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Theorem 3.2 Let \( \{ u\left( 0\right), u\left( 1\right) ,\cdots, u\left( m\right) \} \) is any control sequence, the system (2.3) is set controllable \( {\delta }_{{2}^{\mu n}}^{q} \) from to \( {\delta }_{{2}^{n}}^{p} \) if and only if \( {\left\lbrack {D}_{f}\left\lbrack {2}^{n},{2}^{\left( {\mu - 1}\right) n}\right... | Proof The proof method is similar to (2.11). | No |
Example 4.1 Consider the BCN\n\n\\[ \n\\left\\{ \\begin{array}{l} A\\left( {t + 1}\\right) = u\\left( t\\right) \\land \\left( {\\neg \\left( {A\\left( {t - 2}\\right) \\land B\\left( {t - 1}\\right) }\\right) }\\right) , \\\\ B\\left( {t + 1}\\right) = u\\left( t\\right) \\land \\left( {\\neg \\left( {A\\left( {t - 1}... | Consider the vector form of the logical variables \\( x\\left( t\\right) = A\\left( t\\right) \\ltimes B\\left( t\\right) \\) and the algebraic state-space form is \\( x\\left( {t + 1}\\right) = {L}_{1}u\\left( t\\right) x\\left( t\\right) x\\left( {t - 1}\\right) x\\left( {t - 2}\\right) \\) with\n\n\\[ \n{L}_{1} = {\... | Yes |
Lemma 2.1 \( {}^{\left\lbrack 9\right\rbrack } \) Suppose \( f \in {L}^{1}\left( \bar{\Omega }\right) \) . Let \( h \in {L}^{1}\left( \Omega \right) \) be a solution of (2.1). Then for any \( p \in \left( {0,\frac{2{\pi }^{2}}{\parallel f{\parallel }_{{L}^{1}\left( \Omega \right) }}}\right) \) | \[ {\int }_{\Omega }{\mathrm{e}}^{{3p}\left| h\right| }\mathrm{d}y \leq C\left( {p,\operatorname{diam}\Omega }\right) \] where diam \( \Omega \) denotes the diameter of \( \Omega \) . | Yes |
Theorem 2.1 Suppose \( h \) is a solution of (2.1) with \( f \in {L}^{1}\left( \Omega \right) \) . Then for every constant \( k > 0 \)\n\n\[{\mathrm{e}}^{kh} \in {L}^{1}\left( \Omega \right)\] | Proof Letting \( 0 < \epsilon < \frac{1}{k} \), we split \( f \) as \( f = {f}_{1} + {f}_{2} \) with \( {\begin{Vmatrix}{f}_{1}\end{Vmatrix}}_{1} < \epsilon \) and \( {f}_{2} \in {L}^{\infty }\left( \Omega \right) \) . Denote \( {h}_{i} \) are the solutions of\n\n\[\\left\\{ \\begin{array}{ll} {\\left( -\\Delta \\right... | Yes |
Theorem 2.2 Let \( \\left( {u, v}\\right) \) be a solution of equation \( \\left( *\\right) \) with \( {Q}_{1},{Q}_{2} \\in {L}^{p}\\left( \\Omega \\right) \) and \( {\\mathrm{e}}^{3u},{\\mathrm{e}}^{3v} \\in \) \( {L}^{{p}^{\\prime }}\\left( \\Omega \\right) \) for some \( \\frac{6}{5} < p \\leq \\infty \) . Then \( u... | Proof From Theorem 2.1, we have \( {\\mathrm{e}}^{ku} \\in {L}^{1}\\left( \\Omega \\right) \) for all \( k \), i.e., \( {\\mathrm{e}}^{u} \\in {L}^{r}\\left( \\Omega \\right) \\forall r < \\infty \) . It follows that \( {Q}_{2}{\\mathrm{e}}^{3u} \\in {L}^{p - \\delta }\\forall \\delta > 0 \) if \( p < \\infty \), and \... | Yes |
Corollary 2.1 Let \( \left( {u, v}\right) \) be a solution of\n\n\[ \left\{ \begin{array}{ll} {\left( -\Delta \right) }^{\frac{3}{2}}u = {Q}_{1}{\mathrm{e}}^{3v} + f\left( y\right) , & \text{ in }\Omega , \\ {\left( -\Delta \right) }^{\frac{3}{2}}v = {Q}_{2}{\mathrm{e}}^{3u} + g\left( y\right) , & \text{ in }\Omega , \... | By Theorem 2.2, for any solution \( \left( {u, v}\right) \) of system \( \left( *\right) \), we have \( {\int }_{\Omega }{Q}_{1}\left( y\right) {\mathrm{e}}^{3v}\mathrm{\;d}y < \infty ,{\int }_{\Omega }{Q}_{2}\left( y\right) \) \( \cdot {\mathrm{e}}^{3u}\mathrm{\;d}y < \infty \) . We next show that there is a uniform b... | Yes |
Theorem 2.3 Suppose that \( {Q}_{i}\left( y\right), i = 1,2 \), is continuous function with \( {m}_{i} \leq {Q}_{i} \leq {M}_{i} \) for some positive constants \( {m}_{i} \) and \( {M}_{i},\Omega \) is convex and \( \left( {\mathrm{H}}_{1}\right) \) holds. Then there exists a positive constant \( C \), depending only o... | Proof Step 1 For each \( \left( {u, v}\right) \) solution of system \( \left( *\right) \) we claim that \[ {\int }_{\Omega }{Q}_{1}\left( y\right) {\mathrm{e}}^{3v}{\phi }_{1}\mathrm{\;d}y \leq C,{\int }_{\Omega }{Q}_{2}\left( y\right) {\mathrm{e}}^{3u}{\varphi }_{1}\mathrm{\;d}y \leq C, \] where the constant \( C \) o... | Yes |
Lemma 3.3 Assume that \( \left( {u, v}\right) \) is a solution of \( \left( {* * }\right) \) . Let\n\n\[ \n{w}_{1}\left( y\right) = \frac{1}{2{\pi }^{2}}{\int }_{{\mathbb{R}}^{3}}\ln \frac{\left| y - z\right| }{\left| z\right| + 1}{Q}_{1}\left( z\right) {\mathrm{e}}^{{3v}\left( z\right) }\mathrm{d}z \n\] \n\nand \n\[ \... | Proof For \( \left| y\right| \geq 4 \), decompose \( {\mathbb{R}}^{3} = {A}_{1} \cup {A}_{2} \), where \( {A}_{1} = \left\{ {z\left| \right| z - y\left| {\; \leq \frac{\left| y\right| }{2}}\right. }\right\} \) and \( {A}_{2} = \left\{ {z\left| \right| z - y\left| \right. \geq \frac{\left| y\right| }{2}}\right\} \) . Fo... | Yes |
Lemma 3.4 Assume that \( \left( {u, v}\right) \) is a solution of \( \left( {* * }\right) \) with \( u\left( y\right) = \circ \left( {\left| y\right| }^{2}\right) \) and \( v\left( y\right) = \) \( \circ \left( {\left| y\right| }^{2}\right) \) . Then \( {\Delta u}\left( y\right) \) and \( {\Delta v}\left( y\right) \) c... | Proof Let \( k = u + {w}_{1} \) . It is obvious that \( {\left( -\Delta \right) }^{\frac{3}{2}}k \equiv 0 \) in \( {\mathbb{R}}^{3} \) . Similar to the proof of Lemma 2.2 in [11], we have for any \( {y}_{0} \in {\mathbb{R}}^{3} \) and \( r > 0 \)\n\n\[ \n6{\pi }^{2}{r}^{2}\exp \left( {\frac{{r}^{2}}{2}{\Delta k}\left( ... | Yes |
Lemma 3.5 Suppose that \( \left( {u, v}\right) \) satisfies the assumptions of Theorem 3.2, then\n\n\[ \frac{{w}_{i}\left( y\right) }{\ln \left| y\right| } \rightarrow {\beta }_{i},\text{ uniformly as }\left| y\right| \rightarrow \infty . \] | Proof Here we prove \( {w}_{1}\left( y\right) \rightarrow {\beta }_{1}\ln \left| y\right| \) as \( \left| y\right| \rightarrow \infty \) . We need only to verify that\n\n\[ I = {\int }_{{\mathbb{R}}^{3}}\frac{\ln \left| {y - z}\right| - \ln \left( {\left| z\right| + 1}\right) - \ln \left| y\right| }{\ln \left| y\right|... | Yes |
Lemma 3.6 Suppose that \( \left( {u, v}\right) \) satisfies the assumptions of Theorem 3.2, then\n\n\[ u\left( y\right) = \frac{1}{2{\pi }^{2}}{\int }_{{\mathbb{R}}^{3}}\ln \frac{\left| z\right| + 1}{\left| y - z\right| }{Q}_{1}\left( z\right) {\mathrm{e}}^{{3v}\left( z\right) }\mathrm{d}z + {c}_{0} \]\n\nand\n\n\[ v\l... | Proof From Lemma 3.4, we know \( \Delta \left( {u + {w}_{1}}\right) = 0 \) in \( {\mathbb{R}}^{3} \) . By Theorem 3.1, we get \( {u}^{ + } \in {L}^{\infty } \) . So, combing Lemma 3.3, we have \( u + {w}_{1} \leq c\ln \left| y\right| + c \), since \( u + {w}_{1} \) is harmonic function, by the gradient estimates of har... | Yes |
Lemma 3.7 Suppose that \( \left( {u, v}\right) \) satisfies the assumptions of Theorem 3.2, then \( {u}_{1}\left( y\right) \geq \) \( - {\beta }_{1}\ln \left( {\left| y\right| + 1}\right) - {c}_{1} \) with \( {\beta }_{1} > 1 \) and \( {u}_{2}\left( y\right) \geq - {\beta }_{2}\ln \left( {\left| y\right| + 1}\right) - ... | Proof From Lemma 3.3 and Lemma 3.6, we get\n\n\[ u\left( y\right) > - {\beta }_{1}\ln \left( {\left| y\right| + 1}\right) - {c}_{1} \]\n\nand\n\n\[ v\left( y\right) > - {\beta }_{2}\ln \left( {\left| y\right| + 1}\right) - {c}_{2} \]\n\nBy above inequality, \( {\int }_{{\mathbb{R}}^{3}}{\mathrm{e}}^{3v}\mathrm{\;d}y < ... | Yes |
