text:
• R N SINGH

Articles written in Journal of Earth System Science

• Measurements of reflection coefficients of stratified layers using X-band bistatic scatterometers

Laboratory modelling of earth’s subsurface stratification has been carried out using X-band microwave bistatic scatterometer system. Look angle variation of reflectivity for various subsurface layers under dry and wet conditions have been measured. From measured reflectivity data and the reciprocity theorem the emissivity and the brightness temperature variations have been computed, and the data are in good agreement with reported results. The importance of laboratory and field measurements and its remote sensing application has been discussed.

• Inverse problem for the determination of the bounds on the mantle heat flow

This paper presents a method for determining the nature of the spatial structure of the heat flow anomaly at depth given a few observations of surface heat flow. We use the constraint of the positivity of the heat flow throughout the domain under consideration to arrive at the solution. Both Parker’s ideal body and linear programming approaches have been considered. We find that a few measurements are able to give a reasonable good estimation of the heat flow anomaly at depth. The method has been applied to a real case in the French Massif Central. Results point to the presence of enhanced basal heat flow anomaly.

• Generation of low-frequency oscillations in the subsolar venus ionopause

The steep electron density gradient, the magnetic field curvature and the effect of gravity make the Venus ionopause region unstable. The onset of flute instability in the Venus ionopause plasma has been studied. The dispersion of waves and their growth have been studied using in situ measured data. The existence of low frequency waves in the subsolar magnetic field data ionopause region has been interpreted to arise as the result of flute instability.

• Gravity induced near-surface stresses in long-symmetric ridge-valley systems

Patterns of near-surface gravity and tectonically-induced stresses within ridge-valley systems greatly illuminate our understanding of important geodynamic processes as well as design of experiments to elucidate them. This paper presents analytical results derived for a number of geometrical configurations and range of mechanical properties of a ridge-valley system using the elastic solution of McTigue and Mei. The study reveals (i) the presence of non-zero compressive stresses near the ridge crests, which decrease with increasing Poisson's ratio (μ) and reduce to zero at ridge crests for μ=0·5 and that (ii) the central tensional regime characterizing a valley becomes narrower due to the presence of two ridges; and decreases with increasing μ, becoming compressive at depth. For all geometrical parameters considered, all components of stress show concentration at the outer flanks of the ridges and increase with depth approaching a standard state of stress.

• Rheological thickness and strength of the Indian continental lithosphere

The estimates of rheological thickness and total lithospheric strength for the Indian continental lithosphere have been obtained based on the representative rheological properties of upper crust, lower crust and upper mantle, and some of the available heat flow and heat generation data. The rheological thickness, computed at different locations in the Indian shield, shows lateral variation ranging from 79km in the southern part to 65 km in the northern part for a strain rate of 10-14 s-1. The total strength of the continental lithosphere is of the order of 1013 Nm-1 for the same value of strain rate and decreases northward. The computations carried out for a range of strain rates show an increase in the rheological thickness and strength of the lithosphere with increasing strain rate. These results would be important in understanding the flexural response of the Indian continental lithosphere to surface and subsurface loading, and response to tectonic forces acting on it.

• Effects of transient uplift/erosion on the surface heat flow and heat generation relationship in presence of small scale asthenospheric convection

In this paper the effect of transient uplift/erosion on the relationship between surface heat flow and heat generation for truncated exponential model of radiogenic heat source distribution and basal asthenospheric convection is investigated. Asthenospheric convection is described by a parameterized model, in the form of a nonlinear heat flux boundary condition involving basal temperature and mantle internal temperature. This boundary condition has been linearized and the analytical solution of the problem is obtained by the eigenvalue-eigenfunction expansion method. The analytical solution is used to derive the nature of surface heat flow and heat generation relationship. The results show that the linear relationship is maintained during the uplift/erosion and the estimates of the slope of the linear relationship are different from the depth scale of the exponential model and increase with the rate of uplift/erosion. The estimates of the reduced heat flow also increase with the rate of uplift/erosion. These results would find applications in the interpretation of linear surface heat flow and heat generation relationship which is observed in different tectonic environment.

• Three-dimensional intraplate stress distributions associated with topography and crustal density inhomogeneities beneath the Deccan Volcanic Province

The generation of intraplate earthquakes has been attributed to perturbations in the stress regime, either due to surface and sub-surface loading or strength weakening of the rock mass. The present work aims at estimating the intraplate stresses associated with topography and crustal density inhomogeneities beneath the Deccan Volcanic Province (DVP). A layered crustal model with irregular interfaces of small amplitude has been used for elastostatic stress calculations. The computed principal stress differences show a significant concentration at 5–20 km depths beneath the western side of the region. The maximum magnitude of principal stress difference occurs beneath the Karad at a depth of 10 km with a value of 60 MPa. The deviatoric stress estimates are further superposed on inferred stresses due to the regional plate tectonic forces. These results show principal stress difference concentrations beneath the Koyna, Poona and Karad regions which may thus be more vulnerable to brittle failure. It is also seen that the principal total stress directions point to the strike slip motion at Koyna, similar to that which is associated with the 1967 Koyna earthquake.

• Stability of vegetation growth in a noise-induced system

Vegetation community plays a pivotal role in various geomorphic processes. The growth of vegetation intrinsically depends on the effective shear stresses exerted by the flow of material (e.g., water or soil) along the slope. We comparatively assess the growth and decay of vegetation using linear and logistic growth model coupled with a runoff erosion model. The linear growth model predicts a sharp decrease in the non-dimensional vegetation profile from the upper reach to the lower part along the slope. However, the logistic growth model delivers a smooth gradual decrease in the vegetation extent. Additionally, we propose a stochastic model to capture the role of internal and external factors in the dynamics of vegetation growth using two Gaussian noises. The steady probability distribution functions from the stochastic model provide insights about the role of different noises on the reaction of the system and suggest that bio-environmental factors are difficult to segregate from one another.

$\bf{Highlights}$

$\bullet$ Comparative study between logistic and linear vegetation growth model coupled with a surface erosion model.

$\bullet$ Steady-state equilibrium vegetation profile along any slope suggests that the logistic growth model is more realistic.

$\bullet$ Effect of Gaussian noises in vegetation growth has been demonstrated with the aid of stationary probability distribution.

• # Journal of Earth System Science

Volume 132, 2023
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019

Click here for Editorial Note on CAP Mode

© 2022-2023 Indian Academy of Sciences, Bengaluru.


text:
# A Why doesn't an electron sitting on the Earth radiate?

Tags:
1. May 7, 2018

### craigthone

I have some puzzles when I combine the Equivalence principle and Unruh's effect.
Equivalence principle says that acceleration is equivalent to a uniformly gravitational field, or locally equivalent to gravitational field. Then the electron sitting on earth should have Unruh's effect, i.e. radiate. What is wrong with the argument?

2. May 7, 2018

### Staff: Mentor

Don’t forget to include the detector in your setup. Is it free falling or co-accelerating?

3. May 7, 2018

### craigthone

Sorry, the co-accelerating detector. The observer sits on the surface also.

4. May 8, 2018

### Staff: Mentor

Not quite. It says that accelerating in flat spacetime is locally equivalent to being at rest in a uniform gravitational field.

No, that's not what the Unruh effect says. The Unruh effect says that an accelerating detector should detect radiation from the quantum vacuum. However, the intensity of the radiation for a 1 g acceleration is way, way too small to actually detect.

5. May 8, 2018

### craigthone

Forget the electron, do observers (detector) sitting near earth see Unruh's radiation? I know the answer is no, but I do not have good explainations. Maybe I should ask in flat space for accelerating observer what is the quantum state corresponding to vacuum for static observer in the earth?

6. May 8, 2018

### haushofer

An accelerating observer observes an event horizon, like an observer at rest near a BH. An observer on earth doesn't.

7. May 8, 2018

### Khashishi

There's some discussion on this here.
I think the consensus is no, you don't get Unruh radiation because the Unruh radiation is perceived as coming from the Rindler horizon. But Earth has no horizon. The equivalence principle only applies locally - an accelerating frame looks like gravity. But an accelerating frame is equivalent to a constant gravitational field which extends to infinity. There is a Rindler horizon, which is some distance behind the accelerating frame where not even light can catch up with the center of the frame. But Earth's gravity isn't a constant gravitational field, and there isn't a horizon.

8. May 8, 2018

### Staff: Mentor

Yes.

I already answered this. We don't see Unruh radiation on Earth because it's way too faint for us to detect.

The Rindler horizon is not an event horizon; it's not the boundary of a region that can't send light signals to future null infinity. There are some local similarities between the two, but they're not the same.

[Edit: removed final statement, not correct.]

Last edited: May 8, 2018
9. May 8, 2018

### Staff: Mentor

I know this is a common heuristic, but I'm not sure it's actually true. The actual derivation of the Unruh effect does not rely on the presence of the Rindler horizon. It relies on the fact that, in quantum field theory, which state is the "vacuum" depends on your state of motion. In particular, if the quantum field in flat spacetime is in a vacuum state with respect to inertial observers, that same state will not be a vacuum state with respect to an accelerated observer; it will be a thermal state at the Unruh temperature, and therefore the accelerated observer will observe radiation at that temperature. That is true right at the location of the accelerated observer; there's no need to reach out to the Rindler horizon.

10. May 8, 2018

### sweet springs

OP 's concern is radiation from the event horizon, but as a layman I have a more primitive question of radiation from the charge itself.
Dose a charge sitting on the floor under gravitation of the Earth or accelerating rocket emit light by radiation? I do not think so. I would like to know the condition of radiation more in detail.

11. May 8, 2018

### Staff: Mentor

So am accelerating electron (in an elevator) will not produce any radiation that can be detected by a co-accelerating detector. By the equivalence principle the same is what we expect for the charge and detector at rest in gravity.

12. May 9, 2018

### craigthone

I do not think so.
From QFT in Rinler space, the argument of Unruh's effect takes a special state $\rho=\exp(-2\pi H)$ where $H$ is the Rindler Hamiltonian and the Minkowski boost, this state $\rho=\exp(-2\pi H)$ is Minkowski vacuum. While for quantum fields on the earth, expanding around the surface, we do not have such state, and no compactification of Euclidean time and no temperature. The picture is totally different from near black hole horizon case. So there is no Unruh radiation rather than to weak to detect.

13. May 9, 2018

### stevendaryl

Staff Emeritus
This picture tries to give an intuitive reason that an accelerated charge radiates. Imagine that you have a charged particle moving along the x-axis at a constant speed. The electric field associated with that situation points toward the moving particle. Now, at some point, the particle abruptly halts. Nearby the particle, the electric field points toward the particle. But the information that the particle's motion has changed hasn't reached far-away places, so the electric field continues to point toward where the particle would have been, if it kept traveling at the same speed. Because electric fields have to be continuous in vacuum, there must be a transition region, spreading out spherically at the speed of light, where the electric field rapidly changes from one configuration to the other. This region of rapidly changing electric (and magnetic, since that changes, too) fields is a spherical burst of electromagnetic radiation. This is shown in the following picture:

To the extent that this intuitive picture is accurate, I think we can conclude that there is no radiation as viewed from a reference frame in which the situation is static. If the electric field far away from the charge isn't changing with time, then there shouldn't be a need for the radiation, which sort of acts like a "correction" to the field that propagates at the speed of light.

The interesting case is a charged particle undergoing constant proper acceleration. From the point of view of a Rindler observer, the situation is static, so there should be no radiation. But from the point of view of an inertial observer, the situation is not static, and it's also not the constant-velocity case. I'm not sure what the electric field looks like to the inertial observer.

A charge at rest on a planet shouldn't radiate, according to this heuristic, because the distant electric field is unchanging (in the frame in which the situation is static, which is someone at rest in the Schwarzschild coordinate system).

14. May 9, 2018

### stevendaryl

Staff Emeritus
I have to caution readers that I'm not at all claiming to be an expert in this. I've just given an intuitive heuristic that may reflect a misunderstanding on my part.

