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Lemma 2.9 If for any \( \alpha \in \pi, h \in {H}_{\alpha }, a \in {A}_{\alpha } \), \n\n\[ \n\sum {a}_{\left( \left\lbrack 0\right\rbrack ,\alpha \right) }{\theta }_{\alpha }\left( {{S}_{\alpha }\left( {a}_{\left( {\left\lbrack 0\right\rbrack, i}\right) 1{\alpha }^{-1}}\right) h{a}_{\left( {\left\lbrack 1\right\rbrack... | Proof Indeed for any \( \alpha \in \pi, h \in {H}_{\alpha }, a \in {A}_{\alpha } \), we have \n\n\[ \n{\xi }_{\alpha }\left( {\left\lbrack {m \otimes h}\right\rbrack \cdot a}\right) = {\left( m \cdot {a}_{\left( \left\lbrack 0\right\rbrack, i\right) }\right) }_{\left( \left\lbrack 0\right\rbrack ,\alpha \right) }{\thet... | Yes |
Theorem 1.2 For \( - 1 < \gamma < 1 \), let \( {u}_{\varepsilon } \) be the solution of problem (1.1) with (1.2). Suppose that the initial data satisfy (1.8)-(1.10). Let \( {u}_{1} \) be the solution of the homogenized problem (1.4), then we have the following corrector results: | \[ {\begin{Vmatrix}\widetilde{{u}_{1\varepsilon }^{\prime }} + \widetilde{{u}_{2\varepsilon }^{\prime }} - {u}_{1}^{\prime }\end{Vmatrix}}_{{L}^{2}\left( {0, T;{L}^{2}\left( \Omega \right) }\right) } \rightarrow 0, \] \[ {\begin{Vmatrix}\nabla {u}_{1\varepsilon } - \nabla {u}_{1} - \mathop{\sum }\limits_{{i = 1}}^{n}{\... | Yes |
Theorem 2.6 Let \( {u}_{\varepsilon } = \left( {{u}_{1\varepsilon },{u}_{2\varepsilon }}\right) \) and \( \left\{ {u}_{\varepsilon }\right\} \) be in \( {L}^{\infty }\left( {0, T;{H}_{\gamma }^{\varepsilon }}\right) \) with \( - 1 < \gamma < 1 \) . If\n\n\[ \n{\begin{Vmatrix}{u}_{\varepsilon }\end{Vmatrix}}_{{L}^{\inft... | In fact, the proof can be obtained by following the lines of the proofs of [Theorem 2.12, 14] (see also [Theorem 2.19, 9]) and [Theorem 2.20, 13]. | No |
Theorem 3.1 Under the assumptions of Theorem 1.1, there exist \( {u}_{1} \in {L}^{\infty }\left( {0, T;{H}_{0}^{1}\left( \Omega \right) }\right) \) with \( {u}_{1}^{\prime } \in {L}^{\infty }\left( {0, T;{L}^{2}\left( \Omega \right) }\right) \) and \( {\widehat{u}}_{1} \in {L}^{\infty }\left( {0, T;{L}^{2}\left( {\Omeg... | The proofs of Theorem 3.1 and Theorem 1.1 mainly rely on the periodic unfolding method. Indeed, following the lines of proof of Theorem 3.1 [4], we can use Theorem 2.6 to obtain the proofs of these two theorems. | Yes |
Corollary 4.2 Under the assumptions of Theorem 4.1, we have\n\n(i) \( {\begin{Vmatrix}{u}_{i\varepsilon }^{\prime }\end{Vmatrix}}_{{L}^{2}\left( {0, T;{L}^{2}\left( {\Lambda }_{i\varepsilon }\right) }\right) } \rightarrow 0,{\begin{Vmatrix}\nabla {u}_{1\varepsilon }\end{Vmatrix}}_{{L}^{2}\left( {0, T;{L}^{2}\left( {\La... | To prove this corollary, we need the following classical result.\n\nProposition 4.3 (see [1 | No |
If \( \left( {\delta - r}\right) \left( {\beta + \delta }\right) \frac{{f}_{3}\left( 0\right) }{{f}_{3}^{\prime }\left( 0\right) } + {\beta c} \leq 0 \), then the function\n\n\[ V\left( x\right) = \frac{\beta c}{\left( {\beta + \delta }\right) \left( {\beta + \delta - r}\right) }\left\lbrack {1 - \frac{{f}_{3}\left( x\... | Proof If \( \left( {\delta - r}\right) \left( {\beta + \delta }\right) \frac{{f}_{3}\left( 0\right) }{{f}_{3}\left( 0\right) } + {\beta c} \leq 0 \), it is straightforward to verify that the function given by (4.29) is twice differentiable and satisfies the differential equation (4.2) with boundary condition \( V\left(... | Yes |
Proposition 4.2 The optimal dividend strategy is a barrier strategy. The barrier is 0 if\n\[ \n\\left( {\\delta - r}\\right) \\left( {\\beta + \\delta }\\right) \\frac{{f}_{3}\\left( 0\\right) }{{f}_{3}^{\\prime }\\left( 0\\right) } + {\\beta c} \\leq 0\n\]\nor determined by (4.23) if\n\[ \n\\left( {\\delta - r}\\right... | The functions \( V\\left( x\\right) \) given by (4.29) and (4.30) are the optimal value functions respectively. | No |
Lemma 1 (see [13])\n\n\\[ \nP{I}_{\\Omega }^{a}\\left( {P, Q}\\right) \\approx {t}^{-1}V\\left( t\\right) W\\left( r\\right) \\varphi \\left( \\Theta \\right) \\frac{\\partial \\varphi \\left( \\Phi \\right) }{\\partial {n}_{\\Phi }}, \n\\]\n\n(3.1)\n\n\\[ \n\\text{(resp.}P{I}_{\\Omega }^{a}\\left( {P, Q}\\right) \\app... | \\[ \nP{I}_{\\Omega }^{0}\\left( {P, Q}\\right) \\lesssim \\frac{\\varphi \\left( \\Theta \\right) }{{t}^{n - 1}}\\frac{\\partial \\varphi \\left( \\Phi \\right) }{\\partial {n}_{\\Phi }} + \\frac{{r\\varphi }\\left( \\Theta \\right) }{{\\left| P - Q\\right| }^{n}}\\frac{\\partial \\varphi \\left( \\Phi \\right) }{\\pa... | Yes |
Lemma 2 (see [13])\n\n\[ \n{G}_{\Omega }^{a}\left( {P, Q}\right) \approx V\left( t\right) W\left( r\right) \varphi \left( \Theta \right) \frac{\partial \varphi \left( \Phi \right) }{\partial {n}_{\Phi }}, \n\]\n\n(3.4)\n\n\[ \n\text{(resp.}{G}_{\Omega }^{a}\left( {P, Q}\right) \approx V\left( r\right) W\left( t\right) ... | \[ \n{G}_{\Omega }^{0}\left( {P, Q}\right) \lesssim \frac{\varphi \left( \Theta \right) }{{t}^{n - 2}}\frac{\partial \varphi \left( \Phi \right) }{\partial {n}_{\Phi }} + \frac{{rt\varphi }\left( \Theta \right) }{{\left| P - Q\right| }^{n}}\frac{\partial \varphi \left( \Phi \right) }{\partial {n}_{\Phi }} \n\]\n\n(3.6)... | Yes |
Theorem 3.5 Let \( H \) be a finite-type monoidal Hom-Hopf \( T \) -coalgebra. Then \( D\left( H\right) \) is a crossed monoidal Hom-Hopf \( T \) -coalgebra with the following structures:\n\n- For any \( \alpha \in G \) , \( \alpha \) th component \( {D}_{\alpha }\left( H\right) \) is an associative algebra with the mu... | Proof First, for any \( \alpha \in G \), we will show that \( {D}_{\alpha }\left( H\right) \) is an Hom-associative algebra with unit. Then we will show that \( \Delta \), defined as above, is multipilcative, i.e., that any \( {\Delta }_{\alpha ,\beta } \) is an algebra map. After that, we show that \( \varepsilon \) i... | Yes |
Lemma 2.1 If the Lebasgue measure of the set \( A = \{ \left( {t, s}\right) \in \left\lbrack {0, T}\right\rbrack \times \left\lbrack {0, S}\right\rbrack : a\left( {t, s}\right) < 0\} \) is positive, then we cannot attained the minimum in (2.1) at a function \( a \in A \) . | Proof\n\n\[ E{\left\lbrack {W}^{\alpha ,\beta }\left( t, s\right) - M\left( t, s\right) \right\rbrack }^{2} \]\n\n\[ = {\int }_{0}^{t}{\int }_{0}^{s}{K}_{\alpha }^{2}\left( {t, u}\right) {K}_{\beta }^{2}\left( {s, v}\right) {dudv} - 2{\int }_{0}^{t}{\int }_{0}^{s}{K}_{\alpha }\left( {t, u}\right) {K}_{\beta }\left( {s,... | Yes |
Lemma 3.1 (1) The function \( f\left( {t, s, k}\right) \) admits the following representation: | Proof By the straightforward calculations, we have\n\n\[ f\left( {t, s, k}\right) = {t}^{2\alpha }{s}^{2\beta } - {2kE}\left\lbrack {{W}^{\alpha ,\beta }\left( {t, s}\right) {\int }_{0}^{t}{\int }_{0}^{s}{u}^{\frac{1}{2} - \alpha }{v}^{\frac{1}{2} - \beta }B\left( {{du},{dv}}\right) }\right\rbrack \]\n\n\[ + {k}^{2}{\i... | Yes |
