qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
136,264 | <p>I have a question concerning the stability analysis for a kind of differential equation taking the form
$$\dot x=Ax+Bw,$$
where $A\in \mathbb{R}^{n \times n}$, $B\in \mathbb{R}^{n \times m}$ are constant matrices and
$w \in \mathbb{R}^m$ is a normal random variable, i.e., $w\sim \mathcal{N}(0,W)$ with $W$ ... | juan | 7,402 | <p>I myself am not very fun of rankings. But when the Library of my University
decided to cut down some of the journal subscriptions
(due to the budget crisis in the Eurozone), I gave my personal viewpoint
by means of creating a unbiased ranking of Mathematical Journals.</p>
<p>Mainly what I wanted to measure was the ... |
3,555,084 | <blockquote>
<p>Let
<span class="math-container">$$f(z) = e^z (1+\cos\sqrt{z} ) $$</span>
<span class="math-container">$\Omega=\{z\in\Bbb C: |z|\gt r\}$</span>, <span class="math-container">$r\gt 0$</span>. What is <span class="math-container">$f(\Omega)$</span>?</p>
<p>where <span class="math-container">$... | LHF | 744,207 | <p>To write the complete solution, as per the hint in the comments I gave:</p>
<p><span class="math-container">$$
\begin{aligned}
\lim_{x \to \infty}x^2\left(x^2 - x \cdot \sqrt{x^2 + 6} + 3\right) &= \lim_{x \to \infty}x^2\left(\sqrt{(x^2+3)^2} - \sqrt{x^4 + 6x^2}\right)\\
&=\lim_{x \to \infty}x^2\left(\frac{... |
2,614,969 | <p>I wonder whether there is a general method for accurately estimating the limit of the sequence:</p>
<p>\begin{equation}
x_{n+1} = x_n - x_{n}^{n+1}, \forall x_1 \in (0,1)
\end{equation}</p>
<p>After showing that the limit exists, since $ x_n $ is decreasing and bounded, I managed to derive a lower-bound. In partic... | Community | -1 | <p>The observation of Fede Poncio that the limit of $x_n$ is well-approximated by a quadratic polynomial in terms of $x_1$ is very useful. With a few lines of Python3 code we observe:</p>
<pre><code>import numpy as np
import matplotlib.pyplot as plt
def f(x,N):
X = np.zeros(N,dtype='float64')
X[0] = x
f... |
2,573,487 | <p>I have given this set</p>
<blockquote>
<p>$$ M = \{ x \in [1,2]\times [3,4] ~|~ x\in\mathbb{Q}^2 \} \subset \mathbb{R}^2 $$</p>
</blockquote>
<p>First I have to identify the boundary $\partial M$ and then tell if it is open or closed.</p>
<p>I think that $$ \partial M = \{ (x,y) ~|~ x\not\in\mathbb{Q}^2, 1\leq ... | ajotatxe | 132,456 | <p>The set is not open or closed, because any open ball contains rational and irrational points.</p>
<p>To solve the problem, show that $\overline M=[1,2]\times[3,4]$ and $\overline{M^c}=\Bbb R^2$.</p>
<p>Then $\partial M=[1,2]\times[3,4]\cap \Bbb R^2$</p>
|
93,063 | <p>There is a story I read about tiling the plane with convex pentagons.</p>
<p>You can read about it in this <a href="http://www.ivanrival.com/docs/picturepuzzling_2.pdf">article</a> on pages 1 and 2.</p>
<p>Summary of the story:
A guy showed in his doctorate work all classes of tiling the plane with convex pentagon... | Carl Mummert | 630 | <p>It is not rare, but is uncommon, for false theorems to be published. It is somewhat more common, although still not frequent, for flawed proofs to be published, even though the theorem as stated is correct. One way to find these is to search for "correction", "corrigendum", or "retraction" on MathSciNet. Peer review... |
3,247,563 | <p>Given a segment AB on the plane let <span class="math-container">$r$</span> and <span class="math-container">$s$</span> parallel lines with <span class="math-container">$A \in r$</span> and <span class="math-container">$B \in s $</span>. What is the locus of the circles tangent to <span class="math-container">$AB$</... | Angina Seng | 436,618 | <p>How about
<span class="math-container">$$|\ln(1-x)|=\left|-\sum_{n=1}^\infty\frac{x^n}n\right|
\le\sum_{n=1}^\infty\frac{|x|^n}n\le\sum_{n=1}^\infty|x|^n=\frac{|x|}{1-|x|}$$</span>
etc.?</p>
|
2,217,338 | <p>I am trying to define a simple function that is first concave and then convex as shown in the picture below. Since the resulting equation have to be explained/used by a non-technical audience, the function should be ideally as simple as possible, but I have been unable to find any simple form that matches the requir... | Marco Cantarini | 171,547 | <p>We have that <span class="math-container">$$\begin{align} I\left(a\right) = & \int_{0}^{x}x\sin^{2}\left(ax\right)e^{-x^{2}}dx \\ = &-\frac{1}{4}\int_{0}^{\infty}xe^{-x^{2}-2iax}dx-\frac{1}{4}\int_{0}^{\infty}xe^{-x^{2}+2iax}dx+\frac{1}{2}\int_{0}^{\infty}xe^{-x^{2}}dx \end{align}$$</span> so let us analyze... |
3,137,160 | <blockquote>
<p>Determine which of the following are subspaces of <span class="math-container">$3 \times 3$</span> matrix <span class="math-container">$M$</span>
all <span class="math-container">$3 \times 3$</span> matrices <span class="math-container">$A$</span> such that the trace of <span class="math-container">... | Shri | 442,962 | <p>Take <span class="math-container">$xH \in G/H$</span> of order p by Cauchy theorem then <span class="math-container">$x^p \in H$</span>.</p>
|
2,331,191 | <p>Use either direct proof, proof by contrapositive, or proof by contradiction.</p>
<p>Using proof by contradiction method</p>
<blockquote>
<p>Assume n is a perfect square and n+3 is a perfect square (proof by
contradiction)</p>
<p>There exists integers x and y such that <span class="math-container">$n = x^2$</span> an... | Paolo Leonetti | 45,736 | <p>$$
3=y^2-x^2 \ge (x+1)^2-x^2= 2x+1
$$
which is false if $x>1$.</p>
|
461 | <p>There is a function on $\mathbb{Z}/2\mathbb{Z}$-cohomology called <em>Steenrod squaring</em>: $Sq^i:H^k(X,\mathbb{Z}/2\mathbb{Z}) \to H^{k+i}(X,\mathbb{Z}/2\mathbb{Z})$. (Coefficient group suppressed from here on out.) Its notable axiom (besides things like naturality), and the reason for its name, is that if $a\i... | Eric Wofsey | 75 | <p>Here's one way to understand them. The external cup square $a \otimes a \in H^{2n}(X \times X)$ of $a \in H^n(X)$ induces a map $f:X \times X \to K(Z_2, 2n)$. It can be show that this map factors through a map $g:(X \times X) \times_{Z_2} EZ_2 \to K(2n)$, where $Z_2$ acts on the product by permuting the factors an... |
461 | <p>There is a function on $\mathbb{Z}/2\mathbb{Z}$-cohomology called <em>Steenrod squaring</em>: $Sq^i:H^k(X,\mathbb{Z}/2\mathbb{Z}) \to H^{k+i}(X,\mathbb{Z}/2\mathbb{Z})$. (Coefficient group suppressed from here on out.) Its notable axiom (besides things like naturality), and the reason for its name, is that if $a\i... | Andrew Stacey | 45 | <p>The Steenrod square is an example of a cohomology operation. Cohomology operations are natural transformations from the cohomology functor to itself. There are a few different types, but the most general is an <strong>unstable</strong> cohomology operation. This is simply a natural transformation from $E^k(-)$ to... |
1,643,579 | <p>I've an homework problem that i'm unable to find the right answer.</p>
<p>The problem is:</p>
<p>The line $tx + sy = 2$ goes through point $(2,1)$ and is parallel to line $y = 8 -3x$, find the value of $t^2 + s^2$. </p>
<p>$ A. {32\over49}$ $B.{18\over49}$ $C.{36\over49}$ $D.{25\over49} $ $E.{40\over49} $</p>
