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 |
|---|---|---|---|---|
275,151 | <p><a href="https://i.stack.imgur.com/XxGxa.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XxGxa.png" alt="enter image description here" /></a>I'm a beginner at Mathematica, and I'm trying to figure out how to fill between two lines horizontally. Consider the toy example</p>
<pre><code>Plot[{Conditi... | cvgmt | 72,111 | <ul>
<li>construct two lines to make a closed curve and then use <code>BoundaryDiscretizeGraphics</code> to get a region.</li>
</ul>
<pre><code>Clear[f, g, plot1, plot2, pts1, pts2, reg];
f[x_] = ConditionalExpression[2*x - 2, 2 < x < 6];
g[x_] = ConditionalExpression[2*x + 2, x < 4];
plot1 = Plot[f[x], {x, 0,... |
4,382,739 | <p>I’ve been given the joint density function: f<span class="math-container">$_X$$_,$$_Y$</span>(x,y)=C when (X,Y) is uniform over [-1,1]<span class="math-container">$^2$</span>. I’ve been tasked with finding P{|2X+Y|<span class="math-container">$\le$</span>1} and P{X=Y} however I’m stuck in my question, I’ve deduced a... | Tita | 791,224 | <p>You have <span class="math-container">$f_{X,Y}(x,y) = \frac{1}{4}$</span> for <span class="math-container">$(X,Y)\sim U([-1,1]\times [-1,1])$</span>.</p>
<p>For both the probabilities you want to find you need to integrate twice in the area that you’re given. For the first one you need to integrate where <span class... |
283,776 | <p>I have a curve that looks like this (it's cyclical):</p>
<p><a href="https://i.imgur.com/CRZ7Gbr.png" rel="nofollow noreferrer">Curve</a></p>
<p>I can get a partial fit by fitting a 3rd degree polynomial, but I have a feeling there must be a better fit (something that involves sin & cos). </p>
<p>The fitted ... | Robert Mastragostino | 28,869 | <p>You could try to fit higher-order polynomials, which should give you the flexibility needed to ensure differentiability or some such constraint. Or if you're willing to try parametric curves, you can get a fit relatively easily, like <a href="http://www.wolframalpha.com/input/?i=%7Bsin%28t%29%2Bt%2Csin%28t%29%7D" re... |
27,759 | <p>Suppose $N$ is an RSA modulus (ie, it's the product of two distinct primes), 256 bits long. What is the best method to factor it?</p>
<p>Trial division is out of the question, Pollard's Rho is probably out as well (without significant parallelization). I doubt there are any online tools or math libraries that can... | Juan Joder | 8,407 | <p>Eigendecompose $\mathbf A$ (easily done since you have a symmetric matrix), take the cube root of the eigenvalues, and multiply back the matrix of eigenvectors appropriately.</p>
<p>I get</p>
<p>$$\frac12\begin{pmatrix}\sqrt[3]{5}-1&\sqrt[3]{5}+1\\\sqrt[3]{5}+1&\sqrt[3]{5}-1\end{pmatrix}$$</p>
|
27,759 | <p>Suppose $N$ is an RSA modulus (ie, it's the product of two distinct primes), 256 bits long. What is the best method to factor it?</p>
<p>Trial division is out of the question, Pollard's Rho is probably out as well (without significant parallelization). I doubt there are any online tools or math libraries that can... | Robert Israel | 8,508 | <p>In fact, the only 2 x 2 matrices that do not have cube roots (over the complex numbers) are those with Jordan canonical form $\left[ \matrix{0 & 1\cr 0 & 0\cr}\right]$.<br>
The 3 x 3 matrices with no cube root are those with Jordan form
$\left[ \matrix{0 & 1 & 0\cr 0 & 0 & 1\cr 0 & 0 &am... |
4,039,655 | <p>I'm currently having trouble evaluating the following sum to get a formula in terms of <span class="math-container">$k$</span>:
<span class="math-container">$$\sum_{i=0}^{k-1} 2^i\cdot 4(k-i-1)$$</span></p>
<p>I know that <span class="math-container">$$\sum_{i=0}^n 2^i = 2^{n+1}-1$$</span>
but since my <span class="... | Cesareo | 397,348 | <p>Hint.</p>
<p><span class="math-container">$$
\frac{d}{dx}x^{k-i-1}=(k-i-1)x^{k-i-2}
$$</span></p>
<p>now making <span class="math-container">$x=\frac 12$</span> we have</p>
<p><span class="math-container">$$
\left(\frac12\right)^{k}2^i4(k-i-1)
$$</span></p>
<p>then think on</p>
<p><span class="math-container">$$
\fr... |
2,488,218 | <p>I encountered this problem while practicing for a mathematics competition. </p>
<blockquote>
<p>A cube has a diagonal length of 10. What is the surface area of the cube? <strong>No Calculators Allowed.</strong></p>
</blockquote>
<p>(Emphasis mine)</p>
<p>I'm not even sure where to start with this, so I scribble... | Travis | 404,488 | <p>This problem has a simple solution: $s=2d^2$ where $s$ is the surface area and $d$ is the spacial diagonal.</p>
<p><strong><em>Explanation</em></strong>:</p>
<p>I discovered through Google (while writing the question) that the side length of a cube can be calculated with $d = a\sqrt{3}$ where $d$ is the diagonal o... |
31,539 | <p>I want to learn a bit about Linear Programming. </p>
<p>After some research, I decided to solve the <a href="http://en.wikipedia.org/wiki/Cutting_stock_problem" rel="nofollow">Cutting Stock</a> problem as an example to learn. After doing some more research, I feel like I finally understand Linear Programming enough... | Aaron | 3,027 | <p>Its worth pointing out that if you add integer constraints to a linear program, the problem of solving it becomes NP-hard. There are software packages that will attempt to solve these for you (e.g. CPLEX), and often succeed on very large instances. Nevertheless, for this reason, if you want an exact solution to an i... |
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | Grimm The Opiner | 98,415 | <p>The question doesn't state that the BOY is an expert in maths, so he'd probably go for:</p>
<p>Please say "no" if your number is 1, "don't know" if your number is 2, or "yes" if your number is 3.</p>
|
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | shrinath | 98,437 | <p>Let the boy ask the girl "Subtract 2 from the number your are thinking and then take the square root of the result.Is the result positive?".
If the girl replies "yes" then the number is 3 because square root of 3-2=1 which is positive.
If the girl replies "no" then the number is 2 because square root of 2-2=0 which ... |
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | ND_27 | 98,503 | <p>First of all I ruled out indirect ways of using reference to either of the numbers 1, 2 ,3 to frame a question, as I thought it's implicit in the question that it should challenge your thinking, not your cleverness.
If she answers <em>I don't know</em>, compared to <em>yes</em> or <em>no</em>, it is more likely that... |
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | eskimo | 50,736 | <p>"Using the number of characters in the <strong>first</strong> word of you possible answers (Yes, No, I don't know), which options length corresponds to your selected number?"</p>
|
774,537 | <p>I use <a href="http://en.wikipedia.org/wiki/Polish_notation" rel="nofollow noreferrer">Polish notation</a>. All systems have detachment and uniform substitution as the only primitive rules of the system.</p>
<p>A user named John told me in an <a href="https://philosophy.stackexchange.com/questions/1365/why-is-this... | JohnHalleck | 167,763 | <p>"... (Cpp, CpCqCCpCqrCsr} and {CpCqp, CCpCqrCqCpr} were also suffice to show that there is a single axiom basis for a system. ... I'm not sure if his statement applies to any logical calculus, or just implicational calculi."</p>
<p>Short answer: True of any logical calculus that has the given theorems.</p>
<p>The ... |
774,537 | <p>I use <a href="http://en.wikipedia.org/wiki/Polish_notation" rel="nofollow noreferrer">Polish notation</a>. All systems have detachment and uniform substitution as the only primitive rules of the system.</p>
<p>A user named John told me in an <a href="https://philosophy.stackexchange.com/questions/1365/why-is-this... | Adrian Rezus | 225,751 | <p>Cross ref. to “Examples of mathematical discoveries which were kept as a secret” [Nov 3, 2014]</p>
<p>The Question was: </p>
<p>“What are examples of mathematical discoveries which were kept as a secret when they discovered and then became unfolded after a while by any reasons?” [asked Nov 3, ’14 at 14:17, by Ali ... |
2,173,586 | <p>Let $R$ be an unique factorization domain. Being an integral domain, it has a field of quotients $F$. We can consider $R[x]$ to be a subring of $F[x]$. </p>
<blockquote>
<p>Given any polynomial $f(x)\in F[x]$, then $f(x)=(f_0(x)/a)$, where $f_0(x)\in R[x]$ and where $a\in R$.</p>
</blockquote>
<p>I don't underst... | Kenny Wong | 301,805 | <p>Take this as the definition of infinite unions and intersections:</p>
<p>$x \in A_1 \cup A_2 \cup A_3 \dots $ if and only if $x \in A_i$ for at least one $i \in \mathbb N$.</p>
<p>$x \in A_1 \cap A_2 \cap A_3 \dots $ if and only if $x \in A_i$ for all $i \in \mathbb N$.</p>
<p>This makes perfect sense as it is. T... |
15,013 | <p>I am doing a plot where I have multiple shaded regions, and I want the line that separates the two regions to be dashed with dashes being alternating colors (so the demarcation stands out from both regions).</p>
<p>For example, say I am plotting the two regions shown here</p>
<pre><code>Plot[{1, Abs[BesselJ[1, x]]... | whuber | 91 | <p>A simple but flexible approach might be to plot the function twice, as in this example, with a solid color the first time overlaid by a dashed line the second:</p>
<pre><code>f[x_] := Abs[BesselJ[1, x]];
Plot[{f[x], f[x]}, {x, -10, 10},
PlotStyle -> {Directive[Thick, White],
Directive[Thick, Dashing[{0.1, ... |
421,951 | <p>I did search for whether this question was already answered but couldn't find any.</p>
<p>Does a function have to be "continuous" at a point to be "defined" at the point?</p>
<p>For example take the simple function $f(x) = {1 \over x}$; obviously it is not continuous at $x = 0$. However it does have the $-$ and $+... | Tigran Saluev | 82,730 | <p>No, it <strong>has not</strong>.</p>
<p>You can define a function at any point in any way you wish.</p>
<p>For example, you can define function $\mathrm{sign}(x)$ as
$$
\mathrm{sign}(x) = \left\{\begin{array}{rl}-1, & x < 0 \\ 0, & x = 0 \\ 1, & x > 0\end{array}\right.
