Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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what's the name of the theorem:median of right-triangle hypotenuse is always half of it This question is related to one of my previous questions.
The answer to that question included a theorem: "The median on the hypotenuse of a right triangle equals one-half the hypotenuse".
When I wrote the answer out and showed it a... | Here is a proof without words:
| {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "6",
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Combinatorics problem with negative possibilities I know how to solve the basic number of solutions equations, such as "find the number of positive integer solutions to $x_1 + x_2 + x_3$ = 12, with ". But I have no clue how to do this problem:
Find the number of solutions of $x_1+x_2-x_3-x_4 = 0$ in integers between -4... | Put $x_i+4=:y_i$. Then we have to solve
$$y_1+y_2=y_3+y_4$$
in integers $y_i$ between $0$ and $8$ inclusive. For given $p\geq0$ the equation $y_1+y_2=p$ has $p+1$ solutions if $0\leq p\leq 8$, and $17-p$ solutions if $9\leq p\leq 16$. It follows that the total number $N$ of solutions is given by
$$N=\sum_{p=0}^8(p+1)^2... | {
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How fill in this multiplication table? The following multiplication table was given to me as a class exercise. I should have all the necessary information to fill it completely in. However, I'm not sure how to take advantage of the relations I am given to fill it in?
The Question
A group has four elements $a,b,c$ and $... | Since each element has an inverse, each row and each column of the table must contain all four elements.
After filling in the row and the column of the identity element $c$, we have three $a$'s in the table, and it follows that the only place for the last $a$ is given by $b^2=a$. Trying to distribute four $d$'s in the... | {
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Existence and uniqueness of solution for a seemingly trivial 1D non-autonomous ODE So I was trying to do some existence and uniqueness results beyond the trivial setting. So consider the 1D non-autonomous ODE given by
$\dot{y} = f(t) - g(t) y $ where $f,g \geq 0$ are integrable and $f(t),g(t) \rightarrow 0$ for $t \rig... | Assuming you meant $\dot{y} = f(t) - g(t) y$, this is a linear differential equation and has the explicit solutions $y(t) = \mu(t)^{-1} \int \mu(t) f(t)\ dt$ where
$\mu(t) = \exp(\int g(t)\ dt)$, from which it is clear that you have global existence of
solutions. Uniqueness follows from the standard existence and un... | {
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Coherent sheaves on a non-singular algebraic variety Grothendieck wrote in his letter to Serre(Nov. 12,1957) that every coherent algebraic sheaf on a non-singular algebraic variety(not necessarily quasi-projective) is a quotient of a direct sum of sheaves defined by divisors.
I think "sheaves defined by divisors" means... | This is proved for any noetherian separated regular schemes in SGA 6, exposé II, Corollaire 2.2.7.1 (I learn this result from a comment here: such schemes are "divisorial".) To see that this answers your question, look at op. cit. Définition 2.2.3(ii).
| {
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Help me understand the (continuous) uniform distribution I think I didn't pay attention to uniform distributions because they're too easy.
So I have this problem
*
*It takes a professor a random time between 20 and 27 minutes to walk from his home to school every day. If he has a class at 9.00 a.m. and he leaves... | Your arrival time is at a constant interva; $[a,b]$ and the uniform distribution gives,
$\int_{a}^{x} \frac{dt}{27-20}$
Your starting point, 8:37, is your time 0 and you want to make it to your class by 9. Your minimum walking time is 20 mins which would give make you still on time. But for a walking time of more than ... | {
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Find all complex numbers $z$ satisfying the equation I need some help on this question. How do I approach this question?
Find all complex numbers $z$ satisfying the equation
$$
(2z - 1)^4 = -16.
$$
Should I remove the power of $4$ of $(2z-1)$ and also do the same for $-16$?
| The answer to this problem lies in roots of a polynomial.
From an ocular inspection we know we will have complex roots and they always
Come in pairs! The power of our equation is 4 so we know will have a pair of
Complex conjugates. I believe it is called the Fundamental
Theorem.
Addendum
A key observation is how to rep... | {
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"source": "stackexchange",
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The difference between m and n in calculating a Fourier series I am studying for an exam in Differential Equations, and one of the topics I should know about is Fourier series. Now, I am using Boyce 9e, and in there I found the general equation for a Fourier series:
$$\frac{a_0}{2} + \sum_{m=1}^{\infty} (a_m cos\frac{m... | You should know by now that $n$ and $m$ are just dummy indices. You can interchange them as long as they represent the same thing, namely an arbitrary natural number.
If
$$f(x) = \sum\limits_{n = 1}^\infty {{b_n}\sin \frac{{n\pi x}}{L}}$$
we can multiply both sides by ${\sin \frac{{m\pi x}}{L}}$ and integrate from $x... | {
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What is the meaning of a.s.? What is the meaning of a.s. behind a limit formula (I found this in a paper about stochastic processes) , or sometimes P-a.s.?
| It means almost surely. P-a.s means almost surely with respect to probability measure P. For more details wiki out "almost sure convergence".
Let me give some insights: When working with convergence of sequences of random variables(in general stochastic processes), it is not necessary for convergence to happen for all ... | {
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Solve for variable inside multiple power in terms of the powers. I'm a programmer working to write test software. Currently estimates the values it needs with by testing with a brute force algorithm. I'm trying to improve the math behind the software so that I can calculate the solution(s) instead. I seem to have come ... | Okay so I came up with a that allows me to approximate the answer if $\frac {y_2}{y_1}$ is rational (which it will be in my case because I have limited precision).
I can re-express the original equation as $$(Bx_1+1)^{\frac {y_2}{y_1}}=Bx_2+1$$
If $\frac {y_2}{y_1}$ rational I can change it to $\frac ND$ where $N$ and ... | {
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Counterexample to show that the class of Moscow spaces is not closed hereditary. What is a counterexample to show that the class of Moscow spaces is not closed hereditary?
(A space $X$ is called Moscow if the closure of every open $U \subseteq X$ is the union of a family of G$_{\delta}$-subsets of $X$.)
| This is essentially copied from A. V. Arhangel'skii, Moscow spaces and topological groups, Top. Proc., 25; pp.383-416:
Let $D ( \tau )$ be an uncountable discrete space, and $\alpha D ( \tau )$ the one point compactification of $D ( \tau )$. Then $D ( \tau )$ is a Moscow space, and $D ( \tau )$ is G$_\delta$-dense in... | {
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Getting the Total value from the Nett I can't figure out this formula; I need some help to write it out for a php script.
I have a value of $\$80$.
$\$80$ is the profit from a total sale of $\$100$; $ 20\%$ is the percentage margin for the respective product.
Now I just have the $\$80$ and want to get the total figure... | If you mean that you get $80$ from $100$ after a $20$ per cent discount and want to reverse the process, multiply by $\frac{100}{100-20}=\frac{5}{4}$. The formula you need is $y=\frac{5x}{4}$, so if $x=80$, then $y=100$.
| {
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Finding a High Bound on Probability of Random Set first time user here. English not my native language so I apologize in advance. Taking a final in a few weeks for a graph theory class and one of the sample problems is exactly the same as the $k$-edge problem.
We need prove that if each vertex selected with probabilit... | I was the one who asked the original question that you linked to. Just for fun I tried to see if I could prove a better bound. I was able to prove a bound even better than that. Here's a hint: you can use indicator variables and the second moment method.
| {
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Convergence of alternating series based on prime numbers I've been experimenting with some infinite series, and I've been looking at this one,
$$\sum_{k=1}^\infty (-1)^{k+1} {1\over p_k}$$
where $p_k$ is the k-th prime. I've summed up the first 35 terms myself and got a value of about 0.27935, and this doesn't seem clo... | As mentioned, this series has an expansion given by the OEIS. This series is mentioned in many sources, such as Mathworld, Wells, Robinson & Potter and Weisstein.
These sources all seem to imply that, though the series converges, no known "closed form" for this sum exists.
| {
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Basic theory about divisibility and modular arithmetic I am awfully bad with number theory so if one can provide a quick solution of this, it will be very much appreciated!
Prove that if $p$ is a prime with $p \equiv 1(\mod4) $ then there is an integer $m$ such that $p$ divides $m^2 +1$
| I will assume that you know Wilson's Theorem, which says that if $p$ is prime, then $(p-1)!\equiv -1\pmod{p}$.
