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 |
|---|---|---|---|---|
3,403,855 | <p>Construct an example of a set of real numbers E that has no points of accumulation and yet has the property
that for every ε > 0 there exist points x, y ∈ E so that 0 < |x − y| < ε.</p>
<p>so i know we need a convergent sequence to show that the difference between two elements can be as small as we like, also... | José Carlos Santos | 446,262 | <p>You can take the sequence <span class="math-container">$\left(\sqrt n\right)_{n\in\mathbb N}$</span>, for instance. Can you prove that it works?</p>
|
3,403,855 | <p>Construct an example of a set of real numbers E that has no points of accumulation and yet has the property
that for every ε > 0 there exist points x, y ∈ E so that 0 < |x − y| < ε.</p>
<p>so i know we need a convergent sequence to show that the difference between two elements can be as small as we like, also... | heropup | 118,193 | <p>More generally, all that is required for the existence of <span class="math-container">$x,y \in E$</span> satisfying <span class="math-container">$|x-y| < \epsilon$</span> is that there exist arbitrarily close points in <span class="math-container">$E$</span>, but this does not mean that set <span class="math-con... |
4,092,643 | <p>The solutions manual says</p>
<p><span class="math-container">$$\lim_{x \to \infty}e^{-x^2}\int_{x}^{x+\frac1x}e^{t^2}dt=\lim_{x \to \infty}\frac{e^{(x+\frac1x)^2}-e^{x^2}}{2xe^{x^2}}$$</span></p>
<p>I'm trying to understand how they arrived there. Using L'Hôpital's rule rule, I have</p>
<p><span class="math-contain... | Barry Cipra | 86,747 | <p>Yes, the additional factor <span class="math-container">$(1-1/x^2)$</span> belongs there if you're doing L'Hopital. (Note, whether it's there or not, the limit turns out to be <span class="math-container">$0$</span>, so the solution manual's assertion isn't technically wrong, it could claim to just be skipping a ste... |
1,384,053 | <p>Which number is bigger? $1.01^{101}$ or $2$? and how about $e^{\pi}$ or $\pi^e$?</p>
<p>Tried some algebraic manipulations to no end, so would love some suggestions or some different ways to approach those kind of problem</p>
| Dr. Sonnhard Graubner | 175,066 | <p>HINT: we get $$e^{\pi}>\pi^{e}$$ use the function $$f(x)=\sqrt[x]{x}$$</p>
|
3,688,680 | <p>I know cantor set and rational numbers in <span class="math-container">$\mathbb{R}$</span> are meagre. But they are all zero measure.</p>
<p>So is there any meagre set that is non-zero measure?</p>
| CHAMSI | 758,100 | <p>We have : <span class="math-container">$$ \lim_{x\to 0}{\sqrt{x}\frac{\sin^{2}{x}}{x^{\frac{5}{2}}}}=\lim_{x\to 0}{\left(\frac{\sin{x}}{x}\right)^{2}}=1 $$</span></p>
<p>Thus : <span class="math-container">$$ \frac{\sin^{2}{x}}{x^{\frac{5}{2}}}\underset{x\to 0}{\sim}\frac{1}{\sqrt{x}} $$</span></p>
<p>Since <span ... |
206,318 | <p>For plots like the one shown below, what is the syntax for adding filling between particular lines and the axis, but only in the negative region:</p>
<p><a href="https://i.stack.imgur.com/X4TME.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/X4TME.png" alt="enter image description here"></a></p>
| Lukas Lang | 36,508 | <p>@Kuba's approach works using up-values, which works very nicely for simple cases as the examples given in the question. However, it does not work if the <code>want["..."]</code> expression is nested deeper inside a held expression, as noted in the comments.</p>
<p>Here is an approach using <a href="https://referenc... |
63,723 | <p>How to find $f'(a)$ where $f(x) = \sqrt{1-2x}$ ?</p>
<p>I am not too sure what to do, no matter what I do I can't get the correct answer. I know it is simple algebra but I can't figure it out.</p>
| Bill Dubuque | 242 | <p>My answer to your <a href="https://math.stackexchange.com/questions/60220/finding-lim-limits-h-to-0-frac-sqrt9h-3h/60235#60235">prior question</a> shows how to compute a more general derivative. Namely
if $\rm\ f(x)\: = \ f_0 + f_1\ (x-a) +\:\cdots\:+f_n\ (x-a)^n\:$ and $\rm\: f_0 \ne 0\:$ then rationalizing the nu... |
3,069,244 | <p>Consider the following functional <span class="math-container">$\Phi:\mathbb R^n\to\mathbb R $</span>:
<span class="math-container">$$
\Phi(x)=\sum_{i=1}^{n-1}(1+x_i)(x_i-x_n)^2(2(1+x_i+x_n)+x_i x_n-x_1).
$$</span>
The computer experiments show that it is non-negative for all <span class="math-container">$x_i\geq 0$... | Aphelli | 556,825 | <p>Use the addition formula and simplify the <span class="math-container">$\cosh(\tanh^{-1}(\cdot))$</span>, <span class="math-container">$\sinh(\tanh^{-1}(\cdot))$</span>, your integrand becomes <span class="math-container">$2\cosh(3x)(1-3x^2)+4x\sinh(3x)$</span>.</p>
<p>Note that <span class="math-container">$2\cosh... |
82,745 | <p>Is it possible to simplify the following expression involving instances of Gamma function:</p>
<p>$$E(p)=\frac{\frac{\Gamma(\frac{p+1}{2})}{\Gamma(\frac{p+2}{2})}}
{\left(\frac{\Gamma(\frac{p+1}{p})^2}{\Gamma(\frac{p+2}{p})}\right)^{\frac{p+2}{2}}}$$</p>
<p>where $p$ is rational (or even real) and $p\geq2$. The bo... | Robert Israel | 13,650 | <p>The only parts of this that can be simplified at all are
$ \Gamma \left( {\frac {p+1}{p}} \right) ={\frac {\Gamma \left( \frac{1}{p} \right) }{p}}$, and similarly for $\Gamma\left(\frac{p+2}{p}\right)$ and
$\Gamma\left(\frac{p+2}{2}\right)$</p>
|
82,745 | <p>Is it possible to simplify the following expression involving instances of Gamma function:</p>
<p>$$E(p)=\frac{\frac{\Gamma(\frac{p+1}{2})}{\Gamma(\frac{p+2}{2})}}
{\left(\frac{\Gamma(\frac{p+1}{p})^2}{\Gamma(\frac{p+2}{p})}\right)^{\frac{p+2}{2}}}$$</p>
<p>where $p$ is rational (or even real) and $p\geq2$. The bo... | Eric Naslund | 12,176 | <p>I guess it depends on what you mean by simplify. We could rewrite things in terms of (generalized) central binomial coefficients:</p>
<p>First the denominator: Notice that $$\frac{\Gamma\left(1+\frac{1}{p}\right)^{2}}{\Gamma\left(1+\frac{2}{p}\right)}=\binom{\frac{2}{p}}{\frac{1}{p}}^{-1}=\frac{1}{2p}\frac{\Gamma\... |
3,009,477 | <p>I want to determine <span class="math-container">$f(x) = x+\sin x$</span> is homeomorphic or not on <span class="math-container">$\mathbb{R}$</span>?</p>
<p>A bijective continuous function is homeomorphic if its inverse is also continuous.
