text:
American Institute of Mathematical Sciences

ISSN:
1556-1801

eISSN:
1556-181X

All Issues

Networks & Heterogeneous Media

March 2021 , Volume 16 , Issue 1

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2021, 16(1): 1-29 doi: 10.3934/nhm.2020031 +[Abstract](447) +[HTML](153) +[PDF](681.95KB)
Abstract:

The paper is concerned with a shape optimization problem, where the functional to be maximized describes the total sunlight collected by a distribution of tree leaves, minus the cost for transporting water and nutrient from the base of trunk to all the leaves. In a 2-dimensional setting, the solution is proved to be unique and explicitly determined.

2021, 16(1): 31-47 doi: 10.3934/nhm.2020032 +[Abstract](324) +[HTML](116) +[PDF](1281.65KB)
Abstract:

In this article, we investigate theoretical and numerical properties of the first-order Lighthill-Whitham-Richards (LWR) traffic flow model with time delay. Since standard results from the literature are not directly applicable to the delayed model, we mainly focus on the numerical analysis of the proposed finite difference discretization. The simulation results also show that the delay model is able to capture Stop & Go waves.

2021, 16(1): 49-67 doi: 10.3934/nhm.2020033 +[Abstract](311) +[HTML](106) +[PDF](549.73KB)
Abstract:

In this paper, we investigate the long time behavior of the solution for the nonlinear wave equation with frictional and visco-elastic damping terms in \begin{document}$\mathbb{R}^n_+$\end{document}. It is shown that for the long time, the frictional damped effect is dominated. The Green's functions for the linear initial boundary value problem can be described in terms of the fundamental solutions for the full space problem and reflected fundamental solutions coupled with the boundary operator. Using the Duhamel's principle, we get the pointwise long time behavior of the solution \begin{document}$\partial_{{\bf{x}}}^{\alpha}u$\end{document} for \begin{document}$|\alpha|\le 1$\end{document}.

2021, 16(1): 69-90 doi: 10.3934/nhm.2020034 +[Abstract](312) +[HTML](117) +[PDF](5537.77KB)
Abstract:

The aim of this work is to introduce a two-dimensional macroscopic traffic model for multiple populations of vehicles. Starting from the paper [21], where a two-dimensional model for a single class of vehicles is proposed, we extend the dynamics to a multi-class model leading to a coupled system of conservation laws in two space dimensions. Besides the study of the Riemann problems we also present a Lax-Friedrichs type discretization scheme recovering the theoretical results by means of numerical tests. We calibrate the multi-class model with real data and compare the fitted model to the real trajectories. Finally, we test the ability of the model to simulate the overtaking of vehicles.

2021, 16(1): 91-138 doi: 10.3934/nhm.2021001 +[Abstract](278) +[HTML](104) +[PDF](1922.46KB)
Abstract:

We propose a new model for the emergence of blood capillary networks. We assimilate the tissue and extra cellular matrix as a porous medium, using Darcy's law for describing both blood and interstitial fluid flows. Oxygen obeys a convection-diffusion-reaction equation describing advection by the blood, diffusion and consumption by the tissue. Discrete agents named capillary elements and modelling groups of endothelial cells are created or deleted according to different rules involving the oxygen concentration gradient, the blood velocity, the sheer stress or the capillary element density. Once created, a capillary element locally enhances the hydraulic conductivity matrix, contributing to a local increase of the blood velocity and oxygen flow. No connectivity between the capillary elements is imposed. The coupling between blood, oxygen flow and capillary elements provides a positive feedback mechanism which triggers the emergence of a network of channels of high hydraulic conductivity which we identify as new blood capillaries. We provide two different, biologically relevant geometrical settings and numerically analyze the influence of each of the capillary creation mechanism in detail. All mechanisms seem to concur towards a harmonious network but the most important ones are those involving oxygen gradient and sheer stress. A detailed discussion of this model with respect to the literature and its potential future developments concludes the paper.

2021, 16(1): 139-153 doi: 10.3934/nhm.2021002 +[Abstract](257) +[HTML](114) +[PDF](4743.94KB)
Abstract:

How to understand the dynamical consensus patterns in network systems is of particular significance in both theories and applications. In this paper, we are interested in investigating the influences of distributed processing delay on the consensus patterns in a network model. As new observations, we show that the desired network model undergoes both weak consensus and periodic consensus behaviors when the parameters reach a threshold value and the connectedness of the network system may be absent. In results, some criterions of weak consensus and periodic consensus with exponential convergent rate are established by the standard functional differential equations analysis. An analytic formula is given to calculate the asymptotic periodic consensus in terms of model parameters and the initial time interval. Also, we post the threshold values for some typical distributions included uniform distribution and Gamma distribution. Finally, we give the numerical simulation and analyse the influences of different delays on the consensus.

2019  Impact Factor: 1.053


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# UC San Diego Probability Seminar (2021-2022)

The seminar meets on Thursdays from 11 AM to 12 PM PT in AP&M 6402 with live streaming via Zoom. Please send me an email if you would like to be added to the mailing list or if you have a suggestion for a speaker. Invite links to Zoom streaming will be sent to the mailing list.

## Spring

April 28: Yizhe Zhu (UC Irvine)

Abstract: The stochastic block model has been one of the most fruitful research topics in community detection and clustering. Recently, community detection on hypergraphs have become an important topic in higher-order network analysis. We consider the detection problem in a sparse random tensor model called the hypergraph stochastic block model (HSBM). We prove that a spectral method based on the non-backtracking operator for hypergraphs works with high probability down to the generalized Kesten-Stigum detection threshold conjectured by Angelini et al. We characterize the spectrum of the non-backtracking operator for the sparse random hypergraph, and provide an efficient dimension reduction procedure using the Ihara-Bass formula for hypergraphs. As a result, the community detection problem can be reduced to an eigenvector problem of a non-normal matrix constructed from the adjacency matrix and the degree matrix of the hypergraph. Based on joint work with Ludovic Stephan.

May 19: Ching Wei Ho (Notre Dame)

Abstract: The limiting eigenvalue distributions of two well-known random matrix ensemble, the GUE and the Ginibre ensemble, are respectively the semicircle law (the density is semicircular in the interval $$[-2,2]$$) and the circular law (the uniform measure on the disk of radius $$1$$). These two distributions have a simple relation: the push-forward of the circular law by $$z\to 2\operatorname{Re}(z)$$ is the semicircular law. Recent results in random matrix theory show that this simple relation can be generalized into other Gaussian random matrix models; however, these results work with the (continuous) limiting eigenvalue distributions. In this talk, I will present a recent conjecture with Brian Hall. At the finite-$$N$$ level, the push-forward property can be accomplished by applying the heat operator to the random characteristic polynomial of a random matrix model. For example, if we apply the heat operator at time $$1/N$$ to the characteristic polynomial of the GUE, the roots of the resulting polynomial resemble the circular law.

May 26: Giorgio Cipolloni (Princeton)

Abstract: TBA

June 2: David Renfrew (SUNY Binghamton)

Abstract: TBA

* Available dates:

format: plain


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Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.

It seems to me that the poet has only to perceive that which others do not perceive, to look deeper than others look. And the mathematician must do the same thing (Sofia Kovalevskaya)

How beautiful is this fractal! In previous posts I colored plots using module of complex numbers generated after some iterations. In this occasion I have used the escape-time algorithm, a very well known coloring algorithm which is very easy to implement in R.

Those who want to know more about this fractal can go here. For coloring, I chose a simple scale from red to yellow resulting a fractal interpretation of my country’s flag. You can chose another scale or using a RColorBrewer palette as I did in this previous post. Choosing another x or y ranges you can zoom particular areas of the fractal.

Try yourself and send me your pictures!

library(ggplot2)
xrange <- seq(-8, 8, by = 0.01)
yrange <- seq(-3, 3, by = 0.01)
f  <- function (z) {1/4*(2+7*z-(2+5*z)*cos(pi*z))}
z <- outer(xrange, 1i*yrange,'+')
t <- mat.or.vec(nrow(z), ncol(z))
for (k in 1:10)
{
z <- f(z)
t <- t + (is.finite(z)+0)
}
## Supressing texts, titles, ticks, background and legend.
opt <- theme(legend.position="none",
panel.background = element_blank(),
axis.ticks=element_blank(),
axis.title=element_blank(),
axis.text =element_blank())
z <- data.frame(expand.grid(x=xrange, y=yrange), z=as.vector(t))
ggplot(z, aes(x=x, y=y, color=z)) + geom_tile() + scale_colour_gradient(low="red", high="yellow") + opt


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# 6.4: The Physics of a Wind Turbine

Let’s consider the question: how much energy does wind carry? It turns out that finding the answer is a pretty straightforward task. Suppose that the wind blows with a speed of $$V$$ . Now, let’s put an “imaginary tube” with cross section of $$A$$ parallel to the wind’s velocity direction. Let $$\delta t$$ be an arbitrarily chosen time period. Over ∆t the air particles the wind carries travel the distance of $$V \times \delta t$$, right? So let the length of our tube be $$L = V \times \delta t$$

$volume = A \times L = A \times V \times \delta t$

What we need is the mass m of the air inside the tube. The general relation between the mass and the volume is: mass = volume × density. Denoting the density of the air as $$\rho$$, we get then:

$m=A \times L=A \times V \times \Delta t \times \rho$

The next step is to find the kinetic energy $$K$$ of the air portion inside the tube. All we need is to use the known formula: $$K=m V^{2} / 2$$. Combining it with the expression for $$m$$ we have gotten, we obtain:

$K=\dfrac{A \times \rho \times V^{3}}{2} \Delta t$

But it is not exactly the energy which is of interest to us - we want to find the power the wind carries. The power, as we know, is:

$P=\dfrac{\text { the amount of energy delivered }}{\text { the time it took to deliver it }}$

We have found the energy $$K$$ which the "portion" of the air inside the tube carries. How much time it takes it to leave the pipe through its outlet? The length of the pipe is $$L$$, and the air inside travels with speed $$V$$, so thetime the "portion" in question needs to get completely out through the outlet is:

$\dfrac{L}{V}=\dfrac{V \times \Delta t}{V}=\Delta t$

So, energy $$K$$ passes through the tube outlet in time $$\Delta t$$. Using then $$K$$ from the Eq. 6.1, we the "almost-final" result:

$P=\dfrac{K}{\Delta t}=\dfrac{A \times \rho \times V^{3} \times \Delta t}{2 \Delta t}=\dfrac{A \times \rho \times V^{3}}{2}$

Let's check if the unit of $$P$$ is correct. There is a convention: $$[\mathrm{X}]=$$ the unit of $$\mathrm{X}$$. Therefore, $$[A]=\mathrm{m}^{2} ; \rho]=\mathrm{kg} / \mathrm{m}^{3} ;$$ and $$[V]=\mathrm{m} / \mathrm{s}$$. Hence:

$[P]=\mathrm{m}^{2} * \mathrm{~kg} / \mathrm{m}^{3} * \mathrm{~m}^{3} / \mathrm{s}^{3}=\mathrm{kg} * \mathrm{~m}^{2} / \mathrm{s}^{3}$

which is indeed the unit of power, the Watt, right? ( $$1 \mathrm{~W}=1 \mathrm{~J} / \mathrm{s}=1 \mathrm{~N}^{*} \mathrm{~m} \mathrm{s}$$ $$=1\left(\mathrm{~kg} * \mathrm{~m} / \mathrm{s}^{2}\right)^{*} \mathrm{~m} / \mathrm{s}=1 \mathrm{~kg}^{*} \mathrm{~m}^{2} / \mathrm{s}^{3}-$$ right!)

