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Corollary 3.2 Let \( \left( {X, d}\right) \) be a cone b- metric space with the constant \( s \geq 1 \) . Suppose that mappings \( F, G, H, T : X \rightarrow X \) satisfy following conditions: for all \( x, y \in X \) ,\n\n\[ d\left( {{Fx},{Gy}}\right) \leq {a}_{1}d\left( {{Hx},{Ty}}\right) + {a}_{2}d\left( {{Hx},{Fx}}... | Proof Take \( {a}_{i}\left( {x, y}\right) = {a}_{i}\left( {i = 1,2,3,4,5}\right) \) . Then\n\n\[ {a}_{1}\left( {x, y}\right) + {a}_{4}\left( {x, y}\right) + {a}_{5}\left( {x, y}\right) = {a}_{1} + {a}_{4} + {a}_{5} < \frac{1}{s} \leq 1, \]\n\n\[ {a}_{2}\left( {x, y}\right) + {a}_{5}\left( {x, y}\right) = {a}_{2} + {a}_... | Yes |
Corollary 3.3 Let \( \\left( {X, d}\\right) \) be a cone b- metric space with the constant \( s \\geq 1 \) . Suppose that mappings \( F, G, H, T : X \\rightarrow X \) satisfy following conditions: for all \( x, y \\in X \) ,\n\n\[ \nd\\left( {{Fx},{Gy}}\\right) \\leq {a}_{1}d\\left( {{Hx},{Ty}}\\right) + {a}_{2}d\\left... | Proof Since \( {a}_{3} < {a}_{2},{a}_{5} < {a}_{4} \) or \( {a}_{3} > {a}_{2},{a}_{5} > {a}_{4} \), then \( {a}_{2} + {a}_{3} \\neq 0 \) . We can choose \( \\delta \) such that \( 0 < \\delta < \\min \\left\{ {\\frac{{s}^{3}}{2}\\left( {{a}_{3} - {a}_{2}}\\right) \\left( {{a}_{5} - {a}_{4}}\\right), s\\left( {{a}_{2} +... | Yes |
Corollary 3.5 Let \( \\left( {X, d}\\right) \) be a cone b- metric space with the constant \( s \\geq 1 \) . Suppose that mappings \( F, G, H, T : X \\rightarrow X \) satisfy following conditions: for all \( x, y \\in X \) ,\n\n\[ \nd\left( {{Fx},{Gy}}\\right) \\leq {\\alpha d}\\left( {{Hx},{Ty}}\\right) + \\beta \\lef... | Proof Let \( {a}_{1} = \\alpha ,{a}_{2} = {a}_{3} = \\beta ,{a}_{4} = {a}_{5} = \\gamma \) in Corollary 3.4. | Yes |
Corollary 3.6 Let \( \\left( {X, d}\\right) \) be a cone b- metric space with the constant \( s \\geq 1 \) . Suppose that mappings \( F, G, H, T : X \\rightarrow X \) satisfy following conditions: for all \( x, y \\in X \) ,\n\n\[ d\\left( {{Fx},{Gy}}\\right) \\leq {\\alpha d}\\left( {{Hx},{Ty}}\\right) + \\beta \\left... | Proof Let \( {a}_{1} = \\alpha ,{a}_{2} = {a}_{3} = \\beta ,{a}_{4} = {a}_{5} = 0 \) in Corollary 3.4. | Yes |
Corollary 3.7 Let \( \\left( {X, d}\\right) \) be a cone b-metric space with the constant \( s \\geq 1 \) . Suppose that mappings \( F, G, H, T : X \\rightarrow X \) satisfy following conditions: for all \( x, y \\in X \) ,\n\n\[ d\\left( {{Fx},{Gy}}\\right) \\leq {\\lambda d}\\left( {{Hx},{Fx}}\\right) + {\\kappa d}\\... | Proof Let \( {a}_{1} = {a}_{4} = {a}_{5} = 0,{a}_{2} = \\lambda ,{a}_{3} = \\kappa \) in Corollary 3.4. | Yes |
Corollary 3.8 Let \( \\left( {X, d}\\right) \) be a cone b- metric space with the constant \( s \\geq 1 \) . Suppose that mappings \( F, G, H, T : X \\rightarrow X \) satisfy following conditions: for all \( x, y \\in X \) ,\n\n\[ d\\left( {{Fx},{Gy}}\\right) \\leq {\\lambda d}\\left( {{Hx},{Gy}}\\right) + {\\kappa d}\... | Proof Let \( {a}_{1} = {a}_{2} = {a}_{3} = 0,{a}_{4} = \\lambda ,{a}_{5} = \\kappa \) in Corollary 3.4. | Yes |
Example 1 Let \( X = \{ 1,2,3\}, E = {R}^{2}, P = \{ \left( {x, y}\right) /x \geq 0, y \geq 0\} \) and \( d : X \times X \rightarrow E \) be defined as follows:\n\n\[ d\left( {1,1}\right) = d\left( {2,2}\right) = d\left( {3,3}\right) = \left( {0,0}\right) ; \]\n\n\[ d\left( {1,2}\right) = d\left( {2,1}\right) = \left( ... | \[ \left( {0,0}\right) = d\left( {{F1},{G2}}\right) < \frac{1}{50}d\left( {{H1},{T2}}\right) + \frac{1}{50}d\left( {{H1},{F1}}\right) + \frac{1}{125}d\left( {{T2},{G2}}\right) + \frac{9}{40}d\left( {{H1},{G2}}\right) + \frac{1}{1000}d\left( {{T2},{F1}}\right) , \]\n\n\[ \left( {0,0}\right) = d\left( {{F1},{G3}}\right) ... | Yes |
Theorem 3.3 Let \( \left( {X, d}\right) \) be a cone b- metric space with the constant \( s \geq 1 \) . Suppose that mappings \( F, G, H, J, T, V : X \rightarrow X \) satisfy following conditions: for all \( x, y \in X \) ,\n\n\[ d\left( {{Fx},{Gy}}\right) \leq {a}_{1}\left( {x, y}\right) d\left( {{JTx},{HVy}}\right) +... | Proof From Theorem 3.1, we know that \( F, G,{JT} \) and \( {HV} \) have an unique common fixed point \( q \), that is,\n\n\[ {Fq} = {Gq} = {JTq} = {HVq} = q. \]\n\nSince \( {FJ} = {JF},{JT} = {TJ} \), we have\n\n\[ d\left( {{Jq}, q}\right) = d\left( {{JFq},{Gq}}\right) = d\left( {{FJq},{Gq}}\right) \]\n\n\[ \leq {a}_{... | Yes |
Lemma 1 For problem \( M\left| \text{slotcost, outsourcing}\right| {C}_{\max } + \mathop{\sum }\limits_{j}\mathop{\sum }\limits_{{k \in {B}_{j}}}{f}_{k} \), where \( M \in \{ 1,{O2}\} \) , there exists an optimal schedule without idle between any two outsourced jobs. | Proof Assume that there exists an idle between two outsourced jobs \( i \) and \( j \) where \( i \) is before \( j \) . Let \( t \) be the length of the idle. Moving \( i \) backward \( t \) units will not increase the makespan of all jobs and possibly reduce the slot time costs of \( i \) since \( \left\{ {f}_{k}\rig... | Yes |
Theorem 1 Problem R1 is NP-hard in ordinary sense. | Proof If set \( {f}_{k} = 0 \) for all \( k\mathrm{\;s} \), then R1 can be reduced to the classical scheduling \( {P2}\parallel {C}_{\max } \) . Thus R1 is NP-hard in binary sense. | No |
Theorem 2 Problem R2 is NP-hard in ordinary sense. | Proof We construct an instance of R2 with \( n + 1 \) jobs as follows. Set\n\n\[ \n{p}_{1j} = {a}_{j},\;{p}_{2j} = 0,\;j = 1,2,\cdots, n.\n\]\n\n\[ \n{p}_{1, n + 1} = 0,\;{p}_{2, n + 1} = A.\n\]\n\nFor the time slot costs, let \( {f}_{k} = x \) for \( k = 1,2,\cdots, A \), where \( x < 1 \) and the others \( {f}_{k} = ... | Yes |
Theorem 1 Under assumptions A1-A9, we have\n\n\[ \sqrt{N}\left( {{\widehat{\beta }}_{N} - \beta }\right) \overset{d}{ \rightarrow }N\left( {0,{\sum }_{1}^{-1}\left( \beta \right) \left\lbrack {{V}_{0}\left( \beta \right) + \lambda {V}_{1}\left( \beta \right) }\right\rbrack {\sum }_{1}^{-1}\left( \beta \right) }\right) ... | Proof The proof of Theorem 1 is similar to Theorem 2.3 in Xue [16], so we omit it. | No |
Theorem 2 Under assumptions A1-A9, we have\n\n\[ \sqrt{N + n}\left( {{\widehat{\beta }}_{n, N} - \beta }\right) \overset{d}{ \rightarrow }N\left( {0,{\sum }_{3}^{-1}\left( \beta \right) V\left( \beta \right) {\sum }_{3}^{-1}\left( \beta \right) }\right) ,\] \n\nwhere \( {\sum }_{3}\left( \beta \right) = \frac{\lambda }... | Proof To facilitate the presentation, we give the notations as \( {A}^{\otimes 2} = A{A}^{T} \) for a vector or matrix \( A \) . Define the left side of (12) is \( K\left( \beta \right) \), that is\n\n\[ K\left( \beta \right) = \frac{1}{n + N}\left\{ {\mathop{\sum }\limits_{{j = n + 1}}^{{n + N}}\left( {{Y}_{j} - \wide... | Yes |
