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 |
|---|---|---|---|---|
259,795 | <p>Consider the function $f \colon\mathbb R \to\mathbb R$ defined by
$f(x)=
\begin{cases}
x^2\sin(1/x); & \text{if }x\ne 0, \\
0 & \text{if }x=0.
\end{cases}$</p>
<p>Use $\varepsilon$-$\delta$ definition to prove that the limit $f'(0)=0$.</p>
<p>Now I see that h should equals to delta; and delta should eq... | Alex Youcis | 16,497 | <p>I would post this as a comment, since I need to check it a little more before I am totally ok with it--alas, it is too long. So, an answer it shall stay.</p>
<p>EDIT: I should emphasize that where I am using finite dimesionality is that every linear map in sight is continuous. In particular, a linear isomorphism is... |
2,843,560 | <p>If $\sin x +\sin 2x + \sin 3x = \sin y\:$ and $\:\cos x + \cos 2x + \cos 3x =\cos y$, then $x$ is equal to</p>
<p>(a) $y$</p>
<p>(b) $y/2$</p>
<p>(c) $2y$</p>
<p>(d) $y/6$</p>
<p>I expanded the first equation to reach $2\sin x(2+\cos x-2\sin x)= \sin y$, but I doubt it leads me anywhere. A little hint would be... | Batominovski | 72,152 | <p>Note that $$\sin(x)+\sin(2x)+\sin(3x)=\sin(2x)\,\big(1+2\cos(x)\big)$$
and
$$\cos(x)+\cos(2x)+\cos(3x)=\cos(2x)\,\big(1+2\cos(x)\big)\,.$$
Thus, for the required equalities to be true, we need
$$1+2\cos(x)=s\in\{-1,+1\}\text{ and }2x=\left\{\begin{array}{ll}
2k\pi+y\,,&\text{if }s=+1\\
(2k+1)\pi-y\,,&\text{i... |
1,013,346 | <p>Given a box which contains $3$ red balls and $7$ blue balls. A ball is drawn from the box and a ball of the other color is then put into the box. A second ball is drawn from the box, What is the probability that the second ball is blue? </p>
<p>could anyone provide me any hint? </p>
<p>Please, don't offer a comple... | Felix Marin | 85,343 | <p>$\newcommand{\angles}[1]{\left\langle\,{#1}\,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\half}{{1 ... |
81,267 | <p>I have the following problem: I have a (a lot)*3 table, meaning that I have 3 columns, say X, Y and Z, with real values. In this table some of the rows have the same (X,Y) values, but with different value of Z. For instance</p>
<pre><code>{{12.123, 4.123, 513.423}, {12.123, 4.123, 33.43}}
</code></pre>
<p>have th... | sacratus | 23,055 | <pre><code>DeleteDuplicates[SortBy[data, Last],( #1[[1]]==#2[[1]] && #1[[2]]==#2[[2]] & ) ]
</code></pre>
<p>Explanation:
<code>data</code> is your data of the form <code>{{x1,y1,z1},...,{xn,yn,zn}}</code>. With <code>SortBy[#, Last] &</code> we sort this dataset with respect to the last coordinate, e... |
353 | <p>One of the challenges of undergraduate teaching is logical implication. The case by case definition, in particular, is quite disturbing for most students, that have trouble accepting "false implies false" and "false implies true" as true sentences. </p>
<blockquote>
<p><strong>Question:</strong> What are good poi... | Anschewski | 199 | <p>To supplement Brendan's idea:
I like to connect quantified statements to unquantified in the following way:
Assume a statement like "If X is a dog, then X has a head". Now, once you have found this to be true, you might want the truth not to depend on X. Thus replacing X for anything like "my car" should still give ... |
353 | <p>One of the challenges of undergraduate teaching is logical implication. The case by case definition, in particular, is quite disturbing for most students, that have trouble accepting "false implies false" and "false implies true" as true sentences. </p>
<blockquote>
<p><strong>Question:</strong> What are good poi... | user26872 | 4,665 | <p><em>Here is an example I sometimes use to motivate the truth table for implication. The example taps in to our ability to recognize cheaters.</em> </p>
<p>Suppose you go to the vending machine. The price of a soda is one dollar. </p>
<ol>
<li><p>Suppose you put a dollar in the vending machine and receive a soda. ... |
246,808 | <p>Trying to solve the following PDE with BC <code>T==1</code> on a spherical cap of a unit sphere and <code>T==0</code> at infinity (approximated as <code>r==(x^2 + y^2 + z^2)^0.5==40^0.5</code>) and the flux over the remaining surfaces taken to be zero (only half domains has been specified due to symmetry reasons):</... | Tim Laska | 61,809 | <p>The basic problem appears to be a convective-diffusive heat transfer problem of X-directed fluid flow across a heated spherical cap tip. To study this type of problem, it probably is easier to construct a virtual cuboid wind tunnel. When simulating virtual wind tunnels, the upstream section is typically much shorter... |
1,897,538 | <p>Next week I will start teaching Calculus for the first time. I am preparing my notes, and, as pure mathematician, I cannot come up with a good real world example of the following.</p>
<p>Are there good examples of
\begin{equation}
\lim_{x \to c} f(x) \neq f(c),
\end{equation}
or of cases when $c$ is not in the doma... | B. Goddard | 362,009 | <p>Well, it's your chance to make a point about removable singularities. Students tend to think that the functions $x+3$ and $(x^2-9)/(x-3)$ are the same, but there is that one point where they're different. "Meh! What's one little point among so many?" (I wish) they would ask. All of differential calculus is about ... |
1,897,538 | <p>Next week I will start teaching Calculus for the first time. I am preparing my notes, and, as pure mathematician, I cannot come up with a good real world example of the following.</p>
<p>Are there good examples of
\begin{equation}
\lim_{x \to c} f(x) \neq f(c),
\end{equation}
or of cases when $c$ is not in the doma... | Lubin | 17,760 | <p>A simpler version of the example of @AOrtiz:<br>
Consider a glass of water, and the function $\Delta(h)=$ density at a point $h$ cm above the surface, in gm/ml. It’s $1$ when $h<0$, $0$ (or a very small $\varepsilon$) when $h>0$, and I’m sure that you’ll agree that any reasonable definition of density at a poi... |
588,214 | <p>What is the meaning of $[G:C_G(x)]$ in group theory? Is this equivalent to $\frac{|G|}{|Z_G(x)|}$, or to $|Z_G(x)|$?</p>
| Nicky Hekster | 9,605 | <p>You might want to know that $Z(G) \subseteq C_G(x)$ for all $x \in G$. In fact $Z(G) = \bigcap_{x \in G} C_G(x)$.</p>
|
468,487 | <p>I have calculated the likelihood of an event to be $1$ in $1.07 \times 10^{2867}$.</p>
<p>I'm looking for a way to describe to a layperson how unlikely this event is to occur, but the number is so mind boggling large I can't find a way to put it into words.</p>
<p>Any suggestions would be appreciated</p>
| Hagen von Eitzen | 39,174 | <p>It is the probability of the proverbial monkey hacking a typewriter and producing the first page of Hamlet.</p>
|
907,851 | <p>I am really new into math, why is $-2^2 = -4 $ and $(-2)^2 = 4 $? </p>
| Ahaan S. Rungta | 85,039 | <p>$ -2^2 = - \left( 2^2 \right) = -4 $, whereas $ \left( -2 \right)^2 = \left( -2 \right) \cdot \left( -2 \right) = 4 $, because the negatives cancel. </p>
|
907,851 | <p>I am really new into math, why is $-2^2 = -4 $ and $(-2)^2 = 4 $? </p>
| John Joy | 140,156 | <p>By convention brackets are evaluated first, then exponentiation, division/multiplication, and finally addition/subtraction. Think of the unary negation operator as a shorthand for "multiply by -1". These rules are usually remembered by students with the BEDMAS acronym.</p>
<p>I prefer to think of the rules as those... |
939,212 | <p>Apologies if this isn't at the level of questions expected here!</p>
<p>I've got two simultaneous equations to solve.</p>
<p>(Equation 1): $ x y = 4 $</p>
<p>(Equation 2): $ x + y = 2 $</p>
<p>They produce the following curves:</p>
<p><img src="https://i.stack.imgur.com/dMrNi.png" alt="enter image description h... | Alice Ryhl | 132,791 | <p>So the curves are</p>
<p>$$\frac4x\quad\text{and}\quad2-x$$</p>
<p>To prove that these curves do not meet, it simply means that</p>
<p>$$\frac4x=2-x$$</p>
<p>has no solutions, to solve this we multiply by $x$</p>
<p>$$4=2x-x^2$$</p>
<p>Then multiply with $-1$ and switch the left and right of the equality</p>
... |
939,212 | <p>Apologies if this isn't at the level of questions expected here!</p>
<p>I've got two simultaneous equations to solve.</p>
<p>(Equation 1): $ x y = 4 $</p>
<p>(Equation 2): $ x + y = 2 $</p>
<p>They produce the following curves:</p>
<p><img src="https://i.stack.imgur.com/dMrNi.png" alt="enter image description h... | amWhy | 9,003 | <p>If the two curves intersect (meet), they do so wherever they have a point in common. This will only happen when $y_1=y+2$, \iff $$\frac 4x = 2-x \iff 2x - x^2 = 4 \iff x^2 -2x + 4 = 0$$ for some real $x$.</p>
<p>Use the quadratic equation to show that there is no real solution to this equation Indeed, you need onl... |
570,740 | <p>Hi there I'm having some trouble with the following problem:</p>
<p>I have a $3\times3$ symmetric matrix
$$
A=\pmatrix{1+t&1&1\\ 1&1+t&1\\ 1&1&1+t}.
