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 |
|---|---|---|---|---|
2,909,022 | <p>I don't understand how to get from the first to the second step here and get $1/3$ in front.</p>
<p>In the second step $g(x)$ substitutes $x^3 + 1$.</p>
<p>\begin{align*}
\int_0^2 \frac{x^2}{x^3 + 1} \,\mathrm{d}x
&= \frac{1}{3} \int_{0}^{2} \frac{1}{g(x)} g'(x) \,\mathrm{d}x
= \frac{1}{3} \int_{1}^{... | Keen-ameteur | 421,273 | <p>The derivative of $g(x)$ is $g'(x)=3x^2$. The constant $3$ is not present in the first expression but:</p>
<p>$\frac{1}{3}g'(x)=\frac{1}{3}\cdot 3x^2=x^2$</p>
|
440,082 | <blockquote>
<p><span class="math-container">$$ \int_{1}^{\infty} \frac{\sin^2 (\mu \sqrt{x^2 -1})}{(x+1)^{\frac{9}{2}} (x-1)^{\frac{3}{2}}} \,dx $$</span>
Note: <span class="math-container">$\mu$</span> here is an extremely small constant.</p>
</blockquote>
<p>I have tried:</p>
<ol>
<li>Estimating the integral by Tayl... | Carlo Beenakker | 11,260 | <p>Here is a log-log plot of
<span class="math-container">$$\delta I=I_{\text{appr}}-\int_{1}^{\infty} \frac{\sin^2 (\mu \sqrt{x^2 -1})}{(x+1)^{\frac{9}{2}} (x-1)^{\frac{3}{2}}} \,dx,$$</span>
as a function of <span class="math-container">$\mu$</span>, with
<span class="math-container">$$I_{\text{appr}}=\frac{2 \mu^2}{... |
440,082 | <blockquote>
<p><span class="math-container">$$ \int_{1}^{\infty} \frac{\sin^2 (\mu \sqrt{x^2 -1})}{(x+1)^{\frac{9}{2}} (x-1)^{\frac{3}{2}}} \,dx $$</span>
Note: <span class="math-container">$\mu$</span> here is an extremely small constant.</p>
</blockquote>
<p>I have tried:</p>
<ol>
<li>Estimating the integral by Tayl... | Fred Hucht | 90,413 | <p>I really like the double-series approach by @AccidentalFourierTransform, however I got remaining <span class="math-container">$\Lambda$</span>s in the terms of order <span class="math-container">$O(\mu^8)$</span> onwards. Thinking about this approach, I located the problem in the neglection of higher order terms in ... |
937,064 | <p>The title pretty much says it all:</p>
<p>If supposing that a statement is false gives rise to a paradox, does this prove that the statement is true?</p>
<p><em>Edit:</em> Let me attempt to be a little more precise:</p>
<p>Suppose you have a proposition. Furthermore, suppose that assuming the proposition is false... | user21820 | 21,820 | <p>The whole question boils down to which sentences have a truth value. See <a href="https://math.stackexchange.com/a/1888389/21820">this answer</a> for a detailed analysis of this notion. All modern mathematics is done in a manner that can in fact be formalized in a formal system that does not allow self-reference or ... |
2,348,131 | <p>In our class, we encountered a problem that is something like this: "A ball is thrown vertically upward with ...". Since the motion of the object is rectilinear and is a free fall, we all convene with the idea that the acceleration $a(t)$ is 32 feet per second square. However, we are confused about the sign of $a(t)... | Community | -1 | <p>The gravity force is downward and so is the acceleration (by $F=ma$).</p>
<p>So if you choose a downward axis, the acceleration is positive.</p>
<p>And if you choose an upward axis, the acceleration is negative.</p>
|
221,053 | <p>Given two uncorrelated random variables $X,Y$ with the same variance $\sigma^2 $ I need to compute $\rho= \frac{COV(X,Y)}{\sigma(X)\sigma(Y)}$ between $X+Y$ and $2X+2Y$. I know it should be a number between $-1$ and $1$ and I don't understand how come I get $4$. </p>
<p>Here's what I did:</p>
<p>$COV(X+Y,2X+2Y)... | Did | 6,179 | <p>Show that, for <strong>every</strong> nondegenerate random variable $Z$ and nonzero real number $a$, $\mathrm{var}(aZ)=a^2\cdot\mathrm{var}(Z)$ and $\mathrm{cov}(Z,aZ)=a\cdot\mathrm{var}(Z)$. Deduce that $\varrho(Z,aZ)=\mathrm{sgn}(a)$.</p>
|
3,204,082 | <p>I have a conjecture, but have no idea how to prove it or where to begin. The conjecture is as follows:</p>
<blockquote>
<p>A polynomial with all real irrational coefficients and no greatest common factor has no rational zeros.</p>
</blockquote>
<p>This conjecture excludes the cases where the polynomial does have... | Alexander Gruber | 12,952 | <p>If one were able to prove that, that would imply that <span class="math-container">$\pi x-e=0$</span> has no rational roots. However, it is not known whether <span class="math-container">$e/\pi$</span> is rational. So, I think your conjecture as stated would be difficult to prove.</p>
|
1,076,292 | <p>I wish to use two points say $(x_1$,$y_1)$ and $(x_2$,$y_2)$ and obtain the coefficients of the line in the following form: $$ Ax + By + C = 0$$</p>
<p>Is there any direct formula to compute.</p>
| Ahaan S. Rungta | 85,039 | <p>This is called solving simultaneous equations and is a matter of algebra. There are two ways to go about this but they are almost the same and have the same spirit. </p>
<p>The first way to do it is to solve directly for $A,B,C$ by substituting the points into $Ax+By+C=0$ to get $$ \begin {eqnarray*} Ax_1 + By_1 &a... |
3,738,508 | <p>If <span class="math-container">$G$</span> is order <span class="math-container">$p^2q$</span>, where <span class="math-container">$p$</span>, <span class="math-container">$q$</span> are primes, prove that either a Sylow <span class="math-container">$p$</span>-subgroup or a Sylow <span class="math-container">$q$</sp... | quasi | 400,434 | <p>Here's another proof . . .</p>
<p>
Assume <span class="math-container">$x^4+2$</span> factors in <span class="math-container">$\mathbb{Z}_5[x]$</span> as
<span class="math-container">$$x^4+2=(x^2+ax+b)(x^2+cx+d)$$</span>
Let <span class="math-container">$K$</span> be an algebraic closure of <span class="math-contain... |
2,325,968 | <p>I was trying to calculate : $e^{i\pi /3}$.
So here is what I did : $e^{i\pi /3} = (e^{i\pi})^{1/3} = (-1)^{1/3} = -1$</p>
<p>Yet when I plug : $e^{i\pi /3}$ in my calculator it just prints : $0.5 + 0.866i$</p>
<p>Where am I wrong ? </p>
| José Carlos Santos | 446,262 | <p>That's because your calculator <em>knows</em> that $e^{ix}=\cos(x)+i\sin(x)$, and therefore $e^{i\pi/3}=\cos\left(\frac\pi3\right)+i\sin\left(\frac\pi3\right)=\frac12+\frac{\sqrt3}2i$.</p>
|
5,263 | <p>I recently tried to edit an old question <a href="https://mathoverflow.net/questions/39428/x-th-moment-method">x-th moment method</a> that had got bumped to the front page for other reasons. The post had an equation that was meant to be, and maybe at one point was, struck through, but it no longer is. That is, the... | LSpice | 2,383 | <p>This is not really an answer, just a test in response to @CalvinKhor's <a href="https://meta.mathoverflow.net/questions/5263/mathjax-equivalent-of-strike-strike#comment27024_5264">comment</a>.</p>
<p><code>\require</code> in one answer (or comment?) seems to affect other answers. This may depend on which answer com... |
1,301,476 | <p>(Cross-posted in <a href="https://matheducators.stackexchange.com/q/8173/"><strong>MESE 8173</strong></a>.) </p>
<p>I want to start to do mathematical Olympiad type questions but have absolutely no knowledge on how to solve these apart from my school curriculum. I'm $16$ but know maths up to the $18$ year old level... | Community | -1 | <p>My personal favourite for the mathematical olympiad is "the Mathematical Olympiad Handbook; An Introduction to Problem Solving" by A. Gardiner</p>
<p><a href="http://www.amazon.co.uk/The-Mathematical-Olympiad-Handbook-Introduction/dp/0198501056" rel="nofollow">http://www.amazon.co.uk/The-Mathematical-Olympiad-Handb... |
351,226 | <p>I am trying to brush up on my regular grammar knowledge to prepare for an interview, and I just am not able to solve this problem at all. This is NOT for homework, it is merely me trying to solve this.</p>
<p>I want to give a regular grammar for the language of the finite automaton whose screen shot is below, pleas... | Tara B | 26,052 | <p>If you have a finite automaton for a regular language $L$, you can construct a right regular grammar for $L$ directly from the automaton, by using the states as the variables and the alphabet as the terminals and essentially just writing down the transition function in the form of productions.</p>
<p>So for your au... |
2,433,174 | <p>I'm struggling with the following (<em>is it true?</em>):</p>
<blockquote>
<p>Let <span class="math-container">$X$</span> be a set and denote <span class="math-container">$\aleph(X)$</span> the <em><strong>cardinality</strong></em> of <span class="math-container">$X$</span>. Suppose that <span class="math-container"... | Parcly Taxel | 357,390 | <p>Think of $T$ here as sending all points $(x,y)$ satisfying $x+2y=6$ to $(x',y')=(4x-y,3x-2y)$. Since there is an equation between $x$ and $y$, there must also be one between $x'$ and $y'$, and this latter equation will be the image line.