<p... | ncmathsadist | 4,154 | <p>Use L'Hospital and the fact that
$$- \left(\prod \cos(kx)\right)' =
\sum_{k=1}^n k\sin(kx)\prod_{j\not=k} \cos(jx).$$</p>
|
602,852 | <p>Following Halmos's Naive Set Theory, Russell's Paradox emerges from using the axiom of specification (that for every set $A$ and property $\phi$ there exists a set $Y$ whose elements are those elements $x$ of $A$ for which this property holds) with the property $x \notin x$.</p>
<p>We find then that for any set $A... | MJD | 25,554 | <blockquote>
<p>We find then that for any set A, there is a set B which is not in A. How does this - that everything contains something outside it - entail that 'nothing contains everything',</p>
</blockquote>
<p>Suppose there were a set $U$ that we thought <em>did</em> contain absolutely everything. But we showed,... |
602,852 | <p>Following Halmos's Naive Set Theory, Russell's Paradox emerges from using the axiom of specification (that for every set $A$ and property $\phi$ there exists a set $Y$ whose elements are those elements $x$ of $A$ for which this property holds) with the property $x \notin x$.</p>
<p>We find then that for any set $A... | Dan Christensen | 3,515 | <p>You may be confusing Russell's paradox with the paradox of the universal set. </p>
<p>From the Russell's paradox, we obtain:</p>
<p>$\neg\exists r: \forall x: [x\in r \iff x\notin x]$ </p>
<p>We used this result to resolve the paradox of the universal set, obtaining:</p>
<p>$\neg\exists U: \forall x: x\in U$</p>... |
3,815,640 | <p>what is the most efficient way to calculate the argument of
<span class="math-container">$$
\frac{e^{i5\pi/6}-e^{-i\pi/3}}{e^{i\pi/2}-e^{-i\pi/3}}
$$</span> without calculator ?</p>
<p>i tried to use <span class="math-container">$\arg z_1-\arg z_2$</span> but the argument of <span class="math-container">$e^{i5\pi/6... | Stinking Bishop | 700,480 | <p><strong>Hint</strong>: Sketch the points <span class="math-container">$A=e^{i5\pi/6}$</span>, <span class="math-container">$B=e^{-i\pi/3}$</span> and <span class="math-container">$C=e^{i\pi/2}$</span> on the unit circle, and use geometrical arguments to calculate the angle between <span class="math-container">$\vec{... |
4,458,863 | <p>Let <span class="math-container">$z_1,\;z_2,\;z_3\;$</span> be complex number such that <span class="math-container">$|z_1|=|z_2|=|z_3|=|z_1+z_2+z_3|=2\;\;$</span>. If <span class="math-container">$|z_1-z_3|=|z_1-z_2|\; \;$</span> and <span class="math-container">$z_2 \neq z_3.\; \; $</span> Then Find value of <span... | Dan | 1,374 | <p><a href="https://www.wolframalpha.com/input?i=integral+of+sqrt%28cosh%28x%29%29" rel="nofollow noreferrer">WolframAlpha</a> provides links to <a href="https://reference.wolfram.com/language/ref/EllipticE.html" rel="nofollow noreferrer">EllipticE</a> and <a href="https://reference.wolfram.com/language/ref/Hypergeomet... |
2,762,428 | <p>Excuse me, please, for the initial post, I did not know that someone would apprehend this rudely, and also excuse my English ...<br>
Task:
Find the equivalent to a function, when $ t \to +\infty $<br>
$$ f(t) = \int \limits_{t}^{2t} \frac{x^2}{e^{x^2}} dx $$</p>
<p>My ideas:<br>
1) I tried to find such a function t... | Somos | 438,089 | <p>You want the <a href="https://oeis.org/A007153" rel="nofollow noreferrer">Dedekind numbers (OEIS sequence A007153)</a>. The first few terms of the sequence: $(1, 4, 18, 166, 7579,\dots).$ The key is in the title "number of monotone Boolean functions". This is equivalent to being constructible from AND and OR. Read t... |
2,762,428 | <p>Excuse me, please, for the initial post, I did not know that someone would apprehend this rudely, and also excuse my English ...<br>
Task:
Find the equivalent to a function, when $ t \to +\infty $<br>
$$ f(t) = \int \limits_{t}^{2t} \frac{x^2}{e^{x^2}} dx $$</p>
<p>My ideas:<br>
1) I tried to find such a function t... | Bram28 | 256,001 | <p>Any such formula can be put into DNF by repeated distribution of AND over OR. And, since by Absorption we have that $p + pq=p$, this means that we just need to find the number of ways we can form a collection of terms where no term is part of another term. </p>
<p>For example, for $n=3$ (Say we have $p$, $q$, and $... |
802,014 | <p>For all sets $A$, $B$, $C$, if $A$ is subset of $B$, $B$ is subset of $C$, and $C$ is subset of $A$, then $A = B = C$.</p>
<p>This is a true statement and I need to provide a proof? Thus, when a statement is false I need to provide it with counterexample whereas if it is true then it has to be provided by a proof?<... | Ittay Weiss | 30,953 | <p>Yes, the claim you state is correct, and you thus need to prove it. </p>
<p>In general whatever you claim you need to supply argument for. Whether you call it a proof or a counterexample depends on the situation. To provide a counterexample is to prove that a claim is wrong by exhibiting that it is wrong for a part... |
1,449,776 | <p>I have always known that $a^n=a*a*a*.....$(n times)</p>
<p>Then what exactly is the meaning if $a^0$ and why will it be equal to $1$?</p>
<p>I have checked it in the internet but everywhere the solution is based on the principle that $a^m*a^n=a^{m+n}$ and when $n=0$ it will be $a^m$ and clearly $a^0$ is equal to $... | Brian Tung | 224,454 | <p>It's defined that way (except, usually, for $a = 0$) because it's most consistent to do so. The <em>empty product</em> is defined to be $1$ because $1$ is the multiplicative identity, much as the empty sum is defined to be $0$ because $0$ is the additive identity. Both of these definitions allow for the pattern th... |
4,072,386 | <p>I assume this is a simple proof but i'm stuck here.</p>
<p>I need to prove that if <span class="math-container">$A^3B-B$</span> is invertible then <span class="math-container">$BA-B$</span> is invertible.</p>
<p>So <span class="math-container">$A^3B-B=(A^3-I)B$</span> and then both <span class="math-container">$(A^... | Igor Rivin | 109,865 | <p>Hint <span class="math-container">$$A^3-I = (A-I)(A^2 + A + I).$$</span></p>
|
1,407,700 | <p>I am stuck on solving the following systems of equations with 3 variables. The textbook asks to use the addition method so can we please stick to that.</p>
<p>${5x -y = 3}$</p>
<p>${3x + z = 11}$</p>
<p>${y - 2z = -3}$</p>
<p>I am used to systems of equations where each equation has at least one instance of the... | Riemann | 27,899 | <p>Take <span class="math-container">$$s_n=n+\frac{1}{n},\quad t_n=n,$$</span>
then <span class="math-container">$\lim_{n\to\infty}(s_n-t_n)=0$</span>, but
<span class="math-container">$$\lim_{n\to\infty}(f(s_n)-f(t_n))
=\lim_{n\to\infty}\left[\left(n+\frac{1}{n}\right)^3\sin\frac{1}{n+\frac{1}{n}}
-n^3\sin\frac1n\righ... |
1,217,557 | <p>I was tasked with drawing the contour lines of $ z = \sqrt{xy} $, which I find a bit problematic since I can see no way in which one can plot (by hand, and not with wolfram and others....) the $ z = \sqrt{xy} $ graph in $R^2( x-, y- $ projection} to begin with for this surface...</p>
<p>How can one draw this conto... | Jam | 131,857 | <p>You are right .
