Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Second derivative of a third degree polynomial function Let $f(x) = ax^3 + bx^2 + cx +d $ be a third degree polynomial function.
$$f'' = 6ax + 2b = 0 \Longrightarrow x = \frac{-b}{3a} $$
This is equal to $\frac{1}{3}$ of the sum of the roots of $f(x)$. So my question is: can we say that the root of the second derivativ... | If the zero of $f''(x)$ is $p$. then the polynomial$f(x)$ has root $p$ with multiplicity 3. i.e,$(x-p)^3|f(x)$ and $(x-p)^4$ does not divides$ f(x)$
$(x-p)^3|f(x)$ and $f(x)$ is cubic polynomial $\implies$ $f(x)=k(x-p)^3$ , where $k$ is a real number.
therefore roots of $f(x)$ are $p,p,p$.........................(1)
Su... | {
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"timestamp": "2023-03-29T00:00:00",
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Number of solutions of the function
Let $'f'$ be an even periodic function with period 4 such that $f(x) = 2^x -1$, $0\le x \le2$. The number of solutions of the equation $f(x) = 1$ in $[-10,20]$ are?
Since the given function is even,
$f(x) = f(-x)$
since the given function is periodic,
$f(x+4) = f(x)$
So,
$f(x+4) = ... | First, we observe that as $f$ is even, the solutions $f(x) = 1$ on $[-2,2]$ is 1 and -1. Then, use the periodicity, $f(x+4) = f(x)$ to extend this function to find all the roots in your interval. For instance, we have $f(1) = f(5) = f(9)$ ...
| {
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"url": "https://math.stackexchange.com/questions/2775421",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Induction principle: $n^2-7n+12≥0$ for every $n≥3$ How can I prove that $n^2-7n+12≥0$ for every $n≥3$?
I know that for $n=3$ I have $0≥0$ so the inductive Hypothesis is true.
Now for $n+1$ I have $(n+1)^2-7(n+1)+12=n^2-5n+6$ and now I don't know how to go on...
| Note that
$$n^2-5n+6=n^2-7n+12+2n-6 \stackrel{\color{red}{n^2-7n+12\ge 0}}\ge 0 + 2n-6\ge 0$$
for $n\ge3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2775499",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 3
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Program for drawing geometry What program do math teachers use to draw geometry, or what is used in books?
I want the drawing to look EXACTLY (I mean the same font and the same line thickness etc.) as in the screenshot I send. Found many programs, but not a one to do it in this way. Please help! Here is the image:
| I have not used this program in a few years, so I don't know what changes have occurred lately, but geometer's sketchpad was really nice and probably still is. It is not free, but there is a free trial I think.
http://www.keycurriculum.com/sketchpad.1.html
| {
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Evaluation of a definite integral with square root in denominator The definite integral
$\displaystyle \int_0^{\pi } \frac{x \cos (x)}{\sqrt{\alpha^2-x^2}} \, dx$
is evaluated numerically.
After numerical evaluation with a CAS, it is found that the integral has real numerical values for $\alpha \geq \pi$, and complex n... | If $0<\alpha < \beta$, then $a^2-x^2<0$ for $\alpha < x < \beta$, so you're taking the square root of negative numbers.
| {
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Solving the problem : $z z_x + z_y = 0, \quad z(x,0) = x^2$ Exercise :
For the problem :
$$\begin{cases} zz_x + z_y = 0 \\ z(x,0) = x^2\end{cases}$$
derive the solution :
$$z(x,y) = \begin{cases} x^2, \quad y = 0\\ \frac{1+2xy - \sqrt{1+4xy}}{2y^2}, \quad y \neq 0 \; \text{and} \; 1+4xy >0 \end{cases}$$
When d... | The general solution is :
$$z(x,y) = F(x-zy)\qquad\text{OK}.$$
$F$ is an arbitrary function, to be determines according to the boundary condition.
Condition : $\quad z(x,0)=x^2=F(x-0y)$
So, the function $F$ is determined :
$$F(X)=X^2\qquad\text{any }X$$
We put this function into the general solution where $X=x-zy$.
$$z... | {
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$Z_5$ isomorphic to a subgroup of $S_4$ The question asks me if $Z_5$ is isomorphic to a subgroup of $S_4$.
What I was thinking of doing is writing down all the elements of $S_4$ and then again finding the subgroups generated by every element. But that is just very long. I assume there should be a proper way to check ... | No. First, we know $|S_4|=4!=24,$ and $|\Bbb{Z}_5|=5$. Then it is clear that $5$ does not divide $24,$ and by Lagrange's theorem, the order of any subgroup of $S_4$ must divide $24.$ So no such subgroup can exist.
| {
"language": "en",
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If three points are chosen at random on a circle's edge, what is the probability that the triangle contains the circle's center?
If I created a triangle with 3 random points on the outside edge of a circle, then what’s the probibility that the triangle contains the centerpoint of the circle?
Please answer in as many ... | The center will be included if the three points are not all within the same semicircle. This question shows the chance they are within a semicircle is $\frac 34$ so the chance the center is inside your triangle is $\frac 14$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2776061",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Show that the area ,A,of the rectangle PQRS is given by $2p(16-p^2)$ Show that the area ,A,of the rectangle PQRS is given by $2p(16-p^2)$
For question 10a) I am having trouble finding the $p^2$
I understand the 2p and is able to find the 16 using dy dx=0
Please help
(https://i.stack.imgur.com/GemEv.jpg)
| Let $S(4-p,0)$, then $P(4-p,(4-p)(4+p))$ or $P(4-p,16-p^2)$. Then you have $A=SR\times SP$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Normal linear maps over $\mathbb C$
Prove that if $\alpha: V \to V$ is a normal linear map on a finite-dimensional inner product space $V$ over $\mathbb C$ then $\alpha = \alpha_1 + i\alpha_2$ where $\alpha_1$ and $\alpha_2$ are self-adjoint, and $\alpha_1 \alpha_2 = \alpha_2 \alpha_1$.
Since every normal linear map ... | If we set
$\alpha_1 = \dfrac{\alpha + \alpha^\ast}{2}, \tag 1$
then evidently
$\alpha_1^\ast = \alpha_1, \tag 2$
i.e., $\alpha_1$ is self-adjoint; likewise, setting
$\alpha_2 = \dfrac{\alpha - \alpha^\ast}{2i}, \tag 3$
we see that
$\alpha_2^\ast = \dfrac{\alpha^\ast - \alpha}{-2i} = \dfrac{\alpha- \alpha^\ast}{2i}; \ta... | {
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$X$ a compact Hausdorff space and a sequence $A_1 \supset A_2 \supset \dots$ of closed connected subsets. Show that $\bigcap A_n$ is also connected. X a compact Hausdorff space and a sequence $A_1 \supset A_2 \supset \dots$ of closed connected subsets.
I need to show that the intersection $\bigcap _{n\in\Bbb N}A_n$ is ... | Let $U$ be an open set containing $A=\cap_n A_n$. Then there exists $n_0$ such that $A_n \subset U$ for $n\ge n_0$ ( otherwise the sets $A_n \backslash U$ would be closed, non void and decreasing so their intersection would be nonvoid).
Assume now $\cap_n A_n$ is not connected, $A=A'\cup A''$. Take $U'\supset A'$, $U'... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Binary symmetric channel probability
Given are vectors $x$ and $y$ with $d (x, y) = k ≤ n$, compute $P \left \{Y = y | X = x \right \}$. Does this probability depend on the concrete choice of the vectors x and y?
I'm little bit confused what should I do with d or Hamming distance. Also what formula should I use?
Usual... | If the Hamming distance between $x$ and $y$ is $k$ then there are $k$ positions at which $x$ and $y$ are different.
If $x$ is given and we would like to see $y$ at the output then we would like to see $k$ changes and $n-k$ non-changes.
Assuming that the channel works independently of the input and independently of it... | {
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$A^2=-I_4$. Find possible values of minimal polynomial and characteristic polynomial Let $A\in\mathbb{R}^{4\times 4}$ satisfy $$A^2=-I_4 .$$
(a) Find possible values of $m_a$ (minimal polynomial) and $p_a$ (characteristic polynomial).
(b) Find an example for A satisfying the condition.
Please help me approach the fir... | We have
$$
A^2 = -I\\
A^2 +I = 0
$$
and we direclty read off the polynomial $f(x) = x^2 + 1$, with $f(A) = 0$. Now you just need to figure out how the minimal and characteristic polynomials both relate to any given polynomial where $A$ is a root.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2776654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Show that $g'(x)+g(x)-2e^x=0$
Given a function $g$ which has derivative $g'$ for all x $\in {R}$ and satisfying $g'(0)=2$ and $g(x+y)=e^yg(x)+e^xg(y)$ for all $x,y\in {R}$
Show that $g'(x)+g(x)-2e^x=0$
$\dfrac{g(x+y)}{e^{x+y}}=\dfrac{g(x)}{e^{x}}+\dfrac{g(y)}{e^{y}}$
Putting $x=0$,
$g'(y)=2e^y+g(x)$
Also putting $y=0... | $$g(x+y) = e^y g(x) + e^x g(y)$$
Differentiating w.r.t. y
$$g'(x+y) = e^y g(x) + e^x g'(y)$$
Now put $y=0$
$$g'(x) = e^0 g(x) + e^x g'(0)\implies g'(x) - g(x) - 2e^x = 0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2776776",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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The intuition of the dual space? The dual space of X is defined to be the space of all linear and continuous functionals that map X to R. But, What exactly is a dual space intuitively?
