Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
If B is invertible matrix , then the Row space of A and BA are same I have seen a proof that B is row equivalent to A iff exist invertible matrix C such that :
B = CA , because C is elementary matrix , but i cant find the next step . can B "used" as the elementary matrix that lead from B to BA without changing the r... | Well, since $B=CA$, you see that rows of $B$ can be written as linear combinations of rows of $A$. (To be more explicit, write
$$B=\begin{bmatrix}\mathbf{b_1\\b_2\\ \vdots \\ b_m}\end{bmatrix},\quad A=\begin{bmatrix}\mathbf{a_1\\a_2\\ \vdots, \\ a_m}\end{bmatrix},\quad C=\begin{bmatrix}c_{11} &\cdots & c_{1m}\\
\vdots... | {
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"timestamp": "2023-03-29T00:00:00",
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Can two vectors of 3-Tuples span $\mathbb R^3$? I'm just checking, but when we have $2$ vectors
$$
V_1=\begin{pmatrix}a\\b\\c\end{pmatrix}\text{ and }V_2=\begin{pmatrix}e\\f\\g\end{pmatrix}
$$
We could theoretically span $\mathbb R^3$ real space with just these two vectors right?
| While skyking gave a very elegant explanation for $\mathbb{R}^3$, there is a general fact that $n$ $n+1$ tuples can't span $\mathbb{R}^{n+1}$. One possible proof is this:
1) For $n=1$ this is obviously true.
2) Suppose any element of $\mathbb{R}^{n+1}$ can be represented as a linear combination of your tuples, then you... | {
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Solutions to $100a+10b+c=11a^2+11b^2+11c^2$ and $100a+10b+c=11a^2+11b^2+11c^2+ k$ The problem states that
Find all three digit natural numbers such that when the number is divided by $11$ gives the quotient equal to sum of squares of their digits
Since there is no information about whether remainder is $0$ or not , I... | This might not be the best way but here is how I would do it.
Assume the remainder to be $\lambda$. Remember that $\lambda$ can take any value from 0 to 10.
Note that $100a + 10b +c = 11a^2 + 11b^2 + 11c^2 + \lambda$ can be rewritten as
$$(11a - 50)^2 + (11b - 5)^2 + (11c - \frac{1}{2})^2 = 2525 + \frac{1}{4} - \lambda... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2362528",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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convergence of $\int_1^{\infty}\frac{lnx}{\sqrt[3]x(x+1)}$ I need help checking if the integral:
$$\int_1^{\infty}\frac{\ln x}{\sqrt[3]x(x+1)}$$
converge or diverge.
I tried comparing to bigger integrals using $\ln x \lt x$ or making the denominator smaller but without success.
any suggestions?
| The substitution $x\mapsto z^3$ leads to an absolutely convergent integral:
$$ I = \int_{1}^{+\infty}\frac{\log x}{\sqrt[3]{x}(1+x)}\,dx \stackrel{x\mapsto z^3}{=} \color{blue}{9\int_{1}^{+\infty}\frac{z \log z}{1+z^3}\,dz}\stackrel{z\mapsto t^{-1}}{=}9\int_{0}^{1}\frac{-\log(t)}{1+t^3}\,dt\tag{1}$$
that can be easily ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove the normal approximation of beta distribution How should I prove the normal approximation of beta distribution as follows:
Let $\mathrm B_{r_1, r_2}\sim \mathrm{Beta}(r_1, r_2)$, then prove that
$\sqrt{r_1+r_2} (\mathrm B_{r_1, r_2}- \dfrac{r_1}{r_1+r_2}) \to \mathrm N(0, \gamma(1-\gamma))$
where $r_1, r_2 \to \i... | I finally figured out the answer:
Let $\mathrm B_{r_1, r_2} = \dfrac{V_1}{V_1+V_2}$ where $V_1 \sim \chi^2(2r_1 ) = \mathrm{Gamma}(r_1, 2)$ and $V_2 \sim \chi^2(2r_2 ) = \mathrm{Gamma}(r_2, 2)$ and $V_1, V_2$ are independent.
Now we know that $\dfrac{V_1-2r_1}{\sqrt{4r_1}} \to \mathrm N(0, 1)$ as $r_1\to\infty$ and $\d... | {
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"timestamp": "2023-03-29T00:00:00",
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Question on adjunctions Let $\mathcal{A},\mathcal{B}$ and $\mathcal{C}$ be three categories. Let $F:\mathcal{A}\to \mathcal{B}$ be a functor, and let $G\dashv H$ be an adjunction, where $G:\mathcal{B}\to\mathcal{C}$ and $H:\mathcal{C}\to\mathcal{B}$.
If $G\circ F$ is to have a right adjoint, is it necessary that
i) t... | i) Your assumptions imply that all three categories are either empty or nonempty simultaneously, so that there certainly exists some functor $B\to A.$ But that's not very interesting.
ii) There's no reason $F$ should have a right adjoint. For instance, if $A,C$ are both copies of the category with a single object and ... | {
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Value of this expression If $\alpha$ and $\beta$ are the roots of the equation $$x^2 + x − 3 = 0$$ find the value of the expression $4\beta^2 − \alpha^3$.
I tried using sum of roots and product of roots formulas but could not get the answer.
| We have $$\beta^2=3-\beta\\4\beta^2=12-4\beta$$
Also $$\alpha^2=3-\alpha\\\alpha^3=3\alpha-\alpha^2=3\alpha-3+\alpha=4\alpha-3$$
Thus, $$4\beta^2-\alpha^3=15-4(\alpha+\beta)=15+4=19$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Trace of product of positive matrices Let $A$, $B$ symmetric matrices over $\mathbb{R}$ with the same dimension. If $A$ has only positive eigenvalues and $B$ has only nonnegative eigenvalues, is $\text{trace}(AB)\ge 0?$
If yes, prove it. If no, counterexample it.
| Yes. In fact all eigenvalues of $A B$ are nonnegative. This is because $A^{1/2} B A^{1/2}$ is positive semidefinite, where
$A^{1/2}$ is the positive definite square root of $A$, and
$A B = A^{1/2} (A^{1/2} B)$ and $(A^{1/2} B) A^{1/2}$ have the same
eigenvalues (the products of two matrices in either order always ha... | {
"language": "en",
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Is there any geometric demonstration that proves $m= \frac{y_2 -y_1}{x_2 - x_1}$ (the slope formula)? Is there any geometric demonstration that proves $$m= \frac{y_2 -y_1}{x_2 - x_1}?$$
I know one algebraic demonstration. It's:
1) If you have a function $y=mx+b$ and two points $P_1(x_1,y_1)$ and $P_2(x_2,y_2)$, you ca... | Why did mathematicians decide that the slope was important?
It's important for a lot of reasons, but they mostly come down to differentiation. The wiki page has some beautiful geometrical demonstrations.
| {
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Surface measure $\mathrm{d}S$ of $M$ corresponding to parametrisation I'm reading these notes, and there's a part I don't understand on page 72 (76 in the PDF), right after the equation marked with two stars.
Note that $\sqrt{1+|Dg(y')|^2}dy'$ is the surface measure $\mathrm{d}S$ of $M$ corresponding to the parametris... | This hypersurface $S$ is parametrized by $$y'\mapsto (y',g(y'))\in{\mathbb R}^n\qquad(y'\in{\mathbb R}^{n-1})\ ,$$
hence is considered as graph of the scalar function $g$ defined on ${\mathbb R}^{n-1}$.
If $S$ were a hyperplane $y_n=a_1y_1+a_2y_2+\ldots a_{n-1}y_{n-1}$ then the surface measure on $S$ would be ${1\over\... | {
"language": "en",
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Expressing variable q in terms of p. Where p and q are contents in a quadratic. Suppose that $p$ and $q$ are constants such that the smallest possible value of $x^2+px+q$ is $0$. Express $q$ in terms of $p$.
I am unsure what it is asking. I feel it is asking something very simple. However I am unsure of what it is. Any... | Hint: Apply completing the square
$$x^2+px+q=\left(x+\frac{p}2 \right)^2-\frac{p^2}{4}+q$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Lemma used in Proof of L'Hôpital's Rule for Indeterminate Types of $\infty/\infty$ This question arises from an unproved assumption made in a proof of L'Hôpital's Rule for Indeterminate Types of $\infty/\infty$ from a Real Analysis textbook I am using. The result is intuitively simple to understand, but I am having tro... | Assuming that $a$ is a lower limit point of $A$ \ $\{a\},$ the statement $\lim_{x\to a+}f(x)=\infty$ is equivalent to $$\lim_{c\to a+}\inf \{f(x): x\in (a,c)\cap A\}=\infty.$$ And the statement you wish to derive from $\lim_{x\to a+}f(x)=\infty$ is equivalent to $$\lim_{c\to a+}\sup \{f(x): x\in (a,c)\cap A\}=\infty.$... | {
"language": "en",
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Lengthy integration problem $$ \int\frac{x^3+3x+2}{(x^2+1)^2(x+1)} \, dx$$
I managed to solve the problem using partial fraction decomposition.