$$x'=4x-y$$
$$y'=3x-2y$$
$$-2x'=2y-8x$$
$$y'-2x'=-5x\qquad x=(2x'-y')/5\tag1$$
... |
1,603,272 | <p>I'm trying to figure out if the sequence $e^{(-n)^n}$ where n is a natural number has a convergent subsequence? It's in a past exam paper. I know that obviously I can't apply the Bolzano-Weirstrass theorem because its not a convergence sequence but im not sure how to test for a convergent subsequence if the original... | ncmathsadist | 4,154 | <p>What happens if you look at odd values of $n$?</p>
|
1,591,863 | <p>I'm studying for the sat, and one question was presented as follows:</p>
<p>If $n$ is a positive integer such that the units (ones) digit of $n^2+4n$ is $7$ and the units digit of n is not $7$, what is the units digit of $n+3$?</p>
<p>So I'm trying to find $n$ such that:</p>
<p>$$(n^2+4n) \mod10=7$$</p>
<p>I kn... | James | 81,163 | <p>Firstly, as a point of correct use of MSE, your question should be self-contained. In particular we shouldn't have to read the title to find hypotheses for the question. Anyway the mathematics</p>
<p>Note that a lattice can't have two minimal elements, your tagging is incorrect.</p>
<p>So we have the following fac... |
253,584 | <p>Let $h:\mathbb{R}^n\to\mathbb{R}^m, n>1$ be a twice continuously differentiable function and $J_h:\mathbb{R}^n\to\mathbb{R}^{m\times n}$ be its jacobian matrix. Let us consider the functions $A(x):=J_h^\mathtt{T}(x)J_h(x)\in\mathbb{R}^{n\times n}$ and $B(x):=J_h(x)J_h(x)^\mathtt{T}\in\mathbb{R}^{m\times m}$.</p>
... | Bazin | 21,907 | <p>Let me point out a more specific result for hyperbolic polynomials, known as Bronshtein's theorem (see e.g. the preprint <a href="https://arxiv.org/abs/1309.2150" rel="nofollow noreferrer">https://arxiv.org/abs/1309.2150</a> by A. Parusinski & A. Rainer). Let $p(X,y)$ be a polynomial with degree $m$ in the $X$ v... |
253,584 | <p>Let $h:\mathbb{R}^n\to\mathbb{R}^m, n>1$ be a twice continuously differentiable function and $J_h:\mathbb{R}^n\to\mathbb{R}^{m\times n}$ be its jacobian matrix. Let us consider the functions $A(x):=J_h^\mathtt{T}(x)J_h(x)\in\mathbb{R}^{n\times n}$ and $B(x):=J_h(x)J_h(x)^\mathtt{T}\in\mathbb{R}^{m\times m}$.</p>
... | Benoît Kloeckner | 4,961 | <p>For symmetric matrices, you are good (in fact even in infinite dimension). Let me quote the MathReview of the following reference (itself quoting the article):</p>
<p>Kriegl, Andreas(A-WIEN); Michor, Peter W.(A-ERS)
Differentiable perturbation of unbounded operators. (English summary)
Math. Ann. 327 (2003), no. 1, ... |
1,552 | <p>Closely related: what is the smallest known composite which has not been factored? If these numbers cannot be specified, knowing their approximate size would be interesting. E.g. can current methods factor an arbitrary 200 digit number in a few hours (days? months? or what?).
Can current methods certify that an a... | Aaron Meyerowitz | 8,008 | <p>Yes factoring is an active area. I think there are ranges (maybe $80$ digits?) where factoring is reasonably tractable but many integers have never been examined.</p>
<p>Hueristic tests tell us with extremely high confidence if much larger integers are prime. An actual proof certificate by ECM is never a problem in... |
2,051,555 | <p>I have the following limit to solve.</p>
<p>$$\lim_{x \rightarrow 0}(1-\cos x)^{\tan x}$$</p>
<p>I am normally supposed to solve it without using l'Hôpital, but I failed to do so even with l'Hôpital. I don't see how I can solve it without applying l'Hôpital a couple of times, which doesn't seem practical, nor how ... | Mark Viola | 218,419 | <p>Note that </p>
<p>$$\begin{align}
\left(1-\cos(x)\right)^{\tan(x)}&=\left(2\sin^2(x/2)\right)^{\tan(x)}\\\\
&=2^{\tan(x)}\,\left(\sin(x/2)\right)^{2\tan(x)}\\\\
&=2^{\tan(x)}\,\left(\left(\sin(x/2)\right)^{\sin(x/2)}\right)^{2\tan(x)/\sin(x/2)}\\\\
&=2^{\tan(x)}\,\left(\left(\sin(x/2)\right)^{\sin(x... |
2,051,555 | <p>I have the following limit to solve.</p>
<p>$$\lim_{x \rightarrow 0}(1-\cos x)^{\tan x}$$</p>
<p>I am normally supposed to solve it without using l'Hôpital, but I failed to do so even with l'Hôpital. I don't see how I can solve it without applying l'Hôpital a couple of times, which doesn't seem practical, nor how ... | K Split X | 381,431 | <p>Doing this without L'hopital is tricky, but you have to make estimates.</p>
<p>If the know the graph of $cos(x)/tan(x)$, $cos(0) = 1$, and $tan(0) = 0$</p>
<p>So you have:</p>
<p>$$(1- cos(x))^{tan(x)}$$
$$(0)^{0}$$</p>
<p>But if we estimate (right hand side limit), we have something like:</p>
<p>$$(1- 0.01)^{0... |
92,296 | <p>I trying to review for calculus and I can't figure out how to do $\sqrt{200} - \sqrt{32}$ </p>
| Bill Dubuque | 242 | <p>When simplifying radicals the first step is to expose multiplicative dependencies by normalizing the radicands to be <em>squarefree,</em> i.e. pull out square factors. In your example we have $\rm 200 = 2\cdot 10^2\ $ and $\ 32 = 2\cdot 4^2\ $ so we obtain $\rm \sqrt{200}-\sqrt{32}\ = \sqrt{2\cdot 10^2}-\sqrt{2\cdot... |
3,588,053 | <p>Is this a valid proof that the harmonic series diverges?</p>
<ol>
<li>Assume the series converges to a value, S:</li>
</ol>
<p><span class="math-container">$$S=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+...$$</span></p>
<ol start="2">
<li>Split the series into two, with alternating even and odd denominator... | Simply Beautiful Art | 272,831 | <p>This is almost valid. We need to justify the second step, as mentioned by <a href="https://math.stackexchange.com/questions/3588053/is-this-a-valid-proof-that-the-harmonic-series-diverges#comment7376889_3588053">Ross Millikan</a>, as it is not always valid to split a series into their even and odd terms.</p>
<p>Tak... |
534,500 | <p>Is it true that if a sequence of random matrices $\{X_n\}$ converge in probability to a random matrix $X_n\overset{P}{\to}X$ as $n\to\infty$ that the elements $X_n^{(i,j)}\overset{P}{\to} X^{(i,j)}$ $\forall i,j$ also, or are there additional conditions required?</p>
<p>I think I have proved this using the norm $\|... | Did | 6,179 | <p>Yes: if $Y=\max\limits_{1\leqslant k\leqslant N}|Y_k|$ converges to $0$ in probability, then each $Y_k$ converges to $0$ in probability. </p>
<p><em>Proof:</em> For every $\varepsilon\gt0$, $[Y_k\geqslant\varepsilon]\subseteq[Y\geqslant\varepsilon]$. QED</p>
|
1,470,760 | <p>Okay so $A=0.2, B=0.5$ and the probability that both $A$ and $B$ occur is equal to $0.12$.</p>
<p>What is $P((A \cap B) \cup A^c)$?</p>
<p>What I basically did was $0.12 \times 0.5+0.5+0.2-0.12 = 1.2$. </p>
<p>Am I doing it right?</p>
| Simon S | 21,495 | <p><a href="https://i.stack.imgur.com/lfH6q.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/lfH6q.png" alt="enter image description here" /></a></p>
<p>The set of all events here is <span class="math-container">$X$</span> and hence <span class="math-container">$P(X) = 1$</span>. We are given that <sp... |
58,947 | <p>Let $X$ be a non-compact holomorphic manifold of dimension $1$. Is there a compact Riemann surface $\bar{X}$ suc that $X$ is biholomorphic to an open subset of $\bar{X}$ ?</p>
<p><strong>Edit:</strong> To rule out the case where $X$ has infinite genus, perhaps one could add the hypothesis that the topological space... | André Henriques | 5,690 | <p>No. Take a surface of infinite genus.</p>
|
3,439,223 | <blockquote>
<p>Given the metric space <span class="math-container">$(X,d)$</span>:</p>
<p>If <span class="math-container">$M\subset N \subset X$</span>, <span class="math-container">$M\neq 0$</span>, we have <span class="math-container">$\text{diam}(M)\leq
\text{diam}(N)$</span></p>
</blockquote>