Theorem 3.2 Let \( A = \left\{ {a\left( {t, s}\right) = k{t}^{\frac{1}{2} - \alpha }{s}^{\frac{1}{2} - \beta }, k > 0,\alpha ,\beta \in \left( {\frac{1}{2},1}\right) }\right\} \), then\n\n\[ \mathop{\min }\limits_{{a \in A}}\mathop{\max }\limits_{\substack{{0 \leq t \leq T} \\ {0 \leq s \leq S} }}E{\left\lbrack {W}^{\a... | Proof First of all, we calculate the value of the constant \( k \) which makes \( \mathop{\max }\limits_{\substack{{0 \leq t \leq T} \\ {0 \leq s \leq S} }}f\left( {t, s, k}\right) \)\na minimal value.\n\nFollowing assertion (2) of Lemma 3.1, we have\n\n\[ \mathop{\max }\limits_{\substack{{0 \leq t \leq T} \\ {0 \leq s... | Yes |
Proposition 2.3 (1) If \( M \in A \) -Mod is Gorenstein flat, then \( A\# H{ \otimes }_{A}M \) is Gorenstein flat as a left \( A\# H \) -module. | (1) Since \( M \) is a Gorenstein flat left \( A \) -module, we have an exact sequence\n\n\[ \mathfrak{F} \equiv \cdots \rightarrow {F}^{-2} \rightarrow {F}^{-1} \rightarrow {F}^{0} \rightarrow {F}^{1} \rightarrow \cdots \]\n\nof flat left \( A \) -modules such that \( M = \ker \left( {{F}^{0} \rightarrow {F}^{1}}\righ... | Yes |
Proposition 2.4 Assume that \( A\# H/A \) is separable and \( \varphi : A \rightarrow A\# H \) is a splitting monomorphism of \( \left( {A, A}\right) \) -bimodules. Then \( A \) is a right coherent ring if and only if \( A\# H \) is a right coherent ring. | Proof Let \( {\left\{ {F}_{i}\right\} }_{i \in I} \) be a family of flat left \( A\# H \) -modules, then \( {}_{A}\left( {F}_{i}\right) \) is flat as a left \( A \) -module for every \( i \) . If we consider the adjoint pair \( \left( {A\# H{ \otimes }_{A}-,{}_{A}\left( -\right) }\right) \), we know that \( {}_{A}\left... | Yes |
Lemma 2.6 (1) If \( N \in A \) -Mod is Gorenstein cotorsion, then \( A\# H{ \otimes }_{A}N \) is Gorenstein cotorsion as a left \( A\# H \) -Mod. | Proof (1) Let \( N \) be any Gorenstein cotorsion left \( A \) -module and \( F \) any Gorenstein flat left \( A\# H \) -module. For \( F \) we have an exact sequence \( 0 \rightarrow K \rightarrow P \rightarrow F \rightarrow 0\left( *\right) \) of left \( A\# H \) -modules with \( P \) projective. Since \( {}_{A}\left... | Yes |
Proposition 2.7 Let \( M \in A\# H \) -Mod and \( N \in A \) -Mod. Then\n\n(1) \( {\operatorname{Gcd}}_{A}\left( {{}_{A}M}\right) \leq {\operatorname{Gcd}}_{A\# H}\left( M\right) \) .\n\n(2) \( {\operatorname{Gcd}}_{A\# H}\left( {A\# H{ \otimes }_{A}N}\right) \leq {\operatorname{Gcd}}_{A}\left( N\right) \) . | Proof (1) Assume that \( {Gc}{d}_{A\# H}\left( M\right) = n < \infty \), then there exists an exact sequence of left \( A\# H \) -modules\n\n\[ 0 \rightarrow M \rightarrow {C}^{0} \rightarrow {C}^{1} \rightarrow \cdots \rightarrow {C}^{n} \rightarrow 0 \]\n\nwith every \( {C}^{i} \) being Gorenstein cotorsion. By Lemma... | Yes |
Theorem 2.8 Assume that \( A\# H/A \) is separable and \( \varphi : A \rightarrow A\# H \) is a splitting monomorphism of \( \left( {A, A}\right) -bimodules, then \( l.{Gcd}\left( A\right) = l.{Gcd}\left( {A\# H}\right) \) . | Proof Let \( M \) be any left \( A \) -module. Since \( \varphi : A \rightarrow A\# H \) is a splitting monomorphism of \( \left( {A, A}\right) -bimodules, \( M \) is a direct summand of \( {}_{A}\left( {A\# H{ \otimes }_{A}M}\right) \) . Hence\n\n\[ \n{\operatorname{Gcd}}_{A}\left( M\right) \leq {\operatorname{Gcd}}_{... | Yes |
Corollary 2.9 Let \( A \) be a \( k \) -algebra and \( G \) a finite group with \( {\left| G\right| }^{-1} \in k \) . Then \( l \cdot {Gcd}\left( A\right) = l \cdot {Gcd}\left( {A * G}\right) . | Proof By the definition of the skew group ring, we know that \( A \) is a left \( H \) -module algebra and \( A * G = A\# H \), where \( H = {kG} \) . Since \( G \) a finite group with \( {\left| G\right| }^{-1} \in k, H \) is semisimple. Then from [19], we know that \( A\# H/A \) is separable. By [1, Lemma 4.5], we kn... | Yes |
Proposition 2.1 (Local existence) Under the assumptions of Theorem 1.1, there exists a sufficiently small positive constant \( {t}_{1} \) depending only on \( {m}_{0},{m}_{1},{\begin{Vmatrix}{\rho }_{0} - \bar{\rho }\end{Vmatrix}}_{4} \) and \( {\begin{Vmatrix}{u}_{0}\end{Vmatrix}}_{3} \) such that the Cauchy problems ... | The proof of Proposition 2.1 can be done by using the dual argument and iteration technique, which is similar to that of Theorem 1.1 in [10] and thus omitted here for brevity. | No |
Lemma 2.2 There exists a uniform positive constant \( {c}_{0} \) such that\n\n\[ G\left( {\rho ,\bar{\rho }}\right) \geq {c}_{0}\frac{{\left( \rho - \bar{\rho }\right) }^{2}}{\rho + \bar{\rho }}. \] | Proof Using the L' Hospital rule, we obtain\n\n\[ \mathop{\lim }\limits_{{\rho \rightarrow \bar{\rho }}}\frac{G\left( {\rho ,\bar{\rho }}\right) \left( {\rho + \bar{\rho }}\right) }{{\left( \rho - \bar{\rho }\right) }^{2}} = \mathop{\lim }\limits_{{\rho \rightarrow \bar{\rho }}}\frac{{G}_{\rho }\left( {\rho ,\bar{\rho ... | Yes |
Lemma 2.3 (Lower and upper bounds of \( \rho \left( {t, x}\right) \) for the cases (ii) (a) of Theorem 1.1) Under the assumptions of Proposition 2.2, if the capillarity coefficient \( \kappa \left( \rho \right) \) satisfies the condition (ii) (a) of Theorem 1.1, then there exists a positive constant \( {C}_{3} \) depen... | Proof Let\n\n\[ \n\Psi \left( \rho \right) = \frac{{\left( \rho - \bar{\rho }\right) }^{2}}{\rho + \bar{\rho }},\Upsilon \left( \rho \right) = {\int }_{1}^{\rho }\sqrt{\Psi \left( s\right) }\sqrt{\kappa \left( s\right) }{ds}, \n\]\n\nthen under the condition (ii)(a) of Theorem 1.1, we have\n\n\[ \n\Upsilon \left( \rho ... | Yes |
Lemma 2.4 Let condition (i) of Theorem 1.1 holds and\n\n\[ f\left( \rho \right) = - \frac{2}{3}{\left( \sqrt{\frac{\mu \left( \rho \right) \kappa \left( \rho \right) }{\rho }}{\left( \sqrt{\frac{\mu \left( \rho \right) \kappa \left( \rho \right) }{\rho }}\right) }^{\prime }\right) }^{\prime } + {\left( {\left( \sqrt{\f... | Proof First, by the continuity equation \( {\left( {1.1}\right) }_{1} \), we have\n\n\[ \frac{1}{\rho }{\left( \mu \left( \rho \right) {u}_{x}\right) }_{x} = - \frac{1}{\rho }{\left( \frac{\mu \left( \rho \right) }{\rho }\left( {\rho }_{t} + {\rho }_{x}u\right) \right) }_{x} = - \frac{1}{\rho }{\left( \frac{\mu \left( ... | Yes |
Lemma 2.5 Let conditions (i) and (ii) (b) of Theorem 1.1 hold and \( f\left( \rho \right) \leq 0 \), then there exists a positive constant \( {C}_{5} \) depending only on \( {m}_{0},{m}_{1} \) and \( \begin{Vmatrix}\left( {{\rho }_{0} - \bar{\rho },{u}_{0},{\rho }_{0x}}\right) \end{Vmatrix} \) such that\n\n\[ \n{C}_{5}... | Proof Set\n\n\[ \n\bar{\Psi }\left( \rho \right) = {\int }_{1}^{\rho }\sqrt{\Psi \left( s\right) }\frac{\mu \left( s\right) }{{s}^{3/2}}{ds} \n\]\n\nthen it follows from assumption (ii) (b) of Theorem 1.1 that\n\n\[ \n\bar{\Psi }\left( \rho \right) \rightarrow \left\{ \begin{array}{ll} - \infty , & \rho \rightarrow 0, ... | Yes |
Lemma 2.6 There exists a positive constant \( {C}_{7} \) depending only on \( {m}_{0},{m}_{1} \) and \( \parallel \left( {{\rho }_{0} - }\right. \) \( \bar{\rho },{u}_{0},{\rho }_{0x}\parallel \) such that for \( 0 \leq t \leq T \) , \[ {\begin{Vmatrix}{\rho }_{x}\left( t\right) \end{Vmatrix}}^{2} + {\int }_{0}^{t}{\be... | Proof We derive from Lemmas 2.3-2.5 that \[ {\begin{Vmatrix}{\rho }_{x}\left( t\right) \end{Vmatrix}}^{2} + {\int }_{0}^{t}{\begin{Vmatrix}{\rho }_{x}\left( \tau \right) \end{Vmatrix}}^{2}{d\tau } \leq C{\begin{Vmatrix}\left( {\rho }_{0} - \bar{\rho },{u}_{0},{\rho }_{0x}\right) \end{Vmatrix}}^{2}. \] On the other hand... | Yes |