Let $m=\left(\frac{p-1}{2}\right)^2$. We show that if $p\equiv 1\pmod{4}$, then $m^2\equiv -1\pmod{p}$. This implies that $p$ divides $m^2+1$.
The idea is to pair $1$ with $p-1$, $2$ with $p-2$, $3$ with $p-3$... | {
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Computing $\operatorname{Ext}^1_R(R/x,M)$ How to compute $\operatorname{Ext}^1_R(R/(x),M)$ where $R$ is a commutative ring with unit, $x$ is a nonzerodivisor and $M$ an $R$-module?
Thanks.
| There is an alternative way to doing this problem than taking a projective resolution. Consider the ses of $R$ - modules
$$0 \to R \stackrel{x}{\to} R \to R/(x) \to 0$$
where the multiplication by $x$ map is injective because it is not a zero divisor in $R$. Now we recall a general fact from homological algebra that sa... | {
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prove sum of divisors of a square number is odd Don't know how to prove that sum of all divisors of a square number is always odd.
ex: $27 \vdots 1,3,9,27$; $27^2 = 729 \vdots 1,3,9,27,81,243,729$; $\sigma_1 \text{(divisor
function)} = 1 + 3 + 9 + 27 + 81 + 243 + 729 = 1093$ is odd;
I think it somehow connected to a ... | The divisors $1=d_1<d_2<\cdots <d_k=n^2$ can be partitioned into pairs $(d, \frac {n^2}d)$, except that there is no partner for $n$ itself.
Therefore, the number of divisors is odd.
Thus if $n$ itself is odd (and so are all its divisors), we have the sum of an odd number of odd numebrs, hence the result is odd.
But if... | {
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$p$ an odd prime, $p \equiv 3 \pmod 8$. Show that $2^{(\frac{p-1}{2})}*(p-1)! \equiv 1 \pmod p$ $p$ an odd prime, $p \equiv 3 \pmod 8$. Show that $2^{\left(\frac{p-1}{2}\right)}\cdot(p-1)! \equiv 1 \pmod p$
From Wilson's thm: $(p-1)!= -1 \pmod p$.
hence, need to show that $2^{\left(\frac{p-1}{2}\right)} \equiv -1 \pmo... | All the equalities below are in the ring $\mathbb{Z}/p\mathbb{Z}$.
Note that $-1 = (p-1)! = \prod_{i=1}^{p-1}i = \prod_{i=1}^{\frac{p-1}{2}}(2i-1)\prod_{i=1}^{\frac{p-1}{2}} 2i = 2^{\frac{p-1}{2}} \prod_{i=1}^{\frac{p-1}{2}}(2i-1)\prod_{i=1}^{\frac{p-1}{2}} i$
Now let $S_1, S_2$ be the set of respectively all odd and... | {
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Why the principle of counting does not match with our common sense Principle of counting says that
"the number of odd integers, which is the same as the number of even integers, is also the same as the number of integers overall."
This does not match with my common sense (I am not a mathematician, but a CS student).
C... | Because sets of numbers can be infinitely divisible. See this Reddit comment.
I think his intuition comes from the fact that the world is discrete in practice. You have 2x more atoms in [0, 2cm] than in [0, 1cm]. If you are not looking at something made of atoms, let's say you have 2x more Planck lengths in [0, 2cm] th... | {
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Showing that $\langle p\rangle=\int\limits_{-\infty}^{+\infty}p |a(p)|^2 dp$ How do I show that $$\int \limits_{-\infty}^{+\infty} \Psi^* \left(-i\hbar\frac{\partial \Psi}{\partial x} \right)dx=\int \limits_{-\infty}^{+\infty} p \left|a(p)\right|^2dp\tag1$$
given that $$\Psi(x)=\frac{1}{\sqrt{2 \pi \hbar}}\int \limits_... | The conclusion follows from the Fourier inversion formula (in distribution sense):
$$\begin{align*}
&\int_{-\infty}^{\infty} \Psi^{*} \left( -i\hbar \frac{\partial \Psi}{\partial x} \right) \, dx \\
&= \int_{-\infty}^{\infty} \left( \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} a(p)^{*}e^{-ipx/\hbar} \, dp \right)... | {
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Dihedral group $D_{8}$ as a semidirect product $V\rtimes C_2$? How do I show that the dihedral group $D_{8}$ (order $8$) is a semidirect product $V\rtimes \left\langle \alpha \right\rangle $, where $V$ is Klein group and $%
\alpha $ is an automorphism of order two?
| I like to think about $D_8$ as the group of invariants of a square.
So the our group $V$ is given by the identity, the 2 reflections that have no fix points and their product which is 180°-rotation.
Semidirect product can be characterized as split short exact sequences of groups. This is just a fancy way to say the fo... | {
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How to explain that division by $0$ yields infinity to a 2nd grader How do we explain that dividing a positive number by $0$ yields positive infinity to a 2nd grader? The way I intuitively understand this is $\lim_{x \to 0}{a/x}$ but that's asking too much of a child. There's got to be an easier way.
In response to the... | let us consider that any number divided by zero is undefined.
You can let the kid know in this way:
Division is actually splitting things, for example consider you have 4 chocolates and if u have to distribute those 4 chocolates among 2 of your friends, you would divide it(4) by 2(i.e : 4/2) = 2.
Now consider this, yo... | {
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example of irreductible transient markov chain Can anyone give me a simple example of an irreductible (all elements communicate) and transient markov chain?
I can't think of any such chain, yet it exists (but has to have an infinite number of elements)
thanks
| A standard example is asymmetric random walk on the integers: consider a Markov chain with state space $\mathbb{Z}$ and transition probability $p(x,x+1)=3/4$, $p(x,x-1)=1/4$. There are a number of ways to see this is transient; one is to note that it can be realized as $X_n = X_0 + \xi_1 + \dots + \xi_n$ where the $\x... | {
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How do you compute the number of reflexive relation? Given a set with $n$ elements
I know that there is $2^{n^2}$ relations, because there are $n$ rows and $n$ columns and it is either $1$ or $0$ in each case, but I don't know how to compute the number of reflexive relation. I am very dumb. Can someone help me go throu... | A relation $R$ on $A$ is a subset of $AXA$.
If $A$ is reflexive, each of the $n$ ordered pairs $(a,a)$ belonging to $A$ must be in $R$.
So the remaining $n^2-n$ ordered pairs of the type $(a,b)$ where $a!=b$ may or may not be in R.
So each ordered pair has now 2 choices, to be in $R$ or to not be in $R$.
Hence numb... | {
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Prove that $S_n$ is doubly transitive on $\{1, 2,..., n\}$ for all $n \geqslant 2$.
Prove that $S_n$ is doubly transitive on $\{1, 2,\ldots, n\}$ for all $n \geqslant 2$.
I understand that transitive implies only one orbit, but...
| In fact, $S_n$ acts $n$-fold transitive on $\{1,\ldots,n\}$ (hence the claim follows from $n\ge 2$), i.e. for $n$ different elements (which could that be?) $i_1,\ldots ,i_n$ you can prescribe any $n$ different elements $j_1,\ldots,j_n$ and dan find (exactly) one element $\sigma\in S_n$ such that $\sigma(i_k)=j_k$ for a... | {
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using contour integration I am trying to understand using contour integration to evaluate definite integrals. I still don't understand how it works for rational functions in $x$. So can anyone please elaborate this method using any particular function like say $\int_0^{\infty} \frac {1}{1+x^3} \ dx$. I'ld appreciate th... | What we really need for contour integration by residues to work is a closed contour. An endpoint of $\infty$ doesn't matter so much because we can treat it as a limit as $R \to \infty$, but an endpoint of $0$ is a problem. Fortunately, this integrand is symmetric under rotation by $2 \pi/3$ radians. So we consider a... | {
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Two curious "identities" on $x^x$, $e$, and $\pi$ A numerical calculation on Mathematica shows that
$$I_1=\int_0^1 x^x(1-x)^{1-x}\sin\pi x\,\mathrm dx\approx0.355822$$
and
$$I_2=\int_0^1 x^{-x}(1-x)^{x-1}\sin\pi x\,\mathrm dx\approx1.15573$$
A furthur investigation on OEIS (A019632 and A061382) suggests that $I_1=\frac... | You made a very nice observation! Often it is important to make a good guess than just to solve a prescribed problem. So it is surprising that you made a correct guess, especially considering the complexity of the formula.