I know that <span class="math-container">$f$</span> is bijective. Also <sp... | Robert Lewis | 67,071 | <p>As is pointed out in the comments, the construction which replaces <span class="math-container">$xy$</span> with</p>
<p><span class="math-container">$x \circ y = xy + yx \tag 1$</span></p>
<p>will leave the ring <span class="math-container">$\Bbb Z$</span> unitless under the multiplication operation "<span class="... |
795 | <p>Please observe the following thread <a href="https://math.stackexchange.com/questions/4489/proving-that-the-given-diophantine-equation-has-a-solution">Proving that the given Diophantine equation has a solution</a>.</p>
<p>There is a long boring argument/discussion about whether it should be posted, who should post ... | Larry Wang | 73 | <p>I think that the question of where to draw the line between how much discussion should take place in comments before it deserves to be moved elsewhere may be a difficult one, but <em>if meta-discussion of a specific question on the main site exists already</em>, discussion of things not related to the mathematical c... |
2,352,721 | <h2>Question</h2>
<blockquote>
<p>Four fair six-sided dice are rolled. The probability that the sum of the results being <span class="math-container">$22$</span> is <span class="math-container">$$\frac{X}{1296}.$$</span> What is the value of <span class="math-container">$X$</span>?</p>
</blockquote>
<h2>My Approach</h2... | Dhruv Kohli | 97,188 | <p>The bad combinations criteria is atleast one $x_i \geq 7$.</p>
<p>The number of bad combinations when:</p>
<ol>
<li><p>One of $x_i$ is forced to be greater than or equal to $7$ is $\binom{4}{1}\binom{12 + 4 - 1}{12} = 1820$.</p></li>
<li><p>Two of $x_i$'s are forced to be greater than or equal to $7$ is $\binom{4}... |
2,352,721 | <h2>Question</h2>
<blockquote>
<p>Four fair six-sided dice are rolled. The probability that the sum of the results being <span class="math-container">$22$</span> is <span class="math-container">$$\frac{X}{1296}.$$</span> What is the value of <span class="math-container">$X$</span>?</p>
</blockquote>
<h2>My Approach</h2... | Henry | 6,460 | <p>It is the coefficient of $x^{22}$ in the expansion of the generating function $$(x^6+x^5+x^4+x^3+x^2+x^1)^4=\left(\frac{x(1-x^6)}{1-x}\right)^4$$ which is $$1\cdot{{x}^{24}}+4\cdot {{x}^{23}}+10\cdot {{x}^{22}}+20\cdot {{x}^{21}}+35\cdot {{x}^{20}}+56\cdot {{x}^{19}}+80\cdot {{x}^{18}}\\+104\cdot {{x}^{17}}+125\cdot... |
2,352,721 | <h2>Question</h2>
<blockquote>
<p>Four fair six-sided dice are rolled. The probability that the sum of the results being <span class="math-container">$22$</span> is <span class="math-container">$$\frac{X}{1296}.$$</span> What is the value of <span class="math-container">$X$</span>?</p>
</blockquote>
<h2>My Approach</h2... | G Cab | 317,234 | <p>We have that
$$
\eqalign{
& x_{\,1} + x_{\,2} + x_{\,3} + x_{\,4} = 22\quad \left| {\;1 \le x_{\,k} \le 6} \right.\quad \Rightarrow \cr
& \Rightarrow \quad y_{\,1} + y_{\,2} + y_{\,3} + y_{\,4} = 18\quad \left| {\;0 \le y_{\,k} \le 5} \right. \cr}
$$
and
$$
\eqalign{
& {\rm N}{\rm .... |
2,352,721 | <h2>Question</h2>
<blockquote>
<p>Four fair six-sided dice are rolled. The probability that the sum of the results being <span class="math-container">$22$</span> is <span class="math-container">$$\frac{X}{1296}.$$</span> What is the value of <span class="math-container">$X$</span>?</p>
</blockquote>
<h2>My Approach</h2... | Filip | 764,829 | <p>Although good answers have already been given, I'd like to provide an alternative way of coming up with this, which I found <a href="https://digitalscholarship.unlv.edu/cgi/viewcontent.cgi?article=1025&context=grrj" rel="nofollow noreferrer">here on page 6</a>. </p>
<p>Define <span class="math-container">$f_i(s... |
60,326 | <p>If f is a weight 2 cuspidal newform, then it is common for L(f,1) to vanish. Indeed, the sign of the functional equation of f can force such vanishing. However, if f has weight k>2, then there is no a priori reason why L(f,1) will vanish. </p>
<p>My question: are there known examples where L(f,1)=0 for a newform... | GH from MO | 11,919 | <p>The sign of the functional equation of a self-dual cusp form (regardless its weight) can be changed easily by twisting the form with a quadratic character. In particular, the $L$-function of a self-dual cusp form (of any weight) often vanishes at the center. I don't think there is anything special about weight 2. Se... |
6,741 | <p>I would like to have a plot that is filled to the axis with <code>Green</code> if the <code>y</code> value is greater than 10, and <code>Blue</code> if it is less than 10.</p>
<p>I attempted to do the following:</p>
<p><code>Plot[x, {x, 0, 20}, Filling -> Axis,
ColorFunction ->
Function[{x, y}, Piecewis... | kglr | 125 | <p><code>ParametricPlot</code> with <code>MeshFunctions</code> + <code>MeshShading</code> +<code>Mesh</code> options gives a cleaner picture with no blending of colors at boundaries:</p>
<pre><code>ParametricPlot[{x, v x}, {x, 0, 20}, {v, 0, 1},
MeshFunctions -> {# &}, Mesh -> {{10}},
MeshShading -> {... |
946,973 | <p>After completing the square, what are the solutions to the quadratic equation below?
<span class="math-container">$$x^2 + 2x = 25$$</span></p>
<p><img src="https://i.stack.imgur.com/AoFhV.png" alt="enter image description here" /></p>
<p>Honstely I think it's B. But I'm not sure.</p>
| Michael Hardy | 11,667 | <p>One can got through the process of completing the square (as in "Timbuc"'s posted answer). But one could also check by substitution. So suppose it is proposed that $x=-1+\sqrt{26}$ is a solution. We would then have
\begin{align}
x^2 + 2x & = (-1+\sqrt{26})^2 + 2(-1+\sqrt{26}) \\[10pt]
& = (1-2\sqrt{26}+26... |
1,111,952 | <p><strong>My Try:</strong> </p>
<p>We substitute $y = x^{2/3}$. Therefore, $x = y^{3/2}$ and $\frac{dx}{dy} = \frac{2}{3}\frac{dy}{y^{1/3}}$</p>
<p>Hence, the integral after substitution is: </p>
<p>$$ \frac{3}{2} \int_0^\infty \sin(y)\sqrt{y} dy$$</p>
<p>Let's look at:</p>
<p>$$\int_0^\infty \left|\sin(y)\sqrt{... | Petite Etincelle | 100,564 | <p>By $y = x^{2/3}$, we get $$\int_0^\infty \sin(x^{2/3})dx = \frac{3}{2}\int_0^\infty \sin(y)\sqrt{y}dy$$</p>
<p>Consider the integral over the interval $[2n\pi, 2(n+1)\pi]$, we have
$$
\begin{align}
\int_{2n\pi}^{2(n+1)\pi} \sin(y)\sqrt{y}dy &= \int_{2n\pi}^{(2n+1)\pi}\sin(y)\sqrt{y}dy + \int_{(2n+1)\pi}^{(2n+2... |
1,111,952 | <p><strong>My Try:</strong> </p>
<p>We substitute $y = x^{2/3}$. Therefore, $x = y^{3/2}$ and $\frac{dx}{dy} = \frac{2}{3}\frac{dy}{y^{1/3}}$</p>
<p>Hence, the integral after substitution is: </p>
<p>$$ \frac{3}{2} \int_0^\infty \sin(y)\sqrt{y} dy$$</p>
<p>Let's look at:</p>
<p>$$\int_0^\infty \left|\sin(y)\sqrt{... | Empy2 | 81,790 | <p>$\sin x^{2/3}$ remains above $1/2$ for $x$ between $[(2n+1/6)\pi]^{3/2}$ and $[2n+5/6]^{3/2}$, so the integral rises by more than $\left([2n+5/6]^{3/2}-[2n+1/6]^{3/2}\right)\pi^{3/2}/2$ during that time.<br>
$$[2n+5/6]^{3/2}-[2n+1/6]^{3/2}=\frac{[2n+5/6]^3-[2n+1/6]^3}{[2n+5/6]^{3/2}+[2n+1/6]^{3/2}}\\
>\frac{8n^2}... |
1,111,952 | <p><strong>My Try:</strong> </p>
<p>We substitute $y = x^{2/3}$. Therefore, $x = y^{3/2}$ and $\frac{dx}{dy} = \frac{2}{3}\frac{dy}{y^{1/3}}$</p>
<p>Hence, the integral after substitution is: </p>
<p>$$ \frac{3}{2} \int_0^\infty \sin(y)\sqrt{y} dy$$</p>
<p>Let's look at:</p>
<p>$$\int_0^\infty \left|\sin(y)\sqrt{... | LearnTitan | 209,235 | <p>I will try to look at the problem from a different perspective and to understand exactly why it will <strong>not converge.</strong></p>
<ol>
<li><p>The first thing we should know is that if the integral converges the terms must get <strong>smaller</strong> after each "iteration".</p></li>
<li><p>That means that the... |
1,158,956 | <p>To show that orthogonal complement of a set A is closed.</p>
<p>My try: I first show that the inner product is a continuous map. Let $X$ be an inner product space. For all $x_1,x_2,y_1,y_2 \in X$, by Cauchy-Schwarz inequality we get,
$$|\langle x_1,y_1\rangle - \langle x_2,y_2\rangle| = |\langle x_1- x_2,y_1\rangle... | Alfredo Lozano | 286,630 | <p>I really like your proof, so formalizing it we have:</p>
<p>Let be $\{y_n\}_{n=1}^\infty \in A^\perp$ s.t. $y_n \to y$ and let be $x \in A$.</p>
<p>We now want to show that $y\in A^\perp$.</p>
