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 |
|---|---|---|---|---|
2,964,439 | <p>How do I prove the following by induction?</p>
<blockquote>
<p>Given a finite collection of numbers, the sums/products obtained by adding/multiplying them in any order are all equal.</p>
</blockquote>
| Patrick Stevens | 259,262 | <p>The proof for multiplication is identical to that for addition, so I'll only look at addition; it works for any commutative associative operation.</p>
<p>Inductive hypothesis: any sum of <span class="math-container">$n$</span> numbers is equal to the sum of those same <span class="math-container">$n$</span> numbers... |
242,199 | <p>I was reading the solution to this problem and noticed it used $X = \sum_{i=1}^n X_i$ and $Y = \sum_{i=1}^n Y_i$. I think I understand all other parts except this one. Would you please explain why $X = \sum_{i=1}^n X_i$ and $Y = \sum_{i=1}^n Y_i$?</p>
<p><img src="https://i.stack.imgur.com/hHP8n.png" alt="enter ima... | André Nicolas | 6,312 | <p>Here $X$ is the number of $1$'s when you roll a fair die $n$ times.</p>
<p>Recall that the random variables $X_1, X_2, X_3, \dots, X_n$ were defined as follows. $X_1=1$ if we get a $1$ on the <strong>first</strong> roll, and $X_1=0$ otherwise.</p>
<p>$X_2=1$ if we get a $1$ on the <strong>second</strong> roll, and... |
2,433,438 | <p>Probability </p>
<blockquote>
<p>Question. "$11$ identical balls are distributed in $4$ distinct boxes randomly. Then the probability that any $3$ boxes will together get a greater number of balls than the remaining one is:"</p>
</blockquote>
<p>I am confused about the distribution of balls. Are all the cases i.... | OnoL | 65,018 | <p>The point is to find all combinations of natural numbers $(a,b,c,d)$ such that $\max\{a,b,c,d\}\le 5$ and $a+b+c+d=11$. Without loss of generality, assume $a\ge b\ge c\ge d$. Notice that $\max\{a,b,c,d\}$ can only be $3$, or $4$, or $5$. </p>
<ul>
<li>If $\max\{a,b,c,d\}=3$, then the only possible combination is $(... |
2,633,246 | <p>I need to find how many ways you can arrange the characters in Permutation with the $N$ appearing before all of the vowels but after the $P$. I understand that you can place the $N$ and $P$ $C(6,2)$ ways, but I'm not sure how to accommodate for the vowels. I could place the $N$, the $P$, and the vowels $(6,3)$ ways ... | Ross Millikan | 1,827 | <p>First choose seven slots for $N,P$ and all the vowels. How many ways to do that? $P$ has to go in the first of those and $N$ in the second. How many ways to arrange the vowels in the remaining slots that you chose? How many ways to arrange the remaining consonants in the slots you didn't choose? Note that there... |
354,365 | <p>Let $X_i$ be pairwise-uncorrelated random variables, $\forall\,i \in \mathbf{n} \equiv \{0,\dots,n-1\}$, with identical expectation value $\mathbb{E}(X_i)=\mu$, and identical variance $\mathrm{Var}(X_i)=\sigma^2$. Also, let $\overline{X}$ be their average, $\frac{1}{n}\sum_{i\in \mathbf{n}} X_i$.</p>
<p>Then, as s... | randpan | 822,861 | <p>I just wanted to add one very fast proof that I find easy and useful. Noting <span class="math-container">$r_i$</span> the rows of a matrix <span class="math-container">$A\in \mathbb R^{m\times n}$</span> we can denote as matrix <span class="math-container">$A_{1}$</span>, the matrix <span class="math-container">$A$... |
124,955 | <p>Euler and Lamé are said to have proven FLT for $n=3$ that is, they are believed to have shown that $x^3 + y^3 = z^3$ has no nonzero integer solutions. According to Kleiner they approached this by decomposing $x^3 + y^3$ into $(x + y)(x + y\omega)(x + y\omega^2)$ where $\omega$ is the primitive cube root of unity or ... | Gerry Myerson | 8,269 | <p>The proof takes 5 pages in Hardy and Wright, An Introduction to the Theory of Numbers (pages 248 to 253 in the 6th edition). No doubt it can be found in many other intro Number Theory texts, as well as on the web. </p>
|
290,229 | <p>My textbook says that the product of two ideals $I$ and $J$ is the set of all finite sums of elements of the form $ab$ with $a \in I$ and $b \in J$. What does this mean exactly? Can you give examples?</p>
| ferson2020 | 59,689 | <p>This means that the product $IJ$ is the set of all sums $a_1b_1 + a_2b_2 + ... + a_nb_n$ where $a_1, a_2, ..., a_n \in I$, $b_1, b_2, ..., b_n \in J$.</p>
|
2,087,596 | <p>Find the discontinuity at $f(2)$ of the function $f(x)=\dfrac{x^2-3x+2}{{x^2}+x-6}$.</p>
<p>I am confused. I do not understand that is there discontinuity at $2$ but it has discontinuity at $x=-2$. can you explain it please? For my point of view, there is no discontinuity at $2$ because after factorization I get $f... | Harsh Kumar | 395,886 | <p>Since the anti-log is the inverse of the log: $${antilog}_a(y)=x$$ Taking log both the sides: $$\log_a(x)=\log_a({antilog}_a(y))$$ The $\log$ and $anti\log$ will cross each other $$y=\log_a(x)$$ Hence, $$a^y=x$$</p>
|
4,016,918 | <p><span class="math-container">$$_2F_1(\frac{1}{2},-\frac{1}{2};\frac{1}{2}; \sin^{2}(x))=\cos(x)$$</span></p>
<p>I plug these values into the definition of the hypergeometric function:</p>
<p><span class="math-container">$$_2F_1(a,b,c,x)=\displaystyle\sum_{n=0}^{+\infty}\dfrac{a^{\overline{n}}b^{\overline{n}}}{c^{\ov... | user | 293,846 | <p>First of all we can cancel one parameter:
<span class="math-container">$$\mbox{$_2$}F_1(a,b;a; x)=\mbox{$_1$}F_0(b;;x)$$</span></p>
<p>Further:
<span class="math-container">$$
\mbox{$_1$}F_0\left(-\frac12;;x\right)=\sum_{n\ge0}\frac{\left(-\frac12\right)^\overline{n}}{n!}x^n=\sum_{n\ge0}\binom{\frac12}n(-x)^n=\sqrt{... |
3,314,667 | <h1>Problem</h1>
<p>Given sets <span class="math-container">$\mathcal A$</span>, <span class="math-container">$\mathcal B$</span>, <span class="math-container">$\mathcal Y$</span>, let <span class="math-container">$\mathcal X$</span> be a set with the following properties:</p>
<ul>
<li><span class="math-container">$\... | José Carlos Santos | 446,262 | <p>The answer is negative: If <span class="math-container">$G$</span> is an Abelian group and <span class="math-container">$H$</span> is a subgroup of <span class="math-container">$G$</span>, then <span class="math-container">$H$</span> is Abelian too. In fact, since<span class="math-container">$$(\forall g_1,g_2\in G)... |
328,811 | <p>I already asked a question (<a href="https://math.stackexchange.com/questions/328680/order-of-operations-in-rotation-matrix-notation">Order of operations in rotation matrix notation.</a>) about the order in which a particular equation is "processed" and now I need to generalise that and learn the rules of math notat... | Ross Millikan | 1,827 | <p>There are many conventions here which need to be learned. Parentheses override anything. Horizontal fraction bars (but not diagonal ones) come with parentheses around the stuff above and below them. Then exponents come before multiplication/division, which come before addition/subtraction. We just got (again) th... |
596,598 | <p>A graph with three vertices has a beta index no greater than 1. A beta index of a graph is the ratio of number of edges to the number of vertices. </p>
<p>The answer key says true but I think it's false. If loops are allowed and vertex $A$ connects to vertex $A$ then there could be 6 edges in a graph and 3 vertices... | David Holden | 79,543 | <p>the pre-image of any nonempty open set in $Y$ must be one of $[0,1), (0,1)$ or $(0,1]$, since all of these are open in the usual topology for the closed unit interval $I$, the pre-image of any open set in $Y$ is open in $I$. but this is the condition for continuity.</p>
