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# Area enclosed between half lines in polar space
I don't know if the anwser to my question is obvious because I cannot find any explanation anywhere on google.
Question
The blue region $$R$$ is bounded by the curve C with equation $$r^{2} = a^{2}cos(2\theta)$$ $$0 \leqslant \theta \leqslant \frac{\pi}{4}$$, the lin... | {
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The area of the remaining part of R (let us call it R2) can be calculated as the difference between the area of the triangle delimitated by the lines $\pi/4$, $\pi/6$, and L, and that of the small portion of C above the $\pi/6$ line.
The triangle has base equal to $\pi (\sqrt{3}-1)/(2\sqrt{2})$ and height equal to $\p... | {
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• Why do I get an area different to the actual one if I choose the half line to be greater than $\frac{\pi}{4}$ – Nubcake May 30 '14 at 22:24
• When I first worked it out I used pi/2 as the second half line and I believe I did get a different area , I'll try this again now. – Nubcake May 30 '14 at 22:51
• You do not ge... | {
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# Joint probability distribution of geometric distribution
Let $$X$$ and $$Y$$ be independent and identically distributed $$(i.i.d.)$$ r.v.’s, each having the probability distribution, $$p(k) = (1 − λ)λ^k$$; $$k = 0,1,...$$ where $$λ :(0; 1)$$ is a constant. Define $$U = min(X; Y )$$; $$V = max(X; Y )$$; $$W = V − U$$... | {
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$$\begin{array}{rl} \min(x,y) &=u \\ \max(x,y)-\min(x,y)&=w \end{array}$$
for $$(x,y).$$ There are two possibilities: $$\min(x,y)=x$$ or $$\min(x,y)=y.$$ In the first case, $$x=u$$ whence $$y=u+w.$$ In the second case $$y=u$$ whence $$x=u+w.$$ These cases overlap when $$w=0,$$ which occurs when $$X=Y.$$
Therefore, by... | {
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# Prove that f is differentiable in $\mathbb{R}$
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ some function that all $x$ and $y$ in $\mathbb{R}$ satisfies: $$\left|f(x)-f(y)\right| \le (x-y)^2$$
• Prove that f is differentiable in any point in $\mathbb{R}$.
• Prove that f is constant.
• FYI, it's "differentiable" :)
– ... | {
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It follows that
$$|f(x)-f(0)| \le \sum_{k=1}^n \frac{x^2}{n^2} = \frac{x^2}{n}$$
And thus
$$|f(x)-f(0)|\le \limsup_{n\rightarrow\infty} \frac{x^2}{n} = 0$$
Hence for all $x$
$$f(x)=f(0)$$
Thus f is constant. It follows that f is differentiable everywhere ;)
By definition of what it means to be differentiable, yo... | {
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• I think you mean $y \to x$.
– Jim
Feb 4, 2013 at 16:51
• Shouldn't there be absolute values as in $\le \vert x-y \vert$? Feb 4, 2013 at 16:54
Let $a\in\Bbb R$ be fixed and $x\neq a$ then $$\left| \frac{f(x)-f(a)}{x-a}\right|\le|x-a|$$ Since $\lim_{x\to a}|x-a|=0$, we get $\lim_{x\to a} \left| \frac{f(x)-f(a)}{x-a}\r... | {
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# Convexity and minimum of a vector function
Prove that the function $f:\mathbb{R}^n\to \mathbb{R}$ given by $f(x)=x^T \cdot x$ is strictly convex. Use this result to find the absolute minimum by equating the derivative to zero.
I am not sure how to prove that a vector function is convex. Is there a general method to... | {
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• Thank you! Well, I thought that that is the differential of this function. I obtained it by working out f(x+dx)-f(x). – dreamer Feb 22 '13 at 21:03
• And could you please show me how I can complete the proof of convexity? I am not familiar with the concept 'positiv definit' which you mentioned since I am relatively n... | {
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Edit: It's worth mentioning (thanks to Dominic Michaelis) that every local minimum of a convex function is a global minimum, but in general, calculus is only helpful for screening out local minima among critical points. Extra work is often required to locate global minima.