<p>Negating ... | Clement Yung | 620,517 | <p>For any <span class="math-container">$A,B \subseteq \mathbb{R}$</span>, if <span class="math-container">$\emptyset \neq A \subseteq B$</span>, then <span class="math-container">$\sup{A} \leq \sup{B}$</span> (try to prove this yourself if you're not convinced).</p>
<p>If you agree with this, then since:
<span class=... |
923,235 | <p>Let $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$
be a matrix of complex numbers. Find the characteristic polynomial $\chi_A(t)$ of $A$ and compute $\chi_A(A)$.</p>
<p>I just wanted to confirm that I did this correctly.</p>
<p>Tha answer I have is:
$$\chi_A(t)= \det\begin{pmatrix}a-t&b\\c&d-t\end{pmatri... | Bman72 | 119,527 | <p>In general for a matrix
$$A=\begin{pmatrix}a&b\\c&d\end{pmatrix},\quad \in M(n \times n; \Bbb{C})$$
we have that the characteristic polynomial $\chi_A(t)$ is
\begin{align*}
X_A(t):&=\text{det}(A-t\Bbb{1})\\
&=\det\begin{pmatrix}a-t&b\\c&d-t\end{pmatrix}\\
&=(a-t)(d-t)-bc\\
&=t^2-(a+... |
1,095,870 | <p>How can I solve the following inequality?</p>
<blockquote>
<p>$$\frac{\cos x -\tan^2(x/2)}{e^{1/(1+\cos x)}}>0$$</p>
</blockquote>
| orangeskid | 168,051 | <p>Note that
$$\frac{\cos x -\tan^2(x/2)}{e^{1/(1+\cos x)}} =\frac{ (-1 + 2 \cos x+ \cos^2 x)}{(1 + \cos x)} \cdot e^{-1/(1+\cos x)}$$</p>
<p>$\frac{e^{-1/(1+\cos x)}}{(1 + \cos x)} $ should be taken as a whole, because it is defined everywhere. It is $>0$ everywhere except at $(2k+1)\pi$ where it's zero. Therefor... |
536,128 | <p>I was trying to make sense of a problem when I stumbled upon this on yahoo answers. I was just wondering if it was correct. If it is, can you please maybe explain why?</p>
<p>${\bf r}'(t) = \langle -5 \cos t, -5 \cos t, -4 \sin t \rangle$</p>
<p>${\bf r}''(t) = \langle 5 \sin t, 5 \sin t, -4 \cos t \rangle$. </p>
... | Bill Cook | 16,423 | <p>Yes. This is "legal" for continuous curves. </p>
<p>The heart of the issue is whether your approximation is actually converging to the object in question or not.</p>
<p>For area under a curve, one can show that the Riemann sums (sums of areas of rectangles) better and better approximate the area under the curve as... |
4,043,625 | <p><span class="math-container">\begin{equation}
\left\{\begin{array}{@{}l@{}}
2x\equiv7\mod9 \\
5x\equiv2\mod6
\end{array}\right.\,.
\end{equation}</span>
Can this system of congruences be solved?
I notice that <span class="math-container">$(9,6) = 3 \ne 1$</span> so I can't apply the Chinese theorem of re... | Arthur | 15,500 | <p>Just because they say it's <span class="math-container">$O(\log_2(n))$</span>, that doesn't mean it can't also be <span class="math-container">$O(\log_2(n+1))$</span>. In fact, we have
<span class="math-container">$$
O(\log_2(n)) = O(\log_2(n+1))
$$</span>
and the two complexity categories are entirely equal. We see... |
2,118,761 | <p>How can I show that there are an infinite number of primes by using the Fundamental Theorem of Arithmetic?</p>
| Harnoor Lal | 396,710 | <p>You start by assuming the opposite. Let's say there are a finite amount of prime numbers, in fact, let's write them in a list. <br/></p>
<p>$P_1$, $P_2$, $P_3$, ... $P_n$ <br/></p>
<p>Note, this is a <strong>complete</strong> list. <br/></p>
<p>Now let's form a new number $a$, by multiplying all of our prime numb... |
2,069,392 | <p>Given that $x^4+px^3+qx^2+rx+s=0$ has four positive roots.</p>
<p>Prove that (1) $pr-16s\ge0$ (2) $q^2-36s\ge 0$</p>
<p>with equality in each case holds if and only if four roots are equal.</p>
<p><strong>My Approach:</strong></p>
<blockquote>
<p>Let roots of the equation</p>
<p>$x^4+px^3+qx^2+rx+s=0$ be ... | ajotatxe | 132,456 | <p>We'll need the following:</p>
<blockquote>
<p>If $a_1,\ldots,a_n>0$ then $$\left(\sum_{k=1}^na_k\right)\left(\sum_{k=1}^n\frac1{a_k}\right)\ge
n^2$$
<em>Proof</em>: By AM-GM inequality, $\sum a_k\ge n\sqrt[n]{\prod a_k}$ and $\sum 1/a_k\ge n\sqrt[n]{\prod 1/a_k}$.</p>
</blockquote>
<p>To ease writing, I'll... |
3,821,049 | <blockquote>
<p>Find all complex solutions of <span class="math-container">$$e^{-iz}=\frac{-i+\sqrt 2+1}{-i-\sqrt 2-1}$$</span>
If a solution is <span class="math-container">$z=x+iy$</span> we set <span class="math-container">$\mathfrak{Re}(z)=x$</span> and <span class="math-container">$\mathfrak{Im}(z)=y$</span>.</p>
... | user | 505,767 | <p>We have that</p>
<p><span class="math-container">$$\frac{-i+\sqrt 2+1}{-i-\sqrt 2-1}\frac{i-\sqrt 2-1}{i-\sqrt 2-1}=\frac{-2-2\sqrt 2+i(2+2\sqrt 2)}{4+2\sqrt2}=-\frac{\sqrt 2}2+i\frac{\sqrt 2}2$$</span></p>
<p>then use Euler's formula</p>
<p><span class="math-container">$$e^{-iz}=e^{y-ix}=e^y\left(\cos x-i\sin x\rig... |
126,901 | <p>How to evaluate this determinant $$\det\begin{bmatrix}
a& b&b &\cdots&b\\ c &d &0&\cdots&0\\c&0&d&\ddots&\vdots\\\vdots &\vdots&\ddots&\ddots& 0\\c&0&\cdots&0&d
\end{bmatrix}?$$</p>
<p>I am looking for the different approaches.</p>
| J. M. ain't a mathematician | 498 | <p>Your <em>(upper) arrowhead matrix</em> can be decomposed as follows:</p>
<p>$$\begin{pmatrix}a&b&b&\cdots&b\\c&d&0&\cdots&0\\c&0&d&\ddots&\vdots\\\vdots &\vdots&\ddots&\ddots&0\\c&0&\cdots&0&d\end{pmatrix}=\color{red}{\begin{pmatrix}a-b... |
2,517,469 | <p>Let $P$ be a projective module and $P=P_1+N$, where $P_1$ is a direct summand of $P$ and $N$ is a submodule. Show that there is $P_2\subseteq N$ such that $P=P_1\oplus P_2$. </p>
<p>I know that there is a submodule $P'$ of $P$ such that $P=P_1\oplus P'$. I wanted to consider the projection from this to $P_1$ and us... | Tsemo Aristide | 280,301 | <p>Since $P_1$ is a direct summand, there exists $P_2$ such that $P=P_1\oplus P_2$.
Consider $f:P\rightarrow P_2$ such that for every $x\in P$, write $x=x_1+x_2, x_1\in P_1,x_2\in P_2$, $f(x)=x_2$. Let $x\in P_2$, we can write $x=x_1'+n, x_1'\in P_1, n\in N$, we have $x=f(x)=f(x_1'+n)=f(n)$. This implies that the restr... |
2,284,451 | <blockquote>
<p><span class="math-container">$A$</span> and <span class="math-container">$B$</span> alternately throw a pair of coin. The player who throws head two times first will win.</p>
<p>A has the first throw. The find chance of winning <span class="math-container">$A$</span> is</p>
</blockquote>
<p>Attempt: Let... | Asinomás | 33,907 | <p>Change the game a bit to make it clearer, consider a reasonable probability space on the set of pairs of infinite sequences of $0$ and $1$ $(a_n)$ and $(b_n)$.</p>
<p>Given such a sequence $(a_n)$ we can define the winning time $w(a_n)$ as the first $i$ such that $a_i=a_{i-1}=1$.</p>
<p>We want to find the probabi... |
2,284,451 | <blockquote>
<p><span class="math-container">$A$</span> and <span class="math-container">$B$</span> alternately throw a pair of coin. The player who throws head two times first will win.</p>
<p>A has the first throw. The find chance of winning <span class="math-container">$A$</span> is</p>
</blockquote>
<p>Attempt: Let... | Em. | 290,196 | <p>Yes. Your pattern is correct. Notice that problem says that $A$ and $B$ toss <em>two</em> coins. I assume the game ends when one player flips the two coins and lands two heads. Also, I assume it's a fair coin and that throws are independent of each other. Let $A$ and $B$ be the events that the respective player end... |
1,579,349 | <p>Need help setting this thing up don't really get how to get the derivative is it $0$? If you just plug everything in since there will be no variable.</p>
| d.v | 296,478 | <p>$dy/dx=dy/du×du/dx$ </p>
<p>$dy/dx=3 u^2×2 x=3 (x^2-1)^2×2 x=6 x(x^2-1)^2$</p>
<p>$Then x=2$</p>
<p>$dy/dx_(x=2)=108$</p>
|
1,685,967 | <p>Let $\Omega\subset\mathbb{R}^n $ be bounded smooth domain.
