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 |
|---|---|---|---|---|
194,179 | <p>I want to work with linear expressions involving the formal symbol <span class="math-container">$w[a_1,...,a_n]$</span>, and I would like Mathematica to know that expressions such as</p>
<pre><code>w[a,b,d] + w[a,d,b] = 0
</code></pre>
<p>i.e. that the symbol <code>w</code> is antisymmetric with respect to the swa... | m_goldberg | 3,066 | <p>This is not an answer, but a request for clarification that can't be made in a comment.</p>
<p>I'm confused by your question because it seems to me to be open to several interpretations. Here are three that immediately occur to me.</p>
<p>Suppose that <code>sample</code> is a function, taking no argument, that wil... |
3,289,999 | <p>How can i easily determine the order of
<span class="math-container">$$
[3] \in \mathbb{F}_{11}^\times
$$</span></p>
<p>By the way: <span class="math-container">$\mathbb{F}_{11}^\times =\mathbb{F}_{11}\setminus \{[0]\}$</span>.</p>
<p>Fermat's little theorem states that the order of a group element
has to be a di... | José Carlos Santos | 446,262 | <p>Yes, there is a faster way. Since <span class="math-container">$3^{10}\equiv1\pmod{11}$</span>, the order of <span class="math-container">$3$</span> must divide <span class="math-container">$10$</span>. Therefore, there is no way that, for instance the order is <span class="math-container">$3$</span> or <span class=... |
454,325 | <p>Which is the technically correct definition?</p>
<p>I) An interior point of a set $B$ is a point that is the centre of some $\epsilon$-ball in $B$.</p>
<p>II) An interior point of a set $B$ is a point that is in a set $A\subset B$ in which every point is the centre of some $\epsilon$-ball in $A$.</p>
<p>The two d... | André Nicolas | 6,312 | <p><strong>Hint:</strong> Alll sixth roots of unity are powers of $\alpha$. And $x^6-1=(x^3+1)(x^3-1)$. </p>
|
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 ... | b00n heT | 119,285 | <p>Based on your reasoning, I presume you want to use the fact that the transformation
<span class="math-container">$$f(x)\to f(x+a)$$</span>
corresponds to a horizontal shift of the function by <span class="math-container">$|a|$</span> (where the direction depends on the sign of <span class="math-container">$a$</span>... |
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... | user3716267 | 371,718 | <p>While it is not a "distribution" in the strict sense, there is the notion of an <a href="https://en.wikipedia.org/wiki/Prior_probability#Improper_priors" rel="nofollow noreferrer">improper prior</a>. This corresponds not to a specific random selection procedure, but rather to a general state of ignorance ... |
1,900,333 | <p>If $\frac ab$ rounded to the nearest trillionth is $0.008012018027$, where $a$ and $b$ are positive integers, what is the smallest possible value of $a+b$?</p>
<p>I don't see any strategies here for solving this problem, any help? Thanks in advance!</p>
| Community | -1 | <p>Of course, the question is not exciting; that would be interesting is the following: </p>
<p>I give a decimal expansion with $3k$ or $4k$ digits of a rational $\dfrac{a}{b}\in (0,1)$ where $b<10^k$. Find this rational. Solution: convert the decimal expansion in a continued fraction and write its convergents; sto... |
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... | Rebellos | 335,894 | <p>$$\int_0^pβe^{-βx}dx=\int_0^p[-e^{-βx}]'dx =\big[ e^{-βx}\big]_0^p=-e^{-βp}+1$$</p>
<p>Thus : </p>
<p>$$\lim_{p\to \infty} \int_0^pdx\;\beta e^{-\beta x}= \lim_{p \to \infty} (-e^{-βp}+1)=1$$</p>
|
1,099,885 | <p>How can you calculate</p>
<p>$$\lim_{x\rightarrow \infty}\left(1+\sin\frac{1}{x}\right)^x?$$</p>
<p>In general, what would be the strategy to solving a limit problem with a power?</p>
| Alex Silva | 172,564 | <p>$$\left(1+\sin\frac{1}{x}\right)^x = \left(1+\frac{1}{x} + \mathcal{O}\left(\frac{1}{x^3}\right)\right)^x,$$ as $x \rightarrow \infty$. Thus,
$$ \lim \limits_{x \rightarrow \infty} \left(1+\sin\frac{1}{x}\right)^x = \lim \limits_{x \rightarrow \infty} \left(1+\frac{1}{x}\right)^x = e.$$</p>
|
295,130 | <p>I am taking Abstract Algebra right now and working on the exercises in the introductory section on Set Theory. I am having trouble proving the following question although it makes intuitive sense to me due to the intersection either being occupied by an even number or odd number of sets in the collection, leading to... | Alex Youcis | 16,497 | <p>Hint: If you wanted to find how many things were in a union $\displaystyle \bigcup A_n$ you'd say "ok, well there are $\displaystyle \sum_n |A_n|$ things". You then realize that you've double counted, so you have to get rid of the double counts, so you subtract $\displaystyle \sum_{i,j}|A_i\cap A_j|$. You then reali... |
2,372,925 | <p>A book has 500 pages on which typographical errors could occur. Suppose that there are exactly 10 such errors randomly located on those pages. </p>
<p>$a)$ Find the probability that a random selection of 50 pages will contain no errors. </p>
<p>$b)$ Find the probability that 50 randomly selected pages will contain... | Donald Splutterwit | 404,247 | <p><a href="https://i.stack.imgur.com/mKHLz.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/mKHLz.png" alt="enter image description here"></a></p>
<p>Note that lines of constant $u$ run bottom right to top left & are indicated in red,blue & green.
Lines of constant $v$ run bottom left to top... |
188,880 | <p>Here solutions is array .
I need to know why this code is written.Please help me.</p>
<pre><code>solutionsMod = Mod[solutions, n];
For[j = 1, j <= Length@solutions, j++,
For[i = 1, j <= Length@solution[[1]], i++,
If[solutionsMod[[j, i]] == 0, solutionsMod[[i, i]] =n;
];];];
Export[Tostr... | Henrik Schumacher | 38,178 | <p>For me, the <code>For</code>-loop looks like a very obfuscated and inefficient way of computing</p>
<pre><code>solutionMods = Mod[solutions, n, 1];
</code></pre>
|
2,410,655 | <p><em>Okay so I'm asking this quesion knowing a thing or two about sequences and general terms</em></p>
<p>What is the sum of the series :
$$1+\frac{1\cdot3}{6}+\frac{1\cdot3\cdot5}{6\cdot8}+\cdots$$</p>
<blockquote>
<p><strong>My Try:</strong> <em>I tried calculating the general term $T_{n}$ for the sequence but ... | Raffaele | 83,382 | <p>$$\det\left(
\begin{array}{rrr}
-1 & 2 & 1 \\
3 & a & -2 \\
-1 & 5 & 2 \\
\end{array}
\right)=-3-\alpha$$
if $-3-\alpha\ne 0$ that is $\alpha\ne -3$ there exists <strong>one and only one</strong> solution</p>
<p>$$\left\{\frac{3 a-b+6}{a+3},\frac{b+3}{a+3},\frac{3 (2 a-b+3)}{a+3}\right\}... |
282,889 | <p>I am trying to show that if $f$ is continuous on the interval $[a,b]$ and its upper derivative $\overline{D}f$ is such that $ \overline{D}f \geq 0$ on $(a,b)$, then $f$ is increasing on the entire interval. Here $\overline{D}f$ is defined by
$$
\overline{D}f(x) = \lim\limits_{h \to 0} \sup\limits_{h, 0 < |t| \leq... | N. S. | 9,176 | <p>Probably not the best approach, but here is an idea: show taht MVT holds in this case:</p>
