Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Polynomial with infinitely many zeros. Can a polynomial in $ \mathbb{C}[x,y] $ have infinitely many zeros? This is clearly not true in the one-variable case, but what happens in two or more variables?
| Any nonconstant polynomial $p(x,y)\in\mathbb{C}[x,y]$ will always have infinitely many zeros.
If the polynomial is only a function of $x$, we may pick any value for $y$ and find a solution (since $\mathbb{C}$ is algebraically closed).
If the polynomial uses both variables, let $d$ be the greatest power of $x$ appearing... | {
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"url": "https://math.stackexchange.com/questions/286352",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
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"answer_id": 3
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Example of Matrix in Reduced Row Echelon Form I'm struggling with this question and can't seem to come up with an example:
Give an example of a linear system (augmented matrix form) that has:
*
*reduced row echelon form
*consistent
*3 equations
*1 pivot variable
*1 free variable
The constraints that I'm strugg... | Hint: It's gotta have only three columns, one for each of the variables (1 pivot, 1 free) and one column for the constants in the equations.
| {
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How to integrate $\int_{}^{}{\frac{\sin ^{3}\theta }{\cos ^{6}\theta }d\theta }$? How to integrate $\int_{}^{}{\frac{\sin ^{3}\theta }{\cos ^{6}\theta }d\theta }$?
This is kind of homework,and I have no idea where to start.
| One way is to avoid cumbersome calculations by using s for sine and c for cosine. Split the s^3 in the numerator into s*s^2, using s^2 = 1 - c^2 and putting everything in place you have
s*(1-c^2)/c^6. Since the lonley s will serve as the negative differential of c, the integrand reduces nicely into (1-c^2)*(-dc)/c^6. D... | {
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$x\otimes 1\neq 1\otimes x$ In Bourbaki, Algèbre 5, section 5, one has $A$ and $B$ two $K$-algebras in an extension $\Omega$ of $K$. It is said that if the morphism $A\otimes_K B\to \Omega$ is injective then $A\cap B=K$. I see the reason: if not there would exist $x\in A\cap B\setminus K$ so that $x\otimes 1=1\otimes x... | I hope you know that if $\{v_i\}$ is a basis of $V$ and $\{w_j\}$ is a basis of $W$, then $\{v_i\otimes w_j\}$ is a basis of $V\otimes W$. Now since $x\notin K$, we can extend $\{1,x\}$ to a basis of $A$ and $B$, respectively. Now as a corollary of the above claim you have in particular that $1\otimes x$ and $x\otimes ... | {
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Converting to regular expressions I am really not sure about the following problem, I tried to answer it according to conversion rules the best I can. I was wondering if someone can give me some hints as to whether or not I am on the right track.
Many thanks in advance.
Convert to Regular Expressions
L1 = {w | w begins... | As for $L_1$ you are right, if you would like a more generic approach, it would be $1E^* \cap E^* 0$ which indeed equals $1E^*0$. The thing to remember is that conjunction "and" can be often thought of as intersection of languages.
Regarding $L_2$, it is also ok.
Your answer to $L_3$ is wrong, $EEEEE$ means exactly 5 s... | {
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How would one find other real numbers that aren't in the rational field of numbers? For example, $\sqrt2$ isn't a rational number, since there is no rational number whose square equals two. And I see this example of a real number all the time and I'm just curious about how you can find or determine other numbers like s... | I'm not sure if you're asking about finding a real number, or determining whether a given real number is rational or not. In any case, both problems are (in general) very hard.
Finding a real number
There are lots and lots of real numbers. How many? Well the set of all real numbers which have a finite description as a ... | {
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What is a point? In geometry, what is a point?
I have seen Euclid's definition and definitions in some text books. Nowhere have I found a complete notion. And then I made a definition out from everything that I know regarding maths. Now, I need to know what I know is correct or not.
One book said, if we make a dot on a... | We can't always define everything or prove all facts. When we define something we are describing it according to other well-known objects, so if we don't accept anything as obvious things, we can not define anything too! This is same for proving arguments and facts, if we don't accept somethings as Axioms like "ZFC" ax... | {
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Prove that between every rational number and every irrational number there is an irrational number. I have gotten this far, but I'm not sure how to make it apply to all rational and irrational numbers....
http://i.imgur.com/6KeniwJ.png">
BTW, I'm quite newbish so please explain your reasoning to me like I'm 5. Thanks! ... | Let $p/q$ be a rational number and $r$ be an irrational number.
Consider the number $w = \dfrac{p/q+r}2$ and prove the following statements.
$1$. If $p/q < r$, then $w \in ]p/q,r[$. (Why?)
$2$. Similarly, if $r < p/q$, then $w \in ]r,p/q[$. (Why?)
$3$. $w$ is irrational. (Why?)
| {
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Differential of the exponential map on the sphere I have a problem understanding how to compute the differential of the exponential map. Concretely I'm struggling with the following concrete case:
Let $M$ be the unit sphere and $p=(0,0,1)$ the north pole. Then let $\exp_p : T_pM \cong \mathbb{R}^2 \times \{0\} \to M $ ... | I' ll assume we are talking about the exponential map obtained from the Levi-Civita connection on the sphere with the round metric pulled from $\mathbb R^3$. If so, the exponential here can be understood as mapping lines through the origin of $\mathbb R^2$ to the great circles through the north pole. Its derivative the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/287024",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How did Newton invent calculus before the construction of the real numbers? As far as I know, the reals were not rigorously constructed during his time (i.e via equivalence classes of Cauchy sequences, or Dedekind cuts), so how did Newton even define differentiation or integration of real-valued functions?
| Way earlier, the Greeks invented a surprising amount of mathematics. Archimedes knew a fair amount of calculus, and the Greeks proved by Euclidean geometry that $\sqrt{2}$ and other surds were irrational (thus annoying Pythagorus greatly).
And I can do a moderate amount of (often valid) math without knowing why the the... | {
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Functor between categories with weak equivalance. A homotopical category is category with a distinguished class of morphism called weak equivalence.
A class $W$ of morphisms in $\mathcal{C}$ is a weak equivalence if:
*
*All identities are in $W$.
*for every $r,s,t$ which compositions
$rs$ and $st$ exis... | Maybe you're beginning your journey through model, localized, homotopy categories by a steep way. I would try this short paper first: W. G. Dwyer and J. Spalinski.
| {
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"url": "https://math.stackexchange.com/questions/287151",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Degree of continuous mapping via integral Let $f \in C(S^{n},S^{n})$. If $n=1$ then the degree of $f$ coincides with index of curve $f(S^1)$ with respect to zero (winding number) and may be computed via integral
$$
\deg f = \frac{1}{2\pi i} \int\limits_{f(S^1)} \frac{dz}{z}
$$
Is it possible to compute the degree of... | You could find some useful information (try page 6) here and here.
| {
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"timestamp": "2023-03-29T00:00:00",
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Prove that if $x$ is a non-zero rational number, then $\tan(x)$ is not a rational number and use this to prove that $\pi$ is not a rational number.
Prove that if $x$ is a non-zero rational number, then $\tan(x)$ is not a rational number and use this to prove that $\pi$ is not a rational number.
I heard that this was ... | The proof from a few hundred years ago was done by Lambert and Miklós Laczkovich provided a simplified version later on. The Wikipedia page for "Proof that $\pi$ is irrational" provides this proof (in addition to some other discussion).
http://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational#Laczkovich.27s_proof
... | {
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Evaluate $\int\sin(\sin x)~dx$ I was skimming the virtual pages here and noticed a limit that made me wonder the following
question: is there any nice way to evaluate the indefinite integral below?
$$\int\sin(\sin x)~dx$$
Perhaps one way might use Taylor expansion. Thanks for any hint, suggestion.
| For the maclaurin series of $\sin x$ , $\sin x=\sum\limits_{n=0}^\infty\dfrac{(-1)^nx^{2n+1}}{(2n+1)!}$
$\therefore\int\sin(\sin x)~dx=\int\sum\limits_{n=0}^\infty\dfrac{(-1)^n\sin^{2n+1}x}{(2n+1)!}dx$
Now for $\int\sin^{2n+1}x~dx$ , where $n$ is any non-negative integer,
$\int\sin^{2n+1}x~dx$
$=-\int\sin^{2n}x~d(\cos ... | {
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Finding the Galois group of $\mathbb Q (\sqrt 5 +\sqrt 7) \big/ \mathbb Q$ I know that this extension has degree $4$. Thus, the Galois group is embedded in $S_4$. I know that the groups of order $4$ are $\mathbb Z_4$ and $V_4$, but both can be embedded in $S_4$. So, since I know that one is cyclic meanwhile the other i... | You should first prove that $\mathbf{Q}(\sqrt{5}+\sqrt{7})/\mathbf{Q}$ is a Galois extension. For this it may be useful to verify that $\mathbf{Q}(\sqrt{5}+\sqrt{7}) = \mathbf{Q}(\sqrt{5},\sqrt{7})$. Then you might consider the Galois groups of $\mathbf{Q}(\sqrt{5})/\mathbf{Q}$ and $\mathbf{Q}(\sqrt{7})/\mathbf{Q}$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Evaluate integral with quadratic expression without root in the denominator $$\int \frac{1}{x(x^2+1)}dx = ? $$
How to solve it? Expanding to $\frac {A}{x}+ \frac{Bx +C}{x^2+1}$ would be wearisome.
| You can consider
$\displaystyle \frac{1}{x(x^2 + 1)} = \frac{1 + x^2 - x^2}{x(x^2 + 1)}$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Number of $n$-digit palindromes
How can one count the number of all $n$-digit palindromes? Is there any recurrence for that?
