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# Wave equation

1. Nov 29, 2008

### Confundo

2. Nov 29, 2008

### gabbagabbahey

If $U_0(r,z)=V(r,z)e^{ikz}$, then:

$$\frac{\partial U_0}{\partial z}=\frac{\partial V}{\partial z}e^{ikz}+ikVe^{ikz}$$

due to the product rule.

so.... $$\frac{\partial^2 U_0}{\partial z^2}=\ldots$$???

3. Nov 29, 2008

### Confundo

Oops, I see now I was looking past the obvious. I'll do the calculation in the morning. Is there a reason why they leave out the $$e^{ikz}$$ from the final equation?

Thanks.

edit: $$V = V(r,z)e^{ikz}$$?

Last edited: Nov 29, 2008
4. Nov 30, 2008

### gabbagabbahey

They haven't "left it out"...To see what becomes of that term, work out the derivatives in equation (4) using the assumed form of U_0.

Huh?! ...What are you trying to ask here?

5. Nov 30, 2008

### Confundo

Worked that out now, must learn not to try and do derivatives in my head while tired.

format: plain


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Fit solution paths for linear or logistic regression models penalized by lasso (alpha = 1) or elastic-net (1e-4 < alpha < 1) over a grid of values for the regularization parameter lambda.

COPY_biglasso_main(
X,
y.train,
ind.train,
ind.col,
covar.train,
family = c("gaussian", "binomial"),
alphas = 1,
K = 10,
ind.sets = NULL,
nlambda = 200,
lambda.min.ratio = if (n > p) 1e-04 else 0.001,
nlam.min = 50,
n.abort = 10,
base.train = NULL,
pf.X = NULL,
pf.covar = NULL,
eps = 1e-05,
max.iter = 1000,
dfmax = 50000,
lambda.min = if (n > p) 1e-04 else 0.001,
power_scale = 1,
return.all = FALSE,
warn = TRUE,
ncores = 1
)

## Arguments

family

Either "gaussian" (linear) or "binomial" (logistic).

alphas

The elastic-net mixing parameter that controls the relative contribution from the lasso (l1) and the ridge (l2) penalty. The penalty is defined as $$\alpha||\beta||_1 + (1-\alpha)/2||\beta||_2^2.$$ alpha = 1 is the lasso penalty and alpha in between 0 (1e-4) and 1 is the elastic-net penalty. Default is 1. You can pass multiple values, and only one will be used (optimized by grid-search).

K

Number of sets used in the Cross-Model Selection and Averaging (CMSA) procedure. Default is 10.

ind.sets

Integer vectors of values between 1 and K specifying which set each index of the training set is in. Default randomly assigns these values but it can be useful to set this vector for reproducibility, or if you want to refine the grid-search over alphas using the same sets.

nlambda

The number of lambda values. Default is 200.

lambda.min.ratio

The smallest value for lambda, as a fraction of lambda.max. Default is .0001 if the number of observations is larger than the number of variables and .001 otherwise.

nlam.min

Minimum number of lambda values to investigate. Default is 50.

n.abort

Number of lambda values for which prediction on the validation set must decrease before stopping. Default is 10.

base.train

Vector of base predictions. Model will be learned starting from these predictions. This can be useful if you want to previously fit a model with large-effect variables that you don't want to penalize.

pf.X

A multiplicative factor for the penalty applied to each coefficient. If supplied, pf.X must be a numeric vector of the same length as ind.col. Default is all 1. The purpose of pf.X is to apply differential penalization if some coefficients are thought to be more likely than others to be in the model. Setting SOME to 0 allows to have unpenalized coefficients.

pf.covar

Same as pf.X, but for covar.train. You might want to set some to 0 as variables with large effects can mask small effects in penalized regression.

eps

Convergence threshold for inner coordinate descent. The algorithm iterates until the maximum change in the objective after any coefficient update is less than eps times the null deviance. Default value is 1e-5.

max.iter

Maximum number of iterations. Default is 1000.

dfmax

Upper bound for the number of nonzero coefficients. Default is 50e3 because, for large data sets, computational burden may be heavy for models with a large number of nonzero coefficients.

lambda.min

This parameter has been renamed lambda.min.ratio and is now deprecated.

power_scale

When using lasso (alpha = 1), penalization to apply that is equivalent to scaling genotypes dividing by (standard deviation)^power_scale. Default is 1 and corresponding to standard scaling. Using 0 would correspond to using unscaled variables and using 0.5 is Pareto scaling. If you e.g. use power_scale = c(0, 0.5, 1), the best value in CMSA will be used (just like with alphas).

