text:
Free Version
Difficult

Slope, Correlation When Removing Scatterplot Point: X vs. Y

APSTAT-KCL1TN

A scatterplot of $x$ versus $y$ is shown below:

If the point $(9,49)$ is removed, which of the following statements is TRUE?

A

The slope of the least squares regression line decreases and the correlation coefficient decreases.

B

The slope of the least squares regression line decreases and the correlation coefficient increases.

C

The slope of the least squares regression line decreases and the correlation coefficient does not change.

D

The slope of the least squares regression line increases and the correlation coefficient increases.

E

The slope of the least squares regression line increases and the correlation coefficient decreases.

format: plain


text:
# Computational complexity and implementation of UDU Modified Cholesky Rank 1 Update

I am attempting to increase the performance of a legacy Kalman Filter implementation. The state covariance is factored in terms of UDU, i.e. $\mathbf{P} = \mathbf{U}\mathbf{D}\mathbf{U}^T$.

Many years ago, Bierman showed how scalar measurements may be used to update the covariance. Helpfully, it includes an implementation in FORTRAN that I have reproduced in MATLAB (though it combines certain operations in a loop and thus the algorithm is not particularly transparent).

Ultimately, I would like to replace the routine to take advantage of modern matrix libraries (BLAS, Eigen, etc) to increase performance.

Ultimately, the core of the operation is to perform a rank-1 update (downdate) of the UD-factored covariance in the form of:

$\mathbf{U_+}\mathbf{D_+}\mathbf{U_+}^T = \mathbf{U_-}\mathbf{D_-}\mathbf{U_-}^T - c\mathbf{v}\mathbf{v}^T$

where $\mathbf{U_-}$ and $\mathbf{D_-}$ is my prior known factorisation, c is a non-negative scalar and $\mathbf{v}$ is (known) a column vector. The goal is to determine $\mathbf{U_+}$ and $\mathbf{D_+}$.

So, my questions are:

• For the modified Cholesky update, what is the cost in terms of FLOPS - I understand it is $O(n^2$)?
• Is the a known efficient implementation in a commonly available library, or so I need to try to roll my own?

(For interest, $n$ is about 40 , but is being implemented on an embedded platform).

Yes, rank-one updates of a Cholesky factorization take $O(n^{2})$ time.

Unfortunately, there isn't a low rank update for the Cholesky factorization in LAPACK. There is an update routine for the Cholesky factorization in Linpack, but that routine works with the convention $LL^{T}$ factorization, not the square root free Cholesky factorization.

Unfortunately, there isn't enough data reuse in the process of rank-one updating a Cholesky factorization to get the kind of performance boost from blocknig and cache reuse that you get in various $O(n^3)$ operations that operate on $O(n^{2})$ data. .

• So, it looks like I need to hand-roll a routine. Is there a good reference for this? Any techniques I should watch out for to try to maximise performance? – Damien Aug 27 '13 at 5:41
• Also, is there a reference for a FLOPS count (not just $O(n^2)$ ) for the cholupdate - I haven't managed to find one so far. – Damien Aug 27 '13 at 5:41
• The references that I have at hand are all for updating the LDLT factorization rather than the UDUT factorization. You could rederive this from the Sherman-Morison-Woodbury formula, but I expect that some library work will help you find a version for UDUT. The flops count should be easy to get from the actual code that you use (and you can easily instrument the code to explicitly count the flops used if you want to check your formula.) – Brian Borchers Aug 27 '13 at 20:26

Since a UDU rank 1 update is hard to find, below is a MATLAB implementation (without warranties of any kind)

## Downdate

function [Ubar, Dbar] = ThorntonRank1Downdate(U, D, c, a)

Ubar = U;
Dbar = D;

n = length(a);

for j = n : -1 : 2

s = a(j);
d = D(j) - c * s^2;
b = c / d;
beta = s * b;
c = b * D(j);
Dbar(j) = d;

a( 1:(j-1) ) = a( 1:(j-1) ) - s * U( 1:(j-1) ,j);
Ubar( 1:(j-1) ,j) = U( 1:(j-1) ,j) - beta * a( 1:(j-1) ) ;

end

Dbar(1) = D(1) - c * a(1)^2;

## Update

function [Ubar, Dbar] = ThorntonRank1Update(U, D, c, a)

Ubar = U;
Dbar = D;

n = length(a);

for j = n : -1 : 2

s = a(j);
d = D(j) + c * s^2;
b = c / d;
beta = s * b;
c = b * D(j);
Dbar(j) = d;

for i = j-1 : -1 : 1

a(i) = a(i) - s * U(i,j);
Ubar(i,j) = U(i,j) + beta * a(i);

end

end

Dbar(1) = D(1) + c * a(1)^2;

## Test Program

% Requires symbolic toolbox
syms u12 u13 u14 u23 u24 u34 real
U = [1 u12 u13 u14; 0 1 u23 u24; 0 0 1 u24; 0 0 0 1]
% U = [1 u12 u13; 0 1 u23; 0 0 1]

syms v1 v2 v3 v4 real
v = [v1;v2;v3; v4]
% v = [v1;v2;v3]

syms d1 d2 d3 d4 real
D = diag([d1,d2,d3,d4]);
% D = diag([d1, d2, d3]);

syms c positive

%--------------------------------------------------------------------------

P = U * D * U.' + c * (v * v.');
[Udash, Ddash] = UD(P)

[Ubar, Dbar] = ThorntonRank1Update(U, diag(D), c, v)

update_TestU = simplify(Udash - Ubar, 'seconds', 10)
update_TestD = simplify(Ddash - diag(Dbar), 'seconds', 10)

update_TestPbar  = simplify((Ubar * diag(Dbar) * Ubar.') - P, 'seconds', 10)
update_TestPdash = simplify((Udash * Ddash * Udash.') - P, 'seconds', 10)

%--------------------------------------------------------------------------

P = U * D * U.' - c * (v * v.');
[Udash, Ddash] = UD(P)

[Ubar, Dbar] = ThorntonRank1Downdate(U, diag(D), c, v)

downdate_TestU = simplify(Udash - Ubar, 'seconds', 10)
downdate_TestD = simplify(Ddash - diag(Dbar), 'seconds', 10)

downdate_TestPbar  = simplify((Ubar * diag(Dbar) * Ubar.') - P, 'seconds', 10)
downdate_TestPdash = simplify((Udash * Ddash * Udash.') - P, 'seconds', 10)

%--------------------------------------------------------------------------

## Reference

Page 55 of: Bierman, G. J., "Factorization Methods for Discrete Sequential Estimation", Academic Press, 1977

Also useful was the following paper:

Fletcher, R. & Powell, M. J. D. On the modification of LDLT factorizations Mathematics of Computation, 1974, 28, 1067-1087 (PDF)

## Complexity

Without doing a proper FLOPS count, it looks to be in the order of $n^2 + O(n)$.


text:
#### Previous topic

models.viz2.params

#### Next topic

pools.gearman_cluster

# pools¶

glimpse.pools.MakePool()

Return a new instance of the default worker pool.

Returns: A serializable worker pool.
glimpse.pools.GetClusterPackage(cluster_type=None)

Choose a cluster package by name.

If cluster_type is gearman, for example, then this method returns glimpse.pools.gearman_cluster. The returned module is guaranteed to contain the MakePool() and RunMain() functions.

Parameters: cluster_type (str) – Type of cluster to create (e.g., gearman, zmq, ipython). The default is read from the GLIMPSE_CLUSTER_TYPE environment variable, or is ipython if this variable is unset.
class glimpse.pools.SinglecorePool

A fall-back worker pool that uses a single core of a single machine.

imap(func, iterable, chunksize=1)

Apply a function to an iterable.

imap_unordered(func, iterable, chunksize=1)

Apply a function to an iterable, where elements are evaluated in any order.

map(func, iterable, chunksize=None)

Apply a function to a list.

class glimpse.pools.MulticorePool(*args)

A worker pool that utilizes multiple cores on a single machine.

This class delegates to the corresponding methods of multiprocessing.Pool. Thus, all restrictions — such as limitations on mapping an anonymous function — are preserved.

Note that there are limitations regarding the functions that can be mapped with the methods of this class. Specifically, note that the following example will fail:

>>> pool = glimpse.pools.MulticorePool()
>>> def f(x): return x * 10
>>> pool.map(f, [1, 2, 3])
AttributeError: 'module' object has no attribute 'f'

Here, the AttributeError is thrown because f() is not defined in the worker subprocess. Similarly, an anonymous function also throws an error:

>>> pool.map((lambda x: x * 10), [1, 2, 3])
PicklingError: Can't pickle <type 'function'>: attribute lookup
__builtin__.function failed

Here, it becomes clear that communication between subprocesses uses pickled objects. The PicklingError results from the fact that an anonymous function can not be pickled.

imap(func, iterable, chunksize=1)

Apply a function to an iterable.

imap_unordered(func, iterable, chunksize=1)

Apply a function to an iterable, where elements are evaluated in any order.

map(func, iterable, chunksize=None)

Apply a function to a list.


text:
Converted document
$\newcommand{\lyxlock}{}$
Section 9.2: Configuring Linear Solvers Up Section 9.2: Configuring Linear Solvers Subsection 9.2.2: Skyline

### 9.2.1 Pardiso

The Pardiso solver is an effecient sparse direct linear solver and is the default linear solver. FEBio uses the implementation from the MKL library. It does not require any configuration parameters. It can take symmetric, unsymmetric, and structurally symmetric sparse matrix formats.
Section 9.2: Configuring Linear Solvers Up Section 9.2: Configuring Linear Solvers Subsection 9.2.2: Skyline


text:
## Elementary Statistics (12th Edition)

$0.105.$
We know that $probability=\frac{number \ of \ good \ outcomes}{number\ of\ all\ outcomes}$. Hence here $probability=\frac{2}{17+2}=0.105.$


text:
# Math Help - Determing the ratio of buyers who take up a multi-buy offer (i.e. redemption rate)

1. ## Determing the ratio of buyers who take up a multi-buy offer (i.e. redemption rate)

Hi,

My question is how do i dtermine the % of shoppers who took up a multi buy offer.