<p><strong>Lemma</strong> Let $[c,d]$ be a subinterval of $[a,b]$. Then there exists a point $e \in [c,d]$ so that </p>
<p>$$\frac{f(d)-f(c)}{d-c}=\overline{D}f(e)$$</p>
<p>Proof:</p>
<p>Let $g(x)=f(x)-\frac{f(d)-f(c)}{d-c... |
282,889 | <p>I am trying to show that if $f$ is continuous on the interval $[a,b]$ and its upper derivative $\overline{D}f$ is such that $ \overline{D}f \geq 0$ on $(a,b)$, then $f$ is increasing on the entire interval. Here $\overline{D}f$ is defined by
$$
\overline{D}f(x) = \lim\limits_{h \to 0} \sup\limits_{h, 0 < |t| \leq... | Dave L. Renfro | 13,130 | <p>The following is in response to a recent comment to my answer of <a href="https://math.stackexchange.com/questions/1008559/dini-or-right-or-upper-derivative-of-weierstrass-function">Dini or Right or upper derivative of Weierstrass function</a>.</p>
<p>Corollary 11.4.1 on p. 128 of <strong>[1]</strong> (proved conci... |
282,889 | <p>I am trying to show that if $f$ is continuous on the interval $[a,b]$ and its upper derivative $\overline{D}f$ is such that $ \overline{D}f \geq 0$ on $(a,b)$, then $f$ is increasing on the entire interval. Here $\overline{D}f$ is defined by
$$
\overline{D}f(x) = \lim\limits_{h \to 0} \sup\limits_{h, 0 < |t| \leq... | Sourav Ghosh | 977,780 | <p>Let <span class="math-container">$ f:[a,b]→\Bbb{R}$</span> be a continuous function such that one of Dini's one-sided derivatives exists and positive. Then it can be shown <span class="math-container">$f$</span> is monotonically non-decreasing on <span class="math-container">$[a,b]$</span>.</p>
<p>Assume, <span clas... |
30,728 | <p>Is this graph in the list among the so-called "standard" structures used in <code>GraphData</code>? However, I have not found yet anything like "Carpet" or "Sponge" in the list of the objects that can be built. Maybe, this graph has a different name? </p>
<p>For me, using <code>GraphData</code> helps to save time f... | yode | 21,532 | <p>If we are in <code>11.1</code>, we can use in-built function <code>MengerMesh</code>.Well,I always faced a problem about grid layout.Whatever it is simple enough to plot <em>Sierpinski</em></p>
<pre><code>AdjacencyGraph[Graphics`Region`ToMeshObject[MengerMesh[4]]["AdjacencyMatrix"]]
</code></pre>
<p><img s... |
4,373,464 | <p>Let a and b be a positive integers. Proof that if number <span class="math-container">$ 100ab -1 $</span> divides number <span class="math-container">$ (100a^2 -1)^2 $</span> then also divides number <span class="math-container">$ (100b^2 -1)^2 $</span>.</p>
<p>My attempt:</p>
<p>Let's notice that <span class="math-... | Bill Dubuque | 242 | <p>We show how modular arithmetic allows us to view it as a special case of polynomial <strong>reversal</strong>, i.e. that <span class="math-container">$f(a)=0\Rightarrow \tilde f(a^{-1})=0,\,$</span> where <span class="math-container">$\tilde f$</span> denotes the reverse (reciprocol) polynomial.</p>
<p>Here mod arit... |
3,288,475 | <p>Is there any formula to calculate anti logarithm just using simple calculator.. I already know how to calculate logarithm digit by digit exactly just like this <a href="https://math.stackexchange.com/questions/820094/what-is-the-best-way-to-calculate-log-without-a-calculator">What is the best way to calculate log wi... | DanielWainfleet | 254,665 | <p><span class="math-container">$(4n^2-1)^2+(4n)^2=(4n^2+1)^2.$</span></p>
<p>All Pyth. triplets are of the form <span class="math-container">$(\,k(a^2-b^2),\,2kab,\, k(a^2+b^2)\,)$</span> with positive integers <span class="math-container">$k,a,b.$</span></p>
<p>All primitive triplets have <span class="math-contain... |
3,288,475 | <p>Is there any formula to calculate anti logarithm just using simple calculator.. I already know how to calculate logarithm digit by digit exactly just like this <a href="https://math.stackexchange.com/questions/820094/what-is-the-best-way-to-calculate-log-without-a-calculator">What is the best way to calculate log wi... | poetasis | 546,655 | <p>All non-trivial triplets have values of <span class="math-container">$B$</span> that are multiples of <span class="math-container">$4$</span> where <span class="math-container">$B=2mn$</span>. If <span class="math-container">$m,n$</span> are of like parity, Euclid's formula generates trivial triples, e.g. <span clas... |
1,804,146 | <p>My question is: Give me a field <span class="math-container">$K$</span>. Can we always find two <span class="math-container">$K$</span>-vector space <span class="math-container">$V_{1}$</span>, <span class="math-container">$V_{2}$</span> and a map <span class="math-container">$f:V_{1}\rightarrow V_{2}$</span> such t... | Mariano Suárez-Álvarez | 274 | <p>If the field is a prime field (one of the $F_p$ with $p$ prime, or $\mathbb Q$) then you can't, and this for exactly the reason you mention (which works also for the rationals). If not, you can.</p>
<p>Indeed, in that case the field $K$ contains a proper subfield $L$, and you can find an $L$-linear map which is not... |
1,305,236 | <p>A (probably simple) question I encountered but I don't know how to approach:</p>
<blockquote>
<p>Let $K$ be a field of prime characteristic $p>0$.
Show every $f(x) \in K[x]$ can be represented as $g(x^{p^e})$ for some $e \ge 0$ and $g \in K[x]$ with $g'(x) \neq 0$.</p>
</blockquote>
<p>I saw <a href="https:... | James | 155,690 | <p><strong>Note:</strong> I thought I could use a slightly different method to answer this but alas this ended up looking very similar to the answer already given by @quid. But I spell out some more of the details, so I decided to post this anyway in case it's useful.</p>
<p>For our $f(x) \in K[x]$, let $f(x)=a_0 + a_... |
344,479 | <p>Suppose that $a = 2^kb,$ where $b$ is odd. If $\phi(x) = a,$ prove that $x$ has at most $k$ odd prime divisors.</p>
| Community | -1 | <p>If the prime factorization of $x$ is $2^a \cdot p_1^{a_1} p_2^{a_2}\cdots p_l^{a_l}$, where $p_1,p_2,\ldots,p_l$ are odd primes, then
$$\phi(x)=2^{\max(a-1,0)} p_1^{a_1-1}p_2^{a_2-1}\cdots p_l^{a_l-1}(p_1-1)(p_2-1) \cdots (p_l-1) = 2^k \cdot b$$
Note that $p_j-1$ is even for all $j \in \{1,2,\ldots,l\}$. Hence, if $... |
338,625 | <p>P-adic numbers are complete in one sense and incomplete in another sense. Is it so?</p>
<p>Firstly, does not complete mean connected? I read somewhere that there is not intermediate value theorem for p-adics because they are not connected. (if I am correct).</p>
<p>It seems I need elaboration of this "It can be sh... | Lubin | 17,760 | <p>To see that $\mathbb Q$ is incomplete under the $p$-adic valuation, it suffices to find an element of $\mathbb Q_p$ not in the rationals. For $p=2$, $\sqrt{-7}$ will do, for $p=3$, $\sqrt7$ will do, and for $p>3$, the field $\mathbb Q_p$ contains all $p-1$ roots of unity of order $p-1$. The existence of these irr... |
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 ... | Paresseux Nguyen | 758,600 | <p>Let <span class="math-container">$A_n:=\left[ \frac{n}{n+1},\frac{n+1}{n+2}\right)$</span>.
