qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
3,480,123 | <p>In my master thesis, I'm trying to prove the following limit:
<span class="math-container">$$\lim_{\epsilon \to 0^+}\int_0^1 \frac{\left(\ln\left(\frac{\epsilon}{1-x}+1\right)\right)^\alpha}{x^\beta(1-x)^\gamma}\,\mathrm{d}x=0,$$</span>
where <span class="math-container">$\alpha, \beta, \gamma \in (0,1)$</span>.</p>... | Ethan Bolker | 72,858 | <p>Think of the points as vectors in <span class="math-container">$\mathbb{R}^3$</span>.</p>
<p>If they don't lie on a plane through the origin then they will form a basis, and any point can be written uniquely as a linear combination of the three.</p>
|
65,166 | <p>For a graph $G$, let its Laplacian be $\Delta = I - D^{-1/2}AD^{-1/2}$, where $A$ is the adjacency matrix, $I$ is the identity matrix and $D$ is the diagonal matrix with vertex degrees. I'm interested in the spectral gap of $G$, i.e. the first nonzero eigenvalue of $\Delta$, denoted by $\lambda_{1}(G)$.</p>
<p>Is i... | vanvu | 15,028 | <p>For $d$ fixed and $n$ goes to infinity, the neighborhood of each vertex looks like a tree, with high probability, and I guess you can use the traditional trace method. </p>
|
317,753 | <p>I am taking real analysis in university. I find that it is difficult to prove some certain questions. What I want to ask is:</p>
<ul>
<li>How do we come out with a proof? Do we use some intuitive idea first and then write it down formally?</li>
<li>What books do you recommended for an undergraduate who is studying ... | Community | -1 | <p>I started studying Rudin's <strong>Principles of Mathematical Analysis</strong> book for analysis. I just couldn't understand it because I was thinking that definitions are some decorative blah blahs and proofs are some unintelligible mathematician sorceries. </p>
<p>Then I have decided to learn something about pro... |
1,638,757 | <p>Given $2$ points in 2-dimensional space $(x_s,y_s)$ and $(x_d,y_d)$, our task is to find whether $(x_d,y_d)$ can be reached from $(x_s,y_s)$ by making a sequence of zero or more operations.
From a given point $(x, y)$, the operations possible are:</p>
<pre><code>a) Move to point (y, x)
b) Move to point (x, -y)
c) M... | Hagen von Eitzen | 39,174 | <p>First note that the actions $a, b, c$ can be inverted:
$$ a^{-1}=a,\quad b^{-1}=b,\quad c^{-1}=b\circ c\circ b.$$
Therefore "reachable by zero or more applications of $a, b, c$" is an equivalence relation, which we shall denote by $\sim$.
By applying $b$ if necessary, we see that each $(x,y)\in\Bbb Z$ is equivalent ... |
526,837 | <p>Let $(\Omega, {\cal B}, P )$ be a probability space, $( \mathbb{R}, {\cal R} )$ the usual
measurable space of reals and its Borel $\sigma$- algebra, and $X : \Omega \rightarrow \mathbb{R}$ a random variable.</p>
<p>The meaning of $ P( X = a) $ is intuitive when $X$ is a discrete random variable, because it's the d... | Carlos Eugenio Thompson Pinzón | 99,344 | <p>Let $u=1+2x$ then $du=2dx$ and $2-2u=-4x$. Making the substitution
\begin{align}\int\frac{-4x}{1+2x}dx&=\int\frac{2-2u}{u}\frac{du}2\\&=\int\left(\frac1u-1\right)du\\&=\ln|u|-u+C\end{align}
Substituting back we have:
\begin{align}\int\frac{-4x}{1+2x}dx&=\ln|1+2x|-(1+2x)+C\\&=-2x+\ln|1+2x|+C-1\en... |
4,218,943 | <p>Let <span class="math-container">$A_n$</span> be a sequence of <span class="math-container">$d\times d$</span> symmetric matrices, let <span class="math-container">$A$</span> be a <span class="math-container">$d\times d$</span> symmetric positive definite matrix (matrix entries are assumed to be real numbers). Assum... | Hans Engler | 9,787 | <p>Yes, since the coefficients of the characteristic polynomials of the <span class="math-container">$A_n$</span> converge to those of the char. polynomial of <span class="math-container">$A$</span>, and the roots of a polynomial depend continuously on the coefficients.</p>
|
927,261 | <p>I was doing a presentation on Limits and I was using this $$f(x)=\frac{x^2+2x-8}{x^2-4}$$ to explain different types of limits. </p>
<p>I know that the function is not defined at $x=-2$ or $x=2$. I showed the graph and everyone was ok with the graph at $x=-2$ but one member of the audience didn't like how the grap... | ShakesBeer | 168,631 | <p>Nope, that definitely shouldn't be happening. I think it must be an error with arithmetic with very small numbers, as computers only have so much precision. (You've used $|x-2|<10^{-7}$ so $|x^2-4|<10^{-14}$ which is the range you get problems in).</p>
<p>EDIT: the above is actually slightly wrong. $|x-2|<... |
3,705,580 | <p><span class="math-container">$\mathbf {The \ Problem \ is}:$</span> Let, <span class="math-container">$f,g,h$</span> be three functions defined from <span class="math-container">$(0,\infty)$</span> to <span class="math-container">$(0,\infty)$</span> satisfying the given relation <span class="math-container">$f(x)g(y... | max | 378,372 | <p>The numerator and denominator of your random variable <span class="math-container">$T$</span> are, in fact, independent, since their parameters do not depend on each other. One of the ways to find the distribution of <span class="math-container">$T$</span> is by conditioning on <span class="math-container">$X_3, X_4... |
3,381,939 | <p>Be T tree order n given:</p>
<p><span class="math-container">$(a)$</span> <span class="math-container">$ 95 < n < 100$</span></p>
<p><span class="math-container">$(b)$</span> Just have vertices with degree 1,3,5</p>
<p><span class="math-container">$(c)$</span> T has twice the vertices of degree 3 that degre... | Noah Schweber | 28,111 | <p>There is no such sentence, and this remains true if we allow <span class="math-container">$\wedge,\vee,\leftrightarrow$</span> as well (thus subsuming the previous result you mention). </p>
<p>In fact, fixing a first-order language <span class="math-container">$\Sigma$</span>, the unique (up to isomorphism) one-ele... |
4,272,964 | <p>I want to solve the equation following in a set of complex numbers:</p>
<p><span class="math-container">$$z^2 + \bar z = \frac 1 2$$</span></p>
<p><strong>My work so far</strong></p>
<p>Apparently I have a problem with transforming equation above into form that will be easy to solve. I tried to multiply sides by <sp... | user | 505,767 | <p>We have that</p>
<p><span class="math-container">$$z^2 + \bar{z} = \frac 1 2 \iff \bar z^2 + z = \frac 1 2$$</span></p>
<p>and then subtracting we obtain</p>
<p><span class="math-container">$$z^2-\bar z^2 +\bar z-z=0 \iff (z-\bar z)(z+\bar z-1)=0$$</span></p>
<p>that is</p>
<p><span class="math-container">$$z=\bar z... |
3,460,595 | <p>I am given the following sequence:</p>
<p><span class="math-container">$$a_n = 1^9 + 2^9 + ... + n^9 - an^{10}$$</span></p>
<p>Where <span class="math-container">$a \in \mathbb{R}$</span>. I have to find the value of <span class="math-container">$a$</span> for which the sequence <span class="math-container">$a_n$<... | Z Ahmed | 671,540 | <p><span class="math-container">$$ \sum_{1}^{n} k^p \sim \frac{n^{p+1}}{p+1}~~~~~~(1)$$</span></p>
<p>So <span class="math-container">$a$</span> needs to be <span class="math-container">$\frac{1}{10}.$</span> for <span class="math-container">$$\frac{1^9+2^9+3^9+4^9+...n^9}{n^{10}}$$</span>
to be convergent.</p>
<p><s... |
2,943,461 | <p>I'm stumped on a math puzzle and I can't find an answer to it anywhere!