But that approach is pretty long as it creates five variables. Is there any other shorter method to solve this problem(other than partial fractions)?
I also tried trigonometri... | There are much better ways to find the coefficients in partial fractions than solving five equations in five variables.
Writing your function as
$$ \frac{x^3 + 3 x + 2}{(x^2+1)^2 (x+1)} = \frac{Q(x)}{(x^2+1)^2} + \frac{E}{x+1} $$
multiply both sides by $x+1$ and substitute $x=-1$. We get
$$ \frac{-2}{2^2} = 0 + E$$
so... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2363482",
"timestamp": "2023-03-29T00:00:00",
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How can I express $\frac{dy}{dx}$ if I make the change of variable $x-1 = t$? If $y$ is a function of x and I make the change of variable $x-1=t$. Now what would $\frac{dy}{dx}$ equal in terms of $\frac{dy}{dt}$.
| As: $x-1=t$ then $x(t)=t+1$ so $\frac{dx}{dt}=1$
Now: $$y(x(t))'=\frac{dy}{dt}=\frac{dy}{dx}\frac{dx}{dt}=\frac{dy}{dx}\times1=\frac{dy}{dx}$$
Therefore: $$\frac{dy}{dx}=\frac{dy}{dt}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2363601",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Composition of exponential with an isometry I have trouble understanding the following equation.
We are given an isometry $$F: M \rightarrow M$$ on a Riemannian manifold $M$.
Why does the following hold true?
$$
F \circ exp_p = exp_{F(p)} \circ dF_p
$$
Does someone have any ideas? Thank you.
| Let $M$ be a Riemannian manifold and for $p \in M$, $D(p)$ be the open subset of the tangent space $T_{p}M$ such that :
$$ D(p) = \lbrace v \in T_{p}M, \; \gamma_{v}(1) \; \text{exists} \rbrace $$
where $\gamma_{v}$ is the unique maximal geodesic of $M$ with initial conditions: $\gamma_{v}(0) = p$ and $\dot{\gamma_{v}}... | {
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"timestamp": "2023-03-29T00:00:00",
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Find the limit $ \lim_{x \to (\frac{1}{2})^{-}} \frac{\ln(1 - 2x)}{\tan \pi x} $ Question
$$
\lim_{x \to (\frac{1}{2})^{-}} \frac{\ln(1 - 2x)}{\tan \pi x}
$$
I'm not sure how to go about this limit, I've tried to apply L'Hopital's rule
(as shown).
It seems that the form is going to be forever indeterminate? Unless I... | You do not need de l'Hospital rule for the evaluation of such limit:
$$\lim_{x\to(1/2)^{-}}\frac{\log(1-2x)}{\tan(\pi x)}=\lim_{z\to 0^+}\frac{\log(2z)}{\cot(\pi z)}=\lim_{z\to 0^+}\frac{z\log(2z)}{z\cot(\pi z)} =\frac{0}{\frac{1}{\pi}}=0.$$
It is enough to exploit a substitution $x\mapsto \frac{1}{2}-z$ and the well-k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2363809",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find number of real solutions of $3x^5+2x^4+x^3+2x^2-x-2=0$
Find a number of real roots of $$f(x)=3x^5+2x^4+x^3+2x^2-x-2$$
I tried using differentiation:
$$f'(x)=15x^4+8x^3+3x^2+4x-1=0$$ and I found number of real roots of $f'(x)=0$ by drawing graphs of $g(x)=-15x^4$ and $h(x)=8x^3+3x^2+4x-1$ and obviously from graph... | $f'(x)=15x^4+8x^3+3x^2+4x-1>0$ for $x>\frac{1}{2}$ and $f\left(\frac{1}{2}\right)<0$.
Hence, since $\lim\limits_{x\rightarrow+\infty}f(x)=+\infty$, we see that there is unique root for $x>\frac{1}{2}$.
Now, prove that $f(x)<0$ for all $x\leq\frac{1}{2}$.
For example, for $0\leq x\leq\frac{1}{2}$ we have
$$3x^5+2x^4+x^3... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Expected value of the maximum of binomial random variables Let $X = \{X_1, ..., X_k\}$ be a set of $k$ iid variables drawn from a binomial distribution: $X_i \sim B(n, p)$. How to calculate the upper bound of the expected value of $max(X_i)$?
Several related question (such as: Bounds for the maximum of binomial random ... | $\newcommand\P{\mathbb{P}}\newcommand\E{\mathbb{E}}\newcommand\ol{\overline}$Write $\ol X_{\max} = \max\{\ol X_i\} = \max\{\tfrac{1}{n} X_i\}$. We can compute
$$
\P(\ol X_{\max} > p + t) = \P(\ol X_i > p + t \text{ for some } i=1,\ldots,k) \leq k\P(\ol X_1 > p + t) \leq k\,e^{-2nt^2}
$$
by the union bound and Hoeffding... | {
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"timestamp": "2023-03-29T00:00:00",
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How to remember which function is concave and which one is convex? I always struggle to remember when a function is convex and concave:
Do you have a particular trick to help you remember this?
My trick is based on the Spanish phrase "No cabe", pronounced nô ˈka.βe, which sound just like "concave". "No cabe" means it... | conVex - V looks like the convex function :)
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
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Put $f(x) =\int_0^{t^2} \sin(x^4) dx$ and give a formula for $f'(t)$ $$f\left(t\right) = \int_{0}^{t^2} \sin\left(x^4\right)\ {\rm d}x$$
My answer:
The chain rule is
$$g'(t)f'(g(t))$$
So if
$$g(t) = t^2$$ and
$$f'(x) = \sin(x^4)$$
Then $f'(t)$ must be
$$f'(t) = 2t \sin(t^4)$$
Which was wrong. What is the correct way to... | Hint: Another method is to try to find a change of variables so that you can write
$$f(t) = \int_c^t g(x)\; dx$$
and then you have $f'(t) = g(t)$. The important thing is that you end up with "$t$" as the upper limit.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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integer solutions for $d^2 + 4kT=S^2$ for a given $T$ Consider the following equation:
$d^2 + 4kT=S^2$
We are interested in nonzero integers $d,k,T,S$ that satisfy the above equation. Specifically, we are interested in some values of $T$, for which there exist multiple solutions for $d,k,S$. For example does there ex... | The discriminant of the equation
$$kX^2-dX-T=0$$
is $d^2+4kT$.
Since $k,d,T$ are integers, this discriminant is the square of an integer if and only if the equation has rational solutions, let them be $u$ and $v$.
We know that $uv=-T/k$ and $u+v=d/k$. Then, given $T$, you can just give values to $k$ and choose a pair ... | {
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Graded vector space conditions Wikipedia define the graded ( ring, module, vector space, ...) as here
I noted that in the rings and modules it required the condition of inclusion but in the vector spaces it did not. just the direct sum condition ..why ??
Any hint ?
| Any inclusion V -> W of vector spaces creates a direct sum decomposition $W \cong V \oplus V^{\perp}$.
A graded vector space $W = V_1 \oplus V_2$ would have graded part ($\le 2$) equal to W not $V_2$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Prove $\{f(x,y) \in \mathbb{C}[[x,y]] \mid f(\zeta_n x,\zeta_n^{-1}y) = f(x,y)\}$ is not isomorphic to the formal series ring
Suppose that $\zeta_n$ is a primitive n-th root of $1$.
Let $R$ be $\{f(x,y) \in \mathbb{C}[[x,y]] \mid f(\zeta_n x,\zeta_n^{-1}y) = f(x,y)\}$.
Try to prove that $R$ is not isomorphic to $\math... | Each of the three rings $\mathbb{C}[[x,y]]$, $R$, and $S$ is local, with a unique maximal ideal consisting of power series with constant term $0$. For each ring, I will compute the vector space dimension of the quotient of the ring by the square of the maximal ideal. The three dimensions will be distinct, which proves ... | {
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"source": "stackexchange",
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If the adjoint of an operator is bounded is the operator as well? Let $X$ and $Y$ be normed vector spaces. Let $T: X \to Y$ be a linear operator. Let $T^* : Y^* \to X^*$ be the adjoint of $T$ defined by $T^*(f) =f \cdot T$. Show that if $T^*$ is bounded then $T$ is bounded.
I know that the converse of this statement is... | For any $x\in X$ and $\phi\in Y^*$, we have
$$|T^*\phi(x)|=|\phi(Tx)|\leq\|\phi\|\|T\|\|x\|.$$
It follows that
$$\|T^*\phi\|\leq\|\phi\|\|T\|,$$
thus by definition of the operator norm we have
$$\|T^*\|\leq \|T\|.$$
Similarly we have
$$\|T^{**}\|\leq\|T^*\|,$$
where $T^{**}:X^{**}\to Y^{**}$ is the adjoint operator of... | {
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"url": "https://math.stackexchange.com/questions/2364655",
"timestamp": "2023-03-29T00:00:00",
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$\epsilon$ $\delta$ again! Is the following proof correct?
Theorem. If $\lim_{x\to c}f(x)=L$ and $L>0$ then there is some $\delta>0$ such that for all $x$ such that $0<|x-c|<\delta$, $f(x)>0$.
