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(i) Function \( f \) has at most one \( {\nabla }_{H} \) -derivative at \( \mathrm{t} \) . | Proof (i) The proof is easy and will be omitted. | No |
(i) If \( \mathbb{T} = \mathbb{R} \), then \( f : \mathbb{R} \rightarrow {\mathbb{R}}_{\mathcal{F}} \) is \( {\nabla }_{H} \) -differentiable at \( t \) if and only if the limits\n\n\[ \mathop{\lim }\limits_{{h \rightarrow {0}^{ + }}}\frac{f\left( {t + h}\right) { \ominus }_{g}f\left( t\right) }{h}\text{ and }\mathop{\... | \[ {f}^{{\nabla }_{H}}\left( t\right) = \mathop{\lim }\limits_{{h \rightarrow {0}^{ + }}}\frac{f\left( {t + h}\right) { \ominus }_{g}f\left( t\right) }{h} = \mathop{\lim }\limits_{{h \rightarrow {0}^{ + }}}\frac{f\left( t\right) { \ominus }_{g}f\left( {t - h}\right) }{h}. \] | Yes |
If \( f : \mathbb{T} \rightarrow {\mathbb{R}}_{\mathcal{F}} \) is defined by \( f\left( t\right) = \left\lbrack {0,{t}^{2}}\right\rbrack \) for all \( t \in \mathbb{T} \mathrel{\text{:=}} \left\{ {\frac{n}{2} : n \in {\mathbb{N}}_{0}}\right\} \) , then from Theorem 3.2 (ii), we have that \( f \) is \( {\nabla }_{H} \) ... | \[ {f}^{{\nabla }_{H}}\left( t\right) = \frac{f\left( t\right) { \ominus }_{g}{f}^{\rho }\left( t\right) }{\eta \left( t\right) } = \frac{\left\lbrack {0,{t}^{2}}\right\rbrack - \left\lbrack {0,{t}^{2} - t + \frac{1}{4}}\right\rbrack }{\frac{1}{2}} = \left\lbrack {0,{2t} - \frac{1}{2}}\right\rbrack . \] | Yes |
Theorem 3.5 Assume that \( f, g : \mathbb{T} \rightarrow {\mathbb{R}}_{\mathcal{F}} \) are \( {\nabla }_{H} \) -differentiable at \( t \in {\mathbb{T}}_{k} \) . Then\n\n(i) for any constants \( {\lambda }_{1},{\lambda }_{2} \), the sum \( \left( {{\lambda }_{1}f \oplus {\lambda }_{2}g}\right) : \mathbb{T} \rightarrow {... | Proof (i) Since \( f \) and \( g \) are \( {\nabla }_{H} \) -differentiable at \( t \in {\mathbb{T}}_{k} \), for any \( \epsilon > 0 \), there exist neighborhoods \( {U}_{1} \) and \( {U}_{2} \) of \( t \) with\n\n\[ \n\left\{ \begin{array}{l} D\left\lbrack {{\lambda }_{1}f\left( {t + h}\right) { \ominus }_{g}{\lambda ... | Yes |
Theorem 2.5 Every non-associative superalgebra \( \left( {A, \cdot }\right) \) has an Akivis superalgebra \( \left( {A,\circ ,\left\lbrack {-,-, - }\right\rbrack }\right) \) structure with respect to the operation defined by\n\n\[ x \circ y = x \cdot y - {\left( -1\right) }^{\left| x\right| \left| y\right| }y \cdot x, ... | Proof First, we proceed to verify that \ | No |
Lemma 2.6 Let \( \\left( {A, \\cdot }\\right) \) be a Leibniz superalgebra, and consider on \( \\left( {A, \\cdot }\\right) \) the operation \( \\left\\lbrack {x, y}\\right\\rbrack \\mathrel{\\text{:=}} x \\cdot y - {\\left( -1\\right) }^{\\left| x\\right| \\left| y\\right| }y \\cdot x \) for all \( x, y \\in {hg}\\lef... | Proof (i) Equation (2.1) implies that \n\n\[ \n\\left( {x \\cdot y}\\right) \\cdot z = x \\cdot \\left( {y \\cdot z}\\right) - {\\left( -1\\right) }^{\\left| x\\right| \\left| y\\right| }y \\cdot \\left( {x \\cdot z}\\right) . \n\] \n\n\\left( {2.13}\\right) \n\nLikewise, interchanging \( x \) and \( y \), we have \n\n... | Yes |
Lemma 2.7 Let \( \left( {A, \cdot }\right) \) be a Leibniz superalgebra, \( \left( {A,\circ ,\left\lbrack {-,-, - }\right\rbrack }\right) \) be an Akivis super-algebra associated with Leibniz superalgebra \( \left( {A,}\right) \) . Then\n\n\[ \n{\circlearrowleft }_{x, y, z}{\left( -1\right) }^{\left| x\right| \left| z\... | Proof We get the result from equation (2.10). | No |
Proposition 2.14 Let \( \left( {A, \prec , \succ }\right) \) be a dendriform superalgebra. Define two even bilinear maps \( * ,\left\lbrack {-, - }\right\rbrack : A \times A \rightarrow A \) such that \( x * y = x \prec y + y \succ x,\left\lbrack {x, y}\right\rbrack = {\left( -1\right) }^{\left| x\right| \left| y\right... | Proof Calculate directly,\n\n\[ \left\lbrack {x,\left\lbrack {y, z}\right\rbrack }\right\rbrack = {\left( -1\right) }^{\left| y\right| \left| z\right| + \left| x\right| \left( {\left| y\right| + \left| z\right| }\right) }\left( {z \prec y}\right) \prec x + {\left( -1\right) }^{\left| y\right| \left| z\right| + \left| x... | Yes |
Theorem 3.4 Every (left) Leibniz superalgebra \( \left( {A, \cdot }\right) \) has a Lie-Yamaguti superalgebra structure \( \left( {A,\left\lbrack {-, - }\right\rbrack ,\{ -,-, - \} }\right) \) with respect to the operation defined by\n\n\[ \left\lbrack {x, y}\right\rbrack \mathrel{\text{:=}} x \cdot y - {\left( -1\righ... | Proof Equations (3.2), (2.1) and (2.8) imply\n\n\[ \{ x, y, z\} = - \left( {x \cdot y}\right) \cdot z. \]\n\n(3.3)\n\nMoreover, we have\n\n\[ \left\lbrack {x, y}\right\rbrack \cdot z = \left( {x \cdot y - {\left( -1\right) }^{\left| x\right| \left| y\right| }y \cdot x}\right) \cdot z = 2\left( {x \cdot y}\right) \cdot ... | Yes |
Example 3.6 Let \( A = {A}_{\overline{0}} \oplus {A}_{\overline{1}} \) be a 3-dimensional superspace. \( {A}_{\overline{0}} = \operatorname{span}\left\{ {{e}_{1},{e}_{3}}\right\} \) , \( {A}_{\mathrm{I}} = \operatorname{span}\left\{ {e}_{2}\right\} \) . The nonzero product is given by \( {e}_{2} \cdot {e}_{3} = {e}_{2}... | By Theorem 2.8, when we define the binary operation and the ternary operation by (3.1) and (3.2), we get a Lie-Yamaguti superalgebra \( \left( {A,\left\lbrack {-, - }\right\rbrack ,\{ -,-, - \} }\right) \) with nonzero product\n\n\[ \left\lbrack {{e}_{2},{e}_{3}}\right\rbrack = 2{e}_{2} = - \left\lbrack {{e}_{3},{e}_{2... | No |
Theorem 2.3 Let \( \vartheta \) denote a prime of \( {R}_{d},\delta = \sqrt{d}, D = - d \) . For an arbitrary positive integer \( n \), the equivalence classes of \( {R}_{d}/\left\langle {\vartheta }^{n}\right\rangle \) are of the following types:\n\n(1) \( {R}_{d}/\left\langle {\delta }^{2m}\right\rangle = \left\{ {\l... | Proof (1) As \( {\delta }^{2m} = {d}^{m} \), we get that \( \left\langle {\delta }^{2m}\right\rangle = \left\langle {D}^{m}\right\rangle \) . Suppose \( \alpha = {a}_{1} + {a}_{2}\sqrt{d} \in {R}_{d} \) , where \( {a}_{1},{a}_{2} \in \mathbb{Z} \) . Let \( {a}_{i} = {D}^{m}{k}_{i} + {r}_{i} \) with \( 0 \leq {r}_{i} \l... | Yes |
To illustrate the case \( d = - {19}, q = {23} = \pi \bar{\pi } \) and \( n = 2 \), let \( \gamma = \) \( \frac{1}{2}\left( {{b}_{1} + {b}_{2}\sqrt{-{19}}}\right) \in {R}_{d} \), where \( {b}_{1} = 3 \) and \( {b}_{2} = 1 \). We give the equivalence class in \( {R}_{d}/\left\langle {\pi }^{2}\right\rangle \) which \( \... | Since \( \pi = 2 - \sqrt{-{19}} \) is a proper factor of \( q \) in \( {R}_{d},{\pi }^{2} = - {15} - 4\sqrt{-{19}} = \) \( \frac{-{30}}{2} - \frac{8}{2}\sqrt{-{19}} \) . Denoted by \( s = - {30}, t = - 8 \) . Substituting the values for \( s, t,{b}_{1},{b}_{2}, d, q \) and \( n \) into congruence (2.6), we get that \( ... | Yes |
