text:
## Charged line segment

Line segment with long L has linear density of charge λ. Calculate the electric field:

a) at height z anywhere above the line segment

b) at height z above the centre of the line segment

c) at height z above the line (infinitely long line segment)

Note:: Points b) and c) are special cases of point a)

• #### a) Hint: Picture and Comment

We choose a coordinate system at first and draw a picture.

Note: We suppose in all tasks that the line segment (line) is charged with positive charge. The vector of electric field will be in opposite direction for negative charge.

• #### a) Hint: Connection with a point charge

Positive point charge Q creates electric field in distance r

$E\,=\,\frac{1}{4 \pi \varepsilon_0}\, \frac{Q}{r^2}\,.$

Direction of the vector of electric field is from the charge.

• #### a) Analysis

We choose a coordinate system at first. Put the line segment on axis x and point A on axis z in distance z from the origin of the coordinate system. In this case, the vector of electric field is in a plane xz .

We divide the line segment into small parts which will behave as point charges. We gain the total electric field as an integral of contributions of electric fields from constituent elements.

• #### a) Solution

We choose a coordinate system at first.

Put the line segment on axis x and point A on axis z in distance z from the origin of the coordinate system. Mark end points of the line segment a and b. Simultaneously, these are also x coordinates of end points of the line segment.

Note: If we place the line segment as it is shown in the picture, then the coordinate on its left end point is minus, i.e. value a is minus.

We know, that point charge with value Q creates electric field in distance r:

$E\,=\, \frac{1}{4 \pi \varepsilon_0}\,\frac{Q}{r^2}\,.$

So, we divide the line segment into small parts with length dx which have charge dQ = λdx and which will behave as a point charge.

Such a part of the line segment will in point A (which is in distance r) create electric field:

$\mathrm{d}E\,=\, \frac{1}{4 \pi \varepsilon_0}\,\frac{\mathrm{d}Q}{r^2}\,=\, \frac{1}{4 \pi \varepsilon_0}\,\frac{\lambda\mathrm{d}x}{r^2}\,.\tag{*}$

A vector of the electric field changes its direction for each individual contribution. So, we will calculate the value of the vector in its components. The component of the electric field in direction of y axis (i.e. perpendicularly to the picture) has for all contributions a zero value. We will use a yellow triangle in the next picture to formulate other components of the vector. We mustn´t forget that the component in the x axis is minus.

$\mathrm{d}E_x\,=\,-\, \cos\alpha\,\mathrm{d}E$ $\mathrm{d}E_z\,=\, \sin\alpha\,\mathrm{d}E$

We use sides of the red triangle to formulate sine and cosine

$\cos\alpha\,=\, \frac{x}{r}$ $\sin\alpha\,=\, \frac{z}{r}$

and substitute it to previous equations.

$\mathrm{d}E_x\,=\, -\frac{x}{r}\,\mathrm{d}E$ $\mathrm{d}E_z\,=\,\frac{z}{r}\, \mathrm{d}E$

Now substitute from equation (*) for dE.

$\mathrm{d}E_x\,=\,- \,\frac{1}{4 \pi \epsilon_0}\,\frac{\lambda}{r^2}\,\frac{x}{r} \mathrm{d}x\,=\,- \,\frac{1}{4 \pi \epsilon_0}\,\frac{\lambda x}{r^ 3}\mathrm{d}x$ $\mathrm{d}E_z\,=\, \frac{1}{4 \pi \epsilon_0}\,\frac{\lambda}{r^2}\,\frac{z}{r}\mathrm{d}x\,=\, \frac{1}{4 \pi \epsilon_0}\,\frac{\lambda z}{r^3}\mathrm{d}x$

Formulate a distance r using the red triangle:

$r\,=\, \sqrt{x^2+z^2}$

and substitute it to the formula of both components.

$\mathrm{d}E_x\,=\,- \,\frac{1}{4 \pi \epsilon_0}\,\frac{\lambda x}{\left(x^2+z^2\right)^{\frac{3}{2}} }\mathrm{d}x$ $\mathrm{d}E_z\,=\, \frac{1}{4 \pi \epsilon_0}\,\frac{\lambda z}{\left(x^2+z^2\right)^{\frac{3}{2}} }\mathrm{d}x\tag{**}$

We gain components of the vector of electric field E as an integral of contributions of electric fields from constituent elements over the total length of the line segment.

$E_x\,=\,\int_a^b\mathrm{d}E_x$ $E_z\,=\,\int_a^b\mathrm{d}E_z$

Substitute from equation (**) for constituent elements and integrate it.

$E_x\,=\,- \,\int_a^b\frac{1}{4 \pi \epsilon_0}\,\frac{\lambda x}{\left(x^2+z^2\right)^{\frac{3}{2}} }\mathrm{d}x \,=\,- \,\frac{\lambda}{4 \pi \epsilon_0}\int_a^b\,\frac{x}{\left(x^2+z^2\right)^{\frac{3}{2}} }\mathrm{d}x$ $E_z\,=\,\int_a^b\frac{1}{4 \pi \epsilon_0}\,\frac{\lambda z}{\left(x^2+z^2\right)^{\frac{3}{2}} }\mathrm{d}x\,=\,\frac{\lambda z}{4 \pi \epsilon_0}\int_a^b\,\frac{1}{\left(x^2+z^2\right)^{\frac{3}{2}} }\mathrm{d}x$

A more detailed process of the calculation of integrals is in a separate section at the end of this task.

$E_x\,=\,- \,\frac{\lambda}{4 \pi \epsilon_0}\left[-\frac{1}{\sqrt{x^2+z^2} }\right]_a^b\,=\, \,\frac{\lambda}{4 \pi \epsilon_0}\left[\frac{1}{\sqrt{x^2+z^2} }\right]_a^b$ $E_z\,=\,\frac{\lambda z}{4 \pi \epsilon_0}\left[\frac{x}{z^2\sqrt{x^2+z^2} }\right]_a^b\,=\,\frac{\lambda z}{4 \pi \epsilon_0 z^2}\left[\frac{x}{\sqrt{x^2+z^2} }\right]_a^b$ $E_z\,=\,\frac{\lambda}{4 \pi \epsilon_0 z}\left[\frac{x}{\sqrt{x^2+z^2} }\right]_a^b$

Substitute end points of the line segment and gain a general formulation of components of the electric field.

$E_x\,=\,\frac{\lambda}{4 \pi \epsilon_0}\,\left(\frac{1}{\sqrt{b^2+z^2} }\,-\,\frac{1}{\sqrt{a^2+z^2}}\right)$ $E_y\,=\,0$ $E_z\,=\,\frac{\lambda}{4 \pi \epsilon_0 z}\left(\frac{b}{\sqrt{b^2+z^2} }\,-\,\frac{a}{\sqrt{a^2+z^2} }\right)$
• #### Hint to part b) and c)

You can use the results from part a) for calculation of the intensity above the centre of the line segment and the line, if you replace them by end points of line segment a and b.

If point A is placed above the centre of the line segment, we can use the length of the line segment to formulate end points a, b.

$$a\,=\,-\,\frac{l}{2} \hspace{20px}$$a $$\hspace{20px} b\,=\,\frac{l}{2}\,.$$

A line is actually a line segment, which has end points at infinity, so it is infinitely long, which means $$a \to -\infty$$, $$b \to \infty\,, \qquad$$ or $$\qquad l \to \infty$$ .

• #### b) and c) Analysis

Calculation of the intensity above the centre of the line segment is easier. A line segment is symmetrical with the centre in origin of the coordinate system. For every part of a line segment there is another part, which is symmetrical with axis z. Vectors of intensity from these parts of the line segment are symmetrical with axis z too (see the picture). In that case, the x components of these vectors have the same length and opposite direction so they subtract together. So, x and y components of the vector of the total intensity at point A is zero, which means that its orientation is in direction of axis z. We gain the magnitude of the total intensity by substituing into the result of the previous calculation.

A line is a line segment, which has its end points at infinity. Because a line is symmetrical with axis z, vector of the total intensity has its orientation in the direction of axis z just as the line segment. We gain the magnitude of the vector if we enlarge the lenght of the line segment to infinity.

• #### b) Solution

For calculation of the intensity above the centre of the line segment we use the results from part a)

$E_x\,=\,\frac{\lambda}{4 \pi \epsilon_0}\,\left(\frac{1}{\sqrt{b^2+z^2}}\,-\,\frac{1}{\sqrt{a^2+z^2} }\right)$ $E_y\,=\,0$ $E_z\,=\,\frac{\lambda}{4 \pi \epsilon_0 z}\left(\frac{b}{\sqrt{b^2+z^2} }\,-\,\frac{a}{\sqrt{a^2+z^2} }\right)$

We can formulate end points by using the length of the line segment.

$a\,=\,-\frac{l}{2}$ $b\,=\,\frac{l}{2}$

We calculate x component of intensity.

$E_x\,=\,\frac{\lambda}{4 \pi \epsilon_0}\,\left(\frac{1}{\sqrt{\frac{l^2}{4}+z^2} }\,-\,\frac{1}{\sqrt{\frac{l^2}{4}+z^2}}\right)\,=\,0$

Component is zero, which we deduce in part Analysis qualitatively only.

Now we calculate z component of intensity which is the total magnitude of intensity too, because other two components are zero.

$E_z\,=\,\frac{\lambda}{4 \pi \epsilon_0 z}\left( \frac{\frac{l}{2}}{\sqrt{\frac{l^2}{2^2}+z^2} }\,-\,\frac{-\frac{l}{2}}{\sqrt{\frac{l^2}{2^2}+z^2} }\right)$

We add expressions in brackets and modify the fraction below the root.

$E_z\,=\,\frac{\lambda}{4 \pi \epsilon_0 z} \,\frac{l}{\sqrt{\frac{l^2}{4}+z^2} }\,=\,\frac{\lambda}{4 \pi \epsilon_0 z} \,\frac{l}{\sqrt{\frac{l^2+4z^2}{4}} }\,=\,\frac{\lambda}{4 \pi \epsilon_0 z} \,\frac{l}{\frac{\sqrt{l^2+4z^2}}{2} }$ $E_z\,=\,\frac{\lambda}{2 \pi \epsilon_0 z} \,\frac{l}{\sqrt{l^2+4z^2} }$

So we gain magnitude of the total intensity of the electric field above the centre of the charged line segment.

$E_x\,=\,0$ $E_y\,=\,0$ $E_z\,=\,\frac{\lambda}{2 \pi \epsilon_0 z} \,\frac{l}{\sqrt{l^2+4z^2} }$

The vector of intensity has its orientation in direction of axis z and it has a magnitude

$E\,=\,\frac{\lambda}{2 \pi \epsilon_0 z} \,\frac{l}{\sqrt{l^2+4z^2} }\,.$
• #### c) Solution

For expressing a formula for the intensity above the line we use the result from part b). On the basis of that we know that the vector of the electric intensity above the centre of the charged line segment with length l has its orientation in direction of axis z and its magnitude is

$E\,=\,\frac{\lambda}{2 \pi \epsilon_0 z} \,\frac{l}{\sqrt{l^2+4z^2} }\,.$

Because a line is an infinite line segment we must calculate the value of intensity E for l which approaches infinity.

$E\,=\, \lim_{\small l \tiny \to \infty } \,\frac{\lambda}{2 \pi \epsilon_0 z} \,\frac{l}{\sqrt{l^2+4z^2} }\,=\,\frac{\lambda}{2 \pi \epsilon_0 z} \, \lim_{\small l \tiny \to \infty } \,\frac{l}{\sqrt{l^2+4z^2} }$

Length l is much bigger than distance z, i.e. l >> z. So we can ignore term z2 in denominator of fraction towards l2.

