text:
# Rotating a circle about a larger circle

I've got some wonderful answers this morning on Mathematica Stack Exchange. See Diagonals of a regular octagon and Determine the height of a box packed with spheres. I'm just smiling with the amount I am learning with these responses, so I'd like to ask one more question.

The diagram below shows the circular face of a clock with radius 20 cm and a circular disk with radius 10 cm externally tangent to the clock face at 12 o’clock. The disk has an arrow painted on it, initially pointing in the upward vertical direction. Let the disk roll clockwise around the clock face. At what point on the clock face will the disk be tangent when the arrow is next pointing in the upward vertical direction?

How can I use Mathematica to animate the smaller circle with the arrow rotating about the large circle to show students the answer?

My adaption from the Plotting the Epicycloid

R = 20; r = 10;
fy[\[Theta]_, a_: 1] := (R + r) Cos[\[Theta]] +
a r Cos[(R + r) \[Theta]/r];
fx[\[Theta]_, a_: 1] := (R + r) Sin[\[Theta]] +
a r Sin[(R + r) \[Theta]/r];

plot[max_] :=
ParametricPlot[{fx[\[Theta]], fy[\[Theta]]}, {\[Theta], 0, max},
PlotStyle -> {Red, Thick},
Epilog -> {
Thick,
Blue, Circle[{0, 0}, R],
Black, Circle[{fx[max, 0], fy[max, 0]}, r],
Red, PointSize[Large], Point[{fx[max], fy[max]}],
Green, PointSize[Large], Point[{fx[max, -1], fy[max, -1]}],
Black, Line[{{fx[max, 0], fy[max, 0]}, {fx[max], fy[max]}}]
},
PlotRange -> 40]

Manipulate[
plot[max], {max, 0.000001, 2 Pi}]

Looks like the answer is 4 pm. Sorry about the image, as the number in the entry box should show $2\pi/3$, but it is blocked because of size after I typed it in.

Also, that is what the solution is as posted at The Art of Problem Solving.

• Look up Epicycloid and related demonstrations. – Michael E2 Sep 26 '15 at 19:25
• Look at this Mathematica.SE Q&A. – C. E. Sep 26 '15 at 19:27
• On the face of it, this seems a relatively straightforward task. Are you asking only that somebody do it for you? Or are you running into a more specific problem? – Michael E2 Sep 26 '15 at 19:27
• @Pickett Isn't it close enough to be a duplicate? – Michael E2 Sep 26 '15 at 19:29
• @MichaelE2 Sure, done. – C. E. Sep 26 '15 at 19:32

Just for fun:

Setup:

clock = Graphics[{Circle[{0, 0}, 2],
Table[{Text[ Mod[3 - 6 j/Pi, 12, 1], 1.65 {Cos[j], Sin[j]}],
Line[# {Cos[j], Sin[j]} & /@ {1.8, 2}]}, {j, 0, 23 Pi/12,
Pi/6}], PointSize[0.02], Point[{0, 0}]}];
epi[t_] := 3 {Sin[t], Cos[t]} + {Sin[3 t], Cos[3 t]}

Visualization:

Manipulate[
Show[ParametricPlot[epi[t], {t, 0, 2 Pi},
Epilog -> {Red, PointSize[0.02], Point[epi[j]]},
PlotStyle -> Dashed], clock,
Graphics[{Circle[3 {Sin[j], Cos[j]}, 1],
Text[Rotate[Style["\[DoubleUpArrow]", 40], -3 j],
3 {Sin[j], Cos[j]}]}], Axes -> None,
PlotRange -> {{-4, 4}, {-4, 4}}], {j, 0, 2 Pi}]

• Absolutely wonderful! What fun! I will definitely share this with all our participants at our activity. It so clearly shows the solution. Thanks. :-) – David Sep 27 '15 at 16:28
• For reference: the specific epicycloid being traced here is called a nephroid. – J. M. will be back soon Oct 8 '15 at 14:30

I think you are right, there is a mistake in the reasoning of my now deleted answer. However, I thought some the cosmetic code I wrote for the failed answer might still be of interest to you, so I integrated your code with the cosmetic elements of mine.

R = 20; r = 10;
fy[θ_, a_: 1] := (R + r) Cos[θ] + a r Cos[(R + r) θ/r];
fx[θ_, a_: 1] := (R + r) Sin[θ] + a r Sin[(R + r) θ/r];

origin = {0, 0};
center = {PointSize[Large], Point[origin]};
scale = Range[12];
scalePts = R {Sin @ #, Cos @ #}& /@ N[(30 scale) °];
ticks = Line @ ({#, .9 #}& /@ scalePts);
lbls = Thread[Text[Style[#, 14]& /@ scale, .8 scalePts]];

plot[max_] :=
ParametricPlot[{fx[θ], fy[θ]}, {θ, 0, max},
PlotStyle -> {Red, Thick},
Axes -> False,
PlotRange -> 40]

Manipulate[
Block[{θ = max °},
Show[
If[θ > 0., plot[θ ], Graphics[{}, PlotRange -> 40]],
Graphics[{
Thick, Blue, Circle[origin, R],
Black, center, lbls, ticks, Circle[{fx[θ, 0], fy[θ, 0]}, r],
Red, PointSize[Large], Point[{fx[θ], fy[θ]}],
Green, PointSize[Large], Point[{fx[θ, -1], fy[θ, -1]}],
Black, Line[{{fx[θ, 0], fy[θ, 0]}, {fx[θ], fy[θ]}}]}],
{max, 0., 360., 5., Appearance -> "Labeled"}]

With these changes, your demonstration will look like this

which is more in line with the top image of your question. I hope you find some use for this in your classroom.

• Well it is an epicycloid after all -- what did you mean? Just that you didn't use the word? It still seems a duplicate to me, and David's code you use even comes from Pickett's answer to the duplicate question. – Michael E2 Sep 27 '15 at 3:40

### Major edit

Yes, it is possible to save the phenomena and work only with rotating circles, eschewing any reference to epicycles. (Of course, they are still there.)

I will do it in two steps.

1. A helper function that will generate the graphics primitives for the rolling circle.

roller[r1_, r2_, angle_] :=
Module[{r3 = r1 + r2, center},
center = r3 {Sin @ angle, Cos @ angle};
{Circle[center, r2],
Rotate[Arrow[{center - {0, r2}, center + {0, r2}}, .2], -angle r3/r2]}]

Note that the only change from my previously posted, wrong version is to substitute r3 for r1. roller will work for cicles of any radius, r1 > r2

2. An Manipulate expression that draws the larger circle in the background and uses the helper to rotate the smaller circle.

Manipulate[
With[{r1 = 2., r2 = 1.},
Module[{origin, halfW, center, scale, scalePts, ticks, lbls},
origin = {0, 0};
halfW = r1 + 2 r2;
center = {PointSize[Large], Point[origin]};
scale = Range[12];
scalePts = r1 {Sin @ #, Cos @ #} & /@ N[(30 scale) °];
ticks = Line @ ({#, .925 # } & /@ scalePts);
lbls = Thread[Text[Style[#, 14] & /@ scale, .8 scalePts]];
Graphics[{roller[r1, r2, θ °]},
PlotRange -> {{-#, #}, {-#, #}} &[halfW],
Prolog -> {Circle[origin, r1], center, lbls, ticks}]]],
{θ, 0., 360., 5., Appearance -> "Labeled"},
SaveDefinitions -> True]

The above code produces a demonstration that looks like this as it hits the four o'clock point.

• Wonderful! I want to thank you for your effort. – David Sep 29 '15 at 2:28

format: plain


text:
# What is [HCl] if a 25*mL volume of [NaOH] of 0.250*mol*L^-1 concentration reaches an endpoint with a 22.5*mL volume of the acid?

Jul 8, 2017

You mean to start with a solution of $N a O H \left(a q\right)$

#### Explanation:

$N a O H \left(a q\right)$ reacts with $H C l \left(a q\right)$ with 1:1 stoichiometry.......

$N a O H \left(a q\right) + H C l \left(a q\right) \rightarrow N a C l \left(a q\right) + {H}_{2} O \left(l\right)$

Given that $\text{Concentration"="Moles of titrant"/"Volume of titrant}$ we can work out the molar equivalence......

$\text{Moles of NaOH}$ $=$ $0.250 \cdot m o l \cdot {L}^{-} 1 \times 25 \times {10}^{-} 3 \cdot L = 6.25 \times {10}^{-} 3 \cdot m o l$, and given the 1:1 stoichiometry, this was also the number of moles of base neutralized.

This molar quantity was equivalent to the amount of moles contained in $22.5 \cdot c {m}^{3}$ $H C l$ titrant..........so..........

$\text{Concentration}$ $=$ $\frac{6.25 \times {10}^{-} 3 \cdot m o l}{22.5 \times {10}^{-} 3 \cdot L}$

$\left[H C l\right] = 0.278 \cdot m o l \cdot {L}^{-} 1$.........

Note that $1 \cdot L = 1000 \cdot c {m}^{3} = 1000 \cdot m L$...........


text:
Article | Open | Published:

Tunable Mid IR focusing in InAs based semiconductor Hyperbolic Metamaterial

Abstract

Noble Metals such as Gold and Silver demonstrated for mid IR metamaterials have suffered many obstacles such as: high losses and lack of tunability. The application of doped semiconductors has allowed overcoming the tunability restriction, besides, possessing lower losses as compared to metals. In addition, doped semiconductors have small magnitude of negative real permittivity which is required to realize mid IR Hyperbolic Metamaterials (HMMs). We theoretically demonstrate super focusing based on an all Semiconductor planar HMM using InAs heterostructure. By applying a single slit integrated with doped InAs/InAs HMM, incident light can be coupled to high propagation wave vectors of the HMM modes leading to sub diffraction focusing within the mid IR wave length range. Our proposed structure shows a wide controllable/ tunable operation by changing the doping concentration of InAs. As a consequence, focusing resolution can be tuned over the mid IR ranging from 4.64 μm to 19.57 μm with the maximum achieved resolution is up to 0.045λ at an operating wavelength of 19.57 μm. In addition, we show the effect of substrate refractive index on tuning and enhancing the focusing resolution. Our proposed HMM is an all single based material in which it will not suffer lattice mismatch restrictions during fabrication.

