Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Solution needed for first order ODE $\frac{dy}{dx}$$= \frac{(x+3y-5)}{(x-y-1)}$
The equation is neither homogeneous nor linear. It's not variable separable either.
| Substitute $x = t+2$, $y= u+1$, and it becomes $$u' = \dfrac{t+3u}{t-u}$$
which is homogeneous.
BTW, Maple gives the solution as
$$y \left( x \right) =1-{\frac { \left( x-2 \right) \left( {\rm W}
\left(2\,c \left( x-2 \right) \right)+2 \right) }{{\rm W} \left(2\,c
\left( x-2 \right) \right)}}
$$
where $W$ is the Lam... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2036194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solving $n^a = x(n^x)$ for $x$ Okay. Let me just get straight to the point. I have a formula, $n^a = x(n^x)$. What I'm trying to do is make $x$ the subject of the formula. In other words, I want $x$ to express in terms of $a$ and $n$ only. It occured to me that this problem seemed rather impossible, but I'm no expert, ... | One requires the use of the Lambert W function, which is required in step 3. The solution is given as follows,
$$n^a=xn^x=xe^{x\ln(n)}\tag1$$
$$n^a\ln(n)=x\ln(n)e^{x\ln(n)}\tag2$$
$$W\left(n^a\ln(n)\right)=x\ln(n)\tag3$$
$$x=\frac{W\left(n^a\ln(n)\right)}{\ln(n)}\tag4$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2036334",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A problem in measure theory (outer measure) Show that for every set $A = P(\mathbb{R})$ (power set) there exists $B \in \mathbb{B}(\mathbb{R})$ (Borel set) s.t $A \subset B$ $\lambda^{*}(A) = \lambda (B)$ and $\lambda (N) = 0 \ \forall N \in \mathbb{B}(\mathbb{R})$ with $N = \mathbb{B} \backslash A$
where
$$
\lambda^{... | Hint.
For $n \in \mathbb N$ there exists $\displaystyle B_n =\bigcup_{j \in \mathbb N} I_{j,n}$ with $I_{j,n} \in J$ such that
$$\lambda(B_n)-\lambda^*(A)\le \frac{1}{n}$$ Then define
$$\displaystyle B=\bigcap_{n \in \mathbb N} B_n$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2036430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Example of a locally inverse semigroup which isn't a generalized inverse semigroup I'm studying Howie's Fundamentals of Semigroup Theory.
A semigroup $S$ is locally inverse if $eSe$ is inverse for any idempotent $e$ of $S$. A semigroup is a generalized inverse semigroup if is regular and its idempotents are a normal b... | Take the semigroup $S = \{a, b, c, ab, 0\}$ where $a$, $b$ and $c$ are idempotent and $ca = c$, $ac = a$, $bc = c$, $cb = b$, $abc = a$, $ba = 0$. The non-zero elements form a $\mathcal{D}$-class:
\begin{align}
\hline
|{}^*a &\mid ab| \\
\hline
|{}^*c &\mid{}^*b|\\
\hline
\end{align}
This semigroup is regular, locally ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2036560",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that $(1+\frac{x}{n})^n \rightarrow e^x$ uniformly on any bounded interval of the real line.
Show that $(1+\frac{x}{n})^n \rightarrow e^x$ uniformly on any bounded interval of the real line.
I am trying to argue from the definition of uniform convergence for a sequence of real-valued functions, but am struggling... |
Herein, we present an approach that for any given $\epsilon>0$, produces a number $N$, which depends on $\epsilon$ and not $x$, such that $\displaystyle \left|e^x-\left(1+\frac xn\right)^n\right|<\epsilon$ whenever $n>N$.
To do this we will use the inequalities, which I established in THIS ANSWER using only the limi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2036694",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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"answer_id": 1
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Transformation of a linear independent set is linearly independent Question
Let $v_1,\cdots,v_n$ be vectors in a vector space $V$ and let $T:V→W$ be a linear transformation.
if $T(v_1),\cdots,T(v_n)$ is linearly independent in $W$, show that $v_1,\cdots,v_n$ is linearly independent in $V$.
Here's what i have so far:
i... | You want to show $v_1, \ldots, v_n$ are linearly independent. Suppose they are not. Then there are scalars $c_1, \ldots, c_n$ (not all zero) so that $c_1v_1+\ldots +c_nv_n=0$. Then $$ T(c_1v_1+\ldots +c_nv_n)=T(0)=0.$$ So $c_1T(v_1)+\ldots +c_nT(v_n)=0$, which means that $T(v_1), \ldots, T(v_n)$ are not linearly indepe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2036792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Calculating the sum of $\sum_{k=1}^{\infty}{\frac{(-1)^k}{k}}$ I am trying to find the sum of
$$\sum_{k=1}^{\infty}{\frac{(-1)^k}{k}}$$
I've proven that this converges using the Leibniz test, since
$a_n > 0$ and $\lim_{n\to\infty}{a_n} = 0$.
I am not sure how to go about summing this series up though. Every example I'v... | Using the Taylor series for the natural logarithm,
$$
\ln(x+1)=\sum_{n=1}^\infty\frac{(-1)^{n+1}x^n}{n}
$$
for $-1<x\le 1$. Abel's theorem guarantees that
$$
\ln2=\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}.
$$
Hence,
$$\sum_{n=1}^\infty\frac{(-1)^{n}}{n}=-\ln2.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2037005",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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For a compact set $K\subset \mathbb M_n(\mathbb R)$, the eigenvalues of matrices in $K$ form a bounded set Let $K\subset \mathbb M_n(\mathbb R)$ be a compact subset. Then I have to show that :
All the eigen values of the elements of $K$ form a bounded set.
My work: consider the map $K \to det K$ which is continuous... | Let $||*||$ be any norm on $ \mathbb R^n$ and let $||*||_O$ the matrix norm induced by $||*||$
Since $K$ is compact, $K$ is bounded. Thus, there is $c>0$ such that
$||A||_O \le c$ for all $A \in K$.
Now let $A \in K$ and let $ \lambda$ be an eigenvalue of $A$. Then there is $x \in \mathbb R^n$ with $Ax= \lambda x$ and... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "8",
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Let G be a group such that $(xy)^2 = (yx)^2$ for all x, y ∈ G. Show that $xy^2 = y^2x$ for all x, y ∈ G. I'm not quite sure if I did this right but I had another solution that made no sense to me:
This is the solution I do not understand. I did not solve it this way.
$=>(xy)^{-1}(xy)^2(yx)^{-1} = (xy)^{-1}(yx)^2(yx)^{... | This might be same argument as in previous answer:
$$(x^{-1}\cdot yx)^2=(yx\cdot x^{-1})^2 \,\,\,\, \Rightarrow \,\,\,\, x^{-1}y^2x=y^2.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2037217",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$\int_{\Omega} |f| d\lambda = 0 \Rightarrow \{x \in \Omega: f(x) \neq 0\}$ is a null set
Let $(\Omega, \mathscr A, \lambda)$ be a measure space.
For an $\mathscr A$-measurable numerical function $f: \Omega \rightarrow \Bbb R$, it holds that
$\int_{\Omega} |f| d\lambda = 0 \Rightarrow \{x \in \Omega: f(x) \neq 0\}$ is ... | Reverse of standard machine:
$$\int_{\Omega} |f| d\lambda = 0$$
$$\to \int_{\Omega} f^+ d\lambda + \int_{\Omega} f^- d\lambda = 0$$
$$\to \int_{\Omega} f^+ d\lambda = \int_{\Omega} f^- d\lambda = 0$$
$$\to \sup_{h^+ \le f^+} \int_{\Omega} h^+ d\lambda = \sup_{h^- \le f^-} \int_{\Omega} h^- d\lambda = 0$$
$$\to \int_{\O... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2037324",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Probability of repeated sampling from random draws with replacement I am sampling 552025 patients from a population of 647117 patients. If this sampling is done with replacement, please can someone help tell me :
1) what is the probability that there is any occurrence of repeated sampling of patients?
2) the number of ... | Suppose, more generally, that you have $N$ people in total and wish to sample $m$ with replacement? What is the expected number of distinct people sampled more than once?
Let $X_i$ denote the indicator variable for the $i^{th}$ person. Thus $X_i=1$ if that person is sampled more than once, and $X_i=0$ otherwise. By ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to solve this algebraic problem about remainder of polynomials? Question :
My approach :
Now as I had to obtain a remainder of $\frac{(x-1)^{2017}}{x^2 - x +1}$
So, I could write this as $\frac{(x-1)^{2017}}{(x - 1)^2 + x}$
now I substitute $t = (x-1)$, so $\frac{(x-1)^{2017}}{(x - 1)^2 + x}$ could be written as $... | Hint:
You mean remainder right, not quotient.