A man is filling a pool from 3 hoses. Hose A could fill it in 2 hours, hose B could fill it in 3 hours and hose C can fill it in 6 hours. However, there is a blockage in hose A, so the guy starts by using hoses B and C. When the blockage in hose... | TonyK | 1,508 | <p>Hose <span class="math-container">$A$</span> can fill half a pool per hour.<br>
Hose <span class="math-container">$B$</span> can fill one third of a pool per hour.<br>
Hose <span class="math-container">$C$</span> can fill one sixth of a pool per hour.</p>
<p>So how many pools per hour can hoses <span class="math-co... |
3,433,277 | <blockquote>
<p>It is given
<span class="math-container">$f:\mathbb R \rightarrow \mathbb R$</span>
<span class="math-container">$$f(x):=\tan^{-1}(x+1)+ \cot^{-1}(x)$$</span>
<span class="math-container">$\mathcal R_f=?$</span></p>
</blockquote>
<p>So far, I've learned <span class="math-container">$\tan$</span... | Travis Willse | 155,629 | <p><strong>Hint</strong> Combine the complementarity identity in the question with the arctangent addition identity
<span class="math-container">$$\arctan u \pm \arctan v = \arctan\frac{u \pm v}{1 \mp uv} \pmod \pi .$$</span> (I'm not sure that this approach avoids limits in the strict sense, since finding the range of... |
686,981 | <p>So I have to solve the equation $$y^2=4\tag{1.9.88 unit 3*}$$</p>
<p>I did this: $$y^2=4 \text{ means } \sqrt{y^2}=\sqrt{4}=>y=2$$</p>
<p>But I have a problem, $y$ can be either negative or positive so I need to do: $$\sqrt{y^2}=|y|=2=>y=2- or- y=-2$$</p>
<p>Is it right?</p>
| Anonymous Computer | 128,641 | <p>For this type of problem, I always think, "How would it appear on a graph? What are the x-intercepts?" Of course, I do not have time to actually draw it out. But I can think, "Let $y^2$ be $x^2$. How can $x^2=4$ be turned into a function? I can just move $4$ to the left by subtracting $4$ on both sides, and replacin... |
319,725 | <p>I am trying to prove the following inequality concerning the <a href="https://en.wikipedia.org/wiki/Beta_function" rel="noreferrer">Beta Function</a>:
<span class="math-container">$$
\alpha x^\alpha B(\alpha, x\alpha) \geq 1 \quad \forall 0 < \alpha \leq 1, \ x > 0,
$$</span>
where as usual <span class="math-c... | T. Amdeberhan | 66,131 | <p>This is an attempt to strengthen your claim.</p>
<p>If <span class="math-container">$x$</span> is large then <span class="math-container">$B(x,y)\sim \Gamma(y)x^{-y}$</span> and hence
<span class="math-container">$$B(\alpha x,\alpha)\sim \Gamma(\alpha)(\alpha x)^{-\alpha};$$</span>
where <span class="math-container... |
332,927 | <p>there are two bowls with black olives in one and green in the other. A boy takes 20 green olives and puts in the black olive bowl, mixes the black olive bowl, takes 20 olives and puts it in the green olive bowl. The question is -</p>
<p>Are there more green olives in the black olive bowl or black olive in the green... | André Nicolas | 6,312 | <p>After the two transfers, the number of olives in the left-hand bowl is <strong>unchanged</strong>, and the number of olives in the right-hand bowl is <strong>unchanged</strong>. </p>
<p>So if we look at the left-hand bowl ("green olive bowl") any black olives that end up there must displace exactly as many green ol... |
2,657,053 | <blockquote>
<p>Suppose I know that
$$\sum_{i=1}^n i^2=\frac{n(n+1)(2n+1)}{6}\,\,\,\, \tag{1} $$
How can I prove the the following?
$$
\sum_{i=0}^{n-1} i^2=\frac{n(n-1)(2n-1)}{6}
$$</p>
</blockquote>
<hr>
<p>I have looked up the solution to the other problem but it seems to be a bit confusing to me. Is it pos... | Hypergeometricx | 168,053 | <p>Put $n=N-1$ in $(1)$:</p>
<p>$$\sum_{i=1}^{N-1}i^2=\frac {\overbrace{(N-1)}^n\ \overbrace{N}^{n+1}\ [\overbrace{2(N-1)+1}^{2n+1}]}6=\frac {N(N-1)(2N-1)}6\tag{2}$$
Now write $n$ instead of $N$ in $(2)$ above:
$$\sum_{i=1}^{n-1}i^2=\frac {n(n-1)(2n-1)}6$$
Adding $0^2$ gives
$$\sum_{i=0}^{n-1}i^2=\frac {n(n-1)(2n-1)}... |
3,196,238 | <p>Let <span class="math-container">$ A = \left\{ (x,y) \in \mathbb{R^2} \mid y= \sin ( \frac{1}{x}) , \ 0 < x \leq 1 \right\}$</span> . Find <span class="math-container">$\operatorname{Cl} A$</span> in topological space <span class="math-container">$\mathbb{R^2}$</span> with dictionary order topology.</p>
<p>I ... | Henno Brandsma | 4,280 | <p>The plane in the dictionary order is basically <span class="math-container">$\mathbb{R}$</span> many disjoint topological copies of vertical stalks that are homeomorphic to <span class="math-container">$\mathbb{R}$</span>. So the closure can be taken in each “stalk”. Your <span class="math-container">$A$</span> inte... |
9,345 | <p>On meta.tex.sx, I've asked a question about a class of questions that might get asked over there (and have been) that are (i) ostensibly about maths usage, but (ii) might best be served by an answer that is primarily about how to handle the notation in Latex (See <a href="https://tex.meta.stackexchange.com/questions... | bubba | 31,744 | <p>It seems to me that this site is about mathematics, not about the tools that people use to write mathematics. If we're going to start answering LaTeX questions, then I hope that we will also answer questions about Microsoft Word, and Powerpoint, and give advice about the best pencils, paper, and chalk to use, too.</... |
2,054,949 | <p>From what I understand, these three concepts all describe the points where the function is not continuous. How to tell them apart? Thanks!</p>
| reuns | 276,986 | <ul>
<li><p>If $f(z)$ is holomorphic/analytic on $0 < |z-z_0| < r$ then $z_0$ is an isolated singularity. From the <a href="https://math.stackexchange.com/questions/2038892/residue-of-complex-function/2038898#2038898">Cauchy integral formula in an annulus</a> you have the Laurent series $f(z) = \sum_{n=-\infty}^\... |
194 | <p>In many parts of the world, the majority of the population is uncomfortable with math. In a few countries this is not the case. We would do well to change our education systems to promote a healthier relationship with math. But in the present situation, how can we help the students who come to our classes, which the... | Confutus | 40 | <p>It seems that the first step is to diagnose the problem. What, specifically, causes the fear or anxiety? Getting a history of the student's mathematical experience is a reasonable start. Helpful questions include courses the student has taken, what the student liked best or disliked most, what parts were easiest or ... |
1,447,852 | <p>Compute this sum:</p>
<p><span class="math-container">$$\sum_{k=0}^{n} k \binom{n}{k}.$$</span></p>
<p>I tried but I got stuck.</p>
| Servaes | 30,382 | <p>Here's alternative way to get the result. The first thing to note is that
$$\sum_{k=0}^nk\binom{n}{k}=\sum_{k=0}^nk\cdot\frac{n!}{(n-k)!k!}=\sum_{k=1}^nk\cdot\frac{n!}{(n-k)!k!},$$
because the term with $k=0$ is equal to $0$. Next, cancelling the factor $k$ we find that
$$\sum_{k=1}^nk\cdot\frac{n!}{(n-k)!k!}=\sum_{... |
1,447,852 | <p>Compute this sum:</p>
<p><span class="math-container">$$\sum_{k=0}^{n} k \binom{n}{k}.$$</span></p>
<p>I tried but I got stuck.</p>
| vidhan | 240,536 | <p>$$\large S=\sum_{k=0}^{n} k \binom{n}{k}$$
$$\large S=0\binom{n}{0}+1\binom{n}{1}+2\binom{n}{2}+..+(n-1)\binom{n}{n-1}+n\binom{n}{n}$$</p>
<p>$$\large S=n\binom{n}{n}+(n-1)\binom{n}{n-1}+(n-2)\binom{n}{n-2}+..+1\binom{n}{1}+0\binom{n}{0}$$
Adding the above equations,
$$\large 2S=n(\binom{n}{0}+\binom{n}{1}+\binom{n... |
4,036,761 | <blockquote>
<p>Let G be the additive group of all polynomials in <span class="math-container">$x$</span> with integer coefficients.
