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 |
|---|---|---|---|---|
2,454,663 | <blockquote>
<p>Calculate: $$\int_{-\infty}^{\infty}\frac{(x^4 + 1)^2}{x^{12} + 1}dx.$$</p>
</blockquote>
<p>What I have tried is to divide both numerator and denominator with $x^4 + 1$ and then get two following integrals, because of parity of the integrand (nothing else worked for me): $$2\int_0^\infty \frac{x^4 +... | David H | 55,051 | <hr>
<p><strong>Hint:</strong> For integrals of rational functions of even polynomials over the real line, the following substitution can come in handy:</p>
<p>$$\begin{align}
\mathcal{I}
&=\int_{-\infty}^{\infty}\mathrm{d}x\,\frac{\left(x^{4}+1\right)^{2}}{x^{12}+1}\\
&=\int_{-\infty}^{\infty}\mathrm{d}x\,\f... |
2,454,663 | <blockquote>
<p>Calculate: $$\int_{-\infty}^{\infty}\frac{(x^4 + 1)^2}{x^{12} + 1}dx.$$</p>
</blockquote>
<p>What I have tried is to divide both numerator and denominator with $x^4 + 1$ and then get two following integrals, because of parity of the integrand (nothing else worked for me): $$2\int_0^\infty \frac{x^4 +... | Przemo | 99,778 | <p>Any rational function can always be integrated by decomposing it into partial fractions. In this case this decomposition is particularly simple.
\begin{eqnarray}
\frac{1}{x^4-\sqrt{3} x^2+1} &=& \frac{1}{x^2-\theta_+} \cdot \frac{1}{\theta_+-\theta_-} + \frac{1}{x^2-\theta_-} \cdot \frac{1}{\theta_--\theta_+... |
1,431,556 | <p>I've seen a really interesting result, namely : </p>
<p>$$\underset{s \to 0}{\lim} \Gamma(s) - \frac{\Gamma(\frac{s}{2})}{2} =-\frac{\gamma}{2}$$</p>
<p>This was computed by Wolfram|Alpha, but can someone give some insight on how the $\gamma$ constant appears? What manipulations can one use in cases like this, whe... | Jack D'Aurizio | 44,121 | <p>$$L=\lim_{s\to 0^+}\left(\Gamma(s)-\frac{1}{2}\Gamma\left(\frac{s}{2}\right)\right)=\lim_{s\to 0^+}\frac{\Gamma(s+1)-\Gamma\left(\frac{s}{2}+1\right)}{s}\tag{1}$$
hence by applying De l'Hopital theorem and exploiting $\Gamma'(z)=\psi(z)\,\Gamma(z)$ we have:</p>
<p>$$ L = \psi(1)-\frac{1}{2}\psi\left(1\right)=\frac{... |
783,453 | <p>A set contains 2n+1 elements. Then, what is the number of subsets of the set which contain at most n elements ?</p>
<p>(A) $2n$ (B) $2^{n+1}$ (C) $2^{n−1}$ (D) $2^{2n}$</p>
<p>Any hints as to how to approach this problem ?</p>
| Community | -1 | <p>HINT: (Edited incorporating Carl's comments)</p>
<p>$\sin kx $ doesn't tend to $0$ as $k\rightarrow \infty$, for $x \notin \mathbb{Z} $.</p>
|
532,820 | <p>I've been stuck on this problem for a few hours now and haven't been able to make progress. The original problem is below with my work and progress beneath it. </p>
<blockquote>
<p>$$\frac{dy}{dx}=-\frac{y(x^3-y^3)}{x(2y^3-x^3)}$$</p>
</blockquote>
<p>First:<br>
Let $y=ux$ then $\frac{dy}{dx}=\frac{du}{dx}x + u... | bof | 97,206 | <p>I believe you lost the $+u$ term from $dy/dx=(du/dx)x+u$. If you do the algebra right, I believe you end up with a much easier integral. I get$$\frac{dx}x=\frac{1-2u^3}{u^4}du$$but don't take my word for it, I make lots of mistakes.</p>
<p><strong>P.S.</strong> I guess I got it right this time, because I see Amzoti... |
4,093,732 | <p>The equation of a curve is <span class="math-container">$$ y=8\sqrt x -2x $$</span>
We have to find the values of <span class="math-container">$x$</span> at which the line <span class="math-container">$y = 6$</span> meets the curve</p>
<p>I tried equating them and doing using the quadratic formula like this:
<span c... | user0102 | 322,814 | <p>Differently from the previous comments, I propose another way to approach it for the sake of curiosity.</p>
<p><span class="math-container">\begin{align*}
8\sqrt{x} - 2x = 6 & \Longleftrightarrow x - 4\sqrt{x} + 3 = 0\\\\
& \Longleftrightarrow (x - \sqrt{x}) - (3\sqrt{x} - 3) = 0\\\\
& \Longleftrightarro... |
283,987 | <p>We add $n$ random lines to the unit disc. We do so by adding two points on the disc, transcribing a line between them and extending that line to the boundary of the disc, namely the unit circle. Each line is chosen independently and we do this $n$ times. </p>
<p>After $n$ random lines have been added, what is the p... | Igor Rivin | 11,142 | <p>The distribution of the maximal distance between a pair of random points on the circle is known - when you scale it by $n/\log n$ you get a <a href="https://en.wikipedia.org/wiki/Gumbel_distribution" rel="nofollow noreferrer">Gumbel distribution</a> with scale 1, location 1., see, e.g., </p>
<p><em>Schlemm, Eckhard... |
2,392,494 | <blockquote>
<p>A matrix B is said to be a square root of a matrix A if <span class="math-container">$BB=A$</span></p>
<p>How many different square roots can you find of</p>
<p><span class="math-container">$$A=\begin{pmatrix} 5&0\\0&9\end{pmatrix}.$$</span></p>
</blockquote>
<p>When attempting to answer this qu... | Math Lover | 348,257 | <p>Hints: If $d+a \neq 0$ then $b=0$, $c=0$, $a^2=5$, and $d^2=9$. Compare the first and last equation. Can $d=-a$?</p>
|
4,229,521 | <blockquote>
<p>Let <span class="math-container">$A\in M_{n \times n}(\Bbb R)$</span> and suppose that for every <span class="math-container">$u, v \in \Bbb R^{n}$</span> <span class="math-container">$$(Av,Au) = (v,u)$$</span> where <span class="math-container">$(\cdot,\cdot)$</span> is the standard inner product on <s... | spinosarus123 | 958,184 | <p><span class="math-container">$e_i$</span> is an orthogonal basis. <span class="math-container">$a_i=Ae_i$</span> for each <span class="math-container">$i$</span>. <span class="math-container">$\delta_{i,j}$</span> is called the kroneckerdelta function and is simply <span class="math-container">$1$</span> if <span cl... |
1,426,155 | <p>The question is :</p>
<p>$$\lim_{x\rightarrow 1} \frac{\sqrt{x+3}-2}{x-1}$$</p>
<p>I know I probably have to do some sort of factorisation of the numerator in order to cancel the denominator, but the surd has me stumped I'm afraid.</p>
| juantheron | 14,311 | <p>Let $x+3 = a^2$ and $2 = b\;,$ Then $(x+3)-2^2 = a^2-b^2$</p>
<p>So $$\displaystyle \lim_{x\rightarrow 1} \frac{a-b}{a^2-b^2} = \lim_{x\rightarrow 1}\frac{1}{(a+b)} = \lim_{x\rightarrow 1}\frac{1}{\left(\sqrt{x+3}+2\right)} = \frac{1}{4}$$</p>
|
1,100,857 | <p>If I have a set $S$ with operation $*$ as a monoid. Would I say I have a monoid $S$ with the binary operation $*$ or would I say I have a monoid $(S,*)$ where the binary operation $*$ does blahblahblah?</p>
<hr>
<p>My book referred to $S$ as a monoid, which is strange considering $S$ is a set, and <strong>needs</s... | Learnmore | 294,365 | <p>A set can't have an algebraic structure unless you specify the binary operation given to it.So now you see what you should say</p>
<p>But in some cases authors neglect saying redundant facts.So once $(S,\circ)$ is assumed to be monoid they reduce it to only $S$ is a monoid</p>
|
821,125 | <p>I would like to evaluate the following:
$$\frac{\partial }{\partial \beta }\int _0^{\cos ^{-1}(\beta )}\text{dx} \sqrt{\beta +\cos (x)}$$</p>
<p>given that $0\leq\beta\leq1$ </p>
<p>basically I'd like to find the area under this curve:</p>
<p><img src="https://i.stack.imgur.com/DJ6Lg.png" alt="curve"> </p>
<... | André Nicolas | 6,312 | <p>For the "known function" part, let
$$f(s)=\sum_{k\ge 0} \frac{a^k s^{k+1}}{k!}.$$ Note that $f(s)=se^{as}$.
