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 |
|---|---|---|---|---|
1,995,471 | <p>I'm trying to find all of the the roots to the following polynomial with a variable second coefficient:
$$P(x)=4x^3-px^2+5x+6$$
All of the roots are rational, and $p$ is too. It is also given that the difference of 2 roots equals the third, e.g. $r-s=t$. I would like to solve for the roots using relationships betwee... | James Ko | 385,100 | <p>Turns out it actually was possible to use relationships between roots here to help factor the equation, I just wasn't looking hard enough.</p>
<p>Since $r - s = t$, then $r = s + t$. As mentioned, $\frac 54 = rs + st + rt = st + (s + t)^2$. Now, both $st$ and $s + t$ can be written in terms of $r$, since</p>
<p>$$... |
3,167,062 | <p>Recently friend of mine showed me a way to write <span class="math-container">$f(x)=0$</span>, however I could not find anything about that in the Internet. Is it correct?</p>
<p><span class="math-container">$f(x)=x^2-1\stackrel{set}{=}0 \\
x = 1 \lor x = -1$</span></p>
| Mark Bennet | 2,906 | <p>Geometrically it gives the (signed) area of the parallelogram defined by the two vectors. </p>
<p>If you multiply by the appropriate unit normal to the plane you get the normal three dimensional cross product. You don't get a vector in the plane though.</p>
<p>If you try to define a "cross product" in four dimensi... |
2,159,136 | <p>I arrived to this question while solving a question paper. The question is as follows:</p>
<blockquote>
<p>If $f_k(x)=\frac{1}{k}\left(\sin^kx + \cos^kx\right)$, where $x$ belongs to $\mathbb{R}$ and $k>1$, then $f_4(x)-f_6(x)=?$</p>
</blockquote>
<p>I started as </p>
<p>$$\begin{align}
f_4(x)-f_6(x)&=\f... | Katherine Wu | 817,994 | <p>‘Tis I have a better solution to solve this math problem...</p>
<p>Let <span class="math-container">$a = \cos^2 x$</span> and <span class="math-container">$b = \sin^2 x,$</span> so <span class="math-container">$a + b = 1.$</span> Then
[(a + b)^2 = a^2 + 2ab + b^2 = 1,]so <span class="math-container">$a^2 + b^2 = 1 -... |
58,912 | <p>In the board game <a href="http://en.wikipedia.org/wiki/Hex_%28board_game%29" rel="nofollow">Hex</a>, players take turns coloring hexagons either red or blue. One player tries to connect the top and bottom edges of the board, colored red; the other tries to connect the left and right edges, colored blue. It is known... | t22 | 11,209 | <p>It is even true that $I^{2}$ may be represented as an union of two closed sets
$S_{1}$ and $S_{2}$ none of them containing a topological arc connecting two
opposite sides of the square. Here is a possible example.</p>
<p>Let $D$ be a closed disk in the interior of $I^{2}$ , take 4 disjoint
topological spirals $J_{i... |
2,776,388 | <p>I tried this problem and I first found the lim of $x^x$ as $x$ approaches zero from right to be 1 (I did this by re-writing $x^x$ as an exponential) and when I repeated the same process to find lim of $x^{(x^x)}$, I found 1 again but the final answer should be ZERO. Could I have an explanation on why it's a zero?</p... | Rhys Hughes | 487,658 | <p>As $x\to 0^+$, we see $x^x\to 0^0$, which evidence suggests is $1$ for $x \in \Bbb R$. </p>
<p>Thus for $x^{x^x}$, as $x\to 0^+$ we get $0^1$, which is certainly $0$.</p>
<p>Incidentally, this implies that the limit of $x^{x^{x^x}}=1$, and $x^{x^{x^{x^x}}}=0$, etc...</p>
<p>Graphed for your perusal <a href="http... |
1,299,630 | <p>I've been wondering if there is any use to defining a set that is isomorphic to $\mathbb{Q}^2$ (in the same way that $\mathbb{C}$ is isomorphic to $\mathbb{R}^2$).</p>
<p>I immediately see a problem with <em>e.g.</em> Cauchy's theorem (there's no $\pi$ available) and perhaps even defining a "analytical function" co... | mp8394 | 243,601 | <p>Given a bijective linear isometry $T:X\rightarrow Y$, the dual map $T^*:Y^*\rightarrow X^*$ is also a bijective linear isometry. (This follows from the fact that $T^*$ has inverse $(T^{-1})^*$ and $\|T^*\|=\|T\|$.) From this, we have that $T^{**}:X^{**}\rightarrow Y^{**}$ is a bijective linear isometry as well.</p>
... |
942,846 | <p>Given 5 different green dyes,4 different blue dyes and 3 different red dyes the number of combinations does that can be chosen by taking at least one green and blue dyes is?</p>
| Marc van Leeuwen | 18,880 | <p>You can use inclusion-exclusion here. Count the number of combinations without restrictions (you did not specify the number in a selection, so I can't do the computation for you). Subtract the number of selections without any green dye by redoing the compoutation after removing all gree dyes. Similarly subtract the ... |
942,846 | <p>Given 5 different green dyes,4 different blue dyes and 3 different red dyes the number of combinations does that can be chosen by taking at least one green and blue dyes is?</p>
| Sooraj S | 223,599 | <p>Number of green dyes =5</p>
<p>Number of blue dyes=4</p>
<p>Number of red dyes =3</p>
<p>The required number of ways of choosing the dye is =
$$
[(^2C_1)^5-1]*[(^2C_1)^4-1]*(^2C_1)^3=[2^5-1]*[2^4-1]*2^3\\=(32-1)*(16-1)*8=31*15*8=3720
$$
where each $^2C_1=2$ corresponds to selecting or not selecting a particular ... |
2,753,504 | <p>Where $a,b$ and $c$ are positive real numbers.</p>
<p>So far I have shown that $$a^2+b^2+c^2 \ge ab+bc+ac$$ and that $$a^2+b^2+c^2 \ge a\sqrt{bc} + b\sqrt{ac} + c\sqrt{ab}$$ but I am at a loss what to do next... I have tried adding various forms of the two inequalities but always end up with something extra on the ... | hamam_Abdallah | 369,188 | <p>Since $(a-b)^2\ge 0,$</p>
<p>we have</p>
<p>$$a^2+b^2\ge 2ab$$
$$b^2+c^2\ge 2bc $$
$$c^2+a^2\ge 2ca .$$</p>
<p>the sum gives</p>
<p>$$2 (a^2+b^2+c^2)\ge 2 (ab+bc+ac) $$</p>
|
47,724 | <p>I am looking at a past exam written by a student. There was a question I believed he got correct but received only 1/4. The marker wrote down "4 more compositions, order matters".</p>
<p>This is the problem:</p>
<p>List all 3 part compositions of 5. (recall that compositions have no zeros)</p>
<p>$(1, 1, 3)
(1, ... | mathmath8128 | 11,976 | <p>You're correct... the only possible ways to add 5 from 3 parts is (1,1,2) and (1,2,2)... each one you have 3 choices on where to place the 2 and 1 respectively, so 3*2=6 possible ways.</p>
|
871,730 | <p>Given x and y we define a function as follow : </p>
<pre><code>f(1)=x
f(2)=y
f(i)=f(i-1) + f(i+1) for i>2
</code></pre>
<p>Now given x and y, how to calculate f(n)</p>
<p>Example : If x=2 and y=3 and n=3 then answer is 1</p>
<p>as f(2) = f(1) + f(3), 3 = 2 + f(3), f(3) = 1.</p>
<p>Constraints are : x,y,n all... | Empy2 | 81,790 | <p>Calculate the first dozen numbers. You should find a simple result for this algorithm.</p>
|
3,426,550 | <p>Are there parts in functional analysis that would be easier to learn without real analysis or is this just not possible?</p>
<p>When I say functional analysis I mean</p>
<ol>
<li>Lebesgue measure and integral</li>
<li>spaces of Lebesgue integrable functions</li>
<li>Banach spaces</li>
<li>duality</li>
<li>bounded ... | P Vanchinathan | 28,915 | <p>Items 1 and 2 listed by you specifies Lebesgue measure and integrable functions. It will be unwise to try to understand them without having been exposed Riemann integral which is part of standard first course in real analysis. That Lebesgue theory is logically a generalisation and so usual Riemann integration is... |
2,812,472 | <p>I am trying to show that</p>
<p>$$
\frac{d^n}{dx^n} (x^2-1)^n = 2^n \cdot n!,
$$ for $x = 1$. I tried to prove it by induction but I failed because I lack axioms and rules for this type of derivatives. </p>
<p>Can someone give me a hint?</p>
| W. mu | 369,495 | <p>$$\frac{d^n}{dx^n}(x^2-1)^n=2nx\frac{d^{n-1}}{dx^{n-1}}(x^2-1)^{n-1}+2n\frac{d^{n-2}}{dx^{n-2}}(x^2-1)^{n-1}$$</p>
<p>The second term is $0$ at $x=1$.</p>
<p>So we have the recursion $I_n=2nI_{n-1}$.</p>
|
1,060,213 | <p>I have searched the site quickly and have not come across this exact problem. I have noticed that a Pythagorean triple <code>(a,b,c)</code> where <code>c</code> is the hypotenuse and <code>a</code> is prime, is always of the form <code>(a,b,b+1)</code>: The hypotenuse is one more than the non-prime side. Why is this... | rogerl | 27,542 | <p>All Pythagorean triples are of the form $k(p^2-q^2), 2kpq, k(p^2+q^2)$ where $p$ and $q$ are positive, relatively prime and of opposite parity. Since $a$ is to be prime, it must be $k(p^2-q^2)$, not $2kpq$; this forces $p^2-q^2=1$, which is impossible, or $k=1$ and $p^2-q^2$ is a prime. So the triangle is of the for... |
4,021,297 | <p><span class="math-container">$\Sigma$</span> is a <span class="math-container">$\sigma$</span>-algebra if it satisfies the following three properties:</p>
<ul>
<li><span class="math-container">$X$</span> is in <span class="math-container">$\Sigma$</span>, and <span class="math-container">$X$</span> is considered to ... | Kevin Arlin | 31,228 | <p>This is a pretty confusing question to ask early in one’s logical education! What’s going on is that, if we want to think about the statement “<span class="math-container">$p$</span> implies <span class="math-container">$q$</span>” and we can prove that the converse “<span class="math-container">$q$</span> implies <... |
4,021,297 | <p><span class="math-container">$\Sigma$</span> is a <span class="math-container">$\sigma$</span>-algebra if it satisfies the following three properties:</p>
<ul>
<li><span class="math-container">$X$</span> is in <span class="math-container">$\Sigma$</span>, and <span class="math-container">$X$</span> is considered to ... | user2661923 | 464,411 | <p>If you translate <span class="math-container">$[A \implies B]~~$</span> to <span class="math-container">$~~[(\neg A) \vee B]~~$</span> then the assertion that (in general) <span class="math-container">$~~(P \implies Q)~~$</span> must hold when <span class="math-container">$~~\neg(Q \implies P)~~$</span> ::: is a <st... |
2,279,392 | <p>I could use some help again.