We have <span class="math-container">$\bigcup_{n=1}^{\infty} A_n=\bigcup_{n=1}^{\infty}\left( \bigcup_{k=1}^{n} A_k\right)=\bigcup_{n=1}^{\infty} \left[\frac{1}{2},\frac{n+1}{n+2} \right)=\left[ \frac{1}{2},1\right) $</span... |
562,166 | <p>I am trying to prove that Axiom of choice implies well-ordering principle.</p>
<p>Proof: If $A = \emptyset,$ then, take $\alpha = 0$ and' $\alpha \cong $ </p>
<p>So given a nonempty set $A$, $\exists a_0 \in A$. Define by recursion, the following function $F:ON \rightarrow V$ </p>
<p>$$F(\alpha)=\begin{cases}
\al... | dfeuer | 17,596 | <h3>Hint:</h3>
<p>Every polynomial is either constant or not. If it's not, it must have a certain sort of behavior for arguments with very large absolute values. The local behavior of polynomials also has a useful property.</p>
|
562,166 | <p>I am trying to prove that Axiom of choice implies well-ordering principle.</p>
<p>Proof: If $A = \emptyset,$ then, take $\alpha = 0$ and' $\alpha \cong $ </p>
<p>So given a nonempty set $A$, $\exists a_0 \in A$. Define by recursion, the following function $F:ON \rightarrow V$ </p>
<p>$$F(\alpha)=\begin{cases}
\al... | Robert Lewis | 67,071 | <p>For any polynomial $p(x)$, the function $\vert p(x) \vert$ is continuous and satisfies $\vert p(x) \vert \to \infty$ as $\vert x \vert \to \infty$. Given that this is the case, it is evident that this question is in fact a special case of this one: <a href="https://math.stackexchange.com/questions/422382/problem-ab... |
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... | Bill Dubuque | 242 | <p><strong>Hint</strong> <span class="math-container">$\ \color{#c00}{z\mid x}\Rightarrow\, \underbrace{(z,(x,y))}_{\!\!\!\!\textstyle ((\color{#c00}{z,x}),y)} = (\color{#0a0}{z,y})\ $</span>
so <span class="math-container">$\, \begin{align} \color{#0a0}{z\:\!\mid\:\! y}&\Rightarrow z\mid (x,y)\\
... |
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^... | Mark Bennet | 2,906 | <p>B is $2^{400}$</p>
<p>D is $2^{300}3^{300} \gt 2^{600}$</p>
<p>E is $10^{250} =2^{250}5^{250}$</p>
<p>Then we have $3^6=729, 5^5=3125\gt2^23^6$</p>
<p>So E is $10^{250} =2^{250}5^{250}\gt2^{250}(2^23^6)^{50}=2^{350}3^{300}\gt 2^{300}3^{300}$</p>
|
2,011,261 | <p>A pizza restaurant has 3 crust options, 2 cheese options and 10 choices of toppings. On Saturday nights, the restaurant offers a special deal on 2-toppings pizzas including pizzas with double portions of one toppings. How many distinct special deal pizzas are possible.</p>
<p>My approach:
I assumed (not too sure if... | Rodrigo Dias | 375,952 | <p><strong>Hint:</strong> Write
$$\left(1-\dfrac{1}{n}\right)^n = \dfrac{1}{\left(\dfrac{n}{n-1}\right)^n}$$</p>
|
169,253 | <p>I know the formula for the area of a sector of an arc made by central angle is
$$\text{Area}_\text{Sector}= \frac{\text{Arc Angle} \times \text{Area of Circle} }{360}$$
Now my question is , Is this formula also applicable for Arcs formed by inscribed angles rather than Central Angles ? (I know that angle of an inte... | robjohn | 13,854 | <p>Here are some equal angle inscribed angles (they all subtend half the angle of the red arc). It is clear that the area of the last sector is properly contained in the areas of the others.</p>
<p>$\hspace{2.5cm}$<img src="https://i.stack.imgur.com/RmNED.png" alt="enter image description here"></p>
<p>The area of th... |
169,253 | <p>I know the formula for the area of a sector of an arc made by central angle is
$$\text{Area}_\text{Sector}= \frac{\text{Arc Angle} \times \text{Area of Circle} }{360}$$
Now my question is , Is this formula also applicable for Arcs formed by inscribed angles rather than Central Angles ? (I know that angle of an inte... | David | 134,838 | <p>I'm working on something like this at the moment. In fact, this problem is a special case of what I'm trying to figure out. This is the general formula for these areas, and it's not pretty.</p>
<p>I don't know how to format these posts, but I'll give it a try.</p>
<p>ϕ = angle of vector from the center pointing to... |
317,531 | <p>I wonder why the polynomial $x^p-x$ has $p$ distinct zeros in $\mathbb Z_p$ for any prime $p$, i.e. $x^p-x=x(x-1)\cdots(x-p+1)$.
Do I need to expand the polynomial in order to get the conclusion?</p>
| amWhy | 9,003 | <p>This will help: See <a href="http://en.wikipedia.org/wiki/Fermat%27s_little_theorem" rel="noreferrer"><strong>Fermat's Little Theorem</strong>.</a></p>
<p>Added: ...as I see @anon suggested!</p>
<p>Point being: You don't need to expand the polynomial!</p>
|
996,626 | <p>The simple graphs upto <span class="math-container">$11$</span> vertices do not have <span class="math-container">$5,7,9,...$</span> automorphisms, in other words,
the only odd numbers appearing are <span class="math-container">$1$</span> and <span class="math-container">$3$</span>. Is this true for all graphs ?</p>... | jflipp | 187,123 | <p>I found Frucht's theorem on Wikipedia.
<br>
<a href="http://en.wikipedia.org/wiki/Frucht%27s_theorem" rel="nofollow">http://en.wikipedia.org/wiki/Frucht%27s_theorem</a>
<br>
According to this theorem, any finite group occurs as the automorphism group of a finite, undirected graph.</p>
|
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>
| John Frederick Chionglo | 190,989 | <p>If you are still looking for a way to find solutions to your problem, you might want to consider this.</p>
<p>The “method” is based on introducing parameter variables so that the original variables can be expressed as parameter variables. If such an expression exist, then enumerating integer values for the paramete... |
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[... | bbgodfrey | 1,063 | <p>Here is an answer, but you may not like it. Perform the indefinite integral and evaluate it at its endpoints:</p>
<pre><code>s = Integrate[L[theta], theta]
ans = (s/.theta -> Pi/2) - (s/.theta -> 0)
</code></pre>
<p>The expressions are enormous. To be specific, <code>LeafCount[s]</code> is <code>17739</co... |
82,558 | <p>Given a function $f:\mathbb{R^n}\to \mathbb{R}$ that can be expressed as sum of roots of polynomials, i.e. $f = \sum_{i=0}^k (p_i)^{1/n_i}$ for some polynomials $p_i$ and integers $n_i$. Can one find a polynomial $p:\mathbb{R}^n \to \mathbb{R}$ such that in the domain where both function are defined, we have $p(x_1,... | davidlowryduda | 9,754 | <p>Fortunately, the one you understand can be readily seen in the cell-complex construction.</p>
<p>So, take a rectangle, identify opposite sides. Now, draw a picture of a torus, and draw the rectangle on it. This is very important that you can do this. What does the rectangle look like on the torus? It likes like a s... |
2,047,748 | <p>Question:
suppose $\mathrm{log}_9 X + \mathrm{log}_{27} X = P$. write the value of $\mathrm{log}_3 X + \mathrm{log}_{81} X$ in terms of $P$.</p>
<p>I changed $\mathrm{log}_9 X + \mathrm{log}_{27} X = P$ into $\frac{1}{2} \mathrm{log}_3 X + \frac{1}{3}\mathrm{log}_3 X = P$.</p>
<p>I can't expand $\mathrm{log}_3 X +... | DonAntonio | 31,254 | <p><strong>Hints</strong></p>
<p>Complete, justify and fill in details:</p>
<p>$$P=\log_9x+\log_{27}x=\frac{\log_3x}{\log_39}+\frac{\log_3x}{\log_327}=\left(\frac12+\frac13\right)\log_3x$$</p>
|
2,223,031 | <p>Can someone explain to me how trig functions work in the complex plane? I'm trying to show that $f(z) = \cos(1-\frac{1}{z})$ has an essential singularity at $z=0$, and part of doing that requires I find a sequence $z_n$ so that when $z = z_n$, $\lim\limits_{z \to 0} f(z)=\infty$. Originally I assumed $\cos(z)$ would... | Stella Biderman | 123,230 | <p>You should approach this using the following formula, which is valid for all complex numbers $z$</p>
