sturm–Liouville dynamic equation. We will give several examples. In particular, the q-difference equation which has important applications in quantum theory will be presented. In this paper we obtain a spectral parameter power ser...

In this paper, we prove that Crouzeix-Raviart finite elements of polynomial order p ≥ 5 , odd, are inf-sup stable for the Stokes problem on triangulations. For 4 even, stability was proved by Á. Baran and G. Stoyan in 2007 using macroelement technique, a dimension formula concept critical points triangulation representation corresponding functions . these belong to range divergence operator app...

by elements of Taylor-Hood type. More specifically, for k = 2, 3, the velocity vector u is approximated in the space V k0,h = V k h ∩ H10(Ω), where V kh is the space of continuous piecewise polynomial vectors of degree ≤ k and the pressure p is approximated in the space Qk−1 h consisting of continuous piecewise polynomials of degree ≤ k−1. The stability of these pairs depends on verification of...

Abstract. This paper presents a meshless implementation for the dual analysis of 2D linear elasticity problems. The derivation of the governing system of equations for the discretized equilibrated model is detailed. Crucial implementation issues of the equilibrated algorithm are described. It presents the inf-sup condition for particular cases of essential boundary conditions approximations whe...

Assume a single reaction-diffusion equation has zero as an asymptotically stable stationary point. Then we prove that there exist no localized travelling waves with non-zero speed. That is, if [lim inf |x|→∞ u(x), lim sup|x|→∞ u(x)] is included in an open interval of zero that does not include other stationary points, then the speed has to be zero or the travelling profile u has to be identical...

Consider the equation dx dt +Q(t)G(x(t− σ(t))) = f(t) (∗) where f, σ,Q ∈ C([0,∞), [0,∞)), G ∈ C(R,R),G(−x) = −G(x), xG(x) > o for x 6= 0, G is non-decreasing, t > σ(t), σ(t) is decreasing and t − σ(t) → ∞ as t → ∞. Whenf(t) ≡ 0, a sufficient condition in terms of the constants k = lim inf t→∞ ∫ t t−σ(t) Q(s)ds and L = lim sup t→∞ ∫ t

Using “Fortin operators” we give a new proof of stability for Stenberg’s family of BDMS elements in linear elasticity. Our approach yields the inf-sup condition with respect to the standard norms, which is indispensable for a posteriori error analysis. Furthermore our technique allows the construction of another family of finite elements strongly related to the classical PEERS element. The give...

We consider a random walk S n = n i=1 X i with i.i.d. X i. We assume that the X i take values in Z d , have bounded support and zero mean. For A ⊂ Z d , A = ∅, we define τ A = inf{n ≥ 0 : S n ∈ A}. We prove that there exists a constant C, depending on the common distribution of the X i and d only, such that sup ∅∅ =A⊂Z d P {τ A = n} ≤ C/n, n ≥ 1.