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Proposition 2.3 For all \( c > 1 \n\n\[ \n{\psi }_{1}\left( x\right) = \frac{1}{2\pi i}{\int }_{c - i\infty }^{c + i\infty }\frac{{x}^{s + 1}}{s\left( {s + 1}\right) }\left( {-\frac{{\zeta }^{\prime }\left( s\right) }{\zeta \left( s\right) }}\right) {ds}. \n\] | To make the proof of this formula clear, we isolate the necessary contour integrals in a lemma. | No |
Lemma 2.4 If \( c > 0 \), then\n\n\[ \n\frac{1}{2\pi i}{\int }_{c - i\infty }^{c + i\infty }\frac{{a}^{s}}{s\left( {s + 1}\right) }{ds} = \left\{ \begin{array}{ll} 0 & \text{ if }0 < a \leq 1, \\ 1 - 1/a & \text{ if }1 \leq a. \end{array}\right.\n\]\n\nHere, the integral is over the vertical line \( \operatorname{Re}\l... | Proof. First note that since \( \left| {a}^{s}\right| = {a}^{c} \), the integral converges. We suppose first that \( 1 \leq a \), and write \( a = {e}^{\beta } \) with \( \beta = \log a \geq 0 \) . Let\n\n\[ \nf\left( s\right) = \frac{{a}^{s}}{s\left( {s + 1}\right) } = \frac{{e}^{s\beta }}{s\left( {s + 1}\right) }.\n\... | Yes |
Proposition 1.1 If \( f : U \rightarrow V \) is holomorphic and injective, then \( {f}^{\prime }\left( z\right) \neq 0 \) for all \( z \in U \) . In particular, the inverse of \( f \) defined on its range is holomorphic, and thus the inverse of a conformal map is also holomorphic. | Proof. We argue by contradiction, and suppose that \( {f}^{\prime }\left( {z}_{0}\right) = 0 \) for some \( {z}_{0} \in U \) . Then\n\n\[ f\left( z\right) - f\left( {z}_{0}\right) = a{\left( z - {z}_{0}\right) }^{k} + G\left( z\right) \;\text{ for all }z\text{ near }{z}_{0}, \]\n\nwith \( a \neq 0, k \geq 2 \) and \( G... | Yes |
Theorem 1.2 The map \( F : \mathbb{H} \rightarrow \mathbb{D} \) is a conformal map with inverse \( G : \mathbb{D} \rightarrow \mathbb{H}. \) | Proof. First we observe that both maps are holomorphic in their respective domains. Then we note that any point in the upper half-plane is closer to \( i \) than to \( - i \), so \( \left| {F\left( z\right) }\right| < 1 \) and \( F \) maps \( \mathbb{H} \) into \( \mathbb{D} \) . To prove that \( G \) maps into the upp... | Yes |
Lemma 1.3 Let \( V \) and \( U \) be open sets in \( \mathbb{C} \) and \( F : V \rightarrow U \) a holomorphic function. If \( u : U \rightarrow \mathbb{C} \) is a harmonic function, then \( u \circ F \) is harmonic on \( V \) . | Proof. The thrust of the lemma is purely local, so we may assume that \( U \) is an open disc. We let \( G \) be a holomorphic function in \( U \) whose real part is \( u \) (such a \( G \) exists by Exercise 12 in Chapter 2, and is determined up to an additive constant). Let \( H = G \circ F \) and note that \( u \cir... | No |
Lemma 2.1 Let \( f : \mathbb{D} \rightarrow \mathbb{D} \) be holomorphic with \( f\left( 0\right) = 0 \) . Then\n\n(i) \( \left| {f\left( z\right) }\right| \leq \left| z\right| \) for all \( z \in \mathbb{D} \) .\n\n(ii) If for some \( {z}_{0} \neq 0 \) we have \( \left| {f\left( {z}_{0}\right) }\right| = \left| {z}_{0... | Proof. We first expand \( f \) in a power series centered at 0 and convergent in all of \( \mathbb{D} \)\n\n\[ f\left( z\right) = {a}_{0} + {a}_{1}z + {a}_{2}{z}^{2} + \cdots . \]\n\nSince \( f\left( 0\right) = 0 \) we have \( {a}_{0} = 0 \), and therefore \( f\left( z\right) /z \) is holomorphic in \( \mathbb{D} \) (s... | Yes |
Theorem 2.2 If \( f \) is an automorphism of the disc, then there exist \( \theta \in \) \( \mathbb{R} \) and \( \alpha \in \mathbb{D} \) such that\n\n\[ f\left( z\right) = {e}^{i\theta }\frac{\alpha - z}{1 - \bar{\alpha }z}. \] | Proof. Since \( f \) is an automorphism of the disc, there exists a unique complex number \( \alpha \in \mathbb{D} \) such that \( f\left( \alpha \right) = 0 \) . Now we consider the automorphism \( g \) defined by \( g = f \circ {\psi }_{\alpha } \) . Then \( g\left( 0\right) = 0 \), and the Schwarz lemma gives\n\n\[ ... | Yes |
Corollary 3.2 Any two proper simply connected open subsets in \( \mathbb{C} \) are conformally equivalent. | Clearly, the corollary follows from the theorem, since we can use as an intermediate step the unit disc. Also, the uniqueness statement in the theorem is straightforward, since if \( F \) and \( G \) are conformal maps from \( \Omega \) to \( \mathbb{D} \) that satisfy these two conditions, then \( H = F \circ {G}^{-1}... | No |
Theorem 3.3 Suppose \( \mathcal{F} \) is a family of holomorphic functions on \( \Omega \) that is uniformly bounded on compact subsets of \( \Omega \) . Then:\n\n(i) \( \mathcal{F} \) is equicontinuous on every compact subset of \( \Omega \) . | The theorem really consists of two separate parts. The first part says that \( \mathcal{F} \) is equicontinuous under the assumption that \( \mathcal{F} \) is a family of holomorphic functions that is uniformly bounded on compact subsets of \( \Omega \) . The proof follows from an application of the Cauchy integral for... | Yes |
Lemma 3.4 Any open set \( \Omega \) in the complex plane has an exhaustion. | Proof. If \( \Omega \) is bounded, we let \( {K}_{\ell } \) denote the set of all points in \( \Omega \) at distance \( \geq 1/\ell \) from the boundary of \( \Omega \) . If \( \Omega \) is not bounded, let \( {K}_{\ell } \) denote the same set as above except that we also require \( \left| z\right| \leq \ell \) for al... | No |
Proposition 3.5 If \( \Omega \) is a connected open subset of \( \mathbb{C} \) and \( \left\{ {f}_{n}\right\} \) a sequence of injective holomorphic functions on \( \Omega \) that converges uniformly on every compact subset of \( \Omega \) to a holomorphic function \( f \), then \( f \) is either injective or constant. | Proof. We argue by contradiction and suppose that \( f \) is not injective, so there exist distinct complex numbers \( {z}_{1} \) and \( {z}_{2} \) in \( \Omega \) such that \( f\left( {z}_{1}\right) = \) \( f\left( {z}_{2}\right) \) . Define a new sequence by \( {g}_{n}\left( z\right) = {f}_{n}\left( z\right) - {f}_{n... | Yes |
Proposition 4.1 Suppose \( S\left( z\right) \) is given by (5).\n\n(i) If \( \mathop{\sum }\limits_{{k = 1}}^{n}{\beta }_{k} = 2 \), and \( \mathfrak{p} \) denotes the polygon whose vertices are given (in order) by \( {a}_{1},\ldots ,{a}_{n} \), then \( S \) maps the real axis onto \( \mathfrak{p} - \left\{ {a}_{\infty... | Proof. We assume that \( \mathop{\sum }\limits_{{k = 1}}^{n}{\beta }_{k} = 2 \) . If \( {A}_{k} < x < {A}_{k + 1} \) when \( 1 \leq k \leq n - 1 \), then\n\n\[ \n{S}^{\prime }\left( x\right) = \mathop{\prod }\limits_{{j \leq k}}{\left( x - {A}_{j}\right) }^{-{\beta }_{j}}\mathop{\prod }\limits_{{j > k}}{\left( x - {A}_... | Yes |
Theorem 4.2 If \( F : \mathbb{D} \rightarrow P \) is a conformal map, then \( F \) extends to a continuous bijection from the closure \( \overline{\mathbb{D}} \) of the disc to the closure \( \bar{P} \) of the polygonal region. In particular, \( F \) gives rise to a bijection from the boundary of the disc to the bounda... | The main point consists in showing that if \( {z}_{0} \) belongs to the unit circle, then \( \mathop{\lim }\limits_{{z \rightarrow {z}_{0}}}F\left( z\right) \) exists. To prove this, we need a preliminary result, which uses the fact that if \( f : U \rightarrow f\left( U\right) \) is conformal, then\n\n\[\n\operatornam... | Yes |