$\lim_{\small l \tiny \to \infty } \,\large\frac{l}{\sqrt{l^2+4z^2}}\,=\,\lim_{ \small l \tiny \to -\infty }\large \,\frac{l}{\sqrt{l^2}}$

Calculate the root in denominator of fraction.

$\lim_{ \small l \tiny \to \infty }\, \large \frac{l}{\sqrt{l^2}}\,=\,\lim_{ \small l \tiny \to \infty } \large\,\frac{l}{|l|}\,=\,\lim_{ \small l \tiny \to \infty}\large\, \frac{l}{l}\,=\,\lim_{ \small l \tiny \to \infty }\large \, 1\,=\,1$

We substitute the result of the limit into the formula for magnitude of intensity E and we gain that the charged line creates in distance z electric field with intensity:

$E\,=\,\frac{\lambda}{2 \pi \epsilon_0}\,\frac{1}{ z}\,.$

Comment: We can calculate intensity from the result from part a). The method is very similar but more lengthy.

a) If a line segment is located on x between points with coordinates [a; 0; 0] and [b; 0; 0] and point A, in which we find out the intensity, has coordinates [0; 0; z], then the vector of intensity E has components:

$E_x\,=\,\frac{\lambda}{4 \pi \epsilon_0}\,\left(\frac{1}{\sqrt{b^2+z^2} }\,-\,\frac{1}{\sqrt{a^2+z^2} }\right)$ $E_y\,=\,0$ $E_z\,=\,\frac{\lambda}{4 \pi \epsilon_0 z}\left(\frac{b}{\sqrt{b^2+z^2} }\,-\,\frac{a}{\sqrt{a^2+z^2} }\right)$

b) In distance z above the centre of the line segment the vector of intensity has its direction perpendicularly to the line segment and has a magnitude

$E\,=\,\frac{\lambda}{2 \pi \epsilon_0 z} \,\frac{l}{\sqrt{l^2+4z^2} }\,.$

c) A charged line creates in distance z an electric field with intensity:

$E\,=\,\frac{\lambda}{2 \pi \epsilon_0}\,\frac{1}{ z}\,.$

The vector of intensity has its direction perpendicularly to the line.

• #### A detailed calculation of integrals from part a)

In this section we calculate integrals from section a) Solution.

The first integral  – calculation of component Ex:

$E_x\,=\,-\,\frac{1}{4 \pi \varepsilon_0} \int_a^b \frac{x}{\left( x^2\,+\,z^2\right)^{\frac{3}{2}}}\,\mathrm{d}x$

At first we calculate the integral as indefinite. We will calculate this integral using substitution:

$t\,=\,x^2\,+\,z^2$ $\mathrm{d}t\,=\,2x\,\mathrm{d}x \hspace{20px} \Rightarrow \hspace{20px} x\,\mathrm{d}x\,=\,\frac{\mathrm{d}t}{2}$

Substitute it into integral.

$\int \frac{x}{\left( x^2\,+\,z^2\right)^{\frac{3}{2}}}\,\mathrm{d}x\,=\,\int \frac{\mathrm{d}t}{2t^{\frac{3}{2}}} \,=\,\frac{1}{2}\int t^{-\frac{3}{2}}\,\mathrm{d}t\,=\,-\,\frac{1}{2} \,\frac{t^{-\frac{1}{2}}}{\frac{1}{2}}$

Reduce 1/2 and substitute it instead of t.

$\int \frac{x}{\left( x^2\,+\,z^2\right)^{\frac{3}{2}}}\,\mathrm{d}x\,=\, - \frac{1}{\sqrt{t}}$ $\int \frac{x}{\left( x^2\,+\,z^2\right)^{\frac{3}{2}}}\,\mathrm{d}x\,=\, - \frac{1}{\sqrt{x^2\,+\,z^2}}$

When we substitute inside limits, we gain the result

$E_x\,=\,-\,\frac{1}{4 \pi \varepsilon_0} \left[ - \frac{1}{\sqrt{x^2\,+\,z^2}} \right]_a^b$

The second integral – calculation of component Ez:

The second integral

$E_z\,=\,\frac{\lambda z}{4 \pi \epsilon_0}\int_a^b\,\frac{1}{\left(x^2+z^2\right)^{\frac{3}{2}} }\mathrm{d}x$

At first we calculate the second integral again as indefinite and use the substitution too, but this time the calculation is difficult. A suitable substitution is

$x\,=\,z \,\sinh \,t$ $\mathrm{d}x\,=\,z\,\cosh\, t \,\mathrm{d}t$

Substitute it to integral and edit the integral before the calculation.

$\int\,\frac{1}{\left(x^2+z^2\right)^{\frac{3}{2}} }\mathrm{d}x\,=\, \int\frac{z\,\cosh \,t \,\mathrm{d}t}{\left(z^2\,\sinh^2t+z^2\right)^{\frac{3}{2}}} \,=\, \int\frac{z\,\cosh \, t \,\mathrm{d}t}{\left(z^2\right)^{\frac{3}{2}} \left(\sinh^2t+1\right)^{\frac{3}{2}}} \,=\,$

Reduce z and use formula cosh2t = sinh2t + 1

$\,=\, \int\frac{z\,\cosh t \,\mathrm{d}t}{z^3 \left(\cosh^2t\right)^{\frac{3}{2}} }\,=\, \int\frac{\cosh t \,\mathrm{d}t}{z^2 \,\cosh^3t }\,=$

Now we reduce the expression, remove constants before the integral and then we calculate the integral.

$=\, \frac{1}{z^2}\int\frac{\mathrm{d}t}{ \,\cosh^2 t }\,=\, \frac{1}{z^2} \,\tanh\, t\,=\, \frac{1}{z^2}\, \frac{\sinh\, t}{\cosh\, t} \,=\, \frac{1}{z^2}\frac{\sinh \,t}{\sqrt{1+\sinh^2 t}} \,=$

Put x back from the substitution and edit the complex fraction.

$\,=\, \frac{1}{z^2} \frac{\frac{x}{z}}{\sqrt{1+\frac{x^2}{z^2}}} \,=\, \frac{1}{z^2} \frac{x}{z \sqrt{\frac{x^2+z^2}{z^2}}} \,=\, \frac{1}{z^2}\, \frac{x}{ \sqrt{x^2+z^2}}$

Now we can substitute in limits.

$E_z\,=\, \frac{\lambda z}{4 \pi \epsilon_0}\frac{1}{z^2}\, \left[ \frac{x}{ \sqrt{x^2+z^2}} \right]_a^b \,=\, \frac{\lambda}{4 \pi \epsilon_0\,z}\,\left[ \frac{x}{ \sqrt{x^2+z^2}} \right]_a^b$
• #### Link: How to calculate the intensity in the surroundings of the line differently?

We can calculate the intensity of the electric field in the surroundings of the line using Gauss´ law. This method is publish in task Pole nabité přímky (only in Czech).


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# Difference between revisions of "2009 AMC 8 Problems/Problem 25"

## Problem

A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cut is $\frac{1}{2}$ foot from the top face. The second cut is $\frac{1}{3}$ foot below the first cut, and the third cut is $\frac{1}{17}$ foot below the second cut. From the top to the bottom the pieces are labeled A, B, C, and D. The pieces are then glued together end to end as shown in the second diagram. What is the total surface area of this solid in square feet? $[asy] import three; real d=11/102; defaultpen(fontsize(8)); defaultpen(linewidth(0.8)); currentprojection=orthographic(1,8/15,7/15); draw(unitcube, white, thick(), nolight); void f(real x) { draw((0,1,x)--(1,1,x)--(1,0,x)); } f(d); f(1/6); f(1/2); label("A", (1,0,3/4), W); label("B", (1,0,1/3), W); label("C", (1,0,1/6-d/4), W); label("D", (1,0,d/2), W); label("1/2", (1,1,3/4), E); label("1/3", (1,1,1/3), E); label("1/17", (0,1,1/6-d/4), E);[/asy]$

$[asy] import three; real d=11/102; defaultpen(fontsize(8)); defaultpen(linewidth(0.8)); currentprojection=orthographic(2,8/15,7/15); int t=0; void f(real x) { path3 r=(t,1,x)--(t+1,1,x)--(t+1,1,0)--(t,1,0)--cycle; path3 f=(t+1,1,x)--(t+1,1,0)--(t+1,0,0)--(t+1,0,x)--cycle; path3 u=(t,1,x)--(t+1,1,x)--(t+1,0,x)--(t,0,x)--cycle; draw(surface(r), white, nolight); draw(surface(f), white, nolight); draw(surface(u), white, nolight); draw((t,1,x)--(t+1,1,x)--(t+1,1,0)--(t,1,0)--(t,1,x)--(t,0,x)--(t+1,0,x)--(t+1,1,x)--(t+1,1,0)--(t+1,0,0)--(t+1,0,x)); t=t+1; } f(d); f(1/2); f(1/3); f(1/17); label("D", (1/2, 1, 0), SE); label("A", (1+1/2, 1, 0), SE); label("B", (2+1/2, 1, 0), SE); label("C", (3+1/2, 1, 0), SE);[/asy]$

$\textbf{(A)}\:6\qquad\textbf{(B)}\:7\qquad\textbf{(C)}\:\frac{419}{51}\qquad\textbf{(D)}\:\frac{158}{17}\qquad\textbf{(E)}\:11$

## Solution 1

The areas of the tops of $A$, $B$, $C$, and $D$ in the figure formed has sum $1+1+1+1 = 4$ as do the bottoms. Thus, the total so far is $8$. Now, one of the sides has an area of one, since it combines all of the heights of $A$, $B$, $C$, and $D$, which is $1$. The other side is also the same. Thus the total area now is $10$. From the front, the surface area is half, because if you looked at it straight from the front it would look exactly like a side of $A$, with a surface area of half. From the back, it is the same thing. Thus, the total surface area is $10+\frac{1}{2}+\frac{1}{2}= 11$, or $\boxed{\textbf{(E)}\:11 }$.

## Solution 2 (Faster method that relies on the MCQ choices)

The top parts and the bottom parts sum to 8. The sides add on another 2. Therefore, the only logical answer would be $\boxed{\textbf{(E)}\:11 }$ since it is the only answer greater than 10.

format: plain


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Balance the following chemical equation: F e S ( s ) + H C l ( a q ) F e C l 2 ( a q ) + H 2 S...

Question:

Balance the following chemical equation:

{eq}FeS(s)+HCl(aq) \rightarrow FeCl_2(aq)+H_2S(g) {/eq}.

Balancing Chemical Equations:

Any stoichiometric calculations must be based on the balanced chemical equation. A balanced chemical equation follows the law of conservation of mass. No new atoms must be created and no atoms can be destroyed. In addition, the coefficients from the balanced chemical equation give the ratio by which the chemical species react with one another.

Become a Study.com member to unlock this answer!

The unbalanced chemical equation for the reaction is:

{eq}\rm FeS(s) +HCl(aq) \rightarrow FeCl_2(aq)+H_2S(g) {/eq}

Notice that the atoms of H and...


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# SmoothCircleIC

Circle with a smooth interface

Insert a single disk or sphere with radius (radius) and a smooth interface of width (int_width) with a user specified value range and functional dependence of the interface (profile). (outvalue) is the variable value outside of the disk/sphere and (invalue) is the value on the interior.

(1)

### Hyperbolic tangent profile (TANH)

(2)

In both cases is the initial condition value at , where is the center point (x1, y1, z1) of the circle (making a radius vector).