Lemma 2.8 There exists a positive constant \( {C}_{9} \) depending only on \( {m}_{0},{m}_{1},{\begin{Vmatrix}{\rho }_{0} - \bar{\rho }\end{Vmatrix}}_{2} \) and \( {\begin{Vmatrix}{u}_{0}\end{Vmatrix}}_{1} \) such that for \( t \in \left\lbrack {0, T}\right\rbrack \) , \[{\begin{Vmatrix}{\rho }_{xx}\left( t\right) \end... | Proof Differentiating \( {\left( {1.1}\right) }_{2} \) once with respect to \( x \), then multiplying the resultant equation by \( {\rho }_{xx} \), and using equation \( {\left( {1.1}\right) }_{1} \), we have \[ {\left( \frac{\mu \left( \rho \right) }{2{\rho }^{2}}{\rho }_{xx}^{2} + {u}_{x}{\rho }_{xx}\right) }_{t} + \... | Yes |
Proposition 2.2 We have the following solutions of (2.1), \[ F = \left| y\right| \sqrt{f\left( {\frac{{\left| x\right| }^{2}}{2},\frac{\langle x, y\rangle }{\left| y\right| }}\right) },\;f\left( {t, s}\right) = \sqrt{\frac{{c}_{2}{s}^{2}}{{\left( c + t\right) }^{3}} - \frac{{c}_{2}{s}^{4}}{2{\left( c + t\right) }^{4}}}... | Now let us consider the solution given by \[ f\left( {t, s}\right) = {\left( \mathop{\sum }\limits_{{j = 0}}^{l}{f}_{j}\left( t\right) {s}^{j}\right) }^{\frac{1}{r}},\;{f}_{l} \neq 0,\;r \in N - \{ 0,1\} . \] By a direct calculation, \[ r{f}^{r - 1}{f}_{t} = \mathop{\sum }\limits_{{j = 0}}^{l}{f}_{j}^{\prime }\left( t\... | Yes |
Proposition 3.1 We have the following solutions of (2.1), \[ F = \left| y\right| \sqrt{f\left( {\frac{{\left| x\right| }^{2}}{2},\frac{\langle x, y\rangle }{\left| y\right| }}\right) },\;f\left( {t, s}\right) = \frac{1}{\sqrt{1 - {2t} + {s}^{2}}}\left\lbrack {\lambda \left( t\right) s + \left( {\frac{1}{6}{\lambda }^{\... | Combine Propositions 2.1, 3.1, (3.23) and the fundamental property of the dually flat eq.(1.1), Theorem 1.2 can be achieved. | No |
Consider the \( \left( {{2n} + 1}\right) \) -dimensional Euclidean space \( {\mathbb{R}}^{{2n} + 1} \) equipped with the Cartesian coordinates \( \left( {{x}_{1},\cdots ,{x}_{n},{y}_{1},\cdots ,{y}_{n}, z}\right) \) . Define the almost contact structure \( \left( {\phi ,\xi ,\eta, g}\right) \) by\n\n\[ \phi \left( \fra... | It is clear that there exists a non-zero solution \( {V}^{i} = c,\;{\bar{V}}^{i} = 0,\;{V}^{z} = H\left( {{x}_{1},\cdots ,{x}_{n}}\right) \) , where \( H \) is a smooth function on \( {\mathbb{R}}^{{2n} + 1} \) and \( c \) is a non-zero constant. Hence we see that \( c\mathop{\sum }\limits_{{i = 1}}^{n}\frac{\partial }... | Yes |
Proposition 2.3 For a \( {\phi }^{ * } \) -analytic vector field \( v \) in almost contact manifold \( \left( {M,\phi ,\xi ,\eta, g}\right) \) , the following identity holds: \( g\left( {{\nabla }_{X}v,\xi }\right) + g\left( {{\nabla }_{\xi }v, X}\right) = 0,\forall X \in \mathfrak{X}\left( M\right) \) . | Proof From the definition of the \( {\phi }^{ * } \) -analytic vector field we know \( \left\lbrack {\xi, v}\right\rbrack = 0 \), i.e., \( {\nabla }_{v}\xi = {\nabla }_{\xi }v \) . Using (2.1), a straightforward computation yields\n\n\[ \left\lbrack {{\phi }^{2}X, v}\right\rbrack = {\nabla }_{{\phi }^{2}X}v - {\nabla }... | Yes |
Corollary 2.4 If \( \left( {M,\phi ,\xi ,\eta, g}\right) \) is an almost contact manifold with \( {\phi }^{ * } \) -analytic vector field \( v \), then \( v\left( f\right) = 0 \) . In particular, if \( v \in \mathcal{D} \) the integral curves of \( \xi \) are geodesics. | Proof The first assertion is obvious by making use of Proposition 2.3 with \( X = v \) . If \( v \in \mathcal{D} \), then from Proposition 2.3 with \( X = \xi \), we have \( g\left( {v,{\nabla }_{\xi }\xi }\right) = 0 \), i.e., \( {\nabla }_{\xi }\xi \in \operatorname{Span}\{ \xi \} \) , so \( {\nabla }_{\xi }\xi = 0 \... | Yes |
Corollary 2.5 Let \( v \) be a \( {\phi }^{ * } \) -analytic vector in almost contact manifold \( \left( {M,\phi ,\xi ,\eta, g}\right) \) . If \( \eta \) is closed, then \( f \) is constant. | Proof Applying Proposition 2.3, for any vector field \( X \), we have\n\n\[ \n{d\eta }\left( {X, v}\right) = X\left( {\eta \left( v\right) }\right) - v\left( {\eta \left( X\right) }\right) - \eta \left( \left\lbrack {X, v}\right\rbrack \right) = g\left( {v,{\nabla }_{X}\xi }\right) - g\left( {X,{\nabla }_{\xi }v}\right... | Yes |
Corollary 2.6 Let \( \left( {M,\phi ,\eta ,\xi, g}\right) \) be a contact metric manifold with \( {\phi }^{ * } \) -analytic field \( v \) . Then \( {\nabla }_{v}\xi - \phi {\nabla }_{\phi v}\xi = - {2\phi v} \) and \( \xi \left( f\right) = 0 \) . | Proof Using (2.7) for \( X = v \) and \( X = {\phi v} \), respectively, we have\n\n\[ \n{\nabla }_{v}\xi = - {\phi v} - {Av},\;{\nabla }_{\phi v}\xi = v - {f\xi } - {A\phi v}.\n\]\n\nTherefore it completes the proof the first assertion in view of (2.6) and the above two equations.\n\nOn other hand, in terms of Proposit... | Yes |
Theorem 3.1 A \( {\phi }^{ * } \) -analytic vector field \( v \) in a contact metric manifold \( M \) is Killing. | Proof In view of Proposition 2.3, for any \( X \in \mathfrak{X}\left( M\right) \), we have\n\n\[ \left( {v\lrcorner {d\eta }}\right) \left( X\right) = {d\eta }\left( {v, X}\right) = - X\left( f\right) ,\]\n\nwhich means that \( v\lrcorner {d\eta } = - {df} \) . Thus \( {\mathcal{L}}_{v}{d\eta } = d\left( {v\lrcorner {d... | Yes |
Theorem 3.4 Let \( \left( {M,\phi ,\eta ,\xi, g}\right) \) be a contact metric manifold with \( {\phi }^{ * } \) -analytic field \( v \) . Then \( {\phi v} \) must be not a \( {\phi }^{ * } \) -analytic vector field unless zero vector. | To prove this theorem we need the following two lemmas. | No |
Lemma 3.5 Under the assumption of Theorem 3.4, if \( {\phi v} \) is also a \( {\phi }^{ * } \) -analytic vector field, then \( {\nabla }_{\xi }v = - {\phi v} \) . | Proof Since \( v\lrcorner {d\eta } = - {df} \), using (2.5) we have \( {\phi v} = {Df} \), where \( D \) denotes by the gradient operator. Thus if \( {\phi v} \) is \( {\phi }^{ * } \) -analytic, \( \left\lbrack {{\phi X},{Df}}\right\rbrack = \phi \left\lbrack {X,{Df}}\right\rbrack \) for any \( X \in \mathfrak{X}\left... | Yes |
Lemma 3.6 Under the assumption of Theorem 3.4, if \( {\phi v} \) is also a \( {\phi }^{ * } \) -analytic vector field then\n\n\[ \left( {{\nabla }_{X}\phi }\right) v = g\left( {v, X}\right) \xi - {fX}. \] | Proof For any contact metric manifold the following identity (see [2, Lemma 7.3.2]) holds\n\n\[ {2g}\left( {\left( {{\nabla }_{X}\phi }\right) Y, Z}\right) = g\left( {{N}_{\phi }\left( {Y, Z}\right) ,{\phi X}}\right) + {2d\eta }\left( {{\phi Y}, X}\right) \eta \left( Z\right) - {2d\eta }\left( {{\phi Z}, X}\right) \eta... | Yes |
Theorem 3.7 Let \( v \) be a \( {\phi }^{ * } \) -analytic vector field in a Sasakian manifold. Then \( f = \) \( g\left( {v,\xi }\right) \) is constant and \( v \) is collinear to \( \xi \) with constant length. | Proof In the proof of Theorem 3.1, we have known \( v\lrcorner {d\eta } = - {df} \) . Since \( M \) is Sasakian, \( {N}_{\phi } = 0 \) . Thus it follows from (2.4) and (2.9) that \( {N}_{\phi }\left( {X, v}\right) = {2d\eta }\left( {X, v}\right) \xi = 0 \) for every field \( X \), that means that \( {df} = 0 \) . On th... | Yes |
