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 |
|---|---|---|---|---|
1,793,039 | <p>Given a measure space $(\Bbb R, \mathcal P(\Bbb R), \mu_\Bbb N)$ where $\mathcal P(\Bbb R)$ denotes the power set of $\Bbb R$ and $\mu_\Bbb N$ is defined by $\mu_\Bbb N(A)= \vert {A \cap \Bbb N}\vert$= quantity of natural numbers in $A$, </p>
<p>how can I show that two functions $f,g:\Bbb R \to \mathbb R$ coincide... | ervx | 325,617 | <p>Note that $\{a_{2n}\}$ converges to $1$, while $\{a_{2n-1}\}$ converges to $-1$. If the sequence were to converge, all subsequences would have to converge to the same limit also. </p>
|
1,980,909 | <p>I was playing around with square roots today when I "discovered" this. </p>
<p>$\sqrt{1 + \sqrt{1 + \sqrt{1 + ...}}} = x$</p>
<p>$\sqrt{1 + x} = x$</p>
<p>$1 + x = x^2$ </p>
<p>Which, via the quadratic formula, leads me to the golden ratio. </p>
<p>Is there any significance to this or is it just a random coin... | Sarvesh Ravichandran Iyer | 316,409 | <p>If you play around a little more, you will also notice that:
$$
\frac{1+\sqrt{5}}{2} = 1+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1+ \ldots} } } }
$$</p>
<p>Which simplifies to $x = 1+ \frac 1x \implies x^2=x+1$.</p>
<p>It's no coincidence. I mean to say, it comes directly from the equation itself. </p>
<p>... |
4,383,159 | <p>I am currently working through my course text and one of the examples for a unital, commutative Banach algebra is the space of continuous functions <span class="math-container">$C([a,b])$</span>. Further, in the text, there is a theorem stating:</p>
<blockquote>
<p><span class="math-container">$\sigma(a) \neq 0$</sp... | martini | 15,379 | <p>Note that the identity of the Banach algebra <span class="math-container">$C([0,1])$</span>, which you denote by <span class="math-container">$\mathrm{id}$</span>, is the constant function <span class="math-container">$\mathbf 1 \colon x \mapsto 1$</span>. So, for <span class="math-container">$a \in C([0,1])$</span>... |
3,206,981 | <p>I am doing a transformation problem of getting the graph of <span class="math-container">$\sin (2x – \pi/6)$</span> by applying transformations to <span class="math-container">$F(x) = \sin x$</span></p>
<p>In the process, I let <span class="math-container">$f(x) = F(2x) = \sin 2x$</span>. </p>
<p>Next, I then let ... | 19aksh | 668,124 | <p>Here the graph is plotted as functions of <em>x</em>, taking values of <em>x</em> in the horizontal axis. So there is a shift of <span class="math-container">$\pi/12$</span>.</p>
<p>If we plot it as a function of <em>2x</em>, then we would have a shift of <span class="math-container">$$2.\frac{\pi}{12} = \frac{\pi... |
4,216,113 | <p>I heard that you cannot create a uniform probability distribution on the reals because it breaks the additivity axiom where the individual probabilities of a countable number of disjoint subsets of your space should equal the probability of their union. So how would you mathematically describe an idea such as selec... | GEdgar | 442 | <p>(1) How about: uniform distribution on an interval <span class="math-container">$[a,b]$</span>?</p>
<p>(2) More generally, fix a probability distribution <span class="math-container">$\phi$</span> on <span class="math-container">$\mathbb R$</span>, then use <span class="math-container">$\phi$</span> to specify your ... |
847,266 | <p>Assume a pdf $f(x)$ is continuous along $-\infty$ to $+\infty$.
Does this assumption guarantee that $f(+\infty)=f(-\infty)=0$? How to prove?
Thanks in advance.</p>
| heropup | 118,193 | <p>Refer to the example I constructed <a href="https://math.stackexchange.com/questions/728323/function-whose-limit-does-not-exist-but-the-integral-of-that-function-equals-1/728543#728543">here</a> that easily lends itself to a counterexample.</p>
|
452,011 | <p>Consider the set $A = \{\theta(f) \mid f : \mathbb{R} \rightarrow \mathbb{R},\;f\,\text{non-decreasing}\}$ where $\theta(f)$ denotes the set of functions which are asymptotically within a constant factor of $f$. Define an order on $A$ by setting $\theta(f) \le \theta(g)$ if $f \in O(g)$.</p>
<p>What is the order st... | Mikasa | 8,581 | <p>I did the plot of direction field of your system of ODEs by using Maple. I feel, doing this by hand is a bit hard at least for this system. I hope the codes help you to find the points you are looking for in a convenient way. </p>
<pre><code> [> with(DEtools):
[> dfieldplot([diff(x(t), t) = -y(t)+x(t)*(x(t... |
2,536,791 | <p>I am taking a basic complex analysis course and I'm trying to understand the differences between different forms of convergence.</p>
<p>Specifically, I am trying to distinguish normal convergence from pointwise convergence. I searched around for a similar question, but I was only able to find a comparison between n... | Olivier Oloa | 118,798 | <p>We have
$$
\left(\color{blue}{-e^{-\beta x}}\right)'=\beta e^{-\beta x}
$$ giving
$$
\lim_{p\to \infty} \int_0^pdx\;\beta e^{-\beta x}=\lim_{p\to \infty}\left(\color{blue}{-e^{-\beta x}}\frac{}{}|_0^p\right)=0+ e^{-0}=1
$$</p>
|
2,671,992 | <p>I have had trouble proving that the interior of the following set is empty. I have tried to do it by definition, but haven't managed to figure out the proof.</p>
<p>$$
C := \{ (x,y) \in \mathbb{R^2} \mid x \in (-1,1) \text{ and } y=x^3 \}
$$</p>
| Martín-Blas Pérez Pinilla | 98,199 | <p>Also: because even if we define differentiability in the border points as existence of lateral derivatives, in the <em>proof</em> of the theorem only derivatives in the interior are required.
Sketch of proof:</p>
<p>If $f$ is constant, the conclusion is trivial.</p>
<p>If $f$ isn't constant, max or min are reached... |
217,863 | <p>Find the congruence of $4^{578} \pmod 7$.</p>
<p>Can anyone calculate the congruence without using computer?</p>
<p>Thank you!</p>
| Chandrasekhar | 45,455 | <p>Use the fact that $$4^{3} \equiv 1 \ (\text{mod 7})$$ along with if $a \equiv b \ (\text{mod} \: m)$ then $a^{n} \equiv b^{n} \ (\text{mod}\: m)$.</p>
|
1,124,078 | <p>Is $\ln|x+2|=\ln|2x+4|$? Is this right? I saw something earlier saying this was correct; my first instinct was no.</p>
| Community | -1 | <p>It's almost correct. $\log |2x+4| = \log 2|x+2| = \log|x+2| + \log 2$.</p>
|
1,124,078 | <p>Is $\ln|x+2|=\ln|2x+4|$? Is this right? I saw something earlier saying this was correct; my first instinct was no.</p>
| Mike | 17,976 | <p>Is it possible what you saw before involved calculus? They may not be equal, but they both differ by a constant: $\ln|2x+4|=\ln|x+2|+\ln2$. If one were to evaluate the integral $\int\frac{2dx}{2x+4}$, either $\ln|x+2|+C$ or $\ln|2x+4|+C$ would be considered correct. But this does not mean that the $2$ logarithms a... |
616,393 | <p>Let $z$ be a complex number and $\mathrm{Re}$ denote the real part.</p>
<p>Does there exist a nonconstant entire function $f(z)$ such that $f(z)$ is bounded for $\mathrm{Re}(z)^2 > 1$ ?</p>
| Community | -1 | <p>A more general statement follows from an approximation theorem due to <a href="https://www.agnesscott.edu/lriddle/women/aliceroth.htm" rel="nofollow">Alice Roth</a>, which can be found in the book <a href="https://books.google.com/books?id=20RvBgAAQBAJ" rel="nofollow">Lectures on Complex Approximation</a> by Dieter ... |
148,002 | <p>I'm trying to get my head round the following calculation of the fundamental group of the torus, using Seifert Van-Kampen (I know it's easier to do this by considering covering spaces, but I'm trying to learn the Seifert Van-Kampen method). </p>
<p>Consider the torus $T$ as the unit square in $\mathbb R^2$ with opp... | Connor | 31,063 | <p>Think of the unit square in <span class="math-container">$\mathbb{R}$</span> with the top and bottom edges identified with <span class="math-container">$a$</span> pointing right, and the left and right edges identified with <span class="math-container">$b$</span> pointing down. Now say your generator <span class="ma... |
