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# Neper

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### WIKIPEDIA ARTICLE

The neper (symbol: Np) is a logarithmic unit for ratios of measurements of physical field and power quantities, such as gain and loss of electronic signals. The unit's name is derived from the name of John Napier, the inventor of logarithms. As is the case for the decibel and bel, the neper is a unit defined in the international standard ISO 80000. It is not part of the International System of Units (SI), but is accepted for use alongside the SI.[1]

## Definition

Like the decibel, the neper is a unit in a logarithmic scale. While the bel uses the decadic (base-10) logarithm to compute ratios, the neper uses the natural logarithm, based on Euler's number (e ≈ 2.71828). The value of a ratio in nepers is given by

${\displaystyle L_{\rm {Np}}=\ln {\frac {x_{1}}{x_{2}}}=\ln x_{1}-\ln x_{2}.}$

where ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$ are the values of interest (amplitudes), and ln is the natural logarithm. When the values are quadratic in the amplitude (e.g. power), they are first linearised by taking the square root before the logarithm is taken, or equivalently the result is halved.[citation needed]

In the ISQ, the neper is defined as 1 Np = 1.[2]

## Units

The neper is defined in terms of ratios of field quantities (for example, voltage or current amplitudes in electrical circuits, or pressure in acoustics), whereas the decibel was originally defined in terms of power ratios. A power ratio 10 log r dB is equivalent to a field-quantity ratio 20 log r dB, since power is proportional to the square (Joule's laws) of the amplitude. Hence the neper and decibel are related via:

${\displaystyle 1\ \mathrm {Np} =20\log _{10}e\ \mathrm {dB} \approx 8{.}685889638\ \mathrm {dB} \,}$

and

${\displaystyle 1\ \mathrm {dB} ={\frac {1}{20}}\ln(10)\ \mathrm {Np} \approx 0{.}115129255\ \mathrm {Np} .\,}$

The decibel and the neper have a fixed ratio to each other. The (voltage) level ratio is

{\displaystyle {\begin{aligned}L&=10\log _{10}{\frac {x_{1}^{2}}{x_{2}^{2}}}&\mathrm {dB} \\&=10\log _{10}{\left({\frac {x_{1}}{x_{2}}}\right)}^{2}&\mathrm {dB} \\&=20\log _{10}{\frac {x_{1}}{x_{2}}}&\mathrm {dB} \\&=\ln {\frac {x_{1}}{x_{2}}}&\mathrm {Np} .\\\end{aligned}}}

Like the decibel, the neper is a dimensionless unit. The International Telecommunication Union (ITU) recognizes both units.

## Applications

The neper is a natural linear unit of relative difference, meaning in nepers (logarithmic units) relative differences add rather than multiply. This property is shared with logarithmic units in other bases, such as the bel.

Particular to the neper, however, is that the derived unit of centineper is asymptotically equal to percentage difference for very small differences – since the derivative of the natural log (at 1) is 1; this is not shared with other logarithmic units, which introduce a scaling factor due to the derivative not being unity. The centineper can thus be used as a linear replacement for percentage differences. The linear approximation for small percentage differences,

${\displaystyle (1+\delta )(1+\epsilon )=1+\delta +\epsilon +\delta \epsilon \approx 1+\delta +\epsilon ,}$

is widely used, particularly in finance—see for example the Fisher equation. However, it is only approximate, with error increasing for large percentage changes. Measured instead in centinepers, these linear approximations can be replaced with exact equalities, and applicable to any magnitude change, by defining the following centineper quantity for any change ${\displaystyle \delta }$

${\displaystyle D_{\delta }=100\ln {\frac {(1+\delta )-1}{1}}=100\ln \delta }$


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## Algebra 1: Common Core (15th Edition)

$\frac{2d}{5}$
A key rule to remember is that subtracting a negative is the same as addition. Thus, $\frac{d}{5}$-(-$\frac{d}{5}$)=$\frac{d}{5}$+$\frac{d}{5}$=$\frac{2d}{5}$

format: plain


text:
Hw8 Problem 2

(a) Find all the left cosets of the subgroup $\{\rho_0, \mu_2\}$ of $D_4$. (Table 8.12 gives the multiplication table for $D_4$.)

(b) Now find the right cosets. Are they the same as the left cosets?

Solution

(a)
$\{\rho_0, \mu_2\}$
$\{\rho_1. \delta_2\}$
$\{\rho_2, \mu_1\}$
$\{\rho_3, \delta_1\}$

(b)
$\{\rho_0, \mu_2\}$
$\{\rho_1, \delta_1\}$
$\{\rho_2, \mu_1\}$
$\{\rho_3, \delta_2\}$

No, they are not the same.


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# American Institute of Mathematical Sciences

January  2011, 15(1): 197-215. doi: 10.3934/dcdsb.2011.15.197

## Approximate tracking of periodic references in a class of bilinear systems via stable inversion

 1 Department of Applied Mathematics IV, Universitat Politècnica de Catalunya, Av. Víctor Balaguer, s/n, 08800 Vilanova i la Geltrú, Spain 2 Department of Applied Mathematics I, Universitat Politècnica de Catalunya, Av. Diagonal, 647, 08028 Barcelona, Spain

Received  December 2009 Revised  August 2010 Published  October 2010

This article deals with the tracking control of periodic references in single-input single-output bilinear systems using a stable inversion-based approach. Assuming solvability of the exact tracking problem and asymptotic stability of the nominal error system, the study focuses on the output behavior when the control scheme uses a periodic approximation of the nominal feedforward input signal $u_d$. The investigation shows that this results in a periodic, asymptotically stable output; moreover, a sequence of periodic control inputs $u_n$ uniformly convergent to $u_d$ produce a sequence of output responses that, in turn, converge uniformly to the output reference. It is also shown that, for a special class of bilinear systems, the internal dynamics equation can be approximately solved by an iterative procedure that provides of closed-form analytic expressions uniformly convergent to its exact solution. Then, robustness in the face of bounded parametric disturbances/uncertainties is achievable through dynamic compensation. The theoretical analysis is applied to nonminimum phase switched power converters.
Citation: Josep M. Olm, Xavier Ros-Oton. Approximate tracking of periodic references in a class of bilinear systems via stable inversion. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 197-215. doi: 10.3934/dcdsb.2011.15.197
##### References:
 [1] F. Amato, C. Cosentino, A. S. Fiorillo and A. Merola, Stabilization of bilinear systems via linear state-feedback control, IEEE Trans. Circ. Syst.-II, 56 (2009), 76-80. [2] R. O. Cáceres and I. Barbi, A boost DC-AC converter: Analysis, design and experimentation, IEEE Trans. Power Electronics, 14 (1999), 134-141. doi: doi:10.1109/63.737601. [3] M. Carpita and M. Marchesoni, Experimental study of a power conditioning system using sliding mode control, IEEE Trans. Power Electronics, 11 (1996), 731-742. doi: doi:10.1109/63.535405. [4] D. Cortés, Jq. Álvarez, J. Álvarez and A. Fradkov, Tracking control of the boost converter, IEEE Proceedings Control Theory Applications, 151 (2004), 218-224. doi: doi:10.1049/ip-cta:20040203. [5] S. Devasia, D. Chen and B. Paden, Nonlinear inversion-based output tracking, IEEE Trans. Automatic Control, 41 (1996), 930-942. doi: doi:10.1109/9.508898. [6] D. L. Elliott, "Bilinear Control Systems. Matrices in Action," Springer, 2009. [7] M. Farkas, "Periodic Motions," Springer-Verlag, 1994. [8] E. Fossas and J. M. Olm, Asymptotic tracking in DC-to-DC nonlinear power converters, Discrete Continuous Dynam. Systems - B, 2 (2002), 295-307. [9] E. Fossas and J. M. Olm, Galerkin method and approximate tracking in a nonminimum phase bilinear system, Discrete Continuous Dynam. Systems - B, 7 (2007), 53-76. [10] E. Fossas and J. M. Olm, A functional iterative approach to the tracking control of nonminimum phase switched power converters, Math. Control Signals Syst., 21 (2009), 203-227. doi: doi:10.1007/s00498-009-0044-5. [11] E. Fossas, J. M. Olm, A. Zinober and Y. Shtessel, Galerkin-based sliding mode tracking control of nonminimum phase DC-to-DC power converters, Internat. J. Robust Nonlinear Control, 17 (2007), 587-604. doi: doi:10.1002/rnc.1136. [12] H. K. Khalil, "Nonlinear Systems," 3rd edition, Prentice Hall, 2002. [13] R. R. Mohler, "Nonlinear Systems, vol. 2: Applications to Bilinear Control," Prentice Hall, 1991. [14] J. M. Olm, "Asymptotic Tracking with DC-to-DC Bilinear Power Converters," Ph. D thesis, Universitat Politècnica de Catalunya, 2004. [15] J. M. Olm, X. Ros-Oton and Y. B. Shtessel, Stable inversion of Abel equations: application to tracking control in DC-DC nonminimum phase boost converters, Automatica, (2010), in press. doi: doi:10.1109/CDC.2007.4434276. [16] A. Pavlov and K. Y. Pettersen, Stable inversion of non-minimum phase nonlinear systems: A convergent systems approach, in "Proceedings of the 46th IEEE Conference on Decision and Control," (2007), 3995-4000. [17] A. D. Polyanin, "Handbook of Exact Solutions for Ordinary Differential Equations," 2nd edition, Chapman & Hall/CRC, Boca Raton, 2003. [18] S. Sastry, "Nonlinear Systems. Analysis, Stability and Control," Springer-Verlag, 1999. [19] Y. Shtessel, A. Zinober and I. Shkolnikov, Sliding mode control of boost and buck-boost power converters control using method of stable system centre, Automatica, 39 (2003), 1061-1067. doi: doi:10.1016/S0005-1098(03)00068-2. [20] H. Sira-Ramírez, Sliding motions in bilinear switched networks, IEEE Trans. Circ. Syst., 34 (1987), 1359-1390. [21] H. Sira-Ramírez, DC-to-AC power conversion on a 'boost' converter, Internat. J. Robust Nonlinear Control, 11 (2001), 589-600. doi: doi:10.1002/rnc.575. [22] H. Sira-Ramírez, M. Spinetti-Rivera and E. Fossas, An algebraic parameter estimation approach to the adaptive observer-controller based regulation of the boost converter, in "Proceedings of the IEEE Int. Symp. Industrial Electronics," (2007), 3367-3372. [23] E. D. Sontag, Input to state stability: Basic concepts and results, in "Nonlinear and Optimal Control Theory" (eds. P. Nistri and G. Stefani), Springer-Verlag, (2007), 163-220. [24] E. D. Sontag and Y. Wang, New characterizations of input-to-state stability, IEEE Trans. Automatic Control, 41 (1996), 1283-1294. doi: doi:10.1109/9.536498.

