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 |
|---|---|---|---|---|
3,111,906 | <p>I have to show that for any <span class="math-container">$b >1$</span>, we have
<span class="math-container">$$ b^n > n$$</span>
for all <span class="math-container">$n$</span> sufficiently large, using only very basic analysis (no calculus). My attempt is as follows.</p>
<hr>
<p>We know that <span class="m... | YiFan | 496,634 | <p>We can proceed by induction. Suppose <span class="math-container">$b^x>x$</span> is true for all <span class="math-container">$x= n$</span>. We want to show the claim for <span class="math-container">$x=n+1$</span>. We just need
<span class="math-container">$$b^n(b-1)=b^{n+1}-b^n>1,$$</span>
Because by induct... |
3,111,906 | <p>I have to show that for any <span class="math-container">$b >1$</span>, we have
<span class="math-container">$$ b^n > n$$</span>
for all <span class="math-container">$n$</span> sufficiently large, using only very basic analysis (no calculus). My attempt is as follows.</p>
<hr>
<p>We know that <span class="m... | Calum Gilhooley | 213,690 | <p>(Umberto P.'s argument from first principles is surely the best way to go, but here's another argument, just for the sake of variety.)</p>
<p>Consider the sequence <span class="math-container">$a_n = \frac{b^n}{n}$</span>. We have:
<span class="math-container">$$\frac{a_{n+1}}{a_n} = \frac{b}{1 + \frac{1}{n}} \geqs... |
23,942 | <p>I have tried to resolve the problem of the following link <a href="https://mathematica.stackexchange.com/questions/23931/how-can-i-solve-precision-problem">How can I solve precision problem</a></p>
<p>I can tell the problem described in that link shortly here, It's no mater how many precision is there after decimal... | Spawn1701D | 255 | <p>Use the <code>$PrePrint</code> global parameter:</p>
<pre><code>$PrePrint = If[MatchQ[#, _?NumericQ], NumberForm[#, {4, 2}], #] &;
</code></pre>
<p>Note: if you dont want the way rationals will be represented after setting the global variable (e.g. $3.00/4.00$) then use this instead </p>
<pre><code>$PrePrint ... |
4,228,535 | <p>First the geometric inversion map <span class="math-container">$f:\mathbb{R}^n\setminus \{0\}\rightarrow \mathbb{R}^n$</span> is defined by
<span class="math-container">$$f(x)=\frac{x}{|x|^2}=:X.$$</span>
The one of its properties is following:</p>
<p>For any <span class="math-container">$c\in\mathbb{R}^n, c\not=0$<... | Kavi Rama Murthy | 142,385 | <p>You are making things too commplicated. <span class="math-container">$\psi$</span> is a bounded function and if <span class="math-container">$|\psi| \leq M$</span> we get <span class="math-container">$|\int_{-1/k}^{1/k} \psi (x)dx|\leq \frac M {2k} \to 0$</span>.</p>
|
4,557,576 | <p>Working on a 3U CubeSat as part of a project for a Space Engineering club. To calculate the maximum solar disturbance force, we are trying to calculate the largest shadow a 0.1 * 0.1 * 0.3 rectangular prism can cast.</p>
<p>If the satellite was oriented with the largest side facing the sun directly, the shadow cast ... | Intelligenti pauca | 255,730 | <p>Even if a satisfying answer has been found by the asker, I want to show how the answer can be reached without matrices nor calculus.</p>
<p>If we take a plane surface with area <span class="math-container">$A$</span> and unit normal <span class="math-container">$\vec n$</span>, then the projection of that area along... |
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>Another improved version</p>
<pre><code>TestConstantValuedExpression[expression_,zerovaluetest_]:=Module[{randomvaluestestresult,expressionrandomvalue,previousexpressionrandomvalue,symbolreplacementlist,extremesdifferencetestvalue},
randomvaluestestresult=True;
previousexpressionrandomvalue=False;
Quiet[
Do[
... |
911,584 | <p><em>Disclaimer: This thread is a record of thoughts.</em></p>
<p><strong>Discussion</strong>
Given a compact set.</p>
<blockquote>
<p>Do mere neighborhood covers admit finite subcovers?
$$C\subseteq\bigcup_{i\in I}N_i\implies C\subseteq N_1\cup\ldots N_n$$
<em>(The idea is that neighborhoods are in some sens... | Rebecca J. Stones | 91,818 | <p>This is essentially a counting problem: in the $6^5=7776$ possible outcomes, how many satisfy the condition "the number of rolls resulting in 1 or 2 is greater than the number of rolls resulting in 6"? The probability is this number divided by $6^5$.</p>
<p>We can find this number by filling five initially empty c... |
101,974 | <p>We are familiar with <strong><em>Hurwitz’s theorem</em></strong> which implies there is only the Fibonacci 2-Square, Euler 4-Square, Degen 8-Square, and no more. However, if we relax conditions and allow for <em>rational expressions</em>, then <strong><em>Pfister's theorem</em></strong> states that similar identitie... | Robert Israel | 8,508 | <p>Here's Maple code for getting a 16-square version using the method Conrad outlines. The result is too big to show here (the equation eq has length 4808351).</p>
<blockquote>
<p>C2:= < < c[1] | c[2] >,<-c[2] | c[1]> >:</p>
<p>c2:= c[1]^2 + c[2]^2:</p>
<p>C2b:= subs(seq(c[i]=c[i+2],i=1..2),C2):<... |
3,331,865 | <p>Find the Orthonormal basis of vector space <span class="math-container">$V$</span> of the linear polynomials of the form <span class="math-container">$ax+b$</span> such that <span class="math-container">$\:$</span>, <span class="math-container">$p:[0,1] \to \mathbb{R}$</span>. with inner product</p>
<p><span clas... | Federico Fallucca | 531,470 | <p>An orthonormal base can be </p>
<p><span class="math-container">$\{1,\frac{(x-\frac{1}{2})}{\sqrt{\langle x-\frac{1}{2},x-\frac{1}{2}\rangle}}\}$</span></p>
<p>The idea is simple. The space has dimension <span class="math-container">$2$</span>, so you must choose two polynomials of order at most <span class="math-... |
129,709 | <p>Say we have a continuous function $u(x,y) : \mathbb{R}^2 \rightarrow \mathbb{R}$.</p>
<p>I have seen several textbooks that make the following assertion:</p>
<blockquote>
<p>The length/area element of the zero level set of $u$ is given by
$\lvert\nabla H\left(u\right)\rvert = \delta(u)\lvert\lvert\nabla u\rve... | Christian Blatter | 1,303 | <p>Assume that $\phi:\ (x,y)\mapsto\phi(x,y)$ is a smooth real-valued function, and assume that the set $\Omega\subset{\mathbb R}^2$ defined by
$$\Omega:=\bigl\{(x,y)\ \bigm|\ \phi(x,y)>0\bigr\}$$
is bounded. Then the area $A(\Omega)$ is obviously given by
$$A(\Omega)=\int 1_\Omega(x,y)\ {\rm d}(x,y)=\int H\bigl(\p... |
619,526 | <p>I'm studying up for my algebra exam, and I'm not exactly sure how to solve a problem like the following</p>
<blockquote>
<p>Let $f = X^2 + 1 \in \mathbb{F}_5[X]$, $R = \mathbb{F}_5[X]/\langle f \rangle$ and $\alpha = X + \langle f \rangle \in R$. Show that $\alpha \in R^*$ and that $\vert \alpha \vert = 4$ in $R^... | drhab | 75,923 | <p>$\alpha^{2}=-1_R$ in $R$ so $\alpha^{4}=\left(-1_R\right)^{2}=1_R$
showing that the order of $\alpha$ is $4$ and that $\alpha$
is invertible.</p>
<p>Explaining:</p>
<p>$\alpha^{2}=\left(X+\left(f\right)\right)^{2}=X^{2}+\left(f\right)=-1+\left(f\right)$
since $f\mid X^{2}-\left(-1\right)=X^{2}+1$</p>
|
