Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Axiom of extensionality - why can't we replace $\Rightarrow$ with $\iff$? The axiom of extensionality says that
$$(\forall(x,y))\big(\forall(z)(z\in x\iff z\in y)\Rightarrow x=y\big)$$
Does it not work the other way around?
$$ (x=y)\Rightarrow\forall z(z\in x\iff z\in y) $$
If two sets are equal and $z$ is in one o... | The converse of extensionality is a result of the axioms
of a first order logic with equality. An alternative to
embedding ZF into a first order logic with equality is to
embed it into a first order logic and use the axiom of
extensionality as a definition of equality. Then the
"converse" would be an iteration of the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2471674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Question concerning the notation of path multiplication I'm self-studying topology by using Lee's Introduction to topological manifolds. I've just started reading the chapter on Homotopy and the Fundamental Group. Untill now everything makes perfect sense to me. The only thing bothering me is the notation for the path ... | One reason for the notation is the prevalence of graphical techniques used for gaining intuition, teaching or even as mathematical proof. For instance the following diagram represents the homotopy giving associativity for loop multiplication $(f\cdot g)\cdot h\simeq f\cdot (g\cdot h)$
In the diagram the left-most regi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2471800",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Additive group of integers modulo $6$ I am currently studying Galois Theory and am having trouble understanding group notation.
What does $$\mathbb{Z}/n\mathbb{Z}$$
mean? I understand that its an additive group of modulo $n$ but what would the elements of $$\mathbb{Z}/6\mathbb{Z}$$ be for example?
| The elements are the congruence classes modulo n. For example the elements of the additive group $Z_6$ are $0,1,2,3,4,5$, where $0$=${...-12,-6,0,6,12,...} $ is the set of all integers congruent (mod 6) to $0$ and so on.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2471945",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
Simplifying Quotient of Tensor Products Consider
$$(A\otimes C)/(B\otimes C)$$
where $B$ is a submodule of $A$. ($A,B,C$ are $R$-modules).
Is it true that $$(A\otimes C)/(B\otimes C)\cong(A/B)\otimes C$$?
Thanks. If no, are there any easy counter-examples?
| This is sort of true.
There is a natural map from $B\otimes C$ to $A\otimes C$, but in general
it is not injective, so that we cannot think of
$B\otimes C$ as a submodule of $A\otimes C$. But the image $I$ of
this map is a submodule of $A\otimes C$, and
$(A\otimes C)/I$ is isomorphic to $(A/B)\otimes C$.
What is going ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2472095",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
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"answer_id": 0
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Find a nonempty set $A$ such that $A\cap P(A)=\emptyset$
*
*Find a nonempty set $A$ such that $A\cap P(A)=\emptyset$. (as $P(A)$ is power set of$A$).
My solition. Let $A=${$\emptyset, 1$} be set. Then, $P(A)=${$\emptyset$, {$\emptyset$},{$1$},$A$}. Hence, $A\cap P(A)=\emptyset$. Can you check my solition?
| You need to find a set $A$, none of whose members are subsets of $A$. There's lots of those! The simplest possible example would be a set of one element; can you find one which works?
Another hint: the cardinality of a subset of a finite set is less than or equal to the cardinality of the set. So if $A$ is finite, all ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2472192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Algebraic Geometry: What am I doing wrong? This may be a very stupid question. But please explain what I am doing wrong.
Let $k$ be an algebraically closed field. Let $f\in k[x_1,\dots, x_n]$.
Let $$D(f)=\mathbb{A}^n\setminus Z(f)$$
Then $D(f)\subseteq \mathbb{A}^n$ is open.
We can consider $\mathbb{A}^n$ as a subavar... | The set $Z(fy-1)$ is homeomorphic to $D(f)$, but not equal! While $Z(fy-1)$ is closed in $\mathbb{A}^{n+1}$, $D(f)$ is not, and this is no contradiction since they are not actually the same set.
You may find it helpful to think about the following more familiar example. An open interval $(0,1)$ is homeomorphic to $\m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2472467",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
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Condition for the generic fiber to be dense This is an assertion in the article Groupes Reductifs sur un Corps Locale II by Bruhat and Tits. Here $A$ is an integral domain with field of fractions $K$, and $\mathfrak X$ is an affine $A$-scheme with coordinate ring $B := A[\mathfrak X]$.
I don't think this assertion th... | You are correct: this result is not true without assuming $B$ is reduced. For a simple example, take $A=\mathbb{Z}$ and $B=\mathbb{Z}[x]/(x^2,2x)$. Then $A$ is $B$ mod its nilradical, so the map $\operatorname{Spec} B\to \operatorname{Spec} A$ is a homeomorphism and so the generic fiber is dense in $\operatorname{Spe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2472626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How do we minimize $\left(x+a+b\right)\left(x+a-b\right)\left(x-a+b\right)\left(x-a-b\right)$? Find the minimum value of the following function, where a and b are real
numbers.
\begin{align} f(x)&=\left(x+a+b\right)\left(x+a-b\right)\left(x-a+b\right)\left(x-a-b\right) \end{align}
Note: The solution should not contain ... | By your work
$$f(x)=x^4-2(a^2+b^2)x^2+(a^2-b^2)^2$$ or
$$f(x)=(x^2-a^2-b^2)^2-(a^2+b^2)^2+(a^2-b^2)^2$$ or
$$f(x)=(x^2-a^2-b^2)^2-4a^2b^2.$$
Now, we see that
$$f(x)\geq-4a^2b^2$$ and the equality occurs for $x^2=a^2+b^2$, which is possible.
Thus, $$\min_{\mathbb R}f=-4a^2b^2.$$
Done!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2472757",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Open set in $\Bbb{R}^2$ Topology Consider $\Bbb{R}^2$ with the usual topology. Let $X$ be a subset of $\Bbb{R}^2$. If for every $a \in X$ and $v \in \Bbb R^2$ there exists a d>0, such that $a+vt\in X$, for every $0\leq t<d$, then X is open.
I suppose this theorem is wrong as the choice of the radius of the open ball $d... | For a single point $a$, you may be able to manufacture a sequence $d_n$ depending on $v_n$ such that $d_n$ converges to zero. For that point $a$ there would not be ball around a completely contained within the set because you can always find a smaller $d_n$ than the radius of that ball. The set would probably look sort... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2472843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solving $\dfrac{\partial{u}}{\partial{t}} + u = \dfrac{\partial^2{u}}{\partial{x}^2} + 4\dfrac{\partial{u}}{\partial{x}}$ Using Separation.
Proceeding as follows, use the method of separation of variables to solve
$\dfrac{\partial{u}}{\partial{t}} + u = \dfrac{\partial^2{u}}{\partial{x}^2} + 4\dfrac{\partial{u}}{\part... | Your solution is equivalent to the instructor's solution. Only the symbols chosen for the constants $\lambda$ are different, but related :
$$\lambda_{You}=\lambda_{Instructor}+1$$
It doesn't matter since $\lambda$ is arbitrary.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2472967",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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p-points in topology A point $x_\infty$ in a topological space $X$ is called a p-point if every continuous function on $X$ is constant in a neighborhood
of $x_\infty$.
For example if $X=X_0 \cup x_\infty$ where $X_0$ is an uncountable discrete space and the neighborhoods of $x_\infty$ are co-countable,
then $x... | $X=\omega_1 +1$ in the order topology has a non-isolated $p$-point in your sense (namely $\omega_1$) and is compact Hausdorff.
This is well-known: suppose $f: \omega+1 \to Y$ is continuous and $Y$ is first countable (having countable pseudocharacter will also do). Then let $p = f(\omega_1) \in Y$ and let $U_n$ be a co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2473094",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Positive eigenvectors for nonnegative matrices Let $ A $ a nonnegative matrix (i.e. all the entries of $ A $ are real, nonnegative) of dimensions $ n \times n $. Is it true that the conditions:
*
*$ A x_1 = \lambda_1 x_1 $
*$ A x_2 = \lambda_2 x_2 $
*$ x_1 \gg 0, x_2 \gg 0 $ (i.e. all the components of $ x_1 $ and... | Yes. This is because when $A\ge0$, its spectral radius $\rho(A)$ is the only eigenvalue (over $\mathbb C$) that can possibly possess a positive eigenvector. (That doesn't mean $\rho(A)$ always has a positive eigenvector --- consider e.g. $A=$ the $2\times2$ nontrivial nilpotent Jordan block. Nor does it mean that $\rho... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2473219",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Graph Theory: Tree ($n\geq 4$ nodes) with no vertex degree $2$. Prove there is at least one vertex w/ 2 or 3 leaves as neigbors T: Trees with no vertex of degree 2 have more leaves than internal nodes
So far I have (proof by contradiction).
Consider the opposite. That all nodes have only 1 leaf as a neighbor.
Take some... | I assume you're asking for help with proving such things, and for improving the proof you wrote, and that you're NOT asking for a proof of this particular statement.
As a general hint, when you want to ask a question, it's a good idea to make the question very explicit. A "question mark" can be a big help.