<p>$$\cos(z)=\frac{e^{iz}+e^{-iz}}{2}$$</p>
<p>We also have the cooresponding formula for sine</p>
<p>$$\sin(z)=\frac{e^{iz}-e^{-iz}}{2i}$$</p>
<p>These can both be proven by using $e^{iz}=\cos(z)+i\sin(x)$ and s... |
2,848,177 | <p>Random variables $X$ and $Y$ have joint p.d.f</p>
<p>$$f_{x, y} (x, y) =\begin{cases}
c(x^3 + 2y^3) & 0 \leq x \leq 3, 0 \leq y \leq 4\\
0 & \text{otherwise }\\
\end{cases}
$$</p>
<p>Find the value of c that makes $f_{x, y}$ a joint density function.</p>
<hr>
<p>attempt:</p>
<p>$$1 = \int_{0}^{4}\int_... | John Polcari | 571,558 | <ol>
<li>This is correct.</li>
<li>The appropriate integral is
$$c = \frac{1}{{\int\limits_0^4 {dy\int\limits_0^y {dx} } \left( {{x^3} + 2{y^3}} \right)}}$$</li>
</ol>
|
3,319,010 | <p>I would like to learn more about combinatorics of finite sets (including theorems such as Sperner, Erdos-Ko-Rado theorems, LYM inequality). Is there any good book or article for this topic (if possible with problems and exercises)?</p>
| Arthur | 15,500 | <p>When discussing linear independence of the columns of a matrix, you must allow the coefficients in the linear combinations to come from the same space as the entries in the matrix (otherwise linear algebra doesn't in any way work the way you're used to). And clearly, if we allow <span class="math-container">$a$</spa... |
599,394 | <p>A pack contains $n$ card numbered from $1$ to $n$. Two consecutive numbered cards are removed from</p>
<p>the pack and sum of the numbers on the remaining cards is $1224$. If the smaller of the numbers on</p>
<p>the removing cards is $k$, Then $k$ is.</p>
<p>$\bf{My\; Try}::$ Let two consecutive cards be $k$ and ... | ASpitzer | 114,511 | <p>$(n, k) = (50, 25)$</p>
<p>I think you are almost there. If I understand your question right you just solve for $k$ in terms of $n$. </p>
<p>\begin{align*}
& n(n+1) - (4k + 2) = 2448 \\
\implies & n(n+1) - 2 - 2448 = 4k \\
\implies & n(n+1) - 2450 = 4k \\
\implies & \frac{n(n+1) - 2450} 4 = k
\end... |
2,584,968 | <p>I know that if X is locally compact and Hausdorff, then any non-empty open set $S$ contains a non-empty closed set. I know this to be the case because a locally compact space is a regular space, in which the claim holds.</p>
<p>But why does any open $S$ contain a <em>non-empty open set whose closure is compact and ... | William Elliot | 426,203 | <p>Assume x in open U. By local compactness there is an<br>
open V with x in V subset U and compact K = $\overline V$. </p>
<p>Thus some open W with x in W, L = $\overline W$ subset V.<br>
L is the compact, closed set that answers the question.</p>
|
485,043 | <p>If $x_n \to x$ in a Hilbert space $X$, is it true that $|x_n| \leq C|x|$ for all $n$ for some constant $C$?</p>
<p>It is true for $n$ big enough. But not sure about all $n$.</p>
| Vishal Gupta | 60,810 | <p>Lets answer in detail.</p>
<p>Since $x_{n} \to x$, for every neighbourhood $U$ of $x$, there exists a $m$ such that $m \geq n$ implies that $x_{n} \in U$.</p>
<p>Let $U$ be the ball around $x$ of radius $|x|$ (if $x \neq 0$). Then this implies that $|x_{n}| \leq 2|x|$ and now use the idea suggested by Jim to choos... |
1,800,821 | <p>Consider an $n \times n$ matrix of the form
$$
A = \begin{bmatrix}
a_1 & a_2 & \ldots & a_{n-1} & a_n \\
1 \\
& 1 \\
& & \ddots \\
& & & 1
\end{bmatrix}
$$
for certain $a_1, \ldots, a_n \in \mathbb{R}$ (and zeroes in all blank spaces). Can this matrix ever (for some $n \in \ma... | Community | -1 | <p>The answer is no. Your matrix $A$ is an avatar of a companion matrix. The geometric multiplicity of an eigenvalue of such a matrix is always $1$;</p>
|
5,877 | <p>I've recently run across a series of problems that didn't reflect reality. </p>
<p>For example - </p>
<ul>
<li>An algebra problem with two teens on bicycles. The resulting times showed the bike was moving at 120MPH. </li>
<li>A quadratic equation, "The football follows a path of....." but the equation didn't refl... | quid | 143 | <p>The answers to a word problem should in my opinion make sense (within reasonable limits).
The goal you mentioned that students should check their answers is one shared by many, as also witnessed by our recent question <a href="https://matheducators.stackexchange.com/questions/5818/how-to-award-points-for-sense-maki... |
5,877 | <p>I've recently run across a series of problems that didn't reflect reality. </p>
<p>For example - </p>
<ul>
<li>An algebra problem with two teens on bicycles. The resulting times showed the bike was moving at 120MPH. </li>
<li>A quadratic equation, "The football follows a path of....." but the equation didn't refl... | lukejanicke | 7,046 | <p>Yes. Making sure a solution makes sense in the context of a real-life situation is an essential part of applied mathematics.</p>
<p>But it’s not just enough to have “real-life” problems with solutions that make sense in the context of real-life situations. When a ridiculous context is forced onto a pure mathematics... |
3,020,024 | <blockquote>
<p>Let <span class="math-container">$f: \Bbb R \to \Bbb R$</span> be a function such that <span class="math-container">$f'(x)$</span> exists and is continuous over <span class="math-container">$\Bbb R$</span>. Moreover, let there be a <span class="math-container">$T > 0$</span> such that <span class="... | peterwhy | 89,922 | <p>Consider such $f(x)$ to be non-constant. There must be a global minimum within each period, say at $x=x_0$. Since $f(x)$ is differentiable and continuous, $f'(x_0) = 0$. Hence
$$f(x) \ge f(x_0) = f(x_0)+f'(x_0) \ge 0$$ </p>
|
437,645 | <p>I am trying to gain a better understanding of Schur-Weyl duality specifically applied to symmetric functions. My motivating example is trying to understand the Frobenius character of the multilinear component of the free Lie algebra (Theorem 8.1 in Reutenauer's book on the subject), but my general confusion is more ... | Christopher Ryba | 159,272 | <p>This answer is a response the the prompt</p>
<blockquote>
<p>Any help or general information about the relationship between Schur-Weyl duality and symmetric functions you could provide would be greatly appreciated.</p>
</blockquote>
<p>If you have more questions (e.g. about the free Lie algebra), feel free to ask.</... |
2,118,611 | <p>$$\lim\limits_{x \to 0}\left(\frac{e^2}{(1+ 4x )^{\frac1{2x}}}\right)^{\frac1{3x}}=e^{\frac43}$$</p>
<p>I need help with solving this limit. I don't know how to get to the solution. Thanks.</p>
| Arun | 399,173 | <blockquote>
<p>Hint:</p>
</blockquote>
<p>Take $A=\left(\frac{e^2}{(1+ 4x )^\frac{1}{2x}}\right)^\frac{1}{3x}$ then $\log A=\frac{1}{3x}\left(\log e^2-\log (1+4x)^{\frac{1}{2x}}\right)$ which is in ($\frac{0}{0}$ form) Now apply L'Hospital's rule to get the limit.</p>
|
113,370 | <p>Let $p<q$ be positive integers (with the allowance that $q$ may be $\infty$). How can we show that the sum of $L^p$ and $L^q$ is a Banach space under the norm $\|f\|=\inf\{\|g\|_p+\|h\|_q: g+h=f\}$?</p>
<p>Let $\{f_n\}$ be a sequence in $L^p+L^q$, such that $$\sum_{n=1}^\infty \|f_n\|<\infty.$$</p>
<p>We wou... | Davide Giraudo | 9,849 | <p>You want in fact to show the following result:</p>
<blockquote>
<p>Let <span class="math-container">$(X_1,||\cdot||_1)$</span> and <span class="math-container">$(X_2,||\cdot||_2)$</span> two Banach spaces such that <span class="math-container">$X_i\subset V$</span> where <span class="math-container">$V$</span> is... |
753,231 | <p>Solve $$z^2+2iz-1+2i$$</p>
<p>I tried:</p>
<p>$(z+i)^2-1-1+2i$</p>
<p>$(z+i)^2 = 2-2i$</p>
<p>Which gives me $a^2-b^2 = 2, 2ab = -2, a^2+b^2 = \sqrt(8)$ And this I cannot solve.</p>
