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 |
|---|---|---|---|---|
1,269,738 | <p>I'm looking for problems that due to modern developments in mathematics would nowadays be reduced to a rote computation or at least an exercise in a textbook, but that past mathematicians (even famous and great ones such as Gauss or Riemann) would've had a difficult time with. </p>
<p>Some examples that come to min... | Community | -1 | <p>In the nineteenth century expressing the antiderivative of an elementary function as an elementary function was an open problem.</p>
<p>Nowadays, <a href="https://en.wikipedia.org/wiki/Risch_algorithm" rel="nofollow">Risch algorithm</a>, which can be run on machines, decides whether such operation can be done and, ... |
4,504,080 | <p>If it is given that <span class="math-container">$$\displaystyle \frac{1}{(20-x)(40-x)}+\displaystyle \frac{1}{(40-x)(60-x)}+....+\displaystyle \frac{1}{(180-x)(200-x)}= \frac{1}{256}$$</span> then how to find the maximum value of <span class="math-container">$x$</span> ? I tried solving it with <span class="math-co... | Thomas Preu | 1,011,882 | <p>As <a href="https://math.stackexchange.com/users/16497/alex-youcis">Alex Youcis</a> encouraged me to do so, I try to answer myself.</p>
<p><strong>The mistake</strong> was that <span class="math-container">$A:=\text{Aut}_{\text{var}}(\mathbb{G}_m\times_kK)\cong\left\{a\cdot t^n:a\in K^{\times},n\in\mathbb{Z}^{\times... |
25,917 | <p>$\sqrt{2+\sqrt{2+\sqrt{2+\dots}}}$ </p>
<p>$\dots\sqrt{2+\sqrt{2+\sqrt{2}}}$</p>
<p>Why they are different?</p>
| Bill Dubuque | 242 | <p>This follows from a more general result that's both simpler to prove and more insightful, viz. the result follows immediately by this frequently applicable <strong>multiplicative form of induction</strong>, which shows that for multiplicative sets we need only check the generators (here <span class="math-container">... |
2,374,282 | <p>I am trying to find all connected sets containing $z=i$ on which $f(z)=e^{2z}$ is one to one.
I have no idea how to approach.
Can someone give me some hints?
Thank you</p>
| Grey Matters | 176,262 | <p>If you want to be able to do that in your head, then you'd start by working out the number of the day of both dates (as in, January 2nd would be day #2, December 30th in a non-leap year would be day #364, etc.).</p>
<p>You can do this fairly quickly by memorizing the number of days that occur prior to each month in... |
604,836 | <p>Prove if <span class="math-container">$a \equiv c \pmod{n}$</span> and <span class="math-container">$b \equiv d \pmod n$</span> then <span class="math-container">$ab \equiv cd \pmod{n}$</span>.</p>
<p>I tried to use <span class="math-container">$(a-c)(b-d) = ab-ad-cb+cd$</span>, but it seem doesn't work.</p>
| Stefan4024 | 67,746 | <p>One good way to prove it is to use this fact:</p>
<p>$$a \equiv c \pmod n \implies n|a-c$$
$$b \equiv d \pmod n \implies n|b-d$$</p>
<p>Obviously then $n|b(a-c) + c(b-d)$</p>
<p>So we have: $n|ba - bc + bc - cd$</p>
<p>$$\implies n|ab - cd$$</p>
<p>Q.E.D.</p>
|
2,585,466 | <p>I have two growth curve data sets, A (Martians) and B (Venusians). Data point sets of age (0 (birth) - 250 months, X axis) against height (0 - 200 centimeters, Y axis). The first set (A) contains 67 X Y point pairs, the second set (B) contains 27 point pairs. I have fit both data sets to my favorite version of the L... | ratalan | 506,825 | <p>One possible way to do that is to compute the gradients at a point $(x,y)$ of the functions which define the two curves $f\left(x,y\right)=y^{2}-4ax$ and $g\left(x,y\right)=xy-c^{2}$.
The respective gradients are $\left(\begin{array}{c}
-4a\\
2y
\end{array}\right)$ and $\left(\begin{array}{c}
y\\
x
\end{array}\right... |
3,281,503 | <blockquote>
<p>For natural numbers <span class="math-container">$a$</span> and <span class="math-container">$b$</span>, what is the greatest value of <span class="math-container">$b$</span> so that <span class="math-container">$a^b$</span> has <span class="math-container">$b$</span> digits?</p>
</blockquote>
<p>I k... | Arthur | 15,500 | <p>Proof sketch: First show that <span class="math-container">$a=9$</span> is the best possible value for <span class="math-container">$a$</span> (you don't have to show that it is better than any other value, just that no other value can possibly be better). Then find the maximal <span class="math-container">$b$</span... |
900,884 | <p>If $ A = \{ m^n| \text{ } m, n \in Z \text { and } m, n \ge 2 \} $, then how find $\sum_{k \in A} \frac{1}{k-1} $?</p>
| gtrrebel | 169,563 | <p>The answer is 1.</p>
<p>The result is known as Goldbach-Euler theorem.</p>
<p>See <a href="http://en.wikipedia.org/wiki/Goldbach-Euler_theorem" rel="noreferrer">Wikipedia entry</a> for "proof".</p>
<p>For rigorous proof, you could consider sum of reciprocals of all perfect powers, $S$. Note that sum equals
$$
S =... |
2,072,347 | <p>I was trying to solve this problem, but couldn't figure it out. The solution goes like this:</p>
<p><a href="https://i.stack.imgur.com/1KSWH.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/1KSWH.png" alt="http://www.tkiryl.com/Calculus/Problems/Section%201.4/Calculating%20Limits/Solutions/Calc_S_... | projectilemotion | 323,432 | <p>Alternatively, you could use <a href="https://en.wikipedia.org/wiki/L'H%C3%B4pital's_rule" rel="nofollow noreferrer">L'Hopital's rule</a>:</p>
<p>$\lim \limits_{x \to 0} \frac{\sin(5x)}{\sin(4x)}$=$\lim \limits_{x \to 0} \frac{\frac{d}{dx}\sin(5x)}{\frac{d}{dx}\sin(4x)}$=$\lim \limits_{x \to 0} \frac{5\cos(... |
2,786,656 | <p>I know that if a group is generated by a single element then the group is abelian but does this mean that if a group is abelian then its conjugacy class is composed of a single element?</p>
| Tsemo Aristide | 280,301 | <p>Since $T$ is an isomorphism and an isometry, $T^{-1}$ is also an isometry. We deduce that $T(\{x\in X:\|x\|=1\})=\{y\in Y:\|y\|=1\}$ This implies that $\|T^*(f)\|=sup_{\|x\|=1}||f(T(x))|=sup_{\|y\|=1}|f(y)|=\|f\|$.</p>
|
912,002 | <p>I'm solving some programming puzzle and it has come down to this:</p>
<p>I've a fraction, say 12/13, and I need to multiply it with a smallest possible natural number (say x) to get a whole number. How do I solve for x?</p>
<p>I intuitively feel I need to use LCM to solve this but haven't been able to pin down on ... | Bill Dubuque | 242 | <p><strong>Hint</strong> $\,\ c \dfrac{a}b\in\Bbb Z\iff b\mid ac\iff b\mid ac,bc\iff b\mid(ac,bc)= (a,b)c\iff b/(a,b)\mid c$</p>
|
871,744 | <p>When A and B are square matrices of the same order, and O is the zero square matrix of the same order, prove or disprove:-
$$AB=0 \implies A=0 \text{ or } \ B=0$$</p>
<p>I proved it as follows:-</p>
<p>Assume $A \neq O$ and $ B \neq O$:
then, $$ |A||B| \neq 0 $$
$$ |AB| \neq 0 $$
$$ AB \neq O $$
$$ \therefore A \... | Barath | 316,775 | <p>You aren't wrong! AB = O does not imply that either A or B is zero. In matrices there is no such case. So even when A is not equal to O and B is not equal to zero, you can get AB =O. Voila!</p>
|
3,389,542 | <blockquote>
<p><strong>Proposition.</strong> If <span class="math-container">$\text{Ran}(R) \subseteq \text{Dom}(S)$</span>, then <span class="math-container">$\text{Dom}(S \circ R) = \text{Dom}(R)$</span></p>
</blockquote>
<p>My attempt:</p>
<p>Suppose <span class="math-container">$\text{Ran}(R) \subseteq \text{D... | Nicholas Viggiano | 709,871 | <p>I would argue that the best you can do is using a maximal triangle-free graph (per Turan's construction, this is a complete bipartite graph whose partitions are as balanced as possible). It at least serves as a lower bound: take <span class="math-container">$n$</span> even (for simplicity, I'll handle odd in a bit),... |
2,475,938 | <blockquote>
<p>How can I factor the polynomial <span class="math-container">$x^4-2x^3+x^2-1$</span>?</p>
</blockquote>
<p>This is an exercise in algebra. I have the solution showing that
<span class="math-container">$$
x^4-2x^3+x^2-1=(x^2-x-1)(x^2-x+1).
