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Theorem 3.1 Assume that there exist two positive constants \( c, d \), with \( c < d \), such that\n\n(i) \( f\left( \xi \right) \geq 0,\forall \xi \in \left\lbrack {-c, d}\right\rbrack \) ;\n\n(ii) \( \frac{4{d}^{p}{p}^{-1}}{F\left( d\right) } < \frac{{c}^{p}{\left( p{l}^{p}\right) }^{-1}}{F\left( c\right) } \) ;\n\n(... | Proof We shall apply Theorem 2.1 to prove Theorem 3.1. Take \( X = {W}_{0}^{1, p}\left( \left\lbrack {0,1}\right\rbrack \right) \) endowed with the usual norm \( \parallel u\parallel = {\left( {\int }_{0}^{1}{\left| {u}^{\prime }\left( t\right) \right| }^{p}\mathrm{\;d}t\right) }^{\frac{1}{p}} \) . For each \( u \in X ... | Yes |
Theorem 3.2 Assume that there exist three positive constants \( {c}_{1},{c}_{2}, d \), with \( {c}_{1} < d < \frac{{c}_{2}}{p} \) , such that\n\n(i’) \( f\left( \xi \right) \geq 0,\forall \xi \in \left\lbrack {0,{c}_{2}}\right\rbrack \) ;\n\n(ii’) \( \frac{{pF}\left( d\right) }{6{d}^{p}} > \frac{p{l}^{p}F\left( {c}_{1}... | Proof Fix \( \lambda ,\mu, g \) as in the conclusion and take \( X,\Phi ,\Psi \) as in the proof of Theorem 3.1. Taking into account that the regularity assumptions of Theorem 2.2 on \( \Phi ,\Psi \) are satisfied and owing to the Maximum Principle, \( \left( {\mathrm{b}}_{3}\right) \) holds, our aim is to verify \( \l... | Yes |
Lemma 2.5 There exists a constant \( M > 0 \) such that \( {x}_{1}\left( t\right) \leq \frac{M}{\lambda },{y}_{1}\left( t\right) \leq M,{y}_{2}\left( t\right) \leq M \) for each solution \( \left( {x\left( t\right) ,{y}_{1}\left( t\right) ,{y}_{2}\left( t\right) }\right) \) of system (1.1) with all \( t \) large enough... | Proof Define \( V\left( t\right) = {\lambda x}\left( t\right) + {y}_{1}\left( t\right) + {y}_{2}\left( t\right) \) . For \( t \neq \left( {n + l - 1}\right) T, t \neq {nT} \), calculating the upper right derivative of \( V\left( t\right) \) along the solution of system (1.1), and combining with \( {d}_{2} > {d}_{1} \) ... | Yes |
Theorem 3.1 Let \( \left( {x\left( t\right) ,{y}_{1}\left( t\right) ,{y}_{2}\left( t\right) }\right) \) be any solution of system (1.1), and assume that \( \left( {\mathrm{H}}_{1}\right) \frac{{\lambda \beta }{\delta }_{1}{\mathrm{e}}^{-{d}_{1}\tau }}{a + {\delta }_{1}} < {d}_{2}. \)\n\nThen system (1.1) has a mature p... | Proof From Lemma 2.3 and Lemma 2.4, we can directly obtain the existence of the mature predator-extinction periodic solution \( \left( {{x}^{ * }\left( t\right) ,{y}_{1}^{ * }\left( t\right) ,0}\right) \) of system (1.1). Now we only need to prove the global attractivity of \( \left( {{x}^{ * }\left( t\right) ,{y}_{1}^... | Yes |
Consider the following problem:\n\n\[ \left\\{ \begin{array}{l} {D}_{0 + }^{\frac{7}{2}}u\left( t\right) + {u}^{\frac{t}{3}} = 0,\;t \in \left( {0,1}\right) , \\ u\left( 0\right) = {D}_{0 + }^{\frac{5}{2}}u\left( 0\right) = {D}_{0 + }^{\frac{3}{2}}u\left( 0\right) = 0,{D}_{0 + }^{\frac{1}{2}}u\left( 1\right) = 0, \end{... | So all the conditions in Theorem 3.1 and Corollary 3.2 are satisfied. By Theorem 3.1 and Corollary 3.2, we have problem (3.1) has a unique positive solution on the condition \( \left( {\mathrm{H}}_{1}\right) \left( {\mathrm{H}}_{2}\right) \left( {\mathrm{H}}_{3}\right) \) or \( \left( {\mathrm{H}}_{1}\right) \left( {\m... | Yes |
Consider the boundary value problem\n\n\[ \left\{ \begin{array}{l} {D}_{0 + }^{\frac{7}{2}}u\left( t\right) + \frac{\lambda \arctan \left( {u\left( t\right) }\right) }{\sqrt{t}} = 0,\;t \in \left( {0,1}\right) , \\ u\left( 0\right) = {D}_{0 + }^{\frac{5}{2}}u\left( 0\right) = {D}_{0 + }^{\frac{3}{2}}u\left( 0\right) = ... | We prove the conditions of Theorem 3.3 are satisfied. \( \phi \left( u\right) = \arctan u \) is nondecreasing. Taking into account the formula\n\n\[ \tan \left( {x - y}\right) = \frac{\tan x - \tan y}{1 + \tan x\tan y} \]\n\nwe know \( \tan \left( {x - y}\right) \leq \tan x - \tan y \) . So\n\n\[ \phi \left( u\right) -... | Yes |
Theorem 3.2 Suppose that \( \left. {{\mathrm{A}}_{1}) - {\mathrm{A}}_{2}}\right) \) and (3.1) hold. Furthermore, suppose that there exists a function \( H \in C\left( {D,\mathbb{R}}\right) \), where \( D = \left\{ {\left( {t, s}\right) : t \geq s \geq {t}_{0}}\right\} ,{D}_{0} = \left\{ {\left( {t, s}\right) : t > s \g... | Proof For the sake of contradiction, let \( y\left( t\right) \) be a nonoscillatory solution of (1.1). Without loss of generality, we may assume that \( y \) is an eventually positive solution of (1.1). We proceed as in the proof of Theorem 3.1 to get that \( \left( {{D}_{0 + }^{\alpha }y}\right) \left( t\right) \geq 0... | Yes |
Consider the fractional differential equation\n\n\[ \n{\left\lbrack \frac{1}{{t}^{6}}\left( {D}_{0 + }^{\frac{1}{2}}y\right) \left( t\right) \right\rbrack }^{\prime } + \frac{1}{{t}^{7}}\left( {{D}_{0 + }^{\frac{1}{2}}y}\right) \left( t\right) + {\mathrm{e}}^{t}\left( {{\int }_{0}^{t}{\left( t - s\right) }^{-\frac{1}{2... | where \( \alpha = \frac{1}{2}, r\left( t\right) = {t}^{-6}, p\left( t\right) = {t}^{-7}, q\left( t\right) = {\mathrm{e}}^{t}, f\left( u\right) = u \) . Take \( {t}_{0} > 0, k = 1 \) . Since\n\n\[ \n\omega \left( t\right) = {\mathrm{e}}^{{\int }_{{t}_{0}}^{t}\frac{{r}^{\prime }\left( s\right) + p\left( s\right) }{r\left... | Yes |
Example 4.2 Consider the fractional differential equation\n\n\[ \n{\left\lbrack {t}^{-\frac{3}{2}}\left( {D}_{0 + }^{\frac{1}{2}}y\right) \left( t\right) \right\rbrack }^{\prime } + 3{t}^{-\frac{5}{2}}\left( {{D}_{0 + }^{\frac{1}{2}}y}\right) \left( t\right) + \frac{{\mathrm{e}}^{t}}{t}\left( {3 + {\mathrm{e}}^{{\int }... | where \( \alpha = \frac{1}{2}, r\left( t\right) = {t}^{-\frac{3}{2}}, p\left( t\right) = 3{t}^{-\frac{5}{2}}, q\left( t\right) = \frac{{\mathrm{e}}^{t}}{t}, f\left( u\right) = \left( {3 + {\mathrm{e}}^{u}}\right) u \) . Take \( {t}_{0} > 0, k = 3 \) . Since\n\n\[ \n\omega \left( t\right) = {\mathrm{e}}^{{\int }_{{t}_{0... | Yes |
Theorem 1.1 If \( r \geq 3 \) and \( G \) is a planar graph without cycles of length from 4 to 9, then \( {\chi }_{r}\left( G\right) \leq r + 5 \) . | Let \( G \) be a counterexample to Theorem 1.1 with \( \left| E\right| \) minimized, and \( k = r + 5 \) . We may suppose that \( G \) does not admit \( r \) -hued coloring with the set \( \left\lbrack k\right\rbrack \) but each of its proper subgraph does. Without loss of generality, we can assume that \( G \) is conn... | Yes |
Lemma 3.1 If \( G \) is a counterexample, then each of the following holds.\n\n1) \( G \) is 2-connected.\n\n2) \( G \) has no path \( {v}_{0}{v}_{1}{v}_{2}{v}_{3} \) such that \( d\left( {v}_{1}\right) = d\left( {v}_{2}\right) = 2 \) .\n\n3) \( G \) has no path \( {v}_{0}{v}_{1}{v}_{2}{v}_{3} \) such that \( d\left( {... | Proof 1) Assume that \( G \) is connected and \( v \) is a cut-vertex, so \( G \) has two nontrivial connected subgraphs \( {G}_{1} \) and \( {G}_{2} \) satisfying \( V\left( {G}_{1}\right) \cap V\left( {G}_{2}\right) = \{ v\} \) and \( G = {G}_{1} \cup {G}_{2} \) . Then \( {G}_{i} \) has an \( r \) -hued coloring \( {... | Yes |