$t^3\equiv1\pmod{t^2+t+1}$
As $3|2016, t^{2016}\equiv1$
$\implies t^{2017}\equiv t$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2037565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Representations of symetric group S3 (sorry in advance for my english. Im not sure my terminology is correct)
I'm trying to solve a problem:
Let $e_1$ , $e_2$ , $e_3$ be a base of $C^3$ (3-D vectors with complex elements). Let $A(g)e_i=e_{g(i)}$ be a representation of the symmetric group $S_3$ ($g\in S_3$).
Also let $... | Since $V$ is an invariant subspace of $\mathbb C^3$, $\rho(g)(v) \in V$ for every $v \in V$. Thus you have to compute the $2 \times 2$ matrix associated to every element $ g \in S_3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2037829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Please verify my proof for a continuous function attaining a minimum value on an interval. I was given the following problem in my exam:
A function continuous on $[a,b]$ attains a minimum value on $[a,b]$.
Note: proof should not involve compact sets or sequences.
My proof:
Using the completeness axiom in Real numb... | This sentence is fluff:
"For any closed interval in R ,there exists a greatest lower bound, called the infimum, which is equal to the minimum of that set."
The infimum exists regardless of whether the function attains its infimum
That the image of any continuous function over a closed domain is closed is an interesting... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Show that $K$ is a splitting field of $f(x)$ over $E$.
Let $K$ be a splitting field of $f(x)$ over $F$. If $E$ is a field such that $F\subseteq E\subseteq K$, show that $K$ is a splitting field of $f(x)$ over $E$.
We know that $$f(x) = c(x-u_1)(x-u_2)\dots(x-u_n),$$ where $c \in F \subseteq E \implies c \in E$.
Also ... | Note that if $K = F(u_1,u_2,\ldots,u_n)$ means $K$ is "the smallest subfield of $K$ containing $u_1,u_2,\ldots,u_n$ and $F$. Now $E(u_1,u_2,\ldots,u_n)$ is "the smallest subfield of $K$ containing $(u_1,u_2,\ldots,u_n)$ and $E$. But since $E \subset K$, there is no difference between $E(u_1,u_2,\ldots,u_n)$ and $F(u_1,... | {
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"timestamp": "2023-03-29T00:00:00",
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Write $\frac{1}{i}i$ in the form $xi +y$ This was a test problem that I did not understand at all. I know it is converting complex numbers, but I need help.
How do I write $\frac{1}{i}i$ in the form $xi +y$?
| In general, if you have $$ \frac{a+bi}{c+di} $$
you can multiply by $ (c-di)/(c-di)$ and simplify things nicely... I'll leave the details to you, but can you see how to use this for your problem? But as people have pointed out, it is indeed just equal to 1.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2038208",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Vector subspaces of zero dimension I was confused about zero dimension vector subspaces. Can you please answer the following questions with details/examples.
*
*What is the dimension and basis for vector space that is just composed of the zero vector.
*If the answer to the latter is an empty set, how can I construc... | (1) A vector space that is composed of just the zero vector is zero dimensional and its basis is the empty set.
(2) You can construct a zero vector because the empty sum is defined to be zero (this is somewhat of a cheat). The sum $\sum_{v_i\in\emptyset}a_iv_i$ is an empty sum, and it is defined to be the zero eleme... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Intersection points of two polar curves I was asked to find the area inside $r=3cos\theta$ and outside $r=1+cos\theta$ (see figure)
My question is, how do i find the intersection points, I was taught to make $1+cos\theta = 3cos\theta$ and solving it we get $\theta=\pi/3$ and $\theta =5\pi/3$, but as you can see the cu... | Observe one of the points of intersection is given by
\begin{align}
(r\cos\theta, r\sin\theta)
\end{align}
where $r = 3\cos \frac{\pi}{3}$ and $\theta = \frac{\pi}{3}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2038456",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Completeness and Incompleteness Through browsing questions asked in the past about Godel's completeness and imcompleteness theorems, I've come to see that the sense of completeness in both of the theorems are different. However, I can't see how they are different!
Does this distinction boil down to that of truth and p... | The completeness theorem for first order logic is a general statement about first order languages and associated deductive rules that says if something is true in every model of a theory, then it's a provable consequence of that theory. That is, we do not end up with theories that are consistent but have no model, in w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2038565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 1
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vector space dimension of linear varieties Suppose $Y=Z(f_1,\cdots ,f_r)\subseteq \mathbb{A}^n_k$ where each $f_i$'s are linear homogeneous polynomials which are $k$-linear independent. Then $Y$ is also a vector space over $k$. My question: Is the vector space dimension of $Y$ is same as the dimension of $Y$ as an affi... | Yes.
Note that $Y$ is given as $Ax=0$, where $A$ is a $n \times r$-matrix containing the coefficients of the $f_i$.
Thus $Y= \ker A$.
This should now be clear, maybe depending upon your definition of "dimension".
EDIT Now that we have a definition of dimension to work with: we can at least immediately bound the dimensi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2038688",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How does it derived from LHS term $$\sin\left(\frac{720n\pi}{600}\right) = -\sin\left(\frac{4n\pi}{5}\right).$$
It is a part of derivation I found in an example but this step is not clear to me if I tried to just divide and use reminder but it is not same and $\sin(n\pi) = 0$ so it is not near to above step.
Please ex... | $$\sin{(720/600 n \pi)}$$$$=\sin{(6/5 n \pi)}$$$$=\sin{((2-4/5) n \pi)}$$
$$=\sin{(2 n \pi-4/5n\pi)}$$$$=\sin{(-4/5 n \pi)}$$$$=-\sin{(4/5 n \pi)}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2038843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Is there an analytic form to find the invers of a matrix using jordan form? I know that any cyclic matrix can be inversed by diagonalaizing it $A = PDP^{-1}$ where the columns of $P$ are Fourier basis vectors and the diagonal of the (diagonal matrix) $D$ contains the eigenvalues of $A$ which are the discrete Fourier tr... | $$A=C^{-1}JC \Rightarrow A^{-1}=C^{-1}J^{-1}C$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2038953",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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I have a probability of $\frac{24}{800000}$ to win with one lottery ticket. What if I buy two tickets? Assuming there's a lottery with 800,000 tickets and 24 of these tickets contain a win, my chance to win (if I buy only one ticket) is $\frac{24}{800,000}$ or $\frac{1}{33,333.\overline{3}}$, right?
But what are my c... | Your chances to win at least once are roughly twice as high if you buy two tickets. Not exactly twice, because there is a very small chance both tickets will win, but this is small enough to ignore in an approximation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2039113",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Ito's product rule in higher dimension I'm looking for an analogue version of Ito's famous product rule in higher dimensions. Meaning, let X, Y be $d$-dimensional (Ito-)processes. Then something similar to the following should hold:
$
\left\langle X_{t},Y_{t}\right\rangle =\left\langle X_{0},Y_{0}\right\rangle +\int_{0... | You may use the dot product notation, that is,
\begin{align*}
X_t \cdot Y_t &= \sum_{i=1}^d X_t^i Y_t^i \\
&=\sum_{i=1}^d\left[X_0^i Y_0^i + \int_0^t X_{t-}^i dY_t^i + \int_0^t Y_{t-}^i dX_t^i + [X^i, Y^i]_t\right]\\
&=X_0\cdot Y_0 + \int_0^t X_{t-} \cdot dY_t + \int_0^t Y_{t-} \cdot dX_t + [X, Y]_t,
\end{align*}
where... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2039221",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $(X,d)$ topological space and $f,g:X\to \Bbb{R}$ are continuous, then so is $f+g$ Let $(X,d)$ be topological space and let $C(X)$ denote the set of continuous functions $f:X\to \Bbb{R}$. Show that if $f,g\in C(X)$ then $f+g\in C(X)$. I've seen online references of the continuity of addition topology or something ar... | It suffices* to prove that the pre-images of intervals $ (-\infty, A)$ are open in $X$.
$$ (f+g)^{-1}(-\infty, A)=\{x \in X \ | f(x)+g(x) < A \} = \cup_{B \in \mathbb{R}} (\{x| g(x) < B \} \cap \{x | f(x) < A-B \})$$
$\{x| g(x) < B \}$ and $\{x | f(x) < A-B \}$ are open for any numbers A and B, because $f$ and $g$ ar... | {
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"timestamp": "2023-03-29T00:00:00",
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Modal logic - Show that if $\vdash \Diamond T$ holds, $\vdash \Box A \to \Diamond A$ holds In normal Modal logic, how can I show that if $\vdash \Diamond T$ holds (is derivable), $\vdash \Box A \to \Diamond A$ also holds. I can already prove it by showing that if $\vdash \Diamond T$ holds the frame must be serial and t... | We can use the following propositional tautology $$\vdash (A \land \neg A)\rightarrow \neg \top$$
Then using inverse deduction rule, necessitation rule and deduction we obtain
$$\vdash \Box(A \land \neg A)\rightarrow \Box \neg \top$$
Then by using the fact that $\Box$ distributes over $\land$ we get:
$$\vdash (\Box A \... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Showing n! is greater than n to the tenth power I'd like to show $n!>n^{10} $ for large enough n ( namely $ n \geq 15 $).