Anyhow, as... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2473337",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Total number of factors of a number Can you please explain how can we derive the total number of factors of a composite number using the concept of combinations ? Thanks in advance !
| A composite number $n$ can be written as $n=p_1^{a_1}\cdots p_k^{a_k}$. The number of factors of $n$ is the product of the number of factors of $p_i^{a_i}$ for each $i=1,\ldots,k$. (This is because it is a multiplicative function) It is easy to count the factors of $p^a$: they are simply the powers $p^0,p^1,\ldots,p^a$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2473458",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $(1-\frac x 1 + \frac {x^2} 2 -\cdots)^{-1} = A_0 + \frac{A_1x^2} {1!} + \frac {A_2x^2}{2!}+\cdots$ then $A_n \sim (-1)^{n-1}(n-1)!(\log n)^{-2}$ From G.Pólya "Mathematics and Plausible Reasoning" p.9. Problem 8:
Set $$\biggl(1-\frac x 1 + \frac {x^2} 2 -\frac {x^3} 3 +\cdots \biggr)^{-1} = A_0 + \frac{A_1x^2} {1!} ... | $$ f(z)=\frac{1}{1-\log(1+z)} $$
is an analytic function in a neighbourhood of the origin and the radius of convergence of the Taylor series at the origin equals one. By considering
$$ f(z) = \sum_{n\geq 0}a_n z^n = 1+z+\frac{z^2}{2}+\frac{z^3}{3}+\frac{z^4}{6}+\frac{7 z^5}{60}+\ldots$$
$$ f'(z) = \frac{f(z)^2}{z+1}\q... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2473573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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How to integrate this kind of functions? I was wondering how to evaluate $$\int\frac{sin^4 x}{cos^7 x}dx$$
I tried the usual method of writing the expression in terms of powers of $tan(x)$ and $sec(x)$, but nothing useful came out of it.
My attempt
$$\int\frac{sin^4 x}{cos^7 x}dx$$$$=\int({tan^4x}) ({sec^3x})dx$$$$=\in... | \begin{align*}
\int{\frac{sin^{4}x}{cos^{7}x}}\,dx &= \int{tan^{4}x \cdot sec^{3}x}\,dx\\
&= \int{(sec^{2}x - 1)^2 \cdot sec^3{x}}\,dx\\
&= \int{sec^{7}x}\,dx -2\int{sec^{5}x}\,dx + \int{sec^{3}x}\,dx
\end{align*}
Now,
\begin{align*}
I_{2n+1} &= \int{sec^{2n+1}x}\,dx = \int{sec^{2n-1}x\cdot \sec^{2}x}\,dx\\
&= sec^{2n-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2473676",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Group homology of the rationals Let $\mathbb{Q}$ be the group of rational numbers. How to compute group homology $H_n(\mathbb{Q},\mathbb{Z})=H_n(B\mathbb{Q},\mathbb{Z})$?
I know that $H_0(\mathbb{Q},\mathbb{Z})=\mathbb{Z}$ and $H_1(\mathbb{Q},\mathbb{Z})=\mathbb{Q}_{ab}=\mathbb{Q}$ and I think that $H_n(\mathbb{Q},\ma... | You can explicitly construct a model of $B\mathbb{Q}$ by taking the mapping telescope of the sequence $S^1\stackrel{1}\to S^1\stackrel{2}\to S^1\stackrel{3}\to S^1\stackrel{4}\to\dots$, where $\stackrel{n}\to$ denotes a degree $n$ map. Indeed, if $K$ is such a mapping telescope, we see that the homotopy groups are the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2473827",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Proving that $\frac{ab}{c^3}+\frac{bc}{a^3}+\frac{ca}{b^3}> \frac{1}{a}+\frac{1}{b}+\frac{1}{c}$
Prove that $\dfrac{ab}{c^3}+\dfrac{bc}{a^3}+\dfrac{ca}{b^3}> \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}$, where $a,b,c$ are different positive real numbers.
First, I tried to simplify the proof statement but I got an even mo... | AM-GM helps!
$$\sum_{cyc}\frac{ab}{c^3}=\frac{1}{4}\sum_{cyc}\left(\frac{2ab}{c^3}+\frac{bc}{a^3}+\frac{ca}{b^3}\right)\geq\frac{1}{4}\sum_{cyc}\left(4\sqrt[4]{\left(\frac{ab}{c^3}\right)^2\cdot\frac{bc}{a^3}\cdot\frac{ca}{b^3}}\right)=\sum_{cyc}\frac{1}{c}.$$
Done!
Without $cyc$ we can write the solution so:
$$\frac{a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2473942",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Visible Portion of the Earth's Surface
EDIT: I need help converting the right side to a function of h
Let $A_h$ be the area of the zone corresponding to height h. If we set up a rectangular co-ordinate syustem with the origin at the center A of the spherical Earth with radius R, and if the surface of the earth is obta... | Geometric Approach
In this answer, it is shown that the area of the green strip on the sphere is the same as the area of the red projection onto the cylinder circumscribing the sphere and sharing its axis with the green strip.
We can compute the height of the cap using similar triangles:
Thus, the area of the cap is
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2474288",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
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Suppose that $\sum_{j=1}^\infty a_j$ is conditionally convergent. Show that $\sum_{j=1}^\infty j^{\frac{1}{j}}a_j$ is also convergent.
Suppose that $\sum_{j=1}^\infty a_j$ is conditionally convergent. Use the partial sum formula to show that $$\sum_{j=1}^\infty j^{\frac{1}{j}}a_j$$ is also convergent.
Any hints on ho... | Let $b_j = j^{1/j}$. The sequence $b_j$ is bounded and decreasing (when $j > 2$) with $b_j \to 1$. The sequence of partial sums $S_n = \sum_{j=1}^n a_j$ is convergent and, hence, bounded.
We have convergence of the telescoping series
$$\sum_{j=1}^n (b_j - b_{j+1}) = b_1 - b_{n+1},$$
I'll leave it to you to show th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2474439",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Integral of $\sqrt{ax-x^2}$ Can we integrate a function of the form $$\sqrt{ax-x^2},\text{say for any limits a to b}$$
I was solving a question recently where I had to find the volume and the eqn of base was $$x^2+y^2=ax\implies\,(x-\frac{a}{2})^2+y^2=\frac{a^2}{4}$$
now although it is easy to integrate this when writi... | Yes. Try the following:
\begin{align*}
ax-x^{2} &=-(x^{2}-ax) \\
&= \frac{a^{2}}{4}-(x^{2}-ax+\frac{a^{2}}{4})\\
&= \frac{a^2}{4}-\left(x-\frac{a}{2}\right)^{2}
\end{align*}
Then try substituting $x-\frac{a}{2}=\frac{a}{2}\sin(\theta)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2474542",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Are solutions of the following equation countable : $ \frac{a\exp(ix)}x + \frac{b \exp(iy)}y = c $? I would like to prove that:
For given non zero complex numbers $a,b$ and $c$, the set of positive real numbers $x>0$, $y>0$ satisfying the equation:
$$
\frac{a\exp(ix)}x + \frac{b \exp(iy)}y = c
$$
is countable, where $... | This is a very partial answer attempting to cover all the cases, with a proven case and other cases at the state of conjectures without proofs.
First, have a look at the figures at the bottom ; it suffices to know at present that the common points to the two spirals are in correspondence with solutions. The first figur... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Solution for the inequality: $(x-1)(x-2)(x-3)>0$ Recently I came across an inequality like this:
$$(x-1)(x-2)(x-3)>0$$
The question was: Which solution for this in inequality is correct?
*
*$x>3$ or $x<1$
*$x>3$ or $1<x<2$
*$x<1$ or $2<x<3$
*There are no solutions
There was a timelimit of 3 minutes for solvi... | Draw an axle from left to right, and mark 1, 2, 3 on it. Put a point for x at 0. then move x from left to right.
When x < 1, x is at the left of 1, 2, and 3, therefore (x-1), (x-2), and (x-3) are all negative. We know answer (1) and (3) is not right.
When x > 3, x is at the right of 1, 2, and 3, therefore (x-1), (x-2)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2474802",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 4
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Notation for pointwise exponentiation of a function There exists notation for the pointwise multiplication of a function $j$ or $k$. This is often denoted as $j\cdot k$ or $j \circ k$ using the Hadamard notation. Consider the pointwise exponentiation of some function $f:X \rightarrow Y$ with exponent $n$ denoted by $g:... | For positive $n$, the notation $f^n$ usually means repeated composition as per Wikipedia: Exponential Notation for Function Names.
An exception to this rule is trigonometric and logarithmic functions where the exponent always means repeated multiplication. Thus
$$ \sin^2 x = (\sin x) (\sin x) $$
$$ \log^2 x = (\log x)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2474954",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that $\operatorname{rank}(B) = \operatorname{rank}(AB) + \operatorname{dim}\left(N(A) \cap C(B)\right)$. I am studying for midterm exam, and solving many problems on the textbook. I want to prove the following statement, but I failed to prove it. Please someone help me how to prove the following statement.
If $A... | I consider $A$ and $B$ as linear maps. Denote by $A|_{C(B)}$ the restriction of the linear map $A$ to the subspace $C(B)$. Then $\operatorname{Im}(A|_{C(B)}) = A(C(B)) = \operatorname{Im}(AB)$. Hence, $\operatorname{rank}(AB) = \operatorname{rank}(A|_{C(B)})$. By the rank-nullity theorem, applied to $A|_{C(B)}$, we get... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2475066",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Limit of Function series
Given that $$g(x):=\lim _{ n\rightarrow \infty }{ { (\cos(x)) }^{ 2n } } $$
Find
$$h(x)=\lim _{ m\rightarrow \infty }{ { g(x) }_{ m } } $$
Where
${ g(x) }_{ m }:=g(2\pi m!x)$.