$$
I am trying to determine the values of $t$ for which the vector $b = (1,t,t^2)^\top$ (this is a column vector) is in the column space of $A$... | Community | -1 | <p>Swapping two columns of A will not change its column space. Consider the matrix
$$
A' = \left( \begin{array}{ccc}
1 & 1 & 1+t \\
1 & 1+t & 1 \\
1+t & 1 & 1 \end{array} \right),
$$
which is obtained by interchanging the first and last column of A. Forming the augmented matrix $[A', b]$ and s... |
1,979,876 | <p>Currently I started studying about ray-casting when I came across this following problem based on ray-triangle intersection. The problem was:</p>
<p>You are provided with a triangle with vertices ,<strong>(x1,y1,z1)</strong>, <strong>(x2,y2,z2)</strong> and <strong>(x3,y3,z3)</strong>. A ray with origin <strong>(a1... | Nominal Animal | 318,422 | <p>The way this is done in most raytracers and raycasters is actually quite simple. The intersection distance (from ray origin) is calculated first, and it is typically calculated for several objects for the same ray, so that the objects can be handled in order of increasing distance (unless fully reflected). Then, the... |
467,574 | <p>Using permutation or otherwise, prove that $\displaystyle \frac{(n^2)!}{(n!)^n}$ is an integer,where $n$ is a positive integer.</p>
<p>I have no idea how to prove this..!!I am not able to even start this Can u give some hints or the solution.!cheers.!!</p>
| walcher | 89,844 | <p>Let $H$ be the subgroup of $S_{n^2}$ consisting of all permutations $\sigma$ for which the following holds: if $kn \lt m \le (k+1)n$, then $kn \lt \sigma (m) \le (k+1)n.$ Then $H$ separately permutes the numbers $$1, 2, ..., n; n+1, ..., n+n=2n; 2n+1, ..., 3n; ...; kn+1, kn+2, ..., (k+1)n; ...; n(n-1), ..., n^2,$$ s... |
467,574 | <p>Using permutation or otherwise, prove that $\displaystyle \frac{(n^2)!}{(n!)^n}$ is an integer,where $n$ is a positive integer.</p>
<p>I have no idea how to prove this..!!I am not able to even start this Can u give some hints or the solution.!cheers.!!</p>
| Christoph | 86,801 | <p>Looking up the sequence on <a href="http://oeis.org/A034841" rel="nofollow">OEIS</a> you notice this is the number of arrangements of $1, 2, 3, \ldots, n^2$ in an $n\times n$ matrix such that each row is increasing and thus is an integer.</p>
<p>You can verify this by using combinatorics to calculate the number of ... |
1,290,111 | <p>How one can prove the following statement:</p>
<p>$k(n-1)<n^2-2n$ for all odd $n$ and $k<n$</p>
<p><em>Tried so far</em>: induction on $n$, graphing, and rewriting $n^2−2n$ as $(n−1)^2−1$.</p>
| NickC | 189,951 | <p>This isn't true.</p>
<p>Take $k=4$, $n=5$. We have $4(5-1)=16$ and $5^2-2\cdot 5=15$.</p>
|
2,338,123 | <p>Function $f(z)$ is an entire function such that $$|f(z)| \le |z^{n}|$$ for $z \in \mathbb{C}$ and some $n \in \mathbb{N}$.</p>
<p>Show that the singularities of the function $$\frac {f(z)}{z^{n}}$$ are removable. What can be implied about the function $f(z)$ if moreover $f(1) = i$? Draw a far-reaching conclusion.</... | Community | -1 | <p>Consider $\frac{f(z)}{z^n}$. Since $f$ was entire, we just need to show that the singularity at $0$ is removable. Because of Reimann's theorem(read here <a href="https://en.wikipedia.org/wiki/Removable_singularity#Riemann.27s_theorem" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Removable_singularity#Riem... |
1,988,517 | <p>$$\binom{1}{0},\binom{1}{1},\binom{1}{2}$$
What does this mean, and how do I achieve an numerical value when trying to solve a proof or problem in this form? </p>
| sdd | 179,397 | <p>As you know, in general, $\binom{n}{k}$ is a number with value = the count of ways $k$ objects can be selected out of $n$ objects (when we are not interested in the order of selection, but just in the set of elements that are chosen).</p>
<p>Respectively:</p>
<ul>
<li><p>$\binom{1}{0} = 1$ since there is only one... |
1,988,517 | <p>$$\binom{1}{0},\binom{1}{1},\binom{1}{2}$$
What does this mean, and how do I achieve an numerical value when trying to solve a proof or problem in this form? </p>
| Marble | 302,943 | <p>An expression for ${n}\choose{k}$ that you can use to compute it is $\frac{n}{k!(n-k)!}$.</p>
<p>The actual interpretation of the values is:</p>
<ol>
<li><p>$\binom{1}{0} = 1$ means there is only one way to select 0 elements out of 1element</p></li>
<li><p>$\binom{1}{1} = 1$ means we one element and select the onl... |
1,776,726 | <p>I'm trying to determine whether or not </p>
<blockquote>
<p>$$\sum_{k=1}^\infty \frac{2+\cos k}{\sqrt{k+1}}$$ </p>
</blockquote>
<p>converges or not. </p>
<p>I have tried using the ratio test but this isn't getting me very far. Is this a sensible way to go about it or should I be doing something else?</p>
| SchrodingersCat | 278,967 | <p>Your function will be maximum when the denominator will be minimum.</p>
<p>So, your work will be to calculate $$\frac{d}{dx}\left[\left\{(1-x)-\frac{4}{1-x}+1\right\}^2+1\right]=0$$</p>
<p>It will come down to $$2\left\{(1-x)-\frac{4}{1-x}+1\right\}\left\{-1-\frac{4}{(1-x)^2}\right\}=0$$
This will give rise to $2$... |
908,309 | <p>I'm finding the principal value of $$ i^{2i} $$</p>
<p>And I know it's solved like this:</p>
<p>$$ (e^{ i\pi /2})^{2i} $$ </p>
<p>$$ e^{ i^{2} \pi} $$</p>
<p>$$ e^{- \pi} $$</p>
<p>I understand the process but I don't understand for example where does the $ i $ in $ 2i $ go?</p>
<p>Is this some kind of a prop... | Cookie | 111,793 | <p>The $i$ in $2i$ was combined with the $i$ inside the parentheses. Hence, you got $$i\cdot i = i^2$$ which is due to exponent laws. More applied to your case:
$$(e^{ i\pi /2})^{2i}=e^{i \cdot i \cdot \pi}=e^{i^2\cdot \pi}.$$</p>
|
908,309 | <p>I'm finding the principal value of $$ i^{2i} $$</p>
<p>And I know it's solved like this:</p>
<p>$$ (e^{ i\pi /2})^{2i} $$ </p>
<p>$$ e^{ i^{2} \pi} $$</p>
<p>$$ e^{- \pi} $$</p>
<p>I understand the process but I don't understand for example where does the $ i $ in $ 2i $ go?</p>
<p>Is this some kind of a prop... | syusim | 138,951 | <p>$$\bigl(e^{i\pi /2}\bigr)^{2i} = e^{(i\pi /2) \cdot 2i} = e^{i^2\pi}.$$</p>
<p>This is just an application of the exponent laws. Don't overthink it!</p>
|
1,037,068 | <p>If two sequences converge equally, we have
$$\lim_{n\rightarrow \infty }\left ( a_{n} \right )=\lim_{n\rightarrow \infty }\left ( b_{n} \right )$$</p>
<p>As a follow up, is the following equality also true?
$$\lim_{n\rightarrow \infty }\left ( \ln a_{n} \right )=\lim_{n\rightarrow \infty }\left ( \ln b_{n} \right ... | JohnD | 52,893 | <p>So it is enough to show that if Parseval's Equality holds for all $x\in H$, then any $x\in H$ has a unique set of coefficients $c_n$.</p>
<p>Suppose Parseval's holds for $f,g\in H$ and $f\not=g$ but that they have the same coefficients:
$$
f=\sum_{j=1}^\infty \langle f,v_j\rangle v_j \qquad g=\sum_{j=1}^\infty \lan... |
712,697 | <p>What is the radius of convergence?</p>
<p>$$\sum_{n=0}^{\infty} n^3 (5x+10)^n$$</p>
| Asaf Karagila | 622 | <p>Yes, we can.</p>
<p>Consider $\mathcal U$ as the open cover whose elements are <em>all</em> the open neighborhoods witnessing the local noetherian property. Since every $x\in X$ has such neighborhood, this is certainly an open cover. By quasi-compactness, we have some finite subcover, $V_1,\ldots,V_n$.</p>
<p>Now ... |
3,413,253 | <p>Can anyone give me some examples and non examples of Lindelöf or second countable space and spaces that is Lindelöf but not second countable? And I understand the definition but find it is hard to visualize and imagine.
