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Theorem 3.1 Suppose that \( X \) is a statistical FU-space. Then any subset of \( X \) is a statistical FU-subspace. | Proof Assume that \( Y \) is a nonempty subset of \( X \), and \( A \) is any subset of \( Y \) . Then \( {\bar{A}}^{Y} = Y \cap {\bar{A}}^{X} \) . For any \( x \in {\bar{A}}^{Y} \), it is clear that \( x \in {\bar{A}}^{X} \) . Since \( X \) is a statistical FU-space, there is a sequence in \( A \) statistically conver... | No |
Theorem 3.2 Let \( X \) be a statistical FU-space, and \( f \) be a pseudo-open mapping from \( X \) onto a space \( Y \) . Then \( Y \) is a statistical FU-space. | Proof Suppose that \( A \) is an arbitrary subset of \( Y \) and \( y \in \bar{A} \) . If \( {f}^{-1}\left( y\right) \cap \overline{{f}^{-1}\left( A\right) } = \varnothing \) , then \( {f}^{-1}\left( y\right) \subset X \smallsetminus \overline{{f}^{-1}\left( A\right) } \) . Since \( f \) is pseudo-open, then\n\n\[ y \i... | Yes |
Theorem 3.3 If every subspace of a space \( X \) is s-sequential, then \( X \) is a statistical FU-space. | Proof Suppose that \( A \) is a subset of \( X \) and \( x \in \bar{A} \) . If \( x \in A \), then put \( {x}_{n} = x \) for every \( n \in \mathbb{N} \) . Thus the sequence \( {\left( {x}_{n}\right) }_{n \in \mathbb{N}} \subset A \) converges to \( x \), and hence it is obvious that \( {\left( {x}_{n}\right) }_{n \in ... | Yes |
Theorem 3.4 Let \( X \) be a Hausdorff statistical FU-space. If \( W \) is a weak neighborhood of \( {x}_{0} \in X \), then \( {x}_{0} \in {W}^{ \circ } \) . | Proof Suppose that \( \mathcal{P} = \mathop{\bigcup }\limits_{{x \in X}}{\mathcal{P}}_{x} \) is a weak base for the space \( X \), and then there exists \( P \in {\mathcal{P}}_{{x}_{0}} \) such that \( P \subset W \) . If\n\n\[ \n{x}_{0} \in X \smallsetminus {W}^{ \circ } \subset X \smallsetminus {P}^{ \circ } = \overl... | Yes |
Theorem 2.1 Every graph with \( m \) edges admits a bipartition \( \left( {{V}_{1},{V}_{2}}\right) \) such that\n\n(i) \( e\left( {{V}_{1},{V}_{2}}\right) \geq \frac{m}{2} + \frac{1}{4}h\left( m\right) \) ; and\n\n(ii) \( e\left( {V}_{i}\right) \leq \frac{m}{4} + \frac{1}{8}h\left( m\right) \) for \( i = 1,2 \) . | The bounds in (i) and (ii) are (individually) tight, and the complete graphs \( {K}_{{2n} + 1} \) are the only extremal graphs (modulo isolated vertices). To prove the conclusion, the authors first take a bipartition \( \left( {{V}_{1},{V}_{2}}\right) \) satisfying (i) (such a bipartition does exist following Edwards’ ... | Yes |
Does every graph \( G \) with \( m \) edges have a partition of \( V\left( G\right) \) into \( {V}_{1},{V}_{2},\cdots ,{V}_{k} \) such that both (1.2) and (2.1) hold? | Xu and Yu, by extending and refining the technique used by Bollobás and Scott in [12], proved the following result that affirmatively answers Problem 2.1 up to \( O\left( k\right) \). | No |
Theorem 2.3 \( {}^{\left\lbrack {27}\right\rbrack } \) Let \( k \geq 2 \) be an integer, and \( G \) be a graph with \( m \) edges. Then \( V\left( G\right) \) admits a partition \( \left( {{V}_{1},{V}_{2},\cdots ,{V}_{k}}\right) \) such that\n\n(i) for each \( i \in \{ 1,2,\cdots, k - 1\} \) and for every \( x \in {V}... | By the integrity of \( e\left( {{V}_{1},{V}_{2},\cdots ,{V}_{k}}\right) \), Theorem 2.3 provides an affirmative answer to Problem 2.1 for some values of \( m \) . | No |
Theorem 3.1 \( {}^{\left\lbrack {66}\right\rbrack } \) Let \( G \) be a graph with \( m \) edges, and let \( M \) be a maximum matching of \( G \) . Then \( G \) admits a bisection of size at least \( \frac{m}{2} + \frac{\left| M\right| }{2} \) . | Proof Let \( M = \left\{ {{e}_{1},{e}_{2},\cdots ,{e}_{r}}\right\} \) be a maximum matching of \( G \), and let \( \left( {{V}_{1},{V}_{2}}\right) \) be a balanced bipartition of \( G \) such that \( {e}_{i} \) has one end in \( {V}_{1} \) and the other in \( {V}_{2} \) for each \( i \in \) \( \{ 1,2,\cdots, r\} \), an... | Yes |
Is it possible to remove the tail term \( o\left( m\right) \) in Theorem 3.3? | If the answer is YES, the bound will be sharp as evidenced by the complete bipartite graph \( {K}_{{2k} + 1, n} \) . Compared with the conclusion that graphs with large partitions have good judicious \( {\text{partition}}^{\left\lbrack 2\right\rbrack } \), Lee et al. \( {}^{\left\lbrack {40}\right\rbrack } \) also prov... | Yes |
Theorem 4. \( {\mathbf{4}}^{\left\lbrack {39}\right\rbrack } \) For each positive integer \( l \), there exists integer \( k\left( l\right) \leq {2}^{11} \cdot 3{l}^{2} \) (resp. \( {k}^{\prime }\left( l\right) \leq {2}^{16}{l}^{2} \) ) such that \( \delta \left( G\right) \geq k\left( l\right) \) (resp. connectivity \(... | In [44], the authors improved the bound \( k\left( l\right) \) to \( k\left( l\right) \leq {2}^{4} \cdot {17}{l}^{2} \) . Kühn and Osthus’ theorem is a step toward to a very hard conjecture of Thomassen [63]. | No |
Let \( s \) be a positive integer. Is it true that every graph with minimum degree at least \( {2s} + 1 \) admits a bisection \( \left( {S, T}\right) \) such that \( \min \{ \delta \left( {G\left\lbrack S\right\rbrack }\right) ,\delta \left( {G\left\lbrack T\right\rbrack }\right) \} \geq s \) ? | If the answer is YES, then the bound is best possible as evidenced by the complete graph \( {K}_{{2s} + 1} \) . In the following, we say that a bipartition \( \left( {S, T}\right) \) is \( s \) -good if \( \min \{ \delta \left( {G\left\lbrack S\right\rbrack }\right) ,\delta \left( {G\left\lbrack T\right\rbrack }\right)... | No |
Lemma 1.2 If \( G \) is a finite cyclic group and \( \varphi \) is a skew-morphism of \( G \) with power function \( \pi \), then\n\n(1) \( \varphi \) fixes \( \operatorname{Ker}\pi \) setwise;\n\n(2) the restriction of \( \varphi \) to \( \operatorname{Ker}\pi \) is a group automorphism of \( \operatorname{Ker}\pi \) ... | Proof A direct calculation shows that \( \varphi \left( {\operatorname{Ker}\pi }\right) \) is a subgroup of \( G \) . While for a finite cyclic group, there is only one subgroup of a given order and hence \( \varphi \left( {\operatorname{Ker}\pi }\right) = \operatorname{Ker}\pi \) . The second result follows directly f... | Yes |
Let \( G = {G}_{1}^{S} \circ \left( {{G}_{2}^{V} \cup {G}_{3}^{E}}\right), H = {H}_{1}^{S} \circ \left( {{H}_{2}^{V} \cup {H}_{3}^{E}}\right) \). If \( {G}_{1} \) and \( {H}_{1} \) are \( A \)-cospectral regular graphs, and \( {G}_{2} = {H}_{2},{G}_{3} = {H}_{3} \), then \( G \) is clearly \( A \)-cospectral with \( H ... | For example, we know from \( \left\lbrack {3,\text{ p. }{127}}\right\rbrack \) that \( {X}_{1} \) and \( {Y}_{1} \) are 4-regular \( A \)-cospectral non-isomorphic mates shown in Fig. 2. (In fact, the graph in \( {H}_{1} \) has an edge \( \left( {1,8}\right) \) that is a common edge of three triangles but the graph in ... | Yes |