The Eq. $$6.2$$ is already a useful formula - if we know how big is the area A to which the wind "delivers" its power. For example, is the rotor of a wind turbine is $$R$$, then the area in question is $$A=\pi R^{2}$$. Sometimes, however, we want to know only how much power the wind carries per a unit surface area - denote it as $$p$$. The formula gets even simpler:

$p=\frac{P}{A}=\frac{A \times \rho \times V^{3}}{2 A}=\frac{\rho \times V^{3}}{2}$

The air density ρ is a function of temperature and altitude. At sea level and at temperature 15 (59 F) air has a density of approximately 1.225 kg/m3. The recipe for calculating ρ at other circumstances are given in this Wikipedia article.

If we know the density of air, the speed of wind, and the radius R of a given turbine, is it enough to find out how much power the turbine deliver, using the Eq. 2? Well... almost. There are still some extra factors. One of them is purely of a physical nature. The problem is that a wind turbine cannot “extract” all kinetic energy from the incoming wind. If it did, what would happen? Well, the kinetic energy of the air after passing the turbine would be zero, meaning also that its velocity would be zero – this is clearly not possible, because the air would start “accumulating” behind the rotor and would start blocking the incoming wind! The air behind the rotor must keep moving! So, what happens to the “downstream” wind? In order to accommodate the slower moving air, the stream behind the rotor has to widen considerably.

On Page 158 (check!!!!), there is a link to a video explaining the “anatomy” of a wind energy converter – let’s watch it again, starting from the 4th minute: the reasons why not all energy carried by the “upstream” wind cannot be “captured” by the turbine is very clearly explained.

Then, how much power can be captured from the wind? This question has been answered in a paper published in 1919 by a German physicist Albert Betz who proved that the maximum fraction of the upstream kinetic energy K that can be “absorbed” by an ideal “actuator” – not necessarily a turbine, but any device capable of converting wind energy to another energy form– is $$\frac{16}{27}$$ K, or 59.3% of K. It is known as the Betz Law – and how it can be derived based on the energy, momentum and mass conservation laws is described, for instance, in the Wikipedia article linked.

Let us repeat - $$59.3 \%$$ is the maximum fraction, assuming that the "actuator" is an ideal device. In reality, even the best turbines can capture only a fraction of the "Betz maximum". So, in math language - for a turbine rotor of radius $$R$$ the maximum theoretical power allowed by the Betz Law is:

$P_{\text {Betz } \lim .}=\frac{16}{27} \times \frac{\pi R^{2} \rho V^{3}}{2}$

And in practice, a turbine delivers only a fraction $$\varepsilon$$ of this power, so that the net power obtainable is:

$P_{\text {obtainable }}=\frac{16}{27} \times \frac{\pi R^{2} \rho V^{3}}{2} \times \epsilon$

where $$E$$, to make things worse, is not constant, but is the function of something called the "tip-speed ratio" - we will create a TSR symbol for it:

$\text { tip speed ratio }=T S R=\frac{\text { Linear speed of the blade tip }}{\text { Wind speed }}$

We can therefore write the final result for the net windpower harnessed by a turbine:

$P_{\text {net }}=\frac{8 \pi R^{2} \rho V^{3}}{27} \times \epsilon \quad \text { where } \epsilon=\epsilon(T S R) .$

The bad news is that that there is no simple mathematical expression describing the $$\mathcal{E}(T S R)$$ function. The shape of this function is different for different rotor types in for each type it has to be determined by experimental measurements.

The results of measurements of the net efficiency $$varepsilon_{\text {net }=P_{\text {net }/P$$ plotted versus the Tip Speed Ratio for different types of rotors are presented graphically in a figure uploaded to the ResearchGate.net (a networking site for scientists) by Moshe Zilberman. From the figure it can be seen that the overall efficiency for a modern three-blade rotor has the maximum value of about 46% = 0.46 for the TSR value of about 5.5. By comparing it to the Betz Limit value, 0.593, we find that the value of $$E$$ in Eq. 4 for this STR value is 0.46/0.593 = 0.776 = 77.6%, meaning that the turbine converts nearly $$\frac{3}{4} of the wind power available by the Betz Law to mechanical power. For the American Multiblade farm windmill the maximum net efficiency is as low as 0.31, corresponding to \( E$$ = 52%1. As noted, there is no simple algorithm for calculating the E(TSR) coefficient. However, there is a simple way of dealing with this problem – namely, the power output from a given type of turbine for different wind velocities can be measured experimentally

– and the results can be stored in an easily accessible database. A known Internet tool of this kind is a Swiss Wind Turbine Power Calculator. It con- tains the data for more than 50 types of the most popular turbines. After selecting the type, one gets the measured values of the output power of the turbine for speeds of wind from 1 to 30 m/s, with a 1 m/s increment. Such results constitute what is usually referred to as the “power curve” of the given turbine. For instance, for a popular (over 15,000 installed worldwide!) Vestas-80 with “nametag” output of 2,000 kW, the calculator yields:

Table 6.2: The performance of the Vestas-80 turbine

Speed of

wind (m/s)

Power

output (kW)

Speed of

wind (m/s)

Power

output (kW)

Speed of

wind (m/s)

Power

output (k)W

1

0

11

1652

21

2000

2

0

12

1892

22

2000

3

0

13

1982

23

2000

4

66

14

2000

24

2000

5

156

15

2000

25

2000

6

285

16

2000

26

0

7

467

17

2000

27

0

8

706

18

2000

28

0

8

706

18

2000

28

0

9

1001

19

2000

29

0

10

1329

20

2000

30

0

As follows from the above discussion, the three-blade turbine, which is dominant type used today, can deliver power as high as over $$75 \%$$ of the Betz Limit. It is instructive to confront that with the Power Calculator figures. Let's take, for instant, $$V=10 \mathrm{~m} / \mathrm{s}$$, for which the Calculator gives us $$P=$$ $$1,329 \mathrm{~kW}$$. The rotor diameter of Vestas V8o is 80 meters, so that the radius $$R=40 \mathrm{~m}$$, and $$A=\pi R^{2}$$ is $$5026.5 \mathrm{~m}^{2}$$. The Power Calculator data are for the air density of $$1.225 \mathrm{~kg} / \mathrm{m}^{3}$$. For these numbers from the Eq. $$6.2$$ one gets $$P=3,078,761 \mathrm{~W}=3,078.8 \mathrm{~kW}$$. Correction for the Betz Limit, i.e., multiplication by $$16 / 27$$, yields $$1,824.5 \mathrm{~kW}$$. It means that the efficiency of the V-8o turbine is $$1.329 / 1,824.5=0.7284=72.8 \%$$ - close indeed to the predicted $$75 \%$$. But the Reader is encouraged to make similar checks for several other speed of wind values. It turn out that an efficiency close to $$75 \%$$ is obtained only at favorable wind conditions: not too weak and not too strong.

One reason for the V8o popularity is that it starts generating electric power at wind velocity as low as only $$4 \mathrm{~m} / \mathrm{s}$$. At $$V$$ exceeding $$6 \mathrm{~m} / \mathrm{s}$$ it's output already is above $$70 \%$$ of the Betz Limit (see the rightmost column in the table), but next, for $$V$$ exceeding $$10 \mathrm{~m} / \mathrm{s}$$, the efficiency starts gradually falling. Why? The answer is simple, the maximum output power the generator in the V-8o turbine is capable to deliver is $$2000 \mathrm{~kW}=2 \mathrm{MW}$$. Any electric device has a limit power it can tolerate, otherwise it may overheat or short-circuit. And the power an electric generator delivers depends onhow fast it rotates. Apparently, at wind’s velocity over 13 m/s the generator reaches its maximum allowed speed of rotation. Now, if V keeps increasing, the efficiency of the rotor is artificially lowered, in order not to allow the ro- tor to turn any faster. It can be done by changing the ”pitch” of the blades

– again, we link here the YouTube video where it is shown, 3 minutes and 45 seconds after the beginning, how it is done by rotating the entire blades relative to the rotor hub.

But for wind speed $$\gt 25 \mathrm{~m} / \mathrm{s}$$ it is no longer safe to let the rotor turn – so the blades are set to a neutral position in which they generate no torque and a special electromagnetic brake is engaged to completely immobilize the rotor.

___________________________________________________________

1. It should be noted, however, that for millions of farmers who installed American Multiblade turbines not their power, but the torque offered by those windmills was the most important factor. In contrast to two- and three-bladed turbines, the multiblade rotors produce a high torque right from the moment the wind starts blowing – it’s called the “start-up” torque. And the torque is crucial if the turbine is used, for operating a reciprocating water pump – what was their main task at American farms.

6.4: The Physics of a Wind Turbine is shared under a CC BY 1.3 license and was authored, remixed, and/or curated by Tom Giebultowicz.


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# Solar Wind

#### Philosophaie

Solar Wind is charged plasma ejected from the sun at 400km/s to 750km/s and even as fast as 1000km/s during Solar Flares. It has a 27 day cycle and even can decrease 98% at sparse happenings. It carries a magnetic field with it. It is strong enough to interact with the planets and their magnetic fields to shape magnetospheres, earth's effect from this plasma. The magnetosphere is shaped like a sphere in the upper atmosphere around where the atmosphere faces the sun and has a tail away from the earth.
The solar wind blasts the planet at supersonic speeds producing a bow shock. The sun also has a magnetosphere from the intersteller winds.

My question is can someone estimate the varying magnitude of the magnetic field associated with the Solar Wind.

