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Problem 2.1 What is the maximum constant \( K \) such that all simple CPT-graphs \( G \) have \( {\chi }_{\mathrm{s}}^{\prime }\left( G\right) \geq K \) ? | The foregoing discussion illustrates that \( K \geq 9 \) . We think that the exact value of \( K \) should be 11. The following Fig. 5 presents a CPT-graph \( {G}^{* * } \) with \( {\chi }_{\mathrm{s}}^{\prime }\left( {G}^{* * }\right) = {11} \) . Thus, if our conjecture holds, then the result is the best possible. | No |
Corollary 2.1 Under Assumption 2.1, we assume that the spectral measure \( \mu \) associated with \( \dot{W} \) satisfies Hypothesis 2.1, then with \( \beta \in \left( {0,\frac{1}{4}}\right) \cup \left( {\frac{3}{4},1}\right) \), for fixed \( t \in {\mathbb{R}}_{ + } \), the law of \( u\left( {t, x}\right) \) , the sol... | Proof In order to prove this corollary, according to Márquez-Carreras et al. [20, Theorem 2.1], we need to check that, for fixed \( t \in {\mathbb{R}}_{ + } \) and any \( x \in {\mathbb{R}}^{d} \), there exist \( {\rho }_{1} \) and \( {\rho }_{2} \) such that \( {\rho }_{2} < {\rho }_{1} < 2{\rho }_{2} \), positive con... | Yes |
Lemma 1.4 Let \( \rho \) be a congruence on \( M \) . Then\n\n(i) \( {\rho }_{I} \) is an equivalence on \( I \) ;\n\n(ii) \( {\rho }_{\Lambda } \) is an equivalence on \( \Lambda \) . | Proof It is clear that \( {\rho }_{I} \) is reflexive and symmetric. To show that it is transitive, note first that if \( \left( {i, j}\right) \in {\rho }_{I} \) and \( \left( {j, k}\right) \in {\rho }_{I} \), then for any \( \lambda \in \Lambda ,\left( {i,{p}_{\lambda i}^{-1},\lambda }\right) \rho \left( {j,{p}_{\lamb... | Yes |
Lemma 1.5 If \( \rho \) is a good congruence on \( M \), then \( {\rho }_{T} \) is a good congruence on \( T \) . | Proof Obviously, \( {\rho }_{T} \) is an equivalence relation. If \( a, b, c \in T \) and \( \left( {a, b}\right) \in {\rho }_{T} \), then \( \left( {1, a,1}\right) \rho \left( {1, b,1}\right) \) . Multiplying the relation on the left by \( \left( {1, c{p}_{11}^{-1},1}\right) \) and on the right by \( \left( {1,{p}_{11... | Yes |
Lemma 3.5 If \( S \) is regular cryptic semisuperabundant semigroups, then both \( \widetilde{\mathcal{L}} \) and \( \widetilde{\mathcal{R}} \) are congruences. | Proof By hypothesis \( \mu = \widetilde{\mathcal{H}} \) and \( S/\mu \) is a regular band. Let \( a, b, c \in S \) be such that \( a\widetilde{\mathcal{L}}b \) . Then \( {a}^{0}\mathcal{L}{b}^{0} \) and so \( {a}^{0}\mu \mathcal{L}{b}^{0}\mu \) . It follows that \( {a\mu } = {a}^{0}\mu \mathcal{L}{b}^{0}\mu = {b\mu } \... | Yes |
Proposition 2.1 For a \( * \) -semiring \( R \), the following conditions are equivalent:\n\n(1) \( R \) has \( * \) -IFP;\n\n(2) \( {eRe} \) has \( * \) -IFP for any projection \( e \in R \) ;\n\n(3) \( {eR} \) has \( * \) -IFP for any idempotent \( e \in R \) . | Proof (1) \( \Leftrightarrow \) (2): Assume that \( R \) is a \( * \) -semiring having \( * \) -IFP and \( {e}^{2} = e = {e}^{ * } \) . Let \( a, b \in R \) . Then eae, \( {ebe} \in {eRe} \) . If \( \left( {eae}\right) \left( {ebe}\right) = 0 \), then we have \( \left( {eae}\right) R{\left( ebe\right) }^{ * } = 0 \) si... | Yes |
Theorem 2.1 A function \( f \in {H}^{2}\left( {D,\varphi }\right) \) is minimal with respect to \( w \) if and only if \( f \bot {H}^{2}{\left( D,\varphi \right) }_{w} \), or equivalently \( f \in {H}^{2}{\left( D,\varphi \right) }_{w}^{ \bot } \), where \( {H}^{2}{\left( D,\varphi \right) }_{w}^{ \bot } \) is the orth... | Proof For \( f \in {H}^{2}\left( {D,\varphi }\right) \), we decompose it as \( f = {f}_{1} + {f}_{2} \), with \( {f}_{1} \in {H}^{2}{\left( D,\varphi \right) }_{w}^{ \bot },{f}_{2} \in \) \( {H}^{2}{\left( D,\varphi \right) }_{w} \), then \( {f}_{1}\left( w\right) = f\left( w\right) \) . We have\n\n\[ \parallel f{\para... | Yes |
Theorem 3.1 Given \( w \in {D}^{\left\lbrack m\right\rbrack } \), we have\n\n(1) \( f \in {H}^{2}\left( {D,\varphi }\right) \) is minimal with respect to \( w \) if and only if \( f \bot {H}^{2}{\left( D,\varphi \right) }_{w} \), i.e., \( f \in \) \( {H}^{2}{\left( D,\varphi \right) }_{w}^{ \bot } \) ;\n\n(2) for \( {a... | The proof of Theorem 3.1 is the same as that of Theorem 2.1 and we do not repeat it here. | No |
Lemma 3.1 Let \( H \) be a Hilbert space and \( {\left\{ {V}_{i}\right\} }_{i \in I} \) be a collection of closed linear subspaces of \( H \) . Then we have\n\n(1)\n\n\[ \n{\left( \mathop{\sum }\limits_{{i \in I}}{V}_{i}\right) }^{ \bot } = \mathop{\bigcap }\limits_{{i \in I}}{V}_{i}^{ \bot } \n\]\n\n(2)\n\n\[ \n{\left... | Proof The proof of (1) is trivial, and we try to derive (2) from (1). It suffices to prove that \( \mathop{\bigcap }\limits_{{i \in I}}{V}_{i} = {\left( \overline{\mathop{\sum }\limits_{{i \in I}}{V}_{i}^{ \bot }}\right) }^{ \bot } \) . Note that \( {\left( \overline{\mathop{\sum }\limits_{{i \in I}}{V}_{i}^{ \bot }}\r... | No |
Theorem 3.3 The following statements are equivalent:\n\n(1) For any \( {a}_{1},\cdots ,{a}_{m} \in \mathbb{C} \), there exists \( f \in {H}^{2}\left( {D,\varphi }\right) \) such that \( f\left( {w}^{i}\right) = {a}_{i}, i = 1,\cdots, m \) ;\n\n(2) the matrix \( {K}_{\varphi }\left( w\right) \) defined above is nonsingu... | Proof By Theorem 3.2, finding \( f \in {H}^{2}\left( {D,\varphi }\right) \) satisfying the condition in (1) is equivalent to finding \( {c}_{1},\cdots ,{c}_{n} \in \mathbb{C} \) such that\n\n\[ \n{c}_{1}{K}_{\varphi }\left( {{w}^{j},{w}^{1}}\right) + \cdots + {c}_{m}{K}_{\varphi }\left( {{w}^{j},{w}^{m}}\right) = {a}_{... | Yes |
Corollary 1.1 Let \( M \) be an \( n \) -dimensional compact totally real submanifold with parallel mean curvature in \( {\mathbb{{CP}}}^{n} \) . If \( \tau \left( x\right) \leq 2 \) for all \( x \in M \), then the second fundamental of \( M \) is parallel. | Proof Since \( {\bar{K}}_{\min } = 1 \), from Lemma 1.2 and the assumption, we have\n\n\[ \n{K}_{M} \geq {\bar{K}}_{\min } - \frac{\tau }{2} \geq 0.\n\]\n\nBy the aforementioned results of Ohnita [17] and Urbano [22], we complete the proof of this corollary. | No |
Lemma 1.1 \( {}^{\left\lbrack {10},\text{ Lemma 3.1.1 }\right\rbrack } \) Let \( F \) be a Finsler metric on \( M \) and \( {F}^{ * } \) its dual Finsler metric. For any vector \( y \in {T}_{x}M \smallsetminus \{ 0\}, x \in M \), the covector \( \xi = {g}_{y}\left( {y, \cdot }\right) \in {T}_{x}^{ * }M \) satisfies\n\n... | Naturally, by Lemma 1.1, we define the Legendre transformation \( \mathcal{L} : {TM} \rightarrow {T}^{ * }M \) on Finsler manifold \( \left( {M, F}\right) \) by\n\n\[ \mathcal{L}\left( y\right) \mathrel{\text{:=}} \left\{ \begin{array}{ll} {g}_{y}\left( {y, \cdot }\right) , & y \neq 0 \\ 0, & y = 0 \end{array}\right.\]... | Yes |
Corollary 2.2 If \( H \) is disconnected, then \( {\bar{\pi }}_{\varphi } \) is an automorphism of \( \bar{G} \) . | Proof Obviously \( {\bar{\pi }}_{\varphi } \) is well defined by Lemma 2.4. To show \( {\bar{\pi }}_{\varphi } \) is one-to-one, it suffices to show \( {\bar{\pi }}_{\varphi } \) is surjective for \( \bar{G} \) is finite. Since \( \varphi \) is an automorphism of \( G \circ H \), for every \( \bar{t} \in V\left( \bar{G... | Yes |
