Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
If all paths with the same endpoints are homotopic, then the space is simply connected. Let $X$ be a path connected space such that any two paths in $X$ having the same end points are path homotopic. Then prove that $X$ is simply connected.
I am totally stuck on this problem. Can someone help me please? Thanks for you... | If every two paths having the same end points are path homotopic, then every loop $w: S^1 \longrightarrow X$ can be deformed into a constant path, a point. Which is the definition of simply connected, isn't it?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/541064",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
If P = NP, then 3-SAT can be solved in P Prove that if $P = NP$, then there is an algorithm that can find a boolean assignment for a 3-SAT problem in P time if it exists.
$P = NP$ only says that we can decide whether a 3-SAT problem is satisfiable but it doesn't say anything about how to find a satisfying boolean expr... | Take your formula and check if it is satisfiable. If so, conjoin $x_1$ to your formula, where $x_1$ is your first variable, and check if it is still satisfiable. If so, then there is an assignment where $x_1$ is true; otherwise, there is one where $x_1$ is false.
Conjoin $x_1$ or $\neg x_1$ and repeat for all of the ot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/541114",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Evaluation of $\lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \ln \binom{2n}{n}$
Evaluate
$$\lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \ln \binom{2n}{n}.$$
$\underline{\bf{My\;\;Try}}::$ Let $\displaystyle y = \lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \ln \binom{2n}{n} = \lim_{n\rightarrow \infty}\frac{1}{2n}\cdot ... | Beside the elegant demonstration given by achille hui, I think that the simplest manner to solve this problem is to use Stirling approximation.
At the first order, Stirling's approximation is $n! = \sqrt{2 \pi n} (n/e)^n$. It is very good. Have a look at http://en.wikipedia.org/wiki/Stirling%27s_approximation. They ha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/541232",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Gröbner bases: Polynomial equations. Solution $x$ to $G \cap k[x_1, .., x_i]$ imply solution to $G \cap k[x_1, .., x_i, x_{i+1}]$, $x$ plugged in. I'm have been studying Gröbner bases for a while now and seen a few examples in my textbook / exercises.
Let $\mathcal k$ be a field and $\mathcal k[x_1,..,x_n]$ a polynomia... | For $\mathbb C$, the Extension Theorem tells us when we can extend a partial solution to a complete one.
Theorem 1 (The Extension Theorem)
Let $I=<f_1,...,f_s>\subset \mathbb C[x_1,...,x_n]$. Then for each $1\le i \le s$, write $f_i=g(x_2,...,x_n)x_1^{N_i} + $ terms in which $x_1$ has degree $< N_i$ where $N_i\ge 0$ a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/541316",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Prove one-to-one function Let $S$ be the set of all strings of $0$'s and $1$'s, and define $D:S \rightarrow \mathbb{Z}$ as follows: For all $s\in S$,
$D(s)= \text{the number of}\,\, 1$'s in $s$ minus the number of $0$'s in $s$.
a. Is $D$ one-to-one(injective)? Prove or give counterexample if it is false.
b. Is $D$ ont... | a) Yes, the function is indeed not injective, and your idea is correct!
b) You have to show that, given an arbitrary integer $n$, then you can find a string $S$ such that $D(S)=n$. For example, if $n=3$, then take $S=111$...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/541407",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Extending a connected open set Assume $\emptyset\neq V\subseteq U\subseteq\mathbb{R}^n$ are open and connected sets so that $U\setminus\overline{V}$ is connected as well. Given any point $x\in U$, is there always a connected open set $W\subseteq U$ so that $\{x\}\cup V\subseteq W$ and $U\setminus\overline{W}$ is connec... | There are two cases to consider:
*
*$n\ge 2$. Observe that in this case, for every open connected set $S\subset R^n$, any $a\in A$ and any sufficiently small $r\ge 0$, the complement of the closed ball
$$
A\setminus \overline{B(a, r)}
$$
is still connected. Now, if $V$ is dense in $U$ then the only meaningful answe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/541532",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
logarithm equation with different bases. Why is this like it is? :D $$\dfrac{1}{\log_ae} = \ln(a)$$ I'm solving some exercises and I ran up to this? Maybe it's really banal, but please explain me...
| We can prove $$\log_ab=\frac{\log_cb}{\log_ca}$$ where $a,c >0,\ne1$
$$\implies \log_ab\cdot\log_ba=\cdots=1$$
and conventionally Natural logarithm is written as $\displaystyle \ln a$ which means $\displaystyle \log_ea$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/541604",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
An overview of analysis I'm looking for a book that gives an overview of analysis, a bit like Shafarevich's Basic Notions of Algebra but for analysis. The book I have in mind would give definitions, theorems, examples, and sometimes sketches of proofs. It would cover a broad swathe of analysis (real, complex, functiona... | Loomis & Sternber's Advanced Calculus is available online.
It is a classic that goes well beyond what people normally call calculus
(differential equations, differential geometry, variational principles, ...).
Personally I really like Sternberg's books,
but it is a full blown textbook rather than a survey.
Or maybe Ale... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/541684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 6,
"answer_id": 0
} |
Maximum N that will hold this true Find the largest positive integer $N$ such that
$$\sqrt{64 + 32^{403} + 4^{N+3}}$$
is an integer
Is $N = 1003$?
| Note that with $N=2008$ we have $ (2^{N+3}+8)^2=4^{N+3}+2\cdot 8\cdot 2^{N+3}+64=4^{N+3}+2^{2015}+64=64+32^{403}+4^N,$ so we conjecture that the maximal value is $2008$.
If $2^{2015}+2^6+2^{2N+6}$ is a perfect square then also $\frac1{64}$ of it, i.e. $2^{2009}+1+2^{2N}=m^2$ for some $m\in\mathbb N$.
But if $N> 2008$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/541756",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
basic induction probs Hello guys I have this problem which has been really bugging me. And it goes as follows:
Using induction, we want to prove that all human beings have the same hair colour. Let S(n) be the statement that “any group of n human beings has the same hair colour”.
Clearly S(1) is true: in any grou... | The base case is correct.
Inductive step:
Assume that the result is true for $n = k$, which is to say that everybody in a group of $k$ people has the same hair colour.
For the proof to work, you now have to prove that $P(k)$ implies $P(k+1)$, but it doesn't. Just because it's true that for any group of $k$ people, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/541852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Prove that a group of order 30 has at least three different normal subgroups Prove that a group of order 30 has at least three different normal subgroups.
Prove:
$30=2\cdot3\cdot5$
There are $2$-Sylow, $3$-Sylow and $5$-Sylow subgroups. If $t_p$= number of $p$-Sylow-subgroups. Then $t_2$=$1$, $3$, $5$, $15$ and $t_3$=$... | Let $|G|=30$. Assume that it has no nontrivial normal subgroups. Than there must exist more than one 5-sylow subgroup and more than one 3-sylow subgroup. By the sylow theorems, you can then prove that there must be 10 3-sylow subgroups and 6 5-sylow subgroups. These subgroups intersect trivially, i.e., their intersecti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/541946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Find the coordinates of intersection of a line and a circle
There is a circle with a radius of $25$ ft and origin at $(0, 0)$ and a line segment from (0, -31) to (-37, 8). Find the intersections of the line and circle.
I am asking for somebody to analyze what I am doing wrong in calculating the answer, given the que... | The line equation is $y=(-39/37)x-31.$ To get the closest point on this line to the origin, intersect it with the perpendicular to it from $(0,0),$ which has equation $y=(37/39)x$. The nearest point on the line to $(0,0)$ is then seen to be $(a,b)$ where $a=-44733/2890,\ b=-42439/2890.$ The distance to the origin is th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/542040",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
finding determinant as an function in given matrix Calculate the determinant of the following matrix as an explicit
function of $x$. (It is a polynomial in $x$. You are asked to find
all the coefficients.)
\begin{bmatrix}1 & x & x^{2} & x^{3} & x^{4}\\
x^{5} & x^{6} & x^{7} & x^{8} & x^{9}\\
0 & 0 & 0 & x^{10} & x^{11}... | Another way to look at this: the bottom three rows can't have rank more than $2$, since they have only two nonzero columns, so the whole matrix can't have rank more than $4$, and therefore is singular.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/542148",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Taylor polynomial about the origin Find the 3rd degree Taylor polynomial about the origin of $$f(x,y)=\sin (x)\ln(1+y)$$
So I used this formula to calculate it $$p=f(0,0)+(f_x(0,0)x+f_y(0,0)y)+(\frac{1}{2}f_{xx}(0,0)x^2+f_{xy}(0,0)xy+\frac{1}{2}f_{yy}(0,0)y^2)+(\frac{1}{6}f_{xxx}(0,0)x^3+\frac{1}{2}f_{xxy}(0,0)x^2y+\f... | The answers are the same. $\ln(1) = 0$. And yes, your technique is correct.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/542237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Do there exist functions such that $f(f(x)) = -x$? I am wondering about this. A well-known class of functions are the "involutive functions" or "involutions", which have that $f(f(x)) = x$, or, equivalently, $f(x) = f^{-1}(x)$ (with $f$ bijective).