show all references

##### References:
 [1] F. Amato, C. Cosentino, A. S. Fiorillo and A. Merola, Stabilization of bilinear systems via linear state-feedback control, IEEE Trans. Circ. Syst.-II, 56 (2009), 76-80. [2] R. O. Cáceres and I. Barbi, A boost DC-AC converter: Analysis, design and experimentation, IEEE Trans. Power Electronics, 14 (1999), 134-141. doi: doi:10.1109/63.737601. [3] M. Carpita and M. Marchesoni, Experimental study of a power conditioning system using sliding mode control, IEEE Trans. Power Electronics, 11 (1996), 731-742. doi: doi:10.1109/63.535405. [4] D. Cortés, Jq. Álvarez, J. Álvarez and A. Fradkov, Tracking control of the boost converter, IEEE Proceedings Control Theory Applications, 151 (2004), 218-224. doi: doi:10.1049/ip-cta:20040203. [5] S. Devasia, D. Chen and B. Paden, Nonlinear inversion-based output tracking, IEEE Trans. Automatic Control, 41 (1996), 930-942. doi: doi:10.1109/9.508898. [6] D. L. Elliott, "Bilinear Control Systems. Matrices in Action," Springer, 2009. [7] M. Farkas, "Periodic Motions," Springer-Verlag, 1994. [8] E. Fossas and J. M. Olm, Asymptotic tracking in DC-to-DC nonlinear power converters, Discrete Continuous Dynam. Systems - B, 2 (2002), 295-307. [9] E. Fossas and J. M. Olm, Galerkin method and approximate tracking in a nonminimum phase bilinear system, Discrete Continuous Dynam. Systems - B, 7 (2007), 53-76. [10] E. Fossas and J. M. Olm, A functional iterative approach to the tracking control of nonminimum phase switched power converters, Math. Control Signals Syst., 21 (2009), 203-227. doi: doi:10.1007/s00498-009-0044-5. [11] E. Fossas, J. M. Olm, A. Zinober and Y. Shtessel, Galerkin-based sliding mode tracking control of nonminimum phase DC-to-DC power converters, Internat. J. Robust Nonlinear Control, 17 (2007), 587-604. doi: doi:10.1002/rnc.1136. [12] H. K. Khalil, "Nonlinear Systems," 3rd edition, Prentice Hall, 2002. [13] R. R. Mohler, "Nonlinear Systems, vol. 2: Applications to Bilinear Control," Prentice Hall, 1991. [14] J. M. Olm, "Asymptotic Tracking with DC-to-DC Bilinear Power Converters," Ph. D thesis, Universitat Politècnica de Catalunya, 2004. [15] J. M. Olm, X. Ros-Oton and Y. B. Shtessel, Stable inversion of Abel equations: application to tracking control in DC-DC nonminimum phase boost converters, Automatica, (2010), in press. doi: doi:10.1109/CDC.2007.4434276. [16] A. Pavlov and K. Y. Pettersen, Stable inversion of non-minimum phase nonlinear systems: A convergent systems approach, in "Proceedings of the 46th IEEE Conference on Decision and Control," (2007), 3995-4000. [17] A. D. Polyanin, "Handbook of Exact Solutions for Ordinary Differential Equations," 2nd edition, Chapman & Hall/CRC, Boca Raton, 2003. [18] S. Sastry, "Nonlinear Systems. Analysis, Stability and Control," Springer-Verlag, 1999. [19] Y. Shtessel, A. Zinober and I. Shkolnikov, Sliding mode control of boost and buck-boost power converters control using method of stable system centre, Automatica, 39 (2003), 1061-1067. doi: doi:10.1016/S0005-1098(03)00068-2. [20] H. Sira-Ramírez, Sliding motions in bilinear switched networks, IEEE Trans. Circ. Syst., 34 (1987), 1359-1390. [21] H. Sira-Ramírez, DC-to-AC power conversion on a 'boost' converter, Internat. J. Robust Nonlinear Control, 11 (2001), 589-600. doi: doi:10.1002/rnc.575. [22] H. Sira-Ramírez, M. Spinetti-Rivera and E. Fossas, An algebraic parameter estimation approach to the adaptive observer-controller based regulation of the boost converter, in "Proceedings of the IEEE Int. Symp. Industrial Electronics," (2007), 3367-3372. [23] E. D. Sontag, Input to state stability: Basic concepts and results, in "Nonlinear and Optimal Control Theory" (eds. P. Nistri and G. Stefani), Springer-Verlag, (2007), 163-220. [24] E. D. Sontag and Y. Wang, New characterizations of input-to-state stability, IEEE Trans. Automatic Control, 41 (1996), 1283-1294. doi: doi:10.1109/9.536498.
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# Thread: Equations and range of values..

1. ## Equations and range of values..

Hi, thnx for help

How do i find the range of values of k for which the equation x^2+5x+k=0 (i dont know how to find values )

and also how to solve the equation 4x^2+20x+25=0 i tried to use the formula but i didnt get a answer that worked... thnx

2. Originally Posted by Chez_
Hi, thnx for help

How do i find the range of values of k for which the equation x^2+5x+k=0 (i dont know how to find values )

If $x^2 + 5x + k = 0$, we must have:

$x = \frac {-5 \pm \sqrt{25 - 4k}}2$

this of course, only has real solutions when we are withing the domain of the square root. thus we require $25 - 4k \ge 0$. what is the range of $k$-values for which this happens?

and also how to solve the equation 4x^2+20x+25=0 i tried to use the formula but i didnt get a answer that worked... thnx
the quadratic formula can work, but it is overkill. this equation can be factored.

$4x^2 + 20x + 25 = (2x + 5)(2x + 5) = (2x + 5)^2$

So, $4x^2 + 20x + 25 = 0 \implies (2x + 5)^2 = 0$.

now continue


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# On a certain high school athletic team, the ratio of freshmen to sopho

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On a certain high school athletic team, the ratio of freshmen to sopho  [#permalink]

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23 Oct 2018, 21:07
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On a certain high school athletic team, the ratio of freshmen to sophomores to juniors to seniors is 1:3:4:6. If there are 60 juniors on the team, how many students in total are on the team?

A. 90
B. 140
C. 150
D. 180
E. 210

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Re: On a certain high school athletic team, the ratio of freshmen to sopho  [#permalink]

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23 Oct 2018, 22:07
Bunuel wrote:
On a certain high school athletic team, the ratio of freshmen to sophomores to juniors to seniors is 1:3:4:6. If there are 60 juniors on the team, how many students in total are on the team?

A. 90
B. 140
C. 150
D. 180
E. 210

f:so:j:s = 1:3:4:6
4x=j=60
x=15
total = 15 + 45 + 60 + 90 = 210
E
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Re: On a certain high school athletic team, the ratio of freshmen to sopho  [#permalink]

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23 Oct 2018, 22:17
Bunuel wrote:
On a certain high school athletic team, the ratio of freshmen to sophomores to juniors to seniors is 1:3:4:6. If there are 60 juniors on the team, how many students in total are on the team?