Lemma 2.5 If \( H \) is disconnected, then \( \operatorname{Aut}\left( {G \circ H}\right) /{\operatorname{Fix}}_{\widetilde{R}}\left( {G \circ H}\right) \cong \overline{\operatorname{Aut}\left( G\right) } \) . | Proof Let \( \tau : \operatorname{Aut}\left( {G \circ H}\right) \rightarrow \operatorname{Aut}\left( \bar{G}\right) \) be the map defined by \( \tau \left( \varphi \right) = {\bar{\pi }}_{\varphi } \), where \( \varphi \in \) \( \operatorname{Aut}\left( {G \circ H}\right) \) and \( {\bar{\pi }}_{\varphi } \) is given i... | Yes |
Lemma 1.1 For \( a, b \in A \) ,\n\n\[ \n{\Delta }_{l}\left( {ab}\right) = a \cdot {\Delta }_{l}\left( b\right) + {\Delta }_{l}\left( a\right) \cdot b. \n\] | Proof We have\n\n\[ \na \cdot {\Delta }_{l}\left( b\right) + {\Delta }_{l}\left( a\right) \cdot b = a \cdot \left( {b{l}^{i} \otimes {l}^{i} - {l}^{i} \otimes {l}^{i}b}\right) + \left( {a{l}^{i} \otimes {l}^{i} - {l}^{i} \otimes {l}^{i}a}\right) \cdot b\;(\text{by (1} \n\]\n\n\[ \n= {ab}{l}^{i} \otimes {l}^{i} - a{l}^{... | Yes |
Lemma 1.2 The pair \( \left( {A,{\Delta }_{l}}\right) \) is a coalgebra (without counit). | Proof It is enough to show the coassociative law:\n\n\[ \left( {\mathrm{{id}} \otimes {\Delta }_{l}}\right) {\Delta }_{l}\left( a\right) = \left( {{\Delta }_{l} \otimes \mathrm{{id}}}\right) {\Delta }_{l}\left( a\right) \;\text{ for all }a \in A. \]\n\nApplying (1.2) and \( {l}^{i + 1} = 0 \), we have\n\n\[ {\Delta }_{... | Yes |
Theorem 1.3 The quadruple \( \left( {A,\mathfrak{m},1,{\Delta }_{l}}\right) \) is an \( \varepsilon \) -unitary Hopf algebra of weight zero with the bijective antipode \( S = - \mathrm{{id}} \) . | Proof By Theorem 1.1, \( \left( {A,\mathfrak{m},1,{\Delta }_{l}}\right) \) is an \( \varepsilon \) -unitary bialgebra of weight zero. Further, for \( a \in A \) ,\n\n\[ \left( {\mathrm{{id}} * \mathrm{{id}}}\right) \left( a\right) = \mathfrak{m}\left( {\mathrm{{id}} \otimes \mathrm{{id}}}\right) {\Delta }_{l}\left( a\r... | Yes |
Theorem 1.4 Let \( 0 < p < \infty, q\left( \cdot \right) \in \mathcal{B}\left( {\mathbb{R}}^{n}\right) ,0 < \lambda < \infty , - n{\delta }_{1} + \lambda < \alpha < n{\delta }_{2} + \lambda \) and \( a \in \operatorname{BMO}\left( {\mathbb{R}}_{ + }^{n}\right) \) . If a linear operator \( T \) satisfies (1.3) in Theore... | By [3, Theorem 2.13] we can also obtain the following local version of Theorem 1.4. | No |
Example 1.2 Let \( \mathbb{V} : {V}_{1}\overset{\partial }{ \rightarrow }{V}_{0} \) be a 2-term chain complex of vector spaces, and let \[ {\mathrm{{gl}}}_{0}\left( \mathbb{V}\right) \mathrel{\text{:=}} \left\{ {\left( {{A}_{0},{A}_{1}}\right) \in \mathrm{{gl}}\left( {V}_{0}\right) \oplus \mathrm{{gl}}\left( {V}_{1}\ri... | From the definition, we see that \( {\operatorname{gl}}_{0}\left( \mathbb{V}\right) \) is the space of functors of \( \mathbb{V} \) and \( {\operatorname{gl}}_{1}\left( \mathbb{V}\right) \) is the space of natural transformations, so the functors and natural transformations of a 2-term chain complex constitute a strict... | Yes |
Proposition 1.2 \( {}^{\left\lbrack 6,\text{ Theorem 2.3 }\right\rbrack } \) Let \( \mathfrak{g} \) be a 3-dimensional vector space. There is a one-one correspondence between Lie algebra structures \( \left\lbrack {\cdot , \cdot }\right\rbrack \) on \( \mathfrak{g} \) and compatible pairs \( \left( {k, A}\right) \) suc... | Equation (1.3) gives the expression of the Lie bracket from a compatible pair. Conversely, given a Lie algebra \( \mathfrak{g} \), the compatible pair \( \left( {k, A}\right) \) is constructed as\n\n\[ k \mathrel{\text{:=}} \frac{1}{2}{i}_{\left( P - {P}^{ * }\right) }{\omega }^{ * },\;A \mathrel{\text{:=}} \frac{1}{2}... | Yes |
Lemma 2.4 Let \( n \geq 0 \) be a natural number and \( {\operatorname{pd}}_{{\Lambda }_{\left( 0,0\right) }}\left( {X, Y, f, g}\right) \leq n \) . Then \( {\operatorname{pd}}_{B}Y/\operatorname{Im}f \) \( \leq n \) and \( {\operatorname{pd}}_{A}X/\operatorname{Im}g \leq n \) . | Proof Suppose that \( {\operatorname{pd}}_{{\Lambda }_{\left( 0,0\right) }}\left( {X, Y, f, g}\right) \leq n \) . Then there exists the projective resolution\n\n\[ 0 \rightarrow \left( {{P}_{n}, M{ \otimes }_{A}{P}_{n},1,0}\right) \oplus \left( {N{ \otimes }_{B}{Q}_{n},{Q}_{n},0,1}\right) \rightarrow \cdots \]\n\n\[ \r... | Yes |
Theorem 4.1 Let \( {\Lambda }_{\left( 0,0\right) } = \left( \begin{matrix} A & {}_{A} & {}_{A}{N}_{B} \\ {}_{B} & {M}_{A} & B \end{matrix}\right) \) be a Morita context ring. If \( \left( {X, Y, f, g}\right) \) is a partial tilting \( {\Lambda }_{\left( 0,0\right) } \) -module, then both \( X/\operatorname{Im}g \) and ... | Proof Since \( \left( {X, Y, f, g}\right) \) is partial tilting, we get \( {\operatorname{pd}}_{{\Lambda }_{\left( 0,0\right) }}\left( {X, Y, f, g}\right) \leq 1 \) . So by Lemma \( {2.4},{\operatorname{pd}}_{B}Y/\operatorname{Im}f \leq 1 \) and \( {\operatorname{pd}}_{A}X/\operatorname{Im}g \leq 1 \) . Hence, we have\... | Yes |
(1) If \( f \) is monic and \( \left( {X, Y, f, g}\right) \) is a tilting \( {\Lambda }_{\left( 0,0\right) } \) -module, then \( Y/\operatorname{Im}f \) is tilting and \( M{ \otimes }_{A} \) \( N{ \otimes }_{B}B = 0. \) | Proof We only sketch the proof of (1), and omit the proof for Statement (2) which can be obtained similarly.\n\nBy Theorem 4.1, it is enough to discuss Condition (3) of Definition 1.1. Since \( \left( {X, Y, f, g}\right) \) is a tilting \( {\Lambda }_{\left( 0,0\right) } \) -module, there exists an exact sequence\n\n\[... | Yes |
Lemma 4.1 Let \( {\Lambda }_{\left( 0,0\right) } = \left( \begin{matrix} A & A & {N}_{B} \\ B & {M}_{A} & B \end{matrix}\right) \) be a Morita context ring such that the right modules \( {N}_{B} \) and \( {M}_{A} \) are flat, and let \( X \) be an \( A \) -module and \( Y \) be a \( B \) -module. Then we have:\n\n(1) \... | Proof We only prove (1). The proof for Statement (2) is similar.\n\nSince for every \( {\Lambda }_{\left( 0,0\right) } \) -module \( \left( {{X}^{\prime },{Y}^{\prime },{f}^{\prime },{g}^{\prime }}\right) \) and \( n \geq 0 \), by Lemma 2.3, we have an isomorphism\n\n\[ \n{\operatorname{Ext}}_{{\Lambda }_{\left( 0,0\ri... | No |
(1) If \( \left( {X, M{ \otimes }_{A}X,1,0}\right) \) is tilting, then \( X \) is tilting and \( {N}_{B} = 0 \) . | Proof We only prove (1). The proof for Statement (2) is similar.\n\nBy Lemma 4.1, it is enough to discuss Condition (3) of the Definition 2.1. Since \( \left( {X, M{ \otimes }_{A}}\right. \) \( X,1,0) \) is a tilting \( {\Lambda }_{\left( 0,0\right) } \) -module, there exists an exact sequence\n\n\[ 0 \rightarrow \left... | Yes |
