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In physics, work is what force does.
When a force is applied on an object for a certain amount of time, the work done by the force is defined as the magnitude of the force (its strength) times the displacement of the object from its initial position to its current position (shortest distance between point A and B), times the cosine of the angle between the direction of the force and the actual direction of the displacement of the object. This can be represented by:
{\displaystyle W=Fs\cos \theta }
In this formula, work is W, the force's magnitude is F, the displacement is s, and the angle's cosine is cos θ.
In most circumstances, the last factor (the cosine of the angle) is equal to one because the direction of the force is usually the same direction of the displacement. But in circumstances like pushing a heavy box at an angle not parallel to the ground (let's say pushing it at 20° towards the ground), the direction of the force is not the same as the direction of the displacement. In this example, because of the angle of the applied force, the force is doing less work because it is not as efficient as pushing the box at an angle parallel to the ground.
The more the direction of the displacement gets perpendicular (90°) to the direction of the force, the more the work approaches zero. If the angle is greater than 90°, that indicates the object is overall moving in a backwards direction from what the force intends to do; the force has negative work.
One useful example is a scenario of a group of people playing tug of war. The team that is slowly getting pulled into the middle is exerting a force away from the middle, but despite the best efforts of their forces, there is negative work because the direction of their displacement is in the opposite direction.
Holding a heavy book against its weight produces a force, but if the object is stationary in the air, no work is done. If the object slowly moves upwards, work is positive, but if instead it's slowly descending despite the force's exertion on it, the force has negative work.
We can also look at this from the perspective of the force that is trying to move the book downwards, its weight. The work of the weight on a book being lifted is negative. This is because the downward weight is in the opposite direction to the upward displacement.
Work is considered done when it is positive.
Like energy, it is a scalar quantity, and its SI unit is the joule.[2] Heat conduction is not considered to be a form of work, since there is no macroscopically measurable force, only microscopic forces occurring in atomic collisions. The term work was created in the 1830s by the French mathematician Gaspard-Gustave Coriolis.[3]
According to the work-energy theorem, if an external force acts upon a rigid object, causing its kinetic energy to change from Ek1 to Ek2, then the mechanical work (W) is given by:[4]
{\displaystyle W=\Delta E_{k}=E_{k_{2}}-E_{k_{1}}={\frac {mv_{2}^{2}}{2}}-{\frac {mv_{1}^{2}}{2}}}
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\textcolor[rgb]{0.407843137254902,0.250980392156863,0.36078431372549}{\mathrm{ω}}
\mathrm{ω}
\mathrm{π}:E→M
be a fiber bundle, with base dimension
m
{\mathrm{π}}^{\mathrm{∞}}:{J}^{\mathrm{∞}}\left(E\right) → M
E
({x}^{i}, {u}^{\mathrm{α}}, {u}_{{i}_{}}^{\mathrm{α}}, {u}_{{i}_{}j}^{\mathrm{α}}
{u}_{\mathrm{ij} \cdot \cdot \cdot k}^{\mathrm{α}}, ....)
{\mathrm{Θ}}^{\mathrm{α}} = {\mathrm{du}}^{\mathrm{α}}-{u}_{\mathrm{ℓ}}^{\mathrm{α}}{\mathrm{dx}}^{\mathrm{ℓ}}
{\mathrm{\Omega }}^{\left(n,s\right)}\left({J}^{\infty }\left(E\right)\right)
n
s.
\mathrm{ω} ∈{\mathrm{\Omega }}^{\left(n,s\right)}\left({J}^{\infty }\left(E\right)\right)
{E}_{\mathrm{α}}\left(\mathrm{ω}\right) ∈ {\mathrm{Ω}}^{\left(n-1,s\right)}\left({J}^{\infty }\left(E\right)\right)
\mathrm{ω}
I: {\mathrm{\Omega }}^{\left(n,s\right)}\left({J}^{\infty }\left(E\right)\right)→{\mathrm{\Omega }}^{\left(n,s\right)}\left({J}^{\infty }\left(E\right)\right)
I\left(\mathrm{ω}\right) = \frac{1}{s}{\mathrm{Θ}}^{\mathrm{α} }∧{E}_{\mathrm{α}}\left(\mathrm{ω}\right).
I
\mathrm{η}
\left(n-1, s\right),
I\left({d}_{H}\mathrm{η}\right) = 0
{d}_{H }\mathrm{η}
\mathrm{η}
\mathrm{ω}
\left(n,s\right)
I\left(\mathrm{ω}\right) =0,
\left(n-1, s\right)
\mathrm{ω} = {d}_{H }\mathrm{η}
I
I∘I = I
\textcolor[rgb]{0.407843137254902,0.250980392156863,0.36078431372549}{\mathrm{\omega }}
\left(n, s\right)
I\left(\mathrm{ω}\right)
{J}^{3}\left(E\right)
E
\left(x,u\right)→ x.
{\mathrm{ω}}_{1}
\textcolor[rgb]{0,0,1}{\mathrm{ω1}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)]]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{d}}_{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{c}}_{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{b}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{a}]]]\right)
{\mathrm{ω}}_{2}
\textcolor[rgb]{0,0,1}{\mathrm{ω2}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}]]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{ω3}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\frac{{\textcolor[rgb]{0,0,1}{c}}_{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{b}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{a}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{c}}_{\textcolor[rgb]{0,0,1}{x}}}{\textcolor[rgb]{0,0,1}{2}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{c}]]]\right)
{\mathrm{ω}}_{3}
{\mathrm{ω}}_{3}
\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\frac{{\textcolor[rgb]{0,0,1}{c}}_{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{b}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{a}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{c}}_{\textcolor[rgb]{0,0,1}{x}}}{\textcolor[rgb]{0,0,1}{2}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{c}]]]\right)
{J}^{3}\left(E\right)
E
\left(x,y, u, v\right)→ \left(x,y\right)
{\mathrm{ω}}_{4}.
\textcolor[rgb]{0,0,1}{\mathrm{ω4}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{e}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{8}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{f}]]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{d}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{c}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{a}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{f}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{e}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{b}]]]\right)
{\mathrm{ω}}_{5}.
\textcolor[rgb]{0,0,1}{\mathrm{ω5}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}]]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\frac{{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{x}}}{\textcolor[rgb]{0,0,1}{2}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{12}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{a}}{\textcolor[rgb]{0,0,1}{2}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}\frac{{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{x}}}{\textcolor[rgb]{0,0,1}{2}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{9}]\textcolor[rgb]{0,0,1}{,}\frac{\textcolor[rgb]{0,0,1}{a}}{\textcolor[rgb]{0,0,1}{2}}]]]\right)
{\mathrm{ω}}_{6}
\mathrm{η}.
\textcolor[rgb]{0,0,1}{\mathrm{\eta }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{9}]\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}}]]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{ω6}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{9}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{16}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{9}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{13}]\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{9}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{11}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}}]]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}]]]\right)
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Kerbin/th - Kerbal Space Program Wiki
Kerbin/th
Kerbin as seen from orbit.
Longitude of the ascending node 0 °
Synodic orbital period Not defined
Orbital velocity 9 285 m/s
Standard gravitational parameter 3.5316000×1012 m3/s2
Surface gravity 9.81 m/s2 (1 g)
Escape velocity 3 431.03 m/s
Solar day 21 600.000 s
Sidereal rotational velocity 174.94 m/s
Synchronous orbit 2 863.33 km
Atmospheric pressure 101.325 kPa
Temperaturemin -86.20 °C 186.95 K
Temperaturemax 15 °C 288.15 K
Oxygen present Yes
Splashed 0.4
Lower atmosphere 0.7
Upper atmosphere 0.9
เคอร์บิน เป็นบ้านเกิดของเคอร์บัล, เป็นที่ตั้งของศูนย์อวกาศเคอร์บัล, และเป็นจุดสำคัญของเคอร์บัลสเปซโปรแกรม. มันเป็นตัวแทนของโลกสำหรับเกมแต่มีดวงจันทร์สองดวง ชื่อ Mun และ Minmus.
เคอร์บินเป็นดาวลำดับที่สามในวงโคจรรอบดาวชื่อKerbol. มันเป็นวัตถุอวกาศที่ใหญ่ที่สุดอันดับสามที่โคจร Kerbol, following Jool and Eve. Jool's moon Tylo has the same radius of Kerbin, though it may be classified as larger, as the highest point on Tylo is about 5 km higher than the highest point on Kerbin. However, Tylo has only 80% of Kerbin's mass.
Reaching a stable orbit around Kerbin is one of the first milestones a player might achieve in the game. Doing so with a fuel-optimal ascent[1] requires a delta-v of ≈4500 m/s,[2] (or about 3500 if one is using Ferram Aerospace Research mod) and is the second highest value after Eve. Many interplanetary missions expend over half of their delta-V in reaching Kerbin orbit. The energy required to escape a body from a given altitude is always exactly twice the kinetic energy of a circular orbit around the body at that height, leading one observer to remark:
“ If you can get your ship into orbit, you're halfway to anywhere.
— Robert Heinlein, quoted on page 194 of A Step Farther Out by Jerry Pournelle
3.1 Planetary biome list
3.2 KSC location biome list
4.1 Terminal Velocity Table
“ A unique world, Kerbin has flat plains, soaring mountains and wide, blue oceans. Home to the Kerbals, it has just the right conditions to support a vast, seemingly undepletable population of the eager green creatures.
Reaching a stable orbit around Kerbin is one of the first things budding space programs strive for. It is said that he who can get his ship into orbit is halfway to anywhere.
Topographical representation of Kerbin's surface as of 0.18.2. Click for high resolution. by Zeroignite
Kerbal at Kerbin's highest peak
Kerbin has a roughly equal distribution of surface liquid water and solid land, with polar icecaps and scattered deserts. Some of its mountains exceed 6 km in height, with the tallest peak being 6764 m in altitude at the coordinates 46.3° E 61.6° N. The lowest point is almost 1.4 km deep and about 313° south-west of the Kerbal Space Center.
Terrain model centered on one of Kerbin's most pronounced craters
Unlike other bodies in its system, Kerbin has few visible craters because its environment would erode craters from the few meteors that avoid the gravity or surface of its large moon and survive entry. Nevertheless, some geological formations indicate that bodies have violently collided with Kerbin: two planetary features appear to be impact craters that are coincidentally separated by nearly 180 degrees. The least eroded, and therefore presumably youngest, of the two (both are in excess of 100 km diameter) lies along the coastline. The uplift is easily visible as a series of islands, and the feature has a central peak that pokes up through the water (also known as a rebound peak.) The other, and older of the two, is near the prime meridian in the northern hemisphere and is more easily missed, but its uplift rims are visible, and it has a central rebound peak.
The biomes on Kerbin
Before 0.90 Kerbin was one of the few bodies with multiple biomes, Kerbin was second only to the Mun in number of biomes it has. Following the 0.90 update all celestial bodies have biomes. Science experiments can be performed at all biomes, though Kerbin's low multipliers result in less impressive results than more distant worlds. Kerbin's biomes show a loose correlation with Earth's biomes and geographic features. Uniquely, Kerbin has 33 location biomes at KSC, these are comprised of each building and their props, the crawlerway, the flag, and KSC itself; these give a jumpstart to gathering Science points in Career mode.
Planetary biome list
Kerbin biome map as of 0.90.0
KSC location biome list
SPH Main Building (Roof)
VAB Main Building (Roof)
Kerbin's atmosphere contains oxygen and extends to roughly 69,078 meters. Its atmosphere exponentially rarefies with altitude with a scale height of 5 km.[3] The atmospheric pressure on Kerbin at an altitude expressed in meters, generally is:
{\displaystyle p_{k}=1{\text{atm}}\cdot e^{\frac {-altitude}{5000}}}
The thickness of Kerbin's atmosphere makes it suitable for aerobraking and using parachutes to save fuel during reentry and landing. Since version 0.19, harmless supersonic and shock heating/reentry effects have been applied to objects flying above certain velocities. Debris flying in the lower atmosphere disappears once 2 km from an active craft, but above approx. 23 km (where atmospheric pressure is less than 0.01 atm) debris persists. Spent stages may continue in a stable orbit even if they are going through thick atmosphere that would destabilize the orbit of an active craft[outdated].
The following table gives approximation of terminal velocities at different Kerbin altitudes, which are also the velocities at which a ship should travel for a fuel-optimal vertical ascent from Kerbin, given the game's model of atmospheric drag.[4] The optimal velocity after a gravity turn has been started is less than the corresponding value in the table.[5]
Terminal Velocity Table
A Stayputnik MK2 satellite
A synchronous orbit is achieved with a semi-major axis of 3 463.33 km. Kerbisynchronous Equatorial Orbit (KEO) has a circularly uniform altitude of 2 863.33 km and a speed of 1 009.81 m/s. From a 70 km low equatorial orbit, the periapsis maneuver requires 676.5 m/s and the apoapsis maneuver requires 434.9 m/s. A syncronous Tundra orbit with eccentricity of 0.2864 and inclination of 63 degrees is achieved at 3799.7/1937.7 km. Inclination correlates with eccentricity: higher inclined orbits need to be more eccentric, while equatorial orbit may be circular, essentially KEO.
A semi-synchronous orbit with an orbital period of ½ of Kerbin's rotation period (2 h 59 m 34.7 s or 10774.7 seconds) is achieved at an altitude of 1 581.76 km with an orbital velocity of 1 272.28 m/s. A semi-syncronous Molniya orbit with eccentricity of 0.742[6] and inclination of 63 degrees can not be achieved, because the periapsis would be 36 km below the ground. The highest eccentricity of a semi-synchronous orbit with a periapsis of 70 km is 0.693 with an apoapsis of 3100.36 km.
The Hill sphere (the radius around the planet at which moons are gravitationally stable) of Kerbin is 136 185 km, or roughly 227 Kerbin radii.
From the lowest stable orbit around Kerbin (70 km), the amount of delta-V needed to reach the orbits of other celestials is:
Kerbol escape ~2740 m/s
For comparison, the Δv required to reach geostationary Kerbin orbit from LKO is 1.12 km/s
Topographical map of Kerbin as of version 0.21
A projection map of Kerbin, as of 0.14.1 and before (including the old demo).
An accurate full-colour projection map of Kerbin as of 0.14.2 to 0.17.1
A map displaying the delta V needed from/to Kerbin
Kerbin, Mün and Minmus.
Kerbin with its two natural satellites, the Mun and Minmus, seen from KEO. Minmus can be seen just above the horizon of Kerbin on the left side of the picture.
A topographic heightmap of Kerbin made with the ISA MapSat plugin
Kerbin and the Mun, barely visible from ~500,550,000m
An unmanned probe on escape trajectory, flying past Kerbin, Mün and Minmus
Kerbin's North Pole
A solar eclipse from low Kerbin orbit (LKO).
Kerbin as seen from munar orbit.
Kerbin modified with some texture mods.
The second launch site (currently unused).
Kerbin from above with the Runway Easter Egg
Terrain revised to produce more detailed and interesting landforms.
Fixed ladders on the fuel tanks near the launchpad.
New mesh for the launchpad and area (no launchtower anymore).
New mesh for the runway, with lights and sloping edges for rovers.
Terrain overhaul: Entire planet redo. Deserts, darker and greener grass, islands, darker ocean/water, snow capped mountains. Looks more realistic.
Airport added to island off of KSC coastline. (Not a launching point)
Minmus added.
Mün added.
Terrain overhaul, oceans became wet.
Atmosphere extended from ~34,500 m to ~69,000 m.
Kerbin's continents are derived from libnoise,[8] a coherent noise generating library, though they have been increasingly modified with time.
↑ A fuel-optimal ascent is one which (a) minimizes velocity losses to gravity and atmospheric drag and (b) launches eastward (toward the 90 degree heading) to gain 174.5 m/s of orbital velocity for free, thanks to Kerbin's rotation.
↑ See this challenge on the forum and a popular Kerbin delta-V chart
↑ “Some Major Orbit Types” uses that, the Wikipedia article mentions 0.74105, and “Orbital Parameters of a Molniya Orbit” uses 0.72.
ระบบ Kerbol
Retrieved from "https://wiki.kerbalspaceprogram.com/index.php?title=Kerbin/th&oldid=103019"
Celestials/th
Planets/th
Thai Language pages
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Returns to scale - WikiMili, The Best Wikipedia Reader
Find sources: "Returns to scale" – news · newspapers · books · scholar · JSTOR (July 2016) (Learn how and when to remove this template message)
In economics, returns to scale describe what happens to long run returns as the scale of production increases, when all input levels including physical capital usage are variable (able to be set by the firm). The concept of returns to scale arises in the context of a firm's production function. It explains the long run linkage of the rate of increase in output (production) relative to associated increases in the inputs (factors of production). In the long run, all factors of production are variable and subject to change in response to a given increase in production scale. While economies of scale show the effect of an increased output level on unit costs, returns to scale focus only on the relation between input and output quantities.
There are three possible types of returns to scale: increasing returns to scale, constant returns to scale, and diminishing (or decreasing) returns to scale. If output increases by the same proportional change as all inputs change then there are constant returns to scale (CRS). If output increases by less than the proportional change in all inputs, there are decreasing returns to scale (DRS). If output increases by more than the proportional change in all inputs, there are increasing returns to scale (IRS). A firm's production function could exhibit different types of returns to scale in different ranges of output. Typically, there could be increasing returns at relatively low output levels, decreasing returns at relatively high output levels, and constant returns at some range of output levels between those extremes.[ citation needed ]
When the usages of all inputs increase by a factor of 2, new values for output will be:
Assuming that the factor costs are constant (that is, that the firm is a perfect competitor in all input markets) and the production function is homothetic, a firm experiencing constant returns will have constant long-run average costs, a firm experiencing decreasing returns will have increasing long-run average costs, and a firm experiencing increasing returns will have decreasing long-run average costs. [1] [2] [3] However, this relationship breaks down if the firm does not face perfectly competitive factor markets (i.e., in this context, the price one pays for a good does depend on the amount purchased). For example, if there are increasing returns to scale in some range of output levels, but the firm is so big in one or more input markets that increasing its purchases of an input drives up the input's per-unit cost, then the firm could have diseconomies of scale in that range of output levels. Conversely, if the firm is able to get bulk discounts of an input, then it could have economies of scale in some range of output levels even if it has decreasing returns in production in that output range.
Formally, a production function
{\displaystyle \ F(K,L)}
is defined to have:
Constant returns to scale if (for any constant a greater than 0)
{\displaystyle \ F(aK,aL)=aF(K,L)}
(Function F is homogeneous of degree 1)
Increasing returns to scale if (for any constant a greater than 1)
{\displaystyle \ F(aK,aL)>aF(K,L)}
Decreasing returns to scale if (for any constant a greater than 1)
{\displaystyle \ F(aK,aL)<aF(K,L)}
In a more general set-up, for a multi-input-multi-output production processes, one may assume technology can be represented via some technology set, call it
{\displaystyle \ T}
, which must satisfy some regularity conditions of production theory. [4] [5] [6] [7] [8] In this case, the property of constant returns to scale is equivalent to saying that technology set
{\displaystyle \ T}
is a cone, i.e., satisfies the property
{\displaystyle \ aT=T,\forall a>0}
. In turn, if there is a production function that will describe the technology set
{\displaystyle \ T}
it will have to be homogeneous of degree 1.
The Cobb–Douglas functional form has constant returns to scale when the sum of the exponents is 1. In that case the function is:
{\displaystyle \ F(K,L)=AK^{b}L^{1-b}}
{\displaystyle A>0}
{\displaystyle 0<b<1}
{\displaystyle \ F(aK,aL)=A(aK)^{b}(aL)^{1-b}=Aa^{b}a^{1-b}K^{b}L^{1-b}=aAK^{b}L^{1-b}=aF(K,L).}
Here as input usages all scale by the multiplicative factor a, output also scales by a and so there are constant returns to scale.
But if the Cobb–Douglas production function has its general form
{\displaystyle \ F(K,L)=AK^{b}L^{c}}
{\displaystyle 0<b<1}
{\displaystyle 0<c<1,}
then there are increasing returns if b + c > 1 but decreasing returns if b + c < 1, since
{\displaystyle \ F(aK,aL)=A(aK)^{b}(aL)^{c}=Aa^{b}a^{c}K^{b}L^{c}=a^{b+c}AK^{b}L^{c}=a^{b+c}F(K,L),}
which for a > 1 is greater than or less than
{\displaystyle aF(K,L)}
as b+c is greater or less than one.
Diseconomies of scale and Economies of scale
In microeconomics, economies of scale are the cost advantages that enterprises obtain due to their scale of operation, with cost per unit of output decreasing which causes scale increasing. At the basis of economies of scale there may be technical, statistical, organizational or related factors to the degree of market control.
Growth accounting is a procedure used in economics to measure the contribution of different factors to economic growth and to indirectly compute the rate of technological progress, measured as a residual, in an economy. Growth accounting decomposes the growth rate of an economy's total output into that which is due to increases in the contributing amount of the factors used—usually the increase in the amount of capital and labor—and that which cannot be accounted for by observable changes in factor utilization. The unexplained part of growth in GDP is then taken to represent increases in productivity or a measure of broadly defined technological progress.
In economics, elasticity is the measurement of the percentage change of one economic variable in response to a change in another.
In economics, marginal cost is the change in the total cost that arises when the quantity produced is incremented by one unit; that is, it is the cost of producing one more unit of a good. Intuitively, marginal cost at each level of production includes the cost of any additional inputs required to produce the next unit. At each level of production and time period being considered, marginal costs include all costs that vary with the level of production, whereas other costs that do not vary with production are fixed and thus have no marginal cost. For example, the marginal cost of producing an automobile will generally include the costs of labor and parts needed for the additional automobile but not the fixed costs of the factory that have already been incurred. In practice, marginal analysis is segregated into short and long-run cases, so that, over the long run, all costs become marginal. Where there are economies of scale, prices set at marginal cost will fail to cover total costs, thus requiring a subsidy. Marginal cost pricing is not a matter of merely lowering the general level of prices with the aid of a subsidy; with or without subsidy it calls for a drastic restructuring of pricing practices, with opportunities for very substantial improvements in efficiency at critical points.
In economics and econometrics, the Cobb–Douglas production function is a particular functional form of the production function, widely used to represent the technological relationship between the amounts of two or more inputs and the amount of output that can be produced by those inputs. The Cobb–Douglas form was developed and tested against statistical evidence by Charles Cobb and Paul Douglas between 1927–1947; according to Douglas, the functional form itself was developed earlier by Philip Wicksteed.
A production–possibility frontier (PPF), production possibility curve (PPC), or production possibility boundary (PPB), or Transformation curve/boundary/frontier is a curve which shows various combinations of the amounts of two goods which can be produced within the given resources and technology/a graphical representation showing all the possible options of output for two products that can be produced using all factors of production, where the given resources are fully and efficiently utilized per unit time. A PPF illustrates several economic concepts, such as allocative efficiency, economies of scale, opportunity cost, productive efficiency, and scarcity of resources.
In economics, a production function gives the technological relation between quantities of physical inputs and quantities of output of goods. The production function is one of the key concepts of mainstream neoclassical theories, used to define marginal product and to distinguish allocative efficiency, a key focus of economics. One important purpose of the production function is to address allocative efficiency in the use of factor inputs in production and the resulting distribution of income to those factors, while abstracting away from the technological problems of achieving technical efficiency, as an engineer or professional manager might understand it.
In economics, average cost or unit cost is equal to total cost (TC) divided by the number of units of a good produced :
In economics and in particular neoclassical economics, the marginal product or marginal physical productivity of an input is the change in output resulting from employing one more unit of a particular input, assuming that the quantities of other inputs are kept constant.
In economics, diminishing returns is the decrease in the marginal (incremental) output a production process as the amount of a single factor of production is incrementally increased, while the amounts of all other factors of production stay constant.
In economics, output elasticity is the percentage change of output divided by the percentage change of an input. It is sometimes called partial output elasticity to clarify that it refers to the change of only one input.
The Heckscher–Ohlin model is a general equilibrium mathematical model of international trade, developed by Eli Heckscher and Bertil Ohlin at the Stockholm School of Economics. It builds on David Ricardo's theory of comparative advantage by predicting patterns of commerce and production based on the factor endowments of a trading region. The model essentially says that countries export products that use their abundant and cheap factors of production, and import products that use the countries' scarce factors.
In economics, a cost curve is a graph of the costs of production as a function of total quantity produced. In a free market economy, productively efficient firms optimize their production process by minimizing cost consistent with each possible level of production, and the result is a cost curve. Profit-maximizing firms use cost curves to decide output quantities. There are various types of cost curves, all related to each other, including total and average cost curves; marginal cost curves, which are equal to the differential of the total cost curves; and variable cost curves. Some are applicable to the short run, others to the long run.
Constant elasticity of substitution (CES), in economics, is a property of some production functions and utility functions. Several economists have featured in the topic and have contributed in the final finding of the constant. They include Tom McKenzie, John Hicks and Joan Robinson. The vital economic element of the measure is that it provided the producer a clear picture of how to move between different modes or types of production.
In economics, supply is the amount of a resource that firms, producers, labourers, providers of financial assets, or other economic agents are willing and able to provide to the marketplace or directly to another agent in the marketplace. Supply can be in currency, time, raw materials, or any other scarce or valuable object that can be provided to another agent. This is often fairly abstract. For example in the case of time, supply is not transferred to one agent from another, but one agent may offer some other resource in exchange for the first spending time doing something. Supply is often plotted graphically as a supply curve, with the quantity provided plotted horizontally and the price plotted vertically.
In economics, the marginal product of labor (MPL) is the change in output that results from employing an added unit of labor. It is a feature of the production function, and depends on the amounts of physical capital and labor already in use.
The AK model of economic growth is an endogenous growth model used in the theory of economic growth, a subfield of modern macroeconomics. In the 1980s it became progressively clearer that the standard neoclassical exogenous growth models were theoretically unsatisfactory as tools to explore long run growth, as these models predicted economies without technological change and thus they would eventually converge to a steady state, with zero per capita growth. A fundamental reason for this is the diminishing return of capital; the key property of AK endogenous-growth model is the absence of diminishing returns to capital. In lieu of the diminishing returns of capital implied by the usual parameterizations of a Cobb–Douglas production function, the AK model uses a linear model where output is a linear function of capital. Its appearance in most textbooks is to introduce endogenous growth theory.
↑ Gelles, Gregory M.; Mitchell, Douglas W. (1996). "Returns to scale and economies of scale: Further observations". Journal of Economic Education . 27 (3): 259–261. doi:10.1080/00220485.1996.10844915. JSTOR 1183297.
↑ Frisch, R. (1965). Theory of Production . Dordrecht: D. Reidel.
↑ Ferguson, C. E. (1969). The Neoclassical Theory of Production and Distribution. London: Cambridge University Press. ISBN 978-0-521-07453-7.
Susanto Basu (2008). "Returns to scale measurement," The New Palgrave Dictionary of Economics , 2nd Edition. Abstract.
John Eatwell (1987). "Returns to scale," The New Palgrave: A Dictionary of Economics , v. 4, pp. 165–66.
Zelenyuk, Valentin (2013). "A scale elasticity measure for directional distance function and its dual: Theory and DEA estimation". European Journal of Operational Research. 228 (3): 592–600. doi:10.1016/j.ejor.2013.01.012.
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Order Theory | Brilliant Math & Science Wiki
Noel Lo, Shreyansh Singh Solanki, and Jimin Khim contributed
Mr and Mrs Tan have five children: four boys and a girl. They compared their ages and this is what the children said:
Alfred: "I am the second place, counting either from the oldest or the youngest."
Brenda: "I have at least one younger brother."
Charles: "I am two places older than my sister, Brenda."
(i.e.. there is one sibling older than Brenda but younger than Charles)
Darius: "Brenda is my younger sister."
Eric: "I am the...."
Unfortunately, Eric was interrupted. Can you complete his sentence?
There are three possibilities for the positions (in order of birth) of Charles and Brenda:
\left(1^\text{st}, 3^\text{rd}\right), \left(2^\text{nd}, 4^\text{th}\right), \left(3^\text{rd}, 5^\text{th}\right).
The second possibility is out as we see that Alfred is either
2^\text{nd}
4^\text{th}
so if the second possibility is the case, then there is no possible position for Alfred. The third possibility is also out as Brenda has at least one younger brother so she cannot be the youngest. This leaves us with the first possibility where Charles is the oldest and Brenda is the middle child.
We also see that Darius is older than Brenda, the
3^\text{rd}
child. The first place has already been taken by Charles so Darius must be the second oldest. Alfred is either
2^\text{nd}
4^\text{th}
2^\text{nd}
has been taken by Darius so Alfred is the second youngest. The only vacancy left for Eric is the
5^\text{th}
child. In other words, Eric is the youngest.
_\square
Cite as: Order Theory. Brilliant.org. Retrieved from https://brilliant.org/wiki/order-theory/
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Does cross multiply always work for inequalities? | Brilliant Math & Science Wiki
Pi Han Goh, Calvin Lin, Zandra Vinegar, and
a,b,c,d
b
and
being non-zero, if
\dfrac ab > \dfrac cd
a \times d > c \times b
It looks intuitive: for this problem, cross multiply is the same as multiplying both sides by
b\times d
\dfrac a{\cancel b} \times \cancel bd > \dfrac c{\cancel d} \times b \cancel d ,
a \times d > c\times b
There are other cases to consider besides positive real numbers. In those other cases, this "identity" might fail with one of
b,d
being negative.
\color{#D61F06} {\textbf{false}}
. The claim is true if and only if the denominators are both positive or negative. In particular, for
a=b=c=1
d= -1
, while the constraint of
\dfrac ab > \dfrac cd
is indeed fulfilled, the claim of
a\times d > c\times b
is false because
-1 > 1
is not true.
The reason our initial claim fails is because once we multiply both sides of an inequality by a negative number, the inequality sign must be flipped. As an explicit example, the inequality
-1 > -2
is obviously true. But if we multiply both sides by
-1
, while keeping the inequality sign the same, we have
1 > 2,
which is obviously false.
Rebuttal: Because
a= b= c=1
d=2
makes both
\dfrac ab > \dfrac cd
a\times d > b\times c
Reply: We have only shown that it's true when
a=b=c=1
d=2
. In fact, we should prove (or disprove) that it's true in general. That is, we did not prove that the claim holds for all reals
a,b,c
and
b,d\ne 0
Rebuttal: If we square both sides of the inequality, we get
\dfrac{a^2}{b^2} > \dfrac{c^2}{d^2}
. Then we can cross multiply both sides by
b^2 d^2
, which is a positive number, so the inequality sign does not need to be changed. Thus
a^2 d^2 > b^2 c^2
. Taking the square roots gives the desired claim,
a \times d > b \times c
Reply: Recall that
\sqrt{x^2} = |x|
. So taking the square roots only yields
|ad | > |bc |
ad > bc
0 < \frac{a+c}{b+d} < \frac{a}{b}
\frac{c}{d} < \frac{a+c}{b+d}
Cannot be certain
\frac{a}{b} < \frac{a+c}{b+d} < \frac{c}{d}
\frac{a}{b} + \frac{c}{d} \ne \frac{a+c}{b+d}.
a, b, c, d
are positive integers that satisfy
\frac{a}{b} < \frac{c}{d},
\frac{a+c}{b+d} ?
\frac{a+c}{b+d} < \frac{a}{b} < \frac{ c}{d}
\frac{a}{b} < \frac{a+c}{b+d} < \frac{ c}{d}
\frac{a}{b} < \frac{ c}{d} < \frac{a+c}{b+d}
None of the rest
\frac{a}{b} < \frac{c}{d}
a, b, c, d
(are real variables that) satisfy the above inequality, what can we say about
\frac{a+c}{b+d} ?
Cite as: Does cross multiply always work for inequalities?. Brilliant.org. Retrieved from https://brilliant.org/wiki/does-cross-multiple-always-work-for-inequalities/
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Orbital inclination change - WikiMili, The Best Wikipedia Reader
Find sources: "Orbital inclination change" – news · newspapers · books · scholar · JSTOR (April 2009)
However, maximum efficiency of inclination changes are achieved at apoapsis, (or apogee), where orbital velocity
{\displaystyle v\,}
is the lowest. In some cases, it can require less total delta v to raise the satellite into a higher orbit, change the orbit plane at the higher apogee, and then lower the satellite to its original altitude. [1]
For the most efficient example mentioned above, targeting an inclination at apoapsis also changes the argument of periapsis. However, targeting in this manner limits the mission designer to changing the plane only along the line of apsides.[ citation needed ]
For Hohmann transfer orbits, the initial orbit and the final orbit are 180 degrees apart. Because the transfer orbital plane has to include the central body, such as the Sun, and the initial and final nodes, this can require two 90 degree plane changes to reach and leave the transfer plane. In such cases it is often more efficient to use a broken plane maneuver where an additional burn is done so that plane change only occurs at the intersection of the initial and final orbital planes, rather than at the ends. [2]
In a pure inclination change, only the inclination of the orbit is changed while all other orbital characteristics (radius, shape, etc.) remains the same as before. Delta-v (
{\displaystyle \Delta {v_{i}}\,}
) required for an inclination change (
{\displaystyle \Delta {i}\,}
) can be calculated as follows:
{\displaystyle \Delta {v_{i}}={2\sin({\frac {\Delta {i}}{2}}){\sqrt {1-e^{2}}}\cos(\omega +f)na \over {(1+e\cos(f))}}}
{\displaystyle e\,}
is the orbital eccentricity
{\displaystyle \omega \,}
is the argument of periapsis
{\displaystyle f\,}
is the true anomaly
{\displaystyle n\,}
is the mean motion
{\displaystyle a\,}
is the semi-major axis
For more complicated manoeuvres which may involve a combination of change in inclination and orbital radius, the delta v is the vector difference between the velocity vectors of the initial orbit and the desired orbit at the transfer point. These types of combined manoeuvres are commonplace, as it is more efficient to perform multiple orbital manoeuvres at the same time if these manoeuvres have to be done at the same location.
According to the law of cosines, the minimum Delta-v (
{\displaystyle \Delta {v}\,}
) required for any such combined manoeuvre can be calculated with the following equation
{\displaystyle \Delta {v}={\sqrt {V_{1}^{2}+V_{2}^{2}-2V_{1}V_{2}cos(\Delta i)}}}
{\displaystyle V_{1}}
{\displaystyle V_{2}}
are the initial and target velocities.
Where both orbits are circular (i.e.
{\displaystyle e\,}
= 0) and have the same radius the Delta-v (
{\displaystyle \Delta {v_{i}}\,}
{\displaystyle \Delta {i}\,}
) can be calculated using:
{\displaystyle \Delta {v_{i}}={2v\,\sin \left({\frac {\Delta {i}}{2}}\right)}}
{\displaystyle v\,}
is the orbital velocity and has the same units as
{\displaystyle \Delta {v_{i}}}
In orbital mechanics, a frozen orbit is an orbit for an artificial satellite in which natural drifting due to the central body's shape has been minimized by careful selection of the orbital parameters. Typically, this is an orbit in which, over a long period of time, the satellite's altitude remains constant at the same point in each orbit. Changes in the inclination, position of the lowest point of the orbit, and eccentricity have been minimized by choosing initial values so that their perturbations cancel out., which results in a long-term stable orbit that minimizes the use of station-keeping propellant.
The Clohessy-Wiltshire equations describe a simplified model of orbital relative motion, in which the target is in a circular orbit, and the chaser spacecraft is in an elliptical or circular orbit. This model gives a first-order approximation of the chaser's motion in a target-centered coordinate system. It is very useful in planning rendezvous of the chaser with the target.
1 2 Braeunig, Robert A. "Basics of Space Flight: Orbital Mechanics". Archived from the original on 2012-02-04. Retrieved 2008-07-16.
↑ http://issfd.org/ISSFD_2007/3-1.pdf [ bare URL PDF ]
↑ Owens, Steve; Macdonald, Malcolm (2013). "Hohmann Spiral Transfer With Inclination Change Performed By Low-Thrust System" (PDF). Advances in the Astronautical Sciences. 148: 719. Retrieved 3 April 2020.
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Calculation of lunar orbit anomaly | Planetary Science | Full Text
Original research short communication
Calculation of lunar orbit anomaly
Louise Riofrio1
Studies of the Moon, with thanks to NASA and Johnson Space Center, have quantified an anomaly in measurements of lunar orbital evolution. This finding may have significance for cosmology and the speed of light. The Lunar Laser Ranging Experiment from Apollo reports the Moon’s semimajor axis increasing at a rate of 3.82 ± .07 cm/yr, anomalously high.
Sedimentary data indicates a rate of only 2.9 ± 0.6 cm/yr. From historical eclipse records we can accurately calculate a rate of 2.82 ± .08 cm/yr. A detailed numerical simulation of lunar orbital evolution predicts 2.91 cm/yr. LLRE’s laser light differs from independent experiments by up to 12σ.
Several possible explanations are considered. The author’s hypothesis proposes that the speed of light decreases at rate ċ/c = − 0.24 × 10− 10 yr− 1. This predicts that LLRE will differ by 0.935cm/yr, precisely accounting for the lunar anomaly.
Quantities that appear constant may change over time. The Moon’s orbital distance has long been known to be slowly increasing. Simultaneously Earth’s rotation rate has been slowing, causing increase in length of day. This is interpreted as tidal forces transferring angular momentum from Earth to Moon, causing the orbital semimajor axis to increase. Study of lunar orbital evolution draws data from multiple sources including sediments, eclipse observations and numerical simulation. Results from the Moon have implications for cosmology and light.
The Lunar Laser Ranging Experiment bounces laser light off corner reflectors placed on the Moon’s surface. Reflectors were left behind by the Apollo 11, 14, 15 and Lunokhod missions. LLRE has been used to investigate geophysics of the Earth-Moon system and test Relativity’s equivalence principle. Accuracy has been considered fine enough to rule out significant changes in the gravitational constant G [1–5].
LLRE has measured the Moon’s orbital semimajor axis a =384,402km. Repeated measurements by LLRE over decades, as compiled by Dickey and colleagues, [6] appear to indicate that distance increasing at rate:
\stackrel{˙}{a}=3.82±.07cm/yr
\stackrel{˙}{a}
is derivative of lunar orbital semimajor axis.
This value of
\stackrel{˙}{a}
has been described as anomalously high [7]. As calculated by Bills and Ray, if the Moon were today gaining angular momentum at this rate it would have coincided with Earth about 1.5 Gyr ago. Studies of lunar samples show that the Moon has existed separately from Earth for over 4.5 Gyr. Today the LLRE measurement can be compared with independent experiments, with surprising results.
Geology and paleontology can also tell how the Moon’s orbit has changed. Tidal rhythmites, in particular, carry a record of lunar-induced tides. Rhythmites leave layers on a daily and monthly frequency. Thicknesses of sedimentary layers vary with the height of local tides. As reviewed by Bills and Ray, fossilized rhythmites can determine lunar orbital distance millions of years in the past.
The Mansfield sediment of Indiana, the most recent and accurate to be studied, places the Moon 375,300 ± 1,900 km away at a time 310 Myr ago. For an average recession rate over this period, I subtract from today’s distance and divide by time:
\stackrel{˙}{a}=\frac{384,402km-\left(375,300±1,900km\right)}{310\text{Myr}}=2.9±0.6cm/yr
Independent study by Williams [8] of the older Elatina and Reynella tidal rhythmites also indicates a lower recession rate than LLRE.
Corroborating data may have come from historical astronomers. If the narrow track of total eclipse has been reported over an observatory, it provides an accurate measure of Earth’s slowing rotation rate. When Earth and Moon are considered as a closed system, this tells us how much angular momentum has been transferred between them. For determining lunar recession rate, accuracy of this method can complement LLRE.
Lunar recession rate is thought to vary with change in Earth rotation period. A rate of
\stackrel{˙}{a}=3.82±.07cm/yr
corresponds to change in Earth’s length of day of 2.30 msec/cyr. Observations spanning 2700 yrs compiled by Stephenson and Morrison [9] show change in LOD of 1.70 ± .05 msec/cyr.
When change in LOD varies linearly this author calculates, incorporating the 3% standard deviation:
\frac{\stackrel{˙}{a}}{1.70±.05m\text{sec}/cyr}=\frac{3.82±.07cm/yr}{2.30m\text{sec}/cyr}
\stackrel{˙}{a}=2.82±.08cm/yr
Though eclipse records corroborate Mansfield tidal data, LLRE’s laser light differs by over 12σ.
Transfer of angular momentum from Earth to Moon is subject to many factors. These include height of local tides, ocean depth, location of ocean basins and the slow movement of continental plates. Simple models often fail to account for all these influences. Poliakow [10] has produced a comprehensive numerical simulation of Earth-Moon tidal evolution. The model has been successfully used to solve the independent problem of predicting Earth tides, and may be considered highly robust. In the tidal model, some quantities usually taken as constant become variables.
The simulation predicts for the present:
\stackrel{˙}{a}=2.91cm/yr
A lower lunar recession rate
\stackrel{˙}{a}
, from three independent datasets, is extremely robust.
Different sources have been proposed for lunar orbit anomalies. The lower recession rate found in sedimentary data has been attributed to increasing tides over millions of years. This inference has not been independently verified. For the Moon’s recession to vary so greatly, tidal heights would have to increase enormously over time. Mansfield and other sediments do not show significantly different tidal heights than today.
The recession rate found from eclipse records has been suggested to result from glacial isostatic compensation. This is also an inference, for a very large change would again be required. Glacial reports are also available, and the extent of Earth glaciation over 2700 years has not changed in the amounts required. Sea-level studies suggest that tidal friction has not changed appreciably over this time [11].
These first two explanations may be mutually exclusive. Anomalous tidal changes over millions of years may not explain an anomaly in records just 2700 years old. Conversely, the inference of glacial change could not be maintained over the 310 Gyr lifespan of the youngest Mansfield sediments. The numerical simulation takes into account many factors influencing lunar recession rate, and agrees with experiment. As the lower recession rate is found in three completely independent datasets, alternate explanations must be considered.
Unknown accelerative components may also be considered to affect lunar orbit. Recession rate could then vary non-linearly with change in Earth LOD. An unknown amount of angular momentum may be transferred to the rotational component of the Moon’s momentum. This would likely be a small effect, as the Moon’s mass is large. Such an effect would also result in a lower recession rate, compounding the anomaly.
We may also consider unknown components in Earth’s rotational momentum. If a portion of Earth’s interior were rotating at a faster rate, angular momentum available for transfer to the Moon would be increased. Rapidly rotating regions within the core form a fascinating research subject. However, the inner Earth may not undergo the surface tides to transfer momentum. While a rapidly rotating inner element is an interesting research subject, it may not contribute to the orbital anomaly.
Cosmological origin
Anomalies in orbital measurements of Mercury and the moons of Jupiter are today known to result from Relativity and the speed of light c. A discrepancy of 43″/cyr in Mercury’s 5600″precession is a sign of General Relativity. An anomaly in observations of Jupiter’s moons was predicted by Roemer from the finite speed of light. All possibilities being considered, the lunar anomaly may have a cosmological origin.
This author’s hypothesis may be summarized simply:
GM=t{c}^{3}
where t is age of the Universe, GM is the gravitational constant multiplied by a constant with dimensions of mass.
Speed of light c would then be given by:
c\left(t\right)={\left(GM\right)}^{1/3}{t}^{-1/3}
\stackrel{˙}{c}\left(t\right)=\left(-1/3\right){\left(GM\right)}^{1/3}{t}^{-4/3}
\frac{\stackrel{˙}{c}}{c}=-\frac{1}{3t}=-0.24×{10}^{-10}y{r}^{-1}
where age of Universe t is estimated at 13.7Gyr, the constants GM cancel.
By theory, when t was small c was enormous and the Universe would have expanded like a “Bang.” As age t increases, c would slow due to gravitation and continue to slow at a tiny rate today. This model has been suggested to precisely fit the non-linear redshifts of distant Type Ia supernovae [12], the 4.507034% proportion of bayons and other puzzles. Cosmology makes a surprising but testable prediction: Time for laser light to return would increase each year, making the Moon appear to recede faster as measured by LLRE.
Apparent lunar distance would increase proportional to decrease in c:
\frac{\stackrel{˙}{a}}{a}=-\frac{\stackrel{˙}{a}}{c}=\frac{1}{3t}
\stackrel{˙}{a}=\frac{a}{3t}=\frac{384,400km}{3\left(13.7\text{Gyr}\right)\right)}
\stackrel{˙}{a}=0.935cm/yr
where age of Universe t is estimated at 13.7 Gyr, apparent distance is predicted to increase an additional 0.935cm/yr, precisely accounting for the anomaly.From LLRE and accounting for the speed of light, actual recession rate would be:
\stackrel{˙}{a}=3.82±.07cm/yr-0.935cm/yr
\stackrel{˙}{a}=2.88±.07cm/yr
This value is in 1σ agreement with eclipse records, Mansfield sedimentary data and numerical simulation. If one of these three datasets were found to contain error, the other two would agree with prediction.
Variation of ċ/c = 0.24 × 10− 10 yr− 1 equals − 0.72cm/sec yr, too small to have been detected by previous experiments. For example, a survey by Iorio [13] using data on planetary orbits limits cdot/c to (0.5 ± 2) × 10− 7 yr− 1, more than 3 orders of magnitude greater. Iorio [14] has also written about a change in lunar orbital eccentricity. A change in semimajor axis could be interpreted as a change in eccentricity. However as lunar orbital angular momentum increases the semimajor and semiminor axes would both change proportionately. Iorio finds no known physical basis for an eccentricity change.
An apparent change in the astronomical unit of 15 ± 4m/cyr has been cited by Krasinsky et al. [15] Although planetary observations contain many possible sources of error, the apparent change is of similar order of magnitude to the lunar orbit anomaly. The discrepancy in AU could also be partially due to change in c. On the Moon we have the advantage of laser reflectors, providing a more precise standard of measurement.
Comparison of the Lunar Laser Ranging Experiment with independent datasets shows a very significant anomaly in LLRE measurements of lunar orbital evolution. Independent measurements agree on a slower recession rate. Since LLRE alone relies on the speed of laser light, the anomaly may be precisely predicted by a time-variation in c.
At one time scholars disagreed whether light travelled instantaneously or had a finite speed. Galileo suggested placing lanterns on distant hilltops to time light’s passage, but lacked an accurate clock. Today we have laser lanterns and the distant hilltop of the Moon. Change in c may be detectable in the apparent rate of lunar recession.
The puzzle of “accelerating” redshifts is one of the most interesting in Physics. Lower redshifts increase linearly with distance, indicating expansion of the Universe. Redshifts of distant Type Ia supernovae appear to increase non-linearly, leading to speculation about acceleration and repulsive energies. Since redshifts are related to the speed of light, secular change in c may precisely explain the “accelerating” puzzle.
A changing speed of light has been subject of consideration since at least the first Lord Kelvin [16], recently by Moffat [17], Albrecht and Maguiejo [18]. Applying lunar data to cosmology may shine light on hypothetical “dark” energies. With improved clocks, as will be available on the International Space Station, experimenters will be able to measure c precisely in the laboratory. As with Jupiter’s moons and Mercury, anomalies in orbits may have significance for Physics.
This work was assisted by NASA and the Beyond Einstein program. Great thanks are due to colleagues at the ARES group in Johnson Space Center, Houston. Encouraging communications were made with Marni Dee Sheppeard and Yves-Henry Sanejouand.
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2004 San Sebastian Court, Houston, USA
Correspondence to Louise Riofrio.
LR collected observations of the Moon and drafted the manuscript. The author has read and approved the final manuscript.
Riofrio, L. Calculation of lunar orbit anomaly. planet. sci. 1, 1 (2012). https://doi.org/10.1186/2191-2521-1-1
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Rain Knowpia
Hard rain on a roof
rain falling on a road
Streets in Tampere, Finland watered by night rain.
{\displaystyle d}
{\displaystyle D+dD}
{\displaystyle n(d)=n_{0}e^{-d/\langle d\rangle }dD}
{\displaystyle \langle d\rangle ^{-1}=41R^{-0.21}}
Image of Atlanta, US showing temperature distribution, with blue showing cool temperatures, red warm, and hot areas appearing white.
Updated Köppen–Geiger climate map[82]
One of the main uses of weather radar is to be able to assess the amount of precipitations fallen over large basins for hydrological purposes.[102] For instance, river flood control, sewer management and dam construction are all areas where planners use rainfall accumulation data. Radar-derived rainfall estimates complement surface station data which can be used for calibration. To produce radar accumulations, rain rates over a point are estimated by using the value of reflectivity data at individual grid points. A radar equation is then used, which is
{\displaystyle Z=AR^{b},}
Heavy rain in Glenshaw, Pennsylvania (5:37)
Moderate rain — when the precipitation rate is between 2.5 mm (0.098 in) – 7.6 mm (0.30 in) or 10 mm (0.39 in) per hour[106][107]
The average time between occurrences of an event with a specified intensity and duration is called the return period.[110] The intensity of a storm can be predicted for any return period and storm duration, from charts based on historic data for the location.[111] The return period is often expressed as an n-year event. For instance, a 10-year storm describes a rare rainfall event occurring on average once every 10 years. The rainfall will be greater and the flooding will be worse than the worst storm expected in any single year. A 100-year storm describes an extremely rare rainfall event occurring on average once in a century. The rainfall will be extreme and flooding worse than a 10-year event. The probability of an event in any year is the inverse of the return period (assuming the probability remains the same for each year).[110] For instance a 10-year storm has a probability of occurring of 10 percent in any given year, and a 100-year storm occurs with a 1 percent probability in a year. As with all probability events, it is possible, though improbable, to have multiple 100-year storms in a single year.[112]
A rain dance being performed in Harar, Ethiopia
The northern half of Africa is dominated by the world's most extensive hot, dry region, the Sahara Desert. Some deserts also occupy much of southern Africa: the Namib and the Kalahari. Across Asia, a large annual rainfall minimum, composed primarily of deserts, stretches from the Gobi Desert in Mongolia west-southwest through western Pakistan (Balochistan) and Iran into the Arabian Desert in Saudi Arabia. Most of Australia is semi-arid or desert,[140] making it the world's driest inhabited continent. In South America, the Andes mountain range blocks Pacific moisture that arrives in that continent, resulting in a desertlike climate just downwind across western Argentina.[48] The drier areas of the United States are regions where the Sonoran Desert overspreads the Desert Southwest, the Great Basin and central Wyoming.[141]
Since rain only falls as liquid, it rarely falls when surface temperatures are below freezing, unless there is a layer of warm air aloft, in which case it becomes freezing rain. Due to the entire atmosphere being below freezing most of the time, very cold climates see very little rainfall and are often known as polar deserts. A common biome in this area is the tundra which has a short summer thaw and a long frozen winter. Ice caps see no rain at all, making Antarctica the world's driest continent.
North America 256.0 6,502 Hucuktlis Lake, British Columbia 12 3.66 14
Indian Ocean Foc Foc, La Réunion 71.8 1,820
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Wikiquote has quotations related to Rain.
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Global in Time Stability of Steady Shocks in Nozzles
Jeffrey Rauch1; Chunjing Xie1; Zhouping Xin2
1 Department of Mathematics University of Michigan 530 Church Street Ann Arbor, MI 48109 USA
2 The Institute of Mathematical Sciences and department of mathematics The Chinese University of Hong Kong Hong Kong
Séminaire Laurent Schwartz — EDP et applications (2011-2012), Talk no. 2, 11 p.
We prove global dynamical stability of steady transonic shock solutions in divergent quasi-one-dimensional nozzles. One of the key improvements compared with previous results is that we assume neither the smallness of the slope of the nozzle nor the weakness of the shock strength. A key ingredient of the proof are the derivation a exponentially decaying energy estimates for a linearized problem.
DOI: 10.5802/slsedp.2
Jeffrey Rauch 1; Chunjing Xie 1; Zhouping Xin 2
author = {Jeffrey Rauch and Chunjing Xie and Zhouping Xin},
title = {Global in {Time} {Stability} of {Steady} {Shocks} in {Nozzles}},
doi = {10.5802/slsedp.2},
url = {https://slsedp.centre-mersenne.org/articles/10.5802/slsedp.2/}
TI - Global in Time Stability of Steady Shocks in Nozzles
UR - https://slsedp.centre-mersenne.org/articles/10.5802/slsedp.2/
UR - https://doi.org/10.5802/slsedp.2
DO - 10.5802/slsedp.2
%T Global in Time Stability of Steady Shocks in Nozzles
%U https://doi.org/10.5802/slsedp.2
%R 10.5802/slsedp.2
%F SLSEDP_2011-2012____A2_0
Jeffrey Rauch; Chunjing Xie; Zhouping Xin. Global in Time Stability of Steady Shocks in Nozzles. Séminaire Laurent Schwartz — EDP et applications (2011-2012), Talk no. 2, 11 p. doi : 10.5802/slsedp.2. https://slsedp.centre-mersenne.org/articles/10.5802/slsedp.2/
[1] Courant, R. and Friedrichs, K.O., Supersonic Flow and Shock Waves, Springer-Verlag, 1948. | Zbl: 0041.11302
[2] Elling, V., Regular reflection in self-similar potential flow and the sonic criterion. Commun. Math. Anal. 8 (2010), no. 2, 22-69. | Zbl: 1328.76038
[3] Elling, V., Instability of strong regular reflection and counterexamples to the detachment criterion. SIAM J. Appl. Math. 70 (2009), no. 4, 1330-1340. | Article | MR: 2563517 | Zbl: 1391.76276
[4] Elling, V., Counterexamples to the sonic criterion. Arch. Ration. Mech. Anal. 194 (2009), no. 3, 987-1010. | Article | MR: 2563630 | Zbl: 1255.76049
[5] P. Embid, J. Goodman, and A. Majda,Multiple steady states for
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[6] L. C. Evans, Partial differential equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998. | Article | Zbl: 0902.35002
[7] S.-Y. Ha,
{L}^{1}
stability for systems of conservation laws with a nonresonant moving source, SIAM J. Math. Anal. 33 (2001), no. 2, 411–439. | Article | Zbl: 1002.35086
[8] S.-Y. Ha and T. Yang,
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stability for systems of hyperbolic conservation laws with a resonant moving source, SIAM J. Math. Anal. 34 (2003), no. 5, 1226–1251 | Article | Zbl: 1036.35128
[9] T. Kato, Perturbation theory for linear operators, Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995. | Article | MR: 1335452 | Zbl: 0836.47009
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[12] Wen-Ching Lien, Hyperbolic conservation laws with a moving source, Comm. Pure Appl. Math. 52 (1999), no. 9, 1075–1098. | Article | MR: 1692156 | Zbl: 0932.35142
[13] Tai-Ping Liu, Transonic gas flow in a duct of varying area, Arch. Rational Mech. Anal. 80 (1982), no. 1, 1–18. | Article | MR: 656799 | Zbl: 0503.76076
[14] Tai-Ping Liu, Nonlinear stability and instability of transonic flows through a nozzle, Comm. Math. Phys. 83 (1982), no. 2, 243–260. | Article | MR: 649161 | Zbl: 0576.76053
[15] Tai-Ping Liu, Nonlinear resonance for quasilinear hyperbolic equation, J. Math. Phys. 28 (1987), no. 11, 2593–2602. | Article | MR: 913412 | Zbl: 0662.35068
[16] Tao Luo, Jeffrey Rauch, Chunjing Xie, and Zhouping Xin, Stability of transonic shock solutions for Euler-Poisson equations, Archive Rational Anal. Mech., to appear, arXiv:1008.0378. | Article | MR: 2854670 | Zbl: 1261.76055
[17] Andrew Majda, The existence of multidimensional shock fronts, Mem. Amer. Math. Soc. 43 (1983), no. 281. | Article | MR: 699241 | Zbl: 0517.76068
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[22] Zhouping Xin and Huicheng Yin, The transonic shock in a nozzle, 2-D and 3-D complete Euler systems, J. Differential Equations 245 (2008), no. 4, 1014–1085. | Article | MR: 2427405 | Zbl: 1165.35031
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Electric resistivity surveys - SEG Wiki
Electric resistivity methods are a group composed by a large and diverse range of methods used in prospection geophysics to differentiate the subsurface according to its electromagnetic properties. In the early days, explorations were carried out to detect very conductive base metal massive sulfide ore deposits contained in highly resistive host rocks. Because of the high contrasts, ore exploration surveys have been the prime application of resistivity methods, as is also the case of magnetometry methods, since most of the ore grades are conductive. Later on, where more sensitive measuring configurations were constructed, resistivity methods also started to be used also in other fields like oil and gas prospection.
Current flow I through a piece of conductive material over which a potential difference V has been applied.
An electric current is a flow of electrically charged particles (either electrons or ions). By convention a current is considered to flow from positive (source) to negative (sink), though in the wire the current is due to electrons moving from negative to positive. The SI unit for measuring an electric current is the Ampere [A], which is the flow of electric charge across a surface at the rate of one coulomb per second. Then, a current is the result of a potential difference [V] imposed over a closed loop and the magnitude of its flow depends on the resistance of the loop's material. For most materials, including most rocks, the current trough the resistor increases linearly proportional to the voltage across it, obeying Ohm's law. This proportionality constant R is known as resistance and is a property inherent to the material and, according to Ohm's law can be obtained experimentally as the ratio between the potential difference V and the current I as:
{\displaystyle R={\frac {\Delta V}{I}}}
However, the resistance given by a certain material depends not only on it's electrical properties, namely conductivity, but also on its geometrical structure, as illustrated in the figure on the right.
{\displaystyle R={\frac {L}{\sigma A}}}
2 Array configurations
2.1 Pole-Pole array
2.2 Pole-Dipole array
2.3 Dipole-Dipole array
2.4 Wenner Arrays
2.5 Schlumberger Arrays
Geo-electric resistivity is a geophysical method where two electrodes, known as current electrodes, are used to inject an electric current intro the ground and the potential difference is measured between two distant electrodes, known as potential electrodes. Thus, based on the known input current, the voltage measured and the array geometry, an "apparent" resistivity can be computed.
Standard four electrode configuration in homogeneous half space media.
Pole-Pole array configuration in homogeneous half space media.
The name Pole-Pole already indicates that both, current injection and potential electrodes are modeled as single poles. In practice this is not possible, since current needs a closed loop to flow, but from last equation we may conclude that when the position of one of the electrodes is very large compared to the others, two of the distance terms become very small and can then be neglected.
Therefore, the measured voltage differences is:
{\displaystyle \Delta V={\frac {I}{2\pi \sigma a}}}
In practice the far away electrodes are put more than 20 times the largest electrode spacing away.
As the name indicates, one of the current electrodes is the far away electrode, while the potential electrodes are both used in the line of measurements. Thus, there is an extra degree of freedom to place the other three electrodes in the line of measurements. Let us denote the distance between the two potential electrodes a and the distance between the current electrode and the closest potential electrode na.
Thus, the potential difference measured is:
{\displaystyle \Delta V={\frac {I}{2\pi \sigma an(n+1)}}}
Pole-Dipole array configuration in homogeneous half space media.
In this case, the effect of the distant electrode is negligible and the electric field of the near electrode resembles that of a point source rather than the field of a dipole. A commonly used configuration consists of a combination of two pole-dipole arrays: one forward and one reversed. This configuration is mainly used in profiling, where changes in resistivities are clearly mapped. It is characterized by a high Signal to Noise Ratio.
This is the most sensitive array of those mentioned and therefore also the most prone to the noise. In this configuration, all four electrodes are put in the line at measurable distances from each other. In this array, both current electrodes are next to each other at distance a, while the potential electrodes are also next to each other at distance a and separated from the current electrode pair at a distance na.
Therefore, the potential difference measured is:
{\displaystyle \Delta V={\frac {I}{2\pi \sigma an(n+1)(n+2)}}}
Dipole-Dipole array configuration in homogeneous half space media.
Wenner Arrays
This potential array is characterized by a relatively large distance between the potential electrodes compared with the potential-to-current electrodes distance. This array is suitable for areas with poor grounding conditions or areas where a high amount of noise is expected. The Wenner array has lost most of its popularity because the electrode distances are fixed and the potential electrodes must be at the same distance as the current electrodes. However, it is still used in situations where signal strength is the most important factor, like in situations where the current electrodes are put in a very highly resistive top layer.
The potential difference measured is:
{\displaystyle \Delta V={\frac {I}{2\pi \sigma a}}}
Wenner array configuration: all the distances between electrodes are equal.
Schlumberger Arrays
In this configuration, the potential electrodes are placed at the center of the electrode array with a small separation, typically less than one fifth the current electrodes spacing. The current electrodes separation is gradually increased during the survey while the potential electrodes remain unchanged until the measured voltage becomes too small to be detected. Schlumberger soundings generally have better resolution, greater probing depth, and less time-consuming field deployment compared to Wenner arrays. The two most outstanding disadvantages are: longer current electrode wires are required and the recording instrument needs to be very sensitive.
{\displaystyle \Delta V={\frac {I}{\pi \sigma an(n+1)}}}
Schlumberger array configuration: let the distance between current electrodes be 2L and the distance between potential electrodes be 2a, while the mid-point of both current and potential electrodes coincide.
Retrieved from "https://wiki.seg.org/index.php?title=Electric_resistivity_surveys&oldid=45745"
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Is acceleration the rate of change of speed? | Brilliant Math & Science Wiki
Rohit Gupta, Ashish Menon, and Tim O'Brien contributed
Why some people say it's true: Think of accelerating in a car: when you hit the gas, you speed up, and when you hit the brake, you slow down. Acceleration is generally associated with a change in speed.
Why some people say it's false: In physics, direction matters. If the direction of motion changes, this could be considered acceleration too, even if the speed stays constant.
\color{#20A900}{\text{Reveal the Correct Answer}}
\color{#D61F06}{\textbf{false}}
Acceleration is defined as the rate of change of velocity. Velocity is a vector, which means it contains a magnitude (a numerical value) and a direction. So the velocity can be changed either by changing the speed or by changing the direction of motion (or both). Therefore, it may be possible that the speed is constant, but the velocity is changing because the direction is changing. In this case, acceleration will be non-zero and equal to the rate of change of velocity.
It is a general misconception that rate of change of speed is equal to the magnitude of the rate of change of velocity. However, this is not true in all cases. Consider uniform circular motion: in the case of a uniform circular motion, the particle moves on a circular path with uniform speed. The speed remains constant, but the direction of motion is continuously changing. Due to change in direction of motion, acceleration is non-zero. This acceleration is toward the center of the circle and known as centripetal acceleration.
In general, acceleration can be resolved into two components. One component, which is parallel to the velocity, is known as tangential acceleration. This component changes the speed of the particle and is equal to the rate of change of speed. The other component of acceleration, which is perpendicular to velocity, is known as normal acceleration. This component is responsible for changing the direction of the velocity.
Velocity is the rate of change of displacement, while speed is the rate of change of distance. In other words, velocity is the rate of change of the shortest distance moved by a body from the final position to the initial position, while speed is the rate of change of the total length of the path traveled by a certain body.
\color{#3D99F6}{\text{See Further Discussions}}
Query: What can be said about the acceleration of a particle moving in a zig-zag path with constant speed?
Reply: The acceleration of the particle has to be non-zero, as the particle is changing its direction. However, the tangential component of the acceleration is zero, as the speed remains constant.
Query: If both the speed and direction changes, then is it possible to have zero acceleration?
Reply: No. If speed changes, then tangential acceleration is non-zero. If the direction of motion is changing, then normal acceleration is non-zero. The result of these two accelerations can never be zero as they are perpendicular to each other.
90^\circ
Cite as: Is acceleration the rate of change of speed?. Brilliant.org. Retrieved from https://brilliant.org/wiki/is-acceleration-the-rate-of-change-of-speed/
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Journal of Geographic Information System > Vol.10 No.1, February 2018
Assessment of Potential Aerodrome Obstacles on Flight Safety Operations Using GIS: A Case of Murtala Mohammed International Airport, Lagos-Nigeria ()
Ayeni, A. , Musah, A. and Udofia, S. (2018) Assessment of Potential Aerodrome Obstacles on Flight Safety Operations Using GIS: A Case of Murtala Mohammed International Airport, Lagos-Nigeria. Journal of Geographic Information System, 10, 1-24. doi: 10.4236/jgis.2018.101001.
Rn=\frac{D\left(Obs\right)}{\sqrt[0.5]{\frac{a}{n}}}
[1] Blanchard, G. (1996) Intelligent Transportation System and Highway Infrastucture. Report TP12836E, Special Infrastructure Report, Economic Transport Analysis Canada, Ottawa.
[2] Ayinalem, B. (2015) Wheel/Rail Adhesion under Plastic Bags Contamination Condition. A Master’s Thesis, The School of Mechanical and Industrial Engineering, Addis Ababa University, Addis Ababa Institute of Technology School of Mechanical and Industrial Engineering, The Ashden Trust, London, ICAO 2001.
[3] Whitelegg, J. (2000) Aviation: The Social, Economic and Environmental Impact of Flying. Stockholm Envt’l Institute, University of York England, Ecological Ltd., Lancaster.
[4] International Civil Aviation Organisation ICAO (2009) International Standards and Recommended Practices. Annex 14 to the Convention on International Civil Aviation, Aerodromes, Volume 1 Aerodrome Design and Operations, 5th Edition.
[5] CAR (2006) Aerodrome Design and Operations. Government of India Office of Director General of Civil Aviation Technical Centre. Opp Safdarjang Airport, New Delhi Civil Aviation Requirements Section-4, Aerodrome Standards & Air Traffic Services Series ‘B’, Part I, 31st July 2006.
[6] Civil Aviation Act (2006) Civil Aviation Authority Report and Accounts. CAA, London.
[7] The Nigerian Voice (2012) The Ethnic Origin of Plane Crashes: A Memo on Plane Crashes in Nigeria.
https://webcache.googleusercontent.com/search?q=cache:XV4dFUOn8MAJ:https://
www.thenigerianvoice.com/news/93234/1/the-ethnic-origin-of-plane-crashes-a-memo
-on-plane-crashes-in-nigeria.html+&cd=1&hl=en&ct=clnk&gl=ng
[8] Channels Television (2013) Timeline of Plane Crashes in Nigeria.
https://www.channelstv.com/2013/10/04/timeline-of-plane-crashes-in-nigeria/
[9] Air & Space Magazine (2013) Alaska’s Crash Epidemic: How Technology and an FAA Regional Office Ended It.
http://www.airspacemag.com/flight-today/alaskas-crash-epidemic-70259395/#IFPvM
m7cUztx4Bg8.99
[10] Leadership (2015) 90 Years of Nigeria’s Aviation: Milestones.
http://leadership.ng/news/474259/90-years-of-nigerias-aviation-milestones
[11] Premium Times (2017) 10 Worst Plane Crashes Involving Nigeria.
http://www.premiumtimesng.com/news/5556ten_worst_plane_crashes_involvin
nig eria.html
[12] Fricke, H. and Thiel, C. (2015) A Methodology to Assess the Safety of Aircraft Operations When Aerodrome Obstacle Standards Cannot Be Met. Open Journal of Applied Sciences, 5, 62-81.
[13] Oni, A.O. (2007) Analysis of Accessibility and Connectivity of Ikeja Arterial Roads. 1st National Conference, Lagos, 25-27 September 2007.
[14] Oni, A.F. (2015) Building-Use Conversion and the Perceptual Assessment of Utilities Serviceability in Lagos Metropolis, Nigeria. PhD Thesis, The School of Postgraduate Studies, University of Lagos, Lagos.
[15] Clark, P.J. and Evans, F.C. (1954) Distance to Nearest Neighbour as a Measure of Spatial Relationship in Population. Ecology, 35, 445-453.
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Fernando Garzón1, Aldo Figueroa2
1 Instituto de Investigación en Ciencias Básicas y Aplicadas, Universidad Autónoma del Estado de Morelos, Cuernavaca, México.
2 CONACYT-Centro de Investigación en Ciencias, Universidad Autónoma del Estado de Morelos, Cuernavaca, México.
Abstract: We present an immersed array of four rotors whose promoted flow can be mathematically modeled with a creeping flow solution from the incompressible Navier-Stokes equations. We show that this solution is indeed representative of the two-dimensional experiment and validate such class of solution with experimental data obtained through the Particle Image Velocimetry technique and time-lapsed particles visualizations.
Keywords: Fluid Mechanics, Flettner Rotor, Fundamental Solutions
Boats ships have crossed the oceans around the globe for centuries by the use of paddling, sails, engine propellers or even turbine propellers. An alternative to the latter propulsion systems is the rotor ship. Through this alternative technique, the ship is propelled, at least in part, by large vertical rotors, sometimes known as rotor sails. German engineer Anton Flettner was the first to build a ship which attempted to tap this force for propulsion, and the ships are sometimes known as Flettner ships [1] . A rotor ship is a type of ship designed to use the Magnus effect for propulsion, where the rotation of a solid body can modify its trajectory due to the frictional forces within fluids as consequence of its viscosity [2] . Flettner ships have arrays of large rotorsails that rise from its deck which are rotated via a mechanical linkage to the ship’s propellers. Trials confirm fuel savings of 2.6 percent using a single small Rotor Sail on a route in the North Sea. With these fuel savings, this new wind propulsion technology has a payback period of just four years [3] . Flettner rotor propulsion system presents success in reducing fuel consumption and carbon dioxide (CO2) emissions. Recent examples such as Enercon’s E-ship 1 have proven seaworthy and economically viable along major shipping routes [4] .
Recently, some preliminary assessments of numerical simulations have been conducted by comparison with experimental investigation of Flettner rotors in order to evaluate the functional relationship and the interaction between the control factors [5] , the preliminary design of the Flettner rotor as a ships auxiliary propulsion system [6] , its evaluation with another wind power technology, namely, the towing kite [7] , and characterized in terms of lift and drag coefficient [8] .
In contrast to the previous studies, in this article, a simple model derived from the Navier-Stokes equations is obtained and compared with a simple experimental model that represents the Flettner rotors from the Enercon’s E-Ship 1, with four large rotor sails [9] when the ship is at rest and no flow is incident to it. The objective is the study of the behaviour of the flow patterns due to the rotating cylinders.
