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Corollary 3.8 Let \( \left( {L,\left\lbrack {,,}\right\rbrack }\right) \) be a \( \delta \) Jordan-Lie triple system. \( d \) is a invertible 0 -order derivation of weight \( \lambda \) on \( L \), then \( \left( {L,{\left\lbrack ,,\right\rbrack }_{{d}^{-1}}}\right) \) with \( \left\lbrack {,,}\right\rbrack \) defined ... | Proof By Theorem 3.3, \( {d}^{-1} \) is a Rota-Baxter operator of weight \( \lambda \) on \( \left( {L,\left\lbrack {,,}\right\rbrack }\right) \) . And by Theorem 3.5, \( \left( {L,{\left\lbrack ,,\right\rbrack }_{{d}^{-1}}}\right) \) is a \( \delta \) Jordan-Lie triple system. From Corollary 3.7, \( d \) is a 0-order ... | Yes |
Theorem 3.9 Let \( \left( {L,\left\lbrack {,,}\right\rbrack, p}\right) \) be a Rota-Baxter \( \delta \) Jordan-Lie triple system of weight \( \lambda \) and \( A \) be a linear automorphism on \( \left( {L,\left\lbrack {,,}\right\rbrack, p}\right) \) . If \( A \) and \( p \) are commutative, then \( A \) is also a line... | Proof We need to certify \( A\left( {\left\lbrack x, y, z\right\rbrack }_{p}\right) = {\left\lbrack A\left( x\right), A\left( y\right), A\left( z\right) \right\rbrack }_{p} \), for all \( x, y, z \in L \) . From the definition of \( \left( {L,{\left\lbrack ,,\right\rbrack }_{p}, p}\right) \), we have\n\n\[ A\left( {\le... | Yes |
Corollary 3.10 Let \( \left( {L,\left\lbrack {,,}\right\rbrack, p}\right) \) be a Rota-Baxter \( \delta \) Jordan-Lie triple system of weight \( \lambda \) . Let \( D \) be a k-order Jordan triple \( \theta \) -derivation of weight \( \lambda \) on \( \left( {L,\left\lbrack {,,}\right\rbrack, p}\right) \) and \( A \) b... | Proof Since \( D \) is a k-order Jordan triple \( \theta \) -derivation of weight \( \lambda \) on \( \left( {L,\left\lbrack {,,}\right\rbrack, p}\right) \) , and from Corollary 2.15, we have that \( D \) is a k-order \( \theta \) -derivation of weight \( \lambda \) on \( \left( {L,\left\lbrack {,,}\right\rbrack, p}\ri... | Yes |
Theorem 3.11 Let \( \left( {L,\left\lbrack {\cdot , \cdot }\right\rbrack }\right) \) be a \( \delta \) Jordan-Lie algebra and \( D \) be a 0-order derivation of weight \( \lambda \) on it. Then \( D \) is also a 0 -order derivation of weight \( \lambda \) on \( \delta \) Jordan-Lie triple system \( \left( {L,\left\lbra... | Proof Suppose \( D \) is a 0-order derivation of weight \( \lambda \) on \( \left( {L,\left\lbrack {\cdot , \cdot }\right\rbrack }\right) \), then\n\n\[ D\left( \left\lbrack {x, y, z}\right\rbrack \right) = D\left( \left\lbrack {\left\lbrack {x, y}\right\rbrack, z}\right\rbrack \right) )\]\n\n\[ = \left\lbrack {D\left(... | Yes |
Corollary 3.12 Let \( \left( {L,\left\lbrack {\cdot , \cdot }\right\rbrack, p}\right) \) be a Rota-Baxter \( \delta \) Jordan-Lie algebra of weight \( \lambda \) . Then \( \left( {L,\left\lbrack {,,}\right\rbrack, p}\right) \) be a Rota-Baxter \( \delta \) Jordan-Lie triple system of weight \( \lambda \), where we assu... | Proof It just need to prove that \( p \) is a Rota-Baxter operator of weight \( \lambda \) on \( \left( {L,\left\lbrack {,,}\right\rbrack, p}\right) \) . By Remark \( {3.4},{p}^{-1} \) is a 0 -order derivation of weight \( \lambda \) on \( \left( {L,\left\lbrack {\cdot , \cdot }\right\rbrack, p}\right) \) . From Theore... | Yes |
Corollary 1.1 \( v \) is the pressure, then for any \( {x}_{1},{x}_{2} \in B\left( {\mathcal{O}, R/6}\right) \) and any \( \alpha > 1 \) , | \[ \frac{v\left( {{x}_{2},{t}_{2}}\right) }{v\left( {{x}_{1},{t}_{1}}\right) } \geq {\left( \frac{{t}_{1}}{{t}_{2}}\right) }^{a\alpha }\exp \left( {-\frac{\alpha {\int }_{{t}_{1}}^{{t}_{2}}{\left| {\gamma }^{\prime }\left( s\right) \right| }_{s}^{2}{ds}}{4{v}_{\min }^{\frac{R}{2}, T}}}\right) \times \exp \left\lbrack {... | Yes |
Lemma 2.1\n\n\\[ \n\\frac{\\partial }{\\partial t}{\\Delta v} = \\Delta \\frac{\\partial }{\\partial t}v - 2\\left\\langle {h,{\\nabla }^{2}v}\\right\\rangle - 2\\left\\langle {\\operatorname{div}h - \\frac{1}{2}\\nabla \\left( {{\\operatorname{tr}}_{g}h}}\\right) ,\\nabla v}\\right\\rangle .\n\\] | Proof Recall that \\( \\frac{\\partial }{\\partial t}{\\Gamma }_{ij}^{k} = {g}^{kl}\\left( {{\\nabla }_{i}{h}_{jl} + {\\nabla }_{j}{h}_{il} - {\\nabla }_{l}{h}_{ij}}\\right) \\) . Thus,\n\n\\[ \n\\frac{\\partial }{\\partial t}{\\Delta v} = \\frac{\\partial }{\\partial t}\\left\\{ {{g}^{ij}\\left( {\\frac{{\\partial }^{... | Yes |
Lemma 2.2 Let \( g\left( t\right) \) be a solution to the geometric flow on a Riemannian manifold \( {M}^{n}\left( {n \geq 2}\right) \) for \( \mathrm{t} \) in some time interval \( \left\lbrack {0, T}\right\rbrack \) . Let \( M \) be complete under the initial metric \( g\left( 0\right) \) . Let \( \mathrm{u} \) be a ... | Proof Calculate directly by using the Lemma 2.1. | No |
Lemma 2.3 Suppose that \( u \) is a positive solution to (2.13). Then\n\n\[ L\left( \frac{{v}_{t}}{v}\right) = \left( {m - 1}\right) \frac{{v}_{t}}{v}{\Delta }_{f}v + \frac{{v}_{t}}{v}\frac{{\left| \nabla v\right| }^{2}}{v} + {2m}\nabla v\nabla \left( \frac{{v}_{t}}{v}\right) + 2\left( {m - 1}\right) {h}^{ij}{v}_{i}{f}... | Proof Direct calculation shows that\n\n\[ \nabla \left( \frac{{v}_{t}}{v}\right) = \frac{\nabla {v}_{t}}{v} - \frac{{v}_{t}\nabla v}{{v}^{2}} \]\n\n(2.18)\n\nTherefore, we get\n\n\[ \Delta \left( \frac{{v}_{t}}{v}\right) = \frac{1}{v}\Delta {v}_{t} - \frac{{v}_{t}}{{v}^{2}}{\Delta v} - \frac{2}{{v}^{2}}\nabla v\nabla {... | Yes |
Lemma 2.4 The function \( F \) satisfies the following equation:\n\n\[ L\left( F\right) \leq - \frac{2\left( {m - 1}\right) }{N + n}{\left| {\Delta }_{f}v\right| }^{2} - 2\left( {m - 1}\right) {\operatorname{Ric}}_{f}^{N}\langle \nabla v,\nabla v\rangle + {2m}\nabla v\nabla F - {\left( \left( m - 1\right) {\Delta }_{f}... | Proof For the reader's convenience, we give the details of the proof of Lemma 2.4. By (2.16) and (2.17), we have\n\n\[ L\left( F\right) = L\left( \frac{{\left| \nabla v\right| }^{2}}{v}\right) - {\alpha L}\left( \frac{{v}_{t}}{v}\right) - {\alpha }^{\prime }\frac{{v}_{t}}{v} \]\n\n\[ \leq 2\left( {m - 1}\right) \frac{{... | Yes |
Lemma 2.6 \( \left\lbrack {5,{14},{15}}\right\rbrack \) Let \( \varphi \left( t\right) \) be a nonnegative and nonincreasing function on \( \left\lbrack {{k}_{0}, + \infty }\right) \) satisfying \( \varphi \left( h\right) \leq \frac{c}{{\left( h - k\right) }^{\alpha }}{\left\lbrack \varphi \left( k\right) \right\rbrack... | \[ \varphi \left( h\right) \leq \left\lbrack {{c}^{\frac{1}{1 - \beta }} + {\left( 2{k}_{0}\right) }^{\frac{\alpha }{1 - \beta }}\varphi \left( {k}_{0}\right) }\right\rbrack {2}^{\frac{\alpha }{{\left( 1 - \beta \right) }^{2}}}{\left( \frac{1}{h}\right) }^{\frac{\alpha }{1 - \beta }}; \] (2.3) if \( \beta = 1 \), then ... | Yes |
