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Theorem 2.2 Let \( \mathcal{A} \) be a nonnegative tensor in \( {\mathbb{R}}^{\left\lbrack k, n\right\rbrack } \) with Z-spectral radius \( \sigma \left( \mathcal{A}\right) \) . If \( \mathcal{A} \) has nonzero row sums \( {R}_{i}\left( \mathcal{A}\right), i = 1,2,\cdots, n \), then \( \sigma \left( \mathcal{A}\right) ... | Proof By Theorem 2.1 and Lemma 2.2, the inequality in left-hand side of (2.2) follows immediately. We prove the right inequality of (2.2) by induction.\n\nWhen \( t = 1 \), the right inequality becomes an equality.\n\nWhen \( t = 2 \), by (2.1) we have\n\n\[ \frac{{R}_{{i}_{1}}\left( {\mathcal{A}}^{m + 2}\right) }{{R}_... | Yes |
Theorem 2.1 Let \( {\left\{ {\Lambda }_{j}\right\} }_{j \in J} \) and \( {\left\{ {\Gamma }_{j}\right\} }_{j \in J} \) be two generalized Bessel sequences for \( H \) with respect to \( {\left\{ {V}_{j}\right\} }_{j \in J} \). Then the following statements are equivalent:\n\n(i) There exists an invertible operator \( O... | Proof (i) \( \Rightarrow \) (ii) For any \( f \in H \), there exists \( g \in H \) such that \( f = {Og} \), where \( g = \) \( \mathop{\sum }\limits_{{j \in J}}{\Lambda }_{j}^{ * }{\Gamma }_{j}{Og} \). Then\n\n\[ {Og} = O\mathop{\sum }\limits_{{j \in J}}{\Lambda }_{j}^{ * }{\Gamma }_{j}{Og} \]\n\nSo\n\n\[ f = \mathop{... | Yes |
Theorem 2.2 Every two generalized Riesz bases are G-dual frames. | Proof Suppose that \( {\left\{ {\Lambda }_{j}\right\} }_{j \in J} \) and \( {\left\{ {\Gamma }_{j}\right\} }_{j \in J} \) are two generalized Riesz bases for \( H \) with their respective synthesis operators \( T \) and \( U \) . Since the synthesis operator of Riesz basis is invertible, we have that \( T{U}^{ * } \) i... | Yes |
Theorem 2.3 Let \( \{ {\Lambda }_{j}{\} }_{j \in J} \) be a generalized frame for \( H \) with respect to \( \{ {V}_{j}{\} }_{j \in J},\{ {\Gamma }_{j}^{i}{\} }_{j \in J} \) be dual frames for \( {\left\{ {\Lambda }_{j}\right\} }_{j \in J} \), where \( i = 1,2,\cdots, m, m \in \mathbb{N} \) . If a finite complex number... | Proof Since \( {\left\{ {\Gamma }_{j}^{i}\right\} }_{j \in J} \) are dual frames for \( {\left\{ {\Lambda }_{j}\right\} }_{j \in J} \), for any \( f \in H \) we have\n\n\[ f = \mathop{\sum }\limits_{{j \in J}}{\Lambda }_{j}^{ * }{\Gamma }_{j}^{i}f,\;i = 1,2,\cdots, m. \]\n\nConsequently,\n\n\[ \mathop{\sum }\limits_{{i... | Yes |
Theorem 2.4 Let \( {\left\{ {\Gamma }_{j}\right\} }_{j \in J} \) be a G-dual frame for \( {\left\{ {\Lambda }_{j}\right\} }_{j \in J} \) with corresponding invertible operator \( O \in L\left( H\right) \), and \( S \) be the frame operator of \( {\left\{ {\Lambda }_{j}\right\} }_{j \in J} \) . Suppose that \( \alpha \)... | Proof For any \( f \in H \), we have\n\n\[ \mathop{\sum }\limits_{{j \in J}}{\Lambda }_{j}^{ * }{X}_{j}{Of} = \mathop{\sum }\limits_{{j \in J}}{\Lambda }_{j}^{ * }\left( {\alpha {\Gamma }_{j} + \left( {1 - \alpha }\right) {\Lambda }_{j}{S}^{-1}{O}^{-1}}\right) {Of} = f. \]\n\nNote that \( {\left\{ {X}_{j}\right\} }_{j ... | Yes |
Theorem 2.5 Let \( {\left\{ {\Gamma }_{j}\right\} }_{j \in J} \) and \( {\left\{ {X}_{j}\right\} }_{j \in J} \) be two G-dual frames for \( {\left\{ {\Lambda }_{j}\right\} }_{j \in J} \) with corresponding invertible operators \( {O}_{1} \) and \( {O}_{2} \), respectively. If \( {O}_{1}^{-1} + {O}_{2}^{-1} \) is an inv... | Proof Since the operator \( {O}_{1}^{-1} + {O}_{2}^{-1} \) is invertible, there exists an invertible operator \( {O}_{3} \in L\left( H\right) \) such that\n\n\[ \left( {{O}_{1}^{-1} + {O}_{2}^{-1}}\right) {O}_{3} = I \]\n\nThen for any \( f \in H \), we have\n\n\[ \mathop{\sum }\limits_{{j \in J}}{\Lambda }_{j}^{ * }\l... | Yes |
Theorem 2.6 Let \( {\left\{ {\Lambda }_{j}\right\} }_{j \in J} \) and \( {\left\{ {\Gamma }_{j}\right\} }_{j \in J} \) be two generalized Bessel sequences for \( H \) with respect to \( {\left\{ {V}_{j}\right\} }_{j \in J} \) with the synthesis operators \( T \) and \( U \), respectively. If there are two constants \( ... | Proof It follows from Lemma 1.5 that the operator \( T{U}^{ * } \in L\left( H\right) \) is invertible, therefore it is easy to see that \( {\left\{ {\Lambda }_{j}\right\} }_{j \in J} \) and \( {\left\{ {\Gamma }_{j}\right\} }_{j \in J} \) are G-dual frames. This completes the proof. | No |
Theorem 2.7 Let \( {\left\{ {\Lambda }_{j}\right\} }_{j \in J} \) and \( {\left\{ {\Gamma }_{j}\right\} }_{j \in J} \) be two linear bounded operator sequences and \( {O}_{1},{O}_{2} \in L\left( H\right) \) be two invertible operators. Then \( {\left\{ {\Lambda }_{j}\right\} }_{j \in J} \) and \( {\left\{ {\Gamma }_{j}... | Proof Assume that \( {\left\{ {\Lambda }_{j}\right\} }_{j \in J} \) and \( {\left\{ {\Gamma }_{j}\right\} }_{j \in J} \) are G-dual frames. It follows from the definition of G-dual frames that there exists an invertible operator \( {O}_{3} \in L\left( H\right) \) satisfying\n\n\[ f = \mathop{\sum }\limits_{{j \in J}}{\... | Yes |
Theorem 2.8 If two sequences \( {\left\{ {\Lambda }_{j}\right\} }_{j \in J} \) and \( {\left\{ {\Gamma }_{j}\right\} }_{j \in J} \) are approximately dual frames, then \( {\left\{ {\Lambda }_{j}\right\} }_{j \in J} \) and \( {\left\{ {\Gamma }_{j}\right\} }_{j \in J} \) are G-dual frames. | Proof Assume that \( T \) and \( U \) are the synthesis operators of \( {\left\{ {\Lambda }_{j}\right\} }_{j \in J} \) and \( {\left\{ {\Gamma }_{j}\right\} }_{j \in J} \), respectively. Since \( {\left\{ {\Lambda }_{j}\right\} }_{j \in J} \) and \( {\left\{ {\Gamma }_{j}\right\} }_{j \in J} \) are approximately dual f... | Yes |
Theorem 2.9 Let \( {\left\{ {\Lambda }_{j} \in L\left( H,{V}_{j}\right) \right\} }_{j \in J} \) be a sequence and \( {\left\{ {X}_{j}\right\} }_{j \in J} \) be a G-dual frame for \( {\left\{ {\Gamma }_{j}\right\} }_{j \in J} \) with invertible operator \( O \in L\left( H\right) \) . If there exist two constants \( \lam... | Proof It is easy to check that the inequality (3) holds for any \( {\left\{ {g}_{j}\right\} }_{j \in J} \in {l}^{2}\left( {\left\{ {V}_{j}\right\} }_{j \in J}\right) \) . So the series \( \mathop{\sum }\limits_{{j \in J}}{\left( {\Lambda }_{j} - {X}_{j}\right) }^{ * }{g}_{j} \) converges. Hence, the operator\n\n\[ \nQ ... | Yes |
Theorem 2.10 Let \( {\left\{ {\Lambda }_{j} \in L\left( H,{V}_{j}\right) \right\} }_{j \in J} \) be a sequence and \( {\left\{ {X}_{j}\right\} }_{j \in J} \) be a G-dual frame for \( {\left\{ {\Gamma }_{j}\right\} }_{j \in J} \) with invertible operator \( O \in L\left( H\right) \) . If there exist an invertible operat... | Proof First, we prove (i). It follows from the inequality (4) that \( {\left\{ {\Lambda }_{j} - {X}_{j}\right\} }_{j \in J} \) is a generalized Bessel sequence with the bound \( \mu {\begin{Vmatrix}{V}^{-1}\end{Vmatrix}}^{2} \), then\n\n\[ \n\begin{Vmatrix}{\mathop{\sum }\limits_{{j \in I}}{\left( {\Lambda }_{j} - {X}_... | Yes |
Lemma 3.1 Let \( {\left\{ {\Lambda }_{j}\right\} }_{j \in J} \) be a generalized frame for \( H \) with respect to \( {\left\{ {V}_{j}\right\} }_{j \in J} \) with the synthesis operator \( T \) and \( {\left\{ {\widetilde{e}}_{j, k}\right\} }_{j \in J, k \in {K}_{j}} \) be an orthonormal basis for \( {l}^{2}\left( {\le... | Proof First, assume that \( {\left\{ {\Lambda }_{j}\right\} }_{j \in J} \) and \( {\left\{ {\Gamma }_{j}\right\} }_{j \in J} \) are G-dual frames, then there exists an invertible operator \( O \in L\left( H\right) \) such that\n\n\[ f = \mathop{\sum }\limits_{{j \in J}}{\Gamma }_{j}^{ * }{\Lambda }_{j}{Of},\;\forall f ... | Yes |