Lemma 3.2 If \( G \) is a counterexample, then each of the following holds.\n\n1) \( G \) has no a 3-face \( \left\lbrack {uxy}\right\rbrack \) such that \( d\left( u\right) = 2 \) and \( d\left( x\right) \leq 6 \) . | Proof 1) Assume that \( G \) has a 3-face \( \left\lbrack {uxy}\right\rbrack \) such that \( d\left( u\right) = 2 \) and \( d\left( x\right) \leq 6 \) . Let \( H = G - u \) . By the hypothesis, \( H \) has an \( r \) -hued coloring \( \phi \) with the set \( \left\lbrack k\right\rbrack \) . Obviously, \( \left| {F\left... | Yes |
Lemma 3.3 If \( G \) is a counterexample, then \( G \) does not contain a \( k \) -vertex \( u \) with neighbors \( {u}_{1},{u}_{2},\ldots ,{u}_{k} \) forming one of the configurations 1) to 2).\n\n1) \( k = 3, d\left( {u}_{1}\right) = 2, u \) is incident with a 3 -face \( \left\lbrack {u{u}_{2}{u}_{3}}\right\rbrack \)... | Proof Suppose the lemma is false. If 1) or 2) holds, let \( H = G - {u}_{1} \) . By the hypothesis, \( H \) has an \( r \) -hued coloring \( \phi \) of with the set \( \left\lbrack k\right\rbrack \) . Now we color \( G \) as follows.\n\nIf \( G \) has the configuration 1). Let \( {N}_{G}\left( {u}_{1}\right) = \left\{ ... | Yes |
Theorem 3.1 Under hypotheses (H2.1) and (H2.2), the Cauchy problem (2.1) admits a unique smooth solution \( \left( {{A}_{\delta }^{\epsilon },{v}_{\delta }^{\epsilon }}\right) \left( {x, t}\right) \) on \( {Q}_{T} \) : | By Definition 3.1, we get\n\n\[ {q}_{A} = v{\eta }_{A} + g{h}^{\prime }\left( A\right) {\eta }_{v},\;{q}_{v} = A{\eta }_{A} + v{\eta }_{v}. \]\n\n(3.1)\n\nEliminating \( q\left( {A, v}\right) \) in (3.1), we find that \( \eta \left( {A, v}\right) \) must be a convex solution to following linear wave equation:\n\n\[ {\e... | Yes |
Lemma 3.2 If\n\n\\[ \n\\left( {A - {A}^{ * }}\\right) {f}^{\\prime \\prime }\\left( A\\right) \\geq 0 \n\\]\n\n(3.11)\n\nholds, then \\( \\eta \\left( {A, v}\\right) \\) is the strictly convex entropy. | By (3.3)(3.6), we get\n\n\\[ \n{\\eta }^{ * }\\left( {A, v}\\right) = \\eta \\left( {A, v}\\right) - Y\\left( {A}^{ * }\\right) - {Y}^{\\prime }\\left( {A}^{ * }\\right) \\left( {A - {A}^{ * }}\\right) .\n\\]\n\n(3.12)\n\nNoting (3.6)-(3.8)(3.12), we have\n\n\\[ \n{\\eta }^{ * }\\left( {A, v}\\right) = \\eta \\left( {A... | Yes |
Lemma 3.2 \( {}^{\left\lbrack {16}\right\rbrack } \) Let the function \( m \in P{C}^{\prime }\left\lbrack {{\mathbb{R}}^{ + },\mathbb{R}}\right\rbrack \) satisfies the inequalities\n\n\[ \left\{ \begin{array}{l} {m}^{\prime }\left( t\right) \leq p\left( t\right) m\left( t\right) + q\left( t\right) , \\ t \geq {t}_{0}, ... | \[ m\left( t\right) \leq m\left( {t}_{0}\right) \mathop{\prod }\limits_{{{t}_{0} < {t}_{k} < t}}{d}_{k}\exp \left( {{\int }_{{t}_{0}}^{t}p\left( s\right) \mathrm{d}s}\right) + \mathop{\sum }\limits_{{{t}_{0} < {t}_{k} < t}}\left( {\mathop{\prod }\limits_{{{t}_{k} < {t}_{j} < t}}{d}_{j}\exp \left( {{\int }_{{t}_{0}}^{t}... | Yes |
Lemma 3.3 There exists a constant \( M > 0 \) such that \( x\left( t\right) \leq M, y\left( t\right) \leq M \) for each solution \( \left( {x\left( t\right), y\left( t\right) }\right) \) of (2.2) with all \( t \) large enough. | Proof Define\n\n\[ V\left( t\right) = {ax}\left( t\right) + {by}\left( t\right) \text{, and}\lambda = \min \{ c,\beta \} \text{.} \]\n\nThen \( t \neq {n\tau } \), we have\n\n\[ {D}^{ + }V\left( t\right) + {\lambda V}\left( t\right) = {ar} - a\left( {c - \lambda }\right) x\left( t\right) - {any}\left( t\right) - b\left... | Yes |
Lemma 3.5 The fixed point \( {x}^{ * } \) of (3.2) is globally asymptotically stable. | Proof Making notation as\n\n\[ F\left( {x\left( {n{\tau }^{ + }}\right) }\right) = \frac{1}{c}\left\lbrack {r - \left( {r - {cx}\left( {n{\tau }^{ + }}\right) }\right) {\mathrm{e}}^{-{c\tau })}}\right\rbrack + {r}_{1}, \]\n\n(3.9)\n\nthen,\n\n\[ {\left. \frac{\partial F\left( {x\left( {n{\tau }^{ + }}\right) }\right) }... | Yes |
Lemma 3.6 System (3.2) has a positive periodic solution \( \widetilde{x\left( t\right) } \) . For every solution \( x\left( t\right) \) of (3.2), we have \( x\left( t\right) \rightarrow \widetilde{x\left( t\right) } \) as \( t \rightarrow \infty \) | \[ \widetilde{x\left( t\right) } = \frac{1}{c}\left\lbrack {r - \left( {r - c{x}^{ * }}\right) {\mathrm{e}}^{-c\left( {t - {n\tau }}\right) }}\right\rbrack, t \in ({n\tau },\left( {n + 1}\right) \tau \rbrack , \] (3.11) where \( {x}^{ * } \) is defined as (3.5). | Yes |
Theorem 3.1 If\n\[ \n\frac{ar\tau }{c} < \frac{a\left( {r - c{x}^{ * }}\right) \left( {1 - {\mathrm{e}}^{-{c\tau }}}\right) }{{c}^{2}} + {\beta \tau }\n\]\n\n(3.12)\n\nholds, the oasis vegetation-extinction solution \( \left( {\widetilde{x\left( t\right) },0}\right) \) of (2.2) is globally asymptotically stable. \( {x}... | Proof First, we prove the local stability of the oasis vegetation-extinction solution \( \left( {x\left( t\right) ,0}\right) \) of (2.2). Defining \( {x}_{1}\left( t\right) = x\left( t\right) - \widetilde{x\left( t\right) }, y\left( t\right) = y\left( t\right) \), then we have the following linearly similar system for ... | Yes |
It turns out that the equation\n\n\\[ \n{u}_{tt} = {\\mathrm{e}}^{u}\\left( {{u}_{xx} + {u}_{yy} + {u}_{zz}}\\right) + {\\mathrm{e}}^{u}{u}^{-k}\\left( {1 - k{u}^{k - 1}}\\right) \\left( {{u}_{x}^{2} + {u}_{y}^{2}}\\right) + {c}_{3}k{u}^{{2k} - 1} \n\\]\n\nhas the exact solution | \\[ \nu = \\left\\{ \\begin{array}{l} {c}_{0}{xyz}\\exp {c}_{1}t, k = 1, \\\\ {\\left\\lbrack \\left( 1 - k\\right) \\left( \\ln \\left| xyz\\right| + {c}_{1}t + {c}_{2}\\right) \\right\\rbrack }^{\\frac{1}{k - 1}}, k \\end{array}\\right. \n\\] | Yes |
Assuming \( A = B = C = {\mathrm{e}}^{u}, F = {u}^{k} \), then we get \( D = E = G = - k{\mathrm{e}}^{u}{u}^{-1} \) , and \( Q = {c}_{7}k{u}^{{2k} - 1} - 3{\mathrm{e}}^{u}{u}^{-k} \). Thus, the equation\n\n\[ \n{u}_{tt} = {\mathrm{e}}^{u}\left( {{u}_{xx} + {u}_{yy}}\right) - k{\mathrm{e}}^{u}{u}^{-1}\left( {{u}_{x}^{2}... | has the exact solution\n\n\[ \nu = \left\{ \begin{array}{l} {c}_{0}\exp \left( {\frac{{x}^{2} + {y}^{2} + {z}^{2}}{2} + {c}_{1}t}\right), k = 1, \\ {\left\lbrack \left( 1 - k\right) \left( \frac{{x}^{2} + {y}^{2} + {z}^{2}}{2} + {c}_{1}t + {c}_{2}\right) \right\rbrack }^{\frac{1}{1 - k}}, k \neq 1. \end{array}\right. \... | Yes |
Assuming \( A = B = C = 1/\left( {1 + {u}^{2}}\right), F = {u}^{k} \), we have \( D = E = G = \) \( - k/u\left( {1 + {u}^{2}}\right), Q = {c}_{13}k{u}^{{2k} - 1} \), Thus we have shown that the equation\n\n\[ \n{u}_{tt} = {u}^{m}\left( {{u}_{xx} + {u}_{yy} + {u}_{zz}}\right) - k{u}^{m - 1}\left( {{u}_{x}^{2} + {u}_{y}^... | \[ \nu = \left\{ \begin{array}{l} {c}_{0}\exp \left( {{xyz} + {c}_{1}t}\right), k = 1, \\ {\left\lbrack \left( 1 - k\right) \left( xyz + {c}_{1}t + {c}_{2}\right) \right\rbrack }^{\frac{1}{k - 1}}, k \neq 1. \end{array}\right. \] | Yes |
Assuming \( A = B = C = {u}^{m}, F = {u}^{k} \), we have \( D = E = G = - k{u}^{m - 1}, Q = \) \( {c}_{16}k{u}^{{2k} - 1} \) . Thus we have shown that the equation\n\n\[ \n{u}_{tt} = {u}^{m}\left( {{u}_{xx} + {u}_{yy} + {u}_{zz}}\right) - k{u}^{m - 1}\left( {{u}_{x}^{2} + {u}_{y}^{2} + {u}_{z}^{2}}\right) + {c}_{16}k{u... | \[ \nu = \left\{ \begin{array}{l} {c}_{0}\exp \left( {{x}^{2}y + y - {z}^{2} + {c}_{1}t}\right), k = 1, \\ {\left\lbrack \left( 1 - k\right) \left( {x}^{2}y + y - {z}^{2} + {c}_{1}t + {c}_{2}\right) \right\rbrack }^{\frac{1}{k - 1}}, k \neq 1. \end{array}\right. \] | Yes |
