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 |
|---|---|---|---|---|
2,877,174 | <p>Imagine that we roll two fair six-sided dice (i.e., all six sides have equal probability). Let X1 and
X2 be the random variables representing these outcomes.
Now, imagine we take one of the dice rolls, say X1, and add a (possibly negative) constant c to
the result. If this becomes less than zero, then we set it to z... | Will Jagy | 10,400 | <p>I like to find the actual Jordan form, especially including the matrices that give $R^{-1}A R = J.$ It is not bad when the eigenvalues are integers, and makes concepts concrete. This begins with choosing a column vector i called $w$ such that $A^2 w \neq 0.$ That becomes the far right column.</p>
<p>$$
\frac{1}{8} ... |
1,050,141 | <p>Find the roots of the equation $3^{x+2}$+$3^{-x}$=10 . By inspection the roots are $x=0$ and $x=-2$. But how can I solve this equation otherwise? </p>
| Balbichi | 24,690 | <p>Suppose $y=3^x$. Then $3^{x+2}$+$3^{-x}=10$ is transformed into $$9y+{1\over y}=10\Rightarrow 9y^2-10y+1=0\Leftrightarrow(y-1)(9y-1)=0$$</p>
|
1,450,669 | <p>Find the value of $$\lim\limits_{x\to 0^+}[1+[x]]^{\frac2x}$$where $[x]$ denotes greatest integer function less than or equal to $x$. <br></p>
<hr>
<p><strong>My attempt:</strong> <br>
I calculated $[1+[x]]$ to be $1$ as $x\to 0^+$. <br> Now I am stuck. <br> Please help me.</p>
| Aditya Agarwal | 217,555 | <p>Substitute $y=\frac1x$, so as $x\to0^{+}, y\to\infty$. <br>
So we get $$\lim\limits_{y\to\infty}[1+[\frac{1}{y}]]^{2y}=\lim\limits_{y\to\infty}([1+[\frac1y]]^{y})^2=1^{2.\infty}=1$$ <br></p>
|
85,622 | <p>I'm trying to understand how this works in terms of Galois theory and local class field theory. Assume we have an extension of local fields $E/L/K$ s.t. $E/L$ and $L/K$ are abelian. I'm interested in recognizing when $E/K$ is Galois. Clearly, $E/K$ is Galois if and only if $E$ is always fixed by an extension of an $... | SGP | 11,786 | <p>Not an answer, but perhaps something useful from an apprentice in CFT:</p>
<p>Considerations very closely related to your question led Andre Weil to discover the "Weil group"; see <a href="http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.jmsj/1261734944" rel="nofollow">hi... |
3,563,238 | <p>I try to understand the relation of likelihood to cross-entropy by reading <a href="https://en.wikipedia.org/wiki/Cross_entropy" rel="nofollow noreferrer">cross-entropy</a>.</p>
<p>The problem is I cannot understand the formula for the likelihood in the article. The likelihood is defined as follows</p>
<p><span cl... | WoolierThanThou | 686,397 | <p>It just looks to me like the person who wrote the wiki article isn't phrasing themselves all that greatly.</p>
<p>Your model is that <span class="math-container">$(X_n)_{1\leq n\leq N}$</span> are iid and <span class="math-container">$\mathbb{P}(X_n=i)=q_i$</span>. So given a realisation <span class="math-container... |
306,178 | <p>Given
$$
y_n=\left(1+\frac{1}{n}\right)^{n+1}\hspace{-6mm},\qquad n \in \mathbb{N}, \quad n \geq 1.
$$
Show that $\lbrace y_n \rbrace$ is a decreasing sequence. Anyone can help ? I consider the ratio $\frac{y_{n+1}}{y_n}$ but I got stuck.</p>
| Sugata Adhya | 36,242 | <p><strong>Hint:</strong> Try to use the A.M.- G.M. inequality for positive numbers.</p>
|
306,178 | <p>Given
$$
y_n=\left(1+\frac{1}{n}\right)^{n+1}\hspace{-6mm},\qquad n \in \mathbb{N}, \quad n \geq 1.
$$
Show that $\lbrace y_n \rbrace$ is a decreasing sequence. Anyone can help ? I consider the ratio $\frac{y_{n+1}}{y_n}$ but I got stuck.</p>
| Micheal Brain Hurts | 324,389 | <p><strong>Lemma(1):</strong></p>
<p>Let be $a,b\in\mathbb R^+$</p>
<p>$$\sqrt[n+1]{ab^n}\le \dfrac{a+bn}{n+1}$$</p>
<p><strong>Proof:</strong></p>
<p>$$\sqrt[n+1]{a\underbrace{bbb...b}_{n\;times}}\le \dfrac{a+b+b+b+...+b}{n+1}=\dfrac{a+bn}{n+1}\\ \Box.$$</p>
<p><strong>Lemma(2):</strong></p>
<p>Let be $x_n=\left... |
306,178 | <p>Given
$$
y_n=\left(1+\frac{1}{n}\right)^{n+1}\hspace{-6mm},\qquad n \in \mathbb{N}, \quad n \geq 1.
$$
Show that $\lbrace y_n \rbrace$ is a decreasing sequence. Anyone can help ? I consider the ratio $\frac{y_{n+1}}{y_n}$ but I got stuck.</p>
| Ekadh Singh - Reinstate Monica | 675,447 | <p>the derivative of <span class="math-container">$(1+1/n)^{(n+1)}$</span> is <span class="math-container">$(ln(1+1/n)-1/n)*(1+1/n)^{(n+1)}$</span> so if that is less than <span class="math-container">$0$</span> then <span class="math-container">$(1+1/n)^{(n+1)}$</span> is decreasing. <span class="math-container">$(1+1... |
2,213,047 | <p>Let $F$ be the set of all functions $f : \mathbb{R} \to \mathbb{R}$. A relation $c$ is defined on $F$ by
$f c g$ if and only if $f(x) ≤ g(x)$ for all $x ∈ \mathbb{R} $.
Prove that '$c$' is a partial order.</p>
<p>I have proved that the relation is <em>reflexive</em>. Now I must prove that it is <em>antisymmetric</e... | GoodDeeds | 307,825 | <p>The relation is over the set of functions, $F$, and not on the set of real numbers, $\mathbb R$.</p>
<p>So, here, your $a$s and $b$s are elements of $F$, i.e., functions that map from $\mathbb R$ to $\mathbb R$.</p>
<p>In your case, $a c b$ means $a \le b$, where $a$ and $b$ are functions in $F$. It does not mean ... |
993,385 | <p>Salam,</p>
<p>I would appreciate it if anyone could help me solving this integral:</p>
<p>$$ \int \frac{e^{ax^2+bx}}{\sqrt{1-x^2}}dx
$$</p>
<p>Many thanks.</p>
| k170 | 161,538 | <p>The solution to this integral cannot be found in terms of elementary mathematical functions. </p>
|
993,385 | <p>Salam,</p>
<p>I would appreciate it if anyone could help me solving this integral:</p>
<p>$$ \int \frac{e^{ax^2+bx}}{\sqrt{1-x^2}}dx
$$</p>
<p>Many thanks.</p>
| Lucian | 93,448 | <p>The indefinite integral <a href="http://en.wikipedia.org/wiki/Liouville's_theorem_(differential_algebra)" rel="nofollow">cannot</a> be <a href="http://en.wikipedia.org/wiki/Risch_algorithm" rel="nofollow">expressed</a> in terms of elementary functions. However, the definite integral can, for appropriate limits, ... |
2,916,168 | <p>I am a beginner in proofs and, unfortunately, I cannot wrap my mind around how to prove the simplest things, so I need a bit of help getting started. This is the proof that I am dealing with:</p>
<p>$\text{If }x< y< 0\text{, then }x^{2}> y^{2}\text{.}$</p>
<p>Thank you in advance.</p>
| Paramanand Singh | 72,031 | <p>Multiplying an inequality by a negative number reverses the inequality.</p>
<hr>
<p>We have $x<y$ and multiplying it with negative number $x$ we get $x^2>xy$. Again multiplying the same starting inequality by negative number $y$ we get $xy>y^2$. Now using $x^2>xy$ and $xy>y^2$ we arrive at $x^2>y... |
1,214,042 | <p>"A particle of mass m is attracted towards a fixed point 0 with a force inversely proportional to its instantaneous distance from 0. If the particle is released from rest, at a distance L from 0, find the time for it to reach 0"</p>
<p>My attempt at this question:
Noting the fact $x(0)=L$ and $x'(0)=0$, I tried to ... | Lutz Lehmann | 115,115 | <p>The relation of mass and $k$ is not given, set their fraction to $k/m=1/2$. Solve $\ddot x=-1/(2x)$. Multiply with $\dot x$, integrate once
$$
\frac12(\dot x(t)^2-\dot x(0)^2)= -\frac12\,\ln|x(t)|+\frac12\ln|x(0)|
\implies \dot x(t)=\pm\sqrt{-\ln(|x(t)|/L)}
$$
This is not an easy integral.</p>
|
566,255 | <p>How to evaluate the following integral? </p>
<p>$\int_{0}^{\infty} \sin(kx)dx=\frac 1 k$ </p>