In my current self-guided understanding, I think of a space of function as a set of points( or a region) in infinite dimensional spac... |
In my current self-guided understanding, I think of a space of function as a set of points( or a region) in infinite dimensional space $\Bbb R^\infty$. Let $f$ be a element of a space of functions $X$, can I think of each value $f(x)$ as the magnitude in the dimension $x$?
This is a very good intuition. In fact, the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2776841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
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How would one set up barycentric coordinates for a trapezoid? Barycentric coordinates are great for triangles, but I'm interested in how to construct a barycentric coordinate system for an arbitrary trapezoid.
I've seen this done for an arbitrary quadrilateral, but it ought to be simpler in the case of a trapezoid bec... | The source derives that
$$(\mathbf{c} \times \mathbf{d})\mu^2 + (\mathbf{c} \times \mathbf{b} + \mathbf{a} \times \mathbf{d})\mu + \mathbf{a} \times \mathbf{b} = 0.$$
When $\mathbf{c} \times \mathbf{d}$ is zero, we should just be able to conclude that
$$\mu = \frac{\mathbf{a} \times \mathbf{b}}{\mathbf{c} \times \mathb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2776961",
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Finding integers and the limit of a sequence - from a 100 sequence and more (Russian book) Consider a sequence $a_{1}=1$ and for every $k>1$ integer $a_k=a_{k-1}+\dfrac{1}{a_{k-1}}$.
a) How many positive integers $n$ are there, satisfying that $a_n$ be an integer?
b) Find the limit (if there is) of the $a_k$ sequence!
... | Let's prove that
$$
1 \le a_n \le n
$$
Since $a_1 = 1$ and $a_{n+1} > a_n$ we clearly have $a_n \ge 1$. We can prove that $a_n \le n$ by induction. We have $a_1 = 1 \le 1$ and induction step is
$$
a_{n+1} = a_n + \frac{1}{a_n} \le n + \frac{1}{a_n} \le n + 1,
$$
since $a_n \ge 1$.
About question a). If $a_n$ is an inte... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How is this proof for the scalar product rule of limits valid? If we let $K=\lim\limits_{x \to a} f(x),$
and let $c$ be a constant,
Then in order to show that $\lim\limits_{x \to a} cf(x) = cK$,
we must show that there is an $\epsilon$ for every $\delta$ such that
$\lvert cf(x)-cK \rvert < \epsilon$ whenever $\lvert x-... | The step missing is saying "So take $\delta>0$ such that $|f(x)-K|<\epsilon/|c|$ whenever $|x-a|<\delta$. Thus, that $\epsilon$ and $\delta$ will work to show that $|cf(x)-cK|<\epsilon$ whenever $|x-a|<\delta$."
Most of the time, this step is left out of proof involving limits since it has the same flavor and is largel... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Do Real Symmetric Matrices have 'n' linearly independent eigenvectors? I know that Real Symmetric Matrices have real eigenvalues and the vectors corresponding to each distinct eigenvalue is orthogonal from this answer. But what if the matrix has repeated eigenvalues? Does it have linearly independent (and orthogonal) e... | Real Symmetric Matrices have $n$ linearly independent and orthogonal eigenvectors.
There are two parts here.
1. The eigenvectors corresponding to distinct eigenvalues are orthogonal which is proved here.
2. If some of the eigenvalues are repeated, since the matrix is Real Symmetric, there will exist so many independ... | {
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"url": "https://math.stackexchange.com/questions/2777494",
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Lebesgue outer measure intervals may all be assumed to be open On page 6-7 of Invitation to Ergodic Theory by C. E. Silva
Proposition 2.1.1 Lebesgue outer measure satisfies the following properties.
(1) The interval $I_j$ in the definition of outer measure may all be assumed to be open.
With the proof for (1) being:
L... | The outer measure is defined as an infimum. If you take the infimum over less sets (only the open ones) then the infimum is possibly greater.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2777592",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Gradient of distance vector length Suppose that there is a surface described by:
$\phi(x,y,z)=c$
And suppose that there is a fixed point A:
$\vec{r_A}=(x_A,y_A,z_A)$
Let $\vec r$ be position vector of any point on the surface so that:
$R=|\vec r-\vec r_A|$
Show that $\nabla R$ is a unit vector whose direction is along... | By denoting $\vec r = (x,y,z)$ you can write $R = \sqrt{(x-x_A)^2+(y-y_A)^2+(z-z_A)^2}$. Then as $\nabla = (\frac{d}{dx},\frac{d}{dy},\frac{d}{dz})$ I will just do the calculation for the $x$-coordinate, as it is completely analogous for $y,z$:
$\frac{dR}{dx} = (\frac{1}{2}) R^{-1} 2 (x-x_A) = \frac{x-x_A}{R}$
And by ... | {
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How is the derivative of $\textrm{Trace}\left\{ X^T A X B\right\}$ with respect to $X$ equal to $AXB + A^TXB^T$? How is the derivative of $\mbox{Trace}\left\{ X^T A X B\right\}$ with respect to matrix $X$ equal to $AXB + A^TXB^T$?
\begin{align}
\nabla_X \ \textrm{Trace}\left\{ X^T A X B\right\} = AXB + A^TXB^T
\end{a... | With implicit summation over repeated indices,
$$\frac{\partial\operatorname{tr}X^T AXB}{\partial X_{ij}}=A_{lm}B_{nk}\frac{\partial}{\partial X_{ij}}(X_{lk}X_{mn})=A_{lm}B_{nk}(\delta_{il}\delta_{jk}X_{mn}+X_{lk}\delta_{im}\delta_{jn}),$$where $\delta_{rs}$ is the Kronecker delta ($1$ if $r=s$, $0$ is $r\neq s$). The ... | {
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A common tangent line
The graph of $f(x)=x^4+4x^3-16x^2+6x-5$ has a common tangent line at $x=p$ and $x=q$. Compute the product $pq$.
So what I did is I took the derivative and found out that $p^2+3p+q^2+3q+pq=0$. However when I tried to factorize it I didn't find out an obvious solution. Can someone hint me what to ... | Let the equation of the common tangent be $y=mx+b$.
Solving simultaneously the equations of the common tangent and $f(x)$, we get
$x^4+4x^3-16x^2+(6-m)x-(b+5)=0$
This equation will have two double roots (since a line is touching a curve twice), so let the roots be $p,\,p,\,q,\,q$.
Can you take it from here?
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Surface described by ${\bf r^\top A r + b^\top r}=1$? I was asked to describe the surface described by
$${\bf r}^\top {\bf A} {\bf r} + {\bf b}^\top {\bf r} = 1,$$
where $3 \times 3$ positive definite matrix ${\bf A}$ and vector $\bf b$ are given.
My intuition tells me that it is a rotated ellipsoid with a centre that ... | Note: for convenience we use $2b$ instead of $b$.
$$(x+a)^TA(x+a)+2b^T(x+a)=x^TAx+a^TAx+x^TAa+a^TAa+b^Tx+2b^Ta.$$
Notice that by symmetry of $A$, $x^TAa=a^TAx$. Collecting all the $x$ terms,
$$(2a^TA+2b^T)x$$ can be cancelled with the choice
$$a=-A^{-1}b.$$
Then
$$C=1-a^TAa-2b^Ta=1+b^TA^{-1}b>1.$$
(As $A$ is positive d... | {
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Show that unit circle is not homeomorphic to the real line Show that $S^1$ is not homeomorphic to either $\mathbb{R}^1$ or $\mathbb{R}^2$
$\mathbf{My \ solution}$:
So first we will show that $S^1$ is not homeomorphic to $\mathbb{R}^1$.
To show that they are not homeomorphic we need to find a property that holds in $S^1... | To prove that $\mathbb R^2$ is not compact: Assume that it is. The image of a compact space under a continuous map is compact. The mapping $f:\mathbb R^2 \to \mathbb R, (x,y)\mapsto x$ is continuous and has image $\mathbb R$. Hence $\mathbb R$ is compact. But you yourself showed that $\mathbb R$ is not compact. Contrad... | {
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"timestamp": "2023-03-29T00:00:00",
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Are the probabilities the same? For any problem, is the probability that at least 3 out of 10 people like doing something and the probability that at least 30 out of 100 people like doing something the same? I was wondering if as it grew larger, the probability would grow or shrink. Is it exponential?
| You are probably assuming that each person independently either does it or not with some probability $p$. Fluctuations get smaller as the sample size goes up, so if $p \gt 0.3$ you would expect more chance for $30$ out of $100$ and if $p \lt 0.3$ you would expect more chance for $3$ out of $10$. If $p$ is just slight... | {
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Using expectation/variance algebra in normal distribution A shop sells apples and pears. The masses, in grams, of the apples may be assumed to have a A~$N(180, 12^2)$ distribution and the masses of the pears, in grams, may be assumed to have a P~$N(100, 10^2)$ distribution. Find the probability that the mass of a rando... | Guide:
If assume independence, then we have $A-2P$ is normal distribution with mean $$\mathbb{E}[A-2P]=\mathbb{E}[A]-2\mathbb{E}[P]=180-2(100)$$
and
$$Var[A-2P]=Var[A]+4Va[P]=12^2+4(10^2)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2778369",
"timestamp": "2023-03-29T00:00:00",
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consequence of Hahn-Banach theorem In Wikipedia, it says that Hahn-Banach Theorem shows there are "enough" continuous linear functionals. But, why is that so in a space that is not necessarily normed? How does the statement of Hahn-Banach show this?
| Let $X$ be a normed space and $M$ a finite-dimensional subspace of $X$. Take any linear functional $f : M \to \mathbb{F}$. It will be continuous because $M$ is finite-dimensional.
