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Theorem 0.1 For all \( {S}_{0} \in {H}_{0}^{1}\left( \Omega \right) \cap {H}^{2}\left( \Omega \right) \) and \( b \in C\left( {\bar{Q}}_{{T}_{e}}\right) \) with \( {b}_{t} \in C\left( {\bar{Q}}_{{T}_{e}}\right) \), there exists a global weak solution \( \left( {u, T, S}\right) \) of the problem \( \left( {0.7}\right) -... | The main result is obtained by using the method of compact method. We first study the modified initial boundary value problem and show that it has a Hölder continuous classical solution. Then we prove that the limit of the solution to the modified problem satisfies the problem (0.7)-(0.12). | No |
Theorem 1.1 Let \( \nu ,\kappa > 0,{T}_{e} > 0 \) ; suppose that the function \( b \in C\left( {\bar{Q}}_{{T}_{\mathrm{e}}}\right) \) has the derivative \( {b}_{t} \in C\left( {\bar{Q}}_{{T}_{e}}\right) \) and that the initial data \( {S}_{0} \in {C}^{2 + \alpha }\left( \Omega \right) \) satisfies \( {\left. {S}_{0}\ri... | Proof In [6] it is shown that the unique solution is given by\n\n\[ u\left( {t, x}\right) = {u}^{ * }\left( {{\int }_{a}^{x}\left( {{\chi }_{\kappa } * S}\right) \left( {t, y}\right) \mathrm{d}y - \frac{x - a}{d - a}{\int }_{a}^{d}\left( {{\chi }_{\kappa } * S}\right) \left( {t, y}\right) }\right) + \omega \left( {t, x... | Yes |
Lemma 2.1 There holds for any \( t \in \left\lbrack {0,{T}_{e}}\right\rbrack \) ,\n\n\[ \n{\begin{Vmatrix}{S}_{x}^{\kappa }\left( t\right) \end{Vmatrix}}^{2} + {c\nu }{\int }_{0}^{t}{\int }_{\Omega }{\left| {S}_{x}^{\kappa }\right| }_{\kappa }^{2}{\left| {S}_{xx}^{\kappa }\right| }^{2}\mathrm{\;d}x\mathrm{\;d}\tau \leq... | Proof Observe first that \( {S}_{tx}^{\kappa } \in {L}^{2}\left( {Q}_{{T}_{e}}\right) \) . By Theorem 1.1, we have that for almost all \( t \) ,\n\n\[ \n\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}{\begin{Vmatrix}{S}_{x}^{\kappa }\left( t\right) \end{Vmatrix}}^{2} = {\int }_{\Omega }{S}_{x}^{\kappa }\left( t\right) {S}_{x... | Yes |
Theorem 2.1 There holds for any \( T \in \left\lbrack {0,{T}_{e}}\right\rbrack, n \geq 2 \), and \( n \) being an even number,\n\n\[{\int }_{\Omega }{\left( {S}_{x}^{\kappa }\right) }^{n}\mathrm{\;d}x + {c\nu n}\left( {n - 1}\right) {\int }_{0}^{T}{\int }_{\Omega }{\left| {S}_{x}^{\kappa }\right| }_{\kappa }^{2}{\left|... | Proof We can use the mathematic induction method. From Lemma 2.1, we know that (2.7) holds for \( n = 2 \) . Thus we only prove that if (2.7) holds for any \( n \leq k - 2 \), then (2.7) holds when \( n = k \) . We rewrite (2.7) as\n\n\[{\int }_{\Omega }{\left( {S}_{x}^{\kappa }\right) }^{k}\mathrm{\;d}x + {c\nu k}\lef... | Yes |
Lemma 2.2 There holds\n\n\[ \n2{\int }_{0}^{t}{\int }_{\Omega }\frac{{\left( {S}_{t}^{\kappa }\right) }^{2}}{{\left| {S}_{x}^{\kappa }\right| }_{\kappa }^{2}}\mathrm{\;d}x\mathrm{\;d}\tau + {3c\nu }{\int }_{\Omega }{\left| {S}_{x}^{\kappa }\right| }^{2}\mathrm{\;d}x \leq C.\n\]\n\n\( \left( {2.14}\right) \) | Proof Multiplying (1.4) by \( 2\frac{{S}_{k}^{k}}{{\left| {S}_{x}^{k}\right| }_{k}^{2}} \) and integrating it with respect to \( x \), we obtain\n\n\[ \n2{\int }_{\Omega }\frac{{\left( {S}_{t}^{\kappa }\right) }^{2}}{{\left| {S}_{x}^{\kappa }\right| }_{\kappa }^{2}}\mathrm{\;d}x + {3c\nu }\frac{\partial }{\partial t}{\... | Yes |
Corollary 2.2 There hold for any \( t \in \left\lbrack {0,{T}_{e}}\right\rbrack \), and \( 1 \leq p < 2 \) ,\n\n\[ \n{\int }_{0}^{t}{\int }_{\Omega }{\left( {\left| {S}_{x}^{\kappa }\right| }_{\kappa }^{2}\left| {S}_{xt}^{\kappa }\right| \right) }^{p}\mathrm{\;d}x\mathrm{\;d}\tau \leq C \n\] | Remark It is not difficult to prove this corollary by Poincaré inequality and Sobolev embedding theorem. | No |
Theorem 2.1 If there are two constants \( {r}_{0} > 0 \) and \( c > 1 \) such that\n\n\[ \begin{Vmatrix}{\Phi \left( {s + {r}_{0}, s,\theta }\right) x}\end{Vmatrix} \geq c\parallel x\parallel ,\;\forall \left( {s, x,\theta }\right) \in {\mathbb{R}}_{ + } \times \mathcal{E}, \]\n\nthen the linear skew-evolution semiflow... | Proof Let \( M \geq 1,\omega > 0 \) be given by Definition 1.5 and \( v > 0 \) such that \( c = {\mathrm{e}}^{v{r}_{0}} \) . Let \( t \geq 0 \) . There exist \( n \in \mathbb{N} \) and \( l \in \left\lbrack {0,{r}_{0}}\right) \) such that \( t = n{r}_{0} + l \) . If \( \left( {x,\theta }\right) \in \mathcal{E} \) and \... | Yes |
Theorem 2.3 Let \( \pi = \left( {\Phi ,\sigma }\right) \) be an injective and strongly measurable linear skew-evolution semiflow on \( \mathcal{E} = X \times \Theta \) . Then \( \pi \) is uniformly exponentially expansive if and only if there are a Banach function space \( B \in \mathcal{B}\left( {\mathbb{R}}_{ + }\rig... | Proof Necessity. If \( \pi \) is uniformly exponentially expansive, then by Definition 1.6, there are \( K, v > 0 \) such that\n\n\[ {\int }_{0}^{\infty }\frac{1}{\parallel \Phi \left( {s + \tau, s,\theta }\right) x\parallel }\mathrm{d}\tau \leq \frac{1}{K}{\int }_{0}^{\infty }{\mathrm{e}}^{-{v\tau }}\mathrm{d}\tau = \... | Yes |
Corollary 2.1 Let \( \pi = \left( {\Phi ,\sigma }\right) \) be an injective and strongly measurable linear skew-evolution semiflow on \( \mathcal{E} = X \times \Theta \) . Then \( \pi \) is uniformly exponentially expansive if and only if there exists \( p \geq 1 \) such that\n\n\[ \mathop{\sup }\limits_{{\theta \in \T... | Proof Necessity. It is trivial.\n\nSufficiency. It is immediate by Theorem 2.3 for \( B \mathrel{\text{:=}} {L}^{p}\left( {{\mathbb{R}}_{ + },\mathbb{C}}\right) \) . | No |
Theorem 2.5 Let \( \pi = \left( {\Phi ,\sigma }\right) \) be an injective and strongly measurable linear skew-evolution semiflow on \( \mathcal{E} = X \times \Theta \) . Then \( \pi \) is uniformly exponentially expansive if and only if there are a Banach function space \( B \in \mathcal{B}\left( {\mathbb{R}}_{ + }\rig... | Proof Necessity. It is a simple verification for \( F\left( t\right) = t,\forall t \geq 0 \) and \( B = {L}^{1}\left( {{\mathbb{R}}_{ + },\mathbb{C}}\right) \) .\n\nSufficiency. Let \( \left( {x,\theta, s}\right) \in U \times \Theta \times {\mathbb{R}}_{ + } \), and \( M,\omega \) be given by Definition 1.5. We put \( ... | Yes |
Theorem 1.4 Assume that \( \alpha > 0. \) If \( \mathrm{{Tr}}\left( {{\Gamma }^{-1}{\Gamma }_{H}}\right) \sim O\left( {N}^{-\frac{3}{4}\alpha }\right) , \) \( \mathrm{{Tr}}\left( {\left( {\Gamma }^{-1}{\Gamma }_{H}\right) }^{2}\right) \sim O\left( {N}^{-\frac{3}{2}\alpha }\right) , \) \( r\left( {\left( {\Gamma }^{-1}{... | \[ \sqrt{\frac{{N}^{2\alpha }}{2}\operatorname{Tr}\left( {\left( {\Gamma }^{-1}{\Gamma }_{H}\right) }^{2}\right) - \frac{{N}^{\alpha }}{2}{\left( \operatorname{Tr}\left( {\Gamma }^{-1}{\Gamma }_{H}\right) \right) }^{2}}\left( {{\widehat{\tau }}^{2} - {\tau }^{2}}\right) \xrightarrow[]{\text{ law }}N\left( {0,1}\right) ... | Yes |
Lemma 3.1 Define\n\n\\[ \n{G}_{{N}^{\alpha }} \\mathrel{\\text{:=}} \\frac{1}{{a}^{2}}\\sqrt{\\frac{{N}^{\alpha }}{2}}\\left( {{\\widehat{a}}^{2} - {a}^{2}}\\right) = \\frac{1}{\\sqrt{2{N}^{\alpha }}}{\\left( {\\mathbf{B}}_{\\mathbf{t}} + \\tau {\\mathbf{S}}_{\\mathbf{t}}^{H}\\right) }^{\\mathrm{T}}{\\Gamma }^{-1}\\lef... | Proof Note that \\( {G}_{{N}^{\alpha }} \\) is a random variable living in the second Wiener chaos of an isonor-mal Gaussian process. Now from the definitions of \\( {G}_{{N}^{\alpha }} \\) and its Malliavin derivative, we get\n\n\\[ \nD{G}_{{N}^{\alpha }} = \\frac{2}{\\sqrt{2{N}^{\alpha }}}{\\left( D\\left( {\\mathbf{... | Yes |
