Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
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Theorem 3.3 Let \( Q \) and \( \bar{Q} \) be as above. Then the oriented closed paths in \( \bar{Q} \) form an involutive Lie bialgebra. | Proof The construction of the Lie bialgebra on \( \bar{Q} \) is the same as the string bracket and cobracket in string topology. For the proof of Lie algebras, see [15, Theorem 1.2] and [55, Lemma 3.1], and for the proof of Lie bialgebras, see Schedler [80, §2]. | No |
Suppose that \( \mathfrak{g} \) is an \( n \) -dimensional Lie algebra such that \( \operatorname{Tr}\left( {\operatorname{ad}\left( g\right) }\right) = \) 0, for all \( g \in \mathfrak{g} \) . In literature, such Lie algebras are sometimes also called unimodular Lie algebras. Examples of them are semi-simple Lie algeb... | \[ \langle - , - \rangle : {C}^{ * }\left( \mathfrak{g}\right) \times {C}^{ * }\left( \mathfrak{g}\right) \rightarrow k \] by \[ \langle u, v\rangle = \left( {u \cup v}\right) \left( \Omega \right) \] Then \( \left( {{C}^{ * }\left( \mathfrak{g}\right) ,\cup ,\langle -, - \rangle ,\delta }\right) \) forms a DG cyclic c... | Yes |
Theorem 5.1 Let \( \mathcal{A} \) be a Calabi-Yau \( {A}_{\infty } \) category of degree \( n \) . The cyclic cochain complex \( {\operatorname{Cycl}}^{ * }\left( \mathcal{A}\right) \) has the structure of a DG involutive Lie bialgebra of degree \( \left( {2 - n,2 - n}\right) \), where the Lie algebra and Lie coalgebra... | See [27, Theorem 1] for a proof. | No |
Theorem 5.4 (Lie bialgebra on the noncommutative 0-forms) Let \( V \) be a vector space with a constant symplectic form \( \omega \) . The symplectic form on \( V \) induces a symplectic form on \( {V}^{ * } \) via the identity \( V \rightarrow {V}^{ * } : v \mapsto \omega \left( {v, \cdot }\right) \), which is still d... | For a complete proof, see Schedler [80] or Hamilton [61]. The Lie bracket is usually called the Kontsevich bracket, and is the Poisson bracket on a noncommutative symplectic space. | No |
Theorem 6.1 (Hochschild-Kostant-Rosenberg) Suppose that \( X \) is an algebraic variety. Then there is an isomorphism (the Hochschild-Kostant-Rosenberg isomorphism):\n\n\[ \n{I}_{\mathrm{{HKR}}} : {\mathrm{{HH}}}^{ * }\left( {{\mathcal{O}}_{X},{\mathcal{O}}_{X}}\right) \overset{ \cong }{ \rightarrow }{\mathrm{H}}^{ * }... | Proof See, for example, Calaque and Van den Bergh [23, Theorem 1.3]. | No |
Theorem 7.1 (Pair-of-pants product) Let \( M \) be a symplectic manifold, probably with a contact type boundary. The symplectic chain complex, under the pair-of-pants product, is homotopy associative and commutative, and therefore induces a (graded) commutative algebra structure on the symplectic homology. | \( \\textbf{Proof} : \\) There are several versions of this theorem; see Piunikhin-Salamon-Schwarz [76], Viterbo [91], Weber [94], Abbondandolo-Schwarz [2-3], and Ritter [78]. | No |
Theorem 8.4 (The PSS isomorphism) Let \( N \) be a simply connected manifold as before. Then the Floer cochain complex \( C{F}^{ * }\left( N\right) \) of \( N \) is quasi-isomorphic to its de Rham cochain complex \( {\Omega }^{ \bullet }\left( N\right) \) . | Proof This is the chain level statement of the famous Piunikhin-Salamon-Schwarz (PSS for short) isomorphism, first appeared in [76]. A proof in the most general case (without assuming the result of Fukaya-Seidel-Smith and Nadler cited above), can be found in Abouzaid [5, Theorem 1.1]. | No |
Theorem 1.2 \( {}^{\left\lbrack 3\right\rbrack } \) Let \( G \) be a finite group. If \( M \) is the maximal member in \( \mathcal{{CD}}\left( G\right) \), then | \[ \mathcal{{CD}}\left( G\right) = \mathcal{{CD}}\left( M\right) \] | No |
Theorem 1.5 Let \( G \) be a finite \( p \) -group of nilpotence class 2 such that \( G \in \mathcal{{CD}}\left( G\right) \) . For any positive integer \( k,{A}_{k}/Z\left( G\right) \) denotes \( {\Omega }_{k}\left( {G/Z\left( G\right) }\right) \) and \( {B}_{k} \) denotes \( {\mho }_{k}\left( G\right) Z\left( G\right)... | Proof Let \( a \in {A}_{k} \) . Then \( {a}^{{p}^{k}} \in Z\left( G\right) \) . It follows that \( \left\lbrack {{a}^{{p}^{k}}, g}\right\rbrack = 1 \) for every \( g \in G \) . Since \( c\left( G\right) = 2,\left\lbrack {a,{g}^{{p}^{k}}}\right\rbrack = {\left\lbrack a, g\right\rbrack }^{{p}^{k}} = \left\lbrack {{a}^{{p... | Yes |
Theorem 1.6 Let \( G \) be a finite group. If \( {H}_{1} > {H}_{2} > \cdots > {H}_{k} \) is a chain in \( \mathcal{{CD}}\left( G\right) \), then \( {H}_{i} \vartriangleleft {H}_{i - 1} \) where \( i = 2,3,\cdots, k \) . | Proof Assume the contrary. Then there is an \( x \in {H}_{i - 1} \) such that \( {H}_{i}^{x} \neq {H}_{i} \) for some i. Since \( {C}_{G}\left( {H}_{i}^{x}\right) = {\left( {C}_{G}\left( {H}_{i}\right) \right) }^{x},\left| {H}_{i}^{x}\right| \left| {{C}_{G}\left( {H}_{i}^{x}\right) }\right| = \left| {H}_{i}\right| \lef... | Yes |
Theorem 2.1 Let \( G \) be a finite group such that \( G \in \mathcal{{CD}}\left( G\right) \) . If \( \mathcal{{CD}}\left( G\right) \) is \( {\mathcal{M}}_{m + 2} \) where \( m \geq 3 \), then \( G \) is of nilpotence Class 2 . | Proof Let \( {M}_{1},{M}_{2},{M}_{3} \) be three atoms of \( \mathcal{{CD}}\left( G\right) \) . Then \( {M}_{i} \vartriangleleft G \) by Theorem 1.6 where \( i = 1,2,3 \) . For \( i \neq j, G = {M}_{i}{M}_{j} \) and \( {M}_{i} \cap {M}_{j} = Z\left( G\right) \) . Let \( \bar{G} = G/Z\left( G\right) \) and \( {\bar{M}}_... | Yes |
Theorem 2.2 Let \( G \) be a finite \( p \) -group such that \( G \in \mathcal{{CD}}\left( G\right) \) and \( \mathcal{{CD}}\left( G\right) = {\mathcal{M}}_{m + 2} \) where \( m \geq 3 \) . Assume that \( {M}_{1},{M}_{2},\cdots ,{M}_{m} \) are all atoms of \( \mathcal{{CD}}\left( G\right) \) . Then\n\n(1) Both \( G/Z\l... | Proof (1) By Theorem 2.1, \( c\left( G\right) = 2 \) . Hence, \( {B}_{1} = {\mho }_{1}\left( G\right) Z\left( G\right) \in \mathcal{{CD}}\left( G\right) \) by Theorem 1.5. Our aim is to prove that \( {B}_{1} = Z\left( G\right) \) . If not, \( {B}_{1} = {M}_{i} \) for some \( i \) . For \( j \neq i, G = \left\langle {{B... | Yes |
Theorem 2.3 Suppose that \( G \) is a finite \( p \) -group such that \( G \in \mathcal{{CD}}\left( G\right) \) and \( \mathcal{{CD}}\left( G\right) = \) \( {\mathcal{M}}_{m + 2} \) where \( m \geq 3.{M}_{1},{M}_{2},\cdots ,{M}_{m} \) are all atoms of \( \mathcal{{CD}}\left( G\right) \) . By Theorem 2.2, we may assume ... | Proof Let \( \bar{G} = G/Z\left( G\right) \) and \( {\bar{M}}_{k} = {M}_{k}/Z\left( G\right) \) . Then \( \bar{G} = {\bar{M}}_{1} \times {\bar{M}}_{2} \) . Let \( {\pi }_{s} : \bar{G} \rightarrow {\bar{M}}_{s} \) be the projections. Then, for \( k \geq 3 \) and \( s = 1,2 \), since \( {\bar{M}}_{k} \cap {\bar{M}}_{s} =... | Yes |