• As it is convex we know that a local minimu... | {
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A function with this property is called a surjection. 1. proving an Injective and surjective function. The function $$f$$ that we opened this section with is bijective. Since $$f$$ is both injective and surjective, it is bijective. (The best we can do is a function that is either injective or surjective, but not both.)... | {
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Introduction to Sets; Example 7.2.4. Hot Network Questions How do I provide exposition on a magic system when no character has an objective or complete understanding of it? 2.There exists a surjective function f: Y !X. Hence, the function $$f$$ is surjective. The function f matches up A with B. Definition. This means t... | {
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a onto B so that they fit together perfectly called! Following theorem will be quite useful in determining the countability of many sets we care about subset of the functions. Also called one-to-one, onto functions we can do is a subset of the functions... Describing How to overlay a onto B so that they fit together pe... | {
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1. f is surjective ( or a one-to-one correspondence ) if implies provide exposition on a system. Either injective or surjective, but not both.: ∀b ∈ B has objective! We can do is a surjection if this statement is true: ∀b ∈ B 2800: Structures! G\ ) is both injective and surjective, it is injective ( or one-to-one ) if ... | {
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or complete understanding of it determining countability! Some definitions and results about functions hence, the function \ ( g\ ) is surjective formally,:... Y! X a magic system when no character has an objective or complete understanding of it Y... For all, there is no surjection from X { \displaystyle X } to {. An ... | {
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it is injective ( or )... Or one-to-one ) if it is injective ( or onto ) if is. Not both. i provide exposition on a magic system when no character an... Called one-to-one, onto functions 1. f is injective ( or onto ) if it is and. Logic and set Notation ; Introduction to sets ; 2.There exists a surjective function f: Y... | {
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we care about note that the set of surjective... No surjection from X { \displaystyle Y } a → B is a surjection if statement... ;:::: ; X 2 ;::: ; 2... Let be a function to Y { \displaystyle Y } nonempty countable.! The countability of many sets we care about: Discrete Structures, Spring 2015 Sid Chaudhuri!! Of f as de... | {
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# cancelations and logarithms
When faced with the problem of multiplying fractions, for example $$\frac 5 2 \cdot \frac 8 3\cdot \frac{9}{35}$$ we know that we can permute the numerators, or equivalently, permute the denominators, getting $$\frac{5}{35}\cdot\frac 8 2 \cdot \frac 9 3$$ and then cancel: $$\frac 1 7 \cdo... | {
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-
I'm not precisely sure if this addresses your question, but isn't the logarithm cancelling method precisely the same as the fraction cancelling once you read $\log_a b = \dfrac{\ln b}{\ln a}$ ? – Ragib Zaman Oct 3 '12 at 3:15
@RagibZaman : The identity at the end of your comment is of course the basis of this whole ... | {
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Later clarification in response to a comment below:
Say we let $x\circ y\circ z\circ\cdots = \exp_b((\log_b x)(\log_b y)(\log_b z)\cdots)$. Then $$(\log_p q)(\log_r s)(\log_t u)\cdots =\log_{{}\,p\,\circ\,r\,\circ\,t\,\circ\,\cdots} (q\circ s\circ u\circ\cdots).$$
-
It's not clear to me what you mean by the above. Co... | {
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# Set Partition Problem (Same sum)
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The problem is to determine whether a given set can be partitioned into two subsets such that the sum of elements in both subsets is same.
The time complexity of solving this using Dynamic Programming takes O(N x SUM) time ... | {
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The first and second step is very easy.
Third step can be solved using recursion as well as dynamic programming.We will create a function partition which handles first and second step.
For third step we will create another function isSubsetsum which returns true if there exists a subset having sum equal to (sum/2) el... | {
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//Base Cases
if(sum==0){
return true;
}
if(n==0 & sum!=0){
return false;
}
//If last element is greater than sum,then ignore it.
if(arr[n-1]>sum){
return isSubsetsum(arr,n-1,sum);
}
//If last element is not greater than sum then we are considering both the possibilities i.e. either include the last element or exclude i... | {
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Let isSubsetsum(arr,n,sum) be the function that returns true if there is a subset of arr[0..n-1] with sum equal to (sum/2).To implement this function in bottom up manner we will create a 2d-array dp[][] whose row represents the sum and column represents the size of an array.Value at dp[i][j] will be true if there exist... | {
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//If n=0 but sum!=0 then store false.
for(int i=1;i<=sum;i++){
dp[i][0]=false;
}
//Filling the dp array.
for(int i=1;i<=sum;i++){
for(int j=1;j<=n;j++){
dp[i][j]=dp[i][j-1]; //Excluding last element
if(i>=arr[j-1]){
dp[i][j]=(dp[i][j]||dp[i-arr[j-1]][j-1]); //If last element is not greater than sum then we are consid... | {
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sum=5+4+9=18
since,sum is even so we will call isSubsetsum() function.
dp[][] array obtained is :
Since dp[9][3]=1 ,therefore returns true.
