Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
values |
|---|---|---|
Lemma 4 Suppose that \( {\beta }^{-1}{\alpha }^{n} \in {\mathbb{F}}_{{p}^{u}}^{ * } \), and let \( {s}_{0} = {\alpha }^{\left( {n - 1}\right) t}\left( {{j}_{0},{j}_{1}\alpha ,\cdots ,{j}_{n - 1}{\alpha }^{n - 1}}\right) \) for some \( t\left( {0 \leq t \leq {p}^{m} - 2}\right) ,{j}_{i} \in {\mathbb{F}}_{{p}^{u}}\left( ... | Proof For convenience, we denote \( {s}_{0} = \left( {{a}_{0},{a}_{1},\cdots ,{a}_{n - 1}}\right) \), where \( {a}_{k} = {\alpha }^{\left( {n - 1}\right) t}{j}_{k}{\alpha }^{k},0 \leq \) \( k \leq n - 1 \) . According to relation (1) and \( g\left( x\right) = {x}^{n} + \alpha {x}^{n - 1} + \beta \), we have\n\n\[ \n{a}... | Yes |
Lemma 5 Let \( \alpha \) be a primitive element of \( {\mathbb{F}}_{4} \). If \( n > 2 \) and \( n \equiv 0,2,3,4,6,8,9,{12},{14} \) (mod 15), then the trinomial \( {x}^{n} + \alpha {x}^{n - 1} + \alpha \) is reducible over \( {\mathbb{F}}_{4} \). | Proof Denote \( n = {15k} + t, t = 0,2,3,4,6,8,9,{12},{14} \). For \( t = 0 \), it is easy to verify that \( {\alpha }^{-1} \) is one solution of \( {x}^{15k} + \alpha {x}^{{15k} - 1} + \alpha \) over \( {\mathbb{F}}_{4} \), then \( {x}^{15k} + \alpha {x}^{{15k} - 1} + \alpha \) is reducible over \( {\mathbb{F}}_{4} \)... | Yes |
Theorem 1.2 Let \( n\left( { > 2}\right) \) be even and either \( 3\left| {n - 1\text{or}3}\right| n - 2 \) . If \( N = \left\{ {{\alpha }_{i} \mid i = }\right. \) \( 0,1,\cdots, n - 1\} \) is a normal basis of \( {\mathbb{F}}_{{2}^{n}} \) over \( {\mathbb{F}}_{2} \) satisfying the condition \( \mathbf{P} \) and \( \le... | \[ \beta = \left\{ \begin{array}{ll} \mathop{\sum }\limits_{{i = 0,3 \nmid i + 1}}^{{n - 1}}{\alpha }_{i}, & 3 \mid n - 1 \\ \mathop{\sum }\limits_{{i = 0,3 \nmid i - 1}}^{{n - 1}}{\alpha }_{i}, & 3 \mid n - 2 \end{array}\right. \] | Yes |
Theorem 0.2 Let \( \left( {x, y}\right) = \left( {{x}^{\prime },{y}^{\prime },{x}^{\prime \prime },{y}^{\prime \prime }}\right) = \left( {{z}_{1},{z}_{2}}\right) \in {\mathbb{R}}^{n + m} \), where \( {z}_{1} = \left( {{x}^{\prime },{y}^{\prime }}\right) \in {\mathbb{R}}^{k + l} \) , \( {z}_{2} = \left( {{x}^{\prime \pr... | Using foregoing idea and combining (0.4), we have given a class of CKN type inequality \( {}^{\left\lbrack 8\right\rbrack } \) as follows: | No |
Theorem 0.4 Set \( q, r,\lambda ,\beta ,\gamma ,\sigma, a \) such that\n\n\[ \left\{ \begin{array}{l} q \geq 1,\;r > 0,\;0 \leq a \leq 1 \\ n + \alpha \left( {\lambda - \gamma }\right) r > 0,\;n + \alpha \left( {\lambda - \beta }\right) q > 0 \\ {\gamma r} + Q > 0,\;{\beta q} + Q > 0,\;{2\lambda } + Q > 0 \end{array}\r... | To prove Theorem 0.4, we employ the idea of [8]. So, we first need to establish a class of weighted Hardy-Sobolev type inequality for the generalized Baouendi-Grushin vector fields as follows:\n\nTheorem 0.5 If \( 0 \le | No |
Lemma 3.1 Under hypotheses \( \mathrm{H}\left( f\right) \) and \( \mathrm{H}\left( g\right) \), the energy functional \( \Phi \) is bounded from below, i.e., there is a constant \( c \in \mathbb{R} \), s.t. \( {\Phi x} \geq c \) for all \( x \in {W}_{\text{per }}^{1, p}\left( {T,{\mathbb{R}}^{N}}\right) \) . | Proof From (2.3) and (2.12), we have\n\n\[ \n{\Phi x} = \varphi \left( x\right) + {\psi }_{0}\left( x\right) \n\]\n\n\[ \n= \frac{1}{p}{\int }_{0}^{b}{\left| {x}^{\prime }\left( t\right) \right| }^{p}\mathrm{\;d}t + \frac{1}{p}{\int }_{0}^{b}a\left( t\right) {\left| x\left( t\right) \right| }^{p}\mathrm{\;d}t - \left( ... | Yes |
Proposition 2.1 Let \( F = \frac{{\alpha }^{2}}{\beta } \) be a Kropina metric on an \( n \) -dimensional manifold \( M \) . Assume that \( \left( {h, W}\right) \) is its navigation data. Then \( F \) is a Berwald metric if and only if \( h \) and \( W \) satisfy the following\n\n\[ {R}_{ij} = R{h}_{ij} \]\n\n(12)\n\n\... | Proof From [12], we know that a Kropina metric \( F = \frac{{\alpha }^{2}}{\beta } \) is a Berwald metric if and only if there exist scalar functions \( {f}_{i} \) satisfying\n\n\[ {b}_{i;j} = \left( {{b}^{r}{f}_{r}}\right) {a}_{ij} + {b}_{i}{f}_{j} - {b}_{j}{f}_{i} \]\n\n(14)\n\nwhere \ | Yes |
Theorem 3. \( {1}^{\left\lbrack {14}\right\rbrack }\; \) Let \( \;F = \frac{{\alpha }^{2}}{\beta }\; \) be a Kropina metric on an \( n \) -dimensional manifold \( \;M \) with the navigation representation \( \left( {h, W}\right) \) . Assume that \( n \geq 3 \) . Then \( F \) is of weakly isotropic flag curvature \( {K}... | Proof of Theorem 0.3 Now we prove that for conformally Berwald Kropina metric \( F \) of weakly isotropic flag curvature, the second conclusion (ii) in Theorem 3.1 does not hold.\n\nOtherwise, because \( F \) is a conformally Berwald Kropina metric of weakly isotropic flag curvature, by \( \left\lbrack {{14}\text{, The... | Yes |
Proposition 0.1 For \( r \in \left( {0,1}\right) \), there does not exist a positive integer \( k \) associated with a sequence of vectors \( {x}_{i} \in X,1 \leq i \leq k \), such that (0.1) holds. | Proof By Riesz Lemma, for arbitrarily finitely many vectors \( {x}_{i} \in X,1 \leq i \leq k \), there is a vector \( x \in {S}_{1}\left( 0\right) \) such that\n\n\[ d\left( {x,\operatorname{span}\left\{ {{x}_{i},1 \leq i \leq k}\right\} }\right) \geq r \]\n\nwhich implies that\n\n\[ x \notin \mathop{\bigcup }\limits_{... | Yes |
Lemma 0.1 If \( Y \) is a finite-dimensional linear subspace of \( X \), then there exists a vector \( x \in {S}_{1}\left( 0\right) \) such that \( d\left( {x, Y}\right) = 1 \) . | Proof Set \( {x}_{0} \in X \smallsetminus Y \) and define a set by\n\n\[ \nC \mathrel{\text{:=}} \left\{ {y \in Y : \begin{Vmatrix}{y - {x}_{0}}\end{Vmatrix} \leq {2d}\left( {{x}_{0}, Y}\right) }\right\} .\n\]\n\nClearly, \( C \) is a bounded closed set, and naturally is a compact set, since \( Y \) is a finite-dimensi... | Yes |
Theorem 0.1 For any \( k \in \mathbb{N} \), there does not exist a sequence of vectors \( {x}_{i},1 \leq i \leq k \) , such that\n\n\[ \n{B}_{1}\left( 0\right) \subseteq \mathop{\bigcup }\limits_{{i = 1}}^{k}{U}_{1}\left( {x}_{i}\right) \n\] | Proof By Lemma 0.1, for any finite sequence of vectors \( {x}_{i},1 \leq i \leq k \), there exists a vector \( x \in {S}_{1}\left( 0\right) \) such that\n\n\[ \nd\left( {x,\operatorname{span}\left\{ {{x}_{i} : 1 \leq i \leq k}\right\} }\right) = 1 \n\]\nwhich means that\n\n\[ \nx \notin \mathop{\bigcup }\limits_{{i = 1... | Yes |
Theorem 1.2 The number 3 is an infinitely-packing number with respect to any infinite-dimensional normed linear space \( X \) . If \( 1 \leq R < 2 \), then there do not exist two separated unit balls in \( {B}_{R}\left( 0\right) \) . | Proof We prove the theorem by using the constructive method.\n\nLet \( {x}_{1} \mathrel{\text{:=}} 0 \) and suppose that, for a positive integer \( n \), we have obtained a sequence of vectors \( {x}_{j}, j \in {\mathbb{N}}_{n} \), such that\n\n\[ \n{B}_{1}\left( {x}_{j}\right) \subseteq {B}_{3}\left( 0\right) ,\;j \in... | Yes |