$$L(x)$$
is not linear try not to confuse it because its graph is a line. IF one of the linearity properties do not apply then it is not.IN general
$$F(x)=ax+b$$
is linear iff b=0</p>
|
1,402,953 | <ol>
<li>How can one intuitively understand the definition of a bilinear map? Is there some way of looking at it geometrically? I found the following definition:</li>
</ol>
<p>Let $\mathit{A}$,$\mathit{B}$,$\mathit{C}$ be vector spaces. A map $f:\mathit{A}\times \mathit{B}\to C$ is said to be bilinear if for each fixe... | Sempliner | 122,727 | <p>I think there are many explanations of what it is, but as far as intuition is concerned I think the following is by far the most satisfying. We see bilinear maps as generalizations of the properties of a product, for instance if $K$ is a field or even a ring then the map $\times: K \times K \to K$ is a bilinear map,... |
2,852,550 | <p>I was wondering, Is there Cauchy sequence that are not bounded ? Of course, in complete spaces is not possible. I have a theorem that says that if $(A,d)$ is not complete, then $(\bar A,d)$ is complete. But are there spaces s.t. indeed for all $\varepsilon>0$, there is $N$ s.t. $d(x_n,x_{m})<\varepsilon$ for a... | Hagen von Eitzen | 39,174 | <p>Note that $d(x_n,x_N)<\epsilon$ for all $n\ge N$ and that only finitely many $n<N$ are to be considered beyond that.</p>
|
374,881 | <p>I'd like to know how I can recursively (iteratively) compute variance, so that I may calculate the standard deviation of a very large dataset in javascript. The input is a sorted array of positive integers.</p>
| DolphinDream | 80,405 | <p>Here are the iterative formulas (with derivations) for the <strong>population</strong> (<span class="math-container">$N$</span> normalized) and <strong>sample</strong> (<span class="math-container">$N-1$</span> normalized) standard deviations, which express the <span class="math-container">$\sigma_{n+1}$</span> (<sp... |
19,996 | <p>In 1556, Tartaglia claimed that the sums<br>
1 + 2 + 4<br>
1 + 2 + 4 + 8<br>
1 + 2 + 4 + 8 + 16<br>
are alternative prime and composite. Show that his conjecture is false. </p>
<p>With a simple counter example, $1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 = 255$, apparently it's false. However, I want to prove it in gen... | Brandon Carter | 1,016 | <p>There is simply no such correlation in general. If that were the case, Mersenne primes would not be as interesting to study as they are. You can not prove that $2^n-1$ composite $\Rightarrow 2^{n+1} - 1$ prime, because it isn't true. As Ross mentioned, $2^n-1$ can only be prime when $n$ is prime, so for any prime $p... |
2,521,710 | <p>I am trying to do a proof for convergence. But I am stuck in my proof not getting any further... What is missing to finish that proof?</p>
<p>$$a_n = \frac{1}{(n+1)^2}$$
Show that: $$\lim_{n \to \infty}a_n=0$$</p>
<p>Let $e > 0$ and $\forall n \ge n_0 = \lceil \frac{1}{\sqrt{\epsilon}}\rceil+1 \in \mathbb Z^+... | Community | -1 | <p>You are doing way too much here.</p>
<p>Let $\epsilon > 0$ and choose $n_0 \in \mathbb{N}$ such that $\frac{1}{n_0} < \epsilon$ (note that we used the archimedian property of the real numbers here!)</p>
<p>Then, for $n \geq n_0$, we have:</p>
<p>$$\left|\frac{1}{(n+1)²}\right| = \frac{1}{(n+1)²} \leq \frac{... |
367,116 | <p>I have a question about vacuous true and it always make me confused. If I want to prove that the empty set is the subset of all the set A, the proof is as following:
if x is in empty set, then x is in A. since x is in empty set is always false,, so the conditional statement is always true~
my question is why x is in... | Austin Mohr | 11,245 | <p>You want to prove something like the following:</p>
<p>If $x \in \emptyset$, then $x \in A$.</p>
<p>If you don't like the vacuous nature of the hypothesis, consider the contrapositive (which is logically equivalent to the original statement):</p>
<p>If $x \notin A$, then $x \notin \emptyset$.</p>
<p>Select any $... |
1,179,843 | <p>Proving $\sum_{n=1}^\infty \frac{\xi ^n}{n}$ is not uniformly convergent for $\xi \in (0,1)$.</p>
<p>I am trying to do the above. I have attempted to show it is not a cauchy sequence by considering $||\frac{\xi ^n}{n} ||_{\sup}$ but no avail. Any help please</p>
| user2566092 | 87,313 | <p>Let $f_n$ be the $n$th partial sum function, and consider $|f_{2n} - f_n|_1$. This is equal to</p>
<p>$$H_{2n} - H_n$$</p>
<p>where $H_n = \sum_{k=1}^n 1/k $ is the $n$th harmonic series partial sum. As $n \to \infty,$ we have $H_n \to \ln n + \lambda$ for some constant $\lambda$ independent of $n$ (look up Euler ... |
275,310 | <p>I am a bit confused. What is the difference between a linear and affine function? Any suggestions will be appreciated</p>
| William Balthes | 231,063 | <p>$(1)$<strong>Linear continuous Functional equations of the Form</strong>:
$$F(\alpha x +\delta y)= \alpha F(x) + \delta F(y)$$;</p>
<p>or rather 1- point homogeneity </p>
<p>$(1a)$ (often with Cauchy's equation as well), as in the above post.
However, sometimes Cauchy's equation is not needed in addition to $(1... |
1,530,874 | <p>Is there a case where a function $f$ that is not differentiable at $0$ and a function $g$ that is differentiable at $0$ where $f+g$ is differentiable at $0$?</p>
| Emilio Novati | 187,568 | <p>Hint:</p>
<p>let $f+g=h$, if $h$ is differentiable and $g$ is differentiable what can we say about $f=h-g$ ?</p>
|
77,089 | <p>Fix a field $k$. For a singular variety $X$, I understand that the Grothendieck group $K^0(X)$ of vector bundles on $X$ is not necessarily isomorphic to the Grothendieck group $K_0(X)$ of coherent sheaves on $X$. </p>
<p>I am curious to learn what is known about these two groups in one family of examples: $\mathb... | Steven Landsburg | 10,503 | <p>The identity map from $k$ to itself factors through $D$. Thus, if $K$ represents either $K^0$ or $K_0$, $K(\mathbb P^n_{k})$ is a direct summand of $K(\mathbb P^n_{D})$.</p>
<p>For $M$ a coherent $\mathbb P^n_{D}$-module, we have an exact sequence</p>
<p>$$0\rightarrow \epsilon M \rightarrow M \rightarrow M/\epsi... |
935,707 | <p>I'm having trouble determining this problem.</p>
<p>I need to find the integers in the set {1, ... , 100} that are divisible by 2 or 3 but not both.</p>
<p>The way I tried to approach it was:</p>
<p>If a number is divisible by both 2 and 3 then we can say it is divisible by 6. So we need to exclude integers divis... | Adriano | 76,987 | <p><strong>Hint:</strong> In the set $\{1,\cdots, 100\}$, count the number of multiples of $2$. Then count the number of multiples of $3$, and add the two numbers together. Then subtract twice the number of multiples of $6$. Note that the number of multiples of $6$ is $\lfloor 100/6 \rfloor = 16$, since they are:
\begi... |
935,707 | <p>I'm having trouble determining this problem.</p>
<p>I need to find the integers in the set {1, ... , 100} that are divisible by 2 or 3 but not both.</p>
<p>The way I tried to approach it was:</p>
<p>If a number is divisible by both 2 and 3 then we can say it is divisible by 6. So we need to exclude integers divis... | Community | -1 | <p>Every integer can be written in the form $n=6q+r$ by a division by $6$ ($r$ is the remainder of the division). The term $6q$ is a multiple of both $2$ and $3$, so it suffices to reason on the divisibility of the remainder.</p>
<p>The remainders are periodic,</p>
<p>$$\color{blue}{1,2,3,4,5,0},1,2,3,4,5,0,\color{bl... |
1,475,235 | <p>Why doesn't $e^x$ have an inverse in the complex plane? Can someone please clarify it?</p>
| AnatolyVorobey | 155,893 | <p>Among reals, only $0$ has the property that $e^0 = 1$, but among complex numbers, there are many $z$ such that $e^z=1$, for example, $2\pi i$, $4\pi i$, $6\pi i$ etc. But since $e^{z+w} = e^z*e^w$, you could add any of those numbers to any exponent $w$ and the value of $e^w$ doesn't change. Therefore $e^w$ is not on... |
281,717 | <p>Suppose that $\beta \mathbb{R}$ is Stone–Čech compactification of $\mathbb{R}$. What is the closure of $\mathbb{Q}$? </p>
| Martin Argerami | 22,857 | <p>I would guess it is your first option. A usual terminology in calculus is about absolute and relative (or local) maxima and minima. </p>
<p>The absolute maximum would be then $\max\{f(x):\ x\in[-2,4]\}$. </p>
<p>The phrase "absolute maximum <em>value</em>" probably has to do with the fact that when looking at extr... |
1,406,535 | <p>Let $ f$ be a function such that $|f(u)-f(v)|\leq|u-v|$ for all real $u$ and $v$ in an interval $[a,b]$.Then:<br>
$(i)$Prove that $f$ is continuous at each point of $[a,b]$.<br></p>
<p>$(ii)$Assume that $f$ is integrable on $[a,b]$.Prove that,$|\int_{a}^{b}f(x)dx-(b-a)f(c)|\leq\frac{(b-a)^2}{2}$,where $a\leq c \leq... | PITTALUGA | 94,471 | <p>I believe you got confused. Also in Fermat's little theorem you must put the condition $\gcd(a,p)=1$, otherwise if $p\mid a$ then
$$
a^{p-1}\equiv 0 \mod{p}\;.
$$
Of course also in it's generalization, Euler's theorem, only if $\gcd(a,n)=1$ the you can say
$$
a^{\varphi(n)}\equiv 1 \mod n\;.