The hyperbolic tangent interface is scaled to have the same slope at its midpoint as the cosine interface.

For the COS interface, which is strictly zero outside the particle and strictly inside the particle the interface width int_width denotes exactly the width of the transition region where the order parameter values are in the open interval . The TANH interface function is asymptotic and requires a different definition for the interface width. Here we chose to have the slope at the midpoint match up with the midpoint slope of the COS interface (giving rise to the factor of in eq. Eq. (2)).

## Input Parameters

• invalueThe variable value inside the circle

C++ Type:double

Options:

Description:The variable value inside the circle

• variableThe variable this initial condition is supposed to provide values for.

C++ Type:VariableName

Options:

Description:The variable this initial condition is supposed to provide values for.

C++ Type:double

Options:

• outvalueThe variable value outside the circle

C++ Type:double

Options:

Description:The variable value outside the circle

• y1The y coordinate of the circle center

C++ Type:double

Options:

Description:The y coordinate of the circle center

• x1The x coordinate of the circle center

C++ Type:double

Options:

Description:The x coordinate of the circle center

### Required Parameters

• profileCOSFunctional dependence for the interface profile

Default:COS

C++ Type:MooseEnum

Options:COS TANH

Description:Functional dependence for the interface profile

• rand_seed12345Seed value for the random number generator

Default:12345

C++ Type:unsigned int

Options:

Description:Seed value for the random number generator

• 3D_spheresTruein 3D, whether the objects are spheres or columns

Default:True

C++ Type:bool

Options:

Description:in 3D, whether the objects are spheres or columns

• zero_gradientFalseSet the gradient DOFs to zero. This can avoid numerical problems with higher order shape functions and overlapping circles.

Default:False

C++ Type:bool

Options:

Description:Set the gradient DOFs to zero. This can avoid numerical problems with higher order shape functions and overlapping circles.

• boundaryThe list of boundary IDs from the mesh where this boundary condition applies

C++ Type:std::vector

Options:

Description:The list of boundary IDs from the mesh where this boundary condition applies

• int_width0The interfacial width of the void surface. Defaults to sharp interface

Default:0

C++ Type:double

Options:

Description:The interfacial width of the void surface. Defaults to sharp interface

• z10The z coordinate of the circle center

Default:0

C++ Type:double

Options:

Description:The z coordinate of the circle center

• blockThe list of block ids (SubdomainID) that this object will be applied

C++ Type:std::vector

Options:

Description:The list of block ids (SubdomainID) that this object will be applied

### Optional Parameters

• ignore_uo_dependencyFalseWhen set to true, a UserObject retrieved by this IC will not be executed before the this IC

Default:False

C++ Type:bool

Options:

Description:When set to true, a UserObject retrieved by this IC will not be executed before the this IC

• control_tagsAdds user-defined labels for accessing object parameters via control logic.

C++ Type:std::vector

Options:

Description:Adds user-defined labels for accessing object parameters via control logic.

• enableTrueSet the enabled status of the MooseObject.

Default:True

C++ Type:bool

Options:

Description:Set the enabled status of the MooseObject.


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# Does the serie converge or not?

• June 16th 2012, 11:52 AM
Siron
Does the serie converge or not?
Hi,

I'm not able to check if the following serie converges or not:
$\sum_{n} \frac{1}{\sqrt{n(1+n^2)}}$

The ratio test didn't work, and I think I need a majorant or a minorant but I can't find one.
• June 16th 2012, 12:04 PM
Reckoner
Re: Does the serie converge or not?
Quote:

Originally Posted by Siron
Hi,

I'm not able to check if the following serie converges or not:
$\sum_{n} \frac{1}{\sqrt{n(1+n^2)}}$

Use limit comparison with the convergent p-series $\sum_n\frac1{n^{3/2}}$.

Spoiler:
$\lim_{n\to\infty}\frac{1/\sqrt{n+n^3}}{1/n^{3/2}}$

$=\lim_{n\to\infty}\frac{n^{3/2}}{\sqrt{n+n^3}}$

$=\lim_{n\to\infty}\sqrt{\frac{n^3}{n+n^3}}$

$=\lim_{n\to\infty}\sqrt{\frac1{\frac1{n^2}+1}}$
• June 16th 2012, 12:31 PM
Siron
Re: Does the serie converge or not?
Thanks! ;)


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# SUSY 2016

3-8 July 2016
The University of Melbourne
Australia/Melbourne timezone

4 Jul 2016, 14:20
20m
Alice Hoy Room 309

### Alice Hoy Room 309

Talk Flavour Physics

### Speaker

Olivier Deschamps (Univ. Blaise Pascal Clermont-Fe. II (FR))

### Description

Radiative b-hadron decays are sensitive probes of New Physics through the study of branching fractions, CP asymmetries and measurements of the polarisation of the photon emitted in the decay. During Run 1 of the LHC, the LHCb experiment has collected large samples of radiative b-hadron decays. We present here the latest LHCb measurements, including new results on the time dependence of $B_s \to \phi \gamma$ decays. These results help constrain the size of right-handed currents in extensions of the Standard Model.

### Primary author

Olivier Deschamps (Univ. Blaise Pascal Clermont-Fe. II (FR))


text:
# 94.828 as a fraction

## 94.828 as a fraction - solution and the full explanation with calculations.

If it's not what You are looking for, type in into the box below your number and see the solution.

## What is 94.828 as a fraction?

To write 94.828 as a fraction you have to write 94.828 as numerator and put 1 as the denominator. Now you multiply numerator and denominator by 10 as long as you get in numerator the whole number.
94.828 = 94.828/1 = 948.28/10 = 9482.8/100 = 94828/1000

And finally we have:
94.828 as a fraction equals 94828/1000

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# Preprints 2013

The three-letter code attached to the preprint number indicates the scientific programme during which the paper was written. Click on the code to see the programme details.

Preprint No. Author(s) Title and publication details
NI13001-TOD GV Levina Helical organization of tropical cyclones
NI13002-DAE P Goos and SG Gilmour Testing for Lack of Fit in Blocked and Split-Plot Response Surface Designs
NI13003-AMM M Ainsworth Dispersive behaviour of high order finite element schemes for the one-way wave equation
NI13004-SAS AG Melnikov and A Nies The classification problem for compact computable metric spaces
NI13005-SAS M Lauria, P Pudlák, V Rödl and N Thapen The complexity of proving that a graph is Ramsey
NI13006-GDO I Gálvez-Carrillo, L Lombardi and A Tonks An A$_\infty$ operad in spineless cacti
NI13007-NPA LO Müller and EF Toro A global multi-scale mathematical model for the human circulation with emphasis on the venous system
NI13008-INV J Eckhardt, F Gesztesy, R Nichols and G Teschl Weyl-Titchmarsh Theory for Sturm-Liouville operators with distributional coefficients
NI13009-GDO J Goodman and U Krähmer Untwisting a twisted Calabi-Yau algebra
NI13010-INI JF Toland Energy-minimising parallel flows with prescribed vorticity distribution
NI13011-INI JF Toland Non-existence of global energy-minimisers in Stokes wave problems
NI13012-SAS M Garlík Ajtai's completeness theorem for nonstandard finite structures
NI13013-MLC P Bauman, D Phillips and J Park Existence of solutions to boundary value problems for smectic liquid crystals
NI13014-GDO V Dotsenko, S Shadrin and B Vallette Givental action is homotopy gauge symmetry
NI13015-MLC M Arroyo and A De Simone Shape control of active surfaces inspired by the movement of euglenids
NI13016-TOD RV Buniy and TW Kephart Generalized helicity and Beltrami fields
NI13017-MLC I Busjatskaya and M Monastyrsky Immortality of platonic solids
NI13018-DAE RA Bailey and P Druilhet Optimal cross-over designs for full interaction models
NI13019-POP JB Lasserre Tractable approximations of sets defined with quantifiers
NI13020-CFM C Beaume, H-C Kao, E Knobloch and A Bergeon Localized rotating convection with no-slip boundary conditions
NI13021-TOD R Kerner Discrete groups and internal symmetries of icosahedral viral capsids
NI13022-MLC M Leoni and TB Liverpool Synchronisation and liquid crystalline order in soft active fluids
NI13023-MLC G Dhont and B Zhilinskií The action of the orthogonal group on planar vectors: invariants, covariants, and syzygies
NI13024-CFM HO Jacobs, TS Ratiu and M Desbrun On the coupling between an ideal fluid and immersed particles
NI13025-GDO J Giansiracusa and N Giansiracusa Equations of tropical varieties
NI13026-POP M Kocvara On robustness criteria and robust topology optimization with uncertain loads
NI13027-MLC VL Golo, EI Kats, AA Sevenyuk and DO Sinitsyn Twisted quasiperiodic textures of biaxial nematics
NI13028-CFM C Paterson, SK Wilson and BR Duffy Strongly coupled interaction between a ridge of fluid and an external airflow
NI13030-CFM D Tseluiko, M Galvagno and U Thiele Heteroclinic snaking near a heteroclinic chain in dragged meniscus problems
NI13031-CFM U Thiele Patterned deposition at moving contact lines
NI13032-CFM U Thiele, DV Todorova and H Lopez Gradient dynamics description for films of mixtures and suspensions - the case of dewetting triggered by coupled film height and concentration fluctuations
NI13033-POP IM Bomze Copositivity for second-order optimality conditions in general smooth optimization problems
NI13034-CFM F Gay-Balmaz, TS Ratiu and C Tronci Equivalent theories of liquid crystal dynamics
NI13035-CFM AM Rubio, K Julien, E Knobloch and JB Weiss Upscale energy transfer in three-dimensional rapidly rotating turbulent convection
NI13036-CFM F Gay-Balmaz, DD Holm and TS Ratiu Integrable G-Strands on semisimple Lie groups
NI13037-DAE W Wang, R-B Chen, C-C Huang and WK Wong Particle swarm optimization techniques for finding optimal mixture designs
NI13038-DAE J Qiu, R-B Chen, W Wang and WK Wong Using animal instincts to design efficient biomedical studies
NI13039-DAE R-B Chen, S-P Chang, W Wang, H-C Tung and WK Wong Optimal Minimax Designs via Particle Swarm Optimization Methods
NI13040-DAE H-Q Li, M-L Tang and W-K Wong Confidence intervals for ratio of two Poisson rates using the method of variance estimates recovery
NI13041-DAE BPM Duarte and WK Wong A Semidefinite Programming based approach for finding Bayesian optimal designs for nonlinear models
NI13042-CFM BR Duffy, D Pritchard and SK Wilson The shear-driven Rayleigh problem for generalised Newtonian fluids
NI13043-MLC CL Kane and TC Lubensky Topological Boundary Modes in Isostatic Lattices
NI13044-GDO C Castaño Bernard and TM Gendron Modular invariant of quantum tori
NI13045-CFM G Grün On convergent schemes for diffuse interface models for two-phase flow of incompressible fluids with general mass densities
NI13046-SRO GR Barrenechea, L Boulton and N Boussai Eigenvalue enclosures and applications to the Maxwell operator
NI13047-POP JA De Loera, J Lee, S Margulies and J Miller Weak orientability of matroids and polynomial equations
NI13048-MLC M Trcek, G Cordoyiannis, V Tzitzios, S Kralj and G Nounesis et al Nanoparticle-induced twist-grain boundary phase
NI13049-AMM PS Peixoto and SRM Barros On vector field reconstructions for semi-Lagrangian transport methods on geodesic staggered grids
NI13050-HOL M Blake, D Tong and D Vegh Holographic lattices give the graviton a mass
NI13051-HOL A Buchel and DA Galante Cascading gauge theory on $\it dS_4$ and String Theory Landscape
NI13052-HOL K Kontoudi and G Policastro Flavor corrections to the entanglement entropy
NI13053-CFM D Pritchard, BR Duffy and SK Wilson Shallow flows of generalised Newtonian fluids on an inclined plane
NI13054-TOD J Cantarella and C Shonkwiler The symplectic geometry of closed equilateral random walks in 3-Space
NI13055-MLC A Ranjkesh, M Ambrozic, S Kralj and TJ Sluckin Computational studies of history-dependence in nematic liquid crystals in random environments
NI13056-POP J Fiala, M Kocvara and M Stingl PENLAB: A MATLAB solver for nonlinear semidefinite optimization
NI13057-MFE V Lucarini, R Blender, S Pascale, J Wouters and C Herbert Mathematical and physical ideas for climate science
NI13058-TOD Y Kimura and HK Moffatt Reconnection of skewed vortices
NI13059-MFE C Beck Possible resonance effect of axionic dark matter in S/N/S Josephson junctions
NI13060-MFE V Lucarini, D Faranda, J Wouters and T Kuna Towards a general theory of extremes for observables of chaotic dynamical systems
NI13061-MFE N Glatt-Holtz, V Šverák and V Vicol On inviscid limits for the stochastic Navier-Stokes equations and related models
NI13062-MFE J Földes, N Glatt-Holtz, G Richards and E Thomann Ergodic and mixing properties of the Boussinesq equations with a degenerate random forcing
NI13063-POP IM Bomze and ML Overton Narrowing the difficulty gap for the Celis-Dennis-Tapia problem
NI13064-POP IM Bomze Copositive relaxation beats Lagrangian dual bounds in quadratically and linearly constrained QPs
NI13065-MFE V Lucarini and S Pascale Entropy production and coarse graining of the climate fields in a general circulation model
NI13066-POP M Fampa, J Lee and W Melo On global optimization with indefinite quadratics
NI13067-HOL T Andrade, S Fischetti, D Marolf, SF Ross and M Rozali Entanglement and Correlations near Extremality: CFTs dual to Reissner-Nordström AdS$_5$
NI13068-MQI V Giovannetti, AS Holevo and R García-Patrón A solution of the Gaussian optimizer conjecture
NI13069-POP J Gouveia, PA Parrilo and RR Thomas Approximate cone factorizations and lifts of polytopes