$$</span></p>
<p>But the solution does not show any details.... | Clayton | 43,239 | <p>As @MathLover suggests, use the factorization $$x^4-2x^3+x^2=x^2(x-1)^2=(x(x-1))^2.$$ Now use the fact that it is of the form $y^2-1$ to factor it as the difference of squares.</p>
|
2,081,641 | <p>I figured out that $\lim_{n \to \infty}\frac{(3n^2−2n+1)\sqrt{5n−2}}{(\sqrt{n}−1)(1−n)(3n+2)} $ is $-\sqrt{5}$ but I don't know how to prove it.</p>
| Community | -1 | <p>The dominant terms in the numerator and denominator are both in $n^{5/2}$, hence the limit is finite. The corresponding coefficients are $3\sqrt5$ and $-3$. The ratio gives the answer.</p>
|
114,664 | <p>How would one evaluate $\int_0^1 {\ln(1+x)\over x}\,dx$?</p>
<p>I'd like to do this without approximations. Not quite sure where to start. What really bothers me is that I came across this while reviewing my old intro to calculus book... but I'm fairly certain I've exhausted all the basic methods they teach in th... | josh | 25,488 | <p>$$
I = \int_0^1 \frac{\ln(1+x)}{x}\,dx = \int_0^\infty \ln(1+e^{-t})\,dt\,,
$$
where $x = e^{-t}$. Then expand
$$
\ln(1 + e^{-t}) = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n}\,e^{-tn}\,,
$$
So we find
$$
I = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n}\,\int_0^\infty e^{-tn}\,dt = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^2... |
114,664 | <p>How would one evaluate $\int_0^1 {\ln(1+x)\over x}\,dx$?</p>
<p>I'd like to do this without approximations. Not quite sure where to start. What really bothers me is that I came across this while reviewing my old intro to calculus book... but I'm fairly certain I've exhausted all the basic methods they teach in th... | Alex Becker | 8,173 | <p>The Taylor series for $\frac{\ln(1+x)}{x}$ is $\sum\limits_{n=1}^\infty (-1)^{n-1}\frac{x^{n-1}}{n}$, and this converges absolutely in $(0,1)$ thus we can use it for our integral. This means
$$\int_0^1\frac{\ln(1+x)}{x}=\int_0^1\sum\limits_{n=1}^\infty (-1)^{n-1}\frac{x^{n-1}}{n}=\sum\limits_{n=1}^\infty (-1)^{n-1}... |
2,117,054 | <p>Find all prime solutions of the equation $5x^2-7x+1=y^2.$</p>
<p>It is easy to see that
$y^2+2x^2=1 \mod 7.$ Since $\mod 7$-residues are $1,2,4$ it follows that $y^2=4 \mod 7$, $x^2=2 \mod 7$ or $y=2,5 \mod 7$ and $x=3,4 \mod 7.$ </p>
<p>In the same way from $y^2+2x=1 \mod 5$ we have that $y^2=1 \mod 5$ and $x=0... | Robert Israel | 8,508 | <p>Completing the square and dividing by $5$, we have </p>
<p>$$ (10 x - 7)^2 - 20 y^2 = 29$$</p>
<p>Thus $z = 10 x - 7$ and $w = 2 y$ are solutions of the Pell-type equation</p>
<p>$$ z^2 - 5 w^2 = 29$$</p>
<p>The positive integer solutions of this can be written as</p>
<p>$$\pmatrix{z\cr w\cr} = \pmatrix{9 &... |
3,225,553 | <p>Show that <span class="math-container">$4x^2+6x+3$</span> is a unit in <span class="math-container">$\mathbb{Z}_8[x]$</span>.</p>
<p>Once you have found the inverse like <a href="https://math.stackexchange.com/questions/3172556/show-that-4x26x3-is-a-unit-in-mathbbz-8x">here</a>, the verification is trivial. But how... | Bill Dubuque | 242 | <p>By <a href="https://math.stackexchange.com/a/3224776/242">simpler multiples,</a> to invert <span class="math-container">$\, a - f\,$</span> where <span class="math-container">$\,a\,$</span> is invertible, say <span class="math-container">$\,\color{#0a0}{ab = 1},$</span> and <span class="math-container">$\,f\,$</spa... |
870,583 | <p><strong>Question:</strong>
Each user on a computer system has a password, which is six to eight characters long, where each character is an upper-case letter or a digit. Each password must contain at least one digit. How many possible passwords are there?</p>
<blockquote>
<p>I'm in the <strong>Basic of Counting</... | David | 119,775 | <p>Your calculation for $P_6$ gives the number of words in which the first five characters can be letters or digits, and the last must be a digit. But there is nothing in the rules to say that the <strong>last</strong> must be a digit.</p>
<p>Alternatively, you could interpret your answer to say that, for example, sy... |
1,255,368 | <p>How do I solve this? What steps? I have been beating my head into the wall all evening. </p>
<p>$$ x^2 + y^2 = \frac{x}{y} + 4 $$</p>
| N. F. Taussig | 173,070 | <p>We can multiply both sides of the equation
$$x^2 + y^2 = \frac{x}{y} + 4 \tag{1}$$
by $y$ to obtain
$$x^2y + y^3 = x + 4y \tag{2}$$
Differentiating equation 2 implicitly with respect to $x$ yields
\begin{align*}
2xy + x^2y' + 3y^2y' & = 1 + 4y'\\
(x^2 + 3y^2 - 4)y' & = 1 - 2xy\\
y' & = \frac{1 - 2xy}{x^2... |
1,557,015 | <p>This one looks simple, but apparently there is something more to it.
$$f{(x)=x^x}$$
I read somewhere that the domain is $\Bbb R_+$, a friend said that $x\lt-1, x\gt0$... </p>
<p>I'm really confused, because i don't understand why the domain isn't just all the real numbers.
According to any grapher online the domai... | John Molokach | 90,422 | <p>I am not sure what your mathematics background is, but the function
$f(x)=x^x$ is defined for $\mathbb{R}_+$ as well as a countably infinite set of rational values in $\mathbb{Q}_-$. For example, we can find $f(-\frac n3)$ for all $n\in\mathbb{N}$. In fact, I cannot with confidence write down the entire set of neg... |
2,568,157 | <p>Consider the following:</p>
<p>$$(1^5+2^5)+(1^7+2^7)=2(1+2)^4$$</p>
<p>$$(1^5+2^5+3^5)+(1^7+2^7+3^7)=2(1+2+3)^4$$</p>
<p>$$(1^5+2^5+3^5+4^5)+(1^7+2^7+3^7+4^7)=2(1+2+3+4)^4$$</p>
<p>In General is it true for further increase i.e.,</p>
<p>Is</p>
<p>$$\sum_{i=1}^n i^5+i^7=2\left( \sum_{i=1}^ni\right)^4$$ true $\f... | DanielWainfleet | 254,665 | <p>Let the LHS be $A(n)$ and let the RHS be $2B(n)^4.$ </p>
<p>Then $A(n+1)-A(n)=(n+1)^7+(n+1)^5=(n+1)^5(n^2+2n+2).$ </p>
<p>$$\text {We have }\quad 2B(n+1)^4-2B(n)^4=$$ $$(*)\quad =2(B(n+1)^2+B(n)^2)\cdot (B(n+1)+B(n))\cdot (B(n+1)-B(n)).$$ Since $B(n)=n(n+1)/2$ we have $$B(n+1)^2+B(n)^2=(n+1)^2((n+2)^2+n^2)/4=(n+... |
281,735 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/202452/why-is-predicate-all-as-in-allset-true-if-the-set-is-empty">Why is predicate “all” as in all(SET) true if the SET is empty?</a> </p>
</blockquote>
<p>In don't quite understand this quantification ov... | user58437 | 58,437 | <p>Because there are no members to the set, anything you say about a member can be considered <em>trivially</em> true. It's not really to say that there are actually members for which Q, but rather for all y, Q(y)--i.e. there are no circumstances where y and not Q(y). </p>
|
4,554,831 | <blockquote>
<p>Let <span class="math-container">$(X,d)$</span> be a metric space. Prove that if the point <span class="math-container">$x$</span> is on the boundary of the open ball <span class="math-container">$B(x_0,r)$</span> then <span class="math-container">$d(x_0,x)=r$</span>.</p>
</blockquote>
<p>I find this di... | geetha290krm | 1,064,504 | <p>If <span class="math-container">$d(x,x_0) <r$</span> then <span class="math-container">$x \in B(x_0,r)$</span> so <span class="math-container">$x$</span> is not a boundary point. (<span class="math-container">$B(x_0,r)$</span> is open). If <span class="math-container">$d(x,x_0) >r$</span> and <span class="mat... |
3,884,891 | <p>I am reading an article about "Longest Paths in Digraphs". In the proof there is a step that they considered as a trivial step, it seems easy but I am not being able to write or concieve an exact proof for it.</p>
