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 |
|---|---|---|---|---|
108,277 | <p>If Mochizuki's proof of abc is correct, why would this provide a new proof of FLT?</p>
<p>Edit: In proof of asymptotic FLT, does Mochizuki claim a specific value of n and if so what is this value?</p>
| José Hdz. Stgo. | 1,593 | <p>Let us suppose that <span class="math-container">$x^{n}+y^{n}= z^{n}$</span> with <span class="math-container">$x, y,$</span> and <span class="math-container">$z$</span> relatively prime. By the <em>abc</em> conjecture, <span class="math-container">$|x^{n}|\ll |xyz|^{1+\epsilon}$</span>, <span class="math-container"... |
3,657,751 | <p>Consider the series <span class="math-container">$$\sum_{n=1}^{\infty}\frac{(-1)^{\frac{n(n+1)}{2}+1}}{n}=1+\dfrac12-\dfrac13-\dfrac14+\dfrac15+\dfrac16-\cdots.$$</span> This is clearly not absolutely convergent. On the other hand, obvious choice, alternating series does not work here. Seems like the partial sum seq... | Gary | 83,800 | <p>Observe that
<span class="math-container">$$
\left| {\sum\limits_{n = 1}^N {( - 1)^{T_n + 1} } } \right| \le 2
$$</span>
for any <span class="math-container">$N\geq 1$</span>. Also <span class="math-container">$\frac{1}{n}\to 0$</span> monotonically. Hence, by Dirichlet's test, the series converges. The limit is
<s... |
23,942 | <p>I have tried to resolve the problem of the following link <a href="https://mathematica.stackexchange.com/questions/23931/how-can-i-solve-precision-problem">How can I solve precision problem</a></p>
<p>I can tell the problem described in that link shortly here, It's no mater how many precision is there after decimal... | Silvia | 17 | <p>Is <a href="http://reference.wolfram.com/mathematica/ref/SetAccuracy.html" rel="nofollow"><code>SetAccuracy</code></a> what you want?</p>
<pre><code>a = N[10/3, {∞, 3}]
</code></pre>
<blockquote>
<p>3.33</p>
</blockquote>
<pre><code>b = 3``3
</code></pre>
<blockquote>
<p>3.00</p>
</blockquote>
<pre><code>a ... |
32,150 | <p>I want to test if expressions (mix of variables, functions and numbers) are zero valued, as fast as possible, and <code>PossibleZeroQ</code> is sometimes very slow. One solution I found was to substitute the variables for random reals and test if the value of the substituted expression is less than, say, $0.0001$.</... | Giovanni F. | 4,769 | <p>Ok, posting my version here. It has 0.1 seconds to test, otherwise returns false. I am assuming that if it takes longer than that, then the expression being tested is too complex (for my problem) and is not useful.</p>
<pre><code>TestZeroValuedExpression[expression_,symbolslist_]:=Module[{expressionrandomvalue},
... |
20,802 | <p>Look at the following example:</p>
<p>Which picture has four apples?</p>
<p>A<a href="https://i.stack.imgur.com/Tpm46.png" rel="noreferrer"><img src="https://i.stack.imgur.com/Tpm46.png" alt="enter image description here" /></a></p>
<hr />
<p>B <a href="https://i.stack.imgur.com/AOv29.png" rel="noreferrer"><img src=... | Barmar | 10,551 | <p>When we describe counts in natural language, there's almost always an implicit "exactly" when phrasing like this. We use phrases like "at least 4" when we want a more general description. Most children who have reached a development level where this quiz would be reasonable will probably already... |
2,530,298 | <p>I tried putting y alone and got y=(-6x-5)/5. Which I then put into the distance formula sqrt((x-1)^2+(y+5) and substitute the number above in for y but my answer never comes out correct.. Wondering if I could get some help.</p>
| Nosrati | 108,128 | <p>Another method. Take a circle centered in $(1,-5)$ that is
$$(x-1)^2+(y+5)^2=d^2$$
The slope of tangent line is
$$y'=-\dfrac{f_x}{f_y}=-\dfrac{x-1}{y+5}$$
will be $-\dfrac65$, the slope of given line. Then it's sufficient to solve the system
\begin{cases}
6x+5y+5=0,\\
5x-6y=35.
\end{cases}</p>
|
3,580,258 | <p>Hi: The definition I'll use is this: Let <span class="math-container">$F$</span> be an abelian group and <span class="math-container">$X$</span> a subset of <span class="math-container">$F$</span>. Then <span class="math-container">$F$</span> is a free abelian group on <span class="math-container">$X$</span> if for ... | user3482749 | 226,174 | <p>You are using <span class="math-container">$\langle X \rangle$</span> to mean two different things, and conflating them: </p>
<ol>
<li>You are using it to mean the free abelian group on <span class="math-container">$X$</span>. </li>
<li>You are using it to mean the subgroup of <span class="math-container">$G$</span... |
128,695 | <p>Is there any good guide on covering space for idiots? Like a really dumped down approach to it . As I have an exam on this, but don't understand it and it's like 1/6th of the exam. </p>
<p>So I'm doing Hatcher problem and stuck on 4.</p>
<ol>
<li>Construct a simply-connected covering space of the space $X \subset ... | Ronnie Brown | 28,586 | <p>Books on algebraic topology are usually good on giving invariants to show that spaces are <strong>not</strong> homotopy equivalent, but not so good at showing why spaces <strong>are</strong> homotopy equivalent. In my book "Topology and groupoids" (2006) there is a chapter on cofibrations, which discusses the homot... |
2,439,863 | <p>I was working on the series </p>
<p>$\sum_{n=1}^{\infty}{\frac{(-1)^n}{n}z^{n(n + 1)}}$ and I was to consider when $z = i$. I have that $$\sum_{n=1}^{\infty}{\frac{(-1)^n}{n}i^{n(n + 1)}} = \sum_{n=1}^{\infty}{\frac{(-1)^{\frac{3}{2}n+\frac{1}{2}n^2}}{n}} = 1 - \frac{1}{2} - \frac{1}{3} + \frac{1}{4} + \frac{1}{5} ... | mercio | 17,445 | <p>Let $f(z) = z - \frac 12 z^2 - \frac 13 z^3 + \frac 14 z^4 + \frac 15z^5 - \frac 16 z^6 - \frac 17 z^7 + \frac 18z^8 + \cdots$</p>
<p>Then $(1+z^2)f'(z) = (1+z^2)(1-z-z^2+z^3+z^4-z^5-z^6+z^7+\cdots) = 1-z$</p>
<p>So $f'(z) = \frac {1-z}{1+z^2} = -\frac 12 \frac {2z}{1+z^2} + \frac 1 {1+z^2}$</p>
<p>And $f(z) = - ... |
106,775 | <p>I don't get this, need some help, examples and information</p>
<blockquote>
<p>The linear function $f$ is given by
$$f(x) = 3x - 2 ,\quad -2 \leq x \leq 4.$$</p>
<ol>
<li><p>Enter the independent variable and the dependent variable.</p></li>
<li><p>Determine the function values $f (-2)$, $f (-1)$, $f... | Yeujer | 207,161 | <p>f(2)= 3 times 2 − 2</p>
<p>f(-1)= 3 times -1 - 2</p>
<p>and so on because the -1 in f(-1) is the x-value and if that is the x-value then the the equation 3x-2 it would end up as 3 times -1 subtracted by 2. Which will end up with:
3 times -1=-3
-3 subtracted by 2=-5
So the answer for f(-1) is equal to -5. I hope th... |
3,327,094 | <p>Give an example of a non abelian group of order <span class="math-container">$55$</span>.</p>
<p>To find non abelian group the simplest way is to find one non abelian group whose order divides the order of given group and then we take the group which is the external direct product of the non abelian group and some ... | Chinnapparaj R | 378,881 | <p>Consider the field <span class="math-container">$G=\Bbb Z_{11}$</span>. Now its multiplicative group <span class="math-container">$\Bbb Z_{11}^*$</span> is a group of order <span class="math-container">$10$</span>. Now <span class="math-container">$5$</span> divides <span class="math-container">$10$</span>, so By C... |
306,212 | <p>The only statement I'm sure of is that any hyperbolic or Euclidean manifold is a $K(G,1)$ (i.e. its higher homotopy groups vanish), since its universal cover must be $\mathbb H^n$ or $\mathbb E^n$. But for example, if a complete Riemannian manifold $M$ satisfies one of the following, can I conclude that $M$ is a $K(... | Igor Rivin | 11,142 | <p>For 1-3, yes, by the Cartan-Hadamard Theorem.5-... No. For example, every 3-manifold admits a metric of negative scalar curvature (I think this is actually true for any manifold, due to Lohkamp).</p>
|
316,016 | <p>Could you recommend any approachable books/papers/texts about matroids (maybe a chapter from somewhere)? The ideal reference would contain multiple examples, present some intuitions and keep formalism to a necessary minimum.</p>
<p>I would appreciate any hints or appropriate sources.</p>
| Nemo | 519,978 | <p>I have read chapter 1 of Gordon and
McNultey's book and think it provides an
Excellent clear intro. It will get you to first
Base quickly. </p>
|
1,506,763 | <p>This is not true for infinite measures (<a href="https://math.stackexchange.com/questions/342039/pointwise-convergence-but-not-in-measure">Pointwise convergence, but not in measure</a>). Is it true for a finite measure? Namely, let a finite (probability) measure $\mu(\cdot)$. Does a point-wise convergence of $\mu-$... | Brian Ding | 197,478 | <p>By the reversed Fatou lemma, we have </p>
<p>$\lim \mu( |f_n-f|\geq \epsilon) = \lim \int 1_{\{ |f_n-f|\geq \epsilon \}}d\mu \leq \int \limsup 1_{\{|f_n-f|\geq \epsilon \}}d\mu = 0.$</p>
