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Proposition 2.2 Let \( R \) be an \( \\left( {\\alpha ,\\delta }\\right) \) -compatible ring. Then we have the following:\n\n(1) If \( {ab} = 0 \), then \( a{f}_{i}^{j}\\left( b\\right) = 0 \) for all \( 0 \\leq i \\leq j \) and \( a, b \\in R \) ; | Proof (1) If \( {ab} = 0 \), then \( a{\\alpha }^{i}\\left( b\\right) = a{\\delta }^{j}\\left( b\\right) = 0 \) for all \( i \\geq 0 \) and \( j \\geq 0 \) by Lemma 2.1. Hence \( a{f}_{i}^{j}\\left( b\\right) = 0 \) for all \( 0 \\leq i \\leq j \) . | Yes |
(1) If \( {abc} = 0 \), then \( {a\delta }\left( b\right) c = 0 \) for any \( a, b, c \in R \); | Proof (1) If \( {abc} = 0 \), we have \( \alpha \left( {ab}\right) \delta \left( c\right) = 0,\alpha \left( a\right) \alpha \left( b\right) \delta \left( c\right) = 0 \) and \( {a\alpha }\left( b\right) \delta \left( c\right) = 0 \) . On the other hand, we also have \( {a\delta }\left( {bc}\right) = 0,\;a\left( {\delta... | Yes |
Proposition 2.4 If \( R \) is an \( \left( {\alpha ,\delta }\right) \) -compatible and nil-semicommutative ring, then \( {ab} \in \operatorname{nil}\left( R\right) \) implies \( a{f}_{i}^{j}\left( b\right) \in \operatorname{nil}\left( R\right) \) for all \( 0 \leq i \leq j \) and \( a, b \in R. \) | Proof If \( {ab} \in {nil}\left( R\right) \), we have \( a{\alpha }^{i}\left( b\right) \in {nil}\left( R\right) \) and \( a{\delta }^{j}\left( b\right) \in {nil}\left( R\right) \) for all \( i \geq 0 \) and \( j \geq 0 \) by Proposition 2.2 and Proposition 2.3. This implies that \( a{\delta }^{j}{\alpha }^{i}\left( b\r... | Yes |
Theorem 2.5 Let \( R \) be an \( \left( {\alpha ,\delta }\right) \) -compatible and nil-semicommutative ring, and \( f\left( x\right) = \) \( \mathop{\sum }\limits_{{i = 0}}^{n}{a}_{i}{x}^{i} \in R\left\lbrack {x;\alpha ,\delta }\right\rbrack \) . Then \( f\left( x\right) \in \operatorname{nil}\left( {R\left\lbrack {x;... | Proof \( \left( \Rightarrow \right) \) Suppose that \( f\left( x\right) = \mathop{\sum }\limits_{{i = 0}}^{n}{a}_{i}{x}^{i} \in R\left\lbrack {x;\alpha ,\delta }\right\rbrack \) . Then there exists a positive integer \( k \) such that \( f{\left( x\right) }^{k} = {\left( {a}_{0} + {a}_{1}x + \cdots + {a}_{n}{x}^{n}\rig... | Yes |
Theorem 3.15 Let \( R \) be a nil-semicommutative \( \left( {\alpha ,\delta }\right) \) -compatible ring. Then \( R\left\lbrack {x;\alpha ,\delta }\right\rbrack \) is weak Armendariz. | Proof Let\n\n\[ F\left( y\right) = \mathop{\sum }\limits_{{i = 0}}^{m}{f}_{i}{y}^{i},\;G\left( y\right) = \mathop{\sum }\limits_{{j = 0}}^{n}{g}_{j}{y}^{j} \in R\left\lbrack {x;\alpha ,\delta }\right\rbrack \left\lbrack y\right\rbrack \]\n\nsuch that \( F\left( y\right) G\left( y\right) = 0 \), where\n\n\[ {f}_{i} = \m... | Yes |
Lemma 5 Let \( u, v \) be two fault-free vertices of \( {Q}_{n} \) with \( f \leq n - 1 \) fault vertices and \( H\left( {u, v}\right) = 1 \) . There exists a fault-free path \( P\left\lbrack {u, v}\right\rbrack \) of length at least \( {2}^{n} - {2f} - 1 \) . | Proof We proof the theorem by introduction on \( n \) . When \( n = 3 \), it is clearly true. Assume that it holds for \( n - 1\left( {n \geq 4}\right) \), next we consider \( n \) . We can partite \( {Q}_{n} \) along some component \( i \) into two \( \left( {n - 1}\right) \) -dimensional subcubes \( {Q}_{n - 1}^{i0} ... | Yes |
Theorem 1.1 Let \( f : \left( {{M}^{n}, F}\right) \rightarrow \left( {{\widetilde{M}}^{n + p},\widetilde{F}}\right) \) be an isometric immersion from a Finsler manifold to a Finsler manifold, \( \widetilde{\nabla } \) is the Chern connection of \( \widetilde{M} \) . Then \( \nabla = D \) if and only if | \[ \widetilde{A}\left( {X, Y,{\widetilde{\nabla }}_{Z}{e}_{n}}\right) = A\left( {X, Y,{\nabla }_{Z}{e}_{n}}\right) \] | Yes |
Proposition 3.1 \( D, B, W \) and \( {\nabla }^{ \bot } \) have the following properties:\n\n(1) \( D \) determines a linear torsion-free connection on \( {\pi }^{ * }\left( {TM}\right) \) .\n\n(2) \( B : {\pi }^{ * }{TM} \otimes {\pi }^{ * }{TM} \rightarrow {\left( {\pi }^{ * }TM\right) }^{ \bot } \) is a symmetric bi... | Let \( B\left( {{e}_{i},{e}_{j}}\right) = {B}_{ij}^{\alpha }{e}_{\alpha },{W}_{{e}_{\alpha }}{e}_{i} = {W}_{ij}^{\alpha }{e}_{j} \) . Then we have\n\n\[ {W}_{ij}^{\alpha } = {B}_{ij}^{\alpha } + 2{\widetilde{A}}_{aj\alpha }{\omega }_{n}^{a}\left( {e}_{i}\right) \] | No |
Theorem 3.2 (the Gauss equations) Let \( f : \left( {{M}^{n}, F}\right) \rightarrow \left( {{\widetilde{M}}^{n + p},\widetilde{F}}\right) \) be an isometric immersion from a Finsler manifold to a Finsler manifold. Then we have\n\n\[ \left\{ \begin{array}{ll} {R}_{jkl}^{i} = & {\widetilde{R}}_{jkl}^{i} + {\widetilde{P}}... | Exterior differentiating \( {\omega }_{i}^{\alpha } = {B}_{ij}^{\alpha }{\omega }^{j} \), we have\n\n\[ d{\omega }_{i}^{\alpha } = d\left( {{B}_{ij}^{\alpha }{\omega }^{j}}\right) = d{B}_{ij}^{\alpha }{\omega }^{j} + {B}_{ij}^{\alpha }d{\omega }^{j} \]\n\n\[ = \left( {{B}_{{ij} \mid k}^{\alpha }{\omega }^{k} + {B}_{{ij... | No |
Theorem 3.3 (the Codazzi equations) Let \( f : \left( {{M}^{n}, F}\right) \rightarrow \left( {{\widetilde{M}}^{n + p},\widetilde{F}}\right) \) be an isometric immersion from a Finsler manifold to a Finsler manifold. Then we have\n\n\[ \n\begin{cases} {B}_{{ij} \mid k}^{\alpha } - {B}_{{ik} \mid j}^{\alpha } & = - {\wid... | Set\n\n\[ \nd{\omega }_{\beta }^{\alpha } + {\omega }_{\gamma }^{\alpha } \land {\omega }_{\beta }^{\gamma } \mathrel{\text{:=}} {\Omega }_{\beta }^{ \bot \alpha } = \frac{1}{2}{R}_{\beta kl}^{ \bot \alpha }{\omega }^{k} \land {\omega }^{l} + {P}_{\beta kc}^{ \bot \alpha }{\omega }^{k} \land {\omega }_{n}^{c} + {Q}_{\b... | Yes |
Theorem 3.6 Let \( f : \left( {{M}^{n}, F}\right) \rightarrow \left( {{\widetilde{M}}^{n + p},\widetilde{F}}\right) \) be an isometric immersion from a Finsler manifold to a Finsler manifold. Then we have\n\n\[ K\left( {{e}_{n};{e}_{i}}\right) = \widetilde{K}\left( {{e}_{n};{e}_{i}}\right) + {\widetilde{L}}_{ii\alpha }... | Proof Setting \( j = l = n \) in \( {\left( {3.5}\right) }_{1} \), we obtain\n\n\[ \begin{array}{l} {R}_{nkn}^{i} = {\widetilde{R}}_{nkn}^{i} + {\widetilde{P}}_{nk\alpha }^{i}{B}_{nn}^{\alpha } - {\widetilde{P}}_{nn\alpha }^{i}{B}_{nk}^{\alpha } + {B}_{ik}^{\alpha }{B}_{nn}^{\alpha } \end{array} \]\n\n\[ \left. {-{B}_{... | Yes |
Theorem 4.2 Let \( f : \\left( {{M}^{n}, F}\\right) \\rightarrow \\left( {{\\widetilde{M}}^{n + p},\\widetilde{F}}\\right) \) be an isometric immersion from a Finsler manifold to a Finsler manifold. \( \\widetilde{\\nabla } \) is the Chern connection of \( \\widetilde{M} \) . Then \( \\nabla = D \) if and only if | Proof Recall that the Gauss formula is\n\n\[ \n{\\widetilde{\\nabla }}_{X}Y = {D}_{X}Y + B\\left( {X, Y}\\right) \n\]\n\n(4.2)\n\nIf \( \\nabla = D \), then from (4.1) and (4.2) one gets\n\n\[ \n\\widetilde{A}\\left( {X, Y,{\\widetilde{\\nabla }}_{Z}{e}_{n} - {\\nabla }_{Z}{e}_{n}}\\right) - \\widetilde{A}\\left( {X, Z... | Yes |