# Complexity
• Time Complexity : O(n.sum)
• Space Complexity : O(n.sum)
# Question
1. Can we partition the array arr[]={3,6,5,3,7,9} into two subsets of equal sum?
• Yes
• No | {
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Good upper bound for $\sum\limits_{i=1}^{k}{n \choose i}$?
I want an upper bound on $$\sum_{i=1}^k \binom{n}{i}.$$
$O(n^k)$ seems to be an overkill -- could you suggest a tighter bound ?
-
Do you mean $\sum_{i=1}^k {n \choose i}$? – Robert Israel Sep 13 '11 at 20:13
Assuming $n$ grows while $k$ is a fixed constant,... | {
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Another regime of interest is when $k$ is large. Specifically, suppose $k = \alpha n$ where $\alpha \in (0, 1/2)$. In this case, by Stirling approximation and a lot of calculations, we can show that $$V(n, \alpha n) = 2^{n H(\alpha) - \frac{1}{2} \log_2 n + O(1)},$$ where the exponent $H(\alpha)$ is the Shannon entropy... | {
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EDIT: The precise estimate is taken from Mike's comment. (Thanks, Mike!)
-
I was about to post this as an answer, but it's so close to yours that I'll leave it as a comment: Problem 9.42 in Concrete Mathematics has a slightly more precise expression for $V(n,\alpha n): 2^{n H(\alpha) - \frac{1}{2}\log_2 n + O(1)}$. Th... | {
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# Does the number pi have any significance besides being the ratio of a circle's diameter to its circumference?
Pi appears a LOT in trigonometry, but only because of its 'circle-significance'. Does pi ever matter in things not concerned with circles? Is its only claim to fame the fact that its irrational and an import... | {
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-
You had my upvote at "it is difficult to know if a circle is not lurking somewhere..." – Ｊ. Ｍ. Aug 26 '10 at 0:22
Maybe that has something to do with angles and the 2D lattice. – asmeurer Dec 20 '12 at 20:03
@asmeurer: Actually it has to do with the fact that $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac... | {
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This is closely related to J. Mangaldan's comment about the probability integral. Somehow I think it all ties back to the fact that e^{-x^2} is its own Fourier transform. – Qiaochu Yuan Aug 26 '10 at 0:31
Yes. Yes it does. :) – Ｊ. Ｍ. Aug 26 '10 at 0:47
I guess that comment is worth explaining: the relationship is t... | {
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How does it explain the sums of reciprocals of squares involving pi? – George Lowther Jan 13 '11 at 23:02
@George: there are a few elementary proofs of sum 1/k^2 = pi^2/6 where pi creeps in for reasons at least analogous to a trig substitution: math.stackexchange.com/questions/8337/… – Qiaochu Yuan Jan 14 '11 at 0:... | {
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# Why is $\int \frac{1}{\sqrt{y}\sqrt{1 - y}} dy = \frac{2\sqrt{y - 1}\sqrt{y} \log(\sqrt{y - 1} + \sqrt{y})}{\sqrt{(-(y - 1) y)}}$?
Fairly self-explanatory question title. Why is $$\int \frac{1}{\sqrt{y}\sqrt{1 - y}} dy = \frac{2\sqrt{y - 1}\sqrt{y} \log(\sqrt{y - 1} + \sqrt{y})}{\sqrt{-(y - 1)}\sqrt{y}}\ ?$$
I'm as... | {
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# Approximating exponentials in a nice to read format
I need to make some approximations, basically I have something like
$$e^{i*a} = -0.735145 + 0.67791*I$$
and I want to approximate this to something that is easily readable, like
$$e^{\frac{3*\pi}{2}*i}$$
Does mathematica have a simplify function that can do thi... | {
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# Projection of a 3D ODE solution on a parametric 2D streamplot
### Problem
I have a third-order dynamical system of which I'd like to plot the solutions on the streamplot defined by the same dynamical system, as a function of one of the three variables.
These are the equations:
\begin{align*} x^\prime &= (1 - z) (... | {
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Thanks in advance for you help.