Theorem 2.6 Let \( \bar{R} = {R}_{d}/\left\langle {\left( \sqrt{d}\right) }^{n}\right\rangle, n \) is an arbitrary positive integer. Let \( D = - d \) . Then the unit groups \( U\left( \bar{R}\right) \) of \( \bar{R} \) are as the follows:\n\n(1) Let \( n = 1 \) . Then \( U\left( \bar{R}\right) \cong {\mathbb{Z}}_{D - ... | Proof (1) If \( n = 1 \), by Theorem 2.1 (2), \( \bar{R} \) is a field of order \( D = - d \), so \( \left| {U\left( \bar{R}\right) }\right| = D - 1 \) . Therefore, \( U\left( \bar{R}\right) \) is a cyclic group of order \( D - 1 \) and hence \( U\left( \bar{R}\right) \cong {\mathbb{Z}}_{D - 1} \) . | Yes |
Theorem 2.7 Let \( p \in \mathbb{Z} \) be an odd prime satisfying the Legendre symbol \( \left( \frac{p}{-d}\right) = - 1 \) . Let \( \bar{R} = {R}_{d}/\left\langle {p}^{n}\right\rangle, n \geq 1 \) . Then \( U\left( \bar{R}\right) \cong {\mathbb{Z}}_{{p}^{2} - 1} \times {\mathbb{Z}}_{{p}^{n - 1}} \times {\mathbb{Z}}_{... | Proof For \( \alpha = \left\lbrack {a + b\sqrt{d}}\right\rbrack \in {R}_{d}/\left\langle {p}^{n}\right\rangle \), where \( 0 \leq a, b \leq {p}^{n} - 1 \), it is easy to prove that \( \alpha \) is a unit of \( \bar{R} \) if and only if \( p \nmid \left( {{a}^{2} - d{b}^{2}}\right) \) . So \( \left| {U\left( \bar{R}\rig... | Yes |
Theorem 2.8 Let \( q \in \mathbb{Z} \) be a prime satisfying the Legendre symbol \( \left( \frac{q}{-d}\right) = 1 \) . Suppose that \( \pi \) is a proper factor of \( q \) . Let \( \bar{R} = {R}_{d}/\left\langle {\pi }^{n}\right\rangle, n \geq 1 \) .\n\n(1) Suppose \( q = 2 \) . Then \( U\left( \bar{R}\right) \cong {\... | Proof Applying Theorem 2.1 (4), we derive that \( \bar{R} \cong \mathbb{Z}/\left\langle {q}^{n}\right\rangle \) . So the theorem follows. | No |
Theorem 2.9 Suppose \( d = - 3, - {11}, - {19}, - {43}, - {67}, - {163} \) . Let \( \bar{R} = {R}_{d}/\left\langle {2}^{n}\right\rangle, n \geq 2 \) . Then\n\n(1) \( U\left( \bar{R}\right) = {\bar{R}}_{1} \cup {\bar{R}}_{2} \cup {\bar{R}}_{3} \), where\n\n\( {\bar{R}}_{1} = \left\{ {\left\lbrack {{r}_{1} + {r}_{2}\sqrt... | Proof (1) If \( \alpha = \left\lbrack {{r}_{1} \pm {r}_{2}\sqrt{d}}\right\rbrack \in \bar{R} \), where \( {r}_{1},{r}_{2} \in \mathbb{Z} \), it is easy to show that \( \alpha \in U\left( \bar{R}\right) \) if and only if \( 2 \nmid N\left( \alpha \right) \), i.e., \( 2 \nmid \left( {{r}_{1}^{2} - d{r}_{2}^{2}}\right) \)... | Yes |
Consider the following second-order four-point nonlinear ordinary differential equation \( \left\lbrack {{15},{29}}\right\rbrack \)\n\n\[ x\left( {1 - x}\right) {\mu }^{\prime \prime }\left( x\right) + 6{\mu }^{\prime }\left( x\right) + {2\mu }\left( x\right) + {\mu }^{2}\left( x\right) = f\left( x\right) ,\;0 \leq x \... | The exact solution is given by \( \mu \left( x\right) = \sinh \left( x\right) \) . In Table 1, we list the absolute errors at some different points obtained by the present method with \( k = 3 \) and \( M = 4,6,8,{10} \) . As we see from this table, it is clear that the result obtained by the present method is superior... | Yes |
Consider the following third-order three-point nonlinear boundary value problems (see [13]) \( {\mu }^{\prime \prime \prime }\left( x\right) = {e}^{-x}{\mu }^{2}\left( x\right) ,0 \leq x \leq 1 \) with the following nonlinear conditions\n\n\[ \mu \left( 0\right) + 2{\mu }^{\prime 2}\left( \frac{1}{2}\right) - \mu \left... | The exact solution is \( \mu \left( x\right) = {e}^{x} \). | No |
Consider the following fourth-order three-point linear boundary value problem \( \left\lbrack {{18},{22}}\right\rbrack \)\n\n\[ \n{\mu }^{\left( 4\right) }\left( x\right) = {e}^{x}{\mu }^{\prime \prime \prime }\left( x\right) - \mu \left( x\right) - {e}^{x}\cosh \left( x\right) + 2\sinh \left( x\right) + 1,0 \leq x \le... | The exact solution is \( \mu \left( x\right) = 1 + \sinh \left( x\right) \) . In Table 3, we compare the absolute errors at some different points obtained by the present method and reproducing kernel methods in [18] and differential transform method in [22]. In Figure 3, we show the maximum absolute errors at selected ... | Yes |
Example 4 Consider the following fifth-order four-point boundary value problems [18]\n\n\\[ \n{\\mu }^{\\left( 5\\right) }\\left( x\\right) + \\sin \\left( {2x}\\right) {\\mu }^{\\prime \\prime \\prime }\\left( x\\right) - {\\mu }^{\\prime }\\left( x\\right) + \\cos \\left( {2x}\\right) \\mu \\left( x\\right) = - \\sin... | The exact solution is \\( \\mu \\left( x\\right) = \\sin x \\) . In Table 4, we compare the absolute errors at some different points obtained by the present method and the reproducing kernel method (RKM) in [18]. In Figure 4, we show the maximum absolute errors at selected points with \\( k = 3 \\) and \\( M = 3 \\) th... | Yes |
Example 5 Consider following seventh-order three-point nonlinear boundary value problems [31]\n\n\\[ \n{\\mu }^{\\left( 7\\right) }\\left( x\\right) = \\mu \\left( x\\right) {\\mu }^{\\prime }\\left( x\\right) - {e}^{x}\\left( {6 + x - x{e}^{x} + {x}^{2}{e}^{x}}\\right) ,\\;0 \\leq x \\leq 1 \n\\]\n\nwith boundary cond... | The exact solution is \\( \\mu \\left( x\\right) = \\left( {1 - x}\\right) {e}^{x} \\) . In Table 5, we compare the absolute errors at some different points obtained by the present method and variational iteration method (VIM) in [31]. In Figure 5, we show the maximum absolute errors at selected points with \\( k = 3 \... | Yes |
Lemma 2.2 (Non-homogeneous linear estimate) Let \( s \in \mathbb{R} \), there exists \( C > 0 \) such that, for any \( f \in {X}^{s, b - 1} \) , | \[ {\begin{Vmatrix}{\theta }_{T}\left( t\right) {\int }_{0}^{t}V\left( t - {t}^{\prime }\right) f\left( {t}^{\prime }\right) \mathrm{d}{t}^{\prime }\end{Vmatrix}}_{{X}^{s, b}} \leq C{T}^{\frac{1 - {2b}}{2}}\parallel f{\parallel }_{{X}^{s, b - 1}} \] | No |
Lemma 3.1 (see [4, Propositon 6.1]) Let \( H,{N}_{1},{N}_{2},{N}_{3},{L}_{1},{L}_{2},{L}_{3} > 0 \) obey (3.5)-(3.7) and let the dispersion relations be given by (3.4).\n\n(i) If \( {N}_{\max } \sim {N}_{\min } \) and \( {L}_{\max } \sim H \), then we have\n\n\[ \left( {3.3}\right) \lesssim {L}_{\min }^{\frac{1}{2}}{N}... | (3.8) | No |
Proposition 3.1 For \( s > - 1 \) and \( u, v \in {X}^{s, b} \), there exists \( b \in \left( {1/2,1}\right) \) such that the bilinear inequality holds\n\n\[ \n{\begin{Vmatrix}{u}_{x}{v}_{x}\end{Vmatrix}}_{{X}^{s, b - 1}} \lesssim \parallel u{\parallel }_{{X}^{s, b}}\parallel v{\parallel }_{{X}^{s, b}}, \]\n\n(3.11)\n\... | Proof By Plancherel's formula and duality, it suffices to show that\n\n\[ \n{\begin{Vmatrix}\frac{\langle {\xi }_{1}{\rangle }^{-s}\langle {\xi }_{2}{\rangle }^{-s}\langle {\xi }_{3}{\rangle }^{s}{\xi }_{1}{\xi }_{2}}{\langle i\left( {{\tau }_{1} - {\xi }_{1}^{3}}\right) + {\xi }_{1}^{4} - {\xi }_{1}^{2}{\rangle }^{b}\... | Yes |
Example 2 Let \( X \) be a finite set defined as \( X = \{ 1,2,3,4\} \) . Define \( d : X \times X \rightarrow \lbrack 0,\infty ) \) as\n\n\[ d\left( {1,1}\right) = d\left( {2,2}\right) = d\left( {3,3}\right) = d\left( {4,4}\right) = 0, \]\n\n\[ d\left( {1,2}\right) = d\left( {2,1}\right) = 3 \]\n\n\[ d\left( {2,3}\rig... | The function \( d \) is not a metric on \( X \) . Indeed, note that\n\n\[ 3 = d\left( {1,2}\right) \geq d\left( {1,3}\right) + d\left( {3,2}\right) = 1 + 1 = 2, \]\n\nthat is, the triangle inequality is not satisfied. However \( d \) is a generalized metric on \( X \) , moreover, \( \left( {X, d}\right) \) is a complet... | Yes |