So for example the product normally retails for $4.60 and the offer was buy 2 for$6.00 so if EVERYONE takes up the offer the sell price will be $2.99 right? But not everyone takes up the offer (some ppl prefer to just buy the one unit instead of two). In my data the average sell price for the week of the multi-buy promotion was$3.86

So how do i figure out the % of sales that were made were shoppers took up the multi-buy?

Thanks!

2. ## Re: Determing the ratio of buyers who take up a multi-buy offer (i.e. redemption rate

Originally Posted by agerrrard
Hi,

My question is how do i dtermine the % of shoppers who took up a multi buy offer.

So for example the product normally retails for $4.60 and the offer was buy 2 for$6.00 so if EVERYONE takes up the offer the sell price will be $2.99 right? no$3.00

But not everyone takes up the offer (some ppl prefer to just buy the one unit instead of two). In my data the average sell price for the week of the multi-buy promotion was \$3.86

So how do i figure out the % of sales that were made were shoppers took up the multi-buy?

Thanks!
Let $m$ be the fraction of your $N$ customers that took the multi-buy. Then the total sales were $m\times N\times 2+(1-m)\times N$ and the total they paid was $m\times N \times 2 \times 3.00 + (1-m)\times N \times 4.60$.

So the average price was the total received divided by the total sales:

$p=\frac{m\times N \times 2 \times 3.00 + (1-m)\times N \times 4.60}{m\times N\times 2+(1-m)\times N}$

Now the $N$s cancel and we are told that the average price paid was $\3.86$ so:

$p=\frac{m \times 2 \times 3.00 + (1-m) \times 4.60}{m\times 2+(1-m)}=3.86$

which you now solve for the (decimal) fraction of customers who took up the multi-buy, and multiply that by $100$ to get the percentage.

CB


text:
# representing simple definite integral as convolution in matlab using CONV

A definite integral is a convolution with idial all '1' filter(by the theory) so i tried to test this issue with MATLAB. i have taken a function called fun made an integral over it in 1e-6 - 20*e-6 interval the result is called simbolic_int_result which 37.7 then i have taken fun sampled it in 1e-6 - 20*e-6 interval as shown bellow with 1000 samples and created all "1" filter (idial) and used CONV to make convolution,but its not working, instead of a scalar i get 2000X1 vector Where did i go wrong? Thanks.

c0=2.997*10.^8;% m/s speed of light in vaccum
h=6.625*10.^-34;% J.s Planck constant
k=1.38*10.^-23;%  T/K Boltzmann constant
n=1;

T=307;%  Temperature initiatlisation
fun=@(lambda)(2*pi.*h.*(c0.^2))./((n.^2).*(lambda.^5).*(exp((h.*c0)./(k.*T.*lambda))-1));
simbolic_int_result= integral(fun,1e-6,20e-6)/10;

k=linspace(10.^-6,20*10^-6,1000);
fun_vector=fun(k);

filter=linspace(1,1,1000); % all ones idial filter

convolution_result=conv(filter,fun_vector);


text:
Standard syntax for preconditions, postconditions, and invariants

Alan Falloon Al.Falloon at synopsys.com
Thu Oct 19 09:54:06 EDT 2006

I propose that haskell' include a standard syntax for invariants that
the programmer wants to express. Here is a rough idea (just to get the
point across):

\begin{code}

head xs @ require { not (null xs) }

reverse :: [a] -> [a]
reverse @ ensure { reverse (reverse xs) == xs }
reverse [] = []
reverse (x:xs) = reverse xs ++ x

data RedBlackTree a = RBLeaf a |  RBNode  Bool  (RedBlackTree a) a
(RedBlackTree a)
invariant RedBlackTree a where
RBNode False _ l _ r ==> redDepth l == redDepth r

\end{code}

The intent is not to have standardized checks on the invariants, its
just to supply a common way to specify invariants to that the various
strategies for checking them can all work from the same data. For
example, one tool might use the invariants to generate QuickCheck
properties, a compiler might provide a mode that does run-time checking
of the invariants, and another tool might try to prove the invariants
statically like in ESC [1]. Eventually, we might end up with
sophisticated hybrid approaches that combine static proof with
generating test cases for the unprovable bits. At the very least Haddock
could include the invariants as part of the documentation.

[1] http://www.cl.cam.ac.uk/~nx200/research/escH-hw.ps


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# Give examples of two functions $f: N \to N$ and $g: N \to N$ such that $g\;o\;f$ is onto but $f$ is not onto.

Can you answer this question?

## 1 Answer

0 votes
Toolbox:
• A function $f:A \to B$ is onto if for $y \to B$ then exists unique $x \in A$ such that $f(x)=y$
• We define two function $f:N \to N$ and $g: N \to N$ such that gof is onto $gof(x)=gof(y)$
Given $g(x)= \left\{ \begin{array}{1 1} x-1 & \quad x > 1\\ 0 & \quad x=1 \end{array} \right.$
Consider $x=1 \in$ Codomain of $g$. It is clear that there exists 2 element $x=1, x=2$ such that $g(x)=1$
Therefore $f$ is not onto
Consider $f(x)=x+1$
$\Rightarrow gof(x)=g(f(x))$ $=g(x+1)$ $=x+1-1 = x$
Since $x \in N;x+1 > 1$
$\Rightarrow gof$ is onto. since for $y \in N$ then exists $x \in N$ such that $gof(x)=y$
answered Feb 28, 2013 by
edited Mar 20, 2013

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text:
Reduction numbers, Briançon-Skoda theorems and the depth of Rees rings
Compositio Mathematica, Volume 97 (1995) no. 3, pp. 403-434.
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Aberbach, Ian M.; Huneke, Craig; Trung, Ngô Viêt. Reduction numbers, Briançon-Skoda theorems and the depth of Rees rings. Compositio Mathematica, Volume 97 (1995) no. 3, pp. 403-434. http://www.numdam.org/item/CM_1995__97_3_403_0/

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[HI] Herrmann, M. and Ikeda, S.: On the Gorenstein property of Rees algebras, Manu. Math. 58 (1987) 471-490. | MR | Zbl

[HHR] Herrmann, M., Huneke, C., and Ribbe, J.: On reduction exponents of ideals with Gorenstein form ring, Proc. Edinburgh Math. Soc. (to appear). | Zbl

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[HSV1] Herzog, J., Simis, A., and Vasconcelos, W.V.: On the arithmetic and homology of algebras of linear type, Trans. Amer. Math. Soc. 283 (1984) 661-683. | MR | Zbl

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[Hu1] Huneke, C.: On the associated graded ring of an ideal, Illinois J. Math. 26 (1982) 121-137. | MR | Zbl

[Hu2] Huneke, C.: The theory of d-sequences and powers of ideals, Adv. in Math. 46 (1982) 249-279. | MR | Zbl

[I] Ikeda, S.: On the Gorenstein property of Rees algebras over local rings, Nagoya Math. J. 92 (1986) 135-154. | MR | Zbl

[JK] Johnston, B. and Katz, D.: Castelnuovo regularity and graded rings associated to an ideal, Proc. Amer. Math. Soc. (to appear). | MR | Zbl

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text:
Preparing NOJ

# Happy Prime Number

1000ms 65536K

## Description:

A Happy Number can be defined as follows. From a positive integer n, calculate the sum of square of each digit of n. Then from that sum, repeat the same process over and over again. This cycle terminates if and only if there is 1 in the sequence. Hence, we call the number n a happy number if it generates a finite sequence. Otherwise, the endless cycle occurs (1 never appears in the sequence). We

may call the number generating an endless cycle an unhappy number . Observe the following examples:

700 is a happy number72+02 +02 =49 12 +32 +02 =10

2 is not a happy number
22=4
3
2 +72 =58
1
2 +42 +52=42
4
2 = 16 ... and never terminates

42 +92 =97 12 +02 =1

92 +72 =130

12+62=37 82 +92 =14 22 +02 =4

42 =16
5
2 +82 =89 42 +22 =20

A Prime Number is an integer greater than 1 that can be divided only by 1 and itself. Here are some prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, ...

A Happy Prime Number is a prime number which also satisfies the happy number condition such as 7, 13, 19, ...

Your task is to write a program to show all happy prime numbers less than or equal to a given numbern. (10 n 1000000)

## Input:

A positive number n is the only input of the program

## Output:

the program prints all happy numbers in ascending order, one number in a line.

## Sample Input:

20

## Sample Output:

71319

## Note:

Info

Provider NOJ

Code NOJ1458

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Submitted 1

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AC Rate 100%

Date 04/20/2019 10:03:10

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text:
Question

# By selling 45 lemons for Rs.40, a man loses 20%. How many should he sell for Rs.24 to gain 20% in the transaction?

Verified
157.2k+ views
Hint: Here, we will use the formulae of cost price and selling price with gain and loss percentages to solve the given problem.
Given,
Selling Price of 45 is Rs.40 and he loses 20%. Now, let us find the cost price of these 45 lemons by using the formulae of cost price with loss percentage i.e..,
$C.P = \dfrac{{S.P\times 100}}{{(100 - loss\% )}} \to (1)$
Now, let us substitute the value of S.P and loss% in equation (1), we get
$C.P = \dfrac{{45\times 100}}{{(100 - 20)}} = \dfrac{{45\times 100}}{{80}} = 50$
Therefore, the Cost price of 45 lemons is Rs.50. Now, let us find the Selling price of 45 lemons with
20% gain by using the formulae of selling price with gain percentage i.e..,
$S.P = \dfrac{{C.P\times (100 + gain\% )}}{{100}} \to (2)$
Now, let us substitute the value of C.P and gain% in equation (2), we get
$S.P = \dfrac{{50\times (100 + 20)}}{{100}} = \dfrac{{50\times 120}}{{100}} = 60$
Since, the selling price of 45 lemons with 20% gain is Rs.60, then the selling price of 1 lemon will be $\dfrac{{60}}{{45}} = \dfrac{4}{3}$. Let ‘x’ be the number of lemons then the number of lemons he sell at Rs.24 will be
$\Rightarrow \dfrac{4}{3}\times x = 24 \\ \Rightarrow x = 24\times \dfrac{3}{4} \\ \Rightarrow x = 18 \\$
Hence, the number of lemons he can sell at Rs.24 with 20% gain is 18.
Note: Make sure that percentage of gain should be added to ‘100’ whereas the percentage of loss is subtracted from ‘100’.