<p>From the inner product's continuity we have:</p>
<p>$\forall \epsilon>0\ ,\exists\ \delta>0$ such that:</p>
<p... |
514,912 | <p>I have what may seem a very trivial question, but how it is answered may affect how a proof of mine is structured. It pertains to formatting and convention. When 'recursively' defining a function does it make sense to use quantifiers? </p>
<p>For example would:</p>
<p>$ 5 \in R $</p>
<p>If $ r \in R $, then $ \fo... | DonAntonio | 31,254 | <p>Well, define</p>
<p>$$x=\sqrt{3\sqrt{5\sqrt3\ldots}}=\sqrt{3\sqrt{5x}}\stackrel{\text{square}}\implies x^2=3\sqrt{5x}\stackrel{\text{square again}}\implies x^4=45x\implies$$</p>
<p>$$x^3=45\iff x=\sqrt[3]{45}$$</p>
<p><strong>But</strong>...the above follows at once from arithmetic of limits, on the sequence</p>
... |
3,375,459 | <p>I am trying to study differential geometry.</p>
<p>I am confused with regards to the following function for finding the length of
a curve <span class="math-container">$\gamma$</span> connecting two points <span class="math-container">$p, q ∈ S^2$</span></p>
<p><span class="math-container">$$L(γ) = \int^1_0|\dot{γ}... | Kavi Rama Murthy | 142,385 | <p>Hint: you need <span class="math-container">$a <\frac 1 2$</span> for finiteness of the expectation. To evaluate it just make the substitution <span class="math-container">$x=\sqrt {1-2a}z$</span>. I will let you do the calculations. The answer is <span class="math-container">$e^{c+\frac {b^2} {2(1-2a)}}$</span>... |
418,647 | <p>Sorry if the question is dumb. I am trying to learn representation theory of finite groups from J.P.Serre's book by myself. In section 2.6 on canonical decomposition, he says that let V be a representation of a finite group G, $W_1,...,W_h$ be the distinct irreducible representations of G, and let V = $U_1 \oplus ..... | nakajuice | 22,196 | <p>I assume your graph can't have parallel edges or loops, otherwise statement is true since you can connect two arbitrary vertices with all the 17 edges and 5 vertices left are isolated.</p>
<p>So in this case the maximum number of edges in undirected graph of n vertices is ${n \choose 2}$ (number of pairs of vertice... |
2,942,244 | <p>I need to find unmeasurable dense subset of circle. I think, that I found the unmeasurable set, but I can't show that it is dense. Here is my construction. Let's take <span class="math-container">$\alpha\in\mathbb{R}\setminus\mathbb{Q}$</span>, consider the irrational rotation of the circle on angle <span class="mat... | Henno Brandsma | 4,280 | <p>Let $V$ be a Vitali set in $[0,1)$ ( by AC choose a representative for each class of the relation $x\sim y$ iff $x-y\in\mathbb{Q}$) and take its image under the continuous bijection $t \to (\cos(2\pi t, \sin 2\pi t)$. This is as required. </p>
|
556,150 | <blockquote>
<p>Prove that a metric space is totally bounded if and only if every sequence has a Cauchy subsequence. </p>
</blockquote>
<p>I think I proved the Cauchy subsequence part:</p>
<p>Let $a_{0},a_{1}, a_{2}, a_{3}, a_{4},...\in X$ be a sequence.</p>
<p>For each $k$, let $F \subseteq X$ be a finite $\frac1... | Community | -1 | <p>The stated proof of implication totally bounded $\implies $ Cauchy subsequence is more like a sketch of the approach than a complete proof. Here is how it develops: </p>
<p>Suppose $(a_n)$ is a sequence in a totally bounded metric space $X$. For each $k$, choose a finite $(1/k)$-net in $X$ and call it $F_k$. </p>
... |
1,353,922 | <p>A function $f$ is continuous for all $x \geq 0$ and $f(x) \neq 0$ for all $x >0$. </p>
<p>If $\{f(x)\}^2 = 2\int_0^xf(t)dt $ then $f(x) = x$ for all $x \geq 0$.</p>
<p>But I am stuck with the sum. </p>
| Spenser | 39,285 | <p>By the Fundamental Theorem of Calculus, the function
$$x\mapsto\int_0^xf(t)dt$$
is differentiable since $f$ is continuous. Hence, $(f(x))^2=2\int_0^xf(t)dt$ is differentiable. Since, $f(x)\neq 0$ for $x>0$ and $\sqrt{}$ is differentiable on $(0,\infty)$, the composition
$$\sqrt{(f(x))^2}=|f(x)|=f(x)\tag{1}$$
is d... |
3,406,817 | <p>What intrinsic relationship is there between (riemann) integration and Euclidean geometry that enables one to get area under the curve as a integral? This can only be related to the definition of riemann integral as a limit of sums of areas of rectangles; or is it something else.</p>
| Arthur | 15,500 | <p>A bit tongue-in-cheek, the intrinsic relationship can be seen as "Riemannian integration was constructed specifically for this connection to work the way it does."</p>
|
3,406,817 | <p>What intrinsic relationship is there between (riemann) integration and Euclidean geometry that enables one to get area under the curve as a integral? This can only be related to the definition of riemann integral as a limit of sums of areas of rectangles; or is it something else.</p>
| user376343 | 376,343 | <p>Reffer to <em>method of indivisibles</em> by Bonaventura Cavalieri (1598-1647). This method was used and learned at universities before the discovery of Riemannian integration </p>
|
553,040 | <p>The question is: A card is drawn from an ordinary pack(52 cards) and a gambler bets that either a spade or an ace is going to appear. The probability of his winning are?
I think the answer is $\frac{16}{52} = \frac{4}{13}$. Did I go "probably" go wrong somewhere?</p>
| Satish Ramanathan | 99,745 | <p>There are 13 spades that contain one ace and there are remaining three aces, thus there are altogether 16 choices for the bet and the probability = 16/52 and you are correct.</p>
|
479,851 | <p>I toss a coin a many times, each time noting down the result of the toss. If at any time I have tossed more heads than tails I stop.
I.e. if I get heads on the first toss I stop.
Or if toss T-T-H-H-T-H-H I will stop.
If I decide to only toss the coin at most 2n+1 times, what is the probability that I will get more ... | Kolmo | 20,974 | <p>I will expand my comment.</p>
<p>Set $N=2 n+1$ and $S_k= \sum_{i=1}^k X_i $, where the $X_i$ are i.i.d.r.v. assuming values $+1, -1$ with equal probability ($S_0=0$ a.s.). This is the simple random walk. The idea is that we associate to head a +1 and to tail a -1.</p>
<p>What you are looking for is the probability... |
2,932,139 | <blockquote>
<p>How would I show that the line <span class="math-container">$A=[(x,y,z)=(0,t,t)\mid t\in\mathbb{R}]$</span> is parallel to the plane <span class="math-container">$5x-3y+3z=1$</span>?</p>
</blockquote>
<p>I know the normal vector would be <span class="math-container">$(5,-3,3)$</span>, but how would I... | Bernard | 202,857 | <p>A directing vector of the line is <span class="math-container">$(0,1,1)$</span>since the vector equation of the line is
<span class="math-container">$$\overrightarrow{OM}=t\,(0,1,1).$$</span><br>
This directing vector has to be orthogonal to the normal vector of the plane.</p>
|
3,465,171 | <blockquote>
<p>Let <span class="math-container">$G=[0,\infty) \times [0,\infty)$</span>, <span class="math-container">$\alpha \in (0,1)$</span> and <span class="math-container">$$\phi (x,y)=x^{\alpha} y^{1-\alpha}$$</span> Then <span class="math-container">$\phi$</span> is concave; that is, <span class="math-contain... | Community | -1 | <p>Hint: <span class="math-container">$(z^2+1)=(z+i)(z-i)$</span>. Construct a half circle contour from -R to R in the real axis, where <span class="math-container">$R>1$</span>. Do not differentiate twice. Express your function , <span class="math-container">$f(z)=\frac{g(z)}{(z-i)^2}$</span>. Then observe that g i... |
4,014,768 | <blockquote>
<p>Find the sationary points of the curve and their nature for the equation <span class="math-container">$y=e^x\cos x$</span> for <span class="math-container">$0\le x\le\pi/2$</span>.</p>
</blockquote>
<p>I derived it and got <span class="math-container">$e^x(-\sin x+\cos x)=0$</span>.</p>
<p><span class="... | MathMinded | 881,969 | <p>We can first simplyfy the equation to have;
<span class="math-container">$$\sin{x}=\cos{x}$$</span>
<span class="math-container">$$\implies\sin^2{x}=\cos^2{x}.$$</span></p>
<p>Adding <span class="math-container">$\cos^2{x}$</span> to both sides gives;
<span class="math-container">$$\sin^2{x}+\cos^2{x}=\cos^2{x}+\cos... |
18,136 | <p>I introduced the hypercube (to undergraduate students in the U.S.) in the
context of generalizations of the Platonic solids, explained its
structure, showed it rotating.
