qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
3,112,043 | <p>The first part of the problem is:</p>
<p>Prove that for all integers <span class="math-container">$n \ge 1$</span> and real numbers <span class="math-container">$t>1$</span>,
<span class="math-container">$$ (n+1)t^n(t-1)>t^{n+1}-1>(n+1)(t-1)$$</span></p>
<p>I have done the first part by induction on <span... | Hypergeometricx | 168,053 | <p>Let <span class="math-container">$S=1^n+2^n+3^n+\cdots+m^n$</span>. </p>
<p>From a quick graph sketch it is clear that:
<span class="math-container">$$\begin{align}
\int_0^m x^n dx \;\;&<\qquad \qquad \quad S &&< \int_0^m (x+1)^n dx\\
\left[\frac {x^{n+1}}{n+1}\right]_0^m\;\;&<\qquad \qquad... |
3,466,870 | <p>Suppose </p>
<p><span class="math-container">$$a^2 = \sum_{i=1}^k b_i^2$$</span> </p>
<p>where <span class="math-container">$a, b_i \in \mathbb{Z}$</span>, <span class="math-container">$a>0, b_i > 0$</span> (and <span class="math-container">$b_i$</span> are not necessarily distinct).</p>
<p>Can any positive... | Herman Tulleken | 67,933 | <p>Here is a geometric solution (for <span class="math-container">$k > 5$</span>).</p>
<p>The following are solutions for 6, 7, and 8 squares. </p>
<p><a href="https://i.stack.imgur.com/JGaFA.png" rel="noreferrer"><img src="https://i.stack.imgur.com/JGaFA.png" alt="enter image description here"></a></p>
<p>We can... |
3,466,870 | <p>Suppose </p>
<p><span class="math-container">$$a^2 = \sum_{i=1}^k b_i^2$$</span> </p>
<p>where <span class="math-container">$a, b_i \in \mathbb{Z}$</span>, <span class="math-container">$a>0, b_i > 0$</span> (and <span class="math-container">$b_i$</span> are not necessarily distinct).</p>
<p>Can any positive... | Bill Dubuque | 242 | <p>This holds far more generally. OP is the special case <span class="math-container">$S$</span> = integer squares, which is closed under multiplication <span class="math-container">$\,a^2 b^2 = (ab)^2,\,$</span> and has an element that is a sum of <span class="math-container">$\,2\,$</span> others, e.g. <span class="m... |
649,502 | <p>What do we mean when we talk about a topological <em>space</em> or a metric <em>space</em>? I see some people calling metric topologies metric spaces and I wonder if there is some synonymity between a topology and a space? What is it that the word means, and if there are multiple meanings how can one distinguish t... | Emi Matro | 88,965 | <p>A metric space is a space in which the notion of distance is defined: there is a distance function that exists in that space, and it is true for all points in that space. For example, the distance formula between two points in Euclidean space. </p>
<p>A topological space is a space in which a notion of "closeness" ... |
649,502 | <p>What do we mean when we talk about a topological <em>space</em> or a metric <em>space</em>? I see some people calling metric topologies metric spaces and I wonder if there is some synonymity between a topology and a space? What is it that the word means, and if there are multiple meanings how can one distinguish t... | monroej | 121,862 | <p>In mathematics, you usually call a set (a collection of objects) with some additional structures a space. So for example, a set with a certain distance function is called a metric space, and a set with certain subsets defined to be open is called a topological space. (of course in these two examples the distance f... |
649,502 | <p>What do we mean when we talk about a topological <em>space</em> or a metric <em>space</em>? I see some people calling metric topologies metric spaces and I wonder if there is some synonymity between a topology and a space? What is it that the word means, and if there are multiple meanings how can one distinguish t... | Community | -1 | <p>Probably the best way to answer your question is to describe an observation: </p>
<p>Let $X:=(-1,1)$. Define two metrics on $X$ as follows
$$
d_1(x,y):=|x-y|, \forall x,y\in X,
$$
and
$$
d_2(x,y):=|arctan(x)-arctan(y)|, \forall x,y\in X.
$$</p>
<p>Then you can see that metrics $d_1$ and $d_2$ define the same topo... |
2,238,614 | <p>For $n\ge3$ a given integer, find a Pythagorean Triple having n as one of its members.<br>
Hint: For n an odd integer, consider the triple $$\left(n, \frac 12\left(n^2-1\right), \frac 12(n^2+1)\right);$$ For n even, consider the triple $$\left(n, \left(\frac{n^2}{4}\right)-1, \left(\frac{n^2}{4}\right)+1 \right)$$</... | poetasis | 546,655 | <p><span class="math-container">$\\ \textbf{Matching sides using Euclid's F(m,n)}: $</span> Solving for <span class="math-container">$n$</span>, any values of <span class="math-container">$m$</span> that yield integers provide the <span class="math-container">$F(m,n)$</span> to identify a triple. Examples follow the so... |
2,421,145 | <p>I am practicing problems around NFA and DFA.</p>
<p>I have seen many questions on how to convert NFA to DFA and DFA to Regular expression etc.</p>
<p>But I have seen very different question and I am stuck on how to proceed with the following question? </p>
<p>Given DFA. Convert this DFA to NFA with 5 states.
<a h... | Aflah | 478,577 | <p>You can refer to the following links
<a href="https://www.tutorialspoint.com/automata_theory/dfa_minimization.htm" rel="nofollow noreferrer">https://www.tutorialspoint.com/automata_theory/dfa_minimization.htm</a></p>
<p><a href="https://www.google.co.in/url?sa=t&source=web&rct=j&url=http://web.cs.ucdavi... |
2,005,649 | <p>I am struggling to find a parameterization for the following set : </p>
<p>$$F=\left\{(x,y,z)\in\mathbb R^3\middle| \left(\sqrt{x^2+y^2}-R\right)^2 + z^2 = r^2\right\}
\quad\text{with }R>r$$</p>
<p>I also have to calculate the area. </p>
<p>I know its a circle so we express it in terms of the angle but my prob... | Pieter21 | 170,149 | <p>I think it is a bit easier than what you did if you handle $D$ more freely.</p>
<ol>
<li>Take all combinations: $(A+B+C+D)^8$</li>
<li>Exclude combinations that do not have (at least) one of $A, B, C$ = $ (B+C+D)^8 + (A+C+D)^8 + (A+B+D)^8$</li>
<li>Include combinations that do not have (at least) two of $A, B, C$ =... |
1,548,130 | <p>Let's say that $F$ is a nice well-behaved function. How would I compute the following derivative?</p>
<p>$\frac{\partial}{\partial t} \left\{ \int_{0}^{t} \int_{x - t + \eta}^{x + t - \eta} F(\xi,\eta) d\xi d\eta \right\}$</p>
<p>I'm guessing I need the fundamental theorem of calculus, but the double integral is R... | Ekaveera Gouribhatla | 31,458 | <p>since $gof$ is Surjective for every $y \in A$ $\exists$ $x \in A$ such that</p>
<p>$$gof(x)=y$$ $\implies$</p>
<p>$$g(f(x))=y \tag{1}$$</p>
<p>But since we know that $g$ is surjective, for every $y \in A$ $\exists$ $z \in B$ such that</p>
<p>$$g(z)=y \tag{2}$$ Using $(2)$ in $(1)$ we get</p>
<p>$$g(f(x))=g(z)$$... |
350,747 | <p>Base case: $n=1$. Picking $2n+1$ random numbers 5,6,7 we get $5+6+7=18$. So, $2(1)+1=3$ which indeed does divide 18. The base case holds.
Let $n=k>=1$ and let $2k+1$ be true. We want to show $2(k+1)+1$ is true. So, $2(k+1)+1=(2k+2) +1$....</p>
<p>Now I'm stuck. Any ideas?</p>
| Taladris | 70,123 | <p>A variation on Marvis answer: let $a$ be the middle number. So the numbers are </p>
<p>$$
a-n,\dots,a-2,a-1,a,a+1,a+2,\dots,a+n
$$</p>
<p>So their sum is clearly $(2n+1)a$ which is divisible by $2n+1$.</p>
|
1,955,591 | <p>I have to prove that ' (p ⊃ q) ∨ ( q ⊃ p) ' is a tautology.I have to start by giving assumptions like a1 ⇒ p ⊃ q and then proceed by eliminating my assumptions and at the end i should have something like ⇒(p ⊃ q) ∨ ( q ⊃ p) but could not figure out how to start.</p>
| Doug Spoonwood | 11,300 | <p>Start off by showing the law of the excluded middle (p V $\lnot$ p).</p>
<p>Put the proof of law of excluded middle here such that it ends as follows:</p>
<p>(p V $\lnot$ p)</p>
<pre><code>Hypothesize p
Hypothesize r
.
.