Theorem 2.6 If \( \mathcal{{CD}}\left( G\right) = {\mathcal{M}}_{m + 2} \) where \( m \geq 3 \), then \( m = {p}^{a} + 1 \) for some positive integer \( a \) . | Proof By Theorem 1.2, we may assume that \( G \in \mathcal{{CD}}\left( G\right) \) without loss of generality. It follows from Theorem 2.1 that \( G \) is of nilpotence class 2. By Theorem 1.4, we may assume that \( G \) is a \( p \) -group for some prime \( p \) . By Theorem 2.3 and Remark 2.1, Hypothesis A holds. If ... | Yes |
Theorem 3.1 Let \( m \geq 3 \) be an integer. Then there exists a finite group \( G \) such that \( \mathcal{C}\mathcal{D}\left( G\right) = {\mathcal{M}}_{m + 2} \) if and only if \( m = {p}^{a} + 1 \) for some positive integer \( a \) . | In our coming construction, all atoms of \( \mathcal{{CD}}\left( G\right) \) are self-centralizing. By the proof of Theorem 2.4, there is a division algebra corresponding to the antichain. First, we construct a division algebra of dimension \( a \) . | No |
Theorem 3.2 Suppose that \( a \) is a positive integer, \( F \) is a field containing \( {p}^{a} \) elements and \( {F}^{ * } \) is the multiplication group of \( F \) . Assume that \( {F}^{ * } = \langle g\rangle \) and \( \mathcal{A} : f \mapsto {gf} \) be a linear transformation over \( F \), where \( F \) is regard... | Proof It is obvious that \( \mathcal{A} \) is of order \( {p}^{a} - 1 \) . Let \( \mathbb{W} = \left\{ {I, A,{A}^{2},\cdots ,{A}^{{p}^{a} - 1}}\right\} \) and \( p\left( x\right) \) be the minimal polynomial of \( A \) . Then \( \mathbb{W} \subseteq \mathbb{V} \) . Hence \( \left| \mathbb{V}\right| \geq {p}^{a} \) .\n\... | Yes |
(1) \( {A}^{\mathrm{T}}Z = {ZA} \) if and only if \( {Z}_{i} = {Z}_{1}{A}^{i - 1} \) for \( i \geq 2 \) . | Proof (1) By calculation\n\n\[ {A}^{\mathrm{T}}Z = \left( \begin{matrix} 0 & 1 & \cdots & 0 \\ 0 & 0 & \cdots & 0 \\ \vdots & \vdots & & \vdots \\ {k}_{0} & {k}_{1} & \cdots & {k}_{a - 1} \end{matrix}\right) \left( \begin{matrix} {Z}_{1} \\ {Z}_{2} \\ \vdots \\ {Z}_{a} \end{matrix}\right) = \left( \begin{matrix} {Z}_{2... | Yes |
Lemma 2.1 Suppose that \( A \) is a positive operator on a Hilbert space \( \mathcal{H} \) . Then the following are equivalent.\n\n(1) \( 0 \in \sigma \left( A\right) \) ;\n\n(2) \( \forall \varepsilon > 0,\exists v \in \mathcal{H} \) such that \( \langle {Av}, v\rangle < \varepsilon \parallel v{\parallel }^{2} \) . | Proof If a bounded operator \( T \in \mathcal{B}\left( \mathcal{H}\right) \) is invertible in the algebraic sense, i.e., \( T \) is one-to-one and onto, then \( {T}^{-1} \) is automatically bounded by Open Mapping Theorem. So there are two ways the bounded operator \( T \) can fail to be invertible.\n\n(a) \( T \) is n... | Yes |
Corollary 2.1 Let \( G \) be a unimodular locally compact group acting on a set \( X,\mu \) be an irreducible reversible random walk on \( X \), and \( \nu \) be a stationary measure for \( \mu \) . Assume that Properties (A) and (B) in Section 1 hold. Then the following are equivalent.\n\n(1) \( G \) has Haagerup prop... | Proof \( \;\left( 2\right) \Leftrightarrow \left( 3\right) \; \) It follows from the definitions of \( {\Delta }_{\mu } \) and \( {M}_{\mu } \).\n\n\( \left( 1\right) \Leftrightarrow \left( 2\right) \; \) It follows from Theorems 0.1 and 2.1. | No |
Proposition 2.1 The assumption is the same as that in Corollary 2.1. Let \( k \geq 2 \) be an integer. The following are equivalent.\n\n(1) \( G \) has Haagerup property;\n\n(2) There exists a \( {C}_{0} \) -representation \( \left( {\pi ,\mathcal{H}}\right) \) of \( G \) such that \( {C}_{k} = k \), where\n\n\[ \n{C}_... | Proof From Corollary 2.1 we only need to check that the condition \( {C}_{k} = k \) is equivalent to \( 0 \in \sigma \left( {\Delta }_{\mu }\right) \) . Indeed, since by \( \left\lbrack {2\text{, Proposition 5.2.3}}\right\rbrack \) ,\n\n\[ \n{\Delta }_{\mu * k} = I - {M}_{\mu * k} = \left( {I - {M}_{\mu }}\right) \left... | Yes |
Theorem 2.2 Let \( G \) be a unimodular locally compact group acting on a set \( X \) and \( H \) be a closed subgroup of \( G \) . Let \( \mu \) be an irreducible reversible random walk on \( X \) and \( \nu \) be a stationary measure for \( \mu \) . Assume that Properties (A) and (B) in Section 1 hold. Then the follo... | Proof It follows from the definition of relative Property (T), Theorem 2.1 and the proof of Proposition 2.1. | No |
Corollary 2.2 Let \( \Gamma \) be a group generated by a finite symmetric set \( S \) with \( e \notin S \), and \( \mathcal{G}\left( S\right) \) be connected. If \( \frac{1}{2} \notin \operatorname{co}\left( {\sigma \left( {\Delta }_{{\mu }_{S}}\right) }\right) \), then \( \Gamma \) does not have Haagerup property. | Proof Since \( {\Delta }_{{\mu }_{S}} \) is a positive operator on \( {\Omega }_{\mathcal{H}}^{0}\left( {\mathcal{G}\left( S\right) }\right) \), it follows from [8, Problem 216] that the convex hull of its spectrum \( \sigma \left( {\Delta }_{{\mu }_{S}}\right) \) is the closure of the numerical range \( W\left( {\Delt... | Yes |
Example 2.1 We consider the group \( {\mathbb{Z}}^{2} \rtimes \mathrm{{SL}}\left( {2,\mathbb{Z}}\right) \) . It is well known that the group \( \mathrm{{SL}}\left( {2,\mathbb{Z}}\right) \) is generated by the matrices \( {A}_{1} = \left( \begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right) \) and \( {A}_{2} = \left( \be... | So the set \( S = \left\{ {-I, A, B, C, D, - A, - B, - C, - D,{A}^{-1},{B}^{-1},{C}^{-1},{D}^{-1}, - {A}^{-1}, - {B}^{-1}, - {C}^{-1}}\right. \) , \( \left. \begin{matrix} - {D}^{-1} \\ \\ \\ \end{matrix}\right\} \) is the generating sets of the group \( \;G, \) where \( \;A = \left( \begin{matrix} 1 & 1 & 1 \\ 0 & 1 &... | No |
Theorem 2.1 Let \( S \) be a strongly rpp semigroup. Then the following conditions are equivalent:\n\n(1) \( S \) is a completely \( {\mathcal{J}}^{\left( \ell \right) } \) -simple semigroup;\n\n(2) \( S \) is a locally left cancellative monoid satisfying the regularity condition;\n\n(3) \( S \) satisfies the regularit... | Proof (1) \( \Rightarrow \) (2) Let \( S = \mathcal{M}\left( {M;I,\Lambda ;P}\right) \) be a Rees matrix semigroup over a left can-cellative monoid \( M \) . If \( e \in E\left( S\right) \), then \( e = \left( {{p}_{\lambda i}^{-1}, i,\lambda }\right) \) for some \( i \in I,\lambda \in \Lambda \) . It is easy to see th... | Yes |
Theorem 2.2 Let \( S \) be a strongly rpp semigroup satisfying the regularity condition. Then the following conditions are equivalent:\n\n(1) \( S \) is a completely \( {\mathcal{J}}^{\left( \ell \right) } \) -simple semigroup;\n\n(2) \( S \) is weakly cancellative, that is, for any \( a, x, y \in S \), whenever \( {ax... | Proof (1) \( \Rightarrow \) (2) Suppose that \( S = \mathcal{M}\left( {M;I,\Lambda ;P}\right) \) is a Rees matrix semigroup over the left cancellative monoid \( M \) . Let \( \left( {a, i,\lambda }\right) ,\left( {x, j,\mu }\right) ,\left( {y, k,\kappa }\right) \in S \) . If \( \left( {a, i,\lambda }\right) \left( {x, ... | Yes |