The sum
$$\sum_{k\ge 0} (k+1)\frac{a^ks^k}{k!}$$
is $f'(s)$. This is $(1+as)e^{as}$. </p>
|
1,524,879 | <p>Suppose that we have some function $f: \mathbb R \to \mathbb R$ such that $f$ is integrable (Riemann or Lebesgue, choose one, or some other maybe more general type of integration, if there is such) on some inteval $(a,b)$. </p>
<p>Now suppose that there exist point $x_0 \in (a,b)$ and sequence $\varepsilon_n$ such ... | Simon S | 21,495 | <p>No indeed. $A \Rightarrow \neg B$ is equivalent to $\neg(A \wedge \neg\neg B) = \neg(A \wedge B)$. Hence $\neg(A \Rightarrow \neg B)$ is $A \wedge B$.</p>
<p>For instance: if $A$ is $x = 0$ and $B$ is $x^2 = 0$ then $A \Rightarrow \neg B$ is false and $A\wedge B$ is true.</p>
|
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | J.J. Green | 5,734 | <p>Injective: Semigroups can be <a href="https://msp.org/pjm/1969/31-2/pjm-v31-n2-p11-p.pdf" rel="noreferrer">completely (right/left) injective</a>, while a Banach algebra is injective if the multiplication induces a continuous linear map of the injective tensor product $X\check{\otimes}X$ into $X$ (Varopoulos). I dis... |
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | Thamus Panmegas | 117,559 | <p>Polygons!</p>
<p>Is a polygon a sequence of vertices together with the edges that connect consecutive vertices? If so, can two distinct vertices be the same point? How about two consecutive vertices? Or a pair of vertices two indices apart: can we go from A, to B, and then directly back to A?</p>
<p>Or if not, ... |
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | Ari Brodsky | 27,343 | <p><strong>Linear functions</strong>:</p>
<p>In high-school algebra (sometimes called "pre-calculus"), we are taught that <em>linear functions</em> are those of the form $y=mx+b$, because they are graphed by a straight line in the plane.</p>
<p>Then we study linear algebra in university, and realize that for a functi... |
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | Nephry | 83,561 | <p>The distinction between compact and quasi-compact spaces is another one. If a topological space has the property that any open cover contains a finite subcover, then to some people this space is <em>compact</em>, to others it is <em>quasi-compact</em>. In the latter case, a compact space would then be a quasi-compac... |
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | Michael Bächtold | 745 | <p>"<strong>Function</strong>", prior to about 1910 <em>always</em> meant the $y$ in $y=f(x)$ (Look up any definition form that period). Since roughly 1920 it's officially the $f$. Physicists, engineers and many applied mathematicians still mean $y$ when they talk about functions today. But, since the hijacking of the ... |
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | Fedor Petrov | 4,312 | <p>Is a parallelogram also a trapezoid?</p>
|
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | Robert Furber | 61,785 | <p>In a real vector space $E$, some people use <em>cone</em> to mean a subset $C \subseteq E$ closed under multiplication by positive reals. If it is additionally closed under addition, they call it a <em>convex cone</em>, and if $C \cap -C = \{0\}$, they say it is a <em>proper cone</em>, or that $C$ is <em>proper</em>... |
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | Watson Ladd | 6,084 | <p>Variety has a number of slightly different definitions. Apparently some authors use reduced of finite type over a field, whereas I would want separated to rule out the line with two origins. Most people use scheme-theoretic language to solve this problem.</p>
<p>Conway-Sloane jokes the discriminant of a quadratic f... |
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | Caleb Stanford | 32,499 | <p><strong>The definition of a Turing Machine</strong> is a great example, where multiplied all together there are at least hundreds of possible definitions.</p>
<ul>
<li><p>Is the tape doubly infinite or singly infinite? (If singly infinite, what happens when you move right at the right at the end of the tape?)</p></... |
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | AFarr | 117,809 | <p>In graph theory, there are at least two different meanings of the word "hereditary".</p>
<p>Some definitions first.
Let G be a graph.
If graph H is obtained by deleting 0 or more vertices from G, then H is an induced subgraph of G.
If graph K is obtained by deleting 0 or more vertices, and deleting 0 or more edges... |
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | liuyao | 1,189 | <p>Singular support of a sheaf seems to be a subset of the cotangent bundle, whereas the singular support of a distribution is a subset of the base space. The former is more like the wavefront set, as far as I can intuit.</p>
<p>People try to avoid the confusion by denoting the former by S.S., and the latter by sing s... |
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | Tobias Fritz | 27,013 | <p>Everybody agrees that an <strong><em>isometry</em></strong> is a distance-preserving map. In the context of functional analysis and in particular operator algebras, this is indeed the definition. But in geometry, an isometry is usually required to be bijective, leading e.g. to the <em>isometry group</em> of a metric... |
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | Desiderius Severus | 43,737 | <p>The wide range of choices in the definition of an <strong>automorphic form</strong> is particularly annoying. Depending on the purposes, it could be a meromorphic function fully invariant by a certain discrete group of transformation, a holomorphic function almost-invariant, a differential form, a subrepresentation ... |
64,414 | <p>Show that every prime $p>3$ is either of the form $6n+1$ or of the form $6n+5$, where $n=0,1,2, \dots$</p>
| Dan Brumleve | 1,284 | <p>$6$ divides $6n$, $2$ divides $6n+2$, $3$ divides $6n+3$, $2$ divides $6n+4$, and there are no other cases.</p>
|
64,414 | <p>Show that every prime $p>3$ is either of the form $6n+1$ or of the form $6n+5$, where $n=0,1,2, \dots$</p>
| Stat-R | 26,206 | <p>Copied from <a href="http://primes.utm.edu/notes/faq/six.html" rel="nofollow noreferrer">Are all primes (past 2 and 3) of the forms 6n+1 and 6n-1?</a></p>
<blockquote>
<p>[Considering] $n = 6q + r$</p>
<p>where q is a non-negative integer and the remainder $r$ is one of $0, 1, 2, 3, 4$, or $5$.</p>
<ul>... |
714,259 | <p>How would I solve the following quadratic equation </p>
<p>$$x^2+3x-70=0 $$</p>
<p>This is my attempt below</p>
<p>$$(x-7x) (x+10x)=0 $$</p>
<p>$$ x-7x=0 \implies -6x=0 \implies x=6$$</p>
<p>$$x+10x=0 \implies 11x=0 \implies x=-11$$ </p>
<p>Am I correct in thinking that the two... | amWhy | 9,003 | <p>Your factoring is a bit off:</p>
<p>$$\begin{align}x^2 + 3x-70 = (x - 7)(x+10) = 0 & \iff (x - 7) = 0 \; \text{or}\; (x + 10) = 0 \\ \\ &\iff x = 7 \;\text{or} \;x = -10\end{align}$$</p>
<p>Instead of the above factors, you wrote $$(x - 7x)(x + 10x) = (-6x)(11x) = -66x^2 \neq x^2 + 3x - 70$$</p>
|
250,940 | <p>i found this interesting question on the web but i am not quite sure if my solution is accurate. Honestly i would appreciate few opinions.</p>
<p>Given Question:</p>
<blockquote>
<p>At a subway station, eastbound trains and northbound trains arrive
independently, both according to a Poisson process. On average... | James Fennell | 21,459 | <p>The idea is that the two functions $f(x)=e^x$ and $g(x) = kx$ touch only once; that is, $g$ is tangent to $f$ at some point.</p>
<p>Denote the tangent point $x_0$. Then $f(x_0)=e^{x_0}=kx_0=g(x_0)$. The slopes of the two functions have to be the same at this point as they are tangent:
