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 |
|---|---|---|---|---|
261,410 | <p>Let $z_1,z_2,\dots,z_n\in\Bbb{C}$ be distinct and $w_1,w_2,\dots,w_n\in\Bbb{C}$ be arbitrary. Suppose $f, g$ are two polynomials of degree less than $n$ such that
$$f(z_j)=w_j,\qquad g(z_j)=\bar{w}_j \qquad\text{for $1\leq j\leq n$}.$$
Define $\Omega(z)=\prod_{j=1}^n(z-z_j)$. The following puzzles me.</p>
<blockquo... | Pietro Majer | 6,101 | <p>For $n\ge 3$, let's take $z_j$, for $1\le j\le n$, be the $n$-th roots of unity, and $w_j:=z_j+4\bar{z_j}$, so that the assumption are satisfied by $$f(z):=z+4z^{n-1}$$ $$g(z):=4z+z^{n-1}.$$ However,
$$ |g'(z_k)|=|4+(n-1) z_k^{-2}| \le n+3<\phantom{Z} $$ $$ \phantom{ZZZZ} <4n-5\le|1+4(n-1) z_k^{-2}|= ... |
252,147 | <blockquote>
<p><strong>Problem:</strong> Let $G$ be an infinite abelian group. Show that if $G$ has a nontrivial subgroup $K$ such that $K$ is contained in all nontrivial subgroups of $G$, then $G$ is a $p$- group for some prime $p$. Moreover, $G$ is of type $p^\infty$ (quasicyclic) group.</p>
</blockquote>
<p>I h... | Hagen von Eitzen | 39,174 | <p>Assume $K\le G$ is a non-trivial subgroup of $G$ and that for every non-trivial subgroup $X\le G$, we have $K\le X$.
Then $K$ itself has no nontrivial subgroup. Especially, for any $k\in K\setminus\{1\}$, we have $\langle k\rangle=K$ and thus $K\cong \mathbb Z$ or $K\cong \mathbb Z/n\mathbb Z$. But only the case wit... |
3,149,110 | <p>I am learning algebraic number theory, the exercises are so hard for me, could you please recommend me a book with answers? Many thanks!</p>
| B. Goddard | 362,009 | <p>Reid's book from 1910(-ish) gives a very slow introduction. I don't think it has answers, but it has almost everything worked out in the text. If you don't mind the stilted, 100-year-old language, you mind find it useful. I see it's on Amazon:</p>
<p><a href="https://rads.stackoverflow.com/amzn/click/com/1110380... |
604,824 | <p>So the puzzle is like this:</p>
<blockquote>
<p>An ant is out from its nest searching for food. It travels in a straight line from its nest. After this ant gets 40 ft away from the nest, suddenly a rain starts to pour and washes away all its scent trail. This ant has the strength of traveling 280 ft more then it ... | EuYu | 9,246 | <p>The envelope of the walls sweeps out a circle of radius $40$. Here's one possible method to find the wall (i.e. a path which intersects every tangent of the circle). From the origin, walk to $(0,40)$. Now walk counter-clockwise around the circle three-quarters of the way to $(40,0)$. Now walk vertically upwards to $... |
787,558 | <p>we have $ad-bc >1$ is it true that at least one of $a,b,c,d$ is not divisible by $ad-bc$ ?
Thanks in advance.</p>
<p><strong>Example:</strong>
$a=2$ , $b = 1$, $c = 2$, $d = 2$, $ad-bc = 2$ </p>
<p>so $b$ is not divisible by $ad-bc$</p>
| 355durch113 | 137,450 | <p>For a=2, d=3, b=2, c=1 the statement is not true. Are you sure you wrote it down correctly?</p>
|
361,862 | <p>I would like you to expose and explain briefly some examples of theorems having some hypothesis that are (as far as we know) actually necessary in their proofs but whose uses in the arguments are extremely subtle and difficult to note at a first sight. I am looking for hypothesis or conditions that appear to be almo... | Timothy Chow | 3,106 | <p>This example has been mentioned <a href="https://mathoverflow.net/a/16885">elsewhere on MO</a> but seems worth reproducing here. The abstract of Amnon Neeman's paper <a href="https://doi.org/10.1007/s002220100197" rel="noreferrer">A counterexample to a 1961 “theorem” in homological algebra</a> says:</p>
<blockquot... |
381,011 | <p>I should prove this claim:</p>
<blockquote>
<p>Every undirected graph with n vertices and $2n$ edges is connected.</p>
</blockquote>
<p>If it is false I should find a counterexample.