Lemma 1.3 Assume that \( \left( {u, P}\right) \) is a weak solution of the following stationary Stokes equations:\n\n\[ \left\{ \begin{array}{l} - {\Delta u} + \nabla P = F,\;x \in {\mathbb{R}}^{2}, \\ \operatorname{div}u = 0, \end{array}\right. \] | It follows from the Stokes equations (2.10) that\n\n\[ \parallel \nabla P{\parallel }_{{L}^{2}} \leq C{\begin{Vmatrix}\left( a + 1\right) \left( {u}_{t} + u \cdot \nabla u\right) \end{Vmatrix}}_{{L}^{2}} + C\parallel b \cdot \nabla b{\parallel }_{{L}^{2}} \]\n\n\[ \leq C\left( {\parallel a{\parallel }_{{L}^{\infty }} +... | Yes |
Proposition 2.4 Under the assumptions of Theorem 0.1, the corresponding solution ( \( a, u \) ) of Equations (0.2) admits the following bounds for any \( t > 0 \) :\n\n\[ \n{\int }_{0}^{t}\parallel \nabla u{\parallel }_{{L}^{\infty }}\mathrm{d}\tau \leq {C}_{0}\left( t\right) \n\] \n\n\( \left( {2.25}\right) \)\n\n\[ \... | Proof First, it follows from Lemma 1.3 that\n\n\[ \n{\begin{Vmatrix}{\nabla }^{2}u\end{Vmatrix}}_{{L}^{4}} \leq C{\begin{Vmatrix}\left( a + 1\right) \left( {u}_{t} + u \cdot \nabla u\right) \end{Vmatrix}}_{{L}^{4}} + C\parallel b \cdot \nabla b{\parallel }_{{L}^{4}} \n\] \n\n\[ \n\leq C\left( {\parallel a{\parallel }_{... | Yes |
Theorem 1.1 Let \( D \) be a strongly connected directed pseudograph with at least one arc. Then for any two distinct arcs \( {a}_{1} \) and \( {a}_{m} \) of \( D, L\left( D\right) \) has a Hamiltonian path from \( {a}_{1} \) to \( {a}_{m} \) if and only if \( D \) has a Eulerian trail from arc \( {a}_{1} \) to arc \( ... | Proof Let \( D \) be a strongly connected directed pseudograph with \( V\left( D\right) = \left\{ {{u}_{1},{u}_{2},\cdots ,{u}_{n}}\right\} \) and \( A\left( D\right) = \left\{ {{a}_{1},{a}_{2},\cdots ,{a}_{m}}\right\} \) . If \( D \) has a Eulerian trail from arc \( {a}_{1} \) to arc \( {a}_{m} \), without loss of gen... | Yes |
Lemma 1.2 Let \( D \) be a strongly connected directed pseudograph with at least one arc and let \( {D}^{\prime } \) be the directed pseudograph obtained from \( D \) by adding some loops at each vertex. If there exist two distinct arcs \( {a}_{1} \) and \( {a}_{m} \) of \( D \) such that \( L\left( D\right) \) has no ... | Proof Let \( D \) be a strongly connected directed pseudograph with \( V\left( D\right) = \left\{ {{u}_{1},{u}_{2},\cdots ,{u}_{n}}\right\} \) . For any index \( i \) with \( 1 \leq i \leq n \) and integer \( {\ell }_{i} \geq 0 \), if \( {\ell }_{i} = 0 \), let \( {A}_{i} = \varnothing \) ; if \( {\ell }_{i} \geq 1 \),... | Yes |
Proposition 2.1 The line digraph \( L\left( {D\left\lbrack {U}_{n}\right\rbrack }\right) \) is strongly Hamiltonian-connected, whence it is also weakly Hamiltonian-connected, if \( D\left\lbrack {U}_{n}\right\rbrack \) satisfies one of the following\n\n(i) \( n = 1 \) with \( {\ell }_{1} \geq 1 \) ;\n\n(ii) \( n = 2 \)... | Proof If \( n = 1 \) and \( {\ell }_{1} \geq 1 \), then \( L\left( {D\left\lbrack {U}_{n}\right\rbrack }\right) \cong {K}_{{\ell }_{1}}^{ + } \) . By Proposition 1.1(ii), \( L\left( {D\left\lbrack {U}_{n}\right\rbrack }\right) \) is strongly Hamiltonian-connected. If \( n = 2 \) with \( {\ell }_{1} = {\ell }_{2} = 0 \)... | Yes |
Proposition 2.2 If \( D\left\lbrack {U}_{n}\right\rbrack \) satisfies of the following\n\n(i) \( n = 2 \) with \( {\ell }_{i} \geq 1 \) and \( {\ell }_{3 - i} = 0 \) for \( i \in \{ 1,2\} \) ;\n\n(ii) \( n = 3 \) with \( {\ell }_{1} = {\ell }_{2} = {\ell }_{3} = 0 \) .\n\nThen \( L\left( {D\left\lbrack {U}_{n}\right\rb... | Proof Suppose first that \( n = 2 \) with \( {\ell }_{i} \geq 1 \) and \( {\ell }_{3 - i} = 0 \) for \( i \in \{ 1,2\} \) . Without loss of generality, assume that \( {\ell }_{1} \geq 1 \) and \( {\ell }_{2} = 0 \) . Let \( {D}^{\prime } = L\left( {D\left\lbrack {U}_{n}\right\rbrack }\right) \left\lbrack \left\{ {{a}_{... | Yes |
Corollary 2.2 Let \( {G}^{ * } \) be a based digraph such that each edge is assigned with only one direction. Then the set of regular based loops on \( {G}^{ * } \) and the set of based regular loops on \( F\left( {G}^{ * }\right) \) are in one-to-one correspondence by \[ F : \left\{ {\text{ regular based loop }\phi : ... | Proof By Corollary 2.1, we have that for each regular based loop \( \Phi : {J}_{n}^{ * } \rightarrow F\left( {G}^{ * }\right) \), there exists a unique regular based loop \( \phi : {I}_{n}^{ * } \rightarrow {G}^{ * } \) such that \( F\left( \phi \right) = \Phi \) . On the other hand, let \( \phi : {I}_{n}^{ * } \righta... | Yes |
Lemma 2.2 Let \( {G}^{ * } \) be a digraph. Let \( \phi : {I}_{n}^{ * } \rightarrow {G}^{ * } \) and \( \psi : {I}_{m}^{ * } \rightarrow {G}^{ * }, n \geq m \) be two based line maps. If either of the following holds:\n\n(i) \( \phi \rightarrow \psi \), that is, there is a one-step direct \( C \) -homotopy from \( \phi... | Proof By Definitions 1.2 and 2.2, it can be proved directly. | No |
Example 2.1 Let \( {G}_{1} \) be a based digraph with base vertex \( {v}_{0} \), where\n\n\[ V\left( {G}_{1}\right) = \left\{ {{v}_{0},{v}_{1},{v}_{2},{v}_{3},{v}_{4},{v}_{5},{v}_{6}}\right\} \]\n\n\[ E\left( {G}_{1}\right) = \left\{ {{v}_{0} \rightarrow {v}_{1},{v}_{2} \rightarrow {v}_{1},{v}_{3} \rightarrow {v}_{2},{... | Then there does not exist a shrinking map \( h : {I}_{5}^{ * } \rightarrow {I}_{3}^{ * } \) (in the digraph sense). Hence, \( \phi \) and \( \psi \) are not \( C \) -homotopic. However, there exists a shrinking map \( h : {I}_{5}^{ * } \rightarrow {I}_{3}^{ * } \) (in the graph sense) such that \( h\left( 0\right) = 0,... | Yes |
Proposition 2.1 There is a homomorphism \( f : {\pi }_{1}\left( {G}^{ * }\right) \rightarrow {\pi }_{1}\left( {F\left( {G}^{ * }\right) }\right) \) for any digraph \( {G}^{ * } \) . | Proof By Lemma 2.2, for \( \phi : {I}_{n}^{ * } \rightarrow {G}^{ * } \) and \( \psi : {I}_{m}^{ * } \rightarrow {G}^{ * } \) ,\n\n\( \phi ,\psi \) represent the same element in \( {\pi }_{1}\left( {G}^{ * }\right) \)\n\n\( \Leftrightarrow \phi ,\psi \) are \( C \) -homotopy\n\n\( \Rightarrow F\left( \phi \right), F\le... | Yes |
Proposition 2.2 There is an epimorphism \( g : {\pi }_{1}\left( {N\left( {\Gamma }^{ * }\right) }\right) \rightarrow {\pi }_{1}\left( {\Gamma }^{ * }\right) \) for any based graph \( {\Gamma }^{ * } \) . | Proof By Proposition 2.1, \( g \) is a homomorphism. Hence, it is sufficient to prove that \( g \) is surjective. Suppose that \( \left\lbrack \Phi \right\rbrack = \left\lbrack \Psi \right\rbrack \) represents the same equivalent class in \( {\pi }_{1}\left( {\Gamma }^{ * }\right) \) . Then \( \Phi \) and \( \Psi \) ar... | Yes |
Lemma 3.3 The group homomorphism \( {p}_{ * } \) is injective. The image subgroup \( {p}_{ * }\left( {{\pi }_{1}\left( {\widetilde{G}}^{ * }\right) }\right) \) in \( {\pi }_{1}\left( {G}^{ * }\right) \) consists of the \( C \) -homotopy classes of based loops in \( {G}^{ * } \) whose lifts on \( {\widetilde{G}}^{ * } \... | Proof First, we prove that \( {p}_{ * } \) is injective. An element of the kernel of \( {p}_{ * } \) is a \( C \) -homotopy equivalent class represented by a based loop \( \widetilde{\phi } : {I}_{n}^{ * } \rightarrow {\widetilde{G}}^{ * } \) such that the projected based loop \( \phi = p \circ \widetilde{\phi } \) on ... | Yes |