I already computed $g(x)=0$
but I can't compute $h(x)$ the same way I did with $g(x)$.
Could someone g... | Remember that:
$$\lim_{n\to \infty} x^{2n}=
\begin{cases}
\infty & |x| > 1 \\
1 & |x| = 1 \\
0 & |x| < 1
\end{cases}
$$
In your case, $\cos x\le1$ . Note that $\cos(2\pi m!x)$ is a rather convoluted notation for the simpler $\cos(2\pi zx)$ where $z$ is an integer.
Now what is the value of $\cos(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2475158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving equivalence of norms in $\mathbb{R}^2$
Let $\lVert \cdot \rVert_*:\mathbb{R}^2 \to \mathbb{R}, (x,y) \rightarrow \sqrt{x^2+2xy+3y^2}$ be a norm.
How can I find to constants $k,K \in \mathbb{R}^{>0}$ so that the following equivalence is given:$$k\lVert (x,y) \rVert_2 \leq \Vert (x,y) \rVert_* \leq K\Vert (x,y) ... | Here's a version that's more explicitly geometric, but whose underlying mathematics resemble Roberto's matrix diagonalization.
Rewrite the norm in rotated coordinates $(x', y')$, where $x = x' \cos \theta - y' \sin \theta$ and $y = x' \sin \theta + y' \cos \theta$. We'll choose $\theta$ at our convenience—specifically,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2475271",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Essay on Probability Theory, Machine Learning and Stock Markets. Sorry to bother you, and I'm not entirely sure this is the correct place to be discussing this but I shall try to be brief.
I'm a complete rookie when it comes to anything stochastic/probability based - I only have an undergraduate course in measure theo... | I think I know what you are after, having crossed that bridge myself. Analyzing the stock market is an interesting thing of itself but tread carefully before you think you have a model that will try and predict the stock prices. Here is a large document that talks about modelling high frequency trades. That said, its a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2475353",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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How do I find the equation of a circle given the equation of $3$ tangents I would love some help on finding the equation of the circles tangent to $d_1, d_2$ and $d_3$, given
$$\begin{cases}d_1: y=4x-10
\\
d_2: y=9/4x-15/4
\\
d_3: y=3x-15
\end{cases} $$
My approach: I know that $d_2$ and $d_3$ are parallel. The circl... | The distance to a straight line of equation $$ax+by+c=0$$ is obtained as
$$\frac{|ax+by+c|}{\sqrt{a^2+b^2}}.$$
The center of a circle tangent to the three lines is equidistant to them and you need to solve
$$\dfrac{|ax+by+c|}{\sqrt{a^2+b^2}}=
\dfrac{|a'x+b'y+c'|}{\sqrt{a'^2+b'^2}}=
\dfrac{|a''x+b''y+c''|}{\sqrt{a''^2+b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2475460",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Find a + b + c + bc. My math trainer friend ask my help to look for the shortest possible solution for this problem:
Let $a$, $b$ and $c$ be positive integers such that
$$\left \{\begin{matrix}a + b + ab = 15 \\
b + c + bc = 99 \\
c + a + ca = 399\end{matrix}\right. $$
Find $a + b + c + bc.$
I tried elimination but... | Also elimination leads to the result (as the OP wanted), namely we obtain exactly two solutions over any field of characteristic zero:
$$
(a,b,c)=(-9,-3,-51),(7,1,49).
$$
Then $a+b+c+bc=7+1+49=106$, since only the positive solution was asked.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2475545",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Automorphism group of $\mathbb Q[\sqrt{13}, \sqrt[3]{7}]$. I want to calculate group of $\mathbb Q$-automorphisms of $\mathbb Q[\sqrt{13}, \sqrt[3]{7}]$.
$\mathbb Q[\sqrt{13}, \sqrt[3]{7}]$ is separable as a separable extension of separable extension. Thus $|Aut| = [\mathbb Q[\sqrt{13}, \sqrt[3]{7}]: \mathbb Q] = 6$. ... | The Automorphism group is $\mathbb Z/2\mathbb Z $ because any such automorphism needs to fix $\mathbb Q$ and so must take a root of a polynomial with $\mathbb Q$ coefficients to another such root. Now $\mathbb Q(\sqrt[3]{7},\sqrt11)$ is real so does not contain the other two roots of $x^3-7$. So you have to take $\sqrt... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving $\sqrt{n} \le \sqrt[n]{n!}$ I need to prove one thing:
$$
n \ge 1:
\sqrt{n} \le \sqrt[n]{n!} \le \frac{n + 1}{2}
$$
The second part:
$$
\sqrt[n]{n!} \le \frac{n + 1}{2}
$$
is easy to proof.But the first is more complicated. Help please.
| Hint: Show that, for integers $k$ such that $1\leq k\leq n$, $k(n+1-k)$ has a minimum when $k=1$ or $k=n$.
So: $$n!\cdot n!=(1\cdot n)\cdot(2\cdot (n-1))\cdots((n-1)\cdot 2)\cdots(n\cdot 1)\geq n^{n}$$
[Essentially, $f(x)=x(n+1-x)$ is increasing for $x<\frac{n+1}{2}$ and decreasing for $x>\frac{n+1}{2}$.]
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to find : $\lim_{x \to \pi/2}\frac{1-\sin x}{\sin x+\sin 3x}$ How to find :
$$\lim_{x \to \pi/2}\frac{1-\sin x}{\sin x+\sin 3x}$$
My Try :
$$x-\pi/2=t \to x=t+\frac{\pi}{2}$$
And:
$$ \sin (t+\frac{\pi}{2})=\cos t \\ \sin 3(t+\frac{\pi}{2})=-\cos 3t $$
So :
$$\lim_{t \to 0}\frac{1-\cos t}{\cos t-\cos 3t}=\\\lim_{t... | Adding mine to the pile:
$$\require{cancel}\begin{aligned}\frac{1-\cos t}{\cos t\color{blue}{-\cos 3t}}
&=\frac{1-\cos t}{\color{purple}{\cos t}\color{blue}{-\cos^3 t+3\sin^2 t\cos t}}
\\&=\frac{1-\cos t}{\cos t\left(\color{purple}{\sin^2 t+\cancel{\cos^2 t}}-\cancel{\cos^2 t}+3\sin^2 t\right)}
\\&=\frac{1-\cos t}{\col... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2475832",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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$U_{n+1} = 2U_n^2 - 1$ I need to find the general term for the progression defined by :
$U_{n+1} = 2U_n^2 - 1$
Can any one help me out ? Is it even possible to find the general term ?
| HINT: Notice that
$$2x^2-1=\cos(2\arccos(x))$$
for $x\in [-1,1]$, and
$$2x^2-1=\cosh(2\operatorname{arccosh}(x))$$
for $x\notin [-1,1]$.
Also observe that if $s(x)=2x^2-1$, then
$$U_{n+1}=s(U_n)$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Proof for Linearity of Expectation Question
It is mentioned that the second line of the proof is from the definition of the Union of Probabilities:
I do not understand how that happened. I understand that the first line is the definition of Expectation expanded, but, why are the two random variables being operated b... | It's actually in the first equality where you use (for the first time) that property. From lines 1 to 2 and 2 to 3 they just apply known properties of $\Sigma$ operator. And then from 3 to 4 we apply such a property of probability again. This last one happens because
$$\bigcup_j \{Y=j\}$$
(where as explained in the pro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2476120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Some alternating series converging to values relating to $\pi$. The following series converge to a value relating to $\pi$:
\begin{align}
\frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots&=\frac{\pi}{4},\\
\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+\cdots&=\frac{\pi^2}{8},\\
\frac{1}{1^3}-\frac{1}{3^3... | This method is overkill, but here goes.
For even $n$,
$$\sum_{k=0}^\infty\frac1{(2k+1)^n}=\left(1-\frac1{2^n}\right)\zeta(n).$$
The value of $\zeta(n)$ for even $n$ is well-known. It can be
obtained from the functional equation connecting $\zeta(s)$ and $\zeta(1-s)$.
For odd $n$,
$$\sum_{k=0}^\infty\frac{(-1)^k}{(2k+1)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2476342",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How should one proceed in this trigonometric simplification involving non integer angles? The problem is as follows:
Find the value of this function
$$A=\left(\cos\frac{\omega}{2} +\cos\frac{\phi}{2}\right )^{2} +\left(\sin\frac{\omega}{2} -\sin\frac{\phi}{2}\right )^{2}$$
when $\omega=33^{\circ}{20}'$ and $\phi=... | Hint: You did a miscalculation:
$$A=\left (\cos\frac{\omega}{2}+\cos\frac{\phi}{2} \right )^{2}+\left (\sin\frac{\omega}{2}-\sin\frac{\phi}{2} \right )^{2}$$
$$=\cos ^2\frac{\omega}{2}+2\cos\frac{\omega}{2}\cos\frac{\phi}{2}+\cos^2 \frac{\phi}{2}+\sin ^2\frac{\omega}{2}-2\sin\frac{\omega}{2}\sin\frac{\phi}{2}+\sin^2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2476453",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Compute $\lim_{\theta\rightarrow 0}\frac{\sin{(\tan{\theta})}-\sin{(\sin{\theta})}}{\tan{(\tan{\theta})}-\tan{(\sin{\theta})}}$ I rewrote it by writing the tan as sin/cos and cross multiplying:
$$\frac{\sin{(\tan{\theta})}-\sin{(\sin{\theta})}}{\tan{(\tan{\theta})}-\tan{(\sin{\theta})}}= \frac{\sin{(\tan{\theta})}-\sin... | Using the Mean Value Theorem,
$$
\begin{align}
\lim_{\theta\to0}\frac{\sin(\tan(\theta))-\sin(\sin(\theta))}{\tan(\tan(\theta))-\tan(\sin(\theta))}
&=\lim_{\theta\to0}\frac{\frac{\sin(\tan(\theta))-\sin(\sin(\theta))}{\tan(\theta)-\sin(\theta)}}{\frac{\tan(\tan(\theta))-\tan(\sin(\theta))}{\tan(\theta)-\sin(\theta)}}\\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2476524",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
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Cardinality of a free Boolean algebra on countably many generators The cardinality of a free Boolean algebra on finitely many generators is $2^{2^n}$, where $n$ is the number of generators.