<p>$1_{\{\cdot\}}$ is the indicator function (or characteristic function for analysis people). Since indicator is bounded, it is... |
4,489,898 | <p>After 18 months of studying an advanced junior high school mathematics course, I'm doing a review of the previous 6 months, starting with solving difficult quadratics that are not easily factored, for example:
<span class="math-container">$$x^2+6x+2=0$$</span>
This could be processed via the quadratic equation but t... | G Tony Jacobs | 92,129 | <p>We use a similar trick in teaching first semester calculus. To obtain the well-known product rule for derivatives, we have to deal with the expression:</p>
<p><span class="math-container">$$\frac{f(x+h)g(x+h) - f(x)g(x)}{h}$$</span></p>
<p>This is inconvenient, as the two terms in the numerator have nothing in commo... |
718,850 | <p>I see similar questions asked on here and obviously I did some research and read my book, but it seems like every explanation contradicts another in some way. There are basically infinite scenarios using these and every example problem/scenario I seem to convince myself it could be both!</p>
<p><strong>Here are som... | Nilesh Ingle | 662,954 | <p>Thank you for question and explanations above. Below is the way I try to remember it.</p>
<p>'Does not matter' actually means 'no replacement' AND 'no repetition within arrangement'</p>
<p><em>[Note: In other words <span class="math-container">$[1,2]$</span> or <span class="math-container">$[2,1]$</span> does not... |
1,515,417 | <p>I understand the idea that some infinities are "bigger" than other infinities. The example I understand is that all real numbers between 0 and 1 would not be able to "fit" on an infinite list.</p>
<p>I have to show whether these sets are countable or uncountable. If countable, how would you enumerate the set? If un... | Brian Tung | 224,454 | <p>For Set $2$: Just consider the subset of this set consisting of numbers in the interval $(0, 1)$. Assume you have a complete listing of such numbers, and write them out, padding them with trailing zeros as needed. (Remember that the defining characteristic of Set $2$ is that the numbers can be represented only wit... |
335,258 | <p>Find the domain of the function:
$$f(x)= \sqrt{x^2 - 4x - 45}$$</p>
<p>I'm just guessing here; how about if I square everything and then put it in the graphing calculator?
Thanks,
Lauri</p>
| Sam | 66,646 | <p><em>Hint:</em> The domain of a function is the set of input values for which the function is defined. Do you know of any values for which the square root is not defined?</p>
|
2,493,481 | <p>I'm currently studying calculus of variations. I couldn't find a rigorous definition of a functional on this site.</p>
<ol>
<li>What is the general definition of a functional?</li>
<li>Why for calculus of variations in physics, I must to use for a functional <em>a convex function</em> for the space of the admissib... | supinf | 168,859 | <p>Here is a rigorous definition:
Let $V$ be a vector space over a field $\mathbb K$.
Then every function $F:V\to\mathbb K$ is called a functional.</p>
<p>An important class of functionals are linear functionals.
If $\mathbb K\in \{\mathbb R, \mathbb C\}$ and $V$ is a normed vector space ( or even a topological vector... |
33,646 | <h1>Preamble</h1>
<p>I am a novice SE user, a toddler. In this post I want to criticize some moderator actions, which seems risky and futile, regarding unwritten policies on the meta; however, just for the record, I write this post so that unbiased readers find it helpful.</p>
<p>Please note that I am not throwing a ta... | Tryst with Freedom | 688,539 | <p>Point of this answer: I wish to discuss a more fundamental idea, that is , on why people comment such ways.</p>
<hr />
<p>I think the problem is that, at least culturally, mathematics stack exchange has evolved from a question and answer site into more of a community.</p>
<blockquote>
<p>A community is a social uni... |
30,402 | <p>The envelope of parabolic trajectories from a common launch point is itself a parabola.
In the U.S. soon many will have a chance to observe this fact directly, as the 4th of July is traditionally celebrated with fireworks.</p>
<p>If the launch point is the origin, and the trajectory starts off at angle $\theta$ and... | Arseniy Akopyan | 2,158 | <p>It is easy to see that all these parabolas have the same directrix. Height of a directirix correspond to energy of the body. So you have the family of parabolas with the common point $P$ and the directrix $l$. It is easy to prove, (using just definition of parabola as a locus of points...) that all of the touched th... |
400,715 | <p>Consider the metric space $(\mathbb{Q},d)$ where $\mathbb{Q}$ denotes the rational numbers and $d(x,y)=|x-y|$. Let $$E:=\{x \in\mathbb{Q}:x>0, 2<x^2<3\}$$</p>
<p>Is $E$ closed and bounded in $\mathbb{Q}?$ Is it compact? Justify your answers.</p>
| Community | -1 | <p>It is not closed. You can always find a sequence of $x \in E$ such that converge to a real number.</p>
|
758,950 | <p>I have a pretty straightforward combinatorical problem which is an exercise to one paper about generating functions.</p>
<ol>
<li>How many ways are there to get a sum of 14 when 4 distinguishable dice are rolled? </li>
</ol>
<p>So, one die has numbers 1..6 and as dice are distinguishable then we should use exponen... | epi163sqrt | 132,007 | <p>As <a href="https://math.stackexchange.com/users/205/shreevatsar">ShreevatsaR</a> pointed out it's sufficient to consider ordinary generating functions, since they already take into account that $3,4,3,4$ and $3,3,4,4$ are different. The first is coded as the coefficient of $x^3x^4x^3x^4$, while the second as the co... |
2,078,737 | <p>I will gladly appreciate explanation on how to do so on this matrix:</p>
<p>$$
\begin{pmatrix}
i & 0 \\
0 & i \\
\end{pmatrix}
$$</p>
<p>I got as far as calculating the eigenvalues and came up with $λ = i$.
when trying to find the eigenvectors I came up with the $0$ matrix.<... | B. Goddard | 362,009 | <p>I think most people would say that limits that go to infinity do not exist. But it's a special case of non-existence. Some limit's don't exist because the function bounces around too much. But this limit doesn't exist because it increases without bound. </p>
<p>Some folks will say "What about the extended reals... |
26,192 | <p>I have a list of rules that represents a list of parameters to be applied to a circuit model (Wolfram SystemModeler model): </p>
<pre><code>sk = {
{R1 -> 10080., R2 -> 10080., C1 -> 1.*10^-7, C2 -> 9.8419*10^-8},
{R1 -> 10820., R2 -> 4984.51, R3 -> 10000., R4 -> 10000., C1 -> 1.*10^-7,
C... | Chris Degnen | 363 | <p>This version implements a pair of counters, <code>a</code> & <code>b</code>, while retaining much of the original code :-</p>
<pre><code>a = b = 0;
Table[
If[Length[sk[[i]]] > 4,
cirname = "sallenKey";
(cirname <> ToString[If[a + b != i, ++a, a]] <> "." <>
ToString[sk[[i]][[All,... |
3,664,717 | <p>Let's say there is a function <span class="math-container">$g: B \rightarrow B$</span> and <span class="math-container">$B$</span> is some set.</p>
<p>A relation <span class="math-container">$Rx$</span> over set <span class="math-container">$B$</span> is when </p>
<p><span class="math-container">$a Rx b$</span> </... | SagarM | 142,677 | <p>As far as I understand you are looking for the following kind of example.</p>
<p>let <span class="math-container">$f: N \rightarrow N $</span> be defined as follows
<span class="math-container">$$f(x) = \text{smallest prime larger than }x -x$$</span>
Then,
if f(x)=y, then the relation <span class="math-container">$... |
3,664,717 | <p>Let's say there is a function <span class="math-container">$g: B \rightarrow B$</span> and <span class="math-container">$B$</span> is some set.</p>
<p>A relation <span class="math-container">$Rx$</span> over set <span class="math-container">$B$</span> is when </p>
<p><span class="math-container">$a Rx b$</span> </... | J. C. | 388,924 | <p>Consider the following function (here I'm considering <span class="math-container">$0 \notin \mathbb{N}$</span>, but it can be easily adapted to include <span class="math-container">$0$</span>):
<span class="math-container">$$f: \mathbb{N} \to \mathbb{N} $$</span>
<span class="math-container">$$ f(n)=
\begin{cases}
... |
262,500 | <p>What is the Green’s function of the boundary value problem
$$
\frac{\mathrm d^2 y}{\mathrm d x^2}-\frac{1}{x}\frac{\mathrm dy}{\mathrm dx}=1,\quad y(0)=y(1)=0,
$$</p>
<p>this boundary problem is not self adjoint, so please help me how to solve it.</p>
| user26872 | 26,872 | <p>First note that the solution to the homogeneous problem is
$y(x) = a + b x^2.$</p>
<p>We wish to solve
$$\begin{equation*}
\frac{d^2}{dx^2}G(x,t) - \frac{1}{x} \frac{d}{dx} G(x,t) = \delta(x-t),\tag{1}
\end{equation*}$$
where $G$ satisfies the boundary conditions
$G(0,t) = G(1,t) = 0$.