I'm not sure if my reasoning is right, but I thought that:
For $n=1$, we have $10$ such numbers (including $0$).
For $n=2$, we obviously have $9$ possibilities.
For $n=3$, we can choose 'extrem... | Details depend on whether for example $0110$ counts as a $4$-digit palindrome. We will suppose it doesn't. This makes things a little harder.
If $n$ is even, say $n=2m$, the first digit can be any of $9$, then the next $m-1$ can be any of $10$, and then the rest are determined. So there are $9\cdot 10^{m-1}$ palindrome... | {
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Prove this matrix is neither unipotent nor nilpotent. The question asks me to prove that the matrix,
$$A=\begin{bmatrix}1 & 1\\0 & 1\end{bmatrix}$$
is neither unipotent nor nilpotent. However, can't I simply row reduce this to the identity matrix:
$$A=\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}$$
which shows that it clear... | HINT: As Calvin Lin pointed out in the comments, $(A-I)^2=0$, so $A$ is unipotent. To show that $A$ is not nilpotent, show by induction on $n$ that
$$A^n=\begin{bmatrix}1 & n\\0 & 1\end{bmatrix}\;.$$
| {
"language": "en",
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Find the area of the parallelogram with vertices $K(1, 3, 1), L(1, 6, 3), M(6, 12, 3), N(6, 9, 1)$.
Find the area of the parallelogram with vertices $K(1, 3, 1), L(1, 6, 3), M(6, 12, 3), N(6, 9, 1)$.
I know that I need to get is an equation of the form (a vector) x (a second vector)
But, how do I decide what the two ... | Given a parallelogram with vertices $A$, $B$, $C$, and $D$, with $A$ diagonally opposite $C$, the vectors you want are $A-B$ and $A-D$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/287719",
"timestamp": "2023-03-29T00:00:00",
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Convergence in distribution and convergence in the vague topology From Terrence Tao's blog
Exercise 23 (Implications and equivalences) Let ${X_n, X}$ be random variables taking values in a ${\sigma}$-compact metric space ${R}$.
(ii) Show that if ${X_n}$ converges in distribution to ${X}$, then ${X_n}$ has a tight sequ... | I believe the discussion of this old mathexchange post clarifies what is meant (I don't find the way the exercise was written particularly clear): Definition of convergence in distribution
See in particular the comment by Chris Janjigian. In (iv), the limiting distribution is assumed to be that of a R.V.; in (v), all t... | {
"language": "en",
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Prove $\int_0^\infty \frac{\ln \tan^2 (ax)}{1+x^2}\,dx = \pi\ln \tanh(a)$ $$
\mbox{How would I prove}\quad
\int_{0}^{\infty}
{\ln\left(\,\tan^{2}\left(\, ax\,\right)\,\right) \over 1 + x^{2}}\,{\rm d}x
=\pi
\ln\left(\,\tanh\left(\,\left\vert\, a\,\right\vert\,\right)\,\right)\,{\Large ?}.
\qquad a \in {\mathbb R}\verb*... | Another approach:
$$ I(a) = \int_{0}^{+\infty}\frac{\log\tan^2(ax)}{1+x^2}\,dx$$
first use "lebniz integral differentiation"
and then do change of variable to calculate the integral.
Hope it helps.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Which of these values for $f(12)$ are possible? If $f(10)=30, f'(10)=-2$ and $f''(x)<0$ for $x \geq 10$, which of the following are possible values for $f(12)$ ? There may be more than one correct answer.
$24, 25, 26, 27, 28$
So since $f''(x)<0$ indicates that the graph for $f(x)$ is concave down, and after using slop... | Hint: The second derivative condition tells you that the first derivative is decreasing past $10$, and so is $\lt -2$ past $10$.
By the Mean Value Theorem, $\dfrac{f(12)-f(10)}{12-10}=f'(c)$ for suitable $c$ strictly between $10$ and $12$. Now test the various suggested values.
For example, $\dfrac{27-30}{12-10}=... | {
"language": "en",
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If $f(x)I have this question:
Let $f(x)→A$ and $g(x)→B$ as $x→x_0$. Prove that if $f(x) < g(x)$ for all $x∈(x_0−η, x_0+η)$ (for some $η > 0$) then $A\leq B$. In this case is it always true that $A < B$?
I've tried playing around with the definition for limits but I'm not getting anywhere. Can someone give me a hint o... | To show it is not always the case that $A<B$, you can come up with an example to show it is possible for $A = B$. So if we let $f(x) = (\frac{1}{x})^2$ and $g(x) = (\frac{1}{x})^4$, we know $f(x) < g(x)$ $\forall x \in (-1,1)$. And we know $\lim_{x \to 0} f(x) = \lim_{x \to 0} g(x) = \infty$ so it is possible for $A=B$... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to integrate $\int_{0}^{\infty }{\frac{\sin x}{\cosh x+\cos x}\cdot \frac{{{x}^{n}}}{n!}\ \text{d}x} $? I have done one with $\displaystyle\int_0^{\infty}\frac{x-\sin x}{x^3}\ \text{d}x$, but I have no ideas with these:
$$\begin{align*}
I&=\int_{0}^{\infty }{\frac{\sin x}{\cosh x+\cos x}\cdot \frac{{{x}^{n}}}{n!}\ ... | I can address the second integral:
$$\int_{0}^{\infty }{dx \: \frac{x-\sin x}{\left( {{\pi }^{2}}+{{x}^{2}} \right){{x}^{3}}}}$$
Hint: We can use Parseval's Theorem
$$\int_{-\infty}^{\infty} dx \: f(x) \bar{g}(x) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} dk \: \hat{f}(k) \bar{\hat{g}}(k) $$
where $f$ and $\hat{f}$ are ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/288049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
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Existence of irreducible polynomial of arbitrary degree over finite field without use of primitive element theorem? Suppose $F_{p^k}$ is a finite field. If $F_{p^{nk}}$ is some extension field, then the primitive element theorem tells us that $F_{p^{nk}}=F_{p^k}(\alpha)$ for some $\alpha$, whose minimal polynomial is t... | A very simple counting estimation will show that such polynomials have to exist. Let $q=p^k$ and $F=\Bbb F_q$, then it is known that $X^{q^n}-X$ is the product of all irreducible monic polynomials over$~F$ of some degree$~d$ dividing $n$. The product$~P$ of all irreducible monic polynomials over$~F$ of degree strictly ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/288120",
"timestamp": "2023-03-29T00:00:00",
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removable singularity f(z) is analytic on the punctured disc $D(0,1) - {0}$ and the real part of f is positive. Prove that f has a removable singularity at $0$.
| Instead of looking at $e^{-f(z)}$, I think it's easier to do the following.
First, assume that $f$ is non-constant (otherwise the problem is trivial).
Let $\phi$ be a conformal mapping (you can write down an explict formula for $\phi$ if you want) from the right half-space onto the unit disc, and let $g(z) = \phi(f(z... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/288182",
"timestamp": "2023-03-29T00:00:00",
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Convergence of series $\sum_{n=1}^\infty \ln\left(\frac{2n+7}{2n+1}\right)$? I have the series
$$\sum\limits_{n=1}^\infty \ln\left(\frac{2n+7}{2n+1}\right)$$
I'm trying to find if the sequence converges and if so, find its sum.
I have done the ratio and root test but It seems it is inconclusive.
