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Lemma 2.5 If \( 3\left( 0\right) \) -vertex \( v \) is adjacent to two light \( 3\left( 0\right) \) -vertices \( {v}_{1},{v}_{2} \), then another neighbor of \( v \), namely \( {v}_{3} \), is a \( \Delta \) -vertex; A light \( 3\left( 0\right) \) -vertex is not adjacent to another light \( 3\left( 0\right) \) -vertex. | Proof Suppose to the contrary that \( 3\left( 0\right) \) -vertex \( v \) is adjacent to two light \( 3\left( 0\right) \) -vertex \( {v}_{1},{v}_{2}, \) but \( d\left( {v}_{3}\right) < \Delta \) . By the minimality of \( G,{G}^{\prime } = G - v{v}_{1} \) is \( \left( {\Delta + 4}\right) - 2 \) -distance colorable. We e... | Yes |
Lemma 2.6 For a 4-vertex \( v \), let \( N\left( v\right) = \left\{ {{v}_{1},{v}_{2},{v}_{3},{v}_{4}}\right\} \) be the neighborhood of \( v \) . (1) If \( v \) is adjacent to a 2-vertex \( {v}_{1} \), then \( d\left( {v}_{2}\right) + d\left( {v}_{3}\right) + d\left( {v}_{4}\right) \geq \Delta + 2 \) . | Proof (1) Suppose to the contrary that \( d\left( {v}_{2}\right) + d\left( {v}_{3}\right) + d\left( {v}_{4}\right) \leq \Delta + 1 \) . By the minimality of \( G,{G}^{\prime } = G - {v}_{1} \) is \( \left( {\Delta + 4}\right) \) -2-distance colorable. We erase the colors of \( v \) . Since \( \left| {F\left( {v}_{1}\ri... | Yes |
Lemma 2.7 For a \( k \) -vertex \( v\left( {5 \leq k \leq 9}\right), N\left( v\right) = \left\{ {{v}_{1},{v}_{2},\cdots ,{v}_{k}}\right\} \) and \( d\left( {v}_{1}\right) \leq d\left( {v}_{2}\right) \leq \) \( \cdots \leq d\left( {v}_{k}\right) \)\n\n(1) If \( v \) is adjacent to \( {k2} \) -vertices, then \( d\left( {... | Proof (1) Suppose \( k \) -vertex \( v \) is adjacent to \( k \) 2-vertices, but at least one of \( d\left( {v}_{i1}\right) (1 \leq \) \( i \leq k) \) is smaller than \( \Delta + 4 - \left( {k - 1}\right) \) . Without loss of generality, assume \( d\left( {v}_{11}\right) \leq \Delta + 3 - \left( {k - 1}\right) \) . By ... | Yes |
Lemma 2.8 If \( k \) -vertex \( \left( {5 \leq k \leq 7}\right) v \) is adjacent to \( i \) 2-vertices \( {v}_{1},{v}_{2},\cdots ,{v}_{i}\left( {i < k}\right) \) and \( k - i \) 3-vertices \( {v}_{i + 1},{v}_{i + 2},\cdots ,{v}_{k} \), then \( {v}_{i + 1},{v}_{i + 2},\cdots ,{v}_{k} \) are both heavy 3-vertices. | Proof Assume there is a light 3(0)-vertex in \( {v}_{i + 1},{v}_{i + 2},\cdots ,{v}_{k}. \) Without loss of generality, suppose \( {v}_{i + 1} \) is a light 3-vertex. By the minimality of \( G, G - v{v}_{i + 1} \) is \( \left( {\Delta + 4}\right) \) -2-distance colorable. We erase the colors of \( v,{v}_{i + 1} \) . Si... | Yes |
Example 2.4 Let \( \\left( {L,\\left\\lbrack {\\cdot , \\cdot }\\right\\rbrack ,\\alpha ,\\beta }\\right) \) be a finite dimensional BiHom-Lie superalgebra, \( H \) be a Hopf algebra. Then \( \\left( {H \\otimes L,\\left\\lbrack *\\right\\rbrack ,\\mathrm{{id}} \\otimes \\alpha ,\\mathrm{{id}} \\otimes \\beta }\\right)... | \[ \\left\\lbrack {\\left( {h \\otimes x}\\right) * \\left( {l \\otimes y}\\right) }\\right\\rbrack = h \\otimes l{ \\otimes }_{H}\\left( {1 \\otimes \\left\\lbrack {x, y}\\right\\rbrack }\\right) \] for all \( h, l \\in H \) and homogeneous elements \( x, y \\in L \) . | Yes |
Lemma 2.1 Let \( \\left( {A,*,\\alpha ,\\beta }\\right) \) be a BiHom-associative \( H \) -pseudo-superalgebra. Then\n\n(1) \( \\left( {\\left( {\\sigma \\otimes \\mathrm{{id}}}\\right) { \\otimes }_{H}\\mathrm{{id}}}\\right) \\left( {\\alpha \\left( a\\right) * \\left( {b * c}\\right) }\\right) = \\left( {\\left( {\\s... | Proof We only prove (3), and similarly for (1),(2). For any homogeneous elements \( a, b, c \\in \) \( A \), let\n\n\[ a * b = \\mathop{\\sum }\\limits_{i}{f}_{i} \\otimes {g}_{i} \\otimes {e}_{i},\\;{e}_{i} * \\beta \\left( c\\right) = \\mathop{\\sum }\\limits_{{i, j}}{f}_{i, j} \\otimes {g}_{i, j} \\otimes {e}_{i, j}... | No |
Theorem 2.1 Let \( \\left( {L,\\left\\lbrack *\\right\\rbrack }\\right) \) be a Lie \( H \) -pseudo-superalgebra and let \( \\alpha ,\\beta : L \\rightarrow L \) two commuting linear maps such that \( \\alpha \\left\\lbrack {a, b}\\right\\rbrack = \\left\\lbrack {\\alpha \\left( a\\right) ,\\alpha \\left( b\\right) }\\... | Proof The proof is straightforward and left to the reader. | No |
Proposition 3.1 Let \( Y \) be a Lie \( H \) -differential-superalgebra and \( \left( {L,\left\lbrack *\right\rbrack ,\alpha ,\beta }\right) \) a BiHom-Lie \( H \) -pseudo-superalgebra with bijective \( \alpha ,\beta \) . Then \( \left( {{A}_{Y}L,{\operatorname{id}}_{Y} \otimes \alpha ,{\operatorname{id}}_{Y} \otimes \... | Proof First we shall show that \( {A}_{Y}L \) is an \( H \) -module, but this is easy to check. It remains to verify that the conditions (1) and (2) in Definition 2.3 are satisfied. For this purpose, take \( x{ \otimes }_{H}a, y{ \otimes }_{H}b, z{ \otimes }_{H}c \in {A}_{Y}L \), suppose\n\n\[ \left\lbrack {\beta \left... | No |
Lemma 4.1 With the above notations, for any \( \gamma \in {C}_{\alpha ,\beta }^{n}\left( {L, M}\right) \), we have\n\n\[ \left( {d\gamma }\right) \circ \alpha = {\alpha }_{M} \circ \left( {d\gamma }\right) \]\n\n\[ \left( {d\gamma }\right) \circ \beta = {\beta }_{M} \circ \left( {d\gamma }\right) \] | Proof It is straightforward. | No |
Theorem 4.1 Let \( \left( {L,\left\lbrack *\right\rbrack ,\alpha ,\beta }\right) \) be a multiplicative BiHom-Lie \( H \) -pseudo-superalgebra, and \( \left( {M,{\rho }_{M},{\alpha }_{M},{\beta }_{M}}\right) \) be a BiHom- \( L \) -module, considered as a BiHom-Lie \( H \) -pseudo-superalgebra with respect to the zero ... | Proof Let \( 0 \rightarrow M \rightarrow \widehat{L} \rightarrow L \rightarrow 0 \) be an extension of \( L \) -modules, which is split over \( H \) . Choose a splitting \( \widehat{L} = L \oplus M = \{ l + m \mid l \in L, m \in M\} \) as an \( H \) -module. Denoting the pseudobracket of \( \widehat{L} \) by \( \left\l... | Yes |
Theorem 1.1 Let \( R \) be a ring with unity. Then \( R \) is a weakly clean ring if and only if \( R \) is a weakly clean general ring. | Proof Suppose that \( R \) is a weakly clean ring. For any element \( a \in R \), we have \( a + 1 = u + e \) or \( a + 1 = u - e \), where \( {e}^{2} = e \in \mathrm{{Id}}\left( R\right), u \in U\left( R\right) \) . Then we see that \( a = \left( {u - 1}\right) + e \) or \( a = \left( {u - 1}\right) - e \) and \( u - ... | Yes |
Theorem 1.2 (1) Let \( R \) be a ring with unity. Then the homomorphic image of a weakly clean general ring \( R \) is also a weakly clean general ring. | Proof (1) It is obvious. | No |
Proposition 2.1 Let \( I \) be a general ring and \( S \) be the subring of \( I \) . Then \( \operatorname{wcg}\left( S\right) \leq \operatorname{wcg}\left( I\right) \) . | Proof It is obvious that idempotents of \( S \) are also idempotents of \( I \) . Suppose that \( q \in \) \( Q\left( S\right) \), then there exists \( q \in S \) such that \( p * q = 0 = q * p \) . So we know for any \( q \in Q\left( I\right), e \in {\alpha }_{S}\left( a\right) \) , \( a + e \) or \( a - e \in Q\left(... | Yes |