Lemma 4.2 Let \( G \) be a connected digraph and \( \widetilde{G} \) be the universal cover of \( G \) with covering map \( p \) . Let \( \widetilde{v},{\widetilde{v}}^{\prime } \in {p}^{-1}\left( v\right), v \in V\left( G\right) \) . Then there exists a lifting of identity map on \( G \) which is an automorphism on th... | Proof By [6, Theorem 4.20(iii)], we have that \( {\pi }_{1}\left( {\widetilde{G},\widetilde{v}}\right) \cong {\pi }_{1}\left( {\widetilde{G},{\widetilde{v}}^{\prime }}\right) \) . By Theorem 3.1, there exist digraph maps \( \widetilde{f} : \left( {\widetilde{G},\widetilde{v}}\right) \rightarrow \left( {\widetilde{G},{\... | Yes |
Example 4.1 Let \( G \) be a connected digraph and \( \phi : {I}_{n} \rightarrow G \) a regular loop such that \( \phi \left( i\right) = {v}_{i},0 \leq i \leq n \) and \( {v}_{0} = {v}_{n} \in V\left( G\right) \). | By [17, p. 65] and [12, Section 1.1], the universal cover \( \widetilde{G} \) of \( G \) is an infinite tree as an undirected graph. Hence, by Lemma 3.6, we have that the number of sheets is countable. | Yes |
Lemma 4.3 Let \( G \) be a connected digraph and \( \widetilde{G} \) be the universal cover of \( G \) with covering map \( p \) . Then there exists a homomorphism from \( \mathcal{D}\left( {\widetilde{G}, p}\right) \) to \( {\pi }_{1}\left( G\right) \) . | Proof Let \( \widetilde{f} : \widetilde{G} \rightarrow \widetilde{G} \) be an arbitrary lifting map of identity map on \( G \) such that \( \widetilde{f}\left( \widetilde{v}\right) = {\widetilde{v}}^{\prime } \) . By Lemma 4.1, we know that \( \widetilde{f} \) is determined uniquely. By (A), it follows that \( p\left( ... | Yes |
Proposition 2.1 (1) The Gerstenhaber algebra structure on \( \left( {{\mathcal{A}}_{\Delta },\land ,\left\lbrack {\cdot , \cdot }\right\rbrack }\right) \) can be described by the following way. For any \( {\chi }^{I} \cdot \rho \left( {A}_{I}\right) \in {V}_{I}^{k}\left( \Delta \right) \) and \( {\chi }^{J} \cdot \rho ... | \[ \left( {{\chi }^{I} \cdot \rho \left( {A}_{I}\right) }\right) \land \left( {{\chi }^{J} \cdot \rho \left( {A}_{J}\right) }\right) = {\chi }^{I + J} \cdot \rho \left( {{A}_{I} \land {A}_{J}}\right) ,\] \[ \left\lbrack {{\chi }^{I} \cdot \rho \left( {A}_{I}\right) ,{\chi }^{J} \cdot \rho \left( {A}_{J}\right) }\right\... | Yes |
Example 2.3 Let \( {X}_{\Delta } \) be a smooth compact toric surface. The fan \( \Delta \) has 11 one dimensional cones, which are generated by\n\n\[ \n{e}_{1} = \left( {1,0}\right) ,\;{e}_{2} = \left( {0,1}\right) ,\;{e}_{3} = \left( {1,1}\right) ,\;{e}_{4} = \left( {2,1}\right) ,\;{e}_{5} = \left( {3,1}\right) ,\;{e... | There are three points \( \left( {0,0}\right) ,\left( {0,1}\right) \) and \( \left( {-1,1}\right) \) in \( {S}_{\Delta } \) . See Fig. 4.\n\n\n\nFigure 4 The polytope \( {P}_{\Delta } \) for Example 2.3\n\nWe have\n\n\... | Yes |
Proposition 1.1 Let \( a, b \) be elements of a left GC-lpp semigroup \( S \) . Then we have\n\n(1) \( a{\mathcal{R}}^{ * }b \) if and only if \( {a}^{ + } = {b}^{ + } \) ;\n\n(2) \( {\left( ab\right) }^{ + } = {\left( a{b}^{ + }\right) }^{ + } \) and \( {\left( ea\right) }^{ + } = e{a}^{ + } \) for \( e \in E\left( S\... | Proof (1) It is a direct consequence of the definition of left GC-lpp semigroup.\n\n(2) Noting that \( b{\mathcal{R}}^{ * }{b}^{ + } \), we have \( {ab}{\mathcal{R}}^{ * }a{b}^{ + } \) since \( {\mathcal{R}}^{ * } \) is a left congruence on \( S \) . By (1), it follows that \( {\left( ab\right) }^{ + } = {\left( a{b}^{... | Yes |
Lemma 2.3 Let \( \rho \) be an \( {\mathcal{R}}^{ * } \) -congruence on a left GC-lpp semigroup \( S \) . Then \( \rho \) is idempotent-separating if and only if \( \rho \subseteq {\mathcal{R}}^{ * } \) . | Proof Suppose that \( \rho \) is an idempotent-separating \( {\mathcal{R}}^{ * } \) -congruence on \( S \) . Let \( a, b \in S \) with \( {a\rho b} \) . Since \( \rho \) is an \( {\mathcal{R}}^{ * } \) -congruence, it follows \( {a}^{ + }\rho {b}^{ + } \) from Lemma 2.1. Note that \( \rho \) is idempotent-separating on... | Yes |
Lemma 2.4 Let \( \pi \) be a normal congruence on \( E\left( S\right) \) . Then for all \( a, b \in S \), the followings are equivalent:\n\n(1) \( \left( {a, b}\right) \in {\pi }_{\max } \) ;\n\n(2) \( \left( {a{\pi }_{\min }, b{\pi }_{\min }}\right) \in {\mu }_{S/{\pi }_{\min }} \) . | Proof For \( a, b \in S \) and \( e \in E\left( S\right) \), we have that\n\n\[ \left( {a, b}\right) \in {\pi }_{\max } \Leftrightarrow {a}^{ + }\pi {b}^{ + },{\left( ae\right) }^{ + }\pi {\left( be\right) }^{ + } \]\n\n\[ \Leftrightarrow {a}^{ + }{\pi }_{\min }{b}^{ + },{\left( ae\right) }^{ + }{\pi }_{\min }{\left( b... | Yes |
Theorem 2.2 As above \( {\pi }_{\max } \) is the greatest \( {\mathcal{R}}^{ * } \) -congruence on \( S \) such that \( \operatorname{tr}{\pi }_{\max } = \pi \) . | Proof Clearly, \( {\pi }_{\max } \) is an equivalence on \( S \) . So we will prove that \( {\pi }_{\max } \) is compatible with the multiplication. If \( a{\pi }_{\max }b \), then for any \( e \in E\left( S\right) \), we have \( {\left( ae\right) }^{ + }\pi {\left( be\right) }^{ + } \) and \( {a}^{ + }\pi {b}^{ + } \)... | No |
Proposition 2.7 Let \( \rho \) be an \( {\mathcal{R}}^{ * } \) -congruence on \( S \) . Then we have\n\n(1) \( \operatorname{Ker}{\rho }_{\min } = \left\{ {a \in S : \left( {\exists e \in {a}^{ + }\rho \cap E\left( S\right) }\right) {ea} = e}\right\} \) ;\n\n(2) if \( a \in \operatorname{Ker}{\rho }_{\max } \), then ea... | Proof (1) Let \( a \in \operatorname{Ker}{\rho }_{\min } \) . Then there exist \( g, h \in E\left( S\right) \) such that \( {h\rho }{a}^{ + }{\rho g} \) and \( {ha} = {hg} \) giving \( {gha} = {ghg} = {gh} \) and \( {gh\rho }{a}^{ + } \) . Thus, we can see that \( {gh} \) is the required.\n\nConversely, if \( a \in S, ... | Yes |
Corollary 2.2 With the same notations as Proposition 2.7, we have\n\n(1) \( \operatorname{Ker}{\sigma }_{S} = \{ a \in S : {ea} = e \) for some \( e \in E\left( S\right) \} \) ;\n\n(2) if \( a \in \operatorname{Ker}{\mu }_{S} \), then \( {eae} = {ea} \) for any \( e \in E\left( S\right) \) . | Proof It follows from Proposition 2.7 and the fact that \( {\iota }_{\max } = {\mu }_{S} \) and \( {\omega }_{\min } = {\sigma }_{S} \). | No |
Proposition 2.8 Let \( \rho \) be an \( {\mathcal{R}}^{ * } \) -congruence on \( S \) . Then \( \operatorname{Ker}\rho \) is a normal subsemigroup of \( S \) . | Proof Let \( \operatorname{Ker}\rho = N \) . It is obvious that \( E\left( S\right) \subseteq N \) . Let \( x, y \in S \) and \( n \in N \) . Then there exists \( e \in E\left( S\right) \) such that \( \left( {n, e}\right) \in \rho \), which means \( \left( {{xny},{xey}}\right) \in \rho \) . If \( {xy} \in N \), then\n... | No |
Theorem 2.1 Let \( 1 < p < \infty \) . If there is an \( r > 0 \) such that \( {w}^{r} \in {A}_{p,0},0 < \kappa < 1 \) , \( b \in {\mathrm{{CMO}}}^{{r}^{\prime }\max \left\{ {p,{p}^{\prime }}\right\} } \), then \( {P}_{b} \) is bounded on \( {L}_{0}^{p,\kappa }\left( w\right) \) . | Proof For \( f \in {L}_{0}^{p,\kappa }\left( w\right) \) and \( t > 0 \), we decompose \( f = {f}_{1} + {f}_{2} \), where \( {f}_{1} = f{\chi }_{\left( 0, t\right) }\left( x\right) \) . For \( x \in \left( {0, t}\right) \), we have\n\n\[ \n{P}_{b}{f}_{2}\left( x\right) = \frac{1}{x}{\int }_{0}^{x}\left( {b\left( x\righ... | Yes |