How can I find if the s... | Hint: Note that $$\lim_{n\to+\infty}\frac{\ln\left(\frac{2n+7}{2n+1}\right)}{n^{-1}}\neq0$$ so since the power of $n$ in the denominator is $-1$, so the series diverges.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/288261",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
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Eigenvalues of a $4\times4$ matrix I want to find the eigenvalues of the matrix
$$
\left[
\begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & a & a & 0 \\
0 & a & a & 0 \\
0 & 0 & 0 & b
\end{array}
\right]
$$
Can somebody explain me the theory behind getting the eigenvalues of this $4\times4$ matrix? The way I see it is to take t... | The eigenvalues of $A$ are the roots of the characteristic polynomial $p(\lambda)=\det (\lambda I -A)$.
In this case, the matrix $\lambda I-A$ is made of three blocks along the diagonal.
Namely $(\lambda)$, $\left(\begin{matrix} \lambda-a & -a \\ -a & \lambda -a \end{matrix}\right)$, and $(\lambda -b)$.
The determinant... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Equation in the real world Does a quadratic equation like $x^2 - ax + y = 0$ describe anything in the real world? (I want to know, if there is something in the same way that $x^2$ is describing a square.)
| Though not exactly same, depending upon value of a, following situations count as relevant.
For deep explanation, see wikipedia.
*
*Bernoulli's Effect. This gives relation of velocity of fluid($u$), Pressure($P$), gravitational constant($g$) and height($h$), $$\frac{u^2}{g}+P=h$$
*Mandelbrot Set has Recursive Equa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/288388",
"timestamp": "2023-03-29T00:00:00",
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Questions related to nilpotent and idempotent matrices I have several questions on an assignment that I just can't seem to figure out.
1) Let $A$ be $2\times 2$ matrix. $A$ is nilpotent if $A^2=0$. Find all symmetric $2\times 2$ nilpotent matrices.
It is symmetric, meaning the matrix $A$ should look like $A=\begin{bmat... | For #1, you should also have $b^2 + c^2 = 0$. If you're working over the real numbers, note that the square of a real number is always $\ge 0$, and is $0$ only if the number is $0$.
If complex numbers are allowed, you could have $a = -c = \pm i b$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/288456",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Combinatorial interpretation of the identity: ${n \choose k} = {n \choose n-k}$ What is the combinatorial interpretation of the identity: ${n \choose k} = {n \choose n-k}$?
Proving this algebraically is trivial, but what exactly is the "symmetry" here. Could someone give me some sort of example to help my understanding... | $n \choose k$ denotes the number of ways of picking $k$ objects out of $n$ objects, and specifying the $k$ objects that are picked is equivalent to specifying the $n-k$ objects that are not picked.
To put it differently, suppose you have $n$ objects, and you want to partition them into two sets: a set $A$ of size $k$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/288546",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Question on limit: $\lim_{x\to 0}\large \frac{\sin^2{x^{2}}}{x^{2}}$ How would I solve the following trig equations?
$$\lim_{x\to 0}\frac{\sin^2{x^{2}}}{x^{2}}$$
I am thinking the limit would be zero but I am not sure.
| We use $$\sin^2 x =\frac{1-\cos 2x}{2}$$
$$\lim_{x\to 0}\frac{\sin^2 x^2}{x^2}.=\lim_{x\to 0} \frac {1-\cos 2x^2}{2x^2}=0 $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/288601",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find infinitely many pairs of integers $a$ and $b$ with $1 < a < b$, so that $ab$ exactly divides $a^2 +b^2 −1$. So I came up with $b= a+1$ $\Rightarrow$ $ab=a(a+1) = a^2 + a$
So that:
$a^2+b^2 -1$ = $a^2 + (a+1)^2 -1$ = $2a^2 + 2a$ = $2(a^2 + a)$ $\Rightarrow$
$(a,b) = (a,a+1)$ are solutions.
My motivation is for th... | $(3,8)$ is a possible solution.
This gives us 24 divides 72, and a value of 3 for (b).
Have you considered that if $ab$ divides $a^2+b^2-1$, then we $ab$ divides $a^2 + b^2 -1 + 2ab$?
This gives us $ab$ divides $(a+b+1)(a+b-1)$.
Subsequently, the question might become easier to work with.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding the derivative of an integral $$g(x) = \int_{2x}^{6x} \frac{u+2}{u-4}du $$
For finding the $ g'(x)$, would I require to find first the derivative of $\frac{u+2}{u-4}$
then Replace the $u$ with 6x and 2x and add them ?
(the 2x would have to flip so the whole term is negative)
If the previous statement is true w... | Let $f(u)=\frac{u+2}{u-4}$, and let $F(u)$ be the antiderivative of $f(u)$. Then
$$
g'(x)=\frac{d}{dx}\int_{2x}^{6x}f(u)du=\frac{d}{dx}\left(F(u)\bigg\vert_{2x}^{6x}\right)=\frac{d}{dx}[F(6x)-F(2x)]=6F'(6x)-2F'(2x)
$$
But $F'(u)=f(u)$. So the above evaluates to
$$
6f(6x)-2f(2x)=6\frac{6x+2}{6x-4}-2\frac{2x+2}{2x-4}=\cd... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How many edges? We have a graph with $n>100$ vertices. For any two adjacent vertices is known that the degree of at least one of them is at most $10$ $(\leq10)$. What is the maximum number of edges in this graph?
| Let $A$ be the set of vertices with degree at most 10, and $B$ be the set of vertices with degree at least 11. By assumption, vertices of $B$ are not adjacent to each other. Hence the total number of edges $|E|$ in the graph is equal to the sum of degrees of all vertices in $A$ minus the number of edges connecting two ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Continuity of the (real) $\Gamma$ function. Consider the real valued function
$$\Gamma(x)=\int_0^{\infty}t^{x-1}e^{-t}dt$$
where the above integral means the Lebesgue integral with the Lebesgue measure in $\mathbb R$. The domain of the function is $\{x\in\mathbb R\,:\, x>0\}$, and now I'm trying to study the continuity... | You could also try the basic approach by definition.
For any $\,b>0\,\,\,,\,\,\epsilon>0\,$ choose $\,\delta>0\,$ so that $\,|x-x_0|<\delta\Longrightarrow \left|t^{x-1}-t^{x_0-1}\right|<\epsilon\,$ in $\,[0,b]\,$
:
$$\left|\Gamma(x)-\Gamma(x_0)\right|=\left|\lim_{b\to\infty}\int\limits_0^b \left(t^{x-1}-t^{x_0-1}\righ... | {
"language": "en",
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"source": "stackexchange",
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Sudoku puzzles and propositional logic I am currently reading about how to solve Sudoku puzzles using propositional logic. More specifically, they use the compound statement
$$\bigwedge_{i=1}^{9} \bigwedge_{n=1}^{9} \bigvee_{j=1}^{9}~p(i,j,n)$$
where $p(i,j,n)$ is the proposition that is true when the number
$n$ is in... | Although expressible as propositional logic, for practical solutions, it is computationally more effective to view Sudoku as a Constraint Satisfaction Problem. See Chapter 6 of Russell and Norvig: Artificial Intelligence - A Modern Approach, for example.
| {
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"timestamp": "2023-03-29T00:00:00",
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Show that $(a+b+c)^3 = a^3 + b^3 + c^3+ (a+b+c)(ab+ac+bc)$ As stated in the title, I'm supposed to show that $(a+b+c)^3 = a^3 + b^3 + c^3 + (a+b+c)(ab+ac+bc)$.
My reasoning:
$$(a + b + c)^3 = [(a + b) + c]^3 = (a + b)^3 + 3(a + b)^2c + 3(a + b)c^2 + c^3$$
$$(a + b + c)^3 = (a^3 + 3a^2b + 3ab^2 + b^3) + 3(a^2 + 2ab + ... | In general, $$a^n+b^n+c^n = \sum_{i+2j+3k=n} \frac{n}{i+j+k}\binom {i+j+k}{i,j,k} s_1^i(-s_2)^js_3^k$$
where $s_1=a+b+c$, $s_2=ab+ac+bc$ and $s_3=abc$ are the elementary symmetric polynomials.
In the case that $n=3$, the triples possible are $(i,j,k)=(3,0,0),(1,1,0),$ and $(0,0,1)$ yielding the formula:
$$a^3+b^3+c^3 =... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove about a right triangle How to prove (using vector methods) that the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices.
Defining the right triangle as the one formed by $\vec{v}$ and $\vec{w}$ with hypotenuse $\vec{v} - \vec{w}$, This imply to prove that $||\frac{1}{2}(\vec{v}+\... | If u and v are orthogonal, $u.v=0$.
Then $||u+v||^2 = ||u-v||^2 = ||u||^2+||v||^2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/289021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Leibniz Alternating Series Test Can someone help me find a Leibniz Series (alternating sum) that converges to $5$ ?
