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Theorem 1.1 Let \( \mathcal{N} \) and \( \mathcal{M} \) be nests on a complex separable Hilbert space \( \mathcal{H} \) with \( \dim \mathcal{H} > 2 \) . If \( L : \tau \left( \mathcal{N}\right) \rightarrow \tau \left( \mathcal{M}\right) \) is a linear Lie triple isomorphism, then \( L \) is of the form\n\n\[ L\left( x... | To prove Theorem 1.1, we need some lemmas. The following lemmas can be found in [1] and \( \left\lbrack 2\right\rbrack \) .\n\nLemma \( {1.1}^{\left\lbrack 2,\text{ Theorem }1\right\rbrack } \) Let \( \mathcal{R} \) be a ring, and let \( B : \mathcal{R} \times \mathcal{R} \rightarrow \mathcal{R} \) be a bi-additive map... | Yes |
Theorem 1 For any finite group \( G \) and any \( n,{Q}_{n}\left( G\right) \) is a finite \( \left| G\right| \) -torsion abelian group. | Proof Consider \( G/H \times G/H \) for a subgroup \( H \) of \( G \) . A typical representative element \( \left( {H,{gH}}\right) \) of the \( G \) -orbit \( G\left( {H,{gH}}\right) \) has stabilizer \( H \cap {gH}{g}^{-1} \), a subgroup of \( H \) . \( H \cap {gH}{g}^{-1} = H \) if and only if \( g \in {N}_{G}\left( ... | Yes |
Theorem 2 Suppose that \( G = \left\langle {r, f\left| {\;{r}^{m} = {f}^{2} = e}\right. ,{fr} = {r}^{-1}f}\right\rangle \) is a dihedral group of order \( {2m} \) for an odd \( m \), then\n\n\[{\mathcal{B}}_{1} = \left\{ {\left\langle {r}^{i}\right\rangle \left| i\right| m}\right\} \cup \{ \langle f\rangle \} \cup \lef... | Proof The set of all subgroups of \( G \) can be divided as three classes:\n\n(i) \( \left\langle {r}^{i}\right\rangle, i \mid m \) ;\n\n(ii) \( \left\langle {{r}^{j}f}\right\rangle, j = 0,1,\cdots, m - 1 \) ;\n\n(iii) \( \left\langle {{r}^{i},{r}^{j}f}\right\rangle, i \mid m, j = 0,1,\cdots, m - 1 \) .\n\nSuppose that... | Yes |
Theorem 3 Suppose that \( G = \left\langle {r, f \mid {r}^{{2}^{k}m} = {f}^{2} = e,{fr} = {r}^{-1}f}\right\rangle \) is the dihedral group of order \( {2}^{k + 1}m, k > 0, m \) odd, then\n\n\[{\mathcal{B}}_{2} = \left\{ {\left\langle {r}^{i}\right\rangle \left| i\right| {2}^{k}m}\right\} \cup \left\{ {\left\langle {{r}... | Proof The set of all subgroups of \( G \) can be divided similarly as three classes:\n\n(i) \( \left\langle {r}^{i}\right\rangle, i \mid {2}^{k}m \) ;\n\n(ii) \( \left\langle {{r}^{j}f}\right\rangle, j = 0,1,\cdots ,{2}^{k}m - 1 \) ;\n\n(iii) \( \left\langle {{r}^{i},{r}^{j}f}\right\rangle, i \mid {2}^{k}m, j = 0,1,\cd... | Yes |
Theorem 4 Suppose that \( G = \left\langle {r, f \mid {r}^{m} = {f}^{2} = e,{fr} = {r}^{-1}f}\right\rangle \) is the dihedral group of order \( {2m}, m \) odd, then \( {Q}_{n}\left( G\right) = {I}^{n}\left( G\right) /{I}^{n + 1}\left( G\right) \cong {\mathbb{Z}}_{2} \) . | Proof By Theorem 2,\n\n\[ I\left( G\right) = \mathop{\sum }\limits_{{i \mid m}}\mathbb{Z}{a}_{\left\langle {r}^{i}\right\rangle } + \mathbb{Z}{a}_{\langle f\rangle } + \mathop{\sum }\limits_{{j \mid m, j \neq 1, m}}\mathbb{Z}{a}_{\left\langle {r}^{j}, f\right\rangle }.\]\n\nChoose \( j > 1 \) and \( i \) such that \( \... | Yes |
Theorem 0.1 Let \( V\left( x\right) \in {\mathrm{{RH}}}_{q}\left( {\mathbb{R}}^{d}\right), q > \frac{d}{2} \) and \( b \in {\mathrm{{BMO}}}_{\theta }\left( \rho \right) \) . Then the commutators \( {\mathcal{L}}_{b}^{-\frac{\beta }{2}}\left( f\right) \) is bounded from \( {L}^{\frac{d}{\beta }}\left( {\mathbb{R}}^{d}\r... | \[ {\begin{Vmatrix}{\mathcal{L}}_{b}^{-\frac{\beta }{2}}\left( f\right) \end{Vmatrix}}_{{\mathrm{{BMO}}}_{\mathcal{L}}} \leq C{\left\lbrack b\right\rbrack }_{\theta }\parallel f{\parallel }_{{L}^{\frac{d}{\beta }}\left( {\mathbb{R}}^{d}\right) },\;0 < \beta < d. \] | Yes |
Proposition 1.1 \( {}^{\left\lbrack {13},\text{ Corollary }{1.5}\right\rbrack }\; \) Let \( V\left( x\right) \in {\mathrm{{RH}}}_{q}\left( {\mathbb{R}}^{d}\right), q > \frac{d}{2} \) . For the associated function \( \rho \left( x\right) \) there exists a positive \( C \) and \( {k}_{0} \geq 1 \) such that, for all \( x... | \[ {C}^{-1}\rho \left( x\right) {\left( 1 + \frac{\left| x - y\right| }{\rho \left( x\right) }\right) }^{-{k}_{0}} \leq \rho \left( y\right) \leq {C\rho }\left( x\right) {\left( 1 + \frac{\left| x - y\right| }{\rho \left( x\right) }\right) }^{\frac{{k}_{0}}{{k}_{0} + 1}}. \] | Yes |
Proposition 1.2 \( {}^{\left\lbrack 4,\text{ Lemma 2.3 }\right\rbrack } \) There exists a sequence of points \( {\left\{ {x}_{k}\right\} }_{k = 1}^{\infty } \) in \( {\mathbb{R}}^{d} \), so that the family of critical balls \( \mathcal{Q} = {\left\{ {Q}_{k}\right\} }_{k = 1}^{\infty } \), defined by \( {Q}_{k} \mathrel... | Inequality (1.1) implies that if \( \sigma > 0 \) and \( x, y \in {\sigma Q} \), where \( Q \) is a critical ball, then\n\n\[ \rho \left( x\right) \leq {C}_{\sigma }\rho \left( y\right) \]\n\n\( \left( {1.2}\right) \)\n\nwhere the constant \( {C}_{\sigma } = {C}^{2}{\left( 1 + \sigma \right) }^{\frac{2{k}_{0} + 1}{{k}_... | No |
Proposition 1.4 \( {}^{\left\lbrack 6,\text{ Proposition 4.11 }\right\rbrack } \) If \( V\left( x\right) \in {\mathrm{{RH}}}_{q}\left( {\mathbb{R}}^{d}\right), q > \frac{d}{2} \), then there exist \( \delta = \delta \left( q\right) > \) 0 and \( C > 0 \) such that for every \( N > 0 \) there is a positive constant \( {... | \[ \left| {{k}_{t}\left( {x + h, y}\right) - {k}_{t}\left( {x, y}\right) }\right| \leq {C}_{N}{\left( \frac{\left| h\right| }{\sqrt{t}}\right) }^{\delta }{t}^{-\frac{d}{2}}{\mathrm{e}}^{-\frac{{\left| x - y\right| }^{2}}{Ct}}{\left( 1 + \frac{\sqrt{t}}{\rho \left( x\right) } + \frac{\sqrt{t}}{\rho \left( y\right) }\rig... | Yes |
Lemma 2.1 \( {}^{\left\lbrack 2,\text{ Theorem }{1.1}\right\rbrack } \) Let \( V\left( x\right) \in {\mathrm{{RH}}}_{q}\left( {\mathbb{R}}^{d}\right), q > \frac{d}{2} \) and \( b \in \mathrm{{BMO}} \) . Then for \( 0 < \beta < \) \( d \), the commutator \( {\mathcal{L}}_{b}^{-\frac{\beta }{2}}\left( f\right) \) satisfi... | \[ {\begin{Vmatrix}{\mathcal{L}}_{b}^{-\frac{\beta }{2}}\left( f\right) \end{Vmatrix}}_{{L}^{q}\left( {\mathbb{R}}^{d}\right) } \leq C\parallel b{\parallel }_{\mathrm{{BMO}}}\parallel f{\parallel }_{{L}^{p}\left( {\mathbb{R}}^{d}\right) },\;1 < p < \frac{d}{\beta },\frac{1}{q} = \frac{1}{p} - \frac{\beta }{d}. \] | Yes |
Lemma 2.3 \( \operatorname{Let}\pi : \left( {X, T}\right) \rightarrow \left( {Y, S}\right) \) be a factor map, and \( \mu \in {M}^{e}\left( {X, T}\right) \) with \( {h}_{\mu }\left( {T|\pi }\right) > 0. \) If\n\n\[ \mu = {\int }_{X}{\mu }_{x}^{{P}_{\mu }\left( {{\pi }^{-1}{\mathcal{B}}_{Y}}\right) }\mathrm{d}\mu \left(... | Proof (i) By [11, Lemma 3.7], \( \left( {X,{\mathcal{B}}_{X},\mu, T}\right) \) admits a measurable partition \( \alpha \) with \( \alpha \supset \) \( {\pi }^{-1}{\mathcal{B}}_{Y} \) and \( {P}_{\mu }\left( {{\pi }^{-1}{\mathcal{B}}_{Y}}\right) = \mathop{\bigcap }\limits_{{n = 0}}^{{+\infty }}{T}^{-n}{\alpha }^{ - } \)... | Yes |
Lemma 1.1 For any \( k \geq 2 \), we have\n\n\[ \n{\int }_{0}^{\pi }\left| {\mathop{\sum }\limits_{{j = 1}}^{k}{\log }^{N}j\sin {jx}}\right| \mathrm{d}x \leq M{\log }^{N + 1}k \n\] | This result is quite straightforward by using Abel’s transformation. Details could be found in \( \left\lbrack 5\right\rbrack \) . | No |