Theorem 4.3 Let \( f : \left( {{M}^{n}, F}\right) \rightarrow \left( {{\widetilde{M}}^{n + p},\widetilde{F}}\right) \) be an isometric immersion from a Finsler manifold to a Finsler manifold. If \( M \) is a weakly totally geodesic submanifold of \( \widetilde{M} \) , then \( \nabla = D \) . | Proof If \( M \) is a weakly totally geodesic submanifold, then \( {}^{\left\lbrack 1\right\rbrack } \n\n\[ \n{B}_{nn}^{\alpha } = 0,\forall \alpha . \n\] \n\nSo we have \n\n\[ \n{B}_{{nn} \mid i}^{\alpha }{\omega }^{i} + {B}_{{nn};\lambda }^{\alpha }{\omega }_{n}^{\lambda } = d{B}_{nn}^{\alpha } - 2{B}_{n\lambda }^{\a... | Yes |
Lemma 2.1 (see [6]) Let \( {f}_{i}\left( {i = 1,2,\cdots, n}\right) \) be Lipschitzian near \( x \), and let \( {\lambda }_{i}(i = \) \( 1,2,\cdots, n) \) be scalars. Then \( f \mathrel{\text{:=}} \mathop{\sum }\limits_{{i = 1}}^{n}{\lambda }_{i}{f}_{i} \) is Lipschitzian near \( x \), and we have | \[ \partial \left( {\mathop{\sum }\limits_{{i = 1}}^{n}{\lambda }_{i}{f}_{i}}\right) \left( x\right) \subset \mathop{\sum }\limits_{{i = 1}}^{n}{\lambda }_{i}\partial {f}_{i}\left( x\right) \] | Yes |
Lemma 2.2 (see [9]) Let \( X \) be a real Banach space and \( f : X \rightarrow R \) be a \( p \) -positively homogeneous \( \left( {p > 0}\right) \) and locally Lipschitzian function. Then, for each \( x \in X \) and \( \xi \in \partial f\left( x\right) \) , the following identity holds:\n\n\[ \langle \xi, x\rangle = ... | \[ \langle \partial f\left( x\right), x\rangle = p \cdot f\left( x\right) ,\forall x \in X. \] | No |
Theorem 3.1 Let \( f,{g}_{i}\left( {i \in M}\right) \) be locally Lipschitzian and positively homogeneous functions with degree \( p,{q}_{i}\left( {i \in M}\right) \), respectively. If \( \bar{x} \in K \) is a KKT point of (HOP), then\n\n\[ f\left( \bar{x}\right) = - \mathop{\sum }\limits_{{i = 1}}^{m}{\bar{\lambda }}_... | Proof Let \( \bar{x} \in K \) be a KKT point of (HOP) with associated L-KKT multiplier \( \bar{\lambda } = \) \( \left( {{\bar{\lambda }}_{1},{\bar{\lambda }}_{2},\cdots ,{\bar{\lambda }}_{m}}\right) \in {\mathbb{R}}_{ + }^{m} \) . It follows from (2.2) that there exist \( \xi \in \partial f\left( \bar{x}\right) \) and... | Yes |
Theorem 4.1 Let \( \mathrm{X} \) be a real Banach space and \( \Omega \) be a closed cone of \( X \) . Assume that \( f : \Omega \rightarrow R \) is a locally Lipschitzian and positively homogeneous function with degree \( p \) , and \( {g}_{i} : \Omega \rightarrow R\left( {i \in M}\right) \) are locally Lipschitzian a... | Proof (i) Let \( \bar{x} \in \Omega \) be a KKT point with associated L-KKT multiplier\n\n\[ \bar{\lambda } = \left( {{\bar{\lambda }}_{1},{\bar{\lambda }}_{2},\cdots ,{\bar{\lambda }}_{m}}\right) \in {R}_{ + }^{m} \]\n\nIt follows from (2.2) that there exist \( \xi \in \partial f\left( \bar{x}\right) \) and \( {\zeta ... | Yes |
\[ \text{(HOP) minimize}\;f\left( x\right) \text{,} \]\n\[ \text{ subject to }\;g\left( x\right) \leq 0, \]\n\[ x \in \Omega \]\n\nwhere \( f\left( x\right) = \left| x\right| \) is absolutely 1-homogeneous, \( g\left( x\right) = \max \{ 0, x\} \) is positively 1-homogeneous, \( b = {b}_{1} = 0,\Omega = \left( {-\infty ... | (i) \( 0 \in \partial f\left( 0\right) = \left\lbrack {-1,1}\right\rbrack ,0 \in \partial g\left( 0\right) = \left\lbrack {0,1}\right\rbrack \), take \( {\lambda }_{1} = 1 \), then\n\n\[ 0 \in \partial f\left( 0\right) + 1 \cdot \partial g\left( 0\right) \]\n\nand \( 1 \cdot g\left( 0\right) = 0 \), i.e.,0 is a KKT poi... | Yes |
\[ \text{(HOP) minimize}\;f\left( x\right) \text{,} \]\n\[ \text{ subject to }\;g\left( x\right) \leq 0, \]\n\[ x \in \Omega \]\nwhere \( f : {R}^{2} \rightarrow R, f\left( x\right) = \parallel x\parallel \) is absolutely 1-homogeneous, \( g\left( x\right) = {x}_{1}^{2} + {x}_{2}^{2} \) is positively 2-homogeneous, \( ... | (i) \( \partial f\left( {0,0}\right) = \left\{ {\zeta \in {R}^{2} : {\zeta }_{i} \in \left\lbrack {-1,1}\right\rbrack, i = 1,2}\right\} ,\partial g\left( {0,0}\right) = \{ \left( {0,0}\right) \} \) . Take \( \bar{\lambda } = 1 \), it follows that \( \left( {0,0}\right) \in \partial f\left( {0,0}\right) + 1 \cdot \parti... | Yes |
Theorem 3.2 When \( g = - \frac{15}{64}{c}^{2} \) and \( c > 0 \), equation (1.1) has a kink-like wave solution \( u = {\varphi }_{2}\left( \xi \right) \) and a antikink-like wave solution \( u = {\varphi }_{3}\left( \xi \right) \), where \( \frac{c}{2} < \varphi < \frac{5}{8}c \) and \( \beta \left( \varphi \right) = ... | Proof In Figure 1 (1-6), system (2.4) has four orbits connecting with a saddle point \( {A}_{ + },{L}_{2} \) denotes a orbit lying on the upper-left side of \( {A}_{ + } \), and \( {L}_{3} \) on the lower-left. Note that \( H\left( {A}_{ + }\right) = 0 \), then \( {L}_{2} \) and \( {L}_{3} \) can be respectively descri... | Yes |
Theorem 3.3 When \( - \frac{{c}^{2}}{4} < g < - \frac{15}{64}{c}^{2} \) and \( c > 0 \), equation (1.1) has a soliton solution | Proof In Figure 1 (1-7), system (2.4) has a homoclinic loop consisting of a saddle point \( \left. {{A}_{ + },}\right. \) which can be expressed as \( H\left( {\varphi, y}\right) = {\varphi }^{3}\left( {\frac{1}{3}g + \frac{c}{4}\varphi - \frac{1}{5}{\varphi }^{2} + \frac{3}{2}\varphi {y}^{2}}\right) = H\left( {A}_{ + ... | Yes |
Lemma 2.4 [6, Lemma 1.5] Let \( \varphi \) : \( L \rightarrow {L}^{ * } \) be a derivation and suppose that \( e \in L \) such that \( {\left( \operatorname{ad}e\right) }^{{p}^{r}} = 0 \) . Then \( {e}^{{p}^{r} - 1} \cdot \varphi \left( e\right) \in {\left( {L}^{ * }\right) }^{L} \), where | \[ {\left( {L}^{ * }\right) }^{L} = \left\{ {f \in {L}^{ * } \mid L \cdot f = 0}\right\} = \left\{ {f \in {L}^{ * } \mid f\left( \left\lbrack {L, L}\right\rbrack \right) = 0}\right\} . \] | No |
Lemma 4.1 Let \( \varphi : S\left( {m, n;\underline{t}}\right) \rightarrow S{\left( m, n;\underline{t}\right) }^{ * } \) be a derivation. Then there exists \( f \in S{\left( m, n;\underline{t}\right) }^{ * } \) such that \( \varphi \left( x\right) = {\left( -1\right) }^{p\left( x\right) p\left( f\right) }x \cdot f \) f... | Proof We shall apply Proposition 2.5 to complete the proof. Put \( L : = S\left( {m, n;\underline{t}}\right) \) . Then \( \left\{ {{D}_{1},{D}_{2},\cdots ,{D}_{s}}\right\} \) is a basis of the subalgebra \( {L}^{ - } \) . Consider \( V = S{\left( m, n;\underline{t}\right) }_{q} \) with a basis\n\n\[ \left\{ {{D}_{ij}\l... | Yes |
