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Corollary 2.2 Let \( R \) be a commutative ring and \( {M}_{n}\left( R\right) \) a \( p \) -ring, and \( \varphi \in {M}_{n}\left( R\right) \) and \( x \) , \( v \in \mathbb{N} \) . Then we have the following:\n\n(1) If there exist monic polynomials \( {h}_{0},{h}_{1} \in R\left\lbrack t\right\rbrack \) such that \( {h... | Proof (1) Let \( u = 1 \) . Then we complete the proof by Theorem 2.2.\n\n(2) Let \( u = v = 1 \) . Then it also follows from Theorem 2.2. | Yes |
Corollary 2.3 Let \( R \) be a commutative ring and \( {M}_{n}\left( R\right) \) a \( p \) -ring, \( \varphi \in {M}_{n}\left( R\right) \) and \( k \in \mathbb{N} \) . If there exists a factorization \( \chi \left( {\varphi }^{k}\right) = {h}_{0}{h}_{1} \) such that \( {h}_{0} \in {\mathbb{P}}_{0} \) and \( {h}_{1} \in... | Proof In view of the Cayley-Hamilton Theorem, \( \left( {\chi \left( {\varphi }^{n}\right) }\right) \left( {\varphi }^{n}\right) = {h}_{0}\left( {\varphi }^{n}\right) {h}_{1}\left( {\varphi }^{n}\right) = 0 \) and we are done by Theorem 2.2. | Yes |
Corollary 2.4 Let \( R \) be a commutative ring and \( {M}_{n}\left( R\right) \) a \( p \) -ring, \( m, k \in \mathbb{Z} \), and \( f\left( t\right) = \) \( \mathop{\prod }\limits_{{i = 1}}^{m}\left( {t - {\alpha }_{i}}\right) \in R\left\lbrack t\right\rbrack \), where \( {\alpha }_{i} \in N\left( R\right) \) or \( 1 +... | Proof Case I: \( {\alpha }_{i} \in N\left( R\right) \) for all \( 1 \leq i \leq m \) . Then \( f\left( t\right) \equiv {t}^{m}\left( {{\;\operatorname{mod}\;N}\left( R\right) \left\lbrack t\right\rbrack }\right) \) . Set \( {h}_{0} = f\left( t\right) \) and \( {h}_{1} = 1 \) . Then \( {h}_{0} \in {\mathbb{P}}_{0} \) an... | Yes |
Theorem 2.3 Let \( R \) be a commutative ring and \( {M}_{n}\left( R\right) \) a \( p \) -ring, \( \varphi \in {M}_{n}\left( R\right), k \in \mathbb{N} \) and \( a \) a periodic element of \( R \), and \( {h}_{0},{h}_{a} \) be two monic polynomials such that \( {h}_{0} \in {\mathbb{P}}_{0} \) and \( {h}_{a} \in {\mathb... | Proof Suppose that \( h = {h}_{0}{h}_{a} \) where \( {h}_{0} \in {\mathbb{P}}_{0} \) and \( {h}_{a} \in {\mathbb{P}}_{a} \), where \( a \) is a periodic element. If \( a \in N\left( R\right) \), setting \( {g}_{0} = {h}_{0}{h}_{a} \) and \( {g}_{1} = 1 \), then \( {g}_{0} \in {\mathbb{P}}_{0},{g}_{1} \in {\mathbb{P}}_{... | Yes |
Corollary 2.5 Let \( R \) be a commutative ring and \( {M}_{n}\left( R\right) \) a \( p \) -ring, \( \varphi \in {M}_{n}\left( R\right), k \in \mathbb{N}, a \) a periodic element of \( R \), and \( \chi \left( {\varphi }^{k}\right) = {h}_{0}{h}_{a} \) be such that \( {h}_{0} \in {\mathbb{P}}_{0} \) and \( {h}_{a} \in {... | Proof In view of Cayley-Hamilton Theorem, \( \left( {\chi \left( {\varphi }^{k}\right) }\right) \left( {\varphi }^{k}\right) = {h}_{0}\left( {\varphi }^{k}\right) {h}_{a}\left( {\varphi }^{k}\right) = 0 \), and we are done by Theorem 2.3. | No |
Corollary 2.6 Let \( R \) be a commutative ring, \( \varphi \in {M}_{n}\left( R\right) \) and \( {c}^{2} = e \in R,\left( {{t}^{s}, t - \bar{e}}\right) = \) \( R/N\left( R\right) \left\lbrack t\right\rbrack \) for any \( s \in \mathbb{N} \), and \( h \in R\left\lbrack t\right\rbrack \) be a monic polynomial. If \( h\le... | Proof By analogy with the proof of Theorem 2.3, we have \( {nR} = \ker \left( {{h}_{0}\left( \varphi \right) }\right) \oplus \ker \left( {{h}_{r}\left( \varphi \right) }\right) \) , \( \ker \left( {{h}_{0}\left( \varphi \right) }\right) \) and \( \ker \left( {{h}_{c}\left( \varphi \right) }\right) \) are \( \varphi \) ... | Yes |
Lemma 2.4 Let \( R \) be a commutative ring, \( \varphi \in {M}_{n}\left( R\right), h \in R\left\lbrack t\right\rbrack \) be a monic polynomial and \( k \in \mathbb{N} \) be a positive integer. If \( h\left( {\varphi }^{k}\right) = 0 \) and there exists a factorization \( h = {h}_{0}{h}_{1} \) such that \( {h}_{0} \in ... | Proof By analogy with the proof of Theorem 2.2, we obtain that \( {nR} = \ker \left( {{h}_{0}\left( {\varphi }^{k}\right) }\right) \oplus \) \( \ker \left( {{h}_{1}\left( {\varphi }^{k}\right) }\right) \) . Since \( {h}_{1} = 1 \), we obtain that \( \ker \left( {{h}_{1}\left( {\varphi }^{k}\right) }\right) = 0 \) and s... | Yes |
Proposition 2.1 Let \( R \) be a commutative ring and \( {M}_{n}\left( R\right) \) a \( p \) -ring, \( \varphi \in {M}_{n}\left( R\right), k \in \mathbb{N} \) and \( p \) a periodic element of \( R \), and \( f\left( t\right) = \mathop{\prod }\limits_{{i = 1}}^{n}\left( {t - {\alpha }_{i}}\right) \) where each \( {\alp... | Proof Case I: For all \( 1 \leq i \leq n,{\alpha }_{i} \in N\left( R\right) \) . Then \( f\left( t\right) \equiv {t}^{n}\left( {{\;\operatorname{mod}\;N}\left( R\right) \left\lbrack t\right\rbrack }\right) \) . Set \( {h}_{0} = f\left( t\right) \) , and \( {h}_{1} = 1 \) . Then \( {h}_{0} \in {\mathbb{P}}_{0} \) and \(... | Yes |
Proposition 2.2 Let \( R \) be a projective-free ring and \( {M}_{n}\left( R\right) \) be a \( p \) -ring. Then the following statements are equivalent:\n\n(1) \( \varphi \in {M}_{n}\left( R\right) \) is strongly \( \pi \) nil clean;\n\n(2) There exists a factorization \( \chi \left( {\varphi }^{k}\right) = {h}_{0}{h}_... | Proof (1) \( \Rightarrow \) (2). In view of Theorem 2.1, there exists a factorization \( {nR} = A \oplus B \) and a positive integer \( k \) such that \( A \) and \( B \) are \( \varphi \) -invariant, \( {\left. {\varphi }^{k}\right| }_{A} \in N\left( {\operatorname{end}\left( A\right) }\right) \) and \( 1 - {\left. {\... | No |
Proposition 2.3 Let \( R \) be a projective-free ring, \( {M}_{n}\left( R\right) \) be a \( p \) -ring and \( \varphi \in {M}_{n}\left( R\right) \) . Then the following statements are equivalent:\n\n(1) \( \varphi \) is strongly \( \pi \) nil clean;\n\n(2) There exists some positive integer \( k \) such that \( {\varph... | Proof (1) \( \Rightarrow \) (2). As \( \varphi \in {M}_{n}\left( R\right) \) is strongly \( \pi \) nil clean, there exist a periodic matrix \( E \in N\left( {{M}_{n}\left( R\right) }\right) \) and a nilpotent matrix \( W \in {M}_{n}\left( R\right) \) such that \( \varphi = E + W \) and \( {EW} = {WE} \) . In view of Le... | Yes |
Proposition 2.4 Let \( R \) be a projective-free ring and \( {M}_{2}\left( R\right) \) a \( p \) -ring. Then \( A \in {M}_{2}\left( R\right) \) is strongly \( \pi \) nil clean ring if and only if there exists some \( k \in \mathbb{N} \) such that\n\n(1) \( \det \left( {A}^{k}\right) \in 1 + N\left( R\right) \) and \( \... | Proof By virtue of [5, Proposition 3.5], we obtain that \( {A}^{k} \) is strongly nil clean and so \( {A}^{k} \in {M}_{2}\left( R\right) \) is strongly \( \pi \) nil clean. Then by Proposition 1.3, we complete the proof. | No |