Lemma 1.2 Let \( \\left\\{ {a}_{n}\\right\\} \) be a real sequence satisfying Condition (1). Then for \( n \geq 2 \), it holds that\n\n\[ \n\\left| {a}_{n + 1}\\right| \\log n \\leq {M}_{1}\\left| {a}_{n}\\right| \\log n \\leq M\\mathop{\\sum }\\limits_{{j = \\left\\lbrack \\sqrt{n}\\right\\rbrack }}^{n}\\frac{\\left| ... | Proof From (1), we see that, for \( \\left\\lbrack \\sqrt{n}\\right\\rbrack \\leq j \\leq n \) ,\n\n\[ \n\\frac{\\left| {a}_{n}\\right| }{{\\log }^{N}n} = \\left| {-\\mathop{\\sum }\\limits_{{k = j}}^{{n - 1}}\\Delta \\frac{{a}_{k}}{{\\log }^{N}k} + \\frac{{a}_{j}}{{\\log }^{N}j}}\\right| \\leq \\mathop{\\sum }\\limits... | Yes |
Lemma 1.4 Suppose that a real sequence \( \left\{ {a}_{n}\right\} \) satisfies Condition (1). Then there is a positive constant \( {M}^{ * } \) such that\n\n\[ \mathop{\sum }\limits_{{k = n}}^{{{2n} - 1}}\left| {\Delta {a}_{k}}\right| \leq \frac{{M}^{ * }}{n}\mathop{\sum }\limits_{{k = \frac{n}{2}}}^{{{2n} - 1}}\left| ... | Proof For any \( k \geq 1 \), we have\n\n\[ \left| {\Delta {a}_{k}}\right| \leq \left| {\Delta \frac{{a}_{k}}{{\log }^{N}k}}\right| {\log }^{N}\left( {k + 1}\right) + \frac{\left| {a}_{k}\right| }{{\log }^{N}k}\left( {{\log }^{N}\left( {k + 1}\right) - {\log }^{N}k}\right) . \]\n\nApplying Abel's transformation then by... | Yes |
Lemma 1.5 Suppose that a real sequence \( \left\{ {a}_{n}\right\} \) satisfies Condition (1). Then for sufficiently large \( k \) we have\n\n\[ \mathop{\sum }\limits_{{m \in {J}_{k}^{\left( 2\right) }}}\frac{\left| {a}_{n}\right| }{n} \leq \frac{1}{4}\mathop{\sum }\limits_{{m \in {J}_{k}^{\left( 1\right) }}}\frac{\left... | Proof It is easy to see that, by Lemma 1.4,\n\n\[ \frac{1}{2}\mathop{\sum }\limits_{{{S}_{j} \subseteq {J}_{k}^{\left( 2\right) }}}\left| {a}_{{\mu }_{j}}\right| \leq \frac{1}{2}\mathop{\sum }\limits_{{j = 1}}^{{\kappa }_{k}}\left| {a}_{{\mu }_{j}}\right| \leq \mathop{\sum }\limits_{{n = {2}^{k}}}^{{2}^{k + 1}}\left| {... | Yes |
Corollary 1.1 Let \( \left\{ {a}_{n}\right\} \) satisfy Condition (1). For sufficiently large \( {k}_{0} \) and arbitrary \( {N}_{0} \) with \( {N}_{0} \geq {k}_{0} \) we have\n\n\[ \mathop{\sum }\limits_{{k = {k}_{0}}}^{{N}_{0}}\mathop{\sum }\limits_{{m \in {J}_{k}^{\left( 2\right) }}}\frac{\left| {a}_{m}\right| }{m} ... | Proof The points of \( \{ {2}^{{k}_{0}},{2}^{{k}_{0}} + 1,\cdots {2}^{{N}_{0}} - 1\} \) with \( \{ m : {2}^{k - 1} \leq m < {2}^{k}\} ,\;k = {k}_{0} + 1,{k}_{0} + \) \( 2,\cdots ,{N}_{0} \) obviously repeat twice at most. Therefore, we can draw the conclusion of Corollary 1.1 by a simple calculation. | No |
Lemma 1.6 Under the above notations, \( \left\{ {d}_{m}\right\} \in \) MVBVS. | Proof Since the number of sets \( {S}_{j} \) in \( {J}_{k}^{\left( 1\right) } \) is bounded independent of \( k,\left| {S}_{j}\right| \geq \frac{{2}^{k}}{{32}{M}^{ * }} \) for all \( {S}_{j} \subseteq {J}_{k}^{\left( 1\right) },\left| {d}_{2m}\right| \approx \left| {d}_{m}\right| \) for every \( m \geq {k}_{0} \) and \... | Yes |
Lemma 1.7 Suppose that a real sequence \( \left\{ {a}_{n}\right\} \) satisfies Condition (1) and consider the sine series (2). Let \( g\left( x\right) \in {L}_{2\pi } \) and \( \left\{ {a}_{n}\right\} \) be the Fourier coefficients of \( g\left( x\right) \) . Then for sufficiently large \( {k}_{0} \) and arbitrary \( {... | Proof It is clear to see that\n\n\[ \mathop{\sum }\limits_{{k = {k}_{0}}}^{{N}_{0}}\mathop{\sum }\limits_{{m \in {J}_{k}^{\left( 1\right) }}}\frac{\left| {a}_{m}\right| }{m} = \mathop{\sum }\limits_{{k = {k}_{0}}}^{{N}_{0}}\left( {\mathop{\sum }\limits_{{m \in {J}_{k}^{\left( 1\right) } \cap {I}_{k}^{ + }}}{a}_{m} + \m... | Yes |
Theorem 0.1 Let \( {\mathcal{B}}_{1} \) and \( {\mathcal{B}}_{2} \) be Banach spaces. Suppose that \( T \) given by (2) is a bounded linear operator from \( {L}^{r}\left( {\mathcal{X},{\mathcal{B}}_{1}}\right) \) to \( {L}^{r}\left( {\mathcal{X},{\mathcal{B}}_{2}}\right) \) for some \( r \in \left( {1,\infty }\right) \... | \[ \parallel {Tf}{\parallel }_{{L}^{p\left( \cdot \right) }\left( {\mathcal{X},{\mathcal{B}}_{2}}\right) } \leq C\parallel f{\parallel }_{{L}^{p\left( \cdot \right) }\left( {\mathcal{X},{\mathcal{B}}_{1}}\right) }.\] | Yes |
Corollary 0.1 Let \( p\left( \cdot \right) \in \operatorname{LH}\left( \mathcal{X}\right), q \in (1,\infty \rbrack \) and \( {\mathcal{B}}_{1},{\mathcal{B}}_{2} \) be Banach spaces. Suppose that \( T \) given by (2) is a bounded linear operator from \( {L}^{q}\left( {\mathcal{X},{\mathcal{B}}_{1}}\right) \) to \( {L}^{... | \[ {\begin{Vmatrix}{\left( \mathop{\sum }\limits_{{j \in \mathbb{N}}}{\begin{Vmatrix}T{f}_{j}\end{Vmatrix}}_{{\mathcal{B}}_{2}}^{q}\right) }^{\frac{1}{q}}\end{Vmatrix}}_{{L}^{p\left( \cdot \right) }\left( {\mathcal{X},{\mathcal{B}}_{2}}\right) } \leq C{\begin{Vmatrix}{\left( \mathop{\sum }\limits_{{j \in \mathbb{N}}}{\... | Yes |
Lemma 1.2 \( {}^{\left\lbrack 7,\text{ Lemma 3.13 }\right\rbrack }\; \) Let \( f \in {L}_{\mathrm{{loc}}}^{1}\left( {\mathcal{X},\mathcal{B}}\right) \) such that \( {\oint }_{Q}\parallel f\left( y\right) {\parallel }_{\mathcal{B}}\mathrm{d}\mu \left( y\right) \rightarrow 0 \) as \( \left| Q\right| \rightarrow \infty . ... | \[ {E}_{t} = \{ x \in \mathcal{X} : \mathcal{m}f\left( x\right) > t\} \subseteq \mathop{\bigcup }\limits_{j}3{Q}_{j}. \] | No |
Lemma 1.3 \( {}^{\left\lbrack 7,\text{ Lemma 3.9 }\right\rbrack } \) Let \( f \in {L}_{\mathrm{{loc}}}^{1}\left( {\mathcal{X},\mathcal{B}}\right) \) such that \( {\oint }_{Q}\parallel f\left( y\right) {\parallel }_{\mathcal{B}}\mathrm{d}\mu \left( y\right) \rightarrow 0 \) as \( \left| Q\right| \rightarrow \infty \) . ... | \[ t < {f}_{{Q}_{j}}\parallel f\left( x\right) {\parallel }_{\mathcal{B}}\mathrm{d}\mu \left( x\right) \leq {2}^{n}t \] | No |
Lemma 1.6 Given \( p\left( \cdot \right) \in \mathcal{P}\left( \mathcal{X}\right) \), then \( {L}^{\infty }\left( {\mathcal{X},\mathcal{B}}\right) \subseteq {L}^{p\left( \cdot \right) }\left( {\mathcal{X},\mathcal{B}}\right) \) if and only if \( 1 \in {L}^{p\left( \cdot \right) }\left( {\mathcal{X},\mathcal{B}}\right) ... | Proof If \( 1 \in {L}^{p\left( \cdot \right) }\left( {\mathcal{X},\mathcal{B}}\right) \), then \( \parallel f{\parallel }_{{L}^{p\left( \cdot \right) }\left( {\mathcal{X},\mathcal{B}}\right) } \leq C\parallel f{\parallel }_{{L}^{\infty }\left( {\mathcal{X},\mathcal{B}}\right) }\parallel 1{\parallel }_{{L}^{p\left( \cdo... | Yes |
Lemma 1.7 Given \( p\left( \cdot \right) \in \mathcal{P}\left( \mathcal{X}\right) \), then\n\n\[ \n{L}^{p\left( \cdot \right) }\left( {\mathcal{X},\mathcal{B}}\right) \subseteq {L}^{{p}_{ + }}\left( {\mathcal{X},\mathcal{B}}\right) + {L}^{{p}_{ - }}\left( {\mathcal{X},\mathcal{B}}\right) \n\]\n\nand\n\n\[ \n\parallel f... | Proof By homogeneity, we may assume without loss of generality that \( \parallel f{\parallel }_{{L}^{p\left( \cdot \right) }\left( {\mathcal{X},\mathcal{B}}\right) } = 1 \) . Decompose \( f \) as \( {f}_{1} + {f}_{2} \), where\n\n\[ \n{f}_{1} = f{\chi }_{\left\{ x \in \mathcal{X} : \parallel f\left( x\right) {\parallel... | Yes |
