Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Determining existence of a solution for a system of linear inequalities I have a set of vectors $\mathbf{v}_1,\mathbf{v}_2,...,\mathbf{v}_m\in\mathbb{R}^n$ and I want to know if there exists a nonzero vector $\mathbf{x}$ such that $\mathbf{x}\cdot\mathbf{v}_i\le0$ for any $i$. This is the same as saying the equation $\... | An alternative approach based on Stiemke's theorem would involve the solution of a single somewhat larger LP.
Solve the LP
$\max z$
Subject to
$Vy=0$
$y_{i} \geq z, i=1, 2, \ldots, n$
If the optimal solution has $z=0$, then $Vy=0$, $y > 0$ has no solution.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1879696",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
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Prove that a particular matrix is full column rank
Let $A$ be an $m \times n$ matrix and $\operatorname{rank}(A) =r$. Let $P$ be an $m \times (m-r)$ matrix, such that $\mathbb{C}^m= \mathcal{R}(A) \oplus \mathcal{R}(P)$, then $P$ is of full column rank.
$\mathcal{R}(A)$ : Range of $A$ in $\mathbb{C}^m$
I'm trying to... | $$\begin{cases}\operatorname{rank}P=\dim\mathcal R(P)
\\m=\dim\mathcal R(A)+\dim\mathcal R(p)=r+\dim\mathcal R(P)\\
\operatorname{rank}P\le \min(m,m-r)=m-r\end{cases}$$
What does this yield, in light of the definition of "being full-rank"?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1879771",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $y = W^T x$, what is $\frac{\partial y}{\partial W}$? I would like to derive the derivative of a vector by matrix, i.e. $y = W^Tx$, where $W$ is a matrix, $x,y$ are vectors. What is $\frac{\partial y}{\partial W} = \frac{\partial W^T x}{\partial W}$?
Follow-up:
Define another function $z = a^T y = a^T W^Tx$, so tha... | Consider the full matrix version of this problem, written in terms of the Frobenius (:) Inner Product
$$\eqalign{
Y & = W^TX \cr
z &= A:Y = XA^T:W \cr
}$$
The gradient of $z$ can be evaluated directly
$$\eqalign{
dz &= XA^T:dW \cr
\frac{\partial z}{\partial W} &= XA^T \cr\cr
}$$
It can also be evaluated by the cha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1879900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Why is $P = I_N - \vec{1}\vec{1}^T/N$ a projection matrix, and $P^2=P$? Why is $P = I_N - \vec{1}\vec{1}^T/N$ a projection matrix, and $P^2=P$?
For example, for N=3 :
$$P = I_3 - \vec{1}\vec{1}^T/3 = \begin{pmatrix} 0.67 & 0.33 & 0.33 \\ 0.33 & 0.67 & 0.33 \\ 0.33 & 0.33 & 0.67 \\ \end{pmatrix} $$
and,
$$P^2 = \begin{p... | Your computation of the matrix is wrong.
$$
P = I_3 - \frac{1}{3}\vec{1}\vec{1}^T = \begin{pmatrix} 0.67 & -0.33 & -0.33 \\ -0.33 & 0.67 & -0.33 \\ -0.33 & -0.33 & 0.67 \\ \end{pmatrix}
$$
Note the negative signs. '
This is indeed a projection; it sends $(1,1,1)$ to $(0,0,0)$ and sends $(-1, 1, 0)$ and $(-1, 0, 1)$ to... | {
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"url": "https://math.stackexchange.com/questions/1880010",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Showing there's no maximum in the given interval Let $f$ be function as $f(x) = x^2, x \in [\frac{1}{2}, 3).$ If we want to show there's no maximum in the given interval, is this the right way to do it:
Assume there's maximum $y \in (8, 9)$ for some $x \in [\frac{1}{2}, 3)$. Then show that there's some $y' \in (8, 9)$ ... | Note that $x^2$ is increasing on this interval; thus its supremum is at the right end of the interval which isn't attained from the domain.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1880147",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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If $G\simeq K$ and $H\simeq M$ then is it true that $G/ H \simeq K / M$? Let $G$ and $K$ be groups. Let $H$ be a normal subgroup of $G$ and $M$ be a normal subgroup of $K$ such that $H\simeq M$.
Question: is $ G/H \simeq K / M$?
I am fairly certain that this is tru of the groups are finite. For example, if the groups ... | $$\frac{\mathbb Z}{2\mathbb Z}\not\cong \frac{\mathbb Z }{3\mathbb Z},$$
while $\mathbb Z\cong 2\mathbb Z \cong 3\mathbb Z$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1880247",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Process of finding intersecting points between two functions I have to find the intersecting points of these two functions
$$f(x)=e^x-5x+7$$
and
$$g(x)=2x^2+16x+2$$
I know how to do this with two quadratic equations, by putting the two functions equal to each other, but f(x) is confusing me, because i don't know what... | I don't think you can get a "nice" number by solving this analytically. You may need to use root-finding methods such as Newton's method (or just use the "intersect" feature on your graphing calculator).
| {
"language": "en",
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Under which conditions there exists a solution in linear equations I am having some troubles with an exercise from "Finite dimensional vector spaces" by Halmos. I am not sure if my proof is correct or if my answer is what the author asked for. Thanks for any help!
Suppose that $m < n$ and that $y_{1}, \dots, y_{m}$ ar... | There is loss of generality in your assumption. For example the sum of all may vanish while pairwise they are independent.
You need to find the kernel of the map $\Lambda: c\in {\Bbb C}^m \mapsto \sum_j c_j y_j$. Then clearly if $c\in \ker \Lambda$, a solution to $y_j(x)=\alpha_j$ must verify:
$$ 0=\sum_j c_j y_j(x)= \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1880458",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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LU Decomposition vs. Cholesky Decomposition What is the difference between LU Decomposition and Cholesky Decomposition about using these methods to solving linear equation systems?
Could you explain the difference with a simple example?
Also could you explain the differences between these decomposition methods in:
*
... | Both LU and Cholesky Decomposition is matrices factorization method we use for non-singular( matrices that have inverse) matrices. In general basic different between two method. the later one uses only for square matrices (A = A^T). however LU decomposition we can use any matrices that have inverses. for example see th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1880573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "29",
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Prove "Two parallelograms on the same base and in the same parallels, are equal."
I understand Euclid's way of proving this. But, the book also says that I can prove this by decomposing one parallelogram into pieces, and then forming another parallelogram by combining those pieces together.
I was thinking of dividing... | Here's one way you can do it: you can use the "easier case" in which $e$ is between $b$ and $c$ and $f$ is to the right of $c$ repeatedly. In that case you can simply cut off a triangle from $abcd$ and then reattach it with a translation to get $aefd$. But once you've done this, you can do it again: if you have point... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1881739",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Is travelling salesman problem with integer weight NP-hard? I wonder if travelling salesman problem remains to be NP-hard with an additional constraint that the edge weight is integer.
| The travelling salesman problem already has integer edge weights! For example, in Garey & Johnson, Computers and Intractability, the problem is defined as follows:
TRAVELLING SALESMAN
INSTANCE: A finite set $C=\{c_1,c_2,\ldots,c_m\}$ of "cities", a "distance" $d(c_i,c_j)\in Z^+$ for each pair of cities $c_i,c_j\in C$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1881827",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Convergence/divergence of $\int_{0}^{\infty} \frac{\sqrt[3]x}{1+x}dx$
$$\int_{0}^{\infty} \frac{\sqrt[3]x}{1+x}dx$$
I have read an example on book and they did the following:
$$\frac{\sqrt[3]x}{1+x^2}<\frac{\sqrt[3]x}{x}=\frac{1}{x^{\frac{2}{3}}}$$
and we know that the integral of $\frac{1}{x^{\alpha}}$ converges for... | Your function $\frac{\sqrt[3]x}{1+x}\sim \frac{\sqrt[3]x}{x}=\frac{1}{x^{2/3}}$ as $x\rightarrow +\infty$. We now that the integral $\int_{c}^{\infty}\frac{1}{x^{\alpha}}$ converges for $a>1$ (with $c>0$). Since $2/3 < 1$, your integral diverges
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Find the number of integer solutions of $|x|+|y| \le 10$
Find the solutions of $|x|+|y| \le 10$, where $x$ and $y$ are integers.
My solution:
$$|x|+|y|+z=10$$
Now the solutions if there were no absolute values is:
$$\binom{13-1}{10}=\frac{11*12}{2}=66$$
now subtract that once that have $0$ then multiply the others by... | Pick's Theorem says that the area of a polygon whose vertices have integer coordinates is given by
$$A=I+{B\over2}-1$$
where $I$ is the number of Interior points with integer coordinates and $B$ is the number of Boundary points with integer coordinates. For the given problem the polygon is a square with diagonals of l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1882131",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Is $S_3 \oplus \Bbb Z_2$ isomorphic to $A_4$ or to $D_6$? I know $S_3 \oplus \Bbb Z_2$ is isomorphic either to $A_4$ or to $D_6$, where $S_3$ is the symmetric group of degree $3$, $A_4$ is the alternating group of degree $4$, $D_6$ is the dihedral group of order $12$, and $\oplus$ is the external direct product.