Why is the the free Boolean algebra on countably many generators countable? How to prove it?
| Let $S$ be the set of generators. Define sets $A_n$ recursively:
$$A_0=S\cup\{0,1\}$$
$$A_{n+1}=A_n\cup\{x\vee y:x,y\in A_n\}\cup\{x\wedge y:x,y\in A_n\}\cup\{x':x\in A_n\}$$
(I would have gladly used the notation you are using for the Boolean operations, but you didn't tell me what they are.)
You can easily show by in... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to compute the composition of a function and a piece-wise function? I understand function composition with normal functions. I don't really get it when one of those functions is piecewise.
When do I need to change the condition of the piecewise bit, or do I even do that at all?
Here are the two functions that I wa... | In this case the composition $f(g(n))$ is easier since the piecewise function is $g$, and you only have to split between the cases $n\leq 10$ and $n>10$. More generally, if you have
$$
g(n)=\begin{cases}
g_1(n), &\text{if }n\leq 10,\\
g_2(n), &\text{otherwise},\end{cases}
$$
then you can deduce
$$
f(g(n))=\begin{cases}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2476746",
"timestamp": "2023-03-29T00:00:00",
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Prove that $(A \cap B)^\circ = A^\circ \cap B^\circ$ Let $(X,d)$ be a metric space and let $A, B \subset X$
How can I show that $(A \cap B)^\circ = A^\circ \cap B^\circ$ ?
Please just tell me a Hint. ($A^\circ$ and $B^\circ$ are sets of interior points of A and B)
| Use $x \in A^\circ \iff \exists r >0 : B(x,r) := \{ y\in X \mid d(x,y) < r \} \subset A$.
| {
"language": "en",
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Bolzano theorem and interval solution I have a function $f$ witch is continuous at $[1.4]$. I also have that $f(x)\neq 0,\forall x\in [1,4]$ and $f(1)\cdot f(2)\cdot f(4)=8$. I have already proved that $f(x)>0,\forall x\in [1.4]$.
Now I want:
*
*To prove that the equation $f(x)=2$ has at least one solution at $[1,4]... | I know one standard way to state the intermediate value theorem is that if $f(a)<0$ and $f(b)>0$ then $f(x)=0$ for at least one $x$ in $(a, b)$, but in this case I think you're obscuring the problem by reformulating it that way. The point is that if $f(x)$ is never equal to $2$ on $[1, 4]$, it's either identically less... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2476983",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Proving that: $ | a + b | + |a-b| \ge|a| + |b|$ I am trying to prove this for nearly an hour now:
$$
\tag{$\forall a,b \in \mathbb{R}$}| a + b | + |a-b| \ge|a| + |b|
$$
I'm lost, could you guys give me a tip from where to start, or maybe show a good resource for beginners in proofs ?
Thanks in advance.
| To prove
$$
| a + b | + |a-b| \ge|a| + |b|
$$
Square both the sides. This does not change inequality. We have
$$
| a + b |^2 + |a-b|^2 + 2|a+b||a-b| \ge|a|^2 + |b|^2 + 2|a||b|
$$
$$
(|a|^2 + |b|^2 +2|a||b|cos\theta) + (|a|^2 + |b|^2 -2|a||b|cos\theta) + 2|a+b||a-b| \ge|a|^2 + |b|^2 + 2|a||b|
$$ where $\theta$ is angl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2477064",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Integrate $\arctan{\sqrt{\frac{1+x}{1-x}}}$ I use partial integration by letting $f(x)=1$ and $g(x)=\arctan{\sqrt{\frac{1+x}{1-x}}}.$ Using the formula:
$$\int f(x)g(x)dx=F(x)g(x)-\int F(x)g'(x)dx,$$
I get
$$\int1\cdot\arctan{\sqrt{\frac{1+x}{1-x}}}dx=x\arctan{\sqrt{\frac{1+x}{1-x}}}-\int\underbrace{x\left(\arctan{\sq... | Use change of variable
$$\theta=\arctan\sqrt{\frac{1+x}{1-x}}\in[0,\frac{\pi}{2}).$$
Then we have
$$x=\frac{\tan^2\theta-1}{\tan^2\theta+1}=\sin^2\theta-\cos^2\theta=-\cos2\theta.$$
Therefore
\begin{align}
\int\arctan\sqrt{\frac{1+x}{1-x}}dx&=-\int\theta\,d\cos 2\theta=-\theta\cos 2\theta+\int\cos 2\theta d\theta\\
&=-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2477162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Why are Fibonacci numbers bad for Euclid's Algorithm and how to derive this upper bound on number of steps needed in general? I want to ask two things.
The first is why are consecutive Fibonacci numbers the worst case for Euclid's algorithm? I keep seeing people say it in passing and I understand that it's really bad, ... | You can prove this using induction on the number of gcd() calls.
To prove: If a pair (a,b) takes n steps to compute the gcd, then
*
*a >= F_(k+2)
*b >= F_(k+1)
With one assumption a >= b
Base case n = 1, the lemma is true.
Consider the lemma to be true for n = k - 1, k > 1
Let a pair (a,b) be such that it takes k s... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 4,
"answer_id": 3
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Is a function in the form of $f(x) = (x^2, x + 1)$ injective or surjective? I understand the method used to prove injectivity or surjectivity, however, I am confused as to how to handle a function that presents itself as a set $(m,n)$.
In the example, $f:Z \rightarrow Z \times Z$, $f(x) = (x^2, x + 1)$
We know that $x... | Answering your questions:
*
*You are correct to assume $f$ is injective. A function $f(x)$ is injective iff $f(x) = f(y) \implies x = y$. It is true that $f$ returns two values, but the criterion is the same. If $f(x) = f(y)$, in particular the two second coordinates are the same, but like you said, $x + 1 = y + 1 \... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Image of morphism of projective varieties is projective variety Let $k$ be an algebraically closed field, $X,Y$ projective varieties (irreducible algebraic sets) and $f:X\to Y$ a morphism. Is $f(X)$ a projective variety? I think it is because the image of a morphism is closed and continuity preserves irreducibility. Is... | Yes, this is correct. To be a bit more precise, if $X$ is a projective variety and $Y$ is any variety and $f:X\to Y$ is a morphism, then $f$ is a closed map (in particular its image is closed). And furthermore, the image of an irreducible set under any continuous map is irreducible.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2477575",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Graduate Level Mathematical Logic Textbooks I'm currently a first year mathematics graduate student, and am at an institution which does not have any work being done in mathematical logic, or any logicians on the staff. I've taken a mathematical logic course with the textbook by Leary, 'A Friendly Introduction to Mathe... | Disclaimer: I haven't taken a graduate level logic course.
Peter Smith a retired professor, who used to teach logic at the University of Cambridge put up a guide here with a list of books: http://www.logicmatters.net/tyl/
S. C. Kleene's Introduction to Metamathematics got reviewed by Michael Beeson when it got republis... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2477718",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Number of Divisor of $2^{2}.3^{3}.5^{3}.7^{5}$ Number of Divisor of $2^{2}.3^{3}.5^{3}.7^{5}$ of the form $4n+1$ where n$\in \mathbb{N}$ is ........
My approach is to solve it using remainder theoran like putting $2^{a}\cdot 3^{b}\cdot 5^{c}\cdot 7^{d}$ put $a=0; b=0,2;c=0,1,2,3;d=0,2,4$ but not able to approach
| Guide:
Clearly $a=0$.
$$3 \equiv -1 \pmod 4$$
$$7 \equiv -1 \pmod 4$$
$$5 \equiv 1 \pmod 4$$
Hence we require $b+d$ to be an even number.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2477848",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Prove the Integral of $\frac{1}{x}$ is not a Rational Function The title is pretty self explanatory, I'd like to prove that
\begin{align*}
\int\frac{1}{x}\,dx
\end{align*}
cannot be a rational function. I have attempted a proof by contradiction, but it doesn't seem to lead anywhere. If it is assumed that $F(x)=\frac{p(... | As you wrote in a comment, this is the same thing as proving that $\log$ is not a rational function. Suppose it was. Then we could express $\log$ as $\frac pq$, where $p$ and $q$ are polynomial functions. Furthermore, $\deg p>\deg q$, since $\lim_{x\to+\infty}\log(x)=+\infty$. So $\frac pq=P+R$, where $P$ is a non cons... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2477983",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 1
} |
Finding the smallest positive integers that satisfies given equations Is it possible to find the smallest positive integer/s that satisfy a given equation or some inequality?