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# Angular Acceleration

## Homework Statement

This is not really a question on how to solve the problem, I'm just trying to get clarification on something. For angular acceleration, α, can someone explain to me what αz is? And why does αz = α / R = αy? I understand the rest of the problem, I just don't understand where or what αz is and how αz = α / R = αy. Thanks in advance for any help given!

## Answers and Replies

Related Introductory Physics Homework Help News on Phys.org
TSny
Homework Helper
Gold Member
For angular acceleration, α, can someone explain to me what αz is?
$\alpha_z$ apparently represents the angular acceleration for rotation about the z-axis.
And why does αz = α / R = αy?
The print in the picture is small and hard to read. In equation (2) I think it states that $\alpha_z = \large \frac{a}{R}$, not $\alpha_z = \large \frac{\alpha}{R}.$

$\alpha_z$ apparently represents the angular acceleration for rotation about the z-axis.

The print in the picture is small and hard to read. In equation (2) I think it states that $\alpha_z = \large \frac{a}{R}$, not $\alpha_z = \large \frac{\alpha}{R}.$
Ohhh, you're right about the a/R not α/R. But can you explain to me why does a/R = ay?

TSny
Homework Helper
Gold Member
Ohhh, you're right about the a/R not α/R. But can you explain to me why does a/R = ay?
They aren't claiming that a/R = ay.

The R cancels: (1/2)MR(a/R) = (1/2)May. The "a" on the left is the same as ay.

They aren't claiming that a/R = ay.

The R cancels: (1/2)MR(a/R) = (1/2)May. The "a" on the left is the same as ay.
Ok I see. And just to clarify, if I divide a linear acceleration by the radius R then we get angular acceleration?

Nvm! Figured it out. Thank yoooou! :)

Last edited:


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# Is the number of boundary conditions sufficient for a unique solution?

Consider the following partial differential equation in the bounded domain $\Omega\subset \mathbb{R}^2$ $$\Delta^2 - \nabla\cdot(k(x,y)\nabla u) + \lambda u = f,\text{ in }\Omega$$ where $\lambda >0$ is a constant and $k(x,y) >0$ is a given function of the position. Further, $f(x,y)$ is a given function. The boundary conditions are given by $$u =0,\,\Delta u = 0,\text{ on }\partial\Omega.$$

I need to find the minimisation problem corresponding to this boundary value problem and explain whether the number of boundary conditions is sufficient for a unique solution. I have the following theorem:

Let $L:\sum(\Omega)\to\sum'(\Omega)$, where $\sum(\Omega)$ is a linear space, and suppose that $L$ is linear, self-adjoint, positive and coercive, and let $u_0\in\sum:Lu_0 = f$. Then $u_0\in\sum(\Omega)$ minimises $$F(u) = \int_\Omega\dfrac{1}{2}uLu - ufd\Omega.$$

Hence, after applying integration by parts twice on the first term and once on the second term and using Gauss' theorem, the minimisation problem corresponding to the boundary value problem is $$F(u) = \int_\Omega \dfrac{1}{2}\nabla\cdot\|\nabla u\|^2 + \dfrac{1}{2}k(x,y)\|\nabla u\|^2 + \lambda u^2 - uf\,d\Omega.$$

My Question is: How do I show whether or not the number of boundary conditions is sufficient for a unique solution? I don't have a clue how I should show this.


text:
Gamcdf

Description

Computes the lower/upper tail probability for the gamma distribution with real degrees of freedom, with parameters a and b.

Syntax

double prob = gamcdf(double g, double a, double b[, int tail])

Parameters

g

Input, the value of the gamma variate.

a

Input, the shape parameter $\alpha$ of the gamma distribution, must be positive($a>0$).

b

Input, the scale parameter $\beta$ of the gamma distribution, must be positive($b>0$).

tail

Input, Optional. The tail probability type using for the gamma distribution
• 1 = the for upper tail probability
• 2 = the lower tail probability (Default if not set)

prob

Output, the returned probability.

Example

gamcdf(150,2,2,1)=; //ANS: 2.0357640909741E-31


text:
# Events Archive

Pheno & Vino Seminar, Fiorenza Donato, U of Torino, "Dark matter and the cosmic radiation"
Tue, Apr 19, 2016, 4:00 pm5:30 pm
A viable method for inspecting the presence of Dark Matter particles in our Galaxy is the search for an annihilation signal in the cosmic radiation, and in particular in the rare antimatter fluxes as well as in gamma rays from different regions of sky. We discuss here the recent results from the interpretation of positron and antiproton data…
Pheno & Vino Seminar, Csaba Csaki, Cornell, "Displaced Higgs decays and flavor bounds in twin Higgs models"
Tue, Jan 19, 2016, 4:00 pm5:30 pm
I first examine one of the most interesting potential experimental signals of twin Higgs models, where the ordinary Higgs can decay to long-lived twin sector states, which then in turn decay back to SM states, giving rise to displaced Higgs decays. I will focus on decays in the tracker system, and show that Run I searches already yield some weak…
Pheno & Vino Surjeet Rajendran, Berkeley, "TBA"
Tue, Dec 15, 2015, 4:00 pmTue, Dec 8, 2015, 5:30 pm
TBA

Pheno & Vino Seminar, Surjeet Rajendran, Berkeley, "New Directions in Searching for the Dark Universe"
Tue, Dec 15, 2015, 4:00 pm5:30 pm
In this talk, we discuss ways to search for a wide variety of ultralight bosonic dark matter using precision instruments and probes of ultramassive dark matter using astrophysical probes such as exploding white dwarfs.

Special Pheno & Vino, Clifford Cheung, Caltech, "The Periodic Table of Effective Field Theories"
Mon, Nov 30, 2015, 4:00 pm5:30 pm
The modern S-matrix program has seen immense progress constructing scattering amplitudes in gauge and gravity theories without the aid of a Lagrangian, instead using only physical principles like Lorentz invariance and factorization. In this talk, I discuss how a similar approach is sufficient to construct effective field theories (EFTs),…
Pheno & Vino Seminar, Tien-Tien Yu, Stony Brook U, "Novel Techniques for Detecting sub-GeV Dark Matter"
Tue, Nov 17, 2015, 4:00 pm5:30 pm
Sub-GeV dark matter is a theoretically motivated but largely unexplored paradigm of dark matter. In this talk, I will describe how one can use the scattering of dark matter off of electrons to search for sub-GeV dark matter. I will present some motivated models that can be probed with these techniques as well as projections for current and near…
Pheno & Vino, Gordon Krnjaic, Perimeter, "Discovering Inelastic Thermal-Relic Dark Matter at Colliders"
Tue, Nov 10, 2015, 4:00 pm5:30 pm
Dark Matter particles with inelastic interactions are ubiquitous in extensions of the Standard Model, yet remain challenging to fully probe with existing strategies. We propose a series of powerful searches at hadron and lepton colliders that are sensitive to inelastic dark matter dynamics. In representative models featuring either a massive dark…
Pheno & Vino Seminar, Marco Farina, Cornell, "Asymmetric Twin Dark Matter"
Tue, Oct 27, 2015, 4:00 pm5:30 pm
Can the Asymmetric Dark Matter paradigm be naturally implemented in Twin Higgs models? The mirroring of the Standard Model strong sector suggests that a twin baryon with mass around 5 GeV is a natural dark matter candidate once a twin baryon number asymmetry comparable to the SM asymmetry is generated. The salient features will be discussed,…
Pheno & Vino, Prashant Saraswat, U.MD & Hopkins, "Towards a No-Lose Theorem for Naturalness "
Tue, Oct 20, 2015, 4:00 pm5:30 pm
We derive a phenomenological "no-lose" theorem for naturalness up to the TeV scale, which applies when quantum corrections to the Higgs mass are canceled by perturbative BSM particles (top partners) due to to some symmetry. In particular we focus on neutral top partner scenarios such as Twin Higgs. We classify all perturbative neutral top partner…
Pheno & Vino Seminar, Yoni Kahn, Princeton U., "The Goldstone and Goldstino of Supersymmetric Inflation"
Tue, Oct 6, 2015, 4:00 pm5:30 pm
I construct an effective field theory (EFT) for the minimal degrees of freedom of supersymmetric inflation. These can be viewed as the goldstone of spontaneously broken time translations, and the goldstino of spontaneously broken SUSY, both of which are tied together in an interesting way through the structure of the SUSY algebra. I briefly…
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Tue, Sep 29, 2015, 4:00 pm5:30 pm
Dark matter may be generated by new physics at the TeV scale during an early matter dominated era that ends at temperature much below the TeV scale. Compared to conventional radiation dominated results, yields from both Freeze-Out and Freeze-In processes are greatly suppressed by dilution from entropy production, making Freeze-Out less plausible…
Pheno & Vino Seminar, Matthew Low, IAS, "Composite Spin-1 Resonances at the LHC"
Tue, Sep 22, 2015, 4:00 pm5:30 pm
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Thu, May 14, 2015, 4:00 pm5:30 pm
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Particle Physics Seminar, David Curtin, U. of Maryland, "Probing Colorless Naturalness"
Mon, May 11, 2015, 4:00 pm5:30 pm
Theories of uncolored naturalness solve the (little) hierarchy problem without colored top partners, avoiding stringent LHC bounds. I will classify these theories by the physical properties of their top partners to derive the minimal experimental signatures required for a natural solution to the little Hierarchy problem. All so-far proposed…
Particle Physics Seminar, Sonia El Hedri, U. of Mainz, "Perturbative Unitarity Constraints on Dark Sectors"
Fri, May 1, 2015, 4:00 pm5:30 pm
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Particle Physics Seminar, Eric Kuflik, Cornell, "TBA"
Fri, Apr 17, 2015, 4:00 pm5:30 pm
TBA

Particle Physics Seminar, Illias Cholis, FNAL, "WIMP ante portas? The Fermi galactic center excess."
Fri, Apr 10, 2015, 4:00 pm5:00 pm
The possible gamma-ray excess in the inner Galaxy and the Galactic center suggested by Fermi-LAT observations has triggered great interest in the community. Among its various interpretations have been WIMP dark matter annihilations, gamma-ray emission from a population of millisecond pulsars, or emission from cosmic rays injected in a sequence…
Particle Physics Seminar, Jeremy Mardon, Stanford, "Ultra light vectors -- a journey from inflation to the lab"
Fri, Apr 3, 2015, 4:00 pm5:30 pm
A light vector field can have interesting cosmology and phenomenology. I will trace the vector's cosmological history from inflation, where it is fluctuated into existence, to the universe today, where it takes the form of cold relic matter, and finally to lab experiments searching for this unusual relic.