Corollary 2.2 A strongly rpp semigroup \( S \) is a completely \( {\mathcal{J}}^{\left( \ell \right) } \) -simple semigroup if and only if for any \( a, b \in S,{a}^{\diamond } = {\left( ab\right) }^{\diamond }{a}^{\diamond },{b}^{\diamond } = {b}^{\diamond }{\left( ab\right) }^{\diamond } \) and \( {a}^{\diamond }{\le... | Proof If \( S \) is a completely \( {\mathcal{J}}^{\left( \ell \right) } \) -simple semigroup, then \( S \) is a strongly rpp semigroup satisfying the regularity condition, and so by Theorem 2.2, for any \( a, b \in S, a\overline{\mathcal{R}}{ab}{\mathcal{L}}^{\left( \ell \right) }b \), thus \( {a}^{\diamond }\mathcal{... | Yes |
Theorem 2.3 The following conditions are equivalent for a strongly rpp semigroup \( S \) satisfying the regularity condition:\n\n(1) \( S \) is a completely \( {\mathcal{J}}^{\left( \ell \right) } \) -simple semigroup;\n\n(2) \( S \) satisfies the identity: \( {\left( ab\right) }^{\diamond } = {\left( axb\right) }^{\di... | Proof \( \left( 1\right) \Rightarrow \left( 2\right) \) Suppose that \( S = \mathcal{M}\left( {M;I,\Lambda ;P}\right) \) is a Rees matrix semigroup over a left cancellative monoid \( M \) . For \( a = \left( {u, i,\lambda }\right), b = \left( {v, j,\mu }\right), x \in S \), we have \( {ab} = \left( {u{p}_{\lambda j}v, ... | Yes |
Theorem 3.1 All strongly rpp semigroups form a quasivariety defined by the implications: I. \( {x}^{\diamond }{x}^{\diamond } = {x}^{\diamond } \) ;\n\nL1. \( {ax} = {ay} \Rightarrow {a}^{\diamond }x = {a}^{\diamond }y \) ;\n\nL2. \( {ax} = a \Rightarrow {a}^{\diamond }x = {a}^{\diamond } \) ;\n\nL3. \( a{a}^{\diamond ... | Proof Assume that a semigroup \( S \) satisfies the given conditions. Conditions L1-L3 imply that \( a{\mathcal{L}}^{ * }{a}^{\diamond } \) and Condition I implies that \( {a}^{\diamond } \) is an idempotent. Thus the semigroup \( S \) is rpp. On the other hand, by Condition \( \mathrm{U} \), we know that the idempoten... | Yes |
Theorem 3.2 Let \( \mathfrak{D} \) be a psuedosubvariety of \( \mathfrak{{SRP}} \) . If \( T, U \in \mathfrak{D} \), then\n\n(1) Any \( \diamond \) -subsemigroup of \( T \) is contained in \( \mathfrak{D} \) .\n\n(2) \( T \times U \in \mathfrak{D} \) .\n\n(3) If \( \phi \) is a \( \diamond \) -homomorphism of \( T \) i... | Proof (1) Let \( V \) be a \( \diamond \) -subsemigroup of \( T \) . It suffices to verify that \( V \) is a strongly rpp semigroup. For any \( a \in V \), we have \( {a}^{\diamond } \in V \) . Note that \( {\mathcal{L}}^{{\left( \ell \right) }^{V}} \subseteq {\mathcal{L}}^{{\left( \ell \right) }^{T}} \cap V \times V \... | Yes |
Proposition 3.1 \( \mathfrak{B} = \left\lbrack {a = {a}^{\diamond }}\right\rbrack \) . | Moreover, we have\n\n\[ \mathfrak{{LZB}} = \left\lbrack {a = {a}^{\diamond },{ab} = a}\right\rbrack \;\text{and}\;\mathfrak{{RZB}} = \left\lbrack {a = {a}^{\diamond },{ab} = b}\right\rbrack .\n\nBy Corollary 2.2 and a simple calculation, we can obtain | No |
Proposition 3.4 Denote by \( \mathfrak{{OCJL}} \) the set of completely \( {\mathcal{J}}^{\left( \ell \right) } \) -simple semigroups whose set of idempotents forms a band. Then \( \mathfrak{{OCJL}} = \left\lbrack {{\left( ab\right) }^{\diamond } = {a}^{\diamond }{b}^{\diamond },{a}^{\diamond } = {a}^{\diamond }{b}^{\d... | Proof If \( S \in \mathfrak{{OCJL}} \), then by the arguments before Proposition 3.4, \( S \) satisfies the identity \( {\left( ab\right) }^{\diamond } = {a}^{\diamond }{b}^{\diamond } \) . On the other hand, by Proposition 3.2, \( S \) satisfies the identity \( {a}^{\diamond } = {\left( ab\right) }^{\diamond }{a}^{\di... | Yes |
Proposition 3.5 (C) \( \mathfrak{L} = \mathfrak{{Re}}\mathfrak{B} \vee \mathfrak{L}\mathfrak{{CM}} = \mathfrak{L}\mathfrak{Z}\mathfrak{B} \vee \mathfrak{R}\mathfrak{Z}\mathfrak{B} \vee \mathfrak{L}\mathfrak{{CM}} \) . | \( \textbf{Proof} : \) By the theory of bands (for details, see [15]), it is clear that \( \Re e\mathfrak{B} = \mathfrak{{LZB}} \vee \Re \mathfrak{{ZB}} \) and the second equality is immediate from the first one. Since rectangular bands and left can-cellative monoids are completely \( {\mathcal{J}}^{\left( \ell \right)... | Yes |
Lemma 2.1 Let \( S \) be an ordered semigroup and \( a, b \in S \). (1) If \( S \) satisfies \( \left( {{ab}, a{b}^{2}}\right) \in {\eta }_{l} \), then it satisfies \( \left( {{ab}, a{b}^{k}}\right) \in {\eta }_{l} \) for any \( k \in {\mathbb{Z}}^{ + } \). (2) If \( S \) satisfies \( \left( {{b}^{2},{ab}}\right) \in {... | Proof (1) Assume \( k > 2 \) and suppose that the statement is true for some \( k \), i.e., \( \left( {{ab}, a{b}^{k}}\right) \in \) \( {\eta }_{l} \). Since \( S \) satisfies \( \left( {{ab}, a{b}^{2}}\right) \in {\eta }_{l} \) for any \( a, b \in S \), based on this condition, then for \( k + 1 \) , we can replace th... | Yes |
Lemma 2.4 Let \( \rho \) be a band congruence on a semigroup \( S \) . Then the following conditions are equivalent:\n\n(1) \( \rho \subseteq {\mu }_{l} \) ;\n\n(2) \( \rho \subseteq {\mathcal{M}}_{l} \) ;\n\n(3) Any \( \rho \) -class is a left Archimedean ordered subsemigroup. | Proof (1) \( \Rightarrow \) (3) Let \( A \) be a \( \rho \) -class of \( S \) and let \( a, b \in A \) . First of all, as we have already seen in Remark 1.1, the \( \rho \) -class \( A \) is a subsemigroup of \( S \) . Since \( \left( {{a}^{2}, b}\right) \in \rho \), whence \( \left( {{a}^{2}, b}\right) \in {\mu }_{l} ... | Yes |
Theorem 2.1 The following conditions on an ordered semigroup \( S \) are equivalent:\n\n(1) \( S \) is a band of left Archimedean ordered subsemigroups;\n\n(2) If \( a \leq b \), then \( \left( {{xay},{xaby}}\right) \in {\mu }_{l} \) and \( \left( {{xay},{xbay}}\right) \in {\mu }_{l} \) for all \( x, y \in {S}^{1} \) ;... | Proof \( \left( 1\right) \Rightarrow \left( 2\right) \) Let \( S \) be a band \( B \) of left Archimedean ordered subsemigroups \( {S}_{\alpha }\left( {\alpha \in }\right. \) \( B). \) Assume that \( a, b, x, y \in S \) (in a similar way we can prove the cases with \( x = 1 \) or \( y = 1). \) Let \( a \leq b \) and \(... | Yes |
Theorem 3.1 The following conditions on an ordered semigroup \( S \) are equivalent:\n\n(1) \( S \) is a left normal band of left Archimedean ordered semigroups;\n\n(2) \( \left( {\forall u, v, w \in S}\right) \left( {{uwv},{uvw}}\right) \in {\eta }_{l} \) ;\n\n(3) \( N\left( a\right) = \left\{ {b \in S \mid \left( {b,... | Proof (1) \( \Rightarrow \) (2) Let \( S \) be a left normal band \( B \) of left Archimedean ordered semigroups \( {S}_{\alpha }\left( {\alpha \in B}\right) \) . If \( u \in {S}_{\alpha }, v \in {S}_{\beta }, w \in {S}_{\gamma } \) for some \( \alpha ,\beta ,\gamma \in B \), then \( {uvw} \in {S}_{\alpha \beta \gamma ... | Yes |
