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Proposition 2.5 If (2.7) holds for \( K \), then\n\n\[ \parallel g{\parallel }_{{C}^{s}\left( X\right) } \leq \left( {\kappa + {\kappa }_{2s}}\right) \parallel g{\parallel }_{K},\;\forall g \in {\mathcal{H}}_{K}. \] | The proof of this proposition can be found in [5]. | No |
Proposition 2.10 Let \( c > 0,{q}_{2} \geq 0, t \geq 2 \in \mathbb{Z} \) and \( 0 < {q}_{1} < 1 \), then we have:\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{{t - 1}}{i}^{-{q}_{2}}\exp \left\{ {-c\mathop{\sum }\limits_{{j = i + 1}}^{t}{j}^{-{q}_{1}}}\right\} \leq \left( {\frac{{2}^{{q}_{1} + {q}_{2}}}{c} + {\left( \frac{1 +... | (2.12)\n\nThe elementary inequality can be found in [12]. | No |
Proposition 2.12 If \( {f}_{t} \) is defined by (2.13) and \( {\lambda }_{t} \) decreases with iteration, we can conclude that\n\n\[ \n{\begin{Vmatrix}{f}_{t}\end{Vmatrix}}_{K} \leq \frac{\kappa }{{\lambda }_{t}} \n\]\n\n(2.14) | Proof When \( {f}_{1} = 0 \), it is easy to see that \( \begin{Vmatrix}{f}_{1}\end{Vmatrix} \leq \frac{\kappa }{{\lambda }_{1}} \) . If we assume that \( {\begin{Vmatrix}{f}_{t}\end{Vmatrix}}_{K} \leq \frac{\kappa }{{\lambda }_{t}} \) holds, from the formulation of \( {f}_{t + 1} \), we can show :\n\n\[ \n{\begin{Vmatr... | Yes |
Theorem 2.13 Let \( {f}_{\rho } \in {C}^{s}\left( X\right), K \in {C}^{2s}\left( {X \times X}\right) \) for some \( 0 \leq s \leq \frac{1}{2} \) . Suppose assumptions (2.3) and (2.11) hold, \( {f}_{t} \) is from (2.13) and a=1, if\n\n\[ \n{\lambda }_{t} = {\lambda }_{1}{t}^{-\gamma },{\eta }_{t} = {\eta }_{1}{t}^{-\alp... | The proof will be provided in Section 4. | No |
Theorem 3.14 For the loss function \( {V}^{c}\left( {c > 0}\right) \) and any measurable function \( f : X \rightarrow \) \( \mathbb{R} \), the following holds : | \[ \mathcal{R}\left( {\operatorname{sgn}\left( f\right) }\right) - \mathcal{R}\left( {f}_{\text{bayes }}\right) \leq {C}_{1}\left\{ {{\varepsilon }^{c}\left( f\right) - {\varepsilon }^{c}\left( {f}_{\rho }^{c}\right) }\right\} \] where \( {C}_{1} > 0 \) . | Yes |
Theorem 3.15 For the loss function \( {V}^{{c}_{0}}\left( {{c}_{0} = 0}\right) \) and any measurable function \( f : X \rightarrow \mathbb{R} \), the following holds :\n\n\[ \mathcal{R}\left( {\operatorname{sgn}\left( f\right) }\right) - \mathcal{R}\left( {f}_{\text{bayes }}\right) \leq {C}_{2}{\left\{ {\varepsilon }^{... | The proof of the theorem can be found in [13]. | No |
Lemma 3.16 Let \( {\mathcal{D}}^{c}\left( \lambda \right) = \mathop{\inf }\limits_{{f \in {\mathcal{H}}_{K}}}\left\{ {{\varepsilon }^{c}\left( f\right) - {\varepsilon }^{c}\left( {f}_{\rho }^{c}\right) + \frac{\lambda }{2}\parallel f{\parallel }_{K}^{2}}\right\} \), when the parameter \( a \) in loss function is 1 , we... | Proof First of all we can show that \( {\begin{Vmatrix}{V}^{c} - {V}^{{c}_{0}}\end{Vmatrix}}_{\infty } \leq c \) for any \( c \geq 0 \) . This proof can be found in Appendix.\n\nThen, we have\n\n\[ \n\left| {{\varepsilon }^{{c}_{0}}\left( f\right) - {\varepsilon }^{c}\left( f\right) }\right| = \left| {{\int }_{Z}{V}^{{... | Yes |
Lemma 3.18 Let \( h, g \in {C}^{s}\left( X\right) \) . If (2.4) holds, then\n\n\[ \left| {{\int }_{Z}{V}^{c}\left( {{yh}\left( x\right) }\right) - {V}^{c}\left( {{yg}\left( x\right) }\right) d\left\lbrack {{\rho }^{\left( t\right) } - \rho }\right\rbrack }\right| \]\n\n\[ \leq \left\{ {\left( {\parallel h{\parallel }_{... | The proof of this lemma can be found in [5] because of \( \left| {\partial {V}^{c}\left( {yf}\right) }\right| \leq 1 \) . | No |
Theorem 3.20 \( {V}^{c} \) is the LUM loss function and \( {f}_{\lambda }^{c} \) is defined by (2.9).If \( \vartheta > \lambda > 0 \) , we have\n\n\[ \n{\begin{Vmatrix}{f}_{\lambda }^{c} - {f}_{\vartheta }^{c}\end{Vmatrix}}_{K} \leq \frac{\vartheta }{2}\left( {\frac{1}{\lambda } - \frac{1}{\vartheta }}\right) \left( {{... | We can find the proof of this theorem from [5]. | No |
Lemma 4.21 Define \( \\left\\{ {f}_{t}\\right\\} \) by (2.13), we get\n\n\[ \n{\\mathbb{E}}_{{z}_{t}}\\left( {\\begin{Vmatrix}{f}_{t + 1} - {f}_{{\\lambda }_{t}}^{{c}_{t}}\\end{Vmatrix}}_{K}^{2}\\right) \\leq \\left( {1 - {\\eta }_{t}{\\lambda }_{t}}\\right) {\\begin{Vmatrix}{f}_{t} - {f}_{{\\lambda }_{t}}^{{c}_{t}}\\e... | The proof of the lemma is shown in [6]. | No |
Lemma 3.1 \( u\left( t\right) \) is a solution of the following problem:\n\n\[ \left\{ \begin{array}{l} {}^{C}{D}_{0 + }^{\beta }u\left( t\right) = f\left( {t,{\varphi }_{q}\left( {u\left( t\right) }\right) }\right), t \in \left( {0,1}\right) , \\ u\left( 1\right) = u\left( \delta \right) , \end{array}\right. \] \n\n(3... | Proof If \( u\left( t\right) \) is a solution of problem (3.1) and \( x\left( t\right) = {I}_{0 + }^{\alpha }{\varphi }_{q}\left( {u\left( t\right) }\right) \), then \( u\left( t\right) = \) \( {\varphi }_{p}\left( {{}^{C}{D}_{0 + }^{\alpha }x\left( t\right) }\right) \) and \( {x}^{\left( i\right) }\left( 0\right) = 0,... | Yes |
Lemma 3.2 Let \( L \) be defined by (3.2), then\n\n\[ \operatorname{Ker}L = \{ u \in X \mid u\left( t\right) = c, c \in \mathbb{R},\forall t \in \left\lbrack {0,1}\right\rbrack \} ,\]\n\n(3.4)\n\n\[ \operatorname{Im}L = \left\{ {y \in Y \mid {\int }_{0}^{1}l\left( s\right) y\left( s\right) {ds} = 0}\right\} . \]\n\n(3.... | Proof By Lemma 2.4, we can obtain (3.4). If \( y \in \operatorname{Im}L \), there exists \( u \in \operatorname{dom}L \) such that \( y = {Lu} \in Y \) . From Lemma 2.4, we have\n\n\[ u\left( t\right) = \frac{1}{\Gamma \left( \beta \right) }{\int }_{0}^{t}{\left( t - s\right) }^{\beta - 1}y\left( s\right) {ds} + c, c \... | Yes |
Lemma 3.3 Let \( L \) be defined by (3.2), then \( L \) is a Fredholm operator of index zero. | Proof Clearly, \( \operatorname{Im}P = \operatorname{Ker}L \) and \( P{u}^{2} = {Pu} \). By \( u = \left( {u - {Pu}}\right) + {Pu} \), we have \( X = \operatorname{Ker}P + \operatorname{Ker}L \). By a simple calculation, we obtain \( \operatorname{Ker}L \cap \operatorname{Ker}P = \{ 0\} \). Hence, \( X = \operatorname{... | Yes |
Lemma 3.4 \( {QN} : X \rightarrow Y \) is continuous and bounded and \( {K}_{P}\left( {I - Q}\right) N : \bar{\Omega } \rightarrow X \) is compact, where \( \Omega \subset X \) is bounded. | Proof By the continuity of \( f \), we see that \( {QN}\left( \bar{\Omega }\right) \) and \( {K}_{P}\left( {I - Q}\right) N\left( \bar{\Omega }\right) \) are bounded. That is, there exist constants \( {M}_{1},{M}_{2} > 0 \) such that \( \left| {\left( {I - Q}\right) {Nu})}\right| \leq {M}_{1} \) and \( \mid {K}_{P}(I -... | Yes |