Lemma 1.8 Given \( p\left( \cdot \right) \in \mathcal{P}\left( \mathcal{X}\right) \), then for all \( f \in {L}^{p\left( \cdot \right) }\left( {\mathcal{X},\mathcal{B}}\right) ,{f}_{Q}\parallel f\left( y\right) {\parallel }_{\mathcal{B}}\mathrm{d}\mu \left( y\right) \rightarrow 0 \) as \( \left| Q\right| \rightarrow \i... | Proof Fix \( f \in {L}^{p\left( \cdot \right) }\left( {\mathcal{X},\mathcal{B}}\right) \), by Lemma 1.7 we have that \( f = {f}_{1} + {f}_{2} \), where \( {f}_{1} \in {L}^{{p}_{ + }}\left( {\mathcal{X},\mathcal{B}}\right) \) and \( {f}_{2} \in {L}^{{p}_{ - }}\left( {\mathcal{X},\mathcal{B}}\right) \) . Since \( {p}_{ +... | Yes |
Theorem 1 Let \( F = \frac{{\alpha }^{m + 1}}{{\beta }^{m}} \) with \( m \neq 0, - 1 \) be a general Kropina metric on an \( n \) - dimensional manifold \( M, n \geq 2 \) . Then \( F \) is locally dually flat if and only if there is a 1-form \( \theta = {\theta }_{i}\left( x\right) {y}^{i} \) such that\n\n\[ \n{G}_{\al... | We remark that characterizing equations are similar to one case of results about \( \left( {\alpha ,\beta }\right) \) - metrics in [6]. But the argument in [6] does not apply to our case, since it requires the analyticity of the function \( \phi \) in \( \left( {\alpha ,\beta }\right) \) -metrics which excludes Kropina... | No |
Actually, it is not difficult to verify the following two equalities\n\n\[ \n{\left( {F}^{2}\right) }_{{x}_{i}{y}^{1}}{y}^{i} - 2{\left( {F}^{2}\right) }_{{x}_{1}} = 0,\;{\left( {F}^{2}\right) }_{{x}_{i}{y}^{2}}{y}^{i} - 2{\left( {F}^{2}\right) }_{{x}_{2}} = 0, \n\]\n\nwhich are the equations (4) of a locally dually fl... | By a direct computation,\n\n\[ \n{F}_{{x}_{i}{y}^{1}}{y}^{i} - {F}_{{x}_{1}} = - \frac{4{y}^{2}}{{\left( c - 4{x}_{2}\right) }^{\frac{3}{2}}}\left\{ {1 - {\left\lbrack \frac{{y}^{2}}{\left( {c - 4{x}_{2}\right) {y}^{1} + 4{x}_{1}{y}^{2}}\right\rbrack }^{2}}\right\} \neq 0, \n\]\n\n\[ \n{F}_{{x}_{i}{y}^{2}}{y}^{i} - {F}... | Yes |
Let\n\n\[ A = \frac{1}{100}\left( \begin{array}{ll} {25} & {14} \\ {40} & {12} \end{array}\right) \]\n\nThen its maximal eigenvalue \( \rho \left( A\right) \), the left-eigenvector \( u \), and right-eigenvalue \( g \) are, respectively, | as follows:\n\n\[ \rho \left( A\right) = \frac{{37} + \sqrt{2409}}{200} \]\n\n\[ u = \left( {\frac{5\left( {{13} + \sqrt{2409}}\right) }{7},\;{20}}\right) \approx \left( {{44.34397483},{20}}\right) ,\]\n\n\[ g = {\left( \frac{{13} + \sqrt{2409}}{4},{20}\right) }^{ * }.\] | Yes |
Theorem 1 (Hua's Fundamental Theorem, 1984)\n\n- The optimal choice of \( {x}_{0} \) is \( u \), and it has the fastest grow: \( {x}_{n} = {x}_{0}\rho {\left( A\right) }^{-n} \) .\n\n- Except some very special \( A \), if \( {x}_{0} \neq u \), then the economic system will be collapsed. That is, some component of the p... | Certainly, we do not care if the collapse time is very large, say \( {10}^{4} \) years for instance. However, it is not the case in practice. Table 1 shows the collapse time of Example 1 for the initials different from \( u \) .\n\nTable 1 Input and collapse time\n\n<table><thead><tr><th>\( {x}_{0} \)</th><th>Collapse ... | Yes |
Consider the matrix\n\n\[ \nQ = \left( \begin{matrix} - 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & - 5 & {2}^{2} & 0 & 0 & 0 & 0 & 0 \\ 0 & {2}^{2} & - {13} & {3}^{2} & 0 & 0 & 0 & 0 \\ 0 & 0 & {3}^{2} & - {25} & {4}^{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & {4}^{2} & - {41} & {5}^{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & {5}^{2} & - {61} & {6}^{... | Actually, this matrix is truncated from the corresponding infinite one, in which case we have known that the maximal eigenvalue is \( - \frac{1}{4} \) (refer to Chen (2010) [5, Example 3.6]). | Yes |
Example 3 For the same matrix \( Q \) and \( {\widetilde{v}}_{0} \) as in Example 1, by RQI, we need two iterations only: | \[ {z}_{1} \approx - {0.528215},\;{z}_{2} \approx - {0.525268}. \] This shows not only the power of the second method but also the effectiveness of my \( {v}_{0} \) . For simplicity, from now on, we set \( {\lambda }_{j} \mathrel{\text{:=}} {\lambda }_{j}\left( {-Q}\right) \) . In particular \( {\lambda }_{0} = - \rho ... | No |
Example 4 Let \( Q \) be the same as above and use the uniformly distributed \( {v}_{0} \) . Then\n\n\[ \left( {{z}_{1},{z}_{2},{z}_{3},{\mathbf{z}}_{\mathbf{4}}}\right) \approx \left( {{4.78557},{5.67061},{5.91766},\mathbf{5.{91867}}}\right) .\n\n\]\n\n\[ \left( {{\lambda }_{0},{\lambda }_{1},{\mathbf{\lambda }}_{\mat... | In Example 4, \( {z}_{0} \) is chosen in the automatic way: \( {z}_{0} = {v}_{0}^{ * }\left( {-Q}\right) {v}_{0} \) . If we keep this \( {v}_{0} \) which is not so good, but use a new \( {z}_{0} \), then we come back to our result in two iterations. Here \( {z}_{0} \) is defined by\n\n\[ {z}_{0} = \frac{7}{8{\delta }_{... | Yes |
Theorem 4 The optimal constants \( {\lambda }^{\# } \) in the Poincaré-type inequalities, with/without \( c \) , satisfy\n\n\[ \n{\kappa }^{\# } \leq {\lambda }^{\# - 1} \leq 4{\kappa }^{\# } \n\]\n\nand the optimal constants \( {A}^{\# } \) in the Hardy-type inequalities, with/without \( \mathbb{B} \), satisfy\n\n\[ \... | It is remarkable that the previous proofs for the linear case \( \left( {q = p = 2}\right) \) do not suitable to the present nonlinear situation. To which, we use new analytic proofs (refer to Chen (2013a) [7] and Liao (2016) [17]). | No |
Lemma 1.1 Let \( {\mathcal{M}}^{\langle 1\rangle } = \left\{ {M - {e}_{r}\left( M\right) \mid M \in {\mathcal{M}}^{\left\lbrack 1\right\rbrack }}\right\} \) . Then \[ {\mathcal{M}}^{\langle 1\rangle } = \mathcal{M} \times \mathcal{M} \] where \( \times \) is the Cartesian product. | Proof Because any \( M \in {\mathcal{M}}^{\left\lbrack 1\right\rbrack } \) the root-edge \( {e}_{r}\left( M\right) \) of \( M \) is separable, \( M - {e}_{r}\left( M\right) = \) \( {M}_{1} + {M}_{2},{M}_{1},{M}_{2} \in \mathcal{M} \) . It implies that \( M - {e}_{r}\left( M\right) \in \mathcal{M} \times \mathcal{M} \) ... | Yes |
Lemma 1.2 Let \( {\mathcal{M}}^{\langle 2\rangle } = \left\{ {M - {e}_{r}\left( M\right) \mid M \in {\mathcal{M}}^{\left\lbrack 2\right\rbrack }}\right\} \) . Then \[ {\mathcal{M}}^{\langle 2\rangle } = \mathcal{M} - {\mathcal{M}}_{1} - {\mathcal{M}}_{2} - \vartheta \] where \( {\mathcal{M}}_{i} = \left\{ {M = \left( {... | Proof For a map \( M = \left( {\mathcal{X},\mathcal{J}}\right) \in {\mathcal{M}}^{\langle 2\rangle } \), let \( {M}^{\prime } = \left( {{\mathcal{X}}^{\prime },{\mathcal{J}}^{\prime }}\right) \in {\mathcal{M}}^{\left\lbrack 2\right\rbrack } \) such that \( M = \) \( {M}^{\prime } - {e}_{{r}^{\prime }}\left( {M}^{\prime... | Yes |
Lemma 1.3 Let \( {\mathcal{M}}_{\left( 1\right) } = \left\{ {M - {v}_{\beta r} \mid M \in {\mathcal{M}}_{1}}\right\} \) . Then \[ {\mathcal{M}}_{\left( 1\right) } = \mathcal{M} \] | Proof For any map \( M \in {\mathcal{M}}_{1} \), the vertex \( {v}_{\beta r} \) is an articulate vertex. Thus deleting \( {v}_{\beta r} \) will lead to a map \( M - {v}_{\beta r} \in \mathcal{M} \) . Conversely, for \( M \in \mathcal{M} \), let \( {o}_{1} \) be the rooted vertex of \( M \) . A map \( {M}^{\prime } \in ... | Yes |
