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 |
|---|---|---|---|---|
3,553,233 | <p>I am having difficulty finishing this proof. At first, the proof is easy enough. Here's what I have thus far:<br>
Because <span class="math-container">$5 \nmid n$</span>, we know <span class="math-container">$\exists q \in \mathbb{Z}$</span> such that <span class="math-container">$$n = 5q + r$$</span> where <span cl... | J. W. Tanner | 615,567 | <p>As you showed, <span class="math-container">$n^2=25q^2+10qr+r^2$</span>, where <span class="math-container">$r=1, 2, 3,$</span> or <span class="math-container">$4$</span>.</p>
<p>When <span class="math-container">$r=1$</span>, <span class="math-container">$n^2=25q^2+10qr+1=5(5q^2+2qr)+1=5k+1.$</span></p>
<p>When <... |
1,334,680 | <p>How to apply principle of inclusion-exclusion to this problem?</p>
<blockquote>
<p>Eight people enter an elevator at the first floor. The elevator
discharges passengers on each successive floor until it empties on the
fifth floor. How many different ways can this happen</p>
</blockquote>
<p>The people are <... | Vadeem | 249,634 | <p>Set theory is one of the branch of mathematical logic it clearly means not every mathematical logic can be explained by set theory.Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory. These areas share basic results on logic, particularly first-order lo... |
1,334,680 | <p>How to apply principle of inclusion-exclusion to this problem?</p>
<blockquote>
<p>Eight people enter an elevator at the first floor. The elevator
discharges passengers on each successive floor until it empties on the
fifth floor. How many different ways can this happen</p>
</blockquote>
<p>The people are <... | Stig Hemmer | 224,001 | <p>There are several ways to build up mathematics. One book may do it one way, another book will do it another.</p>
<p>All these methods suffer from the fact that you must start with normal human language for your first definitions. The author try to make their definitions as rigid as possible.</p>
<p>It is very im... |
4,189,614 | <p>I am trying to answer the following question...</p>
<blockquote>
<p>Consider a wall made of brick <span class="math-container">$10$</span> centimeters thick, which separates a room in a house from the outside. The room is kept at <span class="math-container">$20$</span> degrees. Initially the outside temperature i... | user89699 | 85,392 | <p>Strictly speaking, you are correct. But if you stay away from the corners, then you just have 1d heat conduction across the brick only - the transverse gradients can be neglected. Then, the equations are just as you decribe with an error function solution. HTH</p>
|
13,705 | <p>Let $m$ be a positive integer. Define $N_m:=\{x\in \mathbb{Z}: x>m\}$. I was wondering when does $N_m$ have a "basis" of two elements. I shall clarify what I mean by a basis of two elements: We shall say the positive integers $a,b$ generate $N_m$ and denote $N_m=<a,b>$ if every element $x\in N_m$ can be wri... | Robin Chapman | 226 | <p>If $m$ and $n$ are coprime positive integers, then every integer with $>mn-m-n$ is expressible in the form $am+bn$ where $a$ and $b$ are nonnegative integers, but $mn-m-n$ isn't. This is due to <a href="http://en.wikipedia.org/wiki/Coin_problem" rel="nofollow">Sylvester</a>.</p>
<p><strong>Added</strong> The key... |
69,208 | <p>Consider $f:\{1,\dots,n\} \to \{1,\dots,m\}$ with $m > n$.
Let $\operatorname{Im}(f) = \{f(x)|x \in \{1,\dots,n\}\}$.</p>
<p>a.)
What is the probability that a random function will be a bijection when viewed as
$$f':\{1,\dots,n\} \to \operatorname{Im}(f)?$$</p>
<p>b.)
How many different function f are the... | marty cohen | 13,079 | <p>If $\sin x$ is approximately $x$, and $\cos x$ is approximately $1−.5x^2$,
then $(\sin x)^2 + (\cos x)^2$ is approximately
$$x^2 + (1-.5 x^2)^2 = x^2 + 1 - x^2 + .25 x^4 = 1 + .25 x^4.$$</p>
<p>Note that the $x^2$ term cancels out, so the result is $1$ with a term of order $x^4$.</p>
<p>As you use more terms of t... |
1,530,406 | <p>How to multiply Roman numerals? I need an algorithm of multiplication of numbers written in Roman numbers. Help me please. </p>
| John Martin | 301,716 | <p>Take the first number you want to multiply and break it down into parts, any way you choose.
E.G. 43 = XLIII = XL + III, or X + X + X + X + III, etc.
Do the same with the second number in the multiplication.
E.G. 15 = XV = X + V, or V + V + V, etc.</p>
<p>Take any pair of those combinations and set them within a tab... |
2,327,675 | <p>Using the GPU Gems Article <a href="https://developer.nvidia.com/gpugems/GPUGems/gpugems_ch01.html" rel="nofollow noreferrer" title="Effective Water Simulation From Physical Models">Effective Water Simulation From Physical Models</a> I have implemented Gerstner Waves into UE4, I have built the function both on GPU f... | Daniel Wells | 740,611 | <p>Note I'm ignoring a lot of the variables for brevity
Gerstner Waves vaguely simplify to y=sin(x) x = cos(x)</p>
<p>'Sample a vertical offset and send it somewhere that way horizontally' but we want the vertical offset of the point that landed here. We can guess where that point comes from by going backwards by the h... |
3,436,430 | <p>Evaluate <span class="math-container">$$\int_{-2}^{2}\int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}\int_{2-\sqrt{4-x^2-y^2}}^{2+\sqrt{4-x^2-y^2}}(x^2+y^2+z^2)^{3/2} \; dz \; dy \; dx$$</span> by converting to spherical coordinates.<br>
We know that <span class="math-container">$(x^2+y^2+z^2)^{3/2} = (\rho^2)^{3/2} = \rho^3$</s... | Doug M | 317,162 | <p>The boundary of our sphere is <span class="math-container">$x^2 + y^2 + z^2 = 4z$</span></p>
<p>Plugging</p>
<p><span class="math-container">$x = \rho\cos\theta \sin\phi\\
y = \rho\sin\theta\sin\phi\\
z = \rho \cos\phi$</span></p>
<p>We get:</p>
<p><span class="math-container">$\rho^2 = 4\rho\cos\phi\\
\rho = 4\... |
3,660,101 | <p>I want to determine if the series <span class="math-container">$ \sum_{n=2}^{\infty}\frac{\left(-1\right)^{n}}{\left(-1\right)^{n}+n} $</span> converge/diverge. the sequence in the denominator is not monotinic, so I cant use Dirichlet's or Abel's tests. My intuition is that this series converge, becuase its looks cl... | Barry Cipra | 86,747 | <p>It sometimes helps to write out the first few terms, in order to see what you're dealing with, and possibly spot a useful pattern. In this case we have</p>
<p><span class="math-container">$$\begin{align}
\sum_{n=2}^\infty{(-1)^n\over(-1)^n+n}
&={1\over3}-{1\over2}+{1\over5}-{1\over4}+{1\over7}-{1\over6}+\cdots\... |
93,458 | <blockquote>
<p>Let <span class="math-container">$n$</span> be a nonnegative integer. Show that <span class="math-container">$\lfloor (2+\sqrt{3})^n \rfloor $</span> is odd and that <span class="math-container">$2^{n+1}$</span> divides <span class="math-container">$\lfloor (1+\sqrt{3})^{2n} \rfloor+1 $</span>.</p>
</... | Henry | 6,460 | <p>You can use recurrences such as $$f(n)=4f(n-1)-f(n-2)+2$$ or $$f(n)=5f(n-1)-5f(n-2)+f(n-3)$$ starting at $f(0)=1, f(1)=3$.</p>
<p>Then show the various results by induction. </p>
|
1,018,270 | <p>While I was studying about finite differences I came across an article that says "computers can't deal with limit of $\Delta x \to 0$ " in <a href="http://1drv.ms/1sB5P1B" rel="nofollow">finite differences</a>.But if computers can't deal with these equations does anybody know how they compute $ \frac {d}{dx}$ of $x^... | Claude Leibovici | 82,404 | <p>This is not an answer but an illustration of automatic differentiation of a formula (I used Tapenade software online).</p>
<p>The source code I submitted is</p>
<pre><code> SUBROUTINE DUMMY(X,Y)
Y = X ** 2
END
</code></pre>
<p>which was interpreted as</p>
<pre><code> SUBROUTINE DUMMY(x, y)
IMPLICIT NON... |
168,118 | <p>I'm attempting to differentiate an equation in the form</p>
<pre><code>D[sqrt((2*(((a*b*c+Pi*d*e^2+Pi*f*g^2+h*i*j+Pi*k*l^2)/(a*b*c+Pi*d*e^2+Pi*f*g^2))-1)*m)/(n^2 - o^2)/p), a]
</code></pre>
<p>in order to do an error propagation analysis. So I need to differentiate it against a, against b, against c and so on.</p>... | kcr | 49,048 | <p>This is making the assumption that a is Real and positive, and then it calculates the derivative. But the result is zero.</p>