I have tried google it but it turns out I only found some silly examples like finite set or emp... | Henno Brandsma | 4,280 | <p>Lindelöf and second countable are saying that a space is "small" in some sense; so one way to find non-examples is to take products of lots of spaces, such products (or powers) are "big".</p>
<p><span class="math-container">$\Bbb R^I$</span> is not Lindelöf for <span class="math-container">$I$</s... |
72,854 | <p>Hi everybody,</p>
<p>Does there exist an explicit formula for the Stirling Numbers of the First Kind which are given by the formula
$$
x(x-1)\cdots (x-n+1) = \sum_{k=0}^n s(n,k)x^k.
$$</p>
<p>Otherwise, what is the computationally fastest formula one knows?</p>
| joro | 12,481 | <p>Stirling Numbers of the First Kind are treated in the book <a href="http://www.jjj.de/fxt/fxtpage.html#fxtbook" rel="nofollow">"Matters Computational" (was: "Algorithms for Programmers")</a> by Jörg Arndt. A C++ implmentation of Arndt is at <a href="http://www.jjj.de/fxt/demo/comb/stirling1-demo.cc" rel="nofollow">s... |
298,029 | <p>how can we convert sin function into continued fraction ?</p>
<p>for example </p>
<p><a href="http://mathworld.wolfram.com/EulersContinuedFraction.html">http://mathworld.wolfram.com/EulersContinuedFraction.html</a></p>
<p>how can we convert sin to simmilar continued fraction ?? </p>
<p>and what about sinh and co... | robjohn | 13,854 | <p>We will proceed as in <a href="https://math.stackexchange.com/a/287987">this answer</a>.</p>
<p>Define
<span class="math-container">$$
P_n(x)=\sum_{k=0}^\infty\frac{4^{k+n}-\sum\limits_{j=1}^n\binom{2k+2n+1}{2j-1}}{(2k+2n+1)!}(-x^2)^k\tag{1}
$$</span>
Then
<span class="math-container">$$
\begin{align}
\frac{P_{n-1}... |
34,204 | <p>I have several contour lines and one point. How can I find a point in one of those contour lines which is nearest to the given point?</p>
<pre><code>(*Create the implicit curves*)
Data={{10,20,1},{10,40,2},{10,60,3},{10,80,4},{20,25,2},{20,45,3},{20,65,4},{30,30,3},{30,50,4},{40,35,4},{40,55,5},{50,20,4},{50,40,5},... | Michael E2 | 4,999 | <p>My original idea was the same as <a href="https://mathematica.stackexchange.com/users/61/cormullion">cormullion's</a>, and then followed by <code>FindMinimum</code>. But here another, still related way using <code>MeshFunctions</code>. It's not as efficient in general as other ways, perhaps, but as a method, it is... |
619,564 | <p>I have to prove if this function is differentiable.</p>
<p>$$f(x,y)= \begin{cases} \frac{\cos x-\cos y}{x-y} \iff x \neq y \\-\sin x \iff x=y \end{cases}$$</p>
<p>if $x \neq y$ it is continuous, but i want to see if it is continuous in x=y too.</p>
<p>i can rewrite f as
$$ f(x,y)= \begin{cases} \frac{g(x)-g(y)}{... | Christian Blatter | 1,303 | <p>One has
$$f(x,y)=-\int_0^1\sin\bigl((1-t)y+t\,x\bigr)\ dt\qquad\forall\ (x,y)\in{\mathbb R}^2\ .$$
This shows that $f\in C^\infty({\mathbb R}^2)$.</p>
|
1,582,275 | <p>Suppose that $B = S^{-1}AS$ for some $n \times n$ matrices $A$, $B$, and $S$.</p>
<ol>
<li>Show that if $x \in \ker(B)$ then $Sx \in \ker(A)$.</li>
</ol>
<p>Proof: $B = S^{-1}AS$ implies that $SB = AS$ which implies that $SBx = ASx = 0$, that is $Sx \in \ker(A)$.</p>
<ol start="2">
<li>Show that the linear transf... | seeker | 267,945 | <p>$T:Ker\ (B)\rightarrow Ker\ (A)$ is given by $T(x)=S(x)$. First of all by part 1, this is well defined.</p>
<ol>
<li><p>Check that it is linear.</p></li>
<li><p>Let $x\in Ker\ (T)$. Then $T(x)=S(x)=0\implies x\in Ker (S)$. But $S$ is invertible $\implies x=0$. So $T$ is one-one.</p></li>
<li><p>Let $x\in Ker\ (A)\i... |
3,339,647 | <p>I had a class in algebraic topology, our main book is Allen Hatcher, our professor defined a term called "Exponential Law" as the following:</p>
<p><span class="math-container">$Hom (X \times Y, Z) \cong Hom (X, Hom (Y, Z))$</span> </p>
<p><span class="math-container">$\alpha : X \times Y \rightarrow Z $</span></p... | Noah Riggenbach | 482,732 | <p>1.) This has already been answered in the comments, but as an alternative source Davis and Kirk talk about it when they are discussing compactly generated weak hausdorff spaces, which I prefer.</p>
<p>2.) If you write <span class="math-container">$\operatorname{Hom}(X,Y)$</span> as <span class="math-container">$Y^X... |
192,636 | <p>Suppose I have some 3D points, e.g. <code>{{0, 0, 1}, {0, 0, 1.3}, {0, 1, 0}, {1.2, 0, 0}}</code>. Now I want to find the smallest and largest distance between two points.</p>
<p>A trivial way is to find all possible distances, then look for the smallest and largest number.This becomes very much time-consuming for ... | Carl Woll | 45,431 | <p>I think Henrik meant the following approach using <a href="http://reference.wolfram.com/language/ref/Nearest" rel="noreferrer"><code>Nearest</code></a>:</p>
<pre><code>min[pts_] := Min @ Nearest[pts->"Distance", pts, 2][[All, 2]]
</code></pre>
<p>Let's compare the above approach with a simple version based on <... |
4,063,337 | <p>In an exercise I'm asked the following:</p>
<blockquote>
<p>a) Find a formula for <span class="math-container">$\int (1-x^2)^n dx$</span>, for any <span class="math-container">$n \in \mathbb N$</span>.</p>
<p>b) Prove that, for all <span class="math-container">$n \in \mathbb N$</span>: <span class="math-container">$... | DonAntonio | 31,254 | <p>An idea: substitution <span class="math-container">$\;t=\sin x\,,\,\,dt=\cos x\,dx\;$</span> , so</p>
<p><span class="math-container">$$\color{red}{I_n}:=\int_0^1(1-t^2)^ndt=\int_0^{\pi/2}(\cos x)^{2n+1}dx=\int_0^1\left(1-\sin^2x\right)(\cos x)^{2n-1} dx=$$</span></p>
<p><span class="math-container">$$=I_{n-1}-\int_... |
3,162,056 | <p>In Kleene's "Mathematical Logic" and "Introduction to Metamathematics" for a classical predicate calculus the following two rules of inference are chosen.</p>
<p>If <span class="math-container">$A(x) \Rightarrow C$</span> then <span class="math-container">$(\exists xA(x)) \Rightarrow (C)$</span> and
if <span class=... | frabala | 53,208 | <p>An intuitive answer, in a world of match sticks and fire and no magic, i.e. one can not make fire out of nothing.</p>
<p>Consider a variable <span class="math-container">$x$</span> to mean a match stick. Let <span class="math-container">$A(x)$</span> mean "Stick <span class="math-container">$x$</span> smokes" and <... |
4,535,612 | <blockquote>
<p>Is a group of order <span class="math-container">$2^kp$</span> not simple, where <span class="math-container">$p$</span> is a prime and <span class="math-container">$k$</span> is an positive integer?</p>
</blockquote>
<p>I did this for the groups of order <span class="math-container">$2^k 3$</span>. Her... | Viktor Vaughn | 22,912 | <p>Yes, this can be done using elimination theory. See the reference to Cox, Little, and O'Shea's book I've given <a href="https://math.stackexchange.com/a/1318832">here</a> or <a href="https://math.stackexchange.com/a/1490119">here</a>.</p>
<p>We can find the desired polynomial by computing a Gröbner basis using <a hr... |
2,468,155 | <p>This problem is from Challenge and Thrill of Pre-College Mathematics:
Prove that $$ (a^3+b^3)^2\le (a^2+b^2)(a^4+b^4)$$</p>
<p>It would be really great if somebody could come up with a solution to this problem.</p>
| Kenny Lau | 328,173 | <p>$$\begin{array}{rrcl}
& (a^3+b^3)^2 &\le& (a^2+b^2)(a^4+b^4) \\
\iff& a^6 + 2a^3b^3 + b^6 &\le& a^6+a^2b^4+b^2a^4+b^6 \\
\iff& 2a^3b^3 &\le& a^2b^4+b^2a^4 \\
\iff& 2ab &\le& b^2+a^2 \\
\iff& 0 &\le& b^2-2ab+a^2 \\
\iff& 0 &\le& (b-a)^2 \\
\end{a... |
2,468,155 | <p>This problem is from Challenge and Thrill of Pre-College Mathematics:
Prove that $$ (a^3+b^3)^2\le (a^2+b^2)(a^4+b^4)$$</p>
<p>It would be really great if somebody could come up with a solution to this problem.</p>
| Guy Fsone | 385,707 | <p>$$(a^3+b^3)^2 = a^6 + 2a^3b^3 + b^6$$</p>
<p>But we know that $$(X-Y)^2\ge 0\Longleftrightarrow X^2+Y^2 \ge 2XY$$</p>
<p>taking $X= a^3$ and $Y=b^3$ we get
$$2a^3b^3 \le a^2b^4+b^2a^4 $$
so
$$(a^3+b^3)^2 = a^6 + 2a^3b^3 + b^6 \le a^6 + \color{red}{a^2b^4+b^2a^4 } + b^6 = (a^2+b^2)(a^4+b^4) $$</p>
|
16,290 | <p>Hi I am new here and have a calculus question that came up at work.</p>
<p>Suppose you have a $4' \times 8'$ piece of plywood. You need 3 circular pieces all equal diameter. What is the maximum size of circles you can cut from this piece of material?