Example:
$2x^2-3x>24$
Is there a formula for this?
| This is a very broad question. As for your specific equation, it's equivalent to $a(a-1) = 10b$ so you're basically looking for the smallest pair of successive numbers whose product is a multiple of 10. You also know that you need $a\geq5$ otherwise 5 can't divide $a(a-1)$, so you start looking at 5:
$5\times(5-1)=20=2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2478131",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Divergence of $\int_{0}^{1/2} 1/(|\sqrt{x}\ln(x)|)^{p} dx$ The matter of interest is
$$\int_{0}^{1/2} \frac{1}{|\sqrt{x}\ln(x)|^p}\, dx$$
I am aware that this integral converges for $p=2$ (that's not too hard to show). I also believe that this integral diverges for $p>2$...but how can I show that using elementary calcu... | By enforcing the substitution $x=e^{-z}$ we get
$$ \int_{0}^{1/2}\frac{dx}{\left(-\sqrt{x}\log x\right)^p} = \int_{\log 2}^{+\infty}\exp\left[\left(\frac{p}{2}-1\right)z\right]\frac{dz}{z^p} $$
and we clearly need $p\leq 2$ to ensure the (improperly-Riemann or Lebesgue)-integrability of $\exp\left[\left(\frac{p}{2}-1\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2478229",
"timestamp": "2023-03-29T00:00:00",
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Strange symmetry regarding sum $\sum_{n=0}^\infty\frac{n^ne^{-bn}}{\Gamma(n+1)}$ and integral $\int_{0}^\infty\frac{x^xe^{-bx}}{\Gamma(x+1)}dx$ One can show by computation the following for $b>1$
$$\sum_{n=0}^\infty\frac{n^ne^{-b n}}{\Gamma(n+1)}=\frac{1}{1+W_{\color{blue}{0}}(-e^{-b})},\tag{1}$$
(here one assumes that... |
Question 2. Is it possible to alter (1) and (2) to obtain a function
for which sum equals integral?
A simpler form for $z\in[0,\mathrm{e}^{-1})$:
\begin{align}
\sum_{n=0}^\infty
\frac{(z\,n)^n}{\Gamma(n+1)}
&=
\frac1{1+\operatorname{W}_{0}(-z)}
\tag{1}\label{1}
,\\
\int_0^\infty
\frac{(z\,x)^x}{\Gamma(x+1)}\,dx
&=-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2478319",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "26",
"answer_count": 1,
"answer_id": 0
} |
Is there a symbol for antiparallel? I've been doing some work where I've needed to talk about vectors that are parallel and those that are antiparallel, parallel to the negative of the other vector.
Is there a symbol for this?
I can write $A\parallel B$ for parallel, $A \not\parallel B$ for not parallel. Is there a s... | There isn't such a symbol (as far as I know) because if $v\|w$ then $-v\|w$ as well. Indeed each vector is parallel to its own opposite, as you take only the line on which it lies, when you speak of direction.
(To describe that fact you however can write $v\in\mathbb{R}^{-}w$ in an opposite way of "positive" parallelis... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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"answer_id": 2
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Is there a combinatorial interpretation of the triangular numbers? The triangular numbers count the number of items in a triangle with $n$ items on a side, like this:
This can be calculated exactly by the formula $T_n = \sum_{k=1}^n k = \frac{n(n+1)}{2} = {n+1 \choose 2} = {n+1 \choose n-1}$.
Is there any combinatoria... | Here is a combinatorial proof of the identity
$$
1+2+\dotsb+n=\binom{n+1}{2}.
$$
The RHS counts the number of two-element subsets of $\{0,1,\dotsc,n\}$. Let $S _k$ be those two-element subsets of the preceding set with larger element $k$ for $k=1,\dotsb, n$. Then the $S _k$ partition the set of two element subsets of $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2478616",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "26",
"answer_count": 5,
"answer_id": 0
} |
Comparing $\phi(\phi(n))$ and the number of elements in $(\mathbb{Z}/\phi(n)\mathbb{Z})^{\times}$ of order $\le \log(n)$ First notice that $\mid (\mathbb{Z}/\phi(n)\mathbb{Z})^{\times}\mid=\phi(\phi(n))$
Now if $n=p_1^{a_1}...p_k^{a_k}$ then $\phi(n)=p_1^{a_1-1}(p_1-1)...p_k^{a_k-1}(p_k-1)$ with the $p_i \ge 2$ distinc... | We have $\phi(\phi(5)) = \phi(4)=2 > 1.6094... = \log(5)$ but $\phi(\phi(10))=\phi(4)=2 <2.3025...=\log(10)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2478712",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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The Skimpy Donut I've come across this problem on several calculus tutorials but can't find any solutions for it. Can someone please explain how to figure these questions out?
Link to "The Skimpy Donut" problem
For question #1 I found the link below that helped me figure it out:
Volume of a Torus: the Washer Method
A... | I will let you solve for the volume and surface area any way you choose. Here I'll just give the results using Pappus's centroid theorems. The solutions are
$$
V=2\pi R A=2\pi^2 Rr^2\\
S=2\pi R C=4\pi^2 Rr
$$
where $R$ is the centroid of the revolving circle, and $A=\pi r^2$ and $C=2\pi r$ are its area and circumferenc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2478828",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Show that $X_n\to x$ if $\lim\limits_{n\to\infty} \frac{|X_{n+1} - x|}{|X_n - x|}<1.$ Suppose $\{X_n\}$ is a sequence and suppose for some $x \in \mathbb{R}$, the limit
$$L:=\lim_{n\to\infty} \frac{|X_{n+1} - x|}{|X_n - x|}$$
exists and $L <1$. Show that $\{X_n\}$ converges to $x$.
So far I have noted that since eac... | Correct me if wrong:
$a_n:= X_n-x.$
Show:
$ \sum a_n$ is absolutely convergent using the ratio test.
Given :
$\lim_{n \rightarrow \infty }|\dfrac{a_{n+1}}{a_n}| = L \lt 1$.
$\rightarrow:$
There is a $n_0$ such that $a_n \ne 0$ for $n \ge n_0.$
There is a $n_1 (\ge n_0) $ such that for $n \ge n_1$
$|\dfrac{a_{n+1}}{a_n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2478935",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
nonlinear programming with inequality constraint I am confused about a nonlinear programming constrained to the region
$$X = \{(x1,x2) \in\mathbb{R}^2: (x1^2/a^2)+(x2^2/b^2)<=1\}$$
Can anyone show the steps of solving such a problem (for some arbitrary objective function)?
Is there any special property of this constra... | Yes, your feasible set is convex. Whether this helps you will depend on your objective function (and in particular, on whether the objective is also convex.)
Because your constraint is so simple, one general approach is to try both cases where the constraint is active or inactive:
*
*Ignore the constraint and solve ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that a collect of events forms a sigma-algebra.. Let $X$ and $Y$ be random variables defined on some probability space $(\Omega, \mathcal{F},\mathcal{P})$ and let $\mathcal{G}=\sigma (Y)$.
How do I show the following statements?
i) The collection of events $\{ Y \in B\}$, where $B$ runs through the Borel sets $\m... | Let's begin by noting that $\{Y \in B\} =: \{\omega \in \Omega \mid Y(\omega) \in B\} = Y^{-1}(B)$ for any Borel set $B$.
The first part then essentially boils down to noticing that the action of taking preimages under $Y$. (i.e. applying $Y^{-1}$ to Borel sets) "plays nicely" with the relevant set theoretic operation... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to find the area of region $R$? Region $R$ contains all the points $(x,y)$ such that $x^2+y^2\leq100\;$ and $\:\sin(x+y)\geq0$. Find the area of region $R$.
$\:\:\:\:\:\:\:\:\:$$\sin(x+y)\geq0$
$\:\:\:\:\:\:\:\:\:$$\implies2n\pi\leq x+y\leq(2n+1)\pi$
Thus graph would be probably look like this:
$\:\:\:\:\:\:\:\... | For the points in the circle $(a,b)$ if $$\sin(a+b)\leq0$$ then for $(-a,-b)$ $$\sin((-a)+(-b))\geq0$$
Thus if our condition does not hold true for a single point we can take a symmetrically opposite point to make our conditions hold. Thus half the are of our circle will satisfy our condition
(note there will be lin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2479557",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Limit of sums and products
Is there a convenient way to calculate sums like these so you can evaluate the limit? It seems like in most cases you just need to know what it adds up to.
| I always just remember that $$1^k + 2^k + \cdots + n^k$$ is equal to some polynomial of order $k+1$ in $n$, but to get the exact formulas, I have to look them up over and over again.
That said, the two I do remember are $$1+1+\cdots 1 = n$$ (duh) and $$1+2+\cdots + n = \frac{n(n+1)}{2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2479978",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why is $a_{1} > 0 \land a_{n+1}=a_{n}+\frac{1}{a_{n}}$ unbounded? Let $(a_n)$ be a sequence s.t $$a_{1} > 0 \land a_{n+1}=a_{n}+\frac{1}{a_{n}}$$
Prove that $a_{n}$ is unbounded.