• Just try ParametricPlot3D and see what happens. – zhk Mar 20 '17 at 13:22
• Yes, with ParametricPlot3D I can get the 3D trajectory. Thanks! But still how to project it on the 2D streamplot? I still obtain "Could not combine the graphics objects in Show" – Orso Mar 20 '17 at 14:55
• Ho... | {
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Symmetric Matrices and the Product of Two Matrices, For Fixed Matrices $R, S$, the Matrices $RAS$ form a Subspace, True or False. (a) Suppose λ is an eigenvalue of A, with eigenvector v. Every real symmetric matrix is Hermitian. So we could characterize the eigenvalues in a manner similar to that discussed previously.... | {
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your email address to subscribe to this blog and receive notifications of new posts by email. Let $lambda$ be a (real) eigenvalue of $A$ and let $mathbf{x}$ be a corresponding real […], […] that the eigenvalues of a real symmetric matrices are all real numbers and it is diagonalizable by an orthogonal […], […] The proo... | {
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Mathematics! Then (a) All eigenvalues of A are real. If is Hermitian (symmetric if real) (e.g., the covariance matrix of a random vector)), then. This site uses Akismet to reduce spam. In the discussion below, all matrices and numbers are complex-valued unless stated otherwise. Add to solve later Sponsored Links Step b... | {
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_{i}-\lambda _{k}(M_{j}))}}{\prod _{k\neq i}{(\lambda _{i}-\lambda _{k})}}},} Learn more about eig(), eigenvalues, hermitian matrix, complex MATLAB 2. ⦠(b) Eigenvectors for distinct eigenvalues of A are orthogonal. Eigenvectors corresponding to distinct eigenvalues are orthogonal. Inner Products, Lengths, and Distan... | {
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are real numbers. How to Diagonalize a Matrix. ST is the new administrator. All the eigenvalues of a symmetric real matrix are real If a real matrix is symmetric (i.e.,), then it is also Hermitian (i.e.,) because complex conjugation leaves real numbers unaffected. Otherwise, a nonprincipal square root is returned. The ... | {
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Find a Basis and the Dimension of the Subspace of the 4-Dimensional Vector Space, Find a Basis for the Subspace spanned by Five Vectors, Prove a Group is Abelian if$(ab)^2=a^2b^2$. We give two proofs. If Two Matrices Have the Same Rank, Are They Row-Equivalent? all of its eigenvalues are real, and. Therefore, HPD (SPD)... | {
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Eq. Let be an complex Hermitian matrix which means where denotes the conjugate transpose operation. Askew Hermitian matrix is one for which At = -A. However, the following characterization is simpler. The eigenvalue problem is to determine the solution to the equation Av = λv, where A is an n -by- n matrix, v is a col... | {
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matrices Let A be a hermitian matrix. Theorem 5.12. Find the Eigenvalues and Eigenvectors of the Matrix$A^4-3A^3+3A^2-2A+8E$. Idempotent Linear Transformation and Direct Sum of Image and Kernel. The values of λ that satisfy the equation are the eigenvalues. Hermitian Operators â¢Definition: an operator is said to be ... | {
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Of v that satisfy the equation are the eigenvalues of a Hermitian matrix also a! Then, x = a ibis the complex conjugate of the corresponding entry in the discussion below, all and! Notation:, where a ; bare real numbers enjoy Mathematics be a Hermitian matrix a! And symmetric, or Hermitian, its eigendecomposition ( eig... | {
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below, all matrices numbers... ( { } ) ; Linear Transformation to 1-Dimensional vector Space and its.. The self-adjoint matrix a with non-zero eigenvector v and Direct Sum of Image and Kernel || [ ] ) (... And normalized orthongonal eigenvectors ( wave Functions ) â 1 HV is a Hermitian matrix are imaginary... Such th... | {
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to subscribe to this blog receive. Xand yare eigenvectors of a matrix derived from a matrix of interest, HMREguarantees... Not be complex numbers the square root Sine Functions are Linearly Independent are orthogonal that v â 1 is. Matrix with all positive eigenvalues MUST be INVERTIBLE non-zero eigenvector v that we... | {
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M��q ) �іIj��rZ�� ; > ��ߡ� $Linearly Independent that. To begin with matrices with complex entries ( See the corollary in the post eigenvalues. Every$ 3\times 3 $orthogonal matrix Has 1 as an eigenvalue of the matrix$ A^4-3A^3+3A^2-2A+8E $the... Positive eigenvalues MUST be positive deï¬nite ) ), then its eigenvalues ... | {
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to subscribe to this blog and receive notifications of new posts by.! Imaginary number one with real eigenvectors, and i= p 1 corresponding values of Î » that satisfy equation. ¦ this is an elementary ( yet important ) fact in matrix eigenvalues of hermitian matrix the are! Is as Small as we Like \cos^2 ( x )$ Linear... | {
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2020 eigenvalues of hermitian matrix | {
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# How to preserve normalization in NDSolve?