<pre><code>Assuming[ a \[Element] Reals && a > 0,
D[sqrt ((2*(((abc + Pide^2 + Pifg^2 + hij + Pikl^2)/(abc + Pide^2 +
Pifg^2)) - 1)*m)/(n^2 - o^2)/p), a]]
</code>... |
2,106,662 | <p>I'm trying to show that the barycentric coordinate of excenter of triangle ABC, where BC=a, AC=b, and AB=c, and excenter opposite vertex A is Ia, is Ia=(-a:b:c). I've gotten to the point where after a lot of ratio bashing I have that it's (ab/(b+c)):CP:BP, where P is the incenter, but I have no idea where to go from... | marty cohen | 13,079 | <p>Here is my solution
to get the generating functions.</p>
<p>I have shown every step
so any errors
can be readily found.</p>
<p>The method should be valid
even if there are errors.</p>
<p>Let
$A(x)
=\sum_{n=0}^{\infty} a_n x^n
$
and
$B(x)
=\sum_{n=0}^{\infty} b_n x^n
$.</p>
<p>Then
$A(x)-a_0
=\sum_{n=1}^{\infty} ... |
853,308 | <p>I want to calculate the expected number of steps (transitions) needed for absorbtion. So the transition probability matrix $P$ has exactly one (lets say it is the first one) column with a $1$ and the rest of that column $0$ as entries.</p>
<p>$P = \begin{bmatrix} 1 & * & \cdots & * \\ 0 & \vdots &am... | Jakub Konieczny | 10,674 | <p>We do it like that because we have a very good, rigorous grasp on what a set is, so we can speak of a set of pairs representing a function with no ambiguity at all. This is precisely the reason why we often construct <em>everything</em> out of sets - the integers, rationals, reals, and so on. This is mundane, and no... |
2,108,558 | <p>We are started Linear programming problem question. Questions provided in examples are easy. And in exercise starting questions are easy to solve. As we have given conditions to form equations and solve them.</p>
<p>But this question little difficult.</p>
<p>Question - </p>
<p><strong>An aeroplane can carry a max... | Arnaud D. | 245,577 | <p><strong>Hint :</strong> Let $P_A(t)$ be the characteristic polynomial of $A$. $P_A(t)$ can be written as a product of $n$ linear factors $(t-\lambda_i)$; since $A$ is not diagonalizable one of the factors has to appear twice.</p>
|
2,390,538 | <p>The problem says:</p>
<p>If every closed ball in a metric space $X$ is compact, show that $X$ is
separable.</p>
<p>I'm trying to use an equivalence in metric spaces that tells us: let X be the matric space, the following are equivalent</p>
<p>X is 2nd countable</p>
<p>X is Lindeloff</p>
<p>X is separable</p>
<... | Henno Brandsma | 4,280 | <p>$X = \cup_{n \in \mathbb{N}} D(p, n)$, for any $p \in X$ (where $D(x,r)$ denotes the closed ball around $p$ of radius $r$). </p>
<p>So $X$ is $\sigma$-compact hence Lindelöf hence separable. </p>
|
3,528,000 | <p>Given <span class="math-container">$f:\mathbb{R}\rightarrow\mathbb{R}^n$</span> differential and for every <span class="math-container">$x\in\mathbb{R}$</span> <span class="math-container">$\|f\|=1$</span>. </p>
<blockquote>
<p>Prove <span class="math-container">$\langle\,f(x),f'(x)\,\rangle=0$</span> for every... | José Carlos Santos | 446,262 | <p>Note that<span class="math-container">\begin{align}\lVert f\rVert=1&\iff\lVert f\rVert^2=1\\&\iff\langle f,f\rangle=1.\end{align}</span>So, if you differentiate <span class="math-container">$\langle f,f\rangle$</span>, you get <span class="math-container">$0$</span>. But<span class="math-container">\begin{al... |
3,528,000 | <p>Given <span class="math-container">$f:\mathbb{R}\rightarrow\mathbb{R}^n$</span> differential and for every <span class="math-container">$x\in\mathbb{R}$</span> <span class="math-container">$\|f\|=1$</span>. </p>
<blockquote>
<p>Prove <span class="math-container">$\langle\,f(x),f'(x)\,\rangle=0$</span> for every... | BCLC | 140,308 | <p>Let <span class="math-container">$f(x)=[f_1(x), ..., f_n(x)]$</span>, for all <span class="math-container">$x \in \mathbb R$</span> for some unique (scalar) functions <span class="math-container">$f_i: \mathbb R \to \mathbb R$</span>, <span class="math-container">$i=1,...,n$</span></p>
<p>We're given that <span cla... |
1,904,553 | <blockquote>
<p>$$\displaystyle\lim_{x\to0}\left(\frac{1}{x^5}\int_0^xe^{-t^2}\,dt-\frac{1}{x^4}+\frac{1}{3x^2}\right)$$</p>
</blockquote>
<p>I have this limit to be calculated. Since the first term takes the form $\frac 00$, I apply the L'Hospital rule. But after that all the terms are taking the form $\frac 10$. S... | N. S. | 9,176 | <p>By bringing the fractions to the same denominator, start by writing the limit as
$$\displaystyle\lim_{x\to0}\frac{3\int_0^xe^{-t^2}\,dt -3x+x^3 }{3x^5}$$</p>
<p>Now, since this is of the form $0/0$ by L'H and FTC you get
$$\displaystyle\lim_{x\to0}\frac{3e^{-x^2}-3+x^2 }{15x^4}$$</p>
<p>From here it is easy.</p>
|
2,466,855 | <p>We were given this question for homework that the professor couldn't explain how to solve (even in class he had trouble working it out). I'm only aware that we should be using the law of large numbers but I'm not sure how to apply it as the book for the course provides no examples. The answer in class was 10 and the... | Dr Potato | 622,884 | <p>Another approach: Consider this sequence as the direct product of the reciprocal of the factorial function with itself.</p>
<p>Given that the exponential function is the <a href="https://mathworld.wolfram.com/GeneratingFunction.html" rel="nofollow noreferrer">generating function</a> of the reciprocal of the factori... |
4,600,131 | <blockquote>
<p>If <span class="math-container">$$f(x)=\binom{n}{1}(x-1)^2-\binom{n}{2}(x-2)^2+\cdots+(-1)^{n-1}\binom{n}{n}(x-n)^2$$</span>
Find the value of <span class="math-container">$$\int_0^1f(x)dx$$</span></p>
</blockquote>
<p>I rewrote this into a compact form.
<span class="math-container">$$\sum_{k=1}^n\binom... | Dr. Wolfgang Hintze | 198,592 | <p>We consider the more general function</p>
<p><span class="math-container">$$f(x,m,n) =\sum _{k=1}^n (-1)^{k-1} \binom{n}{k} (x-k)^m\tag{1}$$</span></p>
<p>with <span class="math-container">$m=0,1,2,...$</span> and <span class="math-container">$n\ge 1$</span>. The case of interest here has <span class="math-container... |
2,643,900 | <p>for the problem </p>
<p>$$(1-2x)y'=y$$</p>
<p>the BC'S are $y(0)=-1$ and $y(1)=1$ and $0\leq x\leq 1$.</p>
<p>I solved this and got $\ln y =\ln\left(\dfrac 2 {1-2x}\right)+c$.</p>
<p>How do we determine the constant such that $y$ is real and finite everywhere from $0$ to $1$ (both limits included)?</p>
| Dylan | 135,643 | <p>Let's go through the solution again. The equation separates to
$$ \frac{y'}{y} = \frac{1}{1-2x} $$</p>
<p>The equation in undefined on $y=0$ and $x=\frac12$. This means there are 4 possible regions in $\Bbb R^2$ where the solution can exist.</p>
<p>$$ \begin{cases}
\ln y = -\frac12 \ln (1-2x) + c_1, && x ... |
1,699,627 | <p>Why is the following equation true?$$\int_0^\infty e^{-nx}x^{s-1}dx = \frac {\Gamma(s)}{n^s}$$
I know what the Gamma function is, but why does dividing by $n^s$ turn the $e^{-x}$ in the integrand into $e^{-nx}$? I tried writing out both sides in their integral forms but $n^{-s}$ and $e^{-x}$ don't mix into $e^{-nx}$... | DeepSea | 101,504 | <p>The number of ways to select $5$ cards that have $0,1,2,3$ kings are: $ \binom{48}{5}, \binom{4}{1}\binom{48}{4}, \binom{4}{2}\binom{48}{3}, \binom{4}{3}\binom{48}{2}$ respectively. Thus the probability of at most $3$ kings is: $Pr(x \leq 3) =\dfrac{\binom{48}{5}+\binom{4}{1}\binom{48}{4}+\binom{4}{2}\binom{48}{3}+\... |
1,320,112 | <p>Using the following identity $$\int_{0}^{\infty}\frac{f\left ( t \right )}{t}dt= \int_{0}^{\infty}\mathcal{L}\left \{ f\left ( t \right ) \right \}\left ( u \right )du$$
I rewrote $$\int_{0}^{\infty}\frac{\sin^{2}\left ( t \right )}{t^{2}}dt$$
as $$\int_{0}^{\infty}\frac{\sin^{2}\left ( t \right )}{t\cdot t}dt$$
And... | Gappy Hilmore | 219,783 | <p>You seemed to have confused logarithmic identities. The integrand should be</p>
<p>$$\frac{1}{4} \log \left(\frac{4}{s^2}+1\right)$$</p>
<p>which indeed evaluates as $\pi/2$. I am guessing you wrongly factored out an $s^2$</p>
|
118,545 | <p>I gather that the question whether the Bruck-Chowla-Ryser condition was sufficient used to top the list, but now that that's settled - what is considered the most interesting open question?</p>
