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Proposition 6.2.5. 延續上述性質的標號，令 \( T \) 為 \( \mathcal{C} \) 與 \( \left( {ABCPQ}\right) \) 的第四個交點，則 \( T, X,{\mathfrak{Z}}_{P, Q}^{\mathcal{C}}\left( X\right) \) 共線。 | Proof. 令 \( \mathfrak{Z} = {\mathfrak{Z}}_{P, Q}^{\mathcal{C}}\left( X\right) \) 。由上述性质的證明及結論,我們有\n\n\[ T\left( {A,\mathfrak{Z};B, C}\right) = P\left( {A,\mathfrak{Z};B, C}\right) = \left( {A, X;B, C}\right) = T\left( {A, X;B, C}\right) ,\]\n\n即 \( T, X,3 \) 共線。 | No |
Proposition 6.2.7. 令 \( \varphi \) 为 \( \bigtriangleup {ABC} \) 上的一個等共轭變換，且 \( X \in {\mathcal{L}}^{\varphi } \) 。對於任意點 \( P \) ,設 \( \mathfrak{Z} = {\mathfrak{Z}}_{P,\varphi \left( P\right) }^{{\mathcal{L}}^{\varphi }}\left( X\right) \) ,則\n\n\[ \n{P\varphi }\left( P\right) = {\mathcal{S}}_{\varphi \left( 3\right) }^{... | Proof. 設 \( D = {P\varphi }\left( P\right) \cap {BC} \) ,則\n\n\[ \nX\left( {B, C;D, P\mathfrak{Z} \cap {BC}}\right) = P\left( {B, C;\varphi \left( P\right) ,\mathfrak{Z}}\right) \n\]\n\n\[ \n= A\left( {B, C;\varphi \left( P\right) ,3}\right) \n\]\n\n\[ \n\triangleq A\left( {C, B;P,\varphi \left( 3\right) }\right) \n\]\... | Yes |
Proposition 6.2.8. 令 \( \varphi ,\psi \) 为 \( \bigtriangleup {ABC} \) 上的两個等共轭變換。設 \( X \) 为 \( {\mathcal{L}}^{\varphi } \) 和 \( {\mathcal{L}}^{\psi } \) 的第四個交點， \( P \) 為任意一點，則\n\n(i) \( X,{A\varphi }\left( P\right) \cap {\mathcal{L}}^{\varphi },{A\psi }\left( P\right) \cap {\mathcal{L}}^{\psi } \) 共線。\n\n(ii) \( \varp... | Proof. (i) 設 \( {A\varphi }\left( P\right) ,{A\psi }\left( P\right) \) 分别交 \( {\mathcal{L}}^{\varphi },{\mathcal{L}}^{\psi } \) 於 \( \varphi {\left( P\right) }_{A},\psi {\left( P\right) }_{A} \) 。簡單地觀察到\n\n\[ X\left( {A,\varphi {\left( P\right) }_{A};B, C}\right) = A{\left( A,\varphi {\left( P\right) }_{A};B, C\right) ... | Yes |
令 \( \varphi \) 为 \( \bigtriangleup {ABC} \) 上的一等共轭变换,則對於任意兩點 \( P, Q \) ，設 \( {P\varphi }\left( Q\right) \cap \varphi \left( P\right) Q = R,{PQ} \cap \varphi \left( P\right) \varphi \left( Q\right) = S \) ，則 \( \varphi \left( R\right) = S \) 。 | Proof. 定義一等共軛變換將 \( \psi : P \mapsto Q \) ,考慮 \( {\mathcal{L}}^{\varphi },{\mathcal{L}}^{\psi } \) 的交點 \( X \) ,則由 (6.2.8) 的 (ii)。\n\n\[ \n{\mathcal{B}}_{P}^{{\mathcal{L}}^{\psi }}\left( X\right) = {P\varphi }\left( Q\right) = {\mathcal{B}}_{\varphi \left( Q\right) }^{{\mathcal{L}}^{\varphi }}\left( X\right) \]\n\n\[ \... | Yes |
Proposition 6.2.13. 設 \( P,{P}^{ * } \) 为 \( \bigtriangleup {ABC} \) 的一對等角共轭點, \( X \) 为外接圆 \( \Omega \) 上任意點，則\n\n\[ \measuredangle \left( {{\mathfrak{S}}_{P}\left( X\right) ,{BC}}\right) = \measuredangle {AX}{P}^{ * } \] | Proof. 令 \( {P}_{A} = {AP} \cap \Omega ,{P}_{A}^{ * } = A{P}^{ * } \cap \Omega, D = {AP} \cap {BC},{X}_{A} = X{P}_{A} \cap {BC} \) 。由 \( {P}_{A}{P}_{A}^{ * }\parallel {BC} \) ,我們易得 \( \bigtriangleup {X}_{A}{P}_{A}D\overset{ - }{ \sim }\bigtriangleup {AYX} \) 。在 \( {P}_{A}{X}_{A} \) 上取點 \( E \) 使得 \( {DE}\parallel P{X}_... | Yes |
Theorem 6.2.14. 令 \( \Omega = {\mathcal{L}}^{ * } \) 为 \( \bigtriangleup {ABC} \) 的外接圆。對於任意等共轭變換 \( \varphi \) , 設 \( X \) 为 \( {\mathcal{L}}^{\varphi } \) 与 \( \Omega \) 的第四個交點。對於任意點 \( P \) ,設 \( T \) 为 \( {\mathcal{L}}^{\varphi } \) 与 \( \left( {{ABCP\varphi }\left( P\right) }\right) \) 的第四個交點, \( {P}^{ * },\varphi ... | Proof. 由 (6.2.5) 及 (6.2.8) 的 (ii), 我們有\n\n\[ \n{TX} \cap \left( {{ABCP\varphi }\left( P\right) }\right) = {\mathfrak{Z}}_{P,\varphi \left( P\right) }^{{\mathcal{L}}^{\varphi }} = {\mathfrak{S}}_{P}^{{\mathcal{L}}^{\varphi }}\left( X\right) \cap {\mathfrak{S}}_{\varphi \left( P\right) }^{{\mathcal{L}}^{\varphi }}\left( ... | Yes |
Proposition 6.2.16. 若 \( X \) 位於 \( \bigtriangleup {ABC} \) 的外接圆锥曲線 \( \mathcal{C} \) 上, \( P \) 为 \( \bigtriangleup {ABC} \) 与 \( {\mathfrak{p}}_{\mathcal{C}}\left( {\bigtriangleup {ABC}}\right) \) 的透视中心，則 \( {\mathfrak{B}}_{P}^{\mathcal{C}}\left( X\right) \) 为 \( X \) 關於 \( \bigtriangleup {ABC} \) 的三線性極線 \( {\mathfra... | Proof. 令 \( \bigtriangleup {P}_{A}{P}_{B}{P}_{C} \) 为 \( P \) 關於 \( \bigtriangleup {ABC} \) 的 \( \mathcal{C} \) -西瓦三角形,則 \( {AB}{P}_{A}C \) 为 \( \mathcal{C} \) 上的調和四邊形，因此\n\n\[ \left( {B, C;{AX} \cap {BC},{\mathcal{B}}_{P}^{\mathcal{C}}\left( X\right) \cap {BC}}\right) \overset{X}{ = }{\left( B, C;A,{P}_{A}\right) }_{\... | Yes |
Proposition 6.2.20. 設 \( \mathcal{S}t \) 为 \( \bigtriangleup {ABC} \) 的斯坦纳外接椭圆,則對於 \( \mathcal{S}t \) 上一點 \( X \) ,設 \( \left( X\right) \) 为以 \( X \) 为中心的圆锥曲線,设 \( {A}^{\mathrm{p}} = {\mathfrak{p}}_{\left( X\right) }\left( {BC}\right) ,{B}^{\mathrm{p}} = {\mathfrak{p}}_{\left( X\right) }\left( {CA}\right) \) , \( {C}^{... | Proof. 設 \( \mathcal{S}{t}^{\mathfrak{p}} \) 为 \( \bigtriangleup {A}^{\mathfrak{p}}{B}^{\mathfrak{p}}{C}^{\mathfrak{p}} \) 的斯坦納外接橢圓，則\n\n\[ A{\left( A, B;C, X\right) }_{\mathcal{S}t} = \left( {{\infty }_{BC}, B;C,{AX} \cap {BC}}\right) \]\n\n\[ \overset{{\mathfrak{p}}_{\left( X\right) }}{ = }\left( {{A}^{\mathfrak{p}}X... | Yes |
Proposition 6.2.21. 令 \( \varphi \) 为 \( \bigtriangleup {ABC} \) 上的等截共轭變换, \( \mathcal{S}\mathbf{t} = {\mathcal{L}}^{\varphi } \) 为 \( \bigtriangleup {ABC} \) 的斯坦納外接橢圓，則對於任意點 \( P \) ，和一點 \( X \in \mathcal{S}\mathbf{t} \) ,設 \( \left( X\right) \) 為以 \( X \) 為中心的任意锥線,设 \( {A}^{\mathfrak{p}} = {\mathfrak{p}}_{\left( X\ri... | Proof. 令 \( Q = {\mathfrak{p}}_{\left( X\right) }\left( {{\mathcal{B}}_{P}^{\mathcal{S}t}\left( X\right) }\right), R = {\mathfrak{p}}_{\left( X\right) }\left( {{\mathcal{B}}_{\varphi \left( P\right) }^{\mathcal{S}t}\left( X\right) }\right) \) ,則我們等價要證明\n\n\[ {A}^{\mathfrak{p}}{\left( {B}^{\mathfrak{p}},{C}^{\mathfrak{p... | Yes |
Theorem 6.2.22. 延續 (6.2.21) 的標號,設 \( Q = {\mathfrak{p}}_{\left( X\right) }\left( {{\mathcal{B}}_{P}^{St}\left( X\right) }\right), R = {\mathfrak{p}}_{\left( X\right) }\left( {{\mathcal{B}}_{\varphi \left( P\right) }^{St}\left( X\right) }\right) \) , 設 \( {\mathfrak{Z}}_{P,\varphi \left( P\right) }^{\mathcal{S}t}\left( ... | Proof. 設 \( {\infty }_{{P\varphi }\left( P\right) } \) 關於 \( \bigtriangleup {ABC} \) 的等截共軛點為 \( {T}_{\bigtriangleup {ABC}},{\infty }_{QR} \) 關於 \( \bigtriangleup {A}^{\mathfrak{p}}{B}^{\mathfrak{p}}{C}^{\mathfrak{p}} \) 的等截共轭点 \( {T}_{\bigtriangleup {A}^{\mathfrak{p}}{B}^{\mathfrak{p}}{C}^{\mathfrak{p}}} \) ,則\n\n\[ \f... | Yes |
Lemma 2.1. Every nonnegative matrix of the following form\n\n\[ \nT = \left\lbrack \begin{matrix} {\lambda }_{1} & {\beta }_{1} & 0 & \cdots & 0 \\ 0 & {\lambda }_{2} & {\beta }_{2} & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ {\beta }_{n} & 0 & 0 & \cdots & {\lambda }_{n} \end{matrix}\right\rbrack \n\... | Proof. By induction, it is straightforward to show that all entries of \( {T}^{n - 1} \) are positive. | No |