Lemma 2.3 \( G \) has no \( \mathrm{S}\theta \) . | Proof Let \( H \) be an \( \mathrm{S}\theta \) in \( G \), and let \( L \) be a failing list assignment of \( G \) with \( \left| {L\left( v\right) }\right| \geq 3 \) for each vertex \( v \in V\left( G\right) \) . Setting \( {G}^{\prime } = G - V\left( H\right) \), the minimality of \( G \) implies that \( {G}^{\prime ... | Yes |
Lemma 1.3 \( {}^{\left\lbrack 3\right\rbrack } \) If \( n > 3, n = {3}^{\alpha }\mathop{\prod }\limits_{{i = 1}}^{k}{p}_{i}^{{\alpha }_{i}},\left( {{p}_{i},3}\right) = 1\left( {1 \leq i \leq k}\right) \), then | \[ {\varphi }_{3}\left( n\right) = \left\{ \begin{array}{ll} \frac{1}{3}\varphi \left( n\right) + \frac{{\left( -1\right) }^{\Omega \left( n\right) }{2}^{\omega \left( n\right) - \alpha - 1}}{3}, & \text{if }\alpha = 0\text{ or }1,\;{p}_{i} \equiv 2\left( {\;\operatorname{mod}\;3}\right) \left( {1 \leq i \leq k}\right)... | Yes |
Lemma 1.5 If \( n > 6, n = {2}^{\alpha }{3}^{\beta }\mathop{\prod }\limits_{{i = 1}}^{k}{p}_{i}^{{\alpha }_{i}} \), where \( \alpha ,\beta ,{\alpha }_{i} \geq 0,\left( {{p}_{i},6}\right) = 1\left( {1 \leq i \leq k}\right) \), then | Proof If \( n > 6 \), let \( n = {2}^{\alpha }{3}^{\beta }{n}_{1},\left( {6,{n}_{1}}\right) = 1 \) . If \( d \equiv 1\left( {\;\operatorname{mod}\;6}\right) \), then \( \left( \frac{-3}{d}\right) = 1,\left\lbrack \frac{d}{6}\right\rbrack = \frac{d - 1}{6} \) ; if \( d \equiv 5\left( {\;\operatorname{mod}\;6}\right) \),... | Yes |
Lemma 1.6 (a) \( {\varphi }_{4}\left( n\right) \left( {n \geq 1}\right) \) is odd if and only if\n\n<table><thead><tr><th>\( n \)</th><th>4</th><th>8</th><th>\( {p}^{k} \)</th><th>\( 2{p}^{k} \)</th><th>\( 4{p}^{k} \)</th></tr></thead><tr><td>conditions</td><td></td><td></td><td>\( p \equiv 5,7\left( {\;\operatorname{m... | Proof (a) By the definition of the generalized Euler function, it is easy to verify that \( {\varphi }_{4}\left( 1\right) = {\varphi }_{4}\left( 2\right) = {\varphi }_{4}\left( 3\right) = 0 \), and \( {\varphi }_{4}\left( 4\right) = 1 \) . For any \( n \geq 5 \), let \( n = {2}^{\alpha }\mathop{\prod }\limits_{{i = 1}}... | Yes |
Theorem 1.1 Let \( {m}_{1},{m}_{2},\cdots ,{m}_{n} \) be \( n \) non-negative integers. Then for \( {\lambda }_{1}{\lambda }_{2}\cdots {\lambda }_{n} \neq 1 \) , \[ \mathop{\prod }\limits_{{i = 1}}^{n}{H}_{{m}_{i}}\left( {{x}_{i} \mid {\lambda }_{i}}\right) = \frac{1}{{\lambda }_{1}{\lambda }_{2}\cdots {\lambda }_{n} -... | Proof By substituting \( \frac{1}{\lambda } \) for \( \lambda \) in (0.2), we have \[ \frac{\lambda - 1}{\lambda {\mathrm{e}}^{t} - 1}{\mathrm{e}}^{xt} = \mathop{\sum }\limits_{{n = 0}}^{\infty }{H}_{n}\left( {x \mid \frac{1}{\lambda }}\right) \frac{{t}^{n}}{n!}. \] (1.4) It follows from (1.4) that \[ \left\lbrack {\fr... | Yes |
Theorem 2.1 Let \( m \) be a positive integer and \( y = {x}_{1} + {x}_{2} + \cdots + {x}_{m} \) . Then for positive integer \( n \geq m \) , \[ \mathop{\sum }\limits_{\substack{{{k}_{1} + {k}_{2} + \cdots + {k}_{m} = n} \\ {{k}_{1},{k}_{2},\cdots ,{k}_{m} \geq 0} }}\left( \begin{matrix} n \\ {k}_{1},{k}_{2},\cdots ,{k... | Proof Replacing \( \lambda \) by \( - \frac{1}{\lambda } \) in (0.2) gives \[ \frac{\lambda + 1}{\lambda {\mathrm{e}}^{t} + 1}{\mathrm{e}}^{xt} = \mathop{\sum }\limits_{{n = 0}}^{\infty }{H}_{n}\left( {x \mid - \frac{1}{\lambda }}\right) \frac{{t}^{n}}{n!}. \] From the above, we discover \[ \mathop{\sum }\limits_{\subs... | Yes |
Theorem 2.2 Let \( m \) be a positive integer and \( y = {x}_{1} + {x}_{2} + \cdots + {x}_{m} \) . Then for positive integers \( n, r \) with \( n \geq m \) and \( 1 \leq r \leq m - 1 \) , | Proof It is easy from (0.2) to see that | No |
Theorem 3.1 Let \( {m}_{1},{m}_{2}\cdots ,{m}_{n} \) be \( n \) positive integers, and let \( {a}_{1},{a}_{2},\cdots ,{a}_{n} \) be \( n \) positive integers that are relatively prime in pairs. Then for positive integer \( m \) , | \[ \mathop{\sum }\limits_{{i = 1}}^{n}{a}_{i}^{{m}_{1} + {m}_{2} + \cdots + \widehat{{m}_{i}} + \cdots + {m}_{n} + m - n}{s}_{\overrightarrow{\mathbf{m}}, m}\left( {{a}_{1},{a}_{2},\cdots ,\widehat{{a}_{i}},\cdots ,{a}_{n};{a}_{i}}\right) \] \[ = m\mathop{\sum }\limits_{{i = 1}}^{n}\frac{{m}_{1} \cdot {m}_{2}\cdots \wi... | Yes |
Theorem 1 Let \( E \) be the homogeneous Cantor set determined by \( {\left\{ {n}_{k}\right\} }_{k \geq 1},{\left\{ {c}_{k}\right\} }_{k \geq 1} \), and \( \sup {n}_{k} < \infty \) . Then\n\n\[ \n{\mathcal{P}}^{s}\left( E\right) = {2}^{s}\mathop{\limsup }\limits_{{k \rightarrow \infty }}{n}_{k}{\left( \frac{1 - {c}_{k}... | ## 1 Proof of Theorem 1\n\nFor any \( \sigma = \left( {{\sigma }_{1},{\sigma }_{2},\cdots ,{\sigma }_{m}}\right) \in {D}_{m} \), when \( 0 < k \leq m \), we denote \( \sigma |k = \left( {{\sigma }_{1},{\sigma }_{2},\cdots ,{\sigma }_{k}}\right) \) . Let \( {x}_{k} \) be the length of \( k \) -th order basic interval, \... | Yes |
Lemma 1.1 Let \( \\sigma \\in {D}_{k},\\tau \\in {D}_{k + l}\\left( {l > 0}\\right) ,\\tau \\mid k = \\sigma \) . Then\n\n\[ \n\\frac{\\mu \\left( {I\\left( {\\sigma ,\\tau }\\right) }\\right) }{{\\left| I\\left( \\sigma ,\\tau \\right) \\right| }^{s}} \\geq \\min \\left\\{ {{d}_{k + 1}^{-1},{d}_{k + 2}^{-1},\\cdots ,{... | Proof For any \( \\sigma \\in {D}_{k},\\tau \\in {D}_{k + l} \), let \( a\\left( \\sigma \\right) \) be the left endpoint of \( {I}_{\\sigma } \), and \( b\\left( \\tau \\right) \) the right endpoint of \( {I}_{\\tau } \) (see Figure 1).\n\n be the probability measure defined by (1.2), and \( E \) be a homogeneous Cantor set such that \( 0 \leq {\mathcal{P}}^{s}\left( E\right) \leq \infty \) . Then \( {\mathcal{P}}^{s}\left( E\right) = \frac{1}{{\Theta }_{ * }^{s}\left( {\mu, x}\right) } \) for \( \mu \) almost all \( x \in E \), in... | Proof Define \( \nu = {\left( {\mathcal{P}}^{s}\left( E\right) \right) }^{-1} \cdot {\mathcal{P}}^{s}{\left| {}_{E}\text{, where }{\mathcal{P}}^{s}\right| }_{E}\left( A\right) = {\mathcal{P}}^{s}\left( {E \cap A}\right) \) for any \( A \subset \mathbb{R} \) . Then \( \nu \) and \( \mu \) coincide on each \( {I}_{\sigma... | Yes |
Lemma 1.5 Let \( E \) be the homogeneous Cantor set determined by \( {\left\{ {n}_{k}\right\} }_{k > 0} \) and \( {\left\{ {c}_{k}\right\} }_{k > 0} \) , \( \sup {n}_{k} < \infty \), and \( \mu \) be defined as in (1.2). Then\n\n(1) If \( 0 < d < \infty \), then \( {\Theta }_{ * }^{s}\left( {\mu, x}\right) \leq {2}^{-s... | Proof For any \( \sigma \in {D}_{k} \), let \( \tau \in {D}_{k} \) and \( {I}_{\tau } \) be the first \( k \) -th order basic interval to the left of \( {I}_{\sigma } \) . Since \( 0 < {n}_{k}{c}_{k} < 1 \), we have \( a\left( \sigma \right) - b\left( \tau \right) > 0 \) . Hence, there exists \( l = l\left( k\right) > ... | Yes |