.
p
(r -> p)
((p -> q) V (r -> p))
</code></pre>
<p>(p -> (... |
96,468 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/22537/how-many-fixed-points-in-a-permutation">How many fixed points in a permutation</a> </p>
</blockquote>
<p>Suppose we have a collection of n objects, numbered from 1 to n. These objects are placed in ... | yoyo | 6,925 | <p>if a permutation $\pi\in S_n$ has exactly $k$ fixed points, you can consider it a permutation in $S_{n-k}$ with no fixed points (a <a href="http://en.wikipedia.org/wiki/Derangement" rel="nofollow">derangement</a>). so you choose your fixed points ${n \choose k}$ and multiply by the number of derangements on $n-k$... |
281,450 | <p>How can I find the area bound by $\;x=0,\, x=1,\;$ the $\;x$-axis ($y = 0$) and $\;y=x^2+2x\;$ using Riemann sums? </p>
<p>I want to use the right-hand sum. Haven't really found any good resources online to explain the estimation of areas bounded by curves, hoping anyone here can help?</p>
<p>By the way, I would l... | Adi Dani | 12,848 | <p>$$(1 + x + x^{10})^{20}=((1+(x+x^{10}))^{20}=\sum_{k=0}^{20}\binom{20}{k}(x+x^{10})^k=$$
$$=\sum_{k=0}^{20}\binom{20}{k}x^k(1+x^{9})^k=\sum_{k=0}^{20}\binom{20}{k}x^k\sum_{j=0}^{k}\binom{k}{j}x^{9j}=\sum_{k=0}^{20}\sum_{j=0}^{k}\binom{20}{k}\binom{k}{j}x^{k+9j}$$ let $0\leq k+9j=t\leq200\Rightarrow j=\frac{t-k}{9},k... |
121,546 | <p>Consider the 2-Wasserstein distance between probability measures $\mu$ and $\nu$ (on $\mathbb{R}^d$), defined as
$$
d_{W_2}(\mu,\nu) = \inf_{\gamma} \Big[\int \|x-y\|^2 d\gamma(x,y)\Big]^{1/2}
$$
where the $\inf$ is over all couplings $\gamma$ of $\mu$ and $\nu$. Can we define a norm (or something norm-like) on th... | mcuturi | 9,804 | <p>This paper has several links to relevant literature by Kantorovich & Rubinstein who define an OT inspired norm for signed measures.</p>
<p><a href="https://hal.inria.fr/inria-00072186/en" rel="nofollow">https://hal.inria.fr/inria-00072186/en</a></p>
|
121,546 | <p>Consider the 2-Wasserstein distance between probability measures $\mu$ and $\nu$ (on $\mathbb{R}^d$), defined as
$$
d_{W_2}(\mu,\nu) = \inf_{\gamma} \Big[\int \|x-y\|^2 d\gamma(x,y)\Big]^{1/2}
$$
where the $\inf$ is over all couplings $\gamma$ of $\mu$ and $\nu$. Can we define a norm (or something norm-like) on th... | leo monsaingeon | 33,741 | <p>For the Earth-mover <span class="math-container">$W_1$</span> distance (based upon the cost function <span class="math-container">$c(x,y)=\|x-y\|$</span>) this is exactly the purpose of <a href="https://arxiv.org/abs/1910.05105" rel="nofollow noreferrer">this paper</a>. Note however that their construction does <str... |
1,297,863 | <p>Is it possible to write the following function:
$$
f(x) = \begin{cases}
\frac{x-\sin x}{1- \cos x}& x\neq 0\\
0 & x=0
\end{cases}
$$
as a composition of elementary functions (including $\mathrm{sinc} (x) = (\sin x) / x)$ so that I get not large numerical errors for $x$ close to zero?</p>
<p>This is the ... | Claude Leibovici | 82,404 | <p>In the same spirit as Yves Daoust, for more accuracy than using Taylor series, you could use Pade approximants. The simplest one would be $$\frac{x-\sin x}{1- \cos x}\approx \frac{x \left(420-x^2\right)}{45 \left(28-x^2\right)}$$ The error is extremely small : $\approx 10^{-15}$ over the interval $(-0.01,0.01)$. </... |
1,297,863 | <p>Is it possible to write the following function:
$$
f(x) = \begin{cases}
\frac{x-\sin x}{1- \cos x}& x\neq 0\\
0 & x=0
\end{cases}
$$
as a composition of elementary functions (including $\mathrm{sinc} (x) = (\sin x) / x)$ so that I get not large numerical errors for $x$ close to zero?</p>
<p>This is the ... | Doggyshakespeare | 243,086 | <p>You can use
$$\frac{x}{\sin ^2x}+\frac{x}{\tan \left(x\right)\sin \left(x\right)}-\frac{1}{\sin \left(x\right)}-\frac{1}{\tan \left(x\right)}$$
which is equal to the original function, though it is still undefined at 0. It is also equal to
$$\frac{x+x\cos \left(x\right)-\sin \left(x\right)-\sin \left(x\right)\cos \l... |
3,075,979 | <p>Prove that <span class="math-container">$$\frac{k^7}{7}+\frac{k^5}{5}+\frac{2k^3}{3}-\frac{k}{105}$$</span> is an integer using mathematical induction.</p>
<p>I tried using mathematical induction but using binomial formula also it becomes little bit complicated.</p>
<p>Please show me your proof.</p>
<p>Sorry if t... | pwerth | 148,379 | <p>@I like Serena has a great answer but since the OP asked for a proof by induction, I'll show what that would look like. Define
<span class="math-container">$$f(k)=\frac{k^7}{7}+\frac{k^5}{5}+\frac{2k^3}{3}-\frac{k}{105}=\frac{15k^7 + 21k^5+70k^3-k}{105}$$</span></p>
<p>For our base case, let <span class="math-conta... |
7,110 | <p>In 1974, Aharoni proved that every separable metric space (X, d) is Lipschitz isomorphic to a subset of the Banach space c_0.
Thus, for some constant L, there is a map K: X --> c_0 that satisfies the inequality d(u,v) <= || Ku - Kv || <= Ld(u,v) for all u and v in X.
Now, suppose X = l_1 (in this case, L = 2 ... | Bill Johnson | 2,554 | <p>Ady, this could be a hard problem. Why are you interested in the answer?</p>
|
2,461,962 | <p>I'm looking to reproduce
\begin{align}
\partial_{j_m,j_n}\bigg|_{\bf{j}=0}\exp\left(\frac12\bf{j}^\top\bf{B}\bf{j}\right) = B_{mn}
\end{align}
where $B_{mn}=B_{nm}$ is a real, symmetric, positive-definite $N\times N$ matrix. I have tried the following, and I know this is incorrect due to the surplus of indices.