I mentioned <a href="https://matheducators.stackexchange.com/a/1824/511">Alicia Stott</a>, who discovered the <span class="math-container">$6$</spa... | user52817 | 1,680 | <p>I highly recommend "Flatland The Movie." Your institution should be able to purchase it. You can find a free trailer on the internet.</p>
<p>When I was young, I read the book "Flatland: A Romance of Many Dimensions," probably in high school, and it made me "grok" the fourth dimension. </p>
|
1,519,070 | <p>How can I show that the sequence is defined recursively?</p>
<blockquote>
<p>Show that the recursively defined sequence $(x_n)_{n\in\mathbb{N}}$ with
$$x_1=1, \qquad\qquad x_{n+1}=\sqrt{6+x_n}$$
converges and determine its limit </p>
</blockquote>
<p><a href="https://i.stack.imgur.com/cL7BR.png" rel="nofollo... | Gamow | 210,297 | <p>The sequence is defined recursively, with initial value $x_1=1$ and
with recursive relation $x_{n+1}=\sqrt{6+x_n}$. Your task is to prove that it converges and to determine its limit.</p>
<p>Convergence: There is a theorem that says that if a sequence is monotonically increasing and bounded from above, then it has... |
1,328,082 | <p>I've been trying to prove the following inequality, but until now I've had problems coming up with a solution:</p>
<p>$$
2^{mn} \ge m^n
$$</p>
<p>$m$ and $n$ can assume any natural number.</p>
<p>I wasn't able to find any counterexample that would invalidate this inequality, so I am assuming that this statement i... | marty cohen | 13,079 | <p>More a comment than a solution,
but it points to a more general result
(one that I found independently
and am pleased with):</p>
<p>In
<a href="https://math.stackexchange.com/questions/439026/prove-that-nk-2n-for-all-large-enough-n">Prove that $n^k < 2^n$ for all large enough $n$</a>,
I showed that
if $n$ and $... |
2,076,831 | <blockquote>
<p>Is the following set path-connected?</p>
<p><span class="math-container">$A=\{(x,y):y=x\sin \frac{1}{x},x>0\}$</span></p>
</blockquote>
<p>I am unable to understand that should I prove it or disprove it .</p>
<p>Will someone please give me some hints.</p>
| kub0x | 309,863 | <blockquote>
<p>Also, please explain why this isn't one way: $f(x) = x^x$. Thanks in advance.</p>
</blockquote>
<p>The definition of a one-way function is that is easy to perform on every input but hard to invert knowing only the calculations on inputs (images). In your case the calculation on every input becomes im... |
2,585,828 | <p>I saw in an article that for every real number, there exists a Cauchy sequence of rational numbers that converges to that real number. This was stated without proof, so I'm guessing it is a well-known theorem in analysis, but I have never seen this proof. So could someone give a proof (preferably with a source) for ... | Duncan Ramage | 405,912 | <p>One way to define the real numbers is as equivalence classes of Cauchy sequences (see exercise 3.24 in Princples of Mathematical Analysis by Walter Rudin for the details of the (actually, a more general) construction). Under this construction, it is obvious that every real number has such a Cauchy sequence, since ev... |
2,585,828 | <p>I saw in an article that for every real number, there exists a Cauchy sequence of rational numbers that converges to that real number. This was stated without proof, so I'm guessing it is a well-known theorem in analysis, but I have never seen this proof. So could someone give a proof (preferably with a source) for ... | fleablood | 280,126 | <p>Let $x \in \mathbb R$. </p>
<p>Let $q_i \in \mathbb Q$ so that $x - \frac 1i < q_i < x$.</p>
<p>Claim 1: Such a sequence $\{q_i\}$ exists (because the rationals are dense in the reals)</p>
<p>Claim 2: $q_i \to x$ (by the very definition of limits)</p>
<p>Claim 3: $\{q_i\}$ is cauchy. (For $n > \frac 2... |
3,138,450 | <p>Let <span class="math-container">$A \subseteq \mathbb{R}$</span> be bounded above and let <span class="math-container">$c \in \mathbb
{R}$</span>. Define the set <span class="math-container">$c + A = \{c + a : a \in A\} $</span></p>
<p>Now since <span class="math-container">$a \leq \sup A , \forall a\in A$</span>. ... | Theo Bendit | 248,286 | <p>If you wish to use <span class="math-container">$\varepsilon$</span>s instead, suppose <span class="math-container">$\varepsilon > 0$</span>. I claim that <span class="math-container">$c + \sup A - \varepsilon$</span> is <strong>not</strong> and upper bound for <span class="math-container">$c + A$</span>. Since y... |
391,509 | <p>We have $$\dfrac{1+2+3+...+ \space n}{n^2}$$</p>
<p>What is the limit of this function as $n \rightarrow \infty$?</p>
<p>My idea:</p>
<p>$$\dfrac{1+2+3+...+ \space n}{n^2} = \dfrac{1}{n^2} + \dfrac{2}{n^2} + ... + \dfrac{n}{n^2} = 0$$</p>
<p>Is this correct?</p>
| Clement C. | 75,808 | <p>Another method, using Riemann sums applied to $f:x\mapsto x$:
$$
\frac{\sum_{k=1}^n k }{n^2} = \frac{1}{n}\sum_{k=1}^n \frac{k }{n} \xrightarrow[n\to\infty]{}
\int_{0}^{1} xdx
$$</p>
<p>As for your question, each term goes to $0$, <em>but</em> the number of terms grows; you are not in the case where you have the s... |
4,621,227 | <p>This is a very soft and potentially naive question, but I've always wondered about this seemingly common phenomenon where a theorem has some method of proof which makes the statement easy to prove, but where other methods of proof are incredibly difficult.</p>
<p>For example, proving that every vector space has a ba... | Brauer Suzuki | 960,602 | <p>Burnside's theorem that every finite group of order <span class="math-container">$p^aq^b$</span> (where <span class="math-container">$p,q$</span> are primes) is solvable has a short proof using character theory and a much longer proof without characters. See M. Isaacs' books for both proofs.</p>
|
3,091,162 | <p>Let <span class="math-container">$G_1, G_2$</span> be two groups with at least one nontrivial proper subgroup each.</p>
<p>Let <span class="math-container">$S_1, S_2$</span> be the sets of proper subgroups of, respectively <span class="math-container">$G_1, G_2$</span>.</p>
<p>Suppose there exists a bijective func... | the_fox | 11,450 | <p>Certainly not always. I'd be surprised if there is a concrete set of conditions which is both necessary and sufficient to conclude isomorphism between the two groups. (My answer refers to finite groups only.)</p>
<p>There are groups which are called <span class="math-container">$P$</span>-groups in Schmidt's book "... |
3,091,162 | <p>Let <span class="math-container">$G_1, G_2$</span> be two groups with at least one nontrivial proper subgroup each.</p>
<p>Let <span class="math-container">$S_1, S_2$</span> be the sets of proper subgroups of, respectively <span class="math-container">$G_1, G_2$</span>.</p>
<p>Suppose there exists a bijective func... | Lucio Tanzini | 521,474 | <p>Here is the proof in the case of finite abelian groups <span class="math-container">$G_1, G_2$</span> like above.</p>
<p><strong>Lemma 1</strong>
Let <span class="math-container">$G^{(n)}$</span> be the number of elements in <span class="math-container">$G$</span> of order <span class="math-container">$n$</span>. <... |
3,783,186 | <p>I am trying to prove that <span class="math-container">$$2≤\int_{-1}^1 \sqrt{1+x^6} \,dx ≤ 2\sqrt{2} $$</span> I learned that the equation <span class="math-container">$${d\over dx}\int_{g(x)}^{h(x)} f(t)\,dt = f(h(x))h'(x) - f(g(x))g'(x) $$</span> is true due to Fundamental Theorem of Calculus and Chain Rule, a... | Integrand | 207,050 | <p>Surprised not to see this technique yet.</p>