Therefore,
$$G(x,t) =
\be... |
2,941,311 | <p>Given diophantine equation <span class="math-container">$11x+17y +19z =2561$</span> , which <span class="math-container">$x,y,z \geq 1$</span></p>
<p>Find minimum and maximum value of <span class="math-container">$x+y+z$</span></p>
<p>I'm start with reduces equation to <span class="math-container">$11x+17y +1... | hmakholm left over Monica | 14,366 | <p>There are cases where <span class="math-container">$\lim\frac{f(x)}{g(x)}$</span> exists but <span class="math-container">$\lim\frac{f'(x)}{g'(x)}$</span> does <em>not</em> exist. For example, with <span class="math-container">$a=0$</span>:</p>
<p><span class="math-container">$$ f(x) = x^2\sin(1/x) \qquad\qquad g(x... |
3,921 | <p>Say I have a triangle with vertices $(0,0), (2,4), (4,0)$ that I want to rotate along the origin. Rotation by multiples of $90^{\circ}$ is simple. However, I want to rotate by something a bit more complicated, such as $54^{\circ}$. How do I figure out where the vertices would be then?</p>
| Isaac | 72 | <p>One way is to use complex numbers. Multiplying by $\cos\theta+i\sin\theta$ rotates by $\theta$ about 0, so you could multiply $(2+4i)(\cos 54^\circ+i\sin 54^\circ)$ to get the rotation image of (2,4).</p>
|
3,921 | <p>Say I have a triangle with vertices $(0,0), (2,4), (4,0)$ that I want to rotate along the origin. Rotation by multiples of $90^{\circ}$ is simple. However, I want to rotate by something a bit more complicated, such as $54^{\circ}$. How do I figure out where the vertices would be then?</p>
| J. M. ain't a mathematician | 498 | <p>In the answer to <a href="https://math.stackexchange.com/questions/2429">this question</a>, I mentioned the formula for the rotation matrix; one merely takes the product of the rotation matrix with the coordinates (treated as 2-vectors) to get the new rotated coordinates. Note that I gave the matrix for clockwise ro... |
1,749,284 | <p>As part of my homework i've the following question:</p>
<p>The tangent line $ L $ is crossing the graph of $ y = ax^3 + bx $ at point $ x = x_0 $, find another point where the tangent-line $L$ is crossing the graph. Define $ a = 1$ and $b = 0$.</p>
<p>Second part of the question is to graph $y = x^3$ and show the ... | Martín-Blas Pérez Pinilla | 98,199 | <p>Quote from <a href="http://jtra.cz/stuff/essays/math-self-reference/index.html" rel="nofollow">Self Referential Formula in Math</a>:</p>
<blockquote>
Its graph contains all possible bitmaps that fit in region of 17 * 106 grid. So it is not much of wonder that one of those many bitmaps contains meaningful representa... |
3,207,453 | <p>studying the series <span class="math-container">$\sum_\limits{n=2}^\infty \frac{1}{n(\log n)^ {2}}$</span>.</p>
<p>I've tried with the root criterion</p>
<p><span class="math-container">$\lim_{n \to \infty} \sqrt[n]{\frac{1}{n(\log n)^ {2}}}>1$</span>
and the series should diverge.</p>
<p>But I'm not sure
Ca... | Wojowu | 127,263 | <p>Hint: you can also use the <a href="https://en.wikipedia.org/wiki/Cauchy_condensation_test" rel="nofollow noreferrer">Cauchy condensation test</a>.</p>
|
7,247 | <p>Let $f(x) \in L^p(\mathbb{R})$ and $K \in C^m(\mathbb{R})$. Can I then say that $(f \ast K) (x) = \int_{\mathbb{R}} f(t) K(x-t) dt$ is in $C^m$? </p>
<p>I know that this is true if $K$ has compact support, but I was wondering if it is possible to have a stronger result (perhaps $K$ vanishing at $\infty$?). </p>
| Amitesh Datta | 10,467 | <p>I wrote an article on analysis recently and I included the following relevant result (with proof) in the article; I hope it is helpful:</p>
<p><strong>Theorem</strong> Let $f\in L^1(\mathbb{R}^n)\cap L^p(\mathbb{R}^n)$ for some $1\leq p \leq \infty$. Also, let $g\in L^1(\mathbb{R}^n)$ be a function all of whose par... |
4,215,824 | <p>Is there a formal definition of "almost always less than" or "almost always greater than"? I think one could define it using probabilities but not sure how to go about it. If one could show the following, then I think you could say <span class="math-container">$X$</span> is almost always less tha... | tommik | 791,458 | <p>This means that</p>
<p><span class="math-container">$$\mathbb{P}[X\geq x]=0$$</span></p>
<p>Thus there can be some sets where <span class="math-container">$X\geq x$</span> but they are all sets with probability zero</p>
|
4,215,824 | <p>Is there a formal definition of "almost always less than" or "almost always greater than"? I think one could define it using probabilities but not sure how to go about it. If one could show the following, then I think you could say <span class="math-container">$X$</span> is almost always less tha... | Flitwick | 602,312 | <p>If something holds almost surely, it means it is true on a set of probability measure 1. Let <span class="math-container">$X$</span> be a r.v. on some probability space <span class="math-container">$(\Omega,\mathcal{F},\mathbb{P})$</span> and <span class="math-container">$c\in\mathbb{R}$</span>. Then the following a... |
2,354,383 | <p>Why doesn't a previous event affect the probability of (say) a coin showing tails?</p>
<p>Let's say I have a <strong>fair</strong> and <strong>unbiased</strong> coin with two sides, <em>heads</em> and <em>tails</em>.</p>
<p>For the first time I toss it up the probabilities of both events are equal to $\frac{1}{2}$... | D_S | 28,556 | <p>The ratio of heads to tails will approach $1$ as the number of times it's tossed tends towards infinity. If you've tossed the coin $1000$ times, you have hardly tossed it at all.</p>
<p>Another way to think about it: we agree that the first time you toss the coin, the probability of heads is $1/2$. Suppose you t... |
2,354,383 | <p>Why doesn't a previous event affect the probability of (say) a coin showing tails?</p>
<p>Let's say I have a <strong>fair</strong> and <strong>unbiased</strong> coin with two sides, <em>heads</em> and <em>tails</em>.</p>
<p>For the first time I toss it up the probabilities of both events are equal to $\frac{1}{2}$... | Ross Millikan | 1,827 | <p>The <em>assumption</em> for a coin is that is has no memory. That means that the chance of heads is the same on every toss. For a fair coin, that chance is $\frac 12$ <strong>regardless of the history</strong>. If you toss $100$ times and get heads every time (very unlikely, but it could happen) the most probabl... |
3,758,536 | <p>Theorem 3.29: If <span class="math-container">$p>1$</span>,<br />
<span class="math-container">$$
\sum_{n=2}^{\infty}\frac{1}{n(\log\ n)^p}
$$</span>
converges; if <span class="math-container">$p\leq1$</span>, the series diverges.</p>
<p>Proof: The monotonicity of the logarithmic function implies that <span class... | Oliver Díaz | 121,671 | <p>The function <span class="math-container">$g(x)=x\log^p(x)$</span> increases and is positive in the interval <span class="math-container">$(1,\infty)$</span>. From that, it follows that <span class="math-container">$f(x)=\frac{1}{x\log^px}$</span> decreases on <span class="math-container">$(1,\infty)$</span>.</p>
<p... |
3,758,536 | <p>Theorem 3.29: If <span class="math-container">$p>1$</span>,<br />
<span class="math-container">$$
\sum_{n=2}^{\infty}\frac{1}{n(\log\ n)^p}
$$</span>
converges; if <span class="math-container">$p\leq1$</span>, the series diverges.</p>
<p>Proof: The monotonicity of the logarithmic function implies that <span class... | Oliver Díaz | 121,671 | <ul>
<li><p>When <span class="math-container">$p=0$</span> divergence is direct for in such case you get the harmonic series <span class="math-container">$\sum_n\frac1n$</span>.</p>
</li>
