Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Line intersecting spheroid I have two planes $(A): u_{1}x + v_{1}y + w_{1}z = d_{1}$ and $(B): u_{2}x + v_{2}y + w_{2}z = d_{2}$.
They intersect together, then they yield a line $(L)$ that has a direction vector $M (x_{M},y_{M},z_{M})$
$M$ is the cross product of the normal vectors of $A$ and $B$
$M = (u_{1},v_{1},w_{1... | $(x_p, y_p, z_p)$ is any point on the line of intersection.
For example, suppose the two planes are x+ 2y+ z= 1 and 2x- y+ z= 3. Subtracting the first equation from the second gives x- 3y= 2 so x= 3y+ 2. Taking, arbitrarily, y= 0, x= 2. Then both 2+ 0+ z= 1 and 4- 0+ z= 3 give z= -1. (2, 0, -1) is one of the infini... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2925955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Is it possible to determine the largest number $\tau$ such that the spectral radius $\rho(A\pm \tau ee^T) < 1$ Let $A \in M_n(\mathbb R)$ with no particular structure assumed and the spectral radius $\rho(A) < 1$. Let us denote the all $1$ vector by $e = (1, \dots, 1)^T$. I would like to determine a number $\tau \in \m... | In this post, we assume that $A$ is symmetric (otherwise I do not see what to show).
Let $spectrum(A)=\{\lambda_1\geq \cdots\geq\lambda_n\}$. We may assume that $\rho(A)=\lambda_1\geq 0$.
$\textbf{Proposition 1}.$ If $\tau< \dfrac{1-\rho(A)}{n}$, then $\rho(A\pm \tau ee^T)<1$.
$\textbf{Proof}$.
Let $U=ee^T$ (symmetri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2926238",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How to cut a square on $5$ squares? We can cut any square on $n$ squares if $n>5$ and $n=4$.
The proof is easy by induction. Base cases $n=6,7,8$ are easy to find and then since we can cut a square on $4$ squares we get $3$ new squares, so we go from $n\to n+3$ and we are done.
But I can not find a proof that we can't... | If you had glue, @greedoid, it would be too easy:
| {
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"timestamp": "2023-03-29T00:00:00",
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How do I find the equation of a tangent to a hyperbola whose centre is (h,k)? Given that $\frac{(x-3)^2}{9} - \frac{(y-2)^2}{4} = 1$ is equation a hyperbola,
I have to find its tangent at the point $\left(-2,\frac{14}{3} \right)$.
I know about the equations $c^2=(am)^2-b^2$ and $\frac{xx1}{a^2} - \frac{yy1}{b^2} = 1$... | Your hyperbola is $\frac{(x- 3)^2}{9}- \frac{(y- 2)^2}{4}= 1$ and you want to find the tangent line to it "at (-2, 14/3)". The first thing I would do is check to make sure that point is on the hyperbola. With x= -2, $(x- 3)^2= 25$ and $\frac{(x- 3)^2}{9}= \frac{25}{9}$. With $y= \frac{14}{3}$, $(y- 2)^2= \frac{64}{9... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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What is the degree of the following map?
Let $a \notin S^1$, $f:S^1 \rightarrow S^1$, be given by $z \mapsto \frac{z-a}{|z-a|}$. What is the degree of $f$?
Where by degree we mean under the usual definition here
or general case here .
My current thoughts are usuing the definition that it is the value of the unique ... | This is the answer proposed by Steve D.
By the homotopy axiom, the map $H(f):H_1(S^1) \rightarrow H_1(S^1)$ is the same for homotopic maps $g \simeq f$.
When $|a|<1$, we give the homotopy
$$ H_t(z):= \frac{z-ta}{|z-ta|}.$$
So the degree of $f$ is equivalent to that of $g(z)=z$, which is $1$.
When $|a|>1$, we give t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2926571",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Compute $\lim_{x\to0} \ln^x(x)$
$$\lim_{x\to0} \ln^x(x)$$
I know I can use L'Hospital and get an answer equal to one.
But how can he ask a question like this even if the function is not continuous from $[0, 1]$. It has only discrete solutions not a graph
| The given function doesn't have discrete values if you plot it on its natural domain.
Speaking about the natural domain of the function, since you're using a real exponent, which is typically defined as $a^x=e^{a\ln x}$, the natural domain of your function is $[1,+\infty)$. Hence you can't talk about a limit around $0$... | {
"language": "en",
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Find a Jordan form of a non-diagonazable matrix I am trying to find Jordan's form of this matrix:
\begin{pmatrix}
0 & 1 & 0\\
-4& 4& 0\\
-2 & 1& 2
\end{pmatrix}
The only eigenvalue $r$ is 2 and therefore the simplest eigenvector $v_{1}$ is (0, 0, 1)
To get the other two independent vectors (generalized eigenvectors... | I like this method for hand calculations: first, calling your matrix $A,$ let
$$ B = A - 2 I $$
$$
B =
\left(
\begin{array}{ccc}
-2&1&0 \\
-4&2&0 \\
-2&1&0
\end{array}
\right)
$$
A basis for the genuine eigenvectors is given by the convenient
$$
E =
\left(
\begin{array}{cc}
1&0 \\
2&0 \\
0&1
\end{array}
\right)
$$
We ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2926820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How would one prove that a linear combination of convex functions is also convex? As above, how would one mathematically prove that a linear combination of convex functions is also convex?
We know a function defined on a convex set $S$ is convex if:
$$f(tx_1+(1-t)x_2)\leq tf(x_1)+(1-t)f(x_2)$$
where $t$ is from $0$ to ... | hint
You should assume the coefficients positive.
Your sum is finite, so you just need to prove that
$$a_1f_1+a_2f_2$$ is convexe.
$$a_i>0.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2926956",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Solution to this Differential Equation $f''(x)=f(x)f'(x)$ needed I came up with this differential equation and I don't know how to solve it.
$$f''(x)=f(x)f'(x)$$
I attempted to solve it several times, but they were all fruitless. Wolfram Alpha says that the solution is
$$f(x)=\sqrt{2a} \tan\left({\frac{\sqrt{2a}}{2} \... | Let us consider your differential equation:
$$f''(x)=f(x)\cdot f'(x)$$
Integrate with respect to $x$ on both sides. Recognize that $df'(x)=f''(x)\ dx$ and $df(x)=f'(x)\ dx$:
$$\int f''(x)\ dx=\int f(x)\cdot f'(x)\ dx\rightarrow \int df'(x)=\int f(x)\ df(x).$$
It follows that
$$f'(x)=\frac{(f(x))^2}{2}+a=\frac{(f(x))^2+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2927060",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Prove that if the functional sequence $xf(n) + \sum_{k=1}^n f(k)$ is convergent then the sequence $f(n)$ is convergent Let $f:D_f\rightarrow\mathbb{R}$ be a function such that $\mathbb{N}\subseteq D_f$. My question is proving or disprving the following\
"If the functional sequence $xf(n) + \sum_{k=1... | If $g_n(x):=xf(n)+\sum_{k=1}^{n} f(n)$ then:
$$2g_n(x) - g_n(2x) = \sum_{k=1}^{n} f(k)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2927261",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to easily create a polynomial function that gives a desired output? I am looking for an easy way (formula or algorithm) to create a polynomial function that gives the arbitrary preset output for the first values of x. For instance, the desired output can be $y = 1, 2, 3, 4, 5, 6, 100$ for $x = 1, 2, 3, 4, 5, 6, 7$.... | As Gabriel Romon commented: Lagrange interpolation does the trick.
The polynomials form a vector space (which works very similar to the 3D coordinate space) where the coefficients work as coordinates and the monomials including $x^0=1$ as orthogonal standard base. The Lagrange polynomials form another one. They are co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2927511",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
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Minimum of $\left(\frac{1+\sin^2x}{\sin^2x}\right)^n+\left(\frac{1+\cos^2x}{\cos^2x}\right)^n$
I would like to find the minimum of
$$f(x)=\left(\frac{1+\sin^2x}{\sin^2x}\right)^n+\left(\frac{1+\cos^2x}{\cos^2x}\right)^n,$$
where $n$ is a natural number.
I know there is possible by derivate, but
$$f'(x)=n \left(\left(... | Using A.M $\geq G.M.$ on the first and second inequalities yields
$$
\begin{aligned}
f(x) &=\left(\csc ^{2} x+1\right)^{n}+\left(\sec ^{2} x+1\right)^{n} \\
& \geqslant 2 \sqrt{\left[\left(\csc ^{2} x+1\right)\left(\sec ^{2} x+1\right)\right]^{n}} \\
&=2\left[\left(2+\cot ^{2} x\right)\left(2+\tan ^{2} x\right)\right]^... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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prove that $X^2 \equiv 35 \pmod{100}$ has no solutions This problem is from 'A Survey of Modern Algebra' - Garret Birkoff, and Saunders Mac Lane in Section 1.9.