<p>We have a strong digraph <span class="math-container">$D$</span>, that has minimum in degr... | Mike | 544,150 | <p>Let us assume here that <span class="math-container">$k \ge h$</span> as you did and let <span class="math-container">$P$</span> be a <em>path</em> of length <span class="math-container">$c \ge \max(h,k)+1 = k+1$</span>. Your line of reasoning shows that there is indeed such a path <span class="math-container">$P$</... |
90,112 | <p>When reading "Chebyshev centers and uniform convexity" by Dan Amir I encountered the following result which is apparently "known and easy to prove". I'm sure it is, but I can't find a proof and am failing to prove it myself.</p>
<p>The result (slightly simplified) is</p>
<p>If $X$ is a uniformly convex space (i.e.... | Bill Johnson | 2,554 | <p>Another book reference:</p>
<p>This is proved as Lemma 9.2 in ``Functional Analysis and Infinite-dimensional Geometry" by Fabian, Habala, Hajek, Montesinos Santalucia, Pelant, and Zizler.</p>
|
1,775,649 | <p>True or false and explain why?: a matrix with characteristic polynomial $\lambda^3 -3\lambda^2+2\lambda$ must be diagonalizable.</p>
<p>First I found the lambda's that make this zero (eigenvalues) and got $0, 1, 2$ but I don't know if having $0$ as an eigenvalue means that the matrix is not diagonalizable? I know t... | marwalix | 441 | <p>The characteristic polynomial splits as follows $P_M(\lambda)=\lambda(\lambda-1)(\lambda-2)$. The matrix is $3\times 3$ (the degree of the characteristic polynomial) and has three distinct eigenvalues therefore it is diagonalisable.</p>
|
715,809 | <p>I have occasionally come across Leibniz's Law (left to right, ie the indiscernibility of identicals) written with schematic letters in the consequent, and occasionally with a bound predicate-variable taking the place of the schematic letters. What is the relevant difference between these formulations? If there is ... | Peter Smith | 35,151 | <p>Suppose, to fix ideas, we are doing arithmetic, i.e. our first-order variables run over numbers, and suppose our language is countable.</p>
<p>The schematic version of Leibniz's Law, applied in this context, in effect tells us that if the numbers $m$ and $n$ are equal, then a first-order predicate $\varphi$ is true... |
1,811,443 | <p>Let $(a_n)$ be a sequence of rational numbers, where <strong>all rational numbers are terms</strong>. (<em>i.e. enumeration of rational numbers</em>)</p>
<p>Then, is there any convergent sub-sequence of $(a_n)$?</p>
| Jason | 195,308 | <p>Even better - for every real $x$, there is a subsequence of $(a_n)$ converging to $x$.</p>
<p>Indeed, start by letting $a_{n_1}$ be any element of the sequence in $(x-1,x+1)$. Then, given $a_{n_1},\ldots,a_{n_k}$, since there are infinitely many rationals in each interval, there must exist $n_{k+1}>n_k$ such tha... |
2,497,875 | <p>Define $\sigma: [0,1]\rightarrow [a,b]$ by $\sigma(t)=a+t(b-a)$ for $0\leq t \leq 1$. </p>
<p>Define a transformation $T_\sigma:C[a,b]\rightarrow C[0,1]$ by $(T_\sigma(f))(t)=f(\sigma(t))$ </p>
<p>Prove that $T_\sigma$ satisfies the following:</p>
<p>a) $T_\sigma(f+g)=T_\sigma(f)+T_\sigma(g)$</p>
<p>b) $T_\sigma... | Fawad | 369,983 | <p>Hint:</p>
<p>$x-1={\left (\sqrt x \right )}^2 -1^2$</p>
<p>$x-1=(\sqrt x +1)(\sqrt x-1)$</p>
|
2,497,875 | <p>Define $\sigma: [0,1]\rightarrow [a,b]$ by $\sigma(t)=a+t(b-a)$ for $0\leq t \leq 1$. </p>
<p>Define a transformation $T_\sigma:C[a,b]\rightarrow C[0,1]$ by $(T_\sigma(f))(t)=f(\sigma(t))$ </p>
<p>Prove that $T_\sigma$ satisfies the following:</p>
<p>a) $T_\sigma(f+g)=T_\sigma(f)+T_\sigma(g)$</p>
<p>b) $T_\sigma... | Gono | 384,471 | <p>A different solution without needing $\frac{sin(x)}{x}$:
$$\frac{\sin(1-\sqrt{x})}{x-1} = \frac{\sin(1-\sqrt{x}) - \sin(1-\sqrt{1})}{x-1} = -\frac{\cos(1-\sqrt{\xi})}{2\sqrt{\xi}}$$ for a $\xi \in (1,x)$ by the mean value theorem.
And because $\xi \to 1$ if $x\to 1$ we get:</p>
<p>$$\lim_{x\to 1}\frac{\sin(1-\sqrt... |
13,889 | <p><strong>Question:</strong> Are there intuitive ways to introduce cohomology? Pretend you're talking to a high school student; how could we use pictures and easy (even trivial!) examples to illustrate cohomology?</p>
<p><strong>Why do I care:</strong> For a number of math kids I know, doing algebraic topology is fi... | Jjm | 196,095 | <p>There is a nice and, to my opinion, more natural way to motivate cohomology - a geometric one, rather than an analytical one. Please read carefully the following question and answer in math.stackexchange:</p>
<p><a href="https://math.stackexchange.com/questions/1112419/intuitive-approach-to-de-rham-cohomology">Intu... |
2,430,690 | <p>(<a href="https://i.stack.imgur.com/NlAPf.jpg" rel="nofollow noreferrer">https://i.stack.imgur.com/NlAPf.jpg</a>)</p>
<p>This is a question in my school's exercise book
I couldn't think of any equation to be formed to solve it
Please teach me</p>
| George Coote | 445,167 | <p>Remember, </p>
<p>$$\sin^2\theta + \cos^2\theta \equiv 1$$ </p>
<p>For all $\theta$. Given this, and that $\sin\theta = k$, how can we find $\cos\theta$ in terms of $k$? How can we then find $\tan\theta$? Can you think of where to go from there seeing that $w = 360 - \theta$?</p>
|
2,430,690 | <p>(<a href="https://i.stack.imgur.com/NlAPf.jpg" rel="nofollow noreferrer">https://i.stack.imgur.com/NlAPf.jpg</a>)</p>
<p>This is a question in my school's exercise book
I couldn't think of any equation to be formed to solve it
Please teach me</p>
| gen-ℤ ready to perish | 347,062 | <p>First, the diagram makes it clear that $w^\circ = \theta$.</p>
<p>Identically, it is true that $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$. From the Pythagorean theorem, we know that $\cos^2\theta = 1 - \sin^2\theta$.</p>
<p>$$\begin{align}
\tan w^\circ &= \tan\theta \\
&= \frac{\sin\theta}{\cos\theta} \... |
4,444,504 | <p>We have measure theory in this semester.I found the statement of Lusin's theorem on the internet to be:</p>
<blockquote>
<p>Let <span class="math-container">$f:\mathbb{R\to R}$</span> be a Lebesgue measurable function.Then for each <span class="math-container">$\epsilon>0$</span> there exists a closed set <span c... | coudy | 716,791 | <p>The second statement is wrong. There is no compact subset <span class="math-container">$K$</span> in <span class="math-container">${\bf R}$</span> such that <span class="math-container">$|{\bf R} \setminus K| < \varepsilon$</span>. A compact subset of the real line is bounded and his complement is of infinite Leb... |
118,742 | <p>The PDF for $Y$ is
$$f_Y(y) = \begin{cases}
0 & |y|> 1 \\
1-|y| & |y|\leq 1
\end{cases}$$</p>
<p>How do I find the corresponding CDF $F_Y(y)$? I integrated the above piecewise function to get
$$F_Y(y)=\begin{cases}
1/2 -y/2-y^2/2 & [-1,0] \\
1/2-y/2+y^2/2 & [0,1]
\en... | anon | 11,763 | <p>First work with $y\le 0$ to obtain</p>
<p>$$F_Y(y)=\int_{-1}^y 1-|u|du=\int_{-1}^y 1+u du=y-(-1)+\frac{y^2-(-1)^2}{2}=\frac{1}{2}+y+\frac{1}{2}y^2 $$</p>
<p>Now work with $1\ge y\ge0$ by splitting (using the fundamental theorem of calculus)</p>
<p>$$F_Y(y)=\int_{-1}^y f_Y(u)du=F_Y(0)+\int_0^y 1-u du $$</p>
<p>No... |
4,028,534 | <p>I have three points with coordinates: <span class="math-container">$A (5,-1,0),B(2,4,10)$</span>, and <span class="math-container">$C(6,-1,4)$</span>.</p>
<p>I have the following vectors <span class="math-container">$\overrightarrow {CA} = (-1, 0, -4)$</span> and <span class="math-container">$\overrightarrow{CB} = (... | Quanto | 686,284 | <p>Note that <span class="math-container">$a=\sqrt{17}$</span>, <span class="math-container">$b=\sqrt{77}$</span> and <span class="math-container">$\cos C= -\frac{20}{ab}$</span>, which yields the area</p>
<p><span class="math-container">$$A= \frac12 ab \sqrt{1-\cos^2C}=\frac12 \sqrt{17\cdot77-400}=\frac{3\sqrt{101}}2$... |