Show that G is isomorphic to the group <span class="math-container">$\mathbb{Q}$</span>* of all positive rationals (under
multiplication).</p>
</blockquote>
<p>This question is from my ab... | mathcounterexamples.net | 187,663 | <p><strong>Hint</strong></p>
<p>Consider the map</p>
<p><span class="math-container">$$\begin{array}{l|rcl}
\phi : & \mathbb Z[x] & \longrightarrow & \mathbb Q^* \\
& q(x)=q_0 + q_1 x + \dots + q_n x^n& \longmapsto & p_0^{q_0}p_1^{q_0} \cdots p_n^{q_n}\end{array}$$</span></p>
<p>where <span ... |
1,413,363 | <p>The question:</p>
<p>Find values of $a,b,c.$ if $\displaystyle \frac{x^2+1}{x^2+3x+2} = \frac{a}{x+2}+\frac{bx+c}{x+1}$</p>
<p>My working so far:</p>
<p><a href="https://i.imgur.com/VegifVa.jpg" rel="nofollow noreferrer">http://i.imgur.com/VegifVa.jpg</a></p>
<p>How do I isolate $a$, $b$ and $c$?</p>
| Mark Viola | 218,419 | <p>For the right-hand side, let's obtain a common denominator. To that end, we obtain</p>
<p>$$\frac{a}{x+2}+\frac{bx+c}{x+1}=\frac{a(x+1)+(bx+c)(x+2)}{x^2+3x+1}=\frac{bx^2+(a+2b+c)x+(a+2c)}{x^2+3x+1}$$</p>
<p>Equating this last expression to $\frac{x^2+1}{x^2+3x+1}$ we see that</p>
<p>$$b=1$$</p>
<p>$$a+2b+c=0$$<... |
887,327 | <p>I am confused as to how to evaluate the infinite series $$\sum_{n=1}^\infty \frac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n^2+n}}.$$<br>
I tried splitting the fraction into two parts, i.e. $\frac{\sqrt{n+1}}{\sqrt{n^2+n}}$ and $\frac{\sqrt{n}}{\sqrt{n^+n}}$, but we know the two individual infinite series diverge. Now how do I pr... | Hagen von Eitzen | 39,174 | <p>Note that $\frac{\sqrt{n+1}}{\sqrt{n(n+1)}}=\frac1{\sqrt n}$ and $\frac{\sqrt{n}}{\sqrt{n(n+1)}}=\frac1{\sqrt {n+1}}$. KEyword: Telescope</p>
|
887,327 | <p>I am confused as to how to evaluate the infinite series $$\sum_{n=1}^\infty \frac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n^2+n}}.$$<br>
I tried splitting the fraction into two parts, i.e. $\frac{\sqrt{n+1}}{\sqrt{n^2+n}}$ and $\frac{\sqrt{n}}{\sqrt{n^+n}}$, but we know the two individual infinite series diverge. Now how do I pr... | Darth Geek | 163,930 | <p><strong>Hint:</strong></p>
<p>$$\frac{\sqrt{n+1} - \sqrt{n}}{\sqrt{n^2+n}} = \frac{\sqrt{n+1} - \sqrt{n}}{\sqrt{n+1}\sqrt{n}} = \frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n+1}}$$</p>
<p>This kind of series is called <a href="http://en.wikipedia.org/wiki/Telescoping_series">Telescoping Series</a></p>
|
2,426,897 | <p>Let $\mathbb{H}$ be the ring of real quaternions and $Z(\mathbb{H})$ be its center. Of course $Z(\mathbb{H})=\mathbb{R}$. </p>
<p>Suppose $a+bi+cj+dk$, $x+yi+zj+wk \in \mathbb{H}$ such that $(a+bi+cj+dk)(x+yi+zj+wk) \in Z(\mathbb{H})$. </p>
<p>Does it imply $(x+yi+zj+wk)(a+bi+cj+dk) \in Z(\mathbb{H})$?</p>
| eyeballfrog | 395,748 | <p>You could use the root extraction algorithm to find it directly. It's sort of like long division.</p>
<ol>
<li>Starting from the decimal place, divide the number into pairs of digits. So $20\,17$.</li>
<li>Find the largest integer whose square is less than the first pair. $4^2 < 20 < 5^2$. This is the first ... |
883,972 | <p>Let:</p>
<p>$$f(n) = n(n+1)(n+2)/(n+3)$$</p>
<p>Therefore :</p>
<p>$$f∈O(n^2)$$</p>
<p>However, I don't understand how it could be $n^2$, shouldn't it be $n^3$? If I expand the top we get $$n^3 + 3n^2 + 2n$$ and the biggest is $n^3$ not $n^2$.</p>
| Community | -1 | <p>$$ \frac{n(n+1)(n+2)}{n+3} = \frac{\Theta(n^3)}{\Theta(n)} = \Theta(n^2)$$</p>
|
4,500,163 | <blockquote>
<p>Take two positive integers <span class="math-container">$a$</span> and <span class="math-container">$b$</span> that are not multiples of <span class="math-container">$5.$</span> Then, construct a list in the following fashion: let the first term be <span class="math-container">$5,$</span> and starting w... | John Omielan | 602,049 | <p>Note your sequence is a <a href="https://en.wikipedia.org/wiki/Linear_recurrence_with_constant_coefficients" rel="nofollow noreferrer">linear recurrence with constant coefficients</a>, i.e.,</p>
<p><span class="math-container">$$n_0 = 5, \; \; n_k = an_{k-1} + b \; \forall \; k \ge 1 \tag{1}\label{eq1A}$$</span></p>... |
1,074,534 | <p>How can I get started on this proof? I was thinking originally:</p>
<p>Let $ n $ be odd. (Proving by contradiction) then I dont know.</p>
| Mathmo123 | 154,802 | <p><strong>Hint:</strong> Try to construct the smallest number with $k>1$ divisors. If it does not have $2$ as a divisor, can it be the smallest?</p>
|
201,060 | <p>I posted this question on Math, but there has been silence there since. So, I wonder if anyone here can get any closer to the answer to my question using Mathematica. Here is the question:</p>
<p>Suppose I draw <span class="math-container">$N$</span> random variables from independent but identical uniform distribut... | MikeY | 47,314 | <p>Here's an interesting counter-example for a <strong><em>discrete uniform</em></strong> distribution does not tend to your shape as <span class="math-container">$N$</span> grows.</p>
<p>Let your r.v. <span class="math-container">$x$</span> be distributed as per a coin toss, taking value <span class="math-container">... |
201,060 | <p>I posted this question on Math, but there has been silence there since. So, I wonder if anyone here can get any closer to the answer to my question using Mathematica. Here is the question:</p>
<p>Suppose I draw <span class="math-container">$N$</span> random variables from independent but identical uniform distribut... | CElliott | 40,812 | <p>According to "Median." Wikipedia, The Free Encyclopedia. 11 Jun. 2019. Web. 3 Jul. 2019, the median of a uniformly distributed variable is (b-a)/2. In general, for the Uniform, Normal, and Exponential distributions, Mathematica reports the same medians as the Wikipedia article. I did not try the other distribution... |
2,871,729 | <p>Given any $\alpha > 0$, I need to show that for $ x \in [0,\infty)$
\begin{equation}
\lim_{x\to 0} x^{\alpha}e^{|\log x|^{1/2}}=0
\end{equation}</p>
<p>I have tried using L'Hospital's rule. But I am not able to arrive at answer. </p>
<p>Thank you in advance.</p>
| marty cohen | 13,079 | <p>Let
$f(x)
=x^{a}e^{|\log x|^{1/2}}
$.
$\ln f(x)
=a\ln x+|\log x|^{1/2}
$.</p>
<p>Let
$x = 1/y$,
so
$y \to \infty$
as
$x \to 0$.</p>
<p>$\ln f(1/y)
=a\ln (1/y)+|\log (1/y)|^{1/2}
=-a\ln (y)+|\log y|^{1/2}
$.</p>
<p>The key is that
$\dfrac{|\log y|^{1/2}}{\ln(y)}
\to 0$
as $y \to \infty$.</p>
<p>Therefore
$\ln f(1... |
2,871,729 | <p>Given any $\alpha > 0$, I need to show that for $ x \in [0,\infty)$
\begin{equation}
\lim_{x\to 0} x^{\alpha}e^{|\log x|^{1/2}}=0
\end{equation}</p>
<p>I have tried using L'Hospital's rule. But I am not able to arrive at answer. </p>
<p>Thank you in advance.</p>
| user | 505,767 | <p>Let $x=e^{-y}\to 0$ as $y\to \infty$ then</p>
<p>$$\large{x^{\alpha}e^{|\log x|^{1/2}}=e^{-\alpha y}\,e^{\sqrt y}=e^{\sqrt y\,-\,\alpha y}\to 0}$$</p>
<p>indeed</p>
<p>$$\sqrt y-\alpha y=y\left(\frac {\sqrt y}y -\alpha\right)=y\left(\frac 1 {\sqrt y}-\alpha\right) \to -\infty$$</p>
|
3,068,782 | <p>The canonical basis is not a Schauder basis of the space of bounded sequences, but in some way, it uniquely determines every element in the space. Is it a basis in a weaker sense? How is it called?</p>
<p>Thanks a lot.</p>
| Sergio | 633,486 | <p>Any sequence in <span class="math-container">$\ell^\infty$</span> can be understood as a operator of <span class="math-container">$\ell^1$</span> sequences. Then we can take a Schauder basis of <span class="math-container">$\ell^1$</span> and describe the behavior of the <span class="math-container">$\ell^\infty$</s... |
3,224,475 | <p>Let <span class="math-container">$\mathbb{Z}_8$</span> be the ring containing elements integer modulo 8 with operation <span class="math-container">$+$</span> and <span class="math-container">$.$</span> being addition and multiplication modulo 8 resp. I want to find <span class="math-container">$a$</span> for every ... | dgould | 451,573 | <blockquote>
<p>How come he came up the time coomplexity is log in just by breaking off binary tree and knowing height is log n</p>
</blockquote>
<p>I'm guessing this is a key part of the question: you're wondering not just "why is the complexity <em>log(n)</em>?", but "why does knowing that the <em>height of the tr... |
3,385,420 | <p>The question is from <em>Cambridge Admission Test 1983</em></p>
<blockquote>
<p>A room contains m men and w women. They leave one by one at random until only people of the same sex remain. show by a carefully explained inductive argument, or otherwise, that the expected number of people remaining is <span class="... | Frostic | 402,923 | <p>Let X be the number of leavers until only men or women remains. Let <span class="math-container">$S_1=1$</span> if first leaver is a man. </p>
<p><span class="math-container">$\mathbb E[X(m,w)] = \mathbb E[X(m-1,w)|S_1=1]P(S_1 = 1) + \mathbb E[X(m,w-1)|S_1=0]P(S_1=0)$</span>
<span class="math-container">$ =\math... |
1,968,978 | <p>Let $f=(f_0,f_1,f_2...)$ and $g=(g_0,g_1,g_2,...)$ be sequences in $F^{\infty}$. We define multiplication $fg$ by expressing the $n$-th component $(fg)_n=\sum_{i=0}^ng_if_{n-i}$. If $h=(h_0,h_1,h_2,...)$ is also in $F^{\infty}$, we want to show multiplication is associative. Hoffman and Kunze give the following calc... | Simply Beautiful Art | 272,831 | <p>Assuming you meant to solve $\frac{y!}{(y-2)!2!}+10y=108$,</p>
<p>$$\frac{y!}{(y-2)!}=y(y-1)=y^2-y$$</p>
<p>$$\frac{y!}{(y-2)!2!}+10y=0.5y^2+9.5y=108$$</p>
<p>$$y^2+19y=216$$</p>
<p>Quadratic, with solution given as</p>
<p>$$y=8,-27$$</p>
<p>Usually $y>0$, so then $y=8$.</p>
|
612,681 | <p>I am a little confused about the decidablity of validity of first order logic formulas. I have a textbook that seems to have 2 contradictory statements. </p>
<p>1)Church's theorem: The validity of a first order logic statement is undecidable.