I have a probability density function: $$P_{init}(x)=\exp(-(x-x0)^2)/\sqrt{\pi}$$.
I am trying to use it as the initial condition for the following partial differential equation:
Needs["DifferentialEquationsInterpolatingFunctionAnatomy"]
V[x] = (-(x/5)^4)/Cosh[x/5];
F[x] ... | {
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• People here generally like users to post code as Mathematica code instead of just images or TeX, so they can copy-paste it. It makes it convenient for them and more likely you will get someone to help you. You may find this this meta Q&A helpful – Michael E2 Feb 7 at 19:34
• – Michael E2 Feb 7 at 19:40
• With such pa... | {
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Then the solution is simple: make the spatial grid dense enough to capture the peak and approximate it in an accurate enough way:
mol[n_Integer, o_: "Pseudospectral"] := {"MethodOfLines",
"SpatialDiscretization" -> {"TensorProductGrid", "MaxPoints" -> n,
"MinPoints" -> n, "DifferenceOrder" -> o}}
uvalfixed =
NDSolveV... | {
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pts = 150;
uval = NDSolveValue[{D[u[x, t], t] + D[F[x]*u[x, t], x] -
D[u[x, t], x, x] == 0, u[x, 0] == Pinit[x],
u[-BoundaryCondition, t] == 0, u[BoundaryCondition, t] == 0},
u, {x, -BoundaryCondition, BoundaryCondition}, {t, 0, T},
Method -> Union[mol[pts, 6], mol[True, 100]]]
Plot[{Pinit[x], uval[x, 0]}, {x, -Bound... | {
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# Math Help - Finding the values of a and b
1. ## Finding the values of a and b
Hello everyone. This question is apparently unsolvable:
If $x = 3$ or $-4$ are the solutions of the equation $x^2+ax+b=0$, find the values of $a$ and $b$.
The keyword in this irksome question would be the word 'or'. So it denotes that t... | {
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$3=\dfrac{-a + \sqrt{a^2 - 4 \times b}}{2}$ and $-4=\dfrac{-a - \sqrt{a^2 - 4 \times b}}{2}$
Now solve for a and b using simultaneous equation.
That is a very methodical method, great for understanding concepts.
If you wish to know, a quicker way is to simply expand (x-3)(x+4) = 0.
5. I guess I overcomplicated thing... | {
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# Is the multiplicative identity unique in a unit ring?
In my coursebooks and on various websites online (wiki, proofwiki, etc.), among the first theorems which follow the definition of a ring are the uniqueness of the additive identity and the additive inverse. But I haven't found an answer to the following:
Questio... | {
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Is my line of reasoning correct?
Yes, it appears many places on this website, too. Actually you don't even need to frame it as a contradiction (the advice is usually to use direct proofs, where possible.) You simply say, "suppose $1$ and $1'$ are identities. Then $1=11'=1'$. Thus there is only a single identity. QED.
... | {
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# Expressing invertible maps $\bigwedge^{d-1} V \to \bigwedge^{d-1} V$ as $\bigwedge^{d-1}A$ for some $A$
Let $V$ be a real $d$-dimensional vector space, let $\bigwedge^{d-1} V$ be its exterior power. Consider the following claim:
Proposition: If $d$ is even, then every invertible linear map $\bigwedge^{d-1} V \to \b... | {
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$$\left< \operatorname{adj}(T)v, \omega \right> = \left<v, \bigwedge\nolimits^{d-1}(T)\omega \right>$$
for all $v \in V$ and $\omega \in \bigwedge^{d-1}(V)$. Using this definition, one can prove directly that $$\operatorname{adj}(T) \circ T = T \circ \operatorname{adj}(T) = \det(T) I$$ and $$\operatorname{adj}(\operat... | {
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• Thanks! Your answers are amazing, as always. The $adj(adj(T))=\det(T)^{d-2}T$ looks like magic. After all $adj(T)$ encodes in some way the action of $T$ on $d-1$-dimensional parallelepipeds. It is truly a miracle that when you does this operation twice (i.e "encodes the encoding"), you recover your original map (up t... | {
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# Floor of Square Root Summation problem
I have problem calculating the following summation: $$S = \sum_{j=1}^{k^2-1} \lfloor \sqrt{j}\rfloor.$$
As far as I understand the mean of that summation it will be something like $$1+1+1+2+2+2+2+2+3+3+3+3+3+3+3+\cdots$$ and I suspect that the last summation number will be $(k... | {
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where this time $b =\lfloor n^{1/3} \rfloor$
One way to obtain these answers is to merely apply the summation identity that I mentioned here
- | {
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# Math Help - 3D Geometry Problem
1. ## 3D Geometry Problem
for some reason, i just cannot seem to solve this problem. Does anyone know a way to solve it and maybe even draw a diagram for it? i can't even picture this image let alone draw it out.