2,352,527 | <p><em>This is rather a simple problem that I'm posting ; looking forward not for the solution of it but the different ways it could be solved.</em></p>
<p>What is the value of </p>
<p>$\sin\frac{2\pi}{7}\sin\frac{4\pi}{7}+\sin\frac{4\pi}{7}\sin\frac{8\pi}{7}+\sin\frac{8\pi}{7}\sin\frac{2\pi}{7}$ </p>
<p><em>Do I ju... | tattwamasi amrutam | 90,328 | <p>Hint: Take $$g(z)=\frac{f(z)}{z}, z \ne 0$$
and $$g(0)=f'(0)$$.</p>
<p>Show that $g$ is an entire function which is bounded. </p>
|
2,352,527 | <p><em>This is rather a simple problem that I'm posting ; looking forward not for the solution of it but the different ways it could be solved.</em></p>
<p>What is the value of </p>
<p>$\sin\frac{2\pi}{7}\sin\frac{4\pi}{7}+\sin\frac{4\pi}{7}\sin\frac{8\pi}{7}+\sin\frac{8\pi}{7}\sin\frac{2\pi}{7}$ </p>
<p><em>Do I ju... | zhw. | 228,045 | <p>The integral approach you tried is indeed problematic, but let me show you one that works. The key is to look at the integrals of $|f|^2.$ From the given growth estimate we have</p>
<p>$$\tag 1 \int_{-\pi} ^\pi |f(re^{it})|^2\, dt \le 2\pi M^2r^2\,\,\text { for } r\ge 0.$$</p>
<p>On the other hand, we have $f(z) =... |
505,848 | <p>It is known that <a href="http://mathworld.wolfram.com/SomosSequence.html" rel="nofollow noreferrer">the $k$-Somos sequences</a> always give integers for $2\le k\le 7$.</p>
<p>For example, the $6$-Somos sequence is defined as the following : </p>
<p>$$a_{n+6}=\frac{a_{n+5}\cdot a_{n+1}+a_{n+4}\cdot a_{n+2}+{a_{n+3... | njguliyev | 90,209 | <p>Hints:
1. For any $\varepsilon>0$ let $\delta := \frac{a}{N}$ and apply (b).<br>
2. It is obvious that $f(0) = 0$. Since $f$ has a limit at $0$, we can put $\varepsilon = 1$ and write: $\exists \delta > 0,\ \forall x,\ |x|<\delta\colon\ |f(x)|<1$.</p>
|
505,848 | <p>It is known that <a href="http://mathworld.wolfram.com/SomosSequence.html" rel="nofollow noreferrer">the $k$-Somos sequences</a> always give integers for $2\le k\le 7$.</p>
<p>For example, the $6$-Somos sequence is defined as the following : </p>
<p>$$a_{n+6}=\frac{a_{n+5}\cdot a_{n+1}+a_{n+4}\cdot a_{n+2}+{a_{n+3... | Community | -1 | <p>Your Question is completely answered in the following book (Theorem 1.2, page 3). Also you can find some further related topics.</p>
<p>Pl. Kannappan, Functional Equations and Inequalities with Applications, Springer, 2009.</p>
<p>Theorem 1.2. Suppose $A : R\rightarrow R$ satisfies $A(x+y)=A(x)+A(y)$ with $c = A(1... |
2,333,847 | <p>A function $f(x) = k$ and the domain is $\{-2,-1,\dotsc,3\}$. Would I say
$$x = \{-2,-1,\dotsc,3\}\quad\text{or}\quad x \in \{-2,-1,\dotsc,3\} \ ?$$
Thanks. </p>
| Community | -1 | <p>Use $\in$. And be careful with your commas: $\{-2, -1 \color{red}, \dots \color{red}, 3\}$</p>
<p>In general:</p>
<p>$x \in S$ means "$x$ is an element of $S$" or "$x$ is in $S$." (The $\LaTeX$ code to produce "$\in$" is <code>$\in$</code>.)</p>
<p>$x = s$ means "$x$ is $S$" or "$x$ is equal to $S$." In this ... |
174,528 | <p>I am editing my original question, as I have figured out a method of doing what I want.</p>
<p>Now my question is if there is a more elegant, efficient way to do the following:</p>
<pre><code>Options[f] = {
Energy -> energy,
Temperature -> Func[Energy*Frequency, {Energy, Frequency}],
Frequency ... | Henrik Schumacher | 38,178 | <p>Apparently, the only problem was that you scoped <code>f</code> by <code>Module</code> so that <code>OptionValue</code> had a hard time to find the default values for <code>f</code>. (<code>Option</code> stores <code>Option[f]</code> within <code>f</code>, not as <code>DownValue</code> of <code>Option</code>.) </p>
... |
660,315 | <p>Let $A,B,C$ be sets. Identify a condition such that $A \cap C = B \cap C$ together with your condition implies $A=B$. Prove this implication. Show that your condition is necessary by finding an example where $A \cap C = B \cap C$, but $ A \neq B$</p>
<p>Edit: I've read the wrong proposition/definition. UGH! The que... | Asaf Karagila | 622 | <p><strong>HINT:</strong> What happens if both $A=A\cap C$ and $B=B\cap C$? (Note that this generalizes the case of both $A$ and $B$ being empty.)</p>
|
2,747,509 | <p>How would you show that if <span class="math-container">$d\mid n$</span> then <span class="math-container">$x^d-1\mid x^n-1$</span> ?</p>
<p>My attempt :</p>
<blockquote>
<p><span class="math-container">$dq=n$</span> for some <span class="math-container">$q$</span>. <span class="math-container">$$ 1+x+\cdots+x^{d-1}... | fleablood | 280,126 | <p>Notice that for any $x$ and and natural $n$ that $$(x-1)(x^{n-1} + ..... + x + 1) = (x^n + x^{n-1} +....... +x) - (x^{n-1} + x^{n-2} +.... +1) = x^n -1$$ so that $x-1|x^n - 1$ always.</p>
<p><strong>Lemma:</strong> $x-1|x^n-1$ for natural $n$.</p>
<p>Now $d|n$ so let $m = \frac nd$ and let $y= x^d$.</p>
<p>Then $... |
2,747,509 | <p>How would you show that if <span class="math-container">$d\mid n$</span> then <span class="math-container">$x^d-1\mid x^n-1$</span> ?</p>
<p>My attempt :</p>
<blockquote>
<p><span class="math-container">$dq=n$</span> for some <span class="math-container">$q$</span>. <span class="math-container">$$ 1+x+\cdots+x^{d-1}... | Anastassis Kapetanakis | 342,024 | <p>You have:
$$d\mid n \Rightarrow n=ad$$
Then:
$$x^n-1=x^{ad}-1=(x^d)^a-1$$
Setting $x^d=y$ (just for simplifying the process) we have:
$$y^a-1=(y-1)(y^{a-1}+\dots+y+1)=(x^d-1)((x^d)^{a-1}+\dots + x^d+1)$$
In other words we showed that:
$$x^n-1= (x^d-1)((x^d)^{a-1}+\dots + x^d+1) $$
Which obviously implies that:
$$x... |
4,144,083 | <p>I am currently doing an internship in a research laboratory ( I am in my third year of Bachelor ) and I'm really struggling with the things I have to do.</p>
<p>For instance, here's something I'm having trouble with.</p>
<p>Let <span class="math-container">$L$</span> be a finite Galois extension of <span class="math... | reuns | 276,986 | <p>If <span class="math-container">$K$</span> is the subfield of <span class="math-container">$L$</span> fixed by <span class="math-container">$\{ \sigma\in Gal(L/\Bbb{Q}), \sigma(I)\subset I\}$</span>
then try with <span class="math-container">$O_L=\Bbb{Z}[i], I=(1+i)^3,O_K=\Bbb{Z},I\cap O_K=(4)$</span>,</p>
<p>As a g... |
657,047 | <p>So I have $a^n = b$. When I know $a$ and $b$, how can I find $n$?</p>
<p>Thanks in advance!</p>
| lsp | 64,509 | <p>Apply Logarithm on both sides:</p>
<p>$$ \log a^n = \log b$$
$$n \log a = \log b$$
$$n = \frac{\log b}{\log a}$$</p>
<p>If you are a starter in Logarithms, you can refer <a href="http://en.wikipedia.org/wiki/Logarithm" rel="nofollow">here</a>.</p>
|
1,563,044 | <blockquote>
<p>Prove by transfinite induction that there is a function $f:\mathbb R \to\mathbb R$ such that $|f^{-1}(r) \bigcap (a,b)| = 2^{\omega}$ for every $a, b, r \in\mathbb R$ and $a < b$.</p>
</blockquote>
<p>I have:</p>
<p>Let $F = \{(a,b) \times \{r\}\, |\, a, b, r \in\mathbb R\text{ and }a < b\}$ a... | Tormod Haugland | 240,556 | <p>Sadly I don't have enough reputation to comment, so I guess I will post an answer instead. I hope its not inappropriate.</p>
<p>As indicated by the comments, the answer varies on your definition of $e^x$. In fact one definition is implicitly by:</p>
<p>$$ \lim_{h \to 0} \frac{e^h - 1}{h} = 1$$</p>
<p>Defining $e^... |
1,563,044 | <blockquote>
<p>Prove by transfinite induction that there is a function $f:\mathbb R \to\mathbb R$ such that $|f^{-1}(r) \bigcap (a,b)| = 2^{\omega}$ for every $a, b, r \in\mathbb R$ and $a < b$.</p>
</blockquote>
<p>I have:</p>