Now, consider the "anti-involution" equation $f(f(x)) = -x$. It is poss... | Your primary question has been asked and answered already.
Your follow-up question can be answered by similar means. The group structure on $G$ is actually irrelevant: all you're actually using is that each element is paired with an inverse.
Another way to phrase this question is that you have a group action of the cyc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/542312",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
union of two contractible spaces, having nonempty path-connected intersection, need not be contractible show that union of two contractible spaces, having nonempty path-connected intersection, need not be contractible.
can someone give me a proper example please.I could not remind anything.
| Consider sphere $S^2$ with two open subsets $U,V$, s.t. $U$ contains everything but the south pole, $V$ contains everything but the north pole. They are both contractible, their intersection is homotopic to the circle, which is path connected, but their union is $S^2$, not contractible.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/542464",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Multiplicative Inverses in Non-Commutative Rings My abstract book defines inverses (units) as solutions to the equation $ax=1$ then stipulates in the definition that $xa=1=ax$, even in non-commutative rings. But I'm having trouble understanding why this would be true in the generic case.
Can someone help me understand?... | Note: In a ring R with 1, if EVERY non-zero element x has a left inverse, then R is a division ring (so ax=1 implies xa=1).
More generally, if EVERY non-zero element has either a left or a right inverse, then R is a division ring.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/542552",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
absolute value inequalities When answer this kind of inequality
$|2x^2-5x+2| < |x+1|$
I am testing the four combinations when both side are +, one is + and the other is - and the opposite and when they are both -.
When I check the negative options, I need to flip the inequality sign?
Thanks
| Another way - no tricks, just systematically looking at all cases, where we can write the inequality without the absolute value sign, to convince you that all possibilities are covered..
Following the definition of the absolute value function, RHS is easy to rewrite as,
$$|x+1| = \begin{cases} x+1 & x \ge -1\\ -x-1 &x ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/542636",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Why do we subtract 1 when calculating permutations in a ring?
$10$ persons are to be arranged in a ring shape. Number of ways to do that is $9!.$
I wonder why we subtarct $1$ in all such cases.
I can imagine that if A,B,C,D are sitting in a row then B,C,D,A would give me a different combination but had they been sitt... | Imagine that the table is arranged so that one of the seats is due north of the centre of the table. Seat the $n$ people around the table. Let $p_1$ be the person sitting in that north seat, and let the other $n-1$ be $p_2,p_3,\ldots,p_n$ clockwise around the table. Now rotate the table and the seats one place counterc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/542750",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Approximation of pi Given that $\frac{\pi^2}{6}=\sum_{n=1}^{\infty}\left(\frac{1}{n^2}\right)$, I have to write a program in C that finds an approximation of $\pi$ using the formula $S_n=\sum_{i=1}^{n}\left(\frac{1}{i^2}\right)$.
Then the approximation is: $\sqrt{6\cdot S_n}$
Could you tell me the result for $n=100$ so... | Mathematica is a wonderful tool for these kinds of computations; it is worth learning how to use it, and its baby brother Wolfram Alpha.
Here is what I get: link
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/542937",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Difficult Derivative? I'm in a single-variable calculus course, in which we recently covered logarithmic differentiation. The professor proved it that works when $f(x)>0$, and when $f(x)<0$. I've been trying to find a way to derive that kind of function when $f(x)=0$, but I'm not sure if it's possible, or what. I've... | Graph of $f(x)$:
Graph of $f'(x)$:
See both the above graphs.
$f(x)$ is actually not differentiable at $x= 1.5π$. The graph of $f'(x)$ at $x = 1.5π$ is a vertical asymptote. The function's second differential may say that it is increasing/decreasing at $x= 1.5π$ but the first derivative doesn't exist.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/543174",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 8,
"answer_id": 3
} |
Differential equation with infinitely many solutions The problem is to solve for $-1<x<1$ $$y'(x)=\frac{4x^3y(x)}{x^2+y(x)^2}$$ with $y(0)=0$.
I need to show that this equation has infinitely many solutions. Note that $\frac{4y(x)x^3}{x^2+y(x)^2}$ is undefined for $y(0)=0$, but note that $\frac{4x^3y(x)}{x^2+y(x)^2}=2x... | As you observed, the function
$$F(x,y)=\frac{4x^3y}{x^2+y^2},$$
extended by $F(0,0)$, is continuous. Also, its partial derivative with respect to $y$ is bounded near the origin:
$$\left|\frac{\partial F}{\partial y}\right| =
\left|\frac{4x^3(x^2-y^2)}{(x^2+y^2)^2}\right| = 4|x| \frac{x^2|x^2-y^2|}{(x^2+y^2)^2}\le 4|... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/543250",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Any suggestions about good Analysis Textbooks that covers the following topics? I am an undergraduate math major student. I took two courses in Advanced Calculus (Real Analysis): one in Single variable Analysis, and the second in Multivariable Analysis. We basically used Rudin's Book "Principles of Mathematical Analysi... | T. Tao, Analysis II covers the topic that you need.
But for the measure theory, I think "Paul R. Halmos,measure theory" is good.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/543338",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Proving a Property of a Set of Positive Integers I have a question as such:
A set $\{a_1, \ldots , a_n \}$ of positive integers is nice iff
there are no non-trivial (i.e. those in which at least one component
is different from $0$) solutions to the equation $$a_1x_1 + \ldots +
> a_nx_n = 0$$ with $x_1 \ldots x_n \... | Hint: If the sums of two distinct subsets are equal, then the set is not nice.
There are $2^n$ subsets. If the numbers are all $\lt \frac{2^n}{n}$, then the sum of all the numbers is less than $2^n$. Now use the Pigeonhole Principle.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/543446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
$\cos x+\cos 3x+\cos 5x+\cos 7x=0$, Any quick methods? How to solve the following equation by a quick method?
\begin{eqnarray}
\\\cos x+\cos 3x+\cos 5x+\cos 7x=0\\
\end{eqnarray}
If I normally solve the equation, it takes so long time for me.
I have typed it into a solution generator to see the steps. One of the step... | Use product identity $2\cos(x)\cos (y)=\cos(x+y)+\cos(x-y)$
So $\cos x+\cos7x=2\cos4x\cos3x$
And $\cos3x+\cos5x=2\cos4x\cos x$.
Factoring out $2\cos4x$, we get $2\cos4x (\cos3x+\cos x)=0$.
You can solve the 1st factor now right? For the second factor, use the above identity again and you will be done.:)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/543506",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 0
} |
In a non-abelian group, if $C(a)=\langle a\rangle$ then $a\not\in Z(G)$. Suppose $G$ is a non-abelian group and $a∈G$. Prove that if $C(a)= \langle a \rangle$ then $a\not\in Z(G)$. I just don't understand this proof at all. Would someone mind walking me through the entire proof?
| Since $C(a) = \langle a\rangle$ is abelian and $G$ is not, $C(a) \neq G$. Now $a\notin Z(G)$, because otherwise by definition $C(a) = G$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/543567",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
What is the order of the sum of log x? Let
$$f(n)=\sum_{x=1}^n\log(x)$$
What is $O(f(n))$?
I know how to deal with sums of powers of $x$. But how to solve for a sum of logs?
| Using Stirling's formula we have, for $n$ sufficiently large
$$
f(n)=\sum_{k=1}^n\log k=\log(n!)\simeq\log(\sqrt{2\pi}e^{-n}n^{n+1/2}).
$$
Hence
$$
O(f(n))=\log(\sqrt{2\pi}e^{-n}n^{n+1/2}).