Theorem 1.1 The set of strong Szemerédi points is of Hausdorff dimension 1/2. | Proof The upper bound of the Hausdorff dimension is clear by Lemma 2.2 below. The lower bound follows from Theorem 1.1. | No |
Lemma 2.1 \( {}^{\left\lbrack 1\right\rbrack } \) If \( {x}_{1},{x}_{2} \in \lbrack 0,1) \) satisfy that \( {a}_{i}\left( {x}_{1}\right) = {a}_{i}\left( {x}_{2}\right) \) for \( 1 \leq i \leq n \), but \( {a}_{n + 1}\left( {x}_{1}\right) < \) \( {a}_{n + 1}\left( {x}_{2}\right) \), then | \[ \left| {{x}_{1} - {x}_{2}}\right| \geq \frac{1}{{12}{a}_{n + 2}\left( {x}_{2}\right) \cdot {a}_{n + 1}^{2}\left( {x}_{2}\right) \cdot {q}_{n}^{2}\left( {x}_{2}\right) }. \] | Yes |
Proposition 3.1 \( {E}_{c} \subset F \) . | Proof Let \( x \in {E}_{c} \) and let \( y \in E \) be the corresponding point to \( x \) . At first, the partial quotients of \( y \) are strictly increasing. Moveover, by (3.1), for each \( k \geq 1 \) ,\n\n\[ \n{a}_{{n}_{k} + c{n}_{1} + \cdots + c{n}_{k}}\left( x\right) < {a}^{{n}_{k} + 1} \leq {a}_{{n}_{k} + 1}\lef... | Yes |
Lemma 2.2 \( {}^{\left\lbrack 5\right\rbrack } \) Let \( \left\{ {{X}_{n}, n \geq 1}\right\} \) be sequence of \( {\rho }^{ - } \) -mixing with \( \mathrm{E}{X}_{n} = 0,\mathrm{E}{\left| {X}_{i}\right| }^{q} < \infty \) , \( i \geq 1 \), then for \( q \geq 2 \), exist positive real number \( c = c\left( {q,{\rho }^{ - ... | \[ \mathrm{E}\left( {\mathop{\max }\limits_{{1 \leq j \leq n}}{\left| {S}_{j}\right| }^{q}}\right) \leq c\left( {\mathop{\sum }\limits_{{i = 1}}^{n}\mathrm{E}{\left| {X}_{i}\right| }^{q} + {\left( \mathop{\sum }\limits_{{i = 1}}^{n}\mathrm{E}{X}_{i}^{2}\right) }^{q/2}}\right) . \] | Yes |
Theorem 3.1 Let \( \left( {{\mathbf{u}}_{h},{p}_{h},{\phi }_{h}}\right) \) and \( \left( {{\mathbf{u}}^{h},{p}^{h},{\phi }^{h}}\right) \) be defined by Algorithm 2.1 and 2.2. Under the regularity condition assumptions in [16], we have the following error estimate:\n\n\[ \n{\begin{Vmatrix}{\phi }_{h} - {\phi }^{h}\end{V... | Proof See [16]. | No |
Proposition 3.1 In the triangle reference unit, let \( s \) and \( {\left\{ {s}_{i}\right\} }_{i = 1}^{4} \) is defined as in (3.8)- (3.10). Then, we have the following error estimates:\n\n\[ \left| {{s}_{i} - s}\right| \lesssim {xy}\;i = 1,2,3,4,\;\text{ for }\left( {x, y}\right) \in {T}_{i}. \] | Proof Now we use the physical coordinates to present of the basis functions. For instance, we estimate the error between \( {s}_{1} \) and \( s \) on \( {T}_{1} \) . Apparently, we have\n\n\[ \left\{ \begin{array}{l} {\lambda }_{1} = 1 - x - y \\ {\lambda }_{2} = x \\ {\lambda }_{3} = y \end{array}\right. \]\n\n(3.12)\... | Yes |
Theorem 3.2 Let \( {\mathbf{u}}_{H} \) be the solution in (2.11), and \( {\mathbf{u}}_{H}^{ * } \) is the interpolation of \( {\mathbf{u}}_{H} \) with above method. Then, we have the following error estimate:\n\n\[ \n{\begin{Vmatrix}{\mathbf{u}}_{H} - {\mathbf{u}}_{H}^{ * }\end{Vmatrix}}_{{H}^{1}\left( {\Omega }_{1}\ri... | Proof For simplicity, we consider the condition that we only refine the mesh once and denote \( \mathbf{z} = {\mathbf{u}}_{H} - {\mathbf{u}}_{H}^{ * } \) . we have\n\n\[ \n\parallel \mathbf{z}{\parallel }_{{H}^{1}\left( {\Omega }_{1}\right) } = {\left\{ \parallel \mathbf{z}{\parallel }_{{L}^{2}\left( {\Omega }_{1}\righ... | Yes |
Example 4.1 In the following experiments, we apply the MINI elements for Navier-Stokes region and P1 Lagrange elements for Darcy region. The exact solution \( \left( {\mathbf{u},\phi, p}\right) \) is given by \[ \left\{ \begin{array}{l} {u}_{1} = {y}^{2} - {2y} + {2x} \\ {u}_{2} = {x}^{2} - x - {2y} \\ \phi = - {x}^{2}... | Tab.4.1 The errors of Example 4.1 between the exact solution and solutions of Algorithm 2.1 <table><thead><tr><th>\( h \)</th><th>\( {\left| {u}_{1, h} - {u}_{1}\right| }_{1} \)</th><th>\( {\left| {u}_{2, h} - {u}_{2}\right| }_{1} \)</th><th>\( {\left| {\phi }_{h} - \phi \right| }_{1} \)</th><th>\( {\left| {p}_{h} - p\... | Yes |
Example 4.2 In the following experiments, we apply the Q2-Q1 elements for Navier-Stokes region and Q2 elements for Darcy region. The exact solution \( \\left( {\\mathbf{u},\\phi, p}\\right) \) is given by\n\n\\[ \n\\left\\{ \\begin{array}{l} {u}_{1} = \\sin \\left( {\\pi x}\\right) \\sin \\left( {\\pi y}\\right) \\\\ {... | In order to compare the approximation accuracy between Algorithm 2.1 and Algorithm 2.2, the nonlinear system of (10) is solved by Newton iteration: given \( \\left( {{\\mathbf{u}}^{0},{\\phi }^{0},{p}^{0}}\\right) \\in {W}_{h} \\times {Q}_{h} \), for\n\n\( k \\geq 1 \), find \( \\left( {{\\mathbf{u}}^{k + 1},{\\phi }^{... | Yes |
Theorem 2.1 Assume that \( f\left( {t, u}\right) \) satisfies the conditions (A1) and (A2) in Lemma 2.1 with \( f\left( {t,0}\right) \equiv 0 \), and \( {e}_{1}\left( t\right) \) is not identically zero for \( t \in \left\lbrack {0,{2\pi }}\right\rbrack \) . Then PIBVP (1.1)-(1.3) has a unique nontrivial solution gener... | Proof Let \( g\left( {t, u}\right) = f\left( {t, u}\right) - {e}_{1}\left( t\right) \) . Then \( g\left( {t, u}\right) \) satisfies all conditions of Lemma 2.1. Note that \( g\left( {t,0}\right) = {e}_{1}\left( t\right) \) is not identically zero for \( t \in \left\lbrack {0,{2\pi }}\right\rbrack \) . The application o... | Yes |
Theorem 2.2 Assume that \( f\left( {t, u}\right) \) satisfies the conditions (A1)(A2) in Lemma 2.1, \( f\left( {t,0}\right) \equiv 0 \) for \( t \in \left\lbrack {0,{2\pi }}\right\rbrack \), and (A3) holds. Then PIBVP (1.2)-(1.3) has only a trivial solution. | We can prove this theorem in similar way to Theorem 2.1, here is omitted. | No |
Theorem 3.1 Suppose (A4)-(A6) hold. If \( f\left( {t, u}\right) \) and \( F\left( {t, u}\right) \) are both bounded on \( S \mathrel{\text{:=}} \{ \left( {t, u}\right) \in \left\lbrack {0,{2\pi }}\right\rbrack \times \mathbb{R}\} \), then PIBVP (1.1)-(1.3) has at least one solution. | Proof It easy to see \( I\left( u\right) \) is lower semicontinuous \( {}^{\left\lbrack {11}\right\rbrack } \) . Since \( F\left( {t, u}\right) \) is bounded, then there exists \( {M}_{1} > 0 \) such that \( F\left( {t, u}\right) \leq {M}_{1} \) for \( \left( {t, u}\right) \in S \) . Let \( {L}_{{e}_{1}} = {\begin{Vmat... | Yes |
Theorem 3.3 Suppose (A4),(A5) hold and \( {M}_{\left| {e}_{2}\right| } < 1 \) . If \( f\left( {t, u}\right) \) and \( F\left( {t, u}\right) \) are both bounded on \( S \mathrel{\text{:=}} \{ \left( {t, u}\right) \in \left\lbrack {0,{2\pi }}\right\rbrack \times \mathbb{R}\} \) . Then PIBVP (1.2)-(1.3) has at least one w... | Proof It is easy to see \( J\left( u\right) \) is lower semicontious. Because \( F\left( {t, u}\right) \) is bounded, there exists \( {M}_{2} > 0 \) such that \( F\left( {t, u}\right) \leq {M}_{2} \) for \( \left( {t, u}\right) \in S \) . We have\n\n\[ J\left( u\right) = {\int }_{0}^{2\pi }\left( {\frac{1}{2}{u}^{\prim... | Yes |
Theorem 3.4 Suppose (A4),(A5) and (A7) hold and \( {M}_{\left| {e}_{2}\right| } < 1 \) . Then PIBVP (1.2)-(1.3) has at least one weak solution. | The proof of this theorem is similar to that of Theorem 3.3, here is omitted. | No |