Wi... | One way to easily see the isomorphism is to note that we may identify $ C_6 $ in $ S_3 \times C_2 $ as the normal subgroup $ N = A_3 \times C_2 $, and if we denote $ H = \langle ((12), e) \rangle \cong C_2 $ and let $ ((12), e) = h $ then $ NH = S_3 \times C_2 $ and $ N \cap H $ is trivial. It is easily checked that co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1882239",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 3
} |
Minkowski difference of two convex sets is convex? Hi my question is quite straightforward, if we have two disjoint compact convex sets A and B, is their minkowski difference A-B then convex again?
Thanks!
| Each point in $A-B$ is of the form $a-b$, where $a\in A$ and $b\in B$.
Letting $a-b$ and $a^{\prime}-b^{\prime}$ be two points in $A-B$,
we note that for any $\theta\in[0,1]$,
$$
\theta\left(a-b\right)+\left(1-\theta\right)\left(a^{\prime}-b^{\prime}\right)=\left[\theta a+\left(1-\theta\right)a^{\prime}\right]-\left[\t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Help with the limit of an integral I am trying to evaluate this limit
$$\lim \limits_{x \to \infty} \frac {1}{\ln x} \int_{0}^{x^2} \frac{t^5-t^2+8}{2t^6+t^2+4} dt=? $$
Any help will be appreciated.
| Hint: Compare with the integral of just $t^5/(2 t^6)$ as $t$ goes to infinity. Show that the rest really doesn't matter.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1882409",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Sample mean: dependence I have a question that is possibly more about language than math, but still it concerns me a lot. I understand that this question may irritate many (because it's stupid, and apparently because I am stupid too), but still I ask not to hate me too much.
We all remember the definition of sample mea... | I think that, without a more specific qualification of that statement, it is difficult to say what was really meant.
For example, one interpretation could be that, for a simple random sample, the sample mean is an unbiased estimator of the population mean, and this property is independent of the sample size. But this ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1882497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$\operatorname{Hom}_k(k,V)$ is a vector space? Is it true that a vector space is just the set of maps from the underlying field to the space itself. I.e. if $V$ is a vector space of the field $k$ then
$$
V\cong \operatorname{Hom}_k(k,V)
$$ if so then this would make an intuitive understanding of the dual space $V^*$ s... | This works even for infinite-dimensional vector spaces (or for that matter for general modules over unital rings): The map
$$ f \in \operatorname{Hom}_k(k,V) \mapsto f(1) \in V $$
is always vector space isomorphism. You don't need duals for that.
This is clearly injective and a homomorphism; to see that it is surjecti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1882574",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Solve $\int_{0}^{1}\frac{1}{1+x^6} dx$ Let $$x^3 = \tan y\ \ \text{ so that }\ x^2 = \tan^{2/3}y$$
$$3x^2dx = \sec^2(y)dy$$
$$\int_{0}^{1}\frac{1}{1+x^6}dx = \int_{1}^{\pi/4}\frac{1}{1+\tan^2y}\cdot \frac{\sec^2y}{3\tan^{2/3}y}dy = \frac{1}{3}\int_{1}^{\pi/4} \cot^{2/3}y\ dy$$
How should I proceed after this?
EDITED: C... | Continuing where MK12 left off, we proceed as follows:
$$\frac{2-x^2}{x^4-x^2+1} = \frac{1}{2} \times \frac{4-2x^2}{x^4-x^2+1} = \frac{1}{2} \times \frac{1+x^2 + 3(1-x^2)}{x^4-x^2+1}$$
Split the sub-integral into two parts. For one, make the substitution $u=x+\frac{1}{x}$ and the other $v = x-\frac{1}{x}$
Then the rest... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1882650",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Proving that $\log_b(r^s) = s\log_b(r)$ The Question
If $b,r,s \in \mathbb{R^+}$ prove that $\log_b(r^s) = s\log_b(r)$
My Work
1) $\log_b(r^s)$
2) $s$ can be expressed as the sum of an integer part $n$ and a real part $m$: $s = m + n$
3) $\log_b(r^{n+m})$
4) $\log_b(r^nr^m)$
5) $\log_b(r^n) + \log_b(r^m)$
6) $\log_b(r... | Take $\log_b (r^s):=F(s)$ ($r$ an arbitrary constant) and note that $F(0)=\log_b (1)=0$
Show that:
$F(s)=F(s-1)+\log_b r$
Hence $\frac{F(s)-F(s-1)}{s-(s-1)}=\log_b (r)$ and $F$ is of slope $\log_b (r)$:
$$F(s)=(\log_b r)s+c$$
But $F(0)=0$ gives $c=0$.
If we proved it for all positive arbitrary constants $r>0$, then the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1882828",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Proving a ring has only infinite dimensional modules.
Let $R$ be the ring $\mathbb{C}\langle x,y\rangle/(xy-yx-1)$, a quotient of free associative algebra on two generators.
(a) Show every nonzero $R$-module has infinite dimension as a complex vector space.
(b) Let $M$ be an $R$-module with a nonzero element $z$ such ... | (a) $A$ is the ring $C\langle u,v\rangle/(uv-vu-1)$ which means that $[u,v]=1$. Consider some nonempty $A$-module. That is, $M$ is an abelian group under addition and we also have the operation $A\times M \rightarrow M$ satisfying all the module properties. Note that $C\subset A$, and thus we have the operation $C \tim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1882944",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Adapting the Simplex method to use the distance function as the target. Has anyone adapted the simplex method to use the distance from a point as the minimization criteria?
E.g.
Given target vector $\mathbb{t}$, matrix $\mathbb{A}$ and limits $\mathbb{b}$
Minimize $\sqrt{\sum _{i=0}^m \left(\mathbf{x}_i-\mathbf{t}_i\... | See for instance:
Philip Wolfe, The Simplex Method for Quadratic Programming,
Econometrica, Vol. 27, No. 3, (Jul., 1959), pp. 382-398
This is an extension to the Simplex method for a standard Quadratic Programming (QP) problem, so a slightly more general problem than you are stating.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1883024",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Question about Symmetric groups
Given permuations $$
\sigma=
\begin{pmatrix}
\text{1 2 3 4} \\
\text{2 3 4 1}\\
\end{pmatrix}
\qquad\tau=
\begin{pmatrix}
\text{1 2 3 4} \\
\text{4 3 2 1}\\
\end{pmatrix},
$$
show that the subgroup $D_8:=\langle\tau,\sigma\rangle$ of $\operatorname{Sym}(4)$ has order 8 and w... | Note $\sigma$ and $\sigma^2\sigma^3$ are the same since $\sigma^4$ is the identity permutation.
Anyway, if you label the points of a unit square $1,2,3,4$ counterclockwise, then $\sigma$ is a right angle rotation and $\tau$ is a reflection. Evidently $\sigma\tau=\tau\sigma^{-1}$ (this rule defines all dihedral groups $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1883115",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Geodesics on a sphere lie on a plane I have a question concerning closed geodesics on a sphere. I know that non-constant closed geodesics on a sphere are great circles. Hence if choosing one the image lies on a plane. This is easily seen if we know that closed geodesics are great circle. But can one show this without k... | It suffices to show that $\dot{\gamma}$ is normal to a constant vector $n$. Then it has to move in a plane orthogonal to $n$. The vector $n=\gamma\times \dot{\gamma}$ is normal to $\dot{\gamma}$ and $\dot{n}=\gamma\times \ddot{\gamma}= 0$ by the geodesic equation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1883223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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number of elements in subset sum problem I have a set of numbers $i=1,...,100$.
How many combinations exist using numbers from this set that sum to 100 of length 8?
So for example these would be valid solutions:
$(1, 2, 3, 4, 5, 6, 7, 72)$,
$(10,11,22,1,5,8,9,34)$
also, the order is important, that means
$(1, 2, 3, 4, ... | If $(100,0,0,0,0,0,0,0) \neq (0,100,0,0,0,0,0,0)$, for instance, you're looking for the number of solutions $(x_1, x_2, ..., x_8)$ to the equation $x_1 + ... + x_8 = 100$, where each $x_i$ is a non-negative integer.
The solution is given by a stars-and-bars argument. In your particular case, the answer is
$$ \binom{10... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1883315",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Arithmetical function : How can I prove? How can I show that this sum $\sum_{d|n} \mu(d) \log^kd$ is $0$ where $\mu(d)$ is mobius function.
I've expect that this question is solved by induction..!