4,131,916 | <p>Let <span class="math-container">$(a_n)_{n \in \mathbb{Z}}$</span> be some given sequence of positive numbers, such that <span class="math-container">$\lim_{n \to -\infty} a_n=0,\lim_{n \to \infty} a_n=+\infty$</span>.</p>
<p>Let <span class="math-container">$\Omega \subseteq \mathbb{R}^2$</span> be a bounded, con... | Moishe Kohan | 84,907 | <p>Here is an answer to the modified version of your question (when one does not make any assumptions on the radii):</p>
<p><strong>Theorem.</strong> Let <span class="math-container">$\Omega$</span> be a domain (an open and connected subset) in the complex plane. Then there exists a collection of pairwise disjoint open... |
30,728 | <p>Is this graph in the list among the so-called "standard" structures used in <code>GraphData</code>? However, I have not found yet anything like "Carpet" or "Sponge" in the list of the objects that can be built. Maybe, this graph has a different name? </p>
<p>For me, using <code>GraphData</code> helps to save time f... | J. M.'s persistent exhaustion | 50 | <p>The mesh-related functionality in current versions allows for relatively compact code for generating the graph version of the Sierpinski carpet. The following solution is more or less similar to <a href="https://mathematica.stackexchange.com/a/121848">this previous answer</a>:</p>
<pre><code>pos = DeleteCases[Tuple... |
216,815 | <p>I have the following string dataset:</p>
<pre><code>{{22/03 updating, 55.218 (+1.640), 44.321 (+1.640), 4.825 (+0), 6.072 (+0), details},
{21/03, 53.578 (+6.557), 42.681 (+4.821), 4.825 (+793), 6.072 (+943), details},
{20/03, 47.021 (+5.986), 37.860 (+4.670), 4.032 (+627), 5.129 (+689), details},
{19/03, 41.035 (... | kglr | 125 | <pre><code>Cases[{__?PrimeQ}] @ sol
</code></pre>
<blockquote>
<p><code>{{2063, 853}, {2069, 857}}</code></p>
</blockquote>
|
2,071,226 | <p>Assuming $f : \mathbb R \to \mathbb R$ is differentiable, how can one prove that
$$
f'(x) = \lim_{h,k \to 0^+} \frac{f(x+h)-f(x-k)}{h+k},
$$
an alternate expression to the usual limit definition of the derivative $f'(x)=\lim_{h \to 0^+} \frac{f(x+h)-f(x)}h$?</p>
<p>I figured the problem out for the special case of ... | New day rising | 339,847 | <p>This is too long for a comment, so I'm posting here my understanding of answering this problem: </p>
<p>Since by hypothesis $f'(x)=\lim_{h \to 0^+} \frac{f(x+h)-f(x)}h = \lim_{k \to 0^+} \frac{f(x)-f(x-k)}k$...</p>
<p>For all $\epsilon > 0$, there exists $\delta_1 > 0$ such that $0 < h-0<\delta_1$ imp... |
772,997 | <p>My question is pretty much what it says in the headline.<br>
Is $A = f\mathbb{R^2}$ complete, where $f:\mathbb{R^2}\to\mathbb{R^3}$, $f(x,y) = (x,y,x^2-y)$, $(x,y)\in\mathbb{R^2}$.</p>
<p>$f\mathbb{R^2}=$$\{ f(x,y), (x,y) \in \mathbb{R^2} \}$</p>
<p>My initial thought is that as $f\mathbb{R^2}$ is $f$:s image (i.e... | Caleb Stanford | 68,107 | <p>Let $X$ and $Y$ be metric spaces. If $V \subset X$, define the <em>graph</em> of a function $f: V \to Y$ to be the set $\big\{(x, f(x) \; : \; x \in V \big\} \subset X \times Y$. $X \times Y$ is itself a metric space under the metric $d((x_1,y_1),(x_2, y_2)) = d(x_1, y_1) + d(x_2, y_2)$, or $\sqrt{d(x_1, y_1)^2 + ... |
4,172,689 | <p>I have this limit as my question to solve:</p>
<p><span class="math-container">$$\lim_{x\to-1}\frac{2x+\sqrt{3-x}}{x^2+x}$$</span></p>
<p>My procedure:</p>
<p><span class="math-container">$$\lim_{x\to-1}\frac{(2x+\sqrt{3-x})(2x-\sqrt{3-x})}{(x^2+x)(2x-\sqrt{3-x})}$$</span></p>
<p><span class="math-container">$$\lim_... | nonuser | 463,553 | <p>Hint: Your factorisation is not correct.</p>
<p>It is easier to do it if you let <span class="math-container">$t=\sqrt{3-x}$</span>, then <span class="math-container">$x= 3-t^2$</span> and <span class="math-container">$t\to 2$</span>.</p>
|
4,050,571 | <p>Suppose I have a video that plays for 60 minutes at normal "1x" speed. I know that if I set the video to play at "2x" speed, then it should play for 30 minutes.</p>
<p>Now, what if the video is set to play at "1.5x" speed? Intuition leads me to two answers:</p>
<ol>
<li><p>Since 1.5x is... | Joe | 623,665 | <p>Perhaps it helps to change the scenario. Suppose a tortoise had to run a <span class="math-container">$60$</span> metre race. At normal speed, it does one metre per minute, and so the tortoise takes <span class="math-container">$60$</span> minutes. At <span class="math-container">$1.5\times$</span> speed, the tortoi... |
2,799,257 | <p>I have shown that a smooth solution of the problem $u_t+uu_x=0$ with $u(x,0)=\cos{(\pi x)}$ must satisfy the equation $u=\cos{[\pi (x-ut)]}$. Now I want to show that $u$ ceases to exist (as a single-valued continuous function) when $t=\frac{1}{\pi}$.</p>
<p>When $t=\frac{1}{\pi}$, then we get that $u=\cos{(\pi x-u)... | Chee Han | 242,589 | <blockquote>
<p>The solution $u(x,t)$ is implicitly defined by the equation $F(x,t,u) = u - \cos\Big(\pi(x - ut)\Big) = 0$. The Implicit Function Theorem asserts that $F(x,t,u) = 0$ defines $u$ as a function of $x,t$ if $\dfrac{\partial F}{\partial u}\neq 0$, otherwise we expect characteristics to intersect (well, al... |
902,015 | <p>Let $a\in\mathbb Q$ and $a>\dfrac43$. Let $x\in\mathbb R$ and $x^2-ax,x^3-ax\in\mathbb Q$. Prove that $x\in\mathbb Q$.<br/><br/>
<strong>EDIT:</strong><br/>
Thsi is my attempt:<br/>
Let $x^2-ax=b$ and $x^3-ax=q$ for some $b,q\in\mathbb{Q}$. Then I tried to write $x^2$ and $x^3$ in linear terms. I got $x^2=ax+b$ a... | N. S. | 9,176 | <p>Let $x^2-ax=c$ and $x^3-ax=d$. Let $P(y)$ be the minimal polynomial of $x$ over $Q$. </p>
<p>Then $P(Y)| Y^2-aY-c$ and $P(y) | Y^3-aY-d$. In particular the degree of $P$ is at most two.</p>
<p><strong>Case 1</strong> $\deg P=1$, it follows immediately that $x \in \mathbb Q$. </p>
<p><strong>Case 2</strong> $\de... |
4,491,844 | <p>I want to show that every number in <span class="math-container">$[\frac{1}{2},1)$</span> is in a unique interval <span class="math-container">$[\frac{n}{n+1},\frac{n+1}{n+2}]$</span>, where <span class="math-container">$n$</span> is a positive integer. Intuitively, I think this is correct, but I do not know how to ... | Felix B. | 445,105 | <p>You want to use monotonicity, i.e.
<span class="math-container">$$x_0 \le \dots \le x_n \le x_{n+1} \le \dots \le x_\infty,$$</span></p>
<p>where <span class="math-container">$x_\infty=\lim_{n\to\infty} x_n$</span>.</p>
<h4>In at most one interval</h4>
<p>Assume the opposite, i.e. that some number <span class="math-... |
4,491,844 | <p>I want to show that every number in <span class="math-container">$[\frac{1}{2},1)$</span> is in a unique interval <span class="math-container">$[\frac{n}{n+1},\frac{n+1}{n+2}]$</span>, where <span class="math-container">$n$</span> is a positive integer. Intuitively, I think this is correct, but I do not know how to ... | Community | -1 | <p>The proof is very simple. All we need to prove is that</p>
<p><span class="math-container">$\bigcup_{k=1}^{\infty }[\dfrac{k}{k+1},\dfrac{k+1}{k+2})=[\dfrac{1}{2}, 1)$</span>. It is clear that</p>
<p><span class="math-container">$\bigcup_{k=1}^{\infty }[\dfrac{k}{k+1},\dfrac{k+1}{k+2})\subseteq [\dfrac{1}{2}, 1)$</s... |
1,142,568 | <p>Let $A$ be the ring of germs of real analytic functions in $0\in\mathbb R$. Let $x\in A$ be the identity map on $\mathbb R$. How can I show that the map $f\mapsto \sum_n(f^{(n)}(0)/n!)T^n$ from $A$ to $\mathbb R[[T]]$ is injective, and induces an isomorphism from the $xA$-adic completition of $A$ onto $\mathbb R[[T]... | Amitai Yuval | 166,201 | <p><strong>Half answer</strong>: Let $f$ be analytic, and suppose $f$ is carried by the above map to $0$. Then all the derivatives of $f$ at $0$ vanish, and by definition of analytic function, it means $f\equiv0$.</p>
<p>The claim fails to hold when considering $C^\infty$ functions, since there are smooth functions wh... |
2,969,710 | <blockquote>
<p>Use mathematical induction to prove that
<span class="math-container">$$
\frac12 + \frac16 + \ldots + \frac{1}{n(n+1)} = 1 - \frac{1}{n+1}
$$</span></p>
</blockquote>
<p>I am unsure about the prove n+1 step!