(Which I presume means one cannot prove whether a formula is valid or not... | dezakin | 89,487 | <p>First order logic isn't <i>undecidable</i> exactly, but rather often referred to as <i>semidecidable.</i> A valid first order statement is always provably valid. This is a result of the completeness theorems. For all valid statements, there is a decidable, sound and complete proof calculus.</p>
<p>However, satisfia... |
440,583 | <p><strong>Question.</strong> Is there an entire function <span class="math-container">$F$</span> satisfying first two or all three of the following assertions:</p>
<ul>
<li><span class="math-container">$F(z)\neq 0$</span> for all <span class="math-container">$z\in \mathbb{C}$</span>;</li>
<li><span class="math-contain... | Alexandre Eremenko | 25,510 | <p>There is a zero-free entire function bounded in every left half-plane, and such that <span class="math-container">$f-1$</span> is in <span class="math-container">$H^2$</span> in every left half-plane.</p>
<p>Let <span class="math-container">$\gamma$</span> be the boundary of the region <span class="math-container">$... |
2,472,746 | <p>I have that $f_n$ is measurable on a finite measure space.</p>
<p>Define $F_k=\{\omega:|f_n|>k \}$</p>
<p>$F_k$ are measurable and have the property $F_1 \supseteq F_2\supseteq\cdots$</p>
<p>Can I then claim that $m\left(\bigcap_{n=1}^\infty F_n\right) = 0$?</p>
| operatorerror | 210,391 | <p>If $f$ maps to
$\mathbb{R}\cup \{ \pm\infty\}$, the set is still measurable as this intersection is the preimage of the point
$$
+\infty
$$
which is measurable, since it is closed and thus Borel. </p>
<p>In order for the statement that this set be measure zero, you need to restrict your attention to functions whi... |
2,472,746 | <p>I have that $f_n$ is measurable on a finite measure space.</p>
<p>Define $F_k=\{\omega:|f_n|>k \}$</p>
<p>$F_k$ are measurable and have the property $F_1 \supseteq F_2\supseteq\cdots$</p>
<p>Can I then claim that $m\left(\bigcap_{n=1}^\infty F_n\right) = 0$?</p>
| Caleb Stanford | 68,107 | <p>Assuming you meant
$$F_k = \{\omega \;:\; |f_{k}(\omega)| > k\},$$
then this is true, even if $f$ is not measurable (if $f: A \to \mathbb{R}$).
We need only note that for any fixed point $a$,
$f(a)$ will be less than $k$ once $k$ gets large enough, so it will be excluded from $F_k$ for large enough $k$. So
$$
\bi... |
2,005,798 | <p>I have the following equality:
$$
\lim_{k \to \infty}\int_{0}^{k}
x^{n}\left(1 - {x \over k}\right)^{k}\,\mathrm{d}x = n!
$$</p>
<p>What I think is that after taking the limit inside the integral ( maybe with the help of Fatou's Lemma, I don't know how should I do that yet ), then get</p>
<p>$$
\int_{0}^... | Arthur | 15,500 | <p>Hint: For the sum $\sum_{k=0}^{n/2} \binom{n}{2k}$, use the binomial theorem to compare $(1+1)^n$ to $(1+(-1))^n$.</p>
|
2,835,767 | <p>Let $V \subset L^2(\Omega)$ be a Hilbertspace and $\{V_n\}$ a sequence of subspaces such that
\begin{align*}
V_1 \subset V_2 \subset \dots \quad \text{and} \quad \overline{\bigcup_{n \in \mathbb{N}} V_n} = V \, (\text{w.r.t. } V\text{-norm} ).
\end{align*}
For some $f\in L^2(\Omega)$ we define $\phi_n = \sup_{\| v_n... | Sarvesh Ravichandran Iyer | 316,409 | <p>Recall what a subspace must satisfy : it must be closed under addition and scalar multiplication, and contain the zero element (and must be a subset of the vector space you are checking it is a subspace of, but this is clear here). </p>
<p>In none of the conditions does $p(2x)$ come into play. You should explain fr... |
933,487 | <p>How do I find it?</p>
<p>I know that $\mathcal{L}(e^t \cos t) =\frac{s-1}{(s-1)^2+1^2}$ but what is it when multiplied by $t$, as written in the title?</p>
| Matt L. | 70,664 | <p>You need the relation</p>
<p>$$\mathcal{L}\{tf(t)\}\Longleftrightarrow -F'(s)$$</p>
<p>i.e. multiplication in the time domain corresponds to differentiation in the $s$-domain (and a negative sign). Since you know $F(s)$, you can easily derive the result.</p>
|
1,157,007 | <p>I know that $f$ and $g$ have a pole or order $k$ in $z=0$.
$f-g$ is holomorph in $\infty$.</p>
<p>I need to prove that:</p>
<p>$$\oint_{|z|=R} (f-g)' dz = 0$$</p>
<p>Any help?</p>
<p>Note: $f$ and $g$ only have a singularity in $z=0$</p>
| nullUser | 17,459 | <p>If $h$ is holomorphic, then $h'$ exists and is holomorphic. If $h$ is holomorphic, then $\oint_C h dz = 0$ for all circles $C$. Take $h= f-g$, $(f-g)'$ respectively.</p>
|
2,655,075 | <p>How many subsets of the set $\{1, 2, \ldots, 11\}$ have median 6?</p>
<p>So I have split this problem into cases. The first case is if 6 is in the subset and the second is where 6 is not. </p>
<p>In case 1, I did 6 with 0, 1, 2, 3, 4, and 5 numbers surrounding it which yielded 1+16+36+16+1 = 70</p>
<p>My struggle... | WaveX | 323,744 | <p>As far as the sets that DO contain $6$, we have the subset $\{6\}$, and this is the only way with one element in the subset.</p>
<p>Next we have $\{a,6,b\}$ and there are $5$ choices for $a$ and $5$ choices for $b$, giving a total of $25$ ways with three elements.</p>
<p>Following suite, we have $\{a,b,6,c,d\}$ A ... |
572,316 | <p>I am reading a bit about calculus of variations, and I've encountered the following.</p>
<blockquote>
<p>Suppose the given function <span class="math-container">$F(\cdot,\cdot,\cdot)$</span> is twice continuously differentiable with respect to all of its arguments. Among all functions/paths <span class="math-contain... | Avitus | 80,800 | <p>Please, have a look at <a href="https://math.stackexchange.com/questions/445551/assumption-that-delta-q-is-small-in-the-derivation-of-euler-lagrange-equatio/445636#445636">this answer</a> for all details. </p>
<p>In summary, a functional $J$ is a map that takes functions from an appropriate functional space and ret... |
1,046,687 | <p>I have this math problem: "determine whether the series converges absolutely, converges conditionally, or diverges."</p>
<p>I can use any method I'd like. This is the series:$$\sum_{n=1}^{\infty}(-1)^n\frac{1}{n\sqrt{n+10}}$$</p>
<p>I though about using a comparison test. But I'm not sure what series I can compare... | wpm | 514,160 | <p>Note that the summand is alternating, and converges monotonically to zero in absolute value. This implies that the series is convergent.
