Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Product (arbitrary) of open functions is open. Let $f_{\alpha}\colon X_{\alpha}\to Y_{\alpha}$ be open, for all $\alpha \in J$. Then $\prod_{\alpha} f_{\alpha}\colon \prod_{\alpha}X_{\alpha} \to \prod_{\alpha}Y_{\alpha}$ is open? Both $ \prod_{\alpha}X_{\alpha}$ and $ \prod_{\alpha}Y_{\alpha}$ have the product topology... |
2.3.29. Proposition. The Cartesian product $f=\prod_{s\in S} f_s$, where $f_s \colon X_s \to Y_s$ and $X_s\ne\emptyset$ for $s\in S$, is open if and only if all mappings $f_s$ are open and there exists a finite set $S_0\subset S$ such that $f_s(X_S)=Y_S$ for $s\in S\setminus S_0$.
Proof. From 1.4.14 and the equality... | {
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"timestamp": "2023-03-29T00:00:00",
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Cauchy integral formula Can someone please help me answer this question as I cannot seem to get to the answer.
Please note that the Cauchy integral formula must be used in order to solve it.
Many thanks in advance!
\begin{equation*}
\int_{|z|=3}\frac{e^{zt}}{z^2+4}=\pi i\sin(2t).
\end{equation*}
Also $|z| = 3$ is given... | Write integral in the form the function in the form $\frac{e^{zt}}{(z-2i)(z+2i)}$. Then you should split the contour in two parts such that interior of each part contain only one point $2i$ or $-2i$.
Then apply Cauchy integral formula for each contour.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to improve visualization skills (Graphing) Okay, so my problem is, that I have difficulty visualizing graphs of functions.
For example, if we have to calculate the area bounded by multiple curves, I face difficulty in visualizing that how the graph would look like. And in Integral Calculus, it is very important to ... | A friend of mine sent me a link to graph paper site: http://www.printablepaper.net/category/graph
gives you a choice of squares per inch and whether there are bolder lines each inch. Draw stuff. Do not imagine drawing things. Actually do so. If you want to solve $3 |x| = e^x,$ print out some graph paper, draw $y = 3|x|... | {
"language": "en",
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What does a linear equation with more than 2 variables represent? A linear equation with 2 variables, say $Ax+By+C = 0$, represents a line on a plane but what does a linear equation with 3 variables $Ax+By+Dz+c=0$ represent? A line in space, or something else?
On a general note, what does a linear equation with $n$ va... | In Geometry: the linear equation: $Ax+By+Dz+c=0$ in three variables $x$, $y$ & $z$ generally represents a plane in 3-D co-ordinate system having three orthogonal axes X, Y & Z.
The constants $A$, $B$, $D$ shows the $\color{#0ae}{\text{direction ratios}}$ of the vector normal to the plane: $Ax+By+Dz+c=0$. Its direction... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Area of an ellipse. I need to find the area of the image of a circle centred at the origin with radius 3 under the transformation:
$
\begin{pmatrix}
3 & 0\\
0 & \frac{1}{3}
\end{pmatrix}
$
The image is the ellipse $ \frac{x^2}{81}+y^2=1$. It would appear that it has the same area as the original circle i.e. $9\pi$. Is... | It is a known formula that the area enclosed in the ellipse with semi-axes $a$ and $b$ is $\pi ab$, as may be seen from the orthogonal affinity that transforms a circle into an ellipse.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Algebric and geometric multiplicity and the way it affects the matrix Given a matrix $A$. Suppose $A$ has $\lambda_1,\dots,\lambda_n$ eigenvalues each with $g_i$ geometric multiplicity and $r_1,\dots,r_n$ algebric multiplicity, $g_i\leq r_i$.
Given this information alone, can I understand the way the matrix might look ... | The difference between algebraic and geometric comes from the number of linearly independent eigenvectors.
Geometric multiplicity is strictly less than algebraic multiplicity if and only if the number of linearly independent eigenvectors is less than $n$ and some eigenvectors have to be repeated in an eigendecompositio... | {
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$A$ is diagonalizable if $A^8+A^2=I$ Given a matrix $A\in M_{n}(\mathbb{C})$ such that $A^8+A^2=I$, prove that $A$ is diagonalizable.
So let $p(x)=x^8+x^2-1$ and we know that $p(A)=0$.
The next step would be to show that the algebric and geometric multipliciteis of all the eigenvalues are equal.
But this polynomial is ... | It suffices to show that all the eigenvalues are simple. If $\lambda$ is an eigenvalue with multiplicity $\ge2$ then we have
$$\lambda^8+\lambda^2-1=0\tag 1$$
and
$$8\lambda^7+2\lambda=0$$
but clearly $0$ isn't an eigenvalue so
$$\lambda^6=-\frac14\tag2$$
so form $(1)$ we get $$-\frac14\lambda^2+\lambda^2-1=\frac34\lam... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1299681",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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integrate this double integral by any method you can. I'm having trouble with this double integral:
$$\int_0^2\int_0^{2-x} \exp\left(\frac{x−y}{x+y}\right)\text dy\,\text dx$$
| The integral is one over the $2$-simplex $\Delta_2(2) = \{ (x,y ) : x, y \ge 0, x+ y \le 2 \}$.
One standard trick to deal with integral over $d$-simplex of the form
$$\Delta_d(L) = \{ (x_1, x_2, \ldots, x_d ) : x_i \ge 0, \sum_{i=1}^d x_i \le L \}$$
is convert it to one over the $d$-cuboid $[0,L] \times [0,1]^{d-1}... | {
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Show that the subset $S$ in $\mathbb{R}_3$ is a subspace. Show that the subset $S$ in $\mathbb{R}_3$ defined by $S=\{(a,b,c) \in \mathbb{R}_3 \text{ such that } a+b=c \}$ is a subspace.
I'm having trouble adapting the definition of subspace with the part $a+b=c$.
| You want to show, that
$$ S = \left\{ (a,b,c) \in \Bbb R^3 \; : \; a+b = c \right\} \subset \Bbb R^3$$
is a subspace of $\Bbb R^3$.
First, note that $(0,0,0) \in S$, since $0 + 0 = 0$, so $S \neq \emptyset$.
Next, let $v_1 := (a_1, b_1, c_1), \, v_2 := (a_2, b_2, c_2) \in S$. We have to show, that $v_1 + v_2 = (a_1 ... | {
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"timestamp": "2023-03-29T00:00:00",
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Show that $\frac {\sin x} {\cos 3x} + \frac {\sin 3x} {\cos 9x} + \frac {\sin 9x} {\cos 27x} = \frac 12 (\tan 27x - \tan x)$ The question asks to prove that -
$$\frac {\sin x} {\cos 3x} + \frac {\sin 3x} {\cos 9x} + \frac {\sin 9x} {\cos 27x} = \frac 12 (\tan 27x - \tan x) $$
I tried combining the first two or the la... | @MayankJain @user,
I don't know that the case for $n=1$ is trivial, especially for someone in a trig class currently. I will offer a proof without induction using substitution instead. Mayank, to see that this is the case for $n=1$, convert the RHS of the equation as follows:
$\dfrac{1}{2} \cdot \big{[}\dfrac{\sin 3x}{... | {
"language": "en",
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"question_score": "4",
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Is there any cyclic subgroup of order 6 in in $ S_6$? Is there any cyclic subgroup of order 6 in $ S_6$?
Attempt:
$|S_6|=6!=720$
Let $H$ be a subgroup of $S_6$ ,$H$ cyclic $\iff\langle H \rangle=\{e,h,h^2,...,h^{n-1}\}=S_6$
| Yes: no need of great theorems. The subgroup generated by $(1,2,3,4,5,6)$ does the job.
| {
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Area Between Intersecting Lines - Elegant Solution? I am running simulations, and the output will be a line y = mx+b. I am interested in the area below the line between x=0 and x=1. I am only interested in the area that is below the diagonal y = x.
I have figured out how to determine this area by finding areas of trian... | Let $y = mx+b$ be the equation of your line, and then find the point of intersection with $y=x$, which will be $(p,p)$.
Then you need to find two integrals: the integral from $0$ to $p$ of $x-(mx+b)$ and the integral from $p$ to $1$ of the same function. Then pick whichever one is nonnegative!
EDIT: Just realized the l... | {
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How to evaluate $\cot(2\arctan(2))$? How do you evaluate the above?