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/543634",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
factoring cubic polynomial equation using Cramer's rule. 1) I have question about factoring cubic polynomials. In my note it says "Any polynomial equation with positive powers whose coefficients add to 0 will have a root of 1. Another, if sum of the coefficients of the even powers = sum of coefficient of the odd powers... | If the root is 1 the polynomial is divisible by x -1; if the root is -1 the polynomial is divisible by x + 1. Unfortunately I'm getting k = -15 for the root of 1: adding the coefficients we have 1 + 17 -3 + k = 0 so 15 + k = 0. and k = -15. I checked it out by actually dividing through by x -1.
For the second the coe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/543700",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Maximum Likelihood Find the maximum likelihood estimator of $f(x|\theta) = \frac{1}{2}e^{(-|x-\theta|\:)}$,
$-\infty < x < \infty$ ; $-\infty < \theta < \infty$.
I am confused of how to deal with the absolute value here.
| The likelihood function $$L(x;\theta)=\prod\limits_{i=1}^n\frac{1}{2}e^{-|x_i-\theta|}=\frac{1}{2^n}e^{-\sum\limits_{i=1}^n|x_i-\theta|}$$
$$\ln L(x;\theta)=-n\ln2-\sum_{i=1}^{n}|x_i-\theta|$$
$$\frac{\partial\ln L(x;\theta)}{\partial\theta}=\sum_{i=1}^n \text{sign } (x_i-\theta)$$ because $|x|'=\text{sign }x,x\ne0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/543765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Proof by cases, inequality I have the following exercise:
For all real numbers $x$, if $x^2 - 5x + 4 \ge 0$, then either $x \leq 1$ or $x \geq 4$.
I need you to help me to identify the cases and explain to me how to resolve that. Don't resolve it for me please.
| HINT:
If $(x-a)(x-b)\ge0$
Now the product of two terms is $\ge0$
So, either both $\ge0$ or both $\le0$
Now in either case, find the intersection of the ranges of $x$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/543831",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Show there is a closed interval $[a, b]$ such that the function $f(x) = |x|^{\frac1{2}}$ is continuous but not Lipschitz on on $[a, b]$. Hi guys I was given this as an "exercise" in my calculus class and we weren't told what a Lipschitz is so i really need some help, heres the question again:
Show there is a closed i... | Consider the interval $[0,1]$, then clearly $f(x) = x^{1/2}$ is continuous.
Can you show that $f$ is not Lipschitz on $[0,1]$? (Hint: Use the Mean-Value theorem on a closed sub-interval of $(0,1/n]$)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/543936",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Exponential of the matrix I want to calculate the matrix exponential $e^{tA}$ of the matrix with the first row being $(0,1)$ and the second $ (-1,0)$. It would be sufficient if you would me the most important steps.
| Firstly, you should expand an exponent in Taylor series.
Then, you should understand, what happens with matrix, when it is exponentiated with power n.
The last step is to sum up all the matrices and realize, if there are Teylor series of some functions as an entries of the aggregate matrix.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/543992",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 0
} |
$(A\cap B)\cup C = A \cap (B\cup C)$ if and only if $C \subset A$ I have a set identity: $(A \cap B) \cup C = A \cap (B \cup C)$ if and only if $C \subset A$.
I started with Venn diagrams and here is the result:
It is evident that set identity is correct. So I started to prove it algebraic:
1) According to distributiv... | Here is a full algebraic proof. Let's first expand the definitions:
\begin{align}
& (A \cap B) \cup C = A \cap (B \cup C) \\
\equiv & \;\;\;\;\;\text{"set extensionality"} \\
& \langle \forall x :: x \in (A \cap B) \cup C \;\equiv\; x \in A \cap (B \cup C) \rangle \\
\equiv & \;\;\;\;\;\text{"definition of $\;\cap\;$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/544071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
If $R,S$ are reflexive relations, so are $R \oplus S$ and $R \setminus S$?
Suppose $R$ and $S$ are reflexive relations on a set $A$. Prove or disprove each of these statements.
a) $R\oplus S$ is reflexive.
b) $R\setminus S$ is reflexive.
I think both of a) and b) are false, but I'm having trouble with coming up with ... | Hint: What does it mean to say $(x,x) \in R \oplus S$, respectively $(x,x) \in R \setminus S$? Can both $R$ and $S$ be reflexive if this is the case?
The above amounts to a proof by contradiction. But we can avoid this; for example, by the following argument:
Let $x \in A$. We know that $(x,x)\in R$ and $(x,x) \in S... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/544142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Finite Sum $\sum\limits_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}$
Question : Is the following true for any $m\in\mathbb N$?
$$\begin{align}\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}\qquad(\star)\end{align}$$
Motivation : I reached $(\star)$ by using computer. It seems true, but I can't prove it... | (First time I write in a math blog, so forgive me if my contribution ends up being useless)
I recently bumped into this same identity while working on Fourier transforms. By these means you can show in fact that
$$\sum_{k=-\infty}^{+\infty}\frac{1}{(x-k)^2}=\frac{\pi^2}{\sin^2 (\pi x)}.\tag{*}\label{*}$$
Letting $x=\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/544228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "49",
"answer_count": 8,
"answer_id": 1
} |
How many ways can 8 children facing each other in a circle change seats so that each faces a different child. Need some help with this problem. A carousel has eight seats each representing a different animal. Eight children are seated on the carousel but facing inward, so each child is staring at another. In how many ... | I calculate 23040 ways.
The way I see the problem, we have to assign 4 pairs of children (sitting opposite to each other) to 4 distinct slots.
First let us calculate the number of way to seat the children, if we fix the pairs of oppositing children. Then the pairs could be assigned in $4! = 24$ ways to the slots. Sinc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/544317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Cohomological definition of the Chow ring Let $X$ be a smooth projective variety over a field $k$. One can define the Chow ring $A^\bullet(X)$ to be the free group generated by irreducible subvarieties, modulo rational equivalence. Multiplication comes from intersection. The problem is, verifying that everything is wel... | Another result I just ran across. In Hulsbergen's book Conjectures in arithmetic algebraic geometry he mentions the following theorem "of Grothendieck."
For a general ringed space, let
$$
K_0(X)^{(n)} = \{x\in K_0(X)_{\mathbf Q}:\psi^r(x) = r^n x\text{ for all }r\geqslant 1\}
$$
where $\psi^r$ is the $r$-th Adams o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/544383",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
Completely baffled by this question involving putting matrices in matrices This is homework, so only hints please.
Let $A\in M_{m\times m}(\mathbb{R})$
, $B\in M_{n\times n}(\mathbb{R})$
. Suppose there exist orthogonal matrices $P$
and $Q$
such that $P^{T}AP$
and $Q^{T}BQ$
are upper triangular. ... | Take
$$R = \operatorname{diag}(P,Q) = \begin{bmatrix} P \\ & Q \end{bmatrix},$$
and see what happens. Ask if you get stuck.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/544477",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Quotient is principal Let $R$ be a finite commutative ring, let $J$ be a maximal ideal of $R$ and $n$ some positive integer greater or equal than $2$. Is it always true that every ideal of the quotient $R/J^{n}$ is principal?
| No. For instance if $R$ is local then $J^n = 0$ for sufficiently large $n$ and then you are just asking whether a finite, local commutative ring must be principal. The answer is certainly not.
Examples have come up before on this site. One natural one is $R = \mathbb{F}_p[x,y]/\langle x,y \rangle^2$. This is someho... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/544550",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Upper bound for $(1-1/x)^x$ I remember the bound $$\left(1-\frac1x\right)^x\leq e^{-1}$$ but I can't recall under which condition it holds, or how to prove it. Does it hold for all $x>0$?
| Starting from $e^x \geq 1+x$ for all $x \in \mathbb{R}$:
For all $x \in \mathbb{R}$
$$ e^{-x} \geq 1-x. $$
For all $x \neq 0$
$$ e^{-1/x} \geq 1-\frac{1}{x}. $$
And, since $t \mapsto t^x$ is increasing on $[0,\infty)$, for $x \geq 1$
$$ e^{-1} \geq \left(1-\frac{1}{x}\right)^x. $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/544667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Arithmetic sequence in a Lebesgue measurable set Let $A\subseteq[a,b]$ be Lebesgue measurable, such that: $m(A)>\frac{2n-1}{2n}(b-a)$. I need to show that $A$ contains an arithmetic sequence with n numbers ($a_1,a_1+d,...,a_1+(n-1)*d$ for some d).