The series, $S = \sum_{n=1}^{\infty}\frac{(-1)^{n}}{n\sqrt{n+10}}$, is absolutely convergent:
$$|S| =|\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n\sqrt{n+10}}| \leq \sum_{n=1}^{\infty} |\f... |
4,106,933 | <p>I tried <span class="math-container">$\left \vert \frac{\sin x}{x^2} - \frac{\sin c}{c^2}\right \vert \leq \frac{1}{x^2} + \frac{1}{c^2} < \epsilon$</span>, but it doesn't help me much with <span class="math-container">$\vert x - c \vert < \delta$</span>. How can I prove this?</p>
| RRL | 148,510 | <p>Note that for <span class="math-container">$x,y \neq 0$</span>,</p>
<p><span class="math-container">$$\left|\frac{\sin x}{x^2} - \frac{\sin y}{y^2} \right| = \left|\frac{\sin x}{x^2} - \frac{\sin x}{xy} + \frac{\sin x}{xy}- \frac{\sin y}{y^2}\right| \\\leqslant \left|\frac{\sin x }{x}\right|\left|\frac{1}{x} - \frac... |
3,281,828 | <p>I am new to permutation and combination and am looking for guidance in the following example:</p>
<p>We have 3 people - A, B, C</p>
<p>How many ways are there to arrange them into Rank 1,2,3</p>
<p>Looking at the example, it is clear that No repetitions are allowed and that ordering is not important (in the sense... | JMoravitz | 179,297 | <p>To arrange <span class="math-container">$n$</span> items in a row (which can be accomplished in <span class="math-container">$n!$</span> ways) is equivalent to picking <span class="math-container">$k$</span> of <span class="math-container">$n$</span> items to arrange in a row (which can be accomplished in <span clas... |
4,454,630 | <p>Is it true that for integers <span class="math-container">$i+j+k= 3m = n$</span> where <span class="math-container">$i , j, k , m , n\ge 0$</span> the inequality holds ?
<span class="math-container">$$
\binom{n}{i\;j\;k} \le \binom{n}{m\;m\;m}
$$</span>
I tried to show
<span class="math-container">$$
\frac{n!}{m!m!... | Atticus Stonestrom | 663,661 | <p>Let <span class="math-container">$P$</span> be a minimal prime of <span class="math-container">$R$</span>, let <span class="math-container">$K=I+P$</span>, and suppose that <span class="math-container">$K$</span> is a proper ideal of <span class="math-container">$R$</span>. Then, since <span class="math-container">$... |
2,941,854 | <p>I want to determine all the <span class="math-container">$x$</span> vectors that belong to <span class="math-container">$\mathbb R^3$</span> which have a projection on the <span class="math-container">$xy$</span> plane of <span class="math-container">$w=(1,1,0)$</span> and so that <span class="math-container">$||x||... | amd | 265,466 | <p>What you’ve given in your question is a formula for computing an orthogonal projection onto the vector <span class="math-container">$v$</span>, but in this problem you’re projecting onto a <em>plane</em>, so you can’t use that formula, at least not directly. </p>
<p>Think about what it means geometrically to ortho... |
916,794 | <p>I have to find the value of $m$ such that:</p>
<p>$\displaystyle\int_0^m \dfrac{dx}{3x+1}=1.$</p>
<p>I'm not sure how to integrate when dx is in the numerator. What do I do?</p>
<p>edit: I believe there was a typo in the question. Solved now, thank you!</p>
| GuiguiDt | 171,756 | <p>It's like integrating $\int f(x)dx$ with $f(x) = \frac{1}{3x+1}$</p>
|
916,794 | <p>I have to find the value of $m$ such that:</p>
<p>$\displaystyle\int_0^m \dfrac{dx}{3x+1}=1.$</p>
<p>I'm not sure how to integrate when dx is in the numerator. What do I do?</p>
<p>edit: I believe there was a typo in the question. Solved now, thank you!</p>
| Dmoreno | 121,008 | <p>That was probably a typo (meaning the double $\mathrm{d}x$) and your integral is the same as the following:</p>
<p>$$ I = \int^m_0 \frac{1}{3x+1} \, \mathrm{d}x = \int^m_0 \frac{\mathrm{d}x}{3x+1} = \int^m_0 \mathrm{d}x \, \frac{1}{3x+1} $$</p>
<p>You can place $\mathrm{d}x$ in the numerator, or even before t... |
2,419,116 | <p>The problem is:</p>
<p>Prove the convergence of the sequence </p>
<p>$\sqrt7,\; \sqrt{7-\sqrt7}, \; \sqrt{7-\sqrt{7+\sqrt7}},\; \sqrt{7-\sqrt{7+\sqrt{7-\sqrt7}}}$, ....</p>
<p>AND evaluate its limit.</p>
<p>If the convergen is proved, I can evaluate the limit by the recurrence relation</p>
<p>$a_{n+2} = \sqrt{7... | Hagen von Eitzen | 39,174 | <p>We have $a_1=\sqrt 7$, $a_2=\sqrt{7-\sqrt 7}$, and then the recursion $a_{n+1}=f(a_n):=\sqrt{7-\sqrt{7+a_n}}$.</p>
<p>By induction, one quickly shows $0<a_n\le \sqrt 7$.
For $0\le x<y\le\sqrt 7$, we have $$0<\sqrt{7+y}-\sqrt {7+x}=\frac{y-x}{\sqrt{7+x}+\sqrt{7+x}}<\frac{y-x}{2\sqrt 7} $$
and for $0\le x... |
2,736,323 | <blockquote>
<p>Given that $Y \sim U(2, 5)$ and $Z = 3Y - 4$, what is the distribution for $Z$?</p>
</blockquote>
<p>I've worked out that for $Y \sim N(2, 5)$, $Z \sim N(2, 45)$ since </p>
<p>$$\mu=3\cdot2 - 4 = 2$$</p>
<p>and </p>
<p>$$\sigma^2=3^2 \cdot 5 = 45$$</p>
<p>I'm wondering how the working differs whe... | Hendrata | 423,285 | <p>A general technique for this type of problems is to find the CDF for $Y$, then find CDF for $Z$ accordingly, and recognize what that CDF is.</p>
<p>However, for uniform distribution, affine transformation of uniform distribution is still uniform as well. So the lowest value of $Z$ is 2, and the highest value is 11... |
1,242,075 | <p>I have some function $g$. I know that $g \in C^1[a,b]$, so $g'(x)$ exists. I want to know, if $g: [a,b] \to [a,b]$ is onto. How can I find out if this is true or not?</p>
<p>P.S. I am not saying all $g$ have the said property, I want to have some kind of test to distinguish functions with this property from functio... | Cameron Buie | 28,900 | <p>Well, given any $g\in C[a,b],$ we have that $g$ maps $[a,b]$ onto itself if and only if $g(x_1)=a$ for some $x_1\in[a,b],$ $g(x_2)=b$ for some $x_2\in[a,b],$ and $g(x)\in[a,b]$ for all $x\in[a,b].$ Put another way, we have $$\{a,b\}\subseteq g\bigl([a,b]\bigr)\subset[a,b],$$ where $g\bigl([a,b]\bigr)$ denotes the im... |
3,848,179 | <blockquote>
<p>The velocity <span class="math-container">$v$</span> of a freefalling skydiver is modeled by the differential equation</p>
<p><span class="math-container">$$ m\frac{dv}{dt} = mg - kv^2,$$</span></p>
<p>where <span class="math-container">$m$</span> is the mass of the skydiver, <span class="math-container... | Community | -1 | <p>Using hint of @E.H.E, we can see that let <span class="math-container">$x=2\sin(\theta)$</span>, so <span class="math-container">\begin{eqnarray}
\int \sqrt{4-x^{2}}dx&=&4\int \cos^{2}(\theta)d\theta\\
&=&2\theta+2\sin(\theta)\cos(\theta)+C, \quad \theta=\arcsin(x/2) \\
&=&\frac{x}{2}\sqrt{4-... |
118,070 | <p>I have the following code:</p>
<pre><code>a = 6.08717*10^6;
b = a/3;
c = a*1.5;
d = a^2;
matrix={{a, b, c, d},{b, c, d, a},{c, d, a, b},{d, a, b, c}};
matrix // EngineeringForm
</code></pre>
<p>Normally, I use this result by copying (<code>Copy As ► MathML</code>) and pasting into Microsoft Word.</p>
<p>However, ... | Carl Woll | 45,431 | <p>MathML export works by creating <a href="http://reference.wolfram.com/language/ref/TraditionalForm" rel="nofollow noreferrer"><code>TraditionalForm</code></a> boxes and then converting those boxes into MathML. The "Copy As MathML" menu item, on the other hand, takes the boxes in the cell to be copied, and converts t... |
186,726 | <p>Just a soft-question that has been bugging me for a long time:</p>
<p>How does one deal with mental fatigue when studying math?</p>
<p>I am interested in Mathematics, but when studying say Galois Theory and Analysis intensely after around one and a half hours, my brain starts to get foggy and my mental condition d... | GeoffDS | 8,671 | <p>I recently read an <a href="http://www.salon.com/2012/03/14/bring_back_the_40_hour_work_week/">article</a> on the 40 hour work week and I think it is somewhat related. The basic idea of it was that in the mid 20th century, they had a 40 hour work week and they had lots of research on it showing that it was optimal ... |
186,726 | <p>Just a soft-question that has been bugging me for a long time:</p>
<p>How does one deal with mental fatigue when studying math?</p>