Consider the problem\n\n\\[ \n\\left\\{ \\begin{array}{l} {u}^{\\prime \\prime } + {6u} = \\frac{1}{2}{\\mathrm{e}}^{-t},\\;t \\in \\left\\lbrack {0,{2\\pi }}\\right\\rbrack \\\\ u\\left( 0\\right) = u\\left( {2\\pi }\\right) , \\\\ {\\int }_{0}^{2\\pi }u\\left( t\\right) \\mathrm{d}t = 0. \\end{array}\\right.\n\\]\n\n... | It is easy to know that the exact solution of the PIBVP (4.2) is\n\n\\[ \nu\\left( t\\right) = \\frac{1}{14}{\\mathrm{e}}^{-t} + \\frac{\\sqrt{7}\\left( {1 - {\\mathrm{e}}^{-{2\\pi }}}\\right) }{{28}\\sin \\sqrt{6}\\pi }\\sin \\left( {\\sqrt{6}\\left( {t - \\pi }\\right) - \\varphi }\\right) ,\\]\n\nwhere \\( \\varphi ... | Yes |
Example 4.2 Consider the problem\n\n\[ \n\\left\\{ \\begin{array}{l} {u}^{\\prime \\prime } + {6u} = \\frac{1}{t + 1}\\sin t,\\;t \\in \\left\\lbrack {0,{2\\pi }}\\right\\rbrack \\\\ u\\left( 0\\right) = u\\left( {2\\pi }\\right) , \\\\ {\\int }_{0}^{2\\pi }u\\left( t\\right) \\mathrm{d}t = 0. \\end{array}\\right. \n\] | Similar to Example 4.1, we obtain the numerical solution of the PIBVP (4.3) which is present by Figure 4.2.\n\n\n\nFig.4.2 Comparison of two methods for (4.3) | No |
Lemma 2.1 Let \( 1 < p < N \) and \( {\varphi }_{1} > 0 \) be a \( {\lambda }_{1} \) -eigenfunction with \( \begin{Vmatrix}{\varphi }_{1}\end{Vmatrix} = 1 \) and assume \( \left( {\mathrm{H}}_{1}\right) - \left( {\mathrm{H}}_{3}\right) \) and (SCPI) hold. If \( {f}_{0} < {\lambda }_{1} < l \leq + \infty \), then:\n\n(i... | Proof By (SCPI) and \( \left( {\mathrm{H}}_{1}\right) - \left( {\mathrm{H}}_{3}\right) \), if \( l \in \left( {{\lambda }_{1}, + \infty }\right) \), for any \( \varepsilon > 0 \), there exist \( {A}_{1} = {A}_{1}\left( \varepsilon \right) \) , \( {B}_{1} = {B}_{1}\left( \varepsilon \right) \) such that for all \( \left... | Yes |
Lemma 2.3 Let \( p = N \) and \( {\varphi }_{1} > 0 \) be a \( {\lambda }_{1} \) -eigenfunction with \( \begin{Vmatrix}{\varphi }_{1}\end{Vmatrix} = 1 \) and assume \( \left( {\mathrm{H}}_{1}\right) - \left( {\mathrm{H}}_{3}\right) \) and (SCE) hold. If \( {f}_{0} < {\lambda }_{1} < l \leq + \infty \), then:\n\n(i) The... | Proof By (SCE) and \( \left( {\mathrm{H}}_{1}\right) - \left( {\mathrm{H}}_{3}\right) \), if \( l \in \left( {{\lambda }_{1}, + \infty }\right) \), for any \( \varepsilon > 0 \), there exist \( {A}_{1} = {A}_{1}\left( \varepsilon \right) \) , \( {B}_{1} = {B}_{1}\left( \varepsilon \right) ,\kappa > 0 \) and \( q > N \)... | Yes |
Lemma 2.1 \( {}^{\left\lbrack 6\right\rbrack } \) Let \( {X}_{0} \) denote the closed subspace of all constant functions in \( {W}^{1, p}\left( \Omega \right) \) . Let \( X \) be the quotient space \( {W}^{1, p}\left( \Omega \right) /{X}_{0} \) . For \( u \in {W}^{1, p}\left( \Omega \right) \), define the mapping \( P ... | \[ \parallel u - {Pu}{\parallel }_{{L}^{p}\left( \Omega \right) } \leq {k}_{4}\parallel {Du}{\parallel }_{{\left( {L}^{p}\left( \Omega \right) \right) }^{N}}. \] | Yes |
Lemma 2.4 The mapping \( \Phi : {L}^{p}\left( {0, T;{W}^{1, p}\left( \Omega \right) }\right) \rightarrow R \) defined by\n\n\[ \Phi \left( u\right) = {\int }_{0}^{T}{\int }_{\Gamma }{\varphi }_{x}\left( {{\left. u\right| }_{\Gamma }\left( {x, t}\right) }\right) \mathrm{d}\Gamma \left( x\right) \mathrm{d}t \]\n\nfor any... | Proof Similar to the proof of Lemma 3.1 in [5], the result follows.\n\nThis completes the proof. | No |
Lemma 2.5 The mapping \( S \) is linear maximal monotone. | Proof For \( u\left( {x, t}\right), w\left( {x, t}\right) \in D\left( S\right) \), integrating by parts, we have\n\n\[ \left( {w,{Su}}\right) + \left( {u,\frac{{\partial }^{2}w}{\partial {t}^{2}}}\right) = - {\int }_{0}^{T}{\int }_{\Omega }\frac{{\partial }^{2}u}{\partial {t}^{2}}w\left( {x, t}\right) \mathrm{d}x\mathr... | Yes |
Lemma 2.6 Define \( \bar{S} : D\left( \bar{S}\right) = \left\{ {u \in {L}^{p}\left( {0, T;{W}^{1, p}\left( \Omega \right) }\right) : \frac{\partial u}{\partial t} \in {L}^{{p}^{\prime }}\left( {0, T;{\left( {W}^{1, p}\left( \Omega \right) \right) }^{ * }}\right), u\left( {x,0}\right) }\right. \) \( \left. { = u\left( {... | Proof Similar to the proof of Lemma 3.9 in [12], the result follows. | No |
Lemma 2.7 Define the mapping \( C : {L}^{p}\left( {0, T;{W}^{1, p}\left( \Omega \right) }\right) \rightarrow {L}^{{p}^{\prime }}\left( {0, T;{\left( {W}^{1, p}\left( \Omega \right) \right) }^{ * }}\right) \) by\n\n\[ \left( {v,{Cu}}\right) = {\int }_{0}^{T}{\int }_{\Omega }g\left( {x, u,{Du}}\right) v\left( {x, t}\righ... | Proof Step 1 For \( u\left( {x, t}\right) \in {L}^{p}\left( {0, T;{W}^{1, p}\left( \Omega \right) }\right), x \rightarrow g\left( {x, u,{Du}}\right) \) is measurable on \( \Omega \) .\n\nFrom the facts that \( u\left( {x, t}\right) ,\frac{\partial u}{\partial {x}_{i}} \in {L}^{p}\left( \Omega \right), i = 1,2,\cdots, N... | Yes |
Corollary 1.1 Let \( \\left\\{ {{Y}_{ni};1 \\leq i \\leq n, n \\geq 1}\\right\\} \) be an array of rowwise \( \\widetilde{\\rho } \) -mixing identically distributed random variables with \( {\\widetilde{\\rho }}_{1}\\left( 1\\right) < 1 \), such that \( \\mathrm{E}{Y}_{11} = 0 \) and \( \\mathrm{E}{\\left| {Y}_{11}\\ri... | \[ \\mathop{\\sum }\\limits_{{n = 1}}^{\\infty }{n}^{{p\\alpha } - 2}P\\left( {\\mathop{\\max }\\limits_{{1 \\leq i \\leq n}}\\left| {S}_{ni}\\right| > \\varepsilon {n}^{\\alpha }}\\right) < \\infty . \] | Yes |
Corollary 1.3 Let \( \\left\\{ {{Y}_{ni};1 \\leq i \\leq n, n \\geq 1}\\right\\} \) be an array of rowwise \( \\widetilde{\\rho } \) -mixing identically distributed random variables with \( {\\widetilde{\\rho }}_{1}\\left( 1\\right) < 1 \), such that \( \\mathrm{E}{Y}_{11} = 0,\\mathrm{E}{\\left| {Y}_{11}\\right| }^{p}... | \[ \\mathop{\\sum }\\limits_{{n = 1}}^{\\infty }{n}^{{p\\alpha } - 2}l\\left( n\\right) P\\left( {\\mathop{\\max }\\limits_{{1 \\leq i \\leq n}}\\left| {S}_{ni}\\right| > \\varepsilon {n}^{\\alpha }}\\right) < \\infty . \] | Yes |
Lemma 2.1 \( {}^{\left\lbrack 8\right\rbrack } \) Let \( \left\{ {{X}_{n};n \geq 1}\right\} \) be a sequence of \( \widetilde{\rho } \) -mixing random variables with \( \mathrm{E}{X}_{n} = \) 0 and \( \mathrm{E}{\left| {X}_{n}\right| }^{q} < \infty \) for some \( q \geq 2 \) and all \( n \geq 1 \) . Suppose \( N \) is ... | \[ \mathrm{E}\left( {\mathop{\max }\limits_{{1 \leq k \leq n}}{\left| \mathop{\sum }\limits_{{i = 1}}^{k}{X}_{i}\right| }^{q}}\right) \leq D\left( {{\left( \mathop{\sum }\limits_{{i = 1}}^{n}\mathrm{E}{X}_{i}^{2}\right) }^{q/2} + \mathop{\sum }\limits_{{i = 1}}^{n}\mathrm{E}{\left| {X}_{i}\right| }^{q}}\right) . \] | Yes |
Lemma 2.2 \( {}^{\left\lbrack {19}\right\rbrack } \) Let \( A \subset \lbrack 0,\gamma ) \), and \( {\mu A} = t \), we have\n\n(i) For \( t < + \infty ,{\begin{Vmatrix}{\chi }_{A}\end{Vmatrix}}_{\varphi ,\omega }^{ \circ } = {\psi }^{-1}\left( \frac{1}{W\left( t\right) }\right) \cdot W\left( t\right) \) ; (ii) For \( t... | By Lemma 1.1 in [4], and similarly as the proof of Lemma 1.40 in [3], we can get | No |