I know that $\cot(2\tan^{-1}(2)) = \cot(2\cot^{-1}\left(\frac{1}{2}\right))$, but I'm lost as to how to simplify this further.
| Let $y=\cot(2\arctan 2)$. You can use the definition $\cot x\equiv \frac{1}{\tan x} $ and the identity for $\tan 2x\equiv\frac{2\tan x}{1-\tan^2 x}$ to find $\frac 1y=\tan(2 \arctan 2)$ in terms of $\tan(\arctan 2) = 2$.
| {
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How can I simplify $1\times 2 + 2 \times 3 + .. + (n-1) \times n$ progression? I have a progression that goes like this:
$$1\times 2 + 2 \times 3 + .. + (n-1) \times n$$
Is there a way I can simplify it?
| Hint. You have
$$
(n^3-n)-\left((n-1)^3-(n-1)\right)=3\times(n-1)\times n
$$ and the sum is telescoping.
| {
"language": "en",
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How does one prove that $\mathbb{Z}\left[\frac{1 + \sqrt{-43}}{2}\right]$ is a unique factorization domain? By extending the Euclidean algorithm one can show that $\mathbb{Z}[i]$ has unique factorization.
This logic extends to show $\mathbb{Z}\left[\frac{1 + \sqrt{-3}}{2}\right]$, $\mathbb{Z}\left[\frac{1 + \sqrt{-3}}{... | The first thing to keep in mind is that almost all complex quadratic integer rings are non-UFDs. Put another way, if a random $d$ is negative, then you can be almost certain that the ring of algebraic integers of $\mathbb{Q}(\sqrt{d})$ (often denoted $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$) has class number greater than $... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Distance between line and a point Consider the points (1,2,-1) and (2,0,3).
(a) Find a vector equation of the line through these points in parametric form.
(b) Find the distance between this line and the point (1,0,1). (Hint: Use the parametric form of the equation and the dot product)
I have solved (a), Forming:
Vecto... | You can use a formula, although I think it's not too difficult to just go through the steps. I would draw a picture first:
You are given that $\vec{p} = (1,0,1)$ and you already found $\vec{m} = (1, -2, 4)$ and $\vec{l}_0 = (1,2,-1)$. Now it's a matter of writing an expression for $\vec{l}(t) - \vec{p}_0$:
\begin{al... | {
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Showing Uniform convergence of $\frac{n x}{1 + n \sin(x)}$ I want to prove for all $a\in \left(0,\frac{\pi}{2}\right]$, $ \ f_n\to f$ uniformly on $\left[a,\frac{\pi}{2}\right]$.
Also, how is this different from $f_n \to f$ uniformly on $\left(0, \frac{\pi}{2}\right]$ ?
I made the crucial error of omitting that first l... | The non-uniform convergence on $(0,\pi/2]$ was addressed (partially) in another post. It follows from $\displaystyle \lim_{n \to \infty}|f_n(x) - f(x)|= 1$ for any sequence $(x_n)$ with $x_n \to 0$.
Convergence is uniform on $[a,\pi/2]$ for $0 < a < \pi/2$.
Note that as $n \to \infty$ we have for all $x \in [a,\pi/2... | {
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Why do disks on planes grow more quickly with radius than disks on spheres? In the book, Mr. Tompkins in Wonderland, there is written something like this:
On a sphere the area within a given radius grows more slowly with the radius than on a plane.
Could you explain this to me? I think that formulas shows something t... | Theorem: Let $0 < r \leq 2R$ be real numbers, $S_{1}$ a Euclidean sphere of radius $R$, $S_{2}$ a Euclidean sphere of radius $r$ centered at a point $O$ of $S_{1}$, and $D_{r}$ the portion of $S_{1}$ inside $S_{2}$.
The area of $D_{r}$ is $\pi r^{2}$.
Now let $D'_{r}$ be the disk of radius $r$ in $S_{1}$, i.e., the po... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What exactly is wrong with this argument (Lucas-Penrose fallacy) Argument
"For every computer system, there is a sentence which is undecidable for the computer, but the human sees that it is true, therefore proving the sentence via some non-algorithmic method."
| The basic fallacy here comes by regarding the original Gödelian proof. First, one assumes the consistency of a theory T, and from T shows that the sentence G is undecidable. Now, to show (formally) that G is nonetheless true in the standard model (there are models in which G is false!), one requires a more powerful the... | {
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Empty set does not belong to any cartesian product? I am reading from Halmos naive set theory, for Cartesian product, defined as:
$$
A\times B=\left\{x: x \in P(P(A\cup B))\,\wedge\,\exists a \in A,\exists b \in B,\, x=(a,b)\right\}
$$
Empty set belongs to $P(P(AUB))$, but since in $(a,b)$ for some $a$ in $A$ and some ... | With the definition,
$$A \times B = \{ X \in P(P (A \cap B)) : \exists a \in A, \exists b \in B[X = (a,b)]\}$$
One should note that we are defining with $\in$, which does not consider the empty set, but $\{\varnothing\} \in A$ is certainly possible, i.e. the set containing the empty set can be an element (also note tha... | {
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"timestamp": "2023-03-29T00:00:00",
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Distance between points Suppose I have two matrices each containing coordinates of $m$ and $n$ points in 2 D. Is there an easy way using linear algebra to calculate the euclidean distance between all points (i.e., the results should be a $m$ by $n$ matrix)?
| We start with
$$
M = (u_1, \ldots, u_m) \in \mathbb{R}^{3\times m} \\
N = (v_1, \ldots, v_n) \in \mathbb{R}^{3\times n}
$$
and want
$$
D = (d_{ij}) \in \mathbb{R}^{m\times n}
$$
with
\begin{align}
d_{ij}
&= \lVert u_i - v_j \rVert \\
&= \sqrt{(u_i - v_j)\cdot (u_i - v_j)} \\
&= \sqrt{u_i \cdot u_i + v_j \cdot v_j - 2(u... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1301012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to analytically evaluate $\cos(\pi/4+\text{atan}(2))$ This equals $\cos(\pi/4)\cos(\text{atan}(2))-\sin(\pi/4)\sin(\text{atan}(2))$.
I'm just not sure how to evaluate $\cos(\text{atan}(2))$
| Hint: $\sin(\arctan(2))=\frac{\tan(\arctan(2))}{\sec(\arctan(2))}=\frac2{\sqrt5}$ and $\cos(\arctan(2))=\frac1{\sec(\arctan(2))}=\frac1{\sqrt5}$
Recall that $\sec^2(\theta)=\tan^2(\theta)+1$
| {
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$f(n) = n^{\log(n)}$, $g(n) = log(n)^{n}$ is $f\in O(g(n))$? $$f(n) = n^{\log(n)}$$
$$g(n) = \log(n)^n$$
$$f\in O(g(n))\text{ or }f \notin O(g(n))$$
why? I do not seem to get this one in particular
For O (big O)
Thanks!
| Hint: $n^{\log(n)}=e^{(\log n)^2}$ and $\log(n)^n=e^{n \log(\log(n))}$. Which of the two grows faster?
| {
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Solving with integration by parts: $\int \frac 1 {x\ln^2x}dx$
Solving: $$\int \frac 1 {x\ln^2x}dx$$ with parts.
$$\int \frac 1 {x\ln^2x}dx= \int \frac {(\ln x)'} {\ln^2x}dx \overset{parts} = \frac {1} {\ln x}-\int \frac {(\ln x)} {(\ln^2x)'}dx$$
$$\int \frac {(\ln x)} {(\ln^2x)'}dx=\int \frac {(x\ln x)} {(2\ln x)}dx=... | As pointed out in previous answers, this can be found most easily using the substitution $u=\ln x$.
Using integration by parts, though, with $u=(\ln x)^{-2}$ and $dv=\frac{1}{x}dx$, so $du=-2(\ln x)^{-3}dx$ and $v=\ln x$,
gives $\displaystyle\int\frac{1}{x(\ln x)^2}dx=(\ln x)^{-1}-(-2)\int(\ln x)^{-2}\frac{1}{x}dx=(\l... | {
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Prove that $|\sin^{−1}(a)−\sin^{−1}(b)|≥|a−b|$ Question:
Using the Mean Value Theorem, prove that $$|\sin^{−1}(a)−\sin^{−1}(b)|≥|a−b|$$ for all $a,b∈(1/2,1)$. Here, $\sin^{−1}$ denotes the inverse of the sine function.
