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Lemma 3.4 Let \( p \) be an odd prime, and \( n \) the smallest integer such that \( p \mid {T}_{n}\left( m\right) \) . Then \( {2n} \mid \left( {{p}^{2} - 1}\right) \) . | Proof By assumption \( p \mid {T}_{n}\left( m\right) \), we have\n\n\[ \n{\left( \frac{\alpha }{\widetilde{\alpha }}\right) }^{n} \equiv - 1\;\left( {\;\operatorname{mod}\;\mathfrak{P}}\right) ,\text{ hence }{\left( \frac{\alpha }{\widetilde{\alpha }}\right) }^{2n} \equiv 1\;\left( {\;\operatorname{mod}\;\mathfrak{P}}\... | Yes |
Theorem 3.7 Let \( n \in {\mathbb{N}}^{ * } \), and \( p, q \) two different odd primes. Then there exists a positive integer \( {T}_{n, m, p, q} \) such that \[ {\operatorname{ord}}_{q}\left( {{T}_{n{p}^{t}}\left( m\right) }\right) = {\operatorname{ord}}_{q}\left( {{T}_{n{p}^{{T}_{n, m, p, q}}}\left( m\right) }\right)... | Proof Without loss of generality, we may assume \( \left( {n, p}\right) = 1 \) . On the other hand, by (4), we have \[ {T}_{n{p}^{t}}\left( m\right) = \frac{{\widetilde{\alpha }}^{n{p}^{t}}}{2}\mathop{\prod }\limits_{{j = 0}}^{{n - 1}}\left( {1 - {\left( \beta {\zeta }_{2n}^{{2j} + 1}\right) }^{{p}^{t}}}\right) . \] We... | Yes |
Corollary 3.8 Let \( {S}_{n, m, p}\left( t\right) \) be the set of all primes which divide \( {T}_{n{p}^{t}}\left( m\right) \) . Then \( \sharp {S}_{n, m, p}\left( t\right) \rightarrow + \infty \) as \( t \rightarrow + \infty . \) | Proof Suppose that there exists integer \( {t}_{0} \) such that for all \( t \geq {t}_{0},{S}_{n, m, p}\left( t\right) = \) \( {S}_{n, m, p}\left( {t}_{0}\right) \) . By Theorem 3.6 and Theorem 3.7, it would follow that \( {T}_{n{p}^{t}}\left( m\right) \) would be\nequal to a constant times \( {p}^{t} \) for large \( t... | Yes |
Theorem 1.1 Let \( p > 3 \) be a prime, and let \( {N}_{p} \) denote the number of integral solutions to the system of congruence equations\n\n\[ \n\\left\\{ \\begin{array}{l} {x}_{1}^{3} + {x}_{2}^{3} + {x}_{3}^{3} \\equiv {y}_{1}^{3} + {y}_{2}^{3} + {y}_{3}^{3}\\left( {\\;\\operatorname{mod}\\;p}\\right) , \\\\ {x}_{... | Let \( p \) be a prime. For any \( a \\in \\mathbb{Z} \), we clearly have\n\n\[ \n\\mathop{\\sum }\\limits_{{x = 0}}^{{p - 1}}{e}^{{2\\pi i}\\frac{ax}{p}} = \\left\\{ \\begin{array}{ll} p, & \\text{ if }p \\mid a \\\\ 0, & \\text{ otherwise. } \\end{array}\\right.\n\]\n\nThus\n\n\[ \n{p}^{2}{N}_{p} = \\mathop{\\sum }\\... | Yes |
Lemma 2.1 Let \( p \) be an odd prime, and let \( {T}_{n}\left( p\right) \) denote the number of integral solutions to the system of congruence equations\n\n\[ \left\{ \begin{array}{l} \mathop{\sum }\limits_{{j = 1}}^{n}\left( {{x}_{j}^{2} - {y}_{j}^{2}}\right) \equiv 0\left( {\;\operatorname{mod}\;p}\right) \\ \mathop... | Proof For simplicity, we write \( {T}_{n} = {T}_{n}\left( p\right) \) . Clearly, \( {T}_{1} = p \) .\n\nFor \( n \geq 2 \), we write \( {T}_{n} = {T}^{\prime } + {T}^{\prime \prime } \), where \( {T}^{\prime } \) is the number of solutions of (2.6) with \( \mathop{\sum }\limits_{{j = 1}}^{{n - 1}}\left( {{x}_{j} - {y}_... | Yes |
Lemma 2.2 Let \( p > 3 \) be a prime, and let \( {x}_{1},{x}_{2},{x}_{3},{y}_{1},{y}_{2},{y}_{3} \in \{ 0,1,\cdots, p - 1\} \) satisfying \[ \left\{ \begin{array}{l} \mathop{\sum }\limits_{{j = 1}}^{3}\left( {{x}_{j}^{3} - {y}_{j}^{3}}\right) \equiv 0\left( {\;\operatorname{mod}\;p}\right) , \\ \mathop{\sum }\limits_{{... | Proof Clearly, \[ \left( {x - {x}_{1}}\right) \left( {x - {x}_{2}}\right) \left( {x - {x}_{3}}\right) = {x}^{3} - \left( {{x}_{1} + {x}_{2} + {x}_{3}}\right) {x}^{2} + \left( {{x}_{1}{x}_{2} + {x}_{1}{x}_{3} + {x}_{2}{x}_{3}}\right) x - {x}_{1}{x}_{2}{x}_{3}. \] Note that \[ 2\left( {{x}_{1}{x}_{2} + {x}_{1}{x}_{3} + {... | Yes |
Example 2.2 Let \( R \) be an \( \alpha \) -reflexive ring with an endomorphism \( \alpha \) and let\n\n\[ S = \left\{ {\left. \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \right| \;a, b, c, d \in R}\right\} .\n\]\nThen \( S \) is an \( \bar{\alpha } \) -reflexive ring \( {}^{\left\lbrack {17},\text{ Theo... | In fact, let\n\n\[ f\left( x\right) = \left( \begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right) + \left( \begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right) x, g\left( x\right) = \left( \begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right) x,\n\]\n\nand\n\n\[ h\left( x\right) = \left( \begin{array}{ll} 1 & 1 \\ 0 & 0 \e... | Yes |
Example 2.3 Let \( R = {\mathbb{Z}}_{2}\bigoplus {\mathbb{Z}}_{2} \), where \( {\mathbb{Z}}_{2} \) is the ring of integers modulo 2 . Then \( R \) is a commutative reduced ring, and so \( R \) is reflexive. Now, let \( \alpha : R \rightarrow R \) be an endomorphism defined by \( \alpha \left( \left( {a, b}\right) \righ... | \[ g\left( x\right) h\left( x\right) f\left( x\right) = \left( {\left( {1,0}\right) + \left( {1,0}\right) x}\right) \left( {\left( {1,1}\right) + \left( {1,1}\right) x}\right) \left( {\left( {1,0}\right) + \left( {1,0}\right) x}\right) \neq 0. \] | Yes |
Lemma 2.4 Let \( R \) be an Armendariz ring. Then \( R \) is \( \alpha \) -reflexive if and only if \( R \) is strongly \( \alpha \) -reflexive. | Proof It is straightforward from [14, Proposition 3.2]. | No |
Theorem 2.5 Let \( R \) be an Armendariz ring. Then \( R \) is \( \alpha \) -rigid if and only if \( R \) is a reduced strongly \( \alpha \) -reflexive ring with \( \alpha \) a monomorphism. | Proof It is obvious by [14, Proposition 2.1] and Lemma 2.4. | No |
Example 2.7 Let \( {\mathbb{Z}}_{4} \) be the ring of integers modulo 4. Consider the ring\n\n\[ R = \left\{ {\left. \left( \begin{array}{ll} a & b \\ 0 & a \end{array}\right) \right| \;a, b \in {\mathbb{Z}}_{4}}\right\} .\n\nLet \( \alpha : R \rightarrow R \) defined by\n\n\[ \alpha \left( \left( \begin{array}{ll} a &... | But \( T\left( {R, R}\right) \) is not strongly \( \widetilde{\alpha } \) -reflexive since \( T\left( {R, R}\right) \) is not \( \bar{\alpha } \) -reflexive \( {}^{\left\lbrack {17},\text{ Example }{3.6}\right\rbrack } \). | No |
Theorem 2.8 Let \( R \) be a reduced ring. If \( R \) is an \( \alpha \) -reflexive ring, then \( T\left( {R, R}\right) \) is a strongly \( \bar{\alpha } \) -reflexive ring. | Proof It is obvious that \( T\left( {R, R}\right) \) is an Armendariz ring since \( R \) is reduced [11, Proposition 2.5]. Also, it was shown in [14, Proposition 2.2] that \( T\left( {R, R}\right) \) is an \( \bar{\alpha } \) -reflexive ring. It follows that \( T\left( {R, R}\right) \) is a strongly \( \bar{\alpha } \)... | Yes |
Theorem 3.1 Let \( R \) be a ring, \( e \) a central idempotent of \( R \), and \( \Delta \) a multiplicative closed subset consisting of central regular elements of \( R \) . Then the following statements are equivalent.\n\n(1) \( R \) is strongly \( \alpha \) -reflexive;\n\n(2) \( {eR} \) and \( \left( {1 - e}\right)... | Proof (1) \( \Leftrightarrow \) (3) It suffices to show that \( {\Delta }^{-1}R \) is strongly \( \alpha \) -reflexive if \( R \) is strongly \( \alpha \) -reflexive. Let\n\n\[ f\left( x\right) = \mathop{\sum }\limits_{{i = 0}}^{m}{u}_{i}^{-1}{a}_{i}{x}^{i},\;g\left( x\right) = \mathop{\sum }\limits_{{j = 0}}^{n}{v}_{j... | Yes |