<li><p>For <span class="math-container">$p<0$</span> notice that <span class="math-container">$\frac{\log^{-p}}{n}\geq \log^{-p}2... |
73,039 | <p>I have a huge data file which I can't ListPlot.</p>
<p>This code generates similar kind of data:</p>
<pre><code>datatest =RandomSample[Join[RandomReal[{0.5, 15}, 20], RandomReal[.1, 10000]]];
datatest2 = 5 + Riffle[datatest, -datatest];
</code></pre>
<p>I want to filter (delete) the part of the data that is not n... | bill s | 1,783 | <p>A common way to remove outliers is with the Median filter. What you want to do is the opposite: to keep the outliers and remove the inliers. Subtracting the data from the median, then clipping the result and selecting all those larger than a threshold is one way to proceed. </p>
<pre><code>short = Select[Chop[datat... |
73,039 | <p>I have a huge data file which I can't ListPlot.</p>
<p>This code generates similar kind of data:</p>
<pre><code>datatest =RandomSample[Join[RandomReal[{0.5, 15}, 20], RandomReal[.1, 10000]]];
datatest2 = 5 + Riffle[datatest, -datatest];
</code></pre>
<p>I want to filter (delete) the part of the data that is not n... | gpap | 1,079 | <p>I am not an expert (understatemnt of the year!) in signal processing but you can use the band function and create a sparse matrix that has as many 1s as you want around the positions of the peaks. I am not sure the following is the best way to do this but it works:</p>
<pre><code>With[{width = 200},
spArray = Spa... |
5,927 | <p>I have a problem with the binomial coefficient $\binom{5}{7}$. I know that the solution is zero, but I have problems to reproduce that:</p>
<p>${\displaystyle \binom{5}{7}=\frac{5!}{7!\times(5-7)!}=\frac{5!}{7!\times(-2)!}=\frac{120}{5040\times-2}=\frac{120}{-10080}=-\frac{1}{84}}$</p>
<p>Where is my mistake?</p>
| svenwltr | 2,280 | <p>Yes, now I see the problem.</p>
<p>First, (-2)! really isn't defined. And I can't use the factorial method if $n\notin\mathbb{N}$. So I have to go these way:</p>
<p>${\displaystyle \binom{5}{7}=\frac{5\times4\times3\times2\times1\times0\times-1}{7!}=\frac{0}{7!}=0}$</p>
<p>Thus, if $k>n$ the solution will always ... |
5,927 | <p>I have a problem with the binomial coefficient $\binom{5}{7}$. I know that the solution is zero, but I have problems to reproduce that:</p>
<p>${\displaystyle \binom{5}{7}=\frac{5!}{7!\times(5-7)!}=\frac{5!}{7!\times(-2)!}=\frac{120}{5040\times-2}=\frac{120}{-10080}=-\frac{1}{84}}$</p>
<p>Where is my mistake?</p>
| Darsh Ranjan | 2,032 | <p>$(-2)!$ is actually infinite. A more palatable way to phrase that, perhaps, is in terms of the reciprocal factorial: $1/(-2)! = 0$. We only need the recurrence relation $n! = n(n-1)!$, or in terms of reciprocal factorials: $$\frac{1}{(n-1)!} = n\cdot\frac{1}{n!}.$$ That means $\frac{1}{(-2)!} = \frac{0\cdot (-1)}{0!... |
2,356,593 | <blockquote>
<p>Quoting:" Prove: if $f$ and $g$ are continuous on $(a,b)$ and $f(x)=g(x)$ for every $x$ in a dense subset of $(a,b)$, then $f(x)=g(x)$ for all $x$ in $(a,b)$."</p>
</blockquote>
<p>Let $S \subset (a,b)$ be a dense subset such that every point $x \in (a,b)$ either belongs to S or is a limit point of S... | tattwamasi amrutam | 90,328 | <p>Let $S$ be the dense subset of $(a,b)$. Like you wrote, for any $x \in (a,b)$, there exists $x_n \in S$ such that $x_n \to x$. Then $$f(x)=f(\lim_n x_n)=\lim_nf(x_n)=\lim_ng(x_n)=g(x)$$</p>
|
2,839,554 | <p>I would like to formalise some operations I am doing, however it is unclear how I should deal with categorical variables. </p>
<p>Imagine a dataset with 15 distinct couples (<code>ID</code>). Each couple was observed 3 times (<code>time</code>). </p>
<p>Each partner has responded to two questions: <code>p</code> a... | Chinmaya mishra | 445,597 | <p>A system is linear if $A(t),B(t),C(t),D(t)$ are linear i.e $dx/dt=2x+3u$ . If these depend on time as the questions looks like , then (5) and (6) are correct.</p>
<p>If you look at (5) and (6) $A,B,C,D,x,y$ are independent .To decide to what dimension it goes to , just recall the definition of vector field .
For ex... |
2,900,014 | <p>How would solve for $a$ in this equation without using an approximation ?
is it possible?</p>
<p>where $x>0$ and $0<a<\infty$</p>
<p>$x=\Sigma _{i=1}^{n} i^a$</p>
<p>for example $120=\Sigma _{i=1}^{6} i^a$ what is $a$ in this equation?</p>
| David G. Stork | 210,401 | <p>The sum you give yields the <em><a href="https://en.wikipedia.org/wiki/Harmonic_number" rel="nofollow noreferrer">Harmonic number</a></em>, $H_n^{(-a)}$. This can be solved, in <em>Mathematica</em> for instance:</p>
<pre><code>Solve[k == HarmonicNumber[n, -a], a][[1]]
</code></pre>
<p>When you plug in $k=120$ and... |
2,588,408 | <p>A question from <em>Introduction to Analysis</em> by Arthur Mattuck:</p>
<p>Suppose $f(x)$ is continuous for all $x$ and $f(a+b)=f(a)+f(b)$ for all $a$ and $b$. Prove that $f(x)=Cx$, where $C=f(1)$, as follows:</p>
<p>(a)prove, in order, that it is true when $x=n, {1\over n}$ and $m\over n$, where $m, n$ are integ... | Angina Seng | 436,618 | <p>$f(1/2)+f(1/2)=f(1)=C$: what is $f(1/2)$?</p>
<p>$f(1/3)+f(1/3)+f(1/3)=f(2/3)+f(1/3)=f(1)=C$: what is $f(1/3)$?</p>
<p>etc.</p>
|
690,569 | <p>Suppose a function is given by:
$$
f(x)=
\begin{cases}
\cos\left(\dfrac{1}{x}\right) & x\neq 0 \\
0 & x=0
\end{cases}
$$</p>
<p>Show that this function is not continuous. Please help - I don't know how to proceed with formally using the limits.</p>
| Brian Fitzpatrick | 56,960 | <p>There are lots of ways to do this. One way is to use the following result.</p>
<p><strong>Proposition.</strong> A function $f:\mathbb R\to\mathbb R$ is continuous at $p\in\mathbb R$ if and only if for every sequence $\{t_n\}$ with $t_n\to p$ we have $f(t_n)\to f(p)$.</p>
<p>Now, let $t_n=\dfrac{1}{2n\pi}$. Then $... |
690,569 | <p>Suppose a function is given by:
$$
f(x)=
\begin{cases}
\cos\left(\dfrac{1}{x}\right) & x\neq 0 \\
0 & x=0
\end{cases}
$$</p>
<p>Show that this function is not continuous. Please help - I don't know how to proceed with formally using the limits.</p>
| kevin | 131,542 | <p>The problem is at the point $x=0$. Consider one of the two sequences which converge to $0$; $x_n=\frac{1}{2n\pi }$ or $y_n=\frac{1}{(2n+1)\pi }$ for instance, then both sequences go to zero when $n$ goes to $+\infty$ but $f(x_n)=1$ and $f(y_n)=-1$ do not go to zero... thus the function cannot be continuous at $x=0$... |
620,370 | <p>I am tackling a problem which asks:</p>
<p>Find the sum of all the multiples of 3 or 5 below 1000.</p>
<p>My reasoning is that since
Since $\left\lfloor\frac{1000}{3}\right\rfloor = 333$ and $\left\lfloor\frac{1000}{5}\right\rfloor = 200$</p>
<p>This sum can be denoted as:
\begin{equation}
\sum\limits_{n=1}^{333... | nadia-liza | 113,971 | <p>you double added all the multiples of 15</p>
|
2,672,497 | <p>$$\lim _{n\to \infty }\sum _{k=1}^n\frac{1}{n+k+\frac{k}{n^2}}$$ I unsuccessfully tried to find two different Riemann Sums converging to the same value close to the given sum so I could use the Squeeze Theorem. Is there any other way to solve this?</p>
| Paramanand Singh | 72,031 | <p>The sum under limit is not a Riemann sum, but it differs from a Riemann sum by negligible amount.</p>
<p>To setup things let's note that by definition of Riemann integral we have <span class="math-container">$$\int_{0}^{1}f(x)\,dx=\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^{n}f(t_k),\,\frac{k-1}{n}\leq t_k\leq \frac{k... |
1,018,672 | <blockquote>
<p><span class="math-container">$$\int_0^{\infty} \frac{1}{x^3-1}dx$$</span></p>
</blockquote>
<p>What I did:</p>