$$f'(x_0) = g'(x_0)$$
$$e^{x_0... |
301,778 | <p>I am not a mathematician (I am an engineer who is working on improving his mathematics), so I apologize in advance if my question is trivial.</p>
<p>Consider a graph of $N$ nodes, with some defined criterion as to whether two nodes are connected or not. What are some measures of the graph's connectedness? I can thi... | Avi Steiner | 13,487 | <p>The most common measures of connectivity are edge-connectivity and vertex-connectivity. The <strong>vertex-connectivity</strong>, or just <strong>connectivity</strong>, of a graph is the minimum number of vertices you have to remove before you can even hope to disconnect the graph. A graph is called <strong><span cl... |
255,810 | <p>Let $A$ be an $n\times n$ complex matrix, and write $A=X+iY$, where $X$ and $Y$ are real $n\times n$ matricies. Suppose that for every square submatrix $S$ of $A$, $|\mathrm{det}(S)|\leq 1$ (i.e., all minors of $A$ are complex numbers with modulus $\leq 1$). This includes the assumption that $|\mathrm{det}(A)|\leq... | Fedor Petrov | 4,312 | <p>Bound 1 does not hold already for $n=2$. Take a matrix $A=\pmatrix{e^{ia}&e^{ib}\\e^{ic}&e^{id}}$, it satisfies your conditions if $|a+d-b-c|\leqslant \pi/3$. On the other hand, $X=\pmatrix{\cos a&\cos b\\\cos c&\cos d}$ and $$\det X=\cos a\cos d-\cos b\cos c=\frac12\left(\cos(a+d)+\cos(a-d)-\cos(b+c... |
800,125 | <p>prove or find a counterexample: The equation $x^n + y^n = z^n$, where $n$ is a natural number, has no solutions at all where $x, y,z$ are integer.</p>
<p>counterexample: if $n=3$ and $x=1$ and $y=2$ and $z=3$ then
$$1^3 + 2^3 = 3^3$$
$$1 + 8 = 27$$
$9$ is not equal to $27$</p>
<p>Is this right??need help...</p>
| Hakim | 85,969 | <p>The obvious counterexample is: $X=Y=Z=0$. And your approach isn't correct since I could argue that you only proved that $(1,2,3)$ for $n=3$ is not a solution and that there may exist some solution. A counter example would not look for examples in accordance with the original statement, but ones that contradict it, l... |
4,546,654 | <p>Is there any mathematical notation to expand a list/set of values?</p>
<p>I am looking for something that would expand like this:</p>
<p><span class="math-container">$A=(x=0,x<=3)[2x+1]$</span></p>
<p><span class="math-container">$A=[2(0)+1,2(1)+1,2(2)+1,2(3)+1]$</span></p>
<p><span class="math-container">$A=[1,3... | Jam | 161,490 | <p>Yes, the counterpart of sigma or pi notation for sequences is typically notated as <span class="math-container">$\left(a_n\right)_{n\in S}$</span> where <span class="math-container">$a_n$</span> expresses the <span class="math-container">$n$</span>'th term and <span class="math-container">$S$</span> is an index set.... |
2,580,638 | <p>Let $f(x,y) = 4x^2 -3xy -2y + 1 $ . Find $(x , y)$ so that $x , y \in \mathbb{Z} $ and $f(x,y) = 0 $ . I've tried many numbers and got some answers but it's not the solution ! </p>
| SchrodingersCat | 278,967 | <p>$$f(x,y)=0$$
$$\Rightarrow 4x^2-3xy-2y+1=0$$
$$\Rightarrow 4x^2-3xy-y^2+y^2-2y+1=0$$
$$\Rightarrow 4x^2-4xy+xy-y^2+(y-1)^2=0$$
$$\Rightarrow (4x-y)(x-y)+(y-1)^2=0$$
$$\Rightarrow (4x-y)(x-y)+(y-1)^2=0$$</p>
<p>Since $(y-1)^2 \ge 0$, so we can conclude that $(4x-y)(x-y)$ must be negative for $f(x,y)=0$ to hold.</p>
... |
1,612,348 | <p>Are limit points of a sequence and an interval differently defined? I ask because I've been given the following definition of a limit point:</p>
<blockquote>
<p>A limit point is one such that every of its neighborhoods contains infinitely many points of the set other than itself.</p>
</blockquote>
<p>If that is ... | user160110 | 160,110 | <p>A limit point of a sequence $(a_n)_{n\to \infty}$ is defined as the point the sequence itself gets close. An interesting example of this is the sequence $(1)_{n\to \infty}$ approaches $1$. </p>
<p>While the limit point of a set is seen as a point in which every neighborhood of that point (the limit-point) contains ... |
509,349 | <p>The problem I am having is figuring out the way show the following sequence is monotone:</p>
<p>let $x_1 = \frac{3}{2}$ and $x_{n+1} = {x_n}^2-2x_n+2$, show that the sequence $x_n$ is monotone and bounded and find the limit.</p>
<p>I have found the first three terms, and found that the sequence is decreasing, I ha... | Community | -1 | <p>Let $f(x) = x^2 - 2x + 2$. Then it is easy to show that $f([1,2]) \subset [1,2]$. Since $x_1 \in [1,2]$, this implies that
$$x_n = f(x_{n-1}) = (f \circ f)(x_{n-2}) = ... = (f\circ ... \circ f)(x_1) \in [1,2]$$
so $\{x_n\}$ stays inside $[1,2]$ and, in particular, bounded. Now when $1 \le x \le 2$ we have
$$f(x) - x... |
505,059 | <p>I have a question:</p>
<p>If I have to find the commutator $[x, p^2]$ (with $p= {h\over i}{d \over dx} $) the right answer is:</p>
<p>$[x,p^2]=x p^2 - p^2x = x p^2 -pxp + pxp - p^2x = [x,p]p + p[x,p] = 2hip$</p>
<p>But why can't I say:</p>
<p>$[x,p^2]=x p^2 - p^2x = - x h^2{d^2 \over dx^2} + h^2 {d^2 \over dx^2}... | jinawee | 73,833 | <p>Another way to calculate that commutator using $[x,p]=i\hbar$ and introducing intermediate commutators:</p>
<p>$$\begin{align}
[x,p^2]
& =xp^2-px^2=xp^2-ppx+pxp-pxp\\
&=xp^2-pxp+p(xp-px)\\
& =xp^2-pxp+p[x,p]\\
& =xp^2 -pxp+i\hbar p\\
&= xp^2-pxp+xpp-xpp+i\hbar p\\
&=xp^2-xp^2+(xp-... |
505,059 | <p>I have a question:</p>
<p>If I have to find the commutator $[x, p^2]$ (with $p= {h\over i}{d \over dx} $) the right answer is:</p>
<p>$[x,p^2]=x p^2 - p^2x = x p^2 -pxp + pxp - p^2x = [x,p]p + p[x,p] = 2hip$</p>
<p>But why can't I say:</p>
<p>$[x,p^2]=x p^2 - p^2x = - x h^2{d^2 \over dx^2} + h^2 {d^2 \over dx^2}... | shark's pacifier | 449,427 | <p>You can calculate the commutator for 3 different operators, say $\hat{A}$, $\hat{B}$ and $\hat{C}$: $ [\hat{A},\hat{B }\hat{C}]$. Defining the following operator, $\hat{D}\equiv\hat{B}\hat{C}$, so, the original commutator turns out to be:
$$[\hat{A},\hat{D}]=\hat{A}\hat{D}-\hat{D}\hat{A}$$
Coming back to $\hat{B}\ha... |
543,920 | <p>Here is another question from the book of V. Rohatgi and A. Saleh. I would like to ask help again. Here it goes:</p>
<p>Let $\mathcal{A}$ be a class of subsets of $\mathbb{R}$ which generates $\mathcal{B}$. Show that $X$ is an RV on $\Omega\;$ if and only if $X^{-1}(A)$ $\in \mathbb{R}$ for all $A\in \mathcal{A}$.<... | Stefan Hansen | 25,632 | <p>Let $(\Omega,\mathcal{F},P)$ be a probability space and let $X:\Omega\to\mathbb{R}$ be a mapping from $\Omega$ to $\mathbb{R}$. By definition, $X$ is a random variable if $X^{-1}(A)\in\mathcal{F}$ for all $A\in\mathcal{B}$, where $\mathcal{B}$ denotes the Borel sets on $\mathbb{R}$. </p>
<p>Clearly, if $X$ is a ran... |
3,292,629 | <p>My answer of total ways is <span class="math-container">$2^4\dfrac{9\times10}{2}=720$</span> but <span class="math-container">$0$</span> should not be in the first place, number of their ways is <span class="math-container">$9\times2^3=72$</span>. Required answer is <span class="math-container">$720-72=648$</span>. ... | 19aksh | 668,124 | <p>You've substituted <span class="math-container">$\color{red}{u = x^{1/3}} \implies du = \frac{1}{3}x^{-2/3}dx \implies dx = 3x^{2/3}du \implies \color{red}{dx = 3u^2du}$</span></p>
<p>Also, <span class="math-container">$\color{red}{x = u^3}$</span></p>
<p>Hence
<span class="math-container">$$\int\frac{\color{blue}... |
568,923 | <p>what is the name of the curve made up of the points $(x,x^2...x^n)$ in $\mathbb {R}^n$ for all $x\in \mathbb R$??</p>
<p>For example: in $\mathbb R^2$ it would just be a parabola.</p>
