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 |
|---|---|---|---|---|
274,672 | <p>I need a function which initially falls slowly, then quickly and then slowly again. </p>
<p>Shape should be like tan but I want to be able to control the gradient</p>
<p>Properties needed:</p>
<p>$x = 0, y=0$</p>
<p>As $x$ increases $y$ decreases</p>
<p>As $x \rightarrow \infty$, $y \rightarrow -1$</p>
<p>(No ... | Maesumi | 29,038 | <p>You can build on sample formulas on the wiki page you mentioned. For example $y=-kx/\sqrt{1+(kx)^2}$ for different values of $k$. </p>
|
1,238,887 | <p>A special case of the Lagrange multiplier theorem may be stated as: Let $S, T \subset \mathbb{R}^{n}$ be open. Let $f: S \to \mathbb{R}$ be differentiable on $S$ and $g: T \to \mathbb{R}$ differentiable on $T$ such that $S \cap g^{-1}\{ 0 \}$ is not empty. If $f(x)$ is an extremum of $f$ for some $x \in S \cap g^{-1... | Yiorgos S. Smyrlis | 57,021 | <p>Here is a counterexample to your first conjecture, stating that for every unbounded increasing sequence $(a_n)$,
$$\underset{n\rightarrow\infty}{\lim} \frac{a_n}{\sum\limits_{k=1}^n a_k}=K>0
\Rightarrow \exists\ \delta>0: \underset{n\rightarrow\infty}{\lim} \frac{a_n}{e^{\delta n}}=1$$
Take $a_n=n2^n$. Then $\... |
1,238,887 | <p>A special case of the Lagrange multiplier theorem may be stated as: Let $S, T \subset \mathbb{R}^{n}$ be open. Let $f: S \to \mathbb{R}$ be differentiable on $S$ and $g: T \to \mathbb{R}$ differentiable on $T$ such that $S \cap g^{-1}\{ 0 \}$ is not empty. If $f(x)$ is an extremum of $f$ for some $x \in S \cap g^{-1... | Math-fun | 195,344 | <p>Let $a_k=e^{\beta k}$ with $0<\beta<\delta$ so that your assumption holds. Then $$\sum_1^n a_k=\frac{e^{(n+1)\beta}-e^{\beta}}{e^{\beta}-1}$$ now note that $$\lim_{n \to \infty} \frac{e^{\beta n}}{\frac{e^{(n+1)\beta}-e^{\beta}}{e^{\beta}-1}}=1-e^{-\beta}$$</p>
|
1,838,860 | <p>For a given field $k$ and a set $X$ we want to define the ring $k[X]$ of polynomials with $X$ as the set of variables. We do not assume $X$ to be finite. And we want to do this without employing axiom of choice.</p>
<p>Informally, the elements of $k[X]$ will be finite sums of monomials of the form $cx_1^{k_1}\dots ... | Asaf Karagila | 622 | <p>Here is an option which takes place between your two suggestions, without appealing to choice:</p>
<p><strong>Direct limit of finitely generated subrings.</strong></p>
<p>We define $k[X]$ to be the direct limit of the system $\{ k[Y]\mid Y\in[X]^{<\omega}\}$ with inclusions as the embeddings.</p>
<p>Now to che... |
1,838,860 | <p>For a given field $k$ and a set $X$ we want to define the ring $k[X]$ of polynomials with $X$ as the set of variables. We do not assume $X$ to be finite. And we want to do this without employing axiom of choice.</p>
<p>Informally, the elements of $k[X]$ will be finite sums of monomials of the form $cx_1^{k_1}\dots ... | Eric Wofsey | 86,856 | <p>Your "functions with finite support" approach is the standard approach to take to this, and is very likely what any author who considers the question trivial has in mind. The correct notion of "finite" to take when defining monomials is the usual one, as you have done (this is necessary for the polynomial ring you ... |
193,355 | <p>I am taking a calculus course, and in one of the exercises in the book, I am asked to find the limits for both sides of $\sqrt{x}$ where $x \to 0$.</p>
<p>Graph for sqrt(x) from <a href="http://www.wolframalpha.com/input/?i=sqrt%28x%29" rel="nofollow noreferrer">WolframAlpha</a>:<br>
<img src="https://i.stack.imgur... | ncmathsadist | 4,154 | <p>There is undeniably a right-hand limit. You have $\sqrt{x}\to 0$ as $x\downarrow 0$. In fact, since $\sqrt{0} = 0$, you know the square root function is right continuous at zero.</p>
<p>However, there is no possibility limit as $x\to 0-$, since the domain of this function is $[0,\infty)$. To discuss the left-hand... |
3,387,843 | <p>The thing I want to know is that why the term <span class="math-container">$``vector"$</span>.</p>
<p>I had learned that vector is something that has magnitude and direction, and also the the elements of <span class="math-container">$\mathbb{R}^n$</span> i.e. <span class="math-container">$n$</span> tuples can be vi... | Adam Latosiński | 653,715 | <p>The word <em>vector</em> originates from Latin, where it means "a carrier". It was first used in 18th century by astronomers, who were describing the motion of planets. For them, a vector was something that "carries" a point A to point B. It had a specific length and direction. So first vectors in mathematics/physic... |
2,612,538 | <p>I want to find the Eigenvalues and Eigenvectors of the following matrix $$A=\begin{pmatrix}0&1\\-3&0\end{pmatrix}.$$</p>
<p>I'm able to calculate the Eigenvalues, which are simply the zero points of A's characteristic polynom $\lambda^2 + 3$:<br>
$$\lambda_{1,2} = \pm i\sqrt{3}\\$$ </p>
<p>But how do I cal... | Rebellos | 335,894 | <p>Note that : </p>
<p>$$i\sqrt{3}c_1 + c_2 = 0 \Leftrightarrow c_2 = - i\sqrt{3}c_1$$</p>
<p>This means that you've already calculated that simply the coordinate $c_2$ of your eigenvector. This is an equation with infinite solutions, so just pick a simple value for $c_1$. Simply let $c_1 = 1$ and you will derive :</... |
475,267 | <p>Suppose $f(0) =0 $ and $0<f''(x)<\infty (\forall$ $x>0)$, then $\frac{f(x)}{x}$ strictly increases as $x$ increases. </p>
<p>I have shown that $f'(x)-\frac{f(x)}{x} = \frac{1}{2}xf''(c)$, for some $c\in (0,x)$. How do I proceed from here? </p>
| user21820 | 21,820 | <p>I know this is an old question, but here is an intuitive solution. $g=x \mapsto f(x)/x$ is the gradient of the line from the origin to $(x,f(x))$. If $g$ is not [strictly] increasing, we have a clockwise triangle $\triangle OPQ$ where $O,P,Q$ are on the curve with $P$ in the middle. But by mean value theorem on both... |
2,927,779 | <p>I am trying to understand the idea of a function having compact support. I was looking at this <a href="https://math.stackexchange.com/questions/1147407/definition-of-compact-support">post</a> among other places.</p>
<p>If <span class="math-container">$f$</span> has compact support on <span class="math-container">$... | MvG | 35,416 | <p>If I had to pick, I'd go with your reason 1, but both have truth to them.</p>
<p>Yes, area is <em>exactly</em> length times breadth. At least for a rectangle. And most other shapes can be imagined as being decomposed into a (possibly infinite) number of rectangles, and for each of them the same rule holds.</p>
<p>... |
11,267 | <p>John Von Neumann once said to Felix Smith, "Young man, in mathematics you don't understand things. You just get used to them." This was a response to Smith's fear about the method of characteristics. </p>
<p>Did he mean that with experience and practice, one obtains understanding? </p>
| Matt Brenneman | 19,698 | <p>I interpret the quote in a very different way. In general, we understand new ideas based on old ones. In math we can't always do this. I came into math from a applied math background, and when I started to learn math outside of the "plug and chug" engineering math I knew, I had difficulties associating the new conc... |
11,267 | <p>John Von Neumann once said to Felix Smith, "Young man, in mathematics you don't understand things. You just get used to them." This was a response to Smith's fear about the method of characteristics. </p>
