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 |
|---|---|---|---|---|
3,520,269 | <p><img src="https://i.stack.imgur.com/mdM8B.png" alt="enter image description here"></p>
<p>I also know that given the length of 2 sides in a kite and the angle of one of the other angles (which aren't included angles), you can find the area by multiplying these two sides and the sine of the angle. I am unable to fin... | Community | -1 | <p>In fact, <a href="https://en.wikipedia.org/wiki/Quadrilateral#Trigonometric_formulas" rel="nofollow noreferrer">the area of a quadrilateral is <strong>one-half</strong> the product of its diagonals times the sine of their included angle</a>.</p>
<p><a href="https://i.stack.imgur.com/cJd7G.png" rel="nofollow norefer... |
1,994,277 | <p>I am studying linear representation theory for finite groups and came across the claim in title. When $n\geq 5$, $S_n$ does not have an irreducible, $2$- dimensional representation.But I am not sure where to begin with. </p>
<p>Although it seems that this result will follow from <a href="https://math.stackexchange.... | Jeremy Rickard | 88,262 | <p>Here's a hint for one fairly elementary proof.</p>
<p>Suppose $\rho:S_n\to\textrm{GL}(2,\mathbb{C})$ is a representation, where $n\geq5$.</p>
<p>Consider the eigenvalues of $\rho(\sigma)$ for $5$-cycles $\sigma$.</p>
|
1,799,366 | <p>I'm trying to solve the following exercise:</p>
<blockquote>
<p>Let $\mu$ be a probability distribution on $\mathbb{R}$ having second moment $\sigma^2<\infty$ such that if $X$ and $Y$ are independent with law $\mu$ then the law of $(X+Y)/\sqrt{2}$ is also $\mu$. Show that $\mu =\mathcal{N}(0,1)$
Hint: apply ... | drhab | 75,923 | <p>I think it must be proved that $\mu=\mathcal N(0,\sigma^2)$ but for convenience I will also preassume that $\sigma=1$</p>
<p>If $\phi$ denotes the characteristic function then:$$\phi(t)=\phi\left(\frac{t}{\sqrt2}\right)^2$$</p>
<p>Note that this can be repeated to arrive at $\phi(t)=\phi(\frac{t}2)^4$ and can be r... |
25,337 | <p>If you want to compute crystalline cohomology of a smooth proper variety $X$ over a perfect field $k$ of characteristic $p$, the first thing you might want to try is to lift $X$ to the Witt ring $W_k$ of $k$. If that succeeds, compute de Rham cohomology of the lift over $W_k$ instead, which in general will be much e... | user39938 | 39,938 | <p>A general method for for constructing schemes with "arbitrarily bad" deformation spaces (including the non-existence of liftings from char. p to char. 0) is in the following paper:</p>
<p>R. Vakil "Murphy's Law in algebraic geometry: Badly-behaved deformation spaces", Invent. Math. 164 (2006), 569--590. </p>
|
113,446 | <p>Suppose a simple equation in Cartesian coordinate:
$$
(x^2+ y^2)^{3/2} = x y
$$
In polar coordinate the equation becomes $r = \cos(\theta) \sin(\theta)$. When I plot both, the one in polar coordinate has two extra lobes (I plot the polar figure with $\theta \in [0.05 \pi, 1.25 \pi]$ so the "flow" of the curve is cle... | kglr | 125 | <p>A simpler way to restrict to positive radii:</p>
<pre><code>PolarPlot[Max[Sin[θ] Cos[θ], 0], {θ, 0, 2 π}]
</code></pre>
<p><img src="https://i.stack.imgur.com/w4gge.png" alt="Mathematica graphics"></p>
|
1,109,443 | <p>I'm currently learning for an algebra exam and I have some examples of questions from few last years. And I can't find a solution to this one:</p>
<blockquote>
<p>Give three examples of complex numbers where z = -z</p>
</blockquote>
<p>The only complex number I can think of is 0. Because it is a complex number, ... | Hagen von Eitzen | 39,174 | <p>This depends on the model. Instead of arguing that we have only $46$ chromosomes and cross-overs or whatever the mechanism is called are not <em>that</em> common, let us assume a continuous model.
That is, a priori, everybody can be $\alpha$ Cherokee for any $\alpha\in[0,1]$ and the rules are as follows</p>
<ul>
<l... |
1,053,065 | <p>I have a function called $P(t)$ that is the number of the population at time $t$. $t$ being in days.</p>
<p>We know the growth rate is $P'(t) = 2t + 6$</p>
<p>We also know that $P(0) = 100$. How many days till the population doubles?</p>
<p>edit: $P(t) = t^2 + 6t$
edit: $P(t) = t^2 + 6t = 200$
edit: $t^2 + 6t - 2... | asomog | 183,714 | <p>Let $n=\prod_{i=1}^k p_i^{\alpha_i}$, then we want
$$\frac{d(n)}{n}=\frac{\prod_{i=1}^k{1+\alpha_i}}{\prod_{i=1}^k p_i^{\alpha_i}}\leq\frac{1}{2}$$
Look at just one term, and add 1 to the exponent of $p_i$ to observe that:
$$\frac{\alpha+1}{p^\alpha}/\frac{\alpha+2}{p^{\alpha+1}}=\frac{1}{p}\left(1+\frac{1}{\alpha+... |
1,053,065 | <p>I have a function called $P(t)$ that is the number of the population at time $t$. $t$ being in days.</p>
<p>We know the growth rate is $P'(t) = 2t + 6$</p>
<p>We also know that $P(0) = 100$. How many days till the population doubles?</p>
<p>edit: $P(t) = t^2 + 6t$
edit: $P(t) = t^2 + 6t = 200$
edit: $t^2 + 6t - 2... | Esteban Crespi | 3,274 | <p>Let $n>12$ if $a$ is a divisor of $n$ then $a$ is not in the range $n/2 < a < n$ otherwise $n/a$ is an integer in the range $ 1 < n/a < 2$, in the same way $a$ can't be in the range $n/3 < a < n/2$, so the number of integers in this range is at least equal to the number of integers in the interv... |
1,210,285 | <p>Let there be a given function $f \in C([0,1])$, $f(x)>0$; $x\in [0,1]$. Prove </p>
<p>$$\lim_{n\to\infty} \sqrt[n]{f\left({1\over n}\right)f\left({2\over n}\right)\cdots f\left({n\over n}\right)}=e^{\int_0^1 \log f(x) \, dx} $$</p>
<p>All the questions before this required solving an definite integral without N... | Chappers | 221,811 | <p>$f$ is positive, and the logarithm is continuous on $(0,\infty)$, so we can take logarithms of both sides and swap the limit and logarithm to find
$$ \log{\left( \lim_{n\to\infty} \sqrt[n]{ f(1/n) f(2/n) \dotsm f(n/n) }\right)} = \lim_{n\to\infty} \log{\sqrt[n]{f(1/n)f(2/n)\dotsm f(n/n)}} $$
Now applying properties ... |
1,384,752 | <p>I ran across a problem which has stumped me involving existential quantifiers.
Let U, our universe, be the set of all people. Let S(x) be the predicate "x is a student" and I(x) be the predicate "x is intelligent".