Proposition 2.2 The following holds for any general ring \( I \) .\n\n(1) For a nilpotent element \( a \in I,\;\left| {\alpha \left( a\right) }\right| \geq 1 \) ; whereas for an idempotent \( e \in I,\;\left| {\alpha \left( e\right) }\right| \geq 1. | Proof (1) Let \( n \) be a positive integer. If \( a \) is a nilpotent element, then there exists an element \( q \in I \) such that \( a * q = 0 = q * a \), where \( q = \left( {-a + {a}^{2} - \cdots + {\left( -1\right) }^{n - 1}{a}^{n - 1}}\right) \), then \( 0 \in \alpha \left( a\right) . \) It is easy to see that \... | No |
Proposition 2.3 The following holds for any ring \( R \) with unity.\n\n(1) If \( a - b \in J\left( R\right) \), then \( \left| {\alpha \left( a\right) }\right| = \left| {\alpha \left( b\right) }\right| \) . | Proof (1) If \( w = a - b \in J\left( R\right) \) and \( e \in \alpha \left( a\right) \), then we have \( a + e \) or \( a - e \in Q\left( R\right) \).\n\nCase 1: If \( q = a + e \in Q\left( R\right) \), then \( q = w + b + e \) . So we have \( b + e = q - w \), where \( 1 + q \in U\left( R\right) \) , \( w \in J\left(... | Yes |
Theorem 2.1 Let \( R \) be a ring with unity. Then \( \operatorname{wcg}\left( R\right) = 1 \) if and only if \( R \) is abelian and for any \( 0 \neq e = {e}^{2} \in R, e \neq u - v \) for any \( u, v \in Q\left( R\right) \) . | \( \begin{aligned} & \text{ Proof }\left( \Rightarrow \right) \text{ Assume that }\operatorname{wcg}\left( R\right) = 1\text{ and }r \in R, \\ & \text{ then }e = 0 + e = \operatorname{er}\left( {1 - e}\right) + \left\lbrack {e + \operatorname{er}\left( {e - 1}\right) }\right\rbrack . \end{aligned} \) Since \( {\left\lb... | Yes |
Theorem 2.2 Let \( I \) be a general ring. Then \( \operatorname{wcg}\left( I\right) = 1 \) if and only if \( I \) is abelian and \( 0 \neq e = {e}^{2} \in I, e \neq u - v \) for any \( u, v \in Q\left( I\right) . \) | Proof \( \;\left( \Rightarrow \right) \) If \( e = {e}^{2} \), write \( q = {exe} - {ex} \) . Let \( p = - q \) . Then we can see that \( p * q = 0 = q * p \) , so \( q \in Q\left( I\right) \) . Take \( a \in I \) such that \( a = q + e = \left( {q + e}\right) + 0 \) or \( a = q - e = 0 - \left( {e - q}\right) \) . We ... | Yes |
If \( R = \left( \begin{matrix} {\mathbb{Z}}_{2} & {\mathbb{Z}}_{2} \\ 0 & {\mathbb{Z}}_{2} \end{matrix}\right) \), then we have \( \operatorname{wcg}\left( R\right) = 2 \) . | Proof We can check that\n\n\[\n\mathrm{{Id}}\left( R\right) = \left\{ {\;\left( {\frac{\overline{0}}{0}\frac{\overline{0}}{0}}\right) ,\left( {\frac{\overline{1}}{0}\frac{\overline{0}}{0}}\right) ,\left( {\frac{\overline{0}}{0}\frac{\overline{0}}{1}}\right) ,\left( {\frac{\overline{1}}{0}\frac{\overline{1}}{0}}\right) ... | Yes |
Theorem 2.4 If \( R = \left( \begin{array}{ll} A & M \\ 0 & B \end{array}\right) \), where \( {}_{A}{M}_{B} \) is a nontrivial bimodule, \( \operatorname{wcg}\left( A\right) = n \), and \( \operatorname{wcg}\left( B\right) = m \), then the following statements hold.\n\n(1) \( \operatorname{wcg}\left( R\right) \geq \lef... | Proof (1) Let \( T = \left( \begin{matrix} - 1 & 0 \\ 0 & 0 \end{matrix}\right) \in R \) . Then we can find \( \left( \begin{array}{ll} 1 & w \\ 0 & 0 \end{array}\right) \in \alpha \left( T\right) \), where \( w \in M \) . So \( \operatorname{wcg}\left( R\right) \geq \left| {\alpha \left( T\right) }\right| \geq \left| ... | Yes |
Example 2.2 If \( R = {M}_{2}\left( {\mathbb{Z}}_{2}\right) \), then \( \operatorname{wcg}\left( R\right) = 5 \) . | Proof We can check that \( \operatorname{wcg}\left( {\mathbb{Z}}_{2}\right) = 1 \) . Take \( A = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \in {M}_{2}\left( {\mathbb{Z}}_{2}\right) \), where \( {A}^{2} = A \) . Thus\n\n\[\n\mathrm{{Id}}\left( R\right) = \left\{ {\;\left( {\frac{\overline{0}}{\overline{... | Yes |
Lemma 2.1 Let \( \mathbb{F} \) be the finite field of characteristic \( p \) with \( {p}^{f} \) elements, then\n\n\[ \n{\mathbb{W}}_{s}{\left( \mathbb{F}\right) }^{ + } \cong {\bigoplus }_{\begin{matrix} {1 \leq r \leq s} \\ {\gcd \left( {r, p}\right) = 1} \end{matrix}}{W}_{1 + \left\lfloor {{\log }_{p}\frac{s}{r}}\rig... | Let \( \mathcal{B} \) be a basis of \( \mathbb{F} \) over \( {\mathbb{F}}_{p} \) . Then \( \left\{ {1 + b{T}^{r} \mid 1 \leq r \leq s,\gcd \left( {r, p}\right) = 1, b \in \mathcal{B}}\right\} \) is a generating set of \( {\mathbb{W}}_{s}{\left( \mathbb{F}\right) }^{ + } \) . If \( \gcd \left( {r, p}\right) = 1 \), then... | Yes |
Example 2.1 We give a complete correspondence of \( {\mathbb{W}}_{2}{\left( {\mathbb{F}}_{3}\right) }^{ + } \cong {W}_{1}{\left( {\mathbb{F}}_{3}\right) }^{ + } \oplus {W}_{1}{\left( {\mathbb{F}}_{3}\right) }^{ + } = \) \( \mathbb{Z}/3\mathbb{Z} \oplus \mathbb{Z}/3\mathbb{Z} \cong \langle 1 + T\rangle \oplus \left\lang... | \( 1 + T,\;{\left( 1 + T\right) }^{2} = 1 + {2T} + {T}^{2},\;1 + {T}^{2},\;{\left( 1 + {T}^{2}\right) }^{2} = 1 + 2{T}^{2},\;{\left( 1 + T\right) }^{3} = {\left( 1 + {T}^{2}\right) }^{3} = 1, \n\n\( \left( {1 + T}\right) \left( {1 + {T}^{2}}\right) = 1 + T + {T}^{2},\;{\left( 1 + T\right) }^{2}\left( {1 + {T}^{2}}\righ... | No |
Lemma 2.2 Let \( R \) be a ring and \( I \subset R \) a two-sided ideal. Let \( \pi : R \rightarrow R/I \) be the canonical surjection. If there is a ring homomorphism \( \iota : R/I \rightarrow R \) such that \( {\pi \iota } = \mathrm{{id}} \), then for any \( i,{K}_{i}\left( R\right) \cong {K}_{i}\left( {R, I}\right)... | Proof There is a long exact sequence of \( K \) -groups\n\n\[ \cdots \rightarrow {K}_{i}\left( {R, I}\right) \rightarrow {K}_{i}\left( R\right) \overset{{\pi }_{ * }}{ \rightarrow }{K}_{i}\left( {R/I}\right) \rightarrow {K}_{i - 1}\left( {R, I}\right) \rightarrow \cdots . \]\n\nSince \( {K}_{i} \) are functors, we have... | Yes |
Lemma 2.4 \( {}^{\left\lbrack 4\right\rbrack } \) If \( R = k\left\lbrack t\right\rbrack /\left( {t}^{n}\right) \) with \( k \) a commutative ring, then | \[ H{C}_{0}\left( R\right) \cong H{H}_{0}\left( R\right) = R \cong {k}^{n}, \] \[ H{C}_{2m}\left( R\right) \cong {\left( {\operatorname{Ann}}_{k}\left( n\right) \right) }^{m} \oplus {k}^{n} \oplus \left( {{\bigoplus }_{j = 0}^{m - 1}\left( {{\bigoplus }_{a = 1}^{n - 1}{\operatorname{Ann}}_{k}\left( {a + {jn}}\right) }\... | Yes |
Theorem 2.1 Let \( \Lambda = \{ {\Lambda }_{j}{\} }_{j \in \mathbb{J}} \) be a generalized frame for \( H \) with respect to \( \{ {V}_{j}{\} }_{j \in \mathbb{J}} \) with the analysis operator \( {U}_{\Lambda } \), and let \( c = {\left\{ {c}_{j}\right\} }_{j \in \mathbb{J}} \) be a sequence of non-negative numbers. Th... | Proof Assume that \( \Gamma = \{ {c}_{j}{\Lambda }_{j}{\} }_{j \in \mathbb{J}} \) is a generalized frame for \( H \) with respect to \( \{ {V}_{j}{\} }_{j \in \mathbb{J}}. \) Denote its analysis operator by \( {U}_{\Gamma } \), then for any \( x \in H \), we have \[ {U}_{\Gamma }x = {\left\{ {c}_{j}{\Lambda }_{j}x\righ... | Yes |