Introduction

Conventional optical lenses’ performance is known to be constrained by the diffraction limit. This leads to image resolution that is at most about half of the incident wavelength1. The diffraction limited imaging is attributed to the fact that evanescent waves holding detailed information about the imaged object cannot propagate in conventional materials. Therefore, they do not contribute to the formation of the final image. In recent years, the metal-dielectric multilayered structure, hyperbolic metamaterial (HMM), has attracted significant interest for its potential to display dielectric property (ε > 0) in two dimensions while exhibiting metallic properties (ε < 0) in the third dimension2,3, i.e., hyperbolic dispersion relation. This anisotropy has paved the way for various research topics and applications. HMMs have shown negative refraction of light4,5, far field sub wavelength imaging6, and sub diffraction focusing7,8. Due to their unusual dispersion, HMMs are capable of coupling evanescent waves with high spatial frequency components of free space into propagating modes of HMMs2,7,8 and consequently they offer a very promising avenue to surpass the diffraction resolution limit. When covered with a Fresnel zone plate or a diffraction grating that can efficiently excite waves with high wave vectors, subwavelength focusing have been achieved in many HMMs7,8,9,10,11,12,13. Light focusing has been reported in HMM either by negative refraction of light10 or single aperture on HMM7,8. These studies have shown focusing from microwave range7 to UV range11. Focusing in the mid IR range have been reported using metasurfaces or sub-wavelength grating coupler with slight focusing resolution12,13,14,15. A study has shown that hexagonal naturally accruing hydrogen boron nitride crystal with hyperbolic dispersion momentum space has focusing resolution down to $${\rm{0}}\mathrm{.03}{\rm{\lambda }}$$ in the mid IR range16.

On the other hand, common metals such as Gold or Silver that are inherently essential ingredients in building HMMs have their natural plasmon resonance in the optical or deep ultra-violet wavelength range. There are no available metals whose plasmon resonance located within the near or mid-infrared (mid-IR) range, which is an extremely important wavelength range for various detection and sensing applications17. It should be pointed out here that in order to realize hyperbolic dispersion, the magnitude of the real part of metal permittivity and that of the dielectric need to be of small contrast4,18. At mid-IR frequencies, the magnitude of the real permittivity of those metals tends to infinity while there is no dielectric that exhibits such high permittivity value within the same wave length range. This reflects the fact that at these frequencies the electromagnetic properties of noble metals almost resemble those of perfect conductors. This effectively precludes subwavelength metallic structures from supporting localized surface plasmon (LSP) and hence defies any straightforward trials to achieve hyperbolic dispersion in the mid-IR range. Worth mentioning here is that the plasmonic wavelength for a given metal (besides being far away from the mid-IR) is naturally fixed. Hence, having a highly controllable/tunable HMM that is necessary for many applications is not an easy task when using Noble metals4,19,20. Thus, it is very important to look into other alternatives to metals that exhibit some controllability and tunability of the plasma wavelength. A perfect candidate for such purpose is moderately or highly doped semiconductors21. Many studies have reported the use of either doped semiconductors or graphene for tunable mid IR metamaterials21,22,23,24,25. Tuning the Hyperbolic dispersion was studied through inducing field effect in ITO26 or by changing the thickness of the dielectric layers in HMM27. In this work, we demonstrate an all-semiconductor HMM structure that overcomes the above mentioned challenges and achieve subwavelength focusing in the mid IR range using InAs heterostructures. Doped InAs semiconductor offers the shortest plasmon wavelength with relatively wide tunable plasma wave length28. Tuning focusing resolution over the mid IR wavelength range through changing the doping concentration in HMMs has not (to the best of our knowledge) been investigated so far. In this work, we show that the focusing resolution is tunable from 4.64 μm to 19.57 μm by changing the doping concentration. The maximum achieved resolution is up to $${\rm{0}}\mathrm{.045}{\rm{\lambda }}$$ at an operating wavelength of 19.57 μm. Finally, we show the effect of substrate refractive index on tuning the focusing resolution. Our proposed doped InAs/InAs HMM can serve as good application for thermal harvesting and/or for subwavelength thermal imaging.

Results

The structure consists of 20 alternating layers of N-doped InAs acting as the ‘metallic’ layer and undoped InAs as the dielectric layer with negative and positive real parts of the permittivity respectively as shown in Fig. 1. Each layer is of thickness 50 nm. A 500 nm Copper layer with 1 μm wide slit was introduced on top of the HMM. The whole structure is supported on Si substrate. Finite difference time domain (FDTD) Lumerical has been used for simulating the proposed HMM. A TM polarized Gaussian source was incident from the top of the structure with numerical aperture (NA) of 0.2. The propagation wave vector was defined along the z axis. Perfect matched layer (PML) was defined along the x and z directions.

Since the dimensions are sub-wavelength, effective medium approximation (EMA) can be used to study and predict the dispersion behavior of the proposed HMM29. Proper doping concentration was chosen to guarantee negative permittivity in the perpendicular direction. The EMA has been used as follows; Effective permittivity in the perpendicular and the parallel directions are given by $${\varepsilon }_{\perp }={\varepsilon }_{z} < 0$$ and $${\varepsilon }_{\parallel }={\varepsilon }_{x} > 0$$ respectively.

$${\varepsilon }_{\perp }=\frac{{\varepsilon }_{m}{\varepsilon }_{d}}{{f}_{1}{\varepsilon }_{m}+{f}_{2}{\varepsilon }_{d}}$$
(1)
$${\varepsilon }_{\parallel }={f}_{1}{\varepsilon }_{m}+{f}_{2}{\varepsilon }_{d}$$
(2)

Where $${\varepsilon }_{m}$$ and $${\varepsilon }_{d}$$ are the dielectric permittivities of doped InAs and intrinsic InAs respectively. $${f}_{1}$$ and $${f}_{2}$$ are the filling ratios of doped InAs and InAs respectively which according to the design dimensions are found to be 0.5 in all simulations. The permittivity of doped InAs $${\varepsilon }_{doped}$$ for different doping concentration Nd was calculated using Drude model as follows.

$${\varepsilon }_{doped}={\varepsilon }_{\infty }-\frac{{\omega }_{p}^{2}}{1+i{\omega }^{2}{\rm{\Gamma }}}$$
(3)
$${\omega }_{p}^{2}=\frac{{N}_{d}\,.{q}^{2}}{{\varepsilon }_{0}{m}^{\ast }({N}_{{\rm{d}}})}$$
(4)

Where ωp is the plasma frequency, Γ is the damping term and m*(Nd) is the effective mass as function of doping concentration. Experimentally obtained values for epitaxial grown doped InAs for five doping concentrations (scattering rate and mobility) were extracted from previous literature30. For doped InAs, effective mass is obviously highly dependent on the doping concentration30,31. For effective mass calculations, equation (5) is used where ∆E is the band gap calculated by empirical model as follows31:

$${\rm{\Delta }}E=(\frac{{h}^{2}}{2{m}^{\ast }({N}_{d})}){(\frac{3{N}_{d}}{8\pi })}^{\frac{2}{3}}$$
(5)

For N-doping of Nd = 1 × 1019 cm−3, the plasma wavelength occurs at 10.1 μm. Figure 2a shows the hyperbolic dispersion for type one HMM where $${\varepsilon }_{\perp } < 0$$ and $${\varepsilon }_{\parallel } > 0$$ for 10.1 μm < λ < 15.3 μm. The wavelength of operation that is marked by black circle in Fig. 2a was chosen (in addition to exhibiting HMM behavior) to have low imaginary parts of the permittivity in both axes to minimize the system losses. FDTD simulation was first conducted for effective bulk structure of HMM using permittivity calculated from EMA to verify the design performance. The operating wavelength was chosen to be 11.56 μm, at which $${\varepsilon }_{\perp }=-8.6+i3.6$$, and $${\varepsilon }_{\parallel }=6+i0.5$$. Figure 2b shows the electric field distribution $${|E|}^{2}$$ in the XZ plane where focusing is clearly observed in the HMM along the z direction. When light propagates in HMM, high spatial K wave vectors can be excited. Diffraction induced by the existent slit results in interference of these high K wave vectors where they propagate along a definite direction. Figure 2c shows the electric field intensity $${|E|}^{2}$$ which confirms sub-wavelength focusing of incident light in the effective bulk HMM.

Numerical simulation and Discussion

In order to confirm focusing, full wave simulation was performed for the entire structure. Doped InAs has tunable plasma wavelength from 4.64 μm to 17.2 μm. Tuning the plasma wavelength will allow focusing at multiple wavelengths over the mid IR range. A layer of perfect electrical conductor with a slit was used instead of Cu which serves as good approximation for metal of very high negative permittivity.

Five doping concentrations were studied as shown in Table 1. It is shown that for larger doping concentrations, the plasma wavelength is blue shifted. Figure 3 shows out of plane $${\varepsilon }_{\perp }$$ (a – b) and in-plane $${\varepsilon }_{\parallel }$$ (c – d) effective permittivity for different doping concentrations as function of wavelength. It could be observed that for each doping, there is a certain wavelength range that exhibits type one hyperbolic dispersion (where $${\varepsilon }_{\perp } < 0$$ and $${\varepsilon }_{\parallel } > 0$$). Figure 4 shows the electric field distribution $${|E|}^{2}$$ in multilayer doped InAs/InAs HMM at wavelengths 19.57 μm, 11.87 μm, 7.29 μm, 6.5 μm and 5.23 μm for Nd of 2.7 × 1018, 1 × 1019, 3.25 × 1019, 4.4 × 1019 and 7.5 × 1019 cm−3 respectively.

It can be clearly seen that focusing is achievable for all doping concentrations along defined directions with slight variation in focal length. At wavelength 19.57 µm, the focused field is sharp and confined. For lower wavelengths, the field becomes less sharp where it exhibits the least confinement at the smallest studied wavelength of 5.23 μm. This indicates that as the wavelength becomes smaller with respect to the HMM dimensions, the focusing resolution is degraded. The discontinuity within the electric field in HMM layers can be attributed to the electromagnetic field boundary conditions. The tangential component of the electric field is continuous across each metal/dielectric interface in the parallel direction to HMM layer2. In the perpendicular direction, the electric displacement vector is continuous. The electric field will however vary from the metallic to the dielectric layers due to the variation of the permittivity in both layers. Thus, the electric field will experience discontinuity in the perpendicular direction of the HMM to compensate for the change in permittivity. Figure 4b shows the Electric field distribution $${|E|}^{2}$$ for Nd of 1 × 1019 cm−3 and at λ of 11.87 μm which confirms focusing down to 0.049λ as previously demonstrated by EMA analysis. The best achievable focusing is 0.045λ at 19.57 μm. These values are summarized in table one for the five studied doping concentrations.