Lemma 3.1 Let \( C \) be a semiperfect coalgebra and \( M \) be a \( C \) -bicomodule. Then the trivial extension of \( C \) by \( M \) is also semiperfect. | Proof By the assumption, \( C \) is a semiperfect coalgebra that is the category \( {\mathcal{M}}^{C} \) has enough projectives. By [1, Proposition 1.11 and Proposition 1.13], we have the categories \( {\mathcal{M}}^{C} \ltimes {h}_{c}\left( {M, - }\right) ,\left( {-{▱}_{c}M}\right) \rtimes {\mathcal{M}}^{C} \) and \( ... | Yes |
Lemma 3.2 Let \( C \) be a semiperfect coalgebra, \( M \) be a \( C \) -bicomodule and \( \Gamma = C \ltimes M \) be a trivial extension of \( C \) by \( M \) . Let \( X \in {\mathcal{M}}^{C} \) . Then \[ i{d}_{\Gamma }Z\left( X\right) \leq 1 + \max \left\{ {i{d}_{\Gamma }Z\left( {{\Omega }_{c}^{-1}\left( X\right) }\ri... | Proof Let \( \alpha : X \rightarrow E \) be an injective envelope of \( X \) with cokernel \( {\Omega }_{c}^{-1}\left( X\right) \) . Then we have a short exact sequence of \( \Gamma \) -comodules:  where the middle term ... | Yes |
Proposition 3.3 Let \( C \) be a semiperfect coalgebra, \( M \) be a \( C \) -bicomodule and \( \Gamma = \) \( C \ltimes M \) be a trivial extension of \( C \) by \( M \) . Then\n\n\[ \n{gl} \cdot \dim \Gamma \leq {gl} \cdot \dim C + i{d}_{\Gamma }Z\left( M\right) + 1.\n\] | Proof Let \( X \) be a right \( C \) -comodule. We will first prove that\n\n\[ \ni{d}_{\Gamma }Z\left( X\right) \leq i{d}_{c}X + i{d}_{\Gamma }Z\left( M\right) + 1.\n\]\n\nIf \( i{d}_{c}X = \infty \), then the result follows.\n\nAssume that \( i{d}_{c}X = n \) . Applying Lemma 3.2 first to \( Z\left( X\right) \) and th... | Yes |
Corollary 3.4 Let \( C \) be a semiperfect coalgebra, \( M \) be a \( C \) -bicomodule and \( \Gamma = C \ltimes M \) be a trivial extension of \( C \) by \( M \) . If \( i{d}_{c}M = i{d}_{\Gamma }Z\left( M\right) \), then\n\n\[ \n{gl} \cdot \dim \Gamma \leq 2 \cdot {gl} \cdot \dim C + 1 \n\] | Proof It follows from \( i{d}_{\Gamma }Z\left( M\right) = i{d}_{c}M \leq {gl} \) .dim \( C \) and Proposition 3.3. | No |
Lemma 4.1 Let \( C \) be a semiperfect coalgebra, \( M \) be a coflat left \( C \) -comodule and \( \Gamma = C \ltimes M \) . If \( X \in {\mathcal{M}}^{C} \) and \( 0 \rightarrow X \rightarrow {I}_{0} \rightarrow {I}_{1} \rightarrow {I}_{2} \rightarrow \cdots \) is an injective resolution of \( X \) in \( {\mathcal{M}... | Proof It follows from [15, Proposition 1] that \( {I}_{i}{▱}_{c}\Gamma \) is an injective \( \Gamma \) -comodule. | No |
Lemma 4.3 Let \( C \) be a semiperfect coalgebra, \( M \) be a coflat left \( C \) -comodule and \( \Gamma = C \ltimes M \) . Then the functor \( H : {\mathcal{M}}^{C} \rightarrow \left( {-{▱}_{c}M}\right) \rtimes {\mathcal{M}}^{C} \) is exact. | Proof Assume that there is an exact sequence \( 0 \rightarrow X \rightarrow Y \rightarrow Z \rightarrow 0 \) in \( {\mathcal{M}}^{C} \) . Applying the exact functor \( - {▱}_{c}\Gamma \) on the above sequence, we have that the exact sequence\n\n\[ 0 \rightarrow X{▱}_{c}\Gamma \rightarrow Y{▱}_{c}\Gamma \rightarrow Z{▱}... | Yes |
Lemma 4.4 Let \( C \) be a semiperfect coalgebra, \( M \) be a coflat left \( C \) -comodule and \( \Gamma = C \ltimes M \) . If the functor \( H : {\mathcal{M}}^{C} \rightarrow \left( {-{▱}_{c}M}\right) \rtimes {\mathcal{M}}^{C} \) is fully faithfull and \( X \) is a tilting right \( C \) -comodule, then\n\n(1) the se... | Proof (1) The proof follows from that \( H \) and \( U \) are exact functors. | No |
Theorem 4.5 Let \( C \) be a semiperfect coalgebra, \( M \) be a coflat left \( C \) -comodule and \( \Gamma = C \ltimes M \) . If the functor \( H : {\mathcal{M}}^{C} \rightarrow \left( {-{▱}_{c}M}\right) \rtimes {\mathcal{M}}^{C} \) is fully faithfull, then \( X \) is a tilting right \( C \) -comodule if and only if ... | Proof By Lemma 4.4, the sufficiency is easy to know.\n\nConversely, we assume that \( X \) is a right \( C \) -comodule and \( H\left( X\right) \) is a tilting right \( \Gamma \) - comodule. By the assumption, there exists an exact sequence \( 0 \rightarrow H\left( {X}_{2}\right) \rightarrow H\left( {X}_{1}\right) \rig... | Yes |
Lemma 4.6 Let \( C \) be a semiperfect coalgebra, \( M \) be a coflat left \( C \) -comodule and \( \Gamma = C \ltimes M \) . Then\n\n(1) If \( L \in \operatorname{Cogen}X \), then \( L{▱}_{c}\Gamma \in \operatorname{Cogen}\left( {X{▱}_{c}\Gamma }\right) \) ;\n\n(2) If \( L \in \operatorname{Cogen}X \), then \( H\left(... | Proof (1) Since \( L \in \operatorname{Cogen}X \), there exists an index set \( \Lambda \) such that the sequence \( 0 \rightarrow L \rightarrow {X}^{\Lambda } \) is exact. Then we get an exact sequence \( 0 \rightarrow L{▱}_{c}\Gamma \rightarrow {X}^{\Lambda }{▱}_{c}\Gamma \) since the functor \( - {▱}_{c}\Gamma \) is... | Yes |
(1) \( X \) is a tilting injective right \( C \) -comodule if and only if \( H\left( X\right) \) is a tilting injective right \( \Gamma \) -comodule; | Proof (1) \( X \) is a tilting injective right \( C \) -comodule if and only if \( {\operatorname{Ext}}_{c}^{1}\left( {L, X}\right) = 0 \) for any \( L \in \operatorname{Cogen}X \) (see [6, Proposition 3.2]). By Lemma 4.6 and Lemma 4.4(2), for any \( L \in \operatorname{Cogen}X,{\operatorname{Ext}}_{c}^{1}\left( {L, X}... | Yes |
Proposition 5.1 Let \( C \) be a semiperfect coalgebra, \( M \) be a coflat left \( C \) -comodule and \( \Gamma = C \ltimes M \) . If \( X \) is a Gorenstein injective right \( C \) -comodule, then \( X{▱}_{c}\Gamma \) is a Gorenstein injective right \( \Gamma \) -comodule. | Proof By the assumption, \( X \) is a Gorenstein injective right \( C \) -comodule, then there exists an exact sequence\n\n\[ \mathcal{E} \equiv \cdots \rightarrow {E}^{-2} \rightarrow {E}^{-1} \rightarrow {E}^{0} \rightarrow {E}^{1} \rightarrow \cdots \]\n\nof injective right \( C \) -comodules with \( X \cong \ker \l... | Yes |
Corollary 5.3 Let \( C \) be a semiperfect coalgebra, \( M \) be a coflat left \( C \) -comodule and \( \Gamma = C \ltimes M \) . Then \( X{▱}_{c}\Gamma \) is a Gorenstein injective right \( \Gamma \) -comodule if and only if \( X{▱}_{c}\Gamma \) is a Gorenstein coflat right \( \Gamma \) -comodule. | Proof It follows from [13, Proposition 3.4] and Lemma 3.1. | No |
Proposition 2.5 Let \( R \) be a commutative local ring and \( A \in {M}_{2}\left( R\right) \) satisfies that \( A \notin {\left( {M}_{2}\left( R\right) \right) }^{\text{qnil }} \) and \( \det A \in J\left( R\right) \) . The following statements are equivalent.\n\n(1) \( A \) is strongly quasi-nil clean.\n\n(2) \( A \)... | Proof (1) \( \Rightarrow \) (2) Assume that \( A \) is strongly quasi-nil clean. It is clear that \( \operatorname{tr}A \in U\left( R\right) \) . By Lemma 2.4, we may let \( E \doteq {I}_{2} - {A}^{\pi } = \left( \begin{matrix} a & b \\ c & 1 - a \end{matrix}\right) \in {M}_{2}\left( R\right) \), where \( {bc} = a - {a... | Yes |