Theorem 2.1 A positive number \( R \) is an infinitely-packing number with respect to \( {L}^{\infty }\left( \Omega \right) , \) if and only if \( R \geq 2 \) . | To prove Theorem 2.1, we need to introduce the following Lemma 2.1 whose proof is omitted. | No |
Lemma 2.2 Let \( 1 \leq p < + \infty \) . If both \( {\Omega }_{1} \) and \( {\Omega }_{2} \) are two Lebesgue measurable sets and \( {\Omega }_{1} \subset {\Omega }_{2} \), then there exists a subspace of \( {L}^{p}\left( {\Omega }_{2}\right) \) which is isometric isomorphic to \( {L}^{p}\left( {\Omega }_{1}\right) \)... | Proof We can construct an isometric mapping denoted \( I \) from \( {L}^{p}\left( {\Omega }_{1}\right) \) to \( {L}^{p}\left( {\Omega }_{2}\right) \) as follows,\n\n\[ I\left( f\right) \left( x\right) \mathrel{\text{:=}} \left\{ \begin{array}{ll} f\left( x\right) , & \text{ if }x \in {\Omega }_{1}, \\ 0, & \text{ if }x... | Yes |
Lemma 2.3 Suppose that \( \Omega \) is bounded in \( {\mathbb{R}}^{d} \) and \( 1 \leq p \leq 2 \) . If \( {f}_{j} \in {L}^{p}\left( \Omega \right) ,1 \leq j \leq m \) , such that\n\n\[ \n{\begin{Vmatrix}{f}_{j} - {f}_{k}\end{Vmatrix}}_{{L}^{p}\left( \Omega \right) } \geq 2,\;1 \leq j \neq k \leq m,\n\]\n\nthen there h... | Proof Since that \( \Omega \) is a bounded subset of \( {\mathbb{R}}^{d} \), for any fixed \( \varepsilon \in {\mathbb{R}}^{ + } \), there exists a sequence of continuous differentiable functions \( {\widetilde{f}}_{j},1 \leq j \leq m \) such that\n\n\[ \n\max \left\{ {\parallel {f}_{j} - {\widetilde{f}}_{j}{\parallel ... | Yes |
Theorem 2.3 Let \( \Omega \) be bounded in \( {\mathbb{R}}^{d} \) and \( 1 \leq p \leq 2 \) . Then \( R \) is an infinitely-packing number with respect to \( {L}^{p}\left( \Omega \right) \) if and only if \( R \geq {\lambda }_{p} \) . If \( 1 \leq R < {\lambda }_{p} \), the maximum number of pairwise separated unit bal... | Proof By Lemma 2.1 and Theorem B, \( R \) is an infinitely-packing number with respect to \( {L}^{p}\left( \Omega \right) \) if \( R \geq {\lambda }_{p} \) . If there is a sequence of pairwise separated unit balls \( {B}_{1}({f}_{j};{L}^{p}\left( \Omega \right) ),\;1 \leq j \leq m \), contained in the ball \( {B}_{R}\l... | Yes |
Theorem 2.4 Let \( p > 2 \) .\n\n(i) A positive number \( R \) is an infinitely-packing number with respect to \( {L}^{p}\left( \Omega \right) \) if \( R \geq {\lambda }_{p} \).\n\n(ii) If \( {\mu }_{p} \leq R < {\lambda }_{p} \), there exist \( n \) pairwise separated unit balls in \( {B}_{R}\left( 0\right) \) for any... | Proof Statements (i) and (ii) follow directly from Lemma 2.1 and Theorem C.\n\nIf there is a sequence of pairwise separated unit balls \( {B}_{1}\left( {{f}_{j};{L}^{p}\left( \Omega \right) }\right) ,\;1 \leq j \leq m, \) contained in the ball \( {B}_{R}\left( {0;{L}^{p}\left( \Omega \right) }\right) \), by Lemma 2.4, ... | Yes |
For any \( i \in \mathbb{N} \) and \( j \in {\mathbb{N}}^{ * } \), let \( {t}_{i, j} = {t}_{i + j - 1}{t}_{i + j - 2}\cdots {t}_{i + 1}{t}_{i} \). If \( x \in {\widetilde{C}}_{n} \) and \( i \in \mathbb{Z} \) satisfies \( \left( i\right) x - {2n} - 1 > \left( j\right) x \) for any \( j \in \left\lbrack {i + 1, i + a}\r... | \[ \text{(k)}{x}^{\prime \prime } = \left\{ \begin{array}{ll} \left( m\right) x - {2n} - 1, & \text{ if }\langle m\rangle = \langle i\rangle , \\ \left( m\right) x + {2n} + 1, & \text{ if }\langle m\rangle = \langle {2n} + 1 - i\rangle , \\ \left( m\right) x, & \text{ otherwise } \end{array}\right. \] for any \( m \in ... | Yes |
Lemma 4.1 For any \( w \in {E}_{\lambda }^{11} \), we have\n\n(1) \( 0 < {j}_{h} < \cdots < {j}_{1} < {i}_{t + m + s} < \cdots < {i}_{1} < k < {2n} + 1; \)\n\n\( \left( 2\right) \;0 < \left( k\right) w - {2n} - 1 < \left( {j}_{h}\right) w < \;\cdots < \left( {j}_{1}\right) w < \left( {i}_{t + m + s}\right) w < \;\cdots... | Proof Let \( w \in {E}_{\lambda }^{11} \) . We see by \( \left( {j}_{1}\right) w < \left( {i}_{t + m + s}\right) w \) and \( {j}_{1},{i}_{t + m + s} \) being \( w \) -uncomparable that \( {j}_{1} < {i}_{t + m + s} \) . (1) follows the conditions ( \( {\text{a}}_{21} \) ) and ( \( {\text{a}}_{23} \) ) of the condition (... | Yes |
Lemma 4.2 There exists a subset \( {F}_{\lambda }^{11} \) of \( {E}_{\lambda }^{11} \) such that each left-connected component of \( {E}_{\lambda }^{11} \) contains some element in \( {F}_{\lambda }^{11} \) . | Proof Let \( {F}_{\lambda }^{11} \) (see Fig. 2) be the set of all elements in \( {E}_{\lambda }^{11} \) satisfying (1)-(2) in Lemma 4.1 and the following conditions \( \left( {a}_{111}^{\prime }\right) - \left( {a}_{113}^{\prime }\right) \):\n\n\[ \left( {a}_{111}^{\prime }\right) \left( {1,2,\cdots, n}\right) = \left... | Yes |
Lemma 4.3 For any \( w \in {E}_{\lambda }^{21} \), we have\n\n(1) \( 0 < {i}_{t + m + s} < \cdots < {i}_{1} < k < {j}_{h} < \cdots < {j}_{1} < {2n} + 1; \)\n\n\( \left( 2\right) \;0 < \left( k\right) w - {2n} - 1 < \left( {j}_{h}\right) w - {2n} - 1 < \cdots < \left( {j}_{1}\right) - {2n} - 1 < \left( {i}_{t + m + s}\r... | Proof Let \( w \in {E}_{\lambda }^{21} \) . It implies by \( k,{j}_{h} \) with \( \left( k\right) w < \left( {j}_{h}\right) w \) being \( w \) -uncomparable that \( k < {j}_{h} \) . We see from the conditions \( \left( {\mathrm{a}}_{21}\right) \) and \( \left( {\mathrm{a}}_{23}\right) \) of the condition (a) that (1) o... | No |
Lemma 4.4 There exists a subset \( {F}_{\lambda }^{21} \) of \( {E}_{\lambda }^{21} \) such that each left-connected component of \( {E}_{\lambda }^{21} \) contains some element in \( {F}_{\lambda }^{21} \) . | Proof Let \( {F}_{\lambda }^{21} \) (see Fig. 3) be the set of all elements in \( {E}_{\lambda }^{21} \) satisfying (1)-(2) in Lemma 4.3 and the following conditions \( \left( {a}_{211}^{\prime }\right) - \left( {a}_{213}^{\prime }\right) \):\n\n\[ \left( {a}_{211}^{\prime }\right) \left( {1,2,\cdots, n}\right) = \left... | Yes |
Lemma 4.5 Any \( w \in {F}_{\lambda }^{21} \) is in a left-connected component of \( {E}_{\lambda } \) containing some \( {w}^{\prime } \in \) \( {F}_{\lambda }^{11} \) . | Proof For any \( w \in {F}_{\lambda }^{21} \) (see Fig. 3), let \( {w}^{\prime } = {w}_{{I}_{3}}{w}_{{I}_{2}}{w}_{{I}_{1}}w \) (see Fig. 2), where \( {I}_{1} = \) \( \left\{ {{t}_{0},{t}_{1},\cdots ,{t}_{h - 1}}\right\} ,{I}_{3} = \left\{ {{t}_{1},{t}_{2},\cdots ,{t}_{h + r}}\right\} \) and \( {I}_{2} = {I}_{3} \smalls... | Yes |
Lemma 4.6 There exists a subset \( {F}_{\lambda }^{12} \) of \( {E}_{\lambda }^{12} \) such that each left-connected component of \( {E}_{\lambda }^{12} \) contains some element in \( {F}_{\lambda }^{12} \) . | Proof Let \( w \in {E}_{\lambda }^{12} \) . By \( s = h = m = 0 \) and \( s + h + m + t = n - 1 \) we have \( t = n - 1 \) . By the conditions \( \left( {\mathrm{a}}_{21}\right) \) and \( \left( {\mathrm{a}}_{32}\right) \) of the condition \( \left( \mathrm{a}\right) , \) we see that any \( w \in {E}_{\lambda }^{12} \)... | Yes |