As mentioned above, focusing resolution keeps degrading for larger doping concentrations. This can be attributed to the fact that increasing doping concentration enhances the metallic character in the material. Enhancing the excitation of surface plasmon polariton across each metal/dielectric interface increases the dissipation of energy and thus reducing the focusing resolution. This can be clearly seen from the imaginary part of the parallel permittivity in Fig. 3d where damping losses increase for higher doping concentration. Also, the value of the wave length with respect to the slit dimension affects the focusing resolution that we define throughout the manuscript as $$\frac{FWHM}{\lambda }$$. It is worth mentioning here that, for certain doping concentration, focusing does not occur at just particular wavelength but over range of wavelengths. The wavelength with the best focusing was selected to calculate focusing resolution and depicted in Table 1. To illustrate this, each metal slit generate high K modes due to the effect of diffraction generated at the two slit edges. At certain angle, these high k modes could interfere perfectly at an angle $$\tan \,\theta =\frac{\varepsilon //}{\varepsilon \perp }$$ with respect to the normal (perpendicular to HMM surface)7,32. The direction of propagation is dependent on the choice of the permittivity tensor components at specific wave length that causes the focusing effect. Therefore, perfect interference will be occurring at certain permittivity correspondent to certain wavelength which will have better focusing compared to other wavelengths.

Tuning the focusing resolution could be achieved by changing the doping concentration as shown above. Li,G. et al. showed that tuning the FWHM could be done by changing the surrounding medium for layered HMM8. Here, we study the effect of the substrate refractive index on tuning the focusing resolution. We compared four different substrates Ge, Si, GaAs and Al2O3 with different refractive indices of 4, 3.42, 3.2, and 1.47 respectively. We choose Nd of 1 × 1019 cm−3 to compare the focusing resolution for the four different substrates. Figure 5b shows that Al2O3 has the least focusing resolution of $$0.0986\lambda$$ and the focusing is enhanced by increasing the refractive index of the surrounding medium. The maximum achieved resolution is for the highest index substrate Ge which is $$0.0457\lambda$$ that is even better than our previously studied HMM supported on Si.

Conclusion

We theoretically show that sub-diffraction focusing can be achieved in the mid IR range using doped InAs/InAs HMM from 4.64 μm to 19.57 μm. Tuning the operating wavelength and the focusing resolution is achieved by changing the doping concentration. The maximum resolution reached in our study is $$0.045\lambda$$. We also demonstrate that further tuning of the focusing resolution is achievable upon using high refractive index substrates. Finally, choosing material of lower losses and being tunable opens the way to realize focusing over the mid IR range with extremely high resolution limit. In addition, planar hyperbolic metamaterials are easy to fabricate specially that our proposed HMM design is an all single based material and will be immune to lattice mismatch constrains. Focusing in the mid IR wave length range with sub-wavelength resolution limit is an efficient way to realize thermal harvesting and/or thermal imaging applications.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

References

1. 1.

Lee, H., Liu, Z., Xiong, Y., Sun, C. & Zhang, X. Development of optical hyperlens for imaging below the diffraction limit. Opt. Express. 15, 15886–15891 (2007).

2. 2.

Shekhar, P., Atkinson, J. & Jacob, Z. Hyperbolic metamaterials: fundamentals and applications. Nano convergence. 1, 14 (2014).

3. 3.

Drachev, V. P., Podolskiy, V. A. & Kildishev, A. V. Hyperbolic metamaterials: new physics behind a classical problem. Optics express. 21, 15048–15064 (2013).

4. 4.

Naik, G. V., Liu, J., Kildishev, A. V., Shalaev, V. M. & Boltasseva, A. Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials. PNAS. 109, 8834–8838 (2012).

5. 5.

Hoffman, A. J. et al. Midinfrared semiconductor optical metamaterials. J. Appl. Phys. 105, 122411 (2009).

6. 6.

Kildishev, A. V. & Narimanov, E. E. Impedance-matched hyperlens. Optics letters. 32, 3432–3434 (2007).

7. 7.

Kaur, A. et al. Deep subwavelength power concentration-based hyperbolic metamaterials. Int. J. Opt. 2012, 879392 (2012).

8. 8.

Li, G., Li, J. & Cheah, K. W. Subwavelength focusing using a hyperbolic medium with a single slit. Appl. Opt. 50, G27–G30 (2011).

9. 9.

Ma, C. & Liu, Z. Designing super-resolution metalenses by the combination of metamaterials and nanoscale plasmonic waveguide couplers. J. Nanophotonics. 5, 051604 (2011).

10. 10.

Fang, A., Koschny, T. & Soukoulis, C. M. Optical anisotropic metamaterials: Negative refraction and focusing. Phys. Rev. B - Condens. Matter Mater. Phys. 79, 946–950 (2009).

11. 11.

Pan, R., Tang, Z., Pan, J. & Peng, R. Tuning subwavelength-structured focus in the hyperbolic metamaterials. Proc of SPIE. 10154, 101541U (2016).

12. 12.

Abdel-galil, M. & Ismail, Y. Subwavelength Focusing In The Infrared Range Using A Meta Surface. ACES conference, https://doi.org/10.23919/ROPACES.2017.7916019 (2017)

13. 13.

Cheng, Z. et al. Focusing subwavelength grating coupler for mid-infrared suspended membrane waveguide. Optics letters 37, 1217–1219 (2012).

14. 14.

Huang, Z., Bai, X., Zhan, P. & Liu, Y. Dual-Band Light Focusing Using Stacked Graphene Metasurfaces. Acsphotonics. 42, 2094–2097 (2017).

15. 15.

Akın, O. & Demir, H. V. High-efficiency low-crosstalk dielectric metasurfaces of mid-wave infrared focal plane arrays. APL. 110, 143106 (2017).

16. 16.

Dai, S. et al. Subdiffractional focusing and guiding of polaritonic rays in a natural hyperbolic material. ncomm 6, 6963 (2015).

17. 17.

Boltasseva, A. & Atwater, H. Low loss plasmonic metamaterials. Science. 331, 290–291 (2011).

18. 18.

Wei, D. et al. Single-material semiconductor hyperbolic metamaterials. Opt. Express 24, 8735–8745 (2016).

19. 19.

Zhong, Y., Malagari, S. D., Hamilton, T. & Wasserman, D. Review of mid-infrared plasmonic materials. J. Nanophotonics. 9, 093791 (2015).

20. 20.

Naik, G. V. & Boltasseva, A. A comparative study of semiconductor-based plasmonic metamaterials. Metamaterials 5, 1–7 (2011).

21. 21.

Naik, G. V., Shalaev, V. M. & Boltasseva, A. Alternative Plasmonic Materials: Beyond Gold and Silver. Advanced optical materials. 25, 3264–3294 (2013).

22. 22.

Low, T. & Avouris, P. Graphene Plasmonics for Terahertz to Mid-Infrared Applications. Acs nano. 8, 1086 (2014).

23. 23.

Fan, Y. et al. Tunable mid-infrared coherent perfect absorption in a graphene meta-surface. Nat. Sci reports 5, 13956 (2015).

24. 24.

Shen, N. Tunable terahertz meta-surface with graphene cut wires. ACS Photonics 2, 151 (2014).

25. 25.

Fan, Y. et al. Electrically Tunable Goos – Hänchen Effect with Graphene in the Terahertz Regime. Adv. Opt. Mater. 4, 1824 (2016).

26. 26.

Papadakis, G. T. & Atwater, H. A. Field-effect induced tunability in hyperbolic metamaterials. Phys. Rev. B - Condens. Matter Mater. Phys. 92, 184101 (2015).

27. 27.

Gric, T., Cada, M. & Pistora, J. Propagation of surface waves formed at the interface between hyperbolic metamaterial and highly doped semiconductor. Opt. Quantum Electron. 48, 237 (2016).

28. 28.

Law, S., Podolskiy, V. & Wasserman, D. Towards nano-scale photonics with micro-scale photons: The opportunities and challenges of mid-infrared plasmonics. Nanophotonics 2, 103–130 (2013).

29. 29.

Lalanne, P. & Lemercier-Lalanne, D. On the effective medium theory of subwavelength periodic structures. J. Mod. Opt. 43, 2063–2086 (1996).

30. 30.

Law, S., Adams, D. C., Taylor, A. M. & Wasserman, D. Mid-infrared designer metals. Opt. Express 20, 12155–12165 (2012).

31. 31.

ElSayed, M. Y., Ismail, Y. & Swillam, M. A. Semiconductor Plasmonic Gas Sensor Using On-Chip Infrared Spectroscopy. Applied physics (A) 123, 113 (2016).

32. 32.

Kannegulla, A. & Cheng, L. Subwavelength focusing of terahertz waves in silicon hyperbolic metamaterials. Opt. Lett. 41, 3539 (2016).

Author information

Affiliations

1. Department of Physics, School of Sciences and Engineering, The American University in Cairo, Cairo, 11835, Egypt

• Mai Desouky
• , Ahmed M. Mahmoud
•  & Mohamed A. Swillam
2. Electronics and Communications Engineering Department, The American University in Cairo, Cairo, 11835, Egypt

• Ahmed M. Mahmoud

Contributions

M.D. and A.M.M. designed the structure. M.D. conducted the numerical calculations and theoretical modelling. M.S. proposed, directed and supervised the project. All authors have contributed to writing the manuscript.

Competing Interests

The authors declare that they have no competing interests.

Corresponding author

Correspondence to Mohamed A. Swillam.


text:
# Odd digits of $2^n$

Let $u_{b}(n)$ be equal to to number of odd digits of $n$ in base $b$.

For example:
In base $10$, $u_{10}(15074) = 3$
In base $13$, $u_{13}(15610) = u_{13}([7, 1, 4, 10]_{13}) = 2$

What is the value of $$\sum_{n=1}^\infty \frac{u_b(2^n)}{2^n}$$ I don't think there's a nice closed form solution in the general case.
But when the base is even, it seems that the sum evaluates to $\frac{1}{b-1}$.

I have no idea how to approach it. I tried thinking about the generating function. But I can't find a recurrence relation or any nice properties.