We consider \( {\mathbb{R}}^{{2n} + 1} \) equipped with contact structure \( \left( {\phi ,\xi ,\eta, g}\right) \) as in Example 2.2. Let \( v = \mathop{\sum }\limits_{{i = 1}}^{n}\left( {{V}^{i}\frac{\partial }{\partial {x}_{i}} + {\bar{V}}^{i}\frac{\partial }{\partial {x}_{i}}}\right) + {V}^{z}\frac{\partial }{\parti... | \[ {\phi v} = \mathop{\sum }\limits_{{i = 1}}^{n}\left( {-{V}^{i}\frac{\partial }{\partial {y}_{i}} + {\bar{V}}^{i}\frac{\partial }{\partial {x}_{i}}}\right) + \mathop{\sum }\limits_{{i = 1}}^{n}{y}_{i}\frac{\partial }{\partial z}. \] If \( {\phi v} \) is also \( {\phi }^{ * } \) -analytic, we get \[ {y}_{i}\frac{\part... | Yes |
Lemma 2.2 If \( K, L \in {\mathcal{S}}_{o}^{n} \), then for \( {u}_{0} \in {S}^{n - 1} \) | \[ \frac{1}{\left| K\right| \left| L\right| }{\int }_{K}{\int }_{L}\varphi \left( \frac{\left| {u}_{0} \cdot \left( y - z\right) \right| }{{\lambda }_{0}}\right) \mathrm{d}y\mathrm{\;d}z = 1 \] if and only if \( {\rho }_{{I}_{\varphi }\left( {K, L}\right) }^{-1}\left( {u}_{0}\right) = {\lambda }_{0} \). | Yes |
Lemma 2.3 (see [34]) Let \( x = \left( {{x}_{1},{x}_{2},\cdots ,{x}_{n}}\right) \in {\mathbb{R}}^{n} \) and \( y = \left( {{y}_{1},{y}_{2},\cdots ,{y}_{n}}\right) \in {\mathbb{R}}^{n} \) . If \( 0 < {m}_{1} \leq {x}_{k} \leq {M}_{1},0 < {m}_{2} \leq {y}_{k} \leq {M}_{2}, k = 1,\cdots, n \), then | \[ \left( {\mathop{\sum }\limits_{{k = 1}}^{n}{x}_{k}^{2}}\right) \left( {\mathop{\sum }\limits_{{k = 1}}^{n}{y}_{k}^{2}}\right) \leq {\left( \frac{\sqrt{\frac{{M}_{1}{M}_{2}}{{m}_{1}{m}_{2}}} + \sqrt{\frac{{m}_{1}{m}_{2}}{{M}_{1}{M}_{2}}}}{2}\right) }^{2}{\left( \mathop{\sum }\limits_{{k = 1}}^{n}{x}_{k}{y}_{k}\right)... | Yes |
Lemma 2.5 Suppose \( \varphi \in \Phi \) . If \( {K}_{i},{L}_{i} \in {\mathcal{S}}_{o}^{n} \) and \( {K}_{i} \rightarrow K \in {\mathcal{S}}_{o}^{n},{L}_{i} \rightarrow L \in {\mathcal{S}}_{o}^{n} \), then \( {I}_{\varphi }\left( {{K}_{i},{L}_{i}}\right) \rightarrow {I}_{\varphi }\left( {K, L}\right) \) | Proof For \( \varphi \in \Phi \) and \( \varphi \) is convex, continuous, and either strictly increasing on \( \left( {-\infty ,0}\right) \) or strictly decreasing on \( \left( {0,\infty }\right) \), then for \( {u}_{0} \in {S}^{n - 1} \), we will show that\n\n\[ \n{\rho }_{{I}_{\varphi }\left( {{K}_{i},{L}_{i}}\right)... | Yes |
Lemma 2.6 For \( K, L \in {\mathcal{S}}_{o}^{n} \) and \( {\varphi }_{i} \rightarrow \varphi \in \Phi \), then \( {I}_{{\varphi }_{i}}\left( {K, L}\right) \rightarrow {I}_{\varphi }\left( {K, L}\right) \) . | Proof Let \( K, L \in {\mathcal{S}}_{o}^{n} \) and \( {u}_{0} \in {S}^{n - 1} \) . We will show that \( {\rho }_{{I}_{{\varphi }_{i}}\left( {K, L}\right) }^{-1} \rightarrow {\rho }_{{I}_{\varphi }\left( {K, L}\right) }^{-1} \) . For \( \varphi \in \Phi \) with \( \varphi \) is convex, continuous, and either strictly in... | Yes |
Lemma 2.7 Suppose \( \varphi \in \Phi \) . For \( K, L \in {\mathcal{S}}_{o}^{n} \) and a linear transformation \( T \in \mathrm{{GL}}\left( n\right) \) , then \( {I}_{\varphi }\left( {{TK},{TL}}\right) = {T}^{-t}\left( {{I}_{\varphi }\left( {K, L}\right) }\right) \) . | Proof Suppose \( {x}_{0} \in {\mathbb{R}}^{n} \) and\n\n\[ \n{\rho }^{-1}\left( {{I}_{\varphi }\left( {{TK},{TL}}\right) ,{x}_{0}}\right) = {\lambda }_{0} \n\]\n\n(2.11)\n\nLet \( s = {Ty}, t = {Tz} \), then \( \left| {TK}\right| = \left| {\det T}\right| \left| K\right| ,\left| {TL}\right| = \left| {\det T}\right| \lef... | Yes |
Lemma 3.1 Suppose \( \varphi \in \Phi \) is strictly convex and \( K, L \in {\mathcal{K}}_{o}^{n} \) . If \( u \in {S}^{n - 1} \) and \( {x}_{1}^{\prime },{x}_{2}^{\prime } \in {u}^{ \bot } \), then\n\n\[ \n{\rho }_{{I}_{\varphi }\left( {{S}_{u}K,{S}_{u}L}\right) }^{-1}\left( {\frac{{x}_{1}^{\prime } + {x}_{2}^{\prime ... | Proof We only prove (3.1). Inequality (3.2) can be established in the same way.\n\nFor each \( {z}^{\prime } \in {K}_{u},{y}^{\prime } \in {L}_{u} \), let \( {\sigma }_{{z}^{\prime }} = {\sigma }_{{z}^{\prime }}\left( u\right) = \left| {K \cap \left( {{z}^{\prime } + \mathbb{R}u}\right) }\right| \) and \( {\sigma }_{{y... | Yes |
Lemma 3.1 Let \( f \in {C}^{\infty }\left( G\right) \) be a radial function. There holds\n\n(1) \( {\left| {\nabla }_{G}\left( {t}_{n}f\right) \right| }^{2} = {t}_{n}^{2}{\left| {f}^{\prime }\right| }^{2}{\left| {\nabla }_{G}\rho \right| }^{2} + \frac{{\left| x\right| }^{2}}{4}{f}^{2} + \frac{{t}_{n}{\left| x\right| }^... | Proof (1) Since \( f \) is radial, we get, by (2.5),\n\n\[ \n{\left| {\nabla }_{G}\left( {t}_{n}f\right) \right| }^{2} = {\left| {\nabla }_{x}\left( {t}_{n}f\right) \right| }^{2} + \frac{1}{4}{\left| x\right| }^{2}{\left| {\nabla }_{t}\left( {t}_{n}f\right) \right| }^{2} \n\]\n\n\[ \n= {t}_{n}^{2}{\left| {\nabla }_{x}f... | Yes |
Lemma 2.2 Define the mapping \( B : {L}^{p}\left( {0, T;{W}^{1, p}\left( \Omega \right) }\right) \rightarrow {L}^{{p}^{\prime }}\left( {0, T;{\left( {W}^{1, p}\left( \Omega \right) \right) }^{ * }}\right) \) by\n\n\[ \left( {w,{Bu}}\right) = {\int }_{0}^{T}{\int }_{\Omega }\left\langle {{\alpha }_{2}\left( {\left| \nab... | Proof Step \( {1B} \) is everywhere defined. \( \forall u, w \in {L}^{p}\left( {0, T;{W}^{1, p}\left( \Omega \right) }\right) \),\n\n\[ \left| \left( {w,{Bu}}\right) \right| \leq {\int }_{0}^{T}{\int }_{{\Omega }_{m}}{k}_{1}{\left| \nabla u\right| }^{p - 1}\left| {\nabla w}\right| {dxdt} + {\lambda }_{1}{\int }_{0}^{T}... | Yes |
Lemma 2.3 Define\n\n\[ \nS : D\left( S\right) = \left\{ {u\left( {x, t}\right) \in {L}^{p}\left( {0, T;{W}^{1, p}\left( \Omega \right) }\right) : u\left( {x,0}\right) = u\left( {x, T}\right) ,{\alpha }_{1}\left( {\frac{\partial u}{\partial t}\left( {x,0}\right) }\right) = {\alpha }_{1}\left( {\frac{\partial u}{\partial... | Proof It is only need to show that \( S \) is lower-semi continuous on \( {L}^{p}\left( {0, T;{W}^{1, p}\left( \Omega \right) }\right) \) .\n\nFor this, let \( \left\{ {u}_{n}\right\} \) be such that \( {u}_{n} \rightarrow u \) in \( {L}^{p}\left( {0, T;{W}^{1, p}\left( \Omega \right) }\right) \) as \( n \rightarrow \i... | Yes |
Lemma 2.4 Let \( S \) be the same as that in Lemma 2.3. If \( w\left( {x, t}\right) \in \partial S\left( {u\left( {x, t}\right) }\right) \) then\n\n\[ w\left( {x, t}\right) = - \frac{\partial }{\partial t}\left( {{\alpha }_{1}\left( \frac{\partial u}{\partial t}\right) }\right) \text{ a.e. in }\Omega \times \left( {0, ... | Proof Let \( w\left( {x, t}\right) = \frac{\partial \bar{w}\left( {x, t}\right) }{\partial t} \) . In view of the definition of subdifferential, we have if \( w\left( {x, t}\right) \in \partial S\left( {u\left( {x, t}\right) }\right) \), then\n\n\[ {\int }_{0}^{T}{\int }_{\Omega }\left\lbrack {j\left( \frac{\partial u}... | Yes |