By induction, I do not know how to proceed at this step:
$$ (n+1)\cdot n!>(n+1)^{10} $$
As I can't see how to simplify $(n+1)^{10} $.
This seems like such a trivial thing (and it probably is), y... | Replacing $n$ by $n+1$, the LHS is multiplied by $n+1$ while the RHS is multiplied by $\left(1+\frac1n\right)^{10}$, which is bounded (by $1024$ for $n\ge1$, but by $2$ for $n\ge15$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2039592",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 8,
"answer_id": 6
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Prove that $R$ is a ring with division. I'm having problems trying to know if my proof is wrong or not. The problem states:
Let $R$ a ring with 1, not necessary commutative, such that for every $a\in R\setminus\{0\}$, there exists $b\in R\setminus\{0\}$ (which depends on $a$) such that $a\cdot b=1$. Prove that $R$ is ... | All you need to show is that there also exists a left inverse, where for all nonzero $a$, there exists a $b$ so that $ba=1$. Well, you almost had it, the first step was fine, and the idea to "cancel" the $b$ was correct as well, you just need to use the hypothesis on $b$ as well:
let $a$ be nonzero.
Then there exists s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2039720",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Show that $3^{2n+1}-4^{n+1}+6^n$ is never prime for natural n except 1. Show that $3^{2n+1}-4^{n+1}+6^n$ is never prime for natural n except 1. I tried factoring this expression but couldn't get very far. It is simple to show for even n but odd n was more difficult, at least for me.
| You can factor it as $(3^n-2^n)(3^{n+1}+4*2^n)=3^{2n+1}-3*6^n+4*6^n-4*4^n$. Here we juggle between $(ab)^n=a^nb^n$.
Since we can factorise it, to have a prime we need one of these factors to be 1, which only happens when $n$ is one, i.e the first term is $(3-2)$ and the second is 17. Note that the second term can't be ... | {
"language": "en",
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"source": "stackexchange",
"question_score": "4",
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2 queen, $5\times 5$ chess board problem Prove the following:
In $5\times 5$ chess board the least amount of queens you need in order to threaten on each square is 3.
(Square threat: the queens threatens on each square in the diagonals,row and column from the queens position).
I just need to show that you can't threa... | A case analysis is sufficient. Let the top left corner by black.
You need both a B queen and a W queen (we can see this by counting their max range of W and B).
If the W queen is on the boundary, then 3 W boundary squares remain to be covered and their position makes it impossible to be covered by a B queen.
If the W ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2040030",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How many ways to select 3 numbers from $1-30$ so that the sum of them is a multiple of 3? How many ways to select 3 numbers from $1-30$ (each number is used only one time) so that the sum of them is a multiple of 3?
I got the answer is $1360$ by programming (check the sum of every combination), and
I also know the form... | i find this answer for 1-300 and sum is divisible by 3 i hope it can help you
In how many ways can three numbers be selected from the numbers 1,2,…,300
such that their sum is divisible by 3?
So we can choose the first two numbers how we like. The third has to have a definite residue class mod
3
to make the total divis... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the point $P$ on the $x$-axis which orthogonal projection on the line $r:x+y=z-1=0$ is $M=(1,-1,1)$. The points such as their orthogonal projection on the line $r$ is $M=(1,-1,1)$ lay on the plane that passes through that point $M$ and that is orthogonal to $r$.
First of all, then, I determined the directional vec... | The correct solution is in fact, as you say, $(2,0,0)$ and this can be deduced very easily because either all happen in the plane $z=1$ and the point $P$ is on the $x$-axis so $P=(x,0,0)$. Therefore you can act likely as you were in the plane $XOY$ as follow:
The distance of $M$ to $O$ is equal to $\sqrt2$ so you have ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2040271",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to find the remainder of ${289 \times 144^{25}}$ divided by ${71^{71}}$ I am solving this question.
Finding the remainder of
$$\frac{289\times 144^{25}}{71^{71}}$$
This is how I have tried solving it. First it can be simplified to $\frac{17^2 \times 2 ^{50} \times 3^{25}}{71^{71}}$. Now if we use Euler Totient rul... | ${289 \times 144^{25}} \approx 2.6 \times 10^{56} < 2.8 \times 10^{131} \approx 71^{71}$.
So the remainder is ${289 \times 144^{25}}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2040362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Number of arrangements with no consecutive letter the same I was learning the following questions in this site
question 1
question 2
which are indeed similar problems. But, one thing I observed is, in both of these problems, letters are repeating only two times. But, consider the following situation where one letter re... | We shall solve by successively applying the well known "gap" and "subtraction" methods.
Firstly, we shall keep the $E's$ separate by placing them in the gaps of $-A-B-C-D-D-$ and permute the other letters, thus $\binom63\cdot\frac{5!}{2!} = 1200$ ways.
We shall now subtract arrangements with the $D's$ together treating... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2040508",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Laplace expansion of Pfaffian I am reading about Pfaffian, which can be found here https://en.wikipedia.org/wiki/Pfaffian.
We know that the (general) Laplace expansion is very useful to compute determinants of matrices, and I wonder if there is such an expansion to compute Pfaffians (of skew-symmetric matrices). Fortun... | It could be on pages 115 and 116 at the end of Chapter 7 of Northcott's book, Multilinear Algebra (the solution to exercise 8).
| {
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"source": "stackexchange",
"question_score": "1",
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Proving that a function is multiplicative. Let $f(x)$ be a polynomial with integral coefficients, and let $\psi(n)$ denote the numbers of values $f(0), f(1), ..., f(n-1)$ which are coprime to $n$.
I must show that $\psi$ is multiplicative, meaning that:
$$\psi(mn) = \psi(m) \cdot \psi(n)$$
assuming $\gcd(m,n)=1$.
Furt... | Hint:
Use the fact that $\psi(m)$ equals the number of units in the multiplicative group $Z_m^{\times}$. Since $m$ and $n$ are coprime, $
Z_{mn}^{\times}\cong Z_m^{\times}\times Z_n^{\times}$.
Thus there is $
\psi(mn)=\psi(m)\psi(n)$.
| {
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"source": "stackexchange",
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What are some counter-intuitive results in mathematics that involve only finite objects? There are many counter-intuitive results in mathematics, some of which are listed here. However, most of these theorems involve infinite objects and one can argue that the reason these results seem counter-intuitive is our intuitio... | Not sure whether this is the kind of thing you were expecting, but here goes:
Some statements about constructive mathematics can seem very counter-intuitive (at first, this is probably because one is misinterpreting what they mean), e.g.:
*
*the induction principle holds, but on the other hand: that every non-empty ... | {
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Finding $f$ such that $f(x)+f(y)=f\left(\frac{x+y}{1-xy}\right)$ Determine all functions $f$, wich are everywhere differentiable and satisfy
$$f(x)+f(y)=f\left(\frac{x+y}{1-xy}\right)$$
for all real $x$ and $y$ with $x.y \ne 1$.
PS.: The expression sugest some relation with $\tan(x)$ but I can't go further. Any hint?
| With $g:=f\circ \tan$, we have $$g(x)+g(y)=g(x+y) $$
for all $x,y$ with $x+y\ne\frac\pi 2+2k\pi$. This quickly leads to $g(x)=cx$.
| {
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"source": "stackexchange",
"question_score": "4",
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Method to solve a system of differential equations I'm studying systems of linear equations.
I'm now specifically studying systems of linear equations of the 1st order, homogeneous:
$Y' = AY$
$A$ as a constant matrix.
Now I know there are various methods to solve this systems. My professor talked about one of the met... | If there is a $b$ such that $\{b,Ab,...,A^{n-1}b\}$ is a basis (that is,
if the pair $(A,b)$ is completely controllable), then we
can reduce the system to a single differential equation.