<p>(a few remarks added in response to commen... |
1,089,593 | <p>How to solve $\dfrac{dy}{dx}=\cos(x-y)$ ? How do I separate x and y here ?</p>
<p>Please advise.</p>
| shola Otitoju | 203,830 | <p>using $v=x-y$</p>
<p>$\frac{dv}{dx}=1-\frac{dy}{dx}$</p>
<p>Therefore:
$$\frac{dy}{dx}=\cos{(x-y)}v\equiv1-\frac{dv}{dx}=\cos{v}$$ By rearranging, we have:
$$\frac{dv}{1-\cos{v}}=dx$$
therefore:$$\int\frac{dv}{1-\cos{v}}=x+c$$
$$\int\frac{1+\cos{v}}{(1-\cos{v})(1+\cos{v})}dv=x+c$$
$$\int\frac{1+\cos{v}}{\sin^2{v}}... |
31,701 | <p>How do I write a chevron/circum (^) in MathJax? A backslash doesn't work as an escape character.</p>
<p>(Specific context: using <em>x</em>^<em>y</em> to mean XOR as in <span class="math-container">$x\oplus y$</span>. It's not my choice of notation so don't tell me that I can just use a different symbol.)</p>
| Asaf Karagila | 622 | <p>I'd imagine <span class="math-container">$\land$</span> would be the right symbol here as an operator, so <code>\land</code> (logical and) or <code>\wedge</code> will produce <span class="math-container">$\land$</span> (and <span class="math-container">$\wedge$</span>).</p>
<p>But it might not be the same as <span ... |
2,148,976 | <p>I currently work on a problem where I consider a model which requires calculation of the term</p>
<p>$$\frac{1}{U'(0)}$$</p>
<p>with $U'(x)=x^{-0.5}$.</p>
<p>Therefore $U'(0)$ is undefined.</p>
<p>However, $\frac{1}{U'(x)}=x^{0.5}$ is defined for $x=0$ so I get</p>
<p>$$\frac{1}{U'(0)}=0^{0.5}=0$$</p>
<p>Is th... | Community | -1 | <p>No. The domain of $1/U'(x)$ is everything in the domain of $U'(x)$ such that $U'(x) \ne 0$. But $0$ is not in the domain of $U'(x)$.</p>
<p>While it's true that $1/U'(x) = x^{0.5}$, the problem is actually <em>getting</em> to $x^{0.5}$. Consider:
$$\frac1{U'(x)} = \frac1{\frac1{x^{0.5}}} = x^{0.5}$$</p>
<p>That... |
492,408 | <p>solve recurrence relation $a_n = 6 a_{n–1} – 9 a_{n–2}$,
where $a_0 = 1$ and $a_1 = 6$ and Verify, using Principle of Mathematical
Induction, that $a_n = 3^n + n 3^n$.</p>
<p>ans: i have done so far...<br>
put $a_n=b_n$<br>
$a_n-6a_{n-1}+9a_{n-2}=0$<br>
$b_n-6b_{n-2}+9a_{n-2}=0$<br>
$b^2-6b+9=0$, $b=3,3$</p>
<p>ge... | user71352 | 71,352 | <p>If $n=0$ then $a_{0}=1$ and $3^{0}+0=1$ so the formula agrees here.</p>
<p>$a_{1}=6$ by definition and $3^{1}+1\cdot3^{1}=6$ so it is true for $n=1$. Assume this is true for all $k\le n$. We now show this for $n+1$.</p>
<p>$a_{n+1}=6a_{n}-9a_{n-1}=6\big(3^{n}+n3^{n}\big)-9\big(3^{n-1}+(n-1)3^{n-1}\big)$</p>
<p>$=... |
4,075,530 | <p>Let <span class="math-container">$S$</span> be a closed orientable surface. For any positive integer <span class="math-container">$n$</span>, is there a connected <span class="math-container">$n$</span>-sheeted covering space of <span class="math-container">$S$</span>? This is certainly not true if <span class="math... | Connor Malin | 574,354 | <p>Since the first homology is the abelianization of the fundamental group, and <span class="math-container">$H_1(S)=\mathbb{Z}^{2g}$</span>, where <span class="math-container">$g$</span> is the genus, we see that we have a surjective composition <span class="math-container">$\pi_1(S) \rightarrow H_1(S) \rightarrow \ma... |
255,652 | <p>I came across a problem that says:</p>
<p>Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function. If $|f'|$ is bounded, then which of the following option(s) is/are true?</p>
<blockquote>
<p>(a) The function $f$ is bounded.<br>
(b) The limit $\lim_{x\to\infty}f(x)$ exists.<br>
(c) The function $f$ is unifor... | dtldarek | 26,306 | <p><strong>Hints:</strong></p>
<ul>
<li><p>$f(x) = x$,</p></li>
<li><p>$f(x) = \sin(x)$,</p></li>
<li><p>$f$ is Lipschitz continuous $\Rightarrow$ $f$ is uniformly continous,</p></li>
<li><p>$f(x) = 0$.</p></li>
</ul>
|
3,979,912 | <p>Let <span class="math-container">$f:[0,\infty)\rightarrow\mathbb{R}$</span> be a function with a "blow up" in finite time i.e. <span class="math-container">$$\limsup\limits_{t\uparrow T_{max}}|f(t)|=\infty.$$</span>
I don't unterstand the difference between lim and lim sup in this case. How would the defin... | peek-a-boo | 568,204 | <p>Here's the general definition:</p>
<blockquote>
<p>Let <span class="math-container">$X$</span> be a topological space, <span class="math-container">$E\subset X$</span>, <span class="math-container">$a\in\overline{E}$</span> (the topological closure) and <span class="math-container">$f:E\to [-\infty,\infty]$</span> a... |
1,203,799 | <p>Let $m,n,k$ be nonnegative integers. How might I go about evaluating the following integral?</p>
<p>$$ \int_{-\infty}^\infty \left( \frac{\mathrm{d}^m}{\mathrm{d}x^m} e^{-x^2} \right) \left( \frac{\mathrm{d}^n}{\mathrm{d}x^n} e^{-x^2} \right) x^k e^{x^2} \mathrm{d}x $$</p>
| Claude Leibovici | 82,404 | <p>I do not see how a closed form could be found for the integral.</p>
<p>What you can notice is that each derivative term write as a polynomial $$\frac{{d}^m}{{d}x^m} \Big(e^{-x^2}\Big)=P_m(x)~~ e^{-x^2} $$ which makes $$\frac{{d}^m}{{d}x^m} \Big(e^{-x^2}\Big)~~\frac{{d}^n}{{d}x^n} \Big(e^{-x^2}\Big)~~x^k~~e^{x^2}=P_... |
2,035,454 | <p>I am an upcoming year $12$ student, school holidays are coming up in a few days and I've realised I'm probably going to be extremely bored. So I'm looking for some suggestions.</p>
<p>I want a challenge, some mathematics that I can attempt to learn/master. Obviously nothing impossible, but mathematics is my number ... | Vidyanshu Mishra | 363,566 | <p>If you are really seeking for recomandation I can suggest you some books:</p>
<p>$1$.<a href="https://www.google.co.in/url?sa=t&source=web&rct=j&url=http://www-bcf.usc.edu/~lototsky/PiMuEp/PutnamAndBeyond-Andreescu.pdf&ved=0ahUKEwiwgPP55M7QAhULpI8KHbtDDlYQFghXMAg&usg=AFQjCNEAZ8A2q_MnBf7bhnNUtJs8... |
2,035,454 | <p>I am an upcoming year $12$ student, school holidays are coming up in a few days and I've realised I'm probably going to be extremely bored. So I'm looking for some suggestions.</p>
<p>I want a challenge, some mathematics that I can attempt to learn/master. Obviously nothing impossible, but mathematics is my number ... | Community | -1 | <p>1) Establish a formula for the sum of the first $n$ integers. Then for the sum of the first $n$ perfect squares ($n^2$). Then cubes... Go as far as you can.</p>
<p>2) Given $n$ points $(x_i,y_i)$, find a polynomial such that $p(x_i)=y_i$ for all $i$. Try with $n=1$, then $n=2$, then $3$... and try to find a general... |
2,035,454 | <p>I am an upcoming year $12$ student, school holidays are coming up in a few days and I've realised I'm probably going to be extremely bored. So I'm looking for some suggestions.</p>
<p>I want a challenge, some mathematics that I can attempt to learn/master. Obviously nothing impossible, but mathematics is my number ... | John Gowers | 26,267 | <p>When I was exactly your age I spent the summer working out how to solve cubic equations:</p>
<p>$$
ax^3 + bx^2 + cx +d = 0
$$</p>
|
2,035,454 | <p>I am an upcoming year $12$ student, school holidays are coming up in a few days and I've realised I'm probably going to be extremely bored. So I'm looking for some suggestions.</p>
<p>I want a challenge, some mathematics that I can attempt to learn/master. Obviously nothing impossible, but mathematics is my number ... | Marko Riedel | 44,883 | <p>Consider learning the use of the Burnside lemma, the symbolic factorization of permutations into cycles and the computation of cycle indices of common permutation groups (e.g. cyclic, dihedral) and how to apply them to edge / face / vertex colorings of polyhedra, then move on to the Polya Enumeration Theorem. The te... |
