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 |
|---|---|---|---|---|
114,340 | <p>Let $M\subset \mathbb C^2$ be a hypersurface defined by $F(z,w)=0$. Then for some point $p\in M$, I've
$$\text{ rank of }\left(
\begin{array}{ccc}
0 &\frac{\partial F}{\partial z} &\frac{\partial F}{\partial w} \\
\frac{\partial F}{\partial z} &\frac{\partial^2 F}{\partial ^ 2z} &\frac{\... | Robert Israel | 8,508 | <p>Suppose $A$ is the adjacency matrix of graph $G_1$ and $B$ the adjacency matrix of graph $G_2$, where we consider both $G_1$ and $G_2$ to have the same vertices $1,\ldots,n$. Then $(AB)_{ij}$ is the number of ways to get from $i$ to $j$ by going first along an edge of $G_1$ and then along an edge of $G_2$.</p>
|
1,403,228 | <blockquote>
<p><strong>Question:</strong></p>
<p>Let <span class="math-container">$m, n, q, r \in \mathbb Z$</span>. If <span class="math-container">$m = qn + r$</span>, show that <span class="math-container">$\gcd(m, n) = \gcd(n, r)$</span>. Hence justify the Euclidean Algorithm.</p>
</blockquote>
<p>I found this que... | Community | -1 | <p>The following is a demonstration using notions of ring theory.</p>
<p>Let $m,n,q,r\in\mathbb{Z}$ such that $m=qn+r$.</p>
<p>Now consider the ideals $J=m\cdot \mathbb{Z}+n\cdot \mathbb{Z}$ and $J'=n\cdot\mathbb{Z}+r\cdot\mathbb{Z}$. The ring of integer is a principal ideal domain, therefore, since $J$ and $J'$ are ... |
3,170,368 | <p>The limit of <span class="math-container">$f(x,y)=\frac{xy^2}{x^2+y^4}$</span> as <span class="math-container">$(x,y) \longrightarrow(0,0)$</span> is doesn't exist, because if we take two paths:</p>
<p>1) Along the path, <span class="math-container">$x=0$</span>, <span class="math-container">$y\longrightarrow0$</sp... | StackTD | 159,845 | <p>You need to be careful and consider all paths as well, not only paths with constant <span class="math-container">$\theta$</span>.</p>
<blockquote>
<p><span class="math-container">$$=\lim_{r\longrightarrow0}\frac{r\cos(\theta)\sin^2(\theta)}{\cos^2(\theta)+r^2\sin^4(\theta)}$$</span></p>
</blockquote>
<p><span cl... |
2,877,174 | <p>Imagine that we roll two fair six-sided dice (i.e., all six sides have equal probability). Let X1 and
X2 be the random variables representing these outcomes.
Now, imagine we take one of the dice rolls, say X1, and add a (possibly negative) constant c to
the result. If this becomes less than zero, then we set it to z... | José Carlos Santos | 446,262 | <p>Your argument is correct, but your notation is innapropriate. Where you wrote $\operatorname{rank}(\ker\cdots)$, you should have written $\dim(\ker\cdots)$.</p>
|
2,228,328 | <p>Is the function $ f(z)=\dfrac 1 {x^2+y^2} + i \dfrac 1 {x^2+y^2} $ is differentiable and holomorphic somewhere?</p>
<p>We have $z=x+iy$ and $f(z)= \dfrac{1+i}{|z|^2}$ . Now, $f(0)= \lim_{\delta z \rightarrow 0} \frac{f(0+\delta z) - f(0)}{\delta z}=$ undefined. So $f(z)$ is not differentiable at origin. Also sinc... | JMJ | 295,405 | <p>Let the operators $D_1$ and $D_2$ be defined as
$$
D_1(u+iv) = u_x - v_y \ \text{ and } \ D_2(u+iv) = u_y + v_x.
$$
Then $u+iv$ is analytic at $z_0 = x_0 + i y_0$ if and only if $u(x_0,y_0)+iv(x_0,y_0)$ belongs to $\ker D_1 \cap \ker D_2$. </p>
|
189,618 | <p>I'm trying to plot 2D vectors in Mathematica. Built in functions don't really work for me because I want to plot vectors of matrices from the origin to the their coordinates with an arrow on their tips. I made a function </p>
<pre><code>plotMatrixVectors[mat_List] :=
Graphics[Table[Arrow[{origin, i}], {i, mat}]... | m_goldberg | 3,066 | <p>I don't know what went wrong with your code, but you might try this. It is simpler and more robust than your formulation.</p>
<pre><code>plotVectors[pts : {{_?NumberQ, _?NumberQ} __}] :=
Graphics[Arrow[{{0, 0}, #} & /@ pts], Axes -> True]
</code></pre>
<p>Tests</p>
<pre><code>plotVectors[{{1, 5}, {-6, 4... |
478,620 | <p>Let $S$ be a set that satisfies the following property:</p>
<blockquote>
<p>For every countable family $\{U_{i}\}_{i\in \mathbb{N}}$ of subsets of $S$ for which $$S=\bigcup_{i\in \mathbb{N}}U_{i}$$there is a finite set $I\subset \mathbb{N}$ for which $$S=\bigcup_{i \in I}U_{i}$$</p>
</blockquote>
<p>Does it foll... | Moataz | 316,923 | <p>(Hundreds) (Tens) (Units),
Units could be $(1, 3, 7, 9) \rightarrow 4$ numbers,
Tens could be $(0, 1, 2, 3, 4, 6, 7, 8, 9)\rightarrow 9$ numbers,
Hundreds could be $(1, 2, 3, 4, 6, 7, 8, 9) \rightarrow 8$ numbers,
(Hundreds) (Tens) (Units) $\rightarrow
(8) (9) (4) = 288$</p>
|
3,493,729 | <p>Although I looked up the answer on integral calculator com but I still have little to no idea as to how one would proceed to solve this integral.
Integrate <span class="math-container">$\dfrac{e^x(x^4+2)}{(1+x^2)^{5/2}}$</span> wrt <span class="math-container">$x.$</span> I initially tried to convert it to the form ... | Adam Martens | 691,077 | <p>This is not possible. If <span class="math-container">$X,Y$</span> are topological spaces with <span class="math-container">$X$</span> connected and <span class="math-container">$f: X\to Y$</span> is continuous, then <span class="math-container">$f(X)\subset Y$</span> is connected. Since <span class="math-container"... |
380,696 | <p>Show that for any odd $n$ it follows that $n^2 \equiv 1 \mod 8$ and for uneven primes $p\neq 3$ we have $p^2 \equiv 1\mod 24$.</p>
<p>My workings so far: I proceeded by induction. Obviously $1^2 \equiv 1 \mod 8$. Then assume that for a certain uneven number $k$ we have $k^2 \equiv 1 \mod 8$. Then the the next uneve... | vonbrand | 43,946 | <p>If $n$ is odd, $n = 2 k + 1$ and $n^2 = 4 k^2 + 4 k + 1 = 4 k (k + 1) + 1$, and $4 k (k + 1)$ is divisible by 8. Thus $n^2 \equiv 1 \pmod{8}$.</p>
<p>If $p > 3$ is prime, it is odd, and so by the above $p^2 \equiv 1 \pmod{8}$. As it is prime, $p \equiv 1 \pmod{3}$ or $p \equiv 2 \pmod{3}$, so $p^2 \equiv 1 \pmod... |
306,178 | <p>Given
$$
y_n=\left(1+\frac{1}{n}\right)^{n+1}\hspace{-6mm},\qquad n \in \mathbb{N}, \quad n \geq 1.
$$
Show that $\lbrace y_n \rbrace$ is a decreasing sequence. Anyone can help ? I consider the ratio $\frac{y_{n+1}}{y_n}$ but I got stuck.</p>
| Dominic Michaelis | 62,278 | <p>If you don't care about overkilling the Problem you can even do this with calculus, showing the derivative in n is negative.