Particle Physics Seminar, Spencer Chang, U. of Oregon, "Phenomenology of Induced Electroweak Symmetry Breaking"
Fri, Mar 27, 2015, 4:00 pm5:30 pm
In this talk, I will discuss scenarios of induced electroweak symmetry breaking (EWSB), where the Higgs vev is induced by another EWSB source. These additional sources can be due to another Higgs vev or technicolor sector and can mitigate constraints on beyond the Standard Model theories (eg. raising the Higgs mass in supersymmetric models). I…
Special Particle Physics Seminar, Daniel Whiteson, UC Irvine, "Observing Ultra-High Energy Cosmic Rays with Smartphones"
Tue, Mar 10, 2015, 3:00 pm4:30 pm
We propose a novel approach for observing cosmic rays at ultra-high energy ($>10^{18}$~eV) by repurposing the existing network of smartphones as a ground detector array. Extensive air showers generated by cosmic rays produce muons and high-energy photons, which can be detected by the CMOS sensors of smartphone cameras. The small size and low…


text:
0
101 May 31, 2008 at 12:59

Hello,

I’m having some trouble in my prolog homework for some class, and needed some assistance.
I’m kind of newbie in prolog but I’ve been trying to read some tutorials to learn the “mysteries” behind the recursive functions and the backtracking.
My project its kind of simple, i have to make an agent (a) move randomly through a room in 4 directions (up,down,left,right) and he has to find the goal (o), but there are some walls (x), so he can only move to the spots that are free (-) or to the goal.
To accomplish this I’m using 1 fact that’s a list of all the “world”, the list has 16 members and it’s printed has an array of 4x4.

To move the agent, I’m adding or subtracting to his index an amount of 1 (if its to the right or left), or a amount of 4 (if its to up or down).

The problem begins here, i need to make sure that the agent doesn’t move to some values (A1>0 and A1<17, since the values A1=2,A1=10,A1=11 are walls he can’t move there too), for that i’m trying to build up a recursive function that tests the values before the agent move (function test), if the eventual move of the agent its not valid the function gets a new value to the Random move (moves up,down,left,right) and re-tests it until its a valid value. I’ve first built up the function in a .pl document, then inserted it in the project, but there is some problem in the recursive testing.

R is random(4),
teste(A,R),

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

teste(A,R):-
R=0,
A1 is A-4,
valores(A1,R).

teste(A,R):-
R=1,
A1 is A-4,
valores(A1,R).

teste(A,R):-
R=2,
A1 is A-4,
valores(A1,R).

teste(A,R):-
R=3,
A1 is A-4,
valores(A1,R).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

valores(A1,R):-
A1<1,!,
R is random(4),
teste(A1,R).

valores(A1,R):-
A1>16,!,
R is random(4),
teste(A1,R).

valores(A1,R):-
A1=2,!,
R is random(4),
teste(A1,R).

valores(A1,R):-
A1=10,!,
R is random(4),
teste(A1,R).

valores(A1,R):-
A1=11,!,
R is random(4),
teste(A1,R).

Thanks for the help,

Tiago

#### 3 Replies

0
167 May 31, 2008 at 17:50

Sorry, this is not a forum for homework questions. If you are having trouble you should talk to your professor, TA, or classmates.

0
101 May 31, 2008 at 18:09

My professor wants us to use this kind of forum, so there is no problem. My classmates have already gave up on this project!

0
101 Jun 02, 2008 at 17:17

Hello

lanca(A,Z,Nestado):-
R is random(4),
teste(A,R),
...

You first unify R with a random value.
This means that R will keep this value all along the programm with nop possibility of backtrack.

Next, you call teste(A,R).

teste(A,R):-
R=0,
A1 is A-4,
valores(A1,R).

If we suppose that R is 0, you compute and unify A1 with A-4 thne you call valores(A1, 0).

valores(A1,R):-
A1<1,!,
R is random(4),
teste(A1,R).

R is 0 and after you unify R with random(4). It succeed one time on four, and fail three times.
You must change the design of your programm.


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# LC Oscillations - Mathematical Formulation with Working for JEE

## What is LC Oscillator?

Last updated date: 20th Mar 2023
Total views: 81.3k
Views today: 1.59k

An LC oscillation is a circuit that is composed of the capacitor and inductor. In this circuit, the capacitor is fully charged and linked to the uncharged inductor. In LC oscillation, an electric current is set up and undergoes the LC oscillations when a charged capacitor is linked with the inductor. An LC oscillator is a type of circuit comprising  a capacitor and inductor which is designed in such a way that the required oscillations of positive feedback will continue working in the circuit. An oscillator circuit is a type of circuit that is used to convert the DC signal into an AC signal in the form of waves.

In this article, you’’ understand about the concept of LC oscillations and LC oscillator circuits. We will also discuss the types of LC oscillators, frequency of LC oscillator, LC Oscillator derivation, time period of LC oscillation, and working of LC oscillator.

## LC Oscillator Working

This section discusses the working of the LC oscillator in an elaborate manner. The following stages have been mentioned for the understanding of the working of the LC Oscillator:

• This is the beginning stage of the working of the LC Oscillator. In this stage, a fully powered capacitor is connected with the uncharged inductor. Due to this, the Capacitor captures all the energy of the circuit and the inductor remains de-energized. A circuit diagram has been explained for a better understanding of the first stage. The energy stored in the capacitor is depicted by UE and the energy stored in the inductor is depicted by UB

A LC Oscillator Circuit

Depiction of Graphs of Energy Storage

• In stage 2, the capacitor starts energizing the inductor by flowing the electric current to the inductor. The graphs clearly explain that the capacitor starts falling in energy level while the energy of the inductor begins to rise. A circuit diagram along with the graph depicts the current of the inductor and capacitor.

A LC Oscillator Circuit

Depiction of Graphs of Energy Storage

• In Stage 3, the energy stored in the capacitor is transferred by half amount to the inductor by flowing current through it. The energy storage graph depicts the transfer of half amount of energy from the capacitor to the inductor. The graph has shown an equivalent size of the bar graph depicting the half amount of transfer of energy from the capacitor to the inductor.

A LC Oscillator Circuit

Depiction of Graphs of Energy Storage

• In Stage 4, the entire amount of energy that was stored in the capacitor was transferred to the inductor. As the amount of energy that was drained from the capacitor, the transmitted energy to the inductor will be transformed into the magnetic energy. The image of the given circuit diagram will depict the current state of the working LC oscillator.

A LC Oscillator

Depiction of Graphs of Energy Storage

• In stage five, the energy of the capacitor is found to be completely drained and the transferred energy from the capacitor to the inductor makes the inductor fully energized. Now, the inductor will start to charge the capacitor in the same direction as the current.  The current states will be depicted in the following circuit diagram and graphs of the energy storage of the capacitor and inductor.

A LC Oscillator

Depiction of Graphs of Energy Storage

• The sixth stage of the working of the LC Oscillator is the final stage of the working process. In this stage, the energy of the capacitor will become full and it again works as a charged capacitor. On the other hand, the energy of the inductor will again be de-energized. Now, when the current starts flowing in the circuit, the current flowing direction will be opposite to the initial condition.

A LC Oscillator Circuit

Depiction of Graphs of Energy Storage

In conclusion, both the major components of this circuit will again charge twice in the complete cycle of the process of the LC Oscillator.

## A Mathematical Formulation of LC Oscillation

This section derives the LC oscillation formula, derivation of the time period of LC Oscillation, and frequency of LC oscillation. For determination of the frequency and time period of the LC, it is necessary to determine the differential equation of LC oscillation,

A Schematic Diagram of The LC Oscillator

When Kirchhoff's Voltage law is applied to the circuit, the magnitude of the total potential difference across the capacitor and inductor will be zero. Therefore, it means that

\begin{align} &V_{L}+V_{C}=0 \\ &-L \dfrac{d i}{d t}+\dfrac{q}{C}=0 \end{align}

It is known that the current is the flow of charge per unit, it is written as

$i=\dfrac{-dq}{dt}$

As the amount of current will become minimum when current starts flowing from the capacitor to the inductor, substitute the value of current in the above equation. Hence,

\begin{align} &-L \dfrac{d}{d t}\left(\dfrac{-d q}{d t}\right)+\dfrac{q}{C}=0 \\ &L \dfrac{d^{2} q}{d t^{2}}+\dfrac{q}{C}=0 \\ &\dfrac{d^{2} q}{d t^{2}}=-\dfrac{q}{L C} \\ &\dfrac{d^{2} q}{d t^{2}}+\dfrac{q}{L C}=0 \end{align}

This differential equation for LC oscillations is given by the above equation. Therefore, the angular frequency of LC Oscillations will be written as:

$\dfrac{d^{2} q}{d t^{2}}+q \omega^{2}=0$

On comparing the angular frequency will be given as

$\omega=\dfrac{1}{\sqrt{LC}}$

It is known that $\omega=2\pi f$. Therefore,

$f=\dfrac{1}{2\pi\sqrt{LC}}$

Hence, the time period for LC oscillation will be derived,

\begin{align} &T=\dfrac{1}{f} \\ &T=\dfrac{1}{\dfrac{1}{2 \pi \sqrt{L C}}} \\ &T=2 \pi \sqrt{L C} \end{align}

Therefore, the time period of LC oscillation is obtained $2\pi\sqrt{LC}$ and the frequency of LC oscillation is obtained $\dfrac{1}{2\pi\sqrt{LC}}$.

## Types of LC Oscillator

There are five types of LC oscillators which are as shown below:

1. Tuned Collector Oscillator

This is one of the basic types of LC oscillators. This circuit can be easily formed with the capacitor and the transformer which are connected in parallel across the oscillator’s Collector circuit. The tank in this circuit is formed by a transformer and capacitor. The oscillations will be generated by the transformer’s feed within the circuit to the base of the transistors.

1. Tuned Base Oscillator

One of the kinds of LC transistor oscillators in which the location of the circuit lies in between the two terminals of the transistors, i.e ground and the base. This circuit is formed with the help of a capacitor and the main coil of the transformer.

1. Hartley Oscillator

This is a different type of LC oscillator in which the tank circuit includes two inductors and one capacitor. The connections are made in such a way that the capacitor is connected in parallel and the inductors are connected in series and the combination of both is also connected in series. It is known as the Hartley oscillator because it was invented by Ralph Hartley in 1915.

