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Theorem 6.6 (Second Convexity Theorem). Let \( f \) be holomorphic in the strip \( a \leqq \sigma \leqq b \) . For each \( \sigma \) assume that \( f\left( {\sigma + {it}}\right) \) grows at most like a power of \( \left| t\right| \), and let \( \psi \left( \sigma \right) \) be the least number \( \geqq 0 \) for which\... | Proof. The corollary of the Phragmen-Lindelöf theorem shows that there is a uniform \( M \) such that \( f\left( {\sigma + {it}}\right) \ll {\left| t\right| }^{M} \) in the strip. Let \( {L}_{\epsilon }\left( s\right) \) be the formula for the straight line segment between \( \psi \left( a\right) + \epsilon \) and \( \... | Yes |
Lemma 1.1. Let \( \\left\\{ {\\alpha }_{n}\\right\\} \) be a sequence of complex numbers \( {\\alpha }_{n} \\neq 1 \) for all n. Suppose that\n\n\[ \n\\sum \\left| {\\alpha }_{n}\\right| \n\]\n\nconverges. Then\n\n\[ \n\\mathop{\\prod }\\limits_{{n = 1}}^{\\infty }\\left( {1 - {\\alpha }_{n}}\\right) \n\]\n\nconverges ... | Proof. For all but a finite number of \( n \), we have \( \\left| {\\alpha }_{n}\\right| < \\frac{1}{2} \), so\n\n\[ \n\\log \\left( {1 - {\\alpha }_{n}}\\right) \n\]\n\nis defined by the usual series, and for some constant \( \\mathrm{C} \),\n\n\[ \n\\left| {\\log \\left( {1 - {\\alpha }_{n}}\\right) }\\right| \\leqq ... | Yes |
Lemma 1.2. Let \( \\left\\{ {f}_{n}\\right\\} \) be a sequence of analytic functions on an open set U. Let \( {f}_{n}\\left( z\\right) = 1 + {h}_{n}\\left( z\\right) \), and assume that the series \[ \\sum {h}_{n}\\left( z\\right) \] converges uniformly and absolutely on \( U \) . Let \( K \) be a compact subset of \( ... | Proof. By covering \( K \) with a finite number of discs of sufficiently small radius, using the compactness, we may assume that \( K \) is a closed disc. Write \[ f\\left( z\\right) = \\mathop{\\prod }\\limits_{{n = 1}}^{{N - 1}}{f}_{n}\\left( z\\right) \\mathop{\\prod }\\limits_{{n = N}}^{\\infty }{f}_{n}\\left( z\\r... | Yes |
Theorem 2.1. Let \( f \) be an entire function without zeros. Then there exists an entire function \( h \) such that\n\n\[ f\left( z\right) = {e}^{h\left( z\right) } \] | Proof. Since \( \mathbf{C} \) is simply connected, this is merely a restatement of the result of Chapter III, \( §6 \) where we defined the logarithm \( \log f\left( z\right) \) for any function \( f \) which has no zeros. | No |
Lemma 2.2. If \( \left| z\right| \leqq 1/2 \), then\n\n\[ \left| {\log {E}_{n}\left( z\right) }\right| \leqq 2{\left| z\right| }^{n} \] | Proof.\n\n\[ \left| {\log {E}_{n}\left( z\right) }\right| \leqq \frac{{\left| z\right| }^{n}}{n}\mathop{\sum }\limits_{{k = 0}}^{\infty }\frac{1}{{2}^{k}} \leqq 2{\left| z\right| }^{n}. \] | Yes |
Given the sequences \( \left\{ {z}_{n}\right\} ,\left\{ {k}_{n}\right\} ,\left\{ {P}_{n}\right\} \) as above; if the series\n\n\[ \mathop{\sum }\limits_{{n = 1}}^{\infty }{\left( \frac{R}{\left| {z}_{n}\right| }\right) }^{{k}_{n}} \]\n\nconverges for all positive real \( R \) (which is the case if \( {k}_{n} = n \) ), ... | Fix \( R \) . Let \( N \) be such that\n\n\[ \left| {z}_{N}\right| \leqq {2R} < \left| {z}_{N + 1}\right| \]\n\nThen for \( \left| z\right| \leqq R \) and \( n > N \) we have \( \left| {z/{z}_{n}}\right| \leqq 1/2 \), and hence\n\n\[ \left| {\log {E}_{n}\left( {z,{z}_{n}}\right) }\right| \leqq 2{\left( \frac{R}{\left| ... | Yes |
Theorem 3.1. The above canonical product is an entire function of order \( \leqq \rho \) . | Proof. Let \( \epsilon > 0 \) be such that \( \rho + \epsilon < k \) and let \( \lambda = \rho + \epsilon \) . There exists a constant \( C \) such that\n\n\[ \left| {{E}_{k}\left( z\right) }\right| = \left| {\left( {1 - z}\right) {e}^{P\left( z\right) }}\right| \leqq {C}^{{\left| z\right| }^{\lambda }}.\n\]\n\nThis is... | Yes |
Theorem 3.2. Let \( f \) be an entire function of strict order \( \leqq \rho \) . Let \( {v}_{f}\left( R\right) \) be the number of zeros of \( f \) in the disc of radius \( R \) . Then\n\n\[ \n{v}_{f}\left( R\right) \ll {R}^{\rho }\n\] | Proof. Dividing \( f \) by a power of \( z \) if necessary, we may assume without loss of generality that \( f \) does not vanish at the origin. The estimate is an immediate consequence of Jensen's inequality, but corresponds to a coarse form of it, which can be proved ad hoc. Indeed, let \( {z}_{1},\ldots ,{z}_{n} \) ... | Yes |
Theorem 3.3. Let \( f \) have strict order \( \leqq \rho \), and let \( \left\{ {z}_{n}\right\} \) be the sequence of zeros \( \neq 0 \) of \( f \), repeated with their multiplicities, and ordered by increasing absolute value. For every \( \delta > 0 \) the series\n\n\[ \n\sum \frac{1}{{\left| {z}_{n}\right| }^{\rho + ... | Proof. We sum by parts with a positive integer \( R \rightarrow \infty \) :\n\n\[ \n\mathop{\sum }\limits_{{\left| {z}_{n}\right| \leqq R}}\frac{1}{{\left| {z}_{n}\right| }^{\rho + \delta }} \ll \mathop{\sum }\limits_{{r = 1}}^{R}\frac{v\left( {r + 1}\right) - v\left( r\right) }{{r}^{\rho + \delta }} \n\]\n\n\[ \n\leq ... | Yes |
Theorem 3.5 (Hadamard). Let \( f \) be an entire function of order \( \rho \), and let \( \left\{ {z}_{n}\right\} \) be the sequence of its zeros \( \neq 0 \) . Let \( k \) be the smallest integer \( > \rho \) . Let \( P = {P}_{k} \) . Then\n\n\[ f\left( z\right) = {e}^{h\left( z\right) }{z}^{m}\prod \left( {1 - \frac{... | Proof. The series \( \sum 1/{\left| {z}_{n}\right| }^{s} \) converges for \( s > \rho \) . Hence for every \( r \) sufficiently large, there exists \( R \) with \( r \leqq R \leqq {2r} \) such that for all \( n \) the circle of radius \( R \) does not intersect the disc of radius \( 1/{\left| {z}_{n}\right| }^{s} \) ce... | Yes |
Theorem 4.1 (Mittag-Leffler). Let \( \left\{ {z}_{n}\right\} \) be a sequence of distinct complex numbers such that \( \left| {z}_{n}\right| \rightarrow \infty \) . Let \( \left\{ {P}_{n}\right\} \) be polynomials without constant term. Then there exists a meromorphic function \( f \) whose only poles are at \( \left\{... | Proof. Since a principal part at 0 can always be added a posteriori, we assume without loss of generality that \( {z}_{n} \neq 0 \) for all \( n \) . We expand\n\n\[ {P}_{n}\left( \frac{1}{z - {z}_{n}}\right) \]\n\nin a power series of \( z/{z}_{n} \) at the origin. This power series is a linear combination of power se... | Yes |
Theorem 1.1. Let \( P \) be a fundamental parallelogram for \( L \), and assume that the elliptic function \( f \) has no poles on its boundary \( \partial P \) . Then the sum of the residues of \( f \) in \( P \) is 0 . | Proof. We have\n\n\[ \n{2\pi i}\sum \operatorname{Res}f = {\int }_{\partial P}f\left( z\right) {dz} = 0 \n\] \n\nthis last equality being valid because of the periodicity, so the integrals on opposite sides cancel each other (Fig. 2). | Yes |
