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Lemma 3.2 Assume that \( u \) satisfies the hypotheses of Lemma 3.1. Suppose further that \( g = f{g}_{0} \) and \( M \) has a special exhaustion function \( \Phi \) satisfying\n\n\[ \n\\text{(i)}\\mathop{\\liminf }\\limits_{{t \\rightarrow + \\infty }}{\\left| {\\nabla }_{{g}_{0}}\\Psi \\right| }_{{g}_{0}}^{2}\\left( ... | Proof By (2.4) and the definition of Lie differentiation, we have\n\n\[ \n\\left( {{L}_{X}g}\\right) \\left( {{e}_{s},{e}_{t}}\\right) = g\\left( {{\\nabla }_{{e}_{s}}X,{e}_{t}}\\right) + g\\left( {{e}_{s},{\\nabla }_{{e}_{t}}X}\\right)\n\]\n\n\[ \n= \\frac{1}{2}\\left( {\\nabla {\\theta }_{{\\nabla }_{{g}_{0}}\\Psi }}... | Yes |
Lemma 3.3 Maintaining the notation of Lemma 3.2. Denote by \( A\left( x\right) \) (resp. \( B\left( x\right) \) ) the smallest (resp. largest) eigenvalue of \( {\operatorname{Hess}}_{{g}_{0}}\left( \Psi \right) \), that is, the Hessian of \( \Psi \)\n\n\[ A\left( x\right) {g}_{0} \leq {\operatorname{Hess}}_{{g}_{0}}\le... | Proof The proof is a modification of that of Lemma 1.3 in [3] (see also [10]), and we outline it here for completeness.\n\nSince \( {L}_{X}g \) is symmetric, the local orthonormal frame \( \left\{ {e}_{s}\right\} \) may be chosen in such a way that diagonalizes \( {L}_{X}g \) . Let \( {\mu }_{s} \) be the corresponding... | Yes |
Lemma 3.4 Maintaining the notation and assumption of Lemma 3.2. Assume that the functions \( A\left( x\right), B\left( x\right) \) satisfy \( A\left( x\right) \geq \frac{m - 2}{m}B\left( x\right) \) if \( p = 0 \) and \( A\left( x\right) \geq \frac{m - 1}{m + 1}B\left( x\right) \) if \( p \geq 1 \) . Suppose also that\... | Proof We consider the case \( p \geq 1 \) . If \( p = 0 \), the argument is similar. By Lemma 3.3, we have\n\n\[ \frac{1}{2}{\left| du\right| }^{2}\left( {\operatorname{tr}{L}_{X}g - k}\right) - \mathop{\sum }\limits_{{s, t}}{\left\langle {i}_{{e}_{s}}du,{i}_{{e}_{t}}du\right\rangle }_{g}\left( {{L}_{X}g}\right) \left(... | Yes |
Theorem 3.1 Maintaining the notation and assumption of Lemma 3.3. Assume that the functions \( \;A\left( x\right) ,\;B\left( x\right) \; \) satisfy \( \;A\left( x\right) \geq \frac{m - 2}{m}B\left( x\right) \; \) if \( \;p = 0\; \) and \( \;A\left( x\right) \geq \frac{m - 1}{m + 1}B\left( x\right) \; \) if \( \;p \geq ... | Proof The case \( p = 0 \) can be deduced directly from Lemma 3.4. Thus, assume that \( p \geq 1 \), and let \( u \in {L}^{2}\left( {{A}^{p}\left( M\right) }\right) \) be such that \( {\bigtriangleup }^{p}u = {\lambda u} \) with \( \lambda > 0 \) . Then \( v = {\delta u} \) belongs to \( {L}^{2}\left( {{A}^{p - 1}\left... | Yes |
Lemma 4.1 Let \( \left( {M,{g}_{0}}\right) \) be a complete Riemannian manifold with a pole \( o \) and let \( r \) be the distance function relative to \( o \) . Denote by \( X = r{\nabla }_{{g}_{0}}r = r\partial r \) . Assume that the radial sectional curvature of \( \;M\; \) satisfies \( \; - \frac{a}{1 + {r}^{2}} \... | Proof We consider the case \( p \geq 1 \) . The statement relative to the case \( p = 0 \) can be proved in a similar way.\n\nFor \( X = r{\nabla }_{{g}_{0}}r = r\partial r \), we have\n\n\[ {L}_{X}{g}_{0} = {\operatorname{Hess}}_{{g}_{0}}\left( {r}^{2}\right) = {2r}{\operatorname{Hess}}_{{g}_{0}}r + {2dr} \otimes {dr}... | Yes |
Lemma 4.2 Let \( \left( {M,{g}_{0}}\right) \) be a complete Riemannian manifold with a pole \( o \) and let \( r \) be the distance function relative to \( o \) . Assume that the radial sectional curvature of \( M \) satisfies \( - \frac{a}{1 + {r}^{2}} \leq {K}_{r} \leq \frac{b}{1 + {r}^{2}} \) with \( a \geq 0,\;b \i... | Proof If \( \sqrt{1 + {4a}} + \sqrt{1 - {4b}} \geq 2 \), that is, \( {B}_{2} - 1 \geq 1 - {B}_{1} \), then for \( {du} \in {A}^{p + 1}\left( M\right) \), by Lemma 4.1, we have \[ \frac{1}{2}{\left| du\right| }^{2}\left( {\operatorname{tr}{L}_{X}g - k}\right) - \mathop{\sum }\limits_{{s, t}}{\left\langle {i}_{{e}_{s}}du... | Yes |
Theorem 4.1 Let \( \left( {M,{g}_{0}}\right) \) be a complete Riemannian manifold with a pole \( o \) and let \( r \) be the distance function relative to \( o \) . Assume that the radial sectional curvature of \( \;M\; \) satisfies \( \; - \frac{a}{1 + {r}^{2}}\; \leq \;{K}_{r}\; \leq \;\frac{b}{1 + {r}^{2}}\; \) with... | \[ \left| {{f}^{-1}\frac{\partial f}{\partial r}}\right| \leq \frac{1}{2}\left\lbrack {\sqrt{1 - {4b}} - \sqrt{1 + {4a}} + \frac{4}{m - 1}}\right\rbrack {r}^{-1}, \] if \( p = 0 \), \[ \left| {{f}^{-1}\frac{\partial f}{\partial r}}\right| \leq \frac{m - 1}{2\left( {m - {2p} + 1}\right) }\left\lbrack {\sqrt{1 - {4b}} - ... | Yes |
Theorem 2.2 Let \( \\left\\{ {{\\varepsilon }_{i};1 \\leq i \\leq n}\\right\\} \) be identically distributed PA errors, and \( E\\left| {\\varepsilon }_{i}\\right| = \) \( O\\left( {i}^{-\\left( {1 + {2\\rho }}\\right) }\\right), E{\\varepsilon }_{i}^{2} = {\\sigma }^{2} < \\infty \) . Assume that assumptions (A1)-(A3)... | \[ \mathop{\\sup }\\limits_{{0 \\leq t \\leq 1}}\\left| {{g}_{n}\\left( t\\right) - g\\left( t\\right) }\\right| \\overset{p}{ \\rightarrow }0 \] | Yes |
Theorem 2.3 Let \( \\left\\{ {{\\varepsilon }_{i};1 \\leq i \\leq n}\\right\\} \) be PA errors with mean zero, assume that assumptions (A1)-(A3) hold, and \( {2}^{m} = O\\left( {n}^{1 - \\tau }\\right) \) for \( 1/2 < \\tau < 1 \), then | \[ {g}_{n}\\left( t\\right) \\rightarrow g\\left( t\\right) \\text{ a.s.. } \] | No |
Lemma 3.1 Under assumptions (A1)-(A3), we have\n\n(I) \( \mathop{\sup }\limits_{{x, m}}{\int }_{0}^{1}\left| {{E}_{m}\left( {x, y}\right) }\right| {dy} < \infty \) ;\n\n(II) \( {\int }_{0}^{1}\left| {{E}_{m}\left( {t, s}\right) }\right| g\left( s\right) {ds} = g\left( t\right) + O\left( {\eta }_{m}\right) \), where\n\n... | Proof The proofs of (I) and (II) can see Antoniadis et al. (1994), and (III) and (IV) can be found in Lemma 2.1(3) of Sun and Chai (2004). | No |
Lemma 3.2 Let assumptions (A1)-(A3) hold, and \( \left\{ {{\varepsilon }_{i};1 \leq i \leq n}\right\} \) be PA random variables with zero means, then\n\n(I) \( E{g}_{n}\left( t\right) - g\left( t\right) = O\left( {\eta }_{m}\right) + O\left( {n}^{-1}\right) ,\mathop{\lim }\limits_{{n \rightarrow \infty }}E{g}_{n}\left(... | Proof Follows immediately from Lemma 3.1 and the method used for proving Lemma 3.1 of Sun and Chai (2004). | No |
Lemma 3.5 (see Yang (2005)) Let \( \\left\\{ {{X}_{j} : j \\geq 1}\\right\\} \) be a sequence of associated PA random variables, and let \( \\left\\{ {{a}_{j} : j \\geq 1}\\right\\} \) be a real constant sequence, \( 1 = {m}_{0} < {m}_{1} < \\cdots < {m}_{k} = n \) . Denote by \( {Y}_{l} \\mathrel{\\text{:=}} \\mathop{... | \[ \\left| {E\\exp \\left( {{it}\\mathop{\\sum }\\limits_{{l = 1}}^{k}{Y}_{l}}\\right) - \\mathop{\\prod }\\limits_{{l = 1}}^{k}E\\exp \\left( {{it}{Y}_{l}}\\right) }\\right| \\leq 4{t}^{2}\\mathop{\\sum }\\limits_{{1 \\leq s \\leq j \\leq n}}\\left| {{a}_{s}{a}_{j}}\\right| \\left| {\\operatorname{Cov}\\left( {{X}_{s}... | Yes |