Let \( G = {G}_{1}^{S} \circ \left( {{G}_{2}^{V} \cup {G}_{3}^{E}}\right), H = {H}_{1}^{S} \circ \left( {{H}_{2}^{V} \cup {H}_{3}^{E}}\right) \). By Theorem 2.1, we see that if \( {G}_{1} = {H}_{1} \) is regular, \( {G}_{2} \) is \( A \) -cospectral with \( {H}_{2} \) with \( {\Gamma }_{A\left( {G}_{2}\right) }\left( x... | For example, it is know from \( \left\lbrack {{14},\text{ p. }{211}}\right\rbrack \) that \( {X}_{2} \) and \( {Y}_{2} \) are a pair of \( A \) -cospectral graphs shown in Fig. 3. By using MATLAB, we have\n\n\[ \n{\Gamma }_{A\left( {G}_{2}\right) }\left( x\right) = {\Gamma }_{A\left( {H}_{2}\right) }\left( x\right) \n\... | Yes |
Theorem 2.2 Let \( {G}_{1} \) be an \( {r}_{1} \) -regular graph with \( {n}_{1} \) vertices and \( {m}_{1} \) edges, \( {G}_{2} \) and \( {G}_{3} \) be respectively two arbitrary graphs on \( {n}_{2} \) and \( {n}_{3} \) vertices. Then \( G = {G}_{1}^{S} \circ \left( {{G}_{2}^{V} \cup {G}_{3}^{E}}\right) \) has Laplac... | Proof With respect to the partition (6), the Laplacian matrix of \( G \) can be written as \[ L\left( G\right) = \left( \begin{matrix} \left( {{r}_{1} + {n}_{2}}\right) {I}_{{n}_{1}} & - R & - {I}_{{n}_{1}} \otimes {\mathbf{e}}^{\mathrm{T}} & 0 \\ - {R}^{\mathrm{T}} & \left( {2 + {n}_{3}}\right) {I}_{{m}_{1}} & 0 & - {... | Yes |
Example 3 Let \( G = {G}_{1}^{S} \circ \left( {{G}_{2}^{V} \cup {G}_{3}^{E}}\right), H = {H}_{1}^{S} \circ \left( {{H}_{2}^{V} \cup {H}_{3}^{E}}\right) \) . If \( {G}_{1} \) and \( {H}_{1} \) are \( L \) -cospectral regular graphs, and \( {G}_{2} = {H}_{2},{G}_{3} = {H}_{3} \), then \( G \) is \( L \) -cospectral with ... | For example, we take \( {X}_{1} \) and \( {Y}_{1} \) to be the 4-regular graphs in Example 2 (see Fig. 2), then they are also \( L \) -cospectral. Hence, we claim that \( G = {X}_{1}^{S} \circ \left( {{G}_{2}^{V} \cup {G}_{3}^{E}}\right) \) and \( H = {Y}_{1}^{S} \circ \left( {{G}_{2}^{V} \cup {G}_{3}^{E}}\right) \) ar... | Yes |
Let \( G = {G}_{1}^{S} \circ \left( {{G}_{2}^{V} \cup {G}_{3}^{E}}\right), H = {H}_{1}^{S} \circ \left( {{H}_{2}^{V} \cup {H}_{3}^{E}}\right) \). By Theorem 2.2, we see that if \( {G}_{1} = {H}_{1},{G}_{2} \) is \( L \) -cospectral with \( {H}_{2} \), and \( {G}_{3} \) is \( L \) -cospectral with \( {H}_{3} \), then \(... | For example, we know from [12, pp. 383-386] that \( {W}_{2} \) is \( L \) -cospectral with \( {Z}_{2} \) and \( {S}_{3} \) is \( L \) - cospectral with \( {T}_{3} \) shown in Fig. 4. Thus we can construct many infinite families of \( L \) -cospectral graphs \( G = {G}_{1}^{S} \circ \left( {{W}_{2}^{V} \cup {S}_{3}^{E}}... | Yes |
Example 5 Let \( G = {K}_{{n}_{1}}^{S} \circ \left( {{K}_{{n}_{2}}^{V} \cup {K}_{{n}_{3}}^{E}}\right) \) . We know that\n\n\[ t\left( {K}_{{n}_{1}}\right) = {n}_{1}^{{n}_{1} - 2},\;{\Phi }_{L\left( {K}_{{n}_{2}}\right) }\left( x\right) = x{\left( x - {n}_{2}\right) }^{{n}_{2} - 1},\;{\Phi }_{L\left( {K}_{{n}_{3}}\right... | Hence, by Corollary 2.3 and (20)-(21), we have\n\n\[ t\left( G\right) = t\left( {K}_{{n}_{1}}\right) {2}^{{m}_{1} - {n}_{1} + 1}\mathop{\prod }\limits_{{i = 2}}^{{n}_{2}}{\left( 1 + {\mu }_{i}\left( {K}_{{n}_{2}}\right) \right) }^{{n}_{1}}\mathop{\prod }\limits_{{i = 2}}^{{n}_{3}}{\left( 1 + {\mu }_{i}\left( {K}_{{n}_{... | Yes |
Lemma 1.6 Let \( G \) be a finite 2-group. If \( \exp \left( G\right) = 4 \) and \( M\left( G\right) = {44} \), then \( G \) is isomorphic to the following groups: \( {G}_{1},{G}_{2},{G}_{3},{G}_{4},{G}_{5},{G}_{6},{G}_{7},{G}_{8} \) or \( {G}_{9} \) . | Proof If \( G \) is abelian, let \( \left| G\right| = {2}^{t} \) . Then \( {2}^{t - 1} \leq {44} \) by Lemma 1.4. Hence \( t \leq 6 \) and \( \left| G\right| \leq {64} \) . Therefore, \( \left| G\right| = {64} \) . If \( G \) is a nonabelian 2-group with \( \exp \left( G\right) = 4 \) and every \( x \) in \( G \) of or... | Yes |
Lemma 1.7 Suppose that \( G \) is a finite 2-group of exponent 8 and order at least 64. Then \( {n}_{8}\left( G\right) \) is divisible by 8 . | Proof Obviously, \( G \) can not be a cyclic group. Let \( A \) and \( B \) be different maximal subgroups of \( G \) . Set \( H = A \cap B \) . Then \( H \) is a normal subgroup of \( G \), and \( G/H \cong {C}_{2} \times {C}_{2} \) . Let \( C \) denote the third maximal subgroup of \( G \) which contains \( H \) . Th... | Yes |
Lemma 2.2 Let \( w = \left\lbrack {{a}_{1},{a}_{2},\cdots ,{a}_{n}}\right\rbrack \) and \( {w}^{\prime } = {t}_{i}w = \left\lbrack {{a}_{1}^{\prime },{a}_{2}^{\prime },\cdots ,{a}_{n}^{\prime }}\right\rbrack \) with \( i \in \left\lbrack {0, n}\right\rbrack \) . For any \( j \in \left\lbrack n\right\rbrack \) ,\n\n(1) ... | For any \( a \in \mathbb{Z} \), denote by \( \langle a\rangle \) the unique integer in \( \left\lbrack {{2n} + 1}}\right\rbrack \) satisfying \( a \equiv \langle a\rangle \) (mod \( {2n} + 1). \) | Yes |
Theorem 3.1 \( {}^{\left\lbrack 3,8\right\rbrack } \) For any \( i, j \) in \( \mathbb{N} \) with \( j > 0 \), let \( {t}_{i, j} = {t}_{i + j - 1}\cdots {t}_{i + 1}{t}_{i} \) . If \( x \in {\widetilde{C}}_{n} \) and \( i \in \mathbb{Z} \) satisfies \( \left( i\right) x - {2n} - 1 > \left( j\right) x \) for any \( j \in... | \[ \left( k\right) {x}^{\prime \prime } = \left\{ \begin{array}{ll} \left( m\right) x - {2n} - 1, & \text{ if }\langle m\rangle = \langle i\rangle \\ \left( m\right) x + {2n} + 1, & \text{ if }\langle m\rangle = \langle {2n} + 1 - i\rangle \\ \left( m\right) x, & \text{ otherwise. } \end{array}\right. \] | Yes |
Proposition 4.1 The set \( {E}_{{42}^{n - 2}1} \) is infinite. | Proof The result follows from the fact that \( \left\{ {\left\lbrack {{2n} + p\left( {{2n} + 1}\right), n + 1, n + 2,\cdots ,{2n} - 1}\right\rbrack \;|\;p \in }\right. \) \( \mathbb{N}\} \subset {E}_{{\mathbf{{42}}}^{\mathrm{n} - \mathbf{2}}\mathbf{1}} \) | Yes |
Example 4.1 \( n = 3 \) . By Lemma 3.1, we see that for any \( w \in {\widetilde{C}}_{3}, w \in {E}_{421} \) if and only if \( w \) satisfies the following condition (a):\n\n(a) There exist some pairwise not 6-dual \( {i}_{1},{i}_{2}, k \) in [8] with \( {i}_{1},{i}_{2} \) being both \( w \) -tame heads and \( k \) bei... | We see that the element \( w \in {\widetilde{C}}_{3} \) with the form \( w = \left\lbrack {8 + {7p},4,5}\right\rbrack \), where \( p \in \mathbb{N} \), is the element in \( {E}_{\mathbf{{421}}} \) . This implies that the set \( {E}_{\mathbf{{421}}} \) is infinite. | Yes |
Lemma 2.2 Let \( u \) be a smooth solution of the Cauchy problem (0.1)-(0.2). If for some \( B > 0 \), supp \( u\left( t\right) \subseteq \left\lbrack {-B, B}\right\rbrack \), then for all \( \lambda ,\theta \in \mathbb{R} \), we have\n\n\[ \left| {\mathcal{F}\left( {u\left( t\right) }\right) \left( {\lambda + \mathrm{... | Proof Using the Cauchy-Schwarz inequality and (2.1), we have\n\n\[ \left| {\mathcal{F}\left( {u\left( t\right) }\right) \left( {\lambda + \mathrm{i}\theta }\right) }\right| \leq {\int }_{\mathbb{R}}\left| {{\mathrm{e}}^{-\mathrm{i}x\left( {\lambda + \mathrm{i}\theta }\right) }u\left( t\right) \left( x\right) }\right| \... | Yes |