Theorem 3.2 For a ring \( R, R\left\lbrack x\right\rbrack \) is strongly \( \alpha \) -reflexive if and only if \( R\left\lbrack {x,{x}^{-1}}\right\rbrack \) is strongly \( \alpha \) -reflexive. | Proof It suffices to establish the necessity since \( R\left\lbrack x\right\rbrack \) is a subring of \( R\left\lbrack {x,{x}^{-1}}\right\rbrack \) . Suppose that \( R\left\lbrack x\right\rbrack \) is strongly \( \alpha \) -reflexive. Let \( \Delta = \left\{ {1, x,{x}^{2},\cdots }\right\} \), then clearly \( \Delta \) ... | No |
Corollary 3.3 A commutative ring \( R \) is strongly \( \alpha \) -reflexive if and only if so is the total quotient ring of \( R \) . | It was proved in [14, Proposition 3.2] that if \( R \) is an Armendariz ring, then \( R \) is \( \alpha \) -reflexive if and only if \( R\left\lbrack x\right\rbrack \) is \( \alpha \) -reflexive if and only if \( R\left\lbrack {x,{x}^{-1}}\right\rbrack \) is \( \alpha \) -reflexive. Accordingly, we have the equivalence... | No |
Theorem 3.5 Let \( \alpha \) be an endomorphism of a ring \( R \) . Then \( R \) is strongly \( \alpha \) - reflexive if and only if \( R\left\lbrack x\right\rbrack \) is strongly \( \alpha \) -reflexive. | Proof \( \xrightarrow[]{\text{ Assume that }}R \) is strongly \( \alpha \) -reflexive. Let \( f\left( y\right) = {f}_{0} + {f}_{1}y + \cdots + {f}_{m}{y}^{m}, \) \( g\left( y\right) = {g}_{0} + {g}_{1}y + \cdots + {g}_{n}{y}^{n} \) be in \( R\left\lbrack x\right\rbrack \left\lbrack y\right\rbrack \) with \( f\left( y\r... | Yes |
Theorem 3.7 Let \( A\left( {R,\alpha }\right) \) be an Armendariz ring. If \( R \) is \( \alpha \) -reflexive, then \( A\left( {R,\alpha }\right) \) is strongly \( \alpha \) -reflexive. | Proof It is straightforward that \( A\left( {R,\alpha }\right) \) is an \( \alpha \) -reflexive [14, Proposition 3.3]. Then \( A\left( {R,\alpha }\right) \) is strongly \( \alpha \) -reflexive by Lemma 2.4. | Yes |
Lemma 3.8 Let \( R \) be a strongly \( \alpha \) -reflexive ring with \( \alpha \left( 1\right) = 1 \) . Then for any \( f\left( x\right), g\left( x\right) \in R\left\lbrack x\right\rbrack, f\left( x\right) R\left\lbrack x\right\rbrack g\left( x\right) = 0 \) implies \( g\left( x\right) R\left\lbrack x\right\rbrack {\a... | Proof Let \( \;f\left( x\right) R\left\lbrack x\right\rbrack g\left( x\right) = 0 \) with \( f\left( x\right), g\left( x\right) \in R\left\lbrack x\right\rbrack \), then \( g\left( x\right) R\left\lbrack x\right\rbrack \;\alpha \left( {f\left( x\right) }\right) = 0 \) since \( R \) is strongly \( \alpha \) -reflexive. ... | Yes |
Theorem 3.10 Let \( R \) be a reduced ring and \( n \) any positive integer. If \( R \) is strongly \( \alpha \) -reflexive with \( \alpha \left( 1\right) = 1 \), then \( R\left\lbrack x\right\rbrack /\left( {x}^{n}\right) \) is a strongly \( \alpha \) -reflexive ring, where \( \left( {x}^{n}\right) \) is the ideal gen... | Proof It is obvious that \( R\left\lbrack x\right\rbrack /\left( {x}^{n}\right) \) is strongly \( \alpha \) -reflexive since \( R\left\lbrack x\right\rbrack /\left( {x}^{n}\right) \) is both the \( \alpha \) -reflexive ring [14, Proposition 3.5] and Armendariz ring \( {}^{\left\lbrack 2,\text{ Theorem }5\right\rbrack }... | No |
Theorem 1.2 Let \( G \) be a maximal planar graph of order \( n \geq 7 \) with \( \delta \left( G\right) = \delta \) , then \( \bar{G} \) contains all double-star-like trees of order \( n - 2 \) with maximum degree at most \( n - \delta - 1 \) . | By the definition of the double-star-like trees, the maximum degree of \( D{S}_{n - 2}\left( {k, l}\right) \) is \( k + 1 \) and \( k + l \leq n - 4 \) . So, it is sufficient to prove Theorem 1.2 under the assumptions that \( k \leq n - \delta - 2 \) and \( k + l \leq n - 4 \) . Based on these assumptions, we will part... | No |
Lemma 2.1 Let \( G \) be a maximal plane graph of order \( n \geq 6 \) and \( v \in V\left( G\right) \) with \( {d}_{G}\left( v\right) = \delta \left( G\right) = 3 \) . Then \( v \) has no 3-neighbor and at least one \( {5}^{ + } \) -neighbor. | Proof Let the neighbors of \( v \) in clockwise order around \( v \) are \( {v}_{1},{v}_{2},{v}_{3} \) . Since \( G \) is maximal, we have \( {v}_{1}{v}_{2},{v}_{2}{v}_{3},{v}_{3}{v}_{1} \in E\left( G\right) \) . First we show that \( v \) has no 3-neighbor. Assume to the contrary that \( v \) has a 3-neighbor, say \( ... | Yes |
Lemma 2.2 Let \( G \) be a maximal plane graph of order \( n \geq 8 \) and \( v \in V\left( G\right) \) with \( {d}_{G}\left( v\right) = \delta \left( G\right) = 4 \) . Then \( v \) has at most two 4-neighbors. Furthermore, if \( v \) has two 4-neighbors, then \( v \) has at least one \( {6}^{ + } \) -neighbor. | Proof Let the neighbors of \( v \) in clockwise order around \( v \) are \( {v}_{1},{v}_{2},{v}_{3},{v}_{4} \) . Since \( G \) is maximal, we have \( {v}_{1}{v}_{2},{v}_{2}{v}_{3},{v}_{3}{v}_{4},{v}_{4}{v}_{1} \in E\left( G\right) \) . First we show that \( v \) has at most two 4-neighbors. Assume to the contrary that ... | Yes |
Lemma 2.3 Let \( G \) be a plane graph with \( v \in V\left( G\right) \) such that \( {d}_{G}\left( v\right) = d \leq 5 \) . Then there exists at most \( d - 2 \) vertices \( {u}_{1},\ldots ,{u}_{d - 2} \in V\left( G\right) \backslash {N}_{G}\left\lbrack v\right\rbrack \) such that\n\n\[ \left| {{N}_{G}\left( v\right) ... | Proof Let the neighbors of \( v \) in clockwise order around \( v \) are \( {v}_{1},{v}_{2},\ldots ,{v}_{d} \) . Set \( \left\{ {{u}_{1},{u}_{2},\ldots ,\;{u}_{n - 1 - d}}\right\} = V\left( G\right) \backslash {N}_{G}\left\lbrack v\right\rbrack \), where \( n \) is the order of \( G \) . Suppose that \( |{N}_{G}\left( ... | Yes |
Lemma 2.4 Let \( G \) be a maximal plane graph of order \( n \geq 8 \) and \( v \in V\left( G\right) \) with \( {d}_{G}\left( v\right) = \delta \left( G\right) \), then there exists \( w \in V\left( G\right) \smallsetminus {N}_{G}\left\lbrack v\right\rbrack \) such that \( {d}_{G}\left( w\right) \leq 5 \) and \( \left|... | Proof \( \begin{aligned} \text{ Let }{D}_{i}^{\prime } & = {D}_{i} \cap \left( {V\left( G\right) \smallsetminus {N}_{G}\left\lbrack v\right\rbrack }\right) \\ \text{ and }{d}_{i}^{\prime } & = \left| {D}_{i}^{\prime }\right| . \\ \text{ First we assume }\delta \left( G\right) & = 3. \end{aligned} \) By \( \left( *\righ... | Yes |
Lemma 2.5 Let \( G \) be a maximal planar graph of order 8, then \( \bar{G} \) contains \( D{S}_{6}\left( {2,2}\right) \) . | Proof By Lemma 1.3, \( \bar{G} \) contains \( {C}_{4} = {v}_{1}{v}_{2}{v}_{3}{v}_{4}{v}_{1} \) . Since \( G \) contains no \( {K}_{3,3} \) by Theorem 1.1, we have \( {\left( \{ {v}_{1},{v}_{2},{v}_{3}\} ,\{ {v}_{5},{v}_{6},{v}_{7}\} \right) }_{\overline{G}} \neq \varnothing \), say \( {v}_{1}{v}_{5} \in E\left( \overli... | Yes |