I found a solution to the second integral in here, and you can also find a solution to the first ... | {
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An inequality about the gradient of a harmonic function Let $G$ a open and connected set. Consider a function $z=2R^{-\alpha}v-v^2$ with $R$ that will be chosen suitably small, where $v$ is a harmonic function in $G$, and satisfies
$$|x|^\alpha\leqslant v(x)\leqslant C_0|x|^\alpha. \ \ (*)$$
Then,
$$\Delta z+f(z)=-2|... | This is an answer to a question raised in comment to another answer. Since it is of independent interest (perhaps of more interest than the original answer), I post it as a separate answer.
Question: How do we construct homogeneous harmonic functions that are positive on a cone?
Answer. Say, we want a positive homog... | {
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Can a cubic that crosses the x axis at three points have imaginary roots? I have a cubic polynomial, $x^3-12x+2$ and when I try to find it's roots by hand, I get two complex roots and one real one. Same, if I use Mathematica. But, when I plot the graph, it crosses the x-axis at three points, so if a cubic crosses the x... | The three roots are, approximately, $z = -3.545$, $0.167$ or $3.378$.
A corollary of the fundamental theorem of algebra is that a cubic has, when counted with multiplicity, exactly three roots over the complex numbers. If your cubic has three real roots then it will not have any other roots.
I've checked the plot, and ... | {
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coordinate system, nonzero vector field I'm interested in the following result (chapter 5, theorem 7 in volume 1 of Spivak's Differential Geometry):
Let $X$ be a smooth vector field on an $n$-dimensional manifold M with $X(p)\neq0$ for some point $p\in M$. Then there exists a coordinate system $x^1,\ldots,x^n$ for $U$... | As a partial answer: uniqueness of integral curves I believe boils down to the existence and uniqueness theorem from ordinary differential equations.
Furthermore, any coordinate chart can be translated so that 0 maps to the point $p$ on the manifold, whatever point you're interested in. You're just doing a composition... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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$S_n$ acting transitively on $\{1, 2, \dots, n\}$ I am reading Dummit and Foote, and in Section 4.1: Group Actions and Permutation Representations they give the following example of a group action:
The symmetric group $G = S_n$ acts transitively in its usual action as permutations on $A = \{1, 2, \dots, n\}$. Note th... | $\sigma \in \textrm{ Stab } (i)$ for $1\le i \le n$ iff $\sigma $ fixes $i.$ You just have to count the number of permutations that fix $i$ - working in the usual order, there are $n-1$ choices for the image of the first element in the domain, $n-2$ for the second, and so on, so that $|\textrm{ Stab } (i) | = (n-1)!.$ ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is the difference between Tautochrone curve and Brachistochrone curve as both are cycloid? What is the difference between Tautochrone curve and Brachistochrone curve as both are cycloid?
If possible, show some reference please?
| Mathematically, they both are the same curve but they arise from slightly different but related problems.
While the Brachistochrone is the path between two points that takes shortest to traverse given only constant gravitational force, the Tautochrone is the curve where, no matter at what height you start, any mass wil... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/242885",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Are there infinite many integer $n\ge 0$ such that $10^{2^n}+1$ prime numbers? It is clear to see that 11 and 101 are primes which sum of digit is 2. I wonder are there more or infinte many of such prime.
At first, I was think of the number $10^n+1$. Soon, I knew that $n\neq km$ for odd $k>1$, otherwise $10^m+1$ is a f... | Many people wonder the same thing you do. Wilfrid Keller keeps track of what they find out. So far: prime for $n=0$ and $n=1$ only; known to be composite for all other $n$, $2\le n\le23$, and many other values of $n$. The first value for which primality status is unknown is $n=24$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/242949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Mutiple root of a polynomial modulo $p$ In my lecture notes of algebraic number theory they are dealing with the polynomial $$f=X^3+X+1, $$ and they say that
If f has multiple factors modulo a prime $p > 3$, then $f$ and $f' = 3X^2+1$ have a common factor modulo this prime $p$, and this is the linear factor $f − (X/3... | *
*If $f$ has a multiple factor, say $h$ (in any field containing the current base field), then with appropriate $g$, we have
$$f(x)=h(x)^2\cdot g(x)$$
If you take its derivative, it will be still a multiple of $h(x)$, so it is a common factor of $f$ and $f'$.
If polynomials $u$ and $v$ have common factors, then all... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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can open sets be covered with another open set not much bigger? Is that correct that, for any open set $S \subset \mathbb{R}^n$, there exists an open set $D$ such that $S \subset D$ and $D \setminus S$ has measure zero?
I think it is correct and I guess I have seen the proof somewhere before, but I cannot find it in an... | Assume that $S\subset D$ with $S,D$ open and $\mu(D\setminus S)=0$.
If $D$ is not contained in $\overline S$, then the nonempty open set $D\setminus \overline S$ has positive measure. Therefore, $D\subseteq \overline S$. Therefore any open set $S$ with the property that $$\tag1\partial S\subseteq \partial(\mathbb R^n\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Need someone to show me the solution Need someone to show me the solution. and tell me how !
$$(P÷N) × (N×(N+1)÷2) + N×(1-P) = N×(1-(P÷2)) + (P÷2)$$
| \begin{align}
\dfrac{P}{N} \times \dfrac{N(N+1)}2 + N\times (1-P) & = \underbrace{P \times \dfrac{N+1}{2} + N \times (1-P)}_{\text{Cancelling out the $N$ from the first term}}\\
& = \underbrace{\dfrac{PN + P}2 + N - NP}_{\text{$P \times (N+1) = PN + P$ and $N \times (1-P) = N - NP$}}\\
& = \underbrace{\dfrac{PN + P +2N... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Question about of Fatou's lemma in Rick Durrett's book. In Probability Theory and Examples, Theorem $1.5.4$, Fatou's Lemma, says
If $f_n \ge 0$ then
$$\liminf_{n \to \infty} \int f_n d\mu \ge \int \left(\liminf_{n \to \infty} f_n \right) d\mu. $$
In the proof, the author says
Let $E_m \uparrow \Omega$ be sets of f... | At the beginning of section 1.4 where Durrett defines the integral he assumes that the measure $\mu$ is $\sigma$-finite, so I guess it is a standing assumption about all integrals in this book that the underlying measure satisfies this. Remember that it is a probability book, so his main interest are finite measures.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Limit $\lim_{x\rightarrow \infty} x^3 e^{-x^2}$ using L'Hôpital's rule I am trying to solve a Limit using L'Hôpital's rule with $e^x$
So my question is how to find $$\lim_{x\rightarrow \infty} x^3 e^{-x^2}$$
I know to get upto this part here, but I'm lost after that
$$\lim_{x\rightarrow \infty} \frac{x^3}{e^{x^2}}$$
| $$\begin{align}
\lim_{x\rightarrow \infty} \dfrac{x^3}{e^{x^2}}
&=\lim_{x\rightarrow \infty} \dfrac{3x^2}{e^{x^2}2x}\\
&=\lim_{x\rightarrow \infty} \dfrac{3x}{e^{x^2}2}\\
&=\lim_{x\rightarrow \infty} \dfrac{3}{e^{x^2}.2.2x}\\
&=0
\end{align}$$
| {
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"source": "stackexchange",
"question_score": "1",
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The wedge sum of two circles has fixed point property? The wedge sum of two circles has fixed point property?
I'm trying to find a continuous map from the wedge sum to itself, that this property fails, I couldn't find it, I need help.
Thanks
| If by circle you mean $S^1$, and the fixed point property is the claim that every continuous map into itself has a fixed point, for two circles like so:
consider the map that rotates $A$ by 90 degrees, and sends all of $B$ to the image of $x$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/243588",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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On Ceva's Theorem? The famous Ceva's Theorem on a triangle $\Delta \text{ABC}$
$$\frac{AJ}{JB} \cdot \frac{BI}{IC} \cdot \frac{CK}{EK} = 1$$
is usually proven using the property that the area of a triangle of a given height is proportional to its base.