Proposition 3.2 Let \( e \in {C}^{ * } \) be an idempotent. If \( C \) is a basic \( K \) -coalgebra, then \( {eCe} \) is also a basic \( K \) -coalgebra. | Proof Note that \( C \) is basic if and only if \( {C}_{C} = {\bigoplus }_{j \in {I}_{C}}{E}_{j} \) and that \( {Ce} = S\left( {eCe}\right) \) as right \( C \) -comodule. Moreover, \( {Ce} \) is a direct summand of \( {C}_{C} \) . Then \( {eCe} = {\bigoplus }_{j \in {I}_{e}}{\bar{E}}_{j} \) because \( S \) commutes wit... | No |
Proposition 3.3 Let \( C \) be a basic \( K \) -coalgebra and \( e \in {C}^{ * } \) be an idempotent. If \( M \) is a computable right \( C \) -comodule, then \( {eM} \) is a computable right \( {eCe} \) -comodule. | Proof Let us consider the socle filtration of \( M \)\n\n\[ \n{\operatorname{soc}}^{0}M \subseteq {\operatorname{soc}}^{1}M \subseteq \cdots \subseteq {\operatorname{soc}}^{m}M \subseteq \cdots \subseteq M \n\] \n\nwhere \( {\operatorname{soc}}^{0}M = \operatorname{soc}M \) and give \( m \geq 0,{\operatorname{soc}}^{m ... | Yes |
Proposition 3.5 Let \( C \) be a basic \( K \) -coalgebra and \( e \in {C}^{ * } \) be an idempotent such that \( S\left( {S}_{j}\right) = {S}_{j} \) for all \( j \in {I}_{e} \) . If \( N \) is computable right \( {eCe} \) -comodule, then \( S\left( N\right) \) is a computable right \( C \) -comodule. | Proof Let us consider the socle filtration of \( N \)\n\n\[{\operatorname{soc}}^{0}N \subseteq {\operatorname{soc}}^{1}N \subseteq \cdots \subseteq {\operatorname{soc}}^{m}N \subseteq \cdots \subseteq N,\]\n\nwhere \( {\operatorname{soc}}^{0}N = \operatorname{soc}N \) and give \( m \geq 0,{\operatorname{soc}}^{m + 1}N ... | Yes |
Corollary 3.6 Let \( C \) be a bacic \( K \) -coalgebra and \( e \in {C}^{ * } \) be a right semicentral idempotent. If \( N \) is a computable right \( {eCe} \) -comodule, then \( S\left( N\right) \) is a computable right \( C \) -comodule. | Proof By [5], \( e \) is right semicentral if and only if \( S\left( {S}_{j}\right) = {S}_{j} \) for all \( j \in {I}_{e} \) . Then the statement follows from the former result. | No |
Let \( {KQ} \) be the path coalgebra of the quiver\n\n\[ 1\overset{\alpha }{ \rightarrow }2\overset{\beta }{ \rightarrow }3 \]\n\nand \( e \in {C}^{ * } \) be the idempotent associated to the set \( {X}_{e} = \{ 3\} \) . Then \( {E}_{1} = \langle 1\rangle ,{E}_{2} = \) \( \langle 2,\alpha \rangle ,{E}_{3} = \langle 3,\... | \[ S\left( {S}_{1}\right) = {S}_{1}{▱}_{{e}^{\prime }C{e}^{\prime }}C{e}^{\prime } = {e}^{\prime }C{e}^{\prime }{▱}_{{e}^{\prime }C{e}^{\prime }}C{e}^{\prime } \cong C{e}^{\prime } \cong \langle 1\rangle \cong {S}_{1}. \]\n\nThus \( {e}^{\prime } \) is a right semicentral idempotent, by [5]. | Yes |
Consider the quiver \( Q \) , then the Gabriel quiver \( \left( {{Q}_{KQ},{d}_{KQ}}\right) \) of \( {KQ} \) is given by |  | No |
Theorem 3.7 Let \( C \) be a basic \( K \) -coalgebra. \( M \) is a computable right \( C \) -comodule if and only if \( {e}_{U}M \) is a computable right \( {e}_{U}C{e}_{U} \) -comodule for each finite set \( U \subseteq {I}_{C} \) . | Proof It is only to prove if \( S\left( {{e}_{U}M}\right) \) is computable then \( M \) is computable. Let \( S\left( {{e}_{U}M}\right) = N \), we have\n\n\[{\operatorname{soc}}^{0}N \subseteq {\operatorname{soc}}^{1}N \subseteq \cdots \subseteq {\operatorname{soc}}^{m}N \subseteq \cdots \subseteq N.\]\n\nSince \( M \)... | No |
Furthermore, using Definitions 1.3 and 1.4, we have \( A \) is an \( \left( {\alpha ,\beta }\right) - \gamma \) -open set if and only if \( A \) is \( \left( {\alpha ,\gamma }\right) \) -open and \( \left( {\beta ,\gamma }\right) \) -open, the proof follows. | Proof Sufficiency. Suppose \( A \) is an \( \left( {\alpha ,\beta }\right) \) - \( \gamma \) -open set. Let \( x \in A \), then there exist \( \gamma \) -open sets \( U \) and \( V \) such that \( x \in U, x \in V \) and \( \alpha \left( U\right) \cup \beta \left( V\right) \subset A \) . Thus implies \( \alpha \left( U... | Yes |
Theorem 1.1 Let \( \left( {X,\mathcal{T}}\right) \) a topological space and \( \alpha ,\beta ,\gamma \) be operators on \( \mathcal{T} \), then the following statements hold:\n\n(1) Any union of \( \alpha \) -open set is \( \alpha \) -open.\n\n(2) Any union of \( \left( {\alpha ,\beta }\right) \) -open set is \( \left(... | Proof Statement (1) due to [3].\n\n(2) Let \( x \in \mathop{\bigcup }\limits_{{i \in I}}{A}_{i} \) . Then \( x \in {A}_{i} \) for some \( i \) . Since each \( {A}_{i} \) is \( \left( {\alpha ,\beta }\right) \) -open, there exists a \( \beta \) -open set \( {U}_{i} \) such that \( x \in {U}_{i} \) and \( \alpha \left( {... | Yes |
Theorem 1.2 Let \( \\left( {X,\\mathcal{T}}\\right) \) be a topological space and \( \\alpha ,\\beta ,\\gamma \) be operators on \( \\mathcal{T} \), then the following statements hold:\n\n(1) \( \\left( {X,\\mathcal{T}}\\right) \) is \( \\left( {\\alpha ,\\beta }\\right) \) - \( \\gamma \) -regular iff \( {\\mathcal{T}... | Proof (1) Sufficiency. We have \( {\\mathcal{T}}_{\\left( {\\alpha ,\\beta }\\right) - \\gamma } \\subset {\\mathcal{T}}_{\\gamma } \) by Definition 1.5. Now we proof that \( {\\mathcal{T}}_{\\left( {\\alpha ,\\beta }\\right) - \\gamma } \\supset {\\mathcal{T}}_{\\gamma } \) . Let \( A \\in {\\mathcal{T}}_{\\gamma } \)... | Yes |
Proposition 1.1 Let \( \\left( {X,\\mathcal{T}}\\right) \) be a topological space and \( \\alpha ,\\beta ,\\gamma \) be operators on \( \\mathcal{T} \), If \( A \) and \( B \) are \( \\left( {\\alpha ,\\beta }\\right) \) - \( \\gamma \) -open sets, \( \\left( {\\alpha ,\\gamma }\\right) \) and \( \\left( {\\beta ,\\gam... | Proof Suppose \( A \) and \( B \) are \( \\left( {\\alpha ,\\beta }\\right) \) - \( \\gamma \) -open sets. Let \( x \\in A \\cap B \) . By Definition 1.4, there exist \( \\gamma \) -open sets \( H \) and \( K \) such that \( x \\in H, x \\in K,\\alpha \\left( H\\right) \\cup \\beta \\left( K\\right) \\subset A \) . In ... | Yes |
Theorem 1.3 Let \( \\left( {X,\\mathcal{T}}\\right) \) be a topological space and \( \\alpha ,\\beta ,\\gamma \) be operators on \( \\mathcal{T} \) . Then the following statements hold:\n\n(1) \( x \\in c{l}_{\\left( {\\alpha ,\\beta }\\right) - \\gamma }\\left( A\\right) \) iff \( V \\cap A \\neq \\varnothing \) for e... | Proof Proof of (1)-(3) is obvious from Definition 1.4. | No |
Theorem 2.2 Let \( \left( {X,\mathcal{T}}\right) \) be a topological space and \( \alpha ,\beta ,\gamma \) be operators on \( \mathcal{T} \) . Then \( \left( {X,\mathcal{T}}\right) \) is an \( \left( {\alpha ,\beta }\right) - \gamma - {T}_{1} \) space iff every singleton set \( \{ x\} \) is \( \left( {\alpha ,\beta }\r... | Proof Sufficiency. Suppose that \( \left( {X,\mathcal{T}}\right) \) is an \( \left( {\alpha ,\beta }\right) - \gamma - {T}_{1} \) space, then for every pair of distinct points \( x, y \in X \), there exist two \( \gamma \) -open sets \( {P}_{1} \) and \( {P}_{2} \) containing \( x, y \), respectively, such that \( x \n... | Yes |