| Matt L. | 70,664 | <p>(Assuming you mean to solve $z^2+2iz-1+2i=0$)</p>
<p>You were on the right track by completing the square:</p>
<p>$$z^2+2iz-1+2i=(z+i)^2+2i=0$$</p>
<p>From which follows</p>
<p>$$(z+i)^2=-2i\\
z+i=\pm(1-i)\Rightarrow\quad z=\{-1,1-2i\}$$</p>
|
573,619 | <blockquote>
<p>I want to show that the $x$-axis is closed. </p>
</blockquote>
<p>Below is my attempt - I would appreciate any tips on to improve my proof or corrections:</p>
<p>Let $(X,d)$ be a metric space with the usual metric.<br>
Want to Show: $\{(x,y) | x ∈ \Bbb R, y = 0\}$ is closed</p>
<p>Claim: $\{(x,y) |... | Brian M. Scott | 12,042 | <p>Your definition of $\epsilon$ has problems: what are $x$ and $y$? They’ve not been defined up to this point, so $\epsilon$ isn’t defined. The only specific coordinates that are available at this point for you to work with are $a$ and $b$ (and specific real numbers like $0$).</p>
<p>HINT: If $z=\langle a,b\rangle\in... |
1,878,519 | <blockquote>
<p>$$\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... | GAVD | 255,061 | <p>You have $$\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{\sec^2 x}{\sqrt[3]{\tan x}}dx = \int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{1}{\sqrt[3]{\tan x}}d\tan x$$</p>
<p>Replacing $t = \tan x$, you have the integration equals
$$\int_{1}^{\sqrt{3}} \frac{1}{\sqrt[3]{t}}dt = \frac{3}{2}t^{\frac{2}{3}} \mid_{1}^{\sqrt{3}}... |
2,653,503 | <p>How to find a <strong>bijective</strong> function $f: 3\mathbb{N}+1\to 4\mathbb{N}+1$ such that $$f(xy)=f(x)f(y),\forall x,y\in 3N+1$$</p>
<p>If i let $x,y\in 3\mathbb{N}+1$ then there exists $n,m\in \mathbb{N}$ such that $x=3n+1,y=3m+1$ </p>
<p>but I have no idea how I can find a such $f$, Is there a method pleas... | Ng Chung Tak | 299,599 | <p>\begin{align}
a &= \frac{10^{2018}-1}{9} \\
b &= \frac{2(10^{1009}-1)}{9} \\
a-b &= \frac{10^{2018}-2\times10^{1009}+1}{9} \\
&= \frac{(10^{1009}-1)^{2}}{9} \\
\end{align}</p>
<p><strong>Can you proceed?</strong></p>
|
9,840 | <p>The formula for finding the roots of a polynomial is as follows</p>
<p>$$x = \frac {-b \pm \sqrt{ b^2 - 4ac }}{2a} $$
what happens if you want to find the roots of a polynomial like this simplified one
$$ 3x^2 + x + 24 = 0 $$
then the square root value becomes
$$ \sqrt{ 1^2 - 4\cdot3\cdot24 } $$
$$... | Community | -1 | <p>Yeah so have you read about complex numbers. The root will be
$$x = \frac{-1 \pm{i} \sqrt{287}}{2 \times 3}$$
where $i=\sqrt{-1}$. Read more about complex numbers and when a polynomial can have complex roots, that happens when the *Discriminant* factor $b^{2}-4ac <0$. </p>
|
2,220,872 | <p>Let $R \subseteq A \times A$ be a irreflexlive relation($\forall a \in A: (a,a) \notin R)$. I want to proof that if $R$ is dense, then $A$ cannot be finite. A relation $R$ is dense if $\forall (a,b) \in R$: there is $(a,c) \in R$ and $(c,b) \in R$.<br>
I am not exactly sure where to start on this proof. I tried to ... | Apostolos | 1,772 | <p>Your idea on how to prove it is correct, though, as you said, it is a bit informal. There are many ways to make if formal. For example you can show using more or less your argument that there exists an injective function from $\mathbb{N}$ to $R[a]=\{b\in A\mid (a,b)\in R\}$. Another idea is the following: Show via s... |
1,883,047 | <p>I have this logical statement</p>
<p>$$\neg x\lor (x \wedge y)$$</p>
<p>However I do not know what is considered a valid transformation. Normally if there is an $\wedge$ in the middle I treat it like multiplication and pull out some "shared" piece but here I don't know how to use distributive properties. </p>
| Ovi | 64,460 | <p>You can use the distributive law: </p>
<p>$$(\neg x ) \vee (x \wedge y) \equiv (\neg x \vee x) \wedge (\neg x \vee y)$$</p>
<p>Now, the statement $x \vee \neg x \equiv T$ (do you see why?)</p>
<p>$$T \wedge (\neg x \vee y)$$</p>
<p>The proposition $T \wedge p \equiv p$ (again, do you see why?)</p>
<p>So we hav... |
546,809 | <p>The question is :Find the derivative of $f(x)=e^c + c^x$. Assume that c is a constant.</p>
<p>Wouldn't $f'(x)= ce^{c-1} + xc^{x-1}$. It keeps saying this answer is incorrect, What am i doing wrong?</p>
| Claude Leibovici | 82,404 | <p>Since "c" is a constant,the derivative of the first term is zero and then the derivative of f(x) is the same as the derivative of g(x)=c^x. I suggest you go through loagrithms and re-express g(x) in a more familiar form. Are you able to continue with this ?</p>
|
546,809 | <p>The question is :Find the derivative of $f(x)=e^c + c^x$. Assume that c is a constant.</p>
<p>Wouldn't $f'(x)= ce^{c-1} + xc^{x-1}$. It keeps saying this answer is incorrect, What am i doing wrong?</p>
| cgonagu | 89,570 | <p>As stated in the comments, $e^c$ is a constant, so its derivative is zero. As for the second term, $c^x$, the rule for the derivative of a variables as an exponents is better understood if you write the function as</p>
<p>$c^x = e^{x \ln{c}}$</p>
<p>So, clearly the derivative is equal to $c^x\ln{c} $</p>
<p>To su... |
1,910,910 | <p>I am dealing with the following exercise:</p>
<p>Let $u_n$ bounded in $L^\infty[0,1]$ such that, for any continuous function $f: [0,1]\times R$ to $R$
$$\lim_n \int_0^1 f(x, u_n(x))=\int_0^1 f(x, u(x)).$$</p>
<p>Prove that $u_n$ converges to $u$ in $L^1.$
There is a hint saying to prove it first for $u\in C^0$.</p... | yoyo | 349,439 | <p>To approximate $u$ in $L^1$ with continuous functions, first given any $\epsilon >0$, we choose $u_{\epsilon }$ to be a continuous function such that $\|u_{\epsilon }-u\|_1<{\epsilon }$ and setting $f(x, y)=|y- u_{\epsilon }(x)|$ just like your idea, then you will get
$$\lim_n \|u_n- u_{\epsilon }\|_1=\|u- u_... |
391,364 | <p>Let <span class="math-container">$\Sigma_{g,n}$</span> denote an <span class="math-container">$n$</span>-punctured surface of genus <span class="math-container">$g$</span>, with <span class="math-container">$2g+n-2 > 0$</span>. Let <span class="math-container">$\Pi_{g,n}$</span> be its fundamental group (for some... | Dan Petersen | 1,310 | <p>This is surely not the most direct answer. But <span class="math-container">$\mathrm{Out}(\Pi_{g,1}) \cong \mathrm{Out}(F_{2g})$</span> surjects onto <span class="math-container">$\mathrm{GL}(2g,\mathbf Z)$</span>, and the image of <span class="math-container">$\Gamma_{g,1}$</span> lands in <span class="math-contain... |
147,425 | <p>So I've come across the following inequality for probability measures:</p>
<p>$$
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
$$</p>
... | Seamus | 172 | <p>For what it's worth, I convinced myself of the truth of the inequality as follows: If $X$ and $Y$ are both big ($>0.5$) then they must overlap. So, $P(X)+P(Y)-1$ measures the size of the forced overlap. (obviously it will be less than 0 when no overlap is forced.)</p>
|
821,654 | <p>I have a Taylor series problem, well more precisely a Maclaurin series.</p>
<p>I am trying to find convergence of: $f(x) = e^{x^3} + e^{{2x}^3}$</p>
<p>Okay here goes:</p>
<p>$$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+... | abiessu | 86,846 | <p>This particular problem offers another solution angle: factorization. We have</p>
<p>$$380=x^2-x=x(x-1)$$</p>
<p>So now we can see that $380$ is a number multiplied by that number less one. $380$ is between $400=20^2$ and $361=19^2$, so it is reasonable to guess that $x=20$ and it is very easy to see that $20\c... |
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... | Rojo | 109 | <p>Perhaps something like this could help?</p>
<pre><code>SetAttributes[copyAsLatex, HoldFirst];
copyAsLatex[sth_] := CopyToClipboard[ToString[HoldForm[sth] /.