1,130,876 | <p>If the derivatives $f'(x_0)$ and $g'(x_0)$ exist for the functions $f, g: (x_0 - d, x_0 + d)\to\mathbb{R}$, then for $g(x_0) \neq 0$, $\frac{f}{g}$ is also differentiable in $x_0$ and the following applies:</p>
<p>$(\frac{f}{g})'(x_0) = \frac{f'(x_0)g(x_0) - f(x_0)g'(x_0)}{(g(x_0))^2}$</p>
<p>How can I prove this ... | Ben Grossmann | 81,360 | <p>Here's a way of proving this using logarithmic differentiation:
$$
y(x) = f(x)/g(x)\\
\ln(y(x)) = \ln[f(x)/g(x)] = \ln(f(x)) - \ln(g(x))\\
\frac{1}{y(x)} y'(x) = \frac{1}{f(x)}f'(x) - \frac 1{g(x)}g'(x)\\
y'(x) = \left[\frac{f'(x)}{f(x)} - \frac {g'(x)}{g(x)}\right]y(x)\\
y'(x) = \left[\frac{f'(x)}{f(x)} - \frac {g'... |
2,047,579 | <p>Today I was thinking about how to generalize the derivative of $f(x) = x^n$. Of course, we know the following:</p>
<p>$$\frac{d}{dx}x^n = n\cdot x^{n-1}$$</p>
<p>But how can we show that the power rule behaves this way? I took to the limit process and came up with this:</p>
<p>$$\frac{d}{dx}x^n = \lim_{h\to 0}\fr... | user361424 | 361,424 | <p>$$\frac{d}{dx}x^n = \lim_{h\to 0}\frac{(x+h)^n-x^n}{h} = \lim_{h\to 0}\left(\frac{x^n((x+h)^n-x^n)}{x^n\log\left(\frac{(x+h)^n}{x^n}\right)}\cdot\frac{nx\log\left(\frac{x+h}{x}\right)}{xh}\right) = \lim_{h\to0}\left(\frac{x^n}{\log\left(1 + \frac{(x+h)^n-x^n}{x^n}\right)^\frac{x^n}{(x+h)^n-x^n}}\cdot\frac{n\log\left... |
2,047,579 | <p>Today I was thinking about how to generalize the derivative of $f(x) = x^n$. Of course, we know the following:</p>
<p>$$\frac{d}{dx}x^n = n\cdot x^{n-1}$$</p>
<p>But how can we show that the power rule behaves this way? I took to the limit process and came up with this:</p>
<p>$$\frac{d}{dx}x^n = \lim_{h\to 0}\fr... | epi163sqrt | 132,007 | <p>A variation which might be useful</p>
<blockquote>
<p>We obtain</p>
<p>\begin{align*}
\frac{d}{dx}x^n&=\lim_{h\rightarrow 0}\frac{(x+h)^n-x^n}{h}\\
&=\lim_{h\rightarrow 0}\frac{1}{h}\left(\sum_{i=0}^n\binom{n}{i}x^{n-i}h^i-x^n\right)\tag{1}\\
&=\lim_{h\rightarrow 0}\sum_{i=1}^n\binom{n}{i}x^{n-i}... |
4,047,049 | <p>Assuming <span class="math-container">$\log x$</span> is normally distributed. How do I get the distribution of <span class="math-container">$x$</span>? Also how can I get the standard deviation of <span class="math-container">$x$</span> assuming I know some asymmetric error on <span class="math-container">$x$</span... | tommik | 791,458 | <p>This is what your textbook means:</p>
<p>Let's throw 2 fair dice:</p>
<ul>
<li><p>Event <span class="math-container">$E$</span>: the sum of the <em>"2 dice's face up"</em> is even</p>
</li>
<li><p>Event <span class="math-container">$F$</span>: the sum <em>"2 dice's face up"</em> is 2,3 or 4</p>
<... |
87,636 | <p>I'm following the book <em>Measure and Integral</em> of Richard L. Wheeden and Antoni Zygmund. This is problem 4 of chapter 8.</p>
<p>Consider <span class="math-container">$E\subseteq \mathbb{R}^n$</span> a measurable set. In the following all the integrals are taken over <span class="math-container">$E$</span>, <sp... | Community | -1 | <p>I'll add some details on the Minkowski inequality (this question is the canonical Math.SE reference for the equality cases, but almost all of it concerns Hölder's inequality). </p>
<p>The standard proof of the Minkowski inequality begins with
<span class="math-container">$$
\begin{align*}
\int |f+g|^p &\l... |
1,432,729 | <p>I know that $\pi \approx \sqrt{10}$, but that only gives one decimal place correct. I also found an algebraic number approximation that gives ten places but it's so cumbersome it's just much easier to just memorize those ten places.</p>
<p>What's a good approximation to $\pi$ as an irrational algebraic number (or a... | Mr. Brooks | 162,538 | <p>$\root 10 \of {93648}$ is marginally better than $\sqrt{10}$.</p>
<p>But one of the comments has a much better answer, with degree of just $4$.</p>
|
1,432,729 | <p>I know that $\pi \approx \sqrt{10}$, but that only gives one decimal place correct. I also found an algebraic number approximation that gives ten places but it's so cumbersome it's just much easier to just memorize those ten places.</p>
<p>What's a good approximation to $\pi$ as an irrational algebraic number (or a... | Raymond Manzoni | 21,783 | <p>(<em>expanding my comments</em>)</p>
<p>Let's start with the fraction $\;\dfrac{355}{113}\,$ easy to remember with something like :<br>
"doubling the odds to be near the pi" (whatever this may mean...).</p>
<p>It is easy to find starting with the <a href="https://math.stackexchange.com/a/180084/21783">continued fr... |
472,684 | <p>Consider the random walk $X_0, X_1, X_2, \ldots$ on state space $S=\{0,1,\ldots,n\}$ with absorbing states $A=\{0,n\}$, and with $P(i,i+1)=p$ and $P(i,i-1)=q$ for all $i \in S \setminus A$, where $p+q=1$ and $p,q>0$.</p>
<p>Let $T$ denote the number of steps until the walk is absorbed in either $0$ or $n$.</p>
... | steve newman | 382,635 | <p>The answer given only provides an explicit solution for the case p=q=.5</p>
<p>Here is a general solution to the above problem. You can plug in the gamblers starting amount, the final amount, the bias of the coin and get a numerical answer for exactly how many tosses on average it takes for the gambler to win th... |
2,284,376 | <p>Let H be a normal subgroup of G of index 4 show that there are either exactly 3 or exactly 5 subgroups of G containing H( including G and H themselves )</p>
<p>Where do we start from i am completly got no idea..</p>
| Asinomás | 33,907 | <p>Use the fourth isomorphism theorem. The groups containing $H$ are in correspondence with the subgroups of $G/H$.</p>
<p>Hence the number subgroups containing $H$ is equal to the number of subgroups of a group $K$ of order $4$. there are only two such groups.</p>
<p>The first is $\mathbb Z_4$ which has $3$ subgroup... |
3,729,197 | <p>For this problem suppose that the <span class="math-container">$x_i$</span>'s must be non-negative integers, i.e., <span class="math-container">$x_i∈{0,1,2,⋯}$</span> for <span class="math-container">$i=1,2,3$</span>. How many distinct solutions does the following equation have such that at least one of the <span cl... | Favst | 742,787 | <p>By complementary counting and generating functions, this is the number of <a href="https://en.wikipedia.org/wiki/Composition_(combinatorics)" rel="nofollow noreferrer">weak compositions</a> of <span class="math-container">$100$</span> into <span class="math-container">$3$</span> parts (there is a formula in the link... |
3,729,197 | <p>For this problem suppose that the <span class="math-container">$x_i$</span>'s must be non-negative integers, i.e., <span class="math-container">$x_i∈{0,1,2,⋯}$</span> for <span class="math-container">$i=1,2,3$</span>. How many distinct solutions does the following equation have such that at least one of the <span cl... | Mike Earnest | 177,399 | <p>There is no problem, your method is correct. Here is another method which gives the same answer.</p>
<p>The number of solutions where <span class="math-container">$x_1\ge 41$</span> is the same as the number of solutions to <span class="math-container">$x_1+x_2+x_3=59$</span>, which is <span class="math-container">$... |
702,675 | <p>p = False, q = True and r = False.
Is $¬(p∨q)∧(¬p∨r)$ = false?</p>
<p><strong>My reasoning:</strong></p>
<p>$$(p∨q)=T \text{ as it is (F or T)}$$</p>
<p>but its the negation so $¬(p∨q)=F$? </p>
<p>Then, $(¬p∨r)$ as p is F but its the negation again so its T and r=F.