Lemma 1.4 Let \( {\mathcal{M}}_{\langle {21}\rangle } = \left\{ {M - {e}_{r}\left( M\right) \mid M \in {\mathcal{M}}_{21}}\right\} \) . Then \[ {\mathcal{M}}_{\langle {21}\rangle } = \mathcal{M} \times {\mathcal{M}}_{1} \] where \( \times \) is the Cartesian product. | Proof Because any \( M \in {\mathcal{M}}_{21} \) the root-edge \( {e}_{r}\left( M\right) \) is a cut edge of \( M, M - {e}_{r}\left( M\right) = \) \( {M}_{1} + {M}_{2},{M}_{1} \in \mathcal{M},{M}_{2} \in {\mathcal{M}}_{1} \) . It implies that \( M - {e}_{r}\left( M\right) \in \mathcal{M} \times {\mathcal{M}}_{1} \) . C... | Yes |
Lemma 1.5 Let \( {\mathcal{M}}_{\left( {22}\right) } = \left\{ {M - {v}_{0} \mid M \in {\mathcal{M}}_{22}}\right\} ,{v}_{0} = {v}_{\beta \mathcal{J}{\gamma r}} \) . Then\n\n\[ \n{\mathcal{M}}_{\left( {22}\right) } = {\mathcal{M}}^{\left\lbrack 2\right\rbrack } \n\] | Proof For a map \( M \in {\mathcal{M}}_{\left( {22}\right) } \), let \( {M}^{\prime } \in {\mathcal{M}}_{22} \) such that \( M = {M}^{\prime } - {v}_{0}^{\prime } \) . Since \( {v}_{0}^{\prime } \) articulate, \( M \in {\mathcal{M}}^{\left\lbrack 2\right\rbrack } \) .\n\nConversely, for a map \( M = \left( {\mathcal{X}... | Yes |
Corollary 1.1 The enumerating function \( g = {g}_{\mathcal{M}}\left( {y, z}\right) \) satisfies the following equation:\n\n\[ y{g}^{2} - \left( {1 + {2yz} - z}\right) g + 1 + {yz} - z = 0. \] | Now, let \( y = z \) in (1.19). Then we obtain the following result. | No |
Corollary 1.3 The enumerating function \( H = {H}_{\mathcal{M}}\left( y\right) \) satisfies the following equation:\n\n\[ y{H}^{2} - \left( {1 - y + 2{y}^{2}}\right) H + 1 - y + {y}^{2} = 0. \] | \[ H = \frac{1 - y + {y}^{2}}{1 - y + 2{y}^{2} - {yH}} = \frac{\frac{1 - y + {y}^{2}}{1 - y + 2{y}^{2}}}{1 - \frac{yH}{1 - y + 2{y}^{2}}}. \] | Yes |
Theorem 2.1 The enumerating function \( H = {H}_{\mathcal{M}}\left( y\right) \) has the following explicit expression:\n\n\[ \n{H}_{\mathcal{M}}\left( y\right) = \mathop{\sum }\limits_{{n \geq 0}}\mathop{\sum }\limits_{{k = 0}}^{n}\mathop{\sum }\limits_{{i = 0}}^{{\lfloor \frac{n - k}{2}\rfloor }}\mathop{\sum }\limits_... | Proof By employing Lagrangian inversion with one parameter, from (2.1) we have\n\n\[ \n{H}_{\mathcal{M}}\left( y\right) = {\left. \mathop{\sum }\limits_{{k \geq 1}}\frac{{\left( \frac{1 - y + {y}^{2}}{1 - y + 2{y}^{2}}\right) }^{k}}{k!}\frac{{\mathrm{d}}^{k - 1}}{\mathrm{\;d}{H}^{k - 1}}\left\{ {\left( 1 - \frac{yH}{1 ... | Yes |
Theorem 2.2 The enumerating function \( h = {h}_{\mathcal{M}}\left( {x, y}\right) \) has the following explicit expression:\n\n\[ \n{h}_{\mathcal{M}}\left( {x, y}\right) = \mathop{\sum }\limits_{{k, i, t \geq 0}}\mathop{\sum }\limits_{{j = 0}}^{{k + 1}}\frac{{\left( -2\right) }^{i}}{{2k} + i + 1}\left( \begin{matrix} {... | Proof By using Lagrangian inversion with one parameter, from (2.3) one may find that\n\n\[ \n{h}_{\mathcal{M}}\left( {x, y}\right) = {\left. \mathop{\sum }\limits_{{k \geq 1}}\frac{{\left( \frac{{x}^{2} + {x}^{2}{y}^{2} - y}{{x}^{2} + 2{x}^{2}{y}^{2} - y}\right) }^{k}}{k!}\frac{{\mathrm{d}}^{k - 1}}{\mathrm{\;d}{h}^{k ... | Yes |
Theorem 2.3 The enumerating function \( g = {g}_{\mathcal{M}}\left( {y, z}\right) \) has the following explicit expression:\n\n\[ \n{g}_{\mathcal{U}}\left( {y, z}\right) = \mathop{\sum }\limits_{{p, q \geq 0}}\mathop{\sum }\limits_{{i = 0}}^{{\min \{ p, q\} }}\mathop{\sum }\limits_{{j = 0}}^{{\min \{ \lfloor \frac{p - ... | Proof Applying Lagrangian inversion to (2.5), we obtain\n\n\[ \n{g}_{\mathcal{M}}\left( {y, z}\right) = {\left. \mathop{\sum }\limits_{{k \geq 1}}\frac{{\left( \frac{1 + {yz} - z}{1 + {2yz} - z}\right) }^{k}}{k!}\frac{{\mathrm{d}}^{k - 1}}{\mathrm{\;d}{g}^{k - 1}}\left\{ {\left( 1 - \frac{yg}{1 + {2yz} - z}\right) }^{-... | Yes |
Theorem 2.4 The enumerating function \( f = {f}_{\mathcal{M}}\left( {x, y, z}\right) \) has the following explicit expression:\n\n\[ \n{f}_{\mathcal{M}}\left( {x, y, z}\right) = \mathop{\sum }\limits_{{k, i, t \geq 0}}\mathop{\sum }\limits_{{j = 0}}^{{k + 1}}\frac{{\left( -2\right) }^{i}}{{2k} + i + 1}\left( \begin{mat... | Proof By employing Lagrangian inversion with one parameter, from (2.8) we have\n\n\[ \n{f}_{\mathcal{M}}\left( {x, y, z}\right) = {\left. \mathop{\sum }\limits_{{k \geq 1}}\frac{{\left( \frac{{x}^{2} + {x}^{2}{yz} - z}{{x}^{2} + 2{x}^{2}{yz} - z}\right) }^{k}}{k!}\frac{{\mathrm{d}}^{k - 1}}{\mathrm{\;d}{f}^{k - 1}}\lef... | Yes |
Lemma 1.1 Let \( x, y \in {D}_{q} \) and \( \left( {x, y}\right) = 1 \), then \( {\phi }_{q}\left( {xy}\right) = {\phi }_{q}\left( x\right) {\phi }_{q}\left( y\right) \), i.e., \( {\phi }_{q}\left( x\right) \) is multiplicative. | Lemma 1.1 follows immediately from the Chinese Remainder Theorem. | No |
Lemma 1.2 Let \( x \in {D}_{q} \) and \( x = \mathop{\prod }\limits_{{i = 1}}^{k}{p}_{i}^{{\alpha }_{i}} \) be the standard factorization of \( x \) . We have \( {\phi }_{q}\left( x\right) = \mathop{\prod }\limits_{{i = 1}}^{k}{p}_{i}^{{\alpha }_{i} - 1}\left( {{p}_{i} - \left( {q - 1}\right) }\right) . | Proof By Lemma 1.1, we only need to treat the case \( x = {p}^{\alpha } \) . Given \( i\left( {0 \leq i \leq q - 2}\right) \) , define \( {A}_{i} = \left\{ {a\left| p\right| \left( {a + i}\right) ,1 \leq a \leq {p}^{\alpha }}\right\} \), and let \( \left| {A}_{i}\right| \left( {0 \leq i \leq q - 2}\right) \) denote the... | Yes |
Lemma 1.4 Let \( x \in {D}_{q} \) with \( x > q \), then \( {\phi }_{q}^{k}\left( x\right) \equiv 0\left( {\;\operatorname{mod}\;q}\right) \) for all integers \( 1 \leq k \leq \) \( {C}_{q}\left( x\right) \) . | Proof For any \( x \in {D}_{q} \) with \( x > q \), by Lemma 1.2 and definition of \( {D}_{q} \) and \( {C}_{q}\left( x\right) \), we have \( {\phi }_{q}^{k}\left( x\right) \equiv 0\left( {\;\operatorname{mod}\;q}\right) \) for all integers \( 1 \leq k \leq {C}_{q}\left( x\right) . \) | Yes |
Lemma 1.5 If \( {xy} \in {D}_{q} \) with \( x, y > 1 \), then \( x \in {D}_{q} \) and \( y \in {D}_{q} \) . If \( x, y \in {D}_{q} \), then \( {xy} \in {D}_{q} \) . | Proof If \( z \in {D}_{q} \) and \( {z}^{\prime } \mid z \) with \( {z}^{\prime } > 1 \), by definition of \( {D}_{q} \) and Lemma 1.2, it is obvious that \( {z}^{\prime } \in {D}_{q} \) . Hence, if \( {xy} \in {D}_{q} \) with \( x, y > 1 \), then \( x \in {D}_{q} \) and \( y \in {D}_{q} \) .\n\nBy Lemmas 1.1–1.3, we c... | Yes |
Theorem 1.2 If \( x \in {D}_{q} \) and \( q \nmid x \), then \( {C}_{q}\left( {qx}\right) = {C}_{q}\left( x\right) \) . | This is a consequence of the fact that for \( x \in {D}_{q} \) and \( q \nmid x \), by Lemma 1.1, we have \( {\phi }_{q}\left( {qx}\right) = {\phi }_{q}\left( q\right) {\phi }_{q}\left( x\right) = {\phi }_{q}\left( x\right) . | Yes |
Theorem 1.3 If \( x \in {D}_{q} \) and \( q \mid x \), then \( {C}_{q}\left( {qx}\right) = {C}_{q}\left( x\right) + 1 \) . | Proof By Lemma 1.3, we have \( {\phi }_{q}\left( {qx}\right) = q{\phi }_{q}\left( x\right) \) . By Lemma 1.4, we have \( {\phi }_{q}^{k}\left( x\right) \equiv 0({\;\operatorname{mod}\;} \) \( q) \) for all integers \( 1 \leq k \leq {C}_{q}\left( x\right) \) . Let \( m = {C}_{q}\left( x\right) \), then\n\n\[ \n{\phi }_{... | Yes |