I would have expected I could write a function for the area of th... | Listing | 3,123 | <p>Isaac had the right intuition. </p>
<p><img src="https://i.stack.imgur.com/JEswV.png" alt="alt text"></p>
<p>I used Matlab to globally optimize the radius under the constrait that all three circles have the same radius, are in the rectangle and don't intersect each other. What I got is shown above,</p>
<p>R = 1.4... |
4,467,763 | <p>I have the equation</p>
<p><span class="math-container">$A\vec{x} = \vec{b} \tag{1}.$</span></p>
<p>where <span class="math-container">$A$</span> is an <span class="math-container">$m\times n$</span> matrix of rank <span class="math-container">$m$</span>, so that <span class="math-container">$m<n$</span> and the ... | blamocur | 991,086 | <p><span class="math-container">$$|Ax-b|^2 = \left(Ax-b \right)^{T}\cdot\left(Ax-b \right)$$</span>
<span class="math-container">$$ = \left(x^TA^T-b^T \right)\left(Ax-b \right)$$</span>
<span class="math-container">$$ = x^T(A^TA)x - 2b^TAx +|b|^2$$</span> By differentiating the last line over <span class="math-containe... |
4,467,763 | <p>I have the equation</p>
<p><span class="math-container">$A\vec{x} = \vec{b} \tag{1}.$</span></p>
<p>where <span class="math-container">$A$</span> is an <span class="math-container">$m\times n$</span> matrix of rank <span class="math-container">$m$</span>, so that <span class="math-container">$m<n$</span> and the ... | Klaas van Aarsen | 134,550 | <p>We're looking for the Moore-Penrose pseudo inverse <span class="math-container">$A^+$</span> so that we have <span class="math-container">$\vec x = A^+ \vec b$</span>.</p>
<p>Since <span class="math-container">$A$</span> is of full row rank <span class="math-container">$m$</span>, we can have that <span class="math-... |
2,933,753 | <p>Given two finite groups <span class="math-container">$G, H$</span>, we are going to say that <span class="math-container">$G<_oH$</span> if either</p>
<p>a. <span class="math-container">$|G|<|H|$</span></p>
<p>or </p>
<p>b. <span class="math-container">$|G|=|H|$</span> and <span class="math-container">$\dis... | Travis Willse | 155,629 | <p>The following Maple function (code here requires the Maple package <code>GroupTheory</code>) takes as argument a group and returns the sum of the orders of its elements:</p>
<pre><code>sumOfOrders := G -> add(u, u in map(PermOrder, convert(Elements(G), list), G));
</code></pre>
<p>This function takes as argumen... |
114,487 | <p>I have a stack of images (usually ca 100) of the same sample. The images have intrinsic variation of the sample, which is my signal, and a lot of statistical noise. I did a principal components analysis (PCA) on the whole stack and found that components 2-5 are just random noise, whereas the rest is fine. How can I ... | Community | -1 | <p>As requested by Anton. I halved the amount of noise because otherwise some images have barely any signal left. As you can see below, we are still putting in a significant amount of noise.</p>
<p>(To conserve space I'm only visualizing the first ten images in this answer, but the denoising is happening over all 100 ... |
724,045 | <p>I believe I'm not correctly understanding the concept of unique factorization and irreducibles.</p>
<p>Consider $R = \mathbb{F}_7$ and $h \in R[x]$ where $h = x^4 + 4x^3 + 3x^2 + 5x + 6$. Now $h$ has the following factorizations:</p>
<p>$h_1 = (3x^2 + 3x + 4)(5x^2 + x + 5)$</p>
<p>and</p>
<p>$h_2 = (x^2 + x + 6)... | ncmathsadist | 4,154 | <p>You can get in trouble if the convergence is not absolute, i.e. $$\sum_n \|x_n\| = +\infty$$ but
$$\lim_{N\to\infty} \sum_{n\le N} x_n $$
exists.</p>
|
422,084 | <p>$\mathbb{R}^2$ with different topologies on it are homeomorphic as a topological space?
for example with discrete topology and usual topology, what I need is a continous bijection with inverse is continous, from usual to discrete any continous map is finally constant map,so I think they are not homeomorphic.
thank... | Brian M. Scott | 12,042 | <p>No, they are not homeomorphic. $\Bbb R^2$ with the discrete topology is not connected, and $\Bbb R^2$ with the usual topology is connected, so there isn’t even a continuous map from $\Bbb R^2$ with the usual topology onto $\Bbb R^2$ with the discrete topology: continuous maps preserve connectedness. There are many o... |
2,180,102 | <p>If you have 3 labeled points on a surface of a paper. Like </p>
<pre><code> 1
2 3
</code></pre>
<p>This makes a perfect equilateral triangle.</p>
<p>From this perspective I can say that the camera is on top of the paper looking down. We can say the camera is at coordinate $(0,0,100)$. Which is 0 degree... | arctic tern | 296,782 | <p>The proof is valid if you're allowed to use the fact that $\phi$ is multiplicative. Personally, the Chinese Remainder Theorem is what I consider to be <em>the reason</em> that $\phi$ is multiplicative, so this would be circular for me.</p>
<p>Yes, we do have $(\prod_i R_i)^\times=\prod_i R_i^\times$. It is actually... |
1,523,427 | <p>Is it possible to cover all of $\mathbb{R}^2$ using balls $\{ B(x_n,n^{-1/2})\}_{n=1}^\infty$ of decreasing radius $n^{-1/2}$? I know that if we chose e.g. radius $n^{-1}$ it could never work because $\sum \pi (n^{-1})^2 < \infty$. But in this case the balls cover an infinite amount of area, so it seems that it m... | robjohn | 13,854 | <p>With disks centered on an $n{+}1\times n{+}1$ grid, we can cover $[0,1]^2$ with $(n+1)^2$ disks of radius $\frac1{n\sqrt2}$.</p>
<p>Therefore, we can cover $[0,1]^2$ with $\left(\left\lceil\sqrt{\frac n2}\,\right\rceil+1\right)^2$ disks of radius $\frac1{\sqrt{n}}$.</p>
<p>For $n\ge294$, $\left(\left\lceil\sqrt{\f... |
3,708,243 | <p>Equation: </p>
<blockquote>
<p><span class="math-container">$x^2-x-6=0$</span></p>
</blockquote>
<p>The two roots of this equation are <span class="math-container">$3$</span> and <span class="math-container">$-2$</span>. When writing the answer can I also write it as <span class="math-container">$-2, 3$</span> o... | Peter Shor | 2,501 | <p>This isn't the way the simplex algorithm is usually presented, but it is certainly equivalent to the usual presentation. I'm going to use the standard terminology (pivots, phase I, basic feasible solution); if wherever you got this from doesn't use this, feel free to ask about it in comments.</p>
<p>This isn't a re... |
354,986 | <p>I have to show that $h$ is measurable as well as $\int h d(\mu \times \nu) < \infty$ .</p>
<p>I tried showing by contradiction that $\int h$ had to be finite but I'm stuck with showing how it is measurable.</p>
| Julien | 38,053 | <p><strong>1- Measurability:</strong></p>
<p><strong>Fact:</strong> a function is measurable if and only if it is the pointwise limit of a sequence of <a href="http://en.wikipedia.org/wiki/Simple_function" rel="noreferrer">simple functions</a>.</p>
<p>So take $s_n, t_n$ simple such that $f(x)=\lim s_n(x)$ for every $... |
821,768 | <p><img src="https://i.stack.imgur.com/bS4PE.png" alt="enter image description here"></p>
<p>In the rectangle ABCD,
$$1. \, BE = EF = FC = AB$$
$$2. \, \angle AEB = \beta , \angle AFB = \alpha , \angle ACB = \theta. $$
Prove that $\alpha + \theta = \beta$.</p>
<p>I have so far obtained that - $$1. \cos\beta = \sin \... | DSinghvi | 148,018 | <p>You have $\beta= 45degree$ because an isoceles right triangle ABE.