I let
<span class="math-container">$$
\frac12 + \frac16 + \ldots + \frac{1}{n(n+1)} = 1 -... | Doug M | 317,162 | <p>base case n = 1</p>
<p><span class="math-container">$\frac 12 = 1 - \frac 12$</span></p>
<p>Hypothesis:</p>
<p>Suppose <span class="math-container">$\frac 12 + \frac 16 + \cdots + \frac {1}{n(n+1)} = 1 - \frac {1}{n+1}$</span></p>
<p><span class="math-container">$\frac 12 + \frac 16 + \cdots + \frac {1}{n(n+1)}... |
2,969,710 | <blockquote>
<p>Use mathematical induction to prove that
<span class="math-container">$$
\frac12 + \frac16 + \ldots + \frac{1}{n(n+1)} = 1 - \frac{1}{n+1}
$$</span></p>
</blockquote>
<p>I am unsure about the prove n+1 step!
I let
<span class="math-container">$$
\frac12 + \frac16 + \ldots + \frac{1}{n(n+1)} = 1 -... | cansomeonehelpmeout | 413,677 | <p>Compare both expressions (I've named your sum <span class="math-container">$S_n$</span>) and see what is different</p>
<p><span class="math-container">$$S_n=\frac12 + \frac16 + \ldots + \frac{1}{n(n+1)}\tag{1}$$</span></p>
<p><span class="math-container">$$S_{n+1}=\frac12 + \frac16 + \ldots + \frac{1}{n(n+1)} +\fr... |
3,833,257 | <p>I have read about the coupling from the past algorithm that is used for perfect sampling from the stationary distribution of a discrete markov chain. My question is not exactly about this algorithm, but why I can't apply its proof idea as well to "forward coupling", i.e. what is wrong with the following id... | halrankard2 | 819,436 | <p>If you want to show that some number <span class="math-container">$N$</span> is <span class="math-container">$\gcd(x,y)$</span> then you need to show two things:</p>
<ol>
<li><p><span class="math-container">$N$</span> divides <span class="math-container">$x$</span> and <span class="math-container">$y$</span>.</p>
</... |
2,416,597 | <blockquote>
<p>Which of the following is the largest?</p>
<p>A. <span class="math-container">$1^{200}$</span></p>
<p>B. <span class="math-container">$2^{400}$</span></p>
<p>C.<span class="math-container">$4^{80}$</span></p>
<p>D. <span class="math-container">$6^{300}$</span></p>
<p>E. <span class="math-container">$10^... | Michael Rozenberg | 190,319 | <p>$$10^{250}>6^{300}$$ or
$$10^5>6^6$$ or
$$5^5>2\cdot3^6$$ or
$$3125>1458,$$
which is obvious.</p>
<p>Now, it's obvious that $10^{250}$ is a largest number.</p>
|
2,416,597 | <blockquote>
<p>Which of the following is the largest?</p>
<p>A. <span class="math-container">$1^{200}$</span></p>
<p>B. <span class="math-container">$2^{400}$</span></p>
<p>C.<span class="math-container">$4^{80}$</span></p>
<p>D. <span class="math-container">$6^{300}$</span></p>
<p>E. <span class="math-container">$10^... | user577215664 | 475,762 | <p>$$ 10^{250}>6^{300} $$
$$ 10^{5}>6^{6} $$
$$ 5^{5} 2^{5}>2^{6} 3^{6}$$
$$ 5^{5} >2.3^{6}$$</p>
<p>Note that $ 3^2=9<10=5.2 $ so:</p>
<p>$$ 2.3^{6} < 2.10.10.10=2.10^3=2^4.5^3 $$
$$ 2.3^{6} < 2^4.5^3 < 5^5 $$
$$ 2^4 < 5^2 $$
$$ 4^2 < 5^2 $$</p>
|
3,650,114 | <p>Why is there Bx+c term when we try to split partial fraction with irreducible quadratic?</p>
<p>Eg:</p>
<p><span class="math-container">$$\frac{1}{x(x^2+1)} = \frac{A}{x} + \frac{Bx+C}{x^2+1}$$</span></p>
<p>I think that splitting partial fraction is intuition when we directly put it as <span class="math-container">... | Integrand | 207,050 | <p>It's instructive to see what happens if you omit it. Suppose we had
<span class="math-container">$$
\frac{1}{x(x^2+1)} = \frac{A}{x} + \frac{B}{x^2+1}
$$</span>Cross multiplying, we get
<span class="math-container">$$
1 = A (x^2+1) + B(x)
$$</span>Matching the constant terms, we see <span class="math-container">$A=1... |
284,996 | <p>For every $f\in C[0,1]$ there is a sequence of even polynomials which converges uniformly on $[0,1]$ to f ? </p>
<p>What I have tried:</p>
<p>f is continuous on $D:=[0,1]$, let $(x_k)_{k\in \mathbb{N}} \in D$ converge to $y \in D$, then it must hold that (sequence definition of continuity): $$\lim _{k \rightarr... | kahen | 1,269 | <p><strong>Sketch</strong>: </p>
<p>Define $g: [-1,1]\to \mathbb R$ by $g(x) = \begin{cases} f(x) & \text{if } x \geq 0 \cr f(-x) & \text{if } x <0 \end{cases}$.</p>
<p>There is a sequence of even polynomial functions converging uniformly on $[-1,1]$ to $g$ (most proofs of Weierstrass's approximation theor... |
1,088,166 | <p>I'm so confused about cardinalities of some sets. What is the countable infinite product of a two points set $\{0,1\}$? Does it have the same cardinality as the real number $\mathbb R$? Or is the infinite product just countable?</p>
<p>Could anyone give me the answer?</p>
| dalastboss | 194,935 | <p>The countably infinite product of the set $\{0,1\}$ is simply the set of all infinite binary strings which Cantor showed to be uncountable in the classic <a href="http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument#An_uncountable_set" rel="nofollow">diagonalization</a> argument. </p>
|
1,140,527 | <p>How many ways to arrange the books? </p>
<p>I tried to compute by using combination with repetitions that is the 3C22 but I am not so sure.</p>
| Hagen von Eitzen | 39,174 | <p>The number of hands in total is ${40\choose 4}=91390$. The number of best-hands is ${8\choose 3}{8\choose 1}=448$, so the probability of a specific player (Player B, say) being dealt this hand is the quotient of these numbers, i.e., $\approx 0.0049$.</p>
<p>If we want to compute the probaility that <em>at least one... |
1,040,442 | <p>How do I find one value of $x$ in these equations?