546,239 | <p>Myself and my Math teacher are at a disagreement in to what the proper method of solving the question <em>In how many ways can four students be chosen from a group of 12 students?</em> is.</p>
<p>The question comes straight out of a Math revision sheet from a Math book distributed under the national curriculum. The... | Ross Millikan | 1,827 | <p>To justify the $^nC_r$ formula, you can line up all the $n$ boys in $n!$ ways. Then you take the first $r$ for your selection. You can scramble the first $r$ and the last $n-r$ in any way and get the same selection, so you divide by $r!$ and by $(n-r)!$ to get the formula.</p>
|
1,039,487 | <p>Suppose $a$, $b$ are real numbers such that $a+b=12$ and both roots of the equation $x^2+ax+b=0$ are integers. </p>
<p>Determine all possible values of $a$. </p>
<p>I don't know how to go about doing this without long, messy casework. I tries $(x-s)(x-r)=x^2+ax+b$ and got $-r-s=a$ and $rs=b$, but was unable to fi... | André Nicolas | 6,312 | <p>Your method, carried a little further, works. We have $-(r+s)+rs=12$. Rewrite as
$$(r-1)(s-1)=13.$$</p>
<p>There are now not many possibilities, since $13$ is prime. Don't forget about negative numbers.</p>
|
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 $X \subset ... | C.F.G | 272,127 | <p>Here are a list of books where covers covering spaces:</p>
<ul>
<li><p><em>Munkres, James R.</em>, Topology., Upper Saddle River, NJ: Prentice Hall. xvi, 537 p. (2000). <a href="https://zbmath.org/?q=an:0951.54001" rel="nofollow noreferrer">ZBL0951.54001</a>.</p>
</li>
<li><p><em>Lee, John M.</em>, <a href="http://d... |
4,356,938 | <p>I'm reading through <a href="https://press.princeton.edu/books/hardcover/9780691151199/elliptic-tales" rel="nofollow noreferrer">Elliptic Tales</a>.</p>
<p>Addition of 2 points on an elliptic curve is described as follows:</p>
<p><a href="https://i.stack.imgur.com/Hifi9.png" rel="nofollow noreferrer"><img src="https... | Lubin | 17,760 | <p>For complete generality, let’s write the equation of any line in the Cartesian plane thus: <span class="math-container">$ax+by+c=0$</span>, with <span class="math-container">$a$</span> and <span class="math-container">$b$</span> not both zero. Now, remember that points <span class="math-container">$(x,y)$</span> in ... |
2,438,236 | <p>Suppose a finite group $G$ has order smaller than the dimension of a vector space $V$ over any field, how to prove that the representation of $G$ on $V$ is reducible？</p>
| reuns | 276,986 | <p>$\rho : G \to \text{End}(V)$ is a representation, $V$ is a $K$-vector space. Take a non-zero vector $v \in V$ and let $$W = \text{span}(\rho(g_1)v,\rho(g_2)v,\ldots) = \{ \sum_{g \in G} a_g\,\rho(g) v, a_g \in K\}$$</p>
<p>Then $$\rho(h) (\sum_{g \in G} a_g\,\rho(g) v) = \sum_{g \in G} a_g\,\rho(hg) v=\sum_{g \in G... |
1,232,023 | <p>For me, intuitively, integral $2\pi y~dx$ make more sense. I know intuition can not be proof, but by far, most part of math I've learned does match with my intuition. So, I think this one should 'make sense' as well. Probably I didn't understand the way surface area is measured. It will be great if any one could tel... | Mathemagical | 446,771 | <p>OP, if I'm guessing right, the intuition point that you are puzzling over is "how come we can get away with integrating $\pi y^2 dx$ for volume (there too, we are approximating an infinitesimal frustrum by an infinitesimal cylinder) but we are not able to approximate the surface area of the infinitesimal frustrum $2... |
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 ... | Travis Willse | 155,629 | <p>By Sylow's Third Theorem the number <span class="math-container">$n_{11}$</span> of Sylow <span class="math-container">$11$</span>-subgroups of a group <span class="math-container">$G$</span> of order <span class="math-container">$55$</span> divides <span class="math-container">$5$</span> and is congruent to <span c... |
199,199 | <p>Suppose a box contains 5 white balls and 5 black balls.</p>
<p>If you want to extract a ball and then another:</p>
<p>What is the probability of getting a black ball and then a black one?</p>
<p>I think that this is the answer:</p>
<p>Let $A:$ get a black ball in the first extraction, $B:$ get a black ball in th... | osa | 25,167 | <p><strong>Why would you even split it into two events?!</strong></p>
<p>You are just picking two balls. There are <code>C(10,2)=45</code> ways to pick two balls out of 10, but only <code>C(5,2)=10</code> give you two black ones. So <code>10/45 = 2/9</code> is your answer.</p>
|
2,736,295 | <p>Let $f(x,y) = xy^2$ and the domain $D = \lbrace (x,y)| x,y\geq0, x^2 + y^2 \leq 3 \rbrace$</p>
<p>$f_x(x,y) = y^2$ and $f_y(x,y) = 2xy$ </p>
<p>Therefore, the critical points should be $\lbrace (x,y)| y = 0, \sqrt{3} \geq x \geq 0 \rbrace$. </p>
<p>The determinant of the Hessian is
$$\det(Hf(x,y))=
\begin{vmatri... | robjohn | 13,854 | <p>On the circle of radius $r$, we have $x^2+y^2=r^2$. Therefore,
$$
xy^2=r^2x-x^3\tag1
$$
This implies that the interior critical points are at
$$
(x,y)=\frac r{\sqrt3}\left(\pm1,\pm\sqrt2\right)\tag2
$$
with the corresponding values of
$$
xy^2=\pm r^3\frac2{3\sqrt3}\tag3
$$
At the endponts of $x=\pm r$, we get the va... |
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(... | Tim Campion | 2,362 | <p>Let me summarize the information in the comments in a CW post. Feel free to edit.</p>
<ul>
<li><p>For "weaker" notions of curvature, negative curvature seems to not imply that a manifold is a $K(G,1)$. As Deane Yang pointed out, Lohkamp <a href="http://dx.doi.org/10.2307/2118620" rel="nofollow noreferrer">showed</a... |
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>
| azimut | 61,691 | <p>You could have a look at the relatively new book</p>
<p>Gary Gordon, Jennifer McNulty: <em>Matroids. A Geometric Introduction</em>, Cambridge University Press 2012.</p>
<p>From the description: "This book provides the first comprehensive introduction to the field which will appeal to undergraduate students and to ... |
2,682,531 | <p>The definition of a convex set is the following:</p>
<blockquote>
<p>A set $\Omega \subset \mathbb R^n$ is convex if $\alpha x + (1 − \alpha) y \in \Omega, \forall x, y \in \Omega$ and $\forall \alpha \in [0, 1]$.</p>
</blockquote>
<p>With this it should be easy enough to prove that a set is not convex: just fin... | Emilio Novati | 187,568 | <p>Hint:</p>
<p>If the two points $P=(x_P,y_P)$ and $Q=(x_Q,y_Q)$ are in the set than we have $x_P^2+y_P^2=x_Q^2+y_Q^2\leq1$ and to prove that the set is convex we use the definition, that gives:
$$
\alpha(x_P,y_P)+(1-\alpha)(x_Q,y_Q) \in \Omega
$$
that is:
$$
\left[\alpha x_P+(1-\alpha)x_Q \right]^2+\left[\alpha y... |
4,489,898 | <p>After 18 months of studying an advanced junior high school mathematics course, I'm doing a review of the previous 6 months, starting with solving difficult quadratics that are not easily factored, for example:
<span class="math-container">$$x^2+6x+2=0$$</span>