Given a sequence $u_m$ in Sobolev space $H=\left \{v\in H^2(\Omega ):\frac{\partial v}{\partial n}=0 \text{ on } \partial \Omega \right \}$ such that $u_m$ is uniformly bounded i.e. $\|u_m\|_{H^2}\leq M$ and given the function $f(u)=u^3-u$.</p>
<p>If I kn... | Kore-N | 59,827 | <p>I post this as answer,but I am studying myself Sobolev spaces right now, so I am not sure whether it is correct (in fact it may well be completely wrong). The argument will use embedding theorems for Sobolev spaces.</p>
<ol>
<li><p>First we have a sequence that converges weakly in $L^2,$ but is bounded in $H^2 = W^... |
1,685,967 | <p>Let $\Omega\subset\mathbb{R}^n $ be bounded smooth domain.
Given a sequence $u_m$ in Sobolev space $H=\left \{v\in H^2(\Omega ):\frac{\partial v}{\partial n}=0 \text{ on } \partial \Omega \right \}$ such that $u_m$ is uniformly bounded i.e. $\|u_m\|_{H^2}\leq M$ and given the function $f(u)=u^3-u$.</p>
<p>If I kn... | Svetoslav | 254,733 | <p>For simplicity, I consider only the case <span class="math-container">$n=3$</span>.</p>
<p>You have <span class="math-container">$u_m\rightharpoonup u$</span> in <span class="math-container">$L^2$</span>, so it is enough to show that <span class="math-container">$u_m^3\rightharpoonup u^3$</span> in <span class="math... |
15,669 | <p>Borrowing <code>triangularArrayLayout</code> from <a href="https://mathematica.stackexchange.com/questions/9959/visualize-pascals-triangle-and-other-triangle-shaped-lists">here</a>, I have:</p>
<pre><code>triangularArrayLayout[triArray_List, opts___] :=
Module[{n = Length[triArray]},
Graphics[MapIndexed[
T... | Mr.Wizard | 121 | <p>Version 7 does not have Overlay, but one can produce a similar effect from within <code>Graphics</code>. Using your code with this substitution:</p>
<pre><code>layers = {Graphics[{coeffs[[1]], Opacity[0.3], tri[[1]]}],
Graphics[{tri[[1]], Opacity[0.3], coeffs[[1]]}]};
</code></pre>
<p>yields:</p>
<p><... |
1,462,908 | <p>Is it possible to have a set of infinite cardinality as a subset of a set with a finite cardinality? It sounds counter-intuitive, but there are things in math that just are so. Can one definitely prove this using only basic axioms? <br />
The main reason I asked this question is because the book <em>Inverted World</... | NJastro | 273,594 | <p>The proof is very intuitive (as you probably are feeling). But it can be written elaborately as follows, if you wish.</p>
<p>Your claim: For any finite set F, there exists an infinite subset I.</p>
<p>Try to prove:
Let $F$ be a finite set defined as $F = \{f_1, f_2, \ldots , f_n\}$, where $n = 1, 2, \ldots$</p>
<... |
348,614 | <p>Is the following claim true: Let <span class="math-container">$\zeta(s)$</span> be the Riemann zeta function. I observed that as for large <span class="math-container">$n$</span>, as <span class="math-container">$s$</span> increased, </p>
<p><span class="math-container">$$
\frac{1}{n}\sum_{k = 1}^n\sum_{i = 1}^{k} ... | Carlo Beenakker | 11,260 | <p>A variety of formulas of this type (in the sense of a relation between <span class="math-container">$\zeta(s)$</span> and a sum over gcd or lcm) has been derived by Titus Hilberdink and László Tóth in <A HREF="https://arxiv.org/abs/1604.04508" rel="noreferrer">On the average value of the least common multiple of k p... |
844,420 | <p>Given a set containing N numbers, minimize the average where you can take out any string of consecutive numbers in the set.
|N|<=100000</p>
<p>Ex. {5, 1, 7, 8, 2}</p>
<p>You can take out {1,7}, etc. but the way to minimize in this case is just to take out {7,8} which will give a minimum average of (5+2+1)/3=2.6... | emcor | 154,094 | <p>The average is minimized by taking only the smallest value in the set, i.e. take out the largest elements.</p>
<p>By the restriction of first/last element and consecutive string, you would take out the consecutive inner string which has the biggest weighted average (stringlength/n*stringaverage) and bigger than the... |
2,176,081 | <p>I am trying to compute </p>
<blockquote>
<p>$$ \int_0^\infty \frac{\ln x}{x^2 +4}\,dx,$$</p>
</blockquote>
<p>which I find <a href="https://math.stackexchange.com/questions/2173289/integrating-int-0-infty-frac-ln-xx24-dx-with-residue-theorem/2173342">here</a>, without complex analysis. I am consistently getting ... | PM. | 416,252 | <p>One problem is, when you substitute $u=2t$ then $\log t = \log(u/2)$, not $\log(2u)$ as you have stated in your equation $$I = -\frac{1}{2} \int_0^\infty \frac{\ln(2u)}{1+u^2}\,du$$</p>
|
3,785,967 | <p>Let <span class="math-container">$E(R)_X$</span> denote the expected return of asset <span class="math-container">$X$</span>.</p>
<p>Given a market with only 3 assets; <span class="math-container">$A$</span>, <span class="math-container">$B$</span> and <span class="math-container">$C$</span>, the following three thi... | Henry | 6,460 | <p>You need to incorporate the probabilities so</p>
<p><span class="math-container">$$\sigma = \sqrt{0.5(0.4-0.29)^2 + 0.4(0.2-0.29)^2 + 0.1(0.1-0.29)^2}$$</span></p>
<p>This is also</p>
<p><span class="math-container">$$\sigma = \sqrt{0.5(0.4)^2 + 0.4(0.2)^2 + 0.1(0.1)^2 -0.29^2}$$</span></p>
|
73,383 | <p>The problem is:
$$\displaystyle \lim_{(x,y,z) \rightarrow (0,0,0)} \frac{xy+2yz+3xz}{x^2+4y^2+9z^2}.$$</p>
<p>The tutor guessed it didn't exist, and he was correct. However, I'd like to understand why it doesn't exist.</p>
<p>I think I have to turn it into spherical coordinates and then see if the end result depen... | Did | 6,179 | <p>Assume $x=y=z=t$ and $t\to0$, then the ratio converges to $\frac37$.