then $\tan(\alpha + \theta)$=$(\tan\alpha+\tan\theta)$,$1-\tan\alpha\tan\theta
=$(5/6 $\tan\beta$)$1-\tan^2\beta$=$5\tan\beta$ $6-\tan^2\beta$
gives $\tan(\alpha + \theta)=1=\tan\beta$
$\alpha + \theta = \beta$</p>
|
147,378 | <p>I have the following equation:</p>
<p>$$\frac{dx}{dt}+x=4\sin(t)$$</p>
<p>For solving, I find the homogenous part as:
$$f(h)=C*e^{-t}$$</p>
<p>Then finding $f(a)$ and $df(a)$:
$$f(a)=4A\sin(t)+4B\cos(t)$$
$$df(a)=4A\cos(t)-4B\sin(t)$$</p>
<p>Substituting in orginal equation:</p>
<p>$$4A\cos(t)-4B\sin(t)+4A\sin(... | Jon | 20,667 | <p>At the stage</p>
<p>$$4A\cos(t)-4B\sin(t)+4A\sin(t)+4B\cos(t)=4\sin(t)$$</p>
<p>you are done. Collecting common factors you will get</p>
<p>$$4(A+B)\cos(t)+4(A-B-1)\sin(t)=0$$</p>
<p>but this must be independent and so you have a system of equations</p>
<p>$$A-B=1$$</p>
<p>$$A+B=0$$</p>
<p>giving $A=-B=\frac{... |
335,483 | <p>Let $N$ be a set of non-negative integers. Of course we know that $a+b=0$ implies that $a=b=0$ for $a, b \in N$.</p>
<p>How do (or can) we prove this fact if we don't know the subtraction or order?</p>
<p>In other words, we can only use the addition and multiplication.</p>
<p>Please give me advise.</p>
<p>EDIT</... | Ross Millikan | 1,827 | <p>Please set out what you mean by the addition law. You need the axiom that there is no number whose successor is $0$ or this fails. That is what distinguishes the integers from the naturals. It allows you to define order as $x \le y \leftrightarrow \exists (z) x+z=y$</p>
|
215,864 | <p>I run to the following problem which says if you have a smooth curve that is evolving over time (say finite length at the beginning) then </p>
<p>$$\frac{d}{dt}(curve \; length \; at \; time \; t)=-\int_{curve} k\cdot v \; ds,$$</p>
<p>where $k$ is curvature of the curve and $v$ is velocity of point on curve. $ds... | cactus314 | 4,997 | <p>Here's a nice crib-sheet for <a href="http://www.mpi-inf.mpg.de/~ag4-gm/handouts/06gm_curves.pdf" rel="nofollow noreferrer">differential geometry of curves in space</a>.
<hr>
Let $\gamma: [a,b]\times [0,1] \to \mathbb{R}^3$ be a family of curves. The rate of change of arc-length is:
$$ \frac{d}{dt}(\mathrm{arc le... |
388,561 | <p>I am trying to do this question in Bredon's <em>Topology and geometry</em> about using the transversality theorem to show that the intersection of two manifolds is a manifold.</p>
<p>Now it goes as follows:</p>
<p>Let $f(x,y,z)=(2-(x^2+y^2)^{1/2})+z^2$ on $\mathbb{R}^3 - (0,0,z)$. Then one can show that $M=f^{-1}(... | Ted Shifrin | 71,348 | <p>Try writing down a function $g\colon\mathbb R^3-\{z\text{-axis}\}\to \mathbb R^2$ so that, for example, $g^{-1}(1,4) = M\cap N$ and $(1,4)$ is a regular value of $g$.</p>
|
3,501,052 | <p>I want to find the number of real roots of the polynomial <span class="math-container">$x^3+7x^2+6x+5$</span>.
Using Descartes rule, this polynomial has either only one real root or 3 real roots (all are negetive). How will we conclude one answer without doing some long process?</p>
| Ross Millikan | 1,827 | <p>If there are three real roots, the value of the function must be of opposite signs at the points the derivative is zero. </p>
|
3,501,052 | <p>I want to find the number of real roots of the polynomial <span class="math-container">$x^3+7x^2+6x+5$</span>.
Using Descartes rule, this polynomial has either only one real root or 3 real roots (all are negetive). How will we conclude one answer without doing some long process?</p>
| 2'5 9'2 | 11,123 | <p>I note that this is "close" to <span class="math-container">$$(x+5)(x+1)(x+1)=x^3+7x^2+11x+5$$</span> which has a repeated root at <span class="math-container">$-1$</span>, and another root at <span class="math-container">$-5$</span>. The repeated root at <span class="math-container">$-1$</span> is a local minimum, ... |
646,010 | <p>So i kinda think i have figured this out, i'm not very good at math, and need a formula to figure out some stats for a game i'm playing.</p>
<p>I have a Weapon with a reload speed of X sec.. however, i also have a modifier attached, that will make the weapon reload faster by +Y%</p>
<p>i made this formula, mostly ... | mkl314 | 123,304 | <p>The textbook's answer is wrong, and there is no way to prove that this integral diverges. Instead, there are ways to establish its convergence.</p>
<p>Since the integrand is continuous on segment $[0,1]$, it suffices to verify convergence on $(1,\infty)$, which can be established by substituting $x=\sqrt{t}$ follo... |
430,654 | <p>Show that this sequence converges and find the limit.
$a_1 = 0$, $a_{n+1} = \sqrt{5+2a_{n} }$ </p>
| Brian M. Scott | 12,042 | <p>Suppose that we’ve shown — somehow — that the sequence converges to some limit $L$. Finding $L$ is then quite easy. Let $f(x)=\sqrt{5+2x}$; then $f$ is a continuous function on its domain, so</p>
<p>$$L=\lim_{n\to\infty}a_n=\lim_{n\to\infty}a_{n+1}=\lim_{n\to\infty}f(a_n)=f\left(\lim_{n\to\infty}a_n\right)=f(L)\;.$... |
3,278,797 | <p>I tried to solve it and I got answer '3'. But that is just my intuition.I don't have concrete method to prove my answer .I did like this, in order to maximize the fraction, we need to minimize the denominator .So if plug in '1' in expression, denominator becomes '1'.Now denominator is minimalized,the result of ex... | José Carlos Santos | 446,262 | <p>Yes, the maximum is <span class="math-container">$3$</span>. Note that <span class="math-container">$x^2-x+1$</span> is always greater than <span class="math-container">$0$</span> and that<span class="math-container">$$\frac{x^2+x+1}{x^2-x+1}-3=-2\frac{x^2-2x+1}{x^2-x+1}=-2\frac{(x-1)^2}{x^2-x+1}<0,$$</span>unles... |
4,518,908 | <p>For sufficiently large integer <span class="math-container">$m$</span>, in order to prove</p>
<p><span class="math-container">$\frac{(m+1)}{m}<\log(m)$</span></p>
<p>is it sufficient to point out that</p>
<p><span class="math-container">$ \displaystyle\lim_{m \to \infty} \frac{(m+1)}{m}=1 $</span></p>
<p>while</p... | Dark Rebellion | 858,891 | <p><span class="math-container">$\forall x(Px\implies\forall y:P(x+y))$</span></p>
<p><span class="math-container">$\iff \forall x(Px\implies\forall y(True \implies P(x+y)))$</span></p>
<p><span class="math-container">$\iff \forall x(Px\implies\forall y(\exists z:z=x+y \implies P(x+y)))$</span>, this is true because th... |
130,806 | <p><strong>Qusestion:</strong> Let f be a continuous and differentiable function on $[0, \infty[$, with $f(0) = 0$ and such that $f'$ is an increasing function on $[0, \infty[$. Show that the function g, defined on $[0, \infty[$ by $$g(x) = \begin{cases} \frac{f(x)}{x}, x\gt0& \text{is an increasing function.... | Davide Giraudo | 9,849 | <p>$c$ depends on $x$, and what you did doens't prove that if $x_1\leq x_2$ then $c_{x_1}\leq c_{x_2}$. </p>
<p>But we can write for $x>0$, since $f'$ is increasing hence integrable over finite intervals
$$\frac{f(x)}x=\frac{f(x)-f(0)}x=\frac 1x\int_0^xf'(t)dt=\int_0^1f'(xs)ds$$
by the substitution $t... |
40,709 | <p>Wolfram's MathWorld website, at the page on <a href="http://mathworld.wolfram.com/Function.html" rel="nofollow">functions</a>, makes the following claim about the notation $f(x)$ for a function:</p>
<blockquote>
<p>While this notation is deprecated by professional mathematicians, it is the more familiar one for m... | Andrew Stacey | 45 | <p>Vote for this answer if you <strong>disagree</strong> with the statement:</p>
<blockquote>