<p>For <span class="math-container">$|z|< 1$</span> (the most conservative convergence case), the generalized binomial theorem states
<span class="math-container">$$
(1+z)^a = \sum_{k=0}^{\infty} \binom{a}{k}z^{k}
$$</span>In particular, for <span class="math-container... |
643,237 | <p>This statement is suggested as a correction to <a href="https://math.stackexchange.com/questions/643122/splitting-field-containing-nth-root">this question</a>:</p>
<p>If $K$ is the splitting field of the polynomial $P(x)=x^n-a$ over $\mathbb{Q}$, prove that $K$ contains all the $n$th roots of unity.</p>
<p>How to ... | monroej | 121,862 | <p>If $K$ is the splitting field for that polynomial then it contains all $n$th roots of $a$, as you know. So if $\omega$ is an primitive $n$th root of unity, then the roots of $P(x)=x^n-a$ are $\sqrt[n]{a},\sqrt[n]{a}\omega,...,\sqrt[n]{a}\omega^{n-1}$ just as JJ Beck said. We can multiply $\sqrt[n]{a}\omega$ by $\f... |
2,709,878 | <p>How do I know that an equation will have an extraneous solution? </p>
<p>For example in this question:
2log9(x) = log9(2) + log9(x + 24)</p>
| Jyrki Lahtonen | 11,619 | <p>If some step in your process of solving the equation was only a logical implication (rather than an equivalence), then the process may have introduced extraneous solutions, and you need to check all of them to be sure.</p>
<p>Typically the difference is that a step in the solution is not reversible. For example: if... |
632,850 | <blockquote>
<p>Suppose that $f(x)$ is differentiable on $[0,1]$ and $f(0) = f(1) = 0$. It is also
known that $|f''(x)| \le A$ for every $x \in (0,1)$. Prove that
$|f'(x)| \le A/2$ for every $x \in [0,1]$.</p>
</blockquote>
<p>I'll explain what I did so far. First using Rolle's theorem, there is some point $c \i... | Feu | 120,558 | <p>I've already give an answer without Integral to this question in another post. I'll duplicate the answer here. If it's against policy please tell me what I should do.</p>
<blockquote>
<p>I manage an answer using both time a taylor expansion center at a point $x \in \left [0, 1 \right ]$. I get
$$
f(h) = f(x)+f... |
4,474,095 | <p>There is one thing I can't grasp about the proof given in the Linear Algebra Done Right book by Sheldon Axler (attached below).</p>
<p>In the last part it says that <span class="math-container">$(T - \lambda_1I)...(T - \lambda_mI)v = 0$</span>, hence <span class="math-container">$T - \lambda_jI$</span> is not inject... | Acccumulation | 476,070 | <blockquote>
<p>the factors in the equation can be reordered.</p>
</blockquote>
<p>The factors in the characteristic polynomial can be re-ordered, but factors can't in general be, and it's not entirely trivial to prove that they can be in this case.</p>
<blockquote>
<p>hence v is an eigenvector of T corresponding to th... |
1,107,932 | <p>Is the map $f:S_n \to A_{n+2}$ given by </p>
<p>$$f(s)= \begin{cases}
s & s\ \text{is even}\\
s \circ (n+1,\ n+2) & s\ \text{is odd}
\end{cases}$$ </p>
<p>an injective homomorphism? I can show that if it is a homomorphism then it is injective but having difficulty in showing that $f$ is a homomorphism. Pl... | Michael Albanese | 39,599 | <p>A map $f : S_n \to A_{n+2}$ is a homomorphism if $f(\sigma\circ\tau) = f(\sigma)\circ f(\tau)$ for every $\sigma, \tau \in S_n$. As $f$ is defined piecewise, we split this verification into cases.</p>
<ol>
<li>Both $\sigma$ and $\tau$ are even. Note that $\sigma\circ\tau$ is even so </li>
</ol>
<p>$$f(\sigma\circ\... |
1,107,932 | <p>Is the map $f:S_n \to A_{n+2}$ given by </p>
<p>$$f(s)= \begin{cases}
s & s\ \text{is even}\\
s \circ (n+1,\ n+2) & s\ \text{is odd}
\end{cases}$$ </p>
<p>an injective homomorphism? I can show that if it is a homomorphism then it is injective but having difficulty in showing that $f$ is a homomorphism. Pl... | Ben Blum-Smith | 13,120 | <p>One more way to think about it:</p>
<p>Conceive of your map $f$ as actually being from $S_n$ to $S_{n+2}$. The image lies in the subgroup $S_n\times S_2$ consisting of permutations that act independently on the first $n$ and the last $2$ indices. You can see your map as being the identity map to the first factor ($... |
836,753 | <p>$\{X_1, X2, \ldots, X_{121}\}$ are independent and identically distributed random variables such that $E(X_i)= 3$ and $\mathrm{Var}(X_i)= 25$. What is the standard deviation of their average? In other words, what is the standard deviation of $\bar X= {X_1+ X_2+ \cdots + X_{121} \over 121}$?</p>
| anomaly | 156,999 | <p>Since the $X_i$ are independent, their variance is additive: $\text{Var}(X_1 + \cdots + X_n) = \text{Var}(X_1) + \cdots + \text{Var}(X_n)$. Furthermore, $\text{Var}(\lambda X_i) = \lambda^2\text{Var}(X_i)$ for any constant $\lambda$. Thus the mean
$$\overline{X} = \frac{X_1 + \cdots + X_n}{n}$$
satisfies
$$\text{Var... |
3,921,259 | <p>I need to calculate the operator Norm of the linear operator defined as:
<span class="math-container">$$ T:(C([0,1]),||\cdot||_\infty)\rightarrow\mathbb{R} \text{ where } Tf=\sum_{k=1}^n a_kf(t_k)$$</span>
for <span class="math-container">$0\leq t_1<t_2<...<t_n\leq 1$</span> and <span class="math-container"... | copper.hat | 27,978 | <p>It might help to separate <span class="math-container">$T$</span> into the composition of two simpler maps.</p>
<p>Note that the map <span class="math-container">$s:C[0,1] \to \mathbb{R}^n$</span> defined by
<span class="math-container">$s(f) = (f(t_1),...,f(t_n))$</span> has norm one (using the <span class="math-co... |
4,417,353 | <p><a href="https://i.stack.imgur.com/td05G.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/td05G.png" alt="Here is the question. I sent it as a image since I could not find how to write a1 as below." /></a></p>
<p>My first approach is that I need to use induction on that. However the problem is that... | José Carlos Santos | 446,262 | <p>For each <span class="math-container">$n\in\Bbb N$</span>, <span class="math-container">$a_n\leqslant2$</span>. This can be proved by induction: <span class="math-container">$a_1=\sqrt2<2$</span> and if <span class="math-container">$a_n<2$</span>, then <span class="math-container">$a_{n+1}=\sqrt2^{a_n}<\sqr... |
2,414,721 | <p>For all perfect numbers $N$, $\sigma (N) = 2N$, where $\sigma$ is the divisor sigma function.</p>
<p>Let $s$ be a perfect number of the form $3^m 5^n 7^k$, where $m,n,k \geq 1$ are integers.</p>
<p>Then $\sigma (s)= \sigma (3^m 5^n 7^k)$</p>
<p>$ =\sigma (3^m) \sigma (5^n) \sigma (7^k)$ since $3, 5,$ and $7$ are ... | user357980 | 357,980 | <p>Intuitively speaking, the left hand side is generally way smaller than the right hand side because of all the exponentials, so diving by those exponentials, $3^{m+1}, 5^{n+1}, 7^{k+1}$ is a good way to exploit that behavior.</p>
<p>Then you basically play around to find that no values for $m, n$, and $k$ work. (I c... |
464,586 | <blockquote>
<p>Find a basis for $U=\{A\in\mathbb{M}_{22}\mid A^T=-A\}$.</p>
</blockquote>
<p>$\mathbb{M}_{22}$ denotes the set of all $2 \times 2$ matrices. This question appeared on an examination I wrote yesterday. Does a basis even exist? I can't think of any matrices in which $A^T=-A$ except for the zero matrix... | Cameron Buie | 28,900 | <p><strong>Hint</strong>: The diagonal entries must be $0$ and the other entries must be opposite. (Why?) Can you think of a non-zero matrix with this form? What relationship will other elements of $U$ have to this matrix?</p>