I want to write the statement "Some students are intelligent" in the correct logical form. I can see 2... | Graham Kemp | 135,106 | <p>The confusion lies in choice of logical connective to represent a restriction on the domain of discussion. We use conjunction to restrict an existential quantifier, and implication to restrict a universal quantifier.</p>
<p>Here we are restricting the domain, from discussions of all people in the given univ... |
2,618,273 | <p>In integer-base positional numeral systems, the notation of a number in base $n$ uses $n$ numerals. Base 2 uses the symbols 0 and 1, base 10 uses 0123456789, base 16 uses base 10 + ABCDEF. Although the choice of symbols for the numerals is arbitrary, the number of numerals (unique glyphs) is identical to the base.... | fleablood | 280,126 | <p>==== number of numerals that can be digits ====</p>
<p>In the article sited the $k$th digit can but any integer from $0$ to $\frac x{\beta^k}$ which as $x < \beta^{k+1}$ would be any digit greater than equal to $0$ and less than $\beta$. If $\beta$ is not an integer that is from $0$ to $\lfloor \beta \rfloor &l... |
2,515,939 | <p>So, I just need a hint for proving
$$\lim_{n\to \infty} \int_0^1 e^{-nx^2}\, dx = 0$$ </p>
<p>I think maybe the easiest way is to pass the limit inside, because $e^{-nx^2}$ is uniformly convergent on $[0,1]$, but I'm new to that theorem, and have very limited experience with uniform convergence. Furthermore, I don... | zhw. | 228,045 | <p>Letting $y=x/\sqrt n$ shows the integral equals</p>
<p>$$\frac{1}{\sqrt n} \int_0^{\sqrt n} e^{-y^2}\, dy <\frac{1}{\sqrt n} \int_0^{\infty} e^{-y^2}\, dy .$$</p>
<p>Since the last integral converges, the desired limit is $0.$</p>
|
2,329,542 | <p>I looked up wikipedia but honestly I could not make much sense of what I will basically study in Abstract Algebra or what it is all about .</p>
<p>I also looked up a question here :
<a href="https://math.stackexchange.com/questions/855828/what-is-abstract-algebra-essentially">What is Abstract Algebra essentially?</... | EHH | 133,303 | <p>This is a fairly tricky question as Abstract Algebra is one of those things that makes a lot more sense one you've spent some time studying some of it's subject areas.</p>
<p>I will however have a stab at it...</p>
<p>Abstract algebra normally follows the same pattern of taking a set, $Z$ say, and attributing some... |
4,083,697 | <p>I'm thinking about the example <span class="math-container">$f(x)=(x-1)^2$</span> which is clearly symmetric about the line <span class="math-container">$x=1$</span>. The question is really how do you show that it is symmetric about <span class="math-container">$x=1$</span> algebraically? I notice that if you plug i... | user2661923 | 464,411 | <p>Alternative approach:</p>
<p><strong>To Prove</strong>: <span class="math-container">$~|z| \times |w| = |z \times w| ~: z,w \in \Bbb{C}.$</span></p>
<p>Sufficient to show that <span class="math-container">$|z|^2 \times |w|^2 = |z \times w|^2.$</span></p>
<p>Let <span class="math-container">$z = (x + iy), w = (u + iv... |
232,672 | <p>Yesterday, I posted a question that was received in a different way than I intended it. I would like to ask it again by adding some context. </p>
<p>In ZF one can prove $\not\exists x (\forall y (y\in x)).$ This statement can be read in many ways, such as (1) "there is no set of all sets" (2) "the class of all sets... | Thomas Benjamin | 20,597 | <p>Considering what you wrote in your slide presentation "On the definitional character of axioms.", you might be interested in the following preprint by John L. Bell (found on his Homepage) titled "SETS AND CLASSES AS MANY". In it, he initially gives the following naive definition of set (following Cantor in his book... |
353,658 | <p>Let $g : [0, 1] \rightarrow \mathbb{R}$ be twice differentiable with $g^{\prime \prime}(x) > 0$ for all $x \in [0,1]$. Suppose that $g(0) > 0$ and $g(1) = 1$. Prove if $g$ has a fixed point in $(0,1)$, then $g^{\prime}(1) > 1$.</p>
<p>My attempt: Define a function $h(x)=g(x)-x$. Since $g$ has a fixed point... | Sugata Adhya | 36,242 | <p>The Rolle's theorem enables us to get a root of $h'$ on some point lying left to $1$ which helps to conclude the result using the strict monotonicity of $h'$ (Since $h''>0$ on $[0,1]\implies h'$ is strictly increasing on $[0,1]$)</p>
|
2,895,655 | <blockquote>
<p>Four coins of different colour are thrown. If three out of these show heads then find the probability that the remaining one shows tails. </p>
</blockquote>
<p>My approach:</p>
<p>$A$: The event in which 3 heads appear in 3 coins out of 4</p>
<p>$B$: The event in which the 4th coin shows tails</p>
... | N. F. Taussig | 173,070 | <p>You have correctly calculated the probability $\Pr(A \cap B)$. Your error was in the calculation of $\Pr(A)$.</p>
<p>The sample space consists of those events in which at least three of the four coins display heads. The probability that at least three coins display heads is
$$\Pr(A) = \binom{4}{3}\left(\frac{1}{... |
3,459,106 | <p>I have a function
<span class="math-container">$$
\frac{\ln x}{x}
$$</span> and I wonder, is <span class="math-container">$y=0$</span> an asymptote? I mean it is kinda strange that graph is in some place is going through that asymptote. I know it meets the criterium of asymptote, but its kinda strange if you unders... | Marios Gretsas | 359,315 | <p>If <span class="math-container">$f_n$</span> are not continuous then it is not true.</p>
<p>Take <span class="math-container">$f_n=1_{[-\frac{1}{2},\frac{1}{2}]}+\frac{1}{n}$</span> on <span class="math-container">$[-1,1]$</span></p>
<p>where <span class="math-container">$1_A$</span> is the indicator function of ... |
42,957 | <p>I am an "old" programmer used to <em>Fortran</em> and <em>Pascal</em>. I can't get rid of <code>For</code>, <code>Do</code> and <code>While</code> loops, but I know <em>Mathematica</em> can do things much faster!</p>
<p>I am using the following code</p>
<pre><code>SeedRandom[3]
n = 10;
v1 = Range[n];
v2 = RandomRe... | WalkingRandomly | 4,786 | <p>Your version:</p>
<pre><code>SeedRandom[3]
n = 250;
v1 = Range[n];
v2 = RandomReal[250., n];
a = {};
Do[
Do[
AppendTo[a,
(v2[[i]] - v2[[j]])/(v1[[i]] - v1[[j]])],
{j, i - 1, 1, -1}], {i, n, 2, -1}]; // AbsoluteTiming
</code></pre>
<p>on my machine this takes 1.85 seconds</p>
<p>This version</p>
... |
42,957 | <p>I am an "old" programmer used to <em>Fortran</em> and <em>Pascal</em>. I can't get rid of <code>For</code>, <code>Do</code> and <code>While</code> loops, but I know <em>Mathematica</em> can do things much faster!</p>
<p>I am using the following code</p>
<pre><code>SeedRandom[3]
n = 10;
v1 = Range[n];
v2 = RandomRe... | ciao | 11,467 | <p>Just changing it to something like:</p>
<pre><code>Table[(v2[[i]] - v2[[j]])/(v1[[i]] - v1[[j]]), {i, n, 2, -1}, {j, i - 1, 1, -1}] // Flatten
</code></pre>
<p>Should net you a nice boost. Edit - oops, ninja'd</p>
<p>About twice as fast as any so far on large N:</p>
<pre><code>s = Subsets[Range[n, 1, -1], {2}];
... |
4,219,614 | <p>In proving a change of basis theorem in linear algebra, our professor draw this diagram and simply stated that because all the outer squares in this diagram commute, the inner square (green) must also commute (I didn't write the exact mappings, because I think this question is more about diagram chasing and that it... | Troposphere | 907,303 | <blockquote>
<p>because all the outer squares in this diagram commute, the inner square (green) must also commute</p>
</blockquote>
<p>That is not true in general. You can make a counterexample in the category of vector spaces by letting the green square be your favorite <em>non-commuting</em> square and then declare t... |
441,888 | <p>I should clarify that I'm asking for intuition or informal explanations. I'm starting math and never took set theory so far, thence I'm not asking about formal set theory or an abstract hard answer. </p>
<p>From Gary Chartrand page 216 Mathematical Proofs - </p>
<p>$\begin{align} \text{ range of } f & = \{f(x... | Community | -1 | <p>There are two basic ways to "build" sets, aside from listing their elements.</p>
<p>The first is the axiom of subsets: if you already have a set $S$ and you want to create the subset of $S$ of things satisfying some property $P$, then the usual notation is</p>
<p>$$ \{ x \in S \mid P(x) \} $$</p>
<p>For example, ... |
1,272,124 | <p>we know that $1+2+3+4+5.....+n=n(n+1)/2$</p>
<p>I spent a lot of time trying to get a formula for this sum but I could not get it :</p>
<p>$( 2 + 3 + . . . + 2n)$</p>
<p>I tried to write the sum of some few terms.. Of course I saw some pattern between the sums but still the formula I Got didn't give a correct sum... | Adhvaitha | 228,265 | <p>Note that
$$2+4+6+\cdots+2n = 2\left(1+2+3+\cdots+n\right) = 2 \cdot \dfrac{n(n+1)}2 = n(n+1)$$</p>
|
2,593,361 | <p>I’m trying to solve what I’ll call the p-Laplace Equation which is</p>
<p>$$\Delta_p u = 0$$</p>
<p>where $\Delta_p u$ is the p-Laplacian. It is defined as </p>
<p>$$\Delta_p u = \nabla \cdot (|\nabla|^{p-2} \nabla u).$$</p>
<p>Any ideas? I haven’t seen this in a book or anything. I just thought that by analogy,... | fleablood | 280,126 | <p>By archemenian principal there is a unique integer $m$ so that</p>
<p>$mn \le x < (m+1) n$</p>
<p>And,, likewise, there is a unique integer $a$ so that $a \le x < a + 1$.</p>
<p>So $mn\le a \le x < a+1 \le (m+1)n$</p>
<p>And $m \le \frac an \le \frac xn < \frac an + \frac 1 n \le m+1$</p>
<p>From t... |
3,820,929 | <p>When i look at my notes , i realized something i have not realized before.It was as to a modular arithmetic
question.</p>
<p>The question is <span class="math-container">${\sqrt 2} \pmod7$</span></p>
<p>It is very trivial question.The solution is: if <span class="math-container">$x \equiv {\sqrt 2} \pmod7$</span> ,t... | Crostul | 160,300 | <p>You are working in the field of numbers modulo <span class="math-container">$7$</span>, namely <span class="math-container">$\Bbb Z / (7)$</span> which is often denoted by <span class="math-container">$\Bbb F_7$</span>.</p>
<p>Now, <span class="math-container">$2$</span> is an element of <span class="math-container... |
24,927 | <p>Does this mean that the first homotopy group in some sense contains more information than the higher homotopy groups? Is there another generalization of the fundamental group that can give rise to non-commutative groups in such a way that these groups contain more information than the higher homotopy groups? </p>
| Ronnie Brown | 28,586 | <p>These problems puzzled the early topologists: in fact Cech's paper on higher homotopy groups was rejected for the 1932 Int. Cong. Math. at Zurich by Hopf and Alexandroff, who quickly proved they were abelian. We now know this is because group objects in groups are abelian groups. However group objects in the categor... |
1,843,274 | <p>Good evening to everyone. So I have this inequality: $$\frac{\left(1-x\right)}{x^2+x} <0 $$ It becomes $$ \frac{\left(1-x\right)}{x^2+x} <0 \rightarrow \left(1-x\right)\left(x^2+x\right)<0 \rightarrow x^3-x>0 \rightarrow x\left(x^2-1\right)>0 $$ Therefore from the first $ x>0 $, from the second $ x... | Aman Rajput | 307,098 | <p>In inequality questions, first you need to make all the coefficients of $x$ positive.</p>
<p>We can write it as
$$\frac{x-1}{x(x+1)}>0$$</p>
<p>Now , multiply both sides by $(x(x+1))^2$ which is always greater than $0$.