I'm an autodidact and there are no answers in the back so I need you guys to look at my proofs every once in a while to verify them. As you might be able to t... | $X^2=100k+35=4(25k)+35=4(25k+8)\color{red}{+3}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2927957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Extensions of amenable groups Let $1 \to N \to G \to Q \to 1$ be an extension of (discrete) groups, where $N$ and $Q$ are amenable. Using the fixed-point theorem, I know how to show that $G$ is amenable. However, I was wondering if there is a proof which only uses the Følner property, namely $G$ is amenable if and only... | Here's a direct proof.
Let $G$ be a group with $N$ amenable normal subgroup, such that $G/N$ is amenable; write $p:G\to G/N$. Let $S$ be a finite subset of $G$ and $\varepsilon>0$.
There exists a nonempty finite subset $F$ of $G$ such that $p(F)\equiv_\varepsilon p(sF)$ for all $s\in S$, and $p|_F$ is injective. Here $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2928075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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An expression for the sum $\sum\limits _{k=1}^{n-1} k \, (n-k)^2$ I really don't know how to find the sum of the series: $$\sum\limits _{k=1}^{n-1} k \, (n-k)^2 = 1(n-1)^2+2(n-2)^2+3(n-3)^2+\dots+(n-1)1^2.$$
My attempt:
I tried to approach the old school approach of how we find the sum of arithmetic-co geometric progr... | For the series
$$\sum_{k=1}^{n-1} k \, (n-k)^{2}$$
consider
$$\sum_{k=1}^{n-1} k \, (n-k)^{2} = n^{2} \, \sum_{k=1}^{n-1} k - 2n \, \sum_{k=1}^{n-1} k^{2} + \sum_{k=1}^{n-1} k^{3}$$
and use
\begin{align}
\sum_{k=1}^{n} k &= \frac{n(n+1)}{2} \\
\sum_{k=1}^{n} k^{2} &= \frac{n(n+1)(2n+1)}{6} \\
\sum_{k=1}^{n} k^{3} &= ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2928259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 4
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Why isn't this approach in solving $x^2+x+1=0$ valid? There is this question in which the real roots of the quadratic equation have to be found:
$x^2 + x + 1 = 0$
To approach this problem, one can see that $x \neq 0$ because:
$(0)^2 + (0) + 1 = 0$
$1 \neq 0$
Therefore, it is legal to divide each term by $x$:
$x + 1 ... | Transformations you apply to an equation may introduce alien solutions.
Taking an extreme example,
$$x=0\implies 0=0$$ which is satisfied by all $x$ !
So you may apply transformations, but validate the solutions using the original equation.
In your example, you establish
$$x^2+x+1=0\implies x^3-1=0.$$
But as $$x^3-1=0... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Compactness of $\ A= \{f: f$ is power series with infinite radius of convergence and is bounded by $1\}$ Consider the set $\ K=C[0,1]$, the set of continuous functions on $\ [0,1]$, with the supremum norm. Let
$\ A= \{f: f$ is power series with infinite radius of convergence and is bounded by $1\}$.
I am asked to prov... | Let $f_n(x) = x^n, n=1,2,\dots $ Then each $f_n\in A.$ If $A$ were compact, then some subsequence $f_{n_k}$ would converge uniformly to some $f\in A.$ This would imply $f_{n_k}\to f$ pointwise on $[0,1].$ But the full sequence $f_n,$ hence the subsequence $f_{n_k},$ converges pointwise to the function that equals $0$ o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2928505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Does every two variable equation are separable? I am a first year undergraduate student(Not undergone group theory yet) and i describe my situation as follows:
I was trying to trace curve of equation say :
$$x(y-x)^2=ay^2$$
So as we are taught in school by simple steps to work like :
(1) Symmetry
(2) Origin & Tangent ... | In general no, because to put an equation in the form y=f(x) you need for there to only be one y value for each x value.
You can cheat a little on this, as you have in one example, by using ±, but this won't work in general. For example, it has been proven that there is no closed form solution to 5th order (and beyond)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2928626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why can the set of all natural numbers and omega be put in one-to-one correspondence with natural numbers? If $\omega$ comes literally after we've run out of all natural numbers, then why can the set of all natural numbers and omega be put in one-to-one correspondence with natural numbers? I feel the existence of $\om... | There are two distinct notions that are relevant here:
*
*Sets
*Ordered sets
When you talk about "$\omega$ coming after the natural numbers", you are talking about ordered sets — specifically, the ordered set $\omega + 1$. (the underlying set of $\omega + 1$ is is $\mathbb{N} \cup \{ \omega \}$)
There does not ex... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2928720",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Higher differentiability of weak solutions to 2nd order elliptic PDEs with mixed boundary conditions I am interested in regularity results for 2nd order elliptic PDEs with mixed boundary conditions like
$$\left\{\begin{array}{rl}-\text{div}(a\nabla u) =& f &\text{in }\Omega, \\
u=&\varphi &\text{on }\Gamma_D, \\
\fra... | Unfortunately, there is no easy answer to your question. Mixed Dirichlet-Neumann problems have singular solutions even when the boundary conditions are regular. Take $f=\varphi=g=0$ then the function $u(r,\theta)=r^{1/2}\sin\frac\theta2$ is harmonic in the half-space $y>0$ and satisfies the Dirichlet-Neumann boundary c... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find a number which satisfies the equation $|x^6-3x^3+5x| \leq Q$ whenever $|x|$ $\leq$ $2$? NOTE (EDIT): I have noticed that no one noticed that a TRIANGLE INEQUALITY is being used. Hence, part of the confusion in my comments of the solution.
TRIANGLE INEQUALITY:
$|x+y| \leq |x|+|y|$, important part to note that the... | Yes we have that for $|x|\le 2$
$$0-24-10\le x^6-3x^3+5x\le 64+24+10 \implies |x^6-3x^3+5x|\le 98$$
Note that the problem is not asking for the "minimum value $Q$ such that..." but simply for any value $Q$ which satisfy the inequality. Therefore the solution $Q=98$, even if it is not the minimal possible answer, is co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2928977",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A cofinal function into a limit ordinal Let $\alpha$ be a limit ordinal. Let $f\colon\beta\to\alpha$ be a cofinal function, that is, a function so that $f(\beta)$ is unbounded in $\alpha$. That is, we have $(\forall \gamma < \alpha)(\exists \eta < \beta)(\gamma \leq f(\eta))$.
It is true that $\bigcup_{\eta < \beta} f(... | Apparently, yes. It is clear that $\bigcup_{\eta < \beta} f(\eta) \subseteq \alpha$.
Let $\gamma < \alpha$. Since $f$ is cofinal, let $\eta < \beta$ so that $\gamma \leq f(\eta)$. Then $\gamma < f(\eta) + 1$. Note that since $f(\eta) < \alpha$, $f(\eta) + 1 \leq \alpha$. Since $\alpha$ is a limit ordinal, $f(\eta) + 1 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2929062",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to find the equivalence classes on a relation? Suppose we define the relation $∼$ by $v∼w$ (where $v$ and $w$ are arbitrary elements in $R^n$) if there exists a matrix $$A∈ GL_n(R)$$ such that $v=Aw$. What are the equivalence classes for $∼$ in this case? NOTE:$GL_n(R)$ is a set that contains all the $n×n$ matrice... | Given any unit vectors, there is a nonsingular rotation matrix that takes the first vector to the other. Given any non-zero, non-unit vector, there is some non-singular scaling matrix that takes the vector to a unit vector. So given two arbitrary non-zero vectors $v$ and $w$, there are $S_v$ and $S_w$ such that $S_vv$ ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Function transformation: shrink horizontally Write the formula for $f(x)$, if the graph of $f$ can be obtained from the graph of $y = g(x)$ by shrink horizontally by a factor of $5$ then shift left $3$ units
The equation should be
$f(x) = g(5(x+3))$ or $g\left(\frac{1}{5}(x+3)\right)$?
I prefer the second answer but m... | To shrink a function means to make the graph of the function seems narrower.
For example, consider the function
$$f(x)=x^2$$
If you want to make the function shrink horizontally by a factor of 2 you would want the function
$$f(2x) = (2x)^2 = 4x^2$$
On the other hand, you would argue that
$$f\left(\frac{1}{2}x\right)... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How many integers from $1$ through $1000$ are not divisible by any one of $4, 5$, and $6$? I hate to use this resource as an argument settler but my friend and I have come across this question and we cannot agree on an answer.
I got $500$ through the use of inclusion and exclusion and he got $466$ through the use of g... | Inclusion Exclusion makes more sense to me than "the use of gathering LCM's and such"
But you do need LCM to do inclusion exclusion.