The theoretical approach begins with the mass and momentum conservation for real fluids [10]
\nabla \cdot u=0
\frac{\partial u}{\partial t}+\left(u\cdot \nabla \right)u=-\frac{1}{\rho }\nabla p+\nu {\nabla }^{2}u
where the velocity vector is denoted by
u
p
\rho
\nu
is the kinematic viscosity and t represents time. Considering that the surface is flat, and that the motions is laminar, thus the motion of the flow occurs on the x-y plane and the perpendicular velocity is negligible, thus we assume that the flow is two-dimensional. As a first approach, we consider a single rotor. Locating the origin in the geometrical center of the rotor and using polar coordinates, the flow can be considered as symmetric around the origin, that is, independent of the q-direction. Thus the velocity vector is
u={u}_{\theta }\left(r\right)
, which automatically satisfies the continuity Equation (1) and the Navier-Stokes Equation (2) for the r and
\theta
components are
\frac{{u}_{\theta }^{2}}{r}=\frac{1}{\rho }\frac{\text{d}p}{\text{d}r}
\frac{{\text{d}}^{2}{u}_{\theta }}{\text{d}{r}^{2}}+\frac{\text{d}}{\text{d}r}\left(\frac{{u}_{\theta }}{r}\right)=0
Equation (4) is a homogeneous second order differential equation, thus we must impose two boundary conditions. The first one is that the velocity at infinity is zero, that is
{u}_{\theta }\left(\infty \right)=0
. The second one is a non-slip condition, as the cylinder rotates with a uniform angular velocity
\omega
, the tangential velocity at the cylinder’s radius
{R}_{i}
{u}_{\theta }\left({R}_{i}\right)=\omega {R}_{i}
\omega =2\text{π}f
f
is the rotating cylinder frequency. The solution to Equation (4) is
{u}_{\theta }\left(r\right)=\frac{\omega {R}_{i}^{2}}{r}
The stream lines can be obtained with the following relation
\frac{\text{d}\psi }{\text{d}r}={u}_{\theta }
\psi \left(r\right)
is the stream function. The boundary condition is
\psi \left({R}_{i}\right)=0
\psi \left(r\right)=\omega {R}_{i}^{2}\left(\mathrm{ln}\left(r\right)-\mathrm{ln}\left({R}_{i}\right)\right)
Once the velocity field is known, the pressure distribution can be calculated from Equation (3)
p\left(r\right)=-\frac{\rho {\omega }^{2}{R}_{i}^{4}}{2{r}^{2}}+C
where C is a constant that can be obtained by evaluating
p\left(\infty \right)
. The solution (5) and stream line function (7) in Cartesian coordinates are
u\left(x,y\right)=\frac{\omega {R}_{i}^{2}}{\sqrt{{x}^{2}+{y}^{2}}}\mathrm{cos}\left(\mathrm{arctan}\left(y/x\right)\right)
v\left(x,y\right)=\frac{\omega {R}_{i}^{2}}{\sqrt{{x}^{2}+{y}^{2}}}\mathrm{sin}\left(\mathrm{arctan}\left(y/x\right)\right)
\psi \left(x,y\right)=\omega {R}_{i}^{2}\left(\mathrm{ln}\left(\sqrt{{x}^{2}+{y}^{2}}\right)-\mathrm{ln}\left({R}_{i}\right)\right)
As stated previously, Equations (5) and (7) correspond to single rotor located at the origin. However, as the solution is linear, we can sum the solutions to consider the flow of four rotating cylinders, that is we can perform a superposition of rotors. In Cartesian coordinates, the total stream lines solutions
\Psi
for the four contemplated sets of rotation are as follows:
\begin{array}{l}{\Psi }_{1}\left(x,y\right)=\psi \left(x+a,y\right)+\psi \left(x,y+a\right)+\psi \left(x-a,y\right)+\psi \left(x,y-a\right)\\ {\Psi }_{2}\left(x,y\right)=\psi \left(x+a,y\right)+\psi \left(x,y+a\right)-\psi \left(x-a,y\right)+\psi \left(x,y-a\right)\\ {\Psi }_{3}\left(x,y\right)=-\psi \left(x+a,y\right)+\psi \left(x,y+a\right)-\psi \left(x-a,y\right)+\psi \left(x,y-a\right)\\ {\Psi }_{4}\left(x,y\right)=-\psi \left(x+a,y\right)-\psi \left(x,y+a\right)+\psi \left(x-a,y\right)+\psi \left(x,y-a\right)\end{array}
\psi
represents the flow generated by a single rotor with negative (clockwise) or positive (counter clockwise) rotation depending on its sign, and a is the shifting length from the origin. Next sections show the experimental procedure and the comparison between this theoretical model and the experimental measurements.
The experiment set up consist of four rotating cylinders immersed in a fluid layer. The working fluid is water at room temperature (20˚C). The fluid is contained in a
30\times 30\times 10
cm glass box which is open in the upper side, see Figure 1. The measures of the box were calculated so that its boundaries are
Figure 1. Sketch of the experimental set-up, not drawn to scale. (a) Lateral view; (b) Visualized x-y plane. The flow promoted by the rotors is visualized in the x-y plane thorough the PIV technique.
sufficiently far away from the flow. The height of the water level in the glass container is 8 cm. Above the fluid, a rigid table is placed at the top of the box with the four electric DC motors (6 V with gearbox) embedded in it. Considering the geometric center of the rigid table as the origin of coordinates, the electric motors are located at a distance of 2.5 cm from the origin. A rigid cylinder
L=10\text{\hspace{0.17em}}\text{cm}
is coupled to the shaft of every motor so that the cylinder is partially submerged in the liquid. The radius of every cylinder is
{R}_{i}=0.4\text{\hspace{0.17em}}\text{cm}
. The DC motors are connected to a power supply whose output voltage is 6 V. As a single cylinder rotates, it promotes a circular fluid flow around itself. The direction of rotation of the cylinders can be changed by inverting the polarity of each motor. As every cylinder rotates independently, four sets of rotation are studied. Every set of rotation produces a different flow pattern. The frequency of rotation of the rotors is
f=0.667\text{\hspace{0.17em}}\text{Hz}
. The surface of the fluid is seeded with hollow glass spheres, with an approximated diameter of 10 mm. Images of the surface are recorded with a photographic camera Nikon D90 with a AF micro-nikkor 60mm f/2.8D lens placed underneath the glass box. Streamlines of the flow cases are obtained by using an exposition time of 30 s. The Particle Image Velocimetry (PIV) technique was used in order to obtain velocity vector fields. The experimental data gathered was used to perform a comparative with the theoretical results.
In this section, the experimental results are compared with the simple theoretical model obtained from the Navier-Stokes equations. Figure 2 and Figure 3 show
Figure 2. Stream lines of the flows generated by the four sets of rotation. As defined in Equation (12): (a) Set of rotation 1; (b) Set of rotation 2; (c) Set of rotation 3; (d) Set of rotation 4. Experimental results.
Figure 3. Velocity field of the flows generated by the four sets of rotation. As defined in Equation (12): (a) Set of rotation 1; (b) Set of rotation 2; (c) Set of rotation 3; (d) Set of rotation 4. Experimental results.
the experimental stream lines and the vector fields, respectively, for the four sets of rotation. The stream lines allow to easily identify the flow patterns. The first case, Figure 2(a), is characterized by a squared inner vortex among the four rotors sharing a horseshoe at every corner. The second case, Figure 2(b), is an asymmetric flow with two horseshoe patterns. Whereas the third case, Figure 2(c), is a large horseshoe pattern in the geometrical centre. Finally, the fourth case, Figure 2(d), is a jet flow from low-right corner to top-left corner. In turn, the vectorial fields allow us to identify the direction of the flows and their magnitude. Moreover, we can appreciate that every cylinder generates a vortex patter around itself and the interaction of the four vortices rotation in different directions produce a global flow. As the theoretical model departs from the latter assumption, its predictions are close to the experimental results as seen in Figure 4, where the stream lines from Equation (12) are shown with
a=25\text{\hspace{0.17em}}\text{mm}
. Comparing with Figure 2, only the first case does not agree completely, since the inner squared vortex is dismissed by the analytical solution. However, for the rest of the cases the location of the horseshoes, the vortices and the jets are completely predicted. Even more, Figure 5 compares the experimental velocity profiles with the theoretical predictions from Equation (9) for flow cases 1 and 4, see Equation (12). This is a more detailed comparison and even at this level the qualitative comparison is very good. The Reynolds number for the experiments
Figure 4. Stream lines of the flows generated by the four sets of rotation. As defined in Equation (12): (a) Set of rotation 1; (b) Set of rotation 2; (c) Set of rotation 3; (d) Set of rotation 4. Theoretical calculations.
Figure 5. Horizontal velocity u as a function of the y-direction. (a) and (c) experimental profiles of the sets of rotation 1 and 4, respectively; (b) and (d) theoretical profiles of the sets of rotation 1 and 4, respectively.
Re=U{R}_{i}/\nu =8
U
is the maximum velocity in the flow field, which demonstrates that the flow regime is laminar and can be compared with the theoretical model. The theoretical predictions are one order of magnitude higher than experimental measurements. This can be attributed to the friction of the container’s bottom with the fluid. Since the model is two-dimensional this friction is not taken into account, thus the velocity in the theoretical model is higher than the experimental obtained through PIV.
A solution for the incompressible Navier-Stokes equations was derived in the creeping regime for the case of a single rotation rotor. Considering its linearity, it was superposed to consider an array of four rotors in an unbounded domain. An investigation of the validity of this solution was carried out experimentally. It turned out that this solution represents qualitatively the experimental problem. A verification was established through PIV and time-lapse particles visualization.
This research was supported by CONACYT, Mexico, under project 258623. A.F. thanks the Cátedras program from CONACYT.
Cite this paper: Garzón, F. and Figueroa, A. (2017) The Study on the Flow Generated by an Array of Four Flettner Rotors: Theory and Experiment. Applied Mathematics, 8, 1851-1858. doi: 10.4236/am.2017.812132.
[1] Neu Wayne, L. (2013) Flettner Rotor Ship. AccessScience McGraw-Hill Education.
https://doi.org/10.1036/1097-8542.YB130204
[2] Miller, F.P., Vandome, A.F. and John, M.B. (2010) Magnus Effect.
https://books.google.com.mx/books?id=SAPIXwAACAAJ
[3] Marks, H. and Wachtel, B. (2015) Ship Efficiency Technologies Ready to Set Sail.
https://www.greenbiz.com/article/ship-efficiency-technologies-ready-set-sail
[4] Searcy, T. (2017) Harnessing the Wind: A Case Study of Applying Flettner Rotor Technology to Achieve Fuel and Cost Savings for Fiji’s Domestic Shipping Industry. Marine Policy, 86, 164-172. https://doi.org/10.1016/j.marpol.2017.09.020
[5] De Marco, A., Mancini, S., Pensa, C., Scognamiglio, R. and Vitiello, L. (2015) Marine Application of Flettner Rotors: Numerical Study on a Systematic Variation of Geometric Factor. VI International Conference on Computational Methods in Marine Engineering, Rome, 1-12.
[6] De Marco, A., Mancini, S., Pensa, C., Calise, G. and De Luca, F. (2016) Flettner Rotor Concept for Marine Applications: A Systematic Study. International Journal of Rotating Machinery, 2016, 1-12. https://doi.org/10.1155/2016/3458750
[7] Traut, M., Gilbert, P., Walsh, C., Bows, A., Filippone, A., Stansby, P. and Wood, R. (2013) Propulsive Power Contribution of a Kite and a Flettner Rotor on Selected Shipping Routes. Applied Energy, 13, 362-372.
[8] Gadkari, M., Deshpande, V., Mahulkar, S., Khushalani, V., Pardhi, S. and Kedar, A.P. (2017) To Study Magnus Effect on Flettner Rotor. International Research Journal of Engineering and Technology, 4, 1597-1601.
[9] Wikipedia (2017) E-Ship 1. https://en.wikipedia.org/wiki/E-Ship1
[10] Currie, I.G. (1993) Fundamental Mechanics of Fluids. Mechanical Engineering, Marcel Dekker, New York.
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Case_fatality_rate Knowpia
In epidemiology, case fatality rate (CFR) – or sometimes more accurately case-fatality risk – is the proportion of people diagnosed with a certain disease, who end up dying of it. Unlike a disease's mortality rate, the CFR does not take into account the time period between disease onset and death. A CFR is generally expressed as a percentage. It represents a measure of disease lethality and may change with different treatments.[1] CFRs are most often used for with discrete, limited-time courses, such as acute infections.
The mortality rate – often confused with the CFR – is a measure of the relative number of deaths (either in general, or due to a specific cause) within the entire population per unit of time.[2] A CFR, in contrast, is the number of deaths among the number of diagnosed cases only, regardless of time or total population.[3]
From a mathematical point of view, by taking values between 0 and 1 or 0% and 100%, CFRs are actually a measure of risk (case fatality risk) – that is, they are a proportion of incidence, although they don't reflect a disease's incidence. They are neither rates, incidence rates, nor ratios (none of which are limited to the range 0–1). They do not take into account time from disease onset to death.[4][5]
Sometimes the term case fatality ratio is used interchangeably with case fatality rate, but they are not the same. A case fatality ratio is a comparison between two different case fatality rates, expressed as a ratio. It is used to compare the severity of different diseases or to assess the impact of interventions.[6]
Because the CFR is not an incidence rate by not measuring frequency, some authors note that a more appropriate term is case fatality proportion.[7]
If 100 people in a community are diagnosed with the same disease, and 9 of them subsequently die from the effects of the disease, the CFR would be 9%. If some of the cases have not yet resolved (neither died nor fully recovered) at the time of analysis, a later analysis might take into account additional deaths and arrive at a higher estimate of the CFR, if the unresolved cases were included as recovered in the earlier analysis. Alternatively, it might later be established that a higher number of people were subclinically infected with the pathogen, resulting in an IFR below the CFR.
A CFR may only be calculated from cases that have been resolved through either death or recovery. The preliminary CFR, for example, of a newly occurring disease with a high daily increase and long resolution time would be substantially lower than the final CFR, if unresolved cases were not excluded from the calculation, but added to the denominator only.
{\displaystyle {\text{CFR in }}{\%}={\frac {\text{Number of deaths from disease}}{\text{Number of confirmed cases of disease}}}\times 100}
Infection fatality rateEdit
Like the case fatality rate, the term infection fatality rate (IFR) also applies to infectious diseases, but represents the proportion of deaths among all infected individuals, including all asymptomatic and undiagnosed subjects. It is closely related to the CFR, but attempts to additionally account for inapparent infections among healthy people.[9] The IFR differs from the CFR in that it aims to estimate the fatality rate in both sick and healthy infected: the detected disease (cases) and those with an undetected disease (asymptomatic and not tested group).[10] Individuals who are infected, but show no symptoms, are said to have inapparent, silent or subclinical infections and may inadvertently infect others. By definition, the IFR cannot exceed the CFR, because the former adds asymptomatic cases to its denominator.
{\displaystyle {\text{IFR in }}{\%}={\frac {\text{Number of deaths from disease}}{\text{Number of infected individuals}}}\times 100}
A half dozen examples will suggest the range of possible CFRs for diseases in the real world:
The CFR for the Spanish (1918) flu was >2.5%,[11] about 0.1% for the Asian (1956-58) and Hong Kong (1968-69) flus,[12] and <0.1% for other influenza pandemics.[11]
Legionnaires' disease has a CFR of about 15%.[13]: 665
The CFR for yellow fever, even with good treatment, ranges from 20 to 50%.[14]
Bubonic plague, left untreated, will have a CFR of as much as 60%.[15]: 57 With antibiotic treatment, the CFR for septicaemic plague is 45%, pneumonic 29% and bubonic 17%.[16][17]
Zaïre Ebola virus is among the deadliest viruses with a CFR as high as 90%.[18]
Naegleriasis (also known as primary amoebic meningoencephalitis), caused by the unicellular Naegleria fowleri, has a case fatality rate greater than 95%.
Rabies virus has a CFR of almost 100% in unvaccinated individuals.[19]
Mortality rate – Measure of the number of deaths in a population from a given cause, scaled by population, in a set period of time
Pandemic severity index – Proposed measure of the severity of influenza
^ Rebecca A. Harrington, Case fatality rate at the Encyclopædia Britannica
^ For example, a diabetes mortality rate of 5 per 1,000 or 500 per 100,000 characterizes the observation of 50 deaths due to diabetes in a population of 10,000 in a given year, resulting in a yearly diabetes mortality rate of 0.5%, far below the actual diabetic individual's fatality risk. (See Harrington, Op. cit..)
^ "Coronavirus: novel coronavirus (COVID-19) infection" (PDF). Elsevier. 2020-03-25. Archived from the original (PDF) on 2020-03-27. Retrieved 2020-03-27.
^ Entry “Case fatality rate” in Last, John M. (2001), A Dictionary of Epidemiology, 4th edition; Oxford University Press, p. 24.[ISBN missing]
^ Hennekens, Charles H. and Julie E. Buring (1987), Epidemiology in Medicine, Little, Brown and Company, p. 63.[ISBN missing]
^ Bosman, Arnold (2014-05-28). "Attack rates and case fatality". Field Epidemiology Manual Wiki. ECDC. Archived from the original on 2020-03-25. Retrieved 2020-03-25.
^ Peter Cummings: Analysis of Incidence Rates. In: CRC Press (2019).
^ a b "Estimating mortality from COVID-19". www.who.int. Retrieved 2021-12-13.
^ "Infection fatality rate". DocCheck Medical Services GmbH. Retrieved 25 March 2020.
^ "Global Covid-19 Case Fatality Rates". Centre for Evidence-Based Medicine. Retrieved 25 March 2020.
^ a b Taubenberger, Jeffery K.; David M. Morens (January 2006). "1918 influenza: the mother of all pandemics". Emerging Infectious Diseases. Coordinating Center for Infectious Diseases, Centers for Disease Control and Prevention. 12 (1): 15–22. doi:10.3201/eid1201.050979. PMC 3291398. PMID 16494711. Archived from the original on 2009-10-01. Retrieved 2009-04-17.
^ Li, F C K; B C K Choi; T Sly; A W P Pak (June 2008). "Finding the real case-fatality rate of H5N1 avian influenza". Journal of Epidemiology and Community Health. 62 (6): 555–559. doi:10.1136/jech.2007.064030. ISSN 0143-005X. PMID 18477756. S2CID 34200426. Retrieved 2009-04-29.
^ Heymann DL, ed. (2008). Control of Communicable Diseases Manual (19th ed.). Washington, D.C.: American Public Health Association. ISBN 978-0-87553-189-2.
^ "Yellow fever". Fact sheets. World Health Organization. 7 May 2019.
^ USAMRIID (2011). USAMRIID's Medical Management of Biological Casualties Handbook (PDF) (7th ed.). U.S. Government Printing Office. ISBN 9780160900150.
^ WHO guidelines for plague management: revised recommendations for the use of rapid diagnostic tests, fluoroquinolones for case management and personal protective equipment for prevention of post-mortem transmission [Internet]. World Health Organization. 2021.
^ Prentice, Michael B.; Rahalison, Lila (April 7, 2007). "Plague". Lancet. 369 (9568): 1196–1207. doi:10.1016/S0140-6736(07)60566-2. PMID 17416264. S2CID 208790222 – via PubMed.
^ King, John W (April 2, 2008). "Ebola Virus". eMedicine. WebMd. Retrieved 2008-10-06.
^ "Rabies Fact Sheet N°99". World Health Organization. July 2013. Retrieved 28 February 2014.
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Color confinement - Wikipedia
Particle physics phenomenon
The color force favors confinement because at a certain range it is more energetically favorable to create a quark–antiquark pair than to continue to elongate the color flux tube. This is analogous to the behavior of an elongated rubber-band.
An animation of color confinement. If energy is supplied to the quarks as shown, the gluon tube elongates until it reaches a point where it "snaps" and forms a quark–antiquark pair. Thus single quarks are never seen in isolation.
In quantum chromodynamics (QCD), color confinement, often simply called confinement, is the phenomenon that color-charged particles (such as quarks and gluons) cannot be isolated, and therefore cannot be directly observed in normal conditions below the Hagedorn temperature of approximately 2 terakelvin (corresponding to energies of approximately 130–140 MeV per particle).[1][2] Quarks and gluons must clump together to form hadrons. The two main types of hadron are the mesons (one quark, one antiquark) and the baryons (three quarks). In addition, colorless glueballs formed only of gluons are also consistent with confinement, though difficult to identify experimentally. Quarks and gluons cannot be separated from their parent hadron without producing new hadrons.[3]
2 Confinement scale
3 Models exhibiting confinement
4 Models of fully screened quarks
There is not yet an analytic proof of color confinement in any non-abelian gauge theory. The phenomenon can be understood qualitatively by noting that the force-carrying gluons of QCD have color charge, unlike the photons of quantum electrodynamics (QED). Whereas the electric field between electrically charged particles decreases rapidly as those particles are separated, the gluon field between a pair of color charges forms a narrow flux tube (or string) between them. Because of this behavior of the gluon field, the strong force between the particles is constant regardless of their separation.[4][5]
Therefore, as two color charges are separated, at some point it becomes energetically favorable for a new quark–antiquark pair to appear, rather than extending the tube further. As a result of this, when quarks are produced in particle accelerators, instead of seeing the individual quarks in detectors, scientists see "jets" of many color-neutral particles (mesons and baryons), clustered together. This process is called hadronization, fragmentation, or string breaking.
The confining phase is usually defined by the behavior of the action of the Wilson loop, which is simply the path in spacetime traced out by a quark–antiquark pair created at one point and annihilated at another point. In a non-confining theory, the action of such a loop is proportional to its perimeter. However, in a confining theory, the action of the loop is instead proportional to its area. Since the area is proportional to the separation of the quark–antiquark pair, free quarks are suppressed. Mesons are allowed in such a picture, since a loop containing another loop with the opposite orientation has only a small area between the two loops.
Confinement scale[edit]
The confinement scale or QCD scale is the scale at which the perturbatively defined strong coupling constant diverges. This is known as the Landau pole. The confinement scale definition and value therefore depend on the renormalization scheme used. For example, in the MS-bar scheme and at 4-loop in the running of
{\displaystyle \alpha _{s}}
, the world average in the 3-flavour case is given by[6]
{\displaystyle \Lambda _{\overline {MS}}^{(3)}=(332\pm 17)\,{\rm {{MeV}\,.}}}
When the renormalization group equation is solved exactly, the scale is not defined at all. It is therefore customary to quote the value of the strong coupling constant at a particular reference scale instead.
It is sometimes believed that the sole origin of confinement is the very large value of the strong coupling near the Landau pole. This is sometimes referred as infrared slavery (a term chosen to contrast with the ultraviolet freedom). It is however incorrect since in QCD the Landau pole is unphysical[7][8] as it can be seen by the fact that its position at the confinement scale largely depends on the choice renormalization scheme, i.e. on a convention. Most evidences point to a moderately large coupling, typically of value 1-3 [7] depending of the choice of renormalization scheme. In contrast to the simple but erroneous mechanism of infrared slavery, a large coupling is but one ingredient for color confinement, the other one being that gluons are color-charged and can therefore collapse into gluon tubes.
Models exhibiting confinement[edit]
In addition to QCD in four spacetime dimensions, the two-dimensional Schwinger model also exhibits confinement.[9] Compact Abelian gauge theories also exhibit confinement in 2 and 3 spacetime dimensions.[10] Confinement has been found in elementary excitations of magnetic systems called spinons.[11]
If the electroweak symmetry breaking scale were lowered, the unbroken SU(2) interaction would eventually become confining. Alternative models where SU(2) becomes confining above that scale are quantitatively similar to the Standard Model at lower energies, but dramatically different above symmetry breaking.[12]
Models of fully screened quarks[edit]
Besides the quark confinement idea, there is a potential possibility that the color charge of quarks gets fully screened by the gluonic color surrounding the quark. Exact solutions of SU(3) classical Yang–Mills theory which provide full screening (by gluon fields) of the color charge of a quark have been found.[13] However, such classical solutions do not take into account non-trivial properties of QCD vacuum. Therefore, the significance of such full gluonic screening solutions for a separated quark is not clear.
Lund string model
Dual superconducting model
^ Barger, V.; Phillips, R. (1997). Collider Physics. Addison–Wesley. ISBN 978-0-201-14945-6.
^ Greensite, J. (2011). An introduction to the confinement problem. Lecture Notes in Physics. Vol. 821. Springer. Bibcode:2011LNP...821.....G. doi:10.1007/978-3-642-14382-3. ISBN 978-3-642-14381-6.
^ Wu, T.-Y.; Hwang, Pauchy W.-Y. (1991). Relativistic quantum mechanics and quantum fields. World Scientific. p. 321. ISBN 978-981-02-0608-6.
^ Muta, T. (2009). Foundations of Quantum Chromodynamics: An introduction to perturbative methods in gauge theories. Lecture Notes in Physics. Vol. 78 (3rd ed.). World Scientific. ISBN 978-981-279-353-9.
^ Smilga, A. (2001). Lectures on quantum chromodynamics. World Scientific. ISBN 978-981-02-4331-9.
^ "Review on Quantum Chromodynamics" (PDF). Particle Data Group.
^ a b A. Deur, S. J. Brodsky and G. F. de Teramond, (2016) “The QCD Running Coupling” Prog. Part. Nucl. Phys. 90, 1
^ D. Binosi, C. Mezrag, J. Papavassiliou, C. D. Roberts and J. Rodriguez-Quintero, (2017) “Process-independent strong running coupling” Phys. Rev. D 96, no. 5, 054026
^ Wilson, Kenneth G. (1974). "Confinement of Quarks". Physical Review D. 10 (8): 2445–2459. Bibcode:1974PhRvD..10.2445W. doi:10.1103/PhysRevD.10.2445.
^ Schön, Verena; Michael, Thies (2000). "2d Model Field Theories at Finite Temperature and Density (Section 2.5)". In Shifman, M. (ed.). At the Frontier of Particle Physics. pp. 1945–2032. arXiv:hep-th/0008175. Bibcode:2001afpp.book.1945S. CiteSeerX 10.1.1.28.1108. doi:10.1142/9789812810458_0041. ISBN 978-981-02-4445-3. S2CID 17401298.
^ Lake, Bella; Tsvelik, Alexei M.; Notbohm, Susanne; Tennant, D. Alan; Perring, Toby G.; Reehuis, Manfred; Sekar, Chinnathambi; Krabbes, Gernot; Büchner, Bernd (2009). "Confinement of fractional quantum number particles in a condensed-matter system". Nature Physics. 6 (1): 50–55. arXiv:0908.1038. Bibcode:2010NatPh...6...50L. doi:10.1038/nphys1462. S2CID 18699704.
^ Cahill, Kevin (1978). "Example of Color Screening". Physical Review Letters. 41 (9): 599–601. Bibcode:1978PhRvL..41..599C. doi:10.1103/PhysRevLett.41.599.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Color_confinement&oldid=1086623390"
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MultivariatePowerSeries/Add - Maple Help
Home : Support : Online Help : MultivariatePowerSeries/Add
Add multivariate power series or univariate polynomials over power series
Add(P, coefopt)
Add(U)
coefopt
(optional) equation of the form coefficients = C, where coefficients is a keyword and C is a list of polynomials and complex constants
univariate polynomials over power series generated by this package which have the same main variable, power series generated by this package, polynomials, or complex constants
sequence of univariate polynomial over power series generated by this package which have the same main variable, power series generated by this package, polynomials, or complex constants
The command p1 + p2 returns the sum of the terms p1 and p2. The result is a power series.
The command Add(P) returns the sum of the terms in P.
The command Add(P, coefficients = C) returns the sum of the products C[i] * P[i]. The length of the list C must be the same as the number of elements of P.
The command u1 + u2 returns the sum of the terms u1 and u2. The result is a univariate polynomial over power series.
The command Add(U) returns the sum of the entries of U. They are converted to univariate polynomials over power series in the same variable. If this is not possible, an error is raised. This may happen if there are univariate polynomials over power series in different variables. It can also happen if the univariate polynomials over power series all have the same main variable, say x, but one of the other arguments is a power series that is not known to be expressible as a polynomial in x. The same restrictions apply to the calling sequence u1 + u2.
\mathrm{with}\left(\mathrm{MultivariatePowerSeries}\right):
Create three power series.
a≔\mathrm{GeometricSeries}\left([x,y]\right):
b≔\mathrm{PowerSeries}\left(1+x+y+z\right):
c≔\mathrm{PowerSeries}\left(2xy+3{z}^{3}\right):
Create a power series representing the sum of
and
b
a+b
[\textcolor[rgb]{0,0,1}{PowⅇrSⅇrⅈⅇs of}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{:}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\dots }]
a
a+1
[\textcolor[rgb]{0,0,1}{PowⅇrSⅇrⅈⅇs of}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{:}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\dots }]
a
b
c
, and the polynomial
xyz+1
\mathrm{Add}\left(a,b,c,1+xyz\right)
[\textcolor[rgb]{0,0,1}{PowⅇrSⅇrⅈⅇs of}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{:}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\dots }]
a+5b+10c
\mathrm{Add}\left(a,b,c,\mathrm{coefficients}=[1,5,10]\right)
[\textcolor[rgb]{0,0,1}{PowⅇrSⅇrⅈⅇs of}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{30}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{20}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{:}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\dots }]
Create a univariate polynomial over power series, given by a polynomial.
f≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(xz+y{z}^{2}+xy{z}^{3},z\right):
Add a polynomial to
f
. These two calling sequences are equivalent.
f+z+3
[\textcolor[rgb]{0,0,1}{UnⅈvarⅈatⅇPolynomⅈalOvⅇrPowⅇrSⅇrⅈⅇs:}\left(\textcolor[rgb]{0,0,1}{3}\right)\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\dots }\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{3}}]
\mathrm{Add}\left(f,z+3\right)
[\textcolor[rgb]{0,0,1}{UnⅈvarⅈatⅇPolynomⅈalOvⅇrPowⅇrSⅇrⅈⅇs:}\left(\textcolor[rgb]{0,0,1}{3}\right)\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\dots }\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{3}}]
Add a power series to f that is independent of z (and thus trivially polynomial in z).
f+\mathrm{GeometricSeries}\left([x,y]\right)
[\textcolor[rgb]{0,0,1}{UnⅈvarⅈatⅇPolynomⅈalOvⅇrPowⅇrSⅇrⅈⅇs:}\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\dots }\right)\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{3}}]
Create a separate univariate polynomial over power series, and add it to f.
g≔\mathrm{UnivariatePolynomialOverPowerSeries}\left([\mathrm{GeometricSeries}\left([x,y]\right),\mathrm{PowerSeries}\left(3\right)],z\right):
f+g
[\textcolor[rgb]{0,0,1}{UnⅈvarⅈatⅇPolynomⅈalOvⅇrPowⅇrSⅇrⅈⅇs:}\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\dots }\right)\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\dots }\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{3}}]
This will raise an error, because we're trying to add univariate polynomials over power series with different main variables.
h≔\mathrm{UnivariatePolynomialOverPowerSeries}\left([\mathrm{GeometricSeries}\left([x,y]\right),\mathrm{PowerSeries}\left(3\right)],w\right):
f+h
Error, incompatible inputs: expect UnivariatePolynomialOverPowerSeries with the same main variable, but received w <> z
This also will not work, because Maple cannot determine that d is polynomial in z (though actually it is).
d≔\mathrm{PowerSeries}\left(d↦\mathrm{ifelse}\left(d=0,0,\frac{z\cdot {x}^{d-1}}{\left(d-1\right)!}\right),\mathrm{variables}={x,z}\right)
\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{PowⅇrSⅇrⅈⅇs:}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\dots }]
f+d
Error, attempted to convert a power series involving z to a univariate polynomial over power series in z, but it is not known to be polynomial in z
We define e in the same way as d but specify the analytic expression, and then we can successfully add it to f.
e≔\mathrm{PowerSeries}\left(d↦\mathrm{ifelse}\left(d=0,0,\frac{z\cdot {x}^{d-1}}{\left(d-1\right)!}\right),\mathrm{analytic}=z\mathrm{exp}\left(x\right)\right)
\textcolor[rgb]{0,0,1}{e}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{PowⅇrSⅇrⅈⅇs of}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{:}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\dots }]
f+e
[\textcolor[rgb]{0,0,1}{UnⅈvarⅈatⅇPolynomⅈalOvⅇrPowⅇrSⅇrⅈⅇs:}\left(\textcolor[rgb]{0,0,1}{0}\right)\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\dots }\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{3}}]
The MultivariatePowerSeries[Add] command was introduced in Maple 2021.
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Tools Archives - Kan Ouivirach
Author zkanPosted on October 6, 2009 September 14, 2015 Categories ToolsTags Open Source, OpenCV3 Comments on OpenCV 2.0 released
\sqrt 2
Author zkanPosted on September 17, 2009 September 17, 2009 Categories ToolsTags LaTeX, PluginLeave a comment on LaTeX for WordPress
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Tracking error - Wikipedia
In finance, tracking error or active risk is a measure of the risk in an investment portfolio that is due to active management decisions made by the portfolio manager; it indicates how closely a portfolio follows the index to which it is benchmarked. The best measure is the standard deviation of the difference between the portfolio and index returns.
Many portfolios are managed to a benchmark, typically an index. Some portfolios are expected to replicate, before trading and other costs, the returns of an index exactly (e.g., an index fund), while others are expected to 'actively manage' the portfolio by deviating slightly from the index in order to generate active returns. Tracking error is a measure of the deviation from the benchmark; the aforementioned index fund would have a tracking error close to zero, while an actively managed portfolio would normally have a higher tracking error. Thus the tracking error does not include any risk (return) that is merely a function of the market's movement. In addition to risk (return) from specific stock selection or industry and factor "betas", it can also include risk (return) from market timing decisions.
Dividing portfolio active return by portfolio tracking error gives the information ratio, which is a risk adjusted performance measure.
2.1 Index fund creation
If tracking error is measured historically, it is called 'realized' or 'ex post' tracking error. If a model is used to predict tracking error, it is called 'ex ante' tracking error. Ex-post tracking error is more useful for reporting performance, whereas ex-ante tracking error is generally used by portfolio managers to control risk. Various types of ex-ante tracking error models exist, from simple equity models which use beta as a primary determinant to more complicated multi-factor fixed income models. In a factor model of a portfolio, the non-systematic risk (i.e., the standard deviation of the residuals) is called "tracking error" in the investment field. The latter way to compute the tracking error complements the formulas below but results can vary (sometimes by a factor of 2).
The ex-post tracking error formula is the standard deviation of the active returns, given by:
{\displaystyle TE=\omega ={\sqrt {\operatorname {Var} (r_{p}-r_{b})}}={\sqrt {{E}[(r_{p}-r_{b})^{2}]-({E}[r_{p}-r_{b}])^{2}}}={\sqrt {(w_{p}-w_{b})^{T}\Sigma (w_{p}-w_{b})}}}
{\displaystyle r_{p}-r_{b}}
is the active return, i.e., the difference between the portfolio return and the benchmark return.[1] The optimization problem of maximizing the return, subject to tracking error and linear constraints, may be solved using second-order cone programming:
{\displaystyle \max _{w}\;\mu ^{T}(w-w_{b}),\quad {\text{s.t.}}\;(w-w_{b})^{T}\Sigma (w-w_{b})\leq \omega ^{2},\;Ax\leq b,\;Cx=d}
Under the assumption of normality of returns, an active risk of x per cent would mean that approximately 2/3 of the portfolio’s active returns (one standard deviation from the mean) can be expected to fall between +x and -x per cent of the mean excess return and about 95% of the portfolio's active returns (two standard deviations from the mean) can be expected to fall between +2x and -2x per cent of the mean excess return.
Index funds are expected to have minimal tracking errors.
Inverse exchange-traded funds are designed to perform as the inverse of an index or other benchmark, and thus reflect tracking errors relative to short positions in the underlying index or benchmark.
Index fund creation[edit]
Index funds are expected to minimize the tracking error with respect to the index they are attempting to replicate, and this problem may be solved using standard optimization techniques. To begin, define
{\displaystyle \omega ^{2}}
{\displaystyle \omega ^{2}=(w-w_{b})^{T}\Sigma (w-w_{b})}
{\displaystyle w_{b}}
is the benchmark return and
{\displaystyle \Sigma }
is the covariance matrix of assets in an index. While creating an index fund could involve holding all
{\displaystyle N}
investable assets in the index, it is sometimes better practice to only invest in a subset
{\displaystyle K}
of the assets. These considerations lead to the following mixed-integer quadratic programming (MIQP) problem:
{\displaystyle {\begin{aligned}\min _{w}&\quad \omega ^{2}\\{\text{s.t.}}&\quad w_{j}\leq y_{j},\quad \sum _{j=1}^{N}y_{j}\leq K\\&\quad \ell _{j}y_{j}\leq w_{j}\leq u_{j}y_{j},\quad y_{j}\in \{0,1\},\quad \ell _{j},\;u_{j}\geq 0\end{aligned}}}
{\displaystyle y_{j}}
is the logical condition of whether or not an asset is included in the index fund, and is defined as:
{\displaystyle y_{j}={\begin{cases}1,\quad &w_{j}>0\\0,\quad &{\text{otherwise}}\end{cases}}}
^ Cornuejols, Gerard; Tütüncü, Reha (2007). Optimization Methods in Finance. Mathematics, Finance and Risk. Cambridge University Press. pp. 178–180. ISBN 0521861705.
Tracking Error - YouTube
What is the Tracking Error?
Retrieved from "https://en.wikipedia.org/w/index.php?title=Tracking_error&oldid=1082648615"
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Abstract: In view of the lack of patent big data in research on technology foresight in the industrial robot field, this paper introduces an improved method based on patent mining and knowledge map. Firstly, SAO structure is extracted from selected patents, secondly, the similarity between patents is calculated based on extracted SAO structure, thirdly, patent network and patent map are drawn based on calculated patent similarity matrix, technology evolution process and future trends of industrial robot are summarized from patent network, and future potential technology opportunities are predicted based on technological vacancies identified from patent map. Finally, this paper identifies six key technical areas and four potential technical opportunities in the field of the industrial robot.
Keywords: Technology Foresight, Patent Mining, SAO, Knowledge Map, Industrial Robot
Lin\left({W}_{1},{W}_{2}\right)=\frac{2\ast depth\left(lcs\left({W}_{1},{W}_{2}\right)\right)}{depth\left({W}_{1}\right)+depth\left({W}_{2}\right)}
Si{m}^{\prime }\left(X,Y\right)=\frac{2\ast Match\left(X,Y\right)}{|X|+|Y|}
Si{m}^{\prime }\left(X,Y\right)
|X|
|Y|
Si{m}^{\prime }\left(SA{O}_{i},SA{O}_{j}\right)=\left[Si{m}^{\prime }\left({S}_{i},{S}_{j}\right)+Si{m}^{\prime }\left({A}_{i},{A}_{j}\right)+Si{m}^{\prime }\left({O}_{i},{O}_{j}\right)\right]
Si{m}^{\prime }\left({S}_{i},{S}_{j}\right)
Si{m}^{\prime }\left({A}_{i},{A}_{j}\right)
Si{m}^{\prime }\left({O}_{i},{O}_{j}\right)
S{O}_{ij}=\left\{\begin{array}{l}1,Sim\left(S{O}_{i},S{O}_{j}\right)\ge t\\ 0,\text{other}\end{array}
Si{m}_{A,B}=\frac{2\ast Match\left(A,B\right)}{|A|+|B|}
|A|
|B|
Cite this paper: Wen, X. (2019) Technology Foresight Research of Industrial Robot Based on Patent Analysis. Journal of Data Analysis and Information Processing, 7, 74-90. doi: 10.4236/jdaip.2019.72005.
[1] Miles, I. (2010) The Development of Technology Foresight: A Review. Technological Forecasting and Social Change, 77, 1448-1456.
[2] Forecast, J. (2003) Technology Foresight-Past and Future. Journal of Forecast, 22, 79-82. https://doi.org/10.1002/for.846
[3] Gao, H., Wang, D. and Li, Z. (2018) A Review of the Theory and Practice of Technology Foresight. China Management Informationization, 21, 78-82.
[4] Reuters, T. (2016) The World in 2025: 10 Predictions of Innovation.
http://sciencewatch.com/sites/sw/files/m/pdf/World-2025.pdf
[5] Sokolov, A., Chlok, A. and Mesropyan, V. (2013) Long-Term Science and Technology Policy—Russian Priorities for 2030. Social Science Electronic Publishing, New York.
[6] Yang, C. and Wei, H. (2017) Comparative Study on the Text Evaluation of Robotics Technology Roadmap among US, EU, Japan and China. Science of Science and Management of S. & T., 38, 24-34.
[7] National Manufacturing Strategy Advisory Committee (2015) Made in China 2025 Technology Roadmap for Key Areas (2015 Edition).
[8] Ministry of Industry and Information Technology (2017) Industry Key Common Technology Development Guide.
[9] Chen, Y., Tan, J., Wang, Z., et al. (2018) Technological Opportunity Analysis of Industrial Robots from the Perspective of Patents. Science Research Management, 39, 144-156.
[10] Brock-hoff, K.K. (1992) Instruments for Patent Data Analyses in Business Firms. Technovation, 12, 41-59.
[11] Ma, T. (2015) Patent Analysis: Methods, Chart Interpretation and Intelligence Mining. Intellectual Property Press, Beijing, 1.
[12] Zhao, Y., Dong, Y. and Zhu, X. (2006) Patent Analysis and Its Application in Intelligence Research. Library and Intelligence Work, 5, 19-22.
[13] Liu, Y. and Li, D. (2018) Patent Analysis Research Review and Application Suggestions for Scientific and Technological Intelligence. Science and Technology Management Research, 38, 155-160.
[14] Chen, X. (2017) Research on the Status Quo of Patent Strength in Nine Cities in the Pearl River Delta and Countermeasures to Improve It. Technology Management Research, No. 23, 186-191.
[15] Guo, Y., Qian, Y., Zhang, L., et al. (2017) Analysis of China’s Technological Innovation Situation Based on Overseas Patent Layout. Technology Management Research, No. 23, 174-180.
[16] Zhang, H., Long, X. and Hu, J. (2010) Bibliometric Analysis of Domestic Patent Disclosure and Authorization in South China University of Technology. Intelligence Exploration, No. 12, 37-39.
[17] Liu, X., Wen, T. and Yang, Z. (2015) Patent Information Visualization Analysis System Status and Technical Basis. Information Theory and Practice, 38, 1-5.
[18] Li, X., Xie, Q., Huang, L., et al. (2018) Research on the Evolution Trajectory of Emerging Technologies Based on SAO Structural Semantic Mining. Science of Science and Management of Science and Technology, 39, 17-31.
[19] Chen, A., Liu, X. and Gao, G. (2012) Characteristics of the Formation of Highly Cited Patents in Emerging Industries—A Case Study of Fuel Cells. Scientific Research Management, 33, 9-15.
[20] Jeong, C. and Kim, K. (2014) Creating Patents on the New Technology Using Analogy-Based Patent Mining. Expert Systems with Applications, 41, 3605-3614.
[21] Lee, C., Kang, B. and Shin, J. (2014) Novelty-Focused Patent Mapping for Technology Opportunity Analysis. Technological Forecasting and Social Change, 45, 3865-3876.
[22] Yoon, J. and Kim, K. (2012) An Analysis of Property-Function Based Patent Networks for Strategic R&D Planning in Fast-Moving Industries: The Case of Silicon-Based Thin Film Solar Cells. Expert Systems with Applications, 39, 7709-7717.
[23] International Federation of Robotics (2017) Industrial Robots-Definition and Types WR 2016.
https://ifr.org/img/office/Industrial_Robots_2016_Chapter_1_2.pdf
[24] Li, F. (2017) Research on Enterprise Technology Innovation Capability Based on Patent Map. Liaoning University, Liaoning.
[25] Editorial Department of Robotics and Applications (2013) Current Situation and Development of Industrial Robots in China. Robotics and Applications, 1, 3-5.
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Draw the graphs of the equations x + y = 6 and 2x + 3y = 16 on the same - Maths - Linear Equations in Two Variables - 8534611 | Meritnation.com
\mathrm{The} \mathrm{given} \mathrm{equation} \mathrm{is},\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}} x + y = 6\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒ y = 6 - x ...........\left(1\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}
⇒ 3y = 16 - 2x\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒ y = \frac{16 - 2x}{3} ...........\left(2\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}
So, from the graph we see that the two lines intersect at (2, 4).
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Recently Pine announced the new PineNote device, a powerful open-source e-ink device that is more powerful than the PineTab - by a lot! Unfortunately, due to patents, the device will be $400. Whilst still a massive bargain for this class of device, this is greatly outside of my budget.
But this got me thinking, how would I use this device? Is it really the device I actually want?
There have been several discussions recently over on the Pine forum about devices that I am very interested in. In one post there is a discussion about an e-paper variant of the PinePhone. For a while now we have been taunted with the prospect of a PineCom device, but I really suspect this will not be realised for quite some time as yet. I'm sure there are other discussions by members of the Pine community too...
Many of these posts are simply just ideas, or are projects with no real expectation of completion.
Often I find myself wanting a small, portable, low-power Linux device that just does one thing, really well. I want to be able to read my emails, check a web page, make a note, do a calculation - nothing too computationally heavy. Just a series of very simple tasks, one at a time. This seems like a reasonable expectation, and yet this is actually quite hard to find.
High-power solutions exist, such as mobile phones. If you're happy with large binary blobs that side load applications that cannot be deleted, or are willing to pay an arm and leg, then these are great. But mobile phones are almost too capable, they are quite literally mini-desktop machines. I have no doubt there are now mobiles out there more powerful than my daily driver desktop machine!
Worth a mention here is Pine's PinePhone, but as it's called a "smart phone", people expect it to do smart things. It has 4 cores, 3GB of RAM, eats battery, has several cameras, full-fat Linux and a mostly closed-source cell modem.
There are of course low-power solutions, based on the BL602 or the near-infinite devices based on Espressif's ESP8266/ESP32. The problem with these low-power solutions is that they are slightly too limited and cannot build upon the many existing applications and drivers that already exist for Linux. Sure, you could make it do one of these tasks - but you're not going to get many different applications working seamlessly all on the same device.
(Old) Solution
What I guess I'm really looking for is a modern-day PDA. As Wikipedia describes it:
Nearly all modern PDAs have the ability to connect to the Internet. A PDA has an electronic visual display, letting it include a web browser. Most models also have audio capabilities, allowing usage as a portable media player, and also enabling most of them to be used as telephones. Most PDAs can access the Internet, intranets or extranets via Wi-Fi or Wireless WANs. Sometimes, instead of buttons, PDAs employ touchscreen technology. The technology industry has recently recycled the term personal digital assistance. The term is more commonly used for software that identifies a user's voice to reply to the queries.
I think the great thing about a PDA is in the name itself - it's a personal digital assistant - it's entire purpose for being is to assist you. It's not there to track you, it's not there to distract you, it's not there to overwhelm you - it just does one simple task as a time - and it does it well.
I think what we need is a modern-day open-source Linux variant of the PDA concept, with modern ideas and modern hardware. Something that sits halfway between a phone and a "smart" device (such as a PineTime). Something that is designed to sync with the Linux operating system and provide a simplified but coherent user experience.
So now I will propose such a device... I'll start with some hardware, then talking about the operating system and finally talk about the applications. The idea is to make a realistic proposal that can be achieved with existing hardware and software (for the most part).
Ideally I would like to work with Pine on this project, if nothing else because they have a good open-source ethos. A likely candidate base for a Linux PDA would be a PineCube, with the following specifications (which I'll also add notes to):
Allwinner S3 Cortex-A7 - This processor is just about fast enough to run a single application at a time.
128MB DDR3 - One would imagine this would allow a running application to safely use 64MB of RAM.
128Mb SPI Flash - This could be just about viable for having a read-only (for the most part) Linux operating system.
mSD slot - This would then be used for data and could be used for transferring files/applications too and from the device, including loading updates, etc.
5 MPx OV5640 sensor - There has been a great reverse engineering/driver push through the PinePhone community.
10/100M Ethernet - I imagine this to be largely redundant for a PDA.
802.11 b/g/n WiFi - This will be exceptionally useful.
Bluetooth 4.1 - This could be great for syncing and/or connecting to other devices, such as a BT keyboard, PineTime or headphones.
USB 2.0 - Additional option for connecting devices.
26 pin GPIO header - I imagine this would be largely used for connecting the touchscreen LCD.
Microphone - Communications and/or voice recordings.
Passive Power Over Ethernet (PoE) - I imagine this would be largely redundant for a PDA.
Interchangeable M12 lens type (wide, fisheye, zoom, etc.,) - Again, likely largely redundant for a PDA.
IR LEDs for night vision - Could be useful for taking pictures in low-light conditions.
Optional 4.5″ RGB LCD screen - This would of course be exceptionally useful.
Battery support - If we want to really use this device, this will be entirely necessary.
I believe the PineCube offers the minimum of features that such a project may want, and therefore would make a viable base platform. Given this dev-kit ships for $30 USD, one can imagine being able to deliver a complete PDA device for $60 - $80 USD (before shipping and tax) and still make a comfortable margin.
As already discussed, the idea would be to use Linux. There has already been some effort to mainline into the kernel, with a viable distribution likely being Armbian. As the networking appears to be supported, it should be possible to do initial development work via SSH, although a serial terminal is also supported.
The idea for the window manager will be to recycle the oakwm project ideas and build an even lighter user interface from scratch. Many of the original features from wm2, specifically the window decoration, I found to actually be extremely CPU intensive. As it happens though, for the most part, we don't actually need or want window decoration anyway. Rather than hack wm2 to do what we need, it will likely be easier to simply build a minimal user interface from the ground up.
Oakwm window manager
Essentially the user interface will be approximately like the following:
0001 +--------------------------------+
0002 | [=] [app name] [v] [|||] |
0004 | |
0008 | Application |
A space is reserved at the top for a toolbar that is always accessible and on-top, with the application always being 'full-screen' below. The reason for keeping the toolbar always visible is because it will always be running and be the only way to escape some running application and back into the main menu. Only applications in the 'main window' will get some decent processing time, everything else is either closed or opened periodically.
There would also be an on-screen keyboard for touchscreens, so applications would need to be able to handle dynamic resizing:
0025 | Keyboard |
So it's very likely that applications will need to run in under 64MB of RAM when running with a GUI, so they need to be designed really well. This does have benefits for other devices though, for example, the PinePhone which is also very RAM limited. It also couldn't hurt for modern development to be much more resource friendly!
One thing that I am considering is whether to use the JVM on the PDA. Many older Android phones for example would run a very lightweight JVM for all applications, and these were surprisingly fast all things given.
EDIT: This server now operates on 128MB of RAM, including running nginx, a jvm, pandoc and ffmpeg - 128MB should be plenty if handled correctly.
Whilst C/C++ would be both faster and smaller, the JVM offers instant cross-platform capability and the greatest possibility of multiple device support. The benefit of this would be that you could run exactly the same software on your PDA as you do on Linux, Windows and MacOS - with the small RAM footprint! No Linux lock-in.
The greatest consideration would be whether the resources are available. In theory, running applications whilst the JVM is already started should be quite fast. You can even limit the RAM available considerably in order to prevent OOM and swapping.
The last real consideration is the possibility of some kind of 'app-store'. This would just be a collection of PDA specific friendly apps that are able to be accessed from the app menu. Ideally apps that are to be used with the device would need to have several features, such as low-RAM footprints, be touch-input friendly, be able to handle kill signals elegantly, etc. There would be no reason for example to have GIMP in the app store, as there is not a hope in hell of that running on a PDA (nor would it be the purpose or place of a PDA).
This would be the entire device's configuration, such as system theme, language, time/date, internet connection, device switches, etc. Everything that can be adjusted for the PDA and is not application specific would likely find its way in the configuration application.
As there will be so many settings, some form of menu navigation and grouping will be required. It would then make sense to force a reload of the PDA (warm-reset) in order to ensure the entire system state is reset.
A very simple initial application is a calculator - if it can't do this then we really are screwed. Unlike some applications though, as we're running Linux, it should even be possible to make use of a solver and really add some value. You could expect it to answer something like
2x=6
As the PineCube has a camera (it's supposed to be an IP camera after all), it would make sense to make use of it. This would include photos and videos, although it is quite likely this functionality would largely depend on the write speed of the external micro-SD.
This is a basic expectation of a PDA device - to be able to store notes. This will mean not only a text viewer, but a text editor. One should expect that it is also RAM efficient and capable of opening large files, as well as searching for text in the large files.
My initial thinking is that it would make use of markdown for markup of the text, meaning there is some very lightweight rich text, whilst still being very compatible with other formats. With the use of pandoc it could easily be exported to anything the user needs (although this would likely never be run on the PDA itself).
This is notoriously hard to implement, I would most definitely be looking to use a library for this. Time and dates for example are on their own are insanely hard to correctly implement.
The idea essentially would just be to have something that can view an existing calendar - the capability would definitely be quite limited. It's not entirely clear how everything will be synced across multiple applications as so many different formats are used for email calendars.
Again this is something I would want to use a library for. I imagine the easiest way to handle this would be to sync with POP, IMAP or SMTP. This would require the use of several tricks in order to use very little RAM - the shear amount of data alone could make it near impossible to display for example.
This would be text-based communication, initially a lightweight IRC client (as state doesn't need to be tracked). Other chat clients would have some expectation that they are active in the background and require some database to store messages. If we remove the need for logging and just keep some messages in RAM (and of course channels to connect to would be a configuration saved to disk), then in theory this would be extremely lightweight.
This will have to be thought about carefully - the PDA isn't going to be able to use something like Firefox. But a possibility could be Dillo or Netsurf, two 'reasonable' options. If we can use just their libraries, it could be possible to build a very thin single-page wrapper.
Projects like the ESP32-based WiPhone have shown that it's possible to use a little processing power and WiFi to have an internet connected phone, capable of doing calls. This means that it's quite hopeful that an even more powerful machine is capable of handling phone calls quite easily, likely being also able to do much more - such as audio compression, noise reduction and recovery.
Feed Reader / Podcasts
Building a new RAM friendly RSS feed reader has been on my mind for a while now, I even wrote a blog about it. I already found that Thunderbird was ill suited to the task, as is Akregator. A new RSS feed reader that is speed conscious will be a welcomed new piece of software for the open source community.
Given that it can play stuff, why not get it to play music? Perhaps even some small h264 encoded videos? I don't see why this device can't be a very nice little MP3/MP4 player replacement. With BT support, in theory any wireless headphones should be able to play music at acceptable quality.
It could in theory be possible to also watch Youtube videos and the like, possibly through a dedicated wrapper application. This would be something to look at a long-time down the line though.
After using offline maps in FitoTrack as part of my exercise tracking, I have been convinced that offline maps are incredibly useful. Not so long ago I went out for a long walk with a friend late at night, and we ended up using a classical paper map in a petrol garage to find home again. Even if the PDA does not have GPS or any way of locating your position, the ability to just view a map is incredibly useful.
One could imagine a scenario where you are navigating in a town or out in the wilderness with your low-power PDA device, using landmarks to triangulate your position (like they did in the old days).
Given that entire maps of Counties are just a few GBs, it should be quite possible to store this data on an external SD card. Some smart search algorithm could then be used to pull out the relevant data for a given location and then render it on the display.
I am currently in talks in IRC about the PineCube being used for such a purpose. The main points will be in finding out the following:
The current state of the LCD driver. Writing an LCD driver for the Linux kernel from scratch is not something I've done before, so it will be a lot of pain if I end up having to do that.
The current state of the PineCube project overall and ensuring they are still in stock.
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PUCCH format 2 demodulation reference signal - MATLAB ltePUCCH2DRS - MathWorks Deutschland
seq = ltePUCCH2DRS(ue,chs,ack)
[seq,info] = ltePUCCH2DRS(ue,chs,ack)
seq = ltePUCCH2DRS(ue,chs,ack) returns a matrix containing demodulation reference signal (DRS) associated with PUCCH format 2 transmission, given a structure of UE-specific settings, a structure with channel transmission configuration settings, and hybrid ARQ (HARQ) indicator values, ack.
[seq,info] = ltePUCCH2DRS(ue,chs,ack) also returns a PUCCH information structure array, info.
Generate PUCCH Format 2 DM-RS symbols for UE specific settings.
Initialize input configuration structures (ue and chs). Here no HARQ bits will be sent by inputting an empty ack vector. Generate the PUCCH Format 2 DM-RS symbols.
sym = ltePUCCH2DRS(ue,chs,[]);
chs1.ResourceSize = 0;
ack1 = 0;
interferenceNoCoMP = abs(sum(ltePUCCH2DRS(ue1,chs1,ack1).*conj(ltePUCCH2DRS(ue2,chs2,ack2))))
interferenceUsingCoMP = abs(sum(ltePUCCH2DRS(ue1,chs1,ack1).*conj(ltePUCCH2DRS(ue2,chs2,ack2))))
Generate the PUCCH format 2 DM-RS sequences for two transmit antenna paths.
Because there are two antennas, the DM-RS sequences are output as a two- column vector, and the info output structure contains two elements.
Alpha: [2.6180 4.7124 0 0.5236]
{n}_{PUCCH}^{\left(2\right)}
{N}_{RB}^{\left(2\right)}
{N}_{cs}^{\left(1\right)}
binary vector containing 0, 1 or 2 elements
Hybrid ARQ indicator values, specified as nonnegative integer vector. This vector is expected to be the block of bits b(0),...,b(Mbit–1) specified in TS 36.211 [1], Section 5.4.2. An Mbit value of 20, 21, or 22 corresponds to PUCCH format 2, 2a, or 2b, respectively, as described in TS 36.211 [1], Table 5.4-1.
The standard does not support format 2a or 2b transmission with extended cyclic prefix. If the ack setting corresponds to format 2a or 2b transmission and extended cyclic prefix is set for ue.CyclicPrefixUL, the function returns an empty matrix for seq.
PUCCH format 2 information, returned as a structure array with elements corresponding to each transmit antenna and containing these fields. When configured for format 2a or 2b transmission with extended cyclic prefix, the info structure contains all fields, but each field is empty.
{n}_{cs}^{cell}
Orthogonal sequence for each slot, returned as a 4-by-2 numeric matrix. (
\overline{w}
ltePUCCH2DRSDecode | ltePUCCH2DRSIndices | ltePUCCH2 | ltePUCCH2Decode | ltePUCCH2Indices | ltePUCCH1DRS | ltePUCCH3DRS
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Template:Databox equipment bonuses - The RuneScape Wiki
Template:Databox equipment bonuses
This documentation is transcluded from Template:Databox equipment bonuses/doc. [edit] [history] [purge]
Template:Databox equipment bonuses invokes function main in Module:Infobox Bonuses new using Lua.
Standard databox for equippable items.
2.6 degrades
2.16 pvmReduction
2.17 pvpReduction
2.18 attack_range
2.21 altimage
2.22 isrecolour
3.1 mainDamage
3.2 mainAccuracy
3.3 offDamage
3.4 offAccuracy
3.5 mainType and offType
5 Displaying multiple databoxes
Note: All parameter values are case insensitive
The skill level requirements of the equipment; should use {{Skillclickpic}}, with multiple skills being separated by a comma. "None" should be used if no skill requirements exist.
The item's class, this can be easily found by looking at it in game. The possible values are:
The equipment slot in which the item is equipped. The possible values are:
Main or Main-hand or Mainhand or Weapon
Off-hand or shield
Off-hand weapon or OHW
Important: The last two sets of values are not interchangeable. Shields and off-hand weapons must have their parameter values defined separately for categorization purposes.
The tier of the item. This is the value used in the formula here to give the item's stats. By default, rely on the armour value of armour and the accuracy of weapons. However, try to make sure the damage lines up too - remember, if the armour has tier
{\displaystyle t}
Tank armour: armour bonus for tier
{\displaystyle t}
, life points for tier
{\displaystyle t}
{\displaystyle t\geq 80}
, or for shields
{\displaystyle t\geq 70}
Power armour: armour bonus for tier
{\displaystyle t-5}
, damage bonus for tier
{\displaystyle t}
PvP armour: armour bonus for tier
{\displaystyle t}
{\displaystyle t}
Hybrid/all armour: armour bonus for tier
{\displaystyle t-15}
For items with no tier, either N/a, No, or None should be used.
The type of item this is. This accepts a few specific values to define what the item is (all of the following accepted, aliases in the same bullet, comma separated):
Power armour, power
Tank armour, tank
PvP armour, pvp
In addition to being displayed, this adds a category, and for armour ones, allows damage reduction calculation.
If this item degrades. If it does, enter the number of charges it has - see equipment degradation for more info on this. If charges are unknown, "Yes" can be used to show it does degrade. If it doesn't degrade, nothing should be entered (will hide it by default).
The damage of the weapon. This applies to all handednesses of weapons - use the same parameter for main-hand, off-hand, and two-handed weapons. Leave blank if this is not applicable.
The accuracy of the weapon. This applies to all handednesses of weapons. Leave blank if this is not applicable.
The combat style used by a weapon. Fill with the appropriate value or leave it blank / delete the parameter if this is not applicable. This parameter applies to both mainhand and off-hand items. Possible values are:
Crushing or Crush
Slashing or Slash
Stabbing or Stab
Arrows or Arrow
Bolts or Bolt
Magic, Spells or Spell
The armour rating of a piece of equipment. Fill with the appropriate number or leave it blank / delete the parameter if this is not applicable.
The life points boost given by a piece of a equipment. Fill with the appropriate number or leave it blank / delete the parameter if this is not applicable.
The prayer bonus given by a piece of a equipment. Fill with the appropriate number or leave it blank / delete the parameter if this is not applicable.
The Strength bonus given by the piece of equipment. Fill with the appropriate number or leave it blank / delete the parameter if this is not applicable.
The Ranged bonus given by the piece of equipment. Fill with the appropriate number or leave it blank / delete the parameter if this is not applicable.
The Magic bonus given by the piece of equipment. Fill with the appropriate number or leave it blank / delete the parameter if this is not applicable.
pvmReduction
The template will attempt to automatically calculate the damage reduction from the tier, type, and class. Use this parameter if the calculated PvM damage reduction is not correct. If it is correct, delete the parameter.
pvpReduction
The template will attempt to automatically calculate the damage reduction from the tier, type, and class. Use this parameter if the calculated PvP damage reduction is not correct. If it is correct, delete the parameter.
Also accepts attack range (space instead of underscore)
The attack range of the weapon, where melee weapons are range 1, halberds 2, etc. Leave blank/delete if not applicable.
Also accepts aspeed
The attack speed of the weapon. Fill with the appropriate value or leave it blank / delete the parameter if this is not applicable. Possible values are:
An image of a player with the equipment on. Do not include the "File:" prefix and do not resize it. This size is determined through CSS.
| image = Abyssal whip equipped.png
If this parameter is missing, a category is added, except for pocket, ring, and ammo slot items. For other slots, setting it to no will prevent the category being added.
Same usage as image allowing for a second image.
isrecolour
Is the item a cosmetic recolour of another item (e.g. Category:Cosmetic change equipment). Primarily for filtering in the Armoury.
These parameters are still supported by the template, but should ideally be replaced.
Replaced by damage
The damage output of a main-hand or two-handed weapon. Fill with the appropriate number or leave it blank / delete the parameter if this is not applicable.
Replaced by accuracy
The accuracy of a main-hand or two-handed weapon. Fill with the appropriate number or leave it blank / delete the parameter if this is not applicable.
The damage output of an off-hand weapon. Fill with the appropriate number or leave it blank / delete the parameter if this is not applicable.
The accuracy of an off-hand weapon. Fill with the appropriate number or leave it blank / delete the parameter if this is not applicable.
mainType and offType
Replaced by style
The damage type inflicted by that weapon of that hand. Currently does nothing.
Replaced by altimage
Same usage as image allowing for a second image. This can cause issue with switch infoboxes and should be replaced as soon as possible.
{{Databox equipment bonuses
|mainDamage =
|mainAccuracy =
|offDamage =
|offAccuracy =
|armour =
|ranged =
|magic =
|aspeed =
Displaying multiple databoxes
Multiple databoxes can be displayed by identifying version# parameters inside the template.
The parameters that apply to every version box are the same as those on the basic bonuses databox; however, for every parameter there are an additional 20 named paramname#. The unnumbered parameters are the defaults, and will only be used for a version of the corresponding numbered param is not defined. For example, there is name, name1, name2, all the way to name20.
Parameters are only looked for up to the highest "version" defined. The "version" parameters only come as numbered parameters and they are used to define the text in the switch databox buttons.
Parameters can be referenced and reused by using a dollar sign ($) followed by the index. For example: |image4=$1 will use whatever image is being used for image1.
|version1=Level 10|version2=Level 40
|requirements1=10 {{Skill clickpic|Attack}}|requirements2=40 {{Skill clickpic|Attack}}
|class=Melee
|slot=Weapon
|mainDamage1=134|mainDamage2=536
|mainAccuracy1=191|mainAccuracy2=553
|offDamage=0
|offAccuracy=0
|style=Slash
|armour=0
|life=0
|aspeed=Average
|image=Exquisite whip equipped.png
? (edit)? (edit)? (edit)10 40 melee Meleemelee? (edit)? (edit)main hand weapon true? (edit)? (edit)? (edit)? (edit)? (edit)trueSlash? (edit)? (edit)--? (edit)? (edit)--00.00.0000--0--0-- 0 0.0 0.0 0 0.0 0.0 0 0.0 0.00104000PvM: 0%PvP: 0%average
Afalsetrue? (edit)slash0000.00.000000? (edit)? (edit){"ranged":0,"class":"melee","lp":0,"speed":"average","damage":0,"armour":0,"slot":"main hand weapon","strength":0,"style":"slash","magic":0,"prayer":0,"accuracy":0}{"ranged":0,"class":"melee","lp":0,"speed":"average","damage":0,"armour":"0.0","slot":"main hand weapon","strength":0,"style":"slash","magic":0,"prayer":0,"accuracy":0}{"ranged":0,"class":"melee","lp":0,"speed":"average","damage":0,"armour":"0.0","slot":"main hand weapon","strength":0,"style":"slash","magic":0,"prayer":0,"accuracy":0}
Retrieved from ‘https://runescape.wiki/w/Template:Databox_equipment_bonuses?oldid=35491012’
Hunting skill training dummy
1m ago - Mewtew
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Stoichiometry of Gases - Course Hero
General Chemistry/Gases/Stoichiometry of Gases
Stoichiometry of gases relies on the partial pressure of gases—the pressure of one gas in a mixture of gases.
Stoichiometry is the relationship between the amounts of products and reactants in a reaction. The law of conservation of mass states that the mass of the reactants must be equal to the mass of the products. The ideal gas law leads to the understanding that one mole of any gas occupies 22.4 liters of volume at standard temperature and pressure (STP). Together, these laws allow for the calculation of amounts of products and reactants in a chemical reaction involving gases.
For example, consider the combustion of ammonia:
4{\rm{NH}}_{3}(g)+7{\rm O}_{2}( g)\rightarrow4{\rm{NO}}_{2}{(g)}+6{\rm H}_{2}{\rm O}(l)
. The volume of NO2 gas that is produced from the combustion of 12.0 grams of NH3 can be calculated using the volume of gas at STP.
\left(\frac{{12.0\;{\rm g}\;{\rm{NH}}_3}}{1}\right)\!\left(\frac{1\;{\rm{mol}}\;{{\rm{NH}}}_3}{17.04\;\rm g}\right)\!\left(\frac{4\;{\rm{mol}}\;{\rm{NO}}_2}{4\;{\rm{mol}}\;{\rm{NH}}_3}\right)\!\left(\frac{22.4\;{\rm L}}{1\;{\rm{mol}}\;{\rm{NO}}_{2}}\right)=15.8\;{\rm L}\;{\rm{NO}}_{2}
In chemical reactions, it is impossible to keep the gases separate from one another. When more than one gas is present in a container, each gas exerts pressure on the container individually. The partial pressure is the pressure of an ideal gas that contributes to the total pressure of a mixture of gases at constant temperature. Dalton's law of partial pressures states that the total pressure (Ptotal) of a mixture of ideal and nonreacting gases is the sum of the partial pressures of the individual gases.
P_{\rm{total}}=P_1+P_2+P_3\mathellipsis+P_n
For example, consider a gas cylinder of a mixture of oxygen gas (O2) and nitrogen gas (N2). The partial pressure of O2 is 0.65 atm. The partial pressure of N2 is 0.15 atm. Assuming the mixture behaves as an ideal gas, the total pressure of the gas in the cylinder is the sum of the partial pressures.
\begin{aligned}P_{\rm{total}}&=P_{\rm{O_2}}+P_{\rm{N_2}}\\ &=0.65\;\rm{atm}+0.15\;\rm{atm}\\ &=0.80\;\rm{atm}\end{aligned}
The amount of a gas present in a mixture of gases can also be described by its mole fraction (
\chi
), the concentration expressed as the moles of solvent divided by the total number of all moles in a solution. For a gas it is the number of moles of gas (i) divided by the total number of moles in the gas mixture.
\chi=\frac{\text{moles of gas}\;i}{\text{total moles of gas in mixture}}
The partial pressure of a gas can also be stated in terms of its mole fraction.
P_i=\left(P_{\rm{total}}\right)\!\left(\chi\right)
Consider a mixture of 5.0 moles of hydrogen gas (H2) and 4.0 moles of argon (Ar). The total pressure of the mixture is 4.10 atm. The partial pressure of Ar is the product of the total pressure and the mole fraction of Ar.
\begin{aligned}P_{\rm{Ar}}&=\left(P_{\rm{total}}\right)\!\left(\chi_{\rm{Ar}}\right)\\&= \left(P_{\rm{total}}\right)\!\left(\frac {\text{moles }\rm{Ar}} {\text {total moles}}\right)\\&= \left(4.10\;{\rm{atm}}\right)\!\left(\frac {4.0\;{\rm{mol}}} {9.0\;{\rm {mol}} } \right)\\&= 1.8\;{\rm{atm}}\end{aligned}
Partial pressure also contributes to the way a gas dissolves in a liquid. According to Henry's law, the amount of a gas that dissolves in a certain type and volume of liquid is directly proportional to the partial pressure of that gas in equilibrium with that liquid at a specific temperature. This is represented by the equation
C=kP_{\rm{gas}}
, where C is the solubility of the given gas in the given solvent (given in M gas/L or mL gas/L), k is Henry's law constant (usually given in M/atm), and Pgas is the partial pressure of the gas (usually given in atm). This equation can be used to find the concentration of a gas dissolved in a liquid. For example, calculate the concentration of carbon dioxide (CO2) in 1.00 L of water (H2O) at a pressure of 3.10 atm and a temperature of 25.0°C. For CO2 in water, k is
3.36\times10^{-2}\;\rm M/\rm{atm}
\begin{aligned} C&=kP_{\rm{gas}}\\&=\left(3.36\times10^{-2}\,\rm{M/atm}\right)\!(3.10\;\rm{atm})\\&=0.104\;\rm M\end{aligned}
<Ideal Gases>Nonideal Gas Behavior
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Assessment Of Antibiofilm Activity Of Magnesium Fluoride Nanoparticles-Stabilized Nanosized Emulsion After Its Coating On Biomaterial Surfaces | J. Med. Devices | ASME Digital Collection
Tamilvanan Shunmugaperumal,
Shunmugaperumal, T., and Srinivasan, R. (June 3, 2011). "Assessment Of Antibiofilm Activity Of Magnesium Fluoride Nanoparticles-Stabilized Nanosized Emulsion After Its Coating On Biomaterial Surfaces." ASME. J. Med. Devices. June 2011; 5(2): 027502. https://doi.org/10.1115/1.3587101
antibacterial activity, biomedical materials, cellular biophysics, emulsions, magnesium compounds, microorganisms, nanobiotechnology, nanofabrication, nanoparticles, prosthetics
Biomaterials, Coating processes, Coatings, Emulsions, Magnesium (Metal), Nanoparticles, Microorganisms, Artificial limbs, Biophysics, Magnesium compounds, Nanobiotechnology, Nanofabrication, Prostheses
The use of surgically implanted or nonsurgically inserted medical devices has received an escalating interest in modern medical practices. Upon implantation or insertion into patient’s body for exerting the intended purpose like salvage of normal functions of vital organs, the medical devices are unfortunately becoming the sites of competition between host cell integration and microbial adhesion. To control microbial colonization and subsequent biofilm formation onto the medical devices, different approaches either to enhance the efficiency of certain antimicrobial agents or to disrupt the basic physiology of the pathogenic micro-organisms, including novel small molecules and antipathogenic drugs, are being explored. In addition, the various lipid- and polymer-based drug delivery carriers are also investigated for applying antibiofilm coating onto the medical devices especially over catheters. The major objectives of this paper are as follows: (1) to synthesize magnesium fluoride
(MgF2)
nanoparticles; (2) to prepare
MgF2
nanoparticle-stabilized oil-in-water (o/w) nanosized emulsion; (3) to coat biomaterial surfaces (glass coupons) with
MgF2
nanoparticles, and
MgF2
nanoparticle-stabilized emulsion; (4) to challenge the coated and uncoated glass surfaces with fresh bacterial cultures (i.e., Escherichia coli and Staphylococcus aureus) in 24-well plate over 18 h for biofilm formation; and (5) to compare the efficacy of emulsion-coated and emulsion-uncoated glass coupons in restricting the bacterial growth and biofilm formation.