Theorem 1.2. Let \( P \) be a fundamental parallelogram, and assume that the elliptic function \( f \) has no zero or pole on its boundary. Let \( \left\{ {a}_{i}\right\} \) be the singular points (zeros and poles) of \( f \) inside \( P \), and let \( f \) have order \( {m}_{i} \) at \( {a}_{i} \) . Then\n\n\[ \sum {m... | Proof. Observe that \( f \) elliptic implies that \( {f}^{\prime } \) and \( {f}^{\prime }/f \) are elliptic. We then obtain\n\n\[ 0 = {\int }_{\partial P}{f}^{\prime }/f\left( z\right) {dz} = {2\pi }\sqrt{-1}\sum \text{ Residues } = {2\pi }\sqrt{-1}\sum {m}_{i}, \] \n\nthus proving our assertion. | Yes |
Theorem 1.3. Hypotheses being as in Theorem 1.2, we have\n\n\\[ \sum {m}_{i}{a}_{i} \equiv 0\\;\\left( {\\;\\operatorname{mod}\\;L}\\right) \\] | Proof. This time, we take the integral\n\n\\[ {\\int }_{\\partial P}z\\frac{{f}^{\\prime }\\left( z\\right) }{f\\left( z\\right) }{dz} = {2\\pi }\\sqrt{-1}\\sum {m}_{i}{a}_{i} \\]\n\nbecause\n\n\\[ {\\operatorname{res}}_{{a}_{i}}z\\frac{{f}^{\\prime }\\left( z\\right) }{f\\left( z\\right) } = {m}_{i}{a}_{i} \\]\n\nOn t... | Yes |
Theorem 2.3. Let \( {g}_{2} = {g}_{2}\left( L\right) = {60}{s}_{4} \) and \( {g}_{3} = {g}_{3}\left( L\right) = {140}{s}_{6} \) . Then\n\n\[{\wp }^{\prime 2} = 4{\wp }^{3} - {g}_{2}\wp - {g}_{3}\] | Proof. We expand out the function\n\n\[ \varphi \left( z\right) = {\wp }^{\prime }{\left( z\right) }^{2} - 4\wp {\left( z\right) }^{3} + {g}_{2}\wp \left( z\right) + {g}_{3} \]\n\nat the origin, paying attention only to the polar term and the constant term. This is easily done, and one sees that there is enough cancell... | Yes |
Theorem 4.1. The function \( \sigma \) is a theta function, and in fact\n\n\[ \frac{\sigma \left( {z + \omega }\right) }{\sigma \left( z\right) } = \psi \left( \omega \right) {e}^{\eta \left( \omega \right) \left( {z + \omega /2}\right) } \]\n\nwhere\n\n\[ \psi \left( \omega \right) = 1\;\text{ if }\;\omega /2 \in L, \... | Proof. We have\n\n\[ \frac{d}{dz}\log \frac{\sigma \left( {z + \omega }\right) }{\sigma \left( z\right) } = \eta \left( \omega \right) \]\n\nHence\n\n\[ \log \frac{\sigma \left( {z + \omega }\right) }{\sigma \left( z\right) } = \eta \left( \omega \right) z + c\left( \omega \right) \]\n\nwhence exponentiating yields\n\n... | Yes |
Theorem 4.2. For any \( a \in \mathbf{C} \) not in \( L \), we have\n\n\[ \wp \left( z\right) - \wp \left( a\right) = - \frac{\sigma \left( {z + a}\right) \sigma \left( {z - a}\right) }{{\sigma }^{2}\left( z\right) {\sigma }^{2}\left( a\right) }.\] | Proof. The function \( \wp \left( z\right) - \wp \left( a\right) \) has zeros at \( a \) and \( - a \), and has a double pole at 0 . Hence\n\n\[ \wp \left( z\right) - \wp \left( a\right) = C\frac{\sigma \left( {z + a}\right) \sigma \left( {z - a}\right) }{{\sigma }^{2}\left( z\right) }\]\n\nfor some constant \( C \) . ... | Yes |
Lemma 1.1. Let \( I \) be an interval of real numbers, possibly infinite. Let \( U \) be an open set of complex numbers. Let \( f = f\left( {t, z}\right) \) be a continuous function on \( I \times U \) . Assume:\n\n(i) For each compact subset \( K \) of \( U \) the integral\n\n\[ \n{\int }_{I}f\left( {t, z}\right) {dt}... | Proof. Let \( \left\{ {I}_{n}\right\} \) be a sequence of finite closed intervals, increasing to \( I \) . Let \( D \) be a disc in the \( z \) -plane whose closure is contained in \( U \) . Let \( \gamma \) be the circle bounding \( D \) . Then for each \( z \) in \( D \) we have\n\n\[ \nf\left( {t, z}\right) = \frac{... | Yes |
Lemma 2.1.\n\n\\[ \n\\frac{1}{2} + {\\int }_{0}^{\\infty }\\frac{{P}_{1}\\left( t\\right) }{{\\left( 1 + t\\right) }^{2}}{dt} = \\gamma \n\\] | Proof. We apply Euler’s formula to the function \\( f\\left( x\\right) = 1/\\left( {1 + x}\\right) \\) . Then the formula gives\n\n\\[ \n1 + \\frac{1}{2} + \\cdots + \\frac{1}{n + 1} = \\log \\left( {n + 1}\\right) + \\frac{1}{2}\\left( {\\frac{1}{n + 1} + 1}\\right) + {\\int }_{0}^{n}{P}_{1}\\left( t\\right) \\frac{1}... | Yes |
Lemma 2.2. For \( z \) not on the negative real axis, we have\n\n\[ \n{\int }_{0}^{\infty }\frac{{P}_{1}\left( t\right) }{z + t}{dt} = {\int }_{0}^{\infty }\frac{{P}_{2}\left( t\right) }{{\left( z + t\right) }^{2}}{dt} \n\] | Proof. We write\n\n\[ \n{\int }_{0}^{\infty } = \mathop{\sum }\limits_{{n = 0}}^{\infty }{\int }_{n}^{n + 1} \n\]\n\nIntegrating by parts on each interval \( \left\lbrack {n, n + 1}\right\rbrack \) gives the identity of the lemma. The integral involving \( {P}_{2} \) is obviously absolutely convergent, and the differen... | No |
Lemma 2.3. \[ \mathop{\lim }\limits_{{y \rightarrow \infty }}{\int }_{0}^{\infty }\frac{{P}_{1}\left( t\right) }{{iy} + t}{dt} = 0 \] | Proof. The limit is clear from Lemma 2.2. | No |
Lemma 2.4.\n\n\\[ \n1 + {\\int }_{0}^{\infty }\\frac{{P}_{1}\\left( t\\right) }{1 + t}{dt} = \\frac{1}{2}\\log {2\\pi } \n\\] | Proof. From\n\n\\[ \n\\Gamma \\left( z\\right) \\Gamma \\left( {-z}\\right) = \\frac{-\\pi }{z \\cdot \\sin {\\pi z}} \n\\]\n\nwe get\n\n\\[ \n\\left| {\\Gamma \\left( {iy}\\right) }\\right| = \\sqrt{\\frac{2\\pi }{y\\left( {{e}^{\\pi y} - {e}^{-{\\pi y}}}\\right) }}.\n\\]\n\nFrom \\( \\left( *\\right) \\) we get\n\n\\... | Yes |
For \( \operatorname{Re}\left( s\right) > - 1 \) we have an analytic continuation of \( \zeta \left( {s, u}\right) \) given by\n\n\[ \zeta \left( {s, u}\right) = \frac{{u}^{1 - s}}{s - 1} + \frac{{u}^{-s}}{2} - s{\int }_{0}^{\infty }\frac{{P}_{1}\left( t\right) }{{\left( t + u\right) }^{s + 1}}{dt}. \] | Proof. First, for \( \operatorname{Re}\left( s\right) > 1 \), we apply Euler’s summation formula to the function\n\n\[ f\left( t\right) = \frac{1}{{\left( t + u\right) }^{s}} \]\n\nWe write down this summation formula with a finite number of terms \( \sum f\left( k\right) \), with \( 1 \leqq k \leqq n \), and then we l... | Yes |
Theorem 3.2.\n\n\[ \zeta \left( {s, u}\right) = \frac{1}{2} - u - \left( {\log D\left( u\right) }\right) s + O\left( {s}^{2}\right) . \] | Proof. We use the geometric series for \( 1/\left( {s - 1}\right) = - 1/\left( {1 - s}\right) \) and use \( {u}^{-s} = 1 - s\log u + O\left( {s}^{2}\right) \) . The integral on the right of the formula in Theorem 3.1 is holomorphic at \( s = 0 \), and its value at \( s = 0 \) is obtained by substituting \( s = 0 \) . T... | No |
\[ {\zeta }^{\prime }\left( 0\right) = - \frac{1}{2}\log {2\pi }\;\text{ and }\;\zeta \left( s\right) = \frac{1}{s - 1} + \gamma + O\left( {s - 1}\right) . | Proof. For the first expression, put \( u = 1 \) in Theorem 3.2. For the second expression, write \( s = s - 1 + 1 \) in front of the integral of Theorem 3.1. Then the constant term \( \gamma \) drops out by using Lemma 2.1. This concludes the proof. | No |
Corollary 3.4 (Lerch Formula).\n\n\[ \log D\left( u\right) = - {\zeta }^{\prime }\left( {0, u}\right) \] \nor completely in terms of the gamma function, \n\[ \log \Gamma \left( u\right) = {\zeta }^{\prime }\left( {0, u}\right) - {\zeta }^{\prime }\left( 0\right) \] | Proof. Immediate from the power series expansion of Theorem 3.2, and the value of \( {\zeta }^{\prime }\left( 0\right) \) in Corollary 3.3. | No |