Lemma 3.2 Let \( G \) be a maximal planar graph of order \( n \geq {10} \), then \( \bar{G} \) contains \( D{S}_{n - 2}\left( {k, l}\right) \) with \( k + l = n - 6 \) and \( l \geq 2 \) . | Proof By Theorem 1.3, we have that \( \bar{G} \) contains \( T = D{S}_{n - 2}\left( {k + 1, l + 1}\right) \) with \( V\left( T\right) = \left\{ {v, w,{v}_{1},\ldots ,{v}_{k + 1},{w}_{1},\ldots ,{w}_{l + 1}}\right\} \) and \( E\left( T\right) = \left\{ {{vw}, v{v}_{i}, w{w}_{j} \mid 1 \leq i \leq k + 1,1 \leq }\right. \... | Yes |
Lemma 3.4 Let \( G \) be a planar graph of order \( n = {12} \), then \( \overline{G} \) contains \( D{S}_{10}\left( {3,2}\right) . \) | Proof By Theorem 1.3, \( \overline{G} \) contains \( T = D{S}_{10}\left( {4,4}\right) \) with \( V\left( T\right) = \left\{ {v, w,{v}_{1},{v}_{2},{v}_{3},{v}_{4}}\right. \) , \( \left. {{w}_{1},{w}_{2},{w}_{3},{w}_{4}}\right\} \) and \( E\left( T\right) = \{ {vw}, v{v}_{i}, w{w}_{i} \mid 1 \leq i \leq 4\} . \) Since \(... | Yes |
Lemma 4.1 Let \( G \) be a planar graph of order \( n \geq 7 \) . If \( \bar{G} \) contains \( {P}_{k} \), then \( \bar{G} \) contains \( {P}_{k + 1} \) for \( 3 \leq k \leq n - 3 \) . | Proof Let \( P = {v}_{1}{v}_{2}\cdots {v}_{k} \) be a path of order \( k \) in \( \bar{G} \) . If\n\n\[ \n{N}_{\overline{G}}\left( {v}_{1}\right) \cap \left\{ {{v}_{k + 1},{v}_{k + 2},\ldots ,{v}_{n}}\right\} \neq \varnothing ,\n\]\n\nsay \( {v}_{1}{v}_{k + 1} \in E\left( \overline{G}\right) \), then \( P + {v}_{1}{v}_... | Yes |
Lemma 4.2 Let \( G \) be a planar graph of order \( n \geq 7 \), then \( \overline{G} \) contains \( D{S}_{n - 2}\left( {1,1}\right) . \) | Proof If \( 7 \leq n \leq {11} \), then \( \delta \left( G\right) \leq 4 \) by Lemma 1.4, thus \( \Delta \left( \widetilde{G}\right) \geq n - 1 - 4 \geq 2 \) . If \( n \geq {12} \), then \( \Delta \left( \bar{G}\right) \geq n - 1 - 5 \geq 6 \) as \( \delta \left( G\right) \leq 5 \) by Lemma 1.2. So in either case, \( \... | Yes |
Lemma 4.3 Let \( G \) be a planar graph of order \( n \geq 8 \), then \( \overline{G} \) contains \( D{S}_{n - 2}\left( {2,1}\right) . \) | Proof First we assume \( \bar{G} \) contains \( {C}_{n - 3} = {v}_{1}{v}_{2}\cdots {v}_{n - 3}{v}_{1} \) . Since \( G \) contains no \( {K}_{3,3} \) by Theorem 1.1, we have\n\n\[{\left( \left\{ {v}_{1},{v}_{2},{v}_{3}\right\} ,\left\{ {v}_{n - 2},{v}_{n - 1},{v}_{n}\right\} \right) }_{\overline{G}} \neq \varnothing ,\]... | Yes |
Corollary 4.5 Let \( G \) be a maximal planar graph of order \( n \geq 9 \), then \( \bar{G} \) contains \( D{S}_{n - 2}\left( {2,2}\right) \) . | Proof If \( n = {10} \), we have \( D{S}_{8}\left( {2,2}\right) \subseteq \bar{G} \) by Lemma 3.2. If \( n = 9 \) or \( n \geq {11} \), we have \( D{S}_{n - 2}\left( {2,2}\right) \subseteq \bar{G} \) by Lemma 4.4. | Yes |
Lemma 1.4(See \( \\left\\lbrack {5,\\text{ p. }{108}}\\right\\rbrack )\\; \) Let \( \\omega \\left( x\\right) = \\omega \\left( {{x}_{1},{x}_{2},\\cdots ,{x}_{n}}\\right) \) be symmetric with continuous partial derivatives on \( {I}^{n} \), where \( I \) is a subinterval of \( \\left( {0,\\infty }\\right) \) . Then \( ... | \[ \\left( {{x}_{1} - {x}_{2}}\\right) \\left( {{x}_{1}^{2}\\frac{\\partial \\omega }{\\partial {x}_{1}} - {x}_{2}^{2}\\frac{\\partial \\omega }{\\partial {x}_{2}}}\\right) \\geq 0\\;\\text{ for all }\\;x \\in {I}^{n}. \] | Yes |
Lemma 2.2 Suppose that \( f : I \rightarrow \left( {0, + \infty }\right) \) has continuous derivative on \( I \), where \( I \) is a subinterval of \( \left( {0, + \infty }\right) \) . If \( f\left( x\right) \) is monotonic and multiplicatively convex on \( I \) , then\n\n\[ \left( {{x}_{1} - {x}_{2}}\right) \left( {{x... | Proof Without loss of generality, we suppose that \( f\left( x\right) \) is increasing. By Theorem B, it yields that\n\n\[ \left( {{x}_{1} - {x}_{2}}\right) \left( {\frac{{x}_{1}{f}^{\prime }\left( {x}_{1}\right) }{f\left( {x}_{1}\right) } - \frac{{x}_{2}{f}^{\prime }\left( {x}_{2}\right) }{f\left( {x}_{2}\right) }}\ri... | Yes |
Corollary 2.4 Assume that \( 0 < {x}_{i} < 1, i = 1,2,\cdots, n\;\left( {n \geq 2}\right) \), then the symmetric function \( \mathop{\sum }\limits_{{1 \leq {i}_{1} < \cdots < {i}_{r} \leq n}}\frac{\mathop{\prod }\limits_{{j = 1}}^{r}{\left( {x}_{{i}_{j}}\right) }^{\frac{1}{r}}}{1 - \mathop{\prod }\limits_{{j = 1}}^{n}{... | Proof Set \( f\left( x\right) = \frac{x}{1 - x}, x \in \left( {0,1}\right) \) . Simply calculation shows that\n\n\[ \n{f}^{\prime }\left( x\right) = \frac{1}{{\left( 1 - x\right) }^{2}} \geq 0 \]\n\n\( \left( {2.14}\right) \)\n\nand\n\n\[ \nx\left\lbrack {f\left( x\right) {f}^{\prime \prime }\left( x\right) - {f}^{\pri... | Yes |
Corollary 2.5 Assume that \( {x}_{i} > 0, i = 1,2,\cdots, n\left( {n \geq 2}\right) \), then the symmetric function \( \mathop{\sum }\limits_{{1 \leq {i}_{1} < \cdots < {i}_{r} \leq n}}\frac{\mathop{\prod }\limits_{{j = 1}}^{r}{\left( {x}_{{i}_{j}}\right) }^{\frac{1}{r}}}{1 + \mathop{\prod }\limits_{{j = 1}}^{n}{\left(... | Proof \( \begin{aligned} \text{ Set }f\left( x\right) & = \frac{x}{1 - x}\text{,}x \in \left( {0,1}\right) . \\ \text{ It is easy to know that }{f}^{-1} & = \frac{x}{1 + x}\text{,}x \in \left( {0, + \infty }\right) . \end{aligned} \) From Theorem 2.1, directly reduce to the result. | No |
Theorem 3.5 Suppose that \( f : I \rightarrow \left( {0, + \infty }\right) \) has continuous derivative on \( I \) . If \( f\left( x\right) \) is increasing and multiplicatively convex on \( I \), and \( I \) is a subinterval of \( \left( {0, + \infty }\right) \) , then for all \( {x}_{i} \in I\left( {1 \leq i \leq n}\... | \[ \mathop{\sum }\limits_{{1 \leq {i}_{1} < \cdots < {i}_{r} \leq n}}f\left\lbrack {\left( {\frac{nc}{s} - 1}\right) \mathop{\prod }\limits_{{j = 1}}^{r}{\left( \frac{1}{c - {x}_{{i}_{j}}}\right) }^{\frac{1}{r}}}\right\rbrack \leq \mathop{\sum }\limits_{{1 \leq {i}_{1} < \cdots < {i}_{r} \leq n}}f\left\lbrack {\mathop{... | No |
Theorem 3.8 Suppose that \( f : I \rightarrow \left( {0, + \infty }\right) \) has continuous derivative on \( I \) . If \( f\left( x\right) \) is increasing and multiplicatively convex on \( I \), and \( I \) is a subinterval of \( \left( {0, + \infty }\right) \) , then for all \( {x}_{i} \in I\left( {1 \leq i \leq n}\... | Proof It is easy to find that\n\n\[ \left( {\frac{\mathop{\sum }\limits_{{i = 1}}^{n}\frac{1}{{x}_{i}}}{n},\cdots ,\frac{\mathop{\sum }\limits_{{i = 1}}^{n}\frac{1}{{x}_{i}}}{n}}\right) \prec \left( {\frac{1}{{x}_{1}},\cdots ,\frac{1}{{x}_{n}}}\right) . \]\n\n(3.14)\n\nThus (3.13) follows from Definition 1.3, Theorem 2... | No |
Theorem 4.1 Let \( \Omega = \{ {A}_{1},{A}_{2},\cdots ,{A}_{n + 1}\} \) be an \( n \) -dimensional simplex in \( {E}^{n}. \) For a given point \( P \) in \( \Omega \), let \( {B}_{i} \) stand for the intersection point of straight line \( {A}_{i}P \) and hyperplane \( {a}_{i} = \left\{ {{A}_{1}\cdots {A}_{i - 1}{A}_{i ... | Proof By the formula in [19]\n\n\[ {F}_{i}{h}_{i} = {nV},\;\mathop{\sum }\limits_{{i = 1}}^{{n + 1}}{F}_{i}r = {nV},\;\mathop{\sum }\limits_{{i = 1}}^{{n + 1}}\left( {F - 2{F}_{i}}\right) {r}_{i} = {nV} \]\n\n(4.6)\n\nwhere \( F = \mathop{\sum }\limits_{{i = 1}}^{{n + 1}}{F}_{i} \) we get\n\n\[ \mathop{\sum }\limits_{{... | Yes |