The derivative is
$$ \frac{\left(\frac{1}{n}+1\right)^n (n+1) \left(n \log \left(\frac{1}{n}+1\right)-1\right)}{n^2}$$ and so we only need to show that
$$1> n \log(1+\frac{1}{n})$$
By su... |
4,212,199 | <p><span class="math-container">$$\frac15\int_0^5(4+2^{0.1t^2})dt$$</span>
This question is from part (b) of question 1 in the AP Calculus BC 2019 FRQ. I've tried using u-substitution where <span class="math-container">$u = 0.1 t^2$</span>, but this won't work because of the <span class="math-container">$2t$</span> tha... | Oğuzhan Kılıç | 481,167 | <p>There are many ways you can solve this problem, let's first do it in your way. Note that i'm not bothering with units. Because of feet, pound. :( not nice. Also let me note that we must talk about the <span class="math-container">$g$</span> (gravitation constant) because if we don't say anything about it answer wşll... |
2,478,026 | <p><a href="https://i.stack.imgur.com/TyhHX.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/TyhHX.png" alt="enter image description here"></a></p>
<p>It doesn't make sense to me that |E[x] - m| = |E[X-m]|</p>
| bof | 111,012 | <p>$E[X-m]=E[X]-E[m]=E[X]-m$ because $m$ is a constant. For instance, if $m=5,$ then
$E[X-m]=E[X-5]=E[X]-E[5]=E[X]-5.$<br>
Similarly,<br>
$E[X-\mu]=E[X]-\mu=\mu-\mu=0,$ a familiar fact.</p>
|
2,916,168 | <p>I am a beginner in proofs and, unfortunately, I cannot wrap my mind around how to prove the simplest things, so I need a bit of help getting started. This is the proof that I am dealing with:</p>
<p>$\text{If }x< y< 0\text{, then }x^{2}> y^{2}\text{.}$</p>
<p>Thank you in advance.</p>
| user | 505,767 | <p>We have that</p>
<p>$$x<y<0 \iff 0<-y<-x $$</p>
<p>and since $f(a)=a^2$ is strictly increasing for $a>0$</p>
<p>$$0<-y<-x \implies 0<(-y)^2<(-x)^2$$</p>
<p>that is</p>
<p>$$x^2>y^2>0$$</p>
|
566,255 | <p>How to evaluate the following integral? </p>
<p>$\int_{0}^{\infty} \sin(kx)dx=\frac 1 k$ </p>
<p>The book Mathematical Physics by Butkov reads "The sequence $f_N(k)=\int_{0}^{N} \sin(kx) dx=\frac{(1-\cos kN)}k$diverges as N approaches $\infty$, but it is weakly convergent for a suitable chosen set of test functio... | Chris Janjigian | 23,078 | <p>Take $\varphi \in C_c^\infty ((0,\infty))$ and call $f_n(k) = \int_0^n \sin(ky) dy = \frac{1-\cos(nk)}{k}$. We would like to show that
\begin{align*}
\lim_{n \to \infty} \int_{0}^\infty \varphi(k) f_n(k) dk = \int_{0}^\infty \varphi(k) \frac{1}{k} dk
\end{align*}
which would show that as distributions on $C_c^\inft... |
2,462,852 | <p>I want to prove that $f(x)=x^3+x+2$, $f: \mathbb R \rightarrow \mathbb R$ is bijective without calculus. My attempts at showing to prove that it' injective and surjective are written below:</p>
<p>$1)$ Injectivity: </p>
<p>I want to show that $\forall a,b \in \mathbb R$ $f(a)=f(b) \implies a=b$.</p>
<p>I started ... | Peter Szilas | 408,605 | <p>1) Injective:</p>
<p>$f(x) = x^3 +x+2$, $x \in \mathbb{R}$ is strictly monotonically increasing.</p>
<p>Let $x_1< x_2 $, then</p>
<p>$x_1^3 + (x_1+2) \lt x_2^3 + (x_2 +2)$,</p>
<p>is strictly monotonically increasing,</p>
<p>sum of $y_1= x^3$ and a linear function $y_2= x+2$, both strictly monotonically ... |
688,711 | <p>Assume the desity of air <span class="math-container">$\rho$</span> is given by</p>
<p><span class="math-container">$\rho(r)=\rho_0$$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$</span> is the distance from the centre of the earth, <span c... | Julián Aguirre | 4,791 | <p>The limits of iteration in spherical coordinates could be</p>
<ul>
<li>$r>R_0$ (the exterior of the earth)</li>
<li>$0\le\theta<2\,\pi$ (all around the earth)</li>
<li>=$\pi/6<\phi<\pi/2$ (between 30ºN and the equator)</li>
</ul>
|
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... | Phani | 545,039 | <p>To calculate the mass of the atmosphere above the earth's surface you do not need to know the density variation of air vertically nor the extent of the atmosphere above the earth. All one needs to know is the pressure at sea level and the value of acceleration due to gravity (g) at sea level (and an assumption that ... |
213,533 | <p>I am trying to play a sequence of sounds in Mathematica using <code>Table</code>. My real example is much more complicated so that is why I got to using <code>Table</code>, </p>
<p>Why doesn't the follow code work? </p>
<pre><code>frequencies = {{100, 200}, {200, 300}, {400, 600}};
Sound[
Table[
Sound[Play[... | Nasser | 70 | <p>I get something like this</p>
<pre><code>frequencies = {{100, 200}, {200, 300}, {400, 600}};
Sound[Table[
Evaluate@Sound[
Play[Total[Sin[#*2 Pi t] & /@ frequencies[[i]]], {t, 0,
0.5}]], {i, 1, 3, 1}]]
</code></pre>
<p><img src="https://i.stack.imgur.com/8Ltau.png" alt="Mathematica graphics"></p>
... |
2,448,745 | <p>I am trying to derive the general form of the equation of the circle given two points $(x_1,y_1)$, $(x_2,y_2)$and the angle $\theta$ subtended by the chord joining the two points. </p>
<p>So, $\theta$ is basically the angle between two lines
$\implies \tan \theta = |\dfrac{\dfrac{y-y_1}{x-x_1}-\dfrac{y-y_2}{x-x_2}... | M. Winter | 415,941 | <p>A <a href="https://en.wikipedia.org/wiki/Structure_(mathematical_logic)" rel="noreferrer">general algebraic object</a> is a set $X$ together with some specified</p>
<ul>
<li>relations $R_i\subseteq X^n,i\in I$,</li>
<li>functions $f_j:X^n\to X,j\in J$ and</li>
<li>constants $c_k\in X,k\in K$.</li>
</ul>
<p>As you ... |
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... | Montie | 517,270 | <p>Let $\eta$ be a smooth function on $\mathbb{R}^N$ satisfying $\eta \equiv 1$ in $B_1 (0)$ and $\eta (x) = 0$ for all $x \in \mathbb{R}^N \setminus B_2 (0)$. For $R > 0$, define $\eta_R$ by $\eta_R (x) := \eta (\frac{x}{R})$. Then $\eta_R$ is smooth, supported on the ball of radius $2R$, and we have by the chain ... |
1,718,391 | <p>The sequence is </p>
<p>$$\sum_{n=1}^{\infty} \frac{n-1}{n!}$$</p>
<p>I could show that it converges using the Ratio test, but evaluating it seems to be hard. I'm trying to avoid power series and Taylor series and such. Is it possible to solve this some other, simpler way? </p>
<p>I tried writing out the first fe... | Yiyuan Lee | 104,919 | <p>Hint: Note that</p>
<p>$$\frac{n - 1}{n!} = \frac{1}{(n - 1)!} - \frac{1}{n!}$$</p>
<p>A large amount of cancellations will occur between successive terms in the summation.</p>
|
1,718,391 | <p>The sequence is </p>
<p>$$\sum_{n=1}^{\infty} \frac{n-1}{n!}$$</p>
<p>I could show that it converges using the Ratio test, but evaluating it seems to be hard. I'm trying to avoid power series and Taylor series and such. Is it possible to solve this some other, simpler way? </p>
<p>I tried writing out the first fe... | hamid kamali | 208,328 | <p>Consider the function $$f(x)=\frac{\mathbb e^x}{x}=\sum_{n=0}^\infty \frac{x^{n-1}}{n!}$$ And use derivativetion we get that: $$\frac{x\mathbb e^x-\mathbb e^x}{x^2}=\sum_{n=0}^\infty \frac{(n-1)x^{n-2}}{n!}$$ Or: $$\frac{x\mathbb e^x-\mathbb e^x}{x^2}+\frac1{x^2}=\sum_{n=1}^\infty \frac{(n-1)x^{n-2}}{n!}$$ Finally: ... |
72,098 | <p>This is my first time using Stack Exchange, but it looks like a good resource. I am here to ask a couple questions about my homework. We're working from the latest edition of <em>Abstract Algebra</em> by Herstein. This is problem #3 from p. 73 of that book, and it reads as follows:</p>
<blockquote>
<p>Let $G$ be ... | Dylan Moreland | 3,701 | <p>To be clear, $A(G)$ is the set of all <a href="http://en.wikipedia.org/wiki/Bijection" rel="nofollow">bijective</a> set maps $G \to G$. This $A(G)$ is even a group under composition, but an element of $A(G)$ doesn't have to be one of the $L_a$ or pay much attention at all to the group structure of $G$. For example, ... |
2,054,338 | <p>I had this question on my exam and I thought I could solve it using the row echelon method. Well, I couldn't, and I still don't know how to solve it. We were asked to "determine $a$ such the equation system has a unique solution" and to determine $a$ itself. The system had three equations: </p>
<p>$2x - y + az = 3$... | DeepSea | 101,504 | <p><strong>hint</strong>: $1+x+x^2+\cdots+ x^k+\cdots = \dfrac{1}{1-x}, |x| < 1$. Differentiate both sides and plug $x = 1-a$ into the equation to get the answer.</p>
|
2,411,811 | <p>Determine the limit:</p>
<p>$$\lim_{h\to 0}\frac{1-\cos(2h)}{h}$$</p>
<p>I know that the answer is $0$, I am just unsure of how to solve this. Thanks.</p>
| Claude Leibovici | 82,404 | <p>You have different ways for doing it.</p>
<p>The first one is to write $$\cos(2h)=1-2\sin^2(h)\implies 1-\cos(2h)=2\sin^2(h)$$ $$\frac{1-\cos(2h)} h=\frac{2\sin^2(h)}h=2h\frac{\sin^2(h)}{h^2}=2h \left(\frac{\sin(h)}{h} \right)^2$$</p>
<p>The second one would be Taylor expansion
$$\cos(t)=1-\frac{t^2}{2}+\frac{t^4}... |
1,050,664 | <p>Can someone explain why the inverse $4$ modulo $9$ is $7$? What am I missing?