Lemma 3.1 Assume (A1)-(A3). Then \( {\widehat{\beta }}_{n},{\widehat{\theta }}_{n} \) and \( {\widehat{\sigma }}_{n}^{2} \), the QML estimators of \( \beta ,\theta \) and \( {\sigma }^{2} \) in model (1.1)-(1.2) exist. And as \( n \rightarrow \infty \) , \[ \left( {{\widehat{\mathcal{G}}}_{n},{\widehat{\sigma }}_{n}^{2... | Proof See Theorem 3.1 and Theorem 3.2 in Hu [8]. | No |
Lemma 4.1 (i) If \( \alpha < {\delta \eta } \), an optimal strategy does not exist and \( V\left( x\right) = \infty \) . | Proof This lemma can be proved analogously as in the proof of [11, Lemma 5.1].\n\n(i) Let \( {D}^{0} \in \mathcal{D} \) with the barrier \( b = 0 \) . We define the strategy \( {D}_{t}^{\left( 0, a\right) } = {D}_{t}^{0} + {at} \) for some \( a > 0 \) . Now, we have \( {R}_{t}^{\left( 0, a\right) } \leq 0 \) and\n\n\[ ... | Yes |
Theorem 1.4 If \( n = {p}^{\alpha }{q}^{\beta }{n}_{1} > {pq} \), then | \[ {\varphi }_{pq}\left( n\right) = \left\{ \begin{array}{ll} {\varphi }_{pq}\left( {n}_{1}\right) , & \alpha = \beta = 0; \\ {\varphi }_{q}\left( {n}_{1}\right) - {\varphi }_{pq}\left( {n}_{1}\right) , & \alpha = 1,\beta = 0; \\ {\varphi }_{p}\left( {n}_{1}\right) - {\varphi }_{pq}\left( {n}_{1}\right) , & \alpha = 0,... | Yes |
Theorem 1.5 For \( n = {p}^{\alpha }{q}^{\beta }{n}_{1} > {pq} \) . | (1) If \( \alpha = \beta = 0 \), then\n\n\[ \n{\varphi }_{pq}\left( n\right) = \left\{ \begin{array}{ll} \frac{\varphi \left( n\right) }{pq}, & {p}_{i} \equiv 1\left( {\;\operatorname{mod}\;{pq}}\right) \left( {i = 1,\cdots, k}\right) ; \\ \frac{1}{pq}\varphi \left( n\right) + \frac{\left( {{pq} - 2}\right) {\left( -1\... | No |
Lemma 2.4 Let \( \alpha > 0, m \in \mathbb{N} \) and \( D = d/{dx} \) . If the fractional derivatives \( \left( {{D}_{0 + }^{\alpha }y}\right) \left( t\right) \) and \( \left( {{D}_{0 + }^{\alpha + m}y}\right) \left( t\right) \) exist, then | \[ \left( {{D}^{m}{D}_{0 + }^{\alpha }y}\right) \left( t\right) = \left( {{D}_{0 + }^{\alpha + m}y}\right) \left( t\right) \] | Yes |
Lemma 3.1 If \( u\left( {x, t}\right) \) is a positive solution of problems (1.1)-(1.2) in the domain \( D \) , then \( v\left( t\right) \) satisfies the fractional differential inequality\n\n\[ \frac{d}{dt}\left( {{D}_{0 + }^{\alpha }v\left( t\right) \omega \left( t\right) }\right) \leq \frac{\omega \left( t\right) }{... | Proof Let \( u\left( {x, t}\right) \) is a positive solution of problem (1.1)-(1.2) in the domain \( D \), then there exists \( {t}_{0} > 0 \), such that \( u\left( {x, t}\right) > 0 \) in \( \Omega \times \left\lbrack {{t}_{0},\infty }\right) \) . Integrating (1.1) with respect to \( x \) over \( \Omega \) yields\n\n\... | Yes |
Lemma 3.2 If \( u\left( {x, t}\right) \) is a negative solution of problems (1.1)-(1.2) in the domain \( D \) , then \( v\left( t\right) \) satisfies the fractional differential inequality\n\n\[ \frac{d}{dt}\left( {{D}_{0 + }^{\alpha }v\left( t\right) \omega \left( t\right) }\right) \geq \frac{\omega \left( t\right) }{... | Proof Let \( u\left( {x, t}\right) \) is a negative solution of problems (1.1)-(1.2) in the domain \( D \), then there exists \( {\bar{t}}_{0} > 0 \), such that \( u\left( {x, t}\right) < 0 \) in \( \Omega \times \left\lbrack {{\bar{t}}_{0},\infty }\right) \) . Integrating (1.1) with respect to \( x \) over \( \Omega \... | Yes |
Theorem 3.3 If inequality (3.1) has no eventually positive solutions and the inequality (3.3) has no eventually negative solutions, then every solution of problems (1.1)-(1.2) is oscillatory in \( D \) . | Proof Suppose to the contrary that there is a nonoscillatory solution \( u\left( {x, t}\right) \) of problems (1.1)-(1.2). It is obvious that there exists \( {\widetilde{t}}_{0} \) such that \( u\left( {x, t}\right) > 0 \) or \( u\left( {x, t}\right) < 0 \) for \( t \geq {\widetilde{t}}_{0} \). If \( u\left( {x, t}\rig... | Yes |
Lemma 3.4 If\n\n\\[ \n\\mathop{\\liminf }\\limits_{{t \\rightarrow + \\infty }}{\\int }_{{t}_{1}}^{t}\\frac{M + {\\int }_{{t}_{1}}^{\\rho }\\frac{\\omega \\left( s\\right) }{r\\left( s\\right) }G\\left( s\\right) {ds}}{\\omega \\left( \\rho \\right) }{d\\rho } = - \\infty ,\n\\]\n\nthen inequality (3.1) has no eventual... | Proof Suppose to the contrary that (3.1) has a positive solution \\( v\\left( t\\right) \\), then there exists \\( {t}_{1} \\geq {t}_{0} \\) such that \\( v\\left( t\\right) > 0, t \\geq {t}_{1} \\) . Integrating both sides of (3.1) from \\( {t}_{1} \\) to \\( t \\), we obtain\n\n\\[ \n\\left( {{D}_{0 + }^{\\alpha }v\\... | Yes |
Theorem 2.9 Under Assumptions 2.1-2.5, design sliding mode function \( s\left( t\right) = {D}_{t}^{q - 1}\left( {{e}_{1} + }\right. \) \( \left. {{e}_{2} + {e}_{3}}\right) \), choosing controller | Proof When the systems state moving on the sliding mode surface, \( s\left( t\right) = 0,\dot{s}\left( t\right) = 0 \) , then \( s\left( t\right) = {D}_{t}^{q - 1}\left( {{e}_{1} + {e}_{2} + {e}_{3}}\right) = 0 \), so we get \( {D}_{t}^{1 - q}{D}_{t}^{q - 1}\left( {{e}_{1} + {e}_{2} + {e}_{3}}\right) = 0 \), such that ... | Yes |
Theorem 1.1 Suppose \( \\left\\{ {{X}_{n}, n \\geq 0}\\right\\} \) is a countable nonhomogeneous Markov chain taking values in \( S = \\{ 1,2,\\cdots \\} \) with initial distribution of (1.1) and transition matrices of (1.2). Assume that \( f \) is a real function satisfying \( \\left| {f\\left( x\\right) }\\right| \\l... | \[ \\mathop{\\lim }\\limits_{{n \\rightarrow \\infty }}\\mathop{\\sup }\\limits_{{m \\geq 0}}\\frac{1}{n}\\mathop{\\sum }\\limits_{{k = 1}}^{n}\\begin{Vmatrix}{{P}_{k + m} - P}\\end{Vmatrix} = 0 \] (1.4) and \[ \\theta = \\mathop{\\sum }\\limits_{{i \\in S}}\\pi \\left( i\\right) \\left\\lbrack {{f}^{2}\\left( i\\right... | Yes |
Lemma 2.3 Assume that \( \\left\\{ {{X}_{n}, n \\geq 0}\\right\\} \) is a countable nonhomogeneous Markov chain taking values in \( S = \\{ 1,2,\\cdots \\} \) with initial distribution (1.1), and transition matrices (1.2). Suppose that \( P \) is a periodic strongly ergodic stochastic matrix, and \( R \) is matrix each... | \[ \\mathop{\\lim }\\limits_{{n \\rightarrow \\infty }}\\frac{1}{n}{L}_{n}\\left( i\\right) = \\pi \\left( i\\right) \\text{ a.e.. } \] | Yes |
Theorem 3.1 Let \( H \) be a real Hilbert space, and \( K : H \rightarrow {2}^{H} \) be a set-valued mapping such that for each \( u \in H, K\left( u\right) \subset H \) is a closed convex set and \( f : H \rightarrow R \cup \{ + \infty \} \) be proper, convex and lower semicontinuous on \( K\left( u\right) \) . Let \(... | Proof Let \( F : H \rightarrow H \) be defined as follows\n\n\[ F\left( u\right) = u - {ahu} + a{P}_{K\left( u\right) }^{f,\rho }\left\lbrack {{hu} - \rho \left( {M\left( {{Au},{Bu}}\right) - N\left( {{Cu},{Du}}\right) }\right) }\right\rbrack ,\forall u \in H, \]\n\nwhere \( a > 0 \) is a constant. For any \( u, v \in ... | Yes |