| Yuichiro Fujiwara | 27,829 | <p><strong>Edit</strong>: Since this just became available on arXiv, <a href="http://arxiv.org/pdf/1401.3665.pdf" rel="nofollow noreferrer"><strong><em>Peter Keevash solved the existence conjecture of Steiner $t$-designs</em></strong></a>, which means that what I wrote below a year ago as one of the most important open... |
118,545 | <p>I gather that the question whether the Bruck-Chowla-Ryser condition was sufficient used to top the list, but now that that's settled - what is considered the most interesting open question?</p>
| Shahrooz | 19,885 | <p>These four below problems are interesting and still open conjecture in design theory and its related topics:</p>
<p>$1.$ There exist Ucycles for $k$-subsets of $[n]$, provided $k$ divides $Cr(n-1,k-1)$ and $n$ is sufficiently large.</p>
<p>$2.$ For $n≥6$ even, it is not possible to have a length $\frac{n^2}{2}$ c... |
320,355 | <p>Show that $$\nabla\cdot (\nabla f\times \nabla h)=0,$$
where $f = f(x,y,z)$ and $h = h(x,y,z)$.</p>
<p>I have tried but I just keep getting a mess that I cannot simplify. I also need to show that </p>
<p>$$\nabla \cdot (\nabla f \times r) = 0$$</p>
<p>using the first result.</p>
<p>Thanks in advance for any help... | Shuhao Cao | 7,200 | <p>Suppose everything is smooth: consider an arbitrary smooth simply connected domain $D\subset \mathbb{R}^3$, divergence theorem reads:</p>
<p>$$
\iiint_D \nabla \cdot (\nabla f\times \nabla h) \,dV = \iint_{\partial D} \nabla f\times \nabla h\cdot d\mathbf{S} = \iint_{\partial D} \nabla f\times \nabla h\cdot\mathbf{... |
2,658,627 | <p>I am trying to solve the following:</p>
<blockquote>
<p>Suppose the bank paid 12 % per year, but compounded that interest
monthly. That is, suppose 1 % interest was added to your account every
month. Then how much would you have after 30 years and after 60 years
if you started with $100?</p>
</blockquote>
<p>What I ... | Andrew Li | 344,419 | <p>Remember that the compound interest formula is $C = P\left(1+\dfrac{r}{n}\right)^{nt}$ where $r$ is the rate, $n$ is the amount of times during the year it's compounded, $t$ is the number of years, and $P$ is the principle, or initial starting value.</p>
<p>Your rate $r$ is $0.12$, or $12%$. When you divide that by... |
609,770 | <p>We have an empty container and $n$ cups of water and $m$ empty cups. Suppose we want to find out how many ways we can add the cups of water to the bucket and remove them with the empty cups. You can use each cup once but the cups are unique. </p>
<p>The question: In how many ways can you perform this operation.</p... | parsiad | 64,601 | <p>Take $\epsilon=1$. Let $\delta>0$. We can always choose $n$ sufficiently large and $x=\frac{1}{n}$ and $y=\frac{1}{n+1}$ so that
$$
x-y=\frac{1}{n\left(n+1\right)}<\delta.
$$
However,
$$
\left|f\left(\frac{1}{n}\right)-f\left(\frac{1}{n+1}\right)\right|=\left|n-\left(n+1\right)\right|=1.
$$</p>
|
613,961 | <p>I got the following problem:</p>
<p>Let $V$ be a real vector space and let $q: V \to \mathbb R$ be a real quadratic form,<br/> Prove that if the set $L = \{v \in V | q(v) \ge 0\}$ forms a subspace of $V$
then q is definite (meaning $q$ is positive definite, positive semidefinite, negative definite or negative semi... | Eric Auld | 76,333 | <p><strong>Hint:</strong> You want to show that if $L=\{x \mid q(x)\geq 0\}$ is a subspace, then $L=0$ or $L=V$</p>
|
817,680 | <p><strong>Question:</strong></p>
<blockquote>
<p>Assume that $a_{n}>0,n\in N^{+}$, and that $$\sum_{n=1}^{\infty}a_{n}$$ is convergent. Show that
$$\sum_{n=1}^{\infty}\dfrac{a_{n}}{(n+1)a_{n+1}}$$ is divergent?</p>
</blockquote>
<p>My idea: since
$\sum_{n=1}^{\infty}a_{n}$ converges, then there exists $M>... | math110 | 58,742 | <p>Longtime ago,I post my methods</p>
<p>proof:Assmue that: $\displaystyle\sum_{n=1}^{\infty}\dfrac{a_{n}}{(n+1)a_{n+1}}$,then for any $p\in N^{+}$,have
$$\sum_{k=n}^{n+p-1}\dfrac{a_{k}}{(k+1)a_{k+1}}<\dfrac{1}{4}\Longrightarrow\sum_{k=n}^{n+p-1}\dfrac{a_{k}}{(n+p)a_{k+1}}<\dfrac{1}{4}$$
then
$$\dfrac{1}{p}\su... |
1,391,214 | <p>I've been thinking about the differences in numbers so for example: </p>
<p>$\begin{array}{ccccccc} &&&0&&&\\ &&1&&1&&\\&1&&2&&3&\\0&&1&&3&&6
\end{array}$</p>
<p>or with absolute differences:</p>
<p>$\begin{array}{ccc... | rela589n | 892,830 | <p>It seems to be the number of edges in N-dimensional cube:</p>
<pre><code>4 = 1 * 2 + 2 (square)
12 = 4 * 2 + 4 (ordinary cube)
32 = 12 * 2 + 8 (4-D cube)
80 = 32 * 2 + 16
192 = 80 * 2 + 32
448 = 192 * 2 + 64
1024 = 448 * 2 + 128
2304 = 1024 * 2 + 256
5120 = 2304 * 2 + 512
11264 = 5120 * 2 + 1024
24576 = 11264 * 2 + ... |
2,915,735 | <p>So I just got done showing explicitly that an isomorphism exists between these two rings if the $\gcd(m, n) = 1$, and I did not have much trouble with that. For some reason I'm having a lot harder of a time showing that the result <em>is not</em> true if $m$ and $n$ are not relatively prime. Can somebody help me out... | fleablood | 280,126 | <p>If $m$ and $n$ are not relatively prime then they have a least common multiple $L$ which is <em>less</em> than $mn$. (Namely $\frac {mn}{\gcd(n,m)}$)</p>
<p>And if you take any element of $(a,b)\in \mathbb Z_m\times \mathbb Z_n$ and add it to itself $L$ times you get the identity[1]. So there is no element with o... |
9,416 | <p>Say I pass 512 samples into my FFT</p>
<p>My microphone spits out data at 10KHz, so this represents 1/20s.</p>
<p>(So the lowest frequency FFT would pick up would be 40Hz).</p>
<p>The FFT will return an array of 512 frequency bins
- bin 0: [0 - 40Hz)
- bin 1: [40 - 80Hz)
etc</p>
<p>So if my original sound con... | endolith | 2,206 | <p>Generally you need to apply a <a href="http://en.wikipedia.org/wiki/Window_function" rel="noreferrer">windowing function</a> to your signal before performing the FFT, to get clean spikes for the frequency components (without "skirts"). (The only exception is if your frequency components <a href="https://gist.github... |
9,416 | <p>Say I pass 512 samples into my FFT</p>
<p>My microphone spits out data at 10KHz, so this represents 1/20s.</p>
<p>(So the lowest frequency FFT would pick up would be 40Hz).</p>
<p>The FFT will return an array of 512 frequency bins
- bin 0: [0 - 40Hz)
- bin 1: [40 - 80Hz)
etc</p>
<p>So if my original sound con... | Mykie | 832 | <p>You can try a zero-padded FFT ( add zeros to the end of your signal and then take FFT).</p>
|
4,536,320 | <p>Let <span class="math-container">$R$</span> be a ring and <span class="math-container">$A \subseteq R$</span> be finite, say <span class="math-container">$A = \{a\}$</span>. The set <span class="math-container">$$RaR = \{ras\:\: : r,s\in R\}$$</span> Why is this not necessarily closed under addition?</p>
<p>Take <sp... | cluelessmathematician | 798,206 | <p>Brute forcing this (i.e. go through all <span class="math-container">$21^2$</span> combinations of <span class="math-container">$(m,a)\in\{0,...,20\}^2$</span> and tally up the answer) is likely feasible, but tedious and would not work on an exam, especially if the professor chooses anything much higher than 20.</p>... |
1,088,973 | <p>I am posed with the following problem:</p>
<blockquote>
<p>Suppose that $V$ is finite-dimensional and that $S,T\in \mathcal{L}(V)$. Prove that $ST=I$ if and only if $TS=I$, where $I$ is the identity map/operator.</p>
</blockquote>
<p>I have attempted it in the following way.</p>
<p>Consider the proof in one dir... | Ilmari Karonen | 9,602 | <p>Hint: Rewrite your integral as:</p>
<p>$$\int_0^1 \int_y^1 \int_0^y f(x,y,z) \;dz\,dx\,dy
= \int_0^1 \int_0^1 \int_0^1 f(x,y,z) \,\mathbf 1_{x \ge y \ge z} \;dz\,dx\,dy,$$</p>
<p>where the <a href="//en.wikipedia.org/wiki/Indicator_function" rel="nofollow">indicator function</a> $\mathbf 1_{x \ge y \ge z}$ equals ... |
1,088,973 | <p>I am posed with the following problem:</p>
<blockquote>
<p>Suppose that $V$ is finite-dimensional and that $S,T\in \mathcal{L}(V)$. Prove that $ST=I$ if and only if $TS=I$, where $I$ is the identity map/operator.</p>
</blockquote>
<p>I have attempted it in the following way.</p>