Corollary 2.3. Let \( T \) be an \( n \times n \) primitive Markov matrix. Then there is a vector \( \Phi = {\left( {\phi }_{1},{\phi }_{2},\ldots ,{\phi }_{n}\right) }^{t} \) in \( {H}_{2\pi }^{n} \) such that for any \( \Theta \in {H}_{2\pi }^{n},\mathop{\lim }\limits_{{m \rightarrow \infty }}{T}^{m}\Theta = \) \( \P... | Proof. Set \( \mathbf{x} = \frac{1}{2\pi } \cdot \Theta \), then \( \mathbf{x} \) is a positive probability vector and the corollary follows from Theorem 2.2. | No |
Corollary 2.4. Let T be an \( n \times n \) primitive doubly stochastic matrix, and \( \Theta \in {H}_{2\pi }^{n} \). Then \[ \mathop{\lim }\limits_{{m \rightarrow \infty }}{T}^{m}\Theta = {\left( \frac{2\pi }{n},\frac{2\pi }{n},\cdots ,\frac{2\pi }{n}\right) }^{t}. \] | Proof. Since \( T \) is doubly stochastic, its normalized positive eigenvector associated with the eigenvalue 1 is \( \left( {\frac{1}{n},\frac{1}{n},\ldots ,\frac{1}{n}}\right) \). | No |
Theorem 2.5. Let \( {T}_{1},{T}_{2},\cdots ,{T}_{k} \) be primitive doubly stochastic matrices and let \( \Theta \in {H}_{2\pi }^{n} \) . Then\n\n\[ \mathop{\prod }\limits_{{i = 1}}^{\infty }{\widetilde{T}}_{i}\Theta = {\left( \frac{2\pi }{n},\frac{2\pi }{n},\cdots ,\frac{2\pi }{n}\right) }^{t} \]\n\nwhere \( {\widetil... | Proof. Refer to [IM85]. | No |
Theorem 2.6. Let \( {T}_{0} = \left\lbrack {t}_{ij}^{\left( 0\right) }\right\rbrack \) be a primitive Markov matrix and \( {T}_{1} = \left\lbrack {t}_{ij}^{\left( 1\right) }\right\rbrack \) , \( {T}_{2} = \left\lbrack {t}_{ij}^{\left( 2\right) }\right\rbrack ,\cdots ,{T}_{m} = \left\lbrack {t}_{ij}^{\left( m\right) }\r... | Proof. See [IM85]. | No |
Theorem 3.1. Every sequence of midpoint-stretching polygons converges to the regular polygon. | Proof. Let \( P \) be a cyclic polygon inscribed in a circle \( \Gamma \) and let \( {a}_{i} = {z}_{i}{z}_{i + 1} \) be the \( {i}^{\text{th }} \) side of \( P, i = 1,2,\cdots, n \), where \( {z}_{n + 1} = {z}_{1} \) . Also, let \( {\theta }_{i} \) denote the central angle of \( \Gamma \) subtended by \( {a}_{i} \) for... | Yes |
Theorem 3.2. The sequence of \( \Lambda \) -stretching polygons converges to a unique polygon \( P\left( \Phi \right) \), where\n\n(i) \( P\left( \Phi \right) \) is regular if and only if \( {\lambda }_{1} = {\lambda }_{2} = \cdots = {\lambda }_{n} = t \) where \( 0 < t < 1 \) ;\n\n(ii) \( P\left( \Phi \right) \) is de... | (i) When all the components of \( \Lambda \) are equal, the matrix associated to the \( \Lambda \) - stretching is doubly stochastic. This theorem follows from Lemma 2.1 and Corollary 2.4. Note that Theorem 3.1 is a special case where \( t = \frac{1}{2} \) .\n\n(ii) This follows from Lemma 2.1 and Theorem 2.2. | Yes |
Theorem 3.3. Let \( P\left( \Theta \right) \) be a given \( n \) -sided cyclic polygon, and \( {T}_{1},{T}_{2},\cdots ,{T}_{m} \) be a finite number of even \( \Lambda \) -stretchings. Then the sequence of polygons\n\n\[ \n{\left\{ \mathop{\prod }\limits_{{i = 1}}^{k}{\widetilde{T}}_{i}\Theta \right\} }_{k = 1}^{\infty... | Proof. Since every even \( \Lambda \) -stretching is a doubly stochastic matrix, this theorem follows from Theorem 2.5 in the previous section. | No |
Theorem 3.4. Let \( P\left( \Theta \right) \) be a given \( n \) -sided cyclic polygon, and \( {T}_{0} = \left\lbrack {t}_{ij}^{\left( 0\right) }\right\rbrack \) , \( {T}_{1} = \left\lbrack {t}_{ij}^{\left( 1\right) }\right\rbrack ,{T}_{2} = \left\lbrack {t}_{ij}^{\left( 2\right) }\right\rbrack ,\cdots ,{T}_{m} = \left... | Proof. This is a direct consequence of Theorem 2.6. | Yes |
Take any scalene triangle \( \bigtriangleup {A}_{0}{B}_{0}{C}_{0} \) and construct the inscribed circle. The points of tangency form a second triangle, \( \bigtriangleup {A}_{1}{B}_{1}{C}_{1} \) . Then construct the inscribed circle for \( \bigtriangleup {A}_{1}{B}_{1}{C}_{1} \) . The points of tangency on the three si... | The answer is that \( \bigtriangleup {A}_{n}{B}_{n}{C}_{n} \) will approach an equilateral triangle. (Of course, if \( \bigtriangleup {A}_{0}{B}_{0}{C}_{0} \) is equilateral, then every subsequent \( \bigtriangleup {A}_{n}{B}_{n}{C}_{n}, n \geq 1 \), will be equilateral.) To confirm the answer, forming a simple geometr... | Yes |
For a variation of the first example, let \( {T}_{0} = \bigtriangleup {A}_{0}{B}_{0}{C}_{0} \) be any scalene triangle circumscribing a circle \( {\Gamma }_{0} \) with center \( O \) . The line segments \( {AO},{BO} \), and \( {CO} \) (the angle bisectors of \( {T}_{0} \) ) intersect \( {\Gamma }_{0} \) at points \( {A... | To see this, notice that\n\n\[ \n{A}_{n} = \mathop{\sum }\limits_{{k = 1}}^{n}\frac{\pi }{{4}^{k}} + \frac{A}{{4}^{n}},\;\text{ for }\;n \geq 1,\;\text{ so }\mathop{\lim }\limits_{{n \rightarrow \infty }}{A}_{n} = \frac{\pi }{3}. \n\] | No |
Corollary 9. The inradius of \( T \) is at least \( L\sqrt{}3/9 \) . | Proof. Let \( r \) be the inradius of \( T \), and let \( S \) be an incircle. If \( S \cap \partial T \) contains two points \( P \) and \( Q \) that are the ends of a diameter of \( S \), then \( {PQ} = {2r} \geq L/2 \), and so \( r \geq L/4 > L\sqrt{}3/9 \) . If on the other hand \( S \cap \partial T \) contains thr... | Yes |
\[ {\log }_{A}B = \frac{{\log }_{C}B}{{\log }_{C}A};\;A, B, C > 0, A \neq 1 \] | Let \( X = {\log }_{C}B, Y = {\log }_{C}A \), and \( Z = {\log }_{A}B \) . Then, by the definition of logarithms, \( {C}^{X} = B,{C}^{Y} = A \), and \( {A}^{Z} = B \) . Combining these three equalities yields \( {C}^{X} = B = \) \( {\left( {C}^{Y}\right) }^{Z} \) . Therefore, \( X = {YZ} \), which implies \( Z = X/Y \)... | Yes |
Theorem 1.2. \[ \log {AB} = \log A + \log B;A, B > 0 \] | Proof. Let \( X = \log A, Y = \log B \), and \( Z = \log {AB} \) . Then, assuming the default base of 2, \( {2}^{X} = A,{2}^{Y} = B \), and \( {2}^{Z} = {AB} \) . Combining the last three equalities yields \( {2}^{X}{2}^{Y} = \) \( {AB} = {2}^{Z} \) . Therefore, \( X + Y = Z \), which proves the theorem. | Yes |
If \( N \geq 1 \), then \( \mathop{\sum }\limits_{{i = 1}}^{N}{i}^{2} = \frac{N\left( {N + 1}\right) \left( {{2N} + 1}\right) }{6} \) | The proof is by induction. For the basis, it is readily seen that the theorem is true when \( N = 1 \) . For the inductive hypothesis, assume that the theorem is true for \( 1 \leq k \leq N \) . We will establish that, under this assumption, the theorem is true for \( N + 1 \) . We have\n\n\[ \mathop{\sum }\limits_{{i ... | Yes |