Lemma 1.1 The solution \( x\left( t\right) = \left( {{x}_{1}\left( t\right) ,{x}_{2}\left( t\right) ,{x}_{3}\left( t\right) }\right) \) of the system (0.1) with initial value condition (0.2) must be positive, that is, \( {\varphi }_{i}\left( t\right) \geq 0 \) for all \( t \geq 0, i = 1,2,3 \) . | The proof of Lemma 1.1 is similar to that of \( \left\lbrack {3\text{, Lemma 1}}\right\rbrack \), so we omit it here. | No |
Lemma 1.2 If the system (0.1) satisfies\n\n( \( {\mathrm{H}}_{1} \) ) \( {a}_{10}{a}_{21} - {a}_{11}{a}_{20} > 0 \), and\n\n\( \left( {\mathrm{H}}_{2}\right) \;\frac{k\left( {{a}_{10}{a}_{11} - {a}_{20}{a}_{21}}\right) }{n{a}_{21} + \left( {{a}_{10}{a}_{11} - {a}_{20}{a}_{21}}\right) } - {a}_{30} > 0, \)\n\nthen the so... | Proof From the first equation of the system (0.1), it follows that\n\n\[ {x}_{1}^{\prime }\left( t\right) \leq {x}_{1}\left( t\right) \left( {{a}_{10} - {a}_{11}{x}_{1}\left( t\right) }\right) \]\n\nBy applying comparison principle, we have\n\n\[ \mathop{\limsup }\limits_{{t \rightarrow \infty }}{x}_{1}\left( t\right) ... | Yes |
Theorem 3.1 Suppose that the system (0.1) satisfies\n\n( \( {\mathrm{H}}_{7} \) ) \( {p}_{1} > 0 \) and \( {p}_{1}\left( {{p}_{2} + {q}_{2}}\right) - \left( {{p}_{3} + {q}_{1}}\right) > 0 \), and\n\n(H8) \( {p}_{3}^{2} - {q}_{1}^{2} < 0 \) .\n\nThen there exists a number \( {\tau }_{0} \) such that the positive equilib... | Proof If \( \tau = 0 \), then the characteristic equation (3.1) becomes\n\n\[ \n{\lambda }^{3} + {p}_{1}{\lambda }^{2} + \left( {{p}_{2} + {q}_{2}}\right) \lambda + {p}_{3} + {q}_{1} = 0.\n\]\n\nSince \( {p}_{1} > 0 \) and \( {p}_{1}\left( {{p}_{2} + {q}_{2}}\right) - \left( {{p}_{3} + {q}_{1}}\right) > 0 \), all the r... | Yes |
Proposition 1.1 Let \( A \) be a unital separable \( {C}^{ * } \) -algebra, let \( \left\{ {\mathcal{F}}_{n}\right\} \) be a sequence of finite subsets of \( A \), and let \( \{ {\sigma }_{n}\} : A \rightarrow A \) be automorphisms satisfying \( \mathop{\lim }\limits_{{n \rightarrow \infty }}\left( {{\sigma }_{n}^{-1} ... | Proof Since \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{\sigma }_{n}^{-1} \circ \alpha \circ {\sigma }_{n}\left( a\right) = \beta \left( a\right) \) for all \( a \in A \), there exists \( {k}_{n} > {k}_{n - 1} \) such that\n\n\[ \begin{Vmatrix}{{\sigma }_{{k}_{n}}^{-1} \circ \alpha \circ {\sigma }_{{k}_{n}}\left... | Yes |
Theorem 1.1 Let \( A \) be a unital simple A \( \mathbb{T} \) -algebra and let \( B \) be a unital simple \( {C}^{ * } \) -algebra with stable rank one. Then the following conditions are equivalent:\n\n(1) There are increasing sequences of finitely generated subgroups \( \left\{ {F}_{n}\right\} \subseteq {K}_{0}\left( ... | Proof Since \( A \) is a unital simple AT-algebra, by [7, Theorem 1] and [8], we may write \( A = \mathop{\lim }\limits_{{n \rightarrow \infty }}\left( {{A}_{n},{s}_{n}}\right) \), where \( {A}_{n} = {\bigoplus }_{i = 1}^{{r}_{n}}C\left( {\mathbb{T},{M}_{{k}_{{n}_{i}}}}\right) \) for some \( {r}_{n},{k}_{{n}_{1}},{k}_{... | Yes |
Theorem 1.2 Let \( A \) be an AF-algebra and let \( \alpha \) and \( \beta \) be two \( * \) -automorphisms such that \( {A}_{\alpha } \) and \( {A}_{\beta } \) are unital simple AT-algebras. We may write \( {A}_{\alpha } = \mathop{\lim }\limits_{{n \rightarrow \infty }}\left( {{A}_{n},{\phi }_{n}}\right) \) and \( {A}... | Proof \( \;\left( 2\right) \Leftrightarrow \left( 3\right) \) follows from Theorem 1.1, then it is left to show \( \left( 1\right) \Leftrightarrow \left( 3\right) \) .\n\n\( \left( 1\right) \Rightarrow \left( 3\right) \; \) Fixing \( n \), we may assume that \( {A}_{n} = {\bigoplus }_{i = 1}^{{r}_{n}}C\left( {\mathbb{T... | No |
Theorem 1.3 Let \( A \) be a unital simple AF-algebra and let \( \alpha \) be a \( * \) -automorphism on \( A \) . Suppose that \( \alpha \) has the tracial cyclic Rokhlin property. Suppose also that there is an integer \( J \geq 1 \) such that \( \left\lbrack {\alpha }^{J}\right\rbrack = \left\lbrack {\operatorname{id... | Proof By Lemma 1.5, \( A{ \times }_{\alpha }\mathbb{Z} \) is a unital simple \( {C}^{ * } \) -algebra with tracial rank zero. By Lemma 1.6, we have the exact sequence\n\n\[ 0 \rightarrow {K}_{1}\left( {A{ \times }_{\alpha }\mathbb{Z}}\right) \rightarrow {K}_{0}\left( A\right) \xrightarrow[]{{\mathrm{{id}}}_{*0} - {\alp... | Yes |
Lemma 1.7 Let \( A \) be a unital \( {C}^{ * } \) -algebra and let \( \alpha \) be a \( * \) -automorphism on \( A \) with the Rokhlin property. If \( \theta \) is another \( * \) -automorphism on \( A \) , then \( \theta \circ \alpha \circ {\theta }^{-1} \) has the Rokhlin property. | Proof For any \( k \in \mathbb{N} \), any finite subset \( \mathcal{F} \) of \( A \) and any \( \varepsilon > 0 \), since \( \alpha \) has the Rokhlin property, there are positive integers \( {k}_{1},{k}_{2},\cdots ,{k}_{m} \geq k \), and projections \( {e}_{i, j}, i = 1,2,\cdots, m \) , \( j = 0,1,\cdots ,{k}_{i} - 1 ... | Yes |
Theorem 1.4 Let \( A \) be a unital AF-algebra with finitely generated \( {K}_{0} \) group and let \( \alpha ,\beta \) be two \( * \) -automorphisms on \( A \) with the Rokhlin property. If \( {\left( {j}_{\beta }\right) }_{*0} : {K}_{0}\left( A\right) \rightarrow {K}_{0}\left( {A}_{\beta }\right) \) is an injective ho... | Proof \( \left( 1\right) \Rightarrow \left( 2\right) \) By the proof of Theorem 1.1.\n\n\( \left( 2\right) \Rightarrow \left( 1\right) \; \) Since \( {K}_{0}\left( A\right) \) is finitely generated, by Lemma 1.10, there are projections \( {p}_{1},{p}_{2},\cdots , \) \( {p}_{m} \) in \( A \) such that \( \left\{ {\left\... | Yes |
Corollary 1.1 In the situation of Theorem 1.4, if in addition, \( {\left( {j}_{\alpha }\right) }_{*0} : {K}_{0}\left( A\right) \rightarrow {K}_{0}\left( {A}_{\alpha }\right) \) is also an injective homomorphism, then \( \alpha \) and \( \beta \) are weakly approximately conjugate if and only if there exist two sequence... | \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\parallel {\phi }_{n} \circ {j}_{\alpha }\left( a\right) - {j}_{\beta } \circ {\Phi }_{n}\left( a\right) \parallel = 0\;\mathrm{{and}}\;\mathop{\lim }\limits_{{n \rightarrow \infty }}\parallel {\psi }_{n} \circ {j}_{\beta }\left( a\right) - {j}_{\alpha } \circ {\Psi }_{... | Yes |
Lemma 2.1 Let \( \\left( {M,\\varphi ,\\eta ,\\xi, g}\\right) \) be a conformal Sasakian manifold. Then the following formula holds:\n\n\[ \n\\operatorname{div}\\left( {{\\mathcal{L}}_{\\xi }g}\\right) \\left( \\xi \\right) = \\frac{1}{2}{\\Delta f} - \\frac{1}{2}\\xi \\left\\lbrack {\\omega \\left( \\xi \\right) }\\ri... | Proof Since \( M \) is a Sasakian manifold with respect to \( \\widetilde{g} \), Eq. (1.5) implies that \( \\widetilde{\\operatorname{Ric}}\\left( {\\widetilde{\\xi },\\widetilde{\\xi }}\\right) = \) \( {2n} \), thus it follows from (1.12) that\n\n\[ \n{2n}{\\mathrm{e}}^{f} = \\operatorname{Ric}\\left( {\\xi ,\\xi }\\r... | Yes |
Lemma 3.1 For every vector field \( X \in {TM} \), the following equation holds:\n\n\[ 0 = \varphi {V}^{ * } \land \left\lbrack {\left( {\frac{1}{4}\parallel \omega {\parallel }^{2} + {\mathrm{e}}^{f}}\right) {X}^{ * } - \frac{1}{4}\omega \left( X\right) \omega + \frac{1}{2}{\nabla }_{X}\omega }\right\rbrack \n+ {V}^{ ... | Proof Since \( \omega \) is closed and \( V \) is parallel, \n\n\[ {\nabla }_{V}\omega = {\mathcal{L}}_{V}\omega = \mathrm{d}\left( {V\lrcorner \omega }\right) = \mathrm{d}a. \] \n\n(3.2) \n\nIn fact, the relation (1.10) is equivalent to \n\n\[ \left( {{\nabla }_{X}\varphi }\right) {Y}^{ * } = \sqrt{{\mathrm{e}}^{f}}\l... | Yes |