Lemma 3.8 Suppose that \( \left( {u, v}\right) \) satisfies the assumptions of Theorem 3.2, then \( u\left( y\right) \leq \) \( - {\beta }_{1}\ln \left( {\left| y\right| + 1}\right) + {c}_{1} \) and \( v\left( y\right) \leq - {\beta }_{2}\ln \left( {\left| y\right| + 1}\right) + {c}_{2}. \) | Proof In fact, for \( \left| {y - z}\right| \geq 1 \), we get\n\n\[ \left| y\right| \leq \left| {y - z}\right| \left( {\left| z\right| + 1}\right) \]\n\nThen\n\n\[ \ln \left| y\right| - 2\ln \left( {\left| z\right| + 1}\right) \leq \ln \left| {y - z}\right| - \ln \left( {\left| z\right| + 1}\right) . \]\n\nConsequently... | Yes |
Proposition 3.1 For the Sierpiński graph \( {S}_{p}^{n}\left( {n \geq 1\text{and}p \geq 3}\right) ,{Fd}\left( {S}_{p}^{n}\right) = 2\left( {{2}^{n} - 1}\right) \) . | Proof Replace the Fermat vertex of triplet \( u, v, w \) by \( u, v, w \) in turn, we obtain\n\n\[ \mathcal{F}\left( {u, v, w}\right) \leq \frac{2}{3}\left( {d\left( {u, v}\right) + d\left( {v, w}\right) + d\left( {w, u}\right) }\right) \leq {2d}\left( {S}_{p}^{n}\right) = 2\left( {{2}^{n} - 1}\right) ,\]\n\nand the eq... | No |
Lemma 3.2 The Fermat distance of vertex \( u \) and two extreme vertices in \( {S}_{p}^{n} \) satisfies\n\n\[ F\left( {u,{i}^{n},{j}^{n}}\right) = d\left( {u,{i}^{n}}\right) + d\left( {u,{j}^{n}}\right), i, j \in {\left\lbrack p\right\rbrack }_{0}. \]\n\n(3.2) | Eq. (3.2) can be easily derived by induction and Lemma 3.1. The second assertion of Lemma 3.2 comes straightforwardly from Eq. (3.2) and the definition of \( {\varepsilon }_{3}\left( u\right) \) . | No |
Proposition 3.2 Let \( p \geq 3 \) and \( n \in {\mathbb{N}}^{ + } \) . Two types of Fermat radius are given in turn by \[ F{r}_{1}\left( {S}_{p}^{n}\right) = \left\{ {\begin{array}{ll} 2 \cdot \left( {{2}^{n} - 1}\right) , & n < p - 1, \\ 2 \cdot {2}^{n} - 3, & n = p - 1, \\ 3 \cdot {2}^{n - 1}, & n \geq p, \end{array... | Proof We elaborate that the proposition holds in the following three cases. (i) \( 1 \leq n < p - 1 \) . The only admissible value for \( m \) and \( l \) is 0, which yields \( F{r}_{1}\left( {S}_{p}^{n}\right) = \) \( 2 \cdot \left( {{2}^{n} - 1}\right) \) . Let all different letters appear in vertex \( u \) be \( \ma... | Yes |
Lemma 4.1 For \( u \in {F}_{k} \) ,\n\n\[ \n\mathcal{F}\left( {u,{a}_{i},{a}_{j}}\right) = \left\{ \begin{array}{ll} \mathcal{F}\left( {u,{T}_{k}\left( {a}_{i}\right) ,{T}_{k}\left( {a}_{j}\right) }\right) + 1, & k \neq i, j, \\ \mathcal{F}\left( {u,{T}_{k}\left( {a}_{i}\right) ,{T}_{k}\left( {a}_{j}\right) }\right) + ... | Proof For a given vertex \( {u}^{\prime } = k{u}_{n - 1}\cdots {u}_{1} \in k{S}_{p}^{n - 1} \), we have\n\n\[ \n\mathcal{F}\left( {{u}^{\prime },{i}^{n},{j}^{n}}\right) = d\left( {{u}^{\prime },{i}^{n}}\right) + d\left( {{u}^{\prime },{j}^{n}}\right) = \left\{ \begin{array}{ll} d\left( {{u}^{\prime }, k{i}^{n - 1}}\rig... | Yes |
Theorem 4.1 For \( p \geq 3 \), the average Fermat eccentricity of \( S{P}_{p} \) is given by\n\n\[ \n{\bar{\varepsilon }}_{3}\left( {S{P}_{p}}\right) = 2 - \left( {p - 1}\right) {\left( \begin{matrix} {2p} \\ p - 2 \end{matrix}\right) }^{-1}.\n\] | Proof Let\n\n\[ \n{E}_{p - k, p}\left( u\right) = \mathop{\max }\limits_{{u \in S{P}_{p}}}\left\{ {\mathcal{F}\left( {u,{a}_{i},{a}_{j}}\right) : i, j \in {\left\lbrack p - k\right\rbrack }_{0}}\right\}\n\]\n\n\[ \n{M}_{p - k, p} = {\int }_{S{P}_{p}}{E}_{p - k, p}\left( x\right) \mathrm{d}\mu \left( x\right) ,\n\]\n\nw... | Yes |
Proposition 2.1 If a kernel \( K\left( {x, y}\right) \) satisfies Assumption 2, then\n\n\[ \n{D}_{{\mathcal{L}}_{K}}^{r}{\left( f, t\right) }_{{L}^{2}\left( {\mathbb{R}}^{d}\right) } \sim {\omega }_{{\mathcal{L}}_{K}}^{r}{\left( f, t\right) }_{{L}^{2}\left( {\mathbb{R}}^{d}\right) },\;f \in {L}^{2}\left( {\mathbb{R}}^{... | To show (2.7) we need a lemma.\n\nLet \( \left( {\mathcal{B},\parallel \cdot {\parallel }_{\mathcal{B}}}\right) \) be a normed linear space, \( {\left\{ T\left( t\right) : \left( \mathcal{B},\parallel \cdot {\parallel }_{\mathcal{B}}\right) \rightarrow \left( \mathcal{B},\parallel \cdot {\parallel }_{\mathcal{B}}\right... | No |
Corollary 2.2 There holds\n\n\[ \n{D}_{{B}_{4}}^{r}{\left( f, t\right) }_{{L}^{2}\left( {\mathbb{R}}^{d}\right) } \sim {\omega }_{{B}_{4}}^{r}{\left( f, t\right) }_{{L}^{2}\left( {\mathbb{R}}^{d}\right) },\;f \in {L}^{2}\left( {\mathbb{R}}^{d}\right), t > 0, \n\]\n\nwhere\n\n\[ \n{\omega }_{{B}_{4}}^{r}{\left( f, t\rig... | Since \( {\mathrm{e}}^{-\alpha \parallel y{\parallel }^{2}} = {\left( \frac{\pi }{\alpha }\right) }^{\frac{d}{2}}\widetilde{{\mathrm{e}}^{-\frac{{\pi }^{2}}{\alpha }\parallel \cdot {\parallel }^{2}}}\left( y\right) ,\;\alpha > 0, y \in {\mathbb{R}}^{d} \) (see Lemma 4 in Chapter 15 of [15]), \n\nwe have \n\n\[ \n{\math... | Yes |
Corollary 2.3 Let \( r \geq 1 \) be an integer. Then\n\n\[ \n{D}_{\Delta }^{r}{\left( f, t\right) }_{{L}^{2}\left( {\mathbb{R}}^{d}\right) } \sim {\omega }_{\Delta }^{r}{\left( f, t\right) }_{{L}^{2}\left( {\mathbb{R}}^{d}\right) },\;f \in {L}^{2}\left( {\mathbb{R}}^{d}\right), t > 0.\n\] | Proof Replace the \( \lambda \left( \xi \right) \) in (2.6) with \( {\left( 2\pi \parallel \xi \parallel \right) }^{-2} \) . | No |
Corollary 3.3 There is a constant \( C > 0 \) such that for any \( 0 \leq \nu < r \) there hold\n\n\[ \n{I}_{{B}_{2r}\left( {{L}^{2}\left( {\mathbb{R}}^{d}\right) }\right) ,\parallel \cdot {\parallel }_{{B}_{2\left( {r - \nu }\right) }\left( {{L}^{2}\left( {\mathbb{R}}^{d}\right) }\right) } \leq \gamma }{\left( f,\gamm... | In particular we have\n\n\[ \n{I}_{{B}_{2r}\left( {{L}^{2}\left( {\mathbb{R}}^{d}\right) }\right) ,\parallel \cdot {\parallel }_{{B}_{2r}\left( {{L}^{2}\left( {\mathbb{R}}^{d}\right) }\right) } \leq \gamma }{\left( f,\gamma \right) }_{{L}^{2}\left( {\mathbb{R}}^{d}\right) } = \mathop{\inf }\limits_{{g \in {B}_{2r}\left... | Yes |
Corollary 3.4 Let \( \phi \) satisfy Assumption 3. Then there is a constant \( C > 0 \) such that\n\n\[ \n{I}_{{\mathcal{H}}_{{K}^{ * }\left( \phi \right) },\parallel \cdot {\parallel }_{{\mathcal{H}}_{{K}^{ * }\left( \phi \right) } \leq \gamma }{\left( f,\gamma \right) }_{{L}^{2}\left( {\mathbb{R}}^{d}\right) }} = \ma... | Proof Replace the \( \lambda \left( \xi \right) \) in the proof of Proposition 3.1 with \( \widehat{\phi }\left( \xi \right) \) and take \( r = 1 \) . | Yes |
For all \( k \geq 0 \), the search direction sequence \( \left\{ {d}_{k}\right\} \), generated by the HACGP, satisfies the sufficient descent property\n\n\[ \n{F}_{k}^{\mathrm{T}}{d}_{k} \leq - \left( {1 - \frac{1}{\mu }}\right) {\begin{Vmatrix}{F}_{k}\end{Vmatrix}}^{2} \n\]\n\n(2.4)\n\nand the trust region property\n\... | Proof Here \( {F}_{0}^{\mathrm{T}}{d}_{0} = - {\begin{Vmatrix}{F}_{0}\end{Vmatrix}}^{2} \) because \( {d}_{0} = - {F}_{0} \), which satisfies (2.4) and (2.5) for \( k = 0 \) . Next, we prove the case of \( k \geq 1 \) . By the definition of \( {w}_{k - 1} \), one has\n\n\( {d}_{k - 1}^{\mathrm{T}}{w}_{k - 1} = {d}_{k -... | Yes |
Lemma 2.2 Let the sequences \( \left\{ {x}_{k}\right\} \) and \( \left\{ {z}_{k}\right\} \) be generated by the HACGP. For any \( \widehat{x} \in \mathbf{\Omega } \), the sequence \( \left\{ \begin{Vmatrix}{{x}_{k} - \widehat{x}}\end{Vmatrix}\right\} \) is convergent, and sequence \( \left\{ {x}_{k}\right\} \) is bound... | Proof The proof is similar to Lemma 4 in [10], so it is omitted here. | No |
Example 4.1 Let \( {\mathcal{H}}_{1} = {\mathcal{H}}_{2} = {\mathbb{R}}^{m} \) . Let \( \mathcal{A} : {\mathbb{R}}^{m} \rightarrow {\mathbb{R}}^{n},{\mathcal{A}}_{1} : {\mathbb{R}}^{m} \rightarrow {\mathbb{R}}^{m},{\mathcal{A}}_{2} : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n} \) , be matrices generated by using a no... | The initial values \( {x}_{0} \) are randomly generated in \( \left( {0,1}\right) \) and the stopping criterion is \( \begin{Vmatrix}{x}_{n}\end{Vmatrix} < \) \( {10}^{-3} \) . We test Algorithm (1.2) (IPPA), Algorithm 1 (HIPA), and Algorithm 3 (STEGM) and Algorithm 4 (HISVIP) for \( {x}_{0} = {x}_{1} \) and different ... | No |