Hence we are left with this
$$x(x-1)(x+1)>0$$</p>
<p>Solving this critical points are a... |
4,489,675 | <p>When saying that in a small time interval <span class="math-container">$dt$</span>, the velocity has changed by <span class="math-container">$d\vec v$</span>, and so the acceleration <span class="math-container">$\vec a$</span> is <span class="math-container">$d\vec v/dt$</span>, are we not assuming that <span class... | Community | -1 | <p>It is important to be careful when working with infinitesimals. The answer by @mmesser314 is a good answer (+1 from me) which is described in terms of limits and the so-called standard analysis. In that analysis an infinitesimal is not a number. More specifically, an infinitesimal is not a real number.</p>
<p>Howeve... |
299,452 | <p>According to wiki: <a href="https://en.wikipedia.org/wiki/Dedekind_eta_function" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Dedekind_eta_function</a>, Dedekind eta function is defined in many equivalent forms. But none of them is an explicit description (say in algorithmic format) on how to computing it... | André Henriques | 5,690 | <p>Euler's formula</p>
<p>$$
\sum\limits_{n \in \mathbb{Z}} {( - 1)^n q^{\frac{{(3n^2 - n)}}
{2}} } = \prod\limits_{n = 1}^\infty {(1 - q^n ),}
$$</p>
<p>(which can be proven from Jacobi’s triple product identity by using the fact that $\prod\limits_{n = 1}^\infty {(1 - q^{3n} )(1 - q^{3n - 2} )} (1 - q^{3n - 1} ... |
3,197,331 | <p><span class="math-container">$X, Y$</span> are quantities and <span class="math-container">$f : X → Y$</span> a function. Show
the equivalence of the following statements:</p>
<p>(i) <span class="math-container">$f$</span> is injective</p>
<p>(ii) <span class="math-container">$f^{-1}\!\bigl(f(A) \bigr)=A \quad \t... | Ross Millikan | 1,827 | <p>If you compute the expected value, each outcome after an initial tails contributes <span class="math-container">$\frac 14$</span> to the sum, so the sum diverges. This would say I should pay any price to play the game, which is counterintuitive as the chance of winning very much money is very small. The usual reso... |
3,717,144 | <p>Suppose M is a finitely generated non-zero R-module, where R is a commutative unital ring. Show that the tensor product of M with itself is non-zero.</p>
<p>I know one way to show this is to find an R-bilinear map which is nonzero, but am not sure how to find it.</p>
| o.h. | 630,261 | <p>Not sure this is the best proof, but here goes. Note that <span class="math-container">$M\otimes_R M = 0$</span> if and only if <span class="math-container">$(M\otimes_R M)_{\mathfrak p} = 0$</span> for all primes <span class="math-container">$\mathfrak p\subset R$</span> (Atiyah-MacDonald 3.8). But
<span class="mat... |
855,227 | <p>I'm trying to understand Taylor's Theorem for functions of $n$ variables, but all this higher dimensionality is causing me trouble. One of my problems is understanding the higher order differentials. For example, if I have a function $f(x, y)$, then it's first differential is: </p>
<p>$$df = \frac{\partial f}{\part... | Community | -1 | <p>In $$d^2f= \frac{\partial^2 f}{\partial y ^2}dy^2 + 2\frac{\partial^2 f}{\partial y \partial x}dy\:dx + \frac{\partial^2 f}{\partial x ^2}dx^2$$, $\frac{\partial^2 f}{\partial y ^2}dy^2$ is the rate of change of $\frac{\partial f}{\partial y}$ on $y$, same for $\frac{\partial^2 f}{\partial x ^2}dx^2$. $2\frac{\parti... |
855,227 | <p>I'm trying to understand Taylor's Theorem for functions of $n$ variables, but all this higher dimensionality is causing me trouble. One of my problems is understanding the higher order differentials. For example, if I have a function $f(x, y)$, then it's first differential is: </p>
<p>$$df = \frac{\partial f}{\part... | Steven Gubkin | 34,287 | <p>I think all of this makes a lot more sense when you approach it from a multilinear set up.</p>
<p>If $f: \mathbb{R}^2 \to \mathbb{R}$ is a function, then its differential $df$ gives a different linear map at each point. Fundamentally we have </p>
<p>$$
f(\mathbf{p}+\vec{v})\approx f(\mathbf{p})+df(\mathbf{p}, \ve... |
3,479,144 | <p>Let <span class="math-container">$(X,M,\mu)$</span> be a measure space and <span class="math-container">$f \in L^{1}(X,\mu)$</span>. Then show that for <span class="math-container">$E \in M$</span>, <span class="math-container">$\lim_{k \rightarrow \infty} \int_{E} |f|^{1/k} = \mu(E)$</span>. I am able to show this ... | Lubin | 17,760 | <p>If you’re merely asking whether there are infinitely many <span class="math-container">$n$</span> for which the expansion of <span class="math-container">$\sqrt n$</span> is of that form, that’s easy, since the expansion of <span class="math-container">$\sqrt{n^2+2n}$</span> is <span class="math-container">$[n,\over... |
184,699 | <p>First, we make the following observation: let $X: M \rightarrow TM $ be a vector
field on a smooth manifold. Taking the contraction with respect to $X$ twice gives zero, i.e.
$$ i_X \circ i_{X} =0.$$
Is there any "name" for the corresponding "homology" group that one can define
(Kernel mod image)? Has this "homo... | Qiaochu Yuan | 290 | <p>You can get something "interesting" if you couple the $1$-form version with the de Rham differential. Namely, if $\alpha \in \Omega^1(X)$ is a closed $1$-form on a smooth manifold $X$, then the de Rham complex can be equipped with a twisted de Rham differential $d + \alpha$, by which I mean</p>
<p>$$(d + \alpha) \b... |
507,062 | <p>I need your help with a lifelong problem I have always had. There are two things in life I hate most, 1) weddings and 2) Long division. For the life of me I hate division. I am horrible at it. I can multiple large numbers in my head but I can’t divide if my life depended on it. When the division operation is 2 digi... | Patrick Da Silva | 10,704 | <p>Assuming $x \neq 0$, then yes. Otherwise you're dividing by zero.</p>
<p>Hope that helps,</p>
|
507,062 | <p>I need your help with a lifelong problem I have always had. There are two things in life I hate most, 1) weddings and 2) Long division. For the life of me I hate division. I am horrible at it. I can multiple large numbers in my head but I can’t divide if my life depended on it. When the division operation is 2 digi... | amWhy | 9,003 | <p>Notice that the denominators are equal, save for a factor of $-1$: which has no impact since the fraction is squared. $$\; (4 - x^2)^2 = (-(x^2 - 4))^2 = (x^2 - 4)^2$$</p>
<p>So multiply both sides by $(x^2 - 4)^2$, and you'll cancel both denominators. Then there's no need to divide by $5x$.
