where x ∈ R, G : R × R → R is a given nonlinear continuous function in t and x. For a vector x ∈ R we take |x| to be the Euclidean norm of x. Let t0 ≥ 0, then for each continuous function φ : [0, t0] → R, there is at least one continuous function x(t) = x(t, t0, φ) on an interval [t0, I] satisfying (1.1) for t0 ≤ t ≤ I and such that x(t, t0, φ) = φ(t) for 0 ≤ t0 ≤ I. It is assumed that the righ...

This document provides additional description to certain con1 tent of the paper and additional results of the A field-based 2 flow visualization. 3 1. Detailed Derivation of Relation between FTLE and Path4 line Orientation Vector 5 Recall that in Section 5.2 of the paper, the Lagrangian accu6 mulation is applied to collect vector-valued properties. Specif7 ically, we use it to accumulate the fl...

Given a spacelike 2-surface Σ in a spacetime N and a constant future timelike unit vector T0 in R 3,1 , we derive upper and lower estimates of Wang-Yau quasilocal energy E(Σ, X, T0) for a given isometric embedding X of Σ into a flat 3-slice in R 3,1 . The quantity E(Σ, X, T0) itself depends on the choice of X , however the infimum of E(Σ, X, T0) over T0 does not. In particular, when Σ lies in a...

OBJECTIVE To assess, during rapid maxillary expansion, the plaque index (PI) and the gingival index (GI) of patients with unilateral cleft lip and palate(UCLP) using Hyrax (HX) or inverted mini-Hyrax (IMHX) rapid maxillary expanders (RME) considering patients' sex and age. METHODS PI (Quigley Index modified by Turesky et al) and GI (Löe and Silness) of 28 UCLP (11 females; 17 males: aged 8 to...

The partial duplication model with parameters (p, t, t0, Gt0) where p ∈ (0, 1), t, t0 ∈ N, t ≥ t0, and Gt0 is an initial undirected graph is the probability distribution on undirected graphs Gt described by the following recursion: If t = t0, then Gt = Gt0 . Otherwise, select a graph Gt−1 = (Vt−1, Et−1) from the partial duplication model with parameters (p, t − 1, t0, Gt0) and construct Gt, ini...

Abstract We introduce some properties of the new better than used in convex ordering at age t0 (NBUC − t0) and new better than used of second order(2) at age t0 (NBU(2)−t0) classes of life distributions,where the survival probability at age 0 is greater than or equal to the conditional survival probability at specified age t0 > 0. Preservation properties of the two classes under various reliabi...

In this paper, we study the following delay dynamic equation xΔ(t)+ p(t)x(τ(t)) = 0 for t ∈ [t0,∞)T , where t0 ∈ T , supT = ∞ , p ∈ Crd([t0,∞)T,R) alternates in sign infinitely many times and τ ∈ Crd([t0,∞)T,T) is a strictly increasing unbounded function satisfying τ(t) t for all t ∈ [t0,∞)T . Our results extend recent results for arbitrary time scales. Mathematics subject classification (2010)...

and Applied Analysis 3 For 0 < q ≤ 1 we have t0D q t x t 1 Γ ( 1 − q d dt (∫ t t0 x s t − s q ds ) , ( 0 < q ≤ 1, c t0D q t x t 1 Γ ( 1 − q ∫ t t0 x′ s t − s q ds, ( 0 < q ≤ 1. 2.2 Some properties of Riemann-Liouville and Caputo derivatives are recalled below 1–3 . Property 1. When 0 < q ≤ 1, we have c t0D q t x t t0D q t x t − x t0 Γ ( 1 − q t − t0 −q. 2.3 In particular, if x t0 0, we have c t...