The recursive number-printing algorithm is correct for \( n \geq 0 \) . | Proof (by induction on the number of digits in n).\n\nFirst, if \( n \) has one digit, then the program is trivially correct, since it merely makes a call to printDigit. Assume then that printOut works for all numbers of \( k \) or fewer digits. A number of \( k + 1 \) digits is expressed by its first \( k \) digits fo... | Yes |
If \( M > N \), then \( M{\;\operatorname{mod}\;N} < M/2 \) . | ## Proof.\n\nThere are two cases. If \( N \leq M/2 \), then since the remainder is smaller than \( N \), the theorem is true for this case. The other case is \( N > M/2 \) . But then \( N \) goes into \( M \) once with a remainder \( M - N < M/2 \), proving the theorem. | Yes |
The hash family \( H = \left\{ {{H}_{a, b}\left( x\right) = \left( {\left( {{ax} + b}\right) {\;\operatorname{mod}\;p}}\right) {\;\operatorname{mod}\;M}}\right. \), where \( 1 \leq a \leq p - 1 \) , \( 0 \leq b \leq p - 1\} \) is universal. | Let \( x \) and \( y \) be distinct values, with \( x > y \), such that \( {H}_{a, b}\left( x\right) = {H}_{a, b}\left( y\right) \) .\n\nClearly if \( \left( {{ax} + b}\right) {\;\operatorname{mod}\;p} \) is equal to \( \left( {{ay} + b}\right) {\;\operatorname{mod}\;p} \), then we will have a collision. However, this ... | Yes |
For the perfect binary tree of height \( h \) containing \( {2}^{h + 1} - 1 \) nodes, the sum of the heights of the nodes is \( {2}^{h + 1} - 1 - \left( {h + 1}\right) \) . | It is easy to see that this tree consists of 1 node at height \( h,2 \) nodes at height \( h - 1,{2}^{2} \) nodes at height \( h - 2 \), and in general \( {2}^{i} \) nodes at height \( h - i \) . The sum of the heights of all the nodes is then\n\n\[ S = \mathop{\sum }\limits_{{i = 0}}^{h}{2}^{i}\left( {h - i}\right) \]... | Yes |
Any algorithm that sorts by exchanging adjacent elements requires \( \Omega \left( {N}^{2}\right) \) time on average. | The average number of inversions is initially \( N\left( {N - 1}\right) /4 = \Omega \left( {N}^{2}\right) \) . Each swap removes only one inversion, so \( \Omega \left( {N}^{2}\right) \) swaps are required. | Yes |
A binary tree with \( L \) leaves must have depth at least \( \lceil \log L\rceil \) . | Immediate from the preceding lemma. | No |
Any sorting algorithm that uses only comparisons between elements requires \( \Omega \left( {N\log N}\right) \) comparisons. | From the previous theorem, \( \log \left( {N!}\right) \) comparisons are required.\n\n\[ \log \left( {N!}\right) = \log \left( {N\left( {N - 1}\right) \left( {N - 2}\right) \cdots \left( 2\right) \left( 1\right) }\right) \]\n\n\[ = \log N + \log \left( {N - 1}\right) + \log \left( {N - 2}\right) + \cdots + \log 2 + \lo... | Yes |
The decision tree for finding the \( k \) th smallest of \( N \) elements must have at least \( \left( \begin{matrix} N \\ k - 1 \end{matrix}\right) {2}^{N - k} \) leaves. | Observe that any algorithm that correctly identifies the \( k \) th smallest element \( t \) must be able to prove that all other elements \( x \) are either larger than or smaller than \( t \) . Otherwise, it would be giving the same answer regardless of whether \( x \) was larger or smaller than \( t \), and the answ... | Yes |
Any comparison-based algorithm to find the second smallest element must use at least \( N + \lceil \log N\rceil - 2 \) comparisons. | Applying Theorem 7.9, with \( k = 2 \) yields \( N - 2 + \lceil \log N\rceil \) . | Yes |
Any comparison-based algorithm to find the median must use at least \( \lceil {3N}/2\rceil - \) \( O\left( {\log N}\right) \) comparisons. | Apply Theorem 7.9, with \( k = \lceil N/2\rceil \) . | Yes |
When executing a sequence of union instructions, a node of rank \( r > 0 \) must have at least one child of rank \( 0,1,\ldots, r - 1 \) . | By induction. The basis \( r = 1 \) is clearly true. When a node grows from rank \( r - 1 \) to rank \( r \), it obtains a child of rank \( r - 1 \) . By the inductive hypothesis, it already has children of ranks \( 0,1,\ldots, r - 2 \), thus establishing the lemma. | Yes |
Lemma 8.3. \( M = {M}_{t} + {M}_{b} \) | In cases 1 and 3, each original partial find operation is replaced by a partial find on the top half, and in case 2, it is replaced by a partial find on the bottom half. Thus each partial find is replaced by exactly one partial find operation on one of the halves. | No |
Let \( C\\left( {M, N, r}\\right) \) be the number of parent changes for a sequence of \( M \) finds with path compression on \( N \) items, whose maximum rank is \( r \) . Suppose we partition so that all nodes with rank at \( s \) or lower are in the bottom, and the remaining nodes are in the top. Assuming appropriat... | The path compression that is performed in each of the three cases is covered by \( C\\left( {{M}_{t},{N}_{t}, r}\\right) + C\\left( {{M}_{b},{N}_{b}, s}\\right) \) . Node \( w \) in case 3 is accounted for by \( {M}_{t} \) . Finally, all the other bottom nodes on the path are non-root nodes that can have their parent s... | Yes |
Theorem 8.1.\n\n\( C\\left( {M, N, r}\\right) < M + N\\log r. \) | Proof.\n\nWe start with Lemmas 8.5:\n\n\[ C\\left( {M, N, r}\\right) < C\\left( {{M}_{t},{N}_{t}, r}\\right) + C\\left( {{M}_{b},{N}_{b}, s}\\right) + {M}_{t} + N - \\left( {s + 2}\\right) {N}_{t} \]\n\n(8.3)\n\nObserve that in the top half, there are only nodes of rank \( s + 1, s + 2,\\ldots, r \), and thus no node c... | Yes |
Theorem 8.3. \( C\left( {M, N, r}\right) < {2M} + N{\log }^{ * }r \) | Proof.\n\nFrom Lemma 8.5 we have,\n\n\[ C\left( {M, N, r}\right) < C\left( {{M}_{t},{N}_{t}, r}\right) + C\left( {{M}_{b},{N}_{b}, s}\right) + {M}_{t} + N - \left( {s + 2}\right) {N}_{t} \]\n\n(8.9)\n\nand by Theorem 8.1, \( C\left( {{M}_{t},{N}_{t}, r}\right) < {M}_{t} + {N}_{t}\log r \) . Thus,\n\n\[ C\left( {M, N, r... | Yes |
Theorem 8.4. \( C\left( {M, N, r}\right) < {3M} + N{\log }^{* * }r \) | Proof. Following the steps in the proof of Theorem 8.3, we have\n\n\[ C\left( {M, N, r}\right) < C\left( {{M}_{t},{N}_{t}, r}\right) + C\left( {{M}_{b},{N}_{b}, s}\right) + {M}_{t} + N - \left( {s + 2}\right) {N}_{t} \]\n\n(8.15)\n\nand by Theorem 8.3, \( C\left( {{M}_{t},{N}_{t}, r}\right) < 2{M}_{t} + {N}_{t}{\log }^... | Yes |
Any sequence of \( N - 1 \) unions and \( M \) finds with path compression makes at most\n\n\[ \left( {i + 1}\right) M + N{\log }^{\overset{i\text{ times }}{\overbrace{* * * * }}}\left( {\log N}\right) \]\n\nparent changes during the finds. | This follows from the above discussion, and the fact that \( r \leq \log N \) . | No |