Lemma 2.3 Let \( u \in C\left( {\left\lbrack {0, T}\right\rbrack ;{H}^{s}}\right) \left( {s \geq 4}\right) \) be a solution of the Cauchy problem (0.1)-(0.2) and \( B \) as in Theorem 1.2. Then we have\n\n\[ \rho \left( \lambda \right) \preccurlyeq \frac{\sqrt{B}M}{1 + {\lambda }^{4}} \] | Proof Using the Cauchy-Schwarz inequality, we obtain\n\n\[ \mathop{\sup }\limits_{{t \in \mathbb{R}}}{\int }_{\mathbb{R}}\left| {u\left( t\right) \left( x\right) }\right| \mathrm{d}x \leq \sqrt{B}M \]\n\nHence, for all \( t \in I \), we get\n\n\[ \left| {\mathcal{F}\left( {u\left( t\right) }\right) \left( {\lambda }^{\... | Yes |
Lemma 2.7 \( {}^{\left\lbrack 1,\text{ Corollary 2.9 }\right\rbrack } \) Let \( k\left( {\theta ,\lambda }\right) = \mathop{\sup }\limits_{{\left| {\lambda }^{\prime }\right| \geq \lambda > 0}}\left| {\phi \left( {{\lambda }^{\prime } + \mathrm{i}\theta }\right) }\right| \) under the assumptions of Lemma 2.6. If \( \th... | \[ k\left( {\theta ,\lambda }\right) \leq 2\widetilde{\rho }\left( \lambda \right) \] | Yes |
Lemma 2. \( {8}^{\left\lbrack 3\right\rbrack } \) Let \( t \in I,\phi \left( z\right) = \mathcal{F}\left( {u\left( t\right) }\right) \left( z\right) ,\theta \) be as in Lemma 2.7 and \( \rho \) be as in (2.3). Then for fixed \( \left| {\theta }^{\prime }\right| \leq \left| \theta \right| \), we have | \[ \left| {{\phi }^{\prime }\left( {\lambda - {\lambda }^{\prime } + \mathrm{i}{\theta }^{\prime }}\right) }\right| \preccurlyeq B\left\lbrack {\phi \left( \lambda \right) + \phi \left( {\lambda - {\lambda }^{\prime }}\right) }\right\rbrack \left\lbrack {1 + \left| {\log \rho \left( \lambda \right) }\right| }\right\rbr... | Yes |
Theorem 0.2 Let \( \{ {c}_{n}{\} }_{n = 1}^{\infty } \in \) MVBVS in complex sense, and for some \( {\theta }_{1} \in \lbrack 0,\frac{\pi }{2}),\;\{ {c}_{n}\; + \) \( \left. {c}_{-n}\right\} \subset K\left( {\theta }_{1}\right) \) . Then the necessary and sufficient conditions for the uniform convergence of the series ... | \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}n{c}_{n} = 0 \] and \[ \mathop{\sum }\limits_{{n = 1}}^{\infty }\left| {{c}_{n} + {c}_{-n}}\right| < \infty \] | Yes |
Theorem 0.5 Given a complex sequence \( {\left\{ {c}_{n}\right\} }_{n = - \infty }^{\infty } \), suppose that both \( {\left\{ {c}_{n}\right\} }_{n = 1}^{\infty } \) and \( \left\{ {{c}_{n} + {c}_{-n}}\right\} \) satisfy Condition (1). Then the necessary and sufficient conditions for uniform convergence of (3) are that | \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}n{c}_{n} = 0 \] (6) and the series \[ \mathop{\sum }\limits_{{n = 1}}^{\infty }\left( {{c}_{n} + {c}_{-n}}\right) \text{ converges. } \] (7) | Yes |
Lemma 1.1 For a complex sequence \( \left\{ {a}_{n}\right\} \) and all \( n \), we have\n\n\[ \left| {a}_{n}\right| \leq {2M}{b}_{n} \] | Proof This fundamental lemma appeared in [1] for a real sequence \( \left\{ {a}_{n}\right\} \) . Here for the complex sequence the proof is quite similar. Suppose to the contrary that for some \( n \) we have \( \left| {a}_{n}\right| > {2M}{b}_{n} \) . Then for all \( n < k \leq {2n} \) we obtain from Condition (1) for... | Yes |
Lemma 1.3 Let \( f\left( x\right) \) be the sum function of the series (3). If the complex sequence \( \left\{ {c}_{n}\right\} \) satisfies (2) for some \( {\theta }_{0} \in \left\lbrack {0,\frac{\pi }{2}}\right) \), then \( f \in {C}_{2\pi } \) implies (5). | Proof From the expression of \( f\left( x\right) \) we see that\n\n\[ \left| {{c}_{0} + \mathop{\sum }\limits_{{k = 1}}^{\infty }\left( {{c}_{k} + {c}_{-k}}\right) }\right| = \left| {f\left( 0\right) }\right| \leq \parallel f\parallel \]\n\ni.e.,\n\n\[ \left| {\mathop{\sum }\limits_{{k = 1}}^{\infty }\left( {{c}_{k} + ... | Yes |
Lemma 1.4 If a complex sequence \( \left\{ {c}_{n}\right\} \) satisfies Condition (1), then \( \mathop{\lim }\limits_{{n \rightarrow \infty }}n{c}_{n} = 0 \) implies that the series (8) converges uniformly. | Without any difference, the proof of this lemma can be copied word by word from \( \lbrack 4, \) Lemma 3] for the positive case. | No |
Lemma 1.5 If a complex sequence \( \\left\\{ {c}_{n}\\right\\} \) satisfies Conditions (1),(4) and (5), then\n\n\[ \n\\mathop{\\lim }\\limits_{{n \\rightarrow \\infty }}\\begin{Vmatrix}{f - {S}_{n}\\left( f\\right) }\\end{Vmatrix} = 0 \n\] | Proof Similar to the proof of the following identity (15) under Conditions (4) and (5), we can easily see that the sequence \( \\left\\{ {{S}_{n}\\left( {f, x}\\right) }\\right\\} \) is a Cauchy sequence for each \( x \) and therefore it converges at each \( x \) . So we only need to show that\n\n\[ \n\\mathop{\\lim }\... | Yes |
Lemma 2.1 Let \( \left\{ {{c}_{n} + {c}_{-n}}\right\} \) satisfy Condition (1). Then (6) and (7) imply that \( \mathop{\sum }\limits_{{n = 1}}^{\infty }\left( {{c}_{n} + }\right. \) \( \left. {c}_{-n}\right) {\mathrm{e}}^{-\mathrm{i}{nx}} \) convergences uniformly. | Proof Let \( x \in \left( {-\pi ,0}\right) \cup \left( {0,\pi }\right) \), set \( N = \left\lbrack \frac{1}{\left| x\right| }\right\rbrack \) . For any \( m > \max \{ N, n\} \), write \( {}^{ \dagger } \n\n\[ \mathop{\sum }\limits_{{k = n}}^{m}\left( {{c}_{k} + {c}_{-k}}\right) {\mathrm{e}}^{-\mathrm{i}{kx}} = \mathop{... | Yes |
Theorem 1.1 If \( T \) is an algebraically quasi-*- \( n \) -paranormal operator, then \( T \) has the SVEP. | Proof Suppose that \( T \) is an algebraically quasi-*- \( n \) -paranormal operator. Then \( p\left( T\right) \) is a quasi- \( * \) - \( n \) -paranormal operator for some nonconstant complex polynomial \( p \), and hence it follows from [13, Theorem 3.2] that \( p\left( T\right) \) has the SVEP. Therefore, \( T \) h... | Yes |
Theorem 1.2 If \( T \) is an algebraically quasi-*- \( n \) -paranormal operator, then \( T \) is polaroid. | Proof Let \( \lambda \in \operatorname{iso}\sigma \left( T\right) \), where \( T \) is a quasi-*- \( n \) -paranormal operator. Then \( T \) is a polaroid by [13, Theorem 3.4]. If \( T \) is an algebraically quasi-*- \( n \) -paranormal operator, then \( p\left( T\right) \) is a quasi- \( * \) - \( n \) -paranormal ope... | Yes |
Theorem 1.3 Let \( T \) or \( {T}^{ * } \) be an algebraically quasi-*- \( n \) -paranormal operator. Then Weyl’s theorem holds for \( f\left( T\right) \) for every \( f \in H\left( {\sigma \left( T\right) }\right) \) . | Proof From [2, Theorem 2.11], we have that \( T \) is polaroid if and only if \( {T}^{ * } \) is polaroid. We use the fact that if \( T \) is polaroid and \( T \) or \( {T}^{ * } \) has the SVEP then both \( T \) and \( {T}^{ * } \) satisfy Weyl’s theorem in [2, Theorem 3.3]. Suppose that \( T \) or \( {T}^{ * } \) is ... | Yes |