Lemma 3.1 Let \( \\alpha > 0, - 1 < a < 0,0 < b < 1, b < a + 1 \) . Then\n\n\[ \n{\\int }_{0}^{{T}_{n}}\\frac{{X}_{s}}{T - s}d{B}_{s}^{a, b} = {\\int }_{0}^{{T}_{n}}\\frac{{X}_{s}}{T - s}{\\delta }^{a, b}{B}_{s}^{a, b} + {\\beta }_{n} \n\]\n\nwhere\n\n\[ \n{\\beta }_{n} = b{\\int }_{0}^{{T}_{n}}{\\int }_{0}^{r}{\\left(... | Proof By (2.9), we have\n\n\[ \n{\\int }_{0}^{{T}_{n}}\\frac{{X}_{s}}{T - s}d{B}_{s}^{a, b} \n\]\n\n\[ \n= {\\int }_{0}^{{T}_{n}}\\frac{{X}_{s}}{T - s}{\\delta }^{a, b}{B}_{s}^{a, b} + b{\\int }_{0}^{{T}_{n}}{\\int }_{0}^{{T}_{n}}{D}_{s}^{a, b}\\frac{{X}_{r}}{T - r}{\\left( r \\land s\\right) }^{a}{\\left( r \\vee s - ... | Yes |
Lemma 3.2 Let \( - 1 < a < 0,0 < b < 1,{2\alpha } - 1 < b < a + 1 \), then\n\n\[ E{\left( \frac{{X}_{t}}{T - t}\right) }^{2} \leq {2bB}\left( {1 - \alpha, b}\right) B\left( {b - {2\alpha } + 1, a + 1}\right) {\left( T - t\right) }^{{2\alpha } - 2}{T}^{a + b - {2\alpha } + 1},0 \leq t < T. \] | Proof In fact, we have\n\n\[ E{A}_{t}^{2} = b{\int }_{0}^{t}{\int }_{0}^{t}{\left( T - u\right) }^{-\alpha }{\left( T - v\right) }^{-\alpha }{\left( u \land v\right) }^{a}{\left( u \vee v - u \land v\right) }^{b - 1}{dudv} \]\n\n\[ = {2b}{\int }_{0}^{t}{\int }_{0}^{u}{\left( T - u\right) }^{-\alpha }{\left( T - v\right... | Yes |
Lemma 3.3 Assume \( - 1 < a < 0,0 < b < 1,1 - b < {2\alpha } < 1 + b \leq a + 2 \) and let \( {F}_{{T}_{n}} = {\int }_{0}^{{T}_{n}}\frac{{X}_{t}}{T - t}{\delta }^{a, b}{B}_{t}^{a, b} \) . Then \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}E\left( {F}_{{T}_{n}}^{2}\right) = \frac{{b}^{2}}{2}B\left( {1 - \alpha, b}\r... | Proof By the isometry property of the double stochastic integral \( {I}_{2} \), the variance of \( {F}_{{T}_{n}} \) is given by \[ E\left( {F}_{{T}_{n}}^{2}\right) = \frac{{b}^{2}}{2}{I}_{{T}_{n}} \] where \[ {I}_{{T}_{n}} = \;{\int }_{{\left\lbrack 0,{T}_{n}\right\rbrack }^{4}}{\left( T - {t}_{1}\right) }^{\alpha - 1}... | Yes |
Theorem 2.1 Every planar graph without 5-cycles or \( {K}_{4} \) can be decomposed into two forests \( {F}_{1},{F}_{2} \) and a linear forest \( P \) . | Proof We prove the theorem by reduction to absurdity. Let \( G \) be a counter example with \( \left| {V\left( G\right) }\right| + \left| {E\left( G\right) }\right| \) minimum. The following configurations are excluded from \( G \) . These obvious facts will be frequently used.\n\n\( \left( {{C}_{0}1}\right) \) A 5-fac... | No |
Lemma 2.2 \( G \) is connected and has no vertex \( v \) with \( d\left( v\right) \leq 2 \) . | Proof If \( G \) is disconnected, then one of its components is a counter example with fewer vertices, a contradiction. Let \( v \in V\left( G\right) \) with \( d\left( v\right) \leq 2 \) . Consider the graph \( {G}^{\prime } \mathrel{\text{:=}} G - v \) , by the choice of \( G,{G}^{\prime } \) can be decomposed into t... | Yes |
Lemma 2.3 \( G \) has no edge \( {uv} \) with \( d\left( u\right) = 3, d\left( v\right) \leq 4 \) . | Proof Suppose \( {uv} \) is such an edge that \( d\left( u\right) = 3, d\left( v\right) \leq 4 \) . Let \( {w}_{1},{w}_{2} \) be the other two neighbors of \( u \) . Consider the graph \( {G}^{\prime } \mathrel{\text{:=}} G - \left\{ {{uv}, u{w}_{1}, u{w}_{2}}\right\} \), by the choice of \( G \) , \( {G}^{\prime } \) ... | Yes |
Lemma 2.4 \( G \) has no \( \left( {3,5}\right) \) -alternating cycle. | Proof Let \( C = \left\lbrack {{v}_{1}{v}_{2}\cdots {v}_{2r}}\right\rbrack \) be a \( \left( {3,5}\right) \) -alternating cycle in \( G \), i.e., \( d\left( {v}_{i}\right) = 3 \), for \( i = 1,3,\cdots ,{2r} - 1 \) ; and \( d\left( {v}_{i}\right) = 5 \), for \( i = 2,4,\cdots ,{2r} \) . Consider the graph \( {G}^{\prim... | Yes |
Lemma 2.5 The type of \( \left( {5,3,{6}_{2},3}\right) 4 \) -faces are not incident to each other. Therefore, at most two \( \left( {5,3,{6}_{2},3}\right) 4 \) -faces are incident to \( v \) . | Proof Let \( {f}_{1} = \left\lbrack {v{v}_{1}{w}_{1}{v}_{2}}\right\rbrack ,{f}_{2} = \left\lbrack {v{v}_{2}{w}_{2}{v}_{3}}\right\rbrack, d\left( {w}_{1}\right) = d\left( {w}_{2}\right) = 6 \), and \( {w}_{1},{w}_{2} \) are weak of type 2, then \( {w}_{1},{v}_{2},{w}_{2} \) must be contained in one 4 -face, denote by \(... | Yes |
Lemma 2.6 \( d\left( {w}_{2}\right) \geq 6 \) or \( d\left( {v}_{3}\right) \neq 3 \) . Both cases we can get \( c{h}^{\prime }\left( x\right) \geq 0 \) . | Proof Suppose \( d\left( {w}_{2}\right) = 5 \) and \( d\left( {v}_{3}\right) = 3 \), Consider the graph \( {G}^{\prime } \mathrel{\text{:=}} G - E\left( {f}_{1}\right) - E\left( {f}_{2}\right) \) . Since \( \left| {V\left( {G}^{\prime }\right) }\right| + \left| {E\left( {G}^{\prime }\right) }\right| < \left| {V\left( G... | Yes |
Lemma 2.7 For \( \left( {\mathrm{C}1}\right), d\left( {v}_{2}\right) \geq 7, c{h}^{\prime }\left( v\right) > 0 \) . | Proof Suppose \( d\left( {v}_{2}\right) \leq 6 \), Consider the graph \( {G}^{\prime } \mathrel{\text{:=}} G - E\left( {f}_{1}\right) - E\left( {f}_{2}\right) - E\left( {f}_{3}\right) \), since \( \left| {V\left( {G}^{\prime }\right) }\right| + \left| {E\left( {G}^{\prime }\right) }\right| < \left| {V\left( G\right) }\... | Yes |
Lemma 2.8 For \( \left( {\mathrm{C}2}\right), c{h}^{\prime }\left( v\right) \geq 0 \) . | Proof When \( d\left( {v}_{4}\right) = 5 \), if \( d\left( {v}_{1}\right) \geq 6 \) and \( d\left( {v}_{3}\right) \geq 6 \), since \( {v}_{1},{v}_{3} \) cannot be 6 -vertices and weak of type 1 by \( \left( {{C}_{0}3}\right), c{h}^{\prime }\left( v\right) \geq 2 \times 5 - 6 - \left( {1 \times 2 + \frac{3}{2}}\right) >... | Yes |
Lemma 2.9 There is at most one \( \left( {6,3,5,3}\right) 4 \) -face incidenting to \( v \) . | Proof Suppose that there are two \( \left( {6,3,5,3}\right) 4 \) -faces \( {f}_{1},{f}_{2} \) incident to \( v \), let\n\n\[ \n{f}_{1} = \left\lbrack {v{v}_{1}{w}_{1}{v}_{2}}\right\rbrack \n\] \n\nby symmetry, there are three cases to consider, \( {f}_{2} = \left\lbrack {v{v}_{2}{w}_{2}{v}_{3}}\right\rbrack \) or \( {f... | Yes |
Lemma 2.10 For \( \left( {\mathrm{C}1}\right), d\left( {v}_{2}\right) \geq 6 \), so \( c{h}^{\prime }\left( v\right) \geq 0 \) . | Proof If it is not true, then by Lemma 2.3, assume \( d\left( {v}_{2}\right) = 5 \) . Consider the graph\n\n\[ \n{G}^{\prime } \mathrel{\text{:=}} G - E\left( {f}_{1}\right) - E\left( {f}_{2}\right) - E\left( {f}_{3}\right) - E\left( {f}_{4}\right) ,\n\]\n\nsince \( {G}^{\prime } \) has fewer edges than \( G,{G}^{\prim... | Yes |
Lemma 2.11 For (C2), at least one of \( {v}_{1},{v}_{2} \) is a \( {6}^{ + } \) -vertex, So \( c{h}^{\prime }\left( v\right) > 0 \) . | Proof If it is not true, then by Lemma \( {2.3}, d\left( {v}_{1}\right) = d\left( {v}_{3}\right) = 5 \) . Consider the graph \( {G}^{\prime } \mathrel{\text{:=}} G - E\left( {f}_{1}\right) - E\left( {f}_{2}\right) - E\left( {f}_{3}\right) - E\left( {f}_{4}\right) \), then \( {G}^{\prime } \) has fewer edges since \( G,... | Yes |