Proposition 2.1 Let \( \left( {S, X}\right) \) be a dynamical system. Then the following conditions are equivalent.\n\n(1) \( \left( {S, X}\right) \) has dense g-small periodic sets;\n\n(2) For every nonempty open subset \( U \) of \( X \) , there exist \( z \in X \) and a g-syndetic subsemigroup \( T \) of \( S \) suc... | Proof \( \;\left( 1\right) \Rightarrow \left( 2\right) \) . Assume that \( \left( {S, X}\right) \) has dense g-small periodic sets. Let \( U \) be a nonempty open subset of \( X \) . Then there are a closed subset \( Y \subset U \) and a g-syndetic subsemigroup \( T \) of \( S \) such that \( {TY} \subset Y \) . Take a... | Yes |
Proposition 2.2 Let \( \\left( {S, X}\\right) \) be a dynamical system. Consider the following conditions.\n\n(1) \( \\left( {S, X}\\right) \) has dense g-small periodic sets;\n\n(2) For every nonempty open subset \( U \) of \( X \), there exists a g-syndetic subsemigroup \( T \) of \( S \) such that \( U \) contains a... | Proof (1) \( \\Rightarrow \) (2). It follows from Definition 0.3 and Lemma 1.3.\n\n\( \\left( 2\\right) \\Rightarrow \\left( 3\\right) \) . Let \( U \) be a nonempty open subset of \( X \) . Then there exists a g-syndetic sub-semigroup \( T \) of \( S \) such that \( U \) contains a minimal set \( Y \) of \( \\left( {T... | Yes |
Proposition 2.3 Let \( \left( {S, X}\right) \) be a dynamical system, and let \( \mathcal{F} \) be a family of \( S \) . If \( \left( {S, X}\right) \) is transitive and \( {sb}\mathcal{F} \) -point center, then \( {\operatorname{Trans}}_{{sb}\mathcal{F}}\left( {S, X}\right) = \operatorname{Trans}\left( {S, X}\right) \)... | Proof Assume that \( U \) is a nonempty open subset of \( X \) . Since \( \left( {S, X}\right) \) is \( {sb}\mathcal{F} \) -point center, there exists \( y \in U \) such that \( N\left( {y, U}\right) \in {sb}\mathcal{F} \) . By the definition of \( {sb}\mathcal{F} \), there is a subsemigroup \( A \in \mathcal{F} \) suc... | Yes |
Theorem 2.1 Let \( \left( {S, X}\right) \) be a dynamical system. Consider the following conditions.\n\n(1) \( \left( {S, X}\right) \) is transitive and has dense g-small periodic sets;\n\n(2) \( {\operatorname{Trans}}_{{sb}{\mathcal{F}}_{gs}}\left( {S, X}\right) = \operatorname{Trans}\left( {S, X}\right) \) is dense i... | Proof (1) \( \Rightarrow \) (2). Since \( \left( {S, X}\right) \) has dense g-small periodic sets, the system \( \left( {S, X}\right) \) is \( {sb}{\mathcal{F}}_{gs} \) - point center by Proposition 2.2. By Proposition 2.3, \( {\operatorname{Trans}}_{{sb}{\mathcal{F}}_{gs}}\left( {S, X}\right) = \operatorname{Trans}\le... | Yes |
Proposition 3.1 Let \( S \) be an abelian semigroup. If \( A \subset S \) is an \( m \) -set for \( S \), then there exists \( r \in S \) such that for every g-syndetic subsemigroup \( T \) of \( S \), the set \( {N}_{T}\left( {r, A}\right) = \{ t \in T \) : \( {rt} \in A\} \) is syndetic in \( T \) . | Proof Since \( A \subset S \) is an \( m \) -set for \( S \), there exist a minimal system \( \left( {S, X}\right), x \in X \) and a nonempty open subset \( U \) of \( X \) such that \( A \supset N\left( {x, U}\right) \) . Take \( r \in S \) and a neighborhood \( V \) of \( x \) such that \( {rV} \subset U \), then \( ... | Yes |
Lemma 3.2 Let \( \\left( {S, X}\\right) \) be a dynamical system where \( S \) has at most countably many g-syndetic subsemigroups. If \( \\left( {S, X}\\right) \) is g-totally transitive, then there exists \( x \\in X \) such that \( x \\in \\operatorname{Trans}\\left( {T, X}\\right) \) for every g-syndetic subsemigro... | Proof Let \( {\\left\\{ {U}_{i}\\right\\} }_{i = 1}^{\\infty } \) be a countable base of \( X \) . Assume that \( \\Lambda \) is the family of all the g-syndetic subsemigroups of \( S \) . Since \( \\Lambda \) is countable, we may assume that \( \\Lambda = {\\left\\{ {T}_{k}\\right\\} }_{k = 1}^{\\infty } \) . Since \(... | Yes |
Lemma 3.3 Let \( \left( {S, X}\right) \) be a dynamical system where \( S \) is an abelian semigroup and every \( s \) of \( S \) is a surjective map from \( X \) onto itself. If \( T \) is a subsemigroup of \( S \), and \( x \in \operatorname{Trans}\left( {T, X}\right) \) , then \( {rx} \in \operatorname{Trans}\left( ... | Proof Assume that \( x \in \operatorname{Trans}\left( {T, X}\right) \) . Then \( \overline{Tx} = X \) . Choose \( r \in S \) . Since \( S \) is abelian and every \( s \) of \( S \) is a surjective map from \( X \) onto itself, we have \( \overline{T\left( {rx}\right) } = \overline{r\left( {Tx}\right) } = r\left( \overl... | Yes |
Proposition 3.2 Let \( \\left( {S, X}\\right) \) be a dynamical system where \( S \) is a countable abelian semigroup, \( S \) has at most countably many g-syndetic subsemigroups and every \( s \) of \( S \) is a surjective map from \( X \) onto itself. If \( \\left( {S, X}\\right) \) is a g-totally transitive system w... | Proof Assume that \( \\left( {S, X}\\right) \) is a g-totally transitive system with dense g-small periodic sets. By Lemma 3.2, there exists \( x \\in X \) such that \( x \\in \\operatorname{Trans}\\left( {{T}_{1}, X}\\right) \) for every g-syndetic subsemigroup \( {T}_{1} \) of \( S \) . Let \( U \) be a neighborhood ... | Yes |
Theorem 3.1 Let \( \\left( {S, X}\\right) \) be a dynamical system and \( x \\in \\operatorname{Trans}\\left( {S, X}\\right) \) . Then \( \\left( {S, X}\\right) \) is disjoint from all minimal system if and only if \( N\\left( {x, U}\\right) \\cap A \\neq \\varnothing \) for any neighborhood \( U \) of \( x \) and any ... | Proof \( \\left( \\Rightarrow \\right) \) Assume that \( \\left( {S, X}\\right) \) is disjoint from all minimal systems, and \( A \\subset S \) is an \( m \) -set for \( S \) . By Definition 3.1, there exist a minimal system \( \\left( {S, Y}\\right) \), a point \( y \\in Y \) and a nonempty open \( V \) of \( Y \) suc... | Yes |
Proposition 3.3 Let \( \left( {S, X}\right) \) be a transitive dynamical system where \( S \) is a countable abelian semigroup. Let \( x \in \operatorname{Trans}\left( {S, X}\right) \) . If for every neighborhood \( U \) of \( x \) we have the property \( \left( *\right) \), then \( \left( {S, X}\right) \) is disjoint ... | Proof By Theorem 3.1, it suffices to prove that \( N\left( {x, U}\right) \cap A \neq \varnothing \) for any neighborhood \( U \) of \( x \) and any \( m \) -set \( A \) for \( S \) . By Proposition 3.1, there exists \( r \in S \) such that the set \( {N}_{T}\left( {r, A}\right) = \) \( \{ t \in T : {rt} \in A\} \) is s... | Yes |
Theorem 2.1 If a regular \( p \) -group \( G \) has Type \( \left( {{e}_{1},{e}_{2},1}\right) \), where \( {e}_{1} \geq 2{e}_{2},{e}_{2} \geq 3 \), then \( G \) is one of the following groups: | Since \( {\mho }_{1}\left( G\right) \) is regular and \( \exp \left( {G}^{\prime }\right) = {p}^{x},{\left\lbrack {a}^{p},{b}^{p}\right\rbrack }^{{p}^{x - 2}} = 1 \) by Lemma 1.3. It follows that \( \exp \left( {{\mho }_{1}{\left( G\right) }^{\prime }}\right) = {p}^{x - 2} \) . Moreover, \( {\mho }_{1}\left( G\right) \... | No |