Lemma 4.8 For any \( w \in {F}_{\lambda } \), there exists some \( {w}^{\prime } \in \Omega \) with \( T\left( {w}^{\prime }\right) \in {\mathcal{C}}_{\alpha } \) and \( {w}^{\prime }{ \sim }_{L}w \) satisfying\n\n\[ T\left( {w}^{\prime }\right) = \left( \left\{ {\langle \left( \bar{k}\right) w\rangle ,\left\langle {\l... | Proof For any \( w \in {F}_{\lambda } \), let \( {w}^{\prime } = {w}_{{I}_{3}}{w}_{{I}_{4}}{w}_{{I}_{1}}{w}_{{I}_{2}}w \) (see Fig. 6), where \( {I}_{1} = \widetilde{S} \smallsetminus \left\{ {s}_{n}\right\} \) , \( {I}_{2} = {I}_{1} \smallsetminus \left\{ {s}_{0}\right\} ,{I}_{3} = \widetilde{S} \smallsetminus \left\{... | Yes |
Lemma 4.9 For any \( \mathbf{T} = \left( {{\mathbf{T}}_{\mathbf{1}},{\mathbf{T}}_{\mathbf{2}},{\mathbf{T}}_{\mathbf{3}}}\right) \in {\mathcal{C}}_{\alpha },\mathbf{T} \in \mathcal{T}\left( {\mathbf{F}}_{\lambda }\right) \) if and only if the following condition (d) holds:\n\n(d) There exist pairwise not \( {2n} \) -dua... | Proof The result follows from Lemma 4.1 and the fact that \( \langle \left( {{2n} + 1 - i}\right) w\rangle = {2n} + 1 - \langle \left( i\right) w\rangle \) for any \( i \in \left\lbrack {2n}\right\rbrack \) . | Yes |
Proposition 4.2 There exists a subset \( {F}_{\lambda }^{\prime } \) of \( {E}_{\lambda } \) such that each left-connected component of \( {E}_{\lambda } \) contains some \( w \in {F}_{\lambda }^{\prime } \) . | Proof \( \begin{aligned} \text{ Let }{F}_{\lambda }^{\prime } & = \left\{ {\left( {{k}_{1},{k}_{2},\cdots ,{k}_{n}}\right) \mid - {2n} \leq {k}_{i} < {k}_{j} \leq - 1}\right. \text{ and }{k}_{i} + {k}_{j} \neq - {2n} - 1 \\ & \text{ for any }1 \leq \end{aligned} \) \( i < j \leq n\} \) . We see that for any \( {w}^{\pr... | No |
Proposition 4.3 The set \( {F}_{\lambda }^{\prime } \) forms a representative set of the left cells of \( {\widetilde{C}}_{n} \) in \( {E}_{\lambda } \) and each left cells of \( {\widetilde{C}}_{n} \) is left-connected. | Proof The result follows from Lemma 2.7, Lemma 3.2, Proposition 4.2 and Lemma 4.10. | Yes |
Proposition 4.4 The set \( {E}_{\lambda } \) forms a single two-sided cell of \( {\widetilde{C}}_{n} \) . | Proof The result follows from Lemma 2.7 and Lemma 4.11. | No |
Proposition 4.5 There are \( {2}^{n} \) left cells of \( {\widetilde{C}}_{n} \) in \( {E}_{\lambda } \) . | Proof For any \( p : {p}_{1},{p}_{2},\cdots ,{p}_{n} \) in \( {P}_{n} \) and \( j \in \left\lbrack n\right\rbrack \), let \( {q}_{j} = {p}_{{i}_{j}} \), where \( {i}_{1},{i}_{2},\cdots ,{i}_{n} \) is a permutation of \( 1,2,\cdots, n \) satisfying \( {p}_{{i}_{1}} < {p}_{{i}_{2}} < \cdots < {p}_{{i}_{n}} \) . We see th... | Yes |
Lemma 1.1 \( {}^{\left\lbrack 4\right\rbrack } \) Let \( S \) be a semigroup and \( e \) be an idempotent in \( S \) . The following conditions are equivalent:\n\n(i) \( e{\mathcal{L}}^{ * }a \) ;\n\n(ii) \( {ae} = a \) and for all \( x, y \in {S}^{1},{ax} = {ay} \) implies \( {ex} = {ey} \) . | By duality, a similar condition holds for \( {\mathcal{R}}^{ * } \) . A semigroup in which each \( {\mathcal{L}}^{ * } \) -class and each \( {\mathcal{R}}^{ * } \) -class contain an idempotent is called an abundant semigroup (see [4]). The join of the equivalence relations \( {\mathcal{L}}^{ * } \) and \( {\mathcal{R}}... | Yes |
Lemma 1.4 \( {}^{\left\lbrack 2\right\rbrack } \) Let \( S \) be an adequate semigroup. The following conditions are equivalent:\n\n(i) \( {\mathcal{D}}^{ * } = \widetilde{\mathcal{D}} \)\n\n(ii) Every nonempty \( {\mathcal{H}}^{ * } \) -class contains a regular element.\n\nFurthermore, if (i) and (ii) hold, then \( {\... | However, as an example in [1] shows,(i) and (ii) are not necessary conditions for the equality \( {\mathcal{L}}^{ * } \circ {\mathcal{R}}^{ * } = {\mathcal{R}}^{ * } \circ {\mathcal{L}}^{ * } \) | No |
Lemma 2.1 Let \( S \) be a type \( A{\omega }^{2} \) -semigroup and \( E \) be its chain of idempotents. If \( {S}_{i, j} = {e}_{i, j}S{e}_{i, j} \), then\n\n(i) \( {S}_{i, j} \) is a type \( A{\omega }^{2} \) -semigroup with identity \( {e}_{i, j} \) ;\n\n(ii) \( {\mathcal{L}}^{ * }\left( {S}_{i, j}\right) = {\mathcal... | Proof (i) Certainly \( {S}_{i, j} \) is a subsemigroup of \( S \) and \( {e}_{i, j} \) is its identity. Since \( {e}_{i, j}{e}_{k, l} = \) \( {e}_{k, l}{e}_{i, j} = {e}_{k, l} \) for all \( \left( {i, j}\right) \geq \left( {k, l}\right), E\left( {S}_{i, j}\right) = \{ {e}_{k, l} : \left( {k, l}\right) \leq \left( {i, j... | Yes |
Lemma 2.2 Let \( S \) be a type \( A{\omega }^{2} \) -semigroup in which \( {\mathcal{D}}^{ * } = \widetilde{\mathcal{D}} \) . If \( {S}_{i} = \mathop{\bigcup }\limits_{{j, l = 0}}^{\infty }{H}_{\left( {i, j}\right) ,\left( {l, i}\right) }^{ * } \) , \( B = \mathop{\bigcup }\limits_{{i = 0}}^{\infty }{S}_{i},{T}_{p + j... | Proof (i) Let \( S \) be a type \( A{\omega }^{2} \) -semigroup in which \( {\mathcal{D}}^{ * } = \widetilde{\mathcal{D}} \) . Consider the set \( B = \mathop{\bigcup }\limits_{{i = 0}}^{\infty }{S}_{i} \) where \( {S}_{i} = \mathop{\bigcup }\limits_{{j, l = 0}}^{\infty }{H}_{\left( {i, j}\right) ,\left( {l, i}\right) ... | Yes |
Lemma 2.3 Let \( S \) be a type \( A{\omega }^{2} \) -semigroup in which \( {\mathcal{D}}^{ * } = \widetilde{\mathcal{D}} \) such that \( {R}_{m, n}^{ * } \neq \) \( \mathop{\bigcup }\limits_{{j = 0}}^{\infty }{H}_{\left( {m, n}\right) ,\left( {j, m}\right) }^{ * } \) for some \( \left( {m, n}\right) \in {\mathbb{N}}^{... | Proof As \( {R}_{p, q}^{ * } \neq \mathop{\bigcup }\limits_{{j = 0}}^{\infty }{H}_{\left( {p, q}\right) ,\left( {j, p}\right) }^{ * } \), there are two non-negative integers \( k \) and \( l \) where \( k \neq p \) such that \( {H}_{\left( {p, q}\right) ,\left( {l, k}\right) }^{ * } \neq \varnothing \) . Suppose that \... | Yes |
Lemma 2.4 Let \( S \) be a type \( A{\omega }^{2} \) -semigroup in which \( {\mathcal{D}}^{ * } = \widetilde{\mathcal{D}} \) such that \( {R}_{m, n}^{ * } \neq \) \( \mathop{\bigcup }\limits_{{j = 0}}^{\infty }{H}_{\left( {m, n}\right) ,\left( {j, m}\right) }^{ * } \) for some \( \left( {m, n}\right) \in {\mathbb{N}}^{... | Proof By Lemma 2.1, \( {S}_{p, q} = \bigcup \left\{ {{H}_{\left( {a, b}\right) ,\left( {d, c}\right) }^{ * } : \left( {a, b}\right) \leq \left( {p, q}\right) ,\left( {c, d}\right) \leq \left( {p, q}\right) }\right\} \) . Suppose that \( \left( {i, j}\right) > \left( {p, q}\right) ,\;i \neq m \) and \( {H}_{\left( {i, j... | Yes |
Theorem 2.1 Let \( S \) be a type \( A{\omega }^{2} \) -semigroup in which \( {\mathcal{D}}^{ * } = \widetilde{\mathcal{D}} \) . The following conditions are equivalent:\n\n(i) \( S \) has no \( * \) -kernel;\n\n(ii) \( S \) is an \( \omega \) -chain of type \( {A\omega } \) -semigroups in which \( {\mathcal{D}}^{ * } ... | Proof \( {If}\left( i\right) \) holds, then by Lemma 2.4, \( {R}_{m, n}^{ * } = \mathop{\bigcup }\limits_{{j = 0}}^{\infty }{H}_{\left( {m, n}\right) ,\left( {j, m}\right) }^{ * } \) for all \( \left( {m, n}\right) \in {\mathbb{N}}^{0} \times {\mathbb{N}}^{0}. \) Hence, \( S = \mathop{\bigcup }\limits_{{m = 0}}^{\infty... | Yes |