<hr>
<p>It is true that the only element of $U$ having the form $$\begin{bmatrix}a&b\\... |
674,621 | <p>I am trying to figure out what the three possibilities of $z$ are such that </p>
<p>$$
z^3=i
$$</p>
<p>but I am stuck on how to proceed. I tried algebraically but ran into rather tedious polynomials. Could you solve this geometrically? Any help would be greatly appreciated.</p>
| Lubin | 17,760 | <p>The answer of @Petaro is best, because it suggests how to deal with such questions generally, but here’s another approach to the <em>specific</em> question of what the cube roots of $i$ are.</p>
<p>You know that $(-i)^3=i$, and maybe you know that $\omega=(-1+i\sqrt3)/2$ is a cube root of $1$. So the cube roots of ... |
1,443,471 | <p>I have this conjecture: Let <em>a</em> and <em>b</em> be integers and <em>n</em> and <em>m</em> natural numbers. </p>
<p>$$ a \equiv b \bmod n \Rightarrow a^m \equiv b^m \bmod n$$</p>
<p>I think I got the induction proof, but I'm having difficulties on how to proof this with well-ordering principle.</p>
| Cloudscape | 180,668 | <p>Let $k$ be the smallest natural number such that $a^k \not\equiv b^k \mod n$ (such a number exists by the well-ordering principle). We have that $a \equiv b \mod n$ (hence $k \ge 2$). If now $a^{k-1} \equiv b^{k-1} \mod n$, then</p>
<p>$$
a^k \equiv a^{k-1} a \equiv b^{k-1} a \equiv b^{k-1} b \equiv b^k \mod n,
$$<... |
3,828,205 | <blockquote>
<p>Find all integers <span class="math-container">$x$</span> and <span class="math-container">$y$</span> that satisfy <span class="math-container">$$x^4-12x²+x^2y^2+30 < 0$$</span></p>
</blockquote>
<p>Letting <span class="math-container">$a = x^2$</span> and <span class="math-container">$b = y^2$</span... | Michael Rozenberg | 190,319 | <p><span class="math-container">$$x^2y^2<6-(x^2-6)^2\leq6-2^2=2,$$</span> which says <span class="math-container">$$x^2y^2=0$$</span> or <span class="math-container">$$x^2y^2=1.$$</span>
The last is impossible because <span class="math-container">$x^2=4.$</span></p>
<p>Can you end it now?</p>
|
198,049 | <p>Let $S$ be a smooth projective surface over $\mathbb{C}$. (I guess this can be more general—higher dimension, other ground fields, non-projective, maybe even singular?—and I'dd like to hear that.) Let $s \in S$ be a point. Let $\beta \colon X \to S$ be the blowup of $s \in S$. Suppose that $H^{i}(S, T_{S})$ is known... | Francesco Polizzi | 7,460 | <p>The answer is the following and can be found in Hartshorne's book <em>Deformation Theory</em>, see in particular Exercise 10.5 page 83.</p>
<p>We work over an algebraically closed field $k$. Then there is an exact sequence of sheaves $$0 \to \beta_*T_X \to T_S \to k_s \oplus k_s \to 0,$$ inducing an exact sequence... |
3,754,734 | <p>Let <span class="math-container">$ \sum_{n=0}^{\infty}a_{n}x^{n} $</span> be a power series. prove that</p>
<p><span class="math-container">$ \sum_{n=0}^{\infty}a_{n}x^{n},\sum_{n=0}^{\infty}\left(n+1\right)a_{n+1}x^{n} $</span></p>
<p>has the same convergence radius.</p>
<p>So, let <span class="math-container">$ \l... | Eric Towers | 123,905 | <p>You don't say where your indices start, so I pick <span class="math-container">$1$</span>. Altering this is easy.</p>
<p><span class="math-container">$$ \left( \sqrt[n]{|a_n|} \right)_{n \geq 1} = (|a_1|, \sqrt{a_2}, \sqrt[3]{a_3}, \dots ) \text{ and } $$</span>
<span class="math-container">$$ \left( \sqrt[n+1]... |
3,754,734 | <p>Let <span class="math-container">$ \sum_{n=0}^{\infty}a_{n}x^{n} $</span> be a power series. prove that</p>
<p><span class="math-container">$ \sum_{n=0}^{\infty}a_{n}x^{n},\sum_{n=0}^{\infty}\left(n+1\right)a_{n+1}x^{n} $</span></p>
<p>has the same convergence radius.</p>
<p>So, let <span class="math-container">$ \l... | ir7 | 26,651 | <p>Based on <a href="https://math.stackexchange.com/questions/113121/lim-sup-inequality-limsup-a-n-b-n-leq-limsup-a-n-limsup-b-n">this general result</a> (the equality part), we indeed have:</p>
<p><span class="math-container">$$
\limsup_{n\to\infty}\sqrt[n]{(n+1)|a_{n+1}|}=\lim_{n\to\infty}\sqrt[n]{n+1}\limsup_{n\to\... |
189,218 | <p>I have raw images that I can import and view with ImageJ. They are the output of a program which I don't have the source code for, so I'm stuck with the output format.</p>
<p>They are imported as follows with ImageJ:</p>
<blockquote>
<p>Image type: 16 bit Unsigned</p>
<p>Width: 320</p>
<p>Height: 25600</p>
<p>Offset... | Henrik Schumacher | 38,178 | <pre><code>images = Image /@ ArrayReshape[Import[file, "UnsignedInteger16"], {100, 256, 320}];
</code></pre>
|
308,436 | <p>$P$ is probability. We have: $P(A) \ge \frac{2}{3}$, $P(B) \ge \frac{2}{3}$, $P(C) \ge \frac{2}{3}$ and $P(A \cap B \cap C)=0$. We have to find $P(A)$.
How to do it? Of course we have $P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B)- P(A \cap C)- P(B \cap C)$, but what next? Please, help me.</p>
| Did | 6,179 | <p><strong>Hints:</strong> </p>
<ul>
<li>If $\mathbb P(B)\geqslant\frac23$ and $\mathbb P(C)\geqslant\frac23$, then $\mathbb P(B\cap C)\geqslant$ $____$. </li>
<li>If $\mathbb P(A)=\frac23+c$ with $c\gt0$ and $\mathbb P(B\cap C)\geqslant$ $____$, then $\mathbb P(A\cap(B\cap C))\geqslant$ $____$.</li>
<li>But the hypot... |
1,524,109 | <p>Could anyone help me with this proof without using determinant? I tried two ways. </p>
<blockquote>
<p>Let $A$ be a matrix. If $A$ has the property that each row sums to zero, then there does not exist any matrix $X$ such that $AX=I$, where $I$ denotes the identity matrix. </p>
</blockquote>
<p>I then get stuck.... | John Bentin | 875 | <p>If each row of $A$ sums to zero, then each row of the column vector that is the sum of the column vectors constituting $A$ is zero. So the columns of $A$ are not linearly independent, and therefore the matrix is singular (i.e. it has no inverse).</p>
|
1,260,722 | <blockquote>
<p>Prove that <span class="math-container">$f=x^4-4x^2+16\in\mathbb{Q}[x]$</span> is irreducible.</p>
</blockquote>
<p>I am trying to prove it with Eisenstein's criterion but without success: for <strong>p=2</strong>, it divides <strong>-4</strong> and the constant coefficient 16, don't divide the leading ... | marwalix | 441 | <p>$$x^4-4x^2+16=(x^2-(2+\sqrt{-12}))(x^2-(2-\sqrt{-12}))$$</p>
<p>No rational roots and no factorization into quadratics over the rationals. The polynomial is irreducible over the rationals</p>
<p><strong>edit</strong> for those who commented that this is not enough. I factorized over $\mathbb{C}[X]$ and thus proved... |
3,719,575 | <p>For <span class="math-container">$\lambda =0$</span> and <span class="math-container">$\lambda <0$</span> the solution is the trivial solution <span class="math-container">$x\left(t\right)=0$</span></p>
<p>So we have to calculate for <span class="math-container">$\lambda >0$</span></p>
<p>The general solution... | Aditya Dwivedi | 697,953 | <p>By saying <strong><span class="math-container">$x(L) = 0$</span></strong>
you mean <span class="math-container">$\sqrt{\lambda}*L = n \pi$</span></p>
|
2,262,001 | <p>Prove this inequality where $a$, $b$ and $c$ are sides of a triangle and $S$ its Area.