Proof. Let $\epsilon=\frac{L}{2}$ and since $\lim_{x\to c}f(x)=L$ it follows that for some $\delta>0$ it is the case that
$$\f... | It is correct, but too long, since you write$$\forall x(0<|x-c|<\delta\implies |f(x)-L|<\epsilon)$$and, right after that, “It is now apparent that given any arbitrary $x$ such that $0<|x-c|<\delta$ it follows that $|f(x)-L|<\epsilon$.” Therefore, you wrote the same thing twice.
Besides, after making $\epsilon=\frac L2$... | {
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Convex Ideal quadrilaterals are not all the same In hyperbolic geometry, ideal triangles are all congruent to each other.
Convex Ideal quadrilaterals ( a quadrilateral where all 4 points are ideal points) all have the same area $(2\pi)$
But I think for the rest they are not all congruent for example the angle the dia... | I'd suggest thinking about this in the context of the Poincaré disk model. An isometry of the hyperbolic plane is uniquely determined by mapping three ideal points to their images. That's because that isometry is a Möbius transformation, which is uniquely determined by three points and their images. (Actually if the or... | {
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Improper integral $\int_{0}^{\infty} \frac{x}{e^{x}-1} \ dx$
$$\int_{0}^{\infty} \frac{x}{e^{x}-1}\ dx$$
Is there a way to evaluate this integral without using the zeta function?
| $$\text{ if } L(f) = F(s) \implies \int_0^{\infty} \frac f{e^t-1}dt = \sum_{s\ge1} F(s)$$
$$L(t) =\frac 1{s^2}\implies\int_0^{\infty} \frac t{e^t-1}dt = \sum_{s\ge1}\frac 1{s^2} = \zeta(2) $$
| {
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"answer_id": 2
} |
Showing a function can be continuously extended to the unit circle Fix $0 < a < \infty.$ Define for $|z| < 1$ the function $$ f(z) = \sum_{n=0}^\infty 2^{-na} z^{2^n}$$
and show that $f$ extends continuously to the unit circle but can not be analytically continued past the unit circle.
The second part of the problem i... | Finding explicit bounds for $\lvert f(z) - f(z_0)\rvert$ is not nice. It's nicer if one uses the absolute and uniform convergence of the series on the closed unit disk, but then it's still nicer to use the theorem that the uniform limit of continuous functions is continuous.
To see that the series converges absolutely ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2365416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Is the boundary of a manifold always compact? Coming from a background of undergrad physics training, I know that "the boundary of a boundary is zero".
Does this mean that the boundary of a manifold is always compact(or maybe just closed)?
Keep in mind that some of the terminology that mathematicians use might not be... | No, consider $\mathbb R^n $ itself embedded in the " Standard Way" in $\mathbb R^{n+1}$ as $(x_1,x_2,..,x_n,0): x_i \in \mathbb R$. It is itself a manifold with boundary -- and every point is a boundary point -- but it is not compact, since it is not bounded -- you can even let a single coordinate grow as much as you w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2365488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Showing the following sequence of functions are uniformly convergent
Let $f_n:[0,1]\to \mathbb{R}$ be a sequence of continuously differentiable functions such that
$$f_n(0)=0,\:\: |f_n'(x)|\leq 1, \text{for all }n\geq 1, x\in (0,1).$$
Suppose further that $f_n(.)$ is convergent to some function $f(.)$. Show that ... | Let $\varepsilon>0$ be given, and set $\|\, f_k'\| = \sup \left[|\,f_k'(x)|: x \in (0,1) \right]$ $(k=1,2,\ldots)$. Since the collection of open balls $\mathcal{B}: = \{B(\, x, \frac{\varepsilon}{3}) : x \in [0,1] \}$ is a cover for $[0,1]$, we may find a finite subcover, say $B(\,x_1, \frac{\varepsilon}{3}), \, \ldo... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Let S be an infinite set and A a finite subset. Prove that $|S| = |S -A|$. I don't know if my solution is correct. This is what I have so far:
Let $S = \{ s_1, s_2,.....\}$ and $S-A = \{a_1, a_2,.....\}$ where all the elements are arranged in an fixed order.
let $f(x): S \to A , f(s_i) = a_i$
if I prove $f$ is bijectiv... | Let $\mathfrak{a}=|A|$ and $\mathfrak{s}=|S|$, and let $\alpha$ denote an ordinal. We then have that $|S\sim A|=|\{\alpha:\mathfrak{a}<\alpha<\mathfrak{s}\}|$, and from here you should be able to construct a bijective function as you suggest.
As a hint for constructing such a bijection, observe that $$\mathbb{F}=\lan... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the number of 4 digit positive integers if the product of their digits is divisible by 3. Find the number of 4 digit positive integers if the product of their digits is divisible by 3.
Let $abcd$ be the required number.For the product of digits of this number is to be divisible by 3,atleast one digit has to be 3,... | How many have a digit product that is not divisble by $3$? You can then only use $6$ digits for all digits as also $0$ is forbidden for all digits. In total, as you said, there are $9000 =9\times 10^3$ $4$-digit numbers..
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2365813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Evaluate $\lim_{ x\to \infty} \left( \tan^{-1}\left(\frac{1+x}{4+x}\right)-\frac{\pi}{4}\right)x$
Evaluate
$$\lim_{ x\to \infty} \left( \tan^{-1}\left(\frac{1+x}{4+x}\right)-\frac{\pi}{4}\right)x$$
I assumed $x=\frac{1}{y}$ we get
$$L=\lim_{y \to 0}\frac{\left( \tan^{-1}\left(\frac{1+y}{1+4y}\right)-\frac{\pi}{4... | Well, we can start as you, by setting $y=\frac{1}{x}$. Now, our limits transforms to:
$$L=\lim_{y\to0}\frac{\tan^{-1}\left(\frac{1+y}{1+4y}\right)-\frac{\pi}{4}}{y}$$
Now, let $f:\mathbb{R}\to\mathbb{R}$ with
$$f(x)=\tan\left(\frac{1+x}{1+4x}\right)$$
Note that $f(0)=\tan^{-1}(1)=\frac{\pi}{4}$. So, we have:
$$L=\lim_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2365914",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 3
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Finding the maximum value from a derivative function I am having problems understanding how to find the maximum value from a rate of change (derivative) function. The rate of change of Volume with respect to time is $\frac{dv}{dt}=1000- 30t^2 +2t^3$, $0 \le t \le 15$
How do I find the maximum rate of change? the answe... | Relative extrema occur at endpoints or critical points. Critical points occur where a function's derivative is $0$ or undefined.
The maximum of $\frac{dv}{dt}$ is where $\frac{d^2v}{dt^2}=0$ or is undefined.
\begin{align*}
\frac{d^2v}{dt^2} &= -60t + 6t^2 =0 \\
0 &= 6t(t - 10) \\
t&=0, \ t=10
\end{align*}
So $\frac{dv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2366010",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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When is a measure the pushforward of another measure? Let $(X,\Sigma)$ be a measurable space. Let $\mu$ and $\nu$ be two measures on thereon. Are there any reasonable restrictions on these measures to ensure the existence of a (measurable) function, $f: X \to X$ such that
$$ \nu = f_*\mu$$
where $f_*\mu$ is the push f... | If $X$ is any separable completely metrizable space, such as $\mathbb{R}$, a sufficient condition is that $\mu$ is atomless and $\mu(X)=\nu(X)<\infty$, see here for the argument. It should be quite clear that handling atoms of $\mu$ is not easy, the will lead to atoms in $\nu$.
Alternatively, it is sufficient that both... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2366159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Linear Independence I've come across a question in Linear Algebra that I can't quite figure out. I've tried a multitude of things that either don't work or aren't sufficient enough to convince me I understand linear independence well enough.