Theorem 2.2 Let \( T = \mathop{\bigcup }\limits_{{i = 0}}^{{p + q - 1}}{T}_{i} \) be a chain of pairwise-disjoint semigroups \( {T}_{0},{T}_{1},\cdots \) , \( {T}_{p + q - 1} \), where \( \left\{ {{T}_{0},{T}_{1},\cdots ,{T}_{p - 1}}\right\} \) is a set of type \( A\omega \) -semigroups in which \( {\mathcal{D}}^{ * } ... | Proof It is routine to verify that \( \theta \) is a homomorphism and so \( S \) is a semigroup with \( K \) as an ideal.\n\nLet \( E\left( {T}_{i}\right) = \left\{ {{e}_{i, j} : j \in {\mathbb{N}}^{0}}\right\} \left( {i = 0,1,\cdots, p - 1}\right) \) and \( E\left( {T}_{p + j}\right) = \left\{ {e}_{p, j}\right\} \left... | Yes |
Theorem 0.2 Let \( \varphi : \left\lbrack {0,1}\right\rbrack \rightarrow \lbrack 0,\infty ) \) be a function and \( 1 \leq q < \infty , - \frac{n}{q} \leq \lambda < 1 \) . Then \( {U}_{\varphi } \) is a bounded operator on \( {\operatorname{CBLO}}^{q,\lambda }\left( {\mathbb{R}}^{n}\right) \) if and only if | \[ {\int }_{0}^{1}{t}^{\lambda }\varphi \left( t\right) \mathrm{d}t < \infty \] Moreover, \[ {\begin{Vmatrix}{U}_{\varphi }\end{Vmatrix}}_{{\mathrm{{CBLO}}}^{q,\lambda }\left( {\mathbb{R}}^{n}\right) \rightarrow {\mathrm{{CBLO}}}^{q,\lambda }\left( {\mathbb{R}}^{n}\right) } = {\int }_{0}^{1}{t}^{\lambda }\varphi \left(... | Yes |
Corollary 0.1 Let \( 1 \leq q < \infty , - \frac{n}{q} \leq \lambda < 1 \) . Then \( \mathcal{H} \) is a bounded operator on \( {\operatorname{CBMO}}_{ * }^{q,\lambda }\left( {\mathbb{R}}^{n}\right) \) . Moreover, | \[ \parallel \mathcal{H}f{\parallel }_{{\mathrm{{CBMO}}}_{ * }^{q,\lambda }\left( {\mathbb{R}}^{n}\right) \rightarrow {\mathrm{{CBMO}}}_{ * }^{q,\lambda }\left( {\mathbb{R}}^{n}\right) } = \frac{n}{n + \lambda }. \] Especially, for \( \lambda = 0,\;{\left\| \mathcal{H}f\right\| }_{{\mathrm{{CBMO}}}_{ * }^{q,0}\left( {\... | Yes |
Corollary 0.2 Let \( 1 \leq q < \infty , - \frac{n}{q} \leq \lambda < 1 \) . Then \( \mathcal{H} \) is a bounded operator on \( {\mathrm{{CBLO}}}_{ * }^{q,\lambda }\left( {\mathbb{R}}^{n}\right) \) . Moreover, | \[ \parallel \mathcal{H}{\parallel }_{{\mathrm{{CBLO}}}_{ * }^{q,\lambda }\left( {\mathbb{R}}^{n}\right) \rightarrow {\mathrm{{CBLO}}}_{ * }^{q,\lambda }\left( {\mathbb{R}}^{n}\right) } = \frac{n}{n + \lambda }. \] Especially, for \( \lambda = 0,\parallel \mathcal{H}{\parallel }_{{\mathrm{{CBLO}}}_{ * }^{q,\lambda }\le... | Yes |
Lemma 1.1 \( {}^{\left\lbrack 2,4\right\rbrack } \) Let \( T \) be a symmetric \( \left( {0,2}\right) \) -type tensor field and let \( X \) be a vector field. Then\n\n\[ \operatorname{div}\left( {{i}_{X}T}\right) = \left( {\operatorname{div}T}\right) \left( X\right) + \left\langle {T,\nabla {\theta }_{X}}\right\rangle ... | where \( {L}_{X} \) is the Lie derivative of the metric \( g \) in the direction of \( X \) . Indeed, let \( \left\{ {{e}_{1},{e}_{2},\cdots ,{e}_{m}}\right\} \) be a local orthonormal frame field on \( M \) . Then\n\n\[ \frac{1}{2}\left\langle {T,{L}_{X}g}\right\rangle = \mathop{\sum }\limits_{{i, j = 1}}^{m}\frac{1}{... | Yes |
Lemma 1.2 \( {}^{\left\lbrack {12}\right\rbrack } \)\n\n\[ \n{\nabla }_{X}\left( \frac{{\left| \omega \right| }^{2}}{2}\right) = \left\langle {{i}_{X}{\mathrm{\;d}}^{\nabla }\omega + {\mathrm{d}}^{\nabla }{i}_{X}\omega }\right\rangle - \left\langle {\omega \odot \omega ,\nabla {\theta }_{X}}\right\rangle \n\] | \[ \n\langle {\mathrm{d}}^{\nabla }{i}_{X}\omega ,\omega \rangle = \mathop{\sum }\limits_{{{j}_{1} < {j}_{2} < \cdots < {j}_{p - 1};i}}\langle \omega \left( {{e}_{i},{e}_{{j}_{1}},{e}_{{j}_{2}},\cdots ,{e}_{{j}_{p - 1}}}\right) ,\left( {{\nabla }_{{e}_{i}}\omega }\right) \left( {X,{e}_{{j}_{1}},{e}_{{j}_{2}},\cdots ,{e... | No |
Lemma 1.3 Let \( \omega \in {A}^{p}\left( \xi \right) \left( {p \geq 1}\right) \) and let \( {S}_{f,\omega } \) be the stress-energy tensor defined by (2). Then for any vector field \( X \) on \( M \), we have\n\n\[ \left( {\operatorname{div}{S}_{f,\omega }}\right) \left( X\right) = \mathrm{d}f\left( X\right) \frac{{\l... | Proof By using Lemma 1.2 and (5), we derive the following\n\n\[ \left( {\operatorname{div}{S}_{f,\omega }}\right) \left( X\right) = \mathop{\sum }\limits_{{i = 1}}^{m}{\nabla }_{{e}_{i}}{S}_{f,\omega }\left( {{e}_{i}, X}\right) - {S}_{f,\omega }\left( {{e}_{i},{\nabla }_{{e}_{i}}X}\right) \]\n\n\[ = \mathop{\sum }\limi... | Yes |
Theorem 2.2 Suppose that \( u : \left( {M,{\varphi }^{2}{g}_{0}}\right) \rightarrow \left( {N, h}\right) \) is an \( f \) -harmonic map. If \( {C}_{0} - \mu > 0 \) and \( \varphi \) satisfies \( \left( {\varphi }_{1}\right) \) and \( \left( {\varphi }_{2}\right) \), then\n\n\[ \frac{{\int }_{B\left( {\rho }_{1}\right) ... | Proof This follows at once from Theorem 2.1 in which \( p = 1 \) and \( \omega = \mathrm{d}u \) . | Yes |
Lemma 2.2 Let \( \left( {{M}^{m}, g}\right) \) be a complete Riemannian manifold with a pole \( {x}_{0} \) . Denote by \( {K}_{r} \) the radial curvature of \( M \) .\n\n(i) If \( - {\alpha }^{2} \leq {K}_{r} \leq - {\beta }^{2} \) with \( \alpha \geq \beta > 0 \) and \( \left( {m - 1}\right) \beta - {2p\alpha } \geq 0... | Proof If \( {K}_{r} \) satisfies (i), then by Lemma 2.1, we have on \( B\left( r\right) - \left\{ {x}_{0}\right\} \), for every \( r > 0 \) ,\n\n\[ \left\lbrack {\left( {m - 1}\right) {\lambda }_{\min } + 2 - {2p}\max \{ 2,{\lambda }_{\max }\} }\right\rbrack \]\n\n\[ \geq \left( {m - 1}\right) {2\beta r}\coth \left( {\... | Yes |
Theorem 2.4 Let \( \\left( {M, g}\\right) \) be an \( m \) -dimensional complete manifold with a pole \( {x}_{0} \) . Assume that the radial curvature \( {K}_{r} \) of \( M \) satisfies one of the following three conditions:\n\n(i) \( - {\\alpha }^{2} \\leq {K}_{r} \\leq - {\\beta }^{2} \) with \( \\alpha \\geq \\beta ... | Proof From the proof of Theorem 2.1 for \( \\varphi = 1 \) and Lemma 2.2, we have\n\n\[ \n\\frac{\\mathrm{d}}{\\mathrm{d}r}\\frac{{\\int }_{B\\left( r\\right) }f\\frac{{\\left| \\omega \\right| }^{2}}{2}\\mathrm{d}{v}_{g}}{{r}^{\\Lambda - \\mu }} \\geq 0.\n\]\n\nTherefore, we get the monotonicity formula\n\n\[ \n\\frac... | Yes |
Theorem 2.5 Let \( M,{K}_{r} \) and \( \Lambda \) be as in Theorem 2.4. If \( \omega \in {A}^{p}\left( \xi \right) \left( {p \geq 1}\right) \) satisfies the generalized \( f \) -conservation law, and \( \frac{\partial f}{\partial r} \geq 0 \), then\n\n\[ \n\frac{{\int }_{B\left( {\rho }_{1}\right) }f\frac{{\left| \omeg... | Proof From Theorem 2.3 for \( \varphi = 1 \), we know that the formula (18) is true. | No |
Theorem 2.6 Suppose that \( \omega \in {A}^{p}\left( \xi \right) \left( {p \geq 1}\right) \) satisfies the following equation:\n\n\[ \n{\int }_{M}\left( {\operatorname{div}{S}_{f,\omega }}\right) \left( X\right) \mathrm{d}{v}_{g} = {\int }_{M}\mathrm{\;d}f\left( X\right) \frac{{\left| \omega \right| }^{2}}{2}\mathrm{\;... | Proof From (14), we have\n\n\[ \n\left\langle {{S}_{f,\omega },\frac{1}{2}{L}_{X}g}\right\rangle + \frac{{\left| \omega \right| }^{2}}{2}\mathrm{\;d}f\left( X\right) \geq \left( {{C}_{0} - \mu }\right) f\frac{{\left| \omega \right| }^{2}}{2}. \n\]\n\nOn the other hand, taking \( D = B\left( r\right) \) and \( T = {S}_{... | Yes |