Is there any other proof of this theorem (using a different... | I’m sure there is a slick proof lurking in $\mathbb{C}$. This is not it.
We first prove the left implication. Place the origin $O$ at the concurrent point as figured.
Since ratios of lengths are invariant under dilations and rotations, WLOG let the line through $B$ and $K$ be the real axis and scale the triangle such ... | {
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"source": "stackexchange",
"question_score": "9",
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Counterexample of Sobolev Embedding Theorem? Is there a counterexample of Sobolev Embedding Theorem? More precisely, please help me construct a sobolev function $u\in W^{1,p}(R^n),\,p\in[1,n)$ such that $u\notin L^q(R^n)$, where $q>p^*:=\frac{np}{n-p}$.^-^
| Here is how you can do this on the unit ball $\{x | \|x \| \le 1\}$: Set $u(x) = \|x\|^{-\alpha}$. Then $\nabla u$ is easy to find. Now you can compute $\|u\|_{L^q}$ and $\|u\|_{W^{1,p}}$ using polar coordinates. Play around until the $L^q$ norm is infinite while the $W^{1,p}$ norm is still finite.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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The smallest ring containing square root of 2 and rational numbers Can anyone prove why the smallest ring containing $\sqrt{2}$ and rational numbers is comprised of all the numbers of the form $a+b\sqrt{2}$ (with $a,b$ rational)?
| That ring must surely contain all numbers of the form $a+b\sqrt 2$ with $a,b\in\mathbb Q$ because these can be obtained by ring operations. Since that set is closed under addition and multiplication (because $(a+b\sqrt 2)+(c+d\sqrt2)=(a+c)+(b+d)\sqrt 2$ and $(a+b\sqrt2)\cdot(c+d\sqrt 2)=(ac+2bd)+(ad+bc)\sqrt 2$), it is... | {
"language": "en",
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Showing that $ (1-\cos x)\left |\sum_{k=1}^n \sin(kx) \right|\left|\sum_{k=1}^n \cos(kx) \right|\leq 2$ I'm trying to show that:
$$ (1-\cos x)\left |\sum_{k=1}^n \sin(kx) \right|\left|\sum_{k=1}^n \cos(kx) \right|\leq 2$$
It is equivalent to show that:
$$ (1-\cos x) \left (\frac{\sin \left(\frac{nx}{2} \right)}{ \sin \... | Using the identity
$$ 1 - \cos x = 2\sin^2\left(\frac{x}{2}\right), $$
we readily identify the left-hand side as
$$ 2 \sin^2 \left(\frac{nx}{2} \right) \left|\sin((n+1)x)\right|, $$
which is clearly less than or equal to $2$.
| {
"language": "en",
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"source": "stackexchange",
"question_score": "4",
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K critical graphs connectivity and cut vertex Show that a k-critical graph is connected. Furthermore, show that it does not have a vertex whose removal disconnects the graph (such a vertex is known as a cut vertex).
I have managed to proove , I think, the first part
Let's assume G is not connected. Since χ(G) = k, (I... | Your statement is a special case of more general theorem I was researching when I came across your question:
T: Cut in a $k$-critical graph is not clique.
Proof:
Assume that cut $S$ in $k$-critical graph $G=(V,E)$ is clique. Components of $G \setminus S$ are $\{C_1 \dots C_r\}$. For each subgraph of $G$ in the form of ... | {
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"source": "stackexchange",
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Induction: How to prove propositions with universal quantifiers? In my book, they prove with mathematical induction propositions with successions like this:
$$1 + 3 + 5 + \cdots + (2n-1) = n^2$$
In all exercises. However, recently I took some exercises from a different paper and instead of these it told me to prove thi... | You are misunderstanding induction here.
The $\forall$ is always part of it. For example, the real statement for your first result is $$\forall n\in\mathbb N: 1+3+...+(2n-1)=n^2$$
In general, if your equation is: $$\forall n\in\mathbb N: P(n)$$, the principal of mathematical induction says it is enough to show:
$$P(1)$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Basic Counting Problem I was reading a probability book and am having trouble conceptually with one of the examples. The following is a modification.
Let's say that we have $3$ coins that we want to randomly assign into $3$ bins, with equal probability. We can label these coins $a_1$, $a_2$, $a_3$. What is the probabi... | It depends on whether you're thinking of choosing each arrangement of balls with equal probability, or whether you place the balls in each bin with equal probability. I would say that the most natural way to think about this is the latter, since this is most likely how it would happen in real life.
When dealing with e... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to use mathematical induction with inequalities? I've been using mathematical induction to prove propositions like this:
$$1 + 3 + 5 + \cdots + (2n-1) = n^2$$
Which is an equality. I am, however, unable to solve inequalities. For instance, this one:
$$ 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \leq \frac... | I'm not sure what you expect exactly, but here is how I would do the inequality you mention.
We start with the base step (as it is usually called); the important point is that induction is a process where you show that if some property holds for a number, it holds for the next. First step is to prove it holds for the ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How are complex numbers useful to real number mathematics? Suppose I have only real number problems, where I need to find solutions. By what means could knowledge about complex numbers be useful?
Of course, the obviously applications are:
*
*contour integration
*understand radius of convergence of power series
*al... | I have used complex numbers to solve real life problems:
- Digital Signal Processing, Control Engineering: Z-Transform.
- AC Circuits: Phasors. This is a handful of applications broadly labeled under load-flow studies and resonant frequency devices (with electric devices modeled into resistors, inductors, capacitors at... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Poker, number of three of a kind, multiple formulaes I wanted to calculate some poker hands, for a three of a kind I infered, 1) every card rank can form a 'three of a kind' and there are 13 card ranks, 2) there are $\binom{4}{3}$ ways to choose three cards out of the four suits of every card rank, and 3) for the remai... | We can count more or less like you did, using $\dbinom{13}{1}\dbinom{4}{3}\dbinom{48}{2}$ (note the small change), and then subtracting the full houses.
Or else after we have picked the kind we have $3$ of, and the actual cards, we can pick the two "useless" cards. The kinds of these can be chosen in $\dbinom{12}{2}$ w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/244340",
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"source": "stackexchange",
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If $f^2$ is Riemann Integrable is $f$ always Riemann Integrable? Problem: Suppose that $f$ is a bounded, real-valued function on $[a,b]$ such that $f^2\in R$ (i.e. it is Riemann-Integrable). Must it be the case that $f\in R$ ?
Thoughts: I think that this is not necessarily true, but I am having trouble refuting or ... | $$f=2\cdot\mathbf 1_{[a,b]\cap\mathbb Q}-1$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Using Zorn's lemma show that $\mathbb R^+$ is the disjoint union of two sets closed under addition. Let $\Bbb R^+$ be the set of positive real numbers. Use Zorn's Lemma to show that $\Bbb R^+$ is the union of two disjoint, non-empty subsets, each closed under addition.
| Let $\mathcal{P}$ the set of the disjoint pairs $(A,B)$, where $A,B\subseteq\mathbb{R}^+$ are not empty and each one is closed under addition and multiplication by a positive rational number. Note that $\mathcal{P}\neq\emptyset$ because if we consider $X=\mathbb{Q}^+$ and $Y=\{n\sqrt{2}:n\in\mathbb{Q}^+\}$, then $(X,Y)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/244456",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Combinatorial Proof Of Binomial Double Counting Let $a$, $b$, $c$ and $n$ be non-negative integers.
By counting the number of committees
consisting of $n$ sentient beings that can
be chosen from a pool of $a$ kittens, $b$
crocodiles and $c$ emus in two different
ways, prove the identity
$$\sum\limits_{\substack{i,j,k... | $\newcommand{\+}{^{\dagger}}
\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\down}{\dow... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
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Finding generalized eigenbasis
*
*For a complex square matrix $M$, a maximal set of linearly
independent eigenvectors for an eigenvalue $\lambda$ is determined
by solving $$ (M - \lambda I) x = 0. $$ for a basis in the
solution subspace directly as a homogeneous linear system.