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# Newton's Laws

Newton's first and second laws, in Latin, from the original 1687 edition of Principia Mathematica .

In 1687 Isaac Newton's work Philosophiae Naturalis Principia Mathematica ( Latin ; 'Mathematical Principles of Natural Philosophy ') appeared, in which Newton formulated three principles ( laws ) of motion , which were Newtonian axioms , basic laws of motion , Newtonian principles or also Newtonian Laws are known. In Newton's work they are referred to as Lex prima , Lex secunda and Lex tertia ('First / Second / Third Law'), together with axiomata, sive leges motus (' axioms or laws of motion').

These laws form the foundation of classical mechanics . Although they do not apply unreservedly in the context of modern physical theories such as quantum mechanics and relativity, they can be used to make reliable predictions within a wide range of validity .

Usually the following three laws are mentioned, which in a very simplified form read as follows:

1. A body free of force remains at rest or moves in a straight line at a constant speed.
2. Force equals mass times acceleration . ( )${\ displaystyle {\ vec {F}} = m \ cdot {\ vec {a}}}$
3. Force equals opposing force : A force from body A to body B is always accompanied by an equally large but opposing force from body B to body A.
${\ displaystyle {\ vec {F}} _ {A \ to B} = - {\ vec {F}} _ {B \ to A}}$

Newton already assumed that two forces can be combined with a force parallelogram to form one resulting force. The axiom of the parallelogram of forces is also called Newton's fourth law. In modern literature, on the other hand, the superposition principle is usually mentioned as the fourth Newtonian law.

## First Newton's Law

The first Newtonian law is also called lex prima , the principle of inertia , the law of inertia or the inertial law . The principle of inertia makes statements about the movement of physical bodies in inertial systems . There are different versions:

"A body remains in a state of rest or uniform, straight-line movement, unless it is forced to change its state by acting forces ."

Original Latin text:

Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus illud a viribus impressis cogitur statum suum mutare.

The speed is therefore constant in amount and direction. A change in the state of motion can only be achieved by exerting an external force , for example through gravitational force or the force of friction . ${\ displaystyle {\ vec {v}}}$

If there is no force acting on a body, its speed is constant over time.
If no force acts on a mass point, its momentum is constant. The momentum is the product of mass and speed.

In these forms the sentence is only valid when there is no force at all . The reverse, that no force acts when it moves at constant speed, does not follow from this. In this case, several forces can act on him that cancel each other out. In this case there is no resulting force .

For a mass point, Newton's first law corresponds to the equilibrium conditions . For extended bodies, the torque balance must also be checked.

Galileo Galilei was the first to recognize the principle of inertia at the beginning of the 17th century and already formulated that the force-free movement could continue in a straight line as far as desired. He used this for the first correct treatment of the movements of bodies on earth in free fall , inclined throw and on the inclined plane . René Descartes gave the first clear formulation as a general principle of force-free movements in 1644, but it was only Newton that applied the principle of inertia to the movements of extraterrestrial bodies. This is also his special achievement: in the works of antiquity, which were still regarded as correct up to the late Middle Ages, it was believed that the movements on earth and those in the sky obey different laws. Newton recognized them as two special cases of a general law. In addition, Newton declared the straight, unaccelerated movement to be the norm. Only when the movement of a body deviates from this, one has to explain this with the effect of forces. Shortly before Newton it was assumed that circular motion would be the normal case.

The above versions only apply if the movement is described in an inertial frame. Newton's first law is then only a special case of the second. In modern works on theoretical mechanics , the reference system is usually first defined and the first Newtonian axiom is then introduced in one of the following versions or similar versions.

There are reference systems in which the force-free movement takes place at a constant speed. These are inertial systems .
There are coordinate systems in which every force-free mass point moves in a straight line or is at rest. These particularly important coordinate systems are called inertial systems.

It is then determined which properties must apply to inertial systems. (In particular, they must not rotate or be accelerated.) The first Newtonian axiom is thus used as a definition for the concept of the inertial system.

## Second Newton's Law

Newton's second law is also called the lex secunda or principle of action .

It is the basis for many equations of motion in mechanics :

"The change in movement is proportional to the effect of the moving force and happens in the direction of the straight line in which that force acts."

Original Latin text:

"Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur."

Formally, this relationship between force and change in motion is expressed as

${\ displaystyle {\ dot {\ vec {v}}} \ propto {\ vec {F}}}$

The point above a letter is the notation introduced by Newton in another context for the temporal change of a physical quantity . The sign in between means proportional , i.e. standing in a fixed relationship. ${\ displaystyle \ propto}$

In Newton's original work, expressed in modern terms, the generally applicable formulation (with the impulse ) was already described. Newton's writings work with geometric representations of the limit values of distance and area relationships. The representation is also called the momentum set , especially in literature on technical mechanics and fluid mechanics . In words, it means that the change in momentum of a body over time corresponds to the resulting external force acting on this body. This representation is more general than the Euler form mentioned below, since it also applies to bodies with variable mass. The law of momentum is of great importance when calculating impacts from bodies that can no longer be regarded as point-shaped. It can also be represented in integral form with the impulses and at the time or : ${\ displaystyle {\ vec {F}} = {\ dot {\ vec {p}}}}$ ${\ displaystyle {\ vec {p}}}$${\ displaystyle {\ vec {F}} = {\ dot {\ vec {p}}}}$${\ displaystyle {\ vec {p}} _ {0}}$${\ displaystyle {\ vec {p}} _ {1}}$${\ displaystyle t_ {0}}$${\ displaystyle t_ {1}}$

${\ displaystyle {\ vec {p}} _ {1} - {\ vec {p}} _ {0} = \ int _ {t0} ^ {t1} {\ vec {F}} (t) \, \ mathrm {d} t}$

The force can therefore be seen as the cause of the change in momentum. The amount of the impulse changes only by forces that act in the direction of the movement of the body, while the direction of the impulse is changed by forces that are perpendicular to it. If the resulting force is zero, the law of conservation of momentum follows .

The law was first formulated in this form by Leonhard Euler in 1750 . It is the acceleration , that is a measure of the change in speed. ${\ displaystyle {\ vec {F}} = m {\ vec {a}}}$${\ displaystyle {\ vec {a}}}$

This equation is often called the basic equation of mechanics - regardless of whether in Newton's or Euler's formulation .

## Third Newton's Law

Newton's third law, also called lex tertia , interaction principle , counteraction principle , or reaction principle , states:

Forces always appear in pairs. If a body A exerts a force on another body B ( actio ), an equally large but opposing force from body B acts on body A ( reaction ). "

Original Latin text:

" Actioni contrariam semper et aequalem esse reactionem: sive corporum duorum actiones in se mutuo semper esse aequales et in partes contrarias dirigi. "
${\ displaystyle {\ vec {F}} _ {A \ to B} = - {\ vec {F}} _ {B \ to A}}$

The principle of interaction is also referred to as the principle of actio and reactio or, for short, " actio equals reactio " (lat. Actio est reactio ). Newton's third law presupposes an immediate action at a distance . Therefore it has no general validity in the special theory of relativity (and thus in electrodynamics ) and the general theory of relativity - here the momentum conservation of the entire system (particles plus radiation) applies . The principle of interaction can also be formulated in such a way that in a closed system the sum of the forces is zero, which is equivalent to conservation of momentum.

## Superposition principle of forces

In Newton's work, the principle of undisturbed superposition or superposition principle of mechanics is described as an addition to the laws of motion.

If several forces act on a point (or a rigid body ) , these add up vectorially to form a resulting force . ${\ displaystyle {\ vec {F}} _ {1}, {\ vec {F}} _ {2}, \ dots, {\ vec {F}} _ {n}}$${\ displaystyle {\ vec {F}}}$"
${\ displaystyle {\ vec {F}} _ {\ text {res}} = {\ vec {F}} _ {1} + {\ vec {F}} _ {2} + \ dots + {\ vec { F}} _ {n}}$

This superposition principle was later also referred to as lex quarta , Newton's fourth law .

## literature

• Isaac Newton: Philosophiae Naturalis Principia Mathematica . 3. Edition. Innys, Regiae Societatis typographos, London 1726 ( uni-goettingen.de [accessed July 30, 2017]).
• Jerry Marion, Stephen Thornton: Classical Dynamics of Particles and Systems. Harcourt College Publishers, 1995, ISBN 0-03-097302-3 .
• GR Fowles, GL Cassiday: Analytical Mechanics. Saunders College Publishing, 6th Edition 1999, ISBN 0-03-022317-2 .
• Ulrich Hoyer : Is Newton's second axiom of motion a law of nature? In: Journal for general philosophy of science. Volume VIII, 1977, pp. 292-301 doi: 10.1007 / BF01800698

## Individual evidence

1. ^ Philosophiae naturalis principia mathematica. , London, 1726 p. 13 ( GDZ ) - almost also in the edition Geneva 1739, p. 20 ( digitized , 60 of 589): “Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare. "
2. Brandt, Damen: Mechanik - From the mass point to the rigid body. Springer, 2016, p. 12.
3. ^ Gross et al .: Technical Mechanics - Kinetics. 13th edition. Springer, 2015, p. 36.
4. Stillman Drake: Galileo and the Law of Inertia . In: American Journal of Physics . tape 32 , 1964, pp. 601-608 , doi : 10.1119 / 1.1970872 .
5. ^ Roberto Torretti: The Philosophy of Physics . Cambridge University Press, Cambridge 1999, pp. 20-30 .
6. Lichtenegger: Key Concepts in Physics - from Newton's Axioms to Hawking Radiation. Springer, 2015, p. 14.
7. Wilfried Kuhn: History of ideas in physics. Springer, 2nd edition, 2016, p. 218.
8. Bartelmann et al. (Ed.): Theoretical Physics. Springer, 2015, p. 10.
9. Fließbach: Textbook on Theoretical Physics I - Mechanics. 7th edition. Springer, 2015, p. 9.
10. Henz, Langhanke: Paths through Theoretical Mechanics 1. Springer, 2016, p. 42.
Almost identical with Nolting: Basic Course Theoretical Physics 1 - Classical Mechanics. 10th edition. Springer, 2013, p. 173.
11. H. Schrecker: The way to the physical concept of force from Aristotle to Newton. In: Natural sciences in the classroom physics / chemistry. 36, No. 34, 1988, ( abridged version ( Memento from January 20, 2012 in the Internet Archive )).
12. ^ Dreyer: Technical Mechanics - Kinetics, Kinematics. 11th edition. Springer, pp. 123-125.
13. Holzmann, Meyer, Schumpich: Technical Mechanics - Kinetics and Kinematics. 12th edition. Springer, pp. 123-125.
14. Mahnken: Technical Mechanics - Dynamics. 2nd Edition. Springer, p. 329 f.
15. Henz, Langhake: Paths through Theoretical Mechanics 1. Springer, 2016, p. 140
16. Euler: Découverte dun nouveau principe de mécanique. Memoires de l'Academie royal des sciences, Berlin, Volume 6, 1752, p. 185 - Euler Opera Omnia, Series 2, Volume 5, 1957.
17. Lecture notes on electrodynamics and the theory of relativity , p. 4 (PDF; 13.4 MB).