<p>Did he mean that with experience and practice, one obtains understanding? </p>
| hyperpallium | 489,651 | <p>"Getting used to it" is memorization that doesn't feel like memorization, more like speaking a language fluently, or recognizing your grandmother, or knowing how your furniture is arranged.</p>
<p>Knowing axioms and some comsequences of those axioms is a similar memory, but it's not understanding.</p>
<p>When you ... |
389,532 | <p>I'm working out the following problem form Ahlfors' Complex Analysis text:</p>
<p>"Let $X$ and $Y$ be compact sets in a complete metric space $(S,d)$. Prove that there exist $x \in X,y \in Y$ such that $d(x,y)$ is a minimum."</p>
<p>My attempt:</p>
<p>Define $E:=\{d(x,y): x \in X,y \in Y \}$. We will prove that $... | Petra Axolotl | 291,842 | <p>You are right. The completeness of <span class="math-container">$S$</span> is unnecessary.</p>
<p>In your proof we only need <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> to be closed, so that we know hat <span class="math-container">$\bar{x} \in X$</span> and <span class="math-... |
1,578,940 | <p>I am asked to find $[T]_{\alpha}^{\alpha}$ for linear map $T: M_{2\times 2} \rightarrow M_{2\times 2}$ where $\alpha$ is the standard basis and </p>
<p>$T(x) =
\begin{bmatrix}
1 & 1 \\
0 & -1
\end{bmatrix}
x$</p>
<p>How can I approach this... I tried applying T to every vector in the standard ba... | Ben Grossmann | 81,360 | <p>We have
$$
T\pmatrix{a&b\\c&d} = \pmatrix{a+c&b+d\\-c&-d}
$$
Correspondingly, the matrix of $T$ with respect to the standard basis is given by
$$
[T]_\alpha^\alpha = \pmatrix{
1&0&1&0\\
0&1&0&1\\
0&0&-1&0\\
0&0&0&-1}
$$</p>
|
2,607,449 | <p>I'm trying to get intuition about why gradient is pointing to the direction of the steepest ascent. I got confused because I found that directional derivative is explained with help of gradient and gradient is explained with help of directional derivative.</p>
<p>Please explain what are the exact steps that lead fr... | Nick A. | 412,202 | <p>It is nothing more than the familiar chain rule! Let me elaborate.</p>
<p>We are in $\mathbb{R^n}$ and we want to take the diractional derivative to $u=(u_1,u_2,...,u_n)$ of the function $f:\mathbb{R^n}\rightarrow \mathbb{R}$ at some point $p_0=(p_1,...,p_n)$.</p>
<p>By definition $D_uf|_{p_{0}}=\lim_{t\rightarrow... |
17,885 | <p>In most education systems, Mathematics is a compulsory subject from primary school all the way to the start of university. A common reason given is that essential concepts like addition and multiplication are taught to the children. </p>
<p>But for many high school students, especially those who are keen on pursuin... | Joseph O'Rourke | 511 | <p>The OP may be interested in the controversial 2012 article by Andrew Hacker in the NYTimes:
<a href="https://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html" rel="noreferrer">Is Algebra Necessary?</a></p>
<p>Here is one reply, by Peter Flom:
<a href="https://medium.com/q-e-d/is-algebra-necessar... |
473,972 | <p>Let $X$ be an infinite set. Prove that there is a bijective function $f: X \rightarrow X$ with the property that for every $x \in X$ and all $n > 0$: $f^n(x) \neq x$.</p>
<p>I've tried to proved this by considering a bijective function $g: \mathbb{Z} \times X \rightarrow X$ in a certain way (by the composition ... | robjohn | 13,854 | <p>$$
\begin{align}
\int_0^1\int_x^1e^{x/y}\,\mathrm{d}y\,\mathrm{d}x
&=\int_0^1\int_x^1e^t\frac{x}{t^2}\,\mathrm{d}t\,\mathrm{d}x&&y=\frac xt\\
&=\int_0^1\int_0^te^t\frac{x}{t^2}\,\mathrm{d}x\,\mathrm{d}t&&\text{swap order}\\
&=\int_0^1e^t\frac{\frac12t^2}{t^2}\,\mathrm{d}t&&\text{i... |
64,022 | <p><strong>Edit</strong></p>
<p>(As Robert pointed out, what I was trying to prove is incorrect. So now I ask the right question here, to avoid duplicate question)</p>
<p>For infinite independent Bernoulli trials with probability $p$ to success, define a random variable N which equals to the number of successful tri... | Henry | 6,460 | <p>For $p>0$ it is not true. </p>
<p>One way to show this is that the median of a finite binomial random random variable with $n$ trials is $\lfloor np \rfloor$ or $\lceil np \rceil$, which increases without limit as $n$ increases, so your $\Pr \{ N < \infty \}$ must be no more than $0.5$. It is in fact $0$. ... |
3,798,402 | <p>For any natural number <span class="math-container">$n$</span>, how would one calculate the integral</p>
<p><span class="math-container">$$ \int_{0}^{2 \pi} |1 - ae^{i\theta}|^n \ d \theta $$</span></p>
<p>where <span class="math-container">$a$</span> is a complex number such that <span class="math-container">$|a| ... | Community | -1 | <p><strong>Hint:</strong></p>
<p>If <span class="math-container">$|a|=1$</span>, <span class="math-container">$a=e^{i\phi}$</span> and this phase factor can just be dropped (you are integrating over a whole period). Hence WLOG <span class="math-container">$a=1$</span>.</p>
<p>Then for even <span class="math-container">... |
869,114 | <p>I know, stupid question. But humor me for a sec. First off, we know that all real numbers have two numbers which are infinitely close to them, right?
That would seem to be, for any given value of x,</p>
<blockquote>
<p>x ± y</p>
</blockquote>
<p>where </p>
<blockquote>
<p>y = .000...1</p>
</blockquote>
<p>B... | Community | -1 | <p>"Infinitely many zeroes followed by a 1" is actually not a well-defined decimal numeral.</p>
<p>The places in a decimal numeral are all indexed by integers that denote their location relative to the unit place: e.g. the hundred's place has index $2$ and the thousandth's place has index $-3$.</p>
<p>If you have inf... |
2,932,510 | <p><a href="https://i.stack.imgur.com/9kTCs.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/9kTCs.png" alt="enter image description here"></a>
Apostol tells the reader to verify the last sentence.
I've verified the backward implication (<span class="math-container">$a_kx+b_k=0\implies \text{equality}... | Displayname | 597,513 | <p>After you transform the function you obtain:
<span class="math-container">$y=-2(2x+6)^4 -4$</span>, in your answer you are raising <span class="math-container">$2(x+6)$</span> to the power 4, i.e <span class="math-container">$(2x+12)^4.$</span> The rest is all correct though. </p>
|
727,419 | <p>I was trying to answer this question <a href="https://math.stackexchange.com/questions/727378/not-sure-which-test-to-use">Not sure which test to use?</a></p>
<p>Here was my clearly wrong response:</p>
<p>$\ln(1 + \frac{1}{3^k})$ = $\frac{\ln((1+\frac{1}{3k})^k)}{k}$</p>
<p>$\sum_{k=1}^\infty 2^{k-1}\ln((1+3^{-k})... | Martín-Blas Pérez Pinilla | 98,199 | <p><strong>Don't</strong> use the <em>same</em> variable $k$ in <em>both</em> sums!</p>
|
727,419 | <p>I was trying to answer this question <a href="https://math.stackexchange.com/questions/727378/not-sure-which-test-to-use">Not sure which test to use?</a></p>
<p>Here was my clearly wrong response:</p>
<p>$\ln(1 + \frac{1}{3^k})$ = $\frac{\ln((1+\frac{1}{3k})^k)}{k}$</p>
<p>$\sum_{k=1}^\infty 2^{k-1}\ln((1+3^{-k})... | JiK | 98,449 | <p>$$
(a+b)^k = \sum_{i=0}^k \binom{k}{i} a^{k-i} b^{i}
$$</p>
<p>Using this for $(1+3^{-k})^k$, we get (here $a=1$ and $b=3^{-k}$)
$$
(1+3^{-k})^k = \sum_{i=0}^k \binom{k}{i} 3^{-ki}.