In the above basis, $A$ has the form (controllable canonical form)
\begin{bmatrix}
0 & 1 & \cdots & 0 & 0 \\
\vdots & \vdots & \ddots... | {
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Stating that either root is zero in solving a quadratic equation Let's say we have a simple quadratic equation $x^2 - 3x = 0$. To solve, we will factor $x$ out i.e. $x(x-3)=0$, after which we will state $x = 0$ or $(x-3) = 0$. My question is, why is there no third "option" where we say "or both". Isn't it possible for ... | If $x=0$, then $x \ne 3$. If $x=3$ then $x\ne 0$. $x$ can't be two different numbers at the same time.
| {
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"source": "stackexchange",
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Notation: how to denote m-th element of a subsequence $x_{n_k}$ Sorry for probably stupid question, but I was not able to find an answer online. I want to denote m-th element of a subsequence $x_{n_k}$ while still showing that it is subsequence $x_{n_k}$. Is there any notation for this - some sort of $(x_{n_k})_m$ or $... | If $n_1,n_2,n_3,\dotsc$ is a strictly increasing sequence of positive integers, then $x_{n_1},x_{n_2},x_{n_3},\dotsc=(x_{n_k})$ is a subsequence of $x_1,x_2,x_3,\dotsc=(x_n)$. The $m$th term of $(x_{n_k})$ is denoted $x_{n_m}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2041275",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is there a nontrivial oriented link with two components with linking number 0 which is related to the unknot by a single skein relation? While thinking about a problem of determining whether a given link is a slice link or not, I was lead to the following question:
Is there an oriented link with two components (other t... | I found one by myself:
Here the three tiwsts have an effect of cancelling out the -3 linking number made from the double trefoil.
Similarly, every 2-component link with genus 1 can be realized in this way, and we can always cancel out its linking number by giving some number of additional twists.
| {
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"url": "https://math.stackexchange.com/questions/2041391",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Are Jacobi elliptic functions liouvillian? I mean: could a Jacobi elliptic function be expressed in terms of a "finite number of arithmetic operations (+ – × ÷), exponentials, constants, solutions of
algebraic equations (a generalization of nth roots), and indefinite integrals of such elements"?
| I think I have a proof that the non-constant elliptic functions are not Liouvillian (be careful that the elliptic functions are the inverse of the elliptic integrals).
*
*The Liouvillian functions $L$ is a set defined recursively :
$f(z) = 1$ and $g(z) =z$ are in $L$.
If $f(z),g(z)$ are in $L$ then
*
*$a f(z)+b ... | {
"language": "en",
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Applications of Lax Milgram theorem I'm now studying the Lax Milgram theorem and I want to get deep in that topic. What things do you recommend me to study? About generalizations, variational inequalities...
And I also want to study applications of that theorem, especially in weak formulations, do you recommend me an ... | The theorem of Zarantonello goes just like the one for Lax-Milgram. It needs strongly monotone operators, but can handle nonlinear equations.
Also I find it useful to know for which PDEs von Neumann boundary conditions give unique solutions and not.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Number theory related to crypto I have a question related to a piece of coursework that is for cryptography and more for encryption that relies on Number theory, now I have no knowledge of number theory and the tutor did not cover it well enough, and I am starting to learn it slowly, but I have an exam coming up and on... | Since p (here p=61) is a prime you should know that it holds $x^{p-1} = 1 \mod p$.
Therefore $x^{p-1-u} \cdot x^u = 1 \mod p$ and so $x^{p-1-u}$ and $x^u$ are inverses. Therefore $x^{-1} = x^{p-2} \mod p$
To answer the first point we simply have to calculate $53^{59} \mod 61$, which is 38. You can check this by, $53 ... | {
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Fastest method to calculate the integral: $\int^\pi_0 t^2 \cos(nt) dt$ Fastest method to calculate the integral: $\int^\pi_0 t^2 \cos(nt)dt$.
Now I am aware that this is done by doing parts twice, however from inspection I see that terms cancel in the method, is there therefore a straight forward formula I can use for ... | Hint. One may start with
$$
\int_0^\pi e^{(a+in)t}dt=\left[\frac{e^{(a+in)t}}{a+in}\right]_0^\pi=\frac{e^{(a+in)\pi}-1}{a+in},\qquad a,n\in \mathbb{R}^2,\, an\neq0,
$$ then one may differentiate twice with respect to $a$ and take $a=0$ in the real part of each side.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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When integral is diverging? $$ \int f'(x)g(x)dx= f(x)g(x) - \int f(x)g'(x)dx$$
Can I conclude the folowing?
$$ if: f(x)g(x)= \pm \infty , \implies \int f'(x)g(x)dx = \pm \infty $$
| I don't think you can claim that.
Try $f(x) = \ln(x)$ and $g(x) = x$
Then $\int_0^1 f^{\prime}(x) g(x) dx = 1 $ but $x\ln(x)|_0^1 = - \infty$
| {
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"source": "stackexchange",
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Proving identities (mod $pq$) using Fermat's little theorem? I have come across this question, which reminded me of Fermats little theorem, i dont know if the Fermats theorem is actually in use in the following mathematical statements
an integer a is a coprime with p and a coprime with q (p and q are different prime n... | These are really Chinese Remainder Theorem problems. First look mod $p$, then mod $q$. IF the expression is congruent to the same thing mod $p$ and $q$, then CRT says they're also congruent to that "same thing" mod $pq$.
For the second problem $p^{q-1}+q^{p-1} \equiv p^{q-1} +0 \equiv 1 (\bmod{q})$ by Fermat's litt... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to show that there is no analytic function $f$ on $D^*$ such that $Re(f) = log|z|$. How to show that there is no analytic function $f$ on $D^*$ such that $Re(f) = log|z|$. $D^*$ is the unit disk with zero removed. I am trying to show that if such function exists, then it must be $logz$, but $logz$ cannot be defined... | You have the right idea, but the statement
it must be $\log z$, but $\log z$ cannot be defined on $D^{*}$
should be made more precise. One argument to show that $\log$ cannot
be defined as a holomorphic function on $D^{*}$ is that $1/z$
does not have an antiderivative on that domain. You could
therefore argue as foll... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2042456",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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derivatives of Kinetic Energy I read that derivative of Kinetic Energy function = $F.v$ while I got $mv$ when I differentiated it with respect to velocity.
The way I did it is:
$\frac{dK}{dv} = \frac{1}{2} m . \frac{d}{dv} v^2$
So I assumed that the mass is fixed and I differentiated the squared velocity by taking t... | The derivate of kinetic energy respect to the time $t$ is $Fv$:
$$K'=mvv'=mva=Fv$$
In general $v$ depends by time so the total derivative of $K$ is $Fv$, i.d. the instantaneous power.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2042598",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Computing Picard groups by showing invertible modules are uniquely determined I am going to have another go at showing that
the Picard group of $k[x,y]/(xy)$ is trivial.
(see my previous stackexchange posts)
Here I define the Picard group of a ring $R$ as the isomorphism classes of finite locally free modules of ... | From what you said, there exists $u,v\in M$ such that $u$ generates $M$ modulo $x$, $v$ generates $M$ modulo $y$. Then, they both generate the one dimensional $k$ vector space $M/(x,y)M$ and thus $u=av$ for some $0\neq a\in k$ modulo $(x,y)$. Clearly, we may replace $v$ by $av$ without changing anything and thus assume... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving that $E[X]= \sum_{k=0}^{\infty} P(X>k)$ by proving $(n+1)P(X>n) \xrightarrow[]{n \to \infty} 0$
Let $X$ be a random variable with positive integer range and finite mean. To show that $$E[X]= \sum_{k =0}^{\infty} P(X>k).$$
Proof: I showed using induction that $$\sum_{k=0}^n P(X>k) = \sum_{t=1}^n (t \cdot P(X=t... | \begin{align}
E[X]
&= \sum_{k=1}^\infty k\cdot P(X=k)\\
&= \quad P(X=1)\\
&\qquad + P(X=2) + P(X=2)\\
&\qquad + P(X=3) + P(X=3) + P(X=3)\\
&\qquad + P(X=4) + P(X=4) + P(X=4) + P(X=4)\\
&\qquad\vdots\\
&= P(X > 0) + P(X >1) + P(X>2) + P(X >3) + \cdots & \text{add each column}\\
&= \sum_{k=1}^\infty P(X \ge k).
\end{alig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2042896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Let be $f$ a continuous function. Determine the limit $\lim\limits_{h \to 0} \frac{1}{h} \int_{a-h}^{a+h} f(x)\,dx$ $\lim\limits_{h \to 0} \frac{1}{h} \int_{a-h}^{a+h} f(x)\,dx$
I think that this kind of limit should I probably calculate with some kind of epsilon-delta definition.