Does such a series even exist?
Thanks in advance!!!
I've tried looking at a series of the form $ \sum _ 1 ^\infty (-1)^{n} q^n $ which is a geometric series ... But I get $q>1 $ , which is impossible... ... | Take any Leibniz sequence $x_n$, the series of which converges not to zero, say to $c$, and then consider
the sequence $(\frac5c\cdot x_n)_n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/289093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Calculate $20^{1234567} \mod 251$ I need to calculate the following
$$20^{1234567} \mod 251$$
I am struggling with that because $251$ is a prime number, so I can't simplify anything and I don't have a clue how to go on. Moreover how do I figure out the period of $[20]_{251}$? Any suggestions, remarks, nudges in the ri... | If you do not know the little theorem, a painful but -I think- still plausible method is to observe that $2^{10} = 1024 \equiv 20$, and $10^3 = 1000 \equiv -4$. Then, we may proceed like this:
$20^{1234567} = 2^{1234567}10^{1234567} = 2^{123456\times 10 + 7}10^{411522\times 3 + 1} = 1280\times 1024^{123456}1000^{411522... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/289137",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Geometric Distribution $P(X\ge Y)$ I need to show that if $X$ and $Y$ are idd and geometrically distributed that the $P(X\ge Y)$ is $1\over{2-p}$. the joint pmf is $f_{xy}(xy)=p^2(1-p)^{x+y}$, and I think the only way to do this is to use a double sum: $\sum_{y=0}^{n}\sum_{x=y}^m p^2(1-p)^{x+y}$, which leads to me gett... | It is easier to use symmetry:
$$
1 = \mathbb{P}\left(X<Y\right) +\mathbb{P}\left(X=Y\right) + \mathbb{P}\left(X>Y\right)
$$
The first and the last probability are the same, due to the symmetry, since $X$ and $Y$ are iid. Thus:
$$
\mathbb{P}\left(X<Y\right) = \frac{1}{2} \left(1 - \mathbb{P}\left(X=Y\right) \right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/289212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof of equivalence theorem about left invertible matrices I am taking a course in Matrix Theory and we have a theorem that states (among other things) that:
The following conditions on the matrix $A$ of size $m \times n$ are equivalent:
(1) A has left inverse
(2) The system $Ax=b$ has at most one solution for any co... | Existence of left inverse means $A$ is 1-1, i.e., $Ax_1 = Ax_2$ implies $VAx_1 = VAx_2$ , i.e., $x_1=x_2$. So a solution, if it exists, must be unique.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/289266",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Product of pairwise coprime integers divides $b$ if each integer divides $b$ Let $a_1....a_n$ be pairwise coprime. That is $gcd(a_i, a_k) = 1$ for distinct $i,k$, I would like to show that if each $a_i$ divides $b$ then so does the product.
I can understand intuitively why it's true - just not sure how to formulate th... | Use unique prime factorization for each $a_i$ to write it as $$a_i = \prod_{i=1}^k p_i^{\alpha_i}.$$ $k$ is chosen such that it will number all prime factors across the $a_i$, with $\alpha_i = 0$ when $p_i$ is not a factor of $a_i$. In other words, $k$ will be the same number for each $a_i$. By the assumption, $b$ will... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/289337",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Another two hard integrals Evaluate :
$$\begin{align}
& \int_{0}^{\frac{\pi }{2}}{\frac{{{\ln }^{2}}\left( 2\cos x \right)}{{{\ln }^{2}}\left( 2\cos x \right)+{{x}^{2}}}}\text{d}x \\
& \int_{0}^{1}{\frac{\arctan \left( {{x}^{3+\sqrt{8}}} \right)}{1+{{x}^{2}}}}\text{d}x \\
\end{align}$$
| For the second integral, consider the more general form
$$\int_0^1 dx \: \frac{\arctan{x^{\alpha}}}{1+x^2}$$
(I do not understand what is special about $3+\sqrt{8}$.)
Taylor expand the denominator and get
$$\begin{align} &=\int_0^1 dx \: \arctan{x^{\alpha}} \sum_{k=0}^{\infty} (-1)^k x^{2 k} \\ &= \sum_{k=0}^{\infty} (... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that $f = 0$ if $\int_a^b f(x)e^{kx}dx=0$ for all $k$ The problem is show that $f=0$ whenever $f\in C[a,b]$ and
$$\int_a^bf(x)e^{kx}dx =0, \hspace{1cm}\forall k\in\mathbb{N}.$$
Can someone help me?
Thank you!
| First, letting $u=e^x$ we note that $\int_{e^a}^{e^b}f(\ln u)u^{k-1}du=0$ for all $k\in\mathbb N$.
Next, see the following old questions:
Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$
If $f$ is continuous on $[a , b]$ and $\int_a^b f(x) p(x)dx = 0$ then $f = 0$
problem on definite integral
| {
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Proof of a comparison inequality I'm working on a problem that's been giving me some difficulty. I will list it below and show what work I've done so far:
If a, b, c, and d are all greater than zero and $\frac{a}{b} < \frac{c}{d}$, prove that $\frac{a}{b} < \frac{a + c}{b + d} < \frac{c}{d}$.
Alright, so far I think... | As $$\frac ab<\frac cd\implies ad<bc\text { as } a,b,c,d>0$$
$$\frac{a+c}{b+d}-\frac ab=\frac{b(a+c)-a(b+d)}{b(b+d)}=\frac{bc-ad}{b(b+d)}>0 \text{ as } ad<bc \text{ and } a,b,c,d>0 $$
So, $$\frac{a+c}{b+d}>\frac ab$$
Similarly, $$\frac{a+c}{b+d}-\frac cd=\frac{ad-bc}{d(b+d)}<0\implies \frac{a+c}{b+d}<\frac cd$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Average of function, function of average I'm trying to find all functions $f : \mathbb{R} \to \mathbb{R}$ such that, for all $n > 1$ and all $x_1, x_2, \cdots, x_n \in \mathbb{R}$:
$$\frac{1}{n} \sum_{t = 1}^n f(x_t) = f \left ( \frac{1}{n} \sum_{t = 1}^n x_t \right )$$
My intuition is that this is only true if $f$ is ... | What about function f that:
$f(x+y)=f(x)+f(y)$ $\ \ \ \ \ \ x,y\in \mathbb{R}$
$f(a x)=af(x)$ $ \ \ \ \ \ a\in \mathbb{Q}, x\in \mathbb{R}$
This kind of function does not have to be continuous.
(It can be so wild that it does not have to be even measurable. And I guess(not sure at all) if it is measurable than it ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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how to solve this gamma function i know that $\Gamma (\frac {1}{2})=\sqrt \pi$ But I do not understand how to solve these equations
$$\Gamma (m+\frac {1}{2})$$
$$\Gamma (-m+\frac {1}{2})$$
are there any general relation to solve them
for example:
$\Gamma (1+\frac {1}{2})$
$\Gamma (-2+\frac {1}{2})$
| By the functional equation $$\Gamma(z+1)=z \, \Gamma(z)$$ (which is easily proved for the integral definition of $\Gamma$ by parts, and may be used to analytically continue the function to the negative numbers) we find $$\Gamma(1 + \frac{1}{2}) = \frac{1}{2} \Gamma(\frac{1}{2})$$ and $$\Gamma (-2+\frac {1}{2}) = \frac{... | {
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Ring theorem and isomorphic I got a problems as follow
Let $S = \left\{\begin{bmatrix}
a & 0 \\
0 & a \\ \end{bmatrix} | a \in R\right\}$, where $R$ is the set of real numbers. Then $S$ is a ring under matrix addition and multiplication. Prove that $R$ is isomorphic to $S$.
What is the key to prove it? By definition... | Hint: Identify $a$ with \begin{bmatrix} a & 0 \\ 0 & a \\ \end{bmatrix} for each $a$ in $R$.
| {
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Cofactor theorem Shilov on page 12 says determinant $D=a_{1j}A_{1j}+a_{2j}A_{2j}+...+a_{nj}A_{nj}...(I)$ is an identity in the quantities $a_{1j}, a_{2j},..., a_{nj}$. Therefore it remains valid if we replace $a_{ij} (i = 1, 2,. . . , n)$ by any other quantities. The quantities $A_{1j}, A_{2j},..., A_{nj}$ remain unc... | The identity (I) uses the elements $a_{1j}, a_{2j} \ldots$, which of course for the matrix have specific values. What the statement means is that if those values, and only those values, are changed in the matrix, the new determinant is valid for the new matrix.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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socles of semiperfect rings For readers' benefit, a few definitions for a ring $R$.