Corollary 1.3 Let \( T \) be an algebraically quasi-*- \( n \) -paranormal operator. If \( F \) is an operator commuting with \( T \) and \( {F}^{n} \) has a finite rank for some \( n \in \mathbb{N} \), then Weyl’s theorem holds for \( f\left( T\right) + F \) for every \( f \in H\left( {\sigma \left( T\right) }\right) ... | Proof Suppose that \( T \) is an algebraically quasi-*- \( n \) -paranormal operator. By Corollary 1.2 and Theorem 1.3, we have that \( T \) is isoloid and Weyl’s theorem holds for \( f\left( T\right) \) . Notice that \( T \) is isoloid, then \( f\left( T\right) \) is isoloid. The result stems from [9, Theorem 2.4]. | Yes |
Theorem 1.4 Let \( T \in B\left( H\right) \). (i) If \( {T}^{ * } \) is an algebraically quasi-*- \( n \) -paranormal operator, then generalized \( \alpha \) -Weyl’s theorem holds for \( T \). (ii) If \( T \) is an algebraically quasi-*- \( n \) -paranormal operator, then generalized \( \alpha \) -Weyl’s theorem holds ... | Since the SVEP for \( T \) entails that generalized Browder’s theorem holds for \( T \) (see [3, Theorem 3.2]), i.e., \( {\sigma }_{\mathrm{{BW}}}\left( T\right) = {\sigma }_{\mathrm{D}}\left( T\right) \), where \( {\sigma }_{\mathrm{D}}\left( T\right) \) denotes the Drazin spectrum, a sufficient condition for an opera... | No |
Theorem 1.2 For the continuous operator \( T : X \rightarrow Y \) between Banach spaces, the following are equivalent.\n\n(1) \( T \) is an Lcc operator;\n\n(2) \( T \) carries every limited set to relatively compact subset of \( Y \) ;\n\n(3) For an arbitrary Banach space \( Z \) and every limited operator \( S : Z \r... | Statement The equivalence between (1) and (2) is [18, Theorem 2.1]. | No |
Corollary 1.1 Every weakly compact operator is DPcc. | Proof Let \( T : X \rightarrow Y \) be a weakly compact operator between Banach spaces \( X \) and \( Y \) . Since \( T \) carries DP sets into relatively compact sets \( {}^{\left\lbrack 1,\text{ p. }{350}\right\rbrack } \), by Theorem 1.1, \( T \) is a DPcc operator. | No |
Corollary 1.2 For an operator \( T : X \rightarrow Y \), the following are equivalent.\n\n(1) \( T \) is weakly limited.\n\n(2) For each Banach space \( Z \) and each weakly compact operator \( S : Y \rightarrow Z \), the composition operator \( {ST} \) is relatively compact.\n\n(3) For each weakly compact operator \( ... | Proof \( \;\left( 1\right) \Rightarrow \left( 2\right) \; \) If \( \;T \) is weakly limited and \( S : Y \rightarrow Z \) is DPcc, then \( \;T\left( {B}_{X}\right) \) is a DP set of \( Y \) . By Corollary 1.1, every weakly compact operator is DPcc. By Theorem 1.1, \( S\left( {T\left( {B}_{X}\right) }\right) \) is relat... | Yes |
Corollary 1.3 Let \( E \) be a Banach lattice and \( X \) be a Banach space. If \( {E}^{ * } \) is a KB space and \( E \) has the DP property, then each DPcc operator \( T : M \rightarrow X \) is \( M \) -weakly compact. | Proof Let \( T : M \rightarrow X \) be a DPcc operator and let \( \left( {x}_{n}\right) \) be a bounded disjoint sequence in \( E \) . By [1, Theorem 4.59], \( {E}^{ * } \) has an order continuous norm. It follows from [7, Corollary 2.9] that \( {x}_{n}\overset{\mathrm{w}}{ \rightarrow }0 \) . On the other hand, since ... | Yes |
Theorem 2.1 For each two Banach spaces \( X \) and \( Y \), if the closed subspace \( M \) of arbitrary operator ideal \( \mathbb{I}\left( {X, Y}\right) \) has the DPrcP, then all evaluation operators \( {\phi }_{x} \) and \( {\psi }_{{y}^{ * }} \) are DPcc. | Proof Since all \( {\phi }_{x} : M \rightarrow Y \) and \( {\psi }_{{y}^{ * }} : M \rightarrow {X}^{ * } \) are bounded linear operators, it is an easy consequence of Theorem 1.3. | Yes |
Corollary 2.1 Let \( {X}^{ * } \) have the Schur property and \( {Y}^{* * } \) have the DPrcP. Then \( L\left( {X, Y}\right) \) has the DPrcP. | Proof The mapping \( T \rightarrow {T}^{ * } \) maps \( L\left( {X, Y}\right) \) onto a closed subspace of \( L\left( {{Y}^{ * },{X}^{ * }}\right) \), which has the DPrcP by virtue of Theorem 2.5. | Yes |
Theorem 3.2 Let \( E \) and \( F \) be two Banach lattices with \( E \) having the weak DP property. Consider the scheme of operators\n\n\[ E\overset{{S}_{1}}{ \rightarrow }F\overset{{S}_{2}}{ \rightarrow }X \]\n\nwhere \( {S}_{1} \) is a positive operator and dominated by a DPcc oprator. If \( {S}_{2} \) is order weak... | Proof By [1, Theorem 5.58, p. 319], the operator \( {S}_{2} \) admits a factorization through a Banach lattice \( G \) with order continuous norm\n\n\[ F\overset{Q}{ \rightarrow }G\overset{S}{ \rightarrow }X \]\n\nsuch that \( Q \) is a lattice homomorphism and \( {S}_{2} = {SQ} \) . Obviously, the positive operator \(... | Yes |
Lemma 3.1 Every DPcc operator \( T : E \rightarrow X \) from a Banach lattice to a Banach space is order weakly compact. | Proof Let \( \left( {x}_{n}\right) \) be an order bounded disjoint sequence of \( E \) . Then \( {x}_{n}\overset{\mathrm{w}}{ \rightarrow }0 \) holds in \( E \) . On the other hand, \( \left( {x}_{n}\right) \) is a DP set by [3, Theorem 2.5]. So \( \lim \begin{Vmatrix}{T{x}_{n}}\end{Vmatrix} \rightarrow 0 \) . The conc... | Yes |
If \( X = \{ \left\lbrack {a, b}\right\rbrack : a, b \in \mathbb{R}, a \leq b\} \), then \( p(\left\lbrack {a, b}\right\rbrack ,\left\lbrack {c, d}\right\rbrack ) = \max \{ a, b\} - \min \{ c, d\} \) defines a partial metric \( p \) on \( X \) . | Each partial metric \( p \) on \( X \) generates a \( {T}_{0} \) topology \( {\tau }_{p} \) on \( X \) which has as a base the family of open \( p \) -balls \( \left\{ {{B}_{p}\left( {x,\varepsilon }\right) : x \in X,\varepsilon > 0}\right\} \), where \( {B}_{p}\left( {x,\varepsilon }\right) = \{ y \in X : p\left( {x, ... | No |
Lemma 1.2 \( {}^{\left\lbrack {11}\right\rbrack } \) Let \( \left( {X, p}\right) \) be a partial metric space and let \( \left\{ {y}_{n}\right\} \) be a sequence in \( X \) such that \( \mathop{\lim }\limits_{{n \rightarrow \infty }}p\left( {{y}_{n},{y}_{n + 1}}\right) = 0 \) . If \( \left\{ {y}_{2n}\right\} \) is not ... | \[ p\left( {{y}_{{2m}\left( k\right) },{y}_{{2n}\left( k\right) }}\right) ,\;p\left( {{y}_{{2m}\left( k\right) },{y}_{{2n}\left( k\right) + 1}}\right) ,\;p\left( {{y}_{{2m}\left( k\right) - 1},{y}_{{2n}\left( k\right) }}\right) ,\;p\left( {{y}_{{2m}\left( k\right) - 1},{y}_{{2n}\left( k\right) + 1}}\right) . \] | Yes |
Theorem 2.1 Let \( \\left( {X, p, \\preccurlyeq }\\right) \) be a complete ordered partial metric space such that \( X \) is regular. Let \( T : X \\rightarrow X \) be a self-mapping and \( {\\left\{ {f}_{k}\\right\} }_{k = 1}^{\\infty } \) be a sequence of mappings of \( X \) into itself. Suppose that for every \( i, ... | Proof Let \( {x}_{0} \) be an arbitrary point in \( X \). Since \( {f}_{1}X \\subseteq {TX} \), there exists an \( {x}_{1} \\in X \) such that \( T{x}_{1} = {f}_{1}{x}_{0} \). Since \( {f}_{2}X \\subseteq {TX} \), there exists an \( {x}_{2} \\in X \) such that \( T{x}_{2} = {f}_{2}{x}_{1} \). Continuing this process, w... | Yes |
The Euler-Lagrange equation of the volume variational problem with respect to the \( F \) -relative metric is | \[ 0 = \left\lbrack {\frac{{n}^{2}}{8}\frac{{F}^{\prime }}{{\rho }^{2}} + \frac{n\left( {n + 2}\right) }{8}\frac{F}{{\rho }^{3}} - \frac{n\left( {n - 2}\right) \left( {n - 4}\right) }{8\left( {n + 2}\right) }\frac{{F}^{\prime 3}}{{F}^{2}} - \frac{n\left( {n - 2}\right) \left( {n + 6}\right) }{8\left( {n + 2}\right) }\f... | Yes |