x_ /y_ :> Divide[x, y], TeXForm]]
</code></pre>
<p>So</p>
<pre><code>copyAsLatex[
U[x, y] =
Subscript[E, 0]/(4 \[Pi]) E^(I k Subscript[z, 1])/
... |
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, ... | Andrea Mori | 688 | <p>Alternatively, the <span class="math-container">$p$</span>-adic integers <span class="math-container">$\mathbb{Z}_p$</span> can be defined as the projective limit of the sequence
<span class="math-container">$$
\cdots\longrightarrow\frac{\mathbb{Z}}{p^3\mathbb{Z}}
\longrightarrow\frac{\mathbb{Z}}{p^2\mathbb{Z}}
\lon... |
4,628,654 | <blockquote>
<p>Prove the following statement:
<span class="math-container">$$(\forall a,b \in \Bbb R)(a<b\implies(\exists r\in\mathbb Q)(a^7<r-4<b^7))$$</span>
Hint: You may need to use the theorem: Given any two real numbers <span class="math-container">$a < b$</span>, there is some <span class="math-cont... | Community | -1 | <p>Suppose a and b are arbitrary real numbers such that a < b. Our aim is to find a rational number r such that a^7 < r - 4 < b^7. Consider the function f(x) = x^7 - 4. This function is continuous on the interval [a, b], which means that it takes all values between f(a) and f(b). In particular, there exists so... |
1,651,227 | <p>I was searching for pythagorean triples where $b=a+1$, and I found using a python program I made the first 10 integer solutions:</p>
<ol>
<li>$0^2+1^2=1^2$</li>
<li>$3^2+4^2=5^2$</li>
<li>$20^2+21^2=29^2$</li>
<li>$119^2+120^2=169^2$</li>
<li>$696^2+697^2=985^2$</li>
<li>$4059^2+4060^2=5741^2$</li>
<li>$23660^2+236... | Keith Raskin | 692,801 | <p><a href="https://www.linkedin.com/pulse/why-ratios-consecutive-hypotenuses-converge-square-silver-raskin/" rel="nofollow noreferrer">This explains why the ratios converge to the square of the silver ratio.</a></p>
<p><a href="https://i.stack.imgur.com/lArXI.png" rel="nofollow noreferrer"><img src="https://i.stack.i... |
3,487,989 | <p>Slope of line <span class="math-container">$PQ$</span> is
<span class="math-container">$$m=\frac{1}{1-k}$$</span>
The slope perpendicular to it will be
<span class="math-container">$k-1$</span></p>
<p>Since the line is a bisector of PQ it will pass through
<span class="math-container">$(\frac{1+k}{2},\frac 72)$<... | José Carlos Santos | 446,262 | <p>Nothing. Your answer is fine. Note that, when <span class="math-container">$k=0$</span>, <span class="math-container">$Q=(0,3)$</span>. Since <span class="math-container">$P=(1,4)$</span>, the slope of the line passing through <span class="math-container">$P$</span> and <span class="math-container">$Q$</span> is <sp... |
2,340,204 | <p>I searched extensively for an answer, but couldn't find one that specifically explained what I was looking for. In working through a problem in my textbook, part of it involves simplifying an expression using power reduction. This is the step:</p>
<p>$$
\cos^{2}(2\theta) = \frac{1+\cos(2(2\theta))}{2}
$$</p>
<p>I ... | Community | -1 | <p>Let $U=[u_{i,j}]\in M_n(\mathbb{C})$ and let $(E_{i,j})$ be the canonical basis of $M_n(\mathbb{R})$. The first thing to do is to see what is the meaning of </p>
<p>for every real symmetric matrix $A\in M_n(\mathbb{R})$, $\overline{U}AU^T=UAU^*$.</p>
<p>Note that $(E_{i,j}+E_{j,i})$ is a basis over $\mathbb{R}$ of... |
1,951,733 | <blockquote>
<p>Can any polynomial $P\in \mathbb C[X]$ be written as $P=Q+R$ where $Q,R\in \mathbb C[X]$ have all their roots on the unit circle (that is to say with magnitude exactly $1$) ? </p>
</blockquote>
<p>I don't think it's even trivial with degree-1 polynomials... In this supposedly simple case, with $P(X)=... | Community | -1 | <p>The 3 divisible rule works because of the fact that 9,99,999... are all divisible by 3.</p>
<p>For example, 471 is divisible by 3 because 4+7+1 =12 divisible.</p>
<p>Think of this: Any number divisible by 3 is still divisible if you subtract any multiple of 3 by the number itself.</p>
<p>471 = 4*100+7*10+1</p>
<... |
1,821,318 | <p>Generally a function is shown continuous by directly taking left hand or right hand limit.But sometimes the same can be shown continuous by letting h tend to zero.What is the difference,would somebody please explain?</p>
| Virtuoz | 153,521 | <p>Using AM-GM we get
$$
a^4+b^4 + 8 \ge 2(ab)^2 + 8
$$
We want to show that
$$
2(ab)^2 + 8 \ge 8ab
$$
It's equivalent to
$$
2(ab-2)^2\ge 0
$$
which is clearly true</p>
|
1,821,318 | <p>Generally a function is shown continuous by directly taking left hand or right hand limit.But sometimes the same can be shown continuous by letting h tend to zero.What is the difference,would somebody please explain?</p>
| Funktorality | 227,920 | <p>Recall young's inequality for (three) products:
$$abc\leq\frac{a^p}{p}+\frac{b^q}{q}+\frac{c^r}{r}$$
when $1/p+1/q+1/r=1$. If you haven't seen this, you can easily prove it using the concavity of the logarithm. Your result is immediate when you let $c=2$, $p=q=4$, and $r=2$.</p>
<p>Note that in a particular way, yo... |
553,297 | <p>Please help me to evaluate this integral:
$$\large\int_0^{\pi/2}\frac{x}{\sin x}\log^2\left(\frac{1+\cos x-\sin x}{1+\cos x+\sin x}\right)dx$$</p>
| Vladimir Reshetnikov | 19,661 | <p>Your integral can be expressed in terms of certain special functions:
$$\begin{align}
\int_0^{\pi/2}\frac{x}{\sin x}\log^2\left(\frac{1+\cos x-\sin x}{1+\cos x+\sin x}\right)dx&=\frac{\,\pi^2}6K+4\,\beta(4)\\&=\frac{\,\pi^2}6K-\frac{\pi^4}{24}+\frac1{192}\psi^{(3)}\left(\frac14\right)\\&=\frac{\,\pi^2}6K... |
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... | Julián Aguirre | 4,791 | <p>Let $n\ge3$ be an odd integer. Then
$$
(n\,a+b)^2=n^2a^2+2\,n\,a\,b+b^2=a^2+b^2+2\,a\,c
$$
with
$$
c=\frac{n^2-1}{2}\,a+n\,b\in\mathbb{Z}.
$$</p>
|
3,770,004 | <p>Not a duplicate of</p>
<p><a href="https://math.stackexchange.com/questions/2401434/suppose-a-b-and-c-are-sets-prove-that-c-%e2%8a%86-a-b-iff-c-%e2%8a%86-a-%e2%88%aa-b-and">Suppose $A$, $B$, and $C$ are sets. Prove that $C ⊆ A △ B$ iff $C ⊆ A ∪ B$ and $A ∩ B ∩ C = ∅$.</a></p>
<p><a href="https://math.stackexchange.c... | Community | -1 | <p>In the first part, in both the second cases(where it says Case <span class="math-container">$2$</span>) you can simply refer to the similar arguments as in the first cases but with <span class="math-container">$B \setminus A$</span> instead of <span class="math-container">$A\setminus B$</span>.</p>
<p>Since you assu... |
3,770,004 | <p>Not a duplicate of</p>
<p><a href="https://math.stackexchange.com/questions/2401434/suppose-a-b-and-c-are-sets-prove-that-c-%e2%8a%86-a-b-iff-c-%e2%8a%86-a-%e2%88%aa-b-and">Suppose $A$, $B$, and $C$ are sets. Prove that $C ⊆ A △ B$ iff $C ⊆ A ∪ B$ and $A ∩ B ∩ C = ∅$.</a></p>
<p><a href="https://math.stackexchange.c... | halrankard | 688,699 | <p>Your proof is correct. Here is a proof that avoids any mention of specific elements (following the theme of <a href="https://math.stackexchange.com/questions/3763011/suppose-a-b-and-c-are-sets-prove-that-a-delta-b-and-c-are-disjoint/3764438#3764438">my answer to one of your previous questions</a>). The key statemen... |
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... | Batominovski | 72,152 | <p>Using the simple binomial expansion, $$\left(1+\frac{1}{n}\right)^{n(n+1)}=\sum_{k=0}^{n(n+1)}\,\binom{n(n+1)}{k}\,\frac{1}{n^k}\geq 1+\binom{n(n+1)}{1}\,\frac{1}{n}=n+2\,.$$</p>
|
1,733,721 | <p>Determine $\sup E$, $\inf E$, and (where possible) $\max E$, $\min E$ for the set $E = \{ \sqrt[n]{n}: n \in \mathbb{N}\}$.</p>
<p><strong>Attempt:</strong> I've written that $\inf E = 1 = \min E$.</p>
<p>When it comes to finding $\sup E$, I've noticed punching in increasing values of n on my calculator, the eleme... | Clayton | 43,239 | <p>Note that $$\sqrt[n]{n}=n^{1/n}=\exp(\log(n)/n).$$ Thus, $n^{1/n}$ will be maximum when $\log(n)/n$ is maximum (this is a standard property of exponents since the base of the exponent is greater than $1$, i.e., $e>1$).</p>
<p>Consider the more-general setting $f(x)=\log(x)/x$. We can calculate the derivative to ... |
1,543,722 | <p>We are learning about inequalities. I originally assumed it would be the same as equations, except with a different sign. And so far, it has been - except for this.</p>
<p>Take the simple inequality:
$-5m>25$ To solve it, we divide by $-5$ on both sides, as expected.