<p>Let $F = \{(a,b) \times \{r\}\, |\, a, b, r \in\mathbb R\text{ and }a < b\}$ a... | Akiva Weinberger | 166,353 | <p>As I've said fairly often in the last few days (for some reason), one of my favorite equations is:
$$e^x\ge x+1$$
The reason, partly, is that it uniquely defines $e$ without calculus. Hint for a proof: use <a href="https://en.m.wikipedia.org/wiki/Bernoulli%27s_inequality" rel="noreferrer">this</a>. (By the way, do e... |
118,311 | <p>Let $A$ be an $n\times n$ matrix with entries in an arbitrary field $k$. </p>
<p>Is the characteristic polynomial $\det(tI_n-A)$ dependent only on the trace and determinant of $A$?</p>
| Community | -1 | <p>Consider the representation $\tau_3$ (or $\tau_4$) and an eigenvalue $w$ of the the of $x$. Then $w$ is a $7^{th}$ root of unity. Since $x^2$ and $x^4$ are conjugate to $x$, $w^2$ and $w^4$ are also eigenvalues of $x$. All eigenvalues of $x$ cannot be $1$ by column orthonormality. So, $w$, $w^2$ and $w^4$ are distin... |
336,943 | <p>Given $P(A) = 0.5$ and $P(A \cup (B^c \cap C^c)^c)=0.8$.</p>
<p>Determine $P(A^c \cap (B \cup C))$.</p>
<p>I know from DeMorgans law that: $(B^c \cap C^c)^c = (B \cup C)$.</p>
<p>Edit:</p>
<p>Also how can I "prove" that P(X)=P(Y) if and only if $P(X \cap Y^c) = P(X^c \cap Y)$? </p>
| Cameron Buie | 28,900 | <p>Note that $$\begin{align}P\bigl(A^c\cap(B\cup C)\bigr) &= P(A^c\cap A)+P\bigl(A^c\cap(B\cup C)\bigr)\\ &= P\bigl(A^c\cap (A\cup B\cup C)\bigr)\\ &= P\bigl((A\cup B\cup C)-A\bigr),\end{align}$$ and that if $A$ occurs, then $A\cup B\cup C$ occurs. Thus, $$P\bigl(A^c\cap(B\cup C)\bigr)=P(A\cup B\cup C)-P(A)... |
106,396 | <p>An Indian mathematician, Bhaskara I, gave the following amazing approximation of the sine (I checked the graph and some values, and the approximation is truly impressive.)</p>
<p>$$\sin x \approx \frac{{16x\left( {\pi - x} \right)}}{{5{\pi ^2} - 4x\left( {\pi - x} \right)}}$$</p>
<p>for $(0,\pi)$</p>
<p>Here's ... | Community | -1 | <ul>
<li><p>Here is an <a href="http://www.jstor.org/pss/10.4169/math.mag.84.2.098" rel="noreferrer">article</a> which is written by Shailesh Shirali. Unfotunately my university doesn't have access to it.</p></li>
<li><p>Here is one more <a href="http://www.dli.gov.in/rawdataupload/upload/insa/INSA_1/20005af0_121.pdf" ... |
3,632,431 | <blockquote>
<p>Consider the function <span class="math-container">$f: \mathbb{N} \to \mathbb{N}$</span> defined by <span class="math-container">$f(x)=\frac{x(x+1)}{2}$</span>. Show that <span class="math-container">$f$</span> is injective but not surjective.</p>
</blockquote>
<p>So I started by assuming that <span ... | Shaun | 104,041 | <p>To show <span class="math-container">$f(x)=f(y)$</span> implies <span class="math-container">$x=y$</span>, show the contrapositive, namely, </p>
<p><span class="math-container">$$a\neq b\implies f(a)\neq f(b).$$</span></p>
<p>So suppose WLOG that <span class="math-container">$a<b$</span>. Then <span class="math... |
448,694 | <p>Let $k$ be a field, and $A$ be a finitely generated $k$-algebra. Then does $k$ also have to be a finitely generated field?</p>
<p>Motivation: Let $A$ be generated by $\{a_1,a_2,\dots,a_n\}$, and $k$ be generated by $\{k_1,k_2,k_3,\dots\}$. Then the element $k_1k_2k_3\dots a_1\in A$ is not generated by a finite numb... | celtschk | 34,930 | <p>A very simple counter example:</p>
<p>Be $A=\{0\}$ the trivial algebra over $K$: $A$ clearly is finitely generated (it contains just one element!), even if $K$ is not.</p>
|
1,324,062 | <p>Evaluate: </p>
<blockquote>
<p>$$\lim_{h \rightarrow 0} \frac{e^{2h}-1}{h}$$</p>
</blockquote>
<p>Now one way would be using the Maclaurin expansion for $e^{2x}$</p>
<p>However, can we solve it using the definition of the derivative (perhaps considering $f(x)=e^x$)? Many thanks for your help! $$$$
EDIT: I forgo... | wythagoras | 236,048 | <p>Using $e^{2h}-1=(e^h-1)(e^h+1)$, we get </p>
<p>\begin{align*}
\lim_{h \rightarrow 0} \frac{e^{2h}-1}{h} &= 2 \lim_{h \to 0}\frac{
(e^h-1)(e^h+1)}{h} \\&= \lim_{h \to 0}(e^h+1)\lim_{h \to 0}\frac{e^h-1}{h} \\&= 2\lim_{h \to 0}\frac{e^h-1}{h} \\&=2\lim_{h \to 0}\frac{e^{0+h}-e^0}{h} \\&=2f'(0)=2e... |
3,853,980 | <p>Let <span class="math-container">$A$</span> be an <span class="math-container">$n×n $</span> complex matrix such that the three matrices <span class="math-container">$A+I$</span> , <span class="math-container">$A^2+I $</span> , <span class="math-container">$ A^3+I$</span> are all unitary .Prove that<span class="mat... | Michael Hardy | 11,667 | <p>In fact <span class="math-container">$\Pr(\max\{X_1,X_2\}=x) = 0.$</span> I assume you must have meant that the value of the probability <b>density</b> function of <span class="math-container">$\max\{X_1,X_2\}$</span> at <span class="math-container">$x$</span> is <span class="math-container">$2x.$</span></p>
<p><spa... |
3,532,033 | <p>Let <span class="math-container">$(M_1,+,\times)$</span> be an algebraic structure, lets say, for example, a ring. If we have another structure <span class="math-container">$(M_2,+,\times)$</span> isomorphic to the first one does that mean that <span class="math-container">$(M_2,+,\times)$</span> is also a ring ?</p... | 9sven6 | 669,847 | <p>What you say is technically correct, but isomorphic doesn't just mean <span class="math-container">$(M_2,+,\times)$</span> is a ring, but with the exact same structures. <span class="math-container">$M_1$</span> and <span class="math-container">$M_2$</span> can be mapped one-to-one on eachother. </p>
|
35,220 | <p>It is a basic result of group cohomology that the extensions with a given abelian normal subgroup <em>A</em> and a given quotient <em>G</em> acting on it via an action $\varphi$ are given by the second cohomology group $H^2_\varphi(G,A)$. In particular, when the action is trivial (so the extension is a central exten... | David Corwin | 1,355 | <p>You might find the material on the interpretation of $\mathrm{Ext}$ in terms of extensions at <a href="http://en.wikipedia.org/wiki/Ext_functor#Ext_and_extensions" rel="nofollow">Ext and Extensions</a> to be useful. You probably know that $H^n(G,M) = \mathrm{Ext}^n_{\mathbb{Z}[G]}(\mathbb{Z},M)$ (which is not exactl... |
4,348,455 | <p>Each digits of the decimal expansion of the integer <span class="math-container">$2022$</span> (this year) consists of <span class="math-container">$0$</span> or <span class="math-container">$2$</span> and also, each digits of the ternary expansion of the same integer <span class="math-container">$2022$</span> (whic... | paw88789 | 147,810 | <p>I am writing this as an answer rather than as a comment because the formatting is better. In a computer search I found the following thirteen values that satisfy the conditions (all values are given in decimal):</p>
<p><span class="math-container">$0\\
2\\
20\\
222\\
2000\\
2022\\
22220202002202\\
22220202002220\\
2... |
105,071 | <p>As one may know, a <b>dynamical system</b> can be defined with a monoid or a group action on a set, usually a manifold or similar kind of space with extra structure, which is called the <i>phase space</i> or <i>state space</i> of the dynamical system. The monoid or group doing the acting is what I call the <i>time s... | Community | -1 | <p>I have seen talks about this, though somewhat over my head. But, yes absolutely, people do research such things. A. Katok is one name that comes to mind here. See this paper: <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.33.2082" rel="nofollow">http://citeseerx.ist.psu.edu/viewdoc/summary?doi=1... |
148,624 | <p>What does it mean to say that , A bounded linear operator is not "generally" bounded function.