This could be processed via the quadratic equation but t... | KCd | 619 | <p>The method of inserting <span class="math-container">$0$</span> in a clever way by adding and subtracting a term is used many times in analysis. I learned long ago to call this clever use of <span class="math-container">$0$</span> a "propitious zero" and others have been taught that term too: look <a href... |
1,515,417 | <p>I understand the idea that some infinities are "bigger" than other infinities. The example I understand is that all real numbers between 0 and 1 would not be able to "fit" on an infinite list.</p>
<p>I have to show whether these sets are countable or uncountable. If countable, how would you enumerate the set? If un... | John Douma | 69,810 | <p>Set $2$ can be put into one-to-one correspondence with the binary representation of the reals by the map that takes $2$ to $0$ and $3$ to $1$. Thus, this set has the same cardinality as $\mathbb R$ which is uncountable.</p>
|
2,493,481 | <p>I'm currently studying calculus of variations. I couldn't find a rigorous definition of a functional on this site.</p>
<ol>
<li>What is the general definition of a functional?</li>
<li>Why for calculus of variations in physics, I must to use for a functional <em>a convex function</em> for the space of the admissib... | md2perpe | 168,433 | <p>Let $\mathscr F$ be a functional of the form
$$\mathscr F(y) = \int_a^b f(x, y(x), y'(x)) \, dx.$$</p>
<p>We want to find a function $y_0$ that gives a local minimum of $\mathscr F,$ i.e. if we take a "close" function $y_0+\delta y$ then $\mathscr F(y_0+\delta y)$ will not be as small.</p>
<p>The idea is to let $\... |
5,363 | <p>There is something in the definition of the <a href="http://en.wikipedia.org/wiki/Free_product" rel="nofollow noreferrer">free product</a> of two groups that annoys me, and it's this "word" thing:</p>
<blockquote>
<p>If <span class="math-container">$G$</span> and <span class="math-container">$H$</span> are groups... | Martin Brandenburg | 1,650 | <p>In the question above, it seems that the description is far too complicated as it actually is. I will elaborate on the construction of coproducts of groups or monoids a bit. This will go beyond the scope of the specific question, but I hope it's helpful in other situations.</p>
<p>First assume $M,N$ are sets. What ... |
5,363 | <p>There is something in the definition of the <a href="http://en.wikipedia.org/wiki/Free_product" rel="nofollow noreferrer">free product</a> of two groups that annoys me, and it's this "word" thing:</p>
<blockquote>
<p>If <span class="math-container">$G$</span> and <span class="math-container">$H$</span> are groups... | Arturo Magidin | 742 | <p>You can see essentially the same construction in two different ways in <a href="http://math.berkeley.edu/~gbergman/245/">George Bergman's <em>An Invitation to General Algebra and Universal Constructions</em></a> in <a href="http://math.berkeley.edu/~gbergman/245/Ch.2.ps">Chapter 2</a> (link is to a postscript file) ... |
2,906,350 | <p>I have already consulted V.K. Rohatgi, and it has an example where it takes $Y=X^a$ where $a>0$ but the domain of $X$ is positive real values.<br>
Even the theorem for transformation of continuous random values restricts the derivative of $Y$ w.r.t. $X$ to be positive or negative for the entire domain of $X$ wher... | Daman | 519,543 | <p>I 've been doing some questions from Rohtagi also. It's a very good book. And to answer your question.we are give $X\sim N(0,1),f(x)=\dfrac{1}{\sqrt{2\pi}}e^\frac{-x^{2}}{2} ;-\infty<x<\infty$</p>
<p>$Y=X^3$ is a monotone function and for the monotone function, you can directly apply the transformation formul... |
719,056 | <p>I'm writing a Maple procedure and I have a line that is "if ... then ..." and I would like it to be if k is an integer (or if k^2 is a square) - how would onw say that in Maple? Thanks!</p>
| Community | -1 | <p>In terms of differentials (in the single-variable case), $\frac{dy}{dx}$ is the unique scalar with the property that $\frac{dy}{dx}dx = dy$.</p>
<p>$\frac{dy}{du} \frac{du}{dx}$ therefore has the property that</p>
<p>$$\frac{dy}{du} \frac{du}{dx} dx = \frac{dy}{du} du = dy $$</p>
<p>therefore $\frac{dy}{du}\frac{... |
689,546 | <p>I know this problem involves using Cantor's theorem, but I'm not sure how to show that there are more subsets of an infinite enumerable set than there are positive integers. It seems like a lot of these problems are really the same problem, but they require some unique and creative thought to get them just right. An... | João Víctor Melo | 852,373 | <p>Another way to clear it more is to considerer an enumerable set <span class="math-container">$ S = \{s_1, s_2, ..., s_n \}$</span>; now, it's evident that we can take the first digit from <span class="math-container">$s1$</span>, the second digit from <span class="math-container">$s_2$</span>, and so on, such that w... |
2,078,737 | <p>I will gladly appreciate explanation on how to do so on this matrix:</p>
<p>$$
\begin{pmatrix}
i & 0 \\
0 & i \\
\end{pmatrix}
$$</p>
<p>I got as far as calculating the eigenvalues and came up with $λ = i$.
when trying to find the eigenvectors I came up with the $0$ matrix.<... | Community | -1 | <p>$$\lim_{x \to 1^{-}}\tan\left(\frac{\pi x}{2}\right)=\lim_{x \to \frac{\pi}{2}^-}\tan\left(\frac{x}{2}\right)=\lim_{x \to \frac{\pi}{2}^-} \frac{\sin x}{\cos x}.$$</p>
<p>Now, $\lim_{x \to \frac{\pi}{2}^-} \sin x=1.$ But when $\lim_{x \to \frac{\pi}{2}^-},\;~\cos x$ approaches to zero through positive values. Hence... |
4,465,528 | <p>Solving the integral of <span class="math-container">$\cos^2x\sin^2 x$</span>:</p>
<p>My steps are: <span class="math-container">$(\cos x\sin x)^2=\left(\frac{\sin(2x)}{2}\right)^2$</span>. Now we know that</p>
<p><span class="math-container">$$\sin^2(\alpha)=\frac{1-\cos (2x)}{2}\iff\left(\frac{\sin(2x)}{2}\right)^... | gnasher729 | 137,175 | <p>You know exactly 1000 wrong answers were given, and each of 1000 candidates gave at least one incorrect answer. Nobody can have given two wrong answers, or we would have had at least 1001 wrong answers. Therefore everyone gave exactly one wrong answer snd got all the others right.</p>
<p>Of 1000, 300 got answer 3 wr... |
1,468,097 | <p>Below is the problem:</p>
<p>Choose a point uniformly at random from the triangle with vertices (0,0), (0, 30), and (20, 30). Let (X, Y ) be the coordinates of the chosen point. (a) Find the cumulative distribution function of X. (b) Use part (a) to find the density of X.</p>
<p>First, for the first part of the qu... | Yes | 155,328 | <p>A slightly wilder example for 3-dimensional case: If $u:= (1,0,1)$, if $v:= (-1,0,0)$, and if $w := (0,0,-1)$, then $u,v,w$ are pairwisely linearly independent; but $u + v + w = (0,0,0)$. </p>
|
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... | marty cohen | 13,079 | <p>Expanding on
Wojowu's answer,</p>
<p><span class="math-container">$\begin{array}\\
\sum_{k=2^n}^{2^{n+1}-1} \dfrac1{k\ln^2(k)}
&\le \sum_{k=2^n}^{2^{n+1}-1} \dfrac1{k\ln^2(2^n)}\\
&\le 2^n\dfrac1{2^n(n \ln 2)^2}\\
&= \dfrac1{n^2 \ln^2 2}\\
\end{array}
$</span></p>
<p>and the sum of these converges,</p>... |
1,364,417 | <p>Find a real number k such that the limit $$\lim_{n\to\infty}\ \left(\frac{1^4 + 2^4 + 3^4 +....+ n^4}{n^k}\right)$$ has as positive value.