*For a complex square matrix $M$, a g... | Look at the matrix $$M=\pmatrix{1&1\cr0&1\cr}$$ Taking $\lambda=1$, $c=2$, Then $(M-\lambda I)^c$ is the zero matrix, so any two linearly independent vectors will do as a basis for the solution space of $(M-\lambda I)^cu=0$. But that's not what you want: first, you want as many linearly independent eigenvectors as you ... | {
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"timestamp": "2023-03-29T00:00:00",
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Value of $\lim_{n\to \infty}\frac{1^n+2^n+\cdots+(n-1)^n}{n^n}$ I remember that a couple of years ago a friend showed me and some other people the following expression:
$$\lim_{n\to \infty}\frac{1^n+2^n+\cdots+(n-1)^n}{n^n}.$$
As shown below, I can prove that this limit exists by the monotone convergence theorem. I als... | The limit is $\frac{1}{e-1}$. I wrote a paper on this sum several years ago and used the Euler-Maclaurin formula to prove the result. The paper is "The Euler-Maclaurin Formula and Sums of Powers," Mathematics Magazine, 79 (1): 61-65, 2006. Basically, I use the Euler-Maclaurin formula to swap the sum with the correspo... | {
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Can someone show me how to prove the following? I have $f(x)=g(ax+b)$, a and b are constant.
I need to show that $\nabla f(x)=a\nabla g(x)$ and $\nabla^2 f(x)=a^2\nabla^2 g(x)$...
I was thinking that the final answer should have ax+b in it, but apparently it can be shown that the above is true???
| In this problem $f(x) = g(h(x))$, where $h(x) = ax + b$. I'm going to consider the case where $a$ is a matrix rather than a scalar, because it's useful and no more difficult. You can assume $a$ is a scalar if you'd like.
Let's establish some notation. Recall that if $F:\mathbb R^n \to \mathbb R^m$ is differentiable ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to understand $\operatorname{cf}(2^{\aleph_0}) > \aleph_0$ As a corollary of König's theorem, we have $\operatorname{cf}(2^{\aleph_0}) > \aleph_0$
.
On the other hand, we have $\operatorname{cf}(\aleph_\omega) = \aleph_0$.
Why the logic in the latter equation can't apply to the former one?
To be precise, why we ca... | Note that in ordinal arithmetic $2^\omega$ is the supremum of $2^n$ for all $n<\omega$, so it is $\omega$ and thus has cofinality $\aleph_0$.
$\omega$ and $\aleph_0$ are the same set but they nevertheless behave differently in practice -- because tradition is to use the notation $\omega$ for operations where a limiting... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/244790",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Infinitely number of primes in the form $4n+1$ proof Question: Are there infinitely many primes of the form $4n+3$ and $4n+1$?
My attempt: Suppose the contrary that there exist finitely many primes of the form $4n+3$, say $k+1$ of them: $3,p_1,p_2,....,p_k$
Consider $N = 4p_1p_2p_3...p_k+3$, $N$ cannot be a prime of th... | There are infinite primes in both the arithmetic progressions $4k+1$ and $4k-1$.
Euclid's proof of the infinitude of primes can be easily modified to prove the existence of infinite primes of the form $4k-1$.
Sketch of proof: assume that the set of these primes is finite, given by $\{p_1=3,p_2=7,\ldots,p_k\}$, and cons... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/244915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "39",
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Correlated Poisson Distribution $X_1$ and $X_2$ are discrete stochastic variables. They can both be modeled by a Poisson process with arrival rates $\lambda_1$ and $\lambda_2$ respectively.
$X_1$ and $X_2$ have a constant correlation $\rho$.
Is there an analytic equation that describes the probability density function:... | Consider this model that could generate correlated Poisson variables. Let $Y$, $Y_1$ and $Y_2$ be three independent Poisson variable with parameters $r$, $\lambda_1$ and $\lambda_2$. Let $$X_i=Y_i+Y$$ for $i=1,2$. Then $X_1$ and $X_2$ are both Poisson with parameters $\lambda_1$ and $\lambda_2$. They have the correlati... | {
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Example of two dependent random variables that satisfy $E[f(X)f(Y)]=Ef(X)Ef(Y)$ for every $f$ Does anyone have an example of two dependent random variables, that satisfy this relation?
$E[f(X)f(Y)]=E[f(X)]E[f(Y)]$
for every function $f(t)$.
Thanks.
*edit: I still couldn't find an example. I think one should be of two i... | If you take dependent random variables $X$ and $Y$, and set $X^{'} = X - E[X]$ and $Y^{'} = Y - E[Y]$, then $E[f(X^{'})f(Y^{'})]=E[f(X^{'})]E[f(Y^{'})]=0$ as long as $f$ preserves the zero expected value. I guess you cannot show this for all $f$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The graph of a smooth real function is a submanifold Given a function $f: \mathbb{R}^n \rightarrow \mathbb{R}^m $ which is smooth, show that
$$\operatorname{graph}(f) = \{(x,f(x)) \in \mathbb{R}^{n+m} : x \in \mathbb{R}^n\}$$
is a smooth submanifold of $\mathbb{R}^{n+m}$.
I'm honestly completely unsure of where or how ... | The map $\mathbb R^n\mapsto \mathbb R^{n+m}$ given by $t\mapsto (t, f(t))$ has the Jacobi matrix $\begin{pmatrix}I_n\\f'(t)\end{pmatrix}$, which has a full rank $n$ for all $t$ (because of the identity submatrix). This means that its value range is a manifold. Is there anything unclear about it?
How is this a proof tha... | {
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If $u''>0$ in $\mathbf{R}^+$ then $u$ is unbounded? If $u$ is a positive function such that $u''>0$ in the whole $\mathbf{R}^+$ then $u$ is unbounded?
In fact, I know that if $u''>0$ then $u$ is strictly convex. I think that implies $u$ is coercive. I want to prove it.
| $$ u(x) = e^{-x} {}{}{}{}{}{}{}{} $$
EDIT: if you actually meant the entire real line $\mathbb R,$ then any $C^2$ function $u(x)$ really is unbounded. Proof: as $u'' > 0,$ we know that $u'$ cannot always be $0.$ as a result, it is nonzero at some $x=a.$ If $u'(a) > 0,$ then for $x > a$ we have $u(x) > u(a) + (x-a) u'(... | {
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$i,j,k$ Values of the $\Theta$ Matrix in Neural Networks SO I'm looking at these two neural networks and walking through how the $ijk$ values of $\Theta$ correspond to the layer, the node number.
Either there are redundant values or I'm missing how the subscripts actually map from node to node.
$\Theta^i_{jk}$ ... whe... | Yes, $\Theta^i_{jk}$ is the weight that the activation of node $j$ has in the previous input layer $j - 1$ in computing the activation of node $k$ in layer $i$.
| {
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Arc Length: Difficulty With The Integral The question is to find the arc length of a portion of a function.
$$y=\frac{3}{2}x^{2/3}\text{ on }[1,8]$$
I couldn't quite figure out how to evaluate the integral, so I appealed to the solution manual for aid.
I don't quite understand what they did in the 5th step. Could some... | The expression in brackets is precisely what you need for the substitution $u=x^{2/3}+1$ to work.
| {
"language": "en",
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Proof of $\frac{Y^{\lambda}-\lambda}{\sqrt{\lambda}}\to Z\sim N(0,1)$ in distribution as $\lambda\to\infty$? This is an exercise of the Central Limit Theorem:
Let $Y^{\lambda}$ be a Poisson random variable with parameter $\lambda>0$.
Prove that $\frac{Y^{\lambda}-\lambda}{\sqrt{\lambda}}\to Z\sim N(0,1)$ in distribu... | The squeeze theorem in convergence in distribution can be made fully rigorous in the situation you describe--but the shortest proof here might be through characteristic functions.