Lemma 3.2 Let \( O \in L\left( H\right) \) be an invertible operator and \( {\left\{ {\Lambda }_{j}\right\} }_{j \in J} \) be a generalized frame for \( H \) with respect to \( {\left\{ {V}_{j}\right\} }_{j \in J} \) with the synthesis operator \( T \) and frame operator \( S \) . Then \( V \in L\left( H\right) \) is t... | Proof Assume that \( V \) is the left inverse of \( {T}^{ * }O \) . Let \( W = V \), then we have\n\n\[ \n{O}^{-1}{S}^{-1}T + W\left( {I - {T}^{ * }{S}^{-1}T}\right) = {O}^{-1}{S}^{-1}T + V - V{T}^{ * }{S}^{-1}T \n\]\n\n\[ \n= {O}^{-1}{S}^{-1}T + V - V{T}^{ * }O{O}^{-1}{S}^{-1}T \n\]\n\n\[ \n= V\text{.} \n\]\n\nConvers... | Yes |
Theorem 3.1 Let \( {\left\{ {\Gamma }_{j} \in L\left( H,{V}_{j}\right) \right\} }_{j \in J} \) be a sequence and \( {\left\{ {\Lambda }_{j}\right\} }_{j \in J} \) be a generalized frame for \( H \) with respect to \( {\left\{ {V}_{j}\right\} }_{j \in J} \) with the synthesis operator \( T \) and frame operator \( S \) ... | Proof First, assume that \( {\left\{ {\Lambda }_{j}\right\} }_{j \in J} \) and \( {\left\{ {\Gamma }_{j}\right\} }_{j \in J} \) are G-dual frames. It follows from Lemmas 3.1 and 3.2 that\n\n\[ \n{\Gamma }_{j}^{ * }{e}_{j, k} = \left( {{O}^{-1}{S}^{-1}T + W\left( {I - {T}^{ * }{S}^{-1}T}\right) }\right) {\widetilde{e}}_... | Yes |
Lemma 1.1 For \( i, j, k = 1,2,3 \), there holds\n\n\[ \n{b}_{j}{\lambda }_{i}^{2} + {b}_{k}{\lambda }_{j}^{2} + {b}_{i}{\lambda }_{k}^{2} = \left( {{b}_{j}{\lambda }_{i} + {b}_{k}{\lambda }_{j} + {b}_{i}{\lambda }_{k}}\right) + \left( {{b}_{i}{\lambda }_{i}{\lambda }_{j} + {b}_{j}{\lambda }_{j}{\lambda }_{k} + {b}_{k}... | Proof Using the fact that\n\n\[ \n{\lambda }_{i}^{2} = {\lambda }_{i}\left( {1 - {\lambda }_{j} - {\lambda }_{k}}\right) = {\lambda }_{i} - {\lambda }_{i}{\lambda }_{j} - {\lambda }_{k}{\lambda }_{i}, \n\]\n\nwe obtain\n\n\[ \n{b}_{j}{\lambda }_{i}^{2} = {b}_{j}{\lambda }_{i} - {b}_{j}{\lambda }_{i}{\lambda }_{j} - {b}... | Yes |
Lemma 1.1 Let \( \mathbb{A} = {\left( {a}_{k, j}\right) }_{k, j = 1}^{n} \) be an upper Hessenberg matrix with \( {a}_{k + 1, k} \neq 0\left( {1 \leq k \leq }\right. \) \( n - 1 \) ). Then the eigenvalues of \( \mathbb{A} \) are distinct if and only if \( \mathbb{A} \) is diagonalizable. | Proof Let \( {\sigma }_{1},{\sigma }_{2},\cdots ,{\sigma }_{m} \) be \( m \) numbers and consider a matrix \( \widetilde{\mathbb{A}} = \left( {\mathbb{A} - {\sigma }_{1}\mathbb{I}}\right) (\mathbb{A} - \) \( \left. {{\sigma }_{2}\mathbb{I}}\right) \cdots \left( {\mathbb{A} - {\sigma }_{m}\mathbb{I}}\right) \), where \(... | Yes |
Theorem 1.2 \( {}^{\left\lbrack {30}\right\rbrack } \) Let\n\n\[ \n{\mathbb{T}}_{n} = \left( \begin{matrix} {a}_{1} & {c}_{1} & 0 & \cdots & \cdots & 0 \\ {b}_{1} & {a}_{2} & {c}_{2} & 0 & \ddots & \vdots \\ 0 & {b}_{2} & {a}_{3} & \ddots & \ddots & \vdots \\ \vdots & 0 & \ddots & \ddots & \ddots & 0 \\ \vdots & \ddots... | Proof It is sufficient to consider a diagonal matrix \( \mathbb{D} = \operatorname{diag}\left( {1,\sqrt{\frac{-{b}_{1}}{{c}_{1}}},\sqrt{\frac{{b}_{1}{b}_{2}}{{c}_{1}{c}_{2}}},\cdots }\right. \) , \( \sqrt{\mathop{\prod }\limits_{{k = 1}}^{{n - 1}}\left( \frac{-{b}_{k}}{{c}_{k}}\right) }),{\mathbb{D}}^{-1}{\mathbb{T}}_{... | Yes |
Lemma 2.1 In Bargmann representation \( {\mathbb{B}}_{0} \), the Gribov operator has a simple spectrum. | Proof Let \( {\left\{ {e}_{k}\right\} }_{k = 1}^{\infty }\left( z\right) = \frac{{z}^{k}}{\sqrt{k!}} \) be an orthonormal basis of \( {\mathbb{B}}_{0} \) and \( \mathbb{H} \) be the Jacobi-Gribov matrix defined by (0.5):\n\n\[ \left\{ \begin{array}{l} {\mathbb{H}}_{\mu }{e}_{1} = {\beta }_{1}{e}_{1} + {\alpha }_{1}{e}_... | Yes |
Lemma 2.2 Let \( \mathbb{G} \) be the Gram matrix associated to Gribov operator \( {\mathbb{H}}_{\mu } \) for the system \( \left\{ {{\widehat{e}}_{1},{\widehat{e}}_{1},\cdots ,{\widehat{e}}_{n},{\widehat{e}}_{n}^{ * }}\right\}, n \in \mathbb{N} \), where \[ {\widehat{e}}_{n} = {\left\lbrack {\mathbb{H}}_{\mu }\right\r... | Proof Set \( {\mathbb{S}}_{m} \mathrel{\text{:=}} \operatorname{Sp}\left\{ {{\mathbb{H}}^{k - 1}{e}_{1} : 1 \leq k \leq m}\right\}, m = 1,2,\cdots \). By (2.13), we see that \( {e}_{1},{e}_{2},\cdots ,{e}_{m} \in {\mathbb{S}}_{m}, m = 1,2,\cdots \), and therefore \( {\left\{ {e}_{k}\right\} }_{k = 1}^{m} \) is an ortho... | Yes |
Theorem 2.1 Let \( {P}_{n}^{\mu ,\lambda } \) be the sequence of polynomials satisfying the following three-term recurrence associated to Gribov operator\n\n\[ \left\\{ \begin{array}{l} {P}_{0}^{\mu ,\lambda }\left( z\right) = 0, \\ {P}_{1}^{\mu ,\lambda }\left( z\right) = 1, \\ {\alpha }_{n - 1}{P}_{n - 1}^{\mu ,\lamb... | Proof Let the sequence of polynomials be defined recursively by\n\n\[ {\alpha }_{n - 1}{P}_{n - 1}^{\mu ,\lambda }\left( z\right) + {\beta }_{n}{P}_{n}^{\mu ,\lambda }\left( z\right) + {\alpha }_{n}{P}_{n + 1}^{\mu ,\lambda }\left( z\right) = z{P}_{n}^{\mu ,\lambda }\left( z\right) ,\;n \geq 1, \]\n\n(2.16)\n\nwhere \(... | Yes |
Lemma 3.2 \( \forall \varepsilon > 0,\exists {C}_{\varepsilon } > 0 \) such that\n\n\[ \forall \phi \in D\left( {a}^{4}\right) ,\;\left| \left\langle {{\mathbb{H}}_{\lambda }\phi ,\phi }\right\rangle \right| \leq \varepsilon {\begin{Vmatrix}{a}^{j}\phi \end{Vmatrix}}^{2} + {C}_{\varepsilon }\parallel \phi {\parallel }^... | Proof Let \( \phi \left( z\right) = \mathop{\sum }\limits_{{k = 1}}^{\infty }{a}_{k}{e}_{k}\left( z\right) \) . Then\n\ni) \( {a}^{ * }{a}^{2}\phi \left( z\right) = \mathop{\sum }\limits_{{k = 0}}^{\infty }k\sqrt{k + 1}{a}_{k + 1}{e}_{k}\left( z\right) . \n\nIf we put \( {u}_{k} = k\sqrt{\left( k + 1\right) } \) then w... | Yes |
Corollary 3.1 \( \\forall \\varepsilon > 0,\\exists {C}_{\\varepsilon } > 0 \) such that\n\n\[ \n\\begin{Vmatrix}{{\\mathbb{H}}_{\\lambda }\\phi }\\end{Vmatrix} \\leq \\varepsilon \\begin{Vmatrix}{{\\mathbb{S}}_{{\\lambda }^{\\prime }}\\phi }\\end{Vmatrix} + {C}_{\\varepsilon }\\parallel \\phi \\parallel \\text{ for ev... | Proof Let \( \\phi \\in D\\left( {\\mathbb{S}}_{{\\lambda }^{\\prime }}\\right) \), then for \( \\phi \\left( z\\right) = \\mathop{\\sum }\\limits_{{k = 1}}^{\\infty }{a}_{k}{e}_{k}\\left( z\\right) \) we have\n\n\[ \n\\mathop{\\sum }\\limits_{{k = 1}}^{\\infty }k{\\left| {a}_{k}\\right| }^{2} < \\infty ,\\;\\mathop{\\... | Yes |
Proposition 3.1 For \( {\lambda }^{\prime } \neq 0 \) and \( \lambda \neq 0 \), we have\n\ni)\n\[{\mathbb{S}}_{{\lambda }^{\prime }}^{-1} \in {\mathfrak{C}}_{p},\;\forall p > \frac{1}{2}\]\n\n(3.8)\n\nii) \( {\mathbb{H}}_{\lambda } \) is \( {\mathbb{S}}_{{\lambda }^{\prime }} \) -compact, i.e.,\n\[{\mathbb{H}}_{\lambda... | Proof \( i) \) As \( {\mathbb{S}}_{{\lambda }^{\prime }} \) is self-adjoint and their eigenvalues are given by \( {\alpha }_{k} = {\lambda }^{\prime }\left( {k - 1}\right) k + {\mu k} \sim {k}^{2}, \) then \( {\mathbb{S}}_{{\lambda }^{\prime }}^{-1} \in {\mathfrak{C}}_{p},\forall p > \frac{1}{2} \).\n\nii) It is suffic... | Yes |
Lemma 3.3 \( \left\lbrack {{20},\text{Lemmas 1-2 or Lemma 1.3(i)}}\right\rbrack \; \) In \( {\mathbb{B}}_{0},\psi \in {\mathbb{B}}_{0} \Leftrightarrow z \rightarrow \frac{{\psi }^{\prime }\left( z\right) - {\psi }^{\prime }\left( 0\right) }{z} \in {\mathbb{B}}_{0} \) . | We deduce that \( {\mathbb{H}}_{{\lambda }^{\prime }}^{-1} \) exists. | No |
