We will study the Henstock–Kurzweil delta and nabla integrals, which generalize the Henstock–Kurzweil integral. Many properties of these integrals will be obtained. These results will enable time scale researchers to study more general dynamic equations. The Hensock–Kurzweil delta (nabla) integral contains the Riemann delta (nabla) and Lebesque delta (nabla) integrals as special cases.

This article establishes a comparison principle for the nabla fractional difference operator ∇ρ(a)ν, 1<ν<2. For this purpose, we consider two-point boundary value problem with separated conditions and derive corresponding Green’s function. I prove that function satisfies positivity property. Then, deduce relatively general result considered problem.

We tested the hypothesis that behavioral context modulates phase-locking between rhythmic motor activity and concomitant electrical activity induced in primary sensory (S1) cortex. We used exploratory whisking by rat as a model system and recorded two measures: (i) the mystacial electromyogram ( nabla EMG) as a surrogate of vibrissa position, and (ii) the field potential ( nabla LFP) in S1 cort...

Six new compounds [(CpTi)X, Cp=upsilon(nabla)-c(nabla)H(nabla), X=Ge, Ga, B) have been prepared and their Keggin structures determined by elementary analysis, IR, UV, (infinity)H NMR and (infinity for all)WNMR spectrometry. The results show that the complexes retain Keggin structure. The complexes exhibit antitumoral activity in vitro as shown by MTT experiment.

Oscillation criteria are established for a second-order forced dynamic equation on time scales containing both delay and advance arguments. Moreover, the potentials are allowed to change sign. Several nontrivial examples from difference equations are provided to illustrate the easy application of the results. The theory can be applied to second-order dynamic equations regardless of the choice o...

We give a semantics for a classical variant of Dale Miller and Alwen Tiu’s logic FOλ∇. No such semantics seems to have existed for the nabla operator, except for one given by U. Schöpp. Our semantics validates the rule that nabla x implies exists x, but is otherwise faithful to the authors’ original intentions. The semantics is based on category of so-called nabla-sets, which we define as presh...

We conjecture a combinatorial formula for the monomial expansion of the image of any Schur function under the Bergeron–Garsia nabla operator. The formula involves nested labeled Dyck paths weighted by area and a suitable “diagonal inversion” statistic. Our model includes as special cases many previous conjectures connecting the nabla operator to quantum lattice paths. The combinatorics of the i...