For each \( w\left( {x, t}\right) \in {L}^{p}\left( {0, T;{W}^{1 - \frac{1}{p}, p}\left( \Gamma \right) }\right) \) and \( f\left( {x, t}\right) \in {L}^{{p}^{\prime }}\left( {0, T;{\left( {W}^{1, p}\left( \Omega \right) \right) }^{ * }}\right) \) , there exists \( u\left( {x, t}\right) \in {L}^{p}\left( {0, T;{W}^{1, ... | Proof From Lemmas 1.2, 2.3, 1.6 and 2.2, we know that there exists\n\n\[ u\left( {x, t}\right) \in {L}^{p}\left( {0, T;{W}^{1, p}\left( \Omega \right) }\right) ,\]\n\nwhich satisfies\n\n\[ \partial S\left( {u\left( {x, t}\right) }\right) + {Bu}\left( {x, t}\right) = f\left( {x, t}\right) . \]\n\n(2.3)\n\nThen \( \foral... | Yes |
Lemma 2.2 Let \( U, V \) be two \( p \) -divisible \( {kG} \) -modules, \( W \) be a \( {kG} \) -module, and \( P \) be a proper \( p \) -subgroup of \( G \), then\n\n(1) any \( P \) -projective \( {kG} \) -module is \( p \) -divisible, particularly, any projective \( {kG} \) -module is \( p \) -divisible; | Proof (1) It comes from the fact that any direct summand of the \( P \) -projective \( {kG} \) - module (respectively, the projective \( {kG} \) -module) remains to be \( P \) -projective (respectively, projective), and from the fact that the dimension of any indecomposable \( P \) -projective \( {kG} \) - module (resp... | No |
Proposition 2.4 Let \( G \geq H \) and \( V \) be a \( p \) -divisible \( {kG} \) -module, if \( \mathrm{H} \) contains a Sylow \( p \) -subgroup of \( G \), then \( {\operatorname{Res}}_{H}^{G}\left( V\right) \) is a \( p \) -divisible \( {kH} \) -module. | Proof Proof by contradiction. If \( {\operatorname{Res}}_{H}^{G}\left( V\right) \) is not \( p \) -divisible, then \( p \) does not divide \( {\dim }_{k}\left( {V}_{1}\right) \), and then \( k \mid {\operatorname{End}}_{k}\left( {V}_{1}\right) \) for some direct summand \( {V}_{1} \) of \( {\operatorname{Res}}_{H}^{G}\... | Yes |
Proposition 2.5 Let \( N \) be a normal subgroup of \( G \) with \( p\parallel G : N \mid \), then \( V \) is a \( p \) -divisible \( k\left( {G/N}\right) \) -module if and only if \( \inf \left( V\right) \) is a \( p \) -divisible \( {kG} \) -module. | Proof We see that \( V \) is an indecomposable \( k\left( {G/N}\right) \) -module if and only if \( \inf \left( V\right) \) is an indecomposable \( {kG} \) -module, moreover, the dimensions of \( V \) and \( \inf \left( V\right) \) are the same since as \( k \) -module they are the same, then \( V \) is \( p \) -divisi... | Yes |
Corollary 2.6 Let \( U \) be not a \( p \) -divisible \( {kG} \) -module, if \( \mathrm{P} \) is a \( p \) -subgroup of \( G \), then \( U{ \otimes }_{k}V \) is \( P \) -projective if and only if the \( {kG} \) -module \( V \) is \( P \) -projective. | Proof The proof of sufficiency is obvious; for the necessity, we see \( k \mid \left( {{U}^{ * }{ \otimes }_{k}U}\right) \) as the proof of Proposition 2.4, then\n\n\[ k{ \otimes }_{k}V \mid \left( {\left( {{U}^{ * }{ \otimes }_{k}U}\right) { \otimes }_{k}V}\right) ,\]\n\nthat is,\n\n\[ V \mid {U}^{ * }{ \otimes }_{k}\... | Yes |
Proposition 2.7 Let \( G \geq H, V \) be a \( {kG} \) -module, and \( U \) be a \( {kH} \) -module such that \( V = {\operatorname{Ind}}_{H}^{G}\left( U\right) \n\n(1) if the Sylow \( p \) -subgroup of \( \mathrm{H} \) is a proper \( p \) -subgroup of \( G \), then \( V \) is \( p \) -divisible; | Proof (1) By Lemma 2.2(1), \( {\operatorname{Ind}}_{H}^{G}\left( U\right) \) is \( p \) -divisible since \( {\operatorname{Ind}}_{H}^{G}\left( U\right) \) is \( P \) -projective, where \( P \) is a Sylow \( p \) -subgroup of \( H \) . | Yes |
Corollary 2.8 (1) Let \( G \geq H \), then \( {\operatorname{Ind}}_{H}^{G}\left( k\right) \) is \( p \) -divisible if and only if the Sylow \( p \) -subgroup of \( H \) is a proper \( p \) -subgroup of \( G \); | Proof It follows from Proposition 2.7. | No |
Theorem 2.9 Any indecomposable endo- \( p \) -permutation \( {kG} \) -module \( V \) with the vertex \( P \) is \( p \) -divisible if and only if \( \mathrm{P} \) is the proper \( p \) -subgroup of \( G \), moreover, in the case of \( P \) being the Sylow \( p \) -subgroup of \( G, P \) does not divide \( {\dim }_{k}\l... | Proof If \( P \) is the Sylow \( p \) -subgroup of \( G,{\operatorname{Res}}_{P}^{G}\left( V\right) \) is an endo-permutation \( {kP} \) -module, and the source module of \( V \) has the vertex \( P \) and is the direct summand of \( {\operatorname{Res}}_{P}^{G}\left( V\right) \), then \( {\operatorname{Res}}_{P}^{G}\l... | Yes |
Lemma 2.10 Let \( V \) and \( W \) be \( {kG} \) -modules, then for any \( m, n \in Z,{\Omega }^{m}\left( V\right) { \otimes }_{k}{\Omega }^{n}\left( W\right) \cong \) \( {\Omega }^{m + n}\left( {V{ \otimes }_{k}W}\right) { \oplus }_{k} \) (projective), particularly, there exists a projective \( {kG} \) -module \( U \)... | Proof Set \( m = 0, W = k \) in the former, we can obtain the latter. While the latter is also Proposition 11.7.2 of [6], now we prove the former by the latter.\n\nSet \( V{ \otimes }_{k}{\Omega }^{m}\left( k\right) \cong {\Omega }^{m}\left( V\right) { \oplus }_{k}X \) and \( W{ \otimes }_{k}{\Omega }^{n}\left( k\right... | Yes |
Theorem 2.11 Let \( V \) be a \( {kG} \) -module, then for any \( n \in Z, V \) is \( p \) -divisible if and only if \( {\Omega }^{n}\left( V\right) \) is \( p \) -divisible. | Proof If \( V \) is \( p \) -divisible, so is \( V{ \otimes }_{k}{\Omega }^{n}\left( k\right) \), and then \( {\Omega }^{n}\left( V\right) { \oplus }_{k}U \) is also \( p \) -divisible by Lemma 2.10, that is, \( {\Omega }^{n}\left( V\right) \) is \( p \) -divisible for each \( n \in Z \). On the contrary, if \( {\Omega... | Yes |
Corollary 2.12 Let \( 0 \rightarrow U \rightarrow W \rightarrow V \rightarrow 0 \) be a short exact sequence of the \( {kG} \) -modules, if \( W \) is projective, then \( U \) is a \( p \) -divisible \( {kG} \) -module if and only if \( V \) is \( p \) -divisible. | Proof One can check the following \( {kG} \) -module isomorphisms by Schanuel’s lemma\n\n\[ U \cong \Omega \left( V\right) { \oplus }_{k}\text{(projective), \]\n\n\[ V \cong {\Omega }^{-1}\left( U\right) { \oplus }_{k}\text{(projective), \]\n\nthen the result follows from Theorem 2.11 and Lemma 2.2(1)(3). | No |
Proposition 2.13 Let \( V \) be a \( p \) -divisible \( {kG} \) -module and \( W \) be a \( V \) -projective \( {kG} \) - module, then \( W \) is \( p \) -divisible; if moreover, \( V \) is \( Q \) -projective, where \( Q \) is a \( p \) -subgroup of \( G \) , then \( W \) is also \( Q \) -projective. | Proof If \( W \) is indecomposable and \( p \) does not divide \( {\dim }_{k}\left( W\right) \), then \( k \mid \left( {{W}^{ * }{ \otimes }_{k}W}\right) \), and then \( k \mid \left( {{W}^{ * }{ \otimes }_{k}V{ \otimes }_{k}X}\right) \) since \( W \) is \( V \) -projective, where \( X \) is a \( {kG} \) -module; while... | Yes |