Proof:
Consider $a_{n+1}−a_{n}$:
$a_{n+1} - a_{n} = a_{n} + \frac{1}{a_{n}} - a_{n} = \frac{1}{a_{n}}$.
This is greater than $0$. Thus, $a_{n... | Note that $$a_{n+1} - a_n = a_n +\frac{1}{a_n} - a_n = \frac{1}{a_n}>0$$ since $(a_n) > 0 $ for all $n$. Therefore the sequence is strictly increasing.
So we either have:
*
*the sequence converges and is bounded (can you prove this?)
*the sequence does not converge and is unbounded (can you prove this?)
Assume ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2480068",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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Changing the measure of a stochastic process I am trying to understand how to change the measure of a stochastic process using Girsanov's theorem.
In particular, I have the process $dX_t = a dt + dB_t$ for $t \in [0,T]$, and some arbitrary, well-behaved function $v(X)$, where $X$ denotes the path of $X_t$ up to $T$.
I ... | FYI - figured out the answer. The first interpretation is incorrect, while the second interpretation is correct.
When I change the measure in two steps (as in the first interpretation), in the second change of measure, the Brownian motion $B_T$ is with respect to the measure $P^0$, not $P^{\tilde{a}}$. Once I write it... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $P(A) \neq 0$ and $P(B) \neq 0$, then $P(B|A) \geq P(B)$ is equivalent to $P(A|B) \geq P(A)$ I am puzzled by the intuition behind the following fact:
If $P(A) \neq 0$ and $P(B) \neq 0$, then $P(B|A) \geq P(B)$ is equivalent to $P(A|B) \geq P(A)$.
This is easy enough to show by definition of conditional probability,... | Both statements are saying that $P(A \cap B) \ge P(A)\cdot P(B)$. Note that $P(A)\cdot P(B)$ corresponds to $P(A \cap B)$ if $A$ and $B$ were independent. Thus $P(A \cap B) \ge P(A)\cdot P(B)$ means that there is some positive correlation (in a figurative sense) between these two events.
On the other hand, the phrase t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2480490",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 3
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Integrate $f(x)=\sqrt{x^2+2x+3}.$ Completing the square and letting $t=x+1$, I obtain $$\int\sqrt{(x+1)^2+2} \ dx=\int\sqrt{t^2+2}\ dt.$$
Letting $u=t+\sqrt{t^2+2},$ I get
\begin{array}{lcl}
u-t & = & \sqrt{t^2+2} \\
u^2-2ut+t^2 & = & t^2+2 \\
t & = & \frac{u^2-2}{2u} \\
dt &=& \frac{u^2+2}{2u^2}du
\end{array}
Thus ... | Here is how I would work it.
$t = \sqrt 2 \tan \theta\\
dt = \sqrt 2 \sec^2 \theta$
$\int 2\sec^3 \theta\\
\sec\theta\tan\theta + \ln [\sec\theta+\tan\theta]+C\\
\frac {1}{2} t\sqrt {t^2 + 2} + \ln \frac 12 (t+\sqrt{t^2 + 2}+ C\\
\frac {1}{2} (x+1)\sqrt {x^2 + 2x + 3} + \ln [x+1 + \sqrt {x^2 + 2x + 1}]+ C$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2480587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Why is $M \otimes_R N$ not an $R$ module? My professor said in lecture that for $M$ a $(A, R)$ bimodule and $N$ a $(R, B)$ bimodule that $M \otimes_R N$ is not an $R$ module anymore but an abelian group and that it is naturally an $(A, B)$ bimodule. I can see how it is an $(A, B)$ bimodule but I think that it is also a... | In general, $M\otimes N$ is not an $R$-module in any way at all —it is not that one has overlooked a way to do it.
For example, let $R=M_n(k)$, the matrix ring over a field of ome size $n>1$. Then the vector space $V$ of row vector of size $n$ is a right $R$-module in the obvious way, and the vector space $W$ of column... | {
"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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Is a Lipschitz continuous function with this rationality condition piecewise linear? If $f:[0,1] \to \mathbb R$ is Lipschitz, i.e. $|f(x)-f(y)|<K|x-y|$ for fixed $K$, and for every rational $r$ there exists integers $a$, $b$ such that $f(r)=ar+b$, do there exist finitely many intervals $I_n$ such that $[0,1] =\cup I_n$... | Yes, $f$ will be piecewise linear. Specifically, $f$ is contained in the union $U$ of the lines $ax+b$ with $a,b$ integers and $|a|<K.$
The main trick is to notice that the set of slopes between "consecutive" rationals
$$(f(\tfrac{p+1}q)-f(\tfrac p q))q$$
is compact.
Spoiler
Suppose for contradiction that $f$ is not co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2480830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding $\lim_{x\to\pi}\frac{\sin5x}{x-\pi}$ without using L'Hostpital's rule I have the following function which needs to be found. Obviously this function is fairly straight to find the limit using L'hospital's method (which will give $-5$). However, I need to find the limit without L'hospital's rule.
$$\lim_{x \to \... | These kind of problems are typically introduced around the time that it is shown that
$$ \lim_{t\to 0} \frac{\sin(t)}{t} = 1. $$
The trick is to find a way to make that limit appear. In the current context, the first thing that comes to my mind is making it more explicit that the denominator goes to zero as $x \to \pi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Produce a differential equation from the family of curves $x=a\cdot \sin(t+b)$ I'm trying to produce a differential equation from the family of curves:$$x=a\cdot \sin(t+b), ~a,b\in \mathbb{R}$$
I differentiated once with respect to $t$, here $x$ is a function of $t$:
$$x'=a\cos(t+b) \Rightarrow a=\frac{x'}{\cos(t+b)}$$... | If you settle for first order then you have an energy equation with only one retained constant $a$.
$$ x^2 + (\dot x)^2 =a^2 $$
For the second order by differenting this once more you have both constants vanishing resulting in the well known simple harmonic motion differential equation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2481110",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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proof of Box–Muller transform (polar form) There're proofs of Box–Muller transform available online but my book (pattern recognition and machine learning) seems to have put it in a different form.
I didn't follow the derivation of equation 11.12, can anyone please help? Thanks!
EDIT
As mentioned in Nadiels's answer, ... | So first thing there is an error in equations $(11.10)$ and $(11.11)$ and in fact you should have the transformations
$$
y_i = z_i \left( \frac{-2 \ln r^2 }{r^2 } \right)^{1/2}
$$
and in particular we have
\begin{align*}
\exp\left( -\frac{1}{2} \left(y_1^2 + y_2^2 \right) \right) &=\exp\left( \left( z_1^2 +z_2^2\right)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2481237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to scroll over continuous functions? $\newcommand{\RR}{\mathbb{R}}$
$\newcommand{\CR}{\mathcal{C}\left(\RR\right)}$
There are as many continuous functions than real numbers: $\left|\left\{f, f:\CR \rightarrow \RR\right\}\right|=|\RR|$.
It is easy, for a machine, to approximately scroll over every real number from $... | There is no surjective function $s:\mathbb R\to \mathcal C(\mathbb R)$ such that $t_n:r\mapsto s_r(n)$ is continuous for each integer $n.$
There are no continuous surjective maps $[-n,n]\to\mathbb R$. So if $t_n$ is continuous we can pick some $y_n > t_n([-n,n])$ for each positive integer $n.$ Let $f$ be any real funct... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2481360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Bijection from $\{0,1\}^k \rightarrow P(\{1, ..., k\})$ $\{0,1\}^k \rightarrow P(\{1, ..., k\})$
I need this bijection.
I can see both sets have a cardinality of $2^k$, I also noticed that you can perform a (computer) AND-like operation using the bit string, $\{0,1\}^k$, as a mask, to get all the possible combinations ... | Hint:
The simplest way to obtain a bijection between $\{0,1\}^k$ and $P(1,2,...,k)$ is to map
$$(x_1,...,x_k) \mapsto \{i : x_i = 1\}.$$
I'll leave the rest to you (i.e to prove that this is a bijection).
Hope this helps - feel free to ask for clarification.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2481471",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Geodesic over a plane with a "conformal" metric Let be $ds^2=G(x)(dx_1^2+\cdots+dx_n^2)$ a metric over $\mathbb{R}^n$ given by a scalar function $G:\mathbb{R}^n\rightarrow\mathbb{R}$.
Are the geodesic associated to this metric the curves $\gamma$ which satisfies: $\gamma'' =\nabla G$ ?
If yes, how could you prove it?
| The Christoffel symbols for the conformal metric $G\, dx^2= e^{\phi} dx^2$ are
$$\Gamma^i_{jk} = \frac{1}{2}(\delta_{ij} \frac{\partial \phi}{\partial x_k}+\delta_{ik} \frac{\partial \phi}{\partial x_j}-\delta_{jk} \frac{\partial \phi}{\partial x_i })$$ and the system for the geodesics is
$$\frac{d^2 x_i}{dt^2}=\sum_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2481597",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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2 dimensional Smooth manifold of $ℝ^3$ From this post ,
In attempt solve to this problem, I followed Yves Daoust's approach for parametise the torus as followed:
A circle of radius $a$ centered at $(b,0)$ in the plane $xz$ has the
parametric equation
$$x=a\cos(\theta)+b,z=a\sin(\theta),$$ with $\theta$ in the rang... | Two comments: Once you have the parametrization, you could use it to prove that the torus is a $2$-dimensional manifold.