Theorem 2.1 If \( f \in {\mathcal{A}}_{p} \) and \( j \in \{ 1,2,\cdots, s\} \), then\n\n\[ z{\left\lbrack {z}^{1 - p}\mathcal{H}\left( {b}_{j} + 1\right) f\left( z\right) \right\rbrack }^{\prime \prime } = {b}_{j}{\left\lbrack {z}^{1 - p}\mathcal{H}\left( {b}_{j}\right) f\left( z\right) \right\rbrack }^{\prime } - {b}... | Proof From (1.6), we easily get\n\n\[ z{\left\lbrack \mathcal{H}\left( {b}_{j} + 1\right) f\left( z\right) \right\rbrack }^{\prime } + \left( {1 - p}\right) \mathcal{H}\left( {{b}_{j} + 1}\right) f\left( z\right) = {b}_{j}\mathcal{H}\left( {b}_{j}\right) f\left( z\right) + \left( {1 - {b}_{j}}\right) \mathcal{H}\left( ... | Yes |
Theorem 2.2 If \( f \in {\mathcal{A}}_{p} \) and \( {z}^{1 - p}\mathcal{H}\left( {{b}_{j} + 1}\right) f\left( z\right) \) is convex univalent function, then \( {z}^{1 - p}\mathcal{H}\left( {b}_{j}\right) f\left( z\right) \) is close-to-convex of order \( \operatorname{Re}\left( \frac{{b}_{j} - 1}{\left| {b}_{j}\right| ... | Proof From (2.1), we conclude that\n\n\[ \frac{{\left\lbrack {z}^{1 - p}\mathcal{H}\left( {b}_{j}\right) f\left( z\right) \right\rbrack }^{\prime }}{{\left\lbrack {z}^{1 - p}\mathcal{H}\left( {b}_{j} + 1\right) f\left( z\right) \right\rbrack }^{\prime }} = \frac{z{\left\lbrack {z}^{1 - p}\mathcal{H}\left( {b}_{j} + 1\r... | Yes |
Theorem 2.3 If \( f \in {\mathcal{A}}_{p} \) and \( j \in \{ 1,2,\cdots, s\} \), then\n\n\[ \n{z}^{1 - p}\mathcal{H}\left( {{b}_{j} + 1}\right) f\left( z\right) = {\mathcal{J}}_{{b}_{j} - 1}\left\lbrack {{z}^{1 - p}\mathcal{H}\left( {b}_{j}\right) f\left( z\right) }\right\rbrack \n\] \n\nwhere \( {\mathcal{J}}_{{b}_{j}... | Proof From (1.5), we have\n\n\[ \n\mathcal{H}\left( {{b}_{j} + 1}\right) f\left( z\right) = {z}^{p} + \mathop{\sum }\limits_{{k = 1}}^{\infty }\frac{{\left( {a}_{1}\right) }_{k}\cdots {\left( {a}_{q}\right) }_{k}}{{\left( {b}_{1}\right) }_{k}\cdots {\left( {b}_{j} + 1\right) }_{k}\cdots {\left( {b}_{s}\right) }_{k}}\fr... | Yes |
Theorem 2.4 Let \( m \in \mathbb{N} \) and \( j \in \{ 1,2,\cdots, s\} \) . If \( A \in \mathbb{C} \) and \( B \in \left\lbrack {-1,0}\right\rbrack \) satisfy (2.5) or (2.6) with \( \nu = {b}_{j} - 1 \), then \[ {W}_{p}\left( {\mathcal{H}\left( {b}_{j}\right) ;A, B}\right) \subseteq {W}_{p}\left( {\mathcal{H}\left( {{b... | Proof Clearly, it is sufficient to prove (2.8) only for \( m = 1 \) . Let \( f \in {W}_{p}\left( {\mathcal{H}\left( {b}_{j}\right) ;A, B}\right) \) , then from (1.11) we have \[ \frac{z{\left\lbrack {z}^{1 - p}\mathcal{H}\left( {b}_{j}\right) f\left( z\right) \right\rbrack }^{\prime }}{{z}^{1 - p}\mathcal{H}\left( {b}_... | Yes |
Theorem 2.5 If \( f \in {W}_{p}\left( {\mathcal{H}\left( {b}_{j}\right) ;A, B}\right), H\left( z\right) = {z}^{1 - p}\mathcal{H}\left( {b}_{j}\right) f\left( z\right) \in {\mathcal{S}}^{ * } \) and \( G\left( z\right) = \) \( \mathop{\sum }\limits_{{k = 0}}^{\infty }\frac{{\left( {b}_{j}\right) }_{k}}{{\left( {\widetil... | Proof Let \( f \in {W}_{p}\left( {\mathcal{H}\left( {b}_{j}\right) ;A, B}\right) \) . Then by the definition of the class \( {W}_{p}\left( {\mathcal{H}\left( {b}_{j}\right) ;A, B}\right) \) ,\n\nwe have\n\[ \n\frac{z{\left\lbrack {z}^{1 - p}\mathcal{H}\left( {b}_{j}\right) f\left( z\right) \right\rbrack }^{\prime }}{{z... | Yes |
Corollary 3.1 If \( {b}_{j},\widetilde{{b}_{j}} \) are real such that \( 0 < {b}_{j} \leq \widetilde{{b}_{j}} \) and \( \widetilde{{b}_{j}} \geq 2 \) or \( {b}_{j},\widetilde{{b}_{j}} \) are complex \( \left( {{b}_{j},{\widetilde{b}}_{j} \neq 0, - 1, - 2,\cdots }\right) \) such that \( \operatorname{Re}\left\lbrack {{b... | Proof Since \( A, B \) satisfy (1.12), so if \( f \in {W}_{p}\left( {\mathcal{H}\left( {b}_{j}\right) ;A, B}\right) \), then \( H\left( z\right) = {z}^{1 - p}\mathcal{H}\left( {b}_{j}\right) f\left( z\right) \in \) \( {\mathcal{S}}^{ * } \) . By Lemma 3.1, the function\n\n\[ G\left( z\right) = \mathop{\sum }\limits_{{k... | Yes |
Corollary 3.2 Let \( {b}_{j} \in \left( {-1,0}\right) \cup \left( {0,1}\right) \) and \( j \in \{ 1,2,\cdots, s\} \) . If \( \widetilde{{b}_{j}} > 3 + \left| {b}_{j}\right| \), then\n\n\[ \mathop{\sum }\limits_{{k = 0}}^{\infty }\frac{{\left( {b}_{j}\right) }_{k}}{{\left( \widetilde{{b}_{j}}\right) }_{k}}{z}^{k + 1} \i... | Proof If we choose \( b = 1, a = {b}_{j} - 1, c = \widetilde{{b}_{j}} - 1 \) in Lemma 3.2, then we obtain that\n\n\[ F\left( z\right) = \mathop{\sum }\limits_{{k = 0}}^{\infty }\frac{{\left( {b}_{j} - 1\right) }_{k}}{{\left( \widetilde{{b}_{j}} - 1\right) }_{k}}{z}^{k} \]\n\nis convex in \( \mathbb{U} \) for \( {b}_{j}... | Yes |
Corollary 3.3 Let \( {b}_{j} \in \left( {-1,0}\right) \cup \left( {0,1}\right) \) and \( j \in \{ 1,2,\cdots, s\} \) . If \( \widetilde{{b}_{j}} > 3 + \left| {b}_{j}\right| \), then\n\n\[ \n{W}_{p}\left( {\mathcal{H}\left( {b}_{j}\right) ;A, B}\right) \subseteq {W}_{p}\left( {\mathcal{H}\left( \widetilde{{b}_{j}}\right... | Proof The proof follows as the proof of Corollary 3.1 by using Corollary 3.2. | No |
Corollary 3.4 Let \( m \in \mathbb{N} \) and \( j \in \{ 1,2,\cdots, s\} \) . If \( \operatorname{Re}\left( {b}_{j}\right) > 1 \), then\n\n\[ \n{W}_{p}\left( {\mathcal{H}\left( {b}_{j}\right) ;A, B}\right) \subseteq {W}_{p}\left( {\mathcal{H}\left( {{b}_{j} + m}\right) ;A, B}\right) .\n\] | Proof Obviously, it is sufficient to prove this corollary only for \( m = 1 \) . If \( f \in \) \( {W}_{p}\left( {\mathcal{H}\left( {b}_{j}\right) ;A, B}\right) \), then \( H\left( z\right) = {z}^{1 - p}\mathcal{H}\left( {b}_{j}\right) f\left( z\right) \in {\mathcal{S}}^{ * }\left( {A, B}\right) \subseteq {\mathcal{S}}... | Yes |
Theorem 2.1 (see [3, p.54]) Assume that \( \eta \geq 0, m = \lceil \eta \rceil \), and \( f \in {A}^{m}\left\lbrack {a, b}\right\rbrack \) . Then\n\n\[ \n{}_{a}{D}_{t}^{-\eta }\left( {{}_{a}^{c}{D}_{t}^{\eta }f\left( t\right) }\right) = f\left( t\right) - \mathop{\sum }\limits_{{k = 0}}^{{m - 1}}\frac{{D}^{k}f\left( a\... | (2.4) | No |
Theorem 2.2 (see \( \\left\\lbrack {5,\\mathrm{p}{.74}}\\right\\rbrack \) ) The composition of two fractional Riemann-Liouville derivative operators: \( {}_{a}{D}_{t}^{p}\\left( {m - 1 \\leq p < m}\\right) \), and \( {}_{a}{D}_{t}^{q}\\left( {n - 1 \\leq q < n}\\right), m, n \) are both positive integer, | \[ {}_{a}{D}_{t}^{p}\\left( {{}_{a}{D}_{t}^{q}f\\left( t\\right) }\\right) = {}_{a}{D}_{t}^{p + q}f\\left( t\\right) - \\mathop{\\sum }\\limits_{{j = 1}}^{n}{\\left\\lbrack {}_{a}{D}_{t}^{q - j}f\\left( t\\right) \\right\\rbrack }_{t = a}\\frac{{\\left( t - a\\right) }^{-p - j}}{\\Gamma \\left( {1 - p - j}\\right) }.\] | Yes |
Lemma 3.1 In eq. (1.3), let \( \beta = 1 + \alpha ,\alpha \in \left( {0,1}\right) \), and we can have\n\n\[ v\left( t\right) = - \frac{1}{\Gamma \left( \alpha \right) }{\int }_{0}^{t}{\left( t - \tau \right) }^{\alpha - 1}f\left( {\tau, M - {\int }_{\tau }^{+\infty }v\left( s\right) {ds}, v\left( \tau \right) }\right) ... | Proof Using formula (2.5) and the equation \( {}_{0}{D}_{t}^{\beta }u\left( t\right) + f\left( {t, u\left( t\right) ,{u}^{\prime }\left( t\right) }\right) = 0 \), we can get\n\n\[ {}_{0}{D}_{t}^{-\alpha }\left( {{}_{0}{D}_{t}^{\beta }u\left( t\right) }\right) \; = {u}^{\prime }\left( t\right) - \mathop{\sum }\limits_{{... | Yes |