Attempt:
I think I know how to do this but I want to make sure that I am as detailed as possible so ... | we can do this without calculus. first we will show an equivalent inequality that $$|\sin t - \sin s| \le |t-s|.\tag 1$$ you can show $(1)$ using the interpretation that $(\cos t , \sin t)$ is the coordinates of the terminal point on the unit circle corresponding to the signed arc length $t$ measured from $(1.0)$
let... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Uniqueness of basis vectors Say I have 2 vectors $v_1$ and $v_2$ as basis of a subspace. Then is it true that $kv_1$ and $mv_2$ where $k$ and $m$ are real numbers, are also basis for that subspace?
| Start with the definition of a basis of a vector space. A basis is a set of linearly independent vectors that spans the vector space.
Now $\{v_1,v_2\}$ is a spanning set of vectors for a vector space $V$ over a field $F$ if and only if every vector $v\in V$ can be written as a linear combination of the basis vectors.... | {
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Laplace transform of the wave equation I started of with the wave equation $$\frac{\partial^2 u}{\partial x^2}=\frac{\partial^2 u}{\partial t^2}$$
with boundary conditions $u=0$ at $x=0$ and $x=1$ and initial condition $u=sin(\pi x)$ and $\frac{\partial u}{\partial t}=0$ when $t=0$
I have gotten to the stage $$\frac{d^... | $$\frac{d^2 v}{dx^2}-s^2v=-s* sin(\pi x)$$
Once you have this, you have a differential equation of variable $x$. So you will solve it assuming $s$ is a constant. What you have is a linear second order constant coefficient inhomogeneous ordinary differential equation. It will have a complementary (homogeneous) and a par... | {
"language": "en",
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What is $X\cap\mathcal P(X)$? Does the powerset of $X$ contain $X$ as a subset, and thus $X\cap \mathcal{P}(X)=X$, or is $X\cap \mathcal{P}(X)=\emptyset$ since $X$ is a member of the
$\mathcal{P}(X)$, and not a subset?
| It depends.
For "normal" sets, this intersection will be empty, so in general, your second thought is the correct one.
However, if we have something like:
$X = \{1,2,3, \{2,3\}\}$ then intersecting $X$ with its own power set will actually have an element - $\{2,3\}$
The distinction to be made here is subsets vs element... | {
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Why is the chain rule applied to derivatives of trigonometric functions? I'm having trouble to understand why is the Chain rule applied to trigonometric functions, like:
$$\frac{d}{dx}\cos 2x=[2x]'*[\cos 2x]'=-2 \sin 2x$$
Why isn't it like in other variable derivatives? Like in:
$$ \frac{d}{dx} 3x^2=[3x^2]'=6x $$
Does ... | A more appropriate example would be
$$\frac{d}{dx} (3x)^2=2(3x) \cdot (3x)' = 18x$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1301756",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why do we need $\sup$ and $\inf$ when we have $\max$ and $\min$. In my analysis text, it seems that $\max$ and $\min$ are replaced by $\sup$ and $\inf$ for 1D single variable function, why is this the case?
| Well there isn't always a maximum. Consider $S=\{r\in\Bbb Q:r<2\}$. This is a bounded set with no maximum element. But we can say that $2$ is the smallest number which is an upper bound of $S$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1301835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
A bounded sequence in a Banach space Let $X$ be a Banach space and $\langle x_n\rangle $ be a sequence in $X$.
If ( $f(x_n)$ ) is a bounded sequence for any bounded linear functional $f$ on $X$, then ( $x_n$ ) is a bounded sequence in $X$.
I have to prove this fact. I first thought it would be simple, but it turns out ... | As PhoemueX said: the statement follows from the Uniform Boundedness Principle applied to linear maps $\phi_n:X^*\to \mathbb{K}$ that are defined by $\phi_n(f)=f(x_n)$. ($\mathbb{K}$ stands for $\mathbb{R}$ or $\mathbb{C}$.) The principle applies here because $X^*$ is complete even when $X$ isn't.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1301951",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
What is the relationship betweeen a pdf and cdf? I am learning stats. On page 20, my book, All of Statistics 1e, defines a CDF as function that maps x to the probability that a random variable, X, is less than x.
$F_{x}(x) = P(X\leq x)$
On page 23 it gives a function
$P(a < X < b ) = \int_{a}^{b}f_{X}dx$
and then say... | The cumulative distribution function $F_X$ of any random variable $X$ is defined by
$$
F_X(x)=P(X\le x)
$$
for $x\in\mathbb R$. If $X$ is a continuous random variable, then the cumulative distribution function $F_X$ can be expressed as
$$
F_X(x)=\int_{-\infty}^xf(x)\mathrm dx
$$
for $x\in\mathbb R$, where $f$ is the pr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1302015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
} |
Finding the mode of the negative binomial distribution The negative binomial distribution is as follows: $\displaystyle f_X(k)=\binom{k-1}{n-1}p^n(1-p)^{k-n}.$
*
*To find its mode, we want to find the $k$ with the highest probability.
*So we want to find $P(X=k-1)\leq P(X=k) \geq P(X=k+1).$
I'm getting stuck working... | You are on the right track (except for typos in the notation now corrected, see @Henry's Comment and my response). Express binomial coefficients
in terms of factorials. Some factorials will cancel exactly. Others
will have factors in common: for example, 10!/9! = 10.
For a simple start, you might try the case where $p ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1302106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How to not feel bad for doing math? I have a MsC and want to take a PhD in algebraic topology. Probably very few people in the world will have any interest of my thesis. They will pay me for doing my hobby. Its the only job I can think of which doesnt contribute to bettering the world somehow. I feel like I should get... | Simple - you contribute to humankind according to your ability to do your job. Your ability to do your job is fundamentally affected by your enthusiasm for your job. It stands to reason that the people should be taking jobs doing things they want to do anyway, regardless of pay, perks, etc.
If you are making great co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1302318",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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n points can be equidistant from each other only in dimensions $\ge n-1$? 2 points are from equal distance to each other in dimensions 1,2,3,...
3 points can be equidistant from each other in 2,3,... dimensions
4 points can be equidistant from each other only in dimensions 3,4,...
What is the property of number dimens... | One side of this, there is a standard picture. In $\mathbb R^n,$ take the $n$ points
$$ (1,0,0,\ldots,0), $$
$$ (0,1,0,\ldots,0), $$
$$ \cdots $$
$$ (0,0,0,\ldots,1). $$
These are all at pairwise distance $\sqrt 2$ apart.
At the same time, they lie in the $(n-1)$-dimensional plane
$$ x_1 + x_2 + \cdots + x_n = 1. $$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1302395",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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how to prove the sequence based definition of a closure in metric spaces Let $A$ be a subset of a metric space $\Omega$. By definition, the closure of $A$ is the smallest closed set that contains $A$. How to prove that alternativelly, the closure is given by
(1): $\bar A = \{a_* \vert a_* = \lim_{n\rightarrow\infty} a_... | Consider the set $X=\{x\,|\,a_n\to x;\;\forall n\, a_n\in A\}$.
Clearly $A\subset X$. You should be able to prove that $X\subset\Omega$ is closed. Now suppose that $Y\subset\Omega$ is closed, $Y\ne X$ and $A\subset Y\subset X$. Then there must be some $x\in X$ such that $x\ne Y$. Work from there to show that no such $Y... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
How to solve the difference equation $u_n = u_{n-1} + u_{n-2}+1$ Given that:
$$
\begin{equation}
u_n=\begin{cases}
1, & \text{if $0\leq n\leq1$}\\
u_{n-1} + u_{n-2}+1, & \text{if $n>1$}
\end{cases}
\end{equation}
$$
How do you solve this difference equation?
Thanks
EDIT:
From @marwalix's answer:
$$
u_n=v_n... | Write $u_n=v_n+a$ where $a$ is a constant. In that case the recurrence reads as follows
$$v_n+a=v_{n-1}+v_{n-2}+2a+1$$
So if we chose $a=-1$ we are left with
$$v_n=v_{n-1}+v_{n-2}$$
And we're back to a Fibonnacci type and in this case we have $v_0=v_1=2$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Method for proving polynomial inequalities
Let $x\in\mathbb{R}$. Prove that
$\text{(a) }x^{10}-x^7+x^4-x^2+1>0\\
\text{(b) }x^4-x^2-3x+5>0$
Possibly it can be proved in a few different ways, but I have first tried to prove it reducing to a sum of squares. After too many attempts and using a trial-and-error method, ... | Just another way for $(a)$ is using the AM-GMs:
$$\frac12x^{10}+\frac12x^4 \ge x^7, \quad \frac12x^4+\frac12 \ge x^2$$
$$\implies x^{10}-x^7+x^4-x^2+1 \ge \frac12+\frac12x^{10}>0$$
and similarly for $(b)$:
$$\frac12x^4+\frac12 \ge x^2, \quad \frac12x^4+\frac32+\frac32+\frac32 \ge 2\times 3^{3/4}x> 3x$$
Not always appli... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1302670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Eigenvalues of different symmetric $(2n+1)\times(2n+1)$ matrix
I ve looked at other similar post but I could not find help with them
| The rank of your matrix is $2$, which implies that $\lambda=0$ is an eigenvalue with multiplicity at least $2n+1-rank=2n-1$.