Theorem 2.3 Let \( \left( {X,\mathcal{T}}\right) \) be a topological space and \( \alpha ,\beta ,\gamma \) be operators on \( \mathcal{T} \), the following relations hold: \( \left( {\alpha ,\beta }\right) - \gamma - {T}_{5/2} \Rightarrow \left( {\alpha ,\beta }\right) - \gamma - {T}_{2} \Rightarrow \left( {\alpha ,\be... | Proof It is clear by above Theorems 2.1, 2.2 and Remark 2.1. | No |
Theorem 2.4 Let \( \left( {X,\mathcal{T}}\right) \) be a topological space and \( \alpha ,\beta ,\gamma \) be operators on \( \mathcal{T} \), Suppose that \( \left( {\alpha ,\beta }\right) - \gamma : P\left( X\right) \rightarrow P\left( X\right) \) is \( \left( {\alpha ,\gamma }\right) \) -preopen and \( \left( {\beta ... | Proof Sufficiency. Suppose that \( \left( {X,\mathcal{T}}\right) \) is an \( \left( {\alpha ,\beta }\right) - \gamma - {T}_{0}^{ * } \) space, then for every pair of distinct points \( x, y \), there exists an \( \left( {\alpha ,\beta }\right) \) - \( \gamma \) -open set \( P \) such that \( x \in P, y \notin P \) or \... | Yes |
Theorem 2.5 Let \( \left( {X,\mathcal{T}}\right) \) be a topological space and \( \alpha ,\beta ,\gamma \) be operators on \( \mathcal{T} \) . Then \( \left( {X,\mathcal{T}}\right) \) is an \( \left( {\alpha ,\beta }\right) - \gamma - {T}_{1}^{ * } \) space iff every singleton set \( \{ x\} \) is \( \left( {\alpha ,\be... | Proof Sufficiency. Suppose that \( \left( {X,\mathcal{T}}\right) \) is an \( \left( {\alpha ,\beta }\right) - \gamma - {T}_{1}^{ * } \) space, then for every pair of distinct points \( x, y \in X \), there exist two \( \left( {\alpha ,\beta }\right) \) - \( \gamma \) -open sets \( {P}_{1} \) and \( {P}_{2} \) containin... | Yes |
Theorem 2.6 Let \( \left( {X,\mathcal{T}}\right) \) be a topological space and \( \alpha ,\beta ,\gamma \) be operators on \( \mathcal{T} \) . The following relations hold: \( \left( {\alpha ,\beta }\right) \) - \( \gamma \) - \( {T}_{5/2}^{ * } \Rightarrow \left( {\alpha ,\beta }\right) \) - \( \gamma \) - \( {T}_{2}^... | Proof It is clear by above Theorem 2.4, 2.5 and Definition 2.2. | No |
Theorem 2.7 Let \( \left( {X,\mathcal{T}}\right) \) be a topological space and \( \alpha ,\beta ,\gamma \) be operators on \( \mathcal{T} \). The following statements hold:\n\n(1) Every \( \left( {\alpha ,\beta }\right) - \gamma - {T}_{0}^{ * } \) space is an \( \left( {\alpha ,\beta }\right) - \gamma - {T}_{0} \) spac... | Proof (1), (3) proofs are obvious by Definition 2.2. The proof for (2) is obtained by Remark 2.2. | No |
Lemma 2.1 For \( z, w \in \mathbb{U} \) we have\n\n\[{\rho }_{\mathbb{U}}\left( {z, w}\right) = {\ell }_{\mathbb{U}}\left( \gamma \right) = \log \frac{1 + \rho \left( {z, w}\right) }{1 - \rho \left( {z, w}\right) }\]\nwhere \( \gamma \) is the unique arc that joins \( z \) and \( w \), and lies on a circle perpendicula... | Proof See p.153 in [3]. | No |
Lemma 2.2 Let \( \varphi \) be a holomorphic self-map of \( \mathbb{U} \) with a fixed point \( p \), then for any \( z \in \mathbb{U} \) ,\n\n\[{\rho }_{\mathbb{U}}\left( {{\varphi }^{\left\lbrack n\right\rbrack }\left( z\right), p}\right) \leq \parallel \varphi {\parallel }_{\infty }^{n}{\rho }_{\mathbb{U}}\left( {z,... | Proof From Exercises 5-7 on p.171 in [3], we have\n\n\[{\rho }_{\mathbb{U}}\left( {\varphi \left( z\right) ,\varphi \left( p\right) }\right) \leq \parallel \varphi {\parallel }_{\infty }{\rho }_{\mathbb{U}}\left( {z, p}\right)\]\n\nfor all \( z \in \mathbb{U} \) . Since \( p \) is a fixed point of \( \varphi \), we hav... | No |
Lemma 2.3 Let \( v \) be a weight such that \( v \) is radial and satisfying (2.4). For every \( f \in {H}_{v}^{\infty } \) there exists a constant \( C \) (depending on the weight \( v \) ) such that\n\n\[ \left| {f\left( z\right) - f\left( p\right) }\right| \leq {C}_{v}\parallel f{\parallel }_{v}\max \left\{ {\frac{1... | Proof Adaptation of the case \( N = 1 \) in Lemma 3.2 from [10] gives the proof. | No |
Theorem 2.1 Let \( v \) be a weight such that \( v \) is radial and satisfies (2.4). If \( \varphi \in \mathrm{S} \) is a holomorphic self-map of \( \mathbb{U} \) with a fixed point, satisfying\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\parallel \varphi {\parallel }_{\infty }^{n} = 0 \]\n\n(2.8)\n\nthen \( {... | Proof For any \( f\left( z\right) \) holomorphic in \( \mathbb{U} \), denote \( {C}_{p}f = f\left( p\right) \) where \( p \) is the fixed point of \( \varphi \) . It is obviously that \( {C}_{p} \) is a bounded composition operator. We are going to verify\n\n\[ {\begin{Vmatrix}{C}_{\left\lbrack n\right\rbrack } - {C}_{... | Yes |
Theorem 3.2 If \( \varphi \) is a hyperbolic self-map of \( {\mathbb{B}}^{N} \) in the Schur-Agler class, then \( {C}_{\varphi } \) is not mean ergodic in \( {H}_{d,\beta }^{2} \) . | Proof Our proof is an argument by contradiction. If \( {C}_{\varphi } \) were mean ergodic in \( {H}_{d,\beta }^{2} \) , by Corollary 1.5 in [7], we have the following\n\n\[ \begin{Vmatrix}{C}_{\varphi }^{n}\end{Vmatrix} \geq A{\left( 1 - \left| {\varphi }_{n}\left( 0\right) \right| \right) }^{-\beta /2}, \]\n\n\( \lef... | Yes |
Theorem 3.1 If \( H \) is a linear code of length \( n \) over \( {R}_{3} \), then \( \varphi \left( H\right) = {H}^{ + } \otimes {H}^{ - } \), and \( \varphi \left( H\right) \) is linear. | Proof For any \( \left( {{x}_{1},{x}_{2},\cdots ,{x}_{n},{y}_{1},{y}_{2},\cdots ,{y}_{n}}\right) \in \varphi \left( H\right) \), let\n\n\[ \n{h}_{i} = {x}_{i}\left( {1 + v}\right) + {y}_{i}\left( {1 - v}\right) = \left( {{x}_{i} + {y}_{i}}\right) + v\left( {{x}_{i} - {y}_{i}}\right), i = 1,2,\cdots, n \n\] \n\nand let ... | Yes |
Theorem 3.3 Let \( {H}^{ \bot } \) be the dual code of \( H \) . Then \( \varphi \left( {H}^{ \bot }\right) = \varphi {\left( H\right) }^{ \bot } \) . Moreover, if \( H \) is a self-dual code, so is \( \varphi \left( H\right) \) . | Proof To prove the theorem, we first show\n\n\[ \varphi \left( {H}^{ \bot }\right) \subset \varphi {\left( H\right) }^{ \bot } \]\n\ni.e., \( \forall {h}_{1},{h}_{2} \in {R}_{3}^{n} \), \n\n\[ \left\lbrack {{h}_{1},{h}_{2}}\right\rbrack = 0 \Rightarrow \left\lbrack {\varphi \left( {h}_{1}\right) ,\varphi \left( {h}_{2}... | Yes |
Corollary 3.4 Let \( H = \left( {1 + v}\right) {H}^{ + } \oplus \left( {1 - v}\right) {H}^{ - } \) be a linear code of length \( n \) over \( {R}_{3} \) . Then \( \varphi \left( {H}^{ \bot }\right) = {\left( {H}^{ + }\right) }^{ \bot } \otimes {\left( {H}^{ - }\right) }^{ \bot } \) . Moreover, we have \( {H}^{ \bot } =... | Proof Follows by applying Theorem 3.1 and Theorem 3.3. | No |