Corollary 2.8 Let \( R \) be a commutative local ring and \( A \in {M}_{2}\left( R\right) \) . If \( A \) is strongly quasi-nil clean, then one of the following statements holds:\n\n(1) \( A \) is quasipolar and \( {A}^{\pi } = 0 \) or \( {A}^{\pi } = {I}_{2} \) .\n\n(2) \( A \) is quasipolar and \( {A}^{\pi } \) is si... | By Theorem 2.6, we can figure out all strongly quasi-nil clean elements of a ring \( {M}_{2}\left( {\mathbb{Z}}_{2}\right) \) . | No |
Corollary 2.10 Let \( R \) be a commutative local domain with \( J\left( R\right) = {2R} \) and \( R/J\left( R\right) \cong \) \( {\mathbb{Z}}_{2} \) . The following are equivalent for \( A \in {M}_{2}\left( R\right) \):\n\n(1) \( A \) is strongly quasi-nil clean.\n\n(2) \( A \in {\left( {M}_{2}\left( R\right) \right) ... | Proof If \( A \) and \( {I}_{2} - A \) are all not in \( {\left( {M}_{2}\left( R\right) \right) }^{\text{qnil }} \), then \( \operatorname{tr}A \in U\left( R\right) \) and \( \det A \in J\left( R\right) \) . Since \( R/J\left( R\right) \cong {\mathbb{Z}}_{2}, U\left( R\right) = - 1 + J\left( R\right) \) . By [8, Theore... | Yes |
Corollary 2.11 Let \( R \) be a commutative local ring and \( A = \left( \begin{matrix} {a}_{11} & {a}_{12} \\ 0 & {a}_{22} \end{matrix}\right) \in {M}_{2}\left( R\right) \) . The following statements are equivalent:\n\n(1) \( A \) is strongly quasi-nil clean.\n\n(2) One of the following holds:\n\n(i) \( {a}_{11},{a}_{... | Let \( R \) be a commutative local ring. By Corollary 2.11, the matrix \( A = \left( \begin{matrix} {j}_{1} & x \\ 0 & {j}_{2} \end{matrix}\right) \in {M}_{2}\left( R\right) \) where \( {j}_{1},{j}_{2} \in J\left( R\right) \) and \( x \in R \) is strongly quasi-nil clean. | No |
proposition 2.13 Let \( R \) be a ring. The following statements are equivalent:\n\n(1) \( R \) is local and strongly quasi-nil clean.\n\n(2) \( R \) is perfectly J-clean with the only idempotents 0 and 1 . | Proof (1) \( \Rightarrow \) (2) Clearly,0 and 1 are the only idempotents of \( R \) . Since \( R \) is local, \( R \) is quasipolar and \( {R}^{\text{qnil }} = J\left( R\right) \) . Whence \( R \) is perfectly J-clean.\n\n\( \left( 2\right) \Rightarrow \left( 1\right) \) Since \( R \) is perfectly J-clean, \( R/J\left(... | Yes |
Proposition 3.1 If the family of conditional distributions \( {\rho }_{x} \) at \( x \in X \) is Lipschitz-s for some \( s > 0 \), then for any \( 0 \leq \nu < \mu \) . we have\n\n\[ \n{\begin{Vmatrix}{f}_{\lambda }^{\mu } - {f}_{\lambda }^{\nu }\end{Vmatrix}}_{K} \leq \frac{{C}_{\rho }\kappa {\left| \mu - \nu \right| ... | In particular, when \( \lambda > 0 \) and \( {\epsilon }_{t} = {\epsilon }_{1}{t}^{-\beta } \) with \( \beta > 0,\epsilon \geq 0 \), there holds\n\n\[ \n{\begin{Vmatrix}{f}_{\lambda }^{{\epsilon }_{t - 1}} - {f}_{\lambda }^{{\epsilon }_{t}}\end{Vmatrix}}_{K} \leq \frac{{C}_{\rho }\kappa {\epsilon }_{1}^{s}{\beta }^{s}{... | Yes |
Lemma 3.2 Let \( h, g \in {C}^{s}\left( \mathcal{X}\right) \) . If the family of conditional distributions \( {\left\{ {\rho }_{x}\right\} }_{x \in X} \) is Lipschitz- \( s \), then we have\n\n\[ \left| {{\int }_{\mathcal{Z}}{\psi }_{\tau }^{{\epsilon }_{t}}\left( {y - h\left( x\right) }\right) - {\psi }_{\tau }^{{\eps... | The proof of Lemma 3.2 can be found in [5]. | No |
Proposition 3.5 Suppose (2.5), (2.7), (2.8) and (2.9) hold. Take the parameters \( {\eta }_{t},{\lambda }_{t},{\epsilon }_{t} \) as the same form in Theorem 2.6, then we have\n\n\[ \n{\mathbb{E}}_{{z}_{1},\ldots ,{z}_{T}}{\begin{Vmatrix}{f}_{T + 1} - {f}_{{\lambda }_{T}}^{{\epsilon }_{t}}\end{Vmatrix}}_{K}^{2} \leq {C}... | To prove this proposition, we need the following lemma, whose proof can be found in [12].\n\nLemma 3.6 If \( K \) sa | No |
Lemma 3.7 Let \( 0 < \varphi \leq \infty \) and \( \xi > 1 \) . Denote \( r = {\varphi \xi }/\left( {\varphi + 1}\right) > 0 \) . Assume the measure \( \rho \) has a \( \tau \) - quantile of \( \varphi \) -average type \( \xi \), then for any measurable function on \( X \), we have\n\n\[ \n{\begin{Vmatrix}f - {f}_{\rho... | Proof of Theorem 2.8 It is trivial to get the desired conclusion by Lemma 3.7 and Theorem 2.8. | No |
Lemma 2.2 (see [2]) For any \( f, g \in {\mathcal{P}}_{T}\left( {L, M}\right), x, y \in \mathbb{R} \), the following inequalities hold for every positive integer \( n \) . | \[ \left| {{f}^{n}\left( y\right) - {f}^{n}\left( x\right) }\right| \leq {M}^{n}\left| {y - x}\right| \] \[ \begin{Vmatrix}{{f}^{n} - {g}^{n}}\end{Vmatrix} \leq \mathop{\sum }\limits_{{j = 0}}^{{n - 1}}{M}^{j}\parallel f - g\parallel . \] | Yes |
Lemma 2.3 Suppose \( {\lambda }_{i} \in {\mathcal{P}}_{T}\left( {{L}_{i},{M}_{i}}\right) ,\left| {{\lambda }_{1}\left( x\right) }\right| \geq {k}_{1} > 0,\forall x \in \left\lbrack {0, T}\right\rbrack \), then operator \( A \) is continuous and compact on \( {\mathcal{P}}_{T}\left( {L, M}\right) \) . | Proof For any \( f, g \in {\mathcal{P}}_{T}\left( {L, M}\right), x \in \mathbb{R} \), by (2.3) we get\n\n\[ \left| {\left( {Af}\right) \left( x\right) - \left( {Ag}\right) \left( x\right) }\right| \leq \frac{1}{\left| {\lambda }_{1}\left( x\right) \right| }\mathop{\sum }\limits_{{i = 2}}^{n}\left| {{\lambda }_{i}\left(... | Yes |
Theorem 3.1 (i) Suppose that \( A \) is defined by (2.1) and\n\n\[ \alpha \mathrel{\text{:=}} \frac{1}{{k}_{1}}\mathop{\sum }\limits_{{i = 2}}^{n}\mathop{\sum }\limits_{{j = 0}}^{{i - 1}}{L}_{i}{M}^{j} < 1. \]\n\n(3.1)\n\nThen \( A \) is contractive with contraction constant \( \alpha < 1 \) . It has at most one fixed ... | Proof (i) According to (2.4), \( \parallel {Af} - {Ag}\parallel \leq \alpha \parallel f - g\parallel \) and so \( \alpha \) is a contraction constant for \( A \) . Let \( f, g \in {\mathcal{P}}_{T}\left( {L, M}\right) \) be fixed points of \( A \) . Then \( \parallel f - g\parallel = \parallel {Af} - {Ag}\parallel \leq... | Yes |
Consider the equation\n\n\\[ \n\\left( {4 + {\\cos }^{2}\\left( x\\right) }\\right) f\\left( x\\right) + f\\left( {f\\left( x\\right) }\\right) = \\sin \\left( x\\right) ,\\;\\forall x \\in \\mathbb{R}. \n\\] | It corresponds to the case of \\( {\\lambda }_{1}\\left( x\\right) = 4 + {\\cos }^{2}\\left( x\\right) ,{\\lambda }_{2}\\left( x\\right) = 1, F\\left( x\\right) = \\sin \\left( x\\right) \\) . Taking \\( {k}_{1} = \\) 4, \\( {L}_{1} = 5, L = M = {L}^{\\prime } = {M}^{\\prime } = {M}_{1} = {L}_{2} = {M}_{2} = 1 \\) . A ... | Yes |
Example 4.2 Consider the equation\n\n\\[ \n\\left( {4 + {\\cos }^{2}\\left( x\\right) }\\right) f\\left( x\\right) + {\\delta f}\\left( {f\\left( x\\right) }\\right) = \\sin \\left( x\\right) ,\\;\\forall x \\in \\mathbb{R}. \n\\]\n\n(4.2) | As Example 4.1, \\( {\\lambda }_{1}\\left( x\\right) = 4 + {\\cos }^{2}\\left( x\\right) ,{\\lambda }_{2}\\left( x\\right) = \\delta, F\\left( x\\right) = \\sin \\left( x\\right) \\), where \\( \\delta \\) is a parameter. So taking \\( {k}_{1} = 4,{L}_{1} = 5,{L}_{2} = {M}_{2} = \\delta, L = M = {L}^{\\prime } = {M}^{\... | Yes |