\beg... | Jiaqi Li | 480,797 | <p>The way you work with indices are incorrect. It should be
$$\partial_{j_m,j_n}\bigg|_{\mathbf{j=0}}\exp\left(\frac12\bf{j}^\top\bf{B}\bf{j}\right)=\partial_{j_m,j_n}\bigg|_{\mathbf{j=0}}\exp\left(\frac12j_rB_{rs}j_s\right)$$
since when a index appears twice, it means summation over all possible values. See the wikip... |
170,830 | <p>I have two circles with the same radius and I want to calculate the points of tangency. </p>
<p>For example, in the picture below, I want to calculate $(x_3, y_3)$ and $(x_4,y_4)$. I have the radius and the distance between the two circles as shown below:</p>
<p><img src="https://i.stack.imgur.com/tQ2qu.png" alt="... | user29999 | 29,999 | <p>\begin{eqnarray}
y_3 = y_1 + R \cos \theta \\
x_3= x_1 + R \sin \theta \\
y_4= y_2 +R \cos \theta \\
x_4 = x_2 +R \sin \theta \\
\end{eqnarray}</p>
|
458,779 | <p>I need to find the volume of an object restricted with the $x^{2}+z^{2} < 8$ and $0 < y < 2$ planes. It would be easy if the cylinder were "parallel" to the XY plane, because then:</p>
<p>$$0 < r < 2\sqrt{2}$$
$$0 < \phi < 2\pi$$
$$0 < z < 2$$</p>
<p>But well, how should I handle this he... | amWhy | 9,003 | <p>Think of this as a cylinder whose base lies in the $x$-$z$ plane, encompassed by the region $x^{2}+z^{2} < r^2 = 8$ and $0 < y < 2$, so that:</p>
<p>$$0 < r < 2\sqrt{2}$$
$$0 < \phi < 2\pi$$
$$0 < y < 2$$</p>
|
2,554,448 | <p>Beside using l'Hospital 10 times to get
$$\lim_{x\to 0} \frac{x(\cosh x - \cos x)}{\sinh x - \sin x} = 3$$ and lots of headaches, what are some elegant ways to calculate the limit?</p>
<p>I've tried to write the functions as powers of $e$ or as power series, but I don't see anything which could lead me to the righ... | Kenny Lau | 328,173 | <p>$$\begin{array}{cl}
& \displaystyle \lim_{x\to 0} \frac{x(\cosh x - \cos x)}{\sinh x - \sin x} \\
=& \displaystyle \lim_{x\to 0} \frac{xe^x + xe^{-x} - 2x\cos x}{e^x - e^{-x} - 2\sin x} \\
=& \displaystyle \lim_{x\to 0} \frac{x + x^2 + \frac12x^3 + o(x^4) + x - x^2 + \frac12x^3 + o(x^4) - 2x + x^3 + o(x^... |
1,457,623 | <p>A floor is paved with rectangular marble blocks,each of length $a$ and breadth $b$.A circular block of diameter $c(c<a,b)$ is thrown on the floor at random.Show that the chance that it falls entirely on one rectangular block is $\frac{(a-c)(b-c)}{ab}$<br></p>
<p>I thought over this problem,i found total number o... | marty cohen | 13,079 | <p>The center of the disc
must be at least
c/2 from any of the edges.</p>
|
299,405 | <p>(a) In how many ways can the students answer a 10-question true false examination? </p>
<p>(b) In how many ways can the student answer the test in part (a) if it is possible to leave a question unanswered in order to avoid an extra penalty for a wrong answer</p>
<hr>
<p>For part (a) I've got the answer, it is $2^... | Peter Phipps | 15,984 | <p>In part b), for simplicity, let's reduce the problem to just two questions. </p>
<p>There are $2^2$ ways in which both questions may be answered true/false. </p>
<p>If a student does not answer question 1, there are still $2^1$ ways in which they can answer question 2, and vice versa. Thus there are $2\times 2^1... |
3,609,191 | <p>Actually I am not very comfortable with using blocks, I understand the definition that it is a maximal <span class="math-container">$2$</span>-connected graph, though.</p>
<p><strong>My attempt</strong> Suppose not. Then there exists a maximal graph <span class="math-container">$G$</span> which cannot be written as... | Jonas Linssen | 598,157 | <p>I think we need maximality of the blocks rather than maximality of the graph here.</p>
<p>Assume there is a graph, which cannot be written as edge disjoint union of blocks, that is to say there are two <em>distinct</em> blocks <span class="math-container">$A,B$</span>, which share a common edge <span class="math-co... |
84,605 | <p>Let $w$ be a word in letters $x_1,...,x_n$. A value of $w$ is any word of the form $w(u_1,...,u_n)$ where $u_1,...,u_n$ are words. For example, $abaaba$ is a value of $x^2$. A word $u$ is called unavoidable if every infinite word in a finite alphabet contains a value of $u$ as a subword. There is a nice characteriza... | Petar Markovic | 26,899 | <p>I am about to go chair a dissertation defence where the candidate uses an endomorphic image of Zimin words (of his invention) to construct various counterexamples to questions in combinatorics of finite and infinite words, mainly to do with counting palindromes in subwords and scattered subwords of finite and infini... |
1,050,382 | <p>In $\mathbb{R}^5$ there is given vector space $V$. Its dimension is 3. In $\mathbb{R}^{6,5}$ consider the subset $X = \{A \in \mathbb{R}^{6,5} : V \subset \ker A\}$. I have to show that $X$ is a vector space in $\mathbb{R}^{6,5}$ and find its dimension. To show that $X$ is vector space consider $x_1, x_2 \in X$ and ... | dami | 197,371 | <p>I understand $V \subseteq \mathbb{R}^5$ is a subspace, $\dim V = 3$ </p>
<p>$X = \{A \in \mathbb{R}^{6 \times 5} : V \subseteq \ker A \}$</p>
<p>To show that $X$ is a vector space, it suffices to show it is a subspace of $\mathbb{R}^{6 \times 5}$. </p>
<ol>
<li>$0 \in X$, clearly because $V \subseteq \mathbb{R}^... |
2,184,776 | <p>So there's an almost exact question like this here: </p>
<p><a href="https://math.stackexchange.com/questions/576268/use-a-factorial-argument-to-show-that-c2n-n1c2n-n-frac12c2n2-n1#576280">Use a factorial argument to show that $C(2n,n+1)+C(2n,n)=\frac{1}{2}C(2n+2,n+1)$</a></p>
<p>However, I'm getting stuck in just... | Giuseppe | 243,672 | <p>You have it right!</p>
<p>Notice that</p>
<p>$$\frac{(2n)! n + (2n)!(n+1)}{(n+1)n(n-1)!n!} = \frac{(2n)!(2n+1)}{[(n+1)n!][n(n-1)!]}$$</p>
<p>Do you see any simplifications?</p>
|
2,792,770 | <p>I found the following question in a test paper:</p>
<blockquote>
<p>Suppose $G$ is a monoid or a semigroup. $a\in G$ and $a^2=a$. What can we say
about $a$?</p>
</blockquote>
<p>Monoids are associative and have an identity element. Semigroups are just associative. </p>
<p>I'm not sure what we can say about $a... | BCLC | 140,308 | <p>Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space for a fair coin flip $(\{H,T\}, 2^{\Omega}, \mathbb{P}(\omega)=\frac12)$, and let $(\Omega',\mathcal{F}')$ be a (probable?) measurable space for a die roll $(\{1,2,3,4,5,6\}, 2^{\Omega'})$. Let $X(\omega) = 1_{H}(\omega)$, a random variable indicating payo... |
2,450,245 | <p>A set $Q$ contains $0$, $1$ and the average of all elements of every finite non-empty subset of $Q$. Prove that $Q$ contains all rational numbers in $[0,1]$.</p>
<p>This is the exact wording, as it was given to me.
Obviously, the elements that correspond to the average, are rational, since they can be expressed as... | Christian Blatter | 1,303 | <p>By successively taking midpoints we can produce all numbers of the form
$${k\over2^n}\qquad(n\geq0, \ 0\leq k\leq 2^n)\ .$$
Now let two numbers $0<p<q$ be given. Write $q-p=:p'$ for simplicity. Choose a sufficiently large $n$ (see below), and put
$$k_i:=i-1\quad(0< i\leq p'),\qquad k_{q-i}:=2^n-i\quad(0\leq... |
665,596 | <p>Let $b_n$ be the number of lists of length $100$ from the set $\{0,1,2\}$ such that the sum of their entries is $n$. How does $b_{198}$ equal ${100\choose 2}+100$?</p>
| Karthik C | 35,357 | <p>$b_{198}$ calculation:</p>
<p>99 twos and a zero - $100$</p>
<p>98 twos and 2 ones - $^{100}C_2$</p>
|
1,929,698 | <p>Let $f(x)=\chi_{[a,b]}(x)$ be the characteristic function of the interval $[a,b]\subset [-\pi,\pi]$. </p>
<p>Show that if $a\neq -\pi$, or $b\neq \pi$ and $a\neq b$, then the Fourier series does not converge absolutely for any $x$. [Hint: It suffices to prove that for many values of $n$ one has $|\sin n\theta_0|\ge... | ThePhantomE | 820,746 | <p>I see that both of the previous answers to part (b) of this exercise do not fully use the hint in the statement, so I'll attempt to do so. For part (c), the first few paragraphs of <a href="https://math.stackexchange.com/a/3021029/820746">fonini's answer</a> should suffice.</p>
<p>We need to show that the series
<sp... |
2,268,947 | <p><a href="https://i.stack.imgur.com/6hCd2.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/6hCd2.png" alt="enter image description here"></a></p>
<p>($\dot I = \{0,1\}$)</p>
<p>The homotopy I've constructed is:
$$G(t_1, t_2) =
\begin{cases}
\alpha(t_1), & \text{if $(t_1,t_2) \in I \times \{0... | Ronnie Brown | 28,586 | <p>Let $S$ be a simply connected space and let $g: S \to X$ be a map. Let $a,b: [0,1] \to S$ be paths in $S$ with the same end points. Then $ga,gb: [0,1] \to X$ are homotopic rel end points. </p>
<p><strong>Proof</strong> Since $S$ is simply connected there is a homotopy $H_t:a \simeq b$ rel end points. Then $gH_t:ga ... |
2,249,109 | <p><strong>Question:</strong> Three digit numbers in which the middle one is a perfect square are formed using the digits $1$ to $9$.Then their sum is?</p>
<p>$A. 134055$<br>
$B.270540$<br>
$C.170055$<br>
D. None Of The Above</p>
<p>Okay, It's pretty obvious that the number is like $XYZ$ where $X,Z\in[1,9] $ and $Y\i... | Cye Waldman | 424,641 | <p>Here is a formal derivation of your result. The sequence you have found is a generalization of the Fibonacci sequence.</p>
<p>There have been many extensions of the sequence with adjustable (integer) coefficients and different (integer) initial conditions, e.g., $f_n=af_{n-1}+bf_{n-2}$. (You can look up Pell, Jacob... |
2,034,523 | <p>I am sure there is a general and simplified way to solve this problem, I am just unable to figure out the generalized formula (if there is one). </p>
<p>Say we have to write a <strong>code with 4 digits</strong>, the digits can range from <strong>0</strong> to <strong>9</strong>. </p>
<p>All digits in the code <... | barak manos | 131,263 | <p>The total number of combinations is $\binom{10}{4}$.</p>
<p>For each combination there are $4!$ different arrangements.</p>
<p>Exactly $1$ of these arrangements is strictly increasing.</p>
<p>Exactly $1$ of these arrangements is strictly decreasing.</p>
<p>Hence the number of valid arrangements is $\binom{10}{4}... |
277,135 | <p><code>Graphics[{{Blue, Line[{{-5, 3}, {5, 8}}]}, {Dashed, Arrow[{{0, 0}, P}]}, Red, Arrow[{{0, 0}, v}]}, Axes -> True, AxesLabel -> {x, y}]</code>
I've tried combining Graphics and Plot to manipulate my graphic but it says that I can't.</p>
<p><a href="https://i.stack.imgur.com/8slBl.png" rel="nofollow noref... | Daniel Huber | 46,318 | <p>With the answer from user29378 you may get a solution that is close enough to what you want:</p>
<pre><code>d1 = {-1, -1, -1, 1, 1, 1};
d2 = {-2, -2, 2, 2, 2, 2};
d3 = {-4, 4, 4, 4, 4, 4};
dis1 = EmpiricalDistribution[d1];
dis2 = EmpiricalDistribution[d2];
dis3 = EmpiricalDistribution[d3];
</code></pre>
<p>To test t... |
2,876,050 | <p>I have There was a cowboy and 3 donkeys (Donkey A, Donkey B, Donkey C). The cowboy wears an eye cover and shoots randomly. What is the probability of donkey A to still be there after the cowboy shot 2 bullets? There is equal chances that the cowboy hits Donkey A, Donkey B, Donkey C or a Miss.</p>
<p>I am helpless r... | mdnestor | 519,413 | <p>Here is a tree diagram representing the situation. For example, $$\text{Pr}(\text{Hit A on First shot)}=1/4$$ To solve the problem, add up all the cases where $\text{Donkey A}$ survives.