<p>The book Mathematical Physics by Butkov reads "The sequence $f_N(k)=\int_{0}^{N} \sin(kx) dx=\frac{(1-\cos kN)}k$diverges as N approaches $\infty$, but it is weakly convergent for a suitable chosen set of test functio... | daulomb | 98,075 | <p>We should calculate the limit: $\displaystyle\lim_{k\rightarrow\infty}\int_{0}^\infty\sin(kx)\phi(x)dx$ for every $\phi\in C_0^\infty(0, \infty)$, i.e, $\phi$ is a test function. Integration by parts implies that $$\displaystyle\lim_{k\rightarrow\infty}\int_{0}^\infty\sin(kx)\phi(x)dx$$
$$=-\displaystyle\lim_{k\righ... |
3,664,272 | <h2><strong>MOTIVATION</strong></h2>
<p>I am considering investing a significant amount of money into a raffle. In order to decide the number of entries I purchase, I would like to find probability distributions for the number of prizes I will win with respect to the number of entries I purchase.</p>
<h2><strong>HOW ... | lonza leggiera | 632,373 | <ul>
<li>If you buy a total of <span class="math-container">$\ n\ $</span> tickets, the expected number of prizes you win is <span class="math-container">$\ \frac{n}{50}\ $</span>, regardless of which rounds the tickets are in. You're therefore correct that your expected number of wins is always the same for the same n... |
688,711 | <p>Assume the desity of air <span class="math-container">$\rho$</span> is given by</p>
<p><span class="math-container">$\rho(r)=\rho_0$$e^{-(r-R_0)/h_0}$</span> for <span class="math-container">$r\ge R_0$</span></p>
<p>where <span class="math-container">$r$</span> is the distance from the centre of the earth, <span c... | copper.hat | 27,978 | <p>The problem is simplified by the fact that the density is independent of the latitude and longitude. So we can compute the total mass and the multiply this by the fraction of area in the specified region over the total area.</p>
<p>The total enclosed volume at radius $r$ is $V(r) = {4 \over 3} \pi r^3$, hence we ha... |
2,904,359 | <p><strong>Definition.</strong> A formula <span class="math-container">$\phi(x,a)$</span> <em>divides</em> over a set <span class="math-container">$B$</span> if there are <span class="math-container">$k<\mathbb{N}$</span> and a sequence <span class="math-container">$(a_i)_{i<\omega}$</span> such that</p>
<p>(1)... | Alex Kruckman | 7,062 | <p>I hope you accept Mostafa's answer, since he described the canonical example of a formula which forks but does not divide (<span class="math-container">$x=x$</span> in the circular order). I believe this example is originally due to Byunghan Kim, in his thesis on simple theories. </p>
<p>But since you asked for mor... |
1,997,341 | <p>We know that the binomial theorem and expansion extends to powers which are non-integers. </p>
<p>For integer powers the expansion can be proven easily as the expansion is finite. However what is the proof that the expansion also holds for fractional powers? </p>
<p>A simple an intuitive approach would be apprecia... | SirXYZ | 375,099 | <p>Heres a proof:</p>
<p>Let <span class="math-container">$f(a)$</span> denote the binomial expansion of <span class="math-container">$(1+x)^a$</span> and <span class="math-container">$f(b)$</span> denote the same for <span class="math-container">$(1+x)^b$</span>. Where</p>
<p><span class="math-container">$(1+x)^a = ... |
1,883,765 | <p>In the following situation I know how to find the <code>red</code> distance as <code>(diagonal - diamater) / 2</code> however I'm not sure how to find the <code>yellow</code> and <code>green</code> distances.</p>
<p><a href="https://i.stack.imgur.com/RMNLh.png" rel="nofollow noreferrer"><img src="https://i.stack.im... | Daniel | 221,735 | <p>they are equal, if red is $r$ then green\yellow is $r/\sqrt{2}$</p>
|
2,939,605 | <p>The given task goes as follows:</p>
<blockquote>
<p>Show that <span class="math-container">$ f: \mathbb{R} \longrightarrow \mathbb{R}$</span> defined by <span class="math-container">$f(x) = \sqrt{1 + x^2} $</span> is not a polynomial function.</p>
</blockquote>
<p>I tried this approach - if <span class="math-con... | Peter Szilas | 408,605 | <p>Assume <span class="math-container">$\sqrt{x^2+1}= a_nx^n+...a_1x +a_0$</span> , with an <span class="math-container">$a_n \not =0$</span>.</p>
<p>Since <span class="math-container">$\sqrt{x^2+1}$</span> is an even function, the polynomial has to be even, i.e. only even powers occur.</p>
<p>Set <span class="math-c... |
2,634,657 | <p>My question is: For <span class="math-container">$R>0$</span>, Can we choose a family of functions <span class="math-container">$\eta_R\in C_c^1(\mathbf{R}^N)$</span> satisfying
<span class="math-container">$0\leq\eta_R\leq 1$</span> in <span class="math-container">$\mathbf{R}^N$</span>, <span class="math-contain... | username | 948,485 | <p>The usual construction for such fonctions is to define the cut-off as a radial function, that is, <span class="math-container">$$\eta_R(x) = \eta\left(\frac{|x|}R\right)$$</span> where <span class="math-container">$\eta\in c^\infty(\mathbb R)$</span> is given by
<span class="math-container">$$
x\to \begin{cases}1 &a... |
873,224 | <p>I'm not mathematically inclined, so please be patient with my question.</p>
<p>Given </p>
<ul>
<li><p>$(x_0, y_0)$ and $(x_1, y_1)$ as the endpoints of a cubic Bezier curve.</p></li>
<li><p>$(c_x, c_y)$ and r as the centerpoint and the radius of a circle.</p></li>
<li><p>$(x_0, y_0)$ and $(x_1, y_1)$ are on the ci... | robjohn | 13,854 | <p>Let $(x,y)^R=(-y,x)$ represent rotation by $\pi/2$ counterclockwise and
$$
\gamma(t)=(1-t)^3p_0+3t(1-t)^2p_1+3t^2(1-t)p_2+t^3p_3
$$
define a cubic bezier with control points $\{p_0,p_1,p_2,p_3\}$.</p>
<p>Suppose $p_0=(x_0,y_0)$, $p_3=(x_1,y_1)$, and $c=(c_x,c_y)$ are given so that $|p_0-c|=|p_3-c|=r$ ($p_3$ is coun... |
2,655,178 | <p>I am asked to find the equation of a cubic function that passes through the origin. It also passes through the points $(1, 3), (2, 6),$ and $(-1, 10)$. </p>
<p>I have walked through many answers for similar questions that suggest to use a substitution method by subbing in all the points and writing in terms of vari... | Mark Bennet | 2,906 | <p>Given four points $(x_i,y_i)$ consider the functions $$f_1(x)=\frac {(x-x_2)(x-x_3)(x-x_4)}{(x_1-x_2)(x_1-x_3)(x_1-x_4)}$$ so that $f_1(x_1)=1$ and $f_1(x_i)=0, i\neq 1$, and similarly $f_2, f_3, f_4$. Note that the $f_i$ are cubic in $x$.</p>
<p>Then $p(x)=y_1f_1(x)+y_2f_2(x)+y_3f_3(x)+y_4f_4(x)$ is at most a cubi... |
2,329,003 | <p>Parametrise the circle centered at $ \ (1,1,-1) \ $ with radius equal to $ 3 $ in the plane $ x+y+z=1 $ with positive orientation . $$ $$ I have thought the parametriation: </p>
<p>\begin{align} x(t)=1+ 3 \cos (t) \hat j +3 \sin (t) \hat k \\ y(t)=1+3 \cos (t) \hat i+3 \sin (t) \hat k \\ z(t)=-1+3 \cos (t) \hat i... | hamam_Abdallah | 369,188 | <p>The intersection of the sphere
$$(x-1)^2+(y-1)^2+(z+1)^2=9$$
and the plane
$$x+y+z=1$$
can be parametrised by spherical coordinates :
$$x=1+3\sin (\phi)\cos (\theta) $$
$$y=1+3\sin (\phi)\sin (\theta) $$
$$z=-1-3\sin (\phi)(\cos (\theta)+\sin (\theta)) $$.</p>
<p>with $0\le \theta \le 2\pi $ and $0\le \phi \le \pi... |
4,585,589 | <p>In the 4th edition of "Matrix Computations", Golub and Van Loan present "Algorithm 5.1.1 (Householder Vector)". The first couple of lines (translated into MATLAB-syntax) read:</p>
<pre><code>m = length(x); sigma = x(2:m)'*x(2:m); v = [1; x(2:m)];
if sigma == 0 && x(1) >= 0
beta = 0;
... | Jamie Ballingall | 559,730 | <p>In this answer I'll try to provide further context and summarize the consensus I think we achieved in the comments to the accepted answer.</p>
<p>The full algorithm (5.1.1) reads:</p>