$$
I suggest you to loo... |
2,146,508 | <blockquote>
<p>Let $K$ be the algebraic closure of a finite field $k$. Prove that $Gal(K/k) \cong \hat{\mathbb{Z}}$.</p>
</blockquote>
<p>From the definition in the book, here is how $\hat{\mathbb{Z}}$ is defined:
Let $D = Cr(\mathbb{Z}_{p} | \; p \; prime)$, let $\delta: \mathbb{Z} \rightarrow D$ be the map taking... | Dietrich Burde | 83,966 | <p>A detailed proof is, for example, given in James S. Milne's lecture notes on <a href="http://www.jmilne.org/math/CourseNotes/FT.pdf" rel="nofollow noreferrer">Fields and Galois Theory</a>, EXAMPLE 7.16., page $97$. Ingredients are the canonical Frobenius element $\sigma:a\mapsto a^p$, the profinite completion of $\m... |
3,106,550 | <p>Determine the values of a for which the following system of linear equations has no solutions, a unique solution, or infinitely many solutions. You can select 'always', 'never', '<span class="math-container">$a = $</span>', or '<span class="math-container">$a \neq$</span>', then specify a value or comma-separated li... | Dr. Sonnhard Graubner | 175,066 | <p>With <span class="math-container">$$z=\frac{1}{2}(x-3y-8)$$</span> (from the first equation) we get
<span class="math-container">$$x(a-2)+5y=-10$$</span> and <span class="math-container">$$y=-2$$</span> in the third equation, so we obtain
<span class="math-container">$$x(a-2)=0$$</span> Can you finish?</p>
|
3,106,550 | <p>Determine the values of a for which the following system of linear equations has no solutions, a unique solution, or infinitely many solutions. You can select 'always', 'never', '<span class="math-container">$a = $</span>', or '<span class="math-container">$a \neq$</span>', then specify a value or comma-separated li... | Rhys Hughes | 487,658 | <p>Hint:</p>
<p>Let the equations be <span class="math-container">$(1), (2), (3)$</span> in the order you show.</p>
<p>Perform the following three calculations:</p>
<p><span class="math-container">$(1)-(2)$</span></p>
<p><span class="math-container">$(1)-(3)$</span></p>
<p><span class="math-container">$(3)-(2)$</s... |
1,580,270 | <p>Consider the groups $G = \{0,1,2\} = \mathbb Z_3$ and $H = \{a,b,c\}$
given by the following multiplication tables:</p>
<p><a href="https://i.stack.imgur.com/hXgBb.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/hXgBb.jpg" alt="enter image description here"></a></p>
<p>The first one isn't really... | XPenguen | 293,908 | <p><span class="math-container">$O$</span> is the identity in G and <span class="math-container">$b$</span> is the identity in H. Another thing to notice is that the tables are symmetric. Hence <span class="math-container">$G,H$</span> are commutative groups.<br><br></p>
<p>Take a look at <span class="math-container">... |
659,254 | <p>Say $X_1, X_2, \ldots, X_n$ are independent and identically distributed uniform random variables on the interval $(0,1)$.</p>
<p>What is the product distribution of two of such random variables, e.g.,
$Z_2 = X_1 \cdot X_2$?</p>
<p>What if there are 3; $Z_3 = X_1 \cdot X_2 \cdot X_3$?</p>
<p>What if there are $n$ ... | robjohn | 13,854 | <p>An adaptation of <a href="https://math.stackexchange.com/a/2812234">this answer</a> is given here.</p>
<hr>
<p><strong>PDF of a Function of a Random Variable</strong></p>
<p>If $P(X\le x)=F(x)$ is the CDF of $X$ and $P(Y\le y)=G(y)$ is the CDF of $Y$ where $Y=f(X)$, then
$$
F(x)=P(X\le x)=P(Y\le f(x))=G(f(x))\ta... |
2,955,780 | <p>The midpoint of a chord of length <span class="math-container">$2a$</span> is at a distance <span class="math-container">$d$</span> from the midpoint of the minor arc it cuts out from the circle. Show that the diameter of the circle is <span class="math-container">$\frac{a^2+d^2}{d}$</span> .</p>
<p>I know I have t... | Catalin Zara | 317,861 | <p>If the chord is <span class="math-container">$AB$</span> with midpoint <span class="math-container">$M$</span>, the midpoint of the arc is <span class="math-container">$P$</span> and the point diametral opposite to <span class="math-container">$P$</span> is <span class="math-container">$Q$</span>, then the triangles... |
1,495,256 | <p>What are the algebraic rules for solving inequalities with negative signs and fractions like:</p>
<p>$$\frac{1}{x}<-\frac{1}{5}$$</p>
| Yes | 155,328 | <p>If $x \in \mathbb{R}$, then
$\frac{1}{x} < -\frac{1}{5}$ iff $x > -5$ and iff $-x < 5$.</p>
<p>If you are seeking after the rule of thumb, here it is: whenever we take inverse of both sides, the inequality sign reverses; whenever we multiply both sides by $-1$, the inequality sign reverses too.</p>
|
1,495,256 | <p>What are the algebraic rules for solving inequalities with negative signs and fractions like:</p>
<p>$$\frac{1}{x}<-\frac{1}{5}$$</p>
| GambitSquared | 245,761 | <ol>
<li><p>Whenever we take the inverse, and both sides are either positive or both sides are negative then the inequality sign reverses; (if one is side is positive and one side is negative, then the sign stays the same.)</p></li>
<li><p>Whenever we multiply both sides by $-1$, the inequality sign reverses too.</p><... |
1,134,510 | <p>Regarding My Background I have covered stuff like </p>
<p>1.Single Variable Calculus</p>
<p>2.Multivariable Calculus (Multiple Integration,Vector Calculus etc) (Thomas Finney)</p>
<p>3.Basic Linear Algebra Course (Containing Vector spaces,Linear Transformation)</p>
<p>4.Ordinary Differential Equation</p>
<p>5.R... | barak manos | 131,263 | <p>The number of ways to choose $2$ out of $26$ letters is:</p>
<p>$$\binom{26}{2}=325$$</p>
<p>The number of ways to choose $3$ places for the $1$st letter and $4$ places for the $2$nd letter is:</p>
<p>$$\binom{7}{3}\cdot\binom{4}{4}=35$$</p>
<p>The number of ways to choose $4$ places for the $1$st letter and $3$... |
304 | <p>Per <a href="http://blog.stackoverflow.com/2010/07/moderator-pro-tempore/">this post on the SO/SE blog</a> (which, curiously, does not include math.SE in its graphic list), it looks like the admins will choose moderators pro tempore at about 7 days into the public beta. In the roughly 24 hours that we've been in pu... | Robert Cartaino | 69 | <p>The Moderator Pro Tem program is just about complete. <strong>I don't have an objection</strong> to bumping up Math.SE in the schedule a few days (would have been on Tuesday, 8/3 anyway). The meta activity has been pretty solid. That's assuming there isn't any wide-spread objection from here or higher-on up.</p>
<p... |
1,830,989 | <p>so while playing around with circles and triangles I found 2-3 limits to calculate the value of $ \pi $ using the <em>sin, cos and tan</em> functions, I am not posting the formula for obvious reasons.<br>
My question is that is there another infinite series or another way to define the trig functions when the value ... | John Hughes | 114,036 | <p>Well...you could avoid the $\pi$ in the previous answers by substituting for it one of the many infinite series formulas for $\pi$. That'd get rid of the $\pi$, but wouldn't be much practical use, which may be why you're not seeing it all over the internet. :) </p>
<p>I actually <em>do</em> occasionally want to kno... |
431,690 | <p>As far as I know, for any $A$:
$$\mathbf{x}^{T}A\mathbf{y}=0;\forall\mathbf{x},\mathbf{y}\in R^n\Rightarrow A=0$$</p>
<p>Does it mean that
$$\mathbf{x}^{T}A\mathbf{x}=0;\forall\mathbf{x}\in R^n\Rightarrow A=0$$</p>
<p>The condition of the first claim $\forall\mathbf{x},\mathbf{y}\in R^n$ implies that we could take... | Alex Becker | 8,173 | <p>As pointed out in the comments, your conclusion is false. The problem with your reasoning is that the first claim reads (in plain English):</p>
<blockquote>
<p>If, for all $x$ and $y$, $x^TAy=0$, then $A=0$.</p>
</blockquote>
<p>When you set $y=x$ then you are no longer considering all $x$ and $y$, only the pair... |
67,630 | <p>I know of a theorem from Axler's <em>Linear Algebra Done Right</em> which says that if $T$ is a linear operator on a complex finite dimensional vector space $V$, then there exists a basis $B$ for $V$ such that the matrix of $T$ with respect to the basis $B$ is upper triangular.</p>
<p>The proof of this theorem is b... | Harry Altman | 2,884 | <p>The problem is that it doesn't just use the fact that T has an eigenvalue -- it uses that, plus the inductive hypothesis. And in order to prove it for this smaller invariant subspace, you need the fact that T has an eigenvalue on that as well. So there's the problem -- it's not enough to have an eigenvalue, you ne... |
34,795 | <p>Consider torsion free modules over the germ of a fixed isolated algebraic hypersurface singularity {$f=0$}$\subset\mathbb{C}^n$.