$$</p>
<p>What you wrote seems completely different: you use $k$ as the summation index in the new sum, so nobody knows which $k$ is ... |
123,675 | <p>What are some "applications to" / "connections with" topology that one could hope to reasonably cover in a first course on topos theory (for master students)? I have an idea of what parts of the theory I would like to cover, however, I would love some more nice examples and applications. Of course, I have some ideas... | David Roberts | 4,177 | <p>Galois groups of atomic toposes, and relations to fundamental pro-groups of badly-behaved spaces. </p>
|
187,795 | <p>I have run into a problematic behavior, and then came to find that it is a "possible issue" called out in the documentation, but I guess I am just baffled that this IS an issue, and am wondering why. The example from the documentation is:</p>
<p>In the presence of global variables, pattern variables may show unexpe... | Shredderroy | 9,257 | <p>There are many people at this forum who are more knowledgeable about Mathematica than me, but I will attempt an answer anyway. (Believe me, I have done worse things in life.)</p>
<blockquote>
<p>Why is this the behavior? Is there any way to fix the problem without resorting to a delayed assignment?</p>
</blockquo... |
187,795 | <p>I have run into a problematic behavior, and then came to find that it is a "possible issue" called out in the documentation, but I guess I am just baffled that this IS an issue, and am wondering why. The example from the documentation is:</p>
<p>In the presence of global variables, pattern variables may show unexpe... | Anton.Sakovich | 59,438 | <p>Despite some thoughts here, you <em>can</em> localize a pattern variable without switching to <code>SetDelayed</code>:</p>
<pre><code>ClearAll[f, x];
x = 1;
Module[
{y = 0},
f[x_] = x^(1 + 1) + y
];
??f
(*f[x<span class="math-container">$_]=x$</span>^2*)
</code></pre>
<p>You can read more about scoping in <em>Ma... |
2,008,437 | <p>Given four points in the plane, there exists a one-dimensional family of conics through these, often called a pencil of conics. The locus of the centers of symmetry for all of these conics is again a conic. What's the most elegant way of computing it?</p>
<p>I know I could choose five arbitrary elements from the pe... | Futurologist | 357,211 | <p>This is a rough sketch, there might be some imprecision. </p>
<p>What about something like this. Take, let's say, the three by three matrix $A$ representing one degenerate conic from the pencil and let $B$ be another three by three matrix representing another degenerate conic from the pencil (you can form them from... |
3,024,456 | <p>I am working on a problem and I am confused if exponents can be split up in the manner below. </p>
<p><span class="math-container">$$x^{k+2} = x^{k+1}+x^{k+1} = 2x^{k+1} $$</span></p>
<p>I apologize if this is a simple question I tried to look up exponent rules but I couldn't find exponents + a number. </p>
| KM101 | 596,598 | <p>No, they’re not broken up like that. You need to use</p>
<p><span class="math-container">$$a^b\cdot a^c = a^{b+c}$$</span></p>
<p>and vice-versa. In general, you can try to create a simple expression to see if your way is correct. For example, notice</p>
<p><span class="math-container">$$2^{2+2} = 16 \color{red}{... |
90,965 | <p>Im wondering if there's an existing literature on this binary operation involving graphs wherein you identity $n$ vertices from one graph with $n$ vertices from the other such that the resulting structure is still a graph (no loops and multiple edges). For instance, given two paths $[a,b,c]$ and $[d,e,f]$, letting $... | Juan Alvarado | 104,079 | <p>In the book of Lov\'asz </p>
<p><a href="http://www.cs.elte.hu/~lovasz/bookxx/hombook-almost.final.pdf" rel="nofollow noreferrer">http://www.cs.elte.hu/~lovasz/bookxx/hombook-almost.final.pdf</a> </p>
<p>exposes a graph algebra in the sense that you mentioned. </p>
|
2,559,639 | <p>In the book of <em>General Topology</em> by Munkres, on page 104, it is given that</p>
<p><a href="https://i.stack.imgur.com/Pv9MU.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Pv9MU.png" alt="enter image description here"></a></p>
<p>However, as far as I know, the lower limit topology <span c... | Eric Wofsey | 86,856 | <p>Not every open set in the lower limit topology has the form $[a,b)$. Such sets just form a basis for the topology, so every open set is a union of sets of this form. You can write an interval $(a,b)$ as the union of the sets $[c,b)$ for all $c>a$, so $(a,b)$ is open in the lower limit topology.</p>
|
4,370,239 | <p>Verify that
<span class="math-container">\begin{equation}
a.\;x^3+y^3-3xy=0,\; \mathbb{R}_{x\neq2^{2/3}}
\end{equation}</span>
is the solution to
<span class="math-container">\begin{equation}
b.\;(y^2-x)y' - y+x^2=0
\end{equation}</span>
As we know the function g(x) of a. is not easily attainable, therefore we resor... | Jose Arnaldo Bebita Dris | 28,816 | <p><strong>Hint:</strong>
<span class="math-container">$$z = 2xy + x(1 - y)$$</span>
is <strong>not quadratic</strong>, but only <strong>linear</strong> in <span class="math-container">$x$</span>. (Why?)</p>
<p>Can you finish?</p>
|
4,370,239 | <p>Verify that
<span class="math-container">\begin{equation}
a.\;x^3+y^3-3xy=0,\; \mathbb{R}_{x\neq2^{2/3}}
\end{equation}</span>
is the solution to
<span class="math-container">\begin{equation}
b.\;(y^2-x)y' - y+x^2=0
\end{equation}</span>
As we know the function g(x) of a. is not easily attainable, therefore we resor... | BlueElement | 1,020,369 | <p>Thank you to Blue/Arnie/dxiv !</p>
<p>That was just a massive mess up on my end. I know it does not work that way. I was doing it on paper and must have just written it poorly and then let the error flow through.</p>
<p><span class="math-container">$z=(y∗nx)+((1−y)∗x)$</span> <br>
<span class="math-container">$z=ynx... |
1,057,429 | <p>In mathematics statistics. I'm a bit confused by the terminology <em>normal approximation</em>. What is it? Is it just something you say when you approximate, for example the normal distribution? </p>
| Ben Grossmann | 81,360 | <blockquote>
<p>I have defined C as the standard n-dimensional identity matrix.</p>
</blockquote>
<p>Don't do that; choose the real unitary (orthogonal) $C$ that diagonalizes $A$. That is, let $C$ be a matrix whose columns are an orthonormal eigenbasis of $A$, so that $C^T = C^{-1}$ and $D = C^TAC$. Note that $A =... |
3,048,340 | <blockquote>
<p>Suppose <span class="math-container">$f$</span> is a convex function and differentiable everywhere on <span class="math-container">$(0, \infty)$</span> and satisfies
<span class="math-container">$$\lim_{x \rightarrow\infty}{f(x)}=A$$</span>
where <span class="math-container">$A$</span> is an arbit... | Mostafa Ayaz | 518,023 | <p>Hint: once define <span class="math-container">$x=x_0+1$</span> therefore<span class="math-container">$$f(x_0+1)-f(x_0)\ge f'(x_0)$$</span>and tend <span class="math-container">$x_0$</span> to <span class="math-container">$\infty$</span>. Another time let <span class="math-container">$x=x_0-1$</span> therefore <span... |
2,976,057 | <p>I’m doing partial fractions and need to factorize the denominator. They are quadratic. However there are some that aren’t so easy to factorize and my first choice was to use the quadratic equation to find the roots however comparing my answer with the correct one the signs are different. Is the quadratic formula onl... | Steven Alexis Gregory | 75,410 | <p>Factoring in <span class="math-container">$\mathbb Q[x]$</span> is unique up to associates. The units in <span class="math-container">$\mathbb Q[x]$</span> are all nonzero rational numbers. So <span class="math-container">$(x + \frac 12)$</span>, <span class="math-container">$(2x+1), -(2x+1), (4x+2)$</span> and so o... |
279,471 | <p>Is it true that any finite dimensional division algebra over a pseudo-algebraically closed field is trivial? We know that this is true for algebraically closed field.</p>
| Dirk | 109,932 | <p>The fact that every finite dimensional division algebra is trivial is not only true over an algebraically closed field but it is in fact equivalent to the field being algebraically closed. Remember that (extension-)fields are just a special case of division algebras. Thus, if it would be true over pseudo-algebraical... |
2,045,226 | <p>We denote with $\phi$ Euler's Phi Function.</p>
<p><strong>We want to find all $n\in \mathbb{Z^+}: \ \phi (3n)=\phi (4n)=\phi (6n).$</strong></p>
<p><strong>Answer:</strong>
Let $n=2^k3^lm$ : $k,l\in \mathbb{N}, \ m\in \mathbb{Z^+},\ \text{gcd}(m,2)=\text{gcd}(m,3)=1$. Then we have:</p>
<ol>
<li>$\phi (3n)=\phi (... | Erick Wong | 30,402 | <p>The shortest approach I can see uses the following simple principle: $\frac{\phi(n)}n$ is determined by just the set of primes dividing $n$ (and not their exponents). This allows us to make deductions without any explicit calculations regarding exponents.</p>
<p>Suppose that $n$ satisfies the required condition. ... |
593,212 | <p><img src="https://i.stack.imgur.com/4P4gT.jpg" alt=""></p>
<p>This is a part of the group of practice problem I've been working on and I'm just lost. I'm really struggling when it comes to these ring problems. Anybody who could lay out an outline for this problem so I can study the structure to help with other prob... | Dr_Sam | 83,340 | <p>The usual way to treat non-linear ODE is to use iterations to resolve the non-linearity. </p>
<p>Starting from the change of variable $u = y'$, you have indeed</p>
<p>$$u'' + y^2u' -u = 0$$</p>
<p>with the conditions $u(0)=0$ and $u'(1)=1$. To start the iterative method, take a (reasonable) initial value for $y$,... |
3,218,794 | <p>I have <span class="math-container">$f(x)=\cos^{2n}(x)+\sin^{2n}(x),\; n\in \mathbb{N},\; n\geq 2,\;x\in \mathbb{R}$</span>
I need to find the range of the function.