$$\text{Pr}(\text{Miss A on First shot and Second shot})$$
$$=\text{Pr}(\text{Hit B or C on First shot and Miss A on Second Shot})... |
265,067 | <p>$$\lim_{n\to\infty}\frac{(2n-1)!}{3^n(n!)^2}$$</p>
<p>How can I associate limit problem with series? And how can i find limits from series?
Can anyone help?</p>
| doniyor | 32,885 | <p>by ratio rule:<br>
$\dfrac{(2(n+1)-1)!}{3^{n+1}((n+1)!)^2}\cdot\dfrac{3^n(n!)^2}{(2n-1)!}= \dfrac{4n^2+2n}{3n^2+6n+3} \rightarrow \dfrac{4}{3}$</p>
<p>thus the series doesnot converge as the quotient and thus limsup is bigger than 1</p>
|
3,453,175 | <p>If <span class="math-container">$y=\dfrac {1}{x^x}$</span> then show that <span class="math-container">$y'' (1)=0$</span></p>
<p>My Attempt:</p>
<p><span class="math-container">$$y=\dfrac {1}{x^x}$$</span>
Taking <span class="math-container">$\ln$</span> on both sides,
<span class="math-container">$$\ln (y)= \ln \... | Contestosis | 462,389 | <p>As <span class="math-container">$x^x$</span> is <span class="math-container">$\exp(x \ln(x))$</span>, we have <span class="math-container">$y(x) = e ^ {- x \ln(x)}$</span>. It is easy to see that <span class="math-container">$y$</span> is indefinitely derivable over <span class="math-container">$\mathbb{R}^\star_+$<... |
2,807,478 | <p>If $f(x)$ and $g(x)$ are continuous at $x=c$ </p>
<p>then show that:</p>
<p>$h(x)=f(g(x))$ is also continuous at $x=c$.
(Given that $c$ belongs to the Domain of $h$)</p>
| drhab | 75,923 | <p>Let $g:\mathbb R\to\mathbb R$ be the function $x\mapsto x+1$ and let $f:\mathbb R\to\mathbb R$ be prescribed by $x\to x$ if $x<1$ and $x\mapsto x+1$ otherwise.</p>
<p>Then $g$ is continuous at every $0\in\mathbb R$ but can the same be said about $f\circ g$?</p>
|
2,807,478 | <p>If $f(x)$ and $g(x)$ are continuous at $x=c$ </p>
<p>then show that:</p>
<p>$h(x)=f(g(x))$ is also continuous at $x=c$.
(Given that $c$ belongs to the Domain of $h$)</p>
| Badr B | 350,060 | <p>Another counter example: let $f(x)=\frac{1}{x+1}$ and $g(x)=x-1$. Both are continuous at $x=0$, but $f \circ g$ is not. In order for your statement to be always true, then we need both functions to be continuous on $(-\infty, \infty)$ or for $f(c)=c$ and $g(c)=c$. </p>
|
1,703,120 | <p>So I have a vector <span class="math-container">$a =( 2 ,2 )$</span> and a vector <span class="math-container">$b =( 0, 1 )$</span>.<br />
As my teacher told me, <span class="math-container">$ab = (-2, -1 )$</span>.</p>
<p><span class="math-container">$ab = b-a = ( 0, 1 ) - ( 2, 2 ) = ( 0-2, 1-2 ) = ( -2, -1 )$</sp... | DonAntonio | 31,254 | <p>The correct thing is $\;b-a\;$ for the direction vector $\;\vec{ab}\;$. The substraction $\;a-b\;$ gives <em>the opposite</em> direction vector, namely $\;\vec
{ba}\;$</p>
|
66,009 | <p>Hi I have a very simple question but I haven't been able to find a set answer. How would I draw a bunch of polygons on one graph. The following does not work:</p>
<pre><code>Graphics[{Polygon[{{989, 1080}, {568, 1080}, {834, 711}}],
Polygon[{{1184, 1080}, {989, 1080}, {834, 711}, {958, 541}}],
Polygon[{{1379,... | Simon Woods | 862 | <p>Another possible workaround is to wrap <code>Dispatch</code> with a memoized function, so that both expressions <code>a</code> and <code>b</code> contain references to the same internal dispatch table.</p>
<p>i.e. define</p>
<pre><code>mem : disp[x_] := mem = Dispatch[x]
</code></pre>
<p>then use <code>disp</code... |
2,093,720 | <p>$$y~ dy+(2+x^2-y^2)dx$$</p>
<p>I try to solve this equation by putting standard form but becomes more challenge . So your answer is helpful </p>
| Yami Kanashi | 403,840 | <p>Applying AM-GM inequality
We get,
((abc)/(bca))^1/3 ≤ ((a/b) + (b/c) + (c/a))/3</p>
<p>1 ≤ ((a/b) + (b/c) + (c/a))/3</p>
<p>3≤ (a/b) + (b/c) + (c/a)</p>
<p>But since 2017/1000< 3
Therefore there does not exist any such triad.</p>
|
3,109,482 | <p><a href="https://i.stack.imgur.com/awp2x.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/awp2x.png" alt="enter image description here"></a></p>
<p>I'm stumped on determining P(ABC) of Part A. My understanding is:</p>
<ul>
<li><p>Calculate the total number of patients (100)</p></li>
<li><p>Calcul... | John Wayland Bales | 246,513 | <p>You were close, but you chose a straight line segment. You want each point to be a unit distance from the origin. So, provided <span class="math-container">$A\ne-B$</span> you should divide each vector by its magnitude.</p>
<p><span class="math-container">$$R(x)=\frac{A+x(B-A)}{\Vert A+x(B-A)\Vert}$$</span></p>
<p... |
3,353,826 | <p>All the vertices of quadrilateral <span class="math-container">$ABCD$</span> are at the circumference of a circle and its diagonals intersect at point <span class="math-container">$O$</span>. If <span class="math-container">$∠CAB = 40°$</span> and <span class="math-container">$∠DBC = 70°$</span>, <span class="math-c... | Arthur | 15,500 | <p>I have seen the notation <span class="math-container">$[x]$</span>. However, that is some times used as the floor function when TeX is unavailable, or the author is unfamiliar with it (I'm sure there are plenty of examples on this site, for instance).</p>
<p>The safest bet is to say something along the lines of</p>... |
3,353,826 | <p>All the vertices of quadrilateral <span class="math-container">$ABCD$</span> are at the circumference of a circle and its diagonals intersect at point <span class="math-container">$O$</span>. If <span class="math-container">$∠CAB = 40°$</span> and <span class="math-container">$∠DBC = 70°$</span>, <span class="math-c... | Especially Lime | 341,019 | <p>Whatever notation you use (punctured dusk gives some good suggestions), you should <em>always</em> define this explicitly if you are going to use it, since there is no standard way to treat half-integers. (I recently found this out the hard way when I assumed the rounding method I was always taught was standard, but... |
3,353,826 | <p>All the vertices of quadrilateral <span class="math-container">$ABCD$</span> are at the circumference of a circle and its diagonals intersect at point <span class="math-container">$O$</span>. If <span class="math-container">$∠CAB = 40°$</span> and <span class="math-container">$∠DBC = 70°$</span>, <span class="math-c... | Greg Nisbet | 128,599 | <p>It might be too verbose, but something like the following is unlikely to be misinterpreted.</p>
<p><span class="math-container">$$\mathrm{RoundToEven}(5.5) = 6$$</span></p>
<p>If you need another convention such as rounding to the nearest odd number, rounding towards infinity, or rounding towards negative infinity... |
894,152 | <blockquote>
<p>Let $x_{i}\ge 0$ for $i\in\{1,2,\cdots,n\}$ and $x_{1}+x_{2}+\cdots+x_{n}=n$ for $n\ge 3$</p>
<p>Show that for all strictly positive integers $k\ge2$ the following inequality holds :
$$\sum_{i=1}^{n}x^k_{i}\ln{x_{i}}\ln{\dfrac{x_{i}}{n}}\le 0$$</p>
</blockquote>
<p>We consider
$$f(x)=x^k\ln{x}... | RE60K | 67,609 | <p>Firstly, you diffrentiated it wrongly:
$$f(x)=x^k\ln{x}\ln{\dfrac{x}{n}}$$
$$f'(x)=kx^{k-1}\ln{x}\ln{\dfrac{x}{n}}+x^{k-1}\ln{\dfrac{x}{n}}+x^{k-1}\ln{x}$$
$$f'(x)=x^{k-1}\left(k\ln x\ln\frac xn+\ln \frac xn+\ln x\right)$$
$$f''(x)=(k-1)x^{k-2}\left(k\ln x\ln\frac xn+\ln \frac xn+\ln x\right)+x^{k-1}\left(\frac kx\... |
2,428,009 | <p>I want to solve the equation $2^n=2k$ for $n$ even with $n,k \in \Bbb{N}$. I'm not sure how to go about this, using logarithm makes me enter the reals.</p>