$$
and then, <em>after</em>... |
41,707 | <p>Is there a slick way to define a partial computable function $f$ so that $f(e) \in W_{e}$ whenever $W_{e} \neq \emptyset$? (Here $W_{e}$ denotes the $e^{\text{th}}$ c.e. set.) My only solution is to start by defining $g(e) = \mu s [W_{e,s} \neq \emptyset]$, where $W_{e,s}$ denotes the $s^{\text{th}}$ finite approxim... | Ciro Santilli OurBigBook.com | 53,203 | <p><strong>Matrix multiplication is the majority of deep learning and convolutional neural networks</strong></p>
<p>In case you were under a rock, from 2012 on onwards <a href="https://en.wikipedia.org/wiki/Deep_learning" rel="nofollow noreferrer">deep learning</a> algorithms have quickly become the best known algorith... |
3,355,570 | <p>In a recent lecture I attended the following limit was discussed:
<span class="math-container">$$\lim_{(x,y)\to (0,0)} \frac{x^2y^4}{(x^2+y^4)^2}$$</span>
Multiple solutions were used to try and find the limit and illustrate how one would attack similar problems.</p>
<p>In particular we rewrote the expression using... | Allawonder | 145,126 | <p>If you approach along the <span class="math-container">$x$</span>-axis <span class="math-container">$y=0,$</span> with <span class="math-container">$x\ne 0,$</span> you see that the limiting value is <span class="math-container">$0.$</span> However, as you have noted, travelling along the parabola <span class="math-... |
1,832,887 | <p>Consider the conjunction introduction and implication elimination rules of natural deduction:</p>
<p>$$\frac{\Gamma\vdash\alpha \quad \Gamma\vdash\beta}{ \Gamma\vdash \alpha \land \beta} (\land I)
\qquad
\frac{ \Gamma \vdash \alpha \to \beta \quad \Gamma \vdash \alpha} {\Gamma,\vdash\beta} (\to E) \qquad \text{(... | avz2611 | 142,634 | <p>hint : write the polynomial in this form $$f(x)= a(x-1)(x-2)(x-3)(x-4)(x-5)+b(x-1)(x-2)(x-3)(x-4)(x-6) +c(x-1)(x-2)(x-3)(x-5)(x-6)+d(x-1)(x-2)(x-4)(x-5)(x-6)+e(x-1)(x-3)(x-4)(x-5)(x-6)+f(x-2)(x-3)(x-4)(x-5)(x-6)$$ now finding constants are easy </p>
|
991,372 | <p>Consider $f_{k}(x)=(\frac{1}{ x}\frac{d}{d x})^k(\frac{x}{\sinh x})$, $x>0$, $k=0,1,2,\cdots.$ Then, Is $f_{k}(x)$ always a bounded function? </p>
<p>The only thing one need to care is the behavior when $x$ is near $0$, and prove it's bounded. I try the case $k=1,2,$ and I think it's true for general k as well.... | MvG | 35,416 | <p>This is a coordinate-based approach, making heavy use of tools from projective geometry.</p>
<p>Without loss of generality, you can choose the coordinate system in such a way that the inscribed circle is the unit circle. On that you can use a rational parametrization, i.e. choose $a,b,c\in\mathbb R$ such that $A'=(... |
3,796,937 | <p>Prove that <span class="math-container">$2^n+1$</span> is not a cube for any <span class="math-container">$n\in\mathbb{N}$</span>.</p>
<p>I managed to prove this statement but I would like to know if there any other approaches different from mine.</p>
<p>If existed <span class="math-container">$k\in\mathbb{N}$</span... | vonbrand | 43,946 | <p>Suppose <span class="math-container">$2^n + 1 = k^3$</span>. Then <span class="math-container">$2^n = k^3 - 1 = (k^2 + k + 1)(k - 1)$</span>. So both factors are even (<span class="math-container">$k = 2$</span> doesn't work; the first factor is at least <span class="math-container">$3^2 + 3 + 1 = 13$</span>, it can... |
3,796,937 | <p>Prove that <span class="math-container">$2^n+1$</span> is not a cube for any <span class="math-container">$n\in\mathbb{N}$</span>.</p>
<p>I managed to prove this statement but I would like to know if there any other approaches different from mine.</p>
<p>If existed <span class="math-container">$k\in\mathbb{N}$</span... | user | 813,391 | <p>Let's set the cubes to <span class="math-container">$8m^3$</span> and <span class="math-container">$8m^3+12m^2+6m+1$</span>. As <span class="math-container">$8m^3$</span> is even and it doesn't works for <span class="math-container">$n=0$</span>, that's impossible. For the second one, ignoring the <span class="math-... |
2,709,454 | <p>In Basket $A$ we have $2$ Blue and $1$ Red Balls and in Basket $B$ we have $3$ Blue and $3$ Red Balls. we randomly choose 2 balls from each basket (2 from $A$ and 2 from $B$). Then we put the balls from $A$ into $B$ and the balls from $B$ into $A$. Then, We chose one of the baskets randomly and pick a random ball fr... | G Cab | 317,234 | <p>When you randomly pick $2$ balls from B, consider like you pick $2$ balls of a $50\%$ bluish tone,
while those from A will have a $66\%$ of blue.</p>
<p>At the end of the picking and exchange process you have<br>
- in A : 1ball $2/3$+2ball $1/2$ = 3balls with $(2/3+1)/3=5/9$ of tone<br>
- in B : 2ball $2/3$+4ball... |
283,119 | <p>Let $X$ be a Riemannian $n$-manifold with tubular end $\mathbb R^+\times Y$, where $Y$ is a closed $n-1$-manifold. Suppose $L:L^{p,w}_2(X)\to L^{p,w}(X)$ is the Laplacian operator which is translation invariant on the cylindrical end; here $L^{p,w}_i$ is a weighted $L^{p}_i$ space, i.e. $w$ is a function on $X$ suc... | Yoël | 106,906 | <p>Thank you Olivier for this great answer, I wish I could be one of your students ;-) !</p>
<p>Though I still not quite fully understand the adelic setting, here is some "classical" explanation I found, using the theory of ideals in orders of imaginary quadratic fields. I guess this is supposed to be a very well-know... |
1,410,164 | <p>I've found the following identity.</p>
<blockquote>
<p>$$\int_0^1 \frac{1}{1+\ln^2 x}\,dx = \int_1^\infty \frac{\sin(x-1)}{x}\,dx $$ </p>
</blockquote>
<p>I could verify it by using CAS, and calculate the integrals in term of <a href="http://mathworld.wolfram.com/ExponentialIntegral.html">exponential</a> and <a ... | tired | 101,233 | <p>A variant:</p>
<p>$$
\int_1^{\infty}\frac{\sin(x-1)}{x}dx=\Im\int_1^{\infty}\frac{e^{i(x-1)}}{x}dx= \quad (1)\\ \Im\int_1^{\infty}\int_0^{\infty}e^{i(x-1)-xt}dtdx=\Im\int_0^{\infty}\int_1^{\infty}e^{i(x-1)-xt}dxdt=\quad (2) \\
\Im\int_0^{\infty}\frac{e^{-t}}{t-i}dt=-\int_0^{\infty}\frac{e^{-t}}{1+t^2}dt=\quad (3) \... |
3,118,282 | <p>I'm trying to do the following problem in my book, but I don't understand how the book got their answer.</p>
<p>The problem:
Determine whether the following relations are equivalence relations:<span class="math-container">$\newcommand{\relR}{\mathrel{R}}$</span></p>
<p>The relation <span class="math-container">$\... | AmatsukiLove | 261,956 | <p>An equivalence relation <span class="math-container">$\sim$</span> satisfies three axioms.</p>
<ol>
<li><strong>Reflexivity.</strong> <span class="math-container">$x \sim x$</span> for all <span class="math-container">$x$</span>.</li>