<p><span class="math-container">$$\lim_{\epsilon\to0}\int_0^{1-\epsilon} \frac{1}{x^3-1}dx+\lim_{\epsilon\to0}\int_{1+\epsilon}^{\infty} \frac{1}{x^3-1}dx$$</span></p>
<hr />
<p><span class="ma... | 2'5 9'2 | 11,123 | <p>This integral is not defined. You can't write $$\int_0^{\infty} \frac{1}{x^3-1}dx=\lim_{\epsilon\to0^+}\left(\int_0^{1-\epsilon} \frac{1}{x^3-1}dx+\int_{1+\epsilon}^{\infty} \frac{1}{x^3-1}dx\right)$$
(Note that although you initially write two separate limits, you combine them into one limit in a later step, so you... |
251,430 | <p>Consider the measure space $(\mathbb{Z},\mathcal{P}(\mathbb{Z}),\#)$, where $\#$ is the counting measure on $\mathbb{Z}$ and $\mathcal{P}(\mathbb{Z})$ is its power set.</p>
<p>I would like to show that for any measurable function we have $\int f(n)d\#(n)=\sum_{n}f(n)$.</p>
<p>This is what I have done: Let $x\in\ma... | smiley06 | 63,954 | <p>I think the statement holds only for positive measurable functions. If you take $ f(n) = \frac{(-1)^n}{n} $ then $ \sum_n f(n) < \infty $ but $$ \int_\mathbb{Z} f^+ d\# = \int_\mathbb{Z} f^- d\# = \infty $$ So you end up with $\infty-\infty $</p>
|
1,056,045 | <blockquote>
<p>Show that </p>
<p>$2^{3n}-1$ is divisble by $7$ for all $n$ $\in \mathbb N$</p>
</blockquote>
<p>I'm not really sure how to get started on this problem, but here is what I have done so far:</p>
<p>Base case $n(1)$:</p>
<p>$\frac{2^{3(1)}-1}{7} = \frac{8-1}{7} = \frac{7}{7}$</p>
<p>But not sur... | AJY | 192,914 | <p>$2^{3n} = 8^{n}$. Let $s_{n} = 8^{n} - 1$.</p>
<p>$$8^{n + 1} - 1 = 8(8^{n}) - 1 = 8(8^{n} - 8) + 7 = 8(8^{n} - 1) + 7 = 8s_{n} + 7$$</p>
|
1,304,971 | <p>Using a triangular facet approximation of a sphere based on <a href="http://paulbourke.net/geometry/circlesphere/" rel="nofollow">Sphere Generation by Paul Bourke</a>.</p>
<p>We take an octahedron and bisect the edges of its facets
to form 4 triangles from each triangle.</p>
<p><code>
/\ ... | augurar | 85,153 | <p>Observe that each vertex is shared by $6$ faces, except the six vertices of the original octahedron, which are always shared by only $4$ faces in each generation. Each face has $3$ vertices. The number of faces in generation $N$ is $8 \cdot 4^{N}$. Putting this together, we see that the number of vertices in that... |
3,694,661 | <p>I was stuck on a problem from <a href="https://rads.stackoverflow.com/amzn/click/com/0821804308" rel="nofollow noreferrer" rel="nofollow noreferrer" title="Quite a Fun Recreational Math Book">Mathematical Circles: Russian Experience</a>, which reads as follows:</p>
<blockquote>
<p><em>Prove that the number <span ... | Tomita | 717,427 | <p><span class="math-container">$6n^3+3=m^6\tag{1}$</span> </p>
<p><span class="math-container">$n^3\equiv {0,1,6} \pmod{7}$</span> then <span class="math-container">$6n^3+3\equiv {2,3,4} \pmod{7}$</span>.<br>
On the other hand, <span class="math-container">$m^6\equiv {0,1} \pmod{7}$</span><br>
Hence <span class="mat... |
366,415 | <p>Find all the natural numbers where $ϕ(n)=110$ (Euler's totient function)</p>
<p>What the idea behind this kind of questions?</p>
| DonAntonio | 31,254 | <p>Hints: if the prime decomposition of $\,n\,$ is</p>
<p>$$n=\prod_{i=1}^np_i^{a_i}\implies \phi(n)=n\prod_{i=1}^n\left(1-\frac{1}{p_i}\right)\implies$$</p>
<p>$$2\cdot5\cdot 11=110=\phi(n)=n\prod_{i=1}^n\left(1-\frac{1}{p_i}\right)\ldots$$</p>
|
366,415 | <p>Find all the natural numbers where $ϕ(n)=110$ (Euler's totient function)</p>
<p>What the idea behind this kind of questions?</p>
| Warren Moore | 63,412 | <p><strong>Hint.</strong> Assuming you mean Euler's totient $\phi$, if you factor $n=p_1^{e_1}\cdots p_k^{e_k}$, then</p>
<p>$$
\phi(n)=\phi(p_1^{e_1})\cdots\phi(p_k^{e_k})=p_1^{e_1-1}(p_1-1)\cdots p_k^{e_k-1}(p_k-1)
$$</p>
<p>So look for primes $p$ such that $p-1\mid 110$ and go from that.</p>
|
366,415 | <p>Find all the natural numbers where $ϕ(n)=110$ (Euler's totient function)</p>
<p>What the idea behind this kind of questions?</p>
| wendy.krieger | 78,024 | <p>The totient of a prime $p$ is $p-1$. The totient of a prime power $p^n$ is $(p-1)p^{n-1}$.</p>
<p>There can't be more different odd divisors, then the power of $2$, because each new prime brings its own 'supply' of $2$, and an odd factor in a totient comes from odd primes. </p>
<p>So the solutions is a prime of ... |
541,926 | <p>I've been wrecking my brain with this problem and I really hope you can help me. You see I have a triangle that is either an isosceles or equilateral or right and I have to find a way to:
1)Convert it to a right one by moving one of its vertices,
2)Convert it to an isosceles one by moving one of its vertices,
3)Conv... | Stefan4024 | 67,746 | <p>First of all it would be nice to post you thought on the question and what you've already done, this will give everyone a better idea of you knowledge and they can point where is your mistake.</p>
<p>Here are some hint for you questions:</p>
<p><strong>a)</strong> If you want to make a right triangle, use some pro... |
3,668,702 | <p>As the title says, how should I go about finding the shortest distance between all pairs of nodes (Each node has x and y co-odrinates associated with it) on a graph?</p>
<p>A brute force method is to run shortest path finding algorithms between all the pairs of the points. Is there a better way to approach this pro... | Eric Towers | 123,905 | <p>There are several algorithms, differing in the nature of the edge type, edge cost, and running time. Assume a graph <span class="math-container">$G$</span> with vertices, <span class="math-container">$V$</span>, and edge, <span class="math-container">$E$</span>. You do not say if your graph is directed or undirect... |
131,579 | <p>I need some help solving this problem.</p>
<p>A man is about to perform a random walk. He is standing a distance of 100 units from a wall. In his pocket, he has 10 playing cards: 5 red and 5 black.</p>
<p>He shuffles the cards and draws the top card.</p>
<p>If he draws a red card, he moves 50 units (half the di... | Arturo Magidin | 742 | <p>Note that $792=8\times 9\times 11$.</p>
<p>A number is divisible by $8$ if and only if the last three digits are divisible by $8$, so we need $45z$ to be divisible by $8$. That will give you the value of $z$.</p>
<p>A number is divisible by $9$ if and only if the sum of the digits is divisible by $9$. So you need ... |
3,480,890 | <p>I do understand pure mathematical concepts of probability space and random variables as a (measurable) functions. </p>
<p>The question is: what is the real-world meaning of probability and how can we apply the machinery of probability to the real situations?</p>
<p>Ex1: probability of heads for fair coin is 1/2. W... | kccu | 255,727 | <p>This is really more of a philosophy of math question than a math question. You might find this Wikipedia page interesting as a starting point: <a href="https://en.wikipedia.org/wiki/Probability_interpretations" rel="noreferrer">https://en.wikipedia.org/wiki/Probability_interpretations</a>. I don't think there is one... |
3,480,890 | <p>I do understand pure mathematical concepts of probability space and random variables as a (measurable) functions. </p>
<p>The question is: what is the real-world meaning of probability and how can we apply the machinery of probability to the real situations?</p>
<p>Ex1: probability of heads for fair coin is 1/2. W... | Ripi2 | 688,039 | <p>We don't know well how to do the weather forecase, just because we don't know the exact independat variables nor their full relation between them. Our mathematical models are not accurate enough.</p>
<p>But we have zillions of daily measures: wind, temperature, moist, etc.