For each \( 0 < r < 1/2 \), let \( {C}_{r} \) denote the circle centered at \( {z}_{0} \) of radius \( r \) . Suppose that for all sufficiently small \( r \) we are given two points \( {z}_{r} \) and \( {z}_{r}^{\prime } \) in the unit disc that also lie on \( {C}_{r} \) . If we let \( \rho \left( r\right) = \left| {f\... | Proof. If not, there exist \( 0 < c \) and \( 0 < R < 1/2 \) such that \( c \leq \rho \left( r\right) \) for all \( 0 < r \leq R \) . Observe that\n\n\[ f\left( {z}_{r}\right) - f\left( {z}_{r}^{\prime }\right) = {\int }_{\alpha }{f}^{\prime }\left( \zeta \right) {d\zeta } \]\n\nwhere the integral is taken over the arc... | Yes |
Lemma 4.4 Let \( {z}_{0} \) be a point on the unit circle. Then \( F\left( z\right) \) tends to a limit as \( z \) approaches \( {z}_{0} \) within the unit disc. | Proof. If not, there are two sequences \( \left\{ {{z}_{1},{z}_{2},\ldots }\right\} \) and \( \left\{ {{z}_{1}^{\prime },{z}_{2}^{\prime },\ldots }\right\} \) in the unit disc that converge to \( {z}_{0} \) and are so that \( F\left( {z}_{n}\right) \) and \( F\left( {z}_{n}^{\prime }\right) \) converge to two distinct ... | Yes |
Lemma 4.5 The conformal map \( F \) extends to a continuous function from the closure of the disc to the closure of the polygon. | Proof. By the previous lemma, the limit\n\n\[ \mathop{\lim }\limits_{{z \rightarrow {z}_{0}}}F\left( z\right) \]\n\nexists, and we define \( F\left( {z}_{0}\right) \) to be the value of this limit. There remains to prove that \( F \) is continuous on the closure of the unit disc. Given \( \epsilon \), there exists \( \... | Yes |
Theorem 4.7 If \( F \) is a conformal map from the upper half-plane to the polygonal region \( P \) and maps the points \( {A}_{1},\ldots ,{A}_{n - 1},\infty \) to the vertices of \( \mathfrak{p} \), then there exist constants \( {C}_{1} \) and \( {C}_{2} \) such that\n\n\[ F\left( z\right) = {C}_{1}{\int }_{0}^{z}\fra... | Proof. After a preliminary translation, we may assume that \( {A}_{j} \neq 0 \) for \( j = 1,\ldots, n - 1 \) . Choose a point \( {A}_{n}^{ * } > 0 \) on the real line, and consider the fractional linear map defined by\n\n\[ \Phi \left( z\right) = {A}_{n}^{ * } - \frac{1}{z} \]\n\nThen \( \Phi \) is an automorphism of ... | Yes |
Theorem 1.2 An entire doubly periodic function is constant. | Proof. The function is completely determined by its values on \( {P}_{0} \) and since the closure of \( {P}_{0} \) is compact, we conclude that the function is bounded on \( \mathbb{C} \), hence constant by Liouville’s theorem in Chapter 2. | Yes |
Theorem 1.3 The total number of poles of an elliptic function in \( {P}_{0} \) is always \( \geq 2 \) . | Proof. Suppose first that \( f \) has no poles on the boundary \( \partial {P}_{0} \) of the fundamental parallelogram. By the residue theorem we have\n\n\[ {\int }_{\partial {P}_{0}}f\left( z\right) {dz} = {2\pi i}\sum \operatorname{res}f \]\n\nand we contend that the integral is 0 . To see this, we simply use the per... | Yes |
Theorem 1.4 Every elliptic function of order \( m \) has \( m \) zeros in \( {P}_{0} \) . | Proof. Assuming first that \( f \) has no zeros or poles on the boundary of \( {P}_{0} \), we know by the argument principle in Chapter 3 that\n\n\[ \n{\int }_{\partial {P}_{0}}\frac{{f}^{\prime }\left( z\right) }{f\left( z\right) }{dz} = {2\pi i}\left( {{\mathcal{N}}_{\mathfrak{z}} - {\mathcal{N}}_{\mathfrak{p}}}\righ... | Yes |
Lemma 1.5 The two series\n\n\[ \mathop{\sum }\limits_{{\left( {n, m}\right) \neq \left( {0,0}\right) }}\frac{1}{{\left( \left| n\right| + \left| m\right| \right) }^{r}}\;\text{ and }\;\mathop{\sum }\limits_{{n + {m\tau } \in {\Lambda }^{ * }}}\frac{1}{{\left| n + m\tau \right| }^{r}} \]\n\nconverge if \( r > 2 \) . | Recall that according to the Note at the end of Chapter 7, the question whether a double series converges absolutely is independent of the order of summation. In the present case, we shall first sum in \( m \) and then in \( n \) .\n\nFor the first series, the usual integral comparison can be applied. \( {}^{2} \) For ... | Yes |
Theorem 1.6 The function \( \wp \) is an elliptic function that has periods 1 and \( \tau \), and double poles at the lattice points. | Proof. It remains only to prove that \( \wp \) is periodic with the correct periods. To do so, note that the derivative is given by differentiating the series for \( \wp \) termwise so\n\n\[{\wp }^{\prime }\left( z\right) = - 2\mathop{\sum }\limits_{{n, m \in \mathbb{Z}}}\frac{1}{{\left( z + n + m\tau \right) }^{3}}.\]... | Yes |
Theorem 1.7 The function \( {\left( {\wp }^{\prime }\right) }^{2} \) is the cubic polynomial in \( \wp \)\n\n\[{\left( {\wp }^{\prime }\right) }^{2} = 4\left( {\wp - {e}_{1}}\right) \left( {\wp - {e}_{2}}\right) \left( {\wp - {e}_{3}}\right) .\] | Proof. The only roots of \( F\left( z\right) = \left( {\wp \left( z\right) - {e}_{1}}\right) \left( {\wp \left( z\right) - {e}_{2}}\right) \left( {\wp \left( z\right) - {e}_{3}}\right) \) in the fundamental parallelogram have multiplicity 2 and are at the points \( 1/2,\tau /2 \), and \( \left( {1 + \tau }\right) /2 \)... | Yes |
Lemma 1.9 Every even elliptic function \( F \) with periods 1 and \( \tau \) is a rational funcion of \( \wp \) . | Proof. If \( F \) has a zero or pole at the origin it must be of even order, since \( F \) is an even function. As a consequence, there exists an integer \( m \) so that \( F{\wp }^{m} \) has no zero or pole at the lattice points. We may therefore assume that \( F \) itself has no zero or pole on \( \Lambda \) .\n\nOur... | Yes |
Theorem 2.1 Eisenstein series have the following properties:\n\n(i) The series \( {E}_{k}\left( \tau \right) \) converges if \( k \geq 3 \), and is holomorphic in the upper half-plane.\n\n(ii) \( {E}_{k}\left( \tau \right) = 0 \) if \( k \) is odd.\n\n(iii) \( {E}_{k}\left( \tau \right) \) satisfies the following trans... | Proof. By Lemma 1.5 and the remark after it, the series \( {E}_{k}\left( \tau \right) \) converges absolutely and uniformly in every half-plane \( \operatorname{Im}\left( \tau \right) \geq \delta > 0 \) , whenever \( k \geq 3 \) ; hence \( {E}_{k}\left( \tau \right) \) is holomorphic in the upper half-plane \( \operato... | Yes |
Theorem 2.2 For \( z \) near 0, we have\n\n\[ \wp \left( z\right) = \frac{1}{{z}^{2}} + 3{E}_{4}{z}^{2} + 5{E}_{6}{z}^{4} + \cdots \]\n\n\[ = \frac{1}{{z}^{2}} + \mathop{\sum }\limits_{{k = 1}}^{\infty }\left( {{2k} + 1}\right) {E}_{{2k} + 2}{z}^{2k}. \] | Proof. From the definition of \( \wp \), if we note that we may replace \( \omega \) by \( - \omega \) without changing the sum, we have\n\n\[ \wp \left( z\right) = \frac{1}{{z}^{2}} + \mathop{\sum }\limits_{{\omega \in {\Lambda }^{ * }}}\left\lbrack {\frac{1}{{\left( z + \omega \right) }^{2}} - \frac{1}{{\omega }^{2}}... | Yes |
Lemma 2.4 If \( k \geq 2 \) and \( \operatorname{Im}\left( \tau \right) > 0 \), then\n\n\[ \mathop{\sum }\limits_{{n = - \infty }}^{\infty }\frac{1}{{\left( n + \tau \right) }^{k}} = \frac{{\left( -2\pi i\right) }^{k}}{\left( {k - 1}\right) !}\mathop{\sum }\limits_{{\ell = 1}}^{\infty }{\ell }^{k - 1}{e}^{{2\pi i\tau }... | Proof. This identity follows from applying the Poisson summation formula to \( f\left( z\right) = 1/{\left( z + \tau \right) }^{k} \) ; see Exercise 7 in Chapter 4. | No |