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analytic

Let $U$ be a domain in the complex numbers (resp., real numbers). A function $f:U\longrightarrow\mathbb{C}$ (resp., $f:U\longrightarrow\mathbb{R}$) is (resp., real analytic) if $f$ has a Taylor series about each point $x\in U$ that converges to the function $f$ in an open neighborhood of $x$.

1 On Analyticity and Holomorphicity

A complex function is analytic if and only if it is holomorphic. Because of this equivalence, an analytic function in the complex case is often defined to be one that is holomorphic, instead of one having a Taylor series as above. Although the two definitions are equivalent, it is not an easy matter to prove their equivalence, and a reader who does not yet have this result available will have to pay attention as to which definition of analytic is being used.

 Title analytic Canonical name Analytic Date of creation 2013-03-22 12:04:36 Last modified on 2013-03-22 12:04:36 Owner djao (24) Last modified by djao (24) Numerical id 9 Author djao (24) Entry type Definition Classification msc 30B10 Classification msc 26A99 Synonym real analytic Synonym real analytic function Synonym complex analytic Synonym complex analytic function Synonym analytic function Related topic Holomorphic Related topic TaylorSeries Related topic CauchyKowalewskiTheorem


text:
Comment
Share
Q)

A $900 \;pF$ capacitor is charged by $100\; V$ battery. How much electrostatic energy is stored by the capacitor?

$\begin{array}{1 1} 4.5 \times 10^{-6}\;J \\ 4.5 \times 10^{-4}\;J \\ 3.5 \times 10^{-6}\;J \\ 4.5 \times 10^{-4}\;J \end{array}$

$Q= CV=900 \times 10^{-12}F \times 100\;V$
$\quad = 9 \times 10^{-8}\;C$
$\qquad= \large\frac{1}{2}$$CV^2 =(1/2)QV$
$\qquad= (1/2) \times 9 \times 10^{-8}\;C \times 100\;V$
$\qquad= 4.5 \times 10^{-6}\;J$


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The Ohio State University Number Theory Seminar Mondays 4:30 pm MW 154 (building photo) W. Sinnott, D. Goss, J. Cogdell R. Holowinsky, W. Luo

 Scheduled Talks 04/23/2013 - Guillaume Ricotta (Bordeaux) - Fourier coefficients of GL(3) automorphic forms in arithmetic progressions (abstract) We prove that the Fourier coefficients of GL(3) Hecke-Maass cusp forms in arithmetic progression modulo a prime number p have a Gaussian limit distribution as p goes to infinity. This is a joint work with Emmanuel Kowalski. Availability Calendar

 If you would like to give a talk at the OSU Number Theory Seminar, please check the calendar below for available dates and send an e-mail to holowinsky.1 (at) osu.edu for possible scheduling.

Change months
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Availability Calendar

 Past Talks 2012-2013 04/08/2013 - Craig Franze (OSU) - An asymptotic expansion related to the Dickman function. (abstract) 04/01/2013 - Joachim Schwermer (Vienna) - On Lefschetz numbers and arithmetically defined hyperbolic 3-manifolds. (abstract) 02/25/2013 - Bruce Berndt (UIUC) - Unpublished Manuscripts Published with Ramanujan's Lost Notebook. (abstract) 02/04/2013 - Youness Lamzouri (York U) - Large Character Sums. (abstract) 01/14/2013 - Dimitris Koukoulopoulos (Montreal) - When the sieve works. (abstract) 12/03/2012 - Ghaith Hiary (Bristol) - Computing Dirichlet $L$-functions.(abstract) 11/19/2012 - Kevin Ford (UIUC) - Sets $S$ of primes with $p$ in $S$ and $q|(p-1)$ implying $q$ in $S$. (abstract) 10/22/2012 - Micah Milinovich (Ole Miss) - Simple zeros of L-functions of modular forms. (abstract) 10/01/2012 - Tim All (OSU) - On p-adic annihilators of real ideal classes. (abstract) 09/24/2012 - David Goss (OSU) - From Carlitz's module to Euler's Gamma function. (abstract) 09/17/2012 - Roman Holowinsky (OSU) - Hybrid Subconvexity Bounds for Rankin-Selberg Convolutions. (abstract) We will consider various moment averages over special values of Rankin-Selberg convolution $L$-functions $L(\frac{1}{2}, \pi_1 \times \pi_2)$. Each $\pi_j$ will contribute to the size of the associated analytic conductor and we shall take advantage of this fact in order to obtain hybrid subconvexity bounds for individual $L$-values. Topics presented here will be a combination of ongoing joint projects with Nicolas Templier and Ritabrata Munshi. 09/10/2012 - Jim Cogdell (OSU) - The local Langlands correspondence for GL(n) and the symmetric and exterior square $\varepsilon$-factors (abstract) 2011-2012 05/21/2012 - Sun Kim (OSU) - Bessel function series and the circle and divisor problems. (abstract) In approximately 1915, Ramanujan recorded without proofs two identities involving doubly infinite series of Bessel functions. The two identities are connected with the classical circle and divisor problems, respectively. For each of Ramanujan's identities, there are three possible interpretations for the double series. The first identity was proved under those three and the second identity was proved under two of them. In this talk, we briefly review some key aspects of the proofs. Furthermore, we discuss trigonometric analogues of the identities, and a generalization of the first identity in the setting of Riesz sums. This is joint work with Bruce Berndt and Alexandru Zaharescu. 04/30/2012 - Ben Brubaker (MIT) - Demazure operators and Iwahori Whittaker functions. (abstract) Around 1980, Casselman and Shalika gave an elegant proof of a formula (conjectured by Langlands) for the spherical Whittaker function of an unramified principal series. We will begin by reviewing these terms and some key aspects of the proof, which relies on clever manipulations in the Iwahori-Hecke algebra. Then we explore similar formulas for Iwahori-fixed vectors in the unramified principal series, where we "rediscover" a Hecke algebra action due to Lusztig, in the context of equivariant K-theory. (This is joint work with Dan Bump and Tony Licata). 04/23/2012 - Hourong Qin (Nanjing) - The Local $L$-series of CM Elliptic Curves and Quadratic Polynomials Represent Primes. (abstract) Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with complex multiplication. Fix an integer $r$. We give sufficient and necessary conditions for $a_p=r$ for some prime $p$. We show that there are infinitely many primes $p$ such that $a_p=r$ for some fixed integer $r$ if and only if a quadratic polynomial represents infinitely many primes $p$. In particular, for $E: y^2=x^3+x$, our result inplies that there are infinitely many primes $p$ such that $a_p=2$ if and only if there are infinitely many primes $p$ of the form $n^2+1$. 04/16/2012 - Peng Zhao (Princeton) - The Quantum Variance of PSL(2,Z)\PSL(2,R). (abstract) We discuss the quantum variance, which is introduced by Zelditch and describes the fluctuations of a quantum observable, on the phase space of modular surface. We asymptotically evaluate the quantum variance and show that it is equal to the classical variance of the geodesic flow on the phase space after inserting the "correction factor" of certain L-function's central value on each irreducible subspace. It is also very close to the arithmetic variance studied by Luo, Rudnick and Sarnak. This talk is based on a joint work with Peter Sarnak. 03/12/2012 - Henry Kim (Toronto) - Application of the strong Artin conjecture to the class number problem. (abstract) 02/09/2012 - Guillaume Ricotta (Bordeaux) - Height of Heegner points. (abstract) 12/05/2011 - Nathan Conrad Jones (University of Mississippi) - An alternative view of primitivity of Dirichlet characters. (abstract) 10/18/2011 (1:30 pm) - Amy DeCelles (Goshen) - Number theoretic applications of the automorphic spectral theory of higher rank groups. (abstract) 10/10/2011 - Federico Pellarin (St. Etienne) - On the values of certain L-series at even" positive integers. (abstract) 10/03/2011 - Kenneth Ward (Oklahoma State) - An asymptotic relation of class number and genus for abelian extensions of a function field. (abstract) 09/19/2011 - Abhishek Saha (ETH) - Determination of modular forms by Fourier coefficients. (abstract) 2010-2011 05/16/2011 - Sheng-Chi Liu (Texas A&M) - The L^2 restriction norms of Satio-Kurokawa lifts. (abstract) 05/09/2011 - Solomon Friedberg (Boston College) - Eisenstein series, crystals and ice. (abstract) 05/02/2011 - Ling Long (Iowa State) - Recent developments on modular forms for noncongruence subgroups. (abstract) 02/28/2011 - Jeff Hoffstein (Brown) - Shifted multiple Dirichlet series and moments of L-series. (abstract) 02/21/2011 - Gautam Chinta (CUNY) - Orthogonal periods of Eisenstein series. (abstract) 12/03/2010 - Noam Elkies (Harvard) - On the areas of rational triangles. (abstract) 11/22/2010 - Aaron Levin (Michigan State) - Runge's effective method for integral points. (abstract) 11/16/2010 - Lenny Taelman (Leiden) - The Carlitz sheaf, cyclotomic function fields, and Vandiver's conjecture. (abstract) 11/15/2010 - Xiaoqing Li (SUNY Buffalo) - The L^2 restriction of a GL(3) Maass form to GL(2). (abstract) 11/08/2010 - Paul Nelson (Caltech) - Equidistribution of cusp forms in the level aspect. (abstract) 11/01/2010 - Eddie Herman (AIM) - Beyond Endoscopy for the Asai L-function and Quadratic Base Change.(abstract) 10/25/2010 - David Goss (OSU) - The class number formula of Lenny Taelman. (abstract) 10/18/2010 - Jim Cogdell (OSU) - On Local L-functions. (abstract) 10/11/2010 - Yangbo Ye (Iowa) - Resonance of automorphic forms for GL(2) and GL(3). (abstract) 10/04/2010 - Timothy All (OSU) - Mihailescu's Theorem (formerly Catalan's conjecture); an outline of the proof.(abstract) More Number Theory

Mailing

Roman Holowinsky
Department of Mathematics
The Ohio State University
100 Math Tower
231 West 18th Avenue
Columbus, OH 43210-1174

http://www.math.ohio-state.edu

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Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

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## #1 2007-02-11 03:18:17

kylekatarn
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### Forum & Latex - missing features

Is there a way to write double-struck letters like |R, |N, etc...?

latex command is \mathbb{<letter>} but the math tag seems to refuse this syntax.