Theorem 1.1 Let \( \\Gamma = \\left( {X, R}\\right) \) be the distance-regular graph with diameter \( d\\left( \\Gamma \\right) \\geq 2 \) and intersection numbers\n\n\[ \n{p}_{jt}^{h}\\left( {0 \\leq h, j, t \\leq d\\left( \\Gamma \\right) }\\right) .\n\]\n\nThen the following hold.\n\n(i) If \( d\\left( \\Gamma \\rig... | Proof For any \( x, y \\in X \) with \( {\\partial }_{\\Gamma }\\left( {x, y}\\right) = l \), where \( 1 \\leq l \\leq d\\left( \\Gamma \\right) \) . By (1.1) and (1.2), the number of vertices \( z \\in X \) satisfying both \( {\\partial }_{\\bar{\\Gamma }}\\left( {x, z}\\right) = 1 \) and \( {\\partial }_{\\bar{\\Gamm... | Yes |
Lemma 3.2 We denote\n\n\[ \n{\omega }_{i}\left( s\right) = \mathop{\sum }\limits_{{m = 1}}^{p}{a}_{sm}\left( {\theta }_{0}\right) {h}_{im}\left( {\theta }_{0}\right) \n\]\n\nand \( {S}_{n}^{2} = \mathop{\sum }\limits_{{i = 1}}^{n}\mathbf{E}{\omega }_{i}^{2}\left( s\right) \) . Then\n\n\[ \n\mathop{\limsup }\limits_{{n ... | Proof Observe that \( \mathbf{E}\left( {{h}_{im}\left( {\theta }_{0}\right) }\right) = {G}^{\prime }W\mathbf{E}\left\lbrack {g\left( {{X}_{i},{\theta }_{0}}\right) }\right\rbrack = 0 \) and \( \left\{ {{h}_{im}, i \geq 1}\right\} \) is a sequence of independent random variables. Consequently,\n\n\[ \n\mathbf{E}{\omega ... | Yes |
Theorem 2.1 Under the Assumption H1 and H2, \( {\left\{ {L}_{n}\left( t\right), t \in \left\lbrack 0, T\right\rbrack \right\} }_{n \geq 1} \) obeys a large deviation upper bound on \( C\left( {\left\lbrack {0, T}\right\rbrack ,{\mathcal{M}}_{1}\left( \mathbb{R}\right) }\right) \) with speed \( {n}^{2} \) and with good ... | Thanks to the exponential tightness result established in [9], to prove Theorem 2.1, by the usual scheme (see [13]), we only need to show that the rate function is good and a weak large deviation upper bound holds. We first recall the exponential tightness result (see [9] Theorem 3.3): | Yes |
Lemma 1.5 (see [1]) Suppose that \( x\left( t\right) : \lbrack 0, + \infty ) \rightarrow X \) is differentiable in the point of \( x \in \left( {0, + \infty }\right) \), and \( \parallel x\left( t\right) \parallel \) is also differentiable in the point of \( s \), then | \[ \operatorname{Re}{T}_{x\left( t\right) }\left\lbrack {\frac{dx}{dt}\left( s\right) }\right\rbrack = \frac{\parallel {dx}\left( s\right) \parallel }{dt}, s \in \lbrack 0, + \infty ). \] | No |
Lemma 2.6 Let \( f \in {C}^{\left( 2\right) }\left( {\Omega, C\left( {V}_{2,0}\right) }\right) \cap {C}^{\left( 1\right) }\left( {\bar{\Omega }, C\left( {V}_{2,0}\right) }\right) \) and \( {D}^{2}\left\lbrack f\right\rbrack = 0 \) in \( \Omega \), where \( \Omega \) is a bounded domain with smooth boundary in \( R{Q}_{... | Proof By Lemma 2.1 and Stokes' formula (see [5]), the result follows. | No |
Lemma 2.7 (see [9]) Let \( f \in {C}^{\left( 2\right) }\left( {\Omega, C\left( {V}_{2,0}\right) }\right) \cap {C}^{\left( 1\right) }\left( {\bar{\Omega }, C\left( {V}_{2,0}\right) }\right) \) and \( {D}^{2}\left\lbrack f\right\rbrack = 0 \) in \( \Omega \) , where \( \Omega \) is a bounded domain with smooth boundary i... | \[ f\left( \mathbf{x}\right) = {\int }_{\partial \Omega }{H}_{1}\left( {\mathbf{y} - \mathbf{x}}\right) \mathrm{d}{\sigma }_{y}f\left( \mathbf{y}\right) - {\int }_{\partial \Omega }{H}_{2}\left( {\mathbf{y} - \mathbf{x}}\right) \mathrm{d}{\sigma }_{y}D\left\lbrack f\right\rbrack \left( \mathbf{y}\right) . \] | Yes |
Lemma 2.8 Let \( f \in {C}^{\left( 2\right) }\left( {\Omega, C\left( {V}_{2,0}\right) }\right) \cap {C}^{\left( 1\right) }\left( {\bar{\Omega }, C\left( {V}_{2,0}\right) }\right) \) and \( \bigtriangleup \left\lbrack f\right\rbrack = 0 \) in \( \Omega \), where \( \Omega \) is a bounded domain with smooth boundary in \... | Proof By Lemma 2.1, Lemma 2.3 and Stokes' formula, the result follows. | No |
Lemma 2.9 Let \( f \in {C}^{\left( 2\right) }\left( {\Omega, C\left( {V}_{2,0}\right) }\right) \cap {C}^{\left( 1\right) }\left( {\bar{\Omega }, C\left( {V}_{2,0}\right) }\right) \) and \( \bigtriangleup \left\lbrack f\right\rbrack = 0 \) in \( \Omega \), where \( \Omega \) is a bounded domain with smooth boundary in \... | Proof By Lemma 2.1, Lemma 2.3 and Stokes' formula, the result follows. | No |
Lemma 2.10 Let \( f \in {C}^{\left( 2\right) }\left( {\Omega, C\left( {V}_{2,0}\right) }\right) \cap {C}^{\left( 1\right) }\left( {\bar{\Omega }, C\left( {V}_{2,0}\right) }\right) \) and \( \bigtriangleup \left\lbrack f\right\rbrack = 0 \) in \( \Omega \), where \( \Omega \) is a bounded domain with smooth boundary in ... | Proof By Lemma 2.8, it can be similarly proved as in Lemma 2.7. | No |
Lemma 2.11 Let \( f \in {C}^{\left( 2\right) }\left( {\Omega, C\left( {V}_{2,0}\right) }\right) \cap {C}^{\left( 1\right) }\left( {\bar{\Omega }, C\left( {V}_{2,0}\right) }\right) \) and \( \bigtriangleup \left\lbrack f\right\rbrack = 0 \) in \( \Omega \), where \( \Omega \) is a bounded domain with smooth boundary in ... | Proof By Lemma 2.9, it can be similarly proved as in Lemma 2.7. | No |
Lemma 3.13 (see [27]) Let \( f\left( \mathbf{y}\right) \in {\widehat{H}}^{\mu }\left( {\Pi, C\left( {V}_{2,0}\right) }\right) ,{Cf}\left( \mathbf{x}\right) \) be defined as in (3.1), then \( {Cf}\left( \mathbf{x}\right) \) exists and | \[ {Cf}\left( \mathbf{x}\right) = \left\{ \begin{array}{l} \frac{f\left( \infty \right) }{2} - \frac{1}{4\pi }{\iint }_{\Pi }\frac{\overline{\mathbf{y}} - \overline{\mathbf{x}}}{{\left| \mathbf{y} - \mathbf{x}\right| }^{3}}\left( {f\left( \mathbf{y}\right) - f\left( \infty \right) }\right) \mathrm{d}S,\mathbf{x} \in R{... | Yes |
Theorem 3.1 Let \( f \in {C}^{\left( 2\right) }\left( {\Omega, C\left( {V}_{2,0}\right) }\right) \cap {C}^{\left( 1\right) }\left( {\bar{\Omega }, C\left( {V}_{2,0}\right) }\right) \) and \( {D}^{2}\left\lbrack f\right\rbrack = 0 \) in \( \Omega \), where \( \Omega \) is a bounded domain with smooth boundary in \( R{Q}... | Proof By Lemma 2.2, Lemma 2.7 and Stokes' formula, the result follows. | No |
Lemma 3.14 For any \( \mathbf{x} \in R{Q}_{3}^{ + } \) ,\n\n\[ \mathop{\lim }\limits_{{R \rightarrow + \infty }}{\int }_{{\partial }^{ + }B\left( {\mathbf{X}, R}\right) }{H}_{1}\left( {\mathbf{y} - \mathbf{x}}\right) \mathrm{d}{\sigma }_{y} = \mathop{\lim }\limits_{{R \rightarrow + \infty }}{\int }_{{\partial }^{ + }B\... | Proof It can be proved by Lemma 2.5. | No |
Lemma 3.15 Let \( f \in {\widehat{H}}^{\mu }\left( {\overline{R{Q}_{3}^{ + }}, C\left( {V}_{2,0}\right) }\right) \), then for any \( \mathbf{x} \in R{Q}_{3}^{ + } \) , \[ \mathop{\lim }\limits_{{R \rightarrow + \infty }}{\int }_{{\partial }^{ + }B\left( {\mathbf{X}, R}\right) }{H}_{1}\left( {\mathbf{y} - \mathbf{x}}\ri... | Proof It can be proved by Lemma 3.14. | No |
Lemma 3.16 Let \( D\left\lbrack f\right\rbrack \in {\widehat{H}}_{0}^{\mu }\left( {\overline{R{Q}_{3}^{ + }}, C\left( {V}_{2,0}\right) }\right) \), then for any \( \mathbf{x} \in R{Q}_{3}^{ + } \) ,\n\n\[ \mathop{\lim }\limits_{{R \rightarrow + \infty }}{\int }_{{\partial }^{ + }B\left( {\mathbf{x}, R}\right) }{E}_{2}\... | Proof It can be proved by Lemma 2.5. | No |