If I am not mistaken every even $k$ can be the answer. But the answer is 5.</p>
| k170 | 161,538 | <p>First note that
$$\sum\limits_{i=1}^n i^4=\frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}$$
$$=\frac{n^5}{5}+\frac{n^4}{2}+\frac{n^3}{3}-\frac{n}{30}$$
So now we have
$$\lim\limits_{n\to\infty} \left(\frac{1^4 + 2^4 + 3^4 +\cdots + n^4}{n^k}\right)$$
$$=\lim\limits_{n\to\infty} \left(\frac{\frac{n^5}{5}+\frac{n^4}{2}+\frac{n^3}{... |
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}$... | urbano | 462,611 | <p>Even in the 1000th time with a 999 tails, 0 heads, the probability of head is 1/2. But the probability that you get 1000th times tails in a row is 0.5^1000, 9.332636e-302.</p>
|
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}$... | Zdman | 462,609 | <p>The probability will always be 1/2 for each coin toss because the event physically has no tie to the previous throws. It will approach 1 because, as you approach infinity, the difference between the number of heads and tails will become smaller. The difference between heads and tails will become smaller because th... |
316,374 | <p>If <span class="math-container">$a_n$</span> satisfies the linear recurrence relation <span class="math-container">$a_n = \sum_{i=1}^k c_i a_{n-i}$</span> for some constants <span class="math-container">$c_i$</span>, then is there an easy way to find a linear recurrence relation for <span class="math-container">$b_n... | Josiah Park | 118,731 | <p>T. Brown and P.J. Shiue's paper <a href="https://www.fq.math.ca/Scanned/33-4/brown.pdf" rel="nofollow noreferrer">here</a> might be of interest as a first reference. In the introduction they mention that if <span class="math-container">$a_n$</span> is a second-order sequence then the sequence
of squares <span class=... |
3,102,218 | <p>Given a fraction:</p>
<p><span class="math-container">$$\frac{a}{b}$$</span></p>
<p>I now add a number <span class="math-container">$n$</span> to both numerator and denominator in the following fashion:</p>
<p><span class="math-container">$$\frac{a+n}{b+n}$$</span></p>
<p>The basic property is that the second fr... | Martin R | 42,969 | <p>Visually: Consider the <em>slope</em> of the line segment from <span class="math-container">$(0, 0)$</span> to <span class="math-container">$(a+n, b+n$</span>):</p>
<p><a href="https://i.stack.imgur.com/J6y7Z.png" rel="noreferrer"><img src="https://i.stack.imgur.com/J6y7Z.png" alt="enter image description here"></a... |
3,102,218 | <p>Given a fraction:</p>
<p><span class="math-container">$$\frac{a}{b}$$</span></p>
<p>I now add a number <span class="math-container">$n$</span> to both numerator and denominator in the following fashion:</p>
<p><span class="math-container">$$\frac{a+n}{b+n}$$</span></p>
<p>The basic property is that the second fr... | toth | 142,243 | <p>There's a very simple way to see this. Just take the difference between the two fractions and 1. You want to show that this is smaller in modulus for the second fraction.</p>
<p>You get <span class="math-container">$$ \frac{a}{b} - 1 = \frac{a-b}{b} $$</span> and
<span class="math-container">$$ \frac{a+n}{b+n} -1 =... |
3,102,218 | <p>Given a fraction:</p>
<p><span class="math-container">$$\frac{a}{b}$$</span></p>
<p>I now add a number <span class="math-container">$n$</span> to both numerator and denominator in the following fashion:</p>
<p><span class="math-container">$$\frac{a+n}{b+n}$$</span></p>
<p>The basic property is that the second fr... | Bernard | 202,857 | <p>You have to suppose <span class="math-container">$a,b >0$</span>. Now, it is clear that, if <span class="math-container">$a<b,\;$</span> i.e. <span class="math-container">$\:\smash{\dfrac ab}<1$</span>, <span class="math-container">$a+n<b+n$</span>, hence <span class="math-container">$\smash{\dfrac{a+n... |
3,102,218 | <p>Given a fraction:</p>
<p><span class="math-container">$$\frac{a}{b}$$</span></p>
<p>I now add a number <span class="math-container">$n$</span> to both numerator and denominator in the following fashion:</p>
<p><span class="math-container">$$\frac{a+n}{b+n}$$</span></p>
<p>The basic property is that the second fr... | robjohn | 13,854 | <p>If <span class="math-container">$b$</span> and <span class="math-container">$d$</span> have the same sign, both
<span class="math-container">$$
\frac ab-\frac{a+c}{b+d}=\frac1b\frac{ad-bc}{b+d}\tag1
$$</span>
and
<span class="math-container">$$
\frac{a+c}{b+d}-\frac cd=\frac1d\frac{ad-bc}{b+d}\tag2
$$</span>
also ha... |
299,795 | <p>I seem to have completely lost my bearing with implicit differentiation. Just a quick question:</p>
<p>Given $y = y(x)$ what is $$\frac{d}{dx} (e^x(x^2 + y^2))$$</p>
<p>I think its the $\frac d{dx}$ confusing me, I don't what effect it has compared to $\frac{dy}{dx}$. Any help will be greatly appreciated.</p>
| Andreas Caranti | 58,401 | <p>Look, $x = 0$ is definitely a solution. So you want to find solutions $x \ne 0$. Divide by $x^7$...</p>
|
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... | Marios Gretsas | 359,315 | <p>Let $\epsilon >0$ and $x \in (a,b) \cap S^c$.</p>
<p>$\forall t>0, \exists z \in S$ such that $|x-z|<t$ </p>
<p>We have that $f,g$ are continuous at $x$ thus $\exists \delta_1, \delta_2>0$ such that $$|f(y)-f(x)|< \epsilon /2, \forall y: |y-x|< \delta_1 $$ $$|g(y)-g(x)|< \epsilon /2, \forall ... |
592,963 | <p>Find $x^4+y^4$ if $x+y=2$ and $x^2+y^2=8$</p>
<p>So i started the problem by nothing that $x^2+y^2=(x+y)^2 - 2xy$ but that doesn't help!</p>
<p>I also seen that $x+y=2^1$ and $x^2+y^2=2^3$ so maybe $x^3+y^3=2^5$ and $x^4+y^4=2^7$ but i think this is just coincidence</p>
<p>So how can i solve this problem?</p>
<p... | ILoveMath | 42,344 | <p>Notice</p>
<p>$$(x^2 + y^2)^2 = 64 \implies x^4 + y^4 + 2(xy)^2 = 64$$ </p>
<p>and</p>
<p>$$ (x + y )^2 = 4 \implies x^2 + y^2 + 2xy = 4 \implies 2xy = 4 - 8 = -4 \implies xy = -2
$$</p>
<p>$$ \therefore x^4 + y^4 = 64 - 2(xy)^2 = 64 - 2(-2)^2 = 56 $$</p>
|
1,647,517 | <p>I was in need to urgently solve this integral. I already know the result in the closed form, does anybody know how to solve it?
\begin{equation}
\int_{\mathbb{R}}e^{-\frac{x^{2}}{2}}\left(\cos\left(\pi nx\right)\right)dx=\sqrt{2\pi}e^{-\frac{n^{2}\pi^{2}}{2}},
\end{equation}
If anybody already knows it is welcome,... | Joe | 107,639 | <p><strong>HINT:</strong>
$$
\int_{\mathbb{R}}e^{-\frac{x^{2}}{2}}\left(\cos\left(\pi nx\right)\right)dx=
\Re\left[ \int_{\mathbb{R}}e^{-\frac{x^{2}}{2}}e^{i\pi nx}dx\right]
$$
Then observe that
$$
-\frac{x^2}2+i\pi nx=-\left(\frac x{\sqrt2}-\sqrt2i\pi n\right)^2+2\pi^2 n^2
$$
and use the well known integral</p>
<p>$$... |
2,338,321 | <p>Under given standardized random variable $Z = (X-\mu)/\sigma$ Show that </p>
<p>$$C_3(Z) = M_3(Z)\;\;\text{and}\;\; C_4(Z) = M_4(Z) -3$$</p>
<p>where $C_r(Z)$ refer to $r$-th cumulant, $M_r(Z)$ refer to $r$-th moment.</p>
<hr>
<p><strong>Question</strong></p>
<p>I've been required to show above relationship by ... | Sri-Amirthan Theivendran | 302,692 | <p>Note that $Z\sim N(0,1)$. The MGF of a standard normal is
$$
M(t)=\exp\left(\mu t+\frac{1}{2}\sigma^2t^2\right)=\exp\left(\frac{1}{2}t^2\right).
$$
In particular the cumulant generating function of a standard normal is
$$
K(t)=\frac{1}{2}t^2.
$$
It follows that the nth cumulant $\kappa_n=K^{(n)}(0)=0$ for $n>2$.... |
1,594,968 | <p>How much knowledge of group theory is needed in order to begin Galois Theory? Which topics are most relevant?</p>
| Nicky Hekster | 9,605 | <p>Most Galois Theory books are self-explanatory, but you need to familiarize yourself with concepts as <a href="https://en.wikipedia.org/wiki/Solvable_group" rel="nofollow">solvable groups</a> (this relates to equations being solvable by radicals), <a href="https://en.wikipedia.org/wiki/Simple_group" rel="nofollow">si... |
1,270,107 | <p>How many real roots does the below equation have?</p>
<p>\begin{equation*}
\frac{x^{2000}}{2001}+2\sqrt{3}x^2-2\sqrt{5}x+\sqrt{3}=0
\end{equation*}</p>
<p>A) 0 B) 11 C) 12 D) 1 E) None of these</p>
<p>I could not come up with anything.</p>
<p>(Turkish Math Olympiads 2001)</p>
| David K | 139,123 | <p>Consider the discriminant of $f(X) = 2\sqrt{3}x^2-2\sqrt{5}x+\sqrt{3}=0$:</p>
<p>$$(-2\sqrt{5})^2 - 4(2\sqrt{3})\sqrt{3} = 20 - 24 < 0.$$</p>
<p>Therefore $f(x)$ has no real roots.