I was thinking to consider the complete graph with $n$ vertices. Such a graph is connected and contains $\frac{n(n-1)}{2}$ nodes. ... | Mark Bennet | 2,906 | <p>How might you prove or give a counter-example? What does it mean for a graph to be connected?</p>
<p>You observe that the number of edges in a complete graph is $\cfrac {n(n-1)}{2}$, and this grows faster than $2n$. So large complete graphs will have many more edges than $2n$.</p>
<p>You might also know that the m... |
3,628,358 | <p>As stated, I need to prove that, up to isomorphism, the only simple group of order <span class="math-container">$p^2 q r$</span>, where <span class="math-container">$p, q, r$</span> are distinct primes, is <span class="math-container">$A_5$</span> (the alternating group of degree 5).</p>
<p>Now I know the following... | user113019 | 682,920 | <p>Since you asked for an "elementary" proof, I'll try to write one that doesn't use Burnside's transfer theorem.</p>
<p>Let <span class="math-container">$G$</span> be a simple group of order <span class="math-container">$p^2qr$</span>, WLOG <span class="math-container">$q<r$</span>. We call <span class="m... |
54,878 | <p>Consider the 2 parameter family of linear systems </p>
<p>$$\frac{DY(t)}{Dt} = \begin{pmatrix}
a & 1 \\
b & 1 \end{pmatrix} Y(t)
$$</p>
<p>In the ab plane, identify all regions where this system posseses a saddle, a sink, a spiral sink, and so on. </p>
<p>I was able... | Alp Uzman | 169,085 | <p>Here is a slightly more high-brow answer. Given the trace-determinant plane (together with the qualitative behaviors associated to different regions of it), one can determine all cases in one unified argument.</p>
<p>Let us first be explicit on the dependencies of the coefficient matrix to the parameters and put</p>... |
697,984 | <p>I want to check whether the position operator $A$, where $Af(x)=xf(x)$ , is self-adjoint. For this to be true it has to be Hermitian and also the domains of it and its adjoint must be equal. The Hilbert space I'm working with is of course $L^2(\mathbb{R}) $ with the natural inner product. The problem I'm having is w... | homegrown | 125,659 | <p>By the sum rule of probability, we have
$$P[(A\cap B^C)\cup (A^C\cap B)]=P(A\cap B^C)+P(A^C\cap B).$$ Then by noticing that $P(A\cap B^C)=P(A)-P(A\cap B)$ and $P(A^C\cap B)=P(B)-P(A\cap B)$ we arrive at:
$$=P(A)-P(A\cap B)+P(B)-P(A\cap B)=P(A)+P(B)-2P(A\cap B)$$ and we are done.</p>
|
697,984 | <p>I want to check whether the position operator $A$, where $Af(x)=xf(x)$ , is self-adjoint. For this to be true it has to be Hermitian and also the domains of it and its adjoint must be equal. The Hilbert space I'm working with is of course $L^2(\mathbb{R}) $ with the natural inner product. The problem I'm having is w... | Nuduja Mapa | 1,019,214 | <p><strong>Ρ[(Α∩Β′)∪(Α′∩Β)]</strong><br />
<strong>= Ρ(Α∩Β′) + Ρ(Α′∩Β) − Ρ(Α∩Β′∩Α′∩Β)</strong> (Law of GRA)<br />
<strong>= Ρ(Α∩Β′) + Ρ(Α′∩Β) − Ρ(Α∩Α′∩Β′∩Β)</strong> (Commutative)<br />
<strong>= Ρ(Α∩Β′) + Ρ(Α′∩Β) − Ρ()</strong> (Complementary Law)<br />
<strong>= Ρ(Α) − Ρ(Α∩Β) + Ρ(Β) − Ρ(Α∩Β)</strong> (Since, Ρ(Α) = Ρ... |
2,311,979 | <p>Let $A = (a_{i,j})_{n\times n}$ and $B = (b_{i,j})_{n\times n}$</p>
<p>$(AB) = (c_{i,j})_{n\times n}$, where $c_{i,j} = \sum_{k=1}^n a_{i,k} b_{k,j}$, so</p>
<p>$(AB)^T = (c_{j,i})$, where $c_{j,i} = \sum_{k=1}^n a_{j,k}b_{k,i} $, and
