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 |
|---|---|---|---|---|
2,210,372 | <p>I need to identify which of the curves $A_1,...,A_5$ are related to algorithms whose run times are proportional to $n, \log(n), n^2, n^3$ and $1.1^n$:</p>
<p>(Mentioning that the first figure of the $A_5$ column should be $0.015$ (not $0.025$)</p>
<p><a href="https://i.stack.imgur.com/70I2d.png" rel="nofollow nore... | Marcelo Fornet | 427,329 | <p>It can be done at sight I think, but you can try to adjust each function to each curve and pick the one that minimize the error using minimum square deviation. Let's say you want to check if curve A1 is proportional to $f(n)=n^2$. You try to minimize the function $g(a,b)=\Sigma(a f(x_i) + b - y_i)^2$ Here $x_i$ is t... |
674,598 | <p>(a) $f(x) = |x|; -\infty <x < \infty $</p>
<p>The way i approach is to take the derivative for x from 0 to infinity which gives me 1 and take the derivative for -x from 0 to negative infinity which gives me negative 1...therefore, since there is not constant can be 1 and -1 at the same time ....my answer is t... | Ben Grossmann | 81,360 | <p>Not quite. A function can be Lipschitz without being differentiable, and it can be differentiable without being Lipschitz.</p>
<p>For this problem, $1$ will work as a Lipschitz constant.</p>
<p>Why? Note that for any $x,y$, we have
$$
\left||x| - |y|\right| \leq |x-y|
$$</p>
|
1,359,743 | <p>If $\tan \theta = 3\cfrac{15}{16}$, then find $\sin \theta$.</p>
| SF Math | 254,277 | <p>Note that $\tan\theta=\frac{opp}{adj}$. Thus you have a right triangle with side lengths $\text{adj}=16$ and $\text{opp}=63$. I think you can solve it from here.</p>
|
1,403,228 | <blockquote>
<p><strong>Question:</strong></p>
<p>Let <span class="math-container">$m, n, q, r \in \mathbb Z$</span>. If <span class="math-container">$m = qn + r$</span>, show that <span class="math-container">$\gcd(m, n) = \gcd(n, r)$</span>. Hence justify the Euclidean Algorithm.</p>
</blockquote>
<p>I found this que... | John Douma | 69,810 | <p>$m-qn=r$ so if $c$ divides $m$ and $n$ it also divides r. Since this is true for all divisors of $m$ and $n$ it is true for $gcd(m,n)$ and so $gcd(m,n)\le gcd(n,r)$. If $d$ divides $n$ and $r$ then $d$ divides $qn+r$ which equals $m$ so $d$ is a common divisor of $m$ and $n$. Therefore $gcd(n,r)\le gcd(m,n)$ and $gc... |
2,871,084 | <p>Consider that we have a circle drawn a round the <strong>origin (0,0)</strong>. That circle has some points drawn on its circumference. Each of those points has range and azimuth $(r,\theta)$, the <strong>r</strong> and <strong>$\theta$</strong> values of these points are calculate with responding to the <strong>ori... | Quantic_Solver | 581,446 | <p>The map isn't surjective. A counterexample is the <a href="https://en.wikipedia.org/wiki/Weierstrass_function" rel="nofollow noreferrer">Weierstrass function</a> which is continuous.</p>
|
478,620 | <p>Let $S$ be a set that satisfies the following property:</p>
<blockquote>
<p>For every countable family $\{U_{i}\}_{i\in \mathbb{N}}$ of subsets of $S$ for which $$S=\bigcup_{i\in \mathbb{N}}U_{i}$$there is a finite set $I\subset \mathbb{N}$ for which $$S=\bigcup_{i \in I}U_{i}$$</p>
</blockquote>
<p>Does it foll... | qaphla | 85,568 | <p>There are four digits that the number can end with and be odd, not $\frac{9}{2}$, which is what your calculation uses -- that is, there are more even numbers without a five than odd numbers without a five.</p>
<p>More correctly:</p>
<p>$8 * 9 * 4 = 72 * 4 = 288$, as the first digit can be any of $1,2,3,4,6,7,8,9$,... |
478,620 | <p>Let $S$ be a set that satisfies the following property:</p>
<blockquote>
<p>For every countable family $\{U_{i}\}_{i\in \mathbb{N}}$ of subsets of $S$ for which $$S=\bigcup_{i\in \mathbb{N}}U_{i}$$there is a finite set $I\subset \mathbb{N}$ for which $$S=\bigcup_{i \in I}U_{i}$$</p>
</blockquote>
<p>Does it foll... | André Nicolas | 6,312 | <p>There is no reason that there are just as many odd integers that do not contain $5$ as there are even integers that do contain 5. The proper fraction is $\dfrac{4}{9}$.</p>
|
478,620 | <p>Let $S$ be a set that satisfies the following property:</p>
<blockquote>
<p>For every countable family $\{U_{i}\}_{i\in \mathbb{N}}$ of subsets of $S$ for which $$S=\bigcup_{i\in \mathbb{N}}U_{i}$$there is a finite set $I\subset \mathbb{N}$ for which $$S=\bigcup_{i \in I}U_{i}$$</p>
</blockquote>
<p>Does it foll... | Zulqarnain Ansari | 228,238 | <p>out of the nine digits 0,1,2,3,4,6,7,8, and 9. The digit at hundred place may be any digit other than 0, any of the nine digits can occupy tens place and the unit place can be occupied by 1,3,7 and 9. Thus the required number of three digits odd numbers will be 8*9*4=288</p>
|
1,450,669 | <p>Find the value of $$\lim\limits_{x\to 0^+}[1+[x]]^{\frac2x}$$where $[x]$ denotes greatest integer function less than or equal to $x$. <br></p>
<hr>
<p><strong>My attempt:</strong> <br>
I calculated $[1+[x]]$ to be $1$ as $x\to 0^+$. <br> Now I am stuck. <br> Please help me.</p>
| Landon Carter | 136,523 | <p>Recall that if $f(x)\to 1$ and $g(x)\to\infty$ as $x\to0$ then $\lim_{x\to0}f(x)^{g(x)}=e^{\lim_{x\to0}g(x)(f(x)-1)}$.</p>
<p>Here $f(x)=1+[x]\to1$ and $g(x)=2/x\to\infty$ as $x\to0+$. Thus the limit is $e^{\lim_{x\to0+}(2/x)[x]}$</p>
<p>Now as $x$ goes to $0$, $x<1$ must be true after some stage. Then for all ... |
3,493,729 | <p>Although I looked up the answer on integral calculator com but I still have little to no idea as to how one would proceed to solve this integral.
Integrate <span class="math-container">$\dfrac{e^x(x^4+2)}{(1+x^2)^{5/2}}$</span> wrt <span class="math-container">$x.$</span> I initially tried to convert it to the form ... | Henno Brandsma | 4,280 | <p><strong>Proposition</strong>: let <span class="math-container">$X$</span> be a topological space, then TFAE:</p>
<ol>
<li><span class="math-container">$X$</span> is connected.</li>
<li>If <span class="math-container">$O \subseteq X$</span> is clopen (closed and open), <span class="math-container">$O=\emptyset$</spa... |
306,178 | <p>Given
$$
y_n=\left(1+\frac{1}{n}\right)^{n+1}\hspace{-6mm},\qquad n \in \mathbb{N}, \quad n \geq 1.
$$
Show that $\lbrace y_n \rbrace$ is a decreasing sequence. Anyone can help ? I consider the ratio $\frac{y_{n+1}}{y_n}$ but I got stuck.</p>
| Mhenni Benghorbal | 35,472 | <p>Use the first derivative test and prove the function</p>
<p>$$ f(x)=\left(1+\frac{1}{x}\right)^{x+1}\hspace{-6 mm},\qquad \quad x \geq 1, $$</p>
<p>is decreasing on $[1,\infty]$. That is prove $f'(x)<0$ on $[1,\infty]$. </p>
|
3,425,780 | <p>Let <span class="math-container">$H_i$</span> be a <span class="math-container">$\mathbb C$</span>-Hilbert space and <span class="math-container">$T$</span> be a densely-defined linear operator from <span class="math-container">$H_1$</span> to <span class="math-container">$H_2$</span>.</p>
<blockquote>
<p>How can... | IrbidMath | 255,977 | <p>We know that the vertex x coordinates is <span class="math-container">$\frac{-b}{2a} = \frac{1}{4}$</span> solve for <span class="math-container">$b = \frac{-a}{2}$</span> now the point <span class="math-container">$\frac{1}{4} , \frac{-9}{8}$</span> satisfies the parabola so </p>
<p><span class="math-container">$\... |
4,212,199 | <p><span class="math-container">$$\frac15\int_0^5(4+2^{0.1t^2})dt$$</span>
This question is from part (b) of question 1 in the AP Calculus BC 2019 FRQ. I've tried using u-substitution where <span class="math-container">$u = 0.1 t^2$</span>, but this won't work because of the <span class="math-container">$2t$</span> tha... | John Douma | 69,810 | <p>Your integral is wrong. As you know, the total work done is given by the total force exerted over a distance. In this case the force on the cable is variable. We know that the starting weight is <span class="math-container">$200$</span> pounds and decreases by <span class="math-container">$2$</span> pounds for every... |
226,488 | <p>I am currently working on a challenge problem where I need to show that there is a point $x \in \mathbb{R_+}$ such that $\cos(x) = 0$ using only a few properties of the cosine function. In particular, the only properties of the cosine function that I can use are:</p>
<ul>
<li><p>$\cos(x)$ is continuous</p></li>
<li... | Eric Angle | 35,995 | <p>The circle is divided into $n$ wedges with angle
$$
\theta = \frac{2 \pi}{n}.
$$
Consider the triangle formed by the center of the circle (point $A$), one of the two adjacent points (point $B$), and the midpoint of the line joining the two adjacent points (point $C$). Now, $\angle BAC = \theta / 2 = \pi / n$, $AB = ... |
2,673,078 | <p>Generalize the Monty Hall problem where there are $n \geq 3$ doors, of
which Monty opens $m$ goat doors, with $1 \leq m \leq n$.<br><strong>Original Monty Hall Problem:</strong> There are $3$ doors, behind one of which there is a car (which you want), and behind the other two of which there are goats (which you don’... | David K | 139,123 | <p>A correct answer to the problem has been posted.