So its $(T\lor F)$ in this case only one has... | amWhy | 9,003 | <p>Note that simply having successfully worked out the fact that $\lnot(p \lor q)$ is false, you are effectively done. </p>
<p>Why? Because in any conjunction, say $A\land B$, (here, $A, B$ may be any proposition),
by the truth-table definition of conjunction, $A \land B$ is true if and only if <strong>both</strong> $... |
215,846 | <p>Are proofs by induction limited to cases where there is an explicit dependance on a integer, like sums? I cannot grasp the idea of induction being a proof in less explicit cases. What if you have a function that suddenly changes behavior? If a function is positive up to some limit couldn't I prove by induction that ... | Austin Mohr | 11,245 | <p>Here is another example from graph theory.</p>
<p><strong>Claim:</strong> There is a $4$-regular graph on $n$ vertices for all $n \geq 5$.</p>
<p><strong>Base Case:</strong> The complete graph on $5$ vertices is four regular.</p>
<p><strong>Inductive Step:</strong> <em>Assume</em> there is a $4$-regular graph $G$... |
2,770,523 | <p>I'm struggling with the following problem,</p>
<p>Let $g(z)=\sum^k_1 m_{\alpha}(z-z_{\alpha})^{-1}$. Show that if $g(z)=0$, then $z_1,\cdots,z_k$ cannot all lie on the same sie of a straight line through $z$.</p>
<p><strong>What I did:</strong></p>
<p>The book says that I should use the fact that if $z_1,\cdots,z... | Community | -1 | <p>Far from true. $3+5=8,\ 8+7=15{}{}{}{}$</p>
|
2,633,975 | <p>Let $u\in C^2(\Omega)\cap C^0(\overline{\Omega})$, $\Omega\subset\mathbb{R}^n$ open and bounded and $\lambda>0$ sufficiently small so that $2\lambda u<1$. Define $w$ by $w\leq\frac{1}{\lambda}$ and
$$
u(x)=w(x)-\frac{\lambda}{2}w(x)^2.
$$
I have to prove that $w\rightarrow u$ uniformly as $\lambda\rightarrow0$... | frank | 506,630 | <p>Find the general solution of the linear constraint. It will be the least-squares solution plus a contribution from the null space
$$\eqalign{
Bx &= v \cr
B &= vx^+ + C(I-xx^+) \cr
}$$ where $x^+$ is the pseudoinverse of $x$ and $C$ is an arbitrary matrix.</p>
<p>Substituting this expression for $B$ yiel... |
4,341,356 | <p>A French-suited cards pack consist of <span class="math-container">$52$</span> cards where <span class="math-container">$13$</span> are clovers. <span class="math-container">$4$</span> players play a game where every player has <span class="math-container">$13$</span> cards in his/her hands (, in other words the ful... | Mastrem | 253,433 | <p>Using the identity <span class="math-container">$\sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}$</span>, we find that
<span class="math-container">$$
\begin{align*}
\sqrt[n]{\frac{20}{2^{2n+4}+2^{2n+2}}} &= \sqrt[n]{\frac1{2^n}}\cdot\sqrt[n]{\frac{20}{2^4+2^2}}\\
&= \sqrt[n]{\frac1{4^n}}\cdot\sqrt[n]{\frac{20}{16+4}}\\... |
2,694,875 | <p>Let A= {x is reals:x>0} and define a relation on A by x relation y
If xy=0 for x,y in A .</p>
<p>I was wondering if this is reflexive relation. So far I thought
If x=1 and y= 0, then 1*0=0 and
0*1 is also =0. It can be reflexive not sure if I am doing it right by using this counter example . How do I prove wha... | Community | -1 | <p>Let us call R the relation you are considering is. </p>
<p>By definition, </p>
<p>R = { (x,y) belonging to R² | x . y = 0} </p>
<p><strong>Let us think of this relation as a machine that selects pairs of real numbers in the plane R².</strong> In other words, a machine that selects points in the plane R². </p>
<p... |
1,298,970 | <p>I am trying to wrap my head around some integration applications.</p>
<p>I went through the exercise of integrating the circumference of a circle, $2*\pi*r$, to get the area of a circle. I simply used the power rule and get $\pi*r^2$.</p>
<p>However when I extend this to a square, I calculate the length around a... | Barry Cipra | 86,747 | <p>The length of a side, $L$, is analogous to the <em>diameter</em> of a circle, not its radius. So the appropriate variable to use is $\ell=L/2$, in which case the perimeter is $8\ell$, which integrates to $4\ell^2$, and that <em>is</em> correct, since $4\ell^2=(2\ell)^2=L^2$.</p>
<p>(Remark: I posted this before r... |
62,771 | <p>I was just going through a past exam paper for my intro graphs module and the following question came up, and I can't find any notes on it:</p>
<p>Let G = (V,E) be a simple graph. Show that:</p>
<p>$2|E| \leq |V|^2 - |V|$</p>
<p>any ideas?</p>
| Stephan F. Kroneck | 13,462 | <p>Cp. Bondy/Murty "Graph Theory with Applications", Ex. 1.1.3 (so it must be easy), available online ... but you do know what a simple graph is ? (Probably the question does not really belong here, being of such a trivial nature ?) Kind regards, Stephan.</p>
|
3,096,741 | <p><span class="math-container">$X, Y$</span> are two independent <span class="math-container">$\mathcal{N}(0,1)$</span> random variables </p>
<p>this question was a follow up question of this <a href="https://math.stackexchange.com/questions/3096530/show-that-e-fracx22-in-l1-iff-exy-in-l1-iff-exy-in-l1">one</a></p>
... | mathex | 638,227 | <p><span class="math-container">$1+E[XY|X]\leq E[e^{XY}|X]$</span> a.s and <span class="math-container">$E[XY|X]=XE[Y]=0$</span> by independence!!
So we have the desired inequality without any computing</p>
|
2,408,954 | <p>In this webpage, <a href="https://plus.maths.org/content/friends-and-strangers" rel="nofollow noreferrer">https://plus.maths.org/content/friends-and-strangers</a> under the section, FINDING $R(3,4)$,</p>
<p>The author assumes that 10 people/points are necessary and takes out one point, say A. This A is connected to... | Théophile | 26,091 | <p>If there are fewer than $6$ red edges <em>and</em> fewer than $4$ blue edges, then the total number of edges incident to $A$ is strictly less than $5+3=8$, which is a contradiction.</p>
<p>We could prove similar claims by the same reasoning ("there must be at least $5$ red edges or at least $5$ blues edges", or "th... |
1,648,011 | <p>I don't want a proof. I just want verification or correction of the proof I supplied:</p>
<hr>
<p>We start with the fact that between any two real numbers there is a rational number. There is an infinite amount of real numbers between any two real numbers. Hence any real interval can accommodate the whole set of r... | Michael Hardy | 11,667 | <p>Certainly you can find a mapping from $\mathbb N$ into a specified interval $(a,b)$. And in fact you can find a one-to-one correspondence between $\mathbb N$ and the set $(a,b)\cap\mathbb Q$ of all rational numbers in $(a,b)$. But that doesn't prove you cannot also find a mapping from $\mathbb N$ whose image conta... |
1,648,011 | <p>I don't want a proof. I just want verification or correction of the proof I supplied:</p>
<hr>
<p>We start with the fact that between any two real numbers there is a rational number. There is an infinite amount of real numbers between any two real numbers. Hence any real interval can accommodate the whole set of r... | fleablood | 280,126 | <p>"We start with the fact that between any two real numbers there is a rational number."</p>
<p>True. But not just one; there are an infinite number of rational numbers between two real numbers.</p>
<p>"There is an infinite amount of real numbers between any two real numbers."</p>
<p>True. There are an infinite a... |
211,865 | <p>The given matrix is </p>
<p>$$
\begin{pmatrix}
2 & 2 & 2 \\
2 & 2 & 2 \\
2 & 2 & 2 \\
\end{pmatrix}
$$</p>
<p>so, how could i find the eigenvalues and eigenvector without computation?
Thank you</p>
| bartgol | 33,868 | <p>For the eigenvalues, you can look at the matrix and extract some quick informations.</p>
<p>Notice that the matrix has rank one (all columns are the same), hence zero is an eigenvalue with algebraic multiplicity two. For the third eigenvalue, use the fact that the trace of the matrix equals the sum of all its eigen... |
2,943,999 | <p>If <span class="math-container">$a,b,c \in \mathbb{R+, }$</span> Then Prove that <span class="math-container">$$\frac{a^2}{3^3}+\frac{b^2}{4^3}+\frac{c^2}{5^3} \ge \frac{(a+b+c)^2}{6^3}$$</span></p>
<p>My try:</p>
<p>Consider <span class="math-container">$$P=\frac{a}{3\sqrt{3}}+\frac{b}{4\sqrt{4}}+\frac{c}{5\sqrt{... | Kavi Rama Murthy | 142,385 | <p>If <span class="math-container">$X$</span> has uniform distribution on <span class="math-container">$(0,1)$</span>, <span class="math-container">$X_n=X/n$</span> and <span class="math-container">$Y=-X/n$</span> then <span class="math-container">$\int |f_n-g_n|$</span> doe not tend to <span class="math-container">$0... |
62,967 | <p><code>CoefficientRules</code> acts like the following.</p>
<pre><code>In[1]:= CoefficientRules[2 x^3 + 3 x^2 y + 4 x y^2 - 5 x + 1]
Out[1]= {{3, 0} -> 2, {2, 1} -> 3, {1, 2} -> 4, {1, 0} -> -5, {0, 0} -> 1}
</code></pre>
<p>My question is how one can "extend" this function so that it may allow the n... | Fred Simons | 20,253 | <p>Nice to see how much one can do with Groebner bases. This a more elementary solution:</p>
<pre><code>function[pol_] := Module[{vars, v, h, aux},
vars = Variables[pol];
v = Length[vars];
h /: h[arg_]^p_ := h [p *arg];
h /: h[arg1_] h[arg2_] := h[arg1 + arg2];
(List @@ pol) /.
