Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
values |
|---|---|---|
\[ {\int }_{{\mathbb{T}}^{2}}{\left| \widehat{{f}_{A}}\left( \alpha \right) \right| }^{2}\left| {{S}_{N}\left( \alpha \right) }\right| {d\alpha } \geq \frac{1}{2}{\delta }^{2}{q}^{4N}. \] | Proof Write \( \mathrm{I} = \mathop{\sum }\limits_{{d \in {\mathbb{G}}_{N}, m \in {\mathbb{A}}^{2}}}{f}_{A}\left( m\right) {f}_{A}\left( {m + \overrightarrow{d}}\right) \) . By (2.1), we have \[ \mathrm{I} = \mathop{\sum }\limits_{{d \in {\mathbb{G}}_{N}, m, n \in {\mathbb{A}}^{2}}}{f}_{A}\left( m\right) {f}_{A}\left( ... | Yes |
Lemma 12 There exists a monic polynomial \( {b}_{0} \) in \( {\mathbb{G}}_{{2\theta } + 4} \) such that\n\n\[{\int }_{{F}_{{b}_{0}}}{\left| \widehat{{f}_{A}}\left( \alpha \right) \right| }^{2}{d\alpha } \geq c{\delta }^{3}{q}^{3N}\]\n\nwhere \( 0 < c < 1 \) is a constant depending only on \( q \) . | Proof By Proposition 10, we have\n\n\[{\int }_{\mathfrak{m}}{\left| \widehat{{f}_{A}}\left( \alpha \right) \right| }^{2}\left| {{S}_{N}\left( \alpha \right) }\right| {d\alpha } \leq \frac{\delta }{4}{q}^{N}{\int }_{\mathfrak{m}}{\left| \widehat{{f}_{A}}\left( \alpha \right) \right| }^{2}{d\alpha }\n\n\leq \frac{\delta ... | Yes |
Lemma 2.1 For a morphism \( \sigma : B \otimes B \rightarrow K \) in \( {}_{H}^{H}\mathbb{{YD}} \), we can get the following useful formula\n\n\[ \sigma \left( {a,\alpha \left( h\right) \cdot b}\right) = \sigma \left( {{S}^{-1}\left( h\right) \cdot {\beta }^{-2}\left( a\right), b}\right) \]\n\nfor all \( a, b \in B \) ... | Proof We can check that as follows\n\n\[ \sigma \left( {a,\alpha \left( h\right) \cdot b}\right) = \sigma \left( {\left( {{h}_{12}{S}^{-1}\left( {h}_{11}\right) }\right) \cdot {\beta }^{-1}\left( a\right) ,{h}_{2} \cdot b}\right) \]\n\n\[ = \sigma \left( {\alpha \left( {h}_{12}\right) \cdot \left( {{S}^{-1}\left( {h}_{... | Yes |
Theorem 2.5 (i) For a normal left 2-cocycle \( \sigma : B \otimes B \rightarrow K \) in \( {}_{H}^{H}\mathbb{{YD}} \), define \( \bar{\sigma } \) : \( {B}_{ \times }^{\# }H \otimes {B}_{ \times }^{\# }H \rightarrow K \) by\n\n\[ \bar{\sigma }\left( {b \otimes h,{b}^{\prime } \otimes {h}^{\prime }}\right) = \sigma \left... | Proof (i) It is easy to see that \( \bar{\sigma } \) is normal. Since \( {}_{\sigma }B\# H \) is a Hom-algebra, we have\n\n\[ \left( {\left( {\beta \otimes \alpha }\right) \left( {b\# h}\right) }\right) \left( {\left( {{b}^{\prime }\# {h}^{\prime }}\right) \left( {{b}^{\prime \prime }\# {h}^{\prime \prime }}\right) }\r... | Yes |
Proposition 2.7 If \( \sigma \) is a lazy 2-coboundary for \( \left( {B,\beta }\right) \) in \( {}_{H}^{H}\mathbb{{YD}} \), then \( \bar{\sigma } \) is a lazy 2-coboundary for \( \left( {{B}_{ \times }^{\# }H,\beta \otimes \alpha }\right) \), so the group homomorphism \( {Z}_{L}^{2}\left( {B,\beta }\right) \rightarrow ... | Proof It follows immediately from (vi) in Theorem 2.5. | No |
Example 2.8 Let \( A = \operatorname{sp}\left\{ {{1}_{A}, z}\right\} \) and the automorphism \( \beta : A \rightarrow A,\beta \left( {1}_{A}\right) = \) \( {1}_{A},\beta \left( z\right) = - z \) . Then \( \left( {A,\beta }\right) \) is a Hom-algebra with multiplication: \( {1}_{A}{1}_{A} = {1}_{A},{1}_{A}z = z{1}_{A} =... | \[ \Delta \left( {1}_{A}\right) = {1}_{A} \otimes {1}_{A},\;\varepsilon \left( {1}_{A}\right) = {1}_{k}, \] \[ \Delta \left( z\right) = \left( {-z}\right) \otimes {1}_{A} + {1}_{A} \otimes \left( {-z}\right) ,\;\varepsilon \left( z\right) = 0. \] | Yes |
Example 2.9 Let \( K{Z}_{2} = K\{ 1, a\} \) be Hopf group algebra. Then \( \left( {K{Z}_{2},{id}}\right) \) is a Hom-Hopf algebra. | Let \( {T}_{2, - 1} = K\left\{ {1, g, x, y \mid {g}^{2} = 1,{x}^{2} = 0, y = {gx},{gy} = - {yg} = x}\right\} \) be Taft’s Hopf algebra, its coalgebra structure and antipode are given by\n\n\[ \Delta \left( g\right) = g \otimes g,\;\Delta \left( x\right) = x \otimes g + 1 \otimes x,\;\Delta \left( y\right) = y \otimes 1... | Yes |
Lemma 3.4 [5, Lemma 4.5 (i)] Let \( \varphi \in {\mathbb{A}}_{q} \) with \( q \in \lbrack 1,\infty ) \) . Then there exists a positive constant \( C \) such that for any ball \( B \subset {\mathbb{R}}^{n},\lambda \in \left( {1,\infty }\right) \) and \( t \in \left( {0,\infty }\right) \) , | \[ \varphi \left( {{\lambda B}, t}\right) \leq C{\lambda }^{nq}\varphi \left( {B, t}\right) \] | Yes |
For any two functions \( h \) and \( g \), let us define the linearized operator of the weighted \( p \) -Laplacian at point \( v \) ,\n\n\[ \n{L}_{p, f}\left( \psi \right) \doteq {e}^{f}\operatorname{div}\left( {{e}^{-f}{\left| \nabla v\right| }^{p - 2}A\left( {\nabla \psi }\right) }\right) , \n\]\n\nwhere \( {A}^{ij}... | Proof The proof is a direct result by the definition of \( {L}_{p, f} \) and integration by parts. | No |
Lemma 2.2 We have the modified weighted \( p \) -Bochner formula\n\n\[ \n{L}_{p, f}{\left| \nabla v\right| }^{p} = p{\Gamma }_{2, A}\left( v\right) + p{\left| \nabla v\right| }^{p - 2}\left\langle {\nabla v,\nabla {\Delta }_{p, f}v}\right\rangle ,\n\]\n\nwhere\n\n\[ \n{\Gamma }_{2, A}\left( v\right) \doteq {\left| \nab... | Proof According to [9], we have\n\n\[ \n{L}_{p, f}{\left| \nabla v\right| }^{p} = p{\left| \nabla v\right| }^{{2p} - 4}\left( {{\left| \nabla \nabla v\right| }_{A}^{2} + {\operatorname{Ric}}_{f}\left( {\nabla v,\nabla v}\right) }\right) + p{\left| \nabla v\right| }^{p - 2}\left\langle {\nabla v,\nabla {\Delta }_{p, f}v... | Yes |
Lemma 2.3 If \( u, v \) satisfies equations (1.2) and (2.2), we obtain that\n\n\[ \n{\partial }_{t}\left( {{\Delta }_{p, f}v}\right) = {L}_{p, f}\left( {{\partial }_{t}v}\right) \n\]\n\n\( \left( {2.12}\right) \)\n\n\[ \n{\partial }_{t}\left( {uv}\right) = \frac{b}{p - 1}v{L}_{p, f}\left( {uv}\right) \n\]\n\n\( \left( ... | Proof By the definitions of \( A \) and \( {L}_{p, f} \), then\n\n\[ \n{\partial }_{t}\left( {{\Delta }_{p, f}v}\right) = {\partial }_{t}\left\lbrack {{e}^{f}\operatorname{div}\left( {{e}^{-f}{\left| \nabla v\right| }^{p - 2}\nabla v}\right) }\right\rbrack \n\]\n\n\[ \n= {e}^{f}\operatorname{div}\left( {{e}^{-f}\left\l... | Yes |
As a function of \( u \), the expected discounted value of all dividends until ruin \( V\left( {u;b}\right) \) satisfies the following integro-differential equations\n\n\[ \left( {{\delta u} + c}\right) {V}_{1}^{\prime }\left( {u;b}\right) = \left( {\lambda + \beta }\right) {V}_{1}\left( {u;b}\right) - \lambda {\int }_... | Proof For \( - c/\delta < u < 0 \), conditioning on the occurrence of a claim within an infinitesimal time interval \( (0,{\Delta t}\rbrack \), we have\n\n\[ {V}_{1}\left( {u;b}\right) = \left( {1 - {\lambda \Delta t}}\right) {e}^{-{\beta \Delta t}}{V}_{1}\left( {{h}_{\delta }\left( {{\Delta t}, u}\right) ;b}\right) \]... | Yes |
Proposition 3.1 The boundary conditions of \( V\left( {u;b}\right) \) are as follows.\n\n\[ \mathop{\lim }\limits_{{u \downarrow - c/\delta }}{V}_{1}\left( {u;b}\right) = 0 \]\n\n(3.4)\n\n\[ \mathop{\lim }\limits_{{u \uparrow \infty }}{V}_{3}\left( {u;b}\right) = \frac{\gamma }{\beta }. \]\n\n(3.5)\n\nMoreover, by the ... | Proof Condition (3.4) is obviously for the fact that when \( {U}_{\delta }\left( 0\right) = - c/\delta \) the surplus stays at the zero until the next claim and at this time ruin occurs. When \( u \rightarrow \infty \), then the ruin time \( {\tau }_{\delta } = \infty \), and then \( D = {\int }_{0}^{\infty }{e}^{-{\be... | Yes |
