$$\begin{aligned}& \Omega _{1}=\{(u,v)\in C^{1}(I)\times C^{1}(I)\big| ||(u,v)||_{C^{1}}< r \}, \\& \Omega _{2}=\{(u,v)\in C^{1}(I)\times C^{1}(I)\big| ||(u,v)||_{C^{1}}< R \}. \end{aligned}$$ (3.1) Next...
\begin{aligned} u(t)\le W^{-1}\Bigl (W(a)+b\int _{0}^{t}\,\phi (s)\,ds\Bigr ),\; t\in [0,T_1], \end{aligned} (5.3) where T_{1} is such that the range condition \begin{aligned} W_{a,b,\phi }(t):=W(a)+b\int _{0}^{t}\,\phi (s)\,ds \in {...
Write in the file 1-O, the big O notations of the time complexity of the Insertion sort algorithm, with 1 notation per line: in the best case in the average case in the worst case alex@/tmp/sort$ cat 1-main.c #include <stdio.h> #include <stdlib.h> #include "sort.h" /** *...
this value can be made precise, we use the classical big-O notation. The definition below specifies the notion of reductions between query problems. Definition 1. We say that Query(C, L) linearly reduces to Query(C , L ) if ...
$$ \Delta ^{4} x(n-2)=f\bigl(n, x(n)\bigr),\quad n\in [1, N] , $$ (1.1) with Dirichlet boundary conditions $$ x(-1)=x(0)=0=x(N+1)=x(N+2) $$ (1.2) or periodic boundary conditions $$ \Delta ^{i} x(-1)=\Delta ^{i} x(N-1),\quad i=0, 1, ...
On all small stencils and the big stencil we use standard reconstruc- tion, obtaining vi(+0)1/2 = 1 3 v炉i + 5 6 v炉i+1 鈭 1 6 v炉i+2 vi(+1)1/2 = 鈭 1 6 v炉i鈭 1 + 5 6 v炉i + 1 3 v炉i+1 vi(+2)1/2 = 1 3 v炉i鈭 2 鈭 7 6 v炉i鈭 1 + 11...
For computational convenience, we introduce the notations: $$\begin{aligned}& \mathrm{N}_{\mathsf{p}} =\frac{\mathrm{T}^{\mathsf{p}}}{\varGamma(\mathsf {p}+1)}+ \biggl\vert \frac{\beta_{1} \mathrm{T}^{\mathsf{p}}}{(1-\beta_{1})\varGamma (\mathsf{p})} \biggr\ver...
(xix) (xx) Q: the upper bound, a constant numberbig M: a large number This paper showcases an attempt to maximize the net revenue subject to an upper bound of agent B. We consider two versions of the order acceptance two-agent scheduling problems, the LPP problem and th...
We now claim that the class of functions satisfying such conditions implement a regular bouncing pre-big bang scenario exactly satisfying the equations (1.2), and of the type illustrated in figure 2, where: • H(−f1) is the positive maximum of H (the two upper "bulges" of the "...
Z_{h,0}= \bigl\{ u=(u_{i})|u_{-1}=u_{0}=u_{1}=u_{M-1}=u_{M}=u_{M+1}=0 \bigr\} . For clarity, we introduce the following notations: [Math Processing Error] and [Math Processing Error] The discreteL_{2}inner product and corresponding discreteL_{2}-norm for fun...