Assume $x=y=t$, $z=-t$ and $t\to0$, then the ratio converges to $-\frac27$. In both cases, $(x,y,z)\to(0,0,0)$ when $t\to0$. The two limits are not equal hence the limit when $(x,y,z)\to(0,0,0)$ does not exist.</p>
|
2,314,327 | <p>I have a quick question here.</p>
<p>For an exercise, I was asked to factor:</p>
<p>$$11x^2 + 14x - 2685 = 0$$</p>
<p>How do I figure this out quickly without staring at it forever? Is there a quicker mathematical way than guessing number combinations, or do I have to guess until I find the right combination of n... | CopyPasteIt | 432,081 | <p>As mentioned in a comment, you can take all the fun out of this by just using the quadratic equation. But, OK, I'll bite!!!</p>
<p>So we assume that we don't know that formula and that the polynomial factors over the rational numbers (or alternatively, we can't miss a chance of employing twisted logic and factoring ... |
1,305,257 | <p>I do not understand how to use the following information: If $f$ is entire, then </p>
<p>$$\lim _{|z| \rightarrow \infty} \frac{f(z)}{z^2}=2i.$$</p>
<p>So if $f$ is entire, it has a power series around $z_0=0$, so $f(z)=\Sigma_{n=0}^\infty a_nz^n$, and then we get </p>
<p>$$\lim _{|z| \rightarrow \infty} \frac{\S... | HallsofIvy | 244,198 | <p>One could also note that the derivative of an even degree polynomial is an odd degree polynomial. And an odd degree polynomial always has at least one zero.</p>
|
4,216,105 | <p>In the <a href="https://www.feynmanlectures.caltech.edu/I_22.html#Ch22-S5" rel="nofollow noreferrer">Algebra chapter</a> of the Feynman Lectures on Physics, Feynman introduces complex powers:</p>
<blockquote>
<p>Thus <span class="math-container">$$10^{(r+is)}=10^r10^{is}\tag{22.5}$$</span>
But <span class="math-con... | J.G. | 56,861 | <p>As you say, the <strong>only</strong> defining property of <span class="math-container">$i$</span> is <span class="math-container">$i^2=-1$</span>. If you replace <span class="math-container">$i$</span> with <span class="math-container">$-i$</span>, everything still works. Therefore, if <span class="math-container">... |
4,196,185 | <p>Let <span class="math-container">$A$</span> be a <span class="math-container">$n\times n$</span> matrix with minimal polynomial <span class="math-container">$m_A(t)=t^n$</span>, i.e. a matrix with <span class="math-container">$0$</span> in the main diagonal and <span class="math-container">$1$</span> in the diagonal... | jjagmath | 571,433 | <p>For (4), assuming <span class="math-container">$c>0$</span>, we have the solution <span class="math-container">$f(x) = (c^{x/c}\,\Gamma(x/c))^d$</span>. Of course, (3) is the particular case of <span class="math-container">$d=1$</span>.</p>
|
3,287,424 | <p>I have a function <span class="math-container">$$f(z)=\begin{cases}
e^{-z^{-4}} & z\neq0 \\
0 & z=0
\end{cases}$$</span></p>
<p>I have to show cauchy riemann equation is satisfied everywhere. I have shown that it isn't differentiable at <span class="math-container">$z=0$</span>. </p>
<p>Usually I will hav... | Kavi Rama Murthy | 142,385 | <p>This is quite easy. For example, <span class="math-container">$\frac {f(h+i0)-f(0)} h=\frac {e^{-h^{-4}}} h$</span> and the limit as <span class="math-container">$h \to 0$</span> through real values is <span class="math-container">$0$</span>. [ <span class="math-container">$e^{x^{4}} \to \infty$</span> faster than ... |
345,888 | <p>$S$ is vector subspace of $S$ if $S$ is vector space, by hypothesis $S$ is vector space then $S$ is vector subspace of $S$.</p>
<p>But I prove it by contradiction, then $S$ is not vector subspace of $S$, but if $S$ is not vector subspace of $S$ then $S$ is not vector space but I have contraddiction, in fact by hypo... | DonAntonio | 31,254 | <p>$$f'_r=\sum_{-\infty}^\infty |n|c_nr^{|n|-1}e^{in\theta}\;,\;\;f''_{rr}=\sum_{-\infty}^\infty |n|\,(|n|-1)c_n\,r^{|n|-2}e^{in\theta}$$</p>
<p>$$f'_\theta=i\sum_{-\infty}^\infty n\,c_nr^{|n|}e^{in\theta}\;,\;\;\;f''_{\theta\theta}=-\sum_{-\infty}^\infty n^2\,c_nr^{|n|}e^{in\theta}$$</p>
<p>So Laplace's equation in ... |
3,317,728 | <p>Suppose that the moment generating function <span class="math-container">$M_X$$(t)$</span> of a random variable <span class="math-container">$X$</span> is given by </p>
<p><span class="math-container">$$ M_X(t)=\frac{e^t+e^{-t}}{6} + \frac 23 $$</span></p>
<p>I need to find the distribution function <span class="... | Honza | 678,826 | <p>A moment generating function of the form <span class="math-container">$M(e^t)$</span> can be easily converted into a probability generating function of the original, integer-valued random variable, just by replacing <span class="math-container">$e^t$</span> by z. This yields <span class="math-container">$P(z)=\fr... |
3,354,990 | <p>I have points and limits of a function and even the shape of the function and I'm looking for the function, something that very interesting for me how could I control the curve of the function?</p>
<p>(1) <span class="math-container">$\lim\limits_{x \to inf} f(x) = 1 $</span></p>
<p>(2) <span class="math-container... | Fareed Abi Farraj | 584,389 | <p>This function satisfies all conditions except one of them, but maybe it'll help you, try <span class="math-container">$f(x)= -\frac{1}{x+1} +1$</span></p>
|
2,930,413 | <p>The problem is as shown. I tried using gradient and Hessian but can not make any conclusions from them. Any ideas?</p>
<p><span class="math-container">$$\max x_1^{a_1}x_2^{a_2}\cdots x_n^{a_n}$$</span></p>
<p>subject to</p>
<p><span class="math-container">$$\sum_{i=1}^nx_i=1,\quad x_i\geq 0,\quad i=1,2,\ldots,n,$... | Doug M | 317,162 | <p>you could say</p>
<p><span class="math-container">$x'' = kx\\
x = C_1 e^{(\sqrt k)t} + C_2 e^{-(\sqrt k)t}$</span></p>
<p><span class="math-container">$k = -1\\
\sqrt k = i\\
x = C_1 e^{it} + C_2 e^{-it}\\
e^{it} = \cos t+ i\sin t\\
x = A\cos t + B\sin t$</span></p>
<p>Or you could say </p>
<p>let <span class="m... |
743,473 | <p>A long Weierstrass equation is an equation of the form
$$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$
Why are the coefficients named $a_1, a_2, a_3, a_4$ and $a_6$ in this manner, corresponding to $xy, x^2, y, x$ and $1$ respectively? Why is $a_5$ absent?</p>
| Henry | 6,460 | <p>If you find the formulae too mind-bending, suppose you have $1000$ people with these probabilities representing proportions.</p>
<p>Then you would have $100$ left-handed people, of which $45$ would be left-handed males and $55$ left-handed females.</p>
<p>You would also have $900$ right-handed people, of which $45... |
288,051 | <p>In enumerative combinatorics, a <i>bijective proof</i> that $|A_n| = |B_n|$ (where $A_n$ and $B_n$ are finite sets of combinatorial objects of size $n$) is a proof that constructs an explicit bijection between $A_n$ and $B_n$.
Bijective proofs are often prized because of their beauty and because of the insight that ... | FindStat | 113,201 | <p>As mentioned in the comments, the <a href="http://findstat.org" rel="noreferrer">FindStat</a> project is aiming at what you want. Concerning the size: it contains currently about 1000 'combinatorial statistics', that is maps $s:\mathcal C_n\to \mathbb Z$ on some (graded) set of 'combinatorial' objects $\mathcal C_n... |
1,658,284 | <p>So a friend shows me this :</p>
<p>$x^4= x^2+x^2+ \cdots +x^2 $ ( i.e. $x^2$ added $x^2$ times)</p>
<p>Now take the derivative of both side;</p>
<p>$4x^3 = 2x + 2x + \cdots + 2x $;</p>
<p>So $4x^3 = 2x^3 \cdots $(1)</p>
<p>And so dividing by $x^3$ gives $2=1 \cdots $(2).</p>
<p>I know we can't divide by 0 ... | Gregory Grant | 217,398 | <p>You can only do the first step if $x^2$ is an positive integer. So since it doesn't hold for all $x$ you can't take the derivative of both sides like that. It doesn't even make sense to talk about "$x^2$ times" if $x^2$ is not a natural number. To take the derivative of both sides you'd need equality in an entire ... |
1,658,284 | <p>So a friend shows me this :</p>
<p>$x^4= x^2+x^2+ \cdots +x^2 $ ( i.e. $x^2$ added $x^2$ times)</p>
<p>Now take the derivative of both side;</p>
<p>$4x^3 = 2x + 2x + \cdots + 2x $;</p>
<p>So $4x^3 = 2x^3 \cdots $(1)</p>
<p>And so dividing by $x^3$ gives $2=1 \cdots $(2).</p>
<p>I know we can't divide by 0 ... | SchrodingersCat | 278,967 | <ol>
<li>First of all, the statement </li>
</ol>
<blockquote>
<p>$x^2$ added $x^2$ times</p>
</blockquote>
<p>makes sense only if $x^2$ is a positive integer. Else if $x^2$ is not a positive integer, then the statement is meaningless.</p>
<ol start="2">
<li>Moreover, from $(1)$, we have that $4x^3=2x^3 \Rightarrow... |
256,612 | <p>I've found assertions that recognising the unknot is NP (but not explicitly NP hard or NP complete). I've found hints that people are looking for untangling algorithms that run in polynomial time (which implies they may exist). I've found suggestions that recognition and untangling require exponential time. (Untangl... | Carlo Beenakker | 11,260 | <p>This <a href="https://arxiv.org/abs/1211.1079" rel="nofollow noreferrer">2012 report</a> of <em>"A fast branching algorithm for unknot recognition with experimental polynomial time behaviour"</em> by B. Burton and M. Ozlen may well represent the current status of the problem:</p>
<blockquote>
<p>It is a major uns... |
95,314 | <p>To evaluate this type of limits, how can I do, considering $f$ differentiable, and $ f (x_0)> 0 $</p>
<p>$$\lim_{x\to x_0} \biggl(\frac{f(x)}{f(x_0)}\biggr)^{\frac{1}{\ln x -\ln x_0 }},\quad\quad x_0>0,$$</p>
<p>$$\lim_{x\to x_0} \frac{x_0^n f(x)-x^n f(x_0)}{x-x_0},\quad\quad n\in\mathbb{N}.$$</p>
| David Mitra | 18,986 | <p>For the first problem, one may be tempted to use L'Hopital's Rule:</p>
<p>Noting that $f(x)>0$ for $x$ sufficiently close to $x_0$:</p>
<p>$$\ln \biggl[\,\Bigl ({f(x)\over f(x_0)}\Bigr)^{1\over \ln x-\ln x_0}\,\biggr] = {\ln f(x)-\ln f(x_0)\over \ln x-\ln x_0}.$$</p>
<p>We have:
$$\tag{1}\lim_{x\rightarrow x_0... |
4,236,148 | <p><span class="math-container">$R=\{(x,y):x^2=y^2\}$</span> and I have to determine whether its an equivalence relation.</p>
<p>I found that it's reflexive but for the symmetry part I got confused as <span class="math-container">$x=y$</span> is sometimes said to be symmetric others not so I don't know what to take it ... | spinosarus123 | 958,184 | <p>Certainly if <span class="math-container">$x^2=y^2$</span> then <span class="math-container">$y^2=x^2$</span>, so it is symmetric. It is also transitive because if <span class="math-container">$x^2=y^2$</span> and <span class="math-container">$y^2=z^2$</span>, then <span class="math-container">$x^2=z^2$</span>.</p>
|
688,742 | <p>Given $P\colon\mathbb{R} \to \mathbb{R}$ , $P$ is injective (one to one) polynomial function i need to formally prove that $P$ is onto $\mathbb{R}$</p>
<p>my strategy so far .......