$k$ is integer that is a power of $\log$
| For $k=1$ the identity should be the following
$$S_1(n)=\sum_{d|n}\mu(d)\ln(d)=\begin{cases}
0& \mbox{if $n$ is not a power of a prime},\\
-\ln(p)&\mbox{if $n$ is a power of a prime $p$.}
\end{cases}$$
Let $n=p_1^{a_1}\cdots p_r^{a_r}$ with $r>0$ then
$$S_1(n)=-\sum_{1\leq i_1\leq r}\ln(p_{i_1})+\sum_{1\leq i_1<i_2\leq... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A horrid-looking integral $\int_{0}^{5} \frac{\pi(1+\frac{1}{2+\sqrt{x}} )}{\sqrt{10}\sqrt{\sqrt{x}+x}} $
$$
\mathbf{\mbox{Evaluate:}}\qquad
\int_{0}^{5} \frac{\pi(1+\frac{1}{2\sqrt{x}} )}{\sqrt{10}\sqrt{\sqrt{x}+x}}
\,\,\mathrm{d}x
$$
This is a very ugly integral, but appears to have a very simple closed form of: ... | $u=\sqrt{x}$, we have
$$
\int_{0}^{5} \frac{\pi(1+\frac{1}{2\sqrt{x}} )}{\sqrt{10}\sqrt{\sqrt{x}+x}}
\,\,\mathrm{d}x=\frac{\pi}{\sqrt{10}}\int_{0}^{\sqrt{5}} \frac{2u+1} {\sqrt{u+u^2}}du=\frac{2\pi}{\sqrt{10}}\sqrt{u^2+u}\Big{|}_{0}^{\sqrt{5}}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1883532",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
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How do I find out particular solution for my differential equation? How do I find out particular solution for my differential equation?
For example $\ddot{x}+4\dot{x}+4x=t+1+\sin t$.
Can someone explain me why is particular solution here $x_p(t)=At+B+C\sin t+D\cos t$?
| Use Laplace transform:
$$x''(t)+4x'(t)+4x(t)=1+t+\sin(t)\Longleftrightarrow$$
$$\mathcal{L}_t\left[x''(t)+4x'(t)+4x(t)\right]_{(s)}=\mathcal{L}_t\left[1+t+\sin(t)\right]_{(s)}\Longleftrightarrow$$
$$\mathcal{L}_t\left[x''(t)\right]_{(s)}+\mathcal{L}_t\left[4x'(t)\right]_{(s)}+\mathcal{L}_t\left[4x(t)\right]_{(s)}=\math... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1883637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Correction factor for Hyperbolic Curve I have generated several data sets under varying experimental conditions, that are plotted as hyperbolic curves. I have two experiments that were done under identical conditions, but the curve is not the same. I'll call experiment A the "ideal". The equation for this line is: y=(4... | First, to answer your question of how you shift a function like the one you've written, think of the function as a general function $f(x)$. You can shift this function to the right by writing a new function $f(y)$ where $y=x+a$. For your function this substitution is on the variable $S$. If $a>0$ the shift is to the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1883719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Let $A_1, A_2,\dots$ be a sequence of disjoint, finite subsets of $\mathbb{N}$. How can $\bigcup_{n=1}^\infty A_n$ be either finite or infinite? Let $A_n$ be finite subsets of $\mathbb{N}$ that are not $\emptyset$, and $\forall i,j, i\not = j$, $A_i, A_j$ are disjoint, then must
$$\bigcup_{n=1}^\infty A_n = \mathbb{N}... | It must be countable (i.e. in bijection with $\Bbb N$), but not necessarily $=\Bbb N$.
Let $f: \Bbb N \to \bigcup_n A_n$ be given by $f(n) = \min A_n$. It is clear that $f$ is injective, so $\bigcup_n A_n$ is at least as large as $\Bbb N$. Now since the countable union of finite sets is at most countable, it follows t... | {
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Are two compact Hausdorff spaces homeomorphic if their algebras of continuous functions are isomorphic? Suppose that $X$ and $Y$ are two compact Hausdorff spaces and $F\colon C(X) → C(Y)$ is a continuous isomorphism of algebras. Can I say $X$ and $Y$ are homeomorphic?
The key words always lead me to other questions. Ca... | This is the so-called Gelfand-Kolmogorov theorem. It says:
Let $X$ and $Y$ be compact, Hausdorff spaces. Suppose that there exists a ring isomorphism $T\colon C(X)\to C(Y)$. Then there exists a homeomorphism $h\colon Y\to X$ such that $$Tf = f\circ h\text{ for all }f\in C(X).$$ In particular, $T$ is a continuous algeb... | {
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"source": "stackexchange",
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When to use the modulus symbol and when not to use the modulus symbol in integration and differentiation? I am facing this conceptual doubt for quite some time now.
We know $$\frac{d}{dx}{(\sec^{-1}{x})}=\frac{1}{|x|\sqrt{x^2-1}}$$ whereas $$\frac{d}{dx}{(\csc^{-1}{x})}=\frac{-1}{|x|\sqrt{x^2-1}}$$
Now suppose I need t... | The functions $sec^{-1}$ and $-cosec^{-1}$ only differ by a constant in the points where they are both defined. The same is true for $sin^{-1}$ and $-cos^{-1}$. So, when they are both defined, you can choose one or another as you please, as long as you add an additive constant.
This happens because of the formula $\co... | {
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How can I prove: $\sum_{n\leq x}\frac{\phi(n)}{n^2} = \frac{\log x}{\zeta(2)}+\frac{C}{\zeta(2)} + A + O\left(\frac{\log x}{x}\right)$ The problem is that prove that $$\sum_{n\leq x}\frac{\phi(n)}{n^2} = \frac{\log x}{\zeta(2)}+\frac{C}{\zeta(2)} + A + O\left(\frac{\log x}{x}\right)$$
where $C$ is Euler's constant and... | Let we implement the approach suggested by Winther in the comments, with a minor variation.
From
$$ \sum_{n\leq x}\varphi(n) = \frac{x^2}{2\zeta(2)}+O(x\log x) \tag{1}$$
and Abel's summation formula we get:
$$ \sum_{n\geq x}\frac{\varphi(n)}{n^2}=\frac{1}{2\zeta(2)}+O\left(\frac{\log x}{x}\right)+2\int_{1}^{x}\left( \f... | {
"language": "en",
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Integration of trigonometric function $\int\frac{\sin(2x)}{\sin(x)-\cos(x)}dx$
$$\int\frac{\sin(2x)}{\sin(x)-\cos(x)}dx$$
My attempt: Firstly, $\sin(2x)=2\sin(x)\cos(x)$.
After that, eliminate the $\cos(x)$ seen in both the numerator and denominator to get
$$2\int\frac{\sin(x)}{\tan(x)-1}\ dx.$$
From here onwards, sh... | HINT:
$$\int\frac{\sin(2x)}{\sin(x)-\cos(x)}\space\text{d}x=$$
Use $\sin(2x)=2\sin(x)\cos(x)$:
$$2\int\frac{\sin(x)\cos(x)}{\sin(x)-\cos(x)}\space\text{d}x=$$
Sustitute $u=\tan\left(\frac{x}{2}\right)$ and $\text{d}u=\frac{x\sec^2\left(\frac{x}{2}\right)}{2}\space\text{d}x$:
$$-8\int\frac{u(u^2-1)}{(u^2+1)^2(u^2+2u... | {
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Does $2764976 + 3734045\,\sqrt[3]{7} -2707603\,\sqrt[3]{7^2} = 0$? In the process of my numerical computations I have found a very special identity:
*
*$\;\;1264483 + 1707789 \,\sqrt[3]{7} - 1238313\,\sqrt[3]{7^2} = 9.313225746154785 \times 10^{-10}$
*$
-1500493 - 2026256\,\sqrt[3]{7} + 1469290\,\sqrt[3]{7^2}
= 9.... | No. Mathematica tells us that its value is about $-2.0876013027695663896 \times 10^{-9}$.
$$1264483 + 1707789 \times 7^{1/3} - 1238313 \times 7^{2/3} = -3.1767789172657775703*10^{-10}$$ according to Mathematica.
$$-1500493 - 2026256 \times 7^{1/3} + 1469290 \times 7^{2/3} = 1.7699234110429886326 \times 10^{-9}$$
simila... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Help calculating two integrals-- generalized and definite
The first one is to calculate
$$\int_{-\infty}^{\infty} \frac{1}{(1+4x^2)^2} dx.$$
I think this one should be solvable with the method of substitution but I tried using $t=4x^2$ which didn't work well, or where I have miscalculated something.
The second one i... | The first integral can also be evaluated using Glaisher's theorem, which says that if:
$$f(x)=\sum_{n=0}^{\infty}(-1)^n c_n x^{2n}$$
then we have:
$$\int_{0}^{\infty}f(x) dx = \frac{\pi}{2}c_{-\frac{1}{2}}$$
if the integral converges and where an appropriate analytic continuation of the series expansion coefficients ha... | {
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Computing $\int \frac{\ln(x)}{(1+x)^2}dx$ I am trying to solve the following integral
$$\int \frac{\ln(x)}{(1+x)^2}dx.$$
I have found a few similar questions on stackexchange but usually involving boundaries, so the answers have involved methods that don't seem applicable when the integral is unbounded.