$B^T = b_{j,i}$ and $A^T = a_{j,i}$, so </p>
<p>$B^T A^T = d_{j,i}$ where $d_... | Community | -1 | <p>Why are you making things so difficult?</p>
<p>Let $A$ be a $m \times n$ matrix, $B$ be a $n \times p$ matrix.</p>
<p>$$(B^TA^T)_{ij} = \sum_{k=1}^{n}(B^{T})_{ik}(A^T)_{kj}$$</p>
<p>$$= \sum_{k=1}^{n}B_{ki}A_{jk}$$
$$= \sum_{k=1}^{n}A_{jk}B_{ki}$$
$$=(AB)_{ji}$$
$$=((AB)^{T})_{ij}$$</p>
<p>Therefore we conclude<... |
923,000 | <p>I am confused because I have seen implies and equivalent used interchangibly. For instance, I've seen </p>
<p>$$x-y=0 \implies x=y$$</p>
<p>And I've also seen</p>
<p>$$x-y=0 \Longleftrightarrow x=y$$</p>
<p>Are both of these statements correct? Which one am I supposed to use?</p>
<p>I know that implies is tr... | Dave | 174,047 | <p>If you're concerned about time, I don't think reading <em>Calculus</em> by Spivak is the best thing to do.</p>
<p>Either Zorich or Apostol is a great choice. I would say that they're "intermediate" in difficulty. Zorich contains more in-depth discussions of topics, and more examples than does Apostol.</p>
<p>If yo... |
46,462 | <p>Hi I have a simple question. How do I plot the following with Day 1 as my X axis and Day 2 as my Y axis? I need the 22 variances plotted according to the Day they were taken from (these were originally 3D measurements taken over 2 days with the same specimens each day, there were 11 specimens and 22 xyz measurements... | ubpdqn | 1,997 | <p>Significant manual cleaning was required for block of data in post.</p>
<p>The data:</p>
<pre><code>data = {{{"ID", "Day", 1., 2., 3., 4., 5., 6., 7., 8., 9., 10., 11.,
12., 13., 14., 15., 16., 17., 18., 19., 20., 21.,
22.}, {"H. sapiens", 1., 145.7, 153.2, 164.6, 161.1, 170.8,
191.7, 179.2, 178.... |
2,832,311 | <p>Suppose I draw 10 tickets at random with replacement from a box of tickets, each of which is labeled with a number. The average of the numbers on the tickets is 1, and the SD of the numbers on the tickets is 1. Suppose I repeat this over and over, drawing 10 tickets at a time. Each time, I calculate the sum of the n... | Bill Wallis | 350,028 | <p>In addition to the current answer/comment, you are confusing yourself with conflicting notation.</p>
<p>Given the sequence $a_{n} = 1/n$, you then write
$$
a_{3}(n) \geq a_{2}(N) \hspace{20pt}\text{and}\hspace{20pt} a_{2}(n) \geq a_{100}(N)
$$
but these do not make much sense by your definition. Instead, you would ... |
658,168 | <blockquote>
<p>Prove that the set of nonzero real numbers is a group under the operation $*$ defined by
\begin{align}
a*b = \begin{cases} ab &\mbox{if } a > 0 \\
\frac{a}{b} &\mbox{if } a < 0
\end{cases}
\end{align}</p>
</blockquote>
<p>I have trouble proving the <strong>associativity propery</stron... | John Habert | 123,636 | <p>I've highlighted in red the fixes you need. You have to be careful about evaluating $ab$ and $a/b$ to get the sign.</p>
<p>If $a > 0, b < 0$, then $(a*b)*c = \color{red}{(ab)*c = ab/c}$ and $a*(b*c) = a*(b/c) = \color{red}{a(b/c) = ab/c}$.</p>
<p>If $a < 0, b > 0$, then $(a*b)*c = (a/b)*c = \color{red}... |
658,168 | <blockquote>
<p>Prove that the set of nonzero real numbers is a group under the operation $*$ defined by
\begin{align}
a*b = \begin{cases} ab &\mbox{if } a > 0 \\
\frac{a}{b} &\mbox{if } a < 0
\end{cases}
\end{align}</p>
</blockquote>