| Ross Millikan | 1,827 | <p>Have you factored $2401=7^4?$ That is a big clue. You have $7$ choices for $f(a)$, then $7$ choices for $f(b)$ and .... As the choices are independent, you multiply them.</p>
|
568,923 | <p>what is the name of the curve made up of the points $(x,x^2...x^n)$ in $\mathbb {R}^n$ for all $x\in \mathbb R$??</p>
<p>For example: in $\mathbb R^2$ it would just be a parabola.</p>
| john | 79,781 | <p>Each of the four elements of $S$ should be mapped to one of the seven elements of $T$. There is no reason why more than one element of $S$ can't map to the same element of $T$. Hence the total number of ways of doing this is $7^4=2401$.</p>
|
3,464,201 | <p>Let <span class="math-container">$(\Omega, \mathscr{F}, P)$</span> be a measure space with filtration <span class="math-container">$\{\mathscr{F}_n\}_{n \in N}$</span>. Let <span class="math-container">$A_n$</span> be adapted to the filtration <span class="math-container">$\mathscr{F}_{n}$</span>, and <span class="m... | WoolierThanThou | 686,397 | <p>We can't, because the statement is not true.</p>
<p>Assume, that our probability space is <span class="math-container">$[0,1]$</span> with its Borel algebra <span class="math-container">$\mathcal{B}$</span> and the Lesbegue measure <span class="math-container">$m$</span>. Let <span class="math-container">$\mathcal{... |
2,914,976 | <p>Can someone explain to me why $$ \lim\limits_{x \to \infty} x\bigg(\frac{1}{2}\bigg)^x = 0$$
Is it because the $\big(\frac{1}{2} \big)^x$ goes towards zero as $ x $ approaches $\infty$, and anything multiplied by $0 $ included $\infty$ is $0$ ?</p>
<p>Or does this kind of question require using l'hopital's rule be... | maveric | 590,250 | <p>exponential growth is faster than linear growth. hence 0</p>
|
3,952,488 | <p>Let <span class="math-container">$V=\begin{cases}\begin{bmatrix}x\\y\end{bmatrix}:x\in \mathbb{R}^+, y\in \mathbb{R}\end{cases}\bigg\}.$</span>
Then it can be proved that under the operations <span class="math-container">$$\alpha \cdot \begin{bmatrix}x \\ y\end{bmatrix}=\begin{bmatrix}x^{\alpha}\\ \alpha y\end{bmat... | user126154 | 126,154 | <p>The function <span class="math-container">$f(x,y)=(\log x,y)$</span> is an isomorphism between <span class="math-container">$V$</span> and <span class="math-container">$\mathbb R^2$</span>. So a basis is formed by <span class="math-container">$v_1=f^{-1}(1,0)=(e,0)$</span> and <span class="math-container">$v_2=f^{-1... |
2,222,572 | <p>Does there exist a continuous function $f:[0,1]\rightarrow[0,\infty)$ such that $$\int_0^1 \! x^{n}f(x) \, \mathrm{d}x=1$$ for all $n\geq1$?</p>
| NCh | 413,376 | <p>Let me take a look at this question from the point of view of Probability Theory. </p>
<p>We want $\int_0^1 \! x^{n}f(x) \, dx=1$ for any $n\geq 1$ and the function $f(x)$ is assumed to be nonnegative. Take $n=1$ and denote $xf(x)$ by $g(x)$:
$$\int_0^1 \! g(x) \, dx=1, \quad g(x)\geq 0 \text{ for any } x\in[0,1].$... |
1,298,938 | <p>Let $X$ be a CW-complex. Let $\Sigma$ be suspension. Let $R$ be a commutative ring. Is the cup product of
$$
H^*(\Sigma X;R)$$
trivial? How to prove? Where can I find the result?</p>
| Alex Fok | 223,498 | <p>Yes. Actually there is a more general result. If $X=\bigcup_{i=1}^n A_i$ where each subspace $A_i$ is contractible, then the product of any $n$ elements in $H^*(X)$ vanishes. Since $\Sigma X$ is a union of two cones, each of which is contractible, the result follows. </p>
<p>To establish the general result, simply ... |
187,234 | <p>I have the list,</p>
<pre><code>listx = {{0., -0.6}, {0.08, -1.}, {0.16, -0.9}, {0.24, 1.}, {0.32, 0.6}}
</code></pre>
<p>And I want to check if every second element satisfies both conditions (<span class="math-container">$ >= -1$</span> && <span class="math-container">$<= 1$</span>).</p>
<p>I have ... | Thies Heidecke | 47 | <p>I think <code>Thread</code> allows for a really nice syntax here:</p>
<pre><code>If[And @@ Thread[-1 <= listx[[All,2]] <= 1], Print["a"]]
</code></pre>
|
3,126,162 | <p>The goal is to show that <span class="math-container">$$\left(\frac{1}{3}\right)^kn=1 \Rightarrow k = \log_3 n\,.$$</span></p>
<p>So I started with <span class="math-container">$\left(\frac{1}{3}\right)^kn=1 \Leftrightarrow \left(\frac{1}{3}\right)^k=\frac{1}{n}$</span> in order to use the identity <span class="mat... | Haris Gušić | 450,231 | <p>Note that
<span class="math-container">$$-\log_{1/3} n = \frac{\log_{1/3} n}{\log_{1/3}3} = \log_3 n$$</span></p>
|
1,366,206 | <p>I'm self-learning Real Analysis using Real Analysis of Folland, and I got stuck on this problem.</p>
<blockquote>
<blockquote>
<p>Let $(X, \mathcal{M}, \mu)$ be a measure space with $\mu(X) < \infty$, and let $(X, \overline{\mathcal{M}}, \overline{\mu})$ be its completion. Suppose $f: X \rightarrow \mathbb... | John Samples | 172,599 | <p>Since the solution to this appears to be nowhere on the internet - a bit shocking given the ubiquity of the textbook and the (fairly) elementary nature of the problem in the context of Real Analysis - I'll post the whole thing. Hopefully no intractable errors.</p>
<p>Suppose $f$ is $\bar{\mathcal{M}}$-measurable a... |
125,324 | <p>When I enter <code>f + 0</code> where <code>f</code> is an undefined symbol, I get <code>f</code>. So far so good. When I enter <code>f + 0.</code>, I get <code>f + 0.</code>. How can I get Mathematica to simplify 0. just like it does for 0? I tried using <code>Simplify</code>, I got the same result. </p>
| Bruno Le Floch | 39,260 | <p>As @BlacKow pointed out <code>Pi+0.</code> differs from <code>Pi</code> so Mathematica is right not to remove the <code>+0.</code>. One option is to use a replacement rule</p>
<pre><code>f+0. /. {x_+0.:>x}
</code></pre>
|
125,324 | <p>When I enter <code>f + 0</code> where <code>f</code> is an undefined symbol, I get <code>f</code>. So far so good. When I enter <code>f + 0.</code>, I get <code>f + 0.</code>. How can I get Mathematica to simplify 0. just like it does for 0? I tried using <code>Simplify</code>, I got the same result. </p>
| andre314 | 5,467 | <p>From the doc :</p>
<blockquote>
<ul>
<li><p>Chop[expr,delta] replaces numbers smaller in absolute magnitude than
delta by 0. </p></li>
<li><p>Chop uses a default tolerance of 10^-10. </p></li>
</ul>
</blockquote>
<p>Accordingy :</p>
<pre><code>Chop[f + 1. 10^-20]
</code></pre>
<p>gives :</p>
<block... |
109,003 | <p>Let $m,n$ be integers. I want to find the possible values of $m,n$ such that $4(m+n)\over (2m+n)^2+3n^2$ and $4n\over (2m+n)^2+3n^2$ are both integers too. Would someone please help? Of course letting $(2m+n)^2+3n^2=4$ gives some good values, but is this all the $m,n$ I can get? </p>
<p>Added: I can see that the pr... | Math Gems | 75,092 | <p>Hint: put $k = m^2+mn+n^2$. Reducing fractions shows $k\ |\ m+n,n\ \Rightarrow \ k\ |\ (m+n,n) = (m,n)$. </p>
<p>But $\:(m,n)^2\:|\: k\ |\ (m,n)$ so $(m,n) = 1,\ k = \pm 1$. </p>
|
4,184,845 | <p>Let <span class="math-container">$G$</span> be a group and let <span class="math-container">$H \trianglelefteq G$</span>.</p>
<p>Prove that</p>
<p><span class="math-container">$G/H$</span> is a simple group <span class="math-container">$\Longrightarrow$</span> <span class="math-container">$G\neq H$</span> and there ... | John Douma | 69,810 | <p><span class="math-container">$A\cap B = (A\cap B)\cup \phi =(A\cap B)\cup (A\cap\lnot A)=A\cap(\lnot A\cup B)=A\cap\lnot(A\cap\lnot B)=A-(A-B)$</span></p>
|
4,184,845 | <p>Let <span class="math-container">$G$</span> be a group and let <span class="math-container">$H \trianglelefteq G$</span>.</p>