But $f(0) = \sqrt{3} > 0$, so $f(x) > 0$ everywhere.</p>
<p>Now combine this with $$\frac{x^{2000}}{2001} \geq 0.$$</p>
|
4,074,031 | <p>You have been teaching Dennis and Inez math using the Moore method for their entire lives, and you're currently deep into topology class.</p>
<p>You've convinced them that topological manifolds aren't quite the right object of study because you "can't do calculus on them," and you've managed to motivate th... | not all wrong | 37,268 | <p>So if we're going to insist on avoiding integration, I suppose we'd better take some more derivatives. In particular, we want to learn how to take derivatives of tensors. Of course, famously, without some additional structure on even a smooth manifold, there is no such generally well-defined notion. In fact, there a... |
85,288 | <p>For some reason, I'd like to use <code>ParallelTable</code> with a variable number of iterators.</p>
<ul>
<li><code>Table[a[1], {a[1], 0, 10}]</code> works fine:</li>
</ul>
<blockquote>
<p><em>Output:</em> <code>{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}</code></p>
</blockquote>
<ul>
<li><code>ParallelTable[a[1], {a[1], 0, ... | Mr.Wizard | 121 | <p>Using <a href="http://reference.wolfram.com/language/ref/Trace.html"><code>Trace</code></a> we can see that the evaluation of <code>ParallelTable[a[1], {a[1], 0, 10}]</code> becomes:</p>
<pre><code>Parallel`Combine`Private`parallelIterateE[
ParallelTable, Table, Join, Identity, a[1], {a[1], 0, 10}, {Automatic, "G... |
1,381,545 | <p>I have the following problem:</p>
<blockquote>
<p>Show by induction that $F_n \geq 2^{0.5 \cdot n}$, for $n \geq 6$</p>
</blockquote>
<p>Where $F_n$ is the $nth$ Fibonacci number. </p>
<hr>
<h3>Proof</h3>
<p><strong>Basis</strong></p>
<p>$n = 6$. </p>
<p>$F_6 = 8 \geq 2^{0.5 \cdot 6} = 2^{\frac{6}{2}} = 2^3... | Euler88 ... | 252,332 | <p>We have that $F_n>F_{n-1}$ then $$F_{n+1}=F_n+F_{n-1}>2F_{n-1}>2\cdot2^{(n-1)/2}=2^{(n+1)/2}$$</p>
|
3,034,374 | <p>Two cyclists start from the same place to ride in the same direction.A starts at noon with a speed of 8km/hr and B starts at 2pm with a speed of 10km/hr.At what times A and B will be 5km apart ?
My thought process:
As A starts early at 12 so it will have already covered 16km(8*2).
so S relative=V relative*t or say 1... | hamam_Abdallah | 369,188 | <p><span class="math-container">$$x_1(t)=v_1(t-t_1)+x_0=8(t-0)$$</span>
<span class="math-container">$$x_2(t)=v_2(t-t_2)+x_0=10(t-2)$$</span></p>
<p>thus the condition</p>
<p><span class="math-container">$$|x_2(t)-x_1(t)|=|2t-20|=5$$</span>
gives two solutions</p>
<p><span class="math-container">$$t=12,5 \text{ o... |
2,005,365 | <p>For example, p(getting two heads from tossing a coin twice) = 0.5 * 0.5...</p>
<p>I passed my probability course in college, but I am still having trouble getting the intuition for this.</p>
| anonymus | 277,855 | <p>So you have to study the limit when $h\to 0$ of the following quantity : $\frac{\sqrt{x}-\sqrt{x+h}}{h(\sqrt{x}\sqrt{x+h})}$.</p>
<p>Multiplying and dividing by the "conjugate" quantity gives : $\frac{\sqrt{x}-\sqrt{x+h}}{h(\sqrt{x}\sqrt{x+h})} \frac{\sqrt{x}+\sqrt{x+h}}{\sqrt{x}+\sqrt{x+h}} = \frac{-1}{\sqrt{x}\sq... |
2,944,397 | <blockquote>
<p>In one plane <span class="math-container">$Oxy$</span>, given the circle <span class="math-container">$(\!{\rm C}\!): x^{2}+ y^{2}- 2x+ 4y- 4= 0$</span> around point <span class="math-container">$O$</span> for <span class="math-container">$60^{\circ}$</span> and it maps in the circle <span class="math... | user376343 | 376,343 | <p><strong>The question was:</strong></p>
<p><em>Circle <span class="math-container">$\left ( C \right ): x^{2}+ y^{2}- 2\,x+ 4\,y- 4= 0$</span> <span class="math-container">$(1)$</span> in plane <span class="math-container">$Oxy$</span> around point <span class="math-container">$O$</span> for <span class="math-contai... |
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... | ℋolo | 471,959 | <p>For (a):</p>
<p>start with $x=\frac1n$:</p>
<p>$f(1)=f(\sum_{i=1}^n x)=\sum_{i=1}^n f(x)=nf(x)=C\implies f(x)=\frac Cn$</p>
<p>Now do similar thing to $y=\frac mn$:</p>
<p>$f(y)=f(mx)=f(\sum_{i=1}^m x)=\sum_{i=1}^m f(x)=\sum_{i=1}^m \frac Cn=\frac{Cm}n$</p>
<p>For (b) use the fact that all real numbers are the ... |
1,400,876 | <p>Given a measurable space $(\Omega, \cal{F})$, $f:(\Omega, \cal F) \to (\Bbb{B},\cal B)$, where $\cal B$ is the Borel $\sigma$-algebra of $\Bbb R$, is said to be $\cal {F}$-measurable if $f^{-1}(B)\in \cal F $ for any $B\in \cal B$.</p>
<p>Any function $f:\Omega\to\Bbb R$ is trivially $2^\Omega$ measurable. A consta... | ncmathsadist | 4,154 | <p>Consider the symmetric Borel sets
$$\{B\in \mathcal{B}(\mathbb{R})| - B = B\}.$$
The measurable functions in this $\sigma$-algebra are the even measurable functions.</p>
|
690,569 | <p>Suppose a function is given by:
$$
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>
| LeoTheKub | 131,252 | <p>Take $x_n=(2n\pi)^{-1}$ $(n=1,2,3 \ldots)$ and observe that $\displaystyle\lim_{n\to\infty}x_n=0$. Then compute $f(x_n)$ and ask yourself: is $\displaystyle\lim_{n\to\infty}f(x_n)=f(0)\,$?</p>
<p>If the answer is "no", then this shows that $f$ is discontinuous at $x=0$.</p>
|
244 | <p>I know that Hilbert schemes can be very singular. But are there any interesting and nontrivial Hilbert schemes that are smooth? Are there any necessary conditions or sufficient conditions for a Hilbert scheme to be smooth?</p>
| Ben Webster | 66 | <p>A very well-known condition is that the Hilbert scheme of a smooth surface is smooth. As David pointed out below, the Hilbert scheme of a smooth curve is smooth and equal to the symmetric product (since k[t] has only one finite dimension quotient of each dimension). </p>
<p>I don't know of any other examples, but ... |
1,990,105 | <p>Presume that $(x_n)$ is a sequence s. t.</p>
<p>$|x_n-x_{n+1}| \le 2^{-n}$ for all $n \in \mathbb N$</p>
<p>Prove that $x_n$ converges.</p>
<p>What I've tried to think: since $2^{-n}$ converges to 0, and the difference between the terms $x_n$ and $x_{n+1}$ is smaller or equal to it, then $x_n$ must be a cauchy s... | hamam_Abdallah | 369,188 | <p>For each $n,p>0$</p>
<p>by triangular inequality,</p>
<p>$|x_{n+p}-x_n|\leq2^{-n}(2^{-p+1}+2^{-p+2}...+2^{-1}+1)$
$\leq 2^{1-n}(1-2^{-p})\leq 2^{1-n}$.</p>
<p>and since $\lim_{n\to+\infty}2^{1-n}=0$,</p>
<p>$(\forall \epsilon>0) ( \exists N\in \mathbb N) : (\forall n>N) (\forall p>0) |x_{n+p}-x_n... |
125,834 | <p>Is there a general formula for the number of distinguishable arrangements of length $\ell$ of a sequence of size $s$, given the repeated elements? So,</p>
<p>For example, would there be a general formula to solve the problem:</p>
<p>How many distinguishable 5-letter arrangements are there of the word "PROBABILITY"... | André Nicolas | 6,312 | <p>I assume the problem is to find the number of $5$-letter words that can be formed, where we can use a letter no more times than it appears in PROBABILITY.</p>