Lemma 1.7 Let\n\[ L = \\left( \\begin{matrix} A & B \\\\ {B}^{\\mathrm{T}} & D \\end{matrix}\\right) \]\n\nbe the Laplacian matrix of a connected graph. If \( D \) is nonsingular, then\n\n\[ X = \\left( \\begin{matrix} {H}^{\\# } & - {H}^{\\# }B{D}^{-1} \\\\ - {D}^{-1}{B}^{\\mathrm{T}}{H}^{\\# } & {D}^{-1} + {D}^{-1}{B... | Proof Since \( H = A - B{D}^{-1}{B}^{\\mathrm{T}} \) is symmetric, \( {H}^{\\# } \) exists and is symmetric. Since\n\n\[ L = \\left( \\begin{matrix} I & B{D}^{-1} \\\\ 0 & I \\end{matrix}\\right) \\left( \\begin{matrix} H & 0 \\\\ 0 & D \\end{matrix}\\right) \\left( \\begin{matrix} I & 0 \\\\ {D}^{-1}{B}^{\\mathrm{T}} ... | Yes |
Theorem 2.1 Let \( {G}_{1} \) be an \( {r}_{1} \) -regular graph with \( {n}_{1} \) vertices and \( {m}_{1} \) edges, \( {G}_{2} \) and \( {G}_{3} \) be respectively two arbitrary graphs on \( {n}_{2},{n}_{3} \) vertices. Then \( G = {G}_{1}^{S} \circ \left( {{G}_{2}^{V} \cup {G}_{3}^{E}}\right) \) has the resistance d... | Proof Let \( R \) be the incidence matrix of \( {G}_{1} \) . Then with a proper labeling of vertices, the Laplacian matrix of \( G = {G}_{1}^{S} \circ \left( {{G}_{2}^{V} \cup {G}_{3}^{E}}\right) \) can be written as\n\n\[ L\left( G\right) = \left( \begin{matrix} \left( {{r}_{1} + {n}_{2}}\right) {I}_{{n}_{1}} & - R & ... | Yes |
Theorem 3.1 Let \( {G}_{1} \) be an \( {r}_{1} \) -regular graph with \( {n}_{1} \) vertices and \( {m}_{1} \) edges, \( {G}_{2} \) and \( {G}_{3} \) be respectively two arbitrary graphs on \( {n}_{2},{n}_{3} \) vertices. Then the Kirchhoff index of \( G = \) \( {G}_{1}^{S} \circ \left( {{G}_{2}^{V} \cup {G}_{3}^{E}}\r... | Proof Let \( G = {G}_{1}^{S} \circ \left( {{G}_{2}^{V} \cup {G}_{3}^{E}}\right) \), and \( {L}_{G}^{\left( 1\right) } \) be the symmetric \( \{ 1\} \) -inverse of \( {L}_{G} \) given in (2.1). Then\n\n\[ \operatorname{tr}\left( {L}_{G}^{\left( 1\right) }\right) = 2\operatorname{tr}\left( {L}_{{G}_{1}}^{\# }\right) + \o... | Yes |
Lemma 7 Every planar graph \( G \) can be decomposed into two forests \( {F}_{1},{F}_{2} \) and one subgraph \( H \) such that, for every vertex \( v \) in \( G,{d}_{H}\left( v\right) \leq {11} \), and\n\n(1) if \( {d}_{G}\left( v\right) > {11} \), then \( {d}_{{F}_{i}}\left( v\right) \leq \lceil \frac{{d}_{G}\left( v\... | Proof If \( \Delta \left( G\right) \leq 5 \), the result holds trivially. Now suppose that \( \Delta \left( G\right) \geq 6 \) . By Corollary \( 1, G \) has an edge-partition into two forests \( {F}_{1},{F}_{2} \) and a subgraph \( H \) with \( \Delta \left( {F}_{i}\right) \leq \lceil \frac{\Delta \left( G\right) }{2}\... | Yes |
Lemma 3 Let \( k \geq 2 \) be an integer and \( N \) be a unitary \( k \) -perfect number. Then\n\n\[ \omega \left( N\right) > \max \left\{ {s : \mathop{\prod }\limits_{{i = 1}}^{s}\left( {1 + \frac{1}{{P}_{i}}}\right) < k}\right\} \]\n\nwhere \( {P}_{i} \) denotes the \( i \) -th prime. | Proof Let \( N = {p}_{1}^{{\alpha }_{1}}{p}_{2}^{{\alpha }_{2}}\cdots {p}_{t}^{{\alpha }_{t}} \) be the standard prime factorization of \( N \) . Since \( {p}_{i} \geq {P}_{i} \) for \( i = 1,2,\cdots, t \), we have\n\n\[ k = \frac{{\sigma }^{ * }\left( N\right) }{N} = \mathop{\prod }\limits_{{i = 1}}^{t}\frac{1 + {p}_... | Yes |
Lemma 1.1 Let \( R \) be a finite chain ring with maximal ideal \( \langle \gamma \rangle \), index of nilpotency \( e \) even and \( \left| K\right| = \frac{\left| R\right| }{\left| \langle \gamma \rangle \right| } = {p}^{r} = q \) . Then nontrivial self-dual cyclic codes of length \( n \) over \( R \) exist if and on... | Proof of Theorem 0.1 Suppose that (i) is satisfied. Without loss of generality, let \( {\operatorname{ord}}_{{p}_{1}}\left( q\right) = {\widehat{c}}_{1} \) be odd. We need to prove that \( {q}^{i} ≢ - 1\left( {\;\operatorname{mod}\;n}\right) \) for all positive integers \( i \) . Suppose otherwise that there exists som... | Yes |
Theorem 1.1 Let \( {T}_{\theta } \) be a \( B \) -algebra and \( \operatorname{Hom}\left( {{T}_{\theta },{T}_{\theta }}\right) \) be the vector space spanned by \( B \) -algebra homomorphisms of \( {T}_{\theta } \) . Then \( \operatorname{Hom}\left( {{T}_{\theta },{T}_{\theta }}\right) \) is a subalgebra of the general... | Proof The result follows from Definition 1.1, directly. | No |
Theorem 1.2 Let \( L \) be a 3-Lie algebra and \( T\left( L\right) = L\dot{ + }\operatorname{Der}\left( L\right) \) (the direct sum of vector spaces). Then \( \left( {T{\left( L\right) }_{\theta },\left\lbrack {\cdot , \cdot }\right\rbrack }\right) \) is a \( B \) -algebra, where the multiplication \( \left\lbrack {\cd... | Proof From Eq. (1.4), \( \left( {T\left( L\right) ,\left\lbrack {\cdot , \cdot }\right\rbrack }\right) \) is an anticommutative algebra. And for all \( x, y \in \) \( L,{D}_{1},{D}_{2} \in \operatorname{Der}\left( L\right) \) \[ \left\lbrack {\theta \left( {x + {D}_{1}}\right) ,\theta \left( {y + {D}_{2}}\right) }\righ... | Yes |
Theorem 1.3 Let \( {T}_{\theta } \) be a \( B \) -algebra. Then\n\n(1) \( \operatorname{Der}\left( {T}_{0}\right) = \left\{ {D \in \operatorname{Der}\left( T\right) \mid D\left( {T}_{0}\right) \subset {T}_{0}}\right\} \) is a subalgebra of \( \operatorname{Der}\left( T\right) \), \n\n(2) \( \iota \left( {T}_{0}\right) ... | Proof The result follows from the direct computation according to Definition 1.1. | No |
Theorem 2.1 Let \( \left( {T,\left\lbrack {\cdot , \cdot }\right\rbrack ,\theta }\right) \) be a \( B \) -algebra. Then \( \left( {{T}_{0},\left\lbrack {\cdot , \cdot }\right\rbrack }\right) \) is a Lie algebra, and \( {T}_{1} \) is a 3-Lie algebra in the multiplication\n\n\[ \left\lbrack {x, y, z}\right\rbrack = \left... | Proof By Definition 1.1, \( \left( {{T}_{0},\left\lbrack {\cdot , \cdot }\right\rbrack }\right) \) is a Lie algebra, and the ternary multiplication \( \left\lbrack {x, y, z}\right\rbrack \) defined by Eq. (2.1) is skew-symmetric. For any \( x, y, z, u, v \in {T}_{1} \), by Definition 1.1,\n\n\[ \left\lbrack {\left\lbra... | Yes |
Theorem 2.4 Let \( L \) be a 3-Lie algebra. Then \( \left( {\mathrm{{SB}}{\left( L\right) }_{\theta },\pi }\right) \) is an enveloping \( B \) -algebra of \( L \) , where \( \pi : L \rightarrow \mathrm{{SB}}\left( L\right) ,\pi \left( x\right) = - x,\forall x \in L.\mathrm{{SB}}{\left( L\right) }_{\theta } \) is called... | Proof By the above discussion, \( \pi : L \rightarrow \mathrm{{SB}}\left( L\right) ,\pi \left( x\right) = - x,\forall x \in L \) satisfies \( \pi \left( \left\lbrack {x, y, z}\right\rbrack \right) = \) \( \left\lbrack {\left\lbrack {\pi \left( x\right) ,\pi \left( y\right) }\right\rbrack ,\pi \left( z\right) }\right\rb... | Yes |