Recall that if $Y^\lambda$ is Poisson with parameter $\lambda$, $\varphi_\lambda(t)=\mathbb E(\mathrm e^{\mathrm itY^\lambda})$ is simply $... | {
"language": "en",
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How can I give a bound on the $L^2$ norm of this function? I came across this question in an old qualifying exam, but I am stumped on how to approach it:
For $f\in L^p((1,\infty), m)$ ($m$ is the Lebesgue measure), $2<p<4$, let
$$(Vf)(x) = \frac{1}{x} \int_x^{10x} \frac{f(t)}{t^{1/4}} dt$$
Prove that
$$||Vf||_{... | Using Hölder's inequality, we have for $x>1$,
\begin{align}
|V(f)(x)|&\leqslant \lVert f\rVert_{L^p}\left(\int_x^{10 x}t^{-\frac p{4(p-1)}}dt\right)^{\frac{p-1}p}\frac 1x\\
&=\lVert f\rVert_{L^p}A_p\left(x^{-\frac p{4(p-1)}+1}\right)^{\frac{p-1}p}\frac 1x\\
&=A_p\lVert f\rVert_{L^p}x^{\frac{p-1}p-\frac 14}\frac 1x\\
&=... | {
"language": "en",
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Compute the length of an equilateral triangle's side given the area? Given the area of an equilateral triangle, what is an algorithm to determine the length of a side?
| Let $s$ be the side, and $A$ the area. Drop a perpendicular from one vertex to the opposite side. By the Pythagorean Theorem, the height of the triangle is $\sqrt{s^2-\frac{1}{4}s^2}=\frac{s\sqrt{3}}{2}$. It follows that
$$A=\frac{s^2\sqrt{3}}{4}.$$
Thus
$$s^2=\frac{4A}{\sqrt{3}},$$
and therefore
$$s=\sqrt{\frac{4A... | {
"language": "en",
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Find an angle in a given triangle $\triangle ABC$ has sides $AC = BC$ and $\angle ACB = 96^\circ$. $D$ is a point in $\triangle ABC$ such that $\angle DAB = 18^\circ$ and $\angle DBA = 30^\circ$. What is the measure (in degrees) of $\angle ACD$?
| Take $O$ the circumcenter of $\triangle ABD$, see that $\triangle DAO$ is equilateral and, since $\widehat{BAD}=18^\circ$, we get $\widehat{BAO}=42^\circ$, i.e. $O$ is reflection of $C$ about $AB$, that is, $AOBC$ is a rhombus, hence $AD=AO=AC,\triangle CAD$ is isosceles with $\widehat{ADC}=\widehat{ACD}=78^\circ$, don... | {
"language": "en",
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"source": "stackexchange",
"question_score": "3",
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Conditions for Schur decomposition and its generalization Let $M$ be a $n$ by $n$ matrix over a field $F$.
When $F$ is $\mathbb{C}$, $M$ always has a Schur decomposition, i.e. it is always unitarily similar to a triangular matrix, i.e. $M = U T U^H$ where $U$ is some unitary matrix and $T$ is a triangular matrix.
*
... | If the characterisic polynomial factors in linear factors then the Jordan decomposition works as your triangular matrix.
If you have a similar triangular matrix then the characteristic polynomial of $M$ is the characteristic polynomial of $T$ which clearly factors into linear factors.
So, the criterion is exactly the s... | {
"language": "en",
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Must-read papers in Operator Theory I have basically finished my grad school applications and have some time at hand. I want to start reading some classic papers in Operator Theory so as to breathe more culture here. I have read some when doing specific problems but have never systematically study the literature.
I won... | Cuntz - Simple $C^*$-algebras generated by isometries
The Cuntz algebras are very important in various places in C*-algebra theory.
| {
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Prove that $\lim_{x \rightarrow 0} \frac{1}{x}\int_0^x f(t) dt = f(0)$. Assume $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous. Prove that $\lim_{x \rightarrow 0} \frac{1}{x}\int_0^x f(t) dt = f(0)$.
I'm having a little confusion about proving this. So far, it is clear that $f$ is continuous at 0 and $f$ is Riem... | $\def\e{\varepsilon}\def\abs#1{\left|#1\right|}$As $f$ is continuous at $0$, for $\e > 0$ there is an $\delta > 0$ such that $\abs{f(x) - f(0)} \le \e$ for $\abs x \le \delta$. For these $x$ we have
\begin{align*}
\abs{\frac 1x \int_0^x f(t)\, dt - f(0)} &= \abs{\frac 1x \int_0^x \bigl(f(t) - f(0)\bigr)\,dt}\\
&\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Does the Laplace transform biject? Someone wrote on the Wikipedia article for the Laplace trasform that 'this transformation is essentially bijective for the majority of practical uses.'
Can someone provide a proof or counterexample that shows that the Laplace transform is not bijective over the domain of functions fro... | For "the majority of practical uses" it is important that the Laplace transform ${\cal L}$ is injective. This means that when you have determined a function $s\mapsto F(s)$ that suits your needs, there is at most one process $t\mapsto f(t)$ such that $F$ is its Laplace transform. You can then look up this unique $f$ in... | {
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Is there a $SL(2,\mathbb{Z})$-action on $\mathbb{Z}$? Is there a $SL(2,\mathbb{Z})$-action on $\mathbb{Z}$?
I read this somewhere without proof and I am not sure if this is true.
Thank you for your help.
| (This is completely different to my first 'answer', which was simply wrong.)
Denote by $\text{End}(\mathbb{Z})$ the semi-group of group endomorphisms of $\mathbb{Z}$. Since $\mathbb{Z}$ is cyclic, any endomorphism is determined by the image of the generator $1$, and since $1 \mapsto n$ is an endomorphism for any $n\in\... | {
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Complex Analysis and Limit point help So S is a complex sequence (an from n=1 to infinity) has limit points which form a set E of limit points. How do I prove that every limit point of E are also members of the set E. I think epsilons will need to be used but I'm not sure.
Thanks.
| Let $z$ be a limit point of $E$, and take any $\varepsilon>0$. There is some $x\in E$ with $\lvert x-z\rvert<\varepsilon/2$. And since $x\in E$, there are infinitely many members of $S$ within an $\varepsilon/2$-ball around $x$. They will all be within an $\varepsilon$-ball around $z$, and you're done.
| {
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Example 2, Chpt 4 Advanced Mathematics (I)
$$\int \frac{x+2}{2x^3+3x^2+3x+1}\, \mathrm{d}x$$
I can get it down to this:
$$\int \frac{2}{2x+1} - \frac{x}{x^2+x+1}\, \mathrm{d}x $$
I can solve the first part but I don't exactly follow the method in the book.
$$ = \ln \vert 2x+1 \vert - \frac{1}{2}\int \frac{\left(2x... | The post indicates some difficulty with finding $\int \frac{dx}{x^2+x+1}$. We solve a more general problem. But I would suggest for your particular problem, you follow the steps used, instead of using the final result.
Suppose that we want to integrate $\dfrac{1}{ax^2+bx+c}$, where $ax^2+bx+c$ is always positive, or a... | {
"language": "en",
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Find all singularities of$ \ \frac{\cos z - \cos (2z)}{z^4} \ $ How do I find all singularities of$
\
\frac{\cos z - \cos (2z)}{z^4}
\
$
It seems like there is only one (z = 0)?
How do I decide if it is isolated or nonisolated?
And if it is isolated, how do I decide if it is removable or not removable?
If it is non iso... | $$\cos z = 1-\frac{z^2}{2}+\cdots$$
$$\cos2z=1-\frac{(2z)^2}{2}+\cdots$$
$$\frac{\cos z-\cos2z}{z^4}= \frac{3}{2z^2}+\left(\frac{-15}{4!}+a_1z^2 +\cdots\right),$$
hence at $z=0$ there is a pole .