<pre><code>m = length(x); sigma = x(2:m)'*x(2:m); v = [1; x(2:m)];
if sigma == 0 && x(1) >= 0
beta = 0;
elseif sigma == ... |
2,044,362 | <p>(<em>This summarizes scattered results from <a href="https://math.stackexchange.com/questions/879089/prove-2f-1-left-frac13-frac13-frac56-27-right-stackrel-color808080">here</a>, <a href="https://math.stackexchange.com/questions/879854/prove-large-int-11-fracdx-sqrt394-sqrt5-x-left1-x2-right">here</a>, <a href="http... | Nemo | 285,751 | <p>Starting from
$$
\,_2F_1\big(\tfrac{1}{4},\tfrac{1}{4};\tfrac{3}{4};-a\big)=\gamma\,\int_0^1 \frac{dx}{\sqrt{1-x}\,\sqrt[4]{x^3+ax^4}},
$$
$$
(b+\sqrt{b^2-1})^{-1/4}\,_2F_1\big(\tfrac{1}{4},\tfrac{1}{4};\tfrac{1}{2};\tfrac{2\sqrt{b^2-1}}{b+\sqrt{b^2-1}}\big)={\gamma}\,\int_{-1}^1\frac{dx}{\left(1-x^2\right)^{\small... |
2,607,733 | <p>Prove the inlcusion-exclusion principle using the fact that for three pairwise disjoint sets $X$, $Y$, $Z$:</p>
<p>$$|X\cup{Y}\cup{Z|}=|X|+|Y|+|Z|$$</p>
<p>I tried setting $X\cup{Y}=A$, but arrive at $|X\cup{Y|}=|X|+|Y|$ due to the sets being disjoint.</p>
| 5xum | 112,884 | <p>The mapping $$d: (x,y)\mapsto \|x+y\|$$</p>
<p>is <strong>not</strong> a metric, because it does not satisfy the axiom</p>
<p>$$\forall x\in X:d(x,x)=0$$</p>
<p>(except if $X\neq \{0\}$, but then you only have one metrix on $X$ anyway).</p>
|
2,100,793 | <p>Let us say that I have a set $A=\{1,2, 3\}$. Now I need to access, say, the element $3$ of $A$. How do I achieve this?</p>
<p>I know that sets are unordered list of elements but I need to access the elements of a set. Can I achieve this with a tuple? Like $A=(1, 2, 3)$, should I write $A(i)$ to access the i-th elem... | Robert Israel | 8,508 | <p>The fact that a set is "unordered" by definition doesn't prevent you from defining an ordering on it. Indeed, by the Axiom of Choice, any set can be
indexed by an ordinal number. </p>
|
17,819 | <p>$\mathfrak{sl}_2(\mathbb{C})$ is usually given a basis $H, X, Y$ satisfying $[H, X] = 2X, [H, Y] = -2Y, [X, Y] = H$. What is the origin of the use of the letter $H$? (It certainly doesn't stand for "Cartan.") My guess, based on similarities between these commutator relations and ones I have seen mentioned when pe... | Francois Ziegler | 19,276 | <p>The letters <span class="math-container">$\mathrm X$</span> and <span class="math-container">$\mathrm Y$</span> are already used by <strong>Cayley</strong> in what Dieudonné (in <a href="//ams.org/mathscinet-getitem?mr=88c:01020" rel="nofollow noreferrer">MR</a>) calls the first description of all finite-dimensional... |
651,027 | <p><img src="https://i.stack.imgur.com/SIdAQ.jpg" alt="enter image description here"></p>
<p>I'm doing this Solving Abstract Problem but I'm not sure which one it is. I mean from the Series I can see there's a pattern but in the Options I don't see images that link with the Series. Do you have any idea - unless it's B... | Dmoreno | 121,008 | <p>Imagine there are two pendulums, one with an X end and the other with a circle. They are spinning in different "directions". So the answer should be E.</p>
<p>Cheers!</p>
|
2,146,571 | <p>I solved this problem in my textbook but noticed their solution was different than mine. <br/></p>
<p>$1. \ 9e^{-2x}=1$ </p>
<p>$2. \ e^{-2x}=\frac{1}{9}$</p>
<p>$3. -2x=\ln(\frac{1}{9})$</p>
<p>$4. \ x=-\ \frac{1}{2}\ln(\frac{1}{9})$</p>
<p>However, the answer that my textbook gives is $\frac{\ln(9)}{2}$ </p>
... | S.C.B. | 310,930 | <p>Note that $$\ln x +\ln y =\ln xy, \; \ln 1=\ln e^{0}=0$$
If $x, y$ are positive reals, as seen <a href="https://en.wikipedia.org/wiki/List_of_logarithmic_identities#Using_simpler_operations" rel="nofollow noreferrer">here</a>. From this, $$\ln x +\ln \frac{1}{x}=0 \iff \ln x =-\ln \frac{1}{x}$$
So $$\ln \frac{1}{9}=... |
2,146,571 | <p>I solved this problem in my textbook but noticed their solution was different than mine. <br/></p>
<p>$1. \ 9e^{-2x}=1$ </p>
<p>$2. \ e^{-2x}=\frac{1}{9}$</p>
<p>$3. -2x=\ln(\frac{1}{9})$</p>
<p>$4. \ x=-\ \frac{1}{2}\ln(\frac{1}{9})$</p>
<p>However, the answer that my textbook gives is $\frac{\ln(9)}{2}$ </p>
... | BobaFret | 43,760 | <p>$\ln (1/9) = \ln (9^{-1})=-1 \cdot \ln (9)$</p>
|
2,146,571 | <p>I solved this problem in my textbook but noticed their solution was different than mine. <br/></p>
<p>$1. \ 9e^{-2x}=1$ </p>
<p>$2. \ e^{-2x}=\frac{1}{9}$</p>
<p>$3. -2x=\ln(\frac{1}{9})$</p>
<p>$4. \ x=-\ \frac{1}{2}\ln(\frac{1}{9})$</p>
<p>However, the answer that my textbook gives is $\frac{\ln(9)}{2}$ </p>
... | Fede Poncio | 269,098 | <p>$\ln(\frac{1}{9})=\ln(9^{-1})=(-1)\ln(9)$ ;)</p>
|
718,609 | <p>This theorem is the converse of Wilson's theorem:</p>
<blockquote>
<p>If $n$ is composite and $n>4$, then $(n-1)! \equiv 0 \pmod n$</p>
</blockquote>
<p>The question holds up for all the composites I have tried but I'm struggling to form a proof for all composites greater than $4$.</p>
| Michael Hardy | 11,667 | <p>Notice first that </p>
<p>Suppose $p$ is prime and $p\mid n$. Since $n$ is composite, we have $p<n$, so $p\le n-1$.</p>
<p>The multiplicity of the prime factor $p$ in the the factorization of $n$ is the largest integer $k$ such that $p^k\mid n$. Thus $p,p^2,p^3,\ldots,p^{k-1}$ divide $n$ and are all less than $... |
47,753 | <p>What are the Method options Solve command accepts? Solve has the Method option, however documentation contains no methods that it accepts...</p>
| Alexey Popkov | 280 | <p>Undocumented <a href="http://forums.wolfram.com/mathgroup/archive/2010/Nov/msg00787.html" rel="nofollow"><code>Method->"Legacy"</code> option forces <code>Solve</code> in version 8 to use algorithm from versions <=7</a>. </p>
<pre><code>exprs = Together[{(b + d + f)/x - (a + b)/(1 + x) -
2*(c + d + e)/(... |
2,553,470 | <p>The equations of the two chords of the parabola y^2=4ax are to be found such that the pass through the point (-6a,0) and subtends an angle of 45 degree at the vertex of the parabolas. Tried it in many ways including using parametric equations but could not get the two equations</p>
<p>(note. Let the chord interse... | Christian Blatter | 1,303 | <p>If a slot may receive at most one letter then we can just choose the four slots receiving a letter in ${8\choose 4}=70$ ways. It is then determined by the rules which of the chosen slots shall receive which letter.</p>
<p>If the slots may receive any number $\geq0$ of letters then it is a stars and bars problem: We... |
2,996,775 | <p><span class="math-container">$$ \frac {1}{\log_2(x-2)^2} + \frac{1}{\log_2(x+2) ^2} =\frac5{12}.$$</span></p>
<p>I made the graph using wolfram alpha it is giving answer as
6. But how to solve it algebraically?
base of logarithm is 2.</p>
<p>Tried using taking Lcm but then two different log terms
are getting for... | KM101 | 596,598 | <p><span class="math-container">$$\frac{1}{\log_2(x-2)^2}+\frac{1}{\log_2(x+2)^2} = \frac{5}{12}$$</span></p>
<p>Rewrite the logs using <span class="math-container">$$\log_a b^c = c\log_a b$$</span></p>
<p>and factor. Then, simplify both sides.</p>
<p><span class="math-container">$$\frac{1}{2}\cdot\frac{1}{\log_2\ve... |
3,133,831 | <p>I know that supremum means least upper bound. If I have a sequence of events, <span class="math-container">$\{A_n\}_{n=1}^\infty$</span></p>
<p>then <span class="math-container">$$\limsup_{n\rightarrow \infty} A_n = \lim_{n\rightarrow \infty} \sup_{j\geq n} A_j$$</span></p>
<p>I'm having trouble understanding this... | angryavian | 43,949 | <p>I think you are conflating two similarly-notated notions.