Corollary 2.1 Suppose that \( \Lambda = {\left\{ {\Lambda }_{j}\right\} }_{j \in \mathbb{J}} \) is a generalized frame for \( H \) with respect to \( \{ {V}_{j}{\} }_{j \in \mathbb{J}} \) with the analysis operator \( {U}_{\Lambda },\{ {e}_{j, k}{\} }_{k \in {\mathbb{K}}_{j}} \) is an orthonormal basis for \( {V}_{j} \... | Proof Assume that \( \operatorname{Ran}{U}_{\Lambda } \subseteq \operatorname{Dom}{D}_{c},{D}_{c} \) is bounded and \( {\left. {D}_{c}\right| }_{\operatorname{Ran}{U}_{\Lambda }} \) is injective and has closed range. By Theorem 2.1, we obtain immediately that \( \Gamma = \{ {c}_{j}{\Lambda }_{j}{\} }_{j \in \mathbb{J}}... | Yes |
Theorem 2.2 Let \( \Lambda = \{ {\Lambda }_{j}{\} }_{j \in \mathbb{J}} \) be a generalized frame for \( H \) with respect to \( \{ {V}_{j}{\} }_{j \in \mathbb{J}} \) with the analysis operator \( {U}_{\Lambda } \), then the following conditions are equivalent.\n\n(1) \( \Lambda \) is (positively, strictly) scalable.\n\... | Proof (1) \( \Rightarrow \) (2) Assume that \( \Lambda \) is a scalable generalized frame, then there exists a sequence of non-negative numbers \( \;c = \{ {c}_{j}{\} }_{j \in \mathbb{J}}\; \) such that \( \;\Gamma = \{ {c}_{j}{\Lambda }_{j}{\} }_{j \in \mathbb{J}}\; \) is a Parseval generalized frame. Set \( D = {D}_{... | Yes |
Example 2.2 Let \( {\left\{ {h}_{k}\right\} }_{k \in \mathbb{N}} \) and \( {\left\{ {e}_{j}\right\} }_{j \in \mathbb{N}} \) be orthonormal bases for \( H \) and \( V \), respectively. Suppose that \( {V}_{j} = \operatorname{span}\left\{ {{e}_{1},{e}_{2}}\right\}, j = 1,2,3,4 \), and \( {V}_{j} = \operatorname{span}\lef... | Let \( U \) be the analysis operator of \( {\left\{ {\Lambda }_{j}\right\} }_{j \in \mathbb{N}} \) and \( {D}_{c} \) be a strictly positive bounded diagonal operator in \( {l}^{2}\left( {\left\{ {V}_{j}\right\} }_{j \in \mathbb{N}}\right) \) corresponding to a sequence of numbers \( c = {\left\{ {c}_{j}\right\} }_{j \i... | Yes |
Theorem 2.3 Let \( \Lambda = {\left\{ {\Lambda }_{j}\right\} }_{j \in \mathbb{J}} \) be a generalized frame for \( H \) with respect to \( {\left\{ {V}_{j}\right\} }_{j \in \mathbb{J}} \) and let \( U \) be a unitary operator in \( H \), then \( \Lambda \) is scalable if and only if the generalized frame \( \Gamma = {\... | Proof Assume that the generalized frame \( \Lambda \) is scalable with the analysis operator \( {U}_{\Lambda } \) . Then there exists a non-negative operator \( D \) in \( {l}^{2}(\{ {V}_{j}{\} }_{j \in \mathbb{J}}) \) such that \( \overset{―}{{U}_{\Lambda }^{ \ast }D}D{U}_{\Lambda } = {I}_{H}. \) Since \( U \) is a un... | Yes |
Theorem 2.4 Suppose that \( \Lambda = {\left\{ {\Lambda }_{j}\right\} }_{j \in \mathbb{J}} \) is a generalized frame for \( H \) with respect to \( \{ {V}_{j}{\} }_{j \in \mathbb{J}} \) with the analysis operator \( U = {U}_{\Lambda } \), and \( \{ {e}_{j, k}{\} }_{k \in {\mathbb{K}}_{j}} \) is an orthonormal basis for... | Proof Define the operator \( G : H \oplus K \rightarrow {l}^{2}\left( {\left\{ {V}_{j}\right\} }_{j \in \mathbb{J}}\right) \oplus {l}^{2}\left( {\left\{ {V}_{j}\right\} }_{j \in \mathbb{J}}\right) \) by\n\n\[ G\left( {x, y}\right) = U \oplus T\left( {x, y}\right) = \left( {{Ux},{Ty}}\right) ,\;\left( {x, y}\right) \in ... | Yes |
Example 2.3 Let \( {\left\{ {h}_{k}\right\} }_{k \in \mathbb{N}} \) and \( {\left\{ {e}_{j}\right\} }_{j \in \mathbb{N}} \) be orthonormal bases for \( H \) and \( V \), respectively. Suppose that \( {V}_{j} = \operatorname{span}\left\{ {e}_{j}\right\}, j \in \mathbb{N} \) . Define the linear bounded operators \( {\Lam... | If we take the Hilbert space \( K = H \) and the operator \( T = U \), then\n\n\[ \nU{U}^{ * } \oplus T{T}^{ * } = \frac{1}{4}{I}_{{l}^{2}\left( {\left\{ {V}_{j}\right\} }_{j \in \mathbb{J}}\right) } \oplus \frac{1}{4}{I}_{{l}^{2}\left( {\left\{ {V}_{j}\right\} }_{j \in \mathbb{J}}\right) }.\n\]\n\nThis satisfies the c... | Yes |
Theorem 2.5 Suppose that \( \Lambda = {\left\{ {\Lambda }_{j}\right\} }_{j \in \mathbb{J}} \) is a strictly scalable generalized frame for \( H \) with respect to \( {\left\{ {V}_{j}\right\} }_{j \in \mathbb{J}} \) with the analysis operator \( U = {U}_{\Lambda },{\left\{ {e}_{j, k}\right\} }_{k \in {\mathbb{K}}_{j}} \... | Proof Since \( \Lambda \) is strictly scalable, there exists a strictly positive bounded diagonal operator \( D \) in \( {l}^{2}\left( {\left\{ {V}_{j}\right\} }_{j \in \mathbb{J}}\right) \) satifying \( {U}^{ * }{D}^{2}U = {I}_{H} \) . Set \( K = {\left( \operatorname{Ran}DU\right) }^{ \bot } \), then \( K = \operator... | Yes |
Corollary 2.3 Let \( \Lambda = {\left\{ {\Lambda }_{j}\right\} }_{j = 1}^{M} \) be a generalized frame for \( {\mathbb{K}}^{N} \) with respect to \( {\left\{ {\mathbb{K}}^{{\mathbb{I}}_{j}}\right\} }_{j = 1}^{M} \) and let \( U \in {\mathbb{K}}^{P \times N} \) be the matrix representation of its analysis operator, wher... | Proof We prove this result by using the equivalent conditions in Corollary 2.2. Let\n\n\[ H = {\mathbb{K}}^{N},\;{l}^{2}\left( {\left\{ {V}_{j}\right\} }_{j \in \mathbb{J}}\right) = {\bigoplus }_{j = 1}^{M}{\mathbb{K}}^{{\mathbb{I}}_{j}} = {\mathbb{K}}^{P} \]\n\nand \( {u}_{j, k} = {\Lambda }_{j}^{ * }{e}_{j, k}, j = 1... | Yes |
Example 2.4 Let \( {\left\{ {e}_{j}\right\} }_{j = 1}^{4} \) be a standard orthonormal basis for \( {\mathbb{R}}^{4},{\mathbb{K}}^{{\mathbb{I}}_{1}} = {\mathbb{R}}^{4} \) and \( {\mathbb{K}}^{{\mathbb{I}}_{j}} = \) \( {\mathbb{R}}^{2}, j = 2,3,4 \) . Define the linear bounded operators \( {\Lambda }_{j} : {\mathbb{R}}^... | Let \( U \) be the matrix representation of its analysis operator. Then\n\n\[ \nU = \left\lbrack \begin{array}{llll} \frac{1}{2} & 0 & 0 & 0 \\ 0 & \frac{1}{2} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{array}\righ... | Yes |
Proposition 2.2 If \( \mu : {\mathcal{K}}^{1} \rightarrow {\mathcal{K}}^{1} \) is a continuous or monotone valuation with \( \mu \left( {I + t}\right) = \) \( \mu \left( I\right) \), then \( \mu \left( \left\lbrack {a, b}\right\rbrack \right) = \left( {b - a}\right) {I}_{1} + {I}_{2} \) for some intervals \( {I}_{1},{I... | Proof With the help of Lemma 2.2, by a similar argument to that for Proposition 2.1, we have\n\n\[ \sigma \circ \mu \left( \left\lbrack {a, b}\right\rbrack \right) = {\alpha }_{\sigma }\left( {b - a}\right) + {\beta }_{\sigma },\;\tau \circ \mu \left( \left\lbrack {a, b}\right\rbrack \right) = {\alpha }_{\tau }\left( {... | Yes |
Theorem 2.1 Let \( \mu : {\mathcal{K}}^{1} \rightarrow {\mathcal{K}}^{1} \) be a monotone valuation with \( \mu \left( {I + t}\right) = \mu \left( I\right) + t \) for every \( I \in {\mathcal{K}}^{1} \) and \( t \in \mathbb{R} \) . Then the followings are equivalent:\n\n(i) \( \mu \) is idempotent, i.e., \( \mu \circ \... | Proof First, by Proposition 2.1, \( \mu \left( \left\lbrack {a, b}\right\rbrack \right) = \left( {b - a}\right) {I}_{1} + {I}_{2} + a \) for some intervals \( {I}_{1},{I}_{2} \in {\mathcal{K}}^{1} \) with \( \left\lbrack {0,1}\right\rbrack \subset {I}_{1} \) .\n\n\( \left( i\right) \Rightarrow \left( {ii}\right) \; \) ... | Yes |