The left (right) socle of $R$ is the sum of all minimal left (right) ideals of $R$. It may happen that it is zero if no minimals exist.
A ring is semiperfect if all finitely generated modules have projective covers.
Is there a semiperf... | Here is an example due to Bass of a left perfect ring that is not right perfect. See Lam's First Course, Example 23.22 for an exposition (which is opposite to the one I use below).
Let $k$ be a field, and let $S$ be the ring of (say) column-finite $\mathbb{N} \times \mathbb{N}$-matrices over $k$. Let $E_{ij}$ denote ... | {
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Not following what's happening with the exponents in this proof by mathematical induction. I'm not understanding what's happening in this proof. I understand induction, but not why $2^{k+1}=2*2^{k}$, and how that then equals $k^{2}+k^{2}$. Actually, I really don't follow any of the induction step - what's going on with... | I have copied the induction step table format from your question and will reformat it as I had to do in my high school geometry class. I find it very helpful to get away from the table as you gave it.
Set $P(n)$ equal to the statement that 2^n>n^2 for all n>4. Note the basis step 2^4>4^2 is true.
$~~~~~~$Assume that $P... | {
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Find all polynomials $\sum_{k=0}^na_kx^k$, where $a_k=\pm2$ or $a_k=\pm1$, and $0\leq k\leq n,1\leq n<\infty$, such that they have only real zeroes
Find all polynomials $\sum_{k=0}^na_kx^k$, where $a_k=\pm2$ or $a_k=\pm1$, and $0\leq k\leq n,1\leq n<\infty$, such that they have only real zeroes.
I've been thinking ab... | Let $\alpha_1,\alpha_2...\alpha_n$ the real roots. We know:
$$\sum \alpha_i^2=( \sum \alpha_i )^2-2\sum \alpha_i\alpha_j= \left(\frac{a_{n-1}}{a_n}\right)^2-2\left(\frac{a_{n-2}}{a_n}\right)\le 8$$
On the other hand, by AM-GM inequality:
$$\sum \alpha_i^2\ge n \sqrt[n]{|\prod\alpha_i|^2}=n\sqrt[n]{\left|\frac{a_0}{a_n}... | {
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Why doesn't this argument show that every locally compact Hausdorff space is normal? In proving that a locally compact Hausdorff space $X$ is regular, we can consider the one-point compactification $X_\infty$ (this is not necessary, see the answer here, but bear with me). Since $X$ is locally compact Hausdorff, $X_\in... | $A$ and $B$ are closed in $X$. They need not be closed in the compactification $X_\infty$. You could try to fix this by replacing them with their closures in $X_\infty$, but then these need not be disjoint.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Factorization problem Find $m + n$ if $m$ and $n$ are natural numbers such that: $$\frac {m+n} {m^2+mn+n^2} = \frac {4} {49}\;.$$
My reasoning:
Say: $$m+n = 4k$$ $$m^2+mn+n^2 = 49k$$
It follows:$$(m+n)^2 = (4k)^2 = 16k^2 \Rightarrow m^2+mn+n^2 + mn = 16k^2 \Rightarrow mn = 16k^2 - 49k$$
Since: $$mn\gt0 \Rightarrow 16k^... | Observe that $k$ must be a non-zero integer.
We know that $m, n$ are the roots of the quadratic equation
$$X^2 - 4kX + (16k^2 - 49k)$$
The roots, from the quadratic equation, are
$$ \frac { 4k \pm \sqrt{(4k)^2 - 4(16k^2 - 49k) }} {2} = 2k \pm \sqrt{ 49k - 12k^2}$$
The expression in the square root must be a perfect s... | {
"language": "en",
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Classifying mathematical "coincidences" Doing homework a few years ago, I noticed that the sum of the squares of $88$ and $33$ is $8833$. What would this kind of mathematical "curiosity" be called? Does this or any other similar coincidence have any deeper meaning or structure?
| Alf van der Poorten wrote a paper, The Hermite-Serret algorithm and $12^2+33^2$. He notes a number of similar curiosities, such as $$25840^2+43776^2=2584043776$$ and $$1675455088^2+3734621953^2=16754550883734621953$$ and develops the theory behind these things, including a discussion of the way to go from a solution o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/290221",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$9^n \equiv 1 \mod 8$ I would like someone to check this inductive proof (sketch)
The base case is clear. For the inductive step, it follows that $8 \mid 9^{n+1} - 9 = 9(9^n - 1)$ by the indutive hyp. So $9^{n+1} \equiv 9 \equiv 1 \mod 8$.
Feedback would be appreciated.
| I'm assuming you mean what you say when you state your work as a proof "sketch".
The base case is clear. For the inductive step, it follows that $8 \mid 9^{n+1} - 9 = 9(9^n - 1)$ by the indutive hyp. So $9^{n+1} \equiv 9 \equiv 1 \mod 8$.
In your final write up, I'd suggest you "fill in" a bit of detail: e.g., to "... | {
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"timestamp": "2023-03-29T00:00:00",
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Can every diagonalizable matrix be diagonalized into the identity matrix? I'm a chemistry major and I haven't taken much math, but this came up in a discussion of quantum chemistry and my professor said (not very confidently) that if a matrix is diagonalizable, then you should be able to diagonalize it to the identity ... | Take the $0$ $n\times n$ matrix. It's already diagonal (and symmetrical) but certainly can't be diagonalized to the identity matrix.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/290340",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Mulitnomial Coefficient Power on a number using the binomial coefficient $_nC_r=\binom nr$, find the coefficient of $(wxyz)^2$ in the expansion of $(2w-x+3y+z-2)^n$. The answer key says its $n=12$, $r= 2\times2\times2\times2\times4$ in one of the equation for $_nC_r$. Why is there a $4$ there ? is it because there are ... | The coefficient of $a^2b^2c^2d^2e^{n-8}$ in $(a+b+c+d+e)^n$ is the multinomial coefficient $\binom n{2,2,2,2,n-8}$. The $n-8$ is needed because the exponenents need to add up to $n$, anything else would make the multinomial coefficient undefined. For $n=12$ you get $n-8=4$, so I suppose that is where your $4$ comes fro... | {
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"timestamp": "2023-03-29T00:00:00",
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Books for Self learning math from the ground up I am a CSE graduate currently working as a .NET and android developer. I feel my poor basic in mathematics is hampering my improvements as a programmer.
I want to achieve a sound understanding on the basics of mathematics so that i can pick up a book on 3D graphics progra... | Let me propose a different tack since you have a clear goal. Pick up a book on 3D graphics programming or algorithms, and if you come across something that intimidates you too much to get by on your own, ask about it here. We will be able to direct you to exact references to better understand the material in this way. ... | {
"language": "en",
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$n\times n$ board, non-challenging rooks Consider an $n \times n$ board in which the squares are colored black and white in
the usual chequered fashion and with at least one white corner square.
(i) In how many ways can $n$ non-challenging rooks be placed on the white
squares?
(ii) In how many ways can $n$ non-challeng... | We are looking at all permutations of $n$ that are (white squares case) parity-perserving, or (black square case) parity-inversing. If $n$ is even the black squares case is equivalent to the white square case (by a vertical reflection for instance), and if $n$ is odd the black square case has no solutions (the odd numb... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solve the recurrence $y_{n+1} = 2y_n + n$ for $n\ge 0$ So I have been assigned this problem for my discrete math class and am getting nowhere. The book for the class doesn't really have anything on recurrences and the examples given in class are not helpful at all. I seem to be going in circles with the math. Any help ... | Using ordinary generating functions
$$y_{n+1}=2y_n+n$$
gets transformed into
$$\sum_{n=0}^\infty y_{n+1}x^n=2\sum_{n=0}^\infty y_nx^n+\sum_{n=0}^\infty nx^n$$
$$\sum_{n=0}^\infty y_{n+1}x^n=2 y(x)+x\sum_{n=1}^\infty nx^{n-1}$$
$$\sum_{n=0}^\infty y_{n+1}x^n=2 y(x)+x\frac{1}{(1-x)^2}$$
$$\sum_{n=0}^\infty y_{n+1}x^{n+1}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that $\frac{z-1}{\mathrm{Log(z)}}$ is holomorphic off $(-\infty,0]$ Let $f(z)=\frac{z-1}{Log(z)}$ for $z\neq 1$ and $f(1)=1$. Show that $f$ is holomorphic on $\mathbb{C}\setminus(-\infty,0]$.