But you asked where your approach went wrong.</p>
<p>Suppose the event $C_1 \cap X_1$ is given.
In that event, according to the rules of the game, Monty will open
$m$ doors.
Apparently you have no information that would make one set of doors more
likely to be opened ... |
3,129,192 | <p>I've been trying to integrate
<span class="math-container">$$\int\frac{dx}{(x-a)(x-b)}$$</span></p>
<p>By using the substitution
<span class="math-container">$$x=a \cos^2 \theta + b \sin^2 \theta$$</span></p>
<p>The only problem here is I arrived at the result
<span class="math-container">$$\frac{2}{a-b} \ln |\cs... | tomasz | 30,222 | <p><strong>Hint</strong>: Since both spaces are clearly compact Hausdorff, it is enough to find a continuous bijection. Parametrise the annulus by polar coordinates, and then do some scaling.</p>
|
3,712,188 | <blockquote>
<p>Consider the vector field <span class="math-container">$\vec{u}=(xy^2,x^2y,xyz^2)$</span></p>
</blockquote>
<p>The curl of the vector field is <span class="math-container">$$\nabla \times\vec{u}=(xz^2,-yz^2,0)$$</span>
Consider the line integral of <span class="math-container">$\vec{u}$</span> around... | hdighfan | 796,243 | <p>The reason why your approach seems to fail is because this expression is trivially just <span class="math-container">$0$</span>. I'm assuming there's been a transcription error, but if not, then we have <span class="math-container">$$\begin{align}&\lim_{n\to\infty}\frac{1}{n^{n+1}}\sum_{k=1}^nk^p\\
=&\lim_{n... |
3,712,188 | <blockquote>
<p>Consider the vector field <span class="math-container">$\vec{u}=(xy^2,x^2y,xyz^2)$</span></p>
</blockquote>
<p>The curl of the vector field is <span class="math-container">$$\nabla \times\vec{u}=(xz^2,-yz^2,0)$$</span>
Consider the line integral of <span class="math-container">$\vec{u}$</span> around... | Kavi Rama Murthy | 142,385 | <p>Hint: <span class="math-container">$\int_1^{n} x^{p}dx= \sum\limits_{k=2}^{n}\int_{k-1}^{k}x^{p}dx$</span>. Note that <span class="math-container">$\int_{k-1}^{k}x^{p}dx$</span> lies between <span class="math-container">$k^{p}$</span> and <span class="math-container">$(k-1)^{p}$</span>. Using this, conclude that <sp... |
3,664,272 | <h2><strong>MOTIVATION</strong></h2>
<p>I am considering investing a significant amount of money into a raffle. In order to decide the number of entries I purchase, I would like to find probability distributions for the number of prizes I will win with respect to the number of entries I purchase.</p>
<h2><strong>HOW ... | Sirswagger21 | 785,596 | <p>Generally, you are correct in that the expected number of the distribution would more or less be the same. Obviously, going for a split in each is a high risk, high return probability.</p>
<p>The thing is, as you stated earlier, there is no way you will get a sure 100% for both raffles 1 and 2. Therefore, I estimat... |
2,904,359 | <p><strong>Definition.</strong> A formula <span class="math-container">$\phi(x,a)$</span> <em>divides</em> over a set <span class="math-container">$B$</span> if there are <span class="math-container">$k<\mathbb{N}$</span> and a sequence <span class="math-container">$(a_i)_{i<\omega}$</span> such that</p>
<p>(1)... | Mostafa Mirabi | 589,696 | <p>No, forking does not always imply dividing. In simple theories dividing and forking are the same, but they are not the same in general. Look at the following example. </p>
<p><strong>Example.</strong> Let $\mathcal{L}=\{ R^{(3)} \}$ be a language which consists of a ternary relation. Consider the $\mathcal{L}$-stru... |
284,995 | <p>There is a theorem that if a Hopf algebra $H$ is commutative or cocommutative, then $S^2=id_H$, where $S$ denotes the antipode.</p>
<p>May I know if the converse is true?</p>
<p>(i.e. if $S^2=id_H$, does it follow that $H$ is commutative or cocommutative?)</p>
<p>If no, what would be the simplest counter-example?... | JP McCarthy | 19,352 | <p>The Eight Dimensional Kac-Paljutkin Hopf Algebra has an involutive antipode but is neither commutative nor cocommutative.</p>
|
61,174 | <p>I was just thinking about this recently, and I thought of a possible bijection between the natural numbers and the real numbers. First, take the numbers between zero and one, exclusive. The following sequence of real numbers is suggested so that we have bijection. </p>
<p>0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, ... | hmakholm left over Monica | 14,366 | <p>Your function ignores all the real numbers whose decimal representations are not finite, such as </p>
<p>$\dfrac13=0.3333\ldots$</p>
<p>The subset of real numbers that do have finite decimal representations is indeed countable (also because they are all rational and $\mathbb Q$ is countable).</p>
|
61,174 | <p>I was just thinking about this recently, and I thought of a possible bijection between the natural numbers and the real numbers. First, take the numbers between zero and one, exclusive. The following sequence of real numbers is suggested so that we have bijection. </p>
<p>0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, ... | Liam Gavin Murray | 217,943 | <p>The set you have shown is a list of all rationals between $0$ and $1$ that can be written in the form $x /10^n$ with $x\in\mathbb{Z}$, which is countable. But the full set of reals between $0$ and $1$ is bigger.</p>
<p>All reals are the limit of some sub-sequence of this sequence, but not all are in this sequence, ... |
2,939,605 | <p>The given task goes as follows:</p>
<blockquote>
<p>Show that <span class="math-container">$ f: \mathbb{R} \longrightarrow \mathbb{R}$</span> defined by <span class="math-container">$f(x) = \sqrt{1 + x^2} $</span> is not a polynomial function.</p>
</blockquote>
<p>I tried this approach - if <span class="math-con... | dxiv | 291,201 | <p>Alt. hint: following up on OP's idea to use derivatives, note that <span class="math-container">$\,f(x) \cdot f'(x) = x\,$</span>, but the only polynomials which satisfy that identity are <span class="math-container">$\,f(x)=\pm x\,$</span> (<em>why?</em>).</p>
<p>Alternatively, show that there exists no polynomial... |
2,846,700 | <p>Consider the functions $f(x)$, $g(x)$, $h(x)$, where $f(x)$ is neither odd nor even, $g(x)$ is even and $h(x)$ is odd. Is it possible for $f(x) + g(x)$ to be </p>
<ol>
<li>even;</li>
<li>odd?</li>
</ol>
<p>For the second case I can imagine for example $f(x) = x - 1$ and $g(x) = 1$. Then $f$ is neither even nor od... | Florian | 185,854 | <p>Well, by definition $f(x)+g(x)$ is even if and only if $f(x)+g(x)=f(-x)+g(-x)$ for all $x$. If $g(x)$ is even, it satisfies $g(x)=g(-x)$ for all $x$. Inserting this into the previous equation gives $f(x)=f(-x)$ for all $x$, hence $f$ must be even.</p>
<p>Why did it work for odd? Here the definition requires $f(x)+... |
72,098 | <p>This is my first time using Stack Exchange, but it looks like a good resource. I am here to ask a couple questions about my homework. We're working from the latest edition of <em>Abstract Algebra</em> by Herstein. This is problem #3 from p. 73 of that book, and it reads as follows:</p>
<blockquote>
<p>Let $G$ be ... | Zev Chonoles | 264 | <p>Take a look at the problem again:</p>
<p><img src="https://i.stack.imgur.com/CRArC.png" alt="enter image description here"></p>
<p>Part a) says that
$$L_a\in A(G),$$
where the symbol in the middle denotes "is an element of" (<a href="http://en.wikipedia.org/wiki/Element_(mathematics)#Notation_and_terminology" rel=... |
2,735,854 | <p>How many permutations of the word $STRESSLESSNESS$ begin OR end with an $E$? </p>
<p>Correct me if I'm wrong, but you would have to subtract the permutations where $E$ begins AND ends the permutation?</p>
| Dan | 508,768 | <p><em>STRESSLESSNESS</em> is 14 letters long.</p>
<p>If we take away an <strong>e</strong>, it becomes 13 letters long.</p>
<p>The permutations of this 13 letter string is:</p>