There are $1000$ integers. $250$ divisible by $4$ and $200$ by $5$ and $166$ by $6$. There are $1000/20 = 50$ there are divisible by both $4$ and $5$ (divisible by $4$ and $5$ means div... | {
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What is the probability that a randomly chosen $10$-card hand has exactly three three- of-a-kinds (and no four-of-a-kinds)? This is my attempt:
For the first three-of-a-kind:
There are ${13}\choose{1}$ options for the three cards alike and ${4}\choose{3}$ for the suits
For the second three-of-a-kind:
There are ${12}\ch... | We'll divide the number of successful hands by the total number of hands.
The total number of hands is $$\binom{52}{10}$$
The number of successful hands is the number of ways to choose $3$ card values that will be repeated $3$ times, then $1$ card value that will occur once, then the excluded suit for each of the se... | {
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Find all functions $f:\mathbb{N}^+\to\mathbb{N}^+$ such that $f\big(f(n)\big)+f(n)=2n$ for every $n\in\mathbb{N}^+$.
Find all functions $f:\mathbb{N}^+\to\mathbb{N}^+$ such that $$f\big(f(n)\big)+f(n)=2n$$ for every $n\in\mathbb{N}^+$.
I think the answer is $f(n)=n$,We prove this by induction. (at last step I can't i... | At first, we will show that $f$ is injective.
Suppose, for some $m,n$
$f(m)=f(n)\implies f(f(m))=f(f(n))$ therefore, by the given condition we have, $m=n$
Now, we claim, $f(n+1)\ge n+1$
Otherwise,
suppose $f(n+1)=n+1-k (1\ge k\le n)=f(n+1-k)$ (by induction)
Therefore, $n+1=n+1-k$ (as $f$ is injective)
So, a contradict... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2929775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that there exists a number $x$ such that $x^3 = 6$ I want to show that there exists a real number $x$ such that $x^3 = 6$. Here is what I have so far. $\\$
Let $S = \{x \mid x \in \mathbb{R}, x \geq 0, x^3 < 6\}$. By this definition, $S$ is nonempty since $0 \in S$, and also $S$ is bounded above since $2^3 = 8 >... | Note that$$b^3 - 3b^2\varepsilon + 3b\varepsilon^2 - \varepsilon^3>6\iff b^3-6>3b^2\varepsilon-3b\varepsilon^2+\varepsilon^3.$$Now, take $\varepsilon\in\left(0,1\right)$ such that$$\varepsilon<\dfrac{b^3-6}{6b^2\varepsilon}\tag1$$and that$$\varepsilon<\dfrac{b^3-6}2.\tag2$$Then\begin{align}3b^2\varepsilon+\varepsilon^3... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove $\frac{d}{dx} x^n=nx^{n-1} : \forall n\in \mathbb{Z}_{+}$ by induction Problem
Prove $$\frac{d}{dx} x^n=nx^{n-1} : \forall n\in \mathbb{Z}_{+}$$ by mathematical induction.
Attempt to solve
Base case
when $n=1$
$$ \frac{d}{dx} x^1 = 1 \cdot x ^{0}=1 $$
which is true
Induction step
$$\frac{d}{dx}x^{n+1}=(n+1)x^{n... | Assume that we have
$(*) \quad \frac{d}{dx} x^n=nx^{n-1}$
for some $ n\in \mathbb{Z}_{+}$.
For the induction step use $(*)$ and the product rule !
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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What does it mean if the fundamental group is Abelian? On proving the problem that $\pi_1(X,x_0)$ is Abelian iff for every pair $\alpha$ and $\beta$ if paths form $x_0$to$x_1$, we have $\hat \alpha=\hat\beta$ where $X$ is path-connected space
,
I'm curious of meaning for what fundamental group is Abelian. If possible, ... | Consider the wedge sum of two circles (that is, two circles that are connected by one point, let's call it the base point). By Van Kampen's theorem, you can see that the fundamental group of this space is the free product $\mathbb{Z} * \mathbb{Z}$.
Obviously this is not abelian. In fact, if you consider the loop $\alph... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Orthogonal Block Matrix Given $A \in \mathbb{R^{\text{nxn}}}$, $B \in \mathbb{R^{n\text{x}m}}$ and $C \in \mathbb{R^{m\text{x}m}}$ such that
$$ M = \begin{bmatrix}
A & B \\
0 & C
\end{bmatrix} \in \mathbb{R^{(n+m)\text{x}(n+m)}} $$
a block Matrix.
Prove that: $M$ is orthogonal $\iff$ $A$ and $C$ are orthogonal... | The $n+i$-th column of $M$ must be orthogonal to all the first $n$ columns of $M$. If you write the dot product, it's clear that the dot product of the $n+i$-th column of $M$ and the $k$-th column of $M$ (for $k\leq n$) is equal to the dot product of the $i$-th column of $B$ and the $k$-th column of $A$.
Therefore, the... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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General formula for nth element of the sequence 0, 1, 0, 1, ... The sequence is $f = 0, 1, 0, 1, \ldots$
I want to find a general formula for the $n$th element. The sequence starts at $n = 0$ (the $0$ here is not the first element $0$ but rather denotes the $0$th position).
One easy and obvious solution is: $n$th $f = ... | Is the sequence just an eternal alternation of 0 and 1? Then all you need to do is put in a couple dozen alternations in the OEIS, find https://oeis.org/A000035, then just sit back and read until you find a formula you like.
Least significant bit of $n$, lsb(n).
This works even if $n$ is negative, but it gets a littl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2930284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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Find a closed form of $x_{n+2} = x_{n+1}+2x_n +2$ where $x_1 = x_2 = 1$, $n \in \mathbb N$
Find a closed form $x_n$ for the following recurrence relation:
$$
x_{n+2} = x_{n+1} + 2x_n + 2 \\
x_1 = x_2 = 1,\;\;n\in \mathbb N
$$
I'm trying to understand why I get different results for different guesses of a solution ... | If you put $x_n = y_n+c$ so that $y_n$ is a solution to homogenus equation we get $c=-1$.
Since $y_1=y_2= 2$ and $$y_n = a(-1)^n+b2^n$$ we get $$y_n = {2\over 3}(2^n-(-1)^n)$$
or $$x_n = {2\over 3}(2^n-(-1)^n)-1$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
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Understanding the field of fractions of $F[[x]]$ (the ring of formal power series in the indeterminate x with coefficents in F) Let $F[[x]]$ be the ring of formal power series in the indeterminate x with coefficients in F. Show that the field of fractions of $F[[x]]$ is the ring $F((x))$ of formal Laurent series.
I've... | $\sum_{k=-n}^\infty a_kx^k=\frac{\sum_{k=0}^\infty a_{k-n}x^k}{x^n}$, so every Laurent series can be written as a quotient of power series.
Conversely, given a fraction $\frac{F(x)}{G(x)}$, write $G(x)=a_nx^n(1+xH(x)),$ where $H(x)$ is a power series. Then
$$
\frac1{G(x)}=\frac1{a_nx^n}\cdot \frac{1}{1+xH(x)}=\frac1{a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2930591",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving that $\tau = \{N,\varnothing\}\cup \{S_n:n\in\mathbf{N}\}$ is a topology on $\mathbf{N}$. I am required to show that $\tau = \{N,\varnothing\}\cup \{S_n:n\in\mathbf{N}\}$ where $S_n := \{1,\dots,n\},\forall n\in\mathbf{N}$ is a topology on $\mathbf{N}$, the part where i am experiencing some dfficulty is in prov... | The idea is OK, but the proof should be set up for arbitary unions, and then the union is not always $\mathbb{N}$, though it often is:
Let $O_i$, $ i \in I$ be an arbitary union of open sets. We can WLOG omit any $O_i = \emptyset$, because they do not contribute to the union, and if any $O_i = \mathbb{N}$, so is the un... | {
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"source": "stackexchange",
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Show that $\vec a \cdot \vec b = {1 \over 4} |\vec a + \vec b|^2 - {1 \over 4} |\vec a - \vec b|^2$
Show that
$$\vec a \cdot \vec b = {1 \over 4} |\vec a + \vec b|^2 - {1 \over 4} |\vec a - \vec b|^2$$
I have tried:
$$\vec a \cdot \vec b = {1 \over 4} |\vec a + \vec b|^2 - {1 \over 4} |\vec a - \vec b|^2
\\ \vec a... | HINT
Such formula is known as the Polarization Identity. Due to the inner product properties, we have:
\begin{align*}
\lVert\textbf{a}+\textbf{b}\rVert^{2} = \langle\textbf{a}+\textbf{b},\textbf{a}+\textbf{b}\rangle = \langle\textbf{a},\textbf{a}\rangle + 2\langle\textbf{a},\textbf{b}\rangle + \langle\textbf{b},\textbf... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Simple Cubic Polynomial not Yielding Expected Results So I've got this beautiful little piece of math, $n$ in this formula can be substituted with any non-negative number between 1 and 100.
$$
\frac65 n^3 - 15n^2+100n-140
$$
The expected results are kind of as follows...