$$9 = 2\cdot4 + 1$$
$$1 = 9-4\cdot2$$
$$1 = -2\cdot4 + 1\cdot9$$</p>
<p>Isn't then $-2$ inverse of $4$ modulo $9$?</p>
| user141592 | 178,602 | <p>That is correct, but remember that $-2 \equiv 7 \pmod 9$.</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... | Noah Schweber | 28,111 | <p>In a general set, there is no indexing, except the trivial one: a set $A$ is indexed by itself, that is, $A=\{x_a: a\in A\}$ where $x_a=a$.</p>
<p>Indeed, in set theory without the axiom of choice, there can be sets which are impossible to "index nicely," although of course this takes work to be made precise. </p>
... |
1,897,163 | <p>Let $\frac{x^2}{y}\to c\neq 0$ as $(x,y)\to (0,0)^+$. Why does this mean that $y$ goes faster to $0$ as $x$ does?</p>
<p>For values of x and y that are small enough, we surely do not have $x^2/y=1$ so they cannot be identical, okay. Moreover, if the quotient would tend to 0, this would mean that $x$ would be faster... | paf | 333,517 | <p>Let us fix $x\in \Bbb R^*$. We see that
$$f(x)=x\left(\dfrac{1}{1+x^2}\right)^n$$</p>
<p>Since $\left|\dfrac{1}{1+x^2}\right|<1$, we see that $\dfrac{f(x)}{x}$ is a geometric sequence converging to 0. </p>
|
4,232,796 | <p>Consider <span class="math-container">$\sqrt{x^2+y^2}+2\sqrt{x^2+y^2-2x+1}+\sqrt{x^2+y^2-6x-8y+25}$</span>. I need to find global or local minima of this function, but Wolfram Alpha doesn't seem to find one; the answer is that <span class="math-container">$1 + 2\sqrt{5}$</span> is its global minimum</p>
<p>Am I doin... | Hamza | 185,441 | <p>Since <span class="math-container">$f$</span> is entire, we deduce that <span class="math-container">$G=f^{(n)}$</span> is entire to. In particular
<span class="math-container">$$
G(\frac{1}{k+1})=0 \qquad k\ge 1.
$$</span>
and hence, <span class="math-container">$G(0)=G(\lim_k\frac{1}{k+1})=\lim_k G(\frac{1}{k+1})=... |
1,836,785 | <p>Following are the two theorems that Hardy and Wright state in their book</p>
<blockquote>
<p><strong>Theorem A:</strong> The number of primes not exceeding <span class="math-container">$x$</span> is given by <span class="math-container">$\pi(x) \sim \frac{x}{\log{x}}$</span>.</p>
<p><strong>Theorem B</strong>: The ... | tolanj | 349,700 | <p>As a concrete example think of f:x -> (sin x) + 2; and g:x -> 2;
Its clear that f/g does not converge anywhere, ie f !∼ g, but they are of the same order of magnitude as 0.4g < f < 2g.</p>
<p>In general 2. does not imply 1. x and 2x are a trivial example 2x/x is fixed at 2 but they clearly have the same magn... |
1,041,623 | <p>I have been suggested to read the <a href="http://press.princeton.edu/chapters/gowers/gowers_VIII_6.pdf"><em>Advice to a Young Mathematician</em> section </a> of the <em>Princeton Companion to Mathematics</em>, the short paper <a href="http://alumni.media.mit.edu/~cahn/life/gian-carlo-rota-10-lessons.html"><em>Ten L... | Alexandre Eremenko | 110,120 | <p>The best advise I can share was given to me by my mother, (she was a researcher in medicine) when I was a first-year student (of mathematics):
find a good adviser and follow his/her advice.</p>
<p>As a beginner, you usually cannot judge yourself about research areas of mathematics, and what to do and what to learn. ... |
1,041,623 | <p>I have been suggested to read the <a href="http://press.princeton.edu/chapters/gowers/gowers_VIII_6.pdf"><em>Advice to a Young Mathematician</em> section </a> of the <em>Princeton Companion to Mathematics</em>, the short paper <a href="http://alumni.media.mit.edu/~cahn/life/gian-carlo-rota-10-lessons.html"><em>Ten L... | Community | -1 | <p>Here is my answer in short</p>
<ol>
<li><p>Use YouTube videos .Some of them are good and explain basic content.</p>
</li>
<li><p>The Math Sorcerer channel is an excellent for math advice and<br />
inspiration</p>
</li>
<li><p>George Polya’s Mathematics and Plausible Reasoning vol 1 &2 These are great books to ... |
2,377,139 | <p>My question:</p>
<p>Characterize those power series $\sum_{k=0}^\infty a_{k}(x-c)^{k}$ that converge uniformly on ($-\infty, \,\infty$).
What does it mean to characterize a power series?</p>
<p>Let {$a_{k}$} be a sequence of coefficients for a power series. By definition, the radius of convergence $R = \infty$ if ... | Hans Engler | 9,787 | <p>These power series are precisely the polynomials. </p>
<p>The partial sums of a uniformly convergent power series must be a Cauchy sequence in the sup-norm. Let $S_n(x)$ be the $n$-th partial sum. Uniform convergence implies<br>
$$
\forall \epsilon > 0 \, \exists N > 0 \, \ni \, \forall n > N \, \forall x ... |
979,947 | <p>If $(A, B, C)$ are distinct integers $> 1$, and $$f(A, B, C) = \frac{\frac{A^2-1}{A} + \frac{B^2-1}{B}}{\frac{C^2-1}{C}},$$ then for what (if any) triplets $(A, B, C)$ is $f(A, B, C)$ an integer?</p>
<p>UPDATE: I've done some more work and have come up with the following: Based on the equations I combined to arr... | Nameless | 68,482 | <p>If points are awarded as you say, then indeed the expected points you can gain from choosing one of the five answers randomly is
$$Ep=1/5*1-4/5*1/4=4/20-4/20=0.$$
However, you are missing the fact that this happens only in expectation, so it is <em>not</em> the case that</p>
<blockquote>
<p>guessing (blind or oth... |
2,146,571 | <p>I solved this problem in my textbook but noticed their solution was different than mine. <br/></p>
<p>$1. \ 9e^{-2x}=1$ </p>
<p>$2. \ e^{-2x}=\frac{1}{9}$</p>
<p>$3. -2x=\ln(\frac{1}{9})$</p>
<p>$4. \ x=-\ \frac{1}{2}\ln(\frac{1}{9})$</p>
<p>However, the answer that my textbook gives is $\frac{\ln(9)}{2}$ </p>
... | Hushus46 | 240,398 | <p>There exists the following property for logarithms:</p>
<p>$$n \ln{x} = \ln{x^n}$$</p>
<p>So for your problem you have:</p>
<p>$$ -\frac{1}{2} \ln{\left(\frac{1}{9}\right)}=\frac{1}{2}\ln{\left(\left(\frac{1}{9}\right)^{-1}\right)}=\frac{1}{2}\ln{9}= \frac{\ln9}{2}$$</p>
<p>I hope this is sufficient as an explan... |
718,609 | <p>This theorem is the converse of Wilson's theorem:</p>
<blockquote>
<p>If $n$ is composite and $n>4$, then $(n-1)! \equiv 0 \pmod n$</p>
</blockquote>
<p>The question holds up for all the composites I have tried but I'm struggling to form a proof for all composites greater than $4$.</p>
| Olivier Bégassat | 11,258 | <p>If $n$ is composite, then there are $1<a<b<n$ with $ab=n$, unless $n=p^2$ is the square of a prime. In the first case, $a$ and $b$ appear in the product $(n-1)!$ thus $n|(n-1)!$. In the second case, your hypothesis $n>4$ implies $p\geq 3$ and thus $1<p<2p<n-1$ so that $2n=p\times 2p$ divides $(n... |
718,609 | <p>This theorem is the converse of Wilson's theorem:</p>
<blockquote>
<p>If $n$ is composite and $n>4$, then $(n-1)! \equiv 0 \pmod n$</p>
</blockquote>
<p>The question holds up for all the composites I have tried but I'm struggling to form a proof for all composites greater than $4$.</p>
| ajotatxe | 132,456 | <p>If $n=pq$, try to prove that $pq$ divides $(n-1)!$</p>
|
631,348 | <p>Let $\sum a_n$ be a series of non-negative terms and let $$L = \lim_{n\to\infty}n\left(1-\frac{a_{n+1}}{a_n}\right)$$
Prove that the series converges (resp. diverges) if $L > 1$ (resp. $L<1$). I've tried, for example, that when $L<1$, $$n\left(1-\frac{a_{n+1}}{a_n}-L\right)= \frac{n(a_n-a_{n+1}-La_n)}{a_n}\... | user127.0.0.1 | 50,800 | <p>The series even converges <em>absoute</em>, thus set $a_n:=\left|a_n\right|$</p>
<p>Now (if $n$ is big enough)
$$L = \lim_{n\to\infty}n\left(1-\frac{a_{n+1}}{a_n}\right) \Longleftrightarrow \frac{a_{n+1}}{a_n}\leq\frac{n-L}{n}$$</p>
<p>This is equivalent to
$$\left(L-1\right)a_n \leq \left(n-1\right)a_n-na_{n+1}$$... |
2,226,386 | <p>Solve $$\frac{dx}{y+z}=\frac{dy}{z+x}=\frac{dz}{x+y}.$$ Solve the similtaneous equation by using method of multipliers. How can we choose these multipliers? Is there any specific method to know these multipliers?</p>