Given a set <span class="math-container">$A$</span> with an ordering, one might write
<span class="math-container">$$\sup A$$</span>
to denote the supremum of the set <span class="math-container">$A$</span>.</p>
<p>Above, when one writes
<span class="math-co... |
1,248,517 | <p>I try to solve the following problem: <br>
Given a stream of symmetric matrices $A_0, A_1, ...,A_n$ such that $A_i$ is different from $A_{i-1}$ only in one place, I want to compute the eigenvalues of $A_i$. <br>
Since the matrices are very large, computing the eigenvalues from scratch isn't efficient (and since the... | Bill Dubuque | 242 | <p>Note that $\,\underbrace{\gcd(f(n),n\!+\!2) = \gcd(f(-2),n\!+\!2)}\ $ by the Euclidean algorithm, since<br>
$\ {\rm mod}\ n\!+\!2\!:\,\ f(\color{#c00}n)\equiv f(\color{#c00}{-2})\ $ by $\ \color{#c00}{n\equiv -2},\ $ by the $ $ <a href="https://math.stackexchange.com/a/879262/242">Polynomial Congruence Rule</a>, <... |
1,463,258 | <p>Given a cumulative distribution function of the form $P(X\leq x) = 1- e^{-\lambda x^3}$, is there any way to represent it in terms of an exponential or any other distribution? I've thought about exponentials, but don't know how to deal with the cubed term. Thanks!</p>
| André Nicolas | 6,312 | <p>Let $W$ have exponential distribution with parameter $\lambda$. Let $X=W^{1/3}$. Then if $x\gt 0$, we have $\Pr(X\le x)=\Pr(W^{1/3}\le x)=\Pr(W\le x^3)=1-e^{-\lambda x^3}$.</p>
|
3,584,927 | <p>If I have two isomorphic groups, can I write <span class="math-container">$A \xrightarrow{\sim} B$</span> rather than <span class="math-container">$A \cong B$</span> to mean "A is isomorphic to B", or is the arrow notation only used if I have a map <span class="math-container">$\varphi : A \xrightarrow{\sim} B$</spa... | diracdeltafunk | 19,006 | <p>Upgraded to an answer from a comment by request:</p>
<p>I think it's better practice to use <span class="math-container">$A \cong B$</span> when you mean "there exists an isomorphism from A to B" and <span class="math-container">$A \xrightarrow{\sim} B$</span> when you mean "I have a specific isomorphism from A to ... |
3,760,115 | <p>I have these constraints on a cost function</p>
<p><span class="math-container">$$
c = A+Bx=A+B\text{vec}\ (q^*q^\top),
$$</span>
where <span class="math-container">$(c,A)\in\mathbb{R}^{100}$</span>, <span class="math-container">$B\in\mathbb{C}^{100\times 81}$</span>, <span class="math-container">$x\in\mathbb{C}^{81... | greg | 357,854 | <p>The outer product of two vectors can be vectorized in several equivalent ways
<span class="math-container">$$\eqalign{
{\rm vec}(q^*q^T) &= {\rm vec}(q^*q^TI) = {\rm vec}(Iq^*q^T) \\
=q\otimes q^* &= (I\otimes q^*)\,q = (q\otimes I)\,q^* \\
}$$</span>
Use this to rewrite the constraint vector and calculate i... |
2,202,048 | <p>There's a box with $N$ balls in it. One of them is red and the others white. What's the probability of getting the red ball at the $k$th try ( if you're not putting them back in?) where $k = 1,2,3,\ldots,N$</p>
| neofoxmulder | 121,687 | <p>I understand this question a bit differently. Suppose $N = 3 $ then there is $1$ red ball and $2$ white. since $k = 1 , 2 , ... , N$ you must make $3$ picks. </p>
<p>On your first pick the probability of drawing the red ball is $\frac{ 1}{3}$ but on your second pick the probability changes because a ball has been e... |
2,198,293 | <p>Having a bit of trouble in what way I am suppose to go about solving this problem. Any guidance would be great.</p>
<p>Show that there is at least one real solution to:$$x^5 - x^2 - 4 = 0 $$</p>
<p>Thanks in advance.</p>
| A-B-izi | 416,892 | <p>If z is a complex root of a polynomial with real coefficients, then also conjugate of z is a root (that is an easy thing to prove).
Therefore, if a polynomial with real coefficient does not have real roots, it must have an even number of roots.
Now, this polynomial has 5 roots by Fundamental Theorem of Algebra, thu... |
2,198,293 | <p>Having a bit of trouble in what way I am suppose to go about solving this problem. Any guidance would be great.</p>
<p>Show that there is at least one real solution to:$$x^5 - x^2 - 4 = 0 $$</p>
<p>Thanks in advance.</p>
| Emilio Novati | 187,568 | <p>The polynomial has degree $5$ so it has $5$ roots in $\mathbb{C}$. We known that the complex roots are always couple of conjugate numbers if the coefficients are real numbers, so, for the given polynomial, we can have $4$ complex roots and the one other root must be real.</p>
|
3,178,648 | <blockquote>
<p>We assign to every element <span class="math-container">$i$</span> from <span class="math-container">$N=\{1,2,...,n\}$</span> a positive integer <span class="math-container">$a_i$</span>. Suppose <span class="math-container">$$a_1+a_2+...+a_n = 2n-2$$</span> then prove that map <span class="math-conta... | String | 94,971 | <p>Assume that <span class="math-container">$\{a_i\}$</span> contains <span class="math-container">$k$</span> copies of <span class="math-container">$1$</span>. Assume in addition to this that <span class="math-container">$a$</span> is the second lowest number occurring somewhere in the sequence. Then we have:
<span cl... |
2,566,546 | <p>Show that for any $x_1 < x_2$ and $y_1 < y_2$ one has
$P(x_1 < X ≤ x_2, y_1 < Y ≤ y_2) = F(x_2, y_2) + F(x_1, y_1) − F(x_1, y_2) − F(x_2, y_1)$.</p>
<p>Would I just need to split the LHS to something that gives me the right?</p>
| TZakrevskiy | 77,314 | <p>The most trivial example would be the identity matrix - it is a projection on the whole space. The matrices with $\pm 1$ on the main diagonal and $0$ outside this diagonal are square roots of the identity. In other words, you have at least $2^n$ different square roots.</p>
|
4,081,029 | <p>I've been reading <a href="https://ocw.mit.edu/courses/aeronautics-and-astronautics/16-07-dynamics-fall-2009/lecture-notes/MIT16_07F09_Lec03.pdf" rel="nofollow noreferrer">this pdf</a> about vector transformations and I don't quite understand how to implement it in a computer program. On page 10, it shows you how to... | Tristan367 | 901,849 | <p>So <a href="https://computergraphics.stackexchange.com/questions/8562/how-to-convert-from-object-space-into-world-space-exercise-from-3d-math-primer">this page</a> ended up helping me. I simply send the positions of the pixels as if they were a screen slightly in front of the player at 0,0,0 and center them and what... |
1,160,699 | <p>In my discrete math book, I was tasked with finding a counterexample for this:</p>
<blockquote>
<p>If $n$ is prime, then $2^n-1$ is prime.</p>
</blockquote>
<p>Does there exist a counterexample for such a statement? Also, am I wrong in thinking that when something asks for a counterexample, is it looking for som... | user26486 | 107,671 | <p>$2^n-1\in\mathbb P\implies n\in\mathbb P$</p>
<p>This is true because if $n\not\in\mathbb P$, then $n=kl$ for some $k,l\in\mathbb Z_{\ge 2}$ and so </p>
<p>$$2^n-1=2^{kl}-1=(2^k-1)(2^{kl-k}+2^{kl-k-1}+\cdots +1)$$</p>
<p>And this is composite, since both $2^k-1$ and $2^{kl-k}+2^{kl-k-1}+\cdots +1$ are integers la... |
1,160,699 | <p>In my discrete math book, I was tasked with finding a counterexample for this:</p>
<blockquote>
<p>If $n$ is prime, then $2^n-1$ is prime.</p>
</blockquote>
<p>Does there exist a counterexample for such a statement? Also, am I wrong in thinking that when something asks for a counterexample, is it looking for som... | David R. | 158,279 | <p>That's false. But if it makes you feel any better, you're making a mistake very similar to Fermat's. In 1650, Fermat conjectured that all numbers of the form $2^{2^n} + 1$ are prime. They didn't have computers back then, so it was very difficult to find that $2^{2^5} + 1 = 641 \times 6700417$.</p>
<p>But back then ... |
1,025,588 | <p>I just started doing AM-GM inequalities for the first time about two hours ago. In those two hours, I have completed exactly two problems. I am stuck on this third one! Here is the problem:</p>
<p>If $a, b, c \gt 0$ prove that $$ a^3 +b^3 +c^3 \ge a^2b +b^2c+c^2a.$$</p>
<p>I am going crazy over this! A hint or pro... | Macavity | 58,320 | <p>You already have a great answer from @Adriano. To use AM-GM here, in general the idea would be to observe exponents on both sides and try finding a convex combination of $(3, 0, 0), (0, 3, 0)$ and $(0, 0, 3)$ which gives you a term like $(2, 1, 0)$. </p>
<p><em>Muirhead's inequality - if you're familiar with it - ... |
1,696,713 | <p>I am solving exact differential equation, but I am stuck on the step on how to simplify this term or how to rewrite it. </p>
<p>$e^{-2\ln{\sin{x}}}$</p>
| pjs36 | 120,540 | <p>I'd like to expand upon my comment because this is a very interesting question.</p>
<hr>
<p>First, I think the word "causation" is really throwing people off. Causation <em>does</em> have a specific meaning that's been modeled in various ways (I'm thinking statistics and mathematical logic), but none of the ways I... |
484,550 | <p>The problem is as follows: let $n_1, n_2,..., n_t$ be positive integers. Prove that if $n_1+n_2+...+n_t-t+1$ objects are placed into $t$ boxes, then for some $i, i=1, 2, ..., t$, the $i$th box contains at least $n_i$ objects. </p>
<p>I'm having difficulty getting started in developing a proof because I have no in... | Rebecca J. Stones | 91,818 | <p>A counterexample would consist of $t$ boxes with $b_i$ balls, $1 \leq i \leq t$, such that $$b_i \leq n_i-1$$ for all $1 \leq i \leq t$. Now find a upper bound on $\sum_{i=1}^t b_i$ using the above inequality.</p>
|
2,174,454 | <p>My teacher substitutes for this $\sum_{x=0}^{y} ({{y!} \over {x! (y-x)!} })$ by $2^y$, so I tried to use the (Mathematical induction) to prove it (My teacher did not ask me to do that, however I want to do it only to make sure for this statement)</p>
<p>My attempt:</p>
<ul>
<li>when $y=1$ :</li>
</ul>
<p>L.H.S.