15. May 9, 2018

Staff Emeritus
This thread - as is the case for this topic, as well as it's evil twin, radiation of a charge in free fall - is a bit of a mess. But before I respond to the mention of my name, is this really at the A-level? It seems not to be. I am not going to spend a lot of time on writing an A-level answer only to get a reply like "What is a multipole?" or "What 16 components?"

16. May 9, 2018

### craigthone

We need A level answers. It is better to have some equations rather than something else. Thanks in advance.

17. May 9, 2018

### Staff: Mentor

Technically, you're correct, an observer standing on Earth's surface is not in vacuum, there's air around him and ground below him. But imagine, for example, someone standing on an airless planet (or on the Moon). There is vacuum around that person, and they are accelerated, so they should in principle observe Unruh radiation. That's all I was trying to say.

18. May 10, 2018

Staff Emeritus
It's confusing two things. Are you saying this electron should radiate because it is charged? Or are you saying it should radiate because of the Unruh effect, which has nothing to do with charge? Until we get the question straight, it's unlikely we will be able to provide a good answer.

19. May 10, 2018

### craigthone

In the process of discussing, I think the first question (this electron should radiate because it is charged) is answered by Dale using equivalence principle correctly.

The second question is Forget the electron, do observers (detector) sitting near earth see Unruh's radiation? I know the answer is no, but I do not have clear argument.

20. May 10, 2018

Staff Emeritus
The heuristic is that Unruh radition is a consequence of applying the Equivalence Principle to Hawking radiation. If you go the other way, you get Hawking radiation back - i.e. the geometry of a black hole, not the geometry of a planet.

If you are wondering why BH's emit Hawking radiation and planets don't, the easiest way to see this is conservation of energy. A BH can get energy by shrinking its horizon. Since a planet doesn't have a horizon, there's nowhere for the energy to come from.

Last edited: May 10, 2018
21. May 10, 2018

### Staff: Mentor

Very well explained!

22. May 10, 2018

### Staff: Mentor

I would put this the other way around: Hawking radiation is (heuristically) the result of applying the EP to Unruh radiation. Unruh radiation can be derived rigorously using QFT in flat spacetime. Hawking radiation, if you want to try to derive if from first principles, has to be derived using a semi-classical approximation--QFT in a fixed curved background spacetime--which is not rigorous, and nobody AFAIK has figured out how to make it rigorous.

First, I think you mean "A BH can lose energy by shrinking its horizon".

Second, a planet has lots of other ways of losing energy, so I don't think "not being able to lose energy" is a good heuristic for why planets don't emit Hawking radiation. I think a much better heuristic (which is based, not on Hawking's original argument, but on how our understanding has developed in the decades since he first published his result) is that Hawking radiation is associated with trapped surfaces, not event horizons. For a black hole, the two coincide (or almost coincide, since a real hole will be accreting matter), but the difference is that the trapped surface is a local phenomenon--radially outgoing light does not move outward--so it's much easier to see how an object can "tell" whether it's "allowed" to emit Hawking radiation (a planet has no trapped surface, so no Hawking radiation, even though a planet can radiate in lots of other ways).

23. May 10, 2018

Staff Emeritus
I mean it can transfer energy from the horizon to the radiation field. A planet cannot do this.

And the PF quibbling begins!

You know what I meant. Yes, you can make this more complicated. No, that doesn't help with the understanding.

I knew that getting involved in this morass of a thread was a mistake. Lesson learned.

24. May 10, 2018

### Staff: Mentor

I'm not sure the answer is no. See my post #10. In practice, Unruh radiation for an object accelerating at 1 g is many, many orders of magnitude too weak for us to detect, so there's no way of testing the question experimentally.

25. May 10, 2018

format: plain


text:
## Algebra: A Combined Approach (4th Edition)

$\frac{1}{2}$ is an integer $False$
An integer are numbers that do not include a fraction, and as $\frac{1}{2}$ is a fraction it can not be considered an integer.


text:
# Chain rule Differentiation help

Can someone help me differentiate the function $$\frac{\ln(x)+\sin(x)}{x^2}$$ With respect to $x$? I tryed applying the chain rule but I keep geting lost.

-
Where are you stuck? –  Pragabhava May 21 '13 at 18:49
I don't think you need the chain rule here. It seems like we need to apply the quotient rule. –  Jared May 21 '13 at 18:50
Do it in pieces to observe that the chain rule is really not necessary. You have a function $f(x)/g(x)$. To differentiate it, use the quotient rule to get $(f'g-gf')/g^2$. Now you just need to figure out what $f'$ and $g'$ are. $g$ is easy. $f$ is the sum of functions. But this is easy too, just differentiate each one –  Dhruv Ranganathan May 21 '13 at 18:53
It is better to post with an attempt, or plan, of your own. However, you may simply write the function as $x^{-2}(\ln(x) + \sin(x))$ and use the product rule instead. Though, keep in mind, that you must make the assumption $x > 0.$ –  ThisIsNotAnId May 21 '13 at 19:26

Hint:$(\frac{u}{v}) '=\frac{u 'v-uv '}{v^2}$ $$u=\ln(x)+\sin(x) \to u ' =\frac1x+cos(x)$$

$$v=x^2\to v'=2x$$

-

You can use the quotient rule to solve this. Given $f(x)=\frac{u}{v}$, $f'(x)=\frac{u'v-uv'}{v^2}$.

So we have:

$$f'(x)=\frac{(\frac{1}{x}+\cos(x))(x^2)-(\ln(x)+\sin(x))(2x)}{x^4}$$

Which simplifies to:

$$f'(x)=\frac{x+x^2\cos(x)-2x\ln(x)-2x\sin(x)}{x^4}$$

And then:

$$f'(x)=\frac{1+x\cos(x)-2\ln(x)-2\sin(x)}{x^3}$$

Alternatively, you can use the product rule ($f(x)=uv \implies f'(x)=uv'+u'v$) for this on the following function. The product rule is often more intuitive than the quotient rule, so this might work better for you.:

$$f(x)=(\ln(x)+\sin(x))(x^{-2})$$

Which gives:

$$f'(x)=(\ln(x)+\sin(x))(-2)(x^{-3})+x^{-2}(\frac{1}{x}+\cos(x))$$

This simplifies to:

$$f'(x)=(\ln(x)+\sin(x))(-2)(x^{-3})+x^{-3}(1+x\cos(x))$$

And then:

$$f'(x)=\frac{-2\ln(x)-2\sin(x)+1+x\cos(x)}{x^{3}}$$

We can see that the solutions from the applications of the quotient rule and the product rule are equal, so that's a great way to check the answer as well!

-
Why was this downvoted? –  Pedro Tamaroff May 21 '13 at 19:14
@PeterTamaroff I was a bit confused about that too -- I initially answered with just the product rule and then added the quotient rule after. I may have been downvoted before that edit because the quotient rule is a more conventional way to solve the given problem. –  Andy Bromberg May 21 '13 at 19:18

First of all, there's no need to use the chain rule. The chain rule is for when you have a function of a function, for example $\operatorname{e}^{\sin x}$ or $(\sin x - \cos x)^2$. In this case, the quotient rule will suffice. Notice

Recall that the quotient rule says that

$$\left( \frac{\operatorname{f}}{\operatorname{g}} \right)' = \frac{\operatorname{f}'\!\operatorname{g}-\operatorname{f}\!\operatorname{g}'}{\operatorname{g}^2}$$

In your question $\operatorname{f}(x) = \ln x + \sin x$ and $\operatorname{g}(x) = x^2$. Recall that $(\ln x)' = 1/x$, $(\sin x)' = \cos x$ and $(x^2)' = 2x$. Putting all of this together:

\begin{array}{ccc} \left( \frac{\ln x + \sin x}{x^2}\right)' &=& \frac{(1/x+\cos x)x^2-(\ln x + \sin x)(2x)}{(x^2)^2} \\ \\ &=& \frac{x+x^2\cos x - 2x\ln x - 2x\sin x}{x^4} \\ \\ &=& \frac{1}{x^3} + \frac{\cos x}{x^2}-2\frac{\ln x}{x^3} - 2\frac{\sin x}{x^3} \end{array}

-


text:
# Do data structures help to reduce complexity of operations, at the cost of something else? [closed]

I often heard that using data structures help to reduce (time or space) complexity of some operations.

1. I wonder if that is achieved at the cost of something else?
2. When solving a problem, is something still conserved, over all choices of data structures and algorithms, just like the conservation of energy? Such as some kind of overall (time or space) complexity of an algorithms and a data structure being used?

## closed as too broad by Raphael♦Oct 5 '14 at 17:29

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• This question is too broad as it is. The answer to 1. seems to be trivially "yes" (with the knowledge of an introductory course in algorithms/data structures). The answer to 2. may be complexity theory as a field, but it's hard to tell since the question is so vague. Can you focus your question wiht an example? – Raphael Oct 5 '14 at 17:28

To represent a collection of integers, you can use an array with one word of memory per element. For instance, say $A = [0,1,7,10]$. Alternatively, you could use an array $B$ to represent the same collection in the obvious way: $B = [1,1,0,0,0,0,0,1,0,0,1]$. Searching for element $x$ in $A$ takes $\Theta(n)$ time (or perhaps just logarithmic time if $A$ is indeed sorted), but you can do the same in $B$ in $O(1)$ time by just looking at $B[x]$, i.e. just indexing $B$ at position $x$. This is a tradeoff: by using more memory for $B$, you get faster lookups. Time-memory tradeoffs like this are often encountered.
Sometimes, one can show that to solve a problem, you must always no matter what spend some time, memory, or other resource to solve it. In such results, one must always specify the model of computation used. For instance, there is a well-known decision tree lower bound of $\Omega(n \log n)$ for sorting. So in this model, you absolutely can't beat this bound no matter what you do.


text:
yhew Posts: 1, Reputation: 1 New Member #1 Apr 21, 2009, 08:22 AM
General Chemistry equilibrium
An acid-base equilibrium system is created by dissolving 0.10 mol HF in water to a volume of 1.0 L. What is the effect of adding 0.050mol F(aq) to this solution?
 Perito Posts: 3,139, Reputation: 150 Ultra Member #2 Apr 21, 2009, 08:55 AM

The equation for the equilibrium is

$HF\,\rightleftharpoons\,H^+\,+\,F^-$

You know the concentration of HF. You haven't given me any Ka for HF so you can't solve this quantitatively.

Remember that

$K_a=\,\frac{[H^+][F^-]}{[HF]}$

So, from that you can calculate the concentration of [F-] and figure out what happens to the equilibrium (it will drive it to the left).

 Question Tools Search this Question Search this Question: Advanced Search

## Check out some similar questions!

Hi I need to know How I can post reversing etry in to the general ledger ? For example if I have The business last paid a water bill in May, which was for 3 months of water expense up to the end of April. The additional accrual entry for water expense for the last two months is \$2,290. And I...

General Chemistry [ 2 Answers ]

Explain how a liquid can have a boiling point of 80 oC at sea level and a boiling point of 74 oC at an altitude of 1000 m?

General Ledger versus General Journal [ 1 Answers ]

Does the number of entries in the general journal equal the number of entries in the general ledger. Why?


text:
Tic Tac Toe Part 3: The Minimax Algorithm

In two previous posts, I presented code to teach R to play the trivial game of Tic Tac Toe. I thought this was an unbeatable algorithm. Alas, a comment from Alberto shattered my pride as he was able to beat my code.

The reason for the demise of my code was that I didn’t implement a full minimax algorithm, but instead looked only two moves ahead. I thought the common strategy rules (start in the centre and occupy the corners) would make the program unbeatable. When I simulated the game by instructing the computer to play against itself, Alberto’s strategy never arose because the code forces the centre field. Alberto’s code shows that you need to look at least three moves ahead for a perfect game. He has been so kind to share his code and gave me permission to publish it.