I know a set of vectors, S, in vector space V are linearly independent if the... | You just need to use the definition. For the first case you can write:
$$\lambda_1(u+v+w)+\lambda_2(v-2w)+\lambda_3(2u+3w)=0$$
now rewrite like:
$$(\lambda_1+2\lambda_3)u+(\lambda_1+\lambda_2)v+(\lambda_1-2\lambda_2+3\lambda_3)w=0$$
Now use that $u,v,w$ are independent, solve the system and find $\lambda_1,\lambda_2,\l... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Limit of $p_{k+1}(z)/p_k(z)$ with $p_k(z)=\sum\limits_{i=0}^{\lceil zk \rceil}{k\choose i} p^i(1-p)^{k-i}$ and $z
How to find the limit of $\dfrac{p_{k+1}(z)}{p_k(z)}$ as $k$ tends to infinity, where, for every $k$, $$p_k(z)=\sum_{i=0}^{\lceil zk \rceil}{k\choose i} p^i(1-p)^{k-i}$$ for some $z$ and $p$ in $(0,1)$ such... | If $X\sim\operatorname{Bin}(n,p)$, then $\frac{X-np}{\sqrt{np(1-p)}} \xrightarrow{d} N(0,1)$ and one can use it to estimate numerator and denominator. Basically, both are $\Phi(\frac{\sqrt{n}(z-p)}{\sqrt{p(1-p)}})(1+O(\frac1n))$, so the limit is 1.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What does $L(n,\chi_4)$ mean? I was reading some articles related to Euler sums and the Riemann zeta function, when I came across this definition:
$$
L(n,\chi_4) = \sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)^n}
$$
What is this function called and how is it related to the zeta function?
| That author is writing $\chi_4$ for a certain "character", defined by
$$
\chi_4(n) = \begin{cases}
1, &n \equiv 1\pmod 4,\\
-1,&n\equiv 3\pmod 4,\\
0, &\text{otherwise}
\end{cases}
$$
and then
$$
L(s,\chi_4) = \sum_{n=1}^\infty \frac{\chi_4(n)}{n^s} =
\sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)^s}
$$
is the corresponding... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2366504",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How can I solve this limit without L'Hopital rule? I have found this interesting limit and I'm trying to solve it without use L'Hopital's Rule.
$$\lim\limits_{x\rightarrow 0}\frac{\sinh^{-1}(\sinh(x))-\sinh^{-1}(\sin(x))}{\sinh(x)-\sin(x)}$$
I solved it with L'Hopital's rule and I found that the solution is $1$. But if... | Using the Mean Value Theorem, and the fact that $\frac{\mathrm{d}}{\mathrm{d}x}\sinh^{-1}(x)=\frac1{\sqrt{1+x^2}}$, we get that
$$
\frac{\sinh^{-1}(\sinh(x))-\sinh^{-1}(\sin(x))}{\sinh(x)-\sin(x)}=\frac1{\sqrt{1+\xi^2}}\tag{1}
$$
for some $\xi$ between $\sin(x)$ and $\sinh(x)$.
Therefore, since both $\sinh(x)$ and $\si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2366608",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Why is convolution with Fejer kernel enough to prove this result? Exercise 2.9 in Katznelson's Introduction to Harmonic Analysis reads as follows.
Show that for $f \in L^{1}(\mathbb{T})$ the norm of the operator $ F: g \mapsto f * g$ on $L^{1}(\mathbb{T})$ is $||f||_{L^{1}}$.
I can see easily enough how $||f * g)||... | Note that $\|K_n\|_{L^1}=1$ (using Katznelson's normalisation). Let the
operator norm of $g\mapsto f*g$ be $A$. We know $A\le \|f\|_{L^1}$.
Then $\|f\ast K_n\|_{L_1}\le A\|K_n\|_{L^1}=A$. But $\|f\ast K_n\|_{L^1}\to
\|f\|_{L^1}$, so in the limit, $\|f\|_{L^1}\le A$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$\textbf Z[\sqrt{pq}]$ is not a UFD if $\left( \frac{q}p \right) = -1$ and $p \equiv 1 \pmod 4$.
Let $p$ and $q$ be primes such that $p \equiv 1 \pmod 4$ and $\left( \frac q p \right) = -1$. Show that $\textbf Z[\sqrt {pq}]$ is not a UFD.
I tried some examples like $p=5$ and $q = 2$. But I have no clue about the gene... | If $$\left(\frac{q}{p}\right) = -1,$$ that means that $q$ is not a quadratic residue modulo $p$. That much is obvious, right? It also means that $q$ is not a quadratic residue modulo $pq$ either. Thus, in your example, since 2 is not a quadratic residue modulo 5, it can't be a residue modulo 10 either, and is therefore... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
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What is the formal adjoint? Let $L$ a differential operator and consider the equation $Lu=f$ for $f\in L^2$. Then $u$ is a weak solution if $$\left<u,L^*\varphi\right>=\left<f,\varphi\right>,$$
for all $\varphi\in \mathcal C_0^\infty $ where $L^*$ is the formal adjoint.
What is the "formal adjoint" ?
I recall that $\le... | I assume we work with real-valued functions. If $L = \sum_{\alpha} k_{\alpha} D^{\alpha}$ (using multi-index notation), where $k_{\alpha}$ are constants, then $L^{*}$ is given by
$$
L^{*} = \sum_{\alpha} k_{\alpha} (-1)^{|\alpha|} D^{\alpha}.
$$
To see why it makes sense, one may check that for $\phi, \psi \in C_0^{\in... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Let $X$ be a connected scheme. Then $X$ irreducible iff $\forall x, \text{ Spec}(\mathcal{O}_{X,x})$ is I could use some help for the second part of exercise I-34 from Eisenbud and Harris' Geometry of Schemes:
Let $X$ be a connected scheme. Show that $X$ is irreducible if and only if
for all $x \in X$, the stalk lo... | This is false. Indeed, there exists a ring $A$ which has no nontrivial idempotent, which is not a domain, and such that the localization of $A$ at any prime ideal is a domain. See http://stacks.math.columbia.edu/tag/0568 for details of the construction. Note that such a ring automatically is reduced, since all of it... | {
"language": "en",
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Closed form for $\sum_{0\le x\lt\infty}n^{2^{-x}}-1$ Is there a closed form for the expression below?
$$f(n)=\sum_{0\le x\lt\infty}n^{2^{-x}}-1=(n-1)+(\sqrt{n}-1)+(\sqrt{\sqrt{n}}-1)+\cdots$$
Approximations are good as well. This appeared to me while analyzing an algorithm.
| Let $g(x) := f(e^{x/2})$ so that $g(x) = g(x/2) + e^{x/2}-1$. Expanding $g(x)$ in power series we have
$$ g(x) = \sum_{k>0} \frac{x^k}{k!(2^k-1)} =
x + \frac{x^2}{6} + \frac{x^3}{42} + \frac{x^4}{360} + \frac{x^5}{3720} +\cdots $$ which converges everywhere but is unlikely to have closed form.
Approximations of $f(x)$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How do you calculate the expected value of geometric distribution without diffrentiation? Is there any way I can calculate the expected value of geometric distribution without diffrentiation? All other ways I saw here have diffrentiation in them.
Thanks in advance!
| The problem can be viewed in a different perspective to understand more intuitively. Let's see the following definition.
"A person tosses a coin, if head comes he stops, else he passes the coin to the next person. For the next person, he follows the same process: If head comes he stops, else passes the coin to next per... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2368304",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Is there any other function $f(x)$ such that $f$ is Discontinous at $x=a$ but $fof$ is continous Is there any other function $f(x)$ such that $f$ is Discontinous at $x=a$ but $fof$ is continous.
I know the well know example for this as Dirchlet's function. is there any other?
| The function
$$
f(x) = \begin{cases}a - 1 & x\ge a \\ a - 2 & x < a \end{cases}
$$
is discontinuous at $x = a$, but $f(f(x)) = a - 2$ is a continuous (constant) function.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2368533",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is $\bigcup_{n\geq 1}[0,1-1/n] \neq [0,1]$? Sorry but aren't we taking limits $\lim_{m \to \infty} \cup_{n =1 }^{m}[0,1-1/n] = [0,1]$? Why is this supposed to be equal to $[0,1)$?
| This isn't a silly question. It's just that you've discovered a false belief.
Conjecture. Suppose $a : \mathbb{N} \rightarrow \mathbb{R}$ is an order-reversing sequence that's bounded below and $b : \mathbb{N} \rightarrow \mathbb{R}$ is an order-preserving sequence that's bounded above. Then:
$$\bigcup_{i \in \mathb... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Nim game variant Statment is as follow
Given a number of piles in which each pile contains some numbers of stones/coins. In each turn, a player can choose only one pile and remove any number of stones (at least one) from that pile. The player who cannot move is considered to lose the game (i.e., one who take the last ... | We can use the theory of nim-values. Each pile has a nim-value
and the value of the game is the nim-sum (XOR of the binary representations) of the nim-values of the piles. In this variant
the nim-value of a pile is periodic with period $H+1$. The nim-value
of a pile with $k$ stones, $0\le k\le H$ is $H$, and adding $H+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2368702",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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An arithmetic sequence “The sum of the first $13$ terms of an arithmetic sequence is $-234$ and the sum of the first $41$ terms is $-780$. Find the $27^\text{th}$ term of the sequence.”
The $S_n=\frac n2(2a+(n-1)d)$ formula ($S_n$ represents the sum of the first to $n^\text{th}$ terms) only gives me fractional values f... | $a_1$ and $d$ are unknown, but we know:
$\frac{13}{2}(a_1 + a_{13}) = -234 \implies a_1 + a_{13} = - \frac{234\times 2}{13} = -36$
and also $\frac{41}{2}(a_1 + a_{41}) = -780 \implies a_1 + a_{41} = -38\frac{2}{41}$
and also $a_{13}= a_1 + 12d$
and $a_{41} = a_1 + 40d$.