Lemma 3.5 If the condition \( \left( {\mathrm{H}}_{1}\right) \) and \( \left( {\mathrm{H}}_{2}\right) \) hold, the set \[ {\Omega }_{0} = \{ u\left( t\right) \mid {Lu}\left( t\right) = {\lambda Nu}\left( t\right), u\left( t\right) \in C \cap \operatorname{dom}L,\lambda \in \left( {0,1}\right) \} \] is bounded. | Proof For \( u\left( t\right) \in {\Omega }_{0} \), we have \( {QNu}\left( t\right) = 0 \) . By \( \left( {\mathrm{H}}_{1}\right) \) and \( {QNu}\left( t\right) = 0 \), there exists \( {t}_{0} \in \left\lbrack {0,1}\right\rbrack \) such that \( {\varphi }_{q}\left( {u\left( {t}_{0}\right) }\right) \leq {R}_{0} \), i.e.... | Yes |
Lemma 4.1 \( u\left( t\right) \) is a solution of the following problem:\n\n\[ \left\{ \begin{array}{l} {}^{C}{D}_{0 + }^{\beta }u\left( t\right) = f\left( {t,{\varphi }_{q}\left( {u\left( t\right) }\right) }\right), t \in \left( {0,1}\right) , \\ u\left( 1\right) = {\int }_{0}^{1}h\left( t\right) u\left( t\right) {dt}... | Proof The proof process is similar to Lemma 3.1, which is omitted here. | No |
Lemma 4.2. Let \( L \) be defined by (4.2), then\n\n\[ \operatorname{Ker}L = \{ u \in X \mid u\left( t\right) = c, c \in \mathbb{R},\forall t \in \left\lbrack {0,1}\right\rbrack \} ,\]\n\n\[ \operatorname{Im}L = \left\{ {y \in Y \mid {\int }_{0}^{1}\left\lbrack {{\left( 1 - s\right) }^{\beta - 1} - {\int }_{s}^{1}h\lef... | Proof The proof is similar to that of Lemma 3.2, 3.3 and is omitted. | No |
Lemma 4.3 \( {QN} : X \rightarrow Y \) is continuous and bounded and \( {K}_{P}\left( {I - Q}\right) N : \bar{\Omega } \rightarrow X \) is compact, where \( \Omega \subset X \) is bounded. | Proof The proof is similar to that of Lemma 3.4 and is omitted. | No |
Lemma 4.4 If the condition \( \left( {\mathrm{H}}_{1}\right) \) and \( \left( {\mathrm{H}}_{2}\right) \) hold, the set\n\n\[ \n{\Omega }_{0} = \{ u\left( t\right) \mid {Lu}\left( t\right) = {\lambda Nu}\left( t\right), u\left( t\right) \in C \cap \operatorname{dom}L,\lambda \in \left( {0,1}\right) \} \n\]\n\nis bounded... | Proof The proof is similar to that of Lemma 3.5 and is omitted. | No |
Example 5.1 Consider the following problem\n\n\\[ \n\\left\\{ \\begin{array}{l} {}^{C}{D}_{0 + }^{\\frac{1}{2}}{\\varphi }_{2}\\left( {{}^{C}{D}_{0 + }^{\\frac{1}{2}}x\\left( t\\right) }\\right) = \\frac{1}{4} - {\\left. \\frac{1}{20}\\right| }^{C}{D}_{0 + }^{\\frac{1}{2}}x\\left( t\\right) {\\left. \\right| }^{\\frac{... | By Lemma 3.1, we have\n\n\\[ \n\\left\\{ \\begin{array}{l} {}^{C}{D}_{0 + }^{\\frac{1}{2}}u\\left( t\\right) = \\frac{1}{4} - \\frac{1}{20}{\\left| u\\left( t\\right) \\right| }^{\\frac{1}{2}}, \\\\ u\\left( 1\\right) = u\\left( \\frac{1}{2}\\right) . \\end{array}\\right. \n\\]\n\n(5.2)\n\nSo, we get\n\n\\[ \nl\\left( ... | Yes |
Example 5.2 Consider the following problem\n\n\[ \left\\{ \begin{array}{l} {}^{C}{D}_{0 + }^{\frac{1}{2}}{\varphi }_{2}\left( {{}^{C}{D}_{0 + }^{\frac{1}{2}}x\left( t\right) }\right) = \frac{1}{4} - {\left. \frac{1}{20}\right| }^{C}{D}_{0 + }^{\frac{1}{2}}x\left( t\right) {\left. \right| }^{\frac{1}{2}},\;t \in \left( ... | By Lemma 4.1, we have\n\n\[ \left\\{ \begin{array}{l} {}^{C}{D}_{0 + }^{\frac{1}{2}}u\left( t\right) = \frac{1}{4} - \frac{1}{20}{\left| u\left( t\right) \right| }^{\frac{1}{2}}, \\ u\left( 1\right) = {\int }_{0}^{1}u\left( t\right) {dt}. \end{array}\right. \] \n\nSo, we get\n\n\[ {G}_{2}\left( {t, s}\right) = \left\\{... | Yes |
Lemma 2.3 Let \( 0 \leq \tau < p \) and \( 0 \leq i, j \leq 5 \) . Then we have\n\n(i) the autocorrelation of \( {s}_{i}^{\prime } \) is given by\n\n\[ \n{R}_{{s}_{i}^{\prime }}\left( \tau \right) = \left\{ \begin{array}{ll} p, & \text{ if }\tau = 0, \\ - 1 - 2\left( {{\left( -1\right) }^{{s}_{i}\left( \tau \right) } +... | Proof We prove only (iv) and the others can be proved similarly. If \( \tau = 0 \), we have\n\n\[ \n{R}_{{s}_{i}^{\prime },{s}_{j}^{\prime }}\left( 0\right) = \mathop{\sum }\limits_{{t = 0}}^{{p - 1}}{\left( -1\right) }^{{s}_{i}^{\prime }\left( t\right) + {s}_{j}^{\prime }\left( t\right) } \n\]\n\n\[ \n= \mathop{\sum }... | No |
Lemma 2.6 Let \( {s}_{i} \) and \( {s}_{i}^{\prime },0 \leq i \leq 5 \) be Hall’s sextic residue sequences and their modifications defined by (2.1). Then\n\n(i) \( {P}_{{s}_{i}^{\prime }}\left( x\right) = {P}_{{s}_{i}}\left( x\right) + 1 \) ;\n\n(ii) \( {P}_{{s}_{i}}\left( {x}^{4}\right) \equiv 1\left( {{\;\operatornam... | Proof (i) is obvious, so we only prove (ii). Note that \( {x}^{4k} \equiv 1\left( {{\;\operatorname{mod}\;{x}^{4}} - 1}\right) \) for any positive integer \( k \) . Then from the definition of Hall’s sextic residue sequence \( {s}_{i} \) with period \( p \), we get\n\n\[ \n{P}_{{s}_{i}}\left( {x}^{4}\right) = \mathop{\... | Yes |
Theorem 3.1 The first class of binary sequences \( u \) defined by (3.1) has \( {R}_{u}\left( \tau \right) \in \) \( \{ 0, \pm 4, \pm 8\} \) for all \( 1 \leq \tau < {4p} \) . | Proof Writing \( \tau = 4{\tau }_{1} + {\tau }_{2} \), where \( 0 \leq {\tau }_{1} < p \) and \( 0 < {\tau }_{2} < 4 \) or \( 0 < {\tau }_{1} < p \) and \( {\tau }_{2} = 0 \), we calculate the out-of-phase autocorrelation values of the sequence \( u \) in four cases. Case 1. \( {\tau }_{2} = 0,0 < {\tau }_{1} < p \) . ... | Yes |
Example 1 Let \( p = {31} = 4 + {27} \) . To ensure \( 3 \in {D}_{1} \), we use a primitive root \( g = 3 \) of \( p = {31} \) to define the cyclotomic classes of order 6 . Then | \[ {D}_{0} = \{ 1,2,4,8,{16}\} ,\;{D}_{1} = \{ 3,6,{12},{17},{24}\} ,\;{D}_{2} = \{ 5,9,{10},{18},{20}\} , \] \[ {D}_{3} = \{ {15},{23},{27},{29},{30}\} ,{D}_{4} = \{ 7,{14},{19},{25},{28}\} ,{D}_{5} = \{ {11},{13},{21},{22},{26}\} . \] | Yes |
Theorem 1.1 Under the hypothesis \( {H}_{0} \), there exists a sequence \( \left( {d}_{c, k}\right) \) such that, for any \( p > 0 \) and \( T \) large enough, if \( {b}_{1} < {b}_{0} \), for all \( c < \frac{\delta {\left( {b}_{1} - {b}_{0}\right) }^{2}}{8{b}_{0}} \), we have\n\n\[ \n{P}_{\delta ,{b}_{0}}\left( {{V}_{... | The coefficients \( {d}_{c,1},{d}_{c,2},\ldots ,{d}_{c, p} \) may be explicitly given as functions of the derivatives of \( \Lambda \) and \( H \) (see Lemma 2.1) at point \( {a}_{c} \) . For example, the first coefficient \( {d}_{c,1} \) is given by\n\n\[ \n{d}_{c,1} = \frac{1}{{\sigma }_{c}^{2}}\left( {-\frac{{H}_{2}... | Yes |
For all \( \lambda \in {\mathcal{D}}_{\Lambda } \), we have \[ {\Lambda }_{T}\left( \lambda \right) = {T\Lambda }\left( \lambda \right) + H\left( \lambda \right) + {R}_{T}\left( \lambda \right) \] where \[ \Lambda \left( \lambda \right) = - \frac{\delta \left( {\lambda \left( {{b}_{1} - {b}_{0}}\right) + {b}_{0} - \var... | Proof By using Girsanov formula, \[ {\Lambda }_{T}\left( \lambda \right) = \log {E}_{\delta ,\varphi }\left\lbrack {\exp \left( {\lambda \log \frac{d{P}_{\delta ,{b}_{1}}}{d{P}_{\delta ,{b}_{0}}}}\right) \frac{d{P}_{\delta ,{b}_{0}}}{d{P}_{\delta ,\varphi }}}\right\rbrack \] \[ = \log {E}_{\delta ,\varphi }\left\lbrack... | Yes |