Theorem 1.1 For any \( \delta > 0 \), we have\n\n(i) There exists a \( {P}^{ * } > 0 \) such that for any \( P \geq {P}^{ * } \) ,(1.1) has no solution in \( {H}_{0}^{1}\left( \Omega \right) \).\n\n(ii) For any \( 0 < P < {P}^{ * } \), there exists a critical constant \( {\lambda }_{P}^{ * } > 0 \) such that for \( 0 <... | Since our equation (1.1) has both the corner-corrected term and external pressure term, we will use the upper and lower solution method to find the first solution. After showing that the first solution is exactly the local minimum of the corresponding energy functional of (1.1), we want to find the second solution with... | No |
Lemma 2.1 For any solution \( v \) of (2.1), there exists a constant \( {m}_{0} > 0 \) such that \( {\left| v\right| }_{{L}^{\infty }} \leq {m}_{0} \) | Proof We prove this by Proposition 2.5. Let \( L = - \Delta, t = 0, f\left( {x, u}\right) = g\left( v\right) + h\left( v\right) \) . We only need to verify whether the conditions in Proposition 2.5 are satisfied. In fact, we have\n\[ \mathop{\lim }\limits_{{v \rightarrow \infty }}\frac{g\left( v\right) + h\left( v\righ... | Yes |
Lemma 3.1 For any \( \delta > 0 \), if (1.1) has a solution for \( \lambda = {\lambda }_{1} \) with any fixed \( P > 0 \) , then (1.1) has a solution for \( 0 < {\lambda }_{2} < {\lambda }_{1} \) . | Proof Suppose that (1.1) has a solution \( {u}_{{\lambda }_{1}} \) for \( \lambda = {\lambda }_{1} \), and then for any \( 0 < {\lambda }_{2} < {\lambda }_{1} \)\n\nwe have\n\[ \n- \Delta {u}_{{\lambda }_{1}} = \frac{{\lambda }_{1}\left( {1 + \delta {\left| \nabla {u}_{{\lambda }_{1}}\right| }^{2}}\right) }{{\left( 1 -... | Yes |
Lemma 3.2 There exists a \( {\lambda }_{P} > 0 \) such that (1.1) has at least one solution when \( 0 < \lambda < {\lambda }_{P} \) for any fixed \( 0 < P < {P}^{ * } \) . | Proof For any fixed \( 0 < P < {P}^{ * } \), we choose \( \frac{P}{{P}^{ * }} < s < 1 \) such that \( \bar{u} = s{P}^{ * }\varphi \) . We can verify easily \( 0 < \bar{u} < 1 \) and \( \bar{u} \) satisfying\n\n\[ \n- \Delta \bar{u} = P + \frac{1 + \delta {\left| \nabla \bar{u}\right| }^{2}}{{\left( 1 - s\right) }^{p}}{... | Yes |
Lemma 3.3 \( {\lambda }_{P}^{ * } \) defined in (3.2) is bounded. | Proof Suppose that \( 0 < u < 1 \) is the solution of (1.1) for \( {\lambda }_{P} > 0 \) and any fixed \( 0 < P < {P}^{ * } \) . Multiplying (1.1) by \( \varphi \) and integrating on \( \Omega \) on both sides of equation (1.1), we have\n\n\[ \left| \Omega \right| \geq {\int }_{\Omega } - {\Delta u\varphi dx} = {\int }... | Yes |
For any \( \delta > 0,0 < P < {P}^{ * } \) and \( 0 < \lambda < {\lambda }_{P}^{ * } \), the equation (1.1) has at least one upper solution and then at least one solution in \( {H}_{0}^{1}\left( \Omega \right) \) . | According to Lemma 3.2 and Lemma 3.3, we know that when \( 0 < {\lambda }_{1} < {\lambda }_{2} < {\lambda }_{P}^{ * } \) , the equation (1.1) has at least one solution \( {u}_{{\lambda }_{1}, P} \) for \( \lambda = {\lambda }_{1} \) and \( {u}_{{\lambda }_{2}, P} \) for \( \lambda = {\lambda }_{2} \), respectively, for... | Yes |
Lemma 3.5 The solution \( v \) is the local minimum point of the functional \( I \) in \( {H}_{0}^{1}\left( \Omega \right) \) . | Proof We follow the idea in [22] and argue by contradiction. Suppose that \( v \) is not the local minimum point of \( I \) on \( {H}_{0}^{1}\left( \Omega \right) \), and then there exists a sequence \( \left\{ {v}_{n}\right\} \subset {H}_{0}^{1}\left( \Omega \right) \) such that \( \begin{Vmatrix}{{v}_{n} - v}\end{Vma... | No |
Lemma 3.6 The energy functional \( I \) in (2.2) has a mountain pass geometry. | Proof Firstly we take \( {x}_{0} \) in \( \Omega \) arbitrarily, and choose a proper \( R \) such that \( {B}_{R}\left( {x}_{0}\right) \subset \Omega \) . Let \( \psi \in {C}_{0}^{\infty }\left( \Omega \right) \) be a cut-off function satisfying\n\n\[ \left\{ \begin{array}{ll} 0 \leq \psi \leq 1, & \text{ in }\Omega , ... | Yes |
Lemma 3.8 There exists an interval \( J \subset \left\lbrack {\frac{1}{2},1}\right\rbrack \) such that the family of functionals \( {\left\{ {\widetilde{I}}_{\mu }\right\} }_{\mu \in J} \) has a mountain pass geometry. | Proof From Lemma 3.6, we know that \( v \) is the local minimum point of \( I \) and \( I\left( v\right) > \) \( I\left( {t\psi }\right) \) for \( t \) large enough. Here \( v \) is the solution of (2.1). By Lemma 2.1, we know \( {\left| v\right| }_{{L}^{\infty }} \leq {m}_{0} \) . This implies when \( \mu = 1 \), we h... | Yes |
Lemma 3.9 For almost every \( \mu \in \left\lbrack {{\mu }_{0},1}\right\rbrack \), the \( {\left( PS\right) }_{{c}_{\mu }} \) sequence \( \left\{ {v}_{\mu, n}\right\} \) of functional \( {\widetilde{I}}_{\mu } \) satisfies \( {\left( PS\right) }_{{c}_{\mu }} \) condition. | Proof Set \( A\left( u\right) = \frac{1}{2}{\int }_{\Omega }{\left| \nabla u\right| }^{2}{dx}, B\left( u\right) = {\int }_{\Omega }\widetilde{G}\left( u\right) + {\int }_{\Omega }\widetilde{H}\left( u\right) {dx} \) . We see \( A\left( u\right) \rightarrow \infty \) as \( \parallel u\parallel \rightarrow \infty \) in \... | Yes |
Lemma 4.2 The set \( {D}_{1} \) is not empty when \( \lambda \) and \( \mu \) are properly small. | Proof Let \( {B}_{R} \) be a ball of radius \( R \) centered at 0 such that \( \Omega \subset {B}_{R} \) . Denote \( {\beta }_{1} > 0 \) the first eigenvalue of Laplace operator in \( {B}_{R} \) with Dirichlet boundary condition and \( \phi \) the corresponding eigenfunction which satisfies \( 0 < \phi \leq 1 \) in \( ... | Yes |
Lemma 4.3 The set \( {D}_{1} \) is contained in a bounded region. | Proof If the system (1.2) has a solution \( \left( {u, v}\right) \), then\n\n\[ \left\{ \begin{array}{ll} {\beta }_{1}\left| \Omega \right| \geq {\int }_{\Omega } - {\Delta u\phi dx} = {\int }_{\Omega }\frac{\lambda \left( {1 + \delta {\left| \nabla u\right| }^{2}}\right) }{{\left( 1 - v\right) }^{p}}{\phi dx} \geq \la... | Yes |
Lemma 4.4 If the system (1.2) has a solution with the parameter pair \( \left( {\lambda ,\mu }\right) \in {D}_{1} \) , then \( \left( {{\lambda }^{\prime },{\mu }^{\prime }}\right) \) is still in \( {D}_{1} \) for any \( {\lambda }^{\prime } \leq \lambda ,{\mu }^{\prime } \leq \mu \) . | Proof It is easy to verify that the solution \( \left( {u, v}\right) \) with the parameter pair \( \left( {\lambda ,\mu }\right) \) is a upper solution of the system (1.2) with the pair \( \left( {{\lambda }^{\prime },{\mu }^{\prime }}\right) \) . Then by Lemma 4.1 we know the system (1.2) must have at least one soluti... | Yes |