*
*$n = 1 \to 9$
*$n = 2 \to 48$
*$n = 3 \... | If you want to define a cubic polynomial
$$
p(n) = an^3+bn^2+cn+d
$$
satisfying $p(1)=9,p(2)=49,p(3)=p(4)=39$, there is a unique polynomial like that. You can construct the coefficients by solving the Vandermonde system
$$
\begin{pmatrix}
1 & 1 & 1 & 1\\
1 & 2 & 4 & 8\\
1 & 3 & 9 & 27 \\
1 & 4 & 16 & 64
\end{pmatrix}
\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Shortest distance between two lines and find points on each line I'm given the following two lines:
$L_1$: $P_1=(−13, 3, 14)$ with direction vector $d_1=(2, −1, −2)$
$L_2$: $P_2=(5, 4, 4)$ with direction vector $d_2=(−2, 1, 0)$
I'm then asked to find the shortest distance $d$ between these two lines, and then find a po... | HINT
Let consider the plane orthogonal to a line and containing the other one.
| {
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Poker combinatorics: chances opponent has a full house given that you have 2 pairs I've read the answers to some very similar problems to this, but I can't manage to apply the knowledge to this specific one, sorry for this redundance. Here's the question:
What is the probability that your opponent is dealt a full house... | You have 2 pairs
B: The opponent has a full house
$P(B|A) = \frac{P(BA)}{P(A)}$
P(A) is easy enough to calculate, but it's P(BA) that's causing me problems. I'm interpreting it as the probability of the dealer selecting 10 cards that will provide the opponent with a full house and you with 2 pairs. Right now I have:
$P... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2931207",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is the dot product of two vectors a scalar value? I'm having some trouble seeing why dot products are said to give scalar values. As a far as I can see, it just gives another vector that is projected onto one of the 2 original vectors. How, then, is the result a scalar quantity. Can someone please explain this to m... | $$(1,2)\cdot (3,4) = 1 (3) + 2(4) = 11$$
is a scalar.
I think you are confusing dot product with projection.
Suppose $u$ is a unit vector, we can project $v$ onto $u$ and its length would be $|u\cdot v|$ while the projection would be $(u\cdot v) u$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2931370",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Solve $\sin^{3}x+\cos^{3}x=1$
Solve for $x\\ \sin^{3}x+\cos^{3}x=1$
$\sin^{3}x+\cos^{3}x=1\\(\sin x+\cos x)(\sin^{2}x-\sin x\cdot\cos x+\cos^{2}x)=1\\(\sin x+\cos x)(1-\sin x\cdot\cos x)=1$
What should I do next?
| Hint $$\sin^3(x)+\cos^3(x)=\sin^2(x)+\cos^2(x).$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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"answer_id": 4
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Solve $7^x+x^4+47=y^2$
Solve $$7^x+x^4+47=y^2$$ where $x, y \in \mathbb{N}^*$
If $x$ is odd then the left term is congruent with $3$ mod $4$ so it couldn't be a perfect square, so we deduce that $x=2a$ and the relation becomes $$49^a+16a^4+47=y^2$$ and it is easy to see that the left term is divisible by $16$ so we ... | When $x$ is odd then $7^x+x^4+47\equiv 3(\mod 4)$. So, it is not possible $y^2\equiv 3(\mod 4)$.
Let $x=2k$, then for $k\geq 4$
$$(7^k)^2<7^{2k}+(2k)^4+47<(7^k+1)^2$$
This mean we have $k\leq 3$.
if try to $k=1,2,3$ only $x=4$ is a solution.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2931537",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 2
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- Optimization - Standard Grid Search I'm struck into an portfolio opt. problem and the paper I'm replicating (or, better, trying to) is using a "Standard Grid Search".
Since I never encountered it before, I would like to ask you about: what's the intuition behind this numerical method? What's it used for? How could i... | The grid search is, in essence, systematically search through all possible (hyper)parameters to find the best one.
So, for example, if your portfolio depends on two hyperparameters $A,B$ (say taking values in $[0,1]$), then you might search through the $1001^2=1\,002\,001$ possible pairs
$$
(A,B)=(0,0), (0,0.001), (0,0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2931639",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Coding Theory Set Problem Supose that $\mathbb{A}$ is a finite set and take $\overline{u},\overline{v} \in \mathbb{A}^n$. Let:
$$X=\{\overline{x} \in \mathbb{A}^n\mid d(\overline{u},\overline{x})<d(\overline{v},\overline{x})\}$$
$$Y=\{\overline{y} \in \mathbb{A}^n\mid d(\overline{u},\overline{y})>d(\overline{v},\overli... | It's not necessary that the underlying set is a field (or abelian group). Just show that the cardinality of the set $\{x\in A^n\mid d(u,x)=i\}$ with $i\geq 0$ fixed is independent of the choice of $u\in A^n$.
Then we can write $d_i = |\{x\in A^n\mid d(u,x)=i\}|$ for $i\geq 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2931732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How do I solve for $g$ in $12g = 12 \left(\frac{2}{3g} - 1\right) + 11$? For this specific problem, I somehow keep coming up with the wrong answer. Can someone help me?
For the problem, I need to solve for $g$.
$$12g = 12 \left(\frac{2}{3g} - 1\right) + 11$$
Here is how I am trying to solve it:
$$12 \cdot \frac{2}{3g} ... | So you have the equation $12g = 12 (\frac{2}{3g} - 1) + 11$. You want to find $g$. $$12g=\frac{24}{3g}-12+11$$ $$\implies 12g=\frac{8}{g}-1$$ $$\implies 12g^2=8-g$$ $$12g^2+g-8=0$$ $$g=\boxed{\frac{-1\pm\sqrt{385}}{24}}$$
However, if you mean $12g=12(\frac{2}{3g-1})+11$, then we have $$12g=\frac{24}{3g-1}+11$$ $$\impli... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2931898",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Relation between determinant of a Jacobi matrix and its minor Let
$$
A = \begin{bmatrix}
a_1 & b_1 \\
b_1 & a_2 & b_2 \\
& b_2 & \ddots & \ddots \\
& & \ddots & \ddots & b_{n-1} \\
& & & b_{n-1} & a_n
\end{bmatrix},\ B = \begin{bmatrix}
a_2 & b_2 \\
b_2& a_3 & \ddots & \\
&\ddots & \ddots & b_{n-1} \\
& & b_{n-1} & a... | This proposition is not necessarily true in general.
Lemma: If $B$ and $C$ are square matrices and $a \in \mathbb{R}$, then there exists $b \in \mathbb{R}^*$ such that the equation$$
(λ - a) |λI - B| = b^2 |λI - C|
$$
has a real solution.
Proof: It suffices to prove that there exist $λ \in \mathbb{R}$ and $b \in \m... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Commutativity in Mapping Class Groups I am trying to understand the behavior of finite order mapping classes for surfaces of genus g>=2.
After fiddling for a while I started to think that no finite order mapping class commutes with any Dehn twist for g>=2. Is there a simple proof or disproof? Otherwise I'd appreciate ... | The thing to know is that, if $D_\alpha$ denotes the Dehn twist along a simple loop $\alpha$, then for an orientation-preserving homeomorphism $f$ of the surface $S$ we have $f D_\alpha f^{-1}= D_{f(\alpha)}$ (up to isotopy, of course). Thus, the issue reduces to finding a reducible mapping class of finite order $\ge 2... | {
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Confusion regarding the sample space of a experiment
First, of all, I need to realize what sample space is and this is causing me some trouble. Perhaps I am misunderstading the problem? But this is my reasoning:
Since player 1 has two choices at each stage and there is only 4 games he can have provided he wins to play... | There are $5$ discreet numbers which can be randomly arranged in $5! = 120$ ways which is your sample space.
I take it that $P(x=i)$ means $i =$ exactly $0,1,2,3,4$.
In the $120$ different sequences of $5$ numbers, we have $60$ sequences where the second number is larger than the first. ($1,2$ etc)
$P(0) = \frac{60}{1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Using trigonometry to prove $\frac{a}{1-a^2}+\frac{b}{1-b^2}+\frac{c}{1-c^2}=\frac{4abc}{\left(1-a^2\right)\left(1-b^2\right)\left(1-c^2\right)}.$
For the numbers $a,b,c$ with $ab+ac+bc=1$, prove that
$$\frac{a}{1-a^2}+\frac{b}{1-b^2}+\frac{c}{1-c^2}=\frac{4abc}{\left(1-a^2\right)\left(1-b^2\right)\left(1-c^2\right)}.... | Hint:
You probably know that if $x+y+z=\pi$, then
$$\tan\frac{x}{2}\cdot\tan\frac{y}{2}+\tan\frac{x}{2}\cdot\tan\frac{z}{2}+\tan\frac{y}{2}\cdot\tan\frac{z}{2}=1.$$
Now, since $ab+ac+bc=1$, use $a=\tan\frac{x}{2}$....