$... | Ángel Mario Gallegos | 67,622 | <p>Observe that
$$\sum_{x=0}^y\frac{y!}{x!(y-x)!}=\sum_{x=0}^y{y\choose x}$$
This correspond to the develop of $2^y=(1+1)^y\;$ from the Binomial Theorem.</p>
|
3,748,879 | <p><span class="math-container">$g : \mathbb R \to [0,1]$</span> is a non-decreasing and right continuous step function such that <span class="math-container">$g(x)=0$</span> for all <span class="math-container">$x \leq 0$</span> and <span class="math-container">$g(x)=1$</span> for all <span class="math-container">$x \... | Jake Mirra | 278,017 | <p>Convergence of the integral on the interval <span class="math-container">$ [0,1] $</span> always occurs by continuity (thanks to the <span class="math-container">$ 2020 $</span> term). So we can restrict our attention to the interval <span class="math-container">$ [1, \infty) $</span>. We see that the integrand <sp... |
434,685 | <p>Suppose that $X^*$ is the dual space of a normed space $X$. If we renorm the space $X^*$ with a new norm equivalent to the first one, is this new normed space the dual of $X$ as well?
(I think it suffices to prove that a functional $f$ is continuous with a norm1 if and only if it is continuous with norm2 where norm1... | Julien | 38,053 | <p>No. Since you don't change the vector space $X^*$ by renorming it, it remains the dual as the set of continuous linear functionals on $X$. But it is no longer the dual of $X$ as a normed vector space, since that means precisely $\|f\|=\sup_{\|x\|\leq 1} |f(x)|$, which is completely determined by the norm on $X$.</p>... |
4,322,897 | <p>Let <span class="math-container">$\{y_n\}_{n=1}^\infty$</span> be a sequence in <span class="math-container">$\Bbb{R}^n$</span> and <span class="math-container">$z\in\Bbb{R}^n$</span> be given. If <span class="math-container">$\{\|z-y_n\|\}_{n=1}^\infty$</span> is a convergent sequence, what can we say about <span c... | ncmathsadist | 4,154 | <p>Translating a bounded set preserves its boundedness.</p>
<p>Proof.</p>
<p>Suppose <span class="math-container">$B$</span> is bounded and <span class="math-container">$y\in\mathbb{R}^d$</span>. Put <span class="math-container">$b = \sup\{\|x\|: x\in B\}$</span>. Then for <span class="math-container">$x\in B$</spa... |
3,438,043 | <p><strong>Definition:</strong></p>
<p>Let <span class="math-container">$X$</span> be a set.</p>
<p>A set <span class="math-container">$\tau \subset P(X)$</span> is called a topology on <span class="math-container">$X$</span> if:</p>
<p>(a) <span class="math-container">$\emptyset , X\in \tau$</span></p>
<p>(b) <spa... | Vijayakumar Muni | 400,662 | <p>First of all in your definition of a topology <span class="math-container">$\tau$</span> on <span class="math-container">$X,$</span> there is a typo in condition (c).</p>
<p>It is supposed to be <span class="math-container">$``$</span>for <span class="math-container">$\alpha\in J,$</span> where <span class="math-co... |
2,772,190 | <p>The question:</p>
<blockquote>
<p>Determine the solution set (in $\mathbb R$) for the equation $|x^2+2x+2| = |x^2-3x-4|$</p>
</blockquote>
<p>So far, I have determined that for this to be true, $|x^2-3x-4|$ must be greater or equal to $0$, giving $(x-4)(x+1)\ge0$. To find the solution set, $|x^2+2x+2|\ge-1$ as i... | Francesco Carzaniga | 382,649 | <p>Since $|x| = x$ or $-x$, $$|x^2+2x+2| = |x^2-3x-4|$$ if and only if
$$x^2+2x+2 = x^2-3x-4$$ or
$$-(x^2+2x+2) = x^2-3x-4$$ or
$$x^2+2x+2 = -(x^2-3x-4)$$ or
$$-(x^2+2x+2) = -(x^2-3x-4)$$</p>
<p>Some cases are equivalent (can you guess which and why?), so the computations are easier than they look.</p>
|
2,772,190 | <p>The question:</p>
<blockquote>
<p>Determine the solution set (in $\mathbb R$) for the equation $|x^2+2x+2| = |x^2-3x-4|$</p>
</blockquote>
<p>So far, I have determined that for this to be true, $|x^2-3x-4|$ must be greater or equal to $0$, giving $(x-4)(x+1)\ge0$. To find the solution set, $|x^2+2x+2|\ge-1$ as i... | zipirovich | 127,842 | <p>There's a simply fact about absolute values (of real numbers $a$ and $b$) that you should know and understand:
$$|a|=|b| \quad \text{if and only if} \quad a=\pm b.$$
Think about it for a moment, and it should become clear to you.</p>
<p>Then apply this observation to the given equation: you will get two equations, ... |
357,493 | <p>I am searching for a book in commutative algebra with will develop theory from an undergraduate level and lead to areas of current research. Any help is welcome.</p>
| Robert Cardona | 29,193 | <p>You can find '<a href="https://mathoverflow.net/a/385313/175094">A Term of Commutative Algebra</a>' by Allen Altman and Steven Kleiman (introduces Category Theory and has full solutions) which is a more in depth and updated version of 'Introduction to Commutative Algebra' by Atiyah, Macdonald. After that you could ... |
357,493 | <p>I am searching for a book in commutative algebra with will develop theory from an undergraduate level and lead to areas of current research. Any help is welcome.</p>
| Community | -1 | <p>Pinter's <a href="http://www.amazon.ca/Book-Abstract-Algebra-Second-Edition/dp/0486474178/ref=pd_rhf_dp_s_cp_3_1475?ie=UTF8&refRID=1QFKGETYTZ9A7RCCYWQ1" rel="nofollow"><em>A Book of Abstract Algebra (Second Edition)</em></a> is quite good. </p>
|
1,012,985 | <p>Let $A = \left[ \begin{matrix} 3 & 2 & 1\\ 5 & 0 &1\end{matrix}\right]$, </p>
<p>how can I know if there is a matrix $N$ , st. $AN=0$ (N is not a zero matrix) </p>
| user187373 | 187,373 | <p>In this case there is a theorem that guarantees the existence of such an $N$ (of size $3 \times 1)$, because the matrix $A$ has more columns than rows.</p>
|
1,012,985 | <p>Let $A = \left[ \begin{matrix} 3 & 2 & 1\\ 5 & 0 &1\end{matrix}\right]$, </p>
<p>how can I know if there is a matrix $N$ , st. $AN=0$ (N is not a zero matrix) </p>
| peterwhy | 89,922 | <p>Try to solve
$$A\pmatrix{x_1\\x_2\\x_3} = \pmatrix{0\\0}$$</p>
<blockquote>
<p>in the question they demand $N$ be at least two columns.</p>
</blockquote>
<p>And it doesn't matter how many columns $N$ should have. Say, from the above equation, you found a 3-element column vector $\mathbf x \ne \pmatrix{0&0&a... |
4,363,384 | <p>Let <span class="math-container">$A_i$</span>, <span class="math-container">$i \in I$</span> subsets of a space X given topology <span class="math-container">$\tau$</span>. Show that <span class="math-container">$\overline{\bigcup_i A_i }\subset \bigcup_i\overline{ A_i }$</span> does not hold necessarily :</p>
<p>... | Sourav Ghosh | 977,780 | <p>Consider, <span class="math-container">$(\Bbb{R}, \tau_{std}) $</span></p>