Theorem 3.4 \( {\bigwedge }_{\varphi ,\omega }^{ \circ }\lbrack 0,\gamma ) \) is UM if and only if \( \varphi \in {\Delta }_{2},\omega \) is regular, \( \omega \left( t\right) > 0 \) for any \( t \in \lbrack 0,\gamma ) \) and \( {\int }_{0}^{\gamma }\omega \left( t\right) \mathrm{d}t = + \infty \) whenever \( \gamma = ... | Proof Necessity: By theorem 3.1, we only need to prove that \( \omega \) is regular when \( \gamma = + \infty \) . In fact, The UM of \( \mathop{\bigwedge }\limits_{{\varphi ,\omega }}^{ \circ }\lbrack 0,\gamma ) \) yields that there exists \( \eta = \eta \left( \frac{1}{2}\right) \in \left( {0,1}\right) \), such that ... | Yes |
Theorem 4.2 For Orlicz-Lorentz function space \( \mathop{\bigwedge }\limits_{{\varphi ,\omega }}\lbrack 0,\gamma ) \) with the Luxemburg norm, if \( \varphi \notin {\Delta }_{2} \), or \( {t}_{0} \mathrel{\text{:=}} \sup \{ t : \omega \left( t\right) > 0\} \leq \frac{\gamma }{2} \) whenever \( \gamma < + \infty \) or \... | Proof If \( \varphi \notin {\Delta }_{2} \), then for any \( \varepsilon > 0 \), by Lemma 2.3 in [15], we have \( x \in \mathop{\bigwedge }\limits_{{\varphi ,\omega }}\lbrack 0,\gamma ) \) with \( x \geq 0,\parallel x{\parallel }_{\varphi ,\omega } = 1 \) and \( {\rho }_{\varphi ,\omega }\left( x\right) < \varepsilon \... | Yes |
Lemma 2.2 \( {}^{\left\lbrack {12}\right\rbrack } \) Let \( \mathcal{A} = \left( {a}_{{i}_{1}\cdots {i}_{m}}\right) \) be a complex tensor of order \( m \) dimension \( n \) . If \( \mathcal{A} \) is an \( \mathcal{H} \) -tensor, then \( {N}_{2} \neq \varnothing \) . | By Lemma 2.1, if \( {N}_{1} = \varnothing \), then \( \mathcal{A} \) is an \( \mathcal{H} \) -tensor; By Lemma 2.2, if \( \mathcal{A} \) is an \( \mathcal{H} \) -tensor, then \( {N}_{2} \neq \varnothing \) . Hence, we always assume that \( {N}_{1} \neq \varnothing ,{N}_{2} \neq \varnothing \) . In addition, we also ass... | Yes |
Theorem 2.1 Let \( \mathcal{A} = \left( {a}_{{i}_{1}\cdots {i}_{m}}\right) \) be an order \( m \) dimension \( n \) complex tensor. If there exists \( k \in {\mathbb{Z}}^{ + } \) such that \[ \left| {a}_{{ii}\cdots i}\right| > \mathop{\sum }\limits_{\substack{{{i}_{2}\cdots {i}_{m} \in {N}^{m - 1} \smallsetminus {N}^{m... | Proof Let \[ {M}_{i} = \frac{1}{\mathop{\sum }\limits_{{{i}_{2}\cdots {i}_{m} \in {N}_{2}^{m - 1}}}\left| {a}_{i{i}_{2}\cdots {i}_{m}}\right| }\left( {\left| {a}_{{ii}\cdots i}\right| - \mathop{\sum }\limits_{\substack{{{i}_{2}\cdots {i}_{m} \in {N}^{m - 1} \smallsetminus {N}_{2}^{m - 1}} \\ {{\delta }_{i{i}_{2}\ldots ... | Yes |
Theorem 2.2 Let \( \\mathcal{A} = \\left( {a}_{{i}_{1}\\cdots {i}_{m}}\\right) \) be an order \( m \) dimension \( n \) complex tensor. If \( \\mathcal{A} \) is irreducible, and there exists \( k \\in {Z}^{ + } \) such that\n\n\[\\left| {a}_{{ii}\\cdots i}\\right| \\geq \\mathop{\\sum }\\limits_{\\substack{{{i}_{2}\\cd... | Proof Let the matrix \( X = \\operatorname{diag}\\left( {{x}_{1},{x}_{2},\\cdots ,{x}_{n}}\\right) \), where\n\n\[{x}_{i} = \\left\\{ \\begin{matrix} 1, & i \\in {N}_{1}, \\\\ {\\left( {\\delta }_{k + 1, i}\\right) }^{\\frac{1}{m - 1}}, & i \\in {N}_{2}. \\end{matrix}\\right.\]\n\nBy the irreducibility of \( \\mathcal{... | Yes |
Theorem 2.3 Let \( \mathcal{A} = \left( {a}_{{i}_{1}\cdots {i}_{m}}\right) \) be an order \( m \) dimension \( n \) complex tensor. If there exists \( k \in {Z}^{ + } \) such that\n\n\[ \left| {a}_{{ii}\cdots i}\right| \geq \mathop{\sum }\limits_{\substack{{{i}_{2}\cdots {i}_{m} \in {N}^{m - 1} \smallsetminus {N}_{2}^{... | Proof Let the matrix \( X = \operatorname{diag}\left( {{x}_{1},{x}_{2},\cdots ,{x}_{n}}\right) \), where\n\n\[ {x}_{i} = \left\{ \begin{matrix} 1, & i \in {N}_{1}, \\ {\left( {\delta }_{k + 1, i}\right) }^{\frac{1}{m - 1}}, & i \in {N}_{2}. \end{matrix}\right. \]\n\nObviously \( {x}_{i} \neq + \infty \), then \( X \) i... | Yes |
Example 2.1 Consider a tensor \( \mathcal{A} = \left( {a}_{ijk}\right) \) of order 3 dimension 3 defined as follows:\n\n\[ \mathcal{A} = \left\lbrack {A\left( {1, : , : }\right), A\left( {2, : , : }\right), A\left( {3, : , : }\right) }\right\rbrack ,\]\n\n\[ A\left( {1, : , : }\right) = \left( \begin{matrix} 8 & 1 & 0 ... | Obviously,\n\n\[ \left| {a}_{111}\right| = 8,\;{R}_{1}\left( \mathcal{A}\right) = {24},\;\left| {a}_{222}\right| = {12},\;{R}_{2}\left( \mathcal{A}\right) = 4,\;\left| {a}_{333}\right| = {15},\;{R}_{3}\left( \mathcal{A}\right) = 5, \]\n\nso \( {N}_{1} = \{ 1\} ,{N}_{2} = \{ 2,3\} \) . By calculation, we have\n\n\[ {r}_... | Yes |
Lemma 3.1 If \( \frac{\dot{v}}{v} > \frac{\dot{w}}{w} \), the function \( g\left( t\right) \) has a unique zero \( R \) . | Proof Since \( g\left( 0\right) = - w\left( 0\right) < 0 \) and \( \mathop{\lim }\limits_{{t \rightarrow {\Omega }^{ - }}}g\left( t\right) = + \infty \), from intermediate value theorem the function \( g\left( t\right) \) has at least one zero on the interval \( \left( {0,\Omega }\right) \) . Let \[ {c}_{1}\left( t\rig... | Yes |
Lemma 3.2 Under the condition of Lemma 1, the control variable \( \chi \left( t\right) \) satisfies \( \chi \left( t\right) = \) \( 1,0 \leq t < R \) and \( \chi \left( t\right) = 0, R \leq t < \Omega \) . | Proof Since the function \( g\left( t\right) \) has unique zero \( R \) and \( g\left( 0\right) < 0 \), then \( g\left( t\right) < 0 \) when \( 0 \leq t < R \), and\n\n\[ \n{c}_{0}^{\frac{1}{\sigma }}\left( t\right) {\mathrm{e}}^{\left( {r - \rho }\right) t}v\left( {h\left( {t;\Omega }\right) }\right) < {w}_{1}\left( t... | No |
Theorem 4.1 The equation \( {Q}_{1}\left( {R,\Omega }\right) = 0 \) determines a function \( R = R\left( \Omega \right) \) . If\n\n\[ \n\frac{1}{\sigma }\frac{1}{{c}_{0}}\frac{\partial {c}_{0}}{\partial \Omega } + \frac{{v}^{\prime }\left( {h\left( {R;\Omega }\right) }\right) }{v\left( {h\left( {R;\Omega }\right) }\rig... | Proof From (20) and \( \frac{\dot{w}}{w} = \gamma + \frac{\dot{{w}_{2}}}{{w}_{2}} \) ,\n\n\[ \n\frac{\partial {Q}_{1}}{\partial R} = \frac{1}{\sigma }{c}_{0}^{\frac{1}{\sigma } - 1}{\mathrm{e}}^{\left( {r - \rho }\right) R}v\left( {h\left( {R;\Omega }\right) }\right) \frac{\partial {c}_{0}}{\partial R} + {c}_{0}^{\frac... | Yes |
Corollary 4.1 If \( \frac{\partial {c}_{0}}{\partial \Omega } < 0 \), then\n\n\[ \frac{\mathrm{d}R}{\mathrm{\;d}\Omega } = - \frac{\frac{\partial {Q}_{1}}{\partial \Omega }}{\frac{\partial {Q}_{1}}{\partial R}} > 0 \]\n\nand the retiring age increases with respect to the longevity. | Proof In the (21), the first term is less than zero from the given condition, since \( {v}^{\prime }\left( h\right) < 0,\frac{\partial h}{\partial \Omega } > 0 \) and \( {w}_{2}^{\prime }\left( {h;\Omega }\right) > 0 \), the second term is less than zero and the third term is more than zero. These imply \( \frac{\parti... | Yes |
Theorem 3.1 Let self-adjoint boundary value problems be described as (2.1), (2.4)- (2.7). Fix the boundary conditions and the endpoint \( a \) . Fix \( n \in {\mathbf{N}}_{0} \) . Let \( {\lambda }_{n} = {\lambda }_{n}\left( b\right) \) for \( b \in \left( {a, B}\right) \) . Then\n\n1) \( {\lambda }_{n}\left( b\right) ... | Proof 1) The continuity of \( {\lambda }_{n}\left( b\right) \) as a function of \( b \) follows from Theorem 3.1 in [3], although the proof is given there for second-order Sturm-Liouville case, it extends readily to our case. Also we can see [15].\n\n2) The fact that the multiplicity of \( {\lambda }_{n}\left( b\right)... | Yes |