After multiplying bot... |
2,970,787 | <blockquote>
<p>Find <span class="math-container">$\lim_{x\to -\infty} \frac{(x-1)^2}{x+1}$</span></p>
</blockquote>
<p>If I divide whole expression by maximum power i.e. <span class="math-container">$x^2$</span> I get,<span class="math-container">$$\lim_{x\to -\infty} \frac{(1-\frac1x)^2}{\frac1x+\frac{1}{x^2}}$$</... | lisyarus | 135,314 | <p>The point is that the denominator not just tends to zero, but tends to zero from the left, i.e. from being negative.</p>
<p>Alternatively, rewrite like this:</p>
<p><span class="math-container">$$\frac{(x-1)^2}{x+1} = \frac{(x+1)^2-4x}{x+1}=x+1-\frac{4}{1+\frac{1}{x}}$$</span></p>
<p>which clearly tends to <span ... |
3,078,707 | <p>The above question is the equation <span class="math-container">$(2.4)$</span> of the following paper:</p>
<p><a href="http://www.jmlr.org/papers/volume6/tsuda05a/tsuda05a.pdf" rel="nofollow noreferrer">MATRIX EXPONENTIATED GRADIENT UPDATES</a>.</p>
<p>Let <span class="math-container">$M$</span> and <span class="m... | Jack D'Aurizio | 44,121 | <p>Just exploit the principle that an ellipse is a stretched circle.<br>
Let <span class="math-container">$P$</span> our point, <span class="math-container">$\tau$</span> the known tangent through <span class="math-container">$P$</span>, <span class="math-container">$O$</span> the center of the ellipse.</p>
<p><a href... |
3,078,707 | <p>The above question is the equation <span class="math-container">$(2.4)$</span> of the following paper:</p>
<p><a href="http://www.jmlr.org/papers/volume6/tsuda05a/tsuda05a.pdf" rel="nofollow noreferrer">MATRIX EXPONENTIATED GRADIENT UPDATES</a>.</p>
<p>Let <span class="math-container">$M$</span> and <span class="m... | Servaes | 30,382 | <p>A general ellipse, with axes parallel to the coordinate axes, is given by an equation of the form
<span class="math-container">$$u(x-x_0)^2+v(y-y_0)^2=1.$$</span>
The ellipse is symmetric w.r.t. the <span class="math-container">$x$</span>-axis so <span class="math-container">$y_0=0$</span>, and because <span class="... |
192,020 | <p>I suspect this is a duplicate, but I can't seem to find what I'm looking for.</p>
<p>A routine problem I have is the following.</p>
<p>I have a set of data in three (or two, or more) lists:</p>
<pre><code>l1={a1, a2, a3}
l2={b1, b2, b3, b4}
l3={{c1, c2, c3, c4}, {d1, d2, d3, d4}, {e1, e2, e3, e4}}
</code></pre>
... | Coolwater | 9,754 | <pre><code>Join[Tuples[{l1, l2}], ArrayReshape[l3, {Times @@ Dimensions[l3], 1}], 2]
</code></pre>
<blockquote>
<p>{{a1, b1, c1}, {a1, b2, c2}, {a1, b3, c3}, ...}</p>
</blockquote>
<p>Though if the elements of <code>l3</code> are list of equal length, then <code>ArrayReshape</code>/<code>Dimension</code> won't work... |
1,728,097 | <p>So i have this integral : $$ \int_0^\infty e^{-xy} dy = -\frac{1}{x} \Big[ e^{-xy} \Big]_0^\infty$$
The integration part is fine, but I'm not sure what i get with the limits, can someone explain this</p>
<p>Thanks </p>
| Mnifldz | 210,719 | <p>You need to take the limits as $y$ ranges from $0$ to $\infty$. This is simply</p>
<p>$$
\left . - \frac{1}{x} e^{-xy} \right |_0^\infty \;\; =\;\; 0 - -\frac{1}{x} \;\; =\;\; \frac{1}{x}.
$$</p>
|
1,728,097 | <p>So i have this integral : $$ \int_0^\infty e^{-xy} dy = -\frac{1}{x} \Big[ e^{-xy} \Big]_0^\infty$$
The integration part is fine, but I'm not sure what i get with the limits, can someone explain this</p>
<p>Thanks </p>
| Edward Evans | 312,721 | <p>Taking a limit is how many integrals of this form can be evaluated. Let $c$ be a real number and let $I = \int_0^\infty e^{-xy} dy.$ Then,</p>
<p>$$I = \lim_{c \to \infty} \int_0^c e^{-xy} dy = \lim_{c \to \infty} \left[ -\frac{1}{x}e^{-xy}\right]_0^c = 0 -\left(-\frac{1}{x} \right) = \frac{1}{x}.$$</p>
|
341,202 | <p>The divisibility rule for $3$ is well-known: if you add up the digits of $n$ and the sum is divisible by $3$, then $n$ is divisible by three. This is quite helpful for determining if really large numbers are multiples of three, because we can recursively apply this rule:</p>
<p>$$1212582439 \rightarrow 37 \rightar... | Math Gems | 75,092 | <p>$\begin{eqnarray} \rm{\bf Hint}\ \ &&\rm3\ \ divides\ \ a\! +\! 10\,b\! +\! 100\, c\! +\! 1000\,d\! + \cdots\\
\iff &&\rm 3\ \ divides\ \ a\! +\! b\! +\! c\! +\! d\! +\! \cdots +\color{#c00}9\,b\! +\! \color{#c00}{99}\,c\! +\! \color{#c00}{999}\,d\! + \cdots\\
\iff &&\rm3\ \ divides\ \ a\! +\... |
341,202 | <p>The divisibility rule for $3$ is well-known: if you add up the digits of $n$ and the sum is divisible by $3$, then $n$ is divisible by three. This is quite helpful for determining if really large numbers are multiples of three, because we can recursively apply this rule:</p>
<p>$$1212582439 \rightarrow 37 \rightar... | LinAlgMan | 49,785 | <p>Yes, it is true. Let
$$n = a_n a_{n-1} ... a_1 a_0 $$
the integer, where $0 \le a_i \le 9$ are its digits, that is
$$ n = \sum_{i=0}^n a_i \cdot 10^i \ . $$
Since $10^i \equiv 1 \pmod 3$ ($10=3 \cdot 3 + 1$, $100 = 3 \cdot 33 + 1$ $1000 = 3 \cdot 333 + 1$ and so on), we can write
$$ n = \sum_{i=0}^n a_i \cdot 10^i \... |
3,159,199 | <p>I did a question to find relative extrema for the following function:
<span class="math-container">$f(x)=x^2$</span> on <span class="math-container">$[−2,2].$</span></p>
<p>The answer said that there is no relative maxima for this function because relative extrema cannot occur at the end points of a domain.
Why i... | Allawonder | 145,126 | <p>The answer is correct. Why it is so follows immediately from the definition.</p>
<p>There's a difference between an extreme value and a <em>relative</em> extreme value of a function at a point. While your function has a maximum value at <span class="math-container">$\pm2,$</span> it does not have a <em>relative</em... |
1,162,315 | <blockquote>
<p>(b) An electrical circuit comprises three closed loops giving the following equations for the currents $i_1, i_2$ and $i_3$</p>
<p>\begin{align*}
i_1 + 8i_2 + 3i_3 &= -31\\
3i_1 - 2i_2 + i_3 &= -5\\
2i_1 - 3i_2 + 2i_3 &= 6
\end{align*}</p>
</blockquote>
<p>This is the system I need t... | Peter | 82,961 | <p>Hint : Multiply the first equation with $3$ and subtract the second to get one equation
containing only $i_2$ and $i_3$. Multiply the first equation with $2$ and subtract
the third to get another equation containing only $i_2$ and $i_3$.
The result is</p>
<p>$$26i_2+8i_3=-88$$
$$19i_2+4i_3=... |
3,415,331 | <p>It is easy to show, that (for continuous functions f)
<span class="math-container">$$\exists c>0,\exists\alpha>1, \forall x\in \mathbb{R}: |x|^\alpha |f(x)| <c \implies \int |f|dx <\infty$$</span></p>
<p>The question is, whether or not this is also a neccessary condition. I could not come up with any c... | user284331 | 284,331 | <p>If it were, then <span class="math-container">$|f(x)|<c/|x|^{\alpha}$</span> for <span class="math-container">$|x|>1$</span> which entails that <span class="math-container">$f(x)\rightarrow 0$</span> as <span class="math-container">$x\rightarrow\infty$</span>. So we are asking whether integrable functions will... |
4,382,786 | <p>I would like to know what is the following process on the real line called.</p>
<p>Let us fix some <span class="math-container">$X_0$</span> and let <span class="math-container">$X_{i+1} = (1-\gamma)X_i + Y_i$</span> where <span class="math-container">$\gamma$</span> is a fixed real number and <span class="math-cont... | manifolded | 621,937 | <p>This probably doesn't answer your question but I know about ergodic Markov chains that have this form where <span class="math-container">$Y_i$</span> take values on a certain state space and <span class="math-container">$f(x)=(1-\gamma)x$</span> is a bijection on that state space.</p>
<p>Example: Let <span class="ma... |
474,568 | <p>In some books I've seen this symbol $\dagger$, next to some theorem's name, and I don't know what it means. I've googled it with no results which makes me suspect it's not standard.</p>
<p>Does anybody know what it means? One example I'm looking at right now is in a probability book, next to a section about Sitrlin... | Don Larynx | 91,377 | <p>From Wikipedia: "While the asterisk (asteriscus) was used for corrective additions, the obelus was used for corrective deletions of invalid reconstructions". </p>
<p>The obelus, which is the "cross", is similar to the asterisk but is used for making corrections instead of additions.</p>
<p>Edit: I don't know if th... |
572,137 | <p>If $\psi:G \to H$ is a surjective homomorphism, then $|\{g \in G: \psi(g)=h_1\}| = |\{g \in G: \psi(g)=h_2\}|, \forall h_1,h_2 \in H.$ </p>
<p>Could anyone advise on the proof? If $\psi$ is injective, then the result follows. So, what happens if $\psi$ is not injective? By 2nd isomorphism thm, $G/Ker(\psi) \cong H.... | DonAntonio | 31,254 | <p>An idea following egreg's answer: </p>
<p>$$\psi(g)=h_1\in H\;\implies\;\psi(gn)=h_1\;\;\forall\,n\in N:=\ker\psi\implies \psi(gN)=h_1$$</p>
<p>and the other way around:</p>
<p>$$\psi(x)=h_1\implies \psi(g^{-1}x)=\psi(g)^{-1}\psi(x)=h_1^{-1}h_1=1\implies g^{-1}x\in N\iff xN=gN$$</p>
<p>and thus we see that </p>
... |
49,068 | <p>Given lists $a$ and $b$, which represent multisets, how can I compute the complement $a\setminus b$?</p>
<p>I'd like to construct a function <code>xunion</code> that returns the symmetric difference of multisets.