Lemma 2.1 Let \( \left\{ {X}_{k}\right\} \) be the matrix sequence generated by the MALI iteration method, \( \mathcal{R}\left( X\right) = {XCX} - {XD} - {AX} + B \), and \( S \) be the minimal nonnegative solution to (1.1). Then for any \( k \geq 0 \), the following equalities hold\n\n(1) \( \left( {{X}_{k + 1/2} - S}... | Proof The proof is similar to that of in [8], so we omit here. | No |
For the MARE (1.1), if the matrix \( K \) in (1.2) is a nonsingular M-matrix or an irreducible singular M-matrix, \( S \) is the minimal nonnegative solution to (1.1), then for any \( \delta > 0 \) and \( 0 \leq Z \leq S \), the matrices \( {\delta I} + A - {ZC} \) and \( {\delta I} + D - {CZ} \) are nonsingular M-matr... | Proof First, from \( 0 \leq Z \leq S \), we have \( A - {SC} \leq A - {ZC} \) and \( D - {CS} \leq D - {CZ} \) .\n\nIf \( K \) is a nonsingular M-matrix, then \( A - {SC} \) and \( D - {CS} \) are also nonsingular M-matrices by Lemma 1.7. Thus \( {\delta I} + A - {ZC} \) and \( {\delta I} + D - {CZ} \) are nonsingular ... | Yes |
Lemma 2.4 Under the assumption of Lemma 2.2, the following inequalities hold for all \( k \geq 0 \) ,\n\n\[ \n{X}_{k} \leq {X}_{k + 1/2} \leq {X}_{k + 1},\;\mathcal{R}\left( {X}_{k}\right) \geq 0,\;\mathcal{R}\left( {X}_{k + 1/2}\right) \geq 0,\;\mathcal{R}\left( {X}_{k + 1}\right) \geq 0, \n\]\n\n(2.4)\n\nwhere \( \ma... | Proof We prove this lemma by induction.\n\nIn fact, when \( k = 0 \), we have \( \mathcal{R}\left( {X}_{0}\right) = B \geq 0 \) . From Lemma 2.1(2), we have\n\n\[ \n\left( {{X}_{1/2} - {X}_{0}}\right) \left( {{\alpha I} + \left( {D - C{X}_{0}}\right) }\right) = \mathcal{R}\left( {X}_{0}\right) .\n\]\nSince \( {\alpha I... | Yes |
Theorem 2.1 For the MARE (1.1), if \( K \) in (1.2) is a nonsingular M-matrix or an irreducible singular M-matrix, \( S \) is the minimal nonnegative solution to (1.1), and the parameters \( \alpha ,\beta \) satisfy (2.2), then \( \left\{ {X}_{k}\right\} \) is well defined, monotonically increasing and converges to \( ... | Proof Combining Lemma 2.3 with Lemma 2.4, we show that \( \left\{ {X}_{k}\right\} \) is nonnegative, monotonically increasing and bounded from above. Thus there is a nonnegative matrix \( {S}^{ * } \) such that \( \mathop{\lim }\limits_{{k \rightarrow \infty }}{X}_{k} = {S}^{ * } \) . It also holds that \( \mathop{\lim... | Yes |
Corollary 3.1 When \( K \) in (1.2) is a nonsingular M-matrix or an irreducible singular M-matrix with nonzero drift, for any \( \alpha ,\beta \) satisfy (2.2), we have \( \rho \left( {\alpha ,\beta }\right) < 1 \) . In this case, the MALI iteration method has linear convergence rate. When \( K \) is an irreducible sin... | Proof When \( K \) is a nonsingular or an irreducible singular M-matrix with nonzero drift, we know that \( A - {SC} \) and \( D - {CS} \) have at least one which is nonsingular by Lemma 1.9. By Lemma 1.6,\n\n\[ \alpha \geq \max \left\{ {a}_{ii}\right\} \geq \max \left\{ {\left( A - SC\right) }_{ii}\right\} \geq {\lamb... | Yes |
Corollary 3.2 The optimal parameters of the MALI iteration method are\n\n\[ \n{\alpha }_{\text{opt }} = \max \left\{ {a}_{ii}\right\} ,\;{\beta }_{\text{opt }} = \max \left\{ {d}_{jj}\right\} .\n\] | Proof It’s easy to verify that \( \rho \left( {\alpha ,\beta }\right) \) is an increasing function with respect to \( \alpha ,\beta \) . Thus the minimum of \( \rho \left( {\alpha ,\beta }\right) \) is attained at \( \alpha = \max \left\{ {a}_{ii}\right\} ,\beta = \max \left\{ {d}_{jj}\right\} \) . | Yes |
Consider the MARE (1.1) with\n\n\[ D = - {10}{E}_{n \times n} + {180.002}{I}_{n},\;C = {0.001}{E}_{n \times m},\;B = {C}^{T},\;A = {0.018}{I}_{m}, \]\n\nwhere \( {E}_{m \times n} \) is the \( m \times n \) matrix with all ones and \( {I}_{m} \) is the identity matrix of size \( m \) with \( m = 2, n = {18} \). This exa... | The computational results are summarized in Table 1.\n\n<table><thead><tr><th>Method</th><th>IT</th><th>CPU</th><th>RES</th></tr></thead><tr><td>Newton</td><td>3</td><td>0.002008</td><td>\( {7.4339}\mathrm{e} \) - \( {08} \)</td></tr><tr><td>ALI</td><td>-</td><td>-</td><td>-</td></tr><tr><td>MALI</td><td>7</td><td>0.00... | No |
Example 2 Consider the MARE (1.1) with\n\n\[ D = \alpha \left( \begin{matrix} 3 & - 1 & & & \\ - 1 & 4 & - 1 & & \\ & \ddots & \ddots & \ddots & \\ & & - 1 & 4 & - 1 \\ & & & - 1 & 2 \end{matrix}\right) ,\;C = \alpha \left( \begin{matrix} 1 & 1 & & \\ & 1 & \ddots & \\ & & \ddots & 1 \\ & & & 1 \end{matrix}\right) ,\n\... | This example is taken from [5], where we choose \( \alpha = 2 \) and the corresponding \( K \) is an irreducible singular M-matrix with \( \mu > 0 \) . The computational results are summarized in Table 2 for different sizes of \( n \) . | No |
Lemma 2.3 If \( p \geq 6 \), then \( \sum \) is a smooth manifold. | Proof Set\n\n\[ \omega \left( u\right) = \left\langle {{J}^{\prime }\left( u\right), u}\right\rangle ,\;u \in \sum . \]\n\nThen\n\n\[ \left\langle {{\omega }^{\prime }\left( u\right), u}\right\rangle = 2{\int }_{{\mathbb{R}}^{2}}\left( {{\left| \nabla u\right| }^{2} + {u}^{2} + 9{A}_{1}^{2}{u}^{2} + 9{A}_{2}^{2}{u}^{2}... | Yes |
Theorem 2.1 Let \( 1 < \beta < 2 \) and \( f : J \rightarrow X \) be continuous. If \( x \in P{C}_{2 - \beta }\left( {J, X}\right) \) is a solution of (2.1) if and only if \( x \) is a solution of the following the fractional integral equation\n\n\[ x\left( t\right) = \left\{ \begin{array}{l} \frac{1}{\Gamma \left( \be... | Proof For all \( t \in \left( {{t}_{k},{t}_{k + 1}}\right\rbrack \) where \( k = 0,1,\cdots, m \) by Lemma 2.1 and 2.2, we obtain\n\n\[ {D}_{{0}^{ + }}^{\beta }x\left( t\right) = {D}_{{0}^{ + }}^{\beta }\left\lbrack {\frac{1}{\Gamma \left( \beta \right) }{\int }_{0}^{t}{\left( t - s\right) }^{\beta - 1}f\left( s\right)... | Yes |
Lemma 2.3 (see [10]) Assume \( x \in {B}_{v}^{\prime } \), then for \( t \in J,{x}_{t} \in {B}_{v} \) . | Moreover\n\n\[\nl\parallel x\left( t\right) \parallel \leq {\begin{Vmatrix}{x}_{t}\end{Vmatrix}}_{{B}_{v}} \leq \parallel \phi {\parallel }_{{B}_{v}} + l\mathop{\sup }\limits_{{s \in \left\lbrack {0, t}\right\rbrack }}\begin{Vmatrix}{{s}^{2 - \beta }x\left( s\right) }\end{Vmatrix},\n\]\n\nwhere \( l = {\int }_{-\infty ... | No |
Theorem 4.2 Let \( n \) be an nonnegative integer and \( a, b \in \mathcal{F} \) with \( {ab} = {q}^{2n} \) . Then the pair \( \mathbb{A},\mathbb{B} \) defined in (4.1) acts on \( M\left( {n, a, b}\right) \) as a Leonard pair provided \( {\alpha \beta } \) is not among \( {q}^{-2},{q}^{-4}\cdots ,{q}^{-{2n}} \) . | To prove the above theorem, we apply Theorem 2.4. Before do this, we first give some lemmas. | No |
Lemma 4.3 There exists a basis for \( M\left( {n, a, b}\right) \) with respect to which the matrices representing \( \mathbb{A},\mathbb{B} \) have the form of (2.1). | Proof We can obtain this basis by modifying the basis \( \left\{ {{m}_{0},{m}_{1},\cdots ,{m}_{n}}\right\} \) given in Lemma 3.3. For \( 0 \leq i \leq n \), we define \( {u}_{i} = {\alpha }^{i}{m}_{i} \) . We observe \( \left\{ {{u}_{0},{u}_{1},\cdots ,{u}_{n}}\right\} \) is a basis for\n\n\( M\left( {n, a, b}\right) \... | Yes |