$$
\begin{cases}
x \equiv 3 \pmod{5}\\
x \equiv 4 \pmod{7}
\end{cases}
$$</p>
| Mark Reed | 223,915 | <p>Given <em>x ≡ a (mod m)</em> and <em>x ≡ b (mod n)</em>, if there is a solution at all, there are infinitely many of them, all congruent modulo <em>lcm(m,n)</em>.</p>
<p>(If <em>m</em> and <em>n</em> are coprime, a solution must exist per the <a href="https://en.wikipedia.org/wiki/Chinese_remainder_theorem" rel="nof... |
152,116 | <p>Not sure why Mathematica won't integrate the definite or indefinite form of my function. I've tried the indefinite integral and it returns unevaluated. How can I see (and evaluate) what the actual integral looks like?</p>
<h3>Code</h3>
<pre><code>L[theta_] := Sqrt[(a*Sin[theta] - F)^2 + (b*Sin[theta])^2]
Integrate[... | Nasser | 70 | <p>Maple was able to solve this directly</p>
<pre><code>restart;
r:=int(sqrt((a*sin(theta) - F)^2 + (b*sin(theta))^2), theta=0..Pi/2) assuming positive;
r:=simplify(r);
</code></pre>
<p>But the result is large. You can compare this with the Mathematica result given above.</p>
<pre><code>(4*((I*b+a)/((I*b+a)*F-a^2-b^... |
3,226,067 | <p>I have a triangle ABC and I know that <span class="math-container">$\tan\left(\frac{A}{2}\right)=\frac{a}{b+c}$</span>, where <span class="math-container">$a,b,c$</span> are the sides opposite of the angles <span class="math-container">$A,B,C$</span>. Then this triangle is:</p>
<p>a. Equilateral</p>
<p>b. Right tr... | user10354138 | 592,552 | <p>Note that
<span class="math-container">$$
\frac{a}{b+c}=\frac{\sin A}{\sin B+\sin C}=\frac{2\sin(\frac12A)\cos(\frac12A)}{2\sin(\frac12(B+C))\cos(\frac12(B-C))}=\frac{\sin(\frac12A)}{\cos(\frac12(B-C))}
$$</span>
So the given condition is equivalent to
<span class="math-container">$$
\cos\frac{B-C}2=\cos\frac{A}2
$$... |
2,782,492 | <p>Suppose we have the series $\sum a_n$. Define,</p>
<p>$$
L=\lim_{n\to\infty}\frac{a_{n+1}}{a_n}
$$</p>
<p>Then,</p>
<ul>
<li>if $L<1$ the series is absolutely convergent (and hence convergent).</li>
<li>if $L>1$ the series is divergent.</li>
<li>if $L=1$ the series may be divergent, conditionally convergen... | Community | -1 | <p>You can't conclude anything from the fact $\lim_{n \to \infty} a_{n+1}/a_n$ does not exist.</p>
<p>The series may converge in that case: for example, let
$$a_n = \begin{cases}
\frac{1}{2}^n & \text{if $n$ is odd} \\
\frac{1}{3}^n & \text{if $n$ is even} \\
\end{cases}$$
In this case, $\liminf a_{n+1}/a_n =... |
160,741 | <p>$Version</p>
<p>"10.1.0 for Microsoft Windows (64-bit) (March 24, 2015)"</p>
<p>The following integral was nicely and quickly done by Mathematica</p>
<pre><code>i1 = Integrate[1/(1 - x) (PolyLog[3, -x] + 3/4 Zeta[3]), {x, 0, 1}]
(* Out[1027]= (3 Zeta[3])/4 *)
% // N
(* Out[1028]= 0.901543 *)
</code></pre>
<p... | Nasser | 70 | <p>Does this do what you want?</p>
<pre><code>ClearAll[y,x,L0];
op={-y''[x]+NeumannValue[0,x==L0],DirichletCondition[y[x]==0,x==0]};
eig=DEigenvalues[op,y[x],{x,0,L0},6]
(*{Pi^2/(4*L0^2), (9*Pi^2)/(4*L0^2), (25*Pi^2)/(4*L0^2),
(49*Pi^2)/(4*L0^2), (81*Pi^2)/(4*L0^2), (121*Pi^2)/(4*L0^2)}*)
</code></pre>
<p><img src=... |
160,741 | <p>$Version</p>
<p>"10.1.0 for Microsoft Windows (64-bit) (March 24, 2015)"</p>
<p>The following integral was nicely and quickly done by Mathematica</p>
<pre><code>i1 = Integrate[1/(1 - x) (PolyLog[3, -x] + 3/4 Zeta[3]), {x, 0, 1}]
(* Out[1027]= (3 Zeta[3])/4 *)
% // N
(* Out[1028]= 0.901543 *)
</code></pre>
<p... | Bill Watts | 53,121 | <p>Another way to show what is going on.</p>
<pre><code>pde = D[u[x, t], t, t] - D[u[x, t], t] == D[u[x, t], x, x]
</code></pre>
<p>Separate the variables</p>
<pre><code>u[x_, t_] = X[x] T[t];
Thread[pde/u[x, t], Equal] // Apart
(* D[T[t],t,t]/T[t]-D[T[t],t]/T[t]==D[X[x],x,x]/X[x] *)
</code></pre>
<p>The LHS is a ... |
3,640,171 | <p><strong>If <span class="math-container">$A_{1}$</span>,...,<span class="math-container">$A_{m}$</span> are independent events and P(<span class="math-container">$A_{i}$</span>)=p, where (P=probability measure) for i=1,2,...,m find the probability that an even number of <span class="math-container">$A_{i}$</span> occ... | Ben Grossmann | 81,360 | <p><strong>Hint:</strong> Work inductively. If an even number of events occur, then either <span class="math-container">$A_m$</span> occurs and an odd number of the events <span class="math-container">$A_1,\dots,A_{m-1}$</span> occur, or <span class="math-container">$A_m$</span> does not occur and an even number of the... |
2,118,611 | <p>$$\lim\limits_{x \to 0}\left(\frac{e^2}{(1+ 4x )^{\frac1{2x}}}\right)^{\frac1{3x}}=e^{\frac43}$$</p>
<p>I need help with solving this limit. I don't know how to get to the solution. Thanks.</p>
| Joseph Quarcoo | 410,294 | <p>Proceed as follows
Let $\displaystyle y=\lim_{x\rightarrow 0}\bigg(\frac{e^2}{(1+4x)^{1/(2x)}}\bigg)^{\frac{1}{3x}}$. Taking log of both sides we have</p>
<p>\begin{array}
\ \ln y &=&\lim_{x\rightarrow 0}\frac{1}{3x}\bigg[\ln(e^2)-\ln(1+4x)^{1/2x}\bigg]\\
&=&\displaystyle\lim_{x\rightarrow 0}\frac{2... |
222,320 | <p>Let $f(x)$ be a continuous real function s.t $f(x_0) > 0$</p>
<p><strong>Prove</strong>: There is some interval of the form $(x_0 -\delta, x_0 + \delta)$ where $f$ is positive.</p>
<p><strong>Proof</strong>:</p>
<p>Since $f$ is continuous: $\forall \,{\epsilon > 0}\,\, \exists \,{\delta>0}$ s.t. $|x- x_0... | copper.hat | 27,978 | <p>Choose $\epsilon = \frac{f(x_0)}{2}> 0$. Then there exists a $\delta>0$ such that for $|x-x_0| < \delta$, $|f(x)-f(x_0)| < \epsilon = \frac{f(x_0)}{2}$. Then $-\frac{f(x_0}{2} < f(x)-f(x_0)$ from which we get $0 < \frac{f(x_0)}{2} < f(x)$ for all $x$ such that $|x-x_0| < \delta$.</p>
<p>Alte... |
240,637 | <p>I'm trying to use an example to show that Fatou's lemma can not be strengthened to equality. I was given a hint, which I'm not quite sure how to use. I was told that if I look at the one-dimensional case, and let $f_k(x)=\begin{cases}
k, &\quad\text{if } - \frac{1}{k} \leq x \leq \frac{1}{k}\\
0,... | J126 | 2,838 | <p>You already have that
$$
\lim_{k\to\infty}\operatorname{inf}\int f_k\,dm=\lim_{k\to\infty}2=2.
$$
Now you must find
$$
\int \lim_{k\to\infty}\operatorname{inf}f_k\,dm.
$$
But, the limit function is $0$ off a set of measure $0$.</p>
|
1,878,519 | <blockquote>
<p>$$\int_{\frac{\pi}{4}}^{\frac{\pi}{3}}\frac{\sec^2x}{\sqrt[3]{\tan\ x}}dx$$</p>
</blockquote>
<p>$$f(x) = (\tan \ x)^{\frac{2}{3}}, \ f'(x) = \frac{2}{3} \cdot (\tan \ x)^{-\frac{1}{3}} \cdot \sec^2x$$</p>
<p>$$\therefore \int_{\frac{\pi}{4}}^{\frac{\pi}{3}}\frac{\sec^2x}{\sqrt[3]{\tan\ x}}dx = \fra... | BelowAverageIntelligence | 441,199 | <p>Your solution was fine. Just needed to have put $$\frac{3}{2}\int_{\frac{\pi}{4}}^{\frac{\pi}{3}}f'(x)dx$$ instead of $$\frac{3}{2}\int_{\frac{\pi}{4}}^{\frac{\pi}{3}}\frac{f'(x)}{f(x)}dx$$which makes your solution $$\frac{3}{2}(f(\frac{\pi}{3})-f(\frac{\pi}{4}))$$where f(x) is the function you defined, namely $$f(x... |
4,198,478 | <p>Here is the question
<a href="https://i.stack.imgur.com/OcAuN.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/OcAuN.png" alt="enter image description here" /></a>
I am on my second week of learning inferential statistics at a high pace, so I apologize if this is a trivial question.</p>
<p>How exac... | Graham Kemp | 135,106 | <p><span class="math-container">$$f_{X,Y}({x,y})=\begin{cases} 1/\pi, & \text{if $x^2 + y^2 <1$} \\ 0, & \text{o.w.} \end{cases}$$</span></p>
<p>The support is the unit disc, and can be written <span class="math-container">$y^2<1-x^2$</span>. So for a given value of <span class="math-container">$x$</spa... |
1,025,881 | <p>Let $G$ be a finite group and $p$ be the smallest prime divisor of $|G|$ , let $x \in G$ be such that $o(x)=p$ , and suppose for some $h\in G $ , $hxh^{-1}=x^{10}$ , then is it true that $p=3$ ? </p>
| user2345215 | 131,872 | <p>Because the formula for the derivative of a product is
$$g'\!f+gf'$$
Which gives you $c'\!f+cf'=0+cf'=cf'.$</p>
|
147,425 | <p>So I've come across the following inequality for probability measures:</p>
<p>$$
P(X \cap Y) \ge P(X) + P(Y) - 1
$$</p>
<p>I'm trying to work out why it should be true. I'm sure I'm missing something obvious.</p>
<p>I have the following:</p>
<p>$$
P(X \cap Y) = P(X) +P(Y) - P(X \cup Y) \le P(X) +P(Y) - 1
$$</p>
... | Derek Allums | 17,736 | <p>All the pieces are already on this page in one place or another. Here's how I would put them together: $1 \geq P(X \cup Y) = P(X) + P(Y) - P(X\cap Y)$. So, $1-P(X) - P(Y) \geq -P(X\cap Y)$. Then you use what Michael and Hurkly observed (multiply my negative one on each side and flip the inequality) to find that th... |
3,933,241 | <p>I am studying maths as a hobby. I have come across this problem:</p>
<p>Find a general solution for the equation cos 3x = sin 5x</p>
<p>I have said, <span class="math-container">$\sin 5x = \cos(\frac{\pi}{2} - 5x)$</span></p>