Theorem 4.1 Let \( H, K, M, N, A, B, C, D, h \) be same as in Theorem 3.1 and satisfy conditions (i)-(iv) in Theorem 3.1. If the following conditions hold\n\n(a) \( N \) is \( v \) - \( h \) -relaxed Lipschitz with respect to \( A \) and \( B \) ;\n\n(b) there exists \( 0 < k < \frac{\mu + v}{\beta + \gamma } \) such t... | Proof Let \( x = {P}_{K\left( {u}^{ * }\right) }^{f,\rho }\left\lbrack {{hu} - \rho \left( {M\left( {{Au},{Bu}}\right) - N\left( {{Cu},{Du}}\right) }\right) }\right\rbrack \) . Since \( {u}^{ * } \) is the solution of GIMQVI (2.1), then, for all \( \rho > 0 \), we have\n\n\[ \left\langle {\rho \left( {M\left( {A{u}^{ *... | Yes |
Theorem 4.5 Let \( H, K, M, N, A, B, C, D, h \) be same as in Theorem 3.1 and satisfy conditions (i)-(iv) in Theorem 3.1 and condition (a) in Theorem 4.1. If the following condition holds\n\n(b3) there exists \( 0 < k < \frac{\mu + v}{\beta + \gamma } \) such that for any \( \rho > \frac{\alpha \left( {\alpha + {8k}}\r... | Proof Let \n\n\[ x = {hu} - {\rho y}, y = M\left( {{Au},{Bu}}\right) - N\left( {{Cu},{Du}}\right) , \]\n\n\[ {x}^{ * } = h{u}^{ * } - \rho {y}^{ * },{y}^{ * } = M\left( {A{u}^{ * }, B{u}^{ * }}\right) - N\left( {C{u}^{ * }, D{u}^{ * }}\right) . \]\n\nFrom the definition of \( e\left( {u,\rho }\right) \), we have\n\n\[ ... | Yes |
Corollary 2 Let \( f \in {F}_{\alpha }^{2} \), and \( {f}^{\prime },{f}^{\prime \prime } \in {F}_{\alpha }^{2} \), then we have\n\n\[ \n{\begin{Vmatrix}\frac{{f}^{\prime }}{\alpha } + zf\end{Vmatrix}}_{2,\alpha }{\begin{Vmatrix}\frac{{f}^{\prime }}{\alpha } - zf\end{Vmatrix}}_{2,\alpha } \geq {\left( \frac{\parallel f{... | Proof This follows directly from Theorem 3 by setting \( a = b = 0 \) . | No |
Corollary 3 Let \( f \in {F}_{\alpha }^{2} \) and \( {f}^{\prime },{f}^{\prime \prime } \in {F}_{\alpha }^{2} \) . For any \( \delta > 0 \), then we have\n\n\[ \frac{\delta }{2}{\begin{Vmatrix}\frac{{f}^{\prime }}{\alpha } + zf\end{Vmatrix}}_{2,\alpha }^{2} + \frac{1}{2\delta }{\begin{Vmatrix}\frac{{f}^{\prime }}{\alph... | Proof From Corollary 2, we have the following estimates\n\n\[ {\left( \frac{\parallel f{\parallel }_{2,\alpha }^{4}}{{\alpha }^{2}} + {\left| \left\langle \frac{{f}^{\prime \prime }}{{\alpha }^{2}}, f\right\rangle - \left\langle {z}^{2}f, f\right\rangle \right| }^{2}\right) }^{\frac{1}{2}} \leq {\begin{Vmatrix}\frac{{f... | Yes |
Corollary 7 Suppose \( f \) is any function in \( {F}_{\alpha }^{2} \), not identically zero, and \( {f}^{\prime },{f}^{\prime \prime } \in {F}_{\alpha }^{2} \) , then we have\n\n\[ \operatorname{dist}\left( {\frac{{f}^{\prime }}{\alpha } + {zf},\left\lbrack f\right\rbrack }\right) \operatorname{dist}\left( {\frac{{f}^... | Proof This is an equivalent state of Corollary 4, because\n\n\[ \operatorname{dist}\left( {\frac{{f}^{\prime }}{\alpha } + {zf},\left\lbrack f\right\rbrack }\right) = {\begin{Vmatrix}\frac{{f}^{\prime }}{\alpha } + zf\end{Vmatrix}}_{2,\alpha }\left| {\sin \left( {\theta }_{ + }\right) }\right| \]\n\nand\n\n\[ \operator... | Yes |
Corollary 8 If \( f \) is any function in \( {F}_{\alpha }^{2} \), Suppose \( T \) is any operator on \( {F}_{\alpha }^{2} \) such that \( \left\lbrack {T,{T}^{ * }}\right\rbrack = {mI} \), then we have\n\n\[ \n{\begin{Vmatrix}Tf + {T}^{ * }f - af\end{Vmatrix}}_{2,\alpha }{\begin{Vmatrix}Tf - {T}^{ * }f + ibf\end{Vmatr... | Proof This follows from the proofs of Lemma 2 and Theorem 3. | No |
Theorem 7 Let \( f,{f}^{\prime } \in {F}_{\alpha }^{2} \), for any \( a,{b}_{0},{b}_{1} \in \mathbb{C} \), then we have\n\n\[{\left\{ {\begin{Vmatrix}\frac{{f}^{\prime }}{\alpha } - \overline{{b}_{0}}f - \overline{{b}_{1}}zf\end{Vmatrix}}_{2,\alpha }^{2} + {\begin{Vmatrix}zf - {b}_{0}f - \frac{{b}_{1}{f}^{\prime }}{\al... | Proof By (3.10), we get\n\n\[ \langle \left\lbrack {A, B}\right\rbrack f, f\rangle = \langle \left\lbrack {A, B}\right\rbrack f, f\rangle + \langle \left\lbrack {B, V}\right\rbrack f, f\rangle + \langle \left\lbrack {U, A}\right\rbrack f, f\rangle + \langle \left\lbrack {V, U}\right\rbrack f, f\rangle \]\n\n\[ = \left\... | Yes |
Theorem 1.1 (1) Let \( F \in \operatorname{Rat}\left( {{\mathbb{B}}^{n},{\mathbb{B}}^{N}}\right) \) with the geometric rank of \( F \) being \( {\kappa }_{0} \) and \( N = n + \frac{\left( {{2n} - {\kappa }_{0} - 1}\right) {\kappa }_{0}}{2}. \) Suppose that \( \frac{{\kappa }_{0}\left( {{\kappa }_{0} + 1}\right) }{2} <... | (2) Conversely, if \( F \) is defined by (1.1) and (1.2), then the map \( F \) is in \( \operatorname{Rat}\left( {{\mathbb{B}}^{n},{\mathbb{B}}^{N}}\right) \) with \( N = n + \frac{\left( {{2n} - {\kappa }_{0} - 1}\right) {\kappa }_{0}}{2} \) . | Yes |
Lemma 2.1 Let \( F \in {\operatorname{Prop}}_{2}\left( {{\mathbb{H}}_{n},{\mathbb{H}}_{N}}\right) \) with \( 2 \leq n \leq N \) . For each \( p \in \partial {\mathbb{H}}_{n} \), there is an automorphism \( {\tau }_{p}^{* * } \in {\operatorname{Aut}}_{0}\left( {\mathbb{H}}_{N}\right) \) such that \( {F}_{p}^{* * } \math... | Now, we are in a position to the definition of the geometric rank. Write \( \mathcal{A}\left( p\right) \mathrel{\text{:=}} \) \( - {2i}{\left( {\left. \frac{{\partial }^{2}{\left( {f}_{p}\right) }_{l}^{* * }}{\partial {z}_{j}\partial w}\right| }_{0}\right) }_{1 \leq j, l \leq \left( {n - 1}\right) } \) . Then the geome... | Yes |
Theorem 2.2 Suppose that \( F \in {\operatorname{Prop}}_{3}\left( {{\mathbb{H}}_{n},{\mathbb{H}}_{N}}\right) \) has geometric rank \( 1 \leq {\kappa }_{0} \leq n - 2 \) with \( F\left( 0\right) = 0 \) . Then there are \( \sigma \in \operatorname{Aut}\left( {\mathbb{H}}_{n}\right) \) and \( \tau \in \operatorname{Aut}\l... | \[ \left\{ \begin{array}{l} {f}_{l} = \mathop{\sum }\limits_{{j = 1}}^{{\kappa }_{0}}{z}_{j}{f}_{lj}^{ * }\left( {z, w}\right) ,\;l \leq {\kappa }_{0}, \\ {f}_{j} = {z}_{j},\text{ for }{\kappa }_{0} + 1 \leq j \leq n - 1, \\ {\phi }_{lk} = {\mu }_{lk}{z}_{l}{z}_{k} + \mathop{\sum }\limits_{{j = 1}}^{{\kappa }_{0}}{z}_{... | Yes |
Theorem 2.2 [20] Let \( H \in \left( {1/2,1}\right) \) . Under Lipschitzian condition (2.8), for any \( {\mathcal{F}}_{0} \) - adapted function \( {u}_{0} \) such that \( \mathbb{E}\left\lbrack {{\int }_{D}{\left| {u}_{0}\left( x\right) \right| }^{2}{dx}}\right\rbrack < \infty \), eq. (1.1) has a unique adapted mild so... | \[ \mathop{\sup }\limits_{{\left( {t, x}\right) \rbrack \in \left\lbrack {0, T}\right\rbrack \times D}}\mathbb{E}\left\lbrack {\left| u\left( t, x\right) \right| }^{2}\right\rbrack < \infty . \] | No |
Lemma 3.1 If \( K, L \in {\mathcal{K}}_{o}^{n}, p \geq 1,\tau \in \left\lbrack {-1,1}\right\rbrack \) and \( i, j = 0,1,\cdots, n - 1 \), then\n\n\[ \n{W}_{p, i}\left( {K,{\Pi }_{p, j}^{\tau }L}\right) = {W}_{p, j}\left( {L,{\Pi }_{p, i}^{\tau }K}\right) \n\] | Proof According to definitions (2.4) and (1.11), and using Fubini theorem, we get\n\n\[ \n{W}_{p, i}\left( {K,{\Pi }_{p, j}^{\tau }L}\right) = \frac{1}{n}{\int }_{{S}^{n - 1}}h{\left( {\Pi }_{p, j}^{\tau }L, u\right) }^{p}d{S}_{p, i}\left( {K, u}\right) \n\]\n\n\[ \n= \frac{1}{n}{\int }_{{S}^{n - 1}}{\alpha }_{n, p}\le... | Yes |