I took <span class="math-container">$n=2$</span> and I got <span class="math-container">$f(x)=1-\frac{1}{2}\sin^{2}(2x)$</span> and the range of this i... | DINEDINE | 506,164 | <p>Hint:</p>
<p>You have to find the range of the function <span class="math-container">$f(t)=t^n+(1-t)^n,\quad t\in[0,1]$</span></p>
|
2,328 | <p>There is an interesting discussion taking place at MO over the suitability of this question: <a href="https://mathoverflow.net/questions/210089/complex-structure-on-s6-gets-published-in-journ-math-phys">Complex structure on $S^6$ gets published in Journ. Math. Phys</a>. The question is about the correctness of a pap... | Boris Bukh | 806 | <p>A scientific paper is like a politician --- much of its reputation is based on hearsay, and superficial impression, not on cold analytic examination. Very few have willingness and time to investigate thoroughly. Thus, a baseless accusation can easily ruin a reputation. So, it is important to make accusations as prec... |
2,328 | <p>There is an interesting discussion taking place at MO over the suitability of this question: <a href="https://mathoverflow.net/questions/210089/complex-structure-on-s6-gets-published-in-journ-math-phys">Complex structure on $S^6$ gets published in Journ. Math. Phys</a>. The question is about the correctness of a pap... | Community | -1 | <p>The distinction between "published" and "preprint" is fuzzy and not all that relevant. I propose to adopt the <a href="https://meta.mathoverflow.net/a/942/">same policy</a> for both that is: </p>
<blockquote>
<p>[W]e should avoid discussing <strike>preprints</strike> papers in general terms. If there is a spec... |
2,328 | <p>There is an interesting discussion taking place at MO over the suitability of this question: <a href="https://mathoverflow.net/questions/210089/complex-structure-on-s6-gets-published-in-journ-math-phys">Complex structure on $S^6$ gets published in Journ. Math. Phys</a>. The question is about the correctness of a pap... | Timothy Chow | 3,106 | <p>This meta question is phrased as a general question about the appropriateness of discussing published papers on MO. That's a fair question, but I think that it's actually quite a different question from the question of whether Misha Verbitsky's question was appropriate.</p>
<p>The paper by Gabor Etesi has a histor... |
1,820,853 | <p>How to derive the Gregory series for the inverse tangent function?
Why is Gregory series applicable only to the set $ [-\pi/4,\pi/4] $ ?</p>
| Community | -1 | <p>A fast way is by exploiting $\arctan'(x)=\dfrac1{1+x^2}$.</p>
<p>The Taylor series of that derivative is easily established to be the sum of a geometric series of common ratio $-x^2$,</p>
<p>$$\sum_{k=0}^\infty(-x^2)^k=\frac1{1-(-x^2)},$$ which only converges for $x^2<1$.</p>
<p>Then integrating term-wise,</p>... |
1,207,134 | <p>Given the image:
<img src="https://i.stack.imgur.com/EJ3ax.jpg" alt="enter image description here"></p>
<p>and that $x_0 = 1, y_0=0$ and $\text{angles} \space θ_i
, i = 1, 2, 3, · · ·$ can be arbitrarily picked.</p>
<p>How can I derive a recurrence relationship for $x_{n+1}$ and $x_n$?</p>
<p>I actually know what... | sayantankhan | 47,812 | <p>Here's an alternative formulation, and this also serves to illustrate that sometimes a different notation makes things appear simpler. First, define
$$\delta x_n=x_{n+1}-x_n$$
$$\delta y_n=y_{n+1}-y_n$$
Now. from the figure, we have
$$\tan^2 {\theta_{n+1}}=\frac{(x_{n+1}-x_n)^2+(y_{n+1}-y_n)^2}{x_n^2+y_n^2}$$
$$\ta... |
1,472,901 | <h2>Code first :</h2>
<pre><code>sum = 0
for(i=1; i<=n; i++){
for(j=1; j<=i; j++){
if(j%i == 0){
for(k=1; k<=n; k++){
sum = sum + k;
}
}
}
}
</code></pre>
<hr>
<p>Total no. of iterations in $j = 1+2+3+4+ \dot... | Rory Daulton | 161,807 | <p><strong>Critical points</strong> refer to the first derivative. In particular, $x=a$ is a critical point of $f(x)$ if either $f'(a)=0$ or $f'(a)$ is not defined. The importance here is that all maxima or minima are found at critical points or endpoints of a domain. So a common way to find extrema (maxima and minima)... |
1,408,231 | <p>Definition </p>
<p>A function $f$ is <strong>convex</strong> on an interval if for $a,x, \text{and} \;b$ in the interval with $a\lt x\lt b$, we have</p>
<p>$$\frac{f(x)-f(a)}{x-a}\lt \frac{f(b)-f(a)}{b-a}.$$</p>
<p>While reading the proof that if $f$ is convex on some open interval containing $a$, then $f_+'(a)$ ... | Daniel Fischer | 83,702 | <p>Note: I don't assume <em>strict</em> convexity, so I use weak inequalities. For strictly convex $f$, all the inequalities between expressions of the form $\frac{f(y) - f(x)}{y-x}$ are strict.</p>
<p>We have</p>
<p>\begin{align}
\frac{f(b) - f(a)}{b-a} &= \frac{\bigl(f(b) - f(x)\bigr) + \bigl(f(x) - f(a)\bigr)}... |
625,395 | <blockquote>
<p>Let $f:\mathbb{R}^3 \to \mathbb{R}$ be a smooth function rapidly
decreasing to zero as $|(x,y,z)| \to \infty$, and let $D(t)$ denote
$$D(t)=\left\{(x,y,z)\in \mathbb{R}^3 \mid ax+by+cz \leq t\right\},$$
where $a,b,c\in \mathbb{R}$. Compute $$\frac{d}{dt}
\int_{D(t)}f(x,y,z)\,dx\,dy\,dz.$$</p>
<... | Steven Gubkin | 34,287 | <p>I would like to spell out my comment a little more. I do not give a complete solution, but hopefully it will give you some ideas, and maybe lead you to the correct solution. I work in the two dimensional case to make things a little simpler.</p>
<p>Let $I(t) =\int_{D(t)} f(x,y) dxdy$. </p>
<p>By the definition... |
95,343 | <p>I am writing a code for a Lagrange multipliers problem with three variables.