Corollary 3.5 Suppose that \( H = \left( {1 + v}\right) {H}^{ + } \oplus \left( {1 - v}\right) {H}^{ - } \) is a linear code of length \( n \) over \( {R}_{3} \) and let \( {\operatorname{Lee}}_{H}\left( {\bar{X},\bar{Y}}\right) \) denote its Lee weight enumerator as defined above, then | \[ {\operatorname{Lee}}_{{H}^{ \bot }}\left( {\bar{X},\bar{Y}}\right) = {\operatorname{Ham}}_{{\left( {H}^{ + }\right) }^{ \bot }}\left( {\bar{X},\bar{Y}}\right) \cdot {\operatorname{Ham}}_{{\left( {H}^{ - }\right) }^{ \bot }}\left( {\bar{X},\bar{Y}}\right) . \] | Yes |
Theorem 4.1 If \( H = \left( {1 + v}\right) {H}^{ + } \oplus \left( {1 - v}\right) {H}^{ - } \) is a linear code of length \( n \) over \( {R}_{3} \), then \( H \) is a cyclic code over \( {R}_{3} \) if and only if \( {H}^{ + },{H}^{ - } \) are ternary cyclic codes. | Proof For any \( h = \left( {{h}_{0},{h}_{1},\cdots ,{h}_{n - 1}}\right) \in H \), where\n\n\[ \n{h}_{i} = {x}_{i}\left( {v + 1}\right) + {y}_{i}\left( {1 - v}\right) = \left( {{x}_{i} + {y}_{i}}\right) + v\left( {{x}_{i} - {y}_{i}}\right), i = 0,1,\cdots, n - 1.\n\]\n\nTaking \( x = \left( {{x}_{0},{x}_{1},\cdots ,{x}... | Yes |
Theorem 4.4 If \( H = \left( {1 + v}\right) {H}^{ + } \oplus \left( {1 - v}\right) {H}^{ - } \) is a cyclic code of length \( n \) over \( {R}_{3} \), then \( H = \left\langle {\left( {1 + v}\right) {g}_{1}\left( x\right) + \left( {1 - v}\right) {g}_{2}\left( x\right) }\right\rangle \) and \( \left| H\right| = {3}^{{2n... | Proof By Theorem 4.1, we have\n\n\[ \n{H}^{ + } = \left\langle {{g}_{1}\left( x\right) }\right\rangle \subset \frac{{F}_{3}\left\lbrack x\right\rbrack }{\left\langle {x}^{n} - 1\right\rangle },{H}^{ - } = \left\langle {{g}_{2}\left( x\right) }\right\rangle \subset \frac{{F}_{3}\left\lbrack x\right\rbrack }{\left\langle... | Yes |
Theorem 4.7 Let \( {x}^{n} - 1 = \mathop{\prod }\limits_{{i = 1}}^{r}{p}_{i}^{{s}_{i}}\left( x\right) \) be unique representations of \( {x}^{n} - 1 \) as a product of ireducible pairwise-comprime polynomial in \( {F}_{3}\left\lbrack x\right\rbrack \) . Then the number of the cyclic code of length \( n \) over \( {R}_{... | Proof The result directly follows from the fact that the number of ternary cyclic code of length \( n \) is \( \mathop{\prod }\limits_{{i = 1}}^{r}\left( {{s}_{i} + 1}\right) \) . | No |
Theorem 5.1 Let \( H = \left( {1 + v}\right) {H}^{ + } \oplus \left( {1 - v}\right) {H}^{ - } \) be a linear code of length \( n \) over \( {R}_{3} \) . Then \( H \) is a \( v \) -constacyclic code of length \( n \) over \( {R}_{3} \) if and only if \( {H}^{ + },{H}^{ - } \) are cyclic and negacyclic codes of length \(... | Proof For any \( h = \left( {{h}_{0},{h}_{1},\cdots ,{h}_{n - 1}}\right) \in H \), we can write its components as \( {h}_{i} = \) \( {x}_{i}\left( {v + 1}\right) + {y}_{i}\left( {1 - v}\right) \), where \( {x}_{i},{y}_{i} \in {F}_{3},0 \leq i \leq n - 1 \) . Let\n\n\[ x = \left( {{x}_{0},{x}_{1},\cdots ,{x}_{n - 1}}\ri... | Yes |
Theorem 5.2 If \( H = \left( {1 + v}\right) {H}^{ + } \oplus \left( {1 - v}\right) {H}^{ - } \) is a \( v \) -constacyclic code of length \( n \) over \( {R}_{3} \), then \( H = \left\langle {\left( {1 + v}\right) {g}_{1}\left( x\right) ,\left( {1 - v}\right) {g}_{2}\left( x\right) }\right\rangle \) and \( \left| H\rig... | Proof By Theorem 5.1, we have\n\n\[ \n{H}^{ + } = \left\langle {{g}_{1}\left( x\right) }\right\rangle \subset \frac{{F}_{3}\left\lbrack x\right\rbrack }{\left\langle {x}^{n} - 1\right\rangle },{H}^{ - } = \left\langle {{g}_{2}\left( x\right) }\right\rangle \subset \frac{{F}_{3}\left\lbrack x\right\rbrack }{\left\langle... | Yes |
Theorem 5.3 With the notations as in Theorem 5.2. If \( H = \left\langle {\left( {1 + v}\right) {g}_{1}\left( x\right) ,\left( {1 - v}\right) {g}_{2}\left( x\right) }\right\rangle \) , then there is a unique polynomial \( g\left( x\right) \) such that \( H = \langle g\left( x\right) \rangle \) and \( g\left( x\right) \... | Proof Since \( g\left( x\right) = \left( {1 + v}\right) {g}_{1}\left( x\right) + \left( {1 - v}\right) {g}_{2}\left( x\right) ,\langle g\left( x\right) \rangle \subset H \) . Note that\n\n\[- \left( {1 + v}\right) g\left( x\right) = \left( {1 + v}\right) {g}_{1}\left( x\right)\]\n\nand \( - \left( {1 - v}\right) g\left... | Yes |
Theorem 5.5 If \( H \) is a \( v \) -constacyclic code of length \( n \) over \( {R}_{3} \), then its dual code \( {H}^{ \bot } \) is also a \( v \) -constacyclic code over \( {R}_{3} \). | Proof The proof is trivial since \( v = {v}^{-1} \) and the dual of a \( v \) -constacyclic code is a \( {v}^{-1} \) -constacyclic. | Yes |
Example 6.1 Consider all cyclic codes over \( {R}_{3} \) of length 2 . Since \( {x}^{2} - 1 = \left( {x - 1}\right) \left( {x + 1}\right) \) in \( {F}_{3}\left\lbrack x\right\rbrack \), there are 15 nonzero cyclic codes over \( {R}_{3} \) of length 2 . | Table 1 gives the list of all cyclic codes. The ones marked with * denote the optimal ones. | No |
Example 6.2 Consider all \( v \) -constacyclic codes over \( {R}_{3} \) of length 4. | Since\n\n\[ \n{x}^{4} - 1 = \left( {x - 1}\right) \left( {x + 1}\right) \left( {{x}^{2} + 1}\right) \n\]\n\nand \( {x}^{4} + 1 = \left( {{x}^{2} + x - 1}\right) \left( {{x}^{2} - x - 1}\right) \) in \( {F}_{3}\left\lbrack x\right\rbrack \), there are 31 nonzero \( v \) -constacyclic codes over \( {R}_{3} \) of length 4... | Yes |
Proposition 2.4 Let \( \mathcal{H} = \left( {{\mathcal{H}}_{L},{\mathcal{H}}_{R}, S}\right) \) be a Hopf algebroid. Assume that the \( L \) -ring \( \left( {H,{s}_{L}}\right) \) underlying \( {\mathcal{H}}_{L} \) is left semisimple. Consider \( V \) as a left \( A\# H \) -module and \( W \) as an \( A\# H \) -submodule... | Proof Suppose that \( \lambda : V \rightarrow W \) is a left \( A \) -module projection, and \( e \) is a normalized right integral in \( {\mathcal{H}}_{R} \) . Define a map\n\n\[ \widetilde{\lambda } : V \rightarrow W, v \mapsto \sum \left( {{1}_{A}\# S\left( {e}_{\left( 1\right) }\right) }\right) \cdot \lambda \left(... | Yes |
Theorem 2.5 Let \( \mathcal{H} = \left( {{\mathcal{H}}_{L},{\mathcal{H}}_{R}, S}\right) \) be a Hopf algebroid. Assume that the \( L \) -ring \( \left( {H,{s}_{L}}\right) \) underlying \( {\mathcal{H}}_{L} \) is left semisimple.\n\n(1) Let \( V \) be an \( A\# H \) -module. If \( V \) is completely reducible as an \( A... | Proof (1) is immediate from Proposition 2.4.\n\n(2) follows from (1), using the fact that a ring is semisimple if and only if every module is completely reducible. | Yes |
Lemma 3.1 Let \( \mathcal{H} = \left( {{\mathcal{H}}_{L},{\mathcal{H}}_{R}, S}\right) \) be a Hopf algebroid with bijective antipode \( S \) . Then \( A \) is a left and right \( A\# H \) -module via\n\n\[ \left( {a\# h}\right) \rightharpoonup b = a\left( {h \vartriangleright b}\right), b \leftharpoonup \left( {a\# h}\... | Proof By (2.4) in [23], \( A \) is a left \( A\# H \) -module via \ | No |