<p>Consider the proof in one dir... | Mark McClure | 21,361 | <p>The bounds of integration determine equations that bound a solid $S$ in three-dimensional space. After you integrate with respect to the first variable, you should orthogonally project $S$ along the axis specified by that first variable onto the plane spanned by the other two variables. That projection then determi... |
1,532,202 | <p>I want to find out $$\mathcal{L^{-1}}\{\frac{e^{-\sqrt{s+2}}}{s}\}$$
How do you find the inverse Laplace? </p>
<p>thanks</p>
| Jan Eerland | 226,665 | <p>Mathematica 10.0 gives this output:</p>
<p>$$\mathcal{L}_{s}^{-1}\left[\frac{e^{-\sqrt{s+2}}}{s}\right]_{(t)}=\frac{1}{2}e^{-\sqrt{2}}\left(\text{erfc}\left(\frac{1-2t\sqrt{2}}{2\sqrt{t}}\right)+e^{e\sqrt{2}}\space\text{erfc}\left(\frac{1+2t\sqrt{2}}{2\sqrt{t}}\right)\right)$$</p>
|
1,564,962 | <p>I need to calculate the following:</p>
<p>$$x=8 \pmod{9}$$
$$x=9 \pmod{10}$$
$$x=0 \pmod{11}$$</p>
<p>I am using the chinese remainder theorem as follows:</p>
<p>Step 1:</p>
<p>$$m=9\cdot10\cdot11 = 990$$</p>
<p>Step 2:</p>
<p>$$M_1 = \frac{m}{9} = 110$$</p>
<p>$$M_2 = \frac{m}{10} = 99$$</p>
<p>$$M_3 = \fra... | Bernard | 202,857 | <p>Start solving the first two congruences: a <em>Bézout's relation</em> between $9$ and $10$ is $\;9\cdot 9-8\cdot 10=1$, hence
$$\begin{cases}m\equiv\color{red}{8}\mod9 \\ m\equiv \color{red}{9}\mod 10\end{cases}\iff m\equiv\color{red}{9}\cdot 9\cdot 9-\color{red}{8}\cdot8\cdot 10=\color{red}{89}\mod 90 $$
Now solve ... |
2,665,723 | <p>Find value of $k$ for which the equation $\sqrt{3z+3}-\sqrt{3z-9}=\sqrt{2z+k}$ has no solution.</p>
<p>solution i try $\sqrt{3z+3}-\sqrt{3z-9}=\sqrt{2z+k}......(1)$</p>
<p>$\displaystyle \sqrt{3z+3}+\sqrt{3z-9}=\frac{12}{\sqrt{2z+k}}........(2)$</p>
<p>$\displaystyle 2\sqrt{3z+3}=\frac{12}{\sqrt{2z+k}}+\sqrt{2z+k... | Robert Z | 299,698 | <p>We have that
$$n(n-1)=\frac{n^3-(n-1)^3}{3}-\frac{1}{3}.$$
Therefore
$$\begin{align}
(1\cdot 2)+(3\cdot 4)+(5\cdot 6)+\dots +(99\cdot 100)&=\sum_{n=1}^{50}(2n-1)(2n)=4\sum_{n=1}^{50}n(n-1)+2\sum_{n=1}^{50}n\\
&=4\sum_{n=1}^{50}\frac{n^3-(n-1)^3}{3}-\frac{4\cdot 50}{3}+50\cdot 51\\
&=\frac{4}{3}\left(50^3... |
2,665,723 | <p>Find value of $k$ for which the equation $\sqrt{3z+3}-\sqrt{3z-9}=\sqrt{2z+k}$ has no solution.</p>
<p>solution i try $\sqrt{3z+3}-\sqrt{3z-9}=\sqrt{2z+k}......(1)$</p>
<p>$\displaystyle \sqrt{3z+3}+\sqrt{3z-9}=\frac{12}{\sqrt{2z+k}}........(2)$</p>
<p>$\displaystyle 2\sqrt{3z+3}=\frac{12}{\sqrt{2z+k}}+\sqrt{2z+k... | Sri-Amirthan Theivendran | 302,692 | <p>Observe that your sum may be written as
$$
\sum_{n=0}^{99}n(n+1)=2\sum_{n=0}^{99}\binom{n+1}{2}=2\sum_{i=2}^{100}\binom{i}{2}.
$$
Moreover, using Pascal's identity and telescoping, we obtain
$$
2\sum_{i=2}^{100}\left[\binom{i+1}{3}-\binom{i}{3}\right]
=2\binom{101}{3}=\frac{2(101)(100)(99)}{6}=333300.
$$</p>
|
2,665,723 | <p>Find value of $k$ for which the equation $\sqrt{3z+3}-\sqrt{3z-9}=\sqrt{2z+k}$ has no solution.</p>
<p>solution i try $\sqrt{3z+3}-\sqrt{3z-9}=\sqrt{2z+k}......(1)$</p>
<p>$\displaystyle \sqrt{3z+3}+\sqrt{3z-9}=\frac{12}{\sqrt{2z+k}}........(2)$</p>
<p>$\displaystyle 2\sqrt{3z+3}=\frac{12}{\sqrt{2z+k}}+\sqrt{2z+k... | robjohn | 13,854 | <p>$$
\begin{align}
\sum_{k=1}^n(2k-1)2k
&=\sum_{k=1}^n\left[8\binom{k}{2}+2\binom{k}{1}\right]\\
&=8\binom{n+1}{3}+2\binom{n+1}{2}\\
&=\frac43(n+1)n(n-1)+(n+1)n\\[3pt]
&=\frac{n(n+1)(4n-1)}3
\end{align}
$$
Plug in $n=50$ to get
$$
\sum_{k=1}^{50}(2k-1)2k=169150
$$</p>
|
713,732 | <p>I want to know how one would go about solving an <em>unfactorable cubic</em>. I know how to factor cubics to solve them, but I do not know what to do if I cannot factor it. For example, if I have to solve for $x$ in the cubic equation:
$$2x^3+6x^2-x+4=0$$
how would I do it?</p>
<p>Edit: I have heard people telling ... | Batman | 127,428 | <p>Normally, you want to work with codes in standard form - that is, you want to write the generator matrix in the form $[I | P]$. Then, a parity check matrix corresponding to this code is $[-P^T | I]$ (i.e. it is a generator of the dual code - the code consisting of all vectors orthogonal to the original code). </p>
|
1,043 | <p>Hi all,</p>
<p>The short-time fourier transform decomposes a signal window into a sin/cosine series.</p>
<p>How would one approximate a signal in the same way, but using a set of arbitrary basis functions instead of sin/cos? These arbitrary basis functions are likely in my case to be very small discrete chunks of ... | Michael Hoffman | 429 | <p>Wavelets are generally used for nonperiodic signals. They're often used in earthquake detection and things like that. There are many books on the subject, a quick look for "Wavelets" in amazon.com should reveal many.</p>
<p>The Haar Wavelet and Daubchies Wavelet might be good choices. Haar may be better if... |
2,963,324 | <p>I want to prove <span class="math-container">$a \equiv b\;(\text{mod} \;n)$</span> is an equivalence relation then would it be ok to write,</p>
<p>Reflexive as, for all <span class="math-container">$a$</span>, <span class="math-container">$a \equiv a\;(\text{mod} \;n)$</span></p>
<p>Symmetric as, <span class="math... | Mark | 470,733 | <p>It is not ok, because what you did is just wrote what you have to prove, not really proved it. First of all, the relation is on <span class="math-container">$\mathbb{Z}$</span>. Next, what is the relation? <span class="math-container">$a\equiv b$</span>(mod <span class="math-container">$n$</span>) by definition mean... |
1,573,113 | <p>Find the derivative of $\tan$($\sqrt{1 -x}$)</p>
<p>So I know I have to apply the product rule so wouldn't it be </p>
<p>$\sec^x(\sqrt{1-x})+ \sqrt{1-x}$
$\frac{tan(x)}{2}$
but the final answer says $\frac{-sec^2(\sqrt{1-x})}{2\sqrt{1-x}}$.</p>
| Claude Leibovici | 82,404 | <p>It is not the product rule but the chain rule which must be used.</p>
<p>Consider $A=\tan(u)$ with $u=\sqrt{1-x}$. So $$\frac{dA}{dx}=\frac{dA}{du}\times \frac{du}{dx}$$ with $$\frac{dA}{du}=\sec ^2(u)=\sec ^2(\sqrt{1-x})$$ $$\frac{du}{dx}=-\frac{1}{2 \sqrt{1-x}}$$ and finally $$\frac{dA}{dx}=-\frac{\sec ^2\left(\s... |
1,573,113 | <p>Find the derivative of $\tan$($\sqrt{1 -x}$)</p>
<p>So I know I have to apply the product rule so wouldn't it be </p>
<p>$\sec^x(\sqrt{1-x})+ \sqrt{1-x}$
$\frac{tan(x)}{2}$
but the final answer says $\frac{-sec^2(\sqrt{1-x})}{2\sqrt{1-x}}$.</p>
| John Joy | 140,156 | <p>Same as the other answer, with a bit of difference in style.</p>
<p>$$\begin{array}{lll}
\frac{d}{dx}\tan\sqrt{1-x} &=& \frac{d\tan\sqrt{1-x}}{d\sqrt{1-x}}\cdot\frac{d\sqrt{1-x}}{d (1-x)}\cdot\frac{d(1-x)}{dx}\\
&=&\frac{\sec^2\sqrt{1-x}}{1}\cdot\frac{1}{2\sqrt{1-x}}\cdot(-1)\\
&=&\frac{-\se... |
2,498,123 | <blockquote>
<p>Given a $2 \times 2$ matrix $B$ that satisfies $B^2=3B-2I$, find the eigenvalues of $B$.</p>
</blockquote>
<p>My attempt: </p>
<p>Let $v$ be an eigenvector for B, and $\lambda$ it's corresponding eigenvalue. Also, let $T$ be the linear transformation (not that this is exactly necessary for the quest... | Jonas Meyer | 1,424 | <p>Suppose $\lambda $ is an eigenvalue for $B$, with eigenvector $v$. Note that $B^2v=BBv=B(\lambda v)=\lambda^2v$. Apply each side of your equation $B^2=3B-2I$ to the vector $v$ to get $\lambda^2 v= (3\lambda -2)v$, or $(\lambda^2-3\lambda +2)v=0$. If a scalar times a nonzero vector is the zero vector, then the scal... |
3,118 | <p>Can anyone help me out here? Can't seem to find the right rules of divisibility to show this:</p>
<p>If $a \mid m$ and $(a + 1) \mid m$, then $a(a + 1) \mid m$.</p>
| David E Speyer | 448 | <p>An alternate route: We show that, if $ax+by=1$ and $m$ is divisible by $a$ and $b$ then $m$ is divisible by $ab$. (Then apply this with $b=a+1$, $x=-1$ and $y=1$.) </p>
<p><strong>Proof:</strong> Let $m=ak=bl$. Then $ab(xl+ky)=(ax+by)m=m$. QED</p>
<p>The point here is that the hypothesis $\exists_{x,y}: ax+by=1$ i... |
3,385,921 | <p>What does this mean? </p>
<blockquote>
<p>A matrix is diagonizable if and only if its eigenvectors are invertible.</p>
</blockquote>
| Mike | 544,150 | <p>An <span class="math-container">$n \times n$</span> matrix <span class="math-container">$A$</span> is diagonizable i.e., there exists a matrix <span class="math-container">$P$</span> such that <span class="math-container">$A = P^{-1}DP$</span> where <span class="math-container">$D$</span> is a diagonal matrix; iff t... |