Theorem 4.1.\n\n\[ \n{H}_{u}\left( s\right) = - \left( {{e}^{i\pi s} - {e}^{-{i\pi s}}}\right) \Gamma \left( s\right) \zeta \left( {s, u}\right) = - {2i}\sin \left( {\pi s}\right) \Gamma \left( s\right) \zeta \left( {s, u}\right) .\n\] | Proof. We change the variable, putting \( z = - w \) . Then writing \( G = {G}_{u} \) , \( F = {F}_{u} \), we find\n\n\[ \n{H}_{u}\left( s\right) = {e}^{-{i\pi s}}{\int }_{\infty }^{\epsilon }F\left( {-w}\right) {e}^{s\log w}\frac{dw}{w}\n\]\n\[ \n+ {\int }_{-{K}_{\epsilon }}F\left( {-w}\right) {e}^{s\log \left( {-w}\r... | No |
Lemma 4.2. If \( \operatorname{Re}\left( s\right) > 1 \), then\n\n\[{\int }_{-{K}_{\varepsilon }}G\left( w\right) {e}^{s\log \left( {-w}\right) }\frac{dw}{w} \rightarrow 0\;\text{ as }\;\epsilon \rightarrow 0.\] | Proof. The length of \( {K}_{\epsilon } \) and \( \left| {{dw}/w}\right| \) have a product which is bounded. But putting \( r = \left| z\right| \) and \( \sigma = \operatorname{Re}\left( s\right) \), we have\n\n\[{e}^{s\log z} \leq {e}^{\sigma \log r} = {r}^{\sigma }\]\n\nand\n\n\[G\left( z\right) \ll 1/r\;\text{ for }... | Yes |
Theorem 4.3. We have\n\n\\[ \n\\zeta \\left( {s, u}\\right) = - \\frac{1}{2\\pi i}\\Gamma \\left( {1 - s}\\right) {H}_{u}\\left( s\\right) \n\\]\n\nIn particular, if \\( n \\) is a positive integer, then\n\n\\[ \n\\zeta \\left( {1 - n, u}\\right) = - \\frac{1}{2\\pi i}\\Gamma \\left( n\\right) {H}_{u}\\left( {1 - n}\\r... | Proof. Observe that when \\( s = 1 - n \\) in the Hankel integral, then the integrand is a meromorphic function, and so the integrals from \\( - \\infty \\) to \\( - \\epsilon \\) and from \\( - \\epsilon \\) to \\( - \\infty \\) cancel, leaving only the integral over \\( {K}_{\\epsilon } \\) . We can then apply Cauchy... | Yes |
Theorem 4.4. For \( \operatorname{Re}\left( s\right) < 0 \) we have\n\n\[ \n{H}_{u}\left( s\right) = {\left( -2\pi \right) }^{s}\mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{e}^{2\pi iun}{e}^{{i\pi s}/2} - {e}^{-{2\pi iun}}{e}^{-{i\pi s}/2}}{{n}^{1 - s}} \n\]\n\n\[ \n= - {\left( 2\pi \right) }^{s}\mathop{\sum }\limit... | Proof. Let \( m \) be an integer \( \geqq 2 \), and let \( {D}_{m} \) be the path indicated on Fig. 4, consisting of the square and the portion of \( C \) inside the square, with the given orientation.\n\n\n\nFigure ... | Yes |
Theorem 4.5.\n\n\[ \zeta \left( s\right) = {\left( 2\pi \right) }^{s}\Gamma \left( {1 - s}\right) \frac{\sin \left( {{\pi s}/2}\right) }{\pi }\zeta \left( {1 - s}\right) . \] | Observe that the formula of Theorem 4.5 was derived at first when \( \operatorname{Re}\left( s\right) < 0 \) so that the series \( \sum 1/{n}^{1 - s} \) converges absolutely. However, we know from Theorem 4.1 that \( \zeta \left( s\right) \) is a meromorphic function of \( s \), so that quite independently of the serie... | No |
Theorem 4.6. Let \( \xi \left( s\right) = s\left( {s - 1}\right) {\pi }^{-s/2}\Gamma \left( {s/2}\right) \zeta \left( s\right) \) . Then \( \xi \) is an entire function satisfying the functional equation\n\n\[ \xi \left( s\right) = \xi \left( {1 - s}\right) \] | Proof. The term \( s\left( {s - 1}\right) \) remains unchanged under the transformation \( s \mapsto 1 - s \) . That the other factor \( {\pi }^{-s/2}\Gamma \left( {s/2}\right) \zeta \left( s\right) \) remains unchanged fol-\n\nlows at once from Theorem 4.5, using the formulas\n\n\[ \Gamma \left( s\right) \Gamma \left(... | No |
Theorem 1.1. The product\n\n\\[ \n\\mathop{\\prod }\\limits_{p}\\left( {1 - \\frac{1}{{p}^{s}}}\\right) \n\\]\n\nconverges absolutely for \\( \\operatorname{Re}\\left( s\\right) > 1 \\), and uniformly for \\( \\operatorname{Re}\\left( s\\right) \\geqq 1 + \\delta \\) with\n\n\\( \\delta > 0 \\), and we have\n\n\\[ \n\\... | Proof. The convergence of the product is an immediate consequence of the definition given in Chapter XIII, \\( §1 \\) and the same estimate which gave the convergence of the series for the zeta function above. In the same region \\( \\operatorname{Re}\\left( s\\right) \\geqq 1 + \\delta \\), we can use the geometric se... | Yes |
Theorem 1.2. The function\n\n\[ \zeta \left( s\right) - \frac{1}{s - 1} \]\n\nextends to a holomorphic function on the region \( \operatorname{Re}\left( s\right) > 0 \) . | Proof. For \( \operatorname{Re}\left( s\right) > 1 \), we have\n\n\[ \zeta \left( s\right) - \frac{1}{s - 1} = \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{1}{{n}^{s}} - {\int }_{1}^{\infty }\frac{1}{{x}^{s}}{dx} \]\n\n\[ = \mathop{\sum }\limits_{{n = 1}}^{\infty }{\int }_{n}^{n + 1}\left( {\frac{1}{{n}^{s}} - \frac{... | Yes |
Theorem 1.3. The function \( \Phi \) is meromorphic for \( \operatorname{Re}\left( s\right) > \frac{1}{2} \) . Furthermore, for \( \operatorname{Re}\left( s\right) \geqq 1 \), we have \( \zeta \left( s\right) \neq 0 \) and\n\n\[ \Phi \left( s\right) - \frac{1}{s - 1} \]\n\nhas no poles for \( \operatorname{Re}\left( s\... | Proof. For \( \operatorname{Re}\left( s\right) > 1 \), the Euler product shows that \( \zeta \left( s\right) \neq 0 \) . By Chapter XIII, Lemma 1.2, we get\n\n\[ - {\zeta }^{\prime }/\zeta \left( s\right) = \mathop{\sum }\limits_{p}\frac{\log p}{{p}^{s} - 1} \]\n\nUsing the geometric series we get the expansion\n\n\[ \... | Yes |
Proposition 1.4. For \( \operatorname{Re}\left( s\right) > 1 \) we have\n\n\[ \Phi \left( s\right) = s{\int }_{1}^{\infty }\frac{\varphi \left( x\right) }{{x}^{s + 1}}{dx} \] | Proof. To prove this, compute the integral on the right between successive prime numbers, where \( \varphi \) is constant. Then sum by parts. We leave the details as an exercise. | No |
Theorem 2.1 (Chebyshev). \( \varphi \left( x\right) = O\left( x\right) \) . | Proof. Let \( n \) be a positive integer. Then\n\n\[ \n{2}^{2n} = {\left( 1 + 1\right) }^{2n} = \mathop{\sum }\limits_{j}\left( \begin{matrix} {2n} \\ j \end{matrix}\right) \geqq \left( \begin{matrix} {2n} \\ n \end{matrix}\right) \geqq \mathop{\prod }\limits_{{n < p \leqq {2n}}}p = {e}^{\varphi \left( {2n}\right) - \v... | Yes |
Lemma 2.3. The integral\n\n\[ \n{\int }_{1}^{\infty }\frac{\varphi \left( x\right) - x}{{x}^{2}}{dx} \]\n\nconverges. | Proof. Let\n\n\[ \nf\left( t\right) = \varphi \left( {e}^{t}\right) {e}^{-t} - 1 = \frac{\varphi \left( {e}^{t}\right) - {e}^{t}}{{e}^{t}}. \]\n\nThen \( f \) is certainly piecewise continuous, and is bounded by Theorem 2.1. Making the substitution \( x = {e}^{t} \) in the desired integral, \( {dx} = {e}^{t}{dt} \), we... | Yes |
Theorem 2.4. We have \( \varphi \left( x\right) \sim x \) . | Proof. The assertion of the theorem is logically equivalent to the combination of the following two assertions:\n\nGiven \( \lambda > 1 \), the set of \( x \) such that \( \varphi \left( x\right) \geqq {\lambda x} \) is bounded;\n\nGiven \( 0 < \lambda < 1 \), the set of \( x \) such that \( \varphi \left( x\right) \le... | No |