<li><strong>Symmetry.</strong> If <span class="math-container">$x \sim y$</span> ... |
154,818 | <p>Let $G$ be a finitely generated group. The <em>weight</em> $w(G)$ of $G$ is defined to be the minimum number of elements of $G$ whose normal closure in $G$ is the whole of $G$ (this is sometimes also called the <em>normal rank</em>). Obviously, $d(G^{\operatorname{ab}})\leq w(G) \leq d(G)$, where $d(G)$ is the rank ... | Ian Agol | 1,345 | <p>A silly special case is a weight 1 group which is 2-generator. Since it is weight 1, its abelianization is cyclic. For any pair of generators, one may perform <a href="http://en.wikipedia.org/wiki/Nielsen_transformation" rel="noreferrer">Nielsen transformations</a> to get a pair of generators so that one of the gene... |
3,967,862 | <p>If <span class="math-container">$A,B$</span> commute, that <span class="math-container">$e^A\leq e^B$</span> follows from functional calculus.</p>
<p>Is this still true when <span class="math-container">$A,B$</span> do not commute?</p>
<p>Also I wonder if <span class="math-container">$A^{2n+1}\leq B^{2n+1}$</span> i... | Z Ahmed | 671,540 | <p>You may write it as <span class="math-container">$\tan(\pi/4+x)$</span>, which is a single term.</p>
|
533,812 | <p>A field is quadratically closed if each of its elements is a square.</p>
<p>The field <span class="math-container">$\mathbb{F}_2$</span> with two elements is obviously quadratically closed.</p>
<p>However, testing some more finite fields with this property, I didn't find any more. Hence my question is:</p>
<blockquo... | user2902293 | 138,157 | <p>$W^{\wedge}(F)$ is the Witt-Grothendieck ring of $F$, informally it can be thought of set of all formal differences of regular quadratic forms on $F$(this is not exactly right).</p>
<p>$F$ is quadratically closed $\Leftrightarrow$ $dim \colon W^{\wedge}(F) \to \mathbb{Z} $ is an isomorphism $ \Leftrightarrow \lbra... |
65,731 | <p>This was <a href="http://www.quora.com/How-many-topologies-are-there-on-set-of-real-numbers">asked</a> on Quora. I thought about it a little bit but didn't make much progress beyond some obvious upper and lower bounds. The answer probably depends on AC and perhaps also GCH or other axioms. A quick search also failed... | Stefan Geschke | 16,330 | <p>Let $X$ be any set of some infinite size $\kappa$. A topology on $X$ is a set of subsets of $X$. $X$ has $2^\kappa$ subsets and there are $2^{2^\kappa}$ collections of subsets of $X$. This is an upper bound for the number of topologies on $X$. </p>
<p>Now, choose a point $x_0\in X$ and let $Y=X\setminus\{x_0\}$... |
3,825,198 | <p>I have been asked to prove <span class="math-container">$P\land(Q\land R)$</span> from the premises <span class="math-container">$\lnot[P\rightarrow\lnot (Q\land R)]$</span>.</p>
<p>Could someone please point me in the right direction to solve this question, or a question like it?</p>
<p>Premise: <span class="math-c... | user400188 | 400,188 | <p>Consider the definition of <span class="math-container">$A\rightarrow B$</span>, which is <span class="math-container">$\lnot A\lor B$</span>. Also consider de Morgans law, <span class="math-container">$\lnot(A\lor B)=\lnot A\land\lnot B$</span>. Together, you can use these rules to show that <span class="math-cont... |
3,825,198 | <p>I have been asked to prove <span class="math-container">$P\land(Q\land R)$</span> from the premises <span class="math-container">$\lnot[P\rightarrow\lnot (Q\land R)]$</span>.</p>
<p>Could someone please point me in the right direction to solve this question, or a question like it?</p>
<p>Premise: <span class="math-c... | C Squared | 803,927 | <p>Recall that <span class="math-container">$A\rightarrow B$</span> is equivalent to <span class="math-container">$\neg A\vee B$</span>.</p>
<p><span class="math-container">$$\neg[P\rightarrow \neg(Q\wedge R)]\equiv \neg[\neg P\vee \neg(Q\wedge R)]$$</span></p>
<p>Upon distributing the negation, we obtain <span class="... |
1,239,560 | <p>I was told that the relation $\le$ is a total order on R, it is dense, and it has a least upper bound property. I actually have don't understand those 3 properties... :/</p>
| Tom J | 220,971 | <p>In contrast to a partial order the a total order is defined everywhere. This means that given arbitrary a and b either a<=b or b>=a.</p>
|
3,849,649 | <p>Given <span class="math-container">$A^2=A$</span>, <span class="math-container">$2A−B−AB=I$</span>, prove that <span class="math-container">$A−B$</span> is invertible.</p>
<hr />
<p>I have got <span class="math-container">$(I+A)(A-B)$</span>, what is the next step?</p>
<p>Thanks a lot.</p>
| VIVID | 752,069 | <p>You are almost done:
<span class="math-container">$$
\begin{align}
2−− &= \\
A-AB+A-B &= I \\
A^2-AB+A-B &= I \\
A(A-B)+(A-B) &= I \\
(A+I)(A-B) &= I
\end{align}
$$</span></p>
<p>Taking <span class="math-container">$\det$</span> from both sides and using the <a href="https://proofwiki.org/wik... |
2,307,753 | <p>I know that $-\int \tan(t)dt$ = $\ln |\cos t|$ (letting $C=0$). So I would think that $e^{-\int \tan(t)dt}$ would be equal to $e^{\ln |\cos t|} = |\cos t|$. However, my math textbook and Wolfram Alpha both say that $e^{-\int \tan(t)dt}=e^{\ln (\cos t)} = \cos t$. Why can the absolute value be ignored when taking the... | The_Sympathizer | 11,172 | <p><em>This is very subtle!!!!</em></p>
<p>To see what's actually going on, you need to consider first, something you may not have noticed about indefinite integrals. What this arises from is a rather base mistake that seems to be perpetuated in a lot of textbooks, in particular, that ...</p>
<p><span class="math-con... |
3,132,404 | <p>For the purpose of this question, <span class="math-container">$A'$</span> is the derived set of set <span class="math-container">$A$</span>, <span class="math-container">$A^\circ$</span> is the interior of set <span class="math-container">$A$</span>, and <span class="math-container">$A^{c}$</span> is the complement... | Marwan Mizuri | 554,710 | <p><span class="math-container">$\forall A, (A')^{c} = (A^{c})^\circ$</span> does not follow from <span class="math-container">$A^{c} \subset (A')^{c} \iff A^{c} \subset (A^{c})^\circ$</span>. The two sets, <span class="math-container">$(A')^{c}$</span> and <span class="math-container">$(A^{c})$</span>, can be two diff... |
355,801 | <p>The Fubini's Theorem states that for any two $\sigma$-finite measure spaces $(S,\mathcal{S},\mu)$ and $(T,\mathcal{T},\upsilon)$, there exists a unique measure $(\mu \otimes \upsilon)(A\times B)=\mu A \cdot \upsilon B$, $\forall A\in\mathcal{S},B\in\mathcal{T}$.