Lemma 2.2 For \( T \) large enough and for any \( \left( {a, u}\right) \in {\mathbb{R}}^{2} \) such that \( a + {iu} \in {\Delta }_{{\Lambda }_{T}} \) , \[ {\left| \exp \left( {\Lambda }_{T}\left( a + iu\right) - {\Lambda }_{T}\left( a\right) \right) \right| }^{2} \leq {4}^{\delta }{l}^{\delta }\left( a\right) {\left( ... | Proof Step 1: For all \( a \in {\mathcal{D}}_{\Lambda }, u \in \mathbb{R} \), we deduce from (2.2) that \[ \Lambda \left( {a + {iu}}\right) - \Lambda \left( a\right) = - \frac{\delta }{4}\left( {{iu}\left( {{b}_{1} - {b}_{0}}\right) - \varphi \left( {a + {iu}}\right) + \varphi \left( a\right) }\right) , \] which clearl... | Yes |
Lemma 3.1 For all \( c < \frac{\delta {\left( {b}_{1} - {b}_{0}\right) }^{2}}{8{b}_{0}}, T \) tends to infinity, | Proof It follows from Lemma 2.1 that\n\n\[ {A}_{T} = \exp \left( {{\Lambda }_{T}\left( {a}_{c}\right) - c{a}_{c}T}\right) \]\n\n\[ = \exp \left( {{T\Lambda }\left( {a}_{c}\right) + H\left( {a}_{c}\right) + {R}_{T}\left( {a}_{c}\right) - c{a}_{c}T}\right) \]\n\n\[ = \exp \left( {-I\left( c\right) T + H\left( {a}_{c}\rig... | Yes |
Lemma 3.3 For any \( p > 0 \), and for any \( c < \frac{\delta {\left( {b}_{1} - {b}_{0}\right) }^{2}}{8{b}_{0}} \), there exist integers \( q\left( p\right) \) and a polynomial sequence \( \left( {\eta }_{k}\right) \) independent of \( p \), such that, for \( T \) large enough,\n\n\[ \n{\Phi }_{T}\left( u\right) = {e}... | Proof We deduce from (3.2) and (3.6) that there exists \( \xi \in \mathbb{R} \) such that, for any \( p > 0 \n\n\[ \n\log {\Phi }_{T}\left( u\right) = - \frac{{iuc}\sqrt{T}}{{\sigma }_{c}} + \mathop{\sum }\limits_{{k = 1}}^{\left\lbrack 2p + 3\right\rbrack }{\left( \frac{iu}{\sqrt{T}{\sigma }_{c}}\right) }^{k}\left( {\... | Yes |
Lemma 3.4 For all \( c < \frac{\delta {\left( {b}_{1} - {b}_{0}\right) }^{2}}{8{b}_{0}} \), there exists a sequence \( \left( {\psi }_{k}\right) \) such that, for any \( p > 0 \) and \( T \) large enough, \[ {B}_{T} = - \frac{1}{{a}_{c}{\sigma }_{c}\sqrt{2\pi T}}\left( {1 + \mathop{\sum }\limits_{{k = 1}}^{p}\frac{{\ps... | Proof of Theorem 1.1 and 1.2 We complete the proof of Theorem 1.1 by Lemmas 3.1 and 3.4 together with (3.1). The proof of Theorem 1.2 is similar to Theorem 1.1. | No |
Lemma 2.1 Suppose that \( \mathcal{A} \) is a unital \( {C}^{ * } \) -algebra. Let \( T : \mathcal{A} \rightarrow \mathcal{A} \) be a linear mapping satisfying\n\n\[ a \bot b \bot c \Rightarrow \{ a, T\left( b\right), c\} = 0. \]\n\nThen the identity\n\n\[ T\left( p\right) = {pT}\left( p\right) + T\left( p\right) p - {... | Proof Let \( a = p, b = 1 - {p}^{ * }, c = p \), where \( {p}^{2} = p \) in \( \mathcal{A} \) . According to the hypothesis, we have \( {pT}{\left( 1 - {p}^{ * }\right) }^{ * }p = \left\{ {p, T\left( {1 - {p}^{ * }}\right), p}\right\} = 0 \), which gives \( {pT}{\left( 1\right) }^{ * }p = {pT}{\left( {p}^{ * }\right) }... | Yes |
Proposition 2.1 Suppose that \( \mathcal{A} \) is a Banach algebra satisfying property \( \mathbb{B} \) and having a bounded approximate identity \( {\left\{ {\rho }_{i}\right\} }_{i \in I} \) . Let \( \mathcal{M} \) be an essential Banach \( \mathcal{A} \) -bimodule, and let \( T : \mathcal{A} \rightarrow \mathcal{M} ... | Proof Fix \( a, b \in \mathcal{A} \) with \( {ab} = 0 \) . Define a continuous bilinear map \( \varphi : \mathcal{A} \times \mathcal{A} \rightarrow \mathcal{M} \) given by\n\n\[ \n\varphi \left( {x, y}\right) = {aT}\left( {bx}\right) y + {yT}\left( {bx}\right) a.\n\]\n\nWhen \( {xy} = 0 \) in \( \mathcal{A} \), we have... | Yes |
Corollary 2.2 Suppose that \( \mathcal{A} \) is a commutative Banach algebra with the property \( \mathbb{B} \) and having a bounded approximate identity \( {\left\{ {\rho }_{i}\right\} }_{i \in I} \) . Let \( \mathcal{M} \) be an essential Banach \( \mathcal{A} \) - bimodule. Then the following conditions are equivale... | Proof \( \left( 2\right) \Rightarrow \left( 1\right) \) is clear from Proposition 2.1.\n\n\( \left( 1\right) \Rightarrow \left( 2\right) \) If \( T \) is a generalized Jordan derivation, then \( T\left( b\right) = d\left( b\right) + {\xi b} \), for any \( b \in \mathcal{A} \) , where \( d \) is a Jordan derivation from... | Yes |
Corollary 2.3 Suppose that \( \mathcal{A} \) is a commutative Banach \( * \) -algebra with property \( \mathbb{B} \) and having a bounded approximate identity \( {\left\{ {\rho }_{i}\right\} }_{i \in I},\mathcal{M} \) is an essential Banach \( \mathcal{A} * \) -bimodule. Let \( T \) be a continuous linear mapping from ... | Proof If \( \mathcal{A} \) is commutative, \( a \bot b \bot c \) is equivalent to \( a{b}^{ * } = {b}^{ * }c = 0 \) . Let \( d = {b}^{ * } \), then the conditions in the corollary can be replaced by\n\n\[ \n{ad} = {dc} = 0 \Rightarrow {aT}{\left( {d}^{ * }\right) }^{ * }c + {cT}{\left( {d}^{ * }\right) }^{ * }a = 0.\n\... | Yes |
Theorem 2.4 Suppose that \( \mathcal{A} \) is a \( {C}^{ * } \) -algebra, and let \( T : \mathcal{A} \rightarrow \mathcal{A} \) be a bounded linear mapping. Then the following statements are equivalent:\n\n(1) \( \{ a, T\left( b\right), c\} = 0 \), when \( a \bot b \bot c, a, b, c \in \mathcal{A} \) ;\n\n(2) \( T \) is... | Proof \( \left( 1\right) \Rightarrow \left( 2\right) \) . If \( a, b, c \in {\mathcal{A}}_{sa} \) and \( {ab} = {bc} = 0 \) . Hence \( a{b}^{ * } = {b}^{ * }a = 0, b{c}^{ * } = {c}^{ * }b = 0 \) . So we obtain\n\n\[ \n{aT}{\left( b\right) }^{ * }c + {cT}{\left( b\right) }^{ * }a = 0.\n\]\nThen applying \( * \) to both ... | Yes |
Proposition 2.2 Suppose that \( \mathcal{A} \) is a \( {C}^{ * } \) -algebra, \( \mathcal{M} \) is an essential Banach \( \mathcal{A} \) - *-bimodule, and let \( T : \mathcal{A} \rightarrow \mathcal{M} \) be a bounded linear mapping satisfying the following conditions:\n\n\[ \n{aT}{\left( b\right) }^{ * }c + {cT}{\left... | Proof Let \( \mathcal{B} \) denote the abelian \( {C}^{ * } \) -subalgebra of \( \mathcal{A} \) generated by a self-adjoint element \( a \) of \( \mathcal{A} \) . According to Corollary 2.3, we see that \( {\left. T\right| }_{\mathcal{B}} : \mathcal{B} \rightarrow \mathcal{M} \) is a generalized Jordan derivation. Henc... | Yes |
Proposition 2.3 Suppose that \( \mathcal{A} \) is a unital Banach algebra satisfying the property \( \mathbb{B} \), let \( T : \mathcal{A} \rightarrow \mathcal{A} \) be a continuous linear mapping satisfying\n\n\[ \n{ab} = {bc} = 0 \Rightarrow T\left( a\right) T\left( b\right) T\left( c\right) + T\left( c\right) T\left... | Proof Fix \( a, b \in \mathcal{A} \) with \( {ab} = 0 \) . Define a continuous bilinear mapping\n\n\[ \n\varphi : \mathcal{A} \times \mathcal{A} \rightarrow \mathcal{A}\n\]\n\nsuch that\n\n\[ \n\varphi \left( {x, y}\right) = T\left( a\right) T\left( {bx}\right) T\left( y\right) + T\left( y\right) T\left( {bx}\right) T\... | Yes |