<p>Prove that</p>
<p><span class="math-container">$G/H$</span> is a simple group <span class="math-container">$\Longrightarrow$</span> <span class="math-container">$G\neq H$</span> and there ... | ultralegend5385 | 818,304 | <p>Let <span class="math-container">$x\in A\cap B$</span>. Then, <span class="math-container">$x\in A$</span> and <span class="math-container">$x\in B$</span>. This implies <span class="math-container">$x\notin A-B$</span> (think why?). Now, <span class="math-container">$x\in A$</span> and <span class="math-container">... |
4,184,845 | <p>Let <span class="math-container">$G$</span> be a group and let <span class="math-container">$H \trianglelefteq G$</span>.</p>
<p>Prove that</p>
<p><span class="math-container">$G/H$</span> is a simple group <span class="math-container">$\Longrightarrow$</span> <span class="math-container">$G\neq H$</span> and there ... | Graham Kemp | 135,106 | <p><span class="math-container">$A\cap B$</span> contains all elements of <span class="math-container">$A$</span> except for those in <span class="math-container">$A$</span> that are not elements of <span class="math-container">$B$</span>.</p>
<p> <span class="math-container">$\ldots$</span> Thus <span class="math-c... |
2,908,791 | <p>Given integers $x_{0},n$ with $x_{0}^{2}\equiv -1$ (mod $n$) then there are integers $y,b$ with $(y,b)=1,0<b\le\sqrt{n}$ and </p>
<p>$$\left|-\frac{x_{0}}{n}-\frac{y}{b}\right|<\frac{1}{b\sqrt{n}}$$</p>
<p>I tried to solve $|bx_{0}+ny|<\sqrt{n}$ for some integers $y,b$, but it seems do nothing.</p>
<p>Is... | Michael Rozenberg | 190,319 | <p>For $a=2$, $b=0$ and $c=2$ we'll get a value $4$.</p>
<p>We'll prove that it's a maximal value.</p>
<p>Indeed, we need to prove that
$$ab+ac+bc\leq4\left(\frac{a+2b+c}{4}\right)^2$$ or
$$(a-c)^2+4b^2\geq0,$$ which is obvious.</p>
|
1,603,915 | <p>Is my proof for the question in the title correct?</p>
<p>Note that $\mathbb{R} - \{0\} \subset \mathbb{C} - \{0\}$, since every real number is a complex number. Therefore, since any $\phi: \mathbb{R} - \{0\} \to \mathbb{C} - \{0\}$ would map from a set to another set of bigger cardinality, no such $\phi$ could be ... | Mikasa | 8,581 | <p>Generally, $$t(R^*,\cdot)=\{+1,-1\}\neq t(C^*,\cdot)=\{\exp(2ki\pi/n)|k,n\in \mathbb{Z}\}$$ $t(G)$ means the torsion subgroup.</p>
|
96,840 | <p>I want to show the convergence of the following improper integral $\int_0^\infty e^{-x^2}dx$.
I try to use comparison test for integrals
$x≥0$, $-x ≥0$, $-x^2≥0$ then $e^{-x^2}≤1$. So am ending with the fact that $\int_0^\infty e^{-x^2}dx$ converges if $\int_0^\infty dx$ converges but I don’t appreciate this. Thank... | mengdie1982 | 560,634 | <p>Denote
<span class="math-container">$$f(x):=\left(\int_0^{x} e^{-t^2}{\rm d}t\right)^2,~~~g(x):=\int_0^1 \frac{e^{-x^2(1+t^2)}}{1+t^2}{\rm d}t.$$</span>
Then
<span class="math-container">$$f'(x)=2\int_0^{x} e^{-t^2}{\rm d}t\left(\int_0^{x} e^{-t^2}{\rm d}t\right)'=2e^{-x^2}\int_0^{x} e^{-t^2}{\rm d}t,$$</span>
and
<... |
2,581,714 | <p>I have been trying to solve this <a href="https://math.stackexchange.com/q/2581210/144766">recent linear algebra problem</a>:</p>
<blockquote>
<p>Let $A, B$ be $3 \times 3$ matrices such that $(A-B)^2 = 0$. Prove that $\det (AB - BA) = 0$.</p>
</blockquote>
<p>This was my approach:$\DeclareMathOperator{\Tr}{Tr}$... | badjohn | 332,763 | <p>It's unusual and hence clarification would be advisable to avoid misinterpretation. With appropriate clarification then it would be acceptable. You can even invent your own notation provided that you define it. However, when clearer alternatives are easily available, what's the point? </p>
|
2,581,714 | <p>I have been trying to solve this <a href="https://math.stackexchange.com/q/2581210/144766">recent linear algebra problem</a>:</p>
<blockquote>
<p>Let $A, B$ be $3 \times 3$ matrices such that $(A-B)^2 = 0$. Prove that $\det (AB - BA) = 0$.</p>
</blockquote>
<p>This was my approach:$\DeclareMathOperator{\Tr}{Tr}$... | Dr. Sonnhard Graubner | 175,066 | <p>it can be simplified to $$\frac{1}{(10^2)^{1/100}}=\frac{1}{10^{1/50}}$$</p>
|
2,338,819 | <p>I have been stuck on the following question for some time now and would appreciate if someone can provide some guidance on this matter. The question is as follows:</p>
<p>Use predicate logic reasoning techniques to solve the following problem:
All academics who are computer scientists are programmers or mathematici... | enigma | 420,042 | <p>I'm not sure about this statement as being a if Jack is a logistician then he is a philosopher. It seems like you can derive a "reductio ad absurdam" (or contradiction) and get whatever you need as from a contradiction anything follows.</p>
<p>Not sure about this statement: "Any logistician is a philosopher. Jack J... |
2,338,819 | <p>I have been stuck on the following question for some time now and would appreciate if someone can provide some guidance on this matter. The question is as follows:</p>
<p>Use predicate logic reasoning techniques to solve the following problem:
All academics who are computer scientists are programmers or mathematici... | Bram28 | 256,001 | <p>We are told that Jack Jones is not a philosopher, so since all logisiticians are philosophers, it follows that Jack cannot be a logistician either. With that, any statement of the form 'If Jack is a logistician, then X' automatically becomes true, no matter what you fill in for X. That is, you can prove that If Ja... |
1,420,082 | <p>I read the question at <a href="https://math.stackexchange.com/questions/261657/g%C3%B6dels-incompleteness-theorem-meta-reasoning-loophole">Gödel's Incompleteness Theorem -- meta-reasoning "loophole"?</a> about Gödel's incompleteness theorem. My question is little about the contents of that other ... | Zach Stone | 38,565 | <p>There is no absolutely strict terminology. You could say either "inner logic" or "inner system," and we'd know what you mean. But these terms should better mean the same thing! Similarly, "outer logic" or "meta logic" are interchangeable.</p>
<p>There can't be an outermost logic independent of context. You wouldn'... |
3,382,651 | <p>I'm trying to understand one of the steps taken during the process of getting a cnf in Boolean algebra but I just cant understand what is happening here. </p>
<p><span class="math-container">$$\bar A \bar B C + \bar A \bar C \bar D + A \bar C D + \bar A B \bar C$$</span>
<span class="math-container">$$\bar A \b... | Bram28 | 256,001 | <p>Here is why:</p>
<p><span class="math-container">$$A'B'C+A'C'D'+AC'D+A'BC'\overset{Absorption}{=}$$</span></p>
<p><span class="math-container">$$A'B'C+A'C'D'+(AC'D+ABC'D)+(A'BC'D+A'BC'D')\overset{Association, Commutation}{=}$$</span></p>
<p><span class="math-container">$$A'B'C+(A'C'D'+A'BC'D')+AC'D+(ABC'D+A'BC'D)... |
2,562,702 | <blockquote>
<p>The average daily temperature in °F in a certain town is a normal random variable $T$ with $\mu=30$ and $\sigma=12$. The daily heating cost $C$, in dollars, for a building is related to $T$ by $C=-100T+13000$. What is the probability that the daily heating cost for this building on a typical day will ... | Siong Thye Goh | 306,553 | <p>\begin{align}
P((C > 11500) &= P(-100T+13000 > 11500)\\
&= P (-100T>11500-13000)\\
&=P\left(T<\frac{13000-11500}{100}\right)
\end{align}</p>
<p>Hopefully you can finish the question. </p>
<p>As an exercise, you might like to think about what distribution does $C$ follows given that you know... |
282,590 | <p>"Check in detail that $\langle x,y \mid xyx^{-1}y^{-1} \rangle$ is a presentation for $\mathbb{Z} \times \mathbb{Z}$."