Theorem 2.6 Let \( L \) be a 3-Lie algebra over a field \( F \) with \( \operatorname{ch}F \neq 2 \) . If \( I \) is an ideal of the \( B \) -algebra \( \operatorname{SB}{\left( L\right) }_{\theta } \), then\n\n(1) \( {I}_{L} \) and \( {I}_{\mathrm{{Ad}}} \) are an ideal of the 3-Lie algebra \( L \) and the Lie algebra... | Proof If \( x \in {I}_{L} \), then there exists \( D \in \operatorname{ad}\left( L\right) \) such that \( x + D \in I \) . Thus \( \forall y, z \in L \) ,\n\n\[ \left\lbrack {x + D,\operatorname{ad}\left( {y, z}\right) }\right\rbrack = - \left\lbrack {x, y, z}\right\rbrack - \operatorname{ad}\left( {D\left( y\right), z... | Yes |
Lemma 1.3 \( {}^{\left\lbrack {10}\right\rbrack } \) Let \( k \in \mathbb{Z}, h \in {\Lambda }_{1}^{\eta }\left( {\mathbb{R}}^{ + }\right), m \in \mathbb{N} \) and \( R \) be a real-valued polynomial on \( {\mathbb{R}}^{n} \) with \( \deg \left( R\right) \leq m - 1 \) . Suppose that \( P\left( y\right) = \mathop{\sum }... | \[ \left| {{\int }_{{\rho }^{k} \leq \left| y\right| < {\rho }^{k + 1}}{\mathrm{e}}^{-\mathrm{i}P\left( y\right) }\frac{\Omega \left( y\right) h\left( \left| y\right| \right) }{{\left| y\right| }^{n}}\mathrm{\;d}y}\right| \leq C\left( {\log \rho }\right) \parallel h{\parallel }_{{\Lambda }_{1}^{\eta }\left( {\mathbb{R}... | Yes |
Lemma 1.5 Let \( 1 < q \leq 2,\Omega, h \) be as in Lemma 1.4. Let \( {N}_{0},{M}_{0} \in \mathbb{N},{R}_{1} \) and \( {R}_{2} \) be real-valued polynomials on \( {\mathbb{R}}^{n} \) and \( {\mathbb{R}}^{m} \), respectively with \( \deg \left( {R}_{1}\right) \leq {N}_{0} - 1 \) and \( \deg \left( {R}_{2}\right) \leq {M... | \[ \left| {{\int }_{{I}_{k, j}}{\mathrm{e}}^{-\mathrm{i}\left( {P\left( x\right) + Q\left( y\right) }\right) }\frac{\Omega \left( {x, y}\right) h\left( {\left| x\right| ,\left| y\right| }\right) }{{\left| x\right| }^{n}{\left| y\right| }^{m}}\mathrm{\;d}x\mathrm{\;d}y}\right| \] \[ \leq C{\left( \log \rho \right) }^{2}... | Yes |
Lemma 2.2 Let \( \mathcal{P} = \left( {{P}_{1},{P}_{2},\cdots ,{P}_{n}}\right) : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n} \) and \( \mathcal{Q} = \left( {{Q}_{1},{Q}_{2},\cdots ,{Q}_{m}}\right) : {\mathbb{R}}^{m} \rightarrow {\mathbb{R}}^{m} \) be polynomial mappings. Let \( q \in (1,2\rbrack ,\Omega \in {L}^{q}\l... | \[ \parallel {\lambda }_{\mathcal{P},\mathcal{Q}}^{ * }f{\parallel }_{{L}^{p}} \leq C{\left( \log \rho \right) }^{2}\parallel h{\parallel }_{{\widetilde{\Lambda }}_{\gamma }^{\eta }}\parallel \Omega {\parallel }_{{L}^{q}\left( {{\mathbb{S}}^{n - 1} \times {\mathbb{S}}^{m - 1}}\right) }\parallel f{\parallel }_{{L}^{p}}.... | Yes |
Lemma 2.3 Let \( \Phi : {B}_{n}\left( {0,1}\right) \rightarrow {\mathbb{R}}^{n} \) and \( \Psi : {B}_{m}\left( {0,1}\right) \rightarrow {\mathbb{R}}^{m} \) be smooth mappings. Let \( \mathcal{P} = \left( {{P}_{1},{P}_{2},\cdots ,{P}_{n}}\right) : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n} \) and \( \mathcal{Q} = \le... | Proof We shall only prove (2.3). The proof of (2.4) is similar. By Hölder’s inequality, it is easy to check that \[ {\lambda }_{\mathcal{P},\Psi }^{ * }f\left( {x, y}\right) \leq {\left( \mathop{\sup }\limits_{{k \leq {k}_{0}}}{\int }_{{\rho }^{k}}^{{\rho }^{k + 1}}\mathop{\sup }\limits_{{j \leq {j}_{0}}}{\int }_{{\rho... | Yes |
Theorem 1.1 Let \( \varphi \) be an anisotropic growth function as in Definition 1.3. Then\n\n\[ \n{T}_{A}^{\varphi }\left( {{\mathbb{R}}^{n} \times \mathbb{Z}}\right) = {T}_{A}^{\varphi ,\infty }\left( {{\mathbb{R}}^{n} \times \mathbb{Z}}\right)\n\]\n\nwith equivalent (quasi-) norms. | To show Theorem 1.1, we need some technical lemmas. For \( \left( {\varphi ,\infty }\right) \) -atoms, we have the following lemma which is an anisotropic variant of [14, Lemma 3.1]. It is worth mentioning that, we simplify the proof of \( \left\lbrack {{14}\text{, Lemma 3.1}}\right\rbrack \) in our anisotropic setting... | No |
Lemma 1.1 Let \( \varphi \) be an anisotropic growth function as in Definition 1.3. Then, for any \( \left( {\varphi ,\infty }\right) \) -atom \( a \), it holds that \( a \in {T}_{A}^{\varphi }\left( {{\mathbb{R}}^{n} \times \mathbb{Z}}\right) \) . Moreover, there exists a positive constant \( C \) such that, for any \... | Proof Let \( a \) be a \( \left( {\varphi ,\infty }\right) \) -atom. Assume first that (1.8) holds for a moment. By (1.8) with \( \lambda = 1 \) and \( \varphi \left( {B,\frac{1}{\parallel {\chi }_{B}{\parallel }_{{L}^{\varphi }\left( {\mathbb{R}}^{n}\right) }}}\right) = 1 \), we see that there exists a positive consta... | Yes |
Lemma 1.3 There exist positive constants \( \gamma \in \left( {0,1}\right) \) and \( {C}_{\left( \gamma \right) } \), depending on \( \gamma \), such that, for any closed subset \( F \) of \( {\mathbb{R}}^{n} \) whose complement has finite measure, and any nonnegative measurable function \( H \) on \( {\mathbb{R}}^{n} ... | Proof We claim that there exist positive constants \( \gamma \in \left( {0,1}\right) \) and \( {C}_{\left( \gamma \right) } \) such that, for any closed subset \( F \) of \( {\mathbb{R}}^{n} \) and \( \left( {y, k}\right) \in \Gamma \left( {F}_{\gamma }^{ * }\right) \),\n\n\[ {\int }_{F}{\chi }_{y + {B}_{k}}\left( x\ri... | Yes |
Lemma 2.4 Let \( \varphi \) be an anisotropic growth function as in Definition 1.3 satisfying \( q\left( \varphi \right) < \) \( i\left( \varphi \right) \left\lbrack {1 + {\log }_{b}\left( {\lambda }_{ - }\right) }\right\rbrack \), where \( q\left( \varphi \right) \) and \( i\left( \varphi \right) \) are, respectively,... | Proof The proof of Lemma 2.4 is similar to that of [18, Theorem 3.2] with a slight modification, the details being omitted. There is just one remark on the condition of the anisotropic growth function \( \varphi \) satisfying \( q\left( \varphi \right) < i\left( \varphi \right) \left\lbrack {1 + {\log }_{b}\left( {\lam... | No |
Theorem 3.1 (Weak duality) Let \( x \in F,\left( {y,{u}^{ * },{v}^{ * }}\right) \in G \), and \( {u}^{ * } \in {C}^{ * } \smallsetminus \left\{ {0}_{{Y}^{ * }}\right\} \) (or \( {u}^{ * } \in \operatorname{int}{C}^{ * } \) ). Furthermore, suppose that one of the following conditions holds with respect to \( {b}_{0},{b}... | Proof Assume to the contrary that there exist \( \breve{x} \in F,\left( {y,{u}^{ * },{v}^{ * }}\right) \in G \) such that\n\n\[ f\left( \breve{x}\right) { < }_{C}f\left( y\right) \;\left( {\text{ or }f\left( \breve{x}\right) { \leq }_{C}f\left( y\right) }\right) .\n\]\n\nBy \( {u}^{ * } \in {C}^{ * } \smallsetminus \le... | Yes |
Theorem 3.2 (Strong duality) Let \( \bar{x} \) be a weakly efficient solution for the problem (P) at which the Slater regularity condition is satisfied, \( f, g \) be directionally differentiable \( \alpha \) -preinvex on \( \alpha \) -invex set \( K \subset X \) . Moreover, suppose that one of the following conditions... | Proof Since \( \bar{x} \) satisfies all the conditions of Lemma 1.2, there exist \( {\bar{u}}^{ * } \in {C}^{ * },{\bar{u}}^{ * } \neq \) \( {0}_{{Y}^{ * }},{\bar{v}}^{ * } \in {D}^{ * } \) such that \( \left\langle {{\bar{v}}^{ * }, g\left( \bar{x}\right) }\right\rangle = 0 \) and \( \left( {\bar{x},{\bar{u}}^{ * },{\... | Yes |