<p>I am interested in Mathematics, but when studying say Galois Theory and Analysis intensely after around one and a half hours, my brain starts to get foggy and my mental condition d... | Community | -1 | <p>I lighted upon this sterling answer by virtue of user Hepth at <a href="http://www.physicsforums.com/showthread.php?t=677222">http://www.physicsforums.com/showthread.php?t=677222</a>:</p>
<hr>
<p>I too get mental fatigue if I'm working too hard. Usually my problem is if I work on research (read papers/do math/prog... |
1,928,259 | <p>I have the following problem: </p>
<blockquote>
<p>The function $f(x)$ is odd, its period is $5$ and $f(-8) = 1$. What is $f(18)$?</p>
</blockquote>
<p>So, $f(-8) = f(-8 + 5) = 1$. I also know that you could replace $(-8)$ with $(-3)$ and still get the same result of $1$.</p>
<p>I'm just learning about periods.... | Alex Ortiz | 305,215 | <p>The notation
$$
\left(\frac{\partial f}{\partial y}\right)_x \stackrel{\text{def}}{=} \frac{\partial f}{\partial y}
$$
and $x$ is <em>not to be considered constant from thereafter</em>. The use of this notation is to make it explicit which variables are being held constant. I think it is somewhat contrived, but it i... |
2,210,871 | <p>I'm doing some calculus homework and I got stuck on a question, but eventually figured it out on my own. My textbook doesn't have all the answers included (it only gives answers to even numbered questions for some reason). Anyways I got stuck when I needed to solve for x for this function.</p>
<p>$${\ -3x^3+8x-4{\s... | Andre.J | 361,886 | <p>Since you are pursuing a computer science degree, I think you can benefit from learning some Discrete Mathematics. The book by <a href="http://rads.stackoverflow.com/amzn/click/0201199122" rel="nofollow noreferrer">Grimaldi</a> is nice for an introduction, you could also consult the book by <a href="http://rads.sta... |
1,903,416 | <p>Is there a function that can be bijective, with the set of natural numbers as domain and range, other than $f(n) = n$?</p>
| Alex Ortiz | 305,215 | <p>Sure. Consider the function $f: \Bbb N \to \Bbb N$ defined by</p>
<p>$$
f(n) = \cases{
n - 1 & if $n$ is even \\
n + 1 & if $n$ is odd.}
$$</p>
<p>This function is a non-identity bijection. As <em>Hagen von Eitzen</em> notes, depending on your definition of $\Bbb N$, swap $+$ and $-$ in the definition of $... |
25,172 | <p>What would be a good books of learning differential equations for a student who likes to learn things rigorously and has a good background on analysis and topology?</p>
| Dan Fox | 672 | <p>There are many, many books on ODEs, many of them good.</p>
<p>For the basic theory no one seems to have improved much on the book of Coddington and Levinson cited by Gerald Edgar. That book is old-fashioned, has essentially no examples, and could be seen as quite dry. It is clearly written and I was able to learn fr... |
59,888 | <p>Very similar to </p>
<p><a href="https://mathematica.stackexchange.com/questions/11638/parameters-in-plot-titles">Parameters in plot titles</a></p>
<p>in which I want to call a parameter from an array using <code>PlotLabel</code> in my plot using <code>Manipulate</code>. I've tried all of the suggestions in the ab... | halirutan | 187 | <p>What you observe is simply because you use a multiplication without noticing it. Basic example:</p>
<pre><code>"f=" 2
(* 2 "f=" *)
</code></pre>
<p>If you look at the full form of this output with <code>FullForm[%]</code> you see that it is indeed <code>Times[2, "f="]</code>. Because the terms in a multiplication... |
30,081 | <pre><code>For[n = 2500, n < 10000, n += 500,
sum = 0;
pp = N[1/HarmonicNumber[n, 2]];
A = Reverse[IdentityMatrix[n]];
H = IdentityMatrix[n];
For[i = 0, i < n, i++,
For[j = 1, j < n + 1 - i, j++,
H[[j, j + i]] = Sqrt[pp]/(N[Sqrt[n - i]*(i + 1)])]];
T := A.Orthogonalize[N[H].A, Dot];
... | rcollyer | 52 | <p>There is a lot going on with your code, as already pointed out by others. But, I would like to point out a few more things, and this was <em>much</em>, <em>much</em> to long for a comment. </p>
<p>First, I would use <code>SparseArray</code> to generate <code>H</code>, instead of the pre-allocate, fill-in method yo... |
30,081 | <pre><code>For[n = 2500, n < 10000, n += 500,
sum = 0;
pp = N[1/HarmonicNumber[n, 2]];
A = Reverse[IdentityMatrix[n]];
H = IdentityMatrix[n];
For[i = 0, i < n, i++,
For[j = 1, j < n + 1 - i, j++,
H[[j, j + i]] = Sqrt[pp]/(N[Sqrt[n - i]*(i + 1)])]];
T := A.Orthogonalize[N[H].A, Dot];
... | Daniel Lichtblau | 51 | <p>I'm not sure one can get the larger cases due to memory needs. The code below, which uses the array generation of @rcollyer, seems reasonably effective at least for the smaller end.</p>
<pre><code>n = 2500;
Timing[sum = 0;
diags = Sqrt[
1./HarmonicNumber[n, 2]]/(Sqrt[n - #] (# + 1) &@Range[0., n - 1]);
h... |
174,149 | <p>How many seven - digit even numbers greater than $4,000,000$ can be formed using the digits $0,2,3,3,4,4,5$?</p>
<p>I have solved the question using different cases: when $4$ is at the first place and when $5$ is at the first place, then using constraints on last digit.</p>
<p>But is there a smarter way ?</p>
| Sumit Bhowmick | 34,963 | <p>Combinations can be:</p>
<p>Case1#: $5 \{4 3 3 2 4\} 0$, total combinations are: $\frac{5!}{2!2!} = 30$</p>
<p>Case2#: $4 \{5 3 3 2 4\} 0$, total combinations are: $\frac{5!}{2!} = 60$</p>
<p>Case3#: $5 \{4 3 3 0 4\} 2$, total combinations are: $\frac{5!}{2!2!} = 30$</p>
<p>Case4#: $4 \{5 3 3 0 4\} 2$, total com... |
2,061,547 | <p>I am solving for the zeroes of the function:</p>
<blockquote>
<p>$$\frac{\cos(x)(3\cos^2(x)-1)}{(1+\cos^2(x))^2}$$</p>
</blockquote>
<p>The zeroes of the function I found were done by setting $\cos(x)=0$, and $3\cos^2(x)-1=0$</p>
<p>For the $3\cos^2(x)-1=0$ I solved it and got $x=\cos^{-1}(\frac{\sqrt3}{3})$ bu... | Asinomás | 33,907 | <p>One way to do it is to prove that if $x\neq 0$ then $x$ is not a limit point.</p>
<p>Try considering each of the following three cases:</p>
<ul>
<li>$x<0$</li>
<li>$x>1$</li>
<li>$0\leq1$</li>
</ul>
<p>The last case is perhaps the trickiest.</p>
<p>We can divide it into two cases:</p>
<ul>
<li>$x$ is of t... |
145,306 | <p>I had a problem on a program of mine that I could avoid by developing the code through other ways. On the other hand, I still do not know how to solve the simple problem below:</p>
<p>Consider these two definitions:</p>
<p>f = p;
p = 2;</p>
<p>One can use Clear[p] to clear the value of p, which will lead the outp... | Coolwater | 9,754 | <pre><code>f = p; p = 2;
Last[MapAt[Clear, First[OwnValues[f]], 2]]
p
</code></pre>
<blockquote>
<p>p</p>
</blockquote>
<p>Maybe instead of <code>f = p</code> you should use <code>f = Hold[p]</code> or <code>f = Hold[#]&[p]</code> so the above becomes</p>
<pre><code>ReleaseHold[MapAt[Clear, f, 1]]
</code></pre... |
754,301 | <p>Say I have the following maximization.</p>
<p><span class="math-container">$$ \max_{R: R^T R=I_n} \operatorname{Tr}(RZ),$$</span>
where <span class="math-container">$R$</span> is an <span class="math-container">$n\times n$</span> orthogonal transformational, and the SVD of <span class="math-container">$Z$</span> is... | glS | 173,147 | <p>I'll show how to prove the more general case with complex matrices: find the maximum of <span class="math-container">$\operatorname{Tr}(UZ)$</span> over all unitaries <span class="math-container">$U$</span>:
<span class="math-container">$$\max_{U: U^\dagger U=I}|\operatorname{Tr}(UZ)|.$$</span></p>
<p>Leveraging th... |
3,264,693 | <p>For context, I have been relearning a lot of math through the lovely website Brilliant.org. One of their sections covers complex numbers and tries to intuitively introduce Euler's Formula and complex exponentiation by pulling features from polar coordinates, trigonometry, real number exponentiation, and vector space... | Community | -1 | <p>Write <span class="math-container">$z=r_1e^{i\theta_1}$</span>. When multiplying <span class="math-container">$w=r_2e^{i\theta_2}$</span> by <span class="math-container">$z$</span>, we stretch by a factor of <span class="math-container">$r_1$</span>, and rotate by an angle of <span class="math-container">$\theta... |