Now, if $\lambda_1, \lambda_2$ are the remaining eigenvalues, since $tr(A)$ is the sum of the eigenvalues you get $\lambda_1+\lambda_2=0$. [I assume the row and column have the same index]. This ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is cross product not commutative? Why, conceptually, is the cross product not commutative? Obviously I could simply take a look at the formula for computing cross product from vector components to prove this, but I'm interested in why it makes logical sense for the resultant vector to be either going in the negativ... | The magnitude of the resulting vector is a function of the angle between the vectors you are multiplying. The key issue is that the angle between two vectors is always measured in the same direction (by convention, counterclockwise).
Try holding your left thumb and index finger in an L shape. Measuring counterclockwise... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Axiomatic definition of sin and cos? I look for possiblity to define sin/cos through algebraic relations without involving power series, integrals, differential equation and geometric intuition.
Is it possible to define sin and cos through some axioms?
Like:
$$\sin 0 = 0, \cos 0 = 1$$
$$\sin \pi/2 = 1, \cos \pi/2 = 0$... | From what you do not want that pretty much leaves functional equations.
There seems to be a system of two functional equations for sine and cosine: (link)
$$
\Theta(x+y)=\Theta(x)\Theta(y)-\Omega(x)\Omega(y) \\
\Omega(x+y)=\Theta(x)\Omega(y)+\Omega(x)\Theta(y)
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1303044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "36",
"answer_count": 4,
"answer_id": 3
} |
Can two perfect squares average to a third perfect square? My question is does there exist a triple of integers, $a<b<c$ such that $b^2 = \frac{a^2+c^2}{2}$
I suspect that the answer to this is no but I have not been able to prove it yet. I realize this is very similar to the idea of Pythagorean triples but I am not ... | Parametric solution:
$$ a = |x^2 - 2 x y - y^2|, \ b = x^2 + y^2,\ c = x^2 + 2 x y - y^2 $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1303173",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 5,
"answer_id": 4
} |
Find the sequence of partial sums for the series $a_n = (-1)^n$ Does this series converge? Find the sequence of partial sums for the series $$ \sum_{n=0}^\infty (-1)^n = 1 -1 + 1 -1 + 1 - \cdots$$
Does this series converge ?
My answer is that the sequence $= 0.5 + 0.5(-1)^n$. This makes a sequence that alternates betwe... | $$ \sum\limits_{n=0}^\infty (-1)^n = \lim\limits_{m\to\infty}\sum\limits_{n=0}^m (-1)^n $$
$$ = \lim\limits_{m\to\infty} \frac12\left((-1)^m + 1\right) $$
Note that the sequence
$$a_m=\frac12\left((-1)^m + 1\right) $$
is convergent if
$$ \lim\limits_{m\to\infty} a_{2m} = \lim\limits_{m\to\infty} a_{2m+1} = L$$
Howeve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1303265",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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Is there a constructive discontinuous exponential function? It is well-known that the only continuous functions $f\colon\mathbb R\to\mathbb R^+$ satisfying $f(x+y)=f(x)f(y)$ for all $x,y\in\mathbb R$ are the familiar exponential functions. (Prove $f(x)=f(1)^x$ successively for integers $x$, rationals $x$, and then use... | Let $g(x)=\ln f(x)$, so that $g(x+y)=g(x)+g(y)$. The solutions to this equation are precisely the ring homomorphisms from $\mathbb R\to\mathbb R$ and they are in bijection with solutions to your original equation. Since $\mathbb R$ is an infinite dimensional $\mathbb Q$-vector space, there are infinitely many such ring... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Trigonometric relationship in a triangle If in a triangle $ABC$, $$3\sin^2B+4\sin A\sin B+16\sin^2A-8\sin B-20\sin A+9=0$$ find the angles of the triangle. I am unable to manipulate the given expression. Thanks.
| The equation can be rearranged as
$$(16\sin A+2\sin B-10)^2+44(\sin B-1)^2=0\ .$$
The only way this can happen is if
$$16\sin A+2\sin B-10=0\ ,\quad \sin B-1=0\ ;$$
these equations are easily solved to give
$$\sin A=\frac12\ ,\quad \sin B=1\ ,$$
and since $A,B$ are angles in a triangle we have
$$A=\frac\pi2\ ,\quad B=\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1303454",
"timestamp": "2023-03-29T00:00:00",
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What situations should $\oint$ be used? Sometimes people put a circle through the integral symbol: $\oint$
What does this mean, and when should we use this integration symbol?
| This symbol is used to indicate a line integral along a closed loop.
if the loop is the boundary of a compact region $\Omega$ we use also the symbol
$
\int_{\delta \Omega}
$
we can generalize such notation to the boundary of a region in an n-dimensional space and, if $\Omega$ is an orientable manifold we have the gener... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1303631",
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Questions on Kolmogorov Zero-One Law Proof in Williams Here is the proof of the Kolmogorov Zero-One Law and the lemmas used to prove it in Williams' Probability book:
Here are my questions:
*
*Why exactly are $\mathfrak{K}_{\infty}$ and $\mathfrak{T}$ independent? I get that $\mathfrak{K}_{\infty}$ is a countable... | On 1):
Let $A\in \mathfrak{K}_{\infty}$ and $B\in\mathfrak{T}$.
Then $A\in \mathfrak{X}_n$ for some $n$ and since $\mathfrak{X}_n$ and $\mathfrak{T}$ are independent $\sigma$-algebra's we are allowed to conclude that $P(A\cap B)=P(A)\times P(B)$.
This proves that $\mathfrak{K}_{\infty}$ and $\mathfrak{T}$ are indepen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1303735",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Number of points on an elliptic curve and it's twist over $\mathbb{F}_p$. I have another probably very trivial question about elliptic curves. This wikipedia article gives the following formula $|E|+|E^d|=2p+2$ where $E$ is an elliptic curve over $\mathbb{F}_p$ and $E^d$ is it's quadratic twist. At the same time the pa... | If $x^3+ax+b = 0$, then the point $(x,0)$ lies on each of $E$ and $E^d$, for a total of $2$ solutions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1303807",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Convergence of a subsequence in $(C(\mathbb{T}), \|\cdot\|_2)$ Problem: Define, $ \mathbb{T} := \mathbb{R}/{2\pi\mathbb{Z}} $.
Consider a sequence of functions $(g_n)_{n\in \mathbb{N}} \in C^4(\mathbb{T})$ such that, $ \sup_{n \in \mathbb{N}}(\| g_n \|_2 + \| g'_n \|_2 + \| g''_n \|_2) < \infty $. Prove that $(g_n)_{n... | Yes, you need to prove compactness. This is an instance of Rellich–Kondrachov theorem, but presumably the point of the problem is to show compactness directly. To get uniform convergence, it suffices to have Fourier coefficients converging in $\ell^1(\mathbb{Z})$, so let's look at those.
Let $g_n^{(k)}$ be the $k$th c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1303897",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Differentiating both sides of an inequality with monotonic functions If $f(x)\le g(x)$ for all real $x$ for monotonic functions $f$ and $g$ (say, both increasing), does it follow that $f'(x)\le g'(x)$?