Lemma 3.3 Under the assumption that \( {\omega }^{\sharp } \bot V \), we have \( b \neq 0 \) . | Otherwise, let us assume that \( b = 0 \). Applying Lemma 3.2 we get\n\n\[ \n{\nabla }_{\varphi V}\omega = - {2\alpha \varphi }{V}^{ * } + \frac{1}{2}{b\omega } + {2\gamma }\parallel {\varphi V}{\parallel }^{2}\varphi {V}^{ * }.\n\]\n\n(3.13)\n\n\n\nOn the other hand, with Eq. (3.3) we obtain\n\n\[ \n{\nabla }_{\varphi... | Yes |
Example 4.1 On \( {\mathbb{R}}^{3} \) there exists the standard Sasakian structure defined by\n\n\[ \widetilde{\eta } = \mathrm{d}z - y\mathrm{\;d}x,\;\widetilde{\xi } = \mathrm{d}z \]\n\n\[ \widetilde{\varphi } = \left( {\frac{\partial }{\partial x} + y\frac{\partial }{\partial z}}\right) \otimes \mathrm{d}y - \frac{\... | Thus \( g = {\mathrm{e}}^{-f}\widetilde{g} \) . We write \( {x}_{1} = x,{x}_{2} = y,{x}_{3} = z \) and \( {g}_{ij} = g\left( {\frac{\partial }{\partial {x}_{i}},\frac{\partial }{\partial {x}_{j}}}\right) \) for \( i, j \in \{ 1,2,3\} \) . So we\n\nhave\n\[ \left( {g}_{ij}\right) = {\mathrm{e}}^{-f}\left( \begin{matrix}... | Yes |
Lemma 2.1 (1) Let \( s < t \) belong to \( \left\lbrack {0,1}\right\rbrack \) . Then if \( H < \frac{1}{2} \), one has\n\n\[ \left| {\mathbb{E}\left( {{S}_{u}^{H}\left( {{S}_{t}^{H} - {S}_{s}^{H}}\right) }\right) }\right| \leq 2{\left( t - s\right) }^{2H} \]\n\nfor all \( u \in \left\lbrack {0,1}\right\rbrack \) . | Proof To prove (2.7), we just write\n\n\[ \mathbb{E}\left( {{S}_{u}^{H}\left( {{S}_{t}^{H} - {S}_{s}^{H}}\right) }\right) \]\n\n\[ = {t}^{2H} - {s}^{2H} + \frac{1}{2}\left\lbrack {{\left| s - u\right| }^{2H} - {\left| t - u\right| }^{2H}}\right\rbrack + \frac{1}{2}\left\lbrack {{\left| s + u\right| }^{2H} - {\left| t +... | Yes |
Lemma 2.2 Let \( s < t \) belong to \( \left\lbrack {0,1}\right\rbrack \) . (1) Assume that \( H > \frac{1}{2} \) . Then \[ \left| {\mathbb{E}\left( {{S}_{u}^{H}\left( {{S}_{t}^{H} - {S}_{s}^{H}}\right) }\right) }\right| \leq {5H}\left( {t - s}\right) \] for all \( u \in \left\lbrack {0,1}\right\rbrack \) . | Proof We have \[ \mathbb{E}\left( {{S}_{u}^{H}\left( {{S}_{t}^{H} - {S}_{s}^{H}}\right) }\right) = {t}^{2H} - {s}^{2H} + \frac{1}{2}\left\lbrack {{\left| s - u\right| }^{2H} - {\left| t - u\right| }^{2H}}\right\rbrack + \frac{1}{2}\left\lbrack {{\left| s + u\right| }^{2H} - {\left| t + u\right| }^{2H}}\right\rbrack . \... | Yes |
Proposition 2.2 Fix an integer \( q \geq 2 \) . Suppose that the function \( f \) satisfies Assumption \( \left( {\mathbf{H}}_{\mathbf{q}}\right) \) . If \( H \leq \frac{1}{2q} \), then\n\n\[ \mathbb{E}\left( {{V}_{n}^{\left( n\right) }{\left( f\right) }^{2}}\right) = O\left( {2}^{{2n} - {2nHq}}\right) . \]\n\n(2.16) | Proof Using the relation between the Hermite polynomial and multiple stochastic integrals, we have\n\n\[ {H}_{q}\left( {{2}^{nH}\Delta {S}_{k{2}^{-n}}^{H}}\right) = \frac{1}{q!}{2}^{nqH}{I}_{q}\left( {\delta }_{k{2}^{-n}}^{\otimes q}\right) . \]\n\nIn this way, we obtain\n\n\[ \mathbb{E}\left( {{V}_{n}^{\left( n\right)... | Yes |
Proposition 2.3 Let \( q \geq 2 \) be an integer, \( f : \mathbb{R} \mapsto \mathbb{R} \) be a function such that Assumption \( \left( {\mathbf{H}}_{\mathbf{q}}\right) \) holds and assume that \( H \in \left( {\frac{1}{4},\frac{3}{4}}\right) \) . Then as \( n \rightarrow \infty \) , \[ \left( {{S}^{H},{2}^{-\frac{n}{2}... | Proof For \( p = 2,3,\cdots, q \), we set \( {V}_{2, n}^{\left( p\right) }\left( f\right) = {2}^{-\frac{n}{2}}{V}_{n}^{\left( p\right) }\left( f\right) \) . We fix two integers \( m \geq n \), and decompose this sequence as follows, \[ {V}_{2, m}^{\left( p\right) }\left( f\right) = {A}_{p}^{\left( m, n\right) } + {B}_{... | Yes |
Corollary 2.1 Let \( q \geq 2 \) be an integer, \( f : \mathbb{R} \mapsto \mathbb{R} \) be a function such that Assumption \( \left( {\mathbf{H}}_{q}\right) \) holds. Then as \( n \rightarrow \infty \), (1) when \( H > \frac{1}{2} \) and \( q \) is odd, \[ {2}^{-{nH}}\mathop{\sum }\limits_{{k = 1}}^{{2}^{n}}f\left( {S}... | Proof For any \( q \geq 2 \), we have \[ {\left( {2}^{nH}\Delta {S}_{k{2}^{-n}}^{H}\right) }^{q} - {\mu }_{q} = \mathop{\sum }\limits_{{p = 1}}^{q}\left( \begin{array}{l} q \\ p \end{array}\right) {\mu }_{q - p}{2}^{nHp}{I}_{p}\left( {\delta }_{k{2}^{-n}}^{\otimes p}\right) = \mathop{\sum }\limits_{{p = 1}}^{q}p!\left(... | Yes |
Lemma 1.7 Let \( S \) be a completely \( {\mathcal{J}}^{ * } \) -simple semigroup. Then \( E\left( S\right) \times E\left( S\right) \subseteq \mathcal{D} \) . | Proof It follows from Lemmas 1.4 and 1.6. | No |
Lemma 1.9 Assume that sets \( {L}_{S} \) and \( {R}_{S} \) are defined as above. Then,\n\n(1) \( {L}_{S} \) is a left-zero semigroup and for all \( i \in I,{0}_{i} = {0}_{i,\varepsilon } \) ;\n\n(2) \( {R}_{S} \) is a right-zero semigroup and for all \( \lambda \in \Lambda ,{0}_{\lambda } = {0}_{\varepsilon ,\lambda } ... | Proof Since\n\n\[ \n{0}_{i} + {0}_{j} = {r}_{i} + {q}_{\varepsilon } + {r}_{j} + {q}_{\varepsilon } = {r}_{i} + {0}_{\varepsilon ,\varepsilon } + {q}_{\varepsilon } = {r}_{i} + {q}_{\varepsilon } = {0}_{i} \n\]\n\nfor all \( {0}_{i},{0}_{j} \in {L}_{S},{L}_{S} \) is a left-zero semigroup and \( {0}_{i} \) is the idempo... | Yes |
Lemma 1.10 The matrix \( P = \left( {p}_{\lambda, i}\right) \) is normalized. | Proof Since\n\n\[ \n{p}_{\lambda ,\varepsilon } = \left( {{r}_{\varepsilon } + {q}_{\lambda }}\right) + \left( {{r}_{\varepsilon } + {q}_{\varepsilon }}\right) = {r}_{\varepsilon } + \left( {{q}_{\lambda } + {r}_{\varepsilon }}\right) + {q}_{\varepsilon } = {0}_{\varepsilon ,\varepsilon } \n\] \n\nand \n\n\[ \n{p}_{\va... | Yes |
Lemma 2.2 Let \( \\left( {S,+, \\cdot }\\right) \) be an additively completely \( {\\mathcal{J}}^{ * } \) -simple semiring and \( x, y \\in S \) . If \( a \) is an additive regular element in \( {H}_{x}^{ * } \), then for any \( b \\in {H}_{y}^{ * }, - \\left( {ab}\\right) = \\left( {-a}\\right) b \) and \( - \\left( {... | Proof By the hypotheses, \( \\left( {-a}\\right) b \\in {H}_{ab}^{ * } \) . Since \( {ab} + \\left( {-a}\\right) b + {ab} = {ab} \) and \( \\left( {-a}\\right) b + {ab} + \\left( {-a}\\right) b = \\left( {-a}\\right) b,{ab} \) is an additive regular element in \( {H}_{ab}^{ * } \) . Then \( - \\left( {ab}\\right) = \\l... | Yes |
Lemma 2.3 For all \( \left( {i,\lambda }\right) \in I \times \Lambda ,{0}_{i,\lambda }{0}_{j,\mu } = {0}_{{ij},{\lambda \mu }} \) . | Proof Notice that \( {H}_{i,\lambda }^{ + } = {\overrightarrow{R}}_{i}^{ * } \cap {L}_{\lambda }^{ + },{H}_{j,\mu }^{ * } = {\overrightarrow{R}}_{j}^{ * } \cap {L}_{\mu }^{ * },{\overrightarrow{\mathcal{L}}}^{ * },{\overrightarrow{\mathcal{R}}}^{ * } \) and \( {\overrightarrow{\mathcal{H}}}^{ * } \) are congruences on ... | Yes |