Lemma 1 For any symmetric positive-definite matrices \( G \) and \( Z \), of appropriate dimensions, the following inequality holds\n\n\[ - G{Z}^{-1}G \leq Z - {2G}. \] | Proof Since \( Z > 0 \), we have \( \left( {Z - G}\right) {Z}^{-1}\left( {Z - G}\right) \geq 0 \) . The proof follows immediately. | No |
Theorem 1 For given positive scalars \( h,\mu \) and \( \lambda \), if there exist symmetric positive-definite matrices \( X,{S}_{1},{S}_{2}, Z \), appropriately dimensioned matrices \( Y,{U}_{j},{V}_{j}\left( {j = 1,2,3}\right) \), and positive scalars \( {\varepsilon }_{1},{\varepsilon }_{2} \), such that the followi... | Proof Substituting the state feedback controller (2.5) into system (2.1), we obtain the resulting closed-loop system as\n\n\[ {dx}\left( t\right) = f\left( t\right) {dt} + g\left( t\right) {d\omega }\left( t\right) \]\n\nwhere\n\n\[ f\left( t\right) = \left( {A\left( t\right) + {B}_{1}\left( t\right) K}\right) x\left( ... | Yes |
Example 1 Consider the uncertain nonlinear single time-delay system (2.1) with the following parameters\n\n\[ \nA = \left\lbrack \begin{matrix} - 3 & 0 \\ 1 & 4 \end{matrix}\right\rbrack ,{A}_{1} = \left\lbrack \begin{matrix} 1 & 0 \\ - 1 & 3 \end{matrix}\right\rbrack ,{B}_{1} = \left\lbrack \begin{array}{l} 1 \\ 3 \en... | By using matlab solver feasp, for given \( \mu = {0.5},\lambda = {0.2} \), the feasibility upper bound of \( h\left( t\right) \) is 0.3108 . Choosing \( h = {0.3} \), according to Theorem 1, solve LMI in inequality (3.1), and get a set of solutions as follows\n\n\[ \nX = \left\lbrack \begin{array}{ll} {2.0317} & {0.846... | Yes |
Proposition 2.1 Under our assumptions, functional \( \varphi \left( x\right) \) is weakly lower semi-continuous on \( {H}_{2\pi }^{1} \) . | Proof First, it is easy to see that functional \( {\int }_{0}^{2\pi }{e}^{G\left( t\right) }{\left| {x}^{\prime }\left( t\right) \right| }^{2}\mathrm{\;d}t \) is convex continuous. Consequently, by Mazur Theorem, \( {\int }_{0}^{2\pi }{e}^{G\left( t\right) }{\left| {x}^{\prime }\left( t\right) \right| }^{2}\mathrm{\;d}... | Yes |
Lemma 3.1 Let \( \varphi \in {C}^{1}\left( {{H}_{2\pi }^{1}, R}\right) \) and \( M = \left\{ {x \in {H}_{2\pi }^{1} : {\psi }_{j}\left( x\right) = 0, j = 1,\cdots, n}\right\} \) , where \( {\psi }_{j} \in {C}^{1}\left( {{H}_{2\pi }^{1}, R}\right), j = 1,\cdots, n \), and \( {\psi }_{1}^{\prime }\left( x\right) ,\cdots ... | \[ {\varphi }^{\prime }\left( u\right) = \mathop{\sum }\limits_{{j = 1}}^{n}{\lambda }_{j}{\psi }_{j}^{\prime }\left( u\right) \] | Yes |
Lemma 3.6 Under our assumptions, \( x \in {H}_{2\pi }^{1} \) is a critical point of \( \varphi \) if and only if \( x \in M \) and \( x \) is a critical point of \( {\left. \varphi \right| }_{M} \) . | Proof If \( x \in {H}_{2\pi }^{1} \) is a critical point of \( \varphi \), by choosing \( v = 1 \) in (2.2), we have\n\n\[ \n{\int }_{0}^{2\pi }{e}^{G\left( t\right) }f\left( {t, x}\right) \mathrm{d}t - \mathop{\sum }\limits_{{j = 1}}^{p}{e}^{G\left( {t}_{j}\right) }{I}_{j}\left( {x\left( {t}_{j}\right) }\right) = 0, \... | Yes |
Lemma 2.1 Suppose that \( \varphi \) is an analytic self-map of \( \mathbb{D}, g \in H\left( \mathbb{D}\right), Y = {\mathcal{B}}_{\beta } \) or \( {\mathcal{Z}}_{\beta } \) , and the operator \( {J}_{\varphi, g}^{\left( n\right) } : {A}_{u}^{p} \rightarrow Y \) is bounded, then the operator \( {J}_{\varphi, g}^{\left(... | \[ \mathop{\lim }\limits_{{j \rightarrow \infty }}{\begin{Vmatrix}{J}_{\varphi, g}^{\left( n\right) }{f}_{j}\end{Vmatrix}}_{Y} = 0 \] | Yes |
Lemma 2.4 Suppose that \( w \in \mathbb{D} \), then for each fixed \( j \in \{ 1,2,\cdots, n\} \) there exist constants \( {c}_{1},{c}_{2},\cdots ,{c}_{n} \) such that the function\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{{i = 1}}^{n}{c}_{i}{f}_{\varphi \left( w\right), n - 2 + i}\left( z\right) \]\n\nsatisfying... | Proof For a fixed \( w \in \mathbb{D} \) and arbitrary constants \( {d}_{1},{d}_{2},\cdots ,{d}_{n} \), define the function\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{{i = 1}}^{n}\frac{{d}_{i}}{n - 2 + i + \frac{1}{p}}{f}_{\varphi \left( w\right), n - 2 + i}\left( z\right) .\n\nTo finish the proof, we only need to... | Yes |
Theorem 3.1 Suppose that \( \varphi \) is an analytic self-map of \( \mathbb{D} \) and \( g \in H\left( \mathbb{D}\right) \), then the operator \( {J}_{\varphi, g}^{\left( n\right) } : {A}_{u}^{p} \rightarrow {\mathcal{Z}}_{\beta } \) is bounded if and only if the following conditions are satisfied\n\n(i) \( {M}_{0} \m... | Proof First suppose that the operator \( {J}_{\varphi, g}^{\left( n\right) } : {A}_{u}^{p} \rightarrow {\mathcal{Z}}_{\beta } \) is bounded. Take the function \( f\left( z\right) = {z}^{n}/n! \) . Then from Lemma 2.5 it follows that \( f \in {A}_{u}^{p} \) . Since the operator \( {J}_{\varphi, g}^{\left( n\right) } : {... | Yes |
Theorem 3.3 Suppose that \( \\varphi \) is an analytic self-map of \( \\mathbb{D}, g \\in H\\left( \\mathbb{D}\\right) \) and \( {J}_{\\varphi, g}^{\\left( n\\right) } : {A}_{u}^{p} \\rightarrow \) \( {\\mathcal{Z}}_{\\beta } \) is bounded, then\n\n\\[ \n{\\begin{Vmatrix}{J}_{\\varphi, g}^{\\left( n\\right) }\\end{Vmat... | Proof Suppose that \( \\left\\{ {z}_{j}\\right\\} \) is a sequence in \( \\mathbb{D} \) such that \( \\left| {\\varphi \\left( {z}_{j}\\right) }\\right| \\rightarrow {1}^{ - } \) as \( j \\rightarrow \\infty \) . For each \( \\varphi \\left( {z}_{j}\\right) \), taking \( {f}_{j} \) the function in the proof of Theorem ... | Yes |
Corollary 3.6 Suppose that \( \varphi \) is an analytic self-map of \( \mathbb{D}, g \in H\left( \mathbb{D}\right) \) and \( {J}_{\varphi, g}^{\left( n\right) } : {A}_{u}^{p} \rightarrow \) \( {\mathcal{B}}_{\beta } \) is bounded, then \( {J}_{\varphi, g}^{\left( n\right) } : {A}_{u}^{p} \rightarrow {\mathcal{B}}_{\bet... | \[ \mathop{\lim }\limits_{{\left| {\varphi \left( z\right) }\right| \rightarrow {1}^{ - }}}\frac{{\left( 1 - {\left| z\right| }^{2}\right) }^{\beta }}{u\left( \left| {\varphi \left( z\right) }\right| \right) {\left( 1 - {\left| \varphi \left( z\right) \right| }^{2}\right) }^{n + \frac{1}{p}}}\left| {{g}^{\prime }\left(... | Yes |
Theorem 2.3 (Minimal-solution theorem) Under assumptions (H1)-(H3), (i), (ii), equation (1.2) has a minimal solution, that is if \( {Y}^{\prime } \) is another solution of equation (1.2). Then for any given terminal value \( \xi \left( \cdot \right) \in {S}_{\mathcal{F}}^{2}\left( {T, T + K;\mathbb{R}}\right) \), we ha... | Proof Due to for any \( t \in \left\lbrack {0, T + K}\right\rbrack, n \geq m, m, n \in \mathbb{N}, U\left( t\right) \geq {Y}^{n}\left( t\right) \geq {Y}^{m}\left( t\right) \) a.e., there exists a stochastic process \( \{ Y\left( t\right), t \in \left\lbrack {0, T + K}\right\rbrack \} \) such that \( {Y}^{n}\left( t\rig... | No |
Lemma 3.1 If \( u \in {W}_{0} \), then\n\n\[ \parallel u{\parallel }_{2,1} \leq C\parallel {Lu}{\parallel }_{0}\cdots \] | where\n\n\[ {C}^{2} = {336} + {204}{a}_{1}^{2}{\left( {a}_{0} + \lambda {\nu }^{2} - S\sqrt{\lambda }\right) }^{-2} + {C}_{1}\left\lbrack {{n}^{4}{B}^{2}{\nu }^{-2}}\right.\n\n\[ \left. {+{40}{\beta }^{2} + {82}\left( {{2S} + {2\mu }{\gamma }_{1} + {\gamma }_{2}}\right) }\right\rbrack \] | Yes |