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding asymptotes of exponential function and one-sided limit Find the asymptotes of
$$
\lim_{x \to \infty}x\cdot\exp\left(\dfrac{2}{x}\right)+1.
$$
How is it done?
| A related problem. We will use the Taylor series of the function $e^t$ at the point $t=0$,
$$ e^t = 1+t+\frac{t^2}{2!}+\frac{t^3}{3!}+\dots .$$
$$ x\,e^{2/x}+1 = x ( 1+\frac{2}{x}+ \frac{1}{2!}\frac{2^2}{x^2}+\dots )+1=x+3+\frac{2^2}{2!}\frac{1}{x}+\frac{2^3}{3!}\frac{1}{x^2}+\dots$$
$$ = x+3+O(1/x).$$
Now, you can see... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Identity for $\zeta(k- 1/2) \zeta(2k -1) / \zeta(4k -2)$? Is there a nice identity known for
$$\frac{\zeta(k- \tfrac{1}{2}) \zeta(2k -1)}{\zeta(4k -2)}?$$
(I'm dealing with half-integral $k$.) Equally, an identity for
$$\frac{\zeta(s) \zeta(2s)}{\zeta(4s)}$$
would do ;)
| Let $$F(s) = \frac{\zeta(s)\zeta(2s)}{\zeta(4s)}.$$ Then clearly the Euler product of $F(s)$ is $$F(s) = \prod_p \frac{\frac{1}{1-1/p^s}\frac{1}{1-1/p^{2s}}}{\frac{1}{1-1/p^{4s}}}=
\prod_p \left( 1 + \frac{1}{p^s} + \frac{2}{p^{2s}} + \frac{2}{p^{3s}} + \frac{2}{p^{4s}} + \frac{2}{p^{5s}} + \cdots\right).$$
Now introdu... | {
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"source": "stackexchange",
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What is vector division? My question is:
We have addition, subtraction and muliplication of vectors. Why cannot we define vector division? What is division of vectors?
| The quotient of two vectors is a quaternion by definition. (The product of two vectors can also be regarded as a quaternion, according to the choice of a unit of space.) A quaternion is a relative factor between two vectors that acts respectively on the vector's two characteristics length and direction; through its ten... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Fixed point of $\cos(\sin(x))$ I can show that $\cos(\sin(x))$ is a contraction on $\mathbb{R}$ and hence by the Contraction Mapping Theorem it will have a unique fixed point. But what is the process for finding this fixed point? This is in the context of metric spaces, I know in numerical analysis it can be done trivi... | The Jacobi-Anger expansion gives an expression for your formula as:
$\cos(\sin(x)) = J_0(1)+2 \sum_{n=1}^{\infty} J_{2n}(1) \cos(2nx)$.
Since the "harmonics" in the sum rapidly damp to zero, to second order the equation for the fixed point can be represented as:
$x= J_0(1) + 2[J_2(1)(\cos(2x)) + J_4(1)(\cos(4x))]$.
Usi... | {
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Results of dot product for complex functions Suppose we are given a $C^1$ function $f(t):\mathbb{R} \rightarrow \mathbb{C}$ with $f(0) = 1$, $\|f(t)\| = 1$ and $\|f'(t)\| = 1$. I have already proven that $\langle f(t), f'(t)\rangle = 0$ for all $t$. Now I have to show that either $f'(t) = if(t)$ or $f'(t) = -i f(t)$. H... | Presumably by $\langle f(t) , f'(t) \rangle = 0$, you mean that $\text{Re} f(t) \overline{f'(t)} = 0$ (if $z_1,z_2 \in \mathbb{C}$ and $z_1 \overline{z_2} = 0$, then you must have either $z_1 = 0$ or $z_2 = 0$).
If $\text{Re} f(t) \overline{f'(t)} = 0$, then $f(t) \overline{f'(t)} = i \zeta(t)$. where $\zeta$ is real v... | {
"language": "en",
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Probability Problem with $n$ keys A woman has $n$ keys, one of which will open a door.
a)If she tries the keys at random, discarding those that do not work, what is the probability that she will open the door on her $k^{\mathrm{th}}$ try?
Attempt: On her first try, she will have the correct key with probability $\frac1... | For $(a)$, probability that she will open on the first try is $\dfrac1n$. You have this right.
However, the probability that she will open on the second try is when she has failed in her first attempt and succeeded in her second attempt. Hence, the probability is $$\underbrace{\dfrac{n-1}n}_{\text{Prob of failure in he... | {
"language": "en",
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Finding the distance between a line and a vector, given a projection So my question has two parts:
a) Let L be a line given by y=2x, find the projection of $\vec{x}$=$\begin{bmatrix}5\\3\end{bmatrix}$ onto the line L.
So, for this one:
proj$_L$($\vec{x}$) = $\frac{\vec{x}\bullet \vec{y}}{\vec{y}\bullet \vec{y}}$$\times... | Yes, yes, almost done. You need the length of this distance vector, use Pythagorean theorem.
One moment, your line is $y=2x$, then it rather contains $\pmatrix{1\\2}$ than $\pmatrix{2\\1}$ (and its normalvector is $\pmatrix{2\\-1}$)..
| {
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Transition from introduction to analysis to more advanced analysis I am currently studying intro to analysis and learning somethings about
basic topology in metric space and almost finished the course . I am
thinking of taking some more advanced
analysis. Would it be demanding to take some course like
functional anal... | I'm currently taking a graduate functional analysis course having only taken introductory analysis (I majored in physics). It's manageable, but knowing measure theory and lebesgue integration would have definitely helped.
| {
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If $f$ is entire and $z=x+iy$, prove that for all $z$ that belongs to $C$, $\left(\frac{d^2}{dx^2}+\frac{d^2}{dy^2}\right)|f(z)|^2= 4|f'(z)|^2$ I'm kind of stuck on this problem and been working on it for days and cannot come to the conclusion of the proof.
| Let $\frac{\partial}{\partial z} = \tfrac{1}{2}(\frac{\partial}{\partial x} - i \frac{\partial}{\partial y})$ and $\frac{\partial}{\partial \overline{z}} = \tfrac{1}{2}(\frac{\partial}{\partial x} + i \frac{\partial}{\partial y})$. Then $\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} = 4 \frac{\part... | {
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Prove the isomorphism of cyclic groups $C_{mn}\cong C_m\times C_n$ via categorical considerations As the title suggests, I am trying to prove $C_{mn}\cong C_m\times C_n$ when $\gcd{(m,n)}=1$, where $C_n$ denotes the cyclic group of order $n$, using categorical considerations. Specifically, I am trying to show $C_{mn}$ ... | Just follow the definition:
Let $X$ be any group, and $f:X\to C_n$, $g:X\to C_m$ homomorphisms. Now you need a unique homomorphism $h:X\to C_{nm}$ which makes both triangles with $\pi_n$ and $\pi_m$ commute.
And constructing this $h$ requires basically the Chinese Remainder Theorem (and is essentially the same as cons... | {
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Prove $\lfloor \log_2(n) \rfloor + 1 = \lceil \log_2(n+1) \rceil $ This is a question a lecturer gave me. I'm more than willing to come up with the answer. But I feel I'm missing something in logs.
I know the rules, $\log(ab) = \log(a) + \log(b)$ but that's all I have.
What should I read, look up to come up with the a... | Well, even after your edit, this is not an identity in general. It is valid only for integral $n$. For a counter-example of why this is not valid for a positive $n$, take $n = 1.5$. Then,
$$\lfloor \log_2(n) \rfloor + 1 = \lfloor \log_2(1.5) \rfloor + 1 = 1$$
and
$$\lceil \log_2(n + 1) \rceil = \lceil \log_2(2.5) \rcei... | {
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An upper bound for $\sum_{i = 1}^m \binom{i}{k}\frac{1}{2^i}$? Does anyone know of a reasonable upper bound for the following:
$$\sum_{i = 1}^m \frac{\binom{i}{k}}{2^i},$$
where we $k$ and $m$ are fixed positive integers, and we assume that $\binom{i}{k} = 0$ whenever $k > i$.
One trivial upper bound uses the identity ... | Another estimate for $\sum_{i=1}^m\frac1{\sqrt i}$ is
$$2\sqrt{m-1}-2=\int_1^{m-1} x^{-1/2}\, dx \le \sum_{i=1}^m\frac1{\sqrt i}\le1+ \int_1^m x^{-1/2}\, dx=2\sqrt m-1. $$
Thus we can remove the summands with $i<k$ by considering
$$ 4\sqrt m+2 -4\sqrt{k-2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/247261",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 2
} |
Reference about Fredholm determinants I am searching for a reference book on Fredholm determinants. I am mainly interested in applications to probability theory, where cumulative distribution functions of limit laws are expressed in terms of Fredholm determinants. I would like to answer questions like :
*
*How to ex... | Nearly every book on random matrices deals with the subject. For a recent example, see Section 3.4 of An Introduction to Random Matrices by Anderson, Guionnet and Zeitouni.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/247333",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Limiting distribution and initial distribution of a Markov chain For a Markov chain (can the following discussion be for either discrete time or continuous time, or just discrete time?),
*
*if for an initial distribution i.e. the distribution of $X_0$, there
exists a limiting distribution for the distribution of $X... | *
*No, let $X$ be a Markov process having each state being absorbing, i.e. if you start from $x$ then you always stay there. For any initial distribution $\delta_x$, there is a limiting distribution which is also $\delta_x$ - but this distribution is different for all initial conditions.