So we get two linear equations in two variables, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2368788",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Unable to reach the desired answer in trigonometry. The question is:
If $\sin x + \sin y = \sqrt3 (\cos y - \cos x)$
show that $\sin 3x + \sin 3y= 0 $
This is what I have tried:
*
*Squaring of the first equation (Result: Failure)
*Tried to use the $\sin(3x)$ identity but got stuck in the middle steps because I co... | I was trying to find out how the condition was conceived.
$$\sin3x+\sin3y=0\implies\sin3x=\sin(-3y)$$
$\implies3x=180^\circ n+(-1)^n(-3y)$ where $n$ is any integer
$\iff x=60^\circ n+(-1)^{n+1}y$
If $n$ is even $=2m$(say), $x=120^\circ m-y$
$$\implies x+y\equiv\begin{cases}0 &\mbox{if }3\mid m\\120^\circ& \mbox{if } n ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2368879",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What does it mean to say that a statement is independent of a theory T? Let $\phi$ = $\forall x \forall y (x \cdot y = y \cdot x)$. It is sometimes said that $\phi$ is "independent" of the axioms of group theory. What does 'independent' mean $\textit{exactly}$? Is it:
(1) That we can find models of group theory $\mathf... | A sentence $\phi$ is independent of $T$ if there are $M,N\models T$ such that $M\models\phi$ and $N\models\neg\phi$. [This is your (1).]
It is a theorem (Gödel completeness theorem) that $\phi$ is independent of $T$ if and only if $\phi$ and $\neg\phi$ cannot be derived $T$ in some (here not defined) syntactic calcul... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Area of rectangle knowing diagonal and angle between diagonal and edge I found on the web that the area of a rectangle with the diagonal of length $d$, and inner angle (between the diagonal and edge) $\theta$ is $d^2\cos(\theta)\sin(\theta)$. However, I wasn't able to deduce it myself. I tried applying law of sines o... | If you use the formulas for sine and cosine in right-angled triangles, the formula can be proved rather easily: If the width and the height of the rectangle are resp. $w$ and $h$, then the formulas say $\cos(\theta)=w/d$ and $\sin(\theta)=h/d$. If you isolate $w$ and $h$ in these formulas and substitute in the formula ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2369269",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Different ways to state the motivation of the definition of the product topology Suppose for every $i\in\mathscr I,$ $X_i$ is a topological space.
The product space has as its underlying set the product set $X =\prod \limits_{i\,\in\,\mathscr I} X_i$ and as its open sets product sets of the form $\prod\limits_{i\,\in\,... | For the sake of definiteness, I will refer to the name I have seen most commonly used: the product topology is, as others have mentioned, the Initial Topology https://en.wikipedia.org/wiki/Initial_topology with respect to the projections.
A dual concept is that of the Final Topology, which is the finest topology put ... | {
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upper bound for $L^1$ norm of Dirichlet kernel I showed there exists a constant $c$ such that $\|D_N\|_1 \geq c \log N$ and $c$ is independent to $N$
using the fact that $$\| D_N\|_1= \frac 1 \pi \int_{[0,\pi]} \left|\frac{\sin(2N+1)y}{\sin y}\right|\,dt \geq \frac{1}{\pi}\int_{[0,\pi/2]} \left| \frac{\sin(2N+1)y}{\sin... | $D_{n}(x)$ has the form $$D_{n}(x)=\dfrac{1}{2\pi}\dfrac{\sin(n+\frac{1}{2})x}{\sin\frac{1}{2}(x)},$$ so that $$D_{n}(2x)=\dfrac{1}{2\pi}\dfrac{\sin(2n+1)x}{\sin x}.$$
Since $\sin n\alpha\leq n\sin\alpha$, we know that $\sin(2n+1)x\leq (2n+1)\sin x$ and thus
\begin{align*}
(1)\ \ |D_{n}(2x)|=\dfrac{1}{2\pi}\dfrac{|\si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2369473",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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Find hyperbolic area of hyperbolic triangle Given $u=(0,0)$, $v=(\frac12,0)$, $w=(\frac12,\frac12)$. Find the hyperbolic area of the hyperbolic triangle $[u,v,w]$ in $\mathbb{H}^2$.
My approach:
One of the angles, say $\alpha$, is a right angle, so we have that $\alpha=\frac{\pi}2$, so that $[u,v,w]$ is a right triangl... | The area of a hyperbolic triangle is equal to the angle deficit, i.e. to the difference between the hyperbolic sum of interior angles and the Euclidean sum of $\pi$.
Poincaré half plane
Draw the triangle, and it will look like this:
This triangle has two ideal points. I'm not sure I'd call it isosceles, since two legs... | {
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"timestamp": "2023-03-29T00:00:00",
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3 interior points in a grid based polygon Given a polygon with vertices on a grid with 3 interior grid points and no 3 vertices lying on the same line.
Is it true that all vertices are on the same circle?
EDITED
There is also another counter example with convex points:
| Would this qualify as a counterexample (J - I - H - G - L and then the red sides)? It is a polygon, it has vertices on a grid, 3 interior points, and no 3 vertices lying on the same line.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Confusion regarding the domain of a function Let $f(x)=x^2$ and $g(x)=x$. What is the domain of $\frac{f(x)}{g(x)}$?
Evaluating $\frac{f(x)}{g(x)}$ gives us $x$. Does that mean that its domain is all real numbers?
If we evaluate the function at $x=0$, $\frac{f(0)}{g(0)}$, then $g(0)$ will gives us zero. Does that mean ... | The answer to your question is: It depends. Namely on the domain specified for $f$ and the domain specified for $g$.
Attention: A function is not fully specified as long as the domain of the function is not given.
At first you have to check the domain of the function $f$ and the domain of the function $g$. Although i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2369903",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Solving $\sqrt{8+2x-x^2} > 6-3x$.
So I was solving this inequality.
$$\sqrt{8+2x-x^2} > 6-3x$$
*
*First things first, I obtained that the common domain of definition is actually $[-2,4]$.
*Next we would square and solve the quadratic that follows.
But the "solution" seems to have a part, where they took $6-3x \... | Let's say we have an inequality $\sqrt a>b$. We're often interested in getting rid of the square root, so we want to do something along the lines of 'squaring both sides'. But squaring both sides doesn't necessarily preserve the inequality.
Example:
$$\sqrt5>-3$$
But
$$\implies\sqrt5^2>(-3)^2 \implies 5\gt9$$
is clea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2369972",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Picard group of $\Bbb R[x_1,\dots,x_n]/(x_1^2+\dots+x_n^2-1)$ and $\Bbb C[x_1,\dots,x_n]/(x_1^2+\dots+x_n^2-1)$ What is the Picard group of $\Bbb R[x_1,\dots,x_n]/(x_1^2+\dots+x_n^2-1)$, i.e. the coordinate ring of real sphere $S^{n-1}$, and $\Bbb C[x_1,\dots,x_n]/(x_1^2+\dots+x_n^2-1)$?
As $\Bbb R[x_1,x_2]/(x_1^2+x_2^... | For the reals, $n=2$, Picard group is $\mathbb{Z}/2\mathbb{Z}$ and $n>2$, it is trivial. For complex numbers, Picard group is trivial for $n= 2$, equal to $\mathbb{Z}$ when $n=3$ and trivial for $n>3$.
Let us first look at the case of reals. For $n=2$, easy to check that that $I=(x_1, 1-x_2)$ generates the Picard grou... | {
"language": "en",
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"source": "stackexchange",
"question_score": "3",
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Finding the number of elements of a set
Let $S$ be the set of all integers from $100$ to $999$ which are neither divisible by $3$ nor divisible by $5$. The number of elements in $S$ is
*
*$480$
*$420$
*$360$
*$240$
My answer is coming out as $420$, but in the actual answer-sheet the answer is given as $480$. Why... | Solve[{100 <= n <= 999, ! Element[n/3, Integers], !
Element[n/5, Integers]}, n, Integers] // Length
(* 480 *)
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Compilation of proofs for the summation of natural squares and cubes I want to know different proofs for the following formulas,
$$
\sum_{i=1}^n{i^2} = \frac{(n)(n+1)(2n+1)}{6}
$$
$$
\sum_{i=1}^n{i^3} = \frac{n^2(n+1)^2}{2^2}
$$
Please do not mark this as duplicate, since what I specifically want is to be exposed to a ... | By Lagrangian interpolation.
The expression of the sum of the cubes must be a quartic polynomial, because its first order difference is cubic. Furthermore, it has no constant term because the sum of no numbers is zero.
Hence, the average of the cubes is a cubic polynomial, by the four points
$$(1,1),\left(2,\frac{1+2^... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 8
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Does $3$ divide $2^{2n}-1$? Prove or find a counter example of : $3$ divide $2^{2n}-1$ for all $n\in\mathbb N$.