And using the limits:
$\lim\limits_{h... | I will first prove for $h\rightarrow 0^{+}$, for $h\rightarrow 0^{-}$ is treated simialrly. As $f$ is continuous at $x=a$, given $\epsilon>0$, one may find some $\delta>0$ such that $|f(x)-f(a)|<\epsilon$ for every $x$ with $|x-a|<\delta$. For all $h\in(0,\delta)$, we have
$\left|\dfrac{1}{h}\displaystyle\int_{a-h}^{a... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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What is a covering set of a Sierpinski number? What does it do? Recently a new prime number has been discovered, which eliminates one of the six remaining candidates for the smallest Sierpinski numbers. So I was reading the wikipedia article about the Sierpinski number, where I came across what is called a covering set... | A covering set doesn't help in "finding smallest Sierpinski number".
It is merely used in order to show that a given $k\in\mathbb{N}$ is a Sierpinski number, as part of proving that the expression $k\cdot2^n+1$ is composite for every $n\in\mathbb{N}$ (becuse it is divisible by one of the values in the covering set).
In... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Why does $\left\{ \left( \frac{1}{n},\frac{1}{m} \right) : n,m \in Z^+ \right\}$ have Jordan measure $0$?
Let $$A= \left\{ \left( \frac{1}{n},\frac{1}{m} \right) : n,m \in Z^+ \right\}$$
$Z^+$ denotes positive integers. How come this set has a zero area?
Interior is definitely zero area since it doesn't have any int... | Consider the set
$$A:=\left\{\left({1\over m_1},{1\over m_2},\ldots,{1\over m_d} \right)\>\biggm|\>m_i\in{\mathbb N}_{\geq1} \ (1\leq i\leq d)\right\}\subset{\mathbb R}^d\ .$$
Claim. The Jordan content, or $d$-dimensional Jordan measure, of this set is zero.
Proof. Let an $\epsilon>0$ be given. Consider the $d$ plates
... | {
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"question_score": "1",
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Show by definition that $f:[0,1]\to\mathbb R$ is measurable
Show by definition that the function $f:[0,1]\to \mathbb R$ defined by $$f(x)=\begin{cases}\frac{1}{x} & 0<x<1\\3 & x=0\\5 & x=1\end{cases}$$is measurable.
Let $\alpha$ be any real number. Then
$$\{f>\alpha\}=\begin{cases} [0,1] & \alpha<1 \\ [0,1/\alpha]... | Your calculations are still not correct ... since this exercise is not that difficult, this makes me believe that you don't really know what you are doing.
In order to show measurability of $f$, we have to show that the preimages
$$\{f>\alpha\} := \{x \in [0,1]; f(x)>\alpha\}$$
are Borel sets for any $\alpha \in \mathb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2043412",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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set theory (concept of infinity) Let $S$ be a set. $f$ be a function on $S$ into real line $\mathbb{R}$.
Let us define $Af$ as a function from $\mathbb{R}$ into power set of $S$, $PS$
$Af(x) = \{s\mid f(s) \leq x\}$
Question: Is $S$ in the range of $Af$?
(as an example we can chose $S=\mathbb{R}$ and $f(x) = x$)
Th... | $S$ is in the range of $Af$ iff $f$ is bounded from above.
If $f$ isn't bounded from above, there is for every $x\in \mathbb{R}$ a $s_x\in S$ with $f(s_x)>x$ and so $s_x\notin Af(x)$ and so $S\neq Af(x)$ which means $S$ isn't in the range.
If $f$ is bounded from above, there is a $x\in \mathbb{R}$ with $f(s)\leq x$ for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2043522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Computing intersection multiplicity using primary ideals In $\mathbb{P^2}$, $f = x^2-yz, g = (x+z)^2-yz$, Compute the intersection multiplicity of the curves $V(f),V(G)$ at $p = [-2:1:4]$, using the fact that intersection multiplicity of two curves at point $p$ is the hilbert polynomial of the $I(P)$-primary component ... | I think you can save yourself some pain by working in the local ring $\left(\frac{k[X,Y,Z]}{(X^2 - YZ, (X+Z)^2 - YZ)}\right)_{I(p)}$, where $I(p) = (X+2, Y-1, Z-4)$. We can find a more convenient set of generators for the ideal $I = (f,g)_{I(p)}$ as follows.
\begin{align*}
I &= (X^2 - YZ, (X+Z)^2 - YZ) = (X^2 - YZ, (X... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2043617",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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What's wrong with this reasoning that $\frac{\infty}{\infty}=0$?
$$\frac{n}{\infty} + \frac{n}{\infty} +\dots = \frac{\infty}{\infty}$$
You can always break up $\infty/\infty$ into the left hand side, where n is an arbitrary number. However, on the left hand side $\frac{n}{\infty}$ is always equal to $0$. Thus $\frac... | When you learned how to extend arithmetic from the natural numbers to the integers to the rational numbers to the real numbers and to the complex numbers, one of the main motivations was to preserve the laws of arithmetic.
The situation with the extended real numbers (i.e. the real numbers along with $\pm \infty$) is d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2043758",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "62",
"answer_count": 9,
"answer_id": 1
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4th root question/guidance Find all 4th roots of $-8 + 8i\sqrt 3$
$a=-8$
$b=8\sqrt 3$
$r= \sqrt{a^2+b^2}= \sqrt {(-8^2)+(8\sqrt{3})^2)}=\sqrt{64+192}=\sqrt {256} =16$
$\frac ar= cos\theta=\frac{-1}{2}$ $\space $ $\frac br= sin\theta$=$\frac {\sqrt3}{2}$
This gives me a different $\theta$ one being 120 degrees... | Polar form:
$z_1= 2(\cos 30 + i \sin 30)$
$z_2= 2(\cos 120 + i \sin 120)$
$z_3= 2(\cos 210 + i \sin 210)$
$z_4= 2(\cos 300 + i \sin 300)$
Rectangular Form:
$z_1= 2(\frac {\sqrt{3}}{2} + i \frac 12) = \sqrt {3} + i$
$z_2= 2(-\frac 12 + i\frac {\sqrt{3}}{2} ) =-1 +i \sqrt {3}$
$z_3= 2(-\frac {\sqrt{3}}{2} - i \frac 12)=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2043882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Drunk man with a set of keys. I found this problem in a contest of years ago, but I'm not very good at probability, so I prefer to see how you do it:
A man gets drunk half of the days of a month. To open his house, he has a set of keys with $5$ keys that are all very similar, and only one key lets him enter his house.... | I tried focusing instead on the number of times he tries a key and fails. So if he gets it on the 3rd try, he misses $2x$. The probability of doing this, given that he's drunk, is $(4/5) * (4/5) = 16/25$. On the other hand, the probability of him missing twice in a row given that he's sober is $(4/5) * (3/4) = 3/5$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2044007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "78",
"answer_count": 5,
"answer_id": 2
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How to calculate $\frac d{dx}x^x$ without differentiating any logarithms? How to calculate $\frac d{dx}x^x$ without differentiating any logarithms? There isn't any clear derivative rule I can use and the derivative quotients aren't any help:
$$\lim_{h\to0}\frac{f(x+h)-f(x)}h\text{ or }\lim_{h\to x}\frac{f(x)-f(h)}{x-h... | First note that we can write
$$(x+h)^{x+h}=x^{x+h}\left(1+\frac hx\right)^{x+h} \tag 1$$
Expanding the parenthetical term on the right-hand side of $(1)$ in a generalized Binomial Series reveals
$$\left(1+\frac hx\right)^{x+h}=1+\frac{h(x+h)}{x}+O(h^2) \tag 2$$
Putting together $(1)$ and $(2)$ yields
$$\frac{(x+h)^{x+h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2044184",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Midpoint for a triangle area question? In triangle $ABC$ the three midpoints of the sides are $P, Q, R$. The midpoints of sides in triangle $PQR$ are $K, L, M$. What is the area of triangle $ABC$ if the area of triangle $KLM$ is $5$?
I started by drawing a picture with all the information. This gave me a a big triangle... | You can say that the two triangles $ABC$ and $KLM$ are similar by the intercept theorem with similarity ratio $4$, so the ratio between the two areas is $16$ and $ABC(area)=80$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2044282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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There are opens $U,V$ such that $a\in U-V, b\in V-U$ $\iff $ every finite subset is closed Definition: the $T_1$ axiom says that given $2$ distinc points $a,b\in X$, there will exist opens $U$ and $V$ such that $a\in U-V$ and $b\in V-U$.
I need to show that a topological space has the $T_1$ property $\iff$ every finite... | First let $X$ be $T_1$ we show that every finite subset is closed. As Marco suggested it suffices to prove that every singleton set is closed. Now assume this is not the case, say there exists $\{a\}\subseteq X$ which is not closed, this means $X-\{a\}$ is not open, thus by definition there exists $b\in X-\{a\}$ such t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2044390",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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Existence of an onto group homomorphism from $S_4$ to $\Bbb Z_4$
Let $S_n$ be the symmetric group of $n$ letters. Then does there exist an onto group homomorphism
from $S_4$ to $\Bbb Z_4$?