| ksoileau | 480,055 | <p>$2^n=2k$ implies $2^{n-1}=k,$ so the solution set (n,k) is (2 r,2^{2 r-1}) for $r \geqslant 1.$</p>
|
405,087 | <blockquote>
<p>Is $\sum_{n=1}^{\infty} {x^2 e^{-nx}}$ uniformly convergent in $[0,\infty)$?</p>
</blockquote>
<p>So I started by saying that by the geometric series test where $a=x^2$ and $|r| = |\frac{1}{e^x}| \leq 1$, the series converges pointwise.</p>
<p>But how do I exactly prove that it converges uniformly? ... | Community | -1 | <p>Let
$$f_n(x)=x^2e^{-nx}$$
then we have
$$f'_n(x)=e^{-nx}\left(2x-nx^2\right)=0\iff x=0\ \text{or}\ x=\frac{2}{n}$$
so
$$||f_n||_\infty=f_n\left(\frac{2}{n}\right)=\frac{4}{n^2}e^{-2}$$
hence the series $\displaystyle \sum_{n=1}^\infty ||f_n||_\infty$ is convergent and then the series $\displaystyle \sum_{n=1}^\inf... |
405,087 | <blockquote>
<p>Is $\sum_{n=1}^{\infty} {x^2 e^{-nx}}$ uniformly convergent in $[0,\infty)$?</p>
</blockquote>
<p>So I started by saying that by the geometric series test where $a=x^2$ and $|r| = |\frac{1}{e^x}| \leq 1$, the series converges pointwise.</p>
<p>But how do I exactly prove that it converges uniformly? ... | Hans Engler | 9,787 | <p>You can compute the remainder term explicitly, using the formula for the geometric series:
$$ r_N(x) = \sum_{n=N}^\infty x^2 e^{-nx} = x^2e^{-Nx} \sum_{j = 0}^\infty e^{-jx} = \dots
$$
Now find the maximum of $r_N$ on $[0, \infty)$. (It's obviously a positive function.) </p>
<p>If this maximum tends to $0$ as $N... |
964,999 | <p>If A and B are two closed sets of $R$ is A.B closed?
By A.B I mean the set $\sum_{i=1}{^ n} a_ib_i$ where $a_i \in A,b_i\in B,n\in N$
How to view A.B geometrically?
I am new to this subject.Sorry if the question sounds something wrong</p>
| Chris Culter | 87,023 | <p>First, see user5527's answer.</p>
<p>Now, if $A$ and $B$ are closed <em>and bounded</em> subsets of $\mathbb R$, so that they're compact, then the answer is yes. The Cartesian product $A\times B$ is a compact subset of $\mathbb R^2$, and the set of products $A\cdot B$ is its image under a continuous map.</p>
|
377,354 | <p>I'm referencing this page: <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfINTRO.html#sqrtalgsalg" rel="nofollow">An Introduction to the Continued Fraction</a>, where they explain the algebraic method of solving the square root of $14$.</p>
<p>$$\sqrt{14} = 3 + \frac1x$$</p>
<p>So, $x_0 = 3... | genepeer | 50,955 | <p>$\sqrt{14}=3+\sqrt{14}-3=3+\frac{1}{\frac{\sqrt{14}+3}{5}}\implies x_0 = 3$</p>
<p>$\frac{\sqrt{14}+3}{5}=\frac{6+\sqrt{14}-3}{5}=1+\frac{\sqrt{14}-2}{5}=1+\frac{1}{\frac{\sqrt{14}+2}{2}} \implies x_1 = 1$</p>
<p>$\frac{\sqrt{14}+2}{2}=\frac{5+\sqrt{14}-3}{2}=2+\frac{\sqrt{14}-2}{2}=2+\frac{1}{\frac{\sqrt{14}+2}{5... |
2,950,813 | <blockquote>
<p>Take <span class="math-container">$G$</span> to be a group of order <span class="math-container">$600$</span>. Prove that for any element <span class="math-container">$a$</span> <span class="math-container">$\in$</span> G there exist an element <span class="math-container">$b$</span> <span class="math... | Bill Dubuque | 242 | <p>First I show how to reduce it to computing <span class="math-container">$\,7^{-1}\!\pmod{\!600};$</span> then I explain why this reduction works so universally due to <em>persistence</em> of GCDs having linear (Bezout) representation.</p>
<p><span class="math-container">$\langle a\rangle$</span> has order <span cla... |
1,305,935 | <p>Let $f(n)$ be non-negative real valued function defined for each natural number $n$.</p>
<p>If $f$ is convex and $lim_{n\to\infty}f(n)$ exists as a finite number, then can we conclude that $f$ is non-increasing?</p>
| PSPACEhard | 140,280 | <p>Let $\textbf{x}_0$ be a point in the hyperplane $\textbf{wx} - b = -1$, i.e., $\textbf{wx}_0 - b = -1$. To measure the distance between hyperplanes $\textbf{wx}-b=-1$ and $\textbf{wx}-b=1$, we only need to compute the perpendicular distance from $\textbf{x}_0$ to plane $\textbf{wx}-b=1$, denoted as $r$.</p>
<p>Note... |
1,253,475 | <p>I'm trying to prove the following statement: if E is a subspace of V, then dim E + dim $E^{\perp}$ = dim V. I know this is true because when these two subspaces are added, they are equal to V, but I'm not sure how to rigorously say this, could I get a little help?</p>
| Quality | 153,357 | <p>First understand the following</p>
<p>$\mathbf{Thereom:}$ Let $\{v_1,…,v_n\}$ be any basis of an inner product space V. Then there exists an orthonormal basis $\{u_1,…,u_n\}$ of V such that the change of basis matrix from $\{v_i\} to \{u_i\}$ is triangular i.e. for $k=1,2.., n$,</p>
<p>$u_k= a_{k1}v_1+a_{k2}v_2+..... |
176,059 | <p>I asked this question in MSE, but I did not received any answer, so I repeat it here:</p>
<p><a href="https://math.stackexchange.com/questions/858238/a-question-on-fixed-point-property">https://math.stackexchange.com/questions/858238/a-question-on-fixed-point-property</a></p>
<p>Assume that $0<k<n-1$, Note ... | Eric Wofsey | 75 | <p>Here is a partial affirmative answer using mod 2 Steenrod operations; the simplest case of this (for $n$ and $k$ even) is just a correction of the slightly incorrect answer originally posted by Włodzimierz Holsztyński. The result is that if $k+1$ and $n+1$ are both odd multiples of $2^d$ for some integer $d\geq 0$,... |
1,749,340 | <p>I have interesting trigonometric expression for professionals in mathematical science. So, here it is:
$$\sin\dfrac{3\pi}{14}-\sin\dfrac{\pi}{14}-\sin\dfrac{5\pi}{14};$$
Okay! I attempt calculate it:
\begin{gather}
\sin\dfrac{3\pi}{14}-\left(\sin\dfrac{\pi}{14}+\sin\dfrac{5\pi}{14}\right)=\\
=\sin\dfrac{3\pi}{14}-\l... | StackTD | 159,845 | <p>For simplicity, let $x = \frac{\pi}{14}$, then we want to simplify:
$$\sin 3x-\sin x -\sin 5x$$
Multiply by $\cos x$ to get:
$$\color{blue}{\sin 3x\cos x}-\color{green}{\sin x\cos x} -\color{red}{\sin 5x\cos x} \quad (*)$$
With $\sin\alpha\cos\beta = \tfrac{1}{2}\left( \sin(\alpha+\beta)+\sin(\alpha-\beta) \right)$,... |
1,749,340 | <p>I have interesting trigonometric expression for professionals in mathematical science. So, here it is:
$$\sin\dfrac{3\pi}{14}-\sin\dfrac{\pi}{14}-\sin\dfrac{5\pi}{14};$$
Okay! I attempt calculate it:
\begin{gather}
\sin\dfrac{3\pi}{14}-\left(\sin\dfrac{\pi}{14}+\sin\dfrac{5\pi}{14}\right)=\\
=\sin\dfrac{3\pi}{14}-\l... | lab bhattacharjee | 33,337 | <p>Let $14x=\pi$</p>
<p>$$S=\sin3x-\sin x-\sin5x=\sin3x+\sin(-x)+\sin(-5x)$$</p>
<p>Using <a href="https://math.stackexchange.com/questions/17966/how-can-we-sum-up-sin-and-cos-series-when-the-angles-are-in-arithmetic-pro">How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression?</a> ,<... |
302,179 | <p>The question I am working on is:</p>
<blockquote>
<p>"Use a direct proof to show that every odd integer is the difference of two squares."</p>
</blockquote>
<p>Proof:</p>
<p>Let n be an odd integer: $n = 2k + 1$, where $k \in Z$</p>
<p>Let the difference of two different squares be, $a^2-b^2$, where $a,b \in Z... | DonAntonio | 31,254 | <p>Directly:</p>
<p>$$2k+1=k^2+2k+1-k^2=(k+1)^2-k^2\ldots$$</p>
|
1,406,280 | <p>Why does taylor series have ample amount of importance in calculus? </p>
<p>I like to know some insights behind taylor series. </p>
| mathlove | 78,967 | <p>I assume that $a,b,c\ge 1$.</p>