I thought about dividing [a,b] into n equal parts, and show that if I pu... | Hint: You are on the right track. Have you noticed that the length of each of
your sub-intervals is $\frac{b-a}{n}$, while the total length of all the missing pieces is only $\frac{b-a}{2n}$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/544763",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
} |
What is a convex optimisation problem? Objective function convex, domain convex or codomain convex? My teacher in the course Mat-2.3139 did not want to answer this question because it would take too much time. So what does a convex optimisation problem actually mean? Convex objective function? Convex domain or convex c... |
I am not yet sure whether it is a general term for all kind of "something-convex" problems or a specific term to certain mathematical problems.
It could be both: some people, like your teacher, may decide to use it as a general term for "something-convex" in it, while others stick to a precise interpretation. I prefe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/544847",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Example involving Uniform Continuity Question: Could someone give an example of a sequence of uniformly continuous real-valued functions on the reals such that they converge point-wise to a function that is continuous but not uniform continuous.
My attempt so far: I managed to prove this is true in the case of uniform... | I'm not convinced this is the simplest example (I certainly wouldn't want to make it explicit), but it was fun :)
Lines are uniformly continuous and quadratics are not. Also, continuous implies is uniformly continuous on a compact domain. In addition, it's not too hard to show that if you glue two uniformly continuous ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/544941",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Assignment: determining sets are bases of $\mathbb{R}^3$ This is question from an assignment I'm working on:
Which two of the following three sets in $\mathbb{R}^3$ is a basis of $\mathbb{R}^3$?
\begin{align*}
B_1&=\{(1,0,1),(6,4,5),(-4,-4,7)\}\\
B_2&=\{(2,1,3),(3,1,-3),(1,1,9)\}\\
B_3&=\{(3,-1,2),(5,1,1),(1,1,1)\}
\e... | Let V={$v_1,v_2,...,v_n$} be a set of n vectors in $\mathbb{R}^n$. then V is linearly independent iff the matrix A=($v_1,v_2,...,v_n$) is invertible where v_i is the ith column of A . proving this result is an easy exercise
.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/545029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Three Variables-Inequality with $a+b+c=abc$ $a$,$b$,$c$ are positive numbers such that $~a+b+c=abc$
Find the maximum value of $~\dfrac{1}{\sqrt{1+a^{2}}}+\dfrac{1}{\sqrt{1+b^{2}}}+\dfrac{1}{\sqrt{1+c^{2}}}$
| Trigonometric substitution looks good for this, especially if you know sum of cosines of angles in a triangle are $\le \frac32$. However if you want an alternate way...
Let $a = \frac1x, b = \frac1y, c = \frac1z$. Then we need to find the maximum of
$$F = \sum \frac{x}{\sqrt{x^2+1}}$$
with the condition now as $xy + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/545107",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Braid Groups Mapped to Symmetric Groups How can I construct five elements in terms of the Braid Generators $\sigma_1 \sigma_2$ that are in the kernel of the homomorphism from the braid group on three strands to the symmetric group on three letters? I tried, but my braids keep turning out to be the identity braid.
| By definition, the kernel of the canonical homomorphism $B_n\to S_n$ is the pure braid group on $n$ letters $P_n$, which is the group of all braids whose strings start and end on the same point of the disk (they don't permute the end points of the strings). Given this, you should be able to see that elements like $\sig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/545184",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Stochastic differential Im really new in the stochastic procceses please help me.
How can I solve this stochastic differential equation?
$$dX = A(t)Xdt$$
$$X(0) = X_0$$
If $A$:[0,$\infty$]$\to$ $R$ is continous and $X$ is a real random variable.
| The (deterministic) ODE
$$\frac{dx(t)}{dt} = A(t) \cdot x(t) $$
can be solved by transforming it into
$$\log x(t)-c = \int^t \frac{1}{x(s)} dx(s) = \int^t A(s) \, ds$$
The same approach applies to this stochastic differential equation. We use Itô's formula for $f(x) := \log x$ and the Itô process $X_t$ and obtain
$$\lo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/545424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
What do $x\in[0,1]^n$ and $x\in\left\{ 0,1\right\}^n$ mean? $x\in[0,1]^n$
$x\in\{0,1\}^n$
Thank you in advance.
| The first entry is an interval $ [0, 1] $ in the Reals raised to the power n. E. g. for n=2, you can think of this as the unit square. I.e. $ x \in [0, 1]^2 $ iff it is in the unit square
The second entry {0, 1} is simply the set containing only 0 and 1. Picturing our n=2 square from the fist example, this set describe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/545520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Recovering vector-valued function from its Jacobian Matrix Consider a function $f:\Omega\subset\mathbb{R}^n\rightarrow\mathbb{R}^m$, for which the Jacobian matrix
$J_f(x_1,...,x_n)= \left( \begin{array}{ccc}
\frac{\partial f_1}{\partial x_1} & ... & \frac{\partial f_1}{\partial x_n} \\
\vdots & & \vdots \\
\frac{\pa... | Your question is very closely related to:
*
*Frobenius integrability theorem
*Integrability conditions for differential systems
I suspect that the first reference will be of most use.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/545634",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
evaluating $\sum_{n=1}^\infty \frac{n^2}{3^n}$ I am trying to compute the sum $$\sum_{n=1}^\infty \frac{n^2}{3^n}.$$
I would prefer a nice method without differentiation, but if differentiation makes it easier, then that's fine.
Can anyone help me?
Thanks.
| We have
\begin{gather*}
S=\sum_{n=1}^{\infty}\frac{n^2}{3^n}=\frac{1}{3}\sum_{n=0}^{\infty}\frac{(n+1)^2}{3^{n-1}}=\frac{1}{3}\left(\sum_{n=0}^{\infty}\frac{n^2}{3^{n-1}}+2\sum_{n=0}^{\infty}\frac{n}{3^{n-1}}+\sum_{n=0}^{\infty}\frac{1}{3^{n-1}}\right)=\frac{1}{3}\left(S+2 S_1+S_2\right).
\end{gather*}
Then
$$
S_2=\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/545708",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
For sets $A,B,C$, $(A\setminus B)\subset (A\setminus C)\cup (C\setminus B)$ First of all, I am sorry for my bad english, I am from Brazil :-) I have problem with proof for some set theory task.
Here it is:
$A,B,C$ are three sets. Show that:
$$(A\setminus B) \subset (A\setminus C) \cup (C\setminus B)$$
It is clear by ... | If you take out from $A$ things that are in $B$, what is left is certainly things in $A$ not in $C$, except for things in $C$ that were not in $B$, so if you add the latter you are all set. The formula is nothing more than this.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/545794",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
How to prove that $\sum_{k=0}^n \binom nk k^2=2^{n-2}(n^2+n)$ I know that $$\sum_{k=0}^n \binom nk k^2=2^{n-2}(n^2+n),$$ but I cannot find a way how to prove it. I tried induction but it did not work. On wiki they say that I should use differentiation but I do not know how to apply it to binomial coefficient.
Thanks f... | This is equivalent to proving $$\sum_{k=0}^n k^2 {n\choose k}=n(n+1)2^{n-2}.$$ Given $n$ people we can form a committee of $k$ people in ${n\choose k}$ ways. Once the committee is formed we can pick a committee leader and a committee planner. If we allow each person to hold both job titles there are $k$ ways for this t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/545879",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
When is $\binom{n}{k}$ divisible by $n$? Is there any way of determining if $\binom{n}{k} \equiv 0\pmod{n}$. Note that I am aware of the case when $n =p$ a prime. Other than that there does not seem to be any sort of pattern (I checked up to $n=50$). Are there any known special cases where the problem becomes easier... | Well, $n = p^2$ when $k$ is not divisible by $p.$ Also $n=2p$ for $k$ not divisible by $2,p.$ Also $n=3p$ for $k$ not divisible by $3,p.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/545962",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "31",
"answer_count": 3,
"answer_id": 0
} |
Is the set of all sums-of-rationals-that-give-one countable? Some (but not all) sums of rational numbers gives us 1 as a result. For instance:
$$\frac12 + \frac12 = 1$$
$$\frac13 + \frac23 = 1$$
$$\frac37 + \frac{3}{14} + \frac{5}{14} = 1$$
Is the set of all of these sums countable?