• This is an incredible phenomenon – Stella Biderman Jan 30 '16 at 20:15

Here is my solution:

If $a=\sum_{j\geq 0} a_j b^j$, then $u_b(a) = \frac{1}{2}\sum_j (1-(-1)^{a_j})$ because $1-(-1)^m=\begin{cases} 0 & 2\mid m\\ 2 & 2\nmid m\end{cases}$. Note that $a_j$ is the digit directly left of the decimal point in the $b$-adic expansion of $ab^{-j}$, i.e. $a_j = \lfloor ab^{-j} \rfloor \mod b$. Now if $b$ is even we can therefore replace $(-1)^{a_j}$ by $(-1)^{\lfloor ab^{-j} \rfloor}$.

Thus: $$\sum_{n\geq 1} 2^{-n} u_b(2^n) = \sum_{n\geq 1,j\geq 0} 2^{-n} \frac{1}{2} (1-(-1)^{\lfloor 2^n b^{-j}\rfloor})$$ Observe that for $j=0$ we always have $2^n b^{-j} \in 2\mathbb{N}$ so that all summands for $j=0$ vanish. Thus we assume $j\geq 1$ from now on.

Now we have to think about $2^n b^{-j}$. This suggests looking at the $2$-adic expansion of $b^{-j}$: $$b^{-j} =\sum_{n\geq 1} d_n 2^{-n}$$ for some $d_n\in\{0,1\}$ so that $\lfloor 2^n b^{-j} \rfloor \mod 2 = d_n$.

The nice thing about the digits $d$ of any $2$-adic expansion is that $\frac{1}{2}(1-(-1)^d)$ not just indicates whether or not $d$ is odd, it is actually equal to $d$, because $d\in\{0,1\}$. Therefore $\frac{1}{2}(1-(-1)^{\lfloor 2^{n} b^{-j} \rfloor})=\frac{1}{2}(1-(-1)^{d_n})=d_n$.

Thus: $$\sum_{j\geq 1} \sum_{n\geq 1} 2^{-n} \frac{1}{2}(1-(-1)^{\lfloor 2^n b^{-j} \rfloor}) = \sum_{j\geq 1} \sum_{n\geq 1} 2^{-n} d_n = \sum_{j\geq 1} b^{-j} = \frac{1}{b-1}$$ and that's what we wanted to prove.

• That was an amazing proof. Much more simple than I imagined. – Kitegi Jan 31 '16 at 15:57


text:
Brendan Harmon

# Hydrology in GRASS GIS

A tutorial on hydrological modeling and simulation in GRASS GIS.

Contents

## Hydrological Modeling

This tutorial introduces hydrological modeling in GRASS GIS using r.watershed, r.sim.water, and r.lake. Learn more about hydrology in GRASS on the grasswiki page. GRASS GIS includes many modules and addons for hydrological modeling and analysis including:

This tutorial uses the Governor’s Island Dataset for GRASS GIS. Download, extract, and move this geospatial dataset for Governor’s Island in New York City to your grassdata directory. Start GRASS GIS, set the GRASS GIS database directory to grassdata directory, select nyspf_governors_island as your location, and create a new mapset called hydrology.

## Terrain Modeling

Zoom in on the landforms in the southwest of the island. Either set the computation region from the display using the various zoom options dropdown or run g.region and set the boundaries for the region. Save the region. Then set a mask to the vector map shoreline with the module r.mask. The digital elevation model from lidar has substantial noise that would impact hydrological simulations. To reduce this noise smooth the digital elevation model using the module r.neighbors with a moving window size of 5. Optionally make the moving window circular with flag -c.

Visualize the relief of the terrain using direct illumination with r.relief. The light source can be set using the altitude and azimuth settings. Then drape the relief map over the smoothed digital elevation model with r.shade. Alternatively try visualizing relief using diffuse illumination from the skyview factor - the proportion of the sky visible given the surrounding relief. First install the addon module r.skyview with g.extension. Then compute the skyview factor with r.skyview from 16 directions. A composite of the shaded relief and skyview factor will combine direct and diffuse illumination to better visualize relief. Use r.shade to drape the shaded relief map over the skyview factor.

g.region n=189850 s=189100 e=978550 w=976850 save=landforms
r.neighbors -c input=elevation_2017 output=elevation size=5
r.colors -e map=elevation color=elevation
r.relief input=elevation output=relief zscale=2 units=survey
g.extension extension=r.skyview
r.skyview input=elevation output=skyview ndir=16
d.legend raster=elevation at=60,95,2,3.5 font=Lato-Regular fontsize=14

Skyview Factor
Digital Elevation Model

## Flow Accumulation

Flow accumulation - the number of cells that drain through each cell in an elevation raster - can represent the flow of water across a landscape. Compute flow accumulation with the module r.watershed. Then use r.shade to drape the flow accumulation map over the relief map.

r.watershed -a -b --overwrite elevation=elevation accumulation=flow_accumulation
d.legend -l raster=flow_accumulation at=60,95,2,3.5 font=Lato-Regular fontsize=14

Flow Accumulation

To visualize concentrated flow accumulation over the topography, use d.rast to set the range of values to display for the flow accumulation raster. To hide cells with low accumulation values, try setting the range from 100 to 1000000. Alternatively you could use map algebra with r.mapcalc to extract the concentrated flow values. Then layer this flow accumulation map on top of the shaded relief map.

d.rast map=composite_relief
d.rast map=flow_accumulation values=100-1000000

## Shallow Water Flow

Simulate shallow flows of water over the landscape with r.sim.water. First compute the partial derivatives dx and dy of the elevation raster with r.slope.aspect. Then run r.sim.water for a 10 minute rainfall event with a rainfall rate of 150 $$mm/hr$$. Set the rainfall excess rate set to 150 $$mm/hr$$, the iteration time of 10 $$min$$, and the number of walkers to 10000. Setting a higher number of walkers will reduce noise in the solution while increasing computation time. Use r.sim.water to generate maps of water depth $$(m)$$ and discharge $$(m^3/s)$$ for a 10 minute storm. Drape the depth or discharge map over the relief map with r.shade.

r.slope.aspect elevation=elevation dx=dx dy=dy
r.sim.water elevation=elevation dx=dx dy=dy rain_value=150 nwalkers=10000 depth=depth discharge=discharge
d.legend raster=depth_with_landcover at=60,95,2,3.5 font=Lato-Regular fontsize=14

Shallow Water Flow Depth $$(m)$$
Shallow Water Flow Discharge $$(m^3/s)$$

To visualize concentrated flows over the topography, use d.rast to set the range of values to display for either the depth or discharge map To hide cells with low water depth values, try setting the range from 0.03-1.

d.rast map=composite_relief
d.rast map=depth values=0.03-1

## Shallow Water Flow with Landcover

Simulate shallow overland flows of water across different types of landcover with r.sim.water. Landcover can be derived from orthophotography using unsupervised image classification, supervised image classification, or vegetation indices. For simplicity’s sake this example uses unsupervised classification. First derive landcover classes from the 2018 orthophotograph with red, green, blue, and near infrared channels using unsupervised image classification. Create a new imagery group called imagery_2018. Click add, select the PERMANENT mapset, and use the pattern imagery_2018.* to add all the channels for the 2018 orthophotograph to the imagery group. Then create a new imagery subgroup with the same name and check all of the maps. Use module i.cluster with to calculate spectral signatures for 3 landcover classes from the imagery group. Then use the module i.maxlik to classify spectral reflectance based on spectral signatures. Recode the resulting map of landcover classes as Manning’s roughness coefficients using r.recode. Grass should have a Manning’s n value of 0.368, hardscape should be 0.0404, and bare land should be 0.0113. See the appendix at the end of this tutorial for a list of suggested Manning’s n values. For r.recode create a rules file called mannings.txt with the following lines to recode class values to Manning’s n values.

1:1:0.368:0.368
2:2:0.0404:0.0404
3:3:0.0113:0.0113

Then simulate shallow water flow with spatially variable surface roughness using the module r.sim.water. Set the man parameter to your Manning’s roughness map. Drape the depth or discharge map over the relief map with r.shade.

i.cluster group=imagery_2018 subgroup=imagery_2018 signaturefile=signature classes=3
i.maxlik group=imagery_2018 subgroup=imagery_2018 signaturefile=signature output=classification
r.recode input=classification output=mannings rules=mannings.txt
r.slope.aspect elevation=elevation dx=dx dy=dy
r.sim.water elevation=elevation dx=dx dy=dy rain_value=50 man=mannings nwalkers=10000 depth=depth_with_landcover discharge=discharge_with_landcover
d.legend raster=depth_with_landcover at=60,95,2,3.5 font=Lato-Regular fontsize=14

Manning’s Roughness Coefficient
Shallow Water Flow Depth $$(m)$$ with Landcover
Shallow Water Flow Discharge $$(m^3/s)$$ with Landcover

## Shallow Water Flow with Vegetation Indices

Vegetation indices such as Normalized Difference Vegetation Index (NDVI) can be used to classify landcover and derive Manning’s roughness. Derive roughness from NDVI to simulate shallow overland water flow. First use the module i.vi to compute NDVI with the red and near infrared channels of the 2018 orthophotograph.

$NDVI = (NIR - red) / (NIR + red)$

Then recode the NDVI map as Manning’s roughness coefficients using r.recode. To recode class values to Manning’s n values either create a rules file called roughness.txt with the following values or paste these values into the r.recode dialog.