Theorem 3.3 Let \( \mathcal{H} = \left( {{\mathcal{H}}_{L},{\mathcal{H}}_{R}, S}\right) \) be a Hopf algebroid with bijective antipode \( S \) , and \( x \) an \( S \) -fixed left integral in \( {\mathcal{H}}_{L} \), i.e. \( S\left( x\right) = x \) . Then \( \left\lbrack {{A}^{{\mathcal{H}}_{L}},{}_{{A}^{{\mathcal{H}}_... | Proof To satisfy the conditions for a Morita context given in [26], we must check that \( \left\lbrack ,\right\rbrack \) is an \( A\# H \) -bimodule map which is middle \( {A}^{{\mathcal{H}}_{L}} \) -linear, and that (,) is an \( {A}^{{\mathcal{H}}_{L}} \) -bimodule map which is middle \( A\# H \) -linear, and that the... | No |
Theorem 1 Let \( \Omega \) be a semi-disk domain in the complex plane \( \mathbb{C} \) (see Figure 1). Assume that the modified Helmholtz equation has a solution \( q\left( {z,\bar{z}}\right) \) in \( \Omega \) such that it is sufficiently smooth to \( \partial \Omega \) . Then \( q\left( {z,\bar{z}}\right) \) can be e... | Proof First, it sees from [6] that function \( q\left( {z,\bar{z}}\right) \) satisfies the modified Helmholtz equation if and only if the following differential form is closed:\n\n\[ {dW} = {e}^{-{i\beta }\left( {{kz} - \frac{\bar{z}}{k}}\right) }\left\lbrack {\left( {{q}_{z} + {ik\beta q}}\right) {dz} - \left( {{q}_{\... | Yes |
Corollary 1 Under the conditions of Theorem 1, the solution \( q\left( {z,\bar{z}}\right) \) of Laplace equation (see (1.1) with \( \alpha = 0 \) ) can be expressed by\n\n\[ \frac{\partial q}{\partial z} = \frac{1}{2\pi }\mathop{\sum }\limits_{{j = 1}}^{4}{\int }_{{l}_{j}}{e}^{ikz}{\rho }_{j}\left( {x, y, k}\right) {dk... | where\n\n\[ {\rho }_{1}\left( {x, y, k}\right) = \frac{1}{2}\left\lbrack {q\left( {1,0}\right) {e}^{-{ik}} - q\left( {-1,0}\right) {e}^{ik}}\right\rbrack + \frac{i}{2}{\int }_{-1}^{1}{e}^{-{ikx}}\left\lbrack {{kq}\left( {x,0}\right) - i{q}_{y}\left( {x,0}\right) }\right\rbrack {dx}, \]\n\n\[ {\rho }_{2}\left( {x, y, k}... | Yes |
Theorem 2 Let \( \Omega \) be a semi-disk domain in the complex plane \( \mathbb{C} \) described in Figure 1. Assume that the boundary value \( f\left( {x, y}\right) \) has appropriate smoothness and that the Dirichlet boundary problem of the Helmholtz equation (see (2.1) with \( \alpha = {\beta }^{2} \) ) has the solu... | Proof It follows from the global relation (2.16) that\n\n\[ \n{G}_{1}\left( k\right) - {G}_{2}\left( k\right) = F\left( k\right) ,\;k \in \mathbb{C}, \n\] \n\n(3.1)\n\nwhere\n\n\[ \n{G}_{1}\left( k\right) = {\int }_{-1}^{1}{q}_{y}\left( {x,0}\right) {e}^{-{i\beta }\left( {k + \frac{1}{k}}\right) x}{dx},\;{G}_{2}\left( ... | Yes |
Example 2.2 Let \( \mathbb{Z} \) be the ring of integers. Consider the ring\n\n\[ S = \left\{ {\left. \left( \begin{array}{ll} a & b \\ 0 & c \end{array}\right) \right| \;a, b, c \in \mathbb{Z}}\right\} .\n\nLet \( \alpha : S \rightarrow S \) be an endomorphism defined by \( \alpha \left( \left( \begin{array}{ll} a & b... | Then we have\n\n\[ \alpha \left( {f\left( x\right) }\right) g\left( x\right) = \mathop{\sum }\limits_{{k = 0}}^{{m + n}}\left( {\mathop{\sum }\limits_{{i + j = k}}\left( \begin{matrix} {a}_{i} & 0 \\ 0 & 0 \end{matrix}\right) \left( \begin{matrix} {d}_{j} & {e}_{j} \\ 0 & {f}_{j} \end{matrix}\right) }\right) {x}^{k}\n\... | Yes |
Proposition 2.4 Let \( R \) be a ring and \( \alpha \) an endomorphism of \( R \) . Then \( R \) is a right \( \alpha \) -McCoy ring if and only if \( R\left\lbrack x\right\rbrack \) is a right \( \alpha \) -McCoy ring. | Proof Assume that \( R \) is a right \( \alpha \) -McCoy ring. Let \( p\left( y\right) = {f}_{0} + {f}_{1}y + \cdots + {f}_{m}{y}^{m} \) , \( q\left( y\right) = {g}_{0} + {g}_{1}y + \cdots + {g}_{n}{y}^{n} \) be in \( R\left\lbrack x\right\rbrack \left\lbrack y\right\rbrack \) with \( \alpha \left( {p\left( y\right) }\... | Yes |
Proposition 2.6 Let \( R \) be a ring with an endomorphism \( \alpha \) . If \( R \) is right \( \alpha \) -McCoy, then \( {\bigtriangleup }^{-1}R \) is right \( \alpha \) -McCoy. | Proof Assume that \( R \) is right \( \alpha \) -McCoy and let\n\n\[ f\left( x\right) = \mathop{\sum }\limits_{{i = 0}}^{m}{u}_{i}^{-1}{a}_{i}{x}^{i}, g\left( x\right) = \mathop{\sum }\limits_{{j = 0}}^{n}{v}_{j}^{-1}{b}_{j}{x}^{j} \in {\bigtriangleup }^{-1}R\left\lbrack x\right\rbrack \]\n\nwith \( \alpha \left( {f\le... | Yes |
Corollary 2.7 Let \( R \) be a ring. If \( R \) is a right \( \alpha \) -McCoy ring, then \( R\left\lbrack {x;{x}^{-1}}\right\rbrack \) is right \( \alpha \) -McCoy. | Proof Let \( \bigtriangleup = \left\{ {1, x,{x}^{2},\cdots }\right\} \), then clearly \( \bigtriangleup \) is a multipicatively closed subset of \( R\left\lbrack x\right\rbrack \) . Since \( R\left\lbrack {x;{x}^{-1}}\right\rbrack \cong {\bigtriangleup }^{-1}R\left\lbrack x\right\rbrack \), it follows directly from Pro... | Yes |
Proposition 2.9 If \( R \) is an \( \alpha \) -rigid ring, then \( A\left( {R,\alpha }\right) \) is right \( \alpha \) -McCoy. | Proof It follows directly from the fact that \( A\left( {R,\alpha }\right) \) is an \( \alpha \) -rigid ring by [4] and that every \( \alpha \) -rigid ring is right \( \alpha \) -McCoy. | Yes |
Proposition 2.12 Let \( R \) be a right Ore ring with \( Q\left( R\right) \) the classical right quotient ring of \( R \) . If \( \alpha \) is an endomorphism of \( R \), then \( R \) is right \( \alpha \) -McCoy if and only if \( Q\left( R\right) \) is right \( \alpha \) -McCoy. | Proof Let \( F\left( x\right) = \mathop{\sum }\limits_{{i = 0}}^{m}{\delta }_{i}{x}^{i}, G\left( x\right) = \mathop{\sum }\limits_{{j = 0}}^{n}{\beta }_{j}{x}^{j} \) be nonzero polynomials in \( Q\left\lbrack x\right\rbrack \) with \( \alpha \left( {F\left( x\right) }\right) G\left( x\right) = 0 \) . By [6, Proposition... | Yes |
Proposition 3.3 Let \( R \) be a right weak \( \alpha \) -McCoy ring and \( \alpha \) an endomorphism of \( R \) . Then \( {T}_{n}\left( R\right) \) is a right weak \( \alpha \) -McCoy ring. | Proof Let \( f\left( x\right) = \mathop{\sum }\limits_{{i = 0}}^{m}{A}_{i}{x}^{i}, g\left( x\right) = \mathop{\sum }\limits_{{j = 0}}^{n}{B}_{j}{x}^{j} \) be nonzero polynomials in \( {T}_{n}\left( R\right) \left\lbrack x\right\rbrack \) with \( \alpha \left( {f\left( x\right) }\right) g\left( x\right) = 0 \), where \(... | Yes |
Proposition 3.8 Let \( R \) be a ring and \( I \) an ideal of \( R \) such that \( R/I \) is right weak \( \alpha \) -McCoy. If \( I \subseteq \operatorname{nil}\left( R\right) \), then \( R \) is a right weak \( \alpha \) -McCoy ring. | Proof Let \( f\left( x\right) = \mathop{\sum }\limits_{{i = 0}}^{m}{a}_{i}{x}^{i} \) and \( g\left( x\right) = \mathop{\sum }\limits_{{j = 0}}^{n}{b}_{j}{x}^{j} \) be polynomials in \( R\left\lbrack x\right\rbrack \) with \( \alpha \left( {f\left( x\right) }\right) g\left( x\right) = \) 0 . Then we have \( \mathop{\sum... | Yes |