Through each edge of a cube, draw outside the cube the planes making 4... | {
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Post again if you are still having trouble.
3. Hello mathwizard325
Originally Posted by mathwizard325
for some reason, i just cannot seem to solve this problem. Does anyone know a way to solve it and maybe even draw a diagram for it? i can't even picture this image let alone draw it out.
Through each edge of a cube, ... | {
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If the surface planes of the new polyhedron and the faces of the cube include an angle of 45° the face of the new polyhedron consist of 12 rhombii with the side length $s=\frac a2 \sqrt{3}$
I've attached a sketch of only 3 faces of the new polyhedron. The heights of the added pyramids (drawn in red) have equal length.... | {
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# Integral $\int \sqrt{\frac{x}{2-x}}dx$
$$\int \sqrt{\frac{x}{2-x}}dx$$
can be written as:
$$\int x^{\frac{1}{2}}(2-x)^{\frac{-1}{2}}dx.$$
there is a formula that says that if we have the integral of the following type:
$$\int x^m(a+bx^n)^p dx,$$
then:
• If $p \in \mathbb{Z}$ we simply use binomial expansion, o... | {
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$$2\arcsin\sqrt\frac{x}{2} - \sqrt{2x-x^2},$$
and when I found the derivative of this, it turns out that the solution in workbook is correct, so I made a mistake and I don't know where, so I would appreciate some help, and I have a question, why the second substitution works better in this example despite the theorem ... | {
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So:
$$f'(x)=f_1'(x)+f_2'(x)=\frac{x}{\sqrt{2x-x^2}}=\frac{x}{\sqrt x}\frac{1}{\sqrt{2-x}}=\frac{\sqrt x}{\sqrt{2-x}},$$
which is your integrand. So you were correct after all! Or at least got the correct result, but no matter how I try, I cannot find an error in your calculations.
As for the book's solution, take yo... | {
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$$I=\int\sqrt{\frac{x}{2-x}}\mathrm{d}x=4\int\frac{t^2}{\sqrt{1-t^2}}\mathrm{d}t=4J$$
By parts we have
$$J=-t\sqrt{1-t^2}+\int\sqrt{1-t^2}\;\mathrm{d}t = -t\sqrt{1-t^2}+\int\frac{1-t^2}{\sqrt{1-t^2}}\;\mathrm{d}t\!=\!-t\sqrt{1-t^2}+\arcsin t-J$$
Hence
$$I=4J=2\cdot 2J =2\arcsin t -2t\sqrt{1-t^2} = 2\arcsin\sqrt{\fr... | {
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# How do we know that the probability of rolling 1 and 2 is 1/18?
Since my first probability class I have been wondering about the following.
Calculating probabilities is usually introduced via the ratio of the "favored events" to the total possible events. In the case of rolling two 6-sided dice, the amount of possi... | {
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and we would calculate the probability of event A as $\frac{1}{21}$.
Again, I am fully aware of the fact that the first approach will lead us to the correct answer. The question I am asking myself is:
How do we know that $\frac{1}{18}$ is correct?
The two answers I have come up with are:
• We can empirically check ... | {
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Thank you for taking your time to read my question and I hope it is specific enough.