| Doug M | 317,162 | <p>Here is how I approached it.</p>
<p>Translate the whole thing such that the vertex opposite M is the origin.</p>
<p>represent the other vertexes of the triangle with complex numbers $2a,$ and $2b$</p>
<p>Point $M$ is then $a+b$</p>
<p>The center of one square is at $a(1-i)$
and the other is at $b(1+i)$</p>
<p>W... |
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... | Shreyas Pimpalgaonkar | 477,919 | <p>You shouldn't complicate things at this level.</p>
<p>What you are trying to accomplish is the exact definition of the binomial coefficient.</p>
<p>It is the number of ways of choosing n places from a total of m places.</p>
<p>In this case, choose 4 places from 8 places, and for every such selection of places per... |
512,960 | <p>Let's say I have a statement: if p then q.<br>
The converse would be: if q then p.<br>
The inverse would be: if not p then not q.<br>
The contraposition would be: if not q then not p.<br>
What would you call the following? if not p then q.<br>
Thanks.</p>
| Glen O | 67,842 | <p>There's no name for it, because there's no real connection between them. The inverse and converse exist because they still assert a direct correlation between p and q; it's just that, as opposed to the regular and contrapositive forms, the condition for failure is reversed.</p>
<p>On the other hand, "if not p then ... |
245,789 | <p>Let $f$ be a function defined on the interval $(−1, 1)$ such that for all $x, y \in (−1, 1)$, $$f(x + y) = \frac{{f(x) + f(y)}}{{1 - f(x)f(y)}}.$$ Suppose that $f$ is differentiable at $x = 0$, show that $f$ is differentiable on $(−1, 1)$.</p>
<p>Need your kind guidance and help. Thanks in advance!</p>
| DonAntonio | 31,254 | <p>Since $\,f(0)=0\,$, we get:</p>
<p>$$f'(0)=\lim_{x\to 0}\frac{f(x)}{x}\Longrightarrow \,\forall w\in (-1,1)\setminus\{0\}:$$</p>
<p>$$f'(w)=\lim_{h\to 0}\frac{f(w+h)-f(w)}{h}=\lim_{h\to 0}\frac{\frac{f(w)+f(h)}{1-f(w)f(h)}-f(w)}{h}=$$</p>
<p>$$=\lim_{h\to 0}\frac{\left(f(w)^2+1\right)f(h)}{h\left[1-f(w)f(h\right]... |
947 | <p>I'm looking for the algorithm that efficiently locates the "loneliest person on the planet", where "loneliest" is defined as:</p>
<p>Maximum minimum distance to another person — that is, the person for whom the closest other person is farthest away.</p>
<p>Assume a (admittedly miraculous) input o... | Reid Barton | 126,667 | <p>The paper <em>Vaidya, Pravin M.</em>, <a href="https://doi.org/10.1007/BF02187718" rel="noreferrer"><strong>An <span class="math-container">$O(n \log n)$</span> algorithm for the all-nearest-neighbors problem</strong></a>, Discrete Comput. Geom. 4, No. 2, 101-115 (1989), <a href="https://zbmath.org/?q=an:0663.68058"... |
947 | <p>I'm looking for the algorithm that efficiently locates the "loneliest person on the planet", where "loneliest" is defined as:</p>
<p>Maximum minimum distance to another person — that is, the person for whom the closest other person is farthest away.</p>
<p>Assume a (admittedly miraculous) input o... | bhwang | 239 | <p>This is called the "largest empty circle problem" in Computational Geometry, and has an <span class="math-container">$O(n \log n)$</span> solution provided that you are given the convex hull of the points and the corresponding Voronoi diagrams. Here is a very readable paper on the problem:</p>
<blockquote>... |
2,198,293 | <p>Having a bit of trouble in what way I am suppose to go about solving this problem. Any guidance would be great.</p>
<p>Show that there is at least one real solution to:$$x^5 - x^2 - 4 = 0 $$</p>
<p>Thanks in advance.</p>
| Jaideep Khare | 421,580 | <p>Since at $x \rightarrow \infty $ function tends to $\infty$ and at $x \rightarrow -\infty$ function tends to $-\infty$ . Hence it will cut $x$-axis at some point for sure since function is well-continuous.This is true for all polynomials with odd degree.</p>
|
3,790,932 | <p>If <span class="math-container">$f : \bar{\mathbb{Q}} \to \mathbb{Q}$</span> is a continuous function, where <span class="math-container">$\bar{\mathbb{Q}}$</span> denotes the set of algebraic numbers, does such function have to be constant?</p>
| Jyrki Lahtonen | 11,619 | <p>The claim is actually false. Proffering the following counterexample
<span class="math-container">$$f(x)=\begin{cases}1,&\ \text{if the real part of $x$ is greater than $\pi$, and}\\
0,&\ \text{otherwise.}\end{cases}$$</span></p>
|
3,178,648 | <blockquote>
<p>We assign to every element <span class="math-container">$i$</span> from <span class="math-container">$N=\{1,2,...,n\}$</span> a positive integer <span class="math-container">$a_i$</span>. Suppose <span class="math-container">$$a_1+a_2+...+a_n = 2n-2$$</span> then prove that map <span class="math-conta... | Calvin Lin | 54,563 | <p>Here's a direct solution (which I believe gets to the heart of the problem). (Sorry, I just noticed that OP requested a solution based on graph theory in the bounty. I ignored that.)</p>
<p>In <span class="math-container">$ \{ a_i \}$</span>, let there be <span class="math-container">$k$</span> 1's, and let <span c... |
3,178,648 | <blockquote>
<p>We assign to every element <span class="math-container">$i$</span> from <span class="math-container">$N=\{1,2,...,n\}$</span> a positive integer <span class="math-container">$a_i$</span>. Suppose <span class="math-container">$$a_1+a_2+...+a_n = 2n-2$$</span> then prove that map <span class="math-conta... | PkT | 435,267 | <p>Here's a solution based on graph theory. The idea is to remove a "leaf" from the tree and proceed by induction. </p>
<p>Assuming <span class="math-container">$A_n = \{a_1, \dots, a_n\}$</span>, <span class="math-container">$a_1 = 1$</span> and <span class="math-container">$a_2 > 1$</span>, consider the set <span... |
1,842,781 | <p>Suppose we have an alphabet of the following allowed characters: </p>
<ul>
<li>the lowercase letters $a$ through $z$ (26)</li>
<li>the uppercase letters $A$ through $Z$ (26)</li>
<li>the numerals $0$ through $9$ (10)</li>
<li>the common punctuation marks: $.,;:'"()!?-$ ( 11)</li>
<li>the white space (1)</li>
</ul>... | Théophile | 26,091 | <p>Starting off with @heropup's approach, we can speed up the calculations by appealing to symmetry:</p>
<p>\begin{array}{ccccc} & a & & b & \\ 0 & & c & & 1000 \\ & d & & e & \end{array}</p>
<p>Observe that we must have $a=d, b=e$, and $a = 1000-b$. This eliminates all... |
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... | Glärbo | 907,918 | <p>Voxel tracing differs from ordinary ray tracing in that you want to use a coordinate system where the voxels are defined for each unit cell, i.e. for unit (integer) coordinates.</p>
<p>You can use one vector, say <span class="math-container">$\vec{o}$</span> for observer (eye) location (in the voxel coordinates), an... |
3,122,612 | <p>I am doing a presentation on nonlinear optics and I ran into a paper that uses a series to describe a wave equation of a system. The paper will be linked below and the series is:</p>
<p><span class="math-container">$\sum_{n=1}^{\infty}\frac{(-i)^{n-1}}{n!}k^n$</span></p>
<p>I mostly am looking for what i should ca... | Community | -1 | <p><span class="math-container">$$\sum\limits_{n=1}^\infty \frac{(-i)^{n-1}}{n!}k^n=\sum\limits_{n=1}^\infty \frac{(-i)(-i)^{n-1}}{(-i)n!}k^n=\frac{1}{-i}\sum\limits_{n=1}^\infty\frac{(-i)^n}{n!}k^n=i(e^{-ik}-1).$$</span> So I would say yes it is known and there is a name for it -- exponential function.</p>
|
1,696,713 | <p>I am solving exact differential equation, but I am stuck on the step on how to simplify this term or how to rewrite it. </p>
<p>$e^{-2\ln{\sin{x}}}$</p>
| Twiffy | 153,233 | <p>I think the mathematical concept you want is <em>implication</em>.</p>