Alberto recreated two functions, for completeness I have added the complete working code that merges his improvements with my earlier work. The first two functions are identical to the previous post. These functions draw the game board and process the human player’s move by waiting for a mouse click.

```# Draw the game board
draw.board <- function(game) {
xo <- c("X", " ", "O") # Symbols
par(mar = rep(1,4))
plot.new()
plot.window(xlim = c(0,30), ylim = c(0,30))
abline(h = c(10, 20), col="darkgrey", lwd = 4)
abline(v = c(10, 20), col="darkgrey", lwd = 4)
text(rep(c(5, 15, 25), 3), c(rep(25, 3), rep(15,3), rep(5, 3)), xo[game + 2], cex = 4)
# Identify location of any three in a row
square <- t(matrix(game, nrow = 3))
hor <- abs(rowSums(square))
if (any(hor == 3))
hor <- (4 - which(hor == 3)) * 10 - 5
else
hor <- 0
ver <- abs(colSums(square))
if (any(ver == 3))
ver <- which(ver == 3) * 10 - 5
else
ver <- 0
diag1 <- sum(diag(square))
diag2 <- sum(diag(t(apply(square, 2, rev))))
# Draw winning lines
if (all(hor > 0))
for (i in hor) lines(c(0, 30), rep(i, 2), lwd = 10, col="red")
if (all(ver > 0))
for (i in ver) lines(rep(i, 2), c(0, 30), lwd = 10, col="red")
if (abs(diag1) == 3)
lines(c(2, 28), c(28, 2), lwd = 10, col = "red")
if (abs(diag2) == 3)
lines(c(2, 28), c(2, 28), lwd = 10, col = "red")
}

# Human player enters a move
move.human <- function(game) {
text(4, 0, "Click on screen to move", col = "grey", cex=.7)
empty <- which(game == 0)
move <- 0
while (!move %in% empty) {
coords <- locator(n = 1) # add lines
coords\$x <- floor(abs(coords\$x) / 10) + 1
coords\$y <- floor(abs(coords\$y) / 10) + 1
move <- coords\$x + 3 * (3 - coords\$y)
}
return (move)
}
```

Alberto rewrote the functions that analyse the board and determine the move of the computer. The ganador (Spanish for winning) function assesses the board condition by assigning -10 or + 10 for a winning game and 0 for any other situation.

```ganador <- function(juego, player) {
game <- matrix(juego, nrow = 3, byrow = T)
hor <- rowSums(game)
ver <- colSums(game)
diag <- c(sum(diag(game)), sum(diag(apply(game, 1, rev))))
if (-3 %in% c(hor, ver, diag))
return(-10)
if (3 %in% c(hor, ver, diag))
return(10)
else
return(0)
}
```

The next function is the actual minimax algorithm. If the computer starts then the first move ($9!= 362880$ options to assess) takes a little while. The commented lines can be used to force a corner and make the games faster by forcing a random corner.

The minimax function returns a list with the move and its valuation through the ganador function. The function works recursively until it has filled the board and retains the best scoring move using the minimax method. To avoid the computer always playing the same move in the same situation random variables are added.

```minimax <- function(juego, player) {
free <- which(juego == 0)
if(length(free) == 1) {
juego[free] <- player
return(list(move = free, U = ganador(juego, player)))
}
poss.results <- rep(0, 9)
for(i in free) {
game <- juego
game[i] <- player
}
mm <- ifelse(player == -1, "which.min", "which.max")
if(any(poss.results == (player * 10))) {
move <- do.call(mm, list(poss.results))
return(list(move = move, U = poss.results[move]))
}
for(i in free) {
game <- juego
game[i] <- player
poss.results[i] <- minimax(game, -player)\$U
}
random <- runif(9, 0, 0.1)
poss.results[-free] <- 100 * -player
poss.results <- poss.results + (player * random)
move <- do.call(mm, list(poss.results))
return(list(move = move, U = poss.results[move]))
}
```

This final function stitches everything together and lets you play the game. Simply paste all functions in your R console and run them to play a game. The tic.tac.toe function can take two parameters, “human” and/or “computer”. The order of the parameters determines who starts the game.

```# Main game engine
tic.tac.toe <- function(player1 = "human", player2 = "computer") {
game <- rep(0, 9) # Empty board
winner <- FALSE # Define winner
player <- 1 # First player
players <- c(player1, player2)
draw.board(game)
while (0 %in% game & !winner) { # Keep playing until win or full board
if (players[(player + 3) %% 3] == "human") # Human player
move <- move.human(game)
else { # Computer player
move <- minimax(game, player)
move <- move\$move
}
game[move] <- player # Change board
draw.board(game)
winner <- max(eval.game(game, 1), abs(eval.game(game, -1))) == 6 # Winner, winner, chicken dinner?
player <- -player # Change player
}
}

tic.tac.toe()
```

This is my last word on Tic Tac Toe but now that the minimax conundrum is solved I could start working on other similar games such as Connect Four, Draughts or even the royal game of Chess.

R-Cade Games: Simulating the Legendary Game of Pong

Pong is one of the earliest arcade games on the market, first released in 1972. From the day I first saw this miracle box, I wanted to know more about computers.

I learnt how to write code from the 1983 book Dr. C. Wacko’s Miracle Guide to Designing and Programming your own Atari Computer Arcade Games. This book explains in a very clear and humorous way how to write computer games in Atari basic. I devoured this book and spent many hours developing silly games. This article is an ode to Dr Wacko, a computer geek’s midlife-crisis and an attempt to replicate the software I developed thirty years ago.

I showed in a previous post that R can be used for board games. The question is whether we create arcade games in R. My challenge is to recreate the look and feel of 1980s arcade games, or R-Cade games, using R? The code shown below simulates the legendary game of pong.

Playing Pong in R

The code is based on the Wacko’s Boing Program in the above-mentioned book. The R code is fully commented and speaks for itself. Please note that the animation is very clunky when you run it in RStudio. Only the native R Terminal displays the animation correctly.

Perhaps somebody can help me perfect this little ditty. I love to know how to read real-time USB input to control the game, so we get a step closer to the first R-Cade game.

The Pong Code

```# Sound library
library(beepr)

# Game parameters
skill <- 0.87 # Skill (0-1)
score <- 0
high.score <- 0

# Define playing field
par(mar = rep(1,4), bg = "black")
plot.new()
plot.window(xlim = c(0, 30), ylim = c(0, 30))
lines(c(1, 30, 30, 1), c(0, 0, 30, 30), type = "l", lwd = 5, col = "white")

# Playing field boundaries (depend on cex)
xmin <- 0.5
xmax <- 29.4
ymin <- 0.5
ymax <- 29.4

# initial position
x <- sample(5:25, 1)
y <- sample(5:25, 1)
points(x, y, pch = 15, col = "white", cex = 2)

psize <- 4

# Set direction
dx <- runif(1, .5, 1)
dy <- runif(1, .5, 1)

# Game play
while (x > xmin - 1) {
sound <- 0 # Silence
Sys.sleep(.05) # Pause screen. Reduce to increase speed
points(x, y, pch = 15, col = "black", cex = 2) # Erase ball
# Move ball
x <- x + dx
y <- y + dy
# Collision detection
if (x > xmax) {
dx <- -dx * runif(1, .9, 1.1) # Bounce
if (x > xmin) x <- xmax # Boundary
sound <- 10 # Set sound
}
if (y < ymin | y > ymax) {
if (y < ymin) y <- ymin
if (y > ymax) y <- ymax
dy <- -dy * runif(1, .9, 1.1)
sound <- 10
}
if (x < xmin & (y > ypaddle - (psize / 2)) & y < ypaddle + (psize / 2)) {
if (x < xmin) x <- xmin
dx <- -dx * runif(1, .9, 1.1)
sound <- 2
score <- score + 1
}
# Draw ball
points(x, y, pch = 15, col = "white", cex = 2)
if (sound !=0) beep(sound)
if (runif(1, 0, 1) < skill) ypaddle <- ypaddle + dy # Imperfect follow
# Erase back line
lines(c(0, 0), c(0, 30), type = "l", lwd = 8, col = "black")
if (ypaddle > 30 - (psize / 2)) ypaddle <- 30 - (psize / 2)
lines(c(0, 0), c(ypaddle - (psize / 2), ypaddle + (psize / 2)), type = "l", lwd = 8, col = "white")
}
beep(8)
text(15,15, "GAME OVER", cex=5, col = "white")
s <- ifelse(score == 1, "", "s")
text(15,5, paste0(score, " Point", s), cex=3, col = "white")
```

Tic Tac Toe War Games: The Intelligent Minimax Algorithm

In a previous post, I shared how to build a randomised Tic Tac Toe simulation. The computer plays against itself playing at random positions. In this post, I will share how to teach the computer to play the game strategically.

I love the 1983 classic movie War Games. In this film, a computer plays Tic Tac Toe against itself to learn that it cannot win the game to prevent a nuclear war.

Back in those days, I devoured the wonderful book Writing Strategy Games on your Atari by John White which contains an algorithm to play Tic Tac Toe War Games. This is my attempt to relive the eighties using R.

You can find the code on my GitHub page.

Drawing the Board

A previous post describes the function that draws the Tic Tac Toe board. For completeness, the code is replicated below. The game board is a vector of length nine consisting of either -1 (X), 0 (empty field) or 1 (O). The vector indices correspond with locations on the game board:

1 2 3
4 5 6
7 8 9

```draw.board <- function(board) { # Draw the board
xo <- c("X", " ", "O") # Symbols
par(mar = rep(0,4))
plot.new()
plot.window(xlim = c(0,30), ylim = c(0,30))
abline(h = c(10, 20), col="darkgrey", lwd = 4)
abline(v = c(10, 20), col="darkgrey", lwd = 4)
pieces <- xo[board + 2]
text(rep(c(5, 15, 25), 3), c(rep(25, 3), rep(15,3), rep(5, 3)), pieces, cex = 6)
# Identify location of any three in a row
square <- t(matrix(board, nrow = 3))
hor <- abs(rowSums(square))
if (any(hor == 3))
hor <- (4 - which(hor == 3)) * 10 - 5
else
hor <- 0
ver <- abs(colSums(square))
if (any(ver == 3))
ver <- which(ver == 3) * 10 - 5
else
ver <- 0
diag1 <- sum(diag(square))
diag2 <- sum(diag(t(apply(square, 2, rev)))) # Draw winning lines if (hor > 0) lines(c(0, 30), rep(hor, 2), lwd=10, col="red")
if (ver > 0) lines(rep(ver, 2), c(0, 30), lwd=10, col="red")
if (abs(diag1) == 3) lines(c(2, 28), c(28, 2), lwd=10, col="red")
if (abs(diag2) == 3) lines(c(2, 28), c(2, 28), lwd=10, col="red")
}
```

Human Players

This second code snippet lets a human player move by clicking anywhere on the graphic display using the locator function. The click location is converted to a number to denote the position on the board. The entered field is only accepted if it has not yet been used (the empty variable contains the available fields).

```# Human player enters a move
move.human <- function(game) {
text(4, 0, "Click on screen to move", col = "grey", cex=.7)
empty <- which(game == 0)
move <- 0
while (!move %in% empty) {
coords <- locator(n = 1) # add lines
coords\$x <- floor(abs(coords\$x) / 10) + 1
coords\$y <- floor(abs(coords\$y) / 10) + 1
move <- coords\$x + 3 * (3 - coords\$y)
}
return (move)
}
```

Evaluate the Game

This code snippet defines the eval.game function which assesses the current board and assigns a score. Zero means no outcome, -6 means that the X player has won and +6 implies that the O player has won.

```# Evaluate board position
eval.game <- function(game, player) {
# Determine game score
square <- t(matrix(game, nrow = 3))
hor <- rowSums(square)
ver <- colSums(square)
diag1 <- sum(diag(square))
diag2 <- sum(diag(t(apply(square, 2, rev))))
eval <- c(hor, ver, diag1, diag2)
# Determine best score
minimax <- ifelse(player == -1, "min", "max")
best.score <- do.call(minimax, list(eval))
if (abs(best.score) == 3) best.score <- best.score * 2
return (best.score)
}
```

Computer Moves

The computer uses a modified Minimax Algorithm to determine its next move. This article from the Never Stop Building blog and the video below explain this method in great detail.

The next function determines the computer’s move. I have not used a brute-force minimax algorithm to save running time. I struggled building a fully recursive minimax function. Perhaps somebody can help me with this. This code looks only two steps deep and contains a strategic rule to maximise the score.