## #2 2007-02-11 10:03:11

MathsIsFun

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### Re: Forum & Latex - missing features

I am working on a fix for it.

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

## #3 2007-02-11 11:03:45

MathsIsFun

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### Re: Forum & Latex - missing features

Test:

Yay!

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

## #4 2007-02-11 11:30:21

Ricky
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### Re: Forum & Latex - missing features

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

## #5 2007-02-11 12:10:51

mathsyperson
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### Re: Forum & Latex - missing features

Why did the vector cross the road?
It wanted to be normal.

## #6 2007-02-11 23:04:17

kylekatarn
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### Re: Forum & Latex - missing features

#### MathsIsFun wrote:

Test:

Yay!

Very nice! Good job!

## #7 2007-02-12 02:47:57

Ricky
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### Re: Forum & Latex - missing features

Sorry mathsyperson, I think I mean to reply to you post, but instead hit edit....

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

## #8 2007-04-15 02:16:18

ben
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### Re: Forum & Latex - missing features

Is there some trick I'm missing to get LaTex to format in-line, preferably with a smaller font. I've tried every trick I know, without sucess.

Just for example, what I'm objecting to is something like this:

The sum over over all basis vectors

with components
is
, i = 1, 2, ...,n.

Check my raw text to see that what appears is not what I intended.

## #9 2007-04-15 07:40:13

MathsIsFun

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### Re: Forum & Latex - missing features

I could experiment with it.

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman


text:
## Monday, 10 December 2012

### Number 2 History and Mathematics

The number two has many properties in mathematics. An integer is called even if it is divisible by 2. For integers written in a numeral system based on an even number, such as decimal and hexadecimal, divisibility by 2 is easily tested by merely looking at the last digit. If it is even, then the whole number is even. In particular, when written in the decimal system, all multiples of 2 will end in 0, 2, 4, 6, or 8.

Two is the smallest and the first prime number, and the only even one (for this reason it is sometimes called "the oddest prime"). The next prime is three. Two and three are the only two consecutive prime numbers. 2 is the first Sophie Germain prime, the first factorial prime, the first Lucas prime, the first Ramanujan prime, and the first Smarandache-Wellin prime. It is an Eisenstein prime with no imaginary part and real part of the form 3n - 1. It is also a Stern prime, a Pell number, the first Fibonacci prime, and a Markov number—appearing in infinitely many solutions to the Markov Diophantine equation involving odd-indexed Pell numbers.

It is the third Fibonacci number, and the third and fifth Perrin numbers.

Despite being a prime, two is also a highly composite number, because it has more divisors than the number one. The next highly composite number is four.

Vulgar fractions with 2 or 5 in the denominator do not yield infinite decimal expansions, as is the case with all other primes, because 2 and 5 are factors of ten, the decimal base.

Two is the base of the simplest numeral system in which natural numbers can be written concisely, being the length of the number a logarithm of the value of the number (whereas in base 1 the length of the number is the value of the number itself); the binary system is used in computers.

For any number x:

x+x = 2·x addition to multiplication
x·x = x2 multiplication to exponentiation
xx = x↑↑2 exponentiation to tetration

Two also has the unique property that 2+2 = 2·2 = 2²=2↑↑2=2↑↑↑2, and so on, no matter how high the operation is.

Two is the only number x such that the sum of the reciprocals of the powers of x equals itself. In symbols

\sum_{k=0}^{\infin}\frac {1}{2^k}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots=2.

This comes from the fact that:

Powers of two are central to the concept of Mersenne primes, and important to computer science. Two is the first Mersenne prime exponent.

Taking the square root of a number is such a common mathematical operation, that the spot on the root sign where the exponent would normally be written for cubic roots and other such roots, is left blank for square roots, as it is considered tacit.

The square root of two was the first known irrational number.

The smallest field has two elements.

In the set-theoretical construction of the natural numbers, 2 is identified with the set \{\{\emptyset\},\emptyset\}. This latter set is important in category theory: it is a subobject classifier in the category of sets.

Two is a primorial, as well as its own factorial. Two often occurs in numerical sequences, such as the Fibonacci number sequence, but not quite as often as one does. Two is also a Motzkin number, a Bell number, an all-Harshad number, a meandric number, a semi-meandric number, and an open meandric number.

Two is the number of n-Queens Problem solutions for n = 4. With one exception, all known solutions to Znám's problem start with 2.

Two also has the unique property such that

\sum_{k=0}^{n-1} 2^k = 2^{n} - 1

and also

\sum_{k=a}^{n-1} 2^k = 2^n - \sum_{k=0}^{a-1} 2^k - 1

for a not equal to zero

The number of domino tilings of a 2×2 checkerboard is 2.

For any polyhedron homeomorphic to a sphere, the Euler characteristic is

\chi = V-E+F = 2.\

As of 2008, there are only two known Wieferich primes.


text:
#### Nikson420

Joined Dec 4, 2022
4
Found a lot of them but I am not sure about quality. Any suggestions guys?
I need 50 pcs.
Want to buy this one - https://aliexpress.com/e/_DeLuqef

Thanks!

#### dl324

Joined Mar 30, 2015
15,458
Welcome to AAC!
Component in question:

I would be very leary of buying individual components from Ali Express. Especially if you don't have the ability to do quality assurance of enough of the components to convince yourself they aren't counterfeit.

#### SamR

Joined Mar 19, 2019
4,492
I am always leery of any AliX vendor that uses stock photos of their so-called factories and facilities. If they lie about that... Chinese "replica" parts sold by clueless vendors are very iffy at best, especially if you are willing to risk that kind of money on a hope and prayer. I do buy some transistors and chips from china, but I at least give them a basic go/no-go test on arrival and far too often have to file a dispute against the vendor for restitution which is only for the price of the parts and not my time and effort testing and documenting their failure for the dispute. Caveat Emptor and I would never use those parts for a commercial venture or repair. YMMV

#### dl324

Joined Mar 30, 2015
15,458
This is a "store" in Shenzen that specializes in counterfeit parts; from a report by SMT Corp:

If you get a counterfeit part that "works", you're lucky:

#### WBahn

Joined Mar 31, 2012
27,893
Found a lot of them but I am not sure about quality. Any suggestions guys?
I need 50 pcs.
Want to buy this one - https://aliexpress.com/e/_DeLuqef

Thanks!
I don't know about "cheap", at least in terms of price. In quantities of 25 you are looking at about $13 from either DigiKey or Mouser. But... neither currently has stock. If you absolutely need that part and you can't get it anywhere else, then you might buy a few and test them thoroughly before buying more. I would be VERY leery of making a$1k+ order only to discover that they are all the other kind of "cheap", as in worthless counterfeits.

#### DickCappels

Joined Aug 21, 2008
9,512
I have had mixed results with Ali Express, it is a gamble.

For something like that controller that you will probably spend many hours programming and debugging (rinse and repeat) I would recommend buying from an authorized distributor rather than risking many hours or days troubleshooting a defective chip.

#### EnzoEngr

Joined Nov 7, 2022
5
I second the idea of going for an authorized distributor. On the other hand, you may try using search engines such as Octopart or findchips. They are quite useful to make the comparison.

#### Nikson420

Joined Dec 4, 2022
4
Yes guys I think also is better to find good source then go just randomly. Thanks a lot.
Any store or wholeseller is welcome, just reply!

#### MrSalts

Joined Apr 2, 2020
2,619
This is a "store" in Shenzen that specializes in counterfeit parts; from a report by SMT Corp:
View attachment 282293

If you get a counterfeit part that "works", you're lucky:
View attachment 282294
Nice selective blurring. Looks like they are harvesting some shell-fish to me. Why would anyone make a counterfeit chip and them dump them in bulk on a River bank to "dry".

note that SMT Corp is a direct competitor to the companies harvesting chips off of used boards and recovering left- overs from reels and tubes from PCB houses. This must be undercutting into SMT Corps profits.

show Me one of these famous fabs that make "counterfeit" chips. I'd love to take a look at their ultra efficient manufacturing process that makes them so inexpensive. The chips I've ordered work, low noise op amps are low noise, low power op amps are low power. These "counterfeiters" seem to have efficiency knowhow that western manufacturers could use - maybe they are saving a bunch by using that famous trick of dumping the batch on a River bank to dry.

Last edited:


text:
# 【BZOJ 4318】OSU!