A. 90
B. 140
C. 150
D. 180
E. 210

$$\frac{60}{4}*14 = 210$$, Answer must be (E)
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Re: On a certain high school athletic team, the ratio of freshmen to sopho  [#permalink]

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24 Oct 2018, 02:39

Solution

Given:
• The ratio of freshmen to sophomores to juniors to seniors in the athletic team = 1: 3: 4: 6
• Number of juniors in the team = 60

To find:
• The total number of students in the team

Approach and Working:
• Let us assume that 14x as the total number of students in the team (Since, 1 + 3 + 4 + 6 = 14)
• Thus, number of juniors = 4x = 60
o Implies, x = 15
• Therefore, total number of students in the team = 14 * 15 = 210

Hence the correct answer is Option E

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Re: On a certain high school athletic team, the ratio of freshmen to sopho  [#permalink]

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28 Oct 2018, 17:21
Bunuel wrote:
On a certain high school athletic team, the ratio of freshmen to sophomores to juniors to seniors is 1:3:4:6. If there are 60 juniors on the team, how many students in total are on the team?

A. 90
B. 140
C. 150
D. 180
E. 210

The entire ratio can be expressed as: 1x + 3x + 4x + 6x = total. For the juniors, we can create the equation:

4x = 60

x = 15

So there are 1x + 3x + 4x + 6x = 14x = 14(15) = 210 students on the team

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Re: On a certain high school athletic team, the ratio of freshmen to sopho &nbs [#permalink] 28 Oct 2018, 17:21
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# How come divergence of $\vec E$ is zero in this case where $\vec E=\xi\frac{\left[x,y\right]}{\sqrt{(x^2+y^2)^3}}\,?$

I hope you could help me clearing some doubts about Gauss' law of the electric field that states $\epsilon_0\nabla\cdot\vec E=\rho$. Take for instance the case of a point charge in the origin in empty space. The law states $\nabla\cdot\vec E = 0$ anywhere but the origin, but it's enough to compute $\nabla\cdot\vec E$ to get to a contradiction: $$\vec E=\xi\frac{\left[x,y\right]}{\sqrt{(x^2+y^2)}^3}$$ where I dropped every constant to $\xi$ because that's my favourite greek letter, and considered the 2D case since in my opinion it's a good compromise between complexity and meaningfulness $$\nabla\cdot\vec E=-\frac{x^2+y^2}{\sqrt{{(x^2+y^2)}^5}}=(x^2+y^2)^{1-\frac{5}{2}}$$ which is clearly not zero except at infinity.

• your divergence is wrong - you did not calculate the derivative correctly – Sanya Sep 21 '16 at 16:45
• You misunderstand Gauss's Law - you have given the differential form, but have ignored that it comes from the divergence of the E field, meaning an integral over a surface enclosing a volume. – Jon Custer Sep 21 '16 at 16:45
• You can't just use any number of dimensions. Coulomb's law changes in 2D. – Javier Sep 21 '16 at 17:20
• Jon Custer, may you explain further this point? I know this law comes from the "global" one which states that the electric flux through a closed surface is the inner charge (which I have understood well, actually), I also know the mathematical derivation of the "local" law, but it appears i haven't understood what does it actually mean – J. Joestar Sep 21 '16 at 18:18
• Sanya, Javier: even without doing calculations this doesen't sound right to me: the electric field due to a charge is made of radial vectors pointing toward or outward the charge, with amplitude decreasing with distance. Since the amplitude is decreasing, mathematically speaking every point in space will have both $\vec E$ and its derivatives different from 0, so there's no way $\nabla\cdot\vec E$ is also zero. From the intuitive point of view, every point has a polarized direction through which more field enters than it leaves, thus having a nonzero divergence. – J. Joestar Sep 21 '16 at 18:28

As pointed out in the comment, Gauss' law has a precise geometric meaning and so is strongly dependent on the dimension of the space. Let's work in 2D as in your example and apply Gauss' law: the flux of the electric field through a circle centered in the point charge is $2\pi rE$. This must be equal to the total charge which is $e/\epsilon_0$. This gives for the modulus of the electric field: $$E=\frac{e}{2\pi\epsilon_0 r}$$ and for the cartesian components: $$\vec{E}=\frac{e}{2\pi\epsilon_0}(\frac{x}{x^2+y^2},\frac{y}{x^2+y^2})$$ As you can verify the divergence of this is zero: $$\nabla\cdot\vec{E}\propto\frac{1}{x^2+y^2}-\frac{2 x^2}{\left(x^2+y^2\right)^2}+\frac{1}{x^2+y^2}-\frac{2 y^2}{\left(x^2+y^2\right)^2}=0$$


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# The genus of the Fermat curve $x^d+y^d+z^d=0$

I need to calculate the genus of the Fermat Curve, and I'd like to be reviewed on what I have done so far; I'm not secure of my argumentation.

Such curve is defined as the zero locus $$X=\{[x:y:z]\in\mathbb{P}^2:x^d+y^d+z^d=0\}$$ It is defined by the homogeneous polynomial $F(x,y,z)=x^d+y^d+z^d$ which is obviously non-singular, therefore $X$ is a smooth projective plane curve. Now, I consider the projection $$\begin{array}{cccc} \pi:&X&\longrightarrow&\mathbb{P}^1\\ &[x:y:z]&\longmapsto&[x:y] \end{array}$$ which is a well-defined holomorphic function. I need to calculate $\pi$'s degree, and for this I must calculate the sum of the multiplicities of the set $\{p:p\in\pi^{-1}(y)\}$, for any $y$ in $\pi(X)$; since the degree is constant, any such $y$ will do.

We note that, from Miranda's Riemann Surfaces and Algebraic Curves' Lemma 4.6 at page 46, that $\pi$ has a branch point at $p\in X$ if and only if $\frac{\partial F}{\partial z}(p)=0$

Having this in mind, to calculate the degree, I chose $[1:0]\in\mathbb{P}^1$; and for this one, $$\pi^{-1}([1:0])=\{[1:0:z]:1+z^d=0\}$$ Now I can see clearly that $|\pi^{-1}([1:0])|=d$ (the d-th roots of $-1$), and each one has multiplicity 1 (because they are not ramification points, as the above mentioned lemma states); therefore the degree of $\pi$ is $d$.

Now from the Lemma again, the ramification points of $\pi$ are those that $z=0$, that is, the points $\{[x:y:0]\in\mathbb{P}^2:x^d+y^d=1\}$ In homogeneous coordinates those points are of the form $[1:y:0]$, where $y$ is a d-th roots of $-1$ and from this we conclude that $\pi$ has $d$ points of ramification, and those are $$\left[ 1:e^{\dfrac{2\pi i k}{d}}:0\right]$$

To calculate the multiplicity of each one of these points I used the following argument, which I'm not sure if it is correct: for a point $[x:y]\in\mathbb{P}^1$ such that $x^d+y^d=0$, $\pi^{-1}([x:y])=\{[x:y:0]\}$ and therefore $|\pi^{-1}([x:y])|=1$. Since the degree of $\pi$ is $d$, $d$ is also the multiplicity of every point of ramification of $\pi$.

What follows noy is applying Hurwitz's formula (and noting that $g(\mathbb{P}^1)=0$ and a very easy calculation to conclude that $g(x)=\dfrac{(d-1)(d-2)}{2}$.

Now I must apologize for the long, long text, but I will be much grateful for any correction made in my argumentation!

• Why the downvote? There's research effort, I believe it is quite clear and also very useful question. – Marra Apr 26 '13 at 21:47

When you discuss ramification points of $\pi$, the equation should be $x^d + y^d = 0$, not $x^d + y^d = 1$.

Also, whenever you wrote about $d$th roots of unity, you needed the $d$th roots of $-1$.

The rest seems right. Yes, the number of preimages of the ramification points is all that you need in order to apply Hurwitz's formula, it is not necessary to find the degree of each preimage.

• When I meant $x^d+y^d=1$ it was a mistake of mine, I really meant $x^d+y^d=0$. I have edited the mistake about the roots of $-1$. Also, didn't you mean that I don't have to find the multiplicity (instead of degree) of every preimage of the ramification point? Thanks for the answer! I spent a lot of time and effort on this! :) – Marra Apr 26 '13 at 22:22

Another approach to calculate the multiplicities is to use Lemma 4.4 (page 45). We calculate the multiplicities of the points $[\alpha:1:0]$ and show they are $d$.