Sums that differ just by their adde... | Since the set of rationals is countable, the set of all finite sets of rationals is countable, and therefore the set of all finite sums of rationals is countable. In particular, the set of finite sets of rationals with sum $1$ is countable. An explicit bijection between this set and $\Bbb N$ would be messy.
Added: If y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/546043",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
About the Collatz conjecture I worked on the Collatz conjecture extensively for fun and practise about a year ago (I'm a CS student, not mathematician). Today, I was browsing the Project Euler webpage, which has a question related to the conjecture (longest Collatz sequence). This reminded me of my earlier work, so I w... | The answer to your question is yes; one can work the Collatz relation backwards to build numbers that last arbitrarily long (for a simple example, just take powers of 2). However the purpose of the wikipedia records is to see if we can find SMALL numbers that last a long time.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/546140",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Find the cusps for the congruence subgroup $\Gamma_0(p)$ Find the cusps for the congruence subgroup $\Gamma_0(p)$. How does one go about doing this? I know the definition of a cusp - the orbit for the action of $G$, in this case $\Gamma_0(p)$, on $\mathbb Q\cup\{\infty\}$, but I don't see how to get myself started thou... | first of all, you can try to see who is in the orbit of $\infty$.
If $z$ is in the orbit of $\infty$ then there exist a matrix on $\Gamma_0(p)$ say $\gamma=\left(\begin{matrix} a & b \\ pc & d\end{matrix}\right)$ such that $\gamma \infty =z$. This implies $\frac{a}{pc}=z$, then the numbers in the orbit of infinity are... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/546209",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Graph Ramsey Theory for Multiple Copies of Graphs I had the following question from Graph Ramsey theory. Show that if $m \geq 2$, then $$ R((m+1)K _{3},K _{3})\geq R(mK _{3},K _{3}) + 3. $$
Thanks.
| In
S. A. Burr, P. Erdõs, and J. H. Spencer. Ramsey theorems for multiple copies of graphs, Trans. Amer. Math. Soc., 209 (1975), 87-99. MR53 #13015,
it is shown that if $m\ge 2$ and $m\ge n\ge 1$, then $r(mK_3,nK_3)=3m+2n$. This gives your result immediately by taking $n=1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/546270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Evaluating $\lim_{n\rightarrow\infty}x_{n+1}-x_n$
Let $f(x)$ be continuously differentiable on $[0,1]$ and
$$x_n = f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+f\left(\frac{3}{n}\right)+\ldots+f\left(\frac{n-1}{n}\right)$$
Find $$\lim_{n\rightarrow\infty}\left(x_{n+1}-x_n\right)$$
Confusion: I just found a... | Because $f(x)$ is continuously differentiable on $[0,1]$, tben by the mean value theorem:
$$f\left(\frac{i}{n+1}\right)-f\left(\frac{i}{n}\right) = f'(\xi_i)(\frac{i}{n+1} - \frac{i}{n}) \text{ where } \xi_i \in \left[\frac{i}{n+1},\frac{i}{n}\right] \tag{1}$$
Then, by the given formula of $x_n$, we have
\begin{al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/546380",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
At what time is the speed minimum? The position function of a particle is given by $r(t) = \langle-5t^2, -1t, t^2 + 1t\rangle$. At what time is the speed minimum?
| The velocity vector is $\left<-10t,-1,2t+1\right>$. Thus the speed is $\sqrt{(-10t)^2 +(-1)^2+(2t+1)^2}$.
We want to minimize this, or equivalently we want to minimize $104t^2+4t+2$. This is a problem that can even be solved without calculus, by completing the square.
If negative $t$ is allowed, we get $t=-\frac{4}{20... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/546436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Cesaro mean approaching average of left and right limits
Let $f\in L^1(\mathbb{R}/2\pi\mathbb{Z})$, where $\mathbb{R}/2\pi\mathbb{Z}$ means that $f$ is periodic with period $2\pi$. Let $\sigma_N$ denote the Cesaro mean of the Fourier series of $f$. Suppose that $f$ has a left and right limit at $x$. Prove that as $N$ ... | Note that you can write $$(f*F_N)(x)-\frac{f(x^+)+f(x^-)}{2}=$$$$\int_{-\pi}^{0}f(x-y)F_N(y)\,dy-\frac{f(x^+)}{2}+\int_{0}^{\pi}f(x-y)F_N(y)\,dy-\frac{f(x^-)}{2}=$$$$\int_{-\pi}^{0}(f(x-y)-f(x^+)F_N(y)\,dy+\int_{0}^{\pi}(f(x-y)-f(x^-))F_N(y)\,dy,$$ since $F_N$ is symmetric around $0$, so its integral from $-\pi$ to $0$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/546529",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How prove this inequality $\frac{x}{x^3+y^2+z}+\frac{y}{y^3+z^2+x}+\frac{z}{z^3+x^2+y}\le 1$ for $x+y+z=3$ let $x,y,z$ be positive numbers,and such $x+y+z=3$,show that
$$\dfrac{x}{x^3+y^2+z}+\dfrac{y}{y^3+z^2+x}+\dfrac{z}{z^3+x^2+y}\le 1$$
My try:$$(x^3+y^2+z)(\dfrac{1}{x}+1+z)\ge 9$$
so
$$\dfrac{x}{x^3+y^2+z}+\dfrac{y... | Note that $2+x^3=x^3+1+1\geqslant 3x$, $y^2+1\geqslant 2y$, thus$$\dfrac{x}{x^3+y^2+z}=\frac{x}{3+x^3+y^2-x-y}\leqslant\frac{x}{3x+2y-x-y}=\frac{x}{2x+y}.$$Similarly, we can get $$\dfrac{y}{y^3+z^2+x}\leqslant\frac{y}{2y+z},\,\,\,\,\,\dfrac{z}{z^3+x^2+y}\leqslant\frac{z}{2z+x}.$$It suffices to show$$\frac{x}{2x+y}+\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/546675",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
} |
A matrix with given row and column sums There are a set of equations like
$A_x + A_y + A_z = P$
$B_x + B_y + B_z = Q$
$C_x + C_y + C_z = R$
Where the values of only $P, Q, R$ are known.
Also,
we have
$A_x + B_x + C_x = I$
$A_y + B_y + C_y = J$
$A_z + B_z + C_z = K$
where only the values of $I, J$ and $K$ are know... | You are trying to solve
$$\left(\begin{matrix}
1&1&1&0&0&0&0&0&0\\
0&0&0&1&1&1&0&0&0\\
0&0&0&0&0&0&1&1&1\\
1&0&0&1&0&0&1&0&0\\
0&1&0&0&1&0&0&1&0\\
0&0&1&0&0&1&0&0&1
\end{matrix}\right) \left(\begin{matrix} A_x\\A_y\\A_z\\B_x\\B_y\\B_z\\C_x\\C_y\\C_z
\end{matrix}\right) = \left(\begin{matrix} P\\Q\\R\\I\\J\\K \end{matri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/546733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Derivatives question help The question is :Find the derivative of $f(x)=e^c + c^x$. Assume that c is a constant.
Wouldn't $f'(x)= ce^{c-1} + xc^{x-1}$. It keeps saying this answer is incorrect, What am i doing wrong?
| Consider $$f(x)=e^{c}+c^{x}$$ where $c$ is a constant. We know that since $c$ is a constant, $e^c$ is also a constant making ${d\over dx}(e^c)=0$. Also, ${d\over dx}(c^x)=c^x\ln c$. The reason for this is because ${d\over dx}(c^x)={d\over dx}{(e^{\ln c})^x}={d\over dx}({e^{{(\ln c}){x}})}=e^{({\ln c)} x}\cdot {d\over d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/546809",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Need help to prove this by using natural deduction. i m concerned to prove these by using Natural Deduction. And i am also concerned to prove it for both sides. $$\exists x(P (x) \implies A) \equiv \forall xP (x) \implies A$$
I have some difficulties to show it by using Natural Deduction. Any kind of help will be appre... | $(\Longrightarrow)$ Suppose $\exists x(P (x) \implies A)$. This means we can pick an $x_0$ such that $P (x_0) \implies A$. Now suppose $\forall x P (x)$. In particular, this gives us $P(x_0)$. Then, by modus ponens, $A$. So we have shown that $\forall x P(x) \implies A$.