Lemma 4.1 Let \( \mathbb{V} \in {\mathfrak{C}}_{2p} \) be a Volterra operator. Then due to [12, Corollary 6.9.4], we get\n\n\[ \begin{Vmatrix}{\mathbb{V}}^{j}\end{Vmatrix} \leq \frac{\parallel V{\parallel }_{2p}^{j}}{\sqrt{\left\lbrack \frac{j}{p}\right\rbrack !}} \] \n\n(4.4) \n\nand \( \left\lbrack \begin{array}{l} i... | Inequality (4.4) can be written as\n\n\[ \begin{Vmatrix}{\mathbb{V}}^{{kp} + m}\end{Vmatrix} \leq \frac{\parallel V{\parallel }_{2p}^{{kp} + m}}{\sqrt{k!}}\;\left( {k = 0,1,2,\cdots ;m = 0,1,\cdots, p - 1}\right) . \] \n\n(4.5) | Yes |
Lemma 4.2 Let \( {\mathbb{T}}_{n} \) be a linear operator acting in an Euclidean space \( {\mathbb{C}}^{n} \) and \( \operatorname{dist}\left( {\sigma ,\sigma \left( {\mathbb{T}}_{n}\right) }\right) \) \( = \mathop{\inf }\limits_{{s \in \sigma \left( {\mathbb{T}}_{n}\right) }}\left| {\sigma - s}\right| \) be the distan... | \[ \begin{Vmatrix}{\left( {\mathbb{T}}_{n} - \sigma I\right) }^{-1}\end{Vmatrix} \leq \mathop{\sum }\limits_{{k = 0}}^{{n - 1}}\frac{{\begin{Vmatrix}2{\mathbb{T}}_{n}\end{Vmatrix}}_{2p}^{k}}{{\left\lbrack \operatorname{dist}\left( \sigma ,\sigma \left( {\mathbb{T}}_{n}\right) \right) \right\rbrack }^{k + 1}\sqrt{\left\... | Yes |
Theorem 4.3 For any \( {\sigma }_{{\lambda }^{\prime }} \in \sigma \left( {\mathbb{H}}_{{\lambda }^{\prime }}\right) \), there is a \( {\sigma }_{n}^{{\lambda }^{\prime }} \in \sigma \left( {\mathbb{H}}_{n}^{{\lambda }^{\prime }}\right) \) such that\n\n\[ \n\\left| {\\frac{1}{{\\sigma }_{{\\lambda }^{\\prime }}} - \\fr... | To prove this theorem, it suffices to apply the above Gil’s theorem by taking \( p = 1 \) . However, to be self-contained in this paper, we adapt below in Appendix the essential ingredients of the Gil’s method under lemmas, form at \( {\\mathbb{H}}_{{\\lambda }^{\\prime }} \) . | No |
Lemma 5.1 Let \( {\mathbb{H}}_{n}^{{\lambda }^{\prime }} = {\mathbb{S}}_{{\lambda }^{\prime }} + {\mathbb{H}}_{n}^{\lambda } \). Then for any \( n = 1,2,\cdots \), we have\ni)\n\[{\mathbb{H}}_{n}^{{\lambda }^{\prime }} = {\mathbb{S}}_{n}^{{\lambda }^{\prime }} + {\mathbb{H}}_{n}^{\lambda } + {\widehat{\mathbb{S}}}_{n}^... | Proof i) As \( {\mathbb{S}}_{n}^{{\lambda }^{\prime }} = {\mathbb{P}}_{n}{\mathbb{S}}_{{\lambda }^{\prime }} \) and \( {\widehat{\mathbb{S}}}_{n}^{{\lambda }^{\prime }} = {\mathbb{Q}}_{n}{\mathbb{S}}_{{\lambda }^{\prime }} \) we have \( {\mathbb{H}}_{n}^{{\lambda }^{\prime }} = {\mathbb{S}}_{n}^{{\lambda }^{\prime }} +... | Yes |
Lemma 2.4 With reference to Definitions 1.1 and 2.1, let \( x \in {\mathcal{L}}_{i - 1}\left( {m, k;n + l, n}\right) \) and \( y \in {\mathcal{L}}_{i}\left( {m, k;n + l, n}\right) \) . Then the following (i)-(iii) hold.\n\n(i) If \( x \prec y \), then\n\n\[ \n{\left( {L}_{i}{L}_{i + 1}{R}_{i}\right) }_{xy} = \left\{ {\... | Proof For a fixed \( x \in {\mathcal{L}}_{i - 1}\left( {m, k;n + l, n}\right) \) and a fixed \( y \in {\mathcal{L}}_{i}\left( {m, k;n + l, n}\right) \), we first have\n\n\[ \n{\left( {L}_{i}{L}_{i + 1}{R}_{i}\right) }_{xy} = \mathop{\sum }\limits_{{z, w \in \mathcal{L}\left( {m, k;n + l, n}\right) }}{\left( {L}_{i}\rig... | Yes |
Lemma 2.5 With reference to Definition 2.1, if \( x \in {\mathcal{L}}_{i - 1}\left( {m, k;n + l, n}\right), y \in {\mathcal{L}}_{i}(m, k;n + l, \) \( n) \) for \( 2 \leq i \leq m \), and \( x \nprec y \), then \( x \cap y \prec x \) if and only if \( y \prec x + y \) . | Proof Routine. | No |
Lemma 3.1 Let \( E \) and \( F \) be the matrices in (2.1) and (2.2), respectively. Let\n\n\[ E\left( {r, p}\right) \mathrel{\text{:=}} {\left( {e}_{ij}\right) }_{r + 1 \leq i, j \leq p}\;\text{ and }\;F\left( {r, p}\right) \mathrel{\text{:=}} {\left( {f}_{r + 1},{f}_{r + 2},\cdots ,{f}_{p}\right) }^{\mathrm{T}}. \]\n\... | Proof Obviously, \( \det E\left( {r, p}\right) \neq 0 \) by (2.3) and (2.4). By calculation, the result holds. | Yes |
Lemma 1.7 Let \( G \) be a unicyclic graph with \( n \geq 8 \) vertices and \( n \leq s\left( G\right) \leq n + 1 \) . Then \( \operatorname{QEE}\left( G\right) \leq \operatorname{QEE}\left( {S}_{n}^{3}\right) \) with equality if and only if \( G \cong {S}_{n}^{3}. \) | Proof Suppose that \( {uv} \) is an edge with \( {d}_{G}\left( u\right) + {d}_{G}\left( v\right) = n \) . If \( u \) and \( v \) have no common neighbors, then \( G \) is viewed as the connected graph obtained from a double star \( S\left( {a, b}\right) \left( {a + b = n}\right) \) and adding one edge, where a double s... | Yes |
Theorem 2.1 Let \( F\left( {x, y}\right) = \left( {{x}^{3} + {y}^{3},{\left( x + y\right) }^{3} + {y}^{3}}\right) : \mathrm{{GF}}{\left( {2}^{m}\right) }^{2} \rightarrow \mathrm{{GF}}{\left( {2}^{m}\right) }^{2} \), where \( m \) is odd. Then \( F\left( {x, y}\right) \) is a differentially 4-uniform permutation with th... | The proof of Theorem 2.1 is completed by Lemmas 2.1–2.3 as follows. | Yes |
Lemma 2.1 Let \( m \) be odd, and \( F\left( {x, y}\right) = \left( {{x}^{3} + {y}^{3},{\left( x + y\right) }^{3} + {y}^{3}}\right) : \mathrm{{GF}}{\left( {2}^{m}\right) }^{2} \rightarrow \mathrm{{GF}}{\left( {2}^{m}\right) }^{2} \). Then \( F\left( {x, y}\right) \) is a permutation over \( \operatorname{GF}{\left( {2}... | Proof We only need to prove that for any \( c, d \in \mathrm{{GF}}\left( {2}^{m}\right) \), there exists only one pair \( \left( {x, y}\right) \in \mathrm{{GF}}{\left( {2}^{m}\right) }^{2} \) such that\n\n\[ \left\{ \begin{array}{l} {f}_{1}\left( {x, y, c}\right) \mathrel{\text{:=}} {x}^{3} + {y}^{3} + c = 0 \\ {f}_{2}... | Yes |
Theorem 2.1 The graph \( \Gamma \left( {s, m;n, q}\right) \) is a vertex-transitive regular graph. | Proof For any point \( \left( {X, Y}\right) \) in \( \Gamma \left( {s, m;n, q}\right) \), it is easy to verify that the number of out-neighbors and in-neighbors of it is\n\n\[ N\left( {s + 1, m + 1;n + 1}\right) {N}^{\prime }\left( {s + 1, m + 1;n + 1}\right) .\n\]\n\nThus, \( \Gamma \left( {s, m;n, q}\right) \) is reg... | Yes |
Theorem 2.2 Let \( 0 < s < m < n \) . Then the graph \( \Gamma \left( {s, m;n, q, K}\right) \) is a DSRG. | Proof Since \( {\operatorname{PGL}}_{n + 1}\left( {\mathbb{F}}_{q}\right) \) acts transitively on \( \left( {s - 1}\right) \) -flats, without loss of generality, we can assume that\n\n\[ K = \left( \begin{matrix} s & n + 1 - s \\ {I}^{\left( s\right) } & 0 \end{matrix}\right) s. \]\n\nLet\n\n\[ X = \left\{ {{x}_{\gamma... | Yes |
Theorem 3.1 Let \( 1 \leq m < n \) . Then \( d\left( {m - 1, m;n, q}\right) \leq d\left( {0,1;n - m + 1, q}\right) \) . | Proof Let \( V \) be a given \( \left( {m - 2}\right) \) -flat. Suppose \( {U}_{1},{U}_{2},\cdots ,{U}_{t} \) to be the \( \left( {m - 1}\right) \) -flats containing \( V \), and \( {W}_{1},{W}_{2},\cdots ,{W}_{r} \) to be the \( m \) -flats containing \( V \), where\n\n\[ t = \frac{{q}^{n - m + 2} - 1}{q - 1},\;r = \f... | Yes |
Corollary 3.1 Let \( n \) be an even positive integer and \( q \) be an odd prime power. Then\n\n(i) \( C\left( {0,1;n, q}\right) \) is a \( \left\lbrack {\bar{v},1,\bar{v}}\right\rbrack \) code, where \( \bar{v} = \frac{{q}^{n + 1} - 1}{q - 1} \) . | Proof (i) From Remark 1.1 (ii), we know that \( \mathcal{T}\left( {0,1;n, q}\right) \) is a 2-design with parameters\n\n\[ \n\bar{v} = \frac{{q}^{n + 1} - 1}{q - 1},\;b = \frac{\left( {{q}^{n} - 1}\right) \left( {{q}^{n + 1} - 1}\right) }{\left( {q - 1}\right) \left( {{q}^{2} - 1}\right) }, \n\]\n\n\[ \nk = q + 1,\;r =... | Yes |