Let $f,g$ be functions from $\mathbb{N}\to \mathbb{N}$.
Also known is that $f(n) = g(2n)$ for every $n\in\mathbb{N}$.
Assuming that $f$ is a surjective function, how do you prove that $g$ is not a one-to-one function?</p>
<p>Cheers</p>
| CY Aries | 268,334 | <p>Let $m$ be a positive odd integer. Then $g(m)\in\mathbb{N}$.</p>
<p>As $f$ is surjective, there exists $n\in \mathbb{N}$ such that $f(n)=g(m)$.</p>
<p>But this implies that $g(m)=g(2n)$, with $m\ne 2n$.</p>
<p>So $g$ is not one-to-one.</p>
|
57,928 | <p>A function that satisfies both $f(x+1)=f(x)+1$ and $f(x^2)=f(x)^2$ for all real $x$ is known to be the identity over $\mathbb Q$, but is it also the identity over $\mathbb R$? If not, can you provide me an example of such a function? Thanks.</p>
| Steven Stadnicki | 785 | <p>I was just about to supply a counter-example (e.g., set $f(\pi)=e$ and then assign the values at all of the points 'algebraically related' to $\pi$ according to the value of the same algebraic relation applied to $e$) when I realized that this has a clear flaw - for $f$ to be well-defined over $\mathbb{R}$, it must ... |
57,928 | <p>A function that satisfies both $f(x+1)=f(x)+1$ and $f(x^2)=f(x)^2$ for all real $x$ is known to be the identity over $\mathbb Q$, but is it also the identity over $\mathbb R$? If not, can you provide me an example of such a function? Thanks.</p>
| shurtados | 14,682 | <p>if <span class="math-container">$f(x)<x$</span> for some <span class="math-container">$x$</span>, you can suppose <span class="math-container">$x$</span> and <span class="math-container">$f(x)$</span> being greater than <span class="math-container">$1$</span>. Then <span class="math-container">$f\left(x^{2^n}\rig... |
4,412,700 | <p>I've been working on a problem but have been stuck for several hours finishing it.</p>
<p>The problem is to show that
<span class="math-container">$$
\frac{x}{(e+x) \ln(e+x)} \leq \ln(\ln(e+x)) \leq \frac{x}{e}
$$</span>
for all <span class="math-container">$x > 0$</span>.</p>
<p>I proceeded by first integratin... | RRL | 148,510 | <p>Using just the integral definition of <span class="math-container">$\ln$</span>, we have</p>
<p><span class="math-container">$$\ln (\ln(e+x)) = \int_1^{\ln(e+x)} \frac{dt}{t} \geqslant \frac{\ln(e+x) -1}{\ln(e+x)} = \frac{\ln (e \cdot (1+x/e))-1}{\ln(e+x)} = \frac{\ln(1+x/e)}{\ln(e+x)} \\= \frac{1}{\ln(e+x)}\int_1^{... |
2,149,006 | <p>While learning the power rule, one thing popped up in my mind which is confusing me. We know what the power rule states :</p>
<p>$$\frac{\mathrm{d}}{\mathrm{d}x}(x^n) = nx^{n-1}$$ where $n$ is a real number.</p>
<blockquote>
<p>But instead of $n$, if we have a trig function like $\sin(x)$, <strong>will the powe... | Paramanand Singh | 72,031 | <p>It is important to understand the power rule of differentiation $$\frac{d}{dx}x^{n} = nx^{n - 1}\tag{1}$$ The $n$ in exponent is independent of $x$. There is another power rule where $n$ is base namely $$\frac{d}{dx}n^{x} = n^{x}\log n\tag{2}$$ Here also the base $n$ is independent of $x$. Note that there is no powe... |
654,263 | <p>My intention is neither to learn basic probability concepts, nor to learn applications of the theory. My background is at the graduate level of having completed all engineering courses in probability/statistics -- mostly oriented toward the applications without much emphasis on mathematical rigor.</p>
<p>Now I am ve... | broccoli | 50,577 | <p>A good collection of books on probability theory and statistics is listed <a href="http://bayesianthink.blogspot.com/2012/12/the-best-books-to-learn-probability.html" rel="nofollow">here</a>. It is a good collection in the sense it has all ranges of books, from core concepts to lighter reading. Additionally, there i... |
213,513 | <p>I need help to extrapolate these data:</p>
<pre><code>θ = {20.7, 28.62, 32.04};
ω = {5, 6, 7};
</code></pre>
<p>using this equation:</p>
<p>θ(ω)=θ(ω⟶∞)+ c /ω^n to know what the values of c and n</p>
<p>I found a result using the below plot:</p>
<pre><code>ListLinePlot[
Transpose[{ω^-4,#}]&/@{θ},
Fr... | kglr | 125 | <pre><code>ClearAll[mergE]
mergE[f_, red_] := Merge[red]@*KeyValueMap[f[#] -> #2 &];
</code></pre>
<p><strong><em>Examples:</em></strong></p>
<pre><code>a = <|{{a1, a1}, {a2, a2}} -> 1, {{a1, a1}, {a1, a2}} -> 10, {{a1, a2}, {a1, a1}} -> 3|>;
mergE[Sort, Total]@a
</code></pre>
<blockquote>
<... |
4,219,492 | <p>This is a long question/explanation, where I more or less have a method to the madness; however, I've stumbled upon how I perform the calculation and I feel it is quite crude. This question is based off <a href="https://math.stackexchange.com/questions/2631501/finding-the-distribution-of-the-sum-of-three-independent... | Tianyi Miao | 484,971 | <p>The core of this problem is how to systematically convolve 2 piecewise functions, which doesn't necessarily involve probability distributions per se.</p>
<p>This manuscript, which is also motivated by summing uniform random variables, provides such a method: <a href="http://www.mare.ee/indrek/misc/convolution.pdf" r... |
1,689,853 | <p>If I have that $B_t$ is a standard brownian motion process, is $B_t^2 - \frac{t}{2}$ a martingale w.r.t. brownian motion? I know that $B_t^2 - t$ is but can't see it for the latter. </p>
| Marcel | 257,844 | <p>No, it's not. Just calculate the conditional expectation. It is a submartingale.</p>
|
1,353,184 | <p>Let $a$, $b$, and $c$ be positive real numbers. Prove that</p>
<p>$$\sqrt{a^2 - ab + b^2} + \sqrt{a^2 - ac + c^2} \ge \sqrt{b^2 + bc + c^2}$$</p>
<p>Under what conditions does equality occur? That is, for what values of $a$, $b$, and $c$ are the two sides equal?</p>
| Batominovski | 72,152 | <p><strong>An Algebraic Solution:</strong></p>