Proposition 3.4 If \( h = {h}^{ * } \in S\left( {\mathcal{M},\tau }\right) \), then \( h \) is the limit of a sequence of linear combinations of mutually orthogonal projections in measure topology \( \left( {S\left( {\mathcal{M},\tau }\right) = {\overline{\mathcal{P}\left( \mathcal{M}\right) }}^{{t}_{\tau }}}\right) \)... | Proof By [17, Theorem 5.6.18], \( h \) is affiliated with an abelian von Neumann subalgebra \( \mathcal{R} \) of \( \mathcal{M} \) . Hence \( h \) belongs to the \( S\left( {\mathcal{R},{\left. \tau \right| }_{\mathcal{R}}}\right) \) . For an abelian von Neumann algebra, it is well known that \( \mathcal{R} \) can be u... | Yes |
Proposition 3.5 Let \( {\mathcal{M}}_{1} \) and \( {\mathcal{M}}_{2} \) be finite von Neumann algebras, and \( \Phi : {\mathcal{M}}_{1} \rightarrow \) \( {\mathcal{M}}_{2} \) be a unital \( * \) -anti-homomorphism. If \( \Phi \) is normal, then \( \Phi \) is Cauchy-continuous for the measure topologies on \( {\mathcal{... | Proof Let \( {\tau }_{2} \) be a normal tracial state on \( {\mathcal{M}}_{2} \) . Since \( \Phi \) is normal, we note that \( {\tau }_{1} \mathrel{\text{:=}} {\tau }_{2} \circ \Phi \) is a normal tracial state on \( {\mathcal{M}}_{1} \) . For \( \epsilon ,\delta > 0, A \in U\left( {{\tau }_{1},\epsilon ,\delta }\right... | Yes |
Theorem 3.7 Suppose that \( \mathcal{M} \) is a finite von Neumann algebra, and let \( \phi \) be a spectrum-preserving linear mapping from \( S\left( {\mathcal{M},\tau }\right) \) onto itself. Then \( \phi \) is a Jordan \( * \) - isomorphism. | Proof In the proof, we need the [18, Theorem 10] instead of [18, Corollary 11]. The remainder of the proof is similar to that of Theorem 3.6. | No |
Proposition 2.1 Let \( \left( {{x}^{ * },{y}^{ * }}\right) \) be an (possibly local) equilibrium point of EPEC (1.2), if for each \( \nu = 1,\ldots, K \), the MPEC-MFCQ holds at \( \left( {{x}^{\nu , * },{y}^{ * }}\right) \) for \( \operatorname{MPEC}\left( {x}^{*, - \nu }\right) \left( {1.1}\right) \), then there exis... | Proof Since \( \left( {{x}^{ * },{y}^{ * }}\right) \) be an (possibly local) equilibrium point of the EPEC (1.2), it follows that for each \( \nu = 1,\ldots, K \), the point \( \left( {{x}^{\nu , * },{y}^{ * }}\right) \) is a (local) minimizer of the \( \operatorname{MPEC}\left( {x}^{*, - \nu }\right) \left( {1.1}\righ... | Yes |
Proposition 2.2 Let \( \left( {{x}^{ * },{y}^{ * }}\right) \) be an (possibly local) equilibrium point of EPEC (1.2), if for each \( \nu = 1,\ldots, K \), the NNAMCQ holds at \( \left( {{x}^{\nu , * },{y}^{ * }}\right) \) for \( \operatorname{MPEC}\left( {x}^{*, - \nu }\right) \left( {1.1}\right) \), then there exists ... | Proof We can easily obtain the conclusion by applying the Corollary 2.1 of [21]. Since its proof is similar to Proposition 2.1, we omit the proof here. | No |
Theorem 3.1 Assume that the EPEC (1.2) is relatively separable. Then the normalized C-(M-, S-) stationary point conditions of the EPEC (1.2) are equivalent to the C-(M-, S-) stationary point conditions of the following MPEC: | Proof If the EPEC (1.2) is relatively separable and \( \left( {{x}^{ * },{y}^{ * }}\right) \) is its normalized C-(M-, S-) stationary point, we can reformulate the \( \operatorname{EPEC}\left( {1.2}\right) \) to the \( \operatorname{EPEC}\left( {3.1}\right) \) and there exists a vector multipliers \( \left( {{\widetild... | Yes |
\[ \mathop{\min }\limits_{{x}_{1}}{x}_{1}^{2} + a{x}_{1}{x}_{2} \] \[ \text{s.t}\;{x}_{1} + {x}_{2} = c\text{,} \] | \[ 2{x}_{1} + a{x}_{2} - a{\lambda }_{0} = 0 \] \[ 2{x}_{2} + b{x}_{1} - b{\lambda }_{0} = 0 \] \[ {x}_{1} + {x}_{2} = c \] where \( \left( {a{\lambda }_{0}, b{\lambda }_{0}}\right) \) is Lagrange multiplier vectors for shared constraint. This problem have normalized equilibrium \( \left( {\frac{{ac}\left( {b - 2}\righ... | Yes |
Property 2.1 Let \( \Omega \subset {\mathbb{R}}^{n} \) be a domain and \( \Theta \left( x\right) \in {C}^{0}\left( \bar{\Omega }\right) \) with \( \frac{\left( {n - 2}\right) \pi }{2} < \Theta \left( x\right) < \frac{n\pi }{2} \) in \( \bar{\Omega } \) . Suppose \( u \in {C}^{2}\left( \Omega \right) \) is a solution to... | Proof For any \( i = 1,2,\cdots, n \), we can know \( \arctan {\eta }_{i} \in \left( {-\frac{\pi }{2},\frac{\pi }{2}}\right) \), and \( {\eta }_{i} = \mathop{\sum }\limits_{{k \neq i}}{\lambda }_{k} \) , \( \forall i = 1,2,\cdots, n \) . Then we have\n\n\[\\arctan {\eta }_{1} + \\arctan {\eta }_{2} = \\Theta - \\mathop... | Yes |
Lemma 2.7 If \( {\operatorname{essinf}}_{t \in \left\lbrack {0, T}\right\rbrack }b\left( t\right) > - {\lambda }_{1}{}^{\prime } \), where \( {\lambda }_{1}{}^{\prime } = \mathop{\inf }\limits_{{v \in {E}_{0}^{\beta, p}\smallsetminus \{ 0\} }}\frac{{\int }_{0}^{T}{\left| {}_{0}{D}_{t}^{\beta }v\left( t\right) \right| }... | \[ \parallel v{\parallel }_{{L}^{p}} \leq \frac{{T}^{\beta }}{\Gamma \left( {\beta + 1}\right) }\parallel v{\parallel }_{{E}^{\beta, p}} \leq {\Lambda }_{p}{}^{\prime }\parallel v{\parallel }_{\beta }, \] (2.9) \[ \parallel v{\parallel }_{\infty } \leq \frac{{T}^{\beta - \frac{1}{p}}}{\Gamma \left( \beta \right) {\left... | Yes |
Lemma 2.8 ([18]) (Integration by parts) Let \( \\alpha > 0, p \\geq 1, q \\geq 1,1/p + 1/q < 1 + \\alpha \) or \( p \\neq 1, q \\neq 1,1/p + 1/q = 1 + \\alpha \) . If the function \( u \\in {L}^{p}\\left( {\\left\\lbrack {a, b}\\right\\rbrack ,\\mathbb{R}}\\right), v \\in {L}^{q}\\left( {\\left\\lbrack {a, b}\\right\\r... | \[ {\\int }_{a}^{b}\\left\\lbrack {{}_{a}{D}_{t}^{-\\alpha }u\\left( t\\right) }\\right\\rbrack v\\left( t\\right) {dt} = {\\int }_{a}^{b}u\\left( t\\right) \\left\\lbrack {{}_{t}{D}_{b}^{-\\alpha }v\\left( t\\right) }\\right\\rbrack {dt}. \] | No |
Theorem 3.2 Let \( f : \left\lbrack {0, T}\right\rbrack \times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} \) is a function such that \( f\left( {\cdot, u, v}\right) \) is continuous in \( \left\lbrack {0, T}\right\rbrack \) for every \( \left( {u, v}\right) \in {\mathbb{R}}^{2} \) and \( f\left( {t,\cdot , \cd... | Proof The verification process is analogue to Theorem 3.1, which is omitted here. | Yes |
Corollary 3.2 Let \( f : \left\lbrack {0, T}\right\rbrack \times \mathbb{R} \rightarrow \mathbb{R} \) and \( {I}_{j} : \mathbb{R} \rightarrow \mathbb{R}, j = 1,2,\cdots, n \) be continuous. Assuming \( \left( {G}_{1}\right) \) and the following conditions are met.\n\n\( \left( {G}_{3}\right) \) There are \( L,{L}_{1},\... | \[{\gamma }^{ * } \mathrel{\text{:=}} \max \left\{ {\frac{\chi {\int }_{0}^{T}\mathop{\max }\limits_{{\left| x\right| \leq {\Lambda }_{\infty }{\left( pr\right) }^{1/p}}}F\left( {t, x}\right) {dt} - r}{\mathop{\max }\limits_{{\left| x\right| \leq {\Lambda }_{\infty }{\left( pr\right) }^{\frac{1}{p}}}}\mathop{\sum }\lim... | Yes |