Theorem 2.2 Let \( 1 < p < \infty \) . If there is an \( r > 1 \) such that \( {w}^{r} \in {A}_{p,0},0 < \kappa < \) \( \min \left\{ {\frac{1}{{\partial }_{w}{r}^{\prime }} + \frac{{\beta }_{w}r}{{\partial }_{w}r},1}\right\}, b \in {\mathrm{{CMO}}}^{{r}^{\prime }\max \left\{ {p,{p}^{\prime }}\right\} } \), then \( {Q}_... | Proof We only need to prove for any \( t > 0 \) ,\n\n\[{\int }_{0}^{t}{\left| {Q}_{b}f\left( x\right) \right| }^{p}w\left( x\right) \mathrm{d}x \leq C\parallel f{\parallel }_{{L}_{0}^{p,\kappa }\left( w\right) }^{p}w{\left( 0, t\right) }^{\kappa }.\] | No |
Corollary 1.2 Suppose that \( T \) has property \( \left( h\right) \) . If iso \( \sigma \left( T\right) = \varnothing \) and for any \( f \in H\left( T\right) \) , \( f\left( {{\sigma }_{\mathrm{{gk}}}\left( T\right) }\right) \subseteq {\sigma }_{\mathrm{{gk}}}\left( {f\left( T\right) }\right) \), then property \( \le... | Proof By Theorem 1.2, we only need to prove \( f\left( {{\sigma }_{\mathrm{{ea}}}\left( T\right) }\right) \subseteq {\sigma }_{\mathrm{{ea}}}\left( {f\left( T\right) }\right) \) . Let \( {\mu }_{0} \notin {\sigma }_{\mathrm{{ea}}}\left( {f\left( T\right) }\right) \) . Then \( {\mu }_{0} \notin {\sigma }_{\mathrm{{gk}}}... | Yes |
Corollary 2.1 Suppose that \( \mathfrak{F} \) is a solvable saturated formation containing \( \mathfrak{U} \) . If \( E \trianglelefteq G \) such that \( G/E \in \mathfrak{F} \) and \( {F}^{ * }\left( E\right) \) is solvable. Assume that every non-cyclic Sylow subgroup \( P \) of \( {F}^{ * }\left( E\right) \) has a su... | Proof Since \( {F}^{ * }\left( E\right) \) is solvable, \( {F}^{ * }\left( E\right) = F\left( E\right) \) by Lemma 1.6. Take \( P \in {\operatorname{Syl}}_{p}\left( {F\left( E\right) }\right) \) for any \( p \in \pi \left( {F\left( E\right) }\right) \), then \( P \trianglelefteq G \) . If \( P \) is cyclic, take any ch... | Yes |
Theorem 2.1 Assume that there exist constants \( e, b, d > 0 \) with \( e < b < d \) and \( c = \frac{b}{{\tau }_{0}} \) such that the following conditions hold: \( \left( {\mathrm{H}}_{1}\right) f\left( {t, u}\right) \leq \frac{d}{{Q}_{3}},\left( {t, u}\right) \in \left\lbrack {a, T}\right\rbrack \times \left\lbrack {... | Proof First, we show that the operator \( \mathcal{G} \) defined by (2.1) is completely continuous. Observe that the continuity of \( \mathcal{G} \) follows from the continuity of \( f \) . For a positive constant \( \delta \), let \[ \eta : {\mathbb{Z}}^{b} \rightarrow \mathbb{Z},\;\eta \left( {e}_{j}\right) = 1, \] 其... | No |
Theorem 0.2 If \( T \in \mathcal{T}\left( {n, d}\right) \), then the following statements hold.\n\n(i) If \( d = 3 \), then\n\n\[ \mathop{\max }\limits_{{i \in V}}H\left( {\pi, i}\right) \geq \left\{ \begin{array}{ll} \frac{5{n}^{2} - {13n} + 7}{2\left( {n - 1}\right) }, & \text{ if }n - 4\text{ is even; } \\ \frac{5{n... | where the equality holds if and only if \( T \cong T\left( {\left\lfloor \frac{n - 4}{2}\right\rfloor ,\left\lceil \frac{n - 4}{2}\right\rceil }\right) \) . In this case, \( {v}_{0} \) achieves \( \mathop{\max }\limits_{{i \in V}}H\left( {\pi, i}\right) \) . degree at least 3 other than \( w \) . Since \( T \neq {B}_{n... | Yes |
Lemma 3.1 For the symbols above, we have \( \mathop{\max }\limits_{{u \in V}}H\left( {\pi, u}\right) \geq H\left( {\pi, v}\right) \geq {H}^{ * }\left( {{\pi }^{ * }, v}\right) \) . The last equality holds if and only if \( T = {T}^{ * } \) . | Proof We first observe that, by Lemma 1.2, for any \( {v}_{i} \in P, H\left( {{v}_{i}, v}\right) = {H}^{ * }\left( {{v}_{i}, v}\right) = \) \( d{\left( {v}_{i}, v\right) }^{2} + {2j}\mathop{\sum }\limits_{{j = 1}}^{{i - 1}}{p}_{j} + {2i}\left( {\mathop{\sum }\limits_{{j = i}}^{{d - 1}}{p}_{j} + d - i}\right) \) . There... | Yes |
Lemma 2.4 If \( \left( {X, G}\right) \) is a \( G \) -system with the weak specification property, then the set \( \left\{ {x : {M}_{\mathcal{F}, x} = M\left( {X, G}\right) }\right\} \) is residual in \( X \) . | Proof There exist two sequences \( {\left\{ {V}_{i}\right\} }_{i = 1}^{\infty } \) and \( {\left\{ {U}_{i}\right\} }_{i = 1}^{\infty } \) of open balls of \( M\left( X\right) \) such that\n\n(a) \( {V}_{i} \subset \mathcal{C}{\ell }_{X}{V}_{i} \subset {U}_{i} \) for each \( i \geq 1 \) ;\n\n(b) \( \operatorname{diam}\l... | Yes |
Theorem 2.3 Let \( \left( {X, G}\right) \) be a \( G \) -system with the strong specification property. Then the set of proper quasi-weakly almost periodic points relative to \( \mathcal{F} \) of \( \left( {X, G}\right) \) satisfying the result of Theorem 2.1 is a residual subset of \( X \) . | Proof Let \( {\widetilde{\mathrm{{QW}}}}_{\mathcal{F}}\left( {X, G}\right) = \left\{ {x \in {\mathrm{{QW}}}}_{\mathcal{F}}\left( {X, G}\right) : \exists \mu \in {M}_{\mathcal{F}, x}}\right. \), such that \( \left. {{S}_{\mu } = {C}_{\mathcal{F}}\left( x\right) }\right\} \) . By Lemma 2.4, it is enough to show\n\n\[{\wi... | Yes |
Lemma 2.5 \( {C}_{\mathcal{F}}\left( x\right) = \omega \left( {x, G}\right) \Leftrightarrow x \in {\mathrm{{QW}}}_{\mathcal{F}}\left( G\right) \) . | Proof Let \( x \in X \) . By the definitions of the minimal \( \mathcal{F} \) -center of attraction of \( x \) and quasi-weakly almost periodic point relative to \( \mathcal{F} \) of \( \left( {X, G}\right) \), it is easy to see that \( x \in {C}_{\mathcal{F}}\left( x\right) \) if and only if \( x \in {\mathrm{{QW}}}_{... | Yes |
Theorem 0.2 (Folklore) Given a set \( \mathcal{H} \) of graphs and an integer \( k \geq 1 \), if the set of all \( k \) -vertex-critical \( \mathcal{H} \) -free graphs is finite, then there is a polynomial-time algorithm to determine whether an \( \mathcal{H} \) -free graph is \( \left( {k - 1}\right) \) -colorable. | Proof Let \( {H}_{1},{H}_{2},\cdots ,{H}_{r} \) be all \( k \) -vertex-critical \( \mathcal{H} \) -free graphs with \( \left| {H}_{i}\right| = {n}_{i} \), where \( r \) is a finite number. Let \( G \) be an \( n \) -vertex \( \mathcal{H} \) -free graph. Our algorithm for testing whether \( G \) is \( \left( {k - 1}\rig... | No |
Problem 3.4 Are there only finitely many 4-vertex-critical \( \left( {{P}_{t},{C}_{s}}\right) \) -free graphs for \( t \geq 8 \) and \( 3 \leq s \leq t \) with \( s \neq 4 \) ? | Now assume that \( k \geq 5 \) and so \( s \in \{ 3,5\} \) . Since every \( \left( {{P}_{t},{C}_{3}}\right) \) -free graph is \( \left( {t - 2}\right) \) - colorable \( {}^{\left\lbrack {36}\right\rbrack } \), there are no \( k \) -vertex-critical graphs for \( k \geq t - 1 \) . This leaves the following open for \( s ... | No |