I know it looks like an easy problem, but I got stuck and need some clarification. The way I see it, I need to show that $... | If $g$ is holomorphic on an open set $G$, $a\in G$, and $g(a)=0$, then there is a positive integer $n$ and a holomorphic function $h$ on $G$ such that $g(z)=(z-a)^nh(z)$ for all $z\in G$, and $h(a)\neq 0$. (This $n$ is the multiplicity of the zero of $g$ at $a$.)
Since $\mathrm{Log}(1)=0$, and $\mathrm{Log}'(1)=1\neq ... | {
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Is this set closed under addition or multiplication or both and why? $\{-1,0,1\}$
Please give an explanation and also tell me what does closed under addition and multiplication mean.
Different definitions are given everywhere.
| A set $X$ is closed under addition if $x+y\in X$ for any $x,y\in X$. It is closed under multiplication if $x\times y\in X$ for any $x,y\in X$. Note that $x$ and $y$ may or may not be equal.
The set $\{-1,0,1\}$ is closed under multiplication but not addition (if we take usual addition and multiplication between real nu... | {
"language": "en",
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"source": "stackexchange",
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Does $a!b!$ always divide $(a+b)!$ Hello the question is as stated above and is given to us in the context of group theory, specifically under the heading of isomorphism and products. I would write down what I have tried so far but I have made very little progress in trying to solve this over the last few hours!
| The number of ways to choose $a$ objects out of $a+b$ objects if order matters in the selection is $(a+b)\times(a+b-1)\times\cdots\times((a+b)-(a-1))=\frac{(a+b)!}{b!}$, since there are $a+b$ ways to choose the first object, $a+b-1$ ways to choose the second object from the remaining ones, and so on.
However, $a!$ perm... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Inverse of a diagonal matrix plus a Kronecker product? Given two matrices $X$ and $Y$, it's easy to take the inverse of their Kronecker product:
$(X\otimes Y)^{-1} = X^{-1}\otimes Y^{-1}$
Now, suppose we have some diagonal matrix $\Lambda$ (or more generally an easily inverted matrix, or one for which we already know t... | Let $C := \left\{ c_{i,j} \right\}_{i,j=1}^N$ and $A := \left\{ a_{i,j} \right\}_{i,j=1}^n$ be symmetric matrices. The spectral decompositions of the two matrices read $A = O^T D_A O$ and $C = U^T D_C U$ where $D_A := Diag(\lambda_k)_{k=1}^n$ and $D_C := Diag(\mu_k)_{k=1}^N$ and $O \cdot O^T = O^T \cdot O = 1_n$ and $... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Comparing value of two definite integrals ($x^nlnx$) I need compare these two integrals:
$$
(1) \int_{a}^{b}x^n lnx dx\space \space
(2) \int_{a}^{b}x^{n+1} lnx dx
$$
for the following values of [a, b]: (A) [1, 2] for both integrals, (B) [0.5, 1] for both integrals and (C) [0.5, 1] for (1) and [0.3, 1] for (2).
What... | You can give an elegant answer without touching the integrals, using integral properties.
It is know that if $f(x)$ and $g(x)$ are Riemann integrable functions over the closed interval $[a,b]$, and such that $f(x) \geq g(x)$, then $\int_{a}^{b} f(x) dx \geq \int_{a}^{b} g(x) dx$. The same thing goes for $>$ and the opp... | {
"language": "en",
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If the absolute value of a function is Riemann Integrable, then is the function itself integrable? I am trying to check the converses of a few theorems.
I know that that if $g$ is integrable then $|g|$ is integrable. However, if $|g|$ is Riemann Integrable, then is $g$ Rieman integrable?
I know that if $g$ is integrabl... | Let $f(x)=1$ when $x$ is rational, $-1$ when $x$ is irrational. Interval say $[0,1]$.
| {
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"source": "stackexchange",
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Plotting for solution for $y=x^2$ and $x^2 + y^2 = a $ Consider the system $$y=x^2$$ and $$x^2 + y^2 = a $$for $x>0$, $y>0$, $a>0$.
Solving for equations give me $y+y^2 = a$, and ultimately $$y = \frac {-1 + \sqrt {4a+1}} {2} $$ (rejected $\frac {-1 - \sqrt {4a+1}} {2} $ since $y>0$).
The next part is to plot on the $... | Yes, it is insufficient.
You should notice that this equation is "special:"
$$x^2 + y^2 = a$$
This is the graph of a circle, radius $\sqrt{a}$.
So, your graph should contain both the parabola and the part of the circle in the region in question.
Here's a link to a graph from Wolfram Alpha which may help give some intui... | {
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Given an $m$ how to find $k$ such that $k m + 1$ is a perfect square Any way other than trying linearly for all values of $k$.
| If $km+1=n^2$ for some $n$ and $k=am+b$ for some $a,b\in\mathbb{Z}$, $0\le b<m$. Then
$$k=\frac{n^2-1}{m}=a^2m+2b+\frac{b^2-1}{m}$$
So we can know $k$ is integer if and only if $b^2\equiv 1 \pmod{m} $.
For example, if $b=\pm 1$ we get $k=a^2m\pm 2$ and $km+1=(am\pm1)^2$
| {
"language": "en",
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"source": "stackexchange",
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intersection of sylow subgroup I am in need of
*
*Example of a group $G$ a subgroup $A$ which not normal in $G$ and $p$ sylow subgoup $B$ of $G$ such that $A \cap B$ is a $p$-sylow subgroup of $A$.
A detailed solution will be helpful.
| If you want to construct an example which is non-trivial under several points of view, that is $A$ and $B$ and $G$ all distinct, and $A$ is not a $p$-group, you may find it as follows.
Find a finite group $G$ which has a non-normal subgroup $A$ which is not a $p$-group, but whose order is divisible by $p$. Let $S$ be a... | {
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For which $a$ does the equation $f(z) = f(az) $ has a non constant solution $f$ For which $a \in \mathbb{C} -\ \{0,1\}$ does the equation $f(z) = f(az) $ has a non constant solution $f$ with $f$ being analytical in a neighborhood of $0$.
My attempt:
First, we can see that any such solution must satisfy:
$f(z)=f(a^kz)$ ... | Thank you all very much. For completeness, I will write here a sctach of the soultion:
For $|a|=1$ we can take $f(z)=z^k$ with $k = 2\pi/Arg(a) $.
For $|a|>1$, we can notice that actually $f(z)=f(a^kz)$ for all $k \in \mathbb{Z}$, so the solution is similar to the case $|a|<1$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Looking for a simple proof that groups of order $2p$ are up to isomorphism $\Bbb{Z}_{2p}$ and $D_p$ for prime $p>2$. I'm looking for a simple proof that up to isomorphism every group of order $2p$ ($p$ prime greater than two) is either $\mathbb{Z}_{2p}$ or $D_{p}$ (the Dihedral group of order $2p$).
I should note that ... | Use Cauchy Theorem
Cauchy's theorem — Let $G$ be a finite group and $p$ be a prime. If $p$ divides the order of $G$, then $G$ has an element of order $p$.
then you have an element $x\in G$ of order $2$ and another element $y\in G$ of order $p$. Now you have to show that $xy$ is of order $2p$
using commutativity we ge... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Can there be a well-defined set with no membership criteria? Prime numbers are a well-defined set with specific membership criteria. Can the same be said about "numbers"? Aren't numbers (that is, all numbers) a well defined set but without membership criteria? Anybody can say, given a particular object, whether that be... | A set must have membership criteria. Cantor defined sets by (taken from wikipedia)
A set is a gathering together into a whole of definite, distinct objects of our perception [Anschauung] and of our thought – which are called elements of the set.
As the elements in a set therefore are definite, we can describe them in... | {
"language": "en",
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Show that there are no nonzero solutions to the equation $x^2=3y^2+3z^2$ I am asked to show that there are no non zero integer solutions to the following equation
$x^2=3y^2+3z^2$
I think that maybe infinite descents is the key.
So I started taking the right hand side modulo 3 which gives me zero. Meaning that $X^2$ mu... | Assume $\,x,y,z\,$ have no common factor. Now let us work modulo $\,4\,$ : every square is either $\,0\,$ or $\,1\,$ , and since
$$3y^2+3z^2=-(y^2+z^2)=\begin{cases}0\,\;\;,\,y,z=0,2\\{}\\-1=3\,\;\;,\,y=1\,,\,z=0\,\,or\,\,y=0\,,\,z=1\\{}\\-2=2\;\;\,,\,y=z=1\end{cases}$$
so the only possibility is the first one, and thu... | {
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"timestamp": "2023-03-29T00:00:00",
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Holomorphic functions on unit disc Let $f,g$ be holomorphic on $\mathbb{D}:=\lbrace z\in\mathbb{C}:|z|<1\rbrace$, $f\neq0,g\neq0$, such that $$\frac{f^{\prime}}{f}(\frac{1}{n})=\frac{g^{\prime}}{g}(\frac{1}{n}) $$ for all natural $n\geq1$. Does it imply that $f=Cg$, where $C$ is some constant?