Lemma 3.17 Let \( f \in {\widehat{H}}^{\mu }\left( {\Pi, C\left( {V}_{2,0}\right) }\right) \), then for all \( {\mathbf{y}}_{ * },{\mathbf{y}}_{* * } \in \Pi \) ,\n\n\[ \mathop{\lim }\limits_{{R \rightarrow + \infty }}\left( {{\iint }_{D\left( {{\mathbf{y}}_{ * }, R}\right) }{E}_{2}\left( {\mathbf{y} - \mathbf{x}}\righ... | Proof By Lemma 2.5, it can be similarly proved as in Lemma 2.12. | No |
Theorem 3.2 Let \( f \in {C}^{\left( 1\right) }\left( {\overline{R{Q}_{3}^{ + }}, C\left( {V}_{2,0}\right) }\right) \bigcap {C}^{\left( 2\right) }\left( {R{Q}_{3}^{ + }, C\left( {V}_{2,0}\right) }\right), f \in {\widehat{H}}^{\mu }\left( {\overline{R{Q}_{3}^{ + }}, C\left( {V}_{2,0}\right) }\right) \), \( D\left\lbrack... | Proof For any \( \mathbf{x} \in R{Q}_{3}^{ + } \), denote \( \Omega = B\left( {\mathbf{x}, R}\right) \bigcap R{Q}_{3}^{ + } \), by Theorem 3.1, we have \[ f\left( \mathbf{x}\right) = {\int }_{\partial \Omega }{E}_{1}\left( {\mathbf{y} - \mathbf{x}}\right) \mathrm{d}{\sigma }_{y}f\left( \mathbf{y}\right) - {\int }_{\par... | Yes |
Theorem 4.3 Let \( f \in {C}^{\left( 2\right) }\left( {\Omega, C\left( {V}_{2,0}\right) }\right) \cap {C}^{\left( 1\right) }\left( {\bar{\Omega }, C\left( {V}_{2,0}\right) }\right) \) and \( \bigtriangleup \left\lbrack f\right\rbrack = 0 \) in \( \Omega \), where \( \Omega \) is a bounded domain with smooth boundary in... | Proof By Stokes’ formula and Lemma 2.3, for any \( \mathbf{x} \in \Omega \), we have \[ {\int }_{\partial \Omega }{H}_{1}^{ * }\left( {\mathbf{y} - \mathbf{x}}\right) \mathrm{d}{\sigma }_{y}f\left( \mathbf{y}\right) - {\int }_{\partial \Omega }{K}^{ * }\left( {\mathbf{y} - \mathbf{x}}\right) \overline{\mathrm{d}{\sigma... | No |
Theorem 4.4 Let \( f \in {C}^{\left( 2\right) }\left( {\Omega, C\left( {V}_{2,0}\right) }\right) \cap {C}^{\left( 1\right) }\left( {\bar{\Omega }, C\left( {V}_{2,0}\right) }\right) \) and \( \bigtriangleup \left\lbrack f\right\rbrack = 0 \) in \( \Omega \), where \( \Omega \) is a bounded domain with smooth boundary in... | Proof By Stokes’ formula and Lemma 2.3, for any \( \mathbf{x} \in \Omega \), we have\n\n\[ {\int }_{\partial \Omega }\overline{{H}_{1}^{ * }}\left( {\mathbf{y} - \mathbf{x}}\right) \overline{\mathrm{d}{\sigma }_{y}}f\left( \mathbf{y}\right) - {\int }_{\partial \Omega }{K}^{ * }\left( {\mathbf{y} - \mathbf{x}}\right) \m... | Yes |
Lemma 4.18 Let \( f \in {\widehat{H}}^{\mu }\left( {\Pi, C\left( {V}_{2,0}\right) }\right) \), then for all \( {\mathbf{y}}_{ * },{\mathbf{y}}_{* * } \in \Pi \) ,\n\n\[ \mathop{\lim }\limits_{{R \rightarrow + \infty }}\left( {{\iint }_{D\left( {{\mathbf{y}}_{ * }, R}\right) }G\left( {\mathbf{y} - \mathbf{x}}\right) f\l... | Proof It can be similarly proved as in Lemma 2.12. | No |
Theorem 4.5 Let \( f \in {C}^{\left( 1\right) }\left( {\overline{R{Q}_{3}^{ + }}, C\left( {V}_{2,0}\right) }\right) \bigcap {C}^{\left( 2\right) }\left( {R{Q}_{3}^{ + }, C\left( {V}_{2,0}\right) }\right), f \in {\widehat{H}}^{\mu }\left( {\overline{R{Q}_{3}^{ + }}, C\left( {V}_{2,0}\right) }\right) \) , \( D\left\lbrac... | Proof For any \( \mathbf{x} \in R{Q}_{3}^{ + } \), denote \( \Omega = B\left( {\mathbf{x}, R}\right) \bigcap R{Q}_{3}^{ + } \), by Theorem 4.3, we have \[ f\left( \mathbf{x}\right) = {\int }_{\partial \Omega }{E}_{1}\left( {\mathbf{y} - \mathbf{x}}\right) \mathrm{d}{\sigma }_{y}f\left( \mathbf{y}\right) - {\int }_{\par... | Yes |
Corollary 4.1 Let \( f \in {C}^{\left( 1\right) }\left( {\overline{R{Q}_{3}^{ + }}, C\left( {V}_{2,0}\right) }\right) \bigcap {C}^{\left( 2\right) }\left( {R{Q}_{3}^{ + }, C\left( {V}_{2,0}\right) }\right), f \in {\widehat{H}}^{\mu }\left( {\overline{R{Q}_{3}^{ + }}, C\left( {V}_{2,0}\right) }\right) \) and \( D\left\l... | \[ f\left( \mathbf{x}\right) = - {\iint }_{\Pi }{E}_{1}\left( {\mathbf{y} - \mathbf{x}}\right) f\left( \mathbf{y}\right) \mathrm{d}S. \] | Yes |
Theorem 4.6 Let \( f \in {C}^{\left( 1\right) }\left( {\overline{R{Q}_{3}^{ + }}, C\left( {V}_{2,0}\right) }\right) \bigcap {C}^{\left( 2\right) }\left( {R{Q}_{3}^{ + }, C\left( {V}_{2,0}\right) }\right), f \in {\widehat{H}}^{\mu }\left( {\overline{R{Q}_{3}^{ + }}, C\left( {V}_{2,0}\right) }\right) \) , \( \bar{D}\left... | Proof By Theorem 4.4, it can be similarly proved as in Theorem 4.5. | No |
Corollary 4.2 Let \( f \in {C}^{\left( 1\right) }\left( {\overline{R{Q}_{3}^{ + }}, C\left( {V}_{2,0}\right) }\right) \bigcap {C}^{\left( 2\right) }\left( {R{Q}_{3}^{ + }, C\left( {V}_{2,0}\right) }\right), f \in {\widehat{H}}^{\mu }\left( {\overline{R{Q}_{3}^{ + }}, C\left( {V}_{2,0}\right) }\right) \) and \( \bar{D}\... | \[ f\left( \mathbf{x}\right) = - {\iint }_{\Pi }\overline{{E}_{1}}\left( {\mathbf{y} - \mathbf{x}}\right) f\left( \mathbf{y}\right) \mathrm{d}S. \] (4.15) | Yes |
Lemma 2.4 (The Federer-Fleming Theorem) The isoperimetric and Sobolev constants are equal, that is,\n\n\[ \n{\mathfrak{J}}_{\nu }\left( M\right) = {\mathfrak{S}}_{\nu }\left( M\right) \n\] | The detailed proof of the Federer-Fleming theorem can be found in \( \left\lbrack {3,4}\right\rbrack \) and \( \left\lbrack {12}\right\rbrack \) . This elegant result was first proven in [4] by Federer and Fleming, and in [12] independently by Maz'ya in 1960. | No |
Theorem 2.7 Let \( \Omega \) be a connected domain with smooth boundary \( \partial \Omega \) in an \( n \) - dimensional weighted Riemannian manifold \( \left( {M, g,{d\mu }}\right) \) . Assume \( {\lambda }_{p,\varphi }\left( \Omega \right) \) is the first eigenvalue of problem (1.1) for \( \varphi \in {C}^{\infty }\... | Proof For any \( u \in {C}_{0}^{\infty }\left( \Omega \right) \), set\n\n\[ \n\Omega \left( t\right) = \left\{ {x \in \Omega : {\left| u\right| }^{p}{e}^{-\varphi } > t}\right\} \n\]\n\nand\n\n\[ \n\mathrm{V}\left( t\right) = \mathrm{V}\left( {\Omega \left( t\right) }\right) ,\mathrm{A}\left( t\right) = \mathrm{A}\left... | Yes |
Corollary 2.10 Let \( \Omega \) be a connected domain with smooth boundary \( \partial \Omega \) in the Euclidean space \( {\mathbb{R}}^{n} \). Then\n\n\[ \n{\lambda }_{p,\varphi }^{\frac{1}{p}}\left( \Omega \right) \geq \frac{n}{p}\left( {{\left( \frac{{\omega }_{n}}{V\left( \Omega \right) }\right) }^{\frac{1}{n}} - n... | Proof It is well known that\n\n\[ \n{\mathfrak{J}}_{n}\left( \Omega \right) = n{\omega }_{n}^{\frac{1}{n}}\n\]\n\nfor any domain \( \Omega \subseteq {\mathbb{R}}^{n} \), where \( {\omega }_{n} \) denotes the volume of the unit ball in \( {\mathbb{R}}^{n} \). From this fact and (2.4), we can get\n\n\[ \n{\lambda }_{p,\v... | Yes |
If \( \Omega = {B}_{n}\left( R\right) \) is a ball in \( {\mathbb{R}}^{n} \) with radius \( \mathrm{R} \), then the volume of \( \Omega \) is \( V\left( \Omega \right) = {\omega }_{n}{R}^{n} \), and we can get | \[ {\lambda }_{p,\varphi }^{\frac{1}{p}}\left( \Omega \right) \geq \frac{n}{p}\left( {\frac{1}{R} - n{C}_{\varphi }}\right) \] directly by (2.8). Since any ball is trivial Cheeger set (see [2]), by simply calculation, we can obtain \[ h\left( \Omega \right) = \frac{A\left( {\partial \Omega }\right) }{V\left( \Omega \ri... | No |