Corollary 1.1 If \( x \in {D}_{q} \) and \( q \mid x \), then for any integer \( a \geq 1 \), we have \( {C}_{q}\left( {{q}^{a}x}\right) = \) \( {C}_{q}\left( {q}^{a}\right) + {C}_{q}\left( x\right) + 1 \) | Proof (4) is the case when \( x = q \) . Suppose \( x > q \) . Then, since \( {\phi }_{q}^{k}\left( x\right) \equiv 0\left( {\;\operatorname{mod}\;q}\right) \) for all integers \( 1 \leq k \leq {C}_{q}\left( x\right) \) by Lemma 1.4, it follows from repeated applications of Lemma 1.3, that \( {\phi }_{q}^{k}\left( {{q}... | Yes |
Corollary 1.2 If \( x \in {D}_{q} \) and \( q \nmid x \), then for any integer \( a \geq 1 \), we have \( {C}_{q}\left( {{q}^{a}x}\right) = \) \( {C}_{q}\left( {q}^{a}\right) + {C}_{q}\left( x\right) \) | Proof When \( a = 1 \), the result is just Theorem 1.2. So assume that \( a \geq 2 \), then by Lemma \( {1.1} - {1.2},{\phi }_{q}\left( {{q}^{a}x}\right) = {\phi }_{q}\left( {q}^{a}\right) {\phi }_{q}\left( x\right) = {q}^{a - 1}{\phi }_{q}\left( x\right) \) . By Lemma 1.4 and Corollary 1.1, we have\n\n\[ \n{C}_{q}\lef... | Yes |
Theorem 1.4 If \( x \in {D}_{q} \), then \( {C}_{q}\left( {\left( {{2q} - 1}\right) x}\right) = {C}_{q}\left( {{2q} - 1}\right) + {C}_{q}\left( x\right) \) . | Proof Since \( q \in Q,{2q} - 1 \) is also prime. From Lemmas 1.1-1.3, we get\n\n\[ \n{\phi }_{q}\left( {\left( {{2q} - 1}\right) x}\right) = \left( {{2q} - 1}\right) {\phi }_{q}\left( x\right) \text{ or }q{\phi }_{q}\left( x\right) \n\] \n\nand \n\n\[ \n{\phi }_{q}^{k}\left( {\left( {{2q} - 1}\right) x}\right) = \left... | Yes |
Theorem 1.5 If \( x, p \in {D}_{q} \) and \( p > q \) is prime, then \( {C}_{q}\left( {px}\right) = {C}_{q}\left( p\right) + {C}_{q}\left( x\right) \) . | Proof If \( p = {2q} - 1 \), Theorem 1.4 has been established, so assume that the theorem holds for the first \( k - 1 \) primes in \( {D}_{q} \) . Let \( p \) be the \( k \) -th prime in \( {D}_{q} \) . Note that \( {px} \in {D}_{q} \) by Lemma 1.5. Then, from Lemmas 1.1-1.3, we have\n\n\[{\phi }_{q}\left( {px}\right)... | Yes |
Theorem 2.1 The largest number in class \( m \) is \( q{\left( 2q - 1\right) }^{m} \), the largest number not divisible by \( q \) in class \( m \) is \( {\left( 2q - 1\right) }^{m} \) . | Proof Since \( q \in Q,{2q} - 1 \) is a prime, it is straightforward to check that the theorem is true for \( m = 0,1 \) . Now we assume it to be true for all \( {m}^{\prime } < m \), and will prove it for class \( m \) , thus complete the proof of the theorem.\n\nFirst, for any prime \( p > q \) in class \( m \) we ha... | Yes |
Theorem 2.2 The smallest number of class \( m \) that is divisible by \( q \) is \( {q}^{m + 1} \) . | Proof It is straightforward to check that the theorem is true for \( m = 0,1 \) . Now we assume it to be true for all \( {m}^{\prime } < m \), Suppose next that \( {C}_{q}\left( s\right) = m \), where \( s = {q}^{a}r, q \nmid r \) . Now if \( a > 1 \) , by Theorem 1.3, we have \( {C}_{q}\left( s\right) = {C}_{q}\left( ... | Yes |
Theorem 2.3 The smallest number of class \( m \) that is not divisible by \( q \) is greater than \( {q}^{m} \) . | Proof Let \( s \) be the smallest number of class \( m \) that is not divisible by \( q \) . By Theorem 1.2, we have that \( {C}_{q}\left( {qs}\right) = {C}_{q}\left( s\right) \) . Then, by Theorem 2.2, we have \( {qs} > {q}^{m + 1} \), so that \( s > {q}^{m} \) . | Yes |
Theorem 3.1 For any integer \( x \) in Section III of its class, we have that \( x \equiv 0\left( {\;\operatorname{mod}\;q}\right) \) , but \( x ≢ 0 \) (mod \( {q}^{\left\lbrack \left( 2\log \left( 2q - 1\right) - \log q\right) /\left( \log \left( 2q - 1\right) - \log q\right) \right\rbrack } \) ), where \( \left\lbrac... | Proof Let \( x = {q}^{a}s \), where \( s > 1, s ≢ 0\left( {\;\operatorname{mod}\;q}\right), a \geq 1 \) and \( {C}_{q}\left( x\right) = m \) . Then \( {C}_{q}\left( {{q}^{a}s}\right) = \) \( {C}_{q}\left( {q}^{a}\right) + {C}_{q}\left( s\right) = a - 1 + {C}_{q}\left( s\right) = m \) by Corollary \( {}^{ \bullet }{1.2}... | Yes |
Corollary 3.1 If \( \frac{\log x}{\log \left( {{2q} - 1}\right) } > {C}_{q}\left( x\right) \), then \( x \equiv 0\left( {\;\operatorname{mod}\;q}\right) \), but | \[ x ≢ 0\left( {\;\operatorname{mod}\;{q}^{\left\lbrack \frac{2\log \left( {{2q} - 1}\right) - \log q}{\log \left( {{2q} - 1}\right) - \log q}\right\rbrack }}\right) . \] | No |
Theorem 3.2 If \( x \) is an integer in class \( m \geq 2 \) such that\n\n\[ \n{\left( 2q - 1\right) }^{m - 2}\left( {2{q}^{2} - 1}\right) < x < {\left( 2q - 1\right) }^{m},\n\]\n\nthen \( x \equiv 0\left( {\;\operatorname{mod}\;q}\right) \) . | Proof Proceed by induction and assume that the proposition is true for all classes \( {m}^{\prime } < m \) . Suppose that \( x \in {D}_{q} \) with \( x ≢ 0\left( {\;\operatorname{mod}\;q}\right) ,{C}_{q}\left( x\right) = m \) and that \( x \) satisfies (7). If \( x \) is prime, then \( {C}_{q}\left( {x - q + 1}\right) ... | Yes |
Theorem 3.3 If an integer \( x \) is in Section I of its class, then every divisor of \( x \) is in Section I of its class. | Proof The theorem is obviously true for any \( x \) that is prime, so suppose that \( x \) is composite and write \( x = {ds} \) . Since \( x \) is in Section I of its class, we know that \( q \nmid x \), and we can assume that \( q < d < x \) . Thus,\n\n\[ \n{q}^{{C}_{q}\left( {ds}\right) } < {ds} < {q}^{{C}_{q}\left(... | Yes |
Theorem 3.4 Let \( p = {q}^{k}m + q - 1 \) be prime with \( k > 0 \) and \( m ≢ 0\left( {\;\operatorname{mod}\;q}\right) \). Then \( p \) is in Section I of its class if and only if \( m \) is in Section I of its class. | Proof Let \( p = {q}^{k}m + q - 1 \) be prime. Then \( {C}_{q}\left( p\right) = {C}_{q}\left( m\right) + k \). Thus, \( p \) is in Section I of its class if and only if\n\n\[ \n{q}^{{C}_{q}\left( p\right) } = {q}^{{C}_{q}\left( m\right) + k} < {q}^{k}m + q - 1 < {q}^{{C}_{q}\left( m\right) + k + 1} = {q}^{{C}_{q}\left(... | Yes |
Corollary 3.2 Let \( p = {q}^{k}m + q - 1 \) be prime with \( k > 0 \) and \( m ≢ 0\left( {\;\operatorname{mod}\;q}\right) \) . Then \( p \) is in Section II of its class if and only if \( m \) is in Section II of its class. | Proof The result follows from the negation of Theorem 3.4 and the fact that all numbers in Section III are divisible by \( q \) . | No |
Theorem 3.5 Let \( p \in {D}_{q} \) be a prime in Section I of its class, and let \( {p}^{\prime } > q \) be a prime factor of \( p - q + 1 \) . Then \( {p}^{\prime } \) is in Section I of its class. | Proof Write \( p - q + 1 = {q}^{k}m \), where \( k > 0 \) and \( m ≢ 0\left( {\;\operatorname{mod}\;q}\right) \) . By Theorem 3.4, \( m \) is a number in Section I of its class. Also, by Theorem 3.3, we conclude that the prime factors of \( m \) are all in Section I of their respective classes. | Yes |