Theorem 3.2 The following conditions on an ordered semigroup \( S \) are equivalent:\n\n(1) \( S \) is an \( l \) -band of left Archimedean and left \( \pi \) -regular ordered semigroups;\n\n(2) If \( a \leq b \), for each \( x, y \in {S}^{1} \) and \( p, q \in {\mathbb{Z}}^{ + } \), there exist \( m, n \in {\mathbb{Z}... | Proof (1) \( \Rightarrow \) (2) Let \( S \) be an \( l \) -band \( B \) of \( {S}_{\alpha } \) such that \( {S}_{\alpha } \) is a left Archimedean and left \( \pi \) -regular ordered semigroup for every \( \alpha \in B. \) Let \( a \leq b \) and \( a \in {S}_{\alpha },\;b \in {S}_{\beta },\;x \in {S}_{\gamma },\;y \in ... | Yes |
Theorem 3.4 The following conditions on an ordered semigroup \( S \) are equivalent:\n\n(1) \( S \) is a left regular band of left Archimedean and left \( \pi \) -regular ordered semigroups;\n\n\( \left( 2\right) \;\left( {\forall u, v, w \in S}\right) \;\left( {\exists n \in {\mathbb{Z}}^{ + }}\right) \;\left( {\foral... | Proof (1) \( \Rightarrow \) (2) Let \( S \) be a left regular band \( B \) of \( {S}_{\alpha }\left( {\alpha \in B}\right) \) which are left Archimedean and left \( \pi \) -regular ordered semigroups. If \( u \in {S}_{\alpha }, v \in {S}_{\beta }, w \in {S}_{\gamma } \) for some \( \alpha ,\beta ,\gamma \in B \) , then... | Yes |
Lemma 1.1 Let \( G \) be a connected triangle-free graph, \( m \geq 2 \) and \( n \geq 1 \) be two integers. If \( \chi \left( G\right) \geq m + n \) and there exists a vertex \( {v}_{0} \) such that \( d\left( v\right) \geq m + n - 1 \) for any \( v \in V\left( G\right) \backslash \{ {v}_{0}\} , \) then \( G \) contai... | Proof The proof is made by induction on \( m \) . If \( m = 2, B\left( {2, n}\right) \) is the star \( {K}_{1, n + 1} \) . We take a vertex \( {v}_{1} \in N\left( {v}_{0}\right) \), then by the assumption, \( d\left( {v}_{1}\right) \geq n + 1 \) . So, \( {v}_{1} \) has at least \( n \) neighbors different from \( {v}_{... | Yes |
Theorem 1.1 Let \( G \) be a connected triangle-free and \( B\left( {m, n}\right) \) -free graph. Then \( \chi \left( G\right) \leq \) \( m + n - 1 \) . | Proof Suppose that the theorem is false, and \( G \) is a connected triangle-free and \( B\left( {m, n}\right) \) - free graph with minimum order such that \( \chi \left( G\right) \geq m + n. \) Then by Lemma \( {1.2},\delta \left( G\right) \leq m + n - 2. \) We take a vertex \( v \) with \( {d}_{G}\left( v\right) = \d... | Yes |
Lemma 2.1 Let \( T \) be a tree. If \( G \) is triangle-free, \( {C}_{4} \) -free and \( T \) -free, then \( \delta \left( G\right) \leq \left| T\right| - 2 \) . | Proof The proof is made by induction on \( \left| T\right| \) . If \( \left| T\right| = 2 \), then \( T \cong {K}_{2} \) . Hence, if \( G \) is \( {K}_{2} \) -free, it is an empty graph and the result is trivially true.\n\nNow suppose that \( \left| T\right| = k \) and \( \delta \left( G\right) \geq \left| T\right| - 1... | Yes |
Theorem 2.1 Let \( T \) be a tree. If a graph \( G \) is triangle-free, \( {C}_{4} \) -free and \( T \) -free, then \( \chi \left( G\right) \leq \left| T\right| - 1 \) | Proof Let \( H \) be any induced subgraph of \( G \) . Then \( H \) is triangle-free, \( {C}_{4} \) -free and \( T \) -free. By Lemma 2.1, \( \delta \left( H\right) \leq \left| T\right| - 2 \) . Namely, \( \deg \left( G\right) \leq \left| T\right| - 2 \) . Hence, \( \chi \left( G\right) \leq \deg \left( G\right) + 1 \l... | Yes |
Corollary 2.1 Let \( F \) be a forest. If the graph \( G \) is triangle-free, \( {C}_{4} \) -free and \( F \) -free, then \( \chi \left( G\right) \leq \left| F\right| \) | Proof Let \( {T}_{1},{T}_{2},\cdots ,{T}_{r} \) be all the components of \( F \) . Let \( T \) be a tree obtained from \( F \) by adding a new \( {v}_{0} \) and joining it to exactly one vertex of \( {T}_{i} \) for each \( 1 \leq i \leq r \) . Then \( \left| T\right| = \left| F\right| + 1 \) . If \( G \) is \( F \) -fr... | Yes |
Question 0.1 Can Theorem 0.1 be reversed? | In 2007, Ge and Lin solved Question 0.1 for the parenthetic part (see [3, Theorems 2.13 and 3.10]). However, Question 0.1 for the non-parenthetic part is still open up to now. | No |
Lemma 1.3 Let \( f : X \rightarrow Y \) be a mapping, and \( \left\{ {y}_{n}\right\} \) be a sequence converging to \( y \) in \( Y \) . If \( \left\{ {B}_{n}\right\} \) is a decreasing network at some \( x \in {f}^{-1}\left( y\right) \) in \( X \), and \( \left\{ {y}_{n}\right\} \) is eventually in \( f\left( {B}_{n}\... | Proof Let \( \left\{ {B}_{n}\right\} \) be a decreasing network at some \( x \in {f}^{-1}\left( y\right) \) in \( X \), and let \( \left\{ {y}_{n}\right\} \) be eventually in \( f\left( {B}_{k}\right) \) for every \( k \in \mathbb{N} \) . Then, for every \( k \in \mathbb{N} \), there exists \( {n}_{k} \in \mathbb{N} \)... | Yes |
Theorem 1.1 Let \( \left( {f, M, X,\mathcal{P}}\right) \) be a Ponomarev-system. Then \( f \) is a 2-sequence-covering mapping iff \( \mathcal{P} \) is an sof-network of \( X \) . | Proof Sufficiency. Assume that \( \mathcal{P} \) is an \( {sof} \) -network of \( X \) . Let \( x \in X \) and \( b = \left( {\beta }_{n}\right) \in \) \( {f}^{-1}\left( x\right) \) . Then \( \left\{ {P}_{{\beta }_{n}}\right\} \) is a network at \( x \) in \( X \) . For every \( n \in \mathbb{N} \), put \( {B}_{n} = \l... | Yes |
Corollary 1.1 A space \( X \) is a 2-sequence-covering image of a metric space iff \( X \) has an sof-network. | Proof Sufficiency. Assume that \( \mathcal{P} \) is an \( {sof} \) -network of \( X \) . Suppose \( \left( {f, M, X,\mathcal{P}}\right) \) to be a Ponomarev-system. Then \( f : M \rightarrow X \) is 2-sequence-covering from Theorem 1.1 and \( M \) is a metric space. So \( X \) is a 2-sequence-covering image of a metric... | Yes |
Theorem 2.1 Let \( \left( {f, M, X,\left\{ {\mathcal{P}}_{n}\right\} }\right) \) be a Ponomarev-system. Then \( f \) is a 2-sequence-covering mapping iff \( {\mathcal{P}}_{n} \) is an \( {so} \) -cover of \( X \) for every \( n \in \mathbb{N} \) . | Proof Sufficiency. It holds from Theorem 0.1(2).\n\nNecessity. Assume that \( f \) is a 2-sequence-covering mapping. Let \( n \in \mathbb{N} \) . For every \( {P}_{\beta } \in {\mathcal{P}}_{n} \) , let \( x \in {P}_{\beta } \) and \( \{ {x}_{i}\} \) be a sequence converging to \( x \) in \( X. \) It suffices to prove ... | Yes |
Corollary 2.1 A space \( X \) is a 2-sequence-covering \( \pi \) -image of a metric space iff \( X \) has a point-star network consisting of \( {so} \) -covers. | Proof Sufficiency. Assume that \( \left\{ {\mathcal{P}}_{n}\right\} \) is a point-star network consisting of so-covers of \( X \) . Let \( \left( {f, M, X,\left\{ {\mathcal{P}}_{n}\right\} }\right) \) be a Ponomarev-system. Then \( f : M \rightarrow X \) is a 2-sequence-covering \( \pi \) -mapping by Theorem 2.1 and Re... | Yes |
Theorem 1.1 If for any point \( x \in {\mathbb{R}}^{d},\left( {b, A}\right) \) satisfies Hörmander’s condition at point \( x \) and the solution to (1.1) globally exists, then the law of \( {X}_{t}\left( x\right) \) is continuous in variable \( x \) with respect to the total variation distance. In particular, the semig... | \[ x \mapsto \mathbb{E}f\left( {{X}_{t}\left( x\right) }\right) \text{is continuous.} \] | Yes |