$m>-5$.</p>
<p>But, I have been told that... | Community | -1 | <p>Surely you believe that we can add/subtract from inequalities without a problem. I show you why using this.</p>
<p>If you have that $x>y$, then subtract $y$ to get $x-y>0$ and subtract $x$ to get $-y>-x$. That is, multiplying by $-1$ flips the inequality.</p>
|
1,634,454 | <p>I saw this question in a previous year test and it seemed pretty simple, and that can often mean that I am missing something.</p>
<p>If the series $$\sum_{n=0}^{\infty}a_n(x-3)^n$$ converges for $x=-1.1$, then it converges for $x=7$.</p>
<p>So, by definition, I can say that this series converges for $\lvert x-3\rv... | Noble Mushtak | 307,483 | <p>Your reasoning is not quite right because you did not solve the compound inequality correctly and you did not actually prove that $|x-3|<r$ for $x=7$.</p>
<p>Let's say we have the following, as you did:</p>
<p>$$-r+3 < x < r+3$$</p>
<p>Then, we need to solve this compound inequality for $x=-1.1$.</p>
<p... |
1,634,454 | <p>I saw this question in a previous year test and it seemed pretty simple, and that can often mean that I am missing something.</p>
<p>If the series $$\sum_{n=0}^{\infty}a_n(x-3)^n$$ converges for $x=-1.1$, then it converges for $x=7$.</p>
<p>So, by definition, I can say that this series converges for $\lvert x-3\rv... | Lutz Lehmann | 115,115 | <p>The related theorem is that if a power series
$$
\sum a_k(z-z_0)^k
$$
converges for some $z=z_1$ then it also converges for all $z$ with
$$
|z-z_0|<|z_1-z_0|.
$$
Here $z_0=3$, $z_1=-1.1$ so that convergence is guaranteed for $|z-3|<4.1$. $z=7$ satisfies this inequality.</p>
<hr>
<p>Proof idea: The terms of ... |
1,705,736 | <p>Is this $$\sqrt{e^{ix}}=e^{\frac{ix}{2}}=\cos\frac{x}{2}+i\sin\frac{x}{2}$$ true?</p>
<p>Or
$$\sqrt{e^{ix}}=\sqrt{\cos x+i\sin x}$$</p>
<p>How do I express square root of $e^{ix}$ as a non-square root expression?</p>
| Henricus V. | 239,207 | <p>Assume that $x$ is real. You can use De Moivre's formula:
$$ \sqrt{e^{ix}} = e^{ix/2}, e^{ix/2+ i\pi}
$$</p>
|
1,705,736 | <p>Is this $$\sqrt{e^{ix}}=e^{\frac{ix}{2}}=\cos\frac{x}{2}+i\sin\frac{x}{2}$$ true?</p>
<p>Or
$$\sqrt{e^{ix}}=\sqrt{\cos x+i\sin x}$$</p>
<p>How do I express square root of $e^{ix}$ as a non-square root expression?</p>
| Arthur | 15,500 | <p>They're both true. As well as $-\sqrt{\cos x+i\sin x}$ and $e^{ix/2+\pi}$.</p>
<p>While square roots are commonly avoided as much as possible when dealing with complex numbers, using natural exponents give $$\cos (nx)+i\sin(nx)=e^{inx}=(\cos x+i\sin x)^n$$ which is how I remember the double angle formula for sine a... |
4,023,924 | <p>I'm doing the following induction proof and wanted to know if this was valid. I think it is, but I'm seeing more complicated solutions than what I did. What I did seems much easier.</p>
<p>Prove that <span class="math-container">$3^n+4^n<5^n$</span> for all <span class="math-container">$n>2$</span>.</p>
<p>Whe... | Wlod AA | 490,755 | <p>Another proof, NO INDUCTION! -- just for fun.</p>
<p>Let <span class="math-container">$\ n\ge 3.\ $</span> then:</p>
<p><span class="math-container">$$ 3^n + 4^n <\, 3^2\cdot 5^{n-2} + 4^2\cdot 5^{n-2}
\,=\, (3^2+4^2)\cdot 5^{n-2}\, =\ 5^n $$</span></p>
<blockquote>
</blockquote>
<p>REMARK <em>If you look at t... |
825 | <p>What resources are available for any grade level K- 12 that are aligned with the Common Core Mathematics Standards and Mathematical Practices that have sets of problems or problem banks that can be used by teachers for instruction or homework?</p>
| Cameron Christensen | 1,103 | <p>The Khan Academy has been developing a <a href="https://www.khanacademy.org/commoncore/map" rel="nofollow">common core map</a>. They have been working with Smarter Balanced and Illustrative Mathematics on the project.</p>
|
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... | Cesareo | 397,348 | <p>If <span class="math-container">$y(t)$</span> is a known solution them substituting <span class="math-container">$z(t) = \lambda(t)y(t)$</span> into the DE we have</p>
<p><span class="math-container">$$
y\ddot\lambda + (a y+2\dot y)\dot\lambda + \ddot y+a \dot y+b y = 0
$$</span></p>
<p>but</p>
<p><span class="ma... |
4,128,110 | <p>Find the <span class="math-container">$\max$</span> and the <span class="math-container">$\min$</span> with Lagrange multipliers, given <span class="math-container">$$f(x,y,z)=xyz^2,$$</span> <span class="math-container">$$g(x,y,z)=x^2+y^2+z^2-1=0.$$</span></p>
<p><a href="https://i.stack.imgur.com/Nr8G8.jpg" rel="... | Parcly Taxel | 357,390 | <p>We have <span class="math-container">$50$</span> samples for each event, so the quoted event counts can reasonably be taken as representatives of the overall punctuality of each service and a probability extracted. Furthermore, for the purposes of this question bus and train may be safely assumed as independent (you... |
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... | Bernard | 202,857 | <p>Here the factorisation which, I hope, will make you understand why there are really <span class="math-container">$4$</span> roots:
<span class="math-container">$$(m^2+1)^2=\bigr((m-i)(m+i)\bigl)^2=\underbrace{(m-i)}_{\text{two roots}}{}^2\underbrace{(m+i)}_{\text{two roots}}{}^2.$$</span>
You have <span class="math-... |
1,271,580 | <p>If $A \in {M_n}$ has exactly $k$ nonzero eigenvalue and $A$ is normal, why is $\operatorname {rank}(A)=k$?</p>
| Ben Grossmann | 81,360 | <p><strong>Hint:</strong> the number of zero eigenvalues of a matrix is the dimension of the kernel (nullspace) of a matrix.</p>
<p>Now, apply the rank-nullity theorem, noting that $A$ has $n$ eigenvalues in total.</p>
|
2,620,571 | <p>Let $\Omega = \{(x,y)\in (0,\infty)^2 | 4 < x^2+4y^2<16\}$, so the area between two ellipses in the first quadrant. I need to calculate the following integral: $$\int_{\Omega}\frac{xy}{x^2+4y^2}d(x,y)$$ I tried using normal polar coordinates, however the integral gets really messy after the transformation. Doe... | lhf | 589 | <p>Using coordinates, if $O=(0,0), B=(b,-1)$ and the circle has radius $1$, then $A=(1/b,1)$. The vectors $OA$ and $OB$ are clearly orthogonal.</p>
<p>Here is a roadmap.</p>
<p>Let $B=(b,-1)$. A line through $B$ is given by $y+1 = \alpha (x-b)$. Plug this into $x^2+y^2=1$ and get a quadratic equation in $x$. The line... |