The first line stores the value of the players move, the second remainder of the matrix holds the evaluations of all the opponents moves. The code adds a randomised variable, based on the strategic value of a field. The centre has the highest value because it is part of four winning lines. Corners have three winning lines and the rest only two winning lines. This means that the computer will, all things being equal, favour the centre over the corners and favour the other fields least. The randomised variables in the code ensure that the computer does not always pick the same field in a similar situation.

```# Determine computer move
move.computer <- function(game, player) {
empty <- which(game == 0)
eval <- matrix(nrow = 10, ncol = 9, data = 0)
for (i in empty) {
game.tmp <- game
game.tmp[i] <- player
eval[1, i] <- eval.game(game.tmp, player)
empty.tmp <- which(game.tmp ==0)
for (j in empty.tmp) {
game.tmp1 <- game.tmp
game.tmp1[j] <- -player
eval[(j + 1), i] <- eval.game(game.tmp1, -player)
}
}
if (!any(abs(eval[1,]) == 6)) { # When winning, play move
# Analyse opponent move
minimax <- ifelse(player == -1, "max", "min") # Minimax
best.opponent <- apply(eval[-1,], 1, minimax)
eval[1,] <- eval[1,] * -player * best.opponent
}
# Add randomisation and strategic values
board <- c(3, 2, 3, 2, 4, 2, 3, 2, 3) # Strategic values
board <- sapply(board, function(x) runif(1, 0.1 * x, (0.1 * x) + 0.1)) # Randomise
eval[1, empty] <- eval[1, empty] + player * board[empty] # Randomise moves
# Pick best game
minimax <- ifelse(player == -1, "which.min", "which.max") # Minimax
move <- do.call(minimax, list(eval[1,])) # Select best move
return(move)
}
```

This last code snippet enables computers and humans play each other or themselves. The players vector contains the identity of the two players so that a human can play a computer or vice versa. The human player moves by clicking on the screen.

The loop keeps running until the board is full or a winner has been identified. A previous Tic Tac Toe post explains the draw.board function.

```# Main game engine
tic.tac.toe <- function(player1 = "human", player2 = "computer") {
game <- rep(0, 9) # Empty board
winner <- FALSE # Define winner
player <- 1 # First player
players <- c(player1, player2)
draw.board(game)
while (0 %in% game & !winner) { # Keep playing until win or full board
if (players[(player + 3) %% 3] == "human") # Human player
move <- move.human(game)
else # Computer player
move <- move.computer(game, player)
game[move] <- player # Change board
draw.board(game)
winner <- max(eval.game(game, 1), abs(eval.game(game, -1))) == 6 # Winner, winner, chicken dinner?
player <- -player # Change player
}
}
```

You can play the computer by running all functions and then entering tic.tac.toe().

I am pretty certain this simplified minimax algorithm is unbeatable—why don’t you try to win and let me know when you do.

Tic Tac Toe War Games

Now that this problem is solved, I can finally recreate the epic scene from the WarGames movie. The Tic Tac Toe War Games code uses the functions explained above and the animation package. Unfortunately, there are not many opportunities to create sound in R.

```# WAR GAMES TIC TAC TOE
source("Tic Tac Toe/Tic Tac Toe.R")

# Draw the game board
draw.board.wargames <- function(game) {
xo <- c("X", " ", "O") # Symbols
par(mar = rep(1,4), bg = "#050811")
plot.new()
plot.window(xlim = c(0,30), ylim = c(0,30))
abline(h = c(10, 20), col = "#588fca", lwd = 20)
abline(v = c(10, 20), col = "#588fca", lwd = 20)
text(rep(c(5, 15, 25), 3), c(rep(25, 3), rep(15,3), rep(5, 3)), xo[game + 2], cex = 20, col = "#588fca")
text(1,0,"r.prevos.net", col = "#588fca", cex=2)
# Identify location of any three in a row
square <- t(matrix(game, nrow = 3))
hor <- abs(rowSums(square))
if (any(hor == 3))
hor <- (4 - which(hor == 3)) * 10 - 5
else
hor <- 0
ver <- abs(colSums(square))
if (any(ver == 3))
ver <- which(ver == 3) * 10 - 5
else
ver <- 0
diag1 <- sum(diag(square))
diag2 <- sum(diag(t(apply(square, 2, rev)))) # Draw winning lines if (all(hor > 0)) for (i in hor) lines(c(0, 30), rep(i, 2), lwd = 10, col="#588fca")
if (all(ver > 0)) for (i in ver) lines(rep(i, 2), c(0, 30), lwd = 10, col="#588fca")
if (abs(diag1) == 3) lines(c(2, 28), c(28, 2), lwd = 10, col = "#588fca")
if (abs(diag2) == 3) lines(c(2, 28), c(2, 28), lwd = 10, col = "#588fca")
}

library(animation)
player <- -1
games <- 100
saveGIF ({
for (i in 1:games) {
game <- rep(0, 9) # Empty board
winner <- 0 # Define winner
#draw.board.wargames(game)
while (0 %in% game & !winner) { # Keep playing until win or full board
empty <- which(game == 0)
move <- move.computer(game, player)
game[move] <- player
if (i <= 12) draw.board.wargames(game)
winner <- max(eval.game(game, 1), abs(eval.game(game, -1))) == 6
player <- -player } if (i > 12) draw.board.wargames(game)
}
},
interval = c(unlist(lapply(seq(1, 0,-.2), function (x) rep(x, 9))), rep(0,9*94)),
movie.name = "wargames.gif", ani.width = 1024, ani.height = 1024)
```

Tic Tac Toe Simulation — Random Moves

Tic Tac Toe might be a futile children’s game but it can also teach us about artificial intelligence. Tic Tac Toe, or Naughts and Crosses, is a zero-sum game with perfect information. Both players know exactly what the other did and when nobody makes a mistake, the game will always end in a draw.

Tic Tac Toe is a simple game but also the much more complex game of chess is a zero-sum game with perfect information.

In this two-part post, I will build an unbeatable Tic Tac Toe Simulation. This first part deals with the mechanics of the game. The second post will present an algorithm for a perfect game.

Drawing the Board

This first code snippet draws the Tic Tac Toe simulation board. The variable xo holds the identity of the pieces and the vector board holds the current game. Player X is denoted with -1 and player O with +1. The first part of the function draws the board and the naughts and crosses. The second part of the code check for three in a row and draws the corresponding line.

```draw.board <- function(board) { # Draw the board
xo <- c("X", " ", "O") # Symbols
par(mar = rep(0,4))
plot.new()
plot.window(xlim = c(0,30), ylim = c(0,30))
abline(h = c(10, 20), col="darkgrey", lwd = 4)
abline(v = c(10, 20), col="darkgrey", lwd = 4)
pieces <- xo[board + 2]
text(rep(c(5, 15, 25), 3), c(rep(25, 3), rep(15,3), rep(5, 3)), pieces, cex = 6)
# Identify location of any three in a row
square <- t(matrix(board, nrow = 3))
hor <- abs(rowSums(square))
if (any(hor == 3))
hor <- (4 - which(hor == 3)) * 10 - 5
else
hor <- 0
ver <- abs(colSums(square))
if (any(ver == 3))
ver <- which(ver == 3) * 10 - 5
else
ver <- 0
diag1 <- sum(diag(square))
diag2 <- sum(diag(t(apply(square, 2, rev))))
# Draw winning lines
if (hor > 0) lines(c(0, 30), rep(hor, 2), lwd=10, col="red")
if (ver > 0) lines(rep(ver, 2), c(0, 30), lwd=10, col="red")
if (abs(diag1) == 3) lines(c(2, 28), c(28, 2), lwd=10, col="red")
if (abs(diag2) == 3) lines(c(2, 28), c(2, 28), lwd=10, col="red")
}
```

Random Tic Tac Toe

The second part of the code generates ten random games and creates and animated GIF-file. The code adds random moves until one of the players wins (winner <> 0) or the board is full (no zeroes in the game vector). The eval.winner function checks for three in a row and declares a winner when found.

There are 255,168 possible legal games in Tic Tac Toe, 46,080 of which end in a draw. This implies that these randomised games result in a draw 18% of the time.

```eval.winner <- function(board) { # Identify winner
square <- t(matrix(board, nrow = 3))
hor <- rowSums(square)
ver <- colSums(square)
diag1 <- sum(diag(square))
diag2 <- sum(diag(t(apply(square, 2, rev))))
if (3 %in% c(hor, ver, diag1, diag2)) return (1)
else
if (-3 %in% c(hor, ver, diag1, diag2)) return (2)
else
return(0)
}

# Random game
library(animation)
saveGIF ({
for (i in 1:10) {
game <- rep(0, 9) # Empty board
winner <- 0 # Define winner
player <- -1 # First player
draw.board(game)
while (0 %in% game & winner == 0) { # Keep playing until win or full board
empty <- which(game == 0) # Define empty squares
move <- empty[sample(length(empty), 1)] # Random move
game[move] <- player # Change board
draw.board(game)
winner <- eval.winner(game) # Evaulate game
player <- player * -1 # Change player
}
draw.board(game)
}
},
interval = 0.25, movie.name = "ttt.gif", ani.width = 600, ani.height = 600)
```

Tic Tac Toe Simulation

In a future post, I will outline how to program the computer to play against itself, just like in the 1983 movie War Games.

SCADA spikes in Water Treatment Data

SCADA spikes are events in the data stream of water treatment plants or similar installations. These SCADA spikes can indicate problems with the process and could result in an increased risk to public health.

The WSAA Health Based Targets Manual specifies a series of decision rules to assess the performance of filtration processes. For example, this rule assesses the performance of conventional filtration:

“Individual filter turbidity ≤ 0.2 NTU for 95% of month and not > 0.5 NTU for ≥ 15 consecutive minutes.”

Turbidity is a measure for the cloudiness of a fluid because of large numbers of individual particles otherwise invisible to the naked eye. Turbidity is an important parameter in water treatment because a high level of cloudiness strongly correlates with the presence of microbes. This article shows how to implement this specific decision rule using the R language.

Simulation

To create a minimum working example, I first create a simulated SCADA feed for turbidity. The turbidity data frame contains 24 hours of data. The seq.POSIXt function creates 24 hours of timestamps at a one-minute spacing. In addition, the rnorm function creates 1440 turbidity readings with an average of 0.1 NTU and a standard deviation of 0.01 NTU. The image below visualises the simulated data. The next step is to assess this data in accordance with the decision rule.

```# Simulate data
set.seed(1234)
turbidity <- data.frame(DateTime = seq.POSIXt(as.POSIXct("2017-01-01 00:00:00"),
by = "min", length.out=24*60),
Turbidity = rnorm(n = 24*60, mean = 0.1, sd = 0.01)
)
```

The second section simulates five spikes in the data. The first line picks a random start time for the spike. The second line in the for-loop picks a duration between 10 and 30 minutes. In addition, the third line simulates the value of the spike. The mean value of the spike is determined by the rbinom function to create either a low or a high spike. The remainder of the spike simulation inserts the new data into the turbidity data frame.

```# Simulate spikes
for (i in 1:5) {
time <- sample(turbidity\$DateTime, 1)
duration <- sample(10:30, 1)
value <- rnorm(1, 0.5 * rbinom(1, 1, 0.5) + 0.3, 0.05)
start <- which(turbidity\$DateTime == time)
turbidity\$Turbidity[start:(start+duration - 1)] <- rnorm(duration, value, value/10)
}
```

The image below visualises the simulated data using the mighty ggplot. Only four spikes are visible because two of them overlap. The next step is to assess this data in accordance with the decision rule.

```library(ggplot2)
ggplot(turbidity, aes(x = DateTime, y = Turbidity)) + geom_line(size = 0.2) +
geom_hline(yintercept = 0.5, col = "red") + ylim(0,max(turbidity\$Turbidity)) +
```

The following code searches for all spikes over 0.50 NTU using the run length function. This function transforms a vector into a vector of values and lengths. For example, the run length of the vector c(1, 1, 2, 2, 2, 3, 3, 3, 3, 5, 5, 6) is:

• lengths: int [1:5] 2 3 4 2 1
• values : num [1:5] 1 2 3 5 6

The value 1 has a length of 1, the value 2 has a length of 3 and so on. The spike detection code creates the run length for turbidity levels greater than 0.5, which results in a boolean vector. The cumsum function calculates the starting point of each spike which allows us to calculate their duration.