<p>You can use the method you write for
$$a+c=100-2b$$
where $b=1,2,\cdots,49$. </p>
<p>Then, the answer is
$$\sum_{b=1}^{49}\binom{99-2b}{1}=99\times 49-2\cdot\frac{49\cdot 50}{2}=49(99-50)=49^2=\color{red}{2401}.$$</p>
|
4,550,991 | <p>This is question is taken from an early round of a Norwegian national math competition where you have on average 5 minutes to solve each question.</p>
<p>I tried to solve the question by writing every number with four digits and with introductory zeros where it was needed. For example 0001 and 0101 would be the numb... | Abel Wong | 1,090,313 | <p>The answer is correct. I try to do it in another way:</p>
<p>Count the <span class="math-container">$1$</span>'s in thousandth place first.</p>
<p>There is 1000 <span class="math-container">$1$</span>'s from <span class="math-container">$1$</span> to <span class="math-container">$9999$</span>. But we need to exclude... |
4,550,991 | <p>This is question is taken from an early round of a Norwegian national math competition where you have on average 5 minutes to solve each question.</p>
<p>I tried to solve the question by writing every number with four digits and with introductory zeros where it was needed. For example 0001 and 0101 would be the numb... | user2661923 | 464,411 | <p>Your <strong>direct approach</strong> looks good, and may well be the easiest approach for this particular problem. The alternative approach, which generalizes better and is used below is Inclusion-Exclusion.</p>
<p>See <a href="https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle" rel="nofollow nore... |
1,464,143 | <p>$\lim_{n \to \infty} n\ln\left(1+\frac{1}{n}\right)$ using L'Hòpital rule show that this is $1$. Can you do this since there isn't a division and $n$ will obviously tend to infinity and $\ln\left(1+\frac{1}{n}\right)$ will tend to $0$? So there limits aren't matching?</p>
<p>So I set $u=n $</p>
<p>$du=1$</p>
<p>$... | mfl | 148,513 | <p><strong>Hint</strong></p>
<p>$$\lim_{n\to \infty} n\ln \left(1+\frac1n\right)=\lim_{n\to \infty} \frac{\ln \left(1+\frac1n\right)}{\frac 1n}.$$</p>
|
302 | <p>I know that the Fibonacci numbers converge to a ratio of .618, and that this ratio is found all throughout nature, etc. I suppose the best way to ask my question is: where was this .618 value first found? And what is the...significance?</p>
| S. Carnahan | 121 | <p>Golden ratio came first. <a href="http://en.wikipedia.org/wiki/Golden_ratio" rel="noreferrer">Wikipedia</a> has a rather thorough article on it. It's not nearly as pervasive in nature or architecture as people like to say it is. It will show up in anything with regular pentagons, though.</p>
|
255,773 | <p>As it is known that
Integrate[A+B]= Integrate[A] + Integrate[B]</p>
<p>I am facing problem with the following integral, when I integrate</p>
<pre><code>Integrate[(-(1/2) b^2 x^2 (-1 + EulerGamma + Log[(b x)/2]) -
2 (EulerGamma + Log[(b x)/2])) 1/
x ((-1 + x) Log[-1 + x]^2 - 2 (-1 + x) Log[-1 + x] (1 + Log[x])... | Carl Woll | 45,431 | <p>The distributive property only holds when both integrals are convergent. For example:</p>
<pre><code>Integrate[Exp[-x] + Pi/(2x) - ArcTan[x]/x, {x, 1, Infinity}]
</code></pre>
<blockquote>
<p>Catalan + 1/E</p>
</blockquote>
<p>However:</p>
<pre><code>Integrate[Exp[-x] + Pi/(2x), {x, 1, Infinity}]
Integrate[ - ArcTan... |
1,853,846 | <p>Prove that the equation <span class="math-container">$$x^2 - x + 1 = p(x+y)$$</span> has integral solutions for infinitely many primes <span class="math-container">$p$</span>.</p>
<p>First, we prove that there is a solution for at least one prime, <span class="math-container">$p$</span>. Now, <span class="math-cont... | velut luna | 139,981 | <p>$$8\sin\theta=4+\cos\theta$$
$$8\sin\theta-\cos\theta=4$$
$$\sqrt{65}\sin\alpha\sin\theta-\sqrt{65}\cos\alpha\cos\theta=4$$
where $\alpha=\tan^{-1}8$
$$-\sqrt{65}\cos(\theta+\alpha)=4$$
Can you continue from here?</p>
|
542,148 | <p>Calculate the determinant of the following matrix as an explicit
function of $x$. (It is a polynomial in $x$. You are asked to find
all the coefficients.)</p>
<p>\begin{bmatrix}1 & x & x^{2} & x^{3} & x^{4}\\
x^{5} & x^{6} & x^{7} & x^{8} & x^{9}\\
0 & 0 & 0 & x^{10} &... | 2012ssohn | 103,274 | <p>First, note that the 5th column is a multiple of the 4th column. That is,</p>
<p>\begin{bmatrix}
x^4\\
x^9\\
x^{11}\\
x^{13}\\
x^{15}\\
\end{bmatrix}</p>
<p>is $x$ times
\begin{bmatrix}
x^3\\
x^8\\
x^{10}\\
x^{12}\\
x^{14}\\
\end{bmatrix}.</p>
<p>Because the determinant of a matrix does not change when you subtra... |
542,148 | <p>Calculate the determinant of the following matrix as an explicit
function of $x$. (It is a polynomial in $x$. You are asked to find
all the coefficients.)</p>
<p>\begin{bmatrix}1 & x & x^{2} & x^{3} & x^{4}\\
x^{5} & x^{6} & x^{7} & x^{8} & x^{9}\\
0 & 0 & 0 & x^{10} &... | Robert Israel | 8,508 | <p>Another way to look at this: the bottom three rows can't have rank more than $2$, since they have only two nonzero columns, so the whole matrix can't have rank more than $4$, and therefore is singular.</p>
|
8,567 | <p>When highlighting text using <code>Style</code> and <code>Background</code>, as in <code>Style["Test ", White, Background -> Lighter@Blue]</code> is there a way to pad (ie, enlarge) the bounding box? </p>
<p>The bottom of the background seems coincident with the base of the text: <img src="https://i.stack.imgu... | Mr.Wizard | 121 | <p>To answer my own question and further illustrate the kind of operation I am describing, here is a method using <code>Set</code> itself:</p>
<pre><code>SetAttributes[f, HoldAllComplete]
f[args___] :=
Module[{h},
h[args] = 1;
Level[DownValues@h, {4}, HoldComplete]
]
f[own, down[1], sub[1][2], N[n], up]
... |
3,009,112 | <p>I am a geographer/ecologist and I want to know how to accurately calculate volume of a lake or a reservoir? I am not looking for a vague estimate which is generally calculated using surface area and mean height parameters assuming the body is of a certain shape (truncated cone/triangle or circular). Since reservoirs... | the_fox | 11,450 | <p>Corollary <span class="math-container">$(2I)$</span> in the paper "On groups of even order" by Brauer and Fowler says that if <span class="math-container">$G$</span> is a simple group which contains <span class="math-container">$n$</span> involutions and <span class="math-container">$t= \frac{|G|}{n}$</span> then <s... |
3,997,632 | <p>Use the Chain Rule to prove the following.<br />
(a) The derivative of an even function is an odd function.<br />
(b) The derivative of an odd function is an even function.</p>
<p><strong>My attempt:</strong></p>
<p>I can easily prove these using the definition of a derivative, but I'm having trouble showing them us... | Paul Frost | 349,785 | <p>You <em>did</em> use that <span class="math-container">$f$</span> is even. Let us look at the following more general situation that
<span class="math-container">$$ f = v \circ f \circ u .$$</span>
For an even function we have <span class="math-container">$u(x) = -x$</span> and <span class="math-container">$v(x) = x$... |
264,587 | <p><strong>NOTE</strong></p>
<p>I'm sorry, my question was not clear. I want to know all the ways to split a list with a given length simply, <strong>rather than split a cyclic substitution</strong>.