Hahn-Banach lets you extend $f$ to a continuous linear functional defined on all of $X$, so you obtain a nontrivial element of $X^*$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Series 1 - 1/2^2 + 1/3 - 1/4^2 + 1/5 - 1/6^2 + 1/7... This is the exercise 2.7.2 e) of the book "Understanding Analysis 2nd edition" from Stephen Abbott, and asks to decide wether this series converges or diverges:
$$ 1-\dfrac{1}{2^2}+\dfrac{1}{3}-\dfrac{1}{4^2}+\dfrac{1}{5}-\dfrac{1}{6^2}+\dfrac{1}{7}-\dfrac{1}{8^2}\d... | Don't be swayed by the particular case. You have the following
Lemma Let $\sum a_n$ be a series (resp. $\sum b_n$ be a convergent series), then the intertwining $x_{2n}=a_n$ and $x_{2n+1}=b_n$ is convergent iff $\sum a_n$ is so.
Proof Let us, for short, suppose the indexing starting from zero. Then, we get
$$
\sum_{n... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Finding $\int^{1}_{0}\frac{\ln^2(x)}{\sqrt{4-x^2}}dx$
Finding $$\int^{1}_{0}\frac{\ln^2(x)}{\sqrt{4-x^2}}dx$$
Try: Let $$I=\frac{1}{2}\int^{1}_{0}\frac{\ln^2(x)}{\sqrt{1-\frac{x^2}{4}}}=\frac{1}{2}\int^{1}_{0}\sum^{\infty}_{n=0}\binom{-1/2}{n}\bigg(-\frac{1}{4}\bigg)^nx^{2n}\ln^2(x)dx$$
$$I=\frac{1}{2}\sum^{\infty}_{... | This is as far as I've gotten with the integral. I will admit, I underestimated the difficulty of this integral. The four in the denominator proved to be a lot more of a nuisance than I thought. Nevertheless, this integral can be painstakingly computed by hand with the help of our best friend
$$\log\sin x=-\sum\limits... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2778736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
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"answer_id": 2
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How to show that ${(\ln n)}^{\ln n}=n^{\ln(\ln n)}$ How to show that $${(\ln n)}^{\ln n}=n^{\ln(\ln n)}$$
Attempt:
$y={(\ln n)}^{\ln n}$ then $\ln y=\ln n\ln(\ln n)$ what to do next?
| HINT
We have
$${(\ln n)}^{\ln n}=e^{\ln n\cdot \ln (\ln n)}=(e^{\ln n})^{\ln (\ln n)}$$
and recall that by definition $e^{\ln n}=n$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Euler's totient function - prove the product formula Let n $\prod_{i=1}^k p_{i}^{e_i}$ with $k\in \mathbb{N}$, $p_i\neq p_j \in \mathbb{P}, \quad \forall i\neq j$ and $e_i \in \mathbb{N}^+$. Then for Euler's totient function follows:
$$\phi(n)=\prod_{i=1}^k (p_i-1)p_{i}^{e_i-1}$$
I have to prove this theorem in the 3 f... | Good strategy.
It's not quite true that the residue class must be prime, but instead that it must be coprime to $m$.
For $x$ to be a unit in the multiplicative group $(\mathbb{Z}/m\mathbb{Z})^\times$ is the same as saying there exists $y$ such that $xy \equiv 1 \bmod m$. In this phrasing, the question is to show that s... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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show whether $\frac {xy}{x^2+y^2}$ is differentiable in $0$ or not? (multivariable) Q: $f(x,y)=\frac {xy}{x^2+y^2}$ if $(x,y)\not=(0.0)$, and $0$ if $(x,y)=(0,0)$.
Is $f$ differentiable at $(0,0)$?
Attempt: $$\lim_{(x,y) \to (0,0)} \frac {f(x,y)-f(0,0)}{||(x,y)-(0,0)||} = \lim_{(x,y) \to (0,0)} \frac {\frac{xy}{x^2+... | Hint:
AM-GM gives
$$\left|\frac{xy}{x^2+y^2}\right| = \frac{|x||y|}{x^2+y^2} \le \frac{\frac{x^2+y^2}2}{x^2+y^2} = \frac12$$
with equality when $x=y$.
Therefore $\lim_{(x,y)\to (0,0)} f(x,y) \ne 0$ so $f$ is not continuous at $(0,0)$.
| {
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Proving Brooks' Theorem in Graph Theory Okay, so I'm mainly concerned with this lemma we do beforehand (although a similar, albeit less severe, issue comes up in the actual proof of Brooks' Theorem).
"Let $ G $ be a connected graph with maximum degree $ \Delta $ which has a vertex of degree less than $ \Delta $. Then ... | When you color a vertex $v$ in layer $k$ then it is adjacent to at most $\Delta-1$ vertices in total in layers $k,k+1$ (and some yet uncolored vertices in layer $k-1$). Therefore $v$ is adjacent to at most $\Delta-1$ vertices already colored and you have one color available to color $v$ greedily.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Sigmoid functions for a biased random number generator I'm trying to create a random number generator biased toward certain values. To do this I'm using a random number [0,1] as the parameter for a sigmoid curve. I've found the perfect curve to get the results I want,
$c\left( \frac{1}{1+exp(-ax + b(a-16) + 8))}\right)... | I ended up using the function $\frac{1}{a^{|x-m|}}$ where $m$ is the value I'm biased toward and $a = 1 + \frac{b^2}{n}$ where n is the interval of x, and b is the strength of the bias. Using integration, I picked a random area between the min and max $x$ values and solved for $x$.
| {
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"source": "stackexchange",
"question_score": "2",
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I'm not getting Wilson's theorem to work? Wilson's theorem states that:
p is a prime if and only if (p - 1)! $\cong$ -1 (mod p)
Obviously 5 is a prime so this should be true:
(5 - 1)! $\cong$ -1 (mod 5)
But when I tried to test it it doesn't work:
$(5 - 1)! = 4*3*2*1 = 24$
I get the results:
24 % 5 = 4
$24 \cong 4$(m... | Do not worry, you are correct.
Note that $$24 \cong 4(mod5) \cong -1(mod5)$$
Remember, $$24-(-1)=24+1 =25=5k$$
| {
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How I construct a segment $a^2$? The length of a segment is given. How can we construct a segment equal to the square of the given segment?
| Revised attempt:
1) Construct a $\triangle DBC$ with :
$|DB|=1;$ $ \angle BDC =90°$; $|DC|=a$ .
2) Construct a right angle at $C$, with one leg $BC$.
The other leg of this angle intersects $BD$
at $A$, I.e $\angle BCA =90°$.
3) We have :
$|AD| \cdot 1 = a^2.$
| {
"language": "en",
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Is there a shape that can be wrapped perfectly? Wrapping presents in the real world always involves overlapping paper (due to folds, etc).
Is there any shape that can (theoretically) be wrapped by a rectangular piece of paper without any overlap (the shape and the paper have the same surface area)?
If such a thing exis... | One solution is a regular tetrahedron. We can even generalize this to tetrahedrons constructed from regular ones where we just pull two oposing edges apart. The following pictures show a 3D models in blender. The red edges show where we cut the surface apart (called seams) and on the right side we see the unwrapped net... | {
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Linear Algebra - Complete solution for Ax = b Alright, I'm having some trouble understanding the "complete" solution for Ax = b.
For instance, suppose
$$A = \pmatrix{ 1 & 2 & 2 & 2 \\ 2 & 4 & 6 & 8 \\ 3 & 6 & 7 & 10}$$
I can already see that $Ax = \pmatrix{1 \\ 5 \\ 6}$ is a solution for this system but after elimina... | Since $m>n$ the system $Ax=b$ has infinitely many or zero solutions depending upon the augmented RREF
$$A = \pmatrix{ 1 & 2 & 2 & 2 &b_1\\ 0 & 0 & 2 & 4 & b_2 - 2b_1 \\ 0 & 0 & 0 & 0 & b_3 - b_2 - b_1 }$$
Notably if $b_3-b_2-b_1\neq 0$ we have no solution otherwise the general solution is given by $x_P+x_H$ that is th... | {
"language": "en",
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Probability that the millionth decay occurs within 100.2 seconds?
Radioactive decay of an element occurs according to a Poisson process
with rate $10,000$ per second. What is the approximate probability
that the millionth decay occurs within $100.2$ seconds?