I compute $2^{2n}-1$ until $n=5$ and it looks to work, but how can I prove this ? I tried by contradiction, but impossible to conclude. Any idea ?
| HINT: $$2^{2n}-1=4^n-1\equiv 1^n-1=1-1\equiv 0\mod 3$$
| {
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"timestamp": "2023-03-29T00:00:00",
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Does $\partial_t u(x,t) - \partial_x u(x,t) = 0$ have a non-trivial solution? Consider the linear PDE $$\partial_t u(x,t) - \partial_x u(x,t) = 0$$ on the domain $$\Omega=\{(x,t)\;\;|\;\;0\leq x \leq 1,\;0\leq t \leq T\}$$
with boundary conditions $$u(0,t)=u(1,t)=c$$ and the initial condition $$u(x,0)=u_0(x).$$
This is... | Given the boundary conditions, there is only the trivial solution, $u\equiv c$ if $T\ge 1$. However, for $T<1$, non-trivial solutions are possible. This can be seen by using the method of characteristics:
Here, the characteristics are easily found, they are the lines parametrized by $s\mapsto (-s, s)$, along these line... | {
"language": "en",
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"source": "stackexchange",
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Why is this inference invalid? So I purchased a book on logic (for beginners) as the subject interests me, and the author presents the following statement as an example of an invalid inference:
"Everyone wanted to win the prize; so the person who won the race wanted to win the prize."
Symbolised as follows, where xP is... | The notation is difficult to understand, the way I read it is:
$$\frac{\forall x\;xP}{(|x\;xR)P }$$
is equivalent to:
$$\forall x P(x) \rightarrow \exists x (R(x)\land P(x))$$
Which implies:
$$\forall x P(x) \rightarrow \exists x R(x)$$
And this is indeed not the case as $R(x)$ could never be true, for any x.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Symbolic Notation for $\theta "=" \arcsin(-.5)$? I'm teaching PreCalculus and the following issue has always bugged me.
Problem: Solve $\sin\theta = -.5$ for $0 \le \theta \le 2\pi$.
Solution:
\begin{align*}
\sin\theta &= -.5\\
\theta &= \arcsin(-.5) = -\frac\pi6
\end{align*}
But to get this into our desired domain, ou... | The way I always explained it to my students was "First, you find a solution, then you find the solution."
Let $\theta_0 = \arcsin(-\frac 12)$.
The range of $\arcsin$ is $-\frac{\pi}{2} \le \theta_0 \le \frac{\pi}{2}$. Since, on the unit circle, $\sin \theta = y=-\frac 12$, we see quickly that
$\theta_0 = -\frac{\pi}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Third hardest question on a hospital IQ test(Fill the blank)
Place the numbers 1 through 9 in the available input boxes so that the
results of performing the mathematical operations in any row and
column (from left to right and top to bottom) always equals 13.
\begin{bmatrix}\square&+&\square&-&\square\\
-& &+& &... | Label spaces A through I.
A+B, B+E,A+G are all at least 14. So A,B,E, and G are all greater than or equal to 5.
D x E must be less than 13 and E is 5 or more so D = 1 or 2.
If D=2 then E=5. If E = 5 then B+5-H=13 so B=9, H=1. So Gx1 -I =13 is impossible.
So D isn't 2. So D=1. So E+F = 13, so F is at least 4. C x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2370819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Rationalising the denominator surds I have been struggling on this question. I don't understand how to change a negative surd fraction to a positive surd fraction.
Question: Rationalise and simply
$$\frac{2}{1+{\sqrt 6}}$$
What I did:
$\frac{2(1-{\sqrt 6})}{(1+{\sqrt 6)(1-{\sqrt 6})}}$
= $\frac{2{-2\sqrt... | Nothing went wrong
If you take your result $\dfrac{2-2\sqrt 6 }{-5}$
Multiply num and den by $-1$ and collect $2$ in the numerator, you get
$$\dfrac{2(\sqrt 6 -1)}{5}$$ which is the result of the book
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2370915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Cross product in higher dimensions Suppose we have a vector $(a,b)$ in $2$-space. Then the vector $(-b,a)$ is orthogonal to the one we started with. Furthermore, the function $$(a,b) \mapsto (-b,a)$$ is linear.
Suppose instead we have two vectors $x$ and $y$ in $3$-space. Then the cross product gives us a new vector $x... | Yes. It is just like in dimension $3$: if your vectors are $(t_1,t_2,t_3,t_4)$, $(u_1,u_2,u_3,u_4)$, and $(v_1,v_2,v_3,v_4)$, compute the formal determinant:$$\begin{vmatrix}t_1&t_2&t_3&t_4\\u_1&u_2&u_3&u_4\\v_1&v_2&v_3&v_4\\e_1&e_2&e_3&e_4\end{vmatrix}.$$ You then see $(e_1,e_2,e_3,e_4)$ as the canonical basis of $\ma... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "37",
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Prove $ 2<(1+\frac{1}{n})^{n}$ How to prove that $ 2<(1+\frac{1}{n})^{n}$ for every integer $n>1$. I was thinking by induction, it works for $n=2$ but then I couldn't move forward.
| Approach using AM-GM Inequality
First we show that
$$f(n)=\left(1+\frac{1}{n}\right)^n$$
is monotone increasing. To see this, apply the AM-GM inequality to the following $n+1$ terms
$$\left\{1,1+\frac{1}{n},1+\frac{1}{n},\cdots n\mbox{ times}\right\}$$
You get
$$\frac{1+n+1}{n+1} \geq \left(1+\frac{1}{n}\right)^{\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2371118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 5
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Limit for entropy of prime powers defined by multiplicative arithmetic function This question is related to my other question ( Entropy of a natural number ).
Let $f \ge 0$ be a multiplicative arithmetic function and $F(n) = \sum_{d|n}f(d)$.
Define the entropy of $n$ with respect to $f$ to be
$H_f(n) = -\sum_{d|n} \f... | If $f(n)$ is multiplicative and non-zero then let $F(n) = \sum_{d | n} f(d)$ and
$$ h(n) = \frac{\sum_{d | n} f(d) \log f(d)}{F(n)}$$
If $gcd(n,m)=1$ then
$h(n)+h(m)$ $ = \frac{F(m)\sum_{d | n} f(d) \log f(d)+F(n)\sum_{d | m} f(d) \log f(d)}{F(n)F(m)}$ $=\frac{\sum_{d ' | m,d | n} f(dd') \log f(dd')}{F(nm)}=h(nm)$
Th... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Explanation of this theorem on combinations There is this theorem in my book in the chapter of permutations and combinations which states: The total number of combinations of n different things taken any number of them at a time is $2^n$.
Its proof is given as: Each thing may be disposed of in two ways-it may or may no... | Suppose you have some set $S=\{s_1,...,s_n\}$.
You're essentially trying to see how many subsets of $S$ there are (ie. the cardinality of the powerset of $S$).
For each subset $S_i$, assign it the ordered tuple $((x_i)_1,...,(x_i)_n)$ where $(x_i)_k=\begin{cases}0 & s_k\not\in S_i\\1 & s_k\in S_i \end{cases}$
Clearly, ... | {
"language": "en",
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Proof for corresponding eigenvalue If x is an eigenvector of a matrix A, then show that its corresponding eigenvalue is given by $\lambda=\dfrac{Ax\cdot x}{x\cdot x}$
I tried starting from $(A-\lambda{I})x=0$.
$Ax-\lambda{Ix}=0$
$\lambda{Ix}=Ax$
$\lambda=\dfrac{Ax}{Ix}$. This now is a bit confusing. Any help?
| Your intuition starts off right, it is a good idea to begin with the definition of an eigenvalue, i.e. $A x = \lambda x$. Now the problem is that on both sides of the equation you have vectors, so you cannot divide by $x$. The idea is to "transform" the equation into an equation that only involves scalar quantities. A ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2371475",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Minimum radius of curvature of a sinusoidal curve Intuitively, the minimum radius of curvature is obtained at the point of highest amplitude. How do I prove it mathematically?
I proceeded by assuming the curve, $y=A\sin\omega x$. We know that radius of curvature is given by, $$\rho= \frac{[1+(\frac{dy}{dx})^2]^{1.5}}{\... | In full form:
${d \rho \over d x} = \left\{-\left(\cos ^2(x)+1\right)^{1.5} \csc (x)\right\}$
Set it to zero and find (for $x \in \mathbb{R}$) that $x = 1.5708 = \pi/2$, and obvious integral increments.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2371557",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Exercises in comparison geometry After taking an introductory graduate differential geometry course last year and doing a bit of reading about the Ricci flow, I was considering reading Cheeger and Ebin's book on comparison geometry to get some exposure to classical results in Riemannian geometry.
However, the book does... | Petersen's Riemannian Geometry has a few chapters on comparison geometry with quite a few exercises, so it might be worth a look.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove that there exist constants $u,v$ such that $uA+vB$ is positive definite. $A, B$ are $n$ by $n$ symmetric real matrices where $x^TAx=x^TBx=0$ implies $x=0$.
Prove that there are real numbers $u, v$ making $uA+vB$ postive-definite.