Theorem 2.3 Let \( G \) be the groups (II) in Theorem 2.1. If \( {e}_{1} \geq 2{e}_{2} \) and \( {e}_{2} \geq 3 \), then \( G \) are nonabelian regular \( p \) -groups with Type \( \left( {{e}_{1},{e}_{2},1}\right) \) . | Proof By a similar argument as that in Theorem 2.2, we get \( \left| G\right| = {p}^{{e}_{1} + {e}_{2} + 1} \) . The details are omitted.\n\nWe notice that \( \left\langle {{a}^{p},{b}^{p}}\right\rangle \leq {\mho }_{1}\left( G\right) \) and \( \left| \left\langle {{a}^{p},{b}^{p}}\right\rangle \right| \geq {p}^{{c}_{1... | No |
Theorem 0.3 Let \( \left( {{M}^{n};g}\right) \) be a compact Riemannian manifold of dimension \( n \) with \( {\operatorname{Ric}}_{f} \geq \) \( - K \) for a constant \( K \) . Let \( c \geq 0 \) and \( \alpha \geq 1\left( {c < 0\text{and}0 \leq \alpha \leq 1}\right) \) be two constants and \( u \) be a positive solut... | Proof of Theorem 0.2 Let \( \widetilde{u} = \frac{u}{M} \) . Then we have \( 0 < \widetilde{u} \leq 1 \) . From (2), we obtain that \( \widetilde{u} \) satisfies the following equation\n\n\[{\widetilde{u}}_{t} = {\Delta }_{f}\widetilde{u} + a\widetilde{u}\log \widetilde{u} + \widetilde{c}\widetilde{u}\]\n\n(4)\n\nwhere... | No |
Proposition 1.2 \( {}^{\left\lbrack {10}\right\rbrack } \) When \( 2 \leq {2p} < n \), (ii) \( {B}^{\left( p\right) } \) is divergence-free; | Proof (ii) For\n\n\[ \n{B}_{{ijkl}, l}^{\left( p\right) } = \frac{p}{{2}^{p}}{\delta }_{{ij}{j}_{1}{j}_{2}\cdots {j}_{{2p} - 1}{j}_{2p}}^{{kl}{i}_{1}{i}_{2}\cdots {i}_{{2p} - 1}{i}_{2p}}{R}_{{j}_{1}{j}_{2}{i}_{1}{i}_{2}, l}\cdots {R}_{{i}_{{2p} - 1}{i}_{2p}{j}_{{2p} - 1}{j}_{2p}} \]\n\n\[ \n= - \frac{p}{{2}^{p}}{\delta... | Yes |
Lemma 2.1 \( {}^{\left\lbrack {10}\right\rbrack } \) We denote \( {F}^{p} = {\int }_{M}{L}_{p}\mathrm{\;d}v \), then\n\n\[ \nabla {F}_{is}^{p} = \frac{{L}_{p}}{2}{g}_{is} - \frac{p}{2}{B}_{ijkl}^{\left( p - 1\right) }{R}_{s}{}^{jkl} = \frac{1}{2}{A}^{\left( p\right) }.\n\] | Proof\n\n\[ \frac{\mathrm{d}}{\mathrm{d}t}{F}^{p} = {\int }_{M}\frac{\mathrm{d}}{\mathrm{d}t}{L}_{p}\mathrm{\;d}v + \frac{1}{2}{\int }_{M}{L}_{p}\operatorname{tr}h\mathrm{\;d}v.\n\]\n\nFrom the definition of \( {L}_{p} \) and the fact that \( {B}^{\left( p - 1\right) } \) is divergence-free, we have from (1.3) and \( \... | Yes |
Theorem 2.1 Let \( g \) be a critical point of \( {\widetilde{F}}^{p} \) restrict on \( \left\lbrack g\right\rbrack \), and \( g\left( t\right) \) be a smooth variation of \( g \) in \( \left\lbrack g\right\rbrack ,{\left. \frac{\mathrm{d}}{\mathrm{d}t}\right| }_{t = 0}g\left( t\right) = {fg} \) with \( {\int }_{M}f\ma... | Proof From the assumption that \( {\int }_{M}f\mathrm{\;d}v = 0 \), we have \( {V}^{\prime } = 0 \), where \( {\left. {}^{\prime } = \frac{\mathrm{d}}{\mathrm{d}t}\right| }_{t = 0} \) . Since \( g \) is a critical point of \( {\widetilde{F}}^{p}{\left. \right| }_{\left\lbrack g\right\rbrack } \), then \( {\widetilde{F}... | Yes |
Theorem 2.2 Let \( g \) be a critical point of \( {\widetilde{F}}^{2} \), and \( g\left( t\right) \) be a smooth variation of \( g \) in \( \mathcal{M} \) , \( {\left. \frac{\mathrm{d}}{\mathrm{d}t}\right| }_{t = 0}g\left( t\right) = h \) and \( h \in {\mathcal{S}}_{0} = \left\{ {{\operatorname{tr}}_{g}h = 0,\operatorn... | Proof Since \( {\operatorname{tr}}_{g}h = 0 \), then \( {V}^{\prime } = 0 \) . And \( g \) is a critical point of \( {\widetilde{F}}^{2} \), so \( {\widetilde{F}}^{\prime } = {F}^{\prime } = 0 \) . Hence,\n\n\[ {\left. \frac{{\mathrm{d}}^{2}}{\mathrm{\;d}{t}^{2}}\right| }_{t = 0}{\widetilde{F}}^{2} = {\int }_{M}\left\l... | Yes |
Lemma 4.1 On \( \left( {{\mathbb{{CP}}}^{n},{g}_{FS}}\right) \), \[ {A}_{ij}^{\left( p\right) } = {L}_{p}{g}_{ij} - p\left( {{2n} - {2p} + 2}\right) {A}_{ij}^{\left( p - 1\right) } - {2p}{A}^{\left( p - 1\right) }\left( {J{e}_{i}, J{e}_{j}}\right) . \] | Proof Let \( \left\{ {{e}_{1},{e}_{2},\cdots ,{e}_{2n}}\right\} \) be the orthonormal frame, from the definition of \( {A}^{\left( p\right) }\left( {1.1}\right) \) , we have \[ {A}_{ij}^{\left( p\right) } = {L}_{p}{g}_{ij} - \frac{p}{{2}^{p - 1}}{\delta }_{{j}_{1}}^{i}{\delta }_{j{j}_{2}\cdots {j}_{{2p} - 1}{j}_{2p}}^{... | Yes |
Lemma 2.1 Let \( {\alpha }^{\nu },{\beta }^{\nu ij},{\sigma }^{\nu j} \) and \( {X}^{v}\left( s\right) \) be bounded elements in \( {\mathbb{L}}_{G}^{2}\left( {\Omega }_{s}^{d}\right) \) for some \( s \in \) \( \left\lbrack {0, T}\right\rbrack \), where \( \nu = 1,2,\cdots, n \) and \( i, j = 1,2,\cdots, d \) . Conside... | Proof To simplify presentation, we shall prove only the case when \( n = d = 1 \), as the higher dimensional case can be treated in the same way.\n\nWe first consider the case that \( t \in \lbrack s, T) \) . For each positive integer \( N \), set \( \delta = \frac{t - s}{N} \), and take the partition\n\n\[ \n{\pi }_{\... | Yes |
Theorem 2.1 Suppose that \( \widehat{u} \in {\mathbb{C}}_{l}^{1 + \frac{\gamma }{2},2 + \gamma }\left( {\Lambda }^{n}\right) \) for some \( \gamma \in \left( {0,1}\right) \) and \( {\alpha }^{\nu },{\beta }^{\nu ij},{\sigma }^{\nu j} \) are bounded processes in \( {M}_{G}^{2}\left( {0, T}\right) \) for \( \nu = 1,2,\cd... | Proof To simplify presentation, we shall prove only the case when \( n = d = 1 \), as the higher dimensional case can be treated in the same way without substantial difficulty.\n\nWe first consider the case where \( \alpha \left( u\right) ,\beta \left( u\right) ,\sigma \left( u\right) \) are step processes of the form\... | Yes |
Theorem 2.2 Suppose that \( u \in {\mathcal{H}}^{\gamma }\left( {\Omega }^{n}\right) \) for some \( \gamma \in \left( {0,1}\right) \) and \( {\alpha }^{\nu },{\beta }^{\nu ij},{\sigma }^{\nu j} \) are bounded processes in \( {M}_{G}^{2}\left( {0, T}\right) \) for \( \nu = 1,2,\cdots, n \) and \( i, j = 1,2,\cdots, d \)... | Proof \( {It} \) is obvious that Equation \( \left( 6\right) \) holds true for \( u \in {\mathbb{C}}_{l}^{1 + \gamma ,2 + \gamma }\left( {\Omega }^{n}\right) \) by Theorem 2.1. Recalling Theorem 2.2 in Chapter IV of Peng [18] (see also the proof of Steps 1-2 in Theorem 4.1 of Hu et al. [8]), we can also obtain Equation... | Yes |
Corollary 2.1 Suppose that \( B \) is a 1-dimensional \( G \) -Brownian motion. For each \( \gamma \in \left( {0,1}\right) \) and \( u \in {\mathcal{H}}^{\gamma }\left( {\Omega }^{1}\right) \), we have for each \( t \in \left\lbrack {0, T}\right\rbrack \), in \( {\mathbb{L}}_{G}^{2}\left( {\Omega }_{t}^{1}\right) \) | \[ u\left( {B}_{t}\right) = u\left( {B}_{0}\right) + {\int }_{0}^{t}{D}_{t}u\left( {B}_{u}\right) \mathrm{d}u + {\int }_{0}^{t}{D}_{x}u\left( {B}_{u}\right) \mathrm{d}B\left( u\right) + \frac{1}{2}{\int }_{0}^{t}{D}_{xx}^{2}u\left( {B}_{u}\right) \mathrm{d}\langle B\rangle \left( u\right) . \] | Yes |