Theorem 3.3 (Converse duality) Let \( \left( {\bar{y},{\bar{u}}^{ * },{\bar{v}}^{ * }}\right) \) be a weakly efficient (or an efficient) solution for the problem (D). Assume that \( {\bar{u}}^{ * } \in {C}^{ * } \smallsetminus \left\{ {0}_{{Y}^{ * }}\right\} \) (or \( {\bar{u}}^{ * } \in \operatorname{int}{C}^{ * } \) ... | Proof Assume contrary to the result that \( \bar{y} \) is not a weakly efficient (or an efficient) solution for the problem (P), then there exists \( \breve{y} \in F \) such that\n\n\[ f\left( \breve{y}\right) { < }_{C}f\left( \bar{y}\right) \;\left( {\text{ or }f\left( \breve{y}\right) { \leq }_{C}f\left( \bar{y}\righ... | Yes |
Proposition 2.1 (i) If \( x \in K \) is a solution of (WVVI), then it is a solution of (WVICP). | Proof (i) Let \( x \in K \) be a solution of (WVVI). Then\n\n\[ \langle T\left( x\right), y - g\left( x\right) \rangle \notin - \operatorname{int}C\left( x\right) ,\;\forall y \in K. \]\n\n(2.1)\n\nSetting \( y = {\theta }_{X} \) (the origin of \( X \) ) in (2.1), we have\n\n\[ \langle T\left( x\right), g\left( x\right... | Yes |
Proposition 2.2 (SVVI) and (SVICP) have the same solution set. | Proof (I) Let \( x \in K \) be a solution of (SVVI). Then\n\n\[ \langle T\left( x\right), y - g\left( x\right) \rangle \in C\left( x\right) ,\;\forall y \in K. \]\n\n(2.5)\n\nSetting \( y = {\theta }_{X} \) in (2.5), we have\n\n\[ \langle T\left( x\right), g\left( x\right) \rangle \in - C\left( x\right) \]\n\n(2.6)\n\n... | Yes |
Theorem 3.1 Let \( D \) be a nonempty, compact convex subset of Banach space \( X \) . Assume that\n\n(i) mappings \( T : D \rightarrow L\left( {X, Y}\right) \) and \( g : D \rightarrow D \) are continuous;\n\n(ii) \( \forall x \in D,\langle T\left( x\right), x\rangle - \langle T\left( x\right), g\left( x\right) \rangl... | Proof By contradiction, suppose that the (WVVI) is not solvable. Then \( \forall x \in D \), there is some \( y \in D \) such that\n\n\[ \langle T\left( x\right), y - g\left( x\right) \rangle \in - \operatorname{int}C\left( x\right) . \]\n\nDefine set-valued mapping \( S : D \rightarrow {2}^{D} \) as follows. For each ... | Yes |
Corollary 4.1 Let \( X \) and \( K \) be the same as in Theorem 4.1. Assume that the conditions (i)-(iii) in Theorem 4.1 and the following condition (iv)' hold:\n\n(iv)’ there exists a real number \( r > 0 \) such that, \( \forall x \in K,\parallel x\parallel \geq r \), one has\n\n\[ \langle T\left( x\right), g\left( x... | Proof In Theorem 4.1, choose \( {v}_{x} = {\theta }_{X} \) . Then Theorem 4.1 yields the conclusion. This completes the proof. | No |
Corollary 4.2 Let \( X \) and \( K \) be the same as in Theorem 4.1. Assume that the conditions (i)-(iii) in Theorem 4.1 and the following condition (iv)' hold:\n\n(iv)' there exist a real number \( {r}_{0} > 0 \) and some \( {v}_{0} \in K \) such that, \( \forall x \in K,\parallel x\parallel \geq {r}_{0} \), one has\n... | Proof In Theorem 4.1, let \( r = 1 + \max \left\{ {{r}_{0},\begin{Vmatrix}{v}_{0}\end{Vmatrix}}\right\} \), and for each \( x \in K,\parallel x\parallel \geq r \), choose \( {v}_{x} = {v}_{0} \). Then Theorem 4.1 yields the conclusion. This completes the proof. | Yes |
Theorem 4.2 Let \( X \) be a reflexive Banach space, and \( K{\left( {K}_{n}\right) }_{n \in \mathbb{N}} \) be a Galerkin cone of \( X \) . Assume that the following conditions hold:\n\n(i) mappings \( T : K \rightarrow L\left( {X, Y}\right) \) and \( g : K \rightarrow K \) are strongly continuous;\n\n(ii) \( \forall x... | Proof For each \( n \geq {n}_{0} \), by Theorem 4.1, the (WVVI) has a solution \( {x}_{n} \in {K}_{n} \) and \( \begin{Vmatrix}{x}_{n}\end{Vmatrix} < r \) . Since \( X \) is reflexive and the sequence \( \left\{ {x}_{n}\right\} \subseteq K \) is bounded, it has a weakly convergent subsequence (for simplification, we st... | Yes |
Theorem 5.1 Let \( X \) be a Banach space, \( K \subseteq X \) a locally compact convex cone. Assume that\n\n(i) \( C : K \rightarrow {2}^{Y} \) is closed;\n\n(ii) mappings \( T : K \rightarrow L\left( {X, Y}\right) \) and \( g : K \rightarrow K \) are continuous;\n\n(iii) there is a mapping \( h : K \times K \rightarr... | Proof By Proposition 2.2, we only need to show that the (SVVI) is solvable in \( K \) .\n\nIn fact, \( \forall n \in \mathbb{N} \), let \( {D}_{n} = \{ x \in K : \parallel x\parallel \leq n\} \) . It follows from the local compactness of \( K \) that \( {D}_{n} \) is compact and convex. By Theorem 3.2, there is an \( {... | Yes |
Theorem 5.2 Let \( X \) be a reflexive Banach space, and \( K{\left( {K}_{n}\right) }_{n \in \mathbb{N}} \) be a Galerkin cone of \( X \) . Assume that the following conditions hold:\n\n(i) \( C : K \rightarrow {2}^{Y} \) is closed;\n\n(ii) mappings \( T : K \rightarrow L\left( {X, Y}\right) \) and \( g : K \rightarrow... | Proof For \( n \geq {n}_{0} \), by Theorem 5.1, the (SVVI) has a solution \( {x}_{n} \in {K}_{n} \), and \( \begin{Vmatrix}{x}_{n}\end{Vmatrix} < \) \( r \) . Since \( X \) is reflexive and the sequence \( \left\{ {x}_{n}\right\} \subseteq K \) is bounded, it has a weakly convergent subsequence (for simplification, we ... | Yes |
Theorem 0.5 Let \( G \) be a graph with \( \Delta \left( G\right) = 3 \) and \( \mathrm{{mad}}\left( G\right) < \frac{44}{15}. \) Then \( {\mathrm{{ch}}}_{\sum }^{\prime \prime }\left( G\right) \leq 6. | Our proof proceeds by contradiction. Let \( G \) be a counterexample to Theorem 0.5 such that \( \left| {V\left( G\right) }\right| + \left| {E\left( G\right) }\right| \) is as small as possible. Clearly, \( G \) is connected. Let \( {\left( {L}_{x}\right) }_{x \in V \cup E} \) be any given set of lists of real numbers,... | Yes |
Theorem 1.1 Let \( G \) be a cubic graph of order \( n \) . Then \( \nabla \left( G\right) = \left\lceil \frac{n + 2}{4}\right\rceil \) if and only if \( G \) is upper-embeddable. | In fact, Theorem 1.1 not only answers the question proposed by Bau and Beineke, but also generalizes one result of Xuong in [11] where it is proved that if there is a set \( A \) with cardinality \( \left\lfloor \frac{{3n} - 2}{4}\right\rfloor \) in a cubic graph \( G \) such that \( G\left\lbrack A\right\rbrack \) is ... | No |
Lemma 2.1 Let \( G \) be a cubic graph of order \( n \) . Then \( \nabla \left( G\right) \geq \left\lceil \frac{n + 2}{4}\right\rceil \) . | Proof Suppose that \( S \) is a \( \nabla \) -set of \( G \) and there are \( c \) components in \( G - S \) . Then \( \left| {E\left( {G - S}\right) }\right| = n - \left| S\right| - c \) . On the other hand we know \( \left| {{N}_{E}\left( S\right) }\right| \leq 3\left| S\right| \) and equality holds if and only if \(... | Yes |
Lemma 3.1 Let \( G \) be a cubic graph. Then\n\n\[ \nabla \left( G\right) \leq {\gamma }_{M}\left( G\right) + \xi \left( G\right) \] | Proof Let \( {T}_{X} \) be a Xuong tree of \( G \) . By Theorem 1.5,\n\n\[ E\left( G\right) - E\left( {T}_{X}\right) = \left\{ {{e}_{1},{e}_{2}}\right\} \cup \left\{ {{e}_{3},{e}_{4}}\right\} \cup \cdots \cup \left\{ {{e}_{{2m} - 1},{e}_{2m}}\right\} \cup \left\{ {{f}_{1},{f}_{2},\cdots ,{f}_{s}}\right\} \]\n\nhere \( ... | Yes |