Last edited:
Related Astronomy and Astrophysics News on Phys.org

#### Astronuc

Staff Emeritus
NASA's ACE spacecraft makes measurements of the magnetic field in the solar wind.

http://www.swpc.noaa.gov/ace/ace_rtsw_data.html [Broken]
http://www.srl.caltech.edu/ACE/ace_mission.html

http://www-ssc.igpp.ucla.edu/ssc/tutorial/solwind_magsphere_tutorial.pdf

Code:
Parameter        Min  Ave  Max
Flux (cm-2s-1)     1    3  100
Velocity (km/s)  200  400  900
Density (cm-3)   0.4  6.5  100
Helium %          ~0    5   25
B (nT)           0.2    6   80

Last edited by a moderator:

"Solar Wind"

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# TicTacToe with considerations of symmetry

It is not difficult to determine the number of possible games of tic toe, but what about when rotationally symmetric positions are considered equal? Please do not simply give me the number, I would like the intuition of how it is found. IMPORTANT: I am more talking more about arrangements of x's and o's in a 3X3 grid than actual tictactoe games, thus when somebody "wins" the game it continues.

• tell us how do you determine the total number of possible games, so that we can enhance YOUR answer – Thanos Darkadakis Dec 20 '13 at 2:31
• After the first move, of which there are 9 possibilities, there are 8 possibilities for the second move, so 9*8 positions on move 2. There are seven possible turn 3 moves, leading to 9*8*7 positions, and the pattern continues so that eventually there are 9! games. – Platonix Dec 20 '13 at 2:38
• not exactly. games can end on 5th, 6th, 7th, 8th or 9th move. – Thanos Darkadakis Dec 20 '13 at 2:39
• here is an explanation: se16.info/hgb/tictactoe.htm .There is also a link inside the link... – Thanos Darkadakis Dec 20 '13 at 2:42
• Sorry, read my edited question. You are quite correct of course, but I am more interested in the problem of symmetry than the games that "end" – Platonix Dec 20 '13 at 2:53

I'll use Burnside's method: count the number of invariant patterns for each rotation, and average that number over all rotations to get the number of distinguishable patterns.

I assume we're counting patterns of $5$ Xs and $4$ Os. If you're not allowing reflections, there are $4$ rotations of the square in the plane: the identity, $90^o$ degrees either way, or $180^o$.

For the identity rotation, all $\binom94=126$ patterns are invariant.

For a $90^o$ rotation there are $3$ orbits, one of size $1$ (the center) and two of size $4$ (the four corners or the four sides). Since we have $4$ Os, we have to put them in the four corners or the four sides: $2$ invariant patterns.

For the $180^o$ rotation, there is one orbit of size $1$ and there are four orbits of size $2$, consisting of a pair of opposite cells. The Os have to fill two of the size $2$ orbits; the number of invariant patterns is $\binom42=6$.

Thus the number of distinguishable patterns, allowing rotations in the plane but not reflections, is $\dfrac{126+2+2+6}4=34$.

Suppose you also allow the $4$ reflections as symmetries. Each reflection has three orbits of size $1$ (a cell on the axis of reflection) and three orbits of size $2$ (two mirrored cells). The number of invariant patterns for four Os is $\binom30\binom32+\binom32\binom31=12$, and so the number of distinguishable patterns is $\dfrac{126+2+2+6+12+12+12+12}8=23$.

The following MSE post computes the cycle index for the symmetries of an $N\times N$ board, $N$ odd or even, so that we may apply the Polya Enumeration Theorem, which includes Burnside as a special case. For $N$ odd we get that $$Z(H_N) = \frac{1}{8} \left( a_1^{N^2} + 4 a_1^N a_2^{(N^2-N)/2} + 2 a_1 a_4^{(N^2-1)/4}+a_1 a_2^{(N^2-1)/2}\right).$$ Put $N=3$ to obtain $$Z(H_3) = \frac{1}{8} \left(a_1^9 + 4 a_1^3 a_2^3 + 2 a_1 a_4^2 + a_1 a_2^4\right).$$ The substituted cycle index thus becomes $$Z(H_3)(1+z)= 1/8\, \left( 1+z \right) ^{9} +1/2\, \left( 1+z \right) ^{3} \left( 1+{z}^{2} \right) ^{3} +1/4\, \left( 1+z \right) \left( 1+{z}^{4} \right) ^{2}\\ +1/8\, \left( 1+z \right) \left( 1+{z}^{2} \right) ^{4}$$ which is $$Z(H_3)(1+z) = {z}^{9}+3\,{z}^{8}+8\,{z}^{7}+16\,{z}^{6}+23\,{z}^{5}+23\,{z}^{4} +16\,{z}^{3}+8\,{z}^{2}+3\,z+1,$$ so there are $23$ patterns with five cells marked.

I'll look at completely-filled boards with five $X$'s and four $O$'s (say).

I'll assume rotations and reflections are not distinct.

I'll count the number of distinct cases where $O$ occupies each possible number of corners. Call this number $O_{corner}$.

If $O_{corner} = 0$, there are $2$ patterns: one of the $O$'s is in the center, or it isn't. There is at most one vacant side space.

If $O_{corner} = 1$, there are $5$ cases. If $O$ is also in the center, one can fill the two adjacent sides, one adjacent and one "far" side, or the two far sides. If $O$ is not in the center, either an adjacent side or a far side can be blank.

If $O_{corner} = 2$, these two can be in opposite corners, or adjacent corners.

Let's take adjacent corners first. For a second, let's number the squares: upper left is $1$; lower right is $9$. Our two corner $O$'s are in $1$ and $3$. If one of the $O$'s is in the center ($5$) the distinct cases are spaces $2, 4,$ and $8$. If $O$ isn't in the center, the distinct cases for the last two are the pairs of spaces $(2,4), (4,6), (2,8),$ and $(6,8)$. We have $7$ cases.

For opposite corners, if $O$ is in the center, any side square is the same, symmetry-wise. If $O$ is not in the center, the other two $O$'s can be both next to one of the corner $O$'s, one next to each corner $O$ on the same side of the diagonal, one next to each corner $O$ on the opposite side of the diagonal. So $4$ more cases.

Total cases for $O_{corner} = 2$ is $11$.

For $O_{corner} = 3$, the fourth $O$ is on the side in between two corner $O$'s, on the side not between two corner $O$'s, or in the center. That's $3$ cases.

For $O_{corner} = 4$, that's all she wrote ($1$).

So, $22$ cases. I think I got them all.

If you don't want to consider any of the symmetries I've done, or if you want to take all different numbers of $X$'s and $O$'s, then you can go through the same exercise and determine as you go along which ones are distinct.

• I got $23$ patterns using Burnside. Took me a while to figure out which one you missed. With $O_{corner}=1$ and $O$ in the center, there are two ways to fill one adjacent and one far side. – bof Dec 20 '13 at 4:57


text:
# Force vs. Displacement graph question

In the graph below, forces labelled positive act in the direction of motion of the object, forces labelled negative oppose the motion. The object under consideration has a mass of 2.0kg and was initially at rest. Calculate its kinetic energy and speed when
a) d= 2.0m
b) d= 4.0m
c) d=6.0m
d) d= 8.0m

I know W is area under the graph, but not positive how to apply it. I assume the question is only asking for that moment in time, and NOT a time period.
so for (a) it would be W= 4.0N x 2.0m = 8J; which would also be Ek
and V= 2.83m/s
if this is correct then b and c follow the same method, so i got those right aswell.
Not sure of the procedure when the Force is negative.

Last edited:

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No, this is not right. The kinetic energy at a given moment is equal to the work done from the beginning to that moment - given that it starts from rest so in the beginning has zero KE.
The work done can be calculated form the area under the curve, as you said (but did not do).

No, this is not right. The kinetic energy at a given moment is equal to the work done from the beginning to that moment - given that it starts from rest so in the beginning has zero KE.
The work done can be calculated form the area under the curve, as you said (but did not do).
okay, in that case:
a)to calculate work, solve for area under curve
W= l x w + (bXh)/2
W= 2m x 4N + (2m x (6N-4N)/2
W= 8J + 2J
w= 10 J
The Ek is equivilant to w since there is no work lost in the process.
Ek= 10J

now, solve for velocity
Ek= 0.5 (m)v2
v2= 0.5(2.0kg)/10J
v2= 0.1
v= √0.1
v=0.3m/s (to two sig digs)

if this is correct, then (b), (c), and (d) follow the same process

The value of the work looks OK.
Solving for v is not done properly (you can convince yourself calculating the KE with v=0.3m/s and m=2 kg).
Look again at the equation to solve.


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# Theoretically, how does quantum decoherence induce noise?

The decoherence process has allowed us to explain various (classical and decoherence) sources of measurement noise in quantum systems. I intuitively understand this physical concept of decoherence-inducing-noise.

I am trying to understand this concept from an information theoretic perspective. We know that the environment-induced-decoherence takes away state information from a system. That would imply reduction in entropy of the system. Now analyzing the modeling process, the measurement process of a (environment induced) decohering system can be modeled by adding "noise" terms. That would imply addition of entropy to the system. Which is contradicting.

So my question is, from an information perspective, how can the (environment-induced-)decoherence process consistently (theoretically) describe the occurrence of (non-classical) noise? Or, what is the nature of such a noise?

Edit This is an explanation on why I think that decoherence might lead to reduction in entropy. I am using an information theoretic interpretation on entropy for this purpose, which can be interpreted as the amount of information that you lack about the exact physical state of the system (see https://physics.stackexchange.com/q/119515).

The measurement problem has 3 parts (Schlosshauer book on Decoherence). For this discussion only first 2 are relevant: 1) the problem of preferred basis and 2) the problem of non observability of interference.

Consider an unknown system that is initially isolated. Now I (or the environment) make measurements on this system and that makes the system vulnerable to decoherence. So it happens that through einselection I can find a set of robust states that are least prone to decoherence (pointer states) Let's call them $|\Psi_1\rangle$ and $|\Psi_2\rangle$. So this process allowed me to gain more information about the system, then I had before. Hence, the reduction in entropy! Now of course, initially the system was in a pure superposition of these states say $|\Psi\rangle = \alpha |\Psi_1\rangle + \beta |\Psi_2\rangle$, with a definite $\alpha$, $\beta$, therefore zero entropy. But as an observer with no initial knowledge of the system (an alien observer on system earth!) I model my belief on the system as an arbitrary superposition, say $|\Phi\rangle = a|\Phi_1\rangle + b|\Phi_2\rangle$ with unknown $a$, $b$, therefore a positive entropy. Now ofcourse, through the measurement process (and thanks to the physical process of einselection) I can find relationships between $(\alpha, \beta, \Psi_1, \Psi_2)$ and $(a,b,\Phi_1,\Phi_2)$. Hence, this entire process of measurement has 2 competing entropy elements, 1) First entropy element that is reduced by gaining knowledge on the pointer states, hence the basis and coefficients, 2) second entropy element that increases due to the maping of pure states to mixed states.