-1:-0.15:0.001:0.001
-0.15:0:0.0404:0.0404
0:0.2:0.2:0.2
0.2:1:0.368:0.368

Then simulate shallow water flow with spatially variable surface roughness using the module r.sim.water. Set the man parameter to your Manning’s roughness map derived from NDVI. Drape the depth or discharge map over the relief map with r.shade.

i.vi output=ndvi red=imagery_2018.1 nir=imagery_2018.4
d.legend raster=ndvi at=5,45,94,96 font=Lato-Bold fontsize=16
r.recode input=ndvi output=roughness rules=roughness.txt
r.slope.aspect elevation=elevation dx=dx dy=dy
r.sim.water elevation=elevation dx=dx dy=dy rain_value=50 man=roughness nwalkers=10000 depth=depth_with_ndvi discharge=discharge_with_ndvi
d.legend raster=discharge_with_ndvi at=60,95,2,3.5 font=Lato-Regular fontsize=14

Normalized Difference Vegetation Index
Shallow Water Flow Discharge $$(m^3/s)$$ with Landcover from NDVI

## Shallow Water Flow Animation

Simulate and animate water flow over time. First run r.sim.water with the -t flag to generate a time series of water depth or discharge maps. Use g.list with a search pattern of discharge.* to generate a list of all of the discharge maps in the time series. Copy the list, then run g.gui.animation to create an animation from the time series of discharge maps. In the GRASS GIS Animation Tool start by adding a new animation. In the add new animation dialog click the add space-time dataset layer button, set the data type to multiple raster maps, and then paste the list of discharge maps into the dialog box. Optionally check the show raster legend box and set the legend to the last map in the time series. After the animation renders adjust the size of the window and the animation speed and then re-render before exporting it as an animated gif.

r.sim.water -t --overwrite elevation=elevation dx=dx dy=dy depth=depth discharge=discharge nwalkers=10000 niterations=30 output_step=1
g.list type=raster pattern=discharge.* separator=comma
g.gui.animation raster=discharge.01,discharge.02,discharge.03,discharge.04,discharge.05,discharge.06,discharge.07,discharge.08,discharge.09,discharge.10,discharge.11,discharge.12,discharge.13,discharge.14,discharge.15,discharge.16,discharge.17,discharge.18,discharge.19,discharge.20,discharge.21,discharge.22,discharge.23,discharge.24,discharge.25,discharge.26,discharge.27,discharge.28,discharge.29,discharge.30

Animated Shallow Water Flow Discharge

Optionally edit the animation to add the relief raster below the series of discharge maps. Set the opacity of the discharge maps to 80%.

## Manning’s Roughness Coefficients

Manning’s n values are empirical coefficients for surface roughness. Based on literature I recommend the following n values for these types of landcover:

NLCD Class Landcover Category Manning’s n value
11 Open Water 0.001
21 Developed, Open Space 0.0404
22 Developed, Low Intensity 0.0678
23 Developed, Medium Intensity 0.0678
24 Developed, High Intensity 0.0404
31 Barren Land 0.0113
41 Deciduous Forest 0.36
42 Evergreen Forest 0.32
43 Mixed Forest 0.4
52 Shrub/Scrub 0.4
71 Grassland/Herbaceuous 0.368
81 Pasture/Hay 0.325
82 Cultivated Crops 0.325
90 Woody Wetlands 0.086
95 Emergent Herbaceuous Wetlands 0.1825


text:
# Refrigeration Questions and Answers – Air Refrigerator Working on Reverse Carnot Cycle – 1

«
»

This set of Refrigeration Multiple Choice Questions & Answers (MCQs) focuses on “Air Refrigerator Working on Reverse Carnot Cycle – 1”.

1. For a refrigerating system that works on reversed Carnot cycle having vapor as refrigerant, which one of the following is not a process of the cycle in p-v diagram?

a) Isentropic compression process
b) Isothermal compression process
c) Adiabatic expansion process
d) Isothermal expansion process
View Answer

Answer: c
Explanation: The reversed Carnot cycle has 4 processes namely, isothermal compression process, isentropic compression process, isentropic expansion process and isothermal expansion process.
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2. What is the coefficient of performance of the refrigerant system working on reversed Carnot cycle, having Qa as heat absorbed by the air during isothermal expansion per kg of air and Qr as heat rejected during isothermal compression per kg of air?
a) COP=$$\frac{Q_a}{(Q_r – Q_a)}$$
b) COP=$$\frac{Q_r}{(Q_r – Q_a)}$$
c) COP=$$\frac{Q_a}{(Q_r + Q_a)}$$
d) COP=$$\frac{Q_a}{(Q_a – Q_r)}$$
View Answer

Answer: a
Explanation: COP of refrigerator is COP of heat pump – 1
i.e. $$[\frac{Q_r}{Q_r – Q_a}] – 1 = \frac{Q_a}{(Q_r – Q_a)}$$.

3. The COP of refrigerator may be improved by _________
a) decreasing the higher (T2) as well as the lower (T1) temperature
b) decreasing the higher (T2) temperature and increasing the lower (T1) temperature
c) increasing the higher (T2) as well as the lower (T1) temperature
d) keeping the higher (T2) temperature the same while decreasing the lower (T1) temperature
View Answer

Answer: b
Explanation: The COP of refrigerator can be given by :
COP = $$\frac{T_1}{(T_2- T_1)}$$. Hence by decreasing the higher temperature T2 or by increasing the lower temperature T1 the COP may be improved.
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4. What happens to the COP of a Carnot refrigerator in summer and in winter?
a) The COP is more in winter
b) The COP is more in summer
c) The COP remains unaffected
d) The COP fluctuates continuously during winter and summer
View Answer

Answer: a
Explanation: As the higher temperature T2 will always be lower in winter when compared with that of summer i.e. the temperature of air available of heat rejection will be low, the COP in winter will be higher.

5. A refrigerating machine working on Carnot cycle operates between 300K and 250K. Determine the COP.
a) 5
b) 6
c) 4
d) 10
View Answer

Answer: a
Explanation: The COP of refrigerating machine is given by
COP = $$\frac{T_1}{(T_2- T_1)} = \frac{250}{(300-250)}$$=5.
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6. The COPr of a Carnot refrigerating machine is 7.89. What will be the COPp of heat pump?
a) 10.3
b) 7.89
c) 8.89
d) 6.89
View Answer

Answer: c
Explanation: Relation between COP of heat pump and COP of the refrigerating machine is
COPr = COPp-1
7.89 = COPp-1
COPp=7.89+1=8.89.

7. Which one is not a limitation of the Carnot cycle with air or gas as refrigerant?
a) It has low efficiency when working between two fixed temperature limits
b) Machine has to run at high and low speeds during adiabatic and isothermal process. Such variation of speed is not possible
c) Extreme pressure and large volume are developed
d) It is not possible to carry out isothermal heat transfer process in practice
View Answer

Answer: a
Explanation: Carnot cycle has the highest efficiency theoretically when working between two fixed temperature limits. Other limitations of Carnot cycle are machine has to run at high and low speeds during adiabatic and isothermal process. Such variation of speed is not possible, extreme pressure and large volume is developed, it is not possible to carry out isothermal heat transfer process in practice.
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8. For a refrigerating system that works on Carnot cycle having air or gas as refrigerant s, which one of the following is not a process of the cycle in p-v diagram?

a) Adiabatic compression process
b) Isothermal compression process
c) Adiabatic expansion process
d) Isentropic expansion process
View Answer

Answer: d
Explanation: The Carnot cycle having air or gas as refrigerant has 4 processes namely isothermal compression process, adiabatic compression process, isothermal expansion process and adiabatic expansion process.

9. What is the COP of a refrigerator when it’s refrigerating effect R is 6.2755 KW and work consumed is 0.82 KW?
a) 8.8
b) 6.77
c) 7.65
d) 8.56
View Answer

Answer: c
Explanation: COP of refrigerator = Refrigeration effect / work consumed
COP=$$\frac{R}{wc}=\frac{6.2755}{0.82}$$=7.65.
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10. Out of the following reasons which one is responsible for Carnot cycle not being used in practice?
a) It gives low COP
b) It gives high refrigeration effect
c) It gives low refrigeration effect
d) It has low theoretical efficiency
View Answer

Answer: c
Explanation: Carnot cycle is not used in everyday practice and one of the many reasons is that because it has low refrigeration effect, even though it has the highest theoretical efficiency and also it is not possible to carry out isothermal heat transfer in practice.

Sanfoundry Global Education & Learning Series – Refrigeration & Air Conditioning.

To practice all areas of Refrigeration & Air Conditioning, here is complete set of 1000+ Multiple Choice Questions and Answers.

Participate in the Sanfoundry Certification contest to get free Certificate of Merit. Join our social networks below and stay updated with latest contests, videos, internships and jobs!

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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn | Youtube | Instagram | Facebook | Twitter


text:
# Evaluate -1/(8-1/8)

Simplify the denominator.
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Subtract from .
Multiply the numerator by the reciprocal of the denominator.
Multiply by .
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Evaluate -1/(8-1/8)

### Solving MATH problems

We can solve all math problems. Get help on the web or with our math app

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text:
Cryptology ePrint Archive: Report 2018/942

Insured MPC: Efficient Secure Multiparty Computation with Punishable Abort

Carsten Baum and Bernardo David and Rafael Dowsley

Abstract: Fairness in Secure Multiparty Computation (MPC) is known to be impossible to achieve in the presence of a dishonest majority. Previous works have proposed combining MPC protocols with Cryptocurrencies in order to financially punish aborting adversaries, providing an incentive for parties to honestly follow the protocol. This approach also yields privacy-preserving Smart Contracts, where private inputs can be processed with MPC in order to determine the distribution of funds given to the contract. Unfortunately, the focus of existing work is on proving that this approach is possible and they present monolithic and mostly inefficient constructions. In this work, we put forth the first modular construction of Insured MPC'', where the result of the private computation of parties either yields an output describing how to distribute funds or a proof that a set of parties has misbehaved, allowing for financial punishments. Moreover, both the output and the proof of cheating are publicly verifiable, allowing third parties to independently validate an execution.

We present a highly efficient protocol which allows public verification of cheating behavior during the output stage. This scheme is constructed using a publicly verifiable homomorphic commitment scheme, for which we propose an efficient construction. Furthermore, we construct a compiler that uses any such scheme together with a Smart Contract to implement Insured MPC. This compiler requires a standard (non-private) Smart Contract. Our results are proven in the Universal Composability framework using a Global Random Oracle as the setup assumption. From a theoretical perspective, our general results provide the first characterization of sufficient properties that MPC protocols must achieve in order to be efficiently combined with Cryptocurrencies, as well as insights on publicly verifiable protocols. On the other hand, all our constructions and protocols are highly efficient and allow for a fast implementation.

Category / Keywords: cryptographic protocols /


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# Why is the square root of a sum not equal to the square root of each its addends?

Example: Let's presume one was attempting to isolate m below:

A common mistake would be: $k^2 = m^2 + n^2 \to k = m +n$

Even though: $k^2 = m^2 + n^2 \to k \neq m +n$

If you apply a square root to both sides of the equation, you will have an inequality.

Why is this true?

• There is no inequality. Applying the square root gives $k = \pm \sqrt{m^2 + n^2}$, which is the correct way to apply the square root. The important part is to notice that $\sqrt{m^2 + n^2}$ is not the same as $\sqrt{m^2} + \sqrt{n^2}$ – wgrenard Mar 29 '16 at 0:50
• What we need here is a cure for the “law of universal linearity”. – user856 Mar 29 '16 at 1:07
• To me it doesn't make much sense to ask "why" X doesn't equal Y when there's no reason to suggest they're equal in the first place. – arctic tern Mar 29 '16 at 2:50
• The law does hold if the domain of the variables is restricted to the integers modulo 2. – Jan Stout Mar 29 '16 at 6:06

If you put the question in a broader context you may understand it better.