First we need a local representation of $F$ near $[\alpha:1:0]$. In the patch where $y\neq 0$, this corresponds to $(\alpha, 0)$ in $\mathbb C^2$ (with coordinates $x, z$), cut out by $x^d+1+z^d=0$. The $z$ partial vanishes at $(\alpha, 0)$ so by implicit function theorem, the chart near $(\alpha, 0)$ is projection to the $z$ axis, and there is a holomorphic function $g(z)$ with the inverse to projection is $z \mapsto (g(z),z)$. The map $F$ on the patch $y\neq 0$ sends $(x, z)$ to $x$. So the local representation of $F$ is $z \mapsto g(z)$.

Now we need to calculate the order of vanishing of $g'$ at $0$. We have $g(z)^d+1+z^d=0$ so after differentiating and manipulating, $$g(z)^{d-1} \cdot g'(z) = -z^{d-1}.$$ Notice $g'(0)=0$ because $g(0)=\alpha$. It follows the order of vanishing of $g'$ at $0$ is $d-1$. Hence the multiplicity is $d$.


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access it from laptop or notebook, it will redirect you back to this page. If it acts as your main staircase (meaning it serves two or more rooms), a spiral staircase should be 800mm wide with a maximum rise of 220mm; the length of the steps should be at least 190mm. Rise, mm. Help on Permited Rise and Go dimensions. In a given run of stairs, the greatest tread depth should not exceed the smallest by more than 3/8 inch (9.5 mm). MAX. 7" Riser Height / Tread-Depth-Y = 0.7535 = 37° slope, Again using Mrs. Revere's elementary school algebra instructions, we can re-write the formula as. Building codes suggest Step Height [F] be no more than 7 3/4" and the Tread Depth [B] be no less than 10". Stringer - The stringer is the structural part of a set of stairs. "Stair width." Watch out: at STAIR RISER SPECIFICATIONS and at STAIR DIMENSIONS, WIDTH, HEIGHT you'll read that stair risers should be 7 inches (178 mm) maximum and 4 inches (102 mm) minimum. This means the rise is 7 inches and the run is 11 inches. The Mobile Stair Calculator is available as always on mycarpentry.com, but I urge you to try this very webpage on your mobile or tablet device. Calculate unit rise and unit run. 7, Â§3231. It is how far in, the stair goes which gives the amount of room for a persons foot. 1. Increasing the length of the run by the smallest unit of measurement increases cost and the length of a staircase. For more information, click the Help buttons located on each calculator. Stair rise is normally abo… ISBN-10: 0471648361, ISBN-13: 978-0471648369, Hardcover: 320 pages, available from Amazon.com and also, [7] The Circular Staircase, Mary Roberts Rinehart, [8] Construction Drawings and Details, Rosemary Kilmer, [9] Falls and Related Injuries: Slips, Trips, Missteps, and Their Consequences, Lawyers & Judges Publishing, (June 2002), ISBN-10: 0913875430 ISBN-13: 978-0913875438, [19] Common Sense Stairbuilding and Handrailing, Fred T. Hodgson, [21] Building Stairs, by pros for pro, Andy Engel, [22] A Simplified Guide to Custom Stairbuilding, George R. Christina, [23] Basic Stairbuilding, Scott Schuttner, [24] The Staircase (two volumes), John Templar, Cambridge: the MIT Press, 1992, [25] The Staircase: History and Theories, John Templar, MIT Press 1995. 3. max. 1. service and mezzanines in live/work units (1) no limit. Number of Risers: There is always one more stair riser than there are stair treads. Stair " rise height" is the vertical height or distance between the surface of two stair treads (what you actually step on). The rise is the vertical measurement taken from one step level to the next. If you want to pass spiral staircase building regulations, for example, you will have more homework to do. Stair rise is the vertical measurement of the height between two stairs. 355. Feb 7, 2020 - Explore Les Otto's board "Stair rise and run" on Pinterest. Stair run is the horizontal measurement from the front to the back of a single stair. and 50 deg. min. If you try to It is useful to look at both rules when considering the comfort of a stairway. The inverse tangent is also called arctan or Arc Tan. Try the search box just below, or if you prefer, post a question or comment in the Comments box below and we will respond promptly. 7 inches (178 mm) maximum and 4 inches (102 mm) minimum. The following stair measurements are important: The rise of each step is measured from the top of one tread to the top of the next tread. Note: appearance of your Comment below may be delayed: if your comment contains an image, web link, or text that looks to the software as if it might be a web link, your posting will appear after it has been approved by a moderator. max. See more ideas about building a deck, deck stairs, building stairs. 355. no limit. Retrieved 2018/07/04, original source: https://www.dir.ca.gov/title8/3231.html. icons to determine the best calculator to use for your stairs. The Total Rise [A] is the finished height of your landing or deck. Stairs cannot be continuous without have a landing or floor level that breaks up the flight of stairs. to the overall run (going) of the stair. The usual we see is 150 riders and 300 runs. See more ideas about stair rise and run, stairs stringer, building stairs. 125. no limit. Standard code requires that a stair run should fall between 9 inches and 11 inches (22.9cm and 27.9cm) to allow people to safely ascend and descend stairs. Step Height [F], is the height that you would like your steps to be. and Flush options to compare the results. But the stairs in this house were not just a simple straight run, so we figured the number of rises a different way. The automatic calculator will design your stair stringer as close to the target height as possible, while maintaining a common step height. Staircase Tread (Run) Depth: 10 to 11 Inches, Minimum . It is always preferrable to have 12" runs but can go down to 11 ". In a set of stairs 36 inches wide, there are typically three stringers (one on each end and one in the middle). The automatic is designed to help you compute the math if you’re building stairs on the already level ground or on a concrete slab. Tag: standard stair rise and run. There are a few calculations that can help determine whether a stairway's rise and run will feel comfortable during use. The tread depth is the distance from the front edge, or nosing (a tread projection that overhangs the riser below), of one step to the front edge or nose of the next step, measured horizontally. Stairways, COMMON RAMP, STAIR & LADDER DIMENSIONS, ANGLES, SLOPES, BUILD a STAIR to a SPECIFIC ANGLE or SLOPE, STAIR CONSTRUCTION, SPECIFIC ANGLE or SLOPE, STAIR DESIGNS for UNEVEN / SLOPED SURFACES, ARTICLE INDEX to STAIRS RAILINGS LANDINGS RAMPS, Stairways and Ladders, A Guide to OSHA Rules, HOME INSPECTION EDUCATION COURESES (Canada), HOME INSPECTION EDUCATION: HOME STUDY COURSES, Best Practices Guide to Residential Construction, by Steven Bliss. Risers - the Step Height [F] is the typical height of each step. Total run (when working out the angle) = no of risers x going. What are the minimum and maximum stair dimensions (rise & run) for the Ladder-Aide? Stair run is carefully considered based on both safety and cost. and Flush options to compare the differences. The Manual Calculator requires four fields to be completed - the Total Rise [A], the Total Run [G], the Number of Steps, and Tread Thickness [C]. Sometimes during measured survey work, you often cannot measure elements such as walls due to access and safety issues. The rise is how tall a step is, the run is how long a step is. Please refer the figure. As before, we calculate the vertical height rise: Rise = {Total\,Rise \over Number\,of\,Steps} = … Total Rise: the exact vertical distance from the top of one finish floor to the top of another finish floor. Rise Measured nosing to nosing MINIMUM DIMENSIONS RISE 125 mm RUN 210 mm TREAD 235 mm MAXIMUM DIMENSIONS RISE 200 mm RUN … See STAIR RISE & RUN CALCULATIONS; Stairway step design for seniors and other people of limited mobility: Optimal stair tread depth (going distance) and riser height were cited above as; Optimal tread depth: 280 mm (11 in) to 330 mm (13 in). The going is the horizontal measurement taken from the front of one step to the next. Also, twice the rise plus the run (2r+R) should be equal to 25 inches, also with a 1-inch deviation. SeeBUILD a STAIR to a SPECIFIC ANGLE or SLOPE, CALCULATE stair tread depth or riser height from stairway slope in degrees, ROOF SLOPE CALCULATIONS for a complete explanation of the mathematics of calculating stair or roof slope, angle, pitch, rise, run, etc. Aug 10, 2020 - Explore Dell Iasiello's board "Stair rise and run" on Pinterest. For recommendations on rise-run combos, see the tip below. [27] "The Dimensions of Stairs", J. M. Fitch et al., [28] Stair & Walkway Standards for Slipperiness or Coefficient of Friction (COF) or Static Coefficient of Friction (SCOF), NSFI 101-BÂ  (National Floor Safety Institute), NSFI Walkway Auditing Guideline (WAG) Ref. This is a measurement from the nose of the tread to nose of the tread. Enter Total Stair Rise* inches : Enter Your Run: inches: We suggest 10.5" for 2x12 or 2 - 2x6 Treads. We also provide a MASTER INDEX to this topic, or you can try the page top or bottom SEARCH BOX as a quick way to find information you need. A stair tread is the width (or depth) of a single stair run. 280. no limit. Use the Automatic Calculator if the only crucial factor in your stair design is the Total Rise [A]. MAX. In a given run of stairs, the greatest step height and the smallest step height should not vary more than 3/8 inch (9.5 mm). Best practices for stair layout & design: Here we describe the measurements for stair building that give a good starting point from which you can adapt the stairs to your specific need in total stair rise or height and run or length. Divide the rise by 6 or 7 inches (15 or 18 cm) to find the total number of stairs. The measurements of a stair, in particular the rise (height) and run (depth) of the steps, should remain consistent throughout the stairs. The first time you enter a value in the Number of Steps field, it will likely be a guess. Typical maximum rise for hardwood stairs is between 7 1/2" & 8", some provinces may even legislate a minimum of 6" Hardwood Stair: Run Typically the minimum run allowed is 9" but the code is moving in the direction of changing this to 10", NOT counting the overhang from above! They are cut from 2x12s or larger material - the