Let $X$ be the number of decays and the number of expec... | You have a Poisson process with expected value $1,002,000$. The chance that the millionth decay has not happened yet is the sum of the probabilities of exactly $0,1,2,\ldots ,999,999$ decays having happened. I believe you are supposed to use the normal approximation. Based on the figures you quote, a million decays ... | {
"language": "en",
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What is the difference between discrete interval and continous interval I'm looking at my math textbook and it says for discrete distribution where the range is from a to b, $$f(x) = \frac{1}{b-a+1}$$
While for continuous distribution it states that $$f(x) = \frac{1}{b-a}$$
What is it different?
| The discrete interval from $a$ to $b$ (when both are integers) consists of the integers
$$
a, a+1, \ldots, b-1, b;$$
there are $s = b-a + 1$ of these, so with equal weighting, they each get probability $1/s$.
The continuous interval consists of all real numbers $x$ with $a \le x \le b$. Its length is $b-a$, so the uni... | {
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$\frac{\alpha}{\alpha + \beta}\|u\|^2 + \frac{\beta}{\alpha + \beta}\|v\|^2 > \langle u, v\rangle $ is true? I have the following question:
It's true that $$ \frac{\alpha}{\alpha + \beta}\|u\|^2 + \frac{\beta}{\alpha + \beta} \|v\|^2 > \langle u, v\rangle $$
for all $u,v \in \mathbb R^N $ with $\{u, v\} $
linearly... | Unfortunately that is not true. What are you asking is equivalent to
$$\begin{align}
\frac{\alpha}{\alpha + \beta}\|u\|^2 + \frac{\beta}{\alpha + \beta} \|v\|^2 &> \langle u, v\rangle \\
\alpha\|u\|^2 + \beta \|v\|^2 &> (\alpha + \beta)\langle u, v\rangle \\
\alpha\langle u, u\rangle - (\alpha + \beta)\langle u, v\ran... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Is there a function that gives unique values when a unique sequence of numbers is given as input? Consider a random sequence of numbers, like 1, 4, 15, 21, 27, 15... There are no constraints on what numbers may appear in the sequence. Think of it as each element in the sequence is obtained using a random number generat... | Assume each input sequence has finitely many terms, and all terms are nonnegative integers.
Let the function $f$ be given by
$$f\bigl((x_1,...,x_n)\bigr) = p_1^{1+x_1}\cdots p_n^{1+x_n}$$
where $p_k$ is the $k$-th prime number.
Then by the law of unique factorization, the function $f$ has the property you specified.
... | {
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"source": "stackexchange",
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Probability with candies My problem is: A jar containing $1000$ candies, $750$ are red and $250$ are yellow. If you randomly select $10$ candies from the jar, record the amount of red and yellow then replace the candies and repeat this process $50$ times how many times do you think you will get $0$ red candies, $1$ red... | Your formula for all $10$ candies being red is correct. However, the formula you give for exactly $9$ red candies is not correct. What you have written actually calculates the probability that the first $9$ candies are red and the last is yellow - it can equivalently be written as
$$\frac{750}{1000}.\frac{749}{999}.\f... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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matrices $A$ and $B$ such that $AB = -BA$? I was trying to find matrices non-singular $A$ and $B$ such that $AB = -BA$.I tried taking $A$ and $B$ to be general matrices and started with an order of the matrix as $2$ but I go into a bit of lengthy calculation.
This made me think while it was intuitive for me to calculat... | For $2 \times 2$ matrices you can use $$A=
\begin{pmatrix}
a & 0 \\
0 & -a \\
\end{pmatrix},$$
with $a\not =0$ and
$$B=
\begin{pmatrix}
0 & x \\
y & 0 \\
\end{pmatrix}.$$
Then $AB=-BA$ for arbitrary $x,y$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Anti symmetric matrix and rotations Studying the lagrangian formulation of Noether's theorem and came upon how the invariance under rotations gives conservation of angular momentum.
Whilst setting up the problem the notes state that if a potential only depends on the distance between 2 points, namely $V(|r_i-r_j|)$, th... | If you consider the set of $n$-by-$n$ rotation matrices $SO(n)$ as a Lie group, then the corresponding Lie algebra is the set of antisymmetric or skew symmetric $n$-by-$n$ matrices. I.e., in the limit of $\epsilon \to 0$, the any rotation matrix $U$ is equal to $I + \epsilon T$ up to first-order. This is known as an "i... | {
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How to calculate the geometric moments of a log-normal distribution? If I only know geometric mean and geometric standard deviation of a log-normal distribution how can I calculate the $n$-th moment of the distribution?
In the Wikipedia article I can only see a relationship for the $n$-th moment if the (arithmetic) mea... | For simplicity I will call $\mathcal{N}_k$ the $k$-th moment of the normal distribution with parameters $\mu$ and $\sigma$, so for example $\mathcal{N}_1 = \mu$, $\mathcal{N}_2 = \mu^2 + \sigma^2$, $\cdots$
Now if $X$ follows a lognormal distribution, then
\begin{eqnarray}
\mathbb{E}[\ln^k X] &=& \int\frac{{\rm d}x~}{x... | {
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Finding shape preserving positioning of a 'piston' I have a chain of rods that are setup in a default, ideal orientation. Points are labelled with a 'p' prefix, their lengths with an r:
I then stretch them to reach a desired end location, but they lose their shape:
I'd like to see if there's a better 'result' that t... | As in your bottom drawing you know p2 is on the left circle and p3 is on the right. Put p1 at the origin and measure the angle of the p1p2 segment by $\theta$ with $0$ to the right and increasing counterclockwise. The position of p2 is then $(r1\cos \theta, r1\sin \theta)$ You can put p4 on the positive $x$ axis. A... | {
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Proving that a relation R is an equivalence relation While I fully understand what it means to be an equivalence relation, I have a difficulty establishing proof that $R$ is an equivalence relation without just listing all pairs that $R$ creates and testing them.
However this method is greatly time consuming and is not... | $A$ and $B$ are related iff they are equal or complements.
Reflexivity: For every subset $A$ of $S$ we have $A=A$
Symmetry: If $A$ is related to $B$ then either they are equal or complements, so $B$ is also related to $A$
Transitivity: If $A$ is related to $B$ and $B$ is related to $C$, then either $B=A$ or $B$ is co... | {
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Maximal Ideals in $K[X,Y]$ So, given a field $K$ and a polynomial $f(x,y)$ in the ring of polynomials $K[X,Y]$, I am trying to understand why the following statement is true:
If $f(a,b) = 0$ for $(a,b) \in K \times K$, then $(f) \subset (x-a,y-b)$. I know that roots for a polynomial $g(x)$, give linear factors, but tha... | A useful trick is to think about the isomorphism $K[x,y] \cong K[x][y]$, i.e.
a polynomial in $x$ and $y$ can be thought of as a polynomial in $y$ with coefficients that are polynomials in $x$.
For a polynomial $g(x)$, we can use the division algorithm to write $g(x) = q(x)(x-a) + r$, where $r$ is a constant. This... | {
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"source": "stackexchange",
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Calculating Residues using L'Hopital? I am given that the complex function $$f(z)=\frac{(e^{z-1}-1)(\cos(z)-1)}{z^3(z-1)^2}$$ has 2 simple poles, one at $z=0$ and another at $z=1$, and asked to calculate the Residues of the function at the singularities. I know that the residue of a pole $z_0$ of $f(z)$ with order $n$ ... | Hint. Once that we note that both poles are of order $1$ (simple poles) then your computations are correct. What you need now is that
$$\cos(z)=1-\frac{z^2}{2}+o(z^2)\quad\mbox{and}\quad e^{z-1}=1+(z-1)+o(z-1).$$
Or equivalently, by L'Hopital's rule,
$$\lim_{z\rightarrow 0}\frac{\cos(z)-1}{z^2}=\lim_{z\rightarrow 0}\f... | {
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"timestamp": "2023-03-29T00:00:00",
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If $A^*x_n \to y$, there exists a sequence such that $A^*Ay_n \to y$. I'm struggling with a problem from Young's Introduction to Hilbert Space (7.30 to be more specific)
Let $\mathbb{H}$ be a Hilbert space, $A\in B(\mathbb{H})$ (ie a bounded operator) and $(x_n)_{n=1}^{\infty}$ a sequence in $\mathbb{H}$.
Prove that if... | Recall that $\ker A = \ker A^*A$. Namely, clearly $\ker A \subseteq \ker A^*A$. Conversely
$$x \in \ker A^*A \implies A^*Ax = 0 \implies 0 = \langle A^*Ax, x\rangle = \langle Ax, Ax\rangle \implies Ax = 0\implies x \in \ker A$$
Since $(\ker T)^\perp = \overline{\operatorname{Im} T^*}$, taking orthogonal complements in... | {
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Factorial of odds I am trying to find a simple factorial of all the preceding odd numbers. If $9$ were to be picked the equation would read $9\times 7\times 5\times 3\times 1$ (only odd numbers can be picked). Would the following fraction work?
$$\dfrac{x!}{2^{\left(\frac{x-1}{2}\right)}\left(\frac{x-1}{2}\right)!}$$... | Hint:
\begin{align}
9\cdot7\cdot5\cdot3\cdot1&= \frac{9\cdot8\cdot7\cdot6 \cdot5\cdot4\cdot3\cdot2\cdot1}{8\cdot6\cdot4\cdot2}\\&= \frac{9\cdot8\cdot7\cdot6 \cdot5\cdot4\cdot3\cdot2\cdot1}{2^4(4\cdot3\cdot2\cdot1)}\end{align}
| {
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What should I use in order to get the required function $g \in L^1$? Let $f \in L^1$ be such that $f$ is not equivalent to any bounded function. Prove there exists a function $g \in L^1$ such that $fg \notin L^1$.