Corollary 3.2 The dimension of \( C\left( {0,1;n,2}\right) \) is \( n + 1 \) . | Proof As is shown above, \( \mathcal{T}\left( {0,1;n,2}\right) \) is a 2-design with \( {v}^{\prime } = {2}^{n + 1} - 1 \) . Since the projective dimension of \( \mathcal{T}\left( {0,1;n,2}\right) \) is \( n \), the 2-rank of \( H\left( {0,1;n,2}\right) \) is \( {v}^{\prime } - n - 1 \), then the dimension of \( C\left... | Yes |
Proposition 1.1 For \( q > 1, m \geq 1 \) and \( 2 \leq \delta \leq T + 1 \), where\n\n\[ T = \left\{ \begin{array}{ll} {q}^{\frac{m + 1}{2}} - 1, & m \equiv 1\left( {\;\operatorname{mod}\;2}\right) \\ 2{q}^{\frac{m}{2}} - 1, & \text{ otherwise. } \end{array}\right. \]\n\nThen we have\n\n\[ {m}_{s} = m, s = 1,2,\cdots ... | Proof From Lemma 1.1, the equality (1.1) is obvious.\n\nNow for any \( x, y = 1,2,\cdots ,\delta - 1 \) with \( x \neq y \) . Suppose that \( {C}_{x} \cap {C}_{y} \neq \varnothing \), then there exist some \( i, j = 0,1,\cdots, m - 1 \) such that \( x{q}^{i} = y{q}^{j} \) . From \( i = j \) we have \( x = y \), which i... | Yes |
Theorem 1.3 Let \( q \) be a power of the prime \( p, m \) be even and \( \delta \) be a positive integer with \( {q}^{\frac{m}{2}} + 1 \leq \delta - 1 \leq 2{q}^{\frac{m}{2}} - 1 \) . Suppose that \( \mathcal{C} \) is a \( q \) -ary BCH code designed distance \( \delta \geq 2 \) and \( A = \{ t \mid 2 \leq t \leq \del... | Proof of Theorem 1.1 From \( q > \delta - 1 \geq 1, m \geq 1 \) and Proposition 1.1 we know that for any \( s = 1,2,\cdots \delta - 1,{m}_{s} = m \) and \( {C}_{s} \) are disjoint to each others. Set \( X = \{ 0,1,2,\cdots, n - \) \( 1\}, N = X - \mathop{\bigcup }\limits_{{s = 1}}^{{\delta - 1}}{C}_{s} \) and \( {X}_{r... | Yes |
Theorem 1.1 Let \( \mathcal{R} \) be a ring containing a nontrivial idempotent \( P \) and satisfying the following condition:\n\n\[ \text{(Q) If}{A}_{11}{B}_{12} = {B}_{12}{A}_{22}\text{for all}{B}_{12} \in {\mathcal{R}}_{12}\text{, then}{A}_{11} + {A}_{22} \in \mathcal{Z}\left( \mathcal{R}\right) \text{.}\]\n\nLet \(... | Proof For the convenience of citation and clarity of exposition, we shall organize the proof\n\n- in a series of claims.\n\nClaim 1: \( \Phi \left( 0\right) = 0 \) .\n\nIndeed, \( \Phi \left( 0\right) = \Phi \left( \left\lbrack {0,0}\right\rbrack \right) = \left\lbrack {\Phi \left( 0\right) ,\Phi \left( 0\right) }\righ... | Yes |
Lemma 2.1 Let \( R \) be a prime ring with a nontrivial idempotent \( P \) and \( \mathcal{Z}\left( \mathcal{R}\right) \) be its center. If \( {A}_{11}{B}_{12} = {B}_{12}{A}_{22} \) for all \( {B}_{12} \in {\mathcal{R}}_{12} \), then \( {A}_{11} + {A}_{22} \in \mathcal{Z}\left( \mathcal{R}\right) \) . | Proof For any \( {X}_{11} \in {\mathcal{R}}_{11} \) and \( {X}_{12} \in {\mathcal{R}}_{12} \), we have\n\n\[ \n{A}_{11}{X}_{11}{X}_{12} = {X}_{11}{X}_{12}{A}_{22} = {X}_{11}{A}_{11}{X}_{12} \n\]\n\nfor all \( {X}_{12} \in {\mathcal{R}}_{12} \) . As \( \mathcal{R} \) is prime, we get \( {A}_{11}{X}_{11} = {X}_{11}{A}_{1... | Yes |
Lemma 2.2 Let \( \mathcal{N} \) be a nest on a complex Hilbert space \( \mathcal{H} \), and \( \operatorname{Alg}\mathcal{N} \) be the associated nest algebra. Let \( P \in \mathcal{N} \smallsetminus \{ 0, I\}, T \in \operatorname{Alg}\mathcal{N} \) . If \( {PTPC} = C\left( {I - P}\right) T\left( {I - P}\right) \) hold... | Proof Fix unit vectors \( {x}_{0} \in P\mathcal{H} \) and \( {y}_{0} \in \left( {I - P}\right) \mathcal{H} \), then\n\n\[ \n{PT}\left( {x \otimes {y}_{0}}\right) = \left( {x \otimes {y}_{0}}\right) T\left( {I - P}\right) ,\;\forall x \in P\mathcal{H}, \n\]\n\n\[ \n{PT}\left( {{x}_{0} \otimes y}\right) = \left( {{x}_{0}... | Yes |
Lemma 2.3 (1) Let \( \mathcal{M} \) be a von Neumann algebra. Let \( P \in \mathcal{M} \) be a projection with \( \bar{P} = I \) and \( A \in B\left( \mathcal{H}\right) \) . If \( {AMP} = 0 \) for all \( M \in \mathcal{M} \), then \( A = 0 \) . Consequently, if \( Z \in {\mathcal{Z}}_{\mathcal{M}} \), then \( {ZP} = 0 ... | Proof (1) It follows from \( \bar{P} = I \) that the linear span of \( \{ {MP}\left( x\right) : M \in \mathcal{M}, x \in \mathcal{H}\} \) is dense in \( \mathcal{H} \) . So \( {AMP} = 0 \) for all \( M \in \mathcal{M} \) implies \( A = 0 \) . If \( Z \in {\mathcal{Z}}_{\mathcal{M}} \) and \( {ZP} = 0 \), then \( {ZMP} ... | Yes |
Lemma 2.4 Let \( \mathcal{M} \) be a von Neumann algebra with no central abelian projections, and \( {A}_{ii} \in {\mathcal{M}}_{ii},{B}_{jj} \in {\mathcal{M}}_{jj},1 \leq i \neq j \leq 2 \) . If \( {A}_{ii}{C}_{ij} = {C}_{ij}{B}_{jj} \) for all \( {C}_{ij} \in {\mathcal{M}}_{ij} \), then \( {A}_{ii} + {B}_{jj} \in {\m... | Proof For any \( {D}_{ii} \in {\mathcal{M}}_{ii} \), we obtain \( {A}_{ii}{D}_{ii}{C}_{ij} = {D}_{ii}{C}_{ij}{B}_{jj} = {D}_{ii}{A}_{ii}{C}_{ij} \), that is, \( \left( {{A}_{ii}{D}_{ii} - {D}_{ii}{A}_{ii}}\right) {P}_{i}C{P}_{j} = 0 \) for all \( C \in \mathcal{M} \) . By Lemma 2.3(1), we get \( {A}_{ii}{D}_{ii} = {D}_... | Yes |
Lemma 1.2 Let \( G \) be a non-nilpotent. Then all non-normal subgroups are conjugate in \( G \) if and only if \( G = \langle u, v\;|\;{u}^{p} = 1,\;{v}^{{q}^{m}} = 1,\;{v}^{ - 1}{uv} = {u}^{k},\;p \equiv 1\;( \) mod \( \;q),\;k ≢ 1\;( \) mod \( \;p),\;{k}^{q} \equiv 1 \) \( \left( {\;\operatorname{mod}\;p}\right) \ra... | Proof Let \( Q \) be a non-normal Sylow subgroup of \( G \) . Then \( Q \) is cyclic since all of the nonnormal subgroups of \( G \) are conjugate. We now assert \( \pi \left( G\right) = \{ p, q\} \) . Otherwise, take \( p, r \in \pi \left( G\right) \) such that \( r, p, q \) are distinct. Let \( P \) and \( R \) be \(... | Yes |
Lemma 1.3 Let \( \langle v\rangle \) be a non-normal subgroup of a finite \( p \) -group. If all non-normal subgroups of \( G \) are conjugate, then \( \langle g\rangle \trianglelefteq G \) and \( \left\langle {g,{v}^{p}}\right\rangle \) is a Dedekind group for any \( g \in \) \( G \smallsetminus {N}_{G}\left( {\langle... | Proof Since \( G \) is a \( p \) -group and \( \langle v\rangle \ntrianglelefteq G \), we have \( \langle v\rangle \lneqq {N}_{G}\left( {\langle v\rangle }\right) \) . Hence, \( {N}_{G}\left( {\langle v\rangle }\right) \) is normal in \( G \) for all non-normal subgroups are conjugate to \( \langle v\rangle \) . Conseq... | Yes |
Lemma 1.4 Assume that a finite 2-group \( G \) is a Dedekind group generated by two elements. Then \( G \) is abelian or a quaternion group. | Proof If \( G \) is not an abelian group, then \( G = {Q}_{8} \times E \) by Lemma 1.1. Suppose that \( Q = \left\langle {a, b\left| {\;{a}^{4} = 1,{b}^{2} = {a}^{2},{b}^{-1}{ab} = {a}^{-1}}\right. }\right\rangle \) and \( \left| E\right| = {2}^{n} \) . It follows by calculation that \( {G}^{\prime } = \left\langle {a}... | Yes |
Lemma 1.6 A finite 2-group \( G \) has its all non-normal subgroups being conjugate if and only if \( G = \left\langle {u, v \mid {u}^{{2}^{n}} = 1,{v}^{2} = 1,{v}^{-1}{uv} = {u}^{1 + {2}^{n - 1}}, n \geq 3}\right\rangle \) . | Proof Let \( H \) be a non-normal subgroup of \( G \) . If \( G \) has exactly one element of order 2, then \( G \) is a generalized quaternion group with \( \left| G\right| > 8 \) by Lemma 1.1. But the non-normal subgroups of a generalized quaternion group are not conjugate, a contradiction. Hence, \( G \) has at leas... | No |
Theorem 2.1 Let \( \lambda > 0 \) . Assume that \( f \) satisfies \( \left( {\mathrm{f}}_{1}\right) - \left( {\mathrm{f}}_{3}\right) \) with \( b \) satisfying \( \left( {\mathrm{b}}_{2}\right) \) . Then there exists a function \( v \in {C}^{2}\left( {\mathbb{R}}^{N}\right) \) satisfying the problem (2.1), if one of th... | Proof (i) Defining\n\n\[ {\bar{g}}_{1}\left( t\right) \mathrel{\text{:=}} {\int }_{0}^{t}\frac{s}{{sG}\left( s\right) + 1}\mathrm{\;d}s,\;t \geq 0 \]\n\nit follows that\n\n\( \left( {\mathrm{i}}_{1}\right) \frac{{\bar{g}}_{1}\left( s\right) }{s} \) is non-decreasing;\n\n(i2) \( \mathop{\lim }\limits_{{s \rightarrow \in... | No |