Lemma 4.5 The curve \( \Gamma \left( \sigma \right) = \left( {{\lambda }^{ * }\left( \sigma \right) ,{\mu }^{ * }\left( \sigma \right) }\right) \) is continuous. | Proof We prove this by contradiction. Suppose that \( \Gamma \left( \sigma \right) \) is not continuous at some \( {\sigma }_{0} > 0 \) . Then there exists \( {\varepsilon }_{0} > 0 \) such that for any \( \eta > 0 \), when \( 0 < \left| {\sigma - {\sigma }_{0}}\right| < \eta \) we have \( \left| {\Gamma \left( \sigma ... | Yes |
Lemma 4.6 \( {\lambda }^{ * }\left( \sigma \right) \) is decreasing and \( {\mu }^{ * }\left( \sigma \right) \) is increasing with respect to \( \sigma \) . | Proof (i) We first show that \( {\lambda }^{ * }\left( \sigma \right) \) is decreasing. We argue by contradiction. Suppose that \( {\lambda }^{ * }\left( {\sigma }_{1}\right) < {\lambda }^{ * }\left( {\sigma }_{2}\right) \) for \( {\sigma }_{1} < {\sigma }_{2} \) . Then \( {\mu }^{ * }\left( {\sigma }_{1}\right) = {\si... | Yes |
Proposition 2.1 The DCC kernels defined by (2.15) and DOC kernels defined in (2.17) have the following relationships\n\n\[ \n{p}_{n - j}^{\left( n\right) } = \mathop{\sum }\limits_{{l = j}}^{n}{\theta }_{l - j}^{\left( l\right) },\;\forall 1 \leq j \leq n \n\]\n\n(2.19)\n\n\[ \n{\theta }_{n - j}^{\left( n\right) } = {p... | Proof Set \( {q}_{n - j}^{\left( n\right) } = \mathop{\sum }\limits_{{l = j}}^{n}{\theta }_{l - j}^{\left( l\right) }\left( {\forall 1 \leq j \leq n}\right) \) . Then from the definition (2.17), we have\n\n\[ \n\mathop{\sum }\limits_{{j = k}}^{n}{q}_{n - j}^{\left( n\right) }{b}_{j - k}^{\left( j\right) } = \mathop{\su... | Yes |
Lemma 2.4 ([11]) The DOC kernels \( {\theta }_{n - j}^{\left( n\right) } \) have the following properties:\n\n\[ \mathop{\sum }\limits_{{j = 1}}^{n}{\theta }_{n - j}^{\left( n\right) } = {\tau }_{n},\;\text{ for }n \geq 1 \] | \[ \mathop{\sum }\limits_{{k = 1}}^{n}\mathop{\sum }\limits_{{j = 1}}^{k}{\theta }_{k - j}^{\left( k\right) } = {t}_{n},\;\text{ for }n \geq 1. \] | Yes |
Lemma 2.5 ([11]) The DOC kernel \( {\theta }_{n - j}^{\left( n\right) } \) can be explicitly represented by | \[ {\theta }_{n - k}^{\left( n\right) } = \left\{ \begin{array}{l} \frac{{\tau }_{n}\left( {1 + {r}_{k}}\right) }{1 + 2{r}_{k}}\mathop{\prod }\limits_{{i = k + 1}}^{n}\frac{{r}_{i}}{1 + 2{r}_{i}},\;\text{ for }2 \leq k \leq n. \\ {\tau }_{n}\mathop{\prod }\limits_{{i = k + 1}}^{n}\frac{{r}_{i}}{1 + 2{r}_{i}},\;\text{ f... | Yes |
Proposition 2.2 Let \( \tau \) be the maximum time-step size and the time-step ratios satisfy \( 0 < {r}_{k} \leq {r}_{ * } \), where \( {r}_{ * } \) is any given positive constant. The DCC kernels \( {p}_{n - k}^{\left( n\right) } \) defined in (2.15) satisfy\n\n\[ \n{p}_{n - j}^{\left( n\right) } = \frac{1 + {r}_{j}}... | Proof It follows from (2.19) in Proposition 2.1 and Lemma 2.5 that, for \( 2 \leq j \leq n \) ,\n\n\[ \n{p}_{n - j}^{\left( n\right) } = \mathop{\sum }\limits_{{k = j}}^{n}{\theta }_{k - j}^{\left( k\right) } = \frac{1 + {r}_{j}}{1 + 2{r}_{j}}\mathop{\sum }\limits_{{k = j}}^{n}{\tau }_{k}\mathop{\prod }\limits_{{i = j ... | Yes |
Lemma 3.7 Assume \( \lambda > 0 \) and the sequences \( {\left\{ {v}_{j}\right\} }_{j = 1}^{N} \) and \( {\left\{ {\eta }_{j}\right\} }_{j = 0}^{N} \) are nonnegative.\n\nIf\n\n\[ \n{v}_{n} \leq \lambda \mathop{\sum }\limits_{{j = 1}}^{{n - 1}}{\tau }_{j}{v}_{j} + \mathop{\sum }\limits_{{j = 0}}^{n}{\eta }_{j},\;\text{... | The standard induction hypothesis can give the proof of lemma 3.7, which is omitted here. | No |
The truncation error \( {\eta }^{j} \mathrel{\text{:=}} {\mathcal{D}}_{2}u\left( {t}_{j}\right) - {u}_{t}\left( {t}_{j}\right) \left( {1 \leq j \leq N}\right) \) can be expressed by the following form\n\n\[ {\eta }^{j} = \mathop{\sum }\limits_{{l = 1}}^{j}{b}_{j - l}^{\left( j\right) }{G}^{l} + {R}^{j},\;1 \leq j \leq ... | Proof By using the Taylor's expansion (see [16]), one has\n\n\[ {\eta }^{j} = \frac{1}{2}{b}_{0}^{\left( j\right) }{G}^{j} + \frac{1}{2}{b}_{1}^{\left( j\right) }{G}^{j - 1} - \frac{1}{2}{b}_{1}^{\left( j\right) }{\tau }_{j - 1}{\int }_{{t}_{j - 1}}^{{t}_{j}}\left( {2\left( {t - {t}_{j - 1}}\right) + {\tau }_{j - 1}}\r... | Yes |
Theorem 3.10 Let \( u\left( {t, x}\right) \) be the exact solution to problem (1.1). If the condition A1 holds, then the discrete solution \( {u}^{n} \) to BDF2 scheme (1.4) has the second-order convergence in the \( {L}^{2} \) -norm. If \( \kappa > 0 \) and the maximum time-step size \( \tau < 1/\left( {4\kappa }\righ... | Proof From Theorem 3.8 that if \( \kappa > 0 \) and the maximum time step \( \tau \leq \frac{1}{4\kappa } \), it holds\n\n\[ \begin{Vmatrix}{u\left( {t}_{n}\right) - {u}^{n}}\end{Vmatrix} \leq 2\exp \left( {{4\kappa }{t}_{n - 1}}\right) \left( {\begin{Vmatrix}{u\left( 0\right) - {u}^{0}}\end{Vmatrix} + 2\mathop{\sum }\... | Yes |
Proposition 2.2 For any \( x \in {\mathbb{R}}^{n} \) and \( {\left\{ {\theta }_{x, t}\right\} }_{t \in \mathbb{R}} \subset \Theta \), it holds true that\n\n\[ \n{\theta }_{x, t} = {M}_{x, t}\left( {\mathbb{B}}^{n}\right) + x \rightarrow {\mathbb{R}}^{n}\text{ as }t \rightarrow - \infty .\n\] | Proof For any \( y \in {\mathbb{B}}^{n} \) and \( t < 0 \), by (2.2), we obtain\n\n\[ \n\left| {{M}_{x, t}^{-1}y}\right| \leq \begin{Vmatrix}{{M}_{x, t}^{-1}{M}_{x,0}}\end{Vmatrix}\begin{Vmatrix}{M}_{x,0}^{-1}\end{Vmatrix}\left| y\right| \leq {a}_{5}{2}^{{a}_{6}t}\begin{Vmatrix}{M}_{x,0}^{-1}\end{Vmatrix}\left| y\right... | Yes |
Proposition 2.1 Operators \( \mathcal{T} \) and \( {\mathcal{T}}^{\prime } \) satisfy the braided equation | \[ \left( {\mathcal{T} \otimes \iota }\right) \left( {\iota \otimes \mathcal{T}}\right) \left( {\mathcal{T} \otimes \iota }\right) = \left( {\iota \otimes \mathcal{T}}\right) \left( {\mathcal{T} \otimes \iota }\right) \left( {\iota \otimes \mathcal{T}}\right) ,\] \[ \left( {{\mathcal{T}}^{\prime } \otimes \iota }\right... | Yes |
Theorem 3.3 For a coFrobenius Hopf algebra \( H \) , \[{\widehat{M}}_{ \bowtie H\left( {\alpha ,\beta }\right) }\mathcal{M} \cong {}_{H}\mathcal{Y}{\mathcal{D}}^{H}\left( {\alpha ,\beta }\right) \] | Proof The correspondence easily follows from Proposition 3.1 and 3.2. Let \( f : M \rightarrow N \) be a morphism in \( {}_{H}\mathcal{Y}{\mathcal{D}}^{H}\left( {\alpha ,\beta }\right) \), i.e., \( f \) is a module and comodule map. Then in \( {\widehat{H}}_{1 \bowtie H\left( {\alpha ,\beta }\right) }\mathcal{M} \) , \... | Yes |
Corollary 3.4 Let \( H \) be a coFrobenius Hopf algebra and \( \alpha ,\beta ,\gamma \in \operatorname{Aut}\left( H\right) \), then\n\n\[{\widehat{H}}_{ \bowtie H\left( {{\alpha \beta },{\gamma \beta }}\right) }\mathcal{M} \cong {\widehat{H}}_{ \bowtie H\left( {\alpha ,\gamma }\right) }\mathcal{M}.\] | Proof It follows straightforwardly from the fact \( {}_{H}\mathcal{Y}{\mathcal{D}}^{H}\left( {{\alpha \beta },{\gamma \beta }}\right) \cong {}_{H}\mathcal{Y}{\mathcal{D}}^{H}\left( {\alpha ,\gamma }\right) \) . | Yes |