Proposition 1.4 If \( {\Omega }_{1} \supset {\Omega }_{2} \supset \cdots \supset {\Omega }_{n} \supset \cdots \) is a sequence of non-empty compact sets in \( \mathbb{C} \) with the property that\n\n\[ \operatorname{diam}\left( {\Omega }_{n}\right) \rightarrow 0\;\text{ as }n \rightarrow \infty ,\]\n\nthen there exists... | Proof. Choose a point \( {z}_{n} \) in each \( {\Omega }_{n} \) . The condition \( \operatorname{diam}\left( {\Omega }_{n}\right) \rightarrow 0 \) says precisely that \( \left\{ {z}_{n}\right\} \) is a Cauchy sequence, therefore this sequence converges to a limit that we call \( w \) . Since each set \( {\Omega }_{n} \... | Yes |
Theorem 2.1 A continuous function on a compact set \( \Omega \) is bounded and attains a maximum and minimum on \( \Omega \) . | This is of course analogous to the situation of functions of a real variable, and we shall not repeat the simple proof here. | No |
Proposition 2.3 If \( f \) is holomorphic at \( {z}_{0} \), then\n\n\[ \n\frac{\partial f}{\partial \bar{z}}\left( {z}_{0}\right) = 0\;\text{ and }\;{f}^{\prime }\left( {z}_{0}\right) = \frac{\partial f}{\partial z}\left( {z}_{0}\right) = 2\frac{\partial u}{\partial z}\left( {z}_{0}\right) .\n\]\n\nAlso, if we write \(... | Proof. Taking real and imaginary parts, it is easy to see that the Cauchy-Riemann equations are equivalent to \( \partial f/\partial \bar{z} = 0 \) . Moreover, by our earlier observation\n\n\[ \n{f}^{\prime }\left( {z}_{0}\right) = \frac{1}{2}\left( {\frac{\partial f}{\partial x}\left( {z}_{0}\right) + \frac{1}{i}\frac... | Yes |
Theorem 2.4 Suppose \( f = u + {iv} \) is a complex-valued function defined on an open set \( \Omega \) . If \( u \) and \( v \) are continuously differentiable and satisfy the Cauchy-Riemann equations on \( \Omega \), then \( f \) is holomorphic on \( \Omega \) and \( {f}^{\prime }\left( z\right) = \partial f/\partial... | Proof. Write\n\n\[ u\left( {x + {h}_{1}, y + {h}_{2}}\right) - u\left( {x, y}\right) = \frac{\partial u}{\partial x}{h}_{1} + \frac{\partial u}{\partial y}{h}_{2} + \left| h\right| {\psi }_{1}\left( h\right) \]\n\nand\n\n\[ v\left( {x + {h}_{1}, y + {h}_{2}}\right) - v\left( {x, y}\right) = \frac{\partial v}{\partial x... | Yes |
Theorem 2.5 Given a power series \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{z}^{n} \), there exists \( 0 \leq R \leq \infty \) such that:\n\n(i) If \( \left| z\right| < R \) the series converges absolutely.\n\n(ii) If \( \left| z\right| > R \) the series diverges.\n\nMoreover, if we use the convention that \( ... | Proof. Let \( L = 1/R \) where \( R \) is defined by the formula in the statement of the theorem, and suppose that \( L \neq 0,\infty \) . (These two easy cases are left as an exercise.) If \( \left| z\right| < R \), choose \( \epsilon > 0 \) so small that\n\n\[ \left( {L + \epsilon }\right) \left| z\right| = r < 1 \]\... | No |
Corollary 2.7 A power series is infinitely complex differentiable in its disc of convergence, and the higher derivatives are also power series obtained by termwise differentiation. | We have so far dealt only with power series centered at the origin. More generally, a power series centered at \( {z}_{0} \in \mathbb{C} \) is an expression of the form\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{\left( z - {z}_{0}\right) }^{n} \]\n\nThe disc of convergence of \( f \) is n... | Yes |
Proposition 3.1 Integration of continuous functions over curves satisfies the following properties:\n\n(i) It is linear, that is, if \( \alpha ,\beta \in \mathbb{C} \), then\n\n\[ \n{\int }_{\gamma }\left( {{\alpha f}\left( z\right) + {\beta g}\left( z\right) }\right) {dz} = \alpha {\int }_{\gamma }f\left( z\right) {dz... | Proof. The first property follows from the definition and the linearity of the Riemann integral. | No |
Theorem 3.2 If a continuous function \( f \) has a primitive \( F \) in \( \Omega \), and \( \gamma \) is a curve in \( \Omega \) that begins at \( {w}_{1} \) and ends at \( {w}_{2} \), then\n\n\[{\int }_{\gamma }f\left( z\right) {dz} = F\left( {w}_{2}\right) - F\left( {w}_{1}\right)\] | Proof. If \( \gamma \) is smooth, the proof is a simple application of the chain rule and the fundamental theorem of calculus. Indeed, if \( z\left( t\right) : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{C} \) is a parametrization for \( \gamma \), then \( z\left( a\right) = {w}_{1} \) and \( z\left( b\right) ... | Yes |
Corollary 3.3 If \( \gamma \) is a closed curve in an open set \( \Omega \), and \( f \) is continuous and has a primitive in \( \Omega \), then\n\n\[{\int }_{\gamma }f\left( z\right) {dz} = 0\] | This is immediate since the end-points of a closed curve coincide. | Yes |
Corollary 3.4 If \( f \) is holomorphic in a region \( \Omega \) and \( {f}^{\prime } = 0 \), then \( f \) is constant. | Proof. Fix a point \( {w}_{0} \in \Omega \) . It suffices to show that \( f\left( w\right) = f\left( {w}_{0}\right) \) for all \( w \in \Omega \) .\n\nSince \( \Omega \) is connected, for any \( w \in \Omega \), there exists a curve \( \gamma \) which joins \( {w}_{0} \) to \( w \) . Since \( f \) is clearly a primitiv... | Yes |
Corollary 1.2 If \( f \) is holomorphic in an open set \( \Omega \) that contains a rectangle \( R \) and its interior, then\n\n\[ \n{\int }_{R}f\left( z\right) {dz} = 0 \n\] | This is immediate since we first choose an orientation as in Figure 2 and note that\n\n\[ \n{\int }_{R}f\left( z\right) {dz} = {\int }_{{T}_{1}}f\left( z\right) {dz} + {\int }_{{T}_{2}}f\left( z\right) {dz}. \n\] | Yes |
Theorem 2.2 (Cauchy’s theorem for a disc) If \( f \) is holomorphic in a disc, then\n\n\[{\int }_{\gamma }f\left( z\right) {dz} = 0\]\n\nfor any closed curve \( \gamma \) in that disc. | Proof. Since \( f \) has a primitive, we can apply Corollary 3.3 of Chapter 1. | No |
Corollary 2.3 Suppose \( f \) is holomorphic in an open set containing the circle \( C \) and its interior. Then\n\n\[{\int }_{C}f\left( z\right) {dz} = 0\] | Proof. Let \( D \) be the disc with boundary circle \( C \) . Then there exists a slightly larger disc \( {D}^{\prime } \) which contains \( D \) and so that \( f \) is holomorphic on \( {D}^{\prime } \) . We may now apply Cauchy’s theorem in \( {D}^{\prime } \) to conclude that \( {\int }_{C}f\left( z\right) {dz} = 0 ... | Yes |
Theorem 4.1 Suppose \( f \) is holomorphic in an open set that contains the closure of a disc D. If \( C \) denotes the boundary circle of this disc with the positive orientation, then\n\n\[ f\left( z\right) = \frac{1}{2\pi i}{\int }_{C}\frac{f\left( \zeta \right) }{\zeta - z}{d\zeta }\;\text{ for any point }z \in D. \... | Proof. Fix \( z \in D \) and consider the \ | No |