Any sequence of \( N - 1 \) unions and \( M \) finds with path compression makes at most \( {M\alpha }\left( {M, N}\right) + {2M} \) parent changes during the finds. | In Theorem 8.5, choose \( i \) to be \( \alpha \left( {M, N}\right) \) ; thus we obtain a bound of \( \left( {i + 1}\right) M + N\left( {M/N}\right) \) , or \( {M\alpha }\left( {M, N}\right) + {2M} \) . | Yes |
If \( P \) is prime and \( 0 < X < P \), the only solutions to \( {X}^{2} \equiv 1\\left( {\\operatorname{mod}P}\\right) \) are \( X = 1, P - 1 \) . | \( {X}^{2} \equiv 1\\left( {\\operatorname{mod}P}\\right) \) implies that \( {X}^{2} - 1 \equiv 0\\left( {\\operatorname{mod}P}\\right) \). This implies \( \\left( {X - 1}\\right) \\left( {X + 1}\\right) \equiv 0 \) (mod \( P \)). Since \( P \) is prime, \( 0 < X < P \), and \( P \) must divide either \( \\left( {X - 1... | Yes |
The amortized running times of insert, deleteMin, and merge are \( O\left( 1\right), O\left( {\log N}\right) \), and \( O\left( {\log N}\right) \), respectively, for binomial queues. | The potential function is the number of trees. The initial potential is 0 , and the potential is always nonnegative, so the amortized time is an upper bound on the actual time. The analysis for insert follows from the argument above. For merge, assume the two queues have \( {N}_{1} \) and \( {N}_{2} \) nodes with \( {T... | Yes |
Let \( {F}_{k} \) be the Fibonacci numbers defined (in Section 1.2) by \( {F}_{0} = 1,{F}_{1} = 1 \), and \( {F}_{k} = {F}_{k - 1} + {F}_{k - 2} \) . Any node of rank \( R \geq 1 \) has at least \( {F}_{R + 1} \) descendants (including itself). | Let \( {S}_{R} \) be the smallest tree of rank \( R \) . Clearly, \( {S}_{0} = 1 \) and \( {S}_{1} = 2 \) . By Lemma 11.1, a tree of rank \( R \) must have subtrees of rank at least \( R - 2, R - 3,\ldots ,1 \), and 0, plus another subtree, which has at least one node. Along with the root of \( {S}_{R} \) itself, this ... | No |
If \( a + b \leq c \), and \( a \) and \( b \) are both positive integers, then\n\n\[ \log a + \log b \leq 2\log c - 2 \] | Proof.\n\nBy the arithmetic-geometric mean inequality,\n\n\[ \sqrt{ab} \leq \left( {a + b}\right) /2 \]\n\nThus\n\n\[ \sqrt{ab} \leq c/2 \]\n\nSquaring both sides gives\n\n\[ {ab} \leq {c}^{2}/4 \]\n\nTaking logarithms of both sides proves the lemma. | Yes |
EXAMPLE 4 Find the center of mass (centroid) of a thin wire of constant density \( \delta \) shaped like a semicircle of radius \( a \) . | Solution We model the wire with the semicircle \( y = \sqrt{{a}^{2} - {x}^{2}} \) (Figure 6.54). The distribution of mass is symmetric about the \( y \) -axis, so \( \bar{x} = 0 \) . To find \( \bar{y} \), we imagine the wire divided into short subarc segments. If \( \left( {\widetilde{x},\widetilde{y}}\right) \) is th... | Yes |
\[ {\int }_{1}^{\infty }\frac{dx}{1 + {x}^{2}} \] converges by comparison with \( {\int }_{1}^{\infty }\left( {1/{x}^{2}}\right) {dx} \). Find and compare the two integral values. | Solution The functions \( f\left( x\right) = 1/{x}^{2} \) and \( g\left( x\right) = 1/\left( {1 + {x}^{2}}\right) \) are positive and continuous on \( \lbrack 1,\infty ) \). Also,\n\n\[ \mathop{\lim }\limits_{{x \rightarrow \infty }}\frac{f\left( x\right) }{g\left( x\right) } = \mathop{\lim }\limits_{{x \rightarrow \in... | Yes |
EXAMPLE 3 Find the angle \( \theta \) in the triangle \( {ABC} \) determined by the vertices \( A = \left( {0,0}\right), B = \left( {3,5}\right) \), and \( C = \left( {5,2}\right) \) (Figure 12.22). | Solution The angle \( \theta \) is the angle between the vectors \( \overrightarrow{CA} \) and \( \overrightarrow{CB} \) . The component forms of these two vectors are\n\n\[ \overrightarrow{CA} = \langle - 5, - 2\rangle \;\text{ and }\;\overrightarrow{CB} = \langle - 2,3\rangle . \]\n\nFirst we calculate the dot produc... | Yes |
5. A formula for the curvature of the graph of a function in the \( {xy} \) -plane\na. The graph \( y = f\left( x\right) \) in the \( {xy} \) -plane automatically has the parametrization \( x = x, y = f\left( x\right) \), and the vector formula \( \mathbf{r}\left( x\right) = x\mathbf{i} + f\left( x\right) \mathbf{j} \)... | \[ \kappa \left( x\right) = \frac{\left| {f}^{\prime \prime }\left( x\right) \right| }{{\left\lbrack 1 + {\left( {f}^{\prime }\left( x\right) \right) }^{2}\right\rbrack }^{3/2}}. \] | Yes |
EXAMPLE 4 Evaluate \( {\iint }_{S}\sqrt{x\left( {1 + {2z}}\right) }{d\sigma }\; \) on the portion of the cylinder \( z = {y}^{2}/2 \) over the triangular region \( R : x \geq 0, y \geq 0, x + y \leq 1 \) in the \( {xy} \) -plane (Figure 16.48). | Solution The function \( G \) on the surface \( S \) is given by\n\n\[ G\left( {x, y, z}\right) = \sqrt{x\left( {1 + {2z}}\right) } = \sqrt{x}\sqrt{1 + {y}^{2}}. \]\n\nWith \( z = f\left( {x, y}\right) = {y}^{2}/2 \), we use Equation (4) to evaluate the surface integral:\n\n\[ {d\sigma } = \sqrt{{f}_{x}^{2} + {f}_{y}^{... | Yes |
As a simple first example, we calculate and then draw the Penrose diagram of 2-dimensional Minkowski space, \[ d{s}^{2} = - d{t}^{2} + d{x}^{2} \] where \( - \infty < t, x < + \infty \) . | We first make a transformation to \ | No |
For 3-dimensional Minkowski space, the metric is again\n\n\\[ d{s}^{2} = - d{t}^{2} + d{r}^{2}\\left( {+{r}^{2}d{\\theta }^{2}}\\right) \\]\n\n(2.50)\n\nBy dropping the angular dependence (the \\( {r}^{2}d{\\theta }^{2} \\) term) in order to again draw a 2-dimensional diagram, we get the same metric as before, just tha... | Thus\n\n\n\nFigure 2.3 Penrose diagrams: a) Penrose diagram of 2-dimensional Minkowski space; b) Penrose diagram of 3-dimensional Minkowski space; c) Penrose diagram of the Poincaré patch of Anti-de Sitter space; d) Pe... | No |
The commutator \( \left\lbrack {\widehat{a},{\widehat{a}}^{ \dagger }}\right\rbrack \) can be evaluated as follows (remembering that \( \left\lbrack {\widehat{x},\widehat{p}}\right\rbrack = \mathrm{i}\hslash \) ): | \[ \left\lbrack {\widehat{a},{\widehat{a}}^{ \dagger }}\right\rbrack = \frac{m\omega }{2\hslash }\left( {-\frac{\mathrm{i}}{m\omega }\left\lbrack {\widehat{x},\widehat{p}}\right\rbrack + \frac{\mathrm{i}}{m\omega }\left\lbrack {\widehat{p},\widehat{x}}\right\rbrack }\right) \] \[ = \frac{m\omega }{2\hslash }\left( {\fr... | Yes |
We begin by Fourier transforming both \( {x}_{j} \) and \( {p}_{j} \), by writing\n\n\[ \n{x}_{j} = \frac{1}{\sqrt{N}}\mathop{\sum }\limits_{k}{\widetilde{x}}_{k}{\mathrm{e}}^{\mathrm{i}{kja}}, \]\n\n(2.46)\n\n\[ \n{p}_{j} = \frac{1}{\sqrt{N}}\mathop{\sum }\limits_{k}{\widetilde{p}}_{k}{\mathrm{e}}^{\mathrm{i}{kja}}, \... | \[ \n\left\lbrack {{\widetilde{x}}_{k},{\widetilde{p}}_{{k}^{\prime }}}\right\rbrack = \frac{1}{N}\mathop{\sum }\limits_{j}\mathop{\sum }\limits_{{j}^{\prime }}{\mathrm{e}}^{-\mathrm{i}{kja}}{\mathrm{e}}^{-\mathrm{i}{k}^{\prime }{j}^{\prime }a}\left\lbrack {{x}_{j},{p}_{{j}^{\prime }}}\right\rbrack \]\n\n\[ \n= \frac{\... | Yes |