<p>This house believes that the notation $f(x)$ to refer to a function has value in professional mathematics and that there is no need to apologise or feel embarrassed when using it thus.</p>
</blockquote>
<p>(Note: the a... |
2,097,557 | <blockquote>
<p>If $0< \alpha, \beta< \pi$ and $\cos\alpha + \cos\beta-\cos (\alpha + \beta) =3/2$ then prove $\alpha = \beta= \pi/3$</p>
</blockquote>
<p>How do I solve for $\alpha$ and $\beta$ when only one equation is given? By simplification I came up with something like
$$
\sin\frac{\alpha}{2} \sin\frac{... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>Use <a href="http://mathworld.wolfram.com/ProsthaphaeresisFormulas.html" rel="nofollow noreferrer">Prosthaphaeresis</a> Formula on $\cos\alpha,\cos\beta$</p>
<p>and Double angle formula on $$\cos 2\cdot\dfrac{\alpha+\beta}2$$ to get</p>
<p>$$2\cos^2\dfrac{\alpha+\beta}2-2\cos\dfrac{\alpha-\beta}2\cos... |
1,087,080 | <p>By definition, a closed set is a set that contains its limit points. However, by the time the closed set contains its limit points, those points are no longer limit points and become isolated points. For example:</p>
<p>$\mathbf A = \{\frac{1}{n}: n \in \mathbb N \}$. The limit of this set (set $\mathbf A$) is clea... | Sultan of Swing | 144,369 | <p><strong>Regarding the lemma</strong>: Suppose $x$ is a limit point of $A$. Then <em>every</em> <strong>neighborhood</strong> of $x$ contains a point in $A$ that is different than $x$. Since $A \subseteq A'$, then we can clearly see that these <strong>neighborhoods</strong> certainly contain a point in $A'$. The reas... |
1,087,080 | <p>By definition, a closed set is a set that contains its limit points. However, by the time the closed set contains its limit points, those points are no longer limit points and become isolated points. For example:</p>
<p>$\mathbf A = \{\frac{1}{n}: n \in \mathbb N \}$. The limit of this set (set $\mathbf A$) is clea... | egreg | 62,967 | <p>First of all, recall that the notion of limit point has a meaning only for <em>subsets</em> of the real numbers (or, more generally but not too generally, of a metric space).</p>
<p>A point $x$ is a limit point of $A$ if and only if, for every $\varepsilon>0$, the interval $(x-\varepsilon,x+\varepsilon)$ contai... |
4,638,490 | <p>Given functions f and g, as above, what exactly does it mean? Does it mean, for example, that g(n) is <em>exactly</em> equal to <span class="math-container">$2^{h(n)}$</span> for some function h contained in <span class="math-container">$O(f(n))$</span> - or does it rather mean that <span class="math-container">$g(n... | Axo | 1,012,859 | <p>Remember big-Oh <span class="math-container">$(O)$</span> is an asymptotically tight bound. So if for some <span class="math-container">$f,g$</span> you have <span class="math-container">$f(x) = O(g(x)$</span>, then as <span class="math-container">$x$</span> approaches infinity, <span class="math-container">$f$</spa... |
1,102,668 | <p><a href="https://math.stackexchange.com/q/67994/198434">This question</a> shows how dividing both sides of an equation by some $f(x)$ may eliminate some solutions, namely $f(x)=0$. Naturally, all examples admit $f(x)=0$ as a solution to prove the point.</p>
<p>I tried to find a simple example of an equation that co... | user141592 | 178,602 | <p>Well, it depends. In general, $g(x)$ divides $f(x)$ if and only if $g(x) = 0$ is a root of the equation. This happens when you are working over any field, because if $g(x)$ divides $f(x)$, then $f(x) = q(x)g(x)$, so $f(x)$ can only be zero by the null factor law if $g(x) = 0$ or $q(x) = 0$. However, $g(x)=0$ may not... |
42,617 | <p>Let $f(x,y)$ define a surface $S$
in $\mathbb{R}^3$ with a unique local minimum at $b \in S$.
Suppose <a href="http://en.wikipedia.org/wiki/Gradient_descent" rel="noreferrer">gradient descent</a> from any start point $a \in S$
follows a geodesic on $S$ from $a$ to $b$.
(<b>Q1</b>.)
What is the class of functions/sur... | Willie Wong | 3,948 | <p>For (Q1). The tangent space of $S$ is generated by the gradient flow vector field $v = (|\nabla f|^2, \nabla f)$ and the tangents to the level sets $w= (0, \nabla^\perp f)$. The geodesic constraint can be imposed as the condition "no sideways acceleration", which means that $[(\nabla f \cdot \nabla )v] \cdot w = 0$.... |
3,882,457 | <p><a href="https://arxiv.org/pdf/quant-ph/0208163" rel="nofollow noreferrer">These notes</a> are a great introduction to deformation quantization but I failed to check the validity of the statement p.9, right before (5.18).</p>
<p><strong>Context:</strong> let <span class="math-container">$(\mathcal{A},+,\mu)$</span> ... | Cosmas Zachos | 362,193 | <p>Your reference has this explained in its ref [26], namely <a href="https://doi.org/10.1063/1.533395" rel="nofollow noreferrer">Zachos (2000) J Math Phys 41, 5129–5134, hep-th/9912238</a>.</p>
<p>In any case, it is straightforward to prove your (4) through elementary Fourier analysis. That is, use test/sample functio... |
483,442 | <p>I am trying to learn about velocity vectors but this word problem is confusing me.</p>
<p>A boat is going 20 mph north east, the velocity u of the boat is the durection of the boats motion, and length is 20, the boat's speed. If the positive y axis represents north and x is east the boats direction makes an angle o... | André Nicolas | 6,312 | <p>It is little known, but socks have individual identities. There are $\binom{16}{6}$ equally likely ways to choose $6$ socks from the $16$.</p>
<p>Now we find the number of ways to choose $6$ socks, so that there is no pair among them. There are $\dbinom{8}{6}$ ways to choose the "types" of sock we will have. For e... |
182,527 | <p>I have the following question:</p>
<p>Let $X$: $\mu(X)<\infty$, and let $f \geq 0$ on $X$. Prove that $f$ is Lebesgue integrable on $X$ if and only if $\sum_{n=0}^{\infty}2^n \mu(\lbrace x \in X : f(x) \geq 2^n \rbrace) < \infty $.</p>
<p>I have the following ideas, but am a little unsure. For the forward d... | Kevin Arlin | 31,228 | <p>This won't quite do. Your argument doesn't actually use the $2^n$ at all: it would read exactly the same if we asked only for $\sum_{n=0}^{\infty}\mu(\{x \in X: f(x) \geq 2^n\}) < \infty$. Yet $\frac{1}{x}$ satisfies this weaker hypothesis on $[0,1]$: the infinite sum becomes $\sum_{i=0}^\infty \frac{1}{2^i}=2$, ... |
182,527 | <p>I have the following question:</p>
<p>Let $X$: $\mu(X)<\infty$, and let $f \geq 0$ on $X$. Prove that $f$ is Lebesgue integrable on $X$ if and only if $\sum_{n=0}^{\infty}2^n \mu(\lbrace x \in X : f(x) \geq 2^n \rbrace) < \infty $.</p>
<p>I have the following ideas, but am a little unsure. For the forward d... | Did | 6,179 | <p>$$\frac12\left(1+\sum_{n=0}^{+\infty}2^n\,\mathbf 1_{f\geqslant2^n}\right)\leqslant f\lt1+\sum_{n=0}^{+\infty}2^n\,\mathbf 1_{f\geqslant2^n}$$</p>
|
2,206,938 | <p>Context: <a href="http://www.hairer.org/notes/Regularity.pdf" rel="nofollow noreferrer">http://www.hairer.org/notes/Regularity.pdf</a>, section 4.1 (pages 15-16)</p>
<blockquote>
<p>Define
$$(\Pi_x\Xi^0)(y)=1 \qquad (\Pi_x\Xi)(y)=0 \qquad (\Pi_x\Xi^2)(y)=c$$
and
$$(\Pi^{(n)}_x\Xi^0)(y)=1 \qquad (\Pi^{(n)}_x... | 5xum | 112,884 | <blockquote>
<p>It seems to me that these functions have to be dominated by something like $x^\alpha$ for some $\alpha > 0$</p>
</blockquote>
<p>False. Take the function $f(x)=e^x$ as a counterexample.</p>
<hr>
<p>Also, there is no such thing as "Lipschitz continuous at some point". Lipschitz continuity is defi... |
1,333,637 | <p>Where X is a space obtained by pasting the edges of a polygonal region together in pairs. </p>