Theorem 3.12 \( {}^{\left\lbrack {33}\right\rbrack } \) There is exactly one 6-vertex-critical \( \left( {{P}_{6}\text{, diamond}}\right) \) -free graph (see Fig. 5). | In [33], it was proved that every \( \left( {{P}_{6}\text{, diamond}}\right) \) -free graph \( G \) has \( \chi \left( G\right) \leq \max \{ \omega \left( G\right) ,6\} \) . This upper bound on the chromatic number together with Theorem 3.12 implies a polynomial-time algorithm for computing the chromatic number of \( \... | No |
Lemma 2.2 Let \( \mathfrak{T} \) and \( \widetilde{\mathfrak{T}} \) be trees as mentioned in the preceding paragraph. Then \( {\varepsilon }_{1}\left( \mathfrak{T}\right) < \) \( {\varepsilon }_{1}\left( \widetilde{\mathfrak{T}}\right) \) . | Proof It is obvious that \( {e}_{\mathfrak{T}}\left( w\right) = {e}_{\widetilde{\mathfrak{T}}}\left( w\right) \) for each \( w \in V\left( \mathfrak{T}\right) \smallsetminus \{ u\} ,{d}_{\mathfrak{T}}\left( {w,{w}^{\prime }}\right) = {d}_{\widetilde{\mathfrak{T}}}\left( {w,{w}^{\prime }}\right) \) and \( \varepsilon {\... | Yes |
Theorem 2.1 In \( {\mathbb{T}}_{n, d} \), the maximum \( \varepsilon \) -spectral radius is achieved by Dandelion graphs | \[ {\mathrm{{DT}}}_{n, d} = \left( {{k}_{1} \cdot {t}_{1},\cdots ,{k}_{s - 2} \cdot {t}_{s - 2},\frac{d}{2} \cdot {t}_{s - 1},\frac{d}{2} \cdot {t}_{s}}\right) . \] | No |
Lemma 3.1 When \( d \geq 4,{\varepsilon }_{1}\left( {T}_{n, d}^{a, b}\right) \) is the largest root of \( {f}_{a}\left( t\right) \) . | Proof Let \( {P}_{d + 1} = {v}_{0}{v}_{1}{v}_{2}\cdots {v}_{d} \) be a diametrical path of \( {T}_{n, d}^{a, b}, U = \left\{ {{v}_{0},{u}_{1},{u}_{2},\cdots ,{u}_{a}}\right\} \) be the set of pendant neighbors of \( {v}_{1} \), and \( W = \left\{ {{v}_{d},{w}_{1},{w}_{2},\cdots ,{w}_{b}}\right\} \) be the set of pendan... | No |
Lemma 2.1 If \( F : {\;\operatorname{mod}\;A} \rightarrow {\;\operatorname{mod}\;B} \) is a fully faithful functor, then we have \( F\left( {\operatorname{brick}A}\right) \subseteq \) brick \( B \) and \( F\left( {\operatorname{sbrick}A}\right) \subseteq \operatorname{sbrick}B \) . | Proof (1) For any \( S \in \operatorname{brick}A,{\operatorname{End}}_{A}\left( S\right) \) is a division ring. If we want to know \( F\left( S\right) \in \) brick \( B \), we only have to prove that \( {\operatorname{End}}_{B}\left( {F\left( S\right) }\right) \) is also a division ring. Let \( 0 \neq h \in {\operatorn... | No |
Lemma 2.2 \( {}^{\left\lbrack 1\right\rbrack } \) Let \( T : C \rightarrow C \) be a firmly nonexpansive mapping. Then for each \( x, y \in C \) , | \[ \langle \left( {I - T}\right) x - \left( {I - T}\right) y, x - y\rangle \geq \parallel \left( {I - T}\right) x - \left( {I - T}\right) y{\parallel }^{2}. \] | Yes |
Theorem 3.1 If\n\n\[ 0 < \tau < \frac{2}{\mathop{\sum }\limits_{{i = 1}}^{N}{\begin{Vmatrix}{A}_{i}\end{Vmatrix}}^{2}} \]\n\n(3.1) | By Condition (3.1), this yields \( \parallel {Tx} - {Ty}\parallel \leq \parallel x - y\parallel \) . In particular, if \( y \) is a fixed point of \( T \), it then follows that\n\n\[ \parallel {Tx} - y{\parallel }^{2} \leq \parallel x - y{\parallel }^{2} - \tau \left( {2 - \tau \mathop{\sum }\limits_{{i = 1}}^{N}{\begi... | Yes |
Theorem 3.2 If\n\n\\[ \n0 < \\tau < \\frac{2}{1 + \\mathop{\\sum }\\limits_{{i = 1}}^{N}{\\begin{Vmatrix}{A}_{i}\\end{Vmatrix}}^{2}} \n\\]\n\n\\( \\left( {3.3}\\right) \\)\n\nthen the mapping \\( U : H \\rightarrow H \\) defined as\n\n\\[ \nU \\mathrel{\\text{:=}} I - \\tau \\left\\lbrack {I - {P}_{C} + \\mathop{\\sum ... | Proof For convenience, denote by \\( {A}_{0} = I \\) and \\( {Q}_{0} = C \\) . We can rewrite (3.4) in a more compact form:\n\n\\[ \nU = I - \\tau \\left\\lbrack {\\mathop{\\sum }\\limits_{{i = 0}}^{N}{A}_{i}^{ * }\\left( {I - {P}_{{Q}_{i}}}\\right) {A}_{i}}\\right\\rbrack .\n\\]\n\nIt can be obtained in a similar way ... | Yes |
Theorem 1. Let \( \mathcal{Q} \) be a \( * \) -closed subalgebra of compact operators on a Hilbert space \( H \) . Then \( H \) is completely reducible for \( \mathcal{Q} \), and each irreducible subspace occurs with finite multiplicity. | Proof. Let \( \left\{ {E}_{i}\right\} \) be a maximal orthogonal family of \( \mathcal{Q} \) -irreducible subspaces, and let \( F \) be the orthogonal complement of the subspace generated by the \( {E}_{i} \) . Since \( \mathcal{Q} \) is \( * \) -closed, it follows that \( F \) is \( \mathcal{Q} \) -invariant, and ther... | Yes |
Lemma 1. The algebraic sum \( \mathop{\sum }\limits_{{n, m}}{S}_{n, m} \) is \( {L}^{1} \) -dense in \( {C}_{c}\left( G\right) \) . In fact, given \( \epsilon \) and \( f \in {C}_{c}\left( G\right) \), there exists a function \( g \in \sum {S}_{n, m} \) such that the support of \( g \) is contained in \( K\left( {\oper... | Proof. Let\n\n\[ \n{f}_{n, m}\left( y\right) = {\int }_{-\pi }^{\pi }{\int }_{-\pi }^{\pi }f\left( {r\left( \theta \right) {yr}\left( {\theta }^{\prime }\right) }\right) {e}^{in\theta }{e}^{{im}{\theta }^{\prime }}{d\theta d}{\theta }^{\prime } \n\] \n\nbe the \( \left( {n, m}\right) \) Fourier coefficient of \( {f}^{y... | Yes |
Lemma 2. We have:\ni) \( {S}_{n, m} * {S}_{l, q} = 0 \) if \( m \neq l \) .\nii) \( {S}_{n, m}^{ * } = {S}_{m, n} \) .\niii) \( {S}_{n, m} * {S}_{m, q} \subset {S}_{n, q} \) . | Proof. Consider the convolution integral\n\n\[ f * g\left( x\right) = {\int }_{G}f\left( {x{y}^{-1}}\right) g\left( y\right) {dy}. \]\n\nSince \( G \) is unimodular, an integral with respect to \( y \) over \( G \) is invariant under the transformation \( y \mapsto {y}^{-1} \) . Now let \( y \mapsto r\left( \theta \rig... | No |
Lemma 3. The algebra \( {S}_{n, n} \) is commutative. | As we are concerned here with the arbitrary \( {S}_{n, n} \), and not just \( {S}_{0,0} \), we give the proof in a general context. The reader will find it profitable to look at the simpler case of bi-invariant functions given at the beginning of Chapter IV, due to Gelfand. The generalization we give here is due to Sil... | No |
Theorem 1. Let \( G \) be a unimodular locally compact group. Let \( K \) be a compact subgroup. Assume:\n\ni) That there exists an anti-automorphism \( \tau \) of \( G \), of order 2, such that \( {k}^{\tau } = {k}^{-1} \) for all \( k \in K \).\n\nii) If \( S \) is the set of elements \( s \in G \) such that \( {s}^{... | Proof. Define \( {f}^{ * }\left( x\right) = f\left( {x}^{\tau }\right) \). Then\n\n\[ \n{\left( f * g\right) }^{ * } = {g}^{ * } * {f}^{ * } \n\]\n\nOn the other hand, define \( {f}^{\prime }\left( x\right) = f\left( {x}^{\sigma }\right) \). Then\n\n\[ \n{\left( f * g\right) }^{\prime } = {f}^{\prime } * {g}^{\prime } ... | Yes |
Lemma 4. Assume that \( H \) is a Hilbert space and \( \pi \) is unitary on \( K \) . If \( m \neq n \), then \( {H}_{n} \) is perpendicular to \( {H}_{m} \) . | Proof. For \( v \in {H}_{n} \) and \( w \in {H}_{m} \) we have \( \pi {\left( r\left( \theta \right) \right) }^{ * } = \pi \left( {r\left( {-\theta }\right) }\right) \), so\n\n\[ \langle \pi \left( {r\left( \theta \right) }\right) v, w\rangle = {e}^{in\theta }\langle v, w\rangle \]\n\n\[ = \langle v,\pi \left( {r\left(... | Yes |