<p>$$\frac{13!}{7!2!}$$</p>
<p>Since the <strong>E</strong> can occur at the beginning or the end, we multiply this by 2 to get:
$$\frac{1... |
186,310 | <p>I am trying to evaluate: </p>
<pre><code>For[i = 1, i < 100000, i++, h[0] = hs;
h[i]=hi /. Check[FindRoot[Xfnew[a, w, hi, mxs + i*step]*92*10^23) == 1198/10000, {hi, h[i - 1]}]];
If [h[i] == 25, Break[], AppendTo[hin, {mxs + i*step, h[i]}]]];
</code></pre>
<p>where mxs , hs, and step has some values, and hin ... | Titus | 30,703 | <p>An example with Table. <code>f[x_]</code> is a quadratic and every step the constant term changes. The idea is to store the last solution every iteration and use it in next i's solution, THEN store the new solution. I have tried it in various setups (including this) and it works, but feel free to check in your appli... |
2,698,284 | <p>Here is the problem: </p>
<p>Let A be a set with n elements. Find an expression for S(n, 2), the number of partitions of A into exactly two subsets. You can either start with the general recurrence for S(n, k), or count S(n, 2) directly. </p>
<p>I'm having trouble understanding what exactly it wants me to do. So f... | Ross Millikan | 1,827 | <p>Choose one subset of $A$. The other subset in the partition is everything else. You then need to divide by $2$ because you could have chosen the other subset first and gotten the original subset as the everything else.</p>
|
2,698,284 | <p>Here is the problem: </p>
<p>Let A be a set with n elements. Find an expression for S(n, 2), the number of partitions of A into exactly two subsets. You can either start with the general recurrence for S(n, k), or count S(n, 2) directly. </p>
<p>I'm having trouble understanding what exactly it wants me to do. So f... | Mohammad Riazi-Kermani | 514,496 | <p>Suppose your set is $\{ 1,3,8\}$
Our partitions are $$\{\{1\},\{3,8\}\}\\ \{\{3\},\{1,8\}\}\\ \{\{8\},\{1,3\}\} $$</p>
<p>Each partition has 2 disjoint sets whose union is the original set.</p>
|
2,655,178 | <p>I am asked to find the equation of a cubic function that passes through the origin. It also passes through the points $(1, 3), (2, 6),$ and $(-1, 10)$. </p>
<p>I have walked through many answers for similar questions that suggest to use a substitution method by subbing in all the points and writing in terms of vari... | amd | 265,466 | <p>An equation of the cubic that passes through the four points $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$ and $(x_4,y_4)$ is $$\begin{vmatrix} x^3 & x^2 & x & y & 1 \\ x_1^3 & x_1^2 & x_1 & y_1 & 1 \\ x_2^3 & x_2^2 & x_2 & y_2 & 1 \\ x_3^3 & x_3^2 & x_3 & y_3 &... |
212,466 | <p>Consider the list <code>list={1,2,3,4,5,6,7,8}</code> and imagine I wanted to change it to <code>{{1,2,3},{4,5,6},{7,8}}</code>. How can I do this?</p>
<p>Using <code>ArrayReshape</code> in the usual manner</p>
<pre><code>list = {1, 2, 3, 4, 5, 6, 7, 8};
ArrayReshape[list, {3, 3}]
</code></pre>
<p>yields <code>{{... | MarcoB | 27,951 | <p>Using <code>Partition</code> with <code>UpTo</code> is clearly the most expedient answer here, as shown above. However, this got me wondering how one could force <code>ArrayReshape</code> into service anyway, just for fun. For instance, we could twist the use of the padding parameter in <code>ArrayReshape</code> to ... |
1,390,593 | <p>"Since the operation of left multiplication is faithful, $G$ is isomorphic to its image in $\operatorname{Perm}(G)$. If $G$ has order $n$, $\operatorname{Perm}(G)$ is isomorphic to $S_n$." - Artin's Algebra</p>
<p>So left multiplication as an action means $\phi_g(x)=gx$</p>
<p>So proving isomorphism, means bijecti... | Bernard | 202,857 | <p>You're confusing $G$ as a group and $G$ as left-acting on itself. The homomorphism is this:
\begin{align*}
\varphi\colon G&\to \operatorname{Perm}G\\
g&\mapsto (\varphi_g\colon a\mapsto ga)
\end{align*}</p>
<p>It is a homomorphism because $\;\varphi_{gg'}\colon a\mapsto(gg')a=g(g'a)=\varphi_g\bigl(\varphi_{... |
2,054,338 | <p>I had this question on my exam and I thought I could solve it using the row echelon method. Well, I couldn't, and I still don't know how to solve it. We were asked to "determine $a$ such the equation system has a unique solution" and to determine $a$ itself. The system had three equations: </p>
<p>$2x - y + az = 3$... | Robert Israel | 8,508 | <p>Hint: divide by $a$ and integrate.</p>
|
2,411,811 | <p>Determine the limit:</p>
<p>$$\lim_{h\to 0}\frac{1-\cos(2h)}{h}$$</p>
<p>I know that the answer is $0$, I am just unsure of how to solve this. Thanks.</p>
| imranfat | 64,546 | <p>Let's go completely NUTS on this one. Let $2h=x$ so that the limit becomes$$2\lim_{x\to 0}\frac{1-\cos(x)}{x}$$ Now let $x=\pi/2-t$ so that the limit becomes$$2\lim_{t\to \pi/2}\frac{1-\cos(\pi/2-t)}{\pi/2-t}=2\lim_{t\to \pi/2}\frac{sint-1}{t-\pi/2}$$
Now apply the limit definition of the derivative for $sine$ at $t... |
2,607,733 | <p>Prove the inlcusion-exclusion principle using the fact that for three pairwise disjoint sets $X$, $Y$, $Z$:</p>
<p>$$|X\cup{Y}\cup{Z|}=|X|+|Y|+|Z|$$</p>
<p>I tried setting $X\cup{Y}=A$, but arrive at $|X\cup{Y|}=|X|+|Y|$ due to the sets being disjoint.</p>
| Community | -1 | <p>Other answers are pointing to the formal reasons why $d(x,y)=\|x+y\|$ is not a metric. However, I would also suggest that you build your geometrical intuition here.</p>
<p>In geometry, when you have points $A$ and $B$ and the origin $O$ in, say, a plane, you would calculate the vector $\vec{AB}=\vec{OB}-\vec{OA}$ a... |
1,098,587 | <blockquote>
<p>Prove that for a bipartite graph $G$ on $n$ vertices the number of edges in $G$ is at most $\frac{n^2}{4}$. </p>
</blockquote>
<p>I used induction on $n$. </p>
<p>Induction hypothesis: Suppose for a bipartite graph with less than $n$ vertices the result holds true. </p>
<p>Now take a bipartite g... | abhishek kumar | 589,020 | <p>suppose the bipartite graph has m vertices in set1 and n vertices in set2 . the sum of degrees of all vertices would be 2 * e= 2 * m * n
thus, e = m * n
and m+n = v
so
n*(v-n)= e
solve this quadratic equation and discriminant . </p>
|
3,166,201 | <blockquote>
<p>If <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are prime numbers which satisfy <span class="math-container">$x^2-2y^2=1$</span>, solve for <span class="math-container">$x$</span> and <span class="math-container">$y$</span>.</p>
</blockquote>
<p><strong>My attem... | Dr. Mathva | 588,272 | <p>What about </p>
<blockquote>
<p><span class="math-container">\begin{align*}&x^2-2y^2=1\tag{1}\\\iff & x^2-1=(x+1)(x-1)=2y^2\end{align*}</span></p>
</blockquote>
<p>Since <span class="math-container">$2\mid (x+1)(x-1)$</span>, we conclude that both <span class="math-container">$(x+1)$</span> and <span cla... |
356,279 | <p><strong>Definitions</strong> (<a href="https://www.ias.ac.in/article/fulltext/pmsc/122/03/0459-0467" rel="nofollow noreferrer">https://www.ias.ac.in/article/fulltext/pmsc/122/03/0459-0467</a>):</p>
<p>Given a planar convex region <span class="math-container">$C$</span> (could be smooth or polygonal), an <strong>area... | Joseph O'Rourke | 6,094 | <p>This is not an answer, and not even that helpful, but
I wanted to see the central pattern formed by the collection of
perimeter bisectors.
<hr />
<img src="https://i.stack.imgur.com/W8N0K.jpg" width="250" />
<br />
<img src="https://i.stack.imgu... |
1,612,859 | <p>Take $i(G)$ to be the independence number of $G$, i.e. the maximum
number of
pairwise nonadjacent vertices in $G.$ I want to show that if $G$ has
$n$ vertices and $\frac{nk}{2}$ edges where $k \geq 1,$
$i(G) \geq \frac{n}{k+1}.$ I'm having difficulties making the
combinatorial argument on this graph. I get that if t... | Leen Droogendijk | 95,972 | <p>If you are allowed to use Turan's theorem, a simpler solution is possible.</p>
<p>$\overline G$ has $e=\binom n2-\frac{nk}2=\frac n2(n-1-k)$ edges.