Table[vars[[n]] -> h[U... |
123,494 | <p>In <em>Mathematica</em>, almost everything is notebook: your "Untitled-1.nb" is a notebook, Help documentation are a series of notebooks, even those windows helping you to draw things or format your notebooks are, themselves, notebooks.</p>
<p>But I occasionally find some exceptions and I want to know what are they... | Alexey Popkov | 280 | <p>The "Preferences" and "Enter Activation Key" windows are Palette Notebooks located in the directory</p>
<pre><code>FileNameJoin[{$InstallationDirectory, "SystemFiles", "FrontEnd", "SystemResources"}]
</code></pre>
<p>(look at "Preferences.nb" and "ActivationDialog.nb").</p>
<p>But (according to an old MathGroup p... |
68,748 | <p>Among the theorems of early geometric model theory there is one by Lachlan stating that every totally categorical structure with a trivial pregeometry is intrepretable in a dense linear order.</p>
<p>That suggests in particular that there are totally categorical structures with a trivial pregeometry that are not in... | John Baldwin | 31,339 | <p>One example is the theory of a unary functions satisfying $f^2(x) =x$.</p>
<p>As pointed out below by Eric, (with the explicit example of someone else) I was thinking only of 1-dimensional interpretations.</p>
|
1,328,799 | <p>I'm looking for modern graduate-level non-English-language differential geometry textbooks. I'm interested in original works by non-English-language speakers in their native languages, not translations into non-English languages. I'm also not interested in non-English language textbooks which are available in Englis... | Georges Elencwajg | 3,217 | <p><a href="http://www.amazon.fr/Introduction-vari%C3%A9t%C3%A9s-diff%C3%A9rentielles-Jacques-Lafontaine/dp/2759805727" rel="nofollow noreferrer"><strong>Lafontaine: Introduction aux variétés différentielles</strong></a><br>
A rigorous, classical treatise. The second edition is much more complete. </p>
<p><a href="ht... |
4,021,746 | <p>Euclidean distance is not linear in high dimensions. However, in multiple regression the idea is to minimize square distances from data points to a hyperplane.</p>
<p>Other data analysis techniques have been considered problematic for their reliance on Euclidean distances (nearest neighbors), and dimensionality redu... | fleablood | 280,126 | <p>The remaining three colors with be a <em>different</em> set of three colors depending on the color you pick first.</p>
<p>If you pick <span class="math-container">$R$</span> first the remaining three colors are <span class="math-container">$B,G,O$</span> for three signals: <span class="math-container">$RB,RG,RO$</sp... |
4,427,446 | <p>There is a binary number of length <span class="math-container">$N$</span> which consists of a consecutive series of 1s. For example, if <span class="math-container">$N=5$</span> the number is <span class="math-container">$11111$</span>. How many ways are there to intervene on this number (i.e., replacing <span clas... | user2661923 | 464,411 | <blockquote>
<p>Degenerate sequences consisting of a single 1 are also allowed.</p>
</blockquote>
<p>All of the analysis below assumes that such degenerate sequences have been disallowed. Given the stipulation above, the adjustment is simply to
add <span class="math-container">$(n)$</span> to the computation below.</p... |
4,618,932 | <p>The <a href="https://en.wikipedia.org/wiki/Sequence" rel="nofollow noreferrer">wikipedia article</a> for a sequence defines it as such:</p>
<blockquote>
<p>In mathematics, a sequence is an <strong>enumerated</strong> collection of objects in which repetitions are allowed and order matters. [...] The number of elemen... | FShrike | 815,585 | <p>Sequences in a set <span class="math-container">$X$</span> are functions <span class="math-container">$\Bbb N\to X$</span>. The image therefore cannot have cardinality greater than that of <span class="math-container">$\Bbb N$</span>, so sequences are always countable (or finite: the meaning of "countable"... |
2,654,507 | <blockquote>
<p>Find the residue of $\dfrac{z^2}{(z-1)(z-2)(z-3)}$ at $\infty$.</p>
</blockquote>
<p>We know that $\text{Res} (f)_\infty +\text{Res} (f)_{\text{ at other poles}}=0$</p>
<p>Now $f$ has poles at $1,2,3$ of order $1$.</p>
<p>Sum of residues of $f$ at $1,2,3=\dfrac{1}{2}+(-4)+\dfrac{9}{2}=1\implies \t... | José Carlos Santos | 446,262 | <p>Note that\begin{multline}16\cos^6(t)+16\sin^6(t)+48\cos^2(t)\sin^2(t)=\\=16\bigl(\cos^2(t)+\sin^2(t)\bigr)^3-48\bigl(\cos^2(t)+\sin^2(t)\bigr)\cos^2(t)\sin^2(t)+48\cos^2(t)\sin^2(t)=\\=16,\end{multline}since $\cos^2(t)+\sin^2(t)=1$.</p>
|
3,547,816 | <p>I have such an equation where I need to find x</p>
<p><span class="math-container">$$y =\frac{1}{x+1}$$</span></p>
<p>And I know that answer is
<span class="math-container">$$x = \frac{1}{y}-1$$</span></p>
<p>But I did not understand how to get it?</p>
| Michael Hoppe | 93,935 | <p>Take reciprocals an subtract <span class="math-container">$1$</span>.</p>
|
98,402 | <blockquote>
<p><strong>Theorem :</strong></p>
<p>If an odd number <span class="math-container">$n$</span> , <span class="math-container">$n > 1$</span> can be uniquely expressed as : <span class="math-container">$n=x^2-y^2$</span> ; <span class="math-container">$x,y \in \mathbb{Z}^{*}$</span> then</p>
<p><span clas... | Peđa | 15,660 | <p>Thanks to the Dan's and Bill's useful observations I wrote an improved version of the test so here it is : </p>
<p><img src="https://i.stack.imgur.com/yupRK.jpg" alt="enter image description here"></p>
<p>Number of iterations in worst case is given by :</p>
<p>$\left \lceil \frac{p-p^2_{n+1}}{4 \cdot p_{n+1}} \ri... |
172,271 | <p>First a warm-up. Let $\ V\ $ be an arbitrary set of odd natural numbers. Let $\ S(V)\ $ be the generated multiplicative semi-group. What are the necessary and/or sufficient conditions on $\ V\ $ for the property: $\ \exists_{x\ y\in S(V)}\ y-x=2\ $?</p>
<p>Now real questions, all of them open to me. Let $\ \mathbb ... | Emil Jeřábek | 12,705 | <p>For Q1, take $x,y=A\pm1$, where $A$ is the product of all primes below $a+1$. This also gives a (likely badly suboptimal) exponential bound on $b$ for Q3.</p>
|
489,848 | <p>So, I have the following points: $\left( \begin{matrix} 5 \\ 0 \\ 0 \end{matrix} \right), \left( \begin{matrix} 0 \\ 4 \\ -1 \end{matrix} \right), \left( \begin{matrix} -4 \\ 4 \\ 3 \end{matrix} \right)$ and I need to find the equation of the circle passing through them.</p>
<p>Here is how I solved it, but it wa... | copper.hat | 27,978 | <p>Here is another approach:</p>
<p>Let the points be $x_1,x_2,x_3 \in \mathbb{R}^3$. Let $d = (x_1-x_3) \times (x_2-x_3)$, then $d$ is a normal to the plane containing the $x_i$, let $\alpha = \langle d, x_1 \rangle$, then the $x_i$ all lie on the plane $H = \{ x | \langle d, x \rangle = \alpha \}$.</p>
<p>Let $c$ b... |
1,863,151 | <p>I've seen this exercise in several statistics text, but how they get to the final formula is something that I don't quite get. How do two squared terms suddenly become a binomial term? I've been trying to figure out how to get to the final formula but I don't get anywhere near. Hope you can help me.</p>