<p>so</p>
<p><span class="math-container">$\cos 3x = \sin 5x \implies 3x = 2n\pi\pm(\frac{\... | lab bhattacharjee | 33,337 | <p>If you write <span class="math-container">$m$</span> in place of <span class="math-container">$n,$</span> you reached at <span class="math-container">$\dfrac{\pi(1-4m)}4$</span></p>
<p>We <span class="math-container">$$\dfrac{\pi(1-4m)}4=\dfrac{\pi(1+4n)}4\iff m=-n$$</span></p>
<p>In our case <span class="math-conta... |
394,665 | <p>I'm wondering based on the definition of monotonicity:</p>
<blockquote>
<p>A sequence where $a_n\geq a_{n+1}$ for all $n\in\mathbb{N}$ is monotonic. </p>
</blockquote>
<p>So given that the sequence $a_n = 3$ is all the same numbers and is neither increasing or decreasing, is it monotonic? </p>
| amWhy | 9,003 | <p>Yes, a constant sequence (a number repeated indefinitely) is inceed monotonic: it is both monotonic non-decreasing, and monotonic non-increasing.</p>
<p>Hence, one can require that a sequence be <strong>strictly</strong> monotonic increasing or <strong>strictly</strong> monotonic decreasing. Under such a restrictio... |
821,654 | <p>I have a Taylor series problem, well more precisely a Maclaurin series.</p>
<p>I am trying to find convergence of: $f(x) = e^{x^3} + e^{{2x}^3}$</p>
<p>Okay here goes:</p>
<p>$$f'(x) = 3xe^{x^3} + 6x e^{{2x}^3}$$
$$f''(x) = 9x^2e^{x^3} + 3e^{x^3} + 36x^2e^{{2x}^3} + 6e^{{2x}^3}=e^{x^3}(9x^2+3) + e^{{2x}^3}(36x^2+... | Meijin | 145,178 | <p>This is a quadratic polynomial, this can always be solved using the quadratic formula as was mentioned in a comment, or sometimes by factoring the quadratic. Here we can factor like so:</p>
<p>$0 = x^2 - x - 380 = (x+19)(x-20)$</p>
<p>Now on the right hand side we simply have the product of two numbers, this can o... |
821,654 | <p>I have a Taylor series problem, well more precisely a Maclaurin series.</p>
<p>I am trying to find convergence of: $f(x) = e^{x^3} + e^{{2x}^3}$</p>
<p>Okay here goes:</p>
<p>$$f'(x) = 3xe^{x^3} + 6x e^{{2x}^3}$$
$$f''(x) = 9x^2e^{x^3} + 3e^{x^3} + 36x^2e^{{2x}^3} + 6e^{{2x}^3}=e^{x^3}(9x^2+3) + e^{{2x}^3}(36x^2+... | Shahar | 114,474 | <p>$$x^2 - x = 380$$
Rewrite this as:
$$x^2-x-380=0$$
Use quadratic formula to solve it now. It is:
$$x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$$
For:
$$ax^2+bx+c=0$$
Just plug this in:
$$x=\frac{-(-1) \pm \sqrt{(-1)^2-4(1)(-380)}}{2(1)}=\frac{1 \pm \sqrt{1521}}{2}=\frac{1 \pm 39}{2}$$
Which means:</p>
<p>$$\therefore x=20 \... |
24,416 | <p>I am writing my thesis, and when I do copy to LaTeX from Mathematica, it changes the equation variables and also it rearranges the structure of the original equation.</p>
<p>How can I override that ??</p>
<blockquote>
<p><em>New information and further reading : <a href="http://pages.uoregon.edu/noeckel/computer... | Jens | 245 | <p>If you want the exact same formatting as in the <code>StandardForm</code> output of your <em>Mathematica</em> session, then you could do the following:</p>
<pre><code>U[x, y] = (Subscript[E,
0] E^(I k Subscript[z,
1]) E^((I k ((\[Xi] - x)^2 + (\[Eta] - y)^2))/(2 Subscript[z,
1])))/((4 P... |
4,637,604 | <p>When constructing the <span class="math-container">$p$</span>-adic numbers, we proceed for instance as when constructing <span class="math-container">$\mathbb{R}$</span> for the usual distance. Then the integers are king of ``natural", we are used to them (are we can see them as the rational algebraic numbers, ... | KCd | 619 | <p>In complex analysis we can speak about meromorphic functions on the plane being holomorphic at a specific point (say, at <span class="math-container">$0$</span>). But we can also speak about the functions meromorphic or holomorphic near a chosen point without those functions being assumed to come from a meromorphic ... |
18,772 | <p>Suppose $G$ is a semisimple group, and $V_{\lambda}$ is an irreducible finite-dimensional representation of highest weight $\lambda$. Suppose $H \subset G$ is a semisimple subgroup. What is the multiplicity of the trivial representation in $V_{\lambda}|_{H}$? Is there a simple way to read this off from $\lambda$ ... | Ben Webster | 66 | <p>The short answer is: no. In theory, there are explicit formulae for branchings based on the Weyl character formula, but no reasonable person would call these simple.</p>
<p>There are interesting results which give you some information about branching multiplicities, but all involve real work. For instance, if G a... |
162,349 | <p>Is there any conventional notation for variables that can only take the value 0 or 1? (I'm looking for something of the nature of an overbar, a caret, etc.)</p>
| nbubis | 28,743 | <p>What you call a boolean variable in CS, is essentially an element of the finite field or order 2. you could write: $$x \in F_2$$</p>
|
3,786,232 | <p>On page 364 of <em>Elements of information theory, 2nd edition</em>, the set <span class="math-container">$E$</span> is difined as
<span class="math-container">\begin{equation}
E=\Bigl\{P:\sum_{a}P(a)g_j(a)\geq\alpha_j,j=1,2,\ldots,k\Bigr\}.
\end{equation}</span>
To find the closest distribution in <span class="math... | kodlu | 66,512 | <p>The omitted terms in the Lagrange multipliers would drop out after differentiation, and if you are experienced you know that ensuring the <span class="math-container">$P(x)$</span> sum to 1 will be invariant to those extra factors since the expression for <span class="math-container">$P^{\ast}(x)$</span> is homogene... |
3,786,232 | <p>On page 364 of <em>Elements of information theory, 2nd edition</em>, the set <span class="math-container">$E$</span> is difined as
<span class="math-container">\begin{equation}
E=\Bigl\{P:\sum_{a}P(a)g_j(a)\geq\alpha_j,j=1,2,\ldots,k\Bigr\}.
\end{equation}</span>
To find the closest distribution in <span class="math... | Guangyang_ZJU | 654,375 | <p>I have completed the derivation. As stated by the previous answerer, the omitted terms in the Lagrange multipliers would droup out after differentiation and the differentiation is
<span class="math-container">$$
\frac{\partial J}{\partial P(x)}=\log \frac{P(x)}{Q(x)}+1+\sum_{i} \lambda_ig_i(x)+v=0.
$$</span>
Solving... |
1,651,227 | <p>I was searching for pythagorean triples where $b=a+1$, and I found using a python program I made the first 10 integer solutions:</p>
<ol>
<li>$0^2+1^2=1^2$</li>
<li>$3^2+4^2=5^2$</li>
<li>$20^2+21^2=29^2$</li>
<li>$119^2+120^2=169^2$</li>
<li>$696^2+697^2=985^2$</li>
<li>$4059^2+4060^2=5741^2$</li>
<li>$23660^2+236... | poetasis | 546,655 | <p>The values <span class="math-container">$\; 1,5,29,169,\cdots\;$</span> are subset of pell number as shown in the OEIS sequence <a href="https://oeis.org/A000129" rel="nofollow noreferrer">A000129</a>. You are in fact using the odd-value numbers in the sequence.</p>
<p>If we define Euclid's formula for generating Py... |
21,318 | <p>Let $\mathcal V$ be a monoidal category and let $\mathcal C$ be a $\mathcal V$-category. Let's denote the $\mathcal V$-valued hom-functor $[-,-]$. Now for every object $X\in\mathcal C$ we have it's endomorphism object $\mathcal End(X):=[X,X]$ - it is actually a monoid in $\mathcal V$. Can the assignment
$$X\mapsto \... | Emily Riehl | 2,181 | <p>This reminds me of the discussion on pages 264-265 of Eilenberg and Mac Lane's <a href="http://www.ams.org/journals/tran/1945-058-00/S0002-9947-1945-0013131-6/S0002-9947-1945-0013131-6.pdf" rel="noreferrer">General Theory of Natural Equivalences</a> of the "degree of invariance" of various constructions on the categ... |
1,118,269 | <p>I'm reading: <em>Mathematical thought from ancient to modern times by Kline</em>. My question is about this pasasge:</p>
<blockquote>
<p>Beyond its achievements in subject matter, the nineteenth century
reintroduced rigorous proof. No matter what individual mathematicians may
have thought about the soundness ... | timur | 2,473 | <p>There is no distinction between applied math and pure math, especially during the discussed period. It was just math, with less rigour. Also, the idea that applied math has less rigour is an absurd one. This apparent division is not about rigour, it's about traditionally which branches have been called applied and w... |
1,951,733 | <blockquote>
<p>Can any polynomial $P\in \mathbb C[X]$ be written as $P=Q+R$ where $Q,R\in \mathbb C[X]$ have all their roots on the unit circle (that is to say with magnitude exactly $1$) ? </p>
</blockquote>
<p>I don't think it's even trivial with degree-1 polynomials... In this supposedly simple case, with $P(X)=... | Philippe Malot | 39,781 | <p>There are other rules for other bases.</p>
<p>To understand why the rule for $3$ works in base $10$, you need to write your number $x=d_n\times 10^n+\dots+d_1\times 10^1+d_0$ where the $d_i$ are the digits of $x$ in base $10$.