EDIT: The main idea is to check that the mapping $g\colon (0,2\pi)\times (0,2\pi)\to\Bbb R^3$ has rank $2$ everywhere. It follows that, restricting to its image (the torus), the inverse mapping gives... | {
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"question_score": "1",
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Solve the equation $\dfrac{x^3+y^3}{x^3+z^3} = \dfrac{1006}{1001}$
Solve the equation $\dfrac{x^3+y^3}{x^3+z^3} = \dfrac{1006}{1001}$ for $x,y,z \in \mathbb{Z}$.
We must have $x^3+y^3 = 1006d$ and $x^3+z^3 = 1001d$ where $d$ is an integer. This means that $x^3+y^3 \equiv 0 \pmod{1006}$ and $x^3+z^3 \equiv 0 \pmod{100... | Well, at least we have some solutions, like
$$\frac{669^3 + 337^3}{669^3 + 332^3} = \frac{1006}{1001}$$
This should be studied with elliptic curves.
May assume $x$, $y$, $z$, rationals, and then may even assume $x=1$. We get
$$\frac{y^3+1}{z^3+1} = \frac{1006}{1001}\\
z^3 +1 = \frac{1001}{1006}(y^3 + 1)$$
This is an e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2481977",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove that if for all $g \in G$ that $gNg^{-1} \subseteq N$, then we must have for all $g\in G$ that $gNg^{-1}=N$ I feel kind of shaky about this proof so I'd really appreciate some input. I know this is equivalent to some other forms (which I'll prove afterward so please don't spoil it). Here's the statement again:
P... | The statement is true for all groups, not just finite ones. What you've written here is not well explained, it looks like you're showing that conjugation is injective and so it's a bijection. But this style proof will only work when $N$ is finite.
Instead, what happens if you conjugate $N$ by $g^{-1}$ instead of $g$?... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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how to find $\lim_{x\to0}{\frac{|2x-1|-|2x+1|}{x}}=-4$ $$\lim_{x\to0}{\frac{|2x-1|-|2x+1|}{x}}=-4$$
Why? This came from a calculus book, before L'hopital is introduced. I couldn't find the answer myself, so I looked at the answers page. WolframAlpha agrees, and interestingly enough, the function is equal to $-4$ in the... | Hint: Try to see the limits in two cases where once $x \to 0^+$ and other $x \to 0^-$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2482216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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Why is the exponential function used function used when solving linear 2nd order homogeneous differential equations? In my textbook the introduction to solving linear 2nd order homogeneous DE's begins with a general form:
$ay''+by'+ cy =0$
Then they say: "a solution must have the property that its second derivative is... | "a solution must have the property that its second derivative is expressible as a linear combination of its first and zeroth derivatives."
"So what could they mean by that?"
If
$ay^{\prime\prime}+by^\prime+cy=0$
then
$y^{\prime\prime}=-\frac{b}{a}y^\prime-\frac{c}{a}y$
That is all that is meant by that statement.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2482302",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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} |
Solve for real $x$ if $(x^2+2)^2+8x^2=6x(x^2+2)$ Question:
Solve for real $x$ if $(x^2+2)^2+8x^2=6x(x^2+2)$
My attempts:
*
*Here's the expanded form:$$x^4-6x^3+12x^2-12x+4=0$$
*I've plugged this into several online "math problem solving" websites, all claim that "solution could not be determined algebraically, he... | $$(x^2+2)^2+8x^2=6x(x^2+2)$$
$$(x^2+2)^2-6x(x^2+2)+8x^2=0$$
Let $x^2+2=U, x=V$. Then
$$U^2-6UV+8V^2=0$$
Then
$$\left(\frac UV\right)^2-6\left(\frac UV\right)+8=0$$
Then
$\frac UV=2$ or $\frac UV=4$
$\frac {x^2+2}{x}=2$ or $\frac {x^2+2}{x}=4$
$x^2-2x+2=0$ or $x^2-4x+2=0$
$$x=2\pm \sqrt2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2482425",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Finding the basis of a lattice Let $B=(b_1,b_2)$ be the linearly independent vectors and generate a lattice $ L(B)=\{xb_1+yb_2:x,y \in \mathbb{Z}\}$. If any two linearly independent vectors, $b_1^\prime,b_2^\prime$ are taken from the lattice $L(B)$, then $L(B^\prime)$ need not be equal to $L(B^\prime)$. For example,
$... | There is a general answer for modules over any commutative ring:
Let $L$ be a finitely generated free $R$-module with basis $\mathcal B=(b_1,\dots b_n)$, and $b'_1, \dots, b'_n$ be $ n$ vectors in $L$. Then $b'_1, \dots, b'_n$ are a basis of $L$ if and only if
$\;\det_\mathcal{B}(b'_1, \dots, b'_n)$ is a unit in $R... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2482588",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Given equations for the side-lines of a parallelogram, why are these the equations for the diagonal-lines? My book, for a parallelogram $ABCD$ with sides as
$$\begin{align}
AB&\;\equiv\; a\phantom{^\prime}x+b\phantom{^\prime}y +c\phantom{^\prime}=0 \\
BC&\;\equiv\; a^\prime x +b^\prime y +c^\prime=0... | As Michael Rozenberg has written, they should be
$$\small BD\equiv (ax+by+c)(a'x+b'y+c)-(a'x+b'y+c')(ax+by+c')=0\tag1$$$$\small AC\equiv (ax+by+c)(a'x+b'y+c')-(a'x+b'y+c)(ax+by+c')=0\tag2$$
why the equations of diagonals (taking AC for instance) is for the entire line AC and not just points A and C?
$(1)$ can be writ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2482711",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
$\int_{-\infty}^{\infty}f(x+1)-f(x)\text{d}x$
Let $f$ be continuous on $\mathbb R$, $\lim_{x\rightarrow \infty} f(x)=A,\ \lim_{x\rightarrow -\infty} f(x)=B$ . Calculate the integral $$\int_{-\infty}^{\infty}f(x+1)-f(x)\,\text{d}x$$
My intuition says $\frac{A+B}{2}$ (it might be wrong) but I couldn't get close to prov... | You can write the integral as: $$\sum_{n\in\mathbb Z}\left[\int_n^{n+1}f(x+1)dx-\int_n^{n+1}f(x)dx\right]=\sum_{n\in\mathbb Z} [s_{n+1}-s_n]$$ where $$s_n=\int_n^{n+1}f(x)dx$$
Then some telescoping and finding $$\lim_{n\to\infty}(s_n-s_{-n})=A-B$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2482834",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Can the pre-image (under homomorphism) of a subgroup be empty? I'm asked to prove that if $E \leq H, \varphi^{-1}(E) \leq G$ where $\varphi: G \rightarrow H$ is an homomorphism.
I can show that $\varphi^{-1}(E)$ satisfies the group condition of $\forall x, y \in \varphi^{-1}(E), xy^{-1} \in \varphi^{-1}(E)$.
But how d... | Grouphomomorphism $\phi$ sends $\mathsf{id}_G$ to $\mathsf{id}_H\in E$ so $\mathsf{id}_G\in\phi^{-1}(E)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2482908",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
What are the ideals of $\mathbb{Z}/n\mathbb{Z}$? I am trying to find all the ideals of $\mathbb{Z}/2 \times \mathbb{Z}/4$. I've just proven that the ideals of $R$ × $S$ are precisely the sets of the form $\{(x,y):x \in I,y \in J\}$ for $I\subset R,J \subset S$ ideals.
I can see that I need to find the ideals of $\mathb... | Let $\pi : \mathbb{Z}\to \mathbb{Z}/n\mathbb{Z}$ be the canonical projection.
Then by the lattice-isomorphism theorem, all ideals of $\mathbb{Z}/n\mathbb{Z}$ are of the form $\pi(I)$ for $I$ ideal of $\mathbb{Z}$ containing $n\mathbb{Z}$.
Those $I$ are precisely the $d\mathbb{Z}$ for $d\mid n$.
Therefore the ideals of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2483021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Implicit Differentiation of this function [If
$x^2 + xy + y^3 = 1$,
find the value of
$y'''$
at the point where
x = 1.]1
Am i going the right direction, if so how will know what is the values of y' and y''
| First put $x=1$ into $x^2+xy+y^3=1$ to find $y$
$$x=1 \to 1+1y+y^3=1 \\y^3+y=0 \\y(y^2+1)=0 \\y=0\\(1,0)$$ your work point's is $(1,0)$
now
$$x^2+xy+y^2=1 \to \\2x+1y+xy'+3y^2y'=0 \text{ put (1,0) }\\
2(1)+1(0)+y'+3(0)y'=0 \to y'=-2 $$ can you go on ?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2483137",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
$\lim\limits_{N \to +\infty} \sqrt{N+1} \log\left (1+\frac{x}{N+1}\right)$ I have to compute the limit
$$
\lim\limits_{N \to +\infty} \sqrt{N+1} \log \left(1+\frac{x}{N+1}\right)
$$
where $x \ge 0$ is fixed. I tried to see the previous as
$$
\log \lim\limits_{N \to +\infty} \left(1+\frac{x}{N+1}\right)^{\sqrt{N+1}}
$$
... | This one is easy and an immediate consequence of the fundamental inequality satisfied by $\log$ function: $$\log x\leq x-1,x>0\tag{1}$$ For the current question we have $x\geq 0$ and hence $$0\leq \sqrt{N+1}\log\left(1+\frac{x}{N+1}\right) \leq \frac{x} {\sqrt{N+1}}$$ and the result follows via Squeeze Theorem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2483254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 1
} |
How to distinguish two groups $(\mathbb{Z},+)$ , $(\mathbb{Z} \times \mathbb{Z},+)$ using first order logic?