Let $A:=\lbrace\frac{1}{n... | Notice that your last statement is equivalent to ${f'g-g'f}=0$, since $f,g \neq 0$. Now there is no harm in dividing that expression by $g^2$. You get ${f'g-g'f}{g^2}=0$. So, $(f/g)'=0$ and so your result follows.
| {
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"source": "stackexchange",
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Is there a general formula for the antiderivative of rational functions? Some antiderivatives of rational functions involve inverse trigonometric functions, and some involve logarithms. But inverse trig functions can be expressed in terms of complex logarithms. So is there a general formula for the antiderivative of an... | Write the rational function as $$f(z) = \dfrac{p(z)}{q(z)} = \dfrac{p(z)}{\prod_{j=1}^n (z - r_j)}$$
where $r_j$ are the roots of the denominator, and $p(z)$ is a polynomial.
I'll assume $p$ has degree less than $n$ and the roots $r_j$ are all distinct.
Then the partial fraction decomposition of $f(z)$ is
$$ f(z) = \s... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Square of two positive definite matrices are equal then they are equal I have read that if $P, Q$ are two positive definite matrices such that $P^2=Q^2$, then $P=Q$.
I don't know why. Some one can help me? Thanks for any indication.
| It all boils down to this:
Proposition. Suppose $A$ is an $n\times n$ positive definite matrix. Then $A^2$ has an eigenbasis. Furthermore, given any eigenbasis $\{v_1, \ldots, v_n\}$ of $A^2$ such that for each $i$, $A^2v_i=\lambda_iv_i$ for some $\lambda_i>0$, we must have $Av_i=\sqrt{\lambda_i}v_i$.
I will leave th... | {
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$\gcd(m,n) = 1$ and $\gcd (mn,a)=1$ implies $a \cong 1 \pmod{ mn}$ I have $m$ and $n$ which are relatively prime to one another and $a$ is relatively prime to $mn$
and after alot of tinkering with my problem i came to this equality:
$a \cong 1 \pmod m \cong 1 \pmod n$
why is it safe to say that $a \cong 1 \pmod {mn}$?.... | It looks as if you are asking the following. Suppose that $m$ and $n$ are relatively prime. Show that if $a\equiv 1\pmod{m}$ and $a\equiv 1\pmod{n}$, then $a\equiv 1\pmod{mn}$.
So we know that $m$ divides $a-1$, and that $n$ divides $a-1$. We want to show that $mn$ divides $a-1$.
Let $a-1=mk$. Since $n$ divides $a-1$, ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Changing $(1-\cos(x))/x$ to avoid cancellation error for $0$ and $π$? I have to change this formula: $$\frac{1-\cos(x)}{x}$$ so that I can avoid the cancellation error. I can do this for 0 but not for $π$. So I get: $$\frac{\sin^2(x)}{x(1+\cos(x))}$$ which for $x$ close to $0$ gets rid of the cancellation error. But I ... | Another possiblity that avoids cancellation error at both places is
$$ \frac{2 \sin^2(x/2)}{x} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/291823",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why $f(x + x^3, y + y^3) \in L^1(\mathbb{R}^2)$, when $f(x, y) \in L^2(\mathbb{R}^2)$? How show that $f(x + x^3, y + y^3) \in L^1(\mathbb{R}^2)$, when $f(x, y) \in L^2(\mathbb{R}^2)$?
Can someone help me?
Thank you!
| The statement that $f(x, y) \in L^2(\mathbb{R}^2)$ is the same as the statement that $(f(x + x^3, y + y^3))^2(1 + 3x^2)(1 + 3y^2)$ is in $L^1(\mathbb{R}^2)$, which can be seen after a change of variables to $(u,v) = (x + x^3, y + y^3)$ for the latter. Inspired thus, we write
$$\int_{\mathbb{R}^2}|f(x + x^3, y + y^3)| =... | {
"language": "en",
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If $f(x) = x-5$ and $g(x) = x^2 -5$, what is $u(x)$ if $(u \circ f)(x) = g(x)$?
Let $f(x) = x-5$, $g(x) = x^2 -5$. Find $u(x)$ if $(u \circ f)(x) = g(x)$.
I know how to do it we have $(f \circ u) (x)$, but only because $f(x)$ was defined. But here $u(x)$ is not defined. Is there any way I can reverse it to get $u(x... | I think I figured out what my professor did now . . .
$(u \circ f)(x) = g(x)$
$(u \circ f)(f^{-1} (x)) = g( f^{-1}(x)) $
$\big((u \circ f) \circ f^{-1}\big)(x) = (g \circ f^{-1})(x) $
$\big(u \circ (f \circ f^{-1})\big)(x) = (g \circ f^{-1})(x) $
$u(x) = g(f^{-1}(x))$
$u(x) = g(x+5)$
I think this is right. Please co... | {
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definition of left (right) Exact Functors Let $P,Q$ be abelian categories and $F:P\to Q$ be an additive functor. Wikipedia states two definitions on left exact functors (right dually):
*
*$F$ is left exact if $0\to A\to B\to C\to 0$ is exact implies $0\to F(A)\to F(B)\to F(C)$ is exact.
*$F$ is left exact if $0\to ... | Assume 1. holds. First observe that $F$ preserves monomorphisms: If $i : A \to B$ is a monomorphism, then $0 \to A \xrightarrow{i} B \to \mathrm{coker}(i) \to 0$ is exact, hence also $0 \to F(A) \to F(B) \to F(\mathrm{coker}(i))$ is exact. In particular $F(i)$ is a monomorphism.
Now if $0 \to A \xrightarrow{i} B \xrigh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/292037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Finding $n$ such that $\frac{3^n}{n!} \leq 10^{-6}$ This question actually came out of a question. In some other post, I saw a reference and going through, found this, $n>0$.
Solve for n explicitly without calculator:
$$\frac{3^n}{n!}\le10^{-6}$$
And I appreciate hint rather than explicit solution.
Thank You.
| I would use Stirling's approximation $n!\approx \frac {n^n}{e^n}\sqrt{2 \pi n}$ to get $\left( \frac {3e}n\right)^n \sqrt{2 \pi n} \lt 10^{-6}$. Then for a first cut, ignore the square root part an set $3e \approx 8$ so we have $\left( \frac 8n \right)^n \lt 10^{-6}$. Now take the base $10$ log asnd get $n(\log 8 -... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/292122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
} |
Combinatorics Statistics Question The problem I am working on is:
An academic department with five faculty members—Anderson, Box, Cox, Cramer, and Fisher—must select two of its members to serve on a personnel review committee.
Because the work will be time-consuming, no one is anx-ious to serve, so it is decided that t... | most simple way to understand this problem.
(i just did this problem just now just now) lol
A, B, Co, Cr, and F exist.
pick 1 of 5 at random. (1/5)
pick another at random, but this time around there are only 4 choices. so (1/4)
multiply the two values. (1/20)
BUT! that's considering that A is picked at first try then B... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/292187",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Limit of $s_n = \int\limits_0^1 \frac{nx^{n-1}}{1+x} dx$ as $n \to \infty$ Let $s_n$ be a sequence defined as given below for $n \geq 1$. Then find out $\lim\limits_{n \to
\infty} s_n$.
\begin{align}
s_n = \int\limits_0^1 \frac{nx^{n-1}}{1+x} dx
\end{align}
I have written a solution of my own, but I would like to ... | Notice
(1) $\frac{s_n}{n} + \frac{s_{n+1}}{n+1} = \int_0^1 x^{n-1} dx = \frac{1}{n} \implies s_n + s_{n+1} = 1 + \frac{s_{n+1}}{n+1}$.
(2) $s_n = n\int_0^1 \frac{x^{n-1}}{1+x} dx < n\int_0^1 x^{n-1} dx = 1$
(3) $s_{n+1} - s_n = \int_0^1 \frac{d (x^{n+1}-x^n)}{1+x} = \int_0^1 x^n \frac{1-x}{(1+x)^2} dx > 0$
(2+3) $\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/292251",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 5,
"answer_id": 0
} |
Fermat's Little Theorem Transformation I am reading a document which states:
By Fermat's Little Theorem, $a^{p-1}\bmod p = 1$. Therefore, $a^{b^c}\bmod p = a^{b^c\bmod (p - 1)} \bmod p$
For the life of me, I cannot figure out the logic of that conclusion. Would someone mind explaining it? I will be forever in your d... | The key point is that if $\rm\ a^n = 1\ $ then exponents on $\rm\ a\ $ may be reduced mod $\rm\,n,\,$ viz.