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Reduce order of differential equations to first-order - MATLAB odeToVectorField - MathWorks América Latina
\frac{{\mathit{d}}^{2}\mathit{y}}{{\mathit{dt}}^{2}}+{\mathit{y}}^{2}\mathit{t}=3\mathit{t}.
\left(\begin{array}{c}{Y}_{2}\\ 3 t-t {{Y}_{1}}^{2}\end{array}\right)
{{Y}_{i}}^{\prime }
{Y}_{1}=y
\frac{\mathit{d}{\mathit{Y}}_{1}}{\mathit{dt}}={\mathit{Y}}_{2}
\frac{{\mathit{dY}}_{2}}{\mathit{dt}}=3\mathit{t}-\mathit{t}{\mathit{Y}}_{1}^{2}.
\left(\begin{array}{c}{Y}_{2}\\ {Y}_{1}+{Y}_{3}\\ {Y}_{3}-{Y}_{1}\end{array}\right)
\left(\begin{array}{c}f\\ \mathrm{Df}\\ g\end{array}\right)
{{Y}_{i}}^{\prime }
{Y}_{1}=f
{Y}_{2}
{Y}_{3}=g
\frac{{\mathit{d}}^{2}\mathit{y}}{\mathit{d}{\mathit{t}}^{2}}=\left(1-{\mathit{y}}^{2}\right)\frac{\mathit{dy}}{\mathit{dt}}-\mathit{y}.
\left(\begin{array}{c}{Y}_{2}\\ -\left({{Y}_{1}}^{2}-1\right) {Y}_{2}-{Y}_{1}\end{array}\right)
{y}^{\prime }\left(0\right)=2
{y}^{\prime \prime }\left(0\right)=0
y\left(t\right)
t
y\left(t\right)
{y}^{\prime \prime }\left(x\right)=x
y\left(0\right)=a
\left(\begin{array}{c}{Y}_{2}\\ x\end{array}\right)
{a}_{n}\left(t\right){y}^{\left(n\right)}+{a}_{n-1}\left(t\right){y}^{\left(n-1\right)}+\dots +{a}_{1}\left(t\right){y}^{\prime }+{a}_{0}\left(t\right)y+r\left(t\right)=0
\begin{array}{l}{Y}_{1}=y\\ {Y}_{2}={y}^{\prime }\\ {Y}_{3}={y}^{″}\\ \dots \\ {Y}_{n-1}={y}^{\left(n-2\right)}\\ {Y}_{n}={y}^{\left(n-1\right)}\end{array}
\begin{array}{l}{Y}_{1}{}^{\prime }={y}^{\prime }={Y}_{2}\\ {Y}_{2}{}^{\prime }={y}^{″}={Y}_{3}\\ \dots \\ {Y}_{n-1}{}^{\prime }={y}^{\left(n-1\right)}={Y}_{n}\\ {Y}_{n}{}^{\prime }=-\frac{{a}_{n-1}\left(t\right)}{{a}_{n}\left(t\right)}{Y}_{n}-\frac{{a}_{n-2}\left(t\right)}{{a}_{n}\left(t\right)}{Y}_{n-1}-...-\frac{{a}_{1}\left(t\right)}{{a}_{n}\left(t\right)}{Y}_{2}-\frac{{a}_{0}\left(t\right)}{{a}_{n}\left(t\right)}{Y}_{1}+\frac{r\left(t\right)}{{a}_{n}\left(t\right)}\end{array}
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Infinity symbol - Wikipedia
Mathematical symbol, "∞"
"∞" redirects here. For other uses, see infinity (disambiguation) and infinity sign (disambiguation).
{\displaystyle \infty }
U+221E ∞ INFINITY (∞)
U+267E ♾ PERMANENT PAPER SIGN
U+26AD ⚭ MARRIAGE SYMBOL
The infinity symbol (
{\displaystyle \infty }
) is a mathematical symbol representing the concept of infinity. This symbol is also called a lemniscate,[1] after the lemniscate curves of a similar shape studied in algebraic geometry,[2] or "lazy eight", in the terminology of livestock branding.[3]
This symbol was first used mathematically by John Wallis in the 17th century, although it has a longer history of other uses. In mathematics, it often refers to infinite processes (potential infinity) rather than infinite values (actual infinity). It has other related technical meanings, such as the use of long-lasting paper in bookbinding, and has been used for its symbolic value of the infinite in modern mysticism and literature. It is a common element of graphic design, for instance in corporate logos as well as in older designs such as the Métis flag.
Both the infinity symbol itself and several variations of the symbol are available in various character encodings.
2.3 Symbolism and literary uses
First known usage of the infinity symbol, by John Wallis in 1655
John Wallis introduced the infinity symbol
{\displaystyle \infty }
to mathematical literature.
{\displaystyle \infty }
symbol in several typefaces
The lemniscate has been a common decorative motif since ancient times; for instance it is commonly seen on Viking Age combs.[4]
The English mathematician John Wallis is credited with introducing the infinity symbol with its mathematical meaning in 1655, in his De sectionibus conicis.[5][6][7] Wallis did not explain his choice of this symbol. It has been conjectured to be a variant form of a Roman numeral, but which Roman numeral is unclear. One theory proposes that the infinity symbol was based on the numeral for 100 million, which resembled the same symbol enclosed within a rectangular frame.[8] Another proposes instead that it was based on the notation CIↃ used to represent 1,000.[9] Instead of a Roman numeral, it may alternatively be derived from a variant of ω, the lower-case form of omega, the last letter in the Greek alphabet.[9]
Perhaps in some cases because of typographic limitations, other symbols resembling the infinity sign have been used for the same meaning.[7] Leonhard Euler used an open letterform more closely resembling a reflected and sideways S than a lemniscate,[10] and even "O–O" has been used as a stand-in for the infinity symbol itself.[7]
In mathematics, the infinity symbol is used more often to represent a potential infinity,[11] rather than an actually infinite quantity as included in the extended real numbers, the cardinal numbers and the ordinal numbers (which use other notations, such as
{\displaystyle \,\aleph _{0}\,}
and ω, for infinite values). For instance, in mathematical expressions with summations and limits such as
{\displaystyle \sum _{n=0}^{\infty }{\frac {1}{2^{n}}}=\lim _{x\to \infty }{\frac {2^{x}-1}{2^{x-1}}}=2,}
the infinity sign is conventionally interpreted as meaning that the variable grows arbitrarily large towards infinity, rather than actually taking an infinite value, although other interpretations are possible.[12]
The infinity symbol may also be used to represent a point at infinity, especially when there is only one such point under consideration. This usage includes, in particular, the infinite point of a projective line,[13] and the point added to a topological space to form its one-point compactification.[14]
Side view of a camera lens, showing infinity symbol on the focal length indicator
In areas other than mathematics, the infinity symbol may take on other related meanings. For instance, it has been used in bookbinding to indicate that a book is printed on acid-free paper and will therefore be long-lasting.[15] On cameras and their lenses, the infinity symbol indicates that the lens's focal length is set to an infinite distance, and is "probably one of the oldest symbols to be used on cameras".[16]
Symbolism and literary uses[edit]
The infinity symbol appears on several cards of the Rider–Waite tarot deck.[17]
In modern mysticism, the infinity symbol has become identified with a variation of the ouroboros, an ancient image of a snake eating its own tail that has also come to symbolize the infinite, and the ouroboros is sometimes drawn in figure-eight form to reflect this identification—rather than in its more traditional circular form.[18]
In the works of Vladimir Nabokov, including The Gift and Pale Fire, the figure-eight shape is used symbolically to refer to the Möbius strip and the infinite, as is the case in these books' descriptions of the shapes of bicycle tire tracks and of the outlines of half-remembered people. Nabokov's poem after which he entitled Pale Fire explicitly refers to "the miracle of the lemniscate".[19] Other authors whose works use this shape with its symbolic meaning of the infinite include James Joyce, in Ulysses,[20] and David Foster Wallace, in Infinite Jest.[21]
The well-known shape and meaning of the infinity symbol have made it a common typographic element of graphic design. For instance, the Métis flag, used by the Canadian Métis people since the early 19th century, is based around this symbol.[22] Different theories have been put forward for the meaning of the symbol on this flag, including the hope for an infinite future for Métis culture and its mix of European and First Nations traditions,[23][24] but also evoking the geometric shapes of Métic dances,[25], Celtic knots,[26] or Plains First Nations Sign Language.[27]
A rainbow-coloured infinity symbol is also used by the neurodiversity movement, as a way to symbolize the infinite variation of the people in the movement and of human cognition.[28] The Bakelite company took up this symbol in its corporate logo to refer to the wide range of varied applications of the synthetic material they produced.[29] Versions of this symbol have been used in other trademarks, corporate logos, and emblems including those of Fujitsu,[30] Cell Press,[31] and the 2022 FIFA World Cup.[32]
The symbol is encoded in Unicode at U+221E ∞ INFINITY[33] and in LaTeX as \infty:
{\displaystyle \infty }
.[34] An encircled version is encoded for use as a symbol for acid-free paper.
INFINITY PERMANENT PAPER SIGN
GB 18030 161 222 A1 DE 129 55 174 56 81 37 AE 38
Named character reference ∞
OEM-437 (Alt Code)[35] 236 EC
Mac OS Roman[36] 176 B0
Symbol Font encoding[37] 165 A5
Shift JIS[38] 129 135 81 87
EUC-JP[39] 161 231 A1 E7
EUC-KR[40] / UHC[41] 161 196 A1 C4
EUC-KPS-9566[42] 162 172 A2 AC
Big5[43] 161 219 A1 DB
LaTeX[34] \infty \acidfree
CLDR text-to-speech name[44] infinity
The Unicode set of symbols also includes several variant forms of the infinity symbol that are less frequently available in fonts in the block Miscellaneous Mathematical Symbols-B.[45]
INCOMPLETE INFINITY TIE OVER INFINITY INFINITY NEGATED WITH VERTICAL BAR
Unicode 10716 U+29DC 10717 U+29DD 10718 U+29DE
UTF-8 226 167 156 E2 A7 9C 226 167 157 E2 A7 9D 226 167 158 E2 A7 9E
Numeric character reference ⧜ ⧜ ⧝ ⧝ ⧞ ⧞
Named character reference ⧜ ⧝ ⧞
LaTeX[34] \iinfin \tieinfty \nvinfty
Wikimedia Commons has media related to Infinity symbols.
Lazy Eight (disambiguation)
^ Rucker, Rudy (1982). Infinity and the Mind: The science and philosophy of the infinite. Boston, Massachusetts: Birkhäuser. p. 1. ISBN 3-7643-3034-1. MR 0658492.
^ Erickson, Martin J. (2011). "1.1 Lemniscate". Beautiful Mathematics. MAA Spectrum. Mathematical Association of America. pp. 1–3. ISBN 978-0-88385-576-8.
^ Humez, Alexander; Humez, Nicholas D.; Maguire, Joseph (1993). Zero to Lazy Eight: The Romance of Numbers. Simon and Schuster. p. 18. ISBN 978-0-671-74281-2.
^ van Riel, Sjoerd (2017). "Viking Age Combs: Local Products or Objects of Trade?". Lund Archaeological Review. 23: 163–178. See p. 172: "Within this type the lemniscate (∞) is a commonly used motif."
^ Wallis, John (1655). "Pars Prima". De Sectionibus Conicis, Nova Methodo Expositis, Tractatus (in Latin). pp. 4.
^ Scott, Joseph Frederick (1981). The mathematical work of John Wallis, D.D., F.R.S., (1616-1703) (2nd ed.). American Mathematical Society. p. 24. ISBN 0-8284-0314-7.
^ a b c Cajori, Florian (1929). "Signs for infinity and transfinite numbers". A History of Mathematical Notations, Volume II: Notations Mainly in Higher Mathematics. Open Court. pp. 44–48.
^ Maor, Eli (1991). To Infinity and Beyond: A Cultural History of the Infinite. Princeton, New Jersey: Princeton University Press. p. 7. ISBN 0-691-02511-8. MR 1129467.
^ a b Clegg, Brian (2003). "Chapter 6: Labelling the infinite". A Brief History of Infinity: The Quest to Think the Unthinkable. Constable & Robinson Ltd. ISBN 978-1-84119-650-3.
^ Cajori (1929) displays this symbol incorrectly, as a turned S without reflection. It can be seen as Euler used it on page 174 of Euler, Leonhard (1744). "Variae observationes circa series infinitas" (PDF). Commentarii Academiae Scientiarum Petropolitanae (in Latin). 9: 160–188.
^ Barrow, John D. (2008). "Infinity: Where God Divides by Zero". Cosmic Imagery: Key Images in the History of Science. W. W. Norton & Company. pp. 339–340. ISBN 978-0-393-06177-2.
^ Shipman, Barbara A. (April 2013). "Convergence and the Cauchy property of sequences in the setting of actual infinity". PRIMUS. 23 (5): 441–458. doi:10.1080/10511970.2012.753963. S2CID 120023303.
^ Perrin, Daniel (2007). Algebraic Geometry: An Introduction. Springer. p. 28. ISBN 978-1-84800-056-8.
^ Aliprantis, Charalambos D.; Border, Kim C. (2006). Infinite Dimensional Analysis: A Hitchhiker's Guide (3rd ed.). Springer. pp. 56–57. ISBN 978-3-540-29587-7.
^ Zboray, Ronald J.; Zboray, Mary Saracino (2000). A Handbook for the Study of Book History in the United States. Center for the Book, Library of Congress. p. 49. ISBN 978-0-8444-1015-9.
^ Crist, Brian; Aurello, David N. (October 1990). "Development of camera symbols for consumers". Proceedings of the Human Factors Society Annual Meeting. 34 (5): 489–493. doi:10.1177/154193129003400512.
^ Armson, Morandir (June 2011). "The transitory tarot: an examination of tarot cards, the 21st century New Age and theosophical thought". Literature & Aesthetics. 21 (1): 196–212. See in particular p. 203: "Reincarnation is symbolised in a number of cards within the Waite-Smith tarot deck. The primary symbols of reincarnation used are the infinity symbol or lemniscate, the wheel and the circle."
^ O'Flaherty, Wendy Doniger (1986). Dreams, Illusion, and Other Realities. University of Chicago Press. p. 243. ISBN 978-0-226-61855-5. The book also features this image on its cover.
^ Toker, Leona (1989). Nabokov: The Mystery of Literary Structures. Cornell University Press. p. 159. ISBN 978-0-8014-2211-9.
^ Bahun, Sanja (2012). "'These heavy sands are language tide and wind have silted here': Tidal voicing and the poetics of home in James Joyce's Ulysses". In Kim, Rina; Westall, Claire (eds.). Cross-Gendered Literary Voices: Appropriating, Resisting, Embracing. Palgrave Macmillan. pp. 57–73. doi:10.1057/9781137020758_4.
^ Natalini, Roberto (2013). "David Foster Wallace and the mathematics of infinity". In Boswell, Marshall; Burn, Stephen J. (eds.). A Companion to David Foster Wallace Studies. American Literature Readings in the 21st Century. Palgrave Macmillan. pp. 43–57. doi:10.1057/9781137078346_3.
^ Healy, Donald T.; Orenski, Peter J. (2003). Native American Flags. University of Oklahoma Press. p. 284. ISBN 978-0-8061-3556-4.
^ Gaudry, Adam (Spring 2018). "Communing with the Dead: The "New Métis," Métis Identity Appropriation, and the Displacement of Living Métis Culture". American Indian Quarterly. 42 (2): 162–190. doi:10.5250/amerindiquar.42.2.0162. JSTOR 10.5250/amerindiquar.42.2.0162. S2CID 165232342.
^ "The Métis flag". Gabriel Dumont Institute(Métis Culture & Heritage Resource Centre). Archived from the original on 2013-07-24.
^ Racette, Calvin (1987). Flags of the Métis (PDF). Gabriel Dumont Institute. ISBN 0-920915-18-3.
^ Darren R., Préfontaine (2007). "Flying the Flag, Editor's note". New Breed Magazine (Winter 2007): 6. Retrieved 2020-08-26.
^ Barkwell, Lawrence J. "The Metis Infinity Flag". Virtual Museum of Métis History and Culture. Gabriel Dumont Institute. Retrieved 2020-07-15.
^ Gross, Liza (September 2016). "In search of autism's roots". PLOS Biology. 14 (9): e2000958. doi:10.1371/journal.pbio.2000958. PMC 5045192. PMID 27690292.
^ Crespy, Daniel; Bozonnet, Marianne; Meier, Martin (April 2008). "100 years of Bakelite, the material of a 1000 uses". Angewandte Chemie. 47 (18): 3322–3328. doi:10.1002/anie.200704281. PMID 18318037.
^ Rivkin, Steve; Sutherland, Fraser (2005). The Making of a Name: The Inside Story of the Brands We Buy. Oxford University Press. p. 130. ISBN 978-0-19-988340-0.
^ Willmes, Claudia Gisela (January 2021). "Science that inspires". Trends in Molecular Medicine. 27 (1): 1. doi:10.1016/j.molmed.2020.11.001. PMID 33308981. S2CID 229179025.
^ "Qatar 2022: Football World Cup logo unveiled". Al Jazeera. September 3, 2019.
^ "Unicode Character "∞" (U+221E)". Unicode. Compart AG. Retrieved 2019-11-15.
^ a b c Pakin, Scott (May 5, 2021). "Table 294: stix Infinities". The Comprehensive LATEX Symbol List. CTAN. p. 118. Retrieved 2022-02-19.
^ Steele, Shawn (April 24, 1996). "cp437_DOSLatinUS to Unicode table". Unicode Consortium. Retrieved 2022-02-19.
^ "Map (external version) from Mac OS Roman character set to Unicode 2.1 and later". Apple Inc. April 5, 2005. Retrieved 2022-02-19 – via Unicode Consortium.
^ "Map (external version) from Mac OS Symbol character set to Unicode 4.0 and later". Apple Inc. April 5, 2005. Retrieved 2022-02-19 – via Unicode Consortium.
^ "Shift-JIS to Unicode". Unicode Consortium. December 2, 2015. Retrieved 2022-02-19.
^ "EUC-JP-2007". International Components for Unicode. Unicode Consortium. Retrieved 2022-02-19 – via GitHub.
^ "IBM-970". International Components for Unicode. Unicode Consortium. May 9, 2007. Retrieved 2022-02-19 – via GitHub.
^ Steele, Shawn (January 7, 2000). "cp949 to Unicode table". Unicode Consortium. Retrieved 2022-02-19.
^ "KPS 9566-2003 to Unicode". Unicode Consortium. April 27, 2011. Retrieved 2022-02-19.
^ Unicode, Inc. "Annotations". Common Locale Data Repository – via GitHub.
^ "Miscellaneous Mathematical Symbols-B" (PDF). Unicode Consortium. Archived (PDF) from the original on 2018-11-12. Retrieved 2022-02-19.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Infinity_symbol&oldid=1080521487"
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IsConvergent - Maple Help
Home : Support : Online Help : Education : Student Packages : Numerical Analysis : Computation : IsConvergent
determine whether an iterative approximation method converges or not
IsConvergent(A, meth)
\mathrm{nxn}
equation; the method in the form method = one of: gaussseidel, jacobi or SOR(numeric)
The IsConvergent command determines whether the iterative approximation to the linear system A.x=b, using meth as the approximation method, converges to a unique solution or not, for any initial approximate.
The IsConvergent command returns true or false depending on whether the method and system converge to a unique solution or not.
\mathrm{with}\left(\mathrm{Student}[\mathrm{NumericalAnalysis}]\right):
A≔\mathrm{Matrix}\left([[1.3,1.4,5.3],[3.4,7.7,3.1],[4.3,7.4,0.2]]\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1.3}& \textcolor[rgb]{0,0,1}{1.4}& \textcolor[rgb]{0,0,1}{5.3}\\ \textcolor[rgb]{0,0,1}{3.4}& \textcolor[rgb]{0,0,1}{7.7}& \textcolor[rgb]{0,0,1}{3.1}\\ \textcolor[rgb]{0,0,1}{4.3}& \textcolor[rgb]{0,0,1}{7.4}& \textcolor[rgb]{0,0,1}{0.2}\end{array}]
\mathrm{IsConvergent}\left(A,\mathrm{method}=\mathrm{gaussseidel}\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{IsConvergent}\left(A,\mathrm{method}=\mathrm{jacobi}\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
B≔\mathrm{Matrix}\left([[1,0],[0,1]]\right)
\textcolor[rgb]{0,0,1}{B}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}\end{array}]
\mathrm{IsConvergent}\left(B,\mathrm{method}=\mathrm{gaussseidel}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{IsConvergent}\left(B,\mathrm{method}=\mathrm{jacobi}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{IsConvergent}\left(B,\mathrm{method}=\mathrm{SOR}\left(1.25\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
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Without using a calculator, evaluate the following expressions.
\frac { d } { d x }( \int _ { 2 } ^ { x } \sqrt { 9 - x ^ { 3 } } d x )
Mentally integrate first. Will you get an expression or just a constant? What happens when you take the derivative next to the constant?
\frac { d } { d t }(t^{−1}\sin(t))
Product Rule or Quotient Rule, depending how you write
t^{ −1}
\int [ \frac { d } { d x } ( \frac { x + \operatorname { tan } ( x ) } { x - \operatorname { sec } ( x ) } + 5 ) ] d x
Refer to the hint in part (a). But notice that this is an integral of a derivative, not the reverse. That means that this answer will include
+C
\int _ { 0 } ^ { 1 } [ \frac { 1 } { \sqrt { 1 - x ^ { 2 } } } + \operatorname { sec } ^ { 2 } ( x ) ] d x
Both parts of the sum in the integrand have antiderivatives that are trig (or inverse trig) functions.
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Radical Equations Warmup Practice Problems Online | Brilliant
( \sqrt{ 12 } - 1 ) ( \sqrt{ 12 } + 1 ).
and
b,
\left( \sqrt{a} + \sqrt{b} \right) ^2 = a + b .
How many real solutions are there to
x - 4 = 2 \sqrt{ x - 1 } ?
The velocity (V) of an object can be obtained from the kinetic energy (KE) of the object and its mass (M) using the following formula:
V = \sqrt{ \frac{ 2 KE } { M } }.
If the new kinetic energy of the object (with the same mass) is 4 times the old kinetic energy, what can we say about the new velocity of the object?
The new velocity of the object is 2 times the old velocity The new velocity of the object is 4 times the old velocity The new velocity of the object is 8 times the old velocity The new velocity of the object is 16 times the old velocity
\sqrt{ 1 + \frac{ 16}{ 9 } } ?
1 \frac{1}{3}
2 \frac{2}{3}
2 \frac{1}{3}
1 \frac{2}{3}
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What is the best linear approximation of the general function
y = \sin(ax)
x = 0
Recall that a linear approximation is the equation of a secant line on a curvy function when you 'zoom in' on a given point until the graph appears to be linear. This is almost the same as a _________________________________________________.
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Implement lapse rate model for atmosphere - Simulink - MathWorks í•œêµ
Lapse Rate Model
Ï (kg/m3)
Implement lapse rate model for atmosphere
The Lapse Rate Model block implements the mathematical representation of the lapse rate atmospheric equations for ambient temperature, pressure, density, and speed of sound for the input geopotential altitude. You can customize this atmospheric model by specifying atmospheric properties.
The ISA Atmosphere Model and Lapse Rate Model blocks are identical blocks. When configured for ISA Atmosphere Model, the block implements ISA values. When configured for Lapse Rate Model, the block implements the mathematical representation of lapse rate atmospheric equations.
The Lapse Rate Model block icon displays the input and output metric units.
Ï (kg/m3) — Air density
Customize various atmospheric parameters to be different from the lapse rate values. Selecting this check box converts the block from Lapse Rate Model to ISA Atmosphere Model.
Ratio of specific heats γ, specified as a double value.
These equations define the troposphere:
\begin{array}{c}T={T}_{0}âLh\\ P={P}_{0}{\left(\frac{T}{{T}_{0}}\right)}^{\frac{g}{LR}}\\ \mathrm{Ï}={\mathrm{Ï}}_{0}{\left(\frac{T}{{T}_{0}}\right)}^{\frac{g}{LR}â1}\\ a=\sqrt{\mathrm{γ}RT}\end{array}
These equations define the tropopause (lower stratosphere):
\begin{array}{c}T={T}_{0}âLhts\\ P={P}_{0}{\left(\frac{T}{{T}_{0}}\right)}^{\frac{g}{LR}}{e}^{\frac{g}{RT}}{}^{{}^{\left(htsâh\right)}}\\ \mathrm{Ï}={\mathrm{Ï}}_{0}{\left(\frac{T}{{T}_{0}}\right)}^{\frac{g}{LR}â1}{e}^{\frac{g}{RT}}{}^{{}^{\left(htsâh\right)}}\\ a=\sqrt{\mathrm{γ}RT}\end{array}
Absolute temperature at mean sea level in kelvin (K)
Ï 0
Air density at mean sea level in kg/m 3
Static pressure at mean sea level in N/m 2
Height of the troposphere in m
Absolute temperature at altitude h in kelvin (K)
Air density at altitude h in kg/m 3
Static pressure at altitude h in N/m 2
Speed of sound at altitude h in m/s 2
Lapse rate in K/m
Characteristic gas constant J/kg-K
Acceleration due to gravity in m/s 2
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Disjoint-set Data Structure (Union-Find) | Brilliant Math & Science Wiki
Agnishom Chattopadhyay, Debarghya Adhikari, and Jimin Khim contributed
Union-find, as it is popularly called, is a data structure that categorizes objects into different sets and lets checking out if two objects belong to the same set.
The most popular usage of the data structure is to check whether one node in a graph can be reached from another, e.g. in the Kruskal's algorithm to avoid forming cycles.
This data structure is supposed to support two operations:
find(x): Returns some representation of the set to which x belongs.
union(x,y): Merge the sets containing x and y.
Often, it can be equipped with a constructor that organizes every object into its own set.
Here is a very simple (but not all that effective) way to achieve what we want. We keep an array that stores the information about which set the objects are in. The interface is implemented as follows:
find(x): Return the value at position x in the array. This is just O(1).
union(x,y): Scan through the array to check if any of the values are y. If so, update them to x. This is O(n).
class UnionFind{ //Quick Find
UnionFind(int n){ //Set up a union-find data structure with n elements
for (int iii = 0; iii < N; iii++)
sets[iii] = iii;
return sets[x];
void merge(int x, int y){ //We call this merge here. Apparently, union is a keyword in cpp
int root_x = find(x);
int root_y = find(y);
if (sets[iii] == root_x)
sets[iii] = root_y;
Actually, we can do better than that. Let's see how.
This time, we will still use an array for storage but we'll imagine it to be a forest.
We'll keep an array called parents to track of which element is whose parent. Each set forms a tree represented by its node.
Here is an example of a forest where {1,2,5,6,7} form a set and {0,3,4} form another.
find(x): Recursively keep finding the parent of x until an element which is the parent of itself is encountered. Because this is a tree, if the unions were random enough this should do better, but the worst case is
O(N)
, if the tree is very tall.
union(x,y): Find the root of x and make it point towards the root of y.
class UnionFind{ //Quick Union
int root = x;
parent[root_x] = root_y;
The problem with the above data structure is that the trees might become too tall. This problem can be fixed by deciding correctly which tree should go under which.
Would it be a better idea to put Tree 1 under Tree 2 or Tree 2 under Tree 1?
Tree 2 has a height of 4 whereas Tree 1 has a height of 3. If we put Tree 2 under the root of Tree 1, we get a larger tree of height 5. However, putting Tree 1 under the root of Tree 2 still makes a tree of height 4.
In general, when we have two trees of height
m
n
m \leq n,
we should put the tree of height
m
n
and still get a tree of height
n
To implement this, we need to keep an array size[i] that keeps track of the objects in trees rooted at i.
class UnionFind{ //Quick Union with Weighting
for (int iii = 0; iii < N; iii++){
parent[iii] = iii;
size[iii] = 1;
if (size[root_y] > size[root_x]){ //Make sure that the smaller tree goes under the larger tree
size[root_y] += size[root_x];
parent[root_y] = root_x;
size[root_x] += size[root_y];
Now, both find and union work in
O (\log n)
The tree's height increases by at most one node when another tree of greater or equal height is unioned with it.
Since the other tree is at least as large as itself, the resultant tree must have at least double the number of elements.
But there are only
n
elements, so the doubling can happen at most
\log n
Thus, the maximum height of the tree is in
O (\log n),
which is the number of operations we need to approach the root.
Here is another idea: We're already touching all the nodes from x up to the root. Why don't we just as well push them up the tree as we go?
That requires just one line of extra code in the find operation. Check line 22 below.
class UnionFind{ //Quick Union with Weighting and Path Compression
while (parent[root] != root){
parent[root] = parent[parent[root]]; //Push up the node by one level
This practically keeps the tree almost flat. In fact, this makes the operations work in
O (\log ^* n)
time as proved by Hopcroft and Ullman.
\log ^* n
is the number of times one needs to apply
\log
n
to get a value less than or equal to 1. In practice, one could think of it to be almost
O(1)
since it exceeds 5 only after it has reached
2^{65536}.
The bounds were later improved by Tarjan to
O\big(\alpha (n)\big),
\alpha
is the inverse Ackermann function.
Cite as: Disjoint-set Data Structure (Union-Find). Brilliant.org. Retrieved from https://brilliant.org/wiki/disjoint-set-data-structure/
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Point-object trackers assume that each object may give rise to at most one detection per sensor. Therefore, when using point-target trackers for tracking extended objects, features like bounding box detections are first extracted from the sensor measurements at the object-level. These object-level features then get fused with object-level hypothesis from the tracker. A poor object-level extraction algorithm at the sensor level (such as imperfect clustering) thus greatly impacts the performance of the tracker. For an example of this workflow, refer to Track Vehicles Using Lidar: From Point Cloud to Track List (Automated Driving Toolbox).
On the other hand, extended object trackers process the detections without extracting object-level hypothesis at the sensor level. Extended object trackers associate sensor measurements directly with the object-level hypothesis maintained by tracker. To do this, a class of algorithms typically requires complex measurement models of the object extents specific to each sensor modality. For example, refer to Extended Object Tracking with Lidar for Airport Ground Surveillance and Extended Object Tracking of Highway Vehicles with Radar and Camera (Automated Driving Toolbox) to learn how to configure a multi-object PHD tracker for lidar and radar respectively.
In this example, you use the trackerGridRFS System object™ to configure the grid-based tracker. This tracker uses the Random Finite Set (RFS) formulation with Dempster-Shafer approximation [1] to estimate the dynamic map. Further, it uses a nearest neighbor cell-to-track association [2] scheme to track dynamic objects in the scene. To initialize new tracks, the tracker uses the DBSCAN algorithm to cluster unassigned dynamic grid cells.
The scenario used in this example was created using the Driving Scenario Designer (Automated Driving Toolbox) app and was exported to a MATLAB® function. This MATLAB function was wrapped as a helper function helperCreateMultiLidarDrivingScenario. The scenario represents an urban intersection scene and contains a variety of objects that include pedestrians, bicyclists, cars, and trucks.
The ego vehicle is equipped with 6 homogeneous lidars, each with a horizontal field of view of 90 degrees and a vertical field of view of 40 degrees. The lidars are simulated using the lidarPointCloudGenerator (Automated Driving Toolbox) System object. Each lidar has 32 elevation channels and has a resolution of 0.16 degrees in azimuth. Under this configuration, each lidar sensor outputs approximately 18,000 points per scan. The configuration of each sensor is shown here.
The scenario and the data from the different lidars can be visualized in the animation below. For brevity and to make the example easier to visualize, the lidar is configured to not return point cloud from the ground by specifying the HasRoadsInputPort property as false. When using real data or if using simulated data from roads, the returns from ground and other environment must be removed using point cloud preprocessing. For more information, refer to the Ground Plane and Obstacle Detection Using Lidar (Automated Driving Toolbox) example.
{\mathit{P}}_{\mathit{s}}
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Impedance_bridging Knowpia
When the output of a device (consisting of the voltage source VS and output impedance ZS in illustration) is connected to the input of another device (the load impedance ZL in the illustration), these two impedances form a voltage divider:
{\displaystyle V_{L}={\frac {Z_{L}}{Z_{S}+Z_{L}}}V_{S}\,.}
One can maximize the signal level VL by using a voltage source whose output impedance ZS is as small as possible and by using a receiving device whose input impedance ZL is as large as possible. When
{\displaystyle Z_{L}\gg Z_{S}}
(typically by at least ten times), this is called a bridging connection and has a number of effects including:
Increasing the ZL/ZS ratio reduces attenuation of the voltage signal, which helps maintain a high signal-to-noise ratio
Increasing ZL reduces current drawn from the source device, which helps avoid wasting power and helps reduce distortion
Increasing ZL possibly increases environmental noise pickup since to the combined parallel impedance of ZS || ZL (dominated by ZS) increases slightly, which makes it easier for stray noise to drive the signal node
Receiving voltage signals from sources with unchangeable ZSEdit
Bridging is typically used for line or mic level connections where the source device (such as the line-out of an audio player or the output of a microphone) has a unchangeable output impedance ZS. Fortunately, the input impedance of modern op-amp circuits (and many old vacuum tube circuits) is often naturally much higher than the output impedance of these signal sources and thus are naturally-suited for impedance bridging when receiving and amplifying these voltage signals.
Maximizing power at a load with an unchangeable ZLEdit
Given an unchangeable ZL and VS while ZS can be freely changed, one can maximize both the voltage and current (and therefore, the power) at the load through impedance bridging by minimizing ZS. This is because the power delivered to the load in the above circuit (assuming all impedances are purely real) is:
{\displaystyle P_{L}={\frac {V_{S}^{2}R_{L}}{(R_{S}+R_{L})^{2}}}}
This situation is mostly encountered in the interface between an audio amplifier and a loudspeaker. In such cases, the impedance of the loudspeaker is fixed (a typical value being 8 Ω), so to deliver the maximum power to the speaker, the output impedance of the amplifier should be as small as possible (ideally zero).
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Chemical Kinetics, Popular Questions: CBSE Class 12-science ENGLISH, English Grammar - Meritnation
\mathrm{Step} 1: {\mathrm{O}}_{3}\underset{{\mathrm{K}}_{2}}{\overset{{\mathrm{K}}_{1}}{⇌}} {\mathrm{O}}_{2}+\mathrm{O} \left(\mathrm{fast}\right)\phantom{\rule{0ex}{0ex}}\mathrm{Step} 2: {\mathrm{O}}_{3}+\mathrm{O}\underset{\mathrm{K}}{\to }2{\mathrm{O}}_{2} \left(\mathrm{slow}\right)
{N}_{2}{O}_{5}
CC{I}_{4}
{N}_{2}{O}_{5}
{N}_{2}{O}_{5}
2{N}_{2}{O}_{5} \to 4N{O}_{2} + {O}_{2}
{N}_{2}{O}_{5}
\left[N{O}_{2}\right]
{N}_{2}{O}_{5}
\to
C{l}_{2}
C{l}_{2}
C{l}_{2}
Ziyauddin Ahmed asked a question
Question) (i) If slope is equal to
-2×{10}^{-6} se{c}^{-1}
, what will be the value of rate constant?
(ii) How does the half - life of zero order reaction relate to its rate constant?
S{O}_{2}C{l}_{2}\to S{O}_{2}+C{l}_{2}
3.15×{10}^{4}
°
S{O}_{2}C{l}_{2}
°
For the assumed reaction X2 + 2Y2--------> 2XY2, write the rate equation in terms of rate of disappearanc e of Y2. (1 mark)
Neha Stephen asked a question
From the concentration of C4H9Cl at different times given below.calculate the average rate of reaction C4H9Cl + H2O gives C4H9OH + Hcl
during different intervals of time
t/s 0 50 100 150 200 300 400 700 800
C4H9cl 0.100 0.0905 0.0820 0.0741 0.0671 0.0549 0.0439 0.0210 0.017
can u pls explain this question that is how to do it?
If rate constant of a reaction is 1.6×10^(-5) and 6.36×10^(-3) per second at 600K and 700K, calculate activation energy for reaction.
half- life of a first order reaction is 600 seconds. what percentage of the reactant will remain after 30 minutes.
Nitrogen dioxide reacts with fluorine to yield nitroyl fluoride as NO2(g) + F2(g) -- 2 NO2F(g) Write average rate of reaction in terms of (i) rate of formation of NO2F (ii) rate of disappearance of NO2
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Options for training deep learning neural network - MATLAB trainingOptions - MathWorks España
\begin{array}{l}{\mu }^{*}={\lambda }_{\mu }\stackrel{^}{\mu }+\left(1-{\lambda }_{\mu }\right)\mu \\ {\sigma }^{2}{}^{*}={\lambda }_{{\sigma }^{2}}\stackrel{^}{{\sigma }^{2}}\text{}\text{+}\text{}\text{(1-}{\lambda }_{{\sigma }^{2}}\right)\text{}{\sigma }^{2}\end{array}
{\mu }^{*}
{\sigma }^{2}{}^{*}
{\lambda }_{\mu }
{\lambda }_{{\sigma }^{2}}
\stackrel{^}{\mu }
\stackrel{^}{{\sigma }^{2}}
\mu
{\sigma }^{2}
{\theta }_{\ell +1}={\theta }_{\ell }-\alpha \nabla E\left({\theta }_{\ell }\right),
\ell
\alpha >0
\theta
E\left(\theta \right)
\nabla E\left(\theta \right)
{\theta }_{\ell +1}={\theta }_{\ell }-\alpha \nabla E\left({\theta }_{\ell }\right)+\gamma \left({\theta }_{\ell }-{\theta }_{\ell -1}\right),
\gamma
{v}_{\ell }={\beta }_{2}{v}_{\ell -1}+\left(1-{\beta }_{2}\right){\left[\nabla E\left({\theta }_{\ell }\right)\right]}^{2}
{\theta }_{\ell +1}={\theta }_{\ell }-\frac{\alpha \nabla E\left({\theta }_{\ell }\right)}{\sqrt{{v}_{\ell }}+ϵ}
{m}_{\ell }={\beta }_{1}{m}_{\ell -1}+\left(1-{\beta }_{1}\right)\nabla E\left({\theta }_{\ell }\right)
{v}_{\ell }={\beta }_{2}{v}_{\ell -1}+\left(1-{\beta }_{2}\right){\left[\nabla E\left({\theta }_{\ell }\right)\right]}^{2}
{\theta }_{\ell +1}={\theta }_{\ell }-\frac{\alpha {m}_{l}}{\sqrt{{v}_{l}}+ϵ}
E\left(\theta \right)
{E}_{R}\left(\theta \right)=E\left(\theta \right)+\lambda \Omega \left(w\right),
w
\lambda
\Omega \left(w\right)
\Omega \left(w\right)=\frac{1}{2}{w}^{T}w.
\lambda
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Abilities - Ring of Brodgar
Abilities are your Learned skills, and as such are bought using Learning Points. All abilities start at 1, and cost:
{\displaystyle 100LP*(currentlevel+1)}
Learned skills are nonrefundable, so choose wisely. Like Attributes, Abilities can be buffed or debuffed by Wounds and Equipment
2 Non Combat Abilities
Determines the effectiveness of Unarmed Combat moves
Equipment which modifies Unarmed Combat
Grand Troll Helm 10
Determines the effectiveness of Melee Combat moves.
Equipment which modifies Melee Combat
Bone Greaves 5
Snakeskin Boots 2
Increases damage dealt by range weapons by comparing your Perception*Marksmanship with target Agility*Marksmanship
Softcap for crafting the Hunter's Bow , Sling , Bone Arrow and Stone Arrow.
Equipment which modifies Marksmanship
Non Combat Abilities
Determines the visibility of Foragables with
{\displaystyle Perception*Exploration}
Criminal Acts, leave behind a Scent that can be seen if the trackers
{\displaystyle Perception*Exploration}
is more than half that of the perpetrators
{\displaystyle Intelligence*Stealth}
Equipment which modifies Exploration
Determines visibility of scents based on
{\displaystyle Perception*Exploration}
{\displaystyle Intelligence*Stealth}
Makes the tracking pie slice wider/less precise.
Increases cool-down on scents left by the player.
Your own scents do not care that they came from you. So, depending on you attribute levels, you might not see your own scents
Equipment which modifies Stealth
Bandit's Mask 2
Druid's Cloak 15
Determines the softcap of craftable leather/fabric items with
{\displaystyle {\sqrt {Sewing*Dexterity}}}
Equipment which modifies Sewing
Determines the softcap of metal items with
{\displaystyle {\sqrt {Strength*Smithing}}}
Determines the softcap of jewelry with
{\displaystyle {\sqrt {Psyche*Smithing}}}
Smithing does not affect the quality of wrought iron, steel, or bronze.
Equipment which modifies Smithing
Forge Ring 10
hardcaps the quality of clay gathered
Masonry governs all things related to pottery, ceramics, and stoneworking.
Hard caps the quality of stone and ore mined from cave walls, clay and stone dug from the ground, and foraged clay.
Equipment which modifies Masonry
Determines the softcap of wooden items, notably boards with
{\displaystyle {\sqrt {Dexterity*Carpentry}}}
Equipment which modifies Carpentry
Determines the softcap of many cooked foods with
{\displaystyle {\sqrt {Perception*Cooking}}}
Determines the quality of baked goods and other foods with
{\displaystyle {\sqrt {Dexterity*Cooking}}}
Equipment which modifies Cooking
The Perfect Hole 2
Allows the quality of the seeds/plants to increase after planting (with a randomized range of -2 to +5 final quality). If seed quality is greater than farming skill, the crop quality will always be less than or equal to seed quality (randomized range of -2 to 0).
Used as a softcap for many craftable items, such as Straw Doll.
Softcap products from butchering tamed animals ((hides, meat, entrails, bones).
For the unlockable skill and general information, see Farming.
Equipment which modifies Farming:
hardcaps the quality of Forageables a character has picked.
hardcaps the quality of Water and Soil
softcaps the quality of meat, bones and hide when butchering with
{\displaystyle {\sqrt {Survival*ToolQ}}}
softcaps the quality of Spitroast Meat
Used to craft Stone Axe , Bone Saw , and fishing gear
Expand fishing menu when using Primitive Casting-Rod
Equipment which modifies Survival
Lore is a general measure of Wisdom, relevant for "magical" or otherwise involved tasks.
Used to softcap quality of crafted items.
Authority gained equals base LP gained multiplied with
{\displaystyle {\frac {\sqrt {Charisma*Lore}}{10}}}
More uses are intended to be implemented later
Equipment which modifies Lore
In order to advance ability score from 1 to 100, Diligent Hearthling has to amass 504.900 (half a million) Experience points and fill them with equal amount of Learning points.
Retrieved from "https://ringofbrodgar.com/w/index.php?title=Abilities&oldid=94369"
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Forecast responses from Bayesian vector autoregression (VAR) model - MATLAB forecast
Forecast Responses from Posterior Predictive Distribution
Estimate Standard Deviations of Posterior Predictive Distribution
YFStd
Forecast responses from Bayesian vector autoregression (VAR) model
YF = forecast(PriorMdl,numperiods,Y)
[YF,YFStd] = forecast(PriorMdl,numperiods,Y)
forecast is well suited for computing out-of-sample unconditional forecasts of a Bayesian VAR(p) model that does not contain an exogenous regression component. For advanced applications, such as out-of-sample conditional forecasting, VARX(p) model forecasting, missing value imputation, and Gibbs sampler specification for posterior predictive distribution estimation, see simsmooth.
YF = forecast(PriorMdl,numperiods,Y) returns a path of forecasted responses YF over the length numperiods forecast horizon. Each period in YF is the mean of the posterior predictive distribution, which is derived from the posterior distribution of the prior Bayesian VAR(p) model PriorMdl given the response data Y. The output YF represents the continuation of Y.
[YF,YFStd] = forecast(PriorMdl,numperiods,Y) also returns the corresponding standard deviations of the posterior predictive distribution YFStd.
\left[\begin{array}{l}{\text{INFL}}_{t}\\ {\text{UNRATE}}_{t}\\ {\text{FEDFUNDS}}_{t}\end{array}\right]=c+\sum _{j=1}^{4}{\Phi }_{j}\left[\begin{array}{l}{\text{INFL}}_{t-j}\\ {\text{UNRATE}}_{t-j}\\ {\text{FEDFUNDS}}_{t-j}\end{array}\right]+\left[\begin{array}{c}{\epsilon }_{1,t}\\ {\epsilon }_{2,t}\\ {\epsilon }_{3,t}\end{array}\right].
t
{\epsilon }_{t}
\Sigma
. Assume a diffuse prior distribution for the parameters
\left({\left[{\Phi }_{1},...,{\Phi }_{4},\mathit{c}\right]}^{\prime },\Sigma \right)
Load the US macroeconomic data set. Compute the inflation rate, stabilize the unemployment and federal funds rates, and remove missing values.
Create a diffuse prior model. Specify the response series names.
PriorMdl = bayesvarm(numseries,numlags,'SeriesNames',seriesnames);
Forecast Responses
Directly forecast two years (eight quarters) of observations from the posterior predictive distribution. forecast estimates the posterior distribution of the parameters, and then forms the posterior predictive distribution.
YF = forecast(PriorMdl,numperiods,rmDataTable{:,seriesnames});
YF is an 8-by-3 matrix of forecasted responses.
Plot the forecasted responses.
fh = rmDataTable.Time(end) + calquarters(0:8);
for j = 1:PriorMdl.NumSeries
plot(rmDataTable.Time(end - 20:end),rmDataTable{end - 20:end,seriesnames(j)},'r',...
fh,[rmDataTable{end,seriesnames(j)}; YF(:,j)],'b');
legend("Observed","Forecasted",'Location','NorthWest')
title(seriesnames(j))
Consider the 3-D VAR(4) model of Forecast Responses from Posterior Predictive Distribution.
Directly forecast two years (eight quarters) of response observations from the posterior predictive distribution. Return the posterior standard deviations.
[YF,YStd] = forecast(PriorMdl,numperiods,rmDataTable{:,seriesnames});
YF and YStd are 8-by-3 matrices of forecasted responses and corresponding standard deviations, respectively.
Plot the forecasted responses and approximate 95% credible intervals.
fh,[rmDataTable{end,seriesnames(j)}; YF(:,j)],'b',...
fh,[rmDataTable{end,seriesnames(j)}; YF(:,j) + 1.96*YStd(:,j)],'b--',...
fh,[rmDataTable{end,seriesnames(j)}; YF(:,j) - 1.96*YStd(:,j)],'b--');
legend("Observed","Forecasted","Approximate 95% Credible Interval",'Location','NorthWest')
PriorMdl — Prior Bayesian VAR model
conjugatebvarm model object | semiconjugatebvarm model object | diffusebvarm model object | normalbvarm model object
Prior Bayesian VAR model, specified as a model object in this table.
conjugatebvarm Dependent, matrix-normal-inverse-Wishart conjugate model returned by bayesvarm, conjugatebvarm, or estimate
semiconjugatebvarm Independent, normal-inverse-Wishart semiconjugate prior model returned by bayesvarm or semiconjugatebvarm
diffusebvarm Diffuse prior model returned by bayesvarm or diffusebvarm
normalbvarm Normal conjugate model with a fixed innovations covariance matrix, returned by bayesvarm, normalbvarm, or estimate
Y — Presample and estimation sample multivariate response series
Presample and estimation sample multivariate response series, specified as a (numlags + numobs)-by-numseries numeric matrix.
Rows correspond to observations, and the last row contains the latest observation. forecast uses the first numlags = PriorMdl.P observations as a presample to initialize the prior model PriorMdl for posterior estimation. forecast estimates the posterior using the remaining numobs observations and PriorMdl.
numseries is the number of response variables PriorMdl.NumSeries. Columns correspond to individual response variables PriorMdl.SeriesNames.
YF — Path of multivariate response series forecasts
Path of multivariate response series forecasts, returned as a numperiods-by-numseries numeric matrix. YF is the mean of the posterior predictive distribution of each period in the forecast horizon.
YF represents the continuation of the response series Y. Rows correspond to observations; row j is the j-period-ahead forecast. Columns correspond to the columns in Y.
YFStd — Forecast standard deviations
Forecast standard deviations, returned as a numperiods-by-numseries numeric matrix. YFStd is the standard deviation of the posterior predictive distribution of each period in the forecast horizon. Dimensions correspond to the dimensions of YF.
{y}_{t}={\Phi }_{1}{y}_{t-1}+...+{\Phi }_{p}{y}_{t-p}+c+\delta t+Β{x}_{t}+{\epsilon }_{t}.
{y}_{t}={Z}_{t}\lambda +{\epsilon }_{t}.
{y}_{t}={\Lambda }^{\prime }{z}_{t}^{\prime }+{\epsilon }_{t}.
{z}_{t}=\left[\begin{array}{ccccccc}{y}_{t-1}^{\prime }& {y}_{t-2}^{\prime }& \cdots & {y}_{t-p}^{\prime }& 1& t& {x}_{t}^{\prime }\end{array}\right],
\left[\begin{array}{cccc}{z}_{t}& {0}_{z}& \cdots & {0}_{z}\\ {0}_{z}& {z}_{t}& \cdots & {0}_{z}\\ ⋮& ⋮& \ddots & ⋮\\ {0}_{z}& {0}_{z}& {0}_{z}& {z}_{t}\end{array}\right],
\Lambda ={\left[\begin{array}{ccccccc}{\Phi }_{1}& {\Phi }_{2}& \cdots & {\Phi }_{p}& c& \delta & Β\end{array}\right]}^{\prime }
, which is an (mp + r + 2)-by-m random matrix of the coefficients, and the m(mp + r + 2)-by-1 vector λ = vec(Λ).
\ell \left(\Lambda ,\Sigma |y,x\right)=\prod _{t=1}^{T}f\left({y}_{t};\Lambda ,\Sigma ,{z}_{t}\right),
A posterior predictive distribution of a posterior Bayesian VAR(p) model π(Yf|Y,X) is the distribution of the next τ future response variables after the final observation in the estimation sample Yf = [yT+1, yT+2,…,yT+τ] given the following, marginalized over Λ and Σ:
Presample and estimation sample response data Y
Coefficients Λ
Innovations covariance matrix Σ
Estimation and future sample exogenous data X
\pi \left({Y}_{f}|Y,X\right)=\int \pi \left({Y}_{f}|Y,X,\Lambda ,\Sigma \right)\pi \left(\Lambda ,\Sigma |Y,X\right)d\Lambda d\Sigma .
If the posterior predictive distribution is analytically intractable (true for most cases), forecast implements Markov Chain Monte Carlo (MCMC) sampling with Bayesian data augmentation to compute the mean and standard deviation of the posterior predictive distribution. To do so, forecast calls simsmooth, which uses a computationally intensive procedure.
Most Econometrics Toolbox™ forecast functions accept an estimated or posterior model object from which to generate forecasts. Such a model encompasses the parametric structure and data. However, the forecast function of Bayesian VAR models requires presample and estimation sample data to do the following:
Perform Bayesian parameter updating to estimate posterior distributions. forecast implements MCMC sampling with Bayesian data augmentation, which includes a Kalman filter smoothing step that requires the entire observed series.
Predict future responses in the presence of two sources of uncertainty:
Estimation noise ε1,…,εT, which induces parameter uncertainty
Forecast period noise εT+1,…,εT+numperiods
normalbvarm | conjugatebvarm | semiconjugatebvarm | diffusebvarm | empiricalbvarm
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Cerebral blood flow rates in recent great apes are greater than in Australopithecus species that had equal or larger brains
Cerebral blood flow rates in recent great apes are greater than in Australopithecus species that had equal or larger brains. Roger S. Seymour, Vanya Bosiocic, Edward P. Snelling, Prince C. Chikezie, Qiaohui Hu, Thomas J. Nelson, Bernhard Zipfel and Case V. Miller. Volume 286, Issue 1915, November 13 2019. https://doi.org/10.1098/rspb.2019.2208
Abstract: Brain metabolic rate (MR) is linked mainly to the cost of synaptic activity, so may be a better correlate of cognitive ability than brain size alone. Among primates, the sizes of arterial foramina in recent and fossil skulls can be used to evaluate brain blood flow rate, which is proportional to brain MR. We use this approach to calculate flow rate in the internal carotid arteries (Q˙ICA), which supply most of the primate cerebrum. Q˙ICA is up to two times higher in recent gorillas, chimpanzees and orangutans compared with 3-million-year-old australopithecine human relatives, which had equal or larger brains. The scaling relationships between Q˙ICA and brain volume (Vbr) show exponents of 1.03 across 44 species of living haplorhine primates and 1.41 across 12 species of fossil hominins. Thus, the evolutionary trajectory for brain perfusion is much steeper among ancestral hominins than would be predicted from living primates. Between 4.4-million-year-old Ardipithecus and Homo sapiens, Vbr increased 4.7-fold, but Q˙ICA increased 9.3-fold, indicating an approximate doubling of metabolic intensity of brain tissue. By contrast, Q˙ICA is proportional to Vbr among haplorhine primates, suggesting a constant volume-specific brain MR.
[Q with a dot is first derivative of Q (rate of change with time, in this case)]
Brain size is the usual measure in discussions of the evolution of cognitive ability among primates, despite recognized shortcomings [1]. Although absolute brain size appears to correlate better with cognitive ability than encephalization quotient, progression index or neocortex ratio [2,3], an even better correlate might be brain metabolic rate (MR), because it represents the energy cost of neurological function. However, brain MR is difficult to measure directly in living primates and impossible in extinct ones.
One solution to the problem has been to measure oxygen consumption rates and glucose uptake rates on living mammals in relation to brain size and then apply the results to brain sizes of living and extinct primates. Because physiological rates rarely relate linearly to volumes or masses of tissues, any comparison requires allometric analysis. For example, brain MR can be analysed in relation to endocranial volume (≈ brain volume, Vbr) with an allometric equation of the form, MR = aVbrb, where a is the elevation (or scaling factor, indicating the height of the curve) and b is the scaling exponent (indicating the shape of the curve on arithmetic axes). If b = 1.0, then MR is directly proportional to brain size. If b is less than 1, then MR increases with brain size, but the metabolic intensity per unit volume of neural tissue decreases. If b is greater than 1, the metabolic intensity of neural tissue increases. The exponent for brain MR measured as oxygen consumption and glucose use across several mammalian species is approximately 0.86, and the exponent for cortical brain blood flow rate in mammals is between 0.81 and 0.87 [4,5]. The similarity of the exponents indicates that blood flow rate is a good proxy for brain MR in mammals in general. The exponents are less than 1.0, which shows that brain MR and blood flow rate increase with brain size but with decreasing metabolic and perfusion intensities of the neural tissue.
Recent studies show that blood flow rate in the internal carotid artery
\left({\stackrel{˙}{Q}}_{\mathrm{ICA}}\right)
can be calculated from the size of the carotid foramen through which it passes to the brain [6]. The artery occupies the foramen lumen almost entirely [7–9], therefore defining the outer radius of the artery (ro), from which inner lumen radius (ri) can be estimated, assuming that arterial wall thickness (ro – ri) is a constant ratio (w) with lumen radius (w = (ro – ri)/ri), according to the law of Laplace. The haemodynamic equation used to calculate
{\stackrel{˙}{Q}}_{\mathrm{ICA}}
is referred to as the ‘shear stress equation’, and attributed to Poiseuille:
\stackrel{˙}{Q}=\left(\tau \pi {r}_{i}^{3}\right)/\left(4\eta \right)
\stackrel{˙}{Q}
is the blood flow rate (cm3 s−1), τ is the wall shear stress (dyn cm−2), ri is the arterial lumen radius (cm) and η is the blood viscosity (dyn s cm−2) [10]. The technique was validated in mice, rats and humans, but was initially criticized [11], defended [12] and subsequently accepted [13]. However, the calculations involved three questionable assumptions: flow in the cephalic arteries conforms to Poiseuille flow theory, arterial wall shear stress can be calculated accurately from body mass (although there is no clear functional relationship between them) and the arterial wall thickness-to-lumen radius ratio (w) was a certain constant derived from only two values in the literature.
We have now made significant advancements to the initial methodology by replacing the shear stress equation, and its assumptions, with a new equation derived empirically from a meta-analysis of
\stackrel{˙}{Q}
versus ri in 30 studies of seven cephalic arteries of six mammalian genera, arriving at an allometric, so-called ‘empirical equation’,
\stackrel{˙}{Q}
= 155 ri2.49 (R2 = 0.94) [14]. The equation is based on stable cephalic flow rates, which vary little between rest, intense physical activity, mental exercise or sleep [14]. The equation also eliminates reliance on the somewhat tenuous estimation of arterial wall shear stress from body mass. We have also improved the calculation with a more extensive re-evaluation of carotid arterial wall thickness ratio (w = 0.30) from 14 imaging studies on humans (electronic supplementary material, text and table S1 for data and references). The present investigation implements these recent methodological advancements and re-evaluates the scaling of
{\stackrel{˙}{Q}}_{\mathrm{ICA}}
as a function of Vbr in extant haplorhine primates and in fossil hominins. The point of our study is to clarify these relationships between Homo sapiens, Australopithecus and modern great apes (Pongo, Pan, Gorilla) to resolve an apparent allometric conundrum within our previous studies: one analysis based on 34 species of extant Haplorhini, including H. sapiens, resulted in the equation
{\stackrel{˙}{Q}}_{\mathrm{ICA}} =8.82×{10}^{-3}
Vbr0.95 [6], while another analysis of 11 species of fossil hominin, also including H. sapiens, produced the equation
{\stackrel{˙}{Q}}_{\mathrm{ICA}}=1.70×{10}^{-4}
Vbr1.45 [15]. Humans are on both analyses with the largest brains, but the exponents of these equations are markedly different, and the lines converge. The present study confirms that hominin ancestors had lower
{\stackrel{˙}{Q}}_{\mathrm{ICA}}
than predicted from Vbr with the haplorhine equation.
{\stackrel{˙}{Q}}_{\mathrm{ICA}}
in modern great apes is about twice that in Australopithecus species, despite similar or smaller Vbr.
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Variable Step Solvers in Simulink - MATLAB & Simulink - MathWorks Nordic
Variable-Step Discrete Solver
Variable-Step Continuous Solvers
Variable-Step Continuous Explicit Solvers
Variable-Step Continuous Implicit Solvers
Tips for Choosing a Variable-Step Implicit Solver
Variable-step solvers vary the step size during the simulation, reducing the step size to increase accuracy when model states are changing rapidly and increasing the step size to avoid taking unnecessary steps when model states are changing slowly. Computing the step size adds to the computational overhead at each step but can reduce the total number of steps, and hence simulation time, required to maintain a specified level of accuracy for models with rapidly changing or piecewise continuous states.
When you set the Type control of the Solver configuration pane to Variable-step, the Solver control allows you to choose one of the variable-step solvers. As with fixed-step solvers, the set of variable-step solvers comprises a discrete solver and a collection of continuous solvers. However, unlike the fixed-step solvers, the step size varies dynamically based on the local error.
The choice between the two types of variable-step solvers depends on whether the blocks in your model define states and, if so, the type of states that they define. If your model defines no states or defines only discrete states, select the discrete solver. If the model has continuous states, the continuous solvers use numerical integration to compute the values of the continuous states at the next time step.
If a model has no states or only discrete states, Simulink® uses the discrete solver to simulate the model even if you specify a continuous solver.
Use the variable-step discrete solver when your model does not contain continuous states. For such models, the variable-step discrete solver reduces its step size in order to capture model events such as zero-crossings, and increases the step size when it is possible to improve simulation performance.
The model shown in the figure contains two discrete sine wave signals at 0.5 and 0.75 sample times. The graphs below show the signals in the model along with the solver steps for the variable-step discrete and the fixed-step discrete solvers respectively. You can see that the variable-step solver only takes the steps needed to record the output signal from each block. On the other hand, the fixed-step solver will need to simulate with a fixed-step size—or fundamental sample time—of 0.25 to record all the signals, thus taking more steps overall.
Variable-step solvers dynamically vary the step size during the simulation. Each of these solvers increases or reduces the step size using its local error control to achieve the tolerances that you specify. Computing the step size at each time step adds to the computational overhead. However, it can reduce the total number of steps, and the simulation time required to maintain a specified level of accuracy.
You can further categorize the variable-step continuous solvers as one-step or multistep, single-order or variable-order, and explicit or implicit. See One-Step Versus Multistep Continuous Solvers for more information.
The variable-step explicit solvers are designed for nonstiff problems. Simulink provides four such solvers:
ode45 X Medium Runge-Kutta, Dormand-Prince (4,5) pair
ode23 X Low Runge-Kutta (2,3) pair of Bogacki & Shampine
ode113 X Variable, Low to High PECE Implementation of Adams-Bashforth-Moulton
odeN X See Order of Accuracy in Fixed-Step Continuous Explicit Solvers See Integration Technique in Fixed-Step Continuous Explicit Solvers
In general, the ode45 solver is the best to apply as a first try for most problems. The Runge-Kutta (4,5) solver is a fifth-order method that performs a fourth-order estimate of the error. This solver also uses a fourth-order interpolant, which allows for event location and smoother plots.
If the ode45 is computationally slow, the problem may be stiff and thus require an implicit solver.
For problems with stringent error tolerances or for computationally intensive problems, the Adams-Bashforth-Moulton PECE solver can be more efficient than ode45.
The ode23 can be more efficient than the ode45 solver at crude error tolerances and in the presence of mild stiffness. This solver provides accurate solutions by applying a cubic Hermite interpolation to the values and slopes computed at the ends of a step.
odeN The odeN solver uses a nonadaptive Runge-Kutta integration whose order is determined by the Solver order parameter. odeN uses a fixed step size determined by the Max step size parameter, but the step size can be reduced to capture certain solver events, such as zero-crossings and discrete sample hits.
Select the odeN solver when simulation speed is important, for example, when
The model contains lots of zero-crossings and/or solver resets
The Solver Profiler does not detect any failed steps when profiling the model
If your problem is stiff, try using one of the variable-step implicit solvers:
ode15s X Variable, low to medium X X Numerical Differentiation Formulas (NDFs)
ode23s X Low Second-order, modified Rosenbrock formula
ode23t X Low X Trapezoidal rule using interpolant
ode23tb X Low X TR-BDF2
For ode15s, ode23t, and ode23tb a drop-down menu for the Solver reset method appears in the Solver details section of the Configuration pane. This parameter controls how the solver treats a reset caused, for example, by a zero-crossing detection. The options allowed are Fast and Robust. Fast specifies that the solver does not recompute the Jacobian for a solver reset, whereas Robust specifies that the solver does. Consequently, the Fast setting is computationally faster but it may use a small step size in certain cases. To test for such cases, run the simulation with each setting and compare the results. If there is no difference in the results, you can safely use the Fast setting and save time. If the results differ significantly, try reducing the step size for the fast simulation.
For the ode15s solver, you can choose the maximum order of the numerical differentiation formulas (NDFs) that the solver applies. Since the ode15s uses first- through fifth-order formulas, the Maximum order parameter allows you to choose orders 1 through 5. For a stiff problem, you may want to start with order 2.
The following table provides tips for the application of variable-step implicit solvers. For an example comparing the behavior of these solvers, see Exploring Variable-Step Solvers Using a Stiff Model.
Tips on When to Use
ode15s is a variable-order solver based on the numerical differentiation formulas (NDFs). NDFs are related to, but are more efficient than the backward differentiation formulas (BDFs), which are also known as Gear's method. The ode15s solver numerically generates the Jacobian matrices. If you suspect that a problem is stiff, or if ode45 failed or was highly inefficient, try ode15s. As a rule, start by limiting the maximum order of the NDFs to 2.
ode23s is based on a modified Rosenbrock formula of order 2. Because it is a one-step solver, it can be more efficient than ode15s at crude tolerances. Like ode15s, ode23s numerically generates the Jacobian matrix for you. However, it can solve certain kinds of stiff problems for which ode15s is not effective.
The ode23t solver is an implementation of the trapezoidal rule using a “free” interpolant. Use this solver if your model is only moderately stiff and you need a solution without numerical damping. (Energy is not dissipated when you model oscillatory motion.)
ode23tb is an implementation of TR-BDF2, an implicit Runge-Kutta formula with two stages. The first stage is a trapezoidal rule step while the second stage uses a backward differentiation formula of order 2. By construction, the method uses the same iteration matrix in evaluating both stages. Like ode23s, this solver can be more efficient than ode15s at crude tolerances.
For a stiff problem, solutions can change on a time scale that is very small as compared to the interval of integration, while the solution of interest changes on a much longer time scale. Methods that are not designed for stiff problems are ineffective on intervals where the solution changes slowly because these methods use time steps small enough to resolve the fastest possible change. For more information, see Shampine, L. F., Numerical Solution of Ordinary Differential Equations, Chapman & Hall, 1994.
The variable-step solvers use standard control techniques to monitor the local error at each time step. During each time step, the solvers compute the state values at the end of the step and determine the local error—the estimated error of these state values. They then compare the local error to the acceptable error, which is a function of both the relative tolerance (rtol) and the absolute tolerance (atol). If the local error is greater than the acceptable error for any one state, the solver reduces the step size and tries again.
Relative tolerance measures the error relative to the size of each state. The relative tolerance represents a percentage of the state value. The default, 1e-3, means that the computed state is accurate to within 0.1%.
Absolute tolerance is a threshold error value. This tolerance represents the acceptable error as the value of the measured state approaches zero.
The solvers require the error for the ith state, ei, to satisfy:
{e}_{i}\le \mathrm{max}\left(rtol×|{x}_{i}|,ato{l}_{i}\right).
The following figure shows a plot of a state and the regions in which the relative tolerance and the absolute tolerance determine the acceptable error.
Your model has a global absolute tolerance that you can set on the Solver pane of the Configuration Parameters dialog box. This tolerance applies to all states in the model. You can specify auto or a real scalar. If you specify auto (the default), Simulink initially sets the absolute tolerance for each state based on the relative tolerance. If the relative tolerance is larger 1e-3, abstol is initialized at 1e-6. However, for reltol smaller than 1e-3, abstol for the state is initialized at reltol * 1e-3. As the simulation progresses, the absolute tolerance for each state resets to the maximum value that the state has assumed so far, times the relative tolerance for that state. Thus, if a state changes from 0 to 1 and reltol is 1e-3, abstol initializes at 1e-6 and by the end of the simulation reaches 1e-3 also. If a state goes from 0 to 1000, then abstol changes to 1.
Now, if the state changes from 0 to 1 and reltol is set at 1e-4, then abstol initializes at 1e-7 and by the end of the simulation reaches a value of 1e-4.
If the computed initial value for the absolute tolerance is not suitable, you can determine an appropriate value yourself. You might have to run a simulation more than once to determine an appropriate value for the absolute tolerance. You can also specify if the absolute tolerance should adapt similarly to its auto setting by enabling or disabling the AutoScaleAbsTol parameter. For more information, see Auto scale absolute tolerance.
Several blocks allow you to specify absolute tolerance values for solving the model states that they compute or that determine their output:
The absolute tolerance values that you specify for these blocks override the global settings in the Configuration Parameters dialog box. You might want to override the global setting if, for example, the global setting does not provide sufficient error control for all of your model states because they vary widely in magnitude. You can set the block absolute tolerance to:
–1 (same as auto)
real vector (having a dimension equal to the number of corresponding continuous states in the block)
If you do choose to set the absolute tolerance, keep in mind that too low of a value causes the solver to take too many steps in the vicinity of near-zero state values. As a result, the simulation is slower.
On the other hand, if you set the absolute tolerance too high, your results can be inaccurate as one or more continuous states in your model approach zero.
Once the simulation is complete, you can verify the accuracy of your results by reducing the absolute tolerance and running the simulation again. If the results of these two simulations are satisfactorily close, then you can feel confident about their accuracy.
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Fishing - Ring of Brodgar
Enabled Swimming
Hearth Magic(s) Unlocked Fisher's Request for a Catch
Required By (158) A Talking Whale, Abyss Gazer, An Old Boot, Asp, Bass, Boiled River Pearl Mussel, Bone Hook, Bream, Brill, Burbot, Bushcraft Fishingpole, Bushcraft Fishline, Butter-Chived Trout, Carp, Catfish, Cave Sculpin, Cavelacanth, Chitin Hook, Chub, Chum Bait, Cod, Copper Comet, Copperbrush Snapper, Creel, Deep Sea Atavism, Dried Filet of Abyss Gazer, Dried Filet of Asp, Dried Filet of Bass, Dried Filet of Bream, Dried Filet of Brill, Dried Filet of Burbot, Dried Filet of Carp, Dried Filet of Catfish, Dried Filet of Cave Angler, Dried Filet of Cave Sculpin, Dried Filet of Cavelacanth, Dried Filet of Chub, Dried Filet of Cod, Dried Filet of Eel, Dried Filet of Grayling, Dried Filet of Haddock, Dried Filet of Herring, Dried Filet of Ide, Dried Filet of Lavaret, Dried Filet of Mackerel, Dried Filet of Mullet, Dried Filet of Pale Ghostfish, Dried Filet of Perch, Dried Filet of Pike, Dried Filet of Plaice, Dried Filet of Pomfret, Dried Filet of Roach, Dried Filet of Rose Fish, Dried Filet of Ruffe, Dried Filet of Saithe, Dried Filet of Salmon, Dried Filet of Silver Bream, Dried Filet of Smelt, Dried Filet of Sturgeon, Dried Filet of Tench, Dried Filet of Trout, Dried Filet of Whiting, Dried Filet of Zander, Dried Filet of Zope, Eel, Farmer's Fishline, Feather Fly, Fine Fishline, Fishing Net, Fishline, Fishy Eyeball, Gold Spoon-Lure, Grayling, Haddock, Herring, Ide, Lavaret, Lobster, Lobster Pot, Macabre Fishline, Mackerel, Mullet, Opened Oyster, Oyster Pearl, Pale Ghostfish, Pearl Necklace, Perch, Perched Perch, Petrified Seashell, Pike, Pinecone Plug, Plaice, Pomfret, Poppy Wobbler, Primitive Casting-Rod, River Pearl, Roach, Rock Lobster, Rose Fish, Ruffe... further results
The fishing skill allows you to craft and use Bushcraft Fishingpoles and Primitive Casting-Rods and to fish for different types of fish. The fishing skill benefits from completion of the Credos - Fisherman.
2.3 Fish Hook
3 Fishing Trash
4.1 Bait Fishing
4.2 Lure Fishing
5 Fish Data
5.1 Fresh Water Fish
5.2 Ocean Fish
5.3 Cave Fish
"The fathoms give nothing,
Yield not a single thing,
But to the fishermens' snares."
You have mastered some basic fishing techniques, and can craft rods, poles, hooks and lures.
There are two types of fishing: Rod and pole. Pole fishing uses various forms of live or fleshy bait -- such as earthworms, entrails, or ants -- whereas Rod fishing uses lures. Fishing poles catch fish immediately whenever they bite, but the more demanding Rod fishing rather requires you to actually land the Fish by figuring out the many riddles of the sport. The truly rare fish of the world are caught with rods.
To fish you need four categories of items combined together with the fishing skill. These items are:
Craft Either a Bushcraft Fishingpole or a Primitive Casting-Rod.