My code looks like this:</p>
<pre><code> f[x_, y_, z_] := (m1 x + m2 y + m3 z)/Sqrt[m1^2 x + m2^2 y + m3^2 z]
g[x_, y_, z_] := x + y + z - 1
Minimize[{f[x, y, z], g[x, y, z] == 0,
m1 > 0, m2 > 0, m3 > 0, x > 0, y... | Verbeia | 8 | <p>I would first note that the problem you are posing can be reduced to a two-variable one by substituting in your inequality constraint, to eliminate <code>z</code>. I note that your constraint <code>g</code> implies <code>x + y < 1</code>.</p>
<pre><code>Minimize[{f[x, y, 1 - x - y], x + y < 1, m1 > 0, m2 &... |
2,579,074 | <p>When I try to factor the quadratic form, I end up with </p>
<p>$$6x^2+4y^2+2z^2+4xz-4yz = 2((x+z)^2+2x^2+(y-z)^2-z^2)$$ </p>
<p>which does not ensure that $f(x,y,z) \geq 0$ for all $x, y, z \geq 0$ since the $z$ term is negative. How should these kinds of problems be tackled?</p>
| Michael Rozenberg | 190,319 | <p>We need to prove that
$$z^2+2(x-y)z+3x^2+2y^2\geq0$$ for which it's enough to prove that
$$(x-y)^2-(3x^2+2y^2)\leq0$$ or
$$2x^2+2xy+y^2\geq0$$ or
$$x^2+(x+y)^2\geq0,$$ which is obvious.</p>
<p>Done!</p>
|
2,579,074 | <p>When I try to factor the quadratic form, I end up with </p>
<p>$$6x^2+4y^2+2z^2+4xz-4yz = 2((x+z)^2+2x^2+(y-z)^2-z^2)$$ </p>
<p>which does not ensure that $f(x,y,z) \geq 0$ for all $x, y, z \geq 0$ since the $z$ term is negative. How should these kinds of problems be tackled?</p>
| A. Thomas Yerger | 112,357 | <p>I would like to add another solution that works in this setting, and also somewhat more generally, even though it's overkill for your specific question.</p>
<p>This thing is smooth, as it is just a polynomial, so we can look for extrema, and then test if they are maxima or minima. These happen when all partials van... |
2,779,945 | <p>Picture the setup:</p>
<p>A particle of mass $m$ attached with a string of length $R$ spins in a vertical circle around a fixed peg with constant velocity $v>\sqrt{Rg}$ in the $x$-$z$ plane, acting under its weight $mg$. In this model, the probability that the string snaps when the string is at an angle $\theta\... | Phil H | 554,494 | <p>The expected value is a summation of all the positions it could break times the probability of it breaking at that particular position. The definite integral does this summation and the result is an average angle. It should be biased towards the angle of maximum tension but it is an entirely different metric to the ... |
849,510 | <p>Let $f:\mathbb{R}\to\mathbb{R}$ be a $T$-periodic function, that is $f(t+T)=f(t)$ for all $t\in \mathbb{R}$. Assume that
$$\int_0^{+\infty}|f(s)|ds<+\infty.$$
Now if we assume in addition that $f$ is continuous, my intuition tells me that we must have necessarily $f=0$, is this correct ? </p>
| Cameron Williams | 22,551 | <p>This is correct. The way you can see this is by considering the maximum of $|f|$, call it $L$. For any $x$ such that $|f(x)|=L$, we have that $|f(y)| > \frac{L}{2}$ for all $|x-y| < \delta$ (for a sufficient choice of $\delta$). Can you see how to argue it from here?</p>
|
322,598 | <p><a href="https://en.wikipedia.org/wiki/Partially_ordered_set" rel="noreferrer">Partially ordered sets</a> (<em>posets</em>) are important objects in combinatorics (with <a href="https://gilkalai.wordpress.com/2019/02/05/extremal-combinatorics-v-posets/" rel="noreferrer">basic connections to extremal combinatorics</a... | Sam Hopkins | 25,028 | <p>Here's another classic from algebraic combinatorics.</p>
<p>In his PhD thesis <a href="http://www-math.mit.edu/~rstan/pubs/pubfiles/9.pdf" rel="noreferrer">"Ordered Structures and Partitions"</a>, Stanley introduces the <span class="math-container">$(P,\omega)$</span>-partition generating function of a labeled pose... |
322,598 | <p><a href="https://en.wikipedia.org/wiki/Partially_ordered_set" rel="noreferrer">Partially ordered sets</a> (<em>posets</em>) are important objects in combinatorics (with <a href="https://gilkalai.wordpress.com/2019/02/05/extremal-combinatorics-v-posets/" rel="noreferrer">basic connections to extremal combinatorics</a... | Dominic van der Zypen | 8,628 | <p>The <strong>fish-scale conjecture</strong>: In every poset not containing an infinite antichain there exist a chain <span class="math-container">$C$</span> and a decomposition of the vertex set into antichains <span class="math-container">$A_i$</span>, such that <span class="math-container">$C\cap... |
244,875 | <p>\begin{align}
\min_{x} c^Tx \\
s.t.~Ax=b
\end{align}
Note that here $x$ is unrestricted. I need to prove that the dual of this program is given by
\begin{align}
\max_{\lambda} \lambda^Tb \\
s.t.~\lambda^TA\leq c^T
\end{align}</p>
<p>But in the constraint, I always get an equality (using what I learnt)
\begin{align}... | Oily | 303,643 | <p>\begin{align}
\min_{x} c^Tx \\
s.t.~Ax=b \\
Unrestricted
\end{align}</p>
<p>Take $x=x_1-x_2$
\begin{align}
\min c^T(x_1-x_2) \\
s.t.~A(x_1-x_2)=b \\
~ x_1, x_2 \ge 0
\end{align}</p>
<p>This is can be written as
\begin{align}
\min {\begin {pmatrix}c \\ -c \\ \end {pmatrix} }^T \begin {pmatrix} x_1 \\ x_2 \\ \end... |
2,944 | <p>Let $f$ be a diffeomorphism, say from $\mathbb R^n$ to $\mathbb R^n$ , such as the transition map between two coordinate charts on a differentiable manifold.</p>
<p>A differential $n$-form (or rather its coefficient function which is obtained by using the canonical one-chart atlas on $\mathbb R^n$) then transforms ... | kennytm | 171 | <p>You can find a table of these <a href="http://en.wikipedia.org/wiki/Contraposition#Comparisons" rel="nofollow">on Wikipedia</a>:</p>
<p>$$
\begin{array}{c|l} \hline
p \to q & \text{Implication (Conditional)} \\\\
\neg p \to \neg q & \text{Inverse} \\\\
q \to p & \text{Converse} \\\\
\neg q \to \... |
2,509,583 | <p>How is intermediate value theorem valid for $\sin x$ in $[0,\pi]$?</p>
<p>It has max value $1$ in the interval $[0,\pi]$ which doesn't lie between values given by $\sin0$ and $\sin\pi$.</p>
| Martin Argerami | 22,857 | <p>That's not how it works. The Intermediate Value Theorem says that if $f$ is continuous on $[a,b]$, then it achieves every value between $$c=\min\{f(x):\ x\in [a,b]\},$$ and $$d=\max\{f(x):\ x\in [a,b]\}.$$
When $f$ is monotone, it happens that $c,d$ are $f(a),f(b)$, but in general it is not the case. </p>
|
59,105 | <p>Let $n$ be a positive integer. $\; $ Let $f : \mathbb{R}^n \to \mathbb{R}^n$ be everywhere <a href="http://en.wikipedia.org/wiki/Fr%C3%A9chet_derivative">Frechet differentiable</a>.
<br>
Let $S$ be a subset of $\mathbb{R}^n$ with <a href="http://en.wikipedia.org/wiki/Lebesgue_measure">Lebesgue measure</a> zero.</... | Robert Israel | 8,508 | <p>More generally, if $f:\ X \to Y$ is a locally Lipschitz function from one $\sigma$-compact metric space to another and $S \subseteq X$ has $d$-dimensional Hausdorff measure 0, then $f(S)$ also has $d$-dimensional Hausdorff measure 0. This is pretty much immediate from the definition of Hausdorff measures. In your... |
3,603,926 | <p>I came across this elementary counting problem: A coffee shop has 4 different types of coffee. You can order your coffee in a small, medium, or large cup. You can also choose whether you want to add cream, sugar, or milk (any combination is possible, for example, you can choose to add all three). In how many ways ca... | amWhy | 9,003 | <p>Indeed, you're correct! There are <span class="math-container">$4\cdot 3\cdot 2^3$</span> ways to order your coffee. Four options for coffee, three options for size. And the <span class="math-container">$2^3$</span> factor reflects your knowledge that there are two options "yay" or "nay" for each of cream, sugar, ... |
11,299 | <p>How to write this small piece in a functional way (ie. without state variables)?:</p>
<pre><code>test[oldJ_List, newJ_List] := Total[Abs[oldJ - newJ]] > 1;
relax[j_List, x_?NumericQ] := Mean[Nearest[j, x, 4]];
j = Range[100]; (* any numeric list *)
j1 = j/2; (*some initial value for the While[] test to return T... | rm -rf | 5 | <p>Let me first redefine your <code>relax</code> to return a list as:</p>
<pre><code>Clear@relax1
relax1[j_List, i_Integer] := MapAt[Mean[Nearest[j, #, 4]] &, j, i]
</code></pre>
<p>Then, the algorithm can be written in a functional way without state variables using <code>Fold</code> and <code>NestWhile</code> as... |
2,957,440 | <p>Knowing <span class="math-container">$f(1) = 2, f(4)=7,f'(1) = 5,f'(4)=3$</span></p>
<p>Find
<span class="math-container">$\int_1^4xf''(x)dx$</span></p>
<p>It is obvious that if there was no <span class="math-container">$x$</span> the answer would be <span class="math-container">$f'(4) - f'(1)$</span> from the fun... | Parcly Taxel | 357,390 | <p>Use integration by parts.