I have tried u... | Partial integration actually works:
$$\int \frac{\log(x)}{(1+x)^2}dx=-\frac{\log(x)}{(1+x)}+\int\frac{1}{x(1+x)}dx$$
and this last one you can easily solve by writing
$$\frac{1}{x(1+x)}=\frac1{x}-\frac{1}{1+x},$$
thus one obtains
$$\int \frac{\log(x)}{(1+x)^2}dx=-\frac{\log(x)}{(1+x)}+\log(x)-\log(1+x)+C$$
| {
"language": "en",
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support on a triangle and uniform marginal distributions Suppose $X$ and $Y$ are jointly distributed on the support $\operatorname{conv} \{(0,0),(0,1),(1,0)\}$ with the joint PDF $f>0$ everywhere on the support. Is it possible to find $f$ such that the marginal PDFs are given by
$f_X(x)= 1$ for all $x\in[0,1]$ and
$f_Y... | You cannot have an everywhere positive density on the triangle support with the given marginal distributions. To see this, suppse such a distribution exists, and then consider the small triangle to the right of $x=1/2$. The x marginal distribution implies that the probability that x is greater than 1/2 is 1/2. Simil... | {
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Counting strings, need to under stand how this worked Question
Let $N_4 = \{1, 2, ..., 4\}$.
Calculate the number of strings on the set $N_4$ that are of length $8$.
Calculate the number of strings on the set $N_4$ that are of length $8$ and contain exactly five ones.
Answer
For the first part of the question, for eac... | Placing the ones: You have 8 positions to place the first '1', after that 7 positions remain available for the second '1', and so on till the fifth '1' (4 positions left for it). In total you would have 8×7×6×5×4 options; but, because the '1's are indistinguishable from each other, for each placement you are counting t... | {
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concept of canonical forms in PDEs - general question why the concept of deriving the canonical forms in PDEs is useful?
is it because it gives a way to solve some problems in a numerical way which otherwise would be difficult to solve using geometry?
| "The concept of deriving the canonical forms..." I think what is meant is, why is it useful manipulating a general 2nd order PDE into one of the canonical forms useful? To answer my own question, because we can easily solve these canonical form PDE's analytically (at least on reasonable domains). This then allows us ... | {
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If $f(x+1)+f(x-1)=\sqrt 3 f(x), \forall x$ then $f$ is periodic.
If $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$f(x+1)+f(x-1)=\sqrt 3 f(x), \forall x$ then $f$ is periodic.
I tried to replace $x$ by $x+1, x-1$ in the equality,to get something like $f(x + k)=f(x)$ but without success.
Any help is appreciated.... | Fix some $x \in \Bbb R$.
Define $a_n = f(x+n)$, for $n \in \Bbb N$.
Note that $a_n$ satisfies the recurrence relation $a_{n+1} = \sqrt{3} a_n - a_{n-1}$. By the general theory of recurrence relations it follows that $a_n=c_1 r_1^n + c_2 r_2^n$ where $r_1,r_2$ are the roots of the quadratic equation $r^2-\sqrt{3}r+1=0$.... | {
"language": "en",
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Skew lines and what's between them Is it always possible to find a line perpendicular to two skew lines in space?
And how can we visualise the proof geometrically? And if anyone could present the proof that it is always possible to exist a line perpendicular to both skew lines, please elaborate.
| Think about how you would find the shortest distance between the two lines. You know that the shortest distance from a line to a point not on the line is along a segment that’s perpendicular to the line. By symmetry, this means that the shortest distance between two skew lines must be along a segment that’s perpendicul... | {
"language": "en",
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$G$ is an abelian group of order $n$ with the property that $G$ has at most $d$ elements of order $d$, for any $d$ dividing $n$. Then $G$ is cyclic. $G$ is an abelian group of order $n$ with the property that $G$ has at most $d$ elements of order $d$, for any $d$ dividing $n$. Then $G$ is cyclic.
I am not getting any c... | First, decompose $G=C_1\times\cdots\times C_n$, via the Fundamental theorem of finitely generated abelian groups, where the $C_n$ are cyclic. Let's show that the orders of the $C_i$ are pairwise coprime. Let's start, say, with $C_1$ and $C_2$.
Suppose $d$ divides both $|C_1|$ and $|C_2|$. Then $C_1$ has an element $\al... | {
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What is the laymen meaning of "sampling from a binomial distribution"? I know what is binomial distribution but I am not able to realize sampling associated with it. Further a formal statement and applications of sampling from a binomial distribution would be of great help.
Thanks.
| A Binomial Distribution is that of the count of successes among a certain amount of success/failure trials;† each with an identical and independent rate of success.
So if you have a known amount of trials, $n$, all with the same probability of success, $p$, and each is independent of every other, then the count of succ... | {
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Check convergence and find the sum $\sum_{n=1}^{\infty} \frac{1}{9n^2+3n-2}$
$$\sum_{n=1}^{\infty} \frac{1}{9n^2+3n-2}$$
I have starting an overview about series, the book starts with geometric series and emphasizing that for each series there is a corresponding infinite sequence.
For convergence I can look at the pa... | It is a telescopic series:
$$\sum_{n=1}^{N} \frac{1}{9n^2+3n-2}=\sum_{n=1}^{N}\frac{1}{(3n+2)(3n-1)}=\sum_{n=1}^{N}\left(\frac{1/3}{3n-1}-\frac{1/3}{3n+2}\right)\\=\sum_{n=1}^{N}\frac{1/3}{3n-1}-\sum_{m=2}^{N+1}\frac{1/3}{3m-1}=\frac{1/3}{3\cdot 1-1}-\frac{1/3}{3(N+1)-1}\to \frac{1}{6}\quad \mbox{as $N\to+\infty$}.$$
| {
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What is the nth term of the series? $0, 0 , 1 , 3 , 6 , 10 ....$ I am trying to find the relation between the number of nodes and the number of connections possible. So if there are $0$ nodes, that means $0$ connections possible, $1$ node still means $0$ connections possible, $2$ nodes $1$ connection possible, $3$ node... | Hint: If you have graph on $n$ nodes, an edge in the graph corresponds to a choice of $2$ of these nodes (the $2$ which the edge connects). Hence the number of possible edges in the graph is the number of ways you can choose $2$ nodes. Can you continue from here?
| {
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Geometry: Show that two lines are perpendicular I have a homework problem telling me the following;
$ A, B, C, $ and $ D $ are points on a circle. $A_1, B_1, C_1$ and $ D_1$ are midpoints to the arcs $AB, BC, CD$ and $DA$. Show that $A_1C_1$ is perpendicular to $B_1D_1$.
Here's what I drew real quick with Geogebra:
c... | Call the intersection of $A_1C_1$ and $B_1D_1$ point $P$.
$$\measuredangle A_1PB_1=\frac{\measuredangle A_1MB_1+\measuredangle C_1MD_1}{2}$$
We can prove that $\measuredangle A_1MB_1+\measuredangle C_1MD_1=180^{\circ}$ using the following fact:
$$\measuredangle A_1MB_1+\measuredangle C_1MD_1=\frac{\measuredangle A_1M... | {
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Is it possible that the fraction of variance unexplained becomes greater than 1? I read this definition of fraction of variance unexplained from https://en.wikipedia.org/wiki/Fraction_of_variance_unexplained:
$$FVU = \frac{VAR_\text{err}}{VAR_\text{tot}}$$
Support the real data is $x_i$ and the regression estimate is $... | Lets assume a simple model, $Y=\beta_0 + \beta_1X+\epsilon$. The absolute value of Pearson's correlation coefficient is bounded by $1$,
$$
|\rho_{X,Y}| = \left|\frac{cov(X,Y)}{\sigma_X \sigma_X}\right| \le 1
$$
you can easily show it by using Cauchy-Schwartz inequality. To estimate $\rho$ you use the sample equivalent... | {
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Application of 'Total variation of a complex measure is finite.' I have a 'Real and Complex analysis' by Rudin.
And theorem 6.4 says 'Total variation of a complex measure is finite.'
Q: What is an application of this property?
I've read all the exercise in the book and I couldn't find any application of the theorem.
T... | I don't know of any exercises that make use of this result, but I can tell you of a point in the text where he makes crucial use of it: the proof of the theorem of the Brothers Riesz (17.13).
Briefly, to show that the measure is absolutely continuous (with respect to normalized Lebesgue measure on $\mathbb T$), he obt... | {
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0/0 in limit evaluation When evaluating limits we often come across the form 0/0 and we cant get a limiting value. As stated in many textbooks and online resources, the method to still get a limiting value as the input approaches a specific value is to first factorize or simplify the function to cancel out the factors/... | The main thing is:
Taking the limit does not mean evaluating the expression at the point, but rather infinitely close to that point (i.e. approaching the point).