<p>Then, <span class="math-container">$\overline{\cup_{r\in \Bbb{Q}}\{r\}}=\overline{\Bbb{Q}}=\Bbb{R}$</span></p>
<p>And, <span class="math-container">$\cup_{r\in \Bbb{Q}}\
\overline{\{r\}}= \cup_{r\in \Bbb{Q}}\
\{r\} =\Bbb{Q}$</span>... |
1,675,411 | <p>So far I have this:</p>
<p>First consider $n = 5$. In this case $(5)^2 < 2^5$, or $25 < 32$. So the inequality holds for $n = 5$.</p>
<p>Next, suppose that $n^2 < 2^n$ and $n \geq 5$. Now I have to prove that $(n+1)^2 < 2^{(n+1)}$.</p>
<p>So I started with $(n+1)^2 = n^2 + 2n + 1$. Because $n^2 < 2... | Siminore | 29,672 | <p>Remark that
$$2^{n+1} = 2 \cdot 2^n > 2 n^2 = n^2 +n^2> n^2+2n+1 = (n+1)^2.$$
Indeed $n^2-2n-1=(n-1)^2-2>0$ for $n \geq 5$.</p>
|
1,351,458 | <p>Can someone please show me the steps (all of them… yeah, even the obvious ones)
to go from</p>
<p>$$\begin{align}\frac{y+1}{y-1} = 10^{x^2}\end{align}$$</p>
<p>to</p>
<p>$$\begin{align}y=\frac{10^{x^2}+1}{10^{x^2}-1}\end{align}$$</p>
| 3SAT | 203,577 | <p>Solve for $y$:</p>
<blockquote>
<p>$$\frac{y+1}{y-1} = 10^{x^2}$$</p>
</blockquote>
<p>Multiply both sides by $y-1$:
$$y+1=10^{x^2}(y-1)$$
Expand out terms of the right hand side:
$$y+1=10^{x^2}y-10^{x^2}$$
Subtract $1+10^{x^2}y$ from both sides:
$$y(1-10^{x^2})=-1-10^{x^2}$$
Divide both sides by $1-10^{x^2}$
$$... |
915,542 | <p>After working on an ODE I find I am needing to solve the integral </p>
<p>$$\int \frac{u}{b - au - u^2}\mathrm{d}u$$</p>
<p>Trig subs, banging heads against walls, and sobbing have not yielded a solution. Yet. </p>
<p>Could use a hand, thanks.</p>
| Mhenni Benghorbal | 35,472 | <p>Another approach is to use partial fraction</p>
<blockquote>
<p>$$ \frac{u}{b-au-u^2} = \frac{A}{u-\alpha}+\frac{B}{u-\beta} $$</p>
</blockquote>
<p>where $\alpha, \beta$ are the roots of $ b-au-u^2 $. You need to determine $A$ and $B$. The answer will have the form</p>
<blockquote>
<p>$$ I = A\ln(u-\alpha)+B... |
2,509,230 | <p>I was studying Chevalley-Serre relations, which can be summed up to these</p>
<p><span class="math-container">$$\tag{S1}\left[h_{i},\,h_{j}\right]=0$$</span></p>
<p><span class="math-container">$$\tag{S2}\left[e_{i},\,f_{i}\right]=h_{i} \quad \left[e_{i},\,f_{j}\right]=0 \quad\text{for } i\neq j$$</span></p>
... | Hanno | 81,567 | <p>The relations prescribe how the Lie algebra is supposed to decompose when considered as a module over the copy ${\mathfrak s}{\mathfrak l}_2(i)$ of ${\mathfrak s}{\mathfrak l}_2({\mathbb k})$ spanned by $\{e_i,f_i,h_i\}$. Namely, if you know that $\text{ad}(e_i)^{a+1}(e_j)=0$ but $\text{ad}(e_i)^{a}(e_j)\neq 0$, the... |
2,056,979 | <p>I am attempting:
<a href="https://i.stack.imgur.com/FM80Z.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/FM80Z.png" alt="enter image description here"></a></p>
<p>My solution is: But I am not sure where I am going wrong. The answer I get is not divisible by 7.
<a href="https://i.stack.imgur.com/... | Vidyanshu Mishra | 363,566 | <p>HINT:</p>
<p>You have to prove the truth of $p(k+1)$ using $p(k)$, so you have to take out something from $p(k$) and then apply it to $p(k+1)$ to establish its truth.</p>
<p>As you have assumed that $p(k)$ is true. So, $4^{k+1}+5^{2k-1}$ must be divisible by 7 say it is $7m$ where $m$ is an integer. So you get $4^... |
2,056,979 | <p>I am attempting:
<a href="https://i.stack.imgur.com/FM80Z.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/FM80Z.png" alt="enter image description here"></a></p>
<p>My solution is: But I am not sure where I am going wrong. The answer I get is not divisible by 7.
<a href="https://i.stack.imgur.com/... | Bernard | 202,857 | <p>Note $5^2\equiv 4$ and $5^{-1}\equiv 3\mod 7$. Using this, we have:
$$4^{n+1}+5^{2n-1}\equiv 4\cdot 4^n+(5^2)^n\cdot5^{-1}\equiv4\cdot 4^n+3\cdot 4^n=7\cdot4^n\equiv 0\mod 7.$$</p>
|
2,056,979 | <p>I am attempting:
<a href="https://i.stack.imgur.com/FM80Z.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/FM80Z.png" alt="enter image description here"></a></p>
<p>My solution is: But I am not sure where I am going wrong. The answer I get is not divisible by 7.
<a href="https://i.stack.imgur.com/... | barak manos | 131,263 | <p><strong>First, show that this is true for $n=1$:</strong></p>
<p>$4^{1+1}+5^{2-1}=21$</p>
<p><strong>Second, assume that this is true for $n$:</strong></p>
<p>$4^{n+1}+5^{2n-1}=7k$</p>
<p><strong>Third, prove that this is true for $n+1$:</strong></p>
<p>$4^{n+2}+5^{2n+1}=$</p>
<p>$4(4^{n+1})+25(5^{2n-1})=$</p>... |
393,430 | <p><strong>Question:</strong> Is there a simple method for calculating the Fourier transform of a holomorphic complex function <span class="math-container">${f{{\left({z}\right)}}}:\Omega\to{\mathbb{{{C}}}}$</span>?</p>
<p>In order for my question to be well-posed I define a holomorphic function <span class="math-conta... | rpotrie | 5,753 | <p>For certain 3-manifolds (irreducible), if you are willing to take finite index subgroups, the description is relatively easy, and has to do with dehn-twists too.</p>
<p>Let me add the following paper by McCullough which was useful to me when considering this question: <a href="https://projecteuclid.org/journals/jour... |
3,935,443 | <p>Why <span class="math-container">$\sqrt {xy} = \sqrt x \sqrt y$</span> and same for division?
I found question like these on website but I don't anything about precalculus and not even whole algebra so I want to know the proof with basic concepts that can explain it.</p>
| Hagen von Eitzen | 39,174 | <p>To begin with, let us assume that <span class="math-container">$f,g$</span> are injective so that we have no problems with defining the inverses. Then by continuity, they are monotonic. To vahe nno problems with <span class="math-container">$x\to 0^+$</span>, we better assume they are positive (and hence decreasing)... |
3,935,443 | <p>Why <span class="math-container">$\sqrt {xy} = \sqrt x \sqrt y$</span> and same for division?
I found question like these on website but I don't anything about precalculus and not even whole algebra so I want to know the proof with basic concepts that can explain it.</p>
| Edwin Franks | 853,882 | <p>If <span class="math-container">$\lim_{x \to \infty}f(x) = \lim_{x \to \infty}g(x) = 0$</span> and
<span class="math-container">\begin{equation}
\lim_{x \to \infty} \frac{f(x)}{g(x)}=L_1 \text{ and }
\lim_{y \to 0+} \frac{f^{-1}(y)}{g^{-1}(y)}=L_2\tag{1}
\end{equation}</span>
then if <span class="math-container">$... |
1,928,892 | <p>I'm solving a graduate entrance examination problem.