• I'm a little confused about your first point regarding a reduction of entropy. Decoherence is the coupling of a system to a much larger environment and of course adds entropy.
– AGML
Nov 10, 2015 at 19:58
• "Taking away information" = reducing distinguishability = increasing entropy. Nov 10, 2015 at 21:06
• I think "Taking away information" is-not-always-equal-to reducing indistinguishably. Following on the edit above, taking away information also allows the observer (in this case the environment) to know more about the system. Hence, I think there are 2 competing elements here! Nov 10, 2015 at 21:45
• *comment edit: -in- distinguishability Nov 10, 2015 at 21:52
• @PeeyushKumar Information is a measure that is always calculated with reference to a particular observer. So in a measurement process, the information of the observer is increased (ignorance/entropy decreased). In a decoherence process, the information of the observer is decreased (ignorance/entropy increased). I usually find that talking about information as if it is some intrinsic property of a system inevitably leads to confusion. Nov 10, 2015 at 22:02

Better calibrated response:

I'm a little confused about your first argument, that decoherence should somehow reduce entropy. Decoherence is the coupling of a system to a much larger system and of course adds entropy.

Another way of putting it: if one ignores the environmental degrees of freedom, decoherence essentially maps pure states (zero entropy) to mixed states (nonzero entropy). Mixed states can be viewed as pure states plus classical noise. So decoherence adds noise, and thus entropy.

• You make a great point here. So if I view 'decoherence' only as a mapping from pure states to mixed states (ignoring the environmental dof) the explanation completely fits. But there is an inherent assumption on the knowledge of a particular basis. If we suppose an inf-theor-interpretation of entropy which is the amount of information that you lack about the exact physical state of the system ,seelink, then I think there is more to the above question (and my first argument). I will add an edit to the question, to elaborate on this point further. Nov 10, 2015 at 20:50
• You might be confusing uncertainty about the state with uncertainty about the outcomes of measurements. Given a pure state $\psi$ we have complete information about the exact physical state: it is $\psi$. The entropy is zero, because we know the state. Then, we take information away, leaving a mixed state. This can be viewed as multiple pure states with some uncertainty. Now we have only incomplete information about the state. Entropy has been added.
– AGML
Nov 10, 2015 at 21:18
• Yes, I agree with that. I am not confusing these 2 uncertainties. In fact on the contrary, I am thinking about a scenario where both these uncertainties are important and how that relates to noise addition. As you assume in the comment given a pure state ... now suppose there is no initial information on that pure state! Please see the edit on my question, hopefully that will elaborate on this point! Nov 10, 2015 at 21:33

The quantum state is a vector in a Hilbert space. Each basis vector in the Hilbert space is a possible observation one can make. For example, when modelling the position of a free particle, we assign one basis vector to each point in space. The length of the shadow that the state casts upon each vector is the (square root of) the probability that a measurement will detect the state at a given position.

Crucially, if the state is (very nearly) aligned with one of the basis vectors, the measurement is (very nearly) deterministic.

If we had two particles we would need two basis vectors per possible position: one for each particle. Thus, very roughly speaking, the dimension of the Hilbert space associated with a given property is

$N_{properties} * N_{particles}$.

The above equation is obviously nonsense when $N_{properties}$ is infinite (which is usually the case), but the intuition here still holds.

Now suppose we take an initially isolated particle which is in some state very far from the relevant basis vectors. Then measurements on the particle are very undetermined: we have good odds of getting very many different outcomes.

Suppose further we suddenly thrust the poor isolated particle into a system with macroscopic particle number. For example, suppose we open the door on Schrodinger's cat, bringing it into contact with the air in the room, etc. We have suddenly vastly increased $N_{particles}$ in the above equation, and therefore vastly increased the dimensionality of the Hilbert space.

Now think about our original state vector. It will behave classically if it is orthogonal, or nearly orthogonal, to any of the new vectors in the Hilbert space. But we have suddenly added $N_{properties} * N_{particles}$ new vectors - at random orientations w.r.t the state - to be 'almost orthogonal' to!

It is a curious fact about high-dimensional spaces that, as the dimension $D$ of the space is increased, the number of 'nearly' orthogonal vectors to a given vectors scales as $exp(D)$. (See: https://mathoverflow.net/questions/25983/intuitive-crutches-for-higher-dimensional-thinking). Therefore the probability that our state will line up with a basis - that it will behave classically - scales exponentially with particle number. This is decoherence.

• Ah, I didn't read carefully enough - you wanted an information-theoretic analysis. Sorry.
– AGML
Nov 10, 2015 at 19:53
• I like this explanation, gives a great insight. Although I think it's only a partial answer! Nov 10, 2015 at 22:47
• I have been thinking about this... so is in-distinguishability of the environment (or macrosystem)states cause for noise? Nov 12, 2015 at 0:36