Corollary 4.1 Let \( Y\left( {x, t}\right) \) be the exact solution of problem (1.1) with the exact data \( \omega \left( t\right) \) and \( \mu \left( t\right) ,{Y}_{m}^{\delta } \) is the iteration solution with the noisy data \( {\omega }^{\delta },{\mu }^{\delta } \) which satisfy (1.4), and assume that the a-prior... | \[ \begin{Vmatrix}{{Y}_{m}^{\delta }\left( {x, t}\right) - Y\left( {x, t}\right) }\end{Vmatrix} \leq {C}_{3}{E}^{\frac{x}{L}}{\delta }^{1 - \frac{x}{L}}{\left( \ln \frac{1}{\delta }\right) }^{-\frac{px}{L\alpha }}\left( {1 + o\left( 1\right) }\right) ,\text{ for }\delta \rightarrow 0, \] (4.11) where \( {C}_{3} = {C}_{... | Yes |
Theorem 4.4 (The a-posteriori convergence estimate for solving Problem (2.4)) Let \( v \) given by (2.9) be the exact solution of problem (2.4) with the exact data \( \mu \), and \( {v}_{{m}_{2}^{ * }}^{\delta } \) be the iteration solution defined by (3.6) with the measured data \( {\mu }^{\delta } \) which satisfy (1... | Proof Firstly, from the proof process of Theorem 4.2, we know that\n\n\[ \n{J}_{4}^{2} = {\begin{Vmatrix}{\widehat{v}}_{{m}_{2}^{ * }}^{\delta }\left( x,\xi \right) - {\widehat{v}}_{{m}_{2}^{ * }}\left( x,\xi \right) \end{Vmatrix}}^{2} \leq {x}^{2}{E}^{2\frac{x}{L}}{\delta }^{2 - 2\frac{x}{L}}{\left( \ln \frac{1}{\delt... | Yes |
Corollary 4.2 Let \( Y\left( {x, t}\right) \) be the exact solution of problem (1.1) with the exact data \( \omega \left( t\right) \) and \( \mu \left( t\right) \) . The iterative step number \( {m}^{ * } \) denotes the maximum number selected by the a-posteriori rules (4.12) and (4.16), that is to say \( {m}^{ * } = \... | \[ \begin{Vmatrix}{{Y}_{{m}^{ * }}^{\delta }\left( {x, t}\right) - Y\left( {x, t}\right) }\end{Vmatrix} \leq {C}_{6}{E}^{\frac{x}{L}}{\delta }^{1 - \frac{x}{L}}{\left( \ln \frac{1}{\delta }\right) }^{-\frac{px}{L\alpha }}\left( {1 + o\left( 1\right) }\right) \text{, for}\delta \rightarrow 0\text{,} \] (4.20) where \( {... | Yes |
Example 5.1 In the direct problem (5.1), we take \( w\left( t\right) = 1 - {\mathrm{e}}^{-{2t}} \) and \( g\left( t\right) = 2\sin \left( {8\pi t}\right) \). It is clear that \( g\left( t\right) \) is a smooth function, and it belongs to \( {H}^{p}\left( \mathbb{R}\right) \subset {L}^{2}\left( \mathbb{R}\right) \). To ... | Tab. 1 At \( x = 1,\varepsilon = {0.001},\alpha = {0.01} \), the priori bound and the relative errors and iterative step numbers for various \( p \n\n<table><thead><tr><th>\( p \)</th><th>1</th><th>2</th><th>3</th><th>4</th></tr></thead><tr><td>\( E \)</td><td>36</td><td>895</td><td>22469</td><td>564701</td></tr><tr><t... | Yes |
In the forward problem (5.1), we take \( \omega \left( t\right) = 0 \), and\n\n\[ g\left( t\right) = \left\{ \begin{matrix} {0.09} - {\left( t - {0.4}\right) }^{2}, & {0.1} \leq t \leq {0.7} \\ 0, & \text{ others. } \end{matrix}\right. \] | We know that \( g\left( t\right) \) is a continuous function, but it only belongs to \( {H}^{1}\left( \mathbb{R}\right) \subset {L}^{2}\left( \mathbb{R}\right) \) . Note that, \( \parallel Y\left( {L, \cdot }\right) {\parallel }_{{H}^{1}} = \parallel g\left( t\right) {\parallel }_{{H}^{1}} < 1 \), so we choose the a-pr... | No |
Lemma 2.2 Suppose these analytic functions \( a\left( t\right) = \sum {k}_{1p}{t}^{p}, b\left( {{v}_{m}\left( t\right) }\right) = \sum {k}_{2q}{v}_{m}^{q}\left( t\right) \) , \( c\left( {t,{v}_{m}\left( t\right) }\right) = \sum {k}_{3l}{t}^{i}{v}_{m}^{j}\left( t\right), i + j = l \), where \( p, q, l \) are finite posi... | Proof Using known conditions, we can get\n\n\[ a\left( t\right) = \sum {k}_{1p}{t}^{p} \]\n\n\[ = \sum {k}_{1p}{\left( \mathop{\sum }\limits_{{r = 1}}^{m}{\widehat{v}}_{r}{\lambda }_{r}\left( t\right) \right) }^{p} \]\n\n\[ = \sum {k}_{1p}{\left( {\widehat{v}}_{1}{\lambda }_{1}\left( t\right) + {\widehat{v}}_{2}{\lambd... | Yes |
Consider the nonlinear SDE\n\n\[ \n\\mathrm{d}v\\left( t\\right) = {z}^{2}\\cos \\left( {v\\left( t\\right) }\\right) {\\sin }^{3}\\left( {v\\left( t\\right) }\\right) \\mathrm{d}t - z{\\sin }^{2}\\left( {v\\left( t\\right) }\\right) \\mathrm{d}W\\left( t\\right) ,\\;t \\in \\lbrack 0,1), \n\] | the exact solution is\n\n\[ \nv\\left( t\\right) = \\operatorname{arccot}\\left( {{zW}\\left( t\\right) + \\cot \\left( {v}_{0}\\right) }\\right) .\n\] | No |
Example 5.2 \( {}^{\left\lbrack 6\right\rbrack } \) Consider the nonlinear SDE\n\n\[ \n\mathrm{d}v\left( t\right) = {\mathrm{e}}^{-t}\sin \left( {v\left( t\right) }\right) \mathrm{d}t + {\mathrm{e}}^{-t}\cos \left( {v\left( t\right) }\right) \mathrm{d}W\left( t\right) ,\;t \in \lbrack 0,1).\n\] | Let \( {v}_{0} = - 3 \), the simulation results for \( m = {32} \) and \( m = {64} \) are severally given in Fig. 5.3 and Fig. 5.4. These two figs also show that the approximate solution fluctuates around the mean orbit, where the mean solution is obtained by \( {10}^{4} \) trajectories. | No |
Lemma 2.1 For every \( u = {u}_{0}{u}_{1}\cdots {u}_{n - 1} \in {\mathcal{F}}_{\mathbf{c}}\left( n\right) \) with \( n \geq 1 \), its mirror \( \bar{u} = \) \( {u}_{n - 1}{u}_{n - 2}\cdots {u}_{0} \) is also a factor of the Cantor-like sequence. | Proof By the definition of the morphism \( \sigma ,\sigma \left( 1\right) = {10}^{\ell }1 = \overline{\sigma \left( 1\right) } \) and \( \sigma \left( 0\right) = {0}^{\ell + 2} = \overline{\sigma \left( 0\right) } \) . Thus for every \( i \geq 1 \) ,\n\n\[{\sigma }^{i}\left( 1\right) = \overline{{\sigma }^{i}\left( 1\r... | Yes |
Lemma 2.2 For every \( 1 \leq n \leq l \) ,\n\n\[ \mathcal{R}{\mathcal{S}}_{\mathbf{c}}\left( n\right) = \mathcal{L}{\mathcal{S}}_{\mathbf{c}}\left( n\right) = \left\{ {0}^{n}\right\} \]\n\nFor every \( i \geq 0 \) and \( l{\left( l + 2\right) }^{i} < n \leq l{\left( l + 2\right) }^{i + 1} \) ,\n\n\[ \mathcal{R}{\mathc... | Proof The result follows from Theorem 1 in [13] and Lemma 2.1 directly. | No |
Proposition 3.1 For \( i = 1,\cdots, l + 1,{\mathcal{P}}_{\mathbf{c}}^{\left( 1\right) }\left( i\right) = 2 \) and for every \( n \geq 1,1 \leq j \leq l + 1 \) ,\n\n\( {\mathcal{P}}_{\mathbf{c}}^{\left( 1\right) }\left( {\left( {l + 2}\right) n}\right) = 2{\mathcal{P}}_{\mathbf{c}}^{\left( 1\right) }\left( n\right) - 1... | Proof The result follows from Lemma 3.1, Corollary 3.1 and the equality (3.1). | No |
Lemma 4.1 \( {p}_{2}\left( {1,0,0}\right) = 1 \) and for every \( n \geq 2 \) ,\n\n\[ \n{p}_{2}\left( {n,0,0}\right) = {M}_{\mathbf{c}}\left( {n - 2}\right) + 1.\n\]\n\n(4.3)\n\nMoreover, the sequence \( {\left\{ {p}_{2}\left( n,0,0\right) \right\} }_{n \geq 1} \) is \( \left( {l + 2}\right) \) -regular. | Proof The initial values can be verified by enumerating all factors of length at most \( l + 1 \) . Now, let \( n > l + 1 \) and suppose \( n \leq {\left( l + 2\right) }^{i} \) for some \( i \geq 1 \) .\n\nBy the definition of the function \( {M}_{\mathbf{c}}\left( n\right) \), it is obvious that \( {\left| \omega \rig... | Yes |