Example 2.2 Consider the ring \( R = \left\{ {\left. \left( \begin{array}{ll} a & b \\ 0 & c \end{array}\right) \right| \;a, b, c \in \mathbb{Z}}\right\} \), where \( \mathbb{Z} \) is the ring of integers, the endomorphism \( \sigma : R \rightarrow R,\sigma \left( \left( \begin{array}{ll} a & b \\ 0 & c \end{array}\rig... | Let \[ \mathbf{A} = \left( \begin{matrix} {a}_{1} & {b}_{1} \\ 0 & {c}_{1} \end{matrix}\right) ,\mathbf{B} = \left( \begin{matrix} {a}_{2} & {b}_{2} \\ 0 & {c}_{2} \end{matrix}\right) ,\mathbf{C} = \left( \begin{matrix} {a}_{3} & {b}_{3} \\ 0 & {c}_{3} \end{matrix}\right) \in R \] with \( \mathbf{{AB}}\sigma \left( \ma... | Yes |
Example 2.3 Let \( {\mathbb{Z}}_{2} \) be the ring of integers modulo 2. We consider ring \( R = {\mathbb{Z}}_{2}\bigoplus {\mathbb{Z}}_{2} \) with the usual addition and multiplication. Then \( R \) is a commutative reduced ring, and so \( R \) is symmetric. Now let \( \sigma : R \rightarrow R \) given by \( \sigma \l... | For \( A = \left( {1,0}\right), B = \left( {0,1}\right), C = \left( {1,1}\right) \in R \), we have \( {AB\sigma }\left( C\right) = \left( {1,0}\right) \left( {0,1}\right) \left( {1,1}\right) = 0 \) , but \( {AC\sigma }\left( B\right) = \left( {1,0}\right) \left( {1,1}\right) \left( {1,0}\right) = \left( {1,0}\right) \n... | Yes |
Example 2.4 Consider the ring \( R = \left\{ {\left. \left( \begin{array}{ll} a & b \\ 0 & a \end{array}\right) \right| \;a, b \in \mathbb{Z}}\right\} \) and the automorphism \( \sigma : R \rightarrow R \) | \[ \sigma \left( \left( \begin{array}{ll} a & b \\ 0 & a \end{array}\right) \right) = \left( \begin{matrix} a & - b \\ 0 & a \end{matrix}\right) . \] \( R \) is not reduced and hence not \( \sigma \) -rigid. But \( R \) is a symmetric \( \sigma \) -ring. In fact, for any \[ \mathbf{A} = \left( \begin{array}{ll} a & b \... | Yes |
Proposition 2.5 For a nonzero endomorphism \( \sigma \) of a ring \( R \), the following statements are equivalent:\n\n(1) \( R \) is a symmtric \( \sigma \) -ring;\n\n(2) \( {l}_{R}\left( {{b\sigma }\left( c\right) }\right) \subseteq {l}_{R}\left( {{c\sigma }\left( b\right) }\right) \) for any \( a, b, c \in R \) ;\n\... | Proof (1) \( \Leftrightarrow \) (3) Suppose \( {AC\sigma }\left( B\right) = 0 \) for \( A, B, C \subseteq R \) . Then \( {ab\sigma }\left( c\right) = 0 \) for any \( a \in A, b \in B, c \in C \), and hence \( {ac\sigma }\left( b\right) = 0 \) . Therefore, \( {AC\sigma }\left( B\right) = \left\{ {\sum {a}_{i}{c}_{i}\sig... | Yes |
Proposition 2.6 Let \( \sigma \) be a nonzero endomorphism of a ring \( R \) . Then we have the following:\n\n(1) If \( {\sigma }^{2} = i{d}_{R} \), then \( R \) is a right (left) \( \sigma \) -shifting ring if and only if \( R \) is a right (left) semicommutative \( \sigma \) -ring; | Proof (1) Suppose that \( R \) is right \( \sigma \) -shifting and \( {a\sigma }\left( b\right) = 0 \) for \( a, b \in R \) . Then we have \( {b\sigma }\left( a\right) = 0,\sigma \left( b\right) {\sigma }^{2}\left( a\right) = 0 \) and \( \sigma \left( b\right) {\sigma }^{2}\left( a\right) \alpha \left( R\right) = 0 \) ... | No |
Proposition 2.7 Let \( \sigma \) be a monomorphism of a ring \( R \) . If \( R \) is a symmetric \( \sigma \) -ring, then \( R \) is semicommutative. | Proof Assume that \( R \) is a symmetric \( \sigma \) -ring with a monomorphism \( \sigma \) . Since \( 1 \in R, R \) is a right \( \sigma \) -shifting ring. For \( a, b \in R \), if \( {ab} = 0 \), then \( \sigma \left( a\right) \sigma \left( b\right) = 0 \), and hence \( {b\sigma }\left( {\sigma \left( a\right) }\rig... | Yes |
Proposition 2.8 Let \( \sigma \) be an endomorphism of a ring \( R \) with \( \sigma \left( e\right) = e \) for any \( {e}^{2} = e \in R \) . If \( R \) is a symmetric \( \sigma \) -ring, then \( R, R\left\lbrack x\right\rbrack \) and \( R\left\lbrack {x;\sigma }\right\rbrack \) are all abelian. | Proof Assume that \( R \) is a symmetric \( \sigma \) -ring. Then \( R \) is a right \( \sigma \) -shifting ring. For any \( r \in R \), we have\n\n\[ \n{e\sigma }\left( {1 - e}\right) \sigma \left( r\right) = {e\sigma }\left( {\left( {1 - e}\right) r}\right) = 0, \n\]\n\n\[ \n\left( {1 - e}\right) \sigma \left( e\righ... | Yes |
Proposition 2.10 Let \( R \) be a reduced ring with an endomorphism \( \sigma \) . If \( R \) is a symmetric \( \sigma \) -ring, then \( T\left( {R, R}\right) \) is a symmetric \( \bar{\sigma } \) -ring. | Proof Suppose that \( R \) is a symmetric \( \sigma \) -ring. Let \( \mathbf{A} = \left( \begin{matrix} {a}_{1} & {b}_{1} \\ 0 & {a}_{1} \end{matrix}\right) ,\mathbf{B} = \left( \begin{matrix} {a}_{2} & {b}_{2} \\ 0 & {a}_{2} \end{matrix}\right) \), \( \mathbf{C} = \left( \begin{matrix} {a}_{3} & {b}_{3} \\ 0 & {a}_{3}... | Yes |
Proposition 2.12 Let \( \sigma \) be an endomorphism of an abelian ring \( R \) with \( \sigma \left( e\right) = e \) for any \( {e}^{2} = e \in R \) . Then the following statements are equivalent:\n\n(1) \( R \) is a symmetric \( \sigma \) -ring;\n\n(2) \( {eR} \) and \( \left( {1 - e}\right) R \) are symmetric \( \si... | Proof \( \;\left( 1\right) \Rightarrow \left( 2\right) \) Since \( \sigma \left( {eR}\right) \subseteq {eR},\sigma \left( {\left( {1 - e}\right) R}\right) \subseteq \left( {1 - e}\right) R \), it is obvious by the definition.\n\n\( \left( 2\right) \Rightarrow \left( 1\right) \) Let \( a, b, c \in R \) with \( {ab\sigma... | Yes |
Proposition 2.14 Let \( A \) be the corresponding Jordan extension of a ring \( R \) and \( \sigma \) be a monomorphism of \( R \) . Then \( R \) is a symmetric \( \sigma \) -ring if and only if \( A \) is a symmetric \( \sigma \) -ring. | Proof Since \( \sigma \left( R\right) \subseteq R \), it suffices to obtain the necessity.\n\nAssume that \( R \) is a symmetric \( \sigma \) -ring and \( {ab\sigma }\left( c\right) = 0 \) for \( a, b, c \in A \) . By the definition of \( A \), there exists \( n \geq 0 \) such that \( {\sigma }^{n}\left( a\right) ,{\si... | Yes |
Proposition 2.15 Let \( R \) be a ring with an endomorphism \( \sigma, S \) a ring and \( \tau : R \rightarrow S \) a ring isomorphism. Then \( R \) is a symmetric \( \sigma \) -ring if and only if \( S \) is a symmetric \( {\tau \sigma }{\tau }^{-1} \) -ring. | Proof For \( a, b, c \in R \), let \( {a}^{\prime } = \tau \left( a\right) ,{b}^{\prime } = \tau \left( b\right) \) and \( {c}^{\prime } = \tau \left( c\right) \in S \) . Suppose that \( R \) is a symmetric \( \sigma \) -ring and \( {a}^{\prime }{b}^{\prime }{\tau \sigma }{\tau }^{-1}\left( {c}^{\prime }\right) = 0 \) ... | Yes |