Lemma 3.1 If \( \mu : \mathcal{R} \rightarrow {\mathcal{K}}^{1} \) is a continuous (resp., increasing, decreasing) valuation, then both \( \;\mu \circ {\mathrm{R}}_{{J}^{ * }}^{1}\; \) and \( \;\mu \circ {\mathrm{R}}_{{I}^{ * }}^{2}\; \) are continuous (resp., increasing, decreasing) valuations from \( \;{\mathcal{K}}^... | Proof We argue for \( {\mathrm{R}}_{{J}^{ * }}^{1} \) only since the argument for \( {\mathrm{R}}_{{I}^{ * }}^{2} \) is the same.\n\nLet \( {I}_{1} = \left\lbrack {{a}_{1},{b}_{1}}\right\rbrack ,{I}_{2} = \left\lbrack {{a}_{2},{b}_{2}}\right\rbrack ,{I}_{1} \cup {I}_{2} \in {\mathcal{K}}^{1} \) with \( {b}_{1} \leq {b}... | Yes |
Lemma 3.3 If \( \mu : \mathcal{R} \rightarrow {\mathcal{K}}^{1} \) is a monotone valuation with \( \mu \left( {K + t}\right) = \mu \left( K\right) + t \) for all \( t \in V \) and \( \mu \left( {K + t}\right) = \mu \left( K\right) \) for all \( t \in {V}^{ \bot } \), and if \( \mu \left( {\left\lbrack {0,1}\right\rbrac... | Proof By Lemma 3.1, there exist \( {I}_{1},{I}_{2},{I}_{3},{I}_{4} \in {\mathcal{K}}^{1} \) with \( \left\lbrack {0,1}\right\rbrack \subset {I}_{2}, o \in {I}_{1} \cap {I}_{3} \), such that\n\n\[ \mu \left( {\left\lbrack {0, a}\right\rbrack \times \left\lbrack {0, b}\right\rbrack + t}\right) = {ab}{I}_{1} + a{I}_{2} + ... | Yes |
Lemma 2.1 Choosing the different numerical fluxes, the DG spatial discretization operator satisfies the following equalities\n\n\[ \mathcal{D}\left( {z, p;{z}^{\left( \theta \right) }}\right) + \mathcal{D}\left( {p, z;{p}^{\left( \widetilde{\theta }\right) }}\right) = 0 \]\n\n\[ \mathcal{D}\left( {z, p;{z}^{\left( \wid... | Proof These identities can be obtained easily by using integration by parts, the definition of the jump and the weighted averages of the functions, as well as the periodic boundary conditions. Since similar conclusions can be also found in [15], so we omit the details here. | No |
In this example, we solve (2) with exact solution \( u\left( {x, t}\right) = {\mathrm{e}}^{-t}\sin \left( x\right) \) . | The \( {L}^{2} \) -norm numerical errors and order of accuracy can be found in the following Table 1. We clearly observe the optimal convergence rate for all of \( k \) and \( \theta \) . This indicates that the conclusions in Theorem 2.1 and Theorem 2.2 are sharp. | No |
Lemma 1.2 \( {}^{\left\lbrack 6\right\rbrack } \) Let \( 1 < p < q < \infty ,\omega \in {A}_{\left( p, q\right) }, b \in \operatorname{BMO}\left( {\mathbb{R}}^{n}\right), x \in {\mathbb{R}}^{n} \) and \( {r}_{1},{r}_{2} > 0 \) . Then | \[ {\left( \frac{1}{{\omega }^{-{p}^{\prime }}\left( {B\left( {x,{r}_{1}}\right) }\right) }{\int }_{B\left( {x,{r}_{1}}\right) }{\left| b\left( y\right) - {b}_{B\left( {x,{r}_{2}}\right) ,{\omega }^{q}}\right| }^{{p}^{\prime }}\omega {\left( y\right) }^{-{p}^{\prime }}\mathrm{d}y\right) }^{\frac{1}{{p}^{\prime }}} \leq... | Yes |
Theorem 2.1 Let \( \Omega \in {L}^{s}\left( {\mathbb{S}}^{n - 1}\right) \) with \( 1 < s \leq \infty ,0 < \alpha < n,1 < q < \infty \) and \( 1 \leq {s}^{\prime } < \) \( p < \frac{n}{\alpha } \) such that \( \frac{1}{q} = \frac{1}{p} - \frac{\alpha }{n}. \) Suppose that \( f \in {L}_{\mathrm{{loc}}}^{p}\left( {\mathbb... | Proof Let us split \( f \) into two parts \( f = {f}_{1} + {f}_{2} \), where \( {f}_{1} = f{\chi }_{2B} \) and \( {f}_{2} = f{\chi }_{{\left( 2B\right) }^{c}} \) . Since \( {T}_{\Omega ,\alpha } \) is a linear operator, we can write \[ \parallel {T}_{\Omega ,\alpha }\left( f\right) {\parallel }_{{L}_{{\omega }^{q}}^{q}... | Yes |
Theorem 2.2 Let \( \Omega \in {L}^{s}\left( {\mathbb{S}}^{n - 1}\right) \) with \( 1 < s \leq \infty ,\;0 < \alpha < n,\;1 < q < \infty ,\;1 \leq {s}^{\prime } < p < \frac{n}{\alpha } \) such that \( \frac{1}{q} = \frac{1}{p} - \frac{\alpha }{n} \) . Suppose that \( {\omega }^{{s}^{\prime }} \in {A}_{\left( \frac{p}{{s... | Proof For every \( f \in {\mathrm{{VL}}}_{\Pi }^{p,\varphi }\left( {{\mathbb{R}}^{n},{\omega }^{p}}\right) \), we need to show that\n\n\[ \n{\begin{Vmatrix}{T}_{\Omega ,\alpha }\left( f\right) \end{Vmatrix}}_{{\mathrm{{VL}}}_{\Pi }^{q,\psi }\left( {{\mathbb{R}}^{n},{\omega }^{q}}\right) } \lesssim \parallel f{\parallel... | Yes |
Lemma 3. \( {1}^{\left\lbrack 4\right\rbrack }\; \) Let \( \Omega \in {L}^{s}\left( {\mathbb{S}}^{n - 1}\right) \) with \( 1 < s \leq \infty ,\;0 < \alpha < n,\;{s}^{\prime } < p < \frac{n}{\alpha },\;1 < q < \infty \) such that \( \frac{1}{q} = \frac{1}{p} - \frac{\alpha }{n},{\omega }^{{s}^{\prime }} \in {A}_{\left( ... | \[ \parallel {T}_{\Omega ,\alpha }^{m, b}\left( f\right) {\parallel }_{{L}_{{\omega }^{q}}^{q}\left( {\mathbb{R}}^{n}\right) } \leq C\parallel f{\parallel }_{{L}_{{\omega }^{p}}^{p}\left( {\mathbb{R}}^{n}\right) }. \] | Yes |
Theorem 3.2 Suppose that \( \Omega \in {L}^{s}({\mathbb{S}}^{n - 1}) \) with \( 1 < s \leq \infty ,\;0 < \alpha < n,\;1 \leq {s}^{\prime } < p < \frac{n}{\alpha }, \) \( 1 < q < \infty \) such that \( \frac{1}{q} = \frac{1}{p} - \frac{\alpha }{n},{\omega }^{{s}^{\prime }} \in {A}_{\left( \frac{p}{{s}^{\prime }},\frac{q... | Proof The proof is similar to that of Theorem 2.2. So we just have to show that\n\n\[ \n\mathop{\lim }\limits_{{r \rightarrow {0}^{ + }}}\mathop{\sup }\limits_{{x \in \Pi }}\frac{1}{{\psi }^{\frac{1}{q}}\left( {x, r}\right) }{\begin{Vmatrix}{T}_{\Omega ,\alpha }^{m, b}\left( f\right) \end{Vmatrix}}_{{L}_{{\omega }^{q}}... | Yes |
Theorem 2 Let \( p \) be any odd prime with \( 3 \mid p - 1 \) . Then for any character \( \chi {\;\operatorname{mod}\;p} \) with \( {\chi }^{6} \neq {\chi }_{0} \), we have the asymptotic formula | \[ \mathop{\sum }\limits_{{m = 1}}^{{p - 1}}{\left| \mathop{\sum }\limits_{{a = 0}}^{{p - 1}}e\left( \frac{m{a}^{6}}{p}\right) \right| }^{2} \cdot {\left| \mathop{\sum }\limits_{{b = 1}}^{{p - 1}}\chi \left( b\right) e\left( \frac{{mb} + \bar{b}}{p}\right) \right| }^{2} = 5{p}^{3} + R\left( p\right) \] where \( \left| ... | Yes |
Lemma 2 Let \( p \) be an odd prime with \( 3 \mid p - 1 \) . Then for any integer \( m \) with \( \left( {m, p}\right) = 1 \) , we have the identities\n\n\[ \mathop{\sum }\limits_{{a = 0}}^{{p - 1}}e\left( \frac{m{a}^{6}}{p}\right) = \left\{ \begin{array}{ll} {\chi }_{2}\left( m\right) \tau \left( {\chi }_{2}\right) +... | Proof First from the property of Dirichlet character, if \( \chi \) is a third-order character mod \( p \), for any integer \( a \) with \( \left( {a, p}\right) = 1 \), we have\n\n\[ 1 + \chi \left( a\right) + \bar{\chi }\left( a\right) = \left\{ \begin{array}{ll} 3, & \text{if }a\text{ satisfies }a \equiv {b}^{3}{\;(\... | Yes |
Lemma 3 Letting \( p \) be an odd prime with \( 3 \mid p - 1, m \) be any integer with \( \left( {m, p}\right) = 1 \), we have\n\n\[ A\left( {\chi }_{3}\right) = {\chi }_{3}\left( m\right) \tau \left( {\overline{\chi }}_{3}\right) + {\chi }_{3}{\chi }_{2}\left( m\right) \tau \left( {{\overline{\chi }}_{3}{\chi }_{2}}\r... | Proof From the properties of the \( \tau \left( \chi \right) \) and Dirichlet character, we have\n\n\[ {\overline{\chi }}_{3}\left( a\right) = {\chi }_{3}^{2}\left( a\right) = {\chi }_{3}\left( {a}^{2}\right) ,\]\n\n\[\bar{\chi }\left( m\right) \tau \left( \chi \right) = \mathop{\sum }\limits_{{a = 0}}^{{p - 1}}\chi \l... | No |