Lemma 4.2 \( {p}_{2}\left( {1,0,1}\right) = {p}_{2}\left( {1,1,0}\right) = 0 \) and for every \( n \geq 2 \), \[ {p}_{2}\left( {n,0,1}\right) = {p}_{2}\left( {n,1,0}\right) = {M}_{\mathbf{c}}\left( {n - 1}\right) . \] Moreover, the sequences \( {\left\{ {p}_{2}\left( n,0,1\right) \right\} }_{n \geq 1} \) and \( {\left\... | Proof By Lemma 2.1, for every factor \( w \) of \( \mathbf{c} \), its reversal \( \bar{w} \) is also a factor of \( \mathbf{c} \). Therefore, \( {p}_{2}\left( {n,1,0}\right) = {p}_{2}\left( {n,0,1}\right) \). Clearly, for every \( \omega \in {\mathcal{W}}_{n,0,1},1 \leq {\left| \omega \right| }_{1} \leq {M}_{\mathbf{c}... | Yes |
Lemma 4.4 \( {p}_{2}\left( {1,1,1}\right) = 1,{p}_{2}\left( {i,1,1}\right) = 0 \) for \( i = 2,\cdots, l + 1 \) and for every \( n \geq 1,0 \leq \) \( j \leq l + 1 \)\n\n\[ \n{p}_{2}\left( {\left( {l + 2}\right) n + j,1,1}\right) = \left\{ \begin{array}{ll} {p}_{2}\left( {n,1,1}\right) , & \text{ if }j = 0, \\ {p}_{2}\... | Proof The initial values can be proved by enumerating all the factors of length at most \( l + 1 \) . By the recursive formula given in Corollary 4.1, for every \( i \geq 0, g\left( i\right) {\;\operatorname{mod}\;\left( {l + 2}\right) } \in \) \( \{ 0,1,2\} \) . Therefore, for every \( n \geq 1 \) and \( 3 \leq j \leq... | Yes |
Theorem 2 The solution to the multi-dimensional Riemann problem (11) and (12) is decomposed into the six domains \( {\left( {D}_{i}\right) }_{1 \leq i \leq 6} \) in \( {\mathbb{R}}^{2} \times {\mathbb{R}}^{ + } \) as follows: | (i) in \( {D}_{1} = \{ \left( {\alpha, t, x, y}\right) \;|\;t > 0, x\sin \alpha - y\cos \alpha - t > 0, y > t\} ,\n\n\[ \left\{ \begin{array}{l} {u}_{1} = \frac{1}{2}\left( {{u}_{10}\left( {\theta - \varphi }\right) + {u}_{10}\left( {\theta + \varphi }\right) }\right) + \frac{1}{2}{\int }_{\theta - \varphi }^{\theta + ... | Yes |
Corollary 2 If \( {u}_{10}\left( \theta \right) \equiv {u}_{10},{u}_{20}\left( \theta \right) \equiv {u}_{20} \) in Theorem 2, where \( {u}_{10},{u}_{20} \) are constants, then (i) in \( {D}_{1} = \{ \left( {\alpha, t, x, y}\right) \;|\;t > 0, x\sin \alpha - y\cos \alpha - t > 0, y > t\} , \)\n\[ {u}_{1} = {u}_{10},\;{... | Proof We only carry out the calculations for \( {D}_{6} \) because the others can be similarly obtained.\n\nBecause of\n\[ {\int }_{0}^{\alpha }\frac{{u}_{10}\left( \sigma \right) }{{t}^{2} - {r}^{2}{\sin }^{2}\left( {\sigma - \theta }\right) }\mathrm{d}\sigma = {u}_{10}{\left. \arctan \left( \frac{t\sqrt{{t}^{2} - {r}... | Yes |
Theorem 2 If \( A = {\left( {a}_{ij}\right) }_{m \times m} \) in (2) satisfies (7), taking \( T = A, S \) as (8) and \( Q = \) \( {B}^{\mathrm{T}}{SB} \), then all eigenvalues of \( {Q}^{-1}{B}^{\mathrm{T}}{A}^{-1}B \) are real and positive, thus the SOR-like method (4) is convergent with \( \omega \) satisfying (5). | Proof We can see, from Remark 1, that all eigenvalues of \( {Q}^{-1}{B}^{\mathrm{T}}{A}^{-1}B \) are real and positive. Therefore, by Theorem 1, the SOR-like method (4) is convergent if \( 0 < \omega < \) \( \frac{4}{\sqrt{{4\rho } + 1} + 1} \), where \( \rho \) is the spectral radius of \( {Q}^{-1}{B}^{\mathrm{T}}{A}^... | No |
Theorem 1 Suppose that the above assumptions (11) (14) are satisfied, then the initial-boundary value problem (1) \( \sim \) (3) admits a unique global \( {C}^{2} \) solution \( x = x\left( {t,\theta }\right) \) on the domain \( {\mathbb{R}}^{ + } \times {\mathbb{R}}^{ + } \) . | In this section, we will give the proof of Theorem 1. Firstly, we obtain some estimates which play an important role in the proof of Theorem 1. In order to prove Theorem 1, it suffices to prove that under the assumptions of Theorem 1, the initial-boundary value problem (5) has the uniform a priori estimate on the \( {C... | Yes |
Lemma 2 Let \( {R}_{i},{S}_{i} \) be as system (7). Then, \n\n\[ \n\left\{ {\left| {{R}_{i}\left( {t,\theta }\right) }\right| ,\left| {{S}_{i}\left( {t,\theta }\right) }\right| }\right\} \leq C. \n\] | Proof For any fixed point \( \left( {t,\theta }\right) \in {\mathbb{R}}^{ + } \times {\mathbb{R}}^{ + } \), through this point, we draw the forward characteristic \( {\widetilde{C}}_{1} : \theta = {\theta }_{1}\left( t\right) \), i.e., \( \frac{\mathrm{d}{\theta }_{1}\left( t\right) }{\mathrm{d}t} = {\lambda }_{ + }\le... | Yes |
Lemma 2 The slow subsystem \( {I}_{\mathrm{s}} \) is asymptotically stable if and only if there exists a positive definite matrix \( {P}_{\mathrm{s}} \) such that the matrix \( {A}_{0} \) satisfies the following matrix inequality:\n\n\[ \n{P}_{\mathrm{s}}^{\mathrm{T}}{A}_{0} + {A}_{0}^{\mathrm{T}}{P}_{\mathrm{s}} < 0. ... | Proof (Sufficiency) In order to show the asymptotic stability of the slow subsystem \( {I}_{\mathrm{s}} \), choose a Lyapunov function candidate \( {}^{\left\lbrack {17}\right\rbrack } \) as follows:\n\n\[ \n{V}_{\mathrm{s}}\left( {x}_{\mathrm{s}}\right) = {x}_{\mathrm{s}}^{\mathrm{T}}{P}_{\mathrm{s}}{x}_{\mathrm{s}} \... | Yes |
Lemma 3 The fast subsystem \( {I}_{\mathrm{f}} \) is asymptotically stable if and only if there exists a positive definite matrix \( {P}_{\mathrm{f}} \) such that the matrix \( {A}_{22} \) satisfies the following matrix inequality:\n\n\[ \n{P}_{\mathrm{f}}^{\mathrm{T}}{A}_{22} + {A}_{22}^{\mathrm{T}}{P}_{\mathrm{f}} < ... | Proof We choose a Lyapunov function candidate as follows:\n\n\[ \n{V}_{\mathrm{f}}\left( {x}_{\mathrm{f}}\right) = {x}_{\mathrm{f}}^{\mathrm{T}}{P}_{\mathrm{f}}{x}_{\mathrm{f}} \n\]\nand use the following inequality:\n\n\[ \n{\lambda }_{\min }\left( {P}_{\mathrm{f}}\right) \parallel {x}_{\mathrm{f}}{\parallel }_{2}^{2}... | No |
Lemma 4 For a given positive scalar \( {\varepsilon }_{1} \), if the following conditions are satisfied: i) \( a < 0 \) ;\n\nii) \( a + b{\varepsilon }_{1} < 0 \) ;\n\niii) \( a + b{\varepsilon }_{1} + c{\varepsilon }_{1}^{2} < 0 \) ,\n\nwhere \( a, b \), and \( c \) are constants, then\n\n\[ a + {b\varepsilon } + c{\v... | Proof Obviously, from i) and ii), \( a + {b\varepsilon } < 0 \) holds for \( \varepsilon \in \left( {0,{\varepsilon }_{1}}\right\rbrack \) . Consider the following two cases.\n\ni) If \( c > 0 \), consider the following quadratic function of \( \varepsilon \) :\n\n\[ \varphi \left( \varepsilon \right) = c{\varepsilon }... | Yes |
Theorem 2 For a given positive scalar \( {\varepsilon }_{1} > 0 \), if there exist symmetric positive-definite matrices \( {P}_{\mathrm{s}} \) and \( {P}_{\mathrm{f}} \), satisfying the following LMIs:\n\n\[ \n{\Omega }_{0} < 0 \n\]\n\n(7)\n\n\[ \n{\Omega }_{0} + {\varepsilon }_{1}{\Omega }_{1} < 0 \n\]\n\n(8)\n\n\[ \n... | Proof For any non-zero \( n \) -dimension vector \( x \), pre and post multiplying (7) by \( {x}^{\mathrm{T}} \) and its transpose, respectively, we have\n\n\[ \n{x}^{\mathrm{T}}{\Omega }_{0}x < 0 \n\]\n\n(10)\n\n\[ \n{x}^{\mathrm{T}}{\Omega }_{0}x + {\varepsilon }_{1}{x}^{\mathrm{T}}{\Omega }_{1}x < 0 \n\]\n\n(11)\n\n... | Yes |
Lemma 5 The closed-loop slow subsystem\n\n\[ \n{\dot{x}}_{\mathrm{s}}\left( t\right) = \left( {{A}_{0} + {B}_{0}{K}_{0}}\right) {x}_{\mathrm{s}}\left( t\right) \]\n\n(17)\n\nis asymptotically stabilizable, if and only if there exist a positive definite matrix \( {X}_{\mathrm{s}} \) and matrix \( {Y}_{\mathrm{s}} \) sat... | Proof From the inequality (18), pre and post multiplying it by \( {X}_{\mathrm{s}}^{-\mathrm{T}} \) and \( {X}_{\mathrm{s}}^{-1} \), respectively, let \( {X}_{\mathrm{s}} = {\bar{P}}_{\mathrm{s}}^{-1},{Y}_{\mathrm{s}} = {K}_{0}{\bar{P}}_{\mathrm{s}}^{-1} \), then (18) is equivalent to (15). By further using Lemma 2, th... | Yes |