Exercise 27.5 from "Groups and Symmetry" M.A.Armstrong.</p>
<p>This should be an easy exercise but I'm completely unable to answer it. I know the group would be equal to the quotient group $F(X)/N$... | Seirios | 36,434 | <p><strong>Hint:</strong> $\mathbb{Z} \times \mathbb{Z}$ is the abelianization of $\mathbb{F}_2$, so consider the quotient $\mathbb{F}_2/D(\mathbb{F}_2)$.</p>
|
11,743 | <p>As part of many hobbies (origami, sculpting, construction toys) I often find myself building polyhedra from regular polygons. I am intimately familiar with all of the Archimedean and Platonic solids, and can construct most of the other isohedra, deltahedra, and Johnson solids from memory. The smaller prisms, antipr... | lhf | 532 | <p>Try the <a href="http://www.netlib.org/polyhedra/" rel="nofollow">netlib polyhedra database</a>. It does not seem to be listed in Wikipedia polyhedra page.</p>
|
11,743 | <p>As part of many hobbies (origami, sculpting, construction toys) I often find myself building polyhedra from regular polygons. I am intimately familiar with all of the Archimedean and Platonic solids, and can construct most of the other isohedra, deltahedra, and Johnson solids from memory. The smaller prisms, antipr... | rita the dog | 660 | <p>There is George Hart's online <a href="http://www.georgehart.com/virtual-polyhedra/vp.html" rel="nofollow">Encyclopedia of Polyhedra</a>, which is not a database but is very likely something you will want to consult.</p>
<p>Also not a database, but something you should surely know about if you intend to create one,... |
3,078,972 | <p>if <span class="math-container">$n>1$</span> odd number,find <span class="math-container">$$2^n\equiv ?\pmod {12}$$</span></p>
<p>it seem the answer is <span class="math-container">$8$</span>,because
<span class="math-container">$$2^3=8\equiv 8\pmod{12}$$</span>
<span class="math-container">$$2^5=32\equiv 8\pmod... | kelvin hong 方 | 529,398 | <p>To be precise, we want to prove that if <span class="math-container">$n$</span> is an odd number <span class="math-container">$\geq 3$</span>, then <span class="math-container">$$2^n\equiv 8\pmod{12}.$$</span>
Since you've verify the initial case <span class="math-container">$n=3$</span>, we assume if <span class="m... |
3,078,972 | <p>if <span class="math-container">$n>1$</span> odd number,find <span class="math-container">$$2^n\equiv ?\pmod {12}$$</span></p>
<p>it seem the answer is <span class="math-container">$8$</span>,because
<span class="math-container">$$2^3=8\equiv 8\pmod{12}$$</span>
<span class="math-container">$$2^5=32\equiv 8\pmod... | Community | -1 | <p>You can prove it with the Chinese remainder theorem: <span class="math-container">$12=2^2\cdot3$</span>.</p>
<p>We have <span class="math-container">$2^n\cong0\pmod{2^2}$</span>, and<span class="math-container">$2^n\cong2\pmod3$</span> ( since by Fermat's little theorem, <span class="math-container">$2^2\cong1\pmod... |
3,052,512 | <blockquote>
<p><strong>Problem:</strong> Consider the optimization problems
<span class="math-container">$$\min_\beta \|y-X\beta\|^2+\alpha\|\beta\|^2 \tag 1$$</span>
and
<span class="math-container">$$\min_\beta \|\beta\|^2 \text{ subject to } \|y-X\beta\|^2 \le c \tag 2$$</span>
where <span class="math-con... | AugSB | 188,245 | <p>It is true for <span class="math-container">$\alpha>0$</span>. Since <span class="math-container">$\beta^*$</span> is solution of (1), we have:
<span class="math-container">$$\|y-X\beta^*\|^2 + \alpha\|\beta^*\|^2 \le \|y-X\beta\|^2 + \alpha\|\beta\|^2.$$</span>
Reordering:
<span class="math-container">$$\|\beta^... |
883,118 | <p>I think it's true, I just did this demo, please can you help me if I'm missing something or doing it wrong. Thanks.</p>
<p>Let $T\colon V \to W$ a linear transformation. </p>
<p>If $\dim V > \dim W$, then $T$ is not injective. </p>
<p>The contrapositive is: If $T$ is injective, then $\dim V \le \dim W$.</p>
... | leo | 8,271 | <p>Tu prueba es correcta. Una forma más rápida de concluir es recordando que para cualquier espacio vectorial $V$, si $U$ es un subespacio de $V$ entonces $\dim U \leq \dim V$, y, en el contexto de tu pregunta, $\operatorname{Im}(T)$ es siempre un subespacio de $W$.</p>
<hr>
<p>Your proof is correct. A quicker way to... |
2,338,369 | <p>Assume $∃xy∈N$ st. $x^2-y^2 =1$</p>
<p>$(x-y)(x+y) = 1$ so;</p>
<p>$(x-y)∈N$ and $(x+y)∈N$</p>
<p>How do I proceed from the information I know</p>
| Bram28 | 256,001 | <p>You have:</p>
<p>$(x-y)(x+y) = 1$ </p>
<p>and</p>
<p>$(x-y)∈N$ and $(x+y)∈N$</p>
<p>Well, the only way for the product of two natural numbers to be $1$ is for both numbers to be $1$.</p>
<p>So: $x-y=1$ and $x+y=1$</p>
<p>You can take it from there ...</p>
<p>(p.s. $x=1$ and $y=0$ would be a solution, so I ass... |
143,092 | <p>If $N$ is an $n\times n$ nilpotent matrix such that $N^k=0$ for some integer $k$. Is it true that $(DN)^k=0$ for any diagonal matrix $D$?</p>
| André Nicolas | 6,312 | <p>Let $n$ be a positive integer. Then $\varphi(n)$ is defined as the <em>number</em> of integers in the interval $[0,n-1]$ which are relatively prime to $n$, that is, have no factor greater than $1$ in common with $n$. The function $\varphi$ is called the Euler phi-function.
The following result, due to Euler, general... |
143,092 | <p>If $N$ is an $n\times n$ nilpotent matrix such that $N^k=0$ for some integer $k$. Is it true that $(DN)^k=0$ for any diagonal matrix $D$?</p>
| Simon Markett | 30,357 | <p>And yet another proof which also gives an algorithm to compute the (minimal) number of digits and a lower bound. Given an integer $a$ with the desired properties and the allowed digit $0\leq z\leq 9$ we want to find a solution for
$$a\cdot x=z\cdot\sum_{i=0}^n10^i$$
such that x is an integer and $n$ minimal.