### 解题报告

$E_{(i,x^2)}，E_{(i,x)}$分别表示$x^2,x$的期望

$E_{(i,x^3)}=p_i \times E_{(i-1,(x+1)^3)}$
$E_{(i,x^2)}=p_i \times E_{(i-1,(x+1)^2)}$
$E_{(i,x)}=p_i \times (E_{(i-1,x)} + 1)$

$E_{(i,x^3)}=p_i \times (E_{(i-1,x^3)} + 3E_{(i-1,x^2)} + 3E_{(i-1,x)} + 1)$
$E_{(i,x^2)}=p_i \times (E_{(i-1,x^2)} + 2E_{(i-1,x)} + 1)$

### Code

#include<bits/stdc++.h>
#define LL long long
using namespace std;

inline int read() {
char c=getchar(); int f=1,ret=0;
while (c<'0'||c>'9') {if(c=='-')f=-1;c=getchar();}
while (c<='9'&&c>='0') {ret=ret*10+c-'0';c=getchar();}
return ret * f;
}

int main() {
double e1=0,e2=0,e3=0,ans=0,p;
for (int i=1;i<=n;i++) {
scanf("%lf",&p);
ans += e3 * (1 - p);
e3 = p * (e3 + 3 * e2 + 3 * e1 + 1);
e2 = p * (e2 + 2 * e1 + 1);
e1 = p * (e1 + 1);
}
printf("%.1lf\n",ans+e3);
return 0;
}

## 2 thoughts to “【BZOJ 4318】OSU!”

1. Hi, great work. I really appeaciate the information you are providing through your site, i have alwasy find it helpful. Keep up the amazing work.

2. I must thank you for the efforts you’ve put in penning this website. I am hoping to see the same high-grade content by you later on as well. In truth, your creative writing abilities has encouraged me to get my very own blog now 😉


text:
Any company of a decent size and age that creates software at any scale will have gone through a number of “technical evolutions”. These come along as you hire new people, create new products or just better ways of doing things come along. The kind of changes I’m talking about are the adoption of different technologies over time and the introduction of new ones. This cuts right across different aspects of software, from languages to frameworks to techniques to tooling. We end up collecting an number of each for various reasons. It may even be a conscious choice where we see that one technology is better suited to solving a particular problem than another or one system is simply not as important and don’t need the same investment. The reasons here aren’t too important, the fact is you are in this situation and will be for a foreseeable amount of time.

When it comes to deployment software there are often lots of changes that have happened over time and more than likely more than one tool that is currently in use across your teams.

I know within the org I work in we have about 4 (that i know of!)

• Jenkins
• GitLab Pipelines
• Heroku Pipelines
• Octopus Deploy

The issue we have is that we have no insight across the teams into what is deploy, to where, when and how. There is very little reporting built into these tools to see across all of your applications and even if there were you would have to switch between applications to see any of this and implementations would likely differ wildly.

What we wanted to do was to bring all of this information together into one place so that we can have some sort of visibility and enable some level of insight into the applications that are being deployed across teams.

## Web-hooks

Web-hooks are the key to this. Most of these applications support web-hooks right out of the box. Even those that don’t we can provide a pretty simple curl script to inset into any shell script that is run as part of the deployment. Not as nice but it gets the job done.

So we are going to use web-hooks. Great. We need some sort of HTTP endpoint somewhere. And I don’t want to be writing and managing a whole bunch of code to do all this, I especially don’t want to be managing any infrastructure. I also have the issue that some of these systems are on-prem and some are cloud hosted SaaS products. You may think “cloud functions!” and I would agree that solves the infrastructure problem. But I still have to write, test and deploy some code (another deployment to track!).

Azure Logic Apps to the rescue, these are really simple drag and drop style connector apps that are perfect for simple things like this, think of ifttt or zapier on crack. We can have a HTTP trigger to receive JSON data and dump it into a document store for later processing. Think of them as a layer over cloud functions that writes the code for you. They are pretty simple and provide a range of out of the box triggers and connectors, including HTTP and Cosmos DB. So great, we can take in the web-hook data and store it into a document store.

In fact they are so simple in this case that there are only 2 steps

We can use the first request (or any sample data) to generate a JSON schema to help who the shape of the data. (This comes in handy with the Heroku issues later)

The one gotcha here is to make sure that you add in your own id that is generated at the point of the insert, and then nest your data inside there somewhere. Don’t rely on the id data that may come in from the web-hook. It may be a static id for an app or deployment run in which case your data will be overwritten as new event come in for that app or deployment (start, finish, error etc)

At the moment I am dumping all of my data for each hook into different table. Im purposefully not doing any processing or transformation here. I want all the data for using later.

I rinse and repeat this for each of the different data sources. Creating a new, cosmos db collection and logic app for each data source.

## Issues

As you can imagine there is no way that these different systems were going to be sending me any standard format of data. Not even a little bit..

### Octopus

The data coming out of the Octopus subscription is fairly good as you might expect from a deployment specific tool. The biggest problem is that it is all referential. It sends down IDs for projects and environments rather than their names. Which is a little bit of a pain. There are APIs that can be queried to get further information on each of these referenced resources. However, the octopus server sits internally and isn’t accessible from Azure. To get around this i have simply manually dumped out the JSON for both the environment and projects and uploaded them to a Sharepoint site that is accessible by Power BI. I was then able to pull this into a data set, flatten it and link it to the deployment data to get application and environment names rather than just ids. I guess if it becomes a pain I could look at automating this in the future.

### Heroku

Within Heroku each app is tied to a pipeline which has a stage. There is the notion of coupling an app to a pipeline and stage. However none of this information comes over within the release web-hook which is focussed primarily on apps and not pipelines. And at present there doesn’t seem to be any web-hooks associated to pipeline actions specifically. So we would have to query the specific pipeline couplings for an app after we receive the web-hook from the app deployment. Thankfully there is a pipelines API that allows us to query the pipeline couplings from the appId

https://devcenter.heroku.com/articles/platform-api-reference#pipeline-coupling-info-by-app

This looks something like this, with the addition of an Auth header.

$curl -n https://api.heroku.com/apps/$APP_ID_OR_NAME/pipeline-couplings \
-H "Accept: application/vnd.heroku+json; version=3"

This returns the details of the coupling of the pipeline to the app like this

{
"app": {
"id": "01234567-89ab-cdef-0123-456789abcdef"
},
"created_at": "2012-01-01T12:00:00Z",
"id": "01234567-89ab-cdef-0123-456789abcdef",
"pipeline": {
"id": "01234567-89ab-cdef-0123-456789abcdef"
},
"stage": "production",
"updated_at": "2012-01-01T12:00:00Z"
}

This gives is the stage which will be our environment data.

To do this we are able to insert another HTTP request into the logic app. It will make a request to the Heroku API using the app id received in the web-hook to get the pipeline information and attach it to each record when inserting into the db. This way we now have the application data and the pipeline that it is attached to. This logic app now has 3 steps but it was super simple to achieve this and is sill very easy to understand whats going on. The HTTP trigger is omitted from this picture is identical but we have added in another HTTP request step before saving to the DB. We have used the JSON schema to pull out the appid from the incoming JSON and pushed that in as a part of the HTTP request. We then append the body of the response to the document we insert, alongside the web-hook data.

### GitLab

There seem to be a few more issues with the data that comes back from GitLab. Firstly there is no notion of an environment in the pipelines web-hook data. So its a little tricky to figure out what has been deployed and where. Through the teams that use it within our org there seems to be some loose naming conventions for pipeline stages that has allowed me to pull the data apart in processing and gain some insight into what has been deployed and where. The deploy stage is helpfully called deploy and each job is named using the basic schema of deploy:environment and although all the jobs show up in the data only the one that has actually been deployed will be marked as succeeded.

Something like this

{
"id": 467840,
"stage": "deploy",
"name": "deploy:uat",
"status": "success",
"created_at": "2018-03-24 09:50:30 UTC",
"started_at": "2018-03-24 09:54:05 UTC",
"finished_at": "2018-03-24 09:58:01 UTC",
...

I am able to transform this data later when processing it. So i can get the data that i need but it isn’t perfect.

## Rationalising the data

From the data collected above I spent some time thinking about what data was strictly needed to do some basic reporting and insight.

• Date
• Some unique ID
• Environment
• Status
• System

I then used Power BI pull in the data and transform and shape it into the structure above to end up with one stream of data that could be used to visualise. I will go into what i did within Power BI in part 2 of this post.

## Future nice to haves

Throughout doing this i have been thinking about how this data may be used to provide insight into the development and deployment process within teams.

• Deployment length - to help identify long and problematic deploys
• Explicit commit sha to track back to source code
• User that triggered the deploy


text:
Question on a double integral with change of variables

Solve the system $$u = 3x + 2y, v = x + 4y$$ to find expressions for $$x$$ and $$y$$ in terms of $$u$$ and $$v$$.

Use these expressions to find the Jacobian $$∂(x, y)/∂(u, v)$$.

Hence evaluate the integral $$\iint(3x + 2y)(x + 4y) dx dy$$ for the region $$R$$ bounded by the lines $$y = −(3/2)x + 1,\ y = −(3/2)x + 3$$ and $$y = −(1/4)x,\ y = −(1/4)x + 1$$

So I computed the Jacobian = 1/10, I solved the equations to find $$x = h(u,v)$$ and $$y = g(u,v)$$ and then I substituted $$h$$ and $$g$$ in the equations for the boundaries ,which gave me $$u = 2$$ and $$u = 6$$, and $$v = 0$$ and $$v = 4$$. So the integral I have to compute is equal to $$\iint uvJ(u,v)dudv$$ with the above boundaries?

• My comment was the one that was mistaken, sorry. Your notes are probably right. Mar 24, 2019 at 1:55
• So is the integral I have to compute equal to $∬uvJ(u,v)dudv$, with the boundaries I found?
– user600210
Mar 24, 2019 at 1:58

Yes, it is correct. The answer is $$12.8$$.