Theorem 1 If there exist matrices \( P \) and \( Q > 0 \) such that the following LMI conditions are satisfied for all allowable uncertainties,\n\n\[ \n{E}^{\mathrm{T}}P = {P}^{\mathrm{T}}E \geq 0 \n\]\n\n\[ \n{\Omega }_{1} = \left( \begin{matrix} {\widehat{A}}^{\mathrm{T}}P + {P}^{\mathrm{T}}\widehat{A} + Q & P\wideha... | Proof Let \( V\left( x\right) \) be storage function defined as\n\n\[ \nV\left( x\right) = \frac{{x}^{\mathrm{T}}{E}^{\mathrm{T}}{Px}}{2} \n\]\n\nThen,\n\n\[ \n\dot{V}\left( x\right) - {w}^{\mathrm{T}}y + \frac{{x}^{\mathrm{T}}{Qx}}{2} \n\]\n\n\[ \n= \frac{{\dot{x}}^{\mathrm{T}}{Px} + {x}^{\mathrm{T}}{E}^{\mathrm{T}}P\... | Yes |
Theorem 2 If there exist matrices \( P \) and \( Q > 0 \) and positive scalar \( \in \) such that the following LMI conditions are satisfied for all allowable uncertainties:\n\n\[ \n{E}^{\mathrm{T}}P = {P}^{\mathrm{T}}E \geq 0 \]\n\n\[ \n{\Omega }_{2} = \left( \begin{matrix} \left( {1,1}\right) & * & * & * \\ {B}^{\mat... | Proof Let\n\n\[ \n\phi = \left( \begin{matrix} {A}^{\mathrm{T}}P + {P}^{\mathrm{T}}A + Q & {P}^{\mathrm{T}}B - {C}^{\mathrm{T}} \\ {B}^{\mathrm{T}}P - C & - \left( {D + {D}^{\mathrm{T}}}\right) \end{matrix}\right) .\n\nThen, the inequality (11) is equivalent to the following:\n\n\[ \n\phi + {GF}\left( t\right) H + {H}^... | Yes |
Theorem 4 If there exist a nonsingular matrix \( X \) and matrices \( Q > 0, Y \in {\mathbb{R}}^{m \times n} \) and positive scalars \( \;\epsilon > 0, \) such that the following linear matrix inequalities (LMI) conditions hold,\n\n\[ \nX{E}^{\mathrm{T}} = E{X}^{\mathrm{T}} \geq 0 \n\]\n\n(35)\n\n\[ \n{\Omega }_{2} = \... | Proof Substitution of (35) into (34) yields\n\n\[ \n{\Omega }_{3} = \left( \begin{matrix} {\widetilde{\sum }}_{11} & {\widetilde{\sum }}_{12} & \epsilon {G}_{1} & X{H}_{1}^{\mathrm{T}} \\ * & - \left( {D + {D}^{\mathrm{T}}}\right) & - {\epsilon G} & {H}_{2}^{\mathrm{T}} \\ * & - \epsilon {G}_{2}^{\mathrm{T}} & - \frac{... | Yes |
Lemma 1\n\n\[ \left\\{ \begin{array}{l} {p}_{t} - \lambda {p}_{x} = - \frac{1}{4}\left( {{p}^{2} + {3pq}}\right) \frac{{g}^{\prime \prime }\left( \varphi \right) }{{g}^{\prime }\left( \varphi \right) } \\ {q}_{t} + \lambda {q}_{x} = - \frac{1}{4}\left( {{q}^{2} + {3pq}}\right) \frac{{g}^{\prime \prime }\left( \varphi \... | Proof Making use of (15), (16), and (20), we have\n\n\[ {p}_{t} - \lambda {p}_{x} = {v}_{t} + {\left\lbrack {g}^{\prime }\left( \varphi \right) \right\rbrack }^{-\frac{1}{2}}{w}_{t} - \frac{1}{2}w{\varphi }_{t}{\left\lbrack {g}^{\prime }\left( \varphi \right) \right\rbrack }^{-\frac{3}{2}}{g}^{\prime \prime }\left( \va... | Yes |
Lemma 2 If \( {u}_{1}\left( x\right) \) satisfies\n\n\[ \n{u}_{1}\left( x\right) \geq \sqrt{{f}^{\prime }\left( {u}_{0}\right) } \cdot {u}_{0}^{\prime }\left( x\right) \n\]\n\n(28)\n\nthen it holds that\n\n\[ \n0 \leq p\left( {t, x}\right) \leq \mathop{\sup }\limits_{{y \in \mathbb{R}}}{p}_{0}\left( y\right) \n\]\n\n(2... | Proof In fact, passing through any point \( \left( {t, x}\right) \), we can draw two characteristics, defined by \( \xi = {\xi }_{ \pm }\left( {\tau ;t, x}\right) \), which satisfy\n\n\[ \n\left\{ \begin{array}{l} \frac{\mathrm{d}{\xi }_{ \pm }}{\mathrm{d}\tau } = \pm \lambda \left( {\tau ;{\xi }_{ \pm }\left( {\tau ;t... | Yes |
Lemma 3\n\n\[ \left\\{ \begin{array}{l} {r}_{t} - \lambda {r}_{x} = - \frac{1}{4}\left( {{3p} + {2q}}\right) \frac{{g}^{\prime \prime }\left( \varphi \right) }{{g}^{\prime }\left( \varphi \right) } \cdot r - \frac{3p}{4}\frac{{g}^{\prime \prime }\left( \varphi \right) }{{g}^{\prime }\left( \varphi \right) } \cdot s - \... | Proof By a direct calculation, we can get\n\n\[ {r}_{t} - \lambda {r}_{x} = {p}_{tx} - \lambda {p}_{xx} \]\n\n\[ = {\left( {p}_{t} - \lambda {p}_{x}\right) }_{x} + {\lambda }_{x}{p}_{x} \]\n\n\[ = {\lambda }_{x}{p}_{x} - \frac{1}{4}{\left\\lbrack \left( {p}^{2} + 3pq\right) \frac{{g}^{\prime \prime }\left( \varphi \rig... | Yes |
Lemma 4 In the existence domain of the smooth solution, it holds that\n\n\[\n\\left\\{ \\begin{array}{l} - K{\\mathrm{e}}^{Kt} < r < K{\\mathrm{e}}^{Kt} \\\\ - K{\\mathrm{e}}^{Kt} < s < K{\\mathrm{e}}^{Kt} \\end{array}\\right.\n\]\n\nwhere \( K \) is a sufficiently large positive constant. | Proof Denote\n\n\[\nM = \\max \\left\\{ {\\left| {-\\frac{1}{4}\\left( {{3p} + {2q}}\\right) \\frac{{g}^{\\prime \\prime }\\left( \\varphi \\right) }{{g}^{\\prime }\\left( \\varphi \\right) }}\\right| ,\\left| {-\\frac{3p}{4}\\frac{{g}^{\\prime \\prime }\\left( \\varphi \\right) }{{g}^{\\prime }\\left( \\varphi \\right... | Yes |
Corollary 1 Let the conditions of Theorem 2 are fulfilled. Then, the unique linear optimal method of recovery is\n\n\\[ \n\\widehat{\\varphi }\\left( {Ix}\\right) = \\mathop{\\sum }\\limits_{{j = 1}}^{n}{y}_{j}\\frac{\\partial \\omega }{\\partial {t}_{j}}\\left( {M}_{0}\\right) \n\\]\n\nwhere \\( {y}_{1},\\cdots ,{y}_{... | Proof For\n\n\\[ \na = \\left( \\begin{matrix} {a}_{1} \\\\ \\vdots \\\\ {a}_{n} \\end{matrix}\\right) ,\\;b = \\left( \\begin{matrix} {b}_{1} \\\\ \\vdots \\\\ {b}_{n} \\end{matrix}\\right)\n\\]\n\nput\n\n\\[ \n\\left( {a, b}\\right) = \\mathop{\\sum }\\limits_{{j = 1}}^{n}{a}_{j}{b}_{j}\n\\]\n\nSet\n\n\\[ \n\\widehat... | Yes |
Theorem 6 Let \( W \) be a convex and centrally-symmetric set and \( Y \) be a normed linear space. Assume that there exist such linear continuous functionals \( \widehat{\varphi } \) and \( \widehat{x} \in W \) that\n\n(i) \( \mathop{\sup }\limits_{{x \in W}}\left| {{Lx} - \widehat{\varphi }\left( {Ix}\right) }\right|... | Proof It follows from the generalization of Theorem 4 that\n\n\[ E\left( {L, W, I,\delta }\right) = \mathop{\sup }\limits_{\substack{{x \in W} \\ {\parallel {Ix}\parallel \leq \delta } }}\left| {Lx}\right| \geq \left| {L\widehat{x}}\right| = L\widehat{x}. \]\n\nOn the other hand, using the conditions (i) -(iii), for al... | Yes |
Theorem 8 Assume that there exist such \( {\lambda }_{j} \geq 0, j = 1,\cdots, n \), and an element \( \widehat{x} \in W \), for which \( {\begin{Vmatrix}{I}_{j}\widehat{x}\end{Vmatrix}}_{{Y}_{j}} \leq {\delta }_{j}, j = 1,\cdots, n \), and\n\n\[ \n{\begin{Vmatrix}{I}_{0}\widehat{x}\end{Vmatrix}}_{{Y}_{0}} \geq {\left(... | Proof Let \( \varphi : {Y}_{k + 1} \times \cdots \times {Y}_{n} \rightarrow {Y}_{0} \) be an arbitrary method of recovery. Then,\n\n\[ \n2\parallel {I}_{0}\widehat{x}{\parallel }_{{Y}_{0}} = \parallel {I}_{0}\widehat{x} - \varphi \left( 0\right) - \left( {{I}_{0}\left( {-\widehat{x}}\right) - \varphi \left( 0\right) }\... | Yes |
Corollary 2 If \( {s}_{0} > 0 \) and \( \delta < 1 \), then for all \( {\theta }_{j},\left| {\theta }_{j}\right| \leq 1,0 < \left| j\right| \leq N \), the methods | \[ \widehat{\varphi }\left( \widetilde{x}\right) \left( t\right) = \mathop{\sum }\limits_{{0 < \left| j\right| \leq {\lambda }_{2}^{\frac{1}{2k}}}}{\left( \mathrm{i}j\right) }^{k}{\widetilde{x}}_{j}{\mathrm{e}}^{\mathrm{i}{jt}} + \mathop{\sum }\limits_{{{\lambda }_{2}^{\frac{1}{2k}} < \left| j\right| < {\lambda }_{1}^{... | No |
Lemma 2.1(stepsize rule) \( {}^{\left\lbrack {46}\right\rbrack } \) If \( p \) is a local peak selection of \( J \) near \( {v}_{0} \in {S}_{{L}^{ \bot }} \) s.t. \( {J}^{\prime }\left( {p\left( {v}_{0}\right) }\right) \neq 0, p \) is weak-continuous at \( {v}_{0} \) and \( p\left( {v}_{0}\right) = {t}_{0}{v}_{0} + {v}... | \[ J\left( {p\left( {v\left( s\right) }\right) }\right) - J\left( {p\left( {v}_{0}\right) }\right) < - \frac{{t}_{0}s{\begin{Vmatrix}{J}^{\prime }\left( p\left( {v}_{0}\right) \right) \end{Vmatrix}}^{2}}{4{C}_{0}}\text{ (a stepsize rule),} \] (2.3) \( {where}\begin{array}{r} v\left( s\right) = \frac{{v}_{0} - s{d}_{0}}... | Yes |