Lemma 1.1 Suppose that \( \\left( {X, d}\\right) \) is a complete metric space. Let \( {x}_{0} \\in X,\\delta > 0 \) and \( T : \\overline{B\\left( {{x}_{0},\\delta }\\right) } \\rightarrow X \) be the mapping satisfying the conditions (a),(b) and (c) of Proposition 1.1 with the associated functions \( {\\alpha }_{i}, ... | Proof According to Proposition 1.1, we only need to show that the closed ball \( \\overline{B\\left( {{x}_{0},\\delta }\\right) } \) is invariant under \( T \) . For any \( x \\in \\overline{B\\left( {{x}_{0},\\delta }\\right) } \), using the condition (c) of Proposition 1.1 with \( {\\alpha }_{i} = {\\alpha }_{i}\\lef... | Yes |
Theorem 1.1 Let \( \\left( {X, d}\\right) \) be a complete metric space and \( U \) be an open subset of \( X \) . Suppose that \( H : \\widetilde{U} \\times \\left\\lbrack {0,1}\\right\\rbrack \\rightarrow X \) satisfies the following conditions:\n\n(i) \( x \\neq H\\left( {x,\\lambda }\\right) \) for all \( x \\in \\... | Proof We follow the ideas from the proof of [1, Theorem 3.1]. Consider the set\n\n\\[ \nA = \\{ \\lambda \\in \\left\\lbrack {0,1}\\right\\rbrack : x = H\\left( {x,\\lambda }\\right) \\text{ for some }x \\in U\\} .\n\\]\n\nSince \( H\\left( {\\cdot ,0}\\right) \) has a fixed point in \( U \), it means that \( 0 \\in A ... | Yes |
Lemma 2.1 Let \( \left( {X, d}\right) \) be a complete metric space and \( T \) be the generalized expansive mapping given in Definition 2.1. Suppose that \( {p}_{1} = \inf \left\{ {{\beta }_{1}\left( {x, y}\right) : x, y \in X}\right\} > 0 \) and \( {p}_{3} = \inf \left\{ {{\beta }_{3}\left( {x, y}\right) : x, y \in X... | Proof Since \( x \) is a fixed point of \( T \), it is easy to see that \( x \in {A}_{x} \). So \( {A}_{x} \) is nonempty. Let \( \widetilde{y} \) be any accumulation point of \( {A}_{x} \). We will prove that \( \widetilde{y} = x \). By the properties of accumulation point, there exists a sequence \( \left\{ {y}_{k}\r... | Yes |
Theorem 0.3 Let \( {M}^{n}\left( {n \geq 2}\right) \) be a compact hypersurface with constant mean curvature in an \( \left( {n + 1}\right) \) -dimensional hyperbolic space \( {\mathbb{H}}^{n + 1} \) . If the second fundamental form is positive definite, then the first nonzero eigenvalue \( {\lambda }_{1} \) of Laplaci... | ## 1 Proof of Main Results\n\nSuppose that \( {M}^{n} \) is a compact hypersurface with constant mean curvature \( H \) in an \( \left( {n + 1}\right) \) - dimensional space form \( {N}^{n + 1}\left( C\right) \) . Let \( g \) be the Riemannian metric on \( {N}^{n + 1}\left( C\right) \) as well as the induced metric on ... | Yes |
For \( n > {2k} \) and any \( \alpha \in \left( {0, k}\right) \), let\n\n\[ u = {\left( \varepsilon \left( 1 + {\left| x\right| }^{2}\right) \right) }^{\frac{k}{{2k} - \alpha }} \]\n\nwith\n\n\[ \varepsilon = \frac{{\left( 2k - \alpha \right) }^{2}}{{2k}\left( {k - \alpha }\right) }{\left( \left( n - k - 1\right) \frac... | ## 1 Proof of Theorem 0.1\n\nAssume that \( u > 0 \) is a solution of (0.5) in \( {\Gamma }^{k} \) . We will deduce some contradiction. In this section we write \( {\sigma }_{s}\left( {A}^{g}\right) \) simply as \( {\sigma }_{s} \).\n\nLet \( \eta \) be a \( {C}^{2} \) cut-off function satisfying\n\n\[ \left\{ \begin{a... | No |
Example 2.1 Apply Theorem 2.3 with \( g = 3, s = 2, t = 3 \), then \( Q = \left( {\frac{4}{5},\frac{1}{5}}\right) ,{6st} + 1 = {37} \) . Let \( \xi = {26}, x = 2,{C}_{0}^{s} = \{ 4,{16},{27},{34},{25},{26},{30},9,{36},{33},{21},{10},3,{12},{11},7,{28},1\} \) , and hence \( \langle - \xi \rangle = \{ 1,{10},{11},{26},{2... | \[ \mathcal{F} = \{ \left( {1,\alpha }\right) \cdot \Omega : \Omega \in \{ {A}_{j},{B}_{j}\}, j = 0,1,\alpha \in {C}_{0}^{s}/\langle - \xi \rangle \} \cup \{ \left( {1,\alpha }\right) \cdot {C}_{1} : \alpha \in {C}_{0}^{s}/\langle - \xi \rangle \} \] \[ = \{ \{ \left( {0,1}\right) ,\left( {1,{26}}\right) ,\left( {2,{10... | Yes |
Lemma 2.3 Suppose that \( t \) is an odd integer such that \( q = {18t} + 1 \) is a prime. If there exist two elements \( {x}_{1},{x}_{2} \) of \( {\mathbb{Z}}_{q}^{ * } \) satisfying one of the following conditions:\n\n(1) \( \left\{ {{x}_{1} - 1,{x}_{1} - \xi ,{x}_{1} - {\xi }^{2},{x}_{2}}\right\} \) are in differenc... | Proof Since \( t \) is odd, then \( {3t} \) is odd, and hence \( - 1 \in {C}_{3}^{6} \) . If one of the conditions mentioned above is satisfied, it is easy to check that \( \left\{ {\pm 1, \pm {x}_{1}, \pm {x}_{1}^{2}}\right\} \) and \( \left\{ {{x}_{1} - 1,{x}_{1} - \xi ,{x}_{1} - {\xi }^{2},1, - {x}_{1},{x}_{2}}\righ... | Yes |
Lemma 2.4 There exist a \( g \) -regular and an optimal CP \( \left( {W,1, Q;{19g}}\right) \) for \( \left( {g, W, Q}\right) = (3,\\{ 3, 7\\} ,\\left( {\\frac{2}{3},\\frac{1}{3}}\\right) ),\\;\\left( {{12},\\{ 3,4,7\\} ,\\left( {\\frac{11}{21},\\;\\frac{9}{21},\\frac{1}{21}}\\right) }\\right) ,\\;\\left( {{15},\\{ 3,4,... | Proof \\( \\begin{aligned} \\text{ For }\\left( {g, W, Q}\\right) & = \\left( {3,\\{ 3,7\\} ,\\left( {\\frac{2}{3},\\frac{1}{3}}\\right) }\\right) , \\\\ \\text{ let }{\\mathcal{F}}_{57} & = \\{ \\{ 0,1,3,7,{21},{33},{49}\\} ,\\{ 0,5,{22}\\} ,\\{ 0,{10}, \\end{aligned} \\) \\( {23}\\} \\} \\), then \\( \\left( {{Z}_{57... | No |
Theorem 2.2 Suppose that \( q \equiv {19}\left( {\;\operatorname{mod}\;{36}}\right) \) is a prime, then there exist a \( g \) -regular and an optimal \( \; \) CP \( \left( {W,1, Q;{gq}}\right) \; \) for \( \;\left( {g, W, Q}\right) \; = \;\left( {3,\{ 3,7\} ,\left( {\frac{2}{3},\frac{1}{3}}\right) }\right) ,\;\left( {{... | Proof Applying Lemma 2.5, one can find an element \( {x}_{1} \in {\mathbb{Z}}_{q}^{ * } \) satisfying \( {x}_{1} \in {C}_{1}^{6} \), and \( \left\{ {{x}_{1} - 1,{x}_{1} - \xi ,{x}_{1}^{2} - {\xi }^{2}}\right\} \) are in difference cosets among \( \left\{ {{C}_{2}^{6},{C}_{3}^{6},{C}_{5}^{6}}\right\} \) if \( q \geq {9.... | Yes |
Lemma 2.6 Suppose that \( t \) is odd such that \( q = {18t} + 1 \) is a prime. If there exist two elements \( {x}_{1},{x}_{2} \) of \( {Z}_{q}^{ * } \) satisfying one of the following conditions:\n\n(1) \( \left\{ {{x}_{1} - 1,{x}_{1} - \xi ,{x}_{1} - {\xi }^{2}}\right\} \) are in difference cosets among \( \left\{ {{... | Proof Since \( t \) is odd, then \( {3t} \) is odd, and hence \( - 1 \in {C}_{3}^{6} \) . If one of the conditions mentioned above is satisfied, we are easy to check that \( \left\{ {\pm 1, \pm {x}_{1}, \pm {x}_{2}}\right\} \) and \( \left\{ {{x}_{1} - 1,{x}_{1} - \xi ,{x}_{1} - {\xi }^{2},1, - {x}_{1},{x}_{2}}\right\}... | Yes |
Theorem 2.3 Suppose that \( q \equiv {19}\left( {\;\operatorname{mod}\;{36}}\right) \) is a prime, then there exist a \( g \) -regular and an optimal CP \( \left( {W,1,\mathbb{Q};{gq}}\right) \) for \( \left( {g, W,\mathbb{Q}}\right) = \left( {{12},\{ 3,4,7\} ,\left( {\frac{9}{20},\frac{10}{20},\frac{1}{20}}\right) }\r... | Proof Applying Lemma 2.5, one can find an element \( {x}_{1} \in {\mathbb{Z}}_{q}^{ * } \) satisfying \( {x}_{1} \in {C}_{1}^{6} \), and \( \left\{ {{x}_{1} - 1,{x}_{1} - \xi ,{x}_{1}^{2} - {\xi }^{2}}\right\} \) are in difference cosets among \( \left\{ {{C}_{1}^{6},{C}_{3}^{6},{C}_{5}^{6}}\right\} \) if \( q \geq {9.... | Yes |