If \( f \) is holomorphic in an open set \( \Omega \), then \( f \) has infinitely many complex derivatives in \( \Omega \) . Moreover, if \( C \subset \Omega \) is a circle whose interior is also contained in \( \Omega \), then\n\n\[ \n{f}^{\left( n\right) }\left( z\right) = \frac{n!}{2\pi i}{\int }_{C}\frac{f\left( \... | The proof is by induction on \( n \), the case \( n = 0 \) being simply the Cauchy integral formula. Suppose that \( f \) has up to \( n - 1 \) complex derivatives and that\n\n\[ \n{f}^{\left( n - 1\right) }\left( z\right) = \frac{\left( {n - 1}\right) !}{2\pi i}{\int }_{C}\frac{f\left( \zeta \right) }{{\left( \zeta - ... | Yes |
Corollary 4.3 (Cauchy inequalities) If \( f \) is holomorphic in an open set that contains the closure of a disc \( D \) centered at \( {z}_{0} \) and of radius \( R \) , then\n\n\[ \left| {{f}^{\left( n\right) }\left( {z}_{0}\right) }\right| \leq \frac{n!\parallel f{\parallel }_{C}}{{R}^{n}} \]\n\nwhere \( \parallel f... | Proof. Applying the Cauchy integral formula for \( {f}^{\left( n\right) }\left( {z}_{0}\right) \), we obtain\n\n\[ \left| {{f}^{\left( n\right) }\left( {z}_{0}\right) }\right| = \left| {\frac{n!}{2\pi i}{\int }_{C}\frac{f\left( \zeta \right) }{{\left( \zeta - {z}_{0}\right) }^{n + 1}}{d\zeta }}\right| \]\n\n\[ = \frac{... | Yes |
Theorem 4.4 Suppose \( f \) is holomorphic in an open set \( \Omega \) . If \( D \) is a disc centered at \( {z}_{0} \) and whose closure is contained in \( \Omega \), then \( f \) has a power series expansion at \( {z}_{0} \) \n\n\[ \nf\left( z\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{\left( z - {z}_{... | Proof. Fix \( z \in D \) . By the Cauchy integral formula, we have \n\n(10) \n\n\[ \nf\left( z\right) = \frac{1}{2\pi i}{\int }_{C}\frac{f\left( \zeta \right) }{\zeta - z}{d\zeta } \n\] \n\nwhere \( C \) denotes the boundary of the disc and \( z \in D \) . The idea is to write \n\n(11) \n\n\[ \n\frac{1}{\zeta - z} = \f... | Yes |
Corollary 4.5 (Liouville's theorem) If \( f \) is entire and bounded, then \( f \) is constant. | Proof. It suffices to prove that \( {f}^{\prime } = 0 \), since \( \mathbb{C} \) is connected, and we may then apply Corollary 3.4 in Chapter 1.\n\nFor each \( {z}_{0} \in \mathbb{C} \) and all \( R > 0 \), the Cauchy inequalities yield\n\n\[ \left| {{f}^{\prime }\left( {z}_{0}\right) }\right| \leq \frac{B}{R} \]\n\nwh... | Yes |
Corollary 4.6 Every non-constant polynomial \( P\left( z\right) = {a}_{n}{z}^{n} + \cdots + {a}_{0} \) with complex coefficients has a root in \( \mathbb{C} \) . | Proof. If \( P \) has no roots, then \( 1/P\left( z\right) \) is a bounded holomorphic function. To see this, we can of course assume that \( {a}_{n} \neq 0 \), and write\n\n\[ \frac{P\left( z\right) }{{z}^{n}} = {a}_{n} + \left( {\frac{{a}_{n - 1}}{z} + \cdots + \frac{{a}_{0}}{{z}^{n}}}\right) \]\n\nwhenever \( z \neq... | Yes |
Every polynomial \( P\left( z\right) = {a}_{n}{z}^{n} + \cdots + {a}_{0} \) of degree \( n \geq \) 1 has precisely \( n \) roots in \( \mathbb{C} \) . If these roots are denoted by \( {w}_{1},\ldots ,{w}_{n} \) , then \( P \) can be factored as \[ P\left( z\right) = {a}_{n}\left( {z - {w}_{1}}\right) \left( {z - {w}_{2... | Proof. By the previous result \( P \) has a root, say \( {w}_{1} \) . Then, writing \( z = \left( {z - {w}_{1}}\right) + {w}_{1} \), inserting this expression for \( z \) in \( P \), and using the binomial formula we get \[ P\left( z\right) = {b}_{n}{\left( z - {w}_{1}\right) }^{n} + \cdots + {b}_{1}\left( {z - {w}_{1}... | Yes |
Theorem 4.8 Suppose \( f \) is a holomorphic function in a region \( \Omega \) that vanishes on a sequence of distinct points with a limit point in \( \Omega \). Then \( f \) is identically 0. | Proof. Suppose that \( {z}_{0} \in \Omega \) is a limit point for the sequence \( {\left\{ {w}_{k}\right\} }_{k = 1}^{\infty } \) and that \( f\left( {w}_{k}\right) = 0 \). First, we show that \( f \) is identically zero in a small disc containing \( {z}_{0} \). For that, we choose a disc \( D \) centered at \( {z}_{0}... | Yes |
Theorem 5.1 Suppose \( f \) is a continuous function in the open disc \( D \) such that for any triangle \( T \) contained in \( D \)\n\n\[{\int }_{T}f\left( z\right) {dz} = 0\]\n\nthen \( f \) is holomorphic. | Proof. By the proof of Theorem 2.1 the function \( f \) has a primitive \( F \) in \( D \) that satisfies \( {F}^{\prime } = f \) . By the regularity theorem, we know that \( F \) is indefinitely (and hence twice) complex differentiable, and therefore \( f \) is holomorphic. | Yes |
Theorem 5.2 If \( {\left\{ {f}_{n}\right\} }_{n = 1}^{\infty } \) is a sequence of holomorphic functions that converges uniformly to a function \( f \) in every compact subset of \( \Omega \), then \( f \) is holomorphic in \( \Omega \) . | Proof. Let \( D \) be any disc whose closure is contained in \( \Omega \) and \( T \) any triangle in that disc. Then, since each \( {f}_{n} \) is holomorphic, Goursat’s theorem implies\n\n\[{\int }_{T}{f}_{n}\left( z\right) {dz} = 0\;\text{ for all }n\]\n\nBy assumption \( {f}_{n} \rightarrow f \) uniformly in the clo... | Yes |
Theorem 5.3 Under the hypotheses of the previous theorem, the sequence of derivatives \( {\left\{ {f}_{n}^{\prime }\right\} }_{n = 1}^{\infty } \) converges uniformly to \( {f}^{\prime } \) on every compact subset of \( \Omega \) . | Proof. We may assume without loss of generality that the sequence of functions in the theorem converges uniformly on all of \( \Omega \) . Given \( \delta > 0 \) , let \( {\Omega }_{\delta } \) denote the subset of \( \Omega \) defined by\n\n\[ \n{\Omega }_{\delta } = \left\{ {z \in \Omega : \overline{{D}_{\delta }}\le... | Yes |
Theorem 5.4 Let \( F\left( {z, s}\right) \) be defined for \( \left( {z, s}\right) \in \Omega \times \left\lbrack {0,1}\right\rbrack \) where \( \Omega \) is an open set in \( \mathbb{C} \) . Suppose \( F \) satisfies the following properties:\n\n(i) \( F\left( {z, s}\right) \) is holomorphic in \( z \) for each \( s \... | Proof. For each \( n \geq 1 \), we consider the Riemann sum\n\n\[ {f}_{n}\left( z\right) = \left( {1/n}\right) \mathop{\sum }\limits_{{k = 1}}^{n}F\left( {z, k/n}\right) \]\n\nThen \( {f}_{n} \) is holomorphic in all of \( \Omega \) by property (i), and we claim that on any disc \( D \) whose closure is contained in \(... | Yes |
Theorem 5.5 (Symmetry principle) If \( {f}^{ + } \) and \( {f}^{ - } \) are holomorphic functions in \( {\Omega }^{ + } \) and \( {\Omega }^{ - } \) respectively, that extend continuously to \( I \) and\n\n\[ \n{f}^{ + }\left( x\right) = {f}^{ - }\left( x\right) \;\text{ for all }x \in I,\n\]\n\nthen the function \( f ... | Proof. One notes first that \( f \) is continuous throughout \( \Omega \) . The only difficulty is to prove that \( f \) is holomorphic at points of \( I \) . Suppose \( D \) is a disc centered at a point on \( I \) and entirely contained in \( \Omega \) . We prove that \( f \) is holomorphic in \( D \) by Morera’s the... | Yes |