When \( \alpha = \beta = \iota \), then \( \widehat{H} \bowtie H\left( {\iota ,\iota }\right) = D\left( H\right) \) the quantum double of a coFrobenius Hopf algebra. Then we have the following result, which is the main result in [5], i.e., for a coFrobenius Hopf algebra \( H \) , | \[ {}_{H}\mathcal{Y}{\mathcal{D}}^{H} \cong {}_{\widehat{H} \bowtie H}\mathcal{M} \] | No |
Example 3.1 Let \( {E}_{1} = \mathbb{R}, K = \left\lbrack {0,1}\right\rbrack \) and \( P = \lbrack 0, + \infty ) \) be a cone, we define the mappings \( Q : {E}_{1} \rightarrow \mathbb{R} \) and \( \phi : {E}_{1} \rightarrow \mathbb{R} \) by \[ Q\left( x\right) = \sqrt{\left| x\right| } + 1\text{ and }\phi \left( x\rig... | We can check that the solution set of (3.1) is \( \left\lbrack {0,1}\right\rbrack \) . | No |
Theorem 3.2 Let \( {E}_{1} \) be a real Banach space and let \( K \) be a nonempty compact and convex subset of \( {E}_{1} \) . Let \( Q : {E}_{1} \rightarrow \mathbb{R} \) be a mapping satisfies \( Q\left( 0\right) = 0 \) . Assume that:\n\n(i) \( Q \) is concave, continuous on \( {E}_{1} \) ;\n\n(ii) \( \phi : {E}_{1}... | Proof We consider the mapping \( \varphi : K \times K \rightarrow R \) defined by\n\n\[ \varphi \left( {u, v}\right) \mathrel{\text{:=}} - Q\left( {v - u}\right) + \phi \left( u\right) - \phi \left( v\right) . \]\n\nOne can check that \( \varphi \left( {\cdot , \cdot }\right) \) satisfies conditions (i),(ii) and (iii) ... | No |
Example 3.2 Let \( {E}_{1} = \mathbb{R}, K = \left\lbrack {0,1}\right\rbrack \), we define the mappings \( Q : {E}_{1} \rightarrow \mathbb{R} \) and \( \phi : {E}_{1} \rightarrow \mathbb{R} \) by \[ Q\left( x\right) = x\text{ and }\phi \left( x\right) = x\left( {x - 2}\right) . \] | We can check that the solution set of (3.1) is \( \{ 0\} \) . | Yes |
Theorem 4.1 Let \( \left( {{E}_{1}, \preccurlyeq }\right) \) be a totally ordered real Banach space and let \( K \) be a nonempty compact and convex subset of \( {E}_{1}.E \) is a real separable Banach space. The mappings \( Q\left( \cdot \right) \mathrel{\text{:=}} g\left( {t, x, \cdot }\right) \) and \( \phi : {E}_{1... | Proof Theorem 3.1 guarantees that for every \( \left( {t, x}\right) \in \left\lbrack {0, T}\right\rbrack \times E \), the set \( U\left( {t, x}\right) \) is nonempty, convex and compact in \( K \) . Thus the mapping \( U\left( {\cdot , \cdot }\right) \) is well defined.\n\nNow we claim that \( U\left( {\cdot , \cdot }\... | Yes |
Lemma 5.1 [2] Let \( E \) and \( {E}_{1} \) be real Banach space, with \( E \) separable, and let \( K \subseteq {E}_{1} \) be a nonempty compact and convex subset. We assume that the hypotheses of Theorem 4.1 and conditions \( \left( {f1}\right) - \left( {f5}\right) \) are fulfilled. Then we have:\n\n(i) \( F\left( {t... | By Lemma 5.1 we can define the set-valued mapping \( {P}_{F} : C\left( {\left\lbrack {0, T}\right\rbrack ;E}\right) \rightarrow P\left( {{L}^{1}\left( {\left\lbrack {0, T}\right\rbrack ;E}\right) }\right) \) by \[ {P}_{F}\left( q\right) \mathrel{\text{:=}} \{ g \mid g\text{ is strongly measurable and }g\left( t\right) ... | Yes |
For an initial surplus \( u \in \mathbb{N} \), the optimal value functions \( {V}_{m}^{ * }\left( u\right) (m = \) \( 0,1,2,\ldots ) \) satisfy the following discrete HJB equations | Proof Assume that \( \pi \in \Pi \) and the corresponding value functions in different surplus processes are \( {V}_{m}^{\pi }\left( u\right) \left( {m = 0,1,2,\ldots }\right) \) . Then,\n\n\[ {V}_{0}^{\pi }\left( u\right) = \left( {k{\phi }_{0}^{\pi }\left( u\right) - K}\right) + \left( {1 - p\left( 1\right) }\right) ... | Yes |
Theorem 3.2 Assume that \( {\pi }^{ * } \in \Pi ,{W}_{m}^{{\pi }^{ * }}\left( u\right) \) and \( {\phi }_{m}^{{\pi }^{ * }}\left( u\right) \left( {m = 0,1,\ldots }\right) \) are the images of its value functions and the corresponding dividend payment. Then, \( {W}_{m}^{{\pi }^{ * }}\left( u\right) \) are all maximal if... | Proof (\ | No |
Theorem 4.2 Given two positive integers \( n \) and \( l \), assume that \( {W}_{m}\left( {m = 0,1,\ldots }\right) \) are the maximal image functions, \( {\widetilde{W}}_{n + l} \) is an approximation of \( {W}_{n + l} \) such that \( 0 < {\widetilde{W}}_{n + l}\left( u\right) \leq \) \( \frac{r}{1 - r}\left( {k{c}_{0}... | Proof For any positive integer \( m \) such that \( 1 \leq m \leq n + l - 1 \), we get \n\n\[ \nd\left( {{\widetilde{W}}_{m},{W}_{m}}\right) \leq \frac{\bar{P}\left( {m + 1}\right) }{\bar{P}\left( m\right) }{rd}\left( {{\widetilde{W}}_{m + 1},{W}_{m + 1}}\right) , \n\] \n\n(4.8) \n\nfrom which we have \n\n\[ \nd\left( ... | Yes |
(i) If \( n\left( {a - \gamma {u}_{ * }}\right) /a < 1 \), then for any \( \tau > 0 \), all roots of (2.9) are in the unit circle. | Proof By Lemma 2.1, Lemma 2.2 and Corollary 2.4 in [17], we can obtain (i). | No |
Theorem 1.1 Let \( K \) be an imaginary quadratic field and \( \mathcal{O} \) be an order in \( K \) . Let \( p \) be a prime inert in \( K \) and strictly greater than \( \left| {\operatorname{disc}\left( \mathcal{O}\right) }\right| \), and \( {\mathcal{H}}_{p} \) be set of \( {\mathbb{F}}_{p} \) -roots of \( {\wideti... | Moreover, \( {\mathcal{H}}_{p} \neq \varnothing \) if and only if for every prime factor \( \ell \) of disc \( \left( \mathcal{O}\right) \), either condition (i) or (ii) below holds for \( \ell \) depending on its parity:\n\n(i) \( \ell \neq 2 \) and the Legendre symbol \( \left( \frac{-p}{\ell }\right) = 1 \) ;\n\n(ii... | Yes |
Suppose \( {Y}^{\left( m\right) } \sim {MGe}\left( {\mathbf{p}}_{m}\right) \), where \( {\mathbf{p}}_{m} = \left( {{p}_{1},{p}_{2},\cdots ,{p}_{m}}\right) \). The parameters are generated as follows\n\n\[ \n{p}_{i} = \frac{2i}{m\left( {m + 1}\right) }, i = 1,2,\cdots, m.\n\]\n\nLet \( {\mu }_{m} = E\left( {Y}^{\left( m... | Consider four situations when \( m = 5,{10},{15},{20} \), respectively, the expectation and variance of the random variable are calculated as\n\n\[ \n{\mu }_{5} = {18.67},\;{\mu }_{10} = {68.98},\;{\mu }_{15} = {150.61},\;{\mu }_{20} = {263.58};\n\]\n\n\[ \n{\sigma }_{5}^{2} = {169.57},{\sigma }_{10}^{2} = {2420.43},{\... | Yes |
Suppose \( {Y}_{m} \sim \operatorname{MGe}\left( m\right) \), we still use the notation of Example 1, under conditions \( m = {10},{30},{50},{70} \), the expectation and variance are | \[ {\mu }_{10} = {29.29},\;{\mu }_{30} = {119.85},\;{\mu }_{50} = {224.96},\;{\mu }_{70} = {338.29}; \] \[ {\sigma }_{10}^{2} = {125.69},{\sigma }_{30}^{2} = {1331.09},{\sigma }_{50}^{2} = {3837.87},{\sigma }_{70}^{2} = {7652.38}\text{.} \] | Yes |