<p>Alternatively: Show that X is homeomorphic to exactly one of the spaces in the following list: $S^2,T_n, P^2, K_m, P^2\#K_m$, where $K_m$ is the m-fold connected sum of $K$(Klein bottle) with itself and $M \geq 0$.</p>... | goblin GONE | 42,339 | <p>Rather than solving the problem for you, I collect here all relevant facts. Given these facts, actually solving it should be fairly straightforward.</p>
<p><strong>Convention.</strong> Write $T_0$ for the $2$-sphere.</p>
<p><strong>Classification Theorem.</strong> Suppose $X$ is a compact surface.</p>
<ul>
<li>If... |
2,555,463 | <p>Given a line $l$ and two points $p_1$ and $p_2$, identify the point $v$ which is equidistant from $l$, $p_1$, and $p_2$, assuming it exists.</p>
<p>My idea is to: (1) identify the parabolas containing all points equidistant from each point and the line, then (2) intersect these parabolas. As $v$ is equidistant from... | Narasimham | 95,860 | <p>I think you are almost there. The parabola has a property you already know.There are $two$ solutions/points for circle centers, but not one, get detected by a direct procedure as follows:</p>
<p>Intersections of a <em>properly/conveniently placed parabola</em> ( wlog $y=-f$ is chosen directrix) and perpendicular... |
2,178,395 | <p>In how many ways can the letters of the word CHROMATIC be arranged,</p>
<p>find the probability that the string of letters begins with the letter M</p>
<p>I don't understand how to single out M so the possibilities would only begin with M?</p>
| mvw | 86,776 | <p>Consider the number of ways to arrange the letters of CHROATIC, as you have the M already given.</p>
|
767,686 | <p>Let $f:A\rightarrow B$ be a function and let $C_1,C_2\subset A$
Prove that</p>
<p>$f(C_1\cap C_2)=f(C_1)\cap f(C_2) \leftrightarrow$ $f$ is injective</p>
<p>Attempt:</p>
<p>$(\leftarrow)$ Let $f(x)\in f(C_1\cap C_2)$. Then there exists $x\in C_1\cap C_2$ because $f$ is injective. So $x\in C_1$ and $x\in C_2$. So
... | user41281 | 145,477 | <p>Hint: Draw lines of slope $\pm 1$ at $x = 0$ and $x = 2$. By the Mean Value Theorem, no point in the set $\{ (x, y) \, | \, x \in (0, 2), \, y = f(x) \}$ lies outside that parallelogram. </p>
|
767,686 | <p>Let $f:A\rightarrow B$ be a function and let $C_1,C_2\subset A$
Prove that</p>
<p>$f(C_1\cap C_2)=f(C_1)\cap f(C_2) \leftrightarrow$ $f$ is injective</p>
<p>Attempt:</p>
<p>$(\leftarrow)$ Let $f(x)\in f(C_1\cap C_2)$. Then there exists $x\in C_1\cap C_2$ because $f$ is injective. So $x\in C_1$ and $x\in C_2$. So
... | Hagen von Eitzen | 39,174 | <p>Assume $h(1)>1$. Then there exists $\xi\in(0,1)$ with $(1-0)h'(\xi)=h(1)-h(0)$, i.e. $h'(\xi)>1$ contrary to assumption.</p>
<p>Assume $h(1)=1$ and let $h'(1)<1$ then for sufficiently small positive $k$ we have $\frac{h(1-k)-h(1)}{-k}<1$, i.e. $h(1-k)>1-k$, and then for some $\xi\in(0,1-k)$ we find $... |
64,395 | <p>Let G be a directed graph on N vertices chosen at random, conditional on the requirement that the out-degree of each vertex is 1 and the in-degree of each vertex is either 0 or 2. The "periodic" points of G are those contained in a cycle. What do we know about the statistics of G? For instance, what is the mean n... | Derek | 752 | <p>Let $G$ be a directed graph on $N$ vertices such that the out-degree of each vertex is 1 and the in-degree is either 0 or $n$. Letting $N=nt$, there are $t$ vertices with in-degree $n$ and $(n-1)t$ vertices with in-degree 0. Assuming the vertices are labeled, the number of such graphs is
$$
\binom{nt}{t}\frac{(nt)!}... |
1,840,485 | <p>I am an undergraduate really passionate about the mathematics and microbiology. I have few big problems in learning which I would like to seek your advice. </p>
<p>Whenever I study mathematical books (Rudin, Hoffman/Kunze, etc.), I always try to prove every theorem, lemma, corollary, and their relationships in t... | avz2611 | 142,634 | <p>I would suggest you read every problem, and in your head if you can see the direction pretty clearly then no need doing that, generally big texts do have repetition, but concise books meant for only problem solving without any theory do try to make sure each problem is unique.<br>
As far as second part of your resea... |
1,482,205 | <p>Show that $\sigma(AB) \cup \{0\} = \sigma(BA) \cup \{0\}$ in general, and that $\sigma(AB) = \sigma(BA)$ if $A$ is bijective. </p>
<p>I studied the associative statement of this somewhere but it did not include the zeroth element.
If you assume the bijection, how can you show the first part?</p>
<h2>My attempt</h... | NoChance | 15,180 | <blockquote>
<p>"So I got 1b/1b+a/b that simplified is b/b+a/b. So then I added those
two fractions together and got ab/b"</p>
</blockquote>
<p>You have figured out correctly up to this point:</p>
<p>$\frac{b}{b}+\frac{a}{b}$</p>
<p>The next step in your statement is where the error made. Since the denominators ... |
467,268 | <p>Any body knows the meaning of this expectation ($E[g(x)]$) form?</p>
<p>$E[g(x)]=Pr(g(x) >\varepsilon)E[g(x)|g(x) > g(\varepsilon)]+Pr(g(x) \leq\varepsilon)E[g(x)|g(x)\leq g(\varepsilon)]$</p>
| iostream007 | 76,954 | <p>HINT:</p>
<p>equation of line $y-y_1=m(x-x_1)$</p>
<p>now you have m(gradient) and a point ($x_1,y_1$) which is mid point of two given points.just find out mid point of given point.</p>
<p>for mid point: If C(x,y) is mid point of $A(x_1,y_1)$ and $B(x_2,y_2)$</p>
<p>then $x=\dfrac {x_1+x_2}{2}$ and $y=\dfrac {y... |
467,268 | <p>Any body knows the meaning of this expectation ($E[g(x)]$) form?</p>
<p>$E[g(x)]=Pr(g(x) >\varepsilon)E[g(x)|g(x) > g(\varepsilon)]+Pr(g(x) \leq\varepsilon)E[g(x)|g(x)\leq g(\varepsilon)]$</p>
| Avitus | 80,800 | <ul>
<li>A brief introduction</li>
</ul>
<p>A line in $\mathbb R^2$ is described by an equation of the form </p>
<p>$$y=mx+q~~~ (*)$$</p>
<p>(in cartesian coordinates), where the parameters $m$ and $q$ are called the slope (you called it gradient) and the intercept.</p>
<p>Why is the slope called...slope? For any t... |
165,560 | <p>To find the volume of the following region:</p>
<pre><code>fn[x_, y_, z_]:= Abs[0.7*x*Exp[I*y] + 0.3*Sqrt[x^2 + 8*10^-5]
+ Sqrt[x^2 + 3*10^-3]*0.02*Exp[I*z]]
R = ImplicitRegion[fn[x, y, z]<=3*10^-3, {{x, 0, 0.015}, {y, 0, 2*Pi}, {z, 0, 2*Pi}}]
RegionPlot3D[
fn[x, y, z] <= 3*10^-3, {x, 0, 0.015}, {y, 0, 2... | Greg Hurst | 4,346 | <p>I think the main issue is the axes bounds are quite disproportionate and that's effecting the sampling.</p>
<p>Here's your region scaled to the unit cube:</p>
<pre><code>R2 = ImplicitRegion[fn[3 x/200, 2π y, 2π z] <= 3*10^-3, {{x, 0, 1}, {y, 0, 1}, {z, 0, 1}}];
reg = BoundaryDiscretizeRegion[R2, {{0, 1}, {0, 1... |
3,346,676 | <blockquote>
<p><strong>Question.</strong> Find a divergent sequence <span class="math-container">$\{X_n\}$</span> in <span class="math-container">$\mathbb{R}$</span> such that for any <span class="math-container">$m\in\mathbb{N}$</span>,
<span class="math-container">$$\lim_{n\to\infty}|X_{n+m}-X_n|=0$$</span></p>
... | Sangchul Lee | 9,340 | <p>Here is another type of counter-example.</p>
<p>Let <span class="math-container">$x_n = \sin \big(\frac{\pi}{2}\sqrt{n}\big)$</span>. Then <span class="math-container">$x_n$</span> does not converge since <span class="math-container">$x_n$</span> oscillates. This is easily seen by noting that the values of <span cl... |
3,429,623 | <p>Is the union of <span class="math-container">$\emptyset$</span> with another set, <span class="math-container">$A$</span> say, disjoint? Even though <span class="math-container">$\emptyset \subseteq A$</span>?</p>