<p>I have trouble proving the <strong>associativity propery</stron... | Sammy Black | 6,509 | <p>A different answer addresses your question directly, but I am including this answer to offer a slightly different perspective. The approach described below yields the fact that you have a group <em>automatically</em> (without checking various properties), but presumes that you're comfortable with the notions of gro... |
184,266 | <p>Let $a,b,c$ and $d$ be positive real numbers such that $a+b+c+d=4.$ </p>
<p>Prove the inequality </p>
<blockquote>
<p>$$a^2bc+b^2cd+c^2da+d^2ab \leq 4 .$$ </p>
</blockquote>
<p>Thanks :) </p>
| Y.Z | 19,079 | <p>Let $S=a^2bc+b^2cd+c^2da+d^2ab$. We can easily find that:</p>
<p>$$S-(ac+bd)(ab+cd)=-bd(a-c)(b-d);$$
$$S-(bc+ad)(bd+ac)=ac(a-c)(b-d)$$
which implies $$S\le \max\{(ac+bd)(ab+cd),(bc+ad)(bd+ac)\}.$$</p>
<p>By AG mean inequality:</p>
<p>\begin{align*}
(ac+bd)(ab+cd)&\le \left(\frac{(ac+bd)+(ab+cd)}{2}\right)^2... |
1,930,401 | <p>Are there any non-linear real polynomials $p(x)$ such that $e^{p(x)}$ has a closed form antiderivative? If not, is the value of $\int_{0}^{\infty}e^{p(x)}dx$ known for any $p$ with negative leading term other than $-x$ and $-x^2$?</p>
| Takahiro Waki | 268,226 | <p>About convex or concave, it's obviously convex polygon. By repeating this work, we get theorem which area of quadrilateral is max iff four points are on a circle. By using this theorem repeatedly, iff all points are on a circle, we get the area can be max.</p>
<p>When $L_k $ is on a plane of radius R, S is max.
If ... |
3,084,934 | <p>I want to prove or disprove that the Fourier transform <span class="math-container">$\mathcal F \colon (\mathcal S(\mathbb R^d), \lVert \cdot \rVert_1) \to L^1(\mathbb R^d)$</span> is unbounded, where <span class="math-container">$\lVert\cdot \rVert_1$</span> denotes the <span class="math-container">$L^1(\mathbb R^d... | Alexdanut | 629,594 | <p>Alternatively we may use Heine's definition of the limit of a sequence.<br>
Let <span class="math-container">$x_n$</span> be a sequence so that <span class="math-container">$x_n \rightarrow a => \lim_{n\to \infty} f(x_n)=\infty$</span>.
We have: <span class="math-container">$\lim_{x\to a} \frac{1}{f(x)}=\lim_{n\t... |
500,144 | <p>Hello and how you doing today? I just came across a problem which need to use Stokes theorem.</p>
<p>The problems says:</p>
<p>Evaluate the surface integral</p>
<p>$$
\int_{S}\nabla\times\vec{F}\cdot{\rm d}\vec{S}
$$</p>
<p>where F(x,y,z)=$(y^2)i$ + $(2xy)j$+$(xz^2)k$ and S is the surface of the paraboloid $z=
x... | kedrigern | 97,299 | <p>If $$f(x)=(x-a)^3+(x-b)^3+(x-c)^3$$ then $$f'(x)=3(x-a)^2+3(x-b)^2+3(x-c)^2$$ Since $a,b,c$ are distinct real numbers $f'(x) > 0$ for all $x\in\mathbb{R}$ and therefore $f$ is strictly increasing and therefore it has only one real root.</p>
<p>EDIT: The last statement is true since $f$ is a polynomial function o... |
500,144 | <p>Hello and how you doing today? I just came across a problem which need to use Stokes theorem.</p>
<p>The problems says:</p>
<p>Evaluate the surface integral</p>
<p>$$
\int_{S}\nabla\times\vec{F}\cdot{\rm d}\vec{S}
$$</p>
<p>where F(x,y,z)=$(y^2)i$ + $(2xy)j$+$(xz^2)k$ and S is the surface of the paraboloid $z=
x... | Тимофей Ломоносов | 54,117 | <p>Let's substitute the variable: $x=y+\frac{(a+b+c)}{3}$</p>
<p>The equation will now look like </p>