2,043,429 | <p>In my textbook, it states that the general formula for the partial sum </p>
<p>$$\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}$$</p>
<p>My question is, if I have the following sum instead:</p>
<p>$$\sum_{i=1}^n \frac{1}{i^2}$$ </p>
<p>Can I just flip the general formula to get this:</p>
<p>$$\sum_{i=1}^n \frac{1}{i... | Vidyanshu Mishra | 363,566 | <p>The way you are saying it, I remembered the mistake I often used to do when I was in fifth standerd. I used to convert $\frac{1}{\frac{1}{a}+\frac{1}{b}}=a+b$. That is what you are doing here. Just take deep breath and look at what you have written make sense or not.</p>
|
805,341 | <p>Need a bit of help with this question. </p>
<p>We're given two invertible square $n\times n$ matrices $A$ and $B$ with entries in the reals.</p>
<p>We have to show that $AB$ is also invertible and then express $(AB)^{-1}$ in terms of $A$ and $B$. </p>
<p>I've managed to get the first part out. </p>
<p>Since $A$ ... | Cousin | 54,755 | <p>Consider $B^{-1}A^{-1}$. Multiply with this matrix on both sides of $AB$. You'll get the identity each time. Note that since $A$ and $B$ are invertible $A^{-1}$ and $B^{-1}$ are well defined.</p>
|
805,341 | <p>Need a bit of help with this question. </p>
<p>We're given two invertible square $n\times n$ matrices $A$ and $B$ with entries in the reals.</p>
<p>We have to show that $AB$ is also invertible and then express $(AB)^{-1}$ in terms of $A$ and $B$. </p>
<p>I've managed to get the first part out. </p>
<p>Since $A$ ... | tpb261 | 125,795 | <p>Here's another way:
let C be the inverse of AB. then <code>ABC = I</code> and <code>CAB = I</code>. Now try to get C by multiplying on the left and/or multiplying or the right so that you are finally left with expression of the form <code>C = f(A, B)</code>. This is the mathematical form of <a href="https://math.sta... |
1,101,104 | <p>Let $f: A \to B$. How can I show that $f$ is surjective if and only if (for every $C$ and every pair of functions $g, h: B \to C$) when there is the following implication?</p>
<p>$$
g \circ f = h\circ f \to g=h
$$</p>
| Richard | 19,749 | <p>When you multiply the series for $sin(x)$ by $i$, and add it to the series for $cos(x)$, you get the series for $exp(ix)$. In other words, $cos(x)$, is the real part of $exp(ix)$. </p>
<p>Therefore, $cos(x+y)$ is the real part of $exp(i(x+y))$. By laws of exponents, this is $$real( exp(ix)exp(iy))$$</p>
<p>By the ... |
1,101,104 | <p>Let $f: A \to B$. How can I show that $f$ is surjective if and only if (for every $C$ and every pair of functions $g, h: B \to C$) when there is the following implication?</p>
<p>$$
g \circ f = h\circ f \to g=h
$$</p>
| Gabriel | 209,805 | <p>Very bad answer. One cannot use complex analysis to prove the sum of angles formula, one must in fact use the sum of angles formula to prove all of the useful properties of the "complex exponential" (which is based on the definitions of sine and cosine!). Otherwise you are just playing with definitions and giving ... |
41,888 | <p>So I've got this line that contains the solution of a partial, non-extraordinary differential equation (because Mathematica doesn't handle extraordinary partial differential equations):</p>
<pre><code>phi6m = NDSolveValue[{D[u[t, x], t, t] -
D[u[t, x], x, x] == -6 u[t, x]^5 + 10.5 u[t, x]^3 - 4.5 u[t, x], u[0,... | Szabolcs | 12 | <p>There is <code>ValueFunction</code>, documented <a href="http://reference.wolfram.com/mathematica/Experimental/ref/ValueFunction.html">here</a>.</p>
<p>It allows detecting value changes for given symbols.</p>
<p>For example,</p>
<pre><code>In[1]:= Experimental`ValueFunction[x] := Print["x changed"]
In[2]:= x = 6... |
1,439,920 | <blockquote>
<p>So, the question is:<br>
Calculate the probability that 10 dice give more than 2 6s.</p>
</blockquote>
<p>I've calculated that the probability for throwing 3 6s is 1/216.</p>
<p>And by that logic: 1/216 + 1/216 + .. + 1/216 = 10/216.</p>
<p>But I've been told that this isn't the proper way set it... | Zach466920 | 219,489 | <p>$$x^5 − 1102 \cdot x^4 − 2015 \cdot x = 0$$</p>
<p>This factors into,</p>
<p>$$\left( x^4 − 1102x^3 − 2015 \right) \cdot x = 0$$</p>
<p>Therefore $x=0$ is a root. Keep simplifying,</p>
<p>$$x^4 − 1102x^3 − 2015=0$$</p>
<p>Set this equal to $f(x)$,</p>
<p>$$f(x)=x^4 − 1102x^3 − 2015$$</p>
<p>$f(-1)=-912$ and $... |
1,439,920 | <blockquote>
<p>So, the question is:<br>
Calculate the probability that 10 dice give more than 2 6s.</p>
</blockquote>
<p>I've calculated that the probability for throwing 3 6s is 1/216.</p>
<p>And by that logic: 1/216 + 1/216 + .. + 1/216 = 10/216.</p>
<p>But I've been told that this isn't the proper way set it... | Christian Blatter | 1,303 | <p>It's about the real zeros of $x q(x)$ with $q(x):=x^4-1102 x^3-2015$. There is the obvious zero $x=0$. Furthermore $q(0)<0$ and $\lim_{x\to\pm\infty} q(x)=+\infty$ guarantee two more real zeros.</p>
|
785,314 | <p>let $$I_n = \int_{\pi/2}^{x} \frac{\cos^{2n+1}t}{\sin(t)} \ dt, n \geq 0$$</p>
<p>show $$2(n+1)I_{n+1} = 2(n+1)I_n +\cos^{2n+2}x$$</p>
<p>I showed the result by considering $I_{n+1} - I_n$ but I'm wondering how could I do it using integration by parts?</p>
<p>Similarly for $J_n = \int_0^x \frac{\sinh^{2n+1}t}{\co... | Pranav Arora | 117,767 | <p>$$I_n=\int_{\pi/2}^x \frac{\cos^{2n+1}t}{\sin t}\,dt=\int_{\pi/2}^x \frac{\cos^{2n-1}t (1-\sin^2 t)}{\sin t}\,dt= I_{n-1}-\int_{\pi/2}^x \cos^{2n-1}t\sin t\,dt $$</p>
<p>Use the substitution $\cos t=y \Rightarrow -\sin t\,dt=dy$ to get:
$$\int_{\pi/2}^x \cos^{2n-1}t\sin t\,dt =-\int_0^{\cos x} t^{2n-1}\,dt=-\frac{\... |
16,754 | <p>Let $c$ be an integer, not necessarily positive and not a square. Let $R=\mathbb{Z}[\sqrt{c}]$
denote the set of numbers of the form $$a+b\sqrt{c}, a,b \in \mathbb{Z}.$$
Then $R$ is a subring of $\mathbb{C}$ under the usual addition and multiplication.</p>
<p>My question is: if $R$ is a UFD (unique factorization ... | Akhil Mathew | 536 | <p>Yes. If it is a UFD, then it is integrally closed, hence it is a Dedekind domain because it is of dimension one (being contained in the integral closure of $\mathbb{Z}$ in some finite extension of $\mathbb{Q}$). A Dedekind domain is a UFD iff it is a PID: indeed, this is equivalent to every non-zero prime being prin... |
748,489 | <p>I have a measure ($x$) which the domain is $[0, +\infty)$ and measure some sort of variability. I want to create a new measure ($y$) that represents regularity and is inverse related to $x$.</p>
<p>It is easy, just make $y = -x$. However I want this new measure to be positive, to make it more interpretable. Linearl... | Henry | 6,460 | <p>Try something like $y=f(x)= \dfrac{1}{1+x}$.</p>
<p>This has the properties:</p>
<ul>
<li>$f(x)$ decreases as positive $x$ increases </li>
<li>$f(0)=1$</li>
<li>$f(1)=\frac12$</li>
<li>$f(x)\lt 0.01$ for $x \gt 99$ and so also for $x \ge 100$</li>
<li>$f(x) \to 0$ as $x \to +\infty$</li>
</ul>
<p>Its inverse is $... |
3,362,916 | <p>I'm trying to graph <span class="math-container">$|x+y|+|x-y|=4$</span>. I rewrote the expression as follows to get a function that resembles the direction of unit vectors at <span class="math-container">$\pi/4$</span> to the horizontal axis (take it to be <span class="math-container">$x$</span>)<span class="math-co... | Z Ahmed | 671,540 | <p>It is a square with sides as <span class="math-container">$x=\pm 2$</span> and <span class="math-container">$y=\pm 2$</span>,and diagonals as <span class="math-container">$y=\pm x$</span>. for the explanation see my answer in MSE below.</p>
<p><a href="https://math.stackexchange.com/questions/3329862/how-to-draw-gr... |
1,386,307 | <p>If you consider that you have a coin, head or tails, and let's say tails equals winning the lottery. If I participate in one such event, I may not get tails. It's roughly 50%. But if a hundred people are standing with a coin and I or them get to flip it, my chances of having gotten a tail after these ten attempts... | karmalu | 258,239 | <p>From a probability point there is no difference between lottery and coin toss, but there is when you compute the number. Tossing ten coin you have $\frac{1}{2^{10}}=\frac{1}{1024}$ probability of not winning which is $\frac{1023}{1024}$ probability of winning. If you partecipate in ten lotteries, say that in each of... |