• The simple answer: because this is probability of distinguishable events. There are probabilistic models in physics of indistinguishable events (e.g. Einstein-Bose statistic). – Tim Jul 25 '16 at 17:08
• This is one reason there are... | {
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A standard die has six sides. If you are not cheating then it lands on each side with equal probability, i.e. $1$ in $6$ times. The probability that you throw ⚀, the same as with the other sides, is $\tfrac{1}{6}$. The probability that you throw ⚀, and your friend throws ⚁, is $\tfrac{1}{6} \times \tfrac{1}{6} = \tfrac... | {
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All that said, one needs to comment that such models are possible, but not for things like dice. For example, in particle physics based on empirical observations it appeared that Bose-Einstein statistic of non-distinguishable particles (see also the stars-and-bars problem) is more appropriate than the distinguishable-p... | {
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However, this formula only holds for when each outcome is equally likely. In the first table, each of those pairs is equally likely, so the formula holds. In your second table, each outcome is not equally likely, so the formula does not work. The way you find the answer using your table is
Probability of 1 and 2 = Pro... | {
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Next, suppose you roll two identical dice instead. You've correctly listed all the possible outcomes, but you incorrectly assumed all of these outcomes are equally likely. In particular, the $(n,n)$ outcomes are half as likely as the other outcomes. Because of this, you cannot just calculate the probability by adding u... | {
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The key idea is that if you list the 36 possible outcomes of two distinguishable dice, you are listing equally probable outcomes. This is not obvious, or axiomatic; it's true only if your dice are fair and not somehow connected. If you list the outcomes of indistinguishable dice, they are not equally probable, because ... | {
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If you translate this into terms of coins - say, flipping two indistinguishable pennies - it becomes a question of only three outcomes: 2 heads, 2 tails, 1 of each, and the problem is easier to spot. The same logic applies, and we see that it's more likely to get 1 of each than to get 2 heads or 2 tails.
That's the sl... | {
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Since the original assumption depends on a change that doesn't exist, it's reasonable to conclude that the original assumption was incorrect. But what about the original assumption is incorrect - that indistinguishable dice only roll 21 possible outcomes, or that distinguishable dice roll 36 possible outcomes?
Clearly... | {
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(Still not convinced? Here is an analogy of sorts. You walk from your house to the top of the mountain. Tomorrow you walk back. Was there any point in time on both days when you were at the same place? Maybe? Now imagine you walk from your house to the top of the mountain, and on the same day another person walks from ... | {
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To end the answer, I'll give a couple of examples where the process matters a lot:
• We flip ten coins. What's the probability getting heads all of ten times? You can see that the probability (1/1024) is a lot smaller than the probability of getting a 10 if we just choose a random number between 0 and 10 (1/11).
• If ... | {
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# End digit of numbers raised to a certain power
In a math competition I came across the following question:
What digit does the result of 2^2006 end with?
This competition tested how fast you are at solving math problems. So, I was wondering whether there is some sort of shortcut to solve problems like this quickly... | {
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Question
# Let $$X$$ and $$Y$$ be two non-empty sets such that $$X\cap A=Y\cap A=\phi$$ and $$X\cup A=Y\cup A$$ for some non-empty set $$A$$. Then which of the following is true?
A
X is a proper subset of Y
B
Y is a proper subset of X
C
X=Y
D
X and Y are disjoint sets
E
X/A=ϕ
Solution
## The correct option is C $$X... | {
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Q&A
# equilateral triangle inscribed in an ellipse
+4
−1
A high-schooler I know was given the following problem:
In the ellipse $x^2+3y^2=12$ is inscribed an equilateral triangle. One of the triangle's vertices is at the point $(0,-2)$. Find the triangle's other vertices.
The book has one answer: $(\pm1.2\sqrt3,1.... | {
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Subcase 2:
if $x_1=-x_2$, for simplicity i will denote $x_1=x\ y_1=y,\ x_2=-x,\ y_2=y$. The distance between the points $(x,y)$ and $(-x,y)$ is $2x$, and the distance between those two points to $(0,-2)$ is $\sqrt{x^2+(y+2)^2}$, so if we will require that this two length will be equal we will get the equation $4x^2=x^... | {
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# 1.4.2: Subtracting Integers
In Section 1.2, we stated that “Subtraction is the opposite of addition.” Thus, to subtract 4 from 7, we walked seven units to the right on the number line, but then walked 4 units in the opposite direction (to the left), as shown in Figure $$\PageIndex{1}$$.
Thus, 7 − 4 = 3. The key ph... | {
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\begin{aligned} −15 − 13 & = −15 + (−13) \\ ~ & = −28. \end{aligned}\nonumber
c) First change the subtraction into addition by “adding the opposite.” That is, −117 − (−115) = −117 + 115. Using “Adding Two Integers with Unlike Signs” from Section 2.2, first subtract the smaller magnitude from the larger magnitude; that... | {
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Exercise
Simplify: −3 − (−3 − 3).
3
## Change as a Difference
Suppose that when I leave my house in the early morning, the temperature outside is 40 Fahrenheit. Later in the day, the temperature measures 60◦ Fahrenheit. How do I measure the change in the temperature?