Theorem 5.6 (Schwarz reflection principle) Suppose that \( f \) is a holomorphic function in \( {\Omega }^{ + } \) that extends continuously to \( I \) and such that \( f \) is real-valued on \( I \) . Then there exists a function \( F \) holomorphic in all of \( \Omega \) such that \( F = f \) on \( {\Omega }^{ + } \)... | Proof. The idea is simply to define \( F\left( z\right) \) for \( z \in {\Omega }^{ - } \) by\n\n\[ F\left( z\right) = \overline{f\left( \bar{z}\right) }.\]\n\nTo prove that \( F \) is holomorphic in \( {\Omega }^{ - } \) we note that if \( z,{z}_{0} \in {\Omega }^{ - } \), then \( \bar{z},\overline{{z}_{0}} \in {\Omeg... | Yes |
Lemma 5.8 Suppose \( f \) is holomorphic in an open set \( \Omega \), and \( K \subset \Omega \) is compact. Then, there exists finitely many segments \( {\gamma }_{1},\ldots ,{\gamma }_{N} \) in \( \Omega - K \) such that\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{{n = 1}}^{N}\frac{1}{2\pi i}{\int }_{{\gamma }_{n... | Proof. Let \( d = c \cdot d\left( {K,{\Omega }^{c}}\right) \), where \( c \) is any constant \( < 1/\sqrt{2} \), and consider a grid formed by (solid) squares with sides parallel to the axis and of length \( d \) .\n\nWe let \( \mathcal{Q} = \left\{ {{Q}_{1},\ldots ,{Q}_{M}}\right\} \) denote the finite collection of s... | Yes |
Lemma 5.9 For any line segment \( \gamma \) entirely contained in \( \Omega - K \), there exists a sequence of rational functions with singularities on \( \gamma \) that approximate the integral \( {\int }_{\gamma }f\left( \zeta \right) /\left( {\zeta - z}\right) {d\zeta } \) uniformly on \( K \) . | Proof. If \( \gamma \left( t\right) : \left\lbrack {0,1}\right\rbrack \rightarrow \mathbb{C} \) is a parametrization for \( \gamma \), then\n\n\[ \n{\int }_{\gamma }\frac{f\left( \zeta \right) }{\zeta - z}{d\zeta } = {\int }_{0}^{1}\frac{f\left( {\gamma \left( t\right) }\right) }{\gamma \left( t\right) - z}{\gamma }^{\... | Yes |
Lemma 5.10 If \( {K}^{c} \) is connected and \( {z}_{0} \notin K \), then the function \( 1/\left( {z - {z}_{0}}\right) \) can be approximated uniformly on \( K \) by polynomials. | Proof. First, we choose a point \( {z}_{1} \) that is outside a large open disc \( D \) centered at the origin and which contains \( K \) . Then\n\n\[ \frac{1}{z - {z}_{1}} = - \frac{1}{{z}_{1}}\frac{1}{1 - z/{z}_{1}} = \mathop{\sum }\limits_{{n = 1}}^{\infty } - \frac{{z}^{n}}{{z}_{1}^{n + 1}} \]\n\nwhere the series c... | Yes |
Theorem 1.1 Suppose that \( f \) is holomorphic in a connected open set \( \Omega \) , has a zero at a point \( {z}_{0} \in \Omega \), and does not vanish identically in \( \Omega \) . Then there exists a neighborhood \( U \subset \Omega \) of \( {z}_{0} \), a non-vanishing holomorphic function \( g \) on \( U \), and ... | Proof. Since \( \Omega \) is connected and \( f \) is not identically zero, we conclude that \( f \) is not identically zero in a neighborhood of \( {z}_{0} \) . In a small disc centered at \( {z}_{0} \) the function \( f \) has a power series expansion \[ f\left( z\right) = \mathop{\sum }\limits_{{k = 0}}^{\infty }{a}... | Yes |
Theorem 1.2 If \( f \) has a pole at \( {z}_{0} \in \Omega \), then in a neighborhood of that point there exist a non-vanishing holomorphic function \( h \) and a unique positive integer \( n \) such that\n\n\[ f\left( z\right) = {\left( z - {z}_{0}\right) }^{-n}h\left( z\right) \] | Proof. By the previous theorem we have \( 1/f\left( z\right) = {\left( z - {z}_{0}\right) }^{n}g\left( z\right) \) , where \( g \) is holomorphic and non-vanishing in a neighborhood of \( {z}_{0} \), so the result follows with \( h\left( z\right) = 1/g\left( z\right) \) . | Yes |
Theorem 1.3 If \( f \) has a pole of order \( n \) at \( {z}_{0} \), then\n\n\[ f\left( z\right) = \frac{{a}_{-n}}{{\left( z - {z}_{0}\right) }^{n}} + \frac{{a}_{-n + 1}}{{\left( z - {z}_{0}\right) }^{n - 1}} + \cdots + \frac{{a}_{-1}}{\left( z - {z}_{0}\right) } + G\left( z\right) ,\] | Proof. The proof follows from the multiplicative statement in the previous theorem. Indeed, the function \( h \) has a power series expansion\n\n\[ h\left( z\right) = {A}_{0} + {A}_{1}\left( {z - {z}_{0}}\right) + \cdots \]\n\nso that\n\n\[ f\left( z\right) = {\left( z - {z}_{0}\right) }^{-n}\left( {{A}_{0} + {A}_{1}\l... | Yes |
Theorem 1.4 If \( f \) has a pole of order \( n \) at \( {z}_{0} \), then\n\n\[ \n{\operatorname{res}}_{{z}_{0}}f = \mathop{\lim }\limits_{{z \rightarrow {z}_{0}}}\frac{1}{\left( {n - 1}\right) !}{\left( \frac{d}{dz}\right) }^{n - 1}{\left( z - {z}_{0}\right) }^{n}f\left( z\right) .\n\] | The theorem is an immediate consequence of formula (1), which implies\n\n\[ \n{\left( z - {z}_{0}\right) }^{n}f\left( z\right) = {a}_{-n} + {a}_{-n + 1}\left( {z - {z}_{0}\right) + \cdots + {a}_{-1}{\left( z - {z}_{0}\right) }^{n - 1} + \n\]\n\n\[ \n+ G\left( z\right) {\left( z - {z}_{0}\right) }^{n}.\n\] | Yes |
Theorem 2.1 Suppose that \( f \) is holomorphic in an open set containing a circle \( C \) and its interior, except for a pole at \( {z}_{0} \) inside \( C \) . Then\n\n\[ \n{\int }_{C}f\left( z\right) {dz} = {2\pi i}{\operatorname{res}}_{{z}_{0}}f \n\] | Proof. Once again, we may choose a keyhole contour that avoids the pole, and let the width of the corridor go to zero to see that\n\n\[ \n{\int }_{C}f\left( z\right) {dz} = {\int }_{{C}_{\epsilon }}f\left( z\right) {dz} \n\]\n\nwhere \( {C}_{\epsilon } \) is the small circle centered at the pole \( {z}_{0} \) and of ra... | Yes |
Corollary 2.2 Suppose that \( f \) is holomorphic in an open set containing a circle \( C \) and its interior, except for poles at the points \( {z}_{1},\ldots ,{z}_{N} \) inside C. Then\n\n\[{\int }_{C}f\left( z\right) {dz} = {2\pi i}\mathop{\sum }\limits_{{k = 1}}^{N}{\operatorname{res}}_{{z}_{k}}f.\] | For the proof, consider a multiple keyhole which has a loop avoiding each one of the poles. Let the width of the corridors go to zero. In the limit, the integral over the large circle equals a sum of integrals over small circles to which Theorem 2.1 applies. | No |
Corollary 2.3 Suppose that \( f \) is holomorphic in an open set containing a toy contour \( \gamma \) and its interior, except for poles at the points \( {z}_{1},\ldots ,{z}_{N} \) inside \( \gamma \) . Then\n\n\[ \n{\int }_{\gamma }f\left( z\right) {dz} = {2\pi i}\mathop{\sum }\limits_{{k = 1}}^{N}{\operatorname{res}... | The proof consists of choosing a keyhole appropriate for the given toy contour, so that, as we have seen previously, we can reduce the situation to integrating over small circles around the poles where Theorem 2.1 applies. | No |