Lemma 3.1 Assume \( u \) and \( v \) are solutions of (2.1) with \( \lambda = \mu \) and \( \lambda = \nu \), respectively. Then\n\n\[ \n{\left\lbrack u, v\right\rbrack }_{a}^{b} = \left\lbrack {u, v}\right\rbrack \left( b\right) - \left\lbrack {u, v}\right\rbrack \left( a\right) \n\]\n\n\[ \n= \left\lbrack {{\left( p{... | Proof This follows from integration by parts. | No |
Lemma 3.2 Assume a real valued function \( f \in {L}_{\mathrm{{loc}}}\left( {A, B}\right) \) . Then\n\n\[ \mathop{\lim }\limits_{{h \rightarrow 0}}\frac{1}{h}{\int }_{t}^{t + h}f = f\left( t\right) \text{ a.e. in }\left( {A, B}\right) . \] | Proof See the proof given in [3]. | No |
Theorem 3.4 Let (2.2) hold. Consider the boundary value problem (2.1), (2.4)-(2.7) with \( 0 \leq \alpha < \pi ,0 < \beta \leq \pi \), i.e., arbitrary separated conditions at \( a \) and \( b \) . Using the notation of Section 2 and Letting \( \lambda = {\lambda }_{n}, u = {u}_{n} \), we have the following differential... | Proof The proof is more complicated, but consists basically of combining the techniques in the proofs of Theorems 3.2 and 3.3. For small \( h \), we choose \( \mu = \lambda \left( b\right) ,\nu = \lambda \left( {b + h}\right) \), and \( u = u\left( {\cdot, b}\right), v = u\left( {\cdot, b + h}\right) \) . From (3.3) an... | Yes |
Lemma 2.2 Let \( G \) be a plane graph with \( \Delta \left( G\right) \geq 6 \) and without \( 4,5,6,7 \) -cycles, then \( G \) has a vertex \( v \) with \( k \) neighbors \( {v}_{1},{v}_{2},\ldots ,{v}_{k}, d\left( {v}_{1}\right) \leq d\left( {v}_{2}\right) \leq \ldots \leq d\left( {v}_{k}\right) \), and \( v \) satis... | Proof As the proof of Lemma 1, we introduce discharging rules firstly, then check \( {w}^{\prime }\left( x\right) \geq 0 \) for all \( x \in V \cup F \) .\n\nR1 Each \( {8}^{ + } \) -face sends charge of \( \frac{1}{2} \) to every incident vertex;\n\nR2 Each \( {3}^{ + } \) -vertex sends charge of \( \frac{1}{2} \) to ... | Yes |
Lemma 3.2 Let \( {R}_{G} \) and \( {R}_{n} \) are defined by (2.5) and (2.9), respectively. If \( {R}_{G} < \infty \) , for any estimator \( \phi \), we have\n\n\[ \n{R}_{n} - {R}_{G} = {\mathrm{E}}_{ * }\left\lbrack {\frac{{\phi }_{n}\left( x\right) }{{\phi }_{G}\left( x\right) } - \ln \frac{{\phi }_{n}\left( x\right)... | Proof If \( {R}_{G} < \infty \), then\n\n\[ \n{R}_{n} = {R}_{n}\left( {{\phi }_{n}\left( x\right), G\left( \theta \right) }\right) = {\mathrm{E}}_{ * }\left\lbrack {\frac{{\phi }_{n}\left( x\right) }{\theta } - \ln \frac{{\phi }_{n}\left( x\right) }{\theta } - 1}\right\rbrack \n\]\n\n\[ \n= {\mathrm{E}}_{\left( X,\thet... | Yes |
Lemma 3.3 If \( {\mathrm{E}}_{\left( x\right) }\left| {\ln {\phi }_{G}\left( x\right) }\right| < + \infty \), then \( {R}_{G} < \infty \) . | Proof From (3.4),\n\n\[ \n{R}_{G} = {\mathrm{E}}_{\left( x\right) }\left\{ {-\ln {\phi }_{G}\left( x\right) - {\mathrm{E}}_{\left( \theta \mid x\right) }\left( {\ln \theta \mid x}\right) }\right\} \n\]\n\n\[ \n= 2{\mathrm{E}}_{\left( x\right) }\left\lbrack {-\ln {\phi }_{G}\left( x\right) }\right\rbrack \leq 2{\mathrm{... | Yes |
Lemma 3.4 If \( a, b \geq 0 \), then we obtain\n\n\[ \frac{a}{b} - \ln \frac{a}{b} - 1 \leq {\left( a - b\right) }^{2}. \]\n\n(3.5) | Proof From [5], we have \( {\mathrm{e}}^{a - b} - \left( {a - b}\right) - 1 \leq {\left( {\mathrm{e}}^{a} - {\mathrm{e}}^{b}\right) }^{2} \) . Thus,\n\n\[ \frac{a}{b} - \ln \frac{a}{b} - 1 = {\mathrm{e}}^{\left( \ln a - \ln b\right) } - \left( {\ln a - \ln b}\right) - 1 \leq {\left( {\mathrm{e}}^{\ln a} - {\mathrm{e}}^... | Yes |
Lemma 3.5 Let \( {f}_{n}^{\left( r\right) }\left( x\right) \left( {r = 0,1}\right) \) be defined by (2.6), where \( {X}_{1},{X}_{2},\cdots \) be i.i.d. samples sequence. For any \( x \in \Omega \), if the condition (A1) and \( {f}_{G}\left( x\right) \in {C}_{s,\alpha } \) hold and taking \( {b}_{n} = \) \( {n}^{-1/\lef... | Proof By \( {C}_{r} \) - inequality, we have \[ {\mathrm{E}}_{n}{\left| {f}_{n}^{\left( r\right) }\left( x\right) - {f}_{G}^{\left( r\right) }\left( x\right) \right| }^{2\lambda } \leq 2{\left| {\mathrm{E}}_{n}{f}_{n}^{\left( r\right) }\left( x\right) - {f}_{G}^{\left( r\right) }\left( x\right) \right| }^{2\lambda } + ... | Yes |
Lemma 3.6 Let \( {p}_{G}\left( x\right) \) and \( {p}_{n}\left( x\right) \) are defined by (2.2) and (2.7), respectively, where \( {X}_{1},{X}_{2},\cdots \) are independent identically distributed random variable samples sequence, for \( 0 < \lambda \leq 1 \), we have\n\n\[ \n{\mathrm{E}}_{n}{\left| {p}_{n}\left( x\rig... | Proof Since \( {\mathrm{E}}_{n}{p}_{n}\left( x\right) = \mathrm{E}\left\{ {I}_{\left( {X}_{1} > x\right) }\right\} = {\int }_{x}^{\infty }{f}_{G}\left( y\right) \mathrm{d}y = {p}_{G}\left( x\right) ,{p}_{n}\left( x\right) \) is unbiased estimation of \( {p}_{G}\left( x\right) \) . By Jensen’s inequality, we get\n\n\[ \... | Yes |
Lemma 3.7 Let \( \delta > 2 \), if \( {\int }_{\Theta }{\left| \theta \right| }^{\delta }\mathrm{d}G\left( \theta \right) < \infty \), then \( {\mathrm{E}}_{n}{\left| {\phi }_{G}\left( X\right) \right| }^{\delta } < \infty \) . | Proof By Jensen's inequality, we have\n\n\[ \n{\mathrm{E}}_{n}{\left| {\phi }_{G}\left( x\right) \right| }^{\delta } = {\int }_{\Omega }{\left| {\mathrm{E}}_{\left( \theta \mid x\right) }\theta \right| }^{\delta }{f}_{G}\left( x\right) \mathrm{d}x \leq {\int }_{\Omega }\left( {{\mathrm{E}}_{\left( \theta \mid x\right) ... | Yes |
Lemma 2.2 By using Holder inequality, it follows that \( \phi \in {\mathcal{E}}_{m, h} \rightarrow {\int }_{{\mathbb{R}}^{N}}\left( {{\left| \nabla u\right| }^{m - 2}\nabla u\nabla \phi + }\right. \) \( h\left( x\right) {\left| u\right| }^{m - 2}{u\phi })\mathrm{d}x \in \mathbb{R} \) is bounded and | \[ \left| {{\int }_{{\mathbb{R}}^{N}}\left( {{\left| \nabla u\right| }^{m - 2}\nabla u\nabla \phi + h\left( x\right) {\left| u\right| }^{m - 2}{u\phi }}\right) \mathrm{d}x}\right| \leq 2\parallel u{\parallel }_{{\mathcal{E}}_{m, h}}^{m - 1}\parallel \phi {\parallel }_{{\mathcal{E}}_{m, h}}. \] | Yes |
Lemma 2.4 If hypotheses (HF)(iii) and (HF)(vi) hold, then there are \( \rho \geq 0,\eta > 0 \) such that \( \mathop{\inf }\limits_{{\partial {B}_{\rho }}}{I}_{ + }\left( u\right) \geq \eta \), where \( \partial {B}_{\rho } = \{ u \in \mathcal{E} : \parallel u\parallel < \rho \} \) . | Proof In view of (HF)(iii) and (HF)(vi), for \( \forall \epsilon > 0 \), there is \( {l}_{\epsilon } > 0 \) such that\n\n\[ F\left( {x, t}\right) \leq \frac{1}{p}\left( {l\left( x\right) + \epsilon }\right) {\left| t\right| }^{p} + {l}_{\epsilon }{\left| t\right| }^{r}, r \in \left( {p,{p}^{ * }}\right) . \]\n\n(2.26)\... | Yes |
Lemma 2.5 If hypotheses (HF)(iii) and (HF)(iv) hold. Then \( {I}_{ + } \) satisfies a mountain pass geometry. | Proof In view of (HF)(iii) and (HF)(iv), there exist \( M > 0,{C}_{M} > 0 \) such that \( t \in \mathcal{E} \) , we have\n\n\[ F\left( {x, t}\right) \geq \frac{M{\left| t\right| }^{p}}{p} - {C}_{M} \]\n\n(2.30)\n\nLet \( B \subset {\mathbb{R}}^{N} \) be a unit ball, \( u \in {C}_{0}^{1}\left( B\right) \) with \( \paral... | Yes |