Theorem 2.5 (Prime Number Theorem). We have\n\n\[ \pi \left( x\right) \sim \frac{x}{\log x}. \] | Proof. We have\n\n\[ \varphi \left( x\right) = \mathop{\sum }\limits_{{p \leqq x}}\log p \leqq \mathop{\sum }\limits_{{p \leqq x}}\log x = \pi \left( x\right) \log x \] \n\nand given \( \epsilon > 0 \) ,\n\n\[ \varphi \left( x\right) \geqq \mathop{\sum }\limits_{{{x}^{1 - \epsilon } \leqq p \leqq x}}\log p \geqq \matho... | Yes |
Let \( \left\{ {a}_{k}\right\} \left( {k = 0,1,\ldots }\right) \) be a monotone decreasing sequence of real numbers whose limit is 0 . Let \( \left\{ {b}_{k}\right\} \) be a sequence of numbers such that the partial sums \( {B}_{n} \) are bounded. Then \[ \mathop{\sum }\limits_{{k = 0}}^{\infty }{a}_{k}{b}_{k} \] conve... | Let \[ {S}_{n} = \mathop{\sum }\limits_{{k = 0}}^{n}{a}_{k}{b}_{k} \] be the partial sum of the series \( \sum {a}_{k}{b}_{k} \) . We have to estimate \( \left| {{S}_{n} - {S}_{m}}\right| \) for \( m, n \) large. Let \( \left| {B}_{n}\right| \leqq M \) for all \( \bar{n} \) . Then \[ \left| {{B}_{n} - {B}_{m}}\right| \... | Yes |
Proposition 1.3. Let \( \\left\\{ {a}_{k}\\right\\} \) be a sequence of non-negative real numbers, monotone decreasing (not necessarily to 0 ). Let \( \\left\\{ {b}_{k}\\right\\} \) be a sequence of complex numbers such that \( \\sum {b}_{k} \) converges. Then \( \\sum {a}_{k}{b}_{k} \) converges. | Proof. This proposition is a corollary of the preceding one, as follows. We let \( a = \\lim {a}_{k} \), and \( {a}_{k}^{\prime } = {a}_{k} - a \) . Then \( \\left\\{ {a}_{k}^{\prime }\\right\\} \) is a sequence which decreases monotonically to 0 . But\n\n\[ \n\\sum {a}_{k}{b}_{k} = \\sum \\left( {{a}_{k} - a}\\right) ... | Yes |
Theorem 1.5. Let \( \left\{ {a}_{n}\right\} \) be a sequence of complex numbers such that \( \sum {a}_{n} \) converges. Assume that the power series \( \sum {a}_{n}{z}^{n} \) has radius of convergence at least 1. Let \( f\left( x\right) = \sum {a}_{n}{x}^{n} \) for \( 0 \leqq x < 1 \) . Then\n\n\[ \mathop{\lim }\limits... | Proof. Let \( A = \mathop{\sum }\limits_{{k = 1}}^{\infty }{a}_{k},{A}_{n} = \mathop{\sum }\limits_{{k = 1}}^{n}{a}_{k} \) . Consider the partial sums\n\n\[ {s}_{n}\left( x\right) = \mathop{\sum }\limits_{{k = 1}}^{n}{a}_{k}{x}^{k} \]\n\nWe first prove that the sequence of partial sums \( \left\{ {{s}_{n}\left( x\right... | Yes |
Theorem 1.6. If the Dirichlet series \( \sum {a}_{n}/{n}^{s} \) converges for some \( s = {s}_{0} \) , then it converges for any \( s \) with \( \operatorname{Re}\left( s\right) > {\sigma }_{0} = \operatorname{Re}\left( {s}_{0}\right) \), uniformly on any compact subset of this region. | Proof. Write \( {n}^{s} = {n}^{{s}_{0}}{n}^{\left( s - {s}_{0}\right) } \), and sum the following series by parts:\n\n\[ \sum \frac{{a}_{n}}{{n}^{{s}_{0}}}\frac{1}{{n}^{\left( s - {s}_{0}\right) }} \]\n\nIf \( {P}_{n}\left( {s}_{0}\right) = \mathop{\sum }\limits_{{m = 1}}^{n}{a}_{m}/{m}^{{s}_{0}} \), then the tail ends... | Yes |
Theorem 2.1. Suppose that \( P\left( X\right) \) has \( d \) distinct roots \( {\alpha }_{1},\ldots ,{\alpha }_{d} \) . Then: all solutions of \( \left( *\right) \) are of the form\n\n\[ \n{a}_{n} = {b}_{1}{\alpha }_{1}^{n} + \cdots + {b}_{d}{\alpha }_{d}^{n} \n\]\n\nwith arbitrary numbers \( {b}_{1},\ldots ,{b}_{d} \)... | Proof. It will suffice to prove that the solutions \( \left( {\alpha }_{1}^{n}\right) ,\ldots ,\left( {\alpha }_{d}^{n}\right) \) are linearly independent, because then they form a basis for the space \( S \) of solutions.\n\nSuppose there is a relation of linear dependence, that is\n\n\[ \n{b}_{1}{\alpha }_{1}^{n} + \... | Yes |
Theorem 2.2. Assume that \( {\alpha }_{1},\ldots ,{\alpha }_{d} \) are distinct. Then \( F\left( T\right) \) is a rational function. | Proof. We have\n\n\[ F\left( T\right) = {b}_{1}\mathop{\sum }\limits_{{n = 0}}^{\infty }{\left( {\alpha }_{1}T\right) }^{n} + \cdots + {b}_{d}\mathop{\sum }\limits_{{n = 0}}^{\infty }{\left( {\alpha }_{d}T\right) }^{d} \]\n\n\[ = \frac{{b}_{1}}{1 - {\alpha }_{1}T} + \cdots + \frac{{b}_{d}}{1 - {\alpha }_{d}T}. \]\n\nTh... | Yes |
Theorem 4.1. Assuming \( \lambda \neq {\lambda }^{\prime } \) and \( c \neq 0 \), there are exactly two fixed points, namely \( w \) and \( {w}^{\prime } \) . | Proof. Directly from the definition, one verifies that \( M\left( w\right) = w \) and \( M\left( {w}^{\prime }\right) = {w}^{\prime } \), so \( w \) and \( {w}^{\prime } \) are fixed points. Conversely, the condition that \( z \) is a fixed point is expressible as a quadratic equation in \( z \), which has at most two ... | Yes |
Theorem 4.2. Assume \( \left| \lambda \right| < \left| {\lambda }^{\prime }\right| \) and \( c \neq 0 \) . Let \( z \in \mathbf{C} \) and \( z \neq w \) . Then\n\n\[ \mathop{\lim }\limits_{{k \rightarrow \infty }}{M}^{k}\left( z\right) = {w}^{\prime } \] | Proof. Let \( \alpha = \lambda /{\lambda }^{\prime } \) . Then \( \left| \alpha \right| < 1 \) . We have\n\n\[ {S}^{-1}{M}^{k}S\left( \begin{array}{l} z \\ 1 \end{array}\right) = \left( \begin{matrix} {\lambda }^{k}z \\ {\lambda }^{\prime k} \end{matrix}\right) \;\text{ so }\;{S}^{-1}{M}^{k}S\left( z\right) = {\alpha }... | Yes |
Theorem 6.1. Let \( U \) be simply connected (for instance a disc or a rectangle), and let \( F \) be a locally integrable vector field on \( U \) . Then \( F \) has a potential on \( U \) . | Proof. This comes directly from the homotopy form of Cauchy's theorem. The potential is defined by the integral\n\n\[\n g\left( X\right) = {\int }_{{P}_{0}}^{X}F\n\]\n\ntaken from a fixed point \( {P}_{0} \) in \( U \) to a variable point \( X \) . The integral may be taken along any continu curve in \( U \), because t... | Yes |
Theorem 6.2. Let \( U \) be a connected open set in \( {\mathbf{R}}^{2} \), and let \( \gamma \) be a closed chain in \( U \) . Then \( \gamma \) is homologous to a rectangular chain. If \( \gamma \) is homologous to 0 in \( U \), and \( F \) is a locally integrable vector field on \( U \) , then\n\n\[{\int }_{\gamma }... | Proof. Theorem 3.2 of Chapter IV applies verbatim to the present situation, and we know from Theorem 6.1 that the integral of \( F \) around a rectangle is 0 , so the theorem is proved. | No |
Let \( U \) be simply connected, and let \( {P}_{1},\ldots ,{P}_{n} \) be distinct points of \( U \) . Let \( {U}^{ * } \) be the open set obtained from \( U \) by deleting these points. Let \( F \) be a locally integrable vector field on \( {U}^{ * } \) . Let\n\n\[ \n{a}_{k} = \frac{1}{2\pi }{\int }_{{\gamma }_{k}}F \... | Proof. One verifies directly that\n\n\[ \n{\int }_{{\gamma }_{k}}{G}_{{P}_{j}} = \left\{ \begin{array}{ll} {2\pi } & \text{ if }k = j \\ 0 & \text{ if }k \neq j \end{array}\right.\n\]\n\nLet \( \gamma \) be a closed curve in \( {U}^{ * } \) . Then immediately from the definition of \( {\gamma }_{k} \) and Theorem 6.3, ... | Yes |