Conside... |
106,162 | <p><strong>Bug introduced in 7.0 or earlier and fixed in 11.0</strong></p>
<hr>
<p><em>Mathematica</em> 10 gives the following very odd result,</p>
<pre><code>Integrate[Sin[Sqrt[x]]/Sqrt[x], {x, 1, ∞}]
(* 2 Cos[1] *)
</code></pre>
<p>which seems unintuitive. The integrand has an easy to find antiderivative,</p>
<... | Michael E2 | 4,999 | <p>I don't know what <em>Mathematica</em> is doing, but there are two ways to justify the result (if you're willing to accept different formulations of integrability).</p>
<p>In an analogy with <a href="https://en.wikipedia.org/wiki/Ces%C3%A0ro_summation#Ces.C3.A0ro_summability_of_an_integral" rel="nofollow noreferrer... |
1,719,772 | <p>I need to prove that $(-x)(-y) = xy$ using only the field axioms. I tried starting with $ since$ $(-(-x)(-y)) + (-x)(-y) = 0$ by A6, or the additive inverse. And then adding $xy$ to both sides by A2 or transitivity. But I'm sure that to say $(-(-x)(-y))=-(xy)$ is outside the field axioms. So I'm stuck and w... | achille hui | 59,379 | <p>First, we need a lemma $\forall y, 0y = 0$.</p>
<p>$$
\begin{array}{rll}
0y &= 0y + 0 &0 \text{ is additive identity}\\
&= 0y + (0y + (-0y))& -0y \text{ is additive inverse of } 0y\\
&= (0y + 0y) + (-0y)& \text{addition is associative}\\
&= (0+0)y + (-0y) &\text{multiplication is dis... |
3,930,780 | <p>I'm having problem proving that <span class="math-container">$(\log(n))^k=O(n)$</span>. I tried googling it but can't find an answer.</p>
| Luke Collins | 301,095 | <p><span class="math-container">$\log n = O(n^\epsilon)$</span> for any <span class="math-container">$\epsilon > 0$</span>, since
<span class="math-container">$$\log n = \tfrac1\epsilon \log(n^\epsilon) = \tfrac1\epsilon\textstyle\int_1^{n^\epsilon}\frac{dt}{t} <\tfrac1\epsilon \int_1^{n^\epsilon}dt= \tfrac1\epsi... |
798,721 | <blockquote>
<p>Let $a,b,c,d$ positive real numbers with $d= \max(a,b,c,d)$. Proof
that </p>
<p>$$a(d-c)+b(d-a)+c(d-b)\leq d^2$$</p>
</blockquote>
<ul>
<li>I believe that the GM-AM inequality with $n=4$ variables might be helpful. </li>
</ul>
<p>$$\sqrt[n]{x_1 x_2 \dots x_n} \le \frac{x_1+ \dots + x_n}{n}$$... | coffeemath | 30,316 | <p>The inequality, after multiplying it out and moving all to one side, is
$$d^2-ad-bd-cd+ab+ac+bc\ge 0.\tag{1}$$
Since $d=\max(a,b,c,d)$ each of $d-a,\ d-b,\ d-c$ is nonnegative and so
$$(d-a)(d-b)(d-c)\ge 0,
\\ d^3-ad^2-bd^2-cd^2+abd+acd+bcd\ge abc,$$
where at the last step we moved the $-abc$ term over to the right ... |
3,439,581 | <p>I know <span class="math-container">$\overline {A}\subseteq \overline {A\cup B}$</span>, because <span class="math-container">$\overline {A\cup B}$</span> is closed.<br>
Is there a proof for that or is it just a definition thing?</p>
| RGS | 329,832 | <p>Of course there is a proof.</p>
<p><span class="math-container">$A \subset \overline{A \cup B}$</span> follows immediately, right?</p>
<p>What other elements are in <span class="math-container">$\overline A$</span> and what condition do they satisfy? Can you show that such condition can also be satisfied with the ... |
1,871,716 | <blockquote>
<p>Solve the pair $x \equiv 5 \pmod{9}$ and $10x \equiv 6 \pmod{28}$.</p>
</blockquote>
<p>So this means, $10x \equiv 5 \pmod{9}$ and $10x \equiv 6 \pmod{28}$. </p>
<p>So this has a unique solution $10x \equiv a \pmod{252}$</p>
<p>So we have $5 + 9k = 6 + 28n \implies 9k - 28n = 1$</p>
<p>And then I... | boaz | 83,796 | <p>It is not very convenient to prove this assertion for negative numbers. Let us assume
$$
0<a_1<a_2<\cdots<a_n
$$
and consider the function
$$
f(x)=\frac{1}{x-a_1}+\ldots+\frac{1}{x-a_n}-\frac{1}{x}
$$
$f(x)$ is continues at $(-\infty,0)$, $(0,a_1)$, $(a_1,a_2)$ , $\ldots$ , $(a_{n-1},a_n)$ , $(a_n,\inf... |
1,871,716 | <blockquote>
<p>Solve the pair $x \equiv 5 \pmod{9}$ and $10x \equiv 6 \pmod{28}$.</p>
</blockquote>
<p>So this means, $10x \equiv 5 \pmod{9}$ and $10x \equiv 6 \pmod{28}$. </p>
<p>So this has a unique solution $10x \equiv a \pmod{252}$</p>
<p>So we have $5 + 9k = 6 + 28n \implies 9k - 28n = 1$</p>
<p>And then I... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p><span class="math-container">$$\dfrac1x=\sum_{r=1}^n\dfrac1{x+a_r}\iff n=\sum_{r=1}^n\dfrac{a_r}{x+a_r}$$</span></p>
<p>Let <span class="math-container">$p+iq$</span> is a root where <span class="math-container">$p,q$</span> are real</p>
<p><span class="math-container">$$\implies n=\sum_{r=1}^n\dfrac1{p... |
627,464 | <p>Let $I$ be the center of the inscribed sphere of a tetrahedron $ABCD$ and let $I_A$ be the length of the line passing through $I$ from the vertex $A$ to the opposite face. I am looking for a formula for $I_A^2$ which is analogous to the formula for the angle bisector in a triangle $$I_a^2 = \frac {bc}{(b+c)^2}[(b+c)... | JJacquelin | 108,514 | <p>After correction of the typo, the singular points are shown on the joints graphs.</p>
<p>Figure 3 shows the case of isolated singular points.<img src="https://i.stack.imgur.com/67a1j.jpg" alt="enter image description here"> </p>
|
574,812 | <h1>Question</h1>
<ul>
<li>What formula would allow me to predict the center of this circle? </li>
<li>In addition, what attributes of this image must be detected in order
to predict the center? I figured understanding the math first will help me determine what parts of the image are relevant, then I can worry about t... | vadim123 | 73,324 | <p>The hard part is finding three points on the circle using the messy data you're given. Assuming you have done this, $A,B,C$, in that order, then:</p>
<ol>
<li><p>Let $L_1$ be the line that passes through the midpoint of segment $AB$, and is perpendicular to $AB$.</p></li>
<li><p>Let $L_2$ be the line that passes ... |
4,124,652 | <p>A few quick questions from an older text on complex analysis:</p>
<p><span class="math-container">$|e^{it}|^2=e^{it}\cdot\overline{e^{it}}=e^{it}\cdot e^{-it}$</span></p>
<p>I'm not sure about the notation <span class="math-container">$|e^{it}|^2$</span>, but I'm assuming it indicates the magnitude of the vector squ... | user2661923 | 464,411 | <p>For your second question, note that
<span class="math-container">$\displaystyle |z \times w| = |z| \times |w|, ~: ~z,w \in \Bbb{C}.$</span></p>
<p>The above result may be manually verified by setting <br>
<span class="math-container">$\displaystyle z = (x + iy), w = (u + iv)$</span></p>
<p>and then comparing <br>
<s... |
3,512,971 | <p>In a group the ratio of men to women is 5:3
In the same group, the ratio of children to adults is 1:2</p>
<p>What is the ratio of men:women:children?</p>
<p>Through simple trial/error/obviousness, one can see that it is 5:3:4</p>
<p>I can’t wrap my head around a sound algebraic way to solve this.</p>
<p>Please h... | farruhota | 425,072 | <p>Express both ratios in terms of <span class="math-container">$x$</span> and <span class="math-container">$y$</span> as follows:
<span class="math-container">$$m:w=5:3=5x:3x \Rightarrow \color{red}{m+w=8x}\\
c:a=1:2=y:2y \Rightarrow 2y=a=\color{red}{m+w=8x} \Rightarrow c=y=4x$$</span>
Hence:
<span class="math-contain... |
4,518,177 | <p>I have just started learning Linear Algebra and in Vectors there is this concept of standard basis vectors , î and j, and all the vectors can be expressed as the sum of these two basis vectors. I want to know if any two random vectors can also serve as basis vectors ? What is the intuition behind this ?</p>
| copper.hat | 27,978 | <p>You can assume that <span class="math-container">$T$</span> is diagonal, that is <span class="math-container">$T= \operatorname{diag} ( \lambda_1,...,\lambda_n)$</span>.