$$\frac{ab + bc + ca}{4S}\ge \operatorname{ctg} \frac{\pi}{6}$$</p>
| Michael Rozenberg | 190,319 | <p>Since $ab+ac+bc\geq\sum\limits_{cyc}(2ab-a^2)$ it's just $\sum\limits_{cyc}(a-b)^2\geq0$ and
$$\sum_{cyc}(2ab-a^2)=\sum_{cyc}a(b+c-a)>0,$$ it's enough to prove that
$$\sum_{cyc}(2ab-a^2)\geq4\sqrt3S$$ or
$$\sum_{cyc}(2ab-a^2)\geq\sqrt{3\sum_{cyc}(2a^2b^2-a^4)}$$ or
$$\sum_{cyc}(a^4-a^3b-a^3c+a^2bc)\geq0,$$
which ... |
3,366,442 | <p>I am asked to figure out when this limit exists for polynomials <span class="math-container">$f$</span> and <span class="math-container">$g$</span>, if so what the limit is, and then to prove my findings. So far I have gathered that the limit exists if and only if <span class="math-container">$\deg f\leq \deg g$</sp... | Andrew Chin | 693,161 | <p>For rational functions, the limit as <span class="math-container">$x$</span> approaches infinity is based on the limit <span class="math-container">$$\lim\limits_{x\to\infty}\frac1{x}=0.$$</span></p>
<p>Suppose in the following function <span class="math-container">$f(x)$</span>, the numerator is the sum of <span c... |
3,366,442 | <p>I am asked to figure out when this limit exists for polynomials <span class="math-container">$f$</span> and <span class="math-container">$g$</span>, if so what the limit is, and then to prove my findings. So far I have gathered that the limit exists if and only if <span class="math-container">$\deg f\leq \deg g$</sp... | Certainly not a dog | 691,550 | <p>Let the expression be be <span class="math-container">$\sum_{i=0}^n a_ix^i\over\sum_{i=0}^mb_ix^i$</span>.</p>
<p>We may rewrite this as <span class="math-container">$a_nx^n\bigl(1 + \sum_{i=1}^n \frac{a_i}{a_n}x^{-i}\bigr)\over b_mx^m\bigl(1 + \sum_{i=1}^m \frac{a_i}{a_m}x^{-i}\bigr)$</span></p>
<p>Now, consider ... |
6,090 | <p>I have been reading about Riemann Zeta function and have been thinking about it for some time.</p>
<p>Has anything been published regarding upper bound for the real part of zeta function zeros as the imaginary part of the zeros tend to infinity?</p>
<p>Thanks</p>
| T.. | 467 | <p>The term in analytic number theory is "zero-free regions". Any proof of the prime number theorem will produce such a region, and the region is equivalent to the error term in bounds for $\pi(x)$ and to the lower bounds in nonvanishing theorems $|\zeta (1+it)|> 0$. At present, the known zero-free regions are asympto... |
4,202,451 | <p>I have been reading about the Miller-Rabin primality test. So far I think I got it except the part where the accuracy is stated.<br />
E.g from <a href="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test#Accuracy" rel="nofollow noreferrer">wiki</a></p>
<blockquote>
<p>The error made by the primality t... | egglog | 626,167 | <p>Brute-force induction won't work well here, it's better to approach this problem by looking for simplifications. For example, the numerator in the product can be written in the form <span class="math-container">$\frac{(2n)!}{n!}$</span> and can be removed from the product entirely, thus leaving <span class="math-con... |
4,260,645 | <p>I can't find any insights online on how useful the graph of function <span class="math-container">$f(x)$</span> (on the <span class="math-container">$y$</span> axis) versus it's derivative <span class="math-container">$f'(x)$</span>? (on the <span class="math-container">$x$</span> axis) does it provide some useful i... | Ninad Munshi | 698,724 | <p>You can use it to analyze the behavior of differential equations. For example, consider <span class="math-container">$$y' = y^2 - 5x+6$$</span> Plotting <span class="math-container">$y'$</span> vs <span class="math-container">$y$</span> shows us the zeros (constant solutions/asymptotes for non constant solutions) an... |
4,337,320 | <p>Let <span class="math-container">$G$</span> be a group and <span class="math-container">$F:G^n \to G$</span> with the following property: If <span class="math-container">$x_1,…,x_n,h \in G$</span>, then <span class="math-container">$F(hx_1,…,hx_n)=hF(x_1,…,x_n)$</span>. Is there a name for this type of function prop... | Paul | 42,723 | <p>You can say the group operation is left-<a href="https://en.wikipedia.org/wiki/Distributive_property" rel="nofollow noreferrer">distributive</a> over <span class="math-container">$F$</span>.</p>
<p>When writing about this, make sure you include your definition, so it's clear for everyone.</p>
|
3,502,507 | <p>This is very similar to the <a href="https://math.stackexchange.com/questions/3501693/given-that-you-started-with-one-chip-what-is-the-probability-that-you-will-win">question</a> I've just asked, except now the requirement is to gain <span class="math-container">$4$</span> chips to win (instead of <span class="math-... | Tanner Swett | 13,524 | <p>Let <span class="math-container">$p_0, p_1, \ldots, p_4$</span> be the probability of winning if you start with <span class="math-container">$0, 1, \ldots, 4$</span> chips, respectively. Of course, <span class="math-container">$p_0 = 0$</span> and <span class="math-container">$p_4 = 1$</span>.</p>
<p>There are a fe... |
188,503 | <p>A Cayley table of an finite group has to have every element exactly once in every row and exactly once in every column.</p>
<p><strong>Proof</strong> that every element of a group has to be at most once in every row and at most once in every column:</p>
<p>Let $(G, \circ)$ be a group and $a, b, c, d \in G$ with:</... | Brian M. Scott | 12,042 | <p>The term <em>Cayley table</em> is generally restricted to finite groups. However, it’s certainly possible to generalize the idea. For a group $G$ and an element $a\in G$, the $a$ ‘row’ of the table is essentially just the function $$f_a:G\to G:b\mapsto ab\;,$$ and the $a$ ‘column’ is essentially just the function $$... |
3,426,700 | <p>For any fixed <span class="math-container">$C > 0$</span>, prove that <span class="math-container">$\frac{C^{k}}{k!} \to 0$</span> as <span class="math-container">$k \to \infty$</span></p>
<p>The hint given is to choose an integer <span class="math-container">$K>2C$</span> and examine the ratio in the questio... | Bob Krueger | 228,620 | <p>EDIT: this answer is wrong because the part derivative term in the integral is not conjugated. </p>
<p>Since the curve is symmetric about the real axis, the integral is equivalent to <span class="math-container">$\int_C z^2 dz$</span>. Thus by Cauchy's theorem, it is zero.</p>
|
3,426,700 | <p>For any fixed <span class="math-container">$C > 0$</span>, prove that <span class="math-container">$\frac{C^{k}}{k!} \to 0$</span> as <span class="math-container">$k \to \infty$</span></p>
<p>The hint given is to choose an integer <span class="math-container">$K>2C$</span> and examine the ratio in the questio... | Mark Viola | 218,419 | <p>Note that on <span class="math-container">$|z|=1$</span>, <span class="math-container">$\bar z=\frac1z$</span>. Hence, we have</p>
<p><span class="math-container">$$\oint_{|z|=1}(\bar z)^2\,dz=\oint_{|z|=1}\frac1{z^2}\,dz$$</span></p>
<p>Inasmuch as the residue of <span class="math-container">$\frac1{z^2}$</span>... |
28,258 | <p>The <a href="https://math.stackexchange.com/questions/tagged/set-theory" class="post-tag" title="show questions tagged 'set-theory'" rel="tag">set-theory</a> tag is explicitly about mathematics (and metamathematics) in the context of $\mathsf{ZFC}$ and its subsystems and extensions.</p>
<p>Every now and the... | Martin Sleziak | 8,297 | <ol>
<li><p><strong>I am for creating the <a href="https://math.stackexchange.com/questions/tagged/alternative-set-theories" class="post-tag" title="show questions tagged 'alternative-set-theories'" rel="tag">alternative-set-theories</a> tag.</strong> Or some similar name, but a tag intended for the set theorie... |
803,687 | <p>How to prove : </p>
<blockquote>
<p>$A/m^n$ is Artinian for all $n\geq 0$ if $A$ is a Noetherian ring and $m$ maximal ideal.</p>
</blockquote>
<p>Any suggestions ?</p>
| zcn | 115,654 | <p>Hint: Show that a Noetherian ring with only $1$ prime ideal is Artinian. What are the prime ideals of $A/m^n$?</p>
|
30,191 | <p>Does anybody know about software that exactly calculates the tree-width of a given graph and outputs a tree-decomposition? I am only interested in tree-decompositions of reasonbly small graphs, but need the exact solution and a tree-decomposition. Any comments would be great. Thanks!</p>
| András Salamon | 7,252 | <p>For general graphs there are no good algorithms known, as the problem of determining the treewidth of a graph is NP-hard. So if your graphs are not from some special class, and instances are small, then a brute force search over all decompositions of small width is a reasonable approach.</p>
<p>As a previous answe... |
4,404,751 | <p>I want to prove the following: Every symmetric matrix whose entries are calculated as <span class="math-container">$ 1/(n -1) $</span> with <span class="math-container">$n$</span> as the size of the matrix, except for the diagonal which is 0, has a characteristic polynomial with a root at <span class="math-container... | KBS | 532,783 | <p>Let <span class="math-container">$A_n$</span> be the <span class="math-container">$n\times n$</span> matrix such as the off-diagonal entries are <span class="math-container">$1/(n-1)$</span> and 0 on the diagonal. Then <span class="math-container">$A_n$</span> can be written as</p>
<p><span class="math-container">$$... |
2,243,900 | <p>I've researched this topic a lot, but couldn't find a proper answer to this, and I can't wait a year to learn it at school, so my question is:</p>