Consider the convex function \( f\left( x\right) = \frac{x}{2} \) on the interval \( \left\lbrack {0,1}\right\rbrack \), then \( f\left( x\right) \) is a \( \left( {\frac{1}{3},\frac{1}{4}}\right) \) -convex function. And let \( \mu \) be the Lebesgue measure on \( \left\lbrack {0,1}\right\rbrack \) . | In virtue of Proposition 2.3, \( g\left( x\right) = {f}^{2}\left( x\right) = \frac{{x}^{2}}{4} \) is a convex function on \( \left\lbrack {0,1}\right\rbrack \) . In fact, by the convexity of \( f \) ,\n\n\[ g\left( x\right) = {\left( \frac{x}{2}\right) }^{2} = {f}^{2}\left( x\right) = {f}^{2}\left( {x \cdot 1 + \frac{1... | Yes |
Theorem 3.2 Let \( f : \\left\\lbrack {a, b}\\right\\rbrack \\rightarrow \\lbrack 0,\\infty ) \) be \( \\left( {\\alpha, m}\\right) \) -convex function, \( \\left( {\\alpha, m}\\right) \\in {\\left( 0,1\\right) }^{2}, \) \( f\\left( a\\right) \\leq f\\left( b\\right) \) and \( \\mu \) the Lebesgue measure on \( R \) . ... | Proof As \( f \) is an \( \\left( {\\alpha, m}\\right) \) -convex function for \( x \\in \\left\\lbrack {a, b}\\right\\rbrack \), we have\n\n\[ \nf\\left( x\\right) = f\\left( {m\\left( {1 - \\frac{x - {ma}}{b - {ma}}}\\right) a + \\frac{x - {ma}}{b - {ma}}b}\\right)\n\]\n\n\[ \n\\leq m\\left( {1 - {\\left( \\frac{x - ... | Yes |
Theorem 3.3 Let \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \lbrack 0,\infty ) \) be \( \left( {\alpha, m}\right) \) -convex function, \( \left( {\alpha, m}\right) \in {\left( 0,1\right) }^{2} \) , \( f\left( a\right) > f\left( b\right) \) and \( \mu \) the Lebesgue measure on \( R \) . Then,\n\nCase i if \( m ... | Proof Similarly to the proof of Theorem 3.1 and (3) of Proposition 2.1, we consider the function\n\n\[g\left( x\right) = m\left( {1 - {\left( \frac{x - {ma}}{b - {ma}}\right) }^{\alpha }}\right) f\left( a\right) + {\left( \frac{x - {ma}}{b - {ma}}\right) }^{\alpha }f\left( b\right) ,\]\n\nand in this case, we consider ... | Yes |
Example 3.5 Consider the convex function \( f\left( x\right) = {x}^{2} - {3x} + 3 \) on the interval \( \left\lbrack {0,2}\right\rbrack \), then \( f\left( x\right) \) is a \( \left( {\frac{1}{2},\frac{1}{3}}\right) \) -convex function. And let \( \mu \) be the Lebesgue measure on \( \left\lbrack {0,2}\right\rbrack \) ... | By case ii of Theorem 3.2, we have\n\n\[ \text{(S)}{\int }_{0}^{2}{f}^{2}\left( x\right) \mathrm{d}\mu \leq 1\text{.} \] | No |
Lemma 2.3 Let \( \\left( {A\\# H,\\beta \\# \\alpha }\\right) \) be \( L - R \) Hom-smash product, then for all \( a, b \\in \) \( A, h, g \\in H \) the following equalities hold\n\n\\[ \n\\left( {a\\# {1}_{H}}\\right) \\left( {b\\# {1}_{H}}\\right) = {ab}\\# {1}_{H} \n\\]\n\n(12)\n\n\\[ \n\\left( {{1}_{A}\\# h}\\right... | Proof By direct computations. | No |
Theorem 2.4 Let \( \left( {H,\alpha }\right) \) be a monoidal Hom-bialgebra, \( \left( {A,\beta }\right) \) be an \( \left( {H,\alpha }\right) \) - Hom-bimodule bialgebra(that is, \( \left( {A,\beta }\right) \) not only a monoidal Hom-bialgebra, but also a Hom-bimodule algebra and a Hom- bimodule coalgebra ), then the ... | Proof Suppose (16),(17) hold. Then for any \( \left( {a\# h}\right) ,\left( {b\# h}\right) \in \left( {A\# H}\right) \), we have\n\n\[ \n\Delta \left( {a\# h}\right) \Delta \left( {b\# g}\right) = \left( {{a}_{1}\# {h}_{1}}\right) \left( {{b}_{1}\# {g}_{1}}\right) \otimes \left( {{a}_{2}\# {h}_{2}}\right) \left( {{b}_{... | Yes |
Lemma 3.1 Let \( \left( {H,\alpha }\right) \) be a finite monoidal Hom-Hopf algebra with an antipode \( {S}_{H},\left( {A,\beta }\right) \) an \( \left( {H,\alpha }\right) \) -Hom-bimodule algebra, and \( \left( {A\# H,\beta \# \alpha }\right) \) the \( L - R \) Hom-smash product. Then for any \( a \in A, h \in H \), \... | Proof \[ \left( {{1}_{A}\# {h}_{2}}\right) \left( {{S}_{H}^{-1}\left( {{\alpha }^{-1}\left( {h}_{1}\right) }\right) \cdot {\beta }^{-2}\left( a\right) \# {1}_{H}}\right) \;\overset{\left( {15}\right) }{ = }\;\alpha \left( {h}_{21}\right) \cdot \left( {{S}_{H}^{-1}\left( {{\alpha }^{-1}\left( {h}_{1}\right) }\right) \cd... | Yes |
Lemma 3.4 \( \operatorname{Let}\left( {H,\alpha }\right) \) be a finite dimensional simisimple monoidal Hom-Hopf algebra, \( \left( {A,\beta }\right) \) an \( \left( {H,\alpha }\right) \) -Hom-bimodule algebra, and Eq. (16) hold. Let \( \left( {M,\mu }\right) \) be a left \( \left( {A\# H,\beta \# \alpha }\right) \) -H... | Proof Let \( \lambda : \left( {M,\mu }\right) \rightarrow \left( {N,\nu }\right) \) be the canonical projection as \( \left( {A,\beta }\right) \) -Hom-modules.Since \( \left( {H,\alpha }\right) \) is a finite dimensional semisimple monoidal Hom-Hopf algebra, there exists an element \( t \in {\int }_{r}^{H} \) such that... | Yes |
Theorem 2.1 Let \( p \) be an odd prime and let \( a \) be an integer with \( p \nmid a \) . Let\n\n\( b, c \in \mathbb{Z}, m, r \in {\mathbb{Z}}^{ + } = \{ 1,2,3,\ldots \} \) and \( \varepsilon \in \{ \pm 1\} \) . Then\n\n\[{\left( -1\right) }^{r}\mathop{\sum }\limits_{{k = 1}}^{{p - 1}}\frac{\left( \begin{matrix} {2k... | Proof Let \( n = \left( {p - 1}\right) /2 \) . For \( k \in \{ 1,\ldots, p - 1\} \), clearly \( p \mid \left( \begin{matrix} {2k} \\ k \end{matrix}\right) \) if and only if \( k > n \) . By (1.13),\n\n\[k\left( \begin{matrix} {2k} \\ k \end{matrix}\right) \equiv \frac{-{2p}}{\left( \begin{matrix} 2\left( {p - k}\right)... | Yes |
Corollary 2.1 Let \( p > 5 \) be a prime. For any \( b, c \in \mathbb{Z}, m \in {\mathbb{Z}}^{ + } \) and \( \varepsilon \in \{ \pm 1\} \) , we have\n\n\[ \mathop{\sum }\limits_{{k = 1}}^{{p - 1}}\frac{\left( \begin{matrix} {2k} \\ k \end{matrix}\right) }{k}\left( {b\mathop{\sum }\limits_{{j = 1}}^{k}\frac{{\varepsilon... | Proof In view of (1.12) and (1.19),\n\n\[ p\mathop{\sum }\limits_{{k = 1}}^{{p - 1}}\frac{1}{{k}^{2}\left( \begin{matrix} {2k} \\ k \end{matrix}\right) } \equiv \mathop{\sum }\limits_{{k = 0}}^{{\left( {p - 3}\right) /2}}\frac{\left( \begin{matrix} {2k} \\ k \end{matrix}\right) }{\left( {{2k} + 1}\right) {16}^{k}} \equ... | Yes |
Corollary 2.2 Let \( p > 5 \) be a prime. For any \( b, c \in \mathbb{Z}, m \in {\mathbb{Z}}^{ + } \) and \( \varepsilon \in \{ \pm 1\} \) , we have\n\n\[ \mathop{\sum }\limits_{{k = 1}}^{{p - 1}}\frac{{\left( -1\right) }^{k - 1}\left( \begin{aligned} {2k} \\ k \end{aligned}\right) }{{k}^{2}}\left( {b\mathop{\sum }\lim... | Proof In view of (1.12) and (1.20),\n\n\[ p\mathop{\sum }\limits_{{k = 1}}^{{p - 1}}\frac{{\left( -1\right) }^{k - 1}}{{k}^{3}\left( \begin{matrix} {2k} \\ k \end{matrix}\right) } \equiv \mathop{\sum }\limits_{{k = 0}}^{{\left( {p - 3}\right) /2}}\frac{\left( \begin{matrix} {2k} \\ k \end{matrix}\right) }{{\left( 2k + ... | Yes |