(Note: I've seen several questions asking the same thing without the condition of monotonicity, but the counterexampl... | Another more explicit example:
$$
f(x)=x-\frac\pi2+\arctan x,\quad g(x)=x.
$$
$$
f'(x)=1+\frac{1}{1+x^2},\quad g'(x)=1.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1304089",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Distance between two points on a sphere. Say there is a sphere on which there is an ant and the ant wants to go to another point. The ant can't definitely travel through the sphere. So it has to travel along a curve. My question is what is the least distance between the two points i.e. distance between 2 points on a sp... | If $a = (a_{1}, a_{2}, a_{3})$ and $b = (b_{1}, b_{2}, b_{3})$ are points on a sphere of radius $r > 0$ centered at the origin of Euclidean $3$-space, the distance from $a$ to $b$ along the surface of the sphere is
$$
d(a, b) = r \arccos\left(\frac{a \cdot b}{r^{2}}\right)
= r \arccos\left(\frac{a_{1}b_{1} + a_{2}b_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1304169",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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solving second order non-homogeneous differential equation 3 Help me to solve this non-homogeneous differential equation :
$
y''+y=\tan x
$
$
0<x<\dfrac{\pi}{2}
$
I could reach to $y_{c}=c_{1}\cos x + c_{2}\sin x$ but particular solution is where I stopped.
| Knowing a solution of the homogeneous ODE, for example $c\cos(x)$, remplace the constant $c$ by a function $Y(x)$ , i.e.: change of function $y=Y(x)\cos(x)$. This allows to reduce the second order ODE to a first order ODE easier to solve.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1304354",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Are there different ways to find (the) residue(s) for a function with one simple pole vs. a function with several simple poles? Regarding evaluation of residuals for functions with simple poles.
Let's say $m$ represents the order of the pole, then in order to find the residual at each pole/the pole (if only one pole) w... | You should be aware that a simple pole is a point in which the Laurent series of $f$ is
$$
f(z)=\frac{a_{-1}}{z-z_0}+a_0+a_1(z-z_0)+a_2(z-z_0)^2+\ldots
$$
The residual is $a_{-1}$, the coefficient of $1/(z-z_0)$.
Now, we have
$$
(z-z_0)f(z)=a_{-1}+a_0(z-z_0)+a_1(z-z_0)^3+a_2(z-z_0)^3+\ldots
$$
that gives $a_{-1}$ when... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1304444",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$\sin x + c_1 = \cos x + c_2$ While working a physics problem I ran into a seemingly simple trig equation I couldn't solve. I'm curious if anyone knows a way to solve the equation:
$\sin(x)+c_1 = \cos(x)+c_2$
(where $c_1$ and $c_2$ are constants) for $x$ without using Newton's method or some other form of approximatio... | $$
A\sin x + B\cos x = \sqrt{A^2+B^2}\left( \frac A {\sqrt{A^2+B^2}}\sin x+ \frac B {\sqrt{A^2+B^2}}\cos x \right)
$$
Notice that the sum of the squares of the coefficients above is $1$; hence they are the coordinates of some point on the unit circle; hence there is some number $\varphi$ such that
$$
\cos\varphi = \fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1304520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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What are the basic rules for manipulating diverging infinite series? This is something that I played around with in Calc II, and it really confuses me:
$s = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + \ldots = \infty$
$s - s = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + \ldots $
$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - (1 + 2 + 3 + 4 + 5 + 6 + 7 + ... | Basically, you don't.
The only way to manipulate them is if you know precisely what you're doing and using renormalization techniques or analytic continuation or something, but those techniques won't be taught for awhile and have specific rules about them.
Long story short, you don't manipulate them.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Complex analysis, residues
Find the residue at $z=0$ of $f(z)=\dfrac{\sinh z}{z^4(1-z^2)}$.
I did
\begin{align}
\frac{\sinh z}{z^4(1-z^2)} & =\frac{1}{z^4}\left[\left(\sum_{n=0}^\infty \frac{z^{2n+1}}{(2n+1)!)}\right)\left(\sum_{n=0}^\infty z^{2n}\right)\right] \\[8pt]
& =\frac{1}{z^4}\left[\left(z+\frac{z^3}{6}+\cd... | The residue at $z=0$ is the coefficient of the $\frac1z$ term in the expansion
$$
\frac{\sinh(z)}{z^4(1-z^2)}=\frac1{z^4}\left(z+\frac{z^3}6+\frac{z^5}{120}+\dots\right)\left(1+z^2+z^4+\dots\right)
$$
That is the coefficient of $z^3$ term in the expansion
$$
\left(\color{#C00000}{z}+\color{#00A000}{\frac{z^3}6}+\frac{z... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1304709",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Breaking down the equation of a plane Could someone explain the individual parts of a plane equation?
For example:
$3x + y + z = 7$
When I see this I can't imagine what it's supposed to look like.
| Consider the collection of vectors $\{\vec x:\vec x\perp (3\vec i+\vec j+\vec k)\}$. The endpoints of these form a plane through the origin. If you shift this plane upwards $7$ units, you get the plane in question.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1304823",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Why does $\frac{49}{64}\cos^2 \theta + \cos^2 \theta$ equal $\frac{113}{64}\cos^2 \theta $? I have an example:
$$ \frac{49}{64}\cos^2 \theta + \cos^2 \theta = 1 $$
Then what happens next:
$$ \frac{113}{64}\cos^2 \theta = 1 $$
Where has the other cosine disappeared to? What operation happened here? Any hints please.
| $\dfrac{49}{64}\cos^2 \theta + \cos^2 \theta = \left(\dfrac{49}{64} +1\right) \cos^2 \theta = \dfrac{113}{64}\cos^2 \theta$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1304916",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Is $\sqrt{2t \log{\log{\frac{1}{t}}}}$ increasing for $t\in(0,a)$, for a suitable $a>0$? I believe it's obviously, but I tried a lot, and have no clue how can I show that
$$\sqrt{2t \log{\log{\frac{1}{t}}}}$$
is increasing for $t\in(0,a)$ for a suitable $a>0$?
| Taking the derivative of $f(t)=\sqrt{2t\ln{\ln{\dfrac{1}{t}}}}$ with respect to $t$:
$$f'(t)=\dfrac{1}{2\sqrt{2t\ln{\ln{\dfrac{1}{t}}}}}\cdot\left(2\ln{\ln{\dfrac{1}{t}}}+2t\cdot\dfrac{1}{\ln{\dfrac{1}{t}}}\cdot t\cdot\dfrac{-1}{t^2}\right)$$
$$\therefore f'(t)=\dfrac{\ln{\ln{\frac{1}{t}}}-\dfrac{1}{\ln{\frac{1}{t}}}}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1305102",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
Rolling two dice, what is the probability of getting 6 on either of them, but not both? Rolling two dice, what is the probability of getting 6 on one of them, but not both?
| If the first die shows $6$ and the second shows anything but $6$ that can happen $5$ ways (6-1, 6-2, 6-3, 6-4, 6-5). Similarly if the second die shows $6$ and the first anything but $6$ that can happen another $5$ ways (1-6, 2-6, 3-6, 4-6, 5-6).
There are $36$ possible rolls of the two dice in total.
Hence the probab... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1305186",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Homogeneous Markov chains with general state space I found in the book Markov Chains by Revuz the following definition of a Markov chain. In the following $(X_n)_{n \in \mathbb{N}}$ is a sequence of random variables on a probability space $(\Omega,\mathcal{F},P_0)$ and the range is a measurable space $(E,\Sigma)$, and... | Thankfully, YES :)
You can very simply define the needed probability transition by :
\begin{equation}
P : x, \mathcal{A} \mapsto P_0[X_1 \in \mathcal{A} | X_0=x]
\end{equation}
You can then deduce from the definition of the family $(P_n)$ that :
\begin{equation}
P_n ( x, \mathcal{A} ) = P_0[X_{n+1} \in \mathcal{A} | X_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1305247",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Decomposition into irreducibles of a Noetherian topological space I'm struggling with the proof that says a Noetherian topological space $X$ is the finite union of closed irreducible subsets. In particular with this part:
First observe that every nonempty set of closed subsets of $X$ has a minimal element, since other... | Suppose that we have a family of closed sets $\mathcal{F}$ without a minimal element. So pick $F_0 \in \mathcal{F}$. Then $F_1 \in \mathcal{F}$ exists such that $F_1 \subsetneq F_0$, as otherwise $F_0$ would have been minimal. As $F_1$ is not minimal either, some $F_2 \in \mathcal{F}$ exists with $F_2 \subsetneq F_1$ a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1305329",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Why do all parabolas have an axis of symmetry? And if that's just part of the definition of a parabola, I guess my question becomes why is the graph of any quadratic a parabola?