<p>Note that $$\sqrt{a^2-ab+b^2}=\sqrt{\left(a-\frac{b}{2}\right)^2+\left(\frac{\sqrt{3}b}{2}\right)^2}$$ and $$\sqrt{a^2-ac+c^2}=\sqrt{\left(\frac{c}{2}-a\right)^2+\left(\frac{\sqrt{3}c}{2}\right)^2}\,.$$ Using the Triangle Inequality in the form $\sqrt{\sum_{i=1}^nx_i^2... |
271,785 | <p>I have to prove that $f(x)=\log(1+x^2)$ is Uniform continuous in $[0,\infty)$ (with $\epsilon ,\delta$ formulas...)</p>
<p>I wrote the definition: (what I have to prove):</p>
<p>$\forall \epsilon>0 \quad \exists \delta>0 \quad s.t. \quad \forall x,y\in [0,\infty) : \quad |x-y|<\delta \quad \Rightarrow \qu... | Fabian | 7,266 | <p><em>Hint:</em> You can use that $|\log (1+ z)| \leq z$ valid for $z\geq 1$. Thus, for $x>y$
$$\left|\log \left(\frac{1+x^2}{1+y^2}\right)\right| \leq \frac{x^2-y^2}{1+y^2}
\leq \frac{x+y}{1+y^2} (x-y)= \frac{2y +(x-y)}{1+y^2}(x-y) .$$</p>
<p>Moreover, for $x<y$
$$\left|\log \left(\frac{1+x^2}{1+y^2}\right) \... |
3,188,327 | <p>I derived the operator <span class="math-container">$\mathscr{L} = \dfrac{\partial}{\partial{x}} + u\dfrac{\partial}{\partial{y}}$</span> from the PDE <span class="math-container">$u_x + uu_y = 0$</span> in order to figure out whether it is linear. </p>
<p>The textbook solutions take the following steps in finding ... | Trebor | 584,396 | <p>Obviously the notation <span class="math-container">$$\mathscr{L} = \dfrac{\partial}{\partial{x}} + u\dfrac{\partial}{\partial{y}}$$</span> meant <span class="math-container">$$\mathscr{L}[u] = \dfrac{\partial}{\partial{x}} + u\dfrac{\partial}{\partial{y}}.$$</span> So the <span class="math-container">$u$</span> sho... |
1,170,708 | <p>What functions satisfy $f(x)+f(x+1)=x$?</p>
<p>I tried but I do not know if my answer is correct.
$f(x)=y$</p>
<p>$y+f(x+1)=x$</p>
<p>$f(x+1)=x-y$</p>
<p>$f(x)=x-1-y$</p>
<p>$2y=x-1$</p>
<p>$f(x)=(x-1)/2$</p>
| Christian Blatter | 1,303 | <p>This is a linear inhomogeneous problem. The general philosophy for such problems is the following:</p>
<p>(a) Find the general solution $f_h$ of the associated homogeneous problem
$$f(x)+f(x+1)=0\ .$$
To this end write $f_h(x):=e^{i\pi x} g(x)$ with a new unknown function $x\mapsto g(x)$. You will obtain a huge set... |
3,860,199 | <p>What is the integration of <span class="math-container">$\int x^{|x|} dx$</span>?<br />
Actually through several google search finally I have found a solution for the problem <span class="math-container">$\int x^{x} dx$</span> using Gamma function function and I am hardly sure that the same method can be applied to ... | David Kipper | 764,256 | <p><span class="math-container">$x^x$</span> and <span class="math-container">$x^{-x}$</span> are not integrable in elementary terms. See <a href="https://en.wikipedia.org/wiki/Liouville%27s_theorem_%28differential_algebra%29" rel="nofollow noreferrer">This theorem by Liouville.</a> This theorem one way to approach tha... |
174,075 | <p>What is the difference when a line is said to be normal to another and a line is said to be perpendicular to other?</p>
| Samuel Reid | 19,723 | <p>There is no difference between saying two lines are perpendicular and that one line is normal to another line. It is literally a synonymous term, like saying that you take the product of two numbers or you multiply two numbers.</p>
|
4,648,068 | <h1>Background</h1>
<p>There is a source code that generates surface triangles. The isosurface is generated for the iso-value of <code>0</code>. The source code uses a table for <code>2^8=256</code> possible <em>inside/outside</em>, i.e. <em>negative/positive</em>, combinations of <code>8</code> scalar values at <code>... | kabenyuk | 528,593 | <p>This is possible only when <span class="math-container">$-1=1$</span>, otherwise <span class="math-container">$(-1)^4=1\neq-1$</span>.
For the example in the field <span class="math-container">$\mathbb{F}_2$</span> the identity <span class="math-container">$a^4=a$</span> holds.</p>
|
4,648,068 | <h1>Background</h1>
<p>There is a source code that generates surface triangles. The isosurface is generated for the iso-value of <code>0</code>. The source code uses a table for <code>2^8=256</code> possible <em>inside/outside</em>, i.e. <em>negative/positive</em>, combinations of <code>8</code> scalar values at <code>... | Koro | 266,435 | <p>I think that I figured it out:</p>
<p>We have <span class="math-container">$(-a) ^4= a^4$</span> so <span class="math-container">$-a= a$</span> for every <span class="math-container">$a$</span>. It follows that <span class="math-container">$2a=0$</span>,hence the field is of char <span class="math-container">$2$</sp... |
4,566,926 | <p><span class="math-container">$a_n$</span> converges to <span class="math-container">$a \in \mathbb{R}$</span>, find <span class="math-container">$\lim\limits_{n \rightarrow \infty}\frac{1}{\sqrt{n}}(\frac{a_1}{\sqrt{1}} + \frac{a_2}{\sqrt{2}} +\frac{a_3}{\sqrt{3}} + \cdots + \frac{a_{n-1}}{\sqrt{n-1}} + \frac{a_n}{... | Essaidi | 708,306 | <ol>
<li>If <span class="math-container">$a \neq 0$</span> then <span class="math-container">$a_n \sim a$</span> so <span class="math-container">$\dfrac{a_n}{\sqrt{n}} \sim \dfrac{a}{\sqrt{n}}$</span>.<br>
The series <span class="math-container">$\sum \dfrac{a}{\sqrt{n}}$</span> diverges then :
<span class="math-contai... |
90,130 | <p>So now that my term's over, I've been brushing up on my quantum field theory, and I came across the following line in my textbook without any justification:</p>
<p>$$\frac{1}{4\pi^2}\int_m^{\infty}\sqrt{E^2-m^2}e^{-iEt}dE \sim e^{-imt}\text{ as }t\to\infty$$</p>
<p>Well, I can see intuitively that <em>if</em> most... | Brightsun | 118,300 | <p>Considering the integrand as the Fourier transform of a tempered distribution, it makes sense then to write
$$
\int_{m}^{+\infty}\sqrt{E^2-m^2}e^{-iEt}dE = -\frac{\partial^2}{\partial t^2}\int_{1}^{+\infty}\frac{\sqrt{\rho^2-1}}{\rho^2}e^{-imt\rho}d\rho.