Multiplicative penalty factor to apply to variables in the form of 1 / m_j^power_adaptive, where m_j is the marginal statistic for variable j. Default is 0, which effectively disables this option. If you e.g. use power_adaptive = c(0, 0.5, 1.5), the best value in CMSA will be used (just like with alphas).

return.all

Deprecated. Now always return all models.

warn

Whether to warn if some models may not have reached a minimum. Default is TRUE.

## Details

The objective function for linear regression (family = "gaussian") is $$\frac{1}{2n}\textrm{RSS} + \textrm{penalty},$$ for logistic regression (family = "binomial") it is $$-\frac{1}{n} loglike + \textrm{penalty}.$$


text:
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# What is the radius of the inscribed circle to a triangle

Author Message
Senior Manager
Joined: 30 Aug 2003
Posts: 322
Location: dallas , tx
What is the radius of the inscribed circle to a triangle [#permalink]

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27 Jan 2004, 13:19
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This topic is locked. If you want to discuss this question please re-post it in the respective forum.

What is the radius of the inscribed circle to a triangle whose sides measure
21cm, 72cm and 75cm respectively?
(1) 9 cm
(2) 37.5 cm
(3) 28.5 cm
(4) 14.5 cm
_________________

shubhangi

Director
Joined: 28 Oct 2003
Posts: 501
Location: 55405

### Show Tags

27 Jan 2004, 13:25
That's an unusual Pythagorean triple, I say...
Senior Manager
Joined: 30 Aug 2003
Posts: 322
Location: dallas , tx

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27 Jan 2004, 13:36
i know.. ..but what abt radius??
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shubhangi

Manager
Joined: 26 Dec 2003
Posts: 227
Location: India

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27 Jan 2004, 15:48
There is a formula for this but I dont remember. as we know the area of the triangle is 1/2 base * height and so the area is 1/2*21*76 = 756 and the area of the circle is phi r^2 which should be less than 756. So of the choices 37.5 and 28.5 is ruled out. Now left with 9 and 14. These can be the answers, in exam I would opt for 9 as the area is less and also is a multiple of 3 like the sides 21,72 and 75.
Manager
Joined: 11 Oct 2003
Posts: 102
Location: USA

### Show Tags

27 Jan 2004, 18:22
Zhung wrote:

3(24+7-25)/2 = 9

Can you explain?

I eliminated the choices that are greater than 21, but could not proceed further.
Manager
Joined: 25 Jan 2004
Posts: 92
Location: China

### Show Tags

27 Jan 2004, 20:39
c = hypotenuse of a right angled traingle

The sides of the right angled traingle are tangents to the incircle.

cf: intersecting tangents to a circle are equal(use similar traingles)

c = (a-r)+(b-r)
r = (a+b-c)/2
Manager
Joined: 11 Oct 2003
Posts: 102
Location: USA

### Show Tags

27 Jan 2004, 21:14
I wonder how many others got it thru Zhung's way.

Director
Joined: 13 Nov 2003
Posts: 789
Location: BULGARIA

### Show Tags

28 Jan 2004, 04:34
Since this is a right triangle the area is 21x72/2=756. The area of a triangle that has a circle inscribed is S=pr ( p-half the perimeter of triangle, r-radius of inscribed circle).P=168,(21+75+72) ,p=P/2=84 Then 756/84= r =9
Manager
Joined: 11 Oct 2003
Posts: 102
Location: USA

### Show Tags

28 Jan 2004, 05:10
BG,

I knew there was a correlation between perimeter, area and inscribed traingle but could not remember. Thaks for refreshing my memory.

Zhung's approach is very fundemental in case one did not know this correlation.
Intern
Joined: 20 Jan 2004
Posts: 19
Location: KZ

### Show Tags

28 Jan 2004, 08:37
I say for 9 sm.

Guys,
there is quick way to approximate right choice in this particular case.

You see, the diameter of incribed circle CANNOT be more than the least of the sides of a triangle. So the most value of the radius might be is 21/2=10.5 sm. Therefore, I pick (1) w/o any calculations.

General advice. See choices before rush into calculus.
Manager
Joined: 11 Oct 2003
Posts: 102
Location: USA

### Show Tags

28 Jan 2004, 09:17
Solo,

Thank you. I agree with you on this given the choices. I was looking for radius < 21 and that was my error. But, what if both 9 and 10 were given in the choices?