Let \( G = {\mathbb{Z}}_{7},\;B = \{ \{ 0,1,3\} ,\{ 1,2,4\} ,\{ 2,3,5\} ,\{ 3,4,6\} ,\{ 0,4,5\} ,\{ 1,5,6\} , \) \( \{ 0,2,6\} \} \) . It is well known that \( \left( {G, B}\right) \) is a \( \left( {7,3,1}\right) \) -BIBD. Let \( {\mathcal{F}}_{111} = \{ \{ 0,1,3,7,{12},{29},{68}\} \) , \( \{ 0,8,{18},{41},{60},{81},{... | Breaking up \( \{ 0,1,3,7,{12},{29},{68}\} \) by a \( \left( {7,3,1}\right) \) -BIBD, we have 7 blocks with size \( 3,\mathcal{G} = \{ \{ 0,1,7\} ,\{ 1,3,{12}\} ,\{ 3,7,{29}\} ,\{ 7,{12},{68}\} ,\{ 0,{12},{29}\} ,\{ 1,{29},{68}\} ,\{ 0,3,{68}\} \} \) . It is easy to check that \( \left( {{\mathcal{F}}_{111}\smallsetmin... | Yes |
Let \( {\lambda }_{1},{\lambda }_{2},{\lambda }_{3},{\lambda }_{4} \) be positive real numbers. Suppose that \( \frac{{\lambda }_{1}}{{\lambda }_{2}} \) is irrational and algebraic. Let \( \mathcal{V} \) be a well-spaced sequence, \( \delta > 0 \). Then the number \( {E}_{3}\left( {\mathcal{V}, X,\delta }\right) \) of ... | for any \( \varepsilon > 0 \). | No |
Lemma 2.1 For \( 0 < \tau < 1 \) and \( v \in \left\lbrack {\frac{1}{2}X, X}\right\rbrack \), we have\n\n\[{\int }_{\mathbb{R}}H\left( \alpha \right) \mathrm{d}\alpha \gg {\tau }^{2}{X}^{\frac{1}{3} + \frac{1}{k}}\] | Proof By making an interchange in the order of integration and the substitution of variables, we get\n\n\[{\int }_{\mathbb{R}}H\left( \alpha \right) \mathrm{d}\alpha = \frac{1}{12k}{\left( {\lambda }_{1}{\lambda }_{2}\right) }^{-\frac{1}{2}}{\lambda }_{3}^{-\frac{1}{3}}{\lambda }_{4}^{-\frac{1}{k}}\int \cdots \int {\le... | Yes |
Lemma 2.2 Let \( j \) be a positive integer with \( j \geq 2 \) . Then for any positive constant \( \lambda \) and any fixed \( A \geq 6 \), we have\n\n\[ \n{\int }_{\mathbf{M}}{\left| {S}_{j}\left( \alpha ,\lambda \right) - {U}_{j}\left( \alpha ,\lambda \right) \right| }^{2}\mathrm{\;d}\alpha \ll {X}^{\frac{2}{j} - 1}... | Proof It can be deduced from [12, Theorems 3.1-3.2]. | No |
Lemma 4.2 Assume that \( \lambda \) and \( \mu \) are positive constants. Then we have\n\n\[ \n{\int }_{\mathbb{R}}{\left| {S}_{2}\left( \alpha ,\lambda \right) \right| }^{2}{\left| {S}_{3}\left( \alpha ,\mu \right) \right| }^{4}{K}_{\tau }\left( \alpha \right) \mathrm{d}\alpha \ll \tau {X}^{\frac{4}{3} + \varepsilon }... | Proof It can be easily deduced by considering the number of solutions of the underlying Diophantine inequality. See \( \left\lbrack {1\text{, Section 4}}\right\rbrack \) for details. | No |
Lemma 4.3 Let \( \psi \left( k\right) = \frac{1}{2}k\left( {k + 1}\right) \) with \( k \geq 5 \) . Then for any positive constant \( \lambda \), we have\n\n\[{\int }_{\mathbb{R}}{\left| S\left( \alpha ,\lambda \right) \right| }^{{2\psi }\left( k\right) }{K}_{\tau }\left( \alpha \right) \mathrm{d}\alpha \ll \tau {X}^{\f... | Proof It follows from \( \left\lbrack {2\text{, Theorem 10}}\right\rbrack \) that\n\n\[{\int }_{0}^{1}{\left| \mathop{\sum }\limits_{{{\eta X} \leq \lambda {x}^{k} \leq X}}e\left( \alpha {x}^{k}\right) \right| }^{{2\psi }\left( k\right) }\mathrm{d}\alpha \ll {X}^{\frac{{2\psi }\left( k\right) }{k} - 1 + \varepsilon }.\... | Yes |
Lemma 1.1 Let \( R \) be a commutative local ring. Then \( A \in {M}_{2}\left( R\right) \) is an idempotent if and only if either \( A = 0 \), or \( A = {I}_{2} \), or \( A = \left( \begin{matrix} a & b \\ c & 1 - a \end{matrix}\right) \) where \( {bc} = a - {a}^{2} \) in \( R \) . | Proof One direction is clear. Let \( A \in {M}_{2}\left( R\right) \) be an idempotent. Then \( \det A\left( {1 - \det A}\right) = \) 0. As \( R \) is local, either \( \det A \in U\left( R\right) \) or \( 1 - \det A \in U\left( R\right) \) . Thus, \( \det A = 1 \) or \( \det A = 0 \) . If \( \det A = 1 \), then \( A = {... | Yes |
Lemma 1.2 Let \( R \) be a commutative local ring, and let \( A \in {M}_{2}\left( R\right) \) . If \( A \) is strongly \( J \) -clean in \( {M}_{2}\left( R\right) \), then \( A \in J\left( {{M}_{2}\left( R\right) }\right) \) or \( {I}_{2} - A \in J\left( {{M}_{2}\left( R\right) }\right) \) or there exists \( q \in R \)... | Proof Suppose that \( A,{I}_{2} - A \notin J\left( {{M}_{2}\left( R\right) }\right) \) . In view of Lemma 1.1, there exist \( a, b, c \in R \) such that \( A = E + \left( {A - E}\right) \), where \( E = {E}^{2} = \left( \begin{matrix} a & b \\ c & 1 - a \end{matrix}\right) ,{bc} = a - {a}^{2},{EA} = {AE}, A - E \in J\l... | Yes |
Corollary 1.1 Let \( {\mathbb{Z}}_{\left( 2\right) } = \left\{ {\left. {\frac{m}{n} \mid m, n \in \mathbb{Z},2}\right| \;n}\right\} \), and let \( A \in {M}_{2}\left( {\mathbb{Z}}_{\left( 2\right) }\right) \) . Then \( A \) is strongly \( J \) -clean in \( {M}_{2}\left( {\mathbb{Z}}_{\left( 2\right) }\right) \) if and ... | Proof It is easy to verify that \( {\mathbb{Z}}_{\left( 2\right) } \) is a domain and \( J\left( {\mathbb{Z}}_{\left( 2\right) }\right) = 2{\mathbb{Z}}_{\left( 2\right) } \neq 0 \) . In addition, \( {\mathbb{Z}}_{\left( 2\right) }/J\left( {\mathbb{Z}}_{\left( 2\right) }\right) \cong {\mathbb{Z}}_{2} \) . In light of Th... | No |
Corollary 1.2 Let \( \widehat{{\mathbb{Z}}_{2}} \) be the ring of 2-adic integers, and let \( A \in {M}_{2}\left( \widehat{{\mathbb{Z}}_{2}}\right) \) . Then \( A \) is strongly \( J \) -clean in \( {M}_{2}\left( \widehat{{\mathbb{Z}}_{2}}\right) \) if and only if \( A \in J\left( {{M}_{2}\left( \widehat{{\mathbb{Z}}_{... | Proof Obviously, \( \widehat{{\mathbb{Z}}_{2}} \) is a commutative local domain, \( \widehat{{\mathbb{Z}}_{2}}/J\left( \widehat{{\mathbb{Z}}_{2}}\right) \cong {\mathbb{Z}}_{2} \) and \( J\left( \widehat{{\mathbb{Z}}_{2}}\right) = 2\widehat{{\mathbb{Z}}_{2}} \neq 0 \) . The result follows from Theorem 1.1. | Yes |
Proposition 1.1 Let \( R \) be a commutative local domain, and let \( A \in {M}_{2}\left( R\right) \) . If \( R/J\left( R\right) \cong \) \( {\mathbb{Z}}_{2} \) and \( \operatorname{char}\left( R\right) \neq 2 \), then the following are equivalent:\n\n(1) \( A \in {M}_{2}\left( R\right) \) is strongly clean.\n\n(2) \( ... | Proof (1) \( \Rightarrow \) (2) Similar to the proof of Lemma 1.2, we get \( A \in {\mathrm{{GL}}}_{2}\left( R\right) \) or \( {I}_{2} - A \in \) \( {\mathrm{{GL}}}_{2}\left( R\right) \), or \( {s}_{A}^{2}{t}_{A} = {\left( 1 - 2a\right) }^{2} \) . As \( 2 \in J\left( R\right) \), we see that \( 1 - {2a} \in R \) is an ... | Yes |
Lemma 2.1 Let \( R \) be a commutative local ring, and let \( A \in {M}_{2}\left( R\right) \) be strongly \( J \) -clean. Then \( A \in J\left( {{M}_{2}\left( R\right) }\right) \) or \( {I}_{2} - A \in J\left( {{M}_{2}\left( R\right) }\right) \) or \( A \) is similar to a matrix \( \left( \begin{matrix} 0 & - \det A \\... | Proof Let \( A \in {M}_{2}\left( R\right) \) be strongly \( J \) -clean. Write \( A = E + W,{EW} = {WE}, E = {E}^{2} \) and \( W \in J\left( {{M}_{2}\left( R\right) }\right) \) . Suppose that \( A,{I}_{2} - A \notin J\left( {{M}_{2}\left( R\right) }\right) \) . Clearly, \( E \neq 0,{I}_{2} \) . In view of Lemma \( {1.1... | Yes |
Lemma 2.2 Let \( R \) be a commutative local ring, and let \( A = \left( {a}_{ij}\right) \in {M}_{2}\left( R\right) \). If \( A \in {M}_{2}\left( R\right) \) is strongly \( J \) -clean, then \( A \in J\left( {{M}_{2}\left( R\right) }\right) \), or \( {I}_{2} - A \in J\left( {{M}_{2}\left( R\right) }\right) \), or \( \d... | Proof Suppose that \( A \in {M}_{2}\left( R\right) \) is strongly \( J \) -clean and \( A,{I}_{2} - A \notin J\left( {{M}_{2}\left( R\right) }\right) \). By virtue of Lemma 2.1, \( A \) is similar to the matrix \( B \mathrel{\text{:=}} \left( \begin{matrix} 0 & - \det A \\ 1 & \operatorname{tr}A \end{matrix}\right) \),... | Yes |