Further more, for any measurable function $f:S\times T... | Elias Costa | 19,266 | <p>\begin{align}
\Bbb E(\xi^p) =
&
\int \xi^p \, \mathrm d P
\\
=
&
\int \left( \int_0^\infty 1_{\{t < \xi\}}pt^{p-1} \,\mathrm dt\right)\,\mathrm d P
\\
=
&
\int_0^\infty \left( \int 1_{\{t < \xi\}}pt^{p-1} \,\mathrm d P \right) \,\mathrm dt
\\
=
&
\int_0^\infty \left( \int_{\{t < \xi\}}pt^... |
246,827 | <p>Let $A$ and $B$ be two positive definite $n \times n$ matrices. It is, of course, not true that $AB+BA$ is necessarily positive definite. </p>
<p>Consider, though, the results of the following numerical experiment. I generated $A$ by letting its eigenvalues be random in $[0,1]$, and selecting its eigenvectors by ge... | David Zhang | 61,479 | <p>Your question appears to be based on a false premise. In fact $AB+BA$ does <strong>not</strong> tend to be positive definite as $n$ increases, even within the particular distribution you happen to be using.</p>
<p>To demonstrate this, here is a simple piece of <em>Mathematica</em> code that implements precisely the... |
2,915,786 | <p>I started writing a proof using the method of proof by contradiction and encountered a situation which was true. More specifically, the hypothesis that I set out to prove was:</p>
<p>If the first 10 positive integer is placed around a circle, in any order, there exists 3 integer in consecutive locations around the ... | Phil H | 554,494 | <p>Try this. Assuming there is a solution that always sums to less than $17$ for $3$ consecutive numbers, and building around the number $10$, the only possible number combinations adjacent to $10$ will be $4$ numbers from the subset $1,2,3,4,5$. Taking the maximum of these as $1,5$ and $2,4$ $(1,5,10,2,4)$ leaves a mi... |
2,915,786 | <p>I started writing a proof using the method of proof by contradiction and encountered a situation which was true. More specifically, the hypothesis that I set out to prove was:</p>
<p>If the first 10 positive integer is placed around a circle, in any order, there exists 3 integer in consecutive locations around the ... | Arcanist Lupus | 498,762 | <p>A proof by contradiction functions by saying "if A is false, B must be true. I can prove that B is false, so A cannot be false."</p>
<p>You proved that B is true. This means that A <em>could</em> be false, but is not necessarily so because we have made no statements that relate the truth of A to the truth of B.</... |
2,915,786 | <p>I started writing a proof using the method of proof by contradiction and encountered a situation which was true. More specifically, the hypothesis that I set out to prove was:</p>
<p>If the first 10 positive integer is placed around a circle, in any order, there exists 3 integer in consecutive locations around the ... | gandalf61 | 424,513 | <p>You can (more or less) avoid the complications of a proof by contradiction by reasoning as follows:</p>
<p>The total of the $10$ sums of consecutive triplets $a_i+a_{i+1}+a_{i+2}$ is $3 \times 55 = 165$. Therefore the <em>average</em> of the sums of consecutive triplets is $165/10 = 16.5$. But each sum is an intege... |
252,511 | <p>I am interested in the details of Elie Cartan's thesis, and, more specifically the explicit construction of the exceptional Lie groups as groups of symmetries of some specific homogeneous polynomials (according to what I have read in many places). I am interested in the details. For instance, what does one such a po... | Robert Bryant | 13,972 | <p>$\mathrm{G}_2$ is the only one of the exceptional groups that can be defined as the stabilizer of a `generic' tensorial object on a vector space and, over the complex numbers, even this is not quite right.</p>
<p>More precisely, let $V$ be a vector space (over $\mathbb{F}$, which could be $\mathbb{R}$ or $\mathbb{C... |
3,463 | <p>I've been thinking about maps between sets. Injections, surjections and the rest. Often when thinking about some kind of map, it is interesting to say "what about the maps from a set to <em>itself</em>?" Call these maps endomaps. Permutations of elements of the set are a special case of endomaps: they are bijective ... | Arturo Magidin | 742 | <p>Given a set $X$, the collection of all maps $s\colon X\to X$ forms a <em>semigroup</em> under the operation of composition, called the "full transformation semigroup" of $X$, often denoted $\mathcal{T}_X$. Full transformation semigroups play the same role in semigroup theory as symmetric groups do in group theory (t... |
1,579,663 | <p>In the first paragraph of <a href="https://en.wikipedia.org/wiki/Finite_field#Definitions.2C_first_examples.2C_and_basic_properties" rel="nofollow">wikipedia:Finite fields</a> they write</p>
<blockquote>
<p>The identity
$$ (x + y)^p = x^p + y^p $$
is true (for every $x$ and $y$) in a field of characteristic ... | Jack D'Aurizio | 44,121 | <p>It is useful to set $x=az$ for first. That, together with the cosine duplication formula, leads to:
$$ I = 2a\int_{0}^{1}z \sin^2(m\pi z)\,dz =a\int_{0}^{1}z\left(1-\cos(2m\pi z)\right)\,dz,\tag{1}$$
then applying integration by parts:
$$ I = \frac{a}{2}-\int_{0}^{1}az\cos(2m\pi z)\,dz = \frac{a}{2}-\left.\frac{az}{... |
649,073 | <p>Show that there exists a continuous function $f: [-1, 1] \rightarrow \mathbb{R}$ such</p>
<p>$f(0) = 1$ and</p>
<p>$f(x) = \frac{2-x^2}{2} \cdot f(\frac{x^2}{2-x^2})$ </p>
<p>$\forall x \in [-1, 1]$</p>
<p>I tried putting in $x = 1$ and $x = -1$ in the second condition to find that $f(1) = f(-1) = 0$.
I also ... | GEdgar | 442 | <p>Let's find the <strong>most general</strong> solution.</p>
<p>Let continuous function $f : [-1,1] \to \mathbb R$ satisfy
$$
f(x) = \frac{2-x^2}{2} \cdot f\left(\frac{x^2}{2-x^2}\right)\qquad x \in [-1,1].
\tag{$1$}$$
and $f(1)=1$. Observe $f$ is even. Define $G : (0,1] \to \mathbb R$ by
$$
G(y) := \frac{1}{y}\cdo... |
886,097 | <p>First of all, we notice that: $$(a,b)=\bigcup_{a<x<b} [x,b).$$ </p>
<p>Also, we notice that: $$(0,1)-\left\{\frac{1}{n} |\, n\in \mathbb{Z^+}\right\}=\left(\frac{1}{2},1\right)\cup \left(\frac{1}{3},\frac{1}{2}\right)\cup \left(\frac{1}{4},\frac{1}{3}\right)\cup \left(\frac{1}{5},\frac{1}{4}\right)\cup \ldots... | vociferous_rutabaga | 164,345 | <p>The issue lies in the assertion that any interval $(a,b) - K$ can be decomposed as a union of open intervals "in a way similar to $(0,1)$." Indeed, the procedure does not extend if $a<0<b$. Let's assume $b=1$, and $a<0$. Note that $(a,b)-K$ is not equal to $(a,0)\cup\big(\cup_{i=1}^\infty (\frac{1}{i+1},\fr... |
10,992 | <p>Alright, I'm trying to figure out how to calculate a critical value using t-distribution in Microsoft Excel... ex. a one-tailed area of 0.05 with 39 degrees of freedom: t=1.685</p>
<p>I know the answer, but how do I get this? I've tried TDIST() TINV() and TTEST() but they all give me different answers. This web ca... | jarauh | 291,793 | <p>For reference, in EXCEL 2010 there are now two functions: <code>T.INV()</code>, which gives the "left-tailed inverse" (i.e. the inverse of the cumulative distribution function), and there is <code>T.INV.2T()</code>, which assumes as an argument the probability of both tails.</p>
|
566,088 | <p>Prove the following identity.</p>
<p>$$\sum_{i=0}^{n}(-1)^n\binom{-1/2}{i}\binom{-1/2}{n-i} = 1$$</p>
| Matemáticos Chibchas | 52,816 | <p>By generalized binomial theorem we have $(1+x)^{-1/2}=\sum_{n=0}^\infty\binom{-1/2}nx^n$ and this series converges absolutely for $|x|<1$, therefore $\bigl[(1+x)^{-1/2}\bigr]^2$ is the Cauchy product of the series with itself, hence</p>
<p>$$(1+x)^{-1}=\sum_{n=0}^\infty(-1)^nx^n=\sum_{n=0}^\infty\Biggl[\,\sum_{i... |
670,345 | <p>Here's what I came up with:</p>
<p>For this problem, I'm required to use a comparison test to determine if $\Sigma1/ln(n)^n$ converges or diverges. By intuition, I am thinking that $\Sigma1/ln(n)^n$ converges. </p>