Theorem 2.5 If \( k \geq 4 \) is even, and \( \operatorname{Im}\left( \tau \right) > 0 \), then\n\n\[ \n{E}_{k}\left( \tau \right) = {2\zeta }\left( k\right) + \frac{2{\left( -1\right) }^{k/2}{\left( 2\pi \right) }^{k}}{\left( {k - 1}\right) !}\mathop{\sum }\limits_{{r = 1}}^{\infty }{\sigma }_{k - 1}\left( r\right) {e... | Proof. First observe that \( {\sigma }_{k - 1}\left( r\right) \leq r{r}^{k - 1} = {r}^{k} \) . If \( \operatorname{Im}\left( \tau \right) = t \), then whenever \( t \geq {t}_{0} \) we have \( \left| {e}^{2\pi ir\tau }\right| \leq {e}^{-{2\pi r}{t}_{0}} \), and we see that the series in the theorem is absolutely converg... | Yes |
Corollary 2.6 The double sum defining \( F \) converges in the indicated order. We have\n\n\[ F\\left( \\tau \\right) = {2\\zeta }\\left( 2\\right) - 8{\\pi }^{2}\\mathop{\\sum }\\limits_{{r = 1}}^{\\infty }\\sigma \\left( r\\right) {e}^{2\\pi ir\\tau }, \]\n\nwhere \( \\sigma \\left( r\\right) = \\mathop{\\sum }\\limi... | It can be seen that \( F\\left( {-1/\\tau }\\right) {\\tau }^{-2} \) does not equal \( F\\left( \\tau \\right) \), and this is the same as saying that the double series for \( F \) gives a different value \( (\\widetilde{F} \) , the reverse of \( F \) ) when we sum first in \( m \) and then in \( n \) . It turns out th... | No |
Proposition 1.1 The function \( \Theta \) satisfies the following properties:\n\n(i) \( \Theta \) is entire in \( z \in \mathbb{C} \) and holomorphic in \( \tau \in \mathbb{H} \) .\n\n(ii) \( \Theta \left( {z + 1 \mid \tau }\right) = \Theta \left( {z \mid \tau }\right) \) .\n\n(iii) \( \Theta \left( {z + \tau \mid \tau... | Proof. Suppose that \( \operatorname{Im}\left( \tau \right) = t \geq {t}_{0} > 0 \) and \( z = x + {iy} \) belongs to a bounded set in \( \mathbb{C} \), say \( \left| z\right| \leq M \) . Then, the series defining \( \Theta \) is absolutely and uniformly convergent, since\n\n\[ \mathop{\sum }\limits_{{n = - \infty }}^{... | Yes |
For each fixed \( \tau \in \mathbb{H} \), the quotient\n\n\[ \n{\left( \log \Theta \left( z \mid \tau \right) \right) }^{\prime \prime } = \frac{\Theta \left( {z \mid \tau }\right) {\Theta }^{\prime \prime }\left( {z \mid \tau }\right) - {\left( {\Theta }^{\prime }\left( z \mid \tau \right) \right) }^{2}}{\Theta {\left... | Let \( F\left( z\right) = {\left( \log \Theta \left( z \mid \tau \right) \right) }^{\prime } = \Theta {\left( z \mid \tau \right) }^{\prime }/\Theta \left( {z \mid \tau }\right) \) . Differentiating the identities (ii) and (iii) of Proposition 1.1 gives \( F\left( {z + 1}\right) = F\left( z\right) \) , \( F\left( {z + ... | Yes |
Theorem 1.6 If \( \tau \in \mathbb{H} \), then\n\n\[ \Theta \left( {z \mid - 1/\tau }\right) = \sqrt{\frac{\tau }{i}}{e}^{{\pi i\tau }{z}^{2}}\Theta \left( {{z\tau } \mid \tau }\right) \;\text{ for all }z \in \mathbb{C}. \] | Proof. It suffices to prove this formula for \( z = x \) real and \( \tau = {it} \) with \( t > 0 \), since for each fixed \( x \in \mathbb{R} \), the two sides of equation (5) are holomorphic functions in the upper half-plane which then agree on the positive imaginary axis, and hence must be equal everywhere. Also, fo... | Yes |
Corollary 1.7 If \( \operatorname{Im}\left( \tau \right) > 0 \), then \( \theta \left( {-1/\tau }\right) = \sqrt{\tau /i}\theta \left( \tau \right) \). | Note that if \( \tau = {it} \), then \( \theta \left( \tau \right) = \vartheta \left( t\right) \), and the above relation is precisely the functional equation for \( \vartheta \) which appeared in Chapter 4. | No |
Corollary 1.8 If \( \tau \in \mathbb{H} \), then\n\n\[ \theta \left( {1 - 1/\tau }\right) = \sqrt{\frac{\tau }{i}}\mathop{\sum }\limits_{{n = - \infty }}^{\infty }{e}^{{\pi i}{\left( n + 1/2\right) }^{2}\tau } \]\n\n\[ = \sqrt{\frac{\tau }{i}}\left( {2{e}^{{\pi i\tau }/4} + \cdots }\right) . \]\n\nThe second identity m... | Proof. First, we note that \( n \) and \( {n}^{2} \) have the same parity, so\n\n\[ \theta \left( {1 + \tau }\right) = \mathop{\sum }\limits_{{n = - \infty }}^{\infty }{\left( -1\right) }^{n}{e}^{{i\pi }{n}^{2}\tau } = \Theta \left( {1/2 \mid \tau }\right) ,\]\n\nhence \( \theta \left( {1 - 1/\tau }\right) = \Theta \le... | Yes |
Proposition 1.9 If \( \operatorname{Im}\left( \tau \right) > 0 \), then \( \eta \left( {-1/\tau }\right) = \sqrt{\tau /i}\eta \left( \tau \right) \) . | Proof. From the product formula for the theta function, we may write with \( q = {e}^{\pi i\tau } \), \[ \Theta \left( {z \mid \tau }\right) = \left( {1 + q{e}^{-{2\pi iz}}}\right) \mathop{\prod }\limits_{{n = 1}}^{\infty }\left( {1 - {q}^{2n}}\right) \left( {1 + {q}^{{2n} - 1}{e}^{2\pi iz}}\right) \left( {1 + {q}^{{2n... | Yes |
Theorem 2.1 If \( \left| x\right| < 1 \), then \( \mathop{\sum }\limits_{{n = 0}}^{\infty }p\left( n\right) {x}^{n} = \mathop{\prod }\limits_{{k = 1}}^{\infty }\frac{1}{1 - {x}^{k}} \) . | Formally, we can write each fraction as\n\n\[ \frac{1}{1 - {x}^{k}} = \mathop{\sum }\limits_{{m = 0}}^{\infty }{x}^{km} \]\n\nand multiply these out together to obtain \( p\left( n\right) \) as the coefficient of \( {x}^{n} \) . Indeed, when we group together equal integers in a partition of \( n \), this partition can... | Yes |
Proposition 2.2 \( \mathop{\prod }\limits_{{n = 1}}^{\infty }\left( {1 - {x}^{n}}\right) = \mathop{\sum }\limits_{{k = - \infty }}^{\infty }{\left( -1\right) }^{k}{x}^{\frac{k\left( {{3k} + 1}\right) }{2}} \) . | Proof. If we set \( x = {e}^{2\pi iu} \), then we can write\n\n\[ \mathop{\prod }\limits_{{n = 1}}^{\infty }\left( {1 - {x}^{n}}\right) = \mathop{\prod }\limits_{{n = 1}}^{\infty }\left( {1 - {e}^{2\pi inu}}\right) \]\n\nin terms of the triple product\n\n\[ \mathop{\prod }\limits_{{n = 1}}^{\infty }\left( {1 - {q}^{2n}... | Yes |
Theorem 3.1 If \( n \geq 1 \), then \( {r}_{2}\left( n\right) = 4\left( {{d}_{1}\left( n\right) - {d}_{3}\left( n\right) }\right) \) . | To prove the theorem, we first establish a crucial relationship that identifies the generating function of the sequence \( {\left\{ {r}_{2}\left( n\right) \right\} }_{n = 1}^{\infty } \) with the square of the \( \theta \) function, namely\n\n(6)\n\n\[ \theta {\left( \tau \right) }^{2} = \mathop{\sum }\limits_{{n = 0}}... | Yes |
Proposition 3.2 The identity \( {r}_{2}\left( n\right) = 4\left( {{d}_{1}\left( n\right) - {d}_{3}\left( n\right) }\right), n \geq 1 \), is equivalent to the identities (7) \(\\theta {\\left( \\tau \\right) }^{2} = 2\\mathop{\\sum }\\limits_{{n = - \\infty }}^{\\infty }\\frac{1}{{q}^{n} + {q}^{-n}} = 1 + 4\\mathop{\\su... | Proof. We note first that both series converge absolutely since \( \\left| q\\right| < 1 \) , and the first equals the second, because \( 1/\\left( {{q}^{n} + {q}^{-n}}\\right) = {q}^{\\left| n\\right| }/\\left( {1 + {q}^{2\\left| n\\right| }}\\right) \) . Since \( {\\left( 1 + {q}^{2n}\\right) }^{-1} = \\left( {1 - {q... | Yes |