Theorem 7.1. Let \( f \) be a complex harmonic function on a connected open set \( U \) . Let \( S \) be the set of points \( z \in U \) such that \( \partial f/\partial \bar{z} = 0 \) . Suppose that \( S \) has a non-empty interior \( V \) . Then \( V = U \) . | Proof. From p. 92 Theorem 1.6, we know that an open subset of \( U \) which is closed in \( U \) is equal to \( U \) . Thus it suffices to show that \( V \) is closed in \( U \) . Let \( {z}_{0} \) be a point in \( \partial V \cap U \) . Let \( h = \partial f/\partial \bar{z} = u + {iv} \), with \( u \) , \( v \) real.... | Yes |
Theorem 7.2. Assume that \( {f}_{1},{f}_{2} \) satisfy the Cauchy-Riemann equations. Then\n\n\[ \n{f}^{\prime }\left( \gamma \right) {N}_{\gamma } = {N}_{f \circ \gamma } \n\] | Proof. On the one hand, writing vectors vertically, we have\n\n\[ \n{N}_{f \circ \gamma } = \left( \begin{matrix} {\left( {f}_{2} \circ \gamma \right) }^{\prime } \\ - {\left( {f}_{1} \circ \gamma \right) }^{\prime } \end{matrix}\right) = \left( \begin{matrix} {\partial }_{1}{f}_{2}\left( \gamma \right) {\gamma }_{1}^{... | Yes |
Proposition 1.1 \( {}^{\left\lbrack {30}\right\rbrack } \) If \( \gamma \left( {x, t}\right) : {\mathbb{R}}^{2} \rightarrow {\mathbb{R}}^{2} \) is a solution of (1.5), then the curvature \( \kappa \left( {x, t}\right) \) is a solution of the \( \mathrm{{mKdV}} \) equation | \[ {\kappa }_{t} = {\kappa }_{xxx} + \frac{3}{2}{\kappa }^{2}{\kappa }_{x} \] | Yes |
Lemma 1.3 Let \( \psi \) be a nontrivial additive character of \( {\mathbb{F}}_{q} \) and let \( \chi \) be a multiplicative character on \( {\mathbb{F}}_{q} \) of order \( d \) . For polynomials \( f, g \in {\mathbb{F}}_{q}\left\lbrack x\right\rbrack \), if one of the following conditions holds:\n\n(i) \( \deg \left( ... | Proof It can be derived from the complete version in Lemma 1.2 in the standard way. \( ▱ \) | No |
Lemma 1.2 \( {}^{\left\lbrack {11}\right\rbrack } \) If \( X \) is a poset, then the set \( {U}_{x} = X \smallsetminus \downarrow x = \{ z \in X : z \nleq x\} \) is a Scott open set. | It is easy to see that for any \( a \in {K}_{X}, \uparrow a = \{ x : a \leq x\} \) is Scott open. In fact, if \( A \) is any directed subset of \( X \) such that \( \sup \left( A\right) \in \uparrow a \), then \( a \leq \sup \left( A\right) \) . Since \( a \) is compact, then \( \uparrow a \cap A \neq \varnothing \) . ... | No |
Example 2.1 Let \( X = \{ 1,2,3,4\} \) with a topology \[ \tau = \{ \varnothing, X,\{ 1\} ,\{ 2\} ,\{ 1,2\} ,\{ 2,3\} ,\{ 1,4\} ,\{ 1,2,3\} ,\{ 1,2,4\} \} . \] | It is easy to see that \( \mathcal{B} = \{ \{ 1\} ,\{ 2\} ,\{ 1,4\} ,\{ 2,3\} \} \) is the minimal base for \( \tau \) and any element of \( \mathcal{B} \) is not a meet-reducible element, then \( \mathcal{B} \) is a minimal sub-base for \( \tau \) . However, \( \mathcal{B} \) is not the least sub-base. Let \( \varphi ... | Yes |
For \( A \in \mathcal{B} \) and let \( \chi \left( A\right) = \left( {{a}_{i1},\cdots ,{a}_{in}}\right) \), if there exists an index set \( \Delta \) of \( {M}_{\mathcal{B}} \) with \( i \notin \Delta \) such that \( {a}_{ij} = \left\lceil {\mathop{\sum }\limits_{{k \in \Delta }}\frac{{a}_{kj}}{\left| \Delta \right| }}... | Proof We only prove (1). Suppose that \( {B}_{k} \in \mathcal{B} \) corresponds to \( k \in \Delta \) . With hypothesis, \( {a}_{ij} = \left\lceil {\mathop{\sum }\limits_{{k \in \Delta }}\frac{{a}_{kj}}{\left| \Delta \right| }}\right\rceil \) for \( j = 1,\cdots, n \) means that \( x \in \mathop{\bigcup }\limits_{{k \i... | Yes |
Theorem 2.1 Let \( X = \left\{ {{x}_{1},\cdots ,{x}_{n}}\right\} \) be a finite Scott topological space with a base \( \mathcal{B} \) for \( \sigma \left( X\right) \) and \( {M}_{\mathcal{B}} \) be the induced matrix with respect to \( \mathcal{B} \) . For any row \( {\mathbf{r}}_{i} = \left\{ {{a}_{i1},\cdots ,{a}_{in... | Proof We only need to prove that every Scott open set of \( \mathcal{B} \) is not a joint-reducible element. Suppose that there exist \( A \in \mathcal{B} \) and \( \mathcal{F} \subseteq \mathcal{B} \smallsetminus \{ A\} \) such that \( A = \bigcup \mathcal{F} \) . \( \forall x \in A,\exists U \in \mathcal{F} \Rightarr... | Yes |
Theorem 2.2 Let \( X = \left\{ {{x}_{1},\cdots ,{x}_{n}}\right\} \) be a finite Scott topological space and \( \mathcal{B} \subset \sigma \left( X\right) \) . Take \( M \) and \( {M}_{\mathcal{B}} \) as the induced matrices with respect to \( \sigma \left( X\right) \) and \( \mathcal{B} \), respectively. For any \( \le... | Proof Suppose that \( U \) is a Scott open set with containing \( {x}_{j} \) . Denote \( \chi \left( U\right) = \left( {{a}_{1},\cdots ,{a}_{n}}\right) \) and so \( {a}_{j} = 1 \) . There exists \( k \in \Delta ,{\mathbf{r}}_{k} \in {M}_{\mathcal{B}} \) satisfying \( {a}_{kj} = 1 \) . Namely, there exists \( A \in \mat... | Yes |
Theorem 2.3 Let \( X = \left\{ {{x}_{1},\cdots ,{x}_{n}}\right\} \) be a finite Scott topological space with a sub-base \( \mathcal{S} \) for \( \sigma \left( X\right) \) and \( {M}_{\mathcal{S}} \) be the induced matrix with respect to \( \mathcal{S} \) . For any \( {\mathbf{r}}_{i} = \left( {{a}_{i1},\cdots ,{a}_{in}... | Proof Suppose that there are \( A \in \mathcal{S} \) and \( \mathcal{F} \subset \mathcal{S} \smallsetminus \{ A\} \) such that \( A = \bigcap \mathcal{F} \), then \( {A}^{c} = \) \( \mathop{\bigcup }\limits_{{U \in \mathcal{F}}}{U}^{c} \) . It is easy to verify that \( {b}_{j} = \left\lceil {\mathop{\sum }\limits_{{U \... | Yes |
Theorem 2.4 Let \( X = \left\{ {{x}_{1},\cdots ,{x}_{n}}\right\} \) be a finite Scott topological space and \( \mathcal{S} \subset \sigma \left( X\right) \) with the induced matrix \( {M}_{\mathcal{S}} = {\left( {a}_{ij}\right) }_{\left| \mathcal{S}\right| \times n} \) . If \( \left\lceil {\mathop{\sum }\limits_{{k = 1... | Proof Obviously, \( \mathop{\bigcup }\limits_{{A \in \mathcal{S}}}A = X \) if \( \left\lceil {\mathop{\sum }\limits_{{k = 1}}^{\left| \mathcal{S}\right| }\frac{{a}_{kj}}{\left| \mathcal{S}\right| }}\right\rceil = 1 \) for any \( j \leq n \), and so \( \mathcal{S} \) is a sub-base for \( \sigma \left( X\right) \) . | No |
Theorem 2.5 Let \( \left( {X,\tau }\right) \) be a finite topological space with the minimal base \( \mathcal{B} \) . Take \( {M}_{\mathcal{B}} \) to be the induced matrix with respect to \( \mathcal{B} \) . For any row vector \( {\mathbf{r}}_{i} \in {M}_{\mathcal{B}} \) with \( {a}_{ij} = 1 \) and \( \left| {\mathbf{r... | Proof Hypothetically, \( \left\lceil \frac{{\mathbf{r}}_{i} + {\mathbf{r}}_{k}}{2}\right\rceil = {\mathbf{r}}_{i} \) implies \( {U}_{{x}_{k}} \subset {U}_{{x}_{i}} \) since \( \mathcal{B} \) is the minimal base. Consequently, \( \uparrow {x}_{i} = \left\{ {{x}_{j} : {U}_{{x}_{j}} \subset {U}_{{x}_{i}}}\right\} = {U}_{{... | Yes |
Theorem 3.1 Let \( X \) be a finite Scott topological space with the minimal base \( \mathcal{B} \) and \( {\mathcal{F}}_{1} = \left\{ {A \in \sigma \left( X\right) : \chi \left( A\right) \in {R}_{1}}\right\} \) . If \( {\mathcal{F}}_{1} \neq \varnothing \) and \( X \smallsetminus \bigcup {\mathcal{F}}_{1} \neq \varnot... | Proof Obviously, \( X \smallsetminus \bigcup {\mathcal{F}}_{1} \) is a subspace of \( X \) and each member of \( {\mathcal{F}}_{1} \) is not the joint-reducible element, and so \( {\mathcal{F}}_{1} \subset \mathcal{B} \) . Next to prove that \( {\left. \mathcal{B}\right| }_{X \smallsetminus \cup {\mathcal{F}}_{1}} \) i... | Yes |