$(\Longleftarrow)$ Suppose $\forall x P(x) \im... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/546891",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to calculate $\int_0^\frac{\pi}{2}(\sin^3x +\cos^3x) \, \mathrm{d}x$? $$\int_0^\frac{\pi}{2}(\sin^3x +\cos^3x) \, \mathrm{d}x$$
How do I compute this? I tried to do the trigonometric manupulation but I can't get the answer.
| Hint: $\sin^3 x dx = -\sin^2 xd(\cos x) = (\cos^2 x - 1)d(\cos x)$. Do the same for $\cos^3 x dx$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/546986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 7,
"answer_id": 0
} |
Prove $f(x)= e^{2x} + x^5 + 1$ is one to one Prove $$f(x)= e^{2x} + x^5 + 1$$ is one to one.
So my solution is: Suppose $$ f(x_1)=f(x_2),$$
then I am stuck here: $$e^{2x_1}-e^{2x_2}=x^5_1 -x^5_2.$$ How do I proceed?
Also after that I found out that $$(f^-1)'(2)= 4.06$$ but how do I find $$(f^-1)''(2)$$ Differentiat... | You can also prove one-one this way: (Scroll the mouse over the covered region below)
Use the fact : $x_1 \neq x_2 \implies f(x_1) \neq f(x_2)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/547047",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 7,
"answer_id": 4
} |
Existence of an injection Let $A$ and $B$ be two sets. Prove the existence of an injection from $A$ to $B$ or an injection from $B$ to $A$.
I don't know how to proceed, since I don't have any information on $A$ or $B$ to begin with.
Does anybody have a hint ?
| To prove the existence of an injection between two sets $A$ and $B$ from the Axiom of Choice, or from Zorn's Lemma (which is equivalent) the rough idea is that we build a bijection between larger and larger subsets of $A$ and $B$, starting with the empty function.
At every "step" we match one of the remaining elements ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/547118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
system of equation with 3 unknown Solve $$\begin{matrix}i \\ ii \\ iii\end{matrix}\left\{\begin{matrix}x-y-az=1\\ -2x+2y-z=2\\ 2x+2y+bz=-2\end{matrix}\right.$$
For which $a$ does the equation have
*
*no solution
*one solution
*$\infty$ solutions
I did one problem like this and got a fantastic solution from @amzo... | The complete matrix of your system is
$$
\begin{bmatrix}
1 & -1 & -a & 1 \\
-2 & 2 & -1 & 2\\
2 & 2 & b & -2
\end{bmatrix}
$$
and with Gaussian elimination you get
$$
\begin{bmatrix}
1 & -1 & -a & 1 \\
0 & 0 & -1-2a & 4\\
0 & 4 & b+2a & -4
\end{bmatrix}
$$
(sum to the second row the first multiplied by $2$ and to the t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/547233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
are any two vector spaces with the same (infinite) dimension isomorphic? Is it true that any 2 vector spaces with the same (infinite) dimension are isomorphic? I think that it is true, since we can build a mapping from $V$ to $\mathbb{F}^{N}$ where the cardinality of $N$ is the dimension of the vector space - where by ... | Your argument is quite correct.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/547314",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Rotation of 2D polar graph in a 3D space along some fixed axis? Does there exist some systematic way of rotating a 2-D polar graph $r=f(\theta)$ around some axis in a 3D space?
For example: $f(\theta)=cos(\theta)$ in 2-D looks like:
If we want to rotate the above plot along the y-axis (in 3D of-course) the plot shoul... | In spherical coordinates, your $\theta_{2D}$ is given by:
$$\theta_{2D} = \pi/2 - \theta$$
And you have $r = f(\theta)$.
So for your graph you'ld have:
$$r = \cos(\pi/2 - \theta) = \sin(\theta)$$
$\quad\quad\quad\quad\quad\quad\quad$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/547437",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Combinatorics: Mathematic Recurrence
I got
$\textbf{(a)}$ $f(n) = c_12^n + c_2(-10)^n$
and solved $\textbf{(b)}$ similarly.
However, (c)-(f) is not factorizable.
How do I proceed?
| c) (A - 1 + sqrt(6))(A - 1 - sqrt(6))
d) (A + 4)(A - 2)(A - 6)
e) (A + 3)(A - 1)(A - 1)
f) (A + 1)(A + 1)(A + 1)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/547512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Showing the simplifying steps of this equality ... Can someone please show me how these are equivalent in steps$$\frac{(h_2^2 - h_3^2 )}{\dfrac{1}{h_3}-\dfrac{1}{h_2}}=h_2 h_3 (h_2+h_3)$$ I thought it simplifies to $$(h_2^2-h_3^2)(h_3-h_2)$$This would be much appreciated, I cant wrap my head around it.
| Multiplying the numerator & the denominator by $h_2h_3\ne0,$
$$\frac{(h_2^2 - h_3^2 )}{\dfrac{1}{h_3}-\dfrac{1}{h_2}}= \frac{h_2h_3(h_2-h_3)(h_2+h_3)}{(h_2-h_3)}=h_2h_3(h_2+h_3)$$ assuming $h_2-h_3\ne0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/547578",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
"Normalizing" a Function One of our homework problems for my linear algebra course this week is:
On $\mathcal{P}_2(\mathbb{R})$, consider the inner product given by
$$
\left<p, q\right> = \int_0^1 p(x)q(x) \, dx
$$
Apply the Gram-Schmidt procedure to the basis $\left\{1, x, x^2\right\}$ to produce an orthonormal basis ... | The norm in this space is
$$\|u\| = \sqrt{\langle u, u\rangle} = \sqrt{\int_0^1 \left(u(x)\right)^2 dx}$$
So once you have a basis of three functions, compute the norms (i.e. compute the integral of the square, and square root it) and divide the function by the norm. In particular, show that
$$\left\| \frac{u}{\|u\|}\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/547661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proving a Sequence Does Not Converge I have a sequence as such:
$$\left( \frac{1+(-1)^k}{2}\right)_{k \in \mathbb{N}}$$
Obviously it doesn't converge, because it alternates between $0,1$ for all $k$. But how do I prove this fact?
More generally, how do I prove that a sequence does not converge? Are there any neat way... | A sequence of real numbers converges if and only if it's a Cauchy sequence; that is, if for all $\epsilon > 0$, there exists an $N$ such that
$$n, m \ge N \implies |a_n - a_m| < \epsilon$$
If you haven't shown or seen this, I'd strongly suggest trying to prove it (or look it up in pretty much any basic analysis book). ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/547741",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Groups with a R.E. set of defining relations Reading around I found the following two assertion:
1) Every countable abelian group has a recursively enumerable set of defining relations.
2) Every countable locally finite group has a recursively enumerable set of defining relations.
How can we prove this directly?
We nee... | I don't believe these claims. Notice that, for any set $X$ of prime numbers, we can build a countable, abelian, locally finite group in which $X$ is the set of primes that occur as orders of elements. Just take the direct sum (not product, because I want it to be countable and locally finite) of cyclic groups $\mathbb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/547862",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Field and Algebra What is the difference between "algebra" and "field"? In term of definition in Abstract algebra.
(In probability theory, sigma-algebra is a synonym of sigma-field, does this imply
algebra is the same as field?)
| An algebra is a ring that has the added structure of a field of scalars and a coherent (see below) multiplication. Some examples of algebras:
*
*M_n(F), where $F$ is any field.
*$C(T)$, continuous real (or complex)-valued functions on a topological space $T$ (here the scalars could be either the real or the complex... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/547947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
How prove this two function which is bigger? let function
$$f_{n}(x)=\left(1+x+\dfrac{1}{2!}x^2+\cdots+\dfrac{1}{n!}x^n\right)\left(\dfrac{x^2}{x+2}-e^{-x}+1\right)e^{-x},x\ge 0,n\in N^{+}$$
if $\lambda_{1},\lambda_{2},\mu_{1},\mu_{2}$ is postive numbers,and such
$\mu_{1}+\mu_{2}=1$
Question: following which is bigg... | If you are willing to rely on the problem setter to guarantee that there is a solution, you can do the following:
If $\mu_1=\mu_2=\frac 12$ they are equal.