1. Colpitts Oscillator

One type of LC oscillator is the Colpitts oscillator in which the tank circuit is built in such a way that one inductor and two capacitors are used in it. The connection of this circuit is connected in a way where the connections between capacitors are done in series with the parallel connection to the inductor.

1. Clapp Oscillator

This LC Oscillator is a modification of the Colpitts oscillator. In this oscillator, an extra capacitor is connected in the series with the inductor within the tank circuit.  The extra capacitor tries to separate the other two capacitors from the junction of a transistor for advances the frequency

## Conclusion

We have brief information regarding the concept of LC Oscillation and LC oscillator along with the circuit diagram. We also discussed the derivation of the frequency and time period of the LC oscillation, and also classified the types of LC oscillators and explained the working of the LC Oscillator. The article starts with the clarification of the first query, i.e. what is LC Oscillation. This article explained the different types of oscillators along with the connection of the capacitor with the inductor and transistor. It covers the concept of LC Oscillator from a diagram point of view for better clarity of concept.

Competitive Exams after 12th Science

## FAQs on LC Oscillations - Mathematical Formulation with Working for JEE

1. What is the principle of LC Oscillator?

The principle behind the working of the LC oscillator is to use a tank circuit that comprises the capacitor and inductor. In this circuit, whenever the voltage is applied to the circuit, the capacitor of the circuit gets charged. When the battery shuts down, the charging of the capacitor will stop and the stored energy of the capacitor will be used for charging the inductor. When the capacitor gets discharged, the inductor will charge the capacitor in the same way.

2. What is the application area of the LC Oscillator?

The application of the LC Oscillator is distributed in various electrical sectors. These oscillators are broadly applied for the generation of high-frequency signals. Those oscillators which are used for generating such high-frequency signals are known as RF oscillators. By following the practical values of the capacitors and inductors. The maximum range of generated frequency is more than 500MHz. The most applications of the LC oscillators are found in television, radio, RF generators, etc. hence, these are the application areas of the LC Oscillators.

3. What is the weightage of this topic in the JEE exam?

The weightage of the current topic titled LC oscillator is 3%. The chapter on oscillation and waves was found to be an important topic for the JEE. Nearly 2-3 questions can be expected in the given competitive examination of JEE. In this chapter, it is required to keep a check on every topic and practice a maximum number of questions for better performance in this section. Hence, it is required to give importance to this chapter from JEE's point of view.


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# Carbon dioxide sequestration in Limestone

The chemical reaction where dissolved carbon dioxide forms carbonic acid and then reacts with various calcium containing minerals to form calcium carbonate and bicarbonate is rather slow. I know that several years ago there was research into possible ways to speed up the reaction, possibly as a candidate for a truly long term way to scrub $\mathrm{CO}_2$ from the atmosphere. The inspiration for this idea is to speed up the part of the geologic carbon cycle that regulates atmospheric $\mathrm{CO}_2$ concentration.

Does anyone know how the research into catalysts or processes to speed this up are going?

This process does not rely on catalysts, but it addresses the essence of the question. The process described is to inject liquefied carbon dioxide into basalt formations, even if the basalt is very old. This makes the it plausible that we could scrub the excess $\mathrm{CO}_2$ from the atmosphere by:


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This is not exactly what you wanted, but in algebraic geometry it is often easier to prove something for a particular object by considering the moduli space parametrizing such objects. The example I have in mind is the following: Suppose you picked some random elliptic curve over $\mathbb{Q}$ and were wondering if it has a rational point of order $11$. It is possible to answer this for any particular curve with some computational facility, but we don't have to! . We have Mazur's Theorem, which says that the answer is no, it doesn't.' Mazur does this essentially by showing that the corresponding moduli space of elliptic curves with a choice of $11$-torsion point (which is a nice modular curve) has no rational points: so you can never have an elliptic curve over $\mathbb{Q}$ with an $11$-torsion point.
This is not exactly what you wanted, but in algebraic geometry it is often easier to prove something for a particular object by considering the moduli space parametrizing such objects. The example I have in mind is the following: Suppose you picked some random elliptic curve over $\mathbb{Q}$ and were wondering if it has a rational point of order $11$. Treating the curve in isolation, this would amount It is possible to solving answer this for any particular curve with some nasty equations. But, fortunatelycomputational facility, but we don't have to! We have Mazur's Theorem, which says that the answer is no, it doesn't.' Mazur does this essentially by showing that the corresponding moduli space of elliptic curves with a choice of $11$-torsion point (which is a nice modular curve) has no rational points: so you can never have an elliptic curve over $\mathbb{Q}$ with an $11$-torsion point.
This is not exactly what you wanted, but in algebraic geometry it is often easier to prove something for a particular object by considering the moduli space parametrizing such objects. The example I have in mind is the following: Suppose you picked some random elliptic curve over $\mathbb{Q}$ and were wondering if it has a rational point of order $11$. Treating the curve in isolation, this would amount to solving some nasty equations. But, fortunately, we have Mazur's Theorem, which says that the answer is `no, it doesn't.' Mazur does this essentially by showing that the corresponding moduli space of elliptic curves with a choice of $11$-torsion point (which is a nice modular curve) has no rational points: so you can never have an elliptic curve over $\mathbb{Q}$ with an $11$-torsion point.


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# Journal of Operator Theory

Volume 55, Issue 2, Spring 2006  pp. 239-251.

The range of generalized Gelfand transforms on $C^*$-algebras

Authors Eberhard Kirchberg
Author institution: Institut f\"ur Mathematik, Humboldt Universit\"at zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany

Summary:  It is shown that every Dini function on the primitive ideal space of a $C^*$-algebra $A$ is the generalized Gelfand transform of an element of $A$. Here a Dini function on a topological space $X$ means a non-negative lower semi-continuous function $f$ on $X$ with $\sup f\Big(\bigcap\limits _\tau F_\tau\Big)=\inf\limits_\tau\sup f(F_\tau)$ for every downward directed net $\{F_\tau\}_\tau$ of closed subsets of $X$.

Contents    Full-Text PDF


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# One-Shot Object Detection

One-Shot object detection (OSOD) is the task of detecting an object from as little as one example per category. Unlike the Object Detector which requires many varied examples of objects in the real world, the One-Shot Object Detector requires a very small (sometimes even just one) canonical example of the object. The One-shot Object Detector is best suited for two-dimensional objects that have some regularity in the wild. Examples include road signs, logos, playing cards, and clapperboards. The One-Shot Object Detector is not suitable for three-dimensional objects like faces, animals, and cars -- such objects are better suited for the classical Object Detector.

## Introductory Example

In this example, our goal is to localize instances of stop signs in images. To train a model for this application, we supply a single image of a stop sign as a starter image.

Given a starter image (left), a one-shot object detector trained on this starter image will produce instance predictions on a test image (right) which may look like this:

import turicreate as tc

starter_images = tc.SFrame({'image':[tc.Image('stop_sign_starter.png')],
'label':['stop_sign']})

test_images = tc.SFrame({'image':[tc.Image('stop_sign_test1.jpg'),
tc.Image('stop_sign_test2.jpg')]})

# Create a model. This step will take a few hours on CPU and about an hour on GPU
model = tc.one_shot_object_detector.create(starter_images, 'label')

# Save predictions on the test set
test_images['predictions'] = model.predict(test_images)

# Draw prediction bounding boxes on the test images
test_images['annotated_predictions'] = \
tc.one_shot_object_detector.util.draw_bounding_boxes(test_images['image'],
test_images['predictions'])

# To visualize the predictions made on the test set
test_images.explore()

# Save the model for later use in TuriCreate
model.save('mymodel.model')

# Export for use in Core ML
model.export_coreml('MyCustomOneShotDetector.mlmodel')

Examples of test image predictions:

## Deployment to CoreML

To learn more about deploying One-Shot to CoreML details, refer to the Object Detector: Deployment to Core ML chapter.

## How it works

In this section, we will go into detail on what happens under the hood of the create method. A One-Shot Object Detector model is trained in two steps. First, the starter image(s) is used to generate synthetic data and annotations. Second, we train an object detector model on the generated synthetic data.

Stage 1: Automatic synthetic data generation

To begin training, the provided starter image(s) are used to generate a set of synthetic training data. The synthetic images are generated by applying yaw, pitch, and roll rotations on a set of background images. If no background images are provided, a set of default background images are automatically downloaded and used.

Stage 2: Train the Object Detector model

To learn about the training implementation details, refer to the Object Detector: How it Works chapter.

## Troubleshooting

Here is a collection of some known caveats and their respective workarounds while using One-Shot Object Detection.

##### My starter image can appear upside down as well as rotated 90 degrees in the wild. Is there anything I can do to improve the quality of the model?

To ensure a higher quality model, instead of providing one starter image, feel free to provide more than one starter images. For every starter image, we rotate the starter image by up to 20 degrees. As a result, to detect both inverted and erect projections of the same starter image, you can provide 2 starter images with the same label. Here is an example to rotate a Stop Sign starter image by 90 degrees, 180 degrees, and 270 degrees, and construct an SFrame with four starter images:

import turicreate as tc

def pil_to_tc(pil_image):
pixel_data = np.array(pil_image)
FORMAT_RAW = 2
return tc.Image(_image_data=pixel_data.tobytes(),
_width=pixel_data.shape[1],
_height=pixel_data.shape[0],
_channels=pixel_data.shape[2],
_format_enum=FORMAT_RAW,
_image_data_size=pixel_data.size)

image_path = # set image path here
tc_image = tc.Image(image_path)
pil_image = Image.open(image_path)
images = [tc_image]
for rotation_angle in [90, 180, 270]:
rotated = pil_to_tc(pil_image.rotate(rotation_angle, expand=True))
images.append(rotated)

starter_images = tc.SFrame({
'image': images,
'label': ['my_label'] * 4
})
##### While training with the default backgrounds, my starter image does not appear in most of the images in the synthetic data.

This is a known issue (#2341) that we're trying to fix. As a workaround, please try resizing your starter image (while preserving the aspect ratio of the starter image) to get it as close to 500x500 (for the default backgrounds) as possible. You can leverage the following API provided by turicreate.Image:

image = turicreate.Image('starter.png')
turicreate.image_analysis.resize(image, width=image.width//2, height=image.height//2, channels=image.channels)

### Background Images

You can preview the synthetic images the toolkit generates by calling the preview_synthetic_training_data utility. This helps you understand data the model would be trained on:

synthetic_images = \
tc.one_shot_object_detector.util.preview_synthetic_training_data(starter_images, 'label')

synthetic_images.explore()

Some examples of synthetic training data:

Note: In order to view the exact synthetic images used to train the model, the same seed should be used which was used to train the model.