We are required to establish the inequality using the following result:</p>
<p>for $x,y > 0$, $\frac{x}{y} + \frac{y}{x} > 2$ (1), which is easy to prove as it is equivalent to $(x - y)^2 > 0$.</p>
<p>But when it comes to an inequality combining $x, y,... | timon92 | 210,525 | <p>Hint: put $a=x+y, b=y+z. c=z+x$. Then $2x=a-b+c, 2y=a+b-c, 2z=-a+b+c$. Rewrite the inequality in terms of $a,b,c$ and try to apply the lemma you are supposed to use.</p>
|
2,062,671 | <p>How many sequences of positive integer numbers $\{a_n\}$ such that $$a_0 =1, a_1 = 2,|a_{n+2}a_n - a_{n+1}^2| = 1 ?$$</p>
| Hermetically Sealed Halibut | 399,430 | <p>By using the rule given above, we can compute the first few terms of such sequences. It is easily seen that there are at most four sequences of integers which satisfy the conditions.</p>
<p>From looking at the first five terms, my guess is that these can be given in the following ways:
$a_n=n+1$, </p>
<p>$b_{n+2}=... |
2,418,274 | <p>AP=DS=CR=BQ=2a, ABCD and PQRS are squares with a side a
find the angle between AR and BC..
i got $$AR=\sqrt{6}a$$can i use ARQ triangle in order to find the corresponding angle..</p>
<p><a href="https://i.stack.imgur.com/QtRaT.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/QtRaT.jpg" alt="enter... | Community | -1 | <p>Take $R$ as the origin of vectors. Then $\vec{AR}$ is along $a\mathbf{i} + a\mathbf{j} +2a\mathbf{k}$ and $\vec{CB}$ is along $a\mathbf{i}$. The angle between them is hence
$$\cos^{-1}\left( \frac{(\mathbf{i} + \mathbf{j} +2\mathbf{k})\cdot \mathbf{i}}{\sqrt{6}}\right) = \cos^{-1}\frac{1}{\sqrt{6}}$$</p>
|
2,782,338 | <blockquote>
<p>Let $X_1, X_2, \cdots$ be independent and identically distributed random variables with expectation $\mu$. Let $N$ be a positive integer-valued random variable such that $E[N] < \infty$ and such that $I_{N≥n}$ is independent of $X_n$ for all $n$. Prove that $$E\!\left[\sum_{i=1}^NX_i\right]=\mu\, E... | me47 | 353,121 | <p>You have to condition on the event that $N=k$ for some integer $k$;
$$
E \left[ \sum_{i=1}^N X_i \right] = \sum_{k=0}^{+\infty} E \left[ \sum_{i=1}^N X_i \bigg| N = k \right] P(N=k)\\
=\sum_{k=0}^{+\infty} \sum_{i=1}^k E \left[ X_i \right] P(N=k), \quad \text{by independence of } X_i \text{ and } N \\
= \mu \sum_{k=... |
1,348,763 | <p>Using the comparison test I am supposed to figure out whether this integral converges or diverges. what other function should I use? Also, the inequality stating that $1/\sqrt{e^x+1}$ is larger or smaller than another function must be proven.</p>
| JimmyK4542 | 155,509 | <p><strong>Hint</strong>: Clearly, $1+e^x \ge e^x$, so $\sqrt{1+e^x} \ge \sqrt{e^x} = e^{x/2}$, and thus, $\dfrac{1}{\sqrt{1+e^{x}}} \le \dfrac{1}{e^{x/2}} = e^{-x/2}$. </p>
<p>Using the comparison test should be easy now.</p>
|
27,865 | <p>Both the Laplace transform and the Fourier transform in some sense decode the "spectrum" of a function. The Laplace transform gives a power-series decomposition whereas the Fourier transform gives a harmonic (or loop-based) decomposition.</p>
<p>Are there deep connections between these two transforms? <a href="http... | xen | 8,229 | <p>I don't know what answer you are looking for but for example both Laplace and Fourier transform are a so called <a href="http://en.wikipedia.org/wiki/Gelfand_representation">Gelfand Transform</a>.</p>
<p>You can find good introduction to Gelfand Transform in nice book <a href="http://books.google.pl/books?id=q7dR3d... |
1,248,668 | <p>I do have a system of n equations with m variables where m > n with integer coefficients. I wish to find a set of integer solutions to this system (In my case n = 2 and m = 4). Could somebody tell me how I can do it? I already solved this system with Mathematica but I would like to redo these calculations by hand to... | mvw | 86,776 | <p>A linear equation with integer coefficients, where one looks for integer solutions is called a linear Diophantine equation.</p>
<p>The simplest case
$$
a x + b y = c
$$
can be solved systematically and has either no or infinite many solutions.</p>
<p>From here one can move to more variables or more equations. </p>... |
4,556,193 | <p>I meet a series of the form</p>
<p><span class="math-container">$$\sum_{n=0}^{\infty} \frac{x^n}{(2n-1)!!}$$</span>
where <span class="math-container">$(-1)!! = 1$</span>.</p>
<p>I guess it is a Taylor expansion of a function but I don't know what it is. Could anyone here help me?</p>
<p>Remark: The problem comes fr... | C-RAM | 833,331 | <p>Your series is certainly not a trivial one. We shall prove that your series is given by <span class="math-container">$1+f(\sqrt{x})$</span>, where
<span class="math-container">\begin{equation}
f(x)=xe^{x^2/2}\sqrt{\frac{\pi}{2}}\text{erf}\left(\frac{x}{\sqrt{2}}\right)
\end{equation}</span>
and <span class="math-con... |
4,556,193 | <p>I meet a series of the form</p>
<p><span class="math-container">$$\sum_{n=0}^{\infty} \frac{x^n}{(2n-1)!!}$$</span>
where <span class="math-container">$(-1)!! = 1$</span>.</p>
<p>I guess it is a Taylor expansion of a function but I don't know what it is. Could anyone here help me?</p>
<p>Remark: The problem comes fr... | Leucippus | 148,155 | <p>Using
<span class="math-container">$$(2 n -1)!! = \frac{(2 n)!}{2^n \, n!} = 2^n \, \left(\frac{1}{2}\right)_{n}$$</span>
then
<span class="math-container">$$ \sum_{n=0}^{\infty} \frac{x^n}{(2 n - 1)!!} = \sum_{n=0}^{\infty} \frac{\left(\frac{x}{2}\right)^n}{(1/2)_{n}} = {}_{1}F_{1}\left(1 ; \frac{1}{2} ; \frac{x}{2... |
999,000 | <p>Theorem 3.29 If $p >1$ then $\sum_{n=2}^{\infty} \frac{1}{n(\log n)^p}$ converges; if $p \leq 1$, the series diverges. </p>
<p>Proof:</p>
<p>The monotonicity of the logarithmic function implies that $\{\log n\}$ increases. Hence $\frac{1}{n\log n}$ decreases, and we can apply theorem 3.27; this leads us to the ... | Joonas Ilmavirta | 166,535 | <p>Let $r:(0,\infty)\to(0,\infty)$ and $\omega:(0,\infty)\to S^{n-1}$ be smooth smooth functions that define a curve $\gamma$ by $\gamma(s)=r(s)\omega(s)$ in spherical coordinates.
I want to choose $S=\gamma((0,\infty))$, so I need to assume $r(s)\to\infty$ as $s\to\infty$.
I will also assume $r$ to be increasing and $... |
2,341,552 | <p>I am pretty sure that
$$\bigg|\int_{A}e^{it}\,dt\bigg|\leq2$$
for every measurable set $A\subseteq[-\pi,\pi]$,
but I cannot prove this...</p>
| crankk | 202,579 | <p>Hint: $e^{it}=\cos t + i \sin t$ and split $[-\pi,\pi]$ into the regions $[-\pi,-\pi/2],[-\pi/2,0],[0,\pi/2],[\pi/2,\pi]$. How do $\cos$ and $\sin$ behave on these regions? </p>
|
2,341,552 | <p>I am pretty sure that
$$\bigg|\int_{A}e^{it}\,dt\bigg|\leq2$$
for every measurable set $A\subseteq[-\pi,\pi]$,
but I cannot prove this...</p>
| Weaam | 1,746 | <p>We avoid arguments that use the modulus (i.e. triangle inequality) since for some measurable sets, for instance $A = [-\pi, \pi]$, we have an overestimate $$\Big|\int_A e^{it}dt\Big| \leq 2 < \int_A|e^{it}|dt = 2\pi$$</p>
<p>Simply by rotating the value of the integral (which is a complex number) back to the rea... |
554,003 | <p>How can I find a closed form for the following sum?
$$\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$$
($H_n=\sum_{k=1}^n\frac{1}{k}$).</p>
| Lucian | 93,448 | <p>I believe the answer you're looking for is in <a href="http://en.wikipedia.org/wiki/Experimental_mathematics#Applications_and_examples">this Wikipedia article</a> :</p>
<blockquote>
<p>The following identity was first conjectured by <strong>Enrico Au-Yeung</strong> , a student of <a href="http://en.wikipedia.org/... |
554,003 | <p>How can I find a closed form for the following sum?