My try: Suppose that $f:S_4 \to \Bbb Z_4$ is a group homommorphism. Then $S_4/\ker f\cong \Bbb Z_4\implies o(\ker f)=6\implie... | Your argument up to ''$\ker f=S_3\text{ or }\mathbb{Z}_6$'' is correct.
But after this, it is possible but lengthy to continue the arguments; for example if kernel is isomorphic to $S_3$ then you have taken it equal to $\{(1), (123),..\}$; this is correct but needs a justification.
Better is the following: $|\ker f|=6$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2044489",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Separable states Let $U$ and $V$ be state spaces. By one of the axioms of QM we can describe the combined system as $U \otimes V$. If $U$ has basis $\{|\phi_1\rangle, \dots, |\phi_n \rangle\}$ and $V$ has basis $\{|\psi_1\rangle,\dots,\psi_m\}$, then $U \otimes V$ has basis of the form $|\phi_i \rangle \otimes |\psi_j ... | As far as I'm aware this is an open problem. There are multiple measures that do measure degrees of entanglement, see for instance
Plenio, M.B.; Virmani, S. An introduction to entanglement measures. Quant. Inf. Comp. 2007, 7, 1
There is an easy way to verify if a state is maximally entangled: if the partial trace on b... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Weakly convergence and pointwise convergence of $L^2$ How to show the below theorem? In fact ,I feel it is not right , if $f(x)\ne 0 $ at zero measure set , I still have
$$
\int_\Omega f(x)\varphi(x)dx = 0 ~~~\forall \varphi \in C^\infty_0(\Omega)
$$
What is my mistake ?
| Since we work in $L^2$-space, $f=0$ is understood as $f$ belongs to the equivalence class of the function identically equal to $0$ for the equivalence relation "equal almost everywhere".
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2044765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Parametric Form for a General Parabola It is well known that a parametric form of the parabola $y^2=4ax$ is $(at^2, 2at)$.
What are possible parametric forms of the general parabola
$$(Ax+Cy)^2+Dx+Ey+F=0$$
?
| This solution to my other question on the axis of symmetry of a general parabola gives the following:
Axis of symmetry:
$$Ax+Cy+t^*=0$$
Tanget at vertex:
$$(D-2At^*)x+(E-2Ct^*)y+F-{t^*}^2=0$$
where $t^* \left(=\frac {AD+CE}{2(A^2+C^2)}\right)$ is chosen for both lines to be perpendicular.
Solving for the intersection ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2044922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Asymptotic behaviour of $\mathbb{E}[|\chi_n^2 - n|]$. I want to know how $\mathbb{E}[|\chi_n^2 - n|]$ behaves as $n\to \infty$. Simulating this in R suggest that it grows at a rate of $\sqrt{n}$, but I am unable to prove it. Setting $\chi_n^2 =\sum_{i=1}^n Z_i^2$, where $Z_i$ are iid standard normal random variables, t... | Your simulation is correct. Let us observe that
$$
\frac1{\sqrt n}\sum_{i=1}^n(Z_i^2-1)\to\mathcal N(0,2)
$$
in distribution as $n\to\infty$ by the central limit theorem. We can show that
$$
\operatorname E\biggl|\frac1{\sqrt n}\sum_{i=1}^n(Z_i^2-1)\biggr|\to\operatorname E|Y|
$$
as $n\to\infty$, where $Z\sim\mathcal N... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2045002",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Counting points on a binary Edwards curve I want to test an implementation of elliptic curve cryptography over binary Edwards curves as defined here. I want to test it by generating a random curve. Generating the curve itself is trivial (I do not look for secure parameters), I would then like to count the number of poi... | It turned out that my calculations for $a_2$ and $a_6$ was wrong. I did not use GF addition and multiplication modulo the basis polynomial. The correct calculation to go from binary Edwards to binary Weierstrass is the following:
$v^2+uv=u^3+\left(d_1^2+d_2\right)u^2+d_1^4\left(d_1^4+d_1^2+d_2^2\right)$
Using the curv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2045100",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Singletons in coupon collecting problem There are $n$ types of coupons. All types are equally likely to turn up and each "draw" of a coupon is independent of others. If someone collects coupons until they have a complete set of all the $n$ types, what is the expected value of the number of coupons that only appear once... | They are looking for the chance that you get another $T_i$. You can collect it again until you have found all the other types you are looking for. After you get the first $T_i$, they say make a list of the first occurrence after now of $T_i$ and all the coupons you have not found yet. If $T_i$ is the last, you will ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2045183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Find the sum of $\displaystyle\sum_{n=3}^\infty \frac{2^n-1}{3^n}$ My work:$$\sum_{n=3}^\infty \frac{2^n-1}{3^n}$$
$$a_1 = \frac{2^3-1}{3^3}$$
$$a_1 = \frac{7}{27}, r=\frac{2}{3}$$
$$S_N=\frac{\frac{7}{27}}{1-\frac{2}{3}} = \frac{7}{9}$$
The correct answer is $\frac{5}{6}$
| Both geometric series are convergent, so
$$S=\sum_{n=3}^{+\infty}(\frac{2}{3})^n-\sum_{n=3}^{+\infty}\frac{1}{3^n}$$
$$\frac{2^3}{3^3}\frac{1}{1-\frac{2}{3}}-\frac{1}{3^3}\frac{1}{1-\frac{1}{3}}$$
$$=\frac{8}{9}-\frac{1}{18}=\frac{5}{6}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2045321",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Given a series defined by recursion. Prove that there are integers $S$ and $T$ such that $a_{n+1} = S a_n + T a_{n-1}$ I have this problem from an old exam that I can't solve.
Let $\{a_i\}_{i \geq 0}$ be the series define by recursion as:
$a_0 = 2$
$a_1 = 3$
$a_{n+1} = \frac{a_n^2 +5}{a_{n-1}}$ , $\forall n \in \Bbb N... | Define a series $a_{n+1} = \frac{a_n^2 +k}{a_{n-1}}$ with some $a_0, a_1$ for $n\ge2$. Assume there are $S$ and $T$ such that $a_{n+1} = S a_n + T a_{n-1}$.
$$a_{n+1} = \frac{a_n^2+k}{a_{n-1}} \\
S a_n + T a_{n-1} = \frac{a_n^2+k}{a_{n-1}} \\
S a_n a_{n-1} + T a_{n-1}^2 = a_n^2+k \tag{1}
$$
now
$$a_{n+1} = \frac{a_n^2+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2045419",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Number of occurrences of k consecutive 1's in a binary string of length n (containing only 1's and 0's) Say a sequence $\{X_1, X_2,\ldots ,X_n\}$ is given, where $X_p$ is either one or zero ($0 < p < n$). How can I determine the number of strings, which do contain at least one occurrence of consequent $1$'s of length $... | Yet another recurrence-relation based explanation: let $S_n$ be the number of strings of length $n$ which have some run of $k$ consecutive 1s in them, and let $s$ be a string in $S_n$; suppose $s'$ is the string $s$ truncated by one (i.e., with its last character removed). Then either $s'\in S_{n-1}$, or we have that $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2045496",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 5
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Natural Logarithm expressed as Limit In the process of trying to prove the derivative of $f(x)=a^x$ (for $a\in\mathbb{R}$) using the definition of the derivative, one arrives at the following equation:
\begin{align} \frac{df}{dx} = \frac{d}{dx}\left[a^x\right] = \lim_{\Delta x \rightarrow 0}\left[\frac{a^{(x+\Delta x... | In order to show that this limit is $\ln(a)$ you have to bring in the definition of the natural logarithm. And it is not good enough to say that $x = \ln(a) \Leftrightarrow a = e^x$ because that begs the question of how to define $e$.
One typical way to define the natural logarithm is as the integral of $1/x$; but tha... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is $n\int_{0}^{1}f(x)x^{n}dx = n \int_{0}^{1}\left(f(x)-f(1)\right)x^{n}dx + \frac{n}{n+1}f(1)$? I am trying to understand this line in my notes but do not know how this jump is made. Let $f$ be a continuous function on $[0,1]$. Why is
$$n\int_{0}^{1}f(x)x^{n}dx = n \int_{0}^{1}\left(f(x)-f(1)\right)x^{n}dx + \fra... | One may observe that
$$
\begin{align}
n\int_{0}^{1}f(x)x^{n}dx &= n \int_{0}^{1}\left(f(x)-f(1)+f(1)\right)x^{n}dx
\\\\&=n \int_{0}^{1}\left(f(x)-f(1)\right)x^{n}dx+n \int_{0}^{1}f(1)x^{n}dx
\\\\&=n \int_{0}^{1}\left(f(x)-f(1)\right)x^{n}dx+f(1) \cdot n \int_{0}^{1}x^{n}dx
\\\\&=n \int_{0}^{1}\left(f(x)-f(1)\right)x^{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2045706",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is there a constant by which I can multiply to invert one sign? Take the equation $A(x - 7) = x + 7$. Is this possible?