<p>For example, think about mathematical equations that only work sometimes. How about $x^2=x$. If you plug in a random value of $x$, that equation isn't true. But it happens to work when $x=1$. Written mathematically,</p>
<p>$$ x = 1 \quad \Lo... |
4,014,022 | <p>In example <span class="math-container">$1.5$</span> of <em>Cracking the GRE Subject Test</em>, the authors make the following calculation in one step with no additional commentary:</p>
<blockquote>
<p>we interchange <span class="math-container">$x$</span> and <span class="math-container">$y$</span> and solve for <s... | Jessie | 794,754 | <p><span class="math-container">$$y = \frac{x}{1-x^2}$$</span></p>
<p>Replace <span class="math-container">$x$</span> with <span class="math-container">$y$</span></p>
<p><span class="math-container">$$x = \frac{y}{1-y^2}$$</span></p>
<p>and solve <span class="math-container">$y,x=\frac{y}{1-y^2}$</span></p>
<p><span cl... |
73,112 | <blockquote>
<p>Find <span class="math-container">$e^{At}$</span>, where <span class="math-container">$$A = \begin{bmatrix} 1 & -1 & 1 & 0\\ 1 & 1 & 0 & 1\\ 0 & 0 & 1 & -1\\ 0 & 0 & 1 & 1\\ \end{bmatrix}$$</span></p>
</blockquote>
<hr />
<p>So, let me just find <spa... | hmakholm left over Monica | 14,366 | <p>A different, but rather specific, strategy would be to use the ring homomorphism
$${a+bi\in\mathbb C \mapsto \pmatrix{a&-b \\ b&a}\in\mathbb R^{2\times 2}}$$in the block decomposition. Then your problem is equivalent to finding
$$e^{t\pmatrix{1+i & 1\\ 0 & 1+i}}=e^{\pmatrix{t+ti & t\\ 0 & t+t... |
3,748,879 | <p><span class="math-container">$g : \mathbb R \to [0,1]$</span> is a non-decreasing and right continuous step function such that <span class="math-container">$g(x)=0$</span> for all <span class="math-container">$x \leq 0$</span> and <span class="math-container">$g(x)=1$</span> for all <span class="math-container">$x \... | N. S. | 9,176 | <p><strong>Hint</strong>
<span class="math-container">$$ \int_0^\infty \frac{1}{(x^p＋2020)^q} \,dx$$</span>
converges if and only if
<span class="math-container">$$\int_1^\infty \frac{1}{(x^p＋2020)^q} \,dx$$</span></p>
<p>Now apply the Limit Comparison Theorem for <span class="math-container">$$\int_1^\infty \frac{1}{(... |
2,657,571 | <p>Apologies if this is a duplicate. I searched and didn't find anything quite like it.</p>
<p>Suppose I have a drawer with an equal number of N black socks and N white socks. They're all mixed up. So, my chances of picking a matching pair in the first two selections is (N-1)/(2N-1), right? Well, what if, <em>before</... | Bram28 | 256,001 | <p>Of course it shouldn't change!</p>
<p>Let's see if the math works out for a simple example, say $N=3$</p>
<p>First, the 'normal' calculation says that the chances of getting a match $M$ is:</p>
<p>$$P(M)= \frac{N-1}{2N-1}=\frac{2}{5}$$</p>
<p>Now let's see what happens when you randomly partition them.</p>
<p>L... |
3,080,294 | <blockquote>
<p>A cone has its guiding curve to the circle <span class="math-container">$x^2+y^2+2ax+2by=0$</span> and passes through a fixed point <span class="math-container">$(0,0,c)$</span>. If the section of the cone by plane <span class="math-container">$y=0$</span> is a rectangular hyperbola. Prove that the ve... | Intelligenti pauca | 255,730 | <p>Let <span class="math-container">$V=(x_0,y_0,z_0)$</span> be the vertex of the cone. The cone is composed of all lines passing through <span class="math-container">$V$</span> and a point of circle <span class="math-container">$\gamma$</span> of equation <span class="math-container">$x^2+y^2+2ax+2by=0$</span> in the ... |
2,575,156 | <p>I thought of this question through an excercise in algebraic geometry where the rings were $\Gamma(V)$ and $\mathcal{O}_P(V)$, although my question is more general. If $R\subset S$ two (commutative with 1) rings (R subring of S) and I prime ideal in R, then consider I'=(I) the ideal in S, the ideal generated by I. I... | Bernard | 202,857 | <p>As mentioned in @Asquire's answer, you can have counter-examples taking for $R=\mathbf Z$ and for $S$ the ring of integers of some quadratic extension. It is know that an odd prime number $p$ remains prime in $S$ (one says it is <em>inert</em> in this case)if and only if the discriminant of the extension is <em>not<... |
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... | José Carlos Santos | 446,262 | <p>The empty set belongs to <span class="math-container">$\tau$</span> trivially.</p>
<p>The set <span class="math-container">$\mathbb R$</span> belongs to <span class="math-container">$\tau$</span> because is <span class="math-container">$x\in U$</span> and you take <span class="math-container">$\varepsilon=1$</span>... |
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... | Root | 466,361 | <p>Your proof of a) is not right. Note that interval size tending to zero is different from it being zero.
a) is vacuuously true for null set to put it differently there is no x the 'for all x'statement is always true. For <span class="math-container">$\mathbb{R}$</span> it clear that as for any real number say interv... |
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... | Mark | 470,733 | <p>I'm not sure you really understood the definition of <span class="math-container">$\tau$</span>. A set <span class="math-container">$U$</span> is in <span class="math-container">$\tau$</span> if for each <span class="math-container">$x\in U$</span> there is some <span class="math-container">$\epsilon>0$</span> (w... |
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>
| Mythomorphic | 152,277 | <p>For $y\neq1$
$$\frac{y+1}{y-1}=10^{x^2}$$ $$y+1=y\cdot10^{x^2}-10^{x^2}$$ $$1+10^{x^2}=y(10^{x^2}-1 )$$</p>
<p>Then?</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>
| wythagoras | 236,048 | <p>$$\frac{y+1}{y-1} = 10^{x^2}$$</p>
<p>$$y+1 = 10^{x^2}(y-1)$$</p>
<p>$$y+1 = 10^{x^2}y-10^{x^2}$$</p>
<p>$$y+1+10^{x^2} = 10^{x^2}y$$</p>
<p>$$1+10^{x^2} = (10^{x^2}-1)y$$</p>
<p>$$y=\frac{10^{x^2}+1}{10^{x^2}-1}$$</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/... | Community | -1 | <p>Better consider the expression: $$\delta(k+1) = 4^{k+2} + 5^{2k+1} = 4\times 4^{k+1} + 5^2\times 5^{2k-1} = 4[7m-5^{2k-1}] +5^{2k-1}(5^2) = [5^{2k-1} \times (5^2-4) +4(7m)] \mid 7$$ We considered $\delta(k) = 4^{k+1} +5^{2k-1} = 7m$ , since we know $7\mid \delta(k)$. Hope it helps.</p>
|
4,126,527 | <p>Suppose <span class="math-container">$w_1$</span>, <span class="math-container">$w_2$</span>, ..., <span class="math-container">$w_{n-1}$</span> are the complex roots not equal to <span class="math-container">$1$</span> of <span class="math-container">$z^n-1=0$</span>, where <span class="math-container">$n$</span> i... | Community | -1 | <p>From <span class="math-container">$w^n=1$</span> we deduce <span class="math-container">$\bar w^n=1$</span> and <span class="math-container">$(w\bar w)^n=1$</span>. Then as <span class="math-container">$w\bar w$</span> is real and <span class="math-container">$n$</span> is odd, <span class="math-container">$w\bar w=... |
4,126,527 | <p>Suppose <span class="math-container">$w_1$</span>, <span class="math-container">$w_2$</span>, ..., <span class="math-container">$w_{n-1}$</span> are the complex roots not equal to <span class="math-container">$1$</span> of <span class="math-container">$z^n-1=0$</span>, where <span class="math-container">$n$</span> i... | Toby Mak | 285,313 | <p>Since <span class="math-container">$\omega$</span> lies on the unit circle, more generally, let <span class="math-container">$\omega = \cos \theta + i\sin\theta$</span> and <span class="math-container">$\overline{\omega} = \cos \theta-i\sin\theta$</span>:</p>
<p><span class="math-container">$$\text{LHS} = \frac{2-\o... |
3,114,663 | <p>Let there be <span class="math-container">$m$</span> indistinguishable balls, <span class="math-container">$k$</span> bins, <span class="math-container">$C$</span> capacity.