Theorem 3.4 Suppose \( f \) is a holomorphic function in the upper half-plane that satisfies:\n\n(i) \( f\left( {\tau + 2}\right) = f\left( \tau \right) \),\n\n(ii) \( f\left( {-1/\tau }\right) = f\left( \tau \right) \),\n\n(iii) \( f\left( \tau \right) \) is bounded,\n\nthen \( f \) is constant. | For the proof of this theorem, we introduce the following subset of the closed upper half-plane, which is defined by\n\n\[ \mathcal{F} = \{ \tau \in \overline{\mathbb{H}} : \left| {\operatorname{Re}\left( \tau \right) }\right| \leq 1\text{ and }\left| \tau \right| \geq 1\} ,\] | No |
Lemma 3.5 Every point in the upper half-plane can be mapped into \( \mathcal{F} \) using repeatedly one or another of the following fractional linear transformations or their inverses:\n\n\[ \n{T}_{2} : \tau \mapsto \tau + 2,\;S : \tau \mapsto - 1/\tau .\n\]\n\nFor this reason, \( \mathcal{F} \) is called the fundament... | Proof of Lemma 3.5. Let \( \tau \in \mathbb{H} \) . If \( g \in G \) with \( g\left( \tau \right) = \left( {{a\tau } + b}\right) /\left( {{c\tau } + d}\right) \) , then \( c \) and \( d \) are integers, and by (9) we may choose a \( {g}_{0} \in G \) such that \( \operatorname{Im}\left( {{g}_{0}\left( \tau \right) }\rig... | Yes |
Theorem 3.6 Every positive integer is the sum of four squares, and moreover\n\n\[ \n{r}_{4}\left( n\right) = 8{\sigma }_{1}^{ * }\left( n\right) \;\text{ for all }n \geq 1.\n\] | As before, we relate the sequence \( \left\{ {{r}_{4}\left( n\right) }\right\} \) via its generating function to an appropriate power of the function \( \theta \), which in this case is its fourth power. The result is that\n\n\[ \n\theta {\left( \tau \right) }^{4} = \mathop{\sum }\limits_{{n = 0}}^{\infty }{r}_{4}\left... | No |
Proposition 3.7 The assertion \( {r}_{4}\left( n\right) = 8{\sigma }_{1}^{ * }\left( n\right) \) is equivalent to the identity\n\n\[ \theta {\left( \tau \right) }^{4} = \frac{-1}{{\pi }^{2}}{E}_{2}^{ * }\left( \tau \right) ,\;\text{ where }\tau \in \mathbb{H}. \] | Proof. It suffices to prove that if \( q = {e}^{\pi i\tau } \), then\n\n\[ \frac{-1}{{\pi }^{2}}{E}_{2}^{ * }\left( \tau \right) = 1 + \mathop{\sum }\limits_{{k = 1}}^{\infty }8{\sigma }_{1}^{ * }\left( k\right) {q}^{k}. \]\n\nFirst, recall the forbidden Eisenstein series that we considered in the last section of the p... | Yes |
Proposition 3.8 The function \( {E}_{2}^{ * }\left( \tau \right) \) defined in the upper half-plane has the following properties:\n\n(i) \( {E}_{2}^{ * }\left( {\tau + 2}\right) = {E}_{2}^{ * }\left( \tau \right) \).\n\n(ii) \( {E}_{2}^{ * }\left( \tau \right) = - {\tau }^{-2}{E}_{2}^{ * }\left( {-1/\tau }\right) \).\n... | The periodicity (i) of \( {E}_{2}^{ * } \) is immediate from the definition. The proofs of the other properties of \( {E}_{2}^{ * } \) are a little more involved. | No |
Lemma 3.9 The functions \( F \) and \( \widetilde{F} \) satisfy:\n\n(a) \( F\left( {-1/\tau }\right) = {\tau }^{2}\widetilde{F}\left( \tau \right) \),\n\n(b) \( F\left( \tau \right) - \widetilde{F}\left( \tau \right) = {2\pi i}/\tau \),\n\n(c) \( F\left( {-1/\tau }\right) = {\tau }^{2}F\left( \tau \right) - {2\pi i\tau... | Proof. Property (a) follows directly from the definitions of \( F \) and \( \widetilde{F} \) , and the identity\n\n\[{\left( n + m\left( -1/\tau \right) \right) }^{2} = {\tau }^{-2}{\left( -m + n\tau \right) }^{2}.\n\nTo prove property (b), we invoke the functional equation for the Dedekind eta function which was estab... | Yes |
Theorem 1.1 \( {J}_{\nu }\left( s\right) = \sqrt{\frac{2}{\pi s}}\cos \left( {s - \frac{\pi \nu }{2} - \frac{\pi }{4}}\right) + O\left( {s}^{-3/2}\right) \) as \( s \rightarrow \infty \) . | In view of the formula for \( {J}_{\nu }\left( s\right) \) it suffices to investigate\n\n(3)\n\n\[ I\left( s\right) = {\int }_{-1}^{1}{e}^{isx}{\left( 1 - {x}^{2}\right) }^{\nu - 1/2}{dx} \]\n\nand to this end we consider the analytic function \( f\left( z\right) = {e}^{isz}{\left( 1 - {z}^{2}\right) }^{\nu - 1/2} \) i... | Yes |
Proposition 1.2 Suppose a and \( m \) are fixed, with \( a > 0 \) and \( m > - 1 \) . Then as \( s \rightarrow \infty \n\n\[{\int }_{0}^{a}{e}^{-{sx}}{x}^{m}{dx} = {s}^{-m - 1}\Gamma \left( {m + 1}\right) + O\left( {e}^{-{cs}}\right) ,\]\n\nfor some positive \( c \) . | Proof. The fact that \( m > - 1 \) guarantees that \( {\int }_{0}^{a}{e}^{-{sx}}{x}^{m}{dx} = \) \( \mathop{\lim }\limits_{{\epsilon \rightarrow 0}}{\int }_{\epsilon }^{a}{e}^{-{sx}}{x}^{m}{dx} \) exists. Then, we write\n\n\[{\int }_{0}^{a}{e}^{-{sx}}{x}^{m}{dx} = {\int }_{0}^{\infty }{e}^{-{sx}}{x}^{m}{dx} - {\int }_{... | Yes |
Proposition 1.3 Suppose a and \( m \) are fixed, with \( a > 0 \) and \( - 1 < m < 0 \) . Then as \( \left| s\right| \rightarrow \infty \) with \( \operatorname{Re}\left( s\right) \geq 0 \) , \n\n\[ \n{\int }_{0}^{a}{e}^{-{sx}}{x}^{m}{dx} = {s}^{-m - 1}\Gamma \left( {m + 1}\right) + O\left( {1/\left| s\right| }\right) ... | Proof. We begin by showing that when \( \operatorname{Re}\left( s\right) \geq 0, s \neq 0 \) , \n\n\[ \n{\int }_{0}^{\infty }{e}^{-{sx}}{x}^{m}{dx} = \mathop{\lim }\limits_{{N \rightarrow \infty }}{\int }_{0}^{N}{e}^{-{sx}}{x}^{m}{dx} \n\] \n\nexists and equals \( {s}^{-m - 1}\Gamma \left( {m + 1}\right) \) . If \( N \... | Yes |
Proposition 2.1 Under the above assumptions, with \( s > 0 \) and \( s \rightarrow \infty \) ,\n\n\[{\int }_{a}^{b}{e}^{-{s\Phi }\left( x\right) }\psi \left( x\right) {dx} = {e}^{-{s\Phi }\left( {x}_{0}\right) }\left\lbrack {\frac{A}{{s}^{1/2}} + O\left( \frac{1}{s}\right) }\right\rbrack ,\]\n\nwhere\n\n\[A = \sqrt{2\p... | Proof. By replacing \( \Phi \left( x\right) \) by \( \Phi \left( x\right) - \Phi \left( {x}_{0}\right) \) we may assume that \( \Phi \left( {x}_{0}\right) = 0 \) . Since \( {\Phi }^{\prime }\left( {x}_{0}\right) = 0 \), we note that\n\n\[ \frac{\Phi \left( x\right) }{{\left( x - {x}_{0}\right) }^{2}} = \frac{{\Phi }^{\... | Yes |
Proposition 2.2 With the same assumptions on \( \Phi \) and \( \psi \), the relation (8) continues to hold if \( \left| s\right| \rightarrow \infty \) with \( \operatorname{Re}\left( s\right) \geq 0 \) . | Proof. We proceed as before to the equation (9), and obtain the appropriate asymptotic for the first term, by virtue of Proposition 1.3, with \( m = - 1/2 \) . To deal with the rest we start with an observation. If \( \Psi \) and \( \psi \) are given on an interval \( \left\lbrack {\bar{a},\bar{b}}\right\rbrack \), are... | Yes |
Theorem 3.1 Suppose \( u > 0 \) . Then as \( u \rightarrow \infty \) , (i) \( \operatorname{Ai}\left( {-u}\right) = {\pi }^{-1/2}{u}^{-1/4}\cos \left( {\frac{2}{3}{u}^{3/2} - \frac{\pi }{4}}\right) \left( {1 + O\left( {1/{u}^{3/4}}\right) }\right) \) . | To consider the first case, we make the change of variables \( x \mapsto {u}^{1/2}x \) in the defining integral with \( s = - u \) . This gives \[ \operatorname{Ai}\left( {-u}\right) = {u}^{1/2}{I}_{ - }\left( {u}^{3/2}\right) \] where \[ {I}_{ - }\left( t\right) = \frac{1}{2\pi }{\int }_{-\infty }^{\infty }{e}^{{it}\l... | Yes |
Theorem 1.1 A region \( \Omega \) is holomorphically simply connected if and only if \( \Omega \) is simply connected. | Proof. One direction is simply the version of Cauchy's theorem in Corollary 5.3, Chapter 3. Conversely, suppose that \( \Omega \) is holomorphically simply connected. If \( \Omega = \mathbb{C} \), then it is clearly simply connected. If \( \Omega \) is not all of \( \mathbb{C} \), recall that the Riemann mapping theore... | Yes |