Lemma 3.2 If \( K \in {\mathcal{K}}_{o}^{n}, p \geq 1,\tau \in \left\lbrack {-1,1}\right\rbrack \), real \( i \neq n \) and \( j = 0,1,\cdots, n - 1 \), then for any \( M \in {\mathcal{S}}_{o}^{n} \), \[ {W}_{p, j}\left( {K,{\Gamma }_{p, i}^{r}M}\right) = \frac{2{\omega }_{n}}{V\left( M\right) }{\widetilde{W}}_{-p, i}\... | Proof From definitions (2.4),(2.9) and (1.12), and using \( n{c}_{n - 2, p} = \left( {n + p}\right) {c}_{n, p} \), we have \[ {W}_{p, j}\left( {K,{\Gamma }_{p, i}^{\tau }M}\right) = \frac{1}{n}{\int }_{{S}^{n - 1}}{h}_{{\Gamma }_{p, i}^{\tau }M}^{p}\left( v\right) d{S}_{p, j}\left( {K, v}\right) \] \[ = \frac{{\gamma }... | Yes |
Lemma 3.3 If \( K, L \in {\mathcal{S}}_{o}^{n}, p \geq 1,\tau \in \left\lbrack {-1,1}\right\rbrack \) and reals \( i, j \neq n \), then\n\n\[ \frac{{\widetilde{W}}_{-p, j}\left( {K,{\Gamma }_{p, i}^{\tau , * }L}\right) }{V\left( K\right) } = \frac{{\widetilde{W}}_{-p, i}\left( {L,{\Gamma }_{p, j}^{\tau , * }K}\right) }... | Proof Due to considerations (2.9), (1.12), (2.1) and Fubini theorem, we obtain\n\n\[ \frac{{\widetilde{W}}_{-p, j}\left( {K,{\Gamma }_{p, i}^{\tau , * }L}\right) }{V\left( K\right) } \]\n\n\[ = \frac{1}{{nV}\left( K\right) }{\int }_{{S}^{n - 1}}{\rho }_{K}^{n + p - j}\left( u\right) {\rho }_{{\Gamma }_{p, i}^{r, * }L}^... | Yes |
Theorem 1.1 Suppose that \( A \) and \( B \) satisfy either \( \left( {\mathrm{H}}_{1}\right) \) or \( \left( {\mathrm{H}}_{2}\right) . Then problem \( \left( \mathbf{P}\right) \) has a unique optimal control. Moreover,\n\n(i) If \( \mathrm{A} \) and \( \mathrm{B} \) satisfy \( \left( {\mathrm{H}}_{1}\right) ,{\mathbf{... | ## 2 Proof of Theorem 1.1\n\nUnder hypothesis \( \left( {\mathrm{H}}_{1}\right) \) or \( \left( {\mathrm{H}}_{2}\right) \), by the same arguments as those in [16], we can show the existence and uniqueness of the optimal control of problem (P). We omit the proofs here. Next, we continue the proof of Theorem 1.1.\n\n(i) ... | No |
Lemma 2.4 Let \( H \) be a vertex-induced subgraph of \( G \) . Then\n\n(1) \( r\left( H\right) \leq r\left( G\right) ,\;p\left( H\right) \leq p\left( G\right) \) and \( n\left( H\right) \leq n\left( G\right) \) .\n\n(2) If \( r\left( H\right) = r\left( G\right) \), then \( p\left( H\right) = p\left( G\right) \) and \(... | Proof Lemma 2.4 follows from Lemma 2.3 and from the inequality \( r\left( H\right) = p\left( H\right) + \) \( n\left( H\right) \leq p\left( G\right) + n\left( G\right) = r\left( G\right) . | Yes |
Lemma 3.2 Let \( G \) be a connected graph. Then \( n\left( G\right) = 2 \) if and only if \( G \in \mathcal{M}\left( {\Omega }_{3}\right) \) , where \( {\Omega }_{3} = \left\{ {{K}_{3},{C}_{5},{H}_{1},{H}_{2},\cdots ,{H}_{7}}\right\} \), and the graphs \( {H}_{i}\left( {i = 1,2,\cdots ,7}\right) \) are defined in Figu... | Proof It is routine to verify that \( n\left( G\right) = 2 \) for \( G \in \left\{ {{K}_{3},{C}_{5}}\right\} \cup \left\{ {{H}_{i} \mid 1 \leq i \leq 7}\right\} \) . Thus the sufficiency follows from Lemma 2.6.\n\nTo prove the necessity, we note that \( r\left( G\right) > n\left( G\right) = 2 \) . If \( r\left( G\right... | Yes |
Lemma 4.1 If \( H \) is a basic graph, then \( {\left( H \circ m\right) }^{U} \) is also a basic graph. | Proof For any \( i, j \in \{ 1,2,\cdots, n\} \), if \( i \neq j \), as \( H \) is a basic graph, then \( {N}_{H}\left( {v}_{i}\right) \neq \) \( {N}_{H}\left( {v}_{j}\right) \) . So \( {N}_{{\left( H \circ m\right) }^{U}}\left( {v}_{i}^{s}\right) \neq {N}_{{\left( H \circ m\right) }^{U}}\left( {v}_{j}^{t}\right) \left(... | Yes |
Lemma 4.2 Let \( G \) be a connected graph with pendent vertices and \( n\left( G\right) = 3 \) . Then \( G \in \mathcal{M}\left( {\Omega }_{5}\right) \), where \( {\Omega }_{5} = \Gamma \left( {2{K}_{2}}\right) \cup \Gamma \left( {K}_{3}\right) \cup \Gamma \left( {C}_{5}\right) \cup \mathop{\bigcup }\limits_{{i = 1}}^... | Proof Without loss of generality, assume that \( G \) is a basic graph. Let \( H \) be the induced subgraph of \( G \) obtained by deleting the pendant vertex \( x \) together with the vertex \( y \) adjacent to it. By Lemma 2.7, we have \( n\left( H\right) = 2 \) . Furthermore, \( H \) does not have isolated vertices ... | Yes |
Theorem 1.1 Let \( {f}_{0} \) be the initial datum with finite mass, energy and entropy and \( f\left( {t, v}\right) \) be any solution of the Cauchy problem (1.3). Then for all time \( t > 0, f\left( {t, v}\right) \), as a real function of \( v \) variable, is analytic in \( {\mathbb{R}}_{v}^{3} \) . Moreover, for all... | \[ {t}^{\left| \alpha \right| }{\begin{Vmatrix}{\partial }^{\alpha }f\left( t, v\right) \end{Vmatrix}}_{{L}^{2}\left( {\mathbb{R}}_{v}^{3}\right) } \leq {C}^{\left| \alpha \right| + 1}\left\lbrack {\left( {\left| \alpha \right| - 2}\right) !}\right\rbrack \] where \( \alpha \) is an arbitrary multi-indices in \( {\math... | Yes |
Lemma 2.1 For all multi-indices \( \mu \in {\mathbb{N}}^{3},\left| \mu \right| \geq 2 \), we have\n\n\[ \mathop{\sum }\limits_{{1 \leq \left| \beta \right| \leq \left| \mu \right| - 1}}\frac{\left| \mu \right| }{{\left| \beta \right| }^{4}\left( {\left| \mu \right| - \left| \beta \right| }\right) } \leq {24} \]\n\n(2.1... | This lemma was proved in [5]. | No |
Lemma 2.2 There exist positive constants \( B,{C}_{1} \), and \( {C}_{2} > 0 \) with \( B \) depending only on the dimension and \( {C}_{1},{C}_{2} \) depending only on \( {M}_{0},{E}_{0},{H}_{0} \), and \( \gamma \) such that for all multi-indices \( \mu \in {\mathbb{N}}^{3} \) with \( \left| \mu \right| \geq 2 \) and... | \[ \frac{d}{dt}{\begin{Vmatrix}{\partial }^{\mu }f\left( t\right) \end{Vmatrix}}_{{L}^{2}}^{2} + {C}_{1}{\begin{Vmatrix}{\nabla }_{v}{\partial }^{\mu }f\left( t\right) \end{Vmatrix}}_{{L}_{\gamma }^{2}}^{2} \] \[ \leq {C}_{2}{\left| \mu \right| }^{2}{\begin{Vmatrix}{\nabla }_{v}{\partial }^{\mu - 1}f\left( t\right) \en... | Yes |
Proposition 2.3 Let \( {f}_{0} \) be the initial datum with finite mass, energy and entropy and \( f\left( {t, v}\right) \) be any solution of the Cauchy problem (1.3). Then for all \( t \) in the interval \( \left\lbrack {0, T}\right\rbrack \) with \( T \) being an arbitrary nonnegative constant, there exists a consta... | Proof of Proposition 2.3 We use induction on \( \left| \alpha \right| \) to prove estimate (2.3). First, when we take\n\n\[ A = \mathop{\sup }\limits_{{t \in \left\lbrack {0, T}\right\rbrack }}\parallel f\left( {t, v}\right) {\parallel }_{{L}^{2}} + T\mathop{\sup }\limits_{{t \in \left\lbrack {0, T}\right\rbrack }}{\be... | Yes |
Theorem 3.2 Let \( \widetilde{X} = \left\lbrack {{\widetilde{x}}^{ - },{\widetilde{x}}^{ + }}\right\rbrack \) and \( \widetilde{Y} = \left\lbrack {{\widetilde{y}}^{ - },{\widetilde{y}}^{ + }}\right\rbrack \) be two fuzzy random variables. Then \( {\widetilde{d}}_{H}\left( {\widetilde{X},\widetilde{Y}}\right) : \Omega \... | Proof Since \( \widetilde{X} = \left\lbrack {{\widetilde{x}}^{ - },{\widetilde{x}}^{ + }}\right\rbrack \) and \( \widetilde{Y} = \left\lbrack {{\widetilde{y}}^{ - },{\widetilde{y}}^{ + }}\right\rbrack \) are fuzzy random variables, \( {\widetilde{x}}^{ - },{\widetilde{x}}^{ + },{\widetilde{y}}^{ - } \) and \( {\widetil... | Yes |