Theorem 2.1 If \( C \) in (1.4) is nonsingular and \( {\lambda }_{n}/n \rightarrow {\lambda }_{0} \geq n \), then \( {\widehat{\mathbf{B}}}_{n}{ \rightarrow }_{p}\arg \min \left( Z\right) \) where \( Z\left( \Phi \right) = \operatorname{vec}{\left( \Phi - \mathbf{B}\right) }^{T}C\operatorname{vec}\left( {\Phi - \mathbf... | Proof Define \( {Z}_{n} \) as in (2.1). Then let \( {\mathrm{Y}}_{i} = \left\langle {\mathbf{B},{\mathbf{X}}_{i}}\right\rangle + {\epsilon }_{i} \), so we have\n\n\[ \n{Z}_{n}\left( \Phi \right) = \frac{1}{n}\mathop{\sum }\limits_{{i = 1}}^{n}{\left( {\epsilon }_{i} - \operatorname{vec}\left( {\mathbf{X}}_{i}\right) \o... | No |
Theorem 3.2 Let \( {\epsilon }_{i} \) be i.i.d. random variables with \( \mathbb{E}\left| {\epsilon }_{i}\right| < \infty \) and \( \mathbb{E}\left| {\epsilon }_{i}\right| = 0 \) . Assume that (2.1) holds as \( n \rightarrow \infty \) . (a) if \( \frac{{\lambda }_{n}}{n} \rightarrow a \in \left( {0,\infty }\right) \) t... | Proof First consider part (a), let \( {V}_{n}\left( \cdot \right) \) be as in (3.2). Note that \( \parallel \mathbf{B} + U{\parallel }_{ * } - \parallel \mathbf{U}{\parallel }_{ * } \mid \leq \) \( \parallel \mathbf{U}{\parallel }_{ * } \), since \( \frac{{\lambda }_{n}}{n} \rightarrow a \in \left( {0,\infty }\right) \... | Yes |
Theorem 4.1 Suppose \( \Omega \subset {\mathbb{R}}^{n} \) is a \( {C}^{4} \) strictly convex domain and \( \varphi \in {C}^{2}\left( {\partial \Omega }\right) \) . Let \( \Theta \left( x\right) \in {C}^{2}\left( \overline{\Omega }\right) \) with \( \frac{\left( {n - 2}\right) \pi }{2} < \Theta \left( x\right) < \frac{n... | Proof of Theorem 4.1 In the following, we prove Theorem 4.1 by two steps.\n\nStep 1 Prove \( \mathop{\max }\limits_{\bar{\Omega }}\left| {{D}^{2}u}\right| \leq C\left( {1 + \mathop{\max }\limits_{{\partial \Omega }}\left| {{D}^{2}u}\right| }\right) \) .\n\nConsider the auxiliary function\n\n\[ P\left( x\right) = \log {... | Yes |
Theorem 1.1 (Local existence) Let \( \mathbf{\Omega } \) be the whole space \( {\mathbb{R}}^{3} \) or the torus \( {\mathbb{T}}^{3} \) , and the integer \( s \geq 3 \), if the initial energy \( {\mathcal{E}}^{in} < + \infty \), and \( \underline{\rho } < {\rho }^{in} < \bar{\rho } \) for some positive constants \( \und... | We will prove Theorem 1.1 by using the energy methods. We now sketch the main difficulties we have met and the novelties to be used in the proof of the above theorem. The first difficult is how to control the \( {L}^{\infty } \) -bound of density. To overcome this difficulty we use some techniques inspired by the studi... | Yes |
Lemma 2.1 (Basic energy bound) Let \( \left( {c,\rho, u, Q}\right) \) be a sufficiently smooth solution to the system (1.2), then\n\n\[ \frac{1}{2}{\int }_{\mathbf{\Omega }}\left( {{\left| c\right| }^{2} + \rho {\left| u\right| }^{2} + \frac{2a}{\gamma - 1}{\rho }^{\gamma } + {\left| Q\right| }^{2} + K{\left| \nabla Q\... | Proof We multiply the first, third and the forth equations of system (1.2) with \( c, u \) and \( - {K\Delta Q} + Q + {c}_{ * }Q\operatorname{tr}\left( {Q}^{2}\right) \) respectively, and then integrate over the space to get\n\n\[ \frac{1}{2}{\int }_{\mathbf{\Omega }}{\left| c\right| }^{2} + \rho {\left| u\right| }^{2}... | Yes |
Lemma 2.3 Let \( f\left( \rho \right) \) be a smooth function, then for any positive integer \( k \) and \( \rho \in {H}^{k}\left( \mathbb{R}\right) \cup {L}^{\infty }\left( \mathbb{R}\right) \), we have\n\n\[{\partial }_{x}^{k}f\left( \rho \right) = \mathop{\sum }\limits_{{i = 1}}^{k}{f}^{\left( i\right) }\left( \rho ... | Proof we can prove the lemma by induction. Here we omit it for simplicity. | No |
Lemma 3.1 Suppose that the integer \( s \geq 3 \) and the initial data \( \left( {{c}^{in},{\rho }^{in},{u}^{in},{Q}^{in}}\right) \in \) \( \mathbb{R} \times {\mathbb{R}}^{ + } \times {\mathbb{R}}^{3} \times {S}_{0}^{3} \) satisfying\n\n\[ \n{\rho }^{in} \in {\dot{H}}_{\frac{{P}^{\prime }\left( {\rho }^{in}\right) }{{\... | Proof For the iterating system (3.1), the unknown vectors are \( \left( {{c}^{n + 1},{\rho }^{n + 1},{u}^{n + 1},{Q}^{n + 1}}\right) \) in the \( \left( {n + 1}\right) \) -th step. By the construction of the approximation system (3.1), the first equation is a linear equation about \( {c}^{n + 1} \), which admits a uniq... | Yes |
Lemma 4.1 Assume that \( \left( {{c}^{n + 1},{\rho }^{n + 1},{u}^{n + 1},{Q}^{n + 1}}\right) \) is the solution to the iterating equation (3.1), for any fixed positive constant \( M \), we define\n\n\[ \n{T}_{n + 1} = \sup \left\{ {\tau \in \left\lbrack {0,{T}_{n + 1}^{ * }}\right) ;\;\mathop{\sup }\limits_{{t \in \lef... | Proof The proof is almost the same as in Lemma 5.2 of [27]. Here we omit the details for simplicity. | No |
Corollary 2.2 Assume (H1)-(H3) hold. For all \( l \geq \mathbf{0},\frac{1}{n{b}_{n}^{2}}\mathop{\sum }\limits_{{\mathbf{1} \leq \mathbf{k} \leq \mathbf{n}}}\left( {{X}_{\mathbf{k}}{X}_{\mathbf{k} + l} - \mathbb{E}{X}_{\mathbf{k}}{X}_{\mathbf{k} + l}}\right) \) satisfies the LDP on \( \mathbb{R} \) with speed \( {b}_{n}... | \[ {I}^{l}\left( z\right) = \frac{1}{2}\frac{{z}^{2}}{1/{\left( 2\pi \right) }^{2}{\int }_{I}2{\cos }^{2}\left( {l \cdot t}\right) {f}^{2}\left( t\right) {dt} + {k}_{4}{\left( 1/{\left( 2\pi \right) }^{2}{\int }_{I}f\left( t\right) \cos \left( l \cdot t\right) dt\right) }^{2}} \] with the convention that \( a/0 = + \in... | Yes |
Lemma 2.1 Assume that assumptions (H1) and (H2) hold. Then difference system (2.1) has a unique positive equilibrium \( \left( {{X}^{ * },{Y}^{ * }}\right) \) . | Moreover, \[ \left( {{X}^{ * },{Y}^{ * }}\right) = \left( {\frac{{f}_{1}\left( {X}^{ * }\right) + {g}_{1}\left( {Y}^{ * }\right) }{1 - {a}_{1}},\frac{{f}_{2}\left( {X}^{ * }\right) + {g}_{2}\left( {Y}^{ * }\right) }{1 - {a}_{2}}}\right) . \] | Yes |
Corollary 2.1 Suppose that\n\n\\[ \n\\mathop{\\max }\\limits_{{\\left( {s, r}\\right) \\in \\left\\lbrack {0,{X}^{ * }}\\right\\rbrack \\times \\left\\lbrack {0,{Y}^{ * }}\\right\\rbrack }}\\left( {{F}_{1}\\left( r\\right) + {G}_{1}\\left( s\\right) }\\right) \\leq {X}^{ * },\\mathop{\\max }\\limits_{{\\left( {s, r}\\r... | Proof It is easy to see that (2.7) implies that \\( {I}_{0} \\) is the invariant domain of operator T. Moreover,(2.8) and (2.9) imply that \\( { \\cap }_{i \\geq 0}{\\mathbf{T}}^{i}\\left( {I}_{0}\\right) = \\left( {{X}^{ * },{Y}^{ * }}\\right) \\) . Therefore, \\( \\left( {{X}^{ * },{Y}^{ * }}\\right) \\) is the the a... | Yes |
Proposition 2.4 [10, Theorem 2.2.4] (Representation Formula) Let \( f \) be a slice regular function on an axially symmetric \( s \) -domain \( U \subset \mathbb{H} \) . Let \( J \in \mathbb{S} \) and let \( x \pm {yJ} \in U \cap {\mathbb{C}}_{J} \) , then the following equality holds for all \( q = x + {yI} \in U \) , | \[ f\left( {x + {yI}}\right) = \frac{1}{2}\left\lbrack {\left( {1 + {IJ}}\right) f\left( {x - {yJ}}\right) + \left( {1 - {IJ}}\right) f\left( {x + {yJ}}\right) }\right\rbrack . \] | Yes |