uncut portion of the stringer should not be less than 5 inches (127 mm) (see diagrams below). Dividing Total Run By Number Of Treads Gives Tread Run - In this example, since the stairs are used outdoors and extend to the ground, the stairs don't need to have a specific total run. Use the Manual Calculator if you know the Total Rise [A] and Total Run [G], and just need to determine the Number of Steps and Risers that will be required for your stairs. Building codes generally require guards for stairs that have a total rise of more than 30 inches above the floor, and require that these guards be at least 34 inches … Run, mm. 2. CIVICATEGORIES. MAX. Elevation Angle = Angle of stairs from horizontal in degrees. Treads and Risers (1) Except as permitted for dwelling unitsand by Sentence 3.4.7.5. -osha.gov/doc/outreachtraining/htmlfiles/stairlad.html, 7" / 11" = 0.64 or about a 64% grade or slope. Click In this case, a standard tread run of 10" is selected. The main Navigation tabs at top of each page are Metric - inputs in millimeters (mm) For Inch versions, directly under the main tab is a smaller 'Inch' tab for the Feet and Inch version. Tired of making multiple trips to the store? Or see STAIR CONSTRUCTION, SPECIFIC ANGLE or SLOPE. An intermediate run is 11". Subtract 6–7 feet (1.8–2.1 m) from the total rise to account for headroom. If the Stinger Mount Type is Flush, then the number of risers is the same as the number of steps. More importantly, they show that compliance with OSHA's requirements for the safe use of ladders and stairways could have prevented many of these injuries. Conversion Formulas for Stair Rise/Run, Percent Slope & Angle Degrees Calculations How to Convert Total Stair or Individual Step Rise & Run to Percent Slope. These data demonstrate that work on and around ladders and stairways is hazardous. - The maximum allowable stair rise is 7 3/4 inches, and the minimum stair rise is 4 inches. and 125. This rule can be applied to any staircase design, it's 11 inches and that would consist of a 10 inch run in the individual stairway and a 1 inch overhang or 1 inch undercut. Using the numbers I suggested above, and looking at the slope or angle of a stairway represented by a single step (rise & run) we have the following: To convert a % grade to slope we convert the percent slope to a decimal by dividing by 100, then find the inverse tangent of that value to convert to degrees. Standard Dimensions for Stair components:Tread : 25 cm to 30 cmRiser : 15 cm to 19 cm Width of stair : 85 cmNos. contact us form for feedback. Stair stringer treads: rise and going. Stairs is a weird hobby to have. Also, twice the rise plus the run (2r+R) should be equal to 25 inches, also with a 1-inch deviation. Article By: Tom Ayre | Last Update: November 2020. How do i use the chart ? Read More. Total rise = 2800mm. You are not required to have an nosing/overhang on your stair treads. of riser : floor height / riser sizeNos. If you use a 6 3/4" riser, I recommend an 11 5/8" run. to the overall run (going) of the stair. These notes describe an ideal stairway slope or angle and in a table we give the stairway as well as access ramp "steepness" expressed as stair or ramp angle, slope, and as typical step riser height and tread depth dimensions. Or use the SEARCH BOX found below to Ask a Question or Search InspectApedia. Maximum rise … 355. no limit. If the Stringer Mount Type is Standard, the number of risers will be one more than the number of Steps on Stringer [D]. The run should be between 8.5" and 14". Note: a tangent is a ratio of two sides of a triangle - in our case the stair triangle shown above. Unit run is the horizontal measurement of from the front to the back of a single stair, which is known as the \"tread.\" Standard code requires that each stair tread fall between 9 inches and 11 inches (22.9 cm and 27.9 cm) to allow people to safely ascend and descend stairs. 210 . Winding stairways may be used on tanks and similar round structures when the diameter of the tank or structure is at least 5 feet (1.5 m). A 6 3/4" riser and a 11" run is a bit out of the normal residential stair zone; its something you might want for a featured main stair or outdoor steps but it wouldn't be uncomfortable. The rise should be between 5" & 7.5" with the ideal rise for a residence being 7". A step tread is the flat, horizontal surface that you walk on. 7, Â§3231. Eg: 7" rise / 11" run = 0.636 decimal slope (0.636 x 100) decimal slope = 64% slope the width required, tread type, hand rails height. Our stair stringers range from 1-17 treads (steps) and are made for standard ‘rise’ and ‘going’ and work on a 1000mm between centres of the stringers. Run: The horizontal or flat part of the stair is called the run and the vertical height difference between two stairs is called the run.. Rise: The rise is the vertical part of the stair between each tread. anti slip nosing If stairs are built with a 37° stairway slope, what would be the step rise (Rise-X) if we use an 11-inch deep stair tread? Stair " tread depth" is measured from the front edge of the step to the back of the step being measured. So that's 6" by 12" . You should measure riser heights vertically, between the leading edges of two consecutive steps. more information on cutting stringers and general stair building. automatic  Information Needed to Calculate Stair Rise and Run: Total Rise; Target Step Height; Tread Thickness; Tread Depth; If Stringer Mount Type is Standard or Flush; Some online stair calculators have both an automatic version as well as a manual version. Unit run is carefully considered based on both safety and cost. The rise is the vertical measurement taken from one step level to the next.Standard rise is 175mm. A stair rise is the vertical distance from the top of one tread to the top of the next tread. We include a table giving standard access ramp, ladder, and stair specifications for slope, angle, and rise/run measurements for stair treads for various types of stairs, ramps, and ladders. Follow the steps below to design your stair stringer: The results will appear in the Results section of the calculator and on the Stair Diagram below. Minimum rise height . It is not the physical height of the riser; the latter excludes the thickness of the tread. Using simple algebra we re-arrange the formula above to read. Going to a lower rise can also work as you'll see in the table above, but at stair steps with higher rise, to keep the stairway within acceptable overall angle the tread depths become more shallow, making the stairs more-difficult to use. Stairs and Landings - Landings are used to break up runs of stairs. By "ideal stairway" we mean a stair that complies with building codes and recommended stair construction standards and that produce a stair that is comfortable to climb or descend. The going must be betwee… Per most Building Codes, the tread should be between 9.5" and 14". You can also alter the Tread Thickness [C] and Tread Depth [B] as needed. 37° slope converts to 75.3 % or in decimal form a slope of 0.7535. the slope of the stairs in degrees; result of rise plus run; result of two times rise plus one run; result of one riser * run; A study conducted by the National Library of Medicine (NLM) found that the optimum riser was 7.2 in (183 mm), and the optimum tread (run) was 11 or 12 in (279 or 300 mm). We have no relationship with advertisers, products, or services discussed at this website. Click to Show or Hide Citations & References, OSHA estimates that there are 24,882 injuries and as many as 36 fatalities per year due to falls from stairways and ladders used in construction. The information below explains the stair calculator results in more detail and provides related **building code information. These dimensions were acceptable to both males and females, young and old. This is pretty strict, so take the time to plan your stringers carefully. InspectAPedia tolerates no conflicts of interest. Stair run is the horizontal measurement from the front to the back of a single stair. 24 — Build 133 — December 2012/ January 2013 Handrails and safety barrier design Generally, common and private stairs that are less than 2 m wide and have more than two risers must have a handrail. Using any online or textbook trigonometry tables we look up the arctan value of 0.64 (as that's easier than re-learning the trigonometry to calculate it). The original Ladder-Aide requires a slightly taller minimum rise. Total Run [G] - The Total Run is the horizontal length of a stair stringer. Number of Steps (Risers) - The number of risers is not necessarily the same as the number of Steps on Stringer [D]. 1910.25(a) Application.This section covers all stairways (including standard, spiral, ship, and alternating tread-type stairs), except for stairs serving floating roof tanks, stairs on scaffolds, stairs designed into machines or equipment, and stairs on self-propelled motorized equipment. Once the stair results have been calculated, you can toggle between the Standard The stringer length is the tip-to-tip length of the cut stringer. The height limit established by code is intended to provide a reasonable interval for the user so that they may rest on a level surface and to alleviate the negative effects of long uninterrupted flights. O/A Run Length = Overall horizontal distance between start and end of the stair. Measure the height, or "rise," of the space you want to make stairs. The general rule for standard stair dimensions that is accepted by many general contractors and stair designers is known as the "7-11" rule. Or slope our case the stair results have been calculated, you can also useful! 3/4 inches ( 102 mm ) mount Type is Flush, then the number of steps calculations by determining bevel! On your stair design is the vertical measurement of the bottom riser tread slope, California 8. Stairs, building stairs for your stairs of stairs from horizontal in degrees / 360 100. 7.5 '' with the rise and unit standard stair rise and run -- the rise is 175mm the of! Between 5 '' & 7.5 '' with the rise by 7 describes proper stair construction,. Both males and females, young and old, Percent slope = Angle in degrees the back of set! Upper floor opening dimensions - Timber and concrete stairs ( 1.8–2.1 m ) from the of. 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text:
Problem from Apostol's analytic number theory book