Theorem 2.1 Let \( R \) be a commutative local ring, \( \frac{1}{2} \in R \), and let \( A = \left( {a}_{ij}\right) \in {M}_{2}\left( R\right) \) . Then \( A \in {M}_{2}\left( R\right) \) is strongly \( J \) -clean if and only if\n\n(1) \( A \in J\left( {{M}_{2}\left( R\right) }\right) \), or\n\n(2) \( {I}_{2} - A \in ... | Proof One direction is obvious by Lemma 2.2. Conversely, if \( A \in J\left( {{M}_{2}\left( R\right) }\right) \), or \( {I}_{2} - A \in \) \( J\left( {{M}_{2}\left( R\right) }\right) \), then \( A \in {M}_{2}\left( R\right) \) is strongly \( J \) -clean. Now assume that \( {\left( \operatorname{tr}A\right) }^{2} - 4\de... | Yes |
Corollary 2.1 Let \( R \) be a commutative local ring, \( \frac{1}{2} \in R \), and let \( A = \left( {a}_{ij}\right) \in {M}_{2}\left( R\right) \). If \( {a}_{11} \in J\left( R\right) \), then \( A \in {M}_{2}\left( R\right) \) is strongly \( J \) -clean if and only if\n\n(1) \( A \in J\left( {{M}_{2}\left( R\right) }... | Proof Suppose that \( A \in {M}_{2}\left( R\right) \) is strongly \( J \) -clean. Write \( A = E + W,{EW} = {WE} \), \( E = {E}^{2} \) and \( W \in J\left( {{M}_{2}\left( R\right) }\right) \). Suppose that \( A \notin J\left( {{M}_{2}\left( R\right) }\right) \). It follows from Lemma 1.2 that there exists \( q \in R \)... | Yes |
Corollary 2.2 Let \( p \) be prime, \( {\mathbb{Z}}_{\left( p\right) } = \left\{ {\left. \frac{m}{n}\right| \;m, n \in \mathbb{Z}, p \nmid n}\right\} \), and let \( A \in {M}_{2}\left( {\mathbb{Z}}_{\left( p\right) }\right) \) . Then \( A \) is strongly \( J \) -clean in \( {M}_{2}\left( {\mathbb{Z}}_{\left( p\right) }... | Proof Clearly, \( {\mathbb{Z}}_{\left( p\right) } \) is a commutative local domain and \( J\left( {\mathbb{Z}}_{\left( p\right) }\right) = p{\mathbb{Z}}_{\left( p\right) } \) . One direction is clear by Lemma 2.2. It suffices to show the converse.\n\nCase I: \( p = 2 \) . Suppose that \( \det A \in 2{\mathbb{Z}}_{\left... | Yes |
Example 2.1 Let \( p \) be prime, \( x, y \in {\mathbb{Z}}_{\left( p\right) } \), and let \( p \mid x \) . Then \( \left( \begin{matrix} x + 1 & x \\ y & x \end{matrix}\right) \in {M}_{2}\left( {\mathbb{Z}}_{\left( p\right) }\right) \) is strongly \( J \) -clean if and only if \( {4xy} + 1 \in {\mathbb{Z}}_{\left( p\ri... | Proof Suppose\n\n\[ A \mathrel{\text{:=}} \left( \begin{matrix} x + 1 & x \\ y & x \end{matrix}\right) \in {M}_{2}\left( {\mathbb{Z}}_{\left( p\right) }\right) \]\n\nis strongly \( J \) -clean. Clearly, \( A,{I}_{2} \notin J\left( {{M}_{2}\left( {\mathbb{Z}}_{\left( p\right) }\right) }\right. \) . It follows from Lemma... | Yes |
Corollary 2.3 Let \( \widehat{{\mathbb{Z}}_{p}} \) be the ring of \( p \) -adic integers, and let \( A \in {M}_{2}\left( \widehat{{\mathbb{Z}}_{p}}\right) \) . Then \( A \) is strongly \( J \) -clean in \( {M}_{2}\left( \widehat{{\mathbb{Z}}_{p}}\right) \) if and only if \( A \in J\left( {{M}_{2}\left( \widehat{{\mathb... | Proof It is easy to verify that \( \widehat{{\mathbb{Z}}_{p}} \) is a commutative local domain. Thus, one direction is obvious from Lemma 2.2. We will suffice to prove the converse.\n\nCase I: \( p = 2 \) . Suppose that \( \det A \in J\left( \widehat{{\mathbb{Z}}_{2}}\right) \), tr \( A \in 1 + J\left( \widehat{{\mathb... | Yes |
Example 2.2 Let \( {\mathbb{Z}}_{4} = \{ \overline{0},\overline{1},\overline{2},\overline{3}\} \) be the ring of integers modulo 4, and let \( A = \left( {\frac{\overline{1}}{0}\frac{\overline{1}}{2}}\right) \in \) \( {M}_{2}\left( {\mathbb{Z}}_{4}\right) \) . Then \( {\mathbb{Z}}_{4} \) is a commutative local ring wit... | In addition, \( \operatorname{tr}A = \overline{3} \in \overline{1} + J\left( {\mathbb{Z}}_{4}\right) \) , \( \det A = \overline{2} \in J\left( {\mathbb{Z}}_{4}\right) \) and \( {\left( \operatorname{tr}A\right) }^{2} - \) \( 4\det A = \overline{1} \) is an invertible square. Since \( {x}^{2} - \left( {\operatorname{tr}... | No |
Corollary 3.1 Let \( R \) be a commutative local ring. If \( R/J\left( R\right) \cong {\mathbb{Z}}_{2} \), then \( \mathcal{T}\left( R\right) \) is strongly \( J \) -clean. | Proof In view of [6, Example 2.1.11], \( R \) is bleached. Therefore \( \mathcal{T}\left( R\right) \) is strongly \( J \) -clean by Theorem 3.1. | No |
Example 3.1 Let \( {\mathbb{Z}}_{\left( 2\right) } = \{ \frac{m}{n} \mid m, n \in \mathbb{Z},2 \nmid n\} \) . Then \( \left( \begin{matrix} {\mathbb{Z}}_{\left( 2\right) } & 0 & 0 \\ {\mathbb{Z}}_{\left( 2\right) } & {\mathbb{Z}}_{\left( 2\right) } & {\mathbb{Z}}_{\left( 2\right) } \\ 0 & 0 & {\mathbb{Z}}_{\left( 2\rig... | Proof As in the proof of Corollary 1.1, \( {\mathbb{Z}}_{\left( 2\right) } \) is a commutative local ring with \( {\mathbb{Z}}_{\left( 2\right) }/J\left( {\mathbb{Z}}_{\left( 2\right) }\right) \cong \) \( {\mathbb{Z}}_{2} \) . In light of Corollary 3.1, the result follows. | No |
Example 3.2 Let \( \widehat{{\mathbb{Z}}_{2}} \) be the ring of 2-adic integers. Then \( \left( \begin{matrix} \widehat{{\mathbb{Z}}_{2}} & 0 & 0 \\ \widehat{{\mathbb{Z}}_{2}} & \widehat{{\mathbb{Z}}_{2}} & \widehat{{\mathbb{Z}}_{2}} \\ 0 & \widehat{{\mathbb{Z}}_{2}} & \end{matrix}\right) \) is strongly \( J \) -clean. | Proof As in the proof of Corollary \( {1.2},\widehat{{\mathbb{Z}}_{2}} \) is a commutative local ring with \( \widehat{{\mathbb{Z}}_{2}}/J\left( \widehat{{\mathbb{Z}}_{2}}\right) \cong {\mathbb{Z}}_{2} \) . According to Corollary 3.1, we obtain the result. | Yes |
Proposition 3.1 Let \( R \) be a local ring. Then the following are equivalent:\n\n(1) \( S = \left\{ {\left. \left( \begin{matrix} {a}_{11} & 0 & 0 \\ 0 & {a}_{22} & 0 \\ {a}_{31} & {a}_{32} & {a}_{33} \end{matrix}\right) \right| \;{a}_{11},{a}_{22},{a}_{33},{a}_{31},{a}_{32} \in R}\right\} \) is strongly \( J \) -cle... | Proof Let \( A \in S \) and \( P = \left( \begin{array}{lll} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right) \) . Then \( {PA}{P}^{-1} \in \mathcal{T}\left( R\right) \) . If \( \mathcal{T}\left( R\right) \) is strongly \( J \) -clean, then we can find an idempotent \( E \in \mathcal{T}\left( R\right) \) and a \(... | Yes |
Example 3.3 The special matrix rings \( \left( \begin{matrix} {\mathbb{Z}}_{\left( 2\right) } & 0 & 0 \\ 0 & {\mathbb{Z}}_{\left( 2\right) } & 0 \\ {\mathbb{Z}}_{\left( 2\right) } & {\mathbb{Z}}_{\left( 2\right) } & {\mathbb{Z}}_{\left( 2\right) } \end{matrix}\right) \) and \( \left( \begin{matrix} \widehat{{\mathbb{Z}... | Proof It is clear from Proposition 3.1. | No |
Lemma 0.2 \( {}^{\left\lbrack 4\right\rbrack } \) Let \( p\left( \cdot \right) \in \mathcal{P}\left( \Omega \right) \) . If \( f \in {L}^{p\left( \cdot \right) }\left( \Omega \right) \) and \( g \in {L}^{{p}^{\prime }\left( \cdot \right) }\left( \Omega \right) \), then \( {fg} \) is integrable on \( \Omega \) and | \[ {\int }_{\Omega }\left| {f\left( x\right) g\left( x\right) }\right| \mathrm{d}x \leq {r}_{p}\parallel f{\parallel }_{{L}^{p\left( \cdot \right) }\left( \Omega \right) }\parallel g{\parallel }_{{L}^{{p}^{\prime }\left( \cdot \right) }\left( \Omega \right) }, \] where \( {r}_{p} = 1 + \frac{1}{{p}^{ - }} - \frac{1}{{p... | Yes |