polynomial function is continuous and since it one-to-one function it must be strictly monotonic and now i have no idea what to do ..... | Clive Newstead | 19,542 | <p>The limit of a constant is just the value of constant, and when $\lim f(x)$ and $\lim g(x)$ both exist they satisfy
$$\lim(f(x)+g(x)) = \lim f(x) + \lim g(x)$$
$$\lim(f(x)g(x)) = \lim f(x) \cdot \lim g(x)$$
In other words, here, you have
$$\lim \frac{11-e^{-x}}{7} = \lim \frac{1}{7} \cdot \lim(11-e^x) = \lim \frac{1... |
688,742 | <p>Given $P\colon\mathbb{R} \to \mathbb{R}$ , $P$ is injective (one to one) polynomial function i need to formally prove that $P$ is onto $\mathbb{R}$</p>
<p>my strategy so far .......
polynomial function is continuous and since it one-to-one function it must be strictly monotonic and now i have no idea what to do ..... | k170 | 161,538 | <p>First note that
$$ \lim\limits_{x\to\infty} f(x)= f\left(\lim\limits_{x\to\infty} x\right) $$
And
$$ \lim\limits_{x\to\infty} e^{-x}= \lim\limits_{x\to\infty} \frac{1}{e^{x}} =0$$
Therefore
$$\lim\limits_{x\to\infty} \frac{11 - e^{-x}}{7} = \frac{11 - \lim\limits_{x\to\infty} e^{-x}}{7} =\frac{11}{7}$$</p>
|
69,590 | <p>Consider the following code.</p>
<pre><code>f[a_,b_]:=x
x=a+b;
f[1,2]
(* a + b *)
</code></pre>
<p>From a certain viewpoint, one might expect it to return <code>3</code> instead of <code>a + b</code>: the symbols <code>a</code> and <code>b</code> are defined during the evaluation of <code>f</code> and <code>a+b</c... | Basheer Algohi | 13,548 | <p>check : </p>
<pre><code>f[1,2]//Trace
</code></pre>
<p>You will see that 1 & 2 are passed first before replacing x with its value.</p>
<p>If you want to get your result then use <code>Set</code> not <code>SetDelayed</code></p>
<pre><code>Clear[a,b];
x=a+b;
f[a_,b_]=x;
f[1,2]
(*3*)
</code></pre>
|
265,047 | <p>Let $X$ be a Banach space and let $T:X \rightarrow X$ be a bounded linear map. Show that: If $T$ is surrjective then its transpose $T':X' \rightarrow X'$ is bounded below.</p>
<p>My try: We know that $R^\perp_M = N_{M'}$ and since X is surrjective
$R_M = X$ hence $R_M^\perp = N_{M'} = 0$ so $M'$ is invertible and... | André Nicolas | 6,312 | <p>There has been a clarification, which changes the answer. We are looking at
$$\sum_{n=1}^\infty\frac{\ln n+(-1)^nn^{\frac{1}{2}}}{n\cdot n^{\frac{1}{2}}}.$$
This can be split as
$$\sum_{n=1}^\infty\frac{\ln n}{n^{3/2}} +\sum_{n=1}^\infty \frac{(-1)^n}{n}$$</p>
<p>The second is an alternating series, which converge... |
265,047 | <p>Let $X$ be a Banach space and let $T:X \rightarrow X$ be a bounded linear map. Show that: If $T$ is surrjective then its transpose $T':X' \rightarrow X'$ is bounded below.</p>
<p>My try: We know that $R^\perp_M = N_{M'}$ and since X is surrjective
$R_M = X$ hence $R_M^\perp = N_{M'} = 0$ so $M'$ is invertible and... | Mhenni Benghorbal | 35,472 | <p>Note that
$$\sum_{n=1}^\infty\frac{\ln n+(-1)^nn^{\frac{1}{2}}}{n\cdot n^{\frac{1}{2}}}= \sum_{n=1}^\infty\frac{\ln n }{ n^{\frac{3}{2}} } + \sum_{n=1}^{\infty}\frac{ (-1)^n }{n}\,. $$</p>
<p>Now, the first series on the RHS converges by the <a href="http://en.wikipedia.org/wiki/Integral_test_for_convergence" rel=... |
3,186,239 | <p>Two Independent variables have Bernoulli distribution:
<span class="math-container">$X_1$</span> with <span class="math-container">$b(n,p)$</span> and <span class="math-container">$X_2$</span> with <span class="math-container">$b(m,p)$</span>.
How can I find conditional distribution <span class="math-container">$\ma... | callculus42 | 144,421 | <p><strong>Hint:</strong> I assume that <span class="math-container">$X_1$</span> and <span class="math-container">$X_2$</span> are binomial distributed. We have <span class="math-container">$X_2=t-X_1$</span>. Now you can apply the Bayes theorem. Due independency of <span class="math-container">$X_1$</span> and <span ... |
3,186,239 | <p>Two Independent variables have Bernoulli distribution:
<span class="math-container">$X_1$</span> with <span class="math-container">$b(n,p)$</span> and <span class="math-container">$X_2$</span> with <span class="math-container">$b(m,p)$</span>.