While we provide a set of background images by default, you can use your own custom background images (specific to your application) as follows:

# Load background images
my_backgrounds = tc.SArray('my_custom_backgrounds.sarray')

# Create a model using custom background images
model = tc.one_shot_object_detector.create(starter_images, 'label', backgrounds=my_backgrounds)

To preview the synthetic images generated with your custom background images:

synthetic_images = \
tc.one_shot_object_detector.util.preview_synthetic_training_data(starter_images, 'label', my_backgrounds)


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# American Institute of Mathematical Sciences

May  2015, 9(2): 247-253. doi: 10.3934/amc.2015.9.247

## Cryptanalysis of a noncommutative key exchange protocol

 1 Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, Zürich, Switzerland

Received  July 2014 Revised  February 2015 Published  May 2015

In the papers by Alvarez et al. and Pathak and Sanghi a non-commutative based public key exchange is described. A similiar version of it has also been patented (US7184551). In this paper we present a polynomial time attack that breaks the variants of the protocol presented in the two papers. Moreover we show that breaking the patented cryptosystem US7184551 can be easily reduced to factoring. We also give some examples to show how efficiently the attack works.
Citation: Giacomo Micheli. Cryptanalysis of a noncommutative key exchange protocol. Advances in Mathematics of Communications, 2015, 9 (2) : 247-253. doi: 10.3934/amc.2015.9.247
##### References:

show all references

##### References:
 [1] Florian Luca, Igor E. Shparlinski. On finite fields for pairing based cryptography. Advances in Mathematics of Communications, 2007, 1 (3) : 281-286. doi: 10.3934/amc.2007.1.281 [2] Shengtian Yang, Thomas Honold. Good random matrices over finite fields. Advances in Mathematics of Communications, 2012, 6 (2) : 203-227. doi: 10.3934/amc.2012.6.203 [3] Kishan Chand Gupta, Sumit Kumar Pandey, Indranil Ghosh Ray, Susanta Samanta. Cryptographically significant mds matrices over finite fields: A brief survey and some generalized results. Advances in Mathematics of Communications, 2019, 13 (4) : 779-843. doi: 10.3934/amc.2019045 [4] Thomas Westerbäck. Parity check systems of nonlinear codes over finite commutative Frobenius rings. Advances in Mathematics of Communications, 2017, 11 (3) : 409-427. doi: 10.3934/amc.2017035 [5] Angelo Favini, Rabah Labbas, Stéphane Maingot, Maëlis Meisner. Boundary value problem for elliptic differential equations in non-commutative cases. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 4967-4990. doi: 10.3934/dcds.2013.33.4967 [6] Timoteo Carletti. The lagrange inversion formula on non--Archimedean fields, non--analytical form of differential and finite difference equations. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 835-858. doi: 10.3934/dcds.2003.9.835 [7] Leonid Golinskii, Mikhail Kudryavtsev. An inverse spectral theory for finite CMV matrices. Inverse Problems & Imaging, 2010, 4 (1) : 93-110. doi: 10.3934/ipi.2010.4.93 [8] Stefania Fanali, Massimo Giulietti, Irene Platoni. On maximal curves over finite fields of small order. Advances in Mathematics of Communications, 2012, 6 (1) : 107-120. doi: 10.3934/amc.2012.6.107 [9] Igor E. Shparlinski. On some dynamical systems in finite fields and residue rings. Discrete & Continuous Dynamical Systems, 2007, 17 (4) : 901-917. doi: 10.3934/dcds.2007.17.901 [10] Jean-François Biasse, Michael J. Jacobson, Jr.. Smoothness testing of polynomials over finite fields. Advances in Mathematics of Communications, 2014, 8 (4) : 459-477. doi: 10.3934/amc.2014.8.459 [11] Robert Granger, Thorsten Kleinjung, Jens Zumbrägel. Indiscreet logarithms in finite fields of small characteristic. Advances in Mathematics of Communications, 2018, 12 (2) : 263-286. doi: 10.3934/amc.2018017 [12] Francis N. Castro, Carlos Corrada-Bravo, Natalia Pacheco-Tallaj, Ivelisse Rubio. Explicit formulas for monomial involutions over finite fields. Advances in Mathematics of Communications, 2017, 11 (2) : 301-306. doi: 10.3934/amc.2017022 [13] Lorenza D'Elia. $\Gamma$-convergence of quadratic functionals with non uniformly elliptic conductivity matrices. Networks & Heterogeneous Media, 2021  doi: 10.3934/nhm.2021022 [14] Joan-Josep Climent, Elisa Gorla, Joachim Rosenthal. Cryptanalysis of the CFVZ cryptosystem. Advances in Mathematics of Communications, 2007, 1 (1) : 1-11. doi: 10.3934/amc.2007.1.1 [15] Stephen M. Gagola III, Joanne L. Hall. Constructing commutative semifields of square order. Advances in Mathematics of Communications, 2016, 10 (2) : 291-306. doi: 10.3934/amc.2016006 [16] Doston Jumaniyozov, Ivan Kaygorodov, Abror Khudoyberdiyev. The algebraic classification of nilpotent commutative algebras. Electronic Research Archive, , () : -. doi: 10.3934/era.2021068 [17] Fatma-Zohra Benahmed, Kenza Guenda, Aicha Batoul, Thomas Aaron Gulliver. Some new constructions of isodual and LCD codes over finite fields. Advances in Mathematics of Communications, 2019, 13 (2) : 281-296. doi: 10.3934/amc.2019019 [18] Shudi Yang, Xiangli Kong, Xueying Shi. Complete weight enumerators of a class of linear codes over finite fields. Advances in Mathematics of Communications, 2021, 15 (1) : 99-112. doi: 10.3934/amc.2020045 [19] Joseph H. Silverman. Local-global aspects of (hyper)elliptic curves over (in)finite fields. Advances in Mathematics of Communications, 2010, 4 (2) : 101-114. doi: 10.3934/amc.2010.4.101 [20] Zilong Wang, Guang Gong. Correlation of binary sequence families derived from the multiplicative characters of finite fields. Advances in Mathematics of Communications, 2013, 7 (4) : 475-484. doi: 10.3934/amc.2013.7.475

2020 Impact Factor: 0.935


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# X is the product of integers from 1 to 50, inclusive. If 12^N is a fac

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09 Dec 2019, 01:13
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X is the product of integers from 1 to 50, inclusive. If 12^N is a factor of X, what is the greatest possible value of N?

A. 4
B. 12
C. 22
D. 28
E. 57

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X is the product of integers from 1 to 50, inclusive. If 12^N is a fac  [#permalink]

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09 Dec 2019, 01:33
4
The formula for finding the highest power of a prime number in a factorial

If p is a prime number, then the highest power of p in a factorial n is {$$\frac{n}{p}$$} + {$$\frac{n}{p^2}$$} + {$$\frac{n}{p^3}$$}.....
where {$$\frac{a}{b}$$} is the quotient when integer b divides another integer a

Now coming to the problem at hand.

n = $$50$$
p = $$12 = 2^2*3$$

To find the highest power of 12, we need to find the highest power of 3(which is the bigger prime number)
Here, the bigger the prime number the lower will the frequency of that prime be in the factorial.

Therefore, the highest power of 12 in 50 is $${\frac{50}{3}} + {\frac{50}{3^2}} + {\frac{50}{3^3}} = 16 + 5 + 1 = 22$$(Option C)
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X is the product of integers from 1 to 50, inclusive. If 12^N is a fac  [#permalink]

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11 Dec 2019, 05:39
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Bunuel wrote:
X is the product of integers from 1 to 50, inclusive. If 12^N is a factor of X, what is the greatest possible value of N?

A. 4
B. 12
C. 22
D. 28
E. 57

$$50!/{12}^n=50!/(2^23)^n$$
$$quotients(50/[2^1,2^2,2^3…2^x])=25+12+6+3=46$$
$$quotients(50/[3^1,3^2,3^3…3^x])=16+5+1=22$$
$$50!/(12)^n=(2^23)^n=(2^{2n}*3^n)=2^{2(22)}*3^{22}=(12)^{22}$$

Ans (C)
X is the product of integers from 1 to 50, inclusive. If 12^N is a fac   [#permalink] 11 Dec 2019, 05:39
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# Understand the problem

Let f : [0, 2] → R be a continuous function such that $\frac{1}{2} .\int^2_0 f(x)\,dx < f(2)$.
Then which of the following statements must be true?

## Look at the knowledge graph.

##### Source of the problem
I.S.I. B.Stat. Entrance 2017, UGA Problem 20
##### Key Competency
Maximum and minimum property of  function
Medium
##### Suggested Book
Mathematical Circles

Do you really need a hint? Try it first!

See that from the given result we have  $\frac{1}{2}.\int^2_0 f(x)\,dx < f(2) \Rightarrow \int^2_0 f(x)\,dx < 2.f(2) \Rightarrow \int^2_0 f(x)\,dx < \int^2_0 f(2)\,dx$.
Now if f(2) is minimum then f(x)>f(2) for all x belong to [0,2] . Therefore, $\int^2_0 f(x)\,dx > \int^2_0 f(2)\,dx$, which gives contradiction to the given result $\int^2_0 f(x)\,dx < \int^2_0 f(2)\,dx$.
From this we can’t say that f must be strictly increasing or f must attain a maximum value at x = 2.The only thing we can say that  f cannot have a minimum at x = 2.

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#### Vol. 9, No. 8, 2016

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editors’ Interests Scientific Advantages Submission Guidelines Submission Form Editorial Login Contacts Author Index To Appear ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print)
A long $\mathbb{C}^2$ without holomorphic functions

### Luka Boc Thaler and Franc Forstnerič

Vol. 9 (2016), No. 8, 2031–2050
##### Abstract

We construct for every integer $n>1$ a complex manifold of dimension $n$ which is exhausted by an increasing sequence of biholomorphic images of ${ℂ}^{n}$ (i.e., a long ${ℂ}^{n}$), but does not admit any nonconstant holomorphic or plurisubharmonic functions. Furthermore, we introduce new holomorphic invariants of a complex manifold $X$, the stable core and the strongly stable core, which are based on the long-term behavior of hulls of compact sets with respect to an exhaustion of $X$. We show that every compact polynomially convex set $B\subset {ℂ}^{n}$ such that $B=\overline{{B}^{\circ }}$ is the strongly stable core of a long ${ℂ}^{n}$; in particular, holomorphically nonequivalent sets give rise to nonequivalent long ${ℂ}^{n}$’s. Furthermore, for every open set $U\subset {ℂ}^{n}$ there exists a long ${ℂ}^{n}$ whose stable core is dense in $U$. It follows that for any $n>1$ there is a continuum of pairwise nonequivalent long ${ℂ}^{n}$’s with no nonconstant plurisubharmonic functions and no nontrivial holomorphic automorphisms. These results answer several long-standing open problems.

##### Keywords
holomorphic function, Stein manifold, long $\mathbb C^n$, Fatou–Bieberbach domain, Chern–Moser normal form
##### Mathematical Subject Classification 2010
Primary: 32E10, 32E30, 32H02


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#### Vol. 311, No. 2, 2021

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Correction to the article Uniform bounds of the Piltz divisor problem over number fields

### Wataru Takeda

Vol. 311 (2021), No. 2, 505–509
##### Abstract

The article in question gave an upper bound for the Piltz divisor problem over number fields, using estimates of exponential sums. It contained some mistakes. Here we correct these inaccurate claims and give a uniform upper bound for the Piltz divisor problem.

##### Keywords
ideal counting function, exponential sum, Piltz divisor problem
Primary: 11N37
Secondary: 11R42


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##### Dimension Independence in Unconstrained Private ERM via Adaptive Preconditioning
Aug 14, 2020
Peter Kairouz, Mónica Ribero, Keith Rush, Abhradeep Thakurta

In this paper we revisit the problem of private empirical risk minimziation (ERM) with differential privacy. We show that for unconstrained convex empirical risk minimization if the observed gradients of the objective function along the path of private gradient descent lie in a low-dimensional subspace (smaller than the ambient dimensionality of $p$), then using noisy adaptive preconditioning (a.k.a., noisy Adaptive Gradient Descent (AdaGrad)) we obtain a regret composed of two terms: a constant multiplicative factor of the original AdaGrad regret and an additional regret due to noise. In particular, we show that if the gradients lie in a constant rank subspace, then one can achieve an excess empirical risk of $\tilde{O}(1/\epsilon n)$, compared to the worst-case achievable bound of $\tilde{O}(\sqrt{p}/\epsilon n)$. While previous works show dimension independent excess empirical risk bounds for the restrictive setting of convex generalized linear problems optimized over unconstrained subspaces, our results operate with general convex functions in unconstrained minimization. Along the way, we do a perturbation analysis of noisy AdaGrad, which may be of independent interest.