<span class="math-container">$$\int_1^4xf''(x)\,dx=[xf'(x)]_1^4-\int_1^4f'(x)\,dx$$</span>
<span class="math-container">$$=4f'(4)-f'(1)-f(4)+f(1)=4×3-5-7+2=2$$</span></p>
|
1,148,624 | <p>This is for teaching math. I'm wondering if someone knows some striking near-equalities between simple arithmetic expressions. I vaguely remember that such things exist (e.g., numbers that look alike out 10 digits, but that are really different), but I don't remember where I saw them or what they are. I think there ... | d8d0d65b3f7cf42 | 214,217 | <p>additional reference: Section 1.4 (High Precision Fraud) in Borwein, Bailey, Girgensohn: Experimentation in Mathematics, A.K.Peters 2004.</p>
|
1,378,769 | <p>The equation is given by</p>
<p>$$ \sum_{n=1}^N \min(\gamma, \beta a_n)=N$$
where $\beta$ is the variable with $\beta\in[0,\sqrt\gamma/\min(a_n\mid a_n>0)]$, $ \gamma $ is a constant with $1\le\gamma\le N$, and $a_n$ is an nonnegative constant.</p>
<p>I know that the function $f(\beta)=\sum_{n=1}^N \min(\gamm... | Graham Kemp | 135,106 | <blockquote>
<p>Therefore relations makes use of the cartesian product operator, which also makes use of sequences.</p>
</blockquote>
<p>What. Ah... Yes, that is a just <em>wee</em> bit recursive, isn't it now?</p>
<p>The Cartesian product of sets $\rm A, B$ is defined as: $\mathrm A{\times}\mathrm B = \{\la... |
1,221,368 | <p>An inheritance is divided between three people in the ratio $4:7:2$. If the least amount received is $\$\ 2300$ calculate how much the other two people received?</p>
| John Joy | 140,156 | <p>HINT</p>
<p>$$4:7:2 = x:y:2300=x:y:(2\times1150)=\dots$$</p>
<p>so now you multiply 1150 with 4 and 7.</p>
|
359,816 | <p>Let $(f_n)$ and $(g_n)$ be sequences of nonnegative function in $L^1(\mathbb R)$, for which
$$
f_n \to 0, \\
g_n \to 0,
$$
almost everywhere.
Show
$$
\int_A \frac{2f_n g_n}{1+f_n^2+g_n^2} \to 0,
$$
when $A \subset \mathbb R$ is a set of finite measure.</p>
<p>Define
$$
h_n = \frac{2f_n g_n}{1+f_n^2+g_n^2}.
$$
I kno... | Community | -1 | <p>The only suggestion you might still need is: $2f_ng_n \le f_n^2 + g_n^2$. This is what gives the bound ($\frac{a}{1+a} \le 1$).</p>
|
121,760 | <p>I want to find the minimal polynomial (over $\mathbb{Q}$) of: $k:=\sqrt[3]{7-\sqrt{2}}$.</p>
<p>With simple 'tricks' I got that: $P=(x^3-7)^2+2$ is a polynomial such that $P(k)=0$.</p>
<p>But I don't know if, or how to prove that $P$ is the minimal polynomial. How can I prove this/find the minimal polynomial ? </p... | Bill Dubuque | 242 | <p>Any irreducible polynomial having $\rm\:k\:$ as a root necessarily has minimal degree. This proof is easy: the set $\rm\:S\:$ of polynomials $\rm\:f \in \mathbb Q[x]\:$ such that $\rm\:f(k) = 0\:$ is closed under addition, and under multiplication by any $\rm\:g\in \mathbb Q[x],\:$ hence it is closed under gcd, sin... |
2,850,435 | <p>The equation is
$$
\int f(x) = \frac{1}{f'(x)}
$$
I saw it on a shirt so I thought it is probably a joke, but I couldn't get it. So I tried to solve it. On the first look I thought this is never true. Then I thought of differentiating the whole thing.</p>
<p>$$
f(x) = - \frac{f''(x)}{f'(x)^2}
$$ </p>
<p>This is wh... | Paul Frost | 349,785 | <p>I don't think that $Map_0(X \wedge Y, Z) \cong Map_0(X, Map_0(Y,Z))$ is a direct corollary of $Map(X\times Y, Z) \cong Map(X, Map(Y,Z))$. You can easily verify that there is a canonical bijection, but to show that it is a homeomorphism needs extra arguments.</p>
<p>See for example Section 2.4 of</p>
<p>tom Dieck, ... |
3,973,025 | <p>Does the structure of the category of models of a given first order theory where the arrows are embeddings tell us anything interesting about the theory itself?</p>
<p>Two models <span class="math-container">$M$</span> and <span class="math-container">$M'$</span> are isomorphic if and only if there exists an embeddi... | Kyle Gannon | 152,976 | <p>There is a very interesting result by Campion, Cousins, and Ye in the context of elementary embeddings. In particular, the Lascar group of a first order theory <span class="math-container">$T$</span> is isomorphic to the fundamental group of the classifying space of <span class="math-container">$Mod(T)$</span> (wher... |
4,138,346 | <p>I would like someone to verify this exercise for me. Please.</p>
<p>Find the following limit:</p>
<p><span class="math-container">$\lim\limits_{n \to \infty}\left(\dfrac{1}{n+1}+\dfrac{1}{n+2}+...+\dfrac{1}{3n}\right)$</span></p>
<p><span class="math-container">$=\lim\limits_{n \to \infty}\left(\dfrac{1}{n+1}+\dfrac... | FDP | 186,817 | <p><span class="math-container">\begin{align}S_n&=\dfrac{1}{n+1}+\dfrac{1}{n+2}+...+\dfrac{1}{3n}\\
&=\sum_{k=1}^{2n}\frac{1}{n+k}\\
&=\sum_{k=1}^{n}\frac{1}{n+k}+\underbrace{\sum_{k=n+1}^{2n}\frac{1}{n+k}}_{l=k-n}\\
&=\sum_{k=1}^{n}\frac{1}{n+k}+\sum_{k=1}^{n}\frac{1}{2n+l}\\
&=\sum_{k=1}^{n}\left(... |
2,654,702 | <p>I'm working on generating a counter example that shows projections from $X \times Y$ do not always map closed sets to closed sets. I'm working backwards, that is, from the decompositions to the product space. My attempt is to see if I might generate a closed product space whose projection maps to clopen sets. </p>
... | José Carlos Santos | 446,262 | <p>Since $b>1$, the function $t\mapsto b^t$ is increasing. So$$b^r=\sup_{t\leqslant r}b^t=B(r).$$</p>
|
107,425 | <p>Problem 1:</p>
<p>If $n$ people are seated in a random manner in a row containing $2n$ seats, what is the probability that no two people will occupy adjacent seats?</p>
<p>I know that the probability that $n$ people are seated in $2n$ rows is ${2n\choose n}$</p>
<p>I also know that the answer to this problem is $... | Ansgar Esztermann | 23,174 | <p>ad 1: $2n\choose n$ is the number of ways to seat $n$ people without regarding their identities. Now, how many of these will have no two people adjacent? You could occupy all odd seats, leaving the even ones free. Note that this will also leave seat $2n$ free, although this is not necessary in order to "keep the dis... |
107,425 | <p>Problem 1:</p>
<p>If $n$ people are seated in a random manner in a row containing $2n$ seats, what is the probability that no two people will occupy adjacent seats?</p>
<p>I know that the probability that $n$ people are seated in $2n$ rows is ${2n\choose n}$</p>
<p>I also know that the answer to this problem is $... | Gerry Myerson | 8,269 | <p>For problem 2, why not consider the smaller problem where there are only two states? Small enough that you can write down all the possibilities and see what the answer is, and then compare to the formulas you are asking about. </p>
|
2,808,445 | <p>I should mention this up front: this is a question about semantics. It is extremely formal, and a bit pedantic. Most people won't care.</p>
<p>Is det, i.e the determinant function, a single one? Or is it multiple functions, one for each field?</p>
<p>Let $\mbox{Mat}(K)$ denote the set of square matrices over the f... | Eric Wofsey | 86,856 | <p>The meaning that mathematicians usually implicitly have in mind is that there are lots of separate functions written "$\det$". Specifically, for each field $K$ and each $n\in\mathbb{N}$, there is a separate function $\det:M_n(K)\to K$ (which to be completely pedantic perhaps should be written $\det_{n,K}$, but that... |
389,912 | <p>I'm writing a paper and want to cite some references to efficiently prove that over any field <span class="math-container">$k$</span> of characteristic zero, every irreducible representation of a product of symmetric groups, say</p>
<p><span class="math-container">$$ S_{n_1} \times \cdots \times S_{n_p} $$</span></p... | LSpice | 2,383 | <p>Every complex representation of <span class="math-container">$S_n$</span> is defined over <span class="math-container">$\mathbb Z$</span>, but, as suggested in the <a href="https://mathoverflow.net/questions/389912/representations-of-products-of-symmetric-groups#comment993437_389915">comments</a>, it might be more n... |