Proposition 3.3 The following statements are equivalent:\n\n(1) \( R \) is a weak symmetric \( \left( {\sigma ,\delta }\right) \) -ring;\n\n(2) \( {T}_{n}\left( R\right) \) is a weak symmetric \( \left( {\bar{\sigma },\bar{\delta }}\right) \) -ring;\n\n(3) \( {S}_{n}\left( R\right) \) is a weak symmetric \( \left( {\ba... | Proof (1) \( \Rightarrow \) (2) Let \( \mathbf{A} = \left( {a}_{ij}\right) ,\mathbf{B} = \left( {b}_{ij}\right) ,\mathbf{C} = \left( {c}_{ij}\right) \in {T}_{n}\left( R\right) \), where \( {a}_{ij} = 0,{b}_{ij} = \) \( 0,{c}_{ij} = 0 \), for all \( i > j \), with \( \mathbf{{AB}}\sigma \left( \mathbf{C}\right) \in \ope... | Yes |
Corollary 3.4 The trivial extension \( T\left( {R, R}\right) \) of \( R \) by \( R \) is a weak symmetric \( \left( {\bar{\sigma },\bar{\delta }}\right) \) -ring if and only if \( \mathrm{R} \) is a weak symmetric \( \left( {\sigma ,\delta }\right) \) -ring. | Proof By the isomorphism \( T\left( {R, R}\right) \cong {T}_{2}\left( R\right) \), we obtain the proof. | No |
Corollary 3.5 \( R\left\lbrack x\right\rbrack /\left\langle {x}^{n}\right\rangle \) is a weak symmetric \( \left( {\bar{\sigma },\bar{\delta }}\right) \) -ring if and only if \( R \) is a weak symmetric \( \left( {\sigma ,\delta }\right) \) -ring, where \( \left\langle {x}^{n}\right\rangle \) is an ideal of \( R \) gen... | Proof By the isomorphism \( R\left\lbrack x\right\rbrack /\left\langle {x}^{n}\right\rangle \cong {S}_{n}\left( R\right) \), we obtain the proof. | No |
Theorem 3.7 Let \( I \) be a \( \left( {\sigma ,\delta }\right) \) -stable and weak symmetric \( \left( {\sigma ,\delta }\right) \) -ideal of \( R \) . If \( I \subseteq \) \( \operatorname{nil}\left( R\right) \), then \( R/I \) is a weak symmetric \( \left( {\bar{\sigma },\bar{\delta }}\right) \) -ring if and only \( ... | Proof Suppose \( \bar{a}\bar{b}\bar{\sigma }\left( \bar{c}\right) \in \operatorname{nil}\left( {R/I}\right) \) and \( \bar{a}\bar{b}\bar{\delta }\left( \bar{c}\right) \in \operatorname{nil}\left( {R/I}\right) \) . Then there exist some positive integer \( m, n \) such that \( {\left( ab\sigma \left( c\right) \right) }^... | Yes |
Lemma 3.11 If \( R \) is a weakly 2-primal ring and \( f\left( x\right) = {a}_{0} + {a}_{1}x + \cdots + {a}_{n}{x}^{n} \in R\left\lbrack x\right\rbrack \) . Then \( f\left( x\right) \in \operatorname{nil}\left( {R\left\lbrack x\right\rbrack }\right) \) if and only if \( {a}_{i} \in \operatorname{nil}\left( R\right) \) ... | Proof Suppose that \( f\left( x\right) = {a}_{0} + {a}_{1}x + \cdots + {a}_{n}{x}^{n} \in R\left\lbrack x\right\rbrack \in \operatorname{nil}\left( {R\left\lbrack x\right\rbrack }\right) \) . Then by [7], Proposition 1.3, we obtain \( {a}_{i} \in \operatorname{nil}\left( R\right) \) for each \( 0 \leq i \leq n \), and ... | Yes |
Proposition 2.2 (see [5, Proposition 4.1]) There is one-to-one correspondence between \( \mathfrak{t} \) -roots and complex irreducible ad \( \left( {\mathfrak{k}}^{\mathbb{C}}\right) \) -submodules \( {\mathfrak{m}}_{\xi } \) of \( {\mathfrak{m}}^{\mathbb{C}} \) . This correspondence is given by\n\n\[ \n{\mathcal{R}}_... | Since the complex conjugation \( \tau : {g}^{\mathbb{C}} \rightarrow {g}^{\mathbb{C}}, X + {iY} \mapsto X - {iY}\left( {X, Y \in g}\right) \) of \( {g}^{\mathbb{C}} \) with respect to the compact real form \( g \) interchanges the root spaces, i.e. \( \tau \left( {E}_{\alpha }\right) = {E}_{-\alpha } \) and \( \tau \le... | Yes |
Lemma 3.2 (see [11]) Let \( M = G/K \) be a reductive homogeneous space of a compact semisimple Lie group \( G \) and let \( \mathfrak{m} = {\mathfrak{m}}_{1} \oplus \cdots \oplus {\mathfrak{m}}_{s} \) be a decomposition of \( \mathfrak{m} \) into mutually inequivalent irreducible \( {Ad}\left( K\right) \) -submodules.... | \[ {\gamma }_{k} = \frac{1}{2{x}_{k}} + \frac{1}{4{d}_{k}}\mathop{\sum }\limits_{{i, j}}\frac{{x}_{k}}{{x}_{i}{x}_{j}}{c}_{ij}^{k} - \frac{1}{2{d}_{k}}\mathop{\sum }\limits_{{i, j}}\frac{{x}_{j}}{{x}_{k}{x}_{i}}{c}_{ki}^{j}\left( {k = 1,\cdots, s}\right) . \] | Yes |
Lemma 3.3 The non-zero structure constants of generalized flag manifold \( {SO}\left( 8\right) /T \) are given by \( {c}_{1,2}^{5} = {c}_{1,6}^{8} = {c}_{1,7}^{9} = {c}_{1,{10}}^{11} = {c}_{2,3}^{6} = {c}_{2,4}^{7} = {c}_{2,{11}}^{12} = {c}_{3,5}^{8} = {c}_{3,7}^{10} = {c}_{3,9}^{11} = {c}_{4,5}^{9} = {c}_{4,6}^{10} = ... | Proof From the theorem of Lie algebra we can get \( {N}_{\alpha ,\beta }^{2} = \frac{q\left( {p + 1}\right) }{2}\left( {\alpha ,\alpha }\right) ,\left( {\alpha ,\beta }\right) = \) \( - \frac{q - p}{2}\left( {\alpha ,\alpha }\right) \), where \( p, q \) are the largest nonnegative integers such that \( \beta + {k\alpha... | Yes |
Lemma 3.4 The components \( {\gamma }_{i}\left( {i = 1,\cdots ,{12}}\right) \) of Ricci tensor associated to the \( {SO}\left( 8\right) \) - invariant Riemmanian metric \( \mathfrak{g} \) are the following: | From (10) and (11) we get that a \( G \) -invariant Riemmanian metric \( \mathfrak{g} \) on \( M = {SO}\left( 8\right) /T \) is Einstein, if and only if, there is a positive constant \( e \) such that \( {\gamma }_{1} = {\gamma }_{2} = {\gamma }_{3} = {\gamma }_{4} = {\gamma }_{5} = \n\n\( {\gamma }_{6} = {\gamma }_{7}... | Yes |
Theorem 3.5 The full flag manifold \( M = {SO}\left( 8\right) /T \) admits five (up to isometry) \( {SO}\left( 8\right) \) -invariant Einstein metrics. There is a unique Kähler Einstein metric (up to a scale) given by \[\mathfrak{g} = \left( {1,1,1,1,2,2,2,3,3,3,4,5}\right) \] and the other four are non-Kähler (up to a... | Proof We compute \( {H}_{\mathfrak{g}} \) of all the one hundred and sixty positive (real) solutions by formula (16) and obtain five non equal values, the five values are \[ {8.0356},{7.9370},{8.0000},{7.9975},{7.9959}\text{.} \] Thus there are at least five non-isometry Einstein metrics. When \( {H}_{\mathfrak{g}} = {... | Yes |
Theorem 3.2 Let \( \\left( {X,\\land ,\\vee , \\leq }\\right) \) be a lattice and \( \\mu \) be an \( L \) -fuzzy subset of \( X \) . Then \( \\mu \) is a \( {TL} \) -fuzzy ideal of \( X \) if and only if it satisfies the following conditions: for all \( x, y \\in X \) ,\n\n(i) \( \\mu \\left( {x \\vee y}\\right) \\geq... | Proof The proof is straightforward. | No |
Let \( X = \{ 0, a, b, c,1\} \) be a lattice, the partial order on \( X \) is defined as shown in Fig. 1. Let \( L = \{ 1,2,3,4,5,6\} \), the partial order on \( L \) is defined as shown in Fig. 2, \( T = \land \). Define two \( L \)-fuzzy subsets \( \mu \) and \( \nu \) of \( X \) as follows: \( \mu \left( 0\right) = ... | By routine calculations, it is easy to check that \( \mu \) is a \( {TL} \)-fuzzy ideal of \( X \) . | No |
Proposition 3.4 Let \( {\mu }_{i}\left( {i \in I}\right) \) be \( {TL} \) -fuzzy ideals of a lattice \( X \) . Then \( { \cap }_{i \in I}{\mu }_{i} \) is a \( {TL} \) -fuzzy ideal of \( X \) . | Proof The proof is straightforward. | No |