Lemma 2.6 For all \( i, j, k \in I,\lambda ,\mu, v \in \Lambda \) and \( y, z \in T \) ,\n\n\( \left( 1\right) \;{yz} + \left( {-{p}_{{\mu \nu },{ik}}}\right) = {0}_{{\varepsilon \varepsilon },{\varepsilon \varepsilon }} + \left( {-{p}_{{\lambda \nu },{ik}}}\right) + {p}_{{\lambda \nu },{jk}} + \left( {-{p}_{{\varepsil... | Proof Since for all \( i, j, k \in I,\lambda ,\mu, v \in \Lambda \) and \( x, y, z \in T,{0}_{i} + x + {0}_{\lambda },{0}_{j} + y + {0}_{\mu },{0}_{k} + z + {0}_{\nu } \in \) \( S \), we have\n\n\[ \left\lbrack {\left( {{0}_{i} + x + {0}_{\lambda }}\right) + \left( {{0}_{j} + y + {0}_{\mu }}\right) }\right\rbrack \left... | Yes |
Lemma 2.7 The additively cancellative monoid \( {H}_{\varepsilon ,\varepsilon }^{ * } \) with the multiplication \( * \) is a distributive sandwich semiring with a sandwich element \( - {p}_{{\varepsilon \varepsilon },{\varepsilon \varepsilon }} \) . | Proof Let \( x, y, z \in {H}_{\varepsilon ,\varepsilon }^{ * } \) . First,\n\n\[ \left( {x * y}\right) * z = \left( {{0}_{\varepsilon ,\varepsilon } + {xy} + {p}_{{\varepsilon \varepsilon },{\varepsilon \varepsilon }}}\right) * z \]\n\n\[ = {0}_{\varepsilon ,\varepsilon } + \left( {{0}_{\varepsilon ,\varepsilon } + {xy... | Yes |
Theorem 2.1 Let \( \\left( {T,+, \\cdot }\\right) \) be a distributive sandwich semiring with a sandwich element \( t, I \) an additive left-zero semiring, \( \\Lambda \) an additive right-zero semiring, \( P = \\left( {p}_{\\lambda, i}\\right) \) an \( I \\times \\Lambda \) matrix whose every entry is a unit of \( T \... | Proof Assume that \( \\mathcal{M}\\left( {T;I,\\Lambda ;P}\\right) \) is a normalized Rees matrix semigroup with the multiplication which satisfies Conditions (1)-(3). Notice that \( T \) is a distributive sandwich semiring with the sandwich element \( - {p}_{{\\varepsilon \\varepsilon },{\\varepsilon \\varepsilon }} \... | Yes |
Lemma 1.1 For \( \lambda = {0.887041} \), we have\n\n\[ \operatorname{meas}\left( {\mathcal{E}}_{\lambda }\right) \ll {N}_{1}^{-\frac{3}{4} - {10}^{-{10}}}. \] | Proof This is [9, Lemma 2.1]. It is due to Heath-Brown and Puchta [1]. Let\n\n\[ R\left( {{N}_{1},{N}_{2}}\right) = \sum \log {p}_{1}\log {p}_{2}\cdots \log {p}_{8} \]\n\nbe the weighted number of solutions of ( \( {0.2} \) ) in \( \left( {{p}_{1},{p}_{2},\cdots ,{p}_{8},{v}_{1},{v}_{2},\cdots ,{v}_{k}}\right) \) with\... | Yes |
Lemma 1.4 We have\n\n\[ \n{\int }_{0}^{1}\left| {{S}^{4}\left( {\alpha ,{\mathcal{B}}_{i}}\right) {G}^{10}\left( {2\alpha }\right) }\right| \mathrm{d}\alpha \leq {24}\left( {{12} + \varepsilon }\right) {c}_{0}{\mathfrak{J}}^{ + }\left( 0\right) {N}_{i}{L}^{10} + O\left( {{N}_{i}{L}^{9 + \varepsilon }}\right) \]\n\nand\... | Proof See [11, Lemmas 4.4.1, 4.4.2 and 4.5.3]. | No |
Theorem 1 Let \( A = \left( {a}_{ij}\right) \) be an \( n \times n \) nonnegative irreducible matrix and \( c = \left( {{c}_{1},{c}_{2},\cdots }\right. \) , \( {\left. {c}_{n}\right) }^{\mathrm{T}} \) be any vector with positive components. For \( 1 \leq i \leq n \), take\n\n\[ \n{M}_{i} = \frac{1}{{c}_{i}}\mathop{\sum... | Proof Let \( U = \operatorname{diag}\left( {{c}_{1},{c}_{2},\cdots ,{c}_{n}}\right) \) . Obviously, \( A \) and \( B = {U}^{-1}{AU} \) have the same eigenvalues. Thus, applying \( \left\lbrack {9\text{, Theorem 2.1}}\right\rbrack \) to \( B \), one may obtain the required result. | Yes |
Theorem 2 Let \( A = \left( {a}_{ij}\right) \) be an \( n \times n \) nonnegative irreducible matrix and \( c = \left( {{c}_{1},{c}_{2},\cdots }\right. \) , \( {\left. {c}_{n}\right) }^{\mathrm{T}} \) be any vector with positive components. For \( 1 \leq i \leq n \), take\n\n\[ \n{M}_{i} = \frac{1}{{c}_{i}}\mathop{\sum... | Proof Let \( U = \operatorname{diag}\left( {{c}_{1},{c}_{2},\cdots ,{c}_{n}}\right) \) . Obviously, \( A \) and \( B = {U}^{-1}{AU} \) have the same eigenvalues. Thus, applying \( \left\lbrack {9\text{, Theorem 2.2}}\right\rbrack \) to \( B \), one may obtain the required result. | Yes |
Theorem 3 Let \( G \) be a connected graph on \( n \geq 2 \) vertices with generalized average degree \( {\left( {}^{\alpha }m\right) }_{1} \geq {\left( {}^{\alpha }m\right) }_{2} \geq \cdots \geq {\left( {}^{\alpha }m\right) }_{n} \) . Then for \( 1 \leq i \leq n \) ,\n\n\[ \rho \left( {A\left( G\right) }\right) \leq ... | Proof Let \( c = {\left( {d}_{1}^{\alpha },{d}_{2}^{\alpha },\cdots ,{d}_{n}^{\alpha }\right) }^{\mathrm{T}} \) in (7). Obviously, \( c \) is a positive vector as \( G \) is connected. Now apply Theorem 1 to \( A\left( G\right) \) . Notice that \( {M}_{i} = {\left( {}^{\alpha }m\right) }_{i}, M = 0, N = \mathop{\max }\... | Yes |
Theorem 4 Let \( G \) be a connected graph on \( n \geq 2 \) vertices with \( {\left( {}^{\alpha }m\right) }_{1} + {d}_{1} \geq {\left( {}^{\alpha }m\right) }_{2} + {d}_{2} \geq \cdots \geq {\left( {}^{\alpha }m\right) }_{n} + {d}_{n} \). Then for \( 1 \leq i \leq n \), \[ \rho \left( {Q\left( G\right) }\right) \leq \f... | Proof Let \( c = {\left( {d}_{1}^{\alpha },{d}_{2}^{\alpha },\cdots ,{d}_{n}^{\alpha }\right) }^{\mathrm{T}} \). Obviously, \( c \) is a positive vector as \( G \) is connected. Now apply Theorem 1 to \( Q\left( G\right) \). Notice that \( {M}_{i} = {\left( {}^{\alpha }m\right) }_{i} + {d}_{i}, M = \Delta, N = \mathop{... | Yes |
Corollary 5 Let \( G \) be a connected graph on \( n \geq 2 \) vertices with \( {m}_{1} + {d}_{1} \geq {m}_{2} + {d}_{2} \geq \) \( \cdots \geq {m}_{n} + {d}_{n} \) . Then for \( 1 \leq i \leq n \) , \n\n\[ \n\rho \left( {Q\left( G\right) }\right) \leq \frac{{m}_{i} + {d}_{i} + \Delta - N + \sqrt{{\left( {m}_{i} + {d}_... | Proof Taking \( \alpha = 1 \) in (14), the required result follows as \( {\left( {}^{1}m\right) }_{i} = {m}_{i} \) . | Yes |
Theorem 5 Let \( G \) be a connected graph on \( n \geq 2 \) vertices with generalized average transmissions \( {\left( {}^{\alpha }\mathcal{M}\right) }_{1} \geq {\left( {}^{\alpha }\mathcal{M}\right) }_{2} \geq \cdots \geq {\left( {}^{\alpha }\mathcal{M}\right) }_{n} \) . Then for \( 1 \leq i \leq n \) ,\n\n\[ \rho \l... | Proof Let \( c = {\left( {\mathcal{D}}_{1}^{\alpha },{\mathcal{D}}_{2}^{\alpha },\cdots ,{\mathcal{D}}_{n}^{\alpha }\right) }^{\mathrm{T}} \) . Apply Theorem 1 to \( \mathcal{D}\left( G\right) \) . Notice that \( {M}_{i} = {\left( {}^{\alpha }\mathcal{M}\right) }_{i} \) , \( M = 0, N = \mathop{\max }\limits_{{1 \leq i,... | Yes |
Corollary 7 Let \( G \) be a connected graph on \( n \geq 2 \) vertices with average transmissions \( {\mathcal{M}}_{1} \geq {\mathcal{M}}_{2} \geq \cdots \geq {\mathcal{M}}_{n} \) . Then for \( 1 \leq i \leq n \) , | \[ \rho \left( {\mathcal{D}\left( G\right) }\right) \leq \frac{{\mathcal{M}}_{i} - N + \sqrt{{\left( {\mathcal{M}}_{i} + N\right) }^{2} + {4N}\mathop{\sum }\limits_{{k = 1}}^{{i - 1}}\left( {{\mathcal{M}}_{k} - {\mathcal{M}}_{i}}\right) }}{2}, \] (17) where \( N = \mathop{\max }\limits_{{1 \leq i, j \leq n}}\left\{ \fr... | Yes |
Theorem 6 Let \( G \) be a connected graph on \( n \geq 2 \) vertices with generalized average transmissions \( {\left( {}^{\alpha }\mathcal{M}\right) }_{1} \geq {\left( {}^{\alpha }\mathcal{M}\right) }_{2} \geq \cdots \geq {\left( {}^{\alpha }\mathcal{M}\right) }_{n} \) . Then\n\n\[ \rho \left( {\mathcal{D}\left( G\ri... | Proof Apply Theorem 2 to \( \mathcal{D}\left( G\right) \) . The rest proof is similar to that of Theorem 5 and omitted. | No |