Primitive Casting-Rod must be used with Lures. Lures are not consumed as fish are caught. This is called Lure-Fishing
Bushcraft Fishingpole uses bait. Fish are caught automatically and the Bait is Consumed and replaced per fish. This is called Bait-Fishing
As of Fishing for Finery (2021-04-25) update. Strings, as fishing lines, where replaced with dedicated crafted fishing lines.
Bushcraft Fishline
Farmer's Fishline
Fine Fishline
Macabre Fishline
Shepherd's Fishline
Shoreline Fishline
Tanner's Fishline
Woodsman's Fishline
(No specific per-fishline details known)
Fishing line occasionally break, so be sure to have backups. Higher quality fishing line tends to break less often.
A fish hook is added to the fishing pole by left clicking on the hook and then right clicking on the fishing pole. Fish hook types include:
Chitin Hook
Fish hook are also occasionally lost. Using your prized gold nugget passed down by your ancestors might not be wise. Higher quality fish hooks tend to be lost less often.
Bait is expendable meaning you can catch at maximum one fish per bait.
Some types of bait include: Ant Empress, Ant Larvae, Ant Pupae, Ant Queen, Ant Soldiers, Aphids, Bee Larvae, Cave Moth, Chum Bait, Earthworm, Emerald Dragonfly, Entrails, Firefly, Grasshopper, Grub, Ladybug, Leech, Monarch Butterfly, Moonmoth, Raw Crab, Raw Lobster, Ruby Dragonfly, Sand Flea, Silkmoth, Silkworm, Springtime Bumblebee, Stag Beetle, Waterstrider
Lures are durable, but they can be lost. To unlock lures you should craft a Primitive Casting-Rod first.
Some types of lure include: Copper Comet, Copperbrush Snapper, Feather Fly, Gold Spoon-Lure, Pinecone Plug, Poppy Wobbler, Rock Lobster, Steelbrush Plunger, Tin Fly, Woodfish
When something is dropped in a body of water, it is added to a global pool of items that can be fished up anywhere in the world or found buried on the beach. Some people use this feature for charity by throwing their old equipment in the river. You can also drop a written parchment to send a message to a random fisherman as if it were a message in a bottle.
Quoting Jorb: Fish are now a limited and localized resource, working in the same way as any other localized resource. You should notice fish jumping in the water if the area has fish. What you catch when fishing depends on the following factors: Time of day, location, lure, hook, pole and line type as well as the quality level of each piece of fishing equipment.
Bait-Fishing is a semi-automated and a Semi-random activity using a Bushcraft Fishing Pole. Bait, line, and hook combo seem to have no strong correlation between fish types caught. Equip your pole, gather an inventory of bait, and begin fishing. You will automatically catch, recast, and re-bait your hook while bait fishing. The type of fish caught are random, based on the node you are fishing from, however the quality is hard-capped by your Survival. If you are simply looking to catch fish with no real desire for a certain type of fish, this is your easiest option.
Lure-Fishing using a Primitive Casting Rod is a more precise and manual activity. Lure, line, and hook type all play role in what species you will be able to catch. Every fish have certain line, hook, and lure combos that work best for it. If you are having difficulties catching a fish or keeping your lure, be sure to try new combinations. Equip your pole, gather some tackle, and begin fishing. You will be presented a list of fish to "aim" for, with a given % of how likely they are to bite. This list is a function of your Survival and Will, so the higher both stats are, the larger your list will be. Max number of fishes in a list is limited by 10 and apparently is equal to
{\displaystyle \lceil {\sqrt {Will*Surv}}/20\rceil }
. You can choose what kind of fish you want to catch. If you do not pick any, the game will automatically choose the fish with the highest % of biting.
Fish that can be found in rivers and lakes.
Fishing Lure or Bait
Combos listed are for W10, use at your own discretion'
FEP (Based on Q10)
A Talking Whale
2x3 6 ... (PreW11, [Verify: Talking Whale location.] against W11) Shallow Water, Deep Water ...
2x1 2 Copper Comet, Entrails, Grasshopper, Rock Lobster Lure 100%, Lake Depths, River Shallows, River Depths 5 DEX
1x1 1 Earthworm, Rock Lobster Lure, Pinecone Plug Lake Depths, River Depths 2 STR, 2 DEX
1x1 1 Earthworm, Feather Fly, Woodfish, Rock Lobster Lure 100%, Pinecone Plug, Leech, Copper Comet 100% Lake Shallows, Lake Depths 2 INT, 3 CHA
2x1 2 Dragonfly, Earthworm, Entrails, Poppy Wobbler,Copper Comet 100%, Rock Lobster Lure 100%, Pinecone Plug 100% Lake Depths, River Depths 3 CON, 1.5 CHA
3x1 3 Metal Hook, Flax Fibres, Tin Fly, Rock Lobster Lure, Earthworm, Entrails, Leech, Woodfish Lake Depths, River Depths 1.25 AGI, 2.5 INT, 2.5 PSY
2x1 2 Entrails, Woodfish, Tin Fly, Feather Fly Lake Shallows, Lake Depths, River Depths 2.5 CHA, 2.5 CON, 1 PER
1x1 1 Earthworm, Leech, Rock Lobster Lure, Steelbrush Plunger, Tin Fly River Depths, River Shallows 1 CHA, 1 INT
1x2 2 Feather Fly, Rock Lobster Lake Depths, River Shallows 4 AGI
2x1 2 Leech, Entrails, Earthworm, Feather Fly Lake Depths, Lake Shallows 2.5 PER, 1.25 AGI
1x1 1 Woodfish, Waterstrider, Feather Fly Lake Shallows, River Depths 2.0 DEX, 0.5 PSY
2x1 2 Leech, Earthworm, Steelbrush Plunger 100%, Pinecone Plug Lake Depths, River Shallows 4.5 CON, 1.5 AGI
1x1 1 Gold Spoon-Lure, Feather Fly, Woodfish, Poppy Wobbler, Tin Fly, Pinecone Plug, Copper Comet, Entrails, Earthworm, Leech, Waterstrider Lake Depths, River Depths, River Shallows 4 INT
2x1 2 Earthworm, Feather Fly, Woodfish, Tin Fly, Pinecone Plug, Waterstrider, Leech, Entrails, Copper Comet 100% Lake Shallows, Lake Depths, River Shallows 2 STR, 2 INT, 1 CON
1x1 1 Feather Fly, Tin Fly, Firefly, Woodfish, Rock Lobster Lure, Pinecone Plug,Gold Spoon-Lure, Leech, Entrails,Earthworm Lake Shallows, River Depths, River Shallows 1 INT, 3 PER
1x1 1 Earthworm, Woodfish, Rock Lobster Lure, Silkworm, Feather Fly, Pinecone Plug, Leech, Entrails Lake Depths, River Depths 2 INT, 2 DEX
1x1 1 Tin Fly, Dragonfly, Leech, Entrails, Earthworm Lake Depths, River Shallows, River Depths 3 CHA
2x1 2 Rock Lobster Lure, Entrails, Copper Comet, Earthworm Lake Shallows, Lake Depths, River Depths, River Shallows 2 INT, 3 CHA
1x1 1 Feather Fly, Woodfish (100%), Earthworm Lake Depths, Lake Shallows, River Shallows 1 CHAR, 1 DEX
3x1 3 (Flax FibreBone HookWoodfish got me 42x100% on a tile where a rock lobster gave 8x100%), Rock Lobster Lure, Feather Fly Lake Depths, River Shallows 3 STR, 1 PSY
1x1 1 Feather Fly 45%, Woodfish 86%, Pinecone Plug 100%, Entrails Lake Depths, River Depths 2 INT, 1 CON, 1 PER
2x1 2 Entrails, Rock Lobster Lure, Copper Comet, Earthworm,Grasshopper, Feather Fly Lake Depths, River Depths, River Shallows 2 STR, 1.5 PER, 1.5 DEX
1x1 1 Dragonfly, Leech, Entrails, Earthworm, Feather Fly, Woodfish, (best)Tin Fly Lake Depths, River Shallows 2 INT
2x1 2 Dragonfly, Leech, Entrails, Steelbrush Plunger 100%, Tin Fly 100%, Woodfish 100%, Rock Lobster Lure 100%, Earthworm Lake Depths, River Shallows 3 INT
1x1 1 Entrails, Feather Fly 100% Lake Shallows, River Shallows 2 PER, 1 INT
Fish that can be found in Oceans.
2x1 2 Entrails, Rock Lobster Shallow Ocean, Ocean Depths 1 PER, 2 WIL
2x1 2 Entrails, Earthworm, Woodfish, Copper Comet Ocean Depths 2 INT, 3 PER
2x1 2 Entrails, ( Flax Fibre Metal Hook Rock Lobster ) Ocean Depths 3.5 PER, 2 AGI
1x1 1 Entrails, Rock Lobster, Feather Fly, Pinecone Plug Ocean Depths 1 INT, 1 DEX
1x1 1 Entrails, Woodfish Ocean Depths 3 INT
1x1 1 Entrails, Rock Lobster, Woodfish (90%) Shallow Ocean, Ocean Depths 3 STR, 2 PER
1x1 1 Entrails, Copper Comet (100%), Rock Lobster (84%) Ocean Depths 2 INT, 1 WIL
2x1 2 Entrails, Earthworm, Woodfish Ocean Depths 2 WIL, 2 CHA
2x1 2 Entrails, Woodfish 100% Shallow Ocean, Ocean Depths 2 INT, 2 AGI
1x1 1 ... Shallow Ocean, Deep Ocean Curiosity
1x1 1 Entrails, Feather Fly, Rock Lobster Shallow Ocean, Ocean Depths 1 CHA, 2 WIL
Fish that can be found in underground bodies of water only:
2x1 2 Poppy Wobbler, Woodfish (63%) Cave Shallows, Cave Depths 2.5 PER, 2.5 PER+2, 2.5 PSY
Cavelacanth
2x1 2 Woodfish (100%), Bushcraft Fishline, Metal Hook Cave Shallows, Cave Depths 3 PER+2, 3 INT
Cave Sculpin
1x1 1 Woodfish (100%) Cave Shallows, Cave Depths 2 CON+2, 3 WIL
Pale Ghostfish
1x1 1 Woodfish 63%, Rock Lobster 100% Cave Shallows, Cave Depths 1 PSY+2, 2 INT, 2 CHA
Retrieved from "https://ringofbrodgar.com/w/index.php?title=Fishing&oldid=94347"
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On the growth of Sobolev norms for the cubic Szegő equation
Patrick Gérard; Sandrine Grellier
We report on a recent result establishing that trajectories of the cubic Szegő equation in Sobolev spaces with high regularity are generically unbounded, and moreover that, on solutions generated by suitable bounded subsets of initial data, every polynomial bound in time fails for high Sobolev norms. The proof relies on an instability phenomenon for a new nonlinear Fourier transform describing explicitly the solutions to the initial value problem, which is inherited from the Lax pair structure enjoyed by the equation.
author = {Patrick G\'erard and Sandrine Grellier},
title = {On the growth of {Sobolev} norms for the cubic {Szeg\H{o}} equation},
TI - On the growth of Sobolev norms for the cubic Szegő equation
%T On the growth of Sobolev norms for the cubic Szegő equation
Patrick Gérard; Sandrine Grellier. On the growth of Sobolev norms for the cubic Szegő equation. Séminaire Laurent Schwartz — EDP et applications (2014-2015), Talk no. 11, 20 p. doi : 10.5802/slsedp.70. https://slsedp.centre-mersenne.org/articles/10.5802/slsedp.70/
[1] Adamyan, V. M., Arov, D. Z., Krein, M. G., Analytic properties of the Schmidt pairs of a Hankel operator and the generalized Schur-Takagi problem. (Russian) Mat. Sb. (N.S.) 86(128) (1971), 34–75; English transl. Math USSR. Sb. 15 (1971), 31–73.
[2] Bourgain, J., Problems in Hamiltonian PDE’s, Geom. Funct. Anal. (2000), Special Volume, Part I, 32–56.
[3] Bourgain, J., On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, Internat. Math. Res. Notices, (1996), 277-304.
[4] Burq, N., Gérard, P., Tzvetkov, N.: Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds Amer. J. Math. 126, 569–605 (2004).
[5] Colliander J., Keel M., Staffilani G., Takaoka H., Tao, T., Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrodinger equation, Inventiones Math.181 (2010), 39–113.
[6] Dodson, B., Global well-posedness and scattering for the defocusing,
{L}^{2}
-critical, nonlinear Schrödinger equation when
d=2
, arXiv:1006.1375, preprint, 2011.
[7] Gérard, P., Grellier, S., The cubic Szegő equation , Ann. Scient. Éc. Norm. Sup. 43 (2010), 761–810.
[8] Gérard, P., Grellier, S., Invariant Tori for the cubic Szegő equation, Invent. Math. 187 (2012), 707–754.
[9] Gérard, P., Grellier, S., Inverse spectral problems for compact Hankel operators, J. Inst. Math. Jussieu 13 (2014), 273–301.
[10] Gérard, P., Grellier, S., An explicit formula for the cubic Szegő equation, to appear in Trans. A.M.S.
[11] Gérard, P., Grellier, S., Effective integrable dynamics for a certain nonlinear wave equation, Anal. PDEs 5 (2012), 1139–1155.
[12] Gérard, P., Pushnitski, A., An inverse problem for self-adjoint positive Hankel operators, arXiv:1401.2042, to appear in IMRN.
[13] Ginibre, J., Velo, G. Scattering theory in the energy space for a class of nonlinear Schršdinger equations, J. Math. Pures Appl. (9) 64 (1985), no. 4, 363–401.
[14] Guardia, M., Kaloshin, V., Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation, arXiv:1205.5188[math.AP], to appear in J. European Math. Soc.
[15] Grébert, B., Kappeler, T., The defocusing NLS equation and Its Normal Form, EMS series of Lectures in Mathematics, European Mathematical Society, 2014.
[16] Hani, Z., Long-time strong instability and unbounded orbits for some periodic nonlinear Schrödinger equations, to appear Acce in Archives for Rational Mechanics and Analysis. arXiv:1210.7509.
[17] Hani, Z., Pausader, B., Tzvetkov, N., Visciglia, N., Modified scattering for the cubic Schrödinger equations on product spaces and applications, arXiv:1311.2275, 2013.
[18] Hernández B. A., Frías-Armenta M. E., Verduzco F., On differential structures of polynomial spaces in control theory, Journal of Systems Science and Systems Engineering 21 (2012), 372–382.
[19] Kappeler, T., Pöschel, J., KdV & KAM, A Series of Modern Surveys in Mathematics, vol. 45, Springer-Verlag, 2003.
[20] Killip, R., Tao, T., Visan, M., The cubic nonlinear Schršdinger equation in two dimensions with radial data. J. Eur. Math. Soc. 11 (2009), 1203–1258.
[21] Lax, P. : Integrals of Nonlinear equations of Evolution and Solitary Waves, Comm. Pure and Applied Math. 21, 467–490 (1968).
[22] Majda, A., Mc Laughlin, D., Tabak, E., A one dimensional model for dispersive wave turbulence, J. Nonlinear Sci. 7 (1997) 9–44.
[23] Nikolskii, N. K., Operators, functions, and systems: an easy reading. Vol. 1. Hardy, Hankel, and Toeplitz. Translated from the French by Andreas Hartmann. Mathematical Surveys and Monographs, 92. American Mathematical Society, Providence, RI, 2002.
[24] Peller, V.V., Hankel Operators and their applications Springer Monographs in Mathematics. Springer-Verlag, New York, 2003.
[25] Pocovnicu, O. Explicit formula for the solution of the Szegő equation on the real line and applications, Discrete Cont. Dyn. Syst. 31 (2011), 607–649.
[26] Pocovnicu, O. First and second order approximations of a nonlinear wave equation J. Dynam. Differential Equations, article no. 9286 (2013), 29 pp, DOI:10.1007/s10884-013-9286-5.
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. Amer. J. Math. 129 (2007), 1–60.
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Power Mean Inequality (QAGH) | Brilliant Math & Science Wiki
Lawrence Chiou, Hemang Agarwal, Jubayer Nirjhor, and
The QM-AM-GM-HM or QAGH inequality generalizes the basic result of the arithmetic mean-geometric mean (AM-GM) inequality, which compares the arithmetic mean (AM) and geometric mean (GM), to include a comparison of the quadratic mean (QM) and harmonic mean (HM), where
f_{\text{QM}}
denotes the quadratic mean,
f_{\text{AM}}
denotes the arithmetic mean,
f_{\text{GM}}
denotes the geometric mean, and
f_{\text{HM}}
denotes the harmonic mean:
f_{\text{QM}} \geq f_{\text{AM}} \geq f_{\text{GM}} \geq f_{\text{HM}}.
Furthermore, the power mean inequality extends the QM-AM result to compare higher power means and moments.
Comparisons among various means appear frequently in advanced inequality problems. In addition to the AM-GM inequality, the QM-AM-GM-HM and power mean inequalities are important pieces of the inequality problem solving toolkit.
Some Definitions of Means
Arithmetic Mean. Given a list of
k
positive numbers
a_1, \ldots, a_k,
one may be interested in a variety of different types of means. By taking the sum divided by the number of values, the arithmetic mean gives an idea of the mean or "typical" value based on a linear weighting. One might imagine that
k
times the mean gives the sum
k f_{\text{AM}} = a_1 + \cdots + a_k,
which leads to the definition
f_{\text{AM}} = \frac{a_1 + \cdots + a_k}{k}.
Quadratic Mean. One can also apply a quadratic weighting. For a different set of values, one might consider
k
times the square of the mean gives the sum of the squares:
k f^2_{\text{QM}} = a_1^2 + \cdots + a_k^2.
Thus, finding the sum of the squares divided by the number of the values and then taking the square root gives the quadratic mean or root mean square (RMS):
f_{\text{QM}} = \sqrt{\frac{a_1^2 + \cdots + a_k^2}{k}}.
Geometric Mean. Alternatively, one might consider the mean with regard to multiplication, with the
k^\text{th}
power of the mean value equal to the product of the values:
f_{\text{GM}}^k = a_1 \cdots a_k.
This might lead one to find the product of the values and then take the
k^\text{th}
root, which yields the geometric mean
f_{\text{GM}} = \sqrt[k]{a_1 \cdots a_k}.
Harmonic Mean. Finally, one could surmise that
k
times the reciprocal of the mean might equal the sum of the reciprocals of the values:
\frac{k}{f_{\text{HM}}} = \frac{1}{a_1} + \cdots + \frac{1}{a_k}.
f_{\text{HM}} = \frac{k}{\frac{1}{a_1} + \cdots + \frac{1}{a_k}}.
If two numbers have arithmetic mean 2700 and harmonic mean 75, then find their geometric mean.
The arithmetic mean of two numbers
and
b
\frac{a+b}2
The harmonic mean of two numbers
and
b
\frac2{\frac1{a} + \frac1{b}}
The arithmetic mean, geometric mean, and harmonic mean of
a,b,c
are 8, 5, 3, respectively. What is the value of
a^2 + b^2 + c^2
This problem is posed by Matt.
numbers
a_1, a_2, \ldots, a_n
are (respectively)
\frac {\sum_{i=1}^{n} a_n}{n},\quad \sqrt[n]{\prod_{i=1}^{n} a_n},\quad \left( \frac{n} { \sum_{i=1}^{n} \frac{1}{a_n} } \right) .
The arithmetic mean-geometric mean (AM-GM) inequality asserts that the the arithmetic mean is never smaller than the geometric mean:
f_{\text{AM}} \geq f_{\text{GM}}.
It can be used as a starting point to prove the QM-AM-GM-HM inequality.
QM-AM-GM-HM inequality. Given a list of
k
a_1, \ldots, a_k
f_{\text{QM}}
denote the quadratic mean,
f_{\text{AM}}
denote the arithmetic mean,
f_{\text{GM}}
denote the geometric mean, and
f_{\text{HM}}
denote the harmonic mean. Then
f_{\text{QM}} \geq f_{\text{AM}} \geq f_{\text{GM}} \geq f_{\text{HM}}.
Furthermore, equality is achieved if and only if
a_1 = \cdots = a_k.
Starting with the AM-GM inequality
f_{\text{AM}} \geq f_{\text{GM}},
it remains to be proven that
f_{\text{QM}} \geq f_{\text{AM}}
f_{\text{GM}} \geq f_{\text{HM}}.
The latter directly follows from AM-GM with
\frac1{a_1}, \ldots, \frac1{a_k}
\frac{\frac{1}{a_1} + \cdots + \frac{1}{a_k}}{k} \geq \sqrt[k]{\frac{1}{a_1 \cdots a_k}}.
Taking the reciprocal of both sides yields
\sqrt[k]{a_1 \cdots a_k} \geq \frac{k}{\frac{1}{a_1} + \cdots + \frac{1}{a_k}}
To show the former, one can use the Cauchy-Schwarz inequality to write
(a_1 + \cdots + a_k)^2 \leq \big(a_1^2 + \cdots + a_k^2\big)\underbrace{\big(1^2 + \cdots + 1^2\big)}_{k{\text{ times}}}.
k^2
\left(\frac{a_1 + \cdots + a_k}{k}\right)^2 \leq \frac{a_1^2 + \cdots + a_k^2}{k},
\frac{a_1 + \cdots + a_k}{k} \leq \sqrt{\frac{a_1^2 + \cdots + a_k^2}{k}}.\ _\square
The proof of the condition of equality is left as an exercise.
QM-AM-GM-HM for two variables:
a,b > 0,
\sqrt{\dfrac{a^2+b^2}{2}} \geq \dfrac{a+b}{2}\geq \sqrt{ab} \geq \dfrac{2ab}{a+b}.
and
b
are both positive numbers, prove the inequality
\sqrt{\frac{a^2+b^2}{2}} \geq \frac{a+b}{2} .
Squaring the expression on each side, we have
\begin{aligned} \left ( \sqrt{\frac{a^2+b^2}{2}} \right )^2 &= \frac{a^2+b^2}{2} \\ \left ( \frac{a+b}{2} \right )^2 &= \frac{a^2+2ab+b^2}{4}. \end{aligned}
\begin{aligned} \frac{a^2+b^2}{2} - \frac{a^2+2ab+b^2}{4} &= \frac{ 2a^2+2b^2-a^2-2ab-b^2}{4} \\\\ &= \frac{a^2-2ab+b^2}{4} \\\\ &= \frac{(a-b)^2}{4} \geq 0. \end{aligned}
\sqrt{\frac{a^2+b^2}{2}} \geq \frac{a+b}{2}.
_\square
and
b
\frac{a+b}{2} \geq \sqrt{ab} .
Subtracting the right side from the left side gives
\begin{aligned} \frac{a+b}{2} - \sqrt{ab} &= \frac{a+b-2\sqrt{ab}}{2} \\ &= \frac{\big(\sqrt{a}\big)^2-2\sqrt{a}\sqrt{b}+\big(\sqrt{b}\big)^2}{2} \\ & = \frac{\big(\sqrt{a}-\sqrt{b}\big)^2}{2} \geq 0. \end{aligned}
\frac{a+b}{2} \geq \sqrt{ab}.
_\square
and
b
\sqrt{ab} \geq \frac{2ab}{a+b}.
\begin{aligned} \sqrt{ab}-\frac{2ab}{a+b} &= \frac{\sqrt{ab}(a+b)-2ab}{a+b} \\ &= \frac{\sqrt{ab}\left(a+b-2\sqrt{ab}\right)}{a+b} \\ &= \frac{\sqrt{ab}\left(\sqrt{a}-\sqrt{b}\right)^2}{a+b} \geq 0. \end{aligned}
\sqrt{ab} \geq \frac{2ab}{a+b}.
_\square
x
y
are both positive numbers, what is the minimum value of
(2x+3y)\left(\frac{8}{x} + \frac{3}{y}\right) ?
\begin{aligned} (2x+3y)\left(\frac{8}{x} + \frac{3}{y}\right) &= 2x \cdot \frac{8}{x} + 2x \cdot \frac{3}{y} + 3y \cdot \frac{8}{x} + 3y \cdot \frac{3}{y} \\ &= \frac{6x}{y} + \frac{24y}{x} + 25 \\ &\geq 2\sqrt{\frac{6x}{y} \times \frac{24y}{x}}+ 25 \\ &= 24+25=49. \end{aligned}
Thus, the minimum value of
(2x+3y)\left(\frac{8}{x} + \frac{3}{y}\right)
49.
_\square
a>1,
arrange the following three expressions by magnitude:
\begin{array}{c}&1, &\frac{a}{a-1}, &\frac{a+1}{a}. \end{array}
The differences among the three expressions are
\begin{aligned} \frac{a}{a-1}-1 = \frac{a-a+1}{a-1} = \frac{1}{(a-1)} >0 &\Rightarrow \frac{a}{a-1} >1 &\qquad (1) \\ \frac{a+1}{a} - \frac{a}{a-1} = \frac{a^2-1-a^2}{a(a-1)} = -\frac{1}{a(a-1)}<0 &\Rightarrow \frac{a}{a-1} > \frac{a+1}{a} &\qquad (2) \\ \frac{a+1}{a}-1 = \frac{a+1-a}{a} = \frac{1}{a} >0 &\Rightarrow \frac{a+1}{a} > 1. &\qquad (3) \end{aligned}
(1), (2),
(3),
the order relation of the three expressions is
\frac{a}{a-1} > \frac{a+1}{a} > 1.\ _\square
d+d+d < a+b+c
d+d+d = a+b+c
d+d+d > a+b+c
No definite relation can be made
As shown above, the 3 colored squares have the side lengths of
a<b<c
while the blue rectangle has the length of
a+b+c
and the width of
d
If the blue rectangle's area is equal to the sum of 3 squares' areas combined, which of the following statements is correct?
To generalize the arithmetic and quadratic means, one can simply consider any higher power
n
--that is, given an
n^\text{th}
power weighting, the
n^\text{th}
power of the mean might be taken to be
k
times that of the sum of the
n^\text{th}
powers of some
k
k f^n_n = a_1^n + \cdots + a_k^n.
Taking the sum of the
n^\text{th}
powers divided by the number of the values and then taking the
n^\text{th}
root gives the
n^\text{th}
f_n
f_n = \sqrt[n]{\frac{a_1^n + \cdots a_k^n}{k}}.
The power mean inequality asserts that
f_m \geq f_n
m > n.
f_0 = \sqrt[k]{a_1 a_2 a_3 \cdots a_k}
a_1, \ldots, a_k,
f_n
f_n = \sqrt[n]{\frac{a_1^n + \cdots a_k^n}{k}},
m > n,
f_m \geq f_n,
a_1 = \cdots = a_k.
Cite as: Power Mean Inequality (QAGH). Brilliant.org. Retrieved from https://brilliant.org/wiki/power-mean-qagh/
|
UNIBRA/DSEBRA: The German Seismological Broadband Array and Its Contribution to AlpArray—Deployment and Performance
Yield Estimation and Event Discrimination of the 4 August 2020 Beirut Chemical Explosion
S‐Wave Velocity Structure of the Crust and Upper Mantle beneath the North China Craton Determined by Joint Inversion of Rayleigh‐Wave Phase Velocity and Z/H Ratio
Site Amplification at High Spatial Resolution from Combined Ambient Noise and Earthquake Recordings in Sion, Switzerland
Mw
Mw
Recent SRL Focus Sections
93-2A Puerto Rico Seismicity, Tectonics and the 2020 M 6.4 Earthquake Sequence (Guest Editors: Elizabeth Vanacore, Christa von Hillebrandt Andrade, Daniel Edward MaNamara)
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92-1 Monitoring During Crisis (Guest Editors: Kristine L. Pankow, Elizabeth A. Vanacore, and Sergio Barrientos)
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391 MALMI: An Automated Earthquake Detection and Location Workflow Based on Machine Learning and Waveform Migration
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288 SeisBench—A Toolbox for Machine Learning in Seismology
240 Seismic and Infrasound Data Recorded at Regional Seismoacoustic Research Arrays in South Korea from the Six DPRK Underground Nuclear Explosions
|
Calorimetry - Course Hero
General Chemistry/Energy and Calorimetry/Calorimetry
Calorimetry is the study of heat exchange between a system and its surroundings. A calorimeter is a device used to measure the heat exchanged between a system and its surroundings. Calorimeters are often used to measure the net heat released or absorbed by a chemical reaction at a constant pressure, called the heat of reaction (
\bold\Delta H_{\rm{rxn}}
). This is also called enthalpy of reaction. If the substances absorb heat from the surroundings, the reaction is endothermic and
\Delta H_{\rm {rxn}}
is positive. If the substances release heat to their surroundings, the reaction is exothermic and
\Delta H_{\rm {rxn}}
A bomb calorimeter is a constant-volume calorimeter used to measure the heat of a reaction for combustion reactions. The calorimeter contains a bomb chamber into which the reactants are placed, and it is wired to an electric heater that ignites the reactants and causes the combustion to occur. The rest of the calorimeter is filled with water and pressurized oxygen. A stirrer keeps the water mixed so that the heat is evenly distributed, and a thermometer records the temperature of the water.
Combustion is exothermic, so the heat released by the combustion is absorbed by the water inside the calorimeter and causes the temperature of the water to rise. In calorimetry experiments, the heat capacity of the calorimeter, Ccal—not the specific heat—is used. Each calorimeter has a unique Ccal value that must be determined experimentally. The heat capacity of the calorimeter is the heat required to raise the calorimeter by one degree Celsius.
An important caveat is that the entire bomb calorimeter should not lose any heat to its surroundings. Calorimeters are well insulated against heat transfer. A system that does not exchange heat or matter with the surroundings is an isolated system. Thus, a bomb calorimeter is built as an isolated system to ensure that no heat is released to the environment around the device and all of the energy released by the combustion is used to change the temperature inside the calorimeter. Because a bomb calorimeter is an isolated system the sum of the energies of the reactants and the calorimeter must be zero,
q_{\rm{rxn}}+q_{\rm{cal}}=0
where qrxn is the energy released or used by the reaction and qcal is the heat absorbed or released by the calorimeter. The value of qcal can be determined by multiplying the heat capacity of the calorimeter by the change in temperature.
q_{\rm{cal}}=C_{\rm{cal}}\Delta T
\Delta H_{\rm {rxn}}
can be calculated from the
\mathit\Delta T
of the calorimeter. This is because energy is conserved. The energy released by the combustion reaction (qrxn) is equal in magnitude to the energy absorbed by the calorimeter. However, because the calorimeter absorbs heat, its
\mathit\Delta T
and energy change are both positive. The energy change for the exothermic combustion is negative.
q_{\rm{cal}}=-q_{\rm{rxn}}=\Delta H_{\rm{rxn}}
A bomb calorimeter consists of a bomb chamber inside a second chamber of water. The water is stirred and there is a thermometer that measures the water temperature. The test material is placed inside the bomb chamber and ignited by electric leads. The entire system is encased in one or two layers of insulation to ensure that the heat released by the combustion reaction does not escape to the surroundings and is entirely absorbed by the calorimeter.
A bomb calorimeter can be used to find the energy available in food. A calorie (cal) is the amount of energy required to change the temperature of one gram of water by 1°C. It is equal to 4.184 J. The chemical energy of food is usually reported in calories. In the United States, calories in food are described by the unit kilocalorie (kcal), but the kilo part is often not used. Some brands report calories in food as "Cal," with a capital c to distinguish it from the scientific definition of calories.
Bomb Calorimeter Calculation
Given a calorimeter with a heat capacity of 4.2 kJ/°C, calculate the chemical energy in kcal stored in 1.0 g of sucrose (C12H22O11) if the combustion causes the temperature of the calorimeter to rise from 25.0°C to 28.9°C.
The first step is to use the specific heat and
\Delta T
of the calorimeter to calculate the heat absorbed by the calorimeter.
q_{\rm{cal}}=C_{\rm{cal}}\Delta T=\left(4.2\,\frac{\rm{kJ}}{{}\degree\rm C}\right)(28.9\degree\rm C-25.0\degree\rm C)=16.4\;\rm{kJ}
The change in energy of the reaction has the same magnitude but opposite sign as the heat absorbed by the calorimeter.
\Delta H_{\rm{rxn}}=-q_{\rm{cal}}=-16.4\;\rm{kJ}
Thus, the heat released by the combustion of 1.0 g of sucrose—and therefore the heat stored as chemical energy in 1.0 g of sucrose—is 16.4 kJ. To convert this to kcal, use the conversion factor
1\rm\;{cal}=4.184\rm\;{J}
\begin{aligned}\Delta H_{\rm{rxn}}&=(16.4\rm\;{kJ})\!\left(\frac{1\rm{,}000\;{J}}{1\rm\;{kJ}}\right)\!\left(\frac{1\rm\;{cal}}{4.184\rm\;J}\right)\!\left(\frac{1\rm\;{kcal}}{1\rm{,}000\;{cal}}\right)\\&=3.9\rm\;{kcal}\end{aligned}
Therefore 1.0 g of sucrose contains 3.9 kcal of energy.
Bomb calorimeters are useful in industrial laboratories, but the more common device in a general chemistry lab is the simple calorimeter, sometimes called a "coffee cup calorimeter," which is a constant-pressure calorimeter. A simple calorimeter works on the same principle as a bomb calorimeter but is usually made of two nested foam cups. A simple calorimeter is typically used to measure the heats of reaction for other types of processes besides combustion reactions. To measure
\Delta H_{\rm {rxn}}
, the reactants are placed inside the inner cup, usually in an aqueous solution, and a stirrer and thermometer are inserted through a cover. The stirrer ensures the heat is distributed throughout the reaction system, and the thermometer measures the temperature of the solution.
Because foam is a good insulator, there is essentially no heat exchange with the surroundings of the calorimeter. Thus, any temperature change of the solution is a result of energy either being absorbed or released by the chemical reaction.
In a coffee cup calorimeter,
q_{\rm{rxn}}+q_{\rm{soln}}=0
where qrxn is the energy released or used by the reaction (the system), and qsoln is the energy transferred to or from the solution (the surroundings). Therefore,
q_{\rm{rxn}}=-q_{\rm{soln}}
The coffee cup calorimeter measures changes in the solution (the surroundings), not the reaction itself. Because
\Delta H_{\rm{rxn}}
is equal to –qsoln, a negative qsoln means
\Delta H_{\rm{rxn}}
is positive. A positive
\Delta H_{\rm{rxn}}
means the reaction has absorbed energy from the solution, so the reaction is endothermic.
\Delta H_{\rm{rxn}}=-q_{\rm{soln}}
For an exothermic reaction,
\Delta H_{\rm{rxn}}
is negative, so qsoln will be positive and the temperature of the solution will rise.
A simple calorimeter is made of two foam cups, one nested inside the other. The test solution is place in the inner cup, a stirrer keeps the solution mixed, and a thermometer measures the temperature of the solution. Any heat transfer that results from a chemical reaction is entirely absorbed by or released by the solution because the cups act as insulation through which heat cannot pass, so measuring the temperature change of the solution allows for the calculation of the amount of energy that has been transferred.
Simple calorimeters can also measure the heat change of processes other than chemical reactions. For example, in order for salt to dissolve, the lattice structure forming the salt must break down. This dissociation has an energy change associated with it, called the heat of solution, or
\Delta H_{\rm{soln}}
\Delta H_{\rm{soln}}
of NaCl is +3.88 kJ/mol.
Heat Change for a Physical Change
How much NaCl, in grams, is needed to change the temperature of 125 mL of water in a simple calorimeter by 0.50°C? Will the temperature of the water increase or decrease?
First, calculate the heat energy required to change the temperature of 125.0 mL of water by 0.5°C. For this, use the specific heat of water of 4.184 J/(g°C) and assume the density of water is 1.0 g/mL.
\begin{aligned}m_{\rm{water}}&=(125.0\rm\;{mL})\!\left(\frac{1\rm\;{g}}{1\rm\;{mL}}\right)\\&=125.0 \rm\;{g}\\\\q&=mc\Delta T\\&=(125.0\rm\;{g})\!\left(4.184\frac {\rm{J}}{{\rm{(g}}\,\degree\rm{C)}}\right)\!\left(0.50\degree\rm C\right)\\&=261.5\rm\;{J}\\&=0.2615\rm\;{kJ}\end{aligned}
Next, calculate the mass of NaCl required for a
q=2.615\;\rm{kJ}
\Delta H_{\rm{soln}}
is reported in kJ/mol, so the mass required to achieve a given q is as follows:
m_{\rm{NaCl}}=\frac q{\Delta H_{\rm{soln}}}=\frac{0.2615\;\rm{kJ}}{3.88\;{\displaystyle\frac{\rm{kJ}}{\rm{mol}}}}=0.0674\;\rm{mol}
Use the molecular weight of NaCl, 58.44 g/mol, to convert moles to mass.
\begin{aligned}m_{\rm{NaCl}}&=(0.0674\;\rm{mol})(58.44\rm\;{g/mol})\\&=3.9\;\rm{g}\end{aligned}
Thus, if 3.9 g of NaCl is dissolved in 125 mL of water, the temperature of the water will change by 0.50°C. This calculation has taken into account the effect the NaCl has on the specific heat of the water, but that effect is negligible because the mass of the water is much greater than the mass of the NaCl. The second question is whether the water temperature will go up or down by 0.50°C. Because the
\Delta H_{\rm{soln}}\;>\;0
for NaCl, the dissolution process absorbs heat and is endothermic;
q_{\rm{NaCl}}\gt{0}
. This means the NaCl uses heat energy from the water to break its ionic bonds, and
q_{\rm{water}}\lt{0}
. In fact, the heat released by the water has the same magnitude but opposite sign as the heat gained by the NaCl.
q_{\rm{water}}=-q_{\rm{NaCl}}
Heat energy leaves the water and the temperature of the water decreases by 0.50°C.
<Thermochemistry>Chemical Thermodynamics
|
A* Search | Brilliant Math & Science Wiki
Thaddeus Abiy, Hannah Pang, Beakal Tiliksew, and
A* (pronounced as "A star") is a computer algorithm that is widely used in pathfinding and graph traversal. The algorithm efficiently plots a walkable path between multiple nodes, or points, on the graph.
A non-efficient way to find a path [1]
On a map with many obstacles, pathfinding from points
A
B
can be difficult. A robot, for instance, without getting much other direction, will continue until it encounters an obstacle, as in the path-finding example to the left below.
However, the A* algorithm introduces a heuristic into a regular graph-searching algorithm, essentially planning ahead at each step so a more optimal decision is made. With A*, a robot would instead find a path in a way similar to the diagram on the right below.
A* is an extension of Dijkstra's algorithm with some characteristics of breadth-first search (BFS).
An example of using A* algorithm to find a path [2]
Like Dijkstra, A* works by making a lowest-cost path tree from the start node to the target node. What makes A* different and better for many searches is that for each node, A* uses a function
f(n)
that gives an estimate of the total cost of a path using that node. Therefore, A* is a heuristic function, which differs from an algorithm in that a heuristic is more of an estimate and is not necessarily provably correct.
A* expands paths that are already less expensive by using this function:
f(n)=g(n)+h(n),
f(n)
= total estimated cost of path through node
n
g(n)
= cost so far to reach node
n
h(n)
= estimated cost from
n
to goal. This is the heuristic part of the cost function, so it is like a guess.
Using the A* algorithm
In the grid above, A* algorithm begins at the start (red node), and considers all adjacent cells. Once the list of adjacent cells has been populated, it filters out those which are inaccessible (walls, obstacles, out of bounds). It then picks the cell with the lowest cost, which is the estimated f(n). This process is recursively repeated until the shortest path has been found to the target (blue node). The computation of
f(n)
is done via a heuristic that usually gives good results.
h(n)
can be done in various ways:
The Manhattan distance (explained below) from node
n
to the goal is often used. This is a standard heuristic for a grid.
h(n)
= 0, A* becomes Dijkstra's algorithm, which is guaranteed to find a shortest path.
The heuristic function must be admissible, which means it can never overestimate the cost to reach the goal. Both the Manhattan distance and
h(n)
= 0 are admissible.
Using a good heuristic is important in determining the performance of
A^{*}
h(n)
would ideally equal the exact cost of reaching the destination. This is, however, not possible because we do not even know the path. We can, however, choose a method that will give us the exact value some of the time, such as when traveling in a straight line with no obstacles. This will result in a perfect performance of
A^{*}
in such a case.
We want to be able to select a function
h(n)
that is less than the cost of reaching our goal. This will allow
h
to work accurately, if we select a value of
h
that is greater, it will lead to a faster but less accurate performance. Thus, it is usually the case that we choose an
h(n)
that is less than the real cost.
The Manhattan Distance Heuristic
This method of computing
h(n)
is called the Manhattan method because it is computed by calculating the total number of squares moved horizontally and vertically to reach the target square from the current square. We ignore diagonal movement and any obstacles that might be in the way.
h = | x_{start} - x_{destination} | + |y_{start} - y_{destination} |
This heuristic is exact whenever our path follows a straight lines. That is,
A^{*}
will find paths that are combinations of straight line movements. Sometimes we might prefer a path that tends to follow a straight line directly to our destination.
The Euclidean Distance Heuristic
This heuristic is slightly more accurate than its Manhattan counterpart. If we try run both simultaneously on the same maze, the Euclidean path finder favors a path along a straight line. This is more accurate but it is also slower because it has to explore a larger area to find the path.
h = \sqrt{(x_{start} - x_{destination})^2 + (y_{start} - y_{destination})^2 }
Can depth-first search always expand at least as many nodes as A* search with an admissible heuristic?
The answer is no, but depth-first search may possibly, sometimes, by fortune, expand fewer nodes than
A^{*}
search with an admissible heuristic. E.g.. it is logically possible that sometimes, by good luck, depth-first search may reach directly to the goal with no back-tracking.
The main drawback of the
A^{*}
algorithm and indeed of any best-first search is its memory requirement. Since at least the entire open list must be saved, the A* algorithm is severely space-limited in practice, and is no more practical than best-first search algorithm on current machines.
The pseudocode for the A* algorithm is presented with Python-like syntax.
make an openlist containing only the starting node
make an empty closed list
while (the destination node has not been reached):
consider the node with the lowest f score in the open list
if (this node is our destination node) :
put the current node in the closed list and look at all of its neighbors
for (each neighbor of the current node):
if (neighbor has lower g value than current and is in the closed list) :
replace the neighbor with the new, lower, g value
current node is now the neighbor's parent
else if (current g value is lower and this neighbor is in the open list ) :
change the neighbor's parent to our current node
else if this neighbor is not in both lists:
The time complexity of
A^{*}
depends on the heuristic. In the worst case, the number of nodes expanded is exponential in the length of the solution (the shortest path), but it is polynomial when the search space is a tree.
Use A* to find the shortest path from the green square to the yellow square in the grid below.
Let us start by choosing an admissible heuristic. For this case, we can use the Manhattan heuristic. We then proceed to the starting cell. We call it our current cell and then we proceed to look at all its neighbors and compute
f,g,h
for each of them. We then select the neighbor with the lowest
f
cost. This is our new current cell and we then repeat the process above.(populate neighbors and compute
f
g
h
and choose the lowest ). We do this until we are at the goal cell. The image below demonstrates how the search proceeds. In each cell the respective
f
h
g
values are shown. Remember
g
is the cost that has been accrued in reaching the cell and
h
is the Manhattan distance towards the yellow cell while
f
h
g
Patel, A. Introduction to A*. Retrieved April 29, 2016, from http://theory.stanford.edu/~amitp/GameProgramming/concave1.png
Cite as: A* Search. Brilliant.org. Retrieved from https://brilliant.org/wiki/a-star-search/
|
Free Energy - Course Hero
Free energy, which is the capacity of a system to do work, can be used to determine whether a reaction is spontaneous or nonspontaneous under given conditions.
The capacity of a system to do work, or use force to move an object, is its free energy. This is often expressed as Gibbs free energy, named after the American scientist Josiah Willard Gibbs, known for his work in thermodynamics in the late 1800s. Gibbs free energy (G) is the amount of work that can be done by a system, and it is expressed as
G=H-TS
, where H is enthalpy in kilojoules (which is the total internal energy of the system plus the product of the pressure and volume), T is temperature, and S is entropy.
For any chemical reaction, the change in Gibbs free energy (
\Delta G
) can be calculated. Assuming that temperature is standard (25°C, which is 298.15 K), the change in Gibbs free energy is
\Delta G=\Delta H-T\Delta S
, with T = 298.15 K. Importantly, if
\Delta G
is less than 0, then the reaction is spontaneous; it does not need energy input to proceed. If
\Delta G
is greater than 0, then the reaction is nonspontaneous and requires energy input to proceed.
Calculation of Change in Gibbs Free Energy
Calculate the change in Gibbs free energy at 25°C and 1 atm for the reaction
{\rm {N}_{2}}(g)+3{\rm {H}_{2}}( g)\rightarrow2{\rm{NH}_{3}}(g)
, and determine whether the reaction is spontaneous or nonspontaneous.
First, calculate the change in enthalpy (
\Delta H
), which is based on the heat of formation of each molecule (
\Delta H_{f}^\circ
). Elements in their gaseous form have
\Delta H_{f}^\circ=0.
{\rm{NH}}_{3}(g)
\Delta H_{f}^\circ=-45.9\;{\rm{kJ}}
. Subtract the enthalpy of the reactants from the enthalpy of the products.
\begin{aligned}\Delta{H}&=\sum{H}_{\rm{prod}}-\sum{H}_{\rm{reac}}\\&=2\Delta{H}_{\rm{NH}_{3}}-\Delta{H}_{\rm{N}_{2}}-3\Delta{H}_{\rm{H}_{2}}\\&=2(-45.9\;\rm{kJ)}-0\;\rm{kJ}-3(0\;\rm{ kJ})\\&=-91.8\;\rm{kJ}\end{aligned}
Next, calculate the change in molar entropy change (
\Delta{S}
) using the standard molar entropy (S°) for each molecule. The standard molar entropy for N2 is 191.61 J/K·mol, for H2 is 130.68 J/K·mol, and for NH3 is 192.77 J/K·mol.
\begin{aligned}\Delta{S}&=\sum{S}_{\rm{prod}}-\sum{S}_{\rm{reac}}\\&=2\Delta{S}_{\rm{NH}_{3}}-\Delta{S}_{\rm{N}_{2}}-3\Delta{S}_{\rm{H}_{2}}\\&=2(192.77\;\rm{ J/K)}-191.61\;\rm{ J/K}-3(130.68\;\rm{ J/K})\\&=-191.11\;\rm{ J/K}\end{aligned}
When setting up the equation for
\Delta{G}
, it is important to convert all values to the same units.
\Delta{H}
is given in kilojoules, so units for
\Delta{S}
are converted to give –0.198 11 kJ/K. Now the equation can be solved for
\Delta{G}
\begin{aligned}\Delta G&=\Delta H-T\Delta S\\&=-91.8\;{\rm{kJ}}-(298\;{\rm K)}(-0.19811\;{\rm{kJ/K)}}\\&=-32.8\;{\rm{kJ}}\end{aligned}
The change in Gibbs free energy for this reaction is less than 0, so the reaction is spontaneous.
Not all reactions proceed at 298 K. Therefore, the temperature at which a reaction proceeds affects whether the reaction is spontaneous. For the reaction between nitrogen gas and hydrogen gas occurring at a temperature of 550 K, the Gibbs free energy change is:
\begin{aligned}\Delta G&=\Delta H - T\Delta S\\&=-91.8\;\rm{kJ}-(550\rm{ K)}(-0.19811\;\rm{kJ}/\rm K)\\&=17.2\;\rm{kJ}\end{aligned}
This reaction has a
\Delta{G}
greater than 0, so it is not spontaneous. Standard molar free energy change (
\Delta G_{f}^\circ
), given in kL/mol, is the energy change for a reaction at standard temperature and pressure that produces one mole of a substance from its constituent elements. Standard molar free energy change has been calculated for most compounds. It can be used to understand the relationship between free energy (
\Delta{G}
) for a reaction and the equilibrium constant (Keq) for that reaction as a function of temperature. It can also be used to determine whether or not a reaction is at equilibrium, in which the forward and reverse reactions are equal. Keq is the ratio of the concentration of products and reactants of a chemical reaction at equilibrium and indicates whether the reaction favors products or reactants. The relationship between
\Delta{G}
and Keq is given as
\Delta G=\Delta G_{f}^\circ+{R}{T} \;{\rm ln}\;K_{\rm{eq}}
R is the ideal gas constant, 8.314 J/mol⋅K, which relates the kinetic energy of molecules to temperature per mole, and T is temperature.
Using the Equilibrium Constant to Determine the Change in Standard Molar Free Energy
Use the equilibrium constant (Keq) to determine the change in standard molar free energy for the reaction
{\rm {N}_{2}}(g)+3{\rm {H}_{2}}( g)\rightarrow2{\rm{NH}_{3}}(g)
at 25°C (298.15 K) and 2.0 atm.
\Delta G_{f}^\circ
{\rm {N}_{2}}(g)+3{\rm {H}_{2}}( g)\rightarrow2{\rm{NH}_{3}}(g)
at 25°C (298 K) and 2.0 atm has already been found to be –32.8 kJ. To determine Keq considering any reaction
{{a\rm{A}}+{b\rm{B}}}\rightarrow{{c\rm{C}}+{d\rm{D}}}
K_{\rm{eq}}=\frac{\lbrack{\rm C}\rbrack^{c}\lbrack{\rm D}\rbrack^{d}}{\lbrack{\rm A}\rbrack^{a}\lbrack{\rm B}\rbrack^{b}}
where the superscripts for each molecule represent the number of moles of that molecule in the reaction equation.
For a reaction in which the products or reactants are gases, the partial pressure is used rather than the concentration. Therefore, it is possible to calculate Keq for the reaction of N2 and H2 using the coefficients of the balanced equation.
\begin{aligned}K_{\rm{eq}}&=\frac{\lbrack{\rm{NH}_{3}}\rbrack^{2}}{\lbrack{\rm {N}_{2}}\rbrack\lbrack{\rm {H}_{2}}\rbrack^{3}}\\\ &=\frac{\lbrack2.0\rbrack^{2}}{\lbrack2.0\rbrack\lbrack2.0\rbrack^3}\\&=0.25\end{aligned}
Now it is possible to solve for
\Delta{G}
\begin{aligned}\Delta G&=-32.76\;{\rm{kJ}+(8.314\;J/K})(298\;{\rm {K)}{(\rm {ln}}\;0.25)}\\&=-36.20\;\rm{kJ}\end{aligned}
Note that in this step it was necessary to convert J to kJ.
<Laws of Thermodynamics>Suggested Reading
|
Hunting - Ring of Brodgar
LP Cost 60
Required Foraging
Enabled Animal Husbandry, Archery, Tanning
Hearth Magic(s) Unlocked None
Required By (425) A Bloody Mess..., Adder Carcass, Adder Fang, Adder Skeleton, Adderfang Amulet, Ancient Tooth, Ant Chitin, Ant Empress, Ant Farm, Ant Hill, Ant Larvae, Ant Meat, Ant Pupae, Ant Queen, Ant Soldiers, Aphids, Bat Claw, Bat Fang, Bat Wings, Bear Cape, Bear Tooth, Beaver Tail, Beaver Teeth, Bee Larvae, Billygoat Horn, Bird Nest, Black Ribeye Steak, Boar Tusk, Bone Chest, Bone Glue, Bone Marrow, Boreworm Beak, Boreworm Mask, Boreworm Meat, Brimstone Butterfly, Bunny Rabbit, Cachalot Blowhole, Cachalot Tooth, Cave Angler Cape, Cave Angler Light, Cave Centipede, Cave Louse Chitin, Cave Louse Meat, Cave Moth, Cave Skewer, Cave Slime, Caverat Whiskers, Centibab, Chasm Conch Meat, Chick, Chicken Meat, Clean Adder Carcass, Clean Hedgehog Carcass, Clean Mole Carcass, Clean Rabbit Carcass, Clean Squirrel Carcass, Clean Stoat Carcass, Cleaned Bat, Cleaned Bog Turtle, Cleaned Chicken, Cleaned Eagle Owl, Cleaned Golden Eagle, Cleaned Magpie, Cleaned Mallard, Cleaned Pelican, Cleaned Ptarmigan, Cleaned Quail, Cleaned Rock Dove, Cleaned Seagull, Cleaned Swan, Cleaned Wood Grouse Cock, Cleaned Wood Grouse Hen, Crab Claw, Crab Roe, Crabshell, Dark Heart, Dead Adder, Dead Bat, Dead Bog Turtle, Dead Chicken, Dead Eagle Owl, Dead Golden Eagle, Dead Hedgehog, Dead Magpie, Dead Mallard, Dead Mole, Dead Pelican, Dead Ptarmigan, Dead Quail, Dead Rabbit, Dead Rock Dove, Dead Seagull, Dead Squirrel, Dead Stoat, Dead Swan, Dead Winter Coat Stoat, Dead Wood Grouse Cock, Dead Wood Grouse Hen, Dried Filet of Abyss Gazer, Dried Filet of Asp... further results
Necessary for picking up, attacking, killing, skinning and butchering various Creatures across the hearthlands. It is also required to build Drying Frame.
3 Hunting small game
4 Hunting big game
5 Skinning and butchering animals
"In a bloodred moment,
The beast fell still,
Giving life through death."
For what it's worth you have built up enough skill and courage to dare to attack, slay, dress and prepare wild animals. Beware that some fell beasts of the wild kill and eat you and that you might not be competent enough fighter yet to take them on. Start out small, with Ants, and build your skills, before going for bigger prey.
Some small animals, like Rabbits and Frogs, can be picked up without a fight, if you can catch them.
Archery mechanics: When shooting a target with a Hunter's Bow your first click on the animal brings up the aim meter. This is a vertical bar which slowly fills with color the longer you are 'aiming'. When you are ready to shoot, click again. How far you allow the bar to rise before a shot directly affects the accuracy and damage of the attack; waiting for the bar to fill completely gives you the best chance at a successful hit for max damage, whereas shooting immediately will almost guarantee a miss, or low damage on hit. Note that your target will still turn aggressive if your opening shot misses. Enclosing an animal in branch stockpiles to prevent it from fleeing when it takes too much damage is a good idea. It won't prevent it from attacking you though, and it will break the stockpiles if they block it's way to you. You can shoot from boats or over deep water to be safer.
Also note: If several characters surround the same animal at once it will keep moving to try to get at them all and make aiming that much harder and take that much longer. The animal will stand still for longer if only one player attacks it.
See Archery for more information on shooting a bow.
Unarmed/meele combat mechanics: See Sevenless begginer guide to Haven and Hearth to learn some basic combat techniques.
To hunt chickens, rabbits, hedgehogs, moles, adders and squirrels: Have at least a 2x2 inventory slot available.
Chickens, Cocks and Chicks will scatter when you get too close. Setting your speed to running should be enough to catch them.
Rabbits will flee if you come too close or right-click them. Come as close as you can without startling them, and then change to sprint or run and right-click to chase them. Rabbits can be tricky to catch, as they often run faster than you. Either try to catch them on a terrain where you can sprint or ride a Horse to easily do it. Wearing Bunny Slippers makes you run much faster when chasing rabbits.
Squirrels will flee, but are easier to catch. You can easily outrun them at running speed.
Hedgehogs will flee, but are easier to catch. You can easily outrun them at running speed. Beware however, that when picking up a hedgehog you will receive roughly 8-10 HHP damage.
Some things to note: wearing gloves greatly reduces the chance of taking damage. killing the hedgehog via combat will not cause any damage.
Moles will teleport a short distance away. Keep chasing them and you will eventually catch them.
Adders will fight you and their bites can leave you with nerve damage. It is recommended that you run them over with a horse.
Horse Riding Method if you have access to either a Horse or Wild Horse, riding over any of these animals will trample them, allowing you to pick up their corpse.
To butcher: Right click the animal again to get the option to kill it, "wring neck". It then becomes two squares in size vertically. If you wish to go on hunting it is often best to leave it in the form of a dead rabbit/dead chicken because when you continue and butcher it, the animal will then take up more inventory squares. Of course, if you only need the meat, or only the skins, just butcher them and drop the rest of the items.
To hunt toads, frogs and rats: Have a single square empty in your inventory. Right click the animal.
Toads can be used to make the Enthroned Toad curiosity.
Rats make a good early energy rich Rat-on-a-Stick food.
Frogs can be used to make Grilled Frog Legs.
To kill them they must be killed in combat or trampled with a horse. This results in simply a splatter of blood, and no item.
Hunting big game is much different than hunting small game. While small game require no combat skills to hunt and kill, you must be able to kill big game through combat. The intricacies of combat are too great to explain here, so instead some methods to hunt which require little or no combat experience will be discussed
Boat Method
With this method, almost every big game can be hunted safely with very low Unarmed Combat
Once you have access to Boats, acquire two. One boat you will ride in, and the other you carry overhead.
One you have found your big game, initiate combat from the shore and lure the animal to shallow water, preferably somewhere that you can easily block the animal's escape with the boat you are carrying. this can be done by finding a narrow strip of shallow water.
After ensuring the animal cannot escape or flee, attack the animal and then move away before it has a chance to retaliate. Any combat move that creates openings, such as punch, low blow, chop, or kick will work.
After a certain amount of damage is done, the animal will leave combat and begin to flee. however, as the animal cannot flee, you can simply kill it.
If you have a cave nearby you can aggro the animal you want to kill and lead it to the cave entrance.
Attack it to build up openings and when your defenses are getting high, enter the cave and lower them. Rinse and repeat.
It can take a while but eventually you will kill the animal and it rarely runs away.
Be careful with taking too long inside the cave as you may break combat and lose your IP.
You will sometimes be attacked. be sure not to take too much damage.
Animals will attack your boat, so be sure it doesnt break beneath you.
If you butcher animals on shallow water, their bones will be dropped into the water and thus irrecoverable.
Bears will begin to flee after a certain amount of damage, and will then enter rage mode. Be aware of this fact while trying to kill bears.
Mammoths have a ranged attack that can be used at a distance, nullifying any advantage of boat hunting.
If you have access to the Knarr blueprint, you can build a knarr construction site behind an animal using a block of wood and negate its escape.
Other methods to be added later
On how to hunt other animals, see Sevenless' begginer guide to Haven and Hearth
Skinning and butchering animals
You need a Sharp Tool to skin or butcher animals. When a big animal is dead you get the options to skin or to butcher if you right click. If you butcher first the skin is ruined and the skin option is not there. If you want the skin and the meat, you must first skin and then butcher. If you make a mistake and click butcher you can still back away before the job is complete. Then you can skin it and the hide is okay. Also if you do not have space for all the meat you can start to butcher it and then step away in the middle. The rest of the meat will still be there.
If there is even only one piece of meat left it will look like the animal has not yet been done.
Butchering is needed to get fresh hide, raw meat, intestines and entrails from animals corpses
The quality of your animal while butchering is determined by quality of animal corpses and softcapped by your sharp tool quality and survival.
If (ToolQ < HideQ) then
{\displaystyle HideQ=(HideQ+ToolQ)/2}
If (Survival < HideQ) then
{\displaystyle HideQ=(HideQ+Survival)/2}
Tool quality softcap is applied before the survival softcap.
Retrieved from "https://ringofbrodgar.com/w/index.php?title=Hunting&oldid=90745"
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Smoothing Nonmonotone Barzilai-Borwein Gradient Method and Its Application to Stochastic Linear Complementarity Problems
Xiangli Li, "Smoothing Nonmonotone Barzilai-Borwein Gradient Method and Its Application to Stochastic Linear Complementarity Problems", Mathematical Problems in Engineering, vol. 2015, Article ID 425351, 6 pages, 2015. https://doi.org/10.1155/2015/425351
Xiangli Li1
1School of Mathematics and Computing Science, Guangxi Key Laboratory of Cryptography and Information Security, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guangxi Key Laboratory of Automatic Detecting Technology and Instruments, Guilin University of Electronic Technology, Guilin, Guangxi 541004, China
A new algorithm for nonsmooth box-constrained minimization is introduced. The method is a smoothing nonmonotone Barzilai-Borwein (BB) gradient method. All iterates generated by this method are feasible. We apply this method to stochastic linear complementarity problems. Numerical results show that our method is promising.
In this paper, we consider a problemwhere , and . If is differentiable on the feasible set, there are many methods for (1), such as trust region method [1], projected gradient method [2], projected Barzilai-Borwein method [3], Newton method [4], and active-set projected trust region [5]. If is semismooth, there are few methods for (1). In this paper, we only consider that is locally Lipschitzian, but not necessarily differentiable.
In [6], a smoothing projected gradient method (SPG) was introduced for nonsmooth optimization problems with nonempty closed convex set. This algorithm is easy to implement. At each iteration, authors approximate the objective function by a smooth function with a fixed smoothing parameter and employ the classical projected gradient method to obtain a new point. If a certain criterion is satisfied, then authors update the smoothing parameter using the new point for the next iteration.
The main motivation for the current work come from the numerical results of [6]. Through analyzing the algorithm [6], we find that it takes a lot of time to calculate projection when test problems are large-scale. In order to avoid this shortcoming, we propose a smoothing nonmonotone BB gradient method. In our method, we use an active set and a nonmonotone line search strategy. The search direction consists of two parts: some of the components are simply defined; the other components are determined by the Barzilai-Borwein gradient method. We apply it to stochastic linear complementarity problems.
Throughout this paper, will be the Euclidean norm. For all , the orthogonal projection of onto a set will be denoted by . For a given matrix , will be the th row of .
The paper can be outlined as follows. In Section 2, we describe our method. In Section 3, stochastic linear complementarity problems are introduced. In Section 4, we apply our method to stochastic linear complementarity problems and numerical results are illustrated and discussed. Finally, we make some concluding remarks in Section 5.
2. Smoothing Nonmonotone BB Gradient Method
In this section, we propose a smoothing nonmonotone BB gradient method for (1), where is a general locally Lipschitz continuous function.
Definition 1. Let be a locally Lipschitz continuous function. One calls a smoothing function of , if is continuously differentiable in for any , and, for any , and is nonempty and bounded.
Let . Consider the associated set at the stationary point of function : where is a fixed parameter. For , we define index set , , and as follows:where . The set is an estimate of the active set at point . For simplicity, we abbreviate , , and defined by (5) as , , and , respectively. We determine by the following process:It is easy to show .
Now, we state algorithm as follows.
Algorithm 2. Choose positive constants , , a tolerance parameter , integer , constants , and parameters ; choose an initial point , . Set .Stop if . Otherwise, set If , choose , and set . Let , and go to Step 1.Determine , , and according to (5).Compute direction according to (6).Compute the step size by the Armijo line search, satisfying Set , , and .Compute , , and . If , ; otherwise, . If , ; otherwise, . Set whereLet and .Let Go to Step 2.
Remark 3. In Step 2 of Algorithm 2, we use to update the smoothing parameter.
Remark 4. Algorithm 2 employs a two-loop approach: the outer loop updates smoothing parameter , and the inner loop computes a stationary point of . In Step 4, the alternate BB step is used to compute the direction .
Remark 5. Note that , so there exists an infinite subsequence such that This implies that every limit point generated by Algorithm 2 is a stationary point of (1).
The convergence of Algorithm 2 is similar to literature [6, 7], so we omit the process here.
3. Stochastic Linear Complementarity Problems
Let be a probability space with being a subset of . Suppose that the probability distribution is known. The stochastic linear complementarity problem (SLCP) [8] is to find such thatwhere and for are random matrices and vectors. Throughout this paper, we assume that and are measurable functions of and satisfy where stands for the expectation. If only contains a single realization, then (12) reduces to the standard LCP. The LCP has been studied by many researchers [9–14]. Since, in many practical problems, some elements may involve uncertain data, problem (12) has been receiving much attention in recent literature [8, 15–20].
In general, there is no satisfying (12) for almost . A deterministic formulation for the SLCP provides a decision vector which is optimal in a certain sense. Different deterministic formulations may yield different solutions that are optimal in different senses. There are two reformulations of (12) that have been proposed: the expected value (EV) formulation [15] and the expected residual minimization (ERM) formulation [8]. In this paper, we concentrate on ERM. ERM is to find a vector that minimizes the expected residual of the SLCP ; that is,where is defined by denotes the th component of the vector . Here, is an NCP function which has the property In this paper, we choose .
Let ; by [6], we know that is semismooth and is a smoothing function of , where Hence, is a smoothing function of , where and is defined by .
In [6], authors use sample average approximation (SAA) [21], which replaces (14) by its approximationHere, the sample is generated by Monte Carlo sampling method, following the same probability distribution as . So, a smoothing function of is In the next section, we apply Algorithm 2 to (20).
The test problems are randomly generated. The procedure of generating the tested problems is employed from [16, 20]. We omit the procedure here. All these problems are tested in Matlab (version 7.5).
Several parameters are needed to generate the problem: , , and . A vector is randomly generated. When the parameter , then is the unique global solution of the test problems and . If , the global solution is unknown.
Algorithm 2 contains several parameters; we use the values in all numerical experiments. We terminate the iteration if one of the following conditions is satisfied:
We start from the same randomly generated initial point and compare our method with SPG. The numerical results can be seen in Tables 1-2. Let if is differentiable at . In Tables 1-2, denote the number of variables; and present at the finial iterates and , respectively; and denote the value of at the final iterates and , respectively.
(, , , ) Algorithm 2 SPG
(100, 20, 10, 20) 1.450e − 04 1.923e − 11 0.047 2.907e − 04 8.295e − 11 0.281
(100, 20, 10, 0) 6.125e − 04 3.791e − 07 0.313 1.098e − 03 1.144e − 06 2.438
(200, 60, 30, 20) 2.083e − 04 9.858e − 12 1.016 8.388e − 04 1.658e − 10 12.547
(200, 60, 30, 0) 9.381e − 04 1.015e − 06 5.109 1.127e − 03 9.025e − 07 52.625
(200, 80, 40, 0) 6.546e − 04 1.187e − 07 10.453 1.122e − 03 7.939e − 07 60.391
(200, 100, 50, 20) 2.178e − 04 8.540e − 12 3.031 1.552e − 04 4.652e − 12 12.609
(200, 100, 50, 10) 3.605e − 04 7.120e − 11 2.750 2.961e − 04 6.574e − 11 9.297
(200, 100, 50, 0) 9.332e − 04 9.131e − 07 16.828 3.471e − 03 9.422e − 07 117.703
(1000, 50, 25, 10) 1.183e − 04 1.785e − 11 3.078 8.176e − 04 6.966e − 10 49.734
(1000, 50, 25, 0) 7.092e − 04 1.117e − 07 24.359 1.439e − 03 9.236e − 07 137.047
Numerical results for .
(100, 20, 10, 20, 10) 2.384e − 04 5.326e + 02 0.109 6.918e − 04 5.325e + 02 0.469
(100, 20, 10, 20, 5) 6.564e − 05 1.455e + 02 0.078 3.865e − 04 1.455e + 02 0.484
(200, 60, 30, 20, 10) 9.097e − 04 1.752e + 03 1.109 5.129e − 04 1.752e + 03 13.016
(200, 60, 30, 20, 5) 5.947e − 04 4.115e + 02 1.078 4.898e − 04 4.115e + 02 17.047
(200, 100, 50, 20, 10) 8.910e − 04 2.577e + 03 4.922 2.475e − 04 2.577e + 03 20.859
(200, 100, 50, 20, 5) 5.727e − 04 7.772e + 02 3.078 1.726e − 04 7.772e + 02 14.063
(300, 120, 60, 10, 20) 9.449e − 04 7.566e + 03 13.922 4.752e − 04 7.567e + 03 44.484
(1000, 50, 25, 20, 10) 4.506e − 04 1.405e + 03 3.656 4.315e + 00 1.405e + 03 22.391
(1000, 50, 25, 10, 5) 2.947e − 04 3.753e + 02 3.297 5.873e − 04 3.753e + 02 23.734
(1000, 100, 50, 5, 10) 1.118e − 04 2.836e + 03 20.797 3.124e − 04 2.836e + 03 85.547
(1000, 100, 50, 10, 5) 1.700e − 04 8.009e + 02 13.484 9.510e − 05 8.009e + 02 53.516
The results reported in Tables 1-2 show that smoothing nonmonotone BB gradient method is quite promising. From results shown in Tables 1-2, we observe that, firstly, Algorithm 2 has less time for all test problems. Secondly, the function values of Algorithm 2 and SPG are close for most test problems. Through analysis, we find that Algorithm 2 drops faster than SPG in each step. This is mainly attributed to the active set and BB step.
In this paper, we present a smoothing nonmonotone Barzilai-Borwein gradient method for nonsmooth box-constrained minimization. The main idea of our method is to use a parametric smoothing approximation function in nonmonotone BB gradient method. In order to have a high speed of convergence, we calculate two-step size. We use BB method and inexact line search, which make sure that the algorithm has high efficiency. We apply it to stochastic linear complementarity problems. Numerical results show that our method is promising.
This project was supported by the National Natural Science Foundation of China (Grants nos. 61362021 and 11361018), Guangxi Fund for Distinguished Young Scholars (2012GXSFFA060003), Guangxi Natural Science Foundation (no. 2014GXNSFFA118001), the Scientific Research Foundation of the Higher Education Institutions of Guangxi, China (Grant no. ZD2014050), Guangxi Natural Science Foundation (no. 2014GXNSFAA118003), and Guangxi Key Laboratory of Automatic Detecting Technology and Instruments (no. YQ15112).
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Copyright © 2015 Xiangli Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Surface_gravity Knowpia
The surface gravity, g, of an astronomical object is the gravitational acceleration experienced at its surface at the equator, including the effects of rotation. The surface gravity may be thought of as the acceleration due to gravity experienced by a hypothetical test particle which is very close to the object's surface and which, in order not to disturb the system, has negligible mass. For objects where the surface is deep in the atmosphere and the radius not known, the surface gravity is given at the 1 bar pressure level in the atmosphere.
Surface gravity is measured in units of acceleration, which, in the SI system, are meters per second squared. It may also be expressed as a multiple of the Earth's standard surface gravity, g = 9.80665 m/s².[1] In astrophysics, the surface gravity may be expressed as log g, which is obtained by first expressing the gravity in cgs units, where the unit of acceleration is centimeters per second squared, and then taking the base-10 logarithm.[2] Therefore, the surface gravity of Earth could be expressed in cgs units as 980.665 cm/s², with a base-10 logarithm (log g) of 2.992.
The surface gravity of a white dwarf is very high, and of a neutron star even higher. A white dwarf's surface gravity is around 100,000g (9.84 ×105 m/s²) whilst the neutron star's compactness gives it a surface gravity of up to 7×1012 m/s² with typical values of order 1012 m/s² (that is more than 1011 times that of Earth). One measure of such immense gravity is that neutron stars have an escape velocity of around 100,000 km/s, about a third of the speed of light. For black holes, the surface gravity must be calculated relativistically.
Relationship of surface gravity to mass and radiusEdit
Surface gravity of various
Solar System bodies[3]
(1 g = 9.80665 m/s2, the surface gravitational acceleration on Earth)
Moon 0.165 7 g (average)
Mars 0.379 g (midlatitudes)
Phobos 0.000 581 g
Deimos 0.000 306 g
Ceres 0.029 g
Jupiter 2.528 g (midlatitudes)
Io 0.183 g
Europa 0.134 g
Ganymede 0.146 g
Callisto 0.126 g
Saturn 1.065 g (midlatitudes)
Titan 0.138 g
Enceladus 0.012 g
Uranus 0.886 g (equator)
Neptune 1.137 g (midlatitudes)
Triton 0.08 g
Pluto 0.063 g
Eris 0.084 g
67P-CG 0.000 017 g
In the Newtonian theory of gravity, the gravitational force exerted by an object is proportional to its mass: an object with twice the mass produces twice as much force. Newtonian gravity also follows an inverse square law, so that moving an object twice as far away divides its gravitational force by four, and moving it ten times as far away divides it by 100. This is similar to the intensity of light, which also follows an inverse square law: with relation to distance, light becomes less visible. Generally speaking, this can be understood as geometric dilution corresponding to point-source radiation into three-dimensional space.