Now $i(G)\geq\frac n{k+1}$ $\iff$ $\overline G$ has a clique of size at least $\frac n{k+1}$.
From Turan we know that avoiding a clique of size $r+1$ requires $e\leq(1-... |
549,411 | <p>How would I go about generalizing the product rule to the product of $n$ functions $\psi_1(x), \ \psi_2(x), ..., \ \psi_n(x)$? That is, I'm hoping to obtain an expression for</p>
<p>$$
\frac{d}{dx} \prod_{j = 1}^n \psi_j(x)
$$</p>
| Marius Damian | 118,834 | <p>Suppose that exists $m,n\in\mathbb{N^*}$ such that $[n\sqrt{2}]=[(2+\sqrt{2})m]=t\in\mathbb{N^*}.$
Then $t<n\sqrt{2}<n+1,\quad t<(2+\sqrt{2})m<t+1\implies \dfrac{t}{\sqrt{2}}<n<\dfrac{t+1}{\sqrt{2}},\quad \dfrac{t}{2+\sqrt{2}}<m<\dfrac{t+1}{2+\sqrt{2}}\stackrel{(+)}{\implies} t<n+m<t+1,... |
631,348 | <p>Let $\sum a_n$ be a series of non-negative terms and let $$L = \lim_{n\to\infty}n\left(1-\frac{a_{n+1}}{a_n}\right)$$
Prove that the series converges (resp. diverges) if $L > 1$ (resp. $L<1$). I've tried, for example, that when $L<1$, $$n\left(1-\frac{a_{n+1}}{a_n}-L\right)= \frac{n(a_n-a_{n+1}-La_n)}{a_n}\... | ziang chen | 38,195 | <p>If $L>1$, choose $\epsilon\gt 0$, $L-\epsilon>1$ then </p>
<p>$$1-\frac{L-\epsilon}{n} > \frac{a_{n+1}}{a_n}$$</p>
<p>Choose $p$ such that $1\lt p\lt L-\epsilon$, $\sum\frac1{n^p}$ converges. $b_n=\frac1{n^p}$, if $n$ big enough, then</p>
<p>$$\frac{b_{n+1}}{b_n}=(1-\frac{1}{n+1})^p=1-\frac{p}{n}+O(\fra... |
512,960 | <p>Let's say I have a statement: if p then q.<br>
The converse would be: if q then p.<br>
The inverse would be: if not p then not q.<br>
The contraposition would be: if not q then not p.<br>
What would you call the following? if not p then q.<br>
Thanks.</p>
| Rustyn | 53,783 | <p>$$
p \Rightarrow q \equiv \lnot p \lor q $$</p>
<p>$$
\lnot p \Rightarrow q \equiv p\lor q
$$
It is logically equivalent to "$p$ or $q$"</p>
|
2,202,048 | <p>There's a box with $N$ balls in it. One of them is red and the others white. What's the probability of getting the red ball at the $k$th try ( if you're not putting them back in?) where $k = 1,2,3,\ldots,N$</p>
| Sum | 369,413 | <p>The answer will be straightforward,</p>
<p>If the red ball has to come at the $k^{th}$ instance, we must draw white balls till the $(k-1)^{th}$ instance. Probability of getting white balls till $(k-1)^{th}$ instance will be:</p>
<p>$$\frac{N-1}{N} \frac{N-2}{N-1} \cdots \frac{N-k}{N-k+1} = \frac{N-k}{N}$$</p>
<p>... |
947 | <p>I'm looking for the algorithm that efficiently locates the "loneliest person on the planet", where "loneliest" is defined as:</p>
<p>Maximum minimum distance to another person — that is, the person for whom the closest other person is farthest away.</p>
<p>Assume a (admittedly miraculous) input o... | Scott Morrison | 3 | <p>Let's iteratively come up with a candidate, "the loneliest person so far", and at each step weed out lots of people who we can see aren't as lonely.</p>
<p>Pick a person at random, and calculate their "loneliness". Since they're the first, they are by default the loneliest person so far.</p>
<p>Now let's start sea... |
2,566,546 | <p>Show that for any $x_1 < x_2$ and $y_1 < y_2$ one has
$P(x_1 < X ≤ x_2, y_1 < Y ≤ y_2) = F(x_2, y_2) + F(x_1, y_1) − F(x_1, y_2) − F(x_2, y_1)$.</p>
<p>Would I just need to split the LHS to something that gives me the right?</p>
| ajotatxe | 132,456 | <p>Consider
$$Q=\begin{pmatrix}0&-1\\-1&0\end{pmatrix}$$</p>
<p>Let $P=Q^2$. Since $P=I$, $P$ is a projection, but $Q$ is not.</p>
|
1,160,699 | <p>In my discrete math book, I was tasked with finding a counterexample for this:</p>
<blockquote>
<p>If $n$ is prime, then $2^n-1$ is prime.</p>
</blockquote>
<p>Does there exist a counterexample for such a statement? Also, am I wrong in thinking that when something asks for a counterexample, is it looking for som... | mathamphetamines | 126,882 | <p>A counter-example is one instance in which the statement does not hold to be true. Let n=11. $2^{11} - 1$ is a composite number.</p>
<p>$2^{11}-1=2047 = 23 \cdot 89$</p>
|
1,160,699 | <p>In my discrete math book, I was tasked with finding a counterexample for this:</p>
<blockquote>
<p>If $n$ is prime, then $2^n-1$ is prime.</p>
</blockquote>
<p>Does there exist a counterexample for such a statement? Also, am I wrong in thinking that when something asks for a counterexample, is it looking for som... | Robert Soupe | 149,436 | <p>No, it's not. Among the first twenty thousand primes $p$, only thirty-one of them give primes when plugged into the formula $2^p - 1$. In fact, you could have your computer give you a pseudorandom prime number $p < 10^7$ and I would wager you \$100 that $2^p - 1$ is composite.</p>
<p>For a counterexample, at lea... |
3,122,612 | <p>I am doing a presentation on nonlinear optics and I ran into a paper that uses a series to describe a wave equation of a system. The paper will be linked below and the series is:</p>
<p><span class="math-container">$\sum_{n=1}^{\infty}\frac{(-i)^{n-1}}{n!}k^n$</span></p>
<p>I mostly am looking for what i should ca... | Community | -1 | <p><span class="math-container">$$\sum_{n=1}^\infty\frac{(-ik)^n}{(-i)\,n!}=i(e^{-ik}-1)=i(\cos k-1-i\sin k).$$</span></p>
<hr>
<p>This can be rewritten as</p>
<p><span class="math-container">$$2i\cos\frac k2\left(\cos\frac k2-i\sin\frac k2\right)=2i\cos\frac k2e^{-ik/2}$$</span> but there is little benefit.</p>
|
1,025,588 | <p>I just started doing AM-GM inequalities for the first time about two hours ago. In those two hours, I have completed exactly two problems. I am stuck on this third one! Here is the problem:</p>
<p>If $a, b, c \gt 0$ prove that $$ a^3 +b^3 +c^3 \ge a^2b +b^2c+c^2a.$$</p>
<p>I am going crazy over this! A hint or pro... | Adriano | 76,987 | <p>One trick that shows up a lot with AM-GM inequalities is to apply AM-GM several times, then combine them all together somehow. Let's try that. AM-GM tells us that:
$$
x + y + z \geq 3 \sqrt[3]{xyz} \tag{$\star$}
$$
Now for what values of $x,y,z$ would the LHS and RHS of $(\star)$ match up with some of the terms in t... |
1,429,563 | <blockquote>
<p>Let <span class="math-container">$\{F_n\}_{n\geq 0}$</span> be the Fibonacci sequence.</p>
<p>Prove that the number of primes <span class="math-container">$p$</span> so that <span class="math-container">$p\mid F_{p-1}$</span> is infinite.</p>
</blockquote>
<hr />
<p>I tried to use induction, to no avail... | N. S. | 9,176 | <p><strong>Hint 1:</strong> Prove that if the equation
$$x^2-x-1=0 \pmod{p}$$
has two solutions $x_1 \neq x_2$ then there exists constants $C_1, C_2$ such that</p>
<p>$$F_n \equiv C_1x_1^n+ C_2x_2^n \pmod{p}$$</p>
<p><strong>Hint 2:</strong> In this case what can you say about $F_{p-1}$ and $F_0$?</p>
<p><strong>Hi... |
484,550 | <p>The problem is as follows: let $n_1, n_2,..., n_t$ be positive integers. Prove that if $n_1+n_2+...+n_t-t+1$ objects are placed into $t$ boxes, then for some $i, i=1, 2, ..., t$, the $i$th box contains at least $n_i$ objects. </p>
<p>I'm having difficulty getting started in developing a proof because I have no in... | Michael Joyce | 17,673 | <p>Suppose, for the sake of contradiction, that the $i$-th box always contains $< n_i$ objects. Then what is the largest the total number of objects could be? (This occurs when the $i$-th box has $n_i - 1$ objects.) But how many objects are there in total?</p>
|
987,802 | <p>I've solved forming $8$ digit number using $1,1,2,2,2,3,3,3$. We have 8 digits: two 1, three 2 and three 3. First I put three 2's in 8 possible places. Number of putting 2's is $\frac{8!}{3!5!}$ . After putting 2's we have 5 possible empty places left. We put three 3's: $\frac{5!}{3!2!}$ . Then we have 2 empty place... | paw88789 | 147,810 | <p>Line up the violinists say in alphabetical order. </p>
<p>Now pick a violist, a cellist, and a bass for the first violinist ($10^3$ ways).</p>
<p>Now pick a violist, a cellist, and a bass for the second violinist ($9^3$ ways).</p>
<p>And so on.</p>
<p>Do you see what happens from here?</p>
|