$$
<a href="https://i.stack.imgur.com/wovat.png" rel="nofollow... |
158,451 | <p>Suppose that the contents of an urn are $w$ red balls, $x$ yellow balls, $y$ green balls, and $z$ blue balls collectively, where $w \geq 3$, $x\geq 1$, $y\geq 1$, and $z\geq 1$. We draw balls randomly from this urn without replacement.</p>
<p>What is the probability of our having drawn at least 1 yellow ball by (an... | Alex Becker | 8,173 | <p>Let $x=(x_1,\ldots,x_n), y=(y_1,\ldots,y_n), x' = (\bar x,\ldots,\bar x)$ and $y' = (\bar y,\ldots,\bar y)$. Using this, we can simplify notation by writing
$$r_{xy}=\frac{\sum\limits_{i=1}^n(x_i-\bar x)(y_i-\bar y)}{\sqrt{\sum\limits_{i=1}^n(x_i-\bar x)^2\sum\limits_{i=1}^n(y_i-\bar y)^2}}\leq\frac{\sum\limits_{i=1... |
1,494,167 | <p>Using only addition, subtraction, multiplication, division, and "remainder" (modulo), can the absolute value of any integer be calculated?</p>
<p>To be explicit, I am hoping to find a method that does not involve a piecewise function (i.e. branching, <code>if</code>, if you will.)</p>
| Empy2 | 81,790 | <p>EDIT:<br>
$$m=n\%(n^2-n+2)\\
p=m\%(n^2+2)\\
|n|=2p-n$$</p>
<p>If $n\ge0$ then $m=n$ and $p=n$.<br>
If $n<0$ then $m=n^2+2$ and $p=0$. </p>
|
524,073 | <p>Hey can some help me with this textbook question</p>
<p>Let $R^{2×2}$ denote the vector space of 2×2 matrices, and let</p>
<p>$S =\left\{
\left[\begin{matrix}
a \space b \\
b \space c \\
\end{matrix}\right]\mid a,b,c \in \mathbb{R}\right\}$</p>
<p>Find (with justication) a basis for $S$ and determine the dimensi... | rst | 73,496 | <p>$S =\left\{
\left[\begin{matrix}
a \space b \\
b \space c \\
\end{matrix}\right]\mid a,b,c \in \mathbb{R}\right\}$</p>
<p>or [a b b c]---------------(1)</p>
<p>Put values {[1 0 0 0 ],[0 1 0 0],[0 0 1 0],[0 0 0 1]} in (1)
we get</p>
<p>{[1 0 0 0 ],[0 1 1 0],[0 1 1 0],[0 0 0 1]}</p>
<p>... |
2,678,406 | <p>I'm trying to compute the distance between a point and a plane of the form
<span class="math-container">$$
ax+bx+cz = d
$$</span>
not using the standard formula for analytical geometry.</p>
<ul>
<li>I am trying to compute it by coming up with the projection matrix onto the normal of the plane passing through the ori... | Community | -1 | <p>A line perpendicular to the plane, through the point $(u,v,w)$ has the parametric equation</p>
<p>$$x=u+ta,\\y=v+tb,\\z=w+tc.$$</p>
<p>It intersects the plane when</p>
<p>$$a(u+ta)+b(v+tb)+c(w+tc)=d$$ or</p>
<p>$$t=\frac{d-au-bv-cw}{a^2+b^2+c^2}$$</p>
<p>and the distance between the two points is given by</p>
... |
1,090,658 | <p>I'm doing some previous exams sets whilst preparing for an exam in Algebra.</p>
<p>I'm stuck with doing the below question in a trial-and-error manner:</p>
<p>Find all $ x \in \mathbb{Z}$ where $ 0 \le x \lt 11$ that satisfy $2x^2 \equiv 7 \pmod{11}$</p>
<p>Since 11 is prime (and therefore not composite), the Ch... | Did | 6,179 | <p>Hints: </p>
<ul>
<li>$6\cdot2=1\pmod{11}$</li>
<li>$6\cdot7=9\pmod{11}$</li>
<li>$ab=0\pmod{11}\iff (a=0\pmod{11}\vee b=0\pmod{11})$</li>
</ul>
|
88,880 | <p>In a short talk, I had to explain, to an audience with little knowledge in geometry or algebra, the three different ways one can define the tangent space $T_x M$ of a smooth manifold $M$ at a point $x \in M$ and more generally the tangent bundle $T M$:</p>
<ul>
<li>Using equivalent classes of smooth curves through ... | painday | 109,535 | <p>Here is an explanation based on the category-theoretical formulation of tangent spaces.</p>
<p><strong>In short:</strong> To give a definition of tangent spaces, one actually define a functor from the category of (pointed)manifolds to the category of vector spaces that is a <strong>natural extension</strong> of
the ... |
119,561 | <p>I am interested in determining the <strong>minimum</strong> and <strong>maximum</strong> values of the real roots of polynomials of form $P(x)=\sum_{k=0}^n a_{k} x^k$ where $n$ will have a defined value (say 3,4,5...) and $a_k$ are chosen from the set $\{-1,1\}$ with equal probability.</p>
<p>I have tried creating ... | MarcoB | 27,951 | <p>Try something like </p>
<pre><code>x /. Solve[Sum[RandomChoice[{-1, 1}] x^k, {k, 0, 3}] == 0, x, Reals]
</code></pre>
<p>instead of your <code>Roots</code> expression. Use <code>NSolve</code> if you want the numerical value of the roots, rather than a symbolic representation.</p>
<p>For instance:</p>
<pre><code... |
2,576,466 | <p>One says a bounded $f$ is Riemann integrable on [a,b] if the Upper and lower Riemann integrals are equal. Another sufficient condition for Riemann integrablity is that the set of discontinuity of $f$ must be countable set. The following function is continuous only at $x=1/2$ and so the set of discontinuity of $f$ is... | JH vd Walt | 514,810 | <p>Let </p>
<p>$$g(x) = \left\{\begin{array}{lll}
1-x & if & 0\leq x\leq 1/2 \\
x & if & 1/2<x\leq 1 \\
\end{array}\right.$$</p>
<p>and</p>
<p>$$h(x) = \left\{\begin{array}{lll}
x & if & 0\leq x\leq 1/2 \\
1-x & if & 1/2<x\leq 1 \\
\end{array}\right.$$</p>
<p>Both $g$ and $h$ a... |
771,997 | <p>I'm working on a matrix extension of Fermat's Little Theorem, but I'm stuck on trying to show that if $A^p \equiv A$ mod $p$, then $A$ does not have to be diagonalizable.</p>
<p>Any help would be appreciated!</p>
<p><strong>Edit::</strong> I would like to either find a reason why $A$ does not have to be diagonaliz... | Stella Biderman | 123,230 | <p>What people generally think of as the generalization of Fermat's Little Theorem is as follows: </p>
<p>Let $p$ be prime and $A\in GL_n(\mathbb{Z})$. Then $tr(A^p)=tr(A)$ mod $p$, where $tr(A)$ denotes the trace of the matrix $A$. In fact, this holds even more generally: $tr(A^{p^k})=tr(A^{p^{k-1}})$ mod $p^k$ where... |
3,105,482 | <p>For the formula:</p>
<p><span class="math-container">$$1 = \sqrt{x^2 + y^2 + z^2 + w^2}$$</span></p>
<p>How to rewrite it to find <span class="math-container">$w$</span>?</p>
| NicNic8 | 24,205 | <p><span class="math-container">\begin{align}
1 &= \sqrt{x^2 + y^2 + z^2 + w^2} \\
1 &= x^2 + y^2 + z^2 + w^2 \\
w^2 &= 1 - x^2 - y^2 - z^2 \\
w &= \pm\sqrt{1 - x^2 - y^2 - z^2}
\end{align}</span></p>
|
1,173,643 | <p>I saw the following statement written, but I can't understand why it is true.</p>
<p>$$
\dfrac {P(A \text{ and } B)}{P(B)} = \dfrac{P(A)-P(A \text{ and }B^c)}{ 1-P(B^c)}
$$</p>
<p>Any help understanding why these are equivalent would be appreciated.</p>
| Siméon | 51,594 | <p>Since $B$ and $B^c$ satisfy $B \wedge B^c = \emptyset$ and $B \vee B^c = \Omega$, we have$$
P(A \wedge B) + P(A \wedge B^c) = P(A \wedge \Omega) = P(A).
$$
As a consequence, $$\tag{1}P(A \wedge B) = P(A) - P(A \wedge B^c).$$</p>
<p>In addition, the special case $A = \Omega$ gives
$$
\tag{2}P(B) = 1 - P(B^c).