A large object, such as a planet or star, will usually be approximately round, approaching hydrostatic equilibrium (where all points on the surface have the same amount of gravitational potential energy). On a small scale, higher parts of the terrain are eroded, with eroded material deposited in lower parts of the terrain. On a large scale, the planet or star itself deforms until equilibrium is reached.[4] For most celestial objects, the result is that the planet or star in question can be treated as a near-perfect sphere when the rotation rate is low. However, for young, massive stars, the equatorial azimuthal velocity can be quite high—up to 200 km/s or more—causing a significant amount of equatorial bulge. Examples of such rapidly rotating stars include Achernar, Altair, Regulus A and Vega.
The fact that many large celestial objects are approximately spheres makes it easier to calculate their surface gravity. According to the shell theorem, the gravitational force outside a spherically symmetric body is the same as if its entire mass were concentrated in the center, as was established by Sir Isaac Newton.[5] Therefore, the surface gravity of a planet or star with a given mass will be approximately inversely proportional to the square of its radius, and the surface gravity of a planet or star with a given average density will be approximately proportional to its radius. For example, the recently discovered planet, Gliese 581 c, has at least 5 times the mass of Earth, but is unlikely to have 5 times its surface gravity. If its mass is no more than 5 times that of the Earth, as is expected,[6] and if it is a rocky planet with a large iron core, it should have a radius approximately 50% larger than that of Earth.[7][8] Gravity on such a planet's surface would be approximately 2.2 times as strong as on Earth. If it is an icy or watery planet, its radius might be as large as twice the Earth's, in which case its surface gravity might be no more than 1.25 times as strong as the Earth's.[8]
These proportionalities may be expressed by the formula:
{\displaystyle g\propto {\frac {m}{r^{2}}}}
where g is the surface gravity of an object, expressed as a multiple of the Earth's, m is its mass, expressed as a multiple of the Earth's mass (5.976·1024 kg) and r its radius, expressed as a multiple of the Earth's (mean) radius (6,371 km).[9] For instance, Mars has a mass of 6.4185·1023 kg = 0.107 Earth masses and a mean radius of 3,390 km = 0.532 Earth radii.[10] The surface gravity of Mars is therefore approximately
{\displaystyle {\frac {0.107}{0.532^{2}}}=0.38}
times that of Earth. Without using the Earth as a reference body, the surface gravity may also be calculated directly from Newton's law of universal gravitation, which gives the formula
{\displaystyle g={\frac {GM}{r^{2}}}}
where M is the mass of the object, r is its radius, and G is the gravitational constant. If we let ρ = M/V denote the mean density of the object, we can also write this as
{\displaystyle g={\frac {4\pi }{3}}G\rho r}
so that, for fixed mean density, the surface gravity g is proportional to the radius r.
Since gravity is inversely proportional to the square of the distance, a space station 400 km above the Earth feels almost the same gravitational force as we do on the Earth's surface. A space station does not plummet to the ground because it is in a free-fall orbit.
Gas giantsEdit
For gas giant planets such as Jupiter, Saturn, Uranus, and Neptune, the surface gravity is given at the 1 bar pressure level in the atmosphere.[11]
Non-spherically symmetric objectsEdit
Most real astronomical objects are not absolutely spherically symmetric. One reason for this is that they are often rotating, which means that they are affected by the combined effects of gravitational force and centrifugal force. This causes stars and planets to be oblate, which means that their surface gravity is smaller at the equator than at the poles. This effect was exploited by Hal Clement in his SF novel Mission of Gravity, dealing with a massive, fast-spinning planet where gravity was much higher at the poles than at the equator.
To the extent that an object's internal distribution of mass differs from a symmetric model, we may use the measured surface gravity to deduce things about the object's internal structure. This fact has been put to practical use since 1915–1916, when Roland Eötvös's torsion balance was used to prospect for oil near the city of Egbell (now Gbely, Slovakia.)[12], p. 1663;[13], p. 223. In 1924, the torsion balance was used to locate the Nash Dome oil fields in Texas.[13], p. 223.
It is sometimes useful to calculate the surface gravity of simple hypothetical objects which are not found in nature. The surface gravity of infinite planes, tubes, lines, hollow shells, cones, and even more unrealistic structures may be used to provide insights into the behavior of real structures.
In relativity, the Newtonian concept of acceleration turns out not to be clear cut. For a black hole, which must be treated relativistically, one cannot define a surface gravity as the acceleration experienced by a test body at the object's surface because there is no surface. This is because the acceleration of a test body at the event horizon of a black hole turns out to be infinite in relativity. Because of this, a renormalized value is used that corresponds to the Newtonian value in the non-relativistic limit. The value used is generally the local proper acceleration (which diverges at the event horizon) multiplied by the gravitational time dilation factor (which goes to zero at the event horizon). For the Schwarzschild case, this value is mathematically well behaved for all non-zero values of r and M.
The surface gravity
{\displaystyle \kappa }
of a static Killing horizon is the acceleration, as exerted at infinity, needed to keep an object at the horizon. Mathematically, if
{\displaystyle k^{a}}
is a suitably normalized Killing vector, then the surface gravity is defined by
{\displaystyle k^{a}\,\nabla _{a}k^{b}=\kappa k^{b},}
where the equation is evaluated at the horizon. For a static and asymptotically flat spacetime, the normalization should be chosen so that
{\displaystyle k^{a}k_{a}\rightarrow -1}
{\displaystyle r\rightarrow \infty }
{\displaystyle \kappa \geq 0}
. For the Schwarzschild solution, we take
{\displaystyle k^{a}}
to be the time translation Killing vector
{\displaystyle k^{a}\partial _{a}={\frac {\partial }{\partial t}}}
, and more generally for the Kerr–Newman solution we take
{\displaystyle k^{a}\partial _{a}={\frac {\partial }{\partial t}}+\Omega {\frac {\partial }{\partial \varphi }}}
, the linear combination of the time translation and axisymmetry Killing vectors which is null at the horizon, where
{\displaystyle \Omega }
Schwarzschild solutionEdit
{\displaystyle k^{a}}
is a Killing vector
{\displaystyle k^{a}\,\nabla _{a}k^{b}=\kappa k^{b}}
{\displaystyle -k^{a}\,\nabla ^{b}k_{a}=\kappa k^{b}}
{\displaystyle (t,r,\theta ,\varphi )}
{\displaystyle k^{a}=(1,0,0,0)}
. Performing a coordinate change to the advanced Eddington–Finklestein coordinates
{\displaystyle v=t+r+2M\ln |r-2M|}
causes the metric to take the form
{\displaystyle ds^{2}=-\left(1-{\frac {2M}{r}}\right)\,dv^{2}+(\,dv\,dr+\,dr\,dv)+r^{2}\left(d\theta ^{2}+\sin ^{2}\theta \,d\varphi ^{2}\right).}
Under a general change of coordinates the Killing vector transforms as
{\displaystyle k^{v}=A_{t}^{v}k^{t}}
giving the vectors
{\displaystyle k^{a'}=\delta _{v}^{a'}=(1,0,0,0)}
{\displaystyle k_{a'}=g_{a'v}=\left(-1+{\frac {2M}{r}},1,0,0\right).}
Considering the b =
{\displaystyle v}
entry for
{\displaystyle k^{a}\,\nabla _{a}k^{b}=\kappa k^{b}}
gives the differential equation
{\displaystyle -{\frac {1}{2}}{\frac {\partial }{\partial r}}\left(-1+{\frac {2M}{r}}\right)=\kappa .}
Therefore, the surface gravity for the Schwarzschild solution with mass
{\displaystyle M}
{\displaystyle \kappa ={\frac {1}{4M}}(={\frac {c^{4}}{4GM}}}
in SI units).[14]
Kerr solutionEdit
The surface gravity for the uncharged, rotating black hole is, simply
{\displaystyle \kappa =g-k,}
{\displaystyle g={\frac {1}{4M}}}
is the Schwarzschild surface gravity, and
{\displaystyle k:=M\Omega _{+}^{2}}
is the spring constant of the rotating black hole.
{\displaystyle \Omega _{+}}
is the angular velocity at the event horizon. This expression gives a simple Hawking temperature of
{\displaystyle 2\pi T=g-k}
Kerr–Newman solutionEdit
The surface gravity for the Kerr–Newman solution is
{\displaystyle \kappa ={\frac {r_{+}-r_{-}}{2(r_{+}^{2}+a^{2})}}={\frac {\sqrt {M^{2}-Q^{2}-J^{2}/M^{2}}}{2M^{2}-Q^{2}+2M{\sqrt {M^{2}-Q^{2}-J^{2}/M^{2}}}}},}
{\displaystyle Q}
{\displaystyle J}
is the angular momentum, we define
{\displaystyle r_{\pm }:=M\pm {\sqrt {M^{2}-Q^{2}-J^{2}/M^{2}}}}
to be the locations of the two horizons and
{\displaystyle a:=J/M}
Dynamical black holesEdit
Surface gravity for stationary black holes is well defined. This is because all stationary black holes have a horizon that is Killing.[16] Recently there has been a shift towards defining the surface gravity of dynamical black holes whose spacetime does not admit a Killing vector (field).[17] Several definitions have been proposed over the years by various authors. As of current, there is no consensus or agreement of which definition, if any, is correct.[18]
^ Taylor, Barry N., ed. (2001). The International System of Units (SI) (PDF). NIST Special Publication 330. United States Department of Commerce: National Institute of Standards and Technology. p. 29. Retrieved 2012-03-08.
^ Smalley, B. (13 July 2006). "The Determination of Teff and log g for B to G stars". Keele University. Retrieved 31 May 2007.
^ Isaac Asimov (1978). The Collapsing Universe. Corgi. p. 44. ISBN 978-0-552-10884-3.
^ "Why is the Earth round?". Ask A Scientist. Argonne National Laboratory, Division of Educational Programs. Archived from the original on 21 September 2008.
^ Book I, §XII, pp. 218–226, Newton's Principia: The Mathematical Principles of Natural Philosophy, Sir Isaac Newton, tr. Andrew Motte, ed. N. W. Chittenden. New York: Daniel Adee, 1848. First American edition.
^ Astronomers Find First Earth-like Planet in Habitable Zone Archived 2009-06-17 at the Wayback Machine, ESO 22/07, press release from the European Southern Observatory, April 25, 2007
^ a b Valencia, Diana; Sasselov, Dimitar D; O'Connell, Richard J (2007). "Detailed Models of super-Earths: How well can we infer bulk properties?". The Astrophysical Journal. 665 (2): 1413–1420. arXiv:0704.3454. Bibcode:2007ApJ...665.1413V. doi:10.1086/519554. S2CID 15605519.
^ 2.7.4 Physical properties of the Earth, web page, accessed on line May 27, 2007.
^ Mars Fact Sheet, web page at NASA NSSDC, accessed May 27, 2007.
^ "Planetary Fact Sheet Notes".
^ Li, Xiong; Götze, Hans-Jürgen (2001). "Ellipsoid, geoid, gravity, geodesy, and geophysics". Geophysics. 66 (6): 1660–1668. Bibcode:2001Geop...66.1660L. doi:10.1190/1.1487109.
^ a b Prediction by Eötvös' torsion balance data in Hungary Archived 2007-11-28 at the Wayback Machine, Gyula Tóth, Periodica Polytechnica Ser. Civ. Eng. 46, #2 (2002), pp. 221–229.
^ Raine, Derek J.; Thomas, Edwin George (2010). Black Holes: An Introduction (illustrated ed.). Imperial College Press. p. 44. ISBN 978-1-84816-382-9. Extract of page 44
^ Good, Michael; Yen Chin Ong (February 2015). "Are Black Holes Springlike?". Physical Review D. 91 (4): 044031. arXiv:1412.5432. Bibcode:2015PhRvD..91d4031G. doi:10.1103/PhysRevD.91.044031. S2CID 117749566.
^ Wald, Robert (1984). General Relativity. University Of Chicago Press. ISBN 978-0-226-87033-5.
^ Nielsen, Alex; Yoon (2008). "Dynamical Surface Gravity". Classical and Quantum Gravity. 25 (8): 085010. arXiv:0711.1445. Bibcode:2008CQGra..25h5010N. doi:10.1088/0264-9381/25/8/085010. S2CID 15438397.
^ Pielahn, Mathias; G. Kunstatter; A. B. Nielsen (November 2011). "Dynamical surface gravity in spherically symmetric black hole formation". Physical Review D. 84 (10): 104008(11). arXiv:1103.0750. Bibcode:2011PhRvD..84j4008P. doi:10.1103/PhysRevD.84.104008. S2CID 119015033.
Newtonian surface gravity
Exploratorium – Your Weight on Other Worlds
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Celestial_mechanics Knowpia
Modern analytic celestial mechanics started with Isaac Newton's Principia of 1687. The name "celestial mechanics" is more recent than that. Newton wrote that the field should be called "rational mechanics." The term "dynamics" came in a little later with Gottfried Leibniz, and over a century after Newton, Pierre-Simon Laplace introduced the term "celestial mechanics." Prior to Kepler there was little connection between exact, quantitative prediction of planetary positions, using geometrical or arithmetical techniques, and contemporary discussions of the physical causes of the planets' motion.
Johannes KeplerEdit
Johannes Kepler (1571–1630) was the first to closely integrate the predictive geometrical astronomy, which had been dominant from Ptolemy in the 2nd century to Copernicus, with physical concepts to produce a New Astronomy, Based upon Causes, or Celestial Physics in 1609. His work led to the modern laws of planetary orbits, which he developed using his physical principles and the planetary observations made by Tycho Brahe. Kepler's model greatly improved the accuracy of predictions of planetary motion, years before Isaac Newton developed his law of gravitation in 1686.
Isaac Newton (25 December 1642–31 March 1727) is credited with introducing the idea that the motion of objects in the heavens, such as planets, the Sun, and the Moon, and the motion of objects on the ground, like cannon balls and falling apples, could be described by the same set of physical laws. In this sense he unified celestial and terrestrial dynamics. Using Newton's law of universal gravitation, proving Kepler's Laws for the case of a circular orbit is simple. Elliptical orbits involve more complex calculations, which Newton included in his Principia.
Joseph-Louis LagrangeEdit
After Newton, Lagrange (25 January 1736–10 April 1813) attempted to solve the three-body problem, analyzed the stability of planetary orbits, and discovered the existence of the Lagrangian points. Lagrange also reformulated the principles of classical mechanics, emphasizing energy more than force and developing a method to use a single polar coordinate equation to describe any orbit, even those that are parabolic and hyperbolic. This is useful for calculating the behaviour of planets and comets and such. More recently, it has also become useful to calculate spacecraft trajectories.
Simon NewcombEdit
Simon Newcomb (12 March 1835–11 July 1909) was a Canadian-American astronomer who revised Peter Andreas Hansen's table of lunar positions. In 1877, assisted by George William Hill, he recalculated all the major astronomical constants. After 1884, he conceived with A. M. W. Downing a plan to resolve much international confusion on the subject. By the time he attended a standardisation conference in Paris, France, in May 1886, the international consensus was that all ephemerides should be based on Newcomb's calculations. A further conference as late as 1950 confirmed Newcomb's constants as the international standard.
Albert Einstein (14 March 1879–18 April 1955) explained the anomalous precession of Mercury's perihelion in his 1916 paper The Foundation of the General Theory of Relativity. This led astronomers to recognize that Newtonian mechanics did not provide the highest accuracy. Binary pulsars have been observed, the first in 1974, whose orbits not only require the use of General Relativity for their explanation, but whose evolution proves the existence of gravitational radiation, a discovery that led to the 1993 Nobel Physics Prize.
Examples of problemsEdit
Celestial motion, without additional forces such as drag forces or the thrust of a rocket, is governed by the reciprocal gravitational acceleration between masses. A generalization is the n-body problem,[1] where a number n of masses are mutually interacting via the gravitational force. Although analytically not integrable in the general case,[2] the integration can be well approximated numerically.
4-body problem: spaceflight to Mars (for parts of the flight the influence of one or two bodies is very small, so that there we have a 2- or 3-body problem; see also the patched conic approximation)
3-body problem:
Spaceflight to, and stay at a Lagrangian point
{\displaystyle n=2}
case (two-body problem) the configuration is much simpler than for
{\displaystyle n>2}
. In this case, the system is fully integrable and exact solutions can be found.[3]
A binary star, e.g., Alpha Centauri (approx. the same mass)
A binary asteroid, e.g., 90 Antiope (approx. the same mass)
A further simplification is based on the "standard assumptions in astrodynamics", which include that one body, the orbiting body, is much smaller than the other, the central body. This is also often approximately valid.
The Solar System orbiting the center of the Milky Way
A planet orbiting the Sun
A spacecraft orbiting Earth, a moon, or a planet (in the latter cases the approximation only applies after arrival at that orbit)
Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly. (It is closely related to methods used in numerical analysis, which are ancient.) The earliest use of modern perturbation theory was to deal with the otherwise unsolvable mathematical problems of celestial mechanics: Newton's solution for the orbit of the Moon, which moves noticeably differently from a simple Keplerian ellipse because of the competing gravitation of the Earth and the Sun.
Perturbation methods start with a simplified form of the original problem, which is carefully chosen to be exactly solvable. In celestial mechanics, this is usually a Keplerian ellipse, which is correct when there are only two gravitating bodies (say, the Earth and the Moon), or a circular orbit, which is only correct in special cases of two-body motion, but is often close enough for practical use.
The solved, but simplified problem is then "perturbed" to make its time-rate-of-change equations for the object's position closer to the values from the real problem, such as including the gravitational attraction of a third, more distant body (the Sun). The slight changes that result from the terms in the equations – which themselves may have been simplified yet again – are used as corrections to the original solution. Because simplifications are made at every step, the corrections are never perfect, but even one cycle of corrections often provides a remarkably better approximate solution to the real problem.
There is no requirement to stop at only one cycle of corrections. A partially corrected solution can be re-used as the new starting point for yet another cycle of perturbations and corrections. In principle, for most problems the recycling and refining of prior solutions to obtain a new generation of better solutions could continue indefinitely, to any desired finite degree of accuracy.
The common difficulty with the method is that the corrections usually progressively make the new solutions very much more complicated, so each cycle is much more difficult to manage than the previous cycle of corrections. Newton is reported to have said, regarding the problem of the Moon's orbit "It causeth my head to ache."[4]
This general procedure – starting with a simplified problem and gradually adding corrections that make the starting point of the corrected problem closer to the real situation – is a widely used mathematical tool in advanced sciences and engineering. It is the natural extension of the "guess, check, and fix" method used anciently with numbers.
Astrometry is a part of astronomy that deals with measuring the positions of stars and other celestial bodies, their distances and movements.
Celestial navigation is a position fixing technique that was the first system devised to help sailors locate themselves on a featureless ocean.
Developmental Ephemeris or the Jet Propulsion Laboratory Developmental Ephemeris (JPL DE) is a widely used model of the solar system, which combines celestial mechanics with numerical analysis and astronomical and spacecraft data.
Dynamics of the celestial spheres concerns pre-Newtonian explanations of the causes of the motions of the stars and planets.
Numerical analysis is a branch of mathematics, pioneered by celestial mechanicians, for calculating approximate numerical answers (such as the position of a planet in the sky) which are too difficult to solve down to a general, exact formula.
Creating a numerical model of the solar system was the original goal of celestial mechanics, and has only been imperfectly achieved. It continues to motivate research.
An orbit is the path that an object makes, around another object, whilst under the influence of a source of centripetal force, such as gravity.
Orbital elements are the parameters needed to specify a Newtonian two-body orbit uniquely.
Osculating orbit is the temporary Keplerian orbit about a central body that an object would continue on, if other perturbations were not present.
Retrograde motion is orbital motion in a system, such as a planet and its satellites, that is contrary to the direction of rotation of the central body, or more generally contrary in direction to the net angular momentum of the entire system.
Apparent retrograde motion is the periodic, apparently backwards motion of planetary bodies when viewed from the Earth (an accelerated reference frame).
Satellite is an object that orbits another object (known as its primary). The term is often used to describe an artificial satellite (as opposed to natural satellites, or moons). The common noun ‘moon’ (not capitalized) is used to mean any natural satellite of the other planets.
Tidal force is the combination of out-of-balance forces and accelerations of (mostly) solid bodies that raises tides in bodies of liquid (oceans), atmospheres, and strains planets' and satellites' crusts.
Two solutions, called VSOP82 and VSOP87 are versions one mathematical theory for the orbits and positions of the major planets, which seeks to provide accurate positions over an extended period of time.
^ Trenti, Michele; Hut, Piet (2008-05-20). "N-body simulations (gravitational)". Scholarpedia. 3 (5): 3930. Bibcode:2008SchpJ...3.3930T. doi:10.4249/scholarpedia.3930. ISSN 1941-6016.
^ Combot, Thierry (2015-09-01). "Integrability and non integrability of some n body problems". arXiv:1509.08233 [math.DS].
^ Weisstein, Eric W. "Two-Body Problem -- from Eric Weisstein's World of Physics". scienceworld.wolfram.com. Retrieved 2020-08-28.
^ Cropper, William H. (2004), Great Physicists: The life and times of leading physicists from Galileo to Hawking, Oxford University Press, p. 34, ISBN 978-0-19-517324-6 .
John E. Prussing, Bruce A. Conway, Orbital Mechanics, 1993, Oxford Univ. Press
William M. Smart, Celestial Mechanics, 1961, John Wiley.
Doggett, LeRoy E. (1997), "Celestial Mechanics", in Lankford, John (ed.), History of Astronomy: An Encyclopedia, New York: Taylor & Francis, pp. 131–140, ISBN 9780815303220
Encyclopedia:Celestial mechanics Scholarpedia Expert articles
Calvert, James B. (2003-03-28), Celestial Mechanics, University of Denver, archived from the original on 2006-09-07, retrieved 2006-08-21
Newtonian Dynamics Undergraduate level course by Richard Fitzpatrick. This includes Lagrangian and Hamiltonian Dynamics and applications to celestial mechanics, gravitational potential theory, the 3-body problem and Lunar motion (an example of the 3-body problem with the Sun, Moon, and the Earth).
Marshall Hampton's research page: Central configurations in the n-body problem
Celestial Mechanics is a Planetarium Artwork created by D. S. Hessels and G. Dunne
Professor Tatum's course notes at the University of Victoria
Italian Celestial Mechanics and Astrodynamics Association
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Humberto thinks the second derivative has something to do with how well a tangent line will approximate a curve. For example, he is comparing two functions,
f
g
, and he notices that at a certain
x = a
f(a) = g(a)
f^\prime(a) = g^\prime (a)
. However, the second derivatives are different,
f^{\prime\prime}(a) = 0.85
g^{\prime\prime}(a) = 5
. Which tangent line gives a better approximation of its actual function near
x = a
? Use a graph to justify your answer. .
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chebpade - Maple Help
Home : Support : Online Help : Mathematics : Numerical Computations : Approximations : numapprox Package : chebpade
compute a Chebyshev-Pade approximation
chebpade(f, x=a..b, [m, n])
chebpade(f, x, [m, n])
chebpade(f, a..b, [m, n])
procedure or expression representing the function to be approximated
the variable appearing in f, if f is an expression
The function chebpade computes a Chebyshev-Pade approximation of degree
m,n
for the function f.
Specifically, f is expanded in a Chebyshev series on the interval
a..b
a..b
is not specified then the interval
-1..1
is understood), and then the Chebyshev-Pade rational approximation is computed.
n=0
or if the third argument is simply an integer m then the Chebyshev series of degree m is computed.
m,n
Chebyshev-Pade approximation is defined to be the rational function
\frac{p\left(x\right)}{q\left(x\right)}
\mathrm{deg}\left(p\left(x\right)\right)\le m
\mathrm{deg}\left(q\left(x\right)\right)\le n
such that the Chebyshev series expansion of
\frac{p\left(x\right)}{q\left(x\right)}
has maximal initial agreement with the Chebyshev series expansion of f. In normal cases, the series expansion agrees through the term of degree
m+n
a..b
then the first argument is understood to be a Maple operator, and the result will be returned as an operator. If the second argument is an equation
x=a..b
, or a name x, then the first argument is understood to be an expression in the variable x, and the result will be returned as an expression. In all cases, the numerator and denominator will be expressed in terms of the Chebyshev polynomials
T\left(n,x\right)
. See orthopoly[T].
The method used is based on transforming the Chebyshev series to a power series with the same coefficients, computing a Pade approximation for the power series, and then converting back to the appropriate Chebyshev-Pade approximation.
Note that for the purpose of evaluating a rational function efficiently (i.e. minimizing the number of arithmetic operations), the rational function should be converted to a continued-fraction form. See numapprox[confracform].
Various levels of user information will be displayed during the computation if infolevel[chebpade] is assigned values between 1 and 3.
The command with(numapprox,chebpade) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{numapprox}\right):
\mathrm{chebpade}\left(\mathrm{exp}\left(x\right),x=0..1,5\right)
\textcolor[rgb]{0,0,1}{1.75338765437709}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{0.850391653780811}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{0.105208693630937}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{0.00872210473331555}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{0.000543436831150254}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{0.0000271154349130735}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)
\mathrm{chebpade}\left(\frac{\mathrm{sin}\left(x\right)}{x},x=0..2,[2,2]\right)
\frac{\textcolor[rgb]{0,0,1}{0.771073733750623}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{0.221091073962959}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{0.0421244668861024}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)}{\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{0.0836360586596837}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{0.0336007994536882}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)}
\mathrm{chebpade}\left(\mathrm{sin}+\mathrm{cos},-1..1,[1,2]\right)
\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{↦}\frac{\textcolor[rgb]{0,0,1}{0.865073720371733357}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1.11718143936440173}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\right)}{\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{0.266549197507840174}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{0.151597635961381677}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\right)}
r≔\mathrm{confracform}\left(\mathrm{subs}\left(T=\mathrm{orthopoly}[T],\right)\right)
\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{↦}\frac{\textcolor[rgb]{0,0,1}{3.68469281291759376}}{\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{0.104797831264514252}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{2.71705574090406143}}{\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{0.774335922429842616}}}
Check agreement of the Chebyshev series to 7 digits of accuracy.
\mathrm{Digits}≔7
\textcolor[rgb]{0,0,1}{\mathrm{Digits}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{7}
\mathrm{chebpade}\left(r,-1..1,4\right)
\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{↦}\textcolor[rgb]{0,0,1}{0.7651975}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{0.8801012}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{0.2298070}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{0.03912671}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{0.02274805}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\right)
\mathrm{chebpade}\left(\mathrm{sin}+\mathrm{cos},-1..1,4\right)
\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{↦}\textcolor[rgb]{0,0,1}{0.7651975}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{0.8801012}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{0.2298070}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{0.03912671}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{0.004953278}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\right)
Geddes, K.O. "Block Structure in the Chebyshev-Pade Table." SIAM J. Numer. Anal., Vol. 18(5). (Oct. 1981): 844-861.
numapprox[minimax]
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LaTeX for WordPress - Kan Ouivirach
If it works, it will show square root of 2 below:
\sqrt 2
As I'd like to write some math equations on my blog, this LaTeX for WordPress plug-in can help me to do that. And I choose it because it allows me to write in LaTeX format!
Author zkanPosted on September 17, 2009 September 17, 2009 Categories ToolsTags LaTeX, Plugin
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Common collector - 2D PCM Schematics - 3D Model
Common collector (5851 views - Electronics & PCB)
In electronics, a common collector amplifier (also known as an emitter follower) is one of three basic single-stage bipolar junction transistor (BJT) amplifier topologies, typically used as a voltage buffer. In this circuit the base terminal of the transistor serves as the input, the emitter is the output, and the collector is common to both (for example, it may be tied to ground reference or a power supply rail), hence its name. The analogous field-effect transistor circuit is the common drain amplifier and the analogous tube circuit is the cathode follower.
In electronics, a common collector amplifier (also known as an emitter follower) is one of three basic single-stage bipolar junction transistor (BJT) amplifier topologies, typically used as a voltage buffer.
In this circuit the base terminal of the transistor serves as the input, the emitter is the output, and the collector is common to both (for example, it may be tied to ground reference or a power supply rail), hence its name. The analogous field-effect transistor circuit is the common drain amplifier and the analogous tube circuit is the cathode follower.
1 Basic circuit
The circuit can be explained by viewing the transistor as being under the control of negative feedback. From this viewpoint, a common collector stage (Fig. 1) is an amplifier with full series negative feedback. In this configuration (Fig. 2 with β = 1), the entire output voltage VOUT is placed contrary and in series with the input voltage VIN. Thus the two voltages are subtracted according to KVL (the subtractor from the function block diagram is implemented just by the input loop) and their extraordinary difference Vdiff = VIN - VOUT is applied to the base-emitter junction. The transistor continuously monitors Vdiff and adjusts its emitter voltage almost equal (less VBEO) to the input voltage by passing the according collector current through the emitter resistor RE. As a result, the output voltage follows the input voltage variations from VBEO up to V+; hence the name, emitter follower.
Intuitively, this behavior can be also understood by realizing that the base-emitter voltage in the bipolar transistor is very insensitive to bias changes, so any change in base voltage is transmitted (to good approximation) directly to the emitter. It depends slightly on various disturbances (transistor tolerances, temperature variations, load resistance, collector resistor if it is added, etc.) since the transistor reacts to these disturbances and restores the equilibrium. It never saturates even if the input voltage reaches the positive rail.
The common collector circuit can be shown mathematically to have a voltage gain of almost unity:
{\displaystyle {A_{\mathrm {v} }}={v_{\mathrm {out} } \over v_{\mathrm {in} }}\approx 1}
A small voltage change on the input terminal will be replicated at the output (depending slightly on the transistor's gain and the value of the load resistance; see gain formula below). This circuit is useful because it has a large input impedance, so it will not load down the previous circuit:
{\displaystyle r_{\mathrm {in} }\approx \beta _{0}R_{\mathrm {E} }}
and a small output impedance, so it can drive low-resistance loads:
{\displaystyle r_{\mathrm {out} }\approx {R_{\mathrm {E} }}\|{R_{\mathrm {source} } \over \beta _{0}}}
Typically, the emitter resistor is significantly larger and can be removed from the equation:
{\displaystyle r_{\mathrm {out} }\approx {R_{\mathrm {source} } \over \beta _{0}}}
The low output impedance allows a source with a large output impedance to drive a small load impedance; it functions as a voltage buffer. In other words, the circuit has current gain (which depends largely on the hFE of the transistor) instead of voltage gain, because of its characteristics it is preferred in many electronic devices. A small change to the input current results in much larger change in the output current supplied to the output load.
One aspect of buffer action is transformation of impedances. For example, the Thévenin resistance of a combination of a voltage follower driven by a voltage source with high Thévenin resistance is reduced to only the output resistance of the voltage follower (a small resistance). That resistance reduction makes the combination a more ideal voltage source. Conversely, a voltage follower inserted between a small load resistance and a driving stage presents a large load to the driving stage—an advantage in coupling a voltage signal to a small load.
This configuration is commonly used in the output stages of class-B and class-AB amplifiers. The base circuit is modified to operate the transistor in class-B or AB mode. In class-A mode, sometimes an active current source is used instead of RE (Fig. 4) to improve linearity and/or efficiency.[1]
At low frequencies and using a simplified hybrid-pi model, the following small-signal characteristics can be derived. (Parameter
{\displaystyle \beta =g_{m}r_{\pi }}
and the parallel lines indicate components in parallel.)
{\displaystyle {A_{\mathrm {i} }}={i_{\mathrm {out} } \over i_{\mathrm {in} }}}
{\displaystyle \beta _{0}+1\ }
{\displaystyle \approx \beta _{0}}
{\displaystyle \beta _{0}\gg 1}
{\displaystyle {A_{\mathrm {v} }}={v_{\mathrm {out} } \over v_{\mathrm {in} }}}
{\displaystyle {g_{m}R_{\mathrm {E} } \over g_{m}R_{\mathrm {E} }+1}}
{\displaystyle \approx 1}
{\displaystyle g_{m}R_{\mathrm {E} }\gg 1}
{\displaystyle r_{\mathrm {in} }={\frac {v_{\mathrm {in} }}{i_{\mathrm {in} }}}}
{\displaystyle r_{\pi }+(\beta _{0}+1)R_{\mathrm {E} }\ }
{\displaystyle \approx \beta _{0}R_{\mathrm {E} }}
{\displaystyle (g_{m}R_{\mathrm {E} }\gg 1)\wedge (\beta _{0}\gg 1)}
{\displaystyle r_{\mathrm {out} }={\frac {v_{\mathrm {out} }}{i_{\mathrm {out} }}}}
{\displaystyle R_{\mathrm {E} }\parallel \left({r_{\pi }+R_{\mathrm {source} } \over \beta _{0}+1}\right)}
{\displaystyle \approx {1 \over g_{m}}+{R_{\mathrm {source} } \over \beta _{0}}}
{\displaystyle (\beta _{0}\gg 1)\wedge (r_{\mathrm {in} }\gg R_{\mathrm {source} })}
{\displaystyle R_{\mathrm {source} }\ }
is the Thévenin equivalent source resistance.
Figure 5 shows a low-frequency hybrid-pi model for the circuit of Figure 3. Using Ohm's law various currents have been determined and these results are shown on the diagram. Applying Kirchhoff's current law at the emitter one finds:
{\displaystyle (\beta +1){\frac {v_{\mathrm {in} }-v_{\mathrm {out} }}{R_{\mathrm {S} }+r_{\pi }}}=v_{\mathrm {out} }\left({\frac {1}{R_{\mathrm {L} }}}+{\frac {1}{r_{\mathrm {O} }}}\right)\ .}
Define the following resistance values:
{\displaystyle {\frac {1}{R_{\mathrm {E} }}}={\frac {1}{R_{\mathrm {L} }}}+{\frac {1}{r_{\mathrm {O} }}}}
{\displaystyle R={\frac {R_{\mathrm {S} }+r_{\pi }}{\beta +1}}\ .}
Then collecting terms the voltage gain is found as:
{\displaystyle A_{\mathrm {v} }={\frac {v_{\mathrm {out} }}{v_{\mathrm {in} }}}={\frac {1}{1+{\frac {R}{R_{\mathrm {E} }}}}}\ .}
From this result the gain approaches unity (as expected for a buffer amplifier) if the resistance ratio in the denominator is small. This ratio decreases with larger values of current gain β and with larger values of
{\displaystyle R_{\mathrm {E} }}
. The input resistance is found as:
{\displaystyle R_{\mathrm {in} }={\frac {v_{\mathrm {in} }}{i_{\mathrm {b} }}}={\frac {R_{\mathrm {S} }+r_{\pi }}{1-A_{\mathrm {v} }}}\ }
{\displaystyle =\left(R_{\mathrm {S} }+r_{\pi }\right)\left(1+{\frac {R_{\mathrm {E} }}{R}}\right)\ }
{\displaystyle =R_{\mathrm {S} }+r_{\pi }+(\beta +1)R_{\mathrm {E} }\ .}
The transistor output resistance
{\displaystyle r_{\mathrm {O} }}
ordinarily is large compared to the load
{\displaystyle R_{\mathrm {L} }}
{\displaystyle R_{\mathrm {L} }}
{\displaystyle R_{\mathrm {E} }}
. From this result, the input resistance of the amplifier is much larger than the output load resistance
{\displaystyle R_{\mathrm {L} }}
for large current gain
{\displaystyle \beta }
. That is, placing the amplifier between the load and the source presents a larger (high-resistive) load to the source than direct coupling to
{\displaystyle R_{\mathrm {L} }}
, which results in less signal attenuation in the source impedance
{\displaystyle R_{\mathrm {S} }}
as a consequence of voltage division.
Figure 6 shows the small-signal circuit of Figure 5 with the input short-circuited and a test current placed at its output. The output resistance is found using this circuit as:
{\displaystyle R_{\mathrm {out} }={\frac {v_{\mathrm {x} }}{i_{\mathrm {x} }}}\ .}
Using Ohm's law, various currents have been found, as indicated on the diagram. Collecting the terms for the base current, the base current is found as:
{\displaystyle (\beta +1)i_{\mathrm {b} }=i_{\mathrm {x} }-{\frac {v_{\mathrm {x} }}{R_{\mathrm {E} }}}\ ,}
{\displaystyle R_{\mathrm {E} }}
is defined above. Using this value for base current, Ohm's law provides
{\displaystyle v_{\mathrm {x} }}
{\displaystyle v_{\mathrm {x} }=i_{\mathrm {b} }\left(R_{\mathrm {S} }+r_{\pi }\right)\ .}
Substituting for the base current, and collecting terms,
{\displaystyle R_{\mathrm {out} }={\frac {v_{\mathrm {x} }}{i_{\mathrm {x} }}}=R\parallel R_{\mathrm {E} }\ ,}
where || denotes a parallel connection and
{\displaystyle R}
is defined above. Because
{\displaystyle R}
generally is a small resistance when the current gain
{\displaystyle \beta }
{\displaystyle R}
dominates the output impedance which therefore also is small. A small output impedance means the series combination of the original voltage source and the voltage follower presents a Thévenin voltage source with a lower Thévenin resistance at its output node; that is, the combination of voltage source with voltage follower makes a more ideal voltage source than the original one.
AmplifierBipolar junction transistorElectronic componentElectronicsIndicator (distance amplifying instrument)Operational amplifierValve amplifierBuffer amplifierElectrical impedance
This article uses material from the Wikipedia article "Common collector", which is released under the Creative Commons Attribution-Share-Alike License 3.0. There is a list of all authors in Wikipedia
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Solving Systems of Linear Equations - Course Hero
College Algebra/Solving Systems of Equations/Solving Systems of Linear Equations
A system of linear equations can have zero, one, or infinitely many solutions. For two variables, they are represented graphically as parallel lines, intersecting lines, or lines that coincide. For three variables, they are represented as intersections of planes.
A system of equations is a set of equations that are used at the same time. A solution of a system of equations is an ordered pair (triple, etc.) that is a solution of every equation in the system.
\begin{cases}x+y=9 \\\phantom{x+}y=2x \end{cases}
(3,6)
is a solution of the system because when
x=3
y=6
are substituted into both equations, they are both true.
\begin{aligned} x+y&=9\\3+6&=9\\9&=9\end{aligned} \hspace{20pt} \begin{aligned}y&=2x\\6&=2(3)\\6&=6 \end{aligned}
(5,4)
is not a solution of the system because when
x=5
y=4
in both equations, the second equation is not true.
\begin{aligned} x+y&=9\\5+4&=9\\9&=9\end{aligned} \hspace{20pt} \begin{aligned}y&=2x\\4&\overset{?}{=}2(5)\\4&\neq10 \end{aligned}
A system of equations that has at least one solution is called a consistent system. A consistent system with only one solution is an independent system, and a consistent system with an infinite number of solutions is a dependent system. For some systems, there are no values that make each equation true. A system with no solution is called an inconsistent system.
A system of two equations in two variables can be represented by the graphs of the lines. The solutions of the system are points that are on both lines.
Solutions of Systems of Linear Equations in Two Variables
Consistent and independent; one solution
Consistent and dependent; infinite number of solutions
Inconsistent; no solution
Two lines intersecting at one point Two lines that coincide with each other Two parallel lines that do not intersect
A system of three equations in three variables can represented by the graphs of three planes in three dimensions. The solutions of the system are the points that are in all three planes.
Solutions of Systems of Linear Equations in Three Variables
Three planes intersecting at one point Three planes that intersect at one line Three planes that do not intersect at a common point
The solution of a system of linear equations can be estimated from a graph and then confirmed by substituting the values into the equations.
Graphing systems of linear equations is helpful in approximating the solution before using another method to determine the precise solution. If an ordered pair that is graphed makes two equations true, then the graphs of those equations intersect at that point.
\begin{cases}3x+6y=6\\2x-2y=4\end{cases}
Rewrite each equation in slope-intercept form,
y=mx+b
\begin{aligned}3x+6y&=6\\3x+6y-3x&=6-3x\\6y&=6-3x\\\frac{6y}{6}&=\frac{6}{6}-\frac{3x}{6}\\y&=1-\frac{x}{2}\\y&=-\frac{1}{2}x+1\\\phantom{0}\end{aligned}\hspace{20pt}\begin{aligned}2x-2y&=4\\2x-2y+2y&=4+2y\\2x&=4+2y\\2x-4&=4+2y-4\\2x-4&=2y\\\frac{2x}{2}-\frac{4}{2}&=\frac{2y}{2}\\x-2&=y\\y&=x-2\end{aligned}
y=-\frac{1}{2}x+1
y
-intercept is 1. The slope is
-\frac{1}{2}
. The second equation is:
y=x-2
y
-intercept is –2. The slope is 1.
A point of intersection is a solution of both equations. Locate the point where the graphs intersect.
The graphs intersect at a single point, so the system is consistent and independent. The graphs appear to intersect at
(2,0)
To check that a point read from a graph is in fact a solution, substitute the values for
x
y
into both equations and simplify. If they both result in true equations, then the ordered pair is a solution.
\begin{aligned}y&=-\frac{1}{2}x+1\\0&=-\frac{1}{2}(2)+1\\0&=-1+1\\0&=0\end{aligned}\hspace{20pt}\begin{aligned}y&=x-2\\(0)&=(2)-2\\0&=0\\\phantom{0}\\\phantom{0}\end{aligned}
(2,0)
Graph to Confirm a System Has No Solutions
Show that the system of equations has no solutions.
\begin{cases}y=8x-2\\y=8x+5\end{cases}
Compare the slopes and
y
\begin{cases}y=8x-2\\y=8x+5\end{cases}
The slope of each line is 8. The
y
-intercepts are different.
Graph each equation. Locate any points of intersection.
The graphs of the equations are parallel lines that never intersect. So, the system is inconsistent.
The system is inconsistent. So, there are no solutions.
Graph to Confirm a System Has Infinitely Many Solutions
Show that the system of equations has infinitely many solutions.
\begin{cases}\phantom{6}y=x-5\\6y=6x-30\end{cases}
Compare the equations.
\begin{cases}\phantom{6}y=x-5\\6y=6x-30\end{cases}
Each term in the second equation is 6 times a term in the first equation. The equation is a multiple of:
y=x-5
Both equations are equivalent to:
y=x-5
The graphs coincide. So, there are infinitely many points of intersection. The system is consistent and dependent.
The system is consistent and dependent. So, there are infinitely many solutions.
A system of linear equations can be solved by solving an equation for one variable and substituting the resulting expression in the other equation(s).
The substitution method is a method for solving a system of equations by solving for one variable and then substituting the result back into the other equation or equations.
To solve by substitution:
1. Solve one of the equations for one of the variables.
2. Substitute that expression for the variable in the other equation.
4. Write the solution as an ordered pair.
Solve a System of Linear Equations by Substitution
\begin{cases}3x-4y=13\\2x-6y=12\end{cases}
\begin{cases}3x-4y=13\\2x-6y=12\end{cases}
If possible, choose a variable and equation that will not result in the equation containing fractions. Solve the second equation for
x
\begin{aligned}2x-6y&=12\\2x-6y+6y&=12+6y\\2x&=12+6y\\\frac{2x}{2}&=\frac{12}{2}+\frac{6y}{2}\\x&=6+3y\end{aligned}
6+3y
x
\begin{aligned}3x-4y&=13\\3(6+3y)-4y&=13\end{aligned}
\begin{aligned}3(6+3y)-4y&=13\\18+9y-4y&=13\\18+5y-18&=13-18\\5y&=-5\\y&=-1\end{aligned}
Substitute the value from Step 3 back into either of the original equations and solve for the other variable.
y
\begin{aligned}2x-6y&=12\\2x-6(-1)&=12\\2x-(-6)&=12\\2x+6&=12\\2x+6-6&=12-6\\2x&=6\\x&=3\end{aligned}
Write the values of each variable as an ordered pair.
(3, -1)
To check the solution, substitute the values for
x
y
back into both equations and simplify. If they both result in true equations, then the solution is correct.
\begin{aligned}3x-4y&=13\\3(3)-4(-1)&=13\\9-(-4)&=13\\9+4&=13\\13&=13\end{aligned}\hspace{20pt}\begin{aligned}2x-6y&=12\\2(3)-6(-1)&=12\\6-(-6)&=12\\6+6&=12\\12&=12\end{aligned}
Equations in a system can be added, subtracted, and/or multiplied by a nonzero constant in order to eliminate a variable.
Although the substitution method can be used to solve any system of linear equations, this method sometimes results in equations containing fractions. If it is not possible to solve one of the equations in the system for one of the variables without generating fractions, it may be easier to use another algebraic method to solve the system.
The elimination method is a method for solving systems of equations that involves combining multiples of equations to eliminate variables.
To solve by elimination:
1. Multiply both sides of one equation (or both equations, if necessary) by a constant so that one of its terms is the opposite of a term in the other equation.
2. Add the equations to eliminate one of the variables.
4. Substitute the value found in Step 3 into either of the original equations and solve for the other variable.
The elimination method can be expanded and repeated as necessary for a system with two or more equations and variables.
Solve a System of Linear Equations in Two Variables by Elimination
\begin{cases}\phantom{4}x-3y=10\ \ [1]\\4x+9y=19\ \ [2] \end{cases}
\begin{cases}\phantom{4}x-3y=10\ \ [1]\\4x+9y=19\ \ [2] \end{cases}
y
9y
\begin{aligned}3(x-3y)&=3(10)\\3x-9y&=30\hspace{20pt}[3]\end{aligned}
y
\begin{aligned}\begin{cases}3x-9y=30 \hspace{10pt} [3]\\\underline{4x+9y=19} \hspace{10pt}[2]\end{cases}\\\phantom{\lbrack}7x+0y=49 \hspace{10pt}\phantom{[2]} \end{aligned}
x
\begin{aligned}7x&=49\\\frac{7x}{7}&=\frac{49}{7}\\x&=7\end{aligned}
x
found in Step 3 into either of the original equations, then solve for
y
x
x-3y=10
\begin{aligned}x-3y&=10\\(7)-3y&=10\\7-3y-7&=10-7\\-3y&=3\\\frac{-3y}{-3}&=\frac{3}{-3}\\y&=-1\end{aligned}
(7, -1)
x
y
\begin{aligned}x-3y&=10\\(7)-3(-1)&=10\\7-(-3)&=10\\7+3&=10\\10&=10\end{aligned}\hspace{20pt}\begin{aligned}4x+9y&=19\\4(7)+9(-1)&=19\\28+(-9)&=19\\28-9&=19\\19&=19\end{aligned}
Solve a System of Linear Equations in Three Variables by Elimination
\begin{cases}\phantom{5}x+\phantom{2}y+\phantom{6}z=8 & [1]\\ 5x+2y-6z=2 & [2]\\ -x-4y+\phantom{6}z=6 &[3] \end{cases}
Combine two equations to eliminate one of the variables. If necessary, multiply both sides of one equation (or both equations) by a constant so that one of its terms is the opposite of a term in the other equation.
x
-terms in equations [1] and [3] are opposites. Add to eliminate the
x
-term. Label the resulting equation as [4].
\begin{aligned}\begin{cases}\phantom{-}x+\phantom{4}y+z=8\phantom{4}\hspace{10pt}[1]\\\underline{-x-4y+z=6\phantom{4}}\hspace{10pt}[3]\end{cases}\\\phantom{\begin{cases}\end{cases}-}-3y+2z=14\hspace{10pt}[4]\end{aligned}
x
-term from the remaining equation.
Multiply equation [3] by 5 so that its
x
-term is the opposite of the
x
-term in equation [2]. Label the resulting equation as [5].
\begin{cases}5x+2y-6z=2&[2] \\ 5(-x-4y+z=6) &[3] \end{cases} \rightarrow \begin{cases} \phantom{-}5x+2y-6z=2&[2] \\ -5x-20y+5z=30 &[5] \end{cases}
Add equations [2] and [5], and label the resulting equation as [6].
\begin{aligned}\begin{cases}\phantom{-}5x+\phantom{2}2y-6z=2\phantom{0}\hspace{10pt}[2]\\\underline{-5x-20y+5z=30}\hspace{10pt}[5]\end{cases}\\\phantom{\lbrack-5x}-18y-\phantom{5}z=32\hspace{10pt}[6]\end{aligned}
Solve for one of the remaining variables.
z
z
-term in equation [4] from Step 1. Label the resulting equation as [7].
\begin{cases}-3y+2z=14&[4] \\ 2(-18y-z=32) &[6] \end{cases} \rightarrow \begin{cases} -3y+2z=14&[4] \\ -36y-2z=64 &[7] \end{cases}
z
\begin{aligned}\begin{cases}-\phantom{3}3y+2z=14\hspace{10pt}[4]\\\underline{-36y-2z=64}\hspace{10pt}[7]\end{cases}\\\phantom{\begin{cases}\end{cases}}-39y\phantom{-2z}=78\hspace{10pt}[6]\end{aligned}
y
\begin{aligned}-39y&=78\\y&=-2\end{aligned}
Substitute the value found in Step 6 into either of the equations with only two variables, and solve for the other variable.
y
in equation [4].
\begin{aligned}-3y+2z&=14\\-3(-2)+2z&=14\\6+2z&=14\\2z&=8\\z&=4\end{aligned}
Substitute the values that have been found into one of the original equations and solve for the remaining variable.
y
z
\begin{aligned}x+y+z&=8\\x+(-2)+(4)&=8\\x+2&=8\\x&=6\end{aligned}
Write the values of each variable as an ordered triple.
(6,-2,4)
The solution can be checked by substituting the values for
x
y
z
back into the original equations. If all result in true equations, then the solution is correct.
<Vocabulary>Solving Systems of Nonlinear Equations
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From bosonic grand-canonical ensembles to nonlinear Gibbs measures
In a recent paper, in collaboration with Mathieu Lewin and Phan Thành Nam, we showed that nonlinear Gibbs measures based on Gross-Pitaevskii like functionals could be derived from many-body quantum mechanics, in a mean-field limit. This text summarizes these findings. It focuses on the simplest, but most physically relevant, case we could treat so far, namely that of the defocusing cubic NLS functional on a 1D interval. The measure obtained in the limit, which lives over
{H}^{1/2-ϵ}
, has been previously shown to be invariant under the NLS flow by Bourgain.
author = {Nicolas Rougerie},
title = {From bosonic grand-canonical ensembles to {nonlinear~Gibbs~measures}},
TI - From bosonic grand-canonical ensembles to nonlinear Gibbs measures
%T From bosonic grand-canonical ensembles to nonlinear Gibbs measures
Nicolas Rougerie. From bosonic grand-canonical ensembles to nonlinear Gibbs measures. Séminaire Laurent Schwartz — EDP et applications (2014-2015), Talk no. 5, 17 p. doi : 10.5802/slsedp.71. https://slsedp.centre-mersenne.org/articles/10.5802/slsedp.71/
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If F(x) is the antiderivative of f(x), is it true that \(\int_a^b\)f(x)dx=F(b)-F(a)? | Brilliant Math & Science Wiki
\int_a^b
Pi Han Goh, Andrew Hayes, and Jimin Khim contributed
F(x)
is the antiderivative of
f(x),
\displaystyle \int _a^b f(x) \, dx = F(b) -F(a)?
Why some people say it's true: It's what I was taught at school to calculate proper integrals. You just apply the second fundamental theorem of calculus. When I was asked to compute
\int_1^2 x\ dx,
I used the formula and got the right answer, which was
\frac{3}{2}.
Why some people say it's false: The functions might not be continuous over the entire interval.
\color{#D61F06} {\textbf{false}}
. The claim is true if and only if the integrand is continuous over the entire domain of
(a,b)
. This is an important condition to satisfy before you apply the second fundamental theorem of calculus.
Rebuttal: It's true for the following example:
\int_1^2 x \,dx = \left. \frac{1}{2}x^2 \right|_1^2= \frac{3}{2},
so it must always work on all integrals. By the second fundamental theorem of calculus, our claim must be true.
Reply: The integration used in the rebuttal is true only because
f(x) = x
(1,2)
. That is why we can integrate like that. In other words, we have only shown that it's true for a specific case. If a function is not continuous over an interval, like
f(x)=\frac{1}{x}
(-1,1)
\big(
f(0)
\big),
the fundamental theorem of calculus cannot be applied.
Rebuttal: If you can't use the second fundamental theorem of calculus, then how do you calculate integrals that have discontinuities?
Reply: If an integrand contains a discontinuity within the interval, then the integral is undefined. However, the integral can be interpreted as a Cauchy principal value by finding the sum of improper integrals. For example,
\int_{-1}^{1}\frac{1}{x}\ dx.
The integrand has a point of discontinuity at
x=0,
and thus the integral is undefined. However, the Cauchy principal value will be computed as a sum of improper integrals:
\begin{aligned} PV\ \int_{-1}^{1}\frac{1}{x}\ dx &= \int_{-1}^{0}\frac{1}{x}\ dx + \int_{0}^{1}\frac{1}{x}\ dx \\ \\ &= \left. \lim_{a \rightarrow 0^-}\ln|x| \right|_{-1}^{a} + \left. \lim_{a \rightarrow 0^+} \ln|x| \right|_{a}^{1} \\ \\ &= \left(\lim_{x \rightarrow 0^+}\ln(x) \right) - 0 + 0 - \left(\lim_{x \rightarrow 0^+}\ln(x) \right) \\ \\ &= \lim_{x \rightarrow 0^+}\big(\ln(x)-\ln(x) \big) \\ \\ &= 0. \end{aligned}
In this case, the Cauchy principal value matches what we would expect if it was a proper integral. However, this is not always the case. See the improper integrals page for more examples of the Cauchy principal value.
\frac {1} {e}
e
\displaystyle f(x) = \frac 1{1+e^{1/x}}
PV\ \int_{-1}^{1} \frac{d}{dx} f(x) \ dx = \lambda + \frac{1-e}{1+e},
\lambda
is some constant, what is the value of
\lambda?
PV
" before the integral indicates the Cauchy principal value.
Cite as: If F(x) is the antiderivative of f(x), is it true that
\int_a^b
f(x)dx=F(b)-F(a)?. Brilliant.org. Retrieved from https://brilliant.org/wiki/if-the-derivative-of-fx-is-fx-is-it-true-that-int/
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Perform transformation from three-phase (abc) signal to αβ0 stationary reference frame or the inverse - Simulink - MathWorks Nordic
abc to Alpha-Beta-Zero, Alpha-Beta-Zero to abc
Perform transformation from three-phase (abc) signal to αβ0 stationary reference frame or the inverse
The abc to Alpha-Beta-Zero block performs a Clarke transform on a three-phase abc signal. The Alpha-Beta-Zero to abc block performs an inverse Clarke transform on the αβ0 components.
\left[\begin{array}{c}{u}_{\mathrm{α}}\\ {u}_{\mathrm{β}}\\ {u}_{0}\end{array}\right]=\left[\begin{array}{ccc}\frac{2}{3}& â\frac{1}{3}& â\frac{1}{3}\\ 0& \frac{1}{\sqrt{3}}& \frac{â1}{\sqrt{3}}\\ \frac{1}{3}& \frac{1}{3}& \frac{1}{3}\end{array}\right]\left[\begin{array}{c}{u}_{a}\\ {u}_{b}\\ {u}_{c}\end{array}\right]
\left[\begin{array}{c}{u}_{a}\\ {u}_{b}\\ {u}_{c}\end{array}\right]=\left[\begin{array}{ccc}1& 0& 1\\ â\frac{1}{2}& \frac{\sqrt{3}}{2}& 1\\ â\frac{1}{2}& â\frac{\sqrt{3}}{2}& 1\end{array}\right]\left[\begin{array}{c}{u}_{\mathrm{α}}\\ {u}_{\mathrm{β}}\\ {u}_{0}\end{array}\right]
Assume that ua, ub, uc quantities represent three sinusoidal balanced currents:
\begin{array}{l}{i}_{a}=I\mathrm{sin}\left(\mathrm{Ï}t\right)\\ {i}_{b}=I\mathrm{sin}\left(\mathrm{Ï}tâ\frac{2\mathrm{Ï}}{3}\right)\\ {i}_{c}=I\mathrm{sin}\left(\mathrm{Ï}t+\frac{2\mathrm{Ï}}{3}\right)\end{array}
These currents are flowing respectively into windings A, B, C of a three-phase winding, as the figure shows.
In this case, the iα and iβ components represent the coordinates of the rotating space vector Is in a fixed reference frame whose α axis is aligned with phase A axis. Is amplitude is proportional to the rotating magnetomotive force produced by the three currents. It is computed as follows:
{I}_{s}={i}_{a}+jâ
{i}_{\mathrm{β}}=\frac{2}{3}\left({i}_{a}+{i}_{b}â
{e}^{\frac{j2\mathrm{Ï}}{3}}+{i}_{c}â
{e}^{â\frac{j2\mathrm{Ï}}{3}}\right)
|
Add third harmonic or triplen harmonic zero-sequence signal to three-phase signal - Simulink - MathWorks Nordic
Add third harmonic or triplen harmonic zero-sequence signal to three-phase signal
The Overmodulation block increases the linear region of a three-phase PWM generator by adding a third harmonic or triplen harmonic zero-sequence signal V0 to the three-phase original reference signal Uref. This zero-sequence signal does not appear in the line-to-line voltages.
A modulation index of up to 1.1547 (exact value = 2/sqrt(3)) can be used without pulse dropping.
The Overmodulation block implements three overmodulation techniques:
The Third Harmonic overmodulation technique. In this technique the third-harmonic signal V0 subtracted from the original signal is calculated as
{V}_{0}=\frac{|U|}{6}×\mathrm{sin}\left[3\cdot \left(wt+\angle U\right)\right]
The Flat Top overmodulation technique. In this technique the portion of the three-phase input signal exceeding values +/−1 is computed. The three resulting signals are then summed and removed from the original signal Uref. The resulting modified signal Uref* is therefore a flat-top three-phase signal that contains zero-sequence triplen-harmonics. The block outputs a value between −1 and 1.
The Min-Max overmodulation technique. In this technique the minimum and maximum values of the three components of input signal Uref are summed and divided by two, and then subtracted from the input signal. The resulting modified signal Uref* also contains zero-sequence triplen-harmonics. The block outputs a value between −1 and 1.
Overmodulation type
Select the overmodulation technique you want to apply to the Uref signal: Third harmonic (default), Flat top, or Min-Max.
The three-phase reference signal of three-phase PWM generator.
Uref*
The overmodulated three-phase signal of three-phase PWM generator.
Sample Time Inherited
The power_OverModulation example compares the three overmodulation techniques implemented in the Overmodulation block. Choose the overmodulation technique (type 1, 2, or 3 on the first input of the Multiport Switch) and run the simulation. Observe the resulting waveforms in Scope 1.
The model sample time is parameterized with variable Ts (default value of 5e-6). To run a continuous simulation, at the MATLAB® command prompt, enter
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A Formula for the Numerical Range of Elementary Operators
M. Barraa, "A Formula for the Numerical Range of Elementary Operators", International Scholarly Research Notices, vol. 2014, Article ID 246301, 4 pages, 2014. https://doi.org/10.1155/2014/246301
M. Barraa1
1Department of Mathematics, Faculty of Sciences Semlalia, Marrakech, Morocco
Academic Editor: M. Lindstrom
Let be the algebra of bounded linear operators on a complex Hilbert space . For -tuples of elements of and , let denote the elementary operator on defined by . In this paper, we prove the following formula for the numerical range of : , where is the set of unitary operators.
Let be a complex Banach algebra with unit. For -tuples of elements of , and , let denote the elementary operator on defined by This is a bounded linear operator on . Some interesting cases are the generalized derivation and the multiplication for .
The numerical range of is defined by where is the set of normalized states in : See [1–3]. It is well known that is convex and closed and contains the spectrum . For , the algebra of bounded linear operators on a normed space , and , in addition to , we have the spatial numerical range of , given by and it is known that , the closed convex hull of . In the case of a Hilbert space , then is convex but not closed in general and .
Many facts about the relation between the spectrum of and the spectrums of the coefficients and are known. This is not the case with the relation between the numerical range of and the numerical ranges of and . Apparently, the only elementary operator on a Hilbert space for which the numerical range is computed is the generalized derivations [4–8]. It is Fong [4] who first gives the following formula: where is the inner derivation defined by . Shaw [7] (see also [5, 6]) extended this formula to generalized derivations in Banach spaces. For a good survey of the numerical range of elementary operators, you can see [9], where the following problem is posed.
Problem. Determine the numerical range of the elementary operator .
In this paper, we give a formula that answers this problem.
The following theorem is the main result in this paper.
Theorem 1. Let be a complex Hilbert space. Let and be two -tuples of elements in . Then, one has In particular for multiplication and generalized derivation, one has :
From Fong’s formula (see [4, 6, 10]), we can deduce the following.
Corollary 2. For , one has
One of the keys to the proof of our main result is the following lemma.
Lemma 3. Let and be two -tuples of elements in . Then, one has In particular, for , one has
Proof. Let ; by definition, there exists with such that Here, is the linear form trace. Let be the linear form defined by This is a bounded linear form on , with norm being equal to 1, because Since the form is a state on . So, Hence, .
Let be a Banach space. We say that is an isometry if for all . If is an invertible isometry, then its inverse is also an isometry, and we have
In the case of a Hilbert space, an invertible isometry is unitary and its inverse is the adjoint.
Let and be two unitaries operators on ; then with being an invertible isometry and its inverse being .
From this result, we deduce that Now, using Lemma 3, we get
But, the numerical range is closed and the product of two unitaries is also an unitary, hence: So, we have proved the second inclusion of Theorem 1.
For the other inclusion, we will use the two following theorems.
Theorem 4 (See [11]). Let be Banach algebra. For , one has
The norm of an elementary operator is defined by Let be -algebra. An element is said to be unitary if . In the following, denote the set of unitaries in .
Theorem 5 (Russo-Dye [12]). Let be algebra. Let and be two -tuples of elements in . Then, one has
We return now to the proof of the main theorem.
Proof. We need only to show the inclusion “” By Theorem 4, we have And, by Theorem 5, we get But for all and . Hence,
Hence, if , then, for all , Let fixed, there exists a unitary such that Now, using Theorem 4, we have
So, there exists such that . But is arbitrary, . This finishes the proof of the main theorem.
It is well known that, for the spectrum, if , then we have For the numerical range, this not true, but we can deduce the following corollary from the proof of Theorem 1.
Corollary 6. For all , one has
The numerical radius of an operator is denoted by and defined by
Corollary 7. Let and be two -tuples of elements in . Then, one has In particular, for ,
Let be a nonempty subset of the plane and let From Corollary 7 (), one has So, the diameter of the numerical range is equal to the diameter of the -unitary orbit of the operator .
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B. Russo and H. A. Dye, “A note on unitary operators in
{C}^{*}
-algebras,” Duke Mathematical Journal, vol. 33, pp. 413–416, 1966. View at: Publisher Site | Google Scholar | MathSciNet
Copyright © 2014 M. Barraa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Liquid Cooling Performance Capacities | Mikros Technologies
Pressure v. Flow
Resistance v. Flow
Resistance v. Pressure Drop
Cooling Capacity v. Flow
Cooling Capacity v. Pressure Drop
Tabulated Performance Data
High Effectiveness, High Value
With core 1 thermal resistances as low as 0.02 C-cm2/W, low pressure drop and configurable, affordable designs, Mikros microchannels are among the most effective liquid-cooling solutions available. Below are key performance metrics and features to evaluate if a Mikros liquid cold plate is right for your system.
Below are key performance metrics and features to evaluate if a Mikros liquid cold plate is right for your system. With core1 thermal resistances as low as 0.02 C-cm2/W, low pressure drop and configurable, affordable designs, Mikros microchannels are among the most effective liquid-cooling solutions available.
With core1 thermal resistances as low as 0.02 C-cm2/W. Below are key performance metrics and features to evaluate if a Mikros liquid cold plate is right for your system.
Below are key performance metrics and features to evaluate if a Mikros liquid cold plate is right for your system.
With core1 thermal resistances as low as 0.02 C-cm2/W, low pressure drop and configurable, affordable designs, Mikros microchannels are among the most effective liquid-cooling solutions available.
The thermal resistance of Mikros cold plates approaches the lower theoretical limit over a wide range of flows. Our microchannel cold plate matrices can achieve thermal resistances 1-2 orders of magnitude lower than swaged-tube and pin-fin cold plates with comparable pressure drops.
The pressure drop of Mikros microchannel and minichannel cold plates range from 5-35 kPa (1-5 psi). Our patented Normal Flow™ design dramatically reduces pressure drop and provides thermal resistances up to 2 orders of magnitude lower than cold plates at equal pressure drops.
High Cooling Value
As cooling requirements become more stringent, the cost-per-Watt of Mikros’ microchannel cooling quickly becomes less than cold plates with lower effectiveness. Mikros’ high heat transfer capacity can also provide savings in reduced pump and peripheral component size and count.
Mikros cold plates have shown no change in core thermal resistance for nearly 15 years in our endurance test facility using filtered water with corrosion inhibitor and algaecide. Over that time, our clients have incorporated tens of thousands of complex Mikros microchannel cold plates in high-power computing systems where reliability is critical.
Tailored Cooling
Mikros Normal Flow™ microchannels distribute inlet temperature coolant over the entire heated surface. By directing larger flows to those areas requiring higher cooling capacity, the local thermal resistance can be tailored to mitigate “hot spots,” eliminating temperature gradients and reducing the total coolant flow requirements.
Mikros’ Normal Flow™ microchannels can be fabricated in a wide array of complex topologies. Our cold plates range from 4x4mm to 300x500mm assemblies with many cooling areas. We design custom features to accommodate component height and tilt tolerances, CTE-matching, TIM force requirements and non-planar cooling areas.
1 Thermal Resistance Definitions
Total Thermal Resistance:
{R}_{total }={R}_{core}+ {R}_{flow}
Core Thermal Resistance: Rcore =
{R}_{core}={R}_{total}– {R}_{flow}
Flow Thermal Resistance: Rflow = A/
p·cp·Q
As the flow rate increases, flow resistance decreases, and the total thermal resistance approaches the core resistance.
View Mikros Applications
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Within homogeneous isotropic materials, rays are straight lines. By symmetry, they cannot bend in any preferred direction because no such preferred direction exists. Moreover, because the propagation speed is identical in all directions, the spatial separation between two wavefronts, measured along rays, must be the same everywhere. Points at which a single ray intersects a set of wavefronts are called corresponding points, as for example points A, B, and C in Figure 15. Evidently the separation in time between any two corresponding points on any two sequential wavefronts is identical. In other words, if wavefront
{\displaystyle S_{\rm {l}}}
transforms into wavefront
{\displaystyle S_{\rm {2}}}
{\displaystyle \Delta t}
, the distance between corresponding points on any ray will be traversed in the same time
{\displaystyle \Delta t}
. This is true even if the wavefronts travel from one homogeneous isotropic medium into another, and it simply means that every point on
{\displaystyle S_{\rm {1}}}
can be imagined to follow the path of a ray that arrives at
{\displaystyle S_{\rm {2}}}
{\displaystyle \Delta t}
Retrieved from "https://wiki.seg.org/index.php?title=Ray_theory&oldid=164003"
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Speed Knowpia
In everyday use and in kinematics, the speed (commonly referred to as v) of an object is the magnitude of the rate of change of its position with time or the magnitude of the change of its position per unit of time; it is thus a scalar quantity.[1] The average speed of an object in an interval of time is the distance travelled by the object divided by the duration of the interval;[2] the instantaneous speed is the limit of the average speed as the duration of the time interval approaches zero.
Speed has the dimensions of distance divided by time. The SI unit of speed is the metre per second (m/s), but the most common unit of speed in everyday usage is the kilometre per hour (km/h) or, in the US and the UK, miles per hour (mph). For air and marine travel, the knot is commonly used.
The fastest possible speed at which energy or information can travel, according to special relativity, is the speed of light in a vacuum c = 299792458 metres per second (approximately 1079000000 km/h or 671000000 mph). Matter cannot quite reach the speed of light, as this would require an infinite amount of energy. In relativity physics, the concept of rapidity replaces the classical idea of speed.
Italian physicist Galileo Galilei is usually credited with being the first to measure speed by considering the distance covered and the time it takes. Galileo defined speed as the distance covered per unit of time.[3] In equation form, that is
{\displaystyle v={\frac {d}{t}},}
{\displaystyle v}
{\displaystyle d}
{\displaystyle t}
is time. A cyclist who covers 30 metres in a time of 2 seconds, for example, has a speed of 15 metres per second. Objects in motion often have variations in speed (a car might travel along a street at 50 km/h, slow to 0 km/h, and then reach 30 km/h).
Speed at some instant, or assumed constant during a very short period of time, is called instantaneous speed. By looking at a speedometer, one can read the instantaneous speed of a car at any instant.[3] A car travelling at 50 km/h generally goes for less than one hour at a constant speed, but if it did go at that speed for a full hour, it would travel 50 km. If the vehicle continued at that speed for half an hour, it would cover half that distance (25 km). If it continued for only one minute, it would cover about 833 m.
In mathematical terms, the instantaneous speed
{\displaystyle v}
is defined as the magnitude of the instantaneous velocity
{\displaystyle {\boldsymbol {v}}}
, that is, the derivative of the position
{\displaystyle {\boldsymbol {r}}}
{\displaystyle v=\left|{\boldsymbol {v}}\right|=\left|{\dot {\boldsymbol {r}}}\right|=\left|{\frac {d{\boldsymbol {r}}}{dt}}\right|\,.}
{\displaystyle s}
is the length of the path (also known as the distance) travelled until time
{\displaystyle t}
, the speed equals the time derivative of
{\displaystyle s}
{\displaystyle v={\frac {ds}{dt}}.}
In the special case where the velocity is constant (that is, constant speed in a straight line), this can be simplified to
{\displaystyle v=s/t}
. The average speed over a finite time interval is the total distance travelled divided by the time duration.
Different from instantaneous speed, average speed is defined as the total distance covered divided by the time interval. For example, if a distance of 80 kilometres is driven in 1 hour, the average speed is 80 kilometres per hour. Likewise, if 320 kilometres are travelled in 4 hours, the average speed is also 80 kilometres per hour. When a distance in kilometres (km) is divided by a time in hours (h), the result is in kilometres per hour (km/h).
Average speed does not describe the speed variations that may have taken place during shorter time intervals (as it is the entire distance covered divided by the total time of travel), and so average speed is often quite different from a value of instantaneous speed.[3] If the average speed and the time of travel are known, the distance travelled can be calculated by rearranging the definition to
{\displaystyle d={\boldsymbol {\bar {v}}}t\,.}
Using this equation for an average speed of 80 kilometres per hour on a 4-hour trip, the distance covered is found to be 320 kilometres.
Expressed in graphical language, the slope of a tangent line at any point of a distance-time graph is the instantaneous speed at this point, while the slope of a chord line of the same graph is the average speed during the time interval covered by the chord. Average speed of an object is Vav = s÷t
Speed denotes only how fast an object is moving, whereas velocity describes both how fast and in which direction the object is moving.[5] If a car is said to travel at 60 km/h, its speed has been specified. However, if the car is said to move at 60 km/h to the north, its velocity has now been specified.