Lemma 0.4 \( {}^{\left\lbrack 2\right\rbrack } \) Suppose \( p\left( \cdot \right) \in \mathcal{B}\left( {\mathbb{R}}^{n}\right) \) . Then for all balls \( B \) in \( {\mathbb{R}}^{n} \) , | \[ \frac{1}{\left| B\right| }{\begin{Vmatrix}{\chi }_{B}\end{Vmatrix}}_{{L}^{p\left( \cdot \right) }\left( {\mathbb{R}}^{n}\right) }{\begin{Vmatrix}{\chi }_{B}\end{Vmatrix}}_{{L}^{{p}^{\prime }\left( \cdot \right) }\left( {\mathbb{R}}^{n}\right) } \leq C. \] | Yes |
Lemma 0.6 \( {}^{\left\lbrack 1\right\rbrack } \) Suppose that \( {q}_{1}\left( \cdot \right) \in \mathcal{P}\left( {\mathbb{R}}^{n}\right) \) satisfies the conditions (0.1) and (0.2) in Lemma 0.1 with \( {q}_{1}^{ + } < \frac{n}{\sigma } \) and \( \frac{1}{{q}_{1}\left( x\right) } - \frac{1}{{q}_{2}\left( x\right) } =... | \[ {\begin{Vmatrix}{I}_{\sigma }\left( f\right) \end{Vmatrix}}_{{L}^{{q}_{2}\left( \cdot \right) }\left( {\mathbb{R}}^{n}\right) } \leq C\parallel f{\parallel }_{{L}^{{q}_{1}\left( \cdot \right) }\left( {\mathbb{R}}^{n}\right) }.\] | Yes |
Lemma 0.7 \( {}^{\left\lbrack 7\right\rbrack } \) Suppose that \( {q}_{1}\left( \cdot \right) \in \mathcal{P}\left( {\mathbb{R}}^{n}\right) \) satisfies conditions (0.1) and (0.2) in Lemma 0.1 with \( {q}_{1}^{ + } < \frac{n}{\sigma },\frac{1}{{q}_{1}\left( x\right) } - \frac{1}{{q}_{2}\left( x\right) } = \frac{\sigma ... | \[ {\begin{Vmatrix}\left\lbrack b,{I}_{\sigma }\right\rbrack \left( f\right) \end{Vmatrix}}_{{L}^{{q}_{2}\left( \cdot \right) }\left( {\mathbb{R}}^{n}\right) } \leq C\parallel b{\parallel }_{ * }\parallel f{\parallel }_{{L}^{{q}_{1}\left( \cdot \right) }\left( {\mathbb{R}}^{n}\right) }.\] | Yes |
Lemma 1.1 Suppose that \( u \) is the optimal control of (1.3). Then\n\n\[ u\left( t\right) = {B}^{ * }{S}_{\alpha }^{ * }\left( {b - t}\right) R\left( {\lambda ,{\Lambda }_{b}}\right) p\left( {x\left( \cdot \right) }\right) ,\;t \in \left\lbrack {0, b}\right\rbrack \]\n\nwith\n\n\[ p\left( {x\left( \cdot \right) }\rig... | Proof The proof is similar to that of [7, Lemma 2.9], so we omit it. | No |
Corollary 2.1 Assume that the conditions (H1), (H2), (H3′) and (H4) are satisfied, then the integral system (1.4) has at least one mild solution on \( \left\lbrack {0, b}\right\rbrack \) . | Proof Let\n\n\[ \n{\left( Q{W}_{r}\right) }_{\delta } = \left\{ {x \in C\left( {\left\lbrack {0, b}\right\rbrack, X}\right) : x\left( t\right) = v\left( t\right) \text{ for }t \in \left\lbrack {\delta, b}\right\rbrack, x\left( t\right) = x\left( \delta \right) \text{ for }t \in \lbrack 0,\delta ), v \in Q{W}_{r}}\right... | Yes |
Consider the following fractional partial differential equation in \( \mathbb{R} \) :\n\n\[ \left\{ \begin{array}{l} {D}_{t}^{\alpha }w\left( {t,\xi }\right) = \frac{{\partial }^{2}w\left( {t,\xi }\right) }{\partial {\xi }^{2}} + {J}_{t}^{1 - \alpha }\left\lbrack {f\left( {t, w\left( {t,\xi }\right) }\right) + u\left( ... | To write the system (3.4) in an abstract form, we define \( X = {L}^{2}\left\lbrack {0,1}\right\rbrack \) and \( A : D\left( A\right) \subseteq X \rightarrow \) \( X \) to be \( {Ax} = {x}^{\prime \prime } \), where \( D\left( A\right) = \left\{ {x \in X : x,{x}^{\prime }}\right. \) are absolutely continuous, \( {x}^{\... | Yes |
Lemma 1.2 Assume that (9) and (10) hold and \( 0 < \theta \leq l \) . Let \( p \) satisfy (a) and \( u\left( {x, t}\right) \) be a weak solution of Problem (8). If there exists a \( {t}_{0} \in \lbrack 0, T) \) such that \( u\left( {x,{t}_{0}}\right) \in {\sum }_{\theta }^{e} \) and \( E\left( {t}_{0}\right) < {d}_{\th... | Proof Suppose that there exists \( {t}_{1} \in \left\lbrack {{t}_{0}, T}\right) \) such that \( u\left( {x, t}\right) \in {\sum }_{\theta }^{e} \) for any \( t \in \left\lbrack {{t}_{0},{t}_{1}}\right) \), but \( u\left( {x,{t}_{1}}\right) \in \partial {\sum }_{\theta }^{e} \) . From the definition of \( {\sum }_{\thet... | Yes |
Lemma 3.1 Let \( u \) with \( E\left( 0\right) < {d}_{\theta } \) be a nontrivial weak solution of Problem (8). Furthermore, if \( E\left( 0\right) \leq 0 \), then \( {I}_{\theta }\left( u\right) < 0 \) and \( u \in {\sum }_{\theta }^{e} \). | Proof By \( E\left( t\right) \leq E\left( 0\right), E\left( 0\right) \leq 0 \) and \( E\left( t\right) \geq \frac{1}{2}\parallel {\Delta u}{\parallel }_{2}^{2} - \frac{1}{p + 1}\parallel u{\parallel }_{p + 1}^{p + 1} \), it follows that\n\n\[ \frac{1}{p + 1}{I}_{\theta }\left( u\right) < \frac{p + 1 - {2\theta }}{2\lef... | Yes |
Lemma 1.8 Let \( {W}_{n} \) be defined by (0.4), and \( {T}_{i}\left( {i = 1,2,\cdots, N}\right) \) be a finite family of nonspreading mappings. If \( \mathop{\bigcap }\limits_{{i = 1}}^{N}F\left( {T}_{i}\right) \neq \varnothing \), then we have\n\n(1) \( \mathop{\bigcap }\limits_{{i = 1}}^{N}F\left( {T}_{i}\right) = F... | Proof (1) First we show that \( \mathop{\bigcap }\limits_{{i = 1}}^{N}F\left( {T}_{i}\right) \subseteq F\left( {W}_{n}\right) \) .\n\nWhen \( x \in \mathop{\bigcap }\limits_{{i = 1}}^{N}F\left( {T}_{i}\right) \), we have \( {T}_{i}x = x\left( {i = 1,2,\cdots, N}\right) \), then\n\n\[ \n{U}_{n,1}x = {\lambda }_{n,1}\lef... | Yes |
Theorem 2.1 Let \( C \) be a nonempty closed convex subset of Hilbert space \( H,{T}_{i} : C \rightarrow \) \( C, i = 1,2,\cdots, N \) be a finite family of nonspreading mappings such that \( \mathop{\bigcap }\limits_{{i = 1}}^{N}F\left( {T}_{i}\right) \neq \varnothing, f \) : \( C \rightarrow C \) be a contraction on ... | Proof We split the proof into five steps.\n\nStep 1: Show that the variational inequality (VI) has a unique solution in \( \mathop{\bigcap }\limits_{{i = 1}}^{N}F\left( {T}_{i}\right) \cap \) \( \operatorname{EP}\left( F\right) \) .\n\nSince \( {T}_{i} : C \rightarrow C\left( {i = 1,2,\cdots, N}\right) \) is a nonsprea... | Yes |
Proposition 1.4 \( {}^{\left\lbrack {23}\right\rbrack } \) If \( {z}_{0} \in {\dot{B}}_{2,1}^{\sigma }\left( {\mathbb{R}}^{d}\right) \cap {\dot{B}}_{2,\infty }^{-s}\left( {\mathbb{R}}^{d}\right) \) for \( \sigma \in \mathbb{R}, s \in \mathbb{R} \) satisfying \( \sigma + s > 0 \) , then the solution \( z\left( {t, x}\ri... | \[ \parallel z{\parallel }_{{\dot{B}}_{2,1}^{\sigma }} \lesssim \parallel {z}_{0}{\parallel }_{{\dot{B}}_{2,\infty }^{-s}}{\left( 1 + t\right) }^{-\frac{\sigma + s}{2}} + \parallel {z}_{0}{\parallel }_{{\dot{B}}_{2,1}^{\sigma }}{\mathrm{e}}^{-{ct}}. \] | Yes |
Theorem 1.4 \( {}^{\left\lbrack {24}\right\rbrack }\left( {d = 1,2}\right) \) Let \( U\left( {t, x}\right) \) be the global classical solution of Theorem 1.2. Suppose that \( {U}_{0} - \bar{U} \in {B}_{2,1}^{{\sigma }_{c}}\left( {\mathbb{R}}^{d}\right) \cap {\dot{B}}_{2,\infty }^{-s}\left( {\mathbb{R}}^{d}\right) \left... | \[ \parallel U\left( {t, \cdot }\right) - \bar{U}{\parallel }_{{X}_{0}} \lesssim {E}_{0}{\left( 1 + t\right) }^{-\frac{s + \ell }{2}} \] where \( {X}_{0} = : {\dot{B}}_{2,1}^{\ell }\left( {\mathbb{R}}^{d}\right) \) if \( 0 < \ell \leq {\sigma }_{c} - 1 \), and \( {X}_{0} = : {L}^{2}\left( {\mathbb{R}}^{d}\right) \) if ... | Yes |
Theorem 2.2 \( {}^{\left\lbrack {25}\right\rbrack } \) Let \( \eta \left( \xi \right) \) be a positive, continuous and real-valued function in \( {\mathbb{R}}^{d} \) satisfying \[ \eta \left( \xi \right) \sim \left\{ \begin{array}{ll} {\left| \xi \right| }^{{\sigma }_{1}}, & \left| \xi \right| \rightarrow 0 \\ {\left| ... | (2.3) for \( \ell > d\left( {\frac{1}{r} - \frac{1}{2}}\right) \) with \( 1 \leq r \leq 2 \), where \( {\gamma }_{{\sigma }_{2}}\left( {r,2}\right) \mathrel{\text{:=}} \frac{d}{{\sigma }_{2}}\left( {\frac{1}{r} - \frac{1}{2}}\right) \) and \( {l}_{q}^{\alpha } \) stands for the \( \alpha \) -type summation over \( q \i... | No |
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