The code results in a data frame with all spikes higher than 0.50 NTU and longer than 15 minutes. The spike that occurred at 11:29 was higher than 0.50 NTU and lasted for 24 minutes. The other three spikes are either lower than 0.50 NTU. The first high spike lasted less than 15 minutes.

```# Spike Detection
spike.detect <- function(DateTime, Value, Height, Duration) {
runlength <- rle(Value > Height)
spikes <- data.frame(Spike = runlength\$values,
times <- cumsum(runlength\$lengths))
spikes\$Times <- DateTime[spikes\$times]
spikes\$Event <- c(0,spikes\$Times[-1] - spikes\$Times[-nrow(spikes)])
spikes <- subset(spikes, Spike == TRUE & Event > Duration)
return(spikes)
}
spike.detect(turbidity\$DateTime, turbidity\$Turbidity, 0.5, 15)
```

This approach was used to prototype a software package to assess water treatment plant data in accordance with the Health-Based Targets Manual. The finished product has been written in SQL and is available under an Open Source sharing license.

View this code on GitHub.


text:
I would look at Stefan Schwede's articles. I think the answer is yes, see page 7 of Schwede and Shipley: http://www.math.uni-bonn.de/people/schwede/AlgebrasModules.pdf

Perhpas this more explicitly states what you want: http://arxiv.org/PS_cache/math/pdf/9803/9803002v1.pdf It even references the paper above.

The dates in the bibliography may be a bit confusing. It becomes less confusing when you realize that Symmetric Spectra had been around for a year or so before the published version appeared. Certainly people in the know, such as hovey, schwede, and shipley, knew about Symmetric Spectra.

If you tried really hard you could probably find seminar talks about such stuff.

1

I would look at Stefan Schwede's articles. I think the answer is yes, see page 7 of Schwede and Shipley: http://www.math.uni-bonn.de/people/schwede/AlgebrasModules.pdf


text:
# Inverse Laplace Transform Step by Step

1. Jul 5, 2005

### steve2k

Hi - I really need someone to show me step by step how to do an Inverse Laplace transform using a contour integral. The one I would like to understand is the frequency function 1/sqrt(s)

Thank you if you can help me out.

Steve

2. Jul 6, 2005

### lurflurf

This is the contour integral that gives the inverse.
$$f(t)= \frac{1}{2{\pi}i} \int_{a-i\infty}^{a+i\infty} g(s)e^{st} ds$$
or for the specific function.
$$f(t)=\frac{1}{2{\pi}i}\int_{a-i\infty}^{a+i\infty} \frac{e^{st}}{\sqrt{s}} ds$$
We need to take "a" far enough to the right that we avoid problems.
Here we may take a=0, as even though the function has problems at zero, they are not major. You can consider a small right half circle and see the integral is small.
$$f(t)=\frac{1}{2{\pi}i}\int_{-i\infty}^{i\infty} \frac{e^{st}}{\sqrt{s}} ds$$
we can clean the integral up with a substitution i u=s t
$$f(t)=\frac{1}{2{\pi}\sqrt{it}}\int_{-\infty}^{\infty} \frac{e^{iu}}{\sqrt{u}} du$$
This integral can be written in terms of "know" real integrals.
$$\int_0^\infty \frac{sin(x)}{\sqrt{x}} dx=\int_0^\infty \frac{cos(x)}{\sqrt{x}} dx=\sqrt{\frac{\pi}{2}}$$
$$f(t)=\frac{2+2i}{2{\pi}\sqrt{it}}\sqrt{\frac{\pi}{2}}$$

$$f(t)=\frac{1}{\sqrt{{\pi}t}}$$
You can also do a real inversion.
$$f(t)=\lim_{k\rightarrow\infty}\frac{(-1)^k}{k!}g^{(k)}(\frac{k}{t})(\frac{k}{t})^{k+1}$$

Last edited: Jul 6, 2005
3. Jul 6, 2005

### steve2k

Thank you.

Thanks for the reply I really appreciate it!

Steve


text:
# How to find eigenvalues of following block matrices?

Is there a procedure to find the eigenvalues of A?

$$A=\begin{bmatrix}X & I &&&&&&&&& 0\\I & 0 & P &&&&&&&&\\& P^t & 0 & I &&&&&&&\\&&I & 0 & P &&&&&&&\\&&& P^t & 0 & \ddots &&&&&\\&&&& \ddots & \ddots & \ddots &&&&\\&&&&& \ddots & 0 & I &&&\\&&&&&& I & 0 & P &&\\&&&&&&& P^t & 0 & I &0_{k\times y-k}\\&&&&&&&& I &&\\0&&&&&&&& 0_{y-k\times k} && Y_{y\times y}\end{bmatrix}$$ (All rows and columns except the last are $$k \times k$$ blocks.)

where $$A$$ is adjacency matrix of cubic graphs and $$X, P$$ are circulant matrices of order k, Y is a matrix of order y, $$k \not=y$$ and $$I$$ is a identity matrix and $$0$$ is a zero matrix.

For example: How to find eigenvalues of following block matrices?

• – Rodrigo de Azevedo Jun 20 '18 at 7:56
• You mean efficiently or analytically? What's the problem with the regular eig()? – percusse Jun 25 '18 at 16:15
• I want eigenvalues or characteristic polynomial associated to a cubic (3-regular) graphs with adjacency matrix A. Since $y\not= k$, then A is not Jacobian three-diagonal matrix and I cannot find characteristic polynomial of A. – Maryam Hak Jun 25 '18 at 21:12


text:
TITLE

# Three-body nuclei and their astrophysical implications: The case of 6He

AUTHOR(S)
Casal, J.; Rodríguez-Gallardo, M.; Arias, J. M.
PUB. DATE
June 2013
SOURCE
AIP Conference Proceedings;Jun2013, Vol. 1541 Issue 1, p171
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
The analytic THO method [1] is generalized to study three-body nuclei of astrophysical interest, and applied to 6He. Results are consistent with previous publications and experimental data [2, 3].
ACCESSION #
88057062

## Related Articles

• Three-Body Î›NN â†’ nNN Nonmesonic Weak Decay Process of Î› Hypernuclei. Bhang, H.; Ajimura, S.; Aoki, K.; Banu, A.; Fukuda, T.; Hashimoto, O.; Hwang, J.; Kameoka, S.; Kang, B.; Kim, E.; Kim, J.; Kim, M.; Maruta, T.; Miura, Y.; Miyake, Y.; Nagae, T.; Nakamura, M.; Nakamura, S.; Noumi, H.; Okada, S. // Few-Body Systems;Mar2013, Vol. 54 Issue 1-4, p103

The fundamental motivation to study the non-mesonic weak decay (NMWD) of Î› hypernuclei is that it provides the unique channel for the information of the baryon-baryon weak interaction in SU3 group. The relative strength of the two main channels of NMWD whose decay processes are the Î›p...

• Three-Body Treatment of the pep Reaction in the Sun. Irgaziev, B.; Belyaev, V. // Few-Body Systems;Aug2014, Vol. 55 Issue 8-10, p783

The experimental result of the neutrino flux produced by the p + p + e â†’ d + Î½ reaction ( pep) in the Sun at 1 AU was announced for the first time in by the Borexino collaboration. We use the three-body model for the proton-proton-electron system in the initial state for the...

• Experimental Approach to Three Nucleon Forces in Few Nucleon Systems. Sekiguchi, Kimiko // Few-Body Systems;Mar2013, Vol. 54 Issue 1-4, p177

Recent achievements in three nucleon force study via three nucleon scattering at intermediate energies $${(E/A \approx 100-200 {\rm MeV})}$$ are discussed.

• Ab initio calculations in three-body cluster systems. Romero-Redondo, C.; Navrátil, P.; Quaglioni, S. // AIP Conference Proceedings;Jun2013, Vol. 1541 Issue 1, p156

In this work we briefly outline the extension of the ab initio no-core shell model/Resonating group method (NCSM/RGM) to three-body cluster states. We present the results for 6He ground state within a 4He+n+n cluster basis under this approach.

• Two- and Three-Body Coulomb Solution Method in a Momentum Space. Hiratsuka, Y.; Oryu, S. // Few-Body Systems;Mar2013, Vol. 54 Issue 1-4, p209

We present a method to calculate two- and three-body amplitudes including the Coulomb potential in a momentum space. Our aim is to obtain the exact two-body Coulomb amplitudes used in three-body calculations, which reproduce the analytic phase shifts. For the purpose, our theory is based on the...

• Effect of Three Nucleon Forces in p âˆ’ He Scattering. Viviani, M.; Girlanda, L.; Kievsky, A.; Marcucci, L. // Few-Body Systems;Aug2013, Vol. 54 Issue 7-10, p885

The effect of the inclusion of different models of three nucleon (3N) forces in p âˆ’ He elastic scattering at low energies is studied. Two models have been considered: one derived from effective field theory at next-to-next-to-leading order and one derived from a more phenomenological point...

• Î” Isobar Degrees of Freedom in the He Transverse Electron Scattering Response Function. Leidemann, Winfried; Yuan, Luping; Efros, Victor; Orlandini, Giuseppina; Tomusiak, Edward // Few-Body Systems;Aug2013, Vol. 54 Issue 7-10, p1429

The importance of Î”(1232) isobar degrees of freedom is investigated for the transverse ( e, eâ€²) response of He in the quasi-elastic peak region and beyond (considered momentum transfer range: 500 â‰¤ q â‰¤ 1,000 MeV/c). Besides Î” isobar currents ( Î”-IC) meson exchange...

• On the evolution of the ' f' family in the restricted three-body problem. Dutt, Pooja; Sharma, R. // Astrophysics & Space Science;Jun2012, Vol. 340 Issue 1, p63

PoincarÃ© surface of section technique is used to study the evolution of a family ' f' of simply symmetric retrograde periodic orbits around the smaller primary in the framework of restricted three-body problem for a number of systems, actual and hypothetical, with mass ratio varying from 10...

• The effect of photogravitational force and oblateness in the perturbed restricted three-body problem. Abouelmagd, Elbaz // Astrophysics & Space Science;Jul2013, Vol. 346 Issue 1, p51

The model of restricted three-body is generalized to include the effects of the oblateness, the radiation pressure and fictitious forces. The positions of libration points, their stability, the critical mass ratio and periodic orbits emanating from these points are analyzed under the influence...

Share

## Read the Article

Courtesy of VIRGINIA BEACH PUBLIC LIBRARY AND SYSTEM

Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library


text:
## Interval measures of power.(English)Zbl 0918.90051

Summary: Standard power indices, introduced by Shapley and Shubik, Banzhaf, Johnston, and others, assign real numbers to the players in a simple game as a quantitative measure of their influence in the voting situation represented by the game. We consider instead two ‘interval notions of power’ that assign intervals of real numbers to the players. The first of these notions originates in the observation that although a particular choice of weights in a weighted game may not provide an accurate measure of power, the interval of all possible weights for a player is, in fact, a reasonable reflection of the player’s influence. The resulting weight interval is extended to non-weighted games via the observation that every simple game is an intersection of weighted games. For example, we have shown previously [Games Econ. Behav. 5, 170-181 (1993; Zbl 0765.90030)] that the president’s weight interval in the (non-weighted) federal system is ($$15/115$$, $$17/117$$), suggesting that the president has between 13% and 15% of the power. The development of a combinatorial equivalent, the dispersal interval, yields a uniform method for calculating weight intervals. Our second interval index, the market interval, is based on a modification of Peyton Young’s idea that the influence of players can be measured by the sizes of the bribes they can demand when they sell their votes. We show, for example, that the president’s market interval in the federal system is $$(0, 0.2537,\dots)$$; he can reasonably insist on between 0% and 25% of the total bribe. An iterated version of the market interval, based on rationality assumptions about other players’ actions and linked with the notion of a bribery equilibrium, produces potentially smaller intervals that, in some cases, shrink to single points in the limit.