Corollary 2.3 Let \( p > 3 \) be a prime. For any \( b, c \in \mathbb{Z}, m \in {\mathbb{Z}}^{ + } \) and \( \varepsilon \in \{ \pm 1\} \) , we have\n\n\[ \mathop{\sum }\limits_{{k = 1}}^{{p - 1}}\frac{\left( \begin{matrix} {2k} \\ k \end{matrix}\right) }{{k}^{3}}\left( {b\mathop{\sum }\limits_{{j = 1}}^{k}\frac{{\vare... | Proof Let \( n = \left( {p - 1}\right) /2 \) . By (2.3),\n\n\[ \mathop{\sum }\limits_{{k = 0}}^{{n - 1}}\frac{(\frac{2k}{k})}{({2k} + 1{)}^{3}{16}^{k}} \equiv \left( \frac{ - 1}{p}\right) \mathop{\sum }\limits_{{k = 0}}^{{n - 1}}\frac{(\frac{2(n - k)}{n - k})}{({2k} + 1{)}^{3}} = \left( \frac{ - 1}{p}\right) \mathop{\s... | Yes |
Theorem 2.2 Let \( p \) be an odd prime and let \( m \) be an integer with \( p \nmid m\left( {m - 4}\right) \) . Let \( \alpha \) be a positive integer, and let \( {a}_{0},{a}_{1},\cdots ,{a}_{{p}^{\alpha } - 1} \) be \( p \) -adic integers. Define \( {a}_{k}^{ * } = \) \( \mathop{\sum }\limits_{{j = 0}}^{k}\left( \be... | Proof Let \( n = \left( {{p}^{\alpha } - 1}\right) /2 \) . By Lucas’ theorem (see, e.g.,[14]), for each \( k = \) \( n + 1,\ldots ,{p}^{a} - 1 \) we have\n\n\[ \left( \begin{matrix} {2k} \\ k \end{matrix}\right) = \left( \begin{matrix} {p}^{a} + \left( {{2k} - {p}^{a}}\right) \\ 0 \cdot {p}^{a} + k \end{matrix}\right) ... | Yes |
For \( k = 0,1,2,\ldots \) define\n\n\[ \n{f}_{k}\left( x\right) \mathrel{\text{:=}} \mathop{\sum }\limits_{{j = 0}}^{k}{\left( \begin{array}{l} k \\ j \end{array}\right) }^{2}\left( \begin{matrix} {2j} \\ k \end{matrix}\right) {x}^{j}\;\text{ and }\;{g}_{k}\left( x\right) \mathrel{\text{:=}} \mathop{\sum }\limits_{{j ... | Proof By [28, Theorem 2.2],\n\n\[ \n{g}_{k}\left( x\right) = \mathop{\sum }\limits_{{j = 0}}^{k}\left( \begin{array}{l} k \\ j \end{array}\right) {\left( -1\right) }^{j}\left( {{\left( -1\right) }^{j}{f}_{j}\left( x\right) }\right) \;\text{ for all }k = 0,1,2,\ldots\n\]\n\nThus, by applying Theorem 2.2 we obtain that\n... | Yes |
Theorem 2.1 Assume \( {2\alpha } \neq 0, - 1, - 2,\ldots \) Then \( {Q}_{\alpha } \) has a fundamental solution \( {\mathcal{F}}_{\alpha } \) in \( {\mathbb{R}}_{ + }^{2} \) which can be expressed by hypergeometric function as\n\n\[ \n{\mathcal{F}}_{\alpha }\left( {x, s;0,{s}_{0}}\right) = - \frac{{2}^{{2\alpha } - 1}}... | Proof The assumption on \( \alpha \) is needed since otherwise the hypergeometric function is not well defined. The expressions (2.2) and (2.3) of \( V \) and \( W \) and the identity \( {w}_{j} = - {h}_{j}{c}_{j} \) imply in \( {\mathbb{R}}_{ + }^{2} \n\n\[ \n{\mathcal{F}}_{\alpha }\left( {x, s;0,{s}_{0}}\right) = {\l... | Yes |
Theorem 3.2 Assume that \( \left( {\mathrm{H}}_{2}\right) - \left( {\mathrm{H}}_{3}\right) \) hold. If the function \( \widetilde{\rho }\left( x\right) \) satisfies\n\n\[ \widetilde{\rho }\left( x\right) \leq \frac{1}{2\varepsilon }\left( {\mathop{\inf }\limits_{{i = 1,\cdots, r}}{\lambda }_{\min }\left( {Q}_{i}\right)... | Proof Let \( V\left( x\right) = {x}^{T}{Px} \) . Calculate the derivative of \( V\left( x\right) \) along the trajectories of the closed-loop system (3.6), we have\n\n\[ \dot{V}\left( x\right) = \mathop{\sum }\limits_{{i = 1}}^{r}{\eta }_{i}^{2}{x}^{T}\left( {{G}_{ii}^{T}P + P{G}_{ii}}\right) x + 2\mathop{\sum }\limits... | Yes |
Theorem 1.1 Let \( f \in H\left( \mathbb{D}\right) ,\beta ,\gamma \in \left( {-1,\infty }\right) \), and let function \( K \) satisfy(1.1)and (1.2). Then \( f \in {Q}_{K} \) if and only if | \[ \mathop{\sup }\limits_{{a \in \mathbb{D}}}{\int }_{\mathbb{D}}{\int }_{\mathbb{D}}\frac{{\left| f\left( z\right) - f\left( w\right) \right| }^{2}}{{\left| 1 - \overline{z}w\right| }^{4 + \beta + \gamma }}K\left( {1 - {\left| {\varphi }_{a}\left( z\right) \right| }^{2}}\right) \frac{{\left( 1 - {\left| z\right| }^{2}... | Yes |
Lemma 2.1 Suppose \( z \in \mathbb{D}, c < 0, t > - 1 \) . Then\n\n\[ \n{\int }_{\mathbb{D}}\frac{{\left( 1 - {\left| w\right| }^{2}\right) }^{t}}{{\left| 1 - z\bar{w}\right| }^{2 + t + c}}\mathrm{\;d}A\left( w\right) \lesssim 1 \n\] | By [6], we know that if \( K \) satisfies (1.1), then\n\n\[ \n{K}_{1}\left( t\right) = {\int }_{0}^{t}K\left( s\right) \frac{ds}{s} \approx K\left( t\right) ,0 < t < 1 \n\]\n\nand if \( K \) satisfies (1.2), then\n\n\[ \n{K}_{2}\left( t\right) = t{\int }_{0}^{t}K\left( s\right) \frac{ds}{{s}^{2}} \approx K\left( t\righ... | No |
Lemma 2.2 Let (1.1)and(1.2) hold for \( K,\beta > - 1 \) . Then\n\n\[ \n{\int }_{\mathbb{D}}{\varphi }_{K}\left( \frac{1 - {\left| z\right| }^{2}}{{\left| 1 - \bar{z}w\right| }^{2}}\right) {\left( 1 - {\left| z\right| }^{2}\right) }^{\beta }\mathrm{d}A\left( z\right) < \infty .\n\] | Proof Since \( K \) satisfies (1.1) and (1.2), by [6], we obtain that\n\n\[ \n{\varphi }_{K}\left( t\right) \approx {\varphi }_{{K}_{1}}\left( t\right) \lesssim {t}^{{c}_{0}},0 < t \leq 1 \n\]\n\nand\n\n\[ \n{\varphi }_{K}\left( t\right) \approx {\varphi }_{{K}_{2}}\left( t\right) \lesssim {t}^{1 - {c}_{0}}, t \geq 1.\... | Yes |
Lemma 2.4 Suppose \( p > 0 \) and \( \alpha \) is real. For any \( R > 0 \) and \( f \in H\left( \mathbb{D}\right) \), there exists a positive constant \( C \) (depending only on \( R \) ) such that\n\n\[{\left| f\left( z\right) \right| }^{p} \leq \frac{C}{{\left( 1 - {\left| z\right| }^{2}\right) }^{2 + \alpha }}{\int... | Proof See Proposition 4.13 of [14], for example. | No |
Theorem 2.2 \( {}^{\left\lbrack {12}\right\rbrack } \) Let \( \left( {\left\lbrack {0,\infty }\right\rbrack ,\sup , \odot }\right) \) be a semiring, when \( \odot \) is generated with \( g \) , i.e., we have \( x \odot y = {g}^{-1}\left( {g\left( x\right) \cdot g\left( y\right) }\right) \) for every \( x, y \in \left( ... | \[ {\int }^{\sup }f \odot \mathrm{d}\mu = \mathop{\lim }\limits_{{\lambda \rightarrow \infty }}{\int }^{{ \oplus }_{\lambda }}f \odot \mathrm{d}\mu = \mathop{\lim }\limits_{{\lambda \rightarrow \infty }}{\left( {g}^{\lambda }\right) }^{-1}\left( {\int {g}^{\lambda }\left( {f\left( x\right) }\right) \mathrm{{dx}}}\right... | No |
Theorem 2. \( {4}^{\left\lbrack {12}\right\rbrack } \) If \( f : X \rightarrow \left\lbrack {a, b}\right\rbrack \) is a measurable function and \( {\bar{\mu }}_{\mathcal{M}} = \left\lbrack {{\mu }_{l},{\mu }_{r}}\right\rbrack \) is the interval-valued \( \oplus \) -measure, then | \[ {\int }_{X}^{ \oplus }f \odot d{\bar{\mu }}_{\mathcal{M}} = \left\lbrack {{\int }_{X}^{ \oplus }f \odot d{\mu }_{l}{\int }_{X}^{ \oplus }f \odot \mathrm{d}{\mu }_{r}}\right\rbrack \] | No |
Theorem 2.5 \( {}^{\left\lbrack 5\right\rbrack } \) Let \( \left( {\left\lbrack {a, b}\right\rbrack ,\oplus , \odot }\right) \) be a semiring that belongs to one of the three basic classes, where \( \oplus = \sup \) or \( \oplus \) is given by an increasing generator \( g \) . Let \( f : X \rightarrow \left\lbrack {a, ... | \[ {\bar{\mu }}_{\mathcal{M}}^{f} = {\int }_{X}^{ \oplus }f \odot d{\bar{\mu }}_{\mathcal{M}} = \left\lbrack {{\int }_{X}^{ \oplus }f \odot d{\mu }_{l,}{\int }_{X}^{ \oplus }f \odot \mathrm{d}{\mu }_{\mathrm{r}}}\right\rbrack \] where \( A \subseteq X \), has the following properties (i) \( {\overline{\mu }}_{\mathcal{... | Yes |