I am not sure whether it is true or not. I tried to disprove it but found it hard to... | It isn't true. Counterexample:
$$
A=\pmatrix{1&0\\ 0&-1},\ B=\pmatrix{0&1\\ 1&0}.
$$
Clearly, $x^TAx=0$ if and only if $x=(t,\pm t)^T$, and $x^TBx=0$ if and only if $x=(t,0)^T$ or $(0,t)^T$. So, the only solution to $x^TAx=x^TBx=0$ is the zero vector. However, $uA+vB$ is never positive definite because it has a zero tr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to find the coordinates of points on a line perpendicular to a given plane I am given a plane equation $Ax+By+Cz+D=0$ and coordinates $(x,y,z)$ of a point $P$ lying on a plane. I need to determine coordinates $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ of two points $P_1$ and $P_2$ which are located on a line that is passi... | I'll give you an example, hoping it will be better understood
Let the plane be $\pi:\;x+2 y+3 z+4=0$ and the point on it $P(1;\;2;-3)$
The line $r$ passing through $P$ and perpendicular to $\pi$ has parametric equation $r=(1 + t,\; 2 + 2 t,\; -3 - 3 t)$
To find a pair of points $P_1$ and $P_2$ having distance $d=\sqrt{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2372013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Can we Clamp and Clip numbers mathematically? While programming I came up with a the following equation / workflow
clampValue = (value - minValueofRange) / (maxValueofRange - minValueofRange);
clippedValue = min(1, max(0, clampValue));
finalValue = clippedValue * scale;
Now the clipping ultimately causes an if and els... | Using a piecewise function as in T. Linnell's answer or simply writing something like $\min(1,\max(0,c))$ would be clear and absolutely fine in mathematics.
If you just happen to be curious about how to represent $\min$ and $\max$ in a different way, $\max(a,b)=\dfrac{|a-b|+a+b}{2}$ and $\min(a,b)=\dfrac{a+b-|a-b|}{2}$... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Filling a cone part way, find formula for height expressed in V I have an ice cream cone, the pointy side downwards. It's 6 cm tall and the diameter of the opening is 3 centimeter. I fill it to some height $h$ . Now how do I find a formula for its height as a function of volume.
I understand the volume of a cone is cal... | Draw a right angle triangle representing the half of the vertical cross section of the cone. Draw a smaller triangle inside the previous triangle, this represents the partially filled ice cream. By using ratios of $\text{opening radius}:\text{cone height}=\text{radius if ice cream surface}:\text{height of the ice cream... | {
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"timestamp": "2023-03-29T00:00:00",
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Expression for the highest power of 2 dividing $3^a\left(2b-1\right)-1$
Question: I am wondering if an expression exists for the highest power of 2 that divides $3^a\left(2b-1\right)-1$, in terms of $a$ and $b$, or perhaps a more general expression for the highest power of 2 dividing some even $n$?
EDIT: This is equi... | This is only a reply to Daniel's comment but too long for the box
Legend: In the following I mean
*
*$ \{expression,p \} $ the exponent to which primefactor $p$ occurs in $expression$
*$[ m : a ]$ equals $1$ if $a$ divides $m$ otherwise it equals $0$
By analysis of cyclicitiness due to "little Ferm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2372512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Counting the Number of Reps From a Rep Height I'm trying to calculate the number of reps from a rep height.
Example #1
Input: n = 2
Output: Result = 6
Because 1+2+2+1 = 6
--
Example #2
Input: n = 3
Output: Result = 12
Because 1+2+3+3+2+1 = 12
What formula could I use to get from input to output?
| My way before is unnecessarily long. When you add $(1 + 2 + \dots + n) + (n + (n-1) + \dots + 1)$, line up the pairs $(n, 1)$, $(2, n-1)$, $(3, n-2)$, and you'll notice that they all sum to $n+1$. Since you have $n$ pairs, the sum is $n(n+1)$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Describing all holomorphic functions such that $f(n)=n$ for $n \in \mathbb{N}$ This question is inspired by a somewhat simpler one.
The question is: how can we classify all holomorphic functions $f:\mathbb{C}\rightarrow\mathbb{C}$ satisfying the property $\forall n \in \mathbb{N} \quad f(n)=n $?
If we have $g:\mathbb{C... | Let $f$ any such function. Then $g(z)= f(z)/(ze^{2\pi iz})$ is entire and equal to $1$ at the natural numbers.
Therefore $g(z)-1$ is zero at the naturals. Let $\Pi(z)$ be a Weierstrass product giving you an entire function vanishing exactly at the natural numbers.
Therefore all the $g$ are of the form $\Pi(z)h(z)$ for... | {
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"url": "https://math.stackexchange.com/questions/2372771",
"timestamp": "2023-03-29T00:00:00",
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Solving linear equation modulo two pi I have an equation I'd like to solve given as $$s \cdot \alpha \equiv p (\mathrm{mod} 2 \pi) $$ The numbers $s$ and $p$ are known, while $\alpha$ is to be solved for, and additionally is between $0$ and $2 \pi$. I have found a solution using other information, so I know solutions ... | $$\begin{align} s\alpha\, &\equiv\, p\!\!\pmod{\!2\pi \Bbb Z}\\[.3em]
\iff\ s\alpha\, &=\, p\ +\ 2\pi n,\quad {\rm for\ some}\ \ n\in\Bbb Z\\[.3em]
\iff\ \ \ \alpha\, &=\, \dfrac{p}s + \dfrac{2\pi}s\, n,\,\ \ {\rm for\ some}\ \ n\in\Bbb Z\\[.3em]
\iff\ \ \ \alpha\, &\equiv\, \dfrac{p}{s}\!\!\pmod{\!\dfrac{2\pi}s\, \B... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2372886",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Evaluate the integral $\int\limits_{-\infty}^\infty \frac{\cos(x)}{x^2+1}dx$. Evaluate the integral $\displaystyle\int\limits_{-\infty}^\infty \frac{\cos(x)}{x^2+1} dx$.
Hint: $\cos(x) = \Re(\exp(ix))$
Hi, I am confused that if I need to use the Residue Theorem in order to solve this, and I am not sure where I should ... | We may also see that $$I=\int_{-\infty}^{\infty}\frac{\cos\left(x\right)}{1+x^{2}}dx=2\int_{0}^{\infty}\frac{\cos\left(x\right)}{1+x^{2}}dx$$ $$ =\int_{0}^{\infty}\frac{e^{ix}+e^{-ix}}{1+x^{2}}dx=\frac{e^{-1}}{2}\left(\int_{0}^{\infty}\frac{e^{1+ix}+e^{1-ix}}{1+ix}dx+\int_{0}^{\infty}\frac{e^{1+ix}+e^{1-ix}}{1-ix}dx\ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2372949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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$x^5 + y^2 = z^3$ While waiting for my döner at lunch the other day, I noticed my order number was $343 = 7^3$ (surely not the total for that day), which reminded me of how $3^5 = 243$, so that $$7^3 = 3^5 + 100 = 3^5 + 10^2.$$
Naturally, I started wondering about nontrivial integer solutions to $$x^5 + y^2 = z^3 \tag{... | There is a beautiful connection between $a^5+b^3=c^2$ and the icosahedron. Consider the unscaled icosahedral equation,
$$\color{blue}{12^3u v(u^2 + 11 u v - v^2)^5}+(u^4 - 228 u^3 v + 494 u^2 v^2 + 228 u v^3 + v^4)^3 = (u^6 + 522 u^5 v - 10005 u^4 v^2 - 10005 u^2 v^4 - 522 u v^5 + v^6)^2\tag1$$
By scaling $u=12x^5$ an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2373028",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 1,
"answer_id": 0
} |
Can we use chi-square distribution and central limit theorem to find the approximate normal distribution? If $X_1,\ldots,X_i,\ldots,X_n$ are same normal distribution, $X_i \sim \operatorname{Normal}(0,σ^2)$,
and they are independent.
$$
Z = \frac{\sum_{i=1}^n X_i^2} n.
$$
What is the distribution of the square of the ... | If $X_i/\sigma\overset{iid}{\sim}N(0,1)$, then $W_i = (X_i/\sigma)^2\overset{iid}{\sim}\chi^2_1$ and $\mathbb{E}(W_i) = 1$, $\mathbb{Var}(W_i) = 2$. Therefore, if $\bar{W}_n$ is the sample mean, we have by the classic CLT
$$
\sqrt{n}(\bar{W}_n-1)\overset{d}{\rightarrow} N(0,2).
$$
Based on the above we obtain that for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2373140",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Ignoring the constant of integration $C$ in the integrating factor method for solving Linear ODE $$\dfrac{dy}{dx} + p(x)y = f(x)$$
Solving the linear dif. equation, we can use integrating factor method.
We know integrating factor: $exp(\int p(x) dt) = exp(P(x) + C)$.
But we ignore the constant of integration $C$. How c... | The integrating factor is $\exp(P(x)+C)= K\exp(P(x)),$ where $K=\exp(C) > 0$.