### MSC:

 91B12 Voting theory

### Keywords:

voting; power; power index; simple game; weighted voting game

Zbl 0765.90030
Full Text:

### References:

 [1] Banzhaf, J., Weighted voting doesn’t work: A mathematical analysis, Rutgers Law Rev., 19, 317-343 (1965) [2] Bennett, E., The aspiration approach to predicting coalition formation and payoff distribution in sidepayment games, Int. J. Game Theory, 12, No. 1, 1-28 (1983) · Zbl 0504.90093 [3] Brams, S., Rational Politics (1985), CQ Press: CQ Press Washington, DC [4] Brams, S.; Affuso, P.; Kilgour, D. M., Presidential power: A game-theoretical analysis, (Bruce, P.; Harrington, C.; King, G., The Presidency in American Politics (1989), New York Univ. Press: New York Univ. Press New York), 55-74 [5] Cross, J., Some theoretic characteristic of economic and political coalitions, J. Confl. Resol., 11, 184-195 (1967) [6] Dubey, P., On the uniqueness of the Shapley value, Int. J. Game Theory, 4, 131-140 (1975) · Zbl 0352.90085 [7] Dubey, P.; Shapley, L., Mathematical properties of the Banzhaf power index, Math. Op. Res., 4, 99-131 (1975) · Zbl 0409.90008 [8] Einy, E., The desirability relation of simple games, Math. Soc. Sci., 10, 155-168 (1985) · Zbl 0585.90100 [9] Felsenthal, D.; Machover, M., Postulates and paradoxes of relative voting power—a critical re-appraisal, Theory and Decision, 38, 195-229 (1995) · Zbl 0827.90037 [10] Felsenthal, D.; Machover, M.; Zwicker, W., The bicameral postulates and indices of a priori voting power, Theory and Decision (1977), in press · Zbl 0895.90057 [11] Jereslow, R. G., On defining sets of vertices of the hypercube by linear inequalities, Disc. Math., 11, 119-124 (1975) · Zbl 0297.52009 [12] Lambert, J. P., Voting games, power indices, and presidential elections, UMAP Module, 690 (1988) [13] Lucas, W., Measuring power in weighted voting systems, (Brams, S.; Lucas, W.; Straffin, P., Political and Related Models. Political and Related Models, Vol. 2 in Models in Applied Mathematics (1982), Springer-Verlag: Springer-Verlag New York), 182-238, series edited by W. Lucas [14] Ostmann, A., On the minimal representation of homogeneous games, Int. J. Game Theory, 16, 69-81 (1987) · Zbl 0628.90100 [15] Owen, G., Characterization of the Banzhaf-Coleman index, SIAM J. Appl. Math., 35, 315-327 (1978) · Zbl 0404.90103 [16] Owen, G., A note on the Banzhaf-Coleman axioms, (Ordeshook, P., Game Theory and Political Science (1978), New York Univ. Press: New York Univ. Press New York), 451-461 [17] Owen, G., Game Theory (1982), Academic Press: Academic Press Orlando, IL · Zbl 0159.49201 [18] Rosenmüller, J., The structure of homogeneous games, (Working Paper, 137 (1984), Inst. of Math. Ec., University of Bielefeld) · Zbl 0464.90089 [19] Shapley, L., Simple games: An outline of the descriptive theory, Behav. Sci., 7, 59-66 (1962) [20] Shapley, L.; Shubik, M., A method for evaluating the distribution of power in a committee system, Am. Polit. Sci. Rev., 48, 787-792 (1954) [21] Straffin, P., Topics in the Theory of Voting (1980), Birkhäuser: Birkhäuser Boston, MA · Zbl 0456.90002 [22] Straffin, P., Power indices in politics, (Brams, S.; Lucas, W.; Straffin, P., Political and Related Models. Political and Related Models, Vol. 2 in Models in Applied Mathematics (1982), Springer-Verlag: Springer-Verlag New York), 256-321, series edited by W. Lucas [23] Taylor, A., Mathematics and Politics: Strategy, Voting, Power and Proof (1995), Springer-Verlag: Springer-Verlag New York · Zbl 0834.90002 [24] Taylor, A.; Zwicker, W., A characterization of weighted voting, (Proc. Am. Math. Soc., 115 (1992)), 1089-1094, No. 4 · Zbl 0764.90018 [25] Taylor, A.; Zwicker, W., Weighted voting, multicameral representation and power, Games Econom. Behav., 5, 170-181 (1993) · Zbl 0765.90030 [26] Taylor, A.; Zwicker, W., Simple games and magic squares, J. Combinat. Theory. Ser. A, 71, No. 1, 67-88 (1995) · Zbl 0837.05026 [27] Taylor, A.; Zwicker, W., Simple Games: Desirability Relations, Trading, Pseudoweightings (1997), manuscript · Zbl 0943.91005 [28] von Neumann, J.; Morgenstern, O., Theory of Games and Economic Behavior (1944), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0063.05930 [29] Young, H. P., Power, prices, and incomes in voting systems, Math. Program., 14, 129-148 (1978) · Zbl 0384.90115
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.


text:
# Fiscal Stimulus to Guarantee Results

Figure 1. Pareto plot shows Obama’s 2009 stimulus spending was wasteful.

Much of the wasteful spending by Barack Obama – such as his failed stimulus of 2009 – could be prevented through the use of a simple statistical tool

### The Economy’s Performance Gap

Barack Obama’s handling of the economy — America’s most important problem — has been disappointing.

### Obama Defined by Maldistribution Curve

To determine the degree to which the maldistribution of Obama’s stimulus spending is associated with the Pareto distribution with probability density function (PDF)

for

normalized values for the Pareto PDF were plotted as a function of normalized values for spending. Linear regression by method of least squares fitting revealed a near-perfect correlation between the values, with the fitted line’s slope (R) equal to 0.9962 (Figure 2).


text:
Practice Problems -   -  Use 6 Dot Entry   Switch to Nemeth Tutorial

# Lesson 1.5: Symbols of Comparison: Less Than, Greater Than, Less Than or Equal To, Greater Than or Equal To

## Symbols

$<\phantom{\rule{.3em}{0ex}}\text{less than}$
⠈⠣

$>\phantom{\rule{.3em}{0ex}}\text{greater than}$
⠈⠜

$\le \phantom{\rule{.3em}{0ex}}\text{less than or equal to}$
⠸⠈⠣

$\ge \phantom{\rule{.3em}{0ex}}\text{greater than or equal to}$
⠸⠈⠜

## Explanation

One of the design features of UEB is that each print symbol has one and only one braille representation. The symbols for opening angle bracket and closing angle bracket are used in print to represent less than and greater than respectively. The same symbols are used in braille. Like the equals sign, both symbols are formed with a prefix and a root. Less than is formed with a dot four prefix in the first cell and dots one two six root in the second cell. Greater than is formed with a dot four prefix and dots three four five in the root cell.

Less than or equal to and greater than or equal to are each formed with three cells. Less than or equal to is formed with dot four five six in the first cell, dot four in the second cell, and dots one two six in the third cell. Greater than or equal to is formed with dots four five six in the first cell, dot four in the second cell, and dots three four five in the third cell.

A blank space is left before and after the sign of comparison. A space terminates numeric mode. The numeric indicator is required before a numeral that follows a sign of comparison.

### Example 1

$9>5$
⠼⠊⠀⠈⠜⠀⠼⠑

### Example 2

$7<12$
⠼⠛⠀⠈⠣⠀⠼⠁⠃

### Example 3

$10>8$
⠼⠁⠚⠀⠈⠜⠀⠼⠓

The sign of comparison is used whenever the symbol appears in print. Print spacing should be followed.

### Example 4

$\text{Use}\phantom{\rule{.3em}{0ex}}<\phantom{\rule{.3em}{0ex}}\text{or}\phantom{\rule{.3em}{0ex}}>\text{.}$
⠠⠥⠎⠑⠀⠈⠣⠀⠕⠗⠀⠈⠜⠲

### Example 5

$5\phantom{\rule{.3em}{0ex}}\text{dimes}>5\phantom{\rule{.3em}{0ex}}\text{nickels}$
⠼⠑⠀⠙⠊⠍⠑⠎⠀⠈⠜⠀⠼⠑⠀⠝⠊⠉⠅⠑⠇⠎

### Example 6

$5\le 7$
⠼⠑⠀⠸⠈⠣⠀⠼⠛

### Example 7

$7\le 7$
⠼⠛⠀⠸⠈⠣⠀⠼⠛

### Example 8

$9\ge 5$
⠼⠊⠀⠸⠈⠜⠀⠼⠑

### Example 9

$9\ge 9$
⠼⠊⠀⠸⠈⠜⠀⠼⠊

### Example 10

$\text{Write the words for}\phantom{\rule{.3em}{0ex}}<\text{,}\phantom{\rule{.3em}{0ex}}>\text{,}\phantom{\rule{.3em}{0ex}}\le \text{, and}\phantom{\rule{.3em}{0ex}}\ge \text{.}$
⠠⠺⠗⠊⠞⠑⠀⠮⠀⠘⠺⠎⠀⠿⠀⠈⠣⠂⠀⠈⠜⠂⠀⠸⠈⠣⠂⠀⠯⠀⠸⠈⠜⠲


text:
# Dan Cosley, AAAI, 2010

## Citation

Dan Cosley, Daniel Huttenlocher, Jon Kleinberg, Xiangyang Lan, Siddarth Suri. Sequential Influence Models in Social Network. Association for the Advancement of Artificial Intelligence. 2010.

## Summary

In this paper, the authors consider two of the most fundamental deﬁnitions of inﬂuence, one based on a small set of “snapshot” observations of a social network and the other based on detailed temporal dynamics. The former is particularly useful because large-scale social network data sets are often available only in snapshots or crawls. The latter however provides a more detailed process model of how inﬂuence spreads. The authors studied the relationship between these two ways of measuring inﬂuence, in particular establishing how to infer the more detailed temporal measure from the more readily observable snapshot measure. It validates the analysis using the history of social interactions on Wikipedia; the result is the ﬁrst large-scale study to exhibit a direct relationship between snapshot and temporal models of social inﬂuence.

## Brief Description Of The Method

In order to study a method to measure inﬂuence, this paper gives two definitions of social influence: Ordinal-time and Snapshot, then it defines some notation for various sets of tuples. At last, it gives a method about how to Simulating Ordinal Time from Snapshots.

Defination:

• ${\displaystyle B\left(t_{1}\right)=\left\{\left(u,C,k_{1}\right)\right\}}$: u joined C before t1 and u had k1 neighbors in C at t1.
• ${\displaystyle J\left(t_{1},t_{2}\right)=\left\{\left(u,C,k_{1},k_{k2}\right)\right\}}$: u had k1 neighbors in C at t1, u joined C between t1 and t2, u had k2 neighbors in C at t2.
• ${\displaystyle N\left(t_{2}\right)=\left\{\left(u,C,k_{k2}\right)\right\}}$: u did not join C before t2 and u had k2 neighbors in C at t2.

The sets B and N result in shifting no and do upwards with respect to ${\displaystyle n_{s}}$ and ${\displaystyle d_{s}}$. Also, the set J results in stretching ${\displaystyle n_{o}}$ and ${\displaystyle d_{o}}$ when compared ${\displaystyle n_{s}}$ and ${\displaystyle d_{s}}$. Finally, the sets J and N result in ${\displaystyle d_{o}}$ becoming an accumulation or integration of ${\displaystyle d_{s}}$.

## Experimental Result

• For the English Wikipedia data, Figure 3(a) shows the plots of ${\displaystyle n_{o}\left(k\right)}$ and ${\displaystyle d_{o}\left(k\right)}$ on a log-log scale, along with the best linear ﬁt. A linear model accounts relatively well for the data over a large range, suggesting a power law is a reasonable approximation for each of these quantities. This is not surprising, given that ${\displaystyle d_{o}\left(k\right)}$ can be viewed as a variation on the standard degree distribution: it measures the distribution of the “degree” (number of edges) of each node into each community.

• Using the Wikipedia ordinal time data the authors generated two snapshots, choosing November 1, 2005 and November 6, 2006 as two relatively arbitrary moments at which which they measure the full set of community memberships. It then computed p (k) using the snapshot method, shown in Figure 3(b).