Example 2.1 Suppose that \( B \) is a 1-dimensional \( G \) -Brownian motion. For a given \( f \in \) \( {\mathbb{C}}^{1 + \frac{\alpha }{2},2 + \alpha }\left( {\left\lbrack {0, T}\right\rbrack \times \mathbb{R}}\right) \) with \( \alpha \in \left( {0,1}\right) \), we have for each \( t \in \left\lbrack {0, T}\right\rb... | \[ f\left( {t, B\left( t\right) }\right) - f\left( {s, B\left( s\right) }\right) = {\int }_{s}^{t}{\partial }_{u}f\left( {u, B\left( u\right) }\right) \mathrm{d}u + {\int }_{s}^{t}{\partial }_{x}f\left( {B\left( u\right) }\right) \mathrm{d}B\left( u\right) + \frac{1}{2}{\int }_{s}^{t}{\partial }_{xx}^{2}f\left( {B\left... | Yes |
Example 2.2 Suppose\n\n\[ u\left( {\omega }_{t}\right) = {\int }_{0}^{T}g\left( {s,{\omega }_{t, T - t}\left( s\right) }\right) \mathrm{d}s \]\nwhere \( g \in {C}_{b,\text{ Lip }}^{0,2}\left( {\left\lbrack {0, T}\right\rbrack \times \mathbb{R};\mathbb{R}}\right) \) . Then by the definitions of Dupire’s derivatives, | \[ {D}_{t}u\left( {\omega }_{t}\right) = 0,\;{D}_{x}u\left( {\omega }_{t}\right) = {\int }_{t}^{T}{\partial }_{x}g\left( {s,\omega \left( t\right) }\right) \mathrm{d}s,\;{D}_{xx}^{2}u\left( {\omega }_{t}\right) = {\int }_{t}^{T}{\partial }_{xx}^{2}g\left( {s,\omega \left( t\right) }\right) \mathrm{d}s. \]\n\nOne can sh... | Yes |
Theorem 2.3 If \( M\left( t\right) = u\left( {B}_{t}\right) \) for some function \( u \in {\mathcal{H}}^{\gamma }\left( {\Omega }^{d}\right) \) with \( \gamma \in \left( {0,1}\right) \), then\n\n\[ M\left( T\right) = \widehat{\mathbb{E}}\left\lbrack {M\left( T\right) }\right\rbrack + \mathop{\sum }\limits_{i}{\int }_{0... | Proof Applying Theorem 2.2, we obtain for each \( t \in \left\lbrack {0, T}\right\rbrack \) ,\n\n\[ u\left( {B}_{t}\right) = u\left( {B}_{0}\right) + {\int }_{0}^{t}{D}_{t}u\left( {B}_{s}\right) \mathrm{d}s + \mathop{\sum }\limits_{i}{\int }_{0}^{t}{D}_{{x}_{i}}u\left( {B}_{s}\right) \mathrm{d}{B}^{i}\left( s\right) + ... | Yes |
Theorem 3.1 Let \( u \in {\mathcal{H}}^{\gamma }\left( {\Omega }^{n}\right) \) for some \( \gamma \in \left( {0,1}\right) \) be a solution of Equation (11) with\n\n\[ \bar{G}\left( {{\omega }_{t}, r, p, Q}\right) = \left\langle {p,\alpha \left( {\omega }_{t}\right) }\right\rangle + G\left( {\left( \left\langle Q{\sigma... | Proof Applying Theorem 2.2, we get that\n\n\[ \mathrm{d}u\left( {X}_{t}\right) = \left\lbrack {{D}_{t}u\left( {X}_{t}\right) + \left\langle {{D}_{x}u\left( {X}_{t}\right) ,\alpha \left( {X}_{t}\right) }\right\rangle }\right\rbrack \mathrm{d}t + \left\langle {{D}_{x}u\left( {X}_{t}\right) ,\sigma \left( {X}_{t}\right) }... | Yes |
Consider the following PPDE\n\n\[\n\left\{ \begin{array}{l} {D}_{t}u\left( {\omega }_{t}\right) + G\left( {{D}_{xx}^{2}u\left( {\omega }_{t}\right) }\right) = 0,\;{\omega }_{t} \in {\Omega }^{1}, \\ u\left( {\omega }_{T}\right) = \Phi \left( {\omega }_{T}\right) ,\;\omega \in {\Omega }_{T}^{1}, \end{array}\right.\n\]\n... | We first consider the following system of fully nonlinear parabolic partial differential equations, defined on \( \left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{2} \) and parameterized by \( y \in \mathbb{R} \) ,\n\n\[\n\left\{ \begin{array}{l} {\partial }_{s}{v}_{1}\left( {s, x, y}\right) + G\left( {{\partial }... | Yes |
Theorem 1.2 Let \( 0 < \alpha < \frac{1}{2},\gamma = \frac{1}{2} - \alpha, f \in K = \left\{ {f \in {\mathcal{H}}^{d};I\left( f\right) \leq 1}\right\} \) and let \( k\left( \alpha \right) > 0 \) be defined as in (1.1) and \( t \geq 0 \) . Then we have for any \( \tau > 0 \) , | \[ \mathop{\lim }\limits_{{\varepsilon \rightarrow 0}}{\varepsilon }^{\frac{1}{\gamma }}\log {C}_{r, p}\left( {{\begin{Vmatrix}\frac{w\left( {t + h \cdot }\right) - w\left( t\right) }{\sqrt{h}} - \frac{f}{{\varepsilon }^{\frac{1}{2\gamma }}}\end{Vmatrix}}_{\alpha } \leq {\varepsilon \tau }}\right) \] \[ = \mathop{\lim ... | Yes |
Assume that \( {a}_{u} \) satisfies the conditions (i) and (ii), then, for any \( f \in K \) with \( I\left( f\right) < 1 \), we have\n\n\[ \mathop{\liminf }\limits_{{u \rightarrow \infty }}{\left( {\rho }_{u}\right) }^{1 - \alpha }\mathop{\inf }\limits_{{t \in \left\lbrack {0,1 - \frac{{a}_{u}}{u}}\right\rbrack }}\par... | We divide the proof of Theorem 2.1 into the following steps:\n\n(1) Under the conditions (i) and (ii),\n\n\[ \mathop{\liminf }\limits_{{u \rightarrow \infty }}{\left( {\rho }_{u}\right) }^{1 - \alpha }\mathop{\inf }\limits_{{t \in \left\lbrack {0,1 - \frac{{a}_{u}}{u}}\right\rbrack }}\parallel {\beta }_{u}\Delta \left(... | No |
Lemma 2.1 Let \( c > 2 \) . For any \( \eta > 0 \), we have\n\n\[ \n{C}_{r, p}\left( {\mathop{\sup }\limits_{{0 \leq T \leq 1}}\mathop{\sup }\limits_{{0 \leq s < t \leq c}}\frac{\left| w\left( T + t\right) - w\left( t\right) - \left( w\left( T + s\right) - w\left( s\right) \right) \right| }{{\left| t - s\right| }^{\alp... | Proof Similar to that of Lemma 2.2 in [8]. | No |
Lemma 2.1 (Wu [8] (2007)) Assume that the processes \( {X}_{i} = G\left( {F}_{t}\right) \in {L}^{2} \) . Let \( {G}_{n}\left( {F}_{0}\right) = \) \( E\left( {G\left( {F}_{n}\right) \mid {F}_{0}}\right), n \geq 0 \) . Then | \[ \parallel {P}_{0}{X}_{n}\parallel \leq \parallel G\left( {F}_{n}\right) - G\left( {F}_{n}^{ * }\right) \parallel \;\mathrm{{and}}\;\parallel {P}_{0}{X}_{n}\parallel \leq \parallel G\left( {F}_{n}\right) - G\left( {F}_{n}^{ * }\right) \parallel \leq 2\parallel {P}_{0}{X}_{n}\parallel . \] | Yes |
Lemma 2.2 Assume that \( \left( {A}_{1}\right) - \left( {A}_{4}\right) \) hold. Then \[ \mathop{\sum }\limits_{{i = 1}}^{\infty }\psi \left( {e}_{i}\right) {x}_{in} = O\left( {{\log }^{\frac{1}{2}}n}\right) ,\text{ a.s. } \] (2.1) and \[ {A}_{n} = \mathop{\sum }\limits_{{i = 1}}^{\infty }\left( {{\psi }^{\prime }\left(... | Proof For \( k \geq 0 \), let \[ {J}_{k} = \mathop{\sum }\limits_{{i = 1}}^{n}{P}_{i - k}\psi \left( {e}_{i}\right) {x}_{in} \] Then \( \left\{ {{P}_{i - k}\psi \left( {e}_{i}\right) {x}_{in},{F}_{i}}\right\} \) are martingale differences. Therefore, by \( p = \mathop{\sum }\limits_{{i = 1}}^{n}{x}_{in}^{\mathrm{T}}{x}... | Yes |
Lemma 2.3 If \( \left( {A}_{1}\right) \) and \( \left( {A}_{2}\right) \) hold, then\n\n(1) \( h\left( t\right) , - \infty < t < \infty \) is a finite and continuous function everywhere.\n\n(2) \( \psi \left( {e}_{i}\right) \) has finite moments, and \( {E\psi }\left( {e}_{i}\right) = 0 \) .\n\n(3) Given a constant \( L... | Proof (1) By the Taylor expansion, we have\n\n\[ \rho \left( {{e}_{i} \pm t}\right) = \rho \left( {e}_{i}\right) \pm {t\psi }\left( {e}_{i}\right) + {2}^{-1}{t}^{2}{\psi }^{\prime }\left( {{e}_{i} + {\theta }_{ \pm }t}\right) ,\;\left| {\theta }_{ \pm }\right| \leq 1. \]\n\n(2.8)\n\nHence\n\n\[ \rho \left( {{e}_{i} + t... | Yes |
Lemma 2.4 Assume that \( \left( {A}_{1}\right) - \left( {A}_{4}\right) \) hold. If\n\n\[ P\left\{ {\begin{Vmatrix}{{\sum }_{n}^{\frac{1}{2}}{\widehat{\beta }}_{n}}\end{Vmatrix} \geq {n}^{\alpha },\text{ i.o. }}\right\} = 0,\;\forall a > 0. \]\n\n\( \left( {2.14}\right) \) | Proof Let \( J\left( {{l}_{i},\cdots ,{l}_{p}}\right) = \left\{ {\left( {{\mu }_{1},\cdots ,{\mu }_{p}}\right) : {l}_{i}{n}^{-3} \leq {u}_{i} < \left( {{l}_{i} + 1}\right) {n}^{-3},1 \leq i \leq p,{l}_{i} = }\right. \) \( 0, \pm 1,\cdots \} ,{B}_{n} = B \cap \{ \beta : \parallel {\sum }_{n}^{\frac{1}{2}}{\widehat{\beta... | Yes |