For example, if $a=\{1, 1, 2, 1, 1, 3\}$ and $b=\{1, 5, 5, 1\}$, then their symmetric difference is $\b... | ciao | 11,467 | <p>I don't pretend this is the most efficient or pretty, but here's a go at what I think you're after (see latter part of post for faster and simpler realizations):</p>
<pre><code>a = {1, 1, 2, 1, 1, 3};
b = {1, 5, 5, 1};
result = Join[
Flatten[ConstantArray @@@
Flatten[Replace[Cases[GatherBy[Join[Tally[#1... |
300,753 | <p>Revision in response to early comments. Users of set theory need an <em>implementation</em> (in case "model" means something different) of the axioms. I would expect something like this:</p>
<blockquote>
<p>An <em>implementation</em> consists of a "collection-of-elements" $X$, and a relation
(logical pairing)... | Noah Schweber | 8,133 | <p><em>Caveat: it's become clear from comments and revisions that the original portion of this answer - leading up to the horizontal line below - is not really addressing the heart of the OP. I'm leaving it up since I think it is still at least somewhat relevant and potentially useful to readers. See below the horizont... |
2,763,974 | <blockquote>
<p>Find $\displaystyle \lim_{(x,y)\to(0,0)} x^2\sin(\frac{1}{xy}) $ if exists, and find $\displaystyle\lim_{x\to 0}(\lim_{y\to 0} x^2\sin(\frac{1}{xy}) ), \displaystyle\lim_{y\to 0}(\lim_{x\to 0} x^2\sin(\frac{1}{xy}) )$ if they exist.</p>
</blockquote>
<p>Hey everyone. I've tried using the squeeze theo... | user284331 | 284,331 | <p>So the moral of this question is that, one might think that if the double limit exists, then it entails the existence of iterated limits, this question gives a counterexample, but what is true is that,</p>
<blockquote>
<p>Given that $\lim_{(x,y)\rightarrow(a,b)}f(x,y)=L$ exists and that for each $x\in B_{\delta}(... |
354,213 | <p>This is a similar question to the one I have posted <a href="https://math.stackexchange.com/questions/354124/given-u-2-5-3-how-to-find-unit-vectorsu-w-s-t-uv-is-maximal-and-u">before</a>. The problem
is as in the title:</p>
<blockquote>
<p>Given $u=(-2,5,3)$ find a unit vector $v$ s.t $|u\times v|$ is
maximal, ... | user54358 | 54,358 | <p>Let $O$ be an open set in $X\times Y$. If $x\in O$ then there is an open ball $O_x$ containing $x$. You can construct an open rectangle, $R_x$, with rational endpoints to contain $x$ but contained in $O_x$. Now
\begin{eqnarray*}
O=\bigcup_{x\in O} R_x.
\end{eqnarray*}
You assert that it is countable because there ... |
105,723 | <p>I am using MMA do do some algebra and I need to do some simplification. For example, I have the following expression
$$ x^2 \left( \frac{6 \left(2 c_1 x^6+c_2\right)}{x^4} \right)-x \left( 4 c_1 x^3-\frac{2 c_2}{x^3}\right)-8\left(c_1 x^4+\frac{c_2}{x^2}\right ) $$
Can I get intermediate steps to reach the final an... | Jason B. | 9,490 | <p>You have your expression</p>
<pre><code>expression =
x^2 (6 (2 c1 x^6 + c2)/x^4) - x (4 c1 x^3 - 2 c2/x^3) -
8 (c1 x^4 + c2/x^2)
(* -x (-((2 c2)/x^3) + 4 c1 x^3) - 8 (c2/x^2 + c1 x^4) + (
6 (c2 + 2 c1 x^6))/x^2 *)
</code></pre>
<p>for now it hasn't been evaluated to zero yet, but if you try most anything it ... |
985,103 | <p>The set $\{u_{1},u_{2}\cdots,u_{6}\}$
is a basis for a subspace $\mathcal{M}$ of $\mathbb{F}^{m}$ if and
only if $\{u_{1}+u_{2},u_{2}+u_{3}\cdots,u_{6}+u_{1}\}$
is also a basis for $\mathcal{M}$.
So far I have that the two basis are just rearranged sums of each other but don't know where else to go with it.</p>
| TheSilverDoe | 594,484 | <p>No need to write any <span class="math-container">$\varepsilon$</span> here. Let <span class="math-container">$L > 0$</span> such that <span class="math-container">$0 \leq b_n \leq L$</span> for every <span class="math-container">$n$</span>. Then
<span class="math-container">$$0 \leq a_n \leq L \times \frac{a_n}{... |
4,528,059 | <p>The graph of <span class="math-container">$y = f(x)$</span> is as follows:</p>
<p><a href="https://i.stack.imgur.com/vprAu.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/vprAu.png" alt="enter image description here" /></a></p>
<p>Find <span class="math-container">$$\int_{-1}^{1} f(1-x^2) dx$$</sp... | Mike Earnest | 177,399 | <p>Let <span class="math-container">$A$</span> be the matrix whose diagonal entries are <span class="math-container">$q_i-1$</span>, and whose off-diagonal entries are <span class="math-container">$-1$</span>. If you bet <span class="math-container">$x$</span>, then your possible winnings are described by the vector <s... |
1,142,624 | <p>Find a generating function for $\{a_n\}$ where $a_0=1$ and $a_n=a_{n-1} + n$</p>
| Community | -1 | <p>By definition of the generating function,</p>
<p>$$G(z)=\sum_{n=0}^\infty a_nz^n=a_0+\sum_{n=1}^\infty a_nz^n.$$
Applying the recurrence relation,
$$G(z)=a_0+\sum_{n=1}^\infty (a_{n-1}+n)z^n
=a_0+z\sum_{n=1}^{\infty}a_{n-1}z^{n-1}+\sum_{n=1}^\infty nz^n=a_0+zG(z)+\frac z{(1-z)^2}.$$</p>
<p>Solve for $G(z)$.</p>
|
2,431,027 | <p>I am asking about changing the limits of integration. </p>
<p>I have the following integral to evaluate - </p>
<p>$$\int_2^{3}\frac{1}{(x^2-1)^{\frac{3}{2}}}dx$$ using the substitution $x = sec \theta$. </p>
<p>The problem states</p>
<p><strong>Use the substitution to change the limits into the form $\int_a^b$ w... | vik1245 | 265,333 | <p>The book is indeed incorrect. </p>
<p>$$arccos(\frac{1}{3}) \neq \frac{\pi}{3}$$</p>
<p>Note if it was the case that $arccos(\frac{1}{3}) = \frac{\pi}{3}$ as stated in the solutions, </p>
<p>then the integral would total 0. </p>
|
1,424,072 | <p>Im trying to answer this question using the answer given in </p>
<p><a href="https://math.stackexchange.com/questions/795564/let-c-be-a-cube-and-let-g-be-its-rotational-symmetry-group-outline-a-proof-that">Let C be a cube and let G be its rotational symmetry group. Outline a proof that G is isomorphic to Sym(4)</a>... | Hagen von Eitzen | 39,174 | <p>The rotational group of the cube certainly turns a (here always: main) diagonal into a diagonal, hence acts on the set of the four diagonals. This action gives us a homomorphism $\phi\colon G\to \operatorname{Sym}(4)$. On the other hand, $G$ certainly has order $24$: We can pick one of six faces as "ground" face and... |
1,424,072 | <p>Im trying to answer this question using the answer given in </p>
<p><a href="https://math.stackexchange.com/questions/795564/let-c-be-a-cube-and-let-g-be-its-rotational-symmetry-group-outline-a-proof-that">Let C be a cube and let G be its rotational symmetry group. Outline a proof that G is isomorphic to Sym(4)</a>... | David Wheeler | 23,285 | <p>Using the orbit-stabilizer theorem: first of all, we can see that we have "some" homomorphism:</p>
<p>$G \to S_4$ by considering the fact that any such rotation of $G$ permutes the main diagonals.</p>
<p>Now let's consider the action of $G$ on the set of faces of the cube, we have $6$ of these. $G$ clearly always ... |
3,682,900 | <p>I have a hard time solving this one.