Simplifying an expression does change the behaviour at the point (for example getting a value where it previously was indeterminate because of division by 0... | {
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How to show that $\mu$ is $\sigma$-finite This is part of a problem from an old prelim exam in analysis. I am studying to prepare for my own prelim.
Let $\{q_n\}=\mathbb Q$, and let $f_n : \mathbb R \to [0,\infty)$ be a Borel measurable function with $\int f_n d\lambda=1$ and with support $[q_n-2^{-n-1},q_n+2^{-n-1}]$.... | If you've shown $f = \sum f_n <\infty$ on a set $A$ with $\lambda (\mathbb R\setminus A)=0,$ we can do this: For $j,k=1,2, \dots,$ let $B_{jk} = \{x\in [-j,j]\cap A: f(x)\le k\}.$ Then $A\cap [-j,j] = \cup_k B_{jk},$ and $\mu(B_{jk}) = \int_{B_{jk}} f \,d\lambda\le k\cdot 2j<\infty$ for all $j,k.$ We then have $A= \cup... | {
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Simplifying the product of multiple binomial expansions I have a tricky product I'm trying to expand out into a summation, and I'm not sure how to go forward.
I have two sets of numbers, each containing $n$ elements total. The first set, $\{x_k\}$ are real and positive:
$$ x_k \in \mathbb{R}, \; x_k > 0 \; \forall \; k... | Using the same index of summation $m$ in all the sums might be a bit confusing, as these different indices could all take different values in the product. It therefore might be better to put subscripts on those indices, so we would have
$$ f(c) = \prod_{k=1}^n \left( \sum_{m_k = 0}^{y_k} \binom{y_k}{m_k} c^{m_k} x_k^{... | {
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Why aren't vacuous truths just undefined? I am struggling to understand this. According to truth tables, if $P$ is false, it doesn't matter whether $Q$ is true or not: Either way, $P \implies Q$ is true.
Usually when I see examples of this people make up some crazy premise for $P$ as a way of showing that $Q$ can be tr... | Consider the statement:
All multiples of 4 are even.
You would say that statement is true, right?
So let's formulate that in formal logic language:
$\forall x: 4|x \implies 2|x$
(Here "$a|b$" means "$a$ divides $b$", that is, $b$ is a multiple of $a$.)
Now a $\forall$ statement is true if it is true whatever you in... | {
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How does $\left\{f\mid f\colon\mathbb{N}\to\mathbb{R}\right\}$ mean 'the set of all real-valued functions of one natural number variable'? On pg. 83 of Hefferon's Linear Algebra, it says this:
The set $\left \{ f\mid f\colon\mathbb{N}\rightarrow \mathbb{R} \right \}$ of all real-valued functions of one natural number... | If we read it out loud we would say:
The set of functions $f$ such that the domain of $f$ is $\mathbb N$ and the co-domain of $f$ is $\mathbb R$
| {
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"url": "https://math.stackexchange.com/questions/1886415",
"timestamp": "2023-03-29T00:00:00",
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How does one find the integrals of piecewise functions? Of course, I am interested in finding out about how one would find the result of a piecewise function if I simply integrated it.
In other words, how does one integrate a function like:
$f(x)=\{(x^2: x\le0),(x:x>0)\}$
or
$g(x) = sgn(\sin x)$
where $sgn(x)$ is the s... | The first thing to recognize is that "the" antiderivative is a misnomer, because if it exists, it is not unique: we can add any constant and the result will be another antiderivative. In particular, assuming $f$ is continuous, any integral of the form
$$\int_c^x f(t) dt$$
is an antiderivative of $f$. (This is one of t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Examples of problems that are easier in the infinite case than in the finite case. I am looking for examples of problems that are easier in the infinite case than in the finite case. I really can't think of any good ones for now, but I'll be sure to add some when I do.
| A Markov chain is characterized by a discrete probability distribution of the initial state of a system, 0, and a transition matrix P such that pij is the probability of going from state i to state j after any one transition. Under certain assumptions, the probability distribution of states after the first transition i... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "119",
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All real values of $m$ for which $x^2-(m-3)x+m>0\forall x\in \left[1,2\right]$
All real values of $m$ for which $x^2-(m-3)x+m>0\forall x\in \left[1,2\right]$
$\bf{My\; Try::}$ We can write it as $$x^2-(m-3)x+\left(\frac{m-3}{2}\right)^2+m-\left(\frac{m-3}{2}\right)^2>0$$
So $$\left[x-\left(\frac{m-3}{2}\right)\right... | I am sorry but I don't agree with your curves @Roman83.
Here is how some of them look:
for values of parameter $m=0,1,\cdots 10$. If $m=0$ we get the parabola passing through the origin ; if $m=10$ we get the parabola which passes through point $(0,10)$, with increasing order between all of them.
Let us show how it ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A possible Property of Euler's totient function: $n$ such that $\varphi(n)$ and $\varphi(n+1)$ are both powers of two $n$ is an odd positive integer such that $\varphi(n)$ and $\varphi(n+1)$ are both powers of two . Here , $\varphi(n)$ denotes Euler's totient function.
Is it true that $(n+1)$ is either $6$ or a power... | This is not an answer, but might be useful when someone smarter than me tries to prove this.
Write $n=\prod_{i}p_i^{n_i}$ and $n+1=\prod_{i}(p_i')^{m_i}$. Then $\phi(n)=\prod_{i} p_i^{n_i-1}(p_i-1)$ being a power of two implies that if $p_i\neq 2$, then $n_i=1$ and $(p_i-1)$ is a power of two. The same holds for $\phi(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1886835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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Properties of convergent sequences theorem help I am reviewing some material about sequences and I ran across the following theorem.
Theorem: Let $(x_{n})$ and $(y_{n})$ be two convergent sequences with limits $x$ and $y$, respectively. Then $(x_n + y_n)$ converges to $x+y$.
The proof that the text uses is as follows.
... | I would prove it in this way and this proof would stick to the definition of limit:
If $x_n$ and $y_n$ are convergent, let $\lim_{n->\infty} x_n=L_1$ and $\lim_{n->\infty} y_n=L_2$.
Let $\epsilon$ be any given positive number.
Then for the positive number $\frac{\epsilon}{2}$, we can find a positive integer $N_1$ suc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1886950",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A property of dual norm In the convex optimization textbook, page 475, Stepend Boyd defines the normalized steepest descent direction with repect to the norm $||.||$ as
$$ \Delta x_{nsd} = argmin \{ \bigtriangledown f(x)^T v \; | \; ||v|| \leq 1 \}$$
In the Appendix A of this book, the dual norm is defined as
$$ || \b... | HINT It boils down to understanding the dual norm. This dual norm has the nice property that
$$ |\bigtriangledown f(x)^Tv|\leq\|\bigtriangledown f(x)^T\|_*\|v\| $$
for any vector $v$. So when $\|v\|\leq1$ we have
$$ |\bigtriangledown f(x)^Tv|\leq\|\bigtriangledown f(x)^T\|_* $$
Then you need to figure out when equali... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$f$ is differentiable everywhere, does that imply $f'$ is bounded in small nieghborhoods The motivation of this question is when I am looking at Taylor's Remainder theorem, suppose $f$ is $n+1$ differentiable, then the remainder can be written as
$$R_{n+1}(x) = \frac{f^{(n+1)}(c)}{n!} x^{n+1} \quad \text{ where } c\in... | A counterexample to the question in the title is yielded by the function defined by $f(x)=x^2\sin(1/x^2)$ if $x\ne 0$ and $f(0)=0$. This is differentiable everywhere with $f'(0)=0$ (use the definition of the derivative). But if $m\in\mathbb{N}$, then $f'(m^{-1})=-2m\cos(m^2)+2\sin(m^2)/m$ and as $m\to\infty$, $\sin(m^2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Induced isomorphisms on cohomology of double differential complexes Let $K^{\bullet,\bullet}=\bigoplus_{p,q\ge0}K^{p,q}$ be a double differential complex, i.e. we have differential operators $$\cdots\stackrel{d}{\to} K^{p,q-1}\stackrel{d}{\to} K^{p,q}\stackrel{d}{\to} K^{p,q+1}\stackrel{d}{\to}\cdots $$ and $$\cdots\s... | My earlier answer gives a counterexample that ignores your condition that $K$ and $L$ live in the first quadrant. Given that condition, what you want is true:
Let $C^{\bullet,\bullet}$ be the mapping cone of $f$. Then the long exact cohomology sequence for $f^{p,\bullet}$ shows that the horizontal cohomology of
$C^{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1887529",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How would you show that the series $\sum_{n=1}^\infty \frac{(2n)!}{4^n (n!)^2}$ diverges? How would you show that the series $$\sum_{n=1}^\infty \frac{(2n)!}{4^n (n!)^2}$$ diverges? Wolfram Alpha says it diverges "by comparison", but I'd like to know to what you would compare it? I've tried some basic things to no avai... | After the great answer of Adam Hughes one likely needs no more...