Theorem 2.1(LMMP) \( {}^{\left\lbrack {46}\right\rbrack } \) If \( p \) is a local peak selection of \( J \) near \( {v}_{0} \in {S}_{{L}^{ \bot }} \) s.t. \( p \) is weak-continuous at \( {v}_{0}, p\left( {v}_{0}\right) \notin L \) and\n\n\[ \n{v}_{0} = \arg \mathop{\min }\limits_{{v \in {S}_{{L}^{ \bot }}}}J\left( {p... | In the above results, the continuity of \( p \) has been weakened to weak-continuity. LMMP (2.4) characterizes a \( k \) -saddle by a local minimization problem \( \mathop{\min }\limits_{{v \in {S}_{{L}^{ \bot }}}}J\left( {p\left( v\right) }\right) \) . It can be numerically approximated by, e.g., a steepest descent me... | No |
Consider solving an elliptic PDE with nonlinear boundary condition of the form\n\n\[ \left( {-\Delta + {aI}}\right) u\left( x\right) = 0,\;x \in \Omega \text{ and }\frac{\partial u\left( x\right) }{\partial \nu } = g\left( {x, u\left( x\right) }\right) ,\;x \in \partial \Omega . \] | Its energy functional is\n\n\[ J\left( u\right) = {\int }_{\Omega }\frac{1}{2}\left( {{\left| \nabla u\left( x\right) \right| }^{2} + a{u}^{2}\left( x\right) }\right) \mathrm{d}x - {\int }_{\partial \Omega }\mathcal{G}\left( {x, u\left( x\right) }\right) \mathrm{d}{\sigma }_{x},\;u \in {H}^{1}\left( \Omega \right) ,\] ... | Yes |
Theorem 3. \( {1}^{\left\lbrack {77}\right\rbrack } \) If \( p \) is continuous at \( {v}^{ * } \in {S}_{{L}^{ \bot }} \) and \( p\left( {v}^{ * }\right) \notin L \), then \( {u}^{ * } = p\left( {v}^{ * }\right) \) is a critical point of \( J \) if and only if there is \( \mathcal{N}\left( {v}^{ * }\right) \) s.t. | \[ {J}^{\prime }\left( {p\left( {v}^{ * }\right) }\right) \bot p\left( v\right) - p\left( {v}^{ * }\right) ,\;\forall v \in \mathcal{N}\left( {v}^{ * }\right) \cap {S}_{{L}^{ \bot }}. \] | Yes |
Lemma 4.1 Let \( B = L \oplus {L}^{\prime },0 < \theta < 1,{v}_{0} \in {S}_{{L}^{\prime }} \) be given. If \( p \) is a local peak selection of \( J \) w.r.t. \( L \) at \( {v}_{0} \) s.t. \( {J}^{\prime }\left( {p\left( {v}_{0}\right) }\right) \neq 0 \), then there is a PPG \( \Psi \left( {p\left( {v}_{0}\right) }\rig... | (a) \( \Psi \left( {p\left( {v}_{0}\right) }\right) \in {L}^{\prime },0 < \begin{Vmatrix}{\Psi \left( {p\left( {v}_{0}\right) }\right) }\end{Vmatrix} \leq M \) where \( M \geq 1 \) is the bound of the linear projection \( \mathcal{P} \) from \( B \) to \( {L}^{\prime } \) ;\n\n(b) \( \left\langle {{J}^{\prime }\left( {... | Yes |
Example 4.1(eigen-pairs of \( {\Delta }_{p} \) ) Consider finding eigen-pairs \( \left( {\lambda, u}\right) \in \mathbb{R} \times \left( {B\smallsetminus \{ 0\} }\right) \) s.t.\n\n\[ - {\Delta }_{p}u\left( x\right) = \lambda {\left| u\left( x\right) \right| }^{q - 2}u\left( x\right) ,\;u \in {W}_{0}^{1, p}\left( \Omeg... | Case a \( p = q = {1.75} < 2 \), we obtain \( {\lambda }_{1} = {4.2458},{\lambda }_{2} = {9.3173},{\lambda }_{3} = {9.4078} \) , \( {\lambda }_{4} = {14.2805},{\lambda }_{5} = {16.8378},{\lambda }_{6} = {17.2546},{\lambda }_{7} = {23.3660}. \n\nCase \( b\;p = q = {2.5} > 2 \), we obtain \( {\lambda }_{1} = {6.3547},{\l... | Yes |
Theorem 1 Let \( \operatorname{GF}\left( q\right) = \operatorname{GF}\left( {2}^{t}\right), t > 6 \) . Then, \( M = \operatorname{cir}\left( {a, b,1,1}\right) \) is an MDS matrix if one of the conditions occurs:\n\n(i) \( a = x + c\left( {c \in {F}_{2}}\right), b = a + 1 \) ;\n\n(ii) \( a = x + c\left( {c \in {F}_{2}}\... | Proof Let one of conditions (i) wid. We substitute the pair \( \left( {a\left( x\right), b\left( x\right) }\right) \) in the polynomial \( \det \left( {M}_{i}\right) \) for \( \left( {a, b}\right) \) . Then, it is easy to verify that the degree of nonlinearity of any polynomial \( \det \left( {M}_{i}\right) \) will be ... | No |
Theorem 3 Let each of the elements \( - 3,2,5,7 \) be the quadratic non-residue in the field \( \mathrm{{GF}}\left( q\right), q = {p}^{t}, p > 3 \) . Suppose that for all \( a \in \mathrm{{GF}}\left( q\right) \smallsetminus \{ 0, \pm 1\} ,\left( {a + 1}\right) \) is quadratic non-residue and \( b = a + 1 \) . Then, the... | Proof The proof is carried out by checking the above general restrictions on the elements of the MDS matrix \( M \) . We will give some parts of it only.\n\nLet \( a + 1 = {a}^{-1} \) . Then, \( {a}^{2} + a - 1 = 0,4{\left( a + {2}^{-1}\right) }^{2} = {5.5} \) is a quadratic residue.\n\nLet \( a + 1 = {a}^{2} + a - 1 \... | No |
Theorem 1 If \( {\mu }_{0} \) is the largest eigenvalue (in absolute value) of \( {\widehat{L}}_{k}\left( \mu \right) \) with \( u = \) \( {\left\lbrack {u}_{1},{u}_{2},\cdots ,{u}_{k}\right\rbrack }^{\mathrm{T}} \) a corresponding unit eigenvector, letting \( {x}_{1} = {Q}_{k}u \) and \( {\lambda }_{1} = {\lambda }_{0... | Proof From (3) for \( \widehat{L}\left( \mu \right) \), we have\n\n\[ \n\widehat{L}\left( {\mu }_{0}\right) {x}_{1} = {\mu }_{0}{h}_{k + 1, k}{u}_{k}\widehat{A}{q}_{k + 1} + \left\lbrack {\left( {\widehat{C}{Q}_{k} - \widehat{A}{Q}_{k}{G}_{k}}\right) {\mu }_{0} + \left( {\widehat{D}{Q}_{k} - \widehat{A}{Q}_{k}{F}_{k}}\... | Yes |
Consider the inhomogeneous advection partial differential equation\n\n\[ \n{u}_{t} + u{u}_{x} = x + x{t}^{2},\;u\left( {1, t}\right) = t.\n\] | Using the modified asymptotic ADM, we first rewrite the equation (14) into an operator form\n\n\[ \n{L}_{x}{u}^{2} = 2\left( {x + x{t}^{2}}\right) - 2{L}_{t}u.\n\]\n\nApplying the inverse operator \( {L}_{x}^{-1} \) to both sides of (15), we have\n\n\[ \n{u}^{2} = {u}^{2}\left( {0, t}\right) + 2{L}_{x}^{-1}\left( {x + ... | Yes |
For the two-by-two linear system (3), suppose that \( W \) is symmetric positive definite and \( T \) is symmetric semi-positive definite, then the spectral radius of the NPMHSS iteration matrix \( {M}_{\alpha } \) satisfies\n\n\[ \rho \left( {M}_{\alpha }\right) \leq \frac{\sqrt{{\alpha }^{2} + {\mu }_{\max }^{2}}}{\a... | Proof Let\n\n\[ H = \left( \begin{matrix} {V}^{-1}W & 0 \\ 0 & {V}^{-1}W \end{matrix}\right) ,\;S = \left( \begin{matrix} 0 & {V}^{-1}T \\ - {V}^{-1}T & 0 \end{matrix}\right) .\n\nBy direct computation, we have\n\n\[ \rho \left( {M}_{\alpha }\right) = \rho \left( {{\left( \alpha I + H\right) }^{-1}\left( {{\alpha I} - ... | Yes |
Theorem 2 The optimal parameter of the new PMHSS iteration method is given by\n\n\[ \n{\alpha }_{\mathrm{{opt}}} = \frac{{\mu }_{\max }^{2}}{{\lambda }_{\min }} \n\]\n\nwhere \( {\mu }_{\max } \) and \( {\lambda }_{\min } \) are defined as in Theorem 1, and the optimal convergent rate of the NPMHSS iteration method is\... | Proof Let \( f\left( \alpha \right) = \frac{\sqrt{{\alpha }^{2} + {\mu }_{\max }^{2}}}{\alpha + {\lambda }_{\min }} \) . By direct computation, we have\n\n\[ \n{f}^{\prime }\left( \alpha \right) = \frac{\alpha {\lambda }_{\min } - {\mu }_{\max }^{2}}{{\left( \alpha + {\lambda }_{\min }\right) }^{2} \cdot \sqrt{{\alpha ... | No |
Theorem 5 Let \( g\left( t\right) \) be a nonzero sufficiently smooth periodic function. Then, the Caputo fractional derivative \( {}_{\mathrm{{RL}}}{\mathrm{D}}_{0, t}^{\alpha }g\left( t\right) \) is aperiodic. | The proofs of Lemma 7 and Theorem 5 are contained in [23]. | No |
Theorem 6 Let \( f\left( t\right) \) be a nonzero sufficiently smooth periodic function. Then, the Grunwald-Letnikov fractional derivative \( {}_{\mathrm{{GL}}}{\mathrm{D}}_{0, t}^{\alpha }f\left( t\right) \) is aperiodic. | Proof According to (6), there holds\n\n\[ \n{}_{\mathrm{{GL}}}{\mathrm{D}}_{0, t}^{\alpha }f\left( t\right) = {}_{\mathrm{C}}{\mathrm{D}}_{0, t}^{\alpha }f\left( t\right) + \mathop{\sum }\limits_{{k = 0}}^{{m - 1}}\frac{{f}^{\left( k\right) }\left( {0}^{ + }\right) {\left( t\right) }^{k - \alpha }}{\Gamma \left( {k - \... | Yes |