My attempt at explaining:
The way I understand it after some thought is that any quadratic can be written by completing the square as a perfec... | Suppose, $f(x)=(x+a)^2+b$
Then, we have $$f(-a-x)=f(-a+x)=x^2+b$$ for all $x\in \mathbb R$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1305386",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 4,
"answer_id": 3
} |
Book for Module Theory I want a book to cover the following topics in Module Theory:
Modules, Submodules, Quotient modules, Morphisms Exact sequences, three lemma, four lemma, five lemma, Product and Co products, Free modules, Projective modules, Injective modules, Direct sum of Projective modules, Direct product of In... | As it was suggested before, Module Theory: An Approach to Linear Algebra by T. S. Blyth is an awesome title which covers almost every basic topic of Module theory in a very elegant, clear and efficient way. It is hands down my favorite text in the subject, but unfortunately it has been long out of print and therefore i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1305486",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Is there a way to write this expression differently: $\arctan 1/(1+n+n^2)$? $$ \arctan\left(\frac{1}{1+n+n^2}\right)$$
My professor wrote this as
$$\arctan(n+1) - \arctan(n)$$
I don't understand how this expression is right?
| We have $$\tan^{-1}\left( \frac{1}{1+n+n^2}\right)$$ $$=\tan^{-1}\left( \frac{(n+1)-n}{1+n(n+1)}\right)$$ Now, let $\color{blue}{n+1=\tan\alpha}$ ($\implies \color{blue}{\alpha=\tan^{-1}(n+1)}$) & $\color{blue}{n=\tan\beta}$ ($\implies \color{blue}{\beta=\tan^{-1}(n)}$) $\forall \quad \alpha>\beta$
Thus by sustituitng... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1305576",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
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How to show that $\lim_{x \to 0} x^p (\ln x)^r = 0$ I want to show that $\lim_{x \to 0} x^p (\ln x)^r = 0$ if $p > 1$ and $r \in \mathbb{N}$.
To show this, I wanted to use that $\lim_{x \to 0} x \ln x = 0$, and in fact if $p \geq r$ we can write that
$$\lim_{x \to 0} x^p (\ln x)^r = \lim_{x \to 0} x^{p-r}((x \ln x)^r... | First let us examine the following. Given \begin{equation} l > 0 \end{equation} we prove that
\begin{equation}
\lim_{x \to0} ln(x)x^l = 0.
\end{equation}
Doing this comes straight from L'Hospitals' rule. We put the limit in the indeterminate form and get the following.
\begin{equation}
\lim_{x \to \infty}\frac{ln(1/x)}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1305690",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
limits and infinity I'm having trouble wrapping my head around some of the 'rules' of limits. For example,
$$
\lim_{x\to \infty} \sqrt{x^2 -2} - \sqrt{x^2 + 1}
$$
becomes
$$
\sqrt{\lim_{x\to \infty} (x^2) -2} - \sqrt{\lim_{x\to \infty}(x^2) + 1}
$$
which, after graphing, seems to approach zero. My question is how do yo... | Hint: $$\left[\sqrt {x^2 - 2} - \sqrt{x^2 + 1}\right]\dot\, \frac{\sqrt {x^2 - 2} + \sqrt{x^2 + 1}}{\sqrt {x^2 - 2} + \sqrt{x^2 + 1}} = \frac{x^2 - 2 - (x^2 + 1)}{\sqrt {x^2 - 2} + \sqrt{x^2 + 1}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1305782",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Misunderstanding of $\epsilon$-neighborhood I am given the following definition of an $\epsilon$-neighborhood:
Given a real number $a\in \mathbb{R}$ and a positive number $\epsilon>0$, the set
$$\{x \in \mathbb{R}: |x-a|<\epsilon \}$$
is called the $\epsilon$-neighborhood of a.
The book says that the $\epsilon$-neighbo... | You might want to write it this way: $$|x-a|<\epsilon \iff -\epsilon < x-a < \epsilon \iff a-\epsilon < x < a+\epsilon$$And the last expression means that $x \in \left]a-\epsilon,a+\epsilon\right[$. The neighbourhood is centered in $a$, not $x$. The moral of the history is that $|x-a|<\epsilon$ if and only if $x \in \l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1305857",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Abelian group and element order Suppose that $G$ is an Abelian group of order 35 and every element of $G$ satisfies the equation $x^{35} = e.$ Prove that G is cyclic.
I know that since very element of $G$ satisfies the equation $x^{35} = e$ so the order of elements are $1,5,7,35$. I know how to prove the situation when... | A very basic proof:
If $G$ has an element of order $5$ and an element of order $7$, say $a$ and $b$, then since $G$ is abelian, $ab$ has order $35$ and $G$ is cyclic.
So we want to rule out the possibility that $G$ has no elements of order $5$, or alternatively, no elements of order $7$, without recourse to Cauchy's Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1305950",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Pre-calculus - Deriving position from acceleration Suppose an object is dropped from the tenth floor of a building whose roof is 50 feet above the point of release. Derive the formula for the position of the object t seconds after its release is distance is measured from the roof. The positive direction of distance is ... | Yes, add $50$ft to the function, because when $t=0$, the object is $50$ft away from the roof.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1306032",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
$\mathbb{Q}\left ( \sqrt{2},\sqrt{3},\sqrt{5} \right )=\mathbb{Q}\left ( \sqrt{2}+\sqrt{3}+\sqrt{5} \right )$ Prove that
$$\mathbb{Q}\left ( \sqrt{2},\sqrt{3},\sqrt{5} \right )=\mathbb{Q}\left ( \sqrt{2}+\sqrt{3}+\sqrt{5} \right )$$
I proved for two elements, ex, $\mathbb{Q}\left ( \sqrt{2},\sqrt{3}\right )=\mathbb{Q}\... | It is worth noting from alex.jordan and achille hui's responses that if $\theta = \sqrt{2} + \sqrt{3} + \sqrt{5}$, we explicitly have $$\begin{align*} \sqrt{2} &= \tfrac{5}{3} \theta - \tfrac{7}{72} \theta^3 - \tfrac{7}{144} \theta^5 + \tfrac{1}{576} \theta^7, \\ \sqrt{3} &= \tfrac{15}{4} \theta - \tfrac{61}{24} \theta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1306082",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Why the set of all maximum points of a continuous function is a closed set? Suppose $f(x)$ is a continuous function on domain $\Omega$ and its maximum value is $m$. Let $M={\{x|f(x)=m\}}$, then how do I prove $M$ is a closed set? Thank you!
| Suppose $x \notin M$, then $f(x) \neq m$ and since $f$ is continuous, there is some open $U$ containing $x$ such that $f(y) \neq m$ for all $y \in U$, hence $M^c$ is open.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1306170",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Does rotation of a rectangle keep it rectangular? If I rotate a rectangle by 45°, does it stay rectangular or become something else? I mean do 90° angles stay 90°?
I am asking this question because I have some results where the rotated rectangle becomes not so rectangular ... I think I have a problem.
Problem: I think ... | I found the solution.
Well, I used to draw the image using a different scaling between Y-axis and X-axis. The solution is to set DataAspectRatio to [1 1 1].
In order to use I have used axis image but after imagesc(image) !
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1306301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 3
} |
Inequality involving lengths and triangles I was quite sure this would have been asked before but I couldn't find it, so here goes:
If $\displaystyle BC<AC<AB \hspace{5pt} (\alpha<\beta<\gamma)$, show $\displaystyle PA+PB+PC<\beta+\gamma \hspace{3pt}$ (where $P$ arbitrary in the interior of the triangle).