I'm sure there is trick that should be used but if so, I can't spot it.</p>
<p><span class="math-container">$$(3\cdot4^{-x+2}-48)\cdot(2^x-16)\leqslant0$$</span></p>
<p>Here is what I get but I'm anything but confident about this:</p>
<p><span class="math-container">$$3\cdot(2... | Mando | 734,443 | <p>To expound on the previous problem (I can't comment), the interval should be</p>
<p><span class="math-container">$$(-\infty,0]\cup[4,\infty)$$</span></p>
<p>For if </p>
<p><span class="math-container">$$(1-t)(t-16)\le0$$</span></p>
<p>then</p>
<p><span class="math-container">$$(t-1)(t-16)\ge0$$</span></p>
<p>S... |
1,349 | <p>In this question here the OP asks for hints for a problem rather than a full proof.</p>
<p><a href="https://math.stackexchange.com/questions/14477">Proof of subfactorial formula $!n = n!- \sum_{i=1}^{n} {{n} \choose {i}} \quad!(n-i)$</a></p>
<p>Now, while I would like to respect that request, I also feel that questi... | Zev Chonoles | 264 | <p>Here is a sample bit of LaTeX contained in a spoiler block:</p>
<blockquote class="spoiler">
<p> $\sum_{n=1}^\infty\frac{1}{n^2}=\frac{\pi^2}{6}$<br>
$$\sum_{n=1}^\infty\frac{1}{n^2}=\frac{\pi^2}{6}$$ </p>
</blockquote>
<p>This is the code I used to produce it:</p>
<pre><code> >! $\sum_{n=1}^\infty\frac{1}... |
540,135 | <p>$\newcommand{\lcm}{\operatorname{lcm}}$Is $\lcm(a,b,c)=\lcm(\lcm(a,b),c)$?</p>
<p>I managed to show thus far, that $a,b,c\mid\lcm(\lcm(a,b),c)$, yet I'm unable to prove, that $\lcm(\lcm(a,b),c)$ is the lowest such number... </p>
| Community | -1 | <p>The "universal property" of the <span class="math-container">$\newcommand{\lcm}{\operatorname{lcm}}\lcm$</span> is</p>
<blockquote>
<p>if <span class="math-container">$\lcm(a,b) \mid x$</span>, and <span class="math-container">$c \mid x$</span>, then <span class="math-container">$\lcm(\lcm(a,b),c) \mid x$</span>.... |
540,135 | <p>$\newcommand{\lcm}{\operatorname{lcm}}$Is $\lcm(a,b,c)=\lcm(\lcm(a,b),c)$?</p>
<p>I managed to show thus far, that $a,b,c\mid\lcm(\lcm(a,b),c)$, yet I'm unable to prove, that $\lcm(\lcm(a,b),c)$ is the lowest such number... </p>
| Lord_Farin | 43,351 | <p>With this type of problem, it's often useful to be aware of a particular order on <span class="math-container">$\Bbb N_{>0}$</span> -- at least if one is familiar with posets (partially ordered sets).</p>
<p>Namely, the poset <span class="math-container">$(\Bbb N_{>0}, \mid)$</span>, where <span class="math-c... |
572,125 | <p>How to show this function's discontinuity?<br></p>
<p>$ f(n) = \left\{
\begin{array}{l l}
\frac{xy}{x^2+y^2} & \quad , \quad(x,y)\neq(0,0)\\
0 & \quad , \quad(x,y)=(0,0)
\end{array} \right.$</p>
| Clive Newstead | 19,542 | <p>One of the definitions of a cardinal $\kappa$ being regular is that, whenever $\alpha < \kappa$, every function $f : \alpha \to \kappa$ is bounded.</p>
<p>In any case, you can prove this directly, using the fact that a countable union of sets of cardinality $\aleph_1$ has cardinality $\aleph_1$: consider $$\bigc... |
185,478 | <blockquote>
<p>How do I solve for $x$ in $$x\left(x^3+\sin(x)\cos(x)\right)-\big(\sin(x)\big)^2=0$$</p>
</blockquote>
<p>I hate when I find something that looks simple, that I should know how to do, but it holds me up. </p>
<p>I could come up with an approximate answer using Taylor's, but how do I solve this? </... | Ross Millikan | 1,827 | <p>Polynomials and trig functions don't play nice together, so you are usually stuck with numeric solutions. You can start by noting that $x=0$ is a double root, one from the outer $x$ and one from the $x/\sin x$ terms. </p>
|
185,478 | <blockquote>
<p>How do I solve for $x$ in $$x\left(x^3+\sin(x)\cos(x)\right)-\big(\sin(x)\big)^2=0$$</p>
</blockquote>
<p>I hate when I find something that looks simple, that I should know how to do, but it holds me up. </p>
<p>I could come up with an approximate answer using Taylor's, but how do I solve this? </... | N. S. | 9,176 | <p>We prove that $x=0$ is the only solution.</p>
<p>Let </p>
<p>$$f(x)= x^4+x \sin(x) \cos(x)- \sin^2(x) \,.$$</p>
<p>Then $f$ is even, so it is enough to look for roots on $[0, \infty)$.</p>
<p>You cana lso observe that $f(x)\geq x^4 -x-1$, and an easy calculation shows that for all
$x> \sqrt[3]{2}$ we have $x^... |
1,829,342 | <p>So I know that $\sum_{i\geq 0}{n \choose 2i}=2^{n-1}=\sum_{i\geq 0}{n \choose 2i-1}$. However, I need formulas for $\sum_{i\geq 0}i{n \choose 2i}$ and $\sum_{i\geq 0}i{n \choose 2i-1}$. Can anyone point me to a formula with proof for these two sums? My searches thus far have only turned up those first two sums wi... | Felix Marin | 85,343 | <p>$\newcommand{\angles}[1]{\left\langle\,{#1}\,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\half}{{1 ... |
2,609,283 | <p>$u_1 = (2, -1, 3)$ and $u_2 = (0, 0, 0)$</p>
<p>I tried using the cross product of the two but that just gave me the zero vector. I don't know any other methods to get a vector that is orthogonal to two vectors. </p>
<p>The answer is $v = s(1, 2, 0) + t(0, 3, 1)$ , where $s$ and $t$ are scalar values. </p>
| Devendra Singh Rana | 406,845 | <p>Consider any $n \times n$ rank-$1$ matrix $A$ as follows: one of its eigenvalues is its trace and the remaining eigenvalues are zero. Hence, the characteristic polynomial is $$x^{n-1}(x-\mbox{Tr}(A))$$ and the spectrum is $\{0,4\}$.</p>
|
285,548 | <p>I asked the following question on math.SE (<a href="https://math.stackexchange.com/questions/2420298/bvps-for-elliptic-pdos-when-do-green-functions-l2-inverses-define-pseudo-d">https://math.stackexchange.com/questions/2420298/bvps-for-elliptic-pdos-when-do-green-functions-l2-inverses-define-pseudo-d</a>) just over t... | Deane Yang | 613 | <p>Assuming that $P^{-1}$ is a right inverse and $\Omega$ an open subset of $\mathbb{R}^n$ or an open manifold, then you can proceed as follows:</p>
<p>1) An operator $Q: L^2(\Omega) \rightarrow L^2\Omega$ is a pseudodifferential operator, if and only if for any open domain $\Omega'\subset\Omega$, $Q$ restricted to $C... |
2,198,454 | <p>My professor's solution to this is as follows: "Create a 2 x 2 matrix with the first row (corresponding to $X=0$) summing to $P(X=0)=1-p$, the second row summing to $P(X=1)=p$, the first column ($Y=0$) summing to $P(Y=0)=1-q$ and the second column summing to $P(Y=1)=q$. We want to maximize the sum of the diagonal wh... | MR_BD | 195,683 | <p>Here is a simple solution:</p>
<p>Pick a random number <span class="math-container">$z$</span> uniformly in <span class="math-container">$[0,1]$</span></p>
<p>for <span class="math-container">$0 \le z<q$</span> set <span class="math-container">$X=Y=1$</span>;</p>
<p>for <span class="math-container">$q \le z<p$... |
2,560,556 | <p>Let $X,Y,Z$ be topological spaces.