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# Permutation Algorithm

 The textbook Algorithms, 4th Edition by Robert Sedgewick and Kevin Wayne surveys the most important algorithms and data structures in use today. This is ideal for our adoption of stream. A permutation, also called an “arrangement number” or “order”, is a rearrangement of the elements of an ordered list S into a one-to-one correspondence with S itself. Check out this video tutorial on how to solve the Rubik's Cube with the T Permutation. 由上面的输出可知，next_permutation会返回所有这些数字排列的下一个排列，该排列按字典序，紧接着当前的排列。 有123可知，下一个为比123大且最小的那一个排列。. This suggests two high-algorithms to compute the sign of a permutation: Express the permutation as a product of transpositions and count the number of transpositions. In particular questions like: "Does an algorithm exist that only uses iteration (loops) to compute all possible combinations of N distinct items?". You can find the c# source code for the permutation algorithm on this page. Either they implement permutation group algorithms in own code for this fundamental recurring problem or use well-tested. Two concepts that often come up in the study of combinatorics are permutaions and combinations of a set of discrete elements. Permutation group algorithms are one of the workhorses of symbolic algebra systems computing with groups. (arrows attempt to show this) STEP 4: EDGE ORIENTATION. After reading up on Heap's algorithm, it sounds like I am using it. Printing all permutations of a string is a very common interview question. Usually, such algorithms require a careful, manual algorithm engineering effort to reach high performance. Graph Algorithms Using Permutation Groups listed as GRAPE. In the algorithm presented below, exchanges are concentrated within the head of the target array such that one swap produces a new ordered list. Week 4-5: Generating Permutations and Combinations March 1, 2018 1 Generating Permutations The algorithm stops at the permutation. There are several algorithms for enumerating all permutations; one example is the following recursive algorithm: If the list contains a single element, then return the single element. More abstractly, each of the following is a permutation of the letters a,b,c, and d: → → → → → a,b,c,d a,c,d,b b,d,a,c d,c,b,a c,a,d,b. A simple hash function from Robert Sedgwicks Algorithms in C book. > is far easier than any cycling/permutation algorithm. 45: F (R U R' U') F' This is the quickest and shortest OLL case. I have first seen this question in my college exam when we were asked to code the solution using C or C++ language. All permutations can be expressed as the product of transpositions. It uses binary predicate for comparison. But algorithms that use the permutations can sometimes benefit from having the changed values in successive elements always be in consecutive positions. …We created a bunch of permutations with nested loops. The rankings provided by permutation feature importance are often different from the ones you get from Filter Based Feature Selection, which calculates scores before a model is created. Here it is. Therefore, they can be executed on any other hypercubic network and on shuffle-exchange and de Bruijn graphs with only a constant slowdown. Scanner ; public class Permutation { public Permutation() { Scanner input = new Scanner( System. Combinatorial algorithms include algorithms for: Generation: List all structures of a given type, such as combinations and permutations, connected components of a graph. Permutation (nPr) and Combination (nCr) calculator uses total number of objects n and sample size r, r\leq n, and calculates permutations or combinations of a number of objects r, are taken from a given set n. --> Heaps's algorithm (Permutation by interchanging pairs) AppleScript by Jean. This article describes how to build an algorithm to check if the given string is a permutation of palindrome string. \$\begingroup\$ If I understand correctly this exercise was to practice python; I'm not sure how much you want to create an algorithm from scratch. Permutations can be considered as the orbits of a dynamic system p' = succ(p), where p is a permutation and p' is its successor in the lexycographically ordered set of permutations (plus the Nil permutation, whose successor is Nil). If all of the permutations are listed numerically or alphabetically, we call it lexicographic order. , 3 2 1 0), so can't possibly generate random derangements with uniform probability. Given a string, write a function to check if it is a permutation of a palindrome. Here is a diagram of the operational stages of S-DES. Prev_permutation function reorders the range [first, last) into the previous permutation from the set of all permutations that are lexicographically ordered. Of course it can not be exceeded but algorithms which run as fast as it may be write out. Find All Permutations of string - C# // 1. Permutation algorithm In a previous post , I described how I am using model vectors to represent gene structure along a genomic sequence. The easy-handling optical setup to. Symmetric QR Algorithm with Permutations Aravindh Krishnamoorthy Abstract—In this paper, we present the QR Algorithm with Permutations that shows an improved convergence rate compared to the classical QR algorithm. Now, we want to orient the corners in such a manner that when we apply the permutation algorithms, the end up in the correct orientation. This way, if we start with an initial array (which is an even permutation by definition, since zero swaps is an even number of swaps), we will skip over all odd permutations and only return even ones. Proposed Methodologies The methodologies used to determine the best vehicle routing for truck dispatch system (TDS) are Permutation Enumerator Greedy Search Algorithm 1/12/2016 11 11. \$\begingroup\$ If I understand correctly this exercise was to practice python; I'm not sure how much you want to create an algorithm from scratch. Algorithms for Generating Permutations and Combinations Section 6. How OP will achieve that is his/her task. Permutation of String in Java Algorithm To get all the permutations, we will first take out the first char from String and permute the remaining chars. * * This algorithm works correctly because every permutation has a Lehmer code,. Thomas, St. com Pingzhong Tang IIIS, Tsinghua University [email protected] With direct addressing, an element with key k is stored in slot k. Printing all permutations of a string is a very common interview question. The formula of permutation of arranging k elements out of n elements is − n P k = n! / (n - k)! Algorithm. One implementation of shift cursor algorithm gives an optimal solution of permutation generation problem, and one implementation of level algorithm can be used to generate. All the next permutation algorithm needs are iterators which can advance forwards or backwards, step by step. 14 Dec 2014. Print All The Permutations Of a String Dynamic Programming - Count all paths in 2D Matrix with Obstructions in it Stack Data Structure - Introduction and Implementation. To write out all the permutations is usually either very difficult, or a very long task. The formula of permutation of arranging k elements out of n elements is − n P k = n! / (n - k)! Algorithm. Symmetric QR Algorithm with Permutations Aravindh Krishnamoorthy Abstract—In this paper, we present the QR Algorithm with Permutations that shows an improved convergence rate compared to the classical QR algorithm. A permutation is specified as each of several possible ways in which a set or number of things can be ordered or arranged. All these examples have in common that they use diUerent tools to handle the ba-sic symmetry part of their real computational task. Unique Permutations efficient algorithm Showing 1-7 of 7 messages. There are now three randomization plan generators. P yields a 32-bit output from a 32-bit input by permuting the bits of the input block. An algorithm which ensures that each new permutation is created by exchanging only two neighboring elements is called a minimal change algorithm. next_permutation: 32. Going back to our pool ball example, let's say we just want to know which 3 pool balls are chosen, not the order. Thanks for the answer to the question. Permutation of the Last Layer. Algorithms are selected from the literature (work in progress, see "REFERENCES"). permutation algorithm for laser antimissile strategy optimization 3771 When many missiles are coming at the same time from one direction, the radar system detects the targets and measures the positions and computes the velocities of them. This unit covers methods for counting how many possible outcomes there are in various situations. Graph Algorithms Using Permutation Groups - How is Graph Algorithms Using Permutation Groups abbreviated?. Permutation refers number of ways in which set members can be arranged or ordered in some fashion. Each test case contains a single string S in capital letter. Test whether range is permutation of another Compares the elements in the range [first1,last1) with those in the range beginning at first2 , and returns true if all of the elements in both ranges match, even in a different order. Our results also answer an open problem on what can be achieved by permutation manipu- lations. Answer: Here is a recursive solution to print all the permutations of a string. By far, the biggest bulk of sites that I saw. It appears that the permutation used in the sponge to stir the state is more or less c Stack Exchange Network Stack Exchange network consists of 175 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Recently, I received many questions regarding exhaustive combinations using linear data structures such as arrays or character strings. This ordering on the permutations was known to 17th-century English bell ringers. Is there any algorithm to generate all permutations of the integers 1 to n [ Hint: The set of permutations of the integers 1 to n can be obtained from the set of permutations of the integers 1 to n-1 by inserting n in each possible positions of each permutation ]. for algorithm A, choose a sample space S and probability. JAH, Arizona Summer Program 2008 Basic Algorithms for Permutation Groups 3 / 22 Group actions A group G acts (from the right) on a set if!1 = !for all !2 (!g)h = !gh for all !2, g;h 2G. In pseudo-code, then, our algorithm to enumerate even permutations is: Set ParityFlip to False. In Mathematics, a permutation is an arrangement of distinct items in various orders 123,132,213,231,312,321. For example, for the set {a, b, c, T} we can define two different permutations (but there are more, of course} a, c, T, b and T, c, b, a. To calculate permutations with repetition of elements using C#, I recommend starting with the CodeProject article by Adrian Akison entitled ‘ Permutations, Combinations, and. the last permutation, then the next_permutation() will return false and set the permutation to the first permutation the smallest in the ascending order. In order to count how many positions we have for the Rubik’s Cube, we will first learn how to count permutations for our set of marbles. Recently, I've read about 'quantum bogosort' on some wiki. \$\endgroup\$ – BurnsBA Sep 22 '17 at. Repeating the permutation and averaging the importance measures over repetitions stabilizes the measure, but increases the time of computation. 1 Quotient graph example. There are 21 different variations of Last Layer Permutations, and a well-known name for each. We determine a bound for per-formance based on best instantaneous convergence, and develop. 9 Diﬀerential, Linear, and Interpolation Attacks on 56 Block Ciphers 8. Keyword Permutations. The formula for finding the total number of permutations is factorial of number of elements. Permutating the last layer could be tricky, and could get you back to square one if you don't know what you are doing, but don't worry, you'll be able to solve that Cube in every possible PLL situation. For example: If there are 5 people, Jim, Jane, Bob, Susan, and Ralph, and only 3 of them can be on the new PTA committee, how many different combinations are possible?. Paul, USA fsarah. There are computer algorithms and programs to help you with this, and this is probably the best solution. I know that it can be done with multiplying a matrix with a permutation matrix. •Simple recursive method does the job. Stochastic local search methods are at the core of many effective heuristics for tackling different permutation flowshop problems (PFSPs). As an example, if the string is "abc" there are 6 permutations {abc, acb, bac, bca, cab, cba}. Finding all permutations of a String in a Java Program is a tricky question and asked many times in interviews. (Rotating them before replacing elements). Rather, it's generating each permutation on the fly, as it's required. Comparison of the 2 Algorithms The odometer landing is both concise and efficient, so it wins the "style" award, and it as well can hold VAST games (cognate a million balls from a billion!). For instance, Quicksort, Mergesort, and Insertion-sort are all comparison-based sorting algorithms. for algorithm A, choose a sample space S and probability. More abstractly, each of the following is a permutation of the letters a,b,c, and d: → → → → → a,b,c,d a,c,d,b b,d,a,c d,c,b,a c,a,d,b. In this paper we study the a ne equivalence problem, where. Permutation of Last Layer (PLL) PLL, the last step of CFOP, permutes all last-layer pieces while preserving their orientation. java * Execution: java Permutations n * * Enumerates all permutations on n elements. As such, a meta-collection enumerator on any but a trivial set of items will quickly exceed any available computation time. C# Sharp programming, exercises, solution: Write a program in C# Sharp to generate all possible permutations of an array using recursion. •Maintain array a[] where a[i] represents bit i. We'll discuss this problem and some interesting variations of it. Then, a discrete DE algorithm is derived for permutation problems and it is applied to the permutation flowshop scheduling problem with the total flowtime criterion. Optimization problems over permutation matrices appear widely in facility layout, chip design, scheduling, pattern recognition, computer vision, graph matching, etc. Returns true if such permutation exists, otherwise transforms the range into the first permutation (as if by std::sort(first, last)) and returns false. In particular we de ne three re ection operations on ˇ 2S n. The Algorithms & Permutations 2012 workshop (Paris, February 20-21, 2012), brought together people coming from different algorithmic backgrounds working on permutations. Heap’s algorithm constructs all permutations of a given sequence. If no edges are yellow on top, apply first algorithm and then the second. The resulting data structure is called hash table. This article will describe a quick and easy algorithm that gives the full permutation for a natural number. Akgun a,1, Akira Todoroki b a Department of Aerospace Engineering, Mechanics and Engineering Science, University of Florida, Gainesville, FL 32611-6250, USA. How to Find All Permutations of String in Java using Recursion How to find all permutation of a String using recursion is one of the tricky coding questions from Programming job interviews. propose an optimal permutation routing on mesh networks. The algorithm minimizes movement: it generates each permutation from the previous one by interchanging a single pair of elements; the other n−2 elements are not disturbed. You can then bit-fiddle. The formula for finding the total number of permutations is factorial of number of elements. Conventionally, the elements are stored in a mutable array and then randomly swapped a few times. The first time I saw this code was in the original STL published by Alexander Stepanov and Ming. an O(log n) FFT algorithm. my algorithm analysis instructor told about this and stole if from class note (i am a good student, btw!). As such, a meta-collection enumerator on any but a trivial set of items will quickly exceed any available computation time. What are the rules to do that and ensure the single-cycle permutations are produced with uniform probability if the original permutation is produced with uniform probability?. Our results also answer an open problem on what can be achieved by permutation manipu- lations. the largest clique in a permutation graph corresponds to the longest decreasing subsequence in the permutation defining the graph, so the clique problem may be solved in polynomial time for permutation graphs by using a longest decreasing subsequence algorithm. If you want all the permutations with exactly 4 1's, then I would count from 00000000 to 0x11110000 and keep the numbers with 4 1's and throw away the. If all of the permutations are listed numerically or alphabetically, we call it lexicographic order. Stochastic local search methods are at the core of many effective heuristics for tackling different permutation flowshop problems (PFSPs). first energy-efficient permutation routing appeared in [18]. I have first seen this question in my college exam when we were asked to code the solution using C or C++ language. Graph Algorithms Using Permutation Groups - How is Graph Algorithms Using Permutation Groups abbreviated?. Below are the permutations of string ABC. Let's start with permutations, or all possible ways of doing something. We're using the fancy-pants term "permutation", so we're going to care about