*The convergence of distribut... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/247395",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Convergence of series $\sum\limits_{n=2}^\infty\frac{n^3+1}{n^4-1}$
Investigate the series for convergence and if possible, determine its
limit: $\sum\limits_{n=2}^\infty\frac{n^3+1}{n^4-1}$
My thoughts
Let there be the sequence $s_n = \frac{n^3+1}{n^4-1}, n \ge 2$.
I have tried different things with no avail. I su... | $$\frac{n^3+1}{n^4-1}\gt\frac{n^3}{n^4}=\frac1n\;.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/247483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Show that if matrices A and B are elements of G, then AB is also an element of G.
Let $G$ be the set of $2 \times 2$ matrices of the form
\begin{pmatrix} a & b \\ 0 & c\end{pmatrix}
such that $ac$ is not zero. Show that if matrices $A$ and $B$ are elements of $G$, then $AB$ is also an element of $G$.
Do I just ne... | Proving that AB has a non-zero determinant is not enough, because not all 2x2 matrices with non-zero determinant are a element of G.
You need to prove another property of AB. This property is that it has the shape you stated.
This combined with a non-zero determinant guarantees that AB has the prescribed shape with ac ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/247565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Derivatives vs Integration
*
*Given that the continuous function $f: \Bbb R \longrightarrow \Bbb R$ satisfies
$$\int_0^\pi f(x) ~dx = \pi,$$
Find the exact value of
$$\int_0^{\pi^{1/6}} x^5 f(x^6) ~dx.$$
*Let
$$g(t) = \int_t^{2t} \frac{x^2 + 1}{x + 1} ~dx.$$
Find $g'(t)$.
For the first question: Th... | For the first question,
There are infinitely many functions other than $1$ that satisfy
$$\int_0^{\pi} f(x) dx = \pi$$
For instance, couple of other examples are $$f(x) = 2- \dfrac{2x}{\pi}$$
and $$f(x) = \dfrac{2x}{\pi}$$
To evaluate $$\int_0^{\pi^{1/6}} x^5 f(x^6) dx$$ make the substitution $t = x^6$ and see what ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/247634",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Notation for repeated application of function If I have the function $f(x)$ and I want to apply it $n$ times, what is the notation to use?
For example, would $f(f(x))$ be $f_2(x)$, $f^2(x)$, or anything less cumbersome than $f(f(x))$? This is important especially since I am trying to couple this with a limit toward inf... | In the course I took on bifurcation theory we used the notation $$f^{\circ n}(x).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/247710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "35",
"answer_count": 8,
"answer_id": 1
} |
Find a convex combination of scalars given a point within them. I've been banging my head on this one all day! I'm going to do my best to explain the problem, but bear with me.
Given a set of numbers $S = \{X_1, X_2, \dots, X_n\}$ and a scalar $T$, where it is guaranteed that there is at least one member of $S$ that is... | If $X_1<T<X_2$, then $T$ is a weighted average of $X_1$ and $X_2$ with weights $\dfrac{X_2-T}{X_2-X_1}$ and $\dfrac{T-X_1}{X_2-X_1}$, as can be checked by a bit of algebra.
Now suppose $X_3$ is also $>T$. Then $T$ is a weighted average of $X_1$ and $X_3$, and you can find the weights the same way.
Now take $40\%$ of t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/247748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 2
} |
How to prove that $n^{\frac{1}{3}}$ is not a polynomial? I'm reading Barbeau's Polynomials, there's an exercise:
How to prove that $n^{\frac{1}{3}}$ is not a polynomial?
I've made this question and with the first answer as an example, I guess I should assume that:
$$n^{\frac{1}{3}}=a_pn^p+a_{p-1}n^{p-1}+\cdots+ a_0n^... | If $t^{1/3}$ were a polynomial, then its degree would be at least one (because it is not constant). This would imply
$$
\lim_{t\to\infty}\frac{t^{1/3}}t\ne0.
$$
But, precisely, the limit above is indeed zero. So $t^{1/3}$ cannot be a polynomial.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/247800",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 1
} |
how to prove $437\,$ divides $18!+1$? (NBHM 2012) I was solving some problems and I came across this problem. I didn't understand how to approach this problem. Can we solve this with out actually calculating $18!\,\,?$
| Note that $437=(19)(23)$. We prove that $19$ and $23$ divide $18!+1$. That is enough, since $19$ and $23$ are relatively prime.
The fact that $19$ divides $18!+1$ is immediate from Wilson's Theorem, which says that if $p$ is prime then $(p-1)!\equiv -1\pmod{p}$.
For $23$ we need to calculate a bit. We have $22!\equiv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/247879",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 1,
"answer_id": 0
} |
$\mathrm{Spec}(R)\!=\!\mathrm{Max}(R)\!\cup\!\{0\}$ $\Rightarrow$ $R$ is a PID Is the following true:
If $R$ is a commutative unital ring with
$\mathrm{Spec}(R)\!=\!\mathrm{Max}(R)\!\cup\!\{0\}$, then $R$ is a PID.
If yes, how can one prove it?
Since $0$ is a prime ideal, $R$ is a domain. Thus we must prove that ev... | As mentioned, there are easy counterxamples. However, it is true for UFDs since PIDs are precisely the $\rm UFDs$ of dimension $\le 1,\:$ i.e. such that prime ideals $\ne 0$ are maximal. Below is a sketch of a proof of this and closely related results.
Theorem $\rm\ \ \ TFAE\ $ for a $\rm UFD\ D$
$(1)\ \ $ prime ideal... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/247976",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
The sufficient and necessary condition for a function approaching a continuous function at $+\infty$ Problem
Suppose
$f:\Bbb R^+\to\Bbb R$ satisfies
$$\forall\epsilon>0,\exists E>0,\forall x_0>E,\exists\delta>0,\forall x(\left|x-x_0\right|<\delta): \left|f(x)-f(x_0)\right|<\epsilon\tag1$$
Can we conclude that
ther... | Choose $E_0 := 0$, $(E_n)_n \uparrow \infty$ corresponding to $\varepsilon_n := \frac{1}{n}$ for $n \geq 1$ using (1). If $E_n < x \leq E_{n+1}$, choose $\delta_x > 0$ such that $f(y) \in B_{1/n}(f(x))$ for all $y \in B_{\delta_x} (x)$, according to (1). For $n \geq 1$, the balls $(B_{\delta_x}(x))_{x \in [E_n, E_{n+1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/248039",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is $\omega_{\alpha}$ sequentially compact? For an ordinal $\alpha \geq 2$, let $\omega_{\alpha}$ be as defined here.
It is easy to show that $\omega_{\alpha}$ is limit point compact, but is it sequentially compact?
| I think I finally found a solution:
In a well-ordered set, every sequence admits an non-decreasing subsequence. Indeed, if $(x_n)$ is any sequence, let $n_0$ be such that $x_{n_0}= \min \{ x_n : n\geq 0 \}$, and $n_1$ such that $x_{n_1}= \min \{ x_n : n > n_0 \}$, and so on; here, $(x_{n_i})$ is an non-decreasing subse... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/248083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How can it happen to find infinite bases in $\mathbb R^n$ if $\mathbb R^n$ does not admit more than $n$ linearly independent vectors? How can it happen to find infinite bases in $\mathbb R^n$ if $\mathbb R^n$ does not admit more than $n$ linearly independent vectors?
Also considered that each basis of $\mathbb R^n$ has... | Let $E=\{e_1,...,e_n\}$ be the standard basis in $\mathbb{R}^n$.
For each $\lambda\neq 0$, let $E_\lambda = \{\lambda e_1, e_2,...,e_n\}$. Then each $E_\lambda$ is a distinct basis of $\mathbb{R}^n$.
However, each $E_\lambda$ has exactly $n$ elements.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/248179",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
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