There are natural functors (using categories of finitely generated modules): </p>
<p>modules over $\mathbb{C}[x_1,..,x_n]_{(x_1,\dots,x_n)}/(f)$--> modules over $\mathbb{C}${$x_1,..,x_n$}... | Hailong Dao | 2,083 | <p>The most relevant result I am aware of is the following: suppose <span class="math-container">$R$</span> is a local Gorenstein ring, essentially of finite type over a field, with an isolated singularity (so certainly include your case) then the stable categories of maximal Cohen-Macaulay modules over <span class="ma... |
34,795 | <p>Consider torsion free modules over the germ of a fixed isolated algebraic hypersurface singularity {$f=0$}$\subset\mathbb{C}^n$.
There are natural functors (using categories of finitely generated modules): </p>
<p>modules over $\mathbb{C}[x_1,..,x_n]_{(x_1,\dots,x_n)}/(f)$--> modules over $\mathbb{C}${$x_1,..,x_n$}... | Mohan | 9,502 | <p>There are well studied hypersurfaces of dimension 2 which are UFDs whose completions are not. So, any ideal representing a non-trivial element in the class group of the completion will not come from the algebraic ring.</p>
|
34,795 | <p>Consider torsion free modules over the germ of a fixed isolated algebraic hypersurface singularity {$f=0$}$\subset\mathbb{C}^n$.
There are natural functors (using categories of finitely generated modules): </p>
<p>modules over $\mathbb{C}[x_1,..,x_n]_{(x_1,\dots,x_n)}/(f)$--> modules over $\mathbb{C}${$x_1,..,x_n$}... | Mohan | 9,502 | <p>Dear Hailong (and all others reading his), I apologize for using `answer' for a comment, since my openid does not work (I am traveling) or I have forgotten it and MO does not give me a way to comment. </p>
<p>Yes, there are examples over any field. If my memory serves me right, the first example I saw was in Samuel... |
1,098,253 | <p>I have got some trouble with proving that for $x\neq 0$:
$$
\frac{\arctan x}{x }< 1
$$
I tried doing something like $x = \tan t$ and playing with this with no success.</p>
| miracle173 | 11,206 | <p>A basic proof without calculus:</p>
<p>We assume that $x>0$. Then we have to show that</p>
<p>$$\arctan(x)<x \tag{1}$$</p>
<p>But we can concluded this from </p>
<p>$$x \lt \tan(x), \;\; \forall x \in (0,\pi/2) \tag{2}$$
If we substitute all occurrences of $x$ by $\arctan(x)$ in $(2)$ we get</p>
<p>$$\a... |
1,098,253 | <p>I have got some trouble with proving that for $x\neq 0$:
$$
\frac{\arctan x}{x }< 1
$$
I tried doing something like $x = \tan t$ and playing with this with no success.</p>
| Felix Marin | 85,343 | <p>$\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{\ds}[1]{\displaystyle{#1}}
\newcomma... |
1,599,890 | <blockquote>
<p>Let $a_n$ be the number of those permutation $\sigma $ on $\{1,2,...,n\}$ such that $\sigma $ is a product of exactly two disjoint cycles. Then find $a_4$ and $a_5$.</p>
</blockquote>
<p>Calculating $a_4$: Possible cases which can happen are $(12)(34),(13)(24),(14)(23)$, any cycle of the form $(123)... | Christian Blatter | 1,303 | <p>Let $r\in[1\>..\>n-1]$ be the length of the cycle containing the number $n$. Begin the listing of this cycle with $n$. There are $(n-1)(n-2)\cdots(n-r+1)$ ways to choose the remaining entries of this cycle. Now there will be $n-r\geq1$ numbers left over, one of them the largest. Begin the listing of the secon... |
3,159,884 | <p>Prove that if <span class="math-container">$|z+w|=|z-w|$</span> then <span class="math-container">$z\overline{w}$</span> is purely imaginary.</p>
<p>To start off, I said let <span class="math-container">$z=a+bi$</span> and let <span class="math-container">$w=p+qi$</span>. Not sure where to go from here after subbin... | DINEDINE | 506,164 | <p><span class="math-container">$$|z+\omega|^2=|z-\omega|^2 \iff$$</span>
<span class="math-container">$$(z+\omega)(\bar{z}+\bar{\omega})= (z-\omega)(\bar{z}-\bar{\omega})\iff$$</span>
<span class="math-container">$$z\bar{z}+{\omega}\bar{z}+z\bar{\omega}+\bar{\omega}\omega= z\bar{z}-\omega\bar{z}-z\bar{\omega}+\bar{\o... |
298,912 | <p>I was reading some basic information from Wiki about category theory and honestly speaking I have a very weak knowledge about it. As it sounds interesting, I will go into the theory to learn more if it is actually useful in practice.</p>
<p>My question is to know if category theory has some applications in practice... | yoknapatawpha | 108,381 | <p>I know this answer is (very) late, but I think this may be of interest. <a href="http://ames.tamu.edu/A%20categorical%20theory.pdf" rel="noreferrer">This</a> is the Ph.D. thesis of Professor Aaron Ames of Texas A&M which he wrote while a student at UC Berkeley. It applies category theory to hybrid systems and sp... |
41,836 | <p>Nakayama's lemma is as follows:</p>
<blockquote>
<p>Let <span class="math-container">$A$</span> be a ring, and <span class="math-container">$\frak{a}$</span> an ideal such that <span class="math-container">$\frak{a}$</span> is contained in every maximal ideal. Let <span class="math-container">$M$</span> be a finitel... | Martin Brandenburg | 2,841 | <p>There are various forms of the Nakayama lemma. Here is a rather general one; note that it does <em>not</em> involve maximal ideals and is a constructive theorem (Atiyah-MacDonald, Commutative Algebra, Prop. 2.4 ff).</p>
<blockquote>
<p>Let $M$ be a finitely generated $A$-module, $\mathfrak{a} \subseteq A$ be an i... |
275,974 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/264889/how-is-this-called-rationals-and-irrationals">How is this called? Rationals and irrationals</a> </p>
</blockquote>
<p>Please help me prove, that
$$\underset{n\rightarrow\infty}{\lim}\left(\underset{... | zrbecker | 19,536 | <p>Baby Rudin has this as an example.</p>
<p><img src="https://i.stack.imgur.com/DDSHn.png" alt="See Baby Rudin page 145"></p>
|
3,689,096 | <p>This was the Question:- Find all positive integers <span class="math-container">$n$</span> such that <span class="math-container">$\varphi(n)$</span> divides <span class="math-container">$n^2 + 3$</span></p>
<p>What I tried:-</p>
<p>I knew the solution and explanation of all positive integers <span class="math-con... | AT1089 | 758,289 | <p>Note that <span class="math-container">$\phi(n)=1$</span> if and only if <span class="math-container">$n=1,2$</span>, and even if and only if <span class="math-container">$n>2$</span>. Therefore, <span class="math-container">$\boxed{n=1,2}$</span> are both solutions to</p>
<p><span class="math-container">$$ \phi(... |
2,164,823 | <p>Could someone please explain mathematical explanation behind this?</p>
<p>You can win three kinds of basketball points, 1 point, 2 points, and 3 points. Given a total score $n$, print out all the combination to compose $n$.</p>
<p>Examples:<br>
For n = 1, the program should print following:<br>
1 </p>
<p>For n =... | Hazem Orabi | 367,051 | <p>It is a <a href="https://en.wikipedia.org/wiki/Fibonacci_number" rel="nofollow noreferrer">Fibonacci</a> like sequence. <br/>
Let $\,A_{\small n}\,$ be the total number of $\,(1,\,2,\,3)\,$ combinations that compose $\,n\,$, Then:
$$ A_{\small 1}=1,\,A_{\small 2}=2,\,A_{\small 3}=4,\quad\color{red}{A_{\small n}=A_... |
28,892 | <p>I was searching on MathSciNet recently for a certain paper by two mathematicians. As I often do, I just typed in the names of the two authors, figuring that would give me a short enough list. My strategy was rather dramatically unsuccessful in this case: the two mathematicians I listed have written 80 papers toget... | Community | -1 | <p>Sergio Albeverio and Raphael Hoegh-Krohn have <a href="http://www.ams.org/mathscinet/search/publications.html?pg1=INDI&s1=24435%2520and%252086835" rel="noreferrer">98 papers together</a>