Corollary 4.10 Let \( \varphi \in \mathcal{R}\left( \mathbb{H}\right) \) such that \( \varphi \left( {\mathbb{C}}_{I}\right) \subset {\mathbb{C}}_{I} \) for some \( I \in \mathbb{S} \) . Denote \( \varphi \left( p\right) = {p\mu } + c \) with \( \left| \mu \right| \leq 1 \) and \( \mu \in {\mathbb{C}}_{I} \) such that ... | Proof We only need to prove \( {C}_{\varphi } \) is unitary when it is co-isometric. At this time, Theorem 4.9 implies \( \varphi \left( p\right) = {p\mu } \) with \( \left| \mu \right| = 1 \) . For any two functions \( f, g \in {\mathcal{F}}^{2}\left( \mathbb{H}\right) \) writing as\n\n\[ f\left( p\right) = \mathop{\s... | Yes |
Example 4.1 Consider the following fractional differential equation\n\n\[ \n{}_{0}{D}_{t}^{1/5}\left( {u\left( {x, t}\right) + \frac{1}{4}u\left( {x, t - \frac{1}{3}}\right) + u\left( {x, t - \frac{1}{2}}\right) }\right) \n\]\n\n\[ \n= {e}^{t}{u}^{2}{\Delta u}\left( {x, t}\right) + {u}^{4}\left( {x, t - 3}\right) {\Del... | Notice \( \alpha = \frac{1}{5},{r}_{1} = \frac{1}{4},{r}_{2} = 1,{\sigma }_{1} = \frac{1}{3},{\sigma }_{2} = \frac{1}{2}, a\left( t\right) = {e}^{t}, h\left( u\right) = {u}^{2},{h}_{1}\left( u\right) = {u}^{4},{h}_{2}\left( u\right) = {u}^{6} \) ,\n\n\( {b}_{1}\left( t\right) = 1,{b}_{2}\left( t\right) = {t}^{2},{\zeta... | Yes |
Lemma 2.3 Let \( \mu \) be positive measurable on a \( \sigma \) -algebra \( M \) in a set \( \Omega \), so that \( \mu \left( \Omega \right) = 1 \) . If \( f \) is a real function in \( {L}^{1}\left( \mu \right), a < f\left( x\right) < b \) for all \( x \in \Omega \) and \( \phi \) is convex on \( \left( {a, b}\right)... | \[ \phi \left( {{\int }_{\Omega }\left| f\right| {d\mu }}\right) \leq {\int }_{\Omega }\phi \left( \left| f\right| \right) {d\mu } \] | Yes |
Theorem 2.1 Let \( u \in {D}^{\prime }\left( {\Theta ,{ \land }^{\ell }}\right) ,\ell = 0,1,\ldots, n \), be a solution of the nonhomogeneous \( A \) - harmonic equation (1.6) in a bounded domain \( \Theta \subset {\mathbb{R}}^{n} \), and assume that \( 0 < s < \infty \) . Then there exists a constant \( C \), independ... | Proof For any \( s > 0 \), we have\n\n\[ \parallel \mathrm{d}u{\parallel }_{s, O} \leq {C}_{1}{\left| O\right| }^{\frac{p - s}{ps}}\parallel \mathrm{d}u{\parallel }_{p,{\sigma }_{1}O} \]\n\n\[ \leq {C}_{2}{\left| O\right| }^{\frac{p - s}{ps}}\operatorname{diam}{\left( O\right) }^{-1}\parallel u - c{\parallel }_{p,{\sig... | Yes |
Theorem 4.1 The Young function \( \varphi : \lbrack 0,\infty ) \rightarrow \lbrack 0,\infty ) \) belongs to \( {NG}\left( {p, q}\right) \), if \( \varphi \left( t\right) = {t}^{p}{\ln }^{\alpha }(2 + t),\alpha = q - p > 0,1 < p < q < \infty \) . | Proof According to the equivalent definition of \( {NG}\left( {p, q}\right) \), we only need to prove the following facts:\n\n(1) \( \frac{\varphi \left( t\right) }{{t}^{p}} \) is increasing:Let \( f\left( t\right) = \frac{\varphi \left( t\right) }{{t}^{p}} = {\ln }^{q - p}\left( {2 + t}\right) \), then \( {f}^{\prime ... | Yes |
Let \( u\left( {x, y}\right) \) be a function defined in \( {\mathbb{R}}^{2} \) by\n\n\[ u\left( {x, y}\right) = {x}^{3} - 6{x}^{2}y - {3x}{y}^{2} + 2{y}^{3}. \]\n\nIt is easy to check that \( u\left( {x, y}\right) \) is a harmonic function in the upper half plane. Let \( r > 0 \) be a constant, and \( O = \left( {x, y... | we can use Caccioppoli inequality (2.15) with \( c = 0 \), and \( n = 2 \) as follows. First, we know that \( \operatorname{diam}\left( O\right) = {2r},\left| {\sigma O}\right| = \pi {\sigma }^{2}{r}^{2} \), and\n\n\[ \left| {u\left( {x, y}\right) }\right| \leq {\left| x\right| }^{3} + 6{\left| x\right| }^{2}\left| y\r... | Yes |
Example 4.2 Let\n\n\[ \n{a}_{{x}_{1}{x}_{3}} + {b}_{{x}_{2}{x}_{3}} - {c}_{{x}_{1}{x}_{1}} - {c}_{{x}_{2}{x}_{2}} = 0, \]\n\n\[ \n{a}_{{x}_{1}{x}_{2}} + {c}_{{x}_{2}{x}_{3}} - {b}_{{x}_{1}{x}_{1}} - {b}_{{x}_{3}{x}_{3}} = 0, \]\n\n(4.13)\n\n\[ \n{b}_{{x}_{1}{x}_{2}} + {c}_{{x}_{1}{x}_{3}} - {a}_{{x}_{2}{x}_{2}} - {a}_{... | Let \( a = {x}_{2}^{4} + {x}_{3}^{4} + 4\left( {{x}_{2} + {x}_{3}}\right) {x}_{1}^{3} + \left( {{x}_{2}^{3} + 6{x}_{1}{x}_{2}{x}_{3} + {x}_{3}^{3}}\right), b = {x}_{1}^{4} + {x}_{3}^{4} + 4\left( {{x}_{1} + {x}_{3}}\right) {x}_{2}^{3} + \left( {{x}_{1}^{3} + 6{x}_{1}{x}_{2}{x}_{3} + {x}_{3}^{3}}\right) , \) \( c = {x}_... | Yes |
Theorem 3.7 A bi-parameter singular integral operator \( T \) in the sense of Martikainen satisfies weak boundedness condition and the cancellation conditions. Then \( T \) is \( {L}^{p} \) -bounded. | Proof From Proposition 3.5, the bi-parameter singular integral operator \( T \) in the sense of Pott-Villarroya is \( {L}^{p} \) bounded. By Proposition 3.4, the bi-parameter singular integral operators in the sense of Pott-Villarroya and Martikainen are equal. So the bi-parameter singular integral operator \( T \) in ... | Yes |
Lemma 3.8 [14] Let \( T \) be a bi-parameter operator that has partial kernel representation with the kernel \( {K}_{{f}_{i},{g}_{i}}^{i} \) for \( i = 1,2 \) as defined in (2.4) and \( V \) a cube in \( {\mathbb{R}}^{m} \) (resp. in \( {\mathbb{R}}^{n} \) ) | \[ C\left( {{\chi }_{V},{g}_{V}}\right) + C\left( {{g}_{V},{\chi }_{V}}\right) \leq C{\begin{Vmatrix}{g}_{V}\end{Vmatrix}}_{\infty }\left| V\right| \] (3.4) whenever \( {g}_{V} \in {L}_{\infty }\left( V\right) \) | Yes |
Theorem 1.4 Let \( \left( {\mathcal{X}, d,\mu }\right) \) be a space of homogeneous type with the christ system.Let \( T \) be a Calderón-Zygmund operator defined in 3.2, and let \( {V}_{q}T \) be weak type \( \left( {1,1}\right), q > 2 \) . Then for any compactly supported \( f \in {L}^{1}\left( {\mathbb{R}}^{n}\right... | \[ \left| {{V}_{q}{Tf}\left( x\right) }\right| \leq {c}_{\mathcal{X}, q}{\mathcal{A}}_{\mathcal{S}}\left| f\right| \left( x\right) \] | Yes |
Theorem 1.5 Follow the conditions in Theorem 1.4 and let \( w \in {A}_{p},1 < p < \infty \) . Then we have\n\n\[ \n{\begin{Vmatrix}{V}_{q}T\left( f\right) \end{Vmatrix}}_{{L}^{p}\left( w\right) } \leq {c}_{\mathcal{X}, q}{\left\lbrack w\right\rbrack }_{{A}_{p}}^{\max \left( {1,\frac{1}{p - 1}}\right) }\parallel f{\para... | (1.4) | No |
Lemma 4.1 Let \( 2 < q < \infty \) . Then for \( 1 < p < \infty ,{V}_{q}A \) is bounded on \( {L}^{p}\left( \mathcal{X}\right) \) and weak type \( \left( {1,1}\right) \) . | Proof As for the variation of average operators \( {V}_{q}A\left( f\right) \) on an integrable function \( f \) , we can compare it with the variation of the martingale generated by \( f \) . Then we divide the \( {V}_{q}A\left( f\right) \) into three parts, the short variation, the long variation and the martingale va... | No |
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