If a given list has length <span class="math-container">$N$</span> and the rule is <span class="math-container">${m, n, ... | user1066 | 106 | <pre><code>TakeDrop[#,1]&/@NestList[RotateLeft, {a,b,c,d},3]
(* {{{a}, {b, c, d}}, {{b}, {c, d, a}}, {{c}, {d, a, b}},
{{d}, {a, b, c}}} *)
</code></pre>
<p>And</p>
<pre><code>TakeDrop[#,2]&/@NestList[RotateLeft, {a,b,c,d},3]
( {{{a, b}, {c, d}}, {{b, c}, {d, a}}, {{c, d}, {a, b}},
{{d, a}, {b, c}}} ... |
18,444 | <p>I am a student, in my last year of school(17 years old)</p>
<p>When I was about 13 years old I fell into the <a href="https://artofproblemsolving.com/news/articles/avoid-the-calculus-trap" rel="noreferrer">calculus trap</a> by starting off learning trigonometry on my own, when I was supposed to factor equations or ... | paul garrett | 63 | <p>Echoing @AndreasBlass' remark, and having experienced somewhat similar episodes, it is already precarious enough to make such choices _for_oneself_. So, to directly answer your question: I think "no, do not encourage others to (too violently) disconnect from the math curriculum at school". I don't think it's about p... |
18,444 | <p>I am a student, in my last year of school(17 years old)</p>
<p>When I was about 13 years old I fell into the <a href="https://artofproblemsolving.com/news/articles/avoid-the-calculus-trap" rel="noreferrer">calculus trap</a> by starting off learning trigonometry on my own, when I was supposed to factor equations or ... | Community | -1 | <p>The article at artofproblemsolving seems silly to me. The author's idiosyncratic opinion seems to be that students who are ready to take calculus should refrain from taking calculus and instead do math contests. People are all different, and there is not just one appropriate path for a mathematically precocious stud... |
18,444 | <p>I am a student, in my last year of school(17 years old)</p>
<p>When I was about 13 years old I fell into the <a href="https://artofproblemsolving.com/news/articles/avoid-the-calculus-trap" rel="noreferrer">calculus trap</a> by starting off learning trigonometry on my own, when I was supposed to factor equations or ... | Daniel R. Collins | 5,563 | <p>I will also chime in and say that the argument on the linked Art of Problem Solving site is unpersuasive, and somewhat misses a broader point. </p>
<p>Ultimately, the <em>real</em> point of the mathematical discipline is to identify patterns in systems and prove their correctness (hopefully in an insightful, persua... |
2,612,416 | <p>Can you please help me with this limit? I can´t use L'Hopital rule.</p>
<p>$$\lim_{x\to \infty} \frac{\sqrt{4x^2+5}-3}{\sqrt[3]{x^4}-1} $$</p>
| Atmos | 516,446 | <p>Make the ratio of the high degree terms which gives here
$$
\frac{2x}{x^{4/3}} \underset{x \rightarrow +\infty}{\rightarrow}0
$$
EDIT : </p>
<p>I will propose my way of doing things. You are studying the limit around $1$ then, makes it move to $0$. Let $x=1+h$
$$
\frac{\sqrt{4\left(1+h\right)^2+5}-3}{\sqrt[3]{1+h}... |
404,574 | <p>Suppose that:</p>
<p>$Y \pmod B = 0$</p>
<p>$Y \pmod C = X$</p>
<p>I know $B$ and $C$. $Y$ is unknown, it might be an extremely large number, and it does not interest me. </p>
<p>The question is: Is it possible to find $X$, and if so, how?</p>
| André Nicolas | 6,312 | <p>In the case where $b$ aand $c$ are relatively prime, knowing $y\bmod b$ gives <strong>absolutely</strong> no information about $y\bmod c$. It could be any of $0,1,2, \dots,c-1$.</p>
<p>In the general case where $b$ and $c$ are not necessarily prime, let $d=\gcd(b,c)$. Knowing $y\bmod b$ tells us what $y\bmod d$ is.... |
90,459 | <p>I want to find the degree of $\mathbb{Q}(\sqrt{3+2\sqrt{2}})$ over $\mathbb{Q}$. I observe that
$3+2\sqrt{2}=2+2\sqrt{2}+1=(\sqrt{2}+1)^2$ so
$$
\mathbb{Q}(\sqrt{3+2\sqrt{2}})=\mathbb{Q}(\sqrt{2}+1)=\mathbb{Q}(\sqrt{2})
$$
so the degree is 2.</p>
<p>Is there a more mechanical way to show this without noticing the... | Bruno Stonek | 2,614 | <p>Here's how I would do it:</p>
<p>Observe first that there is a tower of fields $\mathbb{Q}\subset \mathbb{Q}(\sqrt2)\subset \mathbb{Q}\left(\sqrt{3+2\sqrt2}\right)$.</p>
<p>Now, the first extension has degree 2. To determine the degree of the extension you're asking, by transitivity of degrees it suffices to compu... |
363,767 | <p>An ellipse is specified $ x^2 + 4y^2 = 4$, and a line is specified $x + y = 4$. I need to find the max/min distances from the ellipse to the line.</p>
<p>My idea is to find two points $(x_1, y_1)$ and $(x_2,y_2)$ such that the first point is on the ellipse and the second point is on the line. Furthermore, the line ... | DonAntonio | 31,254 | <p>The ellipse is</p>
<p>$$x^2+4y^2=4\iff \frac{x^2}{2^2}+y^2=1$$</p>
<p>and the line is $\,y=-x+4\,$ , which is then "above" the ellipse all the time and, in particular, in the first quadrant (draw an approximate sketck of the functions to see why is this relevant).</p>
<p>Thus, we can write down a point on the ell... |
2,266,573 | <p>I am working through some problems about probability and seem to be having trouble working through this one in particular. I'd love some help learning how to go about solving problems such as this.</p>
<p>A website estimates that 19% of people have a phobia regarding public speaking. If three students are assigned ... | The Dead Legend | 433,379 | <p>For this question,:</p>
<p>A) If one kid has a fear then probability of picking him will be $0.19$ while picking one more reduces the probability to $(0.19)*(0.19)$ . For three people: it becomes $(0.19)^3$
This is from selection via independent events.</p>
<p>B)None have fear: $(1-0.19)^3$<br>
This is via noting... |
2,266,573 | <p>I am working through some problems about probability and seem to be having trouble working through this one in particular. I'd love some help learning how to go about solving problems such as this.</p>
<p>A website estimates that 19% of people have a phobia regarding public speaking. If three students are assigned ... | John Doe | 399,334 | <p>Use the fact that these students are chosen independently, and so $$\Bbb P(A\text{ and }B\text{ and }C)=\Bbb P(A)\Bbb P(B)\Bbb P(C)$$ For your problem, you could say $A$ is the event that the first student has a fear of public speaking, etc. You can also use this property for part $B$.</p>
<p>For questions where th... |
1,278,329 | <p>Solve the recurrence $a_n = 4a_{n−1} − 2 a_{n−2}$</p>
<p>Not sure how to solve this recurrence as I don't know which numbers to input to recursively solve?</p>
| k1.M | 132,351 | <p>Hint: Let $r_1,r_2$ be two distinct real roots of the equation
$$
r^2-4r+2
$$
then this recurrence equation has a solution of the form
$$
C_1 r_1^n+C_2 r_2^n
$$
which the constants $C_1,C_2$ can be found by initial values condition.</p>
|
1,902,138 | <p>It's common to see a plus-minus ($\pm$), for example in describing error
$$
t=72 \pm 3
$$
or in the quadratic formula
$$
x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}
$$
or identities like
$$
\sin(A \pm B) = \sin(A) \cos(B) \pm \cos(A) \sin(B)
$$</p>
<p>I've never seen an analogous version combining multiplication with div... | marty cohen | 13,079 | <p>As you indicated, square root can be + or -.
