We have formalized the Sturm-Tarski theorem (also referred as the Tarski theorem): ∑ x∈(a,b),P (x)=0 sign(Q(x)) = Var(SRemS(P, P ′Q; a, b)) where a < b are elements of R ∪ {−∞,∞} that are not roots of P , with P,Q ∈ R[x]. Note, the usual Sturm theorem is an instance of the Sturm-Tarski theorem with Q = 1. The proof is based on [1] and Cyril Cohen’s work in Coq [2]. With the Sturm-Tarki theorem,...

This article is dedicated to establishing a novel approach the cosmological constant, in which it treated as an eigenvalue of certain Sturm–Liouville problem. The key this lies proper formulation physically relevant boundary conditions. Our suggestion regard utilize “holographic condition”, under horizon can only bear natural (i.e., non-fractional) number bits information. Under framework, we s...

the complex-step derivative approximation is applied to compute numerical derivatives. in this study, we propose a new formula of fractional complex-step method utilizing jumarie definition. based on this method, we illustrated an approximate analytic solution for the fractional cauchy-euler equations. application in image denoising is imposed by introducing a new fractional mask depending on s...

Consider the minimal Sturm-Liouville operator A = Amin generated by the differential expression

in this paper, we investigate infinite product representation of the solution of a sturm-liouville equation with an indefinite weight function which has two zeros and/or singularitiesin a finite interval. first, by using of the asymptotic estimates provided in [w. eberhard, g.freiling, k. wilcken-stoeber, indefinite eigenvalue problems with several singular pointsand turning points, math. nachr...

We prove a Filippov-Gronwall type inequality for solutions of a nonconvex secondorder differential inclusion of Sturm-Liouville type.

We derive Povzner–Wienholtz-type self-adjointness results form× m matrix-valued Sturm–Liouville operators T = R [