<blockquote>
<p>What exactly is calculus? </p>
</blockquote>
<p>I know who invented it, the Leibniz controversy, etc., but I'm not exactly sure what it is. I think I ... | B. Goddard | 362,009 | <p>In math, a "calculus" is a set of rules for computing/manipulating some set of objects. For instance the rules $\log AB = \log A+\log B$, etc, are the "logarithm calculus." </p>
<p>But commonly, "calculus" refers to "differential calculus" and "integral calculus." There is a set of rules (product rule, quotient ... |
2,243,900 | <p>I've researched this topic a lot, but couldn't find a proper answer to this, and I can't wait a year to learn it at school, so my question is:</p>
<blockquote>
<p>What exactly is calculus? </p>
</blockquote>
<p>I know who invented it, the Leibniz controversy, etc., but I'm not exactly sure what it is. I think I ... | Paul Sinclair | 258,282 | <p>Another way of understanding calculus is that it is the science of refining approximations. The idea is that if we cannot calculate a value directly, we come up with a scheme that allows us to approximate the value as closely as we want. If that scheme is good enough that by sufficiently refining the approximation, ... |
2,243,900 | <p>I've researched this topic a lot, but couldn't find a proper answer to this, and I can't wait a year to learn it at school, so my question is:</p>
<blockquote>
<p>What exactly is calculus? </p>
</blockquote>
<p>I know who invented it, the Leibniz controversy, etc., but I'm not exactly sure what it is. I think I ... | JeopardyTempest | 354,338 | <p>In algebra, the central focus was lines, which change consistently. Slope was a big focal point, from its use in the fundamental equations of lines (slope-intercept, point-slope, etc), to working on parallel/perpendicular lines, to using lines for models of life and extrapolation/forecasting. While you also should... |
2,243,900 | <p>I've researched this topic a lot, but couldn't find a proper answer to this, and I can't wait a year to learn it at school, so my question is:</p>
<blockquote>
<p>What exactly is calculus? </p>
</blockquote>
<p>I know who invented it, the Leibniz controversy, etc., but I'm not exactly sure what it is. I think I ... | Marco13 | 133,498 | <p>A somewhat broad question. In addition to the existing answers that already go into great detail, here is a more anecdotoral one. </p>
<p>The subject that you asked about is usually referred to by the term "calculus" in English. In German, this subject is called <a href="https://de.wikipedia.org/wiki/Analysis" rel=... |
2,243,900 | <p>I've researched this topic a lot, but couldn't find a proper answer to this, and I can't wait a year to learn it at school, so my question is:</p>
<blockquote>
<p>What exactly is calculus? </p>
</blockquote>
<p>I know who invented it, the Leibniz controversy, etc., but I'm not exactly sure what it is. I think I ... | Tom Russell | 439,282 | <p>It's too simple. The derivative of a function gives its rate of change at a given point. The integral of a function gives the area under the curve at a given point. The fundamental theorem merely says that the rate of change of the area under the curve is the value of the function at a given point.</p>
<p>In other ... |
292,639 | <p>Traditional Fourier analysis picks a period and then describes a function as: $$f(x) = \frac{1}{2} a_0 + \sum_{k=1}^\infty\, (a_k \cos{(\omega \cdot kx)} + b_n \sin{(\omega \cdot kx)})$$</p>
<p>I am wondering whether there is a way to Fourier-analyze a function in a way that the period is dependent on $x$. Let $g$ ... | Clayton | 43,239 | <p>Observe the function is odd (meaning $f(-x)=-f(x)$), so try splitting the integral $$\int_{-4}^4 f(x)dx=\int_{-4}^0 f(x)dx+\int_0^4f(x)dx=\int_0^4f(-x)dx+\int_0^4f(x)dx.$$ What happens next?</p>
|
292,639 | <p>Traditional Fourier analysis picks a period and then describes a function as: $$f(x) = \frac{1}{2} a_0 + \sum_{k=1}^\infty\, (a_k \cos{(\omega \cdot kx)} + b_n \sin{(\omega \cdot kx)})$$</p>
<p>I am wondering whether there is a way to Fourier-analyze a function in a way that the period is dependent on $x$. Let $g$ ... | Inquest | 35,001 | <p>Although the odd function property is a better way: Just to add to the possible solutions:
\begin{align}
\int_{-4}^410x^9+7x^5&=\left[x^{10}+ \dfrac{7x^6}{6} \right] _{-4}^4\\
&=\left[4^{10}+ \dfrac{7\times 4^6}{6} \right]-\left[(-4)^{10}+ \dfrac{7(-4)^6}{6} \right]\\
\end{align}</p>
<blockquote class="spoi... |
2,044,322 | <blockquote>
<p>How can you show that :</p>
<p><span class="math-container">$$D_n=\sum_{k=1}^{n } \frac{1}{k}-\int_{1}^{n+1} \frac{1}{x} \ dx $$</span> is increasing and bounded (and hence convergent). I'm having trouble.</p>
</blockquote>
| Learnmore | 294,365 | <p>$D_{n+1}-D_n=\{\sum_{n=1}^\infty \frac{1}{n+1}-\int_1^{n+2}\frac{1}{x}\}-\{\sum_{n=1}^\infty \frac{1}{n+2}-\int_1^{n+1}\frac{1}{x}\}$</p>
<p>$=\frac{1}{2}-\{\int _{n+1}^{n+2}\frac{1}{x} \}=\frac{1}{2}-\ln{\frac{n+2}{n+1}}$</p>
<p>$=\frac{1}{2}+\ln{\frac{n+1}{n+2}}>0$</p>
|
2,044,322 | <blockquote>
<p>How can you show that :</p>
<p><span class="math-container">$$D_n=\sum_{k=1}^{n } \frac{1}{k}-\int_{1}^{n+1} \frac{1}{x} \ dx $$</span> is increasing and bounded (and hence convergent). I'm having trouble.</p>
</blockquote>
| K. Miller | 264,375 | <p>To show that $D_n$ is bounded above observe that</p>
<p>$$
D_n \leq 1 + \int_1^n \frac{1}{x}\,dx - \int_1^{n+1} \frac{1}{x}\,dx = 1 + \ln\left(\frac{n}{n+1}\right) \leq 1 + \ln(1) = 1
$$</p>
<p>You may obtain this inequality as follows. Observe that $f(x) = 1/x$ is monotonically decreasing on $[1,\infty)$. Thus, $... |
2,044,322 | <blockquote>
<p>How can you show that :</p>
<p><span class="math-container">$$D_n=\sum_{k=1}^{n } \frac{1}{k}-\int_{1}^{n+1} \frac{1}{x} \ dx $$</span> is increasing and bounded (and hence convergent). I'm having trouble.</p>
</blockquote>
| Vim | 191,404 | <p>Why it is increasing:
<a href="https://i.stack.imgur.com/C7fNM.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/C7fNM.png" alt="enter image description here"></a></p>
<p>Why it is bounded: (just note the Red is bounded)
<a href="https://i.stack.imgur.com/MAT47.png" rel="nofollow noreferrer"><img s... |
4,240,598 | <blockquote>
<p>Simplify <span class="math-container">$\dfrac{18 - \dfrac 7 {3x}} {\dfrac 7 {18x} - 3}$</span>?</p>
</blockquote>
<p>I'm having a hard time simplifying this particular expression and am seeking any type of assistance in solving it.</p>
<p>In the expression
<span class="math-container">$$\frac{18 - \dfra... | Peeter Joot | 359 | <p>Yes, <span class="math-container">$1$</span> is meant to be the unit scalar here. For example, assume that you are considering the multivectors generated by a 2D Euclidean vector space with an orthonormal basis <span class="math-container">$ \left\{ { \mathbf{e}_1, \mathbf{e}_2 } \right\} $</span>:
<span class="mat... |
3,319,903 | <p>What is exactly generating function of ordered partitions and how can I get number of ordered partitions from that?</p>
<p>Example:</p>
<p><span class="math-container">$$ 4 = 1+1+1+1 \\ = 2+2 \\ = 1+1+2 \\ = 1+2+1 \\ = 2+1+1 \\ = 1+3 \\ = 3+1 \\ = 4 $$</span>
so we have <span class="math-container">$8$</span> par... | MafPrivate | 695,001 | <p>We can do this by generating functions which you want to.</p>
<p>For partition <span class="math-container">$n$</span>,</p>
<p>If we separate <span class="math-container">$n$</span> into <span class="math-container">$n$</span> numbers, we can generate the function <span class="math-container">$x+x^2+\cdots+x^n$</s... |
523,074 | <p>I have never done integration in my life and I am in first year of university. Is it harder than taking the derivative? I've heard its just going backwards. Also, my high school taught me only differentiation, I don't know why we never touched on integration. I'm going to be starting it next week and I want to know ... | Betty Mock | 89,003 | <p>If you were fine with derivatives, you will be fine with integrals in 1st year calc. It never hurts to pay attention in class (which kind of implies attending class) and to do your homework. In fact, if you have trouble with a problem, you should do more of the same kind as soon as you know the answer. As to diffic... |
506,965 | <p>How can I prove that:
$E_π [ (dQ_X/dπ) S (T)| F_t ]= E_{Q_X} [S(T) | F_t]E_π [ dQ_X/dπ | F_t ]$.
Obviously $E_π [(dQ_X/dπ) S(T) ]= E_{Q_X} [S(T)]$ I know that much, but how to prove when it is conditioned on $F_t$.</p>
| Davide Giraudo | 9,849 | <p>I guess the $Q_X$, $S(T)$ and $\mathcal F_t$ come from a particular context, so we can simplify the notation and prove that if $\mu,\nu$ are two probability measures such that $\nu\ll\mu$ and the random variables $X$ and $\frac{\mathrm d\nu}{\mathrm d\mu}$ are integrable, then
$$\mathbb E_\mu\left(\frac{\mathrm d\n... |
1,361,517 | <p>$K_1 K_2 \dotsb K_{11}$ is a regular $11$-gon inscribed in a circle, which has a radius of $2$. Let $L$ be a point, where the distance from $L$ to the circle's center is $3$. Find
$LK_1^2 + LK_2^2 + \dots + LK_{11}^2$.</p>
<p>Any suggestions as to how to solve this problem? I'm unsure what method to use. </p>
| De ath | 352,964 | <p>Let $\omega = e^{2 \pi i/11}$, a primitive $11^{\text{th}}$ root of unity. We can assume that the circle is centered at the origin. We can also assume that $A_k$ is associated with the complex number $2 \omega^k$
Let $p$ be complex number associated with the point $P$. Then
$PA_1^2 + PA_2^2 + \dots + PA_{11}^2 = \su... |
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