I hope this help you.
| {
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Calculating the value of the limit $$\lim_{(x,y)\to(1,2)}(\sin(y)-\sin(x))$$
My try:
I got as $$\sin(2)-\sin(1)$$
But I cannot calculate the exact value of the given limit. Can anyone please explain this.
| Yes your result is correct, indeed since the function is continuous at the point we have
$$\lim_{(x,y)\to(1,2)}(f(x,y))=f(1,2)$$
that is
$$\lim_{(x,y)\to(1,2)}(\sin(y)-\sin(x))=\sin 2 -\sin 1$$
where $2$ and $1$ are expressed in radians, a numerical evaluation leads to $\approx 0.068$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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In $ \mathbb{Z}_p $ where $ p $ is prime, what is the number of roots of $ x^2+1\equiv 0 \mod p $?
In $ \mathbb{Z}_p $ where $ p $ is prime, what is the number of roots of $ x^2+1\equiv 0 \mod p $?
Since $ x^2\equiv -1 \mod p $, then $ x^4\equiv 1 \mod p $ and we have $ 4 | \phi(p) $. If $ p=5 $, then $ 2,3 $ are two... | Explicitly, when $p \equiv 1 \bmod 4$, the solutions of $x^2 \equiv 1 \bmod p$ are $\pm \left(\frac{p-1}{2}\right)!$. See here.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Simplify $x^\frac{1}{(\log_a x)}$ Simplify $x^\frac{1}{(\log_a x)}$
The solution in my textbook is the following:
Since $\log_a (x^\frac{1}{(\log_a x)}) = \frac{1}{\log_a x}$ $\log_a x
= 1$,
therefore $x^\frac{1}{(\log_a x)} = a^1 = a.$
I understand the law used in the first line is $\log_a (x^y) = y \log_a x$. I al... | First, the expression has sense only for $x > 0$.
So for $x > 0$, the number $y= x^{\frac{1}{\log_a(x)}}$ exists and is $> 0$, so you can compute its $\log_a$. The computation shows that $\log_a(y)=1$.
Now, do you know a lot of numbers whose $\log_a$ are equal to $1$ ?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2932887",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is the standard model for the language of number theory elementarily equivalent to one with a nonstandard element? On page 89 in A Friendly Introduction to Mathematical Logic, the author writes that the standard model $\mathfrak{N}$ for $\mathcal{L}_{NT}$ is elementarily equivalent to a model $\mathfrak{A}$ that has an... | In the notes, I don't see the claim that $c$ is larger than all other numbers of $\mathfrak{A}$. The number $c$ in $\mathfrak{A}$ is larger than $0$, $S(0)$, $S(S(0))$, etc., - so $c$ is greater than every element of $\mathfrak{N}$. But there will be other elements of $\mathfrak{A}$ that are larger than $c$. Not every ... | {
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"url": "https://math.stackexchange.com/questions/2933036",
"timestamp": "2023-03-29T00:00:00",
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} |
How is $\sin^2(x)+\cos^2(x)=1$ where $x$ is an obtuse angle? Ok, so I know that $\sin^2(x)+\cos^2(x)=1$ for all angles. If $x$ is an acute angle in a right angled triangle it's a straightforward proof.
But what about if $x$ is obtuse ? How do I mathematically prove it plus get a visual analysis of the same so you can ... | I can't think of a geometric interpretation right now, but I would argue with the angle reduction formulas.
Let $x \in [\frac{\pi}{2}; \pi]$ be an obtuse angle. This means $x$ can be rewritten in terms of an acute angle $y \in [0; \frac{\pi}{2}]$ plus $\frac{\pi}{2}$, meaning $x=y+\frac{\pi}{2}$. Then, with the reducti... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Does $K[\alpha]$ has an inverse of $\alpha$ if this element is algebraic? I was asking myself a question :
Let $K$ be a field and $K[\alpha]$ the smallest ring containing $K$ and $\alpha$. If $\alpha$ is algebraic, can we always find $\alpha^{-1}$ in $K[\alpha]$.
My thoughts were as following. I $p$ is the minimal poly... | Now let's see . . .
First of all, the method proposed by our OP roi_saumon appears to be correct in its essentials, though lacking in a proof that $\gcd(x, p(x)) = 1$; how may we show this is the case? Well, if
$d(x) \mid x, \; d(x) \mid p(x) \; \text{with} \; x, \; d(x), \; p(x) \in K[x], \tag 1$
there is some $q(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2933242",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Closed form of $\int_0^\infty \left(\frac{\arctan x}{x}\right)^ndx$ I know that for $n=1$ the integral is divergent and that for $n=2$ the integral has a closed form. However, I wonder if the general expression has a closed form.
My attempt: $$\int_0^\infty \left(\frac{\arctan(x)}{x}\right)^ndx=\frac{n}{1-n}\int_0^\inf... | It wasn't requested, but instead of exact representations the OP might want an asymptotic expression for $n \to \infty.$ This one works well with the technique of Depoissonization. Make an exponential power series and analyze it asymptotically:
$$ \sum_{n=0}^\infty \frac{y^n}{n!} C_n = \int_0^\infty \exp{\big(\frac{y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2933377",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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When given a language, does the algebra follow when raising a symbol in the alphabet to 0 or a negative number? For example, I'm given the language {$a^j b a^k | j < k + 4$}. Do I have to worry about cases like $a^0$? Does it come out to 1 or $\epsilon$? What about things like $a^{-1}$?
| $a^0$ means the empty word $\epsilon$, though it's not sure whether they allowed $j=0$ or not.
Negative powers are not defined in this setting, the words only form a semigroup (actually, monoid with $\epsilon$), not a group, so that we don't have inverses.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2933529",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$A\cap G=A\cap H=1$ implies $A=1$? Suppose $G\times H$ is an abelian group and $A\le G\times H$.
Is it true that $A\cap G=A\cap H=1$ implies $A=1$? ($1$ is the trivial subgroup)
| In general, no. Take for example $\langle (1,1) \rangle \leq \mathbb{Z}_2 \times \mathbb{Z}_2$.
| {
"language": "en",
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Evaluate $\lim\big(\frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{2n}\big)$ using sequential methods
Evaluate $$\lim\Big(\frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{2n}\Big)$$ using sequential methods.
Of course: $$\frac{1}{n}\bigg(\frac{1}{1+1/n}+\frac{1}{1+2/n}+\ldots+\frac{1}{1+n/n}\bigg) \rightarrow \int_{0}^{1}\fra... | Using the fact that $\ln(1 + x) \leq x$, we have
$$
\ln(k + 1) - \ln k \leq \frac{1}{k} \leq \ln{k} - \ln(k - 1),
$$
hence
$$
\ln \frac{2n + 1}{n + 1} \leq \sum_{k = {n + 1}}^{2n} \frac{1}{k} \leq \ln \frac{2n}{n} = \ln2,
$$
and the squeeze theorem will give the result.
| {
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"source": "stackexchange",
"question_score": "5",
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About dense subspaces in Banach spaces I have the following problem:
Let $E$ a Banach space and $X_1,X_2$ dense subspaces. Is $X_1\cap X_2$ dense in $X$? What is the answer if $X_1,X_2$ has codimension 1?
I don't know how to start. If anyone can give me a hint it will be appreciated.
Thanks in advance!
| If $X_1$ and $X_2$ are dense, $X_1 \cap X_2$ need not be dense, in fact it might be just $\{0\}$. Consider any infinite-dimensional separable Banach space $E$. Start with a dense countable set $\{w_1, w_2, \ldots\}$. Inductively choose two sequences $x_j$ and $y_j$ such that $\|x_j - w_j\| < 1/j$ and $\|y_j - w_j\| ... | {
"language": "en",
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Isomorphisms in a reflective subcategory Let $S$ be a small family of arrows in a locally presentable category $\mathcal{K}$.
It is known that the category $\mathcal{K}[S^{-1}]$ is reflective in $\mathcal{K}$ and correspond to the solution of the orthogonality problem associated to $S$.
Can I infer that a map is an iso... | No, you cannot infer this. The class of morphisms inverted by any functor must satisfy the two-for-three, even the two-for-six, property. But $S$ is completely arbitrary. Consider, for a dramatic example, $\mathrm{Set}[(\emptyset \to \{*\})^{-1}]$, where inverting a single arrow is equivalent to inverting all arrows. F... | {
"language": "en",
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$18a$ and $25a$ both integers, then so is $a$ Let $a\in \mathbb{Q}$ such that $18a$ and $25a$ are integers, then we wish to prove that $a$ must be an integer itself. What that means is that $a=\frac{p}{1}$ where $p \in \mathbb{Z}$. What we do know is that we can express the $\gcd(18,25)$ as:
$$ \gcd(18,25)=18x +25y$$ ... | Another way to look at this: $18a$ and $25a$ are integers. Therefore, so is $25a-18a = 7a$.