In other words, is there any constant $A$ by which $(x - 7)$ can be multiplied to become $(x + 7)$?
Notes:
*
*$x$ is any real number.
*The original question was $x + 7 = A(x - 7) + B(x + 5)$. I th... | To solve the actual problem consider collecting everything to one side.
You will notice that overall the equation of $(x-Ax-Bx)+(7+7A-5B)=0$ must be 0 independent of choice of $x$. What does that tell you about the relationship between $A$ and $B$? You will also note the constant term must be 0, what does that tell you... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2045823",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $b_j>0$ for every $j$ and if $\sum^{\infty}_{j=1}b_j$ converges then prove that $\sum^{\infty}_{j=1}(b_j)^3$ converges or diverges. If $b_j>0$ for every $j$ and if $\sum^{\infty}_{j=1}b_j$ converges then prove that $\sum^{\infty}_{j=1}(b_j)^3$ converges or diverges.
I believe it to be true if the positivity hypothes... | If $\sum b_j$ converges then $b_j\rightarrow 0$.
For $j$ large enough you have $0<b_j<1 \Rightarrow 0<b_j^2<b_j$
By multiplying with $b_j$ again you get $0<b_j^3<b_j^2\Rightarrow 0<b_j^3<b_j$
Then $\sum b_j^3$ converges by comparison test.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2045949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Rewriting $\sin(a+b) = c$ Given $$\sin(a+b) = c$$
Could it be rewritten as $$a = \arcsin(c) - b$$
For all reals $a$ and $b$?
Sine over reals isn't one-to-one though. Is the above valid?
| This formula doesn't work for all real numbers since the range of $\arcsin(x)$ is $[-\frac{\pi}{2}, \frac{\pi}{2}]$. For example, $$\sin (\frac{3\pi}{2} + \frac{\pi}{4}) = -\frac{\sqrt 2}{2}$$ but $$\arcsin\left(-\frac{\sqrt 2}{2}\right) - \frac{\pi}{4} = -\frac{\pi}{2} \neq \frac{3\pi}{4}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2046080",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Elementary Algebra Problem (in 8th grade) The exercise is to prove that $$ \forall x \in [0,3] $$ : $$ f(x)=\sqrt{18 + 3x -x^{2}} + \sqrt{9-x^{2}} + \sqrt{9-6x+x^{2}} + \sqrt{9x-3x^{2}} \le 12 $$
I notice that when $$ x=0 \implies f(x) = 3\sqrt{2} + 3 + 3 \le 12 $$
and when $$ x=3 \implies f(x) = 3\sqrt{2} \le 12 $$, ... | $$f(x)=\sqrt{(6-x)(3+x)}+\sqrt{(3-x)(3+x)}+\sqrt{(3-x)^2}+\sqrt{3x(3-x)}$$
$\forall x\in [0,3]$, the factors $6-x$, $3+x$, $3-x$ and $3x$ in the radicands are non-negative.
By $GM\le AM$,
$$f(x)\le \frac{(6-x)+(3+x)}{2}+\frac{(3-x)+(3+x)}{2}+3-x+\frac{3x+(3-x)}{2}=12$$
Note that $12$ is not the global maximum, but a r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2046164",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to select at least one mango and one orange? A bag contains $4$ mangoes and $5$ oranges. In how many ways can I make a selection so as to take at least one mango and one orange?
In my book it is given $(2^4-1)(2^5-1)$
I understood $1$ is subtracted because if no mango is chosen. But why is it $2^4$ and $2^5$?
Pleas... | I guess that a selection here means a non-empty subset of fruits.
Now there are $2^9-1$ possible non-empty subsets out of a bag of $9$ fruits, $2^4-1$ contain only mangoes and $2^5-1$ contain only oranges.
What may we conclude?
P.S. Recall that if $S$ is a finite set with $n$ elements, then the number of subsets of $S$... | {
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Find the image and preimage of some functions I have functions
\begin{align}
f: \mathbb{R} \rightarrow \mathbb{R};&\ \ f(x) = x^2-4x-1, \\
g:\mathbb{C} \rightarrow \mathbb{C};&\ \ g(x) = x^5
\end{align}
and the set $D=\{z \in \mathbb{C} | |z|<1\}$.
I need to find $f((1,\infty))$, $f^{-1}((1,\infty))$ and $g(D)$. My i... | I agree with your instructor. His approach is reasonable here. Now a few hints.
1) Draw a graph of the parabola $f(x) = x^2-4x-1=(x-2)^2-5$.
What are your candidates for
$f((1,\infty))$ and $f^{-1}((1,\infty))$?
2) As regards $g(z)=z^5$ try to show that $g(D)=D$.
$g(D)\subseteq D$ because if $|z|<1$ then $|g(z)|=|z^... | {
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"answer_id": 1
} |
Evaluate the integral $\int \sqrt{x^2-2x-3}dx$ I'm asked to evaluate this integral: $\int \sqrt{x^2-2x-3} dx$
I don't see any other way to solve this except by trigonometric substitution, which is precisely what I did once I completed the square and got $\sqrt{4-(x-1)^2}$ as the integrand.
I then performed a substituti... | Another method
Integrate by parts: setting $t=x-1$, we come down to the integral $\;I=\displaystyle\int\sqrt{t^2-4}\,\mathrm d t$.
Set $u=\sqrt{t^2-4}$, $\mathrm dv=\mathrm dt$, whence
$\mathrm du=\dfrac{t\,\mathrm dt}{\sqrt{t^2-4}}, \enspace v=t$, and
\begin{align}
I&=t\sqrt{t^2-4}-\int\dfrac{t^2\,\mathrm dt}{\sqrt{t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2046517",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Constructing a Homothety This exercise has me stumped. I am meant to apply concepts concerning homotheties with circles to solve it. The problem states:
Given halflines k, l starting at a common point (let's call this point V), and a point P inside the angle formed by k and l, construct a circle through P tangent to ... | Consider the following figure:
Here are the steps of construction you have to follow:
*
*Construct the angle bisector of $k$ and $l$
*Take an arbitrary point (red thick) on the angle bisector.
*Drop a perpendicular to $l$ from the red point.
*Draw the red circle.
*Draw a line through $P$ and the intersection of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2046698",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Deep theorem with trivial proof
It is the snobbishness of the young to suppose that a theorem is trivial because the proof is trivial.
-- Henry Whitehead
I have been awestruck by the beauty of this quote.
What is in your opinion a good contender to exemplefy the meaning intended by Whitehead?
Off the top of my head I... | Russell's Paradox, that universal set comprehension is inconsistent with the rest of set theory, can be stated in one elegant line, has no hidden lemmas, and was the cause of arguably the single most profound investigation in the history of mathematics: the quest for formal axioms of set theory.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2046777",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "47",
"answer_count": 11,
"answer_id": 9
} |
$E(x) \equiv 17x + 4 \pmod{26}$ $E(x) \equiv 17x + 4 \pmod{26}$
the numbers 4,7 and 15 need deciphered
My answer spells ART correct me if wrong.
| Correct, $\,17x\!+\!4\,$ has inverse $\,17^{-1}(x\!-\!4)\equiv 12\!-\!3x\ $ by $\ \dfrac{1}{17}\equiv \dfrac{1}{-9}\equiv\dfrac{3}{-27}\equiv \dfrac{3}{-1}$
Therefore $\ 12\!-\!3x\ $ decodes $\, 4,7,15\, \mapsto\, 0,17,19\,\mapsto\, $ A,R,T in English
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2046871",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How can we evaluate this $\prod_{k=1}^n(1+kx)$ $\displaystyle\prod_{k=1}^n(1+kx)=\underbrace{\displaystyle\sum_{k=0}^n a_k x^k}_{\text{I assumed this,it don't have to be like this}}$
I'm investigating what this means, how we can analyse this and get generalized formula.