I haven't ac... | The Fermat-Torricelli point of a triangle $ABC$ is the point $P$ such that $PA+PB+PC$ is minimized. It is usually found through the following lines: Assume, for first, that $ABC$ has no angle greater than $120^\circ$ and take a point $Q$ inside $ABC$. Let $Q'$ the image of $Q$ under a rotation of $60^\circ$ around $B$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1306473",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
existence of holomorphic functions Let $D$= { $z \epsilon \Bbb C :|z|<1$ } . Then there exists a holomorphic function $f:D\to \overline D$ with $f(0)=0$ with the property
a) $f'(0)={1\over 2}$
b) $|f(1/3)|={1\over 4}$
c) $f(1/3)={1\over 2}$
d) $|f'(0)|=\sec(\pi/6)$
when $f(z)={1\over 2}z$ then $a)$ is true
when $f(z... | Hint:
If $f(z)$ is holomorphic function from $D$ onto itself, where $D=\{z\in\mathbb{C}:|z|<1\}$ and $f(0)=0$, then $|f(z)|\le|z|$ and $|f'(z)|\le1$ for all $z\in D$, by Schwartz lemma
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1306581",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Complex analysis, find the residue
Find the residue of $f(z)=\frac{1}{z^2\sin z}$ at $z_0=0$
What I tried
Let $g(z)=1$ and $h(z)=z^2\sin z$, both are analytics but they have zeros of different orders then $f(z)$ don't have removable singularity point at $z_0$
Then I tried to use the fact that if $lim_{z\rightarrow z_... | You may write, as $z \to 0$,
$$
\begin{align}
\frac1{z^2\sin z}&=\frac1{z^2\left(z-\dfrac1{3!}z^3+O(z^5)\right)}\\\\
&=\frac1{z^3\left(1-\dfrac16z^2+O(z^4)\right)}\\\\
&=\frac1{z^3}(1+\frac16z^2+O(z^4))\\\\
&=\frac1{z^3}+\frac1{6z}+O(z)\\\\
\end{align}
$$ and the desired residue is equal to $\dfrac16$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1306654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
How to evaluate $\lim\limits_{n\to +\infty}\frac{1}{n^2}\sum\limits_{i=1}^n \log \binom{n}{i} $ I have to compute:
$$ \lim_{n\to +\infty}\frac{1}{n^2}\sum_{i=1}^{n}\log\binom{n}{i}.$$
I have tried this problem and hit the wall.
| We have:
$$ \prod_{i=1}^{n}\binom{n}{i} = \frac{n!^n}{\left(\prod_{i=1}^{n}i!\right)\cdot\left(\prod_{i=1}^{n-1}i!\right)}\tag{1}$$
hence:
$$ \frac{1}{n^2}\sum_{i=1}^{n}\log\binom{n}{i}=\frac{1}{n^2}\left((n-1)\log\Gamma(n+1)-2\sum_{i=1}^{n-1}\log\Gamma(i+1)\right)\tag{2}$$
and by Stirling's approximation:
$$ \log\Gamm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1306719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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What does the "$\cdot$" mean in an equation I am trying to solve an equation for a project that I am undertaking. The equation is very long and its probably not necessary to show it all here. Most of the equation is fairly straightforward; i.e., $1+(\frac{w}{W})(\frac{d}{t})$, etc. but at the very end it reads $\left(1... | Like this?
$$ \left(1+\frac{w}{W}\right)\left(\frac DT\right)\cdot\frac1T $$
That means multiply.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1306798",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Use Laplace's method with $\int_{0}^{\infty} e^{x(3u-u^3)}du$ as $x\rightarrow \infty$ Use Laplace's method with $\int_{0}^{\infty} e^{x(3u-u^3)}du$ as $x\rightarrow \infty$.
I'm confused about how to taylor expand about u=1? How do I continue? Obviously first of all I have converted it to:
$$\int_{1-\epsilon}^{1+\epsi... | Let's see the where the usual reasoning for Laplace's method leads us.
Let me write the integral as $\exp{(x f(u))}$, where $f(u)$ has a global maximum at $u_* = 1$. We could anticipate that the major contribution to the integral is coming from the neighbourhood of $u = u_*$ as $x \to \infty$. If we expanded the argume... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1306882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that $T$ is a linear operator - Linear Transformations in Linear Algebra We are asked:
Consider the operator $T:\mathbb{R^2} \rightarrow \mathbb{R^2}$ where $T(x_1, x_2) = (x_1 + kx_2, -x_2)$ for every $(x_1, x_2) \in \mathbb{R^2}$. Here $k \in \mathbb{R}$ is fixed.
a) Show that $T$ is a linear operator
b) Show th... | Hint:
there is some confusion between componets and vectors (or points) in your notation.
For part a) you have to prove:
$$
T(\vec x+\vec y)=T(\vec x)+T(\vec y)
$$
with: $\vec x=(x_1,x_2)$ and $\vec y=(y_1,y_2)$
and
$$
T(k \vec x)=kT(\vec x)
$$
with with: $\vec x=(x_1,x_2)$
Use the definition of T for $T \vec x$ and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1306989",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Represent $\cos(x)$ as a sum of sines - where is my mistake Let's look at $f(x)=\cos(x)$ defined on the interval $[0,\pi]$.
We know that for any function $g$ defined on $[0,\pi]$ we have:
$g(x)=\sum_{k=1}^{\infty}B_k\sin(kx)$ where $B_k=\frac{2}{\pi}\int_{0}^{\pi}g(x)\sin(kx)dx$. And $f$ is no different. So in our case... | Your formula after "it can be shown that" is clearly not valid for $k=1$,
so you simply have to compute $B_1$ using some different method. (For example, $\int \cos x \sin x \, dx= \frac12 \sin^2 x + C$.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1307069",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Determine whether or not the limit exists: $\lim_{(x,y)\to(0,0)}\frac{(x+y)^2}{x^2+y^2}$
Determine whether or not the limit $$\lim_{(x,y)\to(0,0)}\frac{(x+y)^2}{x^2+y^2}$$
exists. If it does, then calculate its value.
My attempt:
$$\begin{align}\lim \frac{(x+y)^2}{x^2+y^2} &= \lim \frac{x^2+y^2}{x^2+y^2} + \lim \fr... | Your final answer is correct, but the way there is just slightly misleaded. The main problem is that you seperate your limit into a sum of limits. The first, $$\lim_{(x,y)\to(0,0)} \frac{x^2+y^2}{x^2+y^2}$$
converges and is equal to $1$, as you note. The second though, $$\lim_{(x,y)\to(0,0)} \frac {2xy}{x^2+y^2}$$
dive... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1307149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 6,
"answer_id": 4
} |
Show the set of all functions from $\mathbb{N}$ to $\{0, 1\}$ is uncountable using a contradiction. This is what I have written:
By contradiction, assume it is countable. Write $S=\{\text{all functions } \mathbb{N} \rightarrow \{0,1\} \}$. Then, we can find a bijection $\mathcal{H}: S \rightarrow \mathbb{N}$. Now, I w... | Your idea is correct, Using the binary representation actually makes the explanation way easier. You can always get a binary number that is not in the list and obtain a contradiction using cantor's diagonal method
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1307247",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 1
} |
Finding the square root of a big number, like 676? I am having trouble understanding and finding the square roots of large numbers. How would I go about finding this number efficiently?
| Square root of $676$ is the number such that multiplied by itself will give you 676.
This number is denoted as $\sqrt{676}$.
Note that generally speaking, for a non-zero real number $p$ there are two numbers which squared will give you $p$. They differ by the sign. In order to ensure uniqueness of the square root, the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1307312",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
find the minimum value of $\sqrt{x^2+4} + \sqrt{y^2+9}$ Question:
Given $x + y = 12$, find the minimum value of $\sqrt{x^2+4} + \sqrt{y^2+9}$?
Key:
I use $y = 12 - x$ and substitute into the equation, and derivative it.
which I got this
$$\frac{x}{\sqrt{x^2+4}} + \frac{x-12}{\sqrt{x^2-24x+153}} = f'(x).$$
However, aft... | the constraint is $x + y = 12.$ at a local extremum of $\sqrt{x^2 + 4} + \sqrt{y^2 + 9},$ the critical numbers satisfy $$dx + dy =0,\quad \frac{x\, dx}{\sqrt{x^2 + 4}} + \frac{y\, dy}{\sqrt{y^2 +9}} = 0 \to x\sqrt{y^2 + 9}=y\sqrt{x^2 + 4} $$ squaring the last equation we have $$9x^2 =4y^2 \to y = \pm\frac32x, x + y ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1307398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 5
} |
Fourier series convergence Let $f_n \rightarrow f$ be a sequence of $2\pi$-periodic functions, where the convergence is in $L^1({\mathbb R}/2\pi{\mathbb Z})$.
Then the Fourier-coefficients satisfy $|F(f_n) -F(f)| \rightarrow 0 $ uniformly.