Can anybody explain ? </p>
| Robert Israel | 8,508 | <p>A nonzero linear operator on any vector space (over $\mathbb R$ or $\mathbb C$) is never a bounded function: $\|A(tx)\| = |t| \|Ax\| \to \infty$ as $t \to \infty$ whenever $Ax \ne 0$.</p>
|
148,624 | <p>What does it mean to say that , A bounded linear operator is not "generally" bounded function.
Can anybody explain ? </p>
| Alex Becker | 8,173 | <p>A linear operator $T:V\to W$ is said to be "bounded" if there exists some $M\in\mathbb R$ such that $\|Tx\|\leq M\|x\|$ for all $x\in V$. A function $f:V\to W$ is said to be "bounded" if there exists some $M\in\mathbb R$ such that $\|f(x)\|\leq M$ for all $x\in V$. Note that this is a stronger condition than the pre... |
623,709 | <p>I make the following conjecture: the function
$$
d(x, y):=\frac{||x-y||}{\max(||x||, ||y||)}
$$
is a distance on $H$, where $H$ is a normed vector space or a Hilbert space, and $x, y \in H$ (the function $d$ is defined to be $0$ in the case $x=y=0$). Note that $d$ is scale invariant, i.e., $d(\lambda x, \lambda y)=... | Community | -1 | <p>We know that <span class="math-container">$p(x,y):= \|x-y\|$</span> is a metric. Hence we get for free</p>
<p><span class="math-container">$$\|x-z\|\leq \|x-y\|+\|y-z\|.$$</span></p>
<p>Remark: The case where exactly one of the vectors is the zero vector is trivial because we get an inequality of the form:</p>
<p><s... |
623,709 | <p>I make the following conjecture: the function
$$
d(x, y):=\frac{||x-y||}{\max(||x||, ||y||)}
$$
is a distance on $H$, where $H$ is a normed vector space or a Hilbert space, and $x, y \in H$ (the function $d$ is defined to be $0$ in the case $x=y=0$). Note that $d$ is scale invariant, i.e., $d(\lambda x, \lambda y)=... | coffeemath | 30,316 | <p>Let the norm on the plane be the taxicab norm, norm of $(x,y)$ being $|x|+|y|.$ This induces a metric (satisfying triangle inequality etc.) on the metric space $\mathbb{R}^2$, making it a normed linear space.</p>
<p>But with your metric obtained by division of max norms, take the points $x,y,z$ as $(1,2),(2,1),(2,2... |
1,770,804 | <p>I am a high school student my maths teacher said that if $\,ax+b=cx+d,\,$ then is $\,a=c\,$ and $\,b=d.\,$ Can someone give me a prove of this?</p>
| Community | -1 | <p>By the fundamental theorem of algebra (FTA), a first degree polynomial has exactly $1$ root. Thus if there are $2$ or more values of $x$ for which the equation holds, then $(a-c)x+(b-d)$ is the zero polynomial. So $a=c$ and $b=d$.</p>
|
9,168 | <p>I'm having a doubt about how should we users encourage the participation of new members. So far I have only presented MSE to three of my fellow colleagues in grad school. In an overall way I feel like if MSE becomes too open and wide known, some of the high-rank researchers and top-class grad and undergrads users wi... | Thomas Klimpel | 12,490 | <p>Community growth in the form of people willing to answer question should be encouraged. If these simple questions give some people the opportunity to write nice and clear answers, great. The answers from 10K+ users to these trivial questions might indeed be something worrying. I think the trivial questions themselve... |
4,026,149 | <p>If f is continuous on <span class="math-container">$[a,b]$</span> and <span class="math-container">$f(a)=f(b)$</span> then show that there exists <span class="math-container">$x,y \in (a,b)$</span> such that <span class="math-container">$f(x)=f(y)$</span></p>
<p>It looks obvious if I imagine the graph. But I am not ... | user10354138 | 592,552 | <p><strong>Hint</strong>: Either <span class="math-container">$f$</span> is a constant (so you are done), or there is some <span class="math-container">$t\in(a,b)$</span> such that <span class="math-container">$f(t)\neq f(a)$</span>, in which case employ IVT for <span class="math-container">$f$</span> on <span class="m... |
863,846 | <p>Steven Strogatz has a great informal textbook on Nonlinear Dynamics and Chaos. I have found it to be incredibly helpful to get an intuitive sense of what is going on and has been a great supplement with my much more formal text from Perko.</p>
<p>Anyways I was wondering if anyone knew of any similar informal, intui... | BeaumontTaz | 147,480 | <p>It can be a little dense sometimes, but <em>Numerical Analysis</em> by Burden and Faires is a classic book on the subject.</p>
|
1,823,556 | <p>Let be $X \subset F_1 \cup F_2$, where $F_1$ and $F_2$ are closed. If the function $f\colon X \longrightarrow \mathbb{R}$ is such that $f|_{X \cap F_1}$ and $f|_{X \cap F_2}$ are continuous, so prove that $f$ is continuous. </p>
<p>My attempt:</p>
<p>Suppose that $f$ is discontinuously, so exists $x \in X$ such th... | Brian M. Scott | 12,042 | <p>HINT: Something similar to your approach is workable, but you’ve omitted most of the crucial details. In particular, you’ve not justified the assertion that $f\upharpoonright X\cap F_1$ is discontinuous at $x$. </p>
<p>Suppose that $f$ is not continuous at $x$; then there are an $\epsilon>0$ and a sequence $\lan... |
1,823,556 | <p>Let be $X \subset F_1 \cup F_2$, where $F_1$ and $F_2$ are closed. If the function $f\colon X \longrightarrow \mathbb{R}$ is such that $f|_{X \cap F_1}$ and $f|_{X \cap F_2}$ are continuous, so prove that $f$ is continuous. </p>
<p>My attempt:</p>
<p>Suppose that $f$ is discontinuously, so exists $x \in X$ such th... | Qiyu Wen | 310,935 | <p>We can be a bit more general. Let $X = A\cup B$ be such that $A-B$ and $B-A$ are separated. For any subset $E$ of $X$, its $X$-closure is the union of the $A$-closure of $E\cap A$ and $B$-closure of $E\cap B$. That is,
$$
\bar{E} = \left(\overline{E\cap A} \cap A\right) \cup \left(\overline{E\cap B} \cap B\right).