• The approximation of ordinal time data from snapshot data depends on two factors: the number of snapshots used, and the amount of time between the snapshots. The authors begin by considering the effect of the number of snapshots. They show how the simulation of ordinal-time depends on the number of snapshots taken for English Wikipedia in figure below.

As one would expect, the approximation is becoming increasingly accurate with more snapshots. This is because as the number of snapshots increases the time between them goes to 0. Thus, in the limit, snapshot measurements converge to the ordinal-time measurements. This figure shows that empirically just a few snapshots produce good results for these datasets which means the convergence occurs fairly rapidly as the number of snapshots increases.

## Related papers

This paper uses conclusions in Measuring wikipedia to compare with its experimental result.


text:
Note: Different levels of security can be configured in the Management Pack for Citrix ADC. The least secure configuration is to set the SSL Configuration Advanced Setting to No SSL. The next level of security is to set SSL Configuration to No Verify, and the Management Pack will accept all SSL certificates. For the highest level of security, you can set SSL Configuration to Verify, and the Management Pack will have to verify the SSL certificate on your vROps system. See: Creating an Adapter Instance (Citrix ADC).

To add an SSL certificate to the vROps Truststore:

1. Obtain the SSL certificate for your ADC Appliance server from your Internet browser. Export the certificate as an X.509 Certificate (PEM).
2. Copy the certificate to your vROps machine.
3. Use ‘ssh’ to log in to the vROps machine as the root user, then run the following command:

Linux:

$VCOPS_BASE/jre/bin/keytool -import -alias <product_alias> /tmp/<certfile> -keystore “$VCOPS_DATA_VCOPS/user/conf/ssl/tcserver.truststore” -storepass grep ssltruststorePassword /storage/vcops/user/conf/ssl/storePass.properties | sed s/ssltruststorePassword=// -trustcacerts

Windows:

%VCOPS_BASE%\jre\bin\keytool -import -alias <product_alias> C:\path\to\certfile -keystore “%VCOPS_DATA_VCOPS%\user\conf\ssl\tcserver.truststore” -storepass <truststore_password> -trustcacerts

Parameter Descriptions:

• -alias is a unique name for each key that you add (per host)
• certfile is the location where the cert file was saved
4. Run the reboot command to re-start the vROps machine for the changes to take effect.


text:
Sunday
February 1, 2015

# Homework Help: Algebra

Posted by John on Tuesday, March 12, 2013 at 10:07pm.

Let \phi(n) be the Euler Phi Function. If 1 \leq n \leq 1000, what is the smallest integer value of n that minimizes \frac{\phi(n)}{n}?

Related Questions

Calculus - Let N = \int \int \int_B xyz^2\ dV , where B is the cuboid bounded by...
Math - Let ϕ(n) be the Euler Phi Function. If 1≤n≤1000, what is...
Physics - The electric flux through each of the six sides of a rectangular box ...
Calculus - Evaluate, in spherical coordinates, the triple integral of f(rho,...
Algebra - How many ordered pairs of solutions (a, b) are there to \frac{a}{b...
Algebra - Find the largest possible real value of \sqrt{(x-20)(y-x)} + \sqrt{(...
maths - If sin(theta)=3/5 , tan(phi)= 1/2 and 90¡ã< theta <180¡ã<phi<...
Quantum Physics - Let |\psi\rangle=\frac{1-i}{2}|0\rangle-\frac{1+i}{2}|1\rangle...
Algebra - What is the sum of all integer values of n satisfying 1 \leq n \leq ...
Geometry - Six standard six-sided die are rolled. Let p be the probability that ...

Members


text:
# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
The Lamé wave equation. (English) Zbl 0715.34008

Separation of the three-dimensional Helmholtz equation $\left({\Delta }+{k}^{2}\right)u=0$ in elliptic coordinates leads to the Lamé wave equation

$\left(1\right)\phantom{\rule{1.em}{0ex}}\left[{f}^{1/2}\left(z\right)\frac{d}{dz}{f}^{1/2}\left(z\right)\frac{d}{dz}+\frac{1}{4}q\left(z\right)\right]w=0$

where $f\left(z\right)=\left(z-{a}_{1}\right)\left(z-{a}_{2}\right)\left(z-{a}_{3}\right),\phantom{\rule{1.em}{0ex}}q\left(z\right)=h-\ell z+{k}^{2}{z}^{2},$ h,$\ell$ separation constants. In § 2, the spectrum of equation (1) and the Lamé wave functions are defined and a survey on known results is given. The Klein-Bocher classification of linear second order equations with rational coefficients is presented and the position of equation (1) is indicated. In § 3, the asymptotics of the solutions in the complex z-plane are studied for arbitrary complex parameter values k,h,$\ell$. In § 4, the asymptotics of the spectrum and of the angle wave functions are studied by passing to the complex domain. For the spectral parameters h,$\ell$ a system of equations ${F}_{j}\left(h,\ell \right)=\pi {h}_{j}+{b}_{j}+o\left(1\right)$, $j=1,2$, is obtained where the ${h}_{j}$ are integers which are generalizations of the classical Bohr-Sommerfeld quantization law. The functions ${F}_{j}$ are periods of the hyperelliptic integral $S\left(z\right)={\int }_{{z}_{0}}^{z}{\left[q\left(t\right)/f\left(t\right)\right]}^{1/2}dt$.

##### MSC:
 34M99 Differential equations in the complex domain 34B05 Linear boundary value problems for ODE 33E10 Lamé, Mathieu, and spheroidal wave functions


text:
×

# apology pls!

the last problem i posted area under a curve was wrong. i'm sorry friends. it'll never happen again!!

Note by Samuel Ayinde
2 years, 10 months ago

Sort by:


text:
Question:

[0/1 Points] DETAILS PREVIOUS ANSWERS HARMATHAP12 13.2.007 luate the definite integral; 224z 3 dz Need Help? Read lt Submit Answer [-/1 Points] DETAILS HARMATHAP12 13.2.008. Evaluate the definite integral_ 26w 2 dw Need Help? Ruadl