Therefore so is $18a-2(7a) = 4a.$
Therefore so is $7a-4a = 3a.$
Therefore so is $4a-3a = a.$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding the parametric and vector forms of the line is perpendicular to two lines Finding the parametric and vector forms of the line is perpendicular to lines $(4t,1+2t,3t)$ and $(−1+s,−7+2s,−12+3s)$
And passes through the point of the intersection of two lines
A vector perpendicular to these lines is $$v = (4, 2, 3) ... | HINT
We have found the direction vector $\vec v$ for the perpendicular line, now we need the intersection point $P_0$ to determine the parametric equation
$$P(t)=P_0+t\vec v$$
| {
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How to evaluate contour integral I have a homework problem to evaluate the integral
$$
\oint_{\gamma}\frac{\cos z}{(z+i)^3}dz
$$
along the curve $\gamma(t)=-i+e^{it}, t\in[0,2\pi]$. I proceeded to plug the given information into the definition of a contour integral and got to the expression
$$
\oint_{\gamma}=i\int_{0}^... | Hint: As Jose hints in the comments, we want to use the fact that
$$f^{(k)}(z_0)=\frac{k!}{2\pi i}\oint_{\partial B_r(z_0)}\frac{f(z)}{(z-z_0)^{k+1}}dz$$
with the correct holomorphic function $f(z)$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to solve $y=\tan(θ)$ for $θ$? If $\arctan(\tan(\theta))$ is not necessarily equal to $\theta$, how come if we are given $y=\tan(θ)$ the solution in terms of $\theta$ is $\theta=\arctan(y)$?
I'm trying to intergrate $1/(1+y^2)$ using trig substitution and I am trying to get my solution, $\theta$, in terms of $y$, $y... | Hint
$$\tan x=\tan (x+k\pi), k\in \mathbb{Z}$$
$$\int \frac{dx}{1+x^2}=\arctan x + C.$$
$C$ plays the role of not having $\arctan(\tan \theta)=\theta$ depending on the range.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2934516",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Projectile Envelope
Consider a projectile launched from the origin at velocity $v$ and angle $\theta$. From this other question we see several approaches to arrive at the equation for the envelope of different trajectories with varying $\theta$.
We know from standard projectile formulas that
(i) at $\theta=\frac \pi... | I put together an argument that I think relies only on true facts and avoids actually computing either the envelope's equation or the equation of any trajectory. I suspect that the proofs of the facts used in this argument require (at least in some cases) more sophistication than the calculation using the discriminant ... | {
"language": "en",
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The problem of dividing 3kg flour to three 1kg parts using a balance Suppose you have 3kg flour, and you are asked to divide it to three 1kg parts using a balance scale.
It seems to me that it's impossible to do with a finite number of weighing, but I can't see how to prove it. Is this a known problem? Any hint is app... | Theoretically, assuming it is possible to use the balance scale to divide any flour quantity to two equal parts. Make a continuous sets of two half dividing rounds. At end of each set, sum one side and use the other side to continue the two dividing rounds sets. This shloud give you the sum:
$$ \color{red}{S} =\frac{3}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$A^TA=B^TB$. Is $A=QB$ for some orthogonal $Q$? Suppose that $A$ and $B$ are two real square matrices and $A^TA=B^TB$. Can we say that $A=QB$ for some orthogonal matrix $Q$?
If they are vectors we have $\|a\|^2=a^Ta=b^Tb=\|b\|^2$, so intuitively clear, since we just have to rotate. But it is hard to picture the matrix ... | There is also a nice geometric way to see this.
The geometric interpretation of the singular value decomposition says that a $(n\times n)$-matrix $A$ maps the unit $(n-1)$-sphere in $\mathbb{R}^n$ to a hyperellipsoid. The lengths of the axes of this ellipsoid are the roots of the eigenvalues of $A^T A$.
So if $A^T A =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2934844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solving green function for first quadrant with boundary conditions (dirichlet problem?) Let $\Omega = \{x_1,x_2\in\mathbb{R}^2: x_1>0, x_2>0\}$. Solve the problem $$\Delta u = 0 \mbox{ in $\Omega$ }, u\in C^2(\Omega)\cap C(\overline\Omega) \mbox{ bounded}\\u(x_1,0)=u_0(x_1),x_1\ge 0\\u(0,x_2) = 0, x_2\ge 0$$
Where $u_0... | It is well known that the solution of the Dirichlet problem in a half plain
$$\Delta u=0,\ x_2>0,\quad u(x_1,0)=\psi(x_1),$$
is given by the integral
$$
u(x_1, x_2) = \frac1\pi
\int_{-∞}^∞
\frac{x_2\psi(y)}{(x_1-y)^2+x_2^2}\,dy.
$$
Now extending function $u_0$ to $\tilde u_0$ on $\mathbb R$ as odd:
$\tilde u_0(x_1)=u_... | {
"language": "en",
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Prove that there is no $\{3,5\}$-Hall subgroup in $A_{5}$ My approach.
A $\{3, 5\}$-Hall subgroup $K$ of $A_{5}$ has order $3\cdot 5$ and index $2^{2} = 4$. Note that $A_{5}$ acts in cosets of $K$, with $4$ distincts cosets, this way:
$$A_{5}/K = \{a_{1}K, a_{2}K, a_{3}K, a_{4}K\}.$$
Thus, we have a homomorphism $\varp... | Even easier is to note that all groups of order 15 are cyclic, and $A_5$ has no element of order 15. Thus $A_5$ has no subgroup of order 15.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Let $\mathcal F$ be the set of mappings $f:\Bbb N \to \Bbb N$ for which $f(m) \ge f(n)$ for $m \le n$. Show that $\mathcal F$ is countable
Let $\mathcal F$ be the set of mappings $f:\Bbb N \to \Bbb N$ for which $f(m) \ge f(n)$ for $m \le n$. Show that $\mathcal F$ is countable.
My attempt:
For all $f\in \mathcal F$,... | An alternative approach written for fun and/or clarity:
This produces a sequence with maximum $f(1)$ that eventually becomes constant (as pointed out in the OP). Let us just consider the cases where $f(1) = 3$ for concreteness. This means all sequences that are of one of the following forms:
*
*all $3$s
*finitely m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2935115",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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Finding a bijection from $\{1,2,...,nm\}$ to $X \times Y$ I'm trying to prove that for two finite sets $X,Y$, where $|X|=n$, $|Y|=m$, $|X||Y|=|X \times Y|$. I know that there exists bijections $f:\{1,2,...,n\} \rightarrow X$ and $g:\{1,2,...,m\} \rightarrow Y$ and I'm trying to find a bijection $h:\{1,2,...,nm\} \right... | First let me do a little change: instead of $\{1,2,...,n\},\{1,2,...,m\},\{1,2,...,nm\}$ I will use $\{0,1,...,n-1\},\{0,1,...,m-1\},\{0,1,...,nm-1\}$.
Now, to prove $k$ is bijective I will find the inverse of $k$: $k(x_i,y_j)=(i-1)m + j$, to isolate $i$ I will use the fact that $j<m$, first let's divide by $m$: $\frac... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Whittaker-Shannon-Kotel’nikov theorem in higher dimensions The fundamental result in sampling theory states that if a signal $f(t)$
contains no frequencies higher than $\omega$ cycles per second, then $f(t)$ is completely
determined by its values $f(\frac{k}{2\omega})$ at a discrete sequence of sample
points with spaci... | After a bit of search I have found the answer:
THEOREM 1.6$\dagger$
Let $f(t_1 , t_2 ,\ldots , t_n )$ be a function of $n$ real variables, whose
$n$-dimensional Fourier integral, $g$, exists and is identically zero outside an $n$-dimensional rectangle and is symmetrical about the
origin, that is,
$g(y_1 , y_2 , \ldots ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How can I prove that if $A$ is a singular square matrix, then $\det A = 0$ without using Binet's Theorem? Well, I need to prove that if $A$ is a $n \times n$ matrix and it is singular, then $\det A = 0$, in order to show Binet's Theorem $\det(AB) = \det(A)\cdot\det(B)$ in the case when both $A$ and $B$ are singular squ... | I would use two proof
*
*Permutation change the sign of the determinant
*Multiplying a row by a constant multiply the determinant by this constant
Then, you can just say if you have a singular matrix S, 2 row $r_i$ and $r_j$ are linearly dependent. You can multiply $r_i$ by a constant C to get a new matrix S' where... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Removing factors to make the number ends with'2225' We have the integers 1, 2, ..., 50000.If we multiply them together, the last four digits will be '0000'. At least how many integers must be removed so that the product of the remaining integers ends with '2225'?
I know that if an integer ends with '2225', it's factors... | Suppose the product of the $20000$ with no $2$ or $5$ is $A$, and we choose $5B$ and $5C$. We want to choose $B$ and $C$ so that $ABC=400D+89$. Let $B=1$.
