# sign

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## 1—10 of 101 matching pages

##### 1: Sidebar 5.SB1: Gamma & Digamma Phase Plots
Phase changes around the zeros are of opposite sign to those around the poles. The fluid flow analogy in this case involves a line of vortices of alternating sign of circulation, resulting in a near cancellation of flow far from the real axis.
##### 2: David M. Bressoud
227, in 1980, Factorization and Primality Testing, published by Springer-Verlag in 1989, Second Year Calculus from Celestial Mechanics to Special Relativity, published by Springer-Verlag in 1992, A Radical Approach to Real Analysis, published by the Mathematical Association of America in 1994, with a second edition in 2007, Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, published by the Mathematical Association of America and Cambridge University Press in 1999, A Course in Computational Number Theory (with S. …
##### 3: 14.27 Zeros
• (a)

$\mu<0$, $\mu\notin\mathbb{Z}$, $\nu\in\mathbb{Z}$, and $\sin\left((\mu-\nu)\pi\right)$ and $\sin\left(\mu\pi\right)$ have opposite signs.

• ##### 4: 1.11 Zeros of Polynomials
A monic polynomial of even degree with real coefficients has at least two zeros of opposite signs when the constant term is negative.
###### Descartes’ Rule of Signs
The number of positive zeros of a polynomial with real coefficients cannot exceed the number of times the coefficients change sign, and the two numbers have same parity. A similar relation holds for the changes in sign of the coefficients of $f(-z)$, and hence for the number of negative zeros of $f(z)$. … Both polynomials have one change of sign; hence for each polynomial there is one positive zero, one negative zero, and six complex zeros. …
##### 6: 25.10 Zeros
Because $|Z(t)|=|\zeta\left(\frac{1}{2}+it\right)|$, $Z(t)$ vanishes at the zeros of $\zeta\left(\frac{1}{2}+it\right)$, which can be separated by observing sign changes of $Z(t)$. Because $Z(t)$ changes sign infinitely often, $\zeta\left(\frac{1}{2}+it\right)$ has infinitely many zeros with $t$ real. … By comparing $N(T)$ with the number of sign changes of $Z(t)$ we can decide whether $\zeta\left(s\right)$ has any zeros off the line in this region. Sign changes of $Z(t)$ are determined by multiplying (25.9.3) by $\exp\left(i\vartheta(t)\right)$ to obtain the Riemann–Siegel formula: …
##### 7: 32.11 Asymptotic Approximations for Real Variables
• (c)

If $k_{2}, then $w(x)$ changes sign once, from positive to negative, as $x$ passes from $x_{0}$ to $0$.

• 32.11.10 $w_{k}(x)\sim\operatorname{sign}\left(k\right)\sqrt{\tfrac{1}{2}|x|},$ $x\to-\infty$.
32.11.11 $w_{k}(x)\sim\operatorname{sign}\left(k\right)(x-c_{0})^{-1},$ $x\to c_{0}+$.
32.11.17 $d^{2}=\pi^{-1}\ln\left(1+k^{2}\right),$ $\operatorname{sign}\left(k\right)=(-1)^{n}$.
##### 8: 36.5 Stokes Sets
36.5.7 $X=\dfrac{9}{20}+20u^{4}-\frac{Y^{2}}{20u^{2}}+6u^{2}\operatorname{sign}\left(z% \right),$
36.5.8 $16u^{5}-\frac{Y^{2}}{10u}+4u^{3}\operatorname{sign}\left(z\right)-\frac{3}{10}% |Y|\operatorname{sign}\left(z\right)+4t^{5}+2t^{3}\operatorname{sign}\left(z% \right)+|Y|t^{2}=0,$
36.5.9 $t=-u+\left(\dfrac{|Y|}{10u}-u^{2}-\dfrac{3}{10}\operatorname{sign}\left(z% \right)\right)^{1/2}.$
##### 9: 14.16 Zeros
$\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ has $\max(\lceil\nu-|\mu|\rceil,0)+k$ zeros in the interval $(-1,1)$, where $k$ can take one of the values $-1$, $0$, $1$, $2$, subject to $\max(\lceil\nu-|\mu|\rceil,0)+k$ being even or odd according as $\cos\left(\nu\pi\right)$ and $\cos\left(\mu\pi\right)$ have opposite signs or the same sign. …
• (a)

$\mu>0$, $\mu>\nu$, $\mu\notin\mathbb{Z}$, and $\sin\left((\mu-\nu)\pi\right)$ and $\sin\left(\mu\pi\right)$ have opposite signs.

• ##### 10: 26.13 Permutations: Cycle Notation
For the example (26.13.2), this decomposition is given by ${\left(1,3,2,5,7\right)}{\left(6,8\right)}={\left(1,3\right)}{\left(2,3\right)% }{\left(2,5\right)}{\left(5,7\right)}{\left(6,8\right)}.$The sign of a permutation is $+$ if the permutation is even, $-$ if it is odd. …