Let <span class="math-container">$X_j$</span> denote the total balls in bin <span class="math-container">$j$</span>.
I've seen ways to calcula... | saulspatz | 235,128 | <p>This seems computationally very difficult to me. You might do better with simulation.</p>
<p>It <span class="math-container">$C\ge m$</span> then the constraint has no effect, and we are just looking for the mean and variance of the number of balls in bin <span class="math-container">$j.$</span> This has a binom... |
2,990,400 | <p>I need to find all the continuous functions <span class="math-container">$f$</span> such that
<span class="math-container">$$
[f(t)]^2=\int_0^t f(s) ds
$$</span>
<strong>Attempt:</strong></p>
<p>Since
<span class="math-container">$$
f(t)\leq [f(t)]^2+c, \, \, \forall c \geq \frac14$$</span>
we have</p>
<p><span ... | Lutz Lehmann | 115,115 | <p>Just observe that <span class="math-container">$f(0)=0$</span> and then take the derivative
<span class="math-container">$$
2ff'=f\implies f=0\lor 2f'=1,~f=\frac t2.
$$</span>
As per a hint of Ian in a prior comment, you can also combine a segment where <span class="math-container">$f=0$</span> and then the nullity ... |
4,264,808 | <p>So we know that <span class="math-container">$\frac{n^2}{2} \geq \frac{n}{2}$</span>, but I'm stuck proving that <span class="math-container">$\frac{n^2}{2} \geq \frac{n^2}{2^{\sqrt{\log n}}}\geq \frac{n}{2}$</span>. Am I missing something?</p>
| Alann Rosas | 743,337 | <p>Notice that the expression <span class="math-container">$\frac{x^2}{2^\sqrt{\ln(x)}}$</span> is well-defined if and only if <span class="math-container">$x\geq1$</span>. Division by <span class="math-container">$x$</span> then shows that your inequality is equivalent to</p>
<p><span class="math-container">$$\frac{x}... |
3,245,854 | <p>Hi I am trying to solve an integral problem that involves trig substitution. First I tried completing the square, which gave me <span class="math-container">$1/\sqrt{(x+3)^2+2^2}$</span>. I know I am supposed to use <span class="math-container">$x = \arctan(\theta)$</span>. Does that mean it should be: </p>
<p><spa... | Mohammad Riazi-Kermani | 514,496 | <p>Note that <span class="math-container">$$x^2+6x+13 = (x+3)^2+4$$</span></p>
<p>Your correct substitution is<span class="math-container">$$(x+3)=2\tan(\theta)$$</span></p>
<p><span class="math-container">$$dx=2\sec^2(\theta)d\theta$$</span></p>
<p><span class="math-container">$$(x+3)^2+4=4\tan ^2(\theta)+4 =4(1+\... |
3,245,854 | <p>Hi I am trying to solve an integral problem that involves trig substitution. First I tried completing the square, which gave me <span class="math-container">$1/\sqrt{(x+3)^2+2^2}$</span>. I know I am supposed to use <span class="math-container">$x = \arctan(\theta)$</span>. Does that mean it should be: </p>
<p><spa... | clathratus | 583,016 | <p>So you are trying to find
<span class="math-container">$$I=\int\frac{dx}{\sqrt{x^2+6x+13}}=\int\frac{dx}{\sqrt{(x+3)^2+2^2}}$$</span>
And your goal is to find the right substitution <span class="math-container">$$x+3=a\cdot\tan\theta$$</span>
such that <span class="math-container">$$(x+3)^2+2^2=(a\tan\theta)^2+2^2=... |
3,441,647 | <p>I was asked to find the domain of <span class="math-container">$$\arcsin[\frac{x^2+1}{2x}]$$</span>
My first step was <span class="math-container">$$-1\leq\frac{x^2+1}{2x} \leq1$$</span></p>
<p>What I don't understand is why I cannot cross multiply to get <span class="math-container">$$-2x\le {1+x}^{2} \le 2x$$</sp... | lab bhattacharjee | 33,337 | <p>Let <span class="math-container">$x=\tan y$</span></p>
<p><span class="math-container">$z=\arcsin\dfrac{1+x^2}{2x}=\arcsin(\csc2y)$</span></p>
<p><span class="math-container">$\implies\sin z=\csc2y$</span> which is either <span class="math-container">$\ge1$</span> or <span class="math-container">$\le-1$</span></p>... |
1,435,718 | <blockquote>
<p>Find all pairs of prime numbers $p , q$ for which:
$$p^2 \mid q^3 + 1 \tag{A}$$
and
$$q^2 \mid p^6 − 1 \tag{B}$$</p>
</blockquote>
<p>The question is from the Bulgaria National Olympiad 2014.</p>
<p><em>I'm looking for any solution I may have missed, and generally any alternative method that m... | Keith Backman | 29,783 | <p>The insight to simplify is that $p$ and $q$ must have different parities; they can't both be odd. At least one of them must be odd, because there are not two even primes. Then the other one must be even, because the addition or subtraction of $1$ makes one of the compound expressions even. Hence one of them must be ... |
390,662 | <p>Let <span class="math-container">$X$</span> be a compact Riemann surface. I would like to find a somehow complete reference for the proof of the so called non-Abelian Hodge correspondence relating Dolbeaut, Betti and Higgs bundle moduli spaces.</p>
<p>I've tried to read the original articles by Hitchin (1987) or Sim... | Niels | 11,682 | <p>I recently attented a nice online talk by Pengfei Huang and he indicated two sources:</p>
<ul>
<li><p>the first chapter of his own phd <a href="https://tel.archives-ouvertes.fr/tel-03134917" rel="noreferrer">Non-abelian Hodge theory and some specializations - TEL - Thèses en ligne</a></p>
</li>
<li><p><a href="https... |
12,047 | <p>How to prove the following trigonometric identities ?</p>
<p>1) If $\displaystyle \tan (\alpha) \cdot \tan(\beta) = 1 \text{ then } \alpha + \beta = \frac{\pi}{2}$
</p>
<p>I tried to prove it by using the the formula for $\tan(\alpha + \beta)$ but ain't it valid only when $\alpha + \beta \neq \frac{\pi}{2}$ ?</p>
... | user02138 | 2,720 | <p>When I need to prove trigonometric identities, I tend to use <em>complex exponentials</em>. For instance, to prove your first identity observe that
\begin{align}
\sin \theta = \frac{e^{i \theta} - e^{-i \theta}}{2i} \quad \text{and} \quad \cos \theta = \frac{e^{i \theta} + e^{-i \theta}}{2},
\end{align}
so
\begin{al... |
2,254,694 | <p>I have been stuck on this question for a while now. I have tried many attempts. Here are two that I thought looked promising but lead to a dead end:</p>
<p>Attempt 1:</p>
<p>Write out the terms of $b_n$:</p>
<p>$$b_1=a_{2}-\frac{a_{1}}{2}$$
$$b_2=a_{3}-\frac{a_{2}}{2}$$
$$b_3=a_{4}-\frac{a_{3}}{2}$$
$$\cdots$$
$$... | Paramanand Singh | 72,031 | <p>Let $b_n=a_{n+1}-(a_{n}/2)$ so that $b_{n} \to 0$ as $n\to\infty$. Now we have $a_{2}=b_{1}+(a_{1}/2),a_{3}=b_{2}+b_{1}/2+a_{1}/4$ and continuing in this manner we get $$a_{n+1}=b_{n}+b_{n-1}/2+\cdots+b_{1}/2^{n-1}+a_{1}/2^{n}$$ Let $\epsilon>0$ be given then we can see that there is a positive integer $m$ such t... |