Corollary 1.1 If \( \left( {b, A}\right) \) satisfy Hörmander’s condition at any point \( x \in {\mathbb{R}}^{d} \), and for some Lyapunov function \( H \) and \( c > 0,\left( {1.2}\right) \) holds, then the law of \( {X}_{t}\left( x\right) \) is continuous in variable \( x \) with respect to the total variation distan... | ## 2 Proof of Theorem 1.1\n\nBefore we give the proof of Theorem 1.1, we need some notations and a lemma.\n\nLet \( {B}_{M} \mathrel{\text{:=}} \left\{ {x \in {\mathbb{R}}^{d} : \left| x\right| < M}\right\} ,{\bar{B}}_{M} \mathrel{\text{:=}} \left\{ {x \in {\mathbb{R}}^{d} : \left| x\right| \leq M}\right\} ,{B}_{M}^{c}... | No |
Lemma 2.1 For any \( n \in \mathbb{N} \) and \( x \in {B}_{n} \) , \[ \mathop{\limsup }\limits_{{y \rightarrow x}}{I}_{\left\{ t \geq {\tau }_{n}\left( y\right) \right\} } \leq {I}_{\left\{ t \geq {\tau }_{n - 1}\left( x\right) \right\} }\;\text{ a.s. } \] | Proof Let \( \Gamma \) be a measurable set with \( \mathbb{P}\left( {\Gamma }^{c}\right) = 0 \) and such that \( {X}_{s}^{n}\left( {x,\omega }\right) \) is continuous with respect to \( s \) and \( x \) for \( \omega \in \Gamma \) . For \( \omega \in \Gamma \), the conclusion is apparent if \[ \mathop{\limsup }\limits_... | Yes |
Proposition 3.1 Assume that \( V, U \in {C}^{\infty }\left( \mathbb{R}\right) \) are nonnegative functions such that (H1) and (H2) hold, then the law of \( \left( {{z}_{t},{u}_{t}}\right) \) are continuous with the initial value in the total variation distance. | Proof It can be directly obtained by Theorem 1.1 and [8, Theorem 1.1 and Section 3]. | No |
Lemma 2.2 \( {}^{\left\lbrack {21}\right\rbrack } \) Each left \( \circ \) -monoid contains a unique idempotent. | Proof This follows immediately from the definition of left \( \circ \) -monoids. | No |
Proposition 2.1 Let \( S \) be a \( \mathrm{C} \) -rpp semigroup in which \( {\mathcal{L}}^{ \circ }\left( S\right) \) is a right congruence. Then each \( {\mathcal{L}}^{ \circ } \) -class of \( S \) is a submonoid which contains a unique idempotent. | Proof Let \( S \) be a Co-rpp semigroup. Then each \( {\mathcal{L}}^{ \circ } \) -class contains at least one idempotent. Now, assume that \( {L}_{e}^{ \circ } \) is any \( {\mathcal{L}}^{ \circ } \) -class of \( S \) which contains idempotent \( e \) . For any \( x, y \in {L}_{e}^{ \circ } \) , we have \( x{\mathcal{L... | Yes |
Proposition 3.1 Let \( S \) be a semigroup. Then \( {\mathcal{L}}^{ \circ } \subseteq {\mathcal{L}}^{\circ \circ } \) . | Proof For any \( a, b \in S \), assume that \( \left( {a, b}\right) \in {\mathcal{L}}^{ \circ } \) . For any \( x \in S \), if \( \left( {{ax}, a}\right) \in \mathcal{R} \), then there exists \( u \in {S}^{1} \), such that \( {axu} = a \), and then \( {bxu} = b \) since \( \left( {a, b}\right) \in {\mathcal{L}}^{ \circ... | Yes |
Proposition 3.2 Let \( S \) be a semigroup and \( E\left( S\right) \) be central. For any \( a \in S, e \in E\left( S\right) \) , the following conditions are equivalent:\n\n(1) \( a{\mathcal{L}}^{00}e \)\n\n(2) \( {ae} = a \) and for all \( x \in {S}^{1},\;\left( {{ax}, a}\right) \in \mathcal{R} \) implies \( \left( {... | Proof If (1) holds, then, by the definition of \( {\mathcal{L}}^{\circ \circ } \), for all \( x \in {S}^{1},\left( {{ax}, a}\right) \in \mathcal{R} \) implies \( \left( {{ex}, e}\right) \in \mathcal{R} \) . Also, since \( {e}^{2} = e \), we get \( \left( {{ae}, a}\right) \in \mathcal{R} \) . So there exists \( u \) in ... | Yes |
Proposition 3.3 Each \( \mathcal{R} \) -left \( \circ \) -monoid contains a unique idempotent. | Proof This follows immediately from the definition of \( \mathcal{R} \) -left o-monoid. Assume that \( \mathcal{R} \) -left o-monoid \( S \) has another idempotent \( f \) besides of identity \( e \), then \( {f}^{2} = f \), and we of course have \( {f}^{2}\mathcal{R}f \) . Since \( S \) is an \( \mathcal{R} \) -left \... | Yes |
Proposition 3.4 Let \( {\mathcal{L}}^{\infty }\left( S\right) \) be a right congruence and \( E\left( S\right) \) be central. Then for each \( e \in E\left( S\right) ,{L}_{e}^{\infty } \) is a submonoid of \( S \) with identity \( e \) . | Proof For any \( x, y \in {L}_{e}^{\infty } \), we have \( x{\mathcal{L}}^{\infty }e{\mathcal{L}}^{\infty }y \) . Since \( {\mathcal{L}}^{\infty }\left( S\right) \) is a right congruence, \( E\left( S\right) \) is central, we obtain that \( {xy}{\mathcal{L}}^{\circ \circ }{ey} = y{\mathcal{L}}^{\circ \circ }e \), i.e.,... | Yes |
Proposition 3.5 Let \( S \) be a \( \mathrm{C} \circ \) -wrpp semigroup in which \( {\mathcal{L}}^{\circ \circ }\left( S\right) \) is a right congruence. Then each \( {\mathcal{L}}^{\circ \circ } \) -class of \( S \) contains a unique idempotent. | Proof This follows immediately from Propositions 3.2-3.3. | No |
Lemma 2.1 Let \( L\\left( s\\right) \) be defined in (7) and \( f \) be fixed by an irreducible polynomial \( f\\left( x\\right) = \) \( {x}^{3} + a{x}^{2} + {bx} + c \) . Let \( L\\left( {s, f}\\right) \) and \( L\\left( {s,\\operatorname{sym}^{2}f}\\right) \) denote the Hecke \( L \) -function and the symmetric squar... | Proof This is \( \\left\\lbrack {{11}\\text{, Lemma 2.4}}\\right\\rbrack \) . | No |
Lemma 2.2 For any \( \varepsilon > 0 \), we have the subconvexity bound\n\n\[ L\left( {\frac{1}{2} + \mathrm{i}t, f}\right) { \ll }_{f,\varepsilon }{\left( \left| t\right| + 1\right) }^{\max \left\{ {\frac{2}{3}\left( {1 - \sigma }\right) ,0}\right\} + \varepsilon } \]\n\nuniformly for \( \frac{1}{2} \leq \sigma \leq 2... | Proof \( {Theresult}\left( 5\right) {isduetoGood}\left\lbrack 4\right\rbrack ,{and}\left( 6\right) {wasestablishedbyJutila}\left\lbrack 9\right\rbrack . \) | No |
Lemma 2.3 For any \( \varepsilon > 0 \), we have\n\n\[ L\left( {\sigma + \mathrm{i}t,{\operatorname{sym}}^{2}f}\right) { \ll }_{f,\varepsilon }{\left| t\right| }^{\max \{ \frac{11}{8}\left( {1 - \sigma }\right) ,0\} + \varepsilon } \]\n\nuniformly for \( \frac{1}{2} \leq \sigma \leq 2 \) and \( \left| t\right| \geq 1 \... | Proof This is the result (11) of [11, Lemma 2.3]. | No |
Lemma 2.4 We have\n\n\\[ \n{\\int }_{1}^{T}{\\left| \\zeta \\left( \\frac{1}{2} + \\mathrm{i}t\\right) \\right| }^{12}\\mathrm{\\;d}t \\ll {T}^{2}{\\log }^{17}T \n\\] | Proof This lemma is Theorem 1 in Heath-Brown [5]. | No |
Proposition 1.3 Let \( i, j \in \{ 0,1,\cdots, s - 1\} \) . Then the dimension of \( {\delta }^{i}\left( C\right) \cap \tau {\delta }^{j}\left( {C}^{\prime }\right) \) is \( d \) , where \( d \) is the degree of the polynomial \( {g}_{ij}^{\left( 2\right) }\left( {C,{C}^{\prime }}\right) \) . | Proof Since \( C \) and \( {C}^{\prime } \) are self-dual codes, we regard \( G \) and \( {G}^{\prime } \) as parity check matrices of them respectively. Note that \( \pi \left( G\right) {\mathbf{x}}^{\mathrm{T}} = 0 \) for any \( \mathbf{x} \in \pi \left( C\right) \), where \( \pi \) is any coordinate permutation acti... | Yes |