$$\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$$
($H_n=\sum_{k=1}^n\frac{1}{k}$).</p>
| Ali Shadhar | 432,085 | <p><strong>Different approach:</strong></p>
<p>Start with <a href="https://math.stackexchange.com/questions/3366039/a-group-of-important-generating-functions-involving-harmonic-number">the identity</a></p>
<p><span class="math-container">$$\sum_{n=1}^\infty (H_n^{(2)}-H_n^2)x^{n}=-\frac{\ln^2(1-x)}{1-x}$$</span></p>
... |
554,003 | <p>How can I find a closed form for the following sum?
$$\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$$
($H_n=\sum_{k=1}^n\frac{1}{k}$).</p>
| FDP | 186,817 | <p>This result was a problem in American mathematical monthly in 50s.</p>
<p>Source:</p>
<p>H. F. Sandham and Martin Kneser, <em>The American mathematical monthly</em>, Advanced problem 4305, Vol. 57, No. 4 (Apr., 1950), pp. 267-268</p>
<p><span class="math-container">\begin{align}S&=\sum_{n=1}^\infty \left(\frac{\... |
3,556,334 | <p>Given a rectangle with dimensions <span class="math-container">$10$</span>cm and <span class="math-container">$6$</span>cm, show that for every <span class="math-container">$3$</span> points in the interior of the rectangle, the area of the triangle is less than <span class="math-container">$30$</span> cm<span clas... | Acccumulation | 476,070 | <p>Let <span class="math-container">$x_{min}$</span> be the smallest x coordinate among the three points. Similarly for <span class="math-container">$x_{max}$</span>, <span class="math-container">$y_{min}$</span>, and <span class="math-container">$y_{max}$</span>. Clearly <span class="math-container">$x_{max}-x_{min} &... |
3,081,552 | <p>Is this <code>∅ ⊈ { ∅, 1, 2 }</code> true or false ? </p>
<p>Also, I am confuse since this <code>{ ∅, 1, 2 }</code> has already contain a <code>∅</code>, does it still contain another <code>∅</code> meaning like : <code>{ ∅, ∅, 1, 2 }</code> ? </p>
<p>Is <code>∅ ∈ { ∅, 1, 2 }</code> true ? & <code>{∅} ∈ { ∅,... | DanielWainfleet | 254,665 | <p><span class="math-container">$\neg (A\subset B)$</span> iff there exists a member of <span class="math-container">$A$</span> that is not a member of <span class="math-container">$B$</span>.</p>
<p>Consider the case <span class="math-container">$A=\phi.$</span> </p>
<p>Since <span class="math-container">$\phi$</s... |
63,534 | <p>I found in an article <a href="http://dx.doi.org/10.1103/PhysRev.105.776" rel="nofollow">"Imperfect Bose Gas with Hard-Sphere Interaction"</a>, <em>Phys. Rev.</em> 105, 776–784 (1957) the following integral, but I don't know how to solve it. Any hints?</p>
<p>$$\int_0^\infty {\int_0^\infty {\mathrm dp\mathrm dq\fr... | Sangchul Lee | 9,340 | <p>I will assume that $u, v, w > 0$ and $4vw > u^2$. Let $$
I := \int_{0}^{\infty} \int_{0}^{\infty} \frac{\sinh (upq)}{p^2 - q^2} \, pq \, e^{-vp^2} e^{-wq^2} \; dpdq.
$$ By polar coordinate transform, we obtain $$
\begin{eqnarray*}
I & = & \int_{0}^{\frac{\pi}{2}} \int_{0}^{\infty} \frac{\sinh (r^2 ... |
2,993,958 | <blockquote>
<p>Do there exist three non null vectors <span class="math-container">$a,b,c$</span> with <span class="math-container">$$a.b=a.c$$</span> such that <span class="math-container">$b\ne c$</span>?</p>
</blockquote>
<p>My attempt:</p>
<p>a.(b-c)=0</p>
<p>But is it possible to say b-c=0 since a is non null... | Peter Szilas | 408,605 | <p><span class="math-container">$\vec a\cdot (\vec b-\vec c)=0.$</span></p>
<p><span class="math-container">$\vec a$</span> is perpendicular to <span class="math-container">$(\vec b-\vec c)$</span>.</p>
<p>In 2D: </p>
<p>Let <span class="math-container">$\vec a =(1,0)$</span>, then chose <span class="math-container"... |
3,728,707 | <p>I would like to solve this linear diophantine equation:
<span class="math-container">$$
40x_1+296x_2+945x_3+2048x_4+4500x_5+8640x_6=616103
$$</span>
All the answers have to be an integer number in the interval <span class="math-container">$\{[10] \cup [29,95]\}$</span>.</p>
<p>As a first step, I started by finding a... | Pierre | 401,549 | <p>To find all the answers, I coded my own solver following this algorithm:</p>
<ol>
<li>Find all possible <span class="math-container">$x_6$</span> that fits in the constraints and its corresponding <span class="math-container">$w_6$</span></li>
<li>For each possible <span class="math-container">$x_6$</span>, rewrite ... |
375,025 | <p>How do I solve this simultaneous equation? I know it has multiple solutions, but would anyone be able to show me the exact steps in working these out for future reference? </p>
<p>$$
\begin{cases}
2x +y = 4\\
-6x - 3y = -12
\end{cases}
$$</p>
<p>Thank you...</p>
| Tazwar Sikder | 347,302 | <p>We have: $2x+y=4$ and $-6x-3y=-12$</p>
<p>Let's begin by solving each equation for $y$ and labelling them:</p>
<p>$\Rightarrow y=4-2x \hspace{80 mm}$ (i)</p>
<p>$\Rightarrow y=\dfrac{12-6x}{3}=4-2x \hspace{58.5 mm}$ (ii)</p>
<p>Graphically, these two lines overlap.</p>
<p>Therefore, the system of equations has ... |
1,784,469 | <p>I roll a biased dice 9000 times. the probability of seeing 1,2,3,4,5,6 are $\frac{1}{3}, \frac{1}{12}, \frac{1}{12}, \frac{1}{6}, \frac{1}{6}, \frac{1}{6}$ respectively. I need to use the Central limit theorem to estimate the probability that the total number of 1s that I see is within [2970,3040].</p>
<p>So far, I... | André Nicolas | 6,312 | <p>For $i=1$ to $n$, where $n=9000$, define random variable $X_i$ by $X_i=1$ if we roll a $1$ on the $i$-th roll, and by $X_i=0$ otherwise.</p>
<p>Let $Y=X_1+X_2+\cdots+X_n$. Then $Y$ is the sum of a large number of independent identically distributed "nice" random variables. So $Y$ has a reasonably close to normal di... |
192,391 | <p>I know that this is sometimes the case, but that some matrices are not tensors.
So what is the intuitive and specific demands of a matrix to also be a tensor?
Does it need to be quadratic, singular or something else?</p>
<p>Some sources I read seem to suggest that all rank 2 matrices are tensors while other just cl... | rschwieb | 29,335 | <p>The connection is this: a matrix consists of the coefficients of a (1,1) tensor, but it is not a tensor itself.</p>
<p>Suppose we are talking about a linear transformation $T$ on an $n$ dimensional vector space $V$.</p>
<p>Now $T$ is certainly a tensor (tensors are, after all, multilinear maps on copies of $V$ and... |
192,391 | <p>I know that this is sometimes the case, but that some matrices are not tensors.
So what is the intuitive and specific demands of a matrix to also be a tensor?
Does it need to be quadratic, singular or something else?</p>
<p>Some sources I read seem to suggest that all rank 2 matrices are tensors while other just cl... | Farshad Ashkbous | 347,365 | <p>All matrices are not tensors, although all tensors of rank 2 are matrices.</p>
<p>Example: Matrix T (T11=x , T12=-y , T21=x^2 , T22=-y^2) .This matrix is not tensor rank 2. We test matrix T to rotation matrix A: a11=cos(theta),a12=sin(theta),a21=-sin(theta),a22=cos(theta). Now Expand tensor equation rank2,for examp... |
2,781,702 | <p>Honestly, I have no idea if I put the correct tag on this question, and I don't even know where to begin to solve an equation like this:
$$
f(d,n)=\sum_{i=1}^n\binom{d}{i}.
$$</p>
<p>Could someone explain what the "d" over the "i" inside the parenthesis means? I'm attempting to solve for when d is equal to 20, n is... | fleablood | 280,126 | <p>${d \choose i}$ = "d choose i" is the number of different ways to choose $i$ objects from $d$, total.</p>
<p>So for example if you are given a bag with $a,b,c,d$ in it and you are told to pick two items you can do one of the following: pick $a,b$ ; pick $a,c$; pick $a,d$; pick $b,c$; pick $b,d$ or pick $c,d$. Tho... |
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