Theorem 8 Let \( G \) be a connected graph on \( n \geq 2 \) vertices with \( {\left( {}^{\alpha }\mathcal{M}\right) }_{1} + {\mathcal{D}}_{1} \geq {\left( {}^{\alpha }\mathcal{M}\right) }_{2} + \) \( {D}_{2} \geq \cdots \geq {\left( {}^{\alpha }\mathcal{M}\right) }_{n} + {\mathcal{D}}_{n} \) . Then\n\n\[ \rho \left( {... | where \( S = \mathop{\min }\limits_{{1 \leq i \leq n}}\left\{ {\mathcal{D}}_{i}\right\}, T = \mathop{\min }\limits_{{1 \leq i \neq j \leq n}}\left\{ \frac{{d}_{ij}{\mathcal{D}}_{i}^{\alpha }}{{\mathcal{D}}_{i}^{\alpha }}\right\} \) . Moreover, the equality in (20) holds if and only if \( {\left( {}^{\alpha }\mathcal{M}... | Yes |
Theorem 1.3 Let \( G \) be a maximally triangle-free graph of order \( n \geq 2 \). Then the size \( m \) of \( G \) satisfies\n\n\[ m \leq \left\{ \begin{array}{ll} \lfloor \frac{{\left( \frac{{2n} + 1}{3}\lfloor \frac{{2\delta }\left( G\right) }{3}\rfloor \right) }^{2}}{4}\rfloor + \frac{{n}^{2} - {3n} + {6n\delta }\... | Proof First suppose that \( t \leq \left\lfloor \frac{n}{3}\right\rfloor - \left\lfloor \frac{{2\delta }\left( G\right) }{3}\right\rfloor \), the inequality (1.1) is still valid here. Then \( m \leq f\left( t\right) \leq f\left( {\left\lfloor \frac{n}{3}\right\rfloor - \left\lfloor \frac{{2\delta }\left( G\right) }{3}\... | Yes |
Theorem 1.4 Let \( G \) be a connected maximally irregular graph, \( \delta \left( G\right) \geq 1 \) . Then \( m \geq \) \( \frac{t\left( {t - 1}\right) }{4} + \frac{n}{2}\delta \left( G\right) . | Proof Since \( G \) is maximally irregular, then \( 2 \leq t = \Delta \left( G\right) - \delta \left( G\right) + 1 \) . Let \( A = \left\{ {{v}_{1},{v}_{2},\cdots }\right. \) , \( \left. {{v}_{i} \mid {d}_{G}\left( {v}_{i}\right) = \delta \left( G\right) + i - 1, i = 1,2,\cdots, t}\right\} \), and for any \( v \in V - ... | Yes |
Proposition 1.2 \( {}^{\left\lbrack 5\right\rbrack } \) Let \( q \) be a power of the prime and let \( N = \left\{ {{\alpha }_{i} = {\alpha }^{{q}^{i}} \mid i = 0,1,\cdots, n - }\right. \) 1\} be a normal basis of \( {\mathbb{F}}_{{q}^{n}} \) over \( {\mathbb{F}}_{q} \) . Suppose that the dual basis \( B \) of \( N \) ... | \[ {h}_{i, j} = \left\{ \begin{array}{ll} a + b{t}_{i,0}, & j = 0, \\ - a + b{t}_{0, j}, & i = 0, j = 1,2,\cdots, n - 1, \\ a + b{t}_{i, i}, & i = j = 1,2,\cdots, n - 1, \\ b{t}_{i, j}, & i = 1,2,\cdots, n - 1, j \neq 0, i. \end{array}\right. \] | Yes |
Lemma 2.1 \( {}^{\left\lbrack 8\right\rbrack }\;\left( 1\right) \) Let \( N \) be a Type \( \left( {n,1}\right) \) optimal normal basis of \( {\mathbb{F}}_{{q}^{n}} \) over \( {\mathbb{F}}_{q} \) generated by \( \alpha \), and \( {T}_{\alpha } = \left( {t}_{i, j}\right) \) be the multiplication table of \( N \) . Then ... | \[ {t}_{i, j} = \left\{ \begin{array}{ll} 1, & {q}^{j} \equiv {q}^{i} + 1\left( {{\;\operatorname{mod}\;n} + 1}\right) , \\ 0, & \text{ otherwise. } \end{array}\right. \] | No |
Lemma 3.1 Let \( {M}^{n} \) be an \( n \) -dimensional Lagrangian submanifold in a complex space form \( {\widetilde{M}}^{n}\left( {4\widetilde{c}}\right), n \geq 3, x \in {M}^{n} \) . Then we have\n\n\[ \rho \left( x\right) = {\left\lbrack {\widehat{\delta }}_{c}\left( n - 1\right) \right\rbrack }_{x} + \widetilde{c} ... | Proof If the equality case of (0.3) holds, then inequalities in (2.3),(2.9),(2.10) and (2.13) become equalities. Thus \( {h}_{ij}^{{m}^{ * }} = 0 \), if \( i, j, m \) are all distinct, and there exist functions \( {a}_{1},{a}_{2},\cdots ,{a}_{n} \) such that\n\n\[ {h}_{11}^{{n}^{ * }} = {h}_{22}^{{n}^{ * }} = \cdots = ... | Yes |
Theorem 2.1 Under Assumptions 1.1 and 1.2, there exists at least one solution \( (X, Y \) , \( Z, K) \in {S}^{2} \otimes {S}^{2} \otimes {M}^{2} \otimes {S}_{ci}^{2} \) for RFBSDE (0.1). | To prove it, we need the following result on the approximation of lower semi-continuous functions by Lipschitz ones, first used by Fan and Jiang \( {}^{\left\lbrack 5\right\rbrack } \) in the study of the existence of a solution for BSDEs.\n\nLemma 2.1 Let \( f : { | No |
Theorem 0.2 (Simon's optimal constants) For (0.2) we have the following optimal constants \( \left( {d \geq 3}\right) \) : | [B] \( C = \sqrt{\frac{\pi }{d - 2}} \) for \( \mathfrak{A} = {\left| x\right| }^{-1} \) (i.e., \( a = 0 \) );\n\n\( \left\lbrack C\right\rbrack C = \sqrt{\frac{\pi }{2}} \) for \( \mathfrak{A} = \langle x{\rangle }^{-1}{\left| {\nabla }_{x}\right| }^{\frac{1}{2}}, \) | Yes |
Theorem 0.3 For (0.3) we have the following optimal constants:\n\n(1) \( C = {12}^{-\frac{1}{12}} \) for \( \left( {p, q, d}\right) = \left( {6,6,1}\right) \) ;\n\n(2) \( C = {2}^{-\frac{1}{2}} \) for \( \left( {p, q, d}\right) = \left( {4,4,2}\right) \) ;\n\n(3) \( C = {2}^{-\frac{1}{4}} \) for \( \left( {p, q, d}\rig... | In each case, the isotropic gaussian initial data \( f\left( x\right) = {\mathrm{e}}^{-{\left| x\right| }^{2}} \) is a maximizer and, up to certain invariances, there are no further maximizers. | Yes |
Theorem 1. \( {2}^{\left\lbrack {30}\right\rbrack }\; \) The optimal constant for Type [B] with \( 1 - \frac{d}{2} < a < \frac{1}{2} \) is | \[ C = {\left( \pi {2}^{{2a} - 1}\frac{\Gamma \left( {1 - {2a}}\right) \Gamma \left( {\frac{d}{2} + a - 1}\right) }{\Gamma {\left( 1 - a\right) }^{2}\Gamma \left( {\frac{d}{2} - a}\right) }\right) }^{\frac{1}{2}}, \] and the maximizers are the radial initial data which belong to \( {L}^{2}\left( {\mathbb{R}}^{d}\right)... | Yes |
Theorem 1.5 \( {}^{\left\lbrack 6\right\rbrack } \) Suppose \( d \geq 6 \) and let \( 1 - \frac{d}{2} < {a}^{ * } < {a}_{ * } < \frac{1}{2} \) be the unique solutions of the equations \( {d}^{\frac{1}{2} - {a}_{ * }}\left( {\frac{d}{2} - 1 + {a}^{ * }}\right) = \frac{d}{2} - {a}_{ * } \) and \( \Gamma \left( {\frac{d}{... | (1) For \( 1 - \frac{d}{2} < a \leq {a}^{ * } \), the optimal constants in (1.1) are given by\n\n\[ \nc = {\left( \pi {2}^{{2a} - 1}{d}^{\frac{1}{2} - a}\frac{\Gamma \left( {1 - {2a}}\right) \Gamma \left( {\frac{d}{2} + a}\right) }{\Gamma {\left( 1 - a\right) }^{2}\Gamma \left( {\frac{d}{2} + 1 - a}\right) }\right) }^{... | Yes |
Lemma 3.1 Let \( X, Y \) be Hilbert spaces and \( T : X \rightarrow Y \) a bounded operator. If \( f \) is a maximizer, i.e., \[ \parallel {Tf}{\parallel }_{X} = \parallel T{\parallel }_{X \rightarrow Y} \leq \parallel f{\parallel }_{Y}\;\left( {f \neq 0}\right) , \] then \( f \) is an eigenfunction of the operator \( ... | In fact, by the Schwarz inequality, we have \[ {\begin{Vmatrix}Tf\end{Vmatrix}}_{Y}^{2} = {\left( Tf, Tf\right) }_{Y} = {\left( {T}^{ * }Tf, f\right) }_{X} \] \[ \leq {\begin{Vmatrix}{T}^{ * }Tf\end{Vmatrix}}_{X}{\begin{Vmatrix}f\end{Vmatrix}}_{X} \leq {\begin{Vmatrix}{T}^{ * }T\end{Vmatrix}}_{X \rightarrow X}{\begin{V... | Yes |
As shown in Figure 1 (a), we calculate the illumination direction for a selected region in the egg image. The gradient skewness tensor \( \mathcal{G} \) with \( \mathbf{D} \) is given by \( {\mu }_{3,0} = \) \( {18.4811},\;{\mu }_{2,1} = - {19.0427},\;{\mu }_{1,2} = {13.3917},\;{\mu }_{0,3} = - {15.2338},\;{\mu }_{2,0}... | \[ \varphi \left( \lambda \right) = - {10}^{10}\left( {{0.1652}{\lambda }^{6} - {1.7208}{\lambda }^{4} + {1.3299}{\lambda }^{2} - {0.2674}}\right) \] with roots \( \lambda = {3.0973}, - {3.0973},{0.6414}, - {0.6414},{0.6403}, - {0.6403} \) . Here, the largest D-eigenvalue is \( \lambda = {3.0973} \) with corresponding ... | Yes |