<p>There is no useful general formula. The closest I could get would be to use generating functions, which are certainly overkill for the concrete problem yo... |
2,672,497 | <p>$$\lim _{n\to \infty }\sum _{k=1}^n\frac{1}{n+k+\frac{k}{n^2}}$$ I unsuccessfully tried to find two different Riemann Sums converging to the same value close to the given sum so I could use the Squeeze Theorem. Is there any other way to solve this?</p>
| Mark Viola | 218,419 | <blockquote>
<p>I thought it might be instructive to present a method that relies only on the <strong>squeeze theorem</strong>. To that end, we proceed.</p>
</blockquote>
<hr>
<p>First, it is trivial to see that </p>
<p>$$\sum_{k=1}^n \frac{1}{n+k+k/n^2}\le \sum_{k=1}^n \frac{1}{n+k}\tag 1$$</p>
<hr>
<p>Second,... |
1,018,672 | <blockquote>
<p><span class="math-container">$$\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... | Ari | 185,398 | <p>Remember that in any limit to infinity only the highest power term in the numerator and denominator matter. Thus your expression is equivalent to $\frac{1}{6}ln(1)=0$</p>
|
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... | saz | 36,150 | <p>Hint: Let $f$ be a measurable function, $f \geq 0$. Then there exists a sequence $(f_n)_n$ of step functions such that $f = \sup_n f_n$. Now apply <a href="http://en.wikipedia.org/wiki/Monotone_convergence_theorem#Lebesgue.27s_monotone_convergence_theorem" rel="nofollow">monotone convergence</a> and use the formula ... |
607,917 | <p>Any Help solving this question ?</p>
<blockquote>
<p>a) Find ONE solution <span class="math-container">$\overline x\in\Bbb Z/325\Bbb Z$</span> such that <span class="math-container">$x^2\equiv-1\pmod{325}$</span>. (Hint: CRT and lifting.)</p>
<p>b) How many solutions <span class="math-container">$\overline x$</span... | lab bhattacharjee | 33,337 | <p>We can prove by using <a href="http://mathworld.wolfram.com/DiscreteLogarithm.html" rel="nofollow">Discrete Logarithm</a> and <a href="http://www.proofwiki.org/wiki/Solution_of_Linear_Congruence" rel="nofollow">Linear Congruence</a> Theorem that $x^2\equiv a\pmod m$ has zero or two solutions if $m$ has a primitive... |
182,024 | <p>Following my previous question <a href="https://math.stackexchange.com/q/182000/21813">Relationship between cross product and outer product</a> where I learnt that the Exterior Product generalises the Cross Product whereas the Inner Product generalises the Dot Product, I was wondering if the simple map that I have d... | celtschk | 34,930 | <p>The relation between the inner product of vectors and the interior product is that if you have a metric tensor (and thus a canonical relation between vectors and covectors = $1$-forms), the inner product of two vectors is the interior product of one of the vectors and the $1$-form associated with the other one. That... |
1,394,550 | <p>I have tried to prove the following inequality:</p>
<p>$$
\left(1+\frac{\log n}{n}\right)^n \gt\frac{n+1}{2}, \mbox{for}\;n\in\{2,3,\ldots\}
$$</p>
<p>which seems to be correct (confirmed by numerical result).</p>
<p>Can anyone give me some help or hint? Thanks a lot.</p>
| Björn Friedrich | 203,412 | <p>I just want to give a hint: Since the inequality is claimed to be true for all natural numbers greater than one, I would try a <a href="https://en.wikipedia.org/wiki/Mathematical_induction" rel="nofollow">proof by induction</a>. It consists of four steps:</p>
<ol>
<li>You show that the inequality is true for $n = 2... |
3,324,803 | <p><strong>definition.</strong></p>
<p>The phase flow of the differential equation <span class="math-container">$\dot{x}=\vec{v}\ (x)$</span> is the one-parameter diffeomorphism group for which <span class="math-container">$\vec{v}$</span> is the phase velocity vector field, namely,
<span class="math-container">$$
\ve... | Fareed Abi Farraj | 584,389 | <p>I don't know if this is what you want, but I hope it helps.</p>
<p><span class="math-container">$\log_2(x)=a$</span> means that <span class="math-container">$2^a=x$</span></p>
<p><span class="math-container">$\log_3(x+1)=b$</span> means that <span class="math-container">$3^b=x+1$</span></p>
<p>To get <span class="ma... |
3,324,803 | <p><strong>definition.</strong></p>
<p>The phase flow of the differential equation <span class="math-container">$\dot{x}=\vec{v}\ (x)$</span> is the one-parameter diffeomorphism group for which <span class="math-container">$\vec{v}$</span> is the phase velocity vector field, namely,
<span class="math-container">$$
\ve... | Michael Rozenberg | 190,319 | <p>Let <span class="math-container">$f(x)=\log_3(x+1)+\log_2x.$</span></p>
<p>Thus, since <span class="math-container">$f$</span> increases, our equation has one root maximum.</p>
<p><span class="math-container">$8$</span> is a root, which says that it's an unique root and we are done.</p>
|
3,324,803 | <p><strong>definition.</strong></p>
<p>The phase flow of the differential equation <span class="math-container">$\dot{x}=\vec{v}\ (x)$</span> is the one-parameter diffeomorphism group for which <span class="math-container">$\vec{v}$</span> is the phase velocity vector field, namely,
<span class="math-container">$$
\ve... | B. Goddard | 362,009 | <p>I tried to think of how a middle schooler might solve this:</p>
<p>Let <span class="math-container">$x=2^a$</span>. Then we have</p>
<p><span class="math-container">$$\log_3 (2^a+1) + a = 5.$$</span> So that</p>
<p><span class="math-container">$$2^a+1 = 3^{5-a}.$$</span></p>
<p>Multiply by <span class="math-con... |
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 ... | Favst | 742,787 | <p>Here is some motivation for the choice of <span class="math-container">$7$</span> as the modulus, as you asked. The equation that you want to show that has no solutions in the integers is <span class="math-container">$$6n^3 +3 -m^6=0.$$</span> When it comes to polynomial Diophantine equations, especially of the olym... |
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... | Robert Israel | 8,508 | <p>It's exactly the fact that the conditions can't be duplicated exactly that makes a probability model for the experiment viable.</p>
<p>With some practice, a person with good fine-motor control can become an expert coin-flipper who can fairly reliably produce either "heads" or "tails" as desired. The "fair coin" pr... |
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... | Hymns For Disco | 736,037 | <p>Wikipedia's definition of "randomness" is </p>
<blockquote>
<p>the apparent lack of pattern or predictability in events</p>
</blockquote>
<p>The word "apparent" there is important. Just because something can be predicted in theory, doesn't mean its not random. For example, a pseudo-random number generator in a... |