I know that $m( \{x:|f(x)|>M\} )>0$, $\forall M>0$, but I can't see to come up with anything else useful t... | One can give a simple constructive proof of existence of such $g$.
Assume we have $f\in L^1(0,1)$ which is not $L^\infty$ (we fix $(0,1)$ for the sake of clarity, this does not impose any restriction on the construction below). For each $n\in \mathbb{N}$ define
$$
E_n = \{ x \in (0,1): n\leq |f(x)| \}.
$$
Since $f\no... | {
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"url": "https://math.stackexchange.com/questions/2781573",
"timestamp": "2023-03-29T00:00:00",
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Proof about relatively prime numbers.
Let $a,m,n, \in \mathbb{N}$. I want to show that if $a$ and $mn$ are relatively prime, then $a$ and $m$ are relatively prime.
To start us off, To say $a$ and $mn$ are relatively prime means that gcd($a,mn) = 1.$ I've tried using Bezout's Identity, but have not gotten anywhere. Al... | Try the contrapositive: If $a$ and $m$ are not relatively prime, then $d=\gcd(a,m)>1$ and clearly $d$ divides both $a$ and $mn$, so $\gcd(a,mn)\ge d > 1$, and thus $a$ and $mn$ are not relatively prime.
Since Not B implies Not A is logically equivalent to A implies B, you are done.
| {
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"timestamp": "2023-03-29T00:00:00",
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Under what conditions can the exponent product rule $a^{nm}=(a^n)^m$ be used. Under what conditions can you use the exponent product rule $a^{nm}=(a^n)^{^m}$?
For example $i=\sqrt{-1}=(-1)^{1/2}=(-1)^{2\times 1/4}=((-1)^2)^{1/4}=(1)^{1/4}=1$
What's going wrong here?
| What's wrong in your string of equalities is that:
*
*$i=\sqrt{-1}$ is wrong since $i$ is one of the square roots of $-1$ (the other one being $-i$, of course);
*$(-1)^{2\times(1/4)}=\bigl((-1)^2\bigr)^{1/4}$ is wrong because $(-1)^{2\times(1/4)}$ can be $i$ or $-i$, whereas $\bigl((-1)^2\bigr)^{1/4}$ can be one of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2781858",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Galois group of $\mathbb Q(\sqrt{2+\sqrt{2}})$ I am interested in the elements of the Galois group of $\mathbb Q(\sqrt{2+\sqrt{2}})/\mathbb Q$.
Let $\alpha:=\sqrt{2+\sqrt{2}}$, then the minimal polynomial $m_{\alpha,\mathbb Q}(X)=X^4-4X^2+2$ has roots $$\pm\sqrt{2+\sqrt{2}}=\pm\alpha\\\pm\sqrt{2-\sqrt{2}}=\pm\beta$$ wh... | Everything looks correct. I would add argumentation that the polynomial you got is really irreducible (Eisenstein, for example) and I'd add some calculation to support the claim that, for example, $\alpha\mapsto \beta \implies \beta\mapsto -\alpha$, but I'm guessing you've already done that.
Let me just write this one ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2781979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
When Higher Dimensions Help I'm interested in examples of situations that are easier in higher dimensions. To give a flavor of what I am looking for, here are two of my favorites:
(a) In dimensions 2 and higher, one can characterize the standard normal distribution (up to a constant multiple) as the spherically symmet... | The Poincaré conjecture is much easier to prove for its generalization to higher dimensions. Actually, the first proof was for dimension $n\ge 5$ by Smale in $1960$.
Michael Freedman solved the case $n = 4$ in 1982 and received a Fields Medal in 1986.
Grigori Perelman solved case $n = 3$ in 2003. This was still possib... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2782111",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Is there a correlation between the degree of an extension and it being an algebraic extension? $\Bbb R/ \Bbb Q$ has degree $\infty$ and is not algebraic
$\Bbb C/ \Bbb R$ has degree 2 and is algebraic
Is the degree of $\Bbb Q(x)$ over $\Bbb Q$ infinity or $1$?
The degree of an extension over a finite field is a positive... | Every element of $F(\alpha)$ would be algebraic over $F$.
To see this, suppose that $[F(\alpha):F]=n$ and suppose $a \in F(\alpha)$. Then, for each $i$ $a^i=a_{i,0}+a_{i,1}\alpha+\cdots+a_{i,n-1}\alpha^{n-1}$ for some $a_{i,0},\dots,a_{i,n-1} \in F$. By linear algebra, $a^0,\dots,a^n$ are linearly dependent (when $F(\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2782278",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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What is the $10^5\pmod{35}$? How to evaluate $10^5 \pmod {35}$ ?
I tried this $a=(10^2\cdot 10^3)\pmod{35}$
then again a mod $35$. This is very lengthy please tell me a shorter way?
| $$
10^2 \equiv 30 \left[35\right] \Rightarrow 10^3 \equiv 300 \left[35\right]\equiv 20 \left[35\right]
$$
So
$$
10^4 \equiv 200 \left[35\right] \equiv 25 \left[35\right] \Rightarrow 10^5 \equiv 250 \left[35\right]
$$
So
$$
10^5 \equiv 5 \left[35\right]
$$
( and in fact $ 10^5=2857*35 +5$ )
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2782379",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Invertible matrix of inner product values Let $\{v_1,v_2,\ldots,v_k\}$ be a basis in inner product space $V.$
I need to prove that the matrix
$$A=
\begin{pmatrix}
(v_1,v_1) & (v_1,v_2) & \cdots &(v_1,v_k) \\
\
\vdots & \vdots & \ddots&\vdots\\
(v_k,v_1) & (v_k,v_2) & \cdots&(v_k,v_k)
... | Hint 1
Take a linear combination of the rows which is zero.
Hint 2
So there are coefficients $a_{i}$ such that $\sum_{i=1}^{k} a_{i} (v_{i}, v_{j}) = 0$ for all $j$.
Hint 3
Your aim is to show that all $a_{i}$ are zero.
Hint 4
So for the vector $v = \sum_{i=1}^{k} a_{i} v_{i}$ you have $(v, v_{j}) = 0$ for a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2782491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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How to get all the factors of a number using its prime factorization? For example, I have the number $420$. This can be broken down into its prime factorization of $$2^2 \times3^1\times5^1\times7^1 = 420 $$
Using $$\prod_{i=1}^r (a_r + 1)$$ where $a$ is the magnitude of the power a prime factor is raised by and $r$ is ... | If
$n = \prod_{i=1}^r p_i^{a_i}
$
is the prime factorization on $n$,
there are
$\prod_{i=1}^r (a_i + 1)
$
prime factors.
Look at this
as counting a
$r$-digit number in a variable base,
with the base of the
$i$-th digit being
$a_i+1$,
so that digit goes from
$0$ to $a_i$.
If
$b_i$ is the $i$-th digit,
then the value cor... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2782625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
} |
Finding roots of trigonometric function I have been given the following functions (where $\omega_0$ is some positive constant)
$$\begin{align}
\Gamma(t) &= \frac{1}{t^2} \left(\; \frac{1}{2} \omega_0^2 t^2 - \cos(\omega_0 t) - \omega_0 t \sin(\omega_0 t) + 1 \;\right) \\[4pt]
\gamma(t) = \frac{d \Gamma(t)}{dt} &= \fra... | The general idea goes like this. $\gamma(t)$ behaves like $-\omega_0^2 \cos(\omega_0 t)/t$, and the position of the $k$th root of $\gamma(t)$ is
$$r_k = \frac {\pi (2 k + 1)} {2 \omega_0} + O(k^{-1}).$$
(The next order term is $-2 / (\pi \omega_0 k)$, but we only need the $O$ estimate.) Then
$$\Gamma(r_{2 k}) =
\frac ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2782750",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Why are skew-symmetric matrices of interest? I am currently following a course on nonlinear algebra (topics include varieties, elimination, linear spaces, grassmannians etc.). Especially in the exercises we work a lot with skew-symmetric matrices, however, I do not yet understand why they are of such importance.
So my ... | This is not the area of math you're interested in, but here's an example I might as well write down. In convex optimization we are interested in the canonical form problem
$$
\text{minimize} \quad f(x) + g(Ax)
$$
where $f$ and $g$ are closed convex proper functions and $A$ is a real $m \times n$ matrix. The optimizatio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2782814",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 4,
"answer_id": 0
} |
Let $f_n \to f$ pointwise on $[0,1]$. If $f_n, f$ are continuous, is it true that $\int_0^1 f_n(x) dx \to \int_0^1 f(x)dx$?