Theorem 3.1 If (HF) holds and \( a, b \) are continuous, coercive and positive functions, then (1.1) has at least one nonnegative nontrivial weak solution. | Proof By Lemma 2.3-2.5, we have proved that \( {I}_{ + } \) satisfies a mountain pass geome- \( \operatorname{try}\left\lbrack {18}\right\rbrack \) . Lemma 2.3-2.5 guarantee the existence of \( 0 ≢ u \in \mathcal{W} \) satisfying the following\n\n\[{\int }_{{\mathbb{R}}^{N}}\left( {{\left| \nabla u\right| }^{p - 2}\nab... | Yes |
Theorem 1 For any \( {Y}^{k} = {\left( {Y}_{i}^{k}\right) }_{p \times 1} = {\left\lbrack {L}_{i}^{k},{U}_{i}^{k}\right\rbrack }_{p \times 1} \subseteq {Y}^{0} \), the following conclusions hold:\n\n(i) If \( {\mathrm{{RUB}}}^{k} < \mathrm{{LB}} \), then the \( {Y}^{k} \) does not contain any global optimal solution of ... | Proof (i) If \( {\operatorname{RUB}}^{k} < \mathrm{{LB}} \), then we have\n\n\[ v\left( {Y}^{k}\right) = \mathop{\max }\limits_{{y \in {Y}^{k}, x \in D}}\mathop{\sum }\limits_{{i = 1}}^{p}\frac{{\delta }_{i}\left( x\right) }{{y}_{i}} \leq \mathop{\sum }\limits_{{i = 1}}^{p}\mathop{\max }\limits_{{y \in {Y}^{k}, x \in D... | Yes |
Theorem 2.2 Let \( \Lambda \subseteq { \cup }_{i = 1}^{n}{\Lambda }_{i} \), where \( {\Lambda }_{1},{\Lambda }_{2},\cdots ,{\Lambda }_{n} \) are compact. For each \( \lambda \in \) \( \Lambda \), let \( {\left( {F}_{\lambda, n}\right) }_{n = 1}^{\infty } \) be a sequence of nonzero bounded linear operators on a separab... | Proof Let \( {U}_{1},{U}_{2} \) be two nonempty open set of \( X \) and \( x \in {D}_{1} \) . Choose \( \delta > 0 \) such that \( B\left( {x,\delta }\right) \subset {U}_{2} \) . Take \( \varepsilon = \frac{\delta }{3} \) . Then there exist a dense set \( {D}_{2} \), an increasing sequence \( {\left( {m}_{k}\right) }_{... | Yes |
Proposition 2.1 If all the operators \( {F}_{\lambda, n} = {F}_{\lambda }^{n} \) for a family \( \left\{ {{F}_{\lambda } : \lambda \in \Lambda }\right\} \) and the hypotheses in Theorem 2.3 hold, then \( {\left( {F}_{\lambda }^{n} \oplus {F}_{\lambda }^{n}\right) }_{n = 1}^{\infty } \) is universal. | Proof By theorem 3.4 in [9], it suffice to show for any \( \lambda \in \Lambda \) and any nonempty open sets \( U, V, W \) with \( 0 \in W \), there exists \( n \) for which \( {F}_{\lambda }^{n}\left( U\right) \cap W \neq \varnothing \) and \( {F}_{\lambda }^{n}\left( W\right) \cap V \neq \varnothing \) . Take \( x \i... | Yes |
Theorem 1.1 For every \( {U}_{0} = \left( {{u}_{10},{u}_{20}}\right) \in {L}^{2}\left( {\mathbb{R}}^{N}\right) \times {L}^{2}\left( {\mathbb{R}}^{N}\right) \), there exist \( {T}_{\min },{T}_{\max } \in \) \( (0, + \infty \rbrack \) and a unique, maximal solution \( U \) of (1.1) with each | \[ {u}_{j} \in C\left( {\left( {-{T}_{\min },{T}_{\max }}\right) ,{L}^{2}\left( {\mathbb{R}}^{N}\right) }\right) \cap \left\{ {{L}_{loc}^{\frac{2\left( {N + 2}\right) }{N}}\left( {-{T}_{\min },{T}_{\max }}\right) ;{L}^{\frac{2\left( {N + 2}\right) }{N}}\left( {\mathbb{R}}^{N}\right) }\right\}, j = 1,2. \] | No |
Proposition 2.1 Assume that \( U = \left( {{u}_{1},{u}_{2}}\right) \) is a solution of (1.1). Then there exist positive constants \( {C}_{1} = {C}_{1}\left( N\right) ,{C}_{2} = {C}_{2}\left( N\right) ,{C}_{1}^{\prime } = {C}_{1}^{\prime }\left( {N,\mu ,\gamma ,{\alpha }_{k},{\beta }_{k},{p}_{k},{q}_{k}}\right) \) and \... | Proof Obviously, (2.5) is just the Strichartz's inequality. And we have\n\n\[ \n{\begin{Vmatrix}{\Phi }_{1}\left( {u}_{1},{u}_{2}\right) \end{Vmatrix}}_{{L}^{\frac{2\left( {N + 2}\right) }{N}}\left( {I \times {\mathcal{R}}^{N}}\right) } \n\] \n\n\[ \n\leq C{\left( {\int }_{I}{\int }_{{\mathbb{R}}^{N}}{\left\{ \mu {\lef... | Yes |
Lemma 3.1 Assume that \( U = \left( {{u}_{1},{u}_{2}}\right) \) is the blow-up solution of (1.1) with \( {U}_{0} \in \) \( \left\{ {{L}^{2}\left( {\mathbb{R}}^{N}\right) \times {L}^{2}\left( {\mathbb{R}}^{N}\right) }\right\} \smallsetminus \{ \left( {0,0}\right) \} ,{T}_{\max } < \infty \) and each\n\n\[ \left. {{u}_{j... | Proof Our proof is inspired by that of [3]. Note\n\n\[ {\begin{Vmatrix}{u}_{j}\end{Vmatrix}}_{{L}^{\frac{2\left( {N + 2}\right) }{N}}\left( {\left( {{T}_{0},{T}_{1}}\right) \times {\mathbb{R}}^{N}}\right) } = {\begin{Vmatrix}\mathcal{J}\left( \cdot - {T}_{0}\right) {u}_{j}\left( {T}_{0}\right) + {\Phi }_{j}\left( {u}_{... | Yes |
Lemma 3.1 Let \( B, A \in \mathcal{P}\left( Y\right) \) . If \( B \) and \( A \) are compact, then one has\n\n\[ A + {G}_{k}\left( {B, A}\right) \cdot k \subset B + K. \] | Proof Since \( {G}_{k}\left( {y, z}\right) = {g}_{k}\left( {y, z}\right) \) is continuous for any \( \left( {y, z}\right) \in Y \times Y \) and \( B \) and \( A \) are compact, we conclude that there exist points \( b \in B \) and \( a \in A \) such that \( {G}_{k}\left( {b, a}\right) = {G}_{k}\left( {B, A}\right) \) .... | Yes |
Proposition 3.1 Let \( F : X \rightrightarrows Y \) be \( K \) -quasiconvex set-valued map and \( A \in \mathcal{P}\left( Y\right) \) be any compact subset. Suppose that \( F\left( x\right) \) is a compact set for all \( x \in X \) . Then, \( {G}_{k}\left( {F\left( \cdot \right), A}\right) \) is quasiconvex on \( X \) ... | Proof Let \( {x}_{1},{x}_{2} \in X \) be any points and \( t \in \left\lbrack {0,1}\right\rbrack \) be any real number. It follows from Lemma 3.1 that\n\n\[ A + {G}_{k}\left( {F\left( {x}_{1}\right), A}\right) \cdot k \subset F\left( {x}_{1}\right) + K, \]\n\n\[ A + {G}_{k}\left( {F\left( {x}_{2}\right), A}\right) \cdo... | Yes |
Proposition 3.2 Let \( A \in \mathcal{P}\left( Y\right) \) be any element, \( k \in \operatorname{int}K \) and \( F : X \rightrightarrows Y \) be a set-valued map. If \( {G}_{k}\left( {F\left( \cdot \right), A}\right) \) is quasiconvex over \( X \), then \( F \) is \( K \) -quasiconvex. | Proof We proceed by contradiction. Suppose that \( F \) is not \( K \) -quasiconvex, i.e. there exist \( {x}_{1},{x}_{2} \in X \) and \( \bar{t} \in \left\lbrack {0,1}\right\rbrack \) such that\n\n\[ \left( {F\left( {x}_{1}\right) + K}\right) \cap \left( {F\left( {x}_{2}\right) + K}\right) ⊄ F\left( {\left( {1 - \bar{t... | Yes |
Lemma 3.2 Let the initial datum \( {f}_{0}\left( {x, v}\right) \) be a nonnegative function and satisfy conditions (A1) and (A2). Then there exists one and only one mild solution \( f\left( {t, x, v}\right) \in \) \( C\left( {\lbrack 0,\infty ),{L}^{1}\left( {{\mathbb{R}}_{x}^{3} \times {\mathbb{R}}_{v}^{3},\left( {1 +... | Moreover,\n\n\[ {N}_{m + k}\left( {f\left( t\right) }\right) \leq C\left( T\right) \]\n\n(3.3)\n\n\[ {N}_{m, k}\left( {f\left( t\right) }\right) \leq C\left( T\right) \]\n\n(3.4)\n\n\[ \left( {1 + {\left| x\right| }^{k}}\right) \rho \left( {t, x}\right) \geq C\left( T\right) > 0. \]\n\n(3.5) | No |
Lemma 3.3 Suppose that \( {f}_{n} \in C\left( {\lbrack 0,\infty ),{L}^{1}\left( {{\mathbb{R}}_{x}^{3} \times {\mathbb{R}}_{v}^{3}}\right) }\right) \) is a distribution solution of the BGK equation (3.8)-(3.10) with condition (3.6) and (3.7). Then we have\n\n\[ \n{\begin{Vmatrix}{f}_{n}\left( t\right) \end{Vmatrix}}_{p}... | Proof Firstly, we will prove (3.13). In fact, we notice that the mild solution form of the BGK equation can be written as\n\n\[ \n{f}_{n}\left( {t, x, v}\right) = {\mathrm{e}}^{-t}G\left( t\right) {f}_{0}^{n} + {\int }_{0}^{t}{\mathrm{e}}^{\left( s - t\right) }G\left( {s - t}\right) M\left\lbrack {f}_{n}\right\rbrack \... | Yes |