every last detail, including the order of each item. For example, for the set {a, b, c, T} we can define two different permutations (but there are more, of course} a, c, T, b and T, c, b, a. This Standard also specifies the K ECCAK -p family of mathematical permutations, including the permutation that underlies K ECCAK , in order to facilitate the development of. To make a greater permutation, some element of the list must be replaced by a larger element to its left; to make the next permutation, the replacement must happen as far to the left as possible, in the least significant position, and the replacing element must be as small as possible. Since String is immutable in Java, the idea is to convert the string to character array. While those expression are not unique, if we order the transpositions in order of highest element moved, then that expression is unique. Please try again later. Write a code to generate all permutations of given string in lexicographically sorted order with repetition of characters in the string. Permutation of String in Java Algorithm To get all the permutations, we will first take out the first char from String and permute the remaining chars. Sims introduced algorithms for working with permutation groups. With direct addressing, an element with key k is stored in slot k. And that's all this implementation uses. The next permutation is derived from the difference between permutation p-1 and permutation p-2, and the next, from the difference between permutation p-2 and permutation p-3 and so on back to the start of the subset. I use the left index for the U' and the right thumb for the F'. This article will describe a quick and easy algorithm that gives the full permutation for a natural number. It was created by Stefan Pochmann. This operation is a bit index manipulation and performs any bit complement permutation where k is the value to be xor-ed with the bit index, and is therefore also called the xor permutation which is a subset of bit permute/complement permutations. Based on such bit level permutation, several methods have been proposed in the literature. Initially I needed a permutation algorithm to create all the possible combinations of letters so I could unscramble words. Heap's Algorithm is used to generate all the possible permutation of n-decimals of a number. Algorithm to find Permutations of String using Recursion Find Permutation and Combination of a String , such type of questions can be asked in the written round of the major tech giants like Amazon. Suppose we have a finite sequence of numbers like (0, 3, 3, 5, 8), and want to generate all its permutations. The person who played the last turn (after which the sequence becomes increasing) wins the game. Here is a quick simple Algorithm which computes all Permutations of a String Object in Java. Combinatorial algorithms are algorithms that deal with combinatorial structures, which are sets, ordered n-tuples, and any structures that can be built from them, like graphs. Permutation Group Algorithms. , XOR), and cascade them in a series of stages. Stochastic local search methods are at the core of many effective heuristics for tackling different permutation flowshop problems (PFSPs). As such, a meta-collection enumerator on any but a trivial set of items will quickly exceed any available computation time. We'll also look at how to use these ideas to find probabilities. For example, the permutation matrix will interchange rows 1 and 2 and also interchange rows 3 and 4. Used to code the math Σ and Π symbols. Our results also answer an open problem on what can be achieved by permutation manipu- lations. Initially I needed a permutation algorithm to create all the possible combinations of letters so I could unscramble words. Randomized Algorithm. , 3 2 1 0), so can't possibly generate random derangements with uniform probability. String Permutations Algorithm with example code in C/C++. At the same time is a pretty small problem. Permutations in Python - Part 1. Bit level permutations particularly, are the core of any encryption algorithm. /***** * Compilation: javac Permutations. For each permutation of the response vector y^{*s}, a new forest is grown and the permutation variable importance measure (VarImp^{*s}) for all predictor variables X is computed. JAH, Arizona Summer Program 2008 Basic Algorithms for Permutation Groups 3 / 22 Group actions A group G acts (from the right) on a set if!1 = !for all !2 (!g)h = !gh for all !2, g;h 2G. Gray code, etc. If not, then it transforms the given range into the lexicographi-cally smallest permutation and returns false. If you are familiar with C++, I might recommend the lexicographic order permutations algorithm from the fxtbook (AKA "Matters Computational") as a reference. In particular questions like: "Does an algorithm exist that only uses iteration (loops) to compute all possible combinations of N distinct items?". Combinatorial algorithms are algorithms that deal with combinatorial structures, which are sets, ordered n-tuples, and any structures that can be built from them, like graphs. When describing the reorderings themselves, though, note that the nature of the objects involved is more or. A permutation where no element ends up in its original position is called a derangement. Symmetric QR Algorithm with Permutations Aravindh Krishnamoorthy Abstract—In this paper, we present the QR Algorithm with Permutations that shows an improved convergence rate compared to the classical QR algorithm. Plug-in required. Given a string S. A discrete bat algorithm (DBA) is proposed for optimal permutation flow shop scheduling problem (PFSP). I found Heap's algorithm and implemented that in VBA and it works great however the algorithm is for every permutation of n values. (Rotating them before replacing elements). The easy-handling optical setup to. Read "A discrete differential evolution algorithm for the permutation flowshop scheduling problem, Computers & Industrial Engineering" on DeepDyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. (c) Go to the position with the same allele in P1. C++ Algorithm Library - is_permutation() Function - The C++ function std::algorithm::is_permutation() tests whether a sequence is permutation of other or not. Test whether range is permutation of another Compares the elements in the range [first1,last1) with those in the range beginning at first2 , and returns true if all of the elements in both ranges match, even in a different order. Each permutation is generated by exchanging two adjacent elements of the preceding permutation. 201] give an algorithm for permutations where. After round 16,L 16 and R 16 are swapped,so that the decryption algorithm has the same structure as the encrption algorithm. Digital Video Encryption Algorithms Based on Correlation-Preserving Permutations Daniel Socek, Spyros Magliveras, Dubravko Culibrk, Oge Marques, Hari Kalva and Borko Furht Florida Atlantic University, Boca Raton FL 33431 [email protected] Comparison of the 2 Algorithms The odometer landing is both concise and efficient, so it wins the "style" award, and it as well can hold VAST games (cognate a million balls from a billion!). Algorithm Paradigm: Backtracking Time Complexity: O(n*n!) Note that there are n! permutations and it requires O(n) time to print a a permutation. Cycle Slicer: An Algorithm for Building Permutations on Special Domains Sarah Miracle and Scott Yilek University of St. Heap's Algorithm is used to generate all the permutations of a given object. at any decomposition stage. 📝 Algorithms and data structures implemented in JavaScript with explanations and links to further readings JavaScript Algorithms and Data Structures. English: Output of 9 different algorithms for generating all 4!=24 permutations of length n=4. Prev_permutation function reorders the range [first, last) into the previous permutation from the set of all permutations that are lexicographically ordered. How to generate all Permutations - posted in C/C++ Tutorials: Problem: Given a string, print it's all permutationsThis is a situation that might arise in a programmer's every day project activities. The resulting data structure is called hash table. * * This algorithm works correctly because every permutation has a Lehmer code,. Algorithm Paradigm: Backtracking Time Complexity: O(n*n!) Note that there are n! permutations and it requires O(n) time to print a a permutation. Imagine a fearsomely comprehensive disclaimer of liability. I was looking over this question requesting an algorithm to generate all permutations of a given string. Application of permutation genetic algorithm for sequential model building… 3025 the sequence is an OLH DoE, however, the merged MB– MV DoE, while it is optimised for space ﬁllingness, it does. The explanation for sample 0 does not make sense to me. Mathematica 8 adds new functions and algorithms to work with permutations and permutation groups. 2 Python and Perl Implementations of the Key 46 Expansion Algorithm 8. Randomized Algorithm. algorithms for the random generation of special types of permutations such as invo-lutions or permutations with minimum cycles length would be known. Please see below link for a solution that prints only distinct permutations even if there are. (a) Start with the first allele of P1. Given n and k, return the k-th permutation sequence of permutations of numbers {1,2,. Recursive Algorithm. Print All The Permutations Of a String Dynamic Programming – Count all paths in 2D Matrix with Obstructions in it Stack Data Structure – Introduction and Implementation. This article describes how to build an algorithm to check if the given string is a permutation of palindrome string. Based on such bit level permutation, several methods have been proposed in the literature. A permutation is a rearrangement of letters. One of the ideas that hit my mind was using array cyclic shift. > is far easier than any cycling/permutation algorithm. It's pretty tricky to implement your own generic permutation algorithm, and easy to write one inefficiently or incorrectly, and as such belong in the Swift standard library. ) def permute_next(values): """ Alter the list of values in-place to produce to the next permutation in lexicographical order. Permutation routing on binary hypercube All the greedy permutation routing algorithms on SF all-port end-to-end oBF n are normal hypercube algorithms. JAH, Arizona Summer Program 2008 Basic Algorithms for Permutation Groups 3 / 22 Group actions A group G acts (from the right) on a set if!1 = !for all !2 (!g)h = !gh for all !2, g;h 2G. For example, 54321’s next permutation will be 12345. Below is the syntax highlighted version of Permutations. Conventionally, the elements are stored in a mutable array and then randomly swapped a few times. The numbers of permutations, combinations, and variations all grow exponentially. We'll discuss this problem and some interesting variations of it. Given n and k, return the k-th permutation sequence of permutations of numbers {1,2,. We already know that 3 out of 16 gave us 3,360 permutations. 23 -> 123, 213, 231. Given a string S. For example, given the set of four characters { t, e, s, t }, the algorithm will return { test } twice; once for the permutation { 0123 } and once for { 3120 }. This work describes the theory behind permutation group algorithms, up to the most recent developments based on the classification of finite simple groups. Example: You want to visit the homes of. Here are some examples. This chapter surveys combinatorial properties of permutations (orderings of the numbers 1 through N) and shows how they relate in a natural way to fundamental and widely-used sorting algorithms. Permutation routing on binary hypercube All the greedy permutation routing algorithms on SF all-port end-to-end oBF n are normal hypercube algorithms. How to generate all Permutations - posted in C/C++ Tutorials: Problem: Given a string, print it's all permutationsThis is a situation that might arise in a programmer's every day project activities. Page 3 Genetic Algorithm. Printing all permutations of a string is a very common interview question. The task is to print all permutations of a given string. This is done iteratively for each of the feature columns, one at a time. If any sum fails to match magicSum,. Just stumbled upon this, but found the information useful. The first algorithm we look at is Heap’s algorithm. Early on I shared the forum chatter and algorithm tracking tools to show that it was more widespread than the typical update. Here is one such algorithm, which generates the permutations in Lexicographical order. In order to achieve a stream of permutations, we need Johnson Trotter algorithm and a stream. An Algorithm to Enumerate Sorting Reversals for Signed Permutations ADAM C. [2] The sequence of permutations of n objects generated by Heap's algorithm is the beginning of the sequence of permutations of n +1 objects. Lisppaste pastes can be made by anyone at any time. Due to the parity the amount of the correctly positioned yellow cubelets is limited to three cases : there's no yellow corner piece in the correct position, or there's only one, or all the four pieces are correct. • The chromosome is divided into parts: genes. Home › Math › Navigate a Grid Using Combinations And Permutations Puzzles can help develop your intuition -- figuring how to navigate a grid helped me understand combinations and permutations. The evenness or oddness of a permutation is called its parity. Algorithms for Generating Permutations and Combinations Section 6. In Mathematics, a permutation is an arrangement of distinct items in various orders 123,132,213,231,312,321. 10 Homework Problems 89 2. Theorem (PA = LU; Factorization with Pivoting). The Collected Algorithms (CALGO) is part of a family of publications produced by the ACM. Optimization problems over permutation matrices appear widely in facility layout, chip design, scheduling, pattern recognition, computer vision, graph matching, etc. Permutation Last Layer. Algorithm To Calculate Permutations of a String. 6 seconds and runs out of. Finding all permutations of a String in a Java Program is a tricky question and asked many times in interviews. Question: Print all the permutations of a string. The formula of permutation of arranging k elements out of n elements is − n P k = n! / (n - k)! Algorithm. This algorithm runs in factorial time, and takes quite a while if you run it with an array longer than 10 items. Numeric algorithms ( accumulate ): adds up items in range starting with given initial value, can also do any binary operation if it is given as an function of two arguments( binary_function ). Permutation Checking Algorithm Algorithm CheckPermutation(L) Check is list L contains a permutation of the integers from 1 to N. 6 seconds and runs out of. Algorithms for Generating Permutations and Combinations Section 6. But then I thought: if I know the original string, then the individual bits of the shuffled one can't be independent of each other. Unique Permutations efficient algorithm Showing 1-7 of 7 messages. I decided to whip up a little code to help me solve a word puzzle problem a few days ago (more on this in later parts). A permutation is specified as each of several possible ways in which a set or number of things can be ordered or arranged. Think of stellar constellations and their names as a good example. This tutorial will present three methods of orienting corners. This work describes the theory behind permutation group algorithms, up to the most recent developments based on the classification of finite simple groups. \$\begingroup\$ If I understand correctly this exercise was to practice python; I'm not sure how much you want to create an algorithm from scratch. Algorithm 1. …And let me just bring that file up. There are 24 permutations of an array with four elements and Heap's algorithm should do 23 swaps, but this implementation does 40 swaps. So could anyone find where I am wrong? The expected output differs from input only for the last three elements. And we were just looking at this first file right here. , XOR), and cascade them in a series of stages. practice the algorithm thinking and its implementation. This post describes how I did it. Re: Permutation Algorithm 843853 Mar 25, 2003 5:36 PM ( in response to 843853 ) do u need to keep the results or just print them? it will break ur machine if u run it with an array 280 long! also, it would be easier to recode it to work on general Lists rather than Arrays, so you could call it. All the next permutation algorithm needs are iterators which can advance forwards or backwards, step by step. Theorem (PA = LU; Factorization with Pivoting). edu; [email protected] But then I thought: if I know the original string, then the individual bits of the shuffled one can't be independent of each other. I am not sure what you mean by a "permutation. Below is the syntax highlighted version of Permutations. Each permutation is associated with a letter that loosely corresponds to the permutation pattern to allow easy identification of the algorithms. Find All Permutations of string - C# // 1. Usually, such algorithms require a careful, manual algorithm engineering effort to reach high performance. symmetry computation part translates into computations with and search in permutation groups. It uses binary predicate for comparison. Geometrically, this search corresponds to a simplex-type algorithm on a sub-polytope of the permutohedron, the DAG associahedron.