Lemma 4.4 Referring to Lemma 4.3, the following two equations hold\n\n\[ \n\frac{{\theta }_{i - 2} - {\theta }_{i + 1}}{{\theta }_{i - 1} - {\theta }_{i}} = {q}^{2} + {q}^{-2} + 1,\frac{{\theta }_{i - 2}^{ * } - {\theta }_{i + 1}^{ * }}{{\theta }_{i - 1}^{ * } - {\theta }_{i}^{ * }} = {q}^{2} + {q}^{-2} + 1. \n\] | Proof Immediate from Lemma 4.3 and a simple calculation. | No |
Lemma 4.5 Referring to Lemma 4.3, the scalars \( {\theta }_{i} \) also satisfy the following equation\n\n\[ \mathop{\sum }\limits_{{h = 0}}^{{i - 1}}\frac{{\theta }_{h} - {\theta }_{n - h}}{{\theta }_{0} - {\theta }_{n}} = \frac{{\left( i\right) }_{q}{\left( n - i + 1\right) }_{q}}{{\left( n\right) }_{q}}. \] | Proof Using the sum of the geometric progression, we have\n\n\[ \left( {{q}^{2} - 1}\right) {\left( t\right) }_{q} = {q}^{2t} - 1 \] \n\n(4.4)\n\nThen from (4.4), equation (4.3) holds. | No |
Theorem 2.1 Assume that \( D \subseteq {\mathbb{R}}^{n} \) is an unbounded open set. Let \( 0 < \alpha < n,{T}_{\Omega ,\alpha } \) be defined as in \( \left( {1.1}\right), p\left( \cdot \right), q\left( \cdot \right) \in {\mathcal{P}}_{\infty }^{\log }\left( D\right) \), such that \( 1 < {p}^{ - } \leq {p}^{ + } < \fr... | \[ {\begin{Vmatrix}{T}_{\Omega ,\alpha }\left( f\right) \end{Vmatrix}}_{{L}^{q\left( \cdot \right) }\left( {\widetilde{B}\left( {{x}_{0}, r}\right) }\right) } \lesssim {r}^{\frac{n}{{\theta q}\left( {{x}_{0}, r}\right) }}{\int }_{2r}^{\infty }{t}^{-\frac{n}{{\theta q}\left( {{x}_{0}, r}\right) } - 1}\parallel f{\parall... | Yes |
Theorem 2.2 Assume that \( D \subseteq {\mathbb{R}}^{n} \) is an unbounded open set and \( \Pi \subset D \) . Let \( 0 < \alpha < n,{T}_{\Omega ,\alpha } \) be defined as in (1.1), \( p\left( \cdot \right), q\left( \cdot \right) \in {\mathcal{P}}_{\infty }^{\log }\left( D\right) \), such that \( 1 < {p}^{ - } \leq {p}^... | \[ {c}_{\delta } = : {\int }_{\delta }^{\infty }\frac{\mathop{\sup }\limits_{{x \in \Pi }}{\varphi }^{\frac{1}{{\theta p}\left( {x, t}\right) }}\left( {x, t}\right) {dt}}{{t}^{1 + \frac{n}{{\theta q}\left( {x, t}\right) }}} < \infty \] \( \left( {2.2}\right) \) for each \( \delta > 0 \), and \[ {\int }_{r}^{\infty }\fr... | Yes |
Theorem 2.3 Assume that \( D \subseteq {\mathbb{R}}^{n} \) is an unbounded open set. Let \( 0 < \alpha < \) \( n,\left\lbrack {b,{T}_{\Omega ,\alpha }}\right\rbrack \) be defined as in (1.2) and \( p\left( \cdot \right), q\left( \cdot \right) \in {\mathcal{P}}_{\infty }^{\log }\left( D\right) \), such that \( 1 < {p}^{... | \[ {\begin{Vmatrix}\left\lbrack b,{T}_{\Omega ,\alpha }\right\rbrack \left( f\right) \end{Vmatrix}}_{{L}^{q\left( \cdot \right) }\left( {\widetilde{B}\left( {{x}_{0}, r}\right) }\right) } \lesssim \parallel b{\parallel }_{\mathrm{{BMO}}}{r}^{\frac{n}{{\theta q}\left( {{x}_{0}, r}\right) }}{\int }_{2r}^{\infty }\left( {... | Yes |
Theorem 2.4 Assume that \( D \subseteq {\mathbb{R}}^{n} \) is an unbounded open set and \( \Pi \subset D \) . Let \( 0 < \alpha < n,\left\lbrack {b,{T}_{\Omega ,\alpha }}\right\rbrack \) be defined as \( \left( {1.2}\right), p\left( \cdot \right), q\left( \cdot \right) \in {\mathcal{P}}_{\infty }^{\log }\left( D\right)... | \[ {c}_{\delta } = : {\int }_{\delta }^{\infty }\left( {1 + \ln \frac{r}{t}}\right) \frac{\mathop{\sup }\limits_{{x \in \Pi }}{\varphi }^{\frac{1}{{\theta p}\left( {x, t}\right) }}\left( {x, t}\right) {dt}}{{t}^{1 + \frac{n}{{\theta q}\left( {x, t}\right) }}} < \infty \] (2.5) for each \( \delta > 0 \) and \[ {\int }_{... | Yes |
Lemma 3.1 (see \( \left\lbrack {{19},{20}}\right\rbrack \) ) (Generalized Hölder’s inequality) Let \( D \subseteq {\mathbb{R}}^{n}, p\left( \cdot \right), q\left( \cdot \right) \in \) \( \mathcal{P}\left( {\mathbb{R}}^{n}\right) \) such that \( \frac{1}{p\left( x\right) } + \frac{1}{q\left( x\right) } \equiv 1 \) . If ... | \[ {\int }_{D}\left| {f\left( x\right) g\left( x\right) }\right| {dx} \leq C\parallel f{\parallel }_{{L}^{p\left( \cdot \right) }}\parallel g{\parallel }_{{L}^{q\left( \cdot \right) }}, \] with \( C = \mathop{\sup }\limits_{{x \in D}}\frac{1}{p\left( x\right) } + \mathop{\sup }\limits_{{x \in D}}\frac{1}{q\left( x\righ... | Yes |
Lemma 3.5 Let \( D \subseteq {\mathbb{R}}^{n} \) be an unbounded open set, \( b \in \operatorname{BMO}\left( {\mathbb{R}}^{n}\right) \) and \( p\left( \cdot \right) \in {\mathcal{P}}_{\infty }^{\log }\left( D\right) \) . Then for any \( x \in D,0 < {r}_{1},{r}_{2} < \infty \) and \( \widetilde{B}\left( {x,{r}_{i}}\righ... | Proof Let \( {p}^{ + } < {p}_{1} < \infty ,{p}_{2}\left( x\right) \in {\mathcal{P}}_{\infty }^{\log }\left( D\right) \), such that \( \frac{1}{p\left( x\right) } = \frac{1}{{p}_{1}} + \frac{1}{{p}_{2}\left( x\right) }, x \in D \) . Then by Lemma 3.2, Lemma 3.3 and Lemma 3.4, we obtain\n\n\[ \frac{1}{{\begin{Vmatrix}{\c... | Yes |
Lemma 3.7 (see [23]) Let \( b \in \operatorname{BMO}\left( {\mathbb{R}}^{n}\right), p\left( \cdot \right), q\left( \cdot \right) \in \mathcal{P}\left( {\mathbb{R}}^{n}\right) ,0 < \alpha < n,1 < {p}^{ - } \leq {p}^{ + } < \) \( \frac{n}{\alpha } \) and \( \frac{1}{q\left( x\right) } = \frac{1}{p\left( x\right) } - \fra... | \[ {\begin{Vmatrix}\left\lbrack b,{T}_{\Omega ,\alpha }\right\rbrack \left( f\right) \end{Vmatrix}}_{{L}^{q\left( \cdot \right) }\left( {\mathbb{R}}^{n}\right) } \leq C\parallel b{\parallel }_{\mathrm{{BMO}}}\parallel \Omega {\parallel }_{{L}^{s}\left( {\mathbb{S}}^{n - 1}\right) }\parallel f{\parallel }_{{L}^{p\left( ... | Yes |
Theorem 2.1 Let \( \nu ,\tau \) a and \( b \) be real positive numbers with \( 0 < \nu ,\tau < 1 \), then we have\n\n\[ \n\frac{a{\nabla }_{\nu }b - a{!}_{\nu }b}{a{\nabla }_{\tau }b - a{!}_{\tau }b} \leq \frac{\nu \left( {1 - \nu }\right) }{\tau \left( {1 - \tau }\right) }\text{ for }\left( {b - a}\right) \left( {\tau... | Proof Put \( f\left( v\right) = \frac{1 - v + {vx} - {\left( 1 - v + v{x}^{-1}\right) }^{-1}}{v\left( {1 - v}\right) } \), then we have \( {f}^{\prime }\left( v\right) = \frac{1}{{v}^{2}{\left( 1 - v\right) }^{2}}h\left( x\right) \), where\n\n\[ \nh\left( x\right) = v\left( {1 - v}\right) \left\lbrack {x - 1 + {\left( ... | Yes |
Theorem 2.3 Let \( \nu ,\tau, a \) and \( b \) are real numbers with \( 0 < \nu ,\tau < 1 \), then we have\n\n\[ \frac{{\left( a{\nabla }_{\nu }b\right) }^{2} - {\left( a{!}_{\nu }b\right) }^{2}}{{\left( a{\nabla }_{\tau }b\right) }^{2} - {\left( a{!}_{\tau }b\right) }^{2}} \leq \frac{\nu \left( {1 - \nu }\right) }{\ta... | Proof Put \( f\left( v\right) = \frac{{\left( 1 - v + vx\right) }^{2} - {\left( 1 - v + v{x}^{-1}\right) }^{-2}}{v\left( {1 - v}\right) } \), then we have \( {f}^{\prime }\left( v\right) = \frac{1}{{v}^{2}{\left( 1 - v}\right) }^{2}}h\left( x\right) \), where\n\n\[ h\left( x\right) = {2v}\left( {1 - v}\right) \left\lbr... | Yes |