$$
Fin... |
963,125 | <p>I'm not exactly sure where to start on this one. Any help would be greatly appreciated.</p>
<p>Show that $4$ does not divide $12x+3$ for any $x$ in the integers.</p>
<p>Here's what I have so far:</p>
<p>There exist c in the integers such that 4 | 12x+3. Then 12x+3 = 4y for some y in the integers. This is a contra... | gamma | 88,524 | <p>Note that $12x + 3$ is always odd.</p>
<p>To be divisible by $4$ means to contain at least two factors of $2$.</p>
<p>Odd numbers contain no factors of $2$ so $12x + 3$ can never be divisible by $4$ </p>
|
330,065 | <p>What is an example of a manifold <span class="math-container">$M$</span> with <span class="math-container">$\dim(M)>1$</span> whose Lie algebra <span class="math-container">$\chi^{\infty}(M)$</span> of smooth vector fields admit an elliptic operator <span class="math-container">$D:\chi^{\infty}(M)\to \chi... | Bazin | 21,907 | <p>First a classical result about elliptic operators: in dimension <span class="math-container">$\ge 3$</span> the order of elliptic operators is even. In dimension 2, say in <span class="math-container">$\mathbb R^2$</span> you have elliptic vector fields such as
<span class="math-container">$$
\bar \partial=\frac12(\... |
3,754,225 | <p>Let S be the circle with centre <span class="math-container">$(0,0)$</span>, radius <span class="math-container">$r$</span> units. The chord <span class="math-container">$C$</span> of the circle S subtends an angle of 2π/3 at its center. If R represents the region consisting of all points inside S which are closer t... | g.kov | 122,782 | <p>Using the symmetry, we can consider a upper semicircle
where <span class="math-container">$DE$</span> is a half of the chord.</p>
<p><a href="https://i.stack.imgur.com/XQbzd.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XQbzd.png" alt="enter image description here" /></a></p>
<p>The chord splits... |
139,135 | <p>Suppose $R = \mathbb{Q}[x_1, ..., x_n]/I$, and $J \subset R$ is a given height one ideal. Is there a quick algorithm one could write to determine if $J$ is a principal ideal or necessarily not principal? Is it not possible to do this with Groebner bases?</p>
| Thomas Kahle | 5,495 | <p>In the graded situation the concept of "minimal generators" is well-defined. Just think about the minimal generators as part of the minimal free resolution. Their number is the first total Betti number and those can be computed with Gröbner bases. Macaulay2 has the command <a href="http://www.math.uiuc.edu/Macaul... |
3,271,891 | <p>In <a href="https://en.wikipedia.org/wiki/Boolean_algebra_(structure)" rel="nofollow noreferrer">Wikipedia, the Boolean algebra</a> is defined as a 6-tuple
<span class="math-container">$(A,\wedge,\vee,\neg,0,1)$</span>. In Kuratowski1976, on the other side in the definition on page 34, there is no <span class="math... | amrsa | 303,170 | <p>All Boolean algebras have a top element.</p>
<p>I don't have access to that Kuratowski definition, but tipically, what happens is these alternative definitions have axioms enough to define a partial order relation relative to which the complement operation is order-reversinng. </p>
<p>Given that <span class="math... |
235,940 | <p>Consider a sphere $S$ of radius $a$ centered at the origin. Find the average distance between a point in the sphere to the origin.</p>
<p>We know that the distance $d = \sqrt{x^2+y^2+z^2}$. </p>
<p>If we consider the problem in spherical coordinates, we have a 'formula' which states that the average distance $d_... | anon | 190,539 | <p>All points are equidistant from the origin ... that is why it is a sphere. Therefore, the average distance of a point on the sphere from its center is the radius. </p>
|
24,912 | <p>I have the category-theoretic background of the occasional stroll through MacLane's text, so excuse my ignorance in this regard. I was trying to learn all that I could on the subject of tensor algebras, and higher exterior forms, and I ran into the notion of cohomological determinants. Along this line of inquiry, I ... | Timo Schürg | 473 | <p>I ran across Picard categories in a totally different area of mathematics, but maybe it helps.</p>
<p>In short, a Picard category is a group object in the category of groupoids. </p>
<p>Picard categories come up when you study Picard stacks. Roughly, a Picard stack is a sheaf of Picard categories. The classical e... |
2,067,553 | <p>I'm currently working on another problem: let $x_1,x_2,x_3$ be the roots of the polynomial: $x^3+3x^2-7x+1$, calculate $x_1^2+x_2^2+x_3^2$. Here is what i did: $x^3+3x^2-7x+1=0$ imply $x^2=(7x-x^3-1)/3$. And so $x_1^2+x_2^2+x_3^2= (7x_1-x_1^3-1)+7x_2-x_2^3-1+7x_3-x_3^3-1)/3= 7(x_1+x_2+x_3)/3+(x_1^3+x_2^3+x_3^3)-1$.... | Semiclassical | 137,524 | <p>Hint: Writing your polynomial as $p(x)$, observe that
\begin{align}
p(x)p(-x)
&=(x-x_1)(x-x_2)(x-x_3)\times -(x+x_1)(x+x_2)(x+x_3)\\
&=-(x^2-x_1^2)(x^2-x_2^2)(x^2-x_3^2).
\end{align}</p>
|
785,844 | <p>I've got the following limit to solve:</p>
<p>$$\lim_{s\to 1} \frac{\sqrt{s}-s^2}{1-\sqrt{s}}$$</p>
<p>I was taught to multiply by the conjugate to get rid of roots, but that doesn't help, or at least I don't know what to do once I do it. I can't find a way to make the denominator not be zero when replacing $s$ fo... | Community | -1 | <p>$$\lim_{s\to 1}\dfrac{\sqrt{s}-s^2}{1-\sqrt{s}}=\lim_{s\to 1}\dfrac{\dfrac{1}{2\sqrt{s}}-2s}{-\dfrac{1}{2\sqrt{s}}}=\lim_{s\to 1}\dfrac{\dfrac{1-4s\sqrt{s}}{2\sqrt{s}}}{-\dfrac{1}{2\sqrt{s}}}=\lim_{s\to 1}(-1+4s\sqrt{s})$$
By L'Hopital's rule.</p>
|
2,704,290 | <p>I'm prepared for the competitive exam like <a href="http://csirhrdg.res.in/mathCEN_June2015.pdf" rel="nofollow noreferrer">this</a> (a sample question).</p>
<p>In order to solve the problems, first to familiarize with the prerequisite for the each concept. It's ok! </p>
<p>My problem is: If I'm working certain pro... | polfosol | 301,977 | <p>I think the one good advice that you can get is the fact that the best learning strategy is totally dependent on the learner himself. Joonas has already mentioned some good points, but at the end of the day, it is only you who can evaluate which is the best path to take. In short:</p>
<ul>
<li>Trust yourself and yo... |
1,270,584 | <p>I tried googling for simple proofs that some number is transcendental, sadly I couldn't find any I could understand.</p>
<p>Do any of you guys know a simple transcendentality (if that's a word) proof?</p>
<p>E: What I meant is that I wanted a rather simple proof that some particular number is transcendental ($e$ o... | Yuval Filmus | 1,277 | <p>The easiest proof is via Liouville's criterion.</p>
<p><strong>Lemma.</strong> Suppose $\alpha$ is an irrational algebraic number. There exist integers $C,n$ such that for all integers $p/q$,
$$
\left| \alpha - \frac{p}{q} \right| \geq \frac{C}{q^n}.
$$</p>
<p>You can find a proof of the lemma on <a href="http://e... |
3,291,303 | <p>this very same problem appeared in a different thread but the questions was slightly different. In my case, I'm looking precisely for the answer.</p>
<p>This is how I solved it, I only need confirmation of whether this actually is correct:</p>
<p>So I assumed that 1111... (100 ones) is going to be exactly divided ... | hmakholm left over Monica | 14,366 | <p>Yes, this is a perfectly sound reasoning.</p>
<p>You can also express it as</p>
<p><span class="math-container">$$ 100000010000001\cdots1000000100\cdot 1111111 + 11 = \underbrace{111\cdots111}_{100\text{ ones}} $$</span></p>
|
2,025,069 | <p>$$\sum_{n=0}^{\infty} (-1)^{n}(\frac{x^{2n}}{(2n)!}) * \sum_{n=0}^{\infty} C_{n}x^{n} = \sum_{n=0}^{\infty} (-1)^{n}(\frac{x^{2n+1}}{(2n+1)!})$$</p>
<p>How to find coefficients until the $x^{7} $ ? It is basically sin and written in form of sums but i'm not sure how should i use this form to calculate coefficients ... | hamam_Abdallah | 369,188 | <p>Observe that </p>
<p>$$\sum_{n=0}^{+\infty}C_nx^n=\tan(x)$$</p>
<p>and $C_0=C_2=C_4=C_6=0$.</p>
|
3,163,355 | <p>I found that the Wieferich prime <span class="math-container">$1093$</span> divides <span class="math-container">$3^{1036}-1$</span>.