Similar Solved Questions

Question 2ptsFor the voltaic cell Cr(s) Cr3t (aq,0.2 M) Fe?+ (aq; (?) M) Fels) Ecell is 0.34 V. Calculate the concentration of Fe2* (M) Reduction potential for Cr3t(aq)/Cr(s) is-0.74 V,Fe2*(aq)Fe(s) is-0.44V: Enter number to decimal places:Question 31ptsThe cell potential for the cell . Zn(s) 2H'(M) Zn? (1.6 M) Hzlg) (6.1 atm) is observed to be 0.36 V.Whatis the pH in the H'/Hz half-cell? Reduction potential for Hzlg)/H"(aq) is 0.00 V,for Zn(s)Znz+ (aq) is -0.76 V: Enter number t
Question 2 pts For the voltaic cell Cr(s) Cr3t (aq,0.2 M) Fe?+ (aq; (?) M) Fels) Ecell is 0.34 V. Calculate the concentration of Fe2* (M) Reduction potential for Cr3t(aq)/Cr(s) is-0.74 V,Fe2*(aq)Fe(s) is-0.44V: Enter number to decimal places: Question 3 1pts The cell potential for the cell . Zn(s) ...
3 1U Yov Cross perform Whchofte folowng best exolains these observations? crosses 0f Oo vellow 8 3 repeatedl ' observe progeny
3 1 U Yov Cross perform Whchofte folowng best exolains these observations? crosses 0f Oo vellow 8 3 repeatedl ' observe progeny...
Required information Problem 5-26 CVP Applications; Break-Even Analysis; Graphing (L05-1, LO5-2, L05-4, LO5-5] [The following information...
Required information Problem 5-26 CVP Applications; Break-Even Analysis; Graphing (L05-1, LO5-2, L05-4, LO5-5] [The following information applies to the questions displayed below.] The Fashion Shoe Company operates a chain of women's shoe shops that carry many styles of shoes that are all sold a...
Points C/10 21 y Notes Ask Your Consult Interactive Solution 21.51 to see how this problem...
points C/10 21 y Notes Ask Your Consult Interactive Solution 21.51 to see how this problem can be solved s an area per turn of 3.40 x103 m2. The magnetic field is 0.24 T, and the current in the coll is 0.24 A. A brake shoe is pressed perpendicularly against the shaft normal force that the brake shoe...
3. Why do you consider it necessary to include within the additional notes to the financial...
3. Why do you consider it necessary to include within the additional notes to the financial statements the existing legal demands in which the medical entity is involved?...
The physical properties of ice are different from the physical properties of dry ice (frozen carbon dioxide)
The physical properties of ice are different from the physical properties of dry ice (frozen carbon dioxide). But the physical properties of gaseous water vapor and the physical properties of gaseous carbon dioxide are the same. Why? A.) The valence electrons are more tightly bound to the atom in t...
This is all one question, but had to get all parts of the question in so...
this is all one question, but had to get all parts of the question in so you could understand. [0/3 Points) DETAILS PREVIOUS ANSWERS SERCPWA11 8.WA.010. TUTORIAL MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER You and your classmates wish to stack a number of physics books of width W so that they ha...
Question 1: ** Please Work on R studios** Data set: Firm A, 27 employees out of...
Question 1: ** Please Work on R studios** Data set: Firm A, 27 employees out of 92 work in sales In Firm B, 82 out of 273 employees work in sales. 95% level confidence Suppose Firm B above plans to hire 10 more employees across the entire business. The firm tends to hire in proportion to current sta...
Quasi Newton Methods)What is the purpose of the quasi-Newton methods? Fill in the blanks and the missing right-hand sides in the statement of Rank One Algorithm below:Set k = 0: select To and Ho: Compute 9 = Vf(za): If g =0 STOP: ELSE d ComputeTk+lTk + t dComputetdAgk9k+1Hk+1Set k = k + 1: GO TO StepAtk
Quasi Newton Methods) What is the purpose of the quasi-Newton methods? Fill in the blanks and the missing right-hand sides in the statement of Rank One Algorithm below: Set k = 0: select To and Ho: Compute 9 = Vf(za): If g =0 STOP: ELSE d Compute Tk+l Tk + t d Compute td Agk 9k+1 Hk+1 Set k = k + 1...
Thomas Brothers is expected to pay a $3.5 per share dividend at the end of the... Thomas Brothers is expected to pay a$3.5 per share dividend at the end of the year (that is, D1 = $3.5). The dividend is expected to grow at a constant rate of 5% a year. The required rate of return on the stock, rs, is 19%. What is the stock's current value per share? Round your answer to two ... 1 answer Why is it important to distinguish between the billing provider and the pay-to provider in physician... Why is it important to distinguish between the billing provider and the pay-to provider in physician practices?... 1 answer How do you write the measurement of 12x12x12 in exponential notation how do you write the measurement of 12x12x12 in exponential notation?... 5 answers In the system at right, all capa- citors have the same capaci- tance â‚¬ = 48 uF and a poten- tial difference of Vab is applied across it.VabIf the potential difference across capacitor C; is 45 volts then what is the potential difference Vab ? b (in volts)Cza. 270b. 160540d_ 240e_ 180 In the system at right, all capa- citors have the same capaci- tance â‚¬ = 48 uF and a poten- tial difference of Vab is applied across it. Vab If the potential difference across capacitor C; is 45 volts then what is the potential difference Vab ? b (in volts) Cz a. 270 b. 160 540 d_ 240 e_ 180... 5 answers Solve each formula for the specified variable.$C= rac{5}{9}(F-32)$for$F$(Fahrenheit to Celsius) Solve each formula for the specified variable.$C=\frac{5}{9}(F-32)$for$F$(Fahrenheit to Celsius)... 5 answers Points) Draw the product that would be generated from following reaction:points) Use curved arTows to show how the intermediate below is converted into the product you drew in part A above in just ONE mechanistic stepProduet trom Pant ahove points) Draw the product that would be generated from following reaction: points) Use curved arTows to show how the intermediate below is converted into the product you drew in part A above in just ONE mechanistic step Produet trom Pant ahove... 1 answer Graph the function$g(x)=\log (7 x+21)-4$Graph the function$g(x)=\log (7 x+21)-4$... 1 answer QUESTION 7 As a result of a thorough physical inventory, Greeley Company determined that it had... QUESTION 7 As a result of a thorough physical inventory, Greeley Company determined that it had inventory worth$325,000 at December 31, 2016. This count did not take into consideration the following facts: Walker Consignment currently has goods worth $47,000 on its sales floor that belong to Greele... 5 answers Definition 33 defined bysubsequencesequence (Tn)nzk issequenccin Ihich IL,It+l < nr+2 _Iu is an infinite increasing sequence of integers. Definition 33 defined by subsequence sequence (Tn)nzk is sequencc in Ihich IL, It+l < nr+2 _ Iu is an infinite increasing sequence of integers.... 5 answers PullomermEDR InjuaiNitrate Reduction TestSHEET 5-8OBSERVATIONS AND INTEAPRETATIONS guido, enter YcUr rosuts and nle FreLallara Lalo buoUsing Tabe 5-9 naneiasKcsuitsCololInterpretation possible with Tetpcti Asocdlc posithc) end droducFerAltcrOrganismYINRcagentsUninuculated controiQUESTIONS Couslecr eruiroculated tube: MXnk untol? K 4 Dusifita provided by the uninoculated control What informationSECTION ' Difterential'Teetntt pullomer mEDR Injuai Nitrate Reduction Test SHEET 5-8 OBSERVATIONS AND INTEAPRETATIONS guido, enter YcUr rosuts and nle FreLallara Lalo buo Using Tabe 5-9 naneias Kcsuits Colol Interpretation possible with Tetpcti Asocdlc posithc) end droduc Fer Altcr Organism YIN Rcagents Uninuculated controi QUEST... 1 answer A thin spherical shell of radius 6 . 68 m has a total charge of 9... A thin spherical shell of radius 6 . 68 m has a total charge of 9 . 17 C distributed uniformly over its surface. a) Find the electric field 11 . 3 m from the cen- ter of the shell. The Coulomb constant is 8 . 98755 � 10 9 N � m 2 / C 2 . Answer in units of N / C b)Find the electric fie... 5 answers (x + 2y - l)dr + (3x + 6y)dy = 0. (x + 2y - l)dr + (3x + 6y)dy = 0.... 5 answers Write full sentence interpreting your answers fromn the previous question Make sure include input output and units as needed, Write full sentence interpreting your answers fromn the previous question Make sure include input output and units as needed,... 5 answers 6AI Mltea ulunluttna44 LartEtnan (FIck-eEt AkeAn4lanxKdebes IcELaduid *ta Iha ilnlird dmlalan 0l uuinq Ua4n Ent tumnhn Eenla AE t 4+Ldalechklyvan Muauonapielotody Conpae nal Wdur Du on tid Gon t0 Ktttr 0uImtaaelriCusoeidaainMctttlAiclulne Inpotu7 Cnuuitk tuied 4i6n5{0 < 7 !mnulg:0 42 1 Iauattand thcn dclcr O 6AI Mltea ulunluttna44 LartEtnan (FIck-eEt AkeAn4lanx Kdebes Ic ELaduid *ta Iha ilnlird dmlalan 0l uuinq Ua4n Ent tumnhn Eenla AE t 4+Ldalechklyvan Muauonapielotody Conpae nal Wdur Du on tid Gon t0 Ktttr 0u Imtaaelri Cusoeidaain Mctttl Aiclulne Inpotu7 Cnuuitk tuied 4i6n5 {0 < 7 !mnulg: 0 42 1 ... 1 answer 3 & 4 please write it neat so i can understand please 3. In the deuterium-labeled... 3 & 4 please write it neat so i can understand please 3. In the deuterium-labeled compound below. predict the product(s) of the elimination reaction of the following elimination reaction and tell whether the alkene formed contains deuterium (D) or not. (20 pts) HC CH OH H,SO (aq) heat HCP NaOC... 5 answers Mantc enou er quexaruyeinttQurulonWn-Hr distancemass$ Ke (ailt uder Irce Iau tromn Thn topolabucnRene 1/aLuvane45rmti t In? d40nMoving anoinu quuoneattnl;(rn"MacBook Air
Mantc enou er quexar uyeintt Qurulon Wn-Hr distance mass \$ Ke (ailt uder Irce Iau tromn Thn topolabucnRene 1/a Luvane 45rmti t In? d40n Moving anoinu quuoneattnl;(rn" MacBook Air...
[email protected] 33 .IitzJ6 07 417-> + 2 6r' +r') VFkzLair why:2+the following equation? with What E is wrong 10. (J) =4 + 3explain why the equation pant (4) In view r+r_ lim (x + 3) limg T-2 conect the limit; if it exists 11-32 Evaluate 6x + 5 + 3r 12. lim 11. lim 1 -5 X _*-12 5x + 6 x2 + 3x 13.0 lim 14. lim X-5 x _ _ 12r2 _ 92x2 + 3x + [ 16. lim x _ 2xlim 21" + 7t + 317.(-5 + h)? _ 25(2 + h)' 18. lim Aa
lim @3 33 . Iitz J6 07 417-> + 2 6r' +r') VFkz Lair why: 2+ the following equation? with What E is wrong 10. (J) =4 + 3 explain why the equation pant (4) In view r+r_ lim (x + 3) limg T-2 conect the limit; if it exists 11-32 Evaluate 6x + 5 + 3r 12. lim 11. lim 1 -5 X _*-12 5x + 6 x2 + ...
Calculate the pH and the protonated butanoic acid concentrations of the following aqueous :olution: (the K of butanoic acid i3 86 * 10 *) 1) 0.25 Mbutanoic acid 0.04 Mhydrocbloric acidb) 0.25 M butanoic acid 0.04Msodium bydroxidec) 0.10 M butanoic acidi 0.04 Msodium butanate (the sodium salt of butanoic acid)d) 0.05 M butanoic acid and 0.15 M sodium butanate
Calculate the pH and the protonated butanoic acid concentrations of the following aqueous :olution: (the K of butanoic acid i3 86 * 10 *) 1) 0.25 Mbutanoic acid 0.04 Mhydrocbloric acid b) 0.25 M butanoic acid 0.04Msodium bydroxide c) 0.10 M butanoic acidi 0.04 Msodium butanate (the sodium salt of b...
L 0 1 SubmilAnswer Need Help? L Save Progtess Raad I 1i 3Finco 9 1 3 shaded L region in the SAgung4 5.3.064 L 1 (Round - 1 ustaden= mlld 6304510018 Wlace:014
L 0 1 SubmilAnswer Need Help? L Save Progtess Raad I 1i 3 Finco 9 1 3 shaded L region in the SAgung4 5.3.064 L 1 (Round - 1 ustaden= mlld 6304510018 Wlace: 014...
Euea[egcojte: (-4)Jee j t-(2-Ja jt" ~e ft War {ce~ aln ~ma jc ~-fcv
Euea [egco jte: (-4)Jee j t-(2-Ja jt" ~e ft War {ce~ aln ~ma jc ~-fcv...
Hi, can someone check if I did the chart right? and also the work thanks. Use...
Hi, can someone check if I did the chart right? and also the work thanks. Use the table below to identify the type of metal in the Spede et Water quid Vane Iron Cooper Aluminum 0.09 O.RS Lead 0.214 0.17 Unknown metal Mass of metal ) 29.5g Initial temperature of metalo 180. "C 125 g IF Mass of...
1 point) Math 216 Homework webHW6, Problem 3 Suppose that the mass in a mass-spring-dashpot system...
1 point) Math 216 Homework webHW6, Problem 3 Suppose that the mass in a mass-spring-dashpot system with mass m = 49, damping constant c = 1 12, and spring constant k 185 is set in motion with x(0) 18 and x' (0) 43. (a) Find the position function x(t) in the form x(t) (b) Find the psuedoperiod of...
How do you graph y=-5sqrtx and what is the domain and range?
How do you graph y=-5sqrtx and what is the domain and range?...
2) (3 pts) What is the change in the chemical potential of sucrose when its concentration...
2) (3 pts) What is the change in the chemical potential of sucrose when its concentration in water at 25.0°C is changed from 0.150 mol dm-3 to 1.00 mol dm ? Given R= 8.3145 J K-mor!...


text:
# Divide ( square root of 245)/( square root of 5)

Combine and into a single radical.
Divide by .
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Divide ( square root of 245)/( square root of 5)


text:
# Formula for prediction interval for spatial regression

I have a spatial error model that I have estimated and would like to create a figure with the prediction interval. By spatial error model, I mean that the errors are spatially correlated and thus OLS is unbiased but inefficient (not a spatially auto-regressive model).

I used the geostatistical approach to spatial statistics and estimated a semivariogram for my data and used the semivariogram estimates to create a weighting matrix to account for the spatial autocorrelation in the data. I think the answer to this question though would help anyone using weights in their regression regardless of the exact structure of their weighting matrix.

I see from this question that the formula for a 95% prediction interval for OLS is
$$\hat{y} \pm 1.96 \hat{\sigma} \sqrt{1 + \mathbf{X}^* (\mathbf{X}'\mathbf{X})^{-1} (\mathbf{X}^*)'}.$$

Given my limited intuition, my guess would be that the prediction interval for the case with weights would be this (where $\Omega$ is the weighting matrix):

$$\hat{y} \pm 1.96 \hat{\sigma} \sqrt{1 + \mathbf{X}^* (\mathbf{X}'\mathbf{\Omega}^{-1}\mathbf{X})^{-1} (\mathbf{X}^*)'}$$

and was hoping that I could get some feedback from the community here about the correctness or if I am even in the right ballpark.

• Not quite sure why my formulas aren't rendering on the question page as they render just fine in the preview. If someone has enough reputation and wants to edit so that they display correctly it would be much appreciated. – Samsdram Apr 6 '11 at 21:04

## 2 Answers

Note that @whuber adresses all the points raised in the question, I will provide only the mathematical details. The regression with spatial error assumes that model is the following:

$$y=X\beta+u$$

where $Euu'=\Omega$. The efficient estimate of $\beta$ then is

$$\hat\beta=(X'\Omega X)^{-1}X'\Omega y$$

When you have new observation it also follows the model

$$y^*=X^*\beta+u^*$$

We predict $y^*$ by $X^*\beta$ so the prediction error is

$$y^*-\hat{y^*}=X^*\hat\beta+u^*$$

For linear model $u^*$ will be independent of $u$ so

$$Var(y^*-\hat{y^*})=Var(X^*\hat\beta+u^*)=Var(X^*\hat\beta)+Var(u^*)=\sigma^2+\sigma^2X^{* } (X'X)^{-1})(X^{* })'$$

For the spatial error model we have \begin{align*} Var(X^{* }\hat\beta)= Var(X^{* } (X'\Omega X)^{-1}X'\Omega u) =X^{* } (X'\Omega X)^{-1})(X^{* })' \end{align*}

But now $u^*$ correlates with $u$ and we do not have such convenient decomposition. Thus the formula in your question cannot be used.

The general approach how to calculate the error is describe in this article. It shows how to get the best unbiased linear prediction in linear model when the model errors are correlated. Note that this heavily relies on the fact that you have reliable estimate of $\Omega$, so @whuber answer applies very strongly. Even if you have correct formulas they might give bad results if it is known that $\Omega$ cannot be reliably estimated.

Assuming $\Omega$ has been estimated with no appreciable error--which is rarely the case--the correct formula is given by the equation for the kriging prediction error. It needs to replace everything under the square root sign. What is missing from the second formula you give is any explicit accounting of the covariance between the value at the estimation location and the data values.

Some recent approaches, such as presented in Diggle & Ribeiro's "Model-based Geostatistics" (implemented as an R package) have the additional merit of incorporating the estimation error of $\Omega$ in the prediction uncertainty.

• So really the spatial element of this problem is more of a nuisance rather than the story. If I were interested in the value at a specific location I agree that the kriging prediction would be error would be the ticket. If one were to view the problem as more of a weighted least-squares problem though, how would the prediction interval formula change? – Samsdram Apr 7 '11 at 0:58