Theorem 3.2 Let \( A, B \in {M}_{n}^{+ + } \) and \( 0 < \nu ,\tau < 1 \), then\n\n\[ \tau \left( {1 - \tau }\right) \left( {A{\nabla }_{\nu }B - A{!}_{\nu }B}\right) \leq \nu \left( {1 - \nu }\right) \left( {A{\nabla }_{\tau }B - A{!}_{\tau }B}\right) \]\n\nfor \( \left( {B - A}\right) \left( {\tau - \nu }\right) \geq... | Proof Let \( a = 1 \) in (2.1), for \( \left( {b - 1}\right) \left( {\tau - v}\right) \geq 0 \), then we have\n\n\[ \tau \left( {1 - \tau }\right) \left\lbrack {1 - \nu + {\nu b} - {\left( 1 - \nu + \nu {b}^{-1}\right) }^{-1}}\right\rbrack \]\n\n\[ \leq \nu \left( {1 - \nu }\right) \left\lbrack {1 - \tau + {\tau b} - {... | Yes |
Theorem 4.1 Let \( X \in {M}_{n} \) and \( B \in {M}_{n}^{+ + } \) for \( 0 < v,\tau < 1 \), then we have\n\n\[ \n\frac{\parallel \left( {1 - v}\right) X + {vXB}{\parallel }_{2}^{2} - {\begin{Vmatrix}{\left\lbrack \left( 1 - v\right) {X}^{-1} + v{B}^{-1}{X}^{-1}\right\rbrack }^{-1}\end{Vmatrix}}_{2}^{2}}{v\left( {1 - v... | Proof Since \( B \) is positive definite, it follows by spectral theorem that there exist unitary matrices \( V \in {M}_{n} \) such that \( B = {V\Lambda }{V}^{ * } \), where \( \Lambda = \operatorname{diag}\left( {{\nu }_{1},{\nu }_{2},\cdots ,{\nu }_{n}}\right) \) and \( {\nu }_{i} \) are eigenvalues of \( B \), so \... | Yes |
Example 4.3 Let \( B = \left( \begin{matrix} \frac{1}{2} & 0 \\ 0 & 1 \end{matrix}\right), A = \left( \begin{matrix} \frac{1}{3} & 0 \\ 0 & \frac{1}{2} \end{matrix}\right) \) and \( X = \left( \begin{matrix} 1 & - 1 \\ - 1 & 2 \end{matrix}\right) \), then (4.3) and (4.4) are not true for \( \nu = \frac{1}{2} \) and \( ... | Proof we can compute that\n\n\[ \parallel \left( {1 - \nu }\right) {AX} + {\nu XB}{\parallel }_{2}^{2} = \frac{53}{36} + \frac{23}{9}\nu + \frac{53}{36}{\nu }^{2} \]\n\nand\n\n\[ {\begin{Vmatrix}{\left( \left( 1 - \nu \right) {X}^{-1}{A}^{-1} + \nu {B}^{-1}{X}^{-1}\right) }^{-1}\end{Vmatrix}}_{2}^{2} = \frac{1}{{\left(... | Yes |
Theorem 5.2 Let \( X \in {M}_{n} \) and \( A, B \in {M}_{n}^{+ + } \) for \( 0 < v,\tau < 1 \), then we have for \( \left( {B - A}\right) \left( {\tau - v}\right) \leq 0 \)\n\n\[ \det {\left( A{!}_{v}B\right) }^{\frac{1}{n}} + \frac{v\left( {1 - \nu }\right) }{\tau \left( {1 - \tau }\right) }\det {\left( A{\nabla }_{\t... | Proof We may assume \( 0 < v < \tau < 1 \), then \( 0 < {s}_{j}\left( {{A}^{-\frac{1}{2}}B{A}^{-\frac{1}{2}}}\right) \leq 1 \) for \( \left( {B - A}\right) \left( {\tau - v}\right) \) \( \leq 0 \), so we have \( {A}^{-\frac{1}{2}}B{A}^{-\frac{1}{2}} \leq I \) . By inequality (2.2) and we denote the positive definite ma... | Yes |
Theorem 5.3 Let \( X \in {M}_{n} \) and \( A, B \in {M}_{n}^{+ + } \) for \( 0 < \nu ,\tau < 1 \), then we have for \( \left( {B - A}\right) \left( {\tau - \nu }\right) \leq 0 \) ,\n\n\[ \det {\left( A{!}_{\nu }B\right) }^{\frac{2}{n}} + \frac{\nu \left( {1 - \nu }\right) }{\tau \left( {1 - \tau }\right) }\det {\left( ... | Proof Using the same technique above to (2.4), we can easily get the proof of Theorem 5.3. | No |
Theorem 3.2 Let \( G \) be a plane (weakly) elementary bipartite graph \( G \) with a perfect matching \( M \) and let \( f \) be a proper \( M \)-alternating cell.\n\n(1) If \( G \) has no improper \( M \)-alternating cell (namely, \( M \) is the maximum element of \( \mathcal{M}\left( G\right) \) ), then every (prope... | Proof (1) It is trivial by the definition of \( Z \)-transformation directed graph.\n\n(2) First suppose that the cell \( f \) is a meet-irreducible cell with respect to \( M \), but there is at least one improper \( M \)-alternating cell \( {f}^{\prime } \) such that \( f \) and \( {f}^{\prime } \) are disjoint. Thus ... | Yes |
Theorem 4.2 Let \( L \) be a finite distributive lattice and \( x \in L \) . If \( x \) is covered by at least three elements and covers at least three meet-irreducible elements, then \( L \) is non-matchable. | Proof Suppose to the contrary that \( L \) is matchable. Then there is a plane (weakly) elementary bipartite graph \( G \) such that \( \mathcal{M}\left( G\right) \cong L\left\lbrack 6\right\rbrack \), which implies that a perfect matching \( {M}_{x} \) of \( G \) corresponds to \( x \in L \) . According to the premise... | Yes |
Theorem 4.4 The distributive lattice \( \mathcal{F}\left( \Delta \right) \) is non-matchable, where \( \Delta \) is the poset as shown in Figure 2(a). | Proof Recall that \( \mathcal{F}\left( \Delta \right) \) is a finite distributive lattice. Suppose that \( \mathcal{F}\left( \Delta \right) \) is matchable. Since \( \mathcal{F}\left( \Delta \right) \) has a cut element labeled by 0 (see Figure 2), and is irreducible, there exists a plane elementary bipartite graph \( ... | Yes |
Lemma 2.2 A right \( A \) -module \( M \) is in Tor \( A \) if and only if for each \( x \in M,\varphi \left( x\right) {A}_{Q} \) is finite dimensional over \( Q \) . | For a right \( A \) -module \( M \), let \( {\Gamma }_{A}\left( M\right) = \left\{ {x \in M \mid \varphi \left( x\right) {A}_{Q}}\right. \) is finite dimensional \( \} \) . Then \( {\Gamma }_{A}\left( M\right) \) is a torsion submodule of \( M \) . | No |
Lemma 2.3 With the above notions, we have\n\n(i) \( {\Gamma }_{A}\left( M\right) \) is the largest torsion submodule of \( M \) and \( M/{\Gamma }_{A}\left( M\right) \) is torsion free, which is to say, \( {\Gamma }_{A}\left( {M/{\Gamma }_{A}\left( M\right) }\right) = 0 \) .\n\n(ii) \( {\Gamma }_{A}{\left( M\right) }_{... | Proof Statement (i) is easy to check. We next prove statement (ii). For \( x \in M \) and an nonzero element \( s \in R \), we have \( \left( {x/s}\right) {A}_{Q} = \varphi \left( x\right) {A}_{Q} \) . It follows that \( \left( {x/s}\right) {A}_{Q} \) is finite dimensional if and only if \( \varphi \left( x\right) {A}_... | Yes |
Lemma 2.4 Let \( M \) be a right \( A \) -module. Let \( L \) be an \( {A}_{Q} \) -submodule of \( {M}_{Q} \) such that \( {M}_{Q}/L \) is finite dimensional. Then there is an \( A \) -submodule \( K \) of \( M \) such that \( {M}_{Q}/L \cong \) \( {M}_{Q}/{K}_{Q} \) and \( M/K \) is \( R \) -torsion free. | Proof Let \( \varphi : M \rightarrow {M}_{Q} \) be the localizing map, and \( K = \{ m \in M \mid \varphi \left( m\right) \in L\} \) . Then \( L = {K}_{Q} \) . By the construction, we see that \( M/K \) is \( R \) -torsion free. | Yes |
Proposition 2.1 Let \( M \) be a finitely generated right \( A \) -module. For every \( N \in \) \( \operatorname{Mod}A \), we have\n\n\[ \n{\operatorname{Hom}}_{Q\operatorname{Mod}{A}_{Q}}\left( {{\mathcal{M}}_{Q},{\mathcal{N}}_{Q}}\right) \cong {\operatorname{Hom}}_{Q\operatorname{Mod}A}{\left( \mathcal{M},\mathcal{N... | Proof We have the following computations\n\n\[ \n{\operatorname{Hom}}_{Q\operatorname{Mod}{A}_{Q}}\left( {{\mathcal{M}}_{Q},{\mathcal{N}}_{Q}}\right) = \mathop{\lim }\limits_{ \rightarrow }{\operatorname{Hom}}_{{A}_{Q}}\left( {L,{N}_{Q}/{\Gamma }_{{A}_{Q}}\left( {N}_{Q}\right) }\right)\n\]\n\n\[ \n= \mathop{\lim }\limi... | Yes |
Lemma 3.5 With the notions as above, \( {\left( {A}^{\operatorname{co}H}\right) }_{Q} \cong {\left( {A}_{Q}\right) }^{\operatorname{co}{H}_{Q}} \) . | Proof Let \( \varphi : A \rightarrow {A}_{Q} \) and \( \phi : H \rightarrow {H}_{Q} \) be the localizing maps. Applying \( {\left( -\right) }_{Q} \) to the inclusion map \( {A}^{\operatorname{co}H} \rightarrow A \), we obtain that \( {\left( {A}^{\operatorname{co}H}\right) }_{Q} \) is contained in \( {\left( {A}_{Q}\ri... | Yes |