Lemma 1.1 \( {}^{\lbrack 5\rbrack } \) Let \( F \) be a Finsler metric of scalar flag curvature on a manifold \( M \) . Suppose that the \( S \) -curvature is almost isotropic, \( S = \left( {n + 1}\right) {cF} + \eta \), where \( c = c\left( x\right) \) is a scalar function and \( \eta = {\eta }_{i}{y}^{i} \) is a clo... | \[ K = \frac{3{c}_{{x}^{m}}{y}^{m}}{F} + \sigma \] where \( \sigma = \sigma \left( x\right) \) is a scalar function on \( M \) . | Yes |
Lemma 1.2 \( {}^{\left\lbrack 7\right\rbrack } \) Let \( F = {\alpha \phi }\left( s\right), s = \frac{\beta }{\alpha } \), be an \( \left( {\alpha ,\beta }\right) \) -metric on a manifold and \( b \mathrel{\text{:=}} \parallel \beta {\parallel }_{\alpha } \) . Suppose that \( \phi \neq {k}_{1}\sqrt{1 + {k}_{2}{s}^{2}} ... | (i) \( \beta \) satisfies \( {r}_{j} + {s}_{j} = 0 \), and \( \phi = \phi \left( s\right) \) satisfies \( \Phi = 0 \) . In this case, \( S = 0 \) .\n\n(ii) \( \beta \) satisfies \( {r}_{ij} = \varepsilon ({b}^{2}{a}_{ij} - {b}_{i}{b}_{j}),\;{s}_{j} = 0 \), where \( \varepsilon \) is a scalar function and \( \phi = \phi... | Yes |
Lemma 2.1 \( {}^{\left\lbrack 2\right\rbrack } \) Let \( \left( {M, F}\right) \) be a Finsler metric and let \( R \) be the \( {hh} \) -curvature tensor of the Chern connection. Then the following four statements are equivalent:\n\n(a) \( {R}_{ii} = K{h}_{ii} \), that is, \( \left( {M, F}\right) \) has scalar flag curv... | And, given any of them, we have\n\n\[ \n{\ddot{A}}_{ijk} + K{A}_{ijk} + \frac{1}{3}\left( {{K}_{;i}{h}_{jk} + {K}_{;j}{h}_{ki} + {K}_{;k}{h}_{ij}}\right) = 0, \n\]\n\n(2.1)\n\n\[ \n\left( {n - 2}\right) \left( {\dot{K}{l}_{i} - {K}_{\mid i} + \frac{1}{3}{K}_{;i \mid r}{l}^{r}}\right) = 0. \n\]\n\n(2.2)\n\nwhere \( {R}_... | Yes |
Lemma 2.1 Let \( F \) be a Finsler metric with scalar flag curvature \( K \) on \( M \) . Then the followings hold:\n\n\[ \n{K}_{;j \mid i}{l}^{i} - {\dot{K}}_{;j} + {K}_{\mid j} - \dot{K}{l}_{j} = 0 \]\n\n(2.3)\n\n\[ \n{K}_{\left| j\right| i} - {K}_{\left| i\right| j} = K\left( {{K}_{;i}{l}_{j} - {K}_{;j}{l}_{i}}\righ... | Proof Using Ricci identities on \( K \), we have\n\n(1) \( {K}_{\left| j\right| i} - {K}_{\left| i\right| j} = - {K}_{;s}{R}_{ij}^{s} \) ,\n\n(2) \( {K}_{;j \mid i} - {K}_{\mid i;j} = {K}_{;s}{\dot{A}}_{ij}^{s} \) .\n\nSubstitute \( {R}_{ijl} = K{h}_{ijl} + \frac{1}{3}\left( {{h}_{ij}{K}_{;l} - {h}_{il}{K}_{;j}}\right)... | Yes |
Lemma 2.2 Let \( F \) be a Finsler metric with scalar curvature \( K \) on \( {M}^{n} \) . If \( n \geq 3 \) and \( \dot{K} = 0 \), then \( K = \) const. | Proof Substitute \( \dot{K} = 0 \) into Equations (2.2) and (2.3), we have\n\n\[ \n{K}_{\mid j} = {K}_{;j \mid i}{l}^{i} = 0.\n\]\n\nContracting \( {l}^{i} \) with Equation (2.4), we have \( {K}_{;i} = 0 \), and thus \( K = \) const. | Yes |
For a Finsler metric \( F \) with \( K = \frac{3{c}_{{m}^{i}}{y}^{i}}{F} + \sigma \) on \( {M}^{n} \), if \( n \geq 3 \), then there exists a scalar function on \( M \) such that\n\n\[ \n{c}_{{x}^{i}{x}^{j}}{y}^{i}{y}^{j} - {c}_{{x}^{s}}{G}^{s} = g\left( x\right) {F}^{2} - \mathrm{d}{\sigma F}.\n\] | Proof Substitute the equations (2.5), (2.6) and (2.8) into (2.2), we have\n\n\[ \n{F}_{{y}^{i}}\left( {\mathrm{\;d}{\sigma F} + 2{c}_{{x}^{i}{x}^{j}}{y}^{i}{y}^{j} - 2{c}_{{x}^{s}}{G}^{s}}\right) = F\left( {{\sigma }_{{x}^{i}}F + 2{c}_{{x}^{m}{x}^{i}}{y}^{m} - 2{N}_{i}^{j}{c}_{{x}^{j}}}\right) .\n\]\n\nThat is,\n\n\[ \... | Yes |
Theorem 3.1 Let \( F = {\alpha \phi }\left( s\right) \) be a non-Randers \( \left( {\alpha ,\beta }\right) \) -metric on \( {M}^{n} \), where \( \phi \) is a polynomial of \( s \) . If the flag curvature \( K = \frac{3{c}_{{n}^{i}}{y}^{i}}{F} + \sigma \) and \( n \geq 3 \), then \( K = 0 \) . | Proof From Theorem 2.1, we have\n\n\[ \n{c}_{{x}^{i}{x}^{j}}{y}^{i}{y}^{j} - {c}_{{x}^{s}}{G}^{s} = g\left( x\right) {F}^{2} - \mathrm{d}{\sigma F}.\n\] \n\nChoose a special coordinate system at a point as in [2]. Take a change of coordinates \( \left( {s,{y}^{a}}\right) \rightarrow \) \( \left( {y}^{i}\right) \) by\n\... | Yes |
Lemma 3.1 Suppose that \( F \) is a non-Riemannian \( \left( {\alpha ,\beta }\right) \) -metric. Then the mean Cartan tensor \( {A}_{i} \) and Cartan tensor \( {A}_{ijk} \) is given by: | \[ \begin{array}{l} {A}_{ijk} = {\phi }_{1}\left( s\right) \left( {{A}_{k}{h}_{ij} + {A}_{i}{h}_{jk} + {A}_{j}{h}_{ik}}\right) + {\phi }_{2}\left( s\right) {A}_{i}{A}_{j}{A}_{k}, \end{array} \] where \[ {\phi }_{1}\left( s\right) = - \frac{\phi \left\lbrack {\left( {\phi - s{\phi }^{\prime }}\right) {\phi }^{\prime } -... | Yes |
Theorem 3.3 Suppose that \( F = \alpha + \beta \) is a Randers metric on \( {M}^{n} \) with \( K = \frac{3{c}_{{a}^{i}}{y}^{i}}{F} + \sigma \) and \( {b}^{2} = \) const. If \( n \geq 3 \), then \( F \) is a Minkowski metric. | Proof Since \( K = \frac{3{c}_{{x}^{i}}{y}^{i}}{F} + \sigma \), we have\n\n\[ \n{r}_{00} + {2\beta }{s}_{0} = {2c}\left( {{\alpha }^{2} - {\beta }^{2}}\right) \n\]\n\ni.e.,\n\n\[ \n{r}_{ij} + {b}_{i}{s}_{j} + {b}_{j}{s}_{i} = {2c}\left( {{a}_{ij} - {b}_{i}{b}_{j}}\right) \n\]\n\n(3.5)\n\nContracting \( {b}^{i} \) with ... | Yes |
Proposition 2.1 Let \( \\left( {M, F}\\right) \) be a \( {C}^{\\infty } \) Finsler manifold with almost complex structure \( J \) . Then for each \( p \\in M \) there exists a neighborhood \( U\\left( p\\right) \) and a diffeomorphism \( {\\theta }_{p} : U\\left( p\\right) \\rightarrow U\\left( p\\right) \) such that\n... | Proof Put \( \\theta = - \\frac{1}{2}I + \\frac{\\sqrt{3}}{2}J \) where \( I \) is the identity, then \( {\\theta }^{3} = I \) and for each \( p \\in M \) there exists a neighborhood \( U\\left( p\\right) \) and a diffeomorphism \( {\\theta }_{p} : U\\left( p\\right) \\rightarrow U\\left( p\\right) \) such that \( {\\t... | Yes |
Proposition 2.2 Let \( \\left( {M, F}\\right) \) be a \( {C}^{\\infty } \) Finsler manifold, and \( p \\rightarrow {\\theta }_{p} \) be a family of local cubic diffeomorphisms on \( \\left( {M, F}\\right) \) . Then there exists a \( {C}^{\\infty } \) almost complex structure \( J \) on \( \\left( {M, F}\\right) \) . | Proof Let \( {\\theta }_{{p}^{ * }} : {M}_{p} \\rightarrow {M}_{p} \) denote the induced tangent map of \( {\\theta }_{p} \) at \( p,{\\theta }_{{p}^{ * }} = - \\frac{1}{2}{I}_{p} + \\frac{\\sqrt{3}}{2}{J}_{p} \) where \( {J}_{p}^{2} = - {I}_{p} \) . Furthermore, \( p \\rightarrow {J}_{p} \) is differentiable because \... | Yes |
Proposition 2.3 Let \( \\left( {M, F}\\right) \) be a 3-symmetric Finsler manifold. Then the group \( I\\left( M\\right) \) of holomorphic isometries of \( \\left( {M, F}\\right) \) acts transitively on \( \\left( {M, F}\\right) \) . | Proof The proof is similar to the case of 3-symmetric Riemannian manifold. | No |