2,109,347 | <p>My statistics note states that the variance of the empirical distribution is
$v= \sum_{i=1}^{n}(x_i-\bar x )^2\frac {1} {n}$ which the author then re-writes as
$v= \sum_{i=1}^{n}x_i^2 (\frac {1} {n}) - \bar x^2$. How is this achieved?</p>
| zhoraster | 262,269 | <p>A useful idea is to consider sample mean, variance, moments, quantiles etc as the mean, variance etc with respect to the <em>empirical measure</em>. Namely, the sample mean $\overline x$ is the expectation of empirical measure, which assigns probabilities $1/n$ to the sampled values $\{x_1,x_2,\dots,x_n\}$:
$$
\over... |
2,779,379 | <blockquote>
<p>Let $f:\mathbb R\to \mathbb R$ be a continuous function and $\Phi(x)=\int_0^x (x-t)f(t)\,dt$. Justify that $\Phi(x)$ is twice differentiable and calculate $\Phi''(x)$.</p>
</blockquote>
<p>I'm having a hard time finding the first derivative of $\Phi(x)$. Here's what I tried so far:</p>
<p>Since $f$ ... | chilliefiber | 499,334 | <p>So, this was a silly mistake, but I was really having trouble with it. I'm still not sure I understand exactly what is going on, I'm a bit confused because there are 2 variables (x and t), but we have a multivariable function in $\Phi(x)=\int_0^x (x-t)f(t)\,dt$, so I don't think I can apply the Fundamental Theorem o... |
3,295,318 | <p><span class="math-container">$$\int _{ c } \frac { \cos(iz) }{ { z }^{ 2 }({ z }^{ 2 }+2i) } dz$$</span></p>
<p>I used residue caculus for solving the problem but i am not pretty sure if the approach is right.</p>
<p>The attempt has been annexed in the pictures.</p>
<p><a href="https://i.stack.imgur.com/ZSv27.jpg" r... | herb steinberg | 501,262 | <p>correction:</p>
<p>Let <span class="math-container">$m$</span> be the number. Find the smallest prime factor <span class="math-container">$n$</span> of <span class="math-container">$m$</span> and let <span class="math-container">$k=\frac{m}{n}$</span> The sequence will be <span class="math-container">$k-n+1,k-n+3... |
2,919,841 | <blockquote>
<p><span class="math-container">$$\Large\bigcup\limits_{k\in\bigcup\limits_{i\in I}J_i}A_k=\bigcup\limits_{i\in I}\bigg(\bigcup\limits_{k\in J_i}A_k\bigg)$$</span></p>
</blockquote>
<hr />
<p><strong>My attempt:</strong></p>
<p><span class="math-container">$\large x\in\bigcup\limits_{k\in\bigcup\limits_{i\... | marty cohen | 13,079 | <p>If you assume
the binomial theorem,
you can argue directly
like this:</p>
<p>If
$n >m > a+1$
and
$c = 1+h$
then</p>
<p>$\begin{array}\\
c^n
&=(1+h)^n\\
&=\sum_{k=0}^n \binom{n}{k}h^k\\
&>\binom{n}{m}h^{m}\\
&=\dfrac{\prod_{j=0}^{m-1}(n-j)}{m!}h^{m}\\
&=\dfrac{n^m\prod_{j=0}^{m-1}(1-j/... |
2,919,841 | <blockquote>
<p><span class="math-container">$$\Large\bigcup\limits_{k\in\bigcup\limits_{i\in I}J_i}A_k=\bigcup\limits_{i\in I}\bigg(\bigcup\limits_{k\in J_i}A_k\bigg)$$</span></p>
</blockquote>
<hr />
<p><strong>My attempt:</strong></p>
<p><span class="math-container">$\large x\in\bigcup\limits_{k\in\bigcup\limits_{i\... | user | 505,767 | <p>That nice, as an alternative without binomial theorem we can show that</p>
<p>$$\frac{n^{\alpha}}{c^n}=e^{\alpha \log n-n\log c} \to 0$$</p>
<p>indeed since $\frac{\log x}x\to 0 \implies \frac{\log n}n\to 0$ we have</p>
<p>$$\alpha \log n-n\log c = n\left(\alpha\frac{\log n}n-\log c\right) \to-\infty$$</p>
<p>an... |
2,231,487 | <p>In [Mathematical Logic] by Chiswell and Hodges, within the context of natural deduction and the language of propositions LP (basically like <a href="http://www.cs.cornell.edu/courses/cs3110/2011sp/lectures/lec13-logic/logic.htm" rel="nofollow noreferrer">here</a>) it is asked to show, by counter-example that a certa... | Noah Schweber | 28,111 | <p>There is in fact a small taste of undecidability lurking here, but it's in a very weak form: we're looking at sentences which are undecidable <em>from the empty theory</em>.</p>
<p>The point is that no matter what $p$ is, we can prove - from no axioms at all! - the sentence "$p\vee\neg p$". That is, the sequent $$\... |
1,750,104 | <p>I've had this question in my exam, which most of my batch mates couldn't solve it.The question by the way is the Laplace Transform inverse of </p>
<p>$$\frac{\ln s}{(s+1)^2}$$</p>
<p>A Hint was also given, which includes the Laplace Transform of ln t.</p>
| Salihcyilmaz | 227,487 | <p>$f(s,a) = L(t^a) = \frac{\Gamma(a+1)}{s^{a+1}} $ </p>
<p>Differentiating with respect to a, we get</p>
<p>$L(t^a\cdot lnt) = \frac{\Gamma'(a+1) - \Gamma(a+1)\cdot lns}{s^{a+1}} $<br>
set a = 1.</p>
|
3,978,378 | <p>The question asks Find an efficient proof for all the cases at once by first demonstrating</p>
<p><span class="math-container">$$
(a+b)^2 \leq (|a|+|b|)^2
$$</span></p>
<p>My attempt at the proof:</p>
<p>for <span class="math-container">$a,b\in\mathbb{R}$</span></p>
<p><span class="math-container">$$
\begin{align*}
... | Mohammad Riazi-Kermani | 514,496 | <p>Note that for positive numbers <span class="math-container">$a$</span> and <span class="math-container">$b$</span>, <span class="math-container">$$a^2 \le b^2 \implies a\le b $$</span>
Thus <span class="math-container">$$|a+b|^2 \le (|a|+|b|)^2 \implies |a+b|\le|a|+|b|$$</span></p>
|
496,011 | <p>Suppose that $ p_n(t) $ is the probability of finding n particle at a time t. And the dynamics of the particle is described by this equation : </p>
<p>$$ \frac{d}{dt} p_n(t) = \lambda \Delta p_n(t) $$</p>
<p>Defining one - dimensional lattice translation operator $ E_m = e^{mk} $ with $ km - mk = 1 $ and $\Delta ... | J.H. | 94,278 | <p>Your substitution should be completed by letting
$$\frac12\int (u-1) \sqrt{u}du$$ </p>
<p>By using integration by parts, this can be solved.</p>
|
540,227 | <p>I was try to understand the following theorem:-</p>
<p><strong>Let $X,Y$ be two path connected spaces which are of the same homotopy type.Then their fundamental groups are isomorphic.</strong></p>
<p><strong>Proof:</strong> The fundamental groups of both the spaces $X$ and $Y$ are independent on the base points si... | Pece | 73,610 | <p>In my humble opinion, your proof is too much <em>ad hoc</em>. Just show that $f,f' : (X,x) \to (Y,y)$ induced the same homomorphism of groups $\pi_1(X,x) \to \pi_1(Y,y)$ when they are homotopic : this is sufficient, the rest will come from abstract non sense.</p>
|
3,254,191 | <p>Let <span class="math-container">$V$</span> be a vector space. Determine all linear transformations <span class="math-container">$T:V\rightarrow V$</span> such that <span class="math-container">$T=T^2$</span>.</p>
<p>Suppose <span class="math-container">$x\in V$</span>. Then we can write <span class="math-containe... | Richard Jensen | 658,583 | <p>Given such a <span class="math-container">$T$</span>, notice that <span class="math-container">$T(V)$</span> is a subspace of <span class="math-container">$V$</span>, and that <span class="math-container">$T$</span> acts on <span class="math-container">$T(V)$</span> as the identity. Therefore, such linear transforma... |
3,254,191 | <p>Let <span class="math-container">$V$</span> be a vector space. Determine all linear transformations <span class="math-container">$T:V\rightarrow V$</span> such that <span class="math-container">$T=T^2$</span>.</p>
<p>Suppose <span class="math-container">$x\in V$</span>. Then we can write <span class="math-containe... | Hudson Lima | 680,269 | <p>There are lots of such linear maps. Given a decomposition of the space <span class="math-container">$V=R\oplus K$</span>, we have a unique <span class="math-container">$T:V\to V$</span> such that <span class="math-container">$T(V)=R$</span> and <span class="math-container">$\ker(T)=K$</span> (which is the projection... |
3,765,225 | <p>I have a matrix:
<span class="math-container">$$\left(\begin{array}{lll}
a & 0 & 0 \\
0 & b & 0 \\
0 & 0 & c
\end{array}\right)$$</span>
Which I want to change to:
<span class="math-container">$$\left(\begin{array}{lll}
a & 0 & 0 \\
0 & c & 0 \\
0 & 0 & b
\end{array}\r... | BartBog | 493,336 | <p>To the best of our knowledge, the problem MIN-FORMULA is not in NP.