Theorem 4.10 Let \( A \) be an almost commutative \( R \) -algebra such that \( {gr}\left( A\right) \) is a domain. Assume that \( A \) is a right \( H \) -comodule algebra such that the right \( H \) -coaction preserves the filtration. If \( A/{A}^{\operatorname{co}H} \) is a Hopf dense Galois extension and the canoni... | Proof As before, we write \( {A}_{\mathrm{k}} = A{ \otimes }_{R}\mathbb{k} \) and \( {H}_{\mathrm{k}} = H{ \otimes }_{R}\mathbb{k} \) . Since \( A \) is a filtered \( R \) -algebra, \( {A}_{\mathrm{k}} \) is also a filtered \( \mathbb{k} \) -algebra with the obvious induced filtration. Since the right \( H \) -coaction... | Yes |
Proposition 5.3 Let \( A \) be a filtered algebra with an ascending filtration\n\n\[ 0 \subseteq {F}_{0}A \subseteq {F}_{1}A \subseteq \cdots \subseteq {F}_{i}A \subseteq \cdots, i \in \mathbb{N} \]\n\nsuch that \( {F}_{i}A \) is finite dimensional for all \( i \geq 0 \) . Assume that \( A \) is a right \( H \) -comodu... | Proof Let \( \beta : A{ \otimes }_{{A}^{\infty H}}A \rightarrow A{ \otimes }_{\mathrm{k}}H \) be the canonical map. Similar to the diagram (4.2), we have the following commutative diagram\n\n\[ \begin{array}{l} \operatorname{gr}\left( A\right) { \otimes }_{{gr}\left( {A}^{\operatorname{co}H}\right) }\operatorname{gr}\l... | Yes |
Theorem 3.1 (Verification Theorem) Suppose there exist \( V\left( {t, w, x, y}\right) \) and \( g\left( {t, w, x, y}\right) \in \) \( {C}^{1,2,2,1}\left( {\left\lbrack {0, T}\right\rbrack \times R \times R\times \lbrack - 1,\infty }\right) ) \) satisfying the following conditions: for all \( \left( {t, w, x, y}\right) ... | For an admissible strategy \( {u}^{ * } = {\left\{ {\ell }^{ * }\left( {Z}_{t}, t\right) ,{\pi }_{m}^{ * }\left( t\right) ,{\pi }_{1}^{ * }\left( t\right) ,{\pi }_{2}^{ * }\left( t\right) \right\} }_{t \in \left\lbrack {0, T}\right\rbrack } \), the HJB function (3.1) follows\n\n\[ \mathop{\sup }\limits_{{u \in \Pi }}\l... | Yes |
Theorem 3.1 Assume that \( f \in {C}^{m}\left( I\right) ,\Delta \mathrel{\text{:=}} \left\{ {{x}_{0},{x}_{1},\cdots ,{x}_{N} : {x}_{n - 1} < {x}_{n}, n \in J}\right\} \) be an arbitrary partition of \( I = \left\lbrack {{x}_{0},{x}_{N}}\right\rbrack \) . There are suitable smooth functions \( b \) and \( {\alpha }_{n} ... | Proof For convenience, in the following \( k = 0,1,2,3,\cdots, N - 1 \) . Let \( {S}_{\Delta }\left( x\right) \) be prescribed in eq.(2.1), we choose smooth function \( b \in {C}^{m}\left( I\right) \) to satisfy\n\n\[ \n{b}^{\left( r\right) }\left( {x}_{0}\right) = {S}_{\Delta }^{\left( r\right) }\left( {x}_{0}\right) ... | Yes |
(1) If \( \alpha ,\beta \in {\Lambda }^{1} \) with \( \left\lbrack {{T}_{\alpha },{T}_{\beta }}\right\rbrack \neq 0 \), then \( \alpha \) is connected with \( \beta \) . | Proof (1) Suppose \( \left\lbrack {{T}_{\alpha },{T}_{\beta }}\right\rbrack \neq 0 \), by Lemma 2.8 (1), one gets \( \delta \left( {\alpha + \beta }\right) \in {\Lambda }^{0} \cup \{ 0\} \) . If \( \alpha + \beta = 0 \), then \( \beta = - \alpha \) and so \( \alpha \) is connected with \( \beta \) . Suppose \( \alpha +... | Yes |
Lemma 3.5 Fix \( {\alpha }_{0} \in {\Lambda }^{1} \) and suppose \( {\Lambda }^{0} \) is symmetric. Then the following assertions hold.\n\n(1) If \( \alpha \in {\Lambda }_{{\alpha }_{0}}^{1},\beta \in {\Lambda }^{1},\beta \neq - \alpha \), and \( \left\lbrack {{T}_{0},{T}_{\alpha },{T}_{\beta }}\right\rbrack \neq 0 \) ... | Proof (1) By Lemma 2.8 (5), we have \( \alpha + {\delta \beta } \in {\Lambda }^{1} \) . From here, if \( \left\{ {{\alpha }_{1},\cdots ,{\alpha }_{{2n} + 1}}\right\} \) is a connection from \( {\alpha }_{0} \) to \( \alpha \), then \( \left\{ {{\alpha }_{1},\cdots ,{\alpha }_{{2n} + 1},0,\beta }\right\} \) is a connect... | Yes |
Lemma 3.10 Fix \( {\alpha }_{0} \in {\Lambda }^{1} \) and suppose \( {\Lambda }^{0} \) is symmetric. If \( \alpha ,\beta ,\gamma \in {\Lambda }_{{\alpha }_{0}}^{1} \cup \{ 0\} \) with \( \alpha + \beta + {\delta \gamma } = 0,\xi \in {\Lambda }_{{\alpha }_{0}}^{1} \) and \( \bar{\epsilon },\bar{\rho },\bar{\tau } \in {\... | Proof (1) Suppose \( \left\lbrack {{T}_{\alpha },{T}_{\beta },\left\lbrack {{T}_{\epsilon },{T}_{\bar{\rho }},{T}_{\bar{\tau }}}\right\rbrack }\right\rbrack \neq 0 \), then either \( \left\lbrack {{T}_{\beta },\left\lbrack {{T}_{\epsilon },{T}_{\bar{\rho }},{T}_{\bar{\tau }}}\right\rbrack ,{T}_{\alpha }}\right\rbrack \... | Yes |
Theorem 3.11 Suppose \( {\Lambda }^{0} \) is symmetric. Then for a vector space complement \( U \) of \( {\operatorname{span}}_{\mathbb{K}}\left\{ {\left\lbrack {{T}_{\alpha },{T}_{\beta },{T}_{\gamma }}\right\rbrack : \alpha + \beta + {\delta \gamma } = 0\text{, where}\alpha ,\beta ,\gamma \in {\Lambda }^{1}\cup \{ 0\... | Proof Let us denote \( {\xi }_{0} \mathrel{\text{:=}} {\operatorname{span}}_{\mathbb{K}}\left\{ {\left\lbrack {{T}_{\alpha },{T}_{\beta },{T}_{\gamma }}\right\rbrack : \alpha + \beta + {\delta \gamma } = 0\text{, where}\alpha ,\beta ,\gamma \in {\Lambda }^{1}\cup \{ 0\} }\right\} \) in \( {T}_{0} \) . By Proposition 3.... | Yes |
When we set sparsity to be \( s = {10},{25},{50} \), Figure 1 shows how the successful rate changes in terms of the sample size \( m \) . In this experiment, we fix a number \( K = \lfloor (s(\log n + \) \( \left. {\log \frac{1}{0.01}}\right) )\rbrack \), which is \( {115},{287},{575} \) with respect to the sparsity \(... | It shows when the sample size is in order \( O\left( {s\log n}\right) \) in this setting, we can recover the signal with high possibility. | Yes |
Theorem 3.1 Suppose that \( u \) is the solution of problem (1.1) and \( {u}_{\alpha }^{\delta } \) is the solution of problem (2.6). Let the measured data \( {\varphi }^{\delta } \) satisfy \( \begin{Vmatrix}{{\varphi }^{\delta } - \varphi }\end{Vmatrix} \leq \delta \), and the exact solution \( u \) satisfy\n\n\[ \fr... | Proof Denoting \( {u}_{\alpha } \) as the corresponding solution of problem (2.6) with the exact data \( \varphi \) . Using the triangle inequality, we have\n\n\[ \begin{Vmatrix}{{u}_{\alpha }^{\delta } - u}\end{Vmatrix} \leq \begin{Vmatrix}{{u}_{\alpha }^{\delta } - {u}_{\alpha }}\end{Vmatrix} + \begin{Vmatrix}{{u}_{\... | Yes |
Theorem 3.2 Let \( {u}_{⓫}^{\delta } \) and \( {u}_{⓬}^{\delta } \) be the solutions of problem (2.7) corresponding to the data \( {\varphi }_{1}^{\delta } \) and \( {\varphi }_{2}^{\delta } \), respectively, then for \( \alpha < T \), we have\n\n\[ \begin{Vmatrix}{{u}_{⓫}^{\delta }\left( {\cdot, y}\right) - {u}_{⓬}^{\... | where \( {C}_{3} = \sqrt{8{C}_{4}^{2}{T}_{0}^{2}\left( {1 + {2y}{L}^{2}{C}_{4}^{2}{T}_{0}^{3}{e}^{{2y}{L}^{2}{C}_{4}^{2}{T}_{0}^{3}}}\right) },{C}_{4} = \frac{2}{1 - {e}^{-{2T}}} \). | Yes |