Proposition 2.1 Let \( s \equiv 1\left( {\;\operatorname{mod}\;6}\right) \) and \( C \) be a ternary \( \left\lbrack {{2s} + 2, s + 1}\right\rbrack \) code with generator matrix \( G \) of the form (0.1), where \( A \) is a circulant matrix of size \( s \times s \) with the first row vector \( \mathbf{u} = \left( {{u}_... | Proof The proof is entirely analogous to that of [6, Proposition 4.1], so we omit it. | No |
Proposition 2.2 Let \( i, j \in \{ 0,1,\cdots, s - 1\} \) . Then \( x + 2 \) divides both \( {g}_{ij}^{\left( 1\right) }\left( {C,{C}^{\prime }}\right) \) and \( {g}_{ij}^{\left( 2\right) }\left( {C,{C}^{\prime }}\right) \) . | (1) For any \( i, j \in \{ 0,1,\cdots, s - 1\} \), from the condition (2) in Proposition 2.1, we have \( f\left( \mathbf{u}\right) \equiv f\left( {\mathbf{u}}^{\prime }\right) \equiv 2\left( {\;\operatorname{mod}\;3}\right) \) . Note that the vector corresponding to \( {x}^{i}a\left( x\right) \) is a cyclic shift of u.... | Yes |
Lemma 2.1 (1) \( V = {\delta }^{i}\left( {C}_{e}\right) \cap {\delta }^{j}\left( {C}_{f}\right) \) for any \( e, f \in \{ 1,2,3\}, i, j \in \{ 0,1,\cdots ,6\} \) and \( \left( {i, e}\right) \neq \left( {j, f}\right) \n\n(2) \( \tau \left( V\right) = \tau {\delta }^{i}\left( {C}_{1}\right) \cap \tau {\delta }^{j}\left( ... | Proof For any \( i, j \in \{ 0,1,\cdots ,6\}, C,{C}^{\prime } \in \left\{ {{C}_{e} : e = 1,2,3}\right\} \) and \( i \neq j \) if \( C = {C}^{\prime } \), with the aid of MAGMA it is verified that \( {g}_{ij}^{\left( 1\right) }\left( {C,{C}^{\prime }}\right) = x + 2 \) . By (1) of Proposition 2.2,(1) and (2) follow.\n\n... | Yes |
Lemma 2.2 (1) \( V = {\delta }^{i}\left( {C}_{e}\right) \cap {\delta }^{j}\left( {C}_{f}\right) \) for any \( e, f \in \{ 1,2,\cdots ,{345}\}, i, j \in \{ 0,1,\cdots ,{18}\} \) and \( \left( {i, e}\right) \neq \left( {j, f}\right) \); | Proof For any \( i, j \in \{ 0,1,\cdots ,{18}\}, C,{C}^{\prime } \in \left\{ {{C}_{e} : 1 \leq e \leq {345}}\right\} \) and \( i \neq j \) if \( C = {C}^{\prime } \), with the aid of MAGMA it is verified that \( {g}_{ij}^{\left( 1\right) }\left( {C,{C}^{\prime }}\right) = x + 2 \) . By (1) of Proposition 2.2, the first... | Yes |
Lemma 2.3 (1) \( V = {\delta }^{i}\left( {C}_{e}\right) \cap {\delta }^{j}\left( {C}_{f}\right) \) for any \( e, f \in \{ 1,2,3\}, i, j \in \{ 0,1,\cdots ,{30}\} \) and \( \left( {i, e}\right) \neq \left( {j, f}\right) \n\n(2) \( \tau \left( V\right) = \tau {\delta }^{i}\left( {C}_{1}\right) \cap \tau {\delta }^{j}\lef... | Proof For any \( i, j \in \{ 0,1,\cdots ,{30}\}, C,{C}^{\prime } \in \left\{ {{C}_{e} : e = 1,2,3}\right\} \) and \( i \neq j \) if \( C = {C}^{\prime } \), by MAGMA it is verified that \( {g}_{ij}^{\left( 1\right) }\left( {C,{C}^{\prime }}\right) = x + 2 \) . By (1) of Proposition 2.2, the first two assertions follow.... | Yes |
Lemma 1.1 Let \( {G}^{\prime } \) be a cubic graph that has a good pair \( \left( {{V}_{1}^{\prime },{V}_{2}^{\prime }}\right) \), and let \( G \) be a cubic graph obtained from \( {G}^{\prime } \) by an \( M \) -extension or by an \( N \) -extension. Then \( G \) also has a good pair. | Proof First, suppose that \( G = M\left( u\right) \left( {G}^{\prime }\right) \) is a cubic graph obtained from \( {G}^{\prime } \) by an \( M \) - extension (see Fig. 2). Let \( {V}_{1} = {V}_{1}^{\prime } \cup \{ u\} \) and \( {V}_{2} = V\left( G\right) \smallsetminus {V}_{1} = {V}_{2}^{\prime } \cup \left\{ {{w}_{1}... | Yes |
Lemma 1.2 Let \( G \) be a 2-edge-connected graph. If \( G \) is obtained from \( {\theta }_{i}\left( {i = 1,2,3}\right) \) by a series of \( M \) -extensions or \( N \) -extensions, then \( G \) has a good pair. | Proof By Lemma 1.1, it suffices to verify that each of the \( {\theta }_{i} \) ’s has a good pair.\n\nLet \( {V}_{11} = \{ u\} ,{V}_{12} = \{ v\} ,{V}_{21} = \left\{ {{v}_{1},{v}_{4}}\right\} ,{V}_{22} = \left\{ {{v}_{2},{v}_{3}}\right\} ,{V}_{31} = \left\{ {{u}_{1},{u}_{3},{u}_{5}}\right\} \) and \( {V}_{32} = \) \( \... | Yes |
Lemma 1.1 Let \( G \) be a connected graph with \( Q \) -matrix \( Q\left( G\right) \). If \( \pi \) is an equitable partition of \( V\left( G\right) \), with \( Q \) -divisor matrix \( {Q}_{\pi } \) and characteristic matrix \( {B}_{\pi } \), then\n\n(1) \( Q\left( G\right) {B}_{\pi } = {B}_{\pi }{Q}_{\pi } \)\n\n(2) ... | Proof Suppose that the partition \( \pi \) with cells \( {V}_{1},{V}_{2},\cdots ,{V}_{k} \), and \( \left| {V}_{i}\right| = {n}_{i} \).\n\n(1) It suffices to note that if \( i \in {V}_{s} \), then the \( \left( {i, j}\right) \) -entry of both \( Q\left( G\right) {B}_{\pi } \) and \( {B}_{\pi }{Q}_{\pi } \) is \( {q}_{s... | Yes |
Theorem 1.1 Let \( G \) be a graph of order \( n,{H}_{i}\left( {i = 1,2,\cdots, n}\right) \) be a graph of order \( {n}_{i} \) with all \( Q \) -main eigenvalues \( {q}_{i1},{q}_{i2},\cdots ,{q}_{i{r}_{i}} \), and \( {V}_{i1},{V}_{i2},\cdots ,{V}_{i{r}_{i}} \) be an equitable partition of \( V\left( {H}_{i}\right) \) w... | Proof For short set \( {G}^{\prime } = G\left\lbrack {{H}_{1},{H}_{2},\cdots ,{H}_{n}}\right\rbrack \), the partition chosen is clearly equitable. Denote the corresponding characteristic matrix by \( {B}_{\pi } \). Let \( {\beta }_{i} = {\left( {z}_{1},{z}_{2},\cdots ,{z}_{{n}_{i}}\right) }^{\mathrm{T}} \) be an eigenv... | Yes |
Theorem 1.2 Let \( G \) be a graph of order \( n,{H}_{i} = \left\lbrack {{X}_{i},{Y}_{i}}\right\rbrack \left( {i = 1,2,\cdots, m}\right) \) be an \( \left( {{s}_{i},{t}_{i}}\right) \) - semiregular bipartite graph with order \( {n}_{i} \), and \( {H}_{i}\left( {i = m + 1, m + 2,\cdots, n}\right) \) be an \( {r}_{i} \) ... | \[ {\Phi }_{Q}\left( {G\left\lbrack {{H}_{1},{H}_{2},\cdots ,{H}_{n}}\right\rbrack }\right) = \Phi \left( {Q}_{\pi }\right) \mathop{\prod }\limits_{{i = 1}}^{m}\frac{{\Phi }_{Q}\left( {{H}_{i}, x - {N}_{i}}\right) }{\left( {x - {s}_{i} - {t}_{i} - {N}_{i}}\right) \left( {x - {N}_{i}}\right) }\mathop{\prod }\limits_{{i ... | Yes |
Corollary 1.3 Let \( {G}_{1} \) be an \( r \) -regular graph with order \( {n}_{1} \), and \( {G}_{2} \) be an \( \left( {s, t}\right) \) -semiregular bipartite graph with order \( {n}_{2} \) and size (the number of edges) \( m \) . Then the characteristic polynomial of the matrix \( Q\left( {{G}_{1} \vee {G}_{2}}\righ... | Proof Suppose that the number of vertices in \( {G}_{2} \) with degree \( s \) is \( {x}_{1} \) . Now we use Theorem 1.2 by taking \( G = {K}_{2},{H}_{1} = {G}_{2},{H}_{2} = {G}_{1} \), then \( {N}_{1} = {n}_{1} \) and \( {N}_{2} = {n}_{2} \), the equitable partition matrix \[ {Q}_{\pi } = \left\lbrack \begin{matrix} {... | Yes |