Theorem 3.1 (Riemann's theorem on removable singularities) Suppose that \( f \) is holomorphic in an open set \( \Omega \) except possibly at a point \( {z}_{0} \) in \( \Omega \) . If \( f \) is bounded on \( \Omega - \left\{ {z}_{0}\right\} \), then \( {z}_{0} \) is a removable singularity. | Proof. Since the problem is local we may consider a small disc \( D \) centered at \( {z}_{0} \) and whose closure is contained in \( \Omega \) . Let \( C \) denote the boundary circle of that disc with the usual positive orientation. We shall prove that if \( z \in D \) and \( z \neq {z}_{0} \), then under the assumpt... | Yes |
Corollary 3.2 Suppose that \( f \) has an isolated singularity at the point \( {z}_{0} \) . Then \( {z}_{0} \) is a pole of \( f \) if and only if \( \left| {f\left( z\right) }\right| \rightarrow \infty \) as \( z \rightarrow {z}_{0} \) . | Proof. If \( {z}_{0} \) is a pole, then we know that \( 1/f \) has a zero at \( {z}_{0} \), and therefore \( \left| {f\left( z\right) }\right| \rightarrow \infty \) as \( z \rightarrow {z}_{0} \) . Conversely, suppose that this condition holds. Then \( 1/f \) is bounded near \( {z}_{0} \), and in fact \( 1/\left| {f\le... | Yes |
Theorem 3.3 (Casorati-Weierstrass) Suppose \( f \) is holomorphic in the punctured disc \( {D}_{r}\left( {z}_{0}\right) - \left\{ {z}_{0}\right\} \) and has an essential singularity at \( {z}_{0} \) . Then, the image of \( {D}_{r}\left( {z}_{0}\right) - \left\{ {z}_{0}\right\} \) under \( f \) is dense in the complex p... | Proof. We argue by contradiction. Assume that the range of \( f \) is not dense, so that there exists \( w \in \mathbb{C} \) and \( \delta > 0 \) such that\n\n\[ \left| {f\left( z\right) - w}\right| > \delta \;\text{ for all }z \in {D}_{r}\left( {z}_{0}\right) - \left\{ {z}_{0}\right\} . \]\n\nWe may therefore define a... | Yes |
Theorem 3.4 The meromorphic functions in the extended complex plane are the rational functions. | Proof. Suppose that \( f \) is meromorphic in the extended plane. Then \( f\left( {1/z}\right) \) has either a pole or a removable singularity at 0, and in either case it must be holomorphic in a deleted neighborhood of the origin, so that the function \( f \) can have only finitely many poles in the plane, say at \( {... | Yes |
Theorem 4.3 (Rouché’s theorem) Suppose that \( f \) and \( g \) are holomorphic in an open set containing a circle \( C \) and its interior. If\n\n\[ \left| {f\left( z\right) }\right| > \left| {g\left( z\right) }\right| \;\text{ for all }z \in C, \]\n\nthen \( f \) and \( f + g \) have the same number of zeros inside t... | Proof. For \( t \in \left\lbrack {0,1}\right\rbrack \) define\n\n\[ {f}_{t}\left( z\right) = f\left( z\right) + \operatorname{tg}\left( z\right) \]\n\nso that \( {f}_{0} = f \) and \( {f}_{1} = f + g \) . Let \( {n}_{t} \) denote the number of zeros of \( {f}_{t} \) inside the circle counted with multiplicities, so tha... | Yes |
Theorem 4.4 (Open mapping theorem) If \( f \) is holomorphic and nonconstant in a region \( \Omega \), then \( f \) is open. | Proof. Let \( {w}_{0} \) belong to the image of \( f \), say \( {w}_{0} = f\left( {z}_{0}\right) \) . We must prove that all points \( w \) near \( {w}_{0} \) also belong to the image of \( f \) .\n\nDefine \( g\left( z\right) = f\left( z\right) - w \) and write\n\n\[ g\left( z\right) = \left( {f\left( z\right) - {w}_{... | Yes |
Theorem 4.5 (Maximum modulus principle) If \( f \) is a non-constant holomorphic function in a region \( \Omega \), then \( f \) cannot attain a maximum in \( \Omega \) . | Proof. Suppose that \( f \) did attain a maximum at \( {z}_{0} \) . Since \( f \) is holomorphic it is an open mapping, and therefore, if \( D \subset \Omega \) is a small disc centered at \( {z}_{0} \), its image \( f\left( D\right) \) is open and contains \( f\left( {z}_{0}\right) \) . This proves that there are poin... | Yes |
Corollary 4.6 Suppose that \( \Omega \) is a region with compact closure \( \bar{\Omega } \) . If \( f \) is holomorphic on \( \Omega \) and continuous on \( \bar{\Omega } \) then\n\n\[ \mathop{\sup }\limits_{{z \in \Omega }}\left| {f\left( z\right) }\right| \leq \mathop{\sup }\limits_{{z \in \bar{\Omega } - \Omega }}\... | In fact, since \( f\left( z\right) \) is continuous on the compact set \( \bar{\Omega } \), then \( \left| {f\left( z\right) }\right| \) attains its maximum in \( \bar{\Omega } \) ; but this cannot be in \( \Omega \) if \( f \) is non-constant. If \( f \) is constant, the conclusion is trivial. | Yes |
Theorem 5.2 Any holomorphic function in a simply connected domain has a primitive. | Proof. Fix a point \( {z}_{0} \) in \( \Omega \) and define\n\n\[ F\left( z\right) = {\int }_{\gamma }f\left( w\right) {dw} \]\n\nwhere the integral is taken over any curve in \( \Omega \) joining \( {z}_{0} \) to \( z \) . This definition is independent of the curve chosen, since \( \Omega \) is simply connected, and ... | Yes |
Corollary 5.3 If \( f \) is holomorphic in the simply connected region \( \Omega \) , then\n\n\[{\int }_{\gamma }f\left( z\right) {dz} = 0\]\n\nfor any closed curve \( \gamma \) in \( \Omega \) . | This is immediate from the existence of a primitive. | No |
Theorem 6.1 Suppose that \( \Omega \) is simply connected with \( 1 \in \Omega \), and \( 0 \notin \) \( \Omega \) . Then in \( \Omega \) there is a branch of the logarithm \( F\left( z\right) = {\log }_{\Omega }\left( z\right) \) so that\n\n(i) \( F \) is holomorphic in \( \Omega \) ,\n\n(ii) \( {e}^{F\left( z\right) ... | Proof. We shall construct \( F \) as a primitive of the function \( 1/z \) . Since \( 0 \notin \Omega \), the function \( f\left( z\right) = 1/z \) is holomorphic in \( \Omega \) . We define\n\n\[ \n{\log }_{\Omega }\left( z\right) = F\left( z\right) = {\int }_{\gamma }f\left( w\right) {dw} \n\] \n\nwhere \( \gamma \) ... | Yes |
Theorem 6.2 If \( f \) is a nowhere vanishing holomorphic function in a simply connected region \( \Omega \), then there exists a holomorphic function \( g \) on \( \Omega \) such that\n\n\[ f\left( z\right) = {e}^{g\left( z\right) }.\] | Proof. Fix a point \( {z}_{0} \) in \( \Omega \), and define a function\n\n\[ g\left( z\right) = {\int }_{\gamma }\frac{{f}^{\prime }\left( w\right) }{f\left( w\right) }{dw} + {c}_{0} \]\n\nwhere \( \gamma \) is any path in \( \Omega \) connecting \( {z}_{0} \) to \( z \), and \( {c}_{0} \) is a complex number so that ... | Yes |
Theorem 7.1 The coefficients of the power series expansion of \( f \) are given by\n\n\[ \n{a}_{n} = \frac{1}{{2\pi }{r}^{n}}{\int }_{0}^{2\pi }f\left( {{z}_{0} + r{e}^{i\theta }}\right) {e}^{-{in\theta }}{d\theta }\n\]\n\nfor all \( n \geq 0 \) and \( 0 < r < R \) . Moreover,\n\n\[ \n0 = \frac{1}{{2\pi }{r}^{n}}{\int ... | Proof. Since \( {f}^{\left( n\right) }\left( {z}_{0}\right) = {a}_{n}n \) !, the Cauchy integral formula gives\n\n\[ \n{a}_{n} = \frac{1}{2\pi i}{\int }_{\gamma }\frac{f\left( \zeta \right) }{{\left( \zeta - {z}_{0}\right) }^{n + 1}}{d\zeta }\n\]\n\nwhere \( \gamma \) is a circle of radius \( 0 < r < R \) centered at \... | Yes |
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