<p><a href=... | Rodrigo de Azevedo | 339,790 | <p>$$\begin{array}{rl} n \|\mathrm x - \bar{x} \, 1_n\|_2^2 &= n (\mathrm x^T \mathrm x - 2 \bar{x} 1_n^T \mathrm x + n \bar{x}^2)\\\\ &= n (\mathrm x^T \mathrm x - 2 n \bar{x}^2 + n \bar{x}^2)\\\\ &= n (\mathrm x^T \mathrm x - n \bar{x}^2)\\\\ &= n \|\mathrm x\|_2^2 - (n \bar{x})^2\\\\ &= n \|\math... |
1,337,451 | <p>I know that if $a,b,c$ are in Arithmetic Progression, then $2b=a+c$, but in this case, I am unable to solve for $x$. Hints are appreciated. Thanks.</p>
| ramya msu | 90,617 | <p>Since $2b=a+c$ we have
$2. 2^{x^{2}}=2^{x}+2^{x^{3}}$ </p>
<p>which implies that $2^{x^{2}+1}=2^{x}+2^{x^{3}}$</p>
<p>Now taking log on both sides we get
$x^{2}+1$ log2=log $(2^{x}+2^{x^{3}})$ solving this we can find $x$</p>
|
1,530,118 | <p>I was wondering if anyone could help with this $\epsilon–\delta$ definition of a limit. I have looked it up in my calculus book and online and I just don't understand how to do it.</p>
<p>Prove, using the $\epsilon–\delta$ definition of a limit that</p>
<p>$$\lim_{(x,y)\to(0,0)}\frac{(x^2-y)}{(4x^2+y^2)}$$</p>
| Varun Iyer | 118,690 | <p>You can use taylor series (a lot easier than L'Hopital's):</p>
<p>We have the following limit:
$$\lim_{x\to 0}\frac{\sin x^4-x^4\cos x^4+x^{20}}{x^4(e^{2x}-1-2x^4)}$$
Using taylor series:
$$\lim_{x\to 0}\frac{\sin x^4-x^4\cos x^4+x^{20}}{x^4(e^{2x}-1-2x^4)} = \lim_{x\to 0}\frac{\left(x^4-\frac{(x^4)^3}{3!} + \frac{... |
460,587 | <p>Two cards are drawn without replacement from an ordinary deck, find the probability that the second is a red card, given the first is a red card.</p>
<p>P (2nd Red Card / 1st Red Card) = 13/52 * 12 * 51 = 1/17 - is this correct?</p>
| Nick Peterson | 81,839 | <p>Your answer is not correct, no. (For starters: note that there are 26 red cards in a standard 52-card deck.)</p>
<p>Let $R_1$ be the event that the first card is red, and $R_2$ the event that the second card is red.</p>
<p>If the first card is red, then when you go to draw your second card there are a total of $2... |
3,631,710 | <p>I'm wondering what the best method of sketching a curve is, if you know the coordinates of a point on the line and its intrinsic equation in the form:
<span class="math-container">$$
s=f(\psi)
$$</span>
where s is arc length from the origin to a point on the curve and where <span class="math-container">$$tan(\psi )... | Mathsmerizing | 757,478 | <p>Well, going by the video you posted, I will assume that you wish to explore more competition oriented resources. The one resource for IIT-JEE exam is <a href="https://www.iitmind.com/2018/12/tmh-iitjee-mathematics-pdf.html" rel="nofollow noreferrer">TMH Jee Advanced</a> and this is the book I recommend the most for ... |
3,098 | <p>This is a really newbie question, but it has me confused. Why does this code <strong>work without</strong> <code>// MatrixForm</code> and <strong>doesn't work with</strong> <code>// MatrixForm</code>?</p>
<pre><code>cov = {{0.02, -0.01}, {-0.01, 0.04}} // MatrixForm
W = {w1, w2}; FindMinimum[ W.cov.W, W]
</code></p... | David | 103 | <p><a href="http://reference.wolfram.com/mathematica/ref/MatrixForm.html"><code>MatrixForm</code></a> is a function to prettyprint matrices and cannot be used in computations. Just leave the <a href="http://reference.wolfram.com/mathematica/ref/MatrixForm.html"><code>MatrixForm</code></a> away and you're fine:</p>
<pr... |
3,098 | <p>This is a really newbie question, but it has me confused. Why does this code <strong>work without</strong> <code>// MatrixForm</code> and <strong>doesn't work with</strong> <code>// MatrixForm</code>?</p>
<pre><code>cov = {{0.02, -0.01}, {-0.01, 0.04}} // MatrixForm
W = {w1, w2}; FindMinimum[ W.cov.W, W]
</code></p... | Andrzej Kozlowski | 182 | <p>While the question has been more than answered there are still some things that seem to me worth adding. The first is that, in my opinion, <code>MatrixForm</code> is "essentially" obsolete. If you wish your matrices always look like matrices (in the output) you can set the format type of output cells to <code>Tradit... |
2,144,481 | <p>Let us consider the sequence $(a_n)_{n \ge 1}$ such that
$$a_n=\frac {1}{\sqrt {n^2+1}}+ \frac {1}{\sqrt {n^2+2}} + \dots +\frac {1}{\sqrt {n^2+n}}.$$ Show that for every $k \in \Bbb N, k\gt 0,$ we have $a_n \ge a_k$, for every $n \ge k^2$.</p>
<p>The only method I know is computing the difference $a_{n+1}-a_n$, ... | Community | -1 | <p>HINT</p>
<p>$a_n \ge \frac {n}{\sqrt {n^2+n}}$ and $a_k \le \frac {k}{\sqrt {k^2+1}}$</p>
<p>Can you take it from here?</p>
|
3,294,564 | <blockquote>
<p>There is only one real values of <span class="math-container">$k$</span> for which the quadratic equation <span class="math-container">$kx^2+(k+3)x+k-3=0$</span> has <span class="math-container">$2$</span> positive integer roots. Then the product of these two solutions is</p>
</blockquote>
<p>What i ... | Robert Z | 299,698 | <p>Recall that, for <span class="math-container">$k\not=0$</span>, the product of the roots of the given quadratic equation is the ratio <span class="math-container">$\frac{k-3}{k}=1-\frac{3}{k}$</span> which in your case has to be a positive <strong>integer</strong>.
This means that <span class="math-container">$k\in... |
124,291 | <p>I've been working on this all day long. Here's what I've done until now.The denominator is easy. It's $n^{2n}$. I compute the numerator as follows. </p>
<p>All $n$ bins have at least one ball = $n$ bins must have one of the $2n$ balls each + the remaining $n$ balls are placed in any of the bins in any fashion.</p>
... | Michael Chen | 541 | <p>Consider <a href="https://en.wikipedia.org/wiki/Stars_and_bars_%28probability%29" rel="nofollow">stars and bars</a> as a tool for dealing with these sorts of problems.</p>
|
206,421 | <p>If $4 \tan(\alpha - \beta) = 3 \tan \alpha $, then prove that
$$\tan \beta = \frac{\sin(2 \alpha)}{7 + \cos(2 \alpha)}$$</p>
<p>This is not homework and I've tried everything so I would just like a straight answer thank you in advance. </p>
| Alfred Chern | 42,820 | <p>As $4\tan(\alpha-\beta)=3\tan\alpha$, so $\tan(\alpha-\beta)=\frac{3}{4}\tan\alpha$, then</p>
<p>$$\begin{align*}
\tan\beta&=\tan(\alpha-(\alpha-\beta))\\
&=\frac{\tan\alpha-\tan(\alpha-\beta)}{1+\tan\alpha\tan(\alpha-\beta)}\\
&=\frac{\tan\alpha}{4+3\tan^{2}\alpha}\\
&=\frac{\sin\alpha\cos\alpha}{4... |
1,790,032 | <p>Say you have a number $x^{\sqrt 2}$.</p>
<p>Is there any way to represent this number so that there's no root (or irrational) as the exponent (so that it's easier to understand for me)? I just can't wrap my head around this.</p>
<p>I was thinking something like $$x^{2^{1/2}}$$ and extending on that idea, but I don... | John Joy | 140,156 | <p>Perhaps you could consider it as the root of a function composition.</p>
<p>Suppose that $(f\circ f)(x) = x^2$ then what would $f(x)$ equal? </p>
|