You can notice that $10=3\times 3+1$, $10^2=3\times 33+1$, $10^n=3...3\times 3+1$ (where ... |
239,120 | <blockquote>
<p>The line graph <span class="math-container">$L(G)$</span> of a graph <span class="math-container">$G$</span> is defined in the following way: the vertices of <span class="math-container">$L(G)$</span> are the edges of <span class="math-container">$G$</span>, <span class="math-container">$V(L(G)) = E(G)$... | Gerry Myerson | 8,269 | <p>Concerning the complement of $L(K_5)$, here are some thoughts: </p>
<ol>
<li><p>Given any edge $e$ in $K_5$, show that there are exactly $3$ edges in $K_5$ that don't share a vertex with $e$. </p></li>
<li><p>Call those three edges $a,b,c$, and notice that any two of them share a vertex. </p></li>
<li><p>Deduce tha... |
412,407 | <p>Need help showing that if $f$ is analytic and not identically zero on $A$ then if $f(z_0)=0$, there is an integer $k$ such that $f(z_0) = 0 = \dots = f^{(k-1)}(z_0)$ and $f^{(k)}(z_0) \neq 0$. Any hints on how to get it started? Thanks. </p>
| mrf | 19,440 | <p>Hint: Look at the power series expansion of $f$ around $z = z_0$ (which converges to $f$ at least on a small disc).</p>
|
3,581,475 | <p>We know that for any prime P, the radical R(P)=P. However is the converse of this Statement true. That is, if we know that radical of an ideal I is itself, i.e. R(I)=I, is I prime? I presume it is not but couldn't come with a counterexample. </p>
| red_trumpet | 312,406 | <p>You are right, this is not true. For a counterexample, consider the ideal <span class="math-container">$I = (xy) \subset k[x,y]$</span> for any field <span class="math-container">$k$</span>.</p>
|
134,075 | <p>I am trying to show that,</p>
<p>If $$f\left( x\right) =a_{0}+a_{1}x+\ldots +a_{k}x^{k}$$ then
$$\dfrac {1} {n}\left\{ f\left( x\right) +f\left( wx\right) +\ldots +f\left( w^{n-1}x\right) \right\} =a_{0}+a_{n}x^{n}+a_{2n}x^{2n}+\ldots +a_{\lambda n}x^{\lambda n}$$ $w$ being any root of $x^n=1$(except x= 1), and $\l... | anon | 11,763 | <p>It helps to go the the root of <em>why</em> the formula is true, which is the following orthogonality law$^\dagger$:</p>
<p>$$g(l):=\frac{1}{n}\sum_{j=0} \omega^{jl}=\begin{cases}1 & l\equiv0\bmod n \\ 0 & \rm otherwise. \end{cases} $$</p>
<p>And this is where it comes into play (interpret $f$ as an infini... |
134,075 | <p>I am trying to show that,</p>
<p>If $$f\left( x\right) =a_{0}+a_{1}x+\ldots +a_{k}x^{k}$$ then
$$\dfrac {1} {n}\left\{ f\left( x\right) +f\left( wx\right) +\ldots +f\left( w^{n-1}x\right) \right\} =a_{0}+a_{n}x^{n}+a_{2n}x^{2n}+\ldots +a_{\lambda n}x^{\lambda n}$$ $w$ being any root of $x^n=1$(except x= 1), and $\l... | Jyrki Lahtonen | 11,619 | <p>Hint: Both sides of the equation are linear functions of the polynomial $f$, so it suffices to handle the case, where $f(x)$ is a monomial.</p>
|
4,541,065 | <p>I need help with calculating the sum of this arithmetic series:<br>
<span class="math-container">$9-6+4- \frac 83 + ... + \frac{256}{729}-\frac{512}{2187}$</span> <br><br>
I watched this math video to try to solve it: <a href="https://youtu.be/BA0uxIaMtMs" rel="nofollow noreferrer">https://youtu.be/BA0uxIaMtMs</a> <... | Mark Viola | 218,419 | <p><strong>HINT:</strong></p>
<p>Note that <span class="math-container">$a_n=(-2/3)a_{n-1}$</span> and <span class="math-container">$a_0=9$</span>. Write down the expression for the general term <span class="math-container">$a_n$</span>.</p>
<p>What type of series is <span class="math-container">$\sum_{n=0}^\infty a_n... |
1,959,131 | <p>Hallo :) I am hopeless with this exercise:</p>
<blockquote>
<p>Solve the system of equations over the positive real numbers</p>
<p><span class="math-container">$$\sqrt{xy}+\sqrt{xz}-x=a$$</span></p>
<p><span class="math-container">$$\sqrt{zy}+\sqrt{xy}-y=b$$</span></p>
<p><span class="math-container">$$\sqrt{xz}+\sq... | mathlove | 78,967 | <p>We can solve the system in the following way (though I'm not sure if it is "reasonable") :</p>
<p>We have
$$\sqrt y+\sqrt z-\sqrt x=\frac{a}{\sqrt x}\tag1$$
$$\sqrt z+\sqrt x-\sqrt y=\frac{b}{\sqrt y}\tag2$$
$$\sqrt x+\sqrt y-\sqrt z=\frac{c}{\sqrt z}\tag3$$
From $(1)$,
$$\sqrt z=\sqrt x-\sqrt y+\frac{a}{\sqrt x}\t... |
168,619 | <blockquote>
<p>Can $a^2+b^2+2ac$ be a perfect square if $c\neq \pm b$? </p>
</blockquote>
<p>$a,b,c \in \mathbb{Z}$.<br>
I have tried some manipulations but still came up with nothing. Please help. </p>
<p>Actual context of the question is:<br>
Let say I have an quadratic equation $x^2+2xf(y)+25$ that I have to m... | André Nicolas | 6,312 | <p>A small manipulation changes the problem into a more familiar one. We are interested in the Diophantine equation $a^2+b^2+2ac=y^2$. Complete the square. So our equation is equivalent to $(a+c)^2+b^2-c^2=y^2$. Write $x$ for $a+c$. Our equation becomes
$$x^2+b^2=y^2+c^2.\tag{$1$}$$
In order to get rid of trivial sol... |
270,360 | <p>The <em>Sunday New York Times magazine</em> has a puzzle, <strong>CRAZY EIGHTS</strong> such as this:</p>
<p><a href="https://i.stack.imgur.com/C5fo8.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/C5fo8.png" alt="enter image description here" /></a></p>
<p>The goal is to fill in the <span class="... | cvgmt | 72,111 | <p>Add <code>Dividers</code>?</p>
<pre><code>LegendLayout -> (Column[
Grid[{##}, Alignment -> {Center, Center},
Background -> LightGray] & /@ #, Spacings -> -1,
Dividers -> {None, {3 -> Blue, -3 -> Blue}}] &)
</code></pre>
<p><a href="https://i.stack.imgur.com/QmoqR.png" re... |
425,611 | <p>If $$10^{20} +20^{10}$$ is divided with 4 then what would be its remainder?</p>
| Lord_Farin | 43,351 | <p>It would be zero.</p>
<p>Namely, $10^{20} = (10^2)^{10}$, and since $4$ divides both $10^2 = 100$ and $20$, it will also divide $100^{10}+20^{10}$.</p>
|
2,385,599 | <p>ABC is a triangle. D is the center of BC . AC is perpendicular to AD. prove that $$\cos(A)\cdot \cos(C)=\frac{2(c^2-a^2)}{3ac}$$
problem and my attempts are shown in images. I cannot find the exact way to the answer.</p>
<p><a href="https://i.stack.imgur.com/PUcka.jpg" rel="nofollow noreferrer"><img src="https://i... | Bowditch | 223,831 | <p>$(1+1/n)^n\to e$ as $n\to \infty$, so in particular $\exists N$ so that for all $n>N$, $(1+1/n)^n\geq 2$. This means that $$(1+1/n)^{n(n+1)}\geq 2^{n+1}.$$
Do you know how to finish it off from here?</p>
|
383,478 | <p>I'm trying to develop a reduction formaula for the integral - $\int \sin^n x dx$.
I've successfully developed a formula which is depended on two elements jumps, which is more or less:</p>
<p>$$\int \sin^k x d x = \frac{k-1}{k} \int \sin^{k-2}x d x
- \frac{1}{k} \cos x \sin^{k-1} x
$$</p>
... | TonyB | 80,501 | <p>Vonbrand is correct.
If n is even, reducing in steps of 2 will eventually lead to the $\int \sin ^0x\;dx\;$ (or $\int \sin ^2 x \;dx\;$ if you prefer).
If n is odd, you will end up with $\int \sin ^1x\;dx\;$.