Theorem 3.2 Suppose that \( X = \left\{ {{x}_{1},\cdots ,{x}_{n}}\right\} \) is a finite Scott topological space with the minimal base \( \mathcal{B} \) . If \( {R}_{2} \neq \varnothing \) or \( R \neq \varnothing \), define\n\n\[ \n{\mathcal{F}}_{2} = \left\{ {A \in \sigma \left( X\right) : \chi \left( A\right) \in {R... | Proof For any \( A \in {\mathcal{F}}_{2} \), we are to prove \( A \in \mathcal{B} \) . Obviously, it holds if \( \chi \left( A\right) \in R \) . If \( \chi \left( A\right) \in {R}_{2} \), there exists \( \mathbf{r} \in M \) such that \( \mathbf{r} = \chi \left( A\right) \), then there is \( \mathbf{c} \in M \) satisfyi... | Yes |
Theorem 2.1 If \( {T}_{6, n} = {C}_{6}▱{C}_{n} \) with \( n \geq 6 \), then \( {\chi }_{i}\left( {T}_{6, n}\right) \leq 5 \), and the bound is sharp. | Proof By Corollary 1.1, we know that \( {\chi }_{i}\left( {T}_{6, n}\right) = {\chi }_{i}\left( {T}_{3 \times 2, n}\right) \leq 5 \) if \( n = 6 \) or \( n \geq 8 \) . The remaining case is \( n = 7 \) . The coloring pattern of \( {T}_{6,7} \) is given as follows:\n\n\[ \left( \begin{array}{lllllll} 3 & 2 & 0 & 0 & 1 &... | Yes |
Theorem 0.1 \( {J}_{\mu }\left( v\right) = I\left( v\right) \) for every \( v \equiv 0,1\left( {\;\operatorname{mod}\;8}\right) \) and \( v \geq 8 \) if and only if \( 2 \leq \mu \leq 4 \) . | ## 1 The Necessity of Theorem 0.1\n\nWe shall prove that \( {b}_{v} - 2 \notin {J}_{5}\left( v\right) \) for any \( v \equiv 0,1\left( {\;\operatorname{mod}\;8}\right) \) and \( v \geq 8 \) . This leads to the necessity of Theorem 0.1 straightforwardly.\n\nIf \( {b}_{v} - 2 \in {J}_{5}\left( v\right) \), then there wou... | No |
Lemma 2.1 There are only four possible clusters of 3-faces in \( G \), as depicted in Fig. 2. | \n\nFigure 2 All possible clusters in \( G \) | Yes |
Lemma 1.2 Let \( M \) be a torsion-free module of rank 1 over \( D \). (1) Let \( {M}_{1} \) be a submodule of \( M \) and \( b \) be any nonzero element of \( {M}_{1} \). If the height vectors of \( b \) in \( M \) and \( {M}_{1} \) are \( {\left( l\left( p\right) \right) }_{p} \) and \( {\left( k\left( p\right) \righ... | Proof (1) In view of the exact sequence \[ 0 \rightarrow {M}_{1}/{Db} \rightarrow M/{Db} \rightarrow M/{M}_{1} \rightarrow 0, \] it suffices to prove \( M/{Db} \cong {\bigoplus }_{p}{D}_{{p}^{l\left( p\right) }} \). Note that \( M/{Db} \lesssim Q\left( D\right) /D \) and \( Q\left( D\right) /D \cong {\bigoplus }_{p}{D}... | Yes |
Theorem 1.1 Two torsion-free modules of rank 1 over \( D \) are isomorphic if and only if they have the same type. Moreover every type is the type of some torsion-free module of rank 0 or 1. | Proof It suffices to consider the case that \( D \) contains an infinite number of distinct nonassociate prime elements. Suppose that \( {M}_{1} \) and \( {M}_{2} \) are two torsion-free modules of rank 1 over \( D \) with the same type. Let \( 0 \neq {m}_{1} \in {M}_{1} \) and \( 0 \neq {m}_{2} \in {M}_{2} \) . Then \... | Yes |
Theorem 1.2 Let \( M \) be a torsion-free module of rank 1 over \( D \) and \( \pi \) be the spectrum of \( M \), namely \( \pi = {S}_{p}\left( M\right) \), then\n\n\[{\operatorname{End}}_{D}\left( M\right) \cong Q{\left( D\right) }_{\pi },\;{\operatorname{Aut}}_{D}\left( M\right) \cong Q{\left( D\right) }_{\pi }^{ * }... | Proof Assume \( M \) contains 1 and lies in \( Q\left( D\right) \) . For any element \( a \) of \( M \), there always exist elements \( r, s \) in \( D \) such that \( a = \frac{r}{s} \cdot 1 \) . Let \( \alpha \) be an endomorphism of \( M \) . In view of \( \alpha \left( a\right) = \frac{r}{s} \cdot \alpha \left( 1\r... | Yes |
Theorem 2.1 (1) There exists an LCD \( m \) -th residue code with length \( p \) if and only if\n\n\[ p \equiv 1\;\left( {{\;\operatorname{mod}\;2}m}\right) \] | Proof Note that \( \theta \) is a \( p \) -th primitive root of unity, thus for \( j = 0,1,\cdots, m - 1 \), any root of \( {f}_{j}\left( x\right) \) is in the form \( \beta = {\alpha }^{t} \) for some \( t = 1,2,\cdots, p - 1 \) .\n\n(1) Suppose that \( \mathcal{C} = \left\langle {{f}_{j}\left( x\right) }\right\rangle... | Yes |
Example 2.1 Let \( m = 3 \) and \( n = 7 \) . Then \( n \equiv 1\left( {{\;\operatorname{mod}\;2}m}\right) \) . Suppose that \( q = 8 \) and \( {\theta }^{3} + \theta + 1 = 0 \), then \( \theta \) is primitive elements of \( {\mathbb{F}}_{8} \) and \( 7 \mid \left( {q - 1}\right) ,\theta \) is a fixed 7 th primitive ro... | First, it is easy to see that the set of 3rd residues modulo 7 is\n\n\[ \n{A}_{0} = \left\{ {x\left| {\;{x}^{\frac{7 - 1}{3}} = {x}^{2} \equiv 1\left( {\;\operatorname{mod}\;7}\right) }\right. ,1 \leq x \leq 6}\right\} = \{ 1,6\} , \n\] | No |
Lemma 1.1 \( {}^{\left\lbrack {14}\right\rbrack } \) Let \( i : X \hookrightarrow \mathfrak{M}\left( X\right) \) and \( j : \mathfrak{M}\left( X\right) \hookrightarrow \mathbf{k}\mathfrak{M}\left( X\right) \) be the natural embeddings. Then\n\n(a) the triple \( \left( {\mathfrak{M}\left( X\right), P \mathrel{\text{:=}}... | with \( \lfloor {xyz}\rfloor { \succ }_{\text{lex }}\lfloor x\rfloor { \succ }_{\text{lex }}\lfloor y\rfloor \) . For the word \( \lfloor \left( {x\left( {yz}\right) }\right) \rfloor \lfloor x\rfloor \lfloor y\rfloor \) on \( \{ \lfloor \left( {x\left( {yz}\right) }\right) \rfloor ,\lfloor x\rfloor ,\lfloor y\rfloor \}... | No |
Theorem 1.3 Let \( {S}^{ * } = S\left( {n;{m}_{1},{m}_{2},\cdots ,{m}_{n}}\right) \) be a sun-like graph.\n\n(1) If it is composed of some substructures \( {S}_{A}^{ * },{S}_{B}^{ * },{S}_{C}^{ * } \), one \( {S}_{D}^{ * } \) and one odd path, the center vertex \( {u}_{1} \) in odd path and the center vertex \( {u}_{n}... | Proof We only prove (1). The proof of (2) is similar.\n\nSuppose that sun-like graph \( {S}^{ * } \) is composed of an odd path, substructures \( {S}_{A}^{ * },{S}_{B}^{ * },{S}_{C}^{ * } \) and \( {S}_{D}^{ * } \) (in this order), then graph \( {S}^{ * }▱{P}_{m} \) has a Hamiltonian path \( {\mathrm{{HP}}}_{7} \) as f... | Yes |
Theorem 1.4 Let \( {S}^{ * } = S\left( {n;{m}_{1},{m}_{2},\cdots ,{m}_{n}}\right) \) be a sun-like graph composed of at least two substructures \( {S}_{E}^{ * } \) . If \( m \geq 3 \) is odd, then \( {G}^{ * } = {S}^{ * }▱{P}_{m} \) is not AP. | Proof To prove \( {G}^{ * } = {S}^{ * }▱{P}_{m} \) is not AP, we prove that \( {G}^{ * } \) is not \( \left( {2,2,\cdots ,2}\right) \) -partitionable or \( \left( {1,2,2,\cdots ,2}\right) \) -partitionable, that is, \( {G}^{ * } \) does not have perfect matching or almost perfect matching.\n\nSuppose that graph \( {S}^... | Yes |