Theorem 2.14 Let \( G \geq H \geq {N}_{G}\left( P\right) \), where \( P \) is a Sylow \( p \) -subgroup of \( G \), if \( H \) is strongly \( p \) -embedded, then Green correspondence between the indecomposable \( {kG} \) -modules and the indecomposable \( {kH} \) -modules induces a bijection between the isomorphism cl... | Proof In the case of \( H \) being strongly \( p \) -embedded, for the indecomposable \( p \) -divisible \( {kG} \) -module \( V \) with the vertex \( P,{\operatorname{Res}}_{H}^{G}\left( V\right) \) is a \( p \) -divisible \( {kH} \) -module by Proposition 2.4, it means that the Green correspondent of \( V \) remains ... | Yes |
Lemma 1 Let \( \theta \left( x\right) \in {\mathbb{F}}_{q}\left\lbrack x\right\rbrack \) induce a map from \( {\mathbb{F}}_{q} \) to its subset \( \left\{ {{e}_{1},\cdots ,{e}_{n}}\right\} \) . Define\n\n\[ f\left( x\right) = \mathop{\sum }\limits_{{i = 1}}^{n}{f}_{i}\left( x\right) \left( {1 - {\left( \theta \left( x\... | It is observed from (2.1) that \( f\left( x\right) = {f}_{i}\left( x\right) \) for \( x \in {\theta }^{-1}\left( {e}_{i}\right) \) . In other words, \( f\left( x\right) \) is a piecewise polynomial composed of \( {f}_{i}\left( x\right) \) as pieces. Clearly \( \left\{ {{\theta }^{-1}\left( {e}_{i}\right) \mid i = 1,2,\... | Yes |
Lemma 3 Let \( \xi \) be a primitive element of \( {\mathbb{F}}_{{q}^{2}} \). Then the subfield \[ {\mathbb{F}}_{q} = \{ 0\} \cup \left\{ {{\xi }^{\left( {q + 1}\right) i} \mid i = 1,2,\cdots, q - 1}\right\} . | Proof Since \( \xi \) is a primitive element of \( {\mathbb{F}}_{{q}^{2}},{\xi }^{\left( {q + 1}\right) i} \) are all distinct for \( i \in \{ 1,2,\cdots, q - 1\}. Also \( {\left( {\xi }^{\left( {q + 1}\right) i}\right) }^{q} = {\xi }^{\left( {{q}^{2} + q}\right) i} = {\xi }^{\left( {1 + q}\right) i} = {\xi }^{\left( {... | No |
Lemma 3.1 (see [15]) Assume that \( u\left( {x, t}\right) \) is defined by\n\n\[ \left\{ \begin{array}{ll} \frac{\partial u}{\partial t} = {d}_{1}\bigtriangleup u + {ru}\left( {1 - \frac{u}{K}}\right) , & x \in \Omega, t > 0, \\ \frac{\partial u}{\partial v} = 0, & x \in \partial \Omega, t > 0, \\ u\left( {x,0}\right) ... | From [24], we can obtain that | No |
Example 2.1 Let \( D \) be a domain and \( \alpha \) be the automorphism of the polynomial ring \( R = D\left\lbrack {{x}_{1},{x}_{2},\cdots ,{x}_{m}}\right\rbrack \), with indeterminates \( {x}_{1},{x}_{2}\cdots ,{x}_{m} \), given by \( \alpha \left( {x}_{i}\right) = {x}_{i + 1} \) for \( 1 \leq i \leq m - 1 \) and \(... | Taking\n\n\[ \nA = \left( \begin{matrix} {x}_{1} & {x}_{2} \\ 0 & 0 \end{matrix}\right), B = \left( \begin{matrix} 0 & {x}_{2} \\ 0 & - {x}_{1} \end{matrix}\right) ,\n\]\n\nwe have \( {AB} = \left( \begin{matrix} 0 & {x}_{1}{x}_{2} - {x}_{2}{x}_{1} \\ 0 & 0 \end{matrix}\right) = 0 \), but\n\n\[ \nA\bar{\alpha }\left( B... | Yes |
Proposition 2.2 Let \( R \) be a semicommutative ring. If \( R \) is \( \left( {\alpha ,\delta }\right) \) -weakly rigid, then \( R \) is \( \left( {\alpha ,\delta }\right) \) -compatible. | Proof Suppose \( {ab} = 0 \) for \( a, b \in R \) . Then we have \( {aRb} = 0 \) since \( R \) is semicommutative. This implies that \( {a\delta }\left( b\right) = 0 \) and \( {a\alpha }\left( {Rb}\right) = 0 \) since \( R \) is \( \left( {\alpha ,\delta }\right) \) -weakly rigid. It follows that \( {a\alpha }\left( b\... | Yes |
Theorem 2.3 Let \( \alpha \) be a monomorphism and \( \delta \) an \( \alpha \) -derivation of a ring \( R \), and \( f\left( x\right) = {a}_{0} + {a}_{1}x + \cdots + {a}_{n}{x}^{n} \) . If \( R \) is \( \left( {\alpha ,\delta }\right) \) -weakly rigid and semicommutative, then \( f\left( x\right) \in \) \( \operatorna... | \[ \operatorname{nil}\left( {R\left\lbrack {x;\alpha ,\delta }\right\rbrack }\right) = \operatorname{nil}\left( R\right) \left\lbrack {x;\alpha ,\delta }\right\rbrack . \] | Yes |
Proposition 2.4 Let \( \alpha \) be a monomorphism and \( \delta \) an \( \alpha \) -derivation of a ring \( R \) . Then \( R \) is \( \alpha \) -rigid if and only if \( R \) is \( \left( {\alpha ,\delta }\right) \) -weakly rigid and reduced. | Proof If \( R \) is \( \alpha \) -rigid, then \( R \) is \( \left( {\alpha ,\delta }\right) \) -compatible and reduced from Lemma 2.2 in [4]. Clearly, \( \left( {\alpha ,\delta }\right) \) -compatible rings are \( \left( {\alpha ,\delta }\right) \) -weakly rigid. So \( R \) is \( \left( {\alpha ,\delta }\right) \) -wea... | Yes |
(1) if \( {ab} = 0 \), then \( a{\alpha }^{n}\left( b\right) = 0,{\alpha }^{m}\left( a\right) b = 0 \) for all positive integers \( m, n \) ; | Proof (1) If \( {ab} = 0 \), then \( {aRb} = 0 \) since \( R \) is semicommutative. It follows that \( {a\alpha }\left( {Rb}\right) = 0 \) by the definition of \( \left( {\alpha ,\delta }\right) \) -weakly rigid, and hence \( {a\alpha }\left( b\right) = 0 \) . Since \( R \) is \( \left( {\alpha ,\delta }\right) \) - co... | Yes |
Proposition 2.6 Let \( \\alpha \) be a monomorphism and \( \\delta \) an \( \\alpha \) -derivation of a ring \( R \) . If \( R \) is \( \\left( {\\alpha ,\\delta }\\right) \) -weakly rigid and semicommutative, then we have the following:\n\n(1) \( {ab} = 0 \) implies \( a{f}_{i}^{j}\\left( b\\right) = 0 \) for all \( 0... | Proof (1) \( R \) is \( \\left( {\\alpha ,\\delta }\\right) \) -compatible by Proposition 2.2. If \( {ab} = 0 \), then \( a{\\alpha }^{i}\\left( b\\right) = 0 \) and \( a{\\delta }^{j}\\left( b\\right) = 0 \) for all \( i \\geq 0 \) and \( j \\geq 0 \) . So \( a{f}_{i}^{j}\\left( b\\right) = 0 \) for all \( 0 \\leq i \... | Yes |
Lemma 2.7 Let \( \\alpha \) be a monomorphism and \( \\delta \) an \( \\alpha \) -derivation of a ring \( R \) . If \( R \) is an \( \\left( {\\alpha ,\\delta }\\right) \) -weakly rigid and semicommutative ring, then for\n\n\[ \nf\\left( x\\right) = \\mathop{\\sum }\\limits_{{i = 0}}^{m}{a}_{i}{x}^{i}, g\\left( x\\righ... | Proof Since \( R \) is \( \\left( {\\alpha ,\\delta }\\right) \) -weakly rigid and semicommutative, then \( R \) is \( \\left( {\\alpha ,\\delta }\\right) \) -compatible by Proposition 2.2. By analogy with the proof of Proposition 4.4 in [12], the proofs carry over mutatis mutandis for these conclusions by using of Lem... | No |
Theorem 2.8 Let \( \alpha \) be a monomorphism and \( \delta \) an \( \alpha \) -derivation of a ring \( R \) . If \( R \) is \( \left( {\alpha ,\delta }\right) \) -weakly rigid and semicommutative, then \( R\left\lbrack {x;\alpha ,\delta }\right\rbrack \) is weak symmetric if and only if \( R \) is weak symmetric. | Proof \( \left( \Leftarrow \right) \) Let\n\n\[ f\left( x\right) = \mathop{\sum }\limits_{{i = 0}}^{m}{a}_{i}{x}^{i}, g\left( x\right) = \mathop{\sum }\limits_{{j = 0}}^{n}{b}_{j}{x}^{j}, h\left( x\right) = \mathop{\sum }\limits_{{k = 0}}^{p}{c}_{k}{x}^{k} \in R\left\lbrack {x;\alpha ,\delta }\right\rbrack \]\n\nsuch t... | Yes |
Lemma 2.9 Let \( \alpha \) be a monomorphism and \( \delta \) an \( \alpha \) -derivation of a ring \( R \) . If \( R \) is an \( \left( {\alpha ,\delta }\right) \) -weakly rigid and semicommutative ring, then for any subet \( U \subseteq R \), we have \( {N}_{S}\left( U\right) = {N}_{R}\left( U\right) \left\lbrack {x;... | Proof For any \( f\left( x\right) = {a}_{0} + {a}_{1}x + \cdots + {a}_{n}{x}^{n} \in {N}_{S}\left( U\right) \), we have \( {uf}\left( x\right) \in \operatorname{nil}\left( S\right) \) for all \( u \in U \), and hence \( u{a}_{i} \in \operatorname{nil}\left( R\right) \) for \( 0 \leq i \leq n \) by Theorem 2.3. This imp... | Yes |