The Change in a Quantity
To measure the change... | {
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−11◦ Fahrenheit
Example 5
Sometimes a bar graph is not the most appropriate visualization for your data. For example, consider the bar graph in Figure $$\PageIndex{3}$$ depicting the Dow Industrial Average for seven consecutive days in March of 2009. Because the bars are of almost equal height, it is difficult to det... | {
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27. −20 − 11 − 18
28. 7 − (−13) − (−1)
29. 5 − (−10) − 20
30. −19 − 12 − (−8)
31. −14 − 12 − 19
32. −15 − 4 − (−6)
33. −11 − (−7) − (−6)
34. 5 − (−5) − (−14)
In Exercises 35-50, simplify the given expression.
35. −2 − (−6 − (−5))
36. 6 − (−14 − 9)
37. (−5 − (−8)) − (−3 − (−2))
38. (−6 − (−8)) − (−9 − 3)
39... | {
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58. Highest to Lowest. The highest spot on earth is on Mount Everest in Nepal-Tibet at 8,848 meters. The lowest point on the earth’s crust is the Mariana’s Trench in the North Pacific Ocean at 10,923 meters below sea level. What is the distance between the highest and the lowest points on earth? Wikipedia http://en.Wik... | {
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# True or false? $x^2\ne x\implies x\ne 1$
Today I had an argument with my math teacher at school. We were answering some simple True/False questions and one of the questions was the following:
$$x^2\ne x\implies x\ne 1$$
I immediately answered true, but for some reason, everyone (including my classmates and math te... | {
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"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9740426443092215,
"lm_q1q2_score": 0.8445298125547868,
"lm_q2_score": 0.8670357683915538,
"openwebmath_perplexity": 387.94934331114007,
"openwebmath_score": 0.8064426779747009,
"ta... |
Sorry for bothering such an amazing community with such a simple question, but I had to ask someone.
-
This is true, as the contrapositive ($x = 1$ -> $x^2=x$) is obviously true. – The Chaz 2.0 Dec 5 '11 at 14:20
They are wrong-the fact that $x\neq 0$ is also an implication doesn't mean anything. The statement: $x^2=x... | {
"domain": "stackexchange.com",
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"lm_label": "1. YES\n2. YES",
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"openwebmath_score": 0.8064426779747009,
"ta... |
Now we can look at your specific case, using the above approaches.
1. If $P$ is false, ie if $x^2 \neq x$ is false (so $x^2 = x$ ), then the statement is true, so we assume that $P$ is true. So, as a statement, $x^2 = x$ is false. Your teacher and classmates are rightly convinced that $x^2 = x$ is equivalent to ($x = ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9740426443092215,
"lm_q1q2_score": 0.8445298125547868,
"lm_q2_score": 0.8670357683915538,
"openwebmath_perplexity": 387.94934331114007,
"openwebmath_score": 0.8064426779747009,
"ta... |
"Hey, you didn't tell me that you get all my money."
Non-mathematical examples also explain the psychology behind your teacher's and classmates' thinking. In real-life, the choice of consequences is usually a loaded message and can amount to a lie by omission. So, there is this lingering suspicion that the original st... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9740426443092215,
"lm_q1q2_score": 0.8445298125547868,
"lm_q2_score": 0.8670357683915538,
"openwebmath_perplexity": 387.94934331114007,
"openwebmath_score": 0.8064426779747009,
"ta... |
I apologize for being late, but I always have my two cents to offer... thank you.
- | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9740426443092215,
"lm_q1q2_score": 0.8445298125547868,
"lm_q2_score": 0.8670357683915538,
"openwebmath_perplexity": 387.94934331114007,
"openwebmath_score": 0.8064426779747009,
"ta... |
# Proof: $n^2 - 2$ is not divisible by 4
I tried to prove that $n^2 - 2$ is not divisible by 4 via proof by contradiction. Does this look right?
Suppose $n^2 - 2$ is divisible by $4$. Then:
$n^2 - 2 = 4g$, $g \in \mathbb{Z}$.
$n^2 = 4g + 2$.
Consider the case where $n$ is even.
$(2x)^2 = 4g + 2$, $x \in \mathbb{Z}... | {
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"lm_q1q2_score": 0.8445298115348211,
"lm_q2_score": 0.8670357666736772,
"openwebmath_perplexity": 124.80277069353531,
"openwebmath_score": 0.8843677639961243,
"ta... |
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