Linear speed is the distance travelled per unit of time, while tangential speed (or tangential velocity) is the linear speed of something moving along a circular path.[6] A point on the outside edge of a merry-go-round or turntable travels a greater distance in one complete rotation than a point nearer the center. Travelling a greater distance in the same time means a greater speed, and so linear speed is greater on the outer edge of a rotating object than it is closer to the axis. This speed along a circular path is known as tangential speed because the direction of motion is tangent to the circumference of the circle. For circular motion, the terms linear speed and tangential speed are used interchangeably, and both use units of m/s, km/h, and others.
Rotational speed (or angular speed) involves the number of revolutions per unit of time. All parts of a rigid merry-go-round or turntable turn about the axis of rotation in the same amount of time. Thus, all parts share the same rate of rotation, or the same number of rotations or revolutions per unit of time. It is common to express rotational rates in revolutions per minute (RPM) or in terms of the number of "radians" turned in a unit of time. There are little more than 6 radians in a full rotation (2π radians exactly). When a direction is assigned to rotational speed, it is known as rotational velocity or angular velocity. Rotational velocity is a vector whose magnitude is the rotational speed.
Tangential speed and rotational speed are related: the greater the RPMs, the larger the speed in metres per second. Tangential speed is directly proportional to rotational speed at any fixed distance from the axis of rotation.[6] However, tangential speed, unlike rotational speed, depends on radial distance (the distance from the axis). For a platform rotating with a fixed rotational speed, the tangential speed in the centre is zero. Towards the edge of the platform the tangential speed increases proportional to the distance from the axis.[7] In equation form:
{\displaystyle v\propto \!\,r\omega \,,}
where v is tangential speed and ω (Greek letter omega) is rotational speed. One moves faster if the rate of rotation increases (a larger value for ω), and one also moves faster if movement farther from the axis occurs (a larger value for r). Move twice as far from the rotational axis at the centre and you move twice as fast. Move out three times as far, and you have three times as much tangential speed. In any kind of rotating system, tangential speed depends on how far you are from the axis of rotation.
When proper units are used for tangential speed v, rotational speed ω, and radial distance r, the direct proportion of v to both r and ω becomes the exact equation
{\displaystyle v=r\omega \,.}
Thus, tangential speed will be directly proportional to r when all parts of a system simultaneously have the same ω, as for a wheel, disk, or rigid wand.
metres per second (symbol m s−1 or m/s), the SI derived unit;
Approximate rate of continental drift 0.0000000013 0.0000000042 0.0000000045 0.0000000028 4 cm/year. Varies depending on location.
Speed of a common snail 0.001 0.003 0.004 0.002 1 millimetre per second
Typical suburban speed limit in most of the world 13.8 45.3 50 30
Average peak speed of a cheetah 33.53 110 120.7 75
Speed limit on a French autoroute 36.1 118 130 81
Highest recorded human-powered speed 37.02 121.5 133.2 82.8 Sam Whittingham in a recumbent bicycle[10]
Cruising speed of a Boeing 747-8 passenger jet 255 836 917 570 Mach 0.85 at 35000 ft (10668 m) altitude
Speed of a .22 caliber Long Rifle bullet 326.14 1070 1174.09 729.55
Muzzle velocity of a 7.62×39mm cartridge 710 2330 2600 1600 The 7.62×39mm round is a rifle cartridge of Soviet origin
The fastest recorded speed of the Helios probes 70,220 230,381 252,792 157,078 Recognized as the fastest speed achieved by a man-made spacecraft, achieved in solar orbit.
Orbital speed of the Sun relative to the center of the galaxy 251000 823000 904000 561000
Speed of the Galaxy relative to the CMB 550000 1800000 2000000 1240000
Speed of light in vacuum (symbol c) 299792458 983571056 1079252848 670616629 Exactly 299792458 m/s, by definition of the metre
According to Jean Piaget, the intuition for the notion of speed in humans precedes that of duration, and is based on the notion of outdistancing.[12] Piaget studied this subject inspired by a question asked to him in 1928 by Albert Einstein: "In what order do children acquire the concepts of time and speed?"[13] Children's early concept of speed is based on "overtaking", taking only temporal and spatial orders into consideration, specifically: "A moving object is judged to be more rapid than another when at a given moment the first object is behind and a moment or so later ahead of the other object."[14]
Look up speed or swiftness in Wiktionary, the free dictionary.
Richard P. Feynman, Robert B. Leighton, Matthew Sands. The Feynman Lectures on Physics, Volume I, Section 8–2. Addison-Wesley, Reading, Massachusetts (1963). ISBN 0-201-02116-1.
^ Wilson, Edwin Bidwell (1901). Vector analysis: a text-book for the use of students of mathematics and physics, founded upon the lectures of J. Willard Gibbs. Yale bicentennial publications. C. Scribner's Sons. p. 125. hdl:2027/mdp.39015000962285. This is the likely origin of the speed/velocity terminology in vector physics.
^ a b c Elert, Glenn. "Speed & Velocity". The Physics Hypertextbook. Retrieved 8 June 2017.
^ NASA's Goddard Space Flight Center. "Satellite sea level observations". Global Climate Change. NASA. Retrieved 20 April 2022.
^ Darling, David. "Fastest Spacecraft". Retrieved August 19, 2013.
^ Jean Piaget, Psychology and Epistemology: Towards a Theory of Knowledge, The Viking Press, pp. 82–83 and pp. 110–112, 1973. SBN 670-00362-x
^ Siegler, Robert S.; Richards, D. Dean (1979). "Development of Time, Speed, and Distance Concepts" (PDF). Developmental Psychology. 15 (3): 288–298. doi:10.1037/0012-1649.15.3.288.
^ Early Years Education: Histories and Traditions, Volume 1. Taylor & Francis. 2006. p. 164. ISBN 9780415326704.
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Semi-major axis - Marspedia
In orbital mechanics, the semi-major axis (symbol
{\displaystyle a}
) is half the major axis (distance between apoapsis and periapsis) of an elliptical orbit. It is defined as the mean distance between either focus of the orbit and any point on the orbit and equals the radius if the orbit is circular.
It is equal to the semi-parameter:
{\displaystyle p=a}
For hyperbolic orbits, see semi-transverse axis instead.
Prograde thrust (accelerating the body in the direction in which it is moving) increases the semi-major axis whereas retrograde thrust (in the opposite direction to its movement) decreases it. It is important to note that if no force acts on an orbiting body, it will return to the position where it last performed a maneuver. The opposite side of the orbit is most affected by any thrust whereas the orbit remains almost unchanged close to the point where the maneuver is performed.
Prograde and retorgrade thrust at either apsis of an orbit has the greatest influence on the semi-major axis, by moving the opposite apsis. This means that a spacecraft wishing to go from one circular orbit to a superior or inferior one can do so most efficiently by performing a Hohmann transfer (in the absence of other forces).
Relation to energy
It is important to note that, when working with orbital energies, the energy is not the same as that required to hypothetically take a body to that orbit from either the surface or the center of the body it orbits. In fact, the specific energy of a satellite in elliptical orbit around the Earth is negative.
Since orbital mechanics only concerns itself with changes in orbital energy, the zero could be chosen arbitrarily. It is computationally most convenient to choose the value at escape velocity. This choice makes the semi-major axis inversely proportional to the specific energy and if the mass does not change also to the total orbital energy.
It follows that the orbital energy must be a negative value (since velocity is less than escape velocity) for elliptical orbits.
Retrieved from "https://marspedia.org/index.php?title=Semi-major_axis&oldid=125851"
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The complex plane (also called the Argand plane or Gauss plane) is a way to represent complex numbers geometrically. It is basically a modified Cartesian plane, with the real part of a complex number represented by a displacement along the
x
-axis, and the imaginary part by a displacement along the
y
a+bi
can simply be represented as the point on the Cartesian plane with the coordinates
(a, b)
As we said earlier, the complex plane is basically a modified Cartesian plane where the
x
y
-axis have been dubbed the "real axis" and the "imaginary axis," respectively.
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Stable soliton resolution for equivariant wave maps exterior to a ball
In this report we review the proof of the stable soliton resolution conjecture for equivariant wave maps exterior to a ball in
{ℝ}^{3}
and taking values in the
3
-sphere. This is joint work with Carlos Kenig, Baoping Liu, and Wilhelm Schlag.
author = {Andrew Lawrie},
title = {Stable soliton resolution for equivariant wave maps exterior to a ball},
TI - Stable soliton resolution for equivariant wave maps exterior to a ball
%T Stable soliton resolution for equivariant wave maps exterior to a ball
Andrew Lawrie. Stable soliton resolution for equivariant wave maps exterior to a ball. Séminaire Laurent Schwartz — EDP et applications (2014-2015), Talk no. 3, 11 p. doi : 10.5802/slsedp.66. https://slsedp.centre-mersenne.org/articles/10.5802/slsedp.66/
[1] H. Bahouri and P. Gérard. High frequency approximation of solutions to critical nonlinear wave equations. Amer. J. Math., 121:131–175, 1999.
[2] B. Balakrishna, V. Schechter Sanyuk, J., and A. Subbaraman. Cutoff quantization and the skyrmion. Physical Review D, 45(1):344–351, 1992.
[3] P. Bizoń, T. Chmaj, and M. Maliborski. Equivariant wave maps exterior to a ball. Nonlinearity, 25(5):1299–1309, 2012.
[4] T. Duyckaerts, C. Kenig, and F. Merle. Universality of the blow-up profile for small radial type
\mathrm{II}
blow-up solutions of the energy critical wave equation. J. Eur math. Soc. (JEMS), 13(3):533–599, 2011.
[5] T. Duyckaerts, C. Kenig, and F. Merle. Profiles of bounded radial solutions of the focusing, energy-critical wave equation. Geom. Funct. Anal., 22(3):639–698, 2012.
[6] T. Duyckaerts, C. Kenig, and F. Merle. Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case. J. Eur. Math. Soc. (JEMS), 14(5):1389–1454, 2012.
[7] T. Duyckaerts, C. Kenig, and F. Merle. Classification of radial solutions of the focusing, energy critical wave equation. Cambridge Journal of Mathematics, 1(1):75–144, 2013.
[8] T. Duyckaerts, C. Kenig, and F. Merle. Scattering for radial, bounded solutions of focusing supercritical wave equations. To appear in I.M.R.N, Preprint, 2012.
[9] K. Hidano, J. Metcalfe, H. Smith, C. Sogge, and Y. Zhou. On abstract Strichartz estimates and the Strauss conjecture for nontrapping obstacles. Trans. Amer. Math. Soc., 362(5):2789–2809, 2010.
[10] C. Kenig, A. Lawrie, B. Liu, and W. Schlag. Channels of energy for the linear radial wave equation. Preprint, 2014.
[11] C. Kenig, A. Lawrie, B. Liu, and W. Schlag. Stable soliton resolution for exterior wave maps in all equivariance classes. Preprint, 2014.
[12] C. Kenig, A. Lawrie, and W. Schlag. Relaxation of wave maps exterior to a ball to harmonic maps for all data. Geom. Funct. Anal., 24(2):610–647, 2014.
[13] C. Kenig and F. Merle. Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case. Invent. Math., 166(3):645–675, 2006.
[14] C. Kenig and F. Merle. Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation. Acta Math., 201(2):147–212, 2008.
[15] A. Lawrie and W. Schlag. Scattering for wave maps exterior to a ball. Advances in Mathematics, 232(1):57–97, 2013.
[16] J. Shatah. Weak solutions and development of singularities of the
\mathrm{SU}\left(2\right)
\sigma
-model. Comm. Pure Appl. Math., 41(4):459–469, 1988.
[17] J. Shatah and M. Struwe. Geometric wave equations. Courant Lecture notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York. American Mathematical Society, Providence RI, 1998.
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Specific angular momentum - WikiMili, The Best Wikipedia Reader
In celestial mechanics, the specific relative angular momentum (often denoted
{\displaystyle {\vec {h}}}
{\displaystyle \mathbf {h} }
) of a body is the angular momentum of that body divided by its mass. [1] In the case of two orbiting bodies it is the vector product of their relative position and relative velocity, divided by the mass of the body in question.
Proof of constancy in the two body case
Specific relative angular momentum plays a pivotal role in the analysis of the two-body problem, as it remains constant for a given orbit under ideal conditions. "Specific" in this context indicates angular momentum per unit mass. The SI unit for specific relative angular momentum is square meter per second.
The specific relative angular momentum is defined as the cross product of the relative position vector
{\displaystyle \mathbf {r} }
and the relative velocity vector
{\displaystyle \mathbf {v} }
{\displaystyle \mathbf {h} =\mathbf {r} \times \mathbf {v} ={\frac {\mathbf {L} }{m}}}
{\displaystyle \mathbf {L} }
is the angular momentum vector, defined as
{\displaystyle \mathbf {r} \times m\mathbf {v} }
{\displaystyle \mathbf {h} }
vector is always perpendicular to the instantaneous osculating orbital plane, which coincides with the instantaneous perturbed orbit. It is not necessarily be perpendicular to the average orbital plane over time.
Distance vector
{\displaystyle \mathbf {r} }
, velocity vector
{\displaystyle \mathbf {v} }
, true anomaly
{\displaystyle \theta }
and flight path angle
{\displaystyle \phi }
{\displaystyle m_{2}}
in orbit around
{\displaystyle m_{1}}
. The most important measures of the ellipse are also depicted (among which, note that the true anomaly
{\displaystyle \theta }
is labeled as
{\displaystyle \nu }
Under certain conditions, it can be proven that the specific angular momentum is constant. The conditions for this proof include:
The mass of one object is much greater than the mass of the other one. (
{\displaystyle m_{1}\gg m_{2}}
The coordinate system is inertial.
Each object can be treated as a spherically symmetrical point mass.
No other forces act on the system other than the gravitational force that connects the two bodies.
The proof starts with the two body equation of motion, derived from Newton's law of universal gravitation:
{\displaystyle {\ddot {\mathbf {r} }}+{\frac {Gm_{1}}{r^{2}}}{\frac {\mathbf {r} }{r}}=0}
{\displaystyle \mathbf {r} }
is the position vector from
{\displaystyle m_{1}}
{\displaystyle m_{2}}
with scalar magnitude
{\displaystyle r}
{\displaystyle {\ddot {\mathbf {r} }}}
is the second time derivative of
{\displaystyle \mathbf {r} }
. (the acceleration)
{\displaystyle G}
The cross product of the position vector with the equation of motion is:
{\displaystyle \mathbf {r} \times {\ddot {\mathbf {r} }}+\mathbf {r} \times {\frac {Gm_{1}}{r^{2}}}{\frac {\mathbf {r} }{r}}=0}
{\displaystyle \mathbf {r} \times \mathbf {r} =0}
the second term vanishes:
{\displaystyle \mathbf {r} \times {\ddot {\mathbf {r} }}=0}
It can also be derived that:
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left(\mathbf {r} \times {\dot {\mathbf {r} }}\right)={\dot {\mathbf {r} }}\times {\dot {\mathbf {r} }}+\mathbf {r} \times {\ddot {\mathbf {r} }}=\mathbf {r} \times {\ddot {\mathbf {r} }}}
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left(\mathbf {r} \times {\dot {\mathbf {r} }}\right)=0}
Since the time derivative is equal to zero, the quantity
{\displaystyle \mathbf {r} \times {\dot {\mathbf {r} }}}
is constant. Using the velocity vector
{\displaystyle \mathbf {v} }
in place of the rate of change of position, and
{\displaystyle \mathbf {h} }
for the specific angular momentum:
{\displaystyle \mathbf {h} =\mathbf {r} \times \mathbf {v} }
This is different from the normal construction of momentum,
{\displaystyle \mathbf {r} \times \mathbf {p} }
, because it does not include the mass of the object in question.
{\displaystyle {\ddot {\mathbf {r} }}\times \mathbf {h} =-{\frac {\mu }{r^{2}}}{\frac {\mathbf {r} }{r}}\times \mathbf {h} }
The left hand side is equal to the derivative
{\textstyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\dot {\mathbf {r} }}\times \mathbf {h} \right)}
because the angular momentum is constant.
{\displaystyle -{\frac {\mu }{r^{3}}}\left(\mathbf {r} \times \mathbf {h} \right)=-{\frac {\mu }{r^{3}}}\left(\left(\mathbf {r} \cdot \mathbf {v} \right)\mathbf {r} -r^{2}\mathbf {v} \right)=-\left({\frac {\mu }{r^{2}}}{\dot {r}}\mathbf {r} -{\frac {\mu }{r}}\mathbf {v} \right)=\mu {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\mathbf {r} }{r}}\right)}
Setting these two expression equal and integrating over time leads to (with the constant of integration
{\displaystyle \mathbf {C} }
{\displaystyle {\dot {\mathbf {r} }}\times \mathbf {h} =\mu {\frac {\mathbf {r} }{r}}+\mathbf {C} }
Now this equation is multiplied (dot product) with
{\displaystyle \mathbf {r} }
and rearranged
{\displaystyle {\begin{aligned}\mathbf {r} \cdot \left({\dot {\mathbf {r} }}\times \mathbf {h} \right)&=\mathbf {r} \cdot \left(\mu {\frac {\mathbf {r} }{r}}+\mathbf {C} \right)\\\Rightarrow \left(\mathbf {r} \times {\dot {\mathbf {r} }}\right)\cdot \mathbf {h} &=\mu r+rC\cos \theta \\\Rightarrow h^{2}&=\mu r+rC\cos \theta \end{aligned}}}
Finally one gets the orbit equation [1]
{\displaystyle r={\frac {\frac {h^{2}}{\mu }}{1+{\frac {C}{\mu }}\cos \theta }}}
which is the equation of a conic section in polar coordinates with semi-latus rectum
{\textstyle p={\frac {h^{2}}{\mu }}}
{\textstyle e={\frac {C}{\mu }}}
The second law follows instantly from the second of the three equations to calculate the absolute value of the specific relative angular momentum. [1]
If one connects this form of the equation
{\textstyle \mathrm {d} t={\frac {r^{2}}{h}}\,\mathrm {d} \theta }
with the relationship
{\textstyle \mathrm {d} A={\frac {r^{2}}{2}}\,\mathrm {d} \theta }
for the area of a sector with an infinitesimal small angle
{\displaystyle \mathrm {d} \theta }
(triangle with one very small side), the equation
{\displaystyle \mathrm {d} t={\frac {2}{h}}\,\mathrm {d} A}
Kepler's third is a direct consequence of the second law. Integrating over one revolution gives the orbital period [1]
{\displaystyle T={\frac {2\pi ab}{h}}}
for the area
{\displaystyle \pi ab}
of an ellipse. Replacing the semi-minor axis with
{\displaystyle b={\sqrt {ap}}}
and the specific relative angular momentum with
{\displaystyle h={\sqrt {\mu p}}}
{\displaystyle T=2\pi {\sqrt {\frac {a^{3}}{\mu }}}}
There is thus a relationship between the semi-major axis and the orbital period of a satellite that can be reduced to a constant of the central body.
Classical central-force problem § Specific angular momentum
The Fresnel equations describe the reflection and transmission of light when incident on an interface between different optical media. They were deduced by Augustin-Jean Fresnel who was the first to understand that light is a transverse wave, even though no one realized that the "vibrations" of the wave were electric and magnetic fields. For the first time, polarization could be understood quantitatively, as Fresnel's equations correctly predicted the differing behaviour of waves of the s and p polarizations incident upon a material interface.
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuthal angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system.
In classical mechanics, the two-body problem is to predict the motion of two massive objects which are abstractly viewed as point particles. The problem assumes that the two objects interact only with one another; the only force affecting each object arises from the other one, and all other objects are ignored.
The rigid rotor is a mechanical model of rotating systems. An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space requires three angles, known as Euler angles. A special rigid rotor is the linear rotor requiring only two angles to describe, for example of a diatomic molecule. More general molecules are 3-dimensional, such as water, ammonia, or methane.
In electrodynamics, the Larmor formula is used to calculate the total power radiated by a nonrelativistic point charge as it accelerates. It was first derived by J. J. Larmor in 1897, in the context of the wave theory of light.
Introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788 from his work Mécanique analytique, Lagrangian mechanics is a formulation of classical mechanics and is founded on the stationary action principle.
The Binet equation, derived by Jacques Philippe Marie Binet, provides the form of a central force given the shape of the orbital motion in plane polar coordinates. The equation can also be used to derive the shape of the orbit for a given force law, but this usually involves the solution to a second order nonlinear ordinary differential equation. A unique solution is impossible in the case of circular motion about the center of force.
In classical mechanics, the central-force problem is to determine the motion of a particle in a single central potential field. A central force is a force that points from the particle directly towards a fixed point in space, the center, and whose magnitude only depends on the distance of the object to the center. In many important cases, the problem can be solved analytically, i.e., in terms of well-studied functions such as trigonometric functions.
1 2 3 4 Vallado, David A. (2001). Fundamentals of astrodynamics and applications (2nd ed.). Dordrecht: Kluwer Academic Publishers. pp. 20–30. ISBN 0-7923-6903-3.
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Noncentral F cumulative distribution function - MATLAB ncfcdf - MathWorks Australia
Compute Noncentral F Distribution cdf
Noncentral F cumulative distribution function
p = ncfcdf(x,nu1,nu2,delta)
p = ncfcdf(x,nu1,nu2,delta,'upper')
p = ncfcdf(x,nu1,nu2,delta) computes the noncentral F cdf at each value in x using the corresponding numerator degrees of freedom in nu1, denominator degrees of freedom in nu2, and positive noncentrality parameters in delta. nu1, nu2, and delta can be vectors, matrices, or multidimensional arrays that have the same size, which is also the size of p. A scalar input for x, nu1, nu2, or delta is expanded to a constant array with the same dimensions as the other inputs.
p = ncfcdf(x,nu1,nu2,delta,'upper') returns the complement of the noncentral F cdf at each value in x, using an algorithm that more accurately computes the extreme upper tail probabilities.
The noncentral F cdf is
F\left(x|{\nu }_{1},{\nu }_{2},\delta \right)=\sum _{j=0}^{\infty }\left(\frac{{\left(\frac{1}{2}\delta \right)}^{j}}{j!}{e}^{\frac{-\delta }{2}}\right)I\left(\frac{{\nu }_{1}\cdot x}{{\nu }_{2}+{\nu }_{1}\cdot x}|\frac{{\nu }_{1}}{2}+j,\frac{{\nu }_{2}}{2}\right)
where I(x|a,b) is the incomplete beta function with parameters a and b.
Compare the noncentral F cdf with δ = 10 to the F cdf with the same number of numerator and denominator degrees of freedom (5 and 20 respectively).
p1 = ncfcdf(x,5,20,10);
p = fcdf(x,5,20);
plot(x,p,'-',x,p1,'-')
cdf | ncfpdf | ncfinv | ncfstat | ncfrnd
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Mixture of nitric acid and hydrochloric acid in a 1:3 molar ratio
"🜇" redirects here. For the numismatic abbreviation, see List of numismatic abbreviations.
Aqua regia[note 1]
X3TT5X989E Y
[N+](=O)(O)[O-].Cl.Cl.Cl
Appearance Fuming liquid. Freshly prepared aqua regia is colorless, but it turns yellow, orange or red within seconds.
Density 1.01–1.21 g/cm3
Freshly prepared aqua regia to remove metal salt deposits
Aqua regia (/ˈreɪɡiə, ˈriːdʒiə/; from Latin, literally "regal water" or "royal water") is a mixture of nitric acid and hydrochloric acid, optimally in a molar ratio of 1:3.[note 2] Aqua regia is a fuming liquid. Freshly prepared aqua regia is colorless, but it turns yellow, orange or red within seconds, so named by alchemists because it can dissolve the noble metals gold and platinum, though not all metals.
1 Preparation and decomposition
3.1 Dissolving gold
3.2 Dissolving platinum
3.3 Precipitating dissolved platinum
3.4 Reaction with tin
3.5 Reaction with other substances
Preparation and decomposition[edit]
Upon mixing of concentrated hydrochloric acid and concentrated nitric acid, chemical reactions occur. These reactions result in the volatile products nitrosyl chloride and chlorine gas:
HNO3 + 3 HCl → NOCl + Cl2 + 2 H2O
as evidenced by the fuming nature and characteristic yellow color of aqua regia. As the volatile products escape from solution, aqua regia loses its potency. Nitrosyl chloride (NOCl) can further decompose into nitric oxide (NO) and elemental chlorine (Cl2):
2 NOCl → 2 NO + Cl2
This dissociation is equilibrium-limited. Therefore, in addition to nitrosyl chloride and chlorine, the fumes over aqua regia also contain nitric oxide (NO). Because nitric oxide readily reacts with atmospheric oxygen, the gases produced also contain nitrogen dioxide, NO2 (red fume):
2 NO + O2 → 2 NO2
Aqua regia is also used in etching and in specific analytic procedures. It is also used in some laboratories to clean glassware of organic compounds and metal particles. This method is preferred among most over the more traditional chromic acid bath for cleaning NMR tubes, because no traces of paramagnetic chromium can remain to spoil spectra.[1] While chromic acid baths are discouraged[according to whom?] because of the high toxicity of chromium and the potential for explosions, aqua regia is itself very corrosive and has been implicated in several explosions due to mishandling.[2]
While local regulations may vary, aqua regia may be disposed of by careful neutralization, before being poured down the sink. If there is contamination by dissolved metals, the neutralized solution should be collected for disposal.[3][4]
Dissolving gold[edit]
Aqua regia dissolves gold, although neither constituent acid will do so alone. Nitric acid is a powerful oxidizer, which will actually dissolve a virtually undetectable amount of gold, forming gold(III) ions (Au3+). The hydrochloric acid provides a ready supply of chloride ions (Cl−), which react with the gold ions to produce tetrachloroaurate(III) anions ([AuCl4]−), also in solution. The reaction with hydrochloric acid is an equilibrium reaction that favors formation of tetrachloroaurate(III) anions. This results in a removal of gold ions from solution and allows further oxidation of gold to take place. The gold dissolves to become chloroauric acid. In addition, gold may be dissolved by the chlorine present in aqua regia. Appropriate equations are:
Au + 3 HNO
3 + 4 HCl
{\displaystyle {\ce {<=>>}}}
[AuCl
Au + HNO
{\displaystyle {\ce {<=>>}}}
Solid tetrachloroauric acid may be isolated by evaporating the excess aqua regia, and decomposing the residual nitric acid by repeatedly heating the solution with additional hydrochloric acid. That step reduces nitric acid (see decomposition of aqua regia). If elemental gold is desired, it may be selectively reduced with reducing agents such as sulfur dioxide, hydrazine, oxalic acid, etc.[5] The equation for the reduction of oxidized gold (Au3+) by sulfur dioxide (SO2) is the following:
2 [AuCl4]−(aq) + 3 SO2(g) + 6 H2O(l) → 2 Au(s) + 12 H+(aq) + 3 SO2−4(aq) + 8 Cl−(aq)
Dissolution of gold by aqua regia.
Initial state of the transformation.
Intermediate state of the transformation.
Final state of the transformation.
Dissolving platinum[edit]
Similar equations can be written for platinum. As with gold, the oxidation reaction can be written with either nitric oxide or nitrogen dioxide as the nitrogen oxide product:
Pt(s) + 4 NO−3(aq) + 8 H+(aq) → Pt4+(aq) + 4 NO2(g) + 4 H2O(l)
3 Pt(s) + 4 NO−3(aq) + 16 H+(aq) → 3 Pt4+(aq) + 4 NO(g) + 8 H2O(l)
The oxidized platinum ion then reacts with chloride ions resulting in the chloroplatinate ion:
Pt4+(aq) + 6 Cl−(aq) → [PtCl6]2−(aq)
Experimental evidence reveals that the reaction of platinum with aqua regia is considerably more complex. The initial reactions produce a mixture of chloroplatinous acid (H2[PtCl4]) and nitrosoplatinic chloride ([NO]2[PtCl4]). The nitrosoplatinic chloride is a solid product. If full dissolution of the platinum is desired, repeated extractions of the residual solids with concentrated hydrochloric acid must be performed:
2 Pt(s) + 2 HNO3(aq) + 8 HCl(aq) → [NO]2[PtCl4](s) + H2[PtCl4](aq) + 4 H2O(l)
[NO]2[PtCl4](s) + 2 HCl(aq) ⇌ H2[PtCl4](aq) + 2 NOCl(g)
The chloroplatinous acid can be oxidized to chloroplatinic acid by saturating the solution with molecular chlorine (Cl2) while heating:
H2[PtCl4](aq) + Cl2(g) → H2[PtCl6](aq)
Dissolving platinum solids in aqua regia was the mode of discovery for the densest metals, iridium and osmium, both of which are found in platinum ores and are not dissolved by aqua regia, instead collecting as insoluble metallic powder (elemental Ir, Os) on the base of the vessel.
Dissolution of platinum[note 3] by aqua regia.
Final state of the transformation
(four days later).
Precipitating dissolved platinum[edit]
As a practical matter, when platinum group metals are purified through dissolution in aqua regia, gold (commonly associated with PGMs) is precipitated by treatment with iron(II) chloride. Platinum in the filtrate, as hexachloroplatinate(IV), is converted to ammonium hexachloroplatinate by the addition of ammonium chloride. This ammonium salt is extremely insoluble, and it can be filtered off. Ignition (strong heating) converts it to platinum metal:[6]
3 [NH4]2[PtCl6] → 3 Pt + 2 N2 + 2 [NH4]Cl + 16 HCl
Unprecipitated hexachloroplatinate(IV) is reduced with elemental zinc, and a similar method is suitable for small scale recovery of platinum from laboratory residues.[7]
Reaction with tin[edit]
Aqua regia reacts with tin to form tin(IV) chloride, containing tin in its highest oxidation state:
4 HCl + 2 HNO3 + Sn → SnCl4 + NO2 + NO + 3 H2O
Reaction with other substances[edit]
It can react with iron pyrite to form Iron(III) chloride:
FeS2 + 5 HNO3 + 3 HCl → FeCl3 + 2 H2SO4 + 5 NO + 2 H2O
Aqua regia first appeared in the De inventione veritatis ("On the Discovery of Truth") by pseudo-Geber (after c. 1300), who produced it by adding sal ammoniac (ammonium chloride) to nitric acid.[8] The preparation of aqua regia by directly mixing hydrochloric acid with nitric acid only became possible after the discovery in the late sixteenth century of the process by which free hydrochloric acid can be produced.[9]
The fox in Basil Valentine's Third Key represents aqua regia, Musaeum Hermeticum, 1678
The third of Basil Valentine's keys (ca. 1600) shows a dragon in the foreground and a fox eating a rooster in the background. The rooster symbolizes gold (from its association with sunrise and the sun's association with gold), and the fox represents aqua regia. The repetitive dissolving, heating, and redissolving (the rooster eating the fox eating the rooster) leads to the buildup of chlorine gas in the flask. The gold then crystallizes in the form of gold(III) chloride, whose red crystals Basil called "the rose of our masters" and "the red dragon's blood".[10] The reaction was not reported again in the chemical literature until 1895.[11]
Antoine Lavoisier called aqua regia nitro-muriatic acid in 1789.[12]
When Germany invaded Denmark in World War II, Hungarian chemist George de Hevesy dissolved the gold Nobel Prizes of German physicists Max von Laue (1914) and James Franck (1925) in aqua regia to prevent the Nazis from confiscating them. The German government had prohibited Germans from accepting or keeping any Nobel Prize after jailed peace activist Carl von Ossietzky had received the Nobel Peace Prize in 1935. De Hevesy placed the resulting solution on a shelf in his laboratory at the Niels Bohr Institute. It was subsequently ignored by the Nazis who thought the jar—one of perhaps hundreds on the shelving—contained common chemicals. After the war, de Hevesy returned to find the solution undisturbed and precipitated the gold out of the acid. The gold was returned to the Royal Swedish Academy of Sciences and the Nobel Foundation. They re-cast the medals and again presented them to Laue and Franck.[13][14]
Green death – Aggressive solution used to test the resistance of metals to corrosion
Piranha solution – Oxidizing acid mixture containing sulfuric acid and hydrogen peroxide sometimes also used to clean glassware.
^ The information in the infobox is specific to a molar ratio of 1:3 between nitric acid and hydrochloric acid.
^ The relative concentrations of the two acids in water differ; values could be 65% w/v for nitric acid and 35% w/v for hydrochloric acid – that is, the actual HNO3:HCl mass ratio is less than 1:2.
^ A platinum Soviet commemorative coin to be precise.
^ Hoffman, R. (10 March 2005) How to make an NMR sample, Hebrew University. Accessed 31 October 2006.
^ American Industrial Hygiene Association, Laboratory Safety Incidents: Explosions. Accessed 8 September 2010.
^ Committee on Prudent Practices for Handling, Storage, and Disposal of Chemicals in Laboratories, National Research Council (1995). Prudent Practices in the Laboratory: Handling and Disposal of Chemicals (free fulltext). National Academies Press. pp. 160–161. doi:10.17226/4911. ISBN 978-0-309-05229-0. {{cite book}}: CS1 maint: uses authors parameter (link)
^ "Aqua Regia". Laboratory Safety Manual. Princeton University. Archived from the original on 27 December 2012.
^ Renner, Hermann; Schlamp, Günther; Hollmann, Dieter; Lüschow, Hans Martin; Tews, Peter; Rothaut, Josef; Dermann, Klaus; Knödler, Alfons; et al. "Gold, Gold Alloys, and Gold Compounds". Ullmann's Encyclopedia of Industrial Chemistry. Weinheim: Wiley-VCH. doi:10.1002/14356007.a12_499.
^ Hunt, L. B.; Lever, F. M. (1969). "Platinum Metals: A Survey of Productive Resources to industrial Uses" (PDF). Platinum Metals Review. 13 (4): 126–138.
^ Kauffman, George B.; Teter, Larry A.; Rhoda, Richard N. (1963). Recovery of Platinum from Laboratory Residues. Inorg. Synth. Inorganic Syntheses. Vol. 7. p. 232. doi:10.1002/9780470132388.ch61. ISBN 9780470132388.
^ Karpenko, Vladimír; Norris, John A. (2002). "Vitriol in the History of Chemistry". Chemické listy. 96 (12): 997–1005. p. 1002. As Karpenko & Norris note, the uncertain dating of the pseudo-Geber corpus (which was probably written by more than one author) renders the dating of aqua regia equally uncertain.
^ Multhauf, Robert P. (1966). The Origins of Chemistry. London: Oldbourne. p. 208, note 29; cf. p. 142, note 79.
^ Principe, Lawrence M. (2013). The Secrets of Alchemy. Chicago: University of Chicago Press. ISBN 978-0226682952. pp. 149-153.
^ Rose, Thomas Kirke (1895). "The Dissociation of Chloride of Gold". Journal of the Chemical Society. 67: 881–904. doi:10.1039/CT8956700881. Cf. Principe 2013, p. 152.
^ Lavoisier, Antoine (1790). Elements of Chemistry,. in a New Systematic Order, Containing All the Modern Discoveries. Edinburgh: William Creech. p. 116. ISBN 978-0486646244. .
Aqua Regia at The Periodic Table of Videos (University of Nottingham)
Demonstration of Gold Coin Dissolving in Acid (Aqua Regia)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Aqua_regia&oldid=1089434633"
Oxidizing mixtures
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We study the dynamics of interacting fermionic systems, in the mean-field regime. We consider initial states which are close to quasi-free states and prove that, under suitable assumptions on the inital data and on the many-body interaction, the quantum evolution of the system is approximated by a time-dependent quasi-free state. In particular we prove that the evolution of the reduced one-particle density matrix converges, as the number of particles goes to infinity, to the solution of the time-dependent Hartree-Fock equation. Our theorems allow to describe the dynamics of both pure states (zero temperature states) and mixed states (positive temperature states). Our results hold for all times, and give effective estimates on the rate of convergence towards the Hartree-Fock evolution. The results on pure states are based on joint works with N. Benedikter and B. Schlein, [5, 6]; while those on mixed states are based on a joint work with N. Benedikter, V. Jaksic, C. Saffirio and B. Schlein, [7].
author = {Marcello Porta},
title = {Mean-field evolution of fermionic systems},
TI - Mean-field evolution of fermionic systems
%T Mean-field evolution of fermionic systems
Marcello Porta. Mean-field evolution of fermionic systems. Séminaire Laurent Schwartz — EDP et applications (2014-2015), Talk no. 8, 13 p. doi : 10.5802/slsedp.68. https://slsedp.centre-mersenne.org/articles/10.5802/slsedp.68/
[1] H. Araki and W. Wyss. Representations of canonical anticommutation relations. Helv. Phys. Acta 37 (1964), 136.
[2] A. Athanassoulis, T. Paul, F. Pezzotti and M. Pulvirenti. Strong semiclassical approximation of Wigner functions for the Hartree dynamics. Atti della Accademia Nazionale dei Lincei. Rendiconti Lincei. Matematica e Applicazioni. 22, 525–552 (2011).
[3] V. Bach. Error bound for the Hartree-Fock energy of atoms and molecules. Comm. Math. Phys. 147 (1992), no. 3, 527–548.
[4] C. Bardos, F. Golse, A. D. Gottlieb, and N. J. Mauser. Mean field dynamics of fermions and the time-dependent Hartree-Fock equation. J. Math. Pures Appl. (9) 82 (2003), no. 6, 665–683.
[5] N. Benedikter, M. Porta and B. Schlein. Mean-field evolution of fermionic systems. Comm. Math. Phys. 331, 1087–1131 (2014).
[6] N. Benedikter, M. Porta and B. Schlein. Mean-field dynamics of fermions with relativistic dispersion. J. Math. Phys. 55, 021901 (2014).
[7] N. Benedikter, V. Jakšić, M. Porta, C. Saffirio and B. Schlein. Mean-field evolution of fermionic mixed states. arXiv:1411.0843
[8] W. Braun and K. Hepp. The Vlasov dynamics and its fluctuations in the
1/N
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[9] J. Derezinśki and C. Gérard. Mathematics of quantization and quantum fields. Cambridge University press (2013).
[10] A. Elgart, L. Erdős, B. Schlein, and H.-T. Yau. Nonlinear Hartree equation as the mean field limit of weakly coupled fermions. J. Math. Pures Appl. (9) 83 (2004), no. 10, 1241–1273.
[11] L. Erdős, B. Schlein, and H.-T. Yau. Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems. Inv. Math. 167 (2006), 515–614.
[12] J. Fröhlich and A. Knowles. A microscopic derivation of the time-dependent Hartree-Fock equation with Coulomb two-body interaction. J. Stat. Phys. 145 (2011), no. 1, 23–50.
[13] J. Fröhlich and E. Lenzmann. Dynamical collapse of white dwarfs in Hartree- and Hartree-Fock theory. Comm. Math. Phys. 274 (2007), 737–750.
[14] G. M. Graf and J. P. Solovej. A correlation estimate with applications to quantum systems with Coulomb interactions. Rev. Math. Phys. 6 (1994), 977–997.
[15] C. Hainzl and B. Schlein. Stellar collapse in the time-dependent Hartree-Fock approximation. Comm. Math. Phys. 287 (2009), 705–717.
[16] O. E. Lanford III. The evolution of large classical system. Dynamical Systems, theory and applications. Lecture Notes in Physics 38, 1–111 (1975).
[17] E. H. Lieb, R. Seiringer, J. .P. Solovej and J. Yngvason. The mathematics of the Bose gas and its condensation. Oberwolfach seminars 34, Birkhäuser (2005).
[18] E. H. Lieb and B. Simon. The Thomas-Fermi theory of atoms, molecules and solids. Adv. Math. 23, 22–116 (1977).
[19] P. L. Lions and T. Paul. Sur les mesures de Wigner. Revista matemática iberoamericana 9, 553–618 (1993).
[20] P. A. Markowich and N. J. Mauser. The classical limit of a self-consistent quantum-Vlasov equation in 3D. Mathematical Models and Methods in Applied Sciences 3, 109 (1993).
[21] H. Narnhofer and G. L. Sewell. Vlasov hydrodynamics of a quantum mechanical model. Comm. Math. Phys. 79 (1981), no. 1, 9–24.
[22] S. Petrat and P. Pickl. A New Method and a New Scaling For Deriving Fermionic Mean-field Dynamics. arXiv:1409.0480
[23] J. P. Solovej. Many Body Quantum Mechanics. Lecture Notes. Summer 2007. Available at http://www.mathematik.uni-muenchen.de/~sorensen/Lehre/SoSe2013/MQM2/skript.pdf
[24] H. Spohn. On the Vlasov hierarchy, Math. Methods Appl. Sci. 3 (1981), no. 4, 445–455.
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Asymptotic Stability of Zakharov-Kuznetsov solitons
In this report, we review the proof of the asymptotic stability of the Zakharov-Kuznetsov solitons in dimension two. Those results were recently obtained in a joint work with Raphaël Côte, Claudio Muñoz and Gideon Simpson.
author = {Didier Pilod},
title = {Asymptotic {Stability} of {Zakharov-Kuznetsov~solitons}},
TI - Asymptotic Stability of Zakharov-Kuznetsov solitons
%T Asymptotic Stability of Zakharov-Kuznetsov solitons
Didier Pilod. Asymptotic Stability of Zakharov-Kuznetsov solitons. Séminaire Laurent Schwartz — EDP et applications (2014-2015), Talk no. 13, 12 p. doi : 10.5802/slsedp.73. https://slsedp.centre-mersenne.org/articles/10.5802/slsedp.73/
[1] H. Beresticky and P. L. Lions, Nonlinear scalar field equations, Arch. Rational Mech. Anal., 82 (1983), 313–345.
[2] F. Béthuel, P. Gravejat and D. Smets, Asymptotic stability in the energy space for dark solitons of the Gross-Pitaevskii equation, to appear in Ann. Sci. Éc. Norm. Supér., (2014) arXiv:1212.5027.
[3] R. Côte, “Solitons et Dispersion”, Habilitation à Diriger des Recherches, Université de cergy-Pontoise, 2014.
[4] R. Côte, C. Muñoz, D. Pilod and G. Simpson, Asymptotic stability of high-dimensional Zakharov-Kuznetsov solitons, preprint (2014), arXiv:1406.3196.
[5] A. de Bouard, Stability and instability of some nonlinear dispersive solitary waves in higher dimension, Proc. Royal Soc. Edinburgh, 126 (1996), 89–112.
[6] K. El Dika, Asymptotic Stability of solitary waves for the Benjamin-Bona-Mahony equation, Disc. Cont. Dyn. Syst., 13 (2005), 583–622.
[7] A. V. Faminskii, The Cauchy problem for the Zakharov-Kuznetsov equation, Differential Equations 31 (1995), no. 6, 1002–1012.
[8] P. Gravejat and D. Smets, Asymptotic stability of the black soliton for the Gross-Pitaevskii equation, Proc. London Math. Soc. (2015) doi:10.1112/plms/pdv025.
[9] A. Grünrock and S. Herr, The Fourier restriction norm method for the Zakharov-Kuznetsov equation, Disc. Contin. Dyn. Syst. Ser. A, 34 (2014), 2061–2068.
[10] D. Han-Kwan, From Vlasov-Poisson to Korteweg-de Vries and Zakharov-Kuznetsov, Comm. Math. Phys., 324 (2013), 961–993.
[11] T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in Applied Mathematics, 93-128, Adv. Appl. Math. Suppl. Stud., 8, Academic Press, New York, 1983.
[12] C. E. Kenig and Y. Martel, Asymptotic stability of solitons for the Benjamin-Ono equation, Rev. Mat. Iberoamericana, 25 (2009), no. 3, 909–970.
[13] E. A. Kuznetsov and V. E. Zakharov, On three dimensional solitons, Sov. Phys. JETP., 39 (1974), 285–286.
[14] M. K. Kwong, Uniqueness of positive radial solutions of
\Delta u-u+{u}^{p}
{ℝ}^{n}
, Arch. Rational Mech. Anal., 105 (1989), 243–266.
[15] D. Lannes, F. Linares and J.-C. Saut, The Cauchy problem for the Euler-Poisson system and derivation of the Zakharov-Kuznetsov equation, Prog. Nonlinear Diff. Eq. Appl., 84 (2013), 181–213.
[16] C. Laurent and Y. Martel, Smoothness and exponential decay of
{L}^{2}
-compact solutions of the generalized KdV equations, Comm. Part. Diff. Eq., 29 (2005), 157–171.
[17] F. Linares and A. Pastor, Well-posedness for the two-dimensional modified Zakharov-Kuznetsov equation, SIAM J. Math. Anal., 41 (2009), no. 4, 1323–1339.
[18] Y. Martel, Linear Problems related to asymptotic stability of solitons of the generalized KdV equations, SIAM J. Math. Anal., 38 (2006), 759–781.
[19] Y. Martel and F. Merle, Asymptotic stability of solitons for subcritical generalized KdV equations, Arch. Ration. Mech. Anal., 157 (2001), 219–254.
[20] Y. Martel and F. Merle, Asymptotic Stability of solitons of the subcritical gKdV equations revisited, Nonlinearity, 18 (2005), 55–80.
[21] Y. Martel and F. Merle, Asymptotic stability of solitons of the gKdV equations with general nonlinearity, Math. Ann., 341 (2008), 391–427.
[22] Y. Martel, F. Merle, and T.P. Tsai, Stability and asymptotic stability in the energy space of the sum of
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[23] L. Molinet and D. Pilod, Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications, Ann. Inst. H. Poincaré, Annal. Non., 32 (2015), 347–371.
[24] R. L. Pego and M. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys., 164 (1994), 305–349.
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Solve it Q If 1-ιx1+ιx=a+ιb and a2+b2=1, where a,b∈R and ι=iota, then x is equal to - Maths - Complex Numbers - 12329045 | Meritnation.com
\frac{1-\iota x}{1+\iota x}=a+\iota b and {a}^{2}+{b}^{2}=1
a,b\in R and \iota =iota
https://www.meritnation.com/ask-answer/question/show-that-a-real-value-of-x-will-satify-the-equation-1-ix-1/complex-numbers-and-quadratic-equations/3989385
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Great_circle Knowpia
A great circle, also known as an orthodrome, of a sphere is the intersection of the sphere and a plane that passes through the center point of the sphere. A great circle is the largest circle that can be drawn on any given sphere. Any diameter of any great circle coincides with a diameter of the sphere, and therefore all great circles have the same center and circumference as each other. This special case of a circle of a sphere is in opposition to a small circle, that is, the intersection of the sphere and a plane that does not pass through the center. Every circle in Euclidean 3-space is a great circle of exactly one sphere.
For most pairs of distinct points on the surface of a sphere, there is a unique great circle through the two points. The exception is a pair of antipodal points, for which there are infinitely many great circles. The minor arc of a great circle between two points is the shortest surface-path between them. In this sense, the minor arc is analogous to “straight lines” in Euclidean geometry. The length of the minor arc of a great circle is taken as the distance between two points on a surface of a sphere in Riemannian geometry where such great circles are called Riemannian circles. These great circles are the geodesics of the sphere.
The disk bounded by a great circle is called a great disk: it is the intersection of a ball and a plane passing through its center. In higher dimensions, the great circles on the n-sphere are the intersection of the n-sphere with 2-planes that pass through the origin in the Euclidean space Rn + 1.
Derivation of shortest pathsEdit
Consider the class of all regular paths from a point
{\displaystyle p}
to another point
{\displaystyle q}
. Introduce spherical coordinates so that
{\displaystyle p}
coincides with the north pole. Any curve on the sphere that does not intersect either pole, except possibly at the endpoints, can be parametrized by
{\displaystyle \theta =\theta (t),\quad \phi =\phi (t),\quad a\leq t\leq b}
provided we allow
{\displaystyle \phi }
to take on arbitrary real values. The infinitesimal arc length in these coordinates is
{\displaystyle ds=r{\sqrt {\theta '^{2}+\phi '^{2}\sin ^{2}\theta }}\,dt.}
So the length of a curve
{\displaystyle \gamma }
{\displaystyle p}
{\displaystyle q}
is a functional of the curve given by
{\displaystyle S[\gamma ]=r\int _{a}^{b}{\sqrt {\theta '^{2}+\phi '^{2}\sin ^{2}\theta }}\,dt.}
According to the Euler–Lagrange equation,
{\displaystyle S[\gamma ]}
is minimized if and only if
{\displaystyle {\frac {\sin ^{2}\theta \phi '}{\sqrt {\theta '^{2}+\phi '^{2}\sin ^{2}\theta }}}=C}
{\displaystyle C}
{\displaystyle t}
-independent constant, and
{\displaystyle {\frac {\sin \theta \cos \theta \phi '^{2}}{\sqrt {\theta '^{2}+\phi '^{2}\sin ^{2}\theta }}}={\frac {d}{dt}}{\frac {\theta '}{\sqrt {\theta '^{2}+\phi '^{2}\sin ^{2}\theta }}}.}
From the first equation of these two, it can be obtained that
{\displaystyle \phi '={\frac {C\theta '}{\sin \theta {\sqrt {\sin ^{2}\theta -C^{2}}}}}}
Integrating both sides and considering the boundary condition, the real solution of
{\displaystyle C}
is zero. Thus,
{\displaystyle \phi '=0}
{\displaystyle \theta }
can be any value between 0 and
{\displaystyle \theta _{0}}
, indicating that the curve must lie on a meridian of the sphere. In Cartesian coordinates, this is
{\displaystyle x\sin \phi _{0}-y\cos \phi _{0}=0}
The Funk transform integrates a function along all great circles of the sphere.
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Planck constant - Simple English Wikipedia, the free encyclopedia
A commemoration plaque for Max Planck on his discovery of Planck's constant, in front of Humboldt University, Berlin. English translation: "Max Planck, discoverer of the elementary quantum of action h, taught in this building from 1889 to 1928."
Max Planck, after whom the Planck constant is named
The Planck constant (Planck's constant) says how much the energy of a photon increases, when the frequency of its electromagnetic wave increases by 1 (In SI Units). It is named after the physicist Max Planck. The Planck constant is a fundamental physical constant. It is written as h.
The Planck constant has dimensions of physical action: energy multiplied by time, or momentum multiplied by distance. In SI units, the Planck constant is expressed in joule seconds (J⋅s) or (N⋅m⋅s) or (kg⋅m2⋅s−1). The symbols are defined here.
In SI Units the Planck constant is exactly 6.62607015×10−34 J·s (by definition).[1] Scientists have used this quantity to calculate measurements like the Planck length and the Planck time.
Planck h=WL= Wb/2P 4C/3X=2WbC/3XP. Magnetron W=Wb/2P Electron L=4C/3X = 25e/3 =(13U1d)
1.1 Light: waves or particles?
1.2 Black body radiators
1.2.1 Rayleigh-Jeans Law
1.3 Quantum theory of light
2.3 Colour of light emitting diodes
3 Value of the Planck constant and the kilogram redefinition
3.1 Value of Theoretical Planck constant
λ radiation wavelength
ν radiation frequency
Illustration taken from Newton's original letter to the Royal Society (1 January 1671 [Julian calendar]). S represents sunlight. The light between the planes BC and DE are in colour. These colours are recombined to form sunlight on the plane GH
Between 1670 and 1900 scientists discussed the nature of light. Some scientists believed that light consisted of many millions of tiny particles. Other scientists believed that light was a wave.[2]
Light: waves or particles?[change | change source]
In 1678, Christiaan Huygens wrote the book Traité de la lumiere ("Treatise on light"). He believed that light was made up of waves. He said that light could not be made up of particles because light from two beams do not bounce off each other.[3][4] In 1672, Isaac Newton wrote the book Opticks. He believed that light was made up of red, yellow and blue particles which he called corpusles. Newton explained this by his "two prism experiment". The first prism broke light up into different colours. The second prism merged these colours back into white light.[5][6]
During the 18th century, Newton's theory was given the most attention.[7] In 1803, Thomas Young described the "double-slit experiment".[4] In this experiment, light going through two narrow slits interferes with itself. This causes a pattern which shows that light is made up of waves. For the rest of the nineteenth century, the wave theory of light was given the most attention. In the 1860s, James Clerk Maxwell developed equations that described electromagnetic radiation as waves.
The theory of electromagnetic radiation treats light, radio waves, microwaves and many other types of wave as the same thing except that they have different wavelengths. The wavelength of the light we can see with our eyes is roughly between 400 and 600 nm.[Note 1] The wavelength of radio waves varies from 10 m to 1500 m and the wavelength of microwaves is about 2 cm. In a vacuum, all electromagnetic waves travel at the speed of light. The frequency of the electromagnetic wave is given by:
{\displaystyle \nu ={\frac {c}{\lambda }}}
The symbols are defined here.
Black body radiators[change | change source]
All warm things give off thermal radiation, which is electromagnetic radiation. For most things on Earth this radiation is in the infra-red range, but something very hot (1000 °C or more), gives off visible radiation, that is, light. In the late 1800's many scientists studied the wavelengths of electromagnetic radiation from black-body radiators at different temperatures.
Rayleigh-Jeans Law[change | change source]
Rayleigh-Jeans curve and Planck's curve plotted against photon wavelength.
Lord Rayleigh first published the basics of the Rayleigh-Jeans law in 1900. The theory was based on the Kinetic theory of gasses. Sir James Jeans published a more complete theory in 1905. The law relates the quantity and wavelength of electromagnetic energy given off by a black body radiator at different temperatures. The equation describing this is:[8]
{\displaystyle B_{\lambda }(T)={\frac {2ckT}{\lambda ^{4}}}}
For long-wavelength radiation, the results predicted by this equation corresponded well with practical results obtained in a laboratory. However, for short wavelengths (ultraviolet light) the difference between theory and practice were so large that it earned the nickname "the ultra-violet catastrophe".
Planck's Law[change | change source]
in 1895 Wien published the results of his studies into the radiation from a black body. His formula was:
{\displaystyle B_{\lambda }(T)={\frac {2hc^{2}}{\lambda ^{5}}}e^{-{\frac {hc}{\lambda kT}}}}
This formula worked well for short wavelength electromagnetic radiation, but did not work well with long wavelengths.
In 1900 Max Planck published the results of his studies. He tried to develop an expression for black-body radiation expressed in terms of wavelength by assuming that radiation consisted of small quanta and then to see what happened if the quanta were made infinitely small. (This is a standard mathematical approach). The expression was:[9]
{\displaystyle B_{\lambda }(T)={\frac {2hc^{2}}{\lambda ^{5}}}~{\frac {1}{e^{\frac {hc}{\lambda kT}}-1}}}
If the wavelength of light is allowed to become very large, then it can be shown that the Raleigh-Jeans and the Planck relationships are almost identical.
He calculated h and k and found that
h = 6.55×10−27 erg·sec.
k = 1.34×10−16 erg·deg-1.
The values are close to the modern day accepted values of 6.62606×10−34 and 1.38065×10−16 respectively. The Planck law agrees well with the experimental data, but its full significance was only appreciated several years later.
Quantum theory of light[change | change source]
Solway Conference 1911. Planck, Einstein and Jeans are standing. Planck is second from the left. Einstein is second from the right. Jeans is fifth from the right. Wien is seated, third from the right
It turns out that electrons are dislodged by the photoelectric effect if light reaches a threshold frequency. Below this no electrons can be emitted from the metal. In 1905 Albert Einstein published a paper explaining the effect. Einstein proposed that a beam of light is not a wave propagating through space, but rather a collection of discrete wave packets (photons), each with energy. Einstein said that the effect was due to a photon striking an electron. This demonstrated the particle nature of light.
Einstein also found that electromagnetic radiation with a long wavelength had no effect. Einstein said that this was because the "particles" did not have enough energy to disturb the electrons.[10][11][12]
Planck suggested[9] that the energy of each photon was related to the photon frequency by the Planck constant. This could be written mathematically as:
{\displaystyle E=h\nu ={\frac {hc}{\lambda }}}
Planck received the Nobel Prize in 1918 in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta. In 1921 Einstein received the Nobel Prize for linking the Planck constant to the photoelectric effect.
Application[change | change source]
The Planck constant is of importance in many applications. A few are listed below.
Bohr model of the atom[change | change source]
Bohr's model of the atom. An electron falling from the n=3 shell to the n=2 shell loses energy. This energy is carried away as a single photon.
Visible spectrum of Neon. Each line represents a different pair of energy levels.
In 1913 Niels Bohr published the Bohr model of the structure of an atom. Bohr said that the angular momentum of the electrons going around the nucleus can only have certain values. These values are given by the equation
{\displaystyle L=n{\frac {h}{2\pi }}}
L = angular momentum associated with a level.
n = positive integer.
h = Planck constant.
The Bohr model of the atom can be used to calculate the energy of electrons at each level. Electrons will normally fill up the lowest numbered states of an atom. If the atom receives energy from, for example, an electric current, electrons will be excited into a higher state. The electrons will then drop back to a lower state and will lose their extra energy by giving off a photon. Because the energy levels have specific values, the photons will have specific energy levels. Light emitted in this way can be split into different colours using a prism. Each element has its own pattern. The pattern for neon is shown alongside.
Heisenberg's uncertainty principle[change | change source]
In 1927 Werner Heisenberg published the uncertainty principle. The principle states that it is not possible to make a measurement without disturbing the thing being measured. It also puts a limit on the minimum disturbance caused by making a measurement.
In the macroscopic world these disturbances make very little difference. For example, if the temperature of a flask of liquid is measured, the thermometer will absorb a small amount of energy as it heats up. This will cause a small error in the final reading, but this error is small and not important.
In quantum mechanics things are different. Some measurements are made by looking at the pattern of scattered photons. One such example is Compton scattering. If both the position and momentum of a particle is being measured, the uncertainty principle states that there is a trade-off between the accuracy with which the momentum is measured and the accuracy with which the position is measured. The equation that describes this trade-off is:
{\displaystyle \Delta x\,\Delta p\geqslant {\frac {h}{4\pi }}\qquad \qquad \qquad }
Colour of light emitting diodes[change | change source]
Simple LED circuit that illustrates use of the Planck constant. The colour of the light emitted depends on the voltage drop across the diode. The wavelength of the light can be calculated using the Planck constant.[13]
In the electric circuit shown on the right, the voltage drop across the light emitting diode (LED) depends on the material of the LED. For silicon diodes the drop is 0.6 V. However for LEDs it is between 1.8 V and 2.7 V. This information enables a user to calculate the Planck constant.[13]
The energy needed for one electron to jump the potential barrier in the LED material is given by
{\displaystyle E=Q_{e}V_{L}\,}
Qe is the charge on one electron.
VL is the voltage drop across the LED.
When the electron decays back again, it emits one photon of light. The energy of the photon is given by the same equation used in the photoelectric effect. If these equations are combined, the wavelength of light and the voltage are related by
{\displaystyle \lambda ={\frac {hc}{V_{L}Q_{e}}}\,}
The table below can be calculated from this relationship.
(nm)[Note 2]
red light 650 1.89
green light 550 2.25
blue light 470 2.62
Value of the Planck constant and the kilogram redefinition[change | change source]
Since its discovery, measurements of h have become much better. Planck first quoted the value of h to be 6.55×10−27 erg·sec. This value is within 5% of the current value.
As of 3 March 2014, the best measurements of h in SI units is 6.62606957×10−34 J·s. The equivalent figure in cgs units is 6.62606957×10−27 erg·sec. The relative uncertainty of h is 4.4×10−8.
The reduced Planck constant (ħ) is a value that is sometimes used in quantum mechanics. It is defined by
{\displaystyle \hbar ={\frac {h}{2\pi }}}
Planck units are sometimes used in quantum mechanics instead of SI. In this system the reduced Planck constant has a value of 1, so the value of the Planck constant is 2π.
Plancks constant can now be measured with very high precision. This has caused the BIPM to consider a new definition for the kilogram. The international prototype kilogram is no longer used to define the kilogram. Instead the BIPM defines the Planck constant to have an exact value. Scientists use this value and the definitions of the metre and the second to define the kilogram.[14]
Value of Theoretical Planck constant[change | change source]
The Planck constant can also be mathematically derived:
{\displaystyle h={\frac {\mu _{0}\pi }{12c^{3}}}{[{q_{0}}{[0.9163a_{0}]}^{2}]^{2}}{f_{1r}}^{5}\cdot {s}=6.63\times 10^{-34}J\cdot s}
{\displaystyle \mu _{0}}
is the permeability of free space,
{\displaystyle c}
{\displaystyle q_{0}}
is the electric charge of electron,
{\displaystyle a_{0}}
is the Bohr's radius, and
{\displaystyle f_{1r}}
is the frequency of revolution of the electron in a hydrogen atom
{\displaystyle (f_{1r}=3.29\times 10^{15}rev/s)}
. When these constant values are substituted to the theoretical Planck constant, the theoretical Planck constant value is exactly equal to the experimental value.[15]
The Planck's constant elementary formula in terms of proton-to-electron mass ratio, the charge of electron, speed of light and vacuum permittivity is derived in.[16] It is expressed as follows:
{\displaystyle h={\frac {e^{2}}{c\,\varepsilon _{0}}}{\sqrt {\pi \,{\sqrt {\frac {2}{3}}}\,\,{\frac {m_{p}}{m_{e}}}}}}
wher{\displaystyle e}
is the elementary charge of an electron,
{\displaystyle m_{p}}
the mass of a proton,
{\displaystyle m_{e}}
the mass of an electron,
{\displaystyle \varepsilon _{0}}
the vacuum permittivity, and
{\displaystyle c}
the speed of light.
↑ 0.0004 to 0.0006 mm
↑ 1000 nm = 0.001 mm
↑ O'Connor, John J.; Robertson, Edmund F., "Light through the ages: Ancient Greece to Maxwell", MacTutor History of Mathematics archive, University of St Andrews .
↑ O'Connor, John J.; Robertson, Edmund F., "Christiaan Huygens", MacTutor History of Mathematics archive, University of St Andrews .
↑ 4.0 4.1 Fitzpartick, Richard (14 July 2007). "Wave Optics". Lecture Notes. University of Texas. Retrieved 13 February 2014.
↑ "Isaac Newton's theory on light and colours". The Royal Society. Retrieved 25 February 2014. [permanent dead link]
↑ Duarte, FJ (May 2000). "Newton, Prisms, and the "Optiks" of Tunable Lasers" (PDF). Optics & Photonics News. Optical Society of America. 11 (5): 24–28. Bibcode:2000OptPN..11...24D. doi:10.1364/OPN.11.5.000024. Archived from the original (PDF) on 1 October 2013. Retrieved 13 February 2014.
↑ "Optics". Encyclopeadia Britannica. Edinburgh: A Society of Gentlemen in Scotland. 1771.
↑ Lakoba, T. "Lecture notes and supporting material for MATH 235 - Mathematical Models and Their Analysis" (PDF). University of Vermont. 13. Black-body radiation and Planck's formula. Archived from the original (PDF) on 7 February 2016. Retrieved 25 February 2014.
↑ 9.0 9.1 Planck, Max (1901), "Ueber das Gesetz der Energieverteilung im Normalspectrum" (PDF), Ann. Phys., 309 (3): 553–63, Bibcode:1901AnP...309..553P, doi:10.1002/andp.19013090310 . English translation: "On the Law of Distribution of Energy in the Normal Spectrum".
↑ Arrhenius, S (10 December 1922). The NobelPrize in Physics 1921, ALbert Einstein, Presentation Speech (Speech). Nobel Prize award ceremony. Stockholm.
↑ next ref
↑ Einstein acknowledged Planck
↑ 13.0 13.1 Ducharme, Stephen (2008). "Measuring Planck's Constant with LEDs". Materials Research Science and Engineering Center, University of Nebraska–Lincoln. Retrieved 12 February 2012.
↑ Ian Mills (27 September 2010). "Draft Chapter 2 for SI Brochure, following redefinitions of the base units" (PDF). BIPM. Retrieved 3 March 2014.
↑ "에스비엔과학(주)". www.sbnscience.co.kr. Retrieved 2020-02-08.
↑ Heymann, Y. (2021). Kelvin and the Age of the Universe. Amazon: Centorus Publishing. p. 66. ISBN 978-1838493646.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Planck_constant&oldid=8115368"
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Variable bin size selection for periestimulus time histograms (PSTH) with minimum mean square error criteria | BMC Neuroscience | Full Text
Variable bin size selection for periestimulus time histograms (PSTH) with minimum mean square error criteria
SM Heidarieh1,
M Jahed1 &
A Ghazizadeh2
To date the most common method for extracting neuronal responses is by constructing PSTHs that are time locked to task events. Many parameters of interests such as response magnitude, onset and duration are then calculated from the constructed PSTHs. However the precision of PSTH response estimate critically depends on the choice of bin sizes. This dependence demands objective criteria for bin size selection. A seminal study by Shimazaki and Shinomoto [1] derived an optimal cost function for choosing a fixed bin size for a time varying Poisson process. It is easy to see that using a one-size-fit-all recipe for bin sizes will invariably overestimate and underestimate rate changes for fast and slow fluctuations respectively.
Here we extend previous results by calculating the cost function that minimizes mean square error for variable bin sizes with the same assumptions used previously for time varying Poisson processes
Cost\left(N,\stackrel{⃗}{\Delta }\right)=\frac{1}{{n}^{2}T}{\sum }_{i=1}^{N}\left(\frac{2{k}_{i}-{\left({k}_{i}-{\Delta }_{i}\stackrel{̄}{k}\right)}^{2}}{{\Delta }_{i}}\right)
. To minimize this nonlinear and nonconvex cost function, we utilize an array of methods some of which are widely used for nonlinear optimization, namely: Active set, Simultaneous perturbation stochastic approximation (SPSA), Genetic Algorithm and an approximate heuristic algorithm. Average performance of each algorithm on a typical simulated neuronal firing is calculated using 50 iterations. All methods resulted in a lower cost function compared to fixed bin size as expected. Plotting the final cost vs the algorithm run time shows that the method of 'Active set' overall has the best cost reduction while still being reasonably fast compared to the fixed bin size approach (Figure 1) . Further investigation of the properties of this cost function and developing computationally efficient methods for its minimization will be the basis of future work.
Comparing the costs and time efforts of the algorithms.
Shimazaki H, Shinomoto S: A method for selecting the bin size of a time histogram. Neural computation. 2007, 19 (6): 1503-1527.
Electrical Engineering, Sharif University of Technology, Tehran, Iran
SM Heidarieh & M Jahed
Laboratory of Sensorimotor Research, National Institutes of Health, Bethesda, MD, USA
A Ghazizadeh
SM Heidarieh
M Jahed
Correspondence to SM Heidarieh.
Heidarieh, S., Jahed, M. & Ghazizadeh, A. Variable bin size selection for periestimulus time histograms (PSTH) with minimum mean square error criteria. BMC Neurosci 16, P80 (2015). https://doi.org/10.1186/1471-2202-16-S1-P80
Size Approach
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Check valve in a two-phase fluid network - MATLAB - MathWorks Italia
Check Valve (2P)
Check valve in a two-phase fluid network
Simscape / Fluids / Two-Phase Fluid / Valves & Orifices / Directional Control Valves
The Check Valve (2P) block models a directional control check valve in a two-phase fluid network. The valve maintains the fluid pressure by opening above a specified pressure and allowing flow from port A to port B, but not in the reverse direction. The pressure differential that opens the valve is specified in the Opening pressure specification parameter. This value can be either the pressure difference between ports A and B or the gauge pressure at port A.
Fluid properties inside the valve are calculated from inlet conditions. There is no heat exchange between the fluid and the environment, and therefore phase change inside the valve only occurs due to a pressure drop or a change propagated from another part of the model.
The valve opens when the pressure in the valve, pcontrol, exceeds the cracking pressure, pcrack. The valve is fully open when the control pressure reaches the valve maximum pressure, pmax. The opening fraction of the valve, λ, is expressed as:
\lambda =\left(1-{f}_{leak}\right)\frac{\left({p}_{control}-{p}_{crack}\right)}{\left({p}_{\mathrm{max}}-{p}_{crack}\right)}+{f}_{leak},
pcontrol is the control pressure, which depends on the Opening pressure specification parameter.
When you set Opening pressure specification to Pressure differential, the control pressure is pA ̶ pB.
When you set Opening pressure specification to Gauge pressure at port A, the control pressure is the difference between the pressure at port A and atmospheric pressure.
The cracking pressure and maximum pressure are specified as either a differential value or a gauge value, depending on the setting of the Opening pressure specification. If the control pressure exceeds the maximum pressure, the valve opening fraction is 1.
The mass flow rate depends on the pressure differential, and therefore the open area of the valve. The block calculates this as:
{\stackrel{˙}{m}}_{A}=\lambda {\stackrel{˙}{m}}_{nom}\left[\sqrt{\frac{{v}_{nom}}{2\Delta {p}_{nom}}}\right]\sqrt{\frac{2}{{v}_{in}}}\frac{\Delta p}{{\left(\Delta {p}^{2}+\Delta {p}_{lam}^{2}\right)}^{0.25}},
\Delta {p}_{lam}=\frac{\left({p}_{A}+{p}_{B}\right)}{2}\left(1-{B}_{lam}\right).
{\stackrel{˙}{m}}_{nom}
is the Nominal mass flow rate at maximum valve opening.
Δpnom is the Nominal pressure drop rate at maximum valve opening.
{v}_{in}=\left(1-{x}_{dyn}\right){v}_{liq}+{x}_{dyn}{v}_{vap},
If the inlet vapor quality is a liquid-vapor mixture, the block applied a first-order time lag:
\frac{d{x}_{dyn}}{dt}=\frac{{x}_{in}-{x}_{dyn}}{\tau },
{\stackrel{˙}{m}}_{A}+{\stackrel{˙}{m}}_{B}=0,
{\stackrel{˙}{m}}_{A}
{\stackrel{˙}{m}}_{B}
{\Phi }_{A}+{\Phi }_{B}=0,
Fluid entry port.
Fluid exit port.
Opening pressure specification — Control pressure
When set to Pressure differential, the valve opens when pA ̶ pB exceeds the Cracking pressure differential.
When set to Gauge pressure at port A, the valve opens when pA ̶ patm exceeds the Cracking pressure (gauge).
Cracking pressure differential — Cracking pressure
Valve pressure threshold. When the control pressure, pA ̶ pB, exceeds the opening pressure, the valve begins to open.
To enable this parameter, set Opening pressure specification to Pressure differential.
Cracking pressure (gauge) — Cracking pressure
Valve pressure threshold. When the control pressure, pA ̶ patm, exceeds the opening pressure, the valve begins to open.
To enable this parameter, set Opening pressure specification to Gauge pressure at port A.
Maximum opening pressure differential — Maximum valve operational pressure
Maximum valve operational pressure. The valve begins to open at the cracking pressure value, and is fully open at pmax.
Maximum opening pressure (gauge) — Pressure that fully opens the valve
Valve operational pressure at which the valve is fully open. The valve begins to open at the cracking pressure value, and is fully open at pmax.
Valve inlet pressure in typical, design, or rated conditions. The valve inlet specific volume is determined from the fluid properties tabulated data based on the Nominal inlet pressure and the setting of the Nominal inlet condition specification parameters.
Orifice (2P) | Pressure-Reducing Valve (2P) | Pressure Relief Valve (2P) | Thermostatic Expansion Valve (2P)
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Just a quick random one...
A friend recently proposed the following problem as part of their coursework (which I've changed to not give away the answers to others):
A sequence of numbers are defined as:
{z}_{1}=1
{z}_{x+1}=13{z}_{x}+7
You are to define a function find_x(target) that returns the smallest value of
x
{z}_{x}
is greater than or equal to the target value.
If implemented correctly, you should expect to see:
{z}_{1}=1
{z}_{2}=13×1+7=20
{z}_{3}=13×20+7=267
{z}_{4}=13×267+7=3478
Not too complicated - and I was really just helping with their debugging process as a new Python programmer.