<br>And we do this asertion:
"For the N ti... |
1,326,816 | <p>How to find the Maclaurin series of the function $$f(x)=\frac{1}{(9-x^2)^2}$$
I guess we are gonna use derivatives but i have no idea how the final answer should be formed.</p>
| user84413 | 84,413 | <p>This answer uses ideas similar to the previous answers, but in a different order:</p>
<p>$\displaystyle\frac{1}{9-x^2}=\frac{\frac{1}{9}}{1-\frac{x^2}{9}}=\frac{1}{9}\sum_{n=0}^{\infty}\left(\frac{x^2}{9}\right)^{n}=\sum_{n=0}^{\infty}\frac{1}{9^{n+1}}x^{2n},\;\;$ so differentiating gives</p>
<p>$\displaystyle\fra... |
2,485,447 | <p>My attempt:</p>
<p>3x≡1 mod 7 (1)</p>
<p>4x≡1 mod 9 (2)</p>
<p>Multiply (1) by 5</p>
<p>Multiply (2) by 7</p>
<p>x≡5 mod 7</p>
<p>x≡7 mod 9</p>
<p>So x≡9k+7</p>
<p>9k+7=5(mod7)</p>
<p>k=5(mod7)</p>
<p>k=7j+5</p>
<p>x=9(7j+5)+7</p>
<p>=63j+52</p>
<p>x≡52(mod63)</p>
| lab bhattacharjee | 33,337 | <p>Notice that we need $x+2$ divisible by $7$ and by $9$, hence by LCM$(7,9)$</p>
|
2,239,058 | <blockquote>
<p>Find the spherically symmetric solution to $$\nabla^2u=1$$ in the
region $r=|\mathbf{r}|\le a$ for $a>0$ that satisfies the following
boundary condition at $r=a$:</p>
<p>$\frac{\partial u}{\partial n}=0$</p>
</blockquote>
<p>The solution I have looked at states to begin with $\frac{\parti... | Exodd | 161,426 | <p>You can't prove it because it is false.</p>
<p>$$A = \begin{pmatrix}1 & 0\\ 0 &1\end{pmatrix}$$</p>
<p>$$B = \begin{pmatrix}-1 & 0\\ 0 &-1\end{pmatrix}$$</p>
<p>aren't congruent, since $PAP^T = PP^T$ is definite positive, so can't be $B$. </p>
<p>Another example? let $A$ be the identity matrix, a... |
1,956,855 | <p>I'm doing some math work involving proofs, and one of the definitions is:</p>
<p>|a| = -a when a < 0</p>
<p>Isn't the absolute value of a, positive a no matter what a is in the beginning? Am I looking at this wrong? Could use an explanation.</p>
| Edward Evans | 312,721 | <p>If $a < 0$ then $-a > 0$, so $-a$ is positive.</p>
<p>The point is that if $a = -2$ for instance, then $\lvert -2 \vert = -(-2) = 2.$</p>
|
1,956,855 | <p>I'm doing some math work involving proofs, and one of the definitions is:</p>
<p>|a| = -a when a < 0</p>
<p>Isn't the absolute value of a, positive a no matter what a is in the beginning? Am I looking at this wrong? Could use an explanation.</p>
| fleablood | 280,126 | <p>"Isn't the absolute value of $a$, positive $a$ no matter what $a$ is in the beginning?"</p>
<p>Yes. $-a $ is a positive number. </p>
<p>"Am I looking at this wrong?" Yes. $-a$ is a positive number.</p>
<p>"Could use an explanation?" </p>
<p>$a < 0$. So $a$ is negative. Which means $-a > 0$ and $-a$ is... |
16,797 | <p>Is there any good way to approximate following integral?<br>
$$\int_0^{0.5}\frac{x^2}{\sqrt{2\pi}\sigma}\cdot \exp\left(-\frac{(x^2-\mu)^2}{2\sigma^2}\right)\mathrm dx$$<br>
$\mu$ is between $0$ and $0.25$, the problem is in $\sigma$ which is always positive, but it can be arbitrarily small.<br>
I was trying to expa... | Mose Wintner | 5,523 | <p>How about some good old-fashioned trapezoid rule?</p>
|
2,796,618 | <p>I am trying to</p>
<p>i) determine the infimum</p>
<p>ii) show that there's a function for which $\int_{0}^{1} {f'(x)}^2 dx$ is the infimum</p>
<p>iii) show if such function is unique.</p>
<p>I tried out several functions that suit the given condition, but couldn't see how $\int_{0}^{1} {f'(x)}^2 dx$ changes as ... | Yiorgos S. Smyrlis | 57,021 | <p><strong>Claim.</strong> <em>Let $\,\mathscr X=\{\,g\in C^1[0,1]: g(0)=0\,\,\& \,\,g(1)=1.\}$. Then then functional $\,\varPhi(\,g)=\int_0^1 \big(\,g'(x)\big)^2\,dx$, attains a global minimum at $\,f(x)=x$, i.e.
$$
\min_{g\in\mathscr X}\varPhi(g)=\varPhi(\,f)=1.
$$</em></p>
<p><em>Proof.</em> If $g\in\mathscr X$... |
2,687,932 | <p>Let $x\in\mathbb{R}$. Prove that $x=-1$ if and only if $x^3+x^2+x+1=0$.
This is a bi-conditional statement, thus to prove it we need to prove:
<a href="https://i.stack.imgur.com/PRPSC.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/PRPSC.png" alt="enter image description here"></a></p>
| StackTD | 159,845 | <p>You did the hard(er) part.</p>
<p>To prove the other implication, you simply need to show that $x=-1$ satisfies the cubic equation.</p>
<p>Substitute and simplify!</p>
|
60,081 | <p>I have a stochastic matrix $A \in R^{n \times n}$ whose sum of the entries in each row is $1$. When I found out the eigenvalues and eigenvectors for this stochastic matrix, it always happens that one of the eigenvalues is $1$. </p>
<p>Is it true that for any square <a href="https://en.wikipedia.org/wiki/Stochastic_... | leonbloy | 312 | <p>That is a <a href="http://en.wikipedia.org/wiki/Stochastic_matrix#Definition_and_properties" rel="nofollow noreferrer">basic</a> and important property of stochastic matrices. It's also non-obvious, unless you are aware of the <a href="http://en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem" rel="nofollow nore... |
1,760,148 | <p>If I have a connected metric space $X$, is any ball around a point $x\in X$ also connected?</p>
| Brian M. Scott | 12,042 | <p>No. The <a href="https://en.wikipedia.org/wiki/Knaster%E2%80%93Kuratowski_fan">Knaster-Kuratowski fan</a> is a connected subspace of the plane that becomes totally disconnected when a certain point is removed, so open balls centred at the other points cannot be connected if they are small enough to exclude the explo... |
2,987,071 | <blockquote>
<p>I have to show that the set <span class="math-container">$$\{1, 1 + X, (1 + X)^2 , . . . , (1 + X)^n \}$$</span> is a basis
for <span class="math-container">$\Bbb{R}_n [X]$</span>, where <span class="math-container">$\Bbb{R}_n [X]$</span> denotes the vectorspace of all polynomials of degree less tha... | lhf | 589 | <p>The map <span class="math-container">$p(X) \mapsto p(X+1)$</span> is a bijective linear transformation <span class="math-container">$\Bbb{R}_n [X] \to \Bbb{R}_n [X]$</span>.</p>
<p>The set in question is the image of the canonical basis of <span class="math-container">$\Bbb{R}_n [X]$</span> and so is a basis of <sp... |
483,131 | <p>How can you prove that if $t$ is less than or equal to $1$, that the probability of the sum of a sequence of uniform random variables being less than or equal to $t$ equals $t^k/k!$ ?</p>
<p>In other words:</p>
<p>Prove if $t \leq 1$, $$P(U_1+U_2 + \dots +U_k \leq t)=\frac{t^k}{k!}$$</p>
<p>My thought is that int... | user2566092 | 87,313 | <p>If you are familiar with the Chinese remainder theorem, it gives a one line proof of this, because it implies that the unique solution to $(a-b) = 0 \hbox { mod } p_1 p_2 \cdots p_n$ for $(a-b)$ is the solution to the system $(a-b) = 0 \hbox { mod } p_i$ for each $i$, since the $p_i$ are distinct.</p>
|
1,918,674 | <p>For what value $k$ is the following function continuous at $x=2$?