<p>To prove that it converges by the Direct Comparison Test, I would have to find a convergent serie... | nostrebor | 98,968 | <p>$\sum_{k=1}^n a_nb_n = \sum_{k=1}^n a_nc_n \implies \sum_{k=1}^n a_k(b_k - c_k)$</p>
<p>Extracting a term from the sum gives us the equality:</p>
<p>$(b_1 - c_1) = -[\sum_{k=2}^n(b_k-c_k)]/c_1$, with $ a_k,b_k,c_k \in \mathbb{R}$</p>
<p>Which implies that the summation can take on nonzero values.</p>
|
2,894,315 | <p>Let $a;b;c>0$ such that $a+b+c=6$. Find the maximum value of $A=a^2bc+a^2+2b^2+2c^2$</p>
<hr>
<p>WLOG $b\ge c$. I see maximum value of $A=36$ at $(a;b;c)=(2;1;3)$</p>
<p>So i need to prove $A\le 36$. Or I will prove </p>
<p>$(a+b+c)^4\ge 36a^2bc+(a^2+2b^2+2c^2)(a+b+c)^2$</p>
<p>Or $(2a-b-c)(b^3+c^3+a^2b+a^2... | Michael Rozenberg | 190,319 | <p>For $(a,b,c)\rightarrow(0,6,0)$ we have $A\rightarrow72$ and since
$$72-a^2bc-a^2-2b^2-2c^2\geq0$$ it's
$$2(a+b+c)^4-(a+b+c)^2(a^2+2b^2+2c^2)-36a^2bc\geq0$$ or
$$a^4+6a^3b+6a^3c+9a^2b^2+9a^2c^2-14a^2bc+4ab^3+4ac^3+20b^2ac+20c^2ab+4ab^3+4ac^3+8b^2c^2\geq0,$$
which is obvious, we see that the maximum does not exist, b... |
2,894,315 | <p>Let $a;b;c>0$ such that $a+b+c=6$. Find the maximum value of $A=a^2bc+a^2+2b^2+2c^2$</p>
<hr>
<p>WLOG $b\ge c$. I see maximum value of $A=36$ at $(a;b;c)=(2;1;3)$</p>
<p>So i need to prove $A\le 36$. Or I will prove </p>
<p>$(a+b+c)^4\ge 36a^2bc+(a^2+2b^2+2c^2)(a+b+c)^2$</p>
<p>Or $(2a-b-c)(b^3+c^3+a^2b+a^2... | kvantour | 539,931 | <p>Next to the usage of Lagrange multipliers, one can also solve it directly function-wise by assuming $c(a,b)=6-a-b$, which leads to the following derivatives to find the extrema:</p>
<p>$$\left\{\begin{align}
\frac{\partial A}{\partial a} &= (ab-2)(12-3a-2b)&=0 \\
\frac{\partial A}{\partial b} &= (a^2-4)... |
3,214,255 | <p>How to simplify <span class="math-container">$$\frac{\sqrt{6+4\sqrt{2}}}{4+2\sqrt{2}}?$$</span></p>
<p>Rationalise the denominator </p>
<p><span class="math-container">$$\frac{\sqrt{6+4\sqrt{2}}}{4}(2-\sqrt{2})$$</span></p>
<p>This is still not simplify.</p>
| Robert Z | 299,698 | <p>Hint. Note that
<span class="math-container">$$(2\pm\sqrt{2})=\sqrt{(2\pm\sqrt{2})^2}=\sqrt{6\pm 4\sqrt{2}}.$$</span></p>
|
1,131,323 | <p>I am studying a book on proofs and there are two statements that I don't understand the difference:</p>
<ol>
<li><p>Let $x$ belong to the set of integers. If $x$ has the property that for each integer $m$, $m + x = m$, then $x = 0$.</p></li>
<li><p>Let $x$ belong to the set of integers. If $x$ has the property that... | Robert Soupe | 149,436 | <p>Statement 1 requires $m + x = m$ for all integer $m$, so that $m$ could be $-197$, or $3^{45} + 2$, or any integer whatsoever out of infinitely many. You have to prove that this works for any integer value of $m$ that could possibly be chosen.</p>
<p>Statement 2 only needs a single instance of $m$ such that $m + x ... |
2,386,864 | <p>Six persons P, Q, R, S, T and U play in a tournament called "High Rollers". Every game involved two players. Each of the participants played with every other participant exactly once. In the game both the players rolled an unbiased die each. The player who gets the larger number on the top surface of the die wins th... | Community | -1 | <p>One equivalent way to formulate this conjecture is also the following:</p>
<p>For each natural number $n$ we have:
$ \displaystyle 0 = \prod_{\underset{\gcd(a,b)=1}{2n+1=a+b}} (\Omega(a^2+b^2)-1)$</p>
<p>This is equivalent as to say that the polynomial $\displaystyle f_n(t) = \prod_{\underset{\gcd(a,b)=1}{2n+1=a+b... |
758,135 | <p>A visiting speaker in Economics recently happened to mention that John Maynard Keynes' <a href="http://www.gutenberg.org/ebooks/32625">A Treatise on Probability</a> revolutionized probability theory. I have not heard any such claim before and it struck me as strange. The <a href="http://en.wikipedia.org/wiki/A_Treat... | jaseff | 182,374 | <p>If you have a copy of Kai Lai Chung's Elementary Probability Theory, you will find Keynes' picture next to that of Kolmogorov, Polya, Feller and other probability heavy weights. (I do not know if you would cosider this evidence.) According to my modest understanding, Kolmogorov formalized probability theory and prac... |
3,162,547 | <p>I am trying to convert instances of nested 'for' loops into a summation expression. The current code fragment I have is:</p>
<pre><code>for i = 1 to n:
for j = 1 to n:
if (i*j >= n):
for k = 1 to n:
sum++
endif
</code></pre>
<p>Basically, the 'if' conditional is c... | epi163sqrt | 132,007 | <blockquote>
<p>We obtain
<span class="math-container">\begin{align*}
\color{blue}{\sum_{i=1}^n}&\color{blue}{\sum_{j=1}^n[i j\geq n]\sum_{k=1}^n1}\tag{1}\\
&=n\sum_{i=1}^n\sum_{j=1}^n[i j\geq n]\\
&=n\left(\sum_{i=1}^n\sum_{{j=\left\lfloor n/i \right\rfloor}\atop{i\mid n}}^n1+\sum_{i=1}^n\sum_{{j=\left... |
140,639 | <p><strong>Bug introduced in 11.0 and persisting through 11.3</strong></p>
<hr />
<p>From <a href="https://mathematica.stackexchange.com/a/140460/21532">this answer</a>, I doubt the capability to work on single character. So I give some test to verify this possibility. You can get my test <code>imgs</code> by this code... | Alexey Popkov | 280 | <p>I felt that I miss some simple way to unite closely located components and finally I found it: <code>ImageForestingComponents</code> (thanks to <a href="https://mathematica.stackexchange.com/a/18971/280">this</a> answer)! </p>
<ul>
<li><sub>It is unfortunate that a link to this function isn't included in the "See A... |
2,285,752 | <p>Does $\int_{-\infty}^\infty \sin(t) \,dt $ converge or diverge? How would I prove it? </p>
<p>Should I use 'principle value' to do:
$$\lim_{a \to \infty} \int_{-a}^a \sin(t)\,dt$$</p>
| Sri-Amirthan Theivendran | 302,692 | <p>$$\int_{-a}^a \sin(t)\,dt=0$$
for $a>0$ since sine is an odd function. Hence</p>
<p>$$\lim_{a \to \infty} \int_{-a}^a \sin(t)\,dt=0.$$</p>
|
927,566 | <p>We define measurable function from a measure space to a topological space as space which pulls back open sets to measurable sets. How can we prove measurable functions pulls back Borel set also to measurable sets.
where Borel sets are elements in sigma algebra generated by topology</p>
| PhoemueX | 151,552 | <p>Define</p>
<p>$$
\Sigma_f := \{M \in \mathcal{B} \mid f^{-1}(M) \in \mathcal{A} \},
$$</p>
<p>where $\mathcal{A}$ is the chosen $\sigma$-algebra on your measurable space.</p>
<p>Show that $\Sigma_f$ is a $\sigma$-algebra. Why does that help you?</p>
<p>EDIT: The same proof shows that it suffices to show $f^{-1}(... |
3,488,245 | <p>Define <span class="math-container">$f:\mathbb{R} \to \mathbb{R}$</span> given by <span class="math-container">$f(x)=x^2$</span>. If <span class="math-container">$f^{-1}(x)$</span> is interpreted as a function, it is undefined, since <span class="math-container">$f$</span> is not injective. If <span class="math-cont... | Harshit Agarwal | 661,268 | <p>We define inverse of an element as other element of the same kind such that their product is identity.In other words notion of inverse is understood as defined in case of groups. So this means inverse of a single valued function is expected to be a single valued function only unless otherwise stated. If obviously on... |
23,378 | <p>Franel uses the convergence of</p>
<p>$ \frac{\zeta(s+1)}{\zeta(s)} = \sum \frac{c(n)}{n^s}$</p>
<p>as an equivalent to the Riemann hypothesis.</p>
<p>Does anybody have a citation for this result and/or hints for computing $c(n)$?</p>
<p>Thanks for any insight.</p>
<p>Cheers, Scott</p>
| Gerry Myerson | 3,684 | <p>If I understand correctly, what Scott wants is a citation for Franel's paper on (Farey series and) the Riemann Hypothesis. That would be Les suites de Farey et le problème des nombres premiers, Göttinger Nachr. (1924) 198–201. </p>
|
1,145,466 | <p>$$y=x^2-5, x∈[-2,0]$$
Here's what I did: $$-2≤x≤0$$
$$x^2≤4 ∧ x^2≤0$$
$$x^2≤0$$
$$x^2-5≤0-5$$
$$y≤-5$$ Is it correct?</p>