To deal with this, we define \( \mathbf{z} = \mathbf{x} - \mathbf{y} \) and eliminate \( \mathbf{x} \) to obtain\n\n\[ \widehat{V} = \frac{1}{2{\mathcal{V}}^{2}}\mathop{\sum }\limits_{{{\mathbf{p}}_{1}{\mathbf{p}}_{2}{\mathbf{p}}_{3}{\mathbf{p}}_{4}}}{\widehat{a}}_{{\mathbf{p}}_{1}}^{ \dagger }{\widehat{a}}_{{\mathbf{p... | The last integral gives us a Kronecker delta \( \mathcal{V}{\delta }_{{\mathbf{p}}_{1} + {\mathbf{p}}_{2} - {\mathbf{p}}_{3},{\mathbf{p}}_{4}} \), which can be used to eat up one of the momentum sums (the \( {\mathbf{p}}_{4} \) one here) and sets \( {\mathbf{p}}_{4} = {\mathbf{p}}_{1} + {\mathbf{p}}_{2} - {\mathbf{p}}_... | Yes |
In the continuum limit, how do the Hamiltonian and Lagrangian change as the number of masses goes to infinity while \( \ell \rightarrow 0 \)? | In the continuum limit, these need to be replaced as the number of masses goes to infinity while \( \ell \rightarrow 0 \) . The sums become integrals, so we make the substitution\n\n\[ \mathop{\sum }\limits_{j} \rightarrow \frac{1}{\ell }\int \mathrm{d}x \]\n\n\( \left( {5.38}\right) \)\n\nUsing the substitution in eqn... | Yes |
Show that \( \delta \left( {{p}^{2} - {m}^{2}}\right) \theta \left( {p}_{0}\right) = \frac{1}{2{E}_{\mathbf{p}}}\delta \left( {{p}_{0} - {E}_{\mathbf{p}}}\right) \theta \left( {p}_{0}\right) \) . | We use the identity\n\[ \delta \left\lbrack {f\left( x\right) }\right\rbrack = \mathop{\sum }\limits_{{\{ x \mid f\left( x\right) = 0\} }}\frac{1}{\left| {f}^{\prime }\left( x\right) \right| }\delta \left( x\right) ,\]\nwhere the notation tells us that the sum is evaluated for those values of \( x \) that make \( f\lef... | Yes |
The charge operator is\n\n\\[ \n{\\widehat{Q}}_{\\mathrm{N}} = \\int {\\mathrm{d}}^{3}x{\\widehat{J}}_{\\mathrm{N}}^{0} = \\mathrm{i}\\left\\lbrack {\\left( {{\\partial }^{0}{\\widehat{\\psi }}^{ \\dagger }}\\right) \\widehat{\\psi } - \\left( {{\\partial }^{0}\\widehat{\\psi }}\\right) {\\widehat{\\psi }}^{ \\dagger }... | Of course, there's the ordering ambiguity here, but we'll press on and insert the mode expansion to yield\n\n\\[ \n{\\widehat{Q}}_{\\mathrm{N}} = \\frac{1}{2}\\int {\\mathrm{d}}^{3}p\\left( {-{\\widehat{a}}_{\\mathbf{p}}^{ \\dagger }{\\widehat{a}}_{\\mathbf{p}} + {\\widehat{b}}_{\\mathbf{p}}{\\widehat{b}}_{\\mathbf{p}}... | Yes |
If \( \psi \left( x\right) \rightarrow \psi \left( x\right) {\mathrm{e}}^{\mathrm{i}\alpha \left( x\right) } \), then \( {\partial }_{\mu }\psi \rightarrow \left( {{\partial }_{\mu }\psi }\right) {\mathrm{e}}^{\mathrm{i}\alpha } + \mathrm{i}\left( {{\partial }_{\mu }\alpha }\right) \psi \) and so | \[ {D}_{\mu }\psi = \left( {{\partial }_{\mu } + \mathrm{i}q{A}_{\mu }}\right) \psi \; \rightarrow \;\left( {{\partial }_{\mu }\psi }\right) {\mathrm{e}}^{\mathrm{i}\alpha } + \mathrm{i}\left( {{\partial }_{\mu }\alpha }\right) \psi + \mathrm{i}q{A}_{\mu }\psi {\mathrm{e}}^{\mathrm{i}\alpha } - \mathrm{i}\left( {{\part... | Yes |
The Green's function is given by\n\n\\[ \n{G}^{ + }\left( {x,{t}_{x}, y,{t}_{y}}\right) = \theta \left( {{t}_{x} - {t}_{y}}\right) \left\langle {x\left( {t}_{x}\right) \mid y\left( {t}_{y}\right) }\right\rangle = \theta \left( {{t}_{x} - {t}_{y}}\right) \mathop{\sum }\limits_{n}{\phi }_{n}\left( x\right) {\phi }_{n}^{ ... | Take this in two stages. Stage I: We use the fact that \\( \\frac{\\partial }{\\partial {t}_{x}}\\theta \\left( {{t}_{x} - {t}_{y}}\\right) = \\delta \\left( {{t}_{x} - {t}_{y}}\\right) \\) , to find that the time derivative acting on the retarded Green's function gives us\n\n\\[ \n\\mathrm{i}\\frac{\\partial }{\\parti... | Yes |
We obtain\n\n\\[ \n\\widehat{H} = \\int {\\mathrm{d}}^{3}x\\frac{{\\mathrm{d}}^{3}p{\\mathrm{\\;d}}^{3}q}{{\\left( 2\\pi \\right) }^{3}{\\left( 2{E}_{\\mathbf{p}}2{E}_{\\mathbf{q}}\\right) }^{\\frac{1}{2}}}\\mathop{\\sum }\\limits_{{s, r}}\\left( {{\\bar{u}}^{s}\\left( p\\right) {\\widehat{a}}_{s\\mathbf{p}}^{ \\dagger... | The (normal ordered) Hamiltonian becomes\n\n\\[ \n\\widehat{H} = \\int {\\mathrm{d}}^{3}p\\mathop{\\sum }\\limits_{{s = 1}}^{2}{E}_{\\mathbf{p}}\\left( {{\\widehat{a}}_{s\\mathbf{p}}^{ \\dagger }{\\widehat{a}}_{s\\mathbf{p}} + {\\widehat{b}}_{s\\mathbf{p}}^{ \\dagger }{\\widehat{b}}_{s\\mathbf{p}}}\\right) .\n\\] \n\n(... | Yes |
Tetravalent plutonium hydroxide is dissolved in concentrated hydrochloric acid and diluted to \( {0.5}\mathrm{\;N} \) . HCl. Concentrated aqueous hydrofluoric acid is then added until the solution is \( 1\mathrm{\;N} \) . with respect to HF. The resulting precipitate is separated from the supernatant solution and dried... | \[ {a}_{1} = {12.63} \pm {0.05}\mathrm{\;A}\text{.} \] \[ {a}_{2} = {11.01} \pm {0.05}\mathrm{\;A}\text{.} \] \[ {a}_{3} = {6.98} \pm {0.05}\mathrm{\;A}\text{.} \] The calculated density is 4.89 . | Yes |
The procedure of Example 33 is modified by saturating the hydrochloric acid solution with sulfur dioxide prior to incorporating the hydrofluoric acid. The resulting precipitate is plutonium trifluoride, which is separated from the supernatant solution and dried at \( {300}^{ \circ }\mathrm{C} \) . in a stream of hydrog... | \[ {a}_{1} = {4.087} \pm {0.001}\mathrm{\;A}\text{.} \]\n\[ {a}_{3} = {7.240} \pm {0.001}\mathrm{\;A}\text{.} \]\nThe calculated density is 9.32 . | Yes |
Proposition 1.4 If \( u \in C\\left( \\bar{\\Omega }\\right) \) satisfies the mean value property in \( \\Omega \), then \( u \) assumes its maximum and minimum only on \( \\partial \\Omega \) unless \( u \) is constant. | PROOF: We only prove for the maximum. Set\n\n\[ \n\\sum = \\{ x \\in \\Omega : u\\left( x\\right) = M \\equiv \\mathop{\\max }\\limits_{\\bar{\\Omega }}u\\} \\subset \\Omega .\n\]\n\nIt is obvious that \( \\sum \) is relatively closed. Next we show that \( \\sum \) is open. For any \( {x}_{0} \\in \\sum \), take \( {\\... | Yes |
Proposition 1.13 Suppose \( u \in C\left( {\bar{B}}_{R}\right) \) is harmonic in \( {B}_{R} = {B}_{R}\left( {x}_{0}\right) \) . Then there holds for any multi-index \( \alpha \) with \( \left| \alpha \right| = m \)\n\n\[ \left| {{D}^{\alpha }u\left( {x}_{0}\right) }\right| \leq \frac{{n}^{m}{e}^{m - 1}m!}{{R}^{m}}\math... | Proof: We prove by induction. It is true for \( m = 1 \) by Lemma 1.10. Assume it holds for \( m \) . Consider \( m + 1 \) . For \( 0 < \theta < 1 \), define \( r = \left( {1 - \theta }\right) R \in \left( {0, R}\right) \) . We apply Lemma 1.10 to \( u \) in \( {B}_{r} \) and get\n\n\[ \left| {{D}^{m + 1}u\left( {x}_{0... | Yes |