Lemma 1.3 Let \( \gamma \) be a closed curve in \( \mathbb{C} \) .\n\n(i) If \( z \notin \gamma \), then \( {W}_{\gamma }\left( z\right) \in \mathbb{Z} \) . | Proof. To see why (i) is true, suppose that \( \gamma : \left\lbrack {0,1}\right\rbrack \rightarrow \mathbb{C} \) is a parametrization for the curve, and let\n\n\[ G\left( t\right) = {\int }_{0}^{t}\frac{{\gamma }^{\prime }\left( s\right) }{\gamma \left( s\right) - z}{ds}. \]\n\nThen \( G \) is continuous and, except p... | Yes |
Theorem 1.4 A bounded region \( \Omega \) is simply connected if and only if \( {W}_{\gamma }\left( z\right) = 0 \) for any closed curve \( \gamma \) in \( \Omega \) and any point \( z \) not in \( \Omega \) . | Proof. If \( \Omega \) is simply connected and \( z \notin \Omega \), then \( f\left( \zeta \right) = 1/\left( {\zeta - z}\right) \) is holomorphic in \( \Omega \), and Cauchy’s theorem gives \( {W}_{\gamma }\left( z\right) = 0 \) .\n\nFor the converse, it suffices to prove that the complement of \( \Omega \) is connec... | Yes |
Lemma 1.5 Let \( w \) be any point in \( {F}_{1} \) . Under the above assumptions, there exists a finite collection of closed squares \( \mathcal{Q} = \left\{ {{Q}_{1},\ldots ,{Q}_{n}}\right\} \) that belong to a uniform grid \( \mathcal{G} \) of the plane, and are such that:\n\n(i) \( w \) belongs to the interior of \... | Proof of the lemma. Since \( {F}_{2} \) is closed, the sets \( {F}_{1} \) and \( {F}_{2} \) are at a finite non-zero distance \( d \) from one another. Now consider a uniform grid \( {\mathcal{G}}_{0} \) of the plane consisting of closed squares of side length which is much smaller than \( d \), say \( < d/{100} \), an... | Yes |
Theorem 2.2 Let \( \Gamma \) be a curve in the plane which is simple, closed, and piecewise-smooth. Then, the complement of \( \Gamma \) consists of two disjoint connected open sets. Precisely one of these regions is bounded and simply connected; it is called the interior of \( \Gamma \) and denoted by \( \Omega \) . T... | Moreover, with the appropriate orientation for \( \Gamma \), we have\n\n\[ \n{W}_{\Gamma }\left( z\right) = \left\{ \begin{array}{ll} 1 & \text{ if }z \in \Omega \\ 0 & \text{ if }z \in \mathcal{U} \end{array}\right. \n\] | No |
Lemma 2.5 Suppose \( z \) is a point which does not belong to the smooth curve \( {\Gamma }_{0} \), but that is closer to an interior point of the curve than to either of its end-points. Then \( z \) belongs to \( {\Gamma }_{\epsilon } \) for some \( \epsilon \neq 0 \) . | More precisely, if \( {z}_{0} \in {\Gamma }_{0} \) is closest to \( z \) and \( {z}_{0} = \gamma \left( {t}_{0}\right) \) for some \( {t}_{0} \) in the open interval \( \left( {0, L}\right) \), then \( z = \gamma \left( {t}_{0}\right) + {i\epsilon }{\gamma }^{\prime }\left( {t}_{0}\right) \) for some \( \epsilon \neq 0... | Yes |
Proposition 2.6 Let \( A \) and \( B \) denote the two end-points of a simple smooth curve \( {\Gamma }_{0} \), and suppose that \( K \) is a compact set that satisfies either \[ {\Gamma }_{0} \cap K = \varnothing \;\text{ or }\;{\Gamma }_{0} \cap K = A \cup B. \] If \( z \notin {\Gamma }_{0} \) and \( w \notin {\Gamma... | Proof. By the previous lemma, consider \( {z}_{0} = \gamma \left( {t}_{0}\right) \) and \( {w}_{0} = \gamma \left( {s}_{0}\right) \) that are interior points of \( {\Gamma }_{0} \) closest to \( z \) and \( w \), respectively. Then \[ z = \gamma \left( {t}_{0}\right) + i{\epsilon }_{0}{\gamma }^{\prime }\left( {t}_{0}\... | Yes |
Lemma 2.7 Let \( {\Gamma }_{0} \) be a simple smooth curve. There exists \( {\kappa }_{2} > 0 \) so that the set \( N \), which consists of points of the form \( z = \gamma \left( L\right) + \epsilon {e}^{i\theta }{\gamma }^{\prime }\left( L\right) \) with \( - \pi /2 \leq \theta \leq \pi /2 \) and \( 0 < \epsilon < {\... | Proof. The argument is similar to the one given in the proof of Lemma 2.4. First, we note that\n\n\[ \gamma \left( L\right) + \epsilon {e}^{i\theta }{\gamma }^{\prime }\left( L\right) - \gamma \left( t\right) = {\int }_{t}^{L}\left\lbrack {{\gamma }^{\prime }\left( u\right) - {\gamma }^{\prime }\left( L\right) }\right\... | Yes |
Proposition 2.8 Let \( A \) denote an end-point of the simple smooth curve \( {\Gamma }_{0} \), and suppose that \( K \) is a compact set that satisfies either\n\n\[{\Gamma }_{0} \cap K = \varnothing \;\text{ or }\;{\Gamma }_{0} \cap K = A.\]\n\nIf \( z \notin {\Gamma }_{0} \) and \( w \notin {\Gamma }_{0} \) are close... | We only provide an outline of the argument, which is similar to the proof of Proposition 2.6. It suffices to consider the case when \( z \) and \( w \) lie on opposite sides of \( {\Gamma }_{0} \) and \( A = \gamma \left( 0\right) \) . First, we may join\n\n\[{z}_{\epsilon } = \gamma \left( {t}_{0}\right) + {i\epsilon ... | No |
Theorem 2.9 If a function \( f \) is holomorphic in an open set that contains a simple closed piecewise-smooth curve \( \Gamma \) and its interior, then\n\n\[{\int }_{\Gamma }f = 0\] | Let \( \mathcal{O} \) denote an open set on which \( f \) is holomorphic, and which contains \( \Gamma \) and its interior \( \Omega \) . The idea is to construct a closed curve \( \Lambda \)\nin \( \Omega \) that is so close to \( \Gamma \) that \( {\int }_{\Gamma }f = {\int }_{\Lambda }f \) . Then, the integral on th... | Yes |
Lemma 2.10 Let \( \gamma : \left\lbrack {0,1}\right\rbrack \rightarrow \mathbb{C} \) be a simple smooth curve. Then, for all sufficiently small \( \delta > 0 \) the circle \( {C}_{\delta } \) centered at \( \gamma \left( 0\right) \) and of radius \( \delta \) intersects \( \gamma \) in precisely one point. | Proof. We may assume that \( \gamma \left( 0\right) = 0 \) . Since \( \gamma \left( 0\right) \neq \gamma \left( 1\right) \) it is clear that for each small \( \delta > 0 \), the circle \( {C}_{\delta } \) intersects \( \gamma \) in at least one point. If the conclusion in the lemma is false, we can find a sequence of p... | Yes |
Theorem 3.1 Let \( \\left( {u, p, T}\\right) \) be a nonsingular solution of (1), and \( \\left( {{u}^{h},{p}^{h},{T}^{h}}\\right) \) is a solution of Algorithm 2.4 with mesh \( H \) small enough. Then, there are constants \( {C}_{1},{C}_{2},{C}_{3},{C}_{4} \) such that | Proof As \( \\left( {u, p, T}\\right) \) is a nonsingular solution, it is known \( {}^{\\left\\[ {16}\\right\\]} \) that \( \\left( {{u}^{H},{p}^{H},{T}^{H}}\\right) \\rightarrow \) \( \\left( {u, p, T}\\right) \) strongly in \( X \) as \( {X}^{H} \) becomes dense in \( X \) . For mesh size \( H \) small enough, \( {DF... | No |
Corollary 3.1 Let \( \\left( {u, p, T}\\right) \) is a nonsingular solution of \( \\left( 1\\right) ,\\left( {{u}^{h},{p}^{h},{T}^{h}}\\right) \) is a solution of Algorithm 2.1, and mesh \( h \) small enough. Then, there are constants \( {C}_{1},{C}_{2},{C}_{3},{C}_{4} \) such that\n\n\[ \n{\\left\\{ {\\begin{Vmatrix}u... | Proof Using \( {u}^{H} = {u}^{h},{p}^{H} = {p}^{h} \) and \( {T}^{H} = {T}^{h} \) in (10) if \( {\\Omega }_{H} = {\\Omega }_{h} \), yields Corollary 3.1. We define the local error indicator \( {\\eta }_{K} \) on each \( K \\in {\\Omega }_{h} \) ,\n\n\[ \n{\\eta }_{K} = \\left\\{ {{h}_{K}^{2}\\parallel - \\Pr \\bigtrian... | Yes |