$\pm$ shows this ambiguity.</p>
<p>As far as I know,
there is no similar use case
where the choice is
to multiply or divide
by an expression. </p>
|
1,043,956 | <p>Find a normal vector and a tangent vector to the curve given by the equation: $x^5 + y ^5 =2x^3$ at the point $P(1, 1)$. Find the equation of the tangent line. <br/>
Edit: The notes I have:<img src="https://i.stack.imgur.com/arOee.png" alt="enter image description here"></p>
<p>Taking $f(x, y) = x^5 - 2x^3 + y^5 = ... | ploosu2 | 111,594 | <p>Differentiate explicitly:</p>
<p>$$5x^4 + 5y^4y' = 6x^2$$
Solve for $y'$</p>
|
1,262,174 | <p>I am currently teaching Physics in an Italian junior high school. Today, while talking about the <a href="http://en.wikipedia.org/wiki/Dipole#/media/File:Dipole_Contour.svg" rel="noreferrer">electric dipole</a> generated by two equal charges in the plane, I was wondering about the following problem:</p>
<blockquote>... | Hosein Rahnama | 267,844 | <p>At the first step, I will introduce a proper curve linear coordinates for this problem. This will help to construct the integral for area. We can write the equation of these equi-potential curves as</p>
<p>$$\frac{1}{r_1}+\frac{1}{r_2}=C \tag{1}$$</p>
<p>where $C$ is some real constant and $r_1$ and $r_2$ are defi... |
1,262,174 | <p>I am currently teaching Physics in an Italian junior high school. Today, while talking about the <a href="http://en.wikipedia.org/wiki/Dipole#/media/File:Dipole_Contour.svg" rel="noreferrer">electric dipole</a> generated by two equal charges in the plane, I was wondering about the following problem:</p>
<blockquote>... | Hosein Rahnama | 267,844 | <p>Here is another method based on the curve-linear coordinates introduced by <em>Achille Hui</em>. He introduced the following change of variables</p>
<p>$$\begin{align}
\sqrt{(x+1)^2+y^2} &= u+v\\
\sqrt{(x-1)^2+y^2} &= u-v
\end{align} \tag{1}$$</p>
<p>Then solving for $x$ and $y$ we shall get</p>
<p>$$\beg... |
102,963 | <p>Could someone please explain the difference between the group of all icosahedral symmetries and S5? I know that the former is a direct product, but don't they work the same? Say I have an icosahedron, why wouldn't S5 work as a description of its symmetries? Thank you very much.</p>
<p><strong>Added:</strong> When c... | Arkady | 23,522 | <p>Two groups are treated as same if there is an isomorphism between them.A simple reason why $S_5$ cannot be used to describe the symmetries of an icosahedron(whose group of symmetries we will call $I_h$) is that its structure is fundamentally different from that of $I_h$. For starters, $S_5$ cannot be expressed as a... |
126,553 | <p>In <a href="http://www.icpr2010.org/pdfs/icpr2010_MoAT5.1.pdf" rel="nofollow">this paper</a>, in the Formula at the beginning of 2.2, we have</p>
<p>$B=\{b_i(O_t)\}$</p>
<p>where </p>
<p>$i=0,1$ - the number of probability formula</p>
<p>$O_t$ - the state at moment $t$</p>
<p>$b_i(O_t)$ - two probabilities or e... | joriki | 6,622 | <p>This is a good example of why it makes sense to quote texts with more context, since it's often the context that supplies the clues for interpretation.</p>
<p>$B$ is referred to not just as a likelihood, but as "the likelihood [...] of [...] the frame [...] being a speech or a noise frame". This is rather badly phr... |
16,105 | <p>The answer to this question should be obvious, but I can't seem to figure it out. Suppose we have a surface $F$, and a representation $\rho : \pi_1(F)\to SU(n)$. We can define the homology with local coefficients $H_*(F,\rho)$ straightforwardly as the homology of the twisted complex $$C_*(F,\rho):=C_*(\widetilde{F};... | Paul | 3,874 | <p>What you say is right, and makes sense on any even dimensional manifold. Computing it can be tricky: a useful approach is to use a regular cell complex and the dual complex, then on the chain level the intersection form is given by the identity matrix (see the first couple pages of Milnor's "a duality theorem for Re... |
2,352,313 | <p>If $f_n$ is the number of permutations of numbers $1$ to $n$ that no number is in it's place(I think same as $D_n$)and $g_n$ is the number of the same permutations with exactly one number in it's place Prove that $\mid f_n-g_n \mid =1$.</p>
<p>I need a proof using mosly combinatorics not mostly algebra.I think we s... | Christian Blatter | 1,303 | <p>The <em><a href="https://en.wikipedia.org/wiki/Rencontres_numbers" rel="nofollow noreferrer">rencontres numbers</a></em> $D_n$ satisfy the recursion
$$D_n=(n-1)(D_{n-1}+D_{n-2})\ ,\tag{1}$$
which can be proven as follows: You obtain a derangement $\pi\in{\cal S}_n$ by picking a derangement $\pi'\in{\cal S}_{n-1}$ a... |
2,386,602 | <p>This is a question from an exam I recently failed. </p>
<p>What is the radius of convergence of the following power series? $$(a) \sum_{n=1}^\infty(n!)^2x^{n^2}$$ and $$(b) \sum_{n=1}^\infty \frac {x^{n^2}}{n!}$$</p>
<p>Edit: Here's my attempt at the first one, if someone could tell me if it's any good...</p>
<p>... | hamam_Abdallah | 369,188 | <p><strong>hint</strong></p>
<p>The ration test gives for $a) $,</p>
<p>$$\lim_\infty (n+1)^2e^{(2n+1)\ln (|x|)}$$
$$=0$$ if $|x|<1$.</p>
|
3,062,701 | <p>I want to solve this system by Least Squares method:<span class="math-container">$$\begin{pmatrix}1 & 2 & 3\\\ 2 & 3 & 4 \\\ 3 & 4 & 5 \end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix} =\begin{pmatrix}1\\5\\-2\end{pmatrix} $$</span> This symmetric matrix is singular with one eigenvalue <span ... | Damien | 621,834 | <p>The RLS solution is given by
<span class="math-container">$$ \hat x = A^+ \, b$$</span>
where <span class="math-container">$A^+$</span> is the pseudo inverse of <span class="math-container">$A$</span>.</p>
<p>As <span class="math-container">$A$</span> is not full rank, it is not possible effectively to calculate it... |
1,515,776 | <p>How can I solve something like this?</p>
<p>$$3^x+4^x=7^x$$</p>
<p>I know that $x=1$, but I don't know how to find it. Thank you!</p>
| cr001 | 254,175 | <p>For $x>1$ obviously $(3+4)^x > 3^x+4^x$ by binomial theorem.</p>
<p>For $1>x>0$, we have $(3^x+4^x)^{1\over x} > 3^{x{1\over x}}+4^{x{1\over x}}$ since ${1\over x} > 1$ and hence $3^x+4^x > (3+4)^x$</p>
<p>For $x< 0$, let $y=-x$ then $({1\over 3})^y+({1\over 4})^y>({1\over3})^y > ({1... |
1,515,776 | <p>How can I solve something like this?</p>
<p>$$3^x+4^x=7^x$$</p>
<p>I know that $x=1$, but I don't know how to find it. Thank you!</p>
| Narasimham | 95,860 | <p>By putting the given equation in the form:</p>
<p>$$ {\left( \dfrac {3}{3+4} \right ) } ^x + {\left( \dfrac {4}{3+4} \right ) } ^x =1 $$ </p>
<p>we find it is satisfied by $x=1$. The monotonic nature of exp function gives no other real roots.</p>
|
2,806,858 | <p>There is an equation
$$\sin2\theta=\sin\theta$$
We need to show when the right-hand side is equal to the left-hand side for $[0,2\pi]$.
<hr>
Let's rewrite it as
$$2\sin\theta\cos\theta=\sin\theta$$
Let's divide both sides by $\sin\theta$ (then $\sin\theta \neq 0 \leftrightarrow \theta \notin \{0,\pi,2\pi\}$)
$$2\cos... | Misha Lavrov | 383,078 | <p>Whenever you divide both sides of an equation by something, you are assuming that the thing you're dividing by is nonzero, because dividing by $0$ is not valid.</p>
<p>So going from $2 \sin \theta \cos \theta = \sin \theta$ to $2 \cos\theta = 1$ is only valid when $\sin\theta \ne 0$.</p>
<p>In general, all this me... |
4,090,408 | <p>Show that <span class="math-container">$A$</span> is a whole number: <span class="math-container">$$A=\sqrt{\left|40\sqrt2-57\right|}-\sqrt{\left|40\sqrt2+57\right|}.$$</span>
I don't know if this is necessary, but we can compare <span class="math-container">$40\sqrt{2}$</span> and <span class="math-container">$57$<... | Angelo | 771,461 | <p>Actually it is not necessary to know if <span class="math-container">$\;40\sqrt2\;$</span> is greater or less than <span class="math-container">$\;57$</span>.
Moreover we do not need to square <span class="math-container">$\;A\;$</span> or to solve any system of equations.</p>
<p>The check that the original poster d... |
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