Any of these can be found and the problem is solved.</p>
|
1,397,991 | <p>Consider the real interval $[0,1)$, this is partially ordered set (totally ordered actually). This set has a upper bound like $1$, and according to Zorn's Lemma each partially ordered set with a upper bound should have at least one maximal element. However, in this set there is no maximal element, i.e., element that... | Henno Brandsma | 4,280 | <p>To apply Zorn to a linear order <span class="math-container">$L$</span> is pointless: in order the prove the existence of a maximal element (here: also a maximum), you first have to give a maximum for <span class="math-container">$L$</span> as the condition says we must have an upper bound for <em>every chain</em> ... |
4,323,344 | <p>Let <span class="math-container">$\phi$</span> be Euler's totient function and <span class="math-container">$n$</span> be a positive integer. Let <span class="math-container">$\phi^k(n)$</span> denote <span class="math-container">$k$</span> sucessive applications of the totient function.</p>
<p>Since <span class="ma... | Will Jagy | 10,400 | <p>to check about upper bounds, we print out each time the quantity of interest sets a new record</p>
<p>===================================</p>
<pre><code> n is between 2^(k-1) and 2^k.
diff = n - 2^(k-1)
n: 2 k: 1 factor n 2
n: 3 k: 2 factor n 3 diff 1
n: 5 k: 3 factor n 5 diff 1
n: 11 k: ... |
3,226,815 | <blockquote>
<p>Let <span class="math-container">$y(t)$</span> be a nontrivial solution for the second order differential equation</p>
<p><span class="math-container">$\ddot{x}+a(t)\dot{x}+b(t)x=0$</span></p>
<p>to determine a solution that is linearly independent from <span class="math-container">$y$</span> we set <sp... | Community | -1 | <p>The instructions are given: set <span class="math-container">$z=yv$</span> and obtain an equation in <span class="math-container">$\dot v$</span>, rewritten <span class="math-container">$w$</span>.</p>
<p>The substitutions give</p>
<p><span class="math-container">$$\color{green}{\ddot yv}+2\dot yw+\dot w\color{gre... |
3,290,047 | <p>I understand the solution of <span class="math-container">$m^{2}+1=0$</span> is <span class="math-container">$\iota$</span>. However for sure this solution (<span class="math-container">$(m^{2}+1)^2=0$</span>) should contain four roots. The answer reads <span class="math-container">$\pm \iota$</span> and <span class... | mlchristians | 681,917 | <p>The solution is obtained by solving <span class="math-container">$m^{2} + 1 = 0$</span> which yields <span class="math-container">$m = \pm i$</span>.</p>
<p>That being said, you have effectively a quartic equation which will have exactly four roots counting multiplicities (repititions). </p>
<p>Here, <span class="... |
585,924 | <p>Given a function $$F(x) = \int_0^x \frac{t + 8}{t^3 - 9}dt,$$ is $F'(x)$ different when $x<0$, when $x=0$ and when $x>0$? </p>
<p>When $x<0$, is $$F'(x) = - \frac{x + 8}{x^3 - 9}$$ ... since you can't evaluate an integral going from a smaller number to a bigger number? That's what I initially thought, but ... | user112167 | 112,167 | <p>We have got the identity $\int\limits_{a}^{b}f(x)dx = -\int\limits_{b}^{a}f(x)dx$. Which can easy be verified by writing it out. Im guessing this is what you mean? </p>
|
2,465,064 | <p><a href="https://i.stack.imgur.com/7lBit.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/7lBit.png" alt="enter image description here"></a>
Can anyone help me with this. I honestly have no clue where to begin.</p>
| Vidyanshu Mishra | 363,566 | <p><img src="https://i.stack.imgur.com/ojnka.jpg" alt="enter image description here"></p>
<p>Question in picture deserves an answer in picture. :-P</p>
<p>Credits: Wikipedia</p>
|
1,095,334 | <p>My question is about a cubic graph $G$ that is the edge-disjoint union of subgraphs isomorphic to the graph $H$ that is as below:</p>
<p><img src="https://i.stack.imgur.com/K0fEM.png" alt="enter image description here"></p>
<p>I want to prove that $0$ is an eigenvalue of the adjacency matrix of $G$.</p>
<p>I thin... | Chris Godsil | 16,143 | <p>The edges of the Petersen graph can be partitioned into three copies of your given graph, but its eigenvalues are 3, 1, $-2$. So the claim is false. </p>
<p>What is true is that if there is a partition as described then $-2$ is an eigenvalue. This is because if you pass to the line graph, the vertices
that correspo... |
3,013,384 | <p>Let <span class="math-container">$f(x) = \frac{1}{2}\langle Ax,x\rangle - \langle b,x \rangle + c$</span> with <span class="math-container">$A\in \mathbb{R}^{n\times n}$</span> and <span class="math-container">$b\in \mathbb{R}^n$</span>, <span class="math-container">$c\in \mathbb{R}$</span>. Assume that <span class=... | caverac | 384,830 | <p><span class="math-container">$$
f(x) = \frac{1}{2}\sum_{ij}x_i A_{ij}x_j - \sum_i b_i x_i + c
$$</span></p>
<p>The gradient has components</p>
<p><span class="math-container">\begin{eqnarray}
\frac{\partial f}{\partial x_\alpha} &=& \frac{1}{2}\sum_{ij}(x_iA_{ij}\delta_{\alpha j } + \delta_{\alpha i}A_{ij}... |
4,003,987 | <p>I'm working on some computational mathematics for development in <code>three.js</code> and have been working with out the first parameters of multinormal distributions in Mathematica 12.2. (They're useful in procedural terrain generation because they have nice organic looking properties when mixed together.)</p>
<p... | David G. Stork | 210,401 | <p><span class="math-container">$${\bf \Sigma} =
\begin{pmatrix}
\sigma_x^2, \rho \sigma_x \sigma_y \\ \rho \sigma_x \sigma_y, \sigma_y^2
\end{pmatrix}$$</span></p>
<p>where <span class="math-container">$\rho \in [-1,1]$</span>.</p>
|
4,003,987 | <p>I'm working on some computational mathematics for development in <code>three.js</code> and have been working with out the first parameters of multinormal distributions in Mathematica 12.2. (They're useful in procedural terrain generation because they have nice organic looking properties when mixed together.)</p>
<p... | V. Vancak | 230,329 | <blockquote>
<p>However, I'm wondering what controls the rotation of the spread, and if a top-left, bottom-right orientation is possible?</p>
</blockquote>
<p>Yes, just put minus sign on the off-diagonal terms.</p>
<blockquote>
<p>Or is this a fundamental property of positive and symmetric matrix?</p>
</blockquote>
<p>... |
2,624,703 | <p>I'm having trouble proving these, can anyone help?</p>
<p>The question is as follows:</p>
<p>For each $n \in \mathbb{N}$, let $A_n = \lbrace{ k \in \mathbb{Z} ; k^2 <= n}\rbrace$ </p>
<p>Prove that:</p>
<p>1) $\bigcap A_n$ = $\lbrace 0, 1, -1 \rbrace$</p>
<p>2) $\bigcup A_n = \lbrace ...,-2,-1,0,1,2,...\rbra... | Mostafa Ayaz | 518,023 | <p>All of the singularities are included by $|z|=R$. There is a theorem (I forgot the name) implying that in such case the integral would be:$$I=\int_{|z|=R}f(z)dz=2\pi i Rez_{z=0}\dfrac{1}{z^2}f(\frac{1}{z})$$here we have:$$\dfrac{1}{z^2}f(\frac{1}{z})=\frac{1-3z}{z^3}\sin\frac{z}{1+2z}$$obviously$$\lim_{z\to 0}z^2\fr... |
2,417,736 | <p>Someone told me that the following formula holds for $f$ differentiable and decreasing, with $\lim_{x\rightarrow +\infty}{f(x)}=0$.</p>
<blockquote>
<p>$$\sum_{k=n}^{\infty}{f(k)} = \int_{n}^{\infty}{f(t)dt} +\frac{f(n)}{2}+\mathcal{O}(f'(n))$$</p>
</blockquote>
<p>But I managed to prove only if the function is ... | zhw. | 228,045 | <p>Counterexample: The key idea is that it's easy to arrange for each $f'(n)$ to be $0.$ That implies $O(f'(n))$ is just the zero function for large $n.$ Then you're left with no wiggle room at all and a counterexample can be found.</p>
<p>For $k=0,1,\cdots$ let $g_k:\mathbb R\to [0,\infty)$ be continuous with support... |
211,920 | <blockquote>
<p>Prove that if $S = S^T$ is symmetric and non-singular, then $S^2$ is
positive definite.</p>
</blockquote>
<p>My attempt:</p>