(1) \( E\left( \widetilde{A}\right) = \widetilde{A} \), where \( \widetilde{A} \) denote the fuzzy random variable which have the same outcome \( \widetilde{A} \) for all \( \omega \in \Omega \), i.e., \( \widetilde{A}\left( \omega \right) = \widetilde{A},\forall \omega \in \Omega \) ; | Proof (1) Let \( \widetilde{A} = \left\lbrack {{\widetilde{a}}^{ - },{\widetilde{a}}^{ + }}\right\rbrack \) . If for any \( \omega \in \Omega ,\widetilde{A}\left( \omega \right) = \widetilde{A} \), then \( {\widetilde{a}}^{ - }\left( \omega \right) = {\widetilde{a}}^{ - } \) and \( {\widetilde{a}}^{ + }\left( \omega \r... | Yes |
Lemma 2.5 Let \( h \in C\left( {\left\lbrack {1, e}\right\rbrack ,\mathbb{R}}\right), u \in {C}_{\delta }^{3}\left( {\left\lbrack {1, e}\right\rbrack ,\mathbb{R}}\right) \) . Then the unique solution of the linear Caputo-Hadamard fractional differential equation \[ \left\{ \begin{array}{l} {}_{H}^{C}{D}_{1 + }^{\alpha ... | Proof In view of Lemma 2.4, applying \( {}^{H}{I}_{1 + }^{\alpha } \) to both sides of (2.2), \[ u\left( t\right) = {}^{H}{I}_{1 + }^{\alpha }h\left( t\right) + {c}_{0} + {c}_{1}\left( {\ln t}\right) + {c}_{2}{\left( \ln t\right) }^{2}, \] where \( {c}_{0},{c}_{1},{c}_{2} \in \mathbb{R} \) . The boundary condition \( u... | Yes |
Theorem 3.4 Assume that assumptions of Theorem 3.3 are satisfied. Then fractional nonlinear differential equation (1.1) has at least one solution in \( C\left( {\left\lbrack {1, e}\right\rbrack ,\mathbb{R}}\right) \) . | Proof By the hypotheses and Theorem 3.3, we induct \( {X}_{\left( \underline{u},\bar{u}\right) } \neq \varnothing \), then the solution set of fractional integral equation (2.6) is nonempty in \( C\left( {\left\lbrack {1, e}\right\rbrack ,\mathbb{R}}\right) \) . It follows from the solution set of (2.6) together with L... | Yes |
Consider the following nonlinear Caputo-Hadamard fractional differential equation\n\n\[ \left\{ \begin{array}{l} {}_{H}^{C}{D}_{1 + }^{\frac{5}{2}}u\left( t\right) = \frac{\left( \sqrt{t} + \ln t\right) }{{\left( t + 3\right) }^{3}}\frac{{u}^{2}}{\left| u\right| + 1},1 \leq t \leq e, \\ u\left( 1\right) = {u}^{\prime }... | here \( \alpha = \frac{5}{2},\lambda = 1, f\left( {t, u}\right) = \frac{\left( \sqrt{t} + \ln t\right) }{{\left( t + 3\right) }^{3}}\frac{{u}^{2}}{\left| u\right| + 1},1 \leq t \leq e \) . One can easily calculate \( Q = \) \( \frac{24}{{15}\sqrt{\pi }\left( {3 - e}\right) } \approx {3.2} \) . Clearly \( f \) is a cont... | Yes |
Consider the problem\n\n\\[ \n\\left\\{ \\begin{array}{l} {}_{H}^{C}{D}_{1 + }^{\\frac{5}{2}}u\\left( t\\right) = \\frac{{t}^{4}}{{16}\\sqrt{\\pi }}\\left( {\\left| u\\right| + 1}\\right) , \\\\ u\\left( 1\\right) = {u}^{\\prime }\\left( 1\\right) = 0, u\\left( e\\right) = \\frac{1}{\\left( 6 - 2e\\right) }{\\int }_{1}... | Proof Where \\( \\alpha = \\frac{5}{2},\\;\\lambda = \\frac{1}{\\left( 6 - 2e\\right) },\\;f\\left( {t, u}\\right) = \\frac{{t}^{4}}{{16}\\sqrt{\\pi }}\\left( {\\left| u\\right| + 1}\\right) ,\\;t \\in \\left\\lbrack {1, e}\\right\\rbrack ,\\;f \\) is continuous and nondecreasing with respect to \\( u \\) . Thus\n\n\\[... | Yes |
Lemma 4.1 \( \mathop{\lim }\limits_{{A, B \rightarrow 0}}{\rho }_{ * }^{AB} = + \infty \) . | Proof Eliminating \( {u}_{ * }^{AB} \) in the second equation of (4.1) and (4.2) gives\n\n\[ \n{u}_{ + } - {u}_{ - } = - \sqrt{\frac{{\rho }_{ * }^{AB} - {\rho }_{ - }}{{\rho }_{ * }^{AB}{\rho }_{ - }}\left( {A\left( {{\left( {\rho }_{ * }^{AB}\right) }^{n} - {\rho }_{ - }^{n}}\right) - B\left( {\frac{1}{{\left( {\rho ... | Yes |
Lemma 4.3\n\n\\[ \n\\mathop{\\lim }\\limits_{{A, B \\rightarrow 0}}{u}_{ * }^{AB} = \\mathop{\\lim }\\limits_{{A, B \\rightarrow 0}}{\\sigma }_{1}^{AB} = \\mathop{\\lim }\\limits_{{A, B \\rightarrow 0}}{\\sigma }_{2}^{AB} = \\sigma .\n\\] | Proof From the first equation of (4.1) and (4.2) for \\( {S}_{1} \\) and \\( {S}_{2} \\), by Lemma 4.1, we have\n\n\\[ \n\\mathop{\\lim }\\limits_{{A, B \\rightarrow 0}}{\\sigma }_{1}^{AB} = \\mathop{\\lim }\\limits_{{A, B \\rightarrow 0}}\\frac{{\\rho }_{ * }^{AB}{u}_{ * }^{AB} - {\\rho }_{ - }{u}_{ - }}{{\\rho }_{ * ... | Yes |
\[ \mathop{\lim }\limits_{{A, B \rightarrow 0}}{\int }_{{x}_{1}^{AB}}^{{x}_{2}^{AB}}{\rho }_{ * }^{AB}{dx} = \sqrt{{\rho }_{ + }{\rho }_{ - }}\left( {{u}_{ - } - {u}_{ + }}\right) t \] | Proof Here we only prove the case for \( {\rho }_{ + } \neq {\rho }_{ - } \) . The first equation of the Rankine-Hugoniot condition (3.11) for \( {S}_{1} \) and \( {S}_{2} \) read\n\n\[ \left\{ \begin{array}{l} {\sigma }_{1}^{AB}\left( {{\rho }_{ * }^{AB} - {\rho }_{ - }}\right) = {\rho }_{ * }^{AB}{u}_{ * }^{AB} - {\r... | Yes |
Theorem 4.2 Let \( {u}_{ - } < {u}_{ + } \) and \( \left( {{\rho }_{ + },{u}_{ + }}\right) \in \mathrm{I}\left( {{\rho }_{ - },{\mathrm{u}}_{ - }}\right) \) . For any fixed \( A, B > 0 \), assume that \( \left( {{\rho }^{AB},{u}^{AB}}\right) \) is the two-rarefaction wave Riemann solution of (1.1)-(1.2) with Riemann da... | Indeed, if \( \mathop{\lim }\limits_{{A, B \rightarrow 0}}{\rho }_{ * }^{AB} = K \in \left( {0,\min \left\{ {{\rho }_{ - },{\rho }_{ + }}\right\} }\right) \), then (4.17) leads to \( {u}_{ + } - {u}_{ - } = 0 \), which contradicts with \( {u}_{ - } < {u}_{ + } \) . Thus \( \mathop{\lim }\limits_{{A, B \rightarrow 0}}{\... | Yes |
Lemma 5.1 When \( \left( {{\rho }_{ + },{u}_{ + }}\right) \in \mathrm{V}\left( {{\rho }_{ - },{\mathrm{u}}_{ - }}\right) \), there exists a positive parameter \( {A}_{0} \) such that \( \left( {{\rho }_{ + },{u}_{ + }}\right) \in {S}_{1}{S}_{2}\left( {{\rho }_{ - },{u}_{ - }}\right) \) when \( 0 < A < {A}_{0} \) . | Proof From \( \left( {{\rho }_{ + },{u}_{ + }}\right) \in \mathrm{V}\left( {{\rho }_{ - },{\mathrm{u}}_{ - }}\right) \), we have\n\n\[ \n{u}_{ + } + \sqrt{B}{\rho }_{ + }^{-\frac{\alpha + 1}{2}} \leq {u}_{ - } - \sqrt{B}{\rho }_{ - }^{-\frac{\alpha + 1}{2}} \n\]\n\n(5.1)\n\nthen\n\n\[ \n{\left( {u}_{ - } - {u}_{ + }\ri... | Yes |
Lemma 5.2 \( \mathop{\lim }\limits_{{A \rightarrow 0}}{\rho }_{ * }^{A} = + \infty \) . | Proof Eliminating \( {u}_{ * }^{A} \) in the second equation of (5.7) and (5.8) gives\n\n\[ \n{u}_{ - } - {u}_{ + } = \sqrt{\frac{{\rho }_{ * }^{A} - {\rho }_{ - }}{{\rho }_{ * }^{A}{\rho }_{ - }}\left( {A\left( {{\left( {\rho }_{ * }^{A}\right) }^{n} - {\rho }_{ - }^{n}}\right) - B\left( {\frac{1}{{\left( {\rho }_{ * ... | Yes |
Lemma 5.4 Let \( \mathop{\lim }\limits_{{A \rightarrow 0}}{u}_{ * }^{A} = \widehat{{\sigma }^{B}} \), then\n\n\[ \mathop{\lim }\limits_{{A \rightarrow 0}}{u}_{ * }^{A} = \mathop{\lim }\limits_{{A \rightarrow 0}}{\sigma }_{1}^{A} = \mathop{\lim }\limits_{{A \rightarrow 0}}{\sigma }_{2}^{A} = \widehat{{\sigma }^{B}} \in ... | Proof From the second equation of (5.7) for \( {S}_{1} \), by Lemmas 4.2 and 4.3, we have\n\n\[ \mathop{\lim }\limits_{{A \rightarrow 0}}{u}_{ * }^{A} = {u}_{ - } - \mathop{\lim }\limits_{{A \rightarrow 0}}\sqrt{\frac{{\rho }_{ * }^{A} - {\rho }_{ - }}{{\rho }_{ * }^{A}{\rho }_{ - }}\left( {A\left( {{\left( {\rho }_{ *... | Yes |