Lemma 2.1 For \( N \) -complexes \( X \) and \( Y \), we have\n\n\[{\operatorname{Ext}}_{{C}_{N}\left( \mathcal{C}\right) ,\mathrm{{dw}}}^{1}\left( {X, Y}\right) \cong {\operatorname{Hom}}_{{K}_{N}\left( \mathcal{C}\right) }\left( {X,{\sum Y}}\right)\]\n\nwhere \( {\mathrm{{Ext}}}_{{C}_{N}\left( \mathcal{C}\right) ,\ma... | Proof Let \( {C}_{N}{\left( \mathcal{C}\right) }_{\mathrm{{dw}}} \) be the exact category \( \left( {\mathcal{A},\mathcal{E}}\right) \), where \( \mathcal{A} \) is the category \( {C}_{N}\left( \mathcal{C}\right) \) and \( \mathcal{E} \) is the class of all degreewise split short exact sequences of \( N \) -complexes. ... | Yes |
Theorem 2.2 Let \( \mathcal{M} = \left( {\mathcal{Q},\mathcal{W},\mathcal{R}}\right) \) be a cofibrantly generated, hereditary abelian model structure on the Grothendieck category \( \mathcal{G} \) with enough projectives, core \( \omega \mathrel{\text{:=}} \mathcal{Q} \cap \mathcal{W} \cap \mathcal{R} \) . If \( \math... | Proof The proof is similar to [3, Theorem II.C.3.12]. We begin by showing that all cotorsion pairs are complete. For the top square, it suffices to check that the right-hand sides of the cotorsion pairs listed in it are of the form \( {\mathcal{S}}^{ \bot } \) for a set \( \mathcal{S} \in {C}_{N}\left( \mathcal{G}\righ... | Yes |
Proposition 2.1 Let \( \mathcal{M} = \left( {\mathcal{Q},\mathcal{W},\mathcal{R}}\right) \) be a cofibrantly generated, hereditary abelian model structure on the Grothendieck category \( \mathcal{G} \) with enough projectives. Then there is a diagram of cofibrantly generated abelian model structures on \( {C}_{N}\left(... | Proof Applying Theorem 2.3 to the dashed arrows in Figure 2.1 gives three model structures. Then it suffices to check that their classes of weakly trivial objects all coincide with the class \( {\mathcal{E}}_{N} \) of \( N \) -exact complexes. For the model structures associated to the arrows \( \left( {\widetilde{\mat... | Yes |
Proposition 3.1 Let \( R \) be a ring. Then there is a hereditary abelian model structure \( {\mathcal{M}}^{\text{inj }} = \left( {{\mathcal{E}}_{N},{\operatorname{dg}}_{N}\widetilde{\mathcal{I}}}\right) \) on \( {C}_{N}\left( R\right) \), called the injective model structure. Its homotopy category, Ho \( \left( {\math... | The abelian model structure comes from Hovey’s correspondence and Theorem 2.1. The equivalence follows from the triangle equivalence \( {K}_{N}\left( {\operatorname{dg-inj}R}\right) \simeq {D}_{N}\left( R\right) \; \) by [22, Theorem 2.22], where \( {K}_{N}\left( {\operatorname{dg-inj}R}\right) \) is the homotopy categ... | Yes |
Proposition 3.2 Let \( R \) be a ring. Then there is a hereditary abelian model structure \( {\mathcal{M}}_{\mathrm{{co}}}^{\mathrm{{inj}}} = \left( {{\mathcal{W}}_{\mathrm{{co}}},{\mathrm{{dw}}}_{N}\widetilde{\mathcal{I}}}\right) \) on \( {C}_{N}\left( R\right) \) . Its homotopy category, \( \operatorname{Ho}\left( {\... | Proof The abelian model structure comes from Hovey's correspondence and [31, Theorem 3.10]. It is easy to see the injective Hovey triple \( {\mathcal{M}}_{\mathrm{{co}}}^{\mathrm{{inj}}} = \left( {{\mathcal{W}}_{\mathrm{{co}}},{\mathrm{{dw}}}_{N}\widetilde{\mathcal{I}}}\right) \) is hereditary and \( {\mathcal{W}}_{\ma... | Yes |
Theorem 3.1 Let \( R \) be a ring. Then there is a recollement where the functors \( I \) are inclusions, the notation such as \( E\left( {\mathcal{M}}^{\mathrm{{inj}}}\right) \) means to take a special \( {\operatorname{dg}}_{N}\widetilde{\mathcal{I}} \) - preenvelope by using enough injectives of the cotorsion pair \... | Proof Since \( {\mathrm{{dg}}}_{N}\widetilde{\mathcal{I}} \subseteq {\mathrm{{dw}}}_{N}\widetilde{\mathcal{I}} \) and \( {\mathcal{E}}_{N} \cap {\mathrm{{dw}}}_{N}\widetilde{\mathcal{I}} = {\mathrm{{ex}}}_{N}\widetilde{\mathcal{I}} \), Propositions 3.1,3.2 and 3.3 and \( \left\lbrack {{16},\text{Theorem 4.6}}\right\rbr... | No |
Corollary 3.1 Let \( R \) be a Noetherian ring. Let \( {K}_{N}\left( \mathrm{{GInj}}\right) \) be the homotopy category of all Gorenstein injectives in the category \( {C}_{N}\left( R\right) ,{K}_{{\mathrm{{ex}}}_{N}}\left( \mathrm{{GInj}}\right) \) be the homotopy category of all exact Gorenstein injective \( N \) -co... | Proof \( \begin{aligned} \text{ We set }{\mathcal{M}}_{1} & = \left( {{}^{ \bot }{\mathrm{{dw}}}_{N}\widetilde{\mathcal{G}}\widetilde{\mathcal{I}},{\mathrm{{dw}}}_{N}\widetilde{\mathcal{G}}\widetilde{\mathcal{I}}}\right) , \\ {\mathcal{M}}_{2} & = \left( {{}^{ \bot }{\mathrm{{ex}}}_{N}\widetilde{\mathcal{G}}\widetilde{... | Yes |
Theorem 3.2 Let \( \mathcal{G} \) be any Grothendieck category. Then \( \left( {{\mathcal{E}}_{N}, K\mathcal{I}}\right) \) is a localizing cotor-sion pair in \( {C}_{N}{\left( \mathcal{G}\right) }_{\mathrm{{dw}}} \) . Its induced injective model structure \( \left( {\mathcal{A},{\mathcal{E}}_{N}, K\mathcal{I}}\right) \... | Proof The proof is similar to [17, Theorem 6.3]. We also include a proof for the convenience of the reader. By Theorem 2.1, the cotorsion pair \( \left( {{\mathcal{E}}_{N},{\operatorname{dg}}_{N}\widetilde{\mathcal{I}}}\right) \) is an injective cotorsion pair. First, by the definition of \( K \) -injective \( N \) -co... | Yes |
Corollary 3.2 Let \( \mathcal{G} \) be a Grothendieck category with a projective generator. Then \( \left( {K\mathcal{P},{\mathcal{E}}_{N}, K\mathcal{I}}\right) \) is a localizing cotorsion triple in \( {C}_{N}{\left( \mathcal{G}\right) }_{\mathrm{{dw}}} \) . We have equivalences of triangulated categories\n\n\[ K\math... | Proof This follows from [17, Corollary 4.5] and Theorem 3.2. | No |
Theorem 4.2 For every ring \( R \) with identity, the functor \( \Psi : {C}_{N}\left( R\right) \rightarrow \operatorname{GrMod}\left( A\right) \) maps: (1) \( \mathrm{{dg}} - \mathcal{A}N \) -complexes over \( R \) into \( G\left( \mathcal{A}\right) \mathbb{Z} \) -graded \( A \) -modules; | Proof We only prove (1) and (a), since (2) and (b) follows by duality.\n\nLet \( M \) be a \( \mathrm{{dg}} - \mathcal{A}N \) -complex over \( R \) . We using the fact that \( \left( {{\mathrm{{dg}}}_{N}\widetilde{\mathcal{A}},{}_{N}\widetilde{\mathcal{B}}}\right) \) is complete to find a short exact sequence \( 0 \rig... | No |
Theorem 5.1 Let \( R \) be an Artin algebra. For integers \( M, N \geq 2 \), there is a recollement: | Proof Using the notations as above, by [28], it suffices to prove that \( U \) as a right \( B \) -module has finite projective dimension. To see this, note that \( {U}_{B} \cong {\left( {e}_{1}B\right) }^{\left( N - 1\right) } \), where \( {e}_{1} \in {\mathbb{T}}_{M - 1}\left( R\right) \) is the matrix with 1 in \( \... | No |
Proposition 5.1 Let \( R \) be an Artin algebra. For integers \( M, N \geq 2 \), there is a recollement: | Proof By the proof of Theorem 5.1 and Theorem 5.2, \( U \) as a left \( A \) -module and as a right \( B \) -module is projective, then \( {\operatorname{Hom}}_{A}\left( {M, P}\right) \) is projective for every indecomposable projective \( A \) -module \( P \), and the recollement follows from [26, Theorem 3.12]. | No |