Corollary 3.1 Assume that \( f \) satisfies\n\n(i) \( \mathop{\inf }\limits_{{\Omega \times R}}{f}_{u}^{\prime } > - \lambda \) ;  \( \infty \), then there is a unique solution of the equation \( {Pu} = 0 \) in \( {W}_... | Proof Compare with equations (3.4) and (3.1), we have \( f\left( {x, u}\right) = {au} \), so condition (1) of Theorem 3.1 is satisfied. Denote \( {\omega }_{1}\left( t\right) = {\alpha \omega }\left( t\right) + \beta \), then \( {\int }_{a}^{\infty }\frac{dt}{{\omega }_{1}\left( t\right) } = \infty \) and \( \delta \le... | Yes |
Theorem 2.5 The Euler-Maclaurin method is asymptotically stable if any one of the following conditions is satisfied\n\n\[ \n- \frac{a\left( {R{\left( z\right) }^{3m} + 1}\right) }{\bar{\phi }\left( z\right) } < b < - a,\;a > {a}^{0}, \]\n\n\[ \nb > - \frac{a\left( {R{\left( z\right) }^{3m} + 1}\right) }{\bar{\phi }\lef... | Proof Let\n\[ \n\overline{{\lambda }_{1}} = R{\left( z\right) }^{2m} + \frac{b}{a}\left( {R{\left( z\right) }^{2m} - 1}\right) \]\n\nand\n\[ \n\overline{{\lambda }_{-1}} = R{\left( z\right) }^{-m} + \frac{b}{a}\left( {R{\left( z\right) }^{-m} - 1}\right) , \]\n\nso we need to verify\n\[ \n\left| \frac{\overline{{\lambd... | Yes |
Theorem 2.9 Assume that (1.1) is asymptotically stable, then the Euler-Maclaurin method is asymptotically stable if one of the following conditions is satisfied\n\n(a) \( R{\left( z\right) }^{m} \leq {e}^{a}\;\left( {a \leq \ln {\omega }_{1}}\right) \) ;\n\n(b) \( R{\left( z\right) }^{m} \geq {e}^{a}\;\left( {\ln {\ome... | Proof In view of Theorems 1.2 and 2.5, we will prove that condition (2.5) is satisfied under condition (1.2).\n\nIf (a) holds, then we know from Lemmas 2.3 and 2.4 that \( f\left( r\right) \) is decreasing and \( g\left( \omega \right) \) is increasing. Hence \( \bar{a} < {a}^{0} \) and\n\n\[ \n- \frac{a\left( {R{\left... | Yes |
Proposition 2.1 Denote \( {h}_{0} = H\left( {-\frac{1}{2}A,0}\right) = \frac{1}{24}{A}^{3} \), the points \( P\left( {-\frac{A}{2},0}\right) ,{P}_{1}\left( {0, - \frac{\sqrt{-A}}{3}}\right) \) and \( {P}_{2}\left( {0,\frac{\sqrt{-A}}{3}}\right) \), respectively, then Case I If \( A < 0 \), then \( P \) is a center, \( ... | (a) Smooth periodic wave solutions. First, corresponding to Fig.1(a), when \( A < 0 \), a family of smooth periodic wave solutions of (1.1) exist, which correspond to a family of periodic orbits defined by \( H\left( {u, y}\right) = h \), where \( h \in \left( {{h}_{0},0}\right) \). The numerator of (2.3) can be decomp... | Yes |
Proposition 3.1 Denote\n\n\[ \n{h}_{0} = H\left( {-\frac{1}{2}Q,0}\right) = - \frac{{Q}^{3}}{12} \]\n\nand the points \( P\left( {-\frac{1}{2}Q,0}\right) ,{P}_{1}\left( {0, - \sqrt{{2Q}/9}}\right) \) and \( {P}_{2}\left( {0,\sqrt{{2Q}/9}}\right) \), respectively. Then\n\nCase I If \( Q > 0 \), then \( P \) is a center;... | (a) Smooth periodic wave solutions.\n\nCorresponding to Fig. 3(a), when \( Q > 0 \), a family of smooth periodic wave solutions of (1.1) exist, which correspond to a family of periodic orbits defined by \( H\left( {u, y}\right) = h \in \left( {{h}_{0},0}\right) \), \n\nwe have\n\[ \n{y}^{2} = \frac{8}{{27}{u}^{2}}\left... | Yes |
Example 3.3 The initial value is\n\n\\[ \nu\\left( {x,0}\\right) = {1.0} + \\exp \\left\\{ {-{200}{\\left( x - {0.3}\\right) }^{2}}\\right\\} + \\left\\{ \\begin{array}{ll} 1, & \\text{ if }{0.6} < x < {0.8} \\\\ 0, & \\text{ else. } \\end{array}\\right. \\]\n\n(3.3) | This example is come from [11]. We conduct the simulation on a grid of 200 cells, up to \\( t = 1 \\) , \\( t = {10} \\) and the numerical results are displayed in Fig. 3.3. We see that the numerical solution agrees quite well with the exact one in the smooth region, all the spurious oscillations near discontinuities a... | Yes |
Theorem 2.1 Cyclic codes of length \( {p}^{s} \) over \( R \) are | Type 1 \( \langle 0\rangle ,\langle 1\rangle \) . Type \( {2I} = \left\langle {{u}^{2}{\left( x - 1\right) }^{k}}\right\rangle \), where \( 0 \leq k \leq {p}^{s} - 1 \) . Type \( 3\;I = \left\langle {u{\left( x - 1\right) }^{l} + {u}^{2}\mathop{\sum }\limits_{{j = 0}}^{l}{c}_{2j}{\left( x - 1\right) }^{j}}\right\rangle... | Yes |
Theorem 1.1 Assume (A1) and (A2). Then the non-linear autoregressive model \( {\Phi }_{n} \) is geometrically transient. | ## 2 Proof of Theorem 1.1\n\nThis section is devoted to proving Theorem 1.1 by using the Foster-Lyapunov (or drift) condition for geometric transience.\n\nIt is well known that Foster-Lyapunov conditions were widely used to study the stochastic stability for Markov chains. For examples, Down, Meyn and Tweedie [10-13] s... | No |
Theorem 3.1 If \( P\left( {\widehat{x},\widehat{y},\widehat{z},\widehat{E}}\right) \) exists with \( \frac{r}{k} > \frac{{a}_{1}\widehat{y}}{{\left( n + \bar{x}\right) }^{2}} + \frac{pm}{{\left( p\bar{x} - c\right) }^{2}} \) and \( {t}_{2}\tau + {t}_{1} > 0 \), then \( P\left( {\widehat{x},\widehat{y},\widehat{z},\wide... | Proof With\n\n\[ \frac{r}{k} > \frac{{a}_{1}\widehat{y}}{{\left( n + \widehat{x}\right) }^{2}} + \frac{pm}{{\left( p\widehat{x} - c\right) }^{2}} \]\n\nand\n\n\[ {t}_{2}\tau + {t}_{2} > 0 \]\n\nwe can get that \( {l}_{1} > 0,{l}_{3} > 0 \) . Then \( {t}_{2}\tau + {t}_{1} > 0 \) implies that \( B\left( \tau \right) = {l... | Yes |
Theorem 3.2 If \( P\left( {\widehat{x},\widehat{y},\widehat{z},\widehat{E}}\right) \) exists with \( \frac{r}{k} > \frac{{a}_{1}\widehat{y}}{{\left( n + \bar{x}\right) }^{2}} + \frac{pm}{{\left( p\widehat{x} - c\right) }^{2}} \) and \( \tau < \frac{2f}{\frac{2{a}_{1}^{2}n\widehat{x}\widehat{y}}{\left( {n + \bar{x}}\rig... | Proof We know that the characteristic equation of system (2.5) at \( P\left( {\widehat{x},\widehat{y},\widehat{z},\widehat{E}}\right) \) is given by\n\n\[ \n{v}^{3} + {l}_{1}\left( \tau \right) {v}^{2} + {l}_{2}\left( \tau \right) v + {l}_{3}\left( \tau \right) = 0.\n\]\n\n(3.2)\n\nEquation (3.2) has two purely roots i... | Yes |
Theorem 2.1 Assume that (1.2) and (1.6) hold, if \( \varepsilon \) is small enough, then IBVP (1.1) and (1.3)-(1.5) admits a unique global \( {C}^{1} \) solution. | ## 3 Proof of Main Theorem\n\nBy the local existence theorem of smooth solutions (see [7]), we only need to establish the uniform \( {C}^{1} \) estimates for the solutions of (2.4) a priori. For our purpose, we give the following lemma which play an important role in our analysis.\n\nLemma 3.1 Let \( r\left | No |
Lemma 3.2 Assume that (1.2) holds, if \( \varepsilon \) is small enough, then we have\n\n\[ \left| {{r}_{x}\left( {t, x}\right) }\right| \leq {k}_{3},\left| {{s}_{x}\left( {t, x}\right) }\right| \leq {k}_{3}, \]\n\n(3.14)\n\nwhere\n\n\[ {k}_{3} = \max \left\{ {\left| {{r}_{x}\left( {0, x}\right) }\right| ,\left| {{s}_{... | Proof Noting that (1.2), by the continuity of \( \lambda \), with the help of the local result and a standard continuity argument, for the time being we suppose that\n\n\[ \left| {{\lambda }_{x}\left( {t, x}\right) }\right| \leq {k}_{4} \]\n\n(3.15)\n\nthen we can use the method similar to Lemma 3.1 and easy verify the... | No |