I'd rather know of a property that helps me in the long run.
I agree with Solo   [#permalink] 28 Jan 2004, 09:17
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text:
# Timing of ParallelDo versus Do: simple example [duplicate]

I'm trying to understand why a simple do-loop takes substantially longer when done in parallel than sequentially:

count = 0;
SetSharedVariable[count];
AbsoluteTiming[Do[count++, {nn, 100}]]
Print[count]

Out[493]= {0.000176, Null} (\*Time in seconds*)
Out[494]= 100 (\*The result*)

Whereas if I do

count = 0;
SetSharedVariable[count];
AbsoluteTiming[ParallelDo[count++, {nn, 100}]]
Print[count]

Out[497]= {0.391256, Null}  (\*Time in seconds*)
Out[498]= 100 (\*The result*)

Could someone please explain why ParallelDo takes 0.39 seconds whereas Do takes only 0.00017 seconds?

Thanks!

• Take a look here: Why won't Parallelize speed up my code?. SetSharedVariable forces count to be always evaluated on the main kernel. Thus you gain nothing, but you do lose a lot: evaluation will keep switching between parallel kernels and the main kernel, adding a considerable overhead. – Szabolcs Jun 20 '14 at 14:57
• To see what SetSharedVariable really does, evaluate SetSharedVariable[count] first, then ParallelEvaluate[Information[count]]. It's a very straightforward implementation. – Szabolcs Jun 20 '14 at 15:27
• @Szabolcs, thanks for the helpful comment and link. So then, if I understand correctly, in order for a single variable to be updated by, say, a billion independent loops I'm better off using Do loops instead of ParallelDo loops because of the communication overhead? – MvP Jun 20 '14 at 15:42
• Usually, yes. The best thing is not to use a variable that's changed by multiple threads, but this is not possible for every algorithm. When it is possible, it simplifies parallelization dramatically (in any language). SetSharedVariable is only useful if the variable access is infrequent and takes negligible time compared to other computations. I only found SetSharedVariable useful in very few cases. – Szabolcs Jun 20 '14 at 15:46


text:
# Electric Field and Force

## Homework Statement

A 12 V battery gets connected between the center conductor (black dot) and the sheath of the coaxial cable so that the voltage of the center conductor with respect to the sheath (outer contour) is +12 V. What is the approximate force on a +25C particle located at D.

## Homework Equations

$$F=(Vq)/x$$

## The Attempt at a Solution

Electric field lines extend from the center conductor to the sheath, with field vectors pointing from the center to the outer contour.

Would the average force then just be $$F=(+12V*+12C)/x$$

and x would merely be measured from the center conductor to the position D?

We're told the radius of the circle is 1cm.

## Answers and Replies

Related Introductory Physics Homework Help News on Phys.org
gneill
Mentor
Note that, if you imagine the electric field lines between the inner and outer conductor, that they diverge as they get further from the center. That means that the field won't be a uniform magnitude over all distances as it is with parallel plates.

Have you used Gauss' law before? You should be able to discover the mathematical form of the electric field between the conductors using Gauss' law.


text:
# Implicit Differentiation Calculator

This will be calculated

Mobile Apps

## Implicit differentiation calculator with steps

Implicit differentiation calculator is an online tool used to differentiate the implicit function with respect to its independent variable. This implicit differentiation solver calculates the given implicit function with steps.

## How does implicit derivative calculator work?

Follow the below steps to use our implicit differentiation calculator.

• Input the f(x, y) or write the L.H.S of the implicit equation.
• Then input the g(x, y) or write the R.H.S of the implicit equation.
• Hit the load examples button to use the sample examples.
• Choose the independent variable of the function i.e., x, y, or z.
• Press the calculate key to differentiate the given implicit function.
• Hit the possible intermediate steps to view the solution with steps.
• Press the clear button to recalculate.

## What is implicit differentiation?

Implicit differentiation helps us to differentiate the implicit equations that have both x & y variables. This technique allows us to find the slopes of tangent lines passing through curves that are not considered functions.

In explicit differentiation, equations of multivariable do not define y as a function of x so implicit differentiation is used. Explicit functions are those functions that are written in terms of independent variables only.

While the implicit functions are written in terms of both dependent and independent variables. In this type of derivative, y as a function of x doesn’t consider a constant e.g., the differential of y with respect to x will be dy/dx.

The implicit function can be written as:

$f\left(x,y\right)=g\left(x,y\right)$

## How to differentiate the implicit function?

Follow the below example to learn how to differentiate the implicit function.

Example

Differentiate the implicit function $4x^3+5y^2=2x^2$ with respect to “x” by using the chain rule.

Solution

Step 1: Apply the derivative notation on the given implicit equation.

$\frac{d}{dx}\left(4x^3+5y^2\right)=\frac{d}{dx}\left(2x^2\right)$

Step 2: Use the sum rule of differentiation and apply the derivative notation separately.

$\frac{d}{dx}\left(4x^3\right)+\frac{d}{dx}\left(5y^2\right)=\frac{d}{dx}\left(2x^2\right)$

Step 3: Now use the constant function and power rules to differentiate.