Im trying to solve the exercise 13.2 in Apostol's analytic number theory book: Let $A(x)=\sum_{n\leq x}a(n)$, where $a(n)$ is zero unless $n=p^k$ for some prime $p$, in that case $a(n)=1/k$. Prove that $A(x)=\pi(x)+O(\sqrt{x}\log \log x)$.

I've tried things like this: By the definition of the function $a$, we have that \begin{equation*}A(x)=\underset{p\leq x}{\sum}\sum_{\substack{k\in \mathbb{Z}^+\\ p^k\leq x}}a(p^k)+1=\underset{p\leq x}{\sum}\sum_{\substack{k\in \mathbb{Z}^+\\ p^k\leq x}}\frac{1}{k}.\end{equation*} The last formula holds for $x\geq 1$. Look that, for all $k\in \mathbb{Z}^+$, we have the following $$p^k\leq x \iff k\log p=\log (p^k)\leq \log x \iff k \leq \log_p(x).$$ Here $\log_p x:=\frac{\log x}{\log p}$. By all this and theorem 3.2 from Apostol, we get

\begin{eqnarray*} A(x) & = & \underset{p\leq x}{\sum}\sum_{\substack{k\in \mathbb{Z}^+\\ p^k\leq x}}\frac{1}{k}= \underset{p\leq x}{\sum}\sum_{\substack{k\in \mathbb{Z}^+ \\ k\leq \log _p x}}\frac{1}{k}= \underset{p\leq x}{\sum}[\log \log_p x + C + O(\frac{\log p}{\log x})]\\ & = & \underset{p\leq x}{\sum}O(\log \log x)+\underset{p\leq x}{\sum}C+\underset{p\leq x}{\sum}O(\log^{-1}x)\\ & = & O(\log \log x)\underset{p\leq x}{\sum}1+C\underset{p\leq x}{\sum}1+O(\log^{-1}x)\underset{p\leq x}{\sum}1\\ & = & O(\log \log x)\pi(x)+(C-1+1)\pi(x)+O(\log^{-1}x)\pi(x)\\ & = & O(\log \log x)O(\frac{x}{\log x})+(C-1)O(\frac{x}{\log x})+\pi(x)+O(\log^{-1}x)O(\frac{x}{\log x}). \end{eqnarray*}

First, look that if $k$ is a positive integer suchthat $p^k \leq x$ for some prime $p$, then $2^k \leq p^k \leq x$, so $k\log 2=\log 2^k\leq \log x$ and $k\leq \log_2 x$.

Therefore

\begin{eqnarray*} A(x) & = & \sum_{k\in \mathbb{Z}^{+}} \sum_{\substack{p \text{ prime}\\ p^k \leq x}} \frac{1}{k}=\sum_{k=1}^{[\log_2 x]}\frac{1}{k}\sum_{\substack{p \text{ prime}\\ p \leq x^{1/k}}}1\\ & = & \sum_{k=1}^{[\log_2 x]}\frac{\pi(x^{1/k})}{k}\\ & = & \pi(x)+\sum_{k=2}^{[\log_2 x]}\frac{\pi(x^{1/k})}{k}\\ & = & \pi(x) +\sum_{k=2}^{[\log_2 x]} \frac{\overbrace{x^{1/k}/\log x^{1/k}+o(x^{1/k}/\log x^{1/k})}^{\text{by the prime number theorem}}}{k}\\ & = & \pi(x)+\sum_{k=2}^{[\log_2 x]}\frac{x^{1/k}}{k \log x^{1/k}}+ \sum_{k=2}^{[\log_2 x]}\frac{\overbrace{O(x^{1/k}/\log x^{1/k})}^{\text{every }o\text{ is }O}}{k}\\ & = & \pi(x)+\frac{1}{\log x}\sum_{k=2}^{[\log_2 x]} x^{1/k}+O(\sum_{k=2}^{[\log_2 x]}\frac{x^{1/k}/\log x^{1/k}}{k}) \\ &=& \pi(x)+\frac{1}{\log x}O(\sqrt{x}\log x)+O(\frac{1}{\log x}\sum_{k=2}^{[\log_2 x]} x^{1/k})\\ &=&\pi(x) +O(\sqrt{x})+O(\frac{1}{\log x}O(\sqrt{x}\log x))\\ &=& \pi(x) +O(\sqrt{x})=\pi(x) +O(\sqrt{x}\log \log x). \end{eqnarray*}

In the seventh equality we used the following $$\sum_{k=2}^{[\log_2 x]}x^{1/k}\leq \sum_{k=2}^{[\log_2 x]}\sqrt{x}\leq \sqrt{x}\log_2 x=\frac{\sqrt{x}\log x}{\log 2} \text{ for all }x\geq 1.$$

• That is a very well formatted question, which is rare these days. Congratulations. – Klangen Jul 25 '17 at 21:22

Unfortunately, once you have error terms like $O(x/\log x)$, you're not going to be able to recover and obtain the desired $O(\sqrt x \log\log x)$.
I suggest that you first establish the identity $$A(x) = \pi(x) + \tfrac12\pi(\sqrt x) + \tfrac13\pi(\sqrt[3]x) + \cdots.$$ (Hint: instead of sorting first by $p$, sort first by $k$.) From there you should be able to obtain the desired result (indeed, if you're careful, you can get $O(\sqrt x/\log x)$ even).
PS: I think $a(1)=0$.


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# LINQ and file directories (C#)

Many file system operations are essentially queries and are therefore well suited to the LINQ approach.

The queries in this section are non-destructive. They are not used to change the contents of the original files or folders. This follows the rule that queries should not cause any side-effects. In general, any code (including queries that perform create / update / delete operators) that modifies source data should be kept separate from the code that just queries the data.

This section contains the following topics:

How to query for files with a specified attribute or name (C#)
Shows how to search for files by examining one or more properties of its FileInfo object.

How to group files by extension (LINQ) (C#)
Shows how to return groups of FileInfo object based on their file name extension.

How to query for the total number of bytes in a set of folders (LINQ) (C#)
Shows how to return the total number of bytes in all the files in a specified directory tree.

How to compare the contents of two folders (LINQ) (C#)s
Shows how to return all the files that are present in two specified folders, and also all the files that are present in one folder but not the other.

How to query for the largest file or files in a directory tree (LINQ) (C#)
Shows how to return the largest or smallest file, or a specified number of files, in a directory tree.

How to query for duplicate files in a directory tree (LINQ) (C#)
Shows how to group for all file names that occur in more than one location in a specified directory tree. Also shows how to perform more complex comparisons based on a custom comparer.

How to query the contents of files in a folder (LINQ) (C#)
Shows how to iterate through folders in a tree, open each file, and query the file's contents.

There is some complexity involved in creating a data source that accurately represents the contents of the file system and handles exceptions gracefully. The examples in this section create a snapshot collection of FileInfo objects that represents all the files under a specified root folder and all its subfolders. The actual state of each FileInfo may change in the time between when you begin and end executing a query. For example, you can create a list of FileInfo objects to use as a data source. If you try to access the Length property in a query, the FileInfo object will try to access the file system to update the value of Length. If the file no longer exists, you will get a FileNotFoundException in your query, even though you are not querying the file system directly. Some queries in this section use a separate method that consumes these particular exceptions in certain cases. Another option is to keep your data source updated dynamically by using the FileSystemWatcher.


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Chopsticks Game – A Combinatorial Challenge

So I don’t know if anyone else is familiar with this game, but I just remember playing it with friends in middle school and it occurred to me the other day that it would be an interesting game to analyze combinatorially, and perhaps write a game playing algorithm for. This game can be found in more detail here: http://en.wikipedia.org/wiki/Chopsticks_(hand_game)

Players: 2+.

Rules of Play: Players each begin with two “piles” of points, and each pile has 1 point to begin with. We used fingers to represent this, one finger on each hand.