Example 4.1 \( \Omega = \left\lbrack {0,1}\right\rbrack \times \left\lbrack {0,1}\right\rbrack \), and the chosen functions are added to the righthand side of (1) such that the exact solution of the problem is\n\n\[ u\left( {x, y}\right) = \left( {{u}_{1}\left( {x, y}\right) ,{u}_{2}\left( {x, y}\right) }\right) ,\;{u}... | Table 1, Table 2, and Figure 1 are numerical results of Example 4.1. Table 1 shows the uniform coarse grids \( H \), the fine grids \( h \), the whole approximate errors \( {E}_{1} \) , the addition terms error \ | No |
Lemma 6 Let \( {\phi }_{\tau }\left( z\right) \) and \( {H}_{\tau }\left( z\right) \) be defined by (2) and (3). Then:\n\n(a) \( {\phi }_{\tau }\left( z\right) \) and \( {H}_{\tau }\left( z\right) \) is continuously differentiable at any \( z = \left( {\mu, x, y}\right) \in {\mathbb{R}}_{+ + } \times \) \( {\mathbb{R}}... | Proof (a) Since\n\n\[ \widehat{\omega } = {\left\lbrack {\left( x - y\right) }^{2} + \tau \left( x \circ y\right) + 4{\mu }^{2}e\right\rbrack }^{\frac{1}{2}}\]\n\n\[ = {\left\lbrack {\left( x + \frac{\tau - 2}{2}y\right) }^{2} + \frac{\tau \left( {4 - \tau }\right) }{4}{y}^{2} + 4{\mu }^{2}e\right\rbrack }^{\frac{1}{2}... | Yes |
Lemma 7 Suppose that \( F\left( x\right) \) is a continuously monotone function, and \( {\psi }_{\tau } \) is defined by (15). Let \( \left\{ {z}_{k}\right\} \) be the iteration sequence generated by Algorithm 1. Then the level set\n\n\[ L\left( {z}_{0}\right) = \left\{ {z \mid {\psi }_{\tau }\left( z\right) \leq {\psi... | Proof Similar to the proof of the Lemma 5.3 in [20]. | No |
Theorem 3 Assume that \( F \) is continuously differentiable monotone function and the solution set of SCCP (1) is nonempty and bounded. Let \( {z}_{ * } \) be an accumulation point of the iteration sequence \( \left\{ {z}_{k}\right\} \) generated by Algorithm 1. If all \( V \in \partial {H}_{\tau }\left( {z}_{ * }\rig... | Proof From Lemma 6, we have that \( {H}_{\tau }\left( z\right) \) is strong semismooth on any \( z = \) \( \left( {\mu, x, y}\right) \in {\mathbb{R}}_{ + } \times {\mathbb{R}}^{n} \times {\mathbb{R}}^{n} \) for any \( \tau \in \lbrack 0,4) \) . Therefore, the conclusion of the theorem follows from similar arguments as ... | No |
Example 1 (The second-order cone nonlinear complementarity problem) The test function \( F \) is given as follows\n\n\[ F\\left( x\\right) = \\left( \\begin{matrix} {0.07}{x}_{1}^{3} - 4 \\\\ {0.04}{x}_{2}^{3} - {3.93} \\\\ {0.03}{x}_{3}^{3} - {5.72} \\end{matrix}\\right) . \] | This example has one solution \( {x}_{ * } = {\\left( 5,3,4\\right) }^{T} \) . The different starting vector are listed in Table 1. | "No" |
The one-dimensional Burger's equation is\n\n\[ \left\{ \begin{array}{lll} \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} = \nu \frac{{\partial }^{2}u}{\partial {x}^{2}}, & 0 < x < 1, & 0 < t \leq T, \\ u\left( {x,0}\right) = \sin \left( {\pi x}\right) , & 0 \leq x \leq 1, & \\ u\left( {0, t}\right) = u\... | This equation has an exact solution in the form of the infinite series\n\n\[ u\left( {x, t}\right) = {4\pi \nu }\frac{C}{D + E} \]\n\nwhere\n\n\[ C = \mathop{\sum }\limits_{{j = 1}}^{\infty }j{I}_{j}\left( \frac{1}{2\pi \nu }\right) \sin \left( {j\pi x}\right) \exp \left( {-{j}^{2}{\pi }^{2}{\nu t}}\right) ,\]\n\n\[ D ... | Yes |
Consider Burger's equation with the following forms\n\n\[ \left\{ \begin{array}{lll} \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} = \frac{1}{Re}\frac{{\partial }^{2}u}{\partial {x}^{2}}, & \frac{1}{2} \leq x \leq \frac{3}{2}, & t > 0, \\ u\left( {x,0}\right) = \frac{1}{Re}\left\lbrack {x + \tan \left(... | It possesses the following exact solution\n\n\[ u\left( {x, t}\right) = \frac{1}{{Re} + t}\left\lbrack {x + \tan \frac{xRe}{2\left( {{Re} + t}\right) }}\right\rbrack . \] | Yes |
Theorem 2 In the new higher-order multivariate Markov chain model, there exists a stationary probability \( X \) satisfying \( X = {BX} \) and \( \mathop{\lim }\limits_{{t \rightarrow \infty }}{X}_{t} = X \) . | Proof In the new higher-order multivariate Markov chain model, \( {\lambda }_{jj}^{\left( 1\right) },{\lambda }_{jj}^{\left( n\right) } > 0 \) , \( {P}_{jj}^{\left( 1\right) } \) is irreducible and at least one of them is aperiodic \( \left( {j = 1,2,\cdots, s}\right) \) . As \( B \) is connective, \( B \) is irreducib... | No |
Corollary 2.1 Let \( {X}_{1},{X}_{2},\cdots ,{X}_{n} \) be independent heterogeneous random variables with \( {X}_{i} \sim {IW}\left( {{\alpha }_{i},\beta }\right), i = 1,2,\cdots, n \) . Let \( {Y}_{1},{Y}_{2},\cdots ,{Y}_{n} \) be another set of independent and identically distributional random variables with \( {Y}_... | Proof Note that\n\n\[\n\left( {{\alpha }_{1},{\alpha }_{2},\cdots ,{\alpha }_{n}}\right) \overset{p}{ \succcurlyeq }\left( {\widetilde{\alpha },\cdots ,\widetilde{\alpha }}\right)\n\]\n\nso the result of Corollary 2.1 can be verified easily by Theorem 2.1. | No |
Theorem 2.2 Let \( {X}_{1},{X}_{2},\cdots ,{X}_{n} \) be independent heterogeneous random variables with \( {X}_{i} \sim {IW}\left( {{\alpha }_{i},\beta }}\right), i = 1,2,\cdots, n \) . Let \( {X}_{1}^{ * },{X}_{2}^{ * },\cdots ,{X}_{n}^{ * } \) be another set of independent heterogeneous random variables with \( {X}_... | Proof From (1), the density function of \( {X}_{\left( 1\right) } \) is \[ {f}_{{X}_{\left( 1\right) }}\left( x\right) = \mathop{\prod }\limits_{{i = 1}}^{n}\left\lbrack {1 - {\mathrm{e}}^{-{\left( \frac{{\alpha }_{i}}{x}\right) }^{\beta }}}\right\rbrack \mathop{\sum }\limits_{{i = 1}}^{n}\frac{{\mathrm{e}}^{-{\left( \... | Yes |
Theorem 3.1 Let \( {X}_{1},{X}_{2},\cdots ,{X}_{n} \) be independent heterogeneous random variables with \( {X}_{i} \sim {IW}\left( {{\alpha }_{i},\beta }\right), i = 1,2,\cdots, n \) . Let \( {X}_{1}^{ * },{X}_{2}^{ * },\cdots ,{X}_{n}^{ * } \) be another set of independent heterogeneous random variables with \( {X}_{... | Proof Note that the distribution function of \( {X}_{\left( n\right) } \) is \[ {F}_{{X}_{\left( n\right) }}\left( x\right) = \exp \left( {-\frac{\mathop{\sum }\limits_{{i = 1}}^{n}{\alpha }_{i}^{\beta }}{{x}^{\beta }}}\right) . \] Hence, the reversed hazard rate function of \( {X}_{\left( n\right) } \) can be written ... | Yes |
Corollary 3.1 Let \( {X}_{1},{X}_{2},\cdots ,{X}_{n} \) be independent heterogeneous random variables with \( {X}_{i} \sim {IW}\left( {{\alpha }_{i},\beta }}\right), i = 1,2,\cdots, n \), for \( x > 0,0 < \beta \leq 1 \) ,\n\n\[ P\left\lbrack {{X}_{\left( n\right) } \geq x}\right\rbrack \leq 1 - \exp \left( {-\frac{n{\... | Proof Let \( {Y}_{1},{Y}_{2},\cdots ,{Y}_{n} \) be a set of independent and identically distributional random variables with \( {Y}_{i} \sim {IW}\left( {\bar{\alpha },\beta }}\right) \), where \( \bar{\alpha } = \mathop{\sum }\limits_{{i = 1}}^{n}{\alpha }_{i}/n \) . Recall that\n\n\[ \left( {{\alpha }_{1},{\alpha }_{2... | Yes |