Lemma 4.1 Let \( \mathrm{X} \) be a non-negative random variable. Denote \( {X}_{n} \mathrel{\text{:=}} X \land n, n \geq 1 \) . Then \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{q}_{\alpha }^{ + }\left( {X}_{n}\right) = {q}_{\alpha }^{ + }\left( X\right) \) for any \( \alpha \in \left( {0,1}\right) \) . | Proof Obviously, \( {X}_{n} \uparrow X \) and \( {X}_{n}\xrightarrow[]{\text{ a.s. }}X \) . So \( \left\{ {{q}_{\alpha }^{ + }\left( {X}_{n}\right) : n \geq 1}\right\} \) is an increasing sequence and \( {q}_{\alpha }^{ + }\left( X\right) \) is an upper bound of \( {q}_{\alpha }^{ + }\left( {X}_{n}\right) \) . That is,... | Yes |
Lemma 3.1 If \( {g}_{1}\left( 0\right) < b \), then the function \( g\left( x\right) \) has a unique zero \( {x}_{1} \) on the interval \( \lbrack 0, + \infty ) \) and there exists unique initial consumption \( c\left( 0\right) \) and bequest \( a\left( \Omega \right) \) such the equation (3.8) and (3.10) hold. | Proof Since\n\n\[ \n{g}_{1}^{\prime }\left( x\right) = \frac{{e}^{-{r\Omega }}}{{\int }_{0}^{\Omega }{e}^{\left\lbrack {\sigma \left( {r - \rho }\right) - r}\right\rbrack t}{dt}} - \frac{\sigma {e}^{-{\sigma r\Omega }}{\phi }^{\prime \prime }\left( x\right) }{{\left\lbrack {\phi }^{\prime }\left( x\right) \right\rbrack... | Yes |
Theorem 4.1 Under logarithmic bequest motives, the individual's consumption path and bequest and are given by\n\n\\[ \nc\\left( t\\right) = \\frac{{\\int }_{0}^{R}{e}^{-{rt}}w\\left( t\\right) {dt} + a\\left( 0\\right) }{{\\int }_{0}^{\\Omega }{e}^{-{\\rho t}}{dt} + 1}{e}^{\\left( {r - \\rho }\\right) t},\\;a\\left( \\... | Proof It is only to prove the last part. When \\( \\sigma = 1 \\), from (3.10), \\( c\\left( 0\\right) = {e}^{-{r\\Omega }}a\\left( \\Omega \\right) \\) . Sub-\nstitute it into (3.8), we have \\( c\\left( 0\\right) = \\frac{{\\int }_{0}^{\\dot{R}}{e}^{-{rt}}w\\left( t\\right) {dt} + a\\left( 0\\right) }{{\\int }_{0}^{\... | Yes |
Theorem 4.3 The greater the strength of the bequest motive \( {\theta }_{1} \), the greater the savings rate and bequest. | Proof By (4.3) and (4.4), we have\n\n\[ \frac{\partial c\left( t\right) }{\partial {\theta }_{1}} = - \frac{1}{{\theta }_{1}^{2}}\frac{{e}^{\left( {r - \rho }\right) t}}{{e}^{r\Omega }} < 0,\;\frac{\partial a\left( \Omega \right) }{\partial {\theta }_{1}} = \frac{1}{{\theta }_{1}} = {\int }_{0}^{\Omega }{e}^{-{\rho t}}... | Yes |
Theorem 4.5 The stronger the bequest motive, the greater bequest and the saving rate. | Proof Let\n\n\[ G\left( {x,{\theta }_{1}}\right) = \frac{{\int }_{0}^{R}w\left( t\right) {e}^{-{rt}}{dt} + a\left( 0\right) - {e}^{-{r\Omega }}x}{{\int }_{0}^{\Omega }{e}^{-{\rho t}}{dt}} - \frac{{e}^{-{r\Omega }}{x}^{\eta }}{{\theta }_{1}}, \]\nthen from the proof of Lemma 3.1, there exists a unique \( {x}_{1} > 0 \) ... | Yes |
Theorem 4.7 The stronger the bequest motive, the greater bequest and the saving rate. | Proof Let\n\n\[ G\left( {x,{\theta }_{1}}\right) = \frac{{\int }_{0}^{R}w\left( t\right) {e}^{-{rt}}{dt} + a\left( 0\right) - {e}^{-{r\Omega }}x}{{\int }_{0}^{\Omega }{e}^{-{\rho t}}{dt}} - \frac{{e}^{-{r\Omega }}{\left( {\theta }_{2} + x\right) }^{\eta }}{{\theta }_{1}}, \]\n\nthen from the proof of Lemma 3.1, there e... | Yes |
Theorem 4.8 With the increase of the threshold \( {\theta }_{2} \), consumption will increase, the savings will decrease. | Proof Let\n\n\[ \n{G}_{1}\left( {x,{\theta }_{2}}\right) = \frac{{\int }_{0}^{R}w\left( t\right) {e}^{-{rt}}{dt} + a\left( 0\right) - {e}^{-{r\Omega }}x}{{\int }_{0}^{\Omega }{e}^{-{\rho t}}{dt}} - \frac{{e}^{-{r\Omega }}{\left( {\theta }_{2} + x\right) }^{\eta }}{{\theta }_{1}}, \]\n\nthen from the proof of Lemma 3.1,... | Yes |
Proposition 3.3 By above notation, if \( C \) is an algebra in \( {}_{A}\mathcal{Y}{\mathcal{D}}^{A} \), then \( \bar{C} \) is an algebra in \( {}_{A}\mathcal{Y}{\mathcal{D}}^{A} \) . | Proof For \( c,{c}^{\prime } \in \bar{C} \) and any \( a \in A \) ,\n\n\[ \begin{matrix} a \cdot \left( {c * {c}^{\prime }}\right) & = & a \cdot \left( {{c}_{\left( 0\right) }^{\prime }\left( {{c}_{\left( 1\right) }^{\prime } \cdot c}\right) }\right) = \left( {{a}_{\left( 1\right) } \cdot {c}_{\left( 0\right) }^{\prime... | Yes |
Proposition 3.4 If \( C, D \) are algebras in \( {}_{A}\mathcal{Y}{\mathcal{D}}^{A} \), then \( C \otimes D \) is also an algebra in \( {}_{A}\mathcal{Y}{\mathcal{D}}^{A} \) with the following structures\n\n\[ \n\begin{matrix} a \cdot \left( {c \otimes d}\right) & = & {a}_{\left( 1\right) } \cdot c \otimes {a}_{\left( ... | Proof It is obvious. Indeed, this algebra structure on \( C \otimes D \) given above is just the braided tensor product of \( C \) and \( D \) in the braided tensor category \( {}_{A}\mathcal{Y}{\mathcal{D}}^{A} \). | No |
Proposition 3.5 Let \( \alpha ,\beta \in \operatorname{Aut}\left( A\right) \) and \( M \in {}_{A}\mathcal{Y}{\mathcal{D}}^{A}\left( {\alpha ,\beta }\right) \) be finite dimensional. Then\n\n(1) \( \operatorname{End}\left( M\right) \) is an algebra in \( {}_{A}\mathcal{Y}{\mathcal{D}}^{A} \) with structures\n\n\[ \left(... | Proof We only prove (1) here,(2) is similar. For (1), we first show that \( \operatorname{End}\left( M\right) \) is an object in \( {}_{A}\mathcal{Y}{\mathcal{D}}^{A} \) . In the following, we show the main process: the compatible condition of \( {}_{A}\mathcal{Y}{\mathcal{D}}^{A} \), i.e.,\n\n\[ {\left( a \cdot u\righ... | Yes |
Proposition 3.6 Let \( \alpha ,\beta \in \operatorname{Aut}\left( A\right) \), and \( M \in {}_{A}{\mathcal{{YD}}}^{A}\left( {\alpha ,\beta }\right) \) . Define a new object \( {M}^{\prime } \) as follows: \( {M}^{\prime } \) coincides with \( M \) as left \( A \) -modules, and has a right \( A \) -comodule structure g... | Proof We can get the conclusion by direct computation.\n\n\[ \begin{matrix} {\left( a \cdot m\right) }_{ < 0 > } \otimes {\left( a \cdot m\right) }_{ < 1 > }{a}^{\prime } & = & {\left( a \cdot m\right) }_{\left( 0\right) } \otimes \alpha {\beta }^{-1}\left( {\left( a \cdot m\right) }_{\left( 1\right) }\right) {a}^{\pri... | Yes |
Proposition 3.8 Let \( \alpha ,\beta \in \operatorname{Aut}\left( A\right) \), and \( M \in {}_{A}\mathcal{Y}{\mathcal{D}}^{A}\left( {\alpha ,\beta }\right) \) be finite dimensional. Then \( \operatorname{End}{\left( M\right) }^{op} \cong \operatorname{End}\left( {{}^{\diamond }M}\right) \) as algebras in \( {}_{A}{\ma... | Proof Denote the map \( 1 : \operatorname{End}{\left( M\right) }^{op} \rightarrow \operatorname{End}\left( {{}^{\diamond }M}\right) \) by \( 1\left( u\right) = {u}^{ * } \) for \( u \in \operatorname{End}{\left( M\right) }^{op} \) . It is an algebra isomorphism.\n\nThe map \( 1 \) is \( A \) -linear, the proof is simil... | Yes |
Lemma 1.1 [5] Let the sequence \( \left\{ {{X}_{n}, n \geq 1}\right\} \) of random variables be stochastically dominated by a random variable \( X \) . Then for any \( p > 0, x > 0 \) , | \[ E{\left| {X}_{n}\right| }^{p}I\left( {\left| {X}_{n}\right| \leq x}\right) \leq C\left\lbrack {E{\left| X\right| }^{p}I\left( {\left| X\right| \leq x}\right) + {x}^{p}P\{ \left| X\right| > x\} }\right\rbrack ,\] (1.10) \[ E{\left| {X}_{n}\right| }^{p}I\left( {\left| {X}_{n}\right| > x}\right) \leq {CE}{\left| X\righ... | Yes |