In fact ,I thought $n-$degree equaliton's fo... | Notice that
$$\prod_{k=1}^n(1+kx)=\prod_{k=0}^n(1+kx)=1\times(1+x)\times(1+2x)\times\dots\times(1+nx)\\=(1+nx)\overbrace{!!\dots!!}^x\text{ or }(1+nx)!^{(x)}$$
where the long string of exclamation marks is the multifactorial, the extension of the double factorial. The Wikipedia of the factorial also has a small sectio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2046990",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Prove that $A,B$ have a common eigenvector Let $A,B$ be $2\times2$ real matrices satisfying $\det(A)=\det(B)=1$ and $$\text{tr}(A)>2 , \text{tr}(B)>2, \text{tr}(ABA^{-1}B^{-1})=2$$
Prove that A,B have a common eigenvector.
| One possible approach:
*
*Show that each of $A$ and $B$ has distinct real positive eigenvalues.
*Hence $A$ is diagonalisable over $\mathbb R$. By a change of basis, we may assume that $A=\operatorname{diag}(p,\frac1p)$ for some $p>0$. Let $B=\pmatrix{a&b\\ c&d}$ in this basis.
*Use the given conditions and the ass... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2047101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Proof of $k {n\choose k} = n {n-1 \choose k-1}$ using direct proof I've seen many posts regarding a combinatorial proof of the following question. But for a non-combinatorial proof would the following method work? Also... is this the easiest way to arrive at a proof? It seems to be rather verbose.
Show the formula $k {... | The most straightforward proof uses analysis.
Remember the binomial coefficient $\dbinom nk$ is usually defined as the coefficient as the coefficient of $x^k$ in the expansion of $(1+x)^n$ as a product of $n$ factors:
$$ (1+x)^n=\sum_{k=0}^n\binom nk x^k. $$
Differentiate this relation:
$$n(1+x)^{n-1}=\begin{cases}\d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2047226",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Evaluating limits with several notations $$\lim_{x\to0} \frac{f(e^{5x} - x^2) - f(1)}{x}$$
It is known that $f'(1) = -2$
Given this info, I'm left with many questions. I'm going to assume that I'll want to substitute for something. I'll let $g(x) = e^{5x} - x^2$. But how do I incorporate the fact that $f'(1) = -2$? Sho... | $$\frac{f(e^{5x}-x^{2})-f(1)}{x}=\frac{f(1+e^{5x}-1-x^{2})-f(1)}{e^{5x}-1-x^{2}}\cdot\left(5\cdot\frac{e^{5x}-1}{5x}-x\right)$$ and this tends to $f'(1)\cdot 5=-10$ as $x\to 0$. The use of advanced tools like L'Hospital's Rule and Taylor series are mostly unnecessary for simple limit problems.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2047345",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
How many solutions does $x^2 + 3x +1 \equiv 0\, \pmod{101}$ have? $x^2 + 3x +1 \equiv 0 \pmod{101}$. To solve this I found the determinant $D = 5 \pmod{101}$). Using the Legendre symbol,
$$\left(\frac{5}{101}\right) = \left(\frac{101}{5}\right) \equiv \left(\frac{1}{5}\right) \equiv 1,$$
$\therefore$ The equations have... | If the discriminant of a quadratic is a nonzero square modulo odd prime $p$, then the quadratic has exactly two roots mod $p$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2047575",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Determining isomorphism of ring of fractions/quotients In a homework problem, I was asked to show that if $R=\mathbb{Z}_6$ and $S=\lbrace 2,4 \rbrace$, then $S^{-1}R\cong\mathbb{Z}_3$.
I was able to determine which fractions are equivalent and used that fact to develop the following function $f:\mathbb{Z}_3\to S^{-1}R$... | Hint: Use the Chinese Remainder Theorem. What happens if you localize $\mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z}/3\mathbb{Z}$ at $S$?
Full solution:
By the Chinese Remainder Theorem, we have $\mathbb{Z}/6 \mathbb{Z} \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$.
Since $2 = 0$ in $\mathbb{Z}/2\mathbb{Z}$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2047698",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Solutions of $x^3 + (x+4)^2 = y^2$ I want to solve the above equation in integers. I'm pretty sure the only solutions are $(x,y) = (0, \pm 4)$ but I'm not sure how to prove it.
| A subtle proof technique called proof by SageMath:
E = EllipticCurve([0,1,0,8,16])
E.integral_points(both_signs=True)
which gives:
[(0 : -4 : 1), (0 : 4 : 1)]
where the numbers in the constructor of the EllipticCurve are the Weierstrass coefficients of the curve. For reference, the Weirstrass coefficients are defin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2047812",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Transform permutation to another after at most $\dfrac{n(n-1)}{2}$ moves Let ($a_{1}, a_{2},..., a_{n}$) and ($b_{1}, b_{2}, ..., b_{n}$) be two different permutations of $n$ first natural numbers. Prove that we can transform one permutation to another using at most $\dfrac{n(n-1)}{2}$ transposition operations of two a... | You can use the logic applied in the bubble sort algorithm. First define the relation $$a_i < a_j \iff a_j \text{ appears before } a_i \text{ in the } b \text{ sequence}$$
Then take the first two elements of the $a$ sequence and swap them if $a_2 < a_1$. Eventually after checking $n-1$ pairs (which produce at most $n-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2047974",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Calculating the convergence radius of a power series I've tried to calculate the convergence radius of the following power series:
$$\sum_{n=1}^{\infty}\frac{3^n+4^n}{5^n+6^n}x^n$$
The Cauchy–Hadamard theorem doesn't help in this situation (I think).
So what I did is I tried to apply the d'Alembert ratio test to it and... | $$\begin{align}
\frac{(3^n+4^n)(5^{n+1}+6^{n+1})}{(5^n+6^n)(3^{n+1}+4^{n+1})} &=\frac{6\cdot24^n+6\cdot18^n+5\cdot20^n+5\cdot15^n}{4\cdot24^n+4\cdot20^n+3\cdot18^n+3\cdot15^n}\\
&=\frac{6+6(3/4)^n+5(5/6)^n+5(5/8)^n}{4+4(5/6)^n+3(3/4)^n+3(5/8)^n}\to\frac32
\end{align}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2048204",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
What is the number of integer solutions of the following equation ? Find the solution without computing 9 combinations. The equation is:
$$x_1+x_2+x_3+x_4+x_5+x_6<10$$
with $x_i\geq 0$ for $i=1,2,\dots,6$.
*
*What is the number of integer solutions of the following equation ?
*Find the solution without computing 9 ... | Write
$$x_1+x_2+...+x_6=9-y$$
with $y \ge 0$ and solve
$$x_1+x_2+...+x_6+y=9$$
for integer solution.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2048313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Anyone knows a transformation that can lump non-zero values in a matrix together? Suppose you have a matrix that contains a lot zeros (like spare matrix). Is there any known transformation that can gather non-zero values (e.g. into a corner of the matrix)?I am not sure but I think such transformation is very likely to ... | "Sparse matrix" is a loose term in numerical analysis and I don't think there is a definition for matrices containing "lots of zeros". One can talk about storage of such matrices and each way of storage can be viewed as a desired mapping.
One possible way maybe as follows.
Given an $n\times n$ matrix $(a_{ij})$, one ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2048490",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Find the last digit of $3^{1006}$ The way I usually do is to observe the last digit of $3^1$, $3^2$,... and find the loop. Then we divide $1006$ by the loop and see what's the remainder. Is it the best way to solve this question? What if the base number is large? Like $33^{1006}$? Though we can break $33$ into $3 \time... | $3^{1006}$ or $33^{1006}$
doesn't really matter
$33\equiv 3\pmod {10}\\
33^{1006}\equiv 3^{1006}\pmod {10}$
$3^4 = 81$
You might say this as $3^4\equiv 1 \pmod{10}$
The last digit of $3^n$ is the same last digit as $3^{n+4k}$
that is:
$(3^{n+4k}) = (3^n)(3^{4k})\equiv (3^n)(1) \pmod{10}$
$1006 = 251\cdot 4 + 2$
the las... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2048650",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 1
} |
Proving infinite subsets I have to prove:
An infinite subset of a denumerable set is denumerable.
I understand this has been asked before and I did take the time to read what was said there, but I do not understand still.
I have to prove this using other theorems about denumerable or countable sets, nothing too comp... | Suppose that the subset is not a countable infinite set, so if you prove that your subset $A$ must be more than countable in order to be infinite you have done, because then it can't be a subset of you denumerble set. Suppose that there exists an infinite set $B$ whose cardinality is less than the cardinality of the na... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2048814",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove that a function defined as series is continuous I want to show that the function $f(x) = \sum_{n = 1}^\infty\frac{x^{2n}}{n^24^n}$ is continuous on $(-2, 2)$. I've shown that the series converges on that interval, so how can I show that the function is continuous? Should I proceed straight from the definition? If... | HINT:
If a function is analytic, it is $C^\infty$ according to Taylor's theorem, and if it is smooth or $C^\infty$...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2048900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
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