$4\frac{d}{dx}\left(x^3\right)+5\frac{d}{dx}\left(y^2\right)=2\frac{d}{dx}\left(x^2\right)$

$4\left(3x^2\right)+5\frac{d}{dx}\left(y^2\right)=2\frac{d}{dx}\left(x^2\right)$

$12x^2+5\frac{d}{dx}\left(y^2\right)=2\frac{d}{dx}\left(x^2\right)$

Step 3: Use the chain rule.

$\frac{d}{dx}\left(y^2\right)=\frac{du^2}{du}\frac{du}{dx}where\:u=y\:$

$12x^2+5\left(2y\frac{d}{dx}\left(y\right)\right)=\frac{d}{dx}\left(2x^2\right)$

$12x^2+10y\left(\frac{d}{dx}\left(y\right)\right)=\frac{d}{dx}\left(2x^2\right)$

Step 5: Use the chain rule again.

$12x^2+\left(\frac{d}{dx}\left(x\right)\right)y'\left(x\right)10y=\frac{d}{dx}\left(2x^2\right)$

$12x^2+\left(1\right)y'\left(x\right)10y=\frac{d}{dx}\left(2x^2\right)$

$12x^2+y'\left(x\right)10y=\frac{d}{dx}\left(2x^2\right)$

Step 6: The derivative of $2x^2$ is 4x.

$12x^2+y'\left(x\right)10y=4x$

$y'\left(x\right)10y=4x-12x^2$

$y'\left(x\right)=\frac{4x-12x^2}{10y}$

### References

1. What is implicit differentiation | The Story of Mathematics
2. Example of implicit differentiation | Calculus I - implicit differentiation. (n.d.)


text:
# Math Help - Show that x^3-2 is irreducible in Z/19Z [x]/<x^2-2>

1. ## Show that x^3-2 is irreducible in Z/19Z [x]/<x^2-2>

Please show that x^3-2 is irreducible in Z/19Z [x]/<x^2-2>. There HAS to be a better way than plugging and chugging 361 polynomials.

Thanks!

Julian

2. Originally Posted by aznmaven
Please show that x^3-2 is irreducible in Z/19Z [x]/<x^2-2>. There HAS to be a better way than plugging and chugging 361 polynomials.

Thanks!

Julian
It is sufficient to show $t^3 - 2$ has no roots in $\mathbb{Z}_{19}[x]/(x^2-2)$. Let $\alpha = x + (x^2-2)$. Then we see that $\alpha^2 = 2$.

Now let $a+b\alpha$ be an element in this field with $a,b\in \mathbb{Z}_{19}$. Now show that $(a+b\alpha)^3 - 2 =0$ is impossible by expanding out this the expression and using the fact $\alpha^2 = 2$, and comparing coefficient.


text:
# How many different sets can be formed by two nonempty sets

If A and B are two different nonempty sets, how many distinct sets can be formed with these sets using as many unions,intersections,complements and parentheses as desired.

Four sets are fundamental:$A$,$B$,$A \cup B$,$A \cap B$.

Other sets are $A \cup B'$,$A' \cup B$,$A \cap B'$,$A' \cap B$,$A' \cup B'$,$A' \cap B'$. Any other sets are possible.

-
You could look at cartesian products and power sets. – PEV Jan 17 '11 at 15:27
@Trevor:Sorry,cartesian products and power sets are not allowed. – Vinod Jan 17 '11 at 15:30
To get a clear answer, you have to assume that A and B are not just distinct but also "as independent as possible" in some sense. Otherwise, it could be the case that A = {1} and B = {1,2}, and you only get 8 possible sets. – Rahul Jan 17 '11 at 16:02

$16$. The corresponding Venn diagram has four parts, and you can get any combination of those four parts.
@Vinod $2^{2^n - 1}$ to be accurate. – Sergey Filkin Apr 8 '12 at 8:30


text: The "farthest point distance function" of a non-empty compact set E in the plane is defined by d_{E}(x) := max_{y\in E}  |x-y|. This function arises naturally in the study of sharp inequalities for norms of products of polynomials. The crucial observation which makes it useful is that log d_{E}(x) is a logarithmic potential: it can be written as $\int \log |x-y| d\sigma _{E}(y)$ for some probability measure $\sigma_{E}.$ However, the distribution of the measure $\sigma_{E}$ has proved mysterious. On the basis of various examples Laugesen and Pritsker were led to the unexpected and intriguing conjecture that $\sigma_{E}(E)\leq 1/2$ whenever E contains more than one point. This talk will use a blend of potential theory and convex geometry to explain why this inequality is true, and when equality holds. This is joint work with Ivan Netuka (Charles University, Prague).