Proposition 1.21 There holds for \( x, y \in \Omega \) with \( x \neq y \)\n\n\[ 0 > G\left( {x, y}\right) > \Gamma \left( {x, y}\right) \;\text{ for }n \geq 3 \]\n\n\[ 0 > G\left( {x, y}\right) > \Gamma \left( {x, y}\right) - \frac{1}{2\pi }\log \operatorname{diam}\left( \Omega \right) \;\text{ for }n = 2. \] | Proof: Fix \( x \in \Omega \) and write \( G\left( y\right) = G\left( {x, y}\right) \) . Since \( \mathop{\lim }\limits_{{y \rightarrow x}}G\left( y\right) = - \infty \) then there exists an \( r > 0 \) such that \( G\left( y\right) < 0 \) in \( {B}_{r}\left( x\right) \) . Note that \( G \) is harmonic in \( \Omega \sm... | Yes |
Proposition 1.22 The Green’s function for the ball \( {B}_{R}\left( 0\right) \) is given by\n\n(i)\n\n\[ G\left( {x, y}\right) = \frac{1}{\left( {2 - n}\right) {\omega }_{n}}\left( {{\left| x - y\right| }^{2 - n} - {\left| \frac{R}{\left| x\right| }x - \frac{\left| x\right| }{R}y\right| }^{2 - n}}\right) \;\text{ for }... | Proof: Fix \( x \neq 0 \) with \( \left| x\right| < R \) . Consider \( X \in {\mathbb{R}}^{n} \smallsetminus {\bar{B}}_{R} \) with \( X \) the multiple of \( x \) and \( \left| X\right| \cdot \left| x\right| = {R}^{2} \), that is, \( X = \frac{{R}^{2}}{{\left| x\right| }^{2}}x \) . In other words, \( X \) and \( x \) a... | Yes |
Proposition 1.31 Suppose \( u \) is harmonic in \( {B}_{1} \) . Then there holds\n\n\[ \mathop{\sup }\limits_{{B}_{1/2}}\left| {Du}\right| \leq c\mathop{\sup }\limits_{{\partial {B}_{1}}}\left| u\right| \]\n\nwhere \( c = c\left( n\right) \) is a positive constant. In particular, for any \( \alpha \in \left\lbrack {0,1... | Proof: Direct calculation shows that\n\n\[ \bigtriangleup \left( {\left| Du\right| }^{2}\right) = 2\mathop{\sum }\limits_{{i, j = 1}}^{n}{\left( {D}_{ij}u\right) }^{2} + 2\mathop{\sum }\limits_{{i = 1}}^{n}{D}_{i}u{D}_{i}\left( {\bigtriangleup u}\right) = 2\mathop{\sum }\limits_{{i, j = 1}}^{n}{\left( {D}_{ij}u\right) ... | Yes |
Proposition 1.34 Suppose \( u \in C\left( {\bar{B}}_{1}\right) \) is a harmonic function in \( {B}_{1} = {B}_{1}\left( 0\right) \) . If \( u\left( x\right) < u\left( {x}_{0}\right) \) for any \( x \in {\bar{B}}_{1} \) and some \( {x}_{0} \in \partial {B}_{1} \), then there holds\n\n\[ \frac{\partial u}{\partial n}\left... | Proof: Consider a positive function \( v \) in \( {B}_{1} \) defined by\n\n\[ v\left( x\right) = {e}^{-\alpha {\left| x\right| }^{2}} - {e}^{-\alpha }.\]\n\nIt is easy to see\n\n\[ \bigtriangleup v\left( x\right) = {e}^{-\alpha {\left| x\right| }^{2}}\left( {-{2\alpha n} + 4{\alpha }^{2}{\left| x\right| }^{2}}\right) >... | Yes |
Proposition 2.13 Let \( d \) be a positive number and \( \mathbf{e} \) be a unit vector such that \( \left| {\left( {y - x}\right) \cdot \mathbf{e}}\right| < d \) for any \( x, y \in \Omega \) . Then there exists a \( {d}_{0} > 0 \), depending only on \( \lambda \) and the sup-norm of \( {b}_{i} \) and \( {c}^{ + } \),... | Proof: By choosing \( \mathbf{e} = \left( {1,0,\ldots ,0}\right) \) we suppose \( \bar{\Omega } \) lies in \( \left\{ {0 < {x}_{1} < d}\right\} \) . Assume in addition \( \left| {b}_{i}\right| ,{c}^{ + } \leq N \) for some positive constant \( N \) . We construct \( w \) as follows. Set \( w = {e}^{\alpha d} - {e}^{\al... | Yes |
Proposition 2.15 Suppose \( u \in {C}^{2}\left( \Omega \right) \cap C\left( \bar{\Omega }\right) \) satisfies\n\n\[ \left\{ \begin{array}{ll} {Lu} = f & \text{ in }\Omega , \\ u = \varphi & \text{ on }\partial \Omega , \end{array}\right. \]\n\nfor some \( f \in C\left( \bar{\Omega }\right) \) and \( \varphi \in C\left(... | Proof: We will construct a function \( w \) in \( \Omega \) such that\n\n(i)\n\n\[ L\left( {w \pm u}\right) = {Lw} \pm f \leq 0\;\text{ or }\;{Lw} \leq \mp f\;\text{ in }\Omega ,\]\n\n(ii)\n\n\[ w \pm u = w \pm \varphi \geq 0\;\text{ or }\;w \geq \mp \varphi \;\text{ on }\partial \Omega . \]\n\nDenote \( F = \mathop{\m... | Yes |
Proposition 2.16 Suppose \( u \in {C}^{2}\left( \Omega \right) \cap {C}^{1}\left( \bar{\Omega }\right) \) satisfies \[ \left\{ \begin{array}{ll} {Lu} = f & \text{ in }\Omega , \\ \frac{\partial u}{\partial n} + \alpha \left( x\right) u = \varphi & \text{ on }\partial \Omega , \end{array}\right. \] where \( \mathbf{n} \... | Proof: We prove for a special case and the general case.\n\nCASE 1. Special case: \( c\left( x\right) \leq - {c}_{0} < 0 \) .\n\nWe will show \[ \left| {u\left( x\right) }\right| \leq \frac{1}{{c}_{0}}F + \frac{1}{{\alpha }_{0}}\Phi \;\text{ for any }x \in \Omega . \] Define \( v = \frac{1}{{c}_{0}}F + \frac{1}{{\alpha... | Yes |
Proposition 2.19 Suppose \( u \in {C}^{3}\left( \Omega \right) \) satisfies\n\n\[ \n{a}_{ij}\left( x\right) {D}_{ij}u + {b}_{i}\left( x\right) {D}_{i}u = f\left( {x, u}\right) \;\text{ in }\Omega \n\]\n\nfor \( {a}_{ij},{b}_{i} \in {C}^{1}\left( \bar{\Omega }\right) \) and \( f \in {C}^{1}\left( {\bar{\Omega } \times \... | Proof: We need to take a cutoff function \( \gamma \in {C}_{0}^{\infty }\left( \Omega \right) \) with \( \gamma \geq 0 \) and consider the auxiliary function with the following form:\n\n\[ \nw = \gamma {\left| Du\right| }^{2} + \alpha {\left| u\right| }^{2} + {e}^{\beta {x}_{1}}.\n\]\n\nSet \( v = \gamma {\left| Du\rig... | Yes |
Proposition 2.27 Suppose that \( u \in C\\left( \\bar{\Omega }\\right) \\cap {C}^{2}\\left( \\Omega \\right) \) satisfies\n\n\[ \n{Qu} \\equiv {a}_{ij}\\left( {x, u,{Du}}\\right) {D}_{ij}u + b\\left( {x, u,{Du}}\\right) = 0\\;\\text{ in }\\Omega \n\] \n\nwhere \( {a}_{ij} \\in C\\left( {\\Omega \\times \\mathbb{R} \\ti... | Proof of Proposition 2.27: We prove for subsolutions. Assume \( {Qu} \\geq 0 \) in \( \\Omega \) . Then we have\n\n\[ \n- {a}_{ij}{D}_{ij}u \\leq b\\;\\text{ in }\\Omega . \n\] \n\nNote that \( \\left\\{ {{D}_{ij}u}\\right\\} \) is nonpositive in \( {\\Gamma }^{ + } \) . Hence \( - {a}_{ij}{D}_{ij}u \\geq 0 \), which i... | No |
Corollary 3.11 (Comparison with Harmonic Functions) Suppose \( w \) is as in Lemma 3.10. Then for any \( u \in {H}^{1}\left( {{B}_{r}\left( {x}_{0}\right) }\right) \) there hold for any \( 0 < \rho \leq r \)\n\n\[ \n{\int }_{{B}_{\rho }\left( {x}_{0}\right) }{\left| Du\right| }^{2} \leq c\left\{ {{\left( \frac{\rho }{r... | Proof: We prove this by direct computation. In fact, with \( v = u - w \) we have for any \( 0 < \rho \leq r \)\n\n\[ \n{\int }_{{B}_{\rho }\left( {x}_{0}\right) }{\left| Du\right| }^{2} \leq 2{\int }_{{B}_{\rho }\left( {x}_{0}\right) }{\left| Dw\right| }^{2} + 2{\int }_{{B}_{\rho }\left( {x}_{0}\right) }{\left| Dv\rig... | Yes |