Let $p:X\rightarrow Y$ be a continuous surjection. Let $f:Y\rightarrow Z$ be continuous if and only if $f\circ p:X\rightarrow Z$ is continuous.</p>
<p>I want to prove that this makes $p$ a quotient map. </p>
<p>My thoughts:</p>
<p>Since $p$ is a continuous surjection, all I need... | D_S | 28,556 | <p>This follows from univeral property arguments. If an object satisfies the same universal property as a quotient, product, coproduct etc. then it is that quotient, product, coproduct etc. You can give a much shorter proof of this without universal property arguments, but if you become familiar with such arguments, ... |
1,579,811 | <p>Find general solution for the differential equation $x^3y^{'''}+x^2y^{''}+3xy^{'}-8y=0$</p>
<p>This is the Euler differential equation which can be solved by substitution $x=e^t$. I don't understand the following differential relations:</p>
<p>$$xy^{'}=x\frac{dy}{dx}=\frac{dy}{dt}
$$
$$x^2y^{''}=x^2\frac{d^2y}{dx^... | Jan Eerland | 226,665 | <p>HINT:</p>
<p>$$x^3y'''(x)+x^2y''(x)+3xy'(x)-8y(x)=0\Longleftrightarrow$$
$$x^3\cdot\frac{\text{d}^3y(x)}{\text(d)x^3}+x^2\cdot\frac{\text{d}^2y(x)}{\text(d)x^2}+3x\cdot\frac{\text{d}y(x)}{\text(d)x}-8y(x)=0\Longleftrightarrow$$</p>
<hr>
<p>Assume the solution will be proportional to $x^{\lambda}$ for some constan... |
1,579,811 | <p>Find general solution for the differential equation $x^3y^{'''}+x^2y^{''}+3xy^{'}-8y=0$</p>
<p>This is the Euler differential equation which can be solved by substitution $x=e^t$. I don't understand the following differential relations:</p>
<p>$$xy^{'}=x\frac{dy}{dx}=\frac{dy}{dt}
$$
$$x^2y^{''}=x^2\frac{d^2y}{dx^... | user69468 | 225,040 | <p>It is called Cauchy's equation; what you did was correct. Set the operator $D: = \frac {d} {dt}$. Then form the auxiliary equation and solve it. The equation will be of the form $$(D^3-2D^2+4D-8)y=0,$$ where $x = e^t$. This gives $D = 2, \pm 2i$.</p>
|
2,539,888 | <p>I have an polynomial $x^4+x+1 \in \mathbb{Z}\left\{ x\right\}$ and I want to construct an extension field of $\mathbb{Z}_2$ that include the roots of that polynomial. So is this the right approach?</p>
<p>Let E be the extension field.
$$E= \mathbb{Z}_2 / <x^4+x+1> $$?</p>
<p>If so, how do I find the root of ... | lab bhattacharjee | 33,337 | <p>Hint:</p>
<p>$$2x^2-xy-y^2=2x(x-y)+y(x-y)=?$$</p>
|
2,924,380 | <p><span class="math-container">$\sum_{k=0}^{n}{k\binom{n}{k}}=n2^{n-1}$</span></p>
<p><span class="math-container">$n2^{n-1} = \frac{n}{2}2^{n} = \frac{n}{2}(1+1)^n = \frac{n}{2}\sum_{k=0}^{n}{\binom{n}{k}}$</span></p>
<p>That's all I got so far, I don't know how to proceed</p>
| Mark | 470,733 | <p>There is also a combinatorial proof. Imagine there is a set of <span class="math-container">$n$</span> elements and you want to choose a subset with one special element in it. (it can also be the only element of the subset). How can you do that? You can first choose the special element (<span class="math-container">... |
2,425,337 | <p>What would be an example of a real valued sequence $\{a_{n}\}_{n=1}^{\infty}$ such that $$\frac{a_{n}}{a_{n+1}} = 1 + \frac{1}{n} + \frac{p}{n \ln n} + O\left(\frac{1}{n \ln^{2}n}\right)\ ?$$</p>
| AlgorithmsX | 355,874 | <p>If you scale and flip almost any <a href="https://en.m.wikipedia.org/wiki/Cumulative_distribution_function" rel="nofollow noreferrer">Cumulative Distribution Function</a>, you should get exactly what you're looking for. The simplest CDF is from the <a href="https://en.m.wikipedia.org/wiki/Kumaraswamy_distribution" r... |
673,334 | <p>I used below pseudocode to generate a discrete normal distribution over 101 points.</p>
<pre><code>mean = 0;
stddev = 1;
lowerLimit = mean - 4*stddev;
upperLimit = mean + 4*stddev;
interval = (upperLimit-lowerLimit)/101;
for ( x = lowerLimit + 0.5*interval ; x < upperLimit; x = x + interval) { ... | Community | -1 | <p>You need to re-think how you are "discretizing" the normal distribution. You need to either: (1) partition the real line and set the probability of each discrete value to the probabiity of one of these intervals as calculated by the non-discrete version of the normal distribution. or (2) divide the "probabilties" f... |
3,956,467 | <p>Show that if <span class="math-container">$\alpha\in[0^\circ;45^\circ]:\sin(45^\circ+\alpha)=\cos(45^\circ-\alpha)$</span> and <span class="math-container">$\cos(45^\circ+\alpha)=\sin(45^\circ-\alpha).$</span></p>
<p>I tried to use the unit circle, but I am not sure how to draw the angles <span class="math-container... | Vishu | 751,311 | <p>Hint: <span class="math-container">$$\sin \theta = \cos(90^\circ -\theta) \\ \cos\theta =\sin(90^\circ -\theta)$$</span> The range of <span class="math-container">$\alpha$</span> is irrelevant, as this identity holds for all <span class="math-container">$\theta\in\mathbb R$</span>.</p>
|
3,956,467 | <p>Show that if <span class="math-container">$\alpha\in[0^\circ;45^\circ]:\sin(45^\circ+\alpha)=\cos(45^\circ-\alpha)$</span> and <span class="math-container">$\cos(45^\circ+\alpha)=\sin(45^\circ-\alpha).$</span></p>
<p>I tried to use the unit circle, but I am not sure how to draw the angles <span class="math-container... | Salmon Fish | 955,791 | <p><span class="math-container">$HINT$</span>:You should know that <span class="math-container">$\sin(x)\times \sin(y)=\frac{\cos(x+y)-\cos(x-y)}{-2}$</span> and <span class="math-container">$\sin(x)\times \cos(y)=\frac{\sin(x+y)+\sin(x-y)}{2}$</span></p>
<p>so, try to multiply and divide the former by <span class="ma... |
8,568 | <p>I'm going to be starting teaching a course called algebra COE, which is for students who didn't pass the required state algebra exam to graduate and are now seniors, to do spaced-out exam-like extended problems after extensive support. </p>
<p>I don't want to start the class out with "getting down to business" beca... | MissC | 5,544 | <p>What about the Monty Hall Problem?</p>
<p><a href="https://www.youtube.com/watch?v=mhlc7peGlGg" rel="nofollow">https://www.youtube.com/watch?v=mhlc7peGlGg</a></p>
<p>Not sure if you wanted an algebra related one tho. </p>
|
180,839 | <p>Is there any software which can be used for computing Thurston's unit ball (for second homology of 3-manifolds) of link complements? In particular can I do that with SnapPy?</p>
<p>PS: even a table for Thurston's ball of two component links would be helpful for me.</p>
| Sam Nead | 1,650 | <p>"Better late than never." Stephan Tillmann and William Worden have produced the software package <strong>tnorm</strong>. This can be found here:</p>
<p><a href="https://pypi.org/project/tnorm/" rel="noreferrer">https://pypi.org/project/tnorm/</a></p>
<p>The software should be able to deal with hyperbolic... |
673,385 | <p>Question:</p>
<blockquote>
<p>Determine the multiplicative inverse of $x^2 + 1$ in $GF(2^4)$ with $$m(x) = x^4 + x + 1.$$ </p>
</blockquote>
<p>My confusion is over the $GF (2^4)$.</p>
| robjohn | 13,854 | <p>As shown in <a href="https://math.stackexchange.com/a/86553">this answer</a>,
$$
\int_0^1t^{\alpha-1}\,(1-t)^{\beta-1}\,\mathrm{d}t
=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}
$$
and
$$
\int_0^\infty t^{\alpha-1}\,(1+t)^{-\beta}\,\mathrm{d}t
=\frac{\Gamma(\alpha)\Gamma(\beta-\alpha)}{\Gamma(\beta)}
$$
... |
9,930 | <p>One of the standard parts of homological algebra is "diagram chasing", or equivalent arguments with universal properties in abelian categories. Is there a rigorous theory of diagram chasing, and ideally also an algorithm?</p>
<p>To be precise about what I mean, a diagram is a directed graph $D$ whose vertices are ... | Daniel Litt | 6,950 | <p>Here's an attempt, assuming the target category is finite dimensional vector spaces and the diagram is finite acyclic.</p>
<p>1) Fill in the diagram so it's triangulated (that is, add the compositions of all arrows). Add kernels and cokernels to every arrow in the diagram. Add the natural arrows between these ne... |
9,930 | <p>One of the standard parts of homological algebra is "diagram chasing", or equivalent arguments with universal properties in abelian categories. Is there a rigorous theory of diagram chasing, and ideally also an algorithm?</p>
<p>To be precise about what I mean, a diagram is a directed graph $D$ whose vertices are ... | Greg Kuperberg | 1,450 | <p>This answer is a response to Daniel Litt's answer above. First, let me distill his point. Given a diagram of finite-dimensional vector spaces, every term $A$ has a dimension which is a non-negative integer. In addition, every morphism $f:A \to B$ has a non-negative rank which is at most $\min(\dim A,\dim B)$. If... |
114,438 | <p>I am interested in knowing whether there is a definition for the symbol of a PDO which is NOT linear.