Lemma 5.5 For \( \widehat{{\sigma }^{B}} \) mentioned in Lemma 5.4,\n\n\[ \widehat{{\sigma }^{B}} = {\sigma }^{B} = \frac{{\rho }_{ + }{u}_{ + } - {\rho }_{ - }{u}_{ - } + {\left\{ {\rho }_{ + }{\rho }_{ - }\left( {\left( {u}_{ + } - {u}_{ - }\right) }^{2} - \left( \frac{1}{{\rho }_{ + }} - \frac{1}{{\rho }_{ - }}\righ... | Proof Letting \( \mathop{\lim }\limits_{{A \rightarrow 0}}A{\left( {\rho }_{ * }^{A}\right) }^{n} = L \), by Lemma 5.4, from (5.12) and (5.13) we have\n\n\[ \mathop{\lim }\limits_{{A \rightarrow 0}}{u}_{ * }^{A} = {u}_{ - } - \sqrt{\frac{1}{{\rho }_{ - }}\left( {L + \frac{B}{{\rho }_{ - }^{\alpha }}}\right) } = {u}_{ +... | Yes |
Lemma 5.6\n\n\[ \mathop{\lim }\limits_{{A \rightarrow 0}}{\int }_{{x}_{1}^{A}}^{{x}_{2}^{A}}{\rho }_{ * }^{A}{dx} = {w}_{0}^{B}t \]\n\n(5.19)\n\n\[ \mathop{\lim }\limits_{{A \rightarrow 0}}{\int }_{{x}_{1}^{A}}^{{x}_{2}^{A}}{\rho }_{ * }^{A}{u}_{ * }^{A}{dx} = {w}_{0}^{B}{\sigma }^{B}t. \]\n\n\( \left( {5.20}\right) \) | Proof Here we only prove the case for \( {\rho }_{ + } \neq {\rho }_{ - } \) . Similar to the proof of Lemma 4.4, taking account into (3.11) and (5.18), we have\n\n\[ \mathop{\lim }\limits_{{A \rightarrow 0}}{\rho }_{ * }^{A}\left( {{\sigma }_{2}^{A} - {\sigma }_{1}^{A}}\right) = \mathop{\lim }\limits_{{A \rightarrow 0... | Yes |
Lemma 2.1 Let \( \Omega \) be a bounded Lipschitz domain in \( {R}^{2} \) or \( {R}^{3} \) . Let \( M, K \) be two positive real numbers and \( \rho \) a non-negative function such that\n\n\[ 0 < M \leq {M}_{\rho } = {\int }_{\Omega }{\rho dx},\;{\int }_{\Omega }{\rho }^{\gamma }{dx} \leq K\;\text{ for a certain }\gamm... | Proof see [11]. | No |
Lemma 2.2 Let \( v \in {W}^{1, p}\left( {\Omega }_{\varepsilon }\right) \) . Then we have\n\n\[ \parallel v{\parallel }_{{L}^{q}\left( {\Omega }_{\varepsilon }\right) } \leq C\left( {\left| {{\int }_{{\Omega }_{\varepsilon }}{vdx}}\right| + {\varepsilon }^{n\left( {\frac{1}{q} - \frac{1}{p}}\right) }\parallel \nabla v{... | Proof Obviously, it is enough to show\n\n\[ \parallel v{\parallel }_{{L}^{q}\left( {\Omega }_{\varepsilon }\right) } \leq C{\varepsilon }^{n\left( {\frac{1}{q} - \frac{1}{p}}\right) }\parallel \nabla v{\parallel }_{{L}^{p}\left( {\Omega }_{\varepsilon }\right) },\;v \in {W}^{1, p}\left( {\Omega }_{\varepsilon }\right) ... | Yes |
Lemma 2.1 The energy functional \( {I}_{\mu } \) is coercive and bounded below on \( {N}_{\mu } \) . | Proof For \( u \in {N}_{\mu } \), by the Hölder inequality and Sobolev embedding theorem, we can deduce\n\n\[ \n{I}_{\mu }\left( u\right) = \frac{1}{p}\parallel u{\parallel }^{p} - \frac{1}{q}{\int }_{{\mathbb{R}}^{N}}{f}_{\mu }{\left| u\right| }^{q}{dx} - \frac{1}{r}{\int }_{{\mathbb{R}}^{N}}g{\left| u\right| }^{r}{dx... | Yes |
Lemma 2.2 (i) If \( u \in {N}_{\mu }^{ + } \), then \( {\int }_{{\mathbb{R}}^{N}}{f}_{\mu }\left( x\right) {\left| u\right| }^{q}{dx} > 0 \) . | Proof By (2.1) we can easily derive these results. | No |
Lemma 2.3 (i) For all \( \mu \in \left( {0,{L}_{1}}\right) \), we have \( {N}_{\mu }^{0} = \varnothing \) and \( {\alpha }^{ + } < 0 \) . | Proof (i) Suppose the contrary. We may assume that there exists \( {\mu }_{ * } \in \left( {0,{L}_{1}}\right) \) such that \( {N}_{{\mu }_{ * }}^{0} \neq \varnothing \) . Thus, for each \( u \in {N}_{{\mu }_{ * }}^{0} \), by the Hölder and Sobolev inequalities, we can obtain\n\n\[ 0 = \left\langle {{\Psi }^{\prime }\le... | Yes |
Lemma 2.4 For each \( u \in {\mathbf{W}}_{a}^{1, p}\left( {\mathbb{R}}^{N}\right) \smallsetminus \{ 0\} \), we have\n\n(i) if \( {\int }_{{\mathbb{R}}^{N}}{f}_{\mu }{\left| u\right| }^{q}{dx} \leq 0 \), then there exists a unique \( {t}^{ - } = {t}^{ - }\left( u\right) > {t}_{\max } \) such that \( {t}^{ - }u \in \) \(... | Proof (i) The equation \( {m}_{u}\left( t\right) = {\int }_{{\mathbb{R}}^{N}}{f}_{\mu }{\left| u\right| }^{q}{dx} \) admits a unique solution \( {t}^{ - } > {t}_{\max } \) and \( {m}_{u}^{\prime }\left( {t}^{ - }\right) < 0 \) . Thus \( {t}^{ - }u \in {N}_{\mu }^{ - } \), and (2.9) holds by Lemma 2.3. | Yes |
For \( 0 \leq a < \frac{N - p}{p},0 \leq \lambda < \bar{\lambda } \), problem (3.1) has radially symmetric ground states\n\n\[ \n{u}_{\epsilon }\left( x\right) = {\epsilon }^{-\left( {\frac{N - p}{p} - a}\right) }{v}_{\epsilon }\left( \frac{x}{\epsilon }\right) ,\;\forall \epsilon > 0, \n\]\nsatisfying\n\n\[ \n{\int }_... | Proof As in [19], we can prove that the limiting problem (3.1) has radially symmetric ground states, by which \( {S}_{\lambda } \) can be achieved. Let \( u\left( \xi \right) \) be a radial solution to (3.1). Then we get that\n\n\[ \n{\left( {\xi }^{N - 1 - {ap}}{\left| {u}^{\prime }\right| }^{p - 2}{u}^{\prime }\right... | Yes |
Proposition 3.2 (i) If \( \mu \in \left( {0,{L}_{1}}\right) \), then \( {I}_{\mu } \) has a (P.S.) \( {}_{\alpha } \) -sequence \( {\left\{ {u}_{n}\right\} }_{n \in \mathbb{N}} \subset {N}_{\mu } \) . | Proof The proof is similar to the argument of Proposition 3.3 in [20]. | No |
Lemma 3.4 (i) \( \mathop{\lim }\limits_{{l \rightarrow 0}}{\begin{Vmatrix}{u}_{l}\end{Vmatrix}}^{p} = {S}_{\lambda }^{\frac{r}{r - p}} \) uniformly in \( e \in {\mathbb{S}}^{N - 1} \) ; | We refer to the argument of Lemma 4.2 in He and Yang (see [21]). | No |
Theorem 4.2 For \( \mu \in \left( {0,{L}_{2}}\right) ,\left( {1.1}\right) \) has a positive solution \( {u}_{\mu }^{ - } \in {N}_{\mu }^{ - } \) such that \( {I}_{\mu }\left( {u}_{\mu }^{ - }\right) = {\alpha }^{ - }. \) | Proof By Ekeland's variational principle [22], there exists a minimizing sequence \( {\left\{ {u}_{n}\right\} }_{n \in \mathbb{N}} \subset {N}_{\mu }^{ - } \) such that\n\n\[ \n{I}_{\mu }\left( {u}_{n}\right) = {\alpha }^{ - } + o\left( 1\right) \;\text{ and }\;{I}_{\mu }^{\prime }\left( {u}_{n}\right) = o\left( 1\righ... | Yes |
Lemma 5.2 Assume that \( \left\{ {u}_{n}\right\} \) is a minimizing sequence in \( {N}_{0} \) for \( {I}_{0} \) . Then\n\n(i) \( {\int }_{{\mathbb{R}}^{N}}{f}_{ - }{\left| {u}_{n}\right| }^{q}{dx} = o\left( 1\right) \) ;\n\n(ii) \( {\int }_{{\mathbb{R}}^{N}}\left( {1 - g}\right) {\left| {u}_{n}\right| }^{r}{dx} = o\lef... | Proof For each \( n \), there exists a unique \( {t}_{n} > 0 \) such that \( {t}_{n}{u}_{n} \in {N}^{\infty } \), that is,\n\n\[ \n{t}_{n}^{p}{\begin{Vmatrix}{u}_{n}\end{Vmatrix}}^{p} = {t}_{n}^{r}{\int }_{{\mathbb{R}}^{N}}{\left| {u}_{n}\right| }^{r}{dx}.\n\]\n\nBy Lemma 2.4 (i), we have\n\n\[ \n{I}_{0}\left( {u}_{n}\... | Yes |
Lemma 5.3 There exists \( {d}_{0} < 0 \) such that for \( u \in {N}_{0}\left( {d}_{0}\right) \), we have\n\n\[ \n{\int }_{{\mathbb{R}}^{N}}\frac{x}{{\left| x\right| }^{1 - {ap}}}\left( {{\left| x\right| }^{-{ap}}{\left| \nabla u\right| }^{p} - \frac{\lambda }{{\left| x\right| }^{p\left( {a + 1}\right) }}{u}^{p}}\right)... | Proof Suppose the contrary. We may assume that there exists a sequence \( {\left\{ {u}_{n}\right\} }_{n \in \mathbb{N}} \subset \) \( {N}_{0} \) such that \( {I}_{0}\left( {u}_{n}\right) = \left( {\frac{1}{p} - \frac{1}{r}}\right) {S}_{\lambda }^{\frac{r}{r - p}} + o\left( 1\right) \) and \( {\int }_{{\mathbb{R}}^{N}}\... | Yes |