Does <span class="math-container">$1093$</span> divides infinitely many <span class="math-container">$3^{k}-1$</span>, with k a positive integer? And what features must <span class="m... | fleablood | 280,126 | <p>Notice <span class="math-container">$a^m -1 = (a-1)(a^{m-1} + a^{m-2}+... + a + 1)$</span></p>
<p>So if <span class="math-container">$1093|a^{1036}-1$</span> then <span class="math-container">$1093|a^{1036m} - 1$</span> so there are infinite <span class="math-container">$k = 1036m$</span> where this is true.</p>
<... |
3,714,418 | <p>I need to calculate the angle between two 3D vectors. There are plenty of examples available of how to do that but the result is always in the range <span class="math-container">$0-\pi$</span>. I need a result in the range <span class="math-container">$\pi-2\pi$</span>.</p>
<p>Let's say that <span class="math-conta... | zwim | 399,263 | <p><span class="math-container">$\begin{align}A-(B-A)&=A\cap \overline{(B-A)}=A\cap(\overline{B\cap\overline{A}})=A\cap(A\cup \overline{B})=(A\cap A)\cup(A\cap \overline{B})\\&=A\cup(A-B)\\&=A\end{align}$</span></p>
<p>Thus it is clear from second line that <span class="math-container">$(A-B)$</span> is a ... |
689,591 | <p>Today at class, my teacher stated the following proposition saying it is obvious:</p>
<p>Let $x_0 \in U \subset \mathbb{R}^d$, $U$ open, and $f: U \to \mathbb{R}^m$ differentiable at $x_0$, then for any $v \in \mathbb{R}^d$ we have</p>
<p>$$ D_v f(x_0) = D_f (x_0)(v) $$</p>
<p>It is not obvious to me. Can someone... | Yiorgos S. Smyrlis | 57,021 | <p>Let us first remember what is the meaning of each symbol:</p>
<ul>
<li><p>$D_\nu f(x_0)$ is the derivative of $f$ at $x=x_0$ in the $\nu-$direction, i.e.,
$$
D_\nu f(x_0)=\lim_{h\to 0}\frac{f(x_0+hv)-f(x_0)}{h}. \tag*{(*)}
$$</p></li>
<li><p>On the other hand, we have the following the definition:</p></li>
</ul>
<... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Douglas S. Stones | 2,264 | <p>Simplify (x-a)(x-b)...(x-z).</p>
|
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | matthias beck | 3,193 | <p>(I learned this problem from Persi Diaconis.)
A deck of $n$ different cards is shuffled and laid on
the table by your left hand, face down. An identical deck of cards,
independently shuffled, is laid at your right hand, also face down. You
start turning up cards at the same rate with both hands, first the top
... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Andrea Ferretti | 828 | <p>There is a square with seven monkeys on the floor and seven bananas on the top. Seven ladders go up the square, from one monkey to the banana over it, and the monkeys can climb them. Moreover there are some ropes which connect the ladders.</p>
<p>A monkey will go up towards the bananas, but whenever it meets a rope... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Malik Younsi | 1,162 | <p>I really like the following puzzle, called the blue-eyed islanders problem, taken from Professor Tao's blog :</p>
<p>"There is an island upon which a tribe resides. The tribe consists of 1000 people, with various eye colours. Yet, their religion forbids them to know their own eye color, or even to discuss the topic... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Richard Dore | 27 | <p>A certain rectangle can be covered by 25 coins of diameter 2. Can it always be covered with 100 coins of diameter 1?</p>
|
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Tony Huynh | 2,233 | <p>You have 1000 bottles of wine. Exactly one of the bottles contains a deadly poison, but you don't know which one. The killing time of the poison varies from person to person, but death is imminent in at most $t$ hours after ingestion. You are allowed to use 10 notorious criminals as poison fodder (they are on dea... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Gjergji Zaimi | 2,384 | <p>Here is another of my favorites: Player 1 thinks of a polynomial P with coefficients that are natural numbers. Player 2 has to guess this polynomial by asking only evaluations at natural numbers (so one can not ask for $P(\pi)$). How many questions does the second player need to ask to determine P?</p>
|
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | ACL | 10,696 | <p>When you watch yourself in a mirror, left and right are exchanged. But why aren't top and bottom?</p>
|
4,159,227 | <p>Given the problem of finding the Laplace transform of the function <span class="math-container">$$f(t)=t^ne^{at}$$</span> with <span class="math-container">$n\in\mathbb{N}$</span> and <span class="math-container">$a\in\mathbb{R}$</span>, I realize it can be shown the transform is <span class="math-container">$$\frac... | user577215664 | 475,762 | <p>You can usethe integral definition of the Laplace transform:
<span class="math-container">$$\mathcal{L}\{t^{n+1}e^{at}\}=\int_0^{\infty}t^{n+1}e^{-t(s-a)} dt$$</span>
Integrate by part and use the reccurence for the second integral.
<span class="math-container">$$\mathcal{L}\{t^{n+1}e^{at}\}= \lim_{N\to\infty} \lef... |
1,194,440 | <p>How do I prove if the following formulas are consistent?</p>
<p>∀x$\neg$S(x,x)</p>
<p>∃x P(x)</p>
<p>∀x∃y S(x,y)</p>
<p>∀x(P(x)$\to$∃y S(y,x))</p>
<p>I think I proved part of it...</p>
<p>There is at least one value for x such that P(x) is <em>True</em></p>
<p>This means that for ∀x(P(x)$\to$∃y S(y,x)) ... t... | Graham Kemp | 135,106 | <blockquote>
<p>This means that for $∀x(P(x)→ ∃y S(y,x))$ ... the $P(x)$ part is False because we do not know if it is True for all x.</p>
</blockquote>
<p>No. This statement does not require $P(x)$ to be universally true, only that <em>whenever</em> it is true, <em>then</em> so too is $\exists y\;S(y,x)$.</p>
<p>T... |
2,623,621 | <p>I'm struggling to find the derivative of this function :</p>
<p>$$y=\bigg\lfloor{\arccos\left(\frac{1}{\tan\left(\sqrt{\arcsin x}\right)}\right)}\bigg\rfloor$$</p>
<p>I've been told it should be 0 but how can I find that ?!</p>
| Arnaud Mortier | 480,423 | <p><em>This is a partial solution that works in spaces where closure coincides with sequential closure, such as Frechet spaces.</em> </p>
<p>Pick $$x\in \overline{\bigcup_{i\in I} S_i}$$</p>
<p>Then because $\mathcal{S}$ is locally finite there is a neighbourhood $x\in U$ such that $U$ has a non-empty intersection wi... |
1,534,246 | <p>I'm trying to simplify this boolean expression:</p>
<p>$$(AB)+(A'C)+(BC)$$</p>
<p>I'm told by every calculator online that this would be logically equivalent:</p>
<p>$(AB)+(A'C)$</p>
<p>But so far, following the rules of boolean algebra, the best that I could get to was this: </p>
<p>$(B+A')(B+C)(A+C)$</p>
<p>... | Kevin Zakka | 235,367 | <p>This is known as the consensus rule.</p>
<p>$BC$ is the <em>"consensus"</em> term of $AB$ and $A'C$ and can be removed or added to the boolean expression.</p>
<p><strong>Derivation</strong>
$$(X.Y)+(X'.Z)+(Y.Z)$$
$$=(X.Y)+(X'.Z)+(X+X')(Y.Z)$$
$$=(X.Y)+(X'.Z)+(X.Y.Z)+ (X'.Y.Z)$$
$$=(X.Y)+(X.Y.Z)+(X'.Z)+(X'.Y.Z)$$
$... |
1,534,246 | <p>I'm trying to simplify this boolean expression:</p>
<p>$$(AB)+(A'C)+(BC)$$</p>
<p>I'm told by every calculator online that this would be logically equivalent:</p>
<p>$(AB)+(A'C)$</p>
<p>But so far, following the rules of boolean algebra, the best that I could get to was this: </p>
<p>$(B+A')(B+C)(A+C)$</p>
<p>... | Brian M. Scott | 12,042 | <p>$$\begin{align*}
AB+A'C+BC&=AB(C+C')+A'C(B+B')+BC\\
&=ABC+ABC'+A'BC+A'B'C+BC\\
&=(A+A')BC+BC+ABC'+A'B'C\\
&=BC+BC+ABC'+A'B'C\\
&=BC+ABC'+A'B'C\\
&=(A+A')BC+ABC'+A'B'C\\
&=ABC+A'BC+ABC'+A'B'C\\
&=AB(C+C')+A'(B+B')C\\
&=AB+A'C
\end{align*}$$</p>
|
3,221,547 | <p>I'm trying to prove that there are no simple groups of order <span class="math-container">$240$</span>. So let <span class="math-container">$G$</span> be a simple group such that <span class="math-container">$|G|=240=2^4\cdot3\cdot5$</span>. Then
<span class="math-container">$$n_2\in\{1,3,5,15\}\quad n_3\in\{1,4,10,... | David Leep | 1,025,208 | <p>There is a mistake, or at least a gap, in the provided answer. The case
<span class="math-container">$n_3 = 16$</span> is not considered. In addition, the case <span class="math-container">$n_3 = 40$</span> doesn't seem to be dealt with in the given link. Here is a proof that there are no simple groups of order <s... |
78,946 | <p>I want to find min of the function
$$\frac{1}{\sqrt{2
x^2+\left(3+\sqrt{3}\right)
x+3}}+\frac{1}{\sqrt{2
x^2+\left(3-\sqrt{3}\right)
x+3}}+\sqrt{\frac{1}{3}