Example 2 Let \( {S}^{n} \) be a unit sphere with sectional curvature 1, and \( \Omega \subseteq {S}^{n} \) (small enough) be a relatively compact domain with smooth boundary \( \partial \Omega \) . Then the Ricci curvature of \( {S}^{n} \) is \( n - 1 \) . From [17, Theorem 1.4], we know that for any connected domain ... | \[ \frac{A\left( {\partial \Omega }\right) }{V{\left( \Omega \right) }^{1 - \frac{1}{n}}} \geq n{\omega }_{n}^{\frac{1}{n}}{\left( 1 - \tau V{\left( \Omega \right) }^{\frac{2}{n}}\right) }^{\frac{1}{n}}, \] where \( \tau = \frac{n\left( {n - 1}\right) }{2\left( {n + 2}\right) {\omega }_{n}^{\frac{2}{n}}} \) . According... | Yes |
Lemma 2 [7] Given matrices \( {X}_{1},{B}_{1} \in {R}^{n \times m} \). Let the singular value decompositions of \( {X}_{1} \) and \( {G}_{1} \) be (2.3),(2.4), respectively. Then the matrix equation \( {A}_{1}{X}_{1} = {B}_{1} \) has a symmetric solution \( {A}_{1} \) if and only if | \[ {X}_{1}^{T}{B}_{1} = {B}_{1}^{T}{X}_{1},\;{B}_{1}{X}_{1}^{ + }{X}_{1} = {B}_{1}. \] | Yes |
Theorem 1 Let \( X, B \in {R}^{n \times m} \) be known. Suppose \( {D}_{n} \) with the form of (2.1), \( {D}_{n}^{T}X \) , \( {D}_{n}^{T}B \) have the partition forms of (3.1), and the singular value decompositions of the matrices \( {X}_{2},{X}_{3} \) and \( {G}_{2},{G}_{3} \) are given by (3.2),(3.3) and (3.4),(3.5),... | Proof Suppose the matrix equation (1.1) has a solution \( A \) which is Bisymmetric, then it follows from Lemma 1 that there exist \( {A}_{2} \in S{R}^{\left( {n - k}\right) \times \left( {n - k}\right) },{A}_{3} \in S{R}^{k \times k} \) satisfying \[ A = {D}_{n}\left\lbrack \begin{matrix} {A}_{2} & 0 \\ 0 & {A}_{3} \e... | No |
Lemma 2.2 Let \( f\left( {{x}_{1},{x}_{2},\cdots ,{x}_{n}}\right) \) be a function in \( {R}^{n} \) defined by\n\n\[ f\left( {{x}_{1},{x}_{2},\cdots ,{x}_{n}}\right) = {x}_{1}\mathop{\sum }\limits_{{i = 2}}^{n}{x}_{i} \]\n\nIf \( {x}_{1} + {x}_{2} + \cdots + {x}_{n} = {2\lambda } \), then we have \( f\left( {{x}_{1},{x... | Proof From \( {x}_{1} + {x}_{2} + \cdots + {x}_{n} = {2\lambda } \), we have \( \mathop{\sum }\limits_{{i = 2}}^{n}{x}_{i} = {2\lambda } - {x}_{1} \) . It follows that\n\n\[ f\left( {{x}_{1},{x}_{2},\cdots ,{x}_{n}}\right) = {x}_{1}\left( {{2\lambda } - {x}_{1}}\right) = - {\left( {x}_{1} - \lambda \right) }^{2} + {\la... | Yes |
Theorem 3.1 If \( {M}^{n}\left( {n \geq 3}\right) \) is a submanifold of a Riemannian manifold of quasi-constant curvature \( {N}^{n + p} \), then we have\n\n\[ \delta \left( {{n}_{1},\cdots ,{n}_{k}}\right) \leq c\left( {{n}_{1},\cdots ,{n}_{k}}\right) {H}^{2} + d\left( {{n}_{1},\cdots ,{n}_{k}}\right) a + b\left\lbra... | Proof Let \( x \in {M}^{n} \) and \( \left\{ {{e}_{1},{e}_{2},\cdots ,{e}_{n}}\right\} \) and \( \left\{ {{e}_{n + 1},{e}_{n + 2},\cdots ,{e}_{n + p}}\right\} \) be orthonormal basis of \( {T}_{x}{M}^{n} \) and \( {T}_{x}^{ \bot }{M}^{n} \), respectively, such that the mean curvature vector \( \zeta \) is in the direct... | Yes |
Corollary 4.6 If \( \zeta \left( x\right) = 0 \), then a unit vector \( X \in {T}_{x}M \) satisfies the equality case of (4.2) if and only if \( X \) belongs to the relative null space \( N\left( x\right) \) given by\n\n\[ N\left( x\right) = \left\{ {X \in {T}_{x}M \mid h\left( {X, Y}\right) = 0,\forall Y \in {T}_{x}M}... | Proof Assume \( \zeta \left( x\right) = 0 \) . For each unit vector \( X \in {T}_{x}M \), equality holds in (4.2) if and only if (4.5) and (4.7) hold. Then \( {h}_{1i}^{r} = 0,\forall i, r \), i.e., \( X \in N\left( x\right) \) . | No |
Given \( A \in {R}^{p \times m}, B \in {R}^{p \times n}, C \in {R}^{n \times q}, D \in {R}^{m \times q} \). Let the singular value decompositions of \( A \) be,\n\n\[ A = U\left\lbrack \begin{matrix} \sum & 0 \\ 0 & 0 \end{matrix}\right\rbrack {V}^{T} = {U}_{1}\sum {V}_{1}^{T} \]\n\nwhere \( U = \left( {{U}_{1},{U}_{2}... | Moreover, its general solution can be expressed as\n\n\[ X = D{C}^{ + } + {A}^{ + }B - {A}^{ + }{AD}{C}^{ + } + \left( {I - {A}^{ + }A}\right) Z\left( {I - C{C}^{ + }}\right) ,\forall Z \in {R}^{m \times n}. \] | Yes |
Lemma 2.3 Suppose that matrix equations (1.1) is consistent. Let the singular value decompositions of \( A \) and \( C \) given by (2.3) and (2.4), respectively. Denote by \( X \) the solution of matrix equations (1.1). Then matrix \( {V}^{T}{XP} \) can be partitioned into\n\n\[ \n{V}^{T}{XP} = \left( \begin{array}{ll}... | Proof By (2.6), \( Z \) is arbitrary, we claim from (2.3),(2.4) and (2.7) that \( {X}_{22} \) is arbitrary too. We omit the proof. | No |
Theorem 3.1 Given \( A \in {R}^{p \times m}, B \in {R}^{p \times n}, C \in {R}^{n \times q}, D \in {R}^{m \times q} \) . the singular value decompositions of the matrices \( A, C \) and \( {G}_{1},{H}_{1}^{ + } \) are given by (2.3),(2.4) and (3.2),(3.3), respectively. Then equations (1.1) has a solution \( X \) if and... | Proof Suppose the matrix equation (1.1) has a solution \( X \), then it follows from Lemma 2.2 that (3.4) hold. In this case, let \( \Omega \) be the set of all solutions of equations (1.1). By (3.1), \[ r\left( X\right) = r\left\lbrack \begin{array}{ll} {X}_{11} & {X}_{12} \\ {X}_{21} & {X}_{22} \end{array}\right\rbra... | Yes |
Lemma 1 Fixed point \( {E}_{0} \) is non-hyperbolic if and only if \( a = 1 \) or \( b = 1 \) . Otherwise, \( {E}_{0} \) is one of the types in Table 1. | Proof By (2.2) the Jacobian evaluated at the fixed point \( {E}_{0}\left( {0,0}\right) \) is given by\n\n\[ \n{JF}\left( {0,0}\right) = \left( \begin{array}{ll} a & 0 \\ 0 & b \end{array}\right) \n\]\n\n(2.3)\n\nwhich has eigenvalues \( {\lambda }_{1} = a \) and \( {\lambda }_{2} = b \) . Hence it is easy to obtain the... | Yes |
Lemma 2 The fixed point \( {E}_{1} \) is not hyperbolic if and only if \( a = 3 \) or \( b = 1 + d\left( {a - 1}\right) /a \) . Otherwise, \( {E}_{1} \) is one of the types in Table 2. | Proof Solving \( \left| {\lambda }_{1}\right| = \left| {2 - a}\right| < 1 \) yields \( 1 < a < 3 \) . Obviously \( {\lambda }_{2} > 0 \), from \( {\lambda }_{2} = \) \( {ab}/\left( {{ad} + a - d}\right) < 1 \) we get \( 0 < b < 1 + d\left( {a - 1}\right) /a \) . Hence \( {E}_{1} \) is stable node for \( 1 < a \leq 3 \)... | No |
Lemma 2.2 With the notations as above, we have\n\n\[ T\left( {\alpha ,\beta }\right) = \mathop{\sum }\limits_{{x \in {\mathbb{F}}_{q}}}{\zeta }_{p}^{{\operatorname{Tr}}_{1}^{e}\left( {{Q}_{\alpha ,\beta }\left( x\right) }\right) } \] | \[ = \left\{ \begin{array}{ll} {\eta }_{0}\left( \Delta \right) {\left( -1\right) }^{\left( {e - 1}\right) {r}_{\alpha ,\beta }}{q}_{0}^{s - \frac{{r}_{\alpha ,\beta }}{2}}, & \text{ if }p \equiv 1{\;(\operatorname{mod}\;4)}, \\ {\eta }_{0}\left( \Delta \right) {\left( -1\right) }^{\left( {e - 1}\right) {r}_{\alpha ,\b... | Yes |