Lemma 1 If \( f\left( x\right) \in {\operatorname{Lip}}_{M}\beta \), then for \( n = 1,2,\cdots, i, j, i + j = 0,1,\cdots, n - 1 \) , \[ \left| {{g}_{i + j} - {g}_{i}}\right| \leq M\left\lbrack {\left( {1 - \frac{j}{n - 1}}\right) {\left( \frac{j}{n}\right) }^{\beta } + \frac{j}{n - 1}{\left( \frac{j + 1}{n}\right) }^{... | Proof It easily follows from (1) that \[ \left| {{f}_{i + j + 1} - {f}_{i + 1}}\right| ,\left| {{f}_{i + j} - {f}_{i}}\right| \leq M{\left( \frac{j}{n}\right) }^{\beta },\left| {{f}_{i + j + 1} - {f}_{i + j}}\right| \leq \frac{M}{{n}^{\beta }}. \] It follows from (2) that \[ \left| {{g}_{i + j} - {g}_{i}}\right| = \lef... | Yes |
Lemma 2 If \( f\left( x\right) \) is convex on \( \left\lbrack {0,1}\right\rbrack \), then \( {T}_{n,\alpha }\left( {f;x}\right) \geq f\left( x\right) \) for all \( x \in \left\lbrack {0,1}\right\rbrack \) and \( n \geq 1 \) . | Proof For each \( x \in \left\lbrack {0,1}\right\rbrack \), let us define \( {x}_{r} = \frac{r}{n} \) and \( {\lambda }_{r} = {p}_{n, r}^{\left( \alpha \right) }\left( x\right) ,0 \leq r \leq n \) . We see that \( {\lambda }_{r} \geq 0 \) when \( x \in \left\lbrack {0,1}\right\rbrack \), and note from \( {T}_{n,\alpha ... | Yes |
Lemma 3 For \( f\left( x\right) = {x}^{\beta },0 < \beta \leq 1 \) and for \( n = 1,2,\cdots ,{T}_{n,\alpha }\left( {{x}^{\beta };h}\right) \leq {h}^{\beta },0 \leq h \leq 1 \) . | Proof As a consequence of Lemma 2, we have, since the function \( f\left( x\right) = - {x}^{\beta },0 < \) \( \beta \leq 1 \), is convex on \( \left\lbrack {0,1}\right\rbrack \), that for \( n = 1,2,\cdots ,{T}_{n,\alpha }\left( {-{x}^{\beta };x}\right) \geq - {x}^{\beta } \) . It follows from \( {T}_{n,\alpha }\left( ... | Yes |
Theorem 1.2 Let \( \varphi : {M}^{n} \rightarrow {R}^{n + p} \) be an oriented and compact \( n \) -dimension isometric immersion submanifold without boundary. Then the following integral formulas hold.\n\n\[{\int }_{M}\left( {{H}_{k} + {H}_{k + 1}\langle \varphi ,\xi \rangle }\right) {dM} = 0, k = 0,1,2,\cdots, n - 1,... | Proof Let \( {T}_{\xi } \) be the shape operator of \( {M}^{n} \) along the direction of the unit mean curvature vector field \( \xi \), that is to say, \( {T}_{\xi } \) is a tensor field of type \( \left( {1,1}\right) \) on \( {M}^{n} \) defined by\n\n\[{T}_{\xi }\left( X\right) = - {\left( {\bar{\nabla }}_{X}\xi \rig... | Yes |
Lemma 4.4 Let \( {u}_{n} = u\left( {X,{t}_{n}}\right) ,\forall {v}_{h} \in {V}_{n + 1}^{h} \), then there holds\n\n\[ \left( {{Q}_{n + 1} - {Q}_{n},{v}_{h}}\right) + \left( {\alpha {Q}_{n + \frac{1}{2}},{v}_{h}}\right) {\Delta t} + {a}_{h}\left( {{u}_{n + \frac{1}{2}},{v}_{h}}\right) {\Delta t} + \left( {g\left( {u}_{n... | Proof From (3.2), for all \( v \in {V}_{n + 1}^{h} \) ,\n\n\[ \left( {{Q}_{n + 1} - {Q}_{n},{v}_{h}}\right) + \left( {{\int }_{{t}_{n}}^{{t}_{n + 1}}{\alpha Qd\tau },{v}_{h}}\right) + {a}_{h}\left( {{\int }_{{t}_{n}}^{{t}_{n + 1}}{ud\tau },{v}_{h}}\right) + \left( {{\int }_{{t}_{n}}^{{t}_{n + 1}}g\left( u\right) {d\tau... | Yes |
Theorem 2.1 If \( f\left( z\right) \) given by (1.1) belongs to the class \( \mathcal{Q}{\mathcal{H}}_{\sum {\beta }_{1},\cdots ,{\beta }_{s}}\left( {\eta ;\phi }\right) \), then\n\n\[ \left| {a}_{2}\right| \leq \min \left\{ {\frac{{E}_{1}}{\left( \eta + 1\right) },\frac{2\left( {\left| {{E}_{2} - {E}_{1}}\right| + {E}... | Proof If \( f\left( z\right) \in \mathcal{{QH}} > \mathop{\sum }\limits_{{{\beta }_{1},\cdots ,{\beta }_{s}}}^{{{\alpha }_{1},\cdots ,{\alpha }_{q}}}\left( {\eta ;\phi }\right) \), then by Definition 1.1 and Lemma 1.3, there exist two analytic functions \( u\left( z\right) \) and \( v\left( w\right) \in \mathcal{P} \) ... | Yes |
Theorem 2.2 If \( f\left( z\right) \) given by (1.1) belongs to the class \( \mathcal{Q}\mathcal{H}\mathop{\sum }\limits_{{{\beta }_{1},\cdots ,{\beta }_{s}}}^{{{\alpha }_{1},\cdots ,{\alpha }_{q}}}\left( {\eta ;\phi }\right) \) and \( \delta \in \mathbb{R} \) , then | Proof From (2.25), it follows that\n\n\[ {a}_{3} - \frac{{p}_{2}^{2}\left( {q, s}\right) {a}_{2}^{2}}{{p}_{3}\left( {q, s}\right) } = \frac{{B}_{1}{E}_{1}\left( {{c}_{1} - {d}_{1}}\right) + {B}_{0}{E}_{1}\left( {{c}_{2} - {d}_{2}}\right) }{4\left( {\eta + 2}\right) {p}_{3}\left( {q, s}\right) }.\]\n\nBy (2.24) we easil... | Yes |
Theorem 3.1 If \( f\left( z\right) \) given by (1.1) belongs to the class \( \mathcal{Q}{\mathcal{S}}_{\mathop{\sum }\limits_{{{\beta }_{1},\cdots ,{\beta }_{s}}}\left( {\tau ,\mu ,\lambda ,\gamma ;\phi }\right) }\left( {\tau ,\mu ,\lambda ,\gamma ;\phi }\right) \) , then\n\n\[ \left| {a}_{2}\right| \leq \min \left\{ {... | Proof Here, we follow the method of Theorem 2.1. If \( f\left( z\right) \in \mathcal{Q}{\mathcal{S}}_{\sum {\beta }_{1},\cdots ,{\beta }_{s}}\left( {\tau ,\mu ,\lambda ,\gamma ;\phi }\right) \) , then by Definition 1.2 there exist two analytic functions \( u\left( z\right), v\left( z\right) : \Delta \rightarrow \Delta ... | Yes |
Theorem 3.2 Let \( f\left( z\right) \) given by (1.1) belong to the class \( \mathcal{Q}{\mathcal{S}}_{\sum {\beta }_{1},\cdots ,{\beta }_{s}}^{{\alpha }_{1},\cdots ,{\alpha }_{q}}\left( {\tau ,\mu ,\lambda ,\gamma ;\phi }\right) \) and \( \delta \in \mathbb{R} \) . Then\n\n\[ \left| {{a}_{3} - \delta {a}_{2}^{2}}\righ... | Proof From (3.18), it follows that\n\n\[ {a}_{3} - {a}_{2}^{2} = \frac{{B}_{1}{E}_{1}\tau \left( {{c}_{1} - {d}_{1}}\right) {\left( 1 + \gamma - \gamma \lambda \right) }^{2}}{4{p}_{3}\left( {q, s}\right) \Theta \left( {\mu ,\lambda ,\gamma }\right) } + \frac{{B}_{0}{E}_{1}\tau \left( {{c}_{2} - {d}_{2}}\right) {\left( ... | Yes |
Corollary 0.3 Let \( b \in {\widetilde{\mathbb{L}}}_{x, t}^{p,\infty } \) with \( \operatorname{div}b = 0 \), where \( p \in \left\lbrack {1,\infty }\right\rbrack \cap \left( {\frac{d - 1}{2},\infty }\right\rbrack \) . For any \( T > 0 \) and \( f \in {\widetilde{\mathbb{L}}}_{t, x}^{{q}^{\prime },{p}^{\prime }} \), wh... | \[ \parallel u{\parallel }_{{L}^{\infty }\left( {\left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{d}}\right) } \leq C{\begin{Vmatrix}f{\mathbf{1}}_{\left\lbrack 0, T\right\rbrack }\end{Vmatrix}}_{{\widetilde{\mathbb{L}}}_{t, x}^{{q}^{\prime },{p}^{\prime }}}. \] | Yes |
Theorem 0.6 Under \( \left( {\widetilde{\mathbf{H}}}^{\sigma }\right) \) and \( \left( {\widetilde{\mathbf{H}}}^{b}\right) \), for any \( x \in {\mathbb{R}}^{d} \), there is at least one weak solution \( \left( {\mathfrak{F}, X, W}\right) \) for SDE (15). Moreover, for any \( \left( {p, q}\right) \in {\mathbb{I}}_{{p}_... | \[ \mathbf{E}\left( {{\int }_{\tau }^{\tau + \delta }f\left( {s,{X}_{s}}\right) \mathrm{d}s \mid {\mathcal{F}}_{\tau }}\right) \leq C{\delta }^{\theta }\parallel f{\parallel }_{{\widetilde{\mathbb{L}}}_{t, x}^{q, p}}. \] (18) | Yes |
Example 0.7 Let \( d \geq 3 \) and \( \alpha \in \left( {0,\left( {\frac{d}{2} - 1}\right) \land \left( {\frac{1}{2} + \frac{1}{d - 1}}\right) }\right) ,\beta \in \left( {0,{2\alpha }}\right) \) . For any \( \lambda \geq 0 \) and \( x \in {\mathbb{R}}^{d} \), the following SDE admits a unique strong solution:\n\n\[ \ma... | Note that the starting point can be zero and the uniqueness follows from [8]. | No |
Example 0.8 The following two dimensional degenerate SDE admits a solution:\n\n\[ \left\{ \begin{array}{l} \mathrm{d}{X}_{t}^{1} = {\left| {X}_{t}^{2}\right| }^{\alpha }\mathrm{d}{W}_{t}^{1} + {b}^{1}\left( {X}_{t}\right) \mathrm{d}t \\ \mathrm{\;d}{X}_{t}^{2} = {\left| {X}_{t}^{1}\right| }^{\alpha }\mathrm{d}{W}_{t}^{... | More details can be found in [9]. | No |