In Wikipedia and in the book I am reading (An Introduction to Partial Differential Equations by Renardy-Rogers) I only found the definition for linear PDOs.</p>
<p>Here is the Wikipedia link:</p>
<p><a href="http:... | mdg | 64,184 | <p>See <a href="https://math.stackexchange.com/questions/589522/definition-of-the-principal-symbol-of-a-differential-operator-on-a-real-vector-b/883032#883032">Definition of the principal symbol of a differential operator on a real vector bundle.</a>.</p>
<p>For an example, consider the Ricci curvature operator:
\begi... |
3,255,644 | <p>I need help understanding the definitions and context for a homework question:</p>
<blockquote>
<p>Consider a 3 by 7 matrix A over GF(2) containing distinct columns. The row space C of A is the
subspace over GF(2) generated by the 3 rows. (Extra note: This is a “simplex” code [7,3] with generator
matrix A. It... | xxxxxxxxx | 252,194 | <p>When it says "having distinct columns", it most likely means "distinct nonzero columns" (this is the matrix commonly used in relation to the Hamming code). There are exactly 7 possible nonzero columns in <span class="math-container">$\mathbb{F}_{2}^{3}$</span>, so the matrix should be
<span class="math-container">$$... |
2,251,964 | <p><strong>question(s):</strong></p>
<p>Choose any real or complex clifford algebra $\mathcal{Cl}_{p,q}$. <a href="https://en.wikipedia.org/wiki/Classification_of_Clifford_algebras" rel="nofollow noreferrer">It's known</a> that there is some $A \simeq \mathcal{Cl}_{p,q}$, where $A$ is either a matrix ring $M(n,R)$ or... | benrg | 234,743 | <p>Here is the simplest way I know of getting matrix representations of even or full Clifford algebras of vector spaces over <span class="math-container">$\def\|#1{\mathbb#1}\|R$</span> or <span class="math-container">$\|C$</span> with arbitrary nondegenerate signature. There are just two recursive rules and a trivial ... |
181,940 | <p>I've been unable to find an answer to the following question in the literature
on generalized descriptive set theory. Consider Baire space $\kappa^{\kappa}$
where $\kappa$ is inaccessible. The basic open sets are the $U_f$ where
$f\in\kappa^{<\kappa}$. A perfect set is a nonempty closed set with no
isolated po... | Yair Hayut | 41,953 | <p>For every $\kappa$ of uncountable cofinality there is a tree $T\subseteq 2^{<\kappa}$ such that $[T]=\kappa$. The tree $T$ is the tree of all binary sequences $f\colon \alpha \to 2$, $\alpha <\kappa$ such that $f^{-1} (1)$ is finite. It is clear that for every $f \in T$, $f\frown (1), f\frown (0)$ are both in ... |
3,776,217 | <p>Prove that</p>
<p><span class="math-container">\begin{equation}
y(x) = \sqrt{\dfrac{3x}{2x + 3c}}
\end{equation}</span></p>
<p>is a solution of</p>
<p><span class="math-container">\begin{equation}
\dfrac{dy}{dx} + \dfrac{y}{2x} = -\frac{y^3}{3x}
\end{equation}</span></p>
<p>All the math to resolve this differential ... | Hari krishna Goli | 787,202 | <p>@kira1985 you should not add the p(x).y on both side, here you have to find the answer .</p>
<p>But you are using the answer it self to solve the problem .As the answer is unknown to have to just find the value of the P(x) and Q(x).</p>
<p>which is</p>
<p>/ = /2 − (^3)/3</p>
<p>P(x) = - 1/(2x)</p>
<p>Q(x) = - 1/(3x)... |
2,021,217 | <p>I tried to solve this limit
$$\lim_{x\to 1} \left(\frac{x}{x-1}-\frac{1}{\ln x}\right)$$
and, without thinking, I thought the result was 1. But, using wolfram to verify, I noticed that the limit is $1/2$. </p>
<p>How can I solve it without Hopital/series/integration, just with known limits (link in comments) /sque... | kotomord | 382,886 | <p>So, the overkill proof;</p>
<p>One of the equidistribution theorems:
<a href="https://en.wikipedia.org/wiki/Equidistribution_theorem" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Equidistribution_theorem</a></p>
<p>a is irrational, then sequence $a*p_n$ mod 1 is uniformly distributed.</p>
<p>Lemma:
... |
1,164,037 | <p>The question I propose is this: For an indexing set $I = \mathbb{N}$, or $I = \mathbb{Z}$, and some alphabet $A$, we can define a left shift $\sigma : A^{I} \to A^{I}$ by $\sigma(a_{k})_{k \in I} = (a_{k + 1})_{k \in I}$, because there exists a unique successor $\min \{ i > i_{0} \}$ for all $i \in I$. But one co... | Asaf Karagila | 622 | <p>There's really no issue here, given any <em>partial order</em> $P$, you can take the lexicographic product of $P$ with $\Bbb N$ or $\Bbb Z$, and obtain a partial order where each element has a unique immediate successor.</p>
<p>So there is no limit on the cardinality of such set. If, however, you want every two poi... |
3,246,240 | <p>I have the following problem</p>
<p><span class="math-container">$\frac{d^2y}{dx^2} + \lambda y = 0 , y'(0)=0$</span> and <span class="math-container">$y(3)=0$</span></p>
<p>I'm trying to solve for the eigenvalues <span class="math-container">$\lambda_n$</span> for <span class="math-container">$n=1,2,3...$</span>
... | Kavi Rama Murthy | 142,385 | <p>The Riemann Lebesgue Lemma tells you that <span class="math-container">$\hat {f}(x) \to 0$</span> as <span class="math-container">$x \to \pm \infty$</span>. Any continuous function with this property is uniformly continuous. </p>
|
2,596,700 | <p>We can see, for example, that the years 2009 and 2015 have identical calendars. Similarly, 2000 and 2028.</p>
<p>I read once that given any year X at most 28 years later there will be another year Y, with the calendar identical to that of X.</p>
<p>Here I am referring to our usual calendar, the Gregorian.</p>
<p>... | Ѕᴀᴀᴅ | 302,797 | <p>Since $y = \mathrm{e}^x$ is a continuous function and$$
\left(1 - \frac{1}{a_n}\right)^n = \exp\left(n \ln\left(1 - \frac{1}{a_n}\right)\right),
$$
then$$
\lim_{n \to \infty} \left(1 - \frac{1}{a_n}\right)^n \ \text{exists} \Longleftrightarrow \lim_{n \to \infty} n \ln\left(1 - \frac{1}{a_n}\right) \ \text{exists}.
... |
2,750,790 | <p>A real-valued function has to output a real number or a vector that is composed only of real numbers. </p>
<p>But can a scalar-valued function or vector-valued function output an imaginary number if it is not also labeled as real-valued?</p>
<p>For example, is this a scalar valued function?</p>
<p>$$
f:x \mapsto ... | Arnaud Mortier | 480,423 | <p><em>Scalar</em> means <em>with values in the base field</em>, whatever that field is. It depends on the context. The terminology is derived from the fact that when you multiply a set of vectors by a number, you <em>scale</em> the picture (you change the scale but not the overall shape). Hence numbers are "scalers".<... |
2,750,790 | <p>A real-valued function has to output a real number or a vector that is composed only of real numbers. </p>
<p>But can a scalar-valued function or vector-valued function output an imaginary number if it is not also labeled as real-valued?</p>
<p>For example, is this a scalar valued function?</p>
<p>$$
f:x \mapsto ... | giobrach | 332,594 | <p>A general truth in mathematics is that functions do <em><strong>not</strong></em> exist independently from their domain and codomain. This is a misconception generated by those pre-calc problems where they asked you to "find the domain" of a certain formula. So <span class="math-container">$f$</span> by it... |
1,380,819 | <p>Hi: I'm reading some introductory notes on hilbert spaces and there is a step in a proof that I don't follow. I will put the exact statement below. If someone could explain how it is obtained, it's appreciated. Note that commans between two terms when they have < and > around them denotes the innner product. Also... | ptrsinclair | 247,301 | <p>Note that the first inner product in the second-last line is:
$$ \left\langle x, x - \sum_{n=1}^m \langle x,e_n\rangle e_n\right\rangle = \langle x,x\rangle - \sum_{n=1}^m \langle x,e_n\rangle \langle x,e_n\rangle = \|x\|^2 - \sum_{n=1}^m |\langle x,e_n\rangle|^2 $$
so we would hope that the second is equal to zero.... |
4,642,566 | <p>For <span class="math-container">$x, y ∈$</span> <span class="math-container">$\mathbb{R}$</span>, let <span class="math-container">$x△y = 2(x + y)$</span>. Then <span class="math-container">$△$</span> is a binary operation on <span class="math-container">$\mathbb{R}$</span>.</p>
<p>Show that there is no identity el... | HeroZhang001 | 1,123,708 | <p>Suppose we have an identity <span class="math-container">$e$</span>. Then</p>
<p><span class="math-container">$$0 \triangle e =0\implies2(0+e)=0\implies e=0.$$</span></p>
<p>But</p>
<p><span class="math-container">$$2 \triangle e =2\implies2(2+e)=2\implies e=-1.$$</span></p>
<p>Thus <span class="math-container">$-1=... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.