How can I distinguish two groups $(\mathbb{Z},+)$ and $(\mathbb{Z} \times \mathbb{Z},+)$ in first order logic?
The first one is cyclic and the second one is not but I can't find any thing in first order to pr... | It might have been better to ask for a proof in the first-order theory of groups, instead of "in first-order logic", since after all ZF is a first-order theory. Not to be pedantic, but it seems this did lead to some confusion. Anyway:
Consider the sentence $\exists k \forall x \exists y (x=y+y\lor x=y+y+k)$.
Or, to say... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
} |
Field extension and homomorphism between rings Suppose $M/K$ is a field extension (WLOG we think of $K \subset M$ as sets) and let $a \in M$ be algebraic over $K$. If $\phi:K[x] \rightarrow K[a]$ is a homomorphism such that $x \mapsto a$ then must it be the case that $f(x) \mapsto f(a)$? I suppose this is equivalent to... | It is true that taking $x\mapsto a$ uniquely defines a $K$-homomorphism $K[x]\to K[a]$ (i.e. having the property that $k\mapsto k$ for all $k\in K$), but you could have a more general ring homomorphism (i.e., not a $K$-homomorphism) $K[x]\to K[a]$ taking $x\mapsto a$ which doesn't fix elements of $K$.
For instance, yo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2483550",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Differential equation construction from rate of flow A tank contains V0= 250L of an aqueous solution containing m0=30kg of salt. Water is entering a tank at a rate of 5L/min, and mixture is exiting at a rate of 1L/min. The concentration remains uniform by stirring.
a) Construct a differential equation for the amount of... | Let amount of salt be $S(t)$ with $S(0)=S_0=30$Kg.
Let volume of solution be $V(t)$ with $V(0)=V_0=250$L
Let entry rate be $R=5$L/min and exit rate $r=1$L/min.
Then $V(t+\Delta t)=V(t)+(R-r)\Delta t$ to give $\frac{dV}{dt}=(R-r)$ which solves to give $V(t)=V_0+(R-r)t$.
Now $S(t+\Delta t)=S(t)-r\frac{S(t)}{V(t)}\Delta t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2483836",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Inverse Fourier transform of modified Bessel function I'm relatively new to Fourier transforms so apologize in advance if this problem seems trivial.
In order to solve a second order PDE I have defined the following sine Fourier transform
$$V(r,\lambda)=\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}v(r,z)\sin(\lambda z)\,\textr... | A Fourier-Bessel representation for the ratio
\begin{equation}
\frac{I_1(\lambda r)}{I_1(\lambda)}=\frac{J_1(i\lambda r)}{J_1(i\lambda)}
\end{equation}
can be found in Ederlyi, Higher Transcendental functions II, 7.10.4 (56):
\begin{equation}
\frac{I_1(\lambda r)}{I_1(\lambda)}=2\sum_{m=1}^\infty\frac{\beta_mJ_1(\bet... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2483980",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Proof of an Elementary Result in Linear Algebra Is the Following Proof Correct?
Theorem. If the vectors $\alpha_1,\alpha_2,...,\alpha_n$ constitute a linearly independent list in the Vector space $V$ where as the list of vectors $\alpha_1,\alpha_2,...,\alpha_n,\beta$ is linearly dependent in $V$ then $\beta$ can be uni... | Yes, it is correct. However, you can further explain the assumption $j\in I\backslash \{n+1\}$ and why you don't lose generality.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2484257",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Asymptotics of $3n$ coin flips Suppose we throw a coin $2n$ times. The probability of $n$ times heads (and therefore $n$ times tails) is $$P(\text{"n times heads"}) = \frac{1}{4^n}\binom{2n}{n}.$$
We can use Stirling's formula to get the asymptotics $$\frac{1}{4^n}\binom{2n}{n} \sim \frac{1}{\sqrt{\pi n}} $$
as $n\to \... | Suppose that we toss a fair coin $N$ times. What is the probability that we get $p N$ heads and $(1-p)N$ tails? It is $2^{-N}{N \choose pN}$. Now,
$${N \choose pN} \approx 2^{-H(p) N},$$
where $H(p)$ is the binary entropy function, defined as $H(p) = p \log_2\frac1p + (1-p) \log_2 \frac{1}{1-p}$ (see also https://en.w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2484336",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Every Open Cover of $X$ contains a Countable Subcover I am reading this post where George expresses some worries he has concerning a proof in Munkres' Topology book. I myself have a different worry, although I am sure it is related. Here is the relevant passage:
Let ${B_n}$ be a countable basis and $\mathcal{A}$ an o... | Look at it this way: Take $x \in X$. As $\mathcal{A}$ is an open cover there is some $A_x \in \mathcal{A}$ that contains $x$. As $\{ B_n: n \in \mathbb{N}\}$ is a base we have that there is some $n_x$ such that $x \in B_{n_x} \subseteq A_x$. So at least for that $n_x$ we have such a member of $\mathcal{A}$ that contain... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2484455",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Odd number proof
Prove that for every odd number $n$, it holds that $n^2+3$ is not not divisible
by $8$.
My idea:
Let $n=2k+1$ for $k \in \mathbb{N}$, which implies
$$n^2+3=(2k+1)^2+3=4k^2+4k+1+3=4k^2+4k+4$$
How can I conclude that I cannot divide $4k^2+4k+4$ by $8$?
| Just for fun, a slightly different approach:
If $n^2+3$ is divisible by $8$, then so is $(n-4)^2+3=n^2+3-8n+16$, hence, by induction, $8$ divides $n^2+3$ for at least one $n$ between $-2$ and $2$. But $0^2+3=3$, $(\pm1)^2+3=4$, and $(\pm2)^2+3=7$ are not divisible by $8$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2484716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
$\mathbb{C}^\times$ mod roots of unity isomorphic to $\mathbb{C}^\times$
Let $H = \langle i \rangle =\{ i, -1, -i, 1 \}\le \mathbb{C}^\times$.
Then is $\mathbb{C}^\times/H$ isomorphic to $\mathbb{C}^\times$?
I don't think there exists an isomorphism
$\varphi : \mathbb{C}^\times \to \mathbb{C}^\times/H$
because fo... | Let $n$ be any positive integer, and $\mu_n$ the group of complex $n$-th roots of unity. The map $\Bbb C^\times\to\Bbb C^\times$ given by $z\mapsto z^n$
is a surjective group homomorphism with kernel $\mu_n$. By the First
Isomorphism Theorem, $\Bbb C^\times\cong\Bbb C^\times/\mu_n$.
This is the $n=4$ case.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2484823",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Order of element in the center of $G$ If $G$ is a group of order $p^n$, where $p$ is prime, then by the class equation, the center of $G$, $Z(G)$, is nontrivial. But must the center specifically contain an element of order $p$?
| If $P$ is a $p$-group and $g\in P$ is nontrivial, then $g$ has order $p^k$, in which case $g^{p^{\large k-1}}$ has order $p$. We conclude every $p$-group has an element of order $p$. This applies in particular with the center $Z(P)$ of any $p$-group $P$, which is a nontrivial subgroup.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2484934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
If $N$ is odd, show $\sin^N x$ can be written as a finite sum of the form $\sum _{k=1} ^{N} a_k \sin(kx)$? I am reading a Fourier series book, I got this exercise from the book but have no clue how to prove it. Would you mind kindly giving me some hints?
Thanks!
| As we know in a Fourier series, when $f(x)$ is odd then the series reduces to
$$f(x)=\sum_{n=0}^\infty b_n\sin nx$$
here $\sin x$ is an odd function and for odd $N$, $\sin^Nx$ is odd as well. This means
$$\sin^Nx=\sum_{n=0}^\infty b_n\sin nx$$
for $n>N>0$ we have $b_n=0$ because
\begin{align}
2\pi b_n
&= \int_{-\pi}^{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2485012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Dirac delta function and dirac measure I want to know the relationship between these two things. e.g. What's the results of following integrals? (Let $\mu$ Lebesgue measure, $\nu$ Dirac measure)
$$(1)\int\delta_c(x)\mu(dx)$$
$$(2)\int\delta_c(x)\nu(dx)$$
$$(3)\int f(x)\delta_c(x)\mu(dx)$$
$$(4)\int f(x)\delta_c(x)\nu(d... | In analysis and probability, Dirac's delta $\delta_c$ is commonly seen as a function defined on a space of functions. Here are two examples:
*
*In the Theory of distributions for example $\delta_c$ is defined as $\delta_c:\mathcal{C}^\infty_c(\mathbb{R}^d)\rightarrow\mathbb{R}$ defined as $\delta_c\phi = \phi(c)$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2485145",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Easiest way to see whether these two graphs are isomorphic
I've tested a few isomorphic invariants such as total degrees, total vertices, total edges, total amount of degree $4$ vertices and so on.
It seems that it holds for a lot of the isomorphic invariants.
Is there a good efficient way to check in general if two g... | There's no efficient way known in general. For these small graphs, you could look at the two vertices of degree $2$. In one graph, they share one neighbor, but in the other they share two.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2485314",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
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