However, I like the following comparision-test, at which I arrived after some fiddling.
We compare $\sum a_n $ with $\sum \frac1{2n}$ and show by induction, that always $a_n>\frac1{2n}$ and the series diverges by comparision with the harm... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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Permutations of words from original word. How many distinct 4-letter arrangements can be made with the letters in the word "PARALLEL"?
My approach:
Because we are only looking at how many different permutations there are and not the frequency at which these permutations exist, we can delete the repeated letters and lea... | There are a few cases:
*
*All $4$ letters distinct. There are $\binom{5}{4} = 5$ ways to pick the letters and $4!$ ways to order them.
*Two letters distinct, one letter repeated twice. Either two A's or two L's. If we have two A's, then we have $\binom{4}{2} = 6$ ways to pick the other letters. $4!$ ways to order t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Partial Derivatives Continuous does not guarantee Gradient Function continuous Just want to check my understanding that partial derivatives continuous does not mean that the gradient function $\nabla f$ is continuous. Is that correct?
E.g. $\frac{\partial f}{\partial x}$, $\frac{\partial f}{\partial y}$ continuous does... | We have that $\nabla f$ is continuous. Note that:
\begin{align*}
|| \nabla f(x,y) - \nabla f(x_0,y_0)||^2 = (\frac{\partial f}{\partial x}(x,y)- \frac{\partial f}{\partial x}(x_0,y_0))^2 + (\frac{\partial f}{\partial y}(x,y)- \frac{\partial f}{\partial y}(x_0,y_0))^2
\end{align*}
Thus continuity of the partial derivat... | {
"language": "en",
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Which of the following facts are true of a sequence satisfying $\lim a_n^{\frac{1}{n}}=1$?
Let $a_n$ be a sequence of non-negative numbers such that
$$\lim a_n^{\frac{1}{n}}=1$$
Which of the following are correct?
*
*$\sum a_n$ converges
*$\sum a_nx^n$ converges uniformly on $[-\frac{1}{2},\frac{1}{2}]$
*$\sum a_n... | First of all, way to go for your efforts. As far as I can see, your answers to the first three questions are correct. To refute the last one consider the sequence
$$\{a_n\}=\{1,1,2,1,3,1,4,1,5,1,6,1,....\}=\begin{cases}k,&n=2k-1\\{}\\1,&\text{otherwise}\end{cases}$$
Observe that
$$\left\{\frac{a_{n+1}}{a_n}\right\}=\le... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Proof of definite integral Let $f$ be a continuous function. Without taking an anti-derivative, prove that $$\lim_{a\rightarrow 0^{+}}\int_{0}^{a}f(t)dt=0$$
| There exists $c_a\in [0,a]$ such that $\int_0^af(t)dt=af(c_a)$. Since $f$ is continuous at $0$, there exists $c>0, M>0$, such that for every $x\in [-c,c]. |f(x)|<M$.
For $a\in [-c,c]$, $|\int_0^af(t)dt|\leq |af(c_a)|\leq |a|M$. This implies that $|lim_{a\rightarrow 0}\int_0^af(t)dt|\leq lim_{a\rightarrow 0}M|a|=0$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solution of $ydx-xdy+3x^2y^2e^{x^2}dx=0$ Find the solution of given differential equation:
$$ydx-xdy+3x^2y^2e^{x^2}dx=0$$
I am not able to solve this because of $e^{x^2}$. Could someone help me with this one?
| By dividing both side to $y^2$ we get $$ydx-xdy+3x^{ 2 }y^{ 2 }e^{ x^{ 2 } }dx=0\\ \frac { ydx-xdy }{ { y }^{ 2 } } +3x^{ 2 }e^{ x^{ 2 } }=0\\ d\left( \frac { x }{ y } \right) =3x^{ 2 }e^{ x^{ 2 } }\\ \int { d\left( \frac { x }{ y } \right) =\int { 3x^{ 2 }e^{ x^{ 2 } }dx } } =\frac { 3 }{ 2 } \int { x } d{ e }^{ { ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Inequality involving rearrangement: $ \sum_{i=1}^n |x_i - y_{\sigma(i)}| \ge \sum_{i=1}^n |x_i - y_i|. $ If $x_1 \ge x_2 \ge \cdots \ge x_n$ and $y_1 \ge y_2 \ge \cdots \ge y_n$ are real numbers, and $\sigma$ is any permutation, then
$$
\sum_{i=1}^n |x_i - y_{\sigma(i)}| \ge \sum_{i=1}^n |x_i - y_i|.
$$
This must be a ... | We can prove it in a similar way as the rearangement inequality. There are only finitely many possibilities for $\sigma$, so a minimum is achieved, pick $\sigma$ so that it has the least possible number of inversions among all the permutations that minimize the expression.
Suppose by way of contradiction there is $i<j$... | {
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Construct a vector orthogonal to a given vector in $\mathbb{R}^3$ without singularities I have a unit vector $v = (x_v, y_v, z_v)$ in $\mathbb{R}^3$ and I want to construct another vector $u$ which is orthogonal to $v$. The construction process should be a straight-forward formula without singularities (divisions by ze... |
There is no continuous map $f\colon S^2\to S^2$ such that $f(x)\perp x$ for all $x$.
This is a consequence of the hairy ball theorem.
| {
"language": "en",
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In the expression $3x$, what is the $3$? So this is quite a simple question. I KNOW I learnt this before, but can't for the life of me figure out or find anywhere that refers to the definition I'm looking for. Look at the expression
$$
3x
$$
In this expression, what is the $3$ in this context? The $3$ 'prefixes' the $x... | A "scalar" in contex of Algebra.
Could be also a "coefficient".
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1888965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Cases in which the limiting value of a function $f(x)$ (as $x\to c$) is not equal to $f(c)$? Can anyone state some cases such as mentioned in the title. I have tried to look for them on google and my textbook, but cant find any examples. And incidentally how can this fact be ever true? does it have something to with th... | There are artificial examples created by piecewise definitions. (Richard Feynman once expressed surprise that anybody thought such things are functions.) I'll try to give a more serious example: You are walking past a building with a conventional rectangular shape. It is on your left. The distance from you to the ri... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A functor which takes quasi-isomorphisms to isomorphisms factors through a triangulated functor? Let $\mathcal A$ be an abelian category. Let $\mathbb K(\mathcal A)$ be the homotopy category of complexes of $\mathcal A$. Now. consider a functor $F$ from $\mathbb K(\mathcal A)$ to $\mathbb E$, which is also a triangulat... | If you don't assume that $F$ is a triangulated functor then this may not be true.
For example, let $F:\mathbb{K}(\mathcal{A})\to\mathbb{K}(\mathcal{A})$ be the obvious functor sending a complex to its degree zero homology considered as a complex concentrated in degree zero. Then $F'$ doesn't commute with the translatio... | {
"language": "en",
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Triangular numbers and Wilson's Theorem To explain the process, I was working on some ProjectEuler.net problems and I had just figured out a formula to solve for triangle numbers efficiently. The next problem needed me to solve for factorials. At the time (before I found out there is not a known equation where you can ... | Your two statements are equivalent to saying the following:
$n! = -1 + (n+1)j$ for odd prime $n+1$ and integer $j$.
$n! = n + \frac{n(n+1)k}{2}$ for odd prime $n+1$ and integer $k$.
Let $m=j-1$ and $q = \frac{nk}{2}$ (which will be an integer since $n+1$ is prime so $n$ is an even number and will divide by the denomina... | {
"language": "en",
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Do we still have a category of sets if the inverse image has more than one element? I'm kinda lost at one example in Awodey's: Category Theory.
I am trying to check this example, for that, I made a simple example function:
With this, I mean that there is one function $f$ which associates each element of the codoma... | The main problem is that the composition of functions with at most two preimages is not necessarily a function with at most two preimages.
In this wanna-be category, there does exist an arrow $\{1,2,3,4\}\to\{1,2\}$ (in fact several) and an arrow $\{1,2\}\to\{1\}$, but no arrow from $\{1,2,3,4\}$ to $\{1\}$.
For the re... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1889524",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why in these matrices are $AB=BA$ not equal? What is the logic behind them? We know that in matrices AB=BA.Why in this Matrices $A=\begin{bmatrix} -1 & 3\\
2 & 0\end{bmatrix}$, $B=\begin{bmatrix} 1 & 2\\
-3 & -5\end{bmatrix}$ are not equal to $AB=BA$. WHY? This is matrix of order $2\times 2$ for both $A$ and $B$.
| Matrix multiplication is generally not commutative unless they're both equal or they're inverses (in which case you will obtain the identity).
| {
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"timestamp": "2023-03-29T00:00:00",
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Proving $S^3$ and $\mathbb{R}^3$ are not homeomorphic
Prove $S^3$ and $\mathbb{R}^3$ are not homeomorphic.