Example 3.1 \( {}^{\left\lbrack {101}\right\rbrack } \) . Let \( \mathcal{H} \in {\mathbb{R}}^{{90} \times {90} \times 3} \) be given by\n\n\[ \n{h}_{ijk} = \left\{ \begin{array}{ll} i + j - {60}\left( {k - 1}\right) , & {30k} - {29} \leq i, j \leq {30k} \\ 0, & \text{ otherwise. } \end{array}\right. \n\] | A rank- \( 6\mathrm{{CP}} \) decomposition of \( \mathcal{H} \), as shown in Figure 3(a), achieves a larger fit 0.93 . But it is not a compact representation, since the entries of factor matrices are scattered everywhere, such as the mode-1 factor matrix shown in Figure 4(a). Alternatively, as shown in Figure 3(b), a t... | Yes |
Theorem 3.1 \( {}^{\left\lbrack {50}\right\rbrack }\; \) Any tensor \( \;\mathcal{A} \in {\mathbb{R}}^{{n}_{1} \times {n}_{2} \times \cdots \times {n}_{m}}\; \) can be written as (20), in which (i) \( \;{\mathbf{X}}^{\left( k\right) } \) is an orthonormal \( \;{n}_{k} \times {n}_{k}\; \) matrix for \( \;k \in \llbracke... | The HOSVD of \( \mathcal{A} \) can be computed by executing singular value decomposition on each mode- \( k \) matrix \( {\mathbf{A}}_{\left( k\right) } \) . (i) For \( k = 1 \) to \( m \), compute the left singular matrix of \( {\mathbf{A}}_{\left( k\right) } \), and set it to \( {\mathbf{X}}^{\left( k\right) } \) ; (... | Yes |
Theorem 1.2 As \( n \rightarrow \infty \), we have for each \( x \in \mathbb{R} \) ,\n\n\[ \n{P}_{u, n}\left( {\frac{c}{\sqrt{n}}{\lambda }_{1} - \log \frac{\sqrt{n}}{c} \leq x}\right) \rightarrow {\mathrm{e}}^{-{\mathrm{e}}^{-x}}.\n\] | Note that the limit distribution in the right hand side of (1.2) is the famous Gumbel distribution, which appears widely in the study of extremal statistics for independent random variables. In particular, let \( {\xi }_{n}, n \geq 1 \) be a sequence of independent standard normal random variables, and denote \( {\xi }... | Yes |
Theorem 1.3 As \( n \rightarrow \infty \), we have for \( {x}_{1} > {x}_{2} > \cdots > {x}_{m} \) , | \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}{P}_{u, n}\left( {\frac{c}{\sqrt{n}}{\lambda }_{i} - \log \frac{\sqrt{n}}{c} \leq {x}_{i},1 \leq i \leq m}\right) = {\int }_{-\infty }^{{x}_{1}}{\int }_{-\infty }^{{x}_{2}}\cdots {\int }_{-\infty }^{{x}_{m}}{p}_{0}\left( {u}_{1}\right) \mathop{\prod }\limits_{{i = 2}}^{... | Yes |
Theorem 1.5 Assume that \( {t}_{n}, n \geq 1 \) is a sequence of positive numbers such that\n\n\[ \n{t}_{n} \rightarrow \infty ,\;{t}_{n} - \frac{1}{2c}\log n \rightarrow - \infty .\n\]\n\n(1.5)\n\nLet\n\n\[ \n{X}_{n}\left( {t}_{n}\right) = \frac{{\mathrm{e}}^{\frac{c{t}_{n}}{2}}}{{n}^{\frac{1}{4}}}\left( {{\varphi }_{... | Note that \( \frac{{\mathrm{e}}^{\frac{c{t}_{n}}{2}}}{{n}^{\frac{1}{4}}} \) goes to zero under the assumption (1.5). | No |
Theorem 1.6 (i) With probability \( 1, Y\left( t\right) \) is uniformly continuous on \( \left\lbrack {0,1}\right\rbrack \) . | Proof Begin with the continuity of sample paths of \( Y\left( t\right) \) . Note\n\n\[ E{\left( Y\left( t\right) - Y\left( s\right) \right) }^{2} = {EY}{\left( t\right) }^{2} - {2EY}\left( s\right) Y\left( t\right) + {EY}{\left( s\right) }^{2} \]\n\n\[ = \frac{1}{c}\left( {t - s - {\left( t - s\right) }^{2} - \frac{1}{... | Yes |
Theorem 1.8 Under \( \left( {{\mathcal{P}}_{n},{P}_{u, n}}\right) \) , \[ \frac{1}{{n}^{\frac{3}{4}}}\left( {\log {d}_{\lambda } - \frac{1}{2}n\log n + {An}}\right) \overset{\mathrm{d}}{ \rightarrow }N\left( {0,{\sigma }_{d}^{2}}\right) \] where \( A \) and \( {\sigma }_{d}^{2} \) are given by \[ A = 1 - \log c + \frac... | The complete proof is rather lengthy since it involves precise tail estimates of \( {\lambda }_{k}^{\prime } - {\lambda }_{l}^{\prime } \) for all pairs of \( \left( {k, l}\right) \) . Below we only outline the main steps. It follows easily from (0.8) and (0.7) that \[ {d}_{\lambda } = {d}_{{\lambda }^{\prime }} = n!\f... | Yes |
Lemma 1.1\n\n\\[ \log f\\left( \\mu \\right) = - \\frac{1}{2}n\\log n - n{\\int }_{0}^{\infty }\\frac{t\\log t}{{\\mathrm{e}}^{ct} - 1} + {o}_{p}\\left( {n}^{\\frac{3}{4}}\\right) . \\] | The linearized weighted sum \\( {T}_{n} \\) admits an integral representation up to a negligible error term:\n\n\\[ {T}_{n} = \\frac{{n}^{\\frac{3}{4}}}{c}{\\int }_{0}^{1}\\frac{v\\left( {-\\log \\left( {1 - t}\\right) }\\right) }{t\\left( {1 - t}\\right) }{Y}_{n}\\left( t\\right) \\mathrm{d}t + O\\left( {n}^{\\frac{3}... | No |
Lemma 1.2\n\n\\[ \frac{{T}_{n}}{{n}^{\frac{3}{4}}}\overset{\mathrm{d}}{ \rightarrow }N\left( {0,{\sigma }_{T}^{2}}\right) \\]\n\nwhere \\( {\sigma }_{T}^{2} = {\sigma }_{d}^{2} \\) is given by (1.8). | In combination, we now conclude the proof of Theorem 1.8. | No |
Lemma 1.3 Under \( {Q}_{q},{r}_{1},{r}_{2},\cdots \) is a sequence of independent geometric random variables. In particular, we have\n\n\[ \n{Q}_{q}\left( {\lambda \in \mathcal{P} : {r}_{k} = j}\right) = \left( {1 - {q}^{k}}\right) {q}^{jk},\;j = 0,1,2,\cdots .\n\] | The proof is easy. Indeed, note \( \lambda = \left( {{1}^{{r}_{1}},{2}^{{r}_{2}},\cdots }\right) \), so \( \left| \lambda \right| = \mathop{\sum }\limits_{{k = 1}}^{\infty }k{r}_{k} \) . Thus we have\n\n\[ \n{Q}_{q}\left( \lambda \right) = \mathop{\prod }\limits_{{k = 1}}^{\infty }{q}^{k{r}_{k}}\left( {1 - {q}^{k}}\rig... | Yes |
Theorem 1.9 Under \( \left( {\mathcal{P},{Q}_{{q}_{n}}}\right) ,\left| \lambda \right| \) normally concentrates around \( n \) . Namely\n\n\[ \frac{\left| \lambda \right| - n}{{n}^{\frac{3}{4}}}\overset{\mathrm{d}}{ \rightarrow }N\left( {0,\frac{4}{{c}^{2}}}\right) \] | Moreover, we have the local limit theorem\n\n\[ {Q}_{{q}_{n}}\left( {\left| \lambda \right| = n}\right) = \frac{1}{{96}^{\frac{1}{4}}{n}^{\frac{3}{4}}}\left( {1 + o\left( 1\right) }\right) . \]\n\nBy virtue of (1.9) and (1.10), it suffices to prove\n\n\[ \frac{\left| \lambda \right| - {\mu }_{n}}{{\sigma }_{n}}\overset... | Yes |
Theorem 1.12 For \( {x}_{1} > {x}_{2} > \cdots > {x}_{m} \) , \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}{Q}_{{q}_{n}}\left( {\frac{c}{\sqrt{n}}{\lambda }_{i} - \log \frac{\sqrt{n}}{c} \leq {x}_{i},1 \leq i \leq m}\right) \] | \[ = {\int }_{-\infty }^{{x}_{1}}{\int }_{-\infty }^{{x}_{2}}\cdots {\int }_{-\infty }^{{x}_{m}}{p}_{0}\left( {u}_{1}\right) \mathop{\prod }\limits_{{i = 2}}^{m}p\left( {{u}_{i - 1},{u}_{i}}\right) \mathrm{d}{u}_{1}\mathrm{\;d}{u}_{2}\cdots \mathrm{d}{u}_{m}, \] where \( {p}_{0} \) and \( p \) are as in Theorem 1.3. | Yes |
Lemma 1.6 Assume that \( {K}_{n} \) is a sequence of positive integers such that\n\n\[ \mathop{\sum }\limits_{{k \in {K}_{n}}}\frac{{k}^{2}{q}_{n}^{k}}{{\left( 1 - {q}_{n}^{k}\right) }^{2}} = o\left( {n}^{\frac{3}{2}}\right) \]\n\nThen we have for \( {W}_{n} : \lambda \rightarrow \left( {{r}_{k}\left( \lambda \right), ... | Now we are ready to prove Theorems 1.2, 1.3 and 1.5. We only outline the proof of Theorem 1.2 since the others are similar. Let \( {A}_{n, x} = \frac{\sqrt{n}}{c}\left( {\log \frac{\sqrt{n}}{c} + x}\right) \). Define \( {K}_{n} = \left\{ {k : k \geq {A}_{n, x}}\right\} \), then it is easy to see\n\n\[ \mathop{\sum }\li... | No |