191,262 | <p>I have to fit a Cos^2 function to data I measured. The function is <span class="math-container">$a \cos^2(\frac{bx\pi}{180}+c)+d $</span> and I tried the FindFit and Linear Model Function. I have 4 datasets which I have to fit, the first one worked. The other three only yielded not usable fits. I am pretty new to Ma... | m0nhawk | 2,342 | <p>First, you better use <a href="https://reference.wolfram.com/language/ref/NonlinearModelFit.html" rel="noreferrer"><code>NonlinearModelFit</code></a>, <code>Fit</code> and <code>FindFit</code> haven't updated in a while, so newly introduced <code>LinearModelFit</code> and <code>NonlinearModelFit</code> will have gre... |
1,005,921 | <p>I was just wondering if anyone could shed more light on specific topics in applied mathematics or other skills (programming, etc) commonly used in the following industries: oil and gas, aviation/aeronautics. In particular, I'm interested in how mathematical modelling could be applied in these industries (from the pe... | Jair Taylor | 28,545 | <p>I think the "pseudo-random" nature of the digits of $\pi$ is confusing the issue here. It's perfectly conceivable that the digits of $\pi$ behave in the fashion you indicate - that it would be more likely for a digit to show up if it hadn't been seen in a while. But if this were the case, the digits would not be b... |
370,151 | <p>Let $f: \Bbb R → \Bbb R$ be a continuous function such that $f(x)=x$ has no real solution . Then is it true that $f(f(x))=x$ also has no real solution ? </p>
| roger | 65,388 | <p>Since $f$ is continuous, saying that $f(x) = x$ has no solution means that either $f(x) < x$ for all $x$ or $f(x) > x$ for all $x$. Let's assume wlog that $f(x) > x$, then $f(f(x)) > f(x) > x$ for all $x$ so $f(f(x))=x$ has no solution either.</p>
|
4,005,381 | <p>Why the probability of a single element event is "zero" on the continuous model?, the explanation given is based on probability additivity axioms. but how? some more explanation with the source would be helpful.</p>
<p>cool thanks!</p>
| zwhhh | 880,865 | <p>Assume <span class="math-container">$\xi$</span> is a random variable and <span class="math-container">$p(x)$</span> is the density function of <span class="math-container">$\xi$</span>.</p>
<p>For all <span class="math-container">$t\in R$</span>, we have <span class="math-container">$P(\xi =t)\leq P(t\leq \xi\leq t... |
2,563,779 | <p>Let $(X,d)$ be a metric space. If every subset of $X$ is bounded, does it mean that the space itself is bounded?</p>
| Gribouillis | 398,505 | <p>Yes: $X$ is a subset of $X$, so it is bounded.</p>
|
2,563,779 | <p>Let $(X,d)$ be a metric space. If every subset of $X$ is bounded, does it mean that the space itself is bounded?</p>
| mechanodroid | 144,766 | <p>Perhaps you meant every that every proper subset of $X$ is bounded. In that case, take any $A, B \subsetneq X$ such that $A \cup B = X$. Hence, $X$ is bounded as a finite union of two bounded sets.</p>
<p>Namely $$\operatorname{diam} X = \operatorname{diam}(A\cup B) \le \operatorname{diam} A + d(A, B) + \operatorna... |
299,263 | <p>What is the set of polynomials with integer coefficients look like? Help!</p>
<p>Thanks!</p>
| Herng Yi | 34,473 | <p><strong>Hint:</strong></p>
<p>Here's a "recursive definition of the set of positive rationals".</p>
<ol>
<li>$1$ is in the set.</li>
<li>If $a$ and $b$ are in the set, so is $a + b$.</li>
<li>If $c$ and $d$ are in the set, so is $c/d$.</li>
</ol>
<p>Now, I prove that this "recursive algorithm" covers all of the p... |
3,185,927 | <p><strong>Proposition:</strong> Let <span class="math-container">$X$</span> be a compact Hausdorff space. Suppose there are countable real valued continuous functions <span class="math-container">$\{f_n\}_{n \in \mathbb{Z}_+}$</span> separating <span class="math-container">$X$</span> i.e. for all <span class="math-con... | Romain S | 229,859 | <p>Here's another proof, for fun.</p>
<p>Fix <span class="math-container">$x\in X$</span>. The function <span class="math-container">$d_x(y)=d(x,y)$</span> is continuous as the partial sums
<span class="math-container">$$\sum_{n=1}^N\frac{\min\{|f_n(x)-f_n(y)|, 1\}}{2^n}$$</span>
converge uniformly to <span class="math... |
3,117,139 | <p>I am learning to calculate the arc length by reading a textbook, and there is a question</p>
<p><a href="https://i.stack.imgur.com/Zigqv.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Zigqv.png" alt="enter image description here"></a></p>
<p>However, I get stuck at calculating</p>
<p><span cla... | Hyperion | 636,641 | <p>We can start by making the substitution in question:
<span class="math-container">$$u = x^2, \,du=2x \,dx$$</span>
Then
<span class="math-container">$$\int e^{x^2}\,dx =\int \frac{e^u}{2x} \,du$$</span>
One might think this is easily integrable from here because we have only an <span class="math-container">$e^u$</sp... |
1,908,865 | <p>$$Var\left ( E\left [ Y \mid X \right ] \right )\leqslant Var \left ( Y \right )$$</p>
| Robert Z | 299,698 | <p>We assume that $X$ and $Y$ are random variables on the same probability space, and the variance of $Y$ is finite.
We have that $\mbox{E}[\mbox{Var}[Y \mid X])]\geq 0$ because variance is nonnegative. Therefore by <a href="https://en.wikipedia.org/wiki/Law_of_total_variance" rel="nofollow">Law of total variance</a>,
... |
1,908,865 | <p>$$Var\left ( E\left [ Y \mid X \right ] \right )\leqslant Var \left ( Y \right )$$</p>
| H. H. Rugh | 355,946 | <p>In order to understand the reason behind, an explanation may be given in terms of orthogonal projections: Let our probability space be $L^1(\Omega,{\cal A}, P)$ and let ${\cal B} \subset {\cal A}$ be the $\sigma$-algebra generated by $X$. Then $E(Y|X)$ is the ${\cal B}$ measurable random variable obtained by ortho... |
199,889 | <p>What are applications of the theory of Berkovich analytic spaces? The analytification $X \mapsto X^{\mathrm{an}}$</p>
| Jérôme Poineau | 4,069 | <p>Let me add a few more applications to what has already been mentioned.</p>
<ul>
<li><p>Relation with Bruhat-Tits buildings (Berkovich, then Rémy/Thuillier/Werner). If $G$ is a reductive group over a non-archimedean valued field, then the Bruhat-Tits building $\mathscr{B}(G)$ of $G$ embeds into the analytification $... |
530 | <p>Tau ($\tau = 2 \pi$) has more merits in its application, but pi is the established standard in industry and education. Is the trade-off of teach-ability of circle concepts worth the subsequent confusion due to pi's omnipresence?</p>
<p>How can both be most effectively taught in order to maximize student's understan... | Ruben | 915 | <p>I feel like $\tau$ is very useful, but of course everybody uses $\pi$ in 'the real world'.</p>
<p>I wouldn't hurt to mention $\tau$ one of two times though, and show to students that if they are having trouble with visualizing the length of circle segments or find $\tau$ convenient for other reasons, they can alway... |
240,038 | <p>How can I compute this integral:
$$I=\int\frac{1-(au+1)\exp(-au)}{u^{2}}du$$
$$$$Can anyone help me in this case?</p>
| Community | -1 | <p>$$I(a)=\int\frac{1-(au+1)\exp(-au)}{u^{2}}du$$ then $$\dfrac{dI}{da} = \int \dfrac{u(au+1) \exp(-au) - u\exp(-au)}{u^2} du = \int a \exp(-au) du = - \exp(-ax) + c$$
Hence, $$I(a) = \dfrac{\exp(-ax)}{x} + \underbrace{ca}_{\text{constant}} + d(x)$$
Further $I(0) = 0 \implies d(x) = - \dfrac1x$. Hence, $$I(a)=\int\frac... |
123,102 | <p>As far as I know, it is an open problem to give a formula counting transitive relations on an $n$-element set. Is it easier to count the idempotent relations, that is relations that are both transitive and interpolative? (A relation $\rho$ is interpolative when $x\rho y\implies((\exists z)\ x\rho z \wedge z\rho y).$... | Robert Israel | 13,650 | <p>See OEIS sequence <a href="http://oeis.org/A121337/" rel="noreferrer">A121337</a> and references given there.</p>
|
4,215,226 | <p>Let <span class="math-container">$f$</span> be an integrable function on <span class="math-container">$[a,b]$</span>. Then I have to show that there exists <span class="math-container">$x$</span> in <span class="math-container">$[a,b]$</span> such that <span class="math-container">$\int_a^x f=\int_x^b f$</span>.