| N. F. Taussig | 173,070 | <p>The function $y = x^2 - 5$ is decreasing on the interval $[-2, 0]$. As $x$ decreases from $-2$ to $0$, $x^2$ decreases from $4$ to $0$, so $x^2 - 5$ decreases from $4 - 5 = -1$ to $0 - 5 = -5$. Hence, the range is $[-5, -1]$. </p>
<p><strong>Note:</strong> If you are familiar with calculus, you can demonstrate ... |
3,574,309 | <p>My understanding of a logical formula being valid is that it concludes something, for example:</p>
<p><span class="math-container">$[(A \rightarrow B) \wedge (B \rightarrow C)] \rightarrow [A \rightarrow C]$</span> is logically valid. (I may be mixing this up with rule of inference)</p>
<p>I'm trying to understand... | Bram28 | 256,001 | <p>Two comments.</p>
<p>First, <span class="math-container">$(1-a+ab)(1-b+bc)$</span> works out to <span class="math-container">$1-b+bc-a+ab-abc+ab-ab^2+ab^2c$</span></p>
<p>This only works out to <span class="math-container">$1-a-b+ab+bc$</span> if <span class="math-container">$ab^2 = ab$</span> and <span class="mat... |
2,392,411 | <p>I am in Adv. Algebra 2 and I have a question. Firstly, would like to say I haven't taken algebra in a year due to geometry (stupid order they do but oh well) and I have a question understanding this: $(x+5)^{0}$. That would be $x^{0} + 5^{0}$ which then, wouldn't that be $1 + 1$ since anything that has a power of $0... | Marios Gretsas | 359,315 | <p>$|1- \frac{2}{x}+1|=|\frac{2x-2}{x}|=|\frac{2(x-1)}{x}|$</p>
<p>You have to take an appropriate $\delta$ to keep $x$ ''away'' from $0$</p>
<p>So you must restrict $\delta$ or in case you solve the inequality you have to make some restrictions to $\epsilon$ a priori.</p>
<p>Take $\delta=\min\{ \frac{\epsilon}, \fr... |
8,237 | <p>In this example, I want a series of four buttons to change the value of a variable that is used dynamically to drive a plot. I am trying to figure out why using Table around the buttons causes a problem.</p>
<p>This works:</p>
<pre><code>{Button["1", freq = 1], Button["2", freq = 2], Button["3", freq = 3],
Butt... | Yu-Sung Chang | 820 | <p>The second one does not work, because <code>Button</code> has <a href="http://reference.wolfram.com/mathematica/ref/HoldRest.html" rel="noreferrer"><code>HoldRest</code></a> property:</p>
<pre><code>In[1]:= Attributes[Button]
Out[1]= {HoldRest, Protected, ReadProtected}
</code></pre>
<p>Thus, <code>i</code> in you... |
3,477,152 | <p>I am trying to prove that</p>
<p><span class="math-container">$$\sum_{n=1}^\infty \frac{n}{\sqrt{n+1}}$$</span></p>
<p>diverges without checking the limit, bounds or doing any other lengthy steps, as it should be seen as divergent "immediately", but I have no clue about how I would quickly prove this.</p>
<p>So f... | Dr. Sonnhard Graubner | 175,066 | <p>Use that <span class="math-container">$$\frac{n}{\sqrt{n+1}}\geq \frac{1}{n}$$</span> and this is equivalent to
<span class="math-container">$$n^4-n-1\geq 0$$</span> this is true for <span class="math-container">$$n\geq 2$$</span></p>
|
2,473,089 | <p>I have a question on combinatorics, related to the pigeonhole principle:</p>
<blockquote>
<p>Consider the set $S= \{1,2,3,...,100\}$. Let $T$ be any subset of $S$ with $69$ elements. Then prove that one can find four distinct integers $a,b,c,d$ from $T$ such that $a+b+c=d$. Is it possible for subsets of size $68$... | nonuser | 463,553 | <p>Just an idea.</p>
<p>Divide the set $T=\{a_1<a_2<...<a_{69}\}$ in to two parts. With first $k$ numbers (set $A$) we make all sums (set $A'$) and with the rest of the numbers (set $B$) we make positive differences (set $B'$).</p>
<p>Say $A= \{a_1,a_2,...a_k\}$ (so $a_k\leq 100-(69-k) = 31+k$) and $$A' = \{... |
3,332,927 | <p>An analytic function that maps the entire complex plane into the real axis must map the imaginary axis onto:</p>
<p>A) the entire real axis</p>
<p>B) a point</p>
<p>C) a ray</p>
<p>D) an open finite interval</p>
<p>E) the empty set</p>
<p>I was thinking that it might be a constant function.
Any help would be ... | scsnm | 695,075 | <p>I think this might answer it?</p>
<p>Let f(x+iy) = u(x,y) where u is a real function. Then it is obvious by C-R equations since it is analytic.</p>
<p>Thanks for all the hints provided!</p>
|
2,451,281 | <p>$$y=x+\frac{1}{x-4}$$ I tried to find range of this function as below
$$y(x-4)=x(x-4)+1\\yx-4y=x^2-4x+1\\x^2-x(4+y)+(4y+1)=0\\\Delta \geq 0 \\(4+y)^2-4(4y+1)\geq 0 \\(y-4)^2-4\geq 0\\|y-4|\geq 2\\y\geq 6 \cup y\leq2$$ this usuall way. but I am interested in finall answer... is the other idea to find function range ... | Mark Bennet | 2,906 | <p>Clearly for $|x|$ large, $y$ is large and has the same sign as $x$. For $x=4+h$ where $h$ is small, $y$ has the same sign as $h$ and is as large as we like in absolute value by making $h$ small. We can find the turning points by $$y'=1-\frac 1{(x-4)^2}=0, y''=\frac 2{(x-4)^3}$$ i.e. at $x-4=\pm 1$ with attention to ... |
969,601 | <p>For the function whose graph is a paraboloid given by</p>
<p>$z = x^2 + y^2/4$</p>
<p>I know that the level curve represents an ellipse. I also know that the parametrization of this curve in the form $x = x(t)$, $y = y(t)$ is </p>
<p>$x(t) = \cos(t)$
and<br>
$y(t) = 2\sin(t)$</p>
<p>In class, my professor said t... | Mick A | 153,109 | <p>Sorry for this being a long answer. Actual probabilities are at the end if you want to jump straight there.</p>
<p>To work this out we need the underlying probability of any random square being a mine, knowing nothing about the square. Let's call this probability $q$.</p>
<p>$$q = \dfrac{\text{#mines in the grid}}... |
115,433 | <p>Mathematica has a lot of machinery for working with predefined probability distributions. It is not clear how to make that machinery work with a new distribution.</p>
<p>Suppose I want to define a brand new distribution</p>
<pre><code>MyDistribution[a, b, c]
</code></pre>
<p>What is the minimum I need to specify ... | JimB | 19,758 | <p>I'm going to argue that parameter information and paradise are not lost (until there's a specific counter-example).</p>
<pre><code>(* Define some function *)
someFun[x_, μ_, σ_, a_, b_, c_] := (a/σ) Exp[-(x - μ)^b/(c σ^2)]
(* Define a probability density function that depends on someFun and
some yet to be given... |
1,577,090 | <p>How can I determine if the function is one-to-one .. I know that any odd function is 1-1 and any even function is NOT 1-1 but what about functions that are neither of those, like $x^3+5$ or $x^3+x^2+3$. How can I determine whether it is a one-to-one? </p>
| Will Fisher | 290,619 | <p>The simplest way in my opinion is to take your function, $f(x)$, and set it up so that $y=f(x)$. I'll use your function for example so that we have $y=x^3+x^2+3$. From here we want to perform a simple check for the existence of the inverse function (i.e. the function that when provided a value $f(x)$ will return $x$... |
729,373 | <p>I am to evaluate $\displaystyle\int_0^{\infty} \dfrac{\sin x}{x(x^2+1)}dx$ via contour integration.</p>
<p>Now I used an indented semicircular contour, and the parts lying on the real line and the big arc were no problem, but the small arc is being resistant, and I'm not sure what to do. Usually, on the small arc f... | Daniel Fischer | 83,702 | <blockquote>
<p>My issue is this particular integrand doesn't have a principal part...</p>
</blockquote>
<p>Then the integral over the large semicircle won't work out. Remember that</p>
<p>$$\lvert \sin (x+iy)\rvert^2 = \lvert \sin x \cos (iy) + \sin(iy)\cos x\rvert^2 = \sin^2 x+ \sinh^2 y,$$</p>
<p>so if you keep... |
1,310,460 | <p>Given a finite connected graph $G$, I can make a finite number of cuts on the edges to obtain a tree. What is the most efficient algorithm to perform this procedure? </p>
<p>Thanks,
Vladimir</p>
| Gulab Ghosh | 1,049,095 | <p>SIR MY DOUBT IN 1ST OPT HOW CAN IT BE CORRECT.
f(x)=x ; x€(0,1)
Then f(x)=(0,1) .Here f(x) is bounded in R and also continuous in (0,1)¢R.