987,802 | <p>I've solved forming $8$ digit number using $1,1,2,2,2,3,3,3$. We have 8 digits: two 1, three 2 and three 3. First I put three 2's in 8 possible places. Number of putting 2's is $\frac{8!}{3!5!}$ . After putting 2's we have 5 possible empty places left. We put three 3's: $\frac{5!}{3!2!}$ . Then we have 2 empty place... | Brian M. Scott | 12,042 | <p>Alternatively, after lining up the violinists as in paw88789’s answer, you can choose all of the violists at once by lining them up in front of the violinists; how many ways are there to line up $10$ people? Then do the same with the cellists, and again with the bassists. It’s the same idea, but with the choices mad... |
3,672,814 | <p><span class="math-container">$g_n(y)=\sum_{k=1}^{n} \frac{\sin ky}{k},x \in [0,\pi]$</span>. So i try to find the pointwise limit function first, following this approach in this post <a href="https://math.stackexchange.com/questions/13490/proving-that-the-sequence-f-nx-sum-limits-k-1n-frac-sinkxk-is">Proving that th... | MathematicsStudent1122 | 238,417 | <p><span class="math-container">$\sum_{n \geq 1} \frac{\sin nx}{n}$</span> is exactly the Fourier series of <span class="math-container">$f(x)=\frac{\pi-x}{2}$</span> (<span class="math-container">$x \in [0, 2\pi)$</span>) extended <span class="math-container">$2\pi$</span> periodically to <span class="math-container">... |
183,058 | <p>I'm having difficulty producing a proper animated gif (for inclusion in a PowerPoint presentation). The code I have is this:</p>
<pre><code> HSBhsl[h_, s_, l_] :=
Module[{H, S, B}, H = h; B = l + s (1 - Abs[2 l - 1])/2;
S = (2 B - l)/B; {H, S, B}]
TestCube[\[Theta]_] :=
Graphics3D[Rotate[{EdgeForm[], Hue[HSB... | Piotr Wendykier | 9,239 | <p>To export to an animated PNG with transparency you should convert <code>Graphics</code> to <code>Image</code> first: </p>
<pre><code>list = Table[Image[Show[TestCube[t], ViewPoint -> {1000, 1000, 1000}]], {t,
0, 360, 1}];
</code></pre>
<p>Then, you call <code>Export</code>: </p>
<pre><code>In[9]:= Expo... |
3,080,294 | <blockquote>
<p>A cone has its guiding curve to the circle <span class="math-container">$x^2+y^2+2ax+2by=0$</span> and passes through a fixed point <span class="math-container">$(0,0,c)$</span>. If the section of the cone by plane <span class="math-container">$y=0$</span> is a rectangular hyperbola. Prove that the ve... | Blue | 409 | <p>That the guiding curve is a circle implies that the axis of the cone is perpendicular to the plane of the circle (that is, the <span class="math-container">$xy$</span>-plane, which we'll call "the horizontal") and passes through the circle's center, <span class="math-container">$(-a,-b,0)$</span>. </p>
<p>Also, it ... |
4,322,897 | <p>Let <span class="math-container">$\{y_n\}_{n=1}^\infty$</span> be a sequence in <span class="math-container">$\Bbb{R}^n$</span> and <span class="math-container">$z\in\Bbb{R}^n$</span> be given. If <span class="math-container">$\{\|z-y_n\|\}_{n=1}^\infty$</span> is a convergent sequence, what can we say about <span c... | Literally an Orange | 314,721 | <p>If <span class="math-container">$x_n$</span> is a real-valued sequence then <span class="math-container">$x_n \to x$</span> implies that for some large <span class="math-container">$N$</span>, <span class="math-container">$|x_n-x|<1$</span> for <span class="math-container">$N \leq n$</span>. Now let <span class=... |
2,772,190 | <p>The question:</p>
<blockquote>
<p>Determine the solution set (in $\mathbb R$) for the equation $|x^2+2x+2| = |x^2-3x-4|$</p>
</blockquote>
<p>So far, I have determined that for this to be true, $|x^2-3x-4|$ must be greater or equal to $0$, giving $(x-4)(x+1)\ge0$. To find the solution set, $|x^2+2x+2|\ge-1$ as i... | Servaes | 30,382 | <p>Obviously $|x^2-3x-4|$ is greater than or equal to $0$, because you take the absolute value. This observation tells you nothing. In stead, distinguish cases where the quadratics are positive or negative.</p>
|
2,772,190 | <p>The question:</p>
<blockquote>
<p>Determine the solution set (in $\mathbb R$) for the equation $|x^2+2x+2| = |x^2-3x-4|$</p>
</blockquote>
<p>So far, I have determined that for this to be true, $|x^2-3x-4|$ must be greater or equal to $0$, giving $(x-4)(x+1)\ge0$. To find the solution set, $|x^2+2x+2|\ge-1$ as i... | Cesareo | 397,348 | <p>Knowing that $x^2+2x+2 > 0$</p>
<p>The equation is equivalent to</p>
<p>$$
x^2+2x+2 = \vert x^2-3x-4\vert
$$</p>
<p>which is equivalent to</p>
<p>$$
x^2+2x+2 =
\left\{\begin{array}{lcl}-x^2+3x+4 & \rightarrow & x = \{\frac{1\pm\sqrt{17}}{4}\}\\
x^2-3x-4 & \rightarrow & x = -\frac{6}{5}\end{ar... |
1,012,985 | <p>Let $A = \left[ \begin{matrix} 3 & 2 & 1\\ 5 & 0 &1\end{matrix}\right]$, </p>
<p>how can I know if there is a matrix $N$ , st. $AN=0$ (N is not a zero matrix) </p>
| klimenkov | 161,422 | <p>Let $N = \pmatrix{x_1 & x_2 \\ x_3 & x_4 \\ x_5 & x_6}$.</p>
<p>If you mutiply $A$ and $N$ you will get 4 equations with 6 variables. If you let any 2 of these variables be non-zero, you will get the system of 4 equations with 4 variables. When you solve it, you will get the elements of your matrix and ... |
4,363,384 | <p>Let <span class="math-container">$A_i$</span>, <span class="math-container">$i \in I$</span> subsets of a space X given topology <span class="math-container">$\tau$</span>. Show that <span class="math-container">$\overline{\bigcup_i A_i }\subset \bigcup_i\overline{ A_i }$</span> does not hold necessarily :</p>
<p>... | tomasz | 30,222 | <p>Your example is fine.</p>
<p>The equality <span class="math-container">$\overline{A\cup B}=\bar A\cup \bar B$</span> only tells you about <em>binary</em> unions (and, by induction, finite unions), not arbitrary ones. This is related to the fact that a <em>finite</em> union of closed sets is closed (but not an arbitr... |
1,675,411 | <p>So far I have this:</p>
<p>First consider $n = 5$. In this case $(5)^2 < 2^5$, or $25 < 32$. So the inequality holds for $n = 5$.</p>
<p>Next, suppose that $n^2 < 2^n$ and $n \geq 5$. Now I have to prove that $(n+1)^2 < 2^{(n+1)}$.</p>
<p>So I started with $(n+1)^2 = n^2 + 2n + 1$. Because $n^2 < 2... | MathematicsStudent1122 | 238,417 | <p>The base case is clear. The inductive step is done as follows. Suppose it holds for some $k\geq5$. That is, $k^2<2^k$</p>
<p>We know $(k+1)^2<2k^2 \Longleftrightarrow \frac 12 (k+1)^2 - k^2 < 0$ for $k\geq 5$ (this can be shown rigorously determining the roots of the polynomial and noting that it changes s... |
915,542 | <p>After working on an ODE I find I am needing to solve the integral </p>
<p>$$\int \frac{u}{b - au - u^2}\mathrm{d}u$$</p>
<p>Trig subs, banging heads against walls, and sobbing have not yielded a solution. Yet. </p>
<p>Could use a hand, thanks.</p>
| Jean-Claude Arbaut | 43,608 | <p>You can write $b-au-u^2=\frac{a^2+4b}{4}-(u+\frac a2)^2$</p>
<p>There are thus three cases to consider</p>
<p><strong>A. $a^2+4b=0$</strong></p>
<p>Then your integral is</p>
<p>$$\int\frac{\mathrm{d}u}{b-au-u^2}=-\int\frac{\mathrm{d}u}{(u+\frac a2)^2}=\frac{1}{u+\frac a2}+C$$</p>
<p><strong>B. $a^2+4b>0$</st... |
2,509,230 | <p>I was studying Chevalley-Serre relations, which can be summed up to these</p>
<p><span class="math-container">$$\tag{S1}\left[h_{i},\,h_{j}\right]=0$$</span></p>
<p><span class="math-container">$$\tag{S2}\left[e_{i},\,f_{i}\right]=h_{i} \quad \left[e_{i},\,f_{j}\right]=0 \quad\text{for } i\neq j$$</span></p>
... | Tobias Kildetoft | 2,538 | <p>It might help if we label the generators by the corresponding roots instead (since this is where they come from). When we do this, we get that if $[e_{\alpha_i},e_{\alpha_j}]$ is in the root space corresponding to $\alpha_i + \alpha_j$, so since there are only finitely many roots, at some point, if we keep taking br... |
2,056,979 | <p>I am attempting:
<a href="https://i.stack.imgur.com/FM80Z.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/FM80Z.png" alt="enter image description here"></a></p>
<p>My solution is: But I am not sure where I am going wrong. The answer I get is not divisible by 7.