Lemma 2.10 Let \( \alpha \) be a monomorphism and \( \delta \) an \( \alpha \) -derivation of a ring \( R \) . If \( R \) is an \( \left( {\alpha ,\delta }\right) \) -weakly rigid and semicommutative ring, then \( \phi : N{\operatorname{Ann}}_{R}\left( {2}^{R}\right) \rightarrow N{\operatorname{Ann}}_{S}\left( {2}^{S}\... | Proof We know that \( \phi \) is well defined by Lemma 2.9. Obviously, \( \phi \) is injective. In the following, we show that \( \phi \) is surjective. Let \( {N}_{S}\left( V\right) \in N{\operatorname{Ann}}_{S}\left( {2}^{S}\right) \) and \( g\left( x\right) = \mathop{\sum }\limits_{{j = 0}}^{n}{b}_{j}{x}^{j} \in \) ... | Yes |
For a domain \( R \) and positive integer \( n \), consider the following set of triangular matrices \[ {T}_{n}\left( {R, n}\right) = \left\{ {\left. \left( \begin{matrix} {a}_{0} & {a}_{1} & {a}_{2} & \ldots & {a}_{n - 1} \\ 0 & {a}_{0} & {a}_{1} & \ldots & {a}_{n - 2} \\ 0 & 0 & {a}_{0} & \ldots & {a}_{n - 3} \\ \vdo... | Since \( R \) is a domain, \( {N}_{{T}_{n}\left( {R, n}\right) }\left( X\right) = \) \( \left\{ {\left( {0,{b}_{1},\cdots ,{b}_{n - 1}}\right) \mid {b}_{i} \in R}\right\} = \left( {0,1,0,\cdots ,0}\right) \cdot {T}_{n}\left( {R, n}\right) \), where \( \left( {0,1,0,\cdots ,0}\right) \) is a nilpotent element of \( {T}_... | No |
Theorem 2.15 Let \( R \) be an \( \left( {\alpha ,\delta }\right) \) -weakly rigid and semicommutative ring. Then \( R\left\lbrack {x;\alpha ,\delta }\right\rbrack \) is nilpotent Baer if and only if \( R \) is nilpotent Baer. | Proof ( \( \Leftarrow \) ) Let \( \varnothing \neq X \subseteq S = R\left\lbrack {x;\alpha ,\delta }\right\rbrack \) and \( {N}_{S}\left( X\right) \neq S \) . Suppose \( g\left( x\right) = \mathop{\sum }\limits_{{j = 0}}^{n}{b}_{j}{x}^{j} \in \) \( {N}_{S}\left( X\right) \) . Then \( f\left( x\right) g\left( x\right) \... | No |
Theorem 2.6 Suppose \( A \) is a sectorial operator of type \( \left( {M,\theta ,\alpha ,\mu }\right) \) . If \( f \) satisfies a uniform Hölder condition with exponent \( \beta \in (0,1\rbrack \), then the solution of problem (2.1) is given by | \[ u\left( t\right) = \left\{ \begin{array}{l} {S}_{\alpha }\left( t\right) {u}_{0} + {K}_{\alpha }\left( t\right) {u}_{1} + {\int }_{0}^{t}{T}_{\alpha }\left( {t - \theta }\right) f\left( \theta \right) {d\theta },\;t \in \left\lbrack {0,{t}_{1}}\right\rbrack , \\ {S}_{\alpha }\left( t\right) {u}_{0} + {K}_{\alpha }\l... | Yes |
Lemma 2.7 [9] Let \( A \) be a sectorial operator of type \( \left( {M,\theta ,\alpha ,\mu }\right) \) . If \( f \) satisfies a uniform Hölder condition with exponent \( \beta \in (0,1\rbrack \), then the unique solution of Cauchy problem\n\n\[ \left\{ \begin{array}{l} {D}_{t}^{\alpha }u\left( t\right) = {Au}\left( t\r... | is given by\n\n\[ u\left( t\right) = {S}_{\alpha }\left( t\right) {u}_{0} + {K}_{\alpha }\left( t\right) {u}_{1} + {\int }_{0}^{t}{T}_{\alpha }\left( {t - s}\right) f\left( s\right) {ds}. \] | Yes |
Theorem 3.1 Suppose that conditions (H1)-(H5) are satisfied. If \( \widetilde{M}\left( {b + d}\right) < \frac{1}{2} \), then system (1.1) has at least one mild solution on \( J \) . | Proof We define operator \( \Gamma : {PC}\left( {J, X}\right) \rightarrow {PC}\left( {J, X}\right) \) by\n\n\[ \left( {\Gamma u}\right) \left( t\right) = \left\{ \begin{array}{l} {S}_{\alpha }\left( t\right) \left\lbrack {{u}_{0} - m\left( u\right) }\right\rbrack + {K}_{\alpha }\left( t\right) \left\lbrack {{u}_{1} - n... | Yes |
For each \( k \in \left( {-\frac{1}{2},\frac{1}{2}}\right) \) and given \( b \in \mathbb{R} \), the local time of mBs given by\n\n\[ \n{L}_{k,\varepsilon }\left( {T, b}\right) = {\int }_{0}^{T}{p}_{\varepsilon }\left( {\alpha {B}_{t} + \beta {S}_{t}^{k} - b}\right) {dt} \n\]\n\n\[ \n= \frac{1}{{\left( 2\pi \varepsilon ... | Proof Set\n\n\[ \n{\Phi }_{k,\varepsilon }\left( \overrightarrow{\omega }\right) \equiv {\left( \frac{1}{2\pi \varepsilon }\right) }^{\frac{d}{2}}\exp \left\{ {-\frac{{\left( \alpha {B}_{t} + \beta {S}_{t}^{k} - b\right) }^{2}}{2\varepsilon }}\right\} , \n\]\n\nwhere \( \overrightarrow{\omega } = \left( {{\overrightarr... | Yes |
Theorem 3.2 For each \( k \in \left( {-\frac{1}{2},\frac{1}{2}}\right) \), the local time of mBs\n\n\[ \n{L}_{k}\left( {T, b}\right) \equiv = \frac{1}{2\pi }{\int }_{0}^{T}{\int }_{\mathbb{R}}\exp \left\{ {{i\lambda }\left( {\alpha {B}_{t} + \beta {S}_{t}^{k} - b}\right) }\right\} {d\lambda dt} \n\]\n\nis a Hida distri... | Proof To show this result, we need apply Lemma 2.1 to the \( S \) -transform of the integral with respect to Lebesgue measure \( {dt} \) on \( \left\lbrack {0, T}\right\rbrack \) . For \( \mathbf{f} \in {\mathcal{S}}_{2d}\left( \mathbb{R}\right) \) and any complex number \( z \) ,\n\nby the definition of \( S \) -trans... | Yes |
Theorem 2.2 Let \( G \) be in \( {B}_{n}^{d} \) with the minimum \( {S}_{k}\left( G\right) \).\n\n(i) If \( d = 2, \) then \( G \cong {K}_{\lfloor \frac{n}{2}\rfloor ,\lceil \frac{n}{2}\rceil }.\n\n(ii) If \( d \geq 3 \), then \( G \cong {G}^{ * } \) for odd \( d \) and \( G \in \mathbb{B} \) for even \( d \), where \(... | Proof Choose \( G \in {B}_{n}^{d} \) with bipartition \( \left( {X, Y}\right) \) such that \( {S}_{k}\left( G\right) \) is as small as possible.\n\n(i) If \( d = 2 \), then by Lemma 1.1, \( G \cong {K}_{n - t, t} \), where \( t \geq 2 \) or \( n - t \geq 2 \) . Let \( \left| X\right| = \) \( n - t,\left| Y\right| = t \... | Yes |
Lemma 2.1 Let \( V = \mathop{\sum }\limits_{i}{V}_{i}{e}_{i} \) be a Killing vector field on the \( n \) -dimensional Riemannian manifold \( \left( {M, g}\right) \) . Then we have\n\n\[ \mathop{\sum }\limits_{i}{V}_{i}^{2}\mathop{\sum }\limits_{{i, j}}{V}_{i, j}^{2} \geq 2\mathop{\sum }\limits_{j}{\left( \mathop{\sum }... | Proof It suffices to prove (2.10) for any fixed point \( p \in {\Omega }_{0} \mathrel{\text{:=}} \{ x \in M|V\left( x\right) \neq 0\} \) . Note that on \( {\Omega }_{0},\left( {2.10}\right) \) is equivalent to\n\n\[ {\left| \nabla V\right| }^{2} \geq 2\left| \nabla \right| V{\parallel }^{2} \]\n\n(2.11)\n\nAround \( p ... | Yes |
Lemma 2.4 Let \( A = {\left( {a}_{ij}\right) }_{n \times n} \) be a real symmetric matrix with \( \mathop{\sum }\limits_{i}{a}_{ii} = 0 \) and \( {x}_{1},\cdots ,{x}_{n} \in \mathbb{R} \) . Then\n\n\[ \n- \sqrt{\frac{n - 1}{n}\mathop{\sum }\limits_{{i, j}}{a}_{ij}^{2}}\left( {\mathop{\sum }\limits_{i}{x}_{i}^{2}}\right... | Proof of Theorem 1.1 Let \( V = \mathop{\sum }\limits_{i}{V}_{i}{e}_{i} \) be a non-trivial Killing vector field on \( \left( {M, g}\right) \) . Denote by \( E \) the trace-free part of the Ricci tensor Ric, i.e., \( {E}_{ij} = {R}_{ij} - \left( {R/n}\right) {\delta }_{ij} \) . Then, applying Lemma 2.4, we get that\n\n... | Yes |