Corollary 1.4 Let \( {G}_{i}\left( {i = 1,2}\right) \) be an \( \left( {{s}_{i},{t}_{i}}\right) \) -semiregular bipartite graph with order \( {n}_{i} \) , and the numbers of vertices of the two color classes of \( {G}_{i} \) are \( {x}_{i} \) and \( {n}_{i} - {x}_{i} \), respectively. Then the characteristic polynomial... | Proof By taking \( G = {K}_{2},{H}_{1} = {G}_{1},{H}_{2} = {G}_{2} \) and \( {N}_{1} = {n}_{2},{N}_{2} = {n}_{1} \) in Theorem 1.2, we have the equitable partition matrix\n\n\[ \n{Q}_{\pi } = \left\lbrack \begin{matrix} {s}_{1} + {n}_{2} & {s}_{1} & {x}_{2} & {n}_{2} - {x}_{2} \\ {t}_{1} & {t}_{1} + {n}_{2} & {x}_{2} &... | Yes |
Lemma 1.2 Let \( {G}_{1} \) be an \( r \) -regular graph with order \( {n}_{1} \), and \( {G}_{2} \) be an \( \left( {s, t}\right) \) -semiregular bipartite graph with order \( {n}_{2} \) and size \( m \) . Then\n\n\[ \n{q}_{\min }\left( {{G}_{1} \vee {G}_{2}}\right) = \min \left\{ {{\tau }^{\prime }\left( f\right) ,{q... | Proof Let the number of vertices in \( {G}_{2} \) with degree \( s \) being \( {x}_{1} \) . By Corollary 1.3, \( {q}_{\min }\left( {{G}_{1} \vee }\right. \) \( \left. {G}_{2}\right) \) is the smallest one among \( {q}_{\min }\left( {G}_{1}\right) + {n}_{2},\alpha \left( {G}_{2}\right) + {n}_{1} \) and \( {\tau }^{\prim... | Yes |
Lemma 1.3 Let \( {G}_{i}\left( {i = 1,2}\right) \) be an \( \left( {{s}_{i},{t}_{i}}\right) \) -semiregular bipartite graph with order \( {n}_{i} \), and the numbers of vertices of the two color classes of \( {G}_{i} \) are \( {x}_{i} \) and \( {n}_{i} - {x}_{i} \), respectively. Then\n\n\[ \n{q}_{\min }\left( {{G}_{1}... | Proof Define a function \( h\left( x\right) \) as\n\n\[ \nh\left( x\right) = \left( {{n}_{1}x - {n}_{1}{n}_{2} - \left( {{n}_{1} - 2{x}_{1}}\right) \left( {{s}_{1} - {t}_{1}}\right) }\right) \left( {{n}_{2}x - {n}_{1}{n}_{2} - \left( {{n}_{2} - 2{x}_{2}}\right) \left( {{s}_{2} - {t}_{2}}\right) }\right) . \]\n\nThe two... | Yes |
Theorem 1.5 Let \( G \) be an edge-regular graph, and \( {G}_{i}\left( {i = 1,2}\right) \) be an \( {r}_{i} \) -regular graph with order \( n \) . If \( {r}_{1} \leq {r}_{2} \), then \( q\left( {G \vee {G}_{1}}\right) \leq q\left( {G \vee {G}_{2}}\right) \), with equality holds if and only if \( {r}_{1} = {r}_{2} \) . | Proof The proof is omitted since it is similar to that of Theorem 1.4. | No |
Theorem 2.1 Let \( G \) be a graph of order \( n,{H}_{i}\left( {i = 1,2,\cdots, n}\right) \) be a graph of order \( {n}_{i} \) with main eigenvalues \( {\lambda }_{i1},{\lambda }_{i2},\cdots ,{\lambda }_{i{r}_{i}} \), and \( {V}_{i1},{V}_{i2},\cdots ,{V}_{i{r}_{i}} \) be an equitable partition of \( {H}_{i} \) with exa... | \[ {\Phi }_{A}\left( {G\left\lbrack {{H}_{1},{H}_{2},\cdots ,{H}_{n}}\right\rbrack }\right) = \Phi \left( {A}_{\pi }\right) \mathop{\prod }\limits_{{i = 1}}^{n}\frac{{\Phi }_{A}\left( {{H}_{i}, x}\right) }{\left( {x - {\lambda }_{i1}}\right) \left( {x - {\lambda }_{i2}}\right) \cdots \left( {x - {\lambda }_{i{r}_{i}}}\... | Yes |
Theorem 2.2 Let \( G \) be a graph of order \( n,{H}_{i} = \left\lbrack {{X}_{i},{Y}_{i}}\right\rbrack \left( {i = 1,2,\cdots, m}\right) \) be an \( \left( {{s}_{i},{t}_{i}}\right) \) - semiregular bipartite graph with order \( {n}_{i} \), and \( {H}_{i}\left( {i = m + 1, m + 2,\cdots, n}\right) \) be an \( {r}_{i} \) ... | \[ {\Phi }_{A}\left( {G\left\lbrack {{H}_{1},{H}_{2},\cdots ,{H}_{n}}\right\rbrack }\right) = \Phi \left( {A}_{\pi }\right) \mathop{\prod }\limits_{{i = 1}}^{m}\frac{{\Phi }_{A}\left( {{H}_{i}, x}\right) }{{x}^{2} - {s}_{i}{t}_{i}}\mathop{\prod }\limits_{{i = m + 1}}^{n}\frac{{\Phi }_{A}\left( {{H}_{i}, x}\right) }{x -... | Yes |
Corollary 2.2 Let \( {G}_{1} \) be an \( r \) -regular graph with order \( {n}_{1} \), and \( {G}_{2} \) be an \( \left( {s, t}\right) \) -semiregular bipartite graph with order \( {n}_{2} \) and size (the number of edges) \( m \) . Then the characteristic polynomial of the matrix \( A\left( {{G}_{1} \vee {G}_{2}}\righ... | Proof Suppose that the number of vertices in \( {G}_{2} \) with degree \( s \) is \( {x}_{1} \) . Now we use Theorem 2.2 by taking \( G = {K}_{2},{H}_{1} = {G}_{2},{H}_{2} = {G}_{1} \) . Then we have the equitable partition matrix \[ {A}_{\pi } = \left\lbrack \begin{matrix} 0 & s & {n}_{1} \\ t & 0 & {n}_{1} \\ {x}_{1}... | Yes |
Corollary 2.3 Let \( {G}_{i}\left( {i = 1,2}\right) \) be an \( \left( {{s}_{i},{t}_{i}}\right) \) -semiregular graph with order \( {n}_{i} \) and size \( {m}_{i} \), Then the characteristic polynomial of the matrix \( A\left( {{G}_{1} \vee {G}_{2}}\right) \) is \[ {\Phi }_{A}\left( {{G}_{1} \vee {G}_{2}, x}\right) = \... | Proof By taking \( G = {K}_{2},{H}_{1} = {G}_{1},{H}_{2} = {G}_{2} \) in Theorem 2.2, we have the equitable partition matrix \[ {A}_{\pi } = \left\lbrack \begin{matrix} 0 & {s}_{1} & {x}_{2} & {n}_{2} - {x}_{2} \\ {t}_{1} & 0 & {x}_{2} & {n}_{2} - {x}_{2} \\ {x}_{1} & {n}_{1} - {x}_{1} & 0 & {s}_{2} \\ {x}_{1} & {n}_{1... | Yes |
Lemma 2.2 Let \( {G}_{1} \) be an \( r \) -regular graph with order \( {n}_{1} \), and \( {G}_{2} \) be an \( \left( {s, t}\right) \) -semiregular bipartite graph with order \( {n}_{2} \) and size \( m \) . Then \( {\lambda }_{\min }\left( {{G}_{1} \vee {G}_{2}}\right) = \min \left\{ {{\tau }^{\prime }\left( g\right) ,... | Proof Let the number of vertices in \( {G}_{2} \) with degree \( s \) be \( {x}_{1} \) . One may see that \( {\lambda }_{\min }\left( {{G}_{1} \vee }\right. \) \( \left. {G}_{2}\right) \) is the smallest one among \( {\lambda }_{\min }\left( {G}_{1}\right) ,{\lambda }_{{n}_{2} - 1}\left( {G}_{2}\right) \) and \( {\tau ... | Yes |
Lemma 2.3 Let \( {G}_{i}\left( {i = 1,2}\right) \) be an \( \left( {{s}_{i},{t}_{i}}\right) \) -semiregular bipartite graph with order \( {n}_{i} \), and the numbers of vertices of the two color classes of \( {G}_{i} \) be \( {x}_{i} \) and \( {n}_{i} - {x}_{i} \), respectively. Then \( {\lambda }_{\min }\left( {{G}_{1... | Proof Define \( h\left( x\right) \) as\n\n\[ h\left( x\right) = \left( {{n}_{1}x + 2{s}_{1}{x}_{1}}\right) \left( {{n}_{2}x + 2{s}_{2}{x}_{2}}\right) . \]\n\nThe two roots of \( h\left( x\right) = 0 \) are \( {a}_{1} = - \frac{2{s}_{1}{x}_{1}}{{n}_{1}},{a}_{2} = - \frac{2{s}_{2}{x}_{2}}{{n}_{2}} \) .\n\nIt is easy to s... | Yes |
Lemma 2.3 Let \( 1 < p < \infty ,0 < \alpha < n \) and \( \frac{1}{q} = \frac{1}{p} - \frac{\alpha }{n} \) . Then\n\n(a) There exists constant \( {C}_{1} > 0 \) such that\n\n\[ \n\parallel {Mf}{\parallel }_{{L}^{p}\left( {\mathbb{R}}^{n}\right) } \leq {C}_{1}\parallel f{\parallel }_{{L}^{p}\left( {\mathbb{R}}^{n}\right... | Proof of Theorem 1.1 (a) \( \Rightarrow \) (b) For a ball \( B = B\left( {{x}_{0}, r}\right) \subset {\mathbb{R}}^{n} \) . Let \( b \in {\Lambda }_{\gamma }\left( {\mathbb{R}}^{n}\right) \) , \( f \in {L}^{p}\left( {\mathbb{R}}^{n}\right) \) and \( {f}_{1} = f{\chi }_{2B},{f}_{2} = f - {f}_{1} \) . The fact \( {T}_{b}f... | No |
Proposition 5.1 Suppose that \( A \in {\mathrm{{CBMO}}}_{q}\left( {\mathbb{R}}^{n}\right) ,1 < q < \infty \) . If \( 0 < p < \infty ,1 < {q}_{1},{q}_{2} < \) \( \infty \) and \( \frac{1}{{q}_{2}} = \frac{1}{q} + \frac{1}{{q}_{1}} - \frac{\beta }{n} \), then the following statements are equivalent:\n\n(i) \( {\mathcal{H... | For the case \( \Omega \equiv 1 \), Zhou et al. \( {}^{\left\lbrack {16}\right\rbrack } \) proved a similar result. | No |