Theorem 2.1 Suppose that \( \beta \geq - 1 \) . Let \( \left\{ {{X}_{ni}, i \geq 1, n \geq 1}\right\} \) be an array of rowwise negatively associated random variables which are stochastically dominated by a random variable \( X \) satisfying \( E{\left| X\right| }^{p} < \infty \) for some \( p > 1 \) . Let \( \left\{ {... | Proof Without loss of generality, we can assume that \( {a}_{ni} > 0,1 \leq i \leq n, n \geq 1 \) (otherwise, we use \( {a}_{ni}^{ + } \) and \( {a}_{ni}^{ - } \) instead of \( {a}_{ni} \), resp., and note that \( {a}_{ni} = {a}_{ni}^{ + } - {a}_{ni}^{ - } \) ). From (1.1) and (2.1), without loss of generality, we can ... | Yes |
Corollary 2.1 Suppose that \( \beta \geq - 1 \) . Let \( \left\{ {{X}_{ni}, i \geq 1, n \geq 1}\right\} \) be an array of rowwise negatively associated random variables which are stochastically dominated by a random variable \( X \) . Let \( \left\{ {{a}_{ni}, i \geq 1, n \geq 1}\right\} \) be an array of constants sat... | Proof If \( 1 + \mu + \beta = 0 \), we take \( p = \theta \) in Theorem 2.2. If \( 1 + \mu + \beta > 0 \), we take \( p = \theta + \left( {1 + \mu + \beta }\right) /r, q = \theta \) in Theorem 2.1. Hence (2.3) holds by Theorem 2.1 and Theorem 2.2. | Yes |
Theorem 2.3 Suppose that \( \beta \geq - 1 \) . Let \( \left\{ {{X}_{ni}, i \geq 1, n \geq 1}\right\} \) be an array of rowwise negatively dependent random variables which are stochastically dominated by a random variable \( X \) satisfying \( E{\left| X\right| }^{p} < \infty \) for some \( p > 1 \) . Let \( \left\{ {{... | (2.16) | No |
Theorem 2.5 Suppose that \( \beta \geq - 1 \) . Let \( \left\{ {{X}_{i}, i \geq 1}\right\} \) be a sequence of rowwise \( {\rho }^{ * } \) -mixing random variables which are stochastically dominated by a random variable \( X \) satisfying \( E{\left| X\right| }^{p} < \infty \) for some \( p > 1 \) . Let \( \left\{ {{a}... | Proof For any \( i \geq 1, n \geq 1 \), let \( {X}_{ni} = {X}_{i}I\left( {\left| {{a}_{ni}{X}_{i}}\right| \leq 1}\right) \) . Note that\n\n\[ \mathop{\sum }\limits_{{n = 1}}^{\infty }{n}^{\beta }E{\left( \mathop{\sup }\limits_{{k \geq 1}}\left| \mathop{\sum }\limits_{{i = 1}}^{k}{a}_{ni}{X}_{i}\right| - \epsilon \right... | No |
Theorem 2.6 Suppose that \( \beta \geq - 1 \) . Let \( \left\{ {{X}_{i}, i \geq 1}\right\} \) be a sequence of rowwise \( {\rho }^{ * } \) -mixing random variables which are stochastically dominated by a random variable \( X \) satisfying \( E{\left| X\right| }^{p}\log \left| X\right| < \infty \) for some \( p \geq 1 \... | Proof For any \( i \geq 1, n \geq 1 \), let \( {X}_{ni} = {X}_{i}I\left( {\left| {{a}_{ni}{X}_{i}}\right| \leq 1}\right) \) . Note that\n\n\[ \mathop{\sum }\limits_{{n = 1}}^{\infty }{n}^{\beta }E{\left( \mathop{\sup }\limits_{{k \geq 1}}\left| \mathop{\sum }\limits_{{i = 1}}^{k}{a}_{ni}{X}_{i}\right| - \epsilon \right... | No |
Theorem 3.1 Let \( {M}^{n}, n \geq 2 \), be an \( n \) -dimensional submanifold of a \( {2m} \) -dimensional generalized complex space form \( N\left( {{F}_{1},{F}_{2}}\right) \) endowed with the semi-symmetric metric connection \( \bar{\nabla } \). For each unit vector \( X \in {T}_{x}M \), we have\n\n(1)\n\n\[ \opera... | Proof (1) Let \( X \in {T}_{x}M \) be an unit tangent vector at \( x \). We choose an orthonormal basis \( {e}_{1},\cdots ,{e}_{n},{e}_{n + 1}\cdots ,{e}_{2m} \) such that \( {e}_{1},\cdots ,{e}_{n} \) are tangent to \( M \) at \( x \) and \( {e}_{1} = X \).\n\nWhen we set \( X = W = {e}_{i}, Y = Z = {e}_{j}, i, j = 1,... | Yes |
Corollary 3.2 If the equality case of inequality (3.1) holds for all unit tangent vector \( \mathrm{X} \) of \( {M}^{n} \), then we have\n\n(1) the equality case of inequality (3.1) holds for all unit tangent vector \( \mathrm{X} \) of \( {M}^{n} \) if and only if \( {M}^{n} \) is a totally umbilical submanifold;\n\n(2... | Proof (1) For \( n = 2 \), from Theorem 3.1 we know the equality case of inequality (3.1) holds for all unit tangent vector \( X \) of \( {M}^{2} \) if and only if \( {M}^{2} \) is a totally umbilical submanifold with respect to the semi-symmetric metric connection. Then from Lemma \( {2.2},{M}^{2} \) is a totally umbi... | Yes |
Theorem 4.1 Let \( {M}^{n}, n \geq 3 \), be an \( n \) -dimensional submanifold of a \( {2m} \) -dimensional generalized complex space form \( N\left( {{F}_{1},{F}_{2}}\right) \) endowed with a semi-symmetric connection \( \overline{\nabla } \). Then we have \[ \parallel H{\parallel }^{2} \geq \frac{2\tau }{n\left( {n ... | Proof For \( x \in {M}^{n} \), let \( \left\{ {{e}_{1},\cdots ,{e}_{n}}\right\} \) and \( \left\{ {{e}_{n + 1},\cdots ,{e}_{2m}}\right\} \) be an orthonormal basis of \( {T}_{x}^{M} \) and \( {T}_{x}^{ \bot }M \), respectively, where \( {e}_{n + 1} \) is parallel to the mean curvature vector \( H \). From (3.2), we hav... | Yes |
Theorem 4.2 Let \( {M}^{n}, n \geq 3 \), be an \( n \) -dimensional submanifold of a \( {2m} \) -dimensional generalized complex space form \( N\left( {{F}_{1},{F}_{2}}\right) \) endowed with a semi-symmetric connection \( \overline{\nabla } \) . Then for any integer \( k,2 \leq k \leq n \), and for any point \( x \in ... | Proof Let \( \left\{ {{e}_{1},\cdots ,{e}_{n}}\right\} \) be an orthonormal basis of \( {T}_{x}{M}^{n} \) at \( x \in {M}^{n} \) . The \( k \) -plane section spanned by \( {e}_{{i}_{1}},\cdots ,{e}_{{i}_{k}} \) is denoted by \( {L}_{{i}_{1}\cdots {i}_{k}} \).\n\nThen from (4.6) and (4.12), we have\n\n\[ \parallel H{\pa... | Yes |
Theorem 3.2 Let \( \mathbb{T} \) be a time scale, \( t \in {\mathbb{T}}_{k} \) and \( \alpha \in (0,1\rbrack \) . Then we have the following:\n\n(i) If \( f \) is conformal \( \nabla \) -fractional differentiable of order \( \alpha \) at \( t \), then \( f \) is continuous at \( t \) . | Proof (i) The proof is easy and will be omitted. | No |
(i) If \( \mathbb{T} = \mathbb{R} \), then \( f : \mathbb{R} \rightarrow \mathbb{R} \) is conformable \( \nabla \) -fractional differentiable of order \( \alpha \) at \( t \in \mathbb{R} \) if and only if the limit \( \mathop{\lim }\limits_{{s \rightarrow t}}\frac{f\left( t\right) - f\left( s\right) }{t - s}{t}^{1 - \a... | \[ {\mathbf{T}}_{\alpha }\left( {f}^{\nabla }\right) \left( t\right) = \mathop{\lim }\limits_{{s \rightarrow t}}\frac{f\left( t\right) - f\left( s\right) }{t - s}{t}^{1 - \alpha }. \] If \( \alpha = 1 \), then we have that \( {\mathbf{T}}_{\alpha }\left( {f}^{\nabla }\right) \left( t\right) = {f}^{\nabla }\left( t\righ... | Yes |
If \( f : \mathbb{T} \rightarrow \mathbb{R} \) is defined by \( f\left( t\right) = {t}^{2} \) for all \( t \in \mathbb{T} \mathrel{\text{:=}} \left\{ {\frac{n}{2} : n \in {\mathbb{N}}_{0}}\right\} \), then from Theorem 3.2 (ii) we have that \( f \) is conformable \( \nabla \) -fractional differentiable of order \( \alp... | \[ {\mathbf{T}}_{\alpha }\left( {f}^{\nabla }\right) \left( t\right) = \left( {{2t} - \frac{1}{2}}\right) {\left( t - \frac{1}{2}\right) }^{1 - \alpha }. \] | Yes |
Theorem 3.6 Assume \( f, g : \mathbb{T} \rightarrow \mathbb{R} \) are conformable \( \nabla \) -fractional differentiable of order \( \alpha \) at \( t \in {\mathbb{T}}_{k} \), then\n\n(i) for any constant \( {\lambda }_{1},{\lambda }_{2} \), the sum \( {\lambda }_{1}f + {\lambda }_{2}g : \mathbb{T} \rightarrow \mathbb... | Proof (i) The proof is easy and will be omitted. | No |