Theorem 2.1 Let \( G \) be a biconnected outerplanar graph with diameter 3, then \( \operatorname{rc}\left( G\right) \leq 4 \) and the upper bound is sharp. | Proof We will consider the following five cases to prove Theorem 2.1.\n\nCase 1: The longest induced cycle in \( G \) is \( {C}_{6} \), then \( G \) is \( {C}_{6} \) or one of the following 5 graphs as shown in Fig. 4.\n\n satisfies (2.16) (or (2.18)) and \( {C}_{1}{C}_{3} < 0 \), then the absolute value of energy density of (2.19)-(2.20) satisfies\n\n\[ \left| {w}_{E}\right| \sim \widetilde{C}\left( t\right) {r}^{\frac{n}{2}} \]\n\nas \( r \rightarrow + \infty \) . | From Theorem 2.2 and Corollary 2.1, if \( {C}_{1}{C}_{3} < 0 \) and \( Z \) is the solution of (2.16) or (2.17) (or (2.18)), we see that the energy of (2.20) on \( \lbrack 0, + \infty ) \) is\n\n\[ E\left( \left( {2.20}\right) \right) = {\int }_{0}^{+\infty }{w}_{E}{r}^{n - 1}\mathrm{\;d}r = \infty . \]\n\n(2.31) | Yes |
If the inhomogeneity term is (2.19), then the blowup solution of (0.3) is as follows,\n\n\[ \n\\left\\{ \\begin{array}{l} {S}_{1} = \\frac{\\cos \\left( \\frac{{C}_{2} + Z{C}_{3}}{t - T}\\right) }{\\sqrt{Z - 1}\\sqrt{Z + 1}}, \\\\ {S}_{2} = \\frac{\\sin \\left( \\frac{{C}_{2} + Z{C}_{3}}{t - T}\\right) }{\\sqrt{Z - 1}\... | It is obviously that (2.32) is a blowup solution which blows up at \( t = T \) . Its spatial curvature\n\n\[ \n{\\left| {S}_{r}\\right| }^{2} = \\frac{{Z}_{r}^{2}\\left( {{Z}^{2}{C}_{3}^{2} + {T}^{2} - {2tT} + {t}^{2} - {C}_{3}^{2}}\\right) }{{\\left( Z + 1\\right) }^{2}{\\left( Z - 1\\right) }^{2}{\\left( -t + T\\righ... | Yes |
If \( {C}_{1}{C}_{3} < 0, Z \) is the solution of \( \left( {2.16}\right) ,\left( {2.17}\right) \) and \( \left( {2.18}\right) \), then the absolute value of energy density of (2.32) satisfies | \[ \left| {w}_{E}\right| \sim \widetilde{C}\left( t\right) {r}^{-\frac{n}{2}}\;\text{as}\;r \rightarrow + \infty ,\] and \[{\int }_{\varepsilon }^{C}{w}_{E}{r}^{n - 1}\mathrm{\;d}r = \widetilde{C}\left( t\right) \] where \( \widetilde{C}\left( t\right) \rightarrow + \infty \) as \( t \rightarrow T \). | Yes |
Theorem 2.3 If \( Z \) satisfies (2.17) and \( {C}_{1}{C}_{3} > 0 \), then (2.20) is a finite energy solution of \( r \in \lbrack 0, + \infty ) \) . The absolute value of energy density of (2.20) satisfies\n\n\[ \left| {w}_{E}\right| \sim \bar{C}\left( t\right) {e}^{-\bar{C}{r}^{n}}\;\text{ as }r \rightarrow + \infty .... | In the same time, the energy of (2.20) is\n\n\[ {\int }_{0}^{+\infty }{w}_{E}{r}^{n - 1}\mathrm{\;d}r = \bar{C}\left( t\right) \]\n\nFrom (2.34) and (2.36), we obtain\n\n\[ \mathop{\lim }\limits_{{r \rightarrow + \infty }}{w}_{E} = - \frac{4\left\lbrack {\left( {{k}^{2} - 1}\right) {C}_{3}{}^{2} + {\left( -t + T\right)... | Yes |
If \( Z \) satisfies (2.17) and \( {C}_{1}{C}_{3} > 0 \), then (2.20) is a finite energy solution of \( r \in \lbrack 0, + \infty ) \) . | The absolute value of energy density and energy of (2.32) satisfy\n\n\[ \left| {w}_{E}\right| \sim \bar{C}\left( t\right) {\mathrm{e}}^{-\bar{C}{r}^{n}}\;\text{ as }r \rightarrow + \infty \]\n\nand\n\n\[ {\int }_{0}^{+\infty }{w}_{E}{r}^{n - 1}\mathrm{\;d}r = \bar{C}\left( t\right) \] \n\nrespectively. | Yes |
Theorem 2.4 If \( Z \) satisfies (2.16) (or (2.18)), \( {C}_{1}{C}_{3} > 0 \), and \( {C}_{4} \leq 0 \), then (2.20) (or (2.32)) is a local solution which \( r \in \left\lbrack {0, K}\right\rbrack \) ( \( K \) is a positive constant). This solution is a finite energy solution of \( r \) . The total energy of it satisfi... | If \( Z \) satisfies (2.16) (or (2.18)), \( {C}_{1}{C}_{3} > 0 \), and \( {C}_{4} > 0 \), there is no solution of (2.20) (or (2.32)). | No |
Theorem 0.1 Let \( p \geq {11} \) be a prime and \( k \) be a positive integer. Then for any integer \( 1 \leq a \leq p - 6 \), we have\n\n\[ \left( \begin{matrix} \left( {k + 1}\right) p - a \\ p - a \end{matrix}\right) \equiv 1 - {kp}\left( {k + 1}\right) \left( {{k}^{2} + k - 1}\right) \mathop{\sum }\limits_{{i = 1}... | By Lemma 1.7 in Section 1, from the proof of Theorem 0.1 in Section 2, we have the following | Yes |
Theorem 0.3 Let \( p \geq 7 \) be a prime and \( k \) be a positive integer.\n\n(i) If \( b \) is an integer with \( 0 \leq b \leq \frac{p - 7}{2} \), then\n\n\[ \left( \begin{matrix} {kp} + \frac{p - 1}{2} - b \\ \frac{p - 1}{2} - b \end{matrix}\right) \equiv {4}^{-k\left( {p - 1}\right) } - \frac{1}{12}k\left( {7 + {... | If \( \frac{p - 7}{2} < b < \frac{p - 1}{2} \), replacing the modulus \( {p}^{4} \) with \( {p}^{\frac{p - 1}{2} - b + 1} \) (clearly, \( \frac{p - 1}{2} - b + 1 < 4 \) ), this congruence still holds. | No |
Lemma 1.7 For any prime \( p \geq {11} \), we have\n\n\[ \mathop{\sum }\limits_{{j = 1}}^{6}{\left( kp\right) }^{j}{H}_{j}\left( {p - 1}\right) \equiv - k\left( {k + 1}\right) \left( {{k}^{2} + k - 1}\right) p{R}_{1} + \frac{{k}^{2}{\left( k + 1\right) }^{2}}{2}{p}^{2}\left( {{R}_{1}^{2} - {R}_{2}}\right) \]\n\n\[ \equ... | Proof of Lemma 1.7 By Lemma 1.2, it is easy to check \( {\left( kp\right) }^{j}{H}_{j}\left( {p - 1}\right) \equiv 0\left( {\;\operatorname{mod}\;{p}^{7}}\right) \) \( \left( {j = 5,6}\right) \), which indicates that\n\n\[ S \mathrel{\text{:=}} \mathop{\sum }\limits_{{j = 1}}^{6}{\left( kp\right) }^{j}{H}_{j}\left( {p ... | Yes |
Lemma 1.10 For arbitrary integers \( p, n \) with \( p \geq 3 \) and \( n \geq 1 \), we have\n\n\[ \mathop{\sum }\limits_{{b = 1}}^{{p - 1}}\mathop{\sum }\limits_{{b + 1 \leq {j}_{1} \leq \cdots \leq {j}_{n} \leq p - 1}}\frac{1}{{j}_{1}{j}_{2}\cdots {j}_{n}} = p - 1 - \mathop{\sum }\limits_{{1 \leq {j}_{1} \leq \cdots ... | Proof of Lemma 1.10 If \( n = 1 \), exchanging the order of summations, we see\n\n\[ \mathop{\sum }\limits_{{b = 1}}^{{p - 1}}\mathop{\sum }\limits_{{j = b + 1}}^{{p - 1}}\frac{1}{j} = \mathop{\sum }\limits_{{j = 2}}^{{p - 1}}\frac{1}{j}\mathop{\sum }\limits_{{b = 1}}^{{j - 1}}1 = p - 1 - \mathop{\sum }\limits_{{j = 1}... | Yes |
Example 1.1 Idempotents, units and elements of \( J\left( R\right) \) in a ring \( R \) are radpolar. | Proof Let \( {e}^{2} = e \in R, u \in U\left( R\right) \) and \( j \in J\left( R\right) \) . Write \( p = 1 - e,0,1 \), respectively. Then the result follows from a routine computation. | No |
Example 1.2 A ring \( R \) is local if and only if \( R \) is radpolar with 0,1 the only idempotents. | Proof Assume that \( R \) is radpolar and 0,1 are the only idempotents. Let \( a \in R \) and \( a \notin U\left( R\right) \) . Write \( b = a - 1 \) . Then \( b + {b}^{\Pi } = a - 1 + {b}^{\Pi } \in U\left( R\right) \) implies \( {b}^{\Pi } = 0 \) . So \( 1 - a \in U\left( R\right) \) , and thus \( R \) is local. The ... | No |