96,957 | <p>Stacks, of varying kinds, appear in algebraic geometry whenever we have moduli problems, most famously the stacks of (marked) curves. But these seem to be to be very geometric in motivation, so I was wondering if there are natural examples of stacks that arise in arithmetic geometry or number theory.</p>
<p>To me, ... | stankewicz | 3,384 | <p>Perhaps this doesn't count as "modern" but stacks are ubiquitous in the 1972 Antwerp paper of Deligne and Rapoport. Recall that the $\Gamma_0(N)$ moduli problem is not representable, and so they must frequently work directly with stacks before moving to the coarse moduli scheme we all know and love.</p>
|
390,644 | <p>I'm trying to solve this recurrence relation:</p>
<p>$$
a_n = \begin{cases}
0 & \mbox{for } n = 0 \\
5 & \mbox{for } n = 1 \\
6a_{n-1} - 5a_{n-2} + 1 & \mbox{for } n > 1
\end{cases}
$$</p>
<p>I calculated generator function as:
$$
A = \frac{31x - 24x^2}{1 - 6x + 5x^2} + \frac{x^3}{(... | TZakrevskiy | 77,314 | <p>Let's write your recurrence relation for $n$ and $n+1$:</p>
<p>$a_{n}-6a_{n-1}+5a_{n-2}-1=0$</p>
<p>$a_{n+1}-6a_{n }+5a_{n-1}-1=0$</p>
<p>Now we subtract one from another:
$a_{n+1}-7a_{n }+11a_{n-1}-5a_{n-2}=0$ (relation 2)</p>
<p>Then, from Theorem on <a href="http://en.wikipedia.org/wiki/Recurrence_relation" r... |
390,644 | <p>I'm trying to solve this recurrence relation:</p>
<p>$$
a_n = \begin{cases}
0 & \mbox{for } n = 0 \\
5 & \mbox{for } n = 1 \\
6a_{n-1} - 5a_{n-2} + 1 & \mbox{for } n > 1
\end{cases}
$$</p>
<p>I calculated generator function as:
$$
A = \frac{31x - 24x^2}{1 - 6x + 5x^2} + \frac{x^3}{(... | Adi Dani | 12,848 | <p>$$a_0 =0,a_1=5,a_n=6a_{n-1} - 5a_{n-2} + 1, n > 1$$</p>
<p>$$f(x)=\sum_{n=0}^{\infty}a_nx^n=5x+\sum_{n=2}^{\infty}a_nx^n=$$</p>
<p>$$=5x+\sum_{n=2}^{\infty}(6a_{n-1} - 5a_{n-2} + 1)x^n=$$</p>
<p>$$=5x+6\sum_{n=2}^{\infty}a_{n-1}x^n-5\sum_{n=2}^{\infty}a_{n-2}x^n+\sum_{n=2}^{\infty}x^n=$$</p>
<p>$$=5x+6x\sum_{... |
1,771,279 | <p>Let $B=\{x \in \ell_2, ||x|| \leqslant 1\}$ - Hilbert ball $X \subset B$ - open convex connected set in Hilbert ball, $\bar{X}$ - closure of $X$. $F: \bar{X} \to \bar{X}$ - continuous map that holomorphic into each point of $X$. Is it true in general, that $F$ has fixed point?</p>
| David C. Ullrich | 248,223 | <p>$X=\{x:1/2<||x||<1\}$, $f(x)=-x$. (This was before convexity was added to the hypotheses...)</p>
|
1,541,800 | <p>I happened to stumble upon the following matrix:
$$ A = \begin{bmatrix}
a & 1 \\
0 & a
\end{bmatrix}
$$</p>
<p>And after trying a bunch of different examples, I noticed the following remarkable pattern. If $P$ is a polynomial, then:
$$ P(A)=\begin{bmatrix}
P(a) & P'(a) \... | ASCII Advocate | 260,903 | <p>If $e$ satisfies $Xe=eX$ and $e^k=0$, then $f(X+e)=f(X) + ef'(X)+\frac{e^2}{2}f''(X) + \dots + f^{(k-1}(X)\frac{e^{k-1}}{(k-1)!}$ (a finite sum with $k$ terms). </p>
<p>This is true for polynomials and thus for power series that converge in a neighborhood of $X$.</p>
|
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... | Ross Millikan | 1,827 | <p>If you write y=x^2 and pull the constants out you have $$\frac{1}{2\sqrt{2\pi}\sigma}\int_0^{0.25}\sqrt{y}\cdot \exp(-\frac{(y-\mu )^2}{2\sigma ^2})dy$$ If $\sigma$ is very small, the contribution will all come from a small area in $y$ around $\mu$. So you can set $\sqrt{y}=\sqrt{\mu}$ and use your error function ... |
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... | Community | -1 | <p>Not an answer, but might still be helpful. Using the variable substitution that Ross mentions in his answer, we can treat a simpler case $\mu=0$ more easily. </p>
<p>For the following integral, Wolfram Alpha tells us that (I hope I did not make a transcription error here):</p>
<p>$$\int_0^\infty \sqrt{y}e^{-y^2/(2... |
2,796,618 | <p>I am trying to</p>
<p>i) determine the infimum</p>
<p>ii) show that there's a function for which $\int_{0}^{1} {f'(x)}^2 dx$ is the infimum</p>
<p>iii) show if such function is unique.</p>
<p>I tried out several functions that suit the given condition, but couldn't see how $\int_{0}^{1} {f'(x)}^2 dx$ changes as ... | Botond | 281,471 | <p>So we want to maximize
$$J[y]=\int_{0}^{1} (y'(x))^2 \mathrm{d}x$$
We have that the Lagrange-function is $L(x, y, y')=(y')^2$, and we must have that
$$\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{\partial L}{\partial y'}\right)=\frac{\partial L}{ \partial y}$$
And you just need to solve this differential equation.</p>
|
1,999,352 | <p>so what really is the meaning of a metric space and why is it so important in topology?</p>
| Lee Mosher | 26,501 | <p>There is no single mathematical idea of "space". </p>
<p>Each different kind of space --- vector space; metric space; topological space --- is studied in a different course of mathematics. Googling is not going to get you to a good understanding of all of these different kinds of spaces. You'll have to do some seri... |
82,350 | <p>Consider three circles of radius $1$ in $\mathbb{R}^3$, linked with each other in the same arrangement as three fibers of the Hopf fibration. Now thicken the circles up into non-overlapping standard round Euclidean solid tori of equal thickness. Allowing the tori to move, there will be some maximum thickness (distan... | Joseph O'Rourke | 6,094 | <p>You might look at the work of
Jason Cantarella, Robert B. Kusner, and John Sullivan, particularly
their paper, "<a href="https://doi.org/10.1007/s00222-002-0234-y" rel="nofollow noreferrer" title="zbMATH review at https://zbmath.org/01965446">On the Minimum Ropelength of Knots and Links</a>"
(<em>Invention... |
82,350 | <p>Consider three circles of radius $1$ in $\mathbb{R}^3$, linked with each other in the same arrangement as three fibers of the Hopf fibration. Now thicken the circles up into non-overlapping standard round Euclidean solid tori of equal thickness. Allowing the tori to move, there will be some maximum thickness (distan... | Ian Agol | 1,345 | <p>I think the minimizer should have dihedral symmetry. I'll give a
heuristic explanation for this suggestion.</p>
<p>Consider two linked solid tori of the same shape, such that
the outer diameter is less than twice the inner diameter of
the hole. If two such tori are tangent at a single point,
then one may separate ... |
2,233,185 | <p>I'm finding how many integers under a limit, $L$, only have prime factors from a given set of prime numbers, $P$. The numbers that meet these conditions are called n-smooth numbers. (I've never used sets before so feel free to correct any mistakes I make). Take, for example, $P = \left\{2, 3 \right\}$, and $L = 25$,... | Rafael | 303,887 | <p>Take $a\in[0,1]$ and define $$R\longrightarrow\mathbb{R}\\\ \ \ \ f\longmapsto f(a)$$
This map is a homomorphism surjective which kernel is $M_a$ (an ideal, in particular). So $R/M_a\simeq\mathbb{R}$ and therefore $M_a$ is maximal.</p>
|
1,985,849 | <p>I am really stumped by this equation.