Lemma 1.7 For \( {u}_{1},{u}_{2} \in \mathbb{R} \) , \[ {E}_{u, n}\exp \left( {\frac{{u}_{1}}{{n}^{\frac{1}{4}}}\mathop{\sum }\limits_{{{k}_{1} \leq k < {k}_{2}}}\left( {{r}_{k} - {E}_{{q}_{n}}{r}_{k}}\right) + \frac{{u}_{2}}{{n}^{\frac{1}{4}}}\mathop{\sum }\limits_{{k \geq {k}_{2}}}\left( {{r}_{k} - {E}_{{q}_{n}}{r}_{... | \[ = \exp \left( {\frac{{u}_{1}^{2}}{2{n}^{\frac{1}{2}}}{\sigma }^{2}\left( {{k}_{1},{k}_{2}}\right) + \frac{{u}_{2}^{2}}{2{n}^{\frac{1}{2}}}{\sigma }^{2}\left( {k}_{2}\right) - \frac{c}{4{n}^{2}}{\left( {u}_{1}s\left( {k}_{1},{k}_{2}\right) + {u}_{2}s\left( {k}_{2}\right) \right) }^{2}}\right) \cdot \left( {1 + O\left... | Yes |
Lemma 2.2 Let \( {\mathcal{X}}_{n}, n \geq 1 \) be a sequence of determinantal point processes with kernel \( {K}_{{\mathcal{X}}_{n}} \) on \( \mathbb{R} \), let \( {I}_{n}, n \geq 1 \) be a sequence of intervals on \( \mathbb{R} \). Assume that \( {K}_{{\mathcal{X}}_{n}} \cdot {\mathbf{1}}_{{I}_{n}} \) defines an inte... | \[ \frac{\# {I}_{n} - E\# {I}_{n}}{\sqrt{\operatorname{Var}\left( {\# {I}_{n}}\right) }}\overset{\mathrm{d}}{ \rightarrow }N\left( {0,1}\right) \] | Yes |
Lemma 2.4 Fix \( 0 < x < 2, z \in \mathbb{R} \) and let \( {I}_{\theta } = \left\lbrack {{a}_{\theta },\infty }\right) \) with \( {a}_{\theta } \) given by (2.8). Then as \( \theta \rightarrow \infty \)\n\n\[
{E}_{p,\theta }\left( {\# {I}_{\theta }}\right) = \sqrt{\theta }x - \frac{z}{2\pi }\sqrt{\log \theta } + O\left... | This is a very technical lemma, whose proof is based on Lemma 2.3, see [7]. | No |
Theorem 2.10 Define\n\n\[ \n{\mathcal{X}}_{n, k}\left( \lambda \right) = {\int }_{-\infty }^{\infty }{u}_{k}\left( u\right) \left( {{\Psi }_{\lambda }\left( {\sqrt{n}u}\right) - \sqrt{n}\Omega \left( u\right) }\right) \mathrm{d}u,\;\lambda \in {\mathcal{P}}_{n}. \n\] \n\nThen under \( \left( {{\mathcal{P}}_{n},{P}_{p, ... | Kerov \( {}^{\left\lbrack {29}\right\rbrack } \) first outlined the main idea of the proof of Theorem 2.10, but he did not have time to finish it because of an unexpected death. A complete and rigorous proof was later given by Ivanov and Olshanski \( {}^{\left\lbrack {25}\right\rbrack } \) in 2002. The proof uses essen... | No |
Lemma 2.6 The function \( {\widetilde{p}}_{2},{\widetilde{p}}_{3},\cdots \) belong to the algebra \( \mathbb{A} \) . In particular, we have for any \( \lambda \in \mathcal{P} \) , | \[ \frac{{\widetilde{p}}_{k + 1}\left( \lambda \right) }{k + 1} = \mathop{\sum }\limits_{{j = 0}}^{\left\lbrack \frac{k}{2}\right\rbrack }\frac{1}{{2}^{2j}\left( {{2j} + 1}\right) }\left( \begin{matrix} k \\ {2j} \end{matrix}\right) {\bar{p}}_{k - {2j}}\left( \lambda \right) ,\;k \geq 1. \] (2.13) In a simpler way, (2.... | Yes |
Theorem 2.11 Define\n\n\\[ \n{\\mathcal{Y}}_{n, k}\\left( \\lambda \\right) = {\\int }_{-\\infty }^{\\infty }{u}^{k}\\left( {\\frac{1}{\\sqrt{n}}{\\Psi }_{\\lambda }\\left( {\\sqrt{n}u}\\right) - \\Omega \\left( u\\right) }\\right) \\mathrm{d}u,\\;\\lambda \\in {\\mathcal{P}}_{n}.\n\\]\n\nThen for each \\( k \\geq 0 \\... | To prove the theorem, we note\n\n\\[ \n{\\mathcal{Y}}_{n, k}\\left( \\lambda \\right) = {\\int }_{-\\infty }^{\\infty }{u}^{k}\\left( {\\frac{1}{\\sqrt{n}}{\\Psi }_{\\lambda }\\left( {\\sqrt{n}u}\\right) - \\left| u\\right| }\\right) \\mathrm{d}u - {\\int }_{-\\infty }^{\\infty }{u}^{k}\\left( {\\Omega \\left( u\\right... | Yes |
Theorem 2.12 Define\n\n\\[ \n{\\mathcal{Z}}_{n, k}\\left( \\lambda \\right) = \\frac{{p}_{k}^{\\# }\\left( \\lambda \\right) }{{n}^{\\frac{k}{2}}},\\;\\lambda \\in {\\mathcal{P}}_{n}.\n\\]\n\nThen under \\( \\left( {{\\mathcal{P}}_{n},{P}_{p, n}}\\right) \\) as \\( n \\rightarrow \\infty \\) ,\n\n\\[ \n\\left( {{\\math... | To prove this, we adapt the moment method. Fix \\( l \\geq 1 \\) . We need to check\n\n\\[ \n{E}_{p, n}{\\left( {\\eta }_{n, k}\\right) }^{l} \\rightarrow E{\\xi }_{k}^{l},\\;n \\rightarrow \\infty ,\n\\]\n\nwhere \\( {\\eta }_{n, k} = \\frac{{p}_{k}^{\\# }\\left( \\lambda \\right) }{\\sqrt{k}{n}^{\\frac{k}{2}}} \\) . ... | Yes |
Theorem 2.13\n\n\[ \mathop{\sup }\limits_{{-\infty < x < \infty }}\left| {{P}_{p, n}\left( {{W}_{n}\left( \lambda \right) \leq x}\right) - \Phi \left( x\right) }\right| = O\left( {n}^{-\frac{1}{2}}\right) . \] | The basic strategy of the proof is to construct a \( {W}_{n}^{\prime }\left( \lambda \right) \) such that \( \left( {{W}_{n}\left( \lambda \right) ,{W}_{n}^{\prime }\left( \lambda \right) }\right) \) is an exchange pair and to apply the stein’s method \( \left( {{W}_{n}\left( \lambda \right) ,{W}_{n}^{\prime }\left( \l... | Yes |
Lemma 2.9 (i) \( {p}_{n}\left( {\lambda ,\mu }\right) \) is a well-defined transition probability: | \[ \mathop{\sum }\limits_{{\mu \in {\mathcal{P}}_{n}}}{p}_{n}\left( {\lambda ,\mu }\right) = 1 \] | No |
Lemma 2.10 Let \( {\Delta }_{n}\left( \lambda \right) = {W}_{n}\left( \lambda \right) - {W}_{n}^{\prime }\left( \lambda \right) \). Then\n\n\[ \n{P}_{p, n}\left( {\left| {{\Delta }_{n}\left( \lambda \right) }\right| \geq \frac{4\mathrm{e}\sqrt{2}}{\sqrt{n}}}\right) \leq 2{\mathrm{e}}^{-2\mathrm{e}\sqrt{n}}. \n\] | \[ \n{E}_{p, n}{\left| {\Delta }_{n}\left( \lambda \right) \right| }^{2}{\mathbf{1}}_{\left( \left| {\Delta }_{n}\left( \lambda \right) \right| \geq \frac{4\mathrm{e}\sqrt{2}}{\sqrt{n}}\right) } = O\left( {n}^{-\frac{1}{2}}\right) . \n\] | No |
Theorem 1.2 Suppose that \( \mathcal{A} = \left( {a}_{{i}_{1},{i}_{2},\cdots ,{i}_{m}}\right) \) is a nonnegative tensor in \( {\mathbb{R}}^{\left\lbrack m, n\right\rbrack } \), where \( m \geq 2 \) . Then\n\n\[ \sigma \left( \mathcal{A}\right) \leq {R}_{\max }\left( \mathcal{A}\right) . \] | Proof By Theorem 1.1, we suppose that \( \lambda \) is a Z-eigenvalue of \( \mathcal{A} \) and \( x = \left( {{x}_{1},{x}_{2},\cdots ,{x}_{n}}\right) \) is the corresponding \( \mathrm{Z} \) -eigenvector. Let \( {x}_{p} = \max \left\{ {\left| {x}_{{i}_{1}}\right| : {i}_{1} \in \left\lbrack n\right\rbrack }\right\} \neq... | Yes |
Theorem 2.1 For any integer \( m > 0 \) ,\n\n\[ \n\sigma \left( \mathcal{A}\right) \leq \mathop{\max }\limits_{{1 \leq {i}_{1} \leq n}}\frac{{R}_{{i}_{1}}\left( {\mathcal{A}}^{m + 1}\right) }{{R}_{{i}_{1}}\left( {\mathcal{A}}^{m}\right) }.\n\] | Proof Let the diagonal matrix\n\n\[ \nD = \operatorname{diag}\left( {{R}_{1}\left( {\mathcal{A}}^{m}\right) ,{R}_{2}\left( {\mathcal{A}}^{m}\right) ,\cdots ,{R}_{n}\left( {\mathcal{A}}^{m}\right) }\right)\n\]\n\nfor some integer \( m > 0 \) . Then by Lemma 1.1,\n\n\[ \n\sigma \left( \mathcal{A}\right) = \sigma \left( {... | Yes |
Lemma 2.2 Let \( \mathcal{A} \) be a nonnegative tensor of order \( k \) and \( x \) be a Z-eigenvector of \( \mathcal{A} \) corresponding to the Z-eigenvalue of \( \mathcal{A} \), say \( \lambda \) . Then \( {\lambda }^{\mathop{\sum }\limits_{{i = 1}}^{t}{\left( k - 1\right) }^{i - 1}} \) is a Z-eigenvalue of tensor \... | Proof Let \( {a}_{t} \) be a sequence satisfying \( {a}_{t} = {a}_{t - 1}\left( {k - 1}\right) + 1 \) and \( {a}_{1} = 1 \) . Claim 1: \( {\left( \mathcal{A}\right) }^{t}x = {\lambda }^{{a}_{t}}x \) . Proof of Claim 1: The proof is by induction on \( t \) . For \( t = 1 \), the result is obvious. Now, assume that the r... | Yes |
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