Lemma 1.11 If a \( {5}_{3} \) -vertex \( v \) in \( {H}^{ * } \) is adjacent to two \( {3}_{1} \) -vertices, then every 2-neighbor of \( v \) is a rich 2-vertex. | Proof Suppose that a 5-vertex \( v \) is adjacent to three 2-vertices \( {v}_{1},{v}_{2},{v}_{3} \) and two \( {3}_{1} \) - vertices \( {v}_{4},{v}_{5} \) in \( {H}^{ * } \), where \( {v}_{1} \) is a poor 2-vertex or a semi-poor 2-vertex. Let \( {v}_{1}^{\prime } \) be the neighbor of \( {v}_{1} \) different from \( v ... | Yes |
Lemma 1.12 If a \( {3}_{1} \) -vertex \( v \) is adjacent to two 3-vertices in \( {H}^{ * } \), then all of its 3-neighbors are \( {3}_{0} \) -vertices. | Proof Suppose that a \( {3}_{1} \) -vertex \( v \) is adjacent to two 3-vertices \( {v}_{1},{v}_{2} \) in \( {H}^{ * } \) where \( {v}_{1} \) is a \( {3}_{1} \) -vertex (the case that \( {v}_{1} \) is a \( {3}_{2} \) -vertex is similar). Let \( {v}^{\prime } \) be the 2-neighbor of \( v \), and let \( {v}_{1}^{\prime }... | Yes |
Lemma 1.13 If a \( {3}_{0} \) -vertex \( v \) in \( {H}^{ * } \) is adjacent to three 3-vertices, then one of its 3- neighbors is not a \( {3}_{1} \) -vertex. | Proof Suppose that a 3-vertex \( v \) is adjacent to three \( {3}_{1} \) -vertices \( {v}_{1},{v}_{2},{v}_{3} \) in \( {H}^{ * } \) where \( {v}_{1},{v}_{2},{v}_{3} \) are \( {3}_{1} \) -vertices. Let \( {v}_{1}^{\prime } \) be the 2-neighbor of \( {v}_{1},{v}_{2}^{\prime } \) be the 2-neighbor of \( {v}_{2},{v}_{3}^{\... | Yes |
Lemma 2.1 Let \( \mathcal{A} \) be a family of subsets of \( X \) such that for each \( A \in \mathcal{A} \), there exists a decreasing sequence \( \{ O\left( {n, A}\right) : n \in \mathbb{N}\} \) of open subsets of \( X \) such that \( A = \mathop{\bigcap }\limits_{{n \in \mathbb{N}}}O\left( {n, A}\right) \) . For eac... | Proof (1) We show that \( {f}_{A} \in U\left( {X, P}\right) \) for each \( A \in \mathcal{A} \) . Let \( A \in \mathcal{A} \) . Since \( {a}_{1} \) is an upper bound of \( {f}_{A}\left( X\right) \) and \( P \) is upper-bounded complete, \( {\mathcal{N}}_{x}^{ * }\left( {f}_{A}\right) = {\mathcal{N}}_{x} \neq \varnothin... | Yes |
Theorem 2.2 \( X \) is a regular \( \gamma \) -space if and only if there exists a map \( f : {\mathcal{C}}_{X} \rightarrow U\left( {X, P}\right) \) satisfying \( \left( {a}_{{\mathcal{C}}_{X}}\right) ,\left( {f}_{{\mathcal{C}}_{X}}\right) \) and \( \left( {{f}_{{\mathcal{C}}_{X}}^{\prime }\left( {\tau }^{c}\right) }\r... | Proof Let \( g \) be a \( \gamma \) -function for \( X \) . Then \( K = \mathop{\bigcap }\limits_{{n \in \mathbb{N}}}g\left( {n, K}\right) \) for each \( K \in {\mathcal{C}}_{X} \) . For each \( K \in {\mathcal{C}}_{X} \) and \( x \notin K \), let \( {n}_{K}\left( x\right) = \min \{ n \in \mathbb{N} : x \notin g\left( ... | Yes |
Theorem 2.4 The following are equivalent.\n\n(a) \( X \) is stratifiable.\n\n(b) There exists a map \( f : {\tau }^{c} \rightarrow U\left( {X, P}\right) \) satisfying \( \left( {a}_{{\tau }^{c}}\right) ,\left( {f}_{{\tau }^{c}}\right) \) and \( \left( {{f}_{{\tau }^{c}}^{\prime }\left( {\mathcal{C}}_{X}\right) }\right)... | Proof (a) \( \Rightarrow \) (b): Let \( \rho \) be a stratifiable map which is decreasing with respect to \( n \) . For each \( F \in {\tau }^{c} \) and \( x \notin F \), let \( {n}_{F}\left( x\right) = \min \{ n \in \mathbb{N} : x \notin \rho \left( {n, F}\right) \} \) . For each \( F \in {\tau }^{c} \), define a map ... | Yes |
Theorem 2.6 The following are equivalent.\n\n(a) \( X \) is a Nagata-space.\n\n(b) There exists a map \( f : {\tau }^{c} \rightarrow U\left( {X, P}\right) \) satisfying \( \left( {a}_{{\tau }^{c}}\right) ,\left( {f}_{{\tau }^{c}}\right) ,\left( {{f}_{{\mathcal{S}}_{X}}\left( {\tau }^{c}\right) }\right) \) and \( \left(... | Proof (a) \( \Rightarrow \) (b): Let \( g \) be a Nagata function for \( X \) . It is easy to verify that \( \mathop{\bigcap }\limits_{{n \in \mathbb{N}}}\overline{g\left( {n, F}\right) } = \) \( F \) for each \( F \in {\tau }^{c} \) and that \( \{ g\left( {n, x}\right) : n \in \mathbb{N}\} \) is a neighborhood base of... | Yes |
Theorem 2.1 For any \( T \in \mathcal{T}\left( {\mathcal{L}\left( \mathcal{H}\right) }\right) \), the following conditions are equivalent:\n\n(1) Both \( {H}_{T} \) and \( {H}_{{T}^{ * }} \) are bounded on \( {F}_{\varphi }^{2}\left( {\mathcal{L}\left( \mathcal{H}\right) }\right) \) ;\n\n(2) \( \mathop{\sup }\limits_{{... | Proof The fact that (1) implies (2) comes from Lemma 2.1. According to the remark attached to Lemma 1.2, we can see that\n\n\[{\left\langle {\widehat{\left( {T}^{ * }T\right) }}_{r}\left( z\right) e, e\right\rangle }_{\mathcal{H}} \lesssim {\left\langle \widetilde{{T}^{ * }T}\left( z\right) e, e\right\rangle }_{\mathca... | Yes |
Theorem 3.1 \( T \in {\mathrm{{BMO}}}_{\varphi }^{2} \) belongs to \( {\mathrm{{VMO}}}_{r}^{2} \) if and only if\n\n\[ \mathop{\lim }\limits_{{R \rightarrow \infty }}{\begin{Vmatrix}T \circ {\chi }_{B\left( {0, R}\right) } - T\end{Vmatrix}}_{{\mathrm{{BMO}}}_{\varphi }^{2}} = 0 \]\n\nwhere \( {\chi }_{B\left( {0, R}\ri... | Proof Suppose that \( T \in {\mathrm{{BMO}}}_{\varphi }^{2} \) . It follows that\n\n\[ {\begin{Vmatrix}T \circ {\chi }_{B\left( {0, R}\right) } - T\end{Vmatrix}}_{{\mathrm{{BMO}}}_{\varphi }^{2}}^{2} \]\n\n\[ = \frac{1}{\left| B\left( w, r\right) \right| }{\int }_{B\left( {w, r}\right) }\parallel \left( {T \circ {\chi ... | Yes |
Example 5.2 In this example, we consider \( \alpha = 0,\beta = 1, p = 1 \), that is the following RLW equation \( {}^{\left\lbrack 2\right\rbrack } \)\n\n\[ \n{u}_{t} - {u}_{xxt} + {u}_{x} + u{u}_{x} = 0 \n\] \n\nwith initial condition given by the linear sum of two well separated solitary waves of various amplitudes \... | We choose \( {k}_{1} = {0.4},{k}_{2} = {0.3},{x}_{1} = {15},{x}_{2} = {35}, h = {0.1},\tau = {h}^{2} \) and \( x \in \left\lbrack {0,{120}}\right\rbrack \) . The interactions of these two solitary waves are plotted at different time levels in Fig. 2. The phenomenon of collision of solitons agrees with the result in [3]... | Yes |
Lemma 2.1 For a \( \left( {4,6}\right) \) -fullerene graph \( G,{\operatorname{Fries}}_{6}\left( G\right) \leq \left\lfloor \frac{v\left( G\right) }{3}\right\rfloor \) . | Proof Let \( \mathcal{H} \) be a usual Fries set of \( G \) and \( M \) a perfect matching of \( G \) such that all hexagons in \( \mathcal{H} \) are \( M \) -alternating. Since each edge in \( M \) belongs to at most two hexagons in \( \mathcal{H} \) , we have\n\n\[ \n v\left( G\right) = 2\left| M\right| \geq \mathop{... | Yes |
Theorem 2.2 For a \( \left( {4,6}\right) \) -fullerene graph \( G,{\operatorname{Fries}}_{6}\left( G\right) = \frac{v\left( G\right) }{3} \) if and only if \( G \) is a leapfrog \( \left( {4,6}\right) \) -fullerene. | Proof Let \( G \) be a leapfrog \( \left( {4,6}\right) \) -fullerene. Then \( G \) has a perfect Clar structure \( \mathcal{C} \) containing six squares by Theorem 2.1. So \( \left| \mathcal{C}\right| = 6 + \frac{v\left( G\right) - 4 \times 6}{6} = \frac{v\left( G\right) }{6} + 2 \) . Set \( {M}_{0} \mathrel{\text{:=}}... | Yes |