Suppose $f$ is some function you can apply to numbers. It might be squaring, or square-rooting, or inverting, or raising to some other power, or taking logarithms or $\sin$ or $\cos$ or just adding 15. In none of these cases is $f(x+y)$ the same as $f(x) + f(y)$ (you should check). In general, you would not expect that coincidence.

The special case in which it is true is the function "multiply by a fixed quantity". That's the distributive law:

$$c \times (x + y ) = c \times x + c \times y .$$

Many of the most common errors students make in algebra or precalculus come from thinking that those other functions behave this way too.

Edit: Just in case you missed @Rahul 's comment: What we need here is a cure for the “law of universal linearity”: Pedagogy: How to cure students of the "law of universal linearity"?

• Explained with a simple analogy, thank you. – MacroGuy Mar 29 '16 at 1:07
• If you like the answer you can accept it, and upvote. You can upvote other answers too. – Ethan Bolker Mar 29 '16 at 1:09
• A small point: there exist non-linear functions that satisfy $$f(x+y) = f(x)+f(y)$$ for all $x,y$. This was a bit unintuitive for me. Edit: It is possible to prove that these functions exist, but probably not possible to write one down explicitly ... very strange! – Zubin Mukerjee Mar 29 '16 at 1:56
• @ZubinMukerjee True, but I won't edit "The special case" to "A special case" since the question is tagged algebra-precalculus. – Ethan Bolker Mar 29 '16 at 2:00

"Why is this true?" (that $f(x+y) \ne f(x)+f(y)$).

Because the functions that satisfy $f(x+y)=f(x)+f(y)$ and are not badly behaved (continuous will work) are all linear, so that $f(x) = cx$ for some real $c$.

So if you try $f(x) = \sqrt{x}$, you can not have $f(x+y) = f(x)+f(y)$.

There are (at least) two ways to prove this.

First, if $f(x+y) = f(x)+f(y)$, then $\sqrt{x+y} = \sqrt{x}+\sqrt{y}$, then, squaring, we get $x+y = x+y+2\sqrt{xy}$, so that $2\sqrt{xy} = 0$, so that $xy = 0$, or at least one of $x$ and $y$ is zero.

Second, if we use the result that $f(x) = cx$ for some real $c$, then $\sqrt{x} = cx$. Squaring and dividing by $x$, we get $1=c^2x$, which can not hold for any constant $c$.

Well this can be shown by the triangle inequality. Another way to look at it is for $k^2 = m^2 + n^2$ and $k = m + n$ you have that $(m+n)^2 = m^2 + n^2$ which is only true when either m or n are 0. In other words, the inequality only holds if you assume $m, n$ are nonzero.

Consider $k,m,n$ as the sides of a triangle. Then:

• $k^2 = m^2 + n^2$ means that the triangle is a right triangle.

• $k = m +n$ means that the triangle is a degenerate triangle.

Clearly, no right triangle can be degenerate!

(Unless one of the sides is zero, in which case, yes, $k^2 = m^2$ implies $k=m$.)

• You can have degenerate right triangles if one side is zero, right? – Zubin Mukerjee Mar 29 '16 at 2:07
• @ZubinMukerjee, right. I've edited my answer. – lhf Mar 29 '16 at 10:29

Otherwise $\sqrt n = n$ for all natural number $n$.

More generally $\sqrt q = q$ for all positive rational number $q$.

Homework:

Is it true that: $\sqrt x = x$ for all positive real number $x$?


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# Is it possible to re-insert a LaTeX equation by label?

Is it possible to re-insert an equation in LaTeX via referencing the original equation by label? I already know, it is possible to:

• reuse the original equation numbering via \tag{} as described here
• re-insert an equation by "boxing" the original one as said here

But isn't it possible to reuse the whole equation (i.e. its text and numbering) just using the label? Wouldn't it be natural to do so, instead of artificially creating additional labels by "boxing"? I'm feeling there must be a better way, or is my intuition wrong?

-
No it would not be natural, as you can only reuse something, if it was saved before. Usually you don't need to reuse it, so by default the equation's content is not saved. –  canaaerus Jul 18 '12 at 8:09
I disagree - the whole document is written as text file and hence persistent. It doesn't need saving, it's already saved. I still think it would be natural, or at least could be e.g. a label's attribute whether the whole equation can be reinserted later. Just a plain old academic thought. –  mmm Jul 18 '12 at 8:18
Certainly the text file is persistent. But to access the particular piece of code making up your equation, a lot of parsing has to take place. So the only reasonable approach is saving the equation — either as the original character string,the tokenized input or the final expanded version — and later accessing that saved version. –  canaaerus Jul 18 '12 at 8:28
Always right, huh? :) –  mmm Jul 18 '12 at 8:32
@mmm: the persistency of the equation in the file only helps if you repeatedly re-read the file. LaTeX can't easily do that without disrupting the typesetting, so that persistency is moot. What LaTeX does is to save labels, etc. as macros in an auxiliary file which it reads into memory precisely once (at the beginning of the document). To save every labelled equation for re-use, you could modify the labelling macro so that in a displayed math environment it saves everything until the next newline or the end of the math with the equation number. If you're up for a challenge, try it yourself! –  Niel de Beaudrap Jul 18 '12 at 10:24

As usuallay, nearly everything is possible, but what you want, does not exist yet (AFAIK). What you are asking for is a macro, which goes back to the label, copies the environment completely and expands it at another place, changing the \label into something like \tag in amsmath.
But be warned: Why offers the amsmath package only the \tag-solution? Isn't it more usual to refer to an equation by citing its number than to reprint it completely?
I know this question is old but I thought I'd give my input. It seems like the easiest solution would be to save the equation in a separate file and just \input{} the equation whenever it is needed, using whatever desired means to attach the right equation number to it. Of course this could get messy if you have a lot of equations like this. Maybe then you could save them all in a subdirectory. Not elegant but it should work.


text:
### Continuity decoded

The definition of continuity was framed after decades of deliberation and mathematical squabbling. The current notation we have is due to a Polish mathematician by the name of Weierstrass. It states that

“If $f:\Bbb{R}\to \Bbb{R}$ is continuous at point $a$, then for every $\epsilon>0$, $\exists\delta>0$ such that for $|x-a|<\delta$, $|f(x)-f(a)|<\epsilon$.”

Now let us try and interpret the statement and break it down into simpler statements, in order to give us a strong visual feel.

Can $\epsilon$ be very large? Of course! It can be $1,000,000$ for example. Does there exist a $\delta$ such that $|x-a|<\delta\implies |f(x)-f(a)|<1,000,000$, even if the function is not continuous? Yes. An example would be

$f(x)=x$ for $x\in(-\infty,a)$ and $f(x)=x+1$ for $x\in[a,\infty)$

Does this mean that we have proved a discontinuous function to be continuous? NO.

$\epsilon$ should take up the values of all positive real numbers. So $f(x)$ defined above will fail for $\epsilon$ lower than $0.000\dots01$

Let us suppose for some $\epsilon>0$, we have $|f(x)-f(a)|<\epsilon$ if $|x-a|<\delta$. Let $f(x_1)$ and $f(x_2)$ be two points in $B(f(a),\epsilon)$. Let us now make $\epsilon=\frac{|f(x_1)-f(x_2)|}{2}$. Will the value of $\delta$ also have to decrease? Can it in fact increase?

The value of $\delta$ cannot increase because the bigger interval will contain $x_1$ and $x_2$, and we know that that will violate the condition that for all points in $B(a,\delta)$, the distance between the mappings has to be less than $\frac{|f(x_1)-f(x_2)|}{2}$. Can $\delta$ remain the same? No (for the same reasons, as the interval will still contain $x_1$ and $x_2$). Hence, $\delta$ most definitely has to decrease in this case?

However, does it always have to decrease? No. An example in case is a constant function like $y=b$.

We have now come to the most important aspect of continuity. The smaller we make $\epsilon$, the smaller the value of $\delta$. Does continuity also imply that the smaller we make $\delta$, the smaller the value of $\epsilon$? YES! How? When we decrease $\delta$, $\epsilon$ obviously can’t get bigger. Moreover, we know that there do exist values of $\delta$ which make smaller $\epsilon$ possible. Say, for $|f(x)-f(a)|<\epsilon/2$, it is necessary that $|x-a|<\delta/5$. Hence, if we decrease the radius of the interval on the x-axis from $\delta$ to $\delta/5$, the value of $\epsilon$ (or the bound of the mappings of the points) also decreases to $\epsilon/2$.

In summation, a continuous function is such that

decrease in value of $\epsilon\Longleftrightarrow$ decrease in value of $\delta$

One may ask how does knowing this help?

It has become very easy to prove that differentiable functions are continuous, and a host of other properties of continuous functions.

A doubt that one may face here is does this imply that all continuous functions are differentiable? No. “decrease in value of $\epsilon\Longleftrightarrow$ decrease in value of $\delta$” just implies that the derivative formula at $a$ will have a limit for every cauchy sequence of $x$ converging to $a$. In order for a function to be derivable, all those limits of the different cauchy sequences have to be equal. This is not implied by the aforementioned condition.


text:
# Pairs and alists¶

Associative lists, or alists for shorts, are associative data structures: they map keys to values. This principles has numerous applications. For example, you might define an alist that maps names of composers to their birth dates. LilyPond has an alist that maps language names (like "english" in \language "english") to their note names. These note names are an alist mapping names to pitches. There is another alist to map alterations to glyphs in the music font. In short, there are alists everywhere.

## A word on pairs¶

Alists are built from pairs. These are compound objects which contain two elements exactly, unlike lists, which contain as many of them as wanted. Just like with lists, there are two methods to create a pair:

(cons element1 element2)

or

'(element1 . element2)

The first way is to call the cons function, of which the name is short for construct. Naturally, it expects two arguments. The second way is a form of quoting. The dot between the two elements indicates that the expression is a pair rather than a list. The display of pairs is again similar to how they are quoted: they are printed like lists, but with a dot in the middle.

(cons 1 2)
⇒ (1 . 2)

The spaces around the dot are important. '(1 . 2) is a pair, whereas '(1.2) is a one-element list containing the number “one point two”.