Theorem 7 The time invariant fractional order systems represented by (7) cannot have any periodic solution. | Some one can easily prove this theorem using Lemmas 3~6 and relations (17)~(20). But the proof of this theorem is based on Lemma 5 and we have already discussed in Remark 1 that the proof of Lemma 5 is not correct. In conclusion, it has to be highlight that the statements of Theorems \( 1 \sim 2 \) from [19] and the ma... | No |
Theorem 9 Let \( f\left( x\right) \) satisfy the Lipschitz condition. Then, the solutions of the following fractional order system are memory dependent:\n\n\[ \n{}_{\mathrm{C}}{\mathrm{D}}_{a, t}^{\alpha }x = f\left( x\right) ,\;x\left( a\right) = {x}_{a}.\n\]\n\n(31)\n\nIt means that the solution of (31), which is den... | The above result can be extracted from the non-constant initialization concept of fractional order system in [37]. | No |
Theorem 1 The structures of the Riemann solutions are stable for limits of solutions to the compressible Euler equations as the modified Chaplygin gas pressure tends to the Chaplygin gas pressure, which are illustrated in the following. | (i) When \( \left( {{u}_{ + },{\rho }_{ + }}\right) \in \mathrm{I}\left( {{u}_{ - },{\rho }_{ - }}\right) \), the limit solution of the Riemann problem (1) and (4) with (3) as \( A \rightarrow 0 \) is \( \overleftarrow{R} + \overrightarrow{R} \), which is the solution of the Riemann problem (1) and (4) with (2) (see Fi... | Yes |
Lemma 4 \( \mathop{\lim }\limits_{{A \rightarrow 0}}{\rho }_{ * } = + \infty \) and \( B = \mathop{\lim }\limits_{{A \rightarrow 0}}A{\rho }_{ * }^{\gamma } < + \infty \) . | Proof If \( \mathop{\lim }\limits_{{A \rightarrow 0}}{\rho }_{ * } = + \infty \) is not true, suppose \( \mathop{\lim }\limits_{{A \rightarrow 0}}{\rho }_{ * } = M < + \infty \) . Then, letting \( A \rightarrow 0 \), from (19), we get that\n\n\[ \n{u}_{ - } - {u}_{ + } = \frac{1}{{\rho }_{ - }} + \frac{1}{{\rho }_{ + }... | Yes |
Lemma 5\n\n\[ \mathop{\lim }\limits_{{A \rightarrow 0}}{u}_{ * } = {u}_{ - } - {\left( \frac{1}{{\rho }_{ - }}\left( B + \frac{1}{{\rho }_{ - }}\right) \right) }^{1/2} = {u}_{ + } + {\left( \frac{1}{{\rho }_{ + }}\left( B + \frac{1}{{\rho }_{ + }}\right) \right) }^{1/2} \mathrel{\text{:=}} {u}_{\delta } \]\n\n(22)\n\na... | Proof Equation (22) can be obtained from (17) and (18) directly. From the R-H relation (11) for \( {\overleftarrow{S}}_{A} \) and \( {\overrightarrow{S}}_{A} \), we have\n\n\[ \mathop{\lim }\limits_{{A \rightarrow 0}}{\sigma }_{ - } = \mathop{\lim }\limits_{{A \rightarrow 0}}\frac{{\rho }_{ - }{u}_{ - } - {\rho }_{ * }... | Yes |
Lemma 6\n\n\\[ \n\\mathop{\\lim }\\limits_{{A \\rightarrow 0}}{\\int }_{{\\sigma }_{ - }t}^{{\\sigma }_{ + }t}{\\rho }_{ * }{dx} = \\left( {\\sigma \\left\\lbrack \\rho \\right\\rbrack - \\left\\lbrack {\\rho u}\\right\\rbrack }\\right) t,\\;\\mathop{\\lim }\\limits_{{A \\rightarrow 0}}{\\int }_{{\\sigma }_{ - }t}^{{\\... | Proof From the first equations of the R-H relation (11) for \\( {\\overleftarrow{S}}_{A} \\) and \\( {\\overrightarrow{S}}_{A} \\), we get\n\n\\[ \n\\left\\{ \\begin{array}{l} {\\sigma }_{ - }\\left( {{\\rho }_{ - } - {\\rho }_{ * }}\\right) = {\\rho }_{ - }{u}_{ - } - {\\rho }_{ * }{u}_{ * } \\\\ {\\sigma }_{ + }\\lef... | Yes |
Theorem 3 Let \( \left( {{u}_{ + },{\rho }_{ + }}\right) \in {\mathrm{V}}_{A}\left( {{u}_{ - },{\rho }_{ - }}\right) \) . For any fixed \( A > 0 \), assume \( \left( {u,\rho }\right) \) is the Riemann solution of system (1) and (3) with the initial data (4). Then, as \( A \rightarrow 0,\left( {u,\rho }\right) \) conver... | Proof Set \( \xi = \frac{x}{t} \), then for any fixed \( A > 0 \), the Riemann solutions can be expressed as \[ \left( {u,\rho }\right) \left( \xi \right) = \left\{ \begin{array}{ll} \left( {{u}_{ - },{\rho }_{ - }}\right) , & \xi < {\sigma }_{ - }, \\ \left( {{u}_{ * },{\rho }_{ * }}\right) , & {\sigma }_{ - } < \xi <... | Yes |
Lemma 1 Suppose \( \left( {u, v, w}\right) \left( {x, y}\right) \) is a weak discontinuous solution of (2). Then, if one of \( {\partial }_{ - }u,{\partial }_{ - }v,{\partial }_{ - }w \) is discontinuous at the point \( P\left( {x, y}\right) \), then the others must be discontinuous at \( P \) . | Proof Considering (2), we have\n\n\[ \frac{1}{2}\left( {{\partial }_{ + }u + {\partial }_{ - }u}\right) = {\partial }_{0}u = - {a}_{12}{v}_{y} = - \frac{{a}_{12}}{2\delta }\left( {{\partial }_{ + }v - {\partial }_{ - }v}\right) . \]\n\n(16)\n\nConsider the second equation of (6) and (16) at points \( {P}_{1} \) and \( ... | Yes |
Theorem 1 Suppose \( \\left( {u, v, w}\\right) \\left( {x, y}\\right) \) is a weak discontinuous solution of (2). If one of \( {\\partial }_{ - }u,{\\partial }_{ - }v,{\\partial }_{ - }w \) is discontinuous at the point \( {P}_{0}\\left( {{x}_{0},{y}_{0}}\\right) \), then \( {\\partial }_{ - }u,{\\partial }_{ - }v \), ... | Proof From Lemma 1, we may assume that \( {\\left\\lbrack {\\partial }_{ - }v\\right\\rbrack }_{{P}_{0} - 0}^{{P}_{0} + 0} \\neq 0 \) . At any point \( P\\left( {x, y}\\right) \) on \( {C}_{ + } \) passing through \( {P}_{0} \), from (13), we have\n\n\[ \n{\\partial }_{ + }{\\left\\lbrack {\\partial }_{ - }v\\right\\rb... | Yes |
Lemma 1 Let \( p\left( t\right) = \max \{ t,0\} \) . For any \( \epsilon > 0 \), we have\n\n\[ \mathop{\lim }\limits_{{\epsilon \rightarrow 0}}{p}_{\epsilon }\left( t\right) = p\left( t\right) \] | Proof For any \( \epsilon > 0 \), by the definition of \( {p}_{\epsilon }\left( t\right) \), we have\n\n\[ p\left( t\right) - {p}_{\epsilon }\left( t\right) = \left\{ \begin{array}{lll} 0, & \text{ if } & t < 0, \\ t - \left( {1 + \epsilon }\right) \frac{{t}^{2}}{2\epsilon }, & \text{ if } & 0 \leq t < \epsilon , \\ \f... | Yes |
Lemma 2 For any \( \epsilon > 0,{p}_{\epsilon }\left( t\right) \) is first-order continuously differentiable on \( \mathbb{R} \), where | \[ {p}_{\epsilon }^{\prime }\left( t\right) = \left\{ \begin{array}{lll} 0, & \text{ if } & t < 0 \\ \left( {1 + \epsilon }\right) \frac{t}{\epsilon }, & \text{ if } & 0 \leq t < \epsilon \\ 1 + \frac{{\epsilon }^{2}}{t}, & \text{ if } & t \geq \epsilon \end{array}\right. \] | No |
Lemma 3 For any given \( x \in {\mathbb{R}}^{n},\epsilon > 0 \) and \( \rho > 0 \) , \[ 0 \leq {F}_{1}\left( {x,\rho }\right) - F\left( {x,\rho ,\epsilon }\right) \leq \frac{1}{2}{m\rho \epsilon } \] | Proof For any \( x \in {\mathbb{R}}^{n} \) and \( i = 1,2,\cdots, m \), by the definition of \( p\left( t\right) \) and \( {p}_{\epsilon }\left( t\right) \), we have \[ p\left( {{g}_{i}\left( x\right) }\right) - {p}_{\epsilon }\left( {{g}_{i}\left( x\right) }\right) = \left\{ \begin{array}{lll} 0, & \text{ if } & {g}_{... | Yes |
Theorem 2 Let \( {x}^{ * } \) be an optimal solution of \( \left( {P}_{\rho }\right) \) and \( \bar{x} \in {\mathbb{R}}^{n} \) an optimal solution of \( \left( {\mathrm{{SP}}}_{\rho }\right) \) for some \( \rho \) and \( \epsilon \) . Then, we have that\n\n\[ 0 \leq {F}_{1}\left( {{x}^{ * },\rho }\right) - F\left( {\ba... | Proof From Lemma 3, for \( \rho > 0 \), we have that\n\n\[ 0 \leq {F}_{1}\left( {{x}^{ * },\rho }\right) - F\left( {{x}^{ * },\rho ,\epsilon }\right) \leq \frac{1}{2}{m\rho \epsilon } \]\n\n\[ 0 \leq {F}_{1}\left( {\bar{x},\rho }\right) - F\left( {\bar{x},\rho ,\epsilon }\right) \leq \frac{1}{2}{m\rho \epsilon }.\]\n\n... | Yes |