Theorem 3.7 Let \( c \) be constant and \( m \in \mathbb{N} \) . (i) For \( f \) defined by \( f\left( t\right) = {\left( t - c\right) }^{m} \), we have that \[ {\mathbf{T}}_{\alpha }\left( {f}^{\nabla }\right) \left( t\right) = \rho {\left( t\right) }^{1 - \alpha }\mathop{\sum }\limits_{{i = 0}}^{{m - 1}}{\left( \rho ... | Proof (i) We prove the first formula by induction. If \( m = 1 \), then \( f\left( t\right) = t - c \), and clearly \( {\mathbf{T}}_{\alpha }\left( {f}^{\nabla }\right) \left( t\right) = \rho {\left( t\right) }^{1 - \alpha } \) holds by Example 3.4 and Theorem 3.6(i). Now we assume that \[ {\mathbf{T}}_{\alpha }\left( ... | Yes |
Example 3.8 If \( f : \mathbb{T} \rightarrow \mathbb{R} \) is defined by \( f\left( t\right) = \frac{1}{{t}^{2}} \) for all \( t \in \mathbb{T} \mathrel{\text{:=}} \left\{ {\sqrt{n} : n \in {\mathbb{N}}_{0}}\right\} \), then we have that \( f \) is conformable \( \nabla \) -fractional differentiable of order \( \alpha ... | \[ {T}_{\alpha }\left( {f}^{\nabla }\right) \left( t\right) = - \rho {\left( t\right) }^{1 - \alpha }\left( {\frac{1}{{\left( \rho \left( t\right) \right) }^{2}t} + \frac{1}{\rho \left( t\right) {t}^{2}}}\right) = - {\left( \sqrt{{t}^{2} - 1}\right) }^{-\alpha }\left( {\frac{1}{t\sqrt{{t}^{2} - 1}} + \frac{1}{{t}^{2}}}... | Yes |
Theorem 1.3 Suppose \( V \in {B}_{n/2}, - \infty < \alpha < \infty, p\left( x\right), q\left( x\right) \in \mathcal{B}\left( {\mathbb{R}}^{n}\right) \) satisfy \( {p}_{ + } < \frac{n}{\beta }, \) \( \left. {\frac{1}{q\left( x\right) } = \frac{1}{p\left( x\right) } - \frac{\beta }{n}.}\right. \) If exists \( {q}_{0} \) ... | \[ {\begin{Vmatrix}{I}_{\beta }f\end{Vmatrix}}_{{M}_{\alpha, V}^{q\left( \cdot \right), u}} \leq C\parallel f{\parallel }_{{M}_{\alpha, V}^{p\left( \cdot \right), u}} \] where \( \frac{1}{{\bar{p}}_{B}} = \frac{1}{\left| B\right| }{\int }_{B}\frac{1}{p\left( x\right) }{dx}. \) | Yes |
Lemma 2.7 Suppose that \( p\left( \cdot \right) \in {\mathcal{P}}_{0}^{\log }\left( {\mathbb{R}}^{n}\right) \bigcap {\mathcal{P}}_{\infty }^{\log }\left( {\mathbb{R}}^{n}\right) \) satisfies \( {p}_{ + } < \frac{\beta }{n} \) and \( \frac{1}{p\left( x\right) } - \frac{1}{q\left( x\right) } = \) \( \frac{\beta }{n} \) .... | Using Corollary 2.5 and Corollary 2.10 in [6] and the inequality (1.2), we can get the following result. | No |
Lemma 2.9 [18] Let \( k \) be a positive integer. Then we have that for all \( b \in \operatorname{BMO}\left( {\mathbb{R}}^{n}\right) \) and all \( i, j \in \mathbb{Z} \) with \( i > j \) , | \[ {C}^{-1}\parallel b{\parallel }_{\mathrm{{BMO}}}^{k} \leq \mathop{\sup }\limits_{B}\frac{1}{\parallel {\chi }_{B}{\parallel }_{{L}^{p\left( \cdot \right) }\left( {\mathbb{R}}^{n}\right) }}\parallel {\left( b - {b}_{B}\right) }^{k}{\chi }_{B}{\parallel }_{{L}^{p\left( \cdot \right) }\left( {\mathbb{R}}^{n}\right) } \... | Yes |
Theorem 2.1 For system (2.2)\n\n(i) If \( {\left( {r}_{2} - \mu \frac{2{G}_{0} + c}{{G}_{0}^{2}}\right) }^{2} \geq 4{r}_{1}\frac{{r}_{2}{G}_{0} - \mu }{{G}_{0}} \) and \( \mu < \min \left\{ {\frac{{r}_{2}{G}_{0}^{2}}{2{G}_{0} + c},{r}_{2}{G}_{0}}\right\} \), the positive equilibrium point of system (1.4) is asymptotica... | Proof First, the characteristic equation of the matrix \( E\left( {M}_{0}\right) \) can be written as\n\n\[ \n{\lambda }^{2} + \left( {{r}_{2} - \mu \frac{2{G}_{0} + c}{{G}_{0}^{2}}}\right) \lambda + {r}_{1}\frac{{r}_{2}{G}_{0} - \mu }{{G}_{0}} = 0. \n\]\n\n(2.4)\n\nNow donate \( \Delta \) by\n\n\[ \n\Delta = {\left( {... | Yes |
Theorem 3.1 For the system (2.2), there exist an \( \varepsilon > 0 \) and two small enough neighborhoods \( {P}_{1} \) and \( {P}_{2} \) of \( {\chi }_{0}\left( \mu \right) \), where \( {P}_{1} \subset {P}_{2} \).\n\n(i) If\n\n\[ 2{Y}_{0}\frac{c\mu }{{G}_{0}^{4}}\left( {1 - {\beta c}}\right) + \frac{1}{6} \cdot \frac{... | Proof Theorem 3.1 can be similarly proved as the Hopf bifurcation theorem in [19], so we omit the process here. | No |
Lemma 2.1 Every critical point \( u \in E \) of \( J \) is a solution of (1.1). | Proof We assume that \( u \in E \) is a critical point of \( J \), then \( {J}^{\prime }\left( u\right) = 0 \) . According to\n\n(2.2), this is equivalent to\n\n\[ \mathop{\sum }\limits_{{k \in \mathbb{Z}}}\left\lbrack {a\left( k\right) {\Delta u}\left( {k - 1}\right) {\Delta v}\left( {k - 1}\right) + b\left( k\right) ... | Yes |
Lemma 3.3 \( \left\{ {w}_{n}\right\} \) is a Palais-Smale sequence for \( \Psi \) if and only if \( \left\{ {m\left( {w}_{n}\right) }\right\} \) is a Palais-Smale sequence for \( J \) . | Proof Let \( \left\{ {w}_{n}\right\} \) be a Palais-Smale sequence for \( \Psi \), and let \( {u}_{n} = m\left( {w}_{n}\right) \in \mathcal{N} \) . Since for every \( {w}_{n} \in S \) we have an orthogonal splitting \( E = {T}_{{w}_{n}}S \oplus \mathbb{R}{w}_{n} \), we have\n\n\[ \parallel {\Psi }^{\prime }\left( {w}_{... | Yes |
Example 1 Consider the difference equation\n\n\\[ \n\\Delta \\left\\lbrack {\\left| {\\sin k}\\right| {\\Delta u}\\left( {k - 1}\\right) }\\right\\rbrack - \\left| {\\cos k}\\right| u\\left( k\\right) \n\\]\n\n\\[ \n+ \\left\\lbrack {c\\left( {\\alpha + 2}\\right) u\\left( k\\right) {\\left| u\\left( k\\right) \\right|... | It is easy to show that all the assumptions of Theorem 1.1 are satisfied. Therefore, equation (3.11) has at least one homoclinic solution. | No |
Theorem 1.3 Let \( x : {M}^{m} \rightarrow {\mathbb{S}}_{1}^{m + 1}, m \geq 2 \), be a regular space-like hypersurface. If the Blaschke tensor \( A \) of \( x \) is parallel, then one of the following holds.\n\n1. \( x \) is conformal isotropic and thus is locally conformal equivalent to a maximal space-like regular hy... | Remark 1.1 It is directly verified in Section 3 that each of the regular space-like hypersurfaces stated in the above theorem has a parallel Blaschke tensor. | No |
Lemma 2.1 The conformal position vector \( \overset{\left( 1\right) }{Y} \) of \( \overset{\left( 1\right) }{x} \) is nothing but \( {\left. Y\right| }_{\begin{matrix} \left( 1\right) \\ M \end{matrix}} \), while the conformal position vector \( \overset{\left( 2\right) }{Y} \) of \( \overset{\left( 2\right) }{x} \) is... | \[ \overset{\left( 2\right) }{Y} = \mathbb{T}\left( {\left. Y\right| }_{\begin{matrix} \left( 2\right) \\ M \end{matrix}}\right) ,\text{ where }\mathbb{T} = \left( \begin{matrix} 0 & 1 & 0 \\ 1 & 0 & \\ 0 & & {I}_{m + 1} \end{matrix}\right) . \] | No |
Lemma 4.3 Suppose that \( t \geq 3 \) . If, with respect to an orthonormal frame field \( \left\{ {E}_{i}\right\} \) , (4.5) holds and at a point \( p,{B}_{ij} = {B}_{i}{\delta }_{ij} \), then \( {B}_{i} = {B}_{j} \) in the case that \( {A}_{i} = {A}_{j} \). | Proof By (4.3), for any \( i, j \) satisfying \( {A}_{i} \neq {A}_{j} \), we have \( {\omega }_{j}^{i} = 0 \) . Differentiating this equation, we obtain from (2.13) that \( 0 = {R}_{ijji} = {B}_{ij}^{2} - {B}_{ii}{B}_{jj} + {\overset{.}{A}}_{ii} - {A}_{ij}{\delta }_{ij} + {A}_{jj} - {A}_{ij}{\delta }_{ij}. \) Thus at \... | Yes |