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# Tag Info

15

From what I remember, the MBTI has been compared in some studies to the Big Five (or OCEAN) model of personality. If you've not heard of it, the Big Five is the primary theory of personality that is accepted by researchers who do this sort of thing. Here are some papers comparing the two approaches: Recent comparison and another. The main point is that a ...

11

Facial shape Aggression relates to facial width-to-height ratio (Carré, McCormick, & Mondloch, 2009; Carré & McCormick, 2008). The width-to-height ratio is the distance between the left and right zygion (the outside of the cheek bone) divided by the distance between the top of the upper lip and the mid-brow. Here's a useful image displaying the ...

11

The IPIP may provide what you are looking for. This IPIP Website is intended to provide rapid access to measures of individual differences, all in the public domain, to be developed conjointly among scientists worldwide. In general, the scientific literature tends to focus more on a dimensional approach based on the Big 5 model of personality than ...

10

Digman (1997) performed higher order factor analysis on several Big 5 intercorrelation matrices and proposed that the Big 5 could be partially explained in terms of two higher order factors: Alpha (Agreeableness, Conscientiousness, Emotional Stability) and Beta (Extraversion, Openness). Musek (2007) presented evidence for a Big 1 factor of personality. ...

10

What you may be interested are personality traits caught by some statistical correlations, rather than 'manually' merged by an author's intuition. One of such tools is 16 Personality Factors. The 16 Personality Factors, measured by the 16PF Questionnaire, were derived using factor-analysis by psychologist Raymond Cattell. This article summarizes the ...

10

It's a matter of degree. First of all, "shyness" is not a psychological or psychiatric term, but an everyday English word denoting a commonly observable personality characteristic on a par with courage, cheerfulness, or honesty. The meaning of "shyness" is not exactly defined, and people may use the word "shyness" to refer to different kinds of behaviors, ...

9

Do personality tests predict job performance in general? There is a large academic literature correlating personality test scores with job performance. You might want to check out the meta-analysis by Barrick et al (2001). It reports the meta-analytic correlations often based on hundreds of studies between Big 5 personality test scores and job performance. ...

8

Heritability estimates of Extraversion (and other Big 5 factors) The introduction section of Loehlin et al (1998) provides a narrative review of heritability estimates of big 5 personality traits (i.e., one of which is extraversion). A brief extract gives a flavour of some of the research that has been conducted: A recent heritability analysis of the ...

8

I've been commenting a lot and not answering for quite some time now! My hesitation was largely due to the terms "unfailingly", the mixture of "altruism" and "non-cruelty", and having forgotten my intention to answer. My answer certainly won't be foolproof (haven't yet heard of a trait model that is), and I may not cover both altruism and non-cruelty, but ...

8

A useful model for topics like this comes from McCrae and Costa (1999): There's quite a lot more going on here than is pertinent to your question, but note that influences originate from biological bases on the top left, and from external influences on the upper right. Everything else is modeled as an effect of mediated, dynamic processes between these ...

8

I'm not entirely sure what you mean by "renounce," but you may find some answers to your question in the fields of social psychology and behavioral economics. The first part of your question describes someone who is unwilling to commit or make a decision for fear of losing out on future opportunities. There has been research to indicate that having more ...

8

I was alerted to your post here and would like to respond to a non-technical side of the question. I'm an Intellectual Property Assistant for CPP, Inc., publisher of the MBTI® assessment tool. I help protect the instrument's trademarks and copyrights, and ensure ethical use of the instrument. The names MBTI®, Myers-Briggs®, and Myers-Briggs Type ...

8

Ellis (1974) thought that men prefer rotund features in women and that blonde hair creates such a roundish upper body shape because the blonde hair, which is close in color to (a white woman's) skin, would blend with the skin. In his famous book, The naked ape, Desmond Morris (1967) documented that men prefer adolescent features. In his opinion blonde hair ...

7

The MBTI is based on Carl Jung's work with psychological types. However, Jung's work led to the formation of analytic psychology. This work is often associated with clinical observations and anecdotes instead of controlled scientific study. This means that Jung didn't carry out research that can be considered conclusive and scientifically validated. However, ...

7

I'm guessing you are relating the Big 5 factors of personality to Meyer and Allen's three factor model of commitment (normative, continuance, affective). You certainly could adopt the simple approach of creating observed composite scores for each variable and reporting the correlation matrix and a set of multiple regressions. However, the language of your ...

7

This is an extremely interesting question. I'm going to take a different approach to the question by focusing on both personality traits and leadership theories (e.g. authentic leadership, transformational leadership, servant leadership etc) to answer whether those two distinct areas can influence leaders' children's development. I will admit that I didn't ...

6

There is a large general literature evaluating the degree to which personality tests predict job performance. In particular see for example the review by Barrick et al (2001). In general such reviews find that personality measures provide a small but meaningful prediction of job performance. Ipsative testing So in general, you are asking about how ...

6

This is a primary subject of study in game studies and ludology, which are domains of theory and research unto themselves. Moreover, the question as it pertains to games might be interesting to ask over on Arqade, though I can't guarantee it would be "on topic" enough for their community. You might find the following questions from Arqade interesting, as ...

6

Perhaps check out the Obsessive Beliefs Questionnaire which is designed to measures dysfunctional beliefs related to OCD. There are several papers that examine the factor structure, and perfectionism is typically seen as one of the major aspects of obsessive beliefs. However, it seems to be a particular form of maladaptive perfectionism. See these ...

6

I've edited your question somewhat; I hope I preserved its meaning. Neuroticism relates to personal discomfort largely by definition, and probably to uncomfortable social interaction as well, though somewhat less by definition. Consider this hypothetical, mediated pathway: Neuroticism $\rightarrow$ Anxiety $\rightarrow$ Social anxiety $\rightarrow$ ...

6

Several other disorders relate to introversion, including: Avoidant personality disorder (Morey et al., 2002) Obsessive-compulsive disorder (Samuels et al., 2000) Schizotypal personality disorder (Funder, 1997? Don't have it on-hand, but will try to verify this later when I do.) Schizoid personality disorder (Morey et al., 2002) Generalized anxiety ...

6

Personality studies are typically investigated via the "big five" traits. Some attempts to legitimize the personality traits through biology have been pursued [1]. Following are some excerpts from a particular study that proposed a hypothesis and searched for confirmation in 116 subjects. Of course, this means we should be wary of confirmation bias when ...

6

I am using TRUE (http://www.salleurl.edu/tsenyal/true/features.html) and SAM (Self-Assessment Manikin) interface includes a scale for dominance, along with valence and activation. The scale allows you to rate dominance from 'controlled' to 'in control' on a scale from 1 to 9. References Planet, S., Iriondo, I., Martínez, E., & Montero, J. A. (2008). ...

6

The real question here is whether traditional personality dimensions have such a big influence on behavior as usually thought or, perhaps, if there is such a thing as being “really and truly extroverted from the ground up”. Once you begin to admit that circumstances play an important role or that personality traits are fuzzy types, being “truly extroverted” ...

6

Consider persecutory delusion. They are compulsive in that they manifest as extreme, irrational suspicions with little or no basis in reality. The irrational components often involve absurdly elaborate conspiracy theories, so this is probably close enough to "the entire world", but it often goes much further than "looking down" into more severe forms of ...

Only top voted, non community-wiki answers of a minimum length are eligible


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## Thursday, April 9, 2015

### Physics Links

The recent episode of Skeptics Guide to the Universe podcast had two interesting "basic" physics stories.

1. A new twist to how long it takes to "fall through" a tunnel that stretches from one end of the earth to the other.

The standard answer models the earth is a sphere of uniform density, and resulting trajectory is simple harmonic oscillation, with time period $T = 2 \pi \sqrt{\frac{R}{g}},$ where R is the radius of the earth, and g=9.8 m/s^2 is the gravitational acceleration on the surface.

The required time is T/2 which is approximately 42 minutes.

2. Does a half dead battery bounce more than a full battery when dropped? Neat videos here.
However, “maximum bounce” is reached when the battery is down to about half its charge, at which point the amount of bounce levels off despite the fact that more zinc oxide is still forming. So the bounce technique can reveal that a battery is not fresh, but it is not an indicator that it’s entirely flat. Still, it’s an easy and instant way of checking the profusion of batteries filling our drawers – no multimeter required.


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Why would it be faster to render using multiple canvas elements?

This video suggests that rendering using multiple canvas elements helps performance.

How does this work?

The idea was to use one for the player, one for the enemies, another for the background, though I really don't understand how could it be faster if you use more of the same thing.

Could someone clarify?

• Can you please provide a code example which demonstrates the effect? jsperf.com is good for providing performance-comparisons of javascript code to an online audience. – Philipp Dec 30 '14 at 15:42
• Actually I was just learning about it. Never tried it, but it is simple, you create a number of canvas elements in html then select them with getElementById, use their contexts separately for each player, enemy or whatever. Here is the video I learnt it: youtube.com/watch?v=fjVJ0Fju7tg This video gives an example. So my question is not about the code itself, but more of the theory behind it. – GSquadron Dec 30 '14 at 15:48
• When you have questions about a youtube video with just 1700 views, you might want to ask the author for clarification. – Philipp Dec 30 '14 at 15:51
• But he already is explaining it and I don't understand it in the way he explained. So maybe there could be someone else who might already know, or maybe learn something new. Anyway I will ask him. – GSquadron Dec 30 '14 at 15:53
• – Anko Dec 30 '14 at 16:06

Multiple canvases can help in some cases. Just as a quick example - say you have 2 layers (canvases) to your game.