Theorem 1 (Cauchy’s Theorem) Let \( f \) be holomorphic on a closed disc \( \bar{D}\left( {{z}_{0}, R}\right), R > 0 \) . Let \( {C}_{R} \) be the circle bounding the disc. Then \( f \) has a power series expansion | \[ f\left( z\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }\frac{{\left( z - {z}_{0}\right) }^{n}}{{2\pi }\mathrm{i}}{\int }_{{C}_{R}}\frac{f\left( \zeta \right) }{{\left( \zeta - {z}_{0}\right) }^{n + 1}}\mathrm{\;d}\zeta . \] | Yes |
Theorem 1. 直角三角形斜边的平方和等于两腰的平方和。\n\n可以用符号语言表述为：设直角三角形 \( {ABC} \) ，其中 \( \angle C = {90}^{ \circ } \) ，则有\n\n\( A{B}^{2} = B{C}^{2} + A{C}^{2}\;\left( {1,\text{ eq:tri }}\right) \) | 满足式 公式引用: eq:tri 的整数称为勾股数。 | No |
Let \( a, b, c \) be positive real numbers. Prove that\n\n\[ \frac{a}{b} + \frac{b}{c} + \frac{c}{a} \geq \frac{a + c}{a + b} + \frac{b + c}{b + a} + \frac{c + a}{c + b}. \]\n\nMathlinks Contests | Solution. Without loss of generality, assume that \( c = \min \left( {a, b, c}\right) \) . Note that for \( x, y, z > 0 \) we have\n\n\[ \frac{x}{y} + \frac{y}{z} + \frac{z}{x} - 3 = \frac{1}{xy}{\left( x - y\right) }^{2} + \frac{1}{xz}\left( {x - z}\right) \left( {y - z}\right) .\n\nTherefore the given inequality can ... | Yes |
Let \( a, b, c \) be positive real numbers such that \( a \geq b \geq c \) . Prove that\n\n\[ \n{a}^{2}b\left( {a - b}\right) + {b}^{2}c\left( {b - c}\right) + {c}^{2}a\left( {c - a}\right) \geq 0.\n\] | Solution. We have\n\n\[ \nf\left( {a, b, c}\right) = {a}^{2}b\left( {a - b}\right) + {b}^{2}c\left( {b - c}\right) + {c}^{2}a\left( {c - a}\right) \n\]\n\n\[ \n= \left\lbrack {{a}^{2}b\left( {a - b}\right) - a{b}^{2}\left( {a - b}\right) }\right\rbrack + \left\lbrack {{b}^{2}c\left( {b - c}\right) - a{b}^{2}\left( {b -... | Yes |
Example 3. Let \( a, b, c \) be positive real numbers. Prove that\n\n\[ \n\frac{a + b}{b + c} + \frac{b + c}{c + a} + \frac{c + a}{a + b} + \frac{3\left( {{ab} + {bc} + {ca}}\right) }{{\left( a + b + c\right) }^{2}} \geq 4.\n\] | Solution. Without loss of generality, assume that \( c = \min \left( {a, b, c}\right) \) . We have\n\n\[ \n\frac{a + b}{b + c} + \frac{b + c}{c + a} + \frac{c + a}{a + b} - 3 = \frac{1}{\left( {a + c}\right) \left( {b + c}\right) }{\left( a - b\right) }^{2} + \frac{1}{\left( {a + b}\right) \left( {b + c}\right) }\left(... | Yes |
Theorem 1.1. Let \( Y \) be a \( {C}^{1} \) solution of (1.1) for \( a < t < b \) . Assume that \( \left| {Y\left( t\right) }\right| \leq M \) for \( t \) close to \( b, t < b \) . Then \( Y \) can be continued as a \( {C}^{1} \) solution of (1.1) on an interval \( a < t < b + \varepsilon \) for some \( \varepsilon > 0... | Proof of Theorem 1.1. If \( Y \) is bounded, so is \( F\left( {t, Y}\right) \), hence \( {Y}^{\prime } \) . Thus \( Y\left( t\right) \) has a limit \( Y\left( b\right) \) when \( t \rightarrow b \) ; by the equation, \( {Y}^{\prime } \) has also a limit, and \( Y \) is of class \( {C}^{1} \) for \( a < t \leq b \) . By... | Yes |
Theorem 3.1. Consider the wave equation\n\n\[ \left( {{\partial }_{t}^{2} - {\Delta }_{x}}\right) u = \varepsilon \frac{2\left( {m + 1}\right) }{{\left( m - 1\right) }^{2}}{u}^{m} + \mathop{\sum }\limits_{{-\infty }}^{{m - 1}}{a}_{j}\left( {x, t}\right) {u}^{j},\;\varepsilon = \pm 1 \]\n\nwhere \( m \) is an integer, \... | Proof of Theorem 3.1.\n\na. Set \( \tau = {\left( t - \psi \right) }^{\frac{1}{\left( m - 1\right) }} \) and look for \( u \) of the form \( u = {\tau }^{-2}v\left( {x,\tau }\right) \) . By substitution in (3.1), we find an equation for \( v \) of the form\n\n\[ \left( {3.3}\right) \left( {1 - {\left| \nabla \psi \righ... | Yes |
Proposition 4.1. At the origin, the (scalar) principal symbol \( {\sigma }_{b} \) of the linearized operator of \( {L}_{b} \) on a solution \( \left( {\phi, v}\right) \) is\n\n(4.1)\n\n\[ \n{\sigma }_{b}\left( {0,\zeta }\right) = C{\zeta }_{\kappa }^{N - 1}{\left( \mathop{\sum }\limits_{{j \neq \kappa }}{\partial }_{{\... | Proof.\n\na. The last statement follows from the identity \( \sum {\xi }_{j}{\partial }_{{\xi }_{j}}\lambda = 0,{\partial }_{\xi }\lambda \neq 0 \) and \( {\xi }_{\kappa }^{0} \neq 0 \) .\n\nb. By inspection of (2.5), we have\n\n\[ \n{\sigma }_{b} = \left( {\sum {\partial }_{{\xi }_{j}}\lambda {\zeta }_{j}}\right) \det... | Yes |
Theorem 4.2. Assume that the coefficients \( {A}_{j} \) and \( B \) of \( L \) are analytic near \( \left( {{x}^{0},{u}^{0}}\right) \) . We can choose \( {\partial }_{X}v\left( 0\right) \) such that (2.5) is satisfied at \( X = 0 \), by taking \( {\partial }_{{X}_{\kappa }}v\left( 0\right) \) colinear to \( r \) and th... | The proof of the theorem is obvious, except for the last three technical statements, for which we refer to [Al8]. | No |
Theorem 5.1. Let \( u \) be a blowup solution of \( L \) corresponding to a solution \( \left( {\phi, v}\right) \) of \( {L}_{b} \) for which \( {\partial }_{\kappa }v\left( 0\right) \neq 0 \) . Then, for \( x \in D \) , \[ {\partial }_{x}u\left( x\right) = C\left( x\right) {\left( {\partial }_{\kappa }\phi \right) }^{... | Proof of Theorem 5.1. Let \( \kappa = 1 \) for simplicity. The function \( v \) being a solution of (2.5), we have \( {\partial }_{1}v = {\alpha r} + {\partial }_{1}{\phi w} \) for some \( w \) and \( \alpha \neq 0 \) . Moreover, we have the relations \[ {\partial }_{1}u = {\left( {\partial }_{1}\phi \right) }^{-1}{\pa... | No |
Consider the solution \( u \) of (1.1) with \( f\left( u\right) = {u}^{2} \) and initial value \( {u}_{0} \in {C}_{0}^{2} \). Assume that \( {u}_{0} \) reaches its maximum at a point \( {X}^{0} \) where \( {u}_{0} > 0,{u}_{0}^{\prime \prime } \neq 0 \). Then, if \( u \) exists for \( t < {T}_{0}, u \) remains bounded f... | If we define \( \phi \left( {X, T}\right) \) for \( t < {T}_{0} \) to be the abscissa of the point of ordinate \( T \) on the integral curve of \( L \) starting from \( \left( {X,0}\right) \) and set \( v\left( {X, T}\right) = u\left( {\phi \left( {X, T}\right), T}\right) \), then \( \left( {\phi, v}\right) \) is a sol... | Yes |
Consider the solution \( u \) of (1.1) with \( f\left( u\right) = {u}^{4} \) and initial value \( {u}_{0} \in {C}_{0}^{2} \) . We can choose \( {u}_{0} \) such that \( u \) exists for \( t < {T}_{0} \) and \( \max \left| {u\left( {., t}\right) }\right| \rightarrow \infty \) when \( t \rightarrow {T}_{0}. \) | Proof of Proposition 2.1. We have here\n\n\[ \nv\left( {X, T}\right) = {u}_{0}\left( {X, T}\right) {\left( 1 - 3T{u}_{0}^{3}\left( X, T\right) \right) }^{-\frac{1}{3}}, \]\n\n\[ {\partial }_{X}\phi \left( {X, T}\right) = 1 + {u}_{0}^{\prime }{u}_{0}^{-3}\left\lbrack {{\left( 1 - 3T{u}_{0}^{3}\right) }^{-\frac{1}{3}} - ... | Yes |