If $\mu_1=1, \mu_2=0$, the first is $f_n(1)$ and the second is $f_n(\frac 14(\frac {\lambda_1}{\lambda_2}+ \frac {\lambda_2}{\lambda_1}+2))$, which is $f_n$ of so... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/548033",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Finding number with prime number of divisors? The problem is to find the number of numbers in [l,r] that have prime number of divisors.
Example for $l=1,r=10$
The answer is $6$
since $2,3,4,5,7,9$ are the numbers having prime number of divisors
constraints are $l,r\le10^{12}$
and $r-l\le10^6$.
Can someone help me with ... | Your answer is simply the count of prime numbers plus some extra numbers $p^{2^k},k\geq 1,$ where $p$ is prime, or more generally, all the numbers of the form $p^{2^k},k\geq 0$ and $p$ prime.
See this for more details.
By the way, I am also trying this problem of codechef...but not AC yet :(
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/548299",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Determine run-time of an algorithm Probably a stupid question but I don't get it right now. I have an algorithm with an input n. It needs n + (n-1) + (n-2) + ... + 1 steps to finish. Is it possible to give a runtime estimation in Big-O notation?
| Indeed, it is. Since we have $$n+(n-1)+(n-2)+\cdots+1=\frac{n(n+1)}2,$$ then you should be able to show rather readily that the runtime will be $\mathcal{O}(n^2)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/548394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Automorphism of $\mathbb{Z}\rtimes\mathbb{Z}$ I'm looking for a description of $\mathrm{Aut}(\mathbb{Z}\rtimes\mathbb{Z})$.
I've tried an unsuccessfully combinatorical approac, does anymore have some hints?
Thank you.
| Write the group as $G=\langle x,t\mid txt^{-1}=x^{-1}\rangle$.
Exercise:
the automorphism group is generated by the automorphisms $u,s,z$, where $$u(x)=x,\; u(t)=tx,\quad s(x)=x^{-1},\; s(t)=t, \quad z(x)=x,\; z(t)=t^{-1};$$ check that it is isomorphic to the direct product of an infinite dihedral group (generated by ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/548457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Why is $B\otimes_A C$ a ring, where $B,C$ are $A$-algebra Given $B,C$ be two $A$-algebra, it is said that $B\otimes_AC$ has a ring structure, with the multiplication being defined as:
$$
(b_1\otimes c_1)\cdot(b_2\otimes c_2):=(b_1b_2)\otimes (c_1c_2).
$$
However I don't see an easy way to check it is well-defined.
For,... | That $B$ is an $A$-algebra means that it comes equipped with maps $u_B : A \to B$ (unit) and $m_B : B \otimes_A B \to B$ (multiplication) of $A$-modules such that certain diagrams commute (write them down!). Similarly for $C$. But then $B \otimes_A C$ has the following $A$-algebra structure:
The unit is $A \cong A \oti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/548551",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
transition kernel I've got some trouble with transition kernels.
We look at Markov process with statspace $(S,\mathcal{S})$ and initial distribution $\mu^0$. We have a transition kernel $P:S\times \mathcal{S}\to[0,1]$
Now I have to show that $P^n(x,B):=\int_{S}P^{n-1}(y,B)P(x,dy)$ for $n\geq 2$ and $P^1:=P$ is also a... | We have to show that for each $n\geqslant 1$,
*
*if $x\in S$ is fixed, then the map $S\in\mathcal{S}\mapsto P^n(x,S)$ is a probability measure, and
*if $B\in\cal S$ is fixed, the map $x\in S\mapsto P^n(x,B)$ is measurable.
We proceed by induction. We assume that $P^{n-1}$ is a transition kernel. If $x\in S$ is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/548633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
$x_{n+1}\le x_n+\frac{1}{n^2}$ for all $n\ge 1$ then did $(x_n)$ converges? If $(x_n)$ is a sequence of nonnegative real numbers such that $x_{n+1}\le x_n+\frac{1}{n^2}$ for all $n\ge 1$ then did $(x_n)$ converges?
Can someone help me please?
| Since
$$\sum_{n=1}^{\infty} \frac{1}{n^2} = \zeta(2) = \frac{\pi^2}{6} < \infty,$$
$x_{n}$ is bounded from above by $x_1 + \frac{\pi^2}{6}$. Since $x_n$ is also non-negative, it is a bounded sequence and both hence its lim sup and lim inf exists. Let $L$ be the lim inf of $x_n$.
For any $\epsilon > 0$, pick a $N$ such... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/548705",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Find transformation function when densities are known I need some help with the following probability/statistics problem: Let X be a continuous random variable with density $f_{X}(x) =
\begin{cases}
\mathrm{e}^{-x^{2}} & \text{}x\text{ > 0} \\
0 & \text{}\text{elsewhere}
\end{cases}$. Find the tranformation function $g... |
Find the tranformation function $g$ such that the density of $Y=g(X)$ is...
Assume that $Y=g(X)$ and that $g$ is increasing, then the change of variable theorem yields
$$
g'(x)f_Y(g(x))=f_X(x).
$$
In the present case, one asks that, for every positive $x$,
$$
\frac{g'(x)}{2\sqrt{g(x)}}=2x\mathrm e^{-x^2},
$$
thus,
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/548796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Topology: Example of a compact set but its closure not compact Can anyone gives me an example of a compact subset such that its closure is not compact please? Thank you.
| Write $\tau$ for the standard topology on $\Bbb R$. Consider now $\tau_0=\{U\cup\{0\}\mid U\in\tau\}\cup\{\varnothing\}$.
It's not hard to verify that $\tau_0$ is a topology on $\Bbb R$. It's also not hard to see that $\overline{\{0\}}=\Bbb R$. However one can easily engineer an open cover without a finite subcover.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/548865",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "26",
"answer_count": 6,
"answer_id": 1
} |
How to calculate simple trigonometric problem I tried for an hour or so to solve this but I can't show the way to the solution. How does one solve the below problem?
$\tan(\sin^{-1}(1/3))$?
Is the solution periodic because it is a tangent?
| Please look in the first row of the Wikipedia table on Relationships between trigonometric functions. There you find the relation that
$$\tan(\arcsin(x))=\frac{x}{\sqrt{1-x^2}}$$
This gives you immediately the correct result of
$$\frac{1}{2\sqrt{2}}$$
Now, your task is to understand where this relation comes from.
As ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/548984",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Funtions between sets If $A$ is a set with $m$ elements and $B$ a set with $n$ elements, how many functions are there from $A$ to $B$. If $m=n$ how many of them are bijections?
I got $n^m$ for my first answer.
I wasn't sure for the bijection bit is it just $n$?
| The number of functions from A to B is equal to the number of lists of m elements where each element of the list is an element of b. Since we have n choices for each the answer is $n^m$
For the second one we have that n=m. Call the elements of A: $a_1,a_2...a_n$.
therefore the number of bijections from A to B is the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/549213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Why is a repeating decimal a rational number?
$$\frac{1}{3}=.33\bar{3}$$
is a rational number, but the $3$ keeps on repeating indefinitely (infinitely?). How is this a ratio if it shows this continuous pattern instead of being a finite ratio?