<p>Suppose $S$ is an $m\times n$ symmetric matrix with linearly independent columns, and suppose $q(x) > 0$, then the matrix $q(x) = \mathbf{x}^\mathrm{T}S\mathbf{x}$ is ... | Gerry Myerson | 8,269 | <p>"Find the co-ordinates of $v$ with respect to $F$" is just a fancy way of saying find the coefficients of $v$ as a linear combination of elements of $F$, so it's just asking for real numbers $a_1,a_2,a_3,a_4$ such that $$v=a_1f_1+a_2f_2+a_3f_3+a_4f_4$$ Two matrices are equal if and only if each entry in one equals t... |
456,106 | <p>Was solving some exercise of Number theory, and used this theorem
$$m=[a,b]\Longleftrightarrow \left(\frac{m}{a},\frac{m}{b}\right)=1$$Remembered that the teacher showed it in class, but I do not remember how, and I think I may be in the evaluation, and I am also curious how to proof? $\;\;\;$ Please do not use modu... | Pedro | 23,350 | <blockquote>
<p><strong>CLAIM1</strong> $\hspace{5.5 cm }(a,b)[a,b]=ab$</p>
</blockquote>
<p><strong>P</strong>. Let $d=(a,b), e=\dfrac{ab}{[a,b]}$. We will prove that $d=e$. Recall we define $d$ as the (unique) positive number such that $d$ divides both $a$ and $b$, and if $d'$ is any other common divisor, $d'\mid ... |
456,106 | <p>Was solving some exercise of Number theory, and used this theorem
$$m=[a,b]\Longleftrightarrow \left(\frac{m}{a},\frac{m}{b}\right)=1$$Remembered that the teacher showed it in class, but I do not remember how, and I think I may be in the evaluation, and I am also curious how to proof? $\;\;\;$ Please do not use modu... | lab bhattacharjee | 33,337 | <p>Let $A,B$ be the highest powers of prime $p$ in $a,b$ respectively.</p>
<p>WLOG we can set $A\ge B\ge0$</p>
<p>So, the highest powers of prime $p$ in $m$ will be $A$</p>
<p>So, the highest powers of prime $p$ in $\frac ma$ will be $A-B$</p>
<p>and the highest powers of prime $p$ in $\frac ma$ will be $B-B=0$<... |
4,519,350 | <p>Let
<span class="math-container">$$
I_k=\int_0^\infty (t+a)^k e^{-t}\exp\left(-\frac{(t-\mu)^2}{2\sigma^2}\right)\,\mathrm dt,
$$</span>
with <span class="math-container">$k\in\Bbb N_0$</span> and <span class="math-container">$a>0$</span>. Since <span class="math-container">$k$</span> is an integer we can expand ... | Claude Leibovici | 82,404 | <p><span class="math-container">$$J_\ell=\int_0^\infty t^\ell e^{-t}\exp\left(-\frac{(t-\mu)^2}{2\sigma^2}\right)\,\,dt$$</span></p>
<p>Assuming <span class="math-container">$\Re(\mu )<\sigma ^2$</span>, it seems that
<span class="math-container">$$J_\ell=2^{-\frac{\ell+1}{2}}\, e^{-\frac{\mu ^2}{2 \sigma ^2}} \,\si... |
363,377 | <p>Is there a world where circle is square? (like when triangle can have sum of degrees more than 180 on sphere)
What is the mathematical or at least common-sense proof?</p>
| Ambesh | 64,824 | <p>Yes. If you paint a circle on a rubber sheet, and deform the sheet with a different deformation rate at each point until the circle becomes a square you got one. Now, you could define a bijection between both spaces.</p>
<p>All this assuming that what you meant is an isomorphism in between a topological space (eucl... |
806,015 | <p>The question I'm working on is the following:
Let $C_R$ be a contour in the shape of a wedge starting at the origin, running along the real axis to $x=R$, then along the arc $0 \leq \theta \leq 2\pi/3$, then back down to the origin along the ray $\theta=2\pi/3$. </p>
<p>Evaluate the limit as $R$ approaches infinity... | Ron Gordon | 53,268 | <p>As you noted, the only pole within the contour is at $z=e^{i \pi/3}$. The residue at this pole may be computed as follows:</p>
<p>$$\lim_{z\to e^{i \pi/3}} \frac1{2!} \frac{d^2}{dz^2}\frac{(z-e^{i \pi/3})^3}{(z^3+1)^3}$$</p>
<p>Now,</p>
<p>$$z^3+1 = (z-e^{i \pi/3})(z+1)(z-e^{-i \pi/3}) = (z-e^{i \pi/3}) [z^2+(1... |
2,692,112 | <p>Let $A$ be $m\times n$ matrix with full column rank where $m > n$. Let $P = A(A^TA)^{-1}A^T$. How do we show that $P$ is SPD (symmetric positive definite)? Proving that it is symmetric is trivial, but how can I show it is positive definite?</p>
| user | 505,767 | <p>Note that $P$ is an m-by m <a href="https://en.wikipedia.org/wiki/Projection_matrix#Ordinary_least_squares" rel="nofollow noreferrer">projection matrix</a> on $Col(A)$ thus for $\vec w\neq \vec 0$ and $\vec w \perp Col(A)$ we have $P\vec w=0$ and $w^TPw=0$.</p>
<p>Thus $P$ is a <strong>semi-positive definite</stron... |
1,521,745 | <p>I'm really confused about this math problem. I'm currently taking Calculus, but this problem seems to be like something basic in Algebra that I should understand.... Unfortunately, I don't remember what equivalent expression was used for this. </p>
<p>The problem is:</p>
<p><a href="https://i.stack.imgur.com/cjudA... | Lubin | 17,760 | <p>I’ll do the same thing as @heropup, but without the notation. To save myself typing, I’ll set $c=1/\sqrt2=\cos 45^\circ=\sin45^\circ$. Then you want a rotation of $45^\circ$, and to do this I’ll set
\begin{align}
x&=cX-cY\\y&=cX+cY\,.
\end{align}
Make these substitutions, and if I’m not mistaken, you get a n... |
2,492,071 | <p>There are $3$ boxes, each contains $n$ mangoes. A person takes a mango from one randomly chosen box. This procedure is repeated until one of the boxes becomes empty. Find the probability that two other boxes contain one mango each.</p>
<p>I am new in a probability theory. I know that its conditional probability pro... | Fred | 380,717 | <p>$\sum_{i=0}^n \log \lgroup \lgroup1+\frac{i}{n}\rgroup ^\frac{1}{n}\rgroup= \frac{1}{n}\sum_{i=0}^n \log (1+\frac{i}{n}) \to \int_0^1 \log(1+x) dx$</p>
|
2,148,484 | <p>Find the value of the series $$\sum_{n=0}^\infty \frac{n^{2}}{2^{n}}.$$ I tried the problem but not getting the sum. Please help.</p>
| Maadhav | 416,874 | <p>$$S = \sum_{i=0}^n\frac{i^2}{2^i}$$</p>
<p>$$S -\frac{S}{2} = \sum_{i=0}^n\frac{i^2}{2^i} -\sum_{i=0}^n\frac{i^2}{2^{(i+1)}}$$</p>
<p>$$\frac{S}{2} = \sum_{i=0}^n\frac{i^2}{2^i} -\sum_{i=0}^n\frac{i^2}{2^{(i+1)}}$$</p>
<p>$$\frac{S}{2} = (\frac{1}{2}+\frac{4}{4}+\frac{9}{8}+\frac{16}{16}+ ... +\frac{n^2}{2^n}) - ... |
4,632,471 | <p>I thought this was an interesting concept so I tried to make an impossible map using this.
I also tested it on a solving website and it ended up giving me 6 different colors...<a href="https://i.stack.imgur.com/Imt8I.png" rel="nofollow noreferrer">A picture of it</a></p>
<p><a href="https://i.stack.imgur.com/I8OEL.p... | mathreadler | 213,607 | <p>Here is one way to do it. I don't know how many variations can exist.</p>
<p><a href="https://i.stack.imgur.com/DusYF.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/DusYF.png" alt="enter image description here" /></a></p>
<p><strong>edit</strong> : Reasoning for why this is the only one (up to is... |
271,088 | <p>I am studying the symmetries of a particular function,
$$
f: R^n \rightarrow R
$$
which leave $f(x)$ unchanged (i.e. so $f(Ax) = f(x)$ for some matrix $A \in R^{n \times n}$). I have found that my function is invariant under the action of a matrix group which satisfies the following equation:
$$
A X A^T = X
$$
where... | paul garrett | 15,629 | <p>This is a very degenerate version of an orthogonal group, due to the fact that your $X$ is not of full rank. It is rank one (assuming your $x$ is not $0$), so the quadratic form given by $X$ has signature $(1,0,n-1)$, where the last $n-1$ refers to the dimension of the subspace $K$ consisting of $y$ such that $\lang... |
2,283,230 | <p>Let $f: \Bbb R^n \to R$ be a scalar field defined by</p>
<p>$$ f(x) = \sum_{i=1}^n \sum_{j=1}^n a_{ij} x_i x_j .$$</p>
<p>I want to calculate $\frac{\partial f}{\partial x_1}$. I found a brute force way of calculating $\frac{\partial f}{\partial x_1}$. It goes as follows:</p>
<blockquote>
<p>First, we eliminate... | Arnaud D. | 245,577 | <p>The partial derivative with respect to $x_1$ can be computed as a directional derivative :
$$\frac{\partial f }{\partial x_1}(x) = \frac{d}{dt}(f(x+te_1))|_{t=0}$$
(where $e_1=(1,0,\dots,0)$.)</p>
<p>For $f:x\mapsto \langle x,Ax\rangle$, we obtain
\begin{align}\frac{\partial f }{\partial x_1}(x) & = \frac{d}{d... |
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