Corollary 4.4 If \( t \geq 3 \), then there exists an orthonormal frame field \( \left\{ {E}_{i}\right\} \) such that\n\n\[ \n{A}_{ij} = {A}_{i}{\delta }_{ij},\;{B}_{ij} = {B}_{i}{\delta }_{ij} \n\] \n\n(4.9) \n\nFurthermore, if (4.5) holds, then \n\n\[ \n\left( {B}_{ij}\right) = \operatorname{Diag}\left( {\underset{{\... | Proof Since \( A \) is parallel, we can find a local orthonormal frame field \( \left\{ {E}_{i}\right\} \), such that (4.5) holds. It then suffices to show that, at any point, the component matrix \( \left( {B}_{ij}\right) \) of \( B \) with respect to \( \left\{ {E}_{i}\right\} \) is diagonal. Note that \( {k}_{1},\ld... | Yes |
Lemma 4.5 If \( t \geq 3 \), then all the conformal principal curvatures \( {\mu }_{1},\cdots ,{\mu }_{t} \) of \( x \) are constant, and hence \( x \) is conformal isoparametric. | Proof Without loss of generality, we only need to show that \( {\mu }_{1} \) is constant. To this end, choose a frame field \( \left\{ {E}_{i}\right\} \) such that (4.5) and (4.10) hold. Note that, by (4.3), when \( 1 \leq i \leq {k}_{1} \) and \( j > {k}_{1} \), we have\n\n\[ \n\sum {B}_{ijk}{\omega }^{k} = d{B}_{ij} ... | Yes |
Corollary 4.6 If \( t \geq 3 \), then \( t = 3 \) and \( B \) is parallel. | Proof Indeed, the conclusion that \( B \) is parallel comes from (4.3), Corollary 4.4 and Lemma 4.5.\n\nIf \( t > 3 \), then there exist at least four indices \( {i}_{1},{i}_{2},{i}_{3},{i}_{4} \), such that \( {A}_{{i}_{1}},{A}_{{i}_{2}},{A}_{{i}_{3}},{A}_{{i}_{4}} \) are distinct each other. Then it follows from (4.7... | Yes |
Lemma 4.7 If \( t \leq 2 \) and \( B \) is not parallel, then one of the following cases holds:\n\n(1) \( t = 1 \) and \( x \) is conformal isotropic;\n\n(2) \( t = 2,{\lambda }_{1} + {\lambda }_{2} = 0 \) and \( {B}_{i} = 0 \) either for all \( 1 \leq i \leq {k}_{1} \), or for all \( {k}_{1} + 1 \leq i \leq m \) . | Proof Note that \( \Phi \equiv 0 \) . Thus \( x \) is conformal isotropic if and only if \( t = 1 \) .\n\nIf \( t = 2 \), then for any point \( p \in {M}^{m} \), we can find an orthonormal frame field \( \left\{ {E}_{i}\right\} \) such that (4.9) holds at \( p \) .\n\nBy (4.3), we see that\n\n\[{\omega }_{j}^{i} = 0,\;... | Yes |
Lemma 3.1 Let \( \mathcal{D},{\mathcal{D}}^{\prime },{\mathcal{D}}^{\prime \prime } \) be three triangulated categories, and if there exists a recollement\n\n\[ \n{\mathcal{D}}^{\prime }\underset{{i}^{!}}{\overset{{i}_{ * }}{\overbrace{\overset{{i}_{ * }}{}}}} \rightarrow \mathcal{D}\underset{{j}_{ * }}{\overset{{j}_{!... | Proof From Lemma 2.7 (4), \( \left( {\mathcal{A}\left( {i}^{ * }\right) ,\mathcal{A}\left( {i}_{ * }\right) }\right) ,\left( {\mathcal{A}\left( {i}_{!}\right) ,\mathcal{A}\left( {i}^{!}\right) }\right) ,\left( {\mathcal{A}\left( {j}_{!}\right) ,\mathcal{A}\left( {j}^{!}\right) }\right) \) and \( \left( {\mathcal{A}\lef... | Yes |
Theorem 3.1 Suppose the following conditions hold:\n\n(H1) \( f \in C\left( {\left( {0,1}\right) \times \lbrack 0, + \infty }\right) ,\lbrack 0, + \infty )), f\left( {t, x}\right) \leq g\left( t\right) h\left( x\right), g \in C\left( {\left( {0,1}\right) ,\lbrack 0, + \infty }\right) ), h \in \) \( C\left( {\lbrack 0, ... | Proof Denote \( K = \left\{ {x \in {C}^{ + }\left\lbrack {0,1}\right\rbrack : x\left( t\right) }\right. \) is convex function and \( x\left( t\right) = x\left( {1 - t}\right), t \in \) \( \left\lbrack {0,1}\right\rbrack \} \), then \( K \) is a cone of \( {C}^{ + }\left\lbrack {0,1}\right\rbrack \) .\n\nLet \( u\left( ... | Yes |
Example 4.1 The following boundary value problem:\n\n\[ \left\{ \begin{array}{l} {x}^{\left( 4\right) }\left( t\right) + \frac{h\left( x\right) }{{t}^{2}{\left( 1 - t\right) }^{2}} = 0,0 < t < 1 \\ x\left( 0\right) = x\left( 1\right) = {x}^{\prime }\left( 0\right) = {x}^{\prime }\left( 1\right) = 0 \end{array}\right. \... | Proof Let \( f\left( {t, x}\right) = h\left( x\right) g\left( t\right), g\left( t\right) = \frac{1}{{t}^{2}{\left( 1 - t\right) }^{2}} \) . Obviously \( g\left( t\right) \) is signular at \( t = 0 \) and \( t = 1.h\left( x\right) \in C\lbrack 0, + \infty ) \) . So (H1) holds. Since \n\n\[ G\left( {\tau \left( s\right),... | Yes |
Lemma 3.1 Let \( {\Phi }_{1} \) be a concave Young function with \( {q}_{{\Phi }_{1}} > 0,{\Phi }_{2} \) a concave Young function with \( {q}_{{\Phi }_{2}} > 0 \) or a convex Young function with \( {p}_{{\Phi }_{2}} < + \infty \), and let \( {\Phi }_{1} \preccurlyeq {\Phi }_{2} \) , \( {\Phi }_{1,2}\left( x\right) = {\... | Proof For any \( f \in \mathrm{w}{L}_{{\Phi }_{2}} \) and \( g \in \mathrm{w}{L}_{{\Phi }_{1} \circ {\Psi }_{1,2}} \), if \( \parallel f{\parallel }_{\mathrm{w}{L}_{{\Phi }_{2}}} \cdot \parallel g{\parallel }_{\mathrm{w}{L}_{{\Phi }_{1} \circ {\Psi }_{1,2}}} = 0 \), then (3.1) is obvious. Now we assume that \( \paralle... | Yes |
Theorem 3.1 Let \( {\Phi }_{1} \) be a concave Young function with \( {q}_{{\Phi }_{1}} > 0,{\Phi }_{2} \) a concave Young function with \( {q}_{{\Phi }_{2}} > 0 \) or a convex Young function with \( {p}_{{\Phi }_{2}} < + \infty \), and \( {\Phi }_{1} \preccurlyeq {\Phi }_{2} \). Let \( f = \left( {{f}_{n}, n \in \math... | Proof Setting \( {s}_{0}\left( f\right) = 0 \), for all \( i \geq 1 \), we have \( E\left( {{\left| d{f}_{i}\right| }^{2} \mid {\mathcal{F}}_{i - 1}}\right) = {s}_{i}^{2}\left( f\right) - {s}_{i - 1}^{2}\left( f\right) \), and \[ E\left( {{\left| d\left( T{f}_{i}\right) \right| }^{2} \mid {\mathcal{F}}_{i - 1}}\right) ... | Yes |
Corollary 3.1 Let \( {\Phi }_{1} \) be a concave Young function with \( {q}_{{\Phi }_{1}} > 0,{\Phi }_{2} \) a concave Young function with \( {q}_{{\Phi }_{2}} > 0 \) or a convex Young function with \( {p}_{{\Phi }_{2}} < + \infty \), and \( {\Phi }_{1} \preccurlyeq {\Phi }_{2} \) . Then for any martingale \( f = \left... | Proof From Theorem 3.1 and 3.2, only the inequality (3.4) needs to be proved. In fact, since \( \left( {{\Phi }_{1,2},{\Psi }_{1,2}}\right) \) is a pair of conjugate Young functions, so\n\n\[ \nu{\varphi }_{1,2}\left( \nu\right) = {\Phi }_{1,2}\left( \nu\right) + {\Psi }_{1,2}\left( {{\varphi }_{1,2}\left( \nu\right) }... | Yes |
Proposition 2.7 \( m = {\left( {m}_{\alpha }\right) }_{\alpha \in \pi } \) is a coinvariant of \( M \) if and only if \( \widehat{\operatorname{tr}}\left( m\right) = m \) . | Proof If \( m = {\left( {m}_{\alpha }\right) }_{\alpha \in \pi } \in {M}^{COH} \), then\n\n\[ \widehat{\operatorname{tr}}\left( m\right) = \sum {m}_{\left( \left\lbrack 0\right\rbrack ,\alpha \right) } \cdot {\theta }_{\alpha }\left( {{S}_{\alpha }\left( {m}_{\left( \left\lbrack 1\right\rbrack ,{\alpha }^{-1}\right) }\... | Yes |
Lemma 2.8 For any \( \alpha \in \pi ,{\xi }_{\alpha } \circ {p}_{\alpha } \circ {\rho }_{i,\alpha }^{M} = i{d}_{{M}_{\alpha }} \), where \( {p}_{\alpha } : {M}_{i} \otimes {H}_{\alpha } \rightarrow \overline{{M}_{i} \otimes {H}_{\alpha }} \) is a canonical map. | Proof In fact, for any \( m \in {M}_{\alpha } \), we have\n\n\[ \n{\xi }_{\alpha } \circ {p}_{\alpha } \circ {\rho }_{i,\alpha }^{M}\left( m\right) = \sum {m}_{\left( {\left\lbrack 0\right\rbrack, i}\right) \left( {\left\lbrack 0\right\rbrack ,\alpha }\right) } \cdot {\theta }_{\alpha }\left( {{S}_{\alpha }\left( {m}_{... | Yes |
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