Theorem 3.2 Let \( {X}_{1},{X}_{2},\cdots ,{X}_{n} \) be independent heterogeneous random variables with \( {X}_{i} \sim {IW}\left( {{\alpha }_{i},\beta }\right), i = 1,2,\cdots, n \) . Let \( {Y}_{1},{Y}_{2},\cdots ,{Y}_{n} \) be another set of independent and identically distributional random variables with \( {Y}_{i... | Proof (a) Similar to (5), the reversed hazard rate function of \( {Y}_{\left( n\right) } \) can be written as \[ {\widetilde{r}}_{{Y}_{\left( n\right) }}\left( x\right) = \beta \cdot n{\widetilde{\alpha }}^{\beta } \cdot \frac{1}{{x}^{\beta + 1}}. \] It is easy to see that \[ \mathop{\sum }\limits_{{i = 1}}^{n}{\alpha ... | Yes |
Consider the following fractional descriptor system\n\n\[ \left\lbrack \begin{array}{ll} 1 & 1 \\ 0 & 0 \end{array}\right\rbrack \left\lbrack \begin{array}{l} \left( {{}^{C}{D}_{{0}^{ + }}^{1/2}{x}_{1}}\right) \left( t\right) \\ \left( {{}^{C}{D}_{{0}^{ + }}^{1/2}{x}_{2}}\right) \left( t\right) \end{array}\right\rbrack... | The general solution of this system is given by\n\n\[ {x}_{1}\left( t\right) = {E}_{1/2}\left( {-{t}^{1/2}}\right) {c}_{0} + {t}^{1/2}{E}_{1/2,3/2}\left( {-{t}^{1/2}}\right) - t{E}_{1/2,2}\left( {-{t}^{1/2}}\right) ,\;{x}_{2}\left( t\right) = t,\;t \in \left\lbrack {0, T}\right\rbrack .\n\nwhere \( {x}_{1}\left( 0\righ... | Yes |
Lemma 3.3 Let \( \nu \) be the index of nilpotent of \( N \), i.e., \( {N}^{\nu } = 0 \) and \( {N}^{\nu - 1} \neq 0 \) . Then the algebraic equation (7) has the unique solution\n\n\[ \n{y}_{2}\left( t\right) = - \mathop{\sum }\limits_{{i = 0}}^{{\nu - 1}}{N}^{i}{\left( {}^{C}{D}_{{0}^{ + }}^{\alpha }\right) }^{i}{B}_{... | Proof Because \( N \) is a nilpotent matrix and \( N \) and the operator \( {}^{C}{D}_{{0}^{ + }}^{\alpha } \) commute, by using the Neumann series, we have\n\n\[ \n{y}_{2}\left( t\right) = - {\left( I - {N}^{C}{D}_{{0}^{ + }}^{\alpha }\right) }^{-1}{B}_{2}u\left( t\right) \n\]\n\n\[ \n= - \mathop{\sum }\limits_{{i = 0... | Yes |
Example 4.1 Consider the fractional descriptor system (3) with the following matrices\n\n\[ \nE = \\left\\lbrack \\begin{matrix} 1 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ - 1 & 1 & 0 \\end{matrix}\\right\\rbrack ,\\;A = \\left\\lbrack \\begin{matrix} 1 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & - 1 & 0 \\end{matrix}\\right\\rbrack ,\\;B = \\... | Putting\n\n\[ \n{P}_{1} = {\\left( A - 2E\\right) }^{-1} = \\left\\lbrack \\begin{matrix} - 1 & 0 & 0 \\\\ - 2/3 & 0 & - 1/3 \\\\ 0 & 1 & 0 \\end{matrix}\\right\\rbrack ,\n\]\n\n\[ \n{P}_{2} = \\left\\lbrack \\begin{matrix} - 1 & 0 & 0 \\\\ 1 & - 3 & 0 \\\\ 0 & 0 & 1 \\end{matrix}\\right\\rbrack ,\\;Q = \\left\\lbrack ... | Yes |
Theorem 1 Assume that the time delay constant \( \tau \) is a multiple of the period \( T \) of coefficients of the differential equation (3) and \( \left| {1 + {\gamma }_{k}}\right| \leq 1 \) for \( k \in \{ 1,2,\cdots, q\} \) . If there exists some positive diagonal matrix \( D = \operatorname{diag}\left( {{d}_{1},{d... | Proof Let \( x\left( t\right) = z\left( t\right) - y\left( t\right) \) for all \( t \geq 0 \) . From the relations (2), we derive that\n\n\[ \frac{\parallel x\left( t\right) {\parallel }_{1} - \parallel x\left( {t - s}\right) {\parallel }_{1}}{s} \leq \frac{1}{s}\langle x\left( t\right) - x\left( {t - s}\right) ,\opera... | Yes |
Theorem 2 Suppose that the assumptions \( \left( {\mathrm{H}}_{1}\right) - \left( {\mathrm{H}}_{3}\right) \) hold and the inequalities\n\n\[ \mathop{\sum }\limits_{{j = 1}}^{n}\frac{{d}_{j}}{{d}_{i}}{\int }_{0}^{T}\left\lbrack {\left| {{b}_{ji}\left( t\right) }\right| {L}_{\mathbb{R}}\left( {f}_{i}\right) + \left| {{c}... | Proof Let \( D = \operatorname{diag}\left( {{d}_{1},{d}_{2},\cdots ,{d}_{n}}\right) \) and \( C = \operatorname{diag}\left( {{c}_{1}^{-1},{c}_{2}^{-1},\cdots ,{c}_{n}^{-1}}\right) \), where\n\n\[ {c}_{i} = \frac{1}{T}\left\lbrack {{\int }_{0}^{T}{a}_{i}\left( t\right) \mathrm{d}t - {L}_{\mathbb{R}}\left( {f}_{i}\right)... | Yes |
Consider the following delayed and impulsive CNNs with periodic coefficients\n\n\[ \n{\dot{u}}_{i}\left( t\right) = - {a}_{i}\left( t\right) {u}_{i}\left( t\right) + \mathop{\sum }\limits_{{j = 1}}^{2}{b}_{ij}\left( t\right) {f}_{j}\left( {{u}_{j}\left( t\right) }\right) \n\]\n\n\[ \n+ \mathop{\sum }\limits_{{j = 1}}^{... | In the model (23), the coefficients \( {a}_{i}\left( t\right) ,{b}_{ij}\left( t\right) \) and \( {c}_{ij}\left( t\right) \) are \( \frac{2}{3} \) -periodic for \( i, j = 1,2 \) and \( \tau = 2.L\left( {f}_{j}\right) = 1 \) and \( L\left( {g}_{j}\right) = \frac{1}{4} \) for \( j = 1,2 \) . It is obvious that the assumpt... | Yes |
Theorem 2 In the pure input case, the optimal allocated costs determined by problem \( \left( {P}^{3}\right) \) are equal to \( {c}_{j}^{\prime } \) in (34). | Proof Given that the single outputs of all DMUs are identical, we assume \( {y}_{1j} = \) \( b, j = 1,2,\cdots, n \), where \( b \) is a constant. If we choose allocated costs as those in (34) and choose common weights as follows\n\n\[ \n{v}_{i} = \frac{C{\alpha }_{i}}{\left( {n - 1}\right) \mathop{\sum }\limits_{{j = ... | Yes |
Theorem 3 In the pure output case, the optimal costs determined by problem \( \left( {P}^{3}\right) \) are all equal to \( {c}_{j}^{\prime } \) in (36). | Proof In the pure output case, we have \( m = 0 \) and \( s > 0 \) . Then (23) becomes\n\n\[ \mathop{\sum }\limits_{{r = 1}}^{s}{u}_{r}{y}_{rj} - {c}_{j} = 0,\;j = 1,2,\cdots, n. \]\n\nLet\n\n\[ {u}_{r} = C\frac{{\beta }_{r}}{\mathop{\sum }\limits_{{j = 1}}^{n}{y}_{rj}},\;r = 1,2,\cdots, s. \]\n\n(37)\n\nBy using the d... | Yes |
Example 2.1 Let\n\n\[ P = \left\lbrack \begin{matrix} - & - & 0 & 0 \\ - & 0 & + & 0 \\ 0 & - & - & - \\ 0 & 0 & - & + \end{matrix}\right\rbrack \]\n\nbe an \( 4 \times 4 \) zero-nonzero sign pattern. The graph of \( P \) is a signed digraph \( \Gamma \), see Figure 1. It can be converted into a signed bipartite graph ... | The edges \( {3}^{\prime } - 4 \) and \( {3}^{\prime } - 3 \) are a \( c \) -pair, while \( {3}^{\prime } - 4 \) and \( {3}^{\prime } - 2 \) are not a \( c \) -pair. The cycle \( {3}^{\prime } - 4 - {4}^{\prime } - 3 - {3}^{\prime } \) has one \( c \) -pair, so is an \( o \) -cycle. On the other hand, the cycle \( {1}^... | Yes |
Lemma 3.1 \( {}^{\left\lbrack 1\right\rbrack } \) If \( {\tau }_{1},{\tau }_{2},\cdots ,{\tau }_{m} \) are the cardinality \( m \) cycles in \( M \) -interlacing cycles and \( {\tau }_{i} = \left( {{\tau }_{i1}{\tau }_{i2}\cdots {\tau }_{i{t}_{i}}{\tau }_{i1}}\right) \), then | \[ \operatorname{sign}\left( \mathcal{P}\right) \mathop{\prod }\limits_{{i = 1}}^{n}{P}_{i,\mathcal{P}\left( i\right) } = \mathop{\prod }\limits_{{i = 1}}^{m}{\left( -1\right) }^{{t}_{j} - 1}\mathop{\prod }\limits_{{i = 1}}^{{t}_{j}}{P}_{{\tau }_{ji},{\tau }_{j, i + 1}}\mathop{\prod }\limits_{{k \notin {\tau }_{ji}}}{P... | Yes |
Example 3.1 Let\n\n\[ P = \left\lbrack \begin{array}{llll} + & + & + & + \\ + & - & + & - \\ + & + & - & - \\ + & - & - & + \end{array}\right\rbrack \]\n\nBy diagonal equivalence, we can obtain the sign pattern\n\n\[ {P}^{\prime } = \left\lbrack \begin{array}{llll} - & - & - & - \\ + & - & + & - \\ + & + & - & - \\ - &... | with all negative diagonal entries by Property 3.1, and convert the signed digraph \( \Gamma \) of \( {P}^{\prime } \) into a signed bipartite grap \( G\left( {{U}_{0},{V}_{0}}\right) \), where\n\n\[ {U}_{0} = \{ 1,2,3,4\} ,\;{V}_{0} = \left\{ {{1}^{\prime },{2}^{\prime },{3}^{\prime },{4}^{\prime }}\right\} ,\;\varphi... | Yes |