3,264,693 | <p>For context, I have been relearning a lot of math through the lovely website Brilliant.org. One of their sections covers complex numbers and tries to intuitively introduce Euler's Formula and complex exponentiation by pulling features from polar coordinates, trigonometry, real number exponentiation, and vector space... | Laurel Turner | 677,923 | <p>To briefly address your specific question about "duality" first, there is no "duality" (not sure what is precisely meant by that term here) between complex exponentiation and complex multiplication, unless the arguments of both are real. This is because complex multiplication is commutative; complex exponentiation i... |
1,755,029 | <p>Imagine a cubic array made up of an $n\times n\times n$ arrangement of unit cubes: the cubic array is n
cubes wide, n cubes high and n cubes deep. A special case is a $3\times3\times3$ Rubik’s cube, which
you may be familiar with. How many unit cubes are there on the surface of the $n\times n\times n$ cubic array?</... | John Douma | 69,810 | <p>There are $6$ faces with $n^2$ cubes on each face for a total of $6n^2$ cubes. The eight cubes on the vertices are counted $3$ times each so we must subtract $16$ to get $6n^2-16$ cubes. Likewise, there are $12$ edges, each with $n-2$ cubes that have been double counted so we must subtract $12(n-2)$ to get $6n^2-16-... |
3,862,182 | <p>I encountered this question, and I am unsure how to answer it.</p>
<p>When <span class="math-container">$P(x)$</span> is divided by <span class="math-container">$x - 4$</span>, the remainder is <span class="math-container">$13$</span>, and when <span class="math-container">$P(x)$</span> is divided by <span class="ma... | Bernard | 202,857 | <p>Use the inverse isomorphism of the isomorphism in the <em>Chinese remainder theorem</em>: as <span class="math-container">$x^2-x-12=(x+3)(x-4)$</span>, we have an isomorphism
<span class="math-container">\begin{align}
K[X]/(X^2-X-12)&\xrightarrow[\quad]\sim K[X]/(X+3)\times K[X]/(X-4) \\
P\bmod(X^2-X-12)&\l... |
3,074,901 | <p>Find the rank of the following matrix</p>
<p><span class="math-container">$$\begin{bmatrix}1&-1&2\\2&1&3\end{bmatrix}$$</span></p>
<p>My approach: </p>
<p>The row space exists in <span class="math-container">$R^3$</span> and is spanned by two vectors. Since the vectors are independent of each othe... | eason 曲 | 505,625 | <p>The definition of <em>rank</em> is the number of linearly independent row vectors of a matrix. For a matrix with <span class="math-container">$n$</span> linearly independent col, the max of rank is <span class="math-container">$n$</span>. </p>
<p><em>Span</em> means the linear combination of these vectors includes ... |
3,046,205 | <p>I am trying to figure out the steps between these equal expressions in order to get a more general understanding of product sequences:
<span class="math-container">$$\prod_{k=0}^{n}\left(3n-k\right) + \prod_{k=n}^{2n-3}\left(2n-k\right) = \prod_{j=2n}^{3n}j + \prod_{j=3}^{n}j =\frac{(3n)!}{(2n-1)!}+\frac{n!}{2}$$</s... | Jacky Chong | 369,395 | <p>Using the fact that
<span class="math-container">\begin{align}
a^{\varphi(b)}\equiv 1 \mod b
\end{align}</span>
and the fact that for any <span class="math-container">$n$</span>
<span class="math-container">\begin{align}
n = \varphi(b)k+r
\end{align}</span>
for some <span class="math-container">$0\leq r<\varphi(... |
3,604,388 | <p>Let <span class="math-container">$P_n$</span> be the statement that <span class="math-container">$\dfrac{d^{2n}}{dx^{2n}}(x^2-1)^n = (2n)!$</span> </p>
<p>Base case: n = 0, <span class="math-container">$\dfrac{d^0}{dx^0}(x^2-1)^0 = 1 = 0!$</span></p>
<p>Assume <span class="math-container">$P_m = \dfrac{d^m}{dx^m}... | Rezha Adrian Tanuharja | 751,970 | <p>Try the binomial identity</p>
<p><span class="math-container">$$
\frac{d^{2n}}{dx^{2n}}(x^{2}+1)(x^{2}+1)^{n-1}=\sum_{i=0}^{2n}{\binom{2n}{i}\frac{d^{i}}{dx^{i}}(x^{2}+1)\frac{d^{2n-i}}{dx^{2n-i}}(x^{2}+1)^{n-1}}
$$</span></p>
|
2,483,188 | <p>I am facing this problem: </p>
<p><strong>Turn into cartesian form:</strong></p>
<p>$$\dfrac{1-e^{i\pi/2}}{1 + e^{i\pi/2}}$$</p>
<p>I've tried to operate and I've come up to this:</p>
<p>$$\dfrac{1-2e^{i\pi/2} + e^{i\pi}}{1 - e^{i\pi}}$$</p>
<p>I do not know how to go on, and I've tried to operate with the cart... | copper.hat | 27,978 | <p>Try $F(f) = \int_0^{1 \over 2} f - \int_{1 \over 2}^1 f$. The operator
$F$ is linear & continuous.</p>
<p>The (an) obvious candidate is $g = 1_{[0,{1 \over 2})} - 1_{({1 \over 2},1]}$, but it is not continuous.</p>
|
2,705,980 | <p>I have the following problem:
\begin{cases}
y(x) =\left(\dfrac14\right)\left(\dfrac{\mathrm dy}{\mathrm dx}\right)^2 \\
y(0)=0
\end{cases}
Which can be written as:</p>
<p>$$ \pm 2\sqrt{y} = \frac{dy}{dx} $$</p>
<p>I then take the positive case and treat it as an autonomous, seperable ODE. I get $f(x)=x^2$ as my ... | spaceman | 522,096 | <p>I assume that the $ \mathbf{v}_i $'s are the basis vectors of $ \mathbb{R}^n $. If so, let $ \mathbf{x} \in \mathbb{R^n} $ such that $ A\mathbf{x} = 0 $. Then since $ \mathbf{v}_i $'s form a basis, there exists $ \lambda_i \in \mathbb{R} $ such that $ \mathbf{x} = \sum_{i=1}^{n}\lambda_i \mathbf{v}_i $. Then by line... |
1,488,737 | <blockquote>
<p>Let $A$ be a square matrix of order $n$. Prove that if $A^2=A$ then $\mathrm{rank}(A)+\mathrm{rank}(I-A)=n$.</p>
</blockquote>
<p>I tried to bring the $A$ over to the left hand side and factorise it out, but do not know how to proceed. please help. </p>
| Hamid | 420,694 | <p>Let rank <span class="math-container">$(A) = r$</span> and <span class="math-container">$\lambda$</span> be an eigenvalue of <span class="math-container">$A$</span>. Since <span class="math-container">$A$</span> is idempotent then
<span class="math-container">\begin{align*}
\lambda x = A x = A^2 x = A.Ax = A\lambda ... |
313,025 | <p>I got two problems asking for the proof of the limit: </p>
<blockquote>
<p>Prove the following limit: <br/>$$\sup_{x\ge 0}\ x e^{x^2}\int_x^\infty e^{-t^2} \, dt={1\over 2}.$$</p>
</blockquote>
<p>and, </p>
<blockquote>
<p>Prove the following limit: <br/>$$\sup_{x\gt 0}\ x\int_0^\infty {e^{-px}\over {p+1}} \,... | AJMansfield | 50,951 | <p>The main things I'd add to the other answers is to explicitly apply the definition of a limit, rather than leave it all to higher-level theorems, since the OP said the problem was proving the <em>limits</em>.</p>
<p>$$\lim_{x \rightarrow \infty} f(x) = L \Leftrightarrow \forall \; \varepsilon > 0 \; \exists \; N... |
2,650,628 | <p>The equation $\log_e(x) + \log_e(1+x) =0$ can be written as:</p>
<p>a) $x^2+x-e=0$</p>
<p>b) $x^2+x-1=0$</p>
<p>c) $x^2+x+1=0$</p>
<p>d) $x^2+xe-e=0$</p>
<p>I tried differentiating both sides, then it becomes $\frac{1}{x}+\frac{1}{1+x}=0$, but I dont get any of the answers.</p>
| hamam_Abdallah | 369,188 | <p>For $x>0,$</p>
<p>$$\ln (ex)=\ln (e)+\ln (x)=1+\ln (x) $$</p>
<p>and $$-\ln (x)=\ln \Bigl(\frac {1}{x}\Bigr) $$</p>
<p>the equation will be</p>
<p>$$\ln (x+1)=\ln \Bigl(\frac 1x\Bigr) $$</p>
<p>or</p>
<p>$$x+1=\frac 1x $$
and the answer is $ x^2+x-1=0$</p>
|
2,650,628 | <p>The equation $\log_e(x) + \log_e(1+x) =0$ can be written as:</p>
<p>a) $x^2+x-e=0$</p>
<p>b) $x^2+x-1=0$</p>
<p>c) $x^2+x+1=0$</p>
<p>d) $x^2+xe-e=0$</p>
<p>I tried differentiating both sides, then it becomes $\frac{1}{x}+\frac{1}{1+x}=0$, but I dont get any of the answers.</p>
| Michael Hardy | 11,667 | <p>Recall that $$\log_e x + \log_e(x+1) = \log_e\Big( x(x+1)\Big) = \log_e(x^2+x)$$ and that $$ \text{if } \log_e(x^2+x) = 0 \text{ then } x^2+x = e^0. $$ </p>
|
2,247,498 | <p>Imagine a circle of radius R in 3D space with a line l running threw it's center C in a direction perpendicular to the plane of the circle. Basically, like the axel of a wheel. </p>
<p>From a given point P that is not on the circle or on l, a ray extends to intersect both l and the circle. What would be the equatio... | Futurologist | 357,211 | <p>You should have simply stated your question in terms of orthographic projection from the get go. Try the following algorithm, I hope it works (if, of course, I understand correctly what your goal is).</p>
<p>This time you assume you are given the vector $\vec{p}$ instead of the point $P$ that determines the directi... |
3,959,263 | <p>Let <span class="math-container">$G$</span> be a tree with a maximum degree of the vertices equal to <span class="math-container">$k$</span>.