Example 1.4 If \( R \) is a local ring and \( {C}_{2} \) is the group of order 2, then \( R{C}_{2} \) is radpolar. | Proof If \( 2 \in J\left( R\right) \), then \( R{C}_{2} \) is local by [12, Theorem]. If \( 2 \notin J\left( R\right) \), then \( R{C}_{2} \cong R \oplus R \) by \( \left\lbrack {8\text{, Proposition 3}}\right\rbrack \), and thus \( R{C}_{2} \) is radpolar by Example 1.3. | Yes |
Lemma 1.1 Let \( R \) be a ring. The following statements hold:\n\n(1) If \( a \in R \) is radpolar, then for any \( u \in U\left( R\right) \cap C\left( R\right) \) , \( {au} \) is radpolar and \( {\left( au\right) }^{\Pi } = {a}^{\Pi } \) . | Proof (1) Let \( p = {a}^{\Pi } \) . We show that \( p = {\left( au\right) }^{\Pi } \) . For any \( x \in \operatorname{comm}\left( {au}\right) \), one obtains \( {xa} = {ax} \) since \( u \in U\left( R\right) \cap C\left( R\right) . \) So we have \( {px} = {xp} \) as \( p \in {\operatorname{comm}}^{2}\left( a\right) \... | Yes |
Proposition 1.2 A strongly \( \pi \) -regular ring \( R \) is radpolar if and only if \( {R}^{\mathrm{{nil}}} \subseteq J\left( R\right) \) . | Proof Suppose that \( R \) is radpolar. By Proposition 1.1, we have \( {R}^{\text{nil }} \subseteq {R}^{\text{qnil }} = J\left( R\right) \) .\n\nConversely, given any \( a \in R \), as \( R \) is strongly \( \pi \) -regular, there exists \( {e}^{2} = e \in \operatorname{comm}\left( a\right) \) such that \( a - e = u \i... | Yes |
Proposition 1.3 Every strongly regular element of a ring \( R \) is radpolar. | Proof Let \( a \in R \) be strongly regular. Then \( a = {eu} = {ue} \) for some \( {e}^{2} = e \in R \) and \( u \in U\left( R\right) \) . Write \( p = 1 - e \) . Then \( {ap} = {pa} = 0 \in J\left( R\right) \) . Note that \( \left( {a + p}\right) \left( {e{u}^{-1} + p}\right) = \) \( \left( {e{u}^{-1} + p}\right) \le... | Yes |
Corollary 1.1 A ring \( R \) is strongly regular if and only if \( R \) is radpolar and \( {R}^{\text{qnil}} = 0 \) . | Proof The \ | No |
Proposition 1.4 Let \( R \) be a ring and \( a \in R \). (1) If \( {a}^{k} \) is radpolar for some integer \( k \geq 1 \), then \( a \) is pseudopolar. | Proof (1) Let \( p = {\left( {a}^{k}\right) }^{\Pi } \). By Lemma 1.1, we have \( p \in {\operatorname{comm}}^{2}\left( {a}^{k}\right) ,{a}^{k} - p \in U\left( R\right) \) and \( {a}^{k}p \in J\left( R\right) \). Given \( y \in \operatorname{comm}\left( a\right) \), then \( {a}^{k}y = y{a}^{k} \). So \( p \in {\operato... | Yes |
Proposition 1.5 The center of a radpolar ring is radpolar. | Proof Let \( C\left( R\right) \) be the center of a radpolar ring \( R \) . Given any \( a \in C\left( R\right) \), then there exists \( {p}^{2} = p \in {\text{comm}}_{R}^{2}\left( a\right) \) such that \( p + a \in U\left( R\right) \) and \( {ap} \in J\left( R\right) . \) Note that \( {\text{comm}}_{R}\left( a\right) ... | Yes |
Proposition 1.6 If \( R \) is a radpolar ring, then so is \( {eRe} \) for any \( {e}^{2} = e \in R \) . | Proof For every \( a \in {eRe} \), there exists \( {p}^{2} = p \in {\operatorname{comm}}_{R}^{2}\left( a\right) \) such that \( a + p = u \in U\left( R\right) \) and \( {ap} \in J\left( R\right) \) . Since \( e \in {\operatorname{comm}}_{R}\left( a\right) \), one has \( {pe} = {ep} \) . So \( {\left( epe\right) }^{2} =... | Yes |
Proposition 1.7 Let \( R \) be a ring. Then \( {M}_{n}\left( R\right) \) is not radpolar for any \( n \geq 2 \) . | Proof Let \( A = {E}_{1n} \in {M}_{n}\left( R\right) \) be a matrix with \( \left( {1, n}\right) \) -entry 1 and zeros elsewhere. Then if \( n \geq 2, A \) is nilpotent, and thus quasinilpotent. However, \( A \notin J\left( {{M}_{n}\left( R\right) }\right) \) . In view of Lemma 1.1 (2), \( A \) is not radpolar. Therefo... | Yes |
Theorem 2.1 Let \( R \) be a ring. The following are equivalent for \( a \in R \) :\n\n(1) \( a \) is radpolar in \( R \) .\n\n(2) There exists a unique \( {e}^{2} = e \in \mathbb{R} \) such that \( e \in \) comm \( \left( a\right) ,\;e + a \in U\left( R\right) \) and \( {ae} \in J\left( R\right) . \)\n\n(3) There exis... | Proof \( \left( 1\right) \Leftrightarrow \left( 2\right) \) follows from Lemma 2.1.\n\n(1) \( \Rightarrow \) (3). Write \( p = {a}^{\Pi } \) . Then \( {p}^{2} = p \in {\operatorname{comm}}^{2}\left( a\right), p + a = u \in U\left( R\right) \) and \( {ap} \in J\left( R\right) \) . Multiplying \( p + a = u \) on the righ... | Yes |
Corollary 2.1 A ring \( R \) is radpolar if and only if \( R \) is strongly rad clean and quasipolar. | Proof The \ | No |
Proposition 2.1 A ring \( R \) is radpolar if and only if \( R \) is pseudopolar and \( R/J\left( R\right) \) is strongly regular. | Proof Write \( \bar{R} = R/J\left( R\right) \). For the \ | No |
Corollary 2.2 Let \( R \) be a local ring. The followings are equivalent:\n\n(1) \( R \) is uniquely bleached.\n\n(2) \( {T}_{n}\left( R\right) \) is radpolar for any \( n \geq 1 \) .\n\n(3) \( {T}_{n}\left( R\right) \) is pseudopolar for any \( n \geq 1 \) . | Proof \( \left( 1\right) \Leftrightarrow \left( 3\right) \) follows from \( \left\lbrack {{15}\text{, Theorem }{2.13}}\right\rbrack \) ; and \( \left( 2\right) \Rightarrow \left( 3\right) \) is clear.\n\n\( \left( 3\right) \Rightarrow \left( 2\right) \) . Since \( R \) is local, \( R/J\left( R\right) \) is a division r... | No |
Proposition 2.2 For a ring \( R \), the following hold:\n\n(1) An idempotent \( e \in R \) is \( J \) -quasipolar if and only if \( {2e} \in J\left( R\right) \) . | Proof \( \left( 1\right) \) Let \( {e}^{2} = e \in \mathbb{R}. \) By assumption, there exists \( {p}^{2} = p \in \mathbb{R} \) such that \( p \in {\text{comm}}^{2}\left( e\right) \) and \( e + p = w \in J\left( R\right) \) . Multiplying the equation \( e + p = w \) by \( 1 - p \) and by \( 1 - e \) yield \( e\left( {1 ... | Yes |
Corollary 2.3 A ring \( R \) is \( J \) -quasipolar if and only if \( R/J\left( R\right) \) is Boolean and \( R \) is radpolar. | Proof Suppose that \( R \) is radpolar and \( R/J\left( R\right) \) is Boolean. Given \( a \in R \), write \( b = 1 + a \) and \( p = {b}^{\Pi } \) . Then \( p \in {\operatorname{comm}}^{2}\left( b\right), p + b \in U\left( R\right) \) . Clearly, \( p \in {\operatorname{comm}}^{2}\left( a\right) \) . Since \( R/J\left(... | Yes |
Lemma 3.1 Let \( R \) be a ring and \( a \in R, u \in U\left( R\right) \) . If \( a \) is radpolar, then so is \( {u}^{-1}{au} \) . | Proof Let \( e = {a}^{\Pi } \in R \) . Write \( p = {u}^{-1}{eu} \) . Then \( {p}^{2} = p \), and \( {u}^{-1}{au} + p \in U\left( R\right) \) since \( a + e \in U\left( R\right) \) . For any \( b \in \operatorname{comm}\left( {{u}^{-1}{au}}\right) \), we obtain \( a\left( {{ub}{u}^{-1}}\right) = \left( {{ub}{u}^{-1}}\r... | Yes |