Theorem 3.10 Let \( f, h : \left\lbrack {a, b}\right\rbrack \rightarrow \left\lbrack {a, b}\right\rbrack \) be nonnegative concave measure functions, let a generator \( g : \left\lbrack {a, b}\right\rbrack \rightarrow \left\lbrack {0,\infty }\right\rbrack \) of the pseudo - addition \( \oplus \) and the pseudo - multip... | Proof We use the classical Barnes-Godunova-Levin inequality and then obtain\n\n\[{\left( {\int }_{a}^{b}{\left( g \circ f\right) }^{p}\mathrm{{dx}}\right) }^{\frac{1}{p}}{\left( {\int }_{a}^{b}{\left( g \circ h\right) }^{q}\mathrm{{dx}}\right) }^{\frac{1}{q}} \leq B\left( {p, q}\right) {\int }_{a}^{b}\left( {g \circ f}... | Yes |
Theorem 3.12 Let \( f, h : \left\lbrack {a, b}\right\rbrack \rightarrow {\left\lbrack a, b\right\rbrack }_{ + } \) be nonnegative concave measure function, let a generator \( g : \left\lbrack {a, b}\right\rbrack \rightarrow \left\lbrack {0,\infty }\right\rbrack \) of the pseudo - addition \( \oplus \) and the pseudo - ... | Proof By Theorem 2.5, it holds\n\n\[{\int }_{\left\lbrack a, b\right\rbrack }^{ \oplus }\left( {f \odot h}\right) \odot \mathrm{d}{\bar{\mu }}_{\mathcal{M}} = \left\lbrack {{\int }_{\left\lbrack a, b\right\rbrack }^{ \oplus }\left( {f \odot h}\right) \odot \mathrm{d}{\mu }_{\mathrm{l}},{\int }_{\left\lbrack a, b\right\... | Yes |
Lemma 2.3 \( {\rho }^{\prime \prime }\left( {K}_{n}\right) = \lceil \left( {n + 1}\right) /2\rceil \), where \( n \geq 3 \) is odd. | Proof Let \( n = {2k} + 1 \) where \( k \geq 1 \) and \( V\left( {K}_{n}\right) = \{ u,{v}_{0},{v}_{1},\ldots ,{v}_{{2k} - 1}\} \) . If \( k = 1 \) , then \( {\rho }^{\prime \prime }\left( {K}_{3}\right) = {\rho }^{\prime \prime }\left( {C}_{3}\right) = 2 = \lceil \left( {n + 1}\right) /2\rceil \) by Theorem 1.1. So as... | Yes |
Lemma 2.4 \( {\rho }^{\prime \prime }\left( {K}_{n}\right) = \lceil \left( {n + 1}\right) /2\rceil \), where \( n \geq 2 \) is even. | Proof Let \( n = {2k} \) . Firstly, \( {\rho }^{\prime \prime }\left( {K}_{n}\right) \leq {\rho }^{\prime \prime }\left( {K}_{n + 1}\right) = \lceil \left( {n + 2}\right) /2\rceil = k + 1 \) by Lemma 2.3. On the other hand, \( {\rho }^{\prime \prime }\left( {K}_{n}\right) > k \) . Otherwise, suppose \( {K}_{n} \) has a... | Yes |
Lemma 3.1 \( {\rho }^{\prime \prime }\left( {K}_{n, n}\right) = \lceil \left( {n + 2}\right) /2\rceil \), where \( n \geq 2 \) is even. | Proof \( \Longrightarrow \) Let \( n = {2k}. \) Since \( \Delta \left( {K}_{{2k},{2k}}\right) = {2k}, \) \( {\rho }^{\prime \prime }\left( {K}_{{2k},{2k}}\right) \geq \lceil \frac{{2k} + 1}{2}\rceil = k + 1. \) Next we prove the reverse inequality. Let \( \left( {X, Y}\right) \) be the bipartition of \( {K}_{{2k},{2k}}... | Yes |
Lemma 3.2 \( {\rho }^{\prime \prime }\left( {K}_{n, n}\right) = \lceil \left( {n + 2}\right) /2\rceil \), where \( n \geq 3 \) is odd. | Proof \( \begin{aligned} \text{ Let }n & = {2k} + 1.\text{ Firstly,} \\ {\rho }^{\prime \prime }\left( {K}_{n, n}\right) & \leq {\rho }^{\prime \prime }\left( {K}_{n + 1, n + 1}\right) = \lceil \left( {n + 1 + 2}\right) /2\rceil = k + 2 \end{aligned} \) by Lemma 3.1. Next we prove that \( {\rho }^{\prime \prime }\left(... | Yes |
Lemma 4.1 Let \( G = T \cup C \) be a Halin graph with \( \Delta \left( G\right) \geq 5 \), then \( {\rho }^{\prime \prime }\left( G\right) \leq \) \( {\rho }_{l}^{\prime \prime }\left( G\right) \leq \lceil \left( {\Delta \left( G\right) + 2}\right) /2\rceil \) | Proof It suffices to prove that \( {\rho }_{l}^{\prime \prime }\left( G\right) \leq \lceil \left( {\Delta \left( G\right) + 2}\right) /2\rceil \) . Let \( L\left( x\right) \) be the color list of each \( x \in V\left( G\right) \cup E\left( G\right) \) where \( \left| {L\left( x\right) }\right| = \lceil \left( {\Delta \... | Yes |
Lemma 4.2 Let \( G = T \cup C \) be a Halin graph with \( \Delta \left( G\right) \leq 4 \), then \( {\rho }_{l}^{\prime \prime }\left( G\right) \leq 3 \) . | To prove Lemma 4.2, we need a structural property for Halin graphs which appeared in [16]:\n\nLemma 4. \( {3}^{\left\) | No |
Theorem 1.1 The following are equivalent for any right coherent ring \( R \) . | If any of these conditions holds, then \( \mathcal{G}\mathcal{P} \) is covering. | No |
Lemma 2.1 If \( R \) is a right coherent ring, then \( {M}^{ + } \in \mathcal{F} \cap \mathcal{P}\mathcal{I} \) whenever \( M \) is an injective right \( R \) -module. | Proof The result holds by [11, Theorem 2.2] and [19, Proposition 2.3.5]. | No |
Lemma 2.2 Let \( R \) be a right coherent ring. If \( M \) is a Gorenstein flat left \( R \) - module, then \( {\operatorname{Ext}}_{R}^{ \geq 1}\left( {M, L}\right) = 0 \) for each \( L \in \mathcal{F} \cap \mathcal{P}\mathcal{I} \) and there exists a \( \operatorname{Hom}\left( {-,\mathcal{F} \cap \mathcal{P}\mathcal... | Proof By hypothesis, there is an exact sequence \( \mathbb{F} : \cdots \rightarrow {F}_{1} \rightarrow {F}_{0} \rightarrow {F}^{0} \rightarrow \) \( {F}^{1} \rightarrow \cdots \) of flat left \( R \) -modules with \( M \cong \operatorname{im}\left( {{F}_{0} \rightarrow {F}^{0}}\right) \) such that \( I{ \otimes }_{R}\m... | Yes |
Lemma 2.3 The following are true for any right coherent ring \( R \). (1) \( \mathcal{G}\mathcal{F} \subseteq \mathcal{G}\mathcal{P} \) if and only if \( M \in \mathcal{G}\mathcal{P} \) whenever \( {M}^{ + } \in \mathcal{G}\mathcal{I} \) . | Proof We prove part (1); the proof of (2) is similar. Suppose \( \mathcal{{GF}} \subseteq \mathcal{{GP}} \) . Let \( M \) be a left \( R \) -module with \( {M}^{ + } \in \mathcal{{GI}} \) . Then \( M \in \mathcal{{GF}} \) by [13, Theorem 3.6]. Hence \( M \in \mathcal{{GP}} \) by hypothesis. Conversely, let \( M \) be a... | Yes |
Corollary 2.5 If \( R \) is a right coherent and left perfect ring, then \( \mathcal{{GP}} \) is closed under pure submodules. | Proof The proof is similar to that of Theorem 3.1 in [14] by noting that Theorem 1.1(5) holds. | No |
Corollary 2.6 Let \( R \) be a ring. Then \( \mathcal{G}\mathcal{P} \) is preenveloping if and only if \( \mathcal{G}\mathcal{F} \) is closed under direct products and \( R \) is right coherent and left perfect. | Proof The \ | No |
Theorem 2.3 Assume that (C3) holds. Let \( {\left\{ {s}_{n}\right\} }_{n = 1}^{\infty } \) be determined by \( {s}_{1} = \) \( \tau ,{s}_{2} = \frac{\beta {c}_{1} - \left( {\lambda - \alpha }\right) {a}_{1}}{2{a}_{0}\lambda \left( {\lambda - 1}\right) }{\tau }^{2} \) and \[ {a}_{0}\lambda \left( {\lambda - \alpha }\rig... | Proof Analogously to the proof of Theorem 2.1, let (2.6) be the expansion of a formal solution of (2.5), we also have (2.11) or (2.28). If \( \Phi \left( {{lp} + 1,\lambda }\right) \neq 0 \) for some natural number \( l \), then the equality in (2.28) does not hold for \( n = {lp} + 1 \) since \( 1 - {b}^{-{lp\lambda }... | Yes |