Lemma 5.5 There exists \( {\mu }_{0} \in \left( {0,{L}_{2}}\right) \) such that for each \( \mu \in \left( {0,{\mu }_{0}}\right) \) and \( u \in {N}_{\mu }^{ - }\left( {\alpha }^{ + }\right) \) ,\n\n\[ \n{\int }_{{\mathbb{R}}^{N}}\frac{x}{{\left| x\right| }^{1 - {ap}}}\left( {{\left| x\right| }^{-{ap}}{\left| \nabla u\... | Proof For \( u \in {N}_{\mu }^{ - }\left( {\alpha }^{ + }\right) \), by Lemma 2.4 (i), there exists \( {t}_{0}^{ - }\left( u\right) > 0 \) such that \( {t}_{0}^{ - }\left( u\right) u \in \) \( {N}_{0} \) . Moreover, by Lemma 5.4 and the Hölder inequality and Sobolev embedding theorem, we have\n\n\[ \n{I}_{\mu }\left( u... | Yes |
Lemma 6.5 There exists a sequence \( \left\{ {\sigma }_{l}\right\} \subset {\mathbb{R}}^{ + } \) with \( {\sigma }_{l} \rightarrow 0 \) as \( l \rightarrow \infty \) such that \[ {\varphi }_{\mu }\left( {\mathbb{S}}^{N - 1}\right) \subset \left\lbrack {{I}_{\mu } \leq {\alpha }^{ + } + \left( {\frac{1}{p} - \frac{1}{r}... | Proof By Proposition 4.1, for \( l > {l}_{0} \), we have \( {u}_{\mu }^{ + } + {t}_{ * }{u}_{l} \in {N}_{\mu }^{ - } \) and \[ \mathop{\sup }\limits_{{t \geq 0}}{I}_{\mu }\left( {{u}_{\mu }^{ + } + t{u}_{l}}\right) < {\alpha }^{ + } + \left( {\frac{1}{p} - \frac{1}{r}}\right) {S}_{\lambda }^{\frac{r}{r - p}}\;\text{ un... | Yes |
Lemma 6.7 For \( \mu \in \left( {0,{\mu }_{0}}\right) \) and \( l > {l}_{ * } \), the energy functional \( {I}_{\mu } \) admits at least two critical points in \( \left\lbrack {{I}_{\mu } < {\alpha }^{ + } + \left( {\frac{1}{p} - \frac{1}{r}}\right) {S}_{\lambda }^{\frac{r}{r - p}}}\right\rbrack \) . | Proof It is easy to deduce from Lemmas 6.3, 6.4, 6.6 and Proposition 3.2. | No |
Theorem 1 If \( p \geq 3 \), then there exists a constant \( C > 0 \), depending only on \( q \), such that the following holds: let \( N \in \mathbb{N} \) with \( N \geq 2 \) and \( A \subseteq {\mathbb{G}}_{N}^{2} \) . If \( \left( {A - A}\right) \cap \left\{ {\left( {d,{d}^{2}}\right) : d \in {\mathbb{A}}^{ \times }... | By adapting the lifting argument in [6], we deduce the following analogue of Theorem \( \mathrm{C} \) from Theorem 1. | No |
Lemma 1 For \( M \in {\mathbb{N}}_{ + } \) and \( \omega \in {\mathbb{K}}_{\infty } \), we have\n\n\[ \mathop{\sum }\limits_{{d \in {\mathbb{G}}_{M}}}e\left( {\omega d}\right) = \left\{ \begin{array}{ll} {q}^{M}, & \text{ if }\operatorname{ord}\{ \omega \} < - M \\ 0, & \text{ otherwise. } \end{array}\right. \] | Proof This is \( \left\lbrack {{10}\text{, Lemma 7}}\right\rbrack \) . | No |
Lemma 2 Let \( N \in {\mathbb{N}}_{ + } \) and \( \alpha = \left( {{\alpha }_{1},{\alpha }_{2}}\right) \in {\mathbb{T}}^{2} \) . Let \( b \in {\mathbb{A}}^{ \times } \) and \( m = \left( {{m}_{1},{m}_{2}}\right) \in {\mathbb{A}}^{2} \) with \( \gcd \left( {b,{m}_{1},{m}_{2}}\right) = 1 \) . Suppose that ord \( b \leq N... | Proof Write \( \beta = \left( {{\beta }_{1},{\beta }_{2}}\right) = \alpha - \frac{1}{b}m \) . Then\n\n\[ {S}_{N}\left( \alpha \right) = \mathop{\sum }\limits_{{t \in {\mathbb{G}}_{\text{ord }b}}}e\left( {\frac{1}{b}m\overrightarrow{t}}\right) \mathop{\sum }\limits_{{s \in {\mathbb{G}}_{N - \text{ ord }b}}}e\left( {\bet... | Yes |
Lemma 3 Let \( {r}_{1},{r}_{2} \in \mathbb{N} \) . Then for any \( \alpha = \left( {{\alpha }_{1},{\alpha }_{2}}\right) \in {\mathbb{T}}^{2} \), there exists \( \left( {b,{m}_{1},{m}_{2}}\right) \in {\mathbb{A}}^{3} \) with the following properties\n\n(i) \( b \) is monic and or \( \mathrm{d}b \leq {r}_{1} + {r}_{2} \)... | Proof For \( 1 \leq j \leq 2 \), let \( {\mathbb{T}}_{j} = \left\{ {\omega \in \mathbb{T} : \operatorname{ord}\omega \leq - {r}_{j} - 1}\right\} \) . Then \( {\mathbb{T}}_{j} \) is a subgroup of \( \mathbb{T} \) . Also, \( \left| {\mathbb{T}/{\mathbb{T}}_{j}}\right| = {q}^{{r}_{j}} \).\n\nNote that \( \left| {\mathop{\... | Yes |
Lemma 4 Let \( {a}_{1},{a}_{2},{b}_{1},{b}_{2} \in \mathbb{A} \) with \( {b}_{1},{b}_{2} \neq 0 \) and \( \gcd \left( {{b}_{1},{a}_{1}}\right) = \gcd \left( {{b}_{2},{a}_{2}}\right) = 1 \) . Let \( m = \left( {{m}_{1},{m}_{2}}\right) \in {\mathbb{A}}^{2} \) . Suppose that \( \gcd \left( {{b}_{1},{m}_{1},{m}_{2}}\right)... | Proof Since \( \gcd \left( {{b}_{1},{b}_{2}}\right) = 1,{b}_{2} + {b}_{1}\mathbb{A} \) is invertible in the ring \( {\mathbb{H}}_{1} = \mathbb{A}/{b}_{1}\mathbb{A} \) . Thus,\n\n\[ G\left( {\frac{{a}_{1}}{{b}_{1}}, m}\right) = \mathop{\sum }\limits_{{d + {b}_{1}\mathbb{A} \in {\mathbb{H}}_{1}}}e\left( {\frac{{a}_{1}}{{... | Yes |
Lemma 5 Let \( a, b \in \mathbb{A} \) with \( b \neq 0 \) and \( \gcd \left( {b, a}\right) = 1 \) . Let \( m = \left( {{m}_{1},{m}_{2}}\right) \in {\mathbb{A}}^{2} \) . Suppose that \( \gcd \left( {b,{m}_{1},{m}_{2}}\right) = 1 \) . If \( p \geq 3 \) and \( b \) is irreducible, then we have \[ \left| {G\left( {\frac{a}... | Proof Since \( b \) is irreducible and \( \gcd \left( {b, a}\right) = 1 \), it follows that \( a \neq 0 \) . We divide into two cases.\n\nCase 1 Suppose that \( b \mid {m}_{2} \) . Since \( \gcd \left( {b,{m}_{1},{m}_{2}}\right) = 1, b \nmid {m}_{1} \) . By Lemma 1, we have \[ G\left( {\frac{a}{b}, m}\right) = \mathop{... | Yes |
Proposition 7 Let \( a, b \in \mathbb{A} \) with \( b \neq 0 \) and \( \gcd \left( {b, a}\right) = 1 \) . Let \( m = \left( {{m}_{1},{m}_{2}}\right) \in {\mathbb{A}}^{2} \) . Suppose that \( \gcd \left( {b,{m}_{1},{m}_{2}}\right) = 1 \) . If \( p \geq 3 \), then we have\n\n\[ \left| {G\left( {\frac{a}{b}, m}\right) }\r... | Proof Without loss of generality, we assume that \( a \neq 0 \) and ord \( b \geq 1 \) . Also, \( b \) is monic. There exist \( \iota ,{j}_{1},\cdots ,{j}_{\iota } \in {\mathbb{N}}_{ + } \) and distinct monic irreducible polynomials \( {\sigma }_{1},\cdots ,{\sigma }_{\iota } \) in \( \mathbb{A} \) such that \( b = \ma... | Yes |
Lemma 8 Let \( b,{b}^{\prime } \in \mathcal{B} \). Suppose that \( \left( {{a}_{1},{a}_{2}}\right) \in {\mathcal{A}}_{b} \) and \( \left( {{a}_{1}^{\prime },{a}_{2}^{\prime }}\right) \in {\mathcal{A}}_{{b}^{\prime }} \). If \( \left( {b,{a}_{1},{a}_{2}}\right) \neq \left( {{b}^{\prime },{a}_{1}^{\prime },{a}_{2}^{\prim... | Proof To prove the lemma, we suppose the contrary. Then there exists \[ \left( {{\alpha }_{1},{\alpha }_{2}}\right) \in F\left( {b,{a}_{1},{a}_{2}}\right) \cap F\left( {{b}^{\prime },{a}_{1}^{\prime },{a}_{2}^{\prime }}\right) . \] Let \( 1 \leq j \leq 2 \). Since \[ \left| {\frac{{a}_{j}}{b} - \frac{{a}_{j}^{\prime }}... | Yes |
Proposition 9 If \( b \in \mathcal{B} \), then for any \( \alpha \in {F}_{b} \), we have\n\n\[ \left| {{S}_{N}\left( \alpha \right) }\right| \leq {q}^{N}{\left| b\right| }^{-1/2}. \] | Proof Write \( \left( {{\alpha }_{1},{\alpha }_{2}}\right) = \alpha \) . Take \( a = \left( {{a}_{1},{a}_{2}}\right) \in {\mathcal{A}}_{b} \) such that \( \alpha \in F\left( {b,{a}_{1},{a}_{2}}\right) \) . Since\n\n\[ \left| {{\alpha }_{2} - \frac{{a}_{2}}{b}}\right| < {q}^{-{2M}}{\left| b\right| }^{-1} \leq {q}^{-N}{\... | Yes |
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