Lemma 5. We have:\n\ni) \( {\pi }^{1}\left( {S}_{n, m}\right) H \subset {H}_{n} \) ,\n\nii) \( {\pi }^{1}\left( {S}_{n, m}\right) {H}_{q} = \{ 0\} \) if \( m \neq q \) . | Proof. If \( m \neq q \), then we use the invariance of\n\n\[ \n{\int }_{G}f\left( y\right) \pi \left( y\right) {vdy} \]\n\n\( v \in {H}_{q} \)\n\nunder translations \( y \mapsto {yr}\left( \theta \right) \) . If \( f \in {S}_{n, m} \), we find that the above value of the integral is equal to itself multiplied by \( {e... | Yes |
Lemma 6. Assume that \( \pi \) is irreducible. Then the space \( {H}_{q} \) is irreducible for \( {S}_{q, q} \), and if \( {H}_{q} \neq \{ 0\} \), then \( {\pi }^{1}\left( {S}_{q, q}\right) {H}_{q} \neq \{ 0\} \) . | Proof. Let \( W \) be a proper subspace of \( {H}_{q} \), invariant for \( {\pi }^{1}\left( {S}_{q, q}\right) \) . If \( w \in W \) and \( f \) is a finite sum of functions \( {f}_{n, m} \in {S}_{n, m} \), then by Lemma 5,\n\n\[ \n{\left( {\pi }^{1}\left( f\right) w\right) }_{q} = {\pi }^{1}\left( {f}_{q, q}\right) w \... | Yes |
Theorem 2. Let \( \pi \) be an irreducible representation of \( G \) on a Banach space H. Let \( {H}_{n} \) be the subspace of vectors \( v \) such that\n\n\[ \pi \left( {r\left( \theta \right) }\right) v = {e}^{in\theta }v. \]\n\nIf \( \dim {H}_{n} \) is finite, then \( \dim {H}_{n} = 0 \) or 1 . This is always the ca... | Proof. We know that \( {H}_{n} \) is irreducible for \( {\pi }^{1}\left( {S}_{n, n}\right) \) and finite dimensional linear algebra shows that \( \dim {H}_{n} = 0 \) or 1, since \( {S}_{n, n} \) is commutative. On the other hand, if \( \pi \) is unitary, and \( f \in {S}_{n, n} \), then \( {\pi }^{1}{\left( f\right) }^... | Yes |
Theorem 3. Let \( \pi \) be an irreducible representation of \( G \) on a Banach space \( H \). Then the sum \( \sum {H}_{n} \) is dense in \( H \). If \( H \) is a Hilbert space and \( \pi \) is unitary on \( K \), this sum is an orthogonal decomposition of \( H \). | Proof. Let \( E \) be the (closed) subspace generated by the \( {H}_{n} \). By Lemma 5 and the fact that the sum \( \sum {S}_{m, n} \) is dense in \( {C}_{c}\left( G\right) \), we conclude that \( E \) is \( {C}_{c}\left( G\right) \) -invariant, whence is \( \bar{G} \) -invariant. Since \( \pi \) is irreducible, it fol... | Yes |
Theorem 2. Let \( \pi : K \mapsto \operatorname{Aut}\left( H\right) \) be a unitary irreducible representation of a compact group \( K \) . Then \( H \) is finite dimensional. | Proof. Let \( u \) be a unit vector in \( H \) and let \( P \) be the orthogonal projection on the one-dimensional space \( \left( u\right) \) . Let \( Q : H \rightarrow H \) be the continuous linear map defined by\n\n\[ \n{Qv} = {\int }_{K}\pi {\left( x\right) }^{-1}{P\pi }\left( x\right) {vdx}.\n\]\n\nThen \( Q \) co... | Yes |
Theorem 3. If \( \pi ,\sigma \) are inequivalent irreducible representations of \( K \), then for all \( v \in {H}_{\sigma }, w \in {H}_{\pi }, a \in K \) we have\n\n(2)\n\n\[{\int }_{K}\lambda \left( {\sigma \left( {ax}\right) v}\right) \pi \left( {x}^{-1}\right) {wdx} = 0,\] \ni.e. \( {\pi }^{1}\left( {\sigma }_{\lam... | Proof.\n\n\[{\pi }^{1}\left( {\chi }_{\sigma }^{ - }\right) w = {\int }_{K}\mathop{\sum }\limits_{i}{\lambda }_{i}\left( {\sigma \left( {x}^{-1}\right) {e}_{i}}\right) \pi \left( x\right) {wdx} = 0.\] | Yes |
Theorem 4. Let \( \pi \) be an irreducible representation of \( K \) on \( H \) . Let \( v, w \in H \) and let \( \lambda \) be a functional on \( H \) . Then\n\n\[ \n{\int }_{K}\lambda \left( {\pi \left( {x}^{-1}\right) w}\right) \pi \left( x\right) {vdx} = \frac{1}{d\left( \pi \right) }\lambda \left( v\right) w.\n\] | Proof. For \( v \) fixed, consider the map \( L : H \rightarrow H \) such that \( L\left( w\right) \) is the expression on the left-hand side of the formula to be proved. Then\n\n\[ \nL\left( w\right) = {\int }_{K}\pi \left( x\right) {\varphi }_{\lambda, v}\left\lbrack {\pi \left( {x}^{-1}\right) w}\right\rbrack {dx}.\... | Yes |
For any \( a \in K \) and any functional \( \mu \) on \( H \), we have\n\n\[{\int }_{K}\lambda \left( {\pi \left( a\right) \pi \left( {x}^{-1}\right) w}\right) \pi \left( x\right) {vdx} = \frac{1}{d\left( \pi \right) }\lambda \left( {\pi \left( a\right) v}\right) w\]\n\nand\n\n\[{\int }_{K}\lambda \left( {\pi \left( a\... | Proof. Replace \( v \) by \( \pi \left( a\right) v \) in the theorem, let \( x \mapsto x{a}^{-1} \), and apply the functional \( \mu \) to the relation of the theorem. | No |
Theorem 5. Let \( \pi \) be an irreducible representation of \( K \) . Then\n\n\[ \n{\pi }^{1}\left( {\chi }_{\pi }^{ - }\right) = \frac{1}{d\left( \pi \right) }I \n\] | Proof. We have:\n\n\[ \n{\pi }^{1}\left( {\chi }^{ - }\right) v = {\int }_{K}\chi \left( {x}^{-1}\right) \pi \left( x\right) {vdx} \n\]\n\n\[ \n= {\int }_{K}\mathop{\sum }\limits_{i}{\lambda }_{i}\left( {\pi \left( {x}^{-1}\right) {e}_{i}}\right) \pi \left( x\right) {vdx} \n\]\n\n\[ \n= \mathop{\sum }\limits_{i}{\int }... | Yes |
Theorem 6. Every irreducible representation of \( K \) occurs in the regular representation on \( {L}^{2}\left( K\right) \) . | Proof. By complete reducibility, we know that\n\n\[ \n{L}^{2}\left( K\right) = {\bigoplus }_{\pi }{m}_{\pi }{H}_{\pi } \n\]\n\nLet \( \sigma \) be an irreducible representation and \( \psi \) its character. If \( \sigma \) does not occur, then for all \( \pi \) occurring in \( {L}^{2}\left( K\right) \) we get\n\n\[ \n{... | Yes |
Theorem 7. Let \( \pi ,\sigma \) be irreducible representations of \( K \) . Then:\n\n\[ \n{\chi }_{\sigma } * {\chi }_{\pi } = \left\{ \begin{matrix} 0 & \text{ if } & \sigma \nsim \pi , \\ {d}_{\pi }^{-1}{\chi }_{\pi } & \text{ if } & \sigma \sim \pi . \end{matrix}\right.\n\] | Proof. To avoid subscripts, let \( \chi \) and \( \psi \) be the characters of inequivalent irreducible representations of \( K \), say on spaces \( H \) and \( {H}^{\prime } \) respectively. Let \( \left\{ {e}_{i}\right\} \) be a basis of \( H \) and \( \left\{ {\lambda }_{i}\right\} \) the dual basis, and similarly \... | Yes |
Theorem 1. Let \( K \) be a closed subgroup of \( G \), both assumed unimodular. There exists a unique invariant measure \( {\mu }_{G/K} \) on \( G/K \) such that for any \( f \in {C}_{c}\left( G\right) \) we have\n\n\[ \n{\int }_{G/K}{f}^{K}d{\mu }_{G/K} = {\int }_{G}{fd}{\mu }_{G} \n\] \n\nwhere \( {\mu }_{G} \) is H... | Proof. The uniqueness is obvious. Given \( \varphi \in {C}_{c}\left( {G/K}\right) \), let \( f \in {C}_{c}\left( G\right) \) such that \( {f}^{K} = \varphi \) . The invariant integral on \( G/K \) can be defined by means of the formula in the theorem, provided we show that if \( {f}^{K} = 0 \), then \n\n\[ \n{\int }_{G... | Yes |
Theorem 2. If \( \sigma \) is bounded, then the induced representation \( \pi \) is bounded. If \( \sigma \) is unitary, then the induced representation \( \pi \) is unitary. | Proof. Fix \( y \) . Write \( {ky} = {p}_{k}^{\prime }{k}^{\prime } \), so that\n\n\[ f\left( {ky}\right) = f\left( {{p}_{k}^{\prime }{k}^{\prime }}\right) = \Delta {\left( {p}_{k}^{\prime }\right) }^{1/2}\sigma \left( {p}_{k}^{\prime }\right) f\left( {k}^{\prime }\right) . \]\n\nThen\n\n\[ {\int }_{K}{\left| f\left( k... | Yes |