Theorem 3.6 If \( x \) is the smallest element in its class and \( x ≢ 0\left( {\;\operatorname{mod}\;q}\right) \), then the prime divisors of \( x \) are the smallest elements in their respective classes. | Proof If \( x \) is prime, then the theorem is obvious. So, assume that \( x \) is composite, and write \( x = {ps} \), where \( p \) is prime. If there is some number \( t < p \) with \( {C}_{q}\left( t\right) = {C}_{q}\left( p\right) \), then, since \( x ≢ 0\left( {\;\operatorname{mod}\;q}\right) \), we have by Theor... | Yes |
Theorem 3.7 If \( x = {q}^{s} + q - 1 \) is prime, then \( x \) is the smallest element in its class. | Proof If \( x = {q}^{s} + q - 1 \) is prime, then \( {C}_{q}\left( x\right) = s \) . Since \( {q}^{s} + i \notin {D}_{q},1 \leq i \leq q - 2 \), and since every element \( y \) in class \( s \) is such that \( y > {q}^{s} \) by Corollary 2.2, the result follows. | Yes |
Lemma 2.1 Let \( \mathcal{A} \) be a \( {C}^{ * } \) -algebra and \( {\left( {p}_{n}\right) }_{n \geq 1} \) be a family of projections which is quasicentral in \( \mathcal{A} \) . Let \( a, b \in {\mathcal{A}}^{ + } \) with \( a \preccurlyeq b \) . Then for any \( \varepsilon > 0 \), there are some \( \delta > 0 \) and... | Proof Since \( a \preccurlyeq b \) (we can further assume that \( \parallel b\parallel \leq 1 \) ), there is a sequence \( {\left( {x}_{k}\right) }_{k \geq 1} \subseteq \mathcal{A} \) such that \( \mathop{\lim }\limits_{{k \rightarrow \infty }}{x}_{k}^{ * }b{x}_{k} = a \) . Thus, for any \( \varepsilon > 0 \), we can f... | Yes |
Theorem 2.2 Let \( \mathcal{A} \) be a \( {C}^{ * } \) -algebra and let \( \alpha \in \lbrack 1,\infty ) \) . If \( \mathcal{A} \) has the \( \alpha \) -comparison property, then \( \mathcal{A}/\mathcal{I} \) has the \( \alpha \) -comparison property for any closed two-sided ideal \( \mathcal{I} \) of \( \mathcal{A} \)... | Proof Let \( \pi : \mathcal{A} \rightarrow \mathcal{A}/\mathcal{I} \) be the quotient map. Then there is a morphism induced by \( \pi \) : \( \mathrm{W}\left( \pi \right) : \mathrm{W}\left( \mathcal{A}\right) \rightarrow \mathrm{W}\left( {\mathcal{A}/\mathcal{I}}\right) \) . Suppose \( \langle \bar{a}\rangle ,\langle \... | Yes |
Lemma 1.3 Let \( R \) be a separative exchange ring in which 2 is invertible, and let \( a{ - {a}^{3}} \in R \) be regular. If \( \left( {a - {a}^{3}}\right) R \propto r\left( a\right), R/{aR} \), then \( a \in R \) is unit-regular. | Proof Suppose that \( \left( {a - {a}^{3}}\right) R \propto r\left( a\right), R/{aR} \) . By virtue of Lemma 1.2, \( r\left( a\right) \oplus \left( {a - {a}^{3}}\right) R \cong \) \( R/{aR} \oplus \left( {a - {a}^{3}}\right) R \), where \( r\left( a\right), R/{aR} \) and \( \left( {a - {a}^{3}}\right) \) are all finite... | Yes |
Theorem 1.2 Let \( R \) be a separative exchange ring in which 2 is invertible, and let \( a - {a}^{3} \in \) \( R \) be regular. If \( R\left( {1 - {a}^{2}}\right) R \cap {RaR} = {Rr}\left( a\right) \cap \ell \left( a\right) R \cap {RaR} \), then \( a \in R \) is unit-regular. | Proof Suppose that \( R\left( {1 - {a}^{2}}\right) R \cap {RaR} = {Rr}\left( a\right) \cap \ell \left( a\right) R \cap {RaR} \) . Then we get\n\n\[ R\left( {a - {a}^{3}}\right) R \subseteq R\left( {1 - {a}^{2}}\right) R \cap {RaR} \subseteq \operatorname{Rr}\left( a\right) ,\]\n\nand so \( \left( {a - {a}^{3}}\right) R... | Yes |
Lemma 2.2 Let \( R \) be a ring, and let \( a \in R \) be regular. If\n\n(1) \( {aR}/{ar}\left( {a}^{2}\right) \) is projective;\n\n(2) \( \operatorname{ar}\left( {a}^{2}\right) \cong R/\left( {r\left( a\right) + {aR}}\right) \),\n\nthen \( a \in R \) is special clean. | Proof Construct \( X, Y, K, Z, e, h \) and \( v \) as in the proof of [7, Lemma 15.1.2]. Then \( a - e \in \) \( U\left( R\right) \) . Furthermore, we check that\n\n\[ \n{aR} \cap {eR} \subseteq \left( {K \oplus Z}\right) \cap {hv}\left( R\right) \subseteq \left( {K \oplus Z}\right) \cap \left( {X \oplus Y}\right) = 0.... | Yes |
Lemma 2.3 Let \( R \) be an exchange ring in which 2 is invertible, and let \( a - {a}^{3} \in R \) be regular. If \( {aR}/{ar}\left( {a}^{2}\right) \) and \( R/\left( {{aR} + r\left( a\right) }\right) \) are projective, then\n\n\[ \left( {a - {a}^{3}}\right) R \oplus \operatorname{ar}\left( {a}^{2}\right) \cong \left(... | Proof Let \( b = 1 - a \) and \( c = 1 + a \) . As in the proof of Lemma 1.2, there are \( x, y, z \in R \) such that \( a = {axa}, b = {byb} \) and \( c = {czc} \) . Furthermore, we can find some \( C \subseteq r\left( c\right), D \subseteq {zcR} \) such that \( R = \left( {r\left( a\right) + r\left( b\right) }\right)... | Yes |
Theorem 2.1 Let \( R \) be a separative exchange ring in which 2 is invertible, and let \( a - {a}^{3} \in \) \( R \) be regular. If\n\n(1) \( {aR}/{ar}\left( {a}^{2}\right) \) and \( R/\left( {{aR} + r\left( a\right) }\right) \) are projective;\n\n(2) \( \operatorname{Rar}\left( {a}^{2}\right) = \ell \left( {a}^{2}\ri... | Proof Since \( \operatorname{Rar}\left( {a}^{2}\right) = R\left( {a - {a}^{3}}\right) R \), we get \( a\left( {1 - {a}^{2}}\right) R \subseteq \operatorname{Rar}\left( {a}^{2}\right) \) . Thus we can find some \( {x}_{1},{x}_{2},\cdots ,{x}_{n} \in \mathbb{R};\;{y}_{1},{y}_{2},\cdots ,{y}_{n} \in {ar}\left( {a}^{2}\rig... | Yes |
Corollary 2.1 Let \( R \) be a separative regular ring in which 2 is invertible. Then each \( a \in R \) satisfying \( \operatorname{Rar}\left( {a}^{2}\right) = \ell \left( {a}^{2}\right) {aR} = R\left( {a - {a}^{3}}\right) R \) is special clean. | Proof Since \( R \) is regular, \( {aR}, r\left( a\right) ,{aR} + r\left( a\right) \) and \( K = {ar}\left( {a}^{2}\right) = {aR} \cap r\left( a\right) \) are direct summands of \( {R}_{R} \) . By virtue of Lemma 2.2, we have some \( X \subseteq r\left( a\right) \) such that \( {aR} + r\left( a\right) = {aR} \oplus X \... | Yes |
Example 2.1 Let \( V \) be an infinite-dimensional vector space over a field \( {\mathbb{Z}}_{3} \), and let \( R = \) \( {M}_{3}\left( {{\operatorname{End}}_{{\mathbb{Z}}_{3}}\left( V\right) }\right) \) . Then \( R \) is one-sided unit-regular, i.e., for any \( x \in R \) there exists a right or left invertible \( u \... | \[ a = \left( \begin{array}{lll} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right) \in R. \] Then \( \operatorname{Rar}\left( {a}^{2}\right) \subseteq R\left( {a - {a}^{3}}\right) R \) . Obviously, \( {a}^{2} = {a}^{3} \), and so \( R\left( {a - {a}^{3}}\right) R = {Ra}\left( {1 - a}\right) R \subseteq \operatorna... | Yes |
Theorem 2.2 Let \( R \) be a separative exchange ring in which 2 is invertible, and let \( a - {a}^{3} \in \) \( R \) be regular. If\n\n(1) \( {aR}/{ar}\left( {a}^{2}\right) \) and \( R/\left( {{aR} + r\left( a\right) }\right) \) are projective;\n\n(2) \( \operatorname{Rar}\left( {a}^{2}\right) = \ell \left( {a}^{2}\ri... | Proof By hypothesis, \( R\operatorname{ar}\left( {a}^{2}\right) = \ell \left( {a}^{2}\right) {aR} = R\left( {1 - {a}^{2}}\right) R \) . Construct a map \( \varphi : \left( {1 - {a}^{2}}\right) R \rightarrow \) \( \left( {a - {a}^{3}}\right) R \) given by \( \varphi \left( {\left( {1 - {a}^{2}}\right) r}\right) = \left(... | Yes |