Theorem 3.1 Suppose (I1) or (I2) holds. Then, in a neighborhood of the origin Eq.(3.3) has an analytic solution of the form\n\n\[ g\left( z\right) = {\eta z} + \mathop{\sum }\limits_{{n = 2}}^{\infty }{b}_{n}{z}^{n} \] | Proof If \( \eta = 0 \), then (3.3) has a trivial solution \( \psi \left( z\right) \equiv 0 \) . Assume \( \eta \neq 0 \), we seek a power series solution of (3.3) of the form (3.4). Substituting (3.4) into (3.3) and comparing its coefficients, we obtain\n\n\[ \left\lbrack {{\widetilde{a}}_{0}\left( {\gamma - \alpha }\... | Yes |
Theorem 3.3 Under one of the conditions in Theorem 3.1and Theorems 3.2, Eq.(1.1) has an analytic solution of the form \( x\left( z\right) = \frac{1}{\beta }\lbrack g(\gamma {g}^{-1}\left( z\right) )–{\alpha z}\rbrack \) in a neighborhood of the origin, where \( g \) is an analytic solution of Eq.(3.3) in a neighborhood... | \( \textbf{Proof}\; \) In Theorem 3.1 and Theorems 3.2 we have found a solution \( \psi \) of (3.3) in the form (3.4), which is analytic near 0 . Since \( g\left( 0\right) = 0 \) and \( {g}^{\prime }\left( 0\right) = \eta \neq 0 \) the function \( {g}^{-1} \) is also analytic near \( 0. \) Thus \( \varphi \left( z\righ... | Yes |
Consider the equation\n\n\\[ \frac{3{e}^{z} - 1}{z}{x}^{\prime }\left( z\right) = x\left( {{2z} + {4x}\left( z\right) }\right) + \frac{{e}^{x\left( z\right) }}{x\left( z\right) }.\n\\]\n\n(4.1) | It is in the form of (1.1), where \\( \\alpha = 2,\\beta = 4, G\\left( z\\right) = \\frac{3{e}^{z} - 1}{z}, F\\left( z\\right) = \\frac{{e}^{z}}{z} \\) . Both \\( G \\) and \\( F \\) have a pole at 0. Clearly, \\( \\widetilde{G}\\left( z\\right) = {zG}\\left( z\\right) = 3{e}^{z} - 1 = 2 + \\mathop{\\sum }\\limits_{{n ... | Yes |
Example 3.2 Consider the equation\n\n\[ \cos z \cdot {x}^{\prime }\left( z\right) = x\left( {z + {3x}\left( z\right) }\right) + \frac{1}{2} - {e}^{x\left( z\right) }.\] | It is in the form of (1.1), where \( {\widetilde{a}}_{0} = 1,{\widetilde{c}}_{0} = - \frac{1}{2}, G\left( z\right) = \cos z = 1 - \mathop{\sum }\limits_{{n = 1}}^{\infty }{\left( -1\right) }^{n} \) . \( \frac{{z}^{2n}}{\left( {2n}\right) !}, F\left( z\right) = \frac{1}{2} - {e}^{z} = - \frac{1}{2} - \mathop{\sum }\limi... | Yes |
Proposition 2.1 Let \( \\phi \) be an analytic self-map of \( \\mathbb{D} \) and \( \\phi \\in {\\mathcal{Q}}_{K} \) with\n\n\[ \n\\parallel \\phi {\\parallel }_{\\infty } = \\mathop{\\sup }\\limits_{{z \\in \\mathbb{D}}}\\left| {\\phi \\left( z\\right) }\\right| < 1\n\]\n\nThen the composition operator \( {C}_{\\phi }... | Proof Let \( \\left\\{ {f}_{n}\\right\\} \) be a bounded sequence in \( {\\mathcal{Q}}_{K} \), and converges to 0 uniformly on compact subsets of \( \\mathbb{D} \) . For any \( \\varepsilon > 0 \), since \( \\parallel \\phi {\\parallel }_{\\infty } < 1,\\overline{\\phi \\left( \\mathbb{D}\\right) } \) is a compact subs... | Yes |
Proposition 2.2 Let \( \phi \) be a boundedly valent analytic self-map of \( \mathbb{D} \) such that \( \phi \left( \mathbb{D}\right) \) lies inside a polygon inscribed in the unit circle. Let functions \( {K}_{1},{K}_{2} : \lbrack 0,\infty ) \rightarrow \) \( \lbrack 0,\infty ) \) be increasing functions such that \( ... | Proof (i) \( \Rightarrow \) (ii). Since \( \mathcal{D} \subset {\mathcal{Q}}_{{K}_{1}} \), we see that \( {C}_{\phi } : \mathcal{D} \rightarrow {\mathcal{Q}}_{{K}_{2}} \) is compact. By Theorem \( A,\;\mathop{\lim }\limits_{{\left| a\right| \rightarrow {1}^{ - }}}\| {C}_{\phi }{\sigma }_{a}{\| }_{{\mathcal{Q}}_{{K}_{2}... | Yes |
Lemma 1.4 Let \( R \) be any commutative ring and \( I \) an injective \( R \) -module. For any \( R \) -module \( M \), we have \( {\operatorname{Ext}}_{R \ltimes C}^{i}\left( {{\operatorname{Hom}}_{R}\left( {R \ltimes C, I}\right), M}\right) \cong {\operatorname{Ext}}_{R}^{i}\left( {{\operatorname{Hom}}_{R}\left( {C,... | Proof By Definition 1.1, C has a degreewise finitely generated projective resolution. By Definition 1.3, there exist an \( R \) -module isomorphism \( R \ltimes C \cong R \oplus C \), so as \( R \) -module \( R \ltimes C \) admits a degreewise finitely generated projective resolution. So by \( \lbrack 3 \) , Theorem 3.... | Yes |
Proposition 2.3 \( \; \) Let \( n \) be a non-negative integer. If \( i{d}_{R}\left( C\right) \leq n \), then \( i{d}_{R \ltimes C}\left( P\right) \leq \) \( n \) for every projective \( R \ltimes C \) -module \( P \) . | Proof \( \; \) By [11, Theorem 4.3.2], \( \;i{d}_{R \ltimes C}\left( {R \ltimes C}\right) = i{d}_{R}\left( C\right) .\; \) Hence \( i{d}_{R \ltimes C}\left( {R \ltimes C}\right) \leq n. \) Let \( P \) be any projective \( R \ltimes C \) -module, then \( P \) is a direct summand of some copies of \( R \ltimes C \) . Hen... | Yes |
Proposition 2.4 Let \( C \) be a semidualizing \( R \) -module and \( n \) a non-negative integer.\n\n(1) If \( C - {Gp}{d}_{R}\left( M\right) \leq n \) for every \( R \) -module \( M \), then \( p{d}_{R \ltimes C}\left( E\right) \leq n \) for every injective \( R \ltimes C \) -module \( E \) . | Proof We only prove (1) and the proof of (2) is similar.\n\nTo show that \( p{d}_{R \ltimes C}\left( E\right) \leq n \) for every injective \( R \ltimes C \) -module \( E \), we only need to show \( p{d}_{R \ltimes C}\left( {{\operatorname{Hom}}_{R}\left( {R \ltimes C, I}\right) }\right) \leq n \) for any injective \( ... | Yes |
Theorem 2.5 Let \( C \) a semidualizing module and \( \mathrm{n} \) is a non-negative integer. The following are equivalent\n\n(1) \( i{d}_{R}\left( C\right) \leq n \), i.e., \( C \) is dualizing;\n\n\( \left( 2\right) \) If \( \sup \{ C \) - \( {Gp}{d}_{R}\left( M\right) \mid M \in \mathrm{{Mod}}R\} < \infty \), then ... | Proof (1) \( \Rightarrow \) (2). Since \( i{d}_{R}\left( C\right) \leq n, i{d}_{R \ltimes C}\left( P\right) \leq n \) for every projective \( R \ltimes C \) - module \( P \) by Proposition 2.3. So we have that \( {\operatorname{Ext}}_{R \ltimes C}^{i > n}\left( {M, P}\right) = 0 \) for any \( R \) -module \( M \) . But... | Yes |
Theorem 2.8 Let \( C \) be a semidualizing \( R \) -module. Then \( w{G}_{C} \) -gldim \( \left( R\right) \leq {G}_{C} \) - \( \operatorname{gldim}\left( R\right) \) . | \n\\begin{aligned} \\mathbf{{Proof}}\\text{ If }{G}_{C}\\text{ - }{gldim}\\left( R\\right) & = \\infty , \\\\\n\\text{ it is obviously that }w{G}_{C}\\text{ - }{gldim}\\left( R\\right) & \\leq {G}_{C}\\text{ - }{gldim}\\left( R\\right) . \\end{aligned}\nIf \( {G}_{C} \) - \( {gldim}\\left( R\\right) = n < \\infty \), t... | Yes |
Lemma 2.3 Let \( G \) be a graph. For any induced subgraph \( H \) of \( G, r\left( G\right) \geq r\left( H\right) \) . | Proof If \( H \) is an induced subgraph of \( G \), then \( A\left( H\right) \) is a principle submatrix of \( A\left( G\right) \) . Thus \( r\left( {A\left( G\right) }\right) \geq r\left( {A\left( H\right) }\right) \), i.e. \( r\left( G\right) \geq r\left( H\right) \) . | Yes |