Proposition 3.1 Let \( \\left( {M, F}\\right) \) be a 3-symmetric Finsler manifold. Then\n\n(1) \( t \) is an automorphism of \( G \) and \( {t}^{3} = 1 \) ;\n\n(2) \( {G}_{0}^{t} \\subseteq H \\subseteq {G}^{t} \)\n\n(3) There are \( G \) -invariant Finsler metric and \( G \) -invariant almost complex structure on the... | Proof (1) Let \( g \\in G \), then \( \\theta \\in I\\left( M\\right) \) and \( t\\left( g\\right) \\in G \) . So \( t \) is an automorphism of \( G \) and \( {t}^{3} = 1 \) .\n\n(2) Let \( h \\in H \), then the tangent maps of \( h, t\\left( h\\right) \) and \( {\\theta }_{p} \) at \( p \) satisfy\n\n\\[ \nt{\\left( h... | Yes |
Proposition 3.2 Let \( G \) be a Lie group, \( t : G \rightarrow G \) be an automorphism with \( {t}^{3} = 1 \), and \( H \) be a subgroup of \( G \) with \( {G}_{0}^{t} \subseteq H \subseteq {G}^{\prime } \) . Denote by \( \mathfrak{g} \) and \( \mathfrak{h} \) the Lie algebras of \( G \) and \( H \) , respectively, a... | Proof\n\n\[ \mathfrak{g} \otimes \mathbb{C} = \left( {\mathfrak{h} \otimes \mathbb{C}}\right) \oplus {\mathfrak{m}}^{ + } \oplus {\mathfrak{m}}^{ - }, \]\nwhere \( \mathfrak{h} \otimes \mathbb{C},{\mathfrak{m}}^{ + },{\mathfrak{m}}^{ - } \) are eigenspaces of \( {t}_{ * } \) corresponding to the eigenvalues \( 1,\frac{... | Yes |
Theorem 3.1 Let \( \left( {M, F}\right) \) be a connected 3-symmetric Finsler manifold and \( G \) be the full group of holomorphic isometries of \( \left( {M, F}\right) \) . Then there exists a Riemannian metric \( Q \) on \( M \) which is invariant under the action of \( G \) such that \( \left( {M, Q}\right) \) is a... | Proof The holomorphic isometry group \( G \) acts transitively on \( \left( {M, F}\right) \), so the identity component \( {G}_{0} \) of \( G \) also acts transitively on \( \left( {M, F}\right) \) . For any \( x \in M \), suppose that the isotropy group of \( x \) in \( G \) is \( H \), then the isotropy group of \( x... | Yes |
Theorem 3.2 Let \( G \) be a connected Lie group and \( t : G \rightarrow G \) be an automorphism of order 3. Let \( H \) be a subgroup of \( G \) with \( {G}_{0}^{t} \subseteq H \subseteq {G}^{t} \), we write \( \mathfrak{g} = \mathfrak{h} + \mathfrak{m} \) . Suppose that \( F \) is a Minkowski norm on \( \mathfrak{m}... | Proof Since \( F \) is an Minkowski norm on \( \mathfrak{m} \) with \( \left( {\mathfrak{g},\mathfrak{h}, F}\right) \) being a Minkowski Lie pair, \( F \) can induce an invariant Finsler metric on \( G/H \) (see [5]), then we just prove that for the invariant Finsler metric induced by \( F, G/H \) is a 3-symmetric Fins... | Yes |
Theorem 4.1 Let \( \left( {G/H, F}\right) \) be a 3-symmetric Finsler manifold, \( \mathfrak{g} = \operatorname{Lie}G,\mathfrak{h} = \operatorname{Lie}H \) , \( \mathfrak{g} = \mathfrak{h} + \mathfrak{m} \) with \( \mathfrak{m} \) being the orthogonal complement of \( \mathfrak{h} \) in \( \mathfrak{g} \) and \( \left\... | Proof From \( \left\lbrack {{\mathfrak{m}}_{ + },{\mathfrak{m}}_{ + }}\right\rbrack \subseteq {\mathfrak{m}}_{ - },\left\lbrack {{\mathfrak{m}}_{ - },{\mathfrak{m}}_{ - }}\right\rbrack \subseteq {\mathfrak{m}}_{ + },\left\lbrack {{\mathfrak{m}}_{ + },{\mathfrak{m}}_{ - }}\right\rbrack \subseteq \mathfrak{h} \) and the ... | Yes |
Theorem 4.2 Let \( \\left( {G/H, F}\\right) \) be a 3-symmetric Finsler manifold, \( \\mathfrak{g} = \\operatorname{Lie}G,\\mathfrak{h} = \\operatorname{Lie}H \) , \( \\mathfrak{g} = \\mathfrak{h} + \\mathfrak{m} \) with \( \\mathfrak{m} \) being the orthogonal complement of \( \\mathfrak{h} \) in \( \\mathfrak{g} \) a... | Proof Since \( \\left\\lbrack {{\\mathfrak{m}}_{ + },{\\mathfrak{m}}_{ + }}\\right\\rbrack \\subseteq {\\mathfrak{m}}_{ - },\\left\\lbrack {{\\mathfrak{m}}_{ - },{\\mathfrak{m}}_{ - }}\\right\\rbrack \\subseteq {\\mathfrak{m}}_{ + },\\left\\lbrack {{\\mathfrak{m}}_{ + },{\\mathfrak{m}}_{ - }}\\right\\rbrack \\subseteq ... | Yes |
Theorem 0.1 \( {}^{\left\lbrack 4\right\rbrack } \) A topological space \( G \) is a rectifiable space if and only if there exist \( e \in G \) and two continuous maps \( p : G \times G \rightarrow G, q : G \times G \rightarrow G \) such that for any \( x \in G, y \in G \), the following two identities hold:\n\n\[ p\le... | The above map \( p : G \times G \rightarrow G \) will be called a multiplication on \( G \) . Let \( G \) be a rectifiable space, and let \( p \) be the multiplication on \( G \) . We will write \( {xy} \) instead of \( p\left( {x, y}\right) \) and \( {AB} \) instead of \( p\left( {A, B}\right) \) for any \( A, B \subs... | Yes |
Theorem 1.1 Let \( G \) be a rectifiable space. If every sequentially compact subspace of \( G \) has countable pseudocharacter, then every sequentially compact subspace of \( G \) is metrizable. | Proof Let \( X \) be a non-empty sequentially compact subspace of \( G \) . Since the mapping \( q : G \times G \rightarrow G \) is continuous and the Cartesian product \( X \times X \) is sequentially compact, \( F = \) \( q\left( {X, X}\right) \) is a sequentially compact subset of \( G \) and \( e \in F \) . By the ... | Yes |
Theorem 1.2 If \( G \) is a locally \( \sigma \) -sequentially compact rectifiable space with the Souslin property, then \( G \) is \( \sigma \) -sequentially compact. | Proof For every \( \alpha \in \Gamma \), let \( {\mathcal{A}}_{\alpha } \) be a family of disjoint open subsets of \( G \) such that every element of \( {\mathcal{A}}_{\alpha } \) can be covered by countably many sequentially compact subsets of \( G \), for \( G \) is locally \( \sigma \) -sequentially compact.\n\nThe ... | Yes |
Theorem 1.3 Every locally countably compact and separable rectifiable space is \( \sigma \) - countably compact. | Proof Let \( G \) be a locally countably compact and separable rectifiable space. Then there exists a countable dense subset \( Y \) of \( G \) . Let \( U \) be an open neighborhood of \( e \) which is countably compact. It follows by Lemma 1.1 that \( G = {YU} = \mathop{\bigcup }\limits_{{y \in Y}}{yU} \), which compl... | No |
Lemma 1.2 Suppose that a rectifiable subspace \( H \) of a rectifiable space \( G \) contains a non-empty open neighborhood of \( e \) in \( G \) . Then \( H \) is open in \( G \) . | Proof Let \( U \) be an open neighborhood of \( e \) with \( U \subset H \) . Since the mapping \( p\left( {a, * }\right) \) is open for each \( a \in G, p\left( {a, U}\right) = {aU} \) is open in \( G \) . Therefore, \( H = \mathop{\bigcup }\limits_{{a \in H}}{aU} \) is open in \( G \) . | Yes |
Proposition 1.1 Let \( U \) be an arbitrary open subset of a connected rectifiable space \( G \) . Then \( G = \langle U\rangle \) . | Proof By induction on \( n \), we can define a sequence \( {\left\{ {A}_{n}\right\} }_{n \in \omega } \) of subsets of \( G \) such that\n\n(1) \( {A}_{0} = U \cup p\left( {U, U}\right) \cup q\left( {U, U}\right) \) ;\n\n(2) \( {A}_{1} = p\left( {{A}_{0},{A}_{0}}\right) \cup q\left( {{A}_{0},{A}_{0}}\right) \) ;\n\n(3)... | Yes |