Let $\{f_n\}$ be a sequence of continuous functions on on $[0,1]$. Let $f_n \to f$ pointwise. If $f$ is continuous on $[0,1]$, is it true that $$\int_0^1 f_n(x) dx \to \int_0^1 f(x)dx?$$
I coul... | Let $f_n(x) = n^2x^n(1-x).$ Then $f_n\to 0$ pointwise everywhere in $[0,1],$ but $\int_0^1 f_n(x)\,dx \to 1.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2782921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Integrating integer powers of $\cos(\theta)$ Consider the following integral for $n\in\mathbb{N}$:
$$I_n = \int_0^\pi\cos^n\theta\,d\theta \tag1$$
which, using integration by parts, one can show to be $I_n = 0$ for $n$ odd and to be equal to
$$I_{2m} = \frac{(2m-1)!!}{(2m)!!}\pi \tag2$$
for $n = 2m$ even.
However, I'v... | There seems to be a sign error in your expansion of the odd-power integral. It should be
$$
I_{2m-1} = \frac{1}{2^{2m-1}}\sum_{k=0}^{2m-1}{{2m-1}\choose k}\frac{-2}{i(2k-2m+1)}.
$$
Now notice that
$\binom{2m-1}{2m-1-k} = \binom{2m-1}{k}$
and that
$2(2m-1-k)-2m+1 = -(2k-2m+1),$
and perhaps it will be clearer that the l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2783011",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Finding the intersection of two lines, in polar coordinates The sticking point is figuring out the substitutions for a ratio of cosines of differences.
I have a pair of lines in polar coords:
$$r = \frac{s_1}{\cos(\theta - \alpha_1)} \qquad r = \frac{s_2}{\cos(\theta - \alpha_2)}$$
where
$$\begin{align}
\alpha_1 &= ... | HINT
We have
$$s^1_{val} \cos(\theta - s^2_{ang}) =s^2_{val} \cos(\theta - s^1_{ang})$$
$$s^1_{val} \cos \theta\sin (s^2_{ang})+s^1_{val} \sin \theta\cos (s^2_{ang})=s^2_{val} \cos \theta\sin (s^1_{ang})+s^2_{val} \sin \theta\cos (s^1_{ang})$$
$$s^1_{val} \cos \theta\sin (s^2_{ang})-s^2_{val} \cos \theta\sin (s^1_{ang... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2783118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Gradient In Complex space I have a triangle on the upper half-plane. One corner is at infinity (0,inf), one corner is at zero (0,0) and one corner is at (0,1). However when displayed using the poincare disk model the triangle is equilateral.
I want to put a gradient on the triangle so that (0,1) has a value of red, whi... | Let us denote the Poincaré disk coordinates with $(x_P,y_P)$ and the half-plane coordinates with $(x_H,y_H)$. Some possible solutions:
*
*Inspection of your Poincaré disk picture shows that gradient = $x_P$ clearly gives us the solution. To obtain the same gradient in half-plane coordinates, we need to use the mappi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2783214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If $0Given that $f(x)=e^x(x^2-6x+12)-(x^2+6x+12),\;\;x>0$ is an increasing function. I want to prove that:
If $0<x<\infty$, then $0<\frac{1}{e^x-1}-\frac{1}{x}+\frac{1}{2}<\frac{x}{12}$.
Here is what I have done:
If $0<x<\infty$, then by Mean Value Theorem, $\exists\; c\in(0,x)$ such that
$$f'(c)=\frac{e^x(x^2-6x+... | We'll prove that $$0<\frac{1}{e^x-1}-\frac{1}{x}+\frac{1}{2}$$ or
$$\frac{1}{e^x-1}>\frac{2-x}{2x},$$ which is obvious for $x\geq2$.
But, for $0<x<2$ we need to prove that
$$e^x-1<\frac{2x}{2-x}$$ or $f(x)>0,$ where
$$f(x)=\ln(x+2)-\ln(2-x)-x.$$
Indeed, $$f'(x)=\frac{x^2}{4-x^2}>0,$$ which says
$$f(x)>\lim_{x\rightarro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2783368",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Integration limits problem Here is a part of a problem I have a hard time with:
Let $$f(x)= 10e^{-0.201x}+3$$
Let $$g(x)= -x^2+12x-24$$
Find the area enclosed by the graphs of f and g
Here is the answer as explained by the teacher:
Finding limits $3.8953$ and $8.6940$
Evidence of integrating and subtracting functio... | The limits are found as the intersections of the two curves, a decaying exponential and a downward parabola. There will be no closed-form of the roots and you need to use numerical methods.
To obtain good starting estimates, you can replace the exponential (blue) by its second order development (magenta), to obtain a q... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2783504",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
USAMO 2018: Show that $2(ab+bc+ca) + 4 \min(a^2,b^2,c^2) \geq a^2 + b^2 + c^2$ Here is question 1 from USAMO 2018 Q1 (held in April):
Let $a,b,c$ be positive real numbers such that $a+b+c = 4 \sqrt[3]{abc}$.
Prove that: $$2(ab+bc+ca) + 4 \min(a^2,b^2,c^2) \geq a^2 + b^2 + c^2$$
This question is on symmetric poly... | Without loss of generality, let $a\le b=ax\le c=axy, x\ge 1, y\ge 1$.
Then:
$$a+b+c = 4 \sqrt[3]{abc} \Rightarrow a+ax+axy=4\sqrt[3]{a(ax)(axy)} \Rightarrow 1+x+xy=4x^{\frac23}y^{\frac13} \qquad (1)$$
Also:
$$a+b+c = 4 \sqrt[3]{abc} \Rightarrow a^2+b^2+c^2=16\sqrt[3]{(abc)^2}-2(ab+bc+ca) \qquad (2)$$
Plugging $(2)$ and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2783647",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
find ratio of area of triangle $\Delta{AFG}$ to area of $\Delta {ABC}$ In the figure below, $BD=DE=EC$, $F$ divides $AD$ so that $FA:FD=1:2$ and $G$ divides $AE$ so that $GA:GE=2:1$.
Find ratio of area of triangle $\Delta{AFG}$ to area of $\Delta {ABC}$
My Try:
I noticed that $G$ is centroid of $\Delta{ADC}$ and
$$Ar(... | HINT:
Lemma: Prove that $S_{ABC}=\dfrac{1}{2}AB\times AC\times\sin{BAC}$ and apply similarly for other angles
I will use the lemma above, but I will not prove it here.
Firstly, notice that $S_{ADE}$ has the same altitude as $S_{ABC}$, but $\dfrac{DE}{BC}=\dfrac{1}{3}$, so using the area formula $\dfrac{\text{base}\ti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2783741",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
General request for a book on mathematical history, for a VERY advanced reader. I am aware that there are answered similar questions on here, however I am specifically after a text that would be engaging for a professor of mathematics, also Fellow of the Royal Society (FRS).
He is unwell and in the hospital, and I woul... | I suggest
*
*Mathematical Thought from Ancient to Modern Times, Vol. 1&2, by Morris Kline
*Mathematics and Its History, by John Stillwell
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2783869",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "83",
"answer_count": 29,
"answer_id": 3
} |
Concluding a proof ($\pi$ is irrational) I am making a proof for irrationality of $\pi$ and i proceeded as follows:
Let $\pi=\frac{u}{v}$ for some $u,v\in \Bbb{N}$, define family of integrals:
$$I_n=\frac{v^{2n}}{n!}\int_0^\pi x^n(\pi-x)^n \sin x\,dx$$ By some elementary estimates we have
$$0<I_n\leq\frac{v^{2n}}{n!}\p... | Good question! Given the fact that your estimate and (1) are correct, you should choose (b) rather than (a) because the latter is not correct.
As @InterstellarProbe has mentioned in the comment, there are some sequences $a_n$ tending to $0$, but with $(4n-2)a_n$ still tending to zero$.
The point is how fast $a_n$ tends... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2784007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Proof verification: $x_n \rightarrow a$ and $x_n \rightarrow b$ then $a=b$ Let $(x_n)$ be a sequence in a metric space $S$. Prove: if $x_n \rightarrow a$ and $x_n \rightarrow b$ then $a=b$.
Assume $a \neq b$. Take two balls $B_r(a)$ and $B_r(b)$ with such an $r$ that $B_r(a) \cap B_r(b)=\emptyset$. Then WLOG assume the... | Here is a simpler argument.
Take $\varepsilon>0$.
Then $d(x_n,a) < \varepsilon$ and $d(x_n,b) < \varepsilon$ for all $n$ sufficiently large.
But then $d(a,b) \le d(a,x_n)+ d(x_n,b) = d(x_n,a)+ d(x_n,b) < 2\varepsilon$.
Since $\varepsilon$ is arbitrary, this can only happen if $d(a,b)=0$, which implies $a=b$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2784117",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Doubts about a question I asked a long time ago (eigenvalues) Here I posted a question about the eigenvalues of the matrix $A:=vv^t$ (where $v\in\mathbb{R}^n$).
The question was answered but I think (after some time) that I am not satisfied.
Can someone please expand the answer? I don't understand why $A$ has rank at ... | Note that $$(vv^T)v = v(v^Tv) = \|v\|^2 v$$
so $v$ is an eigenvector with the eigenvalue $\|v\|^2 = \sum_{i=1}^n x_i^2$.