<a href="https://i.stack.imgur.com/... | paw88789 | 147,810 | <p>Hint: $4^{n+2}+5^{2n+1}=4(4^{n+1})+25(5^{2n-1})=4(4^{n+1}+5^{2n-1})+21(5^{2n-1})$.</p>
|
2,056,979 | <p>I am attempting:
<a href="https://i.stack.imgur.com/FM80Z.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/FM80Z.png" alt="enter image description here"></a></p>
<p>My solution is: But I am not sure where I am going wrong. The answer I get is not divisible by 7.
<a href="https://i.stack.imgur.com/... | Bill Dubuque | 242 | <p>Before considering your proof, let's gather some insight from a simpler proof using congruences. Below the inductive step follows very simply by using $\,\rm{\color{#C00}{CPR}} = $ <a href="https://math.stackexchange.com/a/879262/242">Congruence Product Rule</a> to multiply the first two congruences </p>
<p>$$\begi... |
4,126,527 | <p>Suppose <span class="math-container">$w_1$</span>, <span class="math-container">$w_2$</span>, ..., <span class="math-container">$w_{n-1}$</span> are the complex roots not equal to <span class="math-container">$1$</span> of <span class="math-container">$z^n-1=0$</span>, where <span class="math-container">$n$</span> i... | user | 293,846 | <p>Since <span class="math-container">$n$</span> is odd we know: <span class="math-container">$w\ne-1$</span>.
Therefore, using <span class="math-container">$(1+w)(1+\bar w)\ne0$</span> and <span class="math-container">$w\bar w=1$</span>:
<span class="math-container">$$\begin{align}
&\frac{1-\bar{w}}{1+w}+\frac{1-w... |
2,062,671 | <p>How many sequences of positive integer numbers $\{a_n\}$ such that $$a_0 =1, a_1 = 2,|a_{n+2}a_n - a_{n+1}^2| = 1 ?$$</p>
| Will Jagy | 10,400 | <p>Ambiguous question. </p>
<p>Define the rational numbers
$$ b_n = \frac{a_{n+1} + a_{n-1}}{a_n}. $$
IF we always have the constant $C$ with
$$ a_n a_{n+2} - a_{n+1}^2 = C, $$
then a simple calculation shows that
$$ \frac{b_{n+1}}{b_n} = 1, $$ so that we have a constant $\tau$ with
$$ \frac{a_{n+1} + a_{n-1}}{a_... |
4,264,808 | <p>So we know that <span class="math-container">$\frac{n^2}{2} \geq \frac{n}{2}$</span>, but I'm stuck proving that <span class="math-container">$\frac{n^2}{2} \geq \frac{n^2}{2^{\sqrt{\log n}}}\geq \frac{n}{2}$</span>. Am I missing something?</p>
| eyeballfrog | 395,748 | <p>For all <span class="math-container">$t > 0$</span>, we obviously have <span class="math-container">$t > \sqrt{t} - 1$</span>. Therefore we also have
<span class="math-container">$$
e^t > 2^t > 2^{\sqrt{t} - 1},
$$</span>
or equivalently,
<span class="math-container">$$
\frac{e^{2t}}{2^\sqrt{t}} > \fr... |
4,264,808 | <p>So we know that <span class="math-container">$\frac{n^2}{2} \geq \frac{n}{2}$</span>, but I'm stuck proving that <span class="math-container">$\frac{n^2}{2} \geq \frac{n^2}{2^{\sqrt{\log n}}}\geq \frac{n}{2}$</span>. Am I missing something?</p>
| DanielWainfleet | 254,665 | <p>For <span class="math-container">$n\ge 1$</span> we have</p>
<p><span class="math-container">$n^2/2^{\sqrt {\log n}}\ge n/2\iff$</span></p>
<p><span class="math-container">$ n/2^{\sqrt {\log n}}\ge 1/2 \iff $</span></p>
<p><span class="math-container">$ 2n\ge 2^{\sqrt {\log n}}\iff$</span></p>
<p><span class="math-c... |
3,441,647 | <p>I was asked to find the domain of <span class="math-container">$$\arcsin[\frac{x^2+1}{2x}]$$</span>
My first step was <span class="math-container">$$-1\leq\frac{x^2+1}{2x} \leq1$$</span></p>
<p>What I don't understand is why I cannot cross multiply to get <span class="math-container">$$-2x\le {1+x}^{2} \le 2x$$</sp... | lab bhattacharjee | 33,337 | <p>Let <span class="math-container">$y=\arcsin\dfrac{x^2+1}{2x}$</span></p>
<p><span class="math-container">$x^2-2x\sin y+1=0$</span></p>
<p><span class="math-container">$x=\sin y\pm i\cos y$</span></p>
<p>As <span class="math-container">$x$</span> is real, <span class="math-container">$\cos y$</span> must be <span ... |
2,418,274 | <p>AP=DS=CR=BQ=2a, ABCD and PQRS are squares with a side a
find the angle between AR and BC..
i got $$AR=\sqrt{6}a$$can i use ARQ triangle in order to find the corresponding angle..</p>
<p><a href="https://i.stack.imgur.com/QtRaT.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/QtRaT.jpg" alt="enter... | orangeskid | 168,051 | <p>The triangle $\Delta ACR$ has a right angle at $C$. The side $AC = \sqrt{2} CR$, so $\tan \alpha = \frac{1}{\sqrt{2}}$, and you get your $\alpha = \arctan \frac{1}{\sqrt{2}}= \ldots$. </p>
<p>Btw, the large diagonal is $\sqrt{3} \cdot \textrm{edge} = 2\sqrt{3} a$, if you want to use $\arcsin$. </p>
|
1,348,763 | <p>Using the comparison test I am supposed to figure out whether this integral converges or diverges. what other function should I use? Also, the inequality stating that $1/\sqrt{e^x+1}$ is larger or smaller than another function must be proven.</p>
| Marco Cantarini | 171,547 | <p>Note that we can calculate the precise value of the integral, not only the convergence. We have $$\int_{1}^{\infty}\frac{1}{\sqrt{1+e^{x}}}dx\overset{{\scriptstyle e^{x}=u}}{=}\int_{e}^{\infty}\frac{1}{\sqrt{u+1}u}du\overset{{\scriptstyle u+1=v}}{=}\int_{1+e}^{\infty}\frac{1}{\left(v-1\right)\sqrt{v}}dv\overset{{\sc... |
1,556,503 | <blockquote>
<p>Assume that $f$ is $a$-strongly convex and $g$ is $b$-strongly convex. Is the sum $f+g$ strongly convex, and with what constant? </p>
</blockquote>
<p><strong>Definition:</strong> $f$ is $a$-strongly convex if
$$ f(x)-f(y) \le \left<f '(x),x-y\right>-\frac{a}{2}\|x-y\|^2$$
for all $x,y$</p>
| Jing Lu | 295,496 | <p>c-strongly convex, c>=a+b</p>
<p>proof:</p>
<p>f(x+y)>=f(x)+f'(x)y+f''(x)/2 *y^2</p>
<p>g(x+y)>=g(x)+g'(x)y+g''(x)/2 *y^2</p>
<p>where f' is the first order gradient, g' is the first order gradient, f''(x) is some positive number at least a, g''(x) is some positive number at least b, both of them depend on x,</p... |
1,556,503 | <blockquote>
<p>Assume that $f$ is $a$-strongly convex and $g$ is $b$-strongly convex. Is the sum $f+g$ strongly convex, and with what constant? </p>
</blockquote>
<p><strong>Definition:</strong> $f$ is $a$-strongly convex if
$$ f(x)-f(y) \le \left<f '(x),x-y\right>-\frac{a}{2}\|x-y\|^2$$
for all $x,y$</p>
| Guy Fsone | 385,707 | <p>This is totally pretty easy from your definition we have </p>
<p>$$f(x)-f(y) \le \left<f '(x),x-y\right>-\frac{a}{2}\|x-y\|^2$$
and
$$g(x)-g(y) \le \left<g '(x),x-y\right>-\frac{b}{2}\|x-y\|^2$$</p>
<p>summing up both inequality we obtain:
$$f(x) +g(x)-f(y)+g(y) \le\left<f '(x),x-y\right> +\left... |
194,128 | <p>I am trying to solve this problem by induction. The sad part is that I don't have a very strong grasp on solving by inductive proving methods. I understand that there is a base case and that I need an inductive step that will set $k = n$ and then one more step that basically sets $k = n + 1$. </p>
<p>Here is the pr... | Clive Newstead | 19,542 | <p>The way induction works is based on two steps:</p>
<ol>
<li>Check a base case.</li>
<li>Show that if it holds for one number, then it holds for the next.</li>
</ol>
<p>More formally, let $P$ be a property, and let $P(n)$ be the assertion that the property holds true for the natural number $n$; so for instance, $P(... |
12,047 | <p>How to prove the following trigonometric identities ?</p>
<p>1) If $\displaystyle \tan (\alpha) \cdot \tan(\beta) = 1 \text{ then } \alpha + \beta = \frac{\pi}{2}$
</p>