Lemma 3.4 Let \( R \) be a local ring, and \( u \in U\left( R\right), j \in J\left( R\right) \) . The followings are equivalent:\n\n(1) The matrix \( \left( \begin{array}{ll} u & 0 \\ 0 & j \end{array}\right) \) is radpolar in \( {M}_{2}\left( R\right) \) .\n\n(2) The matrix \( \left( \begin{array}{ll} j & 0 \\ 0 & u \... | Proof \( \left( 1\right) \Leftrightarrow \left( 2\right) . \) Since \( \left( {{}_{0}^{u}{}_{j}^{0}}\right) \) is similar to \( \left( {{}_{0}^{j}{}_{u}^{0}}\right) \), the result follows from Lemma 3.1.\n\n\( \left( 2\right) \Rightarrow \left( 3\right) \) . Let \( A = \left( \begin{array}{ll} j & 0 \\ 0 & u \end{array... | Yes |
Theorem 3.1 Let \( R \) be a local ring. Then \( A \in {M}_{2}\left( R\right) \) is radpolar if and only if \( A \) is invertible or \( A \in J\left( {{M}_{2}\left( R\right) }\right) \) or \( A \) is similar to a diagonal matrix \( \left( \begin{matrix} u & 0 \\ 0 & j \end{matrix}\right) \) where \( {l}_{u} - {r}_{j},{... | Proof Suppose that \( A \in {M}_{2}\left( R\right) \) is radpolar. In view of Lemma 3.2, we may assume that \( A \notin U\left( {{M}_{2}\left( R\right) }\right) \) and \( A \notin J\left( {{M}_{2}\left( R\right) }\right) \) . By Lemma 3.3, there exists \( V \in U\left( {{M}_{2}\left( R\right) }\right) \) such that \( {... | Yes |
Proposition 3.1 Let \( R \) be a commutative local ring, and \( A \in {M}_{2}\left( R\right) \) with \( \det A \in J\left( R\right) \) and \( A \notin J\left( {{M}_{2}\left( R\right) }\right) \) . If \( \operatorname{tr}A \in J\left( R\right) \), then \( A \) is not radpolar. | Proof Assume that \( A \) is radpolar in \( {M}_{2}\left( R\right) \) . Since \( \det A \in J\left( R\right) \) and \( A \notin J\left( {{M}_{2}\left( R\right) }\right) \), by Theorem 3.1, \( A \) is similar to a diagonal matrix \( B \) with one entry in \( U\left( R\right) \) and the other in \( J\left( R\right) \) . ... | Yes |
We claim that \( A \) is not radpolar. Otherwise, by Theorem \( {3.1}, \) there exists \( V \in U\left( {{M}_{2}\left( R\right) }\right) \) such that \( B = {V}^{-1}{AV} = \left( \begin{matrix} u & 0 \\ 0 & j \end{matrix}\right) \in {M}_{2}\left( R\right) \) where \( u \in U\left( R\right) \) and \( j \in J\left( R\rig... | As \( 2 = \det A = \det B = {uj} \) and \( 1 = \operatorname{tr}A = \operatorname{tr}B = u + j \), we have \( {u}^{2} - u + 2 = 0 \) . But it is easy to see that the equation \( {u}^{2} - u + 2 = 0 \) has no solution in \( R \) . | Yes |
Proposition 3.2 Let \( R \) be a commutative local ring and let \( A \in {M}_{2}\left( R\right) \) be such that \( \det A \in J\left( R\right) \) and \( \operatorname{tr}A \in U\left( R\right) \) . Then \( A \) is diagonalizable if and only if the equation \( {x}^{2} - \) \( \left( {\operatorname{tr}A}\right) x + \det ... | Proof Assume that the matrix \( A \) is diagonalizable in \( {M}_{2}\left( R\right) \) . So there exists \( V \in \) \( U\left( {{M}_{2}\left( R\right) }\right) \) such that \( {V}^{-1}{AV} = \left( \begin{matrix} u & 0 \\ 0 & j \end{matrix}\right) \) for some \( u, j \in R. \) Write \( B = {V}^{-1}{AV}. \) Then tr \( ... | Yes |
Theorem 3.2 Let \( R \) be a commutative local ring. Then a matrix \( A \in {M}_{2}\left( R\right) \) is radpolar if and only if one of the following holds:\n\n(1) Either \( A \) is invertible or \( A \in J\left( {{M}_{2}\left( R\right) }\right) \) .\n\n(2) The equation \( {x}^{2} - \left( {\operatorname{tr}A}\right) x... | Proof Suppose that \( A \) is radpolar. We may assume that \( \det A \in J\left( R\right) \) and \( A \notin J\left( {{M}_{2}\left( R\right) }\right) \) . Then \( A \) is diagonalizable by Theorem 3.1, and \( \operatorname{tr}A \in U\left( R\right) \) by Proposition 3.1. So (2) follows from Proposition 3.2. For the suf... | Yes |
Proposition 1.2 \( {}^{\left\lbrack {13}\right\rbrack } \) Let \( {C}_{\rho } = {c}_{0}^{2}{2}^{{N}_{0} + 3} \) and \( B = B\left( {{x}_{0}, r}\right) \) be a ball with \( r \geq \rho \left( {x}_{0}\right) \) . Then there exists a subsequence of \( {B}_{j} = B\left( {{x}_{j},\rho \left( {x}_{j}\right) }\right), j \geq ... | \[ B \subset \mathop{\bigcup }\limits_{{j \in {J}_{B}}}{B}_{j} \subset {C}_{\rho }B \] | Yes |
Proposition 0.1 Let \( 0 \leq \lambda < \infty ,0 < q \leq \infty, p\left( \cdot \right) \in \mathcal{P}\left( {\mathbb{R}}^{n}\right) \) and \( \alpha \left( \cdot \right) \in {L}^{\infty }\left( {\mathbb{R}}^{n}\right) \) . If \( \alpha \left( \cdot \right) \) is log-Hölder continuous both at the origin and at infini... | \[ \parallel f{\parallel }_{{\mathrm{M\dot{K}}}_{q, p\left( \cdot \right) }^{\alpha \left( \cdot \right) ,\lambda }\left( {\mathbb{R}}^{n}\right) } \approx \max \left\{ {\mathop{\sup }\limits_{{L < 0, L \in \mathbb{Z}}}{2}^{-{L\lambda }}{\left( \mathop{\sum }\limits_{{k = - \infty }}^{L}{2}^{{k\alpha }\left( 0\right) q... | Yes |
Lemma 1.2 \( {}^{\left\lbrack {13}\right\rbrack }\; \) Let \( p\left( \cdot \right) \in \mathcal{B}\left( {\mathbb{R}}^{n}\right) \), then we have for all measurable subsets \( E \subset B \) ,\n\n\[ \n\frac{\parallel {\chi }_{E}{\parallel }_{{L}^{p\left( \cdot \right) }\left( {\mathbb{R}}^{n}\right) }}{\parallel {\chi... | We remark that Lemmas 1.1-1.2 are due to Izuki ([13, Lemmas 1-2, p. 346]). | No |
Theorem 1.1 (Chow Theorem) Fix a point \( q \in M \) . If the distribution \( D \subset {TM} \) is bracket generating, then the set of points that can be connected to \( q \) by a horizontal curve is the component of \( M \) containing \( q \) . | By Chow Theorem, we know that if \( D \) is bracket generating and \( M \) is connected, then any two points of \( M \) can be joined by a horizontal curve. | Yes |
Lemma 3.1 The causal character of normal sub-Lorentzian geodesics does not depend on time. | Proof Assume that \( \gamma \left( t\right) \) is a normal geodesic which is causal at \( t = 0 \) . Differentiating the sub-Lorentzian Hamiltonian \( H \) \( \left( {3.1}\right) , \) since \( \gamma \left( t\right) \) satisfies \( \left( {3.2}\right) , \) we obtain \( \dot{H}\left( s\right) = 0 \), which implies that ... | Yes |
Lemma 3.2 If \( \eta = 0, \) then the projection of the geodesic of the system (3.5)-(3.6) in \( {\mathbb{R}}^{3} \) is\n\n\[ \n{x}_{1}\left( t\right) = - {\dot{x}}_{1}\left( 0\right) t + {x}_{1}\left( 0\right) ,\;{x}_{2}\left( t\right) = {\dot{x}}_{2}\left( 0\right) t + {x}_{2}\left( 0\right) ,\;z\left( t\right) = z\l... | By Lemma 3.1, we know that if \( - {\dot{x}}_{1}{\left( 0\right) }^{2} + {\dot{x}}_{2}{\left( 0\right) }^{2} < 0\;\left( {x > 0, x = 0}\right) \), then the geodesic (3.7) is time-like (space-like, light-like). | No |
Lemma 3.3 If a normal geodesic is future directed (past directed) at \( t = 0 \), then it remains future directed (past directed) for all \( t \in \lbrack 0, + \infty ) \) . | Proof Since \( {\dot{x}}_{1}\left( 0\right) > 0 \) and \( {x}_{1}\left( t\right) = r\cosh \left( \phi \right) \), then \( \dot{r}\left( 0\right) > 0,\dot{r}\left( t\right) > 0 \) . According to (3.15), \[ {\dot{x}}_{1}\left( t\right) = \dot{r}\cosh \left( \phi \right) + r\sinh \left( \phi \right) \dot{\phi } \] \[ = \s... | Yes |