$... |
109,213 | <p>In classical complex analysis it is easy to prove that a meromorphic function has at most one analytic continuation (on an open connected subset of $\mathbb C$, say).</p>
<p>The problem of non-uniqueness of analytic continuation is one of the reasons why it is not possible (if one wants a good theory) to translate ... | user27056 | 27,056 | <p>No: let $X$ be the union of the coordinate axes in the affine plane. As over $\mathbf{C}$, the answer is affirmative on a connected <em>normal</em> analytic space. Hint: prove in any rigid-analytic space that connected components are witnessed via finite linked chains of connected affinoid opens (and after thereby ... |
3,106,574 | <p>Let <span class="math-container">$(a_n) _{n\ge 0}$</span> <span class="math-container">$a_{n+2}^3+a_{n+2}=a_{n+1}+a_n$</span>,<span class="math-container">$\forall n\ge 1$</span>, <span class="math-container">$a_0,a_1 \ge 1$</span>. Prove that <span class="math-container">$(a_n) _{n\ge 0}$</span> is convergent.<br>
... | Barry Cipra | 86,747 | <p>This is a variant on maxmilgram's answer. If we write <span class="math-container">$a_n=1+u_n$</span> with <span class="math-container">$u_n\ge0$</span> (as the OP observes), the recursion becomes</p>
<p><span class="math-container">$$u_{n+2}^3+3u_{n+2}^2+4u_{n+2}=u_{n+1}+u_n$$</span></p>
<p>This implies <span cla... |
3,927,502 | <p><span class="math-container">$\lim\limits_{n\to\infty}\dfrac{n^2-n+2}{3n^2+2n-4}=\dfrac{1}{3}$</span>.</p>
<p>With epsilon definition I get my answer as <span class="math-container">$N=\left[ \dfrac{5}{9\varepsilon }\right] +1$</span>. But then I thought that how can I evaluate this sequence, in functions <span clas... | zipirovich | 127,842 | <p>How is it any different from what you've already done? Renaming <span class="math-container">$n$</span> into <span class="math-container">$x$</span> and <span class="math-container">$N$</span> into <span class="math-container">$M$</span> doesn't really change anything. The same answer still works. Or you can simplif... |
133,418 | <p>Let $\langle R,0,1,+,\cdot,<\rangle$ be the standard model for R, and let S be a countable model of R (satisfying all true first-order statements in R). Is it true that the set 1,1+1,1+1+1,… is bounded in S? My intuition says "no", but I am yet to find a counter example. I read something about rational functions, ... | hmakholm left over Monica | 14,366 | <p>Let $T$ be the set of all true sentences about $\mathbb R$ and construct $T'$ by adding to $T$ a new constant $c$ together with the axioms $c>1$, $c>1+1$, $c>1+1+1$, ...</p>
<p>Every finite subset of $T$ has $\mathbb R$ as a model, so $T'$ is consistent by the compactness theorem, and has a countable model... |
480,727 | <p>If $$2^x=3^y=6^{-z}$$ and $x,y,z \neq 0 $ then prove that:$$ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0$$</p>
<p>I have tried starting with taking logartithms, but that gives just some more equations.</p>
<p>Any specific way to solve these type of problems?</p>
<p>Any help will be appreciated.</p>
| Mark Bennet | 2,906 | <p>Take logarithms (any base) to obtain $$x\log 2= y\log 3=-z\log 6=-z\log 3-z\log 2$$</p>
<p>Then note that $(y+z)\log 3=-z\log 2$ and $x\log 2 = y\log 3$</p>
<p>Multiply left- and right- hand sides to obtain $$(xy+yz)\log 3 \log 2=-yz\log 2\log 3$$</p>
<p>Whence $xy+yz+zx=0$</p>
<p>Note that we have done no divis... |
3,057,517 | <p>Hy everybody ! </p>
<p>I'm studying population dynamics for my calculus exam, and I don't understand something that seems really easy, so I thought you might be able to help me out ;)</p>
<p>Here's the thing. I have this differential equation <span class="math-container">$\frac{dN}{dt} = \sqrt{N}$</span>.</p>
<p>... | Mohammad Riazi-Kermani | 514,496 | <p>Zero derivative at one point does not mean no growth at other points. </p>
<p>For example the derivative of function <span class="math-container">$$N(t)=\frac {t^2}{4}$$</span> is <span class="math-container">$$N'(t)=t/2$$</span> That is the derivative is zero at <span class="math-container">$t=0$</span> but it is... |
1,032,535 | <p>I know $n \in \mathbb{N}$ and...</p>
<p>$$
a_n = \begin{cases}
0 & \text{ if } n = 0 \\
a_{n-1}^{2} + \frac{1}{4} & \text{ if } n > 0
\end{cases}
$$</p>
<ol>
<li><strong>Base Case:</strong></li>
</ol>
<p>$$a_1 = a^2_0 + \frac{1}{4}$$</p>
<p>$$a_1 = 0^2 + \frac{1}{4} = \frac{1}{... | Adhvaitha | 191,728 | <p>Starting with $a_0 = 0$, it is easier to show the stronger inequality, $0 < a_n < 1/2$ for $n \in \mathbb{Z}^+$. This conclusion immediately falls out from the recurrence relation.</p>
|
199,148 | <p><a href="https://i.stack.imgur.com/9BuHp.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/9BuHp.png" alt="enter image description here"></a> </p>
<pre><code>pbdomains = <|
"Overall " -> Around[2.6, 0.04],
"PB" -> Around[4.25, 0.06]
|>;
BarChart[pbdomains, ChartStyle ... | SelfHorizons Work | 65,424 | <p><a href="https://i.stack.imgur.com/lsGrC.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/lsGrC.png" alt="enter image description here"></a>Based on Rohit Namjoshi solution:</p>
<pre><code>pbdomains = <|"Overall " -> Around[2.6, 0.04],
"PB" -> Around[4.25, 0.06]|>;
BarChart[pbdoma... |
565,046 | <blockquote>
<p>The center of $D_6$ is isomorphic to $\mathbb{Z}_2$.</p>
</blockquote>
<p>I have that
$$D_6=\left< a,b \mid a^6=b^2=e,\, ba=a^{-1}b\right>$$
$$\Rightarrow D_6=\{e,a,a^2,a^3,a^4,a^5,b,ab,a^2b,a^3b,a^4b,a^5b\}.$$
My method for trying to do this has been just checking elements that could be candid... | jasmine | 557,708 | <p>General method to find the centre of <span class="math-container">$D_{2n}$</span></p>
<p>if <span class="math-container">$n=1$</span> or <span class="math-container">$n=2$</span> ,then <span class="math-container">$D_{2n}$</span> is abelian and hence <span class="math-container">$Z(D_{2n})= D_{2n}.$</span></p>
<p>... |
2,172,399 | <p>Equation of the segment : $2x + 4y-3 = 0$
Equation of the hyperbola : $7x^2 - 4y^2 =14$</p>
<p>How do you find the equation of the two linear functions that are both perpendicular to the segment and tangent to the hyperbola?</p>
<p>Thanks</p>
| Mengchun Zhang | 420,459 | <p>The segment has an equation of $\,2x+4y-3=0\,$, hence the equation you are finding should be </p>
<p>$$4x-2y+c=0$$</p>
<p>where $\,c\,$ is a real constant</p>
<p>Now differentiate the hyperbola equation w.r.t. $\,x\,$ </p>
<p>$$14x-8y\frac{dy}{dx}=0$$
$$\Rightarrow\quad\frac{dy}{dx}=\frac{7x}{4y}\qquad$$</p>
<p... |
411,717 | <p>Let $G$ be a group. By an automorphism of $G$ we mean an isomorphism $f: G\to G$
By an inner automorphism of $G$ we mean any function $\Phi_a$ of the following form:
For every $x\in G$, $\Phi_a(x)=a x a^{-1}$.
Prove that every inner automorphism of $G$ is an automorphism of $G$
which means I should prove $\Phi_a$ is... | jim | 44,551 | <p>$\phi_a(xy)=a(xy)a^{-1}=axa^{-1}aya^{-1}=\phi_a(x)\phi_a(y)$</p>
<p>$\phi_a(x)=\phi_a(y)\implies axa^{-1}=aya^{-1}\implies x=y$</p>
<p>$\phi_a$ is also surjective since for each $y \in G $ there exists $x=a^{-1}ya$ s.t. $\phi_a(x)=y$ </p>
|
3,005,842 | <blockquote>
<p>Let <span class="math-container">$(X,Y)$</span> be the coordinates of a point uniformly chosen from a
quadrilateral with vertices <span class="math-container">$(0,0)$</span>, <span class="math-container">$(1,0)$</span>, <span class="math-container">$(1,1)$</span>, <span class="math-container">$(0,2)$</s... | Siong Thye Goh | 306,553 | <p>This is the region:</p>
<p><a href="https://i.stack.imgur.com/kpASM.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/kpASM.png" alt="enter image description here"></a></p>
<p>The area should be <span class="math-container">$\frac32$</span>.