Direct calculation:

$$\hspace{1cm}$$

$$\int_{0}^{0.8}\int_{-\frac32x+1}^{-\frac14x+1} (3x+2y)(x+4y) dydx+\\ \int_{0.8}^{1.6}\int_{-\frac14x}^{-\frac14x+1} (3x+2y)(x+4y) dydx+\\ \int_{1.6}^{2.4}\int_{-\frac14x}^{-\frac32x+3} (3x+2y)(x+4y) dydx=\\ \frac{44}{15}+\frac{104}{15}+\frac{44}{15}=\frac{192}{15}=12.8.$$

Your method: $$\int_{0}^{4}\int_{2}^{6} \frac1{10}uv dudv=\int_0^4 \frac85vdv=12.8.$$

• I got the same result, thanks for the work!
– user600210
Mar 25, 2019 at 22:19
• You are welcome. Good luck. Mar 26, 2019 at 2:11


text:
# Why is Mg(OH)2 sparingly soluble in water but soluble in NH4Cl

• Is $$\ce{Mg(OH)2}$$ sparingly soluble in water because it's hydration enthalpy is not greater than its lattice enthalpy?
• Also, is it soluble in $$\ce{NH4Cl}$$ because it is a stronger base than $$\ce{NH4(OH)}$$ hence it displaces Ammonia and since it is a gas the equilibrium reaction goes forward as the gas is removed?
• Ammonia is basically infinitely soluble in water, and magnesium hydroxide is more soluble in an acidic medium. – Zhe Jan 26 at 13:25

Probably it won't (fully) answer your question, but I spotted a mistake: NH4OH is the common way you use to describe an aqueous solution of ammonia. Therefore, when you write your equations, you consider is as ammonia, not as ammonium hydroxide: it acts as a base. On the other hand, NH4Cl is ammonium chloride, therefore it acts as an acid.

This explains why Mg(OH)2 is more soluble in NH4Cl than NH4OH: in the first case, you mix a (weak) base and a (weak) acid, in the second one you mix two (weak) bases.

With respect to your first question, keep in mind that when the hydration (solvation) energy overcomes the lattice energy the compound is soluble in water (solvent). If they are about the same, then it is mildly soluble (not super sure about it, though). I hope this helps!

The ammonia does not come off as a gas. The reaction lis driven forward, dissolving more magnesium hydroxide, even though the ammonia remaibsin solution.

Magnesium hydroxide comes off a bit strange when we try to classify bases as "strong" or "weak". It looks like a weak base because it is only sparingly soluble in water, but the portion that does dissolve acts like a dilute solution of a strong base (see, for instance, the reaction of magnesium hydroxide with a typical transition metal ion discussed here). So, like other strong bases, the magnesium hydroxide will tend to react with the ammonium ion according to the reaction we already know:

$$\ce{NH4+ + OH- <=> NH3 + H2O}$$

It is this removal of hydroxide ions, not any evolution of ammonia gas, that pulls the dissolution of magnesium hydroxide along.

Because magnesium hydroxide is sparingly soluble, the reaction proceeds only until the solubility equilibrium is re-established. You will typically end up with a mixture of ammonia, ammonium salt and magnesium salt in a slightly alkaline solution. But there will be a lot more magnesium ions in solution with this reaction than with adding magnesium hydroxide to pure water.


text:
tf.nn.log_poisson_loss

tf.nn.log_poisson_loss(
targets,
log_input,
compute_full_loss=False,
name=None
)

Defined in tensorflow/python/ops/nn_impl.py.

See the guide: Neural Network > Losses

Computes log Poisson loss given log_input.

Gives the log-likelihood loss between the prediction and the target under the assumption that the target has a Poisson distribution. Caveat: By default, this is not the exact loss, but the loss minus a constant term [log(z!)]. That has no effect for optimization, but does not play well with relative loss comparisons. To compute an approximation of the log factorial term, specify compute_full_loss=True to enable Stirling's Approximation.

For brevity, let c = log(x) = log_input, z = targets. The log Poisson loss is

  -log(exp(-x) * (x^z) / z!)
= -log(exp(-x) * (x^z)) + log(z!)
~ -log(exp(-x)) - log(x^z) [+ z * log(z) - z + 0.5 * log(2 * pi * z)]
[ Note the second term is the Stirling's Approximation for log(z!).
It is invariant to x and does not affect optimization, though
important for correct relative loss comparisons. It is only
computed when compute_full_loss == True. ]
= x - z * log(x) [+ z * log(z) - z + 0.5 * log(2 * pi * z)]
= exp(c) - z * c [+ z * log(z) - z + 0.5 * log(2 * pi * z)]

Args:

• targets: A Tensor of the same type and shape as log_input.
• log_input: A Tensor of type float32 or float64.
• compute_full_loss: whether to compute the full loss. If false, a constant term is dropped in favor of more efficient optimization.
• name: A name for the operation (optional).

Returns:

A Tensor of the same shape as log_input with the componentwise logistic losses.

Raises:

• ValueError: If log_input and targets do not have the same shape.


text:
# What is the area of a triangle with sides of length 2, 6, and 5?

Mar 14, 2016

Apply Heron's formula to find the area to be

$\frac{\sqrt{351}}{4} \approx 4.6837$

#### Explanation:

Heron's formula states that, given a triangle with side lengths $a , b , c$ and semiperimeter $s = \frac{a + b + c}{2}$ the area $A$ of the triangle is

$A = \sqrt{s \left(s - a\right) \left(s - b\right) \left(s - c\right)}$

In this case, we have $a = 2$, $b = 6$, and $c = 5$. Then, for this triangle we have $s = \frac{2 + 6 + 5}{2} = \frac{13}{2}$. Applying Heron's formula gives us

$A = \sqrt{\frac{13}{2} \left(\frac{13}{2} - 2\right) \left(\frac{13}{2} - 6\right) \left(\frac{13}{2} - 5\right)}$

$= \sqrt{\frac{13}{2} \cdot \frac{9}{2} \cdot \frac{1}{2} \cdot \frac{3}{2}}$

$= \sqrt{\frac{351}{16}}$

$= \frac{\sqrt{351}}{4} \approx 4.6837$


text:
# Some more geometry in the secondary fan

## January 6, 2015

A polyhedral subdivision of finitely many points $A$ in $\mathbb{R}^d$ is a decomposition of its convex hull into non-overlapping polytopes with vertices in $A$. It is regular if the polytopes are projections of facets of a polyhedron in $\mathbb{R}^{d+1}$. Gelfand, Kapranov, and Zelevinsky introduced the secondary fan of $A$, which encodes all regular subdivisions of $A$. I will talk on work with Postinghel and Villamizar that explains another geometric interpretation of the secondary fan. The points $A$ give a curvilinear copy of the convex hull of A in the probability simplex whose vertices correspond to $A$ which we call the real toric variety of $A$. We consider possible Hausdorff limits of translations of this real toric variety. We show that the possible limits correspond to the cones of the secondary fan. Our main tool is a study of sequences in the ambient space of the secondary fan.


text:
I Star formation and heavy elements

Tags:
1. Jan 12, 2018

darkmattermonkey

Hello. First post here so hi all.
My question(s) is regarding the formation of solar masses by accretion of gases and dust. From what I understand, stars are formed by large clouds of gasses and dust particles pulling together and with enough gravity, (magnetic fields), and time. A fusion process begins. Afterwards, planetary bodies, dwarf planets, moons, etc will form (if there is sufficient gas and matter to do so). So everything in a Solar system is comprised of the matter that it originally started *barring anything from outside said system being added to the chemical make up.

Gravity being an effect of mass (to me) means that higher mass equals higher gravity. Simply; Larger bodies attract with greater ability then smaller bodies.

My understanding is that this should be true of elemental density as well. Heavier elements have higher atomic mass and (should) have greater gravitation attraction then lighter elements. So here’s the thing that is confusing me with stars, solar system formation, and accretion process. Our planet has vast amounts of heavier elements e.g. iron. Most of the planets and moons seemed to be comprised of similar matter but If gravity pulls gases and matter towards the center of a forming star, how is it that stars are mostly formed of the smallest elements? Hydrogen & helium but mostly hydrogen? Shouldn’t the heaviest elements (have) pulled to the center of a (developing) gravity well with the lighter elements swirling in the outer layers of rotating cloud? If (as seems to be said a lot these days) the fusion of iron by a star marks the end of a stars (due to iron’s energy absorption) ability to push back against gravitation collapse with consistent/equal energy generation, then how do stars not suffer from rapid fusion decay/energy loss due to already containing heavier metallic elements from the initial formation?

2. Jan 12, 2018

Arman777

Because in the begining of the universe (or in the another words when the first stars formed) there was mostly hydrogen and helium. So In this sense mostly all of the stars contains hydrogen and helium. (If you think more you'll see that our star has a lifetime of 10 billion years). So in this sense I guess its normal to think that the stars that we see around is made by mostly hydorgen because in early times there was only hydrogen and mostly helium and just not enough iron or heavy metals. Of course stars contains heavy elements but not that much respect to the hydrogen due to this reason.

https://briankoberlein.com/2014/02/12/second-generation/

Last edited: Jan 12, 2018
3. Jan 12, 2018

darkmattermonkey

Hello. Thank you for the link (interesting info) to types of stars. It answered some but also raised another. If iron is a "fusion killer", how can some metallic containing stars continue to keep from collapsing within short periods of time? (millions of years vice billions). The article in the link makes sense that first stars formed within the time from the universal starting point (big bang) to when hydrogen/helium/light elements started to form approximately 340,000-390,000 after the big bang. So yes, first population III stars at that point can only be made up of the elements around them. Our sun is a population I star, so shouldn't it have large amount of heavier elements?

I suppose the real problem I have is disinformation or misunderstanding of "why and when" iron can have devastating effects on a stars energy production?
Once iron is fused inside a star, it will suffer gravitational collapse within a short period of time.
So is my understanding of the statement in error?

A. If iron is already present in small traces during protostar formation, its not an issue.
...or...
B. If a star has fused all the lighter elements and starts to fuse iron, that's the beginning of the end.

4. Jan 12, 2018

Arman777

Both A and B are true I think. There are couple reason why I think like that, First reason is even there's some amount of iron in the star, there's also large amount of hydrogen which goes under fusion and produces large amount of energy so at this begining stages nothing happens. But when the star consumes most of the fuel and heavy elements start to produce (like iron), then star starts to die cause there's not much enough energy to keep the star in gravitaitonal balance.

A quote

"When a star is fusing iron in its core, it’s still giving off insane amounts of energy. The helium, hydrogen, carbon, oxygen, and silicon are still there in the star in different shells. Hydrogen is at the surface, still fusing to helium; a little further down, helium fusing to carbon and oxygen; further down we have silicon until the core, where silicon fuses to iron. This is why the star still exists and doesn’t spontaneously explode the moment the first iron atom pops into existence".