Now, I was wondering. Does this imply that the Fourier series $$h_k(x):=\sum_{n... | Let $\{f_n\}$ be the typewriter sequence. $f_n$ converges to $f\equiv0$ in $L^1$ but not pointwise (not even pointwise almost everywhere). The Fourier coefficients of $f$ are all equal to $0$, so its Fourier series converges to $0$. The Fourier series of $f_n$ converges to $f_n$ at all points except two.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1307463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Notation for integral of a vector function over an ellipsoid For a short proof, I need to write a point $\pmb y\in\mathbb{R}^p$ as
the integral of the surface of the ellipse $\pmb x^{\top}\pmb Q\pmb x=c$ where $\pmb Q$ is a $p$ by $p$ PSD matrix (for now $\pmb y$ is defined in words). What is the formal way to write ... | You have lots of options here, but a typical choice is to denote the ellipsoid itself by a symbol, say,
$$E := \{{\bf x} \in \Bbb R^p : {\bf x}^T {\bf Q} {\bf x} = c\},$$
and then write the integral the integral of a real-valued function $f$ as
$$\iint_E f \, dS.$$
Here, $dS$ is the infinitesimal area element of the su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1307537",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to solve inequality for : $|7x - 9| \ge x +3$ How to solve inequality for : $|7x - 9| \ge x + 3$
There is a $x$ on both side that's make me confused...
| We can solve the absolute value inequality by squaring both sides, then solving the resulting quadratic inequality.
\begin{align*}
|7x - 9| & \geq x + 3\\
|7x - 9|^2 & \geq (x + 3)^2\\
(7x - 9)^2 & \geq (x + 3)^2\\
49x^2 - 126x + 81 & \geq x^2 + 6x + 9\\
48x^2 - 132x + 72 & \geq 0\\
4x^2 - 11x + 6 & \geq 0\\
4x^2 - 8x ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1307619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Application of Green's Theorem when undefined at origin Problem:
Let $P={-y \over x^2+y^2}$ and $Q={x \over x^2+y^2}$ for $(x,y)\ne(0,0)$.
Show that $\oint_{\partial \Omega}(Pdx + Qdy)=2\pi$ if $\Omega$ is any open set containing $(0,0)$ and with a sufficiently regular boundary.
Working:
Clearly, we cannot immediately... | If your boundary is smooth, then you may parametrize it by $x(t) = r(t) \cos t$ and $y(t) = r(t) \sin t$, with $t \in [0,2\pi]$. Then your differential form $P \Bbb d x + Q \Bbb d y$ becomes
$$\frac {-r(t) \sin t \Big (r'(t) \cos t - r(t) \sin t \Big) + r(t) \cos t \Big( r'(t) \sin t + r(t) \cos t \Big)} {r^2 (t)} = 1$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1307738",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
6 periods and 5 subjects Question : There are 6 periods in each working day of a school. In how many ways can one organize 5 subjects such that each subject is allowed at least one period? Is the answer 1800 or 3600 ? I am confused.
Initially this appeared as a simple question. By goggling a bit, I am stuck with two an... | 5 subjects on 5 periods
selecting a place for remaining one subject
Final answer
Over-counting check
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1307879",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
How many $3$ digit even numbers can be formed by using digits $1,2,3,4,5,6,7$, if no digits are repeated? How many $3$ digit even numbers can be formed by using digits $1,2,3,4,5,6,7$, if no digits are repeated?
ATTEMPT
There are three places to be filled in _ _ _
I wrote it like this
_ _ $2$
_ _ $4$
_ _ $6$
Now each ... | For the first digit you only have $3$ possibilities, $2,4$ or $6$. For the second digit, having chosen the first digit you only have $6$ possibilities left. For the third digit, having chosen the first two digit you have $5$ possibilities left. In total you have $3 \times 6 \times 5$ possibilities.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1307997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
A compact Hausdorff space It is known that every finite space is compact. Then I am worried whether there exists a compact Hausdorff space $X$ with with ordinal of $X$ is $\omega_0$.
Does anyone know about it?
| Consider the set formed by the sequence {1/n} union the point {0}. This is a closed and bounded subset of the real numbers so it's compact. By definition of a sequence it's countable. It's also Hausdorff since it's a subset of a Hausdorff space (the reals).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1308070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Prove that $G = \langle x,y\ |\ x^2=y^2 \rangle $ is torsion-free.
Prove that $G = \langle x,y\ |\ x^2=y^2 \rangle $ is torsion-free.
Here $x^2$ is central as $x^2y=yx^2$ similarly $y^2$ is central. Apart from this I do not know how to proceed. Taking any arbitrary word in $G$ does not help.
| As noticed here, $G= \langle x,y \mid x^2=y^2 \rangle$ is the fundamental group of the Klein bottle $K$.
Argument 1: Any subgroup $H \leq G$ comes from a cover $C \to K$ where $\pi_1(C) \simeq H$. But $C$ is a surface, and, by looking at the abelianizations of the closed surface groups, we deduce that $C$ cannot be co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1308202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 2
} |
Limit of $n\left(e-\left(1+\frac{1}{n}\right)^n\right)$ I want to find the value of $$\lim\limits_{n \to \infty}n\left(e-\left(1+\frac{1}{n}\right)^n\right)$$
I have already tried using L'Hôpital's rule, only to find a seemingly more daunting limit.
| Let $P_n = (1+1/n)^n$. Then
$$\log{P_n} = n \log{\left ( 1+\frac1{n} \right )} = n \left (\frac1{n} - \frac1{2 n^2} + \frac1{3 n^3} - \cdots \right) = 1-\frac1{2 n} + \frac1{3 n^2} - \cdots$$
$$\therefore P_n = e^1e^{-1/(2 n)+ 1/(3n^2)-\cdots} = e \left (1-\frac1{2 n} + \frac{11}{24 n^2} \cdots \right ) $$
Thus
$$\lim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1308303",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
} |
Is a subset of an inner product space also an inner product space? My question may seem trivial but it's important that I know this. I know for a fact that a subspace of an inner product space is also an inner product space, but how about an arbitrary subset? Could I argue that since we are allowed to pick any subset, ... | A subset of an inner product space does not have to be an inner product space.
As you said, we can pick a subset that is not a subspace, hence not an inner product space.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1308369",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How to compute determinant of $n$ dimensional matrix? I have this example:
$$\left|\begin{matrix}
-1 & 2 & 2 & \cdots & 2\\
2 & -1 & 2 & \cdots & 2\\
\vdots & \vdots & \ddots & \ddots & \vdots\\
2 & 2 & 2 & \cdots & -1\end{matrix}\right|$$
When first row is multiplied by $2$ and added to second, to $nth$ row, determina... | Your recursive approach is fine; just follow it through. Let $D_n(a,b)$ be the determinant of the matrix with diagonal elements $a$ and all other elements $b$; clearly $D_1(a,b)=a$. For $n>1$, multiplying the first row by $-b/a$ and adding it to every other row gives $0$'s in the first column (except for an $a$ in th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1308502",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Compactness of a group with a bounded left-invariant metric Let $G$ be a group equipped with a left-invariant metric $d$: that is, $(G,d)$ is a metric space and $d(xy,xz) = d(y,z)$ for all $x,y,z \in G$. Suppose further that $(G,d)$ is connected, locally compact, and bounded. Must $(G,d)$ be compact?
I can show from ... | The answer is no. Consider $\mathbb R$ with the bounded left invariant metric
$$d(x,y)=\min(|x-y|,1)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1308598",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Statistics: Why doesn't the probability of an accurate medical test equal the probability of you having disease? Suppose there is a test for Disease A that is correct 90% of the time. You had this test done, and it came out positive. I understand that the chance that this test is right is 90%, but I thought this would ... | Let's say 1/1000 people have a disease and we test 1000 people with a 10% fail rate on a test.
In this instance (1000 people) I haven't shown any false negatives. If we were showing 10,000 people we could include one.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1308656",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 12,
"answer_id": 11
} |
Proof of absolute inequality I am new to proofs and would like some help understanding how to prove the following abs inequality.
$$| -x-y | \leq |x| + |y|.$$
I think I should take out the negative in the left absolute value function.? Then prove for the triangle inequailty.
| $$|-x-y| \leq |x| + |y|\\
|-1(x+y)| \leq |x| + |y|\\
|x+y||-1|\leq |x| + |y|\\
1|x+y|\leq |x| + |y|\\
|x+y|\leq |x| + |y|
$$
I think that you can take it from here as long as you use the fact that $|x| = \max(x, -x)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1308798",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Dividing elliptic curve point on integer Can we solve the equation $nP = Q$, where $P$, $Q$ is a rational poins on elliptic curve ($P$ is unknown), and $n$ is integer?
| If $E$ is an elliptic curve the set $E(\Bbb C)$ of its complex points can be identified via Weierstrass theory with the quotient $\Bbb C/\Lambda$ where $\Lambda$ is some lattice, i.e. $\Lambda=\Bbb Zz_1\oplus\Bbb Zz_1$ for some complex numbers $z_1$, $z_2$ which are linearly independent over $\Bbb R$.
Thus, if $Q\in E$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1308897",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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