$$e^x=x$$
I need to prove that this equation has no real roots. But I have no idea how to start.</p>
<p>If you look at the graphs of $y=e^x$ and $y=x$, you can see that those graphs do not meet anywhere. But I am trying to find an algebraic and rigorous proof. Any help is appre... | bulbasaur | 241,800 | <p>At $x=0$ , $e^x>x$ now derivative of $y=x$ equals $1$ while derivative of $e^x$ is $e^x$ which is increasing and always positive thus the two graphs never meet. </p>
|
90,548 | <p>Suppose we are handed an algebra $A$ over a field $k$. What should we look at if we want to determine whether $A$ can or cannot be equipped with structure maps to make it a Hopf algebra?</p>
<p>I guess in order to narrow it down a bit, I'll phrase it like this: what are some necessary conditions on an algebra for ... | M T | 6,481 | <p>A trivial consequence of what Vladimir says is that if $A$ is a Hopf algebra and $k$ is the trivial module (via an augmentation map $\epsilon$, then $\operatorname{Ext}_A(k,k)$ is graded commutative. It's possible to give necessary conditions for this, for example the degree one elements are graded commutative iff ... |
90,548 | <p>Suppose we are handed an algebra $A$ over a field $k$. What should we look at if we want to determine whether $A$ can or cannot be equipped with structure maps to make it a Hopf algebra?</p>
<p>I guess in order to narrow it down a bit, I'll phrase it like this: what are some necessary conditions on an algebra for ... | David E Speyer | 297 | <p>Commutative finitely generated Hopf algebras over a field of characteristic zero are regular. See <a href="http://www.ams.org/mathscinet-getitem?mr=206005" rel="nofollow">Oort</a>.</p>
|
2,666,640 | <p>Assume I have a monoid $M$ and I am not guaranteed that all elements have an inverse.</p>
<p>Say I have the property that:</p>
<p>$a^m = a^{m+n}$</p>
<p>Can I claim than it must be that $i=a^n$?</p>
<p>Why or why not?</p>
| Netchaiev | 517,746 | <p>The answer is <strong>no</strong> : take in $\mathcal{M}_2(\mathbb{R})$ with the regular matrix multiplication
$$ A= \left(
\begin{matrix}
1 & 0 \\
0 & 0
\end{matrix}
\right)$$
$$ A^{m+n}=A^m \neq I , \qquad \forall m,n>0$$</p>
|
4,164,069 | <p>I guess it's true for functions that are Lipshitz or uniformly continuous since we can limit the length of the intervals after the transformation.<br />
However, I don't know if it's true or not, and since <span class="math-container">$1/x$</span> is not one of those, I don't know how to solve this problem.</p>
| Rob Arthan | 23,171 | <p>For the record, here is the proof I had in mind when I wrote my comment. This is intended to address directly the problem implicit in the question, namely that taking reciprocals blows up the length of intervals near <span class="math-container">$0$</span>. It uses two facts: (1) a set <span class="math-container">$... |
3,474,926 | <p>I came across this question:</p>
<p>Find an isomorphism from the group of orientation preserving isometries of the plane to some subgroup of <span class="math-container">$GL_{2}(\mathbb C)$</span>.</p>
<p>I'm having trouble with finding such isomorphism. Mainly, I'm having trouble with finding some representation ... | GSofer | 509,052 | <p>We can represent any orientation preserving isometry with a single angle of rotation and a single vector of translation (or simply a complex number). We can thus use the following monomorphism into <span class="math-container">$GL(2,\mathbb C)$</span>:
<span class="math-container">$$(\alpha,x+iy)\mapsto\begin{bmatri... |
505,617 | <p>Let $a,b,c$ be nonnegative real numbers such that $a+b+c=3$, Prove that</p>
<blockquote>
<p>$$ \sqrt{\frac{a}{a+3b+5bc}}+\sqrt{\frac{b}{b+3c+5ca}}+\sqrt{\frac{c}{c+3a+5ab}}\geq 1.$$</p>
</blockquote>
<p>This problem is from <a href="http://www.artofproblemsolving.com/Forum/viewtopic.php?f=52&t=555716">http:... | Macavity | 58,320 | <p>Let $\displaystyle A = \sqrt{\frac{a}{a+3b+5bc}}+\sqrt{\frac{b}{b+3c+5ca}}+\sqrt{\frac{c}{c+3a+5ab}}$ and $\displaystyle B = \sum_{cyc}a^2(a+3b+5bc)$.</p>
<p>Then by Hölder's inequality we have $A^2B \ge (a+b+c)^3 = 27$.</p>
<p>So it is sufficient to prove that $B \le 27$</p>
<p>$$B = \sum_{cyc}a^3 + 3 \sum_{cyc}... |
2,987,071 | <blockquote>
<p>I have to show that the set <span class="math-container">$$\{1, 1 + X, (1 + X)^2 , . . . , (1 + X)^n \}$$</span> is a basis
for <span class="math-container">$\Bbb{R}_n [X]$</span>, where <span class="math-container">$\Bbb{R}_n [X]$</span> denotes the vectorspace of all polynomials of degree less tha... | Fred | 380,717 | <p>The set <span class="math-container">$B:=\{1, 1 + X, (1 + X)^2 , . . . , (1 + X)^n \}$</span> contains <span class="math-container">$n+1$</span> elements. Since <span class="math-container">$ \dim R_n[X] =n+1$</span>, you have only to show that <span class="math-container">$B$</span> is linearly independent. To this... |
414,432 | <p>We define $f_n:\mathbb{R}\to\mathbb{R}$ by $f_n(x)=\dfrac{x}{1+nx^2}$ for each $n\ge 1$.</p>
<p>I compute that $f(x):= \displaystyle\lim_{n\to \infty}f_n(x) = 0$ for each $x\in\mathbb{R}$.</p>
<p>Now, I want to know in which intervals $I\subseteq \mathbb{R}$ the convergence is uniform.</p>
<p>Any hint? Thanks.</p... | Mhenni Benghorbal | 35,472 | <p>If you use the <a href="https://math.stackexchange.com/questions/370023/how-to-prove-a-sequence-of-a-function-converges-uniformly/370071#370071">technique</a>, then the max over $x\in(-\infty,\infty)$ of the function is achieved at $x=\frac{1}{\sqrt{n}}$ and it equals $\frac{1}{2\sqrt{n}}$. So, we have</p>
<p>$$ \s... |
659,988 | <p>I understood the definition of a $\sigma$-algebra, that its elements are closed under complementation and countable union, but as I am not very good at maths, I could not visualize or understand the intuition behind the meaning of "closed under complementation and countable union". </p>
<p>If we consider the set X... | Clement C. | 75,808 | <p>Its power set, i.e. the set $2^X$ (or $\mathcal{P}(X)$ depending on the notations) of all subsets of $X$.</p>
|
894,917 | <p>I'm stuck with the limit $\lim_{n\to\infty} (2^n)\sin(n) $. I've been trying the squeeze theorem but it doesn't seem to work. I can't think of a second way to tackle the problem. Any push in the right direction would be much appreciated. </p>
<blockquote>
<p>Also, please don't just post the answer up because I wa... | Graham Kemp | 135,106 | <p>No. You cannot take the absolute value when you are looking for the limit. It's not a valid concept.</p>
<p>$2^n\sin n$ oscillates periodically between $-2^n$ and $+2^n$ as $n$ tends to infinitude. It does not converge on a limit. It does not even tend towards infinitude. No limit... |
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 ... | Hagen von Eitzen | 39,174 | <p>An interpretation of eigenvalues and eigenvectors of this matrix makes little sense because it is not in a natural fashion an endomorphism of a vector space: On the "input" side you have (liters of vodka, liters of beer) and on the putput (liters of liquid, liters of alcohol). For example, nothing speaks against swi... |
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 ... | Mohit | 181,749 | <p>Problem is very simple and interesting λ1 ≈ 1.1123 is content of pure beer alcohol volume vise and λ2≈0.2877 is content of pure vodka alcohol volume wise ($8%$ beer in one liter = $80 \mathrm{mL}$ pure beer alcohol).</p>
<p>In Volume direction has no significance so here eigenvector are pure mathematical term.</p>
|
194,867 | <p>Let $f(z)$ be a function meromorphic in a simply connected convex domain $D$ (subset of the complex plane with positive area or the whole complex plane) where $z$ is a complex number.</p>
<p>Are there such functions $f(z)$ where $\Re(f(z))$ is periodic in the domain (no periods larger than the domain please :p ) bu... | Mariano Suárez-Álvarez | 274 | <p>Suppose $f$ is meromorphic in $\mathbb C$ and its real part is periodic of period $p$. Then $z\mapsto f(z)-f(z+p)$ is a meromorphic function whose real part is identically zero. </p>
<p>Can you conclude something from this?</p>
|
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