Multiplying to the equation $$K\exp(P(x))(\dfrac{dy}{dx} + p(x)y) = K\exp(P(x))f(x)$$
$$K\frac{d}{dx}(\exp(P(x)) y)=K\exp(P(x))f(x)$$
We can always divide by $K$ anyway, hence there isn't a need to include the constant.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2373282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 1,
"answer_id": 0
} |
What is an intuitive approach to solving $\lim_{n\rightarrow\infty}\biggl(\frac{1}{n^2} + \frac{2}{n^2} + \frac{3}{n^2}+\dots+\frac{n}{n^2}\biggr)$?
$$\lim_{n\rightarrow\infty}\biggl(\frac{1}{n^2} + \frac{2}{n^2} + \frac{3}{n^2}+\dots+\frac{n}{n^2}\biggr)$$
I managed to get the answer as $1$ by standard methods of so... | "grows much faster than the numerator": you are disregarding the fact that the numerator is actually
$$1+2+3+\cdots n, $$ which doesn't grow slower than $n^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2373357",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 11,
"answer_id": 5
} |
How would I express the series $|1+1+1|+|1+1-1|+|1-1+1|+|1-1-1|+|-1+1+1|+|-1+1-1|+|-1-1+1|+|-1-1-1|$ in summation notation? I tried putting the series
$$
|1+1+1|+|1+1-1|\\+|1-1+1|+|1-1-1|\\+|-1+1+1|+|-1+1-1|\\+|-1-1+1|+|-1-1-1|
$$
into Wolfram Alpha and typing "in summation notation" but it wouldn't tell me what it is ... | $\sum_{i,j,k=0}^1 | (-1)^i + (-1)^j + (-1)^k |$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2373446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 7,
"answer_id": 2
} |
coins on chessboard, who has the winning strategy The game begins with empty $n\times n$ chessboard and a fixed number $m\in\{1,2,\dots,n\}$.
Two players are making moves alternately, each move is placing a coin on one empty square, each row and column can contain at most $m$ coins, the guy who cannot put a coin when h... | The remaining case is $n$ odd, $m$ even. My first intuition was the second player wins but my proof fails.
The maximum number of coins that can fit on a $n \times n$ chessboard without making a $m+1$-alignment is $n*m$, which is even.
If there are less than $n*m$ coins disposed, you can always find at least one row an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2373533",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
} |
Natural numbers large enough can be written as $ab+ac+bc$ for some $a,b,c>0$ Conjecture: Any integer $n>462$ can be written as $n=ab+ac+bc$, where $a,b,c\in\mathbb Z_+$.
Tested for all $n\leq 100,000$, but I would like to see a proof.
The exceptions seems to be $\{1,2,4,6,10,18,22,30,42,58,70,78,102,130,190,210,330... | Not an answer, just a reduction to one particular class of cases.
Claim: If there is a prime $p<2\sqrt{n}$ such that $p\not\mid n$ and $-n$ is a square modulo $p$, then such $a,b,c$ exist.
When $n>1$ is odd, we have $(a,b,c)=\left(1,1,\frac{n-1}{2}\right)$.
When $n>4$ is divisible by $4$, we have $(a,b,c)=\left(2,2,\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2373650",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
Injective ring homomorphism between finite direct product of rings All rings below are commutative with unity.
Let $R$ be a Noetherian ring such that there is an injective ring homomorphism from $R^m$ to $R^n$ , then is it true that $m\le n $ ? If not true in general then can we impose any condition (apart from Artini... | Suppose $\alpha:R^n\times R\to R^n$ is an injective ring homomorphism. Extend the codomain to get a (non-unital) endomorphism $\beta:R^n\times R\to R^n\times R$, where $\beta(x)=(\alpha(x),0)$.
Consider the idempotent element $e_0=(0,1)\in R^n\times R$ and its images $e_n=\beta^n(e_0)$ under powers of $\beta$. Then $\{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2373751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
The length of a gap between the rationals In Terence Tao's book Analysis I, he says
there are still an infinite number of “gaps” or “holes” between the
rationals, although this denseness property does ensure that these
holes are in some sense infinitely small.
I think a gap between the rationals should have zer... | It is possible to prove the greatest lower bound of $A =\left\{ a \middle| a = a_{2} - a_{1}, {\ a}_{1} \in A_{1}{,a}_{2} \in A_{2} \right\}$ is 0 in the rational number system， since if any other positive number $b$ is the greatest lower bound, we could always find a positive number $c=a_{2} - a_{1}<b$ .
I think wha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2373864",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Does anyone know the name of this conjecture? Given $p$ and $q$ are two different prime numbers.
Does there exist a positive integer $n$ such that
$p^n = 1 \pmod q$
Is this conjecture true? If so, any source of the prove. What is the name of this conjecture or theorem (if it is true)?
| Independently of Fermat, it simply results from the fact that the group of units $(\mathbf Z/q\mathbf Z)^\times$ is finite, hence the (multiplicative) subgroup generated by the congruence class $\bar p=p+q\mathbf Z$ is finite.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2374156",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Involution that brings sets to disjoint sets Let $A$ be a collection of subsets of $\{1,2,\dots,n\}$ that is closed under taking subsets (that is, if $U\in A$ and $V\subseteq U$ then $V\in A$). Is there always an involution $f:A\to A$ such that $f(V)\cap V=\emptyset$ for all $V\in A$? I'm guessing yes.
Note that if $A=... | For any $A$ start with the smallest powerset that includes $A$ (e.g. if $A=\{\emptyset,\{1\},\{2\}\}$ start with $\mathcal P(\{1,2\})$). Use the trivial involution you suggested for this power set. Then transform this involution to a new involution by eliminating elements one by one to reach $A$ (start with bigger elem... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2374230",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
} |
Effects of a transformation which resembles a projection For a given symmetric and positive definite matrix $A \in \mathbb{R}^{n\times n}$ having its columns being a basis in $\mathbb{R}^n$ we have:
$$A(A^T A)^{-1} A^T$$
$$ = A(A^2)^{-1} A $$
being a projection matrix. The term $(A^TA)^{-1}$ represents a normalizing fa... | Let $P_n$ denote the set of positive definite $n\times n$ real matrices. The set of matrices of the form you describe,
$$
X=\{AB^{-2}A\mid A,B\in P_n\},
$$
is equal to $P_n$. Indeed if $A,B\in P_n$ then
$$
(AB^{-2}A)^T=A^T(B^T)^{-2}A^T=AB^{-2}A
$$
and
$$
v^T(AB^{-2}A)v=(B^{-1}Av)^T(B^{-1}Av)>0
$$
for any nonzero ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2374318",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Reference request for complete probability text without measure theory I'm looking for a complete probability reference text, covering the majority of standard probability and stochastic process topics that can be covered without the use of measure theory.
I've already had a basic course in probability and in stochast... | A First Course in Probability & Introduction to Probability Models by Sheldon. Ross. The first one is introduction to probability without stochastic processes, while the second one delves into probability models ("Stochastic models") after relatively short brief on basic probability notions, random variables, etc.
In ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2374403",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Why are module preserving elements units? Let $K$ be a real quadratic field and $M \subset K$ be a subgroup of the additive group of $K$ of rank $2$. Why is each $\varepsilon \in K$ with $\varepsilon M=M$ a unit, so an element of $\mathcal O_K^\times$?
What I tried: We can find a basis $b_1,b_2 \in M$ for $M$ such that... | In case that $M$ is a fractional ideal this answer is simple: $ M = \varepsilon M =(\varepsilon) M $, so the principal ideal generated by $\varepsilon$ has to be $\mathcal O_K$. Hence, $\varepsilon \in \mathcal O_K$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2374552",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Functions that are always less than their derivatives I was wondering if there are functions for which $$f'(x) > f(x)$$ for all $x$. Only examples I could think of were $e^x - c$ and simply $- c$ in which $c > 0$. Also, is there any significance in a function that is always less than its derivative?
Edit: Thank you ve... | The inequality $$f'(x) > f(x)$$ is equivalent to $$\left[ f(x) e^{-x} \right]' > 0.$$
So the general solution is to take any differentiable function $g(x)$ with $g'(x) > 0$ and put $f(x) = g(x) e^x$.
Note that nothing is assumed about $f$ except differentiability, which is necessary to ask the question in the first pl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2374685",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 14,
"answer_id": 2
} |
Minimum of the given expression
For all real numbers $a$ and $b$ find the minimum of the following expression.
$$(a-b)^2 + (2-a-b)^2 + (2a-3b)^2$$
I tried expressing the entire expression in terms of a single function of $a$ and $b$. For example, if the entire expression reduces to $(a-2b)^2+(a-2b)+5$ then its mini... | Let $a=\frac{17}{15}$ and $b=\frac{4}{5}$.
Hence, we get a value $\frac{2}{15}$.
Thus, it remains to prove that
$$(a-b)^2 + (2-a-b)^2 + (2a-3b)^2\geq\frac{2}{15}$$ or
$$10(3a-3b-1)^2+3(5b-4)^2\geq0$$
Done!
I got my solution by the following way.
We need to find a maximal $k$ for which the following inequality is true ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2374768",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Marble Probability Problem (Not Putting Marbles Back In Bag) A friend of mine asked this question to me recently. It has been ages since I did probability, but it seems interesting mathematically regardless. Here it is:
Suppose you have $5$ yellow marbles and $135$ green marbles in a bag. $10$ marbles will be pulled ou... | Let $P$ be the probability of pulling at least one yellow marble, and let $P^*$ be the probability of pulling no yellow marbles. Then
$$P=1-P^*$$
The probability of pulling no yellow marbles is
$$p_1p_2...p_{10}$$
where $p_i$ is the probability of pulling a green marble on the $i$th draw. The probability on the first d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2374871",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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