The <strong>complement</strong> of this problem is in NP^SAT, that is a non-deterministic Turing machine with access to a SAT oracle can solve this in polynomial time.</p>
<p>Given input phi, the machine in question would guess a smaller formula psi... |
29,861 | <p>The meta question <em><a href="https://math.meta.stackexchange.com/q/29857/290189">Not actually a question, just a rant!</a></em> has inspired me to ask for <em>what</em> an answerer can do in case of self-deletion by the question asker while the answerer is typing the answer.</p>
<p>Since per se site is supposed t... | GNUSupporter 8964民主女神 地下教會 | 290,189 | <p>Due to <a href="https://creativecommons.org/licenses/by-sa/3.0/" rel="nofollow noreferrer">CC-BY-SA 3.0</a> license on SE network, once a post is published to the SE network, then it's released to the community. Therefore, <em>everyone has the right to use the contents of every post provided that the post owner is ... |
1,343,995 | <p>We know that $i^i$ is real. But how to explain it geometrically maybe in terms of rotation. like we can explain geometrically multiplication of two complex numbers and so on. Can someone show me a little bit about geometric interpretation of $i^i$ and tell me if what I think below is true?</p>
<p>Note :(Some info... | user21820 | 21,820 | <p>Your note is precisely the point. There is a geometric meaning for addition and multiplication and real powers, but imaginary exponents don't have any because they must be defined in terms of $\exp$ and $\ln$ or something equivalent. And there is no geometric interpretation of $\exp$.</p>
|
2,992,416 | <blockquote>
<p>A pendulum of length <span class="math-container">$1$</span> m and mass <span class="math-container">$100$</span> g attached to the end. Another 100 g mass move horizontally with speed 2 m/s. When collision happens this ball sticks with the pendulum and move together. Find the initial linear speed of ... | Arthur | 15,500 | <p>There is a bijection from <span class="math-container">$\Bbb Z^2$</span> to <span class="math-container">$\Bbb N$</span>, inducing a bijection from <span class="math-container">$\{-1,1\}^{\Bbb Z^2}$</span> to <span class="math-container">$\{-1,1\}^{\Bbb N}$</span>. And <span class="math-container">$\{-1,1\}^{\Bbb N}... |
2,992,416 | <blockquote>
<p>A pendulum of length <span class="math-container">$1$</span> m and mass <span class="math-container">$100$</span> g attached to the end. Another 100 g mass move horizontally with speed 2 m/s. When collision happens this ball sticks with the pendulum and move together. Find the initial linear speed of ... | badjohn | 332,763 | <p>Consider the simpler and, naively, smaller set <span class="math-container">$\{0, 1\} ^ \mathbb{N}$</span>. There is a clear near bijection to the interval <span class="math-container">$[0, 1]$</span> by writing the reals in binary. So, this might make it more clear that this set is uncountable and yours also. </... |
4,168,223 | <p>Let <span class="math-container">$R$</span> be the row reduced echelon form of a <span class="math-container">$4 \times 4$</span> real matrix <span class="math-container">$A$</span> and
let the
third column of <span class="math-container">$R$</span> be <span class="math-container">$\left[\begin{array}{l}0 \\ 1 \\ 0 ... | THIRUMAL 5688 | 469,196 | <p>For <span class="math-container">$P$</span> <span class="math-container">\begin{aligned}
&{\left[\begin{array}{llll}
1 & 0 & 0 & 0 \\
0 & 1 & 1 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0
\end{array}\right]\left[\begin{array}{l}
0 \\
-2 \\
2 \\
0
\end{array}\right]=\left... |
132,591 | <p>Let $f(x)$ be a positive function on $[0,\infty)$ such that $f(x) \leq 100 x^2$. I want to bound $f(x) - f(x-1)$ from above. Of course, we have $$f(x) - f(x-1) \leq f(x) \leq 100 x^2.$$ This is not good for me though. I need a bound which is linear (or at worst linear-times-root) in $x$.</p>
<p>Is there an inequali... | Yongyi Chen | 12,810 | <p>There is no such bound. Let $c$ be a real number, and let
$$f(x)=\begin{cases} 0&\text{if $x<c$}\\100x^2&\text{if $x\geq c$}.\end{cases}$$
Then for $x\in [c,c+1)$ the inequality $f(x)-f(x-1)\leq 100x^2$ is the best bound possible. So we cannot make a better bound for a general function satisfying $f(x)\leq... |
569,300 | <p>Let $G$ be a finite group and assume it has more than one Sylow $p$-subgroup.</p>
<p>It is known that order of intersection of two Sylow p-subgroups may change depending on the pairs of Sylow p-subgroups.</p>
<blockquote>
<p>I wonder whether there is a condition which guarantees that intersection of any two Sylo... | Nicky Hekster | 9,605 | <p>I can prove the following, which provides a criterion to start with.<p>
<strong>Theorem</strong> Let $G$ be a finite group. Then the following are equivalent.<p>
$(a)$ For all $S,T \in Syl_p(G)$ with $S \neq T$, $S \cap T=1$.<br>
$(b)$ For each non-trivial $p$-subgroup $P$ of $G$, $N_G(P)$ has a <em>unique</em> Sylo... |
1,504,433 | <p>I was wondering if you could help me with this question, in discrete math.</p>
<p>Prove that gcd(m, n) | lcm(n, m) for any non-zero integers m, n</p>
<p>any help is appreciated!</p>
| Ian Miller | 278,461 | <p><strong>General Case</strong></p>
<p>For the general problem (which sounds more what you need) consider several inputs $i_1$, $i_2$, $i_3$, etc. If you use a proportion $p_1$, $p_2$, $p_3$, etc of each then the output, $o$, will be:</p>
<p>$$o=i_1\times p_1+i_2\times p_2+i_3\times p_3+\cdots$$</p>
<p>Also $p_1+p_... |
222,480 | <p>How many $10$-digit numbers have two digits $1$, two digits $2$, three digits $3$, three digits $4$ so that between the two digits $1$ it has at least <strong>other two digits</strong> and between two digits $2$ it has at least <strong>other two digits</strong> (not necessarily distinct)? Thanks!</p>
| Lutz Lehmann | 115,115 | <p>The order reduction method seeks a second basis solution in the form <span class="math-container">$y=y_1u$</span>, where <span class="math-container">$y_1(x)=\frac{x}{(1-x)^2}$</span> is the already found basis solution.