<p>I tried to prove it by using the the formula for $\tan(\alpha + \beta)$ but ain't it valid only when $\alpha + \beta \neq \frac{\pi}{2}$ ?</p>
... | user3180 | 67,758 | <p>1) tan(a)tan(b) = 1 is equivilant to sin(a)sin(b) = Cos(a)cos(b) **using the fact that tan is sine divided by cosine. Next use
cos(a+b) = cos(a)cos(b) -sin(a)sin(b)</p>
<p>which implies sin(a)sin(b) = cos(a)cos(b) -cos(a+b)</p>
<p>Use this on the left hand side of ** to get </p>
<pre><... |
12,047 | <p>How to prove the following trigonometric identities ?</p>
<p>1) If $\displaystyle \tan (\alpha) \cdot \tan(\beta) = 1 \text{ then } \alpha + \beta = \frac{\pi}{2}$
</p>
<p>I tried to prove it by using the the formula for $\tan(\alpha + \beta)$ but ain't it valid only when $\alpha + \beta \neq \frac{\pi}{2}$ ?</p>
... | Américo Tavares | 752 | <p>Added (and corrected twice):</p>
<p>$$\tan \alpha \tan \beta =1\Leftrightarrow \dfrac{\sin \alpha }{\cos \alpha }%
\dfrac{\sin \beta }{\cos \beta }=1$$
</p>
<p>Multiplying by $\cos\alpha\cos\beta\ne 0$, gives</p>
<p>$$\sin \alpha \sin \beta -\cos \alpha\cos \beta =0 \Leftrightarrow \cos (\alpha +\beta )=0$$
</p>... |
2,254,694 | <p>I have been stuck on this question for a while now. I have tried many attempts. Here are two that I thought looked promising but lead to a dead end:</p>
<p>Attempt 1:</p>
<p>Write out the terms of $b_n$:</p>
<p>$$b_1=a_{2}-\frac{a_{1}}{2}$$
$$b_2=a_{3}-\frac{a_{2}}{2}$$
$$b_3=a_{4}-\frac{a_{3}}{2}$$
$$\cdots$$
$$... | Sangchul Lee | 9,340 | <p>Let $b_n = a_{n+1} - \frac{1}{2}a_n$. Then</p>
<p>$$ a_n = b_{n-1} + \frac{b_{n-2}}{2} + \cdots + \frac{b_1}{2^{n-2}} + \frac{a_1}{2^{n-1}}. $$</p>
<p>Since $(b_n)$ converges, there exists $M$ such that $|a_1| \leq M$ and $|b_n| \leq M$ for all $n$. Thus for any <em>fixed</em> $m$ and for any $n > m$, we have</... |
1,248,668 | <p>I do have a system of n equations with m variables where m > n with integer coefficients. I wish to find a set of integer solutions to this system (In my case n = 2 and m = 4). Could somebody tell me how I can do it? I already solved this system with Mathematica but I would like to redo these calculations by hand to... | marekszpak | 147,160 | <p>Well, finally I have used old good Hermite Normal Form. I knew this method before but I thought about something which could be easier for bigger systems to solve by hand.</p>
<p>If something check paper <a href="https://www.math.uwaterloo.ca/~wgilbert/Research/GilbertPathria.pdf" rel="nofollow">Linear Diophantine E... |
2,765,570 | <p>Let $E = \{\frac{1}{n} | n \in \mathbb{N}\}$. Show that the function $f(x) = 1 \text{ if } x \in E \text{ and } f(x) = 0 \text{ if } x \notin E$ is integrable on $[0,1]$.</p>
<p>I know of the definition of integrable but I'm having a hard time applying it to this function.</p>
| Olecram | 558,042 | <p>You can note that, the inferior sum and superior sum be equal to 0, because for every partition of interval [0,1], $$P=\{0=x1<...<xn=1\}$$ $$\inf_{x \in [x_k,x_{k+1}]} f(x)=0$$ because you always can take x that is not of form $$\frac{1}{n}$$,the same for sup, thus, f is integrable and your integral is equal t... |
495,064 | <p>I'm currently tackling the following problems:</p>
<p>a) Let $a,b \in \mathbb{Z}$ and let $m$ be a nonnegative integer. Prove that $(a,b)=1$ if and only if $(a^m,b)=1$.</p>
<p>If $(a^m,b)=1$ then $pa^m+qb=1$ for some integers $p$ and $q$. It follows that $(pa^{m-1})a+qb=1$, which shows that $(a,b)=1$. This took ca... | André Nicolas | 6,312 | <p>The Bezout Theorem is I think not the best approach. I prefer the approach of some1.new4U. </p>
<p>But Bezout will work. Since $a$ and $b$ are relatively prime, there exist integers $x$ and $y$ such that $ax+by=1$.</p>
<p>Take the $m$-th power of $ax+by$, using the Binomial Theorem. The first term is $a^mx^m$. The... |
2,341,552 | <p>I am pretty sure that
$$\bigg|\int_{A}e^{it}\,dt\bigg|\leq2$$
for every measurable set $A\subseteq[-\pi,\pi]$,
but I cannot prove this...</p>
| Richard | 191,878 | <p>We have
$$ I=\int_a^be^{it}dt=\frac{1}{i}(e^{ib}-e^{ia}) $$
Then
$$ |I|=|-i||e^{ib}-e^{ia}|=|e^{ib}||1-e^{i(b-a)}|\le |1|+|e^{i(b-a)}|\le 1+1=2$$</p>
|
554,003 | <p>How can I find a closed form for the following sum?
$$\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$$
($H_n=\sum_{k=1}^n\frac{1}{k}$).</p>
| Cody | 13,295 | <p>SOS always has the most clever and ingenious solutions, but if I may contribute something I have found interesting. A fun method of evaluating a whole slew of Euler sums is by using the residues of digamma. </p>
<p>By noting the identity, $\displaystyle \sum_{n=1}^{\infty}\frac{(H_{n})^{2}}{n^{2}}=2\sum_{n=1}^{\in... |
554,003 | <p>How can I find a closed form for the following sum?
$$\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$$
($H_n=\sum_{k=1}^n\frac{1}{k}$).</p>
| Ali Shadhar | 432,085 | <p>Using the generating function:</p>
<p><span class="math-container">\begin{gather}
\sum_{n=1}^\infty\frac{H_{n}^2}{n^2}x^{n}=\operatorname{Li}_4(x)-2\operatorname{Li}_4(1-x)+2\ln(1-x)\operatorname{Li}_3(1-x)+\frac12\operatorname{Li}_2^2(x)\nonumber\\
\qquad-\ln^2(1-x)\operatorname{Li}_2(1-x)-\frac13\ln(x)\ln^3(1-x)+2... |
167,228 | <p>I have shown several plots by Show function:</p>
<pre><code>Show[p1, p2, p3, p4, p5, p6, p7, p8, PlotRange -> All]
</code></pre>
<p>I need to save data of all plots in one <code>txt</code> format, so that they can be recognizable from each other. How can I do that?</p>
<p>Each plot is a 2D plot say, $f_i$ as a... | Carl Woll | 45,431 | <p>Polynomials with real coefficients have either real roots, or complex roots that come in conjugate pairs. Suppose $r$ and $\bar{r}$ are complex conjugates, then:</p>
<pre><code>(x - r)(x - Conjugate[r]) == x^2 - 2 Re[r] x + Abs[r]^2 //FullSimplify
</code></pre>
<blockquote>
<p>True</p>
</blockquote>
<p>The func... |
3,081,552 | <p>Is this <code>∅ ⊈ { ∅, 1, 2 }</code> true or false ? </p>
<p>Also, I am confuse since this <code>{ ∅, 1, 2 }</code> has already contain a <code>∅</code>, does it still contain another <code>∅</code> meaning like : <code>{ ∅, ∅, 1, 2 }</code> ? </p>
<p>Is <code>∅ ∈ { ∅, 1, 2 }</code> true ? & <code>{∅} ∈ { ∅,... | Michael Rozenberg | 190,319 | <p><span class="math-container">$$\oslash\subset\{\oslash,1,2\}$$</span> and <span class="math-container">$$\oslash\in\{\oslash,1,2\}.$$</span>
Also, <span class="math-container">$$\{\oslash\}\subset\{\oslash,1,2\}.$$</span></p>
|
479,671 | <p>Let $p$ and $q$ be two distinct primes. Prove that $$p^{q-1} + q^{p-1} =1 \mod pq$$ I try to used Fermat little theorem and I obtain the congruence $p^q +q^p=0 \mod pq$. From this I don know how to relate to the congruence on the question.</p>
| pritam | 33,736 | <p>By Fermat's little theorem $p^{q-1}\equiv 1\pmod{q}$ and $q^{p-1}\equiv 0\pmod{q}$, now adding these two $$p^{q-1}+q^{p-1}\equiv1\pmod{q}$$ Similarly, the above also holds modulo $p$. Now since $\gcd (p,q)=1$, this also holds modulo $pq$.</p>
|
4,531,507 | <p><a href="https://i.stack.imgur.com/rJmjH.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/rJmjH.jpg" alt="enter image description here" /></a></p>
<blockquote>
<p>Find the area of <span class="math-container">$ABCD$</span> if <span class="math-container">$PSC$</span> is a semicircle.</p>
</blockqu... | gus f | 1,025,519 | <p>We can see in triangles PQC and QBC that</p>
<p><span class="math-container">$∠QPC = ∠BQC,∠QCP$</span> is common.