according to MathSciNet. </p>
|
28,892 | <p>I was searching on MathSciNet recently for a certain paper by two mathematicians. As I often do, I just typed in the names of the two authors, figuring that would give me a short enough list. My strategy was rather dramatically unsuccessful in this case: the two mathematicians I listed have written 80 papers toget... | S. Okada | 36,665 | <p>We get 135 matches for "Author=(Jimbo, Michio and Miwa, Tetsuji)" in mathscinet.</p>
|
606,843 | <p><strong>Definitions:</strong></p>
<ol>
<li><p>$[x]$ is the integer value of $x$. For example: $[4.3]=4$.</p></li>
<li><p>$\{x\}=x-[x]$.</p></li>
</ol>
<p>Can someone help me calculate the derivative of the following functions, and determine where the derivative are not defined?</p>
<ol>
<li><p>$[x^2]\sin^2(\pi x)... | jimbo | 115,363 | <p>Derivate Fourier expansion </p>
<p>$$\{x\}= \frac{1}{2} - \frac{1}{\pi} \sum_{k=1}^\infty \frac{\sin(2 \pi k x)} {k}$$</p>
|
374,619 | <p>In <a href="https://math.stackexchange.com/a/373935/752">this recent answer</a> to <a href="https://math.stackexchange.com/q/373918/752">this question</a> by Eesu, Vladimir
Reshetnikov proved that
$$
\begin{equation}
\left( 26+15\sqrt{3}\right) ^{1/3}+\left( 26-15\sqrt{3}\right) ^{1/3}=4.\tag{1}
\end{equation}
$$</p... | Joseph G. | 73,740 | <p>Using the fact that: $$x^3+y^3=(x+y)(x^2-xy+y^2)$$ and letting $x=m^{\frac{1}{3}}$ and $y=n^{\frac{1}{3}}$, we get the statement: $$m+n=(m^{\frac{1}{3}}+n^{\frac{1}{3}})(m^\frac{2}{3}-(mn)^{\frac{1}{3}}+n^{\frac{2}{3}})$$Maybe this will help.</p>
|
374,619 | <p>In <a href="https://math.stackexchange.com/a/373935/752">this recent answer</a> to <a href="https://math.stackexchange.com/q/373918/752">this question</a> by Eesu, Vladimir
Reshetnikov proved that
$$
\begin{equation}
\left( 26+15\sqrt{3}\right) ^{1/3}+\left( 26-15\sqrt{3}\right) ^{1/3}=4.\tag{1}
\end{equation}
$$</p... | N. S. | 9,176 | <p>Yikes. I am hopind I didn't do any mistake, but that would be a miracle :)</p>
<p>Let $x= \left( p+q\sqrt{3}\right) ^{1/3}\,;\, y= \left(
p-q\sqrt{3}\right) ^{1/3}$.</p>
<p>Then </p>
<p>$$xy= (p^2-3q^2)^\frac{1}{3} $$</p>
<p>Hence</p>
<p>$$2p=x^3+y^3=(x+y)^3-3xy(x+y)=n^3-3n\sqrt[3]{p^2-3q^2}$$</p>
<p>This show... |
2,666,409 | <blockquote>
<p>If $SL(2)=\{A\in \mathcal{M}_2: \det(A)=1\}$, find a parametrization around the identity matrix $I_2$ and find the first fundamental form. </p>
</blockquote>
<p>I've proved that $SL(2)$ is an hypersurface in $\mathbb{R}^4$, so it has dimension $3$. However, I don't understand very well what does "aro... | K B Dave | 534,616 | <p>Since $\mathrm{SL}_2$ is a quadric hypersurface, one way to get a parametrization near the point $I_2$ is to <a href="https://en.wikipedia.org/wiki/Stereographic_projection#Generalizations" rel="nofollow noreferrer">stereographically project</a> $\mathrm{SL}_2$ onto the tangent hyperplane at $I_2$ from some other po... |
685,918 | <p>I'm doing exercises in Real Analysis of Folland and got stuck on this problem. I don't know how to calculate limit with the variable on the upper bound of the integral. Hope some one can help me solve this. I really appreciate.</p>
<blockquote>
<blockquote>
<p>Show that $\lim\limits_{k\rightarrow\infty}\int_0... | Community | -1 | <p>We have</p>
<p>$$\int_0^k x^n(1-k^{-1}x)^kdx=\int_0^\infty x^n(1-k^{-1}x)^k\chi_{(0,k)}dx$$
then since
$$x^n(1-k^{-1}x)^k\chi_{(0,k)}\le x^n e^{-x},\;\forall k$$
and the function
$$x\mapsto x^n e^{-x}$$ is integrable on the interval $(0,\infty)$ then by the dominated convergence theorem we have
$$\lim_{k\to\infty}\... |
226,449 | <p>Many counting formulas involving factorials can make sense for the case $n= 0$ if we define $0!=1 $; e.g., Catalan number and the number of trees with a given number of vetrices. Now here is my question:</p>
<blockquote>
<p>If $A$ is an associative and commutative ring, then we can define an
unary operation on ... | ncmathsadist | 4,154 | <p>Here are two resons.</p>
<ol>
<li><p>We can define $n!$ to be the number of rearrangements of $n$ distinct objects in a list. The empty list has one rearrangement: itself. </p></li>
<li><p>We can define $n!$ as the product of all positive integers $k$ with $1\le k \le n$. If $n$ is zero, we have an empty product. ... |
1,873,194 | <p>Entropy of random variable is defined as:</p>
<p>$$H(X)= \sum_{i=1}^n p_i \log_2(p_i)$$</p>
<p>Which as far as I understand can be interpreted as how many yes/no questions one would have to ask on average, to find out the value of the random variable $X$.</p>
<p>But what if the log base is changed to for example ... | Olivier Oloa | 118,798 | <p>One has
$$\log_a(x)=\frac{\log b}{\log a}\:\log_b(x)$$
thus
$$H(X)= \sum_{i=1}^n p_i \log_a(p_i)=\frac{\log b}{\log a}\sum_{i=1}^n p_i \log_b(p_i)$$ or
$$
H_a(X)=\frac{\log b}{\log a}H_b(X).
$$</p>
|
3,104,051 | <p>I have the work of the proof done, but at the end after showing
<span class="math-container">$3^{2(n+1)} - 1=9(3^{2n} - 1)+8$</span>
I make the statement that since <span class="math-container">$9(3^{2n} - 1)$</span> is a multiple of 8 and 8 is a multiple of 8 then <span class="math-container">$3^{2n} - 1$</span> is... | B. Goddard | 362,009 | <p>If you want to add a detail, you're assuming <span class="math-container">$3^{2n}-1$</span> is a divisible by <span class="math-container">$8$</span>, so it equals <span class="math-container">$8w$</span> for some integer <span class="math-container">$w$</span>. Then your last expression is</p>
<p><span class="mat... |
537,968 | <p>Let $|\alpha|<1$ and $\psi_{\alpha}(z)=(\alpha-z)/(1-\bar\alpha z)$. I want to prove that $$\frac 1 {\pi} \int\int_{\mathbb{D}}|{\psi_{\alpha}}^{'}|dxdy = \frac{1-|\alpha|^2}{|\alpha|^2}\log\frac{1}{1-|\alpha|^2}$$</p>
<p>I calculated ${\psi_\alpha}^{'}(z)=(|\alpha|^2-1)/(1-\bar\alpha z)^2$. I substituted it and... | Daniel Fischer | 83,702 | <p>Without loss of generality, you can assume $0 < \alpha < 1$. Let us call $\rho := \alpha r$. Then</p>
<p>$$\frac{1}{\lvert 1 - \rho e^{i\theta}\rvert^2} = \frac{1}{1+\rho^2 - 2\rho\cos\theta},$$</p>
<p>and you can integrate that using one of the classical methods, like the residue theorem.</p>
|
4,552,955 | <p>I'm solving a probability problem, and I've ended up with this sum:</p>
<p><span class="math-container">$$\sum\limits_{k=0}^{n-a-b}\binom{n-a-b}{k}(a+k-1)!(n-a-k)!$$</span></p>
<p>WolframAlpha says I should get the answer <span class="math-container">$\frac{n!}{a\binom{a+b}{a}}$</span>, but I don't see how to get th... | Marko Riedel | 44,883 | <p>We seek to find a closed form of</p>
<p><span class="math-container">$$(n-1)! \sum_{k=0}^{n-a-b}
{n-a-b\choose k} {n-1\choose n-a-k}^{-1}$$</span></p>
<p>where <span class="math-container">$n\gt a+b$</span> and <span class="math-container">$a,b\ge 1.$</span></p>
<p>Recall from <a href="https://math.stackexchange.co... |
2,329,600 | <p>I haven't studied any maths since I was at university 20 years ago. Yesterday, however, I came across a pair of equations in an online article about gaming and I couldn't understand how they'd been derived. </p>
<p>Here's the scenario. If we make a single trial of generating a number between 1 and 20, there's an ev... | Matti P. | 432,405 | <p>If I interpret correctly, the question you're asking is: Choosing randomly two numbers between 1 and 20, what is the probability that at least one of them is at least 11?</p>
<p>To answer this, we can look at the three cases where this occurs:</p>
<ul>
<li>The first number is less than $11$ and the second is at le... |
2,150,832 | <p>I don't understand this equation $\int_0^t ds \int_0^{t'} ds' \delta(s-s')= \min(t,t')$.