<span class="math-container">$$
x(1-x)[y_1u''+2y_1'u']-3x[y_1u']=0
\implies \frac{u''}{u'}=\fr... |
33,389 | <p>Consider Schrödinger's <em>time-independent</em> equation
$$
-\frac{\hbar^2}{2m}\nabla^2\psi+V\psi=E\psi.
$$
In typical examples, the potential $V(x)$ has discontinuities, called <em>potential jumps</em>.</p>
<p>Outside these discontinuities of the potential, the wave function is required to be twice differentiable... | Piero D'Ancona | 7,294 | <p>Since you talk about 'jump' discontinuities, I guess you are interested in a one dimensional Schroedinger equation, i.e., $x\in\mathbb{R}$. In this situation a nice theory can be developed under the sole assumption that $V\in L^1(\mathbb{R})$ (and real valued of course). By a nice theory I mean that the operator $-d... |
2,239,203 | <p>Why does $\lim\limits_{x\to\infty}(x!)^{1/x}\neq 1?$</p>
<p>As far as I know, anything to the power of $0$ is $1$.</p>
<p>We have a factorial raised to $1/\infty=0$, but the limit is not $1$? I don't even know what the limit is. But it seems like infinity? Why is this?</p>
<p><a href="https://i.stack.imgur.com/hY... | Trevor Gunn | 437,127 | <p>That doesn't follow: $1/x$ is not the same as $0$ even for very large values of $x$. Keep in mind that "the limit as $x$ goes to infinity" is not the same thing as "plug in infinity wherever you see $x$". It is about how the function is behaving for larger and larger $x$. No matter how large $x$ gets, $1/x$ is still... |
2,239,203 | <p>Why does $\lim\limits_{x\to\infty}(x!)^{1/x}\neq 1?$</p>
<p>As far as I know, anything to the power of $0$ is $1$.</p>
<p>We have a factorial raised to $1/\infty=0$, but the limit is not $1$? I don't even know what the limit is. But it seems like infinity? Why is this?</p>
<p><a href="https://i.stack.imgur.com/hY... | DanielWainfleet | 254,665 | <p>Method (1). Prove the following: If $f(x)\to \infty$ as $x\to \infty$ then $G(n)=\frac {1}{n}\sum_{j=1}^nf(j)\to \infty$ as $n\to \infty.$</p>
<p>With $f(x)=\ln x$ we have $G(n)=\ln (n!^{(1/n)}).$</p>
<p>Method (2). $\ln x$ is monotonic increasing. So for $n\geq 2$ we have $\ln n>\int_{n-1}^n \ln x\;dx .$ So $... |
2,239,203 | <p>Why does $\lim\limits_{x\to\infty}(x!)^{1/x}\neq 1?$</p>
<p>As far as I know, anything to the power of $0$ is $1$.</p>
<p>We have a factorial raised to $1/\infty=0$, but the limit is not $1$? I don't even know what the limit is. But it seems like infinity? Why is this?</p>
<p><a href="https://i.stack.imgur.com/hY... | Claude Leibovici | 82,404 | <p>As Salahamam_ Fatima commented, think about Stirling approximation.</p>
<p>$$y=(x!)^{1/x}\implies \log(y)=\frac 1x \log(x!)$$ Now, using Stirling approximation $$\log(x!)=x (\log (x)-1)+\frac{1}{2} \left(\log (2 \pi )+\log
\left({x}\right)\right)+O\left(\frac{1}{x}\right)$$ $$\log(y)=(\log (x)-1)+\frac{\log (2 \... |
1,600,051 | <blockquote>
<p>If $x_1,x_2,\ldots,x_n$ are real numbers larger than $1$, prove that $$\dfrac{1}{1+x_1}+\dfrac{1}{1+x_2}+\cdots+\dfrac{1}{1+x_n} \geq \dfrac{n}{\sqrt[n]{x_1x_2\cdots x_n}+1}$$</p>
</blockquote>
<p><strong>Attempt</strong></p>
<p>AM-GM doesn't work here since we will get an upper bound. I don't see C... | Community | -1 | <p>Consider the function $f(x)=\frac{1}{1+e^x}$ which is convex for $x>0$ .</p>
<p>Now use Jensen's inequality : </p>
<p>$$f( \ln x_1)+f( \ln x_2)+\ldots+f( \ln x_n) \geq n f \left (\frac{\ln x_1+\ln x_2+\ldots+\ln x_n}{n} \right)$$ </p>
<p>This is exactly your inequality : </p>
<p>$$\frac{1}{1+x_1}+\frac{1}{1+x... |
1,600,051 | <blockquote>
<p>If $x_1,x_2,\ldots,x_n$ are real numbers larger than $1$, prove that $$\dfrac{1}{1+x_1}+\dfrac{1}{1+x_2}+\cdots+\dfrac{1}{1+x_n} \geq \dfrac{n}{\sqrt[n]{x_1x_2\cdots x_n}+1}$$</p>
</blockquote>
<p><strong>Attempt</strong></p>
<p>AM-GM doesn't work here since we will get an upper bound. I don't see C... | openspace | 243,510 | <p>Use Cauchy :
$$\frac{1}{1+x_{1}} + ... >= \frac{n}{((1+x_{1})...)^{\frac{1}{n}}}$$</p>
<p>Consider the:
$$(1+x_{1})...)^{\frac{1}{n}} <= (x_{1}...)^{\frac{1}{n}}+1$$</p>
<p>The last one you could prove by yourself(use induction).</p>
|
1,717,149 | <p>Is it true or false that if $V$ is a vector space and $T:V \to W$ is a linear transformation such that $T^2 = 0$, then $Im(T) \subseteq Ker(T)$ ?<br>
I don't understand it that much. It doesn't seem related... I can have a vector $v$ from $V$ that its power by 2 equals zero but $T(v) \neq 0_{v}$ </p>
| lhf | 589 | <p>Try <em>The Fascination of Groups</em> by Budden.</p>
|
1,717,149 | <p>Is it true or false that if $V$ is a vector space and $T:V \to W$ is a linear transformation such that $T^2 = 0$, then $Im(T) \subseteq Ker(T)$ ?<br>
I don't understand it that much. It doesn't seem related... I can have a vector $v$ from $V$ that its power by 2 equals zero but $T(v) \neq 0_{v}$ </p>
| p Groups | 301,282 | <p><em>Groups and Symmetry: A Guide to <strong>Discovering Mathematics</em></strong>, by David W. Farmer.</p>
<p>The highlighted title may convince that it assumes not too much mathematics for the learner. It is very little book, not of the type <em>Definition-Theorem-Proof</em>.</p>
<p>At least (in on-line preview) ... |
1,095,918 | <p><img src="https://i.stack.imgur.com/EST8r.jpg" alt="my problem is in prop 27, cannot prove it. Can use definition before. Notice that p-closure is the closure of G in the open point topolgy"></p>
<p>For extra notations: C(E,F) is the set of all continuous functions from E to F (topological spaces).
Can anybody help... | Robert Israel | 8,508 | <p>What is $p$-closure? If a neighbourhood $V$ of $x$ works for $G$, does it also work for $\overline{G}$?</p>
|
468 | <p>Textbook writers are blessed with only solving problems with neat answers. Numerical coefficients are small integers, many terms cancel, polynomials split into simple factors, angles have trigonometric functions with known values. Pure bliss.</p>
<p>The "real life" is different (as any of us knows).</p>
<p>Giving ... | Confutus | 40 | <p>It may be useful to present at least one application that is solvable but does not have a simple answer. As an example, quadratic equations with rational coefficients that cannot be easily solved by factoring and require use of the quadratic formula arise in chemistry.</p>
|
4,551,407 | <p>Here's the question:<br />
If we have m loaves of bread and want to divide them between n people equally what is the minimum number of cuts we should make?<br />
example:<br />
3 loaves of bread and 15 people the answer is 12 cuts.<br />
6 loaves of bread and 10 people the answer is 8 cuts.</p>
<p>for example 1, I f... | user577215664 | 475,762 | <p><span class="math-container">$$x^2y''+4xy'+2y=f(x)$$</span>
Rewrite the DE as:
<span class="math-container">$$(x^2y)''=f(x)$$</span></p>
|
1,851,209 | <p>Let $L:X\to Y$ an linear operator. I saw that $L$ is bounded if $$\|Lu\|_Y\leq C\|u\|_X$$
for a suitable $C>0$. This definition looks really weird to me since such application is in fact not necessary bounded as $f:\mathbb R\to \mathbb R$ defined by $f(x)=x$. So, is there an error in <a href="https://en.wikipedia... | Thomas | 128,832 | <p>The term "bounded" has a special meaning (the one you wrote down) when it comes to linear operators on topological vector spaces (like normed spaces). It's a common definition in Functional analysis. It is not equivalent to the usual notion of a bounded map. (And it's not really helping to say this that it's an unfo... |
1,366,372 | <p>In this <a href="https://math.stackexchange.com/questions/1365989/testing-pab-using-2-dice">question</a>
: </p>
<p>$$ P_r(a\cap b)=P_r(a,b)=P_r(a)P_r(b)$$</p>
<p>However in this <a href="https://stats.stackexchange.com/questions/156852/what-do-did-you-do-to-remember-bayes-rule/156866#156866">question</a>: </p>
<p... | Conrad Turner | 201,962 | <p>No because the two dice are independent the outcome on one has no effect on the other so: $p(a,b)=p(a)p(b)=p(b|a)p(a)$ hence $p(b|a)=p(b)$ (note the first equality is the definition of independence in this context: RVs $A$ and $B$ are independent iff $\mbox{prob}(A=a,B=b)=\mbox{prob}(A=a)\times \mbox{prob}(B=b)$ )</... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.