Theorem 1.4 Every connected locally \( \sigma \) -compact rectifiable space \( G \) is \( \sigma \) -compact. | Proof Take an open neighborhood \( V \) of \( e \) in \( G \) such that \( F = \bar{V} = \mathop{\bigcup }\limits_{{i \in \mathbb{N}}}{K}_{i} \) is \( \sigma \) -compact, where each \( {K}_{i} \) is compact. Put\n\n(1) \( {A}_{0} = F \cup p\left( {F, F}\right) \cup q\left( {F, F}\right) \) ;\n\n(2) \( {A}_{1} = p\left(... | Yes |
Theorem 1.5 Every connected locally \( \sigma \) -sequentially compact rectifiable space \( G \) is \( \sigma \) - sequentially compact. | Proof Take an open neighborhood \( V \) of \( e \) in \( G \) such that \( F = \bar{V} = \mathop{\bigcup }\limits_{{i \in \mathbb{N}}}{K}_{i} \) is \( \sigma \) - sequentially compact, where each \( {K}_{i} \) is sequentially compact. Put\n\n(1) \( {A}_{0} = F \cup p\left( {F, F}\right) \cup q\left( {F, F}\right) \) ;\... | Yes |
Proposition 1.2 The product of countably many sequentially compact rectifiable spaces is a sequentially compact rectifiable space. | The product of countably many sequentially compact spaces is sequentially compact (see [5, Theorem 3.10.35]). It is easy to verify that the product of arbitrary family of rectifiable spaces with the Tychonoff product topology is also a rectifiable space. Therefore, the product of countably many sequentially compact rec... | Yes |
Theorem 2.2 If \( G \) is a \( \kappa \) -Fréchet-Urysohn rectifiable space, then it is strongly \( \kappa \) -Fréchet-Urysohn. | Proof Let \( \left\{ {{A}_{n} : n \in \omega }\right\} \) be a decreasing open family of \( G \) and let \( e \in \mathop{\bigcap }\limits_{{n \in \omega }}\overline{{A}_{n}} \) . If \( G \) is discrete, the conclusion is trivially true. Assume that \( G \) is non-discrete. Then there is a sequence \( \{ {a}_{n}{\} }_{... | Yes |
Theorem 2.3 The product of a \( \kappa \) -Fréchet-Urysohn rectifiable space \( G \) with a first countable space \( M \) is \( \kappa \) -Fréchet-Urysohn. | Proof Let \( j \) be the natural projection of \( G \times M \) onto \( G \) . For any open subset \( A \) of \( G \times M \) and any point \( \left( {x, y}\right) \in \bar{A} \subset G \times M \), let \( \left\{ {{U}_{n} : n \in \omega }\right\} \) be a decreasing countable base of the space \( M \) at the point \( ... | Yes |
Lemma 2.2 If \( 1 < c < 2 \), then we have\n\n\[ \n{\int }_{-\tau }^{\tau }\left| {I\left( x\right) }\right| \mathrm{d}x \ll {X}^{1 - c}\log X,\;{\int }_{-\tau }^{\tau }{\left| I\left( x\right) \right| }^{2}\mathrm{\;d}x \ll {X}^{2 - c}.\n\] | Proof By the first derivative estimate for trigonometric integrals (see [13, Lemma 4.3]),\n\nwe have\n\[ \nI\left( x\right) \ll \min \left( {X,\frac{1}{\left| x\right| {X}^{c - 1}}}\right) .\n\]\n\n(2.4)\n\nWe can safely deduce (2.3) from this estimate. We omit the details. | No |
Lemma 2.3 If \( 1 < c < 2 \), then we have\n\n\[ \n{\int }_{-\tau }^{\tau }{\left| S\left( x\right) \right| }^{2}\mathrm{\;d}x \ll {X}^{2 - c}\log X,\;{\int }_{-\tau }^{\tau }{\left| {S}_{1}\left( x\right) \right| }^{2}\mathrm{\;d}x \ll {X}^{2 - c}\log X, \]\n\n(2.5)\n\n\[ \n{\int }_{n}^{n + 1}{\left| S\left( x\right) ... | Proof It is in the same manner as in the proof of [14, Lemma 7]. | No |
Lemma 2.4 We have\n\n\[ \n{I}_{1}\left( x\right) \ll \frac{X}{\log X},\;{S}_{1}\left( x\right) \ll \frac{X}{\log X} \n\] | Proof This can be easily deduced from the prime number theorem. | No |
Lemma 2.5 Let\n\[ J\\left( X\\right) = {\\int }_{-\\infty }^{\\infty }{I}^{4}\\left( x\\right) \\Phi \\left( x\\right) e\\left( {-{Nx}}\\right) \\mathrm{d}x. \]\n\nIf \( 1 < c < 2 \), then we have\n\n\[ J\\left( X\\right) \\ll \\varepsilon {X}^{4 - c},\\;J\\left( X\\right) \\geq \\varepsilon {X}^{4 - c}. \] | Proof By (2.4) and (2.2),\n\n\[ J\\left( X\\right) = {\\int }_{\\left| x\\right| < {X}^{-c}}{I}^{4}\\left( x\\right) \\Phi \\left( x\\right) e\\left( {-{Nx}}\\right) \\mathrm{d}x + {\\int }_{\\left| x\\right| \\geq {X}^{-c}}{I}^{4}\\left( x\\right) \\Phi \\left( x\\right) e\\left( {-{Nx}}\\right) \\mathrm{d}x \]\n\n\[ ... | Yes |
Lemma 2.6 Let \( 1 < c < 2 \) . Then\n\n\[ \n{\int }_{\tau }^{H}{\left| S\left( x\right) \right| }^{2}\left| {\Phi \left( x\right) }\right| \mathrm{d}x \ll {X}^{1 + \eta } \n\] | Proof By Lemmas 2.1 and 2.3, we have\n\n\[ \n{\int }_{\tau }^{H}{\left| S\left( x\right) \right| }^{2}\left| {\Phi \left( x\right) }\right| \mathrm{d}x \ll \varepsilon {\int }_{0}^{1}{\left| S\left( x\right) \right| }^{2}\mathrm{\;d}x + \mathop{\sum }\limits_{{1 \leq n \leq H}}\frac{1}{n}{\int }_{n}^{n + 1}{\left| S\le... | Yes |
Lemma 4.1 Let \( \alpha \) and \( \beta \) be real, \( {\alpha \beta }\left( {\alpha - 1}\right) \left( {\beta - 1}\right) \left( {\alpha - 2}\right) \left( {\beta - 2}\right) \neq 0, X > 0, M, N \geq 1 \) , \( \left| {a\left( m\right) }\right| \leq 1,\left| {b\left( n\right) }\right| \leq 1 \) . Then\n\n\[ \n{\left( X... | Proof See \( \left\lbrack {{11}\text{, Theorem 9}}\right\rbrack \) . | No |
Lemma 4.2 Let \( \mathcal{D} \) be a subdomain of the rectangle \[ \{ \left( {x, y}\right) : M < x \leq {2M}, N < y \leq {2N}\} \;\left( {M \geq N}\right) \] such that any line parallel to any coordinate axis intersects it in \( O\left( 1\right) \) line segments. Let \( \alpha ,\beta \) be real numbers, \( {\alpha \bet... | Proof This is a version of Kolesnik's AB-Theorem. For the proof see [9, Lemma 1]. | No |
Lemma 4.3 Assume that \( x \) is a real number with \( {X}^{\frac{1}{2} - c - \eta } < \left| x\right| < \tau \), and that \( a\left( m\right), b\left( k\right) \) are complex numbers of modulars \( \leq 1 \) . Assume further that \( {MK} \asymp X \) and \( {X}^{\eta } \ll K \ll {X}^{\frac{1}{2} - \eta }. \) Then we ha... | Proof Denote the given sum by \( U \) . By Cauchy’s inequality and Weyl-Van der Corput inequality (see [5, Lemma 2.5]), we get\n\n\[ {\left| U\right| }^{2} \ll \frac{{X}^{2}}{Q} + \frac{X}{Q}\mathop{\sum }\limits_{{1 \leq q \leq Q}}\mathop{\sum }\limits_{{k \sim K}}\left| {\mathop{\sum }\limits_{{m \sim M}}e\left( {f\l... | Yes |
Lemma 4.4 Assume that \( x \) is a real number with \( {X}^{\frac{1}{2} - c - \eta } < \left| x\right| < \tau \), and that \( a\left( m\right) \) are complex numbers of modulus \( \leq 1 \) . Assume further that \( {MK} \asymp X \) and \( K \geq {X}^{\frac{1}{3} + \eta } \) . Then we have\n\n\[ \mathop{\sum }\limits_{{... | Proof Denote the given sum by \( U \) . The exponent pair \( \left( {\frac{1}{6},\frac{2}{3}}\right) \) gives\n\n\[ \left| U\right| \ll \mathop{\sum }\limits_{{m \sim M}}\left| {a\left( m\right) }\right| \left| {\mathop{\sum }\limits_{{k \sim K}}\mathrm{e}\left( {x{m}^{c}{k}^{c}}\right) }\right| \]\n\n\[ \ll M\left( {{... | Yes |