Lemma 2.5 Let \( s \) be odd and \( k/e \) be even. Then the following identities hold.\n\n(i) \( \mathop{\sum }\limits_{{\alpha ,\beta \in {\mathbb{F}}_{q}}}S\left( {\alpha ,\beta }\right) = {p}^{2m} \) ; | Proof (i) We observe that\n\n\[ \mathop{\sum }\limits_{{\alpha ,\beta \in {\mathbb{F}}_{q}}}S\left( {\alpha ,\beta }\right) = \mathop{\sum }\limits_{{\alpha ,\beta \in {\mathbb{F}}_{q}}}\mathop{\sum }\limits_{{x \in {\mathbb{F}}_{q}}}\chi \left( {\alpha {x}^{\frac{{p}^{k} + 1}{2}} + \beta {x}^{\frac{{p}^{3k} + 1}{2}}}\... | Yes |
Theorem 3.1 The value distribution of the multiset\n\n\[ \left\{ {S\left( {\alpha ,\beta }\right) = \mathop{\sum }\limits_{{x \in {\mathbb{F}}_{q}}}\chi \left( {\alpha {x}^{\frac{{p}^{k} + 1}{2}} + \beta {x}^{\frac{{p}^{3k} + 1}{2}}}\right) \mid \alpha ,\beta \in {\mathbb{F}}_{q}}\right\} \]\n\nis described as shown in... | Proof It is clear that \( S\left( {\alpha ,\beta }\right) = {p}^{m} \) if \( \left( {\alpha ,\beta }\right) = \left( {0,0}\right) \) . For \( \left( {\alpha ,\beta }\right) \in {\mathbb{F}}_{q}^{2} \smallsetminus \{ \left( {0,0}\right) \} \), by Lemmas 2.2, 2.3 and 2.4, we have\n\n\[ S\left( {\alpha ,\beta }\right) \in... | Yes |
Theorem 3.2 Let \( p \) be an odd prime, \( m \) and \( k \) be two positive integers with \( e = \) \( \gcd \left( {m, k}\right), m \geq 3 \) . If \( m/e \) is odd and \( k/e \) is even, then the weight distribution of the code\n\n\[ \mathcal{C} = \left\{ {c\left( {\alpha ,\beta }\right) = {\left( {\operatorname{Tr}}_... | Proof The Hamming weight of the codeword \( c = c\left( {\alpha ,\beta }\right) \) in \( \mathcal{C} \) is given by\n\n\[ \begin{matrix} {w}_{H}\left( c\right) & = & \# \left\{ {x \in {\mathbb{F}}_{q}^{ * }|{\mathrm{{Tr}}}_{1}^{m}\left( {\alpha {x}^{\frac{{p}^{k} + 1}{2}} + \beta {x}^{\frac{{p}^{3k} + 1}{2}}}\right) \n... | Yes |
For \( 0 < p < \infty \), there exists a positive constant \( C \) depending only on \( p \) such that for any \( f \in {F}_{p} \), \n\n\[ \n\left| {{f}^{\prime }\left( z\right) }\right| \leq \frac{C}{{\left( 1 - {\left| z\right| }^{2}\right) }^{1 + \frac{1}{p}}}\parallel \left| f\right| {\parallel }_{{F}_{p}} \n\] | Proof If we replace \( f \) by \( {f}^{\prime } \) in Lemma 2.1 and let \( q = \infty \), then for every \( f \in {F}_{p} \), there exists \( C > 0 \) depending only on \( p \) such that \n\n\[ \n\left| {{f}^{\prime }\left( z\right) }\right| \leq \mathop{\sup }\limits_{{0 \leq t \leq {2\pi }}}\left| {{f}^{\prime }\left... | Yes |
Corollary 2.3 Suppose that \( 0 < p < \infty \), then there is a positive constant \( C \) satisfying such that for any \( f \in {F}_{p} \) and \( z \in \mathbb{D} \) ,\n\n\[ \left| {f\left( z\right) }\right| \leq \frac{C}{{\left( 1 - {\left| z\right| }^{2}\right) }^{\frac{1}{p}}}\parallel \left| f\right| {\parallel }_... | Proof Suppose \( f \in {F}_{p} \), we have\n\n\[ f\left( z\right) = f\left( 0\right) + z{\int }_{0}^{1}{f}^{\prime }\left( {tz}\right) {dt} \]\n\nThen by Corollary 2.2 and \( \left| {f\left( 0\right) }\right| \leq \parallel \left| F\right| {\parallel }_{{F}_{p}} \), we can easily get\n\n\[ \left| {f\left( z\right) }\ri... | Yes |
Corollary 2.4 For any \( z \in \mathbb{D},\mathop{\lim }\limits_{{\left| z\right| \rightarrow {1}^{ - }}}{\begin{Vmatrix}{\delta }_{z}\end{Vmatrix}}_{{F}_{p,0}^{ * }} \rightarrow \infty \) . Moreover, \( \frac{{\delta }_{z}}{{\begin{Vmatrix}{\delta }_{z}\end{Vmatrix}}_{{F}_{p,0}^{ * }}} \rightarrow 0{\text{weak}}^{ * }... | Proof For each \( \zeta \in \mathbb{T} \), we consider the function \( {f}_{\zeta }\left( z\right) = \log \left( {1 - \bar{\zeta }z}\right) \) which belongs to \( \mathcal{B} \) . Recall that \( {\begin{Vmatrix}{\delta }_{z}\end{Vmatrix}}_{{\mathcal{B}}^{ * }} \approx \log \left( \frac{2}{1 - {\left| z\right| }^{2}}\ri... | Yes |
Lemma 2.5 For \( 0 < p < \infty \), the map \( \delta : \mathbb{D} \rightarrow {F}_{p}^{ * } \), defined by \( \delta : z \mapsto {\delta }_{z} \), is continuous with respect to pseudo-hyperbolic distance metric. | Proof By Corollary 2.3, we obtain that \( {\left( 1 - {\left| z\right| }^{2}\right) }^{\frac{1}{p}}\left| {f\left( z\right) }\right| \lesssim \parallel \parallel f\parallel {\parallel }_{{F}_{p}} \), then \( {F}_{p} \subset {H}_{{v}_{1}}^{\infty } \) and \( \parallel f{\parallel }_{{H}_{{v}_{1}}^{\infty }} \lesssim \pa... | Yes |
Theorem 3.8 Let \( 0 < p < \infty \) and \( \varphi \) be a holomorphic self-map of \( \mathbb{D} \) such that \( {C}_{\varphi } : {F}_{p} \rightarrow \mathcal{B} \) is bounded. Then the following statements are equivalent.\n\n(i) \( \mathop{\lim }\limits_{{\left| {\varphi \left( z\right) }\right| \rightarrow 1}}F\left... | Proof (i) \( \Rightarrow \) (ii) Since \( {C}_{\varphi } : {F}_{p} \rightarrow \mathcal{B} \) is bounded, we have that \( \mathop{\sup }\limits_{{z \in \mathbb{D}}}F\left( {p,\varphi, z}\right) < \infty \) from Theorem 3.6. Let \( \left\{ {f}_{n}\right\} \) be a bounded sequence in \( {F}_{p} \) converging to 0 uniform... | Yes |
Lemma 2.1 (Log Sobolev inequality, see [3]) Let \( \\left( {{M}^{n}, g}\\right) \) be a compact Riemannian manifold. For any \( a > 0 \), there exists a constant \( C\\left( {a, g}\\right) \) such that if \( \\varphi > 0 \) satisfies \( {\\int }_{M}{\\varphi }^{2}{dVol} = 1 \), then | \[ {\\int }_{M}{\\varphi }^{2}\\log {\\varphi d}\\operatorname{Vol} \\leq a{\\int }_{M}{\\left| \\nabla \\varphi \\right| }^{2}d\\operatorname{Vol} + C\\left( {a, g}\\right) \] | Yes |
Lemma 2.2 Let \( \\left( {{M}^{n}, g}\\right) \) be a compact Riemannian manifold, \( F : M \\rightarrow \\mathbb{R} \) be a smooth function and \( \\lambda \) be a positive constant, then there exists a smooth function \( f : M \\rightarrow \\mathbb{R} \) satisfies the equation\n\n\[ F + 2\\bigtriangleup f - {\\left| ... | Proof Define a functional \( W \)\n\n\[ W\\left( {g, f}\\right) = {\\int }_{M}\\left( {F + 2\\bigtriangleup f - {\\left| \\nabla f\\right| }^{2} + {2\\lambda f}}\\right) {e}^{-f}{dVol} \]\n\nand\n\n\[ \\mu \\left( g\\right) = \\inf \\left\\{ {W\\left( {g, f}\\right) : f \\in {C}^{\\infty }\\left( M\\right) \\text{ with... | Yes |
Proposition 2.1 For the compact shrinking Ricci-harmonic soliton (2.3), the potential function \( f \) equals a Hodge-de Rham potential up to a constant. | Proof By the Hodge-de Rham decomposition theorem, there exists a divergence-free vector field \( Y \) and a function \( b \) on \( {M}^{n} \), such that\n\n\[ \nX = Y + \nabla b \n\]\n\n(2.5)\n\nwe deduce \( \operatorname{div}X = \bigtriangleup b \) . By Theorem 1.1, we can find a potential function \( f \) to \( \left... | Yes |