Theorem 2.3 Let \( G \) be a \( \left( {4,6}\right) \) -fullerene graph that is not leapfrog. Then \[ {\operatorname{Fries}}_{6}\left( G\right) = \left\{ \begin{array}{ll} \left\lfloor \frac{v\left( G\right) }{3}\right\rfloor - 1, & \text{ if }v\left( G\right) \equiv 4\left( {\;\operatorname{mod}\;6}\right) \\ \left\lf... | \( {G}^{M} - E\left( {\mathcal{H}}^{M}\right) \) has exactly two edges. But this is impossible by Claim 2. Hence \( {\operatorname{Fries}}_{6}\left( G\right) = \) \( \left\lfloor \frac{v\left( G\right) }{3}\right\rfloor - 1 \) . | No |
Theorem 1.2 Let \( G \) be a connected graph with \( n \) vertices and \( m \) edges. Then\n\n\[{\rho }_{U}\left( G\right) \geq \frac{\left( {{\gamma }_{A}\Delta + 2{\gamma }_{I} + {\gamma }_{J}n - {\gamma }_{A} + {\gamma }_{D}\delta }\right) + \sqrt{{\left( {\gamma }_{A}\Delta + 2{\gamma }_{I} + {\gamma }_{J}n - {\gam... | Proof By Lemma 1.4, we have\n\n\[{R}_{v}\left( {U}^{2}\right) = - \left( {{\gamma }_{A}^{2} + {\gamma }_{A}{\gamma }_{D}}\right) \mathop{\sum }\limits_{\substack{{u \sim v} \\ {u \neq v} }}{d}_{u} + \left( {{\gamma }_{A}{\gamma }_{D} + {\gamma }_{D}^{2}}\right) {d}_{v}^{2}\]\n\n\[ + \left( {2{\gamma }_{A}{\gamma }_{I} ... | Yes |
Corollary 2.4 Let \( G \) be a tree with \( n \) vertices, and let \( {\Lambda }_{1} = \left( {1 - \alpha }\right) \left( {\Delta - 1}\right) + {\alpha \delta } \) , \( {\Lambda }_{2} = \left( {1 - \alpha }\right) \left( {\delta - 1}\right) + {\alpha \Delta } \) . Then \[ \frac{{\Lambda }_{1} + \sqrt{{\Lambda }_{1}^{2}... | If \( G \) is a connected semi-regular bipartite graph, by Proposition 2.1, we directly get the following results. | No |
Corollary 2.1 Suppose that \( \left( {{\left\{ {\mathcal{G}}_{j}\right\} }_{j \in \mathbb{J}},{\left\{ {\mathcal{F}}_{j}\right\} }_{j \in \mathbb{J}}}\right) \) is an alternate dual HS-frame pair for \( \mathcal{H} \) w.r.t. \( \mathcal{K} \). Then, for each \( \lambda \in \mathbb{R} \), for any \( \mathbb{I} \subset \... | Proof Since \( {\left\{ {\mathcal{F}}_{j}\right\} }_{j \in \mathbb{J}} \) is an alternate dual of \( {\left\{ {\mathcal{G}}_{j}\right\} }_{j \in \mathbb{J}},{S}_{\mathcal{{FG}}} = {\mathrm{I}}_{\mathcal{H}} \). For each \( j \in \mathbb{J} \), let\n\n\[ \n{\mathcal{E}}_{j} = \left\{ \begin{array}{ll} 0, & j \in \mathbb... | Yes |
Corollary 2.2 Suppose that \( {\left\{ {\mathcal{G}}_{j}\right\} }_{j \in \mathbb{J}} \) is an HS-frame for \( \mathcal{H} \) w.r.t. \( \mathcal{K} \) with HS-frame operator \( {S}_{\mathcal{G}} \), and that \( {\widetilde{\mathcal{G}}}_{j} = {\mathcal{G}}_{j}{S}_{\mathcal{G}}^{-1} \) for every \( j \in \mathbb{J} \) .... | Proof Let \( {\mathcal{F}}_{j} = {\mathcal{G}}_{j}{S}_{\mathcal{G}}^{-\frac{1}{2}} \) . Then \( {S}_{\mathcal{{FG}}} = {S}_{\mathcal{G}}^{\frac{1}{2}} \) . For any \( j \in \mathbb{J} \), take\n\n\[ {\mathcal{E}}_{j} = \left\{ \begin{array}{ll} 0, & j \in \mathbb{I}; \\ {\mathcal{F}}_{j}, & j \in {\mathbb{I}}^{c}. \end... | No |
Theorem 2.2 Under Assumptions \( \left( {\mathrm{{H1}}}^{\prime }\right) ,\left( {\mathrm{H}2}\right) \) and \( \left( {\mathrm{H}3}\right) \), the mixed stochastic differential equation (0.1) has a unique solution \( X \) such that\n\n\[ \parallel X{\parallel }_{\beta ,\infty ;\left\lbrack {0, T}\right\rbrack } \leq {... | Proof Obviously, Assumption \( \left( {\mathrm{{H1}}}^{\prime }\right) \) implies Assumption (H1). Thus, Theorem 2.1 ensures the existence of solution to (0.1). On the other hand, by defining some appropriate stopping time and using the method of \( \left\lbrack {{13}\text{, Theorem 7}}\right\rbrack \), we can easily p... | No |
Theorem 1.1 Let \( \mathcal{A} \) and \( \mathcal{B} \) be two factor von Neumann algebras and let \( \eta \in \mathbb{C} \smallsetminus \{ 0\} \) . Suppose that \( \phi \) is a bijective map from \( \mathcal{A} \) to \( \mathcal{B} \) with \( \phi \left( {\left\lbrack {\left\lbrack A, B\right\rbrack }^{\eta }, C\right... | Proof If \( \eta = 1 \), based on the result of \( \left\lbrack {9\text{, Theorem 2.1}}\right\rbrack \), then \( \phi \) is additive. In the following we assume \( \eta \neq 1 \) .\n\nClaim 1: \( \phi \left( 0\right) = 0 \) .\n\nSince \( \phi \) is surjective, there exists \( A \in \mathcal{A} \) such that \( \phi \lef... | No |
Lemma 5.1 Any Eulerian ribbon graph has a checkerboard colorable partial Petrial. | Proof Let \( G \) be a Eulerian ribbon graph. We first color vertex line segments and half edge line segments at each vertex as shown in Fig. 27(1). Let \( A \) be the set of incoherent edges as shown in Fig. 27(2)(b). Then we obtain a checkerboard coloring of \( {G}^{\tau \left( A\right) } \) . | Yes |
Theorem 5.3 \( {}^{\left\lbrack {46}\right\rbrack } \) Any embedded graph has a checkerboard colorable twisted dual. | Proof For any ribbon graph \( G \), we can obtain a checkerboard colorable twisted dual of \( G \) as follows. Let \( T \) be a spanning quasi-tree of \( G \) . We can obtain a bouquet \( {G}^{\delta \left( T\right) } \) . And \( {G}^{\delta \left( T\right) } \) is clearly Eulerian. Then by Lemma 5.1, \( {G}^{\delta \l... | Yes |
Lemma 2.2 Let \( G \) be a connected bipartite graph of diameter 3, and let \( {P}_{4} = {abcd} \) be a fixed diametrical path of \( G \) . If \( {n}_{D}\left( G\right) = 4 \), then\n\n(i) \( \forall u \in V\left( G\right) ,{N}_{G}\left( u\right) \) is an independent set;\n\n(ii) if \( {S}_{\{ b\} } \neq \varnothing \)... | Proof Since \( G \) is bipartite,(i) holds. For (ii), by the symmetry of \( b \) and \( c \), we only need to prove the result for \( {S}_{\{ b\} } \neq \varnothing \) . In this case, by Lemma 2.1 (ii), we have \( \left| {S}_{\{ b\} }\right| = 1 \) . If \( \left| {S}_{\{ a\} }\right| \geq 2 \) , take \( u \in {S}_{\{ b... | Yes |
Lemma 2.3 Let \( G \) be a connected bipartite graph of diameter 4, and let \( {P}_{5} = {abcde} \) be a fixed diametrical path of \( G \) . If \( {n}_{D}\left( G\right) = 4 \), we have\n\n(i) \( {V}_{3} = {V}_{4} = \varnothing \) ;\n\n(ii) \( {S}_{\{ c\} } = \varnothing \) ;\n\n(iii) \( \left| {S}_{\{ a\} }\right| \le... | Proof Let \( {M}_{0} \) be the principal submatrix of \( G \) corresponding to \( {P}_{5} = {abcde} \) . Then\n\n\[ \n{M}_{0} = \left( \begin{array}{lllll} 0 & 1 & 2 & 3 & 4 \\ 1 & 0 & 1 & 2 & 3 \\ 2 & 1 & 0 & 1 & 2 \\ 3 & 2 & 1 & 0 & 1 \\ 4 & 3 & 2 & 1 & 0 \end{array}\right) .\n\]\n\nFirst, we consider (i). If \( {V}_... | Yes |
Theorem 1.1 Let \( G = {C}_{3n} \cup {nT} \) be a CPT-graph. Then \( {\chi }_{s}^{\prime }\left( G\right) \leq {19} \) . | Proof Assume that \( {nT} = {T}_{1} \cup {T}_{2} \cup \cdots \cup {T}_{n} \) . Then \( {T}_{1},{T}_{2},\cdots ,{T}_{n} \) are \( n \) vertex-disjoint triangles, and \( {C}_{3n} \) is a Hamiltonian cycle. If \( n \leq 3 \), then \( \left| {E\left( G\right) }\right| = {6n} \leq {18} \) . By assigning distinct colors to e... | Yes |
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