Corollary 1.1 It holds that\n\n\[ \mathop{\sup }\limits_{n}E\left\lbrack {{\int }_{0}^{T}{\int }_{\mathcal{O}}n\left| {{u}_{t}^{n} - \pi \left( {u}_{t}^{n}\right) }\right| \mathrm{d}x\mathrm{\;d}t}\right\rbrack \]\n\n\[ < {CE}\left\lbrack {\parallel \xi {\parallel }^{2} + {\int }_{0}^{T}\parallel h\left( {s,\cdot ,0,0}... | Proof Since there is a constant \( \delta > 0 \) ,\n\n\[ \delta {\int }_{0}^{T}{\int }_{\mathcal{O}}\left| {{u}_{t}^{n} - \pi \left( {u}_{t}^{n}\right) }\right| \mathrm{d}x\mathrm{\;d}t \leq {\int }_{0}^{T}\left( {\pi \left( {u}_{t}^{n}\right) ,{u}_{t}^{n} - \pi \left( {u}_{t}^{n}\right) }\right) \mathrm{d}t \]\n\n\[ \... | Yes |
Lemma 1.4 For almost every \( \omega \in \Omega \), the measures \( \left\{ {L}^{n}\right\} \) is tight. | Proof We will prove that, for every \( \varepsilon > 0 \), there is a positive number \( M > 0 \) such that for every \( n \), \[ E\left\lbrack {{\int }_{0}^{T}{\int }_{\mathcal{O}}{I}_{\{ \left| x\right| > M\} }\left| {L}^{n}\right| \left( {\mathrm{d}x,\mathrm{\;d}t}\right) }\right\rbrack < \varepsilon . \] In fact, \... | Yes |
Proposition 0.1 Let \( p < q \) be odd primes. Then\n\n\[ g\left( {\Phi }_{pq}\right) = p - 1 \] | In 2014, Moree \( {}^{\left\lbrack 5\right\rbrack } \) used numerical semigroups to provide a conceptual proof of Proposition 0.1. Kaplan also gave an elegant proof of this result, see Camburu et al. [2]. | No |
Lemma 1.2 Let \( p < q < r \) be odd primes. Let \( n \geq 0 \) be an integer and \( f\left( i\right) \) be the unique value \( 0 \leq f\left( i\right) \leq {pq} - 1 \) such that\n\n\[ \n{rf}\left( i\right) + i \equiv n\;\left( {{\;\operatorname{mod}\;p}q}\right) .\n\]\n\nPut\n\[ \n{a}^{ * }\left( {{pq}, m}\right) = \l... | Proof See Kaplan [4]. | No |
Theorem 1.1 Let \( \;\mathcal{M} \) be a factor von Neumann algebra with \( \dim \mathcal{M} > 1 \), and \( \;L : \mathcal{M} \rightarrow \mathcal{M} \) be the second nonlinear mixed Lie triple derivation. Then \( L \) is an additive \( * \) -derivation. | Now we prove Theorem 1.1 by several lemmas.\n\nLemma 1.1 \( L\left( 0\) | No |
Lemma 1.1 \( L\left( 0\right) = 0 \) and \( L\left( I\right) \in \mathbb{C}I \) . | Proof It is clear that\n\n\[ L\left( 0\right) = L\left( {\left\lbrack \left\lbrack 0,0\right\rbrack ,0\right\rbrack }_{ * }\right) = {\left\lbrack \left\lbrack L\left( 0\right) ,0\right\rbrack ,0\right\rbrack }_{ * } + {\left\lbrack \left\lbrack 0, L\left( 0\right) \right\rbrack ,0\right\rbrack }_{ * } + {\left\lbrack ... | Yes |
Lemma 1.6 For every \( {A}_{ii},{B}_{ii} \in {\mathcal{M}}_{ii} \) with \( i = 1,2 \), we have\n\n(1) \( L\left( {{A}_{11} + {B}_{11}}\right) - \left( {L\left( {A}_{11}\right) + L\left( {B}_{11}\right) }\right) \in \mathbb{C}I \) ;\n\n(2) \( L\left( {{A}_{22} + {B}_{22}}\right) - \left( {L\left( {A}_{22}\right) + L\lef... | Proof Write \( T = L\left( {{A}_{11} + {B}_{11}}\right) - L\left( {A}_{11}\right) - L\left( {B}_{11}\right) \) . It is clear that\n\n\[ L\left( {\left\lbrack \left\lbrack {P}_{k},{A}_{11} + {B}_{11}\right\rbrack ,{P}_{k}\right\rbrack }_{ * }\right) = L\left( {\left\lbrack \left\lbrack {P}_{k},{A}_{11}\right\rbrack ,{P}... | Yes |
Lemma 1.7 \( L \) is an additive map. | Proof It follows from Lemmas 1.4–1.6 that \( L\left( {A + B}\right) \) – \( L\left( A\right) \) – \( L\left( B\right) \in \mathbb{C}I \) for all \( A, B \in \mathcal{M}. \) Then there exists the map \( h : \mathcal{M} \times \mathcal{M} \rightarrow \mathbb{C}I \) such that \( h\left( {A, B}\right) = L\left( {A + B}\rig... | Yes |
Lemma 1.8 \( L\left( {\mathrm{i}A}\right) = \mathrm{i}L\left( A\right) \) for every \( A \in \mathcal{M} \) . | Proof It is clear that\n\n\[ L\left( {\left\lbrack \left\lbrack \mathrm{i}A, B\right\rbrack, C\right\rbrack }_{ * }\right) = {\left\lbrack \left\lbrack L\left( \mathrm{i}A\right), B\right\rbrack, C\right\rbrack }_{ * } + {\left\lbrack \left\lbrack \mathrm{i}A, L\left( B\right) \right\rbrack, C\right\rbrack }_{ * } + {\... | Yes |
Lemma 1.1 \( {}^{\left\lbrack 1\right\rbrack } \) Let \( 1 < q < \infty ,0 < {r}_{2} < {r}_{1},0 \leq \varsigma < \frac{1}{n} \) and \( b \in {\mathrm{{LC}}}_{q,\varsigma }^{\left\{ {x}_{0}\right\} } \) . Then\n\n\[ \n{\left( \frac{1}{{\left| B\left( {x}_{0},{r}_{1}\right) \right| }^{1 + \varsigma q}}{\int }_{B\left( {... | And, from this inequality, we have\n\n\[ \n\left| {{b}_{B\left( {{x}_{0},{r}_{1}}\right) } - {b}_{B\left( {{x}_{0},{r}_{2}}\right) }}\right| \leq C\left( {1 + \ln \frac{{r}_{1}}{{r}_{2}}}\right) {\left| B\left( {x}_{0},{r}_{1}\right) \right| }^{\varsigma }\parallel b{\parallel }_{{\mathrm{{LC}}}_{q,\varsigma }^{\{ {x}_... | Yes |
Theorem 2.1 Let \( \Omega \in {L}^{s}\left( {\mathbb{S}}^{n - 1}\right) \left( {1 < s \leq \infty }\right) \) satisfy the condition (0.1), \( \lambda > 1 + \frac{2}{n} \) and \( \max \{ 2,{s}^{\prime }\} < q < \infty \), where \( {s}^{\prime } = \frac{s}{s - 1} \) is the conjugate exponent of \( s \) . Then the inequal... | Proof Let \( B = B\left( {{x}_{0}, r}\right) \) . We write \( f = {f}_{1} + {f}_{2} \), where \( {f}_{1} = f{\chi }_{2B} \) and \( {f}_{2} = f{\chi }_{{\left( 2B\right) }^{c}} \) . Thus, we have\n\n\[ \n{\begin{Vmatrix}{g}_{\lambda }^{ * }\left( f\right) \end{Vmatrix}}_{{L}^{q}\left( B\right) } \leq {\begin{Vmatrix}{g}... | Yes |
Theorem 2.2 Let \( \Omega \in {L}^{s}({\mathbb{S}}^{n - 1})\;\left( {1 < s \leq \infty }\right) \) satisfy the condition \( \left( {0.1}\right) ,\;\lambda > 1 + \frac{2}{n} \) and \( \max \{ 2,{s}^{\prime }\} < q < \infty . \) If functions \( \varphi ,\psi : {\mathbb{R}}^{n} \times \left( {0,\infty }\right) \rightarrow... | Proof Take \( {v}_{1}\left( l\right) = \varphi {\left( {x}_{0}, l\right) }^{-1}{l}^{-\frac{n}{q}},{v}_{2}\left( l\right) = \psi {\left( {x}_{0}, l\right) }^{-1}, g\left( l\right) = \parallel f{\parallel }_{{L}^{q}\left( {B\left( {{x}_{0}, l}\right) }\right) } \) and \( w\left( l\right) = \) \( {l}^{-\frac{n}{q} - 1} \)... | Yes |
Theorem 3.1 Let \( \Omega \in {L}^{s}({\mathbb{S}}^{n - 1})\;\left( {1 < s \leq \infty }\right) \) satisfy the condition \( \left( {0.1}\right) ,\;\lambda > 1 + \frac{2}{n} \) and \( \max \{ 2,{s}^{\prime }\} < q < \infty \) . Let \( 1 < p,{q}_{1},{q}_{2},\cdots ,{q}_{m} < \infty \), such that \( \frac{1}{q} = \frac{1}... | Proof Without loss of generality, it is sufficient for us to show that the conclusion holds for \( m = 2 \) . Let \( B = B\left( {{x}_{0}, r}\right) \) . And, we write \( f = {f}_{1} + {f}_{2} \), where \( {f}_{1} = f{\chi }_{2B},{f}_{2} = f{\chi }_{{\left( 2B\right) }^{c}} \) . Thus, we have\n\n\[ \n\parallel {g}_{\la... | Yes |
Theorem 3.2 Let \( \Omega \in {L}^{s}({\mathbb{S}}^{n - 1})\;\left( {1 < s \leq \infty }\right) \) satisfy the condition \( \left( {0.1}\right) ,\;\lambda > 1 + \frac{2}{n} \) and \( \max \{ 2,{s}^{\prime }\} < q < \infty \) . Let \( 1 < p,{q}_{1},{q}_{2},\cdots ,{q}_{m} < \infty \), such that \( \frac{1}{q} = \frac{1}... | Proof Taking \( {v}_{1}\left( l\right) = \varphi {\left( {x}_{0}, l\right) }^{-1}{l}^{-\frac{n}{p}},{v}_{2}\left( l\right) = \psi {\left( {x}_{0}, l\right) }^{-1}, g\left( l\right) = \parallel f{\parallel }_{{L}^{q}\left( {B\left( {{x}_{0}, l}\right) }\right) } \) and \( w\left( l\right) = \) \( {\left( 1 + \ln \frac{l... | Yes |