<p>I would say, yes - vacuously. But some confirmation would be great.</p>
| William Elliot | 426,203 | <p>There is no such thing as a disjoint union of two sets.<br>
Two sets are disjoint when their intersection is disjoint.<br>
By thus definition, the empty set and any set are disjoint.<br>
Usually disjoint is limited to not empty sets. </p>
<p>A collection K of sets is collectively disjoint when <span class="math-co... |
767,304 | <p>Prove that there are no real numbers $x$ such that</p>
<p>$$\sum_{n\,=\,0}^\infty \frac {(-1)^{n + 1}} {n^x} = 0$$</p>
<p>Can I have a hint please?</p>
| Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}
\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\down}{\... |
139,417 | <p>I have a polygon defined by a list of nodes (x,y). I want to cut the polygon by a horizontal line at position y = a and get the new polygon above the position y = a. I am using the RegionIntersect function, but it seems very slow if I want to combine the function with Manipulate function as well. Is there any way to... | Marchi | 29,455 | <p>On my machine doing everything in one step and discretizing the regions appears to smooth things out a bit.</p>
<pre><code>Manipulate[
RegionPlot[
DiscretizeRegion@
RegionIntersection[
DiscretizeRegion@
Polygon[{{0, 0}, {300, 0}, {300, 500}, {0, 750}}],
DiscretizeRegion@
ImplicitRegion[{0 &... |
2,558,988 | <blockquote>
<p>Let us consider a function $f(x,y)=4x^2-xy+4y^2+x^3y+xy^3-4$. Then find the maximum and minimum value of $f$.</p>
</blockquote>
<p>My attempt. $f_x=0$ implies $8x-y+3x^2y+y^3=0$ and $f_y=0$ implies $-x+8y+x^3+3xy²=0$ and $f_{xy}=3x^2+3y^2-1$. Now $f_x+f_y=0$ implies $(x+y)((x+y)^2+7)=0$ implies $x=-y... | José Carlos Santos | 446,262 | <p>Fix a $x_0\in X$. Consider these subsets of $X\times\mathbb R$:</p>
<ul>
<li>$\{x_0\}\times(0,+\infty)$;</li>
<li>$\left\{\bigl(x,f(x)+k\bigr)\right\}$ ($k>0$).</li>
</ul>
<p>They are all connected. So, for each $k>0$ the set$$G_k=\{x_0\}\times(0,+\infty)\cup\left\{\bigl(x,f(x)+k\bigr)\right\},$$since it's t... |
2,073,230 | <p>I thought I'd might use induction, but that seems too hard, then I tried to take the derivative and show that that's positive $\forall$n. But I can't figure out how to do that either, I've tried induction there too.</p>
| user399601 | 399,601 | <p>You can do it directly with the binomial formula: since \begin{align*} \Big(\frac{n+0.06}{n} \Big)^n &= 1 + \frac{0.06}{n} \binom{n}{1} + ... + \frac{0.06^n}{n^n} \binom{n}{n} \\ &= \sum_{k=0}^n \frac{0.06^k}{k!} \Big( 1 - \frac{1}{n}\Big) \cdot ... \cdot \Big( 1 - \frac{k-1}{n}\Big) \end{align*} and each $1... |
3,337,210 | <p>I am struggling with the following equation, which I need to proof by induction:</p>
<p><span class="math-container">$$\sum_{k=1}^{2n}\frac{(-1)^{k+1}}{k}= \sum_{k=n+1}^{2n}\frac{1}{k}$$</span></p>
<p><span class="math-container">$n\in \mathbb{N}$</span>.<br/> I tried a few times and always got stuck.</p>
<p>Help... | poetasis | 546,655 | <p>I don't know if this helps but I offer a proof from a paper I am writing in hopes that it shows that such can be presented intuitively. Some authors seem to think the reader has the the same background knowledge from [sometimes] years of research (and insights gained) that when into developing a proof. My proof coul... |
3,337,210 | <p>I am struggling with the following equation, which I need to proof by induction:</p>
<p><span class="math-container">$$\sum_{k=1}^{2n}\frac{(-1)^{k+1}}{k}= \sum_{k=n+1}^{2n}\frac{1}{k}$$</span></p>
<p><span class="math-container">$n\in \mathbb{N}$</span>.<br/> I tried a few times and always got stuck.</p>
<p>Help... | NazimJ | 533,809 | <p>What I like to do when a proof is long and abstract, is I break it up into chunks that I can describe intuitively in 1 sentence. Then these sentences form the outline of the proof which is an explanation. The skill here is to decide how much detail to include in each sentence</p>
<p>For example I will do this for ... |
3,337,210 | <p>I am struggling with the following equation, which I need to proof by induction:</p>
<p><span class="math-container">$$\sum_{k=1}^{2n}\frac{(-1)^{k+1}}{k}= \sum_{k=n+1}^{2n}\frac{1}{k}$$</span></p>
<p><span class="math-container">$n\in \mathbb{N}$</span>.<br/> I tried a few times and always got stuck.</p>
<p>Help... | David K | 139,123 | <p>For the law of cosines, we know that two sides and the included angle determine a triangle, so <span class="math-container">$a$</span> is definitely determined by <span class="math-container">$b,$</span> <span class="math-container">$c,$</span> and <span class="math-container">$\alpha.$</span></p>
<p>If you had no ... |
2,131,679 | <p>Let $f: \mathbb{R} \to \mathbb{R}$ be continuous and $D \subset \mathbb{R}$ be a dense subset of $\mathbb{R}$. Furthermore, $\forall y_1,y_2 \in D \ f(y_1)=f(y_2)$. Should $f$ be a constant function?</p>
<p>My attempt:
Since $f$ is continuous
$$\forall x_0 \ \forall \varepsilon >0 \ \exists \delta>0 \ \forall... | Futurologist | 357,211 | <p>Triangles say $ABD$ and $BCE$ are congruent because $AB = BC, \,\, AD = \frac{1}{3} AC = \frac{1}{3} AB = BE$ and $\angle \, BAD = \angle \, CBE = 60^{\circ}$. Hence $\angle \, ADO = \angle \, ADC = \angle \, CEB = \angle \, OEB = \theta$. This implies that $\angle \, ADO + \angle \, AEO = \angle \, ADO + 180^{\circ... |
3,384,280 | <p>I'm trying to solve the limit of this sequence without the use an upper bound o asymptotic methods:</p>
<p><span class="math-container">$$\lim_{n\longrightarrow\infty}\frac{\sqrt{4n^2+1}-2n}{\sqrt{n^2-1}-n}=\left(\frac{\infty-\infty}{\infty-\infty}\right)$$</span></p>
<p>Here there are my differents methods:</p>
... | user | 505,767 | <p>From here</p>
<p><span class="math-container">$$\frac{\sqrt{4n^2+1}-2n}{\sqrt{n^2-1}-n}=\frac{\sqrt{4+\dfrac{1}{n^2}}-2}{\sqrt{1-\dfrac{1}{n^2}}-1}$$</span></p>
<p>we can use that</p>
<p><span class="math-container">$$\sqrt{4+\dfrac{1}{n^2}}=2\sqrt{1+\dfrac{1}{4n^2}}\sim 2\left(1+\dfrac{1}{8n^2}\right)=2+\dfrac{1... |
300,435 | <p>Let $V,V',V''$ and $W$ be vector spaces over $k$. Then, it is known that $\operatorname{Hom}(\cdot,V)$ is a contravariant exact functor, i.e. for each exact sequence</p>
<p>$0\to V'\to V\to V'' \to 0$,</p>
<p>and each $W$, the induced sequence</p>
<p>$0\to \operatorname{Hom}(V'',W)\to\operatorname{Hom}(V,W)\to\op... | Martin | 49,437 | <p>To say that$\DeclareMathOperator{Hom}{Hom}$
$$
0 \to V' \xrightarrow{i} V \xrightarrow{p} V'' \to 0
$$
is exact is equivalent to saying that $i$ is a kernel of $p$ and $p$ is a cokernel of $i$. This amounts to the automatic exactness of
$$
0 \to \Hom(W,V') \xrightarrow{i_\ast}\Hom(W,V) \xrightarrow{p_\ast} \Hom(W,V'... |
300,435 | <p>Let $V,V',V''$ and $W$ be vector spaces over $k$. Then, it is known that $\operatorname{Hom}(\cdot,V)$ is a contravariant exact functor, i.e. for each exact sequence</p>
<p>$0\to V'\to V\to V'' \to 0$,</p>
<p>and each $W$, the induced sequence</p>
<p>$0\to \operatorname{Hom}(V'',W)\to\operatorname{Hom}(V,W)\to\op... | Nate Eldredge | 822 | <p>I'm going to adjust your notation, because $V'$ looks too much like a dual space to me.</p>
<p>In the category of Banach spaces, where the morphisms are the continuous linear maps, one should perhaps interpret "image" in the categorical sense, as the <em>closure</em> of the image. (See the comments on Martin's ans... |
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