Quasiquoting can also be used for pairs.

(define birth 1756)
(age . ,(- 2021 birth))
⇒ (age . 265)

The two elements of a pair are accessed using the functions car and cdr.

(car '(1 . 2))
⇒ 1
(cdr '(1 . 2))
⇒ 2

It is regrettable that the names “car” and “cdr” have not been chosen to be more explicit. They are actually technical acronyms dating back to the first implementations of Lisp (they mean “Contents of the Address part of the Register” and “Contents of the Decrement part of the Register”).

## Alists¶

An alist is simply a list of pairs. The car of each pair is a key, and the cdr is the associated value. Keys are normally unique, i.e., no two pairs in the alist have the same car. However, there are legitimate cases where this rule of thumb can be waived.

Here is a first alist:

(define birth-dates
'(("Mozart" . 1756)
("Beethoven" . 1778)
("Schumann" . 1810)
("Hába" . 1893)
("Weill" . 1900)
("Ligeti" . 1923)))

Guile (reminder: the Scheme implementaton used by LilyPond) has an array of functions devoted to alists.

### Retrieving a value¶

First, here are those to retrieve a value. There is not just one, but six, all with differences! They are grouped in two families:

• assq, assv, assoc,

• assq-ref, assv-ref, assoc-ref.

Functions from the first family are called like this (using assoc as an example):

(assoc key alist)

The return value is the pair (key . value), if it exists in the alists, and the boolean #f otherwise.

Annoyingly, functions from the second family are called with these arguments swapped:

(assoc-ref alist key)

These only return the value associated with the key, rather than the pair with the key and the value.

The difference between functions from the same family is how they compare keys. All of these functions use the same algorithm. They start with the first pair in the alist, compare its car to the key being looked up, and return the cdr if the test is successful, else, they continue with the second pair, etc. As you can notice, a function is needed to compare the wanted key to keys from the alist. The assq and assq-ref functions use eq?. assoc and assoc-ref use equal?. As for assv and assv-ref, they use a third predicate, eqv?, which is much less common; it is similar to eq? but compares numbers and characters by value (though it still compares strings by identity).

Therefore, if the keys in the alist are symbols, you should use assq or assq-ref. In the general case, better use assoc and assoc-ref.

Our alist of birth dates has strings for the keys, so we are going to use assoc and assoc-ref.

(assoc "Schumann" birth-dates)
⇒ ("Schumann" . 1810)
(assoc-ref birth-dates "Schumann")
⇒ 1810
(assoc-ref birth-dates "Bartók") ; not in the alist
⇒ #f

Clearly, assoc-ref is the most practical. With assoc, you need to explicitly take the cdr of the returned pair. So why does Scheme have assoc? The necessity becomes clear when considering alists where the values can be #f. Let us consider the same set of composers, and build the alist which tells, for every of them, whether one (at least) of his parents was a musician.

(define parents-musicians
'(("Mozart" . #t)
("Beethoven" . #t)
("Schumann" . #f)
("Weill" . #t)))
(assoc-ref parents-musicians "Mozart")
⇒ #t
(assoc-ref parents-musicians "Bach")
⇒ #f

assoc-ref returns #f for unfound keys. This does not allow to distinguish whether Bach’s parents were not musicians, or Bach is simply not in the alist. assoc allows to draw this difference.

(assoc "Mozart" parents-musicians)
⇒ ("Mozart" . #t)
(assoc "Bach" parents-musicians)
⇒ #f

If Bach had been in the alist, the result of assoc would have been a pair. This time, #f is unambiguous.

With assoc-ref, or functions of the same family, it is common to utilize the fact that any value is considered true except #f to give default values thanks to or (cf. On universal truth).

(or (assoc-ref birth-dates "Bach")
"birth date unknown")
⇒ "birth date unknown"

With assoc, you need to store the pair in a variable.

(let ((pair (assoc "Bath" parents-musicians)))
(if pair
(cdr pair)
"no entry for this composer"))
⇒ "no entry for this composer"

Be aware that LilyPond defines yet another functions, assoc-get. Like assoc and assoc-ref, it uses equal? to test equality of keys. Like assoc-ref, it returns the value associated with a key, not the (key . value) pair. However, its arguments are in the same order as assoc, namely the key first, then the alist. Finally, assoc-get has an optional third parameter specifying the value returned if the key is not found. This parameter defaults to #f, in line with the standard alist functions.

(assoc-get "Bach" parents-musicians "no entry for this composer")
⇒ "no entry for this composer"

### Modifying values¶

Again, there is not a single function to modify alists. There are three, assq-set!, assv-set! and assoc-set!. Like assq-ref, assv-ref and assoc-ref, they use the predicates eq?, eqv? and equal?, respectively. They are called as

(assxxx-set! alist key value)

They return a new alist.

(define birth-dates
'(("Mozart" . 1756)
("Beethoven" . 1778)
("Schumann" . 1810)
("Hába" . 1893)
("Weill" . 1900)
("Ligeti" . 1923)))
(assoc-set! birth-dates "Schönberg" 1874)
⇒ (("Schönberg" . 1874) ("Mozart" . 1756) ("Beethoven" . 1778) ("Schumann" . 1810) ("Hába" . 1893) ("Weill" . 1900) ("Ligeti" . 1923))
(assoc-set! birth-dates "Weill" 1901) ; let's lie on history
⇒ (("Mozart" . 1756) ("Beethoven" . 1778) ("Schumann" . 1810) ("Hába" . 1893) ("Weill" . 1901) ("Ligeti" . 1923))

The exclamation mark at the end of the names of these functions means that they are allowed to modify the alist contents in place, by mutating the alist. In this example, if they do so, it means that birth-dates after the first assoc-set! contains a new key. However, they are not guaranteed to do so! You must always save the return value of assxxx-set! and work on that.

There is also acons, which is designed to add an entry to an alist which does not already contain the key, or for cases where repeated keys are not a problem. acons just adds a new pair at the start of the alist. The order of arguments is different from the assxxx-set! functions:

(acons key value alist)

For example:

(acons "Weill" 1901 birth-dates)
⇒ (("Weill" . 1901) ("Mozart" . 1756) ("Beethoven" . 1778) ("Schumann" . 1810) ("Hába" . 1893) ("Weill" . 1900) ("Ligeti" . 1923))
`


text:
## Physics of Dissipative Kerr Solitons in Optical Microresonators and Application to Low-noise Frequency Synthesis

Optical-frequency combs, that is spectra of equidistant coherent optical lines, have revolutionized the precision measurements of time and frequency. In 2007 a new method to generate optical frequency combs was discovered. In contrast to conventional generation methods based on pulsed laser sources, these Kerr combs' or microcombs' are generated entirely via nonlinear frequency conversion in a microresonator pumped by a continuous-wave laser. More recently, the discovery of dissipative soliton formation in these cavities has enabled the generation of low-noise comb states with reproducible spectral envelopes, required in applications. Solitons are pulses of light which retain their shape as they circulate in the resonator, owing to the balance between counter-acting effects. On the one hand, the tendency of the pulse to spread due to anomalous group velocity dispersion is counteracted by the nonlinear self-phase modulation. On the other hand, the losses in the cavity are lifted by the nonlinear parametric gain provided by the driving laser. These states are robust attractors of the nonlinear cavity system under specific driving conditions. In this thesis, the properties and dynamics of dissipative soliton states are studied experimentally in crystalline magnesium fluoride whispering gallery mode resonators. Several methods are developed to accurately determine and control the driving parameters as well as to improve the comb stability. The observations provide an accurate verification of the Lugiato-Lefever equation commonly used to describe the system. Furthermore, unexpected deviations from this canonical model are observed and accounted for with an enriched framework. The improved fundamental understanding and control of the system is applied for the generation of an ultralow-noise microcomb driven with an ultra-stable laser. In combination with a novel transfer oscillator method, this comb is used to synthesize ultralow-noise microwaves via optical frequency division. Lastly, a novel method for synthesizing multiple distinct frequency combs from a single resonator and with a single laser is devised. It relies on multiplexing solitons in different spatial modes of the microresonator. Up to three combs are generated simultaneously from a single device for the first time.

Kippenberg, Tobias
Year:
2019
Publisher:
Lausanne, EPFL
Keywords:
Laboratories:
LPQM1

Record appears in:


text:
Thursday, November 19, 2015

# K-means clustering with Python

## Introduction

K-means is one of the simplest unsupervised learning algorithms that solve the well known clustering problem. The procedure follows a simple and easy way to classify a given data set through a certain K number of clusters. The main idea is to define K centroids, one for each cluster. The next step is to take each point belonging to a given data set and associate it to the nearest centroid. At this point we need to re-calculate K new centroids of the clusters resulting from the previous step. After we have these K new centroids, a new binding has to be done between the same data set points and the nearest new centroid. As a result of this loop we may notice that the K centroids change their location step by step until no more changes are done.

## Implementation

Scikit-learn provides with full implementation of K-means algorithm though KMeans class. Let's have a look at several interesting situations, which might occur during data clustering:

import numpy as np
import matplotlib.pyplot as plt
from sklearn.cluster import KMeans
from sklearn.datasets import make_blobs

plt.figure(figsize=(12, 12))

n_samples = 1500
random_state = 170
X, y = make_blobs(n_samples=n_samples, random_state=random_state)

# Incorrect number of clusters
y_pred = KMeans(n_clusters=2, random_state=random_state).fit_predict(X)

plt.subplot(221)
plt.scatter(X[:, 0], X[:, 1], c=y_pred)
plt.title("Incorrect Number of Blobs")

# Anisotropicly distributed data
transformation = [[ 0.60834549, -0.63667341], [-0.40887718, 0.85253229]]
X_aniso = np.dot(X, transformation)
y_pred = KMeans(n_clusters=3, random_state=random_state).fit_predict(X_aniso)

plt.subplot(222)
plt.scatter(X_aniso[:, 0], X_aniso[:, 1], c=y_pred)
plt.title("Anisotropicly Distributed Blobs")

# Different variance
X_varied, y_varied = make_blobs(n_samples=n_samples,
cluster_std=[1.0, 2.5, 0.5],
random_state=random_state)
y_pred = KMeans(n_clusters=3, random_state=random_state).fit_predict(X_varied)

plt.subplot(223)
plt.scatter(X_varied[:, 0], X_varied[:, 1], c=y_pred)
plt.title("Unequal Variance")

# Unevenly sized blobs
X_filtered = np.vstack((X[y == 0][:500], X[y == 1][:100], X[y == 2][:10]))
y_pred = KMeans(n_clusters=3, random_state=random_state).fit_predict(X_filtered)

plt.subplot(224)
plt.scatter(X_filtered[:, 0], X_filtered[:, 1], c=y_pred)
plt.title("Unevenly Sized Blobs")

plt.show()

At first we wrongly assume that there are 3 clusters in the data and make K-means find them. Later we transform the data anisotropicly. Since K-means uses Euclidian distance to to associate points to clusters, it does not work well with non-globular clusters. The last two datasets introduce blobs of different variance and size, but this makes no difference and the classification succeeds.

