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References in periodicals archive ?
Table 1 Quandt-Andrews unknown breakpoint test Full sample: 1948Q1-2017Q4, T = 280 observations Trimmed sample (1958Q3-2007Q3): left trim (15%), right trim (15%) Estimated break date: 2004Q1 Number of breaks compared: 197 Ho: no structural break Varying parameters [[alpha].sub.1] [[alpha].sub.1] and [[alpha].sub.2] Test F statistic p value Statistic p value Supremum Wald 18.58 0.0004 18.58 0.0023 Average Wald 6.16 0.0021 7.68 0.0056 Exponential Wald 6.10 0.0000 6.22 0.0027 Table 2 Bai-Perron multiple breakpoint tests Tests of l + 1 versus l globally determined breaks.
The prime objective of this article is to develop a new hybrid scheme on a piecewise-uniform Shishkin mesh for solving the singularly perturbed BVPs of the form (1.1)-(1.2) so that the method is at least second-order uniformly convergent with respect to [epsilon] in the discrete supremum norm.
Now if we define [??] as [mathematical expression not reproducible] for [lambda]([member of] A) [not equal to] v and [mathematical expression not reproducible], then [??] [not member of] S; otherwise, it contradicts that [??] is the supremum of the maximal chain [bar.S].
(a) Set the supremum value from interval solution of the best optimum problem
Caption: Figure 1: [alpha]-cut of membership function: infimum and supremum.
Therefore, the supremum of this map is attained at the value of [lambda] such that
In the case when L = 0.5, the supremum is reached for all subsets [omega] of the form [omega] = {(r, [theta]) [member of] [0,1] x [0,2[pi]] | [theta] [member of] [[omega].sub.0]} of measure [pi]/2, where [[omega].sub.[theta]] is any measurable subset of [0,2[pi]] such that [omega] and its symmetric image are complementary in [0,2[pi]].
Let b [member of] [Lip.sub.[alpha]], subsequently, if b satisfies [mathematical expression not reproducible], in which the supremum is taken over all x, y [member of] [R.sup.n] and x [not equal to] y.
Taking the supremum with respect to the past time s [less than or equal to] t in (51) and (50), we get (42) and (43).
Both of them are Banach spaces with the supremum norm.
where supremum can be replaced with maximum in engineering practice.
We note that iterative techniques combined with lower and upper solutions are applied in the literature to approximately solve various problems in ordinary differential equations [19], for second order periodic boundary value problems [10], for differential equations with maxima [3], [14], for difference equations with maxima [7], for impulsive differential equations [9], [12], for impulsive integro-differential equations [15], for impulsive differential equations with supremum [17], for differential equations of mixed type [18], for Riemann-Liouville fractional differential equations [8], [25], [27], for Caputo impulsive fractional differential equations [11] and for non- instantaneous impulsive differential equation [6].