I understand that $\pi$ is irrational because it extends infinitely with... | $$\begin{align}
0.3333333333333\ldots &= 0.3 +0.03 +0.003 +0.0003+ \ldots\\
&=\frac{3}{10} + \frac{3}{100} + \frac{3}{1000}+ \frac{3}{10000} +\ldots
\end{align}$$
If you know the sum of a geometric sequence, that is:
$$a+aq+aq^2+aq^3+\ldots = \frac{a}{1-q} \quad\text{ if $|q| < 1$}$$
you can use it to conclude that for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/549254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 7,
"answer_id": 1
} |
Probability task (Find probability that the chosen ball is white.) I have this task in my book:
First box contains $10$ balls, from which $8$ are white. Second box contains $20$ from which $4$ are white. From each box one ball is chosen. Then from previously chosen two balls, one is chosen. Find probability that the ch... | In case of $(w,a)$ or $(a,w)$ you need to consider that one of these two balls is chosen (randomly as by coin tossing, we should assume). Therefore these cases have to be weighted by a factor of $\frac 12$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/549329",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Intersection of $\{ [n\sqrt{2}]\mid n \in \mathbb{N}^* \}$ and $\{ [n(2+\sqrt{2})]\mid n \in \mathbb{N}^* \}$ Find the intersection of sets $A$ and $B$ where
$$A = \{ [n\sqrt{2}]\mid n \in \mathbb{N}^* \}$$
$$B = \{ [n(2+\sqrt{2})]\mid n \in \mathbb{N}^* \}.$$
([$x$] is the integer part of $x$)
Using the computer, we f... | Suppose that exists $m,n\in\mathbb{N^*}$ such that $[n\sqrt{2}]=[(2+\sqrt{2})m]=t\in\mathbb{N^*}.$
Then $t<n\sqrt{2}<n+1,\quad t<(2+\sqrt{2})m<t+1\implies \dfrac{t}{\sqrt{2}}<n<\dfrac{t+1}{\sqrt{2}},\quad \dfrac{t}{2+\sqrt{2}}<m<\dfrac{t+1}{2+\sqrt{2}}\stackrel{(+)}{\implies} t<n+m<t+1,\;\text{false}\implies A\cap B=\e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/549395",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
How to find integrals using limits? How to find integrals using limits?
The question arise when I see that to find the derivative of a function $f(x)$ we need to find: $$\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$$ and it works fine for finding derivatives of every function you can give. But is there a similar approach to fi... | Yes.
The simplest way to write
$\int_a^b f(x)\, dx$
as a limit
is to divide the range of integration
into $n$ equal intervals,
estimate the integral over each interval
as the value of $f$
at either the start, end, or middle
of the interval,
and sum those.
Mathematically,
$$\int_a^b f(x)\, dx
= \lim_{n \to \infty} \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/549479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Can we have $x^n\equiv (x+1)^n \pmod m$ for large enough $n$? $x^n\equiv (x+1)^n$ For what values of m and n can we find an x that solves this?
| It is not clear from your question what is fixed and what varies. To illustrate: suppose you want a solution with $n=10$ --- is that a large enough $n$? Very well, then --- pick your favorite $x$, say, $x=42$. Then calculate $43^{10}-42^{10}$ and call it $Q$. Then if $m$ is $Q$, or any factor of $Q$, you will have $(x+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/549541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Any group of order $12$ must contain a normal Sylow subgroup This is part of a question from Hungerford's section on Sylow theorems, which is to show that any group with order 12, 28, 56, or 200 has a normal Sylow subgroup. I am just trying the case for $|G| = 12$ first.
I have read already that one can't conclude in g... | Hint: If $n_3 = 4$, then how many elements of order 3 are there in the group? How many elements does that leave for your groups of order 4?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/549806",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 0
} |
Bayes' Theorem with multiple random variables I'm reviewing some notes regarding probability, and the section regarding Conditional Probability gives the following example:
$P(X,Y|Z)=\frac{P(Z|X,Y)P(X,Y)}{P(Z)}=\frac{P(Y,Z|X)P(X)}{P(Z)}$
The middle expression is clearly just the application of Bayes' Theorem, but I can... | We know
$$P(X,Y)=P(X)P(Y|X)$$
and
$$P(Y,Z|X)=P(Y|X)P(Z|X,Y)$$
(to understand this, note that if you ignore the fact that everything is conditioned on $X$ then it is just like the first example).
Therefore
\begin{align*}
P(Z|X,Y)P(X,Y)&=P(Z|X,Y)P(X)P(Y|X)\\
&=P(Y,Z|X)P(X)
\end{align*}
Which derives the third expression... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/549887",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "29",
"answer_count": 3,
"answer_id": 0
} |
Proving that $A \cap (A \cup B) = A$ . Please check solution For homework I need to prove the folloving:
$$
A \cap (A \cup B) = A
$$
I did that in the following manner:
$$
A \cap (A \cup B)\\
x \in A \land (x \in A \lor x \in B)\\
(x \in A\ \land x \in A) \lor (x \in A \land x \in B)\\
x \in A \lor (x \in A \land x \in... | Using fundamental laws of Set Algebra
$$\begin{cases}A \cap (A \cup B) & Given\\
=(A \cap A) \cup (A \cap B) & \text{Distributive Law}\\
=A \cup (A \cap B) & A \cap A = A\\
=A & A \cap B \subset A
\end{cases}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/550076",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Prove that $ \int_{0}^{1}\frac{\sqrt{1-x^2}}{1-x^2\sin^2 \alpha}dx = \frac{\pi}{4\cos^2 \frac{\alpha}{2}}$. Prove that $\displaystyle \int_{0}^{1}\frac{\sqrt{1-x^2}}{1-x^2\sin^2 \alpha}dx = \frac{\pi}{4\cos^2 \frac{\alpha}{2}}$.
$\bf{My\; Try}::$ Let $x = \sin \theta$, Then $dx = \cos \theta d\theta$
$\displaystyle = \... | Substitute $x=\frac{t}{\sqrt{\cos a+t^2}}$. Then
$dx= \frac{\cos a\ dt}{(\cos a+t^2)^{3/2}}$ and
\begin{align}\int_{0}^{1}\frac{\sqrt{1-x^2}}{1-x^2\sin^2 a}dx
=& \int_{0}^{\infty}\frac{\sqrt{\cos a}\ \frac1{t^2} }{(t^2+ \frac1{t^2} )\cos a +(1+\cos^2a)}\overset{t\to 1/t}{dt}\\
=& \ \frac{\sqrt{\cos a}}2\int_{0}^{\inft... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/550145",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
Is the empty set always part of the result of an intersection? From very basic set theory we have that:
"The empty set is inevitably an element of every set."
Then, is it correct to assume that the intersection of
$A = \{1, 2, 3\}$ and $B = \{3, 4, 5\}$ is actually $\{\emptyset, 3\}$, and not just $\{3\}$?
Thank you
| While it is true that for all sets $A$, we have $\emptyset \subseteq A$, it is not true that for all sets $A$, $\emptyset \in A$.
The intersection of $A,B$ is defined by $A \cap B = \{x | x \in A$ and $x \in B\}$.
What is important here is the difference between being a subset and being an element of a set. Do you now ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/550244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Show that interval $(a, b)$ is not open in $\mathbb{R}^2$ I know that interval $(a, b)$ is open in $\mathbb{R}$. To show that interval $(a,b)$ is open in $\mathbb{R}$, I have done so: Let it be $x\in (a,b)$. Enough to find an open ball containing the point $x$, and that is included in the interval $(a,b)$. Suffice to g... | As kahen pointed out, what you want to say is that $$(a,b)\times\{0\}$$ is not open in $\Bbb R^2$. Now, pick any point ${\bf x}=(x,0)$ with $a<x<b$. Then $B({\bf x},\varepsilon)$ contains elements of the form $(x_1,x_2)$ with $x_2\neq 0$ (prove this!), so $B({\bf x},\varepsilon)$ cannot be contained in $(a,b)\times \{0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/550350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Intuitive reasoning why are quintics unsolvable I know that quintics in general are unsolvable, whereas lower-degree equations are solvable and the formal explanation is very hard. I would like to have an intuitive reasoning of why it is so, accessible to a bright high school student, or even why it should be so. I hav... | @Sawarnik, what are you referring to exactly when you write that "the formal explanation is very hard"? The basic idea is actually quite simple. I looked through the comments above and none seem to mention the smallest nonabelian simple group which happens to be the group $A_5$ of order $60$. This group is not contain... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/550401",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "51",
"answer_count": 4,
"answer_id": 3
} |
Norm of random vector plus constant Suppose that $w$ is a multivariate standard normal vector and $c$ a real vector of the same size. I know that for positive x
$$P(||w+c||^2\geq x)\ \geq \ P(||w||^2\geq x)$$
but I cannot prove it. We use the euclidean norm , in dimension 2 or 3 drawings show that the inequality above... | Very rough sketch for a general case: Perhaps the equivalent $P(\|w+c\|^2\leq x)\ \leq \ P(\|w\|^2\leq x)$ is more intuitive to deal with. For both probabilities, one evaluates the probability that $w$ is in a subset of $R^d$ of $d$-dimensional Lebesgue measure (let us call this measure $\mu$) $m$, where $m$ is the vo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/550477",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.