But domain is not compact.</p>
|
2,019,049 | <p><a href="https://i.stack.imgur.com/SULGd.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/SULGd.png" alt="enter image description here"></a></p>
<blockquote>
<p>Let $\theta \in \mathbb{R}$. Prove: if $\prod\limits_{k=1}^{20} (\cos(k\theta) + i \sin(k\theta)) = i$, then there exists an integer $... | Jan Eerland | 226,665 | <p>when $\text{z}\in\mathbb{C}$:
$$\text{z}=\Re\left[\text{z}\right]+\Im\left[\text{z}\right]i=\left|\text{z}\right|e^{\left(\arg\left(\text{z}\right)+2\pi\text{n}\right)i}=\left|\text{z}\right|\cos\left(\arg\left(\text{z}\right)\right)+\left|\text{z}\right|\sin\left(\arg\left(\text{z}\right)\right)i$$
Where $\left|\te... |
2,518,523 | <p>So I've stumbled upon this problem:</p>
<blockquote>
<p>Compare the numbers:</p>
<p><span class="math-container">$$9^{8^{8^9}} \text{ and }\,8^{9^{9^8}}$$</span></p>
</blockquote>
<p>I got this in a test and I had no idea what the answer to the problem is... Can someone give me the answer and how you can find it? It... | Doug M | 317,162 | <p>$8^9 > 9^8$</p>
<p>In fact it is more than $3$ times greater. </p>
<p>Which would suggest that
$8^{8^9} \gg 9^{9^8}$ ($\gg$ means significantly greater)</p>
<p>So then the next level</p>
<p>$9^{8^{8^9}}$ is a larger base to a larger exponent than $8^{9^{9^8}}$</p>
|
143,721 | <p>The exercise is about convex functions:</p>
<p>How to prove that $f(t)=\int_0^t g(s)ds$ is convex in $(a,b)$ whenever $0\in (a,b)$ and $g$ is increasing in $[a,b]$?</p>
<p>I proved that </p>
<p>$$f(x)\leq \frac{x-a'}{b'-x}f(b')+\left(1-\frac{x-a'}{b'-x}\right)f(a')$$</p>
<p>when we have </p>
<p>$$x=\left(1-\fra... | copper.hat | 27,978 | <p>A slightly different approach:</p>
<p>We need to show $f(x+\lambda(y-x)) \leq f(x) + \lambda (f(y)-f(x))$, with $\lambda \in (0,1)$. Suppose $x<y$. Then
$$f(x+\lambda(y-x)) - f(x) = \int_{x}^{x+\lambda(y-x)} g(s) \; ds$$
Using the change of variables $t=\frac{s-x}{\lambda}+x$, we get
$$\int_{x}^{x+\lambda(y-x)} ... |
4,488,740 | <p>It is known that <span class="math-container">$$\sin^{−1}x+\sin^{−1}y = \sin^{-1}\left[x\sqrt{1 – y^2} + y\sqrt{1 – x^2}\right] $$</span> if <span class="math-container">$x, y ≥ 0$</span> and <span class="math-container">$x^2+y^2 ≤ 1.$</span></p>
<p>I know that the given condition makes sure that <span class="math-c... | Guillermo García Sáez | 696,501 | <p>Hint: <span class="math-container">$A^3-B^3=(A-B) (A^2+AB+B^2) $</span>.</p>
<p>Apply it for <span class="math-container">$A=\sqrt[3]{(x+h)^2}$</span> and <span class="math-container">$B=\sqrt[3]{x^2}$</span></p>
|
66,187 | <p>I would like to remove facets from an <a href="http://mathworld.wolfram.com/Octahedron3-Compound.html" rel="nofollow noreferrer">octahedron 3-compound</a> - like in the picture below.</p>
<p><img src="https://i.stack.imgur.com/xvvj4.png" alt="octahedron 3-compound with holes"></p>
<p>I tried to combine these two g... | J. M.'s persistent exhaustion | 50 | <p>Using Heike's <a href="https://mathematica.stackexchange.com/a/5891">reimplementation of the old routine <code>PerforatePolygons[]</code></a>:</p>
<pre><code>PerforatePolygons[PolyhedronData["OctahedronThreeCompound"], 3/4]
</code></pre>
<p><img src="https://i.stack.imgur.com/eNn99.png" alt="octahedron 3-compound ... |
2,753,812 | <p>So far I have been looking for a possible divisor such that the sum of $7$ seventh powers will never leave a certain remainder but there seems not to be such a divisor below $400$ and the search is too computationally heavy to continue.</p>
| asdf | 436,163 | <p>This is a case of Waring's problem: <a href="https://en.wikipedia.org/wiki/Waring%27s_problem" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Waring%27s_problem</a></p>
<p>It can be proven (Ben Green) that it is not possible to solve this using modular arithmetic - for $7$ seventh powers a combinatorial so... |
3,483,017 | <p>Is there a nice or simple form for a sum of the following form?
<span class="math-container">$$ 1 + \sum_{i=1}^k \binom{n-1+i}{i} - \binom{n-1+i}{i-1}$$</span></p>
<p>Motivation: Due to a computation in the formalism of Schubert calculus the above sum with <span class="math-container">$k = \lceil n/2 \rceil -1$</s... | Paras Khosla | 478,779 | <p><strong>Hint</strong></p>
<p>The following method does not use telescopic series but it makes use of elementary binomial theorem along with some geometric progression to arrive at the answer.</p>
<p><span class="math-container">$$\begin{aligned}S&=\sum_{i=1}^{k}{n-1+i\choose n-1}-\sum_{i=1}^{k}{n-1+i\choose n}... |
162,613 | <p>How can I apply a <code>LabelingFunction</code> with an if else condition?
Suppose you have the following plot:</p>
<pre><code>BarChart[{4, 3, 2, 1}, BarOrigin -> Left, LabelingFunction ->(Placed[#,Left] &)]
</code></pre>
<p><a href="https://i.stack.imgur.com/2o9wE.png" rel="noreferrer"><img src="https:... | Ulrich Neumann | 53,677 | <pre><code>BarChart[{4, 3, 2, 1}, BarOrigin -> Left,LabelingFunction -> (Placed[#, If[# <= 2, Left, Right]] &)]
</code></pre>
<p>will answer your question! Bold might be realized similar...</p>
|
1,131,814 | <p>The question is probably obvious, but is there a sense to say that for exemple $$f:[1,2]\cup[3,4]\longrightarrow \Bbb R$$
is continuous on $[1,2]\cup [3,4]$ or not really ?</p>
| Loreno Heer | 92,018 | <p>Yes, the same definition of continuity applies also to this set.</p>
<p>Note that $f:[1,2] \cup [3,4] \to \mathbb{R}$ is continuous if for every open set $U \in \mathbb{R}$ the set $f^{-1}(U)$ is an open subset of $[1,2] \cup [3,4]$.</p>
|
3,502,123 | <p>I've seen many proofs of this theorem. But, unable to think any example showing this.</p>
<p>Suppose, how to write (0,1) as a countable Union of disjoint open intervals. </p>
<p>No idea! I'm stuck </p>
<p>Plz help!</p>
| kimchi lover | 457,779 | <p>Assuming the OP meant to <span class="math-container">$\mu^k$</span> to mean the <span class="math-container">$k$</span>-th convolution product of <span class="math-container">$\mu$</span>:</p>
<p>If the convolution of <span class="math-container">$k$</span> copies of <span class="math-container">$\mu$</span>, for... |
3,502,123 | <p>I've seen many proofs of this theorem. But, unable to think any example showing this.</p>
<p>Suppose, how to write (0,1) as a countable Union of disjoint open intervals. </p>
<p>No idea! I'm stuck </p>
<p>Plz help!</p>
| Kuldeep Guha Mazumder | 423,627 | <p>Okay so I will go along Kimchi Lover's line, with some more elaborations. I shall be using the property of characteristic functions that states that if <span class="math-container">$\mu$</span> and <span class="math-container">$\nu$</span> are two Borel probability measures on <span class="math-container">$\mathbb{R... |
3,371,516 | <p>The question is as follows:</p>
<p>Suppose that E, F, and G are events with P(A) = 19/100, P(B) = 7/25, P(C) = 3/10. Furthermore, suppose A andB are mutually exclusive, A and C are independent, and P(B | C) = 11/15. Find P(A ∪ B ∪ C).</p>
<p>My attempt is as follows:</p>
<p>P(A ∪ B ∪ C) = P(A) + P(B)... | Chappers | 221,811 | <p>You've lost the <span class="math-container">$P(A \cap C)$</span> term when expanding <span class="math-container">$P(A \cup B \cup C)$</span>, which is equal to <span class="math-container">$P(A)P(C)$</span> since <span class="math-container">$A$</span> and <span class="math-container">$C$</span> are independent.</... |
43,650 | <p>Consider the following problem:</p>
<blockquote>
<p>Let $f:{\mathbb R}^3 \to{\mathbb R}$ be $$f(x,y,z)=x+4z$$ where $$x^2+y^2+z^2\leq 2.$$ Find the minimum of $f$. </p>
</blockquote>
<p>This is similar to the question <a href="https://math.stackexchange.com/q/41385/9464">here</a>. However, since this is not an a... | davidlowryduda | 9,754 | <p>You should do the standard derivative test and see if any results come up within your region. You should then perform a Lagrange Multiplier Method on the boundary, i.e. on $x^2 + y^2 + z^2 = 2$. Because it's a compact place, you will have a max and a min, and it will either be on the interior or on the boundary.</p>... |
89,176 | <p>The Hahn–Banach theorem states that: Given a sublinear functional $S: V \rightarrow \mathbb R$, if $T: U \rightarrow \mathbb R$ is a linear functional on a linear subspace $U \subseteq V$ that is dominated by $S$ on $U$, then there exists a linear extension of $T$ to $V$ that is dominated by $S$ on $V$.</p>
<p>Now,... | Yemon Choi | 763 | <p>This is not really an answer, more of an extended comment, but it was getting too long. Besides, if I am mistaken then it is easier to downvote or correct something posted as an answer.