Given how quickly something like this can be coded, I decided to whip something up so that I could test the correctness of their solution:
0001 def find_x(target) :
0002 x = 1
0003 z_x = 1
0004 while z_x < target :
0005 z_x = (13 * z_x) + 7
0006 x += 1
0007 return x
0008 print(find_x(1))
0009 print(find_x(10))
0010 print(find_x(100))
0011 print(find_x(1000))
0012 print(find_x(10000))
0013 print(find_x(100000))
And the results of the program:
So that is kind of boring, but probably of help for others. Let's dissect this a little and make it more interesting...
Firstly, what problems does this current version have? I.e. What traps are there for new players?
Negative values - We are not able to handle small values, we should gracefully fail. The sequence is undefined for
i\le 0
Calculation too large - We should be careful of overflows of large integer values during the calculation.
Slow for large values - This will be quite painful for really large values, but fortunately this won't be a problem for Python and its limitations for the maximum integer size.
For the first two, we can do something like (we'd likely pre-calculate max_z_x):
0020 def find_x_v2(target) :
0021 max_z_x = (sys.maxint - 5) / 13
0022 assert(target >= 1)
0026 assert(z_x < max_z_x)
0030 print(find_x_v2(1)) # 1
0031 print(find_x_v2(10)) # 2
0032 print(find_x_v2(100)) # 3
0033 print(find_x_v2(1000)) # 4
0034 print(find_x_v2(10000)) # 5
0035 print(find_x_v2(100000)) # 6
0036 print(find_x_v2(1000000)) # 7
0037 print(find_x_v2(10000000)) # 8
0038 print(find_x_v2(100000000)) # 9
0039 print(find_x_v2(1000000000)) # 10
0040 print(find_x_v2(10000000000)) # 11
0041 print(find_x_v2(100000000000)) # 12
0042 print(find_x_v2(1000000000000)) # 13
0043 print(find_x_v2(10000000000000)) # 14
0044 print(find_x_v2(100000000000000)) # 15
0045 print(find_x_v2(1000000000000000)) # 16
0046 print(find_x_v2(10000000000000000)) # 17
0047 print(find_x_v2(100000000000000000)) # 18
0048 print(find_x_v2(1000000000000000000)) # 19
0049 print(find_x_v2(10000000000000000000)) # 20
0050 print(find_x_v2(100000000000000000000)) # 21
And as a result we get:
0070 Traceback (most recent call last):
0071 File "main.py", line 41, in <module>
0072 print(find_x_v2(10000000000000000000)) # 20
0073 File "main.py", line 17, in find_x_v2
0075 AssertionError
So it appears that Python correctly crashes for suitably large integer values.
Start off with the following formula:
{z}_{x+1}=13{z}_{x}+7
For the third case (skipping the second) we can write:
{z}_{x+1}=13\left(13{z}_{x-1}+7\right)+7
{z}_{x+1}={13}^{2}{z}_{x-1}+\left(13×7\right)+7
For the fourth case we can write:
{z}_{x+1}={13}^{2}\left(13{z}_{x-2}+7\right)+\left(13×7\right)+7
{z}_{x+1}={13}^{3}{z}_{x-2}+\left({13}^{2}×7\right)+\left(13×7\right)+7
{z}_{x+1}={13}^{x}{z}_{1}+{13}^{x-1}7+...+{13}^{0}7
{z}_{6}={13}^{5}+{13}^{4}7+{13}^{3}7+{13}^{2}7+{13}^{1}7+{13}^{0}7=587880
Whilst it's annoying that we have this long term that expands, actually for the most part we can ignore it. The first term
{13}^{x-1}7
is much larger than all the summed terms down to
{13}^{0}7
. For the search we can therefore simplify to:
{z}_{x+1}={13}^{x}+{13}^{x-1}7
We are trying to find something around the target,
t
, so now we search for:
t={13}^{x}+{13}^{x-1}7
13t=20×{13}^{x}
\frac{13t}{20}={13}^{x}
lo{g}_{13}\left(\frac{13t}{20}\right)=x
And now we have something for approximating the location of some positive value of
x
given some target
I'm not going to go to the effort of programming this (because I don't care so much), but it's interesting to see that it's possible to find an approximate solution very quickly, without being required to do a large loop. In theory you can now find arbitrary values by performing just a 'few' calculations either side of the predicted value 1.
I would need to think about exactly a search strategy.↩
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In this note, we discuss the nonlinear stability in exponential time of Minkowski space-time with a translation space-like Killing field, proved in [13]. In the presence of such a symmetry, the
3+1
vacuum Einstein equations reduce to the
2+1
Einstein equations with a scalar field. We work in generalized wave coordinates. In this gauge Einstein equations can be written as a system of quasilinear quadratic wave equations. The main difficulty in [13] is due to the decay in
1/\sqrt{t}
of free solutions to the wave equation in
2
dimensions, which is weaker than in
3
dimensions. As in [21], we have to rely on the particular structure of Einstein equations in wave coordinates. We also have to carefully choose an approximate solution with a non trivial behaviour at space-like infinity to enforce convergence to Minkowski space-time at time-like infinity.
author = {C\'ecile Huneau},
title = {Stability in exponential time of {Minkowski} space-time with a space-like translation symmetry},
TI - Stability in exponential time of Minkowski space-time with a space-like translation symmetry
%T Stability in exponential time of Minkowski space-time with a space-like translation symmetry
Cécile Huneau. Stability in exponential time of Minkowski space-time with a space-like translation symmetry. Séminaire Laurent Schwartz — EDP et applications (2014-2015), Talk no. 19, 14 p. doi : 10.5802/slsedp.77. https://slsedp.centre-mersenne.org/articles/10.5802/slsedp.77/
[1] S. Alinhac – « The null condition for quasilinear wave equations in two space dimensions I », Invent. Math. 145 (2001), no. 3, p. 597–618.
[2] S. Alinhac – « An example of blowup at infinity for a quasilinear wave equation », Astérisque (2003), no. 284, p. 1–91, Autour de l’analyse microlocale.
[3] A. Ashtekar, J. Bičák & B. G. Schmidt – « Asymptotic structure of symmetry-reduced general relativity », Phys. Rev. D (3) 55 (1997), no. 2, p. 669–686.
[4] R. Bartnik & J. Isenberg – « The constraint equations », in The Einstein equations and the large scale behavior of gravitational fields, Birkhäuser, Basel, 2004, p. 1–38.
[5] G. Beck – « Zur Theorie binärer Gravitationsfelder », Zeitschrift für Physik 33 (1925), no. 14, p. 713–728.
[6] B. K. Berger, P. T. Chruściel & V. Moncrief – « On “asymptotically flat” space-times with
{G}_{2}
-invariant Cauchy surfaces », Ann. Physics 237 (1995), no. 2, p. 322–354.
[7] Y. Choquet-Bruhat & R. Geroch – « Global aspects of the Cauchy problem in general relativity », Comm. Math. Phys. 14 (1969), p. 329–335.
[8] Y. Choquet-Bruhat & V. Moncrief – « Nonlinear stability of an expanding universe with the
{S}^{1}
isometry group », in Partial differential equations and mathematical physics (Tokyo, 2001), Progr. Nonlinear Differential Equations Appl., vol. 52, Birkhäuser Boston, Boston, MA, 2003, p. 57–71.
[9] D. Christodoulou & S. Klainerman – The global nonlinear stability of the Minkowski space, Princeton Mathematical Series, vol. 41, Princeton University Press, Princeton, NJ, 1993.
[10] P. Godin – « Lifespan of solutions of semilinear wave equations in two space dimensions », Comm. Partial Differential Equations 18 (1993), no. 5-6, p. 895–916.
[11] A. Hoshiga – « The existence of global solutions to systems of quasilinear wave equations with quadratic nonlinearities in 2-dimensional space », Funkcial. Ekvac. 49 (2006), no. 3, p. 357–384.
[12] C. Huneau – « Constraint equations for 3 + 1 vacuum Einstein equations with a translational space-like Killing field in the asymptotically flat case II », arXiv:1410.6061.
[13] —, « Stability in exponential time of Minkowski Space-time with a translation space-like Killing field », arXiv:1410.6068.
[14] F. John – « Blow-up for quasilinear wave equations in three space dimensions », Comm. Pure Appl. Math. 34 (1981), no. 1, p. 29–51.
[15] S. Klainerman – « The null condition and global existence to nonlinear wave equations », in Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N.M., 1984), Lectures in Appl. Math., vol. 23, Amer. Math. Soc., Providence, RI, 1986, p. 293–326.
[16] S. Klainerman – « Uniform decay estimates and the Lorentz invariance of the classical wave equation », Comm. Pure Appl. Math. 38 (1985), no. 3, p. 321–332.
[17] H. Kubo & K. Kubota – « Scattering for systems of semilinear wave equations with different speeds of propagation », Adv. Differential Equations 7 (2002), no. 4, p. 441–468.
[18] H. Lindblad – « Global solutions of nonlinear wave equations », Comm. Pure Appl. Math. 45 (1992), no. 9, p. 1063–1096.
[19] —, « Global solutions of quasilinear wave equations », Amer. J. Math. 130 (2008), no. 1, p. 115–157.
[20] H. Lindblad & I. Rodnianski – « The weak null condition for Einstein’s equations », C. R. Math. Acad. Sci. Paris 336 (2003), no. 11, p. 901–906.
[21] —, « The global stability of Minkowski space-time in harmonic gauge », Ann. of Math. (2) 171 (2010), no. 3, p. 1401–1477.
[22] R. Wald – General Relativity, The University of Chicago press, 1984.
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"Welded" redirects here. For the play, see Welded (play).
In addition to melting the base metal, a filler material is typically added to the joint to form a pool of molten material (the weld pool) that cools to form a joint that, based on weld configuration (butt, full penetration, fillet, etc.), can be stronger than the base material. Pressure may also be used in conjunction with heat or by itself to produce a weld. Welding also requires a form of shield to protect the filler metals or melted metals from being contaminated or oxidized.
3.2.2 Arc welding power supplies
3.4 Energy beam welding
3.5 Solid-state welding
5.2 Lifetime extension with after treatment methods
7 Unusual conditions
9 Costs and trends
10 Glass and plastic welding
10.1 Glass welding
10.2 Plastic welding
The term weld is of English origin, with roots from Scandinavia. It is often confused with the Old English word weald, meaning 'a forested area', but this word eventually morphed into the modern version, wild. The Old English word for welding iron was samod ('to bring together') or samodwellung ('to bring together hot', with hot more relating to red-hot or a swelling rage; in contrast to samodfæst, 'to bind together with rope or fasteners').[1] The term weld is derived from the Middle English verb well (wæll; plural/present tense: wælle) or welling (wællen), meaning 'to heat' (to the maximum temperature possible); 'to bring to a boil'. The modern word was probably derived from the past-tense participle welled (wællende), with the addition of d for this purpose being common in the Germanic languages of the Angles and Saxons. It was first recorded in English in 1590, from a version of the Christian Bible that was originally translated into English by John Wycliffe in the fourteenth century. The original version, from Isaiah 2:4, reads, "...thei shul bete togidere their swerdes into shares..." (they shall beat together their swords into plowshares), while the 1590 version was changed to, "...thei shullen welle togidere her swerdes in-to scharris..." (they shall weld together their swords into plowshares), suggesting this particular use of the word probably became popular in English sometime between these periods.[2]
The word is derived from the Old Swedish word valla, meaning 'to boil'. Sweden was a large exporter of iron during the Middle Ages, and many other European languages used different words but with the same meaning to refer to welding iron, such as the Illyrian (Greek) variti ('to boil'), Turkish kaynamak ('to boil'), Grison (Swiss) bulgir ('to boil'), or the Lettish (Latvian) sawdrit ('to weld or solder', derived from wdrit, 'to boil'). In Swedish, however, the word only referred to joining metals when combined with the word for iron (järn), as in valla järn (literally: 'to boil iron'). The word possibly entered English from the Swedish iron trade, or possibly was imported with the thousands of Viking settlements that arrived in England before and during the Viking Age, as more than half of the most common English words in everyday use are Scandinavian in origin.[3][4]
The history of joining metals goes back several millennia. The earliest examples of this come from the Bronze and Iron Ages in Europe and the Middle East. The ancient Greek historian Herodotus states in The Histories of the 5th century BC that Glaucus of Chios "was the man who single-handedly invented iron welding".[5] Welding was used in the construction of the Iron pillar of Delhi, erected in Delhi, India about 310 AD and weighing 5.4 metric tons.[6]
In 1800, Sir Humphry Davy discovered the short-pulse electrical arc and presented his results in 1801.[8][9][10] In 1802, Russian scientist Vasily Petrov created the continuous electric arc,[10][11][12] and subsequently published "News of Galvanic-Voltaic Experiments" in 1803, in which he described experiments carried out in 1802. Of great importance in this work was the description of a stable arc discharge and the indication of its possible use for many applications, one being melting metals.[13] In 1808, Davy, who was unaware of Petrov's work, rediscovered the continuous electric arc.[9][10] In 1881–82 inventors Nikolai Benardos (Russian) and Stanisław Olszewski (Polish)[14] created the first electric arc welding method known as carbon arc welding using carbon electrodes. The advances in arc welding continued with the invention of metal electrodes in the late 1800s by a Russian, Nikolai Slavyanov (1888), and an American, C. L. Coffin (1890). Around 1900, A. P. Strohmenger released a coated metal electrode in Britain, which gave a more stable arc. In 1905, Russian scientist Vladimir Mitkevich proposed using a three-phase electric arc for welding. Alternating current welding was invented by C. J. Holslag in 1919, but did not become popular for another decade.[15]
World War I caused a major surge in the use of welding, with the various military powers attempting to determine which of the several new welding processes would be best. The British primarily used arc welding, even constructing a ship, the "Fullagar" with an entirely welded hull.[19][20] Arc welding was first applied to aircraft during the war as well, as some German airplane fuselages were constructed using the process.[21] Also noteworthy is the first welded road bridge in the world, the Maurzyce Bridge in Poland (1928).[22]
Arc welding[edit]
Main articles: Arc welding, Shielded metal arc welding, Gas tungsten arc welding, Gas metal arc welding, Flux-cored arc welding, Submerged arc welding, and Electroslag welding
Arc welding processes[edit]
Arc welding power supplies[edit]
Resistance welding[edit]
Energy beam welding[edit]
Solid-state welding[edit]
Other solid-state welding processes include friction welding (including friction stir welding and friction stir spot welding),[46] magnetic pulse welding,[47] co-extrusion welding, cold welding, diffusion bonding, exothermic welding, high frequency welding, hot pressure welding, induction welding, and roll bonding.[45]
Many distinct factors influence the strength of welds and the material around them, including the welding method, the amount and concentration of energy input, the weldability of the base material, filler material, and flux material, the design of the joint, and the interactions between all these factors.[51]
For example, the factor of welding position influences weld quality, that welding codes & specifications may require testing—both welding procedures and welders—using specified welding positions: 1G (flat), 2G (horizontal), 3G (vertical), 4G (overhead), 5G (horizontal fixed pipe), or 6G (inclined fixed pipe).
To test the quality of a weld, either destructive or nondestructive testing methods are commonly used to verify that welds are free of defects, have acceptable levels of residual stresses and distortion, and have acceptable heat-affected zone (HAZ) properties. Types of welding defects include cracks, distortion, gas inclusions (porosity), non-metallic inclusions, lack of fusion, incomplete penetration, lamellar tearing, and undercutting.
The metalworking industry has instituted codes and specifications to guide welders, weld inspectors, engineers, managers, and property owners in proper welding technique, design of welds, how to judge the quality of welding procedure specification, how to judge the skill of the person performing the weld, and how to ensure the quality of a welding job.[51] Methods such as visual inspection, radiography, ultrasonic testing, phased-array ultrasonics, dye penetrant inspection, magnetic particle inspection, or industrial computed tomography can help with detection and analysis of certain defects.
{\displaystyle Q=\left({\frac {V\times I\times 60}{S\times 1000}}\right)\times {\mathit {Efficiency}}}
Welding can be dangerous and unhealthy if the proper precautions are not taken. However, using new technology and proper protection greatly reduces risks of injury and death associated with welding.[58] Since many common welding procedures involve an open electric arc or flame, the risk of burns and fire is significant; this is why it is classified as a hot work process. To prevent injury, welders wear personal protective equipment in the form of heavy leather gloves and protective long-sleeve jackets to avoid exposure to extreme heat and flames. Synthetic clothing such as polyester should not be worn since it may burn, causing injury.[59] Additionally, the brightness of the weld area leads to a condition called arc eye or flash burns in which ultraviolet light causes inflammation of the cornea and can burn the retinas of the eyes. Goggles and welding helmets with dark UV-filtering face plates are worn to prevent this exposure. Since the 2000s, some helmets have included a face plate which instantly darkens upon exposure to the intense UV light. To protect bystanders, the welding area is often surrounded with translucent welding curtains. These curtains, made of a polyvinyl chloride plastic film, shield people outside the welding area from the UV light of the electric arc, but cannot replace the filter glass used in helmets.[60]
^ A Concise Anglo-Saxon Dictionary by John R. Clark Hall, Herbert T. Merritt, Herbert Dean Meritt, Medieval Academy of America -- Cambridge University Press 1960 Page 289
^ An Etymological Dictionary of the English Language by Walter William Skeat -- Oxford Press 1898 Page 702
^ A Dictionary of English Etymology by Hensleigh Wedgwood -- Trubner & Co. 1878Page 723
^ A History of the English Language by Elly van Gelderen -- John Benjamins Publishing 2006
^ Herodotus. The Histories. Trans. R. Waterfield. Oxford: Oxford University Press. Book One, 25.
^ a b Lincoln Electric, p. 1.1-1
^ Lincoln Electric, The Procedure Handbook Of Arc Welding 14th ed., page 1.1-1
^ a b Hertha Ayrton. The Electric Arc, pp. 20, 24 and 94. D. Van Nostrand Co., New York, 1902.
^ a b c A. Anders (2003). "Tracking down the origin of arc plasma science-II. early continuous discharges" (PDF). IEEE Transactions on Plasma Science. 31 (5): 1060–9. Bibcode:2003ITPS...31.1060A. doi:10.1109/TPS.2003.815477.
^ Lazarev, P.P. (December 1999), "Historical essay on the 200 years of the development of natural sciences in Russia" (PDF), Physics-Uspekhi, 42 (1247): 1351–1361, doi:10.1070/PU1999v042n12ABEH000750, archived from the original (Russian) on 2011-02-11
^ Nikołaj Benardos, Stanisław Olszewski, "Process of and apparatus for working metals by the direct application of the electric current" patent nr 363 320, Washington, United States Patent Office, 17 may 1887.
^ a b c d e Weman, p. 26
^ "Lesson 3: Covered Electrodes for Welding Mild Steels". Retrieved 18 May 2017.
^ A History of Welding. weldinghistory.org
^ The Engineer (6 February 1920) p. 142
^ Lincoln Electric, p. 1.1–5
^ Sapp, Mark E. (February 22, 2008). "Welding Timeline 1900–1950". WeldingHistory.org. Archived from the original on August 3, 2008. Retrieved 2008-04-29.
^ Kazakov, N.F (1985). "Diffusion Bonding of Materials". University of Cambridge. Archived from the original on 2013-09-01. Retrieved 2011-01-13.
^ Mel Schwartz (2011). Innovations in Materials Manufacturing, Fabrication, and Environmental Safety. CRC Press. p. 300. ISBN 978-1-4200-8215-9.
^ Lincoln Electric, pp. 1.1–10
^ a b c d Weman, p. 63
^ a b Cary & Helzer 2005, p. 103
^ Lincoln Electric, p. 5.4-3
^ Weman, p. 53
^ a b c Weman, p. 31
^ Weman, pp. 37–38
^ Kalpakjian and Schmid, p. 780
^ a b c d e f Weman, pp. 80–84
^ Weman, pp. 95–101
^ a b c d Weman, pp. 89–90
^ Stephan Kallee (August 2006) "NZ Fabricators begin to use Friction Stir Welding to produce aluminium components and panels". New Zealand Engineering News.
^ Stephan Kallee et al. (2010) Industrialisation of Electromagnetic Pulse Technology (EMPT) in India 38th Anniversary Issue of PURCHASE India.
^ Hicks, John (1999). Welded Joint Design. New York: Industrial Press. pp. 52–55. ISBN 0-8311-3130-6.
^ Cary & Helzer 2005, pp. 19, 103, 206
^ a b Weman, pp. 60–62
^ Lincoln Electric, pp. 6.1-5–6.1–6
^ Kalpakjian and Schmid, pp. 821–22
^ Weman, p. 5
^ How To Weld By Todd Bridigum - Motorbook 2008 Page 37
^ a b c d e f g h Lancaster, J.F. (1999). Metallurgy of welding (6th ed.). Abington, Cambridge: Abington Pub. ISBN 1-85573-428-1.
^ ANSI/AWS Z49.1: "Safety in Welding, Cutting, and Allied Processes" (2005)
^ "Safety and Health Injury Prevention Sheets (SHIPS) | Process: Hot Work - Welding, Cutting and Brazing - Hazard: Burns and Shocks | Occupational Safety and Health Administration". www.osha.gov. Retrieved 2019-10-12.
^ a b Cary & Helzer 2005, pp. 52–62
^ Welding and Manganese: Potential Neurologic Effects. The inhalation of nano particles National Institute for Occupational Safety and Health. March 30, 2009.
^ James D Byrne; John A Baugh (2008). "The significance of nano particles in particle-induced pulmonary fibrosis". McGill Journal of Medicine. 11 (1): 43–50. PMC 2322933. PMID 18523535.
^ a b c Weman, pp. 184–89
^ ASM International (2003). Trends in Welding Research. Materials Park, Ohio: ASM International. pp. 995–1005. ISBN 0-87170-780-2.
^ Gregory L. Snow and W. Samuel Easterling (October 2008) Strength of Arc Spot Welds Made in Single and Multiple Steel Sheets Archived 2014-06-11 at the Wayback Machine , Proceedings of the 19th International Specialty Conference on Cold-Formed Steel Structures, Missouri University of Science and Technology.
^ Freek Bos, Christian Louter, Fred Veer (2008) Challenging Glass: Conference on Architectural and Structural Applications. JOS Press. p. 194. ISBN 1-58603-866-4
^ Bernard D. Bolas (1921) A handbook of laboratory glassblowing. London, G. Routledge and sons
^ Plastics and Composites: Welding Handbook By David A. Grewell, A. Benatar, Joon Bu Park – Hanser Gardener 2003
^ Handbook of Plastics Joining: A Practical Guide By Plastics Design Library – PDL 1997 Page 137, 146
Kalpakjian, Serope; Schmid, Steven R. (2001). Manufacturing Engineering and Technology. Prentice Hall. ISBN 0-201-36131-0.
Milestones in the History of Welding
Retrieved from "https://en.wikipedia.org/w/index.php?title=Welding&oldid=1088905418"
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A man observes that when he moves up a distance c metres on a slope, the angle of depression of - Maths - Coordinate Geometry - 10699945 | Meritnation.com
A man observes that when he moves up a distance c metres on a slope, the angle of depression of a point on the horizontal plane from the base of the slope is 30°, and when he moves up further a distance c metres, the angle of depression of that point is 45°. The angle of inclination of the slope with the horizontal is:-
Figure to represent the given information will be:
Applying m - n theorem of trigonometry, we get:\phantom{\rule{0ex}{0ex}}\left(c + c\right) cot\left(\Theta - 30°\right) = c cot 15° - c cot 30°\phantom{\rule{0ex}{0ex}}⇒cot\left(\Theta - 30°\right) = \frac{1}{2}\frac{\mathrm{sin}\left(30° - 15°\right)}{\mathrm{sin} 15° \mathrm{sin} 30°}\phantom{\rule{0ex}{0ex}}⇒cot\left(\Theta - 30°\right) = \frac{1}{2}\frac{1}{\mathrm{sin} 30°} = 1 = cot 45°\phantom{\rule{0ex}{0ex}}⇒\Theta - 30° = 45°\phantom{\rule{0ex}{0ex}}\therefore \Theta = 75°
So, angle of inclination if the slope with horizontal will be 75°.
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Identity and One-to-One Functions - Course Hero
College Algebra/Composite and Inverse Functions/Identity and One-to-One Functions
f(x) = x
The identity function is the function for which the output is equal to the input. If the input is
x
, then the identity function is
f(x)=x
Although composition of functions is not generally commutative, composition with
f(x)=x
is commutative. This is because any function composed with the identity function is itself, regardless of the order of the composition. For
f(x)=x
g(x)
f(g(x))=g(f(x))=g(x)
The graph of the identity function is a line with a slope of 1 and a
y
-intercept of zero. For every point on the line, the
x
-value is the same as the
y
Each output value of the identity function
f(x)=x
is the same as the corresponding input value. So the
y
-coordinate of each point on the graph of the function shown is always the same as the
x
For all functions, each input corresponds to exactly one output, which can be determined by a vertical line test. In a one-to-one function, each output corresponds to exactly one input.
In a mapping diagram, this means that every element of the domain is mapped to only one element in the range, and vice versa. On a graph, the horizontal line test uses horizontal lines to determine whether a function is one-to-one; if any horizontal line intersects the graph in more than one point, the graph is not one-to-one.
For example, a simple function is:
f(x)=x^2
Any two opposite values, such as 1 and –1, will have the same output. So, the function is not one-to-one. For other functions, it may be easier to determine whether the function is one-to-one using the horizontal line test on a graph.
Function, Not One-to-One
The mapping shows that one domain value has two different range values.
The mapping shows that every domain value has exactly one range value, but a range value has two different domain values.
The mapping shows that every domain value has exactly one range value and that each range value only has one domain value.
The graph shows that an
x
-value has two different
y
The graph shows that every
x
-value has exactly one
y
-value but that a
y
x
x
y
-value and that each
y
-value has only one
x
<Decomposing Composite Functions>Inverse Functions
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Simulation-based optimization - Wikipedia
Simulation-based optimization (also known as simply simulation optimization) integrates optimization techniques into simulation modeling and analysis. Because of the complexity of the simulation, the objective function may become difficult and expensive to evaluate. Usually, the underlying simulation model is stochastic, so that the objective function must be estimated using statistical estimation techniques (called output analysis in simulation methodology).
Once a system is mathematically modeled, computer-based simulations provide information about its behavior. Parametric simulation methods can be used to improve the performance of a system. In this method, the input of each variable is varied with other parameters remaining constant and the effect on the design objective is observed. This is a time-consuming method and improves the performance partially. To obtain the optimal solution with minimum computation and time, the problem is solved iteratively where in each iteration the solution moves closer to the optimum solution. Such methods are known as ‘numerical optimization’ or ‘simulation-based optimization’.[1]
In simulation experiment, the goal is to evaluate the effect of different values of input variables on a system. However, the interest is sometimes in finding the optimal value for input variables in terms of the system outcomes. One way could be running simulation experiments for all possible input variables. However, this approach is not always practical due to several possible situations and it just makes it intractable to run experiments for each scenario. For example, there might be too many possible values for input variables, or the simulation model might be too complicated and expensive to run for suboptimal input variable values. In these cases, the goal is to find optimal values for the input variables rather than trying all possible values. This process is called simulation optimization.[2]
Specific simulation–based optimization methods can be chosen according to Figure 1 based on the decision variable types.[3]
Fig.1 Classification of simulation based optimization according to variable types
Optimization exists in two main branches of operations research:
Optimization parametric (static) – The objective is to find the values of the parameters, which are “static” for all states, with the goal of maximizing or minimizing a function. In this case, one can use mathematical programming, such as linear programming. In this scenario, simulation helps when the parameters contain noise or the evaluation of the problem would demand excessive computer time, due to its complexity.[4]
Optimization control (dynamic) – This is used largely in computer science and electrical engineering. The optimal control is per state and the results change in each of them. One can use mathematical programming, as well as dynamic programming. In this scenario, simulation can generate random samples and solve complex and large-scale problems.[4]
1 Simulation-based optimization methods
1.1 Statistical ranking and selection methods (R/S)
1.2 Response surface methodology (RSM)
1.6 Dynamic programming and neuro-dynamic programming
1.6.2 Neuro-dynamic programming
Simulation-based optimization methods[edit]
Some important approaches in simulation optimization are discussed below. [5] [6]
Statistical ranking and selection methods (R/S)[edit]
Ranking and selection methods are designed for problems where the alternatives are fixed and known, and simulation is used to estimate the system performance. In the simulation optimization setting, applicable methods include indifference zone approaches, optimal computing budget allocation, and knowledge gradient algorithms.
Response surface methodology (RSM)[edit]
In response surface methodology, the objective is to find the relationship between the input variables and the response variables. The process starts from trying to fit a linear regression model. If the P-value turns out to be low, then a higher degree polynomial regression, which is usually quadratic, will be implemented. The process of finding a good relationship between input and response variables will be done for each simulation test. In simulation optimization, response surface method can be used to find the best input variables that produce desired outcomes in terms of response variables.[7]
Heuristic methods change accuracy by speed. Their goal is to find a good solution faster than the traditional methods, when they are too slow or fail in solving the problem. Usually they find local optimal instead of the optimal value; however, the values are considered close enough of the final solution. Examples of these kinds of methods include tabu search and genetic algorithms.[4]
Metamodels enable researchers to obtain reliable approximate model outputs without running expensive and time-consuming computer simulations. Therefore, the process of model optimization can take less computation time and cost.[8]
Stochastic approximation[edit]
Stochastic approximation is used when the function cannot be computed directly, only estimated via noisy observations. In these scenarios, this method (or family of methods) looks for the extrema of these function. The objective function would be:[9]
{\displaystyle {\underset {{\text{x}}\in \theta }{\min }}f{\bigl (}{\text{x}}{\bigr )}={\underset {{\text{x}}\in \theta }{\min }}\mathrm {E} [F{\bigl (}{\text{x,y}})]}
{\displaystyle y}
is a random variable that represents the noise.
{\displaystyle x}
is the parameter that minimizes
{\displaystyle f{\bigl (}{\text{x}}{\bigr )}}
{\displaystyle \theta }
is the domain of the parameter
{\displaystyle x}
Derivative-free optimization methods[edit]
Derivative-free optimization is a subject of mathematical optimization. This method is applied to a certain optimization problem when its derivatives are unavailable or unreliable. Derivative-free methods establish a model based on sample function values or directly draw a sample set of function values without exploiting a detailed model. Since it needs no derivatives, it cannot be compared to derivative-based methods.[10]
For unconstrained optimization problems, it has the form:
{\displaystyle {\underset {{\text{x}}\in \mathbb {R} ^{n}}{\min }}f{\bigl (}{\text{x}}{\bigr )}}
The limitations of derivative-free optimization:
1. Some methods cannot handle optimization problems with more than a few variables; the results are usually not so accurate. However, there are numerous practical cases where derivative-free methods have been successful in non-trivial simulation optimization problems that include randomness manifesting as "noise" in the objective function. See, for example, the following [5] .[11]
2. When confronted with minimizing non-convex functions, it will show its limitation.
3. Derivative-free optimization methods are relatively simple and easy, but, like most optimization methods, some care is required in practical implementation (e.g., in choosing the algorithm parameters).
Dynamic programming and neuro-dynamic programming[edit]
Dynamic programming[edit]
Dynamic programming deals with situations where decisions are made in stages. The key to this kind of problems is to trade off the present and future costs.[12]
One dynamic basic model has two features:
1) It has a discrete time dynamic system.
2) The cost function is additive over time.
For discrete features, dynamic programming has the form:
{\displaystyle x_{k+1}=f_{k}(x_{k},u_{k},w_{k}),k=0,1,...,N-1}
{\displaystyle k}
represents the index of discrete time.
{\displaystyle x_{k}}
is the state of the time k, it contains the past information and prepares it for future optimization.
{\displaystyle u_{k}}
is the control variable.
{\displaystyle w_{k}}
is the random parameter.
For the cost function, it has the form:
{\displaystyle g_{N}(X_{N})+\sum _{k=0}^{N-1}g_{k}(x_{k},u_{k},W_{k})}
{\displaystyle g_{N}(X_{N})}
is the cost at the end of the process.
As the cost cannot be optimized meaningfully, it can be used the expect value:
{\displaystyle E\{g_{N}(X_{N})+\sum _{k=0}^{N-1}g_{k}(x_{k},u_{k},W_{k})\}}
Neuro-dynamic programming[edit]
Neuro-dynamic programming is the same as dynamic programming except that the former has the concept of approximation architectures. It combines artificial intelligence, simulation-base algorithms, and functional approach techniques. “Neuro” in this term origins from artificial intelligence community. It means learning how to make improved decisions for the future via built-in mechanism based on the current behavior. The most important part of neuro-dynamic programming is to build a trained neuro network for the optimal problem.[13]
Simulation-based optimization has some limitations, such as the difficulty of creating a model that imitates the dynamic behavior of a system in a way that is considered good enough for its representation. Another problem is complexity in the determining uncontrollable parameters of both real-world system and simulation. Moreover, only a statistical estimation of real values can be obtained. It is not easy to determine the objective function, since it is a result of measurements, which can be harmful to the solutions.[14][15]
^ Nguyen, Anh-Tuan, Sigrid Reiter, and Philippe Rigo. "A review on simulation-based optimization methods applied to building performance analysis."Applied Energy 113 (2014): 1043–1058.
^ Carson, Yolanda, and Anu Maria. "Simulation optimization: methods and applications." Proceedings of the 29th Winter Simulation Conference. IEEE Computer Society, 1997.
^ Jalali, Hamed, and Inneke Van Nieuwenhuyse. "Simulation optimization in inventory replenishment: a classification." IIE Transactions 47.11 (2015): 1217-1235.
^ a b c Abhijit Gosavi, Simulation‐Based Optimization: Parametric Optimization Techniques and Reinforcement Learning, Springer, 2nd Edition (2015)
^ a b Fu, Michael, ed. (2015). Handbook of Simulation Optimization. Springer.
^ Spall, J.C. (2003). Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control. Hoboken: Wiley.
^ Rahimi Mazrae Shahi, M., Fallah Mehdipour, E. and Amiri, M. (2016), Optimization using simulation and response surface methodology with an application on subway train scheduling. Intl. Trans. in Op. Res., 23: 797–811. doi:10.1111/itor.12150
^ Yousefi, Milad; Yousefi, Moslem; Ferreira, Ricardo Poley Martins; Kim, Joong Hoon; Fogliatto, Flavio S. (2018). "Chaotic genetic algorithm and Adaboost ensemble metamodeling approach for optimum resource planning in emergency departments". Artificial Intelligence in Medicine. 84: 23–33. doi:10.1016/j.artmed.2017.10.002. PMID 29054572.
^ Powell, W. (2011). Approximate Dynamic Programming Solving the Curses of Dimensionality (2nd ed., Wiley Series in Probability and Statistics). Hoboken: Wiley.
^ Conn, A. R.; Scheinberg, K.; Vicente, L. N. (2009). Introduction to Derivative-Free Optimization. MPS-SIAM Book Series on Optimization. Philadelphia: SIAM. Retrieved 2014-01-18.
^ Fu, M.C., Hill, S.D. Optimization of discrete event systems via simultaneous perturbation stochastic approximation. IIE Transactions 29, 233–243 (1997). https://doi.org/10.1023/A:1018523313043
^ Cooper, Leon; Cooper, Mary W. Introduction to dynamic programming. New York: Pergamon Press, 1981
^ Van Roy, B., Bertsekas, D., Lee, Y., & Tsitsiklis, J. (1997). Neuro-dynamic programming approach to retailer inventory management. Proceedings of the IEEE Conference on Decision and Control, 4, 4052-4057.
^ Prasetio, Y. (2005). Simulation-based optimization for complex stochastic systems. University of Washington.
^ Deng, G., & Ferris, Michael. (2007). Simulation-based Optimization, ProQuest Dissertations and Theses
Retrieved from "https://en.wikipedia.org/w/index.php?title=Simulation-based_optimization&oldid=1067355915"
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Tree - Ring of Brodgar
Skill(s) Required Lumberjacking
Object(s) Required Soil x4, Tree Seed*, Water (1.0 L)
Produced By Treeplanter's Pot and Herbalist Table
Required By Branch, Log, Scratch-Marked Bark, Stump, Treebark
Tree is a generic term which can refer to any of the 72 species. Following is a list of links to their pages: Alder Tree, Almond Tree, Apple Tree, Ash Tree, Aspen Tree, Bay Willow Tree, Beech Tree, Birch Tree, Birdcherry Tree, Black Pine Tree, Buckthorn Tree, Carob Tree, Cedar Tree, Chaste Tree, Checker Tree, Cherry Tree, Chestnut Tree, Conker Tree, Cork Oak Tree, Crabapple Tree, Cypress Tree, Dogwood Tree, Dwarf Pine, Elm Tree, Fig Tree, Fir Tree, Gloomcap, Gnome's Cap, Goldenchain Tree, Hazel Tree, Hornbeam Tree, Juniper Tree, King's Oak Tree, Larch Tree, Laurel Tree, Lemon Tree, Linden Tree, Lote Tree, Maple Tree, Mayflower Tree, Medlar Tree, Mound Tree, Mulberry Tree, Oak Tree, Olive Tree, Osier Tree, Pear Tree, Persimmon Tree, Pine Tree, Plane Tree, Plum Tree, Poplar Tree, Quince Tree, Rowan Tree, Sallow Tree, Silver Fir Tree, Sorb Tree, Spruce Tree, Stone Pine Tree, Sweetgum Tree, Sycamore Tree, Tamarisk Tree, Terebinth Tree, Towercap, Tree Heath Tree, Trumpet Chantrelle, Walnut Tree, Warty Birch Tree, Whitebeam Tree, Willow Tree, Wood Strawberry Tree, Yew Tree.
See the Tables/Trees page for specific per-tree data.
1.3 Skills Affecting Treeplanting
2 Ways to Improve the Quality of Farmed Trees
There are two methods currently available to farm trees: Treeplanter's Pot method' and Direct Method
Treeplanter's Pot Method:
To grow a tree, one needs a Treeplanter's Pot, a Herbalist Table, 1.0 liter of Water, 4 units of Soil (or Soil-like material), and a Seed of Tree or Bush of the plant you wish to grow. Place the Soil, Water, and seed into the pot, and then place the filled pot on an herbalist table. After 1 hour and 15 minutes, the tree will either sprout or die. If it is dead, all you can do is empty the pot and start over. If it has sprouted, you will have 24 real-life hours to plant it before it dies.
Take the pot, find a spot where you wish to plant it, and right click on the spot while holding the pot to get the option to plant it. The tree will steadily increase its size and material, but its quality, and the quality of the products gathered from it, stays the same as is determined when it is first grown (See below section: Ways to Improve...) If you chop a or harvest a tree or bush before it has matured fully, all materials harvested will have lower quality than the quality of the tree.
Gather the seeds you wish to plant. With a shovel equipped, pick up the seed from your inventory with your cursor and right click the ground where you wish to plant the tree. This process takes around 10 seconds and is very fast. Trees planted using this method will be half the quality of the seed used, capped downwards at quality 10.
Most, if not all trees seem to take about 150 hours (a little over 6 days) to fully mature on any forest terrain. If planted on other terrains, such as grassland, they grow much slower.
As they grow, trees will naturally convert the ground tiles around them into the native forest biome they're planted on. If it was not planted on a forest tile, it will instead create a generic type of forest terrain called Wald.
Tree seeds that are planted directly into the ground have a random chance to become stunted during growth. The chance to become stunted at some point during growth is determined upon planting, and is not affect by any nearby items or outside forces, which means that trees can be planted close together and close to other objects. Skills related to trees affect the chance of stunting (See Skills Below)
Stunted trees give wood of a lesser quality, limited by their growth percentage. Other products such as seeds are unaffected by this and are always of full tree quality. [Verify] Logs from stunted trees also produce less blocks and boards.
Stunted trees do regenerate their resources.
If a tree is planted with a Treeplanter's Pot, it will never be stunted, unless picked from. [Verify]
Skills Affecting Treeplanting
These skills affect the chance for a seeding to sprout from its treepot, as well as the chance for a tree planted directly in the ground to stunt before maturity
Plant Lore "greater success"
Woodsmanship "greater success"
Forestry "significantly greater success"
Druidic Rite "greatest possible"
Note: The seed's chances of sprouting are increased solely on the above skills.
Inspecting a growing tree will return its growth percentage, and its final quality. (check system-chat for the latter one for growing trees)
Non user-planted trees will not return any info when inspected.
New tree spawning also works on p-claimed ground.
Ways to Improve the Quality of Farmed Trees
High quality Herbalist Table: You will need high quality source lumber and Finer Plant Fibre to craft a high quality Herbalist Table.
High quality Treeplanter's Pot: Quality is softcapped by Dexterity, Masonry and Farming and dependent on the quality of the Clay, Fuel, and Kiln used. Pottery products (such as the Treeplanter's Pot) will also have their quality halved without the use of a Potter's Wheel. The wheel also affects the quality of the product (for better or for worse).
High quality Water: Water quality depends on the location at which you gather it.
High quality Soil: Just as water, Soil quality is dependent on the location you gather it at. See Finding high quality water, clay, and soil for more information.
Bat Guano or Mulch can be used instead of soil, which is generally of a higher quality.
High quality seed: Same quality as the tree of which it was taken.
Farming skill: The quality of the tree is softcapped by the Farming skill of the player who plants the seed in the Treeplanter's Pot.
Best matching Tree quality formula so far:
{\displaystyle {\frac {_{q}Soil*2+_{q}Water*2+_{q}Pot*3+_{q}Table*3+_{q}Seed*15}{25}}(-5,+5)}
Harvest Skis (2021-10-21) >"Saplings now die after 24 RL hours, rather than 8."
Retrieved from "https://ringofbrodgar.com/w/index.php?title=Tree&oldid=92669"
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Inverse scattering without phase information
R.G. Novikov
We report on non-uniqueness, uniqueness and reconstruction results in quantum mechanical and acoustic inverse scattering without phase information. We are motivated by recent and very essential progress in this domain.
author = {R.G. Novikov},
title = {Inverse scattering without phase information},
TI - Inverse scattering without phase information
%T Inverse scattering without phase information
R.G. Novikov. Inverse scattering without phase information. Séminaire Laurent Schwartz — EDP et applications (2014-2015), Talk no. 16, 13 p. doi : 10.5802/slsedp.74. https://slsedp.centre-mersenne.org/articles/10.5802/slsedp.74/
[ABR] N.V. Alexeenko, V.A. Burov, O.D. Rumyantseva, Solution of the three-dimensional acoustical inverse scattering problem. The modified Novikov algorithm, Acoust. J. 54(3), 2008, 469-482 (in Russian); English transl.: Acoust. Phys. 54(3), 2008, 407-419.
[AS] T. Aktosun, P.E. Sacks, Inverse problem on the line without phase information, Inverse Problems 14, 1998, 211-224.
[AW] T. Aktosun, R. Weder, Inverse scattering with partial information on the potential, J. Math. Anal. Appl. 270, 2002, 247-266.
[BAR] V.A. Burov, N.V. Alekseenko, O.D. Rumyantseva, Multifrequency generalization of the Novikov algorithm for the two-dimensional inverse scattering problem, Acoust. J. 55(6), 2009, 784-798 (in Russian); English transl.: Acoustical Physics 55(6), 2009, 843-856.
[Ber] Yu.M. Berezanskii, On the uniqueness theorem in the inverse problem of spectral analysis for the Schrödinger equation, Tr. Mosk. Mat. Obshch. 7, 1958, 3-62 (in Russian).
[BS] F.A. Berezin, M.A. Shubin, The Schrödinger Equation, Vol. 66 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, 1991.
[Buc] A.L. Buckhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl. 16(1), 2008, 19-33.
[ChS] K. Chadan, P.C. Sabatier, Inverse Problems in Quantum Scattering Theory, 2nd edn. Springer, Berlin, 1989.
[DT] P. Deift, E.Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math. 32, 1979, 121-251.
[E] G. Eskin, Lectures on Linear Partial Differential Equations, Graduate Studies in Mathematics, Vol.123, American Mathematical Society, 2011.
[EW] V.Enss, R.Weder, Inverse potential scattering: a geometrical approach,
[F1] L.D. Faddeev, Uniqueness of the solution of the inverse scattering problem, Vest. Leningrad Univ. 7, 1956, 126-130 (in Russian).
[F2] L.D. Faddeev, Inverse problem of quantum scattering theory. II, Itogy Nauki i Tekh. Ser. Sovrem. Probl. Mat. 3, 1974, 93-180 (in Russian); English transl.: Journal of Soviet Mathematics 5, 1976, 334-396.
[FM] L.D. Faddeev, S.P. Merkuriev, Quantum Scattering Theory for Multi-particle Systems, Nauka, Moscow, 1985 (in Russian); English transl: Math. Phys. Appl. Math. 11 (1993), Kluwer Academic Publishers Group, Dordrecht.
[G] P.G. Grinevich, The scattering transform for the two-dimensional Schrödinger operator with a potential that decreases at infinity at fixed nonzero energy, Uspekhi Mat. Nauk 55:6(336),2000, 3-70 (Russian); English transl.: Russian Math. Surveys 55:6, 2000, 1015-1083.
[GS] F. Gesztesy, B. Simon, Inverse spectral analysis with partial information on the potential. I. The case of an a.c. component in the spectrum, Helv. Phys. Acta 70, 1997, 66-71.
[HH] P. Hähner, T. Hohage, New stability estimates for the inverse acoustic inhomogeneous medium problem and applications, SIAM J. Math. Anal., 33(3), 2001, 670-685.
[HN] G.M. Henkin, R.G. Novikov, The
\overline{\partial }
-equation in the multidimensional inverse scattering problem, Uspekhi Mat. Nauk 42(3), 1987, 93-152 (in Russian); English transl.: Russ. Math. Surv. 42(3), 1987, 109-180.
[I] M.I. Isaev, Exponential instability in the inverse scattering problem on the energy interval, Funkt. Anal. Prilozhen. 47(3), 2013, 28-36 (in Russian); English transl.: Funct. Anal Appl. 47, 2013, 187-194.
[IN] M.I. Isaev, R.G. Novikov, New global stability estimates for monochromatic inverse acoustic scattering, SIAM J. Math. Anal. 45(3), 2013, 1495-1504.
[K] M.V. Klibanov, Phaseless inverse scattering problems in three dimensions, SIAM J. Appl. Math. 74, 2014, 392-410.
[KR1] M.V. Klibanov, V.G. Romanov, The first solution of a long standing problem: Reconstruction formula for a 3-d phaseless inverse scattering problem for the Schrödinger equation, J. Inverse Ill-Posed Probl. doi:10.1515/jiip-2015-0025; arXiv:1412.8210v1, December 28, 2014.
[KR2] M.V. Klibanov, V.G. Romanov, Reconstruction procedures for two inverse scattering problems without the phase information, arXiv:1505.01905v1, May 8, 2015.
[KS] M.V. Klibanov, P.E. Sacks, Phaseless inverse scattering and the phase problem in optics, J. Math. Phys. 33, 1992, 3813-3821.
[L] B.M. Levitan, Inverse Sturm-Liuville Problems, VSP, Zeist, 1987.
[Mar] V.A. Marchenko, Sturm-Liuville Operators and Applications, Birkhäuser, Basel, 1986.
[Mel] R.B. Melrose, Geometric scattering theory, Cambridge University Press, 1995.
[Mos] H.E. Moses, Calculation of the scattering potential from reflection coefficients, Phys. Rev. 102, 1956, 559-567.
[N1] R.G. Novikov, Multidimensional inverse spectral problem for the equation
-\Delta \psi +\left(v\left(x\right)-Eu\left(x\right)\right)\psi =0
, Funkt. Anal. Prilozhen. 22(4), 1988, 11-22 (in Russian); English transl.: Funct. Anal. Appl. 22, 1988, 263-272.
[N2] R.G. Novikov, The inverse scattering problem at fixed energy level for the two-dimensional Schrödinger operator, J. Funct. Anal., 103, 1992, 409-463.
[N3] R.G. Novikov, The inverse scattering problem at fixed energy for Schrödinger equation with an exponentially decreasing potential, Comm. Math. Phys. 161, 1994, 569-595.
[N4] R.G. Novikov, Inverse scattering up to smooth functions for the Schrödinger equation in dimension 1, Bull. Sci. Math. 120, 1996, 473-491.
[N5] R.G. Novikov, Approximate inverse quantum scattering at fixed energy in dimension 2, Proc. Steklov Inst. Math. 225, 1999, 285-302.
[N6] R.G. Novikov, The
\overline{\partial }
-approach to monochromatic inverse scattering in three dimensions, J. Geom. Anal. 18, 2008, 612-631.
[N7] R.G. Novikov, Absence of exponentially localized solitons for the Novikov-Veselov equation at positive energy, Physics Letters A 375, 2011, 1233-1235.
[N8] R.G. Novikov, An iterative approach to non-overdetermined inverse scattering at fixed energy, Mat. Sb. 206(1), 2015, 131-146 (in Russian); English transl.: Sbornik: Mathematics 206(1), 2015, 120-134.
[N9] R.G. Novikov, Explicit formulas and global uniqueness for phaseless inverse scattering in multidimensions, J. Geom. Anal. doi:10.1007/s12220-014-9553-7; arXiv:1412.5006v1, December 16, 2014.
[N10] R.G. Novikov, Formulas for phase recovering from phaseless scattering data at fixed frequency, Bull. Sci. Math. doi:10.1016/j.bulsci.2015.04.005; arXiv:1502.02282v2, February 14, 2015.
[N11] R.G. Novikov, Phaseless inverse scattering in the one-dimensional case, Eurasian Journal of Mathematical and Computer Applications 3(1), 2015, 63-69; arXiv:1503.02159, March 7, 2015.
[New] R.G. Newton, Inverse Schrödinger scattering in three dimensions, Springer, Berlin, 1989.
[NM] N.N. Novikova, V.M. Markushevich, On the uniqueness of the solution of the inverse scattering problem on the real axis for the potentials placed on the positive half-axis. Comput. Seismology 18, 1985, 176-184 (in Russian).
[R] T. Regge, Introduction to complex orbital moments, Nuovo Cimento 14, 1959, 951-976.
[S] P. Stefanov, Stability of the inverse problem in potential scattering at fixed energy, Annales de l’Institut Fourier, tome 40(4), 1990, 867-884.
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f(t) = 2 \cdot t .
Signal with function
f(t) = 2 \cdot t
u(t) = \begin{cases} 1 & \mbox{if } t \geq 0 \\\\ 0 & \mbox{if } t \lt 0. \end{cases}
\sigma(t) = \begin{cases} 1 & \mbox{if } t = 0 \\\\ 0 & \mbox{if } t \neq 0.\end{cases}
r(t) = \begin{cases} t & \mbox{if } t \geq 0 \\\\ 0 & \mbox{if } t \lt 0. \end{cases}
x(t) = \begin{cases} \frac{t^{2}}2 & \mbox{if } t \geq 0 \\\\ 0 & \mbox{if } t \lt 0.\end{cases}
This signal follows the path of a sinusoid
\big(
\sin(x)
\cos(x)\big).
These signals are very important because they are the basis for Fourier analysis and Fourier series. They are used often in electrical engineering:
x(t) = \begin{cases} A \cdot \cos(w_0 \cdot t\pm \phi) \text{ or}\\\\ A \cdot \sin(w_0 \cdot t\pm \phi). \end{cases}
A signal is even if it satisfies the equation
f(t) = f(-t)
. In other words, even function are symmetric about the
y
-axis. A signal is odd if it satisfies the equation
f(t) = -f(-t)
. In other words, about the
y
-axis, the signal is mirrored about the
x
-axis. For example,
\cos(t)
is an even signal, and
\sin(t)
is an odd signal.
A signal is periodic if it satisfies the equation
f(t) = f(t + T)
T
is the fundamental time period. What this is really saying is if the signal
f(t)
repeats every
T
time, it is periodic. An aperiodic function does not repeat. An example of a period signal is any sinusoidal signal like
\cos(x)
. An example of an aperiodic signal is the unit impulse signal. It signals a pulse at
f(0)
but then nothing afterwards.
Operations on signals that affect amplitude follow the basic mathematical operations: addition and multiplication. The addition of two signals results in a signal that has their summed amplitudes at every time step. So, summing two signals,
X_0
X_1
X_2
. That means at every possible time step,
X_2
has a signal value that is the sum of the values of
X_0
X_1
at that time step.
c \cdot \big(X_0(t) + X_1(t)\big) = cX_0(t) + cX_1(t) .
It's possible to change a signal's time by shifting it, scaling it, or reversing it. Time shifting takes a signal
X_0(t)
and shifts it by a value,
t_0
. This shift can be positive or negative to indicate which direction in time the signal is being shifted. The result is
X_0(t \pm t_0)
Time scaling will either compress or expand the signal. For example, if a signal is defined from
t = -2
t = 2
, and its time is scaled by a factor of 2, the new signal will be defined from
t = -4
t = 4
Time reversal is an operation where the signal
f(t)
is turned into
f(-t)
. This new signal is the input signal, mirrored about the
y
Difference equations are mathematical ways of expressing systems, and they are often recursive (it depends on earlier states). As an example, let's look at the Fibonacci sequence. At each step of this sequence, we add the two previous inputs. So, if we define
x[n]
as the current input and
y[n]
as the output at a given time step, then
y[n] = y[n-1] + y[n-2] + x[n]
describes the Fibonacci sequence. Here,
x[n]
is a signal whose value equals 1 at time = 0 and zero thereafter. It is the exact same as the unit impulse signal from above. The difference equation, also sometimes called the recurrence relation, is important because it allows us to mathematically manipulate these elements.
Scale: This triangular device will scale the incoming signal by a scalar term,
c
(also sometimes called a buffer).
Y = X + \mathcal{R}X + \mathcal{R}^2X .
\mathcal{R}
is used to denote delay on an input signal. These operator equations (also sometimes called system functions) are used often in predicting system behavior.
A system is linear if it satisfies the following property, where signals
x_1(t)
x_2(t)
y_1(t)
y_2(t)
T\big[a_1x_1(t) + a_2x_2(t)\big] = a_1T\big[x_1(t)\big] + a_2T\big[x_2(t)\big] = a_1y_1(t) + a_2y_2(t) .
Linear systems are typically much simpler than their non-linear counterparts. They are used in automatic control theory, signal processing, and telecommunications. Specifically, wireless communication can be modeled by linear systems.
A system is time-variant if its input and output relationship varies with time. The equations that define these classes are as follows: when
y(n, t) = T[x(n-t)] = \mbox{input change}
y(n - t) = \mbox{output change}
\begin{aligned} y(n, t) &= y(n-t) \ \ \mbox{for time-invariant systems}\\ y(n, t) &\neq y(n-t) \ \ \mbox{for time-variant systems}. \end{aligned}
Time-variant systems are interesting to study because its output depends more on time because the system itself changes over time. The human vocal chords are time-variant because the vocal organs change as the tongue and the velum move. Time-invariant systems are much easier to reason about and model.
Static systems are memory-less systems. An example equation for a static system might be
y[t] = 2^{x[t]} .
This is because the output at the current time step
y[0]
is dependent only on the input from the current time step
x[t]
. On the other hand, a dynamic system (a system with memory) might have the following system equation:
y[t] = 2 \cdot x[t - 1] .
Here, the output at the current time step
y[t]
is 2 times the input from the previous time step
x[t-1]
, so the system must remember that input.
Similar to the distinction between static and dynamic systems, a causal system is one that depends on only present and past inputs. So,
y[t] = 2 \cdot x[t - 1]
still described a causal system. A non-causal system depends on future inputs.
y[t] = x[t+1]
is a non-causal system.
A stable system is one that has bounded outputs for bounded inputs. In other words, for a bounded signal, the output amplitude is finite. So, the system described by
y[n] = 2 \cdot x[n]
Systems can be operated on and combined just like the signals they use. Cascading two systems, for example, combined two systems in a simple way. Cascading system
S_0
S_1
lets the output of
S_0
Y
, be the input for
S_1
W
. This operation is commutative as long as both systems are initially at rest (equal to 0). So, if the original systems have system functions
\begin{aligned} S_0\mbox{ : }Y &= \Phi_1 X \\ S_1\mbox{ : }Z &= \Phi_2 W, \end{aligned}
then we set the equation for
S_0
equal to the input for
S_1
S_1\mbox{ : }Z = \Phi_2 \cdot \Phi_1 X.
Parallel addition is another way to combine systems. Let's say
S_0
Y = \Phi_1X
S_1
Z = \Phi_2X
. It's important that the input signal is the same for this operation. So, the addition of these two output signals will simply be
W = (\Phi_0 + \Phi_2)X ,
W
is the summed output of the two systems.
\begin{aligned} S_0\mbox{ : }A &= \Phi_1X \\ S_1\mbox{ : }B &= \Phi_2X \\ S_2\mbox{ : }C &= \Phi_3W. \end{aligned}
S_0
S_1
and cascade the resulting system with
S_2
. What is the output of these systems configured in this way?
The output,
Z
Z = \Phi_3(\Phi_2 + \Phi_1)X .
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Variance | Brilliant Math & Science Wiki
Mei Li, Andy Hayes, Lathan Liou, and
Variance is a statistic that is used to measure deviation in a probability distribution. Deviation is the tendency of outcomes to differ from the expected value.
Studying variance allows one to quantify how much variability is in a probability distribution. Probability distributions that have outcomes that vary wildly will have a large variance. Probability experiments that have outcomes that are close together will have a small variance.
The variance explored on this page is different from sample variance, which is the variance of a sample of data.
X
is a numerical discrete random variable with distribution
p(x)
and expected value
\mu = \text{E}(X)
, the variance of
X
\sigma^2
\text{Var}[X]
\sigma^2=\text{Var}[X] = \text{E}\big[ (X - \mu)^2\big] = \sum_x (x - \mu)^2 p(x).
Note that from the definition, the variance is always non-negative, and if the variance is equal to zero, then the random variable
X
takes a single constant value, which is its expected value
\mu.
In the rest of this summary, it is assumed
X
is a discrete numerical random variable with distribution
p(x)
\mu = E(X).
The following theorem gives another method to calculate the variance.
The variance of random variable
X
\text{Var}[X] = \text{E}\big[X^2\big] - \mu^2.
\begin{aligned} \text{Var}[X] &= \text{E}\big[ (X - \mu)^2\big] \\ &= \text{E} \big[X^2 - 2 \mu X + \mu^2 \big]\\ &= \text{E}\big[X^2\big] - 2 \mu \text{E}[X] + \text{E}\big[\mu^2\big]\\ &= \text{E}\big[X^2\big] - 2\mu \cdot \mu + \mu^2\\ &= \text{E}\big[X^2\big] - \mu^2, \end{aligned}
where the third line follows from linearity of expectation.
_\square
What is the variance of a fair, six-sided die roll?
X
be the random variable that represents the result of the die roll. It is known that
\text{E}[X]=\frac{7}{2}
\text{E}[X^2]
\text{E}\big[X^2\big]=\sum\limits_{k=1}^6{\dfrac{1}{6}k^2}=\dfrac{1}{6}\times\dfrac{(6)(7)(13)}{6}=\dfrac{91}{6}.
Then, the variance can be calculated:
\text{Var}[X]=\text{E}\big[X^2\big]-\mu^2=\dfrac{91}{6}-\dfrac{49}{4}=\dfrac{35}{12}\approx 2.917.\ _\square
The following properties of variance correspond to many of the properties of expected value. However, some of these properties have different results.
c
\text{Var}[c]=0.
\begin{aligned} \text{Var}[c]&=\text{E}\Big[\big(c-E[c]\big)^2\Big] \\ &=\text{E}\big[(c-c)^2\big] \\ &=0.\ _\square \end{aligned}
X
c
\text{Var}[cX] = c^2 \text{Var}[X].
By the properties of expectation,
\text{E}[cX] = c \text{E}[X] = c\mu
\begin{aligned} \text{Var}[cX] &= \text{E}\big[ ( cX - c \mu)^2 \big]\\ &= \text{E}\big[c^2(X- \mu)^2\big]\\ &= c^2 \text{E}\big[(X- \mu)^2\big]\\ &= c^2\text{Var}[X].\ _\square \end{aligned}
X
c
\text{Var}[X + c] = \text{Var}[X] .
\begin{aligned} \text{Var}[X + c] &= \text{E}\big[(X+c)^2\big] - (\mu+c)^2\\ &= \text{E}\big[X^2 + 2cX + c^2\big] - \big(\mu^2 + 2 c\mu + c^2\big)\\ &= \text{E}\big[X^2\big] + 2c\text{E}[X] + \big(c^2 - \mu^2 - 2c\mu - c^2\big) \\ &= \text{E}\big[X^2\big] - \mu^2\\ &= \text{Var}[X]. \ _\square \end{aligned}
The above two theorems show how translating or scaling the random variable by a constant changes the variance. The first theorem shows that scaling the values of a random variable by a constant
c
scales the variance by
c^2
. This makes sense intuitively since the variance is defined by a square of differences from the mean. The second theorem shows that translating all variables by a constant does not change the variance. This also makes intuitive sense, since translating all variables by a constant also translates the expected value, and the spread of the translated values around the translated expected value remains unchanged.
From the linearity property of expected value, for any two random variables
X
Y
E(X+Y) = E(X) + E(Y).
However, this does not hold for variance in general. One special case for which this does hold is the following:
X
Y
be independent random variables. Then
\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y).
\begin{aligned} \text{Var}(X+Y) &= E\big( (X+Y)^2 \big) - \big(E(X +Y)\big)^2\\ &= E( X^2 + 2XY + Y^2 ) - \big(E(X) + E(Y)\big)^2\\ &= E(X^2) + 2E(XY) + E(Y^2) - \big(E(X)^2 + 2E(X)E(Y) + E(Y)^2\big)\\ &= E(X^2) + 2E(X)E(Y) + E(Y^2) - E(X)^2 - 2E(X)E(Y) - E(Y)^2\\ &= E(X^2) - E(X)^2 + E(Y^2) - E(Y)^2\\ &= \text{Var}(X) + \text{Var}(Y), \end{aligned}
where the calculation
E(XY) = E(X) E(Y)
in the fourth line follows from the independence of random variables
X
Y
_\square
The following is a generalization of the above theorem.
X_1, X_2, \ldots, X_k
\text{Var}(X_1 + X_2 + \cdots + X_k) = \text{Var}(X_1) + \text{Var}(X_2) + \cdots + \text{Var}(X_k).
For non-independent random variables
X
Y
\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) + 2\text{Cov}(X,Y).
\begin{aligned} \text{Var}(X+Y) &= \text{Cov}(X+Y, X+Y)\\ &= \text{Cov}(X,X) + \text{Cov}(Y,Y) + 2\text{Cov}(X,Y)\\ &= \text{Var}(X) + \text{Var}(Y) + 2\text{Cov}(X,Y). \end{aligned}
In order to calculate the variance of the sum of dependent random variables, one must take into account covariance.
\mu
The standard deviation of a random variable, denoted
\sigma
, is the square root of the variance, i.e.
\sigma(X) = \sqrt{\text{Var}(X)}.
Note that the standard deviation has the same units as the data. The variance of a random variable is also denoted by
\sigma^2
2
bags, each containing balls numbered
1
5
. From each bag,
1
ball is removed. What are the variance and standard deviation of the total of the two balls?
X
be the random variable denoting the sum of these values. Then, the probability distribution of
X
\begin{array} { | l | l | l | l | l | l | l | l | l | l l | } \hline x & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & \\ \hline P(X=x) & \frac{1}{25} & \frac{2}{25} & \frac{3}{25} & \frac{4}{25} & \frac{5}{25} & \frac{4}{25} & \frac{3}{25} & \frac{2}{25} & \frac{1}{25} & \\ \hline \end{array}
Previously, the expected value was calculated,
E[X] = 6.
\begin{aligned} \text{Var}(X) =& E\big[(X - \mu)^2\big] \\ =& (2-6)^2 \times \frac{1}{25} + (3-6)^2 \times \frac{2}{25} + (4-6)^2 \times \frac{3}{25} \\ & + (5-6)^2 \times \frac{4}{25} + (6-6)^2 \times \frac{5}{25} + (7-6)^2 \times \frac{4}{25} \\ & + (8-6)^2 \times \frac{3}{25} + (9-6)^2 \times \frac{2}{25} + (10-6)^2 \times \frac{1}{25} \\ =& 4. \end{aligned}
Then the standard deviation is
\sigma(X) = \sqrt{4} = 2.\ _\square
3
six-sided dice are rolled. What are the variance and standard deviation for the number of times a
3
Y
be the random variable representing the number of times a
3
is rolled. The table below lists the probabilities of rolling different numbers of
3
\begin{array}{|c|cccc|} \hline \mbox{num 3s} & 0 & 1 & 2 & 3\\ \hline \mbox{probability } & \frac{125}{216} & \frac{75}{216} & \frac{15}{216} & \frac{1}{216}\\ \hline \end{array}
The expected number of times a
3
is rolled is
\frac{1}{2}
E(Y) = \frac{1}{2}
. Now the goal is to calculate
E(Y^2):
E(Y^2) = \frac{0^2 \times 125 + 1^2 \times 75 + 2^2 \times 15 + 3^2 \times 1}{216} = \frac{144}{216} = \frac{2}{3}.
\text{Var}(Y) = \frac{2}{3} - \left(\frac{1}{2}\right)^2 = \frac{5}{12}
\sigma(Y) = \sqrt{\frac{5}{12}}
_\square
p
X_i
X_i=1
i^\text{th}
X_i=0
i^\text{th}
Y
be a random variable indicating the number of trials until the first flip of heads in the sequence of coin flips. What are the variance and standard deviation of
Y?
In the Expected Value wiki, it was demonstrated that
Y
is a geometrically distributed random variable with
E(Y) = \frac{1}{p}
\text{Var}(Y) = E(Y^2) - \frac{1}{p^2},
\begin{aligned} E(Y^2) & = 1 \cdot p + 2^2 (1-p) p + 3^2 (1-p)^2 p + 4^2 (1-p)^3 p + \cdots\\\\ &= p\big[1 + 4(1-p) + 9(1-p)^2 + 16(1-p)^3 + \cdots \big]. \end{aligned}
To compute this sum, consider the algebraic identity
1+ x + x^2 + x^3 + \cdots = \frac{1}{1-x}.
By first differentiating this equation, then multiplying throughout by
x
, and then differentiating again,
\begin{aligned} 1 + 2x + 3x^2 + \cdots &= \frac{1}{(1-x)^2}\\ x + 2x^2 + 3x^3 + \cdots &= \frac{x}{(1-x)^2}\\ 1 + 4x + 9x^2 + \cdots &= \frac{1+x}{(1-x)^3}. \end{aligned}
\begin{aligned} E(Y^2) &= p\frac{1+(1-p)}{\big(1-(1-p)\big)^3} \\ &= \frac{1+(1-p)}{p^2} \\ &= \frac{2-p}{p^2}\\ \\ \text{Var}(Y) &= E(Y^2) - \big(E(Y)\big)^2\\ &= \frac{2-p}{p^2} - \frac{1}{p^2} \\ &= \frac{1-p}{p^2} \\ \\ \sigma(Y) &= \sqrt{ \frac{1-p}{p^2} }. \ _\square \end{aligned}
Cite as: Variance. Brilliant.org. Retrieved from https://brilliant.org/wiki/variance-definition/
|
Nesbitts Inequality | Brilliant Math & Science Wiki
Nesbitts Inequality
Daniel Liu, Akshat Sharda, P C, and
Nesbitt's inequality is a special case of the Shapiro inequality. It states that for positive real numbers
a, b,
c
\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\ge \dfrac{3}{2}.
Nesbitt's inequality is a famous inequality with many unique solutions. It states that
a,b
c,
\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\ge \dfrac{3}{2}. \ _\square
Below is a list of proofs of Nesbitt's inequality. Feel free to add your own proofs.
Prove Nesbitt's Inequality
Clearing denominators and full expansion gives that it is equivalent to
2a^3+2b^3+2c^3\ge a^2b+a^2c+b^2a+b^2c+c^2a+c^2b.
However, by AM-GM
\dfrac{a^3+a^3+b^3}{3}\ge a^2b,
so summing the inequality symmetrically gives the desired inequality.
_\square
\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b} = \dfrac{a^2}{ab+ac}+\dfrac{b^2}{bc+ba}+\dfrac{c^2}{ca+cb}\ge \dfrac{(a+b+c)^2}{2(ab+bc+ca)}.
By Titu's lemma, it remains to prove
\begin{array}{c}&\dfrac{(a+b+c)^2}{2(ab+bc+ca)}\ge \dfrac{3}{2} &\text{ or } &(a+b+c)^2\ge 3(ab+bc+ca), \end{array}
which is true after full expansion and the use of
a^2+b^2+c^2\ge ab+bc+ca
_\square
Without losing generality we assume that
a\geq b\geq c
\frac{1}{b+c}\geq\frac{1}{a+c}\geq\frac{1}{a+b}.
Now, applying Chebyshev's inequality, we have
3\bigg(\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\bigg)\geq(a+b+c)\bigg(\frac{1}{b+c}+\frac{1}{a+c}+\frac{1}{a+b}\bigg).
Applying Titu's lemma, we get
\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{a+c}\geq\frac{9}{2(a+b+c)}
\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\geq\frac{3}{2}.\ _\square
S
denote the left side of the inequality, then we can transform
S
\begin{aligned} \text{LHS} =S &=\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} \\ &= 1+\frac{a}{b+c}+1+\frac{b}{c+a}+1+\frac{c}{a+b}-3 \\ &=\frac{a+b+c}{b+c} +\frac{a+b+c}{c+a}+\frac{a+b+c}{a+b}-3 \\ &= \frac{1}{2}\Big[ (a+b)+(b+c)+(c+a) \Big] \left( \frac{1}{b+c}+\frac{1}{c+a} +\frac{1}{a+b} \right) -3. \end{aligned}
x=a+b, y=b+c, z=c+a,
\begin{aligned} 2S & = (x+y+z)\left(\frac{1}{y}+\frac{1}{z}+\frac{1}{x} \right) - 6 \\ & = \underbrace{\frac{x}{y}+\frac{y}{x}}_{\geq 2}+\underbrace{\frac{x}{z} + \frac{z}{x}}_{\geq2}+\underbrace{\frac{y}{z}+\frac{z}{y}}_{\geq2} - 3 \geq 3 \\ \Rightarrow S&= \frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\geq\frac{3}{2}. \ _\square \end{aligned}
Cite as: Nesbitts Inequality. Brilliant.org. Retrieved from https://brilliant.org/wiki/nesbitts-inequality/
|
Vandermonde's Identity | Brilliant Math & Science Wiki
Kishlaya Jaiswal, Sandeep Bhardwaj, Samir Khan, and
Vandermonde's identity (or Vandermonde's convolution), named after Alexandre-Théophile Vandermonde, states that any combination of
k
objects from a group of
(m+n)
objects must have some
0\ \leq\ r\ \leq\ k
m
objects and the remaining
(k-r)
n
{m+n \choose k} = \sum_{r=0}^k {m \choose r}{n \choose k-r}.
This identity can be useful when evaluating certain other sums, or doing difficult combinatorics problems.
We consider the binomial expansion of
(1+x)^{m+n}
(1+x)^{m+n} = \sum_{k=0}^{m+n} {m+n \choose k}x^k.