Lemma 0.1 Assume the time step ratios \( {r}_{k} \) satisfy A1. For any real sequence \( {\left\{ {w}_{k}\right\} }_{k = 1}^{n} \) , it holds that\n\n\[ 2{w}_{k}\mathop{\sum }\limits_{{j = 1}}^{k}{b}_{k - j}^{\left( k\right) }{w}_{j} \geq \frac{{r}_{k + 1}}{\left( 1 + {r}_{k + 1}\right) }\frac{{w}_{k}^{2}}{{\epsilon }_... | The positive semi-definiteness of \( {\theta }_{n - k}^{\left( n\right) } \) is derived by the positive semi-definiteness of \( {b}_{n - k}^{\left( n\right) } \) . | No |
Lemma 0.4 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 |
The truncation error \( {\eta }^{j} \mathrel{\text{:=}} {\mathcal{D}}_{2}u\left( {t}_{j}\right) - {\partial }_{t}u\left( {t}_{j}\right) \left( {1 \leq j \leq N}\right) \) can be expressed the following form | \[ {\eta }^{j} = \mathop{\sum }\limits_{{l = 1}}^{j}{b}_{j - l}^{\left( j\right) }{G}^{l} + {R}^{j},\;1 \leq j \leq N. \] | Yes |
Theorem 0.4 Let \( u\left( {t, x}\right) \) be the solution to problem (1). If the BDF2 kernels \( {b}_{n - k}^{\left( n\right) } \) defined in (4) are positive semi-definite (or the condition A1 holds), then the solution \( {u}^{n} \) to BDF2 scheme (2) is convergent in the \( {L}^{2} \) -norm. If \( \kappa > 0 \) and... | \[ \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\left( {{\tau }_{1} + \tau }\right) {\int }_{0}^{{t}_{1}}\begin{Vmatrix}{{\partial }_{tt}u}\end{Vmatrix}\mathrm{d}t + \mathop{\sum }\limi... | Yes |
Lemma 3.3 [31] (a) The space of all Lebesgue integrable functions and the spaces of restricted Denjoy and wide Denjoy integrable functions are dense in \( {D}_{HK} \) . | Since the primitive \( F \) of a HK integrable function \( f \) is continuous and \( {F}^{\prime }\left( x\right) = f\left( x\right) \) is almost everywhere. It is easy to see that \( {HK} \subset {D}_{HK} \) . By Lemma 2.5 and Lemma 3.3, the following corollary holds. | No |
Theorem 4.1 Let \( f \in {D}_{HK} \) . Then there exists a sequence \( \left\{ {f}_{n}\right\} \) of HK integrable functions such that\n\n(1) for every \( g \in \mathcal{B}\mathcal{V},\mathop{\lim }\limits_{{n \rightarrow \infty }}{\int }_{a}^{b}{f}_{n}g = {\int }_{a}^{b}{fg} \) ;\n\n(2) \( \mathop{\lim }\limits_{{n \r... | Proof (1) By Lemma 3.3, the space \( {D}_{HK} \) is a separable Banach space. By Corollary 3.4, the space \( {HK} \) is dense in \( {D}_{HK} \) . So, for \( f \in {D}_{HK} \), there exists a sequence \( \left\{ {f}_{n}\right\} \) of \( \mathrm{{Hk}} \) integrable functions satisfying\n\n\[ \begin{Vmatrix}{{f}_{n} - f}\... | Yes |
Theorem 4.2 Assume that \( \left\{ {f}_{n}\right\} \) is a sequence of HK integrable functions satisfying\n\n(1) for every \( g \in \mathcal{B}\mathcal{V},{\int }_{a}^{b}{f}_{n}g \) converges;\n\n(2) \( {\int }_{a}^{b}{f}_{n}g \) uniformly converges on \( O\left( \mathcal{{BV}}\right) \) .\n\nThen there exists \( f \in... | Proof For each \( n \in \mathbb{N} \), the fact that \( {f}_{n} \in {HK} \) and \( {HK} \subset {D}_{HK} \) implies \( {f}_{n} \in {D}_{HK} \) . Since \( {\int }_{a}^{b}{f}_{n}g \) converges for every \( g \in \mathcal{B}\mathcal{V} \) then \( \left\{ {{\int }_{a}^{b}{f}_{n}g}\right\} \) is a Cauchy sequence. Denote\n\... | Yes |
Theorem 4.4 A distribution \( f \in {D}_{HK} \) with the primitive \( F \) if and only if there exists a sequence \( \left\{ {f}_{n}\right\} \) of HK integrable functions with the primitives \( {F}_{n} \) satisfying\n\n(1) \( \left\{ {F}_{n}\right\} \) is bounded in \( C\left( \left\lbrack {a, b}\right\rbrack \right) \... | Proof (Necessity) Assume that \( f \in {D}_{HK} \) . Then \( F \in {C}_{0} \) . By Theorem 4.3, there exists a sequence \( \left\{ {f}_{n}\right\} \) of Henstock-Kurzweil integrable functions satisfying (1) and (2) in Theorem 4.2. This is, for every \( g \in \mathcal{B}\mathcal{V},{\int }_{a}^{b}{f}_{n}g \) converges t... | Yes |
Example 5.2 Suppose that \( x \in \left\lbrack {0,1}\right\rbrack \) . Let\n\n\[ F\left( x\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{\sin {n}^{2}{x\pi }}{{n}^{2}},\;\text{ and }\;{F}_{n}\left( x\right) = \mathop{\sum }\limits_{{k = 1}}^{n}\frac{\sin {k}^{2}{x\pi }}{{k}^{2}}. \]\n\nThen\n\n\[ F\left( x\rig... | Since for each \( n \in \mathbb{N} \), the function \( {F}_{n}\left( x\right) \) is continuous on \( \left\lbrack {0,1}\right\rbrack \) and \( {F}_{n}\left( x\right) \) uniformly converges to \( F\left( x\right) \), so \( F \) is continuous on \( \left\lbrack {0,1}\right\rbrack \) and \( F\left( 0\right) = 0 \), which ... | Yes |
Lemma 6.1 Assume that \( \left\{ {f}_{n}\right\} \) is a sequence of the HK integrable functions on \( \left\lbrack {a, b}\right\rbrack \) satisfying\n\n(1) for every \( g \in \mathcal{B}\mathcal{V},\left\{ {\left( {HK}\right) {\int }_{a}^{b}{f}_{n}g}\right\} \) converges;\n\n(2) the sequence \( \left\{ {F}_{n}\right\}... | Proof Taking \( g\left( x\right) = {\chi }_{\left\lbrack a, x\right\rbrack } \in \mathcal{{BV}} \) . Since \( {f}_{n} \in {HK} \) and \( \left\{ {\left( {HK}\right) {\int }_{a}^{b}{f}_{n}g}\right\} \) converges for \( g \in \mathcal{B}\mathcal{V} \), one has\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}{F}_{n}\... | Yes |
Lemma 6.2 Assume that \( f \in \mathcal{H}{\mathcal{K}}_{\left\lbrack a, b\right\rbrack } \) . Then there exists a sequence \( \left\{ {f}_{n}\right\} \) of HK integrable functions on \( \left\lbrack {a, b}\right\rbrack \) satisfying\n\n(1) for every \( g \in \mathcal{B}\mathcal{V},\left\{ {\left( {HK}\right) {\int }_{... | Proof Let \( f \in {HK} \) with primitive \( F \in {AC}{G}^{ * } \) . According to [26, p198, Exercise 5.7], there is a sequence \( \left\{ {\varphi }_{n}\right\} \) of step functions such that \( {\varphi }_{n} \rightarrow f \) is almost everywhere and the primitives \( {\Phi }_{n} \) of \( {\varphi }_{n} \) are unifo... | Yes |
Theorem 6.4 Assume that \( \mathcal{A} \subset {HK} \) and \( \mathcal{B} = \left\{ {F : F\left( t\right) = {\int }_{a}^{t}f, f \in \mathcal{A}}\right\} \) . Then \( \mathcal{A} \) is relatively compact in \( {HK} \) if and only if for every sequence \( \left\{ {f}_{n}\right\} \subset \mathcal{A} \), there exists a sub... | Proof The proof is directly deduced from Theorem 6.3. | No |
Theorem 3.4 Let \( \\left( {X, M,\\Delta , \\leq }\\right) \) be a complete ordered fuzzy metric space with \( \\Delta \) a triangular norm of H-type. Let \( \\Phi = \\left( {{\\sigma }_{1},{\\sigma }_{2},\\cdots ,{\\sigma }_{n},\\tau }\\right) \) be \( \\left( {n + 1}\\right) \) -tuple of mappings from \( {\\Lambda }_... | Proof Let \( Y = {X}^{n} \) . For \( \\left( {{x}_{1},{x}_{2},\\cdots ,{x}_{n}}\\right) ,\\left( {{y}_{1},{y}_{2},\\cdots ,{y}_{n}}\\right) \\in {X}^{n}, t > 0,{M}^{ * } \) and binary relation \( \\preccurlyeq \) on \( Y \) are defined as\n\n\[ {M}^{ * }\\left( {\\left( {{x}_{1},{x}_{2},\\cdots ,{x}_{n}}\\right) ,\\lef... | No |