$\newcommand{\norm}[1]{\Vert#1\Vert}
\newcommand{\Real}{{\bf R}}
\newcommand{\ptp}{\hat{\otimes}}$</p>
<hr>
<p>I think that in th... |
89,176 | <p>The Hahn–Banach theorem states that: Given a sublinear functional $S: V \rightarrow \mathbb R$, if $T: U \rightarrow \mathbb R$ is a linear functional on a linear subspace $U \subseteq V$ that is dominated by $S$ on $U$, then there exists a linear extension of $T$ to $V$ that is dominated by $S$ on $V$.</p>
<p>Now,... | Yves | 21,759 | <p>First of all, I would like to thank you for your detailed answers.</p>
<p>On the following web page, I found a thesis entitled "Sur les opérateurs multisouslinéaires " by TALLAB Abdelhamid :</p>
<p><a href="http://www.univ-msila.dz/theses/index.php?option=com_docman&task=cat_view&gid=46&limit=5&ord... |
271,886 | <p>I am interested in explicit generators of the cohomology $H^\bullet(SU(n),\mathbb{Z})$. Let $\omega = g^{-1} dg$ be the Maurer-Cartan form on $SU(n)$. The forms $\alpha_3,\alpha_5,\dots,\alpha_{2n-1}$, defined by
$$ \alpha_k := \text{Tr}(\omega^{k})$$
are bi-invariant and define classes in de Rham cohomology.</p>
<... | Jeremy Daniel | 34,256 | <p>OK, I have understood how it works. This is classical material but it seems difficult to find an answer in the literature for the non-specialist; so let me explain.</p>
<p>We consider the universal bundle $E U(n) \rightarrow B U(n)$ with fiber $U(n)$. Since $E U(n)$ is contractible, the Serre spectral sequence of t... |
2,544,544 | <p>There is a theorem stated in my textbook as following and my question is below the proof :</p>
<p>Let <span class="math-container">$0<p<q<\infty$</span> and let <span class="math-container">$f$</span> in <span class="math-container">$L^{p,\infty}(X,\mu) \cap L^{q,\infty}(X,\mu)$</span>, where <span class="... | user284331 | 284,331 | <p>This is a classical technique in harmonic analysis with the title: let some constant to be determined later. You may try to put $B=1$ to see what happen, of course, you cannot conclude anything actually. So the strategy is to keep $B$ fixed and we will config the value of $B$ while needed.</p>
<p>If we want $\min\{... |
4,080,680 | <p>If <span class="math-container">$\sum a_n$</span> diverges does <span class="math-container">$\sum \frac{a_n}{\ln n}$</span> necessarily diverge for <span class="math-container">$a_n>0$</span>?</p>
<p>I tried <span class="math-container">$a_n=\frac{\ln n}{n^2}$</span> to try bag an easy counterexample but it turn... | José Carlos Santos | 446,262 | <p>A counter example would be <span class="math-container">$\displaystyle\sum_{n=2}^\infty\frac1{n\log(n)}$</span>. It diverges, but the series <span class="math-container">$\displaystyle\sum_{n=2}^\infty\frac1{n\log^2(n)}$</span> converges.</p>
|
2,076,656 | <p>As a homework, I was asked to solve this <strong>equation</strong>, $$(3x-4\lfloor x\rfloor=0),x\in \Bbb R$$ For $x\in \Bbb Z:\lfloor x\rfloor=x \implies x=0$ But for $x\not\in\Bbb Z : \lfloor x\rfloor=\frac 34x$ So now we know that $\frac 34x\in\Bbb Z$ and $x\in\Bbb R-\Bbb Z$, so maybe ? define a function such that... | David Quinn | 187,299 | <p>Let $x=n+\epsilon$, where $n\in\mathbb{Z}$ and $0\leq\epsilon<1$</p>
<p>Then the equation becomes $$3n+3\epsilon=4n\implies 3\epsilon=n$$</p>
<p>Therefore, $0\leq\frac n3<1\implies n=0,1,2$ and correspondingly, $\epsilon=0, \frac 13,\frac 23$</p>
<p>Therefore the solutions are $$x=0,\frac 43,\frac 83$$</p>
|
1,665,931 | <p>Let $a,c \in \mathbb{R}$ and $b \in \mathbb{C}$. We consider the equation $$a \bar{z}z+ b\bar{z} + \bar{b}z+c=0.$$ What curve it represents in the complex plane?</p>
<p>I think it a circle, but I am not able to conclude. Any help?</p>
| copper.hat | 27,978 | <p>This is a case where writing $z=x+iy$ for $x,y$ real helps.</p>
<p>Let $b=b_x+i b_y$.</p>
<p>If $a=0$, then the equation is $xb_x + y b_y + c = 0$ which is easily seen to be a line.</p>
<p>If $a \neq 0$, we can take $a=1$ without loss of generality. Then the
equation is $x^2+y^2 +xb_x + y b_y + c = 0$. To find a ... |
348,276 | <p>I am not an expert, thus I apologize if my question is very naive. Let <span class="math-container">$\mathsf{M}$</span> be a model category (I do not assume any functoriality on the factorization),</p>
<blockquote>
<p><strong>Q1.</strong> Is there a reference where it is proven that (all) homotopy colimits exist?</p... | Simone Virili | 24,891 | <p>Let me start first answering your second question (this is quite close to the idea of John Klein).</p>
<p><strong>Definition.</strong>
A pair <span class="math-container">$(\mathbf C,\mathcal W)$</span>, with <span class="math-container">$\mathbf C$</span> a category and <span class="math-container">$\mathcal W$</s... |
176,055 | <p>I heard teachers say [cosh x] instead of saying "hyperbolic cosine of x".</p>
<p>I also heard [sinch x] for "hyperboic sine of x". Is this correct?</p>
<p>How would you pronounce tanh x? Instead of saying "hyperbolic tangent of x"?</p>
<p>Thank you very much in advance.</p>
| Reg Dodds | 66,265 | <p>My maths professor Siegfried Goeldner who got his PhD in mathematics at the Courant Institute at New York University under one of the German refugees from Goetingen, in 1960, pronounced sinh as /ʃaɪn/, cosh as /kɒʃ/ ("cosh") and tanh as /θæn/, i.e., as shine, cosh and than with a soft th like in theta---the same pro... |
2,906,282 | <p>I want calculate the reflection from a straight <strong>R</strong> on the surface from the straight <strong>BC</strong>: </p>
<p><a href="https://i.stack.imgur.com/bOINy.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/bOINy.png" alt="enter image description here"></a></p>
<p>The triangle with th... | Bernard | 202,857 | <p>It has nothing to do with linear algebra. It is a general fact that in any unital ring, a product of invertible elements: $a_1a_2\dots a_n$, is invertible, and its invertible, and its inverse is the product of the inverses of the factors, in reverse order. In other words:
$$(a_1a_2\dots a_n)^{-1}=a_n^{-1}\dots a_2^... |
3,224,765 | <p>The following question was asked on a high school test, where the students were given a few minutes per question, at most:</p>
<blockquote>
<p>Given that,
<span class="math-container">$$P(x)=x^{104}+x^{93}+x^{82}+x^{71}+1$$</span>
and,
<span class="math-container">$$Q(x)=x^4+x^3+x^2+x+1$$</span>
what is t... | Nightgap | 506,645 | <p>I think if the candidates know what a geometric series is, the question is okay. Indeed, one uses exactly this trick to find the formula for the geometric series, i.e. one writes
<span class="math-container">$$(x-1)\sum_{k=1}^nx^k=x^{n+1}-1$$</span> to find that <span class="math-container">$$\sum_{k=1}^\infty x^k=... |
162,812 | <p>I have an interesting calc question here but im not sure how to solve it. Can someone perhaps give me a helping hand or guide me through steps?</p>
<blockquote>
<p>A balloon that takes images of the earth is shot up in the sky with
rockets from 0 ft off the ground is given by the height of the function s(t)=
... | Jonathan Dewein | 24,181 | <p>Well, I believe that the velocity of an object is its derivative.</p>
<p>So, find the derivative of your initial function.</p>
<p>$$\
s(t) = -18t^2 + 120t
$$</p>
<p>With a simple application of the power rule, we arrive at</p>
<p>$$\
s'(t) = -36t + 120
$$</p>
<p>$$or~ v(t) = -36t+120$$</p>
<p>Now, you have a f... |
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