Example 3.2 Let\n\n\[ \nP = \left\lbrack \begin{matrix} 0 & + & - & 0 \\ 0 & - & - & 0 \\ + & + & + & - \\ 0 & + & - & 0 \end{matrix}\right\rbrack .\n\]\n\nBy permutation we obtain the sign pattern\n\n\[ \n{P}^{\prime } = \left\lbrack \begin{matrix} 0 & + & - & 0 \\ 0 & - & - & 0 \\ 0 & + & - & 0 \\ + & + & + & - \end{... | We search for a maximum sub-signed bipartite \( \operatorname{graph}G\left( {{U}_{1},{V}_{1}}\right) \) with \( {U}_{1} = \{ 2,3,4\} \) and \( {V}_{1} = \left\{ {{2}^{\prime },{3}^{\prime },{4}^{\prime }}\right\} \) relative to a maximum matching \( {M}_{1} : \left\{ {2 - {2}^{\prime },3 - {3}^{\prime },4 - {4}^{\prime... | No |
Theorem 3.1 Assume that for some \( T > 0 \), and \( s > \frac{3}{2}, u \in C\left( {\left\lbrack {0, T}\right\rbrack ;{H}^{s}\left( \mathbb{R}\right) }\right) \) is a strong solution of the initial value problem (1) with the initial data \( {u}_{0}\left( x\right) = u\left( {x,0}\right) \) . If \( {u}_{0}\left( x\right... | Proof For simplicity, we introduce the following notations\n\n\[ M = \mathop{\sup }\limits_{{0 \leq t \leq T}}\parallel u\left( t\right) {\parallel }_{{H}^{s}},\;{\varphi }_{N}\left( x\right) = \left\{ \begin{array}{ll} 1, & x \leq 0, \\ {\mathrm{e}}^{\theta x}, & x \in \left( {0, N}\right) , \\ {\mathrm{e}}^{\theta N}... | Yes |
Theorem 1 Under Assumption 1 and Assumption 2, and suppose that constants \( {h}_{i} \) are given. For arbitrary delays \( {d}_{i}\left( t\right) \), systems (1) and (2) can achieve hybrid projective synchronization in the sense of mean square, if there exist \( N \times N \) -order matrices \( {Q}_{1i} > 0,{Q}_{2i} > ... | Proof Let \( e\left( t\right) = {\left\lbrack {e}_{1}^{T}\left( t\right) ,{e}_{2}^{T}\left( t\right) ,\cdots ,{e}_{N}^{T}\left( t\right) \right\rbrack }^{T} \) . For arbitrary \( r\left( t\right) = i, i \in S \) . We choose a Lyapunov-Krasovskii function as\n\n\[ V\left( {{e}_{t}, t, i}\right) = {V}_{1}\left( t\right) ... | Yes |
Under Assumption 1 and Assumption 2, and suppose that constants \( {h}_{i} \) are given. For arbitrary delays \( {d}_{i}\left( t\right) \), systems (1) and (2) will achieve complete synchronization in the sense of mean square, if there exist \( N \times N \) -order matrices \( {Q}_{1i} > \) \( 0,{Q}_{2i} > 0,{Q}_{1} > ... | Since the following proof is the same as that of Theorem 1, we omit it. The response system in the network (2) and the driving system (1) can realize complete synchronization in the sense of mean square. | No |
Corollary 2 Under Assumption 1 and Assumption 3, and suppose that a constant \( h \) is given. For a given delay \( d\left( t\right) \), systems (1) and (16) will achieve hybrid projective synchronization, if there exist \( N \times N \) -order matrices \( {Q}_{1} > 0,{Q}_{2} > 0 \), such that the following matrix ineq... | Proof Choose the Lyapunov-Krasovskii function as\n\n\[ V\left( t\right) = {V}_{1}\left( t\right) + {V}_{2}\left( t\right) \]\n\n(19)\n\nwhere\n\n\[ {V}_{1}\left( t\right) = {e}^{T}\left( t\right) e\left( t\right) \]\n\n\[ {V}_{2}\left( t\right) = {\int }_{t - d\left( t\right) }^{t}{e}^{T}\left( s\right) \left( {{Q}_{1}... | Yes |
Theorem 2 Let Assumption 1 and Assumption 2 hold. Suppose that there exist constants \( {\alpha }_{j} > 0, j = 1,2,\cdots, n \), such that\n\n\[ \n- {2r}{c}_{i} + \left( {{2r} - 1}\right) \mathop{\sum }\limits_{{j = 1, j \neq i}}^{n}\left| {a}_{ij}\right| + \frac{1}{{\alpha }_{i}}\mathop{\sum }\limits_{{j = 1, j \neq i... | Proof Let \( {a}_{ii} = 0 \) in the proof of Theorem 1, the theorem can be easily proved. | No |
Theorem 3 Let Assumption 1 and Assumption 2 hold. Suppose that there exist constants \( {\alpha }_{j} > 0,{a}_{jj} < 0, j = 1,2,\cdots, n \), such that \[ - {2r}{c}_{i} + {2r}{a}_{ii}{u}_{i} + \left( {{2r} - 1}\right) \mathop{\sum }\limits_{{j = 1, j \neq i}}^{n}\left| {a}_{ij}\right| + \frac{1}{{\alpha }_{i}}\mathop{\... | Proof Let \( C = C\left( {\left\lbrack {-\tau ,0}\right\rbrack ,{\mathbb{R}}^{n}}\right) \) be the Banach space of continuous functions which map \( \left\lbrack {-\tau ,0}\right\rbrack \) into \( {\mathbb{R}}^{n} \) with the topology of uniform convergence. For any \( \varphi \in \mathbb{C} \), we define \[ \parallel ... | Yes |
Theorem 4 Let Assumption 1 and Assumption 2 hold. Suppose that there exist constants \( {\alpha }_{j} > 0, j = 1,2,\cdots, n \), such that\n\n\[ - {2r}{c}_{i} + \left( {{2r} - 1}\right) \mathop{\sum }\limits_{{j = 1, j \neq i}}^{n}\left| {a}_{ij}\right| + \frac{1}{{\alpha }_{i}}\mathop{\sum }\limits_{{j = 1, j \neq i}}... | Proof Let \( {a}_{ii} = 0 \) in the proof of Theorem 3, the conclusion in Theorem 4 can be easily obtained. | No |
Example 2 Let \( n = 2,{f}_{j}\left( x\right) \equiv f\left( x\right) = x - \arctan x/2 \), It is obvious that the response functions \( {f}_{j}\left( x\right), j = 1,2 \) satisfy \( \left| {f\left( x\right) - f\left( y\right) }\right| \leq \left| {x - y}\right| \), then \( {u}_{1} = 1,{u}_{2} = 1 \) . Let\n\n\[ \n{\su... | By taking \( r = 1,{c}_{1} = {c}_{2} = 8,{a}_{11} = {a}_{22} = - 3,{a}_{12} = {a}_{21} = {b}_{12} = {b}_{21} = 2,{b}_{11} = {b}_{22} = 2 \) in Corollary 3, then\n\n\[ \n2{c}_{1} - 2{a}_{11} > \left| {a}_{21}\right| + \left| {a}_{21}\right| + \left| {b}_{11}\right| + \left| {b}_{12}\right| + \left| {b}_{11}\right| + \le... | Yes |
Corollary 1 Suppose that there is a \( T \) -periodic function \( w \in \mathbb{C}\left( {\lbrack 0,\infty }\right) ,\left( {0,\infty }\right) ) \) such that (2) holds and\n\n\[ \n{\int }_{t}^{t - \tau \left( t\right) }\left\lbrack {p\left( s\right) + w\left( s\right) q\left( s\right) }\right\rbrack \mathrm{d}s = \ln w... | Proof The condition (9) is exactly the condition (3) by taking \( k = 1 \) . | No |
Consider the delay differential equation\n\n\[ \n{x}^{\prime }\left( t\right) = \left( {{\mathrm{e}}^{-{2t}} + \cos t}\right) x\left( t\right) - {\mathrm{e}}^{-{2t}}{\mathrm{e}}^{4\sin t}{x}^{3}\left( {t - \pi }\right) ,\;t \geq 0. \n\] | To prove the validity of the hypotheses (2) and (3), we let \( w\left( t\right) = {\mathrm{e}}^{-4\sin t} \) and obtain\n\n\[ \n{\int }_{t}^{t + T}\left\lbrack {p\left( u\right) + w\left( u\right) q\left( u\right) }\right\rbrack \mathrm{d}u = {\int }_{t}^{t + {2\pi }}\cos u\mathrm{\;d}u = 0, \n\]\n\nand\n\n\[ \nk{\int ... | Yes |
Consider the delay differential equation\n\n\[ \n{x}^{\prime }\left( t\right) = \left( {\cos t}\right) x\left( t\right) + \left( {3\cos {3t} - \cos t}\right) {\mathrm{e}}^{\sin {3t} - 2\sin {2t}}{x}^{2}\left( \frac{2t}{3}\right) ,\;t \geq 0, \n\] \n\nwhere the periodic \( T = {2\pi }, k = 2 \) and \( \tau \left( t\righ... | There exists a \( w\left( t\right) = {\mathrm{e}}^{-\sin {3t} + 2\sin {2t}} \) such that \( p\left( t\right) + w\left( t\right) q\left( t\right) = 3\cos {3t} \) . The hypotheses (2) and (3) are valid, since\n\n\[ \n{\int }_{t}^{t + T}\left\lbrack {p\left( s\right) + w\left( s\right) q\left( s\right) }\right\rbrack \mat... | Yes |
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