Lemma 4.5 Let \( 1 < c < \\frac{97}{81} \). Assume that \( x \) is a real number with \( \\tau < \\left| x\\right| < H \), and that \( a\\left( m\\right), b\\left( k\\right) \) are complex numbers of modulus \( \\leq 1,{MK} \\asymp X \). Assume further that \( K \) satisfies\n\n\[ B \\ll K \\ll C.\\]\n\nThen we have\n\... | Proof \( \\; \) Denote the given sum by \( U.\\; \) This follows immediately from Lemma 4.1 with \( \\left( {m, n}\\right) = \\left( {k, m}\\right) \). So we complete the proof of Lemma 4.5. | No |
Lemma 4.6 Let \( 1 < c < \\frac{97}{81} \) . Assume that \( x \) is a real number with \( \\tau < \\left| x\\right| < H \), and that \( a\\left( m\\right), b\\left( k\\right) \) are complex numbers of modulus \( \\leq 1 \), Assume further that \( {MK} \\asymp X \) and\n\n\[ K \\gg {X}^{\\frac{1}{2}} \]\n\nThen we have\... | Proof Denote the given sum by \( U \) . If \( K \\geq {X}^{\\frac{3}{5}} \), in the same manner as in (4.2), we have\n\n\[ \\left| U\\right| \\ll M\\left( {{\\left( \\left| x\\right| {X}^{c}{K}^{-1}\\right) }^{\\frac{1}{6}}{K}^{\\frac{2}{3}} + {\\left( \\left| x\\right| {X}^{c}{K}^{-1}\\right) }^{-1}}\\right) \\ll {X}^... | Yes |
Lemma 5.1 \( {}^{\left\lbrack 6\right\rbrack } \) Let \( v > 0 \) be fixed, \( x \geq {x}_{0}\left( v\right) ,{x}^{v} \leq z \leq x \) . Let \( \omega \left( x\right) \) also be the continuous solution of the differential-difference equation\n\n\[ \left\{ \begin{array}{ll} \omega \left( x\right) = \frac{1}{x}, & \text{... | \[ \omega \left( x\right) = \frac{1 + \log \left( {x - 1}\right) }{x},\;\text{ if }2 \leq x \leq 3. \]\n\n\[ \omega \left( x\right) = \frac{1 + \log \left( {x - 1}\right) }{x} + \frac{1}{x}{\int }_{2}^{x - 1}\frac{\log \left( {t - 1}\right) }{t}\mathrm{\;d}t,\;\text{ if }3 \leq x \leq 4. \]\n\n\[ \omega \left( x\right)... | Yes |
Lemma 5.4 Let \( 1 < c < \frac{97}{81},{MK} \asymp X \), and \( K \) satisfies the inequality (4.3). Let \( I, J \) be positive integers and \( {\mathcal{I}}_{i},{\mathcal{J}}_{j} \) be intervals for \( 1 \leq i \leq I,1 \leq j \leq J \) . Write \[ a\left( {m, k}\right) = \mathop{\sum }\limits_{\substack{{r{p}_{1}{p}_{... | Proof The proof is similar to that of [8, Lemma 11]. | No |
Let \( 1 < c < \frac{97}{81} \) and \( u \geq 1 \) . Suppose that \( K \) satisfies the condition (4.3), there exists a \( \mathcal{D} \subset \) \( \{ 1,2,\cdots, u\} \) with\n\n\[ \mathop{\prod }\limits_{{j \in \mathcal{D}}}{p}_{j} \sim K \]\n\nThen\n\n\[ \mathop{\sum }\limits_{{{p}_{1},{p}_{2},\cdots ,{p}_{u}}}S\lef... | Proof Set\n\n\[ k = \mathop{\prod }\limits_{{j \in \mathcal{D}}}{p}_{j},\;m = \left( {\mathop{\prod }\limits_{{j \notin \mathcal{D}}}{p}_{j}}\right) n. \]\n\nBy (5.13), we have\n\n\[ \mathop{\sum }\limits_{{{p}_{1},{p}_{2},\cdots ,{p}_{u}}}S\left( {{\mathcal{A}}_{{p}_{1}{p}_{2}\cdots {p}_{u}},{p}_{1}}\right) = \mathop{... | Yes |
Lemma 1.1 \( {}^{\left\lbrack {24}\right\rbrack } \) For any two chains in \( {LM} \) that are in general position, the binary operation\n\n\[ \bullet : {C}_{p}\left( {LM}\right) \otimes {C}_{q}\left( {LM}\right) \rightarrow {C}_{p + q - n}\left( {LM}\right) \]\n\nrespects the boundary map, and hence passes to the homo... | For a proof, see [24, Lemma 2.3]. Note that \( \bullet \) has degree \( - n \) . If we shift the degree of \( {\mathrm{H}}_{ * }\left( {LM}\right) \) down by \( n \), namely, let \( {\mathbb{H}}_{ * }\left( {LM}\right) \mathrel{\text{:=}} {\mathrm{H}}_{* + n}\left( {LM}\right) \), then \( \left( {{\mathbb{H}}_{ * }\lef... | No |
Lemma 1.2 The loop product on \( {C}_{ * }\left( {LM}\right) \) is commutative up to homotopy. Namely, there exists a binary operation\n\n\[ h : {C}_{ * }\left( {LM}\right) \otimes {C}_{ * }\left( {LM}\right) \rightarrow {C}_{ * }\left( {LM}\right) \]\n\nsuch that for any \( \mathbf{a},\mathbf{b} \in {C}_{ * }\left( {L... | Proof See [24, Lemma 3.2]. A pictorial explanation is given in Figure 3.\n\n\n\nFigure 3 Homotopy for the commutativity of the loop product. For two chains a and b in \( {C}_{ * }\left( {LM}\right) \) , fixing a time \( ... | No |
Example 2.1 (Chevalley-Eilenberg complex) Let \( \left( {\mathfrak{g},\left\lbrack {, \cdot }\right\rbrack }\right) \) be a (graded) Lie algebra. Denote by \( \widetilde{\mathfrak{g}} \) the desuspension of \( \mathfrak{g} \) (i.e., shifting the degrees of \( \mathfrak{g} \) down by one). Let \( {C}_{ * }\left( \mathfr... | \[ \Delta \left( {{\bar{a}}_{1}{\bar{a}}_{2}\cdots {\bar{a}}_{n}}\right) = \mathop{\sum }\limits_{{i < j}}{\left( -1\right) }^{i + j - 1}\overline{\left\lbrack {a}_{i},{a}_{j}\right\rbrack } \cdot {\bar{a}}_{1}{\bar{a}}_{2}\cdots {\widehat{\bar{a}}}_{i}\cdots {\widehat{\bar{a}}}_{j}\cdots {\bar{a}}_{n} \] where \( {}^{... | Yes |
Example 2.2 (Calabi-Yau manifolds) Let \( X \) be a compact, simply-connected Kähler manifold with \( {c}_{1}\left( X\right) = 0 \) . According to Yau’s solution to the Calabi conjecture, there exists a nowhere vanishing holomorphic volume form \( \Omega \) on \( X \), and \( X \) is called a Calabi-Yau manifold. Let \... | \[ \begin{matrix} \iota : & {\Omega }^{0, \bullet }\left( {X,\mathop{\bigwedge }\limits^{ \bullet }{\mathcal{T}}_{X}}\right) & \rightarrow & {\Omega }^{n - \bullet , \bullet }\left( X\right) \\ & \Phi & \mapsto & \Phi \dashv \Omega , \end{matrix} \] where \( \dashv \) is the interior product (contraction). Let \[ \begi... | No |
Lemma 2.1 (Tian-Todorov) Let \( X \) be a Calabi-Yau manifold. Then \( \left( {{\Omega }^{0, \bullet }\left( {X,\mathop{\bigwedge }\limits^{ \bullet }{\mathcal{T}}_{X}}\right) ,\land }\right. \) , \( \Delta \) ) is a Batalin-Vilkovisky algebra. | For a proof of this lemma, see, for example, [84, Lemma 3.1]. | No |
Example 2.3 (Riemannian manifolds) Let \( \left( {{M}^{n}, g}\right) \) be a Riemannian \( n \) -manifold with Riemannian metric \( g \) . Note that \( g \) induces a volume form on \( M \) . Let \( {\Omega }^{ \bullet }\left( M\right) \) be the differential forms on \( M \), which is a DG algebra with differential \( ... | Theorem 2.1 Let \( \lef | No |
Theorem 3.1 (Chas-Sullivan) There is a graded Lie algebra on the loop homology \( {\mathbb{H}}_{ * }^{{S}^{1}}\left( {LM}\right) \) . | Proof For the free loop space \( {LM} \), we have a fibration\n\n\[ \n{S}^{1} \rightarrow {S}^{\infty } \times {LM} \rightarrow {S}^{\infty }{ \times }_{{S}^{1}}{LM} \n\]\n\nwhich gives the Gysin long exact sequence\n\n\[ \n\cdots \rightarrow {\mathrm{H}}_{ * }\left( {LM}\right) \overset{E}{ \rightarrow }{\mathrm{H}}_{... | Yes |
Lemma 3.2 There is a Lie coalgebra structure on \( {\mathbb{H}}_{ * }^{{S}^{1}}\left( {{LM}, M}\right) \), where \( M \) is the set of constant loops. | Proof See Chas-Sullivan [25]. The idea of constructing the bracket \( \delta \) is the same as the bracket, namely, taking a family of loops and breaking it into two parts over the self-intersection points.\n\nHowever, different from the Lie bracket case, there is a small technical difficulty: the co-bracket does not r... | No |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.