Let \( E \in \mathcal{M}\left( {I,\left\{ {n}_{k}\right\} ,\left\{ {c}_{k, j}\right\} }\right) \) with \( {c}_{ * } > 0 \) . If \( {n}_{k} \geq 3 \) for all \( k \geq 1 \), then \( {r}_{ * } \geq {c}_{ * } > 0,\mathop{\sup }\limits_{k}{n}_{k} < \infty \) and \( \mathop{\sum }\limits_{{i = 1}}^{{n}_{k}}\left( {\mid {I}_... | Therefore Theorem 1 extends the results of Theorem 2 in [11]. | No |
Example 2 Let \( E \) be an uniform Cantor set (see [12]) with \( {c}_{ * } > 0 \) . If \( {n}_{k} \geq 3 \), then \( {r}_{ * } = {c}_{ * } > 0,\mathop{\sup }\limits_{k}{n}_{k} < \infty \) and \( \mathop{\sum }\limits_{{i = 1}}^{{n}_{k}}\left( {\mid {I}_{\sigma * i} \mid }\right) \geq \left( {1 + {c}_{ * }}\right) \lef... | Therefore Theorem 1 extends the results of Theorem 1.2 in [12] when \( {c}_{ * } > 0 \) . | Yes |
Lemma 3 Suppose \( E \) is a Moran set satisfying the following conditions\n\n(1) \( \mathop{\sup }\limits_{k}{n}_{k} < \infty \) ;\n\n(2) \( 0 < \mathop{\inf }\limits_{k}{D}_{k} \leq \mathop{\sup }\limits_{k}{D}_{k} < 1 \) .\n\nThen we have \( {\dim }_{P}E = {s}^{ * } \) . | It is easy to verify that if for a Moran set \( E \in \mathcal{M}\left( {I,\left\{ {n}_{k}\right\} ,\left\{ {c}_{k, j}\right\} }\right) \), the conditions of Theorem 1 hold, then \( E \) satisfies \( \mathop{\sup }\limits_{k}{n}_{k} < \infty \) and \( 0 < \mathop{\inf }\limits_{k}{D}_{k} \leq \mathop{\sup }\limits_{k}{... | Yes |
Proposition 1 Suppose \( E \) is the Moran set satisfying \( \mathop{\sup }\limits_{k}{n}_{k} < \infty \) and\n\n\[ \mathop{\lim }\limits_{{k \rightarrow \infty }}\frac{\operatorname{card}\left\{ {1 \leq i \leq k : {D}_{i} \leq \alpha }\right\} }{k} = 1 \]\n\n(3.3)\n\nfor some constant \( \alpha \in \left( {0,1}\right)... | By Proposition 1, we have the corollary below. | No |
Corollary 1 Suppose \( E \) is the Moran satisfies the conditions of Theorem 1. If \( {\dim }_{P}E = 1 \), then there exists a subsequence \( {\left\{ {k}_{t}\right\} }_{t} \) and a constant \( c > 0 \) such that\n\n\[{\mu }_{d}\left( {J}_{\sigma }\right) \leq c{\left| {J}_{\sigma }\right| }^{d}\]\n\nfor any basic inte... | Proof Since \( {n}_{k} \geq 2,{r}_{ * } > 0 \), we have \( {D}_{k} \leq 1 - {r}_{ * } < 1 \) . Take \( \alpha = 1 - {r}_{ * } \), we have\n\n\[ \mathop{\lim }\limits_{{k \rightarrow \infty }}\frac{\operatorname{card}\left\{ {1 \leq i \leq k : {D}_{k} \leq 1 - {r}_{ * }}\right\} }{k} = 1 \]\n\nnotice that \( {\dim }_{p}... | No |
Assume that \( f \in {SB}{C}_{0}\left( {R \times {L}^{2}\left( {P, B}\right) ,{L}^{2}\left( {P, B}\right) }\right) \) satisfying that for any \( \varepsilon > 0 \), there exists \( \delta > 0 \) and stochastic process \( L\left( t\right) \in {SBC}\left( {R,{L}^{2}\left( {P, B}\right) }\right) \) such that\n\n\[ E\paral... | Proof For any given \( \varepsilon > 0 \), let \( \delta \) and \( L\left( t\right) \) be as in the assumptions. Take \( {\varepsilon }^{\prime } = \min \{ \varepsilon ,\delta \} \) . Since \( K \) is compact, there exists \( {x}_{1},\cdots ,{x}_{k} \in K \) such that \( K \subset \mathop{\bigcup }\limits_{{i = 1}}^{k}... | Yes |
Lemma 3.1 If \( {\left\{ {\mathbf{X}}_{\lambda }\right\} }_{\lambda \in \Lambda } \) is a collection of complete \( {\mathcal{P}}_{C} \) -resolution, then \( { \coprod }_{\Lambda }{\mathbf{X}}_{\lambda } \) is a complete \( {\mathcal{P}}_{C} \) -resolution. Thus, the class of Gorenstein \( C \) -projective \( R \) -mod... | Proof For each projective \( R \) -module \( Q \), there is an isomorphism\n\n\[ \n{\operatorname{Hom}}_{R}\left( {{ \coprod }_{\Lambda }{\mathbf{X}}_{\lambda }, C{ \otimes }_{R}Q}\right) \cong { \coprod }_{\Lambda }{\operatorname{Hom}}_{R}\left( {{\mathbf{X}}_{\lambda }, C{ \otimes }_{R}Q}\right) .\n\]\n\nSo if the co... | Yes |
Lemma 3.3 Let \( M \) be an \( R \) -module. Then \( M \) has a \( {\mathcal{P}}_{C}\left( R\right) \) -resolution which is \( {\operatorname{Hom}}_{R}\left( {-,{\mathcal{P}}_{C}\left( R\right) }\right) \) -exact if and only if \( M \) has a \( {\mathcal{{GP}}}_{C}\left( R\right) \) -resolution which is \( {\operatorna... | Proof It is enough to show the \ | No |
Lemma 3.4 Let \( M \) be an \( R \) -module. Then \( M \) has a \( {\mathcal{P}}_{C}\left( R\right) \) -coresolution which is \( {\operatorname{Hom}}_{R}\left( {-,{\mathcal{P}}_{C}\left( R\right) }\right) \) -exact if and only if \( M \) has a \( {\mathcal{{GP}}}_{C}\left( R\right) \) -coresolution which is \( {\operat... | Proof It is enough to show the \ | No |
Theorem 3.9 \( \mathcal{G}{\mathcal{I}}_{C}\left( R\right) = {\mathcal{G}}^{2}{\mathcal{I}}_{C}\left( R\right) \) . | Let \( \;\mathcal{G}\left( {\mathcal{G}{\mathcal{I}}_{C}\left( R\right) }\right) = \{ M \in R \) -Mod \( \mid \) there exists a \( {\text{Hom}}_{R}\left( {H, - }\right) \) -exact exact sequence\n\n\[ G : \cdots \rightarrow {G}_{1} \rightarrow {G}_{0} \rightarrow {G}^{0} \rightarrow {G}^{1} \rightarrow \cdots \]\n\nin \... | Yes |
Lemma 3.10 Let \( M \) be an \( R \) -module. If \( \mathcal{G}{\mathcal{F}}_{C}\left( R\right) \) is closed under extensions, then \( M \) has an \( {\mathcal{F}}_{C}\left( R\right) \) -resolution which is \( {\mathcal{I}}_{C}\left( R\right) { \otimes }_{R} \) -exact if and only if \( M \) has a \( {\mathcal{{GF}}}_{C... | Proof It is enough to show the \ | No |
Theorem 3.12 If \( {\mathcal{{GF}}}_{C}\left( R\right) \) is closed under extensions, then \( {\mathcal{{GF}}}_{C}\left( R\right) = {\mathcal{G}}^{2}{\mathcal{F}}_{C}\left( R\right) \) . | Let \( \mathcal{G}\left( {\mathcal{G}{\mathcal{F}}_{C}\left( R\right) }\right) = \{ M \in R \) -Mod \( \mid \) there exists a \( H{ \otimes }_{R} \) -exact exact sequence\n\n\[ \mathbf{G} : \cdots \rightarrow {G}_{1} \rightarrow {G}_{0} \rightarrow {G}^{0} \rightarrow {G}^{1} \rightarrow \cdots \]\n\nin \( R \) -Mod wi... | Yes |
Theorem 3.2 Let \( m, n \) be non-negative integers. Then for positive integer \( l \) with \( l \geq m + n + 1 \)\n\n\[ \mathop{\sum }\limits_{{k = 0}}^{n}\left( \begin{array}{l} n \\ k \end{array}\right) {\left( -1\right) }^{k}\left( {m + n - k}\right) !\frac{{\operatorname{Li}}_{-\left( {l - k}\right) }\left( z\righ... | It follows that we show some special cases of Theorem 3.2. Taking \( l = m + n + 1 \) in Theorem 3.2, we obtain that for non-negative integers \( m, n \), \n\n\[ \mathop{\sum }\limits_{{k = 0}}^{n}\left( \begin{array}{l} n \\ k \end{array}\right) {\left( -1\right) }^{n - k}\frac{{\operatorname{Li}}_{-\left( {m + k + 1}... | Yes |
Proposition 3.1 Let \( \\left( {H,\\beta }\\right) \) be a Hom-bialgebra, \( \\left( {A,\\vartriangleright ,\\alpha }\\right) \) an \( \\left( {H,\\beta }\\right) \) -module Hom-algebra and \( m \\in \\mathcal{Z} \) . Then \( \\left( {A{\\sharp }^{m}H,\\alpha \\otimes \\beta }\\right) \\left( {A{\\sharp }^{m}H = A \\ot... | Proof It is straightforward by the definition of Hom-algebra. | No |
Proposition 3.2 Let \( \\left( {H,\\beta }\\right) \) be a Hom-bialgebra, \( \\left( {C,\\rho ,\\alpha }\\right) \) an \( \\left( {H,\\beta }\\right) \)-comodule Hom-coalgebra and \( n \\in \\mathcal{Z} \). Then \( \\left( {C\\natural H,\\alpha \\otimes \\beta }\\right) \\left( {C\\natural H = C \\otimes H}\\right. \) ... | Proof Straightforward. | No |
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