I tried to work with the property of the dirac delta function that $\int_a^b \delta(x-c)dx = 1$ if $c \in [a,b]$, but I can't see how I can obtain the minimum. Can someone help me? </p>
<p>Thank you in advance!</p>
| David Holden | 79,543 | <p>let $\chi_t(x)$ be the indicator function for the interval $[0,t]$. then, as you point out:
$$
\int_0^{t'}ds'\delta(s-s') = \chi_{t'}(s)
$$
but now:
$$
\int_0^t\chi_{t'}(s) ds = \int_0^{\infty} \chi_{t}(s)\chi_{t'}(s) ds = \int_0^{\infty} \chi_{\min(t,t')}(s) ds = \min(t,t')
$$</p>
|
307,545 | <p>If $\gcd(a,b)=1$, how can I find the values that $\gcd(a+b,a^2+b^2)$ can possibly take? I can't find a way to use any of the elemental divisibility and gcd theorems to find them. </p>
| Pedro | 23,350 | <p>Note that $a^2+b^2=(a+b)^2-2ab$. Thus $a^2+b^2\equiv -2ab\mod a+b$ and you need to find $(a+b,-2ab)=(a+b,2ab)$. Can you move on?</p>
<p><strong>ADD</strong> Note that $a,b$ cannot be both even. If one is odd and the other is even, then $2\not\mid a+b$, so $(a+b,2ab)=(a+b,ab)$. But if $p>2$ is a prime with $... |
7,223 | <p>I want to produce a <em>Mathematica</em> Computable Document in which <code>N</code> appears as a variable in my formulae. But <code>N</code> is a reserved word in the <em>Mathematica</em> language. Is there a way round this other than using a different symbol? It seems a severe limitation if you cannot use <em>Math... | Rojo | 109 | <p>Ok, I'm late here, but the first thing I would have answered hasn't been answered already.</p>
<p>If it's a one-cell-er, you can use <code>Module</code></p>
<pre><code>Module[{N},
N = 8;
N + 3]
</code></pre>
<p>This can never bring you trouble with internal definitions since that N you see is not actually N.</... |
2,454,895 | <p>I don't know how to solve this equation:$$(1)\quad e^ {-x} = -\ln x$$</p>
<p>$x$ should be the abscissa of the point $P$ where the two functions meet on the plan and $$ P \in f(x) :y=x$$</p>
<p>so $(1)$ should be equal to $$ e^{-x}=x=-\ln x$$ </p>
<p>How do I solve this?</p>
| Stu | 460,772 | <p>$\quad e^ {-x} = -\ln x\iff e^ {-x}+\ln x=0 $</p>
<p>Let $g(x)=e^ {-x}+\ln x$</p>
<p>$g(1)=\dfrac{1}{e}>0$ and $\displaystyle \lim_{x\to 0} g(x)=-\infty$</p>
<p>The Intermediate value theorem guaranties there exists $c\in (0,1)$ such that $g(c)=0 \iff \exists c\in (0,1) $ such that $e^ {-c} = -\ln c$</p>
|
2,291,852 | <p>$$\int_\pi^\infty{\frac{x \cos x}{x^2-1}dx}$$</p>
<p>So the only think I came up with was to take an absolute value of ${\frac{x \cos x}{x^2-1}}$ and by comparison test the integral does not converge. </p>
<p>But I see it's not very close to the solution, so what should I do?</p>
| Joe | 107,639 | <p>Let's call your set $A$.</p>
<p>Being your $A\subseteq\Bbb R$, you know that it is compact iff it is closed and bounded.</p>
<p>Now $A$ is closed since it is counterimage of the singleton $\{1\}$ (which is closed) thru the continous function $f(x)=x^2+e^x$.</p>
<p>Finally, $A$ is bounded since</p>
<p>$$
\lim_{x\... |
2,291,852 | <p>$$\int_\pi^\infty{\frac{x \cos x}{x^2-1}dx}$$</p>
<p>So the only think I came up with was to take an absolute value of ${\frac{x \cos x}{x^2-1}}$ and by comparison test the integral does not converge. </p>
<p>But I see it's not very close to the solution, so what should I do?</p>
| MooS | 211,913 | <p>Since you also asked to prove that the set is finite (which is of course much stronger than compact):</p>
<p>If $f(x)=x^2+e^x-1$ has infinitely many roots, then by Rolle's theorem the same holds for $f'$ and thus also $f''$. But $f''(x)=2+e^x$ has no roots at all.</p>
|
2,163,494 | <p>Let $f: A\to B; \ g,h:B\to A$ and $f\circ g = I_B$ and $h \circ f = I_A$</p>
<p>I want to simply state that for any function $f$ if $f \circ h = I_A$ then it must be that $h = f^{-1}$ but that seems incomplete to me. What can I do for fixing this?</p>
| Nigel Overmars | 96,700 | <p>To put the other answer a bit differently: We have that $A \subseteq \overline{A}$, and since $A$ is bounded, $\overline{A}$ is compact. It follows that $f(\overline{A})$ is compact and hence bounded. Since we have that $f(A) \subseteq f(\overline{A})$, the result follows.</p>
|
3,518,719 | <p>Evaluate <span class="math-container">$$\lim_{n \to \infty} \sqrt[n^2]{2^n+4^{n^2}}$$</span></p>
<p>We know that as <span class="math-container">$n\to \infty$</span> we have <span class="math-container">$2^n<<2^{2n^2}$</span> and therefore the limit is <span class="math-container">$4$</span></p>
<p>In a more... | Ted Shifrin | 71,348 | <p>Of course it's true for a plane curve that <span class="math-container">$\dot b = -\tau n$</span>. Since <span class="math-container">$b$</span> is constant, we deduce that <span class="math-container">$\tau = 0$</span>. There's no contradiction here.</p>
|
2,452,777 | <p>Let X be a topological space, $\mathcal{U} = \{U_\alpha\}_\alpha$ an open cover of $X$ and $\mathcal{F}$ a presheaf of abelian groups on $X$. Then one can define the Čech cohomology groups of $\mathcal{U}$ with values in $\mathcal{F}$:
\begin{equation}
\check{H}^k(\mathcal{U}, \mathcal{F})
\end{equation}
The Čech co... | Daniel Robert-Nicoud | 60,713 | <p>You can give the index set $A$ a poset structure by $a\le b$ if $U_a\subseteq U_b$. Then taking the limit over the associated category $A$ of the functor
$$\mathcal{U}:A\longrightarrow P(X)$$
given by $\mathcal{U}(a):=U_a$ is the same as taking the limit over the cover ordered by refinement.</p>
|
300,531 | <p>Prove that : $$ \gamma=-\int_0^{1}\ln \ln \left ( \frac{1}{x} \right) \ \mathrm{d}x.$$</p>
<p>where $\gamma$ is Euler's constant ($\gamma \approx 0.57721$).</p>
<hr>
<p>This integral was mentioned in <a href="http://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant">Wikipedia</a> as in <a href="http://mathw... | Chappers | 221,811 | <p>Another way, from another definition: by the Dominated Convergence Theorem,
$$ \int_0^{\infty} e^{-u} \log{u} \, du = \lim_{n \to \infty} \int_0^n \left( 1 - \frac{u}{n} \right)^{n-1} \log{u} \, du. $$
Then, changing variables to $v=1-u/n$,
$$ \begin{align}
\int_0^n \left( 1 - \frac{u}{n} \right)^{n-1} \log{u} \, du... |
1,722,443 | <p>I am unsure how to solve the following problem. I was able to find similar questions, but had trouble understanding them since they did not show full solutions.</p>
<p>The question:</p>
<p>Find ALL solutions (between $1$ & $40$) to the equation $25x \equiv 10 \pmod{40}$.</p>
| N. F. Taussig | 173,070 | <p>Let's use the definition of congruence. $a \equiv b \pmod{n} \iff a = b + kn$ for some integer $k$. Hence, $25x \equiv 10 \pmod{40}$ means $$25x = 10 + 40k$$ for some integer $k$. Dividing each side of the equation $25x = 10 + 40k$ by $5$ yields $$5x = 2 + 8k$$
for some integer $k$. Thus,
$$5x \equiv 2 \pmod{8... |
1,908,820 | <p>The question is: </p>
<p>If $$\int_3^9f(x) dx = 7$$
evaluate
$$\int_3^9 2f(x)+1 dx$$</p>
<p>I know that you can factor the 2 outside of the integral. But, then I am still left with a '+ 1' inside the integral that when I take the integral of becomes $x$. So then would I proceed to stating this:
$$=2\int_3^9 f(x... | ncmathsadist | 4,154 | <p>Look at your algebra; you have extended the parenthesis too far. The factor of $2$ does not extend to the second integral.</p>
|
1,908,820 | <p>The question is: </p>
<p>If $$\int_3^9f(x) dx = 7$$
evaluate
$$\int_3^9 2f(x)+1 dx$$</p>
<p>I know that you can factor the 2 outside of the integral. But, then I am still left with a '+ 1' inside the integral that when I take the integral of becomes $x$. So then would I proceed to stating this:
$$=2\int_3^9 f(x... | Enrico M. | 266,764 | <p>$$\int_3^9 2\ f(x) + 1\ \text{d}x = 2\int_3^9 f(x)\ \text{d}x + \int_3^9 1\ \text{d}x$$</p>
<p>Since you already know what the $f(x)$ integral is, you just have to evaluate the other trivial one.</p>
<p>$$ = 2\cdot 7 + x\bigg|_3^9 = 14 + (9 - 3) = 20$$</p>
<p>The $2$ factor does multiply $f(x)$ only.</p>
<p>You ... |
3,499,352 | <p>Define a function <span class="math-container">$f:X\to Y$</span> to be a strict contraction, if there exists a <span class="math-container">$c\in [0,1[$</span> such that
<span class="math-container">$$\forall x,y \in X: d_Y\left( f(x),f(y)\right)\le cd_X(x,y)$$</span>
Now, consider a metric space <span class="math-c... | Fimpellizzeri | 173,410 | <p><span class="math-container">$f_c(x) = cx$</span> is a strict contraction whenever <span class="math-container">$|c|<1$</span>.