Also, explicitly
$$vv^T = \begin{pmatrix} x_1^2 & x_1x_2 & \ldots & x_1x_n \\
x_2x_1 & x_2^2 & \ldots & x_2x_n\\
\vdots & \vdots & \ddots & \vdots \\
x_nx_1 & x_nx_2 & \ldots & x_n^2\end{pmatrix}$$
s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2784186",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 2
} |
Misunderstanding Caley-Hamilton Theorem - Characteristic Polynomial in the Standard/Factorized Form so my question is about the Caley-Hamilton theorem. Consider the following Matrix A.
$$A =\begin{pmatrix}
-1 & 0 & 4 \\
2 & -1 & 0 \\
3 & 2 & -1
\end{pmatrix}$$
The characteristic polynomial is (according to WolframAlpha... | You have to check your algebra. The same manipulations that show that $$-x^3-3x^2+9X+27=-(x-3)(x+3)^2$$ will work when writing $A$ instead of $x$.
In WA, the matrix product is denoted with a period. And the square is interpreted entrywise. Here is the computation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2784539",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Determinant of the matrix associated with the quadratic form If A is the matrix associated with the quadratic form $4x^2+9y^2+2z^2+8yz+6zx+6xy$ then what is the determinant of A? I don't know how to solve quadratic form of a matrix pls help me
| Consider the matrix $$A= \begin{pmatrix}
4 & 3 & 3 \\
3 & 9 & 4 \\
3 & 4 & 2 \\
\end{pmatrix}.$$
Note that the coefficients relating to $x^2,y^2$ and $z^2$ lie on its diagonal. The other entries correspond to half of the coefficients of the interaction terms. Multiplying this matrix with $\begin{pmatrix}
x&y&z \\
\en... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2784685",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Is it only the generator of the group that commutes with all the other elements? If a group is generated by an element does that mean the generator commutes with all the other elements or does it mean that because the group is cyclic(as it has a generator) that all elements commute with each other.
For example, I am t... | The short answer is: the idea of "generators" and "commutativity" are completely disjoint. I don't really know what else to say...
In $D_4$, the generators $a$ and $b$ do not commutate with each other, so cannot each commute with every element. For example, $ba=a^3b\neq ab$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2784813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Inverse projective transformation: given $\varphi_A:\mathbb{P^3}\to \mathbb{P^3}$ and a line $r$, find $\varphi_A^{-1}(r)$ The transformation $A:\mathbb{R^4}\to \mathbb{R^4}$, represented by the matrix:
$A$ =$
\begin{bmatrix}
3 & 0 & 1 &0 \\
0 & -3 & 0 & 1 \\
-1 & 0 & 1 & 0 \\ 0&-1&0&-1
\end{bmatrix... | Your method works. Another method is to use $A$ directly: If you have a point transform given by the invertible homogeneous matrix $A$, i.e., $\mathbf p' = A\mathbf p$, then planes transform as $\mathbf\pi'=A^{-T}\mathbf\pi$ because $$\mathbf\pi^T\mathbf p = 0 \iff \mathbf\pi^T(A^{-1}\mathbf p') = (A^{-T}\mathbf\pi)^T\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2785008",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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How can I compute $\mathbb P\{T_nLet $(X_n)_{n}$ a random walk over $\mathbb Z$ starting at $0$, i.e. $\mathbb P\{X_0=0\}=1$. I denote $T_k=\inf\{n\geq 1\mid X_n=k\}$. I suppose that $$\mathbb P\{X_{n+1}=X_n+1\mid X_n,...,X_0\}=p\quad \text{and}\quad \mathbb P\{X_{n+1}=X_n-1\mid X_n,...,X_0\}=q.$$
Remark that $q=1-p$. ... | Good question! For your three examples, I think you shouldn't condition on $X_0 =0$ because it does not change anything ($P(X_0=0)=1$). Instead we should condition on the value of $X_1$ (for $n\geq 2$) and use Bayesian formula. $P(T_1<T_0) = P(X_1 = 1) = p$ is obvious. For $T_2$, we have
$$
P(T_2<T_0) = P(T_2<T_0|X_1=1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2785131",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Derivative of L-1 norm of matrix Assume you want to find the derivative respect to X ($p \times p$) matrix of
$$
\frac{\partial}{\partial X} || X - A ||_1
$$
where A is ($p \times p$) matrix.
How can I do it?
| Define
$$\eqalign{
Y &= (X-A) \cr
B &= {\rm abs}(Y) \cr
G &= {\rm signum}(Y) \cr
B &= Y\odot G \cr
}$$ where the functions are applied element-wise.
Then find the differential and gradient of the norm as
$$\eqalign{
\phi &= 1:B = 1:Y\odot G = G:Y \cr
d\phi &= G:dY = G:dX \cr
\frac{\partial\phi}{\partial X} &= G =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2785215",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
About a continuous function satisfying a given integral equation
Question: Let $f(x):[0,2] \to \mathbb{R}$ be a continuous function, satisfying the equation
$$
\int_{0}^{2} f(x)(x-f(x)) \,dx = \frac{2}{3}.
$$
Find $2f(1)$.
The solution took $f(x)=\frac {x}{2}$. Yes, I know it does not contradict the condition but... | By A.M-G.M inequality we have $$f(x)(x-f(x))\leq \Big({f(x)+x-f(x)\over 2}\Big)^2 = {x^2\over 4}$$
so we always have $$
\int_{0}^{2} f(x)(x-f(x)) \,dx \leq \int_{0}^{2} {x^2\over 4} \,dx=\frac{2}{3}.
$$
Since we have equality we have $f(x)=x-f(x)$ for each $x$ and we are done (remember that we have equality in A.M-G.M ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2785349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Confused about diagonal matix notation Reading a book of physics I found the following definition of diagonal matrix:
$$A_{ij}= A_{ii}\delta_{ij}$$
I understand a diagonal matrix has only diagonal elements nonzero, but is the previous notation correct?
I'm somehow confused because if we choose $A_{ij}$ with $i\neq j $... | It's correct, if $i=j$, $A_{ij}=A_{ii}\delta_{ij}=A_{ii}\cdot 1 = A_{ii}$
if $i\ne j$, $A_{ij}=A_{ii}\delta_{ij}=A_{ii}\cdot 0 = 0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2785462",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Bayes theorem P(A) for second step Here is the problem. Alice went to doctor and doctor said test will produce 99% true result for ill people and 99% true negative for non-ill people. For this particular illness there is 1 per 1000 who get ill.
So I implemented this in bayes theorem formula and found out Alice chance t... | $\newcommand{\ill}{\text{ill}}\newcommand{\well}{\text{well}}$
First we'll do it by turning the crank. Then further below, we'll see a simpler way.
\begin{align}
\Pr(\ill\mid +) & = \frac{\Pr(+\mid \ill)\Pr(\ill)}{\Pr(+\mid\ill)\Pr(\ill) + \Pr(+\mid\well)\Pr(\well)} \\[10pt]
& = \frac{99\times 1}{(99\times 1) + (1\time... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2785579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
AP Calculus Fundamental Theorem of Calculus Please help me go over this problem; I am a bit confused.
Find ${\displaystyle \frac{\mathrm d}{\mathrm dt} \int_2^{x^2}e^{x^3}\mathrm dx}$.
| The Fundamental Theorem of Calculus states that if
$$g(x) = \int_{a}^{f(x)} h(t)~{\rm d}t$$
where $a$ is any constant, then
$$g'(x) = h(f(x)) \cdot f'(x)$$
Using this with the integral, $g(x) = g(x)$, $f(x) = x^2$, and $h(x) = e^{x^3}$.
So, $$\frac{\mathrm d}{\mathrm dx} \int_2^{x^2}e^{x^3}\mathrm dx=(2x)e^{x^6}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2785698",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Is there a proof for $\lim_{x \to a} \frac{1}{x-a} = \infty$? I am an adult software developer who is trying to do a math reboot. I am working through the exercises in the following book.
Ayres, Frank , Jr. and Elliott Mendelson. 2013. Schaum's Outlines Calculus Sixth Edition (1,105 fully solved problems, 30 problem-so... | A function is not defined properly unless you specify its domain. When we say the function $\frac{1}{x}$ we usually mean the function $f: \mathbb{R}\setminus \lbrace 0 \rbrace \rightarrow \mathbb{R} $ defined by $f(x)=\frac{1}{x}$ for all $x \in \mathbb{R} \setminus \lbrace 0 \rbrace $. Here $\lim_{x \to a} f(x) $ does... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2785912",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Transformation of a square Having fun with some integrals, I caught myself thinking about transforming of regions. So I have the following questions.
Suppose we have the square determined by inequalities $0<x<1, 0<y<1$ and a transformation rule $u=xy,v=x+y$.