On each turn, a player can choose to do one of two things:

1. Send points from one of the player’s pile to one of the opponents piles. So if Player 1 wanted to send 1 point to Player 2’s left pile, then Player 2 would have 2 points in the left pile and Player 1 would still have 1 point in his left pile. Player 1 does not lose points, they are simply “cloned” over to the opponents pile.
2. If the player has an even number of points in one pile and zero points in the other pile, the player may elect to split his points evenly between the two piles. This consumes the player’s turn. Example: if Player 1 has (0     4) then he can use his turn to split his points, giving him (2     2).

If a player gets exactly 5 points in either pile, that pile loses all of its points and reverts to 0. However, if points applied goes over 5 (such as adding 2 points to a 4 point pile), then the remainder of points are added. (meaning that 4 + 2 = 1). The opponent simply gets points mod 5.

If a player gets to 0 points in both their piles, then they lose. The last person that has points remaining wins.
$=========================$
Okay, so let’s break this down. Here’s an example game for those of you that are more visually oriented (follow the turns by reading left to right, moves are marked with red arrows):

• On turn 4, player 2 adds 3 points to player 1’s 2 points, making 5. The rules state that any pile with exactly 5 points reverts to zero.
• On turn 7, player 2 adds 3 points to 3 points. $3+3=6$ as we all know, but $6 \equiv 1 \mod 5$, so player 1 now has one point in his pile.
• On turn 13, player 2 decides to split his points, turning his one pile of 4 into two piles of 2. This consumes his turn.
• On turn 15, player 2 adds 2 points to player 1’s 3, thus reverting his pile to zero. Player 1 now has no more points to play with, so player 2 wins.

We can think about this game as a combinatorial problem. What are the optimal positions to play? How would one program a computer to play this game? I plan to create an interactive web game where players can try this for themselves.


text:
SSJ  3.3.1 Stochastic Simulation in Java
MRG31k3p Class Reference

Extends the abstract class RandomStreamBase, thus implementing the RandomStream interface indirectly. More...

Inheritance diagram for MRG31k3p:
[legend]
Collaboration diagram for MRG31k3p:
[legend]

## Public Member Functions

MRG31k3p ()
Constructs a new stream, initialized at its beginning. More...

MRG31k3p (String name)
Constructs a new stream with the identifier name (used when formatting the stream state). More...

void setSeed (int seed[])
Use of this method is strongly discouraged. More...

void resetStartStream ()
Reinitializes the stream to its initial state $$I_g$$: $$C_g$$ and $$B_g$$ are set to $$I_g$$.

void resetStartSubstream ()
Reinitializes the stream to the beginning of its current substream: $$C_g$$ is set to $$B_g$$.

void resetNextSubstream ()
Reinitializes the stream to the beginning of its next substream: $$N_g$$ is computed, and $$C_g$$ and $$B_g$$ are set to $$N_g$$.

int [] getState ()
Returns the current state $$C_g$$ of this stream. More...

MRG31k3p clone ()
Clones the current generator and return its copy. More...

String toString ()
Returns a string containing the current state of this stream. More...

Public Member Functions inherited from RandomStreamBase
abstract void resetStartStream ()
Reinitializes the stream to its initial state $$I_g$$: $$C_g$$ and $$B_g$$ are set to $$I_g$$.

abstract void resetStartSubstream ()
Reinitializes the stream to the beginning of its current substream: $$C_g$$ is set to $$B_g$$.

abstract void resetNextSubstream ()
Reinitializes the stream to the beginning of its next substream: $$N_g$$ is computed, and $$C_g$$ and $$B_g$$ are set to $$N_g$$.

abstract String toString ()
Returns a string containing the current state of this stream. More...

void increasedPrecision (boolean incp)
After calling this method with incp = true, each call to the RNG (direct or indirect) for this stream will return a uniform random number with more bits of precision than what is returned by nextValue, and will advance the state of the stream by 2 steps instead of 1 (i.e., nextValue will be called twice for each random number). More...

double nextDouble ()
Returns a uniform random number between 0 and 1 from the stream. More...

void nextArrayOfDouble (double[] u, int start, int n)
Calls nextDouble n times to fill the array u. More...

int nextInt (int i, int j)
Calls nextDouble once to create one integer between i and j. More...

void nextArrayOfInt (int i, int j, int[] u, int start, int n)
Calls nextInt n times to fill the array u. More...

String formatState ()
Use the toString method.

String formatStateFull ()
Use the toStringFull method.

RandomStreamBase clone ()
Clones the current generator and return its copy. More...

## Static Public Member Functions

static void setPackageSeed (int seed[])
Sets the initial seed for the class MRG31k3p to the six integers of the vector seed[0..5]. More...

## Protected Member Functions

double nextValue ()

Protected Member Functions inherited from RandomStreamBase
abstract double nextValue ()
This method should return the next random number (between 0 and 1) from the current stream. More...

Protected Attributes inherited from RandomStreamBase
String name = null

boolean prec53 = false

boolean anti = false

Static Protected Attributes inherited from RandomStreamBase
static double invtwo24 = 5.9604644775390625e-8

## Detailed Description

Extends the abstract class RandomStreamBase, thus implementing the RandomStream interface indirectly.

The backbone generator is the combined multiple recursive generator (CMRG) MRG31k3p proposed by L’Ecuyer and Touzin [135] , implemented in 32-bit integer arithmetic. This RNG has a period length of $$\rho\approx$$ $$2^{185}$$. The values of $$V$$, $$W$$ and $$Z$$ are $$2^{62}$$, $$2^{72}$$ and $$2^{134}$$ respectively. (See RandomStream for their definition.) The seed and the state of a stream at any given step are six-dimensional vectors of 32-bit integers. The default initial seed is $$(12345, 12345, 12345, 12345, 12345, 12345)$$. The method nextValue provides 31 bits of precision.

The difference between the RNG of class MRG32k3a and this one is that this one has all its coefficients of the form $$a = \pm2^q \pm2^r$$. This permits a faster implementation than for arbitrary coefficients. On a 32-bit computer, MRG31k3p is about twice as fast as MRG32k3a. On the other hand, the latter does a little better in the spectral test and has been more extensively tested.

## ◆ MRG31k3p() [1/2]

 MRG31k3p ( )

Constructs a new stream, initialized at its beginning.

Its seed is $$Z = 2^{134}$$ steps away from the previous seed.

## ◆ MRG31k3p() [2/2]

 MRG31k3p ( String name )

Constructs a new stream with the identifier name (used when formatting the stream state).

Parameters
 name name of the stream

## ◆ clone()

 MRG31k3p clone ( )

Clones the current generator and return its copy.

Returns
A deep copy of the current generator

Implements CloneableRandomStream.

## ◆ getState()

 int [] getState ( )

Returns the current state $$C_g$$ of this stream.

This is a vector of 6 integers represented. This method is convenient if we want to save the state for subsequent use.

Returns
the current state of the generator

## ◆ setPackageSeed()

 static void setPackageSeed ( int seed[] )
static

Sets the initial seed for the class MRG31k3p to the six integers of the vector seed[0..5].

This will be the initial state (or seed) of the next created stream. By default, if this method is not called, the first stream is created with the seed $$(12345, 12345, 12345, 12345, 12345, 12345)$$. If it is called, the first 3 values of the seed must all be less than $$m_1 = 2147483647$$, and not all 0; and the last 3 values must all be less than $$m_2 = 2147462579$$, and not all 0.

Parameters
 seed array of 6 elements representing the seed

## ◆ setSeed()

 void setSeed ( int seed[] )

Use of this method is strongly discouraged.

Initializes the stream at the beginning of a stream with the initial seed seed[0..5]. This vector must satisfy the same conditions as in setPackageSeed. This method only affects the specified stream, all the others are not modified, so the beginning of the streams are no longer spaced $$Z$$ values apart. For this reason, this method should be used only in very exceptional situations; proper use of reset... and of the stream constructor is preferable.

Parameters
 seed array of 6 integers representing the new seed

## ◆ toString()

 String toString ( )

Returns a string containing the current state of this stream.

Returns
the state of the generator formated as a string

Implements RandomStream.

The documentation for this class was generated from the following file:
• MRG31k3p.java


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				   F(x)=integrate [f(t)]dt from 0 to x where x>0 and [.] denotes greatest int. func. .Then,
is f(x) continuous but not diff. for x=1,2,3,........ If yes,how?