• Let $C_1$ be whatever digit makes $AC_1$ end in $9$.
• Let $C_2$ be whatever digit makes $A(C_1+10C_2)$ end in 89.
• Let $C_3$ be whichever of $1,2,3,4$ makes $... | {
"language": "en",
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Name and derivation of the approximation $\frac{1+x}{1+y} - 1 \approx x-y$? I am wondering if there is a name and way to derive the following approximation:
$$\frac{1+x}{1+y} - 1 \approx x-y$$
I'm essentially interested in how to refer to this.
| I would call this a Taylor approximation. When $|y|\lt1$,
$$
\begin{align}
\frac{1+x}{1+y}-1
&=-1+(1+x)\left(1-y+y^2-y^3+\dots\right)\\
&=x-y-xy+y^2+xy^2-y^3-xy^3+\dots\\[6pt]
&=x-y+O\!\left(\max(|x|,|y|)^2\right)
\end{align}
$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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I can't seem to undersatnd how did we compute $\frac{}{}$ from the function $\mathbf T=\frac{v }{1+uvw}$ $$ \mathbf T = \frac{v}{1+uvw}$$
Solution:
$$d\mathbf T =
\frac{\partial\mathbf T} {\partial u}\,du +\frac{\partial\mathbf T}{\partial v} \, dv +\frac{\partial\mathbf T}{\partial w} \, dw $$
$$d\mathbf T =
\frac{−v... | Quotient rule:
\begin{align}
& \frac\partial {\partial v} \, \frac v {1+uvw} = \frac{(1+uvw) \dfrac\partial{\partial v} v - v \dfrac\partial {\partial v} (1+uvw)}{(1+uvw)^2} \\[12pt]
= {} & \frac{(1+uvw) - v(uw)}{(1+uvw)^2} = \frac 1 {(1+uvw)^2}.
\end{align}
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find coefficients to a quadratic equation knowing roots and a point... Given a standard quadratic equation:
$$p(z) = az^2 + bz + c$$
We know that $-10$ and $10-i$ are roots.
We know that $p(i)=-10$
What are $a$, $b$ and $c$?
| Try $p(z)=a(z-i)(z-10+i)$
Multiply and find coefficients using the information at $z=i$
| {
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Does $3-4+1$ equal $0$ or $-2$.. or maybe $2$? Does $3 - 4 + 1 = 0$ or $3 - 4 + 1 = -2$?
Makes sense that $(3 - 4) + 1 = 0$ and $3 - (4 + 1) = -2$, but what if there are no parenthesis?
Also, if I have $4$ apples and I add $1$ more apple, then I have $5$ apples, but if I eat $3$ apples, then I have $2$ left.
Any ideas?... | Add positives together, and you will have $3+1=4$
The only negative that you have is $-4$
The sum is then $4-4=0$
If you do not have parenthesis you add positives and keep the result.
Then you add negatives and keep the result. Then find the total result by the algebraic sum of those two.
For example $$1-5-3+12-23=?$$
... | {
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A box has three coins. One has two heads, another two tails and the last is a fair coin. I am stuck on this question:
A box has three coins. One has two heads, one has two tails, and the other is a fair coin with one head and one tail. A coin is chosen at random, and comes up head.
a) What is the probability that th... | Your answer to part (a) is incorrect. If you pull out a random coin and flip, you have six scenarios:
*
*Head 1 of 2-headed coin
*Head 2 of 2-headed coin
*Head of fair coin
*Tail of fair coin
*Tail 1 of 2-tailed coin
*Tail 2 of 2-tailed coin
3 of those are "heads", and 2 of those 3 correspond to the 2 headed ... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "7",
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Understand a dynamical figure: stable manifold, node, domain of attraction Could someone please point out where the node, saddle and stable manifold are in this figure (as indicated in the caption)? I can see there is a focus, but I am not sure about the others. Thank you in advance.
Source: http://www.staff.science.uu... | The focus is at the center where the flow is spiraling in.
The stable manifold is the slant line coming down from the focus.
The saddle is the intersection of the slant line and the middle trajectory.
The node is intersection of the slant with the lower trajectory.
| {
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Solving cos(x)=x Algebraically I think a graphical approach is the best way, but can algebra be used? With IVT, from x:[0,1] a solution must exist.
| As said in comments and answers, you need a numerical method such as Newton.
Consider the function $$f(x)=x-\cos(x)\qquad f'x)=1+\sin(x)$$ The iterates will be given by
$$x_{n+1}=\frac{x_n \sin (x_n)+\cos (x_n)}{1+\sin (x_n)}$$ You can have a very good starting point building the $[2,2]$ Padé approximant of the functio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2936354",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Subcategories of T1, Hausdorff and TD spaces are reflective subcategories of Category *Top*. How do I construct a T1, Hausdorff and TD reflections to show that subcategories of T1, Hausdorff and TD spaces are reflective subcategories of Category Top?
The T0 reflection is just the T0 quotient of a space X. What about t... | This question and its answers talk in detail about the Hausdorff reflection.
The $T_1$ reflection of $X$ is the quotient under the intersection of all equivalence relations on $X$ that have closed equivalence classes, see the post here with an attempted proof (the previous one has the statement).
My details: Let $X$ b... | {
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"url": "https://math.stackexchange.com/questions/2936481",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $b_k \to 1/a$ if $a_k \to a$ where $b_0 = 0$ and $b_k = 1/a_k$ for $k>0$
Let $a_n$ where $n \in \mathbb {N}$ be a sequence of rational numbers converging to $a$. Suppose $a \neq 0$, for $k = 1, 2, ...$ let
$$b_k=\begin{cases} 0 & \text{if}\;a_k=0\\\\\frac{1}{a_k} &\text{if}\;a_k \neq 0\end{cases}$$
Pro... | Hint
Because $a\not=0$ (let's say $a>0$), show that $\exists n_0\in\mathbb{N}: \forall n\geq n_0\quad a_n>0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2936608",
"timestamp": "2023-03-29T00:00:00",
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7 fishermen caught exactly 100 fish and no two had caught the same number of fish. Then there are three who have together captured at least 50 fish.
$7$ fishermen caught exactly $100$ fish and no two had caught the same number of fish. Prove that there are three fishermen who have captured together at least $50$ fish.... | If the maximum number of fish caught is $m$, then the total number of fish caught is no more than $m+(m-1)+...+(m-6)$. So there is one fisherman that caught at least 18 fish. Repeat this process for the second and third highest number of fish caught and you should be good.
I should add that this is a common proof techn... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove this problem using Mean - Value theorem let $m$ and $n$ are in $\mathbb{N}$ with $m>n$ . Use the Mean Value Theorem, to prove that $(1+x)^\frac{m}{n}\ge 1+\frac{m}{n}x, x \ge -1$
since $(1+x)^\frac{m}{n}$ is continuous on $[-1,x]$ and differentiable on $(-1,x)$ so by mean value theorem $\frac{f(x)-f(-1)}{x+1}=f... | Let $f(x) = (1+x)^{\frac mn}$. Then
$$f'(x) = \frac mn (1+x)^{\frac mn -1} =\frac mn (1+x)^{\frac {m-n}n} $$
$f'(x)$ is increasing for $x\ge -1$, and in particular, $f'(0) = \frac mn$.
Case 1: $x > 0$. Consider $f(0)$ and $f(x)$. By mean value theorem, there exists a $c\in (0, x)$ that satisfies
$$\begin{align*}
f'(c) ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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} |
Evaluating $\lim_{n \to \infty}\left({^n\mathrm{C}_0}{^n\mathrm{C}_1}\dots{^n\mathrm{C}_n}\right)^{\frac{1}{n(n+1)}}$
$\lim_{n \to \infty}\left({^n\mathrm{C}_0}{^n\mathrm{C}_1}\dots{^n\mathrm{C}_n}\right)^{\frac{1}{n(n+1)}}$ is equal to:
a) $e$
b) $2e$
c) $\sqrt e$
d) $e^2$
Though it looks really innocent at first si... | $$\prod_{k=0}^{n}\binom{n}{k}=\frac{n!^{n}}{\prod_{k=0}^{n}k!^2}=\frac{n!^n}{\left[\prod_{k=1}^{n}k^{n+1-k}\right]^2}=\frac{n!^n}{n!^{2n+2}}\prod_{k=1}^{n}k^{2k} \tag{1}$$
hence the outcome depends on the asymptotic behaviour of the hyperfactorial.Since by Riemann sums
$$ \lim_{n\to +\infty} \frac{1}{n}\sum_{k=1}^{n}\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2936916",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Are there uncountably many injective functions from $\mathbb N$ to $\mathbb N$? I think there are, but I haven't been able to prove this. I tried to make two injections, but I get stuck on trying to map all functions f from $\mathbb N$ to $\mathbb N$ onto the injective ones. How do you make sure f becomes injective, wh... | Each injection from $\mathbb{N} \to \mathbb{N}$ can be thought of as an infinite sequence. Now use Cantor's diagonal argument to show uncountably many functions.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
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Limit of a sum using complex analysis. I'm trying to find the limit of this sum: $$S_n =\frac{1}{n}\left(\frac{1}{2}+\sum_{k=1}^{n}\cos(kx)\right)$$
I tried to find a formula for the inner sum first and I ended up getting zero as an answer.