1,248,668 | <p>I do have a system of n equations with m variables where m > n with integer coefficients. I wish to find a set of integer solutions to this system (In my case n = 2 and m = 4). Could somebody tell me how I can do it? I already solved this system with Mathematica but I would like to redo these calculations by hand to... | k1.M | 132,351 | <p>There is no way, as I know, to determine all integer solutions to the system. You should at first find all real solutions to the system, this gives you a subspace $V\subset\mathbb R^m$ of dimension at least $m-n$. Now for finding all integer solutions you should find the lattice points of the subspace $V$.</p>
|
495,064 | <p>I'm currently tackling the following problems:</p>
<p>a) Let $a,b \in \mathbb{Z}$ and let $m$ be a nonnegative integer. Prove that $(a,b)=1$ if and only if $(a^m,b)=1$.</p>
<p>If $(a^m,b)=1$ then $pa^m+qb=1$ for some integers $p$ and $q$. It follows that $(pa^{m-1})a+qb=1$, which shows that $(a,b)=1$. This took ca... | user66733 | 66,733 | <p>HINT: How about thinking this way: $(a,b) \neq 1 \iff (a^m,b) \neq 1$?</p>
<p>So, for example, if $(a^m,b)=d>1$ then you can find a prime number that $p \mid d$, and this is guaranteed by the prime factorization theorem in integers. Now see what happens in this case.</p>
|
2,008,243 | <p>How to find the exact value of the following limit?</p>
<p>$\lim_\limits{x \to 0}\frac{e^{x\cos{x^2}}-e^{x}}{x^5}$</p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>$$\dfrac{e^{x\cos x^2}-e^x}{x^5}=e^x\cdot\dfrac{e^{x(\cos x^2-1)}-1}{x^5}$$</p>
<p>$$\dfrac{e^{x(\cos x^2-1)}-1}{x^5}=\dfrac{e^{x(\cos x^2-1)}-1}{x(\cos x^2-1)}\cdot\dfrac{x(\cos x^2-1)}{x^5}$$</p>
<p>$$\dfrac{\cos x^2-1}{x^4}=-\left(\dfrac{\sin x^2}{x^2}\right)^2\cdot\dfrac1{(1+\cos x^2)}$$</p>
<p>... |
2,303,339 | <p>How does this proof work?</p>
<p><strong>Theorem.</strong>$\quad$Let $G$ be a group. Then $G$ has a unique identity. </p>
<p><strong>Proof.</strong>$\quad$Assume that $e$ and $f$ are two identities in $G$. Since $e$ is
an identity, $ef=f$; and since $f$ is an identity, $ef=e$. Thus $e=ef=f$. </p>
<p>I think ne... | HK Lee | 37,116 | <p>We use Caley-Hamilton theorem, i.e. $X^2-{\rm tr}\ X \cdot X+{\rm
det}\ XI=0$, for $A+B$ :</p>
<p>For direct computation we have a claim $$ {\rm }
{\rm tr }A \ {\rm tr}B- {\rm tr} (AB) =
{\rm det} (A+B)- {\rm det}A- {\rm det} B $$</p>
<p>If $A=\left(
\begin{array}{cc}
a & b \\
c &... |
2,033,834 | <blockquote>
<p>Evaluate the work integral where $F(x,y)=\langle-y,x\rangle$ over a triangle with vertices $A(-2,-2)$, $B(2,-2)$, $C(0,1)$.</p>
</blockquote>
<p>I am not sure how to approach this problem.
I tried setting $AB(4,0)$, $BC(-2,3)$ and $CA(-2,-3)$ but I am not sure how to proceed.</p>
<p>Without using G... | Akarsh Verma | 392,733 | <p>You will want to split this line integral into three. The first being the line integral from A to B, second from B to C, and third from C to A. The first line integral is along the line $y=-2$ and it is from $ x =-2$ to $x=2$. Thus, it can be parametrized as $x=t$ and $y=-2$ where $-2 \le t \le 2$. From this, we get... |
644,325 | <p>Let $M$ be an $R$-module and $N$ be a submodule of $M$. Can we always write $M \cong N \oplus M/N$ as $R$-modules ? If not, then under what conditions on $M$, the direct sum holds ?</p>
| zcn | 115,654 | <p>(This question has almost certainly been asked before, but it seemed easier to just type this up than search for a previous answer). </p>
<p>There is always an exact sequence</p>
<p>$$0 \to N \to M \to M/N \to 0$$</p>
<p>I will state when this sequence is split exact (as pointed out in the comments, this is not t... |
644,325 | <p>Let $M$ be an $R$-module and $N$ be a submodule of $M$. Can we always write $M \cong N \oplus M/N$ as $R$-modules ? If not, then under what conditions on $M$, the direct sum holds ?</p>
| Wei Zhou | 106,010 | <p>No, it is not true. Consider the cyclic of order 4. It is a Z-module. Let N be the subgroup of order 2, then you get a counter example.</p>
|
999,000 | <p>Theorem 3.29 If $p >1$ then $\sum_{n=2}^{\infty} \frac{1}{n(\log n)^p}$ converges; if $p \leq 1$, the series diverges. </p>
<p>Proof:</p>
<p>The monotonicity of the logarithmic function implies that $\{\log n\}$ increases. Hence $\frac{1}{n\log n}$ decreases, and we can apply theorem 3.27; this leads us to the ... | Dap | 467,147 | <p>(This is more of a "sketch proof" at the moment.)</p>
<p>The pairs are exactly <span class="math-container">$(t,n)$</span> with <span class="math-container">$t\leq n.$</span> The necessity was posted in a comment which I'll quote here:</p>
<blockquote>
<p>For <span class="math-container">$t>n$</span> there is... |
2,963,560 | <blockquote>
<p>Given <span class="math-container">$a,b,c\in \Bbb Z$</span>, pairwise distinct, and <span class="math-container">$n\in \Bbb N\setminus\{0\}$</span> prove that
<span class="math-container">$$S(n)=\frac{a^n}{(a-b)(a-c)}+\frac{b^n}{(b-a)(b-c)}+\frac{c^n}{(c-a)(c-b)}\in \Bbb Z.$$</span></p>
<p>Sou... | metamorphy | 543,769 | <p>For <span class="math-container">$n > 1$</span>, <span class="math-container">$S(n)=\displaystyle\sum_{\substack{n_1, n_2, n_3 \geqslant 0 \\ n_1 + n_2 + n_3 = n - 2}}a^{n_1}b^{n_2}c^{n_3}$</span> (not the most elegant, but nice I think).</p>
<p>For a motivation: the <a href="https://en.wikipedia.org/wiki/Comple... |
28,586 | <p>Imagine I have two lists:</p>
<pre><code>List1 = {0, 1, 0.6, 0.5, 1.2, 0.4};
List2 = {"a", "b", "c", "d", "e", "f"};
</code></pre>
<p>How can I use <code>Pick</code> to return a third list consisting of items in List2 that have a array position corresponding to an array position with a value $\leq N$ in the first ... | Mike Honeychurch | 77 | <p>Several ways to create the stencil. Perhaps most intuitive is just to run a test:</p>
<pre><code>stencil = # <= 0.5 & /@ List1
Pick[List2, stencil]
(* {"a", "d", "f"} *)
</code></pre>
<p><strong>Edit</strong></p>
<p>To answer the comment below, <code>Pick</code> "picks out those elements of <em>list</em... |
554,003 | <p>How can I find a closed form for the following sum?
$$\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$$
($H_n=\sum_{k=1}^n\frac{1}{k}$).</p>
| Ali Shadhar | 432,085 | <p>From <a href="https://math.stackexchange.com/questions/3263927/prove-frac-partial-partial-m-textbn-m-textbn-m-sum-k-0n-1-f/3271177#3271177">here</a> we have </p>
<p><span class="math-container">$$\displaystyle\int_0^1 x^{n-1}\ln^2(1-x)\ dx=\frac1n\left({H_n^2}+H_n^{(2)}\right)$$</span></p>
<p>dividing both sides ... |
554,003 | <p>How can I find a closed form for the following sum?