Lemma 1.1 Let \( f : X \rightarrow Y \) . If \( f \) is an open mapping (resp., \( f \) preserves sequentially open sets) and \( y \in {NI}\left( Y\right) \) (resp., \( y \in {NS}\left( Y\right) \) ), then \( \partial {f}^{-1}\left( y\right) = {f}^{-1}\left( y\right) \) . | Proof Let \( y \in Y \) with \( \partial {f}^{-1}\left( y\right) \neq {f}^{-1}\left( y\right) \) . Then \( {\left( {f}^{-1}\left( y\right) \right) }^{ \circ } \neq \varnothing \) . Since the mapping \( f \) is an open mapping (resp., \( f \) preserves sequentially open sets), \( \{ y\} = f\left( {\left\lbrack {f}^{-1}\... | Yes |
Theorem 1.1 Let \( f : X \rightarrow Y \) be an open mapping. If \( Y \) is a sequential space, then \( f \) is an almost \( s \) -mapping \( \Leftrightarrow f \) is a near \( s \) -mapping \( \Leftrightarrow f \) is a boundary- \( s \) -mapping. | Proof Because \( Y \) is a sequential space, \( I\left( Y\right) = S\left( Y\right) \) . Hence, the mapping \( f \) is an almost \( s \) -mapping \( \Leftrightarrow f \) is a near \( s \) -mapping. For each \( y \in I\left( Y\right) \), it follows from the fact that \( {f}^{-1}\left( y\right) \) is a clopen set of \( X... | Yes |
Lemma 1.2 If \( f : M \rightarrow X \) is an open boundary- \( s \) -mapping, then there is a subset \( D \) of \( X \) such that \( {\left. f\right| }_{D} : D \rightarrow X \) is an open boundary-countable mapping. | Proof If \( x \in {NI}\left( X\right) \), since \( f \) is a boundary- \( s \) -mapping, there is a countable dense subset \( {D}_{x} \) of \( \partial {f}^{-1}\left( x\right) \) . Let\n\n\[ D = {f}^{-1}\left( {I\left( X\right) }\right) \cup \mathop{\bigcup }\limits_{{x \in {NI}\left( X\right) }}{D}_{x},\;g = {\left. f... | Yes |
Lemma 1.3 Let \( \\left( {f, M, X,\\mathcal{P}}\\right) \) be a Ponomarev’s system and \( x \\in X \) . Then \( \\mathcal{P} \) is point-countable at \( x \) if and only if \( {f}^{-1}\\left( x\\right) \) is separable in \( M \) . | Proof Let \( \\mathcal{P} = \\left\\{ {{P}_{\\alpha } : \\alpha \\in \\Lambda }\\right\\} \) and \( {\\Gamma }_{x} = \\left\\{ {\\alpha \\in \\Lambda : x \\in {P}_{\\alpha }}\\right\\} \). Sufficiency. Suppose that \( {f}^{-1}\\left( x\\right) \) is separable. Let \( {\\pi }_{1} \) be the projection from the Cartesian ... | Yes |
Lemma 2.2 The followings are equivalent for a space \( X \) :\n\n(1) \( X \) has an so-network which is point-countable at non-sequentially open points.\n\n(2) \( X \) is a 2-sequence-covering near \( s \) -image of a metric space.\n\n(3) \( X \) is a 2-sequence-covering boundary- \( s \) -image of a metric space. | Proof \( \;\left( 1\right) \Rightarrow \left( 2\right) .\; \) Let \( \;X\; \) have an \( \;{so} \) -network \( \;\mathcal{P}\; \) which is point-countable at non-sequentially open points. We may assume that \( \{ \{ x\} : x \in S\left( X\right) \} \subseteq \mathcal{P} \) . Let \( \left( {f, M, X,\mathcal{P}}\right) \)... | Yes |
Theorem 2.1 The followings are equivalent for a space \( X \) :\n\n(1) \( X \) has a base which is point-countable at non-isolated points.\n\n(2) \( X \) is an open almost \( s \) -image (resp., near \( s \) -image, boundary- \( s \) -image) of a metric space.\n\n(3) \( X \) is an open boundary-countable image of a met... | Proof It is obvious that an open image of a metrizable space is a first-countable space \( {}^{\left\lbrack 4\right\rbrack } \) , and every first-countable space is a sequential space \( {}^{\left\lbrack 4\right\rbrack } \) . By Theorem 1.1, almost \( s \) -mappings, near \( s \) -mappings and boundary- \( s \) -mappin... | Yes |
Theorem 3.1 The followings are equivalent for a space \( X \) :\n\n(1) \( X \) is a first-countable space.\n\n(2) \( X \) is an almost-open boundary- \( s \) -image of a metric space.\n\n(3) \( X \) is a countably bi-quotient boundary- \( s \) -image of a metric space. | Proof We only need to prove that \( \left( 3\right) \Rightarrow \left( 1\right) \Rightarrow \left( 2\right) \) .\n\n(3) \( \Rightarrow \) (1). Let \( f : M \rightarrow X \) be a countably bi-quotient boundary- \( s \) -mapping, where \( M \) is a metrizable space. Suppose that \( \mathcal{B} \) is a point-countable bas... | Yes |
There is an open almost \( s \) -image of a metric space which is not an open \( s \) -image of a metric space. | Let \( \psi \left( \mathbb{N}\right) = \mathcal{A} \cup \mathbb{N} \) be Mrówka’s space, where \( \mathcal{A} \) is a maximal almost disjoint family of \( \mathbb{N} \) (see [15]). Since \( \psi \left( \mathbb{N}\right) \) has not a point-countable base, it is not an open \( s \) -image of a metric \( {\text{space}}^{\... | Yes |
There is a 2-sequence-covering \( s \) -image of a metric space which is not a countably bi-quotient boundary- \( s \) -image of a metric space. | Let \( \beta \mathbb{N} \) be the Čech-Stone compactification of \( \mathbb{N} \) (see [4]). Since \( \beta \mathbb{N} \) is not a first-countable space, by Theorem 3.1, \( \beta \mathbb{N} \) is not a countably bi-quotient boundary- \( s \) -image of a metric space. Let \( M \) be the set \( \beta \mathbb{N} \) endowe... | Yes |
There is an almost-open and near \( s \) -mapping which is not an almost \( s \) - mapping. | The first uncountable ordinal is denoted by \( {\omega }_{1} \) . For each \( \alpha < {\omega }_{1} \), let \( {X}_{\alpha } \) be a copy of \( \beta \mathbb{N} \) . Put \( X = {\bigoplus }_{\alpha < {\omega }_{1}}{X}_{\alpha } \), and define a mapping \( f : X \rightarrow \beta \mathbb{N} \) such that every \( {\left... | Yes |
Example 3.4 There is a countably bi-quotient boundary- \( s \) -image of a metric space which is not an open boundary- \( s \) -image of a metric space. | Let \( X \) be the Euclidean plane endowed with the butterfly topology \( {}^{\left\lbrack {15}\right\rbrack } \) . Then \( X \) is a first-countable regular space without any non-isolated point. By Theorem 3.1, \( X \) is a countably bi-quotient boundary- \( s \) -image of a metric space. Since the compact subset \( \... | Yes |
Theorem 1.1 Let \( G \) be a cubic graph. Then \( \nabla \left( G\right) = {\gamma }_{M}\left( G\right) + \xi \left( G\right) \), where \( {\gamma }_{M}\left( G\right) \) and \( \xi \left( G\right) \) are the maximum genus and Betti deficiency of \( G \), respectively. | Proof Suppose that \( {T}_{X} \) and \( T \) are the Xuong-tree and the spanning tree of a cubic graph \( G \), respectively. By Lemma 1.2, both of the co-trees \( G - E\left( {T}_{X}\right) \) and \( G - E\left( T\right) \) have the following edge-partitions:\n\n\[ G - E\left( {T}_{X}\right) = \{ {e}_{1},{e}_{2}\} \cu... | Yes |
Corollary 3.1 Let \( G \) be a cubic graph except for \( {K}_{4} \) . Then \( G \) is 3-colorable. Meanwhile, there exists a vertex partition \( \left( {{V}_{1},{V}_{2}}\right) \) of \( G \) such that \( e\left( {{V}_{1},{V}_{2}}\right) \geq 3\nabla \), where \( e\left( {{V}_{1},{V}_{2}}\right) = \) \( \left| {E\left( ... | Proof By Theorem 2.1, there exists a vertex set \( A \) of a cubic graph \( G\left( {G \neq {K}_{4}}\right) \) such that \( A \) is an independent set and \( V\left( G\right) - A \) induces a forest. Then we may color vertices of \( V\left( G\right) - A \) by 1,2 and color vertices of \( A \) by 3 . Therefore, \( G \) ... | Yes |