<span class="math-container">$∠PQC = ∠QBC = 90$</span>
Therefore both triangles are similar.</p>
<p><span class="math-container">$\frac{QC}{PC}=\frac{BC}{QC}$</span>,
<span class="math-container">$PC∗BC=... |
4,531,507 | <p><a href="https://i.stack.imgur.com/rJmjH.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/rJmjH.jpg" alt="enter image description here" /></a></p>
<blockquote>
<p>Find the area of <span class="math-container">$ABCD$</span> if <span class="math-container">$PSC$</span> is a semicircle.</p>
</blockqu... | Doug M | 317,176 | <p>Lets call lengths:<br />
<span class="math-container">$\overline {BC} = x\\
\overline {PB} = y\\
\overline {BQ} = z$</span></p>
<p><span class="math-container">$\frac {x+y}2$</span> is the raduis of the circle.<br />
<span class="math-container">$\frac {x^2+xy}{2}$</span> is the area of the rectangle.</p>
<p><span ... |
2,390,804 | <p>How does one compute the following integral?
$$\int_{0}^{\infty} \frac{\sqrt{x}\sin(x)}{1+x^2} dx$$</p>
<hr>
<p>I have tried extending $x$ to the complex plane then evaluating the following contour integral
$$\oint_C \frac{\sqrt{x}e^{ix}}{1+x^2} dx$$
with the contour $C$ running along the whole real axis and then... | JJacquelin | 108,514 | <p>For information :</p>
<p>$\int_{0}^{\infty} \frac{\sqrt{x}\sin(y\:x)}{1+x^2} dx$ is the Fourier Sine Transform of $\frac{\sqrt{x}}{1+x^2}\quad $ In the Harry Bateman's Tables of Integral Transforms an even more general formula can be seen on page 71, Eq.28 , the Fourier Sine Transform of $x^{2\nu}(x^2+a^2)^{-\mu-1}... |
2,993,958 | <blockquote>
<p>Do there exist three non null vectors <span class="math-container">$a,b,c$</span> with <span class="math-container">$$a.b=a.c$$</span> such that <span class="math-container">$b\ne c$</span>?</p>
</blockquote>
<p>My attempt:</p>
<p>a.(b-c)=0</p>
<p>But is it possible to say b-c=0 since a is non null... | nonuser | 463,553 | <p>Take isosceles triangle <span class="math-container">$ABC$</span> such that <span class="math-container">$AB = AC$</span> and let <span class="math-container">$M$</span> be a midpoint of <span class="math-container">$BC$</span>. Then <span class="math-container">$$\vec{AM} \cdot \vec{AB} = \vec{AM} \cdot \vec{AC}$$<... |
805,902 | <p>I'm struggling to figure out what I'm exactly required to do. The problem states</p>
<p>"Compute which lines through the point $(1, 0)$ that are tangent to the parabola defined by $y = x^2$."</p>
<p>I believe it's a simple question however I've been going around this for quite a bit. </p>
<p>I'll appreciate any k... | mfl | 148,513 | <p>The equation of a tangent line to the parabola at $(x_0,x_0^2)$ is given by $y-x_0^2=2x_0(x-x_0).$ If $(1,0)$ is a point of this line we have $-x_0^2=2x_0(1-x_0),$ or equivalently, $x_0^2=2x_0.$ So $x_0=0$ or $x_0= 2.$ That is, $(1,0)$ lies in the tangents to the parabola at points $(0,0)$ and $(2,4).$ Their equatio... |
805,902 | <p>I'm struggling to figure out what I'm exactly required to do. The problem states</p>
<p>"Compute which lines through the point $(1, 0)$ that are tangent to the parabola defined by $y = x^2$."</p>
<p>I believe it's a simple question however I've been going around this for quite a bit. </p>
<p>I'll appreciate any k... | Joel Hernandez | 120,476 | <p>Some comments were really helpful but I believe none gave me the intuition for this, which is what I really didn't understand , however I'm truly grateful for everyone that commented or attempted to help me.</p>
<p>Here's my approach after dreaming about it the entire night. (Not just the intuition)</p>
<p>We will... |
805,902 | <p>I'm struggling to figure out what I'm exactly required to do. The problem states</p>
<p>"Compute which lines through the point $(1, 0)$ that are tangent to the parabola defined by $y = x^2$."</p>
<p>I believe it's a simple question however I've been going around this for quite a bit. </p>
<p>I'll appreciate any k... | André Nicolas | 6,312 | <p>We can use basic "algebra." First draw a picture. That will pick up the fact that the $x$-axis is such a tangent line, and that there appears to be another one. </p>
<p>Now for the algebra. The tangent line "kisses" the parabola, so meets it at a "double-point." </p>
<p>The generic (non-vertical) line through $(1,... |
692,383 | <p>Let $p, q$ be prime numbers which may or may not be distinct.
Let $\mathbb{Q}_p$ be the field of $p$-adic numbers.
Let $\mathbb{Z}_p$ be the ring of $p$-adic integers.
We define similarly $\mathbb{Q}_q$ and $\mathbb{Z}_q$.</p>
<p>Let $A = \mathbb{Q}_p\otimes_{\mathbb{Q}} \mathbb{Q}_q$.
Let $\lambda\colon \mathbb{Z}... | Martin Brandenburg | 1,650 | <p>Since $A$ is the localization of $B$ at the elements $p$ and $q$ (by formal nonsense), the question is if $p$ and $q$ are regular elements of $B=\mathbb{Z}_p \otimes \mathbb{Z}_q$. But this is because $p$ is a regular element of $\mathbb{Z}_p$ and $\mathbb{Z}_q$ is torsion-free and hence flat over $\mathbb{Z}$; simi... |
3,841 | <p>Taking tori in symmetric products and "miraculously" proving that the Floer homology is independent of choices always seemed, well, miraculous. Some time ago Max Lipyanski explained to me the origins of this construction from gauge theory on surfaces, a la Atiyah-Floer conjecture, which I have then forgotten. What ... | Mike Usher | 424 | <p>From Szabó's delightfully understated <a href="http://www.ams.org/notices/200704/comm-veblen-web.pdf" rel="nofollow noreferrer">response</a> (pdf) to receiving the Veblen prize:</p>
<blockquote>
<p>The joint work with Peter Ozsváth
which is noted here grew out of our
attempts to understand Seiberg–Witten
moduli spac... |
47,728 | <p>Function <code>CreateTemporary</code> can generate a unique file name like</p>
<pre><code>file = CreateTemporary[]
(* "C:\Documents and Settings\kkouptsov\Local \Settings\Temp\m-263a0380-3a03-49ac-95cb-d21390b2d3fd" *)
</code></pre>
<p>How do we generate a unique file name such as above example? The reason I don't... | Kuba | 5,478 | <p>Rather ugly but should work</p>
<pre><code>Block[{m},
While[FileExistsQ[
m = FileNameJoin[{Directory[], FromCharacterCode[RandomInteger[25, 35] + 97]}]]];
m]
</code></pre>
|
1,284,086 | <p>Given a state $(|1\rangle \otimes |0\rangle \otimes |1\rangle) + (|0\rangle \otimes |0\rangle \otimes |1\rangle)$, is it possible to factorise out the $|0\rangle$ in the middle of both of them?</p>
<p>Alternatively written as $|101\rangle + |001\rangle$ for convenience.</p>
<p>For a state such as the following it ... | Simon S | 21,495 | <p>$(|1\rangle \otimes |0\rangle \otimes |1\rangle) + (|0\rangle \otimes |0\rangle \otimes |1\rangle)$ is clearly a product state. It's tricky to find good notation to denote this and I don't think there's an established standard way. Here's one option:</p>
<p>Let $\sigma_{ij}$ be an automorphism that flips the $i$th ... |
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