Let <span class="math-container">$c_n = 1-1/n$</span> so that <span class="math-container">$c_n\to 1$</span>.</p>
<p>With <span class="math-container">$f(x) = x$</span> being the limit ... |
64,881 | <p>I am having trouble with this problem from my latest homework.</p>
<p>Prove the arithmetic-geometric mean inequality. That is, for two positive real
numbers $x,y$, we have
$$ \sqrt{xy}≤ \frac{x+y}{2} .$$
Furthermore, equality occurs if and only if $x = y$.</p>
<p>Any and all help would be appreciated.</p>
| lhf | 589 | <p>$\phantom{Proof without words.........}$
<a href="https://i.stack.imgur.com/9OXeY.png" rel="noreferrer"><img src="https://i.stack.imgur.com/9OXeY.png" alt="enter image description here"></a></p>
|
2,864,585 | <p>I tried to calculate the Hessian matrix of linear least squares problem (L-2 norm), in particular:</p>
<p>$$f(x) = \|AX - B \|_2$$
where $f:{\rm I\!R}^{11\times 2}\rightarrow {\rm I\!R}$</p>
<p>Can someone help me?<br>
Thanks a lot.</p>
| mathreadler | 213,607 | <p>Yep squared norm is better. </p>
<p>$$\|AX-B\|_F^2 = (AX-B)^T(AX-B) = \Big/\text{ simplify }\Big/ = X^TA^TAX + \text{linear & const terms}$$</p>
<p>Now you should see what the Hessian is. If you still don't you can check out <a href="https://en.wikipedia.org/wiki/Hessian_matrix#Use_in_optimization" rel="nofoll... |
2,864,585 | <p>I tried to calculate the Hessian matrix of linear least squares problem (L-2 norm), in particular:</p>
<p>$$f(x) = \|AX - B \|_2$$
where $f:{\rm I\!R}^{11\times 2}\rightarrow {\rm I\!R}$</p>
<p>Can someone help me?<br>
Thanks a lot.</p>
| Rodrigo de Azevedo | 339,790 | <p>Let $f : \mathbb R^{m \times n} \to \mathbb R$ be defined by</p>
<p>$$f (\mathrm X) := \frac 12 \| \mathrm A \mathrm X - \mathrm B \|_{\text{F}}^2 = \frac 12 \| (\mathrm I_n \otimes \mathrm A) \, \mbox{vec} (\mathrm X) - \mbox{vec} (\mathrm B) \|_2^2$$</p>
<p>where $\mbox{vec}$ is the <a href="https://en.wikipedia... |
858,716 | <p>I'm self-studying real analysis using Abbott's text "Understanding Analysis." I'm trying to think out/prove as much on my own as I can, so I am working on proving the Nested Interval Property (Theorem 1.4.1 in the book) using "just" the Axiom of Completeness. The author does prove it in the book, but as I say, I lik... | Nannes | 153,221 | <p>$f(x,y)=x/(x-y)$</p>
<p>Just use the definition</p>
<p>$\bigtriangledown f(x,y)=(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y})$</p>
<p>$\frac{\partial f}{\partial x}=-y/(x-y)^2$</p>
<p>$\frac{\partial f}{\partial y}=x/(x-y)^2$</p>
<p>so</p>
<p>$\bigtriangledown f(x,y)=(-y/(x-y)^2,x/(x-y)^2)$</p>... |
252,767 | <p>I'm looking for a tangible example of a free abelian group whose quotient with a subgroup is not free abelian. There's a theorem that says that every abelian group is a quotient of some free group, but I'm looking for a more exact example.</p>
| Cameron Buie | 28,900 | <p>The most obvious example of a free group is $\Bbb Z$--which is the free group of one generator. (Can you see why?)</p>
<p>Now consider any finite cyclic group--necessarily abelian but not a free abelian group, and (isomorphic with) a quotient of $\Bbb Z$.</p>
|
4,302,855 | <p>I have tried setting up multiple systems of equations using many known volumes but I always seem to come up short. My last attempt was a hollow cylinder but that leaves you with three unknowns in only two sim. equations (for V and S.A). Can anyone help?</p>
| Second Person Shooter | 3,901 | <p>There should be many closed curves satisfying the given constraints.</p>
<p><a href="https://i.stack.imgur.com/PRaRn.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/PRaRn.png" alt="enter image description here" /></a>
If <span class="math-container">$A$</span>, <span class="math-container">$C$</sp... |
186,240 | <p>I need some notion about topology(I'm very interested in boundary points, open sets) and few examples of solved exercises about limits of functions($f:\mathbb{R}^{n}\rightarrow \mathbb{R}^m$) using $\epsilon, \delta$ and also some theory for continous functions.
Please give me some links or name of the books which ... | rschwieb | 29,335 | <p><a href="http://www.math.psu.edu/katok_a/TOPOLOGY/Chapter1.pdf" rel="nofollow">Here is a fast free internet resource</a> to get you started, while your text is in the mail.</p>
|
2,308,770 | <p>I have an equation of the form $A*i*j + B*i +C*j = N$
where I have the values of $A,B,C$ and $N$ and I want to solve for integer values of $i$ and $j$.</p>
<p>How would I approach this? I could try trial and error but the numbers I'm working with are relatively large (eg $>10^{40}$). But I'm also happy to work ... | CY Aries | 268,334 | <p>Multiply the whole equation by $A$.</p>
<p>\begin{align}
A^2ij+ABi+ACj&=AN\\
(Ai+C)(Aj+B)&=AN+BC
\end{align}</p>
<p>So $Ai+C$ and $Aj+B$ are factors of $AN+BC$.</p>
<p>For example, consider $2ij+3i+7j=336$. the equation can be written as</p>
<p>\begin{align}
2ij+3i+7j&=336\\
4ij+6i+14j&=672\\
(2i... |
253,359 | <p>I'm trying to prove by induction the following statement without success:<br>
$$\forall n \ge 2, \;\forall d \ge 2 : d \mid n(n+1)(n+2)...(n+d-1) $$</p>
<p>For the base case: $n = 2$, $d = 2$<br>
$2\mid 2(2+1)$ which is true.<br></p>
<p>Now, the confusion begins! I assume I would need to use the second induction p... | wisefool | 51,807 | <p>Fixed $d$, it is easy to show that the statement holds for every $n$, by induction. The inductive step is as follows:
suppose $d|n(n+1)\cdots(n+d-1)$, then put $m=gcd(d,n)$ and observe that $gcd(d,n+d)=m$ as well, therefore, by assumption
$$\frac{d}{m}\vert (n+1)\cdots(n+d-1)$$
so
$$d=\frac{d}{m}m\vert (n+1)\cdots(n... |
1,458,975 | <p>I'm having a issue with solving this problem. I know that the answer is $ a=3, b=1 $. But i'm not sure how to get to that conclusion.</p>
<p>Given that $(a+i)(2-bi)=7-i$, find the value of $a$ and of $b$, where $a,b \in \mathbb{Z}$.</p>
| John Hughes | 114,036 | <p>You're halfway on the right track. But you're not going to be able to prove it's rational. </p>
<p>Suppose that you DO have $\sqrt{3} = a + b \sqrt{2}$. Try squaring both sides and simplifying. Then do something like that again, until you've got no square roots. And then argue ( by looking at factors of 2 and 3 in... |
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