Theorem 2.2 Let \( s \geq 1,{G}_{1} \) be a connected nonbipartite graph containing \( {2s} \) distinct vertices \( {v}_{1},{v}_{2},\cdots ,{v}_{s},{v}_{s + 1},{v}_{s + 2},\cdots ,{v}_{2s} \) . Let \( {T}_{1} \) be a nontrivial tree containing a vertex \( {u}_{1} \) and \( {B}_{i} \) be a connected bipartite graph cont... | Proof Let \( {G}^{ * } = {G}_{1}\left( {{v}_{1},{v}_{2}\cdots ,{v}_{s}}\right) \diamond \left( {{T}_{1}\left( {u}_{1}\right) ,{B}_{2}\left( {u}_{2}\right) ,\cdots ,{B}_{s}\left( {u}_{s}\right) }\right) \) . Without loss of generality, we assume that \( x \) is a unit vector. Let \( \left( {{U}_{1},{W}_{1}}\right) \) be... | Yes |
Proposition 1.2 With the above notations, we have\n\n\[ \bar{\beta } \circ {d}^{s} = {d}^{s + 1} \circ \bar{\beta }.\] | Proof For \( \eta \in {C}^{l}\left( {\mathfrak{g};V}\right) \), we have\n\n\[ \bar{\beta } \circ {d}^{s}\eta \left( {{x}_{1},\cdots ,{x}_{l + 1}}\right) = \mathop{\sum }\limits_{{i = 1}}^{{l + 1}}{\left( -1\right) }^{i + 1}{\beta }^{l + 2 + s}\rho \left( {x}_{i}\right) {\beta }^{-l - 2 - s}\eta \left( {\alpha \left( {x... | Yes |
Theorem 1.1 For \( s = 0,1,2,\cdots \), we have \( {H}^{k}\left( {d}^{s}\right) \cong {H}^{k}\left( {d}^{s + 1}\right) \) . | Proof By \( {d}^{s + 1} \circ \bar{\beta } = \bar{\beta } \circ {d}^{s} \), for \( \eta \in {Z}^{k}\left( {d}^{s}\right) \), we have \( \bar{\beta }\left( \eta \right) \in {Z}^{k}\left( {d}^{s + 1}\right) \) . On the other hand, for \( {\eta }_{1} \in {B}^{k}\left( {d}^{s}\right) \), there is \( \omega \in {C}^{k - 1}\... | Yes |
Proposition 2.1 For \( {\xi }_{1} \in { \land }^{k}{\mathfrak{g}}^{ * },{\xi }_{2} \in { \land }^{l}{\mathfrak{g}}^{ * } \), we have\n\n\[ d\left( {{\xi }_{1} \land {\xi }_{2}}\right) = d{\xi }_{1} \land \bar{\alpha }\left( {\xi }_{2}\right) + {\left( -1\right) }^{k}\bar{\alpha }\left( {\xi }_{1}\right) \land d{\xi }_{... | Proof This proof is similar to Proposition 3.2 in [21]. | No |
Proposition 2.3 With the above notations, we have\n\n\[ \bar{\alpha } \circ {d}^{s} = {d}^{s + 1} \circ \bar{\alpha } \] | Proof For any \( \eta \in {C}^{l}\left( {\mathfrak{g};V}\right) \), by \( \rho \left( {\alpha \left( {x}_{i}\right) }\right) = \beta \circ \rho \left( {x}_{i}\right) \circ {\beta }^{-1} \), we have\n\n\[ \bar{\alpha } \circ {d}^{s}\eta \left( {{x}_{1},\cdots ,{x}_{l + 1}}\right) = {d}^{s}\eta \left( {\alpha \left( {x}_... | Yes |
Theorem 2.1 Let \( V \) be a vector space, \( \beta \in \mathrm{{GL}}\left( V\right) \) . Then \( \left( {\mathfrak{g},\left\lbrack {\cdot , \cdot }\right\rbrack ,\alpha }\right) \) is a Hom-Lie algebra, and \( \rho : \mathfrak{g} \rightarrow \mathfrak{{gl}}\left( V\right) \) is a representation of \( \left( {\mathfrak... | Proof With the propositions we have proved above, we just need to prove the sufficient conditions.\n\nStep 1: for a fixed map \( \beta \in \mathrm{{GL}}\left( V\right) \) . We define \( \rho : \mathfrak{g} \rightarrow \mathfrak{{gl}}\left( V\right) \) as follows,\n\n\[ \n{d}^{s}v\left( x\right) = {\beta }^{1 + s}\rho \... | Yes |
Lemma 2.3 \( {}^{\left\lbrack {25},\text{ Lemma 3.3}\right\rbrack }\; \) Let \( {p}_{1}\; \in \;\left( {0,1}\right) ,\;\delta \; \mathrel{\text{:=}} \;\frac{n}{{p}_{1}} - \frac{n + 1}{2} \) and \( \varphi \) be a growth function as in Definition 1.2, which is of uniformly lower type \( {p}_{1} \) . Suppose that \( b \)... | \[ {\left( {T}_{R}^{\delta }\left( b\right) \right) }_{m}^{ * }\left( x\right) \leq C\parallel b{\parallel }_{{L}^{\infty }}{\left( \frac{r}{\left| x - {x}_{0}\right| }\right) }^{\frac{n}{{p}_{1}}}. \] | Yes |
Lemma 1.1 For \( r > 1 \), let\n\n\[ \n{\phi }^{\left( \frac{1}{r}\right) } : \operatorname{Int}\partial {A}_{r} \rightarrow \operatorname{Int}\partial {A}_{r} \]\n\n\[ \n{\phi }^{\left( r\right) } : \operatorname{Ext}\partial {A}_{r} \rightarrow \operatorname{Ext}\partial {A}_{r} \]\n\nbe two quasisymmetric maps commu... | The basic idea of the proof is to lift by universal covering to parallel strips, where one can just use linear interpolation. The whole construction can be made in a symmetric way, so the resulting map will be symmetric with respect to the unit circle. For more details of the proof the reader can refer to \( \left\lbra... | No |
Theorem 2.1 Let \( 0 \leq \alpha < 1,0 \leq \beta < 1 \) and \( \alpha \neq 1 - \beta \) . Then there exists a degree 3 Blaschke product\n\n\[ \n{F}_{\alpha ,\beta }\left( z\right) = \frac{{\mathrm{e}}^{-{2\pi }\mathrm{i}\beta }}{AB}z\left( \frac{z - A}{1 - \bar{A}z}\right) \left( \frac{z - B}{1 - \bar{B}z}\right) \n\]... | Now let \( 0 < \alpha ,\beta < 1 \) be irrationals of Brjuno type, and \( F = {F}_{\alpha ,\beta } \) . By a brief computation, we can get \( {F}^{\prime }\left( 0\right) = {\mathrm{e}}^{-{2\pi }\mathrm{i}\beta } \), so by the Brjuno-Yoccoz theorem \( {}^{\left\lbrack {14}\right\rbrack } \) the origin is the center of ... | No |
Theorem 1.1 If a convex \( \nu \) -gon can be tiled with finitely many isosceles right triangles, then \( \nu \in \{ 3,4,5,6,7,8\} \) . | Proof Suppose that a convex \( \nu \) -gon is dissected into finitely many isosceles right triangles. Then its inner angles have sizes of \( \frac{\pi }{4},\frac{\pi }{2} \), or \( \frac{3\pi }{4} \), and the sizes of the respective outer angles are \( \frac{3\pi }{4},\frac{\pi }{2} \) and \( \frac{\pi }{4} \) . Since ... | Yes |
Theorem 1.2 For \( \nu = 3,4 \) we have\n\n(1) \( {\mathcal{T}}_{3} = \left\{ {{p}^{2},2{p}^{2} : p \in \mathbb{N}}\right\} \). | Proof (1) If a triangle \( T \) can be dissected into finitely many isosceles right triangles, then the leg length of \( T \) is \( x + \sqrt{2}y \), where \( x, y \in {\mathbb{Z}}^{ + } \) . Comparing the area of \( T \) and of its tiles shows that there exists a positive integer \( k \) such that \( \frac{{\left( x +... | Yes |
Lemma 2.2 \( {\mathcal{T}}_{5} = \{ {p}^{2} - {q}^{2} - {r}^{2},\;2\left( {{p}^{2} - {q}^{2} - {r}^{2}}\right) : p, q, r \in \mathbb{N},\;p > q + r\} \cup \{ {p}^{2} - 2{q}^{2} - 2{r}^{2} : p, q, r \in \mathbb{N},\;p > {2q},\;p > {2r}\} \cup \{ 2{p}^{2} - {q}^{2} - {r}^{2} : p, q, r \in \mathbb{N},\;p > q,\;p > r\} \cu... | Proof Suppose that a convex pentagon \( \mathcal{P} \) can be cut into finitely many isosceles right triangles. Then by a simple computation we know that \( \mathcal{P} \) has the following configurations.\n\nCase 1: \( \mathcal{P} \) has three inner angles of size \( \frac{\pi }{2} \) and two of size \( \frac{3\pi }{4... | Yes |
Theorem 2.1 \( \left\{ {1,2,\cdots ,{2.5} \cdot {10}^{10} - 1}\right\} \smallsetminus {\mathcal{T}}_{5} = \{ 1,2,3,4,8\} \) . | Proof From Lemma 2.1 we have known \( \{ 1,2,\cdots ,{2.5} \cdot {10}^{10} - 1\} \smallsetminus \{ {p}^{2} - {q}^{2} - {r}^{2} : p, q, r \in \mathbb{N}, p > q + r\} \subseteq \mathcal{L} \) and\n\n\[ \n\{ 1,2,\cdots ,{2.5} \cdot {10}^{10} - 1\} \backslash \left\{ {{p}^{2} - {q}^{2} - {r}^{2} : p, q, r \in \mathbb{N},\;... | Yes |
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