For <span class="math-container">$x \in (0,1)$</span>,
<... |
7,761 | <p>Our undergraduate university department is looking to spruce up our rooms and hallways a bit and has been thinking about finding mathematical posters to put in various spots; hoping possibly to entice students to take more math classes. We've had decent success in finding "How is Math Used in the Real World"-type po... | Marius Kempe | 369 | <p>The German organization <em>Imaginary</em> made a lovely set of posters showing algebraic surfaces. You can <a href="http://www.imaginary-exhibition.com/poster.php" rel="noreferrer">buy a set</a>, though they're in German. Rice University maths department has them on hanging in the corridors. Here's an example:</p>
... |
7,761 | <p>Our undergraduate university department is looking to spruce up our rooms and hallways a bit and has been thinking about finding mathematical posters to put in various spots; hoping possibly to entice students to take more math classes. We've had decent success in finding "How is Math Used in the Real World"-type po... | Karl | 4,668 | <p>This is a slightly left of field answer but have you thought of no posters but something interactive instead? I'd go for installing whiteboards with a hanging pen and stick a problem of the week on it. The questions could be made to be colourful, inspiring and relevant to what you're teaching. The most creative sol... |
2,629,744 | <p>I have done the sum by first plotting the graph of the function in the Left Hand Side of the equation and then plotted the line $y=k$. For the equation to have $4$ solutions, both these two curves must intersect at $4$ different points, and from the two graphs, I could see that for the above to occur, the value of $... | csar | 446,038 | <p>The closed form is $a_n=\begin{cases}a\in\mathbb{R},&n=1\\1,&n\geq2\end{cases}$</p>
<p>It can be verified by substituting the first few $n$'s in the formula (also very easy to show by induction).</p>
|
4,375,994 | <blockquote>
<p>Question:</p>
<p>Show that, <span class="math-container">$$\pi =3\arccos(\frac{5}{\sqrt{28}}) +
3\arctan(\frac{\sqrt{3}}{2}) ~~~~~~ (*)$$</span></p>
</blockquote>
<p><em>My proof method for this question has received mixed responses. Some people say it's fine, others say that it is a verification, inst... | José Carlos Santos | 446,262 | <p>You have<span class="math-container">$$3\arccos\left(\frac5{\sqrt{28}}\right)\in[0,3],$$</span>since <span class="math-container">$\frac{\sqrt3}2<\frac5{\sqrt{28}}<1$</span>, and therefore<span class="math-container">$$3>3\frac\pi6>3\arccos\left(\frac5{\sqrt{28}}\right)>0.$$</span>You also have<span c... |
4,601,727 | <p>I'm aware that the title might be a bit off, I am unsure on how to describe this.</p>
<p>For <span class="math-container">$n\in \mathbb{N}$</span>, define <span class="math-container">$n+1$</span> independent random variables <span class="math-container">$X_0, \ldots , X_n$</span> which are uniformly distributed ove... | Snoop | 915,356 | <p>The random variables <span class="math-container">$Y_\ell=\mathbf{1}_{[0,X_0)}(X_\ell),\,1\leq \ell \leq n$</span> indicate if <span class="math-container">$X_\ell<X_0$</span>. Then by total expectation:
<span class="math-container">$$P(|S|=k)=E\bigg[P\bigg(\sum_{1\leq \ell \leq n}Y_\ell=k\bigg|X_0\bigg)\bigg]=\i... |
3,137,599 | <p>I actually have a doubt about the solution of this question given in my book. It uses the equations tan 2A = - tan C (from A=B, A+B+C = 180 degrees) and 2 tan A + tan C = 100, thereby formulating the cubic equation <span class="math-container">$x^3 - 50x^2 + 50=0$</span>. The discriminant is <span class="math-contai... | J.G. | 56,861 | <p>Since <span class="math-container">$A$</span> is the repeated angle, <span class="math-container">$A$</span> is acute so <span class="math-container">$x:=\tan A>0$</span>. We must thus count the positive roots of <span class="math-container">$x^3-50x^2+50=0$</span>. The roots have a negative product, <span class=... |
3,378,004 | <p>If <span class="math-container">$H$</span> and <span class="math-container">$K$</span> are abelian subgroups of a group <span class="math-container">$G$</span>, then <span class="math-container">$H\cap K$</span> is a normal subgroup of <span class="math-container">$\left\langle H\cup K\right\rangle$</span>.</p>
<p>... | egreg | 62,967 | <p>Suppose <span class="math-container">$L$</span> is a subgroup of <span class="math-container">$\langle X\rangle$</span>, where <span class="math-container">$X$</span> is a subset of a group <span class="math-container">$G$</span>. In order to show that <span class="math-container">$L$</span> is normal in <span class... |
153,902 | <p>Let $A_i$ be open subsets of $\Omega$. Then $A_0 \cap A_1$ and $A_0 \cup A_1$ are open sets as well.</p>
<p>Thereby follows, that also $\bigcap_{i=1}^N A_i$ and $\bigcup_{i=1}^N A_i$ are open sets.</p>
<p>My question is, does thereby follow that $\bigcap_{i \in \mathbb{N}} A_i$ and $\bigcup_{i \in \mathbb{N}} A_i$... | Cameron Buie | 28,900 | <p>Any union of a set of open sets is again open. However, infinite intersections of open sets need not be open. For example, the intersection of intervals $(-1/n,1/n)$ on the real line (for positive integers $n$) is precisely the singleton $\{0\}$, which is not open.</p>
|
46,631 | <p>I'm writing a program to play a game of <a href="http://en.wikipedia.org/wiki/Pente" rel="noreferrer">Pente</a>, and I'm struggling with the following question:</p>
<blockquote>
<p>What's the best way to detect patterns on a two-dimensional board?</p>
</blockquote>
<p>For example, in Pente a pair of neighboring ... | Victor K. | 1,351 | <p>Here is my own <strong>rough</strong> answer - it turns out that asking a question on SE helps clarifying one's thinking! I would still appreciate if some of the experts can weigh in.</p>
<p>First, we'll store the board as a square matrix of symbols <code>B</code>, <code>W</code> and <code>"."</code>:</p>
<pre><co... |
4,288,188 | <p>I am trying to obtain a formulae for a summation problem under section (d) given in a solutions manual for "Data Structures and Algorithm Analysis in C - Mark Allen Weiss", here's the screen shot</p>
<p><a href="https://i.stack.imgur.com/fMie6.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur... | Thomas Andrews | 7,933 | <p>The technique given in the question gives the recurrence:</p>
<p><span class="math-container">$$3S_N=\sum_{i}\frac{(i+1)^N-i^N}{4^i}=\sum_{j=0}^{N-1} \binom Nj S_j$$</span></p>
<p>This follows from the binomial theorem: <span class="math-container">$$(i+1)^N-i^n=\sum_{j=0}^{N}\binom Nj i^j$$</span></p>
<p>But “solvi... |
3,521,534 | <p>I tried solving a calculus problem and I got the right result, but I don't understand the solution provided at the end of the exercise. Even though I got the same answer, I would like to understand what's happening in the given solution aswell.</p>
<blockquote>
<p>Consider the function: <span class="math-containe... | Martin Argerami | 22,857 | <p>At zero, the function <span class="math-container">$f$</span> is either continuous or has a jump; by considering a small enough interval around <span class="math-container">$0$</span>, you can conclude that <span class="math-container">$f$</span> has to be continuous. That gives you <span class="math-container">$b=-... |
3,624,662 | <p><a href="https://i.stack.imgur.com/kwAMn.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/kwAMn.png" alt=" c"></a></p>
<p>In my mind, I can think of below example which seems to work.</p>
<p>If <span class="math-container">$(X,T) = \mathbb{R}$</span>, and <span class="math-container">$A = (0,\inf... | Qurultay | 338,156 | <p>Noting that any bounded closed subset of <span class="math-container">$\mathbb{R}$</span> is compact, so we need to think of unbounded closed subsets like: <span class="math-container">$$[a,\infty), \mathbb{N}, \ldots$$</span></p>
|
1,356,545 | <p>Given a fair 6-sided die, how can we simulate a biased coin with P(H)= 1/$\pi$ and P(T) = 1 - 1/$\pi$ ?</p>
| Asinomás | 33,907 | <p>Throwing a dice $n$ times gives you a space of $6^n$ outcomes, so take an $n$ so that you can approximate $\frac{1}{\pi}$ by $\frac{m}{6^{n}}$ as precisely as required.</p>
<p>After this just pick $m$ of the $6^m$ results to represent heads and the other results represent tails.</p>
<p>The probability of getting t... |
365,631 | <p>Suppose we want to prove that among some collection of things, at least one
of them has some desirable property. Sometimes the easiest strategy is to
equip the collection of all things with a measure, then show that the set
of things with the desired property has positive measure. Examples of this strategy
appear in... | R W | 8,588 | <ol>
<li>The proofs of existence of expanders by Barzdin - Kolmogorov and Pinsker,</li>
</ol>
<p>and (somewhat related)</p>
<ol start="2">
<li>Gromov's proof of the existence of groups with no coarse embedding into a Hilbert space.</li>
</ol>
|
365,631 | <p>Suppose we want to prove that among some collection of things, at least one
of them has some desirable property. Sometimes the easiest strategy is to
equip the collection of all things with a measure, then show that the set
of things with the desired property has positive measure. Examples of this strategy
appear in... | zeraoulia rafik | 51,189 | <p><a href="https://en.wikipedia.org/wiki/Zeta_function_universality" rel="nofollow noreferrer">Universality of Riemann zeta function</a> , Which related to the approximation of every Holomorphic function <span class="math-container">$f(z)$</span> by Riemann zeta function in the strip .</p>
<blockquote>
<p><strong>Cor... |
365,631 | <p>Suppose we want to prove that among some collection of things, at least one
of them has some desirable property. Sometimes the easiest strategy is to
equip the collection of all things with a measure, then show that the set
of things with the desired property has positive measure. Examples of this strategy
appear in... | Ian Agol | 1,345 | <p><a href="https://arxiv.org/abs/1405.6410" rel="nofollow noreferrer">Lubotzky, Maher, and Wu showed</a> for any <span class="math-container">$n\in \mathbb{Z}, g\in \mathbb{N}$</span> the existence of homology 3-spheres of <a href="https://en.wikipedia.org/wiki/Heegaard_splitting" rel="nofollow noreferrer">Heegaard ge... |
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