Lemma 3.4 \( \mu \left( x\right) \) satisfies the ordinary differential equation\n\n\[ \n{\mu }^{\prime }\left( x\right) = 2 - 4\coth \left( {3x}\right) \mu \left( x\right) \n\]\n\n(3.29)\n\nIf \( {x}_{c} \) is a critical point for \( \mu \left( x\right) \), then\n\n\[ \n\mu \left( {x}_{c}\right) = \frac{1}{2}\tanh \le... | Proof Differentiating (3.27),\n\n\[ \n{\mu }^{\prime }\left( x\right) = 2 - \frac{8}{3}\coth \left( {3x}\right) \frac{{\int }_{0}^{3x}{\sinh }^{\frac{4}{3}}\left( v\right) \mathrm{d}v}{{\sinh }^{\frac{4}{3}}\left( {3x}\right) } \n\]\n\n\[ \n= 2 - 4\coth \left( {3x}\right) \mu \left( x\right) \text{. } \n\]\n\n(3.31)\n\... | Yes |
Lemma 3.5 If \( {x}_{c} > 0 \) is a critical point for \( \mu \left( x\right) \), then | \[ {\mu }^{\prime \prime }\left( {x}_{c}\right) > 0 \] (3.34) and all the positive critical points are local minima. | Yes |
Lemma 3.6 \( \mu \left( x\right) \) is a strictly increasing function in \( \left( {-\infty , + \infty }\right) \), whose range is \( \left( {-\frac{1}{2},\frac{1}{2}}\right) \) . | Proof Since \( \mathop{\lim }\limits_{{x \rightarrow + \infty }}\tanh \left( x\right) = 1 \) ,\n\n\[ \mathop{\lim }\limits_{{x \rightarrow + \infty }}\mu \left( x\right) = \mathop{\lim }\limits_{{x \rightarrow + \infty }}\frac{1}{2}\frac{\sinh \left( {3x}\right) }{\cosh \left( {3x}\right) } = \frac{1}{2}. \]\n\n(3.35)\... | Yes |
Theorem 3.1 Let \( A = \left( {{x}_{1},{x}_{2}, z}\right) \) be a point such that \( {x}_{1} > 0\left( {{x}_{1} < 0}\right) , - {x}_{1}^{2} + {x}_{2}^{2} < 0 \) , \( z \neq 0 \)\n\n\[ \n- \frac{1}{2} < \frac{z}{{\left( -{x}_{1}^{2} + {x}_{2}^{2}\right) }^{2}} < \frac{1}{2} \n\]\n\n(3.37)\n\nThen there is a unique time-... | Proof Following from Lemma 3.6, there is a unique time-like geodesic joining the origin with the point \( A \) . According to (3.19)-(3.23), we can get (3.39) and (3.40). | No |
Theorem 3.2 Let \( B = \left( {{x}_{1},{x}_{2}, z}\right) \) be a point such that \( - {x}_{1}^{2} + {x}_{2}^{2} > 0, z \neq 0 \) , \[ - \frac{1}{2} < \frac{z}{{\left( -{x}_{1}^{2} + {x}_{2}^{2}\right) }^{2}} < \frac{1}{2} \] Then there is a unique space-like geodesic joining the origin with the point \( B \) . | The geodesic is parametrized by the solution \( \phi \) of \[ \frac{z}{{\left( -{x}_{1}^{2} + {x}_{2}^{2}\right) }^{2}} = \mu \left( {\phi - {\phi }_{0}}\right) \] The length is given by \[ {l}^{4} = \nu \left( {\phi - {\phi }_{0}}\right) \left( {z + {R}^{4}}\right) = \nu \left( {\phi - {\phi }_{0}}\right) \left( {z + ... | Yes |
Theorem 3.3 Let \( C = \left( {{x}_{1},{x}_{2}, z}\right) \) be a point such that \( - {x}_{1}^{2} + {x}_{2}^{2} = 0, z = 0 \) . Then there is a light-like geodesic jointing the origin with \( C \) . | Proof We suppose that geodesics are parametrized on the interval \( \left\lbrack {0,1}\right\rbrack \) joining the origin with \( C \) . When \( \eta = 0 \), since \( - {x}_{1}^{2}\left( 1\right) + {x}_{2}^{2}\left( 1\right) = 0 \), it follows that \( - {\dot{x}}_{1}{\left( 0\right) }^{2} + {\dot{x}}_{2}{\left( 0\right... | Yes |
Theorem 3.4 Let us define the following three sets\n\n\[ \n{R}_{t} = \left\{ {-{x}_{1}^{2} + {x}_{2}^{2} < 0,\frac{\left| z\right| }{{\left( -{x}_{1}^{2} + {x}_{2}^{2}\right) }^{2}} < \frac{1}{2}}\right\} ,\;{R}_{s} = \left\{ {-{x}_{1}^{2} + {x}_{2}^{2} > 0,\frac{\left| z\right| }{{\left( -{x}_{1}^{2} + {x}_{2}^{2}\rig... | Proof The proof of Theorem 3.4 is contained in Theorems 3.1-3.3, and Remarks 3.1 and\n\nFrom the proof of Theorem 3.3, we can get that there are no light-like geodesics joining the origin with \( D = \left( {{x}_{1},{x}_{2}, z}\right) \), where \( - {x}_{1}^{2} + {x}_{2}^{2} = 0, z \neq 0 \) . According to Theorems 3.1... | Yes |
Corollary 0.2 If \( m - 1 \) and \( n \) are positive integers such that \( \left( {n,6}\right) = 1 \) and \( t = \left( {\phi \left( {n}^{2}\right) - }\right. \) 1) \( m, s = \left( {\phi \left( {n}^{2}\right) - 1}\right) \left( {m - 1}\right) \), then | \[ \mathop{\sum }\limits_{{\sigma \in {A}_{2}\left( n\right) }}\mathop{\sum }\limits_{\underset{\left( {k, n}\right) = 1}{k = 1}}^{n}\frac{\sigma \left( k\right) }{{k}^{m}} \equiv \left\{ \begin{array}{l} \frac{{n}^{2}\phi \left( n\right) }{2}\mathop{\prod }\limits_{{p \mid n}}\left( {1 - {p}^{t - 1}}\right) {B}_{t} \\... | Yes |
Corollary 0.4 If \( m - 1 \) and \( n \) are positive integers such that \( \left( {n,6}\right) = 1 \) and \( t = \left( {\phi \left( {n}^{2}\right) - }\right. \) | 1) \( m, s = \left( {\phi \left( {n}^{2}\right) - 1}\right) \left( {m - 1}\right) \), then\n\n\[
\mathop{\sum }\limits_{{\sigma \in {A}_{3}\left( n\right) }}\mathop{\sum }\limits_{\substack{{k = 1} \\ {\left( {k, n}\right) = 1} }}^{n}\frac{\sigma \left( k\right) }{{k}^{m}} \equiv \left\{ \begin{array}{l} \frac{{n}^{2}\... | Yes |
Lemma 1.2 \( {}^{\left\lbrack 6 - 8\right\rbrack }\; \) If \( s \) and \( n \) are positive integers such that \( \left( {n,6}\right) = 1 \) and \( t = \left( {\phi \left( {n}^{2}\right) - 1}\right) s, \) where \( \phi \left( \cdot \right) \) denotes the Euler’s totient function, then | \[ \mathop{\sum }\limits_{\substack{{k = 1} \\ {\left( {k, n}\right) = 1} }}^{n}\frac{1}{{k}^{s}} \equiv \left\{ \begin{array}{l} \frac{t}{2}{n}^{2}\mathop{\prod }\limits_{{p \mid n}}\left( {1 - {p}^{t - 2}}\right) {B}_{t - 1}\left( {\;\operatorname{mod}\;{n}^{2}}\right) ,\;\text{ if }s\text{ is od } \\ n\mathop{\prod ... | Yes |
Lemma 1.3 If \( p \) is an odd prime and \( m \leq p - 2 \) is a positive integer, then\n\n\[ \mathop{\sum }\limits_{{k = 1}}^{{p - 1}}{k}^{m}H\left( k\right) \equiv {B}_{m}\left( {\;\operatorname{mod}\;p}\right) \] | Proof \( \triangleq \) Note that \( \mathop{\sum }\limits_{{k = 1}}^{{i - 1}}{k}^{m} = \frac{{B}_{m + 1}\left( i\right) - {B}_{m + 1}}{m + 1} \) and \( {B}_{n}\left( x\right) = \mathop{\sum }\limits_{{j = 0}}^{n}\left( \begin{aligned} n \\ j \end{aligned}\right) {B}_{n - j}{i}^{j}. \) By Lemma 1.1 , we have\n\n\[ \math... | Yes |
Proposition 0.1 Let \( B = \left\{ {{\mathbf{b}}_{1},{\mathbf{b}}_{2},\cdots ,{\mathbf{b}}_{n}}\right\} \) and \( C = \left\{ {{\mathbf{c}}_{1},{\mathbf{c}}_{2},\cdots ,{\mathbf{c}}_{n}}\right\} \) be two sets of linearly independent vectors in \( {\mathbb{R}}^{n} \). Then they generate the same lattice (i.e., \( C \) ... | \[ {\mathbf{b}}_{i} = \mathop{\sum }\limits_{{j = 1}}^{n}{u}_{ij}{\mathbf{c}}_{j}\;\text{ and }\;{\mathbf{c}}_{i} = \mathop{\sum }\limits_{{j = 1}}^{n}{v}_{ij}{\mathbf{b}}_{j}\;\left( {1 \leq i \leq n}\right) . \] | Yes |
Proposition 1.1 Every lattice in \( {\mathbb{R}}^{1} \) is standard. | Proof Let \( {\mathbf{b}}_{1} \) be a basis of a lattice \( \Lambda \subseteq {\mathbb{R}}^{1} \) : Then\n\n\[ \Lambda = L\left( {\mathbf{b}}_{1}\right) = \left\{ {k{\mathbf{b}}_{1} \mid k \in \mathbb{Z}}\right\} \]\n\nObviously,\n\n\[ {\lambda }_{1} = \inf \left\{ {\begin{Vmatrix}{k{\mathbf{b}}_{1}}\end{Vmatrix} \mid ... | Yes |
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