Theorem 3. The spaces \( H\left( \sigma \right) \) and \( H\left( {\sigma }^{-1}\right) \) are dual to each other under this symmetric product. For \( y \in G, f \in H\left( \sigma \right), g \in H\left( {\sigma }^{-1}\right) \), we have\n\n\[ \left\lbrack {\pi \left( y\right) f, g}\right\rbrack = \left\lbrack {f,\pi \... | Proof. This is an easy computation:\n\n\[ {\int }_{K}\pi \left( y\right) f\left( k\right) g\left( k\right) {dk} = {\int }_{K}f\left( {ky}\right) g\left( k\right) {dk} \]\n\n\[ = {\int }_{K}\Delta {\left( {p}_{k}^{\prime }\right) }^{1/2}\sigma \left( {p}_{k}^{\prime }\right) f\left( {k}^{\prime }\right) g\left( k\right)... | Yes |
Theorem 4. For \( \psi \in {C}_{c}\left( {G, K}\right) \) and \( s \in \mathbf{C} \) we have\n\n\[ \operatorname{tr}{\pi }_{s}^{1}\left( \psi \right) = {\int }_{A}\mathbf{H}\psi \left( a\right) \rho {\left( a\right) }^{s}{da}. \] | In the sequel, the integral as above with \( s \) as a parameter will be called a Mellin transform \( \mathbf{M} \), and thus we could abbreviate still further the formula for the trace by\n\n\[ \operatorname{tr}{\pi }_{s}^{1}\left( \psi \right) = \mathbf{{MH}}\psi \left( s\right) \]\n\nIf \( \psi \) is bi-invariant un... | Yes |
Theorem 1. Let \( G \) be locally compact unimodular, and let \( K \) be a compact subgroup. Let \( \tau \) be an anti-automorphism of \( G \) of order 2 such that given \( x \in G \) there exist \( {k}_{1},{k}_{2} \in K \) satisfying\n\n\[ \n{x}^{\tau } = {k}_{1}x{k}_{2} \n\]\n\nThen the algebra \( {C}_{c}\left( {G//K... | Proof. Haar measure is invariant under \( x \mapsto {x}^{\tau } \) because\n\n\[ \n1 = \Delta \left( {\tau }^{2}\right) = \Delta \left( \tau \right) \Delta \left( \tau \right) \n\]\n\nso \( \Delta \left( \tau \right) = 1 \) . Also \( f\left( x\right) = f\left( {x}^{\tau }\right) \) for any \( f \in {C}_{c}\left( {G//K}... | Yes |
Theorem 2. Let \( \pi : G \rightarrow {GL}\left( H\right) \) be a representation which is star closed on \( G \) and \( K \) . Assume \( {H}^{K} \neq 0 \) and \( H \) is equal to the closure of \( \pi \left( G\right) {H}^{K} \) . Then \( {H}^{K} \) is \( {C}_{c}\left( {G//K}\right) \) -irreducible if and only if \( H \... | Proof. \( \Rightarrow \) : Let \( W \) be a closed \( G \) -stable subspace \( \neq 0 \) of \( H \), so that \( {W}^{ \bot } \) is also \( G \) -stable (because of star closure). Let \( P = {P}_{K} \) be the orthogonal projection on \( {H}^{K} \) . We consider two cases. First, suppose that \( {PW} = 0 \) . From\n\n\[ ... | Yes |
Theorem 3. Let \( \pi : G \rightarrow {GL}\left( H\right) \) be a unitary irreducible representation. If \( {C}_{c}\left( {G//K}\right) \) is commutative, then dim \( {H}^{K} \leq 1 \) . | Proof. Suppose \( \dim {H}^{K} \neq 0 \) . By Theorem 2 we know that \( {H}^{K} \) is irreducible for \( {C}_{c}\left( {G//K}\right) \) which is a commutative star closed algebra of operators. Schur’s lemma shows that \( \dim {H}^{K} = 1 \), as desired. | Yes |
Theorem 4. Assume that \( G = {PK} \), where \( P \) is a closed subgroup, and \( P \times K \rightarrow {PK} = G \) is a topological isomorphism. Let\n\n\[ \n\rho : P \rightarrow {\mathbf{C}}^{ * }\n\]\n\nbe a character (continuous homomorphism), which we extend to a function\non \( G \) by setting \( \rho \left( {pk}... | Proof. Write \( x = {p}_{1}{k}_{1} \) . Then for \( \psi \in {C}_{c}\left( {G//K}\right) \) we get\n\n\[ \n\rho * \psi \left( x\right) = {\int }_{G}\rho \left( {x{y}^{-1}}\right) \psi \left( y\right) {dy}\n\]\n\n\[ \n= {\int }_{G}\rho \left( {{p}_{1}y}\right) \psi \left( {y}^{-1}\right) {dy}.\n\]\n\nWriting \( y = {pk}... | Yes |
Theorem 6. Let \( f \in C\left( {G//K}\right) \) . Then \( f \) is spherical if and only if the map\n\n\[ L : \varphi \mapsto {\int }_{G}\varphi \left( x\right) f\left( x\right) {dx} \]\n\nis an algebra homomorphism of \( {C}_{c}\left( {G//K}\right) \) into \( \mathbf{C} \) . | Proof. By definition,\n\n\[ L\left( {\varphi * \psi }\right) = {\int }_{G}{\int }_{G}\varphi \left( {x{y}^{-1}}\right) \psi \left( y\right) f\left( x\right) {dydx}. \]\n\nInterchange \( {dydx} \) to \( {dxdy} \), let \( x \mapsto {xy} \), get the right-hand side\n\n\[ = {\int }_{G}{\int }_{G}\varphi \left( x\right) \ps... | Yes |
Theorem 7. Any continuous algebra homomorphism of \( {L}^{1}\left( {G//K}\right) \) into \( \mathbf{C} \) is of the form\n\n\[ \varphi \mapsto \left( {f * \varphi }\right) \left( e\right) \]\n\nfor some bounded spherical function \( f \) . | Proof. By measure theory, given a character \( L \neq 0 \) of the algebra \( {L}^{1}\left( {G//K}\right) \), there exists a bounded measurable function \( f \) such that\n\n\[ L\left( \varphi \right) = {\int }_{G}\varphi \left( x\right) f\left( x\right) {dx} \]\n\nReplace \( \varphi \left( x\right) \) by \( \varphi \le... | Yes |
Theorem 8. Let \( \pi : G \rightarrow \operatorname{Aut}\left( H\right) \) be a unitary representation and assume that there exists a unit vector \( u \in {H}^{K} \) which generates \( H \) topologically under \( \pi \) . Then\n\n\[ \n\dim {H}^{K} = 1 \Leftrightarrow \text{the function}f\left( x\right) = \langle \pi \l... | Proof. We have seen that \( f\left( e\right) = 1 \) and \( f \) is bi-invariant. Assume that \( {H}^{K} = \mathbf{C}u \) has dimension 1. For any \( \varphi \in {C}_{c}\left( {G//K}\right) ,{\pi }^{1}\left( \varphi \right) u \) is fixed by \( K \), so \( {\pi }^{1}\left( \varphi \right) u = \lambda \left( \varphi \righ... | Yes |
Theorem 9. The association\n\n\[ \n\varphi \mapsto \left( {{\pi }_{\varphi },{H}_{\varphi },\varphi }\right) \n\]\n\nis a bijection from the set of positive definite functions on \( G \) to the isomorphism classes of triples \( \left( {\pi, H, u}\right) \) consisting of a unitary representation\n\n\[ \n\pi : G \rightar... | Proof. This is an immediate consequence of Theorem 8, §4 and the irreducibility theorem of \( \$ 2 \) . | No |
Theorem 1. Hf is invariant under the Weyl group, i.e.\n\n\[ \mathbf{H}f\left( a\right) = \mathbf{H}f\left( {a}^{-1}\right) . \] | Proof. By continuity it suffices to prove the assertion when \( D\left( a\right) \neq 0 \) , and so \( \left| {D\left( a\right) }\right| = \left| {D\left( {a}^{-1}\right) }\right| \) . Note that\n\n\[ x \mapsto {wx}{w}^{-1} \]\n\nis an inner automorphism of \( G \), of order 2, sending \( a \mapsto {a}^{-1} \) . This m... | Yes |
Theorem 2. If \( f, g \in {C}_{c}\left( {G//K}\right) \), then\n\n\[ \mathbf{H}\left( {f * g}\right) = \mathbf{H}f * \mathbf{H}g \]\n\ni.e. on \( {C}_{c}\left( {G//K}\right) \), the Harish transform is an algebra homomorphism. | Proof. We have\n\n\[ \mathbf{H}\left( {f * g}\right) \left( a\right) = \rho \left( a\right) {\int }_{N}\left( {f * g}\right) \left( {an}\right) {dn} \]\n\n\[ = \rho \left( a\right) {\int }_{N}{\int }_{G}f\left( {any}\right) g\left( {y}^{-1}\right) {dydn}\;\left( {\text{ by }y \mapsto {y}^{-1}}\right) \]\n\n\[ = \rho \l... | Yes |
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