94,213 | <p>You want to access a particular smartphone which has a 4-digit numeric pin, entered by tapping the screen. One day you see the owner wipe the screen, unlock the device, and then get distracted and walk off. You rush over, grab the phone but alas it has auto-locked again. The screen is clean except for smudge marks o... | Mark Bennet | 2,906 | <p>This ought to be solvable by thinking about it, so here is a method to work on.</p>
<p>To crack the case of an n digit pin known to contain precisely m distinct digits, compute the number of possibilities using up to m digits (without worrying about whether all the digits are used) and then subtract the number of p... |
136,121 | <p>I would like an efficient algorithm for square root of a positive integer. Is there a reference that compares various square root algorithms?</p>
| Will Jagy | 3,324 | <p>In case what is desired is the floor of the square root of an integer, there were two surprises when i wrote this in C++ for numbers up to a bound, in my case $2^{31} - 1.$ There is no impediment to doing Newton's method with integers. The first trick is that you can get an infinite loop where Newton's method cycles... |
2,110,079 | <p>Is it true that every infinite finitely generated group contains an element of infinite period?</p>
<p>I think it is true, but I can't prove it.</p>
| Rene Schipperus | 149,912 | <p>"This was once a paradox but now time gives it proof". </p>
<p>The simplest example is a consequence of the Golod-Shafarevitch theorem:</p>
<p>Theorem:For each prime $p$ there is an infinite group generated by three elements where each element has finite order a power of $p$.</p>
<p>See Herstein, Noncommutative R... |
36,792 | <p>I need to play with a lot of powers such as <code>10^-3</code>. <code>1E-3</code> does not work for it. Is there any short form for it?</p>
| Joe | 44,710 | <p>I ran into this as well teaching some solubility chemistry (it's logs all the way down). The notation can get in the way, but most students can keep up with 1×10^-5 syntax - just don't forget to wrap it in brackets i.e.:</p>
<pre><code>1*^-10/1*^-5 == (1×10^-10)/(1×10^-5)
(* True *)
</code></pre>
<p>NB: the <code... |
3,269,080 | <p>There are a lot of functions that look wobbly.</p>
<p>For example <span class="math-container">$x^4 + x^3$</span> looks a little wobbly when it gets near the x axis. The function <span class="math-container">$\sin(x)$</span> is extremely wobbly. The function <span class="math-container">$\sin(x) + x$</span> is also... | Adrian Keister | 30,813 | <p>Perhaps you're thinking of <a href="https://en.wikipedia.org/wiki/Total_variation" rel="noreferrer">total variation</a>. You can think of it as "vertical distance traveled". You could, perhaps, over some region, divide the range by the total variation. That is, you could define your "wobbliness" as
<span class="math... |
3,546,801 | <p>I have often heard (both online and in person) people say that "<span class="math-container">$\mathbb{R}^2$</span> can't be totally ordered." I would like to understand this statement. </p>
<p>Of course, on the face of it, this is false: Pick your favorite bijection <span class="math-container">$f:\mathbb{R}^2 \to ... | Henno Brandsma | 4,280 | <p>There is a notion of a GO-space (generalised ordered space), i.e. a topological space <span class="math-container">$(X,\tau)$</span> such that <span class="math-container">$X$</span> has a linear order such that the order topology induced by <span class="math-container">$<$</span> is a subset of <span class="math... |
49,015 | <p>This question might be astoundingly naive, because my understanding of modular forms is so meek. It occurred to me that the reason I was never able to penetrate into the field of modular forms, automorphic forms, the Langland's program and so forth was because my appeal is to things that have the feel of SGA1, and t... | François Brunault | 6,506 | <p>I don't know any published work by Grothendieck specifically on modular forms.</p>
<p>It seems however that Grothendieck has spent some time thinking about moduli spaces of elliptic curves, for example. He has written a very long manuscript "La longue marche à travers la théorie de Galois", which is about what is n... |
49,015 | <p>This question might be astoundingly naive, because my understanding of modular forms is so meek. It occurred to me that the reason I was never able to penetrate into the field of modular forms, automorphic forms, the Langland's program and so forth was because my appeal is to things that have the feel of SGA1, and t... | descenso | 110,305 | <p>Grothendieck was aware (at least) of modular forms and their relations to motives by J P Serre in a letter dated December 31, 1986.</p>
<p><a href="https://i.stack.imgur.com/YgaPe.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/YgaPe.jpg" alt="enter image description here"></a></p>
<p>Grothendie... |
1,641,589 | <p>I have a theorem for the hamming bound or the sphere packing bound.</p>
<p>A q-ary $(n, m, 2e+1)$ code satisfies
$$M \bigg\{ \binom {n}{0} + \binom{n}{1} (q-1)+...+\binom{n}{e}(q-1)^e\bigg\} \leq q^n $$</p>
<p>What is $q^n$?</p>
<p>What is this theorem trying to say?</p>
| Will Orrick | 3,736 | <p>Here $q$ stands for the number of symbols in the alphabet and $q^n$ is the number of $n$-letter words that can be constructed from those symbols. The sum
$$
\binom {n}{0} + \binom{n}{1} (q-1)+...+\binom{n}{e}(q-1)^e
$$
is the volume of a "Hamming ball", that is, it's the number of $n$-letter words that differ in no... |
878,237 | <p>Given N cards where if ith card has number x on its front side then it will have -x on back side and a single operation that can be done only once that is to flip any number of cards in consecutive order only once.</p>
<p>Now we need to flip cards in such a way that sum of number of upper face of cards is maximum.<... | Henno Brandsma | 4,280 | <p>The binomial theorem for $n=7$ says:</p>
<p>$(a + b)^7 = {7 \choose 0}a^0b^7 + {7 \choose 1}a^1b^6 + {7 \choose 2}a^2b^5 \ldots {7 \choose 7}a^7b^0$</p>
<p>Take $a = 3$, $b = 2x^2$. We need $x^{10}$, so we take the term with ${7 \choose 2}$, which is ${7 \choose 2} \cdot 3^2 \cdot (2x^2)^5$, which is the required ... |
869,892 | <p>The polynomials $p(x) = 5x^3 - 27x^2 + 45x - 21$ and $q(x) = x^4 - 5x^3 + 8x^2 - 5x + 3$ both interpolate the points $(1,2) , (2,1) , (3,6), (4,47)$. Even though these polynomials are of different degree, I do not understand how this is possible when the Lagrange interpolation theorem states there is only one polyno... | Bill Dubuque | 242 | <p><strong>Hint</strong> $\,f(x)\,$ is a solution $\!\iff\! f(x) + (x\!-\!1)(x\!-\!2)(x\!-\!3)(x\!-\!4)g(x)\,$ is a solution, for any $\,g(x),\,$ since both take the same values for $\,x\in\{1,2,3,4\}.\,$ The least degree polynomial of RHS form is the remainder of $\,f(x)\,$ when divided by $\,h(x) = (x\!-\!1)\cdots(x\... |
1,830,799 | <p>1There are $\frac{21!}{2!3!} = 120$ total positions (disregarding order within same colour). I imagine labelling the people Y (yellow) and NY (not yellow), so I imagine I have $4$ copies of the letter Y and $5$ of NY. So I draw out $9$ slots and want to arrange so that at least $2$ Y are together.
