and Applied Analysis 3 Lemma 7. Assume that conditions (H1), (H2) hold. Suppose that x(t, t0, x0) is a solution of (1) satisfying initial value x(t + 0 ) = x0. Then −x(t + T, t0, x0) is also a solution of (1), and −x (t + T, t0, x0) = x (t, t0, −x (t0 + T, t0, x0)) , t ∈ R. (5) Proof. Let φ(t) ≡ −x(t + T, t0, x0), ψ(t) ≡ x(t, t0, −x(t0 + T, t0, x0)). Then for t ̸ = τk, k ∈ Z, we have by (H2) tha...

and Applied Analysis 3 see 10 and in terms of the weights. For a weight v the associated weight ṽ is defined as follows ṽ z : ( sup {∣∣f z ∣∣ : f ∈ H∞ v ,∥∥f∥∥H∞ v ≤ 1 })−1 . 1.7 For a typical weight v the associated weight ṽ is also typical. Furthermore, for each z ∈ B there is an fz ∈ H∞ v , ‖fz‖H∞ v ≤ 1, such that fz z 1/ṽ z , and the same holds for the space H0 v. It is known that H ∞ ṽ ∼ H...

BACKGROUND Efficacy and safety of various treatments using fractional laser or radiofrequency depend, to a large extent, on precise movement of equipment head across the patient's skin. In addition, they both depend on uniform distribution of emitted pulses throughout the treated skin area. The pulses should be closely adjacent but they should not overlap. Pulse overlapping results in amplifica...

OBJECTIVE The aims of the present laboratory study were twofold: a) to investigate the suitability of Knoop and Vickers surface microhardness (SMH) in comparison to transverse microradiography (TMR) to investigate early enamel caries lesion formation; b) to compare the kinetics of caries lesion initiation and progression between human and bovine enamel. DESIGN Specimens (90×bovine and 90×huma...

We previously demonstrated that a swept-source optical coherence tomography system (SS-OCT) without polarizationsensing could quantitatively estimate the progress of cavitated dentin lesions in vitro. The purpose of the current study was to investigate the ability to estimate the lesion development in non-cavitated dentin lesions with a surface layer. Bovine root dentin specimens were subjected...

PURPOSE An in situ study evaluated the remineralization potential of 225 ppm fluoride (F) rinses with and without a calcium phosphate agent (TCP-Si-Ur) on eroded enamel. METHODS 20 human patients participated in this IRB approved study. Enamel blocks extracted from 20 human molars were assigned to each of the three study phases (G1, G2, G3). Each block was eroded using 1% citric acid (pH = 2....

State-of-the-art airborne lidar data of passive scalars have shown that the spatial stratification of the atmosphere is scaling: the vertical extent (Δz) of structures is typically ≈ z H x ! where Δx is the horizontal extent and Hz is a stratification exponent. Assuming horizontal isotropy, the volumes of the structures therefore vary as !x!x!x H z = !x D s where the “elliptical dimension” Ds c...