Corollary 2.2 Let \( R \) be a separative regular ring in which 2 is invertible. Then each \( a \in R \) satisfying \( \operatorname{Rar}\left( {a}^{2}\right) = \ell \left( {a}^{2}\right) {aR} = R\left( {1 - {a}^{2}}\right) R \) is special clean. | Proof As in the proof of Corollary 2.1, we prove that \( {aR}/{ar}\left( {a}^{2}\right) \) and \( R/\left( {{aR} + r\left( a\right) }\right) \) are projective. Therefore the result follows by Theorem 2.2. | No |
Theorem 2.1 If \( {\bar{E}}_{ik} \) and \( {\bar{F}}_{ik} \) are of type \( m \) for any \( i \in I \), then \( H\acute{ \otimes }{wU} \) is a weak Hopf superalgebra with the weak antipode \( T \), but not a Hopf superalgebra. In general, \( H\acute{ \otimes }{wU} \) is a graded bialgebra. | The theorem follows from Lemmas 2.1-2.2 below. | No |
Corollary 2.2 The algebra \( {wU} \) is a graded algebra with the comultiplication and the counit given by | \n\[
\Delta \left( J\right) = J \otimes J,\;\Delta \left( {K}_{i}\right) = {K}_{i} \otimes {K}_{i},\;\Delta \left( {\bar{K}}_{i}\right) = {\bar{K}}_{i} \otimes {\bar{K}}_{i},
\]\n\n\( \left( {2.14}\right) \)\n\n\[
\Delta \left( {D}_{i}\right) = {D}_{i} \otimes {D}_{i},\;\Delta \left( {\bar{D}}_{i}\right) = {\bar{D}}_{i... | Yes |
Proposition 3.1 \( \mathcal{U} = \mathcal{U}{\bar{J}}^{m} \oplus \mathcal{U}\left( {1 \otimes 1 - {\bar{J}}^{m}}\right) \) is a direct sum of algebras. Furthermore, if all \( {\widetilde{b}}_{ik} = {\widetilde{c}}_{jl} = {\widetilde{g}}_{ik} = {\widetilde{h}}_{jl} = 1 \), then the subalgebra \( \mathcal{U}{\bar{J}}^{m}... | Proof Since \( {\bar{J}}^{m} \) is a central idempotent element of \( \mathcal{U} \), then \( \mathcal{U} = \mathcal{U}{\bar{J}}^{m} \oplus \mathcal{U}\left( {1 \otimes 1 - {\bar{J}}^{m}}\right) \) . Note that\n\n\[ \nx{J}^{m} = {J}^{m}x = x,\;x = {K}_{i},{\bar{K}}_{i},{D}_{i},{\bar{D}}_{i},{E}_{ik},{F}_{ik} \]\n\nfor ... | Yes |
Proposition 3.2 Let \( \widehat{J} = \frac{1}{m}\mathop{\sum }\limits_{{r = 1}}^{m}{\bar{J}}^{r} \) . If all \( {\widetilde{b}}_{ik} = {\widetilde{c}}_{jl} = {\widetilde{g}}_{ik} = {\widetilde{h}}_{jl} = 1 \), then \( \mathcal{U}\widehat{J} \) is isomorphic to \( H \otimes {U}_{q}\left( \mathcal{G}\right) \) as algebra... | Proof It is easy to prove that \( \widehat{J} \) is a central idempotent element, and \( \mathcal{U}\widehat{J} \) becomes a subalgebra of \( \mathcal{U} \) . Similar to Proposition 3.1, there is a map \( \rho \) from \( H \otimes {U}_{q}\left( \mathcal{G}\right) \) to \( \mathcal{U}\widehat{J} \) defined by\n\n\[ \rho... | Yes |
Proposition 4.1 The multiplicative set \( \mathcal{L} \) is a right Ore set and\n\n\[ \operatorname{Ass}\left( \mathcal{L}\right) = \left( {1 \otimes 1 - {\bar{J}}^{m}}\right) \] | Proof It is obvious that \( \mathcal{L} \) is a right Ore set. Then under the right Ore condition Ass \( \left( \mathcal{L}\right) \) turns out a two-sided ideal. Since \( \bar{J}\left( {1 \otimes 1 - {\bar{J}}^{m}}\right) = 0,\left( {1 \otimes 1 - {\bar{J}}^{m}}\right) \subseteq \operatorname{Ass}\left( \mathcal{L}\ri... | Yes |
Proposition 4.2 The algebra \( {U}_{q}\left( {H,\mathcal{G}}\right) \) is a graded bialgebra with the comultiplication, the counit given by\n\n\[ \n\Delta \left( {h \otimes 1}\right) = \sum \left( {{h}_{1} \otimes 1}\right) \left( {{h}_{2} \otimes 1}\right) ,\;\text{ for }h \in H, \n\]\n\n\[ \n\Delta \left( \bar{J}\rig... | Proof It is obvious that \( {U}_{q}\left( {H,\mathcal{G}}\right) \) is isomorphic to the right quotient ring \( \mathcal{U}/\mathcal{L} \) . Then \( T \) is an antipode of this Hopf algebra \( {U}_{q}\left( {H,\mathcal{G}}\right) \) . | Yes |
Proposition 4.3 (i) Suppose all \( {\widetilde{b}}_{ik} = {\widetilde{c}}_{jl} = {\widetilde{g}}_{ik} = {\widetilde{h}}_{jl} = 1 \) and \( {t}_{i} = 0 \), then \( {U}_{q}\left( {H,\mathcal{G}}\right) \) becomes a Hopf superalgebra. Moreover, the Hopf superalgebra \( {U}_{q}\left( {H,\mathcal{G}}\right) \) is isomorphic... | Proof It is easy to verify that \( T \) is an antipode of \( {U}_{q}\left( {H,\mathcal{G}}\right) \) if \( H\acute{ \otimes }{wU} \) is a weak Hopf algebra. The principal ideal \( \left( \bar{J}\right) \) is a cyclic group with order \( m \), and \( \left( \bar{J}\right) \cong {\mathbb{Z}}_{m} \) . | No |
Lemma 1.1 Suppose that the assumptions \( \left( {\mathrm{A}}_{2}\right) ,\left( {\mathrm{A}}_{4}\right) \) and \( \left( {\mathrm{A}}_{5}\right) \) hold. Then any \( {\left( \mathrm{{PS}}\right) }_{c} \) - sequence \( \left\{ {u}_{n}\right\} \subset X \) of \( I \) is bounded in \( X \) . | Proof Suppose that \( \left\{ {u}_{n}\right\} \subset X \) is a \( {\left( \mathrm{{PS}}\right) }_{c} \) -sequence of \( I \), that is,\n\n\[ I\left( {u}_{n}\right) = c + o\left( 1\right) ,\;\left\langle {{I}^{\prime }\left( {u}_{n}\right) ,{u}_{n}}\right\rangle = o\left( 1\right) . \]\n\n(1.1)\n\nAssume \( \| {u}_{n}\... | Yes |
Lemma 1.3 Suppose that \( \left( {\mathrm{A}}_{1}\right) ,\left( {\mathrm{A}}_{2}\right) ,\left( {\mathrm{A}}_{4}\right) \) and \( \left( {\mathrm{A}}_{6}\right) \) hold. Then any \( {\left( \mathrm{{PS}}\right) }_{c} \) -sequence \( \left\{ {u}_{n}\right\} \subset X \) of \( I \) has a convergent subsequence in \( X \... | Proof Suppose that \( \left\{ {u}_{n}\right\} \subset X \) is a (PS)_c-sequence of \( I \), then (1.1) holds. We now show that \( \{ {u}_{n}\} \) is bounded in \( X. \) Assume \( \| {u}_{n}\| \rightarrow \infty \) by contradiction. Let \( {\widehat{u}}_{n} = \frac{{u}_{n}}{\| {u}_{n}\| }. \) Then \( \| {\widehat{u}}_{n... | Yes |
Lemma 2.2 Assume that \( \left( {\mathrm{A}}_{2}\right) \) and \( \left( {\mathrm{A}}_{4}\right) \) are fulfilled. Then for any finite dimensional subspace \( \widetilde{X} \subset X \), there holds\n\n\[ \mathop{\lim }\limits_{{\parallel u\parallel \rightarrow \infty }}I\left( u\right) = - \infty ,\;\forall u \in \wid... | Proof We prove (2.1) by contradiction. Assume that for some sequence \( \left\{ {u}_{n}\right\} \subset \widetilde{X} \) with \( \begin{Vmatrix}{u}_{n}\end{Vmatrix} \rightarrow \infty \), there exists \( M > 0 \) such that \( I\left( {u}_{n}\right) \geq - M \) for all \( n \in \mathbb{N} \) . Let \( {\widehat{u}}_{n} =... | Yes |
Lemma 2.3 If \( 1 \leq s < {p}^{ * } \), then\n\n\[ \n{\beta }_{k}\left( s\right) \mathrel{\text{:=}} \mathop{\sup }\limits_{{u \in {Z}_{k},\parallel u\parallel = 1}}\parallel u{\parallel }_{s} \rightarrow 0,\;\text{ as }k \rightarrow \infty .\n\] | Proof Using the fact that \( {Z}_{k + 1} \subset {Z}_{k} \), we can easily deduce \( 0 \leq {\beta }_{k + 1} \leq {\beta }_{k} \) . Thus, \( {\beta }_{k} \rightarrow {\beta }_{0} \geq 0 \), as \( k \rightarrow \infty \) . If \( {\beta }_{0} > 0 \), by the definition of \( {\beta }_{k} \), there exists \( {u}_{k} \in {Z... | Yes |
Lemma 2.4 There exist constant \( \rho ,\alpha > 0 \) such that \( {\left. I\right| }_{\partial {B}_{\rho } \cap {Z}_{k}} \geq \alpha \) . | Proof By (1.4) and (2.2), we have\n\n\[ I\left( u\right) = \frac{a}{p}\parallel u{\parallel }^{p} + \frac{\lambda }{p\left( {\tau + 1}\right) }\parallel u{\parallel }^{p\left( {\tau + 1}\right) } - {\int }_{\Omega }F\left( {x, u}\right) \mathrm{d}x \]\n\n\[ \geq \frac{a}{p}\parallel u{\parallel }^{p} - \frac{{C}_{1}}{p... | Yes |