Also, notice that, we can consider the expansion by observing that
(1+x)^{m+n} = (1+x)^m(1+x)^n:
\begin{aligned} (1+x)^{m+n} & = (1+x)^{m}(1+x)^{n} \\ & = \left(\sum_{i=0}^m {m \choose i} x^i\right)\left(\sum_{j=0}^n {n \choose j} x^j\right) \\ & = \left({m \choose 0}+{m \choose 1}x+{m \choose 2}x^2+\cdots \right) \times \left({n \choose 0}+{n \choose 1}x+{n \choose 2}x^2+\cdots\right) \\ & = \left({m \choose 0}{n \choose 0}\right)x^0 +\left({m \choose 0}{n \choose 1}+{m \choose 1}{n \choose 0}\right)x +\left({m \choose 0}{n \choose 2}+{m \choose 1}{n \choose 1}+{m \choose 2}{n \choose 0}\right)x^2+\cdots. \end{aligned}
Thus, we can conclude that the coefficient of
x^k
in the above expansion is
{m \choose 0}{n \choose k}+{m \choose 1}{n \choose k-1}+\cdots+{m \choose k}{n \choose 0} = \sum_{r=0}^k {m \choose r}{n \choose k-r}.
Therefore, by comparing the coefficients of
x^k
\sum_{r=0}^k {m \choose r}{n \choose k-r} = {m+n \choose k}.\ _\square
Note: In particular, Vandermonde's identity holds for all binomial coefficients, not just the non-negative integers that are assumed in the combinatorial proof.
m
n
girls in a class and you're asked to form a team of
k
pupils out of these
m+n
students, with
0 \le k \le m+n.
You can do this in
{m+n \choose k}
ways. But, now we count in rather a different manner. To form the team, you can choose
i
k-i
girls for some fixed
i
{m \choose i}{n \choose k-i}
ways to do this. Now, either you can have
0
k
girls, or
1
boy and
k-1
2
k-2
\ldots
. That is, there are
\sum_{r=0}^k {m \choose r}{n \choose k-r}
ways to form the team.
Thus, we derive at our result
\sum_{r=0}^k {m \choose r}{n \choose k-r} = {m+n \choose k}.
{2n \choose n} = \sum_{r=0}^n {n \choose r}^2.
The above sum is a special case of Vandermonde's identity where
m=k=n.
{n+n \choose n} = \sum_{r=0}^n {n \choose r}{n \choose n-r}.
Therefore, because of the identity
{n \choose n-r}={n \choose r}
{2n \choose n} = \sum_{r=0}^n {n \choose r}^2.\ _\square
I'll leave the combinatorial proof of this identity as an exercise for you to work out.
In the algebraic proof of the above identity, we multiplied out two polynomials to get our desired sum. Similarly, by multiplying out
polynomials, you can get the generalized version of the identity, which is
\sum_{k_1+\dots +k_p = m}^m {n\choose k_1} {n\choose k_2} {n\choose k_3} \cdots {n \choose k_p} = { pn \choose m}.
You can also think of it combinatorially, by considering
p
bags with each bag consisting of
n
balls. Thus, you have a total of
pn
balls. Now you need to pick up
m
balls in total. There are
{pn \choose m}
On a similar argument as stated above, we can also pick up
m
balls by picking
i_1
balls from bag #1,
i_2
balls from bag #2, ..., and
i_p
balls from bag #p. Thus for a fixed set of
(i_1,i_2,\ldots,i_p)
{n\choose k_1} {n\choose k_2} {n\choose k_3} \cdots {n \choose k_p}
ways to do it, and hence in total
\sum_{k_1+\dots +k_p = m}^m {n\choose k_1} {n\choose k_2} {n\choose k_3} \cdots {n \choose k_p}
This leads us to our desired result
\sum_{k_1+\dots +k_p = m}^m {n\choose k_1} {n\choose k_2} {n\choose k_3} \cdots {n \choose k_p} = { pn \choose m}.\ _\square
Moving the LHS (in Vandermonde's identity) to RHS in the denominator yields
\begin{aligned} 1 & = & \frac{\sum_{r=0}^k {m \choose r}{n \choose k-r}}{{m+n \choose k}} \\ & = & \sum_{r=0}^k \frac{{m \choose r}{n \choose k-r}}{{m+n \choose k}} \\ & = & \frac{{m \choose 0}{n \choose k}}{{m+n \choose k}}+\frac{{m \choose 1}{n \choose k-1}}{{m+n \choose k}}+\frac{{m \choose 2}{n \choose k-2}}{{m+n \choose k}}+\cdots . \end{aligned}
Each term in the above sum can be interpreted as a probability, that is, the probability distribution of the number of blue balls in
k
draws without replacement from a bag containing
n
blue and
m
green balls. The resulting distribution is better known as hypergeometric probability distribution.
Chu-Vandermonde's Identity:
The identity was extended to non-integer arguments, by Wenchang Chu, and is known by the name Chu-Vandermonde Identity, which is stated as follows:
For general complex-valued
x
y
and any non-negative integer
n
{x+y \choose n}=\sum_{k=0}^n {x \choose k}{y \choose n-k}.
It can be re-written in terms of falling Pochhammer symbol as
(x+y)_n = \sum_{k=0}^n {n \choose k} (x)_k (y)_{n-k} ,
(x)_{k}=x(x-1)(x-2)\ldots(x-k+1)
and is known as the falling Pochhammer symbol (or descending factorial).
Rothe-Hagen Identity:
The Rothe-Hagen identity, named after Heinrich August Rothe and Johann Georg Hagen, is a further generalization of Vandermonde's identity, which extends for all complex numbers
(a,b,c)
\sum_{k=0}^n\frac{a}{a+bk}{a+bk \choose k}{c-bk \choose n-k}={a+c \choose n}.
How many 4-digit numbers are there such that the thousands digit is equal to the sum of the other 3 digits?
12=012
As the well-known song goes, on the first day of Christmas my true love gave to me a partridge in a pear tree. On the second day of Christmas, my true love gave to me 2 turtle doves and a partridge in a pear tree. On day
n
she gives me 1 of something, 2 of something else, ...,
n
of something else. At the end of the first 16 days, how many gifts has my true love given to me in total?
1,000,000,000
have the product of their digits equal to 49?
At Peter's school, to progress to the next year, he has to pass an exam every summer. Every exam is out of 50 and the pass mark is always 25. To graduate from school, he must pass 10 exams. Since Peter's work ethic increases with age, his scores never decrease.
N
be the number of different series of marks Peter could have achieved, given that he left school without ever failing an exam. What are the last 3 digits of
N
Cite as: Vandermonde's Identity. Brilliant.org. Retrieved from https://brilliant.org/wiki/vandermondes-identity/
|
transform(deprecated)/tallyinto - Maple Help
Home : Support : Online Help : transform(deprecated)/tallyinto
stats[transform, tallyinto]
group together data into a given pattern
stats[transform, tallyinto](data,partition)
transform[tallyinto](data,partition)
stats[transform, tallyinto['extra']](data, partition)
list showing how the data is to be split
The function tallyinto of the subpackage stats[transform, ...] groups the items of data into the pattern given by partition.
The values (see transform[statvalue]) in the result are the values that were in partition.
If there are data items that do not fit into the given pattern and the parameter extra gives a name, then they are put into the name specified by extra. If the parameter extra is not specified, an error condition is raised which gives the non-conforming items.
Remember that for the stats package, the upper boundary of a class (or range) excludes the boundary. So the class
3..4
represents points
x
3<=x<4
. Refer to stats[data] for more information.
To regroup data, use stats[transform, tally] and stats[transform, statsort].
\mathrm{with}\left(\mathrm{stats}\right):
\mathrm{data1}≔[7,11,2,19,13,5,7,10,15,16]
\textcolor[rgb]{0,0,1}{\mathrm{data1}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{11}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{19}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{13}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{10}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{15}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{16}]
\mathrm{transform}[\mathrm{tallyinto}]\left(\mathrm{data1},[1..5,5..10,10..15,15..20]\right)
[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Weight}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{10}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Weight}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{10}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{15}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Weight}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{15}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{20}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\right)]
\mathrm{data2}≔[1,2,3,\mathrm{missing},3,4..5,4..5,\mathrm{Weight}\left(4..5,6\right),6..7,6..7]
\textcolor[rgb]{0,0,1}{\mathrm{data2}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{missing}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Weight}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{7}]
\mathrm{transform}[\mathrm{tallyinto}]\left(\mathrm{data2},[1..3,3..4,4..8]\right)
[\textcolor[rgb]{0,0,1}{\mathrm{Weight}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Weight}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Weight}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{8}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{10}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{missing}}]
Note, ranges are inclusive at the lower point, but excluding at the higher point, see what happens when 3..4 is omitted.
\mathrm{infolevel}[\mathrm{stats}]≔1
{\textcolor[rgb]{0,0,1}{\mathrm{infolevel}}}_{\textcolor[rgb]{0,0,1}{\mathrm{stats}}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{1}
\mathrm{transform}[\mathrm{tallyinto}]\left(\mathrm{data2},[1..3,4..8]\right)
Error, (in `stats/abort`) [[transform[tallyinto], `these data items do not fit in target pattern`, 3, 3]]
\mathrm{exceptions}≔'\mathrm{exceptions}':
\mathrm{transform}[\mathrm{tallyinto}['\mathrm{exceptions}']]\left(\mathrm{data2},[1..3,4..8]\right)
[\textcolor[rgb]{0,0,1}{\mathrm{Weight}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Weight}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{8}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{10}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{missing}}]
\mathrm{exceptions}
[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]
|
Search results for: Jacques Verstraëte
Counting hypergraphs with large girth
Sam Spiro, Jacques Verstraëte
Morris and Saxton used the method of containers to bound the number of ‐vertex graphs with edges containing no ‐cycles, and hence graphs of girth more than . We consider a generalization to ‐uniform hypergraphs. The girth of a hypergraph is the minimum such that there exist distinct vertices and hyperedges with for all ...
Jiaxi Nie, Jacques Verstraëte
Random Structures & Algorithms > 59 > 1 > 79 - 95
In this paper, we consider a randomized greedy algorithm for independent sets in r‐uniform d‐regular hypergraphs G on n vertices with girth g. By analyzing the expected size of the independent sets generated by this algorithm, we show that , where converges to 0 as g → ∞ for fixed d and r, and f(d, r) is determined by a differential equation. This extends earlier results...
Victor Falgas‐Ravry, Klas Markström, Jacques Verstraëte
Let be a graph of density p on n vertices. Following Erdős, Łuczak, and Spencer, an m‐vertex subgraph H of G is called full if H has minimum degree at least . Let denote the order of a largest full subgraph of G. If is a nonnegative integer, define Erdős, Łuczak, and Spencer proved that for , In this article, we prove the following lower bound: for , Furthermore,...
Journal of Combinatorial Theory, Series B > 2017 > 122 > C > 457-478
The expansion G+ of a graph G is the 3-uniform hypergraph obtained from G by enlarging each edge of G with a vertex disjoint from V(G) such that distinct edges are enlarged by distinct vertices. Let exr(n,F) denote the maximum number of edges in an r-uniform hypergraph with n vertices not containing any copy of F. The authors [10] recently determined ex3(n,G+) when G is a path or cycle, thus settling...
Stability in the Erdős–Gallai Theorems on cycles and paths
Zoltán Füredi, Alexandr Kostochka, Jacques Verstraëte
The Erdős–Gallai Theorem states that for k≥2, every graph of average degree more than k−2 contains a k-vertex path. This result is a consequence of a stronger result of Kopylov: if k is odd, k=2t+1≥5, n≥(5t−3)/2, and G is an n-vertex 2-connected graph with at least h(n,k,t):=(k−t2)+t(n−k+t) edges, then G contains a cycle of length at least k unless G=Hn,k,t:=Kn−E(Kn−t).In this paper we prove a stability...
Cycles in triangle-free graphs of large chromatic number
Alexandr Kostochka, Benny Sudakov, Jacques Verstraëte
More than twenty years ago Erdős conjectured [4] that a triangle-free graph G of chromatic number k≥k0(ε) contains cycles of at least k2−ε different lengths as k→∞. In this paper, we prove the stronger fact that every triangle-free graph G of chromatic number k≥k0(ε) contains cycles of 1/64(1 − ε)k2 logk/4 consecutive lengths, and a cycle of length at least 1/4(1 − ε)k2logk. As there exist triangle-free...
Disjoint Cycles: Integrality Gap, Hardness, and Approximation
Mohammad R. Salavatipour, Jacques Verstraete
Lecture Notes in Computer Science > Integer Programming and Combinatorial Optimization > 51-65
In the edge-disjoint cycle packing problem we are given a graph G and we have to find a largest set of edge-disjoint cycles in G. The problem of packing vertex-disjoint cycles in G is defined similarly. The best approximation algorithms for edge-disjoint cycle packing are due to Krivelevich et al. [16], where they give an
O\sqrt{\rm log n}
-approximation for undirected graphs and an $O(\sqrt{n})$...
On sets of integers with restrictions on their products
Michael Tait, Jacques Verstraëte
A product-injective labeling of a graph G is an injection χ:V(G)→Z such that χ(u)χ(v)≠χ(x)χ(y) for any distinct edges uv,xy∈E(G). Let P(G) be the smallest N≥1 such that there exists a product-injective labeling χ:V(G)→[N]. Let P(n,d) be the maximum possible value of P(G) over n-vertex graphs G of maximum degree at most d. In this paper, we determine the asymptotic value of P(n,d) for all but a small...
On coupon colorings of graphs
Bob Chen, Jeong Han Kim, Michael Tait, Jacques Verstraete
Discrete Applied Mathematics > 2015 > 193 > C > 94-101
Let G be a graph with no isolated vertices. A k-coupon coloring of G is an assignment of colors from [k]≔{1,2,…,k} to the vertices of G such that the neighborhood of every vertex of G contains vertices of all colors from [k]. The maximum k for which a k-coupon coloring exists is called the coupon coloring number of G, and is denoted χc(G). In this paper, we prove that every d-regular graph G has χc(G)≥(1−o(1))d/logd...
Probabilistic constructions in generalized quadrangles
Jeroen Schillewaert, Jacques Verstraete
We study random constructions in incidence structures, and illustrate our techniques by means of a well-studied example from finite geometry. A maximal partial ovoid of a generalized quadrangle is a maximal set of points no two of which are collinear. The problem of determining the smallest size of a maximal partial ovoid in quadrangles has been extensively studied in the literature. In general, theoretical...
A counterexample to sparse removal
Craig Timmons, Jacques Verstraëte
European Journal of Combinatorics > 2015 > 44 > Part A > 77-86
The Turán number of a graph H, denoted ex(n,H), is the maximum number of edges in an n-vertex graph with no subgraph isomorphic to H. Solymosi (2011) conjectured that if H is any graph and ex(n,H)=O(nα) where α>1, then any n-vertex graph with the property that each edge lies in exactly one copy of H has o(nα) edges. This can be viewed as conjecturing a possible extension of the removal lemma to...
Journal of Combinatorial Theory, Series A > 2015 > 129 > Complete > 57-79
A k-path is a hypergraph Pk={e1,e2,…,ek} such that |ei∩ej|=1 if |j−i|=1 and ei∩ej=∅ otherwise. A k-cycle is a hypergraph Ck={e1,e2,…,ek} obtained from a (k−1)-path {e1,e2,…,ek−1} by adding an edge ek that shares one vertex with e1, another vertex with ek−1 and is disjoint from the other edges.Let exr(n,G) be the maximum number of edges in an r-graph with n vertices not containing a given r-graph G...
Turán numbers of bipartite graphs plus an odd cycle
Peter Allen, Peter Keevash, Benny Sudakov, Jacques Verstraëte
Journal of Combinatorial Theory, Series B > 2014 > 106 > Complete > 134-162
For an odd integer k, let Ck={C3,C5,…,Ck} denote the family of all odd cycles of length at most k and let C denote the family of all odd cycles. Erdős and Simonovits [10] conjectured that for every family F of bipartite graphs, there exists k such that ex(n,F∪Ck)∼ex(n,F∪C) as n→∞. This conjecture was proved by Erdős and Simonovits when F={C4}, and for certain families of even cycles in [14]. In this...
On independent sets in hypergraphs
The independence number of a hypergraph H is the size of a largest set of vertices containing no edge of H. In this paper, we prove that if Hn is an n‐vertex ‐uniform hypergraph in which every r‐element set is contained in at most d edges, where , then where satisfies as . The value of cr improves and generalizes several earlier results that all use...
Hypergraph Ramsey numbers: Triangles versus cliques
Alexandr Kostochka, Dhruv Mubayi, Jacques Verstraete
Journal of Combinatorial Theory, Series A > 2013 > 120 > 7 > 1491-1507
A celebrated result in Ramsey Theory states that the order of magnitude of the triangle-complete graph Ramsey numbers R(3,t) is t2/logt. In this paper, we consider an analogue of this problem for uniform hypergraphs. A triangle is a hypergraph consisting of edges e,f,g such that |e∩f|=|f∩g|=|g∩e|=1 and e∩f∩g=∅. For all r⩾2, let R(C3,Ktr) be the smallest positive integer n such that in every red–blue...
Peter Keevash, Benny Sudakov, Jacques Verstraëte
Let F be a family of graphs. A graph is F-free if it contains no copy of a graph in F as a subgraph. A cornerstone of extremal graph theory is the study of the Turán number ex(n,F), the maximum number of edges in an F-free graph on n vertices. Define the Zarankiewicz number z(n,F) to be the maximum number of edges in an F-free bipartite graph on n vertices. Let C k denote a cycle of length k, and...
The de Bruijn–Erdős theorem for hypergraphs
Noga Alon, Keith E. Mellinger, Dhruv Mubayi, Jacques Verstraëte
Designs, Codes and Cryptography > 2012 > 65 > 3 > 233-245
Fix integers n ≥ r ≥ 2. A clique partition of $${{[n] \choose r}}$$ is a collection of proper subsets $${A_1, A_2, \ldots, A_t \subset [n]}$$ such that $${\bigcup_i{A_i \choose r}}$$ is a partition of $${{[n]\choose r}}$$ . Let cp(n, r) denote the minimum size of a clique partition of $${{[n] \choose r}}$$ . A classical theorem of de Bruijn and Erdős states that cp(n, 2) = ...
On the threshold for k-regular subgraphs of random graphs
Paweł Prałat, Jacques Verstraëte, Nicholas Wormald
The k-core of a graph is the largest subgraph of minimum degree at least k. We show that for k sufficiently large, the threshold for the appearance of a k-regular subgraph in the Erdős-Rényi random graph model G(n,p) is at most the threshold for the appearance of a nonempty (k+2)-core. In particular, this pins down the point of appearance of a k-regular subgraph to a window for p of width roughly...
On decompositions of complete hypergraphs
Sebastian M. Cioabă, André Kündgen, Jacques Verstraëte
We study the minimum number of complete r-partite r-uniform hypergraphs needed to partition the edges of the complete r-uniform hypergraph on n vertices and we improve previous results of Alon.
Two-regular subgraphs of hypergraphs
Dhruv Mubayi, Jacques Verstraëte
Journal of Combinatorial Theory, Series B > 2009 > 99 > 3 > 643-655
We prove that the maximum number of edges in a k-uniform hypergraph on n vertices containing no 2-regular subhypergraph is (n−1k−1) if k⩾4 is even and n is sufficiently large. Equality holds only if all edges contain a specific vertex v. For odd k we conjecture that this maximum is (n−1k−1)+⌊n−1k⌋, with equality only for the hypergraph described above plus a maximum matching omitting v.
INDEPENDENT SETS (2)
PROBABILISTIC METHOD (2)
STEINER SYSTEMS (2)
TURÁN NUMBERS (2)
BIPARTITE TURAN PROBLEM (1)
CLIQUE PARTITIONS (1)
COUPON COLORING (1)
DE BRUIJN-ERDŐS (1)
EXCLUDED CYCLES (1)
EXPANSIONS OF GRAPHS (1)
GENERALIZED 4-CYCLE (1)
GENERALIZED QUADRANGLES (1)
GRAHAM–POLLAK (1)
GRAPH DISCREPANCY (1)
GRAPH PARTITIONS (1)
HAMMING GRAPH (1)
HYPERGRAPH TURÁN NUMBER (1)
HYPERGRAPH TURÁN NUMBERS (1)
HYPERGRAPHS WITH LARGE GIRTH (1)
INJECTIVE COLORING (1)
KNESER GRAPH (1)
LOOSE TRIANGLE (1)
PROPERTY B (1)
R-GRAPH (1)
RAINBOW GRAPH (1)
RANDOMIZED GREEDY ALGORITHM (1)
REGULAR SUBGRAPHS (1)
TRIPLE SYSTEM (1)
TURÁN FUNCTION (1)
TURÁN PROBLEM (1)
UNIFORM HYPERGRAPHS (1)
ZARANKIEWICZ PROBLEM (1)
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The keyboard we discuss here is a TeckNet mechanical keyboard, X705 UK layout. I believe it is called a 65% keyboard, although it doesn't actually say. I personally never use the numpad and have no purpose for it on my desk, so I can save some space and money by simply having a keyboard without it. From memory it cost about £40 when it was originally purchased.
Generally I have found the keyboard to be okay. The blue key switches are nice and uniform, and day-to-day it doesn't tend to miss a click. I was never particularly happy with the RGB light selection, it always appeared immature to me - but it did the trick.
The keyboard has done pretty well so far, having done about 4 years of service every day for at least 12 hours a day. It's travelled around the world multiple times and has survived many a spill. I think it's fair to say that I got my money's worth from this keyboard, and if they still sold them, I would definitely recommend picking one up for abuse.
Recently though, the n and m keys begun to stick, not responding some times and getting stuck other times. I tried taking off the caps and cleaning them the best I could, but generally this indicates that the switch itself has allowed the ingress of dirt or has failed entirely.
This isn't the first time that it would have failed either, some time ago the w key stopped pressed correctly. Back then I took it apart and switched the Pause switch (which I never use) with the w key - and all was good again.
The first job is to take it apart. The screws are placed under the key caps, so all the key caps need to come off. Unfortunately it is not easy to see in the picture where the screws are as they are recessed into the chassis and the camera is awful.
Without key-caps
The next job is to figure out which switches you want to replace and mark them (it's easy to make mistakes). You need to keep the "new" (less worn) switches separate from the screwed ones, as once they are out of the keyboard you will have zero way to tell which is which.
To de-solder the switches, you'll likely want a solder sucker. I personally had the soldering iron set to 300 degrees, heat the solder to a liquid and then suck it away. Do this a few times for each pin and all is good. The switch can then be easily lifted out of the PCB.
Then you'll want to swap the switches over and solder them back into the keyboard. This is just normal soldering. Foreshadowing: Just be careful not to get the pad too hot as it could lift, and not to use too much pressure. Also bare in mind that the pin is seated in plastic and won't take too much heat to warp the switch itself.
I kinda forgot this process and really screwed up. I forgot that I needed to use a solder sucker and went with the "more heat" solution whilst lifting. The switch did eventually come out, with part of the PCB with it!
I knew this would be an issue, but decided to go ahead with the repair anyway and just ignore it. I put the keyboard back together, tested it, and of course the entire top row (Esc to F11) had stopped working. Oops. That lifted pad was going to be a problem...
Knowing just enough electronics to cause massive headaches, I believed that I had broken a trace, such that the connection was completely broken. Given that the PCB is simple and most connections are visible, I could likely just bypass the trace manually with some wire.
0001 A B C
0002 | | |
0003 +---+---+
0007 D E F
If each key had its own pin on the controller to detect whether it is pressed, for 100 or so keys you would need 100 or so pins on a microcontroller. This is pretty insane, so it's simply not done. Instead, keyboards utilize a concept called scan lines, where each key is individually tested many times a second to see if it is pressed.
0008 A--+---+---+
0010 B--+---+---+
0012 C D E
It turns out that you go from needing
N
pins on your keyboard controller to needing
m×n
pins. For the matrix example, you see we reduce from needing 6 pins to needing 5 - the savings get greater as the number of keys is increased.
Hacked keyboard fix
Given that the keys exist on a matrix and that the entire row had stopped working, I suspect I had burnt out the trace around the F11 key. In the picture you can see I soldered a bypass wire across the PCB - and it worked!
The last time I had issues with my keyboard I said to myself "if this happens again, I will buy a new keyboard". I really don't think this keyboard will survive any more repair jobs. I'm now even unsure which switches I swapped!
I'm clearly a heavy keyboard user, enough that I can actually wear out a 50 million rated switch in just a few years. I believe my next keyboard needs removable switches, otherwise I'll be doomed to having to keep soldering the bloody thing when they inevitably wear out.
I have been thinking about the SK61 with replaceable mechanical optical switches for about $80 (NZD), but I really user arrow keys a lot when typing in vim for example. I also use the Home and End keys a hell of a lot when navigating around lines.
I did actually try ordering one some months ago, but unfortunately my order was cancelled and it never arrived.
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Vapor pressure - Wikipedia
Pressure exterted by a vapor in thermodynamic equilibrium
The pistol test tube experiment. The tube contains alcohol and is closed with a piece of cork. By heating the alcohol, the vapors fill in the space, increasing the pressure in the tube to the point of the cork popping out.
Vapor pressure (or vapour pressure in English-speaking countries other than the US; see spelling differences) or equilibrium vapor pressure is defined as the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases (solid or liquid) at a given temperature in a closed system. The equilibrium vapor pressure is an indication of a liquid's evaporation rate. It relates to the tendency of particles to escape from the liquid (or a solid). A substance with a high vapor pressure at normal temperatures is often referred to as volatile. The pressure exhibited by vapor present above a liquid surface is known as vapor pressure. As the temperature of a liquid increases, the kinetic energy of its molecules also increases. As the kinetic energy of the molecules increases, the number of molecules transitioning into a vapor also increases, thereby increasing the vapor pressure.
2 Estimating vapor pressures with Antoine equation
4 Liquid mixtures: Raoult's law
6 Boiling point of water
9 Estimating vapor pressure from molecular structure
10 Meaning in meteorology
Measurement and units[edit]
Estimating vapor pressures with Antoine equation[edit]
The Antoine equation[2][3] is a pragmatic mathematical expression of the relation between the vapor pressure and the temperature of pure liquid or solid substances. It is obtained by curve-fitting and is adapted to the fact that vapor pressure is usually increasing and concave as a function of temperature. The basic form of the equation is:
{\displaystyle \log P=A-{\frac {B}{C+T}}}
{\displaystyle T={\frac {B}{A-\log P}}-C}
{\displaystyle P}
{\displaystyle T}
{\displaystyle A}
{\displaystyle B}
{\displaystyle C}
{\displaystyle \log }
{\displaystyle \log _{10}}
{\displaystyle \log _{e}}
{\displaystyle \log P=A-{\frac {B}{T}}}
{\displaystyle T={\frac {B}{A-\log P}}}
Sublimations and vaporizations of the same substance have separate sets of Antoine coefficients, as do components in mixtures.[2] Each parameter set for a specific compound is only applicable over a specified temperature range. Generally, temperature ranges are chosen to maintain the equation's accuracy of a few up to 8–10 percent. For many volatile substances, several different sets of parameters are available and used for different temperature ranges. The Antoine equation has poor accuracy with any single parameter set when used from a compound's melting point to its critical temperature. Accuracy is also usually poor when vapor pressure is under 10 Torr because of the limitations of the apparatus[citation needed] used to establish the Antoine parameter values.
Relation to boiling point of liquids[edit]
Liquid mixtures: Raoult's law[edit]
{\displaystyle P_{\rm {tot}}=\sum _{i}Py_{i}=\sum _{i}P_{i}^{\rm {sat}}x_{i}\,}
{\displaystyle P_{\rm {tot}}}
{\displaystyle x_{i}}
{\displaystyle i}n the liquid phase and
{\displaystyle y_{i}}
{\displaystyle i}n the vapor phase respectively.
{\displaystyle P_{i}^{\rm {sat}}}
{\displaystyle i}
{\displaystyle \ln \,P_{\rm {s}}^{\rm {sub}}=\ln \,P_{\rm {l}}^{\rm {sub}}-{\frac {\Delta _{\rm {fus}}H}{R}}\left({\frac {1}{T_{\rm {sub}}}}-{\frac {1}{T_{\rm {fus}}}}\right)}
{\displaystyle P_{\rm {s}}^{\rm {sub}}}
{\displaystyle T_{\rm {sub}}<T_{\rm {fus}}}
{\displaystyle P_{\rm {l}}^{\rm {sub}}}
{\displaystyle T_{\rm {sub}}<T_{\rm {fus}}}
{\displaystyle \Delta _{\rm {fus}}H}
{\displaystyle R}
{\displaystyle T_{\rm {sub}}}
{\displaystyle T_{\rm {fus}}}
Graph of water vapor pressure versus temperature. At the normal boiling point of 100 °C, it equals the standard atmospheric pressure of 760 Torr or 101.325 kPa.
{\displaystyle \log _{10}\left({\frac {P}{1{\text{ Torr}}}}\right)=8.07131-{\frac {1730.63\ {}^{\circ }{\text{C}}}{233.426\ {}^{\circ }{\text{C}}+T_{b}}}}
{\displaystyle T_{b}={\frac {1730.63\ {}^{\circ }{\text{C}}}{8.07131-\log _{10}\left({\frac {P}{1{\text{ Torr}}}}\right)}}-233.426\ {}^{\circ }{\text{C}}}
{\displaystyle T_{b}}
is the boiling point in degrees Celsius and the pressure
{\displaystyle P}
Dühring's rule[edit]
The following table is a list of a variety of substances ordered by increasing vapor pressure (in absolute units).
Octaethylene glycol[10] 9.2×10−8 Pa 9.2×10−13 6.9×10−10 89.85
Glycerol 0.4 Pa 0.000004 0.003 50
Mercury 1 Pa 0.00001 0.0075 41.85
Tungsten 1 Pa 0.00001 0.0075 3203
Water (H2O) 2.3 kPa 0.023 17.5 20
Propanol 2.4 kPa 0.024 18.0 20
Ethanol 5.83 kPa 0.0583 43.7 20
Freon 113 37.9 kPa 0.379 284 20
Carbonyl sulfide 1.255 MPa 12.55 9412 25
Nitrous oxide[12] 5.660 MPa 56.60 42453 25
Carbon dioxide 5.7 MPa 57 42753 20
Estimating vapor pressure from molecular structure[edit]
Several empirical methods exist to estimate the vapor pressure from molecular structure for organic molecules. Some examples are SIMPOL.1 method,[13] the method of Moller et al.,[9] and EVAPORATION (Estimation of VApour Pressure of ORganics, Accounting for Temperature, Intramolecular, and Non-additivity effects).[14][15]
Meaning in meteorology[edit]
In meteorology, the term vapor pressure is used to mean the partial pressure of water vapor in the atmosphere, even if it is not in equilibrium,[16] and the equilibrium vapor pressure is specified otherwise. Meteorologists also use the term saturation vapor pressure to refer to the equilibrium vapor pressure of water or brine above a flat surface, to distinguish it from equilibrium vapor pressure, which takes into account the shape and size of water droplets and particulates in the atmosphere.[17]
Raoult's law: vapor pressure lowering in solution
^ Růžička, K.; Fulem, M. & Růžička, V. "Vapor Pressure of Organic Compounds. Measurement and Correlation" (PDF). Archived from the original (PDF) on 2010-12-26. Retrieved 2009-10-18.
^ a b What is the Antoine Equation? (Chemistry Department, Frostburg State University, Maryland)
^ a b Sinnot, R.K. (2005). Chemical Engineering Design] (4th ed.). Butterworth-Heinemann. p. 331. ISBN 978-0-7506-6538-4.
^ Wagner, W. (1973), "New vapour pressure measurements for argon and nitrogen and a new method for establishing rational vapour pressure equations", Cryogenics, 13 (8): 470–482, Bibcode:1973Cryo...13..470W, doi:10.1016/0011-2275(73)90003-9
^ Dreisbach, R. R. & Spencer, R. S. (1949). "Infinite Points of Cox Chart Families and dt/dP Values at any Pressure". Industrial and Engineering Chemistry. Vol. 41, no. 1. p. 176. doi:10.1021/ie50469a040.
^ a b Moller B.; Rarey J.; Ramjugernath D. (2008). "Estimation of the vapour pressure of non-electrolyte organic compounds via group contributions and group interactions". Journal of Molecular Liquids. 143: 52–63. doi:10.1016/j.molliq.2008.04.020.
^ Krieger, Ulrich K.; Siegrist, Franziska; Marcolli, Claudia; Emanuelsson, Eva U.; Gøbel, Freya M.; Bilde, Merete (8 January 2018). "A reference data set for validating vapor pressure measurement techniques: homologous series of polyethylene glycols" (PDF). Atmospheric Measurement Techniques. Copernicus Publications. 11 (1): 49–63. doi:10.5194/amt-11-49-2018. ISSN 1867-1381. Retrieved 7 April 2022.
^ "Thermophysical Properties Of Fluids II – Methane, Ethane, Propane, Isobutane, And Normal Butane" Archived 2016-12-21 at the Wayback Machine (page 110 of PDF, page 686 of original document), BA Younglove and JF Ely.
^ Pankow, J. F.; et al. (2008). "SIMPOL.1: a simple group contribution method for predicting vapor pressures and enthalpies of vaporization of multifunctional organic compounds". Atmos. Chem. Phys. 8 (10): 2773–2796. Bibcode:2008ACP.....8.2773P. doi:10.5194/acp-8-2773-2008.
^ "Vapour pressure of Pure Liquid Organic Compounds: Estimation by EVAPORATION". Tropospheric Chemistry Modelling at BIRA-IASB. 11 June 2014. Retrieved 2018-11-26.
^ Compernolle, S.; et al. (2011). "EVAPORATION: a new vapour pressure estimation method for organic molecules including non-additivity and intramolecular interactions". Atmos. Chem. Phys. 11 (18): 9431–9450. Bibcode:2011ACP....11.9431C. doi:10.5194/acp-11-9431-2011.
^ Glossary Archived 2011-04-15 at the Wayback Machine (Developed by the American Meteorological Society)
^ A Brief Tutorial. jhuapl.edu (An article about the definition of equilibrium vapor pressure)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Vapor_pressure&oldid=1089626021"
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Solve system of differential equations - MATLAB dsolve - MathWorks América Latina
\frac{\mathit{dy}}{\mathit{dt}}=\mathit{ay}
{C}_{1} {\mathrm{e}}^{a t}
\frac{{\mathit{d}}^{2}\mathit{y}}{{\mathit{dt}}^{2}}=\mathit{ay}
{C}_{1} {\mathrm{e}}^{-\sqrt{a} t}+{C}_{2} {\mathrm{e}}^{\sqrt{a} t}
\frac{\mathit{dy}}{\mathit{dt}}=\mathit{ay}
y\left(0\right)=5
5 {\mathrm{e}}^{a t}
\frac{{\mathit{d}}^{2}\mathit{y}}{{\mathit{dt}}^{2}}={\mathit{a}}^{2}\mathit{y}
y\left(0\right)=b
{y}^{\prime }\left(0\right)=1
\frac{{\mathrm{e}}^{a t} \left(a b+1\right)}{2 a}+\frac{{\mathrm{e}}^{-a t} \left(a b-1\right)}{2 a}
\begin{array}{l}\frac{\mathit{dy}}{\mathit{dt}}=\mathit{z}\\ \frac{\mathit{dz}}{\mathit{dt}}=-\mathit{y}.\end{array}
{C}_{1} \mathrm{cos}\left(t\right)+{C}_{2} \mathrm{sin}\left(t\right)
{C}_{2} \mathrm{cos}\left(t\right)-{C}_{1} \mathrm{sin}\left(t\right)
{C}_{1} \mathrm{cos}\left(t\right)+{C}_{2} \mathrm{sin}\left(t\right)
{C}_{2} \mathrm{cos}\left(t\right)-{C}_{1} \mathrm{sin}\left(t\right)
\frac{\partial }{\partial t}y\left(t\right)={e}^{-y\left(t\right)}+y\left(t\right)
\frac{\partial }{\partial t}\mathrm{ }y\left(t\right)={\mathrm{e}}^{-y\left(t\right)}+y\left(t\right)
{\mathrm{W}\text{lambertw}}_{0}\left(-1\right)
F\left(y\left(t\right)\right)=g\left(t\right)
\left(\begin{array}{c}\left({\int \frac{{\mathrm{e}}^{y}}{y {\mathrm{e}}^{y}+1}\mathrm{d}y|}_{y=y\left(t\right)}\right)={C}_{1}+t\\ {\mathrm{e}}^{-y\left(t\right)} \left({\mathrm{e}}^{y\left(t\right)} y\left(t\right)+1\right)=0\end{array}\right)
F\left(y\left(x\right)\right)=g\left(x\right)
{\mathrm{e}}^{y\left(x\right)}+\frac{{y\left(x\right)}^{2}}{2}={C}_{1}+{\mathrm{e}}^{-x}+\frac{{x}^{2}}{2}
\frac{\mathit{dy}}{\mathit{dt}}=\frac{\mathit{a}}{\sqrt{\mathit{y}}}+\mathit{y}
y\left(a\right)=1
{\left({\mathrm{e}}^{\frac{3 t}{2}-\frac{3 a}{2}+\mathrm{log}\left(a+1\right)}-a\right)}^{2/3}
a
\begin{array}{l}\left\{\begin{array}{cl}\left\{\begin{array}{cl}\left\{{\sigma }_{1}\right\}& \text{ if }-\frac{\pi }{2}<{\sigma }_{2}\\ \left\{{\sigma }_{1},-{\left(-a+{\mathrm{e}}^{\frac{3 t}{2}-\frac{3 a}{2}+\mathrm{log}\left(a+{\left(-\frac{1}{2}+{\sigma }_{3}\right)}^{3/2}\right)+2 \pi {C}_{2} \mathrm{i}}\right)}^{2/3} \left(\frac{1}{2}+{\sigma }_{3}\right)\right\}& \text{ if }{\sigma }_{2}\le -\frac{\pi }{2}\end{array}& \text{ if }{C}_{2}\in \mathbb{Z}\\ \varnothing & \text{ if }{C}_{2}\notin \mathbb{Z}\end{array}\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\sigma }_{1}={\left(-a+{\mathrm{e}}^{\frac{3 t}{2}-\frac{3 a}{2}+\mathrm{log}\left(a+1\right)+2 \pi {C}_{2} \mathrm{i}}\right)}^{2/3}\\ \\ \mathrm{ }{\sigma }_{2}=\text{angle}\left({\mathrm{e}}^{\frac{3 {C}_{1}}{2}+\frac{3 t}{2}}-a\right)\\ \\ \mathrm{ }{\sigma }_{3}=\frac{\sqrt{3} \mathrm{i}}{2}\end{array}
\left({x}^{2}-1{\right)}^{2}\frac{{\partial }^{2}}{\partial {x}^{2}}y\left(x\right)+\left(x+1\right)\frac{\partial }{\partial x}y\left(x\right)-y\left(x\right)=0
{C}_{2} \left(x+1\right)+{C}_{1} \left(x+1\right) \int \frac{{\mathrm{e}}^{\frac{1}{2 \left(x-1\right)}} {\left(1-x\right)}^{1/4}}{{\left(x+1\right)}^{9/4}}\mathrm{d}x
x=-1
\left(\begin{array}{c}x+1\\ \frac{1}{{\left(x+1\right)}^{1/4}}-\frac{5 {\left(x+1\right)}^{3/4}}{4}+\frac{5 {\left(x+1\right)}^{7/4}}{48}+\frac{5 {\left(x+1\right)}^{11/4}}{336}+\frac{115 {\left(x+1\right)}^{15/4}}{33792}+\frac{169 {\left(x+1\right)}^{19/4}}{184320}\end{array}\right)
\infty
\left(\begin{array}{c}x-\frac{1}{6 {x}^{2}}-\frac{1}{8 {x}^{4}}\\ \frac{1}{6 {x}^{2}}+\frac{1}{8 {x}^{4}}+\frac{1}{90 {x}^{5}}+1\end{array}\right)
\left(\begin{array}{c}x-\frac{1}{6 {x}^{2}}-\frac{1}{8 {x}^{4}}-\frac{1}{90 {x}^{5}}-\frac{37}{336 {x}^{6}}\\ \frac{1}{6 {x}^{2}}+\frac{1}{8 {x}^{4}}+\frac{1}{90 {x}^{5}}+\frac{37}{336 {x}^{6}}+\frac{37}{1680 {x}^{7}}+1\end{array}\right)
\frac{\mathit{dy}}{\mathit{dx}}=\frac{1}{{\mathit{x}}^{2}}{\mathit{e}}^{-\frac{1}{\mathit{x}}}
{C}_{1}+{\mathrm{e}}^{-\frac{1}{x}}
\mathit{y}\left(0\right)=1
{\mathrm{e}}^{-\frac{1}{x}}+1
{\mathit{e}}^{-\frac{1}{\mathit{x}}}
x=0
\underset{\mathit{x}\to {0}^{+}}{\mathrm{lim}}\text{\hspace{0.17em}}{\mathit{e}}^{-\frac{1}{\mathit{x}}}=0
\underset{\mathit{x}\to {0}^{-}}{\mathrm{lim}}\text{\hspace{0.17em}}{\mathit{e}}^{-\frac{1}{\mathit{x}}}=\infty
\mathrm{lim}\text{\hspace{0.17em}}\mathit{x}\to {\mathit{x}}_{0}^{+}
O\left({\mathrm{var}}^{n}\right)
O\left({\mathrm{var}}^{-n}\right)
\mathrm{lim}x\to {x}_{0}^{+}\text{ }
|
Finding the right representation for your NLP data | Tryolabs
When considering what information is important for a certain decision procedure (say, a classification task), there's an interesting gap between what's theoretically ---that is, actually--- important on the one hand and what gives good results in practice as input to machine learning (ML) algorithms, on the other.
Let's look at sentiment analysis tools as an example. Expression of sentiment is a pragmatic phenomenon. To predict it correctly, we need to know both the meaning of the sentences and the context in which those sentences appeared. How do you get the meaning of a sentence? Well, you need to know the meaning of the lexical items and the sentence's syntactic structure. So, the relevant data points are the following.
The meaning of the lexical items used.
The syntactic structure of the sentences.
The context in which those sentences appeared.
It might be surprising, then, to learn that sentiment analysis tools traditionally use Bag of Words (BoW) to represent texts. The only thing that counts is which words appear in the text and how frequently. So, "Dancing Monkeys in a Tuxedo is better than any other movie" has the same representation as "any other movie is better than Dancing Monkeys in a Tuxedo". They will yield the same result, regardless of the size and quality of the training set. But we use BoW representations because they give excellent results in sentiment analysis tools. So why does it work anyway? I'll get to that in a second.
Imagine you're presented with a relatively novel natural language processing (NLP) problem. Something that hasn't been explored extensively in the literature. For example, say your client wants to predict how likely it is for a user to order food to some restaurant called Rest-a-uraunt based on users' textual reviews of restaurants (including Rest-a-uraunt) on some food ordering site.
You know what to do. You'll use ML techniques. You can use a supervised algorithm if you can find the following information.
Textual reviews of restaurants.
For each review, whether or not the user later ordered to Rest-a-uraunt.
There's still a few things you have to decide. One is which ML algorithm to use. Another one is how to represent user reviews. In some cases, you can use deep learning techniques to learn the representation automatically so that you don't have to answer this second question. In practice, though, even when using neural networks the first step is finding the right representation for your data. Especially in NLP.
Few NLP problems are tackled with ML techniques using actual text (i.e. a sequence of characters) as the algorithm's input. Most normally, you'll get an intermediate representation of your text that has the information that your algorithm needs and doesn't have too much noise. So, one of the big questions for developers working in NLP is how do you decide what representation to use? In other words, how do you know what data is relevant?
This is the structure of the problem:
What is the best way to represent it?
Considerations specific to the algorithm used.
You know that what's theoretically important isn't always what's useful as input to your ML algorithm, but there must be other criteria. There's no hard-and-fast rule but I'll give you some pointers.
Tips when choosing a representation for your data
Your algorithm isn't as intelligent as it seems. It gives impressive results because it's fed well-structured data. For example, most classifiers work by grouping elements that are close together in a vector space. Their job amounts to finding the boundaries between classes. They won't work if the elements of the same class are far away from each other and intercepted by members of a different class.
Your main objective when deciding how to represent your data is to make sure that 'distance' in your vector space is a meaningful relation. In other words, that things that are close together behave similarly with respect to what you need to know about the data. BoW has that characteristic. The distance relationship is obviously meaningful. If two BoWs are close together in the vector space, that means they have similar amounts of the same words.
For certain features, it's trivial to make the distance relation meaningful. Say, if your feature is the length of the longest word or the linear distance between two words, then that will be an integer, and 'close' values in that feature will mean you have similar lengths. Simple. On the other hand, embedding syntactic structures in a vector space while making the distance relation meaningful is not quite as easy. Embedding word meanings in vector space is also not trivial, but we've gotten pretty good at it with Word2Vec, GloVe, etc.
Word2Vec gets distance right!
Things got much easier with recurrent neural networks and recursive neural networks. With them, you can shift the focus from representing entire sentences and paragraphs as vectors and instead treat them by dealing with individual words iteratively or recursively. You can use an actual tree of word embeddings as a representation. So, for your Rest-a-uraunt project you can parse all sentences in the review and use a recursive neural network to visit all nodes in the tree bottom-up, calculating probability-of-a-future-order-to-Rest-a-uraunt scores. This is likely not what you want to do, as it will take a very long time to train and perhaps a prohibitively long time to calculate scores after it's been trained. But something similar has been done before at Stanford NLP, and it's nice to know you could do it this way if everything else fails.
Funnily enough, when I tried the two Dancing Monkeys in a Tuxedo sentences with Stanford's recursive sentiment analysis tool, it classified both sentences as negative. When I changed 'Dancing Monkeys in a Tuxedo' to 'this movie', it classified both as positive.
Linear algorithms have special needs
It's a well known fact that most traditional linear classification algorithms approximate your classes to shapes with straight boundaries in your input vector space (i.e. the boundaries are hyperplanes) . In some cases of binary classification, they will try to find a single straight boundary dividing your two classes. This fact often goes unexplained, so let's give a brief explanation.
Linear ML algorithms learn a vector of weights w of the same length as the input vector x. In each prediction, they calculate the dot product of x and w. The result is the input's score. This score may be given as the final result, or used as input to some other (often simple) rule. In the case of classification algorithms often a rule is used to relate score intervals to classes. Your rule may be 'if the score is bigger than 10, then the user will likely order to Rest-a-uraunt later'.
Let's look at the case where we use a vector with two features
(x_1, x_2)
as input and
(w_1, w_2)
as the vector of weights. The score then is
x_1 w_1 + x_2 w_2
If our rule for classification is 'if the score is bigger than k, then we classify it as A', then our boundary for class A is
x_1 w_1 + x_2 w_2 = k
x_2 = -\dfrac{w_1}{w_2} x_1 + \dfrac{k}{w_2}
That's the equation of a line! If you start with three features instead of two you'll get a plane, and so on. Every time you use score intervals to classify, you'll get straight boundaries.
So, if you represent your data in such a way that classes can easily be separated by straight lines (or planes, etc.), your linear algorithm will do a better job at separating your classes.
Let me show you an example of how you can get it wrong. Imagine Rest-a-uraunt specializes in pasta. You may decide to use the following approach to predict whether or not a certain user will in the future order to Rest-a-uraunt:
Extract reviews made by that user to other restaurants.
Use a sentiment analysis tool to classify reviews into positive and negative ones.
Use a linear classifier to get a score of each review indicating how likely it is that the user will order to Rest-a-urant based on a BoW of the review and the pre-computed sentiment.
Aggregate the scores you computed for all of the user's reviews (you could take the average, for example).
You will train the classifier in point 3 with reviews tagged with a sentiment and an indication of whether or not the user later ordered to Rest-a-uraunt.
The idea is that if the user uses pasta-related words a lot in their review and they show a positive sentiment at the same time, they're likely to order to Rest-a-uraunt in the near future. You're thinking your linear algorithm may be able to figure that out. Well, if you don't structure your input correctly, it likely won't.
Remember your linear algorithm learns a weight for each feature in the input. That means that if you represent sentiment with a single feature, a positive sentiment will impact the score either always positively or always negatively. But you want your algorithm to interpret a positive sentiment as a sign that the user is likely to order to Rest-a-uraunt only when combined with a high frequency of certain words ('pasta', 'Italian', etc.) but to interpret it as a sign that it's unlikely they'll order when other words are frequent ('celiac', for example).
What you can do in this case is to restructure your input vector so that instead of having a unique, separate feature for the sentiment of the review, you use feature combinations (also called feature crosses) so that all word frequency features include information about the sentiment.
For example, you can have two features for each word. The first one will be 0 if the sentiment is negative, and have the number of occurrences (or a tf-idf score) of the word if the sentiment is positive. The other one will work the other way around, 0 if the sentiment is positive, the number if it's negative. This way your algorithm will be able to assign a weight to the combination of a word's frequency and a positive or negative sentiment, which is what you wanted. When you need the combination of two or more features to be especially meaningful for your linear algorithm, you'll need to cross your features like this.
Your algorithm is guessing
Part of the reason why ML techniques are so successful is that they aren't only doing the logical, theoretically sound, inferences that you were thinking about when you implemented your algorithm, if they're doing them at all. They're also finding statistical regularities that don't make sense theoretically but work in practice.
Remember the example where you used a BoW and a pre-computed sentiment analysis for your Rest-a-uraunt project? Okay. You implemented the algorithm following my advice about how to structure your features and it's giving good results when tested with your cross-validation set. Congrats! Now, if you check how it works (i.e. what words combined with which sentiment have a high weight, etc.) you might be surprised.
You were thinking pasta-related words combined with a positive sentiment was indicative of a possible future order to Rest-a-uraunt. You were probably right. What you may not have foreseen is that your algorithm is also using all kinds of general-purpose words to make inferences. Words that don't seem to have any relationship with pasta, sentiments, or whatever.
What's going on? Well, your algorithm is using 'hidden' statistical correlations that don't really make much sense in the grand scheme of things but happen to work. This is good! ML algorithms solve impressive problems because they do this. It's one of the ways they can (often) beat rule-based systems. They see the correlations your engineer can't see.
That means sometimes information that seems irrelevant ends up helping your algorithm. Sometimes is key here. Adding everything you can find won't work. That said, it's a good idea to fiddle around with your features a bit, to see what works. Some things you can't plan in advance. Remain flexible!
Too much guessing is dangerous, though. It can lead to overfitting. It can also trick you into thinking your algorithm works better than it does. For example, you might test your algorithm with a small data set and find out it works reasonably well. Yet, if your algorithm is mostly just guessing, it won't scale. Guessing has a very low ceiling.
So, if you're too surprised when you check your algorithm's learned weights, you should take it as an indication that it might not be doing a good job.
In conclusion, when using ML techniques in NLP, you should always pay attention to what information you need to feed your algorithm and how you can represent that information to get the best results. To sum up the issue, let's repeat what I called 'the structure of the problem' above.
Data representation problems will sometimes lead you to the right ML algorithm. Recurrent and recursive neural networks allow you to represent your data in ways you can't with other algorithms. Linear algorithms have special needs that you may not be able to satisfy easily. You can't get the best results if you pick your algorithm without thinking about the way you want to represent your data.
Here are some links if you want to learn more about this topic. The keyword here is 'feature engineering':
Learning data science: feature engineering
Feature engineering for deep learning
Non-mathematical feature engineering techniques
NLP-specific:
Text mining and feature engineering in R
Stanford NLP - features for text
|
Charge and Electric Fields | Brilliant Math & Science Wiki
Aditya Narayan Sharma, Chung Kevin, Abhijeet Vats, and
God Ly
Z Heywood
When you get shocked from touching a metal doorknob after walking across a rug, what you feel are electrons jumping from your hand to the door metal. The electrons jump because they're attracted to the door by their electric charge. Measures of charge are commonly symbolized as
q
, and charge is a characteristic property of all matter.
The standard unit of charge in the SI system is the Coulomb. Some types of matter, like neutrons, have no charge and are electrically neutral (hence their name). Other particles like protons and electrons, have positive or negative electric charge, respectively. Despite the unit of charge being the Coulomb, the "unit" particles of charge, the proton and the electron, each have a charge of
\pm 1.6\times10^{-19}\text{ C}.
The SI unit of charge is the Coulomb, abbreviated as
\text{C}:
1\textrm{ C}\approx 6.25\times 10^{18} q_e.
Particle colliders have revealed particles called quarks, which come together in particular combinations to form other particles such as the neutron and proton, and they exhibit charges of
\{+\frac23 q_p, +\frac13 q_p, -\frac13q_p, -\frac23 q_p\}
They are more fundamental than protons and neutrons, but because of their strong tendency to join up into more stable aggregate particles, they are rarely seen outside the special situation of the particle collider.
The proton consists of three quarks (up, up, and down) of charge
\{+\frac23 q_p, +\frac23 q_p, -\frac13 q_p\}
q_p = \left(\frac23+\frac23-\frac13\right)q_p=q_p
Charges wouldn't be all that interesting if they didn't enable some kind of motion. In fact, charged matter is seen to undergo peculiar phenomena that are not experienced by uncharged matter. Some examples of these are the well-known aurora borealis, electrons flowing en masse from cloud to cloud during storms (a.k.a. lightning), and when your hair stands on end after riding down a slide.
One of the simplest interactions that a charged particle can have is with an electric field. The electric field is essentially a 3D grid that fills all of space, and records a value and direction at every point corresponding to the force that a charged particle would experience if it were placed at that point. Hence, if a positively charged particle is in an electric field, it experiences a push along the local direction of the field while a negatively charged particle will experience a push along the direction opposite the local direction of the field. This is an important definition which should be noted in problem solving.
An electric field
\vec{E}(\vec{r})
is defined at every point in space
\vec{r}
, and acts on positively charged particles in the direction
\frac{\vec{E}}{\left|E\right|}
. Similarly, it acts upon negatively charged particles in the direction
-\frac{\vec{E}}{\left|E\right|}
In two dimensions, we can visualize the electric field as a lattice of arrows:
The length and width of each arrow corresponds to the field strength at the underlying point. While this representation is discrete, the electric field is continuous, and therefore can be interpolated between the arrows shown in the grid.
\begin{aligned} \vec{F} &= q\vec{E} \\ &= m\vec{a}. \end{aligned}
This relation is a special case of the Lorentz force law with
\vec{B}=0
(when magnetic fields are present, there is an extra term that is beyond the focus of this article).
For the purposes of classical electrodynamics, an electric field is more or less defined by this relationship in that we can put a charge at various locations, measure the force it feels, and use the Lorentz force law to calculate the electric field at each point.
Stop that muon
A high energy muon (of charge
-q_p
) enters the upper atmosphere along a trajectory straight through the center of the Earth. There are two very large clouds (one directly above the other) separated by a distance of
l=100\text{ km}
, with an electric field of constant strength between them, that is aligned vertically.
If the muon has an incoming kinetic energy of
\text{KE}_i = 6.4\times10^{-16}\text{ kJ}
, how strong would the electric field between the two clouds have to be in order to bring the muon to rest before it passes the bottom cloud? (Ignore breakdown reactions and assume the muon to be a stable particle.)
In the electric field, the muon will experience a force of
F=q_pE
. At most, the muon can travel
100\text{ km}
before coming to rest, before which the field will have performed work in the amount
W=F\cdot d = q_pEl
on the muon.
Therefore, we can say that
W = \text{KE}_i
, and the minimal electric field strength is given by
E = \frac{\text{KE}_i}{q_pl} \approx 40\textrm{ N/C}.
A circular ring of radius
a
is uniformly charged with charge
+Q\text{ C}
at its circumference. Find the electric field acting at a point distance
x
from the center of the ring.
The figure clearly illustrates the idea. Let the point be
P
x
units from the center of the circular ring
A
a
The electric field acting
\vec{E}
P
\vec{DF}
making angle
\theta
with the horizontal can be divided into two components, namely
E\cos\theta
E\sin\theta
The vertical component however doesn't contribute to the electric field of the point, because for any two elementary portions of the circular loop that are opposite to each other and equal will cancel. So the net electric field will be towards the
x
The distance of the point from the ring is
\sqrt{x^2+a^2}.
Now consider an elementary length
dl
of the loop, the charge on which is given by
dq=\frac{q}{2\pi a}dl.
\begin{aligned} dE|\vec{dE}|&= \frac{dq}{4\pi{\varepsilon}_{0}}\frac{1}{\left(\sqrt{x^2+a^2}\right)^2}\\ dE_x &=dE\cos\theta \\ &= dE \cdot \frac{x}{(x^2+a^2)^{1/2}} \\ &= \frac{1}{4\pi{\varepsilon}_{0}}\frac{q\cdot x\cdot dl}{2\pi a(x^2+a^2)^{3/2}}, \end{aligned}
\cos\theta
is derived from the right triangle
ADP.
Now, the electric field due to the entire circular loop is given by
\begin{aligned} E&=\int_{whole loop}^{} dE_x \\ &= \int_{whole loop}^{} \frac{1}{4\pi{\varepsilon}_{0}}\frac{q\cdot x\cdot dl}{2\pi a(x^2+a^2)^{3/2}}\\ &= \frac{1}{4\pi{\varepsilon}_{0}}\frac{q\cdot x}{2\pi a(x^2+a^2)^{3/2}} \int_{whole loop}^{}dl\\ &=\frac{1}{4\pi{\varepsilon}_{0}}\frac{q\cdot x\cdot (2\pi a)}{2\pi a(x^2+a^2)^{3/2}} \qquad \text{(the entire length of the loop is } 2\pi a) \\ &=\frac{1}{4\pi{\varepsilon}_{0}}\frac{q\cdot x}{(x^2+a^2)^{3/2}}. \end{aligned}
P
lies at the center of the loop: in this case
x=0
E=0.
P
x>>a,
a^2
can be neglected in comparison to
x^2
E=\frac{1}{4\pi{\varepsilon}_{0}}\frac{q}{x^2}.
In addition to responding to external electric fields, charged particles give rise to electric fields of their own. Knowing the behavior of these particle fields allows us to build up our understanding to more complicated arrangements of charge, which are crucial in applying our knowledge to engineering problems.
As such, experimental physicists of the
18^\text{th}
century like Priestley and Coulomb carefully measured the electric field of charges in the lab. They used clever arrangements to measure the force felt by an object of charge
q
(test charge) when placed in the vicinity of another object of charge
Q
(source charge). One way to do this is to hold
Q
fixed (e.g. mount it on an electrically neutral stick) and measure the force felt by
q
as it is placed at various locations within the field of
Q
. After measuring enough points, an approximate picture emerged like the arrow diagram below.
One thing they found is that the force felt by
q
was independent of direction, i.e. the force is spherically symmetric and points directly along the vector connecting
q
Q
\vec{F}(\vec{r}) = \vec{F}(r)
. Another was a strong dependence on the distance away from
Q
. The figure below shows data for the relative strength of the force felt by
q
as a function of the distance from
Q
, as well as the curves
r^\alpha
\alpha = 1,-1,-2,-3
It is clear that the force decreases with the inverse square of the distance, and therefore
F \propto \frac{1}{r^2}
. The force was also found to increase linearly as
q
Q
increased, and therefore
F\propto qQ
. Finally, the force was attractive if
q
Q
had opposite signs, and it was repulsive for like sign.
Putting all these observations together, we can say
F \sim \frac{qQ}{r^2}
. According to the Lorentz force law, the force felt by the test charge is given by
F=qE_Q
F=qE_Q\sim\frac{qQ}{r^2}
E_Q\sim \frac{Q}{r^2}
E_Q=k\frac{Q}{r^2},
k
is a constant of proportionality that has been determined experimentally as
9\times 10^9\text{ Nm}^2\text{/C}^2
. This is what's known as Coulomb's law.
The strength of the electric field at position
r
due to a point charge
Q
E = k\frac{Q}{r^2}.
The field points along the vector from position
r
to the charge. By convention, field lines point inward toward negatively charged particles like electrons and outward for positively charged particles like protons.
While Coulomb's law is strictly true only for point charges, it is still an excellent approximation for the electric field far away from more complicated arrangements of particles. Close-up, an arbitrary arrangement of charges can have an electric field of great detail that isn't easily visualized without a computer.
However, as the scale increases, these local variations will drop off quickly, and at great distances the field will tend to look like
\displaystyle E(x) = k \sum_i\frac{q_i}{(r_i-x)^2} \approx k\frac{Q_\text{net}}{(\bar{r}-x)^2}.
Q_\text{net}>0
, the field lines will point radially outward, and if
Q_\text{net}<0
, the field lines will point radially inward toward the center of the distribution.
Two charges of strength
q
r_-=-\epsilon
r_+=+\epsilon
. The field close to either charge is given by
E(r) = k\frac{q}{\left(r+\epsilon\right)^2}+k\frac{q}{\left(r-\epsilon\right)^2}.
What does the field look like at very far distances?
r \gg \epsilon
E(r)
2k\frac{q}{r^2} + 6k\frac{q\epsilon^2}{r^4}
\epsilon^2/r^2\approx 0
, we can ignore the second term.
Thus, the field looks like
\displaystyle 2k\frac{q}{r^2}
far away from the charges, which is equal to
\displaystyle k\frac{Q_\text{net}}{r^2}
Below, we show the same dipole field as seen zoomed out by a factor of ten. The local structure of the field around the dipoles is no longer visible, and the arrangement looks roughly the same as the field for one charge of strength
2q
\frac{3}{2}d
2d
\frac{5}{2}d
3d
A particle with electric charge
-q
enters a uniform electric field at the point
P=(0, 3d).
The direction of the electric field is the
+y
direction. The charged particle moves along a projectile path inside the electric field. After exiting the electric field, it shows a uniform motion, arriving at
Q=(4d, 0).
If another charged particle with the same mass but a different electric charge of
-2q
enters the electric field in the same way as above, what will be the destination point on the
x
Ignore the gravitational force and the sizes of the charged particles.
In the last example, we exploited a property of electric fields called superposition. The principle of superposition states that in the presence of multiple sources of electric field, the resultant field is simply the sum of the individual fields at each point.
In the presence of multiple fields
E_1(r),E_2(r),\ldots, E_n(r)
, the field strength at
r
E_\text{tot}(r) = \sum_i E_i(r).
Near or far, flux remains.
Coulomb's law suggests a curious quantity that ought to be the same for any spherical surface centered on a particle. If we multiply the strength of the electric field everywhere on the surface, by the surface area of the small patch of surface that the field penetrates, we get
\begin{aligned} \Phi_E &= \sum E(r_i) \times \Delta A(r_i) \\ &= k\frac{q}{r^2}\times 4\pi r^2 \\ &= 4\pi k q. \end{aligned}
Regardless of how big or how little the encapsulating sphere is made to be, this quantity, the field everywhere on the surface, times the surface area, will always be equal to
4\pi k
times the total charge enclosed by the surface.
In fact, the surface doesn't need to be centered on the charge, and the surface need not be spherical. The relation
\Phi_E = 4\pi k Q_\text{enc}
holds for any closed surface whatsoever that encloses the charge
Q_\text{enc}
This is a rather curious observation. Could this flux coincidence have fundamental importance?
To close, we'll compare the strength of gravity with the strength of the Coulomb force.
Comparison with gravity
To make the comparison fair, we'll compare the gravitational attraction of two protons with their electric repulsion. The gravitational attraction between them is given by
G\frac{m_p^2}{r^2}
\begin{aligned} F_E/F_G &= k\frac{q_p^2}{r^2} \frac{1}{G}\frac{r^2}{m_p^2} \\ &= \frac{kq_p^2}{Gm_p^2} \\ &\approx \frac{9\times 10^9}{6.6\times 10^{-11}}\frac{\left(1.6\times^{-19}\right)^2}{\left(1.7\times10^{-27}\right)^2} \\ &\approx 10^{36}. \end{aligned}
It is safe to say that the gravitational attraction of two protons is utterly insignificant next to their electric repulsion.
Given this enormous discrepancy, we might wonder when, if ever, gravitational interactions are worth considering. The saving grace for gravity is that because Coulomb forces are so strong that charged objects tend to pair up in such a way that most macroscopic objects are electrically neutral. As charge-neutral objects don't participate in Coulomb interactions, there is little Coulomb force to speak of.
On the other hand, gravity has no "negative particle," so all masses contribute to the attractive force. For very massive objects gravitational interactions can be quite significant.
Cite as: Charge and Electric Fields . Brilliant.org. Retrieved from https://brilliant.org/wiki/charge-and-electric-fields/
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Mensuration, Studymaterial: CBSE Class 8 ENGLISH, English Grammar - Meritnation
Consider a container, which is cylindrical in shape. Let us consider that 10 litres of milk can be stored in this container.
If the container is half-filled with milk, can we find the quantity of milk in the container?
Yes, we can find it. When the container is half-filled with milk, then the quantity of milk in the container is 5 litres.
Here, the quantity of milk in the container is the volume of the milk which is 5 litres.
And the container can store a maximum of 10 litres of milk, which is the capacity of the container.
If the container is completely filled with milk, then
Capacity of container = volume of milk = 10 litres
Thus we can say that,
“Volume is the amount of space occupied by an object, while capacity refers to the quantity that a container holds".
The units of volume of solid material are cm3, m3, dm3 etc and the unit of volume of liquid and capacity is litre.
Let us discuss some examples based on volume and capacity.
A cubical container has each side measuring 20 cm. The container is half-filled with water. Metal stones are dropped in the container till the water comes up to the brim. Each stone is of volume 10 cm3. Calculate the number of stones and the capacity of the container.
We know that volume of cube = (side) 3
∴ Volume of cubical container = (20)3 cm3
∴ Capacity of container = 8000 cm3
= 8 litres (
\because
1 litre = 1000 cm3)
The container is half-filled with water.
∴ Volume of water in the container = 4 litres
and, volume of metal stones = 4 litres
= 4 × 1000 cm3
Volume each metal stone = 10 cm3
∴ Number of stones =
= 400 stones
Thus, the capacity of the container is 8 litres and the number of stones is 400.
An oil tank is in the form of a cuboid whose dimensions are 60 cm, 30 cm, and 30 cm respectively. Find the quantity of oil that can be stored in the tank.
∴ Quantity of oil = Capacity of tank
∴ Quantity of oil that can be stored in the tank = 54 litres
Water is pouring in a cubical reservoir at a rate of 50 litres per minute. If the side of the reservoir is 1 metre, then how much time will it take to fill the reservoir?
Side of reservoir = 1 m
∴ Capacity of reservoir = 1 m × 1 m × 1 m
∴ Capacity of reservoir = 1000 litres
50 litres of water is filled in 1 minute.
1 litre of water is filled in .
⇒
1000 litres of water will be filled in =
∴ Thus, the reservoir is filled in 20 minutes.
Orange juice is available in two packs − a tin cylinder of radius 2.1 cm and height 10 cm and a tin can with rectangular base of length 4 cm, width 3 cm, and height 12 cm. Which of the two packs has a greater capacity?
For tin cylinder,
And, height (h) = 10 cm
Capacity of cylinder = πr2h
For tin can with rectangular base,
Width (b) = 3 cm
Therefore, the tin can with a rectangular base has greater capacity than the tin cylinder.
Surface Areas of a Cube and a Cuboid
We give gifts to our friends and relatives at one time or another. We usually wrap our gifts in nice and colourful wrapping papers. Look, for example, at the nicely wrapped and tied gift shown below.
Clearly, the gift is packed in box that is cubical or shaped like a cube . Suppose you have a gift packed in a similar box. How would you determine the amount of wrapping paper needed to wrap the gift? You could do so by making an estimate of the surface area of the box. In this case, the total area of all the faces of the box will tell us the area of the wrapping paper needed to cover the box.
Knowledge of surface areas of the different solid figures proves useful in many real-life situations where we have to deal with them. In this lesson, we will learn the formulae for the surface areas of a cube and a cuboid . We will also solve some examples using these formulae.
The word ‘cuboid’ is made up of ‘cube’ and ‘-oid’ (which mea…
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Dot product - MATLAB dot - MathWorks América Latina
Dot Product of Complex Vectors
Dot Product of Multidimensional Arrays
C = dot(A,B,dim)
C = dot(A,B) returns the scalar dot product of A and B.
If A and B are vectors, then they must have the same length.
If A and B are matrices or multidimensional arrays, then they must have the same size. In this case, the dot function treats A and B as collections of vectors. The function calculates the dot product of corresponding vectors along the first array dimension whose size does not equal 1.
C = dot(A,B,dim) evaluates the dot product of A and B along dimension, dim. The dim input is a positive integer scalar.
Create two simple, three-element vectors.
B = [2 -2 -1];
Calculate the dot product of A and B.
C = A(1)*B(1) + A(2)*B(2) + A(3)*B(3)
Create two complex vectors.
A = [1+i 1-i -1+i -1-i];
B = [3-4i 6-2i 1+2i 4+3i];
C = 1.0000 - 5.0000i
The result is a complex scalar since A and B are complex. In general, the dot product of two complex vectors is also complex. An exception is when you take the dot product of a complex vector with itself.
Find the inner product of A with itself.
D = dot(A,A)
The result is a real scalar. The inner product of a vector with itself is related to the Euclidean length of the vector, norm(A).
Create two matrices.
Find the dot product of A and B.
The result, C, contains three separate dot products. dot treats the columns of A and B as vectors and calculates the dot product of corresponding columns. So, for example, C(1) = 54 is the dot product of A(:,1) with B(:,1).
Find the dot product of A and B, treating the rows as vectors.
D = dot(A,B,2)
In this case, D(1) = 46 is the dot product of A(1,:) with B(1,:).
Create two multidimensional arrays.
A = cat(3,[1 1;1 1],[2 3;4 5],[6 7;8 9])
B = cat(3,[2 2;2 2],[10 11;12 13],[14 15; 16 17])
Calculate the dot product of A and B along the third dimension (dim = 3).
C = dot(A,B,3)
The result, C, contains four separate dot products. The first dot product, C(1,1) = 106, is equal to the dot product of A(1,1,:) with B(1,1,:).
Dimension to operate along, specified as a positive integer scalar. If no value is specified, the default is the first array dimension whose size does not equal 1.
dot(A,B,1) treats the columns of A and B as vectors and returns the dot products of corresponding columns.
dot(A,B,2) treats the rows of A and B as vectors and returns the dot products of corresponding rows.
dot returns conj(A).*B if dim is greater than ndims(A).
The scalar dot product of two real vectors of length n is equal to
u\text{\hspace{0.17em}}·\text{\hspace{0.17em}}v=\sum _{i=1}^{n}{u}_{i}{v}_{i}={u}_{1}{v}_{1}+{u}_{2}{v}_{2}+...+{u}_{n}{v}_{n}\text{\hspace{0.17em}}.
This relation is commutative for real vectors, such that dot(u,v) equals dot(v,u). If the dot product is equal to zero, then u and v are perpendicular.
For complex vectors, the dot product involves a complex conjugate. This ensures that the inner product of any vector with itself is real and positive definite.
u\text{\hspace{0.17em}}·\text{\hspace{0.17em}}v=\sum _{i=1}^{n}{\overline{u}}_{i}{v}_{i}\text{\hspace{0.17em}}.
Unlike the relation for real vectors, the complex relation is not commutative, so dot(u,v) equals conj(dot(v,u)).
When inputs A and B are real or complex vectors, the dot function treats them as column vectors and dot(A,B) is the same as sum(conj(A).*B).
When the inputs are matrices or multidimensional arrays, the dim argument determines which dimension the sum function operates on. In this case, dot(A,B) is the same as sum(conj(A).*B,dim).
For the syntax dot(A,B), the arrays A and B must have the same size, even if they are vectors.
cross | sum | conj | norm | tensorprod
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