I u... | Bertrand Einstein IV | 824,733 | <p>The solution to this problem is almost identical to the post you linked:</p>
<p>Consider the case where we only have 4 cards and all 4 are aces, how many would we expect to draw before getting all aces? Trivially 4.</p>
<p>For <span class="math-container">$\color{red}4$</span> cards with <span class="math-container"... |
4,213,335 | <p>Consider this equation where <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are positive integers.</p>
<p><span class="math-container">$$k = \frac{2^a - 1}{2^{a+b} - 3^b}$$</span></p>
<p>This equation has the trivial solution <span class="math-container">$k=1, a=1, b=1$</span>.</... | Gottfried Helms | 1,714 | <p>There is another representation of this formula, which connects it with an -as well unsolved- "detail in the Waring-problem" (see <a href="https://mathworld.wolfram.com/PowerFractionalParts.html" rel="nofollow noreferrer">mathworld.com</a>) namely the distance of <span class="math-container">$(3/2)^N$</spa... |
2,588,271 | <blockquote>
<p>Show that $\mathcal{L}\{t^{1/2}\}=\sqrt{\pi}/(2s^{3/2}), \: s>0$</p>
</blockquote>
<p>By the definition of Laplace transform we get:</p>
<p>$$\mathcal{L}\{t^{1/2}\} = \int_0^\infty t^{1/2}e^{-st} \, dt = \{x = \sqrt{st} \} = \dfrac{2}{s^{3/2}} \int_0^\infty e^{-x^2} x^2 \, dx. $$</p>
<p>A known ... | Michael Hardy | 11,667 | <p>$$
\int_0^\infty x^2 e^{-x^2} \, dx = \int_0^\infty \frac x 2 e^{-x^2} \Big( 2x\, dx \Big) = \int_0^\infty \frac {\sqrt u} 2 e^{-u} \, du = \frac 1 2 \Gamma\left( \frac 3 2 \right).
$$</p>
|
2,588,271 | <blockquote>
<p>Show that $\mathcal{L}\{t^{1/2}\}=\sqrt{\pi}/(2s^{3/2}), \: s>0$</p>
</blockquote>
<p>By the definition of Laplace transform we get:</p>
<p>$$\mathcal{L}\{t^{1/2}\} = \int_0^\infty t^{1/2}e^{-st} \, dt = \{x = \sqrt{st} \} = \dfrac{2}{s^{3/2}} \int_0^\infty e^{-x^2} x^2 \, dx. $$</p>
<p>A known ... | Mohammad Riazi-Kermani | 514,496 | <p>You can try integration by parts with $u=x$ and $dv=xe^{-x^2}dx$ to evaluate$$ \int_0^\infty e^{-x^2} x^2 \, dx.$$ Note that,$du=dx$ and $v=-(1/2)e^{-x^2}.$ Therefore; $$\int e^{-x^2} x^2 \, dx=-(1/2)xe^{-x^2} +1/2\int e^{-x^2}\,dx.$$ Upon evaluating from $0$ to $\infty$,the first term vanishes and you get $$\int_... |
2,869,189 | <p>Is the method of calculating determinant of $3\times 3$ matrix by diagonals,
apply also on $4\times 4$ matrix?</p>
<p>for example:</p>
<p>$$\begin{matrix}2&2&1&3|\\1&4&4&5|\\5&1&1&6|\\7&1&4&5|\end{matrix}\begin{matrix}2&2&1\\1&4&4\\5&1&1\\7&am... | Siong Thye Goh | 306,553 | <p>No, it is not valid. My computation using Octave shows that the determinant is negative.</p>
<pre><code>octave:1> A = [ 2 2 1 3; 1 4 4 5; 5 1 1 6; 7 1 4 5]
A =
2 2 1 3
1 4 4 5
5 1 1 6
7 1 4 5
octave:2> det(A)
ans = -114.00
</code></pre>
|
1,970,082 | <p>How would you go about proving that:</p>
<p>$\lim_{x\to2^-}\frac{1}{x-2}= -\infty$
(i.e. x approaching 2 from left hand side)</p>
<p>Been struggling to make much headway on this. Not sure how I'm supposed to use Epsilon in this case. So far, I have: <br/>
Given any negative number M there exists $\delta$ s.t. $f(x... | Ethan Alwaise | 221,420 | <p>First note that if $\epsilon > 0$ is given, $1/q \geq \epsilon$ implies $q > 0$ and $q \leq 1/\epsilon$, so there are only finitely many integers such $q$. Can you conclude that for each of these choices of $q$, there are only finitely many integers $p$ such that $p/q \in (a,b)?$</p>
|
2,764,586 | <p>Let $S$ be the set of all complex numbers $z$ satisfying the rule $$|z-i|=\sqrt{2}|\bar{z}+1|$$</p>
<p>Show that $S$ contains points on a circle.</p>
<p>My attempt,</p>
<p>By substituting $z = x + yi$, and squaring both sides. But I can't get the circle form. </p>
| Siong Thye Goh | 306,553 | <p>Altenative if you are familiar with <a href="https://en.wikipedia.org/wiki/Circles_of_Apollonius" rel="nofollow noreferrer">Circles of Apollonius</a>,</p>
<p>$$|z-i| = \sqrt{2}|\bar{z}+1|=\sqrt{2}|z+1|=\sqrt{2}|z-(-1)|$$</p>
<p>Here $i$ and $-1$ are the foci.</p>
|
2,764,586 | <p>Let $S$ be the set of all complex numbers $z$ satisfying the rule $$|z-i|=\sqrt{2}|\bar{z}+1|$$</p>
<p>Show that $S$ contains points on a circle.</p>
<p>My attempt,</p>
<p>By substituting $z = x + yi$, and squaring both sides. But I can't get the circle form. </p>
| trancelocation | 467,003 | <p>Another way is using the complex equation of a circle and complex square completion:</p>
<ul>
<li>$|z-a| = r$ is the circle with radius $r$ around $a = a_x + ia_y$
$$|z-a|^2 = |z|^2 + |a|^2 - 2Re(\bar z a)$$</li>
<li>Furthermore, note that $|\bar z+1| = |z+1|$</li>
<li>$|z-i|=\sqrt{2}|\bar{z}+1|$ becomes
$$|z|^2 + ... |
1,972,079 | <p>I would like to split splines of DXF files to lines and arcs in 2D for a graphic editor. From DXF file, I have extracted the following data:</p>
<ul>
<li>degree of spline curve</li>
<li>number of knots and knot vectors</li>
<li>number of control points and their coordinates</li>
<li>number of fit points and their c... | bubba | 31,744 | <p>See <a href="https://math.stackexchange.com/questions/1679920/curve-fitting-using-circles/1681399#1681399">this question</a>. </p>
<p>The answers include several useful links.</p>
<p><a href="http://www.ryanjuckett.com/programming/biarc-interpolation/" rel="nofollow noreferrer">One of the links</a> provides a very... |
3,315,516 | <p>A convex function is locally Lipschitz on a closed and bounded set. This implies that a nonconvex function may not be locally Lipschitz. Is every nonconvex function not locally Lipschitz? </p>
| WalterJ | 344,100 | <p>No, take for example <span class="math-container">$\sin(x)$</span>. </p>
|
3,315,516 | <p>A convex function is locally Lipschitz on a closed and bounded set. This implies that a nonconvex function may not be locally Lipschitz. Is every nonconvex function not locally Lipschitz? </p>
| Acccumulation | 476,070 | <p>If this were true, why would we need different terms "Lipschitz" and "convex"? A convex function is a function that is concave upward, so convexity is not preserved by vertical reflection. But Lipschitz continuity is preserved by relections, so given any function <span class="math-container">$x \rightarrow f(x)$</sp... |
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