Theorem 0.1 Let \( n = d \) and \( \Gamma \left( y\right) = P\left( {\varphi \left( {\rho \left( y\right) }\right) }\right) \otimes {y}^{\prime } = ({P}_{1}\left( {\varphi \left( {\rho \left( y\right) }\right) }\right) {y}_{1}^{\prime },{P}_{2}\left( {\varphi \left( {\rho \left( y\right) }\right) }\right) {y}_{2}^{\pri... | \[ \parallel {T}_{h,\Omega ,\Gamma }\left( f\right) {\parallel }_{{L}^{p}\left( {\mathbb{R}}^{n}\right) } \leq {C}_{p}{\left( q - 1\right) }^{-1}{\left( \gamma - 1\right) }^{-1}\parallel \Omega {\parallel }_{{L}^{q}\left( {\mathbb{S}}^{n - 1}\right) }\parallel h{\parallel }_{{\Delta }_{\gamma }\left( {\mathbb{R}}^{ + }... | Yes |
Corollary 0.1 Let \( n = d \) and \( \Gamma \left( y\right) = \left( {{P}_{1}\left( {\rho \left( y\right) }\right) {y}_{1}^{\prime },{P}_{2}\left( {\rho \left( y\right) }\right) {y}_{2}^{\prime },\cdots ,{P}_{n}\left( {\rho \left( y\right) }\right) {y}_{n}^{\prime }}\right) \) with \( {P}_{i} \) being real-valued polyn... | \[ {\begin{Vmatrix}{T}_{h,\Omega ,\Gamma }\left( f\right) \end{Vmatrix}}_{{L}^{p}\left( {\mathbb{R}}^{n}\right) } \leq {C}_{p}\left( {1 + {N}_{1}\left( h\right) }\right) \left( {1 + \parallel \Omega {\parallel }_{L{\log }^{ + }L\left( {S}^{n - 1}\right) }}\right) \parallel f{\parallel }_{{L}^{p}\left( {\mathbb{R}}^{n}\... | Yes |
Corollary 0.2 Let \( n = d \) and \( \Gamma \left( y\right) = {A}_{d}^{{P}_{N}\left( \varphi \right) }\left( y\right) \) with \( \varphi \in \mathfrak{F} \) and \( {P}_{N}\left( t\right) = \mathop{\sum }\limits_{{i = 1}}^{N}{a}_{i}{t}^{i} \) and \( {P}_{N}\left( t\right) > 0 \) if \( t \neq 0 \) . Suppose that \( \Omeg... | \[ {\begin{Vmatrix}{T}_{h,\Omega ,\Gamma }\left( f\right) \end{Vmatrix}}_{{L}^{p}\left( {\mathbb{R}}^{n}\right) } \leq {C}_{p}\left( {1 + {N}_{1}\left( h\right) }\right) \left( {1 + \parallel \Omega {\parallel }_{L{\log }^{ + }L\left( {S}^{n - 1}\right) }}\right) \parallel f{\parallel }_{{L}^{p}\left( {\mathbb{R}}^{n}\... | Yes |
Lemma 1.2 Let \( A > 0, l \in \mathbb{N} \smallsetminus \{ 0\} \) and \( \left\{ {{\sigma }_{k, s} : 0 \leq s \leq l\text{and}k \in \mathbb{Z}}\right\} \) be a family of uniformly bounded Borel measures on \( {\mathbb{R}}^{n} \) with \( {\sigma }_{k,0}\left( \xi \right) = 0 \) for every \( k \in \mathbb{Z} \) and \( \x... | Proof We only prove Lemma 1.2 for the case \( {\left\{ {a}_{k, s}\right\} }_{k \in \mathbb{Z}} \) satisfying the condition (a). The other case can be obtained similarly. For \( s \in \{ 1,2,\cdots, l\} \), let \( {r}_{s} = \operatorname{rank}\left( {L}_{s}\right) \) . There exist two nonsingular linear transformations ... | Yes |
Theorem 0.1 Let \( \left( {{M}^{n}, g}\right) \) be an \( n \) -dimensional Riemannian manifold \( n \geq 2 \) with \( \operatorname{Ric}\left( M\right) \geq \) \( - k \) for some \( k \geq 0 \) . Suppose that \( u \) is any positive solution to the equation (0.1) in \( {Q}_{R, T} \equiv \) \( B\left( {{x}_{0}, R}\righ... | \[ \frac{\left| \nabla u\right| }{u} \leq C\left( {\frac{1}{R} + \frac{1}{\sqrt{T}} + \sqrt{k}}\right) \left( {1 + \ln \frac{\mathcal{D}}{u}}\right) \;\text{ in }{Q}_{\frac{R}{2},\frac{T}{2}}. \] | Yes |
Theorem 0.2 Let \( \\left( {{M}^{n}, g}\\right) \) be an \( n \) -dimensional non-compact Riemannian manifold with \( \\operatorname{Ric}\\left( M\\right) \\geq - k \) for some constant \( k \\geq 0 \) . Suppose that \( u\\left( {x, t}\\right) \) is a positive smooth solution to the parabolic equation (0.1) in \( {Q}_{... | \[ \\frac{\\left| \\nabla u\\right| }{u} \\leq \\left( {\\frac{\\widetilde{c}}{R}\\beta + \\frac{c\\left( {\\alpha ,\\delta }\\right) }{R} + \\frac{c\\left( \\delta \\right) }{\\sqrt{T}} + c\\left( \\delta \\right) {\\left( \\left| a\\right| + k\\right) }^{\\frac{1}{2}} + c\\left( \\delta \\right) {\\left| a\\right| }^... | Yes |
Theorem 0.3 Let \( \left( {{M}^{n}, g}\right) \) be an \( n \) -dimensional complete Riemannian manifold \( n \geq 2 \) with \( \operatorname{Ric}\left( M\right) \geq - k \) for some constant \( k \geq 0 \) . Suppose that \( u\left( {x, t}\right) \) is a positive smooth solution to the equation (0.1) in \( {Q}_{R, T} \... | \[ \frac{\left| \nabla u\right| }{u} \leq c\left( {\frac{1}{R} + \frac{1}{\sqrt{T}} + \sqrt{k} + \sqrt{\left| a\right| }{\tau }^{\frac{\gamma }{4}}}\right) \left( {1 + \ln \frac{\mathcal{D}}{u}}\right) \] in \( {Q}_{\frac{R}{2},\frac{T}{2}} \), where \( \tau = \max \{ 1,{\left( \ln \mathcal{D}\right) }^{2}\} \) | Yes |
Corollary 0.1 Let \( \left( {{M}^{n}, g}\right) \) be an \( n \) -dimensional complete Riemannian manifold \( n \geq 2 \) with \( \operatorname{Ric}\left( M\right) \geq - k \) for some constant \( k \geq 0 \) . Suppose that \( u \) is a positive and bounded solution to the equation (0.1) and \( 1 \leq \gamma < \frac{3}... | \[ u\left( {{x}_{1}, t}\right) \leq {u}^{\eta }\left( {{x}_{2}, t}\right) {e}^{\alpha \left( {1 - \eta }\right) } \] where \( \alpha = 1 + \ln \mathcal{D},\eta = \exp \left\{ {-c\left( n\right) \rho \left( {\frac{1}{{t}^{\frac{1}{2}}} + \sqrt{k} + \sqrt{\left| a\right| }{\tau }^{\frac{\gamma }{4}}}\right) }\right\} \),... | Yes |
Theorem 1.1 Let \( G \) be a planar graph and \( t \) be an integer. If \( t \geq \Delta \) and \( G \) contains no \( 3,4,6 \) -cycles and no intersecting 5-cycles, then \( {\chi }_{s}^{\prime }\left( G\right) \leq {3t} + 1 \) . | Proof We proceed by contradiction. Let \( G \) be a counterexample to the theorem that minimizes \( \left| {E\left( G\right) }\right| + \left| {V\left( G\right) }\right| \) . So a graph obtained from \( G \) by removing any edge or vertex has a strong \( \left( {{3t} + 1}\right) \) -edge-coloring \( \sigma \) . By mini... | Yes |
Corollary 1.1 Let \( {G}_{1} \) be an \( {r}_{1} \) -regular graph on \( {n}_{1} \) vertices and \( {m}_{1} \) edges, and \( {G}_{2} \) be an \( {r}_{2} \) -regular graph on \( {n}_{2} \) vertices. Then the \( A \) -spectrum of \( {G}_{1}\bar{ \star }{G}_{2} \) consists of\n\n(i) 0, repeated \( {m}_{1} - {n}_{1} \) tim... | Proof Since \( {G}_{2} \) is \( {r}_{2} \) -regular, then by \( \left\lbrack {3\text{, Proposition 2}}\right\rbrack \) ,\n\n\[ \n{\Gamma }_{A\left( {G}_{2}\right) }\left( x\right) = \frac{{n}_{2}}{x - {r}_{2}} \n\]\n\nThe only pole of \( {\Gamma }_{A\left( {G}_{2}\right) }\left( x\right) \) is the maximal eigenvalue \(... | Yes |
Let \( G \) be an \( r \) -regular graph on \( n \) vertices and \( m \) edges, and \( {K}_{p, q} \) be a complete bipartite graph with \( p, q \geq 1 \) . Then the \( A \) -spectrum of \( G\bar{ \star }{K}_{p, q} \) consists of\n\n(i) 0, repeated \( \left( {p + q - 3}\right) n + m \) times;\n\n(ii) four roots of the e... | Proof By [14, Proposition 8], we have\n\n\[ \n{\Gamma }_{A\left( {K}_{p, q}\right) }\left( x\right) = \frac{\left( {p + q}\right) x + {2pq}}{{x}^{2} - {pq}}. \n\]\n\nNote that the \( A \) -spectrum of \( {G}_{2} = {K}_{p, q} \) is \( \sigma \left( {A\left( {K}_{p, q}\right) }\right) = \left( {{0}^{\left( p + q - 2\righ... | Yes |
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