The question is: what form will this square have in new coor... | Perhaps you can get some insight from mapping some specific sets
$(x, y_0)\mapsto (x y_0, x + y_0)$
where $0\leq y_0 \leq 1$ is a constant number. In the $xy$-plane this a horizontal line. In the $uv$-plane a couple of things can happen, if $y_0 =0$, we obtain a vertical line running through the origin. If $y_0\not ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2786100",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Name for geometry that differs by a translation I have a simplistic question: If I have two triangles, and there exists a translation that makes them equivalent (all their vertices would be the same after the translation), then is there a special term in geometry that I would use to describe the relationship between t... | They say the translate.
See https://en.wikipedia.org/wiki/Translation_(geometry):
"If $T$ is a translation, then the image of a subset $A$ under the function $T$ is the translate of $A$ by $T$. The translate of $A$ by $T_v$ is often written $A + v$."
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2786178",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
A formula for higher order derivatives of inverse function The formula for higher order derivatives of compound functions is known as Faà di Bruno's formula. Does there exist a similar formula for higher order derivatives of an inverse function, i.e. $D^k(f^{-1}(x))$?
I would be most interested in a non-recursive formu... | As requested by the OP, I add a link to a paper containing the formula and the bibliographic data (in Japanese).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2786280",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
multivariable quadratic form I see the multivariable quadratic form given by 2 different expressions:
$$f(X)=X^{T}AX$$
versus
$$f(X)=\frac{1}{2}X^{T}AX +B^{T}X+C$$
Which is right? and crucially why the difference? What affect does it have?
| $X^{T}AX$ is a quadratic form.
$\frac{1}{2}X^{T}AX +B^{T}X+C$ is a quadratic polynomial, of which the quadratic term, $\frac{1}{2}X^{T}AX$, is a quadratic form.
The factor of $\frac{1}{2}$ in the quadratic polynomial is there "for convenience" because it makes the Hessian of the quadratic polynomial equal to $A$ (pres... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2786426",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How do I show that this set is compact and not compact at the same time? Given:
1.) For each $n \in \mathbb{N}, L_n$ is a line segment from $(0,0)$ to $(1, \frac{1}{n}) $
2.)$ L_\infty$ is a line segment from $(0,0)$ to $(1,0) $
3.) Both $L_n$ and $L_\infty$ are equipped with the subspace topology induced on $\mathbb{R... | In the "defined topology", $\{(1,1/n):n\in\Bbb N\}$ is a closed
discrete subset. Compact spaces cannot have closed
infinite discrete subsets.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2786502",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
How to get the exact value of $\sin(x)$ if $\sin(2x) = \frac{24}{25}$ How to get the exact value of $\sin(x)$ if $\sin(2x) = \frac{24}{25}$ ?
I checked various trigonometric identities, but I am unable to derive $\sin(x)$ based on the given information.
For instance:
$\sin(2x) = 2 \sin(x) \cos(x)$
| Refer to the diagram below.
$AD$ is the angle bisector of the right-angled triangle $\Delta ABC$.
Given $BC=24$ and $AC=25$.
Let $\angle DAB = x$.
From $\Delta ABC$ we see that $\sin(2x) = \dfrac{24}{25}$.
Now, by angle bisector theorem, $BD:DC = 7:25$.
Therefore, $BD = \dfrac{7}{7+25} \times 24 = \dfrac{21}4$.
Observi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2786868",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 4
} |
Maximizing $3\sin^2 x + 8\sin x\cos x + 9\cos^2 x$. What went wrong?
Let $f(x) = 3\sin^2 x + 8\sin x\cos x + 9\cos^2 x$. For some $x \in \left[0,\frac{\pi}{2}\right]$, $f$ attains its maximum value, $m$. Compute $m + 100 \cos^2 x$.
What I did was rewrite the equation as $f(x)=6\cos^2x+8\sin x\cos x+3$. Then I let $\m... | Using the identity $$\cos^2 x=\frac {1+\cos 2x}{2}$$ and $$2\sin x\cos x=\sin 2x$$ the question changes to finding minimum value of the function
$$6+3\cos 2x+4\sin 2x$$
And now using a standard result that the range of a function $a\sin \alpha\pm b\cos \alpha$ is $[-\sqrt {a^2+b^2},\sqrt {a^2+b^2}]$
Hence the range of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2787031",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Summation of $\sum_{r=1}^{n} \frac{\cos (2rx)}{\sin((2r+1)x \sin((2r-1)x}$ Summation of $$S=\sum_{r=1}^{n} \frac{\cos (2rx)}{\sin((2r+1)x) \sin((2r-1)x)}$$
My Try:
$$S=\sum_{r=1}^{n} \frac{\cos (2rx) \sin((2r+1)x-(2r-1)x)}{\sin 2x \:\sin((2r+1)x \sin((2r-1)x}$$
$$S=\sum_{r=1}^{n} \frac{\cos (2rx) \left(\sin((2r+1)x \co... | Good question! Here is one possible approach. First we rewrite the numerator as
$$
\cos(2rx) = \cos[(r+\frac{1}{2})x+(r-\frac{1}{2})x] = \cos\frac{2r+1}{2}x\cos\frac{2r-1}{2}x - \sin\frac{2r+1}{2}x\sin\frac{2r-1}{2}x
$$
and the denominator as
$$
4\sin\frac{2r+1}{2}x\cos\frac{2r+1}{2}x\cdot \sin\frac{2r-1}{2}x\cos\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2787215",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Valuations and different ideal in local fields We know that if we have a totally ramified finite extension of local fields $L/K$ then the different ideal is $D_{L/K} = ( h'(\pi))$ i.e. the ideal generated by te derivative of the minimal polinomial of $\pi$ (a uniformizer of L). My question is: since the uniformizer is... | Another uniformizer has the form $\tilde{\pi} = u \pi$ for some unit $u \in O_L^{\times}$. The minimal polynomial $\tilde h$ is just $\tilde h(X) = h(u^{-1}X)$, and in particular we get
$$\tilde h'(\tilde{\pi}) = u^{-1} h'(u^{-1} \tilde{\pi}) = u^{-1} h(\pi).$$
Thus the ideal generated by $\tilde h'(\tilde{\pi})$ in $O... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2787316",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to show $e^t$ is the unique solution to the integral equation $\int_0^1 e^{ts} x(s) ds = \frac{e^{t+1}-1}{t+1}$? How to show $x(t) = e^t$ is the unique solution to the following integral equation? $$\int_0^1 e^{ts} x(s) ds = \frac{e^{t+1}-1}{t+1}$$
My thoughts:
$$(t+1)\int_0^1 e^{ts} x(s) ds = e^{t+1}-1 $$
Integrat... | Suppose $x(s)$ and $y(s)$ both satisfy the integral equation. Then $\int_0^1 e^{ts}(x(s)-y(s))ds=0$ for all $t$. Then the same is true if we replace $e^{ts}$ with a finite linear combination of elements of the family
$\left\{e^{ts}\right\}_{t\in \mathbb{R}}$. Thus
$$\int_0^1 f(s)(x(s)-y(s))ds,\qquad \forall f\in \op... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2787421",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Probability approach in the expected payoff of a dice game I am trying to understand the problem of expected payoff of a dice game explained here. I can roll the dice up to three times, but after each one I decide if I want to try once again or not. The idea is to find an optimal strategy that maximizes the expected pa... | Using your method:
$1$-roll game:
$$E(X)=\sum_{k=1}^6 kP(X=k)=1\cdot \frac{1}{6}+2\cdot \frac16+3\cdot \frac16+4\cdot \frac16+5\cdot \frac16+6\cdot \frac16=3.5.$$
$2$-roll game:
$$E(X)=\sum_{k=1}^6 kP(X=k)=1\cdot \frac{3}{36}+2\cdot \frac{3}{36}+3\cdot \frac{3}{36}+\\
4\cdot \left(\frac16+\frac{3}{36}\right)+5\cdot \l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2787564",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Topological "closure" of a binary relation Let $f$ be a binary relation on a set $U$.
Topology $T f = \{ E \in \mathscr{P} U \mid
f [E] \subseteq E \}$ (here $f[E]$ is the image of a set $E$ by binary relation $f$).
Conjecture Closure operator $\operatorname{cl}$ of $T f$ is equal to $E \mapsto ( \operatorname{id}_U \c... | $E \mapsto ( \operatorname{id}_U \cup f \cup f^2 \cup f^3 \cup \ldots ) [E]$ maps open sets to itself. So, it can be closure only if all open sets are closed.
For a counterexample for the conjecture take $f = \{ (0, 0), (1, 1), (0, 1) \}$. Open sets are $\{ \}$, $\{ 1 \}$, $\{ 0, 1 \}$. $\{ 1 \}$ is open but not closed... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2787679",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proving a set is bounded. Let $E = \{x \in\mathbb R^p : \sum_{i = p}^p X_i^2/\alpha_i^2 \leq 1\}$
Prove that $E$ is closed and bounded.
To prove that $E$ is closed I used the fact that the boundary of the set $E$ is equal to $\{x \in\mathbb R^p : \sum_{i = p}^p X_i^2/\alpha_i^2 = 1\}$ and the boundary is contained in... | Try to get a lower bound on $\sum\frac{x_i^2}{\alpha_i^2}$ involving the Euclidean norm of $x$. For example:
$$\sum\frac{x_i^2}{\alpha_i^2}\geq\min_i\left(\frac1{\alpha_i^2}\right)||x||_2^2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2787757",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
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