I've encountered this question on a PhD exam in topology. This is at a level where we are expected to understand cohomology already, so there are already a lot of obvious one line proofs I could give (e.g. they ... | $S^3$ is compact, while $\mathbb{R}^3$ is not. Since any continuous function $f:S^3\rightarrow \mathbb{R}^3$ maps compact subsets of $S^3$ to compact subsets of $\mathbb{R}^3$, it can't be surjective (or else $f(S^3)=\mathbb{R}^3$ is also compact).
| {
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"timestamp": "2023-03-29T00:00:00",
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How to visualize the one point compactification of a plane with finitely points removed and $\mathbb{R}_{\text{discrete}}$ I wish to visualize the one point compactification of $\mathbb{R}^2$ with finitely points removed and that of $\mathbb{R}_{\text{discrete}}$
For the first question, I can picture it with one point ... | Judging from your description, you’re probably not visualizing the one-point compactification of $\Bbb R^2\setminus\{p\}$ correctly. The one-point compactification of $\Bbb R^2\setminus\{p\}$ can be visualized as a horn torus, a torus with inner radius $0$. It’s what you get if you start with $S^2$ and identify the nor... | {
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Multiplying sparse matrices If I have two sparse matrices, $A$ and $B$. Let's say $A$ has $k$ non-zero entries and $B$ has $j$ non-zero entries. Let's assume all I know is the amount of non-zero entries each matrix has, I don't know where they are or what their value is. The dimensions of the matrices are known and are... | It cannot be k+j as it can be seen in the next counterexample:
We define k = 3 and j = 2 and create the next two matrices, with dimension that agree for the multiplication as you stated:
$A=\left( \begin{array}{ccc}
1 & 0 & \cdots & 0 \\
1 & 0 & \cdots & 0\\
1 & 0 & \cdots & 0 \\
0 & 0 & \cdots & 0 \\
\vdots & \vdots &... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1890015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Evaluate the reciprocal of the following infinite product I hae to evaluate the reciprocal of the following product to infinity
$$\frac{1 \cdot 3 \cdot 5 \cdot 7 \cdot 9 \cdot 11 \cdot 13 \cdot 15 \cdot 17 \cdot 19}{2 \cdot 2 \cdot 6 \cdot 6 \cdot 10 \cdot 10 \cdot 14 \cdot 14 \cdot 18 \cdot 18}\cdot\ldots $$
I am gues... | Note that $$1-\frac{1}{\left(4n-2\right)^{2}}=\frac{\left(4n-3\right)\left(4n-1\right)}{\left(4n-2\right)\left(4n-2\right)}
$$ and $$P=\prod_{n\geq1}\left(1-\frac{1}{\left(4n-2\right)^{2}}\right)^{-1}=\prod_{n\geq1}\frac{\left(4n-2\right)\left(4n-2\right)}{\left(4n-3\right)\left(4n-1\right)}
$$ $$=\prod_{n\geq0}\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1890127",
"timestamp": "2023-03-29T00:00:00",
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Perpendicular Medians Medians $\overline{AX}$ and $\overline{BY}$ of $\triangle ABC$ are perpendicular at point $G$. Prove that $AB = CG$.
In your diagram, $\angle AGB$ should appear to be a right angle.
I've drawn the diagram, but I don't have anything in mind.
|
Let $M -$ midpoint $AB$.
The median on the hypotenuse of a right triangle equals one-half the hypotenuse.
In triangle $AGB$ $\angle AGB=90^{\circ}$ then $GM=MA=MB=\frac12AB \Rightarrow AB=2GM$
The centroid divides each median into two segments, the segment joining the centroid to the vertex multiplied by two is equal ... | {
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"timestamp": "2023-03-29T00:00:00",
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Show that $\int_0^1 \frac{\ln^3(x)}{1+x} dx = \sum_{k=0}^{\infty}(-1)^{k+1} \frac6{(k+1)^4}$ Show that;
$$ \int_0^1 \frac{\ln^3(x)}{1+x} dx = \sum_{k=0}^{\infty}(-1)^{k+1} \frac6{(k+1)^4}$$
I arrived to the fact that
$$ \int_0^1 \frac{\ln^3(x)}{1+x} dx = \sum_{k=0}^{\infty}(-1)^k\int_0^1 x^k \ln^3(x)dx$$
But I am u... | Just integrate by parts:
\begin{align*}
\int_0^1 x^k \ln^3(x) \, dx &= \frac{x^{k + 1}}{k + 1} \ln^3 x \big|_0^1 - \int_0^1 \frac{x^{k + 1}}{k + 1} 3 \ln^2(x) \frac 1 x \, dx \\
&= -\frac{3}{k + 1} \int_0^1 x^k \ln^2(x) \, dx
\end{align*}
The next application reverses the sign, picks up factor $2/(k + 1)$, and drops th... | {
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} |
Intuitively understanding Fatou's lemma I learnt Fatou's lemma a while ago. I am able to prove it and use it. I know examples showing that the inequality may be strict. But I don't really have an intuitive way to understand it. Any good thoughts?
| Fatou's lemma tells you that in the limit "mass" can only be lost but not generated. Let's recall the satement. If $f_n,f\geq 0$ are measurable and $f_n\to f$ pointwise a.e., then we have $\int f \leq \liminf_{n\to\infty} \int f_n$.
A classical example is $f_n= n \chi_{[0,1/n]}$ where $\int f_n=1$ for all $n$, but in t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1890542",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 3,
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Let $B_n = H_{n+1} \cap \dots \cap H_{n+ [\log_2\log_2 n]}$, $H_n$ probability $n$th coin heads. How are $\{B_n\}_n^\infty$ not independent? I think title is pretty self explanatory. Consider infinite fair coin tossing. Let $H_n$ be the the event that the $n^\text{th}$ coin comes up heads. Define
$$B_n = H_{n+1} \cap H... | I'll assume the $H_n$'s are independent with $0 < P(H_n) = p < 1$
You can show the $B_n$'s are not independent by showing that they are not pairwise independent.
$$B_n = H_{n+1} \cap H_{n+2} \cap \cdots \cap H_{n+ [\log_2\log_2 n]}$$
$$B_{n+1} = H_{n+2} \cap H_{n+3} \cap \cdots \cap H_{n+1+ [\log_2\log_2 (n+1)]}$$
$$P(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1890719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to approach this complex numbers question? I have started to approach this problem from writing the point on the perpendicular bisector as (z2+z3)/2 + ix(z2-z3), then was thinking of equaling the distance or something to find x. But I am not able to proceed, I am new to complex number, and don't know the real trick... | Since the circum circle is $|z|=1$, the orthocenter is given by $z_1+z_2+z_3$. If $z_4$ represents the point $P$, then
\begin{align*}
\frac{z_1+z_2+z_3 - z_1}{\overline{z_1+z_2+z_3} - \bar{z_1}} &= \frac{z_4-z_1}{\bar{z_4}-\bar{z_1}}\\
z_2z_3 &= -z_4z_1
\end{align*}
where we have used $\bar{z} = \frac{1}{z}$ when $|z|... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1890809",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
proving or refuting the convergence of a digital sequence Let the following digital sequence; $\Sigma_{n=2}^\infty \dfrac{sin(nx)}{log(n)}$
Dirichlet's criteria says that if $b_n$ decreases and $lim$ $b_n =0$ and if the partial sums of a sequence $a_n \in \mathbb{R}$ then $\Sigma_{n=1}^\infty a_n b_n$ converges.
*
... | In order to apply the Dirichlet's Test you still have show that the sequence $\{a_n\}_{n\geq 2}$ is bounded.
Now by the addition formula $\cos(x+y)=\cos x \cos y -\sin x \sin y$, for $x\not=2 m\pi$ with $m\in\mathbb{Z}$ (otherwise $a_n=0$), it is easy to obtain
$$\sin(kx)=\frac{\cos\bigl((k-\frac{1}{2})x\bigr)-\cos\bi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1890927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Is Rayo's number really that big? I was reading about large numbers, and came across Rayo's Number which is defined to be the smallest integer that is not nameable by any expression in the language of set theory that contains less than $10^{100}$ symbols.
Now, my question is: Is this number really that large?
If we pic... | No, a number like 10^(10^100) is much smaller than Rayo(10^100), because there are not a Googol digits, but a Googol symbols in first order set-theory, and it is pretty efficient to write down big numbers such as TREE(3), which can be expressed with MUCH LESS than a Googol symbols, you don't need anything like a Googol... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1891030",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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How to show $\log \cosh(\sqrt x)$ is concave? I know the definition of convex and concave functions and the second order condition to justify convexity (concavity). But still, I do not know how to show $\log \cosh(\sqrt x)$ is concave.