Theorem 1.1. Consider a quasilinear system (1.1), which is assumed to be symmetrizable hyperbolic. For some integer \( s > \frac{n}{2} + 1 \), assume that\n\n(1.3)\n\n\[ \n{u}_{0} \in {H}^{s},{u}_{0}\left( x\right) \in {G}_{0} \]\n\nwhere \( {G}_{0} \) a relatively compact subset of \( G \) . Then, for any compact neig... | For symmetric hyperbolic systems, we refer to [Ma] for a proof; for strictly hyperbolic systems or equations, the corresponding statement is also true, and we refer to [Me]. | No |
Theorem 2.3. Consider \( u \) a \( {C}^{1} \) solution of (1.1) for \( 0 \leq t < T < \infty \) with initial value \( {u}_{0} \in {H}^{s} \) (s integer, \( s > \frac{n}{2} + 1 \) ). Assume that there exist a constant \( M \) and a relatively compact subset \( {G}_{1} \) of \( G \) such that, for \( 0 \leq t < T \) ,\n\... | Proof of Theorem 2.3.\n\na. Let \( {T}_{s} \) be the lifespan corresponding to \( {u}_{0} \) . For \( t < {T}_{s} \) and \( \left| \alpha \right| \leq s \) , we consider \( {u}_{\alpha } = {\partial }_{x}^{\alpha }u \), which is a solution of \( \left( {{\partial }_{t} + \sum {A}_{j}\left( u\right) {\partial }_{j}}\rig... | Yes |
Theorem 3.1. Let \( u \) be a solution of (3.1) with initial data\n\n\[ u\left( {x,0}\right) = f\left( x\right) ,{\partial }_{t}u\left( {x,0}\right) = g\left( x\right), f, g \in {C}_{0}^{\infty }\n\]\n\nsupported in \( \left| x\right| \leq R \) . Assume \( 1 < p < 1 + \sqrt{2} \) and \( \int {gdx} > 0 \) . Then \( \bar... | Proof of Theorem 3.1.\n\na. Set \( F\left( t\right) = \int u\left( {x, t}\right) {dx} \) ; using the equation and the fact that the support of \( u \) is contained in \( \left| x\right| \leq R + t \), we have \( {F}^{\prime \prime }\left( t\right) = \int {\left| u\right| }^{p}{dx} \) ; hence, by the Hölder inequality a... | Yes |
Theorem 1.1. Consider a semilinear system (1.1). For some integer \( s > \frac{n}{2} \), assume that\n\n(1.2)\n\n\[{u}_{0} \in {H}^{s}\text{.}\]\n\nThen, for any \( M \geq {\left| {u}_{0}\right| }_{s} \), there exists \( T > 0 \), depending only on \( M \) and \( B \), and a unique \( u \) solution of (1.1) for \( 0 \l... | This theorem can be proved exactly as its homologue (Theorem 2.2) of Chapter I. | No |
Theorem 1.2. Consider \( u \) a \( {C}^{1} \) solution of (1.1) for \( 0 \leq t < T < \infty \) with initial value \( {u}_{0} \in {H}^{s}, s > \frac{n}{2} \) . Assume that there exists \( M \) such that, for \( 0 \leq t < T \) ,\n\n\[ \left| {u\left( {x, t}\right) }\right| \leq M \]\n\nThen\n\n\[ u \in {C}^{0}\left( {\... | The proof is a simple modification of that of Theorem 2.3, using the above Theorem 1.1. | No |
Theorem 3.1. Assume \( {u}_{0} \in \mathcal{W}\left( {\mathbb{R}}^{4}\right) \) . Then there exists a unique maximal influence domain \( {\Omega }_{\max } \) containing \( \{ t \leq 0\} \) and a unique solution \( u \in \mathcal{W}\left( {\Omega }_{\max }\right) \) of (3.2). If \( \left( {{x}^{0},{t}^{0}}\right) \in \p... | The proof will be divided into three steps.\n\nStep 1: Local existence and uniqueness in a cone\n\na. We prove first a uniqueness result.\n\nUniqueness lemma.\n\n(i) Let \( \Omega \) be an influence domain, \( {u}_{1} \in {L}^{\infty }\left( \Omega \right) ,{u}_{1} = 0 \) for \( t < 0 \) and \( {u}_{2} = E * {u}_{1} \)... | Yes |
Theorem 4.1. Suppose (4.2), (4.3) and (4.5). Then there exists a classical solution \( u \) of \( \left( {4.1}\right) \) in \( \Omega = \{ \left( {x, t}\right) ,\left| x\right| \leq R + T,0 \leq t < \phi \left( x\right) \} , \) satisfying\n\n(a)\n\n(i) \( 0 < \phi \left( x\right) < T \) for \( \left| x\right| \leq R,0 ... | Proof of Theorem 4.1.\n\nA. Construction of \( u,\phi ,\Omega \) .\n\na. As usual, the solution \( u \) is obtained by a fix point argument. Define \( {u}_{0} = 0 \) and \( {u}_{n + 1} \) by\n\n(4.7) \( \left( {{\partial }_{t}^{2} - \Delta }\right) {u}_{n + 1} = F\left( {u}_{n}\right) ,{u}_{n + 1}\left( {x,0}\right) = ... | Yes |
Theorem 3.1. Consider a \( 2 \times 2 \) system in diagonal form\n\n\[ \n{\partial }_{t}w + \Lambda \left( w\right) {\partial }_{x}w = 0 \n\] \n\nwith initial data \( {w}^{0} \in {C}_{0}^{\infty }\left\lbrack {a, b}\right\rbrack \) . Denote by \( \left\lbrack {{m}_{i},{M}_{i}}\right\rbrack \) the range of \( {w}_{i}^{0... | Proof of Theorem 3.1.\n\na. Assume \( \bar{T} = \infty \) . Then the values of \( {w}_{i} \) are always contained in the interval \( \left\lbrack {{m}_{i},{M}_{i}}\right\rbrack \) . Let \( {\Gamma }_{x}^{j} \) the \( j \) -characteristic issued from \( \left( {x,0}\right) \) . On \( {\Gamma }_{b}^{1} \), we have \( {w}... | Yes |
Theorem 3.2. Consider \( w \in {C}^{\infty } \) a solution satisfying the assumptions \( {H}_{1} \) and \( {H}_{2} \) . Let \( \left( {\phi, v}\right) \) be the corresponding solution of (3.1) in \( \Omega \) . Then \( \left( {\phi, v}\right) \) can be extended as a smooth solution \( \left( {\widetilde{\phi },\widetil... | Proof of Theorem 3.2.\n\n1. Let us denote by \( A = \left( {{X}_{A},{T}_{A}}\right) \) the point of image \( \alpha \) by the application\n\n\[ \left( {X, T}\right) \mapsto \left( {\phi \left( {X, T}\right), T}\right) \]\n\nand so on. Since \( {\partial }_{T}{v}_{1} = 0 \) and \( {v}_{1}\left( {X, - {T}_{0}}\right) = {... | Yes |
Theorem 4.1. Let \( M > 0 \) be some fixed number and assume that the solution \( u \) is smooth for \( 0 \leq t < T\left( \varepsilon \right) \), with \( {\varepsilon T}\left( \varepsilon \right) \leq M \) . Then there are constants \( {I}_{0},{S}_{0},{\delta }_{0},{\varepsilon }_{1} > 0 \) such that, for \( 0 \leq t ... | Proof of Theorem 4.1.\n\n1. We will argue as follows ( a procedure called \ | No |
Theorem 1.1. For all \( p \geq 1 \) and \( 1 \leq j \leq N \) we have\n\n\[ {u}_{j}^{\left( p\right) }\left( {x, t}\right) = \mathop{\sum }\limits_{{0 \leq q \leq p - 1}}{t}^{q}{v}_{jq}^{\left( p\right) }\left( {{\sigma }_{j}\left( {x, t}\right) }\right) + {r}_{j}^{\left( p\right) }\left( {x, t}\right) ,{\sigma }_{j}\l... | Proof of Theorem 1.1.\n\na. We have immediately\n\n\[ {u}_{j}^{\left( 1\right) } = {v}_{j0}^{\left( 1\right) }\left( {\sigma }_{j}\right) ,\;{v}_{j0}^{\left( 1\right) } = {\left( {u}_{0}^{\left( 1\right) }\right) }_{j},\;{r}_{j}^{\left( 1\right) } \equiv 0, \]\n\nwhich is (1.5) for \( p = 1 \).\n\nb. Assume the theorem... | No |
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