Example 3.5 Let \( X = \{ 0, a, b,1\} \) be a lattice, the partial order on \( X \) is defined as shown in Fig. 3 in Example 3.3 (2). Let \( L = \left\lbrack {0,1}\right\rbrack \) and \( T = {T}_{M} \) . Define two \( L \) -fuzzy subsets \( \mu \) and \( \nu \) of \( X \) as follows: \( \mu \left( 0\right) = \frac{3}{5... | Since \( \left( {\mu \cup \nu }\right) \left( {a \vee b}\right) = \left( {\mu \cup \nu }\right) \left( 1\right) = \frac{1}{5} < \left( {\mu \cup \nu }\right) \left( a\right) \land \left( {\mu \cup \nu }\right) \left( b\right) = \frac{3}{10},\mu \cup \nu \) is not a \( {TL} \) -fuzzy ideal of \( X \) . | Yes |
Theorem 5.2 Let \( {\mu }_{i} \) be a \( {TL} \) -fuzzy ideal of a lattice \( {X}_{i}, i = 1,2 \) . Then \( {\mu }_{1}{ \times }_{T}{\mu }_{2} \) is a \( {TL} \) -fuzzy ideal of \( {X}_{1} \times {X}_{2} \) . | Proof Assume that \( {\mu }_{i} \) be a \( {TL} \) -fuzzy ideal of a lattice \( {X}_{i}, i = 1,2 \) . For any \( \left( {{x}_{1},{x}_{2}}\right) \) , \( \left( {{y}_{1},{y}_{2}}\right) \in {X}_{1} \times {X}_{2} \), then we have that\n\n\[ \n{\mu }_{1}{ \times }_{T}{\mu }_{2}\left( {\left( {{x}_{1},{x}_{2}}\right) \vee... | Yes |
Theorem 5.4 Let \( {X}_{1} \) and \( {X}_{2} \) be two lattices and \( \mu \) be a \( {TL} \) -fuzzy ideal of \( {X}_{1} \times {X}_{2} \) . Then \( {\mu }_{{X}_{i}} \) is a \( {TL} \) -fuzzy ideal of \( {X}_{i}, i = 1,2 \) . | Proof Assume that \( \mu \) be a \( {TL} \) -fuzzy ideal of a lattice \( {X}_{1} \times {X}_{2} \) . For any \( x, y \in {X}_{1} \) , then we have that\n\n\[ \n{\mu }_{{X}_{1}}\left( {x{ \vee }_{1}y}\right) = \mathop{\bigvee }\limits_{{b \in {X}_{2}}}\mu \left( {x{ \vee }_{1}y, b}\right) \geq \mathop{\bigvee }\limits_{... | Yes |
Theorem 5.6 Let \( {X}_{1} \) and \( {X}_{2} \) be two lattices and \( \mu \) be a \( {TL} \) -fuzzy ideal of \( {X}_{1} \times {X}_{2} \) , let \( a \in {X}_{1}, b \in {X}_{2} \) . Then \( {\left. {\mu }_{1}\right| }_{b} \) is a \( {TL} \) -fuzzy ideal of \( {\left. {X}_{1}\text{and}{\mu }_{2}\right| }_{a} \) is a \( ... | Proof The proof is straightforward. | No |
Lemma 5.7 Let \( {X}_{1} \) and \( {X}_{2} \) be two lattices and \( \mu \) be a \( {TL} \) -fuzzy ideal of \( {X}_{1} \times {X}_{2} \) such that \( \operatorname{Im}\mu \subseteq {D}_{T} \), let \( a \in {X}_{1}, b \in {X}_{2} \) . Then \( {\left. {\left. {\mu }_{1}\right| }_{b}{ \times }_{T}{\mu }_{2}\right| }_{a} \... | Proof Assume that \( \mu \) is a \( {TL} \) -fuzzy ideal of a lattice \( {X}_{1} \times {X}_{2} \) . First, we prove that \( \mu \subseteq {\mu }_{{X}_{1}}{ \times }_{T}{\mu }_{{X}_{2}} \) . For any \( \left( {x, y}\right) \in {X}_{1} \times {X}_{2} \), we have that \[ \mu \left( {x, y}\right) \leq \mathop{\bigvee }\li... | Yes |
Theorem 5.8 Let \( {X}_{1} \) and \( {X}_{2} \) be two lattices with the bottom element 0 and \( \mu \) be a \( {TL} \) -fuzzy ideal of \( {X}_{1} \times {X}_{2} \) such that \( {Im\mu } \subseteq {D}_{T} \) . Then \( \mu \) is the \( T \) -product of a \( {TL} \) -fuzzy ideal of \( {X}_{1} \) and a \( {TL} \) -fuzzy i... | Proof Assume that \( \mu = {\mu }_{1}{ \times }_{T}{\mu }_{2} \), where \( {\mu }_{1} \) and \( {\mu }_{2} \) are \( {TL} \) -fuzzy ideals of \( {X}_{1} \) and \( {X}_{2} \) , respectively. Then \( {\mu }_{1}\left( x\right) \leq {\mu }_{1}\left( 0\right) \) for any \( x \in {X}_{1} \) and \( {\mu }_{2}\left( y\right) \... | Yes |
Lemma 3.1 Let \( \{ \xi \left( t\right), t \geq 0\} \) be a real-valued random process with continuous and bounded sample paths, if \( {w}_{t},{W}_{T} \) are defined as in (2.2), and\n\n\[ \operatorname{Var}\left( {{\int }_{1}^{T}{w}_{t}\xi \left( t\right) \mathrm{d}t}\right) \ll {\left( {W}_{T}\right) }^{2}{\left( \ln... | Proof The proof can refer to Appendix. | No |
Lemma 3.2 Suppose \( \{ X\left( t\right), t \geq 0\} \) is a continuous mean square differentiable stationary Gaussian process with covariance function \( r\left( \cdot \right) \) satisfying conditions (1.1),(1.5) and (2.1). Let \( q = {u}_{t}^{-1}{\left( \ln t\right) }^{-\beta \left( {1 + \varepsilon }\right) } \), we... | Proof The proof can refer to Appendix. | No |
Lemma 1.1 (see [9]) The Schwarz problem for Cauchy-Riemann equation in \( \Omega \)\n\n\[ \n{w}_{\bar{z}} = f\text{ in }\Omega ,\;\operatorname{Re}w = \gamma \text{ on }\partial \Omega ,\n\] | \[ \n\frac{\alpha }{\pi }{\int }_{0}^{\frac{\pi }{\alpha }}\operatorname{Im}w\left( {e}^{i\varphi }\right) \mathrm{d}\varphi = c,\;c \in \mathbb{R} \n\]\n\nfor \( f \in {L}_{p}\left( {\Omega ;\mathbb{C}}\right), p > 2,\gamma \in C\left( {\partial \Omega ;\mathbb{R}}\right) \), is uniquely solvable by\n\n\[ \nw\left( z\... | Yes |
Lemma 2.2 For \( f \in {L}_{p}\left( {\Omega ;C}\right), p > 2 \) ,\n\n\[ \n\frac{{\partial }^{n}{T}_{n}\left\lbrack f\right\rbrack \left( z\right) }{\partial {\bar{z}}^{n}} = f\left( z\right) ,\;z \in \Omega ;\;{\left\{ \operatorname{Re}{T}_{n}\left\lbrack f\right\rbrack \right\} }^{ + }\left( t\right) = 0,\;t \in \pa... | Proof Since for \( l = 1,2,\cdots, n \) ,\n\n\[ \n{T}_{l}\left\lbrack f\right\rbrack \left( z\right) = \frac{{\left( -1\right) }^{l - 1}}{\left( {l - 1}\right) !}\mathop{\sum }\limits_{{k = 0}}^{{l - 1}}\left( \begin{matrix} l - 1 \\ k \end{matrix}\right) {\left( -z - \bar{z}\right) }^{l - k - 1}{T}_{1}\left\lbrack {{\... | Yes |
Theorem 2.1 The Schwarz problem for polyanalytic equation in \( \Omega \) , \[ \left\{ \begin{array}{ll} {\partial }_{\bar{z}}^{n}w\left( z\right) = f\left( z\right) , & z \in \Omega ,\;f \in {L}_{p}\left( {\Omega ,\mathbb{C}}\right), p > 2; \\ {\left\{ \operatorname{Re}\left( {\partial }_{\bar{z}}^{k}w\right) \right\}... | is solvable by \[ w\left( z\right) = {S}_{n}\left\lbrack {{\gamma }_{0},{\gamma }_{1},\cdots ,{\gamma }_{n - 1}}\right\rbrack \left( z\right) + {T}_{n}\left\lbrack f\right\rbrack \left( z\right) + i\mathop{\sum }\limits_{{k = 0}}^{{n - 1}}{\left( z + \bar{z}\right) }^{k}{c}_{k}, \] where \( {c}_{k} \in \mathrm{R},{S}_{... | Yes |
Proposition 2 Under DT (2.4), the matrix \( \bar{V} \) in (2.5) has the same form as V, that is\n\n\[ \bar{V} = \left( \begin{matrix} {\bar{V}}_{11} & {\bar{V}}_{12} & {\bar{V}}_{13} \\ {\bar{V}}_{21} & {\bar{V}}_{22} & {\bar{V}}_{23} \\ {\bar{V}}_{31} & {\bar{V}}_{32} & {\bar{V}}_{33} \end{matrix}\right) \]\n\nin whic... | Proof In a way similar to Proposition 1, we denote \( {T}^{-1} = {T}^{ * }/\det T \) and\n\n\[ \left( {{T}_{t} + {TV}}\right) {T}^{ * } = \left( \begin{array}{lll} {g}_{11}\left( \lambda \right) & {g}_{12}\left( \lambda \right) & {g}_{13}\left( \lambda \right) \\ {g}_{21}\left( \lambda \right) & {g}_{22}\left( \lambda ... | Yes |
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