Hint $\rm\quad a^n = 1\ \,\Rightarrow\,\ a^i = a^j\ \ { if} \ \ i\equiv j\,\ (mod\ n)\:$
Proof $\rm\ \ i = j\!+\!nk\:$ $\Rightarrow$ $\rm\:a^i = a^{j+nk} = a^j (a^n)^k = a^j 1^k = a^j\ \ $ QED
Yours is the speci... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/292365",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Proving the sum of the first $n$ natural numbers by induction I am currently studying proving by induction but I am faced with a problem.
I need to solve by induction the following question.
$$1+2+3+\ldots+n=\frac{1}{2}n(n+1)$$
for all $n > 1$.
Any help on how to solve this would be appreciated.
This is what I hav... | Think of pairing up the numbers in the series. The 1st and last (1 + n) the 2nd and the next to last (2 + (n - 1)) and think about what happens in the cases where n is odd and n is even.
If it's even you end up with n/2 pairs whose sum is (n + 1) (or 1/2 * n * (n +1) total)
If it's odd you end up with (n-1)/2 pairs who... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/292423",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
} |
Proving the equality in weak maximum principle of elliptic problems This one is probably simple, but I just can't prove the result.
Suppose that $\mathop {\max }\limits_{x \in \overline \Omega } u\left( x \right) \leqslant \mathop {\max }\limits_{x \in \partial \Omega } {u^ + }\left( x \right)$ and $\mathop {\min }\li... | $\left. \begin{gathered}
\mathop {\max }\limits_{x \in \overline \Omega } u\left( x \right) \leqslant \mathop {\max }\limits_{x \in \partial \Omega } {u^ + }\left( x \right)\mathop \leqslant \limits^{{u^ + } \leqslant \left| u \right|} \mathop {\max }\limits_{x \in \partial \Omega } \left| {u\left( x \right)} \righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/292510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is there a good repository for mathematical folklore knowledge? Among mathematicians there is lot of folklore knowledge for which it is not obvious how to find original sources. This knowledge circulates orally.
An example: Among math competition folks, a common conversation is the search for a function over the reals ... | One place that contains a lot of mathematics is the nLab. It is largely centered on higher category theory/homotopy theory but also contains a lot of general stuff. It is certainly not aiming to only contain folklore knowledge but it does contain a lot of it.
Wikipedia will also certainly contain folklore knowledge em... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/292565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
$Q=\Sigma q_i$ and its differentiation by one of its variables Suppose that $Q = q_1 + ... +q_n$.
why is
$$\frac{dQ}{dq_i} = \Sigma_{j=1}^{n}\frac{\partial q_j}{\partial q_i}$$?
Is it related to each $q_i$ being independent to other $q_i$'s?
| No, it is because differentiation is linear, ie, if $h=f+g$, then $\frac{d h}{d x} = \frac{d f}{d x} + \frac{d g}{d x}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/292626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove that $f:[0,\infty)\to\mathbb{R}$ where $f(x) := {1\over x}\cos({1\over x}),x>0$ ,does $f$ has the intermediate value property on $[0,\infty)$? Prove that $f:[0,\infty)\to\mathbb{R}$ where $f(x) := {1\over x}\cos({1\over x}),x>0$ does $f$ has the intermediate value property on $[0,\infty)$?
Attempts: In $\mathbb{R... | The function $f\colon [0,+\infty)\to \mathbb R$
$$
f(x) = \begin{cases}
\frac {1}{x} \cos \frac 1 x & \text{if $x>0$}\\\\
0 & \text {if $x=0$}
\end{cases}
$$
is not continuous but has the intermediate value property. In fact given two points $a,b \in [0,\infty)$ with $a<b$ you have two possibilities:
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/292686",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
What's the difference between $|\nabla f|$ and $\nabla f$? what's the difference between $|\nabla f|$ and $\nabla f$ for example in :
$$\nabla\cdot{\nabla f\over|\nabla f|}$$
| $|\nabla f|$ is the magnitude of the $\nabla f$ vector. The expression $\frac{\nabla f}{|\nabla f|}$ is thus a unit vector.
$\nabla$ (gradient) acts on a scalar to give a vector, and $\nabla\cdot$ (divergence) acts on a vector to give a scalar.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/292766",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
newton: convergence when calculating $x^2-2=0$ Find $x$ for which $x^2-2=0$ using the newton algorithm and $x_0 = 1.4$.
Then you get $x_{k+1} = x_k + \frac{1}{x_k} - \frac{x_k}{2}$.
How to show that you need 100 steps for 100 digits precision?
So I need to show for which $N$ it is $|x_N-\sqrt{2}| \leq 10^{-100}$ and th... | Try to show that $|x_{k+1}-\sqrt2|\leqslant\frac12(x_k-\sqrt2)^2$ hence $|x_k-\sqrt2|\leqslant2\delta_k$ implies that $|x_{k+1}-\sqrt2|\leqslant2\delta_{k+1}$ with $\delta_{k+1}=\delta_k^2$.
Then note that $\delta_0\lt10^{-2}$ hence $\delta_k\leqslant10^{-2^{k+1}}$ for every $k$, in particular $2\delta_6\lt2\cdot10^{-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/292891",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find $\lim_{x\to 0^+} \ln x\cdot \ln(1-x)$ Find $$\lim_{x\to 0^+} \ln x\cdot \ln(1-x)$$
I've been unable to use the arithmetic rules for infinite limits, as $\ln x$ approaches $-\infty$ as $x\to 0^+$, while $\ln(1-x)$approaches $0$ as $x\to 0^+$, and the arithmetic rules for the multiplication of infinite limits only a... | Hint: Another approach which is similar to @Mhenni's is:
When $x\to 0$ and we know that $\alpha(x)$ is very small then $$\ln(1+\alpha(x))\sim\alpha(x)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/293025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 5,
"answer_id": 3
} |
Euler graph - a question about the proof I have a question about the proof of this theorem.
A graph is Eulerian $\iff$ it is connected and all its vertices have even degrees.
My question concerns "$\Leftarrow$"
Let $T=(v_0, e_1, v_1, ..., e_m, v_m)$ be a trip in Eulerian graph G=(V, E) where vertices can repeat but edg... | You have a typo: it’s when $v_0\ne v_m$ that you can conclude that $v_0$ is incident to an odd number of edges of $T$. It’s incident to $v_1$, and any other vertices of $T$ to which it is incident must come in pairs, one just before it and one just after in the tour. But then, as you say, $T$ would not be maximal, so t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/293106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Linear algebra eigenvalues and diagonalizable matrix Let $A$ be an $n\times n$ matrix over $\mathbb{C}$.
First I don't understand why $AA^*$ can be diagnosable over $\mathbb{C}$.
And why $i+1$ can't be eigenvalue of $AA^*$?
Hope question is clear enough and I don't have any spelling mistake and used right expressions.
| Any Hermitian matrix is diagonalizable. All eigenvalues of a Hermitian matrix are real. These two facts (that you probably learnt) solve the question: Show your matrix is Hermitian and note that $1+i$ is not real.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/293172",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Compute $\sum_{k=1}^{n} \frac 1 {k(k + 1)} $ More specifically, I'm supposed to compute $\displaystyle\sum_{k=1}^{n} \frac 1 {k(k + 1)} $ by using the equality $\frac 1 {k(k + 1)} = \frac 1 k - \frac 1 {k + 1}$ and the problem before which just says that, $\displaystyle\sum_{j=1}^{n} a_j - a_{j - 1} = a_n - a_0$.
I can... | It means: find how much it sums to. In fact, you have already said everything you need to solve the problem. You only have to put 1 and 1 together to obtain 2.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/293244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Dirac's theorem question Give an example to show that the condition $\deg(v) \geq n/2$ in the statement of Dirac's theorem cannot be replaced by $\deg(v) \geq (n-1)/2$
The textbook gives the solution:
The complete bipartite graph $K_{n/2 - 1, n/2 + 1}$ if $n$ is even, and $K_{(n-1)/2, (n+1)/2}$ if $n$ is odd.
If anyone... | The complete bipartite graph $K_{(n-1)/2, ~ (n+1)/2}$ has $(n-1)/2 + (n+1)/2 = n$ vertices.
Each vertex has degree greater than or equal to $(n-1)/2$ but this graph does not contain any Hamiltonian cycles, so the conclusion of Dirac's theorem does not hold.
You may wish to consider, for example, $K_{1,2}$ or $K_{2,3}$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/293396",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
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