$$f(x) = \begin{cases}
\frac{\sqrt{2x+5}-\sqrt{x+7}}{x-2} & x \neq 2 \\
k & x = 2
\end{cases}$$</p>
<p>I was thinking about multiplying the numerator by it's conjugate, but that makes the denominator very messy, so I don't rlly know what to d... | Claude Leibovici | 82,404 | <p>Beside the simple solutions already given,you could change variable $x=y+2$ which makes
$$\frac{\sqrt{2x+5}-\sqrt{x+7}}{x-2}=\frac{\sqrt{2 y+9}-\sqrt{y+9}}{y}$$ Now, using the generalized binomial theorem or Taylor series, you should find $$\frac{\sqrt{2 y+9}-\sqrt{y+9}}{y}=\frac{1}{6}-\frac{y}{72}+O\left(y^2\right)... |
1,319,761 | <p>I'm stuck trying to figure out how to solve the following integral:</p>
<p>$\int_{C(0,1)^+}\sin(z)dz$</p>
<p>I've tried parameterizing z(t) but then I get</p>
<p>$\int_0^{2\pi}\sin(e^{it})ie^{it}$ which I don't know how to integrate.</p>
<p>So then I'm looking to use Cauchy's Integral formula but I'm not sure if... | Timbuc | 118,527 | <p><strong>First way:</strong></p>
<p>Parametrize the unit circle as $\;\gamma(t):=e^{it}\;,\;\;0\le t<2\pi\;\implies \gamma'(t)=ie^{it}dt $</p>
<p>so that your integral becomes</p>
<p>$$i\int_0^{2\pi}e^{it}\sin e^{it}dt=\left.-i\cos e^{it}\right|_0^{2\pi}=0$$</p>
<p><strong>Second way:</strong></p>
<p>The func... |
692,376 | <p>Can a vector space over an infinite field be a finite union of proper subspaces ?</p>
| George Turcas | 119,544 | <p>Denote by $K$ the field over which $V$ is a vector space.
Suppose the answer is true and consider write $V=\bigcup\limits_{i=1}^{n} V_i$, where $V_i$ are proper subspaces of $V$ and $n \in \mathbb{N}$ is minimal.</p>
<p>Because $n$ is minimal, there exists an element $v_n \in V_n \setminus \bigcup\limits_{i=1}^{n-1... |
441,448 | <p><strong>Contextual Problem</strong></p>
<p>A PhD student in Applied Mathematics is defending his dissertation and needs to make 10 gallon keg consisting of vodka and beer to placate his thesis committee. Suppose that all committee members, being stubborn people, refuse to sign his dissertation paperwork until the ... | awkward | 76,172 | <p>You don't really need eigenvectors to solve this problem. Just treat it as a linear programming problem and let the Simplex Algorithm (or your algorithm of choice) come up with a feasible solution. Since you need an objective function, you can choose one arbitrarily; for example, you might choose to minimize the t... |
2,882,678 | <p>Why are the morphisms of the category of sets functions?
Shouldn't the morphisms take an object in a category and turn it into another object of the category, i.e. map Set to Set. I don't understand how e.g.
$f(x)=x^2$
is a map from a set to a set. </p>
<p>If the morphism had been the image of a set under a functio... | BallBoy | 512,865 | <blockquote>
<p>I don't understand how e.g. $f(x)=x^2$ is a map from a set to a set.</p>
</blockquote>
<p>In fact it is a map from a set to a set: you haven't given a precise definition of the function, so there are multiple options what the sets are, but $f$ may, e.g., map from the domain $\mathbb R$ (the set of al... |
1,021,631 | <p>Does anybody know how to solve the equation</p>
<p>$\mathbf{a} + \mathbf{b} \times \hat{\mathbf{v}} = c \hat{\mathbf{v}},$</p>
<p>where $\mathbf{a}$ and $\mathbf{b}$ are given real vectors, for the unit vector $\hat{\mathbf{v}}$ and the real number $c$?</p>
| Janak | 184,121 | <p>Here is an illustration of how to use Cauchy Scwarz inequality in this case:</p>
<p>You want to show: $\sum(X_i-\bar X)(Y_i-\bar Y) \leq\sqrt{\sum(X_i-\bar X)^2\sum(Y_i -\bar Y)^2}$ </p>
<p>CS inequality gives us:
$$\sum a_ib_i \leq\sqrt{\sum a_i^2\sum b_i^2},$$ here equality occurs if $a_i=c_1+c_2b_i$ $\forall... |
1,045,941 | <p>Usually this is just given as a straight up definition in a calculus course. I am wondering how you prove it?
I tried using the limit definition, $$\lim\limits_{h\rightarrow 0} \dfrac{\log(x+h)-\log(x)}{h}$$
but this led to no developments.</p>
| Steven Gubkin | 34,287 | <p>This depends on how you define $\log(x)$. As you say, this could be taken to be the definition of $\log(x)$: define $y=\log(x)$ as the unique solution to the first order diff EQ $y'=\frac{1}{x}$ satisfying $y(1)=0$.</p>
<p>You might define $e^x$ as a solution to a diff EQ, and $\log$ as its inverse. In this case... |
2,025,007 | <p>One can show that if $n \geq 3$ is a positive integer, $d=n^2-4$, and $\varepsilon = 1$ if $n$ is odd and $\varepsilon = 0$ if $n$ is even, then the continued fraction expansion of $\frac{\sqrt{d}+\varepsilon}{2}$ has period of even length of the form $(1,n-2)$. One can show that such a continued fraction has perio... | Will Jagy | 10,400 | <p>We need consider only $d = n^2 - 4$ when $n = 12 w + 3.$ That is, if $n \neq 0 \pmod 3,$ then one of $(n+2)(n-2)$ is divisible by $3.$ Next, if $n \equiv 1 \pmod 4,$ then both of $(n+2),(n-2) \equiv 3 \pmod 4.$</p>
<p>The cycle of Gauss reduced forms equivalent to $x^2 - d y^2$ is of length 6, as you have found. T... |
1,329,214 | <p>I'm having difficulty solving a linear algebra problem:<br>
Let $A,B,C,D$ be real $n \times n$ matrices. Show that there is a non-zero $n \times n$ matrix $X$ such that $AXB$ and $CXD$ are both symmetric. </p>
<p>There is an accompanying hint:<br>
Show that the set of all matrices $X$ for which $AXB$ is symmetric... | Meni Rosenfeld | 153,429 | <p>Let $T$ map every matrix $X$ to $AXB$.</p>
<ol>
<li><p>Show that $T$ is a linear transformation.</p></li>
<li><p>What is the preimage of the space of symmetric matrices under $T$?</p></li>
<li><p>What is the dimension of the space of symmetric matrices?</p></li>
<li><p>What can you deduce from 1 and 2 about the dim... |
989,740 | <p>How do I prove the following statement?</p>
<blockquote>
<p>If $x^2$ is irrational, then $x$ is irrational. The number $y = π^2$ is irrational. Therefore, the number $x = π$ is irrational</p>
</blockquote>
| André Nicolas | 6,312 | <p>To prove the first assertion, we can use a proof by contradiction.</p>
<p>Suppose to the contrary that $x$ is rational:
$$\exists (a, b) : b\ne 0\land x=\frac{a}{b}$$
$$\implies x^2=\frac{a^2}{b^2}$$
$$\therefore x^2 \text{ is rational. }$$
This contradicts the given fact that $x^2$ is irrational.</p>
<p><strong>A... |
4,204,282 | <p>How do I calculate <span class="math-container">$$\int_{-\infty}^{\infty} \frac{dw}{1+iw^3}$$</span> using complex path integrals?</p>
<p>I just need a hint on how to start, not the actual computation, because I need to understand how to deal with similar questions.</p>
<p><strong>Edit:</strong> Following @Tavish's ... | Svyatoslav | 869,237 | <p>I would recommend to implement some transformations of the integral first.</p>
<p>For example,
<span class="math-container">$$I=\int_{-\infty}^\infty\frac{dw}{1+iw^3}=-i\int_{-\infty}^\infty\frac{dw}{w^3-i}=-i\int_{0}^\infty\frac{dw}{1+iw^3}-i\int_{-\infty}^0\frac{dw}{1+iw^3}$$</span>
Making change in the second int... |
1,517,086 | <p>I spent a long time trying to find a natural deduction derivation for the formula $\exists x(\exists y A(y) \rightarrow A(x))$, but I always got stuck at some point with free variables in the leaves. Could someone please help me or give me some hints to find a proof. </p>
<p>Thanks.</p>
| BrianO | 277,043 | <p>You can derive it this way:</p>
<ol>
<li>$\exists y\,A(y) \qquad\qquad\textsf{assumption}$</li>