Then <span class="math-container">$\sum_k |\lambda - \lambda_k|^2 v_k^2 < \epsilon^2$</span>. If <span class="math-container">$|\lambda - \lambd... |
2,414,620 | <p>Lets have two known vectors $x_1'$ and $x_2'$ in XY ($\mathbb{R}^2$) representing the orthogonal projection of two unknown vectors $x_1$ and $x_2$ in $\mathbb{R}^3$, with known lengths $|x_1|=l_1$, $|x_1|=l_2$, and $x_1\cdot x_2=0$. </p>
<p>Let be the orthogonal projection made through $\hat z$, thus projecting ort... | amd | 265,466 | <p>(I’m going to use slightly different notation from yours so as not to confuse coordinates with vectors.) </p>
<p>For each of the vectors $\mathbf v_1'=(x_1,y_1,0)$ and $\mathbf v_2'=(x_2,y_2,0)$ there are two possible pre-images, even in your simple example: $\mathbf v_1=\left(x_1,y_1,\pm\sqrt{l_1^2-x_1^2-y_1^2}\r... |
1,632,169 | <p>Hi I am trying to prove the following the formula and this is what I have so far</p>
<pre><code>false ∨ p ≡ p
</code></pre>
<p>This is what I have do so far</p>
<pre><code>false ∨ p
(p ∧ ¬p) v p ≡ negation
¬p ∧ P ≡ absorbtion
</code></pre>
<p>This is where I am stuck, I do not know how to proceed forward</p>
| Clement C. | 75,808 | <p>Take $y\in\mathbb{R}$. Then, there exists a unique $k\in\mathbb{Z}$ such that $$y\in [2\pi k +\frac{\pi}{2}, 2\pi(k+1)+\frac{\pi}{2}) = [y_1, y_2).$$</p>
<p>Now, $f(2\pi k+\frac{\pi}{2}) = 2\pi k+\frac{\pi}{2}$, and $f(2\pi(k+1))+\frac{\pi}{2}) = 2\pi(k+1))+\frac{\pi}{2}$. Use the IVT on $[y_1, y_2]$ to show that $... |
467,354 | <p>How many positive integers are there that can be written in the form
$$\frac{m^3+n^3}{m^2+n^2+m+n+1}$$
where $m$ and $n$ are positive integers.</p>
<p>I invented this problem and was stuck with it for a long time. It is really interesting to know if there is a solution for this problem. </p>
| Dan Rust | 29,059 | <p>I'll assume $T$ is a group. An invariant set can be partitioned in to its orbits (as any set with a group action on it can be) and also any union of orbits is an invariant set. In particular, there is a bijective correspondence between the set of all sets of orbits of the system and the set of invariant subsets of t... |
3,612,010 | <p>Proof:<br>
Let <span class="math-container">$S = (1,3)\cup(5,8)$</span><br>
Clearly <span class="math-container">$\forall x \in S,\;x<8$</span>. Hence <span class="math-container">$8$</span> is an upper bound of <span class="math-container">$S$</span>.<br>
To prove that 8 is the least upper bound suppose there ex... | Z Ahmed | 671,540 | <p><span class="math-container">$$L=\lim_{n\to \infty} {n \choose k}\frac{1}{n^k}\left(1-\frac{1}{n}\right)^{n-k}$$</span>
<span class="math-container">$$L=\lim_{n \to \infty} F(n)= \lim_{n \infty} \frac{n(n-1)(n-2)(n-3)...(n-k+1)}{n^k~k!} \lim_{n \to \infty}\left(1-\frac{1}{n}\right)^{-k} \lim_{n \to \infty}\left(1-\f... |
1,928,515 | <p>This subject is very foreign to me and has me kind of confused. This problem seems very easy, but because I'm new to this, its not to me. I'm not sure of the name of the law I'm using, or if its correct at all. I'm supposed to prove that:</p>
<pre><code>A \ (A ∩ B) = A \ B
x ∈ A\(A∩B) ⇔ (x ∈ A\A) ∩ (x ∈ A\B)) ... | Robert Israel | 8,508 | <p>As far as I know, characteristic equations are used for linear recurrence relations. This one is nonlinear.</p>
<p>EDIT: There is, however, a nice closed-form solution. </p>
<p>Hint: $$\coth(2x) = \dfrac{1}{2} \left(\coth(x) + \frac{1}{\coth(x)}\right) $$</p>
|
415,928 | <p>Determine whether $X^4-16X^2+4$ is irreducible in $\mathbb{Q}[X]$.</p>
<p>To solve this problem, I reasoned that since $X^4-16X^2+4$ has no rational roots hence irreducible.</p>
<p>But there is a hint to this question that uses different approach:<br>
Try supposing it is reducible, then it must factor into a produ... | Hagen von Eitzen | 39,174 | <ol>
<li><p>A polynomial is irreducible if it cannot be written as product of two polynomials of smaller degree. It is not necessarily the case that one of these factors has degree 1. (Though, if the original polynomial is quadratic or cubic, then it suffices to check against linear factors).</p></li>
<li><p>If we coul... |
3,398,566 | <p>The number of integers in the range of 'c' such that there exists a line which intersects the curve <span class="math-container">$ y = x^4 – 6x^3 + 12x^2 + cx + 1$</span> at four distinct points. </p>
<p>My approach we need to intersect with line <span class="math-container">$y=mx+C$</span></p>
<p>Substituting w... | Vasily Mitch | 398,967 | <p>The second derivative <span class="math-container">$y''(x)=12x^2-36x+24=12(x-1)(x-2)$</span> has roots <span class="math-container">$\{1,2\}$</span> independently of <span class="math-container">$c$</span>. Thus, <span class="math-container">$y(x)$</span> has two points of inflection at <span class="math-container">... |
928,735 | <p>I attempt to solve the equation </p>
<p>$(z+1)^5=z^5$.</p>
<p>My first approach is to expand the left hand side but ı get more complicated equation. So I couldn't go further. Secondly, I write equation as, since $z\neq0$, </p>
<p>$(\frac{z+1}{z})^5=1$, put $\xi=\frac{z+1}{z}$ </p>
<p>and attempt to solve equiva... | Jonas Meyer | 1,424 | <p>$(z+1)^5=z^5$ implies that $|z+1|^2=|z|^2$, which implies that $x=\mathrm{Re}(z)=-\frac12$. Then $\left(\frac12+iy\right)^5=\left(-\frac12+iy\right)^5$ is a quadratic equation in $y^2$.</p>
|
928,735 | <p>I attempt to solve the equation </p>
<p>$(z+1)^5=z^5$.</p>
<p>My first approach is to expand the left hand side but ı get more complicated equation. So I couldn't go further. Secondly, I write equation as, since $z\neq0$, </p>
<p>$(\frac{z+1}{z})^5=1$, put $\xi=\frac{z+1}{z}$ </p>
<p>and attempt to solve equiva... | Hypergeometricx | 168,053 | <p>To exploit symmetry, put $z=w-\frac 12$, which gives $z+1=w+\frac 12$. </p>
<p>Solving:</p>
<p>$$\begin{align}(z+1)^5&=z^5\\
\left(w+\frac 12\right)^5&=\left(w-\frac 12\right)^5\\
2\left[5w^4\left(\frac12\right)+10w^2\left(\frac 12\right)^3+\left(\frac 12\right)^5\right]&=0\\
80w^4+40w^2+1&=0\\
w^2... |
42,879 | <p>Let $G$ be a finite group and $\chi$ a character of $G$. The values of $\chi$ generate an abelian Galois extension $K$ of $\mathbb{Q}$, and so one can consider the conjugate $\sigma(\chi)$ of $\chi$ by any element $\sigma$ of the Galois group. What's the shortest way to prove that $\sigma(\chi)$ is also a character ... | David E Speyer | 448 | <p>OK, I've thought about it a bit more and I'll give an alternate proof below the first horizontal line. However, I highly deny that this proof is better, and I am not sure it is actually shorter.
To my mind, the morally correct proof is to show that, if $K \subseteq L$ with $L$ algebraically closed, and a system of p... |
3,399,628 | <p>Let the sample space be <span class="math-container">$[0,1]$</span> with the Borel sigma algebra and the probability dx:</p>
<p><span class="math-container">$X(\omega)=1/(1-\omega)$</span>. </p>
<p>The support is then <span class="math-container">$[1, +\infty)$</span></p>
<p>How do I compute the expected value of... | NCh | 413,376 | <p>Look at <a href="https://en.wikipedia.org/wiki/Expected_value#General_case" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Expected_value#General_case</a> :
<span class="math-container">$$
\mathop{\text E}[X] = \int_\Omega X(\omega) \,d\,{\text P}(\omega)
$$</span>
Since <span class="math-container">$\Omeg... |
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