Theorem 1 Let\n\n\\[ \n{S}_{k}\left( x\right) = \mathop{\sum }\limits_{\substack{{1 \leq {n}_{1},{n}_{2} \leq {x}^{\frac{1}{2}}} \\ {1 \leq {n}_{3} \leq {x}^{\frac{1}{k}}} }}d\left( {{n}_{1}^{2} + {n}_{2}^{2} + {n}_{3}^{k}}\right) ,\\;3 \leq k \in \mathbb{N}. \n\\] \n\nThen we have\n\n\\[ \n{S}_{k}\left( x\right) = {C}... | We establish Theorem 1 by means of the circle method. In order to estimate the sum of divisor effectively, we employ the results obtained by Guo and Zhai (see [2, Lemma 7.1]).\n\n## 1 Preliminaries\n\nThroughout this paper, \\( x \\) is a large positive integer. For any \\( \\alpha \\in \\mathbb{R} \\), put\n\n\\[ \n{f... | Yes |
Lemma 1.1 For any \( a, q \in \mathbb{Z} \) with \( \left( {q, a}\right) = 1 \) and \( q > 0 \), let\n\n\[ \n{S}_{k}\left( {q, a}\right) = \mathop{\sum }\limits_{{r = 1}}^{q}e\left( {\frac{a}{q}{r}^{k}}\right) .\n\]\n\nThen we have\n\n\[ \n{S}_{k}\left( {q, a}\right) \ll {q}^{\frac{k - 1}{k}}.\n\] | Proof See \( \left\lbrack {4\text{, Theorem 4.2}}\right\rbrack \) . | No |
Lemma 1.2 For \( k \geq 1 \), we have\n\ni)\n\n\[ \n{\int }_{0}^{1}e\left( {\beta {\mu }^{k}}\right) \mathrm{d}\mu \ll \frac{1}{{\left( 1 + \left| \beta \right| \right) }^{\frac{1}{k}}}\n\]\n\nii)\n\n\[ \n{\int }_{0}^{3}e\left( {\beta \mu }\right) \log \mu \mathrm{d}\mu \ll \frac{\log \left( {2 + \left| \beta \right| }... | Proof For i), it follows from integration by parts together with trivial bounds. For ii), when \( \left| \beta \right| > 1 \), we have\n\n\[ \n{\int }_{0}^{3}e\left( {\beta \mu }\right) \log \mu \mathrm{d}\mu = \left( {{\int }_{0}^{\frac{1}{\left| \beta \right| }} + {\int }_{\frac{1}{\left| \beta \right| }}^{3}}\right)... | Yes |
Lemma 1.3 Suppose that \( \left( {a, q}\right) = 1 \) and \( \alpha = \frac{a}{q} + \beta \) . Then\n\n\[ \n{f}_{k}\left( \alpha \right) = {V}_{k}\left( {\alpha, q, a}\right) + O\left( {{q}^{\frac{1}{2} + \varepsilon }{\left( 1 + x\left| \beta \right| \right) }^{\frac{1}{2}}}\right) .\n\]\n\nIf further \( \left| \beta ... | Proof See [4, Theorem 4.1]. | No |
Lemma 1.4 Suppose that \( \alpha = \frac{a}{q} + \beta \in \mathfrak{M} \) and \( {Q\tau } \leq x,\tau > {x}^{\frac{1}{2} + \varepsilon } \). Then\n\n\[ f\left( {-\alpha }\right) = \frac{x\log x}{q}{\int }_{0}^{3}e\left( {-{\mu x\beta }}\right) \mathrm{d}\mu + \frac{x}{q}{\int }_{0}^{3}e\left( {-{\mu x\beta }}\right) \... | Proof See [2, Lemma 7.1]. | No |
Lemma 1.5 Suppose \( M > 0, N > 0,{u}_{m} > 0,{v}_{n} > 0,{A}_{m} > 0,{B}_{n} > 0(1 \leq m \leq \) \( M,1 \leq n \leq N \) ), and let \( {Q}_{1} \) and \( {Q}_{2} \) be given non-negative numbers, \( {Q}_{1} \leq {Q}_{2} \) . Then there is a \( q \) such that \( {Q}_{1} \leq q \leq {Q}_{2} \) and\n\n\[ \mathop{\sum }\l... | Proof This is \( \left\lbrack {3\text{, Lemma 3}}\right\rbrack \) . | Yes |
Lemma 1.6 Let \( j \) be an integer with \( j \geq 2 \) . Suppose that there exist integers \( a, q \) with \( q \geq 1,\left( {a, q}\right) = 1 \) such that \( \left| {\alpha - \frac{a}{q}}\right| \leq {q}^{-2} \) and \( q \leq x \) . Then one has \[ {f}_{j}\left( \alpha \right) \ll {x}^{\frac{1}{j} + \varepsilon }{\l... | Proof See [5, Theorem 1.5]. This conclusion is superior to Weyl’s inequality for \( j \geq 8 \) . | No |
Lemma 1.1 If \( u \) is a solution of \( \left( {0.3}\right) –\left( {0.4}\right) \) with \( {2n} \) - \( 2 \) interior zeros with \( {u}^{\prime }\left( 0\right) > 0,\;u\left( 0\right) > \) \( 0,{u}^{\prime }\left( 1\right) < 0 \) and \( u\left( 1\right) > 0 \), then for \( \alpha \geq 0,\beta \geq 0 \) ,\n\n\[{\lambd... | Proof Let \( u \) be a solution of (0.3)-(0.4) with \( {2n} - 2 \) interior zeros with \( {u}^{\prime }\left( 0\right) = m > 0 \) , \( u\left( 0\right) > 0,{u}^{\prime }\left( 1\right) = - r < 0 \) and \( u\left( 1\right) > 0 \) . Denote \( {x}_{0} \) as the first critical point, \( {x}_{1} \) and \( {x}_{2} \) as the ... | Yes |
Lemma 1.3 Given \( n \in \mathbb{N},\alpha ,\beta \in \left( {0,\infty }\right) \) and \( \rho \in \left( {\theta ,\infty }\right) \). (a) If \( u\left( x\right) \) is a sign-changing solution with \( {2n} - 1 \) interior zeros with \( u\left( 0\right) < 0,{u}^{\prime }\left( 0\right) < 0,{u}^{\prime }\left( 1\right) <... | \[ {\lambda }^{\frac{1}{p}} = {G}_{{2n} - 1}\left( {\alpha ,\beta ,\rho }\right) = {\left( \frac{p - 1}{p}\right) }^{\frac{1}{p}}\left\{ {{2n}{\int }_{0}^{\rho }\frac{\mathrm{d}u}{{\left( F\left( \rho \right) - F\left( u\right) \right) }^{\frac{1}{p}}} + {2n}{\int }_{q}^{0}\frac{\mathrm{d}u}{{\left( F\left( q\right) - ... | Yes |
Theorem 2.1 Let\n\n\\[ \n{\\lambda }_{n} = \\left( \\frac{p - 1}{p}\\right) {\\left\\{ 2n{\\int }_{0}^{\\theta }\\frac{\\mathrm{d}u}{{\\left( -F\\left( u\\right) \\right) }^{\\frac{1}{p}}} - {\\int }_{0}^{\\alpha {m}^{ * }}\\frac{\\mathrm{d}u}{{\\left( -F\\left( u\\right) \\right) }^{\\frac{1}{p}}} - {\\int }_{0}^{\\be... | Proof According to the quadrature technique, a solution with \\( {2n} - 2 \\) interior zeros exists if for \\( \\lambda > 0 \\) there exists \\( \\rho \\in \\left( {\\theta ,\\infty }\\right) \\) such that \\( {\\lambda }^{\\frac{1}{p}} = {G}_{{2n} - 2}\\left( {\\alpha ,\\beta ,\\rho }\\right) \\) . To prove this we wi... | Yes |
Theorem 2.2 Let\n\n\[ \n{\lambda }_{n} = \left( \frac{p - 1}{p}\right) {\left\{ 2n{\int }_{0}^{\theta }\frac{\mathrm{d}u}{{\left( -F\left( u\right) \right) }^{\frac{1}{p}}} - {\int }_{0}^{\beta {r}^{ * }}\frac{\mathrm{d}u}{{\left( -F\left( u\right) \right) }^{\frac{1}{p}}}\right\} }^{p};\;n \in \mathbb{N}. \]\n\nThen f... | Proof It follows by analyzing \( {G}_{{2n} - 1}\left( {\alpha ,\beta ,\rho }\right) \) defined in (1.21) instead of \( {G}_{{2n} - 2}\left( {\alpha ,\beta ,\rho }\right) \) in the proof of Theorem 2.1. | Yes |
Theorem 2.3 Let\n\n\\[ \n{\\lambda }_{n} = \\left( \\frac{p - 1}{p}\\right) {\\left\\{ 2n{\\int }_{0}^{\\theta }\\frac{\\mathrm{d}u}{{\\left( -F\\left( u\\right) \\right) }^{\\frac{1}{p}}}\\right\\} }^{p},\\;n \\in \\mathbb{N}. \n\\]\n\nThen for each \\( \\lambda \\in \\left( {0,{\\lambda }_{n}}\\right) \\) there exist... | Proof It follows by analyzing \\( {G}_{2n}\\left( {\\alpha ,\\beta ,\\rho }\\right) \\) defined in (1.25) instead of \\( {G}_{{2n} - 2}\\left( {\\alpha ,\\beta ,\\rho }\\right) \\) in the proof of Theorem 2.1. | Yes |
Theorem 2.4 Let\n\n\[ \n{\lambda }_{n} = \left( \frac{p - 1}{p}\right) {\left\{ 2n{\int }_{0}^{\theta }\frac{\mathrm{d}u}{{\left( -F\left( u\right) \right) }^{\frac{1}{p}}} - {\int }_{0}^{\alpha {m}^{ * }}\frac{\mathrm{d}u}{{\left( -F\left( u\right) \right) }^{\frac{1}{p}}}\right\} }^{p},\;n \in \mathbb{N}. \]\n\nThen ... | Proof It follows by analyzing \( {G}_{{2n} - 1}^{1}\left( {\alpha ,\beta ,\rho }\right) \) defined in (1.29) instead of \( {G}_{{2n} - 2}\left( {\alpha ,\beta ,\rho }\right) \) in the proof of Theorem 2.1. | Yes |
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