<li>$A(a) \qquad\quad\qquad\textsf{$\exists$ new parameter introduction}\text{ ($a$)}$</li>
<li>$\exists y\,A(y) \to A(a) \quad\quad\textsf{discharge 1.}$</li>
<li>$\exists x\,(\exists y\,A(y) \to A(x)) \quad\textsf{$\ex... |
1,260,945 | <p>$\textbf{My understanding of divergence:}$ Consider any vector field $\textbf{u}$, then $\operatorname{div}(u) = \nabla \cdot u$. More conceptually, if I place an arbitrarily small sphere around any point of the vector field $\textbf{u}$, divergence measures the amount of "particles" exiting the sphere, i.e. positi... | Elaqqad | 204,937 | <p>Let:
$$A=\{x^2|x\in \Bbb Z_p\},\ \ \ \ \ \ \ B=\{-(1+y^2)|y\in \Bbb Z_p\}$$</p>
<p>it is known that $$|A|=|B|=\frac{p+1}{2}$$ (maybe you can try to prove this ), if $A\cap B=\varnothing$ then $|A\cup B|=|A|+|B|=p+1>|\Bbb Z_p |$ which is impossible. as a conclusion $A\cap B$ is not empty and you're done.</p>
|
2,400,900 | <p>Let $f(x) : \mathbb{R}^n \rightarrow \mathbb{R}$ be a convex and differentiable function, and let $P$ be a point in $\mathbb{R}^n$. </p>
<p>Define a function $g(m): R \rightarrow R$ to be the distance between point $P$ and the sub-level set $ K_m = \{ x \in \mathbb{R}^n \mid f(x) \le m\}$, i.e., $g(m) = d(P, K_m)$... | orangeskid | 168,051 | <p>HINT: the function $g$ is convex. </p>
<p>Indeed, we have </p>
<p>$$\lambda_1 K_{m_1} + \lambda_2 K_{m_2} \subset K_{\lambda_1 m_1 + \lambda_2 m_2}$$ since $f$ is convex, and</p>
<p>$$d(P, \lambda_1 K_{m_1} + \lambda_2 K_{m_2}) \le \lambda_1 d( P, K_{m_1}) + \lambda_2 d(P, K_{m_2} ) $$ since $Q\mapsto d(P, Q)$ is... |
2,400,900 | <p>Let $f(x) : \mathbb{R}^n \rightarrow \mathbb{R}$ be a convex and differentiable function, and let $P$ be a point in $\mathbb{R}^n$. </p>
<p>Define a function $g(m): R \rightarrow R$ to be the distance between point $P$ and the sub-level set $ K_m = \{ x \in \mathbb{R}^n \mid f(x) \le m\}$, i.e., $g(m) = d(P, K_m)$... | haydn_c | 472,778 | <p>Another way to look at this is note that $ h(x, \lambda, m) = d(x, P) + \lambda (f(x) - m)$ is an affine function of $m$ and a convex function for $x$, for each given $\lambda$. </p>
<p>Hence $h_1(x, \lambda) = max_{\lambda \ge 0} h(x, \lambda, m)$ is a convex function of $m$ and $x$ since point-wise supremum prese... |
4,417,325 | <p>When I say "divisibility trick" I mean "a recursive algorithm designed to show that, after multiple iterations, if the final output is a multiple of the desired number, then the original was also a multiple of the same number." Here's an example for a divisibility trick for 17.</p>
<blockquote>
<... | mjqxxxx | 5,546 | <p>I wouldn't say that it's generally true that these linear reductions preserve divisibility; specific choices are being made for that to work out. For the first relation, where <span class="math-container">$n=10q+r$</span>:
<span class="math-container">$$
q-5r = q-5(n-10q) = 51q-5n\equiv-5n\;\text{(mod 17)}.
$$</spa... |
4,417,325 | <p>When I say "divisibility trick" I mean "a recursive algorithm designed to show that, after multiple iterations, if the final output is a multiple of the desired number, then the original was also a multiple of the same number." Here's an example for a divisibility trick for 17.</p>
<blockquote>
<... | Bill Dubuque | 242 | <p>No, divisibility tests are <em>not</em> restricted to <em>linear</em> forms. As explained <a href="https://math.stackexchange.com/a/16015/242">here</a> & <a href="https://math.stackexchange.com/a/2989299/242">here</a> the rule for casting out nines: <span class="math-container">$\,9\mid 10a+b\!\iff\! 9\mid a+b\,... |
2,375,023 | <p>So I have to find an interval (in the real numbers) such that it contains all roots of the following function:
$$f(x)=x^5+x^4+x^3+x^2+1$$</p>
<p>I've tried to work with the derivatives of the function but it doesn't give any information about the interval, only how many possible roots the function might have.</p>
| Χpẘ | 309,642 | <p>HINT: Determine how many real roots there are. If there is only one root, how big is the interval?</p>
|
2,375,023 | <p>So I have to find an interval (in the real numbers) such that it contains all roots of the following function:
$$f(x)=x^5+x^4+x^3+x^2+1$$</p>
<p>I've tried to work with the derivatives of the function but it doesn't give any information about the interval, only how many possible roots the function might have.</p>
| David K | 139,123 | <p>Note that
$$ f(x) = x^5+x^4+x^3+x^2+1 = (x + 1)(x^4 + x^2) + 1. $$</p>
<p>Now try answering some question about $f(x)$ for numbers that are relatively easy to work with:</p>
<p>Can $f(x)$ be zero if $x > 0$?
Can $f(x)$ be zero if $x < 0$?</p>
<p>Can $f(x)$ be zero if $x > -1$?
Can $f(x)$ be zero if $x &... |
1,109,918 | <p>Is it always possible to add terms into limits, like in the following example? (Or must certain conditions be fulfilled first, such as for example the numerator by itself must converge etc)</p>
<p>$\lim_{h \to 0} {f(x)} = \lim_{h \to 0} \frac{e^xf(x)}{e^x}$</p>
| abel | 9,252 | <p>i will use the fundamental theorem of calculus and integration by parts to derive
$\lim_{h \to 0} \frac{f(x + h) + f(x - h) - 2f(x)}{h^2}= f''(x)$</p>
<p>the left hand side limit may exist even for a function that is twice differentiable. take for example $f(x) = |x|, x = 0.$ the left hand side is $2$ and right ha... |
1,929,977 | <p>Let $f\colon\mathbb R\to\mathbb R$ satisfy the Lipschitz condition: there exists $K\geq 0$ such that for all $x,y\in\mathbb R$, we have $|f(x)-f(y)|\leq K\cdot |x-y|$. Is it true that $f$ has one-sided derivatives everywhere? I.e., that the limits
$$\lim_{h\nearrow 0}\frac{f(x+h)-f(x)}{h}\quad\text{and}\quad\lim_{h\... | Community | -1 | <p>No. Counterexample: </p>
<p>Let $g : [1,2] \to \mathbb R$ be the graph which linearly connects the points
$$(1, 1), (1.5, -1.5), (2, 2).$$</p>
<p>Define $f [0,1] \to \mathbb R$ be </p>
<p>$$f(x) = \begin{cases} 0 & \text{if } x=0,\\ \frac{1}{2^{n+1}}\ g(2^{n+1} x) &\text{if }x\in [2^{-(n+1)}, 2^{-n}), n... |
1,015,264 | <p>This is a worked out example in my book, but I am having a little trouble understanding it:</p>
<p>Consider the system of equations:</p>
<p>$$x'=y+x(1-x^2-y^2)$$
$$y'=-x+y(1-x^2-y^2)$$</p>
<p>The orbits and limit sets of this example can be easily determined by using polar coordinates. (My question: what is the m... | Mark Fischler | 150,362 | <p>When you see an $x^2 + y^2$ in a problem that is probably going to be tractable, one good first thing to try is to see how the problem looks transformed to polar coordinates. (Even in a problem that comes about in real life, where you don't know the solution will be possible to obtain, this is a good first shot.)</... |
2,077,275 | <p>Let $U$ be the set $U$ of quaternions of unit length. I know that $U\times S^1$ is compact, connected and is a $2n$ manifold in a $2n+1$ dimensional vector space $V$.</p>
<blockquote>
<p>How can I construct a differentiable tangent vector field on $U\times S^1$ that has no zeroes?</p>
</blockquote>
<p>What I kno... | Jean Marie | 305,862 | <p>We assume that the initial system has a non-zero determinant (see remark below for a discussion).</p>
<p>Let us consider the following system:</p>
<p>$$\tag{1}\left\{
\begin{aligned}
x e^{ia} + y e^{2ia} + ze^{3ia}&=e^{4ia}\\
x e^{ib} + y e^{2ib} + ze^{3ib}&=e^{4ib}\\
x e^{ic} + y e^{2ic} + ze^{3ic}&=e... |
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