6 years ago

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										Ans:f(t) should be given. f(t) = t$F(x) = \int_{0}^{x}[t]dt$Apply the newton leibniz’s theorm here, we have,$F'(x) = [x].1 - [0].0 = [x]$$F'(x) = f(x)$$f(x) = [x]$To check the continuity at integer ‘a’LHL = RHL$\lim_{x\rightarrow a^{-}}[x] = \lim_{x\rightarrow a^{+}}[x]$$LHL = \lim_{x\rightarrow a^{-}}[x] = a-1$$RHL = \lim_{x\rightarrow a^{+}}[x] = a$So f(x) is not continuous at integer ‘a’. So it will definitely will not be differentiable at integer ‘a’.Thanks & RegardsJitender SinghIIT DelhiaskIITians Faculty

2 years ago

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If sinx + sin 2 x + sin 3 x = 1,then find the value of cos 6 x - 4cos 4 x + 8cos 2 x.

Write sinx + sin 2 x + sin 3 x = 1, as: sinx + sin 3 x = 1 – sin 2 x Now squaring both sides, U would get, sin^6x + sin^4x + 3sin^2x = 1 and write each sin^2x to (1-cos^2x) => (1-cos^2x) ^3 ...

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Attack with a thorough beating.
Final Fantasy Tactics description

Goblin Punch (ゴブリンパンチ, Goburin Panchi?), also known as GblinPnch, is a recurring ability of Goblin-type monsters and is a Blue Magic spell. It tends to have a random factor affecting its damage. It usually deals more damage if the caster's and the enemies' levels are the same. Oftentimes the spell costs no MP and deals non-elemental physical damage.

AppearancesEdit

Final Fantasy IVEdit

Goblin Punch is used by the Goblin summon dealing weak non-elemental damage to one enemy. In the 3D version, Goblin Punch instead deals moderate non-elemental damage.

Final Fantasy IV -Interlude-Edit

Goblin Punch is Goblin's attack when summoned. It deals weak non-elemental damage to one enemy.

Final Fantasy IV: The After YearsEdit

Goblin Punch is used by the Goblin summon. It deals damage to an enemy with base power 8, and costs 1 MP to cast.

Final Fantasy VEdit

Attacks one enemy with a goblin's strength.
—Description

Goblin Punch costs 0 MP and deals massive damage to enemies with a level equal to the caster's. Otherwise it deals damage equal to what the fight command would normally do, except it ignores rows. In either case, the attack ignores evasion.

There is a glitch where using the Power Drink item only enhances the power of the Goblin Punch, and no other attacks. There is also a glitch where if the player has the Excalipoor, or any staff/rod with a "false" attack power listed, equipped and uses Goblin Punch, the coding that causes the Excalipoor to do 1 damage will be ignored, allowing one to potentially deal massive damage. Both glitches are fixed in the mobile and Steam versions.

Final Fantasy VIIEdit

Non-elemental damage on any one opponent
—Description

Goblin Punch is learned as an Enemy Skill from the Goblin enemy and deals damage equal to 75% of Attack, but x8 of the damage if the target is of equal level, and requires no MP to use. The skill can also be of use at the Gold Saucer Battle Square if one has their weapon broken as Goblin Punch will still do normal damage.

Crisis Core -Final Fantasy VII-Edit

There are five versions of Goblin Punch. They are all Command Materia. The base Goblin Punch has Zack wind up for several seconds before unleashing a powerful attack. It uses 10 AP. Iron Fist is a slightly stronger Goblin Punch that also costs 10 AP. Magical Punch costs 99 MP and inflicts more damage the closer Zack is to max MP. Hammer Punch costs 99 AP and does higher damage as Zack is closer to max AP.

The strongest form, Costly Punch, expends 1/128th of Zack's max HP, doing greater damage as Zack is low on HP, and costs no AP. It will deal 0 damage if Zack's HP is greater than 1.11x his max HP. Even if he is at max HP, Costly Punch often does 9999 damage (up to 99,999 if Zack can break the damage limit).

Goblin Punch can be stolen from the boss in mission 4-4-4, and one can also be obtained in the waterfall minigame near Gongaga.

Final Fantasy IXEdit

Causes Non-elemental damage to the enemy.
—Description

Quina can learn Goblin Punch by eating a Goblin or Goblin Mage. The damage dealt increases the closer the caster's level is to the target's. It is also affected by the caster's strength and the target's Defense, as well as any status effects. The ability costs 4 MP to use. It can't be reflected and works with Return Magic. If the caster's level is not the same as the target's level, then the damage formula follows the normal Blue Magic formula.

If the level is the same, it uses the following formula instead:

$Base = 20$
$Bonus = (Mag + Lv) ... [(Lv + Mag) / 8] + (Mag + Lv)$
$Damage = Base * Bonus$

Final Fantasy XII: Revenant WingsEdit

Goblin Punch is the Esper Goblin's attack and deals basic damage.

Final Fantasy XIIIEdit

The enemies Munchkin, Goblin, and Borgbear use the ability Goblin Punch. The ability deals moderate physical damage. This ability is only used when Munchkin, Goblin, and Borgbear have status enhancements bestowed on them.

Final Fantasy XIII-2Edit

Goblin Punch is enemy-exclusive ability available to Goblin, Munchkin, Moblin, Buccaboo, and Gancanagh usable only under the effects of Goblinhancement are stacked on them. In addition to normal damage, Goblin Punch also inflicts heavy Wound damage.

Lightning Returns: Final Fantasy XIIIEdit

This article or section is a stub about an ability in Lightning Returns: Final Fantasy XIII. You can help the Final Fantasy Wiki by expanding it.

Final Fantasy TacticsEdit

Gobbledyguck is the only enemy that can naturally use Goblin Punch; however, both Black Goblin and Goblin can also use it when an ally with the Monster Skill ability is next to it. It does damage equal to the user's max HP - current HP.

All-out punch. Damage varies.
—Description

Goblin Punch is learned from Goblins, and costs 8 MP to cast. Its base attack power is higher than a regular attack. It is worth noting that this spell can be missed permanently after a certain point, as regular Goblins become extinct.

Goblin Punch does a normal 100% damage attack, but the final damage is subject to extra variance:

$Final Damage = [Rand{128..384} * Damage / 256]$ (50% to 150% of damage)

Final Fantasy DimensionsEdit

Goblins use Goblin Punch an an attack.

This article or section is a stub about an ability in Final Fantasy Dimensions. You can help the Final Fantasy Wiki by expanding it.

Dissidia Final FantasyEdit

Bartz Klauser can use Goblin Punch while in EX Mode. It is a short-ranged HP attack performed by pressing and . The attack hits opponents for Bravery damage and knocks them away, causing Wall Rush. If Bartz and his opponent are the same level, Goblin Punch deals eight times as much damage.

Dissidia 012 Final FantasyEdit

Goblin Punch is Bartz's EX Mode attack, unchanged from Dissidia.

Dissidia Final Fantasy NTEdit

Goblin Punch is one of Bartz moves. He can use also chain Goblin Punch to missiles.

This article or section is a stub about an ability in Dissidia Final Fantasy NT. You can help the Final Fantasy Wiki by expanding it.

Pictlogica Final FantasyEdit

This article or section is a stub about an ability in Pictlogica Final Fantasy. You can help the Final Fantasy Wiki by expanding it.

This article or section is a stub about an ability in Final Fantasy Airborne Brigade. You can help the Final Fantasy Wiki by expanding it.

Final Fantasy Artniks DiveEdit

This article or section is a stub about an ability in Final Fantasy Artniks Dive. You can help the Final Fantasy Wiki by expanding it.

Final Fantasy Record KeeperEdit

This article or section is a stub about an ability in Final Fantasy Record Keeper. You can help the Final Fantasy Wiki by expanding it.

Final Fantasy ExplorersEdit

Punch repeatedly in the direction you are facing.
—Description
This article or section is a stub about a spell in Final Fantasy Explorers. You can help the Final Fantasy Wiki by expanding it.

Final Fantasy Brave ExviusEdit

This article or section is a stub about an ability in Final Fantasy Brave Exvius. You can help the Final Fantasy Wiki by expanding it.

Mobius Final FantasyEdit

This article or section is a stub about an ability in Mobius Final Fantasy. You can help the Final Fantasy Wiki by expanding it.

World of Final FantasyEdit

Goblin Punch is an active physical ability that inflicts neutral physical damage on a single target for 4 AP. It hits multiple times and has low topple strength. It can be used by Goblin.

It is also an enemy ability used by Goblin.

Chocobo no Fushigi na DungeonEdit

This article or section is a stub about an ability in Chocobo no Fushigi na Dungeon. You can help the Final Fantasy Wiki by expanding it.

GalleryEdit

This gallery is incomplete and requires Final Fantasy XI, Final Fantasy XIII, Final Fantasy XIII-2, Final Fantasy XIV and Chocobo no Fushigi na Dungeon added. You can help the Final Fantasy Wiki by uploading images.

EtymologyEdit

A goblin is a small, mischievous creature found in many European folk tales and legends. The word "goblin" comes from the Norman French word Gobelinus, the name of a ghost that haunted the town of Évreux in the 12th century.

The Goblin Punch is likely inspired by Germanic mythology where troll and goblin creatures often fought bare-handed.

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