Theorem 3 Let \( {x}^{ * } \) be an optimal solution of \( \left( {\mathrm{P}}_{\rho }\right) \) and \( \bar{x} \in {\mathbb{R}}^{n} \) an optimal solution of \( \left( {\mathrm{{SP}}}_{\rho }\right) \) for some \( \rho \) and \( \epsilon \) . Furthermore, let \( {x}^{ * } \) be feasible to \( \left( \mathrm{P}\right) ... | Proof It is easy to see that\n\n\[ \mathop{\sum }\limits_{{i \in I}}p\left( {{g}_{i}\left( {x}^{ * }\right) }\right) = 0 \]\n\nand by Theorem 2, we have that\n\n\[ 0 \leq {F}_{1}\left( {{x}^{ * },\rho }\right) - F\left( {\bar{x},\rho ,\epsilon }\right) = f\left( {x}^{ * }\right) + \rho \mathop{\sum }\limits_{{i \in I}}... | Yes |
Theorem 4 Let \( f \) and \( {g}_{i}\left( {i \in I}\right) \) in (P) be convex. Let \( {x}^{ * } \) be an optimal solution to (P) and \( {y}^{ * } \in {\mathbb{R}}^{m} \) a Lagrange multiplier vector corresponding to \( {x}^{ * } \) . Then, for some \( \epsilon > 0 \) ,\n\n\[ \n{F}_{1}\left( {{x}^{ * },\rho }\right) -... | Proof By the convexity of \( f \) and \( {g}_{i}, i = 1,2,\cdots, m \), we have\n\n\[ \n\left\{ \begin{array}{ll} f\left( x\right) \geq f\left( {x}^{ * }\right) + \nabla f{\left( {x}^{ * }\right) }^{\mathrm{T}}\left( {x - {x}^{ * }}\right) , & \forall x \in {\mathbb{R}}^{n}, \\ {g}_{i}\left( x\right) \geq {g}_{i}\left(... | Yes |
Consider the example in \( \left\lbrack {4,7,9}\right\rbrack \) , \( \left( {P4.1}\right) \;{min}\;f\left( x\right) = {x}_{1}^{2} + {x}_{2}^{2} + 2{x}_{3}^{2} + {x}_{4}^{2} - 5{x}_{1} - 5{x}_{2} - {21}{x}_{3} + 7{x}_{4} \) \( \text{s.t.}{g}_{1}\left( x\right) = 2{x}_{1}^{2} + {x}_{2}^{2} + {x}_{3}^{2} + 2{x}_{1} + {x}_... | Let \( {x}^{0} = \left( {0,0,0,0}\right) ,\epsilon = {10}^{-6},{\epsilon }_{0} = {0.1},{\rho }_{0} = {10},\eta = {0.01} \) and \( N = 4 \) . We use Algorithm 1 to solve (P4.1). Numerical results are given in Table 1. Therefore, we get an approximate solution \( {x}^{ * } = \left( {{0.169928},{0.834624},{2.008881}, - {0... | Yes |
\[ \text{(P4.2)}\min f\left( x\right) = {\left( {x}_{1} - 2\right) }^{2} + {\left( {x}_{2} - 1\right) }^{2} + 1 \] \[ \text{s.t.}{g}_{1}\left( x\right) = {x}_{1} + {x}_{2} - 2 \leq 0\text{,} \] \[ {g}_{2}\left( x\right) = {x}_{1}^{2} - {x}_{2} \leq 0. \] | Let \( {x}^{0} = \left( {1,2}\right) ,\epsilon = {10}^{-6},{\epsilon }_{0} = {0.5},{\rho }_{0} = {10},\eta = {0.05} \) and \( N = {10} \) . We use Algorithms 1,2 and 3 to solve (P4.2). Numerical results are given in Table 4. The solution generated by Algorithm 1 at the 2-nd iteration. The approximate solution is \( {x}... | Yes |
Proposition 1 The functions \( {S}^{\left( i, j\right) } \) and \( S\left( {a, b}\right) \) defined in (5) and (6) where the elements satisfy (4) have the following symmetric property:\n\n\[ \n{S}^{\left( i, j\right) } = {S}^{\left( j, i\right) },\;S\left( {a, b}\right) = S\left( {b, a\right) .\n\] | The proof is not simple but similar to the autonomous case (see the section 4.3 in [3]). Here, we skip it. | No |
Theorem 3 The \( N \) -soliton solution of Q3 (31), which we denote by \( {u}^{\left( N\right) } = {u}_{n, m}^{\left( N\right) } \), is given by the formula\n\n\[ \n{u}^{\left( N\right) } = {AF}\left( {a, b}\right) \left\lbrack {1 - \left( {a + b}\right) S\left( {a, b}\right) }\right\rbrack + {BF}\left( {a, - b}\right)... | Proof The proof is similar to the autonomous case. We need three steps. Let us proceed them step by step.\n\n\( \textbf{Step 1}\; \) We first introduce a new associated dependent variable \( \;{U}^{\left( N\right) } = {U}_{n, m}^{\left( N\right) }\; \) given by\n\n\[ \n{U}^{\left( N\right) } = \left( {a + b}\right) {AF... | Yes |
Theorem 1 \( \; \) Under the conditions that \( t \in (0,1\rbrack ,\widehat{u} \in {H}^{1 + t}\left( \Omega \right) ,{\widehat{u}}_{1} \in {H}^{1 + t}\left( {\Omega }_{1}\right) \) and \( {u}_{0} \in \) \( {H}^{\frac{1}{2} + t}\left( \Gamma \right) \), there exists a constant \( c \) satisfies\n\n\[ \n{\begin{Vmatrix}\... | Proof According to [10], there exists a constant \( C > 0 \) independent with \( h \), such that\n\n\[ \n{\begin{Vmatrix}\left( {\widehat{u}}_{h} - \widehat{u},{\widehat{u}}_{1h} - {\widehat{u}}_{1}\right) \end{Vmatrix}}_{E}^{2} \leq C\left\{ {\mathop{\inf }\limits_{{\left( {{u}_{h},{u}_{1h}}\right) \in {D}_{h}}}\left(... | Yes |
Lemma 2 Denote \( {e}_{h\Gamma } \mathrel{\text{:=}} {\phi }_{h\Gamma } - {\phi }_{h\Gamma }^{n} \). Then, it holds\n\n\[ \mathrm{D}G\left( {{\widehat{u}}_{h},{\bar{e}}_{h\Gamma }}\right) - \mathrm{D}G\left( {{\widehat{u}}_{h}^{n},{\bar{e}}_{h\Gamma }}\right) \geq \mathrm{D}G\left( {{\widehat{u}}_{h},{\widehat{u}}_{h} ... | Proof Set \( {\phi }_{h0} = {\phi }_{h\Gamma } + {u}_{0}^{h},{\phi }_{h0}^{n} = {\phi }_{h\Gamma }^{n} + {u}_{0}^{h} \). According to the linearity of \( \mathrm{D}G\left( {\cdot , \cdot }\right) \) in its second argument, there holds\n\n\[ \mathrm{D}G\left( {{\widehat{u}}_{h},{\bar{e}}_{h\Gamma }}\right) - \mathrm{D}G... | Yes |
Theorem 2 There exists a constant \( c \) independent with \( h \) such that Algorithm 1 converges when \( 0 < \theta < 2/c \) . When \( \theta = 1/c \), the algorithm converges fastest, and\n\n\[{\begin{Vmatrix}{e}_{h}^{n + 1}\end{Vmatrix}}_{{S}_{h}}^{2} \leq \left( {1 - \frac{1}{c}}\right) {\begin{Vmatrix}{e}_{h}^{n}... | This proof can be obtained by [11], we will not dwell on it. Without loss of generality, let the center of circle \( {\Gamma }_{0} \) be the origin and the radius of \( {\Gamma }_{0} \) be \( R \), the minimum circle have the same center as \( {\Gamma }_{0} \) and the radius \( r \), which contains \( \Omega \) . Then,... | No |
Theorem 1 Let \( A \in {\mathbb{C}}^{n \times n} \) be a positive-definite matrix, \( H = \frac{1}{2}\left( {A + {A}^{ * }}\right) \) and \( S = \) \( \frac{1}{2}\left( {A - {A}^{ * }}\right) \) be its Hermitian and skew-Hermitian parts, and \( \alpha \) be a non-zero constant. Then, the spectral radius \( \rho \left( ... | Proof Let\n\n\[ \widehat{L}\left( \alpha \right) = {TL}\left( \alpha \right) {T}^{-1} \]\n\nwhere\n\n\[ T = \left( {{\alpha I} + {P}^{-\frac{1}{2}}S{P}^{-\frac{1}{2}}}\right) {P}^{\frac{1}{2}}. \]\n\nThen,\n\n\[ \rho \left( {L\left( \alpha \right) }\right) = \rho \left( {\widehat{L}\left( \alpha \right) }\right) \]\n\n... | Yes |
Corollary 1 Let \( A, H \), and \( S \) be defined as in Theorem \( 1,{\lambda }_{\max } \) and \( {\lambda }_{\min } \) be the largest and the smallest eigenvalues of the matrix \( {P}^{-\frac{1}{2}}H{P}^{-\frac{1}{2}} \), and \( {\sigma }_{\max } \) be the largest singular value of the matrix \( {P}^{-\frac{1}{2}}S{P... | Proof We can obtain\n\n\[ \n\delta \left( \alpha \right) = \frac{{\sigma }_{\max }}{\sqrt{{\alpha }^{2} + {\sigma }_{\max }^{2}}}\mathop{\max }\limits_{{{\lambda }_{i} \in \lambda \left( {{P}^{-\frac{1}{2}}H{P}^{-\frac{1}{2}}}\right) }}\left| \frac{\alpha - {\lambda }_{i}}{{\lambda }_{i}}\right| \n\]\n\n\[ \n= \max \{ ... | Yes |
Theorem 1 Let \( {A}_{i},{B}_{i} \), and \( {C}_{i}\left( {i = 1,2,3}\right) \) be given, and \( X \in {\mathbb{H}}^{{m}_{1} \times {n}_{1}}, Y, Z \), and \( W \in {\mathbb{H}}^{{p}_{1} \times {q}_{1}} \) are variables. Suppose that the mixed triplets of the system of the Sylvester matrix equations (2) is consistent. D... | Proof From Lemma 1, the expression of \( X \) in the system of the coupled generalized Sylvester quaternion matrix equations (2) can be expressed as\n\n\[ \nX = {X}_{0} - {L}_{{A}_{1}}{U}_{2} - {T}_{3}{R}_{D}{B}_{1} - {L}_{M}{L}_{S}{T}_{1}{B}_{1} - {L}_{M}{T}_{2}{R}_{N}{B}_{1} \n\]\n\nwhere \( {X}_{0} \) is a special s... | Yes |
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