Lemma 3.1 Let \( A = M - N \) be a splitting of the non-Hermitian positive definite matrix \( A \) . Assume that \( M, N \) are given by (2.2). Then\n\n\[ \begin{Vmatrix}{{\left( \alpha I + {A}_{H}\right) }^{-1}N{M}^{-1}\left( {{\alpha I} + {A}_{H}}\right) }\end{Vmatrix} < 1. \]\n\n(3.1)\n\nFurthermore, if \( A \) is a... | Proof Since \( {\left( \alpha I + {A}_{H}\right) }^{-1}N{M}^{-1}\left( {{\alpha I} + {A}_{H}}\right) = {\left( \alpha I + {A}_{H}\right) }^{-1}\left( {{\alpha I} - {A}_{H}}\right) \left( {{\alpha I} - {A}_{S}}\right) ({\alpha I} + \) \( {\left. {A}_{S}\right) }^{-1} \), then\n\n\[ \begin{Vmatrix}{{\left( \alpha I + {A}... | Yes |
Theorem 3.1 Let \( A = M - N \) be a splitting of the non-Hermitian positive definite matrix \( A \) . Assume that \( M, N \) are given by (2.2). \( {\omega }_{k} \) is solved by the quadratic programming (2.12). Let \( z = {Ax} - b \) . If \( \left\langle {{\left( \alpha I + {A}_{H}\right) }^{-1}{\bar{z}}_{k + 1},{\le... | \[ r \geq - \left( {\ln \begin{Vmatrix}{{\left( \alpha I + {A}_{H}\right) }^{-1}N{M}^{-1}\left( {{\alpha I} + {A}_{H}}\right) }\end{Vmatrix} + \ln \left( {\sin \varphi }\right) }\right) . \] \( \left( {3.3}\right) \) If \( A \) is a normal matrix, then \[ r \geq - \left( {\ln \rho \left( {{M}^{-1}N}\right) + \ln \left(... | Yes |
Lemma 3.2 Let \( A = M - N \) be a splitting of the non-Hermitian positive definite matrix \( A \) . Assume that \( M, N \) are given by (2.7). Then\n\n\[ \begin{Vmatrix}{{\left( \alpha I + {A}_{G}\right) }^{-1}N{M}^{-1}\left( {{\alpha I} + {A}_{G}}\right) }\end{Vmatrix} < 1. \]\n\n\( \left( {3.5}\right) \)\n\nFurtherm... | Proof Since \( {\left( \alpha I + {A}_{G}\right) }^{-1}N{M}^{-1}\left( {{\alpha I} + {A}_{G}}\right) = {\left( \alpha I + {A}_{G}\right) }^{-1}\left( {{\alpha I} - {A}_{G}}\right) \left( {{\alpha I} - {A}_{S} - {A}_{K}}\right) ({\alpha I} + \) \( {\left. {A}_{S} + {A}_{K}\right) }^{-1} \), then\n\n\[ \begin{Vmatrix}{{\... | Yes |
Theorem 3.3 Let \( A = M - N \) be a splitting of the non-Hermitian positive definite matrix \( A \) . Assume that \( M, N \) are given by (2.5) or (2.9). \( {\omega }_{k} \) is obtained by the quadratic programming (2.14). Let \( z = {Ax} - b \) . If \( \left\langle {{\left( \alpha P + {A}_{H}\right) }^{-1}{\bar{z}}_{... | \[ r \geq - \left( {\ln \begin{Vmatrix}{{\left( \alpha P + {A}_{H}\right) }^{-1}N{M}^{-1}\left( {{\alpha P} + {A}_{H}}\right) }\end{Vmatrix} + \ln \left( {\sin \varphi }\right) }\right) . \] (3.11) | Yes |
Lemma 2.1 Let \( \\alpha = \\min \\left\\{ {{H}_{1}\\left( {s}^{\\prime }\\right) ,{H}_{1}\\left( s\\right) }\\right\\} \) and \( \\beta = \\min \\left\\{ {{H}_{2}\\left( {t}^{\\prime }\\right) ,{H}_{2}\\left( t\\right) }\\right\\} \), where \( s,{s}^{\\prime }, t,{t}^{\\prime } \\in \\) \( \\left\\lbrack {0,1}\\right\... | Proof From (2.1), we obtain\n\n\[ \n{\\Delta }_{s, t}{X}^{H}\\left( {{s}^{\\prime },{t}^{\\prime }}\\right)\n\]\n\n\[ \n= {X}^{H\\left( {{s}^{\\prime },{t}^{\\prime }}\\right) }\\left( {{s}^{\\prime },{t}^{\\prime }}\\right) - {X}^{H\\left( {{s}^{\\prime }, t}\\right) }\\left( {{s}^{\\prime }, t}\\right) - {X}^{H\\left... | Yes |
Lemma 3.1 \( {}^{\left\lbrack 4\right\rbrack } \) Let \( {X}^{n} = \left\{ {{X}^{n}\left( {s, t}\right) : \left( {s, t}\right) \in {\left\lbrack 0,1\right\rbrack }^{2}}\right\} \) be a sequence of continuous adapted processes. Assume that there exists \( a = \left( {{a}_{1},{a}_{2}}\right) ,{a}_{1},{a}_{2} > 0 \) and c... | \[ \mathrm{E}{\left| {\Delta }_{s, t}{X}^{n}\left( {s}^{\prime },{t}^{\prime }\right) \right| }^{p} \leq {\mathrm{C}}_{p}{\left| {s}^{\prime } - s\right| }^{{a}_{1}p}{\left| {t}^{\prime } - t\right| }^{{a}_{2}p}, \] (3.1) and \[ \mathrm{E}{\left| {\Delta }_{s,1}{X}^{n}\left( {s}^{\prime }, t\right) \right| }^{p} \leq {... | Yes |
Theorem 3.1 Under the assumptions 1-5 in Section 6, if \( {\mathbf{\beta }}_{0} \) is the true value of \( \mathbf{\beta } \), we have \( {l}_{n}\left( {\mathbf{\beta }}_{0}\right) \overset{D}{ \rightarrow }{\chi }_{p}^{2} \), where \( {\chi }_{p}^{2} \) is a \( {\chi }^{2} \) -distribution with \( p \) degrees of free... | Theorem 3.1 enables us to make inference about the parametric components \( \mathbf{\beta } \) . Specifically, we can test the hypothesis \( {H}_{0} : \mathbf{\beta } = {\mathbf{\beta }}_{0} \) and construct confidence regions for \( \mathbf{\beta } \) . Define \( {\mathbf{I}}_{\alpha }\left( \mathbf{\beta }\right) = \... | Yes |
Lemma 6.1 Let \( \left( {{\mathbf{X}}_{1},{\mathbf{Y}}_{1}}\right) ,\cdots ,\left( {{\mathbf{X}}_{n},{\mathbf{Y}}_{n}}\right) \) be i.i.d random vectors, where the \( {\mathbf{Y}}_{i}s \) are scalar random variables. Further assume that \( \mathrm{E}{\left| {Y}_{i}\right| }^{s} < \infty \) and \( \mathop{\sup }\limits_... | Proof This lemma can be proved by Mack and Silverman \( {}^{\left\lbrack {17}\right\rbrack } \) . | No |
Lemma 6.2 Under the assumptions 1-5, we have\n\n\[ \n{n}^{-1}\mathop{\sum }\limits_{{i = 1}}^{n}{\delta }_{i}\left( {{\mathbf{X}}_{i} - {\overline{\mathbf{X}}}_{i}}\right) {\left( {\mathbf{X}}_{i} - {\overline{\mathbf{X}}}_{i}\right) }^{\mathrm{T}}\overset{P}{ \rightarrow }\mathbf{\sum }.\n\] | Proof By Lemmas 3.1 and 3.2 in Opsomer and Ruppert (1997), we have\n\n\[ \n{\mathbf{S}}_{1}^{ * } = {\mathbf{S}}_{1} - {\mathbf{1}}_{n}{\mathbf{1}}_{n}^{\mathrm{T}}/n + {o}_{p}\left( {{\mathbf{1}}_{n}{\mathbf{1}}_{n}^{\mathrm{T}}/n}\right) ,\;{\left( {\mathbf{I}}_{n} - {\mathbf{S}}_{1}^{ * }{\mathbf{S}}_{2}^{ * }\right... | Yes |
Lemma 6.3 Under the Assumptions 1-5, we have \( \mathop{\max }\limits_{{1 \leq i \leq n}}\begin{Vmatrix}{{\mathbf{\eta }}_{in}\left( \mathbf{\beta }\right) }\end{Vmatrix} = {o}_{p}\left( {n}^{1/2}\right) \), where \( \parallel \cdot \parallel \) is the Euclidean norm with \( \parallel \mathbf{a}\parallel = {\left( {a}_... | Proof The proof of the Lemma is similar to that of lemma A.6. in WANG et al. \( {}^{\left\lbrack {18}\right\rbrack } \) and Lemma 4.2. in WEI et al. \( {}^{\left\lbrack {19}\right\rbrack } \) . We omit the details here. | No |
Lemma 6.4 Under Assumptions 1-5, we have\n\n\[ \frac{1}{\sqrt{n}}\mathop{\sum }\limits_{{i = 1}}^{n}{\mathbf{\eta }}_{in}\left( \mathbf{\beta }\right) \overset{D}{ \rightarrow }N\left( {\mathbf{0},\mathbf{\Omega }}\right) \;\text{ as }\;n \rightarrow \infty . \] | Proof Let \( \mathbf{\Delta } = \operatorname{diag}\left\{ {{\delta }_{1},\cdots ,{\delta }_{n}}\right\} \), from the definition of \( {\mathbf{\eta }}_{\text{in }}\left( \mathbf{\beta }\right) \), we have that\n\n\[ \frac{1}{\sqrt{n}}\mathop{\sum }\limits_{{i = 1}}^{n}{\mathbf{\eta }}_{in}\left( \mathbf{\beta }\right)... | Yes |
Lemma 6.5 Under the assumptions 1-5, we have\n\n\\[ \n\\frac{1}{n}\\mathop{\\sum }\\limits_{{i = 1}}^{n}{\\mathbf{\\eta }}_{in}\\left( \\mathbf{\\beta }\\right) {\\mathbf{\\eta }}_{in}{\\left( \\mathbf{\\beta }\\right) }^{\\mathrm{T}}\\overset{p}{ \\rightarrow }\\mathbf{\\sum },\\text{ as }\\;n \\rightarrow \\infty .\n... | Proof By the same argument as that in the proof of Lemma 6.4, we can prove Lemma 6.5 by the law of large numbers. We omit the details here. | No |
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