Lemma 3.1 Let \( \\left( {{M}^{n}, g}\\right) \) be the complete noncompact gradient shrinking Ricci-harmonic soliton structure (3.1), we have the following four equalities\n\n\[ R - \\alpha {\\left| \\nabla \\phi \\right| }^{2} + \\bigtriangleup f = \\frac{n}{2} \]\n\n(3.2)\n\n\[ R - \\alpha {\\left| \\nabla \\phi \\r... | Proof Taking trace of the first equation of (3.1), (3.2) is obtained.\n\nTaking covariant derivatives and using the commutation formula for the covariant derivatives, we have\n\n\[ {\\nabla }_{i}{R}_{jk} - {\\nabla }_{j}{R}_{ik} - \\alpha \\left( {{\\nabla }_{j}\\phi {\\nabla }_{i}{\\nabla }_{k}\\phi - {\\nabla }_{i}\\... | Yes |
Lemma 3.2 Let \( \left( {{M}^{n}, g}\right) \) be the complete noncompact shrinking Ricci-harmonic soliton structure (3.1) and \( \mathbf{V}\left( r\right) \mathrel{\text{:=}} {\int }_{\{ f < r\} }{dV} \) and \( {\mathbf{V}}_{R}\left( r\right) \mathrel{\text{:=}} {\int }_{\{ f < r\} }R - \alpha {\left| \nabla \phi \rig... | Proof Integrating by parts and using eq. (3.2),\n\n\[ \frac{n}{2}\mathbf{V}\left( r\right) - {\mathbf{V}}_{R}\left( r\right) = {\int }_{\{ f < r\} }{\Delta fdV} = {\int }_{\{ f = r\} }\nabla f \cdot \frac{\nabla f}{\left| \nabla f\right| }{dA} = {\int }_{\{ f = r\} }\left| {\nabla f}\right| {dA} \]\n\nwhich implies\n\n... | Yes |
Lemma 2.4 (see [4]) Let \( q \) be univalent in \( \mathbb{U} \) . Then there exists a point \( \xi \) with \( \left| \xi \right| = r \) such that for all \( z,\left| z\right| = r \) , | \[ \left| {z - \xi }\right| \left| {q\left( z\right) }\right| \leq \frac{2{r}^{2}}{1 - {r}^{2}} \] | Yes |
Theorem 3.1 Let \( f \in {R}_{m}^{s}\left( {b, k,\lambda }\right) \) . Then \[ \left( {1 - \lambda }\right) {f}_{k}\left( z\right) + {\lambda z}{f}_{k}^{\prime }\left( z\right) \in {R}_{m}\left( {1 - b}\right) \;\left( {z \in \mathbb{U}}\right) . | Proof Let \[ F\left( z\right) = \left( {1 - \lambda }\right) f\left( z\right) + {\lambda z}{f}^{\prime }\left( z\right) \] and \[ {F}_{k}\left( z\right) = \left( {1 - \lambda }\right) {f}_{k}\left( z\right) + {\lambda z}{f}_{k}^{\prime }\left( z\right) \] Then condition (1.2) can be written as \[ 1 + \frac{1}{b}\left( ... | Yes |
Theorem 3.3 Let \( f \in {R}_{m}^{s}\left( {b, k,\lambda }\right) \) with \( 0 < \lambda \leq 1 \) . Then\n\n\[ f\left( z\right) = \frac{1}{\lambda }{z}^{1 - \frac{1}{\lambda }}{\int }_{0}^{z}{\int }_{0}^{u}\exp \left( {\frac{b}{k}\mathop{\sum }\limits_{{\mu = 0}}^{{k - 1}}{\int }_{0}^{{\varepsilon }^{\mu }\xi }\frac{p... | Proof Suppose that \( f \in {R}_{m}^{s}\left( {b, k,\lambda }\right) \) . From (1.2) and (3.16), we have\n\n\[ \left( {1 - \lambda }\right) {f}^{\prime }\left( z\right) + \lambda {\left( z{f}^{\prime }\left( z\right) \right) }^{\prime } = \frac{\left( {1 - \lambda }\right) {f}_{k}\left( z\right) + {\lambda z}{f}_{k}^{\... | Yes |
Theorem 3.4 Let \( f \in {R}_{m}^{s}\left( {b, k,\lambda }\right) \) with \( k \geq 2 \) . Then \[ \left| {a}_{2}\right| \leq \frac{m\left| b\right| }{2\left( {1 + \lambda }\right) }.\] | Proof Suppose that \( f \in {R}_{m}^{s}\left( {b, k,\lambda }\right) \) . In view of Theorem 3.1, there exists a function \( \phi \in {R}_{m}\left( {1 - b}\right) ,\phi \left( z\right) = \left( {1 - \lambda }\right) {f}_{k}\left( z\right) + {\lambda z}{f}_{k}^{\prime }\left( z\right) \) such that \[ z{f}^{\prime }\left... | Yes |
Theorem 3.5 Let \( f \in {R}_{m}^{s}\left( {b, k,\lambda }\right) \) with \( k \geq 2 \) . Then the unit disk \( \mathbb{U} \) is mapped by every univalent function \( f \) onto a domain that contains the disk \( \left| \omega \right| < {r}_{1} \), where\n\n\[ \n{r}_{1} = \frac{2\left( {1 + \lambda }\right) }{4\left( {... | Proof Suppose that \( f \in {R}_{m}^{s}\left( {b, k,\lambda }\right) \) . Also, let \( {\omega }_{0} \) be any complex number such that \( f\left( z\right) \neq {\omega }_{0} \) for \( z \in \mathbb{U} \), then \( {\omega }_{0} \neq 0 \) and\n\n\[ \n\frac{{\omega }_{0}f\left( z\right) }{{\omega }_{0} - f\left( z\right)... | Yes |
Theorem 3.7 Let \( f \in {K}_{m}^{s}\left( {\alpha, b, k,\lambda ,\delta }\right) \) with \( 0 < \lambda \leq 1 \) and \( 0 < b \leq 1 \) . Then\n\n\[ \left| {a}_{n}\right| \leq \left\{ \begin{array}{ll} {C}_{1}\left( {\alpha, b,\delta, m}\right) M{\left( n\right) }^{1 - \alpha }{n}^{\frac{{4\alpha b} - 1}{2}} & \left(... | Proof Suppose that \( f \in {K}_{m}^{s}\left( {\alpha, b, k,\lambda ,\delta }\right) \) . For \( n \geq 1 \) and \( z = r{e}^{i\theta } \), Cauchy’s Theorem gives that\n\n\[ n{a}_{n} = \frac{1}{{2\pi }{r}^{n}}{\int }_{0}^{2\pi }z{f}^{\prime }\left( z\right) {e}^{-{in\theta }}{d\theta }. \]\n\n(3.42)\n\nUsing Theorem 3.... | Yes |
Theorem 3.8 Let \( f \in {K}_{m}^{s}\left( {\alpha, b, k,\lambda ,\delta }\right) \) with \( 0 < \lambda \leq 1 \) and \( 0 < b \leq 1 \) . Then\n\n\[ \n\begin{Vmatrix}{{a}_{n + 1}\left| -\right| {a}_{n}}\end{Vmatrix} \leq \left\{ \begin{array}{ll} {C}_{2}\left( {\alpha, b,\delta, m}\right) M{\left( r\right) }^{1 - \al... | Proof It is known that for \( \xi \in \mathbb{U}, z = r{e}^{i\theta } \) and \( n \geq 1 \), one has\n\n\[ \n\left| {\left( {n + 1}\right) \xi {a}_{n + 1} - n{a}_{n}}\right| \leq {\int }_{0}^{2\pi }\left| {z - \xi }\right| \left| {z{f}^{\prime }\left( z\right) }\right| {d\theta }.\n\]\n\n(3.46)\n\nSince \( f \in {K}_{m... | Yes |
Lemma 2.1 If conditions (1.2) and (1.3) hold, then the weight \( K \) has the following proposition:\n\n(1) \( K\left( t\right) /{t}^{c} \) is non-decreasing for some \( c > 0 \) .\n\n(2) \( K\left( t\right) /{t}^{p} \) is non-increasing for some \( 0 < p < 1 \) .\n\n(3) \( K\left( {2t}\right) \approx K\left( t\right) ... | The above result can be found in [6]. | No |
Lemma 3.1 Let \( \left\{ {z}_{n}\right\} \) be a sequence in the unit disc such that (1.5) holds. Then there is a constant \( C \) such that whenever \( \left\{ {a}_{n}\right\} \in {l}^{\infty } \), there exists \( f \in {H}^{\infty } \) such that\n\n\[ f\left( {z}_{n}\right) = {a}_{n},\;n = 1,2,\cdots \]\n\n(3.3)\n\na... | For any given \( \left\{ {a}_{n}\right\} \in {l}^{\infty } \), by Lemma 3.1 there exists a function \( f \in {H}^{\infty } \) such that (3.3) holds. We now show that \( {d\mu }\left( z\right) = \sum {\delta }_{{\xi }_{n}} \) is a \( K \) -Carleson measure. For any \( a \in \mathbb{D} \), by Lemma 1.4 in [7], we have\n\... | Yes |
Lemma 3.1 Let \( M \) be a complete Riemannian manifold with a special exhaustion function \( \Phi \) . Assume \( u \in {A}^{p}\left( M\right) \) satisfies \( {\bigtriangleup }^{p}u = {\lambda u} \) and \( {\delta u} = 0 \) . Let also \( k \in R \) and \( X \in \Gamma \left( {TM}\right) \) be a given vector field, then... | Proof By (2.3) and (2.6), we have\n\n\[ \n{\int }_{\partial {B}_{\Phi }\left( R\right) }{S}_{\omega }\left( {X,\nu }\right) d{\sigma }_{R} = {\int }_{{B}_{\Phi }\left( R\right) }\left\lbrack {\left\langle {{S}_{\omega },\nabla {\theta }_{X}}\right\rangle + \left\langle {{\delta \omega },{i}_{X}\omega }\right\rangle + \... | Yes |
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