Thanks for your help.
| For $x>0$ the second derivative is:
$$\frac{\text{sech}^2\left(\sqrt{x}\right)}{4x}-\frac{\tanh \left(\sqrt{x}\right)}{4x^{3/2}}=\frac{1}{4x\cosh(\sqrt{x})} \left(\frac{1}{\cosh(\sqrt{x})}-\frac{\sinh \left(\sqrt{x}\right)}{\sqrt{x}}\right)$$
Hence the second derivative is negative (and our function is concave) a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1891131",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving that $\sin(54°)\sin(66°) = \sin(48°)\sin(96°)$ I'm trying to prove that $\sin(54°)\sin(66°) = \sin(48°)\sin(96°)$ but I don't really have a way to approach it. Most of what I tried was replacing $\sin(2x)$ with $2\sin(x)\cos(x)$ or changing sines with cosines but none of that has really simplified it.
Would app... | As $\sin54^\circ=\cos36^\circ$ and $$\sin48^\circ=2\cos24^\circ\sin24^\circ$$
the proposition reduces to $$\cos36^\circ=2\sin24^\circ\sin96^\circ=\cos(96-24)^\circ-\cos(96+24)^\circ=\cos72^\circ-\left(-\dfrac12\right)$$
Now use How do we prove $\cos(\pi/5) - \cos(2\pi/5) = 0.5$ ?.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1891212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Number of bit strings of length 8 that do not contain "$100$"? I am thinking the total number of possible strings is $2^8$ and the number of strings with $100$ at the beginning would be $2^8 - 2^3 = 2^5$. Now "$100$" can shift across the string $5$ times going to the right. Is the answer then $2^8 - 2^5 \times 5$?
| As discussed in the comments, the straight forward approach as proposed in the question won't work because it multiply counts the bad strings in which $100$ appears more than once (indeed, it counts bad strings once for each appearance of $100$).
For short strings (like length $8$) a more careful count via the princip... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1891318",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 0
} |
Solve $2\ddot{y}y - 3(\dot{y})^2 + 8x^2 = 0$ Solve differential equation
$$2\ddot{y}y - 3(\dot{y})^2 + 8x^2 = 0$$
I know that we have to use some smart substitution here, so that the equation becomes linear.
The only thing I came up with is a smart guessed particular solution: $y = x^2$. If we plug this function in, w... | Hint:
Let $y=\dfrac{1}{u^2}$ ,
Then $y'=-\dfrac{2u'}{u^3}$
$y''=\dfrac{6(u')^2}{u^4}-\dfrac{2u''}{u^3}$
$\therefore\dfrac{2}{u^2}\left(\dfrac{6(u')^2}{u^4}-\dfrac{2u''}{u^3}\right)-3\left(-\dfrac{2u'}{u^3}\right)^2+8x^2=0$
$\dfrac{12(u')^2}{u^6}-\dfrac{4u''}{u^5}-\dfrac{12(u')^2}{u^6}=-8x^2$
$\dfrac{4u''}{u^5}=8x^2$
$u... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1891382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
} |
Decompose rotation matrix to plane of rotation and angle I would like to decompose an $n$-dimensional orthogonal rotation matrix (restricting to simple rotation with a single plane of rotation) to the two basis vectors of the plane of rotation, and an angle of rotation.
The common method is decomposing the rotation mat... | This is the same answer as given by "arctic tern," but expressed differently.
If $R$ is an orthogonal rotation matrix (i.e. $R^{-1} = R^T$ and $\det(R) = 1$), then it can be diagonalized via a unitary similarity matrix. The eigenvalues have absolute value $1$, and come in conjugate pairs. And their product must be $1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1891466",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 1
} |
Triangle inequality in complex numbers let $|z|=R$, by using the triangle inequality, find a lower bound for $$|z^4+5z^2+4|$$
approachh: $$|z^4+5z^2+4|\geq|z^4|-|5z^2+4| \geq |z^4|-(|5z^2|+4)=|z^4|-|5z^2|-4=R^4-5R^2-4$$
but the solution is $$|z^4+5z^2+4| \geq R^4-5R^2+4$$
what went wrong?
| You get a sharper lower bound if you render
$z^4+5z^2+4=(z^2+1)(z^2+4)$
and find the lower bounds for the absolute values of the factors on the right side.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1891551",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Negative/Positive Index Numbers I came across a question that I was unable to solve, it involves positive and negative index numbers in the same fraction and I'm not sure how to solve that, if anyone could help me please??
$$\dfrac{\left(\frac{7a^5b^3}{5a^6b^2}\right)}{\left(\frac{7b^3a^2}{5b^5a^4}\right)}$$
Also some... | $$\dfrac{\left(\frac{7a^5b^3}{5a^6b^2}\right)}{\left(\frac{7b^3a^2}{5b^5a^4}\right)}=\frac{7a^5b^3}{5a^6b^2}\times\frac{5b^5a^4}{7b^3a^2}
=\frac{a^5b^3b^5a^4}{a^6b^2b^3a^2}=\frac{a^9b^8}{a^8b^5}=ab^3.$$
The division of a fraction is equivalent to the multiplication of its reciprocal, which is obtained by swapping the n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1891628",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Does there exist such a subgroup of a linear group? Could anybody give an example of the following case : $F$ is a field with nonzero characteristic $p$, $V$ is a $F$-vector space with finite dimension, $G$ is a subgroup of $GL(V)$, $G$ is infinite but has finite exponent and its (least) exponent is not divisible by $p... | This does not exist. With no loss of generality, you can suppose that the field is algebraically closed. Let $H$ be the Zariski closure of $G$ and $H_0$ its unit component. Then $H$ is a connected positive-dimensional algebraic group, of finite exponent. Since tori have infinite exponent, $H_0$ has to be unipotent. Hen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1891722",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Approximation using the Taylor series
$$\sqrt{1+2t\sigma\cos\theta+\frac12t^2\sigma^2(3+\cos(2\theta))}=1+t\sigma\cos\theta+\frac12t^2\sigma^2+O(t^3)$$
What is the omitted step between these two equations? The parameter $t$ lies between $0$ and $1$ and it is said that approximation by using Taylor series is used.
| Hint. One may use the Taylor series expansion, as $u \to 0$,
$$
\sqrt{1+u}=1+\frac u2-\frac{u^2}{8}+O(u^3)
$$ applying it to $u=2t\sigma \cos \theta+\dfrac 12t^2\sigma^2 (3+ \cos 2\theta) $ as $t \to 0$, observing that
$$
\frac u2-\frac{u^2}{8}=t\sigma \cos \theta+\dfrac 14t^2\sigma^2 (3+ \cos 2\theta)-\frac{4t^2\sigma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1891830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Transitive sets: problem in proof of Lemma I.8.6 of Kunen's 'Foundations of Mathematics' I've been studying Kunen's notes titled 'The Foundations of Mathematics'.
Definition I.8.1 in Kunen says
$z$ is a transitive set iff $\forall y \in z\, [y \subseteq z]$
In the proof of Lemma I.8.6, $\alpha$ is a transitive set an... | I agree with you that the end of the proof of Lemma I.8.6 in Kunen is misleading and incomplete. It can be fixed as below.
Let $\alpha$ be an ordinal, that is, $\alpha$ is a transitive set (every element of it is a subset of it) such that the relation $\in$ on $\alpha$ is a well-order (i.e., a (strict) total order for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1891943",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Can probability of getting a ball from a box be greater than $1$
A box has three black balls and three red balls. Therefore, probability of
getting a black ball is $\dfrac{1}{2}$ (so is the case for red ball
also)
Suppose one person takes a ball and put it back in the box. Then
second person takes a ball and put ... | In your first (wrong) approach you use the rule:
$\Pr(A\cup B\cup C)=\Pr(A)+\Pr(B)+\Pr(C)\tag1$
However, this rule can only be used if the events $A,B,C$ are mutually exclusive which is not the case here.
A rule that always works is:
$\Pr(A\cup B)=\Pr(A)+\Pr(B)-\Pr(A\cap B)\tag2$
$A,B$ are by definition mutually exclus... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1892020",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
If $f:X\to X$ is an epimorphism, which of these statements are true?
$X$ is a linear space, $\dim X = n < \infty$. $f:X\to X$ is epimorphism. Then:
a. $f$ is monomorphism.
b. there is exists base such that matrix of $f$ is diagonal.
c. there is exists base such that matrix of $f$ is symmetric.
a. is true, be... | For (a), your argument isn't really complete; I'd say you haven't fully justified how you know $A$ is a change-of-basis matrix.
But there's actually no need to talk about matrices at all, just use the rank-nullity theorem, the fact that $X$ is finite-dimensional, and the fact that
*
*$f$ is an epimorphism $\iff$ $\o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1892195",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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What applications require the study of nonhomogenous partial differential equations I've been working on a bit of abstract calculus which allows me to solve various PDEs in a somewhat novel fashion. It occurs to me, I can also solve nonhomogeneous PDEs by my method. For example,
$$ u_{xx}+u_{yy} = x^2-y^2$$
I could pro... | The first application I could think of would be geometric and/or optical modeling. Multivariate polynomials describe or approximate geometries and if you differentiate and put constraints on their level sets you can be getting (systems of) differential equations of the type you have there.
And if you can use it to desc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1892290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
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