Theorem 3.5 Let \( \left( {X, M,\Delta , \leq }\right) \) be a complete ordered fuzzy metric space with \( \Delta \) a triangular norm of H-type. Let \( \Phi = \left( {{\sigma }_{1},{\sigma }_{2},\cdots ,{\sigma }_{n},\tau }\right) \) be \( \left( {n + 1}\right) \) -tuple of mappings from \( {\Lambda }_{n} \) into itse... | \[ M\left( {F\left( {{x}_{1},{x}_{2},\cdots ,{x}_{n}}\right), F\left( {{y}_{1},{y}_{2},\cdots ,{y}_{n}}\right) ,\varphi \left( t\right) }\right) \geq \mathop{\min }\limits_{{1 \leq i \leq n}}M\left( {g{x}_{i}, g{y}_{i}, t}\right) \] for which \( g{x}_{\tau \left( i\right) }{ \leq }_{i}g{y}_{\tau \left( i\right) } \) fo... | Yes |
Corollary 3.6 Let \( \\left( {X, M,\\Delta , \\leq }\\right) \) be a complete ordered fuzzy metric space with \( \\Delta \) a triangular norm of H-type. \( \\varphi : \\lbrack 0, + \\infty ) \\rightarrow \\lbrack 0, + \\infty ),\\varphi \\in {\\Psi }_{\\omega } \), Let \( F : {X}^{2} \\rightarrow X \) and \( g : X \\ri... | \[ M\\left( {F\\left( {{x}_{1},{x}_{2}}\\right), F\\left( {{y}_{1},{y}_{2}}\\right) ,\\varphi \\left( t\\right) }\\right) \\geq \\min \\left\{ {M\\left( {g{x}_{1}, g{y}_{1}, t}\\right), M\\left( {g{x}_{2}, g{y}_{2}, t}\\right) }\\right\} \] for which \( g{x}_{1} \\leq g{y}_{1} \) and \( g{x}_{2} \\geq g{y}_{2} \) . If ... | Yes |
Corollary 3.9 Let \( \\left( {X, M,\\Delta , \\leq }\\right) \) be a complete ordered fuzzy metric space with \( \\Delta \) a triangular norm of H-type. Let \( \\Phi = \\left( {{\\sigma }_{1},{\\sigma }_{2},\\cdots ,{\\sigma }_{n},\\tau }\\right) \) be \( \\left( {n + 1}\\right) \) -tuple of mappings from \( {\\Lambda ... | \[ M\\left( {F\\left( {{x}_{1},{x}_{2},\\cdots ,{x}_{n}}\\right), F\\left( {{y}_{1},{y}_{2},\\cdots ,{y}_{n}}\\right) ,{kt}}\\right) \\geq \\mathop{\\min }\\limits_{{1 \\leq i \\leq n}}M\\left( {g{x}_{i}, g{y}_{i}, t}\\right) \] for which \( g{x}_{\\tau \\left( i\\right) }{ \\leq }_{i}g{y}_{\\tau \\left( i\\right) } \)... | Yes |
Corollary 3.10 Let \( \\left( {X, M,\\Delta , \\leq }\\right) \) be a complete ordered fuzzy metric space with \( \\Delta \) a triangular norm of H-type. Let \( \\Phi = \\left( {{\\sigma }_{1},{\\sigma }_{2},\\cdots ,{\\sigma }_{n},\\tau }\\right) \) be \( \\left( {n + 1}\\right) \) -tuple of mappings from \( {\\Lambda... | \[ M\\left( {F\\left( {{x}_{1},{x}_{2},\\cdots ,{x}_{n}}\\right), F\\left( {{y}_{1},{y}_{2},\\cdots ,{y}_{n}}\\right) ,{kt}}\\right) \\geq \\mathop{\\min }\\limits_{{1 \\leq i \\leq n}}M\\left( {g{x}_{i}, g{y}_{i}, t}\\right) \] for which \( g{x}_{\\tau \\left( i\\right) }{ \\leq }_{i}g{y}_{\\tau \\left( i\\right) } \)... | Yes |
Corollary 3.11 Let \( \left( {X, M,\Delta , \leq }\right) \) be a complete ordered fuzzy metric space with \( \Delta \) a triangular norm of H-type. Let \( \Phi = \left( {{\sigma }_{1},{\sigma }_{2},\cdots ,{\sigma }_{n},\tau }\right) \) be \( \left( {n + 1}\right) \) -tuple of mappings from \( {\Lambda }_{n} \) into i... | \[ M\left( {F\left( {{x}_{1},{x}_{2},\cdots ,{x}_{n}}\right), F\left( {{y}_{1},{y}_{2},\cdots ,{y}_{n}}\right) ,{kt}}\right) \geq \mathop{\min }\limits_{{1 \leq i \leq n}}M\left( {g{x}_{i}, g{y}_{i}, t}\right) \] for which \( g{x}_{\tau \left( i\right) }{ \leq }_{i}g{y}_{\tau \left( i\right) } \) for all \( i \in {\Lam... | No |
Let \( R \) be a 3-torsion free ring and let \( L \) be a \( \delta \) Jordan-Lie triple system over ring \( R \) . Let \( D : L \rightarrow L \) be a k-order generalized Jordan triple \( \left( {\theta ,\varphi }\right) \) -derivation of weight \( \lambda \) with respect to \( \alpha \), where \( \alpha \) is a k-orde... | By (2.2) and (2.4), we have\n\n\[ \left( {D - \alpha }\right) \left( \left\lbrack {x, y, x}\right\rbrack \right) = D\left( \left\lbrack {x, y, x}\right\rbrack \right) - \alpha \left( \left\lbrack {x, y, x}\right\rbrack \right) \]\n\n\[ = {\delta }^{k}\left\lbrack {\alpha \left( x\right) ,\theta \left( y\right) ,\varphi... | Yes |
Proposition 2.10 If \( D : L \rightarrow L \) is a k-order generalized Jordan triple \( \left( {\theta ,\varphi }\right) \) - derivation of weight \( \lambda \) with respect to \( \alpha \) satisfying (2.7), where \( \alpha \) is a k-order Jordan triple \( \left( {\theta ,\varphi }\right) \) -derivation of weight \( \l... | Proof If \( \lambda = 0 \), then \( D\left( \left\lbrack {x, y, x}\right\rbrack \right) = \alpha \left( \left\lbrack {x, y, x}\right\rbrack \right) \) . By (1.3) and (1.4)\n\n\[D\left( \left\lbrack {y, x, x}\right\rbrack \right) = \alpha \left( \left\lbrack {y, x, x}\right\rbrack \right) ,\;D\left( \left\lbrack {x, x, ... | Yes |
Theorem 2.12 Let \( R \) be a 3-torsion free ring and let \( D : L \rightarrow L \) be a k-order generalized Jordan triple \( \left( {\theta ,\varphi }\right) \) -derivation of weight \( \lambda \) with respect to the k-order Jordan triple \( \left( {\theta ,\varphi }\right) \) -derivation \( \alpha \) of weight \( \la... | Proof It follows from Theorem 2.11 that \( \alpha \) is a k-order \( \left( {\theta ,\varphi }\right) \) -derivation of weight \( \lambda \) . Applying Proposition 2.9, we get from (2.6) that\n\n\[B\left( {\lambda, k}\right) \left( {x, y, z}\right) + B\left( {\lambda, k}\right) \left( {y, z, x}\right) + B\left( {\lambd... | Yes |
Corollary 2.14 Let \( R \) be a 3-torsion free ring. Then \( D : L \rightarrow L \) is a k-order Jordan triple \( \theta \) -derivation of weight \( \lambda \) if and only if \( D \) is a k-order \( \theta \) -derivation of weight \( \lambda \) . | Proof It is clear that condition (2.6) of Corollary 2.13 is valid when \( D = \alpha \) . | No |
Corollary 2.15 Let \( R \) be a 3-torsion free ring. Then \( D : L \rightarrow L \) is a k-order Jordan triple derivation of weight \( \lambda \) if and only if \( D \) is a k-order derivation of weight \( \lambda \) . | Proof This is a special case that \( \theta = {I}_{L} \) in Corollary 2.14, where \( {I}_{L} \) is the identity map on \( L \) . | Yes |
Corollary 2.17 Let \( R \) be a 3-torsion free ring. Let \( A \) be a linear automorphism of \( L \) . Then \( D : L \rightarrow L \) is a k-order Jordan triple derivation of weight \( \lambda \) if and only if \( {A}^{n}D{A}^{-n} \) is a k-order derivation of weight \( \lambda \) for all positive integer \( n \) . | Proof If \( D \) is a k-order Jordan triple derivation of weight \( \lambda \), it follows Corollary 2.15 that \( D \) is a k-order derivation of weight \( \lambda \) . And from Theorem 2.16, \( {AD}{A}^{-1} \) is also a k-order derivation of weight \( \lambda \) . By mathematical induction, \( {A}^{n}D{A}^{-n} \) is a... | Yes |
Theorem 2.18 If \( D \) is a k-order derivation of \( \delta \) Jordan-Lie triple system \( L, Z\\left( L\\right) \) is the center of \( L \), then \( D\\left( {Z\\left( L\\right) }\\right) \\subseteq Z\\left( L\\right) \) . | Proof For arbitrary element \( x \) in \( Z\\left( L\\right) \) and for all \( y, z \\in L \), we have \( \\left\\lbrack {x, y, z}\\right\\rbrack = 0 \). Since \( D \) is a k-order derivation, we have\n\n\\[ \n{\\delta }^{k}\\left\\lbrack {D\\left( x\\right), y, z}\\right\\rbrack = D\\left( \\left\\lbrack {x, y, z}\\ri... | Yes |
Corollary 3.7 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 \( \left( {\theta ,\varphi }\right) \) -derivation of weight \( \lambda \) on \( L \) satisfying the relation any two of \( D, p,\alpha ,\... | Proof It is the direct results of Theorem 3.6. | Yes |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.