How can I find conditional distribution <span class="math-container">$\ma... | P. Quinton | 586,757 | <p>Assuming you meant Binomial random variables and that they are independents, you can always write <span class="math-container">$X_1=\sum_{i=1}^n Y_i$</span> and <span class="math-container">$X_2=\sum_{i=n+1}^{n+m} Y_i$</span> where the random variables are independent Bernoulli random variable with parameter <span ... |
3,014,438 | <p>Find Number of Non negative integer solutions of <span class="math-container">$x+2y+5z=100$</span></p>
<p>My attempt: </p>
<p>we have <span class="math-container">$x+2y=100-5z$</span> </p>
<p>Considering the polynomial <span class="math-container">$$f(u)=(1-u)^{-1}\times (1-u^2)^{-1}$$</span></p>
<p><span class=... | xpaul | 66,420 | <p>Note that the number of non-negative integer solutions of the following equation
<span class="math-container">$$ x+y=n $$</span>
is <span class="math-container">$n+1$</span>. Here <span class="math-container">$n$</span> is a non-negative integer. Clearly <span class="math-container">$5|(x+2y)$</span>. Let
<span cla... |
3,014,438 | <p>Find Number of Non negative integer solutions of <span class="math-container">$x+2y+5z=100$</span></p>
<p>My attempt: </p>
<p>we have <span class="math-container">$x+2y=100-5z$</span> </p>
<p>Considering the polynomial <span class="math-container">$$f(u)=(1-u)^{-1}\times (1-u^2)^{-1}$$</span></p>
<p><span class=... | farruhota | 425,072 | <p>Given: <span class="math-container">$x+2y=100-5z$</span>, tabulate:
<span class="math-container">$$\begin{array}{c|c|c}
z&x&\text{count}\\
\hline
0&100,98,\cdots, 0&\color{red}{51}\\
1&\ \ 95,93,\cdots, 1&\color{blue}{48}\\
2&\ \ 90,88,\cdots, 0&\color{red}{46}\\
3&\ \ 85,83,\cdot... |
1,512,528 | <p>As the title says, I'm looking to find all solutions to $$x^2 \equiv 4 \pmod{91}$$ and I am not exactly sure how to proceed.</p>
<p>The hint was that since 91 is not prime, the Chinese Remainder Theorem might be useful.</p>
<p>So I've started by separating into two separate congruences:
$$x^2 \equiv 4 \pmod{7}$$ $... | Clément Guérin | 224,918 | <p>By the Chinese remainder theorem you get an isomorphism between those two rings :</p>
<p>$$\psi:\frac{\mathbb{Z}}{91\mathbb{Z}}\rightarrow \frac{\mathbb{Z}}{7\mathbb{Z}}\times \frac{\mathbb{Z}}{13\mathbb{Z}} $$</p>
<p>$$x\mapsto (x\text{ mod } 7,x\text{ mod } 13) $$</p>
<p>it means that to solve $4=s^2$ mod $91$ ... |
1,862,232 | <p>I'm studying basic Ring Theory. And in my textbook, the author states the definition of Euclidean domain:<br>
An integral domain $R$ is called to be a <em>Euclidean domain</em> precisely when there is a function $f: R\setminus\{0\}\rightarrow\Bbb N_0$, called degree function of $R$, such that:<br>
(i) If $a,b \in R\... | Sol He | 998,296 | <p>Proving all units having degree 0 somehow means for any function f, they have degree 0. But I can always define a new degree function by adding 1 to an existing one. Now no one has degree 0.</p>
|
3,349,630 | <p>If a,b,c are positive real numbers,prove that
<span class="math-container">$$ \frac{a}{b+2c} + \frac{b}{c+2a} + \frac{c}{a+2b} \ge 1 $$</span>
I tried solving and i have no idea how to proceed I mechanically simplified it it looks promising but im still stuck. This is from the excersice on Cauchy Schwartz Inequality... | IamKnull | 610,697 | <p>As we have, <span class="math-container">$2+a^3=a^3+1+1\geqslant 3a$</span>, <span class="math-container">$b^2+1\geqslant 2b$</span>, thus<span class="math-container">$$\dfrac{a}{a^3+b^2+c}=\frac{a}{3+a^3+b^2-a-b}\leqslant\frac{a}{3a+2b-a-b}=\frac{a}{2a+b}.$$</span>Similarly, we can get <span class="math-container">... |
8,023 | <p>I'm looking for an easily-checked, local condition on an $n$-dimensional Riemannian manifold to determine whether small neighborhoods are isometric to neighborhoods in $\mathbb R^n$. For example, for $n=1$, all Riemannian manifolds are modeled on $\mathbb R$. When $n=2$, I believe that it suffices for the scalar c... | Danny Calegari | 1,672 | <p>Greg's comment on Deane's answer is sort of correct (given suitable hypotheses), but maybe a bit misleading in the context of this discussion. Since the character count doesn't allow it, I'm adding this comment as an "answer" (though it is not an answer to the original question).</p>
<p>There are non-isometric 2-sp... |
4,383,800 | <p>I can already see that the <span class="math-container">$\lim_\limits{n\to\infty}\frac{n^{n-1}}{n!e^n}$</span> converges by graphing it on Desmos, but I have no idea how to algebraically prove that with L’Hopital’s rule or induction. Where could I even start with something like this?</p>
<p>Edit: For context, I came... | Salcio | 821,280 | <p>First of, note that from the A-G inequality one gets <span class="math-container">$(1+1/(n+1))^{n+1} > (1+/1/n)^n$</span>
In words, that the sequence <span class="math-container">$a_n = (1+1/n)^n$</span> is increasing. One can see this by substituting <span class="math-container">$b_1=1$</span> and <span class="... |
225,866 | <p>If I define, for example,</p>
<pre><code>f[OptionsPattern[{}]] := OptionValue[a]
</code></pre>
<p>Then the output for <code>f[a -> 1]</code> is 1.</p>
<p>However, in my code, I have a function that must be called using the syntax <code>f[some parameters][some other parameters]</code>, and I want to add options to... | flinty | 72,682 | <p>I can see at least three ways:</p>
<pre><code>Through[{x, y, z}[#]] & /@ vl
</code></pre>
<p>Or alternatively:</p>
<pre><code>Transpose[# /@ vl & /@ {x, y, z}]
</code></pre>
<p>Or alternatively:</p>
<pre><code>Outer[#2[#1]&, vl, {x, y, z}]
</code></pre>
|
2,733,728 | <p>How can one find a general form for $\int_0^1 \frac {\log(x)}{(1-x)} dx=-\zeta(2)
\,?$ Namely $\int_0^1 \frac {\log^n(x)}{(1-x)^m} dx\,$ where $n,m\ge1$ Similar to the original integral I let $1-x=u\,$ which gives $$\int_{-1}^0 \frac {\log^n(1+x)}{x^m} dx$$ and expanding into series we have: $\int_{-1}^0x^{-m}(\s... | gandalf61 | 424,513 | <blockquote>
<p>"... if an action takes up $0$ time, then surely this action never happened ..."</p>
</blockquote>
<p>This is where your error lies. In classical physics we assume that space and time are continuous and can be divided into parts that are as small as we like. Under these assumptions it is possible for... |
441,404 | <p>If H is a Hilbert space, Is B(H) under the operator norm a Hilbert space?
If not, is there exists any norm on B(H) that makes it a Hilbert space?</p>
| bradhd | 5,116 | <p>To call something a Hilbert space means that it is equipped with a complete <em>inner product</em>, not just a norm. A complete normed vector space is called a <em>Banach space</em>, and indeed $B(H)$ with the operator norm is a Banach space. Indeed, it has some additional structure (multiplication given by composit... |
441,404 | <p>If H is a Hilbert space, Is B(H) under the operator norm a Hilbert space?
If not, is there exists any norm on B(H) that makes it a Hilbert space?</p>
| Josse van Dobben de Bruyn | 246,783 | <p>For your first question: we have that $B(H)$ is a Hilbert space if and only if $\dim(H) \leq 1$ holds.</p>
<ul>
<li>If $\dim(H) \leq 1$ holds, then it is clear that $B(H)$ is a Hilbert space.</li>
<li>If $\dim(H) > 1$ holds, then we may choose $x,y\in H$ with $\lVert x \rVert = \lVert y \rVert = 1$ and $\langle ... |
428,530 | <p>Let $\Omega := [0, 1] \times [0,\pi]$. We are searching for a function $u$ on $\Omega$ s.t.
$$
\Delta u =0
$$
$$
u(x,0) = f_0(x), \quad u(x,1) = f_1(x), \quad u(0,y) = u(\pi,y) = 0
$$ with
$$
f_0(x) = \sum_{k=1}^\infty A_k \sin kx \quad, f_1(x) = \sum_{k=1}^\infty B_k \sin kx
$$
If I use seperation of variables, ... | Avitus | 80,800 | <p>Hint: which is the most general solution of </p>
<p>$$f''(x)=-\lambda f(x)$$</p>
<p>and</p>
<p>$$g''(y)=\lambda g(y)?$$</p>
<p>You need to consider linear combinations of exponentials. Such exponentials have real or complex exponents depending on the sign of $\lambda$, i.e. $\lambda >0$, $\lambda<0$ (not n... |
76,600 | <p>The group of three dimensional rotations $SO(3)$ is a subgroup of the Special Euclidean Group $SE(3) = \mathbb{R}^3 \rtimes SO(3)$. The manifold of $SO(3)$ is the three dimensional real projective space $RP^3$. Does $RP^3$ cause a separation of space in the manifold of $SE(3)$? </p>
<p>(edit) Sorry about lack of cl... | Sai | 14,667 | <p>I cannot post comments yet, but I am interested in the answer to these questions. It appears $R^2 \times SO(3)$ will not partition $SE(3)$ into disconnected pieces because $R^2 \times SO(3)$ is not compact. What about the set $M \times RP^3$ where $M$ is the Mobius strip? That is a five dimensional surface. Does it... |
2,542,184 | <p>Is $A = \begin{bmatrix}
1&1&0\\
0&1&0\\
0&0&1\\
\end{bmatrix}$
and $B = \begin{bmatrix}
1&1&0\\
0&1&1\\
0&0&1\\
\end{bmatrix}$ similar? Please justify your answer.</p>
<p>So far what I've done is to check rank, det, trace, and characteristic polynomial to maybe dispr... | Community | -1 | <p>If $A$ and $B$ are similar, $A-I$ and $B-I$ must be similar too, which means that they must have the same rank. This is not the case here, thus $A$ and $B$ are not similar.</p>
|
2,130,658 | <p>How would I go about proving this mathematically? Having looked at a proof for a similar question I think it requires proof by induction. </p>
<p>It seems obvious that it would be even by thinking about the first few cases. As for $n=0$ there will be no horizontal dominoes which is even, and for $n=1$ there can onl... | Chirantan Chowdhury | 337,567 | <p>Mathematically you can design $t_n$ to be the number of horizontal dominoes and then solve the recursion $ t_n = 2 + t_{n-2}$ for with inital values $ t_1 = 0 , t_2 = 2$.</p>
|
1,088,338 | <p>There are at least a few things a person can do to contribute to the mathematics community without necessarily obtaining novel results, for example:</p>
<ul>
<li>Organizing known results into a coherent narrative in the form of lecture notes or a textbook</li>
<li>Contributing code to open-source mathematical softw... | Steven Gubkin | 34,287 | <p>As @Hennobrandsma said, you can teach mathematics well.</p>
<p>In addition to developing tools which aid mathematics research (as you mention), you can also work to develop tools which aid teaching mathematics. </p>
<p>Without coming up with "new mathematics", you can apply mathematics in novel ways to other fiel... |
1,088,338 | <p>There are at least a few things a person can do to contribute to the mathematics community without necessarily obtaining novel results, for example:</p>
<ul>
<li>Organizing known results into a coherent narrative in the form of lecture notes or a textbook</li>
<li>Contributing code to open-source mathematical softw... | Federico Poloni | 65,548 | <p>Another way to contribute that hasn't been mentioned yet is <strong>advocacy</strong>. It is important to raise awareness on the importance of mathematics and scientific literacy in general. For instance, here are a few concepts that it would be useful to disseminate to the general public:</p>
<ul>
<li>that some kn... |
3,541,947 | <p>How do you pronounce <span class="math-container">$\mathbb{F}_2, \mathbb{F}_2^n, \mathbb{N}^k, [n] = \{1,\ldots,n\},$</span> and <span class="math-container">$S \subseteq [n]$</span> when you're reading a text?</p>
<p>I've just started reading more advanced math textbooks and these are appearing all the time. </p>
| PrincessEev | 597,568 | <p>It's ultimately a matter of preference and taste. At some point in learning mathematics, it helps to "break away" the notation from spoken language to avoid distracting questions such as this. For instance, you know <span class="math-container">$\Bbb N$</span> refers to the natural numbers (and presumably know what ... |
4,231,509 | <p>I'm trying to prove that the group <span class="math-container">$(\mathbb{R}^*, \cdot)$</span> is not cyclic (similar to [1]). My efforts until now culminated into the following sentence:</p>
<blockquote>
<p>If <span class="math-container">$(\mathbb{R}^*,\cdot)$</span> is cyclic, then <span class="math-container">$\... | Sahan Manodya | 937,663 | <p>Suppose <span class="math-container">$(\mathbb{R}^*,\cdot)$</span> is cylclic. Now <span class="math-container">$\langle2,3\rangle$</span> is a subgroup of <span class="math-container">$(\mathbb{R}^*,\cdot)$</span>, but it is cyclic since it is a subgroup of a cyclic group and therefore <span class="math-container">... |
3,842,739 | <p>Let <span class="math-container">$H$</span> be a group and <span class="math-container">$H^m=\{ h^m \mid h\in H\}$</span>.</p>
<p>I know that this is a subgroup of <span class="math-container">$H$</span> when <span class="math-container">$H$</span> is abelian.