Theorem 1.1 Assume \( {\left( 2q\right) }^{\prime } \leq p < \infty, a\left( x\right) \in \operatorname{BMO}\left( {\mathbb{R}}^{n}\right) \), and \( V\left( x\right) \in {\operatorname{RH}}_{q}\left( {\mathbb{R}}^{n}\right), q \geq \frac{n}{2} \) . Let \( w\left( {x, y}\right) = \frac{{c}_{k}}{{\left( 1 + \frac{\left|... | Proof Since the kernel is positive, we can also assume \( f \geq 0 \) . Also, we may assume \( q > \frac{n}{2} \) because of the property \( {\mathrm{{RH}}}_{q} \Rightarrow {\mathrm{{RH}}}_{q + \varepsilon } \) for some \( \varepsilon > 0 \) . According to proof of [10, Theorem 5.10], we have the following pointwise bo... | Yes |
Corollary 1.1 Let \( {\left( 2q\right) }^{\prime } \leq p < \infty \) and \( B \) an open ball of \( {\mathbb{R}}^{n} \) . Suppose \( V\left( x\right) \in {\mathrm{{RH}}}_{q}\left( B\right) \) , \( q \geq \frac{n}{2}, a\left( x\right) \in \operatorname{BMO}\left( B\right) \) . If \( w\left( {x, y}\right) \) is a real f... | \[ \parallel {S}_{k}^{ * }\left( f\right) {\parallel }_{{L}^{p}\left( B\right) } \leq C\parallel f{\parallel }_{{L}^{p}\left( B\right) },\;\parallel {S}_{k, a}^{ * }\left( f\right) {\parallel }_{{L}^{p}\left( B\right) } \leq C\parallel a{\parallel }_{\mathrm{{BMO}}}\parallel f{\parallel }_{{L}^{p}\left( B\right) }.\] | Yes |
Lemma 2.2 Assume that (0.2)-(0.4) hold, \( V\left( x\right) \in {\mathrm{{RH}}}_{q}\left( {B}_{r}\right), q \geq \frac{n}{2} \) and \( {a}_{ij} \) are smooth for \( i, j = 1,2,\cdots, n \) . Then, for any \( p \in (1,{2q}\rbrack \), there exists a constant \( C > 0 \) such that for any \( {B}_{r} \subset \subset \Omega... | Proof \( {\text{ Denote }}_{i}{\Gamma }_{i}\left( {x, t}\right) = \frac{\partial }{\partial {t}_{i}}\Gamma \left( {x, t}\right) , \) we might as well assume \( u \in {C}_{0}^{\infty }\left( {{B}_{r}\left( {x}_{0}\right) }\right) .\)\n\nFor \( {\mathcal{L}}_{0}u = \left( {{\mathcal{L}}_{0} - \mathcal{L}}\right) u + \mat... | Yes |
Proposition 1.3 \( \;\partial {p}_{\mathcal{I}}\left( e\right) \subset \partial \parallel e{\parallel }_{0} \subset {S}_{{\ell }_{\infty }^{ * }}^{ + } \) . | Proof Since \( \mathcal{I} \) is a proper statistical nontrivial ideal, \( {c}_{0} \subset {X}_{\mathcal{I}} \) . This entails that \( {p}_{\mathcal{I}} \leq \parallel \cdot {\parallel }_{0} \) on \( {\ell }_{\infty } \) . According to [1] (see, for instance, Lemma 2.9), \( {p}_{\mathcal{I}}\left( e\right) = \operatorn... | Yes |
Proposition 1.4 Suppose that \( {x}^{ * } \in \operatorname{ext}\partial {p}_{\mathcal{I}}\left( e\right) \) . Then \( {\mathcal{F}}_{{x}^{ * }} \) is a free ultrafilter which contains \( {\mathcal{F}}_{\mathcal{I}} \) . | Proof According to Proposition 1.2 and Theorem 1.1, it is easy to check that \( {\mathcal{F}}_{{x}^{ * }} \) is a free ultrafilter. For any \( A \in \mathcal{I} \), by Propositions 1.1 and \( {1.3},0 \leq \left\langle {{x}^{ * },{\chi }_{A}}\right\rangle \leq {p}_{\mathcal{I}}\left( {\chi }_{A}\right) = 0 \) . This ent... | Yes |
Theorem 1.2 Let \( \mathcal{I} \subset {2}^{\mathbb{N}} \) be a proper statistical nontrivial ideal, \( \left( {x}_{n}\right) \subset X, x \in X \) . Then \( \left( {x}_{n}\right) \) is \( \mathcal{I} \) -almost convergent to \( x \) if and only if there is a subsequence \( \left( {x}_{{n}_{k}}\right) \) of \( \left( {... | Proof Necessity. Assume that \( \left( {x}_{n}\right) \) is \( \mathcal{I} \) -almost convergent to \( x \) . Then there exists a strictly increasing subset \( G = \left\{ {n}_{k}\right\} \) of \( \mathbb{N} \) such that \( {x}_{{n}_{k}} \rightarrow x \) as \( k \rightarrow \infty \) and that \( \left\langle {{x}^{ * }... | Yes |
Lemma 2.1 Let \( X \) be a topological space. Then the following are equivalent:\n\n(1) \( A \) is s-sequentially open;\n\n(2) \( X \smallsetminus A \) is s-sequentially closed. | Proof \( \;\left( 1\right) \Rightarrow \left( 2\right) \; \) Suppose that a subset \( A \) of \( X \) is s-sequentially open. For any sequence \( \left( {x}_{n}\right) \subset X \smallsetminus A \) with \( {x}_{n}\overset{\mathrm{s}}{ \rightarrow }x \), then we claim that \( x \in X \smallsetminus A \) . If not, suppos... | Yes |
Theorem 2.1 Let \( X \) be a topological space. Then the following are equivalent:\n\n(1) \( X \) is an s-sequential space;\n\n(2) Every s-sequentially open subset of \( X \) is open;\n\n(3) Every s-sequentially closed subset of \( X \) is closed. | Proof Obviously, \( \left( 2\right) \Leftrightarrow \left( 3\right) \) by Lemma 2.1. Therefore, it suffices to show that \( \left( 1\right) \Rightarrow \left( 3\right) \) and \( \left( 3\right) \Rightarrow \left( 1\right) \) .\n\n(1) \( \Rightarrow \) (3) Assume that \( A \) is an s-sequentially closed subset of \( X \... | Yes |
There exists an s-sequential space which is not a sequential space. | Proof Let \( S = \left\{ {{x}_{n} : n \in \mathbb{N}}\right\} \) be a sequence such that \( {x}_{n} \neq {x}_{m} \) if \( n \neq m \) . Take \( x \notin S \) and put \( X = S \cup \{ x\} \) . The topology on \( X \) is defined in the following way:\n\n(1) Each point \( {x}_{n} \) is isolated;\n\n(2) Each open neighborh... | Yes |
Theorem 2.2 Let \( X \) be an s-sequential space and \( f : X \rightarrow Y \), where \( Y \) is an arbitrary topological space. Then \( f \) is continuous if and only if it preserves limits of statistical sequences, i.e., whenever \( {\left( {x}_{n}\right) }_{n \in \mathbb{N}} \) is a sequence statistically converging... | Proof Assume that \( f \) is continuous, and \( {\left( {x}_{n}\right) }_{n \in \mathbb{N}} \) is a sequence in \( X \) such that \( {x}_{n}\overset{\mathrm{s}}{ \rightarrow }x \) in \( X \) . Let \( U \) be an arbitrary open neighborhood of \( f\left( x\right) \) in \( Y \) . Since \( f \) is continuous, \( {f}^{-1}\l... | Yes |
Theorem 2.3 Each statistically sequentially open (closed) subspace of an s-sequential space is s-sequential. | Proof Let \( X \) be an s-sequential space. Suppose that \( Y \) is a statistically sequentially open subset of \( X \) . Then it follows from Theorem 2.1 that \( Y \) is open in \( X \) . Next we shall show that \( Y \) is s-sequential. By Theorem 2.1, it suffices to show that each s-sequentially open subset in \( Y \... | Yes |
Theorem 2.4 Let \( X \) be an s-sequential space, and \( f \) be a quotient mapping from \( X \) onto a space \( Y \) . Then \( Y \) is an s-sequential space. | Proof By Theorem 2.1, we shall show that each s-sequentially open set in \( Y \) is open. Assume that \( U \) is s-sequentially open in \( Y \) . Since \( f \) is a quotient mapping and \( X \) is an s-sequential space, it suffices to show that \( {f}^{-1}\left( U\right) \) is s-sequentially open in \( X \) . Take an a... | Yes |
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