## Conclusion

K-means provides us with easy to use clustering algorithms. It's fast, easy to follow and is used vastly in various fields including Vector Quantization.


text:
# Interaction Hamiltonian

1. Feb 5, 2010

### The thinker

1. The problem statement, all variables and given/known data

Write out:

$$H_{SE}(\left|\right\beta,i_{\beta}\rangle\otimes\left|\right e_{j}\rangle)$$

and

$$exp(-iH_{SE}t)(\left|\right\beta,i_{\beta}\rangle\otimes\left|\right e_{j}\rangle)$$

Where:

$$H_{SE}=\sum_{\alpha,j}\gamma(\alpha,j)P^{(\alpha)}\otimes\left|e_{j}\right\rangle\left\langle e_{j}\right|$$

and

$$P^{(\alpha)}=\sum_{i_{\alpha}}\left|i_{\alpha}\right\rangle\left\langle i_{\alpha}\right|$$

($$\left|i_{\alpha}\right\rangle$$ can be written $$\left|\right\alpha,i_{\alpha}\rangle$$ where alpha is a quantum number indexed by $$i_{\alpha}$$ )

3. The attempt at a solution

For the first part I'm fairly sure it comes out as:

$$\sum_{\beta,j}\gamma(\beta,j)\left|\right\beta,i_{\beta}\rangle\otimes\left|\right e_{j}\rangle$$

But the second part I am not sure of, is it something like:

$$(Cos(t)-i\gamma(\alpha,j)Sin(t))(\left|\right\beta,i_{\beta}\rangle\otimes\left|\right e_{j}\rangle)$$

Thanks!

2. Feb 5, 2010

### xepma

In the first you should not summate over $j$ (and you need to explain why ;))

For the second you first apply the Taylor expansion for the exponential. After that, compute:

$H_{SE}^2$ followed by generalizing this to $H_{SE}^n$.

3. Feb 5, 2010

### The thinker

Thanks for that.

I'll have a bash at that.. although I honestly can't see why you wouldn't sum over j

4. Feb 5, 2010

### The thinker

Oh wait... is it because the $$e_{j}$$ basis correspond to different alpha's but not i's?

Edit: Actually on second thought that doesn't make sense because we are summing over alpha(beta).

5. Feb 7, 2010

### The thinker

Can anyone else offer some more help?

-I've been teaching myself dirac notation as part of my project this year. This is the first time I've looked at interaction Hamiltonians.


text:
Statistics[Distributions] - Maple Programming Help

Home : Support : Online Help : Statistics and Data Analysis : Statistics Package : Distributions : Statistics/Distributions/EmpiricalDistribution

Statistics[Distributions]

 EmpiricalDistribution
 Empirical distribution

 Calling Sequence EmpiricalDistribution(sample, opts)

Parameters

 sample - a list or rtable of samples opts - (optional) an equation of the form probabilities = plist

Description

 • The empirical distribution is a discrete probability distribution with probability function generated from the given sample.
 • If opts is not given, then Maple assigns a probability to each variate present within the sample equal to the fraction of occurrences of that value within the entire sample. If opts is specified, then plist should be a list or rtable of probabilities (nonnegative real numbers adding up to 1), having equally many entries as sample; the ith entry of plist specifies the probability for the ith entry of sample.
 • If all the values in sample are positive integers, you can use the ProbabilityTable distribution instead.
 • Note that the EmpiricalDistribution command is inert and should be used in combination with the RandomVariable command.

Examples

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$
 > $P≔\mathrm{Array}\left(\left[1,1,1,2,4,4,5.5\right]\right):$
 > $X≔\mathrm{RandomVariable}\left(\mathrm{EmpiricalDistribution}\left(P\right)\right):$
 > $\mathrm{ProbabilityFunction}\left(X,1\right)$
 $\frac{{3}}{{7}}$ (1)
 > $\mathrm{ProbabilityFunction}\left(X,5\right)$
 ${0}$ (2)
 > $\mathrm{ProbabilityFunction}\left(X,5.5\right)$
 $\frac{{1}}{{7}}$ (3)
 > $\mathrm{CDF}\left(X,4\right)$
 $\frac{{6}}{{7}}$ (4)
 > $\mathrm{Mean}\left(X\right)$
 ${2.64285714285714}$ (5)
 > $\mathrm{Variance}\left(X\right)$
 ${2.908163265}$ (6)
 > $Y≔\mathrm{RandomVariable}\left(\mathrm{EmpiricalDistribution}\left(⟨1,2.5,4,5⟩,'\mathrm{probabilities}'=\left[\frac{1}{3},\frac{1}{4},\frac{13}{60},\frac{1}{5}\right]\right)\right)$
 ${Y}{≔}{\mathrm{_R0}}$ (7)
 > $\mathrm{ProbabilityFunction}\left(Y,\frac{5}{2}\right)$
 $\frac{{1}}{{4}}$ (8)
 > $\mathrm{CDF}\left(Y,4.5\right)$
 ${0.8000000000}$ (9)
 > $\mathrm{Median}\left(Y\right)$
 $\frac{{5}}{{2}}$ (10)
 > $\mathrm{Tally}\left(\mathrm{Sample}\left(Y,{10}^{6}\right)\right)$
 $\left[{5.}{=}{199920}{,}{4.}{=}{216982}{,}{1.}{=}{332622}{,}{2.50000000000000}{=}{250476}\right]$ (11)

References

 Evans, Merran; Hastings, Nicholas; and Peacock, Brian. Statistical Distributions. 3rd ed. Hoboken: Wiley, 2000.
 Johnson, Norman, L.; Kotz, Samuel; and Balakrishnan, N. Continuous Univariate Distributions. 2nd ed. 2 vols. Hoboken: Wiley, 1995.
 Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.

Compatibility

 • The probabilities option was introduced in Maple 16.


text:
Kingman's coalescent and Brownian motion

Abstract : We describe a simple construction of Kingman's coalescent in terms of a Brownian excursion. This construction is closely related to, and sheds some new light on, earlier work by Aldous and Warren. Our approach also yields some new results: for instance, we obtain the full multifractal spectrum of Kingman's coalescent. This complements earlier work on Beta-coalescents by the authors and Schweinsberg. Surprisingly, the thick part of the spectrum is not obtained by taking the limit as $\alpha \to 2$ in the result for Beta-coalescents mentioned above. Other analogies and differences between the case of Beta-coalescents and Kingman's coalescent are discussed.
Keywords :
Document type :
Preprints, Working Papers, ...
2009
Domain :

https://hal.archives-ouvertes.fr/hal-00375053
Contributor : Julien Berestycki <>
Submitted on : Sunday, April 12, 2009 - 1:00:45 PM
Last modification on : Wednesday, October 12, 2016 - 1:03:51 AM

Identifiers

• HAL Id : hal-00375053, version 1
• ARXIV : 0904.1526

Citation

J. Berestycki, N. Berestycki. Kingman's coalescent and Brownian motion. 2009. <hal-00375053>

Metrics

Consultations de la notice


text:
# Right Triangles and Trigonometry

## Objective

Define angles in standard position and use them to build the first quadrant of the unit circle.

## Common Core Standards

### Core Standards

?

• F.TF.A.2 — Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

?

• 4.G.A.1

## Criteria for Success

?

1. Identify an angle in standard position if the vertex is at the origin and one ray is on the positive $x$-axis.
2. Orient the initial side with the ray on the $x$-axis and the terminal side as the other ray.
3. Describe the unit circle as a circle where the radii are one unit and each radius is the hypotenuse of triangles formed inside the unit circle.
4. Describe the importance of the unit circle as it relates to trigonometric ratios. This unit circle will be used throughout higher math.
5. Define reference angles as positive and only in the first quadrant.
6. Know that $(x_\theta,y_\theta)=(\mathrm{cos}\theta,\mathrm{sin}\theta)$

## Tips for Teachers

?

• This lesson and the next lesson cover key concepts in the unit by reinforcing and reviewing G-SRT.6 and G-SRT.7.
• The unit circle is used throughout calculus and is built on trigonometric ratios.

## Anchor Problems

?

### Problem 1

Given a blank coordinate plane, draw a 45° angle with a vertex at the origin.

#### Guiding Questions

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

### Problem 2

Suppose that point $P$ is the point on the unit circle obtained by rotating the initial ray 30°. Find ${\mathrm{sin}(30^\circ)}$ and ${\mathrm{cos}(30^\circ)}$

#### Guiding Questions

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

#### References

EngageNY Mathematics Algebra II > Module 2 > Topic A > Lesson 4Example 1

Algebra II > Module 2 > Topic A > Lesson 4 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by The Match Foundation, Inc.

### Problem 3

Each of the points in Quadrant I represent THE point of intersection of a ray that forms a 30° angle, a 45° angle, and a 60° angle with the $x$-axis in the initial position. What are the coordinate points where the rays intersect the circle at 30°, 45°, and 60°?

#### Guiding Questions

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

#### References

Unit Circle with Reference Triangles by Match Foundation, Inc. is made available by Desmos. Copyright © 2017 Desmos, Inc. Accessed March 13, 2017, 4:27 p.m..

## Problem Set

?

The following resources include problems and activities aligned to the objective of the lesson that can be used to create your own problem set.

• Include problems where students are asked to identify and fix the error. For example, give the sine of an angle and a drawing on the unit circle with the angle not in standard position.
• Include problems that require students to find the coordinate point based on the angle provided. For example, given an angle, find the coordinate point. Make sure to vary the information given and what is requested for an answer.
• Include problems where students need to make a drawing based on the parameters provided. For example, “Draw an angle that has a sine of… Is there only one angle that will give this sine?”

## Target Task

?

Draw an angle in standard position that has an intersection point on the unit circle of ${\left(\frac{1}{2},\frac{\sqrt3}{2}\right)}$. What is the sine of this angle? The cosine?