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# Why would the brain flip the images perceived by your eyes?

The following is a common scientific statement, which you don't have to google long for to find:

The eye views images upside-down in the manner of a camera lens, but our brains reinterpret this input to allow us to see things the correct way up.

My first question is quite straightforward: Is this statement valid?

I don't understand why you should come to the conclusion that your brain should 'flip' the image. Wouldn't your brain be able to cope just fine without flipping it? If anything I would suspect the brain to not flip the image, if there is no reason to do so.

In order for this statement to be valid, I would expect a scientific theory/experiment from which can be concluded that the brain does process the vision in such a way that, after it being processed all subsequent processing occurs on the 'inverted' image. That seems to me to be the most logical interpretation of 'to flip'.

• I wanted to keep the question short and concise, but as a sidenote, I do know about the experiment where a person wears glasses which inverts their vision after which their vision adapts to it. But this only proves your vision is able to adapt to what you are used to. It doesn't prove that your brain inverts the image initially. Feb 9, 2012 at 0:21
• how would you ever hope to tell apart if the brain flips the image or works with it upside down? It is obvious that the projection on your retina is upside down... but there is functionally no difference between 'right'-side up and 'upside'-down in processing... except for your subjective experience. Feb 9, 2012 at 0:27
• I have often thought about this, and surely the most obvious answer should be that the brain doesn't flip the image at all, but that the whole world is in fact in 'reality' 'upside down' (that is what we consider to be upside down). What we 'see' is the upside down image of an 'upside down' world.
– user7394
Jan 5, 2015 at 12:12
• @Tomp24 Welcome to Stack exchange and thank you for sharing your thoughts! However, 'upside down' is a subjective percept; it is constructed by and with respect to the viewer. The world cannot be upside down 'in reality'. The world is. And we perceive it.
– AliceD
Jan 5, 2015 at 12:18
• Also of interest: cogsci.stackexchange.com/questions/1125/… Apr 18, 2016 at 23:06

It is not meaningful to talk about your brain processing something as 'right-side up"' or 'upside-down'. The 'images' in your brain are just collections of neural activations, and not actual pictures. Thus they cannot have an orientation. The only meaningful way to test your question is to try flipping the input the brain receives and seeing if it can cope.

Fortunately, the brain is capable of flipping your visual field if required as measured through perceptual adaptation experiments using inversion glasses. This has been demonstrated very drastically in studies, by for instance requiring a participant to wear inversion glasses for a long time. At first they are confused, and unable to orient themselves and do basic tasks, however after enough time the brain can adopt enough to even do activities like riding a bike. This suggests that from the only way you can measure things (i.e. behaviorally) the brain is capable of adapting to an upside down world (some participants even reported that after extended use the world even seemed "right side up"). This is functionally equivalent to the brain being able to process your visual information in either orientation. If it is capable of processing in either orientation, the question of "does my brain flip the image" becomes a pseudo-question and unanswerable.

### References

• Taylor, J. G. (1962). The behavioral basis of perception. New Haven: Yale University Pres

• Harris, C.S. (1965) "Perceptual adaptation to inverted, reversed, and displaced vision." Psychological Review 72(6): 419-444. [pdf]]

• Di Paolo, E.A. (2003) "Organismically-inspired robotics: homeostatic adaptation and teleology beyond the closed sensorimotor loop", {Dynamical systems approach to embodiment and sociality: 19-42 [pdf

• As in my comment, I know about this study. In informal discussions it is most often used as a proof your brain 'does' flip images. As you state, I never understood the reasoning behind this, and your answer confirms my suspicion it is just an urban myth. Feb 9, 2012 at 0:40
• Thanks for the paper! "Spatial relations are not originally perceived by the eye, but are the result of the association of visual sensations with previous muscular and tactual experiences." "This belief in the primacy of touch is so ingrained that experimental results are sometimes flagrantly misinterpreted in order to support it." .... Finally I have a conversation stopper! Such a fun feeling when your intuition turns out to be right. :) Feb 9, 2012 at 0:51
• Nice answer.I'm not an expert but i'd like to share a personal experience attached to this, I'm myopic with cylindrical power,when I started wearing specs, in those days, I felt as if my height had increased and it was very difficult for me to make the adjustments, my steps used to falter, later on, I believe my eye-brain co-ordination have made reasonable progress, I don't have that issue anymore. physics.stackexchange.com/q/228077. Why can't we say that the new born baby see the world very differently, Babies learn to see over a period of time, much like they learn to walk and talk. Sep 29, 2016 at 6:29
• Reminds me of a lyric in "Clint Eastwood" by Gorillaz - "Y'all can see me now 'cause you don't see with your eye. You perceive with your mind". Nov 27, 2016 at 10:01

I think part of what makes this question confusing is the use of expressions like "what the eye sees", "what the brain sees" and "what the frog's eye tells the frog's brain". Nobody sees anything except the experiencing subject. When one stops thinking that the brain (or some visual-system part of the brain) observes the image on the retina, then the question of whether anything is being flipped becomes meaningless.

As regards perceptual adaptation, it is interesting to note that it is not universal. In Sperry's famous frog-eye-inversion experiment, the frogs never adapted.

References

• Roger W. Sperry (1943). Effect of 180 Degree Rotation of the Retinal Field on Visuomotor Coordination. The Journal of Experimental Zoology 92 (3): 263–279

The fact that the image does not appears upside-down has to do with the way visual information is processed in the brain. In his book, Jeff Hawkins argues that the low-level visual features on the retina (being upside down, distorted, and changing rapidly) are lost in the process of forming invariant representation. And it's those representations that we experience consciously.

From On Intelligence (official site, pdf)

The light receptors in your retina are unevenly distributed. They are densely concentrated in the fovea at the center, and get gradually sparser out in the periphery. In contrast, the cells in the cortex are evenly distributed. The result is that the retinal image relayed onto the primary visual area, V1, is highly distorted [and upside-down, if you will]. When your eyes fixate on the nose of a face versus on an eye of the same face, the visual input is very different, as though it is being viewed through a distorting fisheye lens that is jerking violently to and fro. Yet when you see the face, it doesn't appear distorted, and it doesn't appear to be jumping around. Most of the time you aren't even aware that the retinal pattern has changed at all, let alone so dramatically. You just see "face." (Figure 2b shows this effect on a view of a beach landscape.) This is a restatement of the mystery of invariant representation we talked about in chapter 4, on memory. What you "perceive" is not what V1 sees. How does your brain ever know it is looking at the same face, and why don't you know the inputs are changing and distorted?

The process of forming invariant representations is explained in the book, but I won't quote it there because its quite long.

There are many transformations between the light hitting your retina and your perception of the world.

The signals from your retina initially travel through the visual pathways to reach your visual cortex, where visual information is processed. The representation of this visual information in your brain is also shaped by other brain activity representing your other senses (sound, touch, etc) and also internal processes like your current mood or expectations.

By the time all of this information is incorporated into your conscious perception, the orientation of the photons as they hit your retina is gone and all you have is your brain's interpretation of the world around you.

It is not the case that the brain flips the retinal image, nor does it have to, nor are there any images in the brain like there are on the retina. It is just as meaningless to say that the retinal image is upside down in relation to the orientation of our perceptions.

Richard L. Gregory gives a nice explanation:

It is generally accepted that this does not need a special compensation mechanism because retinal images are not seen, as objects are seen […]. A compensating mechanism is not needed as they are not objects of perception but rather one stage of processing lying between objects and vision. […] When the head is tilted, the world remains upright. This extends to standing on one's head, when the retinal image is reversed and yet up and down remain normal. (Richard L. Gregory (2004): Illusions, In: The Oxford Companion to the Mind, 2nd Edition, p. 429).

Another way to grasp this issue is by the difference of physical space and phenomenal space, as Norbert Bischof does:

We thus have to distinguish a physical space, in which our body is located and the brain processes take place, and a space experience which the phenomenal world is embedded in. [… Both] cannot be localized in relation to one another. There is no superordinate coordinate system which they can be jointly fit into. They are […] »incommensurable«, which literally means there is no common scale that could be applied to both. […] (Norbert Bischof (2009): Psychologie, 2nd Edition, pp. 48–49, my translation).

Bischof goes on to state that the retinal image and the consciously experienced phenomenon can not possibly be integrated within one and the same space in the first place, and thus don't stand in any spatial relation whatsoever.


text: It is distinct from the Robotic System Toolbox (tag: robotics-system-toolbox) which is proprietary software developed and maintained by the MathWorks.


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Poster
Root Cause Analysis of Failures in Microservices through Causal Discovery
Azam Ikram · Sarthak Chakraborty · Subrata Mitra · Shiv Saini · Saurabh Bagchi · Murat Kocaoglu

Tue Nov 29 09:00 AM -- 11:00 AM (PST) @ Hall J #506
Most cloud applications use a large number of smaller sub-components (called microservices) that interact with each other in the form of a complex graph to provide the overall functionality to the user. While the modularity of the microservice architecture is beneficial for rapid software development, maintaining and debugging such a system quickly in cases of failure is challenging. We propose a scalable algorithm for rapidly detecting the root cause of failures in complex microservice architectures. The key ideas behind our novel hierarchical and localized learning approach are: (1) to treat the failure as an intervention on the root cause to quickly detect it, (2) only learn the portion of the causal graph related to the root cause, thus avoiding a large number of costly conditional independence tests, and (3) hierarchically explore the graph. The proposed technique is highly scalable and produces useful insights about the root cause, while the use of traditional techniques becomes infeasible due to high computation time. Our solution is application agnostic and relies only on the data collected for diagnosis. For the evaluation, we compare the proposed solution with a modified version of the PC algorithm and the state-of-the-art for root cause analysis. The results show a significant improvement in top-$k$ recall while significantly reducing the execution time.

#### Author Information

##### Saurabh Bagchi (Purdue University)

Saurabh Bagchi is a Professor in the School of Electrical and Computer Engineering and the Department of Computer Science at Purdue University in West Lafayette, Indiana. He is the founding Director of a university-wide resilience center at Purdue called CRISP (2017-present). He is the recipient of the Alexander von Humboldt Research Award (2018), the Adobe Faculty Award (2017), the AT&T Labs VURI Award (2016), the Google Faculty Award (2015), and the IBM Faculty Award (2014). He was elected to the IEEE Computer Society Board of Governors for the 2017-19 term and re-elected in 2019. He is an ACM Distinguished Scientist (2013), a Senior Member of IEEE (2007) and of ACM (2009), and a Distinguished Speaker for ACM (2012). He is a co-lead on the \$39M WHIN-SMART center at Purdue. Saurabh's research interest is in dependable computing and distributed systems. He is proudest of the 21 PhD students and 50 Masters thesis students who have graduated from his research group and who are in various stages of building wonderful careers in industry or academia. In his group, he and his students have way too much fun building and breaking real systems. Along the way this has led to 10 best paper awards or nominations at IEEE/ACM conferences. Saurabh received his MS and PhD degrees from the University of Illinois at Urbana-Champaign and his BS degree from the Indian Institute of Technology Kharagpur, all in Computer Science. He was selected as the inaugural International Visiting Professor at IIT Kharagpur in 2018.