But I want to know that what happens if <span class="ma... | lulu | 252,071 | <p>For any finite simple group of even order, the squares are not a subgroup.</p>
<p>To see this, note the following:</p>
<p>Lemma: If the squares from a group <span class="math-container">$H$</span> are contained in a subgroup <span class="math-container">$G$</span> then <span class="math-container">$G$</span> is nor... |
2,965,459 | <p>Some curves defined by polynomial equations are disconnected over reals but not over complexes, e.g., <span class="math-container">$x y - 1 = 0$</span>. How can we convince someone with background only on equations over reals that the curve drawn by above equation is connected over complexes? Is a plot or something... | Ricardo Buring | 23,180 | <p>A solution over <span class="math-container">$\mathbb{C}$</span> is a pair <span class="math-container">$(z, 1/z)$</span> with <span class="math-container">$z \neq 0$</span>. Consider another solution <span class="math-container">$(w, 1/w)$</span>. There is a path from <span class="math-container">$z$</span> to <spa... |
1,262,322 | <p>Suppose that virus transmision in 500 acts of intercourse are mutually independent events and that the probability of transmission in any one act is $\frac{1}{500}$. What is the probability of infection?</p>
<p>So I do know that one way to solve this is to find the probability of complement of the event we are tryi... | zoli | 203,663 | <p>The general answer is independent of viruses and intercouses. </p>
<p>Let $C_1, C_2$ be two events of the same probability $p$ and the question is the probability that at least one of them occurs.</p>
<p>One can say that</p>
<ol>
<li>$$P(C_1 \cup C_2)=P(C_1)+P(C_2)-P(C_1 \cap C_2).$$
or that</li>
<li>$$P(C_1\cup ... |
696,848 | <p>$\DeclareMathOperator{\rank}{rank}$
First off I'm sorry I'm still not able to make of use the built in formula expressions, I don't have time to learn it now, I'll do it before my next question.</p>
<p>I have a couple of questions regarding eigenvectors and generalized eigenvectors. To some of these questions I kno... | Barry Cipra | 86,747 | <p>Finding an $x$ that violates the given inequality is only one way to disprove the Riemann Hypothesis. Another way would simply be to find a (nontrivial) zero of the zeta function with real part not equal to $1/2$. So far we've "only" computed about ten trillion zeros. The first counterexample could be the very ne... |
85,470 | <p>We decided to do secret Santa in our office. And this brought up a whole heap of problems that nobody could think of solutions for - bear with me here.. this is an important problem.</p>
<p>We have 4 people in our office - each with a partner that will be at our Christmas meal.</p>
<p>Steve,
Christine,
Mark,
Mary,... | Martin Eden | 182,077 | <p>See this algorithm here: <a href="http://weaving-stories.blogspot.co.uk/2013/08/how-to-do-secret-santa-so-that-no-one.html" rel="nofollow">http://weaving-stories.blogspot.co.uk/2013/08/how-to-do-secret-santa-so-that-no-one.html</a>. It's a little too long to include in a Stack Exchange answer.</p>
<p>Essentially, w... |
126,052 | <p>I have no doubt that the following observation is quite well known. Let $\varphi:[0,1]\to [0,1]$ be a continuous map. Assume that the iterates $\varphi^n$ converge pointwise to some continuous map $\varphi_\infty$. Then the convergence is in fact uniform. However, I was unable to locate a reference. Does anybody kno... | gerw | 32,507 | <p>This result seems to be a consequence of Dini's theorem (as noted by nonlinearism):</p>
<p>If $\varphi^n$ converges to a continuous function $\varphi_\infty$, we have $\varphi_\infty = 0$. Hence, $\varphi < 1$ on $[0,1]$. By compactness, we have $\varphi < p$ for some $p < 1$. Therefore, the convergence is... |
3,459,532 | <p>I have a pretty straightforward linear programming problem here:</p>
<p><span class="math-container">$$ maximize \hskip 5mm -x_1 + 2x_2 -3x_3 $$</span></p>
<p>subject to</p>
<p><span class="math-container">$$ 5x_1 - 6x_2 - 2x_3 \leq 2 $$</span>
<span class="math-container">$$ 5x_1 - 2x_3 = 6 $$</span>
<span class... | Ben Grossmann | 81,360 | <p>Another approach, to add to the existing list. Let's suppose that you insist on calculating eigenvalues by finding <span class="math-container">$|A - \lambda I_d|$</span>. We can do so using the <a href="https://en.wikipedia.org/wiki/Weinstein%E2%80%93Aronszajn_identity" rel="nofollow noreferrer">Weinstein-Aronsza... |
592,560 | <p>Let G be an abelian group. Show that, if G is not cyclic, then for all $x\in G$, there is a divisor $d$ of $n = |G|$ which is strictly smaller than n satisfying $x^d=1$. </p>
<p>I'm guessing that this is a consequence of Lagrange's Theorem. We can have that G is a disjoint union of left cosets that all have the sam... | Rodney Coleman | 73,128 | <p>If G is a finite group and x is an element of G, then o(x) divides card G, so there exists d dividing card G such that x^d=1. As here G is not cyclic, d is strictly less than card G. (Remark. "abelian" is superfluous in the statement of your pb.)</p>
|
2,837,281 | <p>I know the splitting field is generated by $2^{1/4}$ and $i$, I could show $\mathbb{Q} [ 2^{1/4}, i] = \mathbb{Q}[i+2^{1/4}]$ using some algebra. </p>
<p>For the non trivial direction $\mathbb{Q} [ 2^{1/4}, i] \subset \mathbb{Q}[i+2^{1/4}]$. Let us call $\alpha = i+2^{1/4}$, then we know
$$(\alpha-i)^4 - 2 = 0$$
e... | Eric Wofsey | 86,856 | <p>Let $K=\mathbb{Q}[2^{1/4},i]$, $L=\mathbb{Q}[i+2^{1/4}]$, and $G=Gal(\mathbb{Q}[2^{1/4},i]/\mathbb{Q})\cong D_8$. By Galois theory, in order to show that $L=K$, it suffices to show that the subgroup $H\subseteq G$ of automorphisms that fix $L$ is trivial. Since $i+2^{1/4}$ generates $L$, we can also describe $H$ a... |
3,756,436 | <p>Recently I was doing a physics problem and I ended up with this quadratic in the middle of the steps:</p>
<p><span class="math-container">$$ 0= X \tan \theta - \frac{g}{2} \frac{ X^2 \sec^2 \theta }{ (110)^2 } - 105$$</span></p>
<p>I want to find <span class="math-container">$0 < \theta < \frac{\pi}2$</span> ... | G. Smith | 573,507 | <p>You were on the right track; you just didn't go far enough. Your first equation relating <span class="math-container">$X$</span> and <span class="math-container">$\theta$</span> is</p>
<p><span class="math-container">$$0=X\tan\theta-\frac{g}{2v^2}X^2\sec^2\theta-h\tag1$$</span></p>
<p>where I've written <span class=... |
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