The background of your game is not going to change very frequently, for example. So one canvas will be dedicated to just rendering the background of your game. This means you only have to render the background once, and only re-render it when it changes. The other layer of your game can render anything that moves constantly.

If all were rendered on the same canvas, the background would be re-drawn every frame along with everything that moves, and this can cause a more slow rendering time.

I hope this was somewhat clear - I can elaborate more if needed.

• Good answer in my opinion. Another example will be a player which is always in the center of screen and the background moving – Erez Dec 30 '14 at 16:43
• Thanks for the good answer. If you can elaborate this more, I think all would greatly benefit. For example, how do I know when to use multiple or just one canvas, if the background canvas is going to change or move in intervals of seconds. – GSquadron Dec 30 '14 at 17:32
• You can clear and redraw a rectangle in the middle of your canvas which is faster than doing so for the whole canvas. So if say we draw the hud in another canvas and your hp reduced, just redraw the hp section – Erez Dec 30 '14 at 19:04
• So to sum it up, use multiple canvases and region clearing to do the least unnecessary drawing you can have. Of course it's not a must, just a good practice – Erez Dec 30 '14 at 19:09
• I don't think there is a "formula" to determining if multiple should be used or not, but just think of "how often will this part be updated?" and "is this part only a small portion of the screen, so I can use a smaller canvas?" – CodeNMore Dec 30 '14 at 23:02


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## Numerical solutions of stochastic differential delay equations under local Lipschitz condition.(English)Zbl 1015.65002

Under certain hypotheses, which include the less restrictive assumption that $$f,g$$ satisfy a local (rather than global) Lipschitz condition, a theorem is proved establishing convergence of Euler-Maruyama approximate solutions to the solution of the stochastic differential delay equation with variable delay $dx(t)= f\biggl(x,\bigl( \delta(t)\bigr)\biggr) dt+g\biggl(x(t), x \bigl(\delta(t) \bigr)\biggr) dB(t)$ where $$B$$ is an $$m$$-dimensional Brownian motion.

### MSC:

 65C30 Numerical solutions to stochastic differential and integral equations 65L20 Stability and convergence of numerical methods for ordinary differential equations 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 34F05 Ordinary differential equations and systems with randomness 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
Full Text:

### References:

 [1] Baker, C.T.H.; Buckwar, E., Numerical analysis of explicit one-step methods for stochastic delay differential equations, LMS J. comput. math., 3, 315-335, (2000) · Zbl 0974.65008 [2] Buckwar, E., Introduction to the numerical analysis of stochastic delay differential equations, J. comput. appl. math., 125, 297-307, (2000) · Zbl 0971.65004 [3] Hale, J.K.; Lunel, V.S.M., Introduction to functional differential equations, (1993), Springer Berlin [4] D.J. Higham, X. Mao, A.M. Stuart, Strong convergence of numerical methods for nonlinear stochastic differential equations, SIAM J. Num. Anal., 2002, to appear. · Zbl 1026.65003 [5] Y. Hu, S.E.A. Mohammed, F. Yan, Numerical solution of stochastic differential systems with memory, Preprint, Southern Illinois University, 2001. [6] Kloeden, P.E.; Platen, E., Numerical solution of stochastic differential equations, (1992), Springer Berlin · Zbl 0925.65261 [7] Kolmanovskii, V.; Myshkis, A., Applied theory of functional differential equations, (1992), Kluwer Academic Publishers Dordrecht · Zbl 0917.34001 [8] Komori, Y.; Saito, Y.; Mitsui, T., Some issues in discrete approximate solution for stochastic differential equations, Comput. math. appl., 28, 269-278, (1994) · Zbl 0810.65145 [9] Küchler, U.; Platen, E., Strong discrete time approximation of stochastic differential equations with time delay, Math. comput. simulation, 54, 189-205, (2000) [10] Mao, X., Approximate solutions for a class of stochastic evolution equations with variable delays—part II, Numer. funct. anal. optim., 15, 1 &2, 65-76, (1994) · Zbl 0796.60068 [11] Mao, X., Exponential stability of stochastic differential equations, (1994), Marcel Dekker New York · Zbl 0851.93074 [12] Mao, X., Stochastic differential equations and applications, (1997), Horwood England · Zbl 0874.60050 [13] Milstein, G.N., Numerical integration of stochastic differential equations, (1995), Kluwer Academic Publishers Group Dordrecht · Zbl 0810.65144 [14] Platen, E., An introduction to numerical methods for stochastic differential equations, Acta numer., 8, 195-244, (1999) [15] Saito, Y.; Mitsui, T., T-stability of numerical schemes for stochastic differential equations, World sci. ser. appl. anal., 2, 333-344, (1993) · Zbl 0834.65146 [16] Saito, Y.; Mitsui, T., Stability analysis of numerical schemes for stochastic differential equations, SIAM J. numer. anal., 33, 2254-2267, (1996) · Zbl 0869.60052
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


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# Help me please.... Mixed Partials

• Sep 8th 2009, 04:42 PM
tiga killa
Use the "mixed partials" check to see if the following differential equation is exact

(-1x^2 + 2y)dx + (2x - 3y^1)dy = 0

My(x,y) = ?
Nx(x,y) = ?
F(x,y) = ?
• Sep 8th 2009, 05:00 PM
luobo
Quote:

Originally Posted by tiga killa
Use the "mixed partials" check to see if the following differential equation is exact

(-1x^2 + 2y)dx + (2x - 3y^1)dy = 0

My(x,y) = ?
Nx(x,y) = ?
F(x,y) = ?

May not be what you want, but interesting to solve
$-x^2 dx+ (2ydx+2xdy) -3ydy= 0$
$\Rightarrow -x^2 dx+ 2d(xy) -3ydy= 0$
$\Rightarrow -\frac{x^3}{3}+2xy-\frac{3y^2}{2}=C$
• Sep 8th 2009, 05:05 PM
tiga killa
what is My(x,y)??
• Sep 18th 2009, 08:02 AM
mr fantastic
Quote:

Originally Posted by tiga killa
what is My(x,y)??

Go back to your class notes or textbook and refer to the definitions that you will find there.


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# Multidimensional Kalman Filter

After reading the "Kalman Filter in one dimension" section, you should be familiar with the concepts of the Kalman Filter. In this section, we derive the multidimensional (multivariate) Kalman Filter equations.

Until now, we've dealt with one dimensional processes, like estimating the liquid temperature. But many dynamic processes have two, three, or even more dimensions.

For instance, the state vector that describes the airplane's position in space is three-dimensional:

$\left[ \begin{matrix} x\\ y\\ z\\ \end{matrix} \right]$

The state vector that describes the airplane position and velocity is six-dimensional:

$\left[ \begin{matrix} x\\ y\\ z\\ \dot{x}\\ \dot{y}\\ \dot{z}\\ \end{matrix} \right]$

The state vector that describes the airplane position, velocity, and acceleration is nine-dimensional:

$\left[ \begin{matrix} x\\ y\\ z\\ \dot{x}\\ \dot{y}\\ \dot{z}\\ \ddot{x}\\ \ddot{y}\\ \ddot{z}\\ \end{matrix} \right]$

Assuming a constant acceleration dynamic model, we can describe the extrapolated airplane state at time $$n$$ by nine motion equations:

$\begin{cases} x_{n} = x_{n-1} + \dot{x}_{n-1} \Delta t+ \frac{1}{2}\ddot{x}_{n-1} \Delta t^{2}\\ y_{n} = y_{n-1} + \dot{y}_{n-1} \Delta t+ \frac{1}{2}\ddot{y}_{n-1} \Delta t^{2}\\ z_{n} = z_{n-1} + \dot{z}_{n-1} \Delta t+ \frac{1}{2}\ddot{z}_{n-1} \Delta t^{2}\\ \dot{x}_{n} = \dot{x}_{n-1} + \ddot{x}_{n-1} \Delta t\\ \dot{y}_{n} = \dot{y}_{n-1} + \ddot{y}_{n-1} \Delta t\\ \dot{z}_{n} = \dot{z}_{n-1} + \ddot{z}_{n-1} \Delta t\\ \ddot{x}_{n} = \ddot{x}_{n-1}\\ \ddot{y}_{n} = \ddot{y}_{n-1}\\ \ddot{z}_{n} = \ddot{z}_{n-1}\\ \end{cases}$

It is common practice to describe a multidimensional process with a single equation in matrix form.

First, it is very exhausting to write all these equations; representing them in matrix notation is much shorter and more elegant.

Second, computers are highly efficient at matrix calculations. Implementing the Kalman Filter in matrix form yields faster computation run time.

The following chapters describe the Kalman Filter equations in matrix form. And, of course, the theoretical part is followed by fully solved numerical examples.

The final chapter includes two numerical examples. In the first example, we design a six-dimensional Kalman Filter without control input. In the second example, we design a two-dimensional Kalman Filter with a control input.


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# 2.3.3 Resistors and Heating

### Resistors

Basic requirements for resistors (still one of the most numerous component in circuits) are:

• Large region of R values (= device resistance in ) within one production technology.
• Small (ideally vanishing) temperature coefficient .
• Minimal noise.
• Small dependence of ρ on production parameters (good repeatability).
• No Ageing.
• Small thermoelectrical coefficients to Cu (you want a resistor, not a thermoelement).

Materials of choice include

• Ta, Ta based alloys, and in particular "Constantan" (55% Cu, 44% Ni, 1% Mn), a resistor material with an especially small temperature coefficient αρ, but a large thermoelectric coefficient).
• Strange mixtures of conductors and insulators including "Cermet" (short for Ceramics - Metals), e.g. Cr - SiO2.

Details and data in the (future) link.

### Heating

Basic requirements for heating elements are:

• High melting point.
• Chemical stability at high temperatures and in potentially corrosive environments.
• Mechanical strength at high temperatures.

The choice of a materials depends significantly on the range of temperatures envisioned. We have:

• FeNiCr, FeNiAl alloys.
• Pt, W, Ta, Mo - stable elements with a high melting point.
• MoSi2 Among more industrial applications also used as heaters in dish washers - this is very aggressive environment!
• Graphite (up to 3000 K in non-oxidizing gas).

Some details and data can be found in the links.

2.3.3 Resistors and Heating is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.