\left(2 x^2+2 x+1\right)}.$$
I know, the exact value minimum is $\sqrt{3}$ at $x = 0$. With <em>Mathematica</em>, I tried </p>
<pre><code>A = 1... | kirma | 3,056 | <p>In this case you can find minimum by comparing values at zero derivates:</p>
<pre><code>TakeSmallestBy[{A /. #, #} & /@ Solve[D[A, x] == 0, x, Reals], First, 1]
</code></pre>
<blockquote>
<p>{{Sqrt[3], {x -> 0}}}</p>
</blockquote>
<p><code>TakeSmallestBy</code> is a v10.1 function similar to <code>MinimalBy... |
3,298,282 | <p>Find the number of terms in the expansion <span class="math-container">$(1+a^3+a^{-3})^{100}$</span></p>
<p>I used the concept <span class="math-container">$a^3+a^{-3}=T$</span>, while using this I have 101 terms, from <span class="math-container">$T^2$</span> to <span class="math-container">$T^{100}$</span> how do... | farruhota | 425,072 | <p>It is equivalent to finding the number of terms in:
<span class="math-container">$$(1+a^3+a^{-3})^{100}=a^{-300}(a^6+a^3+1)^{100} \Rightarrow (a^6+a^3+1)^{100},$$</span>
whose terms will have powers (since the signs are positive and exponents are multiples of <span class="math-container">$3$</span>):
<span class="ma... |
2,742,497 | <p>When two events $A$ and $B$ are independent, $P(A)\cdot P(B)$ equals the probability of both events occurring together.</p>
<p>$$P(A|B) = \frac{P(A\cap B)}{P(B)}.$$</p>
<p>If $A$ and $B$ are not independent, $P(A\cap B)$ is either greater than or less than $P(A)\cdot P(B)$.</p>
<ol>
<li><p>I want to understand t... | farruhota | 425,072 | <p>Consider the following events $M$={pass midterm exam}, $F$={pass final exam} and the probabilities:
$$P(M)=0.8, P(F)=0.7,P(M\cap F)=0.6.$$
Then:
$$P(M|F)=\frac{P(M\cap F)}{P(F)}=\frac{0.6}{0.7}=\frac67>0.8=P(M); \\
P(F|M)=\frac{P(F\cap M)}{P(M)}=\frac{0.6}{0.8}=\frac34>0.7=P(F).$$
Hence, passing one exam incre... |
1,930,933 | <blockquote>
<p>Does there exist an $n \in \mathbb{N}$ greater than $1$ such that $\sqrt[n]{n!}$ is an integer?</p>
</blockquote>
<p>The expression seems to be increasing, so I was wondering if it is ever an integer. How could we prove that or what is the smallest value where it is an integer?</p>
| Winther | 147,873 | <p>If $\sqrt[n]{n!} = k \in \mathbb{N}$ then $n! = k^n$. When $n\geq 2$ we have $2\mid n!$ so we must also have $2\mid k$ which means that we can write $k = 2^{m} \ell$ for some integers $m$ and $\ell$. This again means that</p>
<p>$$n! = 2^{mn}\ell^n \implies 2^{mn} \mid n!$$</p>
<p>so the power of two that divides ... |
482,010 | <p>Sorry for my bad english And also my bad math literature in asking my first question.</p>
<p>Assume we have 15 glass full of water.</p>
<p>I need to count the number of possible ways which I can empty 5 glass in a way that no 2 empty glass remain in sequence.</p>
| Accidental Statistician | 88,206 | <p>If the glasses are in a line, say we draw a full glass as a star $*$, and an empty glass as a vertical bar $|$. Then a possible way to empty five glasses is drawn as a line of five stars and ten bars, where the bars are not next to each other. One such way could be drawn as</p>
<p>$|**|**|**|**|**$</p>
<p>This is ... |
1,030,050 | <p>Assignment:</p>
<blockquote>
<p>Let $f$ be Lebesgue - measurable and $a,b \in \mathbb{R}$ with the property:
$$\frac{1}{\lambda(M)} \cdot \int_Mf\ d\lambda \in [a,b]$$
for all Lebesgue - measurable sets $M \subset \mathbb{R}^n$ with $0 < \lambda(M) < \infty$.</p>
<p>Show that: $f(x) \in [a,b]$ almo... | RHP | 13,463 | <p>Suppose $N = \{x:f(x)\notin[a,b]\}$ is not a null set.</p>
<p>Then at least one of $N_1 = \{x:f(x)<a\}$ or $N_2 = \{x:f(x)>b\}$ is not a null set since $N = N_1\cup N_2$</p>
<p>If $N_1$ is not a null set, then (since $f(x)<a$ on $N_1$)</p>
<p>$$\frac{1}{\lambda(N_1)}\int_{N_1}f~d\lambda<\frac{1}{\lamb... |
3,353,483 | <p>I have just begun studying finite fields today, and it is clear in GF(2) why 1+1=0. (I just show that 1+1 can't equal 1, or 1=0, which contradicts an axiom that states that 1 is not 0).</p>
<p>If we interpreted these symbols "1", "+", "1", "0" as we would in primary school, clearly this breaks arithmetic rules in R... | Lubin | 17,760 | <p>There’s really no big deal. You are really working in integers (<em>never</em> in the reals), and whenever you get an answer that’s too big or too negative, subtract or add a multiple of your prime number <span class="math-container">$n$</span> (conventionally, we call the modulus <span class="math-container">$p$</s... |
3,353,483 | <p>I have just begun studying finite fields today, and it is clear in GF(2) why 1+1=0. (I just show that 1+1 can't equal 1, or 1=0, which contradicts an axiom that states that 1 is not 0).</p>
<p>If we interpreted these symbols "1", "+", "1", "0" as we would in primary school, clearly this breaks arithmetic rules in R... | fleablood | 280,126 | <p>Beneath it all... we count.</p>
<p>And it doesn't matter what we count; we just count.</p>
<p>So we define <span class="math-container">$4$</span> as what we get if we add <span class="math-container">$1$</span> a "tick-tick-tick-tick" number of times. And we define <span class="math-container">$5$</span> as what... |
2,779,841 | <p>I'm wondering how the following recursion for <span class="math-container">$D(n)$</span>, the derangement of <span class="math-container">$[n]$</span> distinct elements, can be proved</p>
<p><span class="math-container">$$D(n) = nD(n-1) + (-1)^n$$</span></p>
<hr>
<p>I see the argument for </p>
<p><span class="ma... | sku | 341,324 | <p>Algebraic proof:</p>
<p>$$D_{n+1} = nD_n + nD_{n-1} = (n+1)D_n - D_n + nD_{n-1}$$</p>
<p>We want to prove that $nD_{n-1} - D_n = (-1)^n$</p>
<p>$$nD_{n-1} - D_n = (n-1)D_{n-1} + D_{n-1} - ((n-1)D_{n-1} + (n-1)D_{n-2}) = D_{n-1} - (n-1)D_{n-2}$$</p>
<p>$$D_{n-1} - (n-1)D_{n-2} = D_{n-1} - (n-2)D_{n-2} - D_{n-2}$$... |
3,361,489 | <p><strong>Question:</strong></p>
<p>Do there exist functions <span class="math-container">$f$</span> and <span class="math-container">$g$</span> such that
<span class="math-container">$$\lim_{x \to c} f(x) = 1 \text{ and } \lim_{x \to c} f(x) g(x) - g(x) \neq 0 \, ?$$</span>
(Allowing, of course, for <span class="ma... | Bernard | 202,857 | <p><em>Counter-example</em>:</p>
<p><span class="math-container">$\lim_{x\to 0}\cos x=1$</span>, and
<span class="math-container">$$\lim_{x\to0}\cos x\,\frac1{x^2}-\frac1{x^2}=\lim_{x\to 0}\frac{\cos x-1}{x^2}=-\frac12.$$</span></p>
|
434,614 | <p>Is there a name for a type of grid you might find in Battleship? Where coordinates don't relate to points on a grid but rather the squares themselves?</p>
| Patrick Hew | 190,548 | <p><em>Cell-centered grid</em> may be a slightly better name, for when the coordinates refer to the cell. By contrast in a <em>node-centered grid</em>, the coordinates refer to the points on the grid. </p>
<p>The distinction is important in applications such as <a href="https://au.mathworks.com/help/images/image-coord... |
1,103,478 | <p>$ r = 2\cos(\theta)$ has the graph<img src="https://i.stack.imgur.com/yvLb1.png" alt="enter image description here"></p>
<p>I want to know why the following integral to find area does not work $$\int_0^{2 \pi } \frac{1}{2} (2 \cos (\theta ))^2 \, d\theta$$</p>
<p>whereas this one does:</p>
<p>$$\int_{-\frac{\pi}{... | abel | 9,252 | <p>the limit is from $0$ to $\pi$ not $0$ to $2\pi$ if you do the latter, you will go around the circle twice.</p>
<p>at $\theta = 0$ you are $(2,0)$ at $\theta = \pi/3$ you are at $x = 1, y = 1$ at $\theta = \pi/2$ you are at $(0,0)$ and then you move in the lower semicircle for the next $\pi/2$ to $\pi$ if you go fa... |
4,297,177 | <p>I have a polynomial:</p>
<p><span class="math-container">$f(n)=\frac{1}{4}n^2-24n-16$</span></p>
<p>I am supposed to show that it is <span class="math-container">$\Omega(n^2)$</span> and <span class="math-container">$O(n^2)$</span>. Once I prove those, it should be easy to show that it is also <span class="math-cont... | Gary | 83,800 | <p>It is easy to show that
<span class="math-container">$$
a_n: n \mapsto \frac{{f(n)}}{{n^2 }}
$$</span>
is an increasing sequence and <span class="math-container">$\lim_{n \to +\infty}a_n=1/4$</span>. Thus, <span class="math-container">$a_n <1/4$</span> for all <span class="math-container">$n\geq 1$</span>. Inpart... |
3,443,226 | <blockquote>
<p>Prove that
<span class="math-container">$$
\int_{0}^{2\pi}\frac{d\theta}{(a+\cos\theta)^2}=\frac{2\pi a}{(a^2-1)^{\frac{3}{2}}}.