<strong>At least</strong> how many vertices with a degree of <span class="math-container">$1$</span> can be in <span class="math-container">$G$</span> and why?</p>
<p>I think ... | FFjet | 597,771 | <p><strong>HINT.</strong> Let <span class="math-container">$G$</span> be a smallest tree that meets the requirements. Could adding nodes to <span class="math-container">$G$</span> decrease the answer?</p>
|
159,585 | <p>This is a kind of a plain question, but I just can't get something.</p>
<p>For the congruence and a prime number $p$: $(p+5)(p-1) \equiv 0\pmod {16}$.</p>
<p>How come that the in addition to the solutions
$$\begin{align*}
p &\equiv 11\pmod{16}\\
p &\equiv 1\pmod {16}
\end{align*}$$
we also have
$$\begin{... | M Turgeon | 19,379 | <p>(If you know a little abstract algebra.) </p>
<p>The equation $$(p+5)(p−1)\equiv 0\pmod{16}$$ implies that either $p+5$ or $p-1$ is a zero divisor in $\mathbb{Z}/16\mathbb{Z}$; these are $$\{0,2,4,6,8,10,12,14\}.$$ However, you have some restrictions, e.g. if $p+5\equiv 2$, then $p-1$ should be congruent to $8$, wh... |
259,808 | <p>For example, suppose I wanted to determine which of the following has the fastest asymptotic growth:</p>
<ol>
<li><p>$n^2\log(n)+(\log(n))^2$</p></li>
<li><p>$n^2+\log(2^n)+1$</p></li>
<li><p>$(n+1)^3+(n-1)^3$</p></li>
<li><p>$(n+\log(n))^22^{100}$</p></li>
</ol>
<p>Are there any straightforward methods to tell wh... | Jair Taylor | 28,545 | <p>Some tips:</p>
<ul>
<li><p>Keep the following asymptotic inequalities in your head, where $f(n) \ll g(n)$ means that $\lim_{n\rightarrow \infty} \frac{f(n)}{g(n)} = 0$:$$1 \ll \log n \ll \sqrt{n} \ll n\ll n^2 \ll n^3 \ll \ldots \ll e^n \ll n!$$ and more generally, $n^a \ll n^b$ whenever $0 \leq a < b$ and $a^n ... |
2,547,508 | <p>I am trying to prove that various expressions are real valued functions. Is it possible to state that, because no square roots (or variants such as quartic roots etc) are in that function, it is a real valued function?</p>
| Community | -1 | <p>"Function" is such an incredibly broad notion that it is wholly implausible to give a general answer to this sort of question that isn't something trivial like "Applying a real-valued function always gives a real number".</p>
<p>More meaningful questions need to restrict the scope; for example, one might ask what s... |
112,320 | <p>What is the number of strings of length $235$ which can be made from the letters A, B, and C, such that the number of A's is always odd, the number of B's is greater than $10$ and less than $45$ and the Number of C's is always even?</p>
<p>What I can think of is </p>
<p>$$\left(\binom{235}{235} - \left\lfloor235 -... | Daniel Pietrobon | 17,824 | <p>If there is an odd number of A's and an even number of C's, it follows that there must be an even number of B's.</p>
<p>Therefore, to construct a string of length 235, satisfying the properties above, it is sufficient to specify the position of the A's and the B's, because then the C's will be forced.</p>
<p>There... |
761,726 | <p>We know that the Banach space $X$ is infinite-dimensional,</p>
<p>theconclusion we want to show is: then $X'$ is also infinite-dimensional. </p>
<p>$X'$: the space of linear bdd functions</p>
| Vincent Boelens | 94,696 | <p>Use contraposition: if $X'$ is finite dimensional, then $X''$ is as well. Since $X$ can be embedded isometrically in $X''$, it must be finite dimensional.</p>
|
1,432,040 | <p>My approach is to unite the group consisting of Alice ($A$), Bob ($B$) and $k$ persons between them into one new "person".
So we can permute $n-(k+2)$ and $k$ persons between $A$ and $B$ separately.</p>
<p>It seems like the answer should be $(n-(k+2))!\times k!$, but it is suspicious because it does not take into... | Hagen von Eitzen | 39,174 | <p>Your idea is fine. But so is your suspicion. You can remedy that by multiplying with $2$ in the end. Apart from that, you made a little mistake: When replacing the $k+2$ persons with one new "person", you end up with $n-(k+2)+1$ persons.</p>
|
1,432,040 | <p>My approach is to unite the group consisting of Alice ($A$), Bob ($B$) and $k$ persons between them into one new "person".
So we can permute $n-(k+2)$ and $k$ persons between $A$ and $B$ separately.</p>
<p>It seems like the answer should be $(n-(k+2))!\times k!$, but it is suspicious because it does not take into... | Javier | 241,291 | <p>You have to consider that people between A and B and people outside them are also interchangeable, so the answer can be simplified:</p>
<p>Consider A in the position $1$ of $n$. Now B is in the position $k+2$. There are $n-2$ anonymous people who can change positions, so in this configuration there are $(n-2)!$ pos... |
714,000 | <p>Lets say:</p>
<p>$X = \{x_1, x_2, x_3, ... \} $ be a set of Real numbers in range $(R_1, R_2)$ and $m =$ mean of $x$</p>
<p>If I have to increase mean of set $X$ by $3$, each number in the set has to be increase by $3$.
But how to increase mean of set $X$ by $3$, by only changing a subset of X. Is there any mathem... | AnonSubmitter85 | 33,383 | <p>Let $x_1, \dots, x_k$ be the subset that does not change and let $x_{k+1},\dots, x_N$ be the subset that does. We have by hypothesis that</p>
<p>$$
{1 \over N} (x_1 + \cdots + x_k) + {1 \over N} (x_{k+1} + \cdots + x_N) = m.
$$</p>
<p>And you want to find some function $f$ such that</p>
<p>$$
{1 \over N} (x_1 + \... |
273,127 | <p>Let $X$ be the set of all harmonic functions external to the unit sphere on $\mathbb R^3$ which vanish at infinity, so if $V \in X$, then $\nabla^2 V(\mathbf{r}) = 0$ on $\mathbb R^3 - S(2)$ and $\lim_{r \rightarrow \infty} V(r) = 0$. Now consider a function $f: X \rightarrow \mathbb R$, defined by
$$
f(V)(\mathbf{r... | Karl Fabian | 20,804 | <p>This problem has an important background in geomagnetism.
When planning the MAGSAT satellite mission (1979/80) to determine the spherical harmonic coefficients of the Earth's magnetic field from space,
Backus (JGR, 1970) showed that a measurement of the total field intensity <span class="math-container">$||\nabla ... |
331,859 | <p>I need to find the antiderivative of
$$\int\sin^6x\cos^2x \mathrm{d}x.$$ I tried symbolizing $u$ as squared $\sin$ or $\cos$ but that doesn't work. Also I tried using the identity of $1-\cos^2 x = \sin^2 x$ and again if I symbolize $t = \sin^2 x$ I'm stuck with its derivative in the $dt$.</p>
<p>Can I be given a h... | IcyFlame | 63,288 | <p>If you only want a hint that will be simple:</p>
<p>Whenever we have even powers of sin and cos multiplied then we must convert the integral into higher angles.</p>
|
2,343,993 | <blockquote>
<p>Find the limit -$$\left(\frac{n}{n+5}\right)^n$$</p>
</blockquote>
<p>I set it up all the way to $\dfrac{\left(\dfrac{n+5}{n}\right)}{-\dfrac{1}{n^2}}$ but now I am stuck and do not know what to do.</p>
| Jaideep Khare | 421,580 | <p>$$\mathrm L=\lim_{n \to \infty}\left(\frac{n}{n+5}\right)^n \implies \ln {\mathrm L}=\lim_{n \to \infty} n \underbrace{\left(\frac{\ln \left( 1-\frac{5}{n+5} \right)}{\frac{-5}{n+5}} \right)}_{\text{This limit is 1}} \times \frac{-5}{n+5}$$$$ \implies \ln{\mathrm L}=\lim_{n \to \infty}n \cdot \left(\frac{-5}{n+5}\ri... |
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