Lemma 2.4 For the vectors \( {\alpha }_{1},{\alpha }_{2},\ldots ,{\alpha }_{n} \) and matrix \( A \) defined above, we have \( A{\alpha }_{1} = {\alpha }_{2}, A{\alpha }_{n} = {\alpha }_{n - 1} \) and \( A{\alpha }_{i} = {\alpha }_{i - 1}^{\left( 2\right) } \) for \( 2 \leq i \leq n - 1; | Proof It is easy to check these equaitons by the definition of \( {\alpha }_{i} \) and \( {\alpha }_{i}^{\left( j\right) } \). | No |
Lemma 2.5 For \( 2 \leq i \leq \frac{n + 1}{2} \), each of the following holds:\n\n(i) for \( 3 \leq j \leq i,{\alpha }_{j}^{\left( i + 1 - j\right) } - A{\alpha }_{j - 1}^{\left( i + 2 - j\right) } + {\alpha }_{j - 2}^{\left( i + 3 - j\right) } = 0 \) ;\n\n(ii) \( {\alpha }_{2}^{\left( i - 1\right) } - A{\alpha }_{1}^... | Proof We prove the equations by the definition of \( {\alpha }_{i}^{\left( j\right) } \) and Lemma 2.4.\n\n(i) \( {\alpha }_{j}^{\left( i + 1 - j\right) } + {\alpha }_{j - 2}^{\left( i + 3 - j\right) } = \mathop{\sum }\limits_{{k = 1}}^{{i + 1 - j}}{\alpha }_{j + 2\left( {k - 1}\right) } + \mathop{\sum }\limits_{{k = 1... | Yes |
Corollary 3.1 Every grid graph \( {P}_{n} \times {P}_{m}\left( {n \leq m}\right) \) , | \[ \eta \left( {{P}_{n} \times {P}_{m}}\right) = \left\{ \begin{matrix} n & \text{ if } & m = n \\ 0 & \text{ if } & m = n + 1 \\ i & \text{ if } & m \geq n + 2\text{ and }{P}_{n} \times {P}_{m}\text{ could be ruduced to a }{P}_{i} \times {P}_{i} \\ 0 & \text{ if } & m \geq n + 2\text{ and }{P}_{n} \times {P}_{m}\text{... | No |
Theorem 1.2 Let \( a, b, c, d \in {\mathbb{Z}}^{ + } \) with \( a \leq b \leq c \leq d. \) Suppose that the polynomial \( {u}^{3} + a{v}^{3} + b{x}^{3} + c{y}^{3} + d{z}^{3} \) is universal over \( \mathbb{N} \). Then \( \left( {a, b, c, d}\right) \) must be among the following 32 quadruples: | \[ \left( {1,2,2,3}\right) ,\left( {1,2,2,4}\right) ,\left( {1,2,3,4}\right) ,\left( {1,2,4,5}\right) ,\left( {1,2,4,6}\right) ,\left( {1,2,4,9}\right) , \] \[ \left( {1,2,4,{10}}\right) ,\left( {1,2,4,{11}}\right) ,\left( {1,2,4,{18}}\right) ,\left( {1,3,4,6}\right) ,\left( {1,3,4,9}\right) ,\left( {1,3,4,{10}}\right)... | Yes |
Problem 4.1 Determine \( s\left( k\right) \) and \( t\left( k\right) \) for any integer \( k > 1 \), where \( s\left( k\right) \) is the smallest positive integer \( s \) such that\n\n\[ \n\\left\\{ {{a}_{1}{x}_{1}^{k} + {a}_{2}{x}_{2}^{k} + \\ldots + {a}_{s}{x}_{s}^{k} : {x}_{1},\\ldots ,{x}_{s} \\in \\mathbb{N}}\\rig... | Clearly, \( s\left( k\right) \\leq t\left( k\right) \\leq g\left( k\right) \) for all \( k = 2,3,4,\\ldots \) It is easy to see that \( s\left( 2\right) = \) \( t\left( 2\right) = 4 \) . By Theorem 1.1 or 1.2, we have \( s\left( 3\right) \\geq 5 \) . In view of Conjecture \( {1.2},{u}^{3} + {v}^{3} + \) \( 2{x}^{3} + 2... | Yes |
Theorem 5.1 Let \( a > 2 \) be an integer. If \( a \) is even or \( a \) is composite with a prime divisor congruent to 3 mod 4 , then\n\n\[ \left\{ {{x}^{2} + {y}^{2} - {z}^{a} : x, y, z \in \mathbb{Z}}\right\} \neq \mathbb{Z}. \] | Proof If \( a \) is even, \( m, z \in \mathbb{Z} \) and \( m \equiv 6\left( {\;\operatorname{mod}\;8}\right) \), then \( m + {z}^{a} \equiv 6,7\left( {\;\operatorname{mod}\;8}\right) \) and hence \( m + {z}^{a} \) is not the sum of two squares.\n\nNow suppose that \( a = {pn} \) with \( n > 1 \) odd and \( p \) a prime... | Yes |
Theorem 3.2 Let \( 1 < p < \infty \) and \( \left( {A, D\left( A\right) }\right) \) be a Hille-Yosida operator. The function \( f \) satisfies the conditions \( \left( {H}_{1}\right) \) and \( \left( {H}_{2}\right) \) . If \( 0 < \frac{2M}{{e}^{\omega } - 1}{\left( \frac{{e}^{\omega q} - 1}{\omega q}\right) }^{\frac{1}... | Proof Define the operator \( \Gamma \) acting on \( {P}_{TA}\left( {R,{X}_{0}}\right) \) by \[ {\Gamma u}\left( t\right) = {\int }_{-\infty }^{t}{P}_{S, - 1}{T}_{-1}\left( {t - s}\right) f\left( {s, u\left( s\right) }\right) \mathrm{d}s - {\int }_{t}^{+\infty }{P}_{U, - 1}{T}_{-1}\left( {t - s}\right) f\left( {s, u\lef... | Yes |
Theorem 3.3 Let \( p = 1 \) and \( \left( {A, D\left( A\right) }\right) \) be a Hille-Yosida operator. The function \( f \) satisfies the conditions \( \left( {H}_{1}\right) \) and \( \left( {H}_{2}\right) . \) If \( 0 < \frac{{2M}\parallel L{\parallel }_{{S}^{1}}}{1 - {e}^{-\omega }} < 1 \), then Eq.(1) has a unique \... | Proof Proceeding as in the proof of Theorem 3.2, we define the operator \( \Gamma \) on \( {P}_{TA}\left( {R,{X}_{0}}\right) \) by\n\n\[ \n{\Gamma u}\left( t\right) = {\int }_{-\infty }^{t}{P}_{S, - 1}{T}_{-1}\left( {t - s}\right) f\left( {s, u\left( s\right) }\right) \mathrm{d}s - {\int }_{t}^{+\infty }{P}_{U, - 1}{T}... | Yes |
Theorem 3.4 Let \( \left( {A, D\left( A\right) }\right) \) be a Hille-Yosida operator. The function \( f \) satisfies the condition \( \left( {H}_{1}\right) \) and the Lipschitz condition\n\n\[ \parallel f\left( {t, x}\right) - f\left( {t, y}\right) \parallel \leq L\parallel x - y\parallel \]\n\nfor all \( t \in R, x, ... | Proof Similar as the proof of Theorem 3.2 and Theorem 3.3, we define the operator \( \Gamma \) on \( {P}_{TA}\left( {R,{X}_{0}}\right) \) by\n\n\[ {\Gamma u}\left( t\right) = {\int }_{-\infty }^{t}{P}_{S, - 1}{T}_{-1}\left( {t - s}\right) f\left( {s, u\left( s\right) }\right) \mathrm{d}s - {\int }_{t}^{+\infty }{P}_{U,... | Yes |
Theorem 2.1 Assume \( {\mathbf{H}}_{1} - {\mathbf{H}}_{3} \) hold, then in space \( C\left( {0, T;\mathbb{R}}\right) \) as \( \epsilon \rightarrow 0 \) . | Remark Here notice the extra term in the limit equation including two parts\n\n\[\n\frac{1}{2}{\left\lbrack \frac{\sigma \left( {{x}^{\epsilon }\left( t\right) }\right) }{\alpha \left( {{x}^{\epsilon }\left( t\right) }\right) }\right\rbrack }_{x}^{\prime }\frac{\sigma \left( {{x}^{\epsilon }\left( t\right) }\right) }{\... | Yes |
Lemma 2.2 Assume \( {\mathbf{H}}_{1} - {\mathbf{H}}_{3} \), Moreover for any \( 0 \leq t \leq T \)\n\n\[ \mathbb{E}\left| {\epsilon {\dot{x}}^{\epsilon }\left( t\right) }\right| \rightarrow 0,\;\text{ as }\;\epsilon \rightarrow 0 \]\n\nand\n\n\[ {\epsilon }^{3/2}\mathbb{E}{\int }_{0}^{t}{\left\lbrack {\dot{x}}^{\epsilo... | Proof Let \( {y}^{\epsilon } = {\dot{x}}^{\epsilon } \), then equations (1)-(2) is rewritten as\n\n\[ {\dot{x}}^{\epsilon } = {y}^{\epsilon },\;{x}^{\epsilon }\left( 0\right) = {x}_{0} \]\n\n(7)\n\n\[ \epsilon {\dot{y}}^{\epsilon } = - \alpha \left( {x}^{\epsilon }\right) {y}^{\epsilon } + f\left( {x}^{\epsilon }\right... | Yes |
Lemma 3.1\n\n\\[ \n{A}^{\\epsilon }{X}^{\\epsilon }\\left( t\\right) \n\\]\n\n\\[ \n= {\\varphi }^{\\prime }\\left( {{x}^{\\epsilon }\\left( t\\right) }\\right) \\frac{f\\left( {{x}^{\\epsilon }\\left( t\\right) }\\right) }{\\alpha \\left( {{x}^{\\epsilon }\\left( t\\right) }\\right) } + {\\varphi }^{\\prime \\prime }\... | Proof This is a direct computation. First by the definition of \\( {A}^{\\epsilon } \\) and equations (7)-(8) we have\n\n\\[ \n{A}^{\\epsilon }\\varphi \\left( {{x}^{\\epsilon }\\left( t\\right) }\\right) = - {\\varphi }^{\\prime }\\left( {{x}^{\\epsilon }\\left( t\\right) }\\right) \\frac{\\epsilon {\\dot{y}}^{\\epsil... | Yes |
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