The sum is supposed to converge to $\cot(x\over 2)$ and that appears in my last... | Just write $\cos (kx)$ as the real part of $e^{ikx}$.
| {
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"url": "https://math.stackexchange.com/questions/2937135",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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What is the formal definition of a Mackey Topology? I believe the question is self-explanatory. After browsing in the web the following definitions stand out:
Assume we have a dual pair $(X,X')$
*
*The topology of uniform convergence on the convex balanced subsets of $X'$ that are compact in the weak topology $\sigm... | Both definitions are perfectly `formal', provided you know what all the terms mean. The wikipedia definition links to polar topology where you find an explanation how the topology is generated.
Both the wikipedia page and the Encyclopedia page point to textbooks on topological vector spaces where you can learn more.
Pr... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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Proving $x$ between $\alpha - \epsilon$, if $\alpha$ is supremum Let $S\subseteq\Bbb R$ and $\alpha \in \Bbb R$. If $\alpha = \sup(S)$, then show that for any $\epsilon > 0$, there is some $x \in S$ such that $\alpha - \epsilon < x$.
What I have done :
Since $\alpha$ is the supremum, $x<\alpha$ and $\alpha - \epsilon <... | We want to show the existence of such $x$.
Suppose it doesn't exist, then we have $\alpha - \epsilon$ being an upper bound of $S$ which contradicts that $\alpha$ is the least upper bound.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2937454",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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If Rolle's Theorem is assumed to be true, doesn't that prove the MVT? If we assume that Rolle's Theorem is true is it practical to say that it also proves the MVT?
My reasoning is that even though Rolle's Theorem is the special case for when $f(a)=f(b)$ and the secant line between $(a,f(a))$ and $(b,f(b))$ is horizonta... | A proof of MVT used Rolle's theorem explicitly.
Consider a continuous function $f:[a,b]\to\Bbb R$ that is differentiable on $(a,b)$. We are to prove MVT, i.e. to prove there exists a point $c\in(a,b)$ such that $f'(c)=\frac{f(b)-f(a)}{b-a}$, assuming that Rolle's theorem is true.
Define a new function $g:[a,b]\to\Bbb R... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Solution of a system of congruence equations Consider the following system out of which I want to find all possible values of $\lambda$.
\begin{eqnarray}
1287\lambda\equiv0 (\mathrm{mod}\ 6)\label{eq1}\\
165\lambda\equiv0 (\mathrm{mod}\ 4)\label{eq2}\\
9\lambda\equiv0 (\mathrm{mod}\ 2)\label{eq3}
\end{eqnarray}
Since I... | You are correct that $t$ must be a multiple of $9$ and $\lambda$ must be even, but this actually only uses the third equation. You also have the condition that $r=\frac{55t}6$ is an integer, and this means $t$ also has to be a multiple of $6$, so $t$ is a multiple of $\operatorname{lcm}(9,6)=18$. (You also need the fir... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2937686",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Counterexamples Of The Infinite Dimensional Linear Space Let $V$ and $W$ be two finite dimensional linear spaces over the field $\mathbb{F}$ and $\mathscr{A} :V\rightarrow W$ be a linear map between $V$ and $W$. Then we have
*
*$\mathscr{A}$ is an injective linear map if and only if there exists a linear map $\mat... | Exercise (1):
$\Leftarrow$: Assume $A$ was not injective, i.e. we had $v, v'$ with $Av = Av'$. Then $BAv = v = v' = BAv'$, which is a contradiction.
$\Rightarrow$: Define the linear map $B: Im(A) \to V$ by
$$B(w) \in A^{-1}(w)$$
for every $w \in Im(A)$. This definition is unique and well-defined since $A$ is injective.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2937812",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Calculate limit of $\frac{1}{n}\cdot (1+\sqrt{2}+\sqrt{3}+\cdots+\sqrt{n})$ How to prove that
$$\lim_{n\to\infty}(\frac{1}{n}\cdot(1+\sqrt{2}+\sqrt{3}+\cdots+\sqrt{n})) = +\infty$$
using only basic limit operations and theorems?
| Creative telescoping does the job nicely. We have
$$(n+1)\sqrt{n+1}-n\sqrt{n} = \sqrt{n+1}+\frac{n}{\sqrt{n}+\sqrt{n+1}} \leq \frac{3}{2}\sqrt{n+1}$$
hence
$$\sum_{k=1}^{n}\sqrt{k}\geq \frac{2}{3}\sum_{k=1}^{n}\left[n\sqrt{n}-(n-1)\sqrt{n-1}\right] = \frac{2}{3}n\sqrt{n}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2938083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
Prove that upper triangular matrices are closed under inversion Prove that the group of upper triangle matrices $G\subset GL_n(F)$ is also a subgroup of $GL_n(F)$
Contains identity: $I_n\in G$ since $I_{ij}=0$ when $i>j$
Closed under multi: $C=AB, c_{ij}=\sum_{k=1}^na_{ik}b_{kj},i=1,2..,n,j=1,2..,n$ for all $c_{ij},i>j... | Conceptually, the algebra of upper triangular matrices $UT_n(F)$ is the set of transformations which fixes a complete flag of subspaces of $F^n$.
That is, for all upper triangular matrices, there is a chain of subspaces of $F^n$ written as $0\subseteq V_1\subseteq V_2\subseteq\ldots\subseteq V_n$, where $\dim_F(V_i)=i$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2938222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Is $\mathbb{Z}_p [x]$ finite? Seems to me like it is. There are only finitely many distinct powers of $x$ modulo $p$, by Fermat's Little Theorem (they are $\{1, x, x^2, ..., x^{p-2}\}$), and the coefficient that I choose for each of these powers can only be taken from $\{0,1,2,..., p-1\}$. So essentially I'm choosin... | $\Bbb Z_p[x]$ is indeed infinite, since it contains polynomials, with coefficients in $\Bbb Z_p$, of arbitrary high degree, e.g. $x^n \in \Bbb Z_p[x]$, where $n \in \Bbb N$; also, $x^n \ne x^m$ if $m \ne n$.
Fermat's Little Theorem, that $a^p = a$ for $a \in \Bbb Z_p$, does not apply to the indeterminate $x \in \Bbb Z_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2938372",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 1
} |
Partial sum of divergent series I am trying to find the nth partial sum of this series:
$S(n) = 2(n+1)^2$
I found the answer on WolframAlpha:
$\sum_{n=0}^m (1+2n)^2 =\frac{1}{3}(m+1)(2m+1)(2m+3)$
How can I calculate that sum, without any software?
| $\sum_\limits{i=0}^n 2(i + 1)^2 = 2\sum_\limits{i=1}^{n+1} i^2$
Which gets to the meat of the question, what is $\sum_\limits{i=1}^n i^2$?
There are a few ways to do this. I think that this one is intuitive.
In the first triangle, the sum of $i^{th}$ row equals $i^2$
The next two triangles are identical to the first... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2938478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Sum involving Hermite polynomials I am wondering if there is a simpler form of the following summation involving the (physicists') Hermite polynomials:
$$\sum_{k=0}^{n}\frac{H_k(x)}{(2i)^k},$$
where $i=\sqrt{-1}$ is the imaginary unit. I would love to find a "closed form" solution for this summation, perhaps in terms o... | An operator form, or symbolic for computation, can be seen as
$$\sum_{k=0}^{n} \frac{H_{k}(x)}{(2 \, i)^{n}} = (-i)^{n} \, e^{- D^{2}/4} \, \left( \frac{x^{n+1} - i^{n+1}}{x - i} \right).$$
This is obtained by using
$$H_{n}(x) = e^{- D^{2}/4} \, (2 x)^{n}$$ and then summing the resulting series.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2938597",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Bit strings of length $n$ in which any two consecutive $1$'s are separated by an even number of $0$’s I want to find the number of bit strings of length $n$ in which any two consecutive $1$'s are separated by an even number of $0$’s
My attempt:
Let $a_n$ be the required number
Take two cases:
*
*The string ends with... | Form a graph with $6$ nodes: $S, A, B, C, D, E$.
*
*$S$ the starting state (we're here when the string is empty)
*$A$ a state for beginning zeros
*$B$ a state where you have just gotten a one (and the string isn't already rejected)
*$C$ counting zeros after a one, have odd amount
*$D$ counting zeros after a one,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2939104",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
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