I get $30$ arrang... | David Quinn | 187,299 | <p>Hint...consider when <strong>none</strong> of the yellows are next to each other, which can happen in $\binom 52 \times \binom 64$ ways...can you finish?</p>
|
355,262 | <p>Is there a closed-form expression for the sum $\sum_{k=0}^n\binom{n}kk^p$ given positive integers $n,\,p$? Earlier I thought of this series but failed to figure out a closed-form expression in $n,\,p$ (other than the trivial case $p=0$).</p>
<p>$$p=0\colon\,\sum_{k=0}^n\binom{n}kk^0=2^n$$</p>
<p>I know that $\sum_... | Mhenni Benghorbal | 35,472 | <p>A <a href="https://math.stackexchange.com/questions/225908/why-m-power-by-n-equals-sum-of-n-numbrs">relate problem</a>. Try this formula
$$ \sum_{k=0}^n\binom{n}kk^p= 2^n\sum_{k=0}^{p}\begin{Bmatrix} p\\k \end{Bmatrix} {n\choose k}2^{-k}k!, $$</p>
<p>where $p \in \mathbb{N}$ and $\begin{Bmatrix} p\\k \end{Bmatrix}$... |
355,262 | <p>Is there a closed-form expression for the sum $\sum_{k=0}^n\binom{n}kk^p$ given positive integers $n,\,p$? Earlier I thought of this series but failed to figure out a closed-form expression in $n,\,p$ (other than the trivial case $p=0$).</p>
<p>$$p=0\colon\,\sum_{k=0}^n\binom{n}kk^0=2^n$$</p>
<p>I know that $\sum_... | Fabian Pereyra | 748,338 | <p>We have:</p>
<p><span class="math-container">$$\sum_{k=0}^n {n\choose k} k^p =
\sum_{k=0}^n {n\choose k} \left[\sum_{j=0}^p j! {k\choose j} {p\brace j}\right] =
\sum_{j=0}^p j! \left[\sum_{k=j}^n {n\choose k} {k\choose j} \right] {p\brace j} =
\sum_{j=0}^p j! 2^{n-j}{n\choose j} {p\brace j} ,$$</span></p>
<p>with ... |
2,717,264 | <blockquote>
<p>$$\int{\theta \tan^3{(\theta^2)}\sec^4{(\theta^2)}d\theta}$$</p>
</blockquote>
<p>I thought of the method of splitting up the $\tan^3 \theta^2$ to $\tan^2 \theta^2$ and $\tan\theta^2$.</p>
<p>And then using trig identity $1+\tan^2\theta=\sec^2\theta$ to express the whole integral in terms of $\sec\t... | GNUSupporter 8964民主女神 地下教會 | 290,189 | <p>Use Cauchy's condensation test with $2^n > n$. (Easily proved by induction/binomial theorem.)</p>
<p>$$2^n a_{2^n} = \frac{2^n}{\ln 2^n} = \frac{2^n}{n \ln 2} > \frac{1}{\ln2}$$</p>
<p>So the series diverges.</p>
|
58,914 | <p>Question:</p>
<p>If a square matrix $A$ satisfies $A^2=I$ and $\det A>0$, show that $A+I$ is non-singular.</p>
<p>I have tried to suppose a non-zero vector $x$ s.t. $Ax=x$ but fail to make a contradiction.</p>
<p>And I tried to find the inverse matrix of $A+I$ directly, suppose $(A+I)^{-1}=\alpha I +\beta A$, ... | Pete L. Clark | 299 | <p>This seems to be false: consider the $3 \times 3$ diagonal matrix with diagonal entries $1,-1,-1$. Similarly, taking an $n \times n$ diagonal matrix with two entries $-1$ and all the rest equal to $1$ gives a counterexample for any $n \geq 2$.</p>
<p>(A comment on how I came up with this: the matrix $A+I$ is singu... |
4,616,445 | <p>Could anyone enlighten me on how to go about expanding the following function around <span class="math-container">$x_0 = 0$</span>:</p>
<p><span class="math-container">$$
f(x):= \log(1+x)e^{x}
$$</span></p>
<p>I have tried using Cauchy Product Series and bruteforce computation of the coefficients but I always find m... | Stephen Donovan | 869,084 | <p>The product rule solution shown works very well (and thank you to the other answerers, you've done some lovely work) but for the sake of completeness I thought it would be good to show that your original idea of using the Cauchy product formula does work out, and gives the same result.</p>
<p>We can start with our b... |
955,721 | <p>Okay, so last time I got help figuring out a simple binomial coefficient misunderstanding. Now I'm trying to figure out what happens if the following scenario occurs:</p>
<p>Player <span class="math-container">$1$</span> gets a <span class="math-container">$5$</span>-hand of poker cards and tells everyone that he di... | imranfat | 64,546 | <p>Or... Ad and subtract an e-power in the numerator, then divide the fraction. You will integrate $1$ and $\frac{-e^{x}}{1+e^x}$ The latter one is an $ln$ term. </p>
|
1,292,836 | <p>I am trying to evaluate $$\oint _C \frac{-ydx+xdy}{x^2+y^2}$$</p>
<p>clockwise around the square with vertices (−1,−1), (−1,1), (1,1), and (1,−1).</p>
<p>So from the question,
$$\vec{F}=<\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2}>$$</p>
<p>I first conducted the gradient test $\frac{\partial F_2}{\partial x}=\fra... | Matematleta | 138,929 | <p>Put a circle of radius $r$ centered about the origin, so that the circle fits inside the rectangle. Then you know by Green's Thm that the integral over the rectangle is the same as the integral over the circle, which when you switch to polar coordinates, will give you $-2\pi $.</p>
|
1,292,836 | <p>I am trying to evaluate $$\oint _C \frac{-ydx+xdy}{x^2+y^2}$$</p>
<p>clockwise around the square with vertices (−1,−1), (−1,1), (1,1), and (1,−1).</p>
<p>So from the question,
$$\vec{F}=<\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2}>$$</p>
<p>I first conducted the gradient test $\frac{\partial F_2}{\partial x}=\fra... | Christian Blatter | 1,303 | <p>Your field $\vec F$ is nothing else but $\nabla\arg$. </p>
<p>The argument (polar angle) function $$\arg: \>\dot{\mathbb R}^2\to{\mathbb R}/(2\pi{\mathbb Z})$$ is locally a nice real function, but is globally only defined up to multiples of $2\pi$. Therefore on the one hand its gradient is well defined. For $x&... |
156,511 | <p>Suppose I have a simple function like $\cos(kx)$, and I am plotting a number of such functions with different values for $k$ as in:</p>
<pre><code>Plot[Table[Cos[k x], {k, 1, 4, 1}], {x, 0, π}]
</code></pre>
<p>How can I color them depending on the $k$ parameter without setting it manually myself, as in:</p>
<pre... | Syd Geraghty | 53 | <p>It may help to use <code>PlotLabels</code> and <code>PlotLegends</code> options to identify the plotted functions.</p>
<pre><code>Plot[Evaluate[
Table[Style[Cos[k x], Blend[{Blue, Red}, k/4]], {k, 1, 4, 1}]], {x, 0, π},
PlotLabels -> Placed[Automatic, {Above, Below, After, Below}],
PlotLegends -> "Expressio... |
4,186,977 | <p>The challenge is to prove <span class="math-container">$$\left(1-\dfrac{a}{b}\right)\left(1+\dfrac{c}{d}\right)=4.$$</span></p>
<p><a href="https://i.stack.imgur.com/mTFF9.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/mTFF9.jpg" alt="enter image description here" /></a></p>
<p>Apart from the Pyt... | Blue | 409 | <p>It seems like the form of the target relation is trying to tell us something deeper about the configuration that I'm just not seeing. Nevertheless, here's a solution that amounts to a few direct calculations followed by a tedious change of parameters.</p>
<hr />
<p>With different notation: Let <span class="math-cont... |
81,435 | <p>Normally we consider simple arithmetic to be related to the world of objects. So the sum $3+2=5$ means $3$ three apples and $2$ apples gives $5$ apples. But is there an alternative interpretation which does not have anything to do with discrete objects?</p>
| sdcvvc | 12,523 | <blockquote>
<p>But is there an alternative interpretation which does not have anything to do with discrete objects?</p>
</blockquote>
<p>If you have a rope 2 meters long, and join it with 3 meter long rope, you get 5 meter long rope. "meter" is not a discrete quality here, the same interpretation gives 0.2 m + 0.5 ... |
2,688,808 | <p><span class="math-container">$$\frac{1}{u^2}+\frac{1}{v^2}=\frac{1}{w^2}$$</span></p>
<p>I want to generate all primitive solutions up to <span class="math-container">$u \le N$</span>. Is there a parametric solution?</p>
<p>By brute force, I got these solutions:</p>
<p><span class="math-container">$(15, 20, 12),(20,... | Will Jagy | 10,400 | <p>With $A$ being even, $r>s>0,$ $\gcd(r,s) = 1$ and $r+s$ odd,</p>
<p>$$ A = 2rs\left( r^2 + s^2 \right) \; , \; \; B = r^4 - s^4 \; , \; \; C = 2rs\left( r^2 - s^2 \right) \; . \; \; $$</p>
<p>The proof starts with $\gcd(A,B,C) = 1,$ goes on with $\gcd(A,B) = g, \; \;$ $\gcd(g,C) = 1,$ $A = ga, \; \; B =... |
153,772 | <p>Is there any relation between the chromatic number of a graph $G$ and its complement $G'$ that are always true?</p>
<p>I saw these ones: $\chi(G)\chi(G')\geq n$ and $\chi(G)+\chi(G')\geq 2n$,</p>
<p>but I'm not pretty sure about them.</p>
| Dmitri Ilushin | 65,090 | <p>You're actually not quite right. so your inplication that every vertex of H has degree less than or equal to q - 1 is not true. as long as one of these has degree less than or equal to q - 1, the complement graph's degree would be equal to n - q.</p>
|
4,474,386 | <p>According to the definition, <span class="math-container">$f(x) = a·x^n$</span> is a power function. If we shift it to <span class="math-container">$f(x) = a·(x - c)^n$</span> or, more general, to <span class="math-container">$f(x) = a·(x - c)^n + d$</span>, it becomes a polynomial function (not a power function any... | Arthur | 15,500 | <p>Power functions must have a single term, but are allowed to have fractional or negative exponent.</p>
<p>Polynomials can have more than one term ("poly", meaning "many", refers to exactly this), but all exponents must be natural numbers.</p>
<p>A single term with a natural number exponent is both... |
4,474,386 | <p>According to the definition, <span class="math-container">$f(x) = a·x^n$</span> is a power function. If we shift it to <span class="math-container">$f(x) = a·(x - c)^n$</span> or, more general, to <span class="math-container">$f(x) = a·(x - c)^n + d$</span>, it becomes a polynomial function (not a power function any... | Stinking Bishop | 700,480 | <p>In the natural languages, we often use the word "is" ambiguously, which I think is in the root of the confusion here.</p>
<p>When you say "the sky is blue" it may mean "the sky is one of blue things" but it may also mean "the 'colour of the sky' is blue".</p>
<p>The difference... |
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