3,117,111 | <p>For a given <span class="math-container">$n \times n$</span>-matrix <span class="math-container">$A$</span>, the characteristic polynomial of <span class="math-container">$A$</span> is <span class="math-container">$\lambda^n+a_{n-1}\lambda^{n-1}+\cdots+a_1\lambda+a_0$</span>. I am curious to know if we can upper bou... | arbitUser1401 | 51,241 | <p>Here's an answer for <span class="math-container">$a_{n-2} = \sum_{i<j}\lambda_i\lambda_j$</span>:</p>
<p><span class="math-container">$$
\begin{align}
\sum_{i<j}\lambda_i\lambda_j \leq &\sum_{i<j} (\lambda_{i}^{2}+\lambda_{j}^{2})/2\\
= & \frac{n-1}{2}\sum_{i} \lambda_{i}^{2}\\
\leq & \frac{n-... |
2,041,534 | <p>I think I have a method to solve the problem. I am aware that its NP complete and its so tempting to solve. I am aware that I can be wrong 99.99% but I wanted to give a shot at it. I want to put it to test.</p>
<p>Given 2 Graphs : A, B (No Self loops / No Multi-edges)<a href="https://i.stack.imgur.com/PZUUN.png" re... | Will Orrick | 3,736 | <p>Two three-regular graphs on six vertices are shown below. One is planar, the other is not, which implies they are non-isomorphic. As far as I understand, your procedure would produce the same signature for these graphs.</p>
<p><a href="https://i.stack.imgur.com/dRZRq.png" rel="nofollow noreferrer"><img src="https... |
4,224,471 | <p>These two statements are equivalent.</p>
<p>a) x(a + b) = 8 + 5a</p>
<p>b) xa + xb = 8 + 5a</p>
<p>So why is that if we solve X for both of them... They both have different answers. For example a) will equal to x = 8 + 5a / a + b</p>
<p>But b) will equal to x + x = 8 + 5a / a + b, because essentially in b), we will ... | John Dawkins | 189,130 | <p>(In reply to your question — this wouldn't fit in a comment.)</p>
<p>Yes, <span class="math-container">$\mathcal L_\lambda$</span> (the Yoshida approximation of <span class="math-container">$\mathcal L$</span>) is the generator of a Markov process <span class="math-container">$X_\lambda(t)$</span> that approximates ... |
99,239 | <p>Suppose $\mathcal{H}$ is a separable Hilbert space over $\mathbb{C}$ (countable dimensions) with inner product $\langle,\rangle$. Let $A$ be a bounded linear operator on $\mathcal{H}$, i.e, in $B(\mathcal{H}$). Suppose further that $A$ is not a multiple of the identity
operator. Then is it true that there exist t... | Andreas Thom | 8,176 | <p>The answer is yes, this is true (assuming that the Hilbert space is complex).</p>
<p>If $\langle \xi,A\xi \rangle = \sigma$ for some $\sigma \in \mathbb C$ and all $\xi$, then $B:=A - \bar \sigma 1_H$ has the property that $\langle \xi,B\xi \rangle =0$ for all $\xi \in H$. We need to show $B=0$. Let $\xi \in H$ be ... |
12,787 | <p>My son is taking algebra and I'm a little rusty. Not using a calculator or the internet, how would you find the roots of $2x^4 + 3x^3 - 11x^2 - 9x + 15 = 0$. Please list step by step. Thanks, Brian</p>
| Adrián Barquero | 900 | <p>You can try first finding the rational roots using the <a href="http://en.wikipedia.org/wiki/Rational_root_theorem" rel="nofollow">rational root theorem</a> in combination with the <a href="http://en.wikipedia.org/wiki/Factor_theorem" rel="nofollow">factor theorem</a> in order to reduce the degree of the polynomial ... |
754,392 | <p>Consider the following integrals in variables $x,y$ over the whole $\mathbb{R}$, where $a,b\in\mathbb{R}/0$ are constants:</p>
<p>$$\int dx \int dy ~\delta(x-a)\delta(y-b\,x)=\int dy ~\delta(y-b\,a)=1$$</p>
<p>In the following we will evaluate the above integrals in a slightly different way and obtain a completely... | izœc | 83,639 | <p>Recall that the power series for $e^x$ is
$$
\sum_{n=0} ^\infty { \frac{x^n}{n!} } .
$$
Thus, the power series for $e^{-x^2}$ is
$$
\sum_{n=0} ^\infty { \frac{(-x^2)^n}{n!} } = \sum_{n=0} ^\infty { \frac{(-1)^n x^{2n}}{n!} } .
$$
Integrating term by term yields
$$
F(x) = \int F'(x) = \int \sum_{n=0} ^\infty { \fr... |
2,498,424 | <p>Consider double sequences $a_{n,m}\in\mathbb R$ where $n,m\in\mathbb Z,$ satisfying</p>
<ol>
<li>$a_{n,m}=a_{n-1,m}+a_{n,m-1}$ for all $n,m\in\mathbb Z,$ and</li>
<li>$\sup_\limits{m\in\mathbb Z}|a_{n,m}|<\infty$ for all $n\in\mathbb Z.$</li>
</ol>
<p>An example solution is $a_{n,m}=(-1)^m2^{-n}.$ A more genera... | Josse van Dobben de Bruyn | 246,783 | <p>The answer is <strong>no</strong>: not every such double sequence is of the form you describe. I will prove the following things:</p>
<ol>
<li>Different (complex) measures give rise to different (complex) double sequences;</li>
<li>The domain of definition is larger than the set of measures specified by OP;</li>
<l... |
2,498,424 | <p>Consider double sequences $a_{n,m}\in\mathbb R$ where $n,m\in\mathbb Z,$ satisfying</p>
<ol>
<li>$a_{n,m}=a_{n-1,m}+a_{n,m-1}$ for all $n,m\in\mathbb Z,$ and</li>
<li>$\sup_\limits{m\in\mathbb Z}|a_{n,m}|<\infty$ for all $n\in\mathbb Z.$</li>
</ol>
<p>An example solution is $a_{n,m}=(-1)^m2^{-n}.$ A more genera... | Josse van Dobben de Bruyn | 246,783 | <p>As a by-product of my other answer, I found a neat topological argument why there must be more than conjectured by OP.</p>
<h2>Prerequisites and notation.</h2>
<ul>
<li>Let <span class="math-container">$\mathbb{N} = \{0,1,2,\ldots\}$</span> denote the natural numbers with zero.</li>
<li>Once again let <span class="m... |
3,379,893 | <p><a href="https://i.stack.imgur.com/pfWuG.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/pfWuG.png" alt="enter image description here"></a></p>
<p>I am completely lost, I have no idea where to even start. I am sure someone here would be able to do this easily which is why I'm posting this here. B... | grand_chat | 215,011 | <p>Labeling your diagram as below, here are a few hints:
<img src="https://i.stack.imgur.com/cWZqJ.png" width="400" /></p>
<ol>
<li><p>Explain why <span class="math-container">$\theta=\alpha$</span>.</p></li>
<li><p>Inspect the right triangle at the lower left to deduce:
<span class="math-container">$$\cos \alpha =\fr... |
2,580,232 | <p>Suppose that $A$ and $B$ are vector subspaces of $V$, and let $C$ and $D$ be bases for $A$ and $B$, respectively.</p>
<p>Then is it true that </p>
<ol>
<li>$C \cup D$ is a basis for $A+B$?</li>
<li>$\operatorname{dim}(A+B) \le |C| + |D|$ (where $|\cdot|$ denotes cardinality)?</li>
</ol>
<p>I am really bad at dime... | John Doe | 399,334 | <p>Hints:</p>
<p>a) This can be rearranged as $$\frac{df}{dx}-f=e^x$$
You can then use the method of <a href="http://weber.itn.liu.se/~krzma/DS2017/Integrating%20factor%20method.pdf" rel="nofollow noreferrer">integrating factors</a> to solve it.</p>
<p>b) This can be rearranged as $$\frac1f\frac{df}{dx}=e^x$$Then int... |
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