$$</span></p>
</blockquote>
<p>This is an exercise in Stein's <em>Complex Analysis</em>.</p>
<p>By letting <span class="math-container">$z=e^{i\theta}$</span>, we have
... | Jack D'Aurizio | 44,121 | <p>Without complex analysis, for any <span class="math-container">$a>1$</span> we have
<span class="math-container">$$ J(a)=\int_{0}^{2\pi}\frac{d\theta}{a+\cos\theta}=2\int_{0}^{\pi}\frac{d\theta}{a+\cos\theta}=2\int_{0}^{\pi/2}\left(\frac{1}{a+\cos\theta}+\frac{1}{a-\cos\theta}\right)\,d\theta $$</span>
or
<span c... |
710,045 | <p><a href="https://i.stack.imgur.com/7rJAB.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/7rJAB.png" alt="question" /></a></p>
<p>For (a) I have said</p>
<p>By MVT:</p>
<p><span class="math-container">$|f(x)-f(y)|\le K|x-y|$</span></p>
<p>Choose <span class="math-container">$\delta=\epsilon/K$</sp... | Tom Cooney | 8,580 | <p>In finite dimensions, all norms are equivalent.
(For example, see <a href="https://math.stackexchange.com/q/599824/8580">Any two norms on finite dimensional space are equivalent</a>.)</p>
<p>This means the following: if $\Vert \cdot \Vert_1$ and $\Vert \cdot \Vert_2$ are the two norms on the space $X$, there are c... |
301,724 | <p>There's an example in my textbook about cancellation error that I'm not totally getting. It says that with a $5$ digit decimal arithmetic, $100001$ cannot be represented.</p>
<p>I think that's because when you try to represent it you get $1*10^5$, which is $100000$. However it goes on to say that when $100001$ is r... | ncmathsadist | 4,154 | <p>That is correct. You have what is called type overflow. Want to be amused? If you have a C compiler run this program</p>
<pre><code>#include<stdio.h>
int main(void)
{
int x = 1;
while (x > 0)
{
x = x + 1;
}
printf("x = %d\n", x);
}
</code></pre>
<p>It's educational. This is ... |
811,535 | <p>I want to show that every prime power $p^k$ that divides $\binom{2m}{m}$ is smaller than or equal to $2m$.</p>
<p>As a first step, I looked at
$$\binom{2m}{m}
= \frac{(2m)!}{(m!)^2}
= \frac{2m(2m-1) \ldots (m+2)(m+1)}{m!} \, .$$
Here I'm essentially stuck. I can apply the prime factorization to numerator and denomi... | Chen Yihan | 763,624 | <p>Let <span class="math-container">$c_k$</span> be the integer such that <span class="math-container">$p^{c_k}\leq k\leq p^{c_k+1}$</span>. For integer <span class="math-container">$n$</span>, the number of elements in <span class="math-container">$[n]=\{1,2,\dots,n\}$</span> divided by <span class="math-container">$p... |
458,472 | <p>Just doing some revision for ODEs and came across this problem. Find the general solution to $$u''+4u=0.$$</p>
<p>So far I've applied the characteristic polynomial:
$$\begin{array}{r c l}
\lambda^2 +4 & = & 0 \\
\lambda^2 & = & -4 \\
\lambda & = & i\sqrt{4} \\
\lambda & = & 2i, -2i. ... | Amzoti | 38,839 | <p>If we wrote:</p>
<p>$$\tag 1 e^{a+ 2i} = e^{a t}(\cos 2t + i \sin 2t)v_1$$</p>
<p>where $v_1$ is the eigenvector, and for your problem $a = 0$.</p>
<p>When we expand $(\cos 2t + i \sin 2t)v_1$, we get an expression of the form:</p>
<p>$$(\alpha) + i(\beta)$$</p>
<p>Because we know that the real imaginary parts ... |
866,144 | <p>Prove that
$$1-2^{-x}\geq \frac{\sqrt 2}{2}\sin\left(\frac{\pi}{4} x\right)$$
for $x\in[0,1]$.
Any suggestions please?</p>
| Robert Israel | 8,508 | <p>Let $f(x) = 1 - 2^{-x} - \frac{\sqrt{2}}{2} \sin(\frac{\pi}{4} x)$. Note that
$f(0) = f(1) = 0$ and $f'(0) > 0$ while $f'(1) < 0$. </p>
<ol>
<li>Show that $f'' < 0$ for
$0 \le x \le 0.8$, and note that $f'(0.8) < 0$ and $f(0.8) > 0$.</li>
<li>Show that $f' < 0$ for $0.8 \le x \le 0.9$.</li>
<li>... |
158,483 | <p>When I started mathematica, this message popped up.</p>
<pre><code>Part::partw: Part 5 of PacletManager`Utils`Private`$taskData[2] does not exist.
</code></pre>
<p>Does anyone know what this is? My version is mma 11.2.</p>
| Dmitriy Sh. | 79,885 | <p>If you encounter this message at the MMA's startup, please check init.m located in %AppData%\Mathematica\FrontEnd. I solved the same problem (I use MMA 11.3) just by deleting entire text/commands in the %AppData%\Mathematica\FrontEnd\init.m file. This init.m is automatically initialized at every system startup and a... |
3,065,331 | <p>In <a href="https://rads.stackoverflow.com/amzn/click/0199208255" rel="nofollow noreferrer">Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers</a> one of the very first stated equations are, as in the title of the question,</p>
<p><span class="math-container">$$
\ddot{x} = \fr... | WarreG | 519,466 | <p>Try putting <span class="math-container">$$ \frac{dx}{dt} = y(x,t).$$</span>
Notice that <span class="math-container">$y$</span> is a function of <span class="math-container">$x$</span> <em>and</em> <span class="math-container">$t$</span>. You can then use the chain rule for <span class="math-container">$$\frac{dy(x... |
1,423,728 | <p>The definition of a limit in <a href="http://rads.stackoverflow.com/amzn/click/0321888545" rel="nofollow noreferrer">this book</a> stated like this </p>
<p><a href="https://i.stack.imgur.com/tKjaa.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/tKjaa.png" alt="enter image description here"></a><... | Eric M. Schmidt | 48,235 | <p>It certainly would be possible to define limits in a more general situation, provided that you're careful enough. The given definition has the advantage that you don't have to worry about the function being undefined, provided $\delta$ is small enough. If you wanted to be more general, you could drop the open interv... |
131,435 | <p>Wikipedia is a widely used resource for mathematics. For example, there are hundreds of mathematics articles that average over 1000 page views per day. <a href="http://en.wikipedia.org/wiki/Wikipedia:WikiProject_Mathematics/Popular_pages" rel="noreferrer">Here is a list of the 500 most popular math articles</a>. The... | Waldemar | 34,050 | <p>"(...)most academics – despite goodwill to contribute – still perceive major barriers to participation, which typically include a general lack of time to contribute, but also barriers of a technical, social and cultural nature. These encompass the lack of incentives from the perspective of a professional career, the... |
131,435 | <p>Wikipedia is a widely used resource for mathematics. For example, there are hundreds of mathematics articles that average over 1000 page views per day. <a href="http://en.wikipedia.org/wiki/Wikipedia:WikiProject_Mathematics/Popular_pages" rel="noreferrer">Here is a list of the 500 most popular math articles</a>. The... | Federico Poloni | 1,898 | <p>My feeling towards Wikipedia is that it is great as a quick reference to check facts and results, but it doesn't work as well for explaining well introductory material. The reasons are various.</p>
<p>It is difficult to provide a good explanation for <em>everyone</em>. Often, different intended targets conflict: yo... |
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