At this point, the energy process is just no longer exothermic but endothermic. Iron cannot be fused into anything heavier because of the insane amounts of energy and force required to fuse iron atoms. The atomic structure of iron is very stable, more so than most other elements. I’m not saying all other elements are radioactive or unstable, just that iron is slightly more stable than the previous elements.

Stars this massive can turn into several things; it depends on how heavy it is. They can explode into supernova, collapse into various types of neutron stars, or even form a black hole. The iron in the star’s core isn’t the reason why the star went supernova, its overall mass made it explode. But, the iron in its core caused it to die."

https://futurism.com/what-happens-when-stars-produce-iron/

If you are asking why not much higher then the current value, well I dont know why, simply maybe there wasn't enough heavy metal in the nebula or maybe some of them created planets (a small ratio compared to sun's mass but still..)

5. Jan 12, 2018

rootone

Our Sun and similar stars formed from gas and dust that was still primarily made up of hydrogen and helium.
There would have been also some amount of heavier elements, those having been cooked up in earlier stars that ended their life as as supernovae.
I don't think the term 'a large amount' is really appropriate, since in total, elements heavier than helium amount to around 2%.
Most of it being still fairly light elements, Carbon, Oxygen, Nitrogen.
The truely heavier elements may seem like a lot because the Earth and other rocky planets and objects exist,
but these comprise still only a tiny amount of the whole content of the solar system.
98% of the solar system is hydrogen and helium in the Sun.

Last edited: Jan 12, 2018
6. Jan 12, 2018

Staff: Mentor

Metallicity is the term astronomers use. It is confusing. In this context it means: the content in a star of every element on the table of elements above hydrogen and helium.

https://en.wikipedia.org/wiki/Stellar_population

It is a measure of the "age" of a star, as well as whether the star was formed out of "recycled" matter or formed very early on in the age of the universe. Population I stars are newbies, made from "old" hydrogen and helium and newer heavier elements that came from a supernova. Population II stars are "oldies" formed a long time ago when there was not much in the way of heavier elements. Let's ignore Population II stars for now.

Next. Iron can be in a star and NOT stop fusion. It may have gotten there from the remnants of a supernova because it is a population I star, for example.
What the "iron thing" is about is that fusion of iron into heavier element cannot occur in normal stars like our sun. So when lots of iron builds up the star becomes increasingly hotter, and if the star is big enough it becomes a supernova - then iron gets fused into elements like gold.

So. We have elements in us humans and in the Earth's crust that came from a supernova, which is kind of cool. And our sun does have iron in it and and other heavier elements, too, just in incredibly small amounts. It seems happy to continue fusion.

7. Jan 12, 2018

darkmattermonkey

Thank you (again). This has helped my understanding a lot and I appreciate the time and patience you have taken to explain it. The evolution of stars has fascinated me for years and the various stages they go through are interestingly complex yet simple when looked at from a novice POV. The final stages of a stars life are extremely curious as they can (as you said in the previous response) become a number of different and altogether extraordinary objects. Stars that with potential to become Super/Hypernovae, destructive power to the Nth degree plus new material for future stars and planets. But the truly massive stars which can become; Black Hole's, Quasar's, magnetar's, neutron stars, or even just a stellar core.
My person favorite is the magnetar {Babcock's star specifically}. My interest in the sun lead me to websites like SOHO, SDO, and NASA. These lead to wanting to know more about astronomy and astrophysics. The question I asked was a result of knowing slightly more and starting to question the deeper mechanics of it.
Thank you again.

Respectfully
DMM

P.s.
Now if they can just figure out what is causing GRB's ((unknown source) assumed to be from the birth of black holes or neutron star collisions).

8. Jan 12, 2018

darkmattermonkey

Hello and thank you for the response.

As I am also happy that it (the Sun) continues to fuse. :)
{There is no fusion 'Halfinum'}
So (If I am understanding you correctly sir) it isn't necessarily the content OR the fusion process of iron. It (end of a stars life) is dependent of the physical size/fusion ability of any given star and the element that's fused isn't the factor. The characteristics of the star that come into play more so then what elements it is producing via nuclear fusion.

Larger stars then ours can produce iron without ill effect, smaller stars may never achieve fusing iron due to the limitations of size and mass.

Essentially, there isn't any "death metal" specific to every star, there are only the limits of the individual star to fuse heavier elements until they cannot fuse any higher.

Respectfully,
DMM

9. Jan 12, 2018

Staff: Mentor

10. Jan 12, 2018

sophiecentaur

There is another relevant factor and that is the bigger a star, the faster its reactions are and the shorter its lifetime, before it actually goes Nova (producing heavy elements). The early biggest stars, that formed a supernova will have supplied the Universe's heavy elements relatively quickly, compared with the more modest sized stars that can be composed of this 'second hand' material in nebulae resulting from supernovae.

11. Jan 12, 2018

Staff: Mentor

You have to be careful about what exactly you are comparing.

If you have a single atom of hydrogen and a single atom of iron, then yes, the iron will have stronger gravity than the hydrogen. Or if you have, say, $10^{45}$ atoms of hydrogen and $10^{45}$ atoms of iron, the iron will have stronger gravity than the hydrogen. But that's not the right comparison to be making for the scenario you are concerned with. See below.

Because there are vastly more atoms of hydrogen than of heavy elements. For example, the Earth has roughly $10^{45}$ atoms, mostly of heavier elements like iron. But the Sun has roughly $10^{58}$ atoms (actually nuclei, since the Sun is too hot for atoms to exist, it's a plasma), mostly of hydrogen. That's why the Sun is so much more massive than the Earth and is the primary source of gravity in the solar system. The same was true of the cloud of matter that originally formed the solar system; yes, iron atoms are heavier than hydrogen atoms, but there are so many more hydrogen atoms that the latter are what count as far as gravity is concerned.

12. Jan 13, 2018

stefan r

Suppose you started with a 0.1 solar mass ball or pure iron. Then you dump hydrogen on it. The iron would stay in the center. The hydrogen layer would increase in density until fusion started. Then it would puff directly into something similar to a red giant star. Fusion would occur at or very near the boundary between the iron and the hydrogen layers. The energy from the fusion will blow material out. The density of hydrogen would drop and the fusion would slow down. As hydrogen reached the surface it would continue fusing. A solar mass star with an iron inner core would burn hydrogen much faster than the sun. "Fusion slowing down" means burning hydrogen thousands of times as fast as the sun instead of tens of thousands.

The only places that have a stellar mass ball of iron is the core of huge stars. Large stars collapse into neutron stars or black holes. When stars form the gas and dust is mixed. The core of large stars fuse to become iron but started out with the same mix as other stars.

Building a star is not within NASA's budget. It is only hypothetical.

13. Jan 15, 2018

Staff: Mentor

Moderator's note: A number of off topic posts have been deleted.

14. Jan 15, 2018

darkmattermonkey

Yes, that is something I had originally considered. The state of different elements existing as plasma, it would simplify (as well as complicate) many questions of how a star works.

I decided against thinking that about stars (nuclear fusion process) operating this way as scientist can determine what (elements) stars are fusing based on their radiated color/light. So (attempting) to think along these rationals, stars internally have both conditions for complete atomic nuclei of elements AND matter which has become plasma(?). There could be both at different layers of stars. Much in the same way that the planet Jupiter is a Gas Giant but internally is solid. Different states of matter at different layers. The idea that Jupiter has enough pressure to compress hydrogen into a metal-like material at its core for me raised the question of what EXACTLY is the physical state of atomic and sub-atomic matter in a stars core?
(This isn’t a question for response, only one other thing I question and hope to understand one day).

Thank you for replying and correcting my usage of element/mass and the gravitational comparisons.

I appreciate the knowledge, “boot to the head”.

Respectfully & Gratefully,
DMM

15. Jan 15, 2018

darkmattermonkey

:)
They cannot make a (fusioning) star but a donut (Tokamak)? Sure. Order us up that Magnetically-bottled ring please.

Thank you for the input, I appreciate it. Yourself and others here have provided some truly insightful information and some links to even more to help me understand solar process’s.

Respectfully & Gratefully,
DMM

16. Jan 16, 2018

stefan r

We only see radiation from the photosphere. A thin outer layer. Fusion only occurs in the core. Many stars have convection zones but most do not reach all the way from the surface to the core. Convection does not transport a pure sample of a deeper layer.

You can get the temperature and pressure required to fuse elements in a lab on earth. Stars go through "dredge ups" later in their life cycle. From the elements/isotopes that emerge at various stages we can learn a lot about what happened and when.

Degenerate matter. Electron degenerate matter should be in the core of Jupiter, most of a white dwarf, and the core of red giant stars.

The sun is plasma all the way down.

17. Mar 31, 2018

alantheastronomer

You were wondering why all the heavy elements didn't all sink to the center of the sun, and instead made the inner planets? Well, previous supernovae infused the interstellar medium ( clouds of hydrogen gas) with the heavy elements. These elements permeated the solar nebula so they were distributed throughout the solar system. As the sun and planets formed, these heavier elements indeed migrated toward the inner part of the solar system due to the gravity of the sun, and these are what formed the inner planets, so your intuition was correct!

18. Apr 2, 2018

snorkack

No, they did not.
Silicon, a lighter element, is abundant in inner planets. Argon, a heavier element, is not.

19. Apr 2, 2018

stefan r

Silicate rock is denser than argon gas at room temperature. In a distillation apparatus oxygen (O2) and argon separate from nitrogen as a group. Argon is separated from oxygen in a second more difficult step. Chemically pure argon is produced by burning something because separating O2 from Ar by distillation is painful. Argon will liquefy after methane and carbon monoxide but slightly before nitrogen.

I had the impression that a new star blows gas away. Much of Earth's iron was already in the inner solar system.

20. Apr 2, 2018

snorkack

And low pressure. Like that of solar nebula.
Yes.
Sun blew away what was gas, whether light elements (hydrogen and helium) or heavy elements (xenon), including stuff which is not gas at room temperature and atmospheric pressure but is vapour at room temperature and space vacuum (water). What was solid dust grains remained - whether it was heavy elements (uranium) or light elements (lithium, which becomes chemically bonded to silicate rocks).