$$\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$$
($H_n=\sum_{k=1}^n\frac{1}{k}$).</p>
| Ali Shadhar | 432,085 | <p><strong>I think this is the shortest solution</strong> </p>
<p>Using the generating function</p>
<p><span class="math-container">$$\frac12\ln^2(1-x)=\sum_{n=1}^\infty\frac{H_n}{n+1}x^{n+1}=\sum_{n=1}^\infty\frac{H_{n-1}}{n}x^n$$</span></p>
<p>Multiply both sides by <span class="math-container">$\frac{\ln(1-x)}{x}... |
13,467 | <p>I will present two problems alongside solutions, student is doing problems of type I like a cakewalk but has several issues with the problems of type 2;</p>
<p><strong>Type I</strong></p>
<blockquote>
<p>Consider an experiment of rolling two dice:</p>
<p>Sample space $$ S = \{(1,1),(1,2),(1,3), \cdots, (6,6... | guest | 9,298 | <p>The question is very confusingly stated. (Hopefully it is more clearly done in native language.) I can't help with the logic barrier for the students, but if they are having a hard time with the logic and then you add on confusing language, this makes things worse. Or may be more the problem than the logic barrie... |
4,141,812 | <p>Considering the vector space <span class="math-container">$\mathbb{R}^3$</span>, find all values of <span class="math-container">$k$</span> such that
<span class="math-container">\begin{align*}
\begin{bmatrix}
2k^2 \\
-3k \\
1
\end{bmatrix} \in \text{ span }
\left\{
... | gfppoy | 390,474 | <p>By the definition of a vertical asymptote it suffices to find <span class="math-container">$a$</span> such that <span class="math-container">$e^x+x \rightarrow 0$</span> as <span class="math-container">$x \rightarrow a^+$</span> or <span class="math-container">$x \rightarrow a^-$</span>. But since <span class="math-... |
216,040 | <p>A model for the movement of a stock price supposes that if the present price is S then after one period, say one second, it will either go up to uS with probability p or go down to dS with probability q = 1 - p. Assuming that successive movements are independent,approximate the probability that the stock will be up ... | André Nicolas | 6,312 | <p>Let the random variable $W$ be our net gain in one throw of the three dice. </p>
<p>With probability $\dfrac{1}{6^3}$, our number comes up $3$ times, and $W=3$.</p>
<p>With probability $\dbinom{3}{2}\dfrac{1}{6^2}\dfrac{5}{6}$ our number comes up twice, and $W=2$.</p>
<p>With probability $\dbinom{3}{1}\dfrac{1}{6... |
4,619,164 | <p>So one way to solve this ordinary differential equation is by computing the integral of both sides of <span class="math-container">$$\frac{y'}{y} = 1$$</span>
However, I did it another way and I think I made a mistake, but where is it? What I did is:
<span class="math-container">$$y' = y $$</span>
<span class="math-... | Trinity-Slifer | 1,139,301 | <p>If <span class="math-container">$y'=\frac{dy}{dx}$</span> as you wrote <span class="math-container">$y$</span> is a function in the variable <span class="math-container">$x$</span> so <span class="math-container">$\int y(x) dx$</span> is not <span class="math-container">$yx$</span>. I think the only way is to integr... |
1,148,674 | <p>I have always asked myself why this happens. </p>
<p>If $x = 4$, then $\sqrt{x} = 2$, but if I search for the $\sqrt{4}$, I get $2$ & $-2$.</p>
| Akshay Bodhare | 213,722 | <p>There is a difference.
$$\sqrt {x^2}=|x|$$</p>
<p>And,If
$$x^2=k \implies x=\pm\sqrt{k}$$</p>
|
1,342,951 | <p>I was thinking what function I should compare it to. If I say whether a function is smaller or bigger than this one, then I must prove that. I was thinking of (x+1)^2 but I realized that this converges so that won't prove the integral diverges. So, what function should I use?</p>
| Jahan Claes | 247,555 | <p>You define conditional probability by the Kolmogorov definition. From the Kolmogorov definition, you can prove $P(A\cap B) = P(A\mid B)P(B)$. Then you just have to define what you mean when you write $P(A,B)$. You've defined it to mean $P(A,B)=P(A\cap B)$. Thus, from the definition of conditional probability and the... |
1,284,086 | <p>Given a state $(|1\rangle \otimes |0\rangle \otimes |1\rangle) + (|0\rangle \otimes |0\rangle \otimes |1\rangle)$, is it possible to factorise out the $|0\rangle$ in the middle of both of them?</p>
<p>Alternatively written as $|101\rangle + |001\rangle$ for convenience.</p>
<p>For a state such as the following it ... | Fallen Apart | 201,873 | <p>If you consider the definition of tensor product by <a href="http://en.wikipedia.org/wiki/Tensor_product#Universal_property" rel="nofollow">universal property</a>, which is unavoidably equal to one you have in mind, then you have two natural isomorphisms $V\otimes W\cong W\otimes V$ and $(V_1\otimes V_2)\otimes V_3\... |
83,236 | <p>I need a <code>compare-operation</code> for two lists of same length (usually > 100000) which does the following:</p>
<blockquote>
<p>{2,3,5,4,1,8,7} <code>compare-operation</code> {1,4,6,3,2,8,8} = {2,4,6,4,2,8,8}</p>
</blockquote>
<p>The resulting list has at each position the greater (or equal element in case... | yode | 21,532 | <p>If you are in 11.1 or later verion</p>
<pre><code>ThreadingLayer[Max][{{2, 3, 5, 4, 1, 8, 7} , {1, 4, 6, 3, 2, 8, 8}}]
</code></pre>
<blockquote>
<p><code>{2.,4.,6.,4.,2.,8.,8.}</code></p>
</blockquote>
<p>Of course,we have build-in method if you are in old verion</p>
<pre><code>Internal`MaxAbs[{2, 3, 5, 4, 1,... |
4,340,560 | <p>I have to teach limits to infinity of real functions of one variable.
I would like to start my course with a beautiful example, not simply a basic function like <span class="math-container">$1/x.$</span> For instance, I thought of using the functions linked to the propagation of covid-19 and show that, under the bas... | Paul Sinclair | 258,282 | <p>The problem with "real-life" applications of limits at infinity is that at a fundamental level, there are none. Every place you see something relating to the infinite in the real world, it is actually not the real world, but a mathematical model of the real world. And, without fail, if you examine the mode... |
3,009,351 | <p>Consider the vectors <span class="math-container">$q_1=(1,1,1)$</span> and <span class="math-container">$q_3=(1,1,-2)$</span>. I need to find a third vector <span class="math-container">$q_2$</span> such that <span class="math-container">$\{q_1,q_2,q_3\}$</span> is a arthogonal basis for <span class="math-container... | MathIsNice1729 | 274,536 | <p>None of <span class="math-container">$q_1$</span> and <span class="math-container">$q_2$</span> are normalized. Hence, the formula would be
<span class="math-container">$$ q_2=v-{\langle v|q_1\rangle \over \langle q_1|q_1\rangle} q_1-{\langle v|q_3\rangle \over \langle q_3|q_3\rangle} q_3 = ({1 \over 2},-{1 \over 2}... |
1,238,887 | <p>A special case of the Lagrange multiplier theorem may be stated as: Let $S, T \subset \mathbb{R}^{n}$ be open. Let $f: S \to \mathbb{R}$ be differentiable on $S$ and $g: T \to \mathbb{R}$ differentiable on $T$ such that $S \cap g^{-1}\{ 0 \}$ is not empty. If $f(x)$ is an extremum of $f$ for some $x \in S \cap g^{-1... | SiXUlm | 58,484 | <p>Not true.</p>
<p>Take $a_n = n * (\frac{e}{2})^{n}$. It's easy to verify that $\lim \frac{a_n}{e^n} = 0$</p>
<p>On the other hand, $\frac{a_n}{\sum a_n} = \frac{1}{\sum a_i/a_n}$, where i runs from 1 to n</p>
<p>Now consider the denominator:</p>
<p>$\sum \frac{a_i}{a_n} = \sum \frac{i}{n}*(\frac{e}{2})^{i-n} $ &... |
1,238,887 | <p>A special case of the Lagrange multiplier theorem may be stated as: Let $S, T \subset \mathbb{R}^{n}$ be open. Let $f: S \to \mathbb{R}$ be differentiable on $S$ and $g: T \to \mathbb{R}$ differentiable on $T$ such that $S \cap g^{-1}\{ 0 \}$ is not empty. If $f(x)$ is an extremum of $f$ for some $x \in S \cap g^{-1... | Yiorgos S. Smyrlis | 57,021 | <p>Not true.</p>
<p>Take the sequence $\{a_n\}$ of the form:
$$
a_n=b_j,\quad \text{for}\,\,\,k_j\le n< k_{j+1},
$$
such that</p>
<p>a. The $b_j$ are positive and form an increasing sequence,</p>
<p>b. $b_j=a_1+\cdots+a_{k_j-1}$, and hence $\dfrac{a_{k_j}}{a_1+\cdots+a_{k_j}}=\dfrac{1}{2}$,</p>
<p>c. $k_j$ is ch... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.