<p>$$3y^3+(2a^2-2ab-2ac+2b^2-2bc+2c^2)y+\frac{(a+b-2c)(a-3b+c)(b-2a+c)}{9}=0$$</p>
<p>Now we apply Cardano's method.</p>
<p>$$Q=\left(\frac{2a^2-2ab-2ac+2b^2-2bc+2c^2}{3}\right)^3+\left(\frac{\frac{(a+b-2c)(a-3b+c)... |
500,144 | <p>Hello and how you doing today? I just came across a problem which need to use Stokes theorem.</p>
<p>The problems says:</p>
<p>Evaluate the surface integral</p>
<p>$$
\int_{S}\nabla\times\vec{F}\cdot{\rm d}\vec{S}
$$</p>
<p>where F(x,y,z)=$(y^2)i$ + $(2xy)j$+$(xz^2)k$ and S is the surface of the paraboloid $z=
x... | egreg | 62,967 | <p>Set $m=(a+b+c)/3$, $A=a-m$, $B=b-m$, $C=c-m$ and $x=y+m$. Then your equation becomes
$$
(y-A)^3 + (y-B)^3 + (y-C)^3 = 0
$$
and, since $A+B+C=0$, your expansion applies to give
$$
y^3+(A^2+B^2+C^2)-\frac{A^3+B^3+C^3}{3}=0
$$
which is a suppressed cubic, whose discriminant is
$$
\frac{1}{4}\biggl(-\frac{A^3+B^3+C^3}{3... |
104,335 | <p>I am implementing a code that works correctly but it takes too much time. I did not see how to optimize it to be run quickly. Here is my code:</p>
<pre><code>data=RandomInteger[{1,400},{5000,2}];
c=10;
r=60;
pts=c + r {Cos[#], Sin[#]} & /@ Range[0, 2 π, 2 π/16];
newCoord = Table[(# - pts[[i]]) & /@ data,... | Searke | 144 | <p>Most of your time is spent in defining PolarCoords. Let's take a look at your code. </p>
<p>It looks like you've tried to optimize it already. Let's try to simplify it first:</p>
<pre><code>PolarCoords =
Map[Function[i,
ToPolarCoordinates /@
newCoord[[i]] /. {x_, y_} /; y < 0 -> {x, y + 2 \[Pi]}]... |
2,007,403 | <p>Determine the convergence or divergence of </p>
<p>$$\sqrt[n]{n!}$$</p>
<p>I was trying to use the propriety $\lim_{n\to\infty}\sqrt[n]{n}=1$, maybe I can write this</p>
<p>$\lim_{n\to\infty}\sqrt[n]{n!}=\lim_{n\to\infty}\sqrt[n]{n \ \times(n-1)\times(n-2)\times(n-3)\cdots2\ \times \ 1}=\lim_{n\to\infty}\sqrt[n]n... | Claude Leibovici | 82,404 | <p>Whenever I see a factorial somewhere in a problem of limits, I automatically thing about <a href="https://en.wikipedia.org/wiki/Stirling%27s_approximation" rel="nofollow noreferrer">Stirling approximation</a></p>
<p>$$n!\sim \sqrt{2\pi n}\,\left(\frac n e \right)^n$$ $$\sqrt[n]{n!}\sim (2 \pi n)^{\frac 1{2n}}\,\fra... |
428,843 | <p>Consider the lines in the image below:</p>
<p><img src="https://i.stack.imgur.com/AWmrd.png" alt="enter image description here"></p>
<p>Given a set of arbitrary points $p1$ and $p2$ where the direction of travel is from the former to the latter, I want to be able to directional arrow marks as in the image above.</... | DonAntonio | 31,254 | <p>Hints: Induction on</p>
<p>$$\bullet\;\;x_n<x_{n+1}\iff 1+\sqrt{2+\sqrt{3\ldots+\sqrt n}}<1+\sqrt{2+\sqrt{3+\ldots+\sqrt{n+\sqrt{n+1}}}}\iff$$</p>
<p>$$2+\sqrt{3+\ldots+\sqrt n}<2+\sqrt{3+\ldots\sqrt{n+1}}\iff\ldots$$</p>
<p>$$\bullet\bullet\;x_{n+1}^2=1+\sqrt{2+\sqrt{3+\ldots+\sqrt{n+1}}}\le 1+\sqrt2\le... |
428,843 | <p>Consider the lines in the image below:</p>
<p><img src="https://i.stack.imgur.com/AWmrd.png" alt="enter image description here"></p>
<p>Given a set of arbitrary points $p1$ and $p2$ where the direction of travel is from the former to the latter, I want to be able to directional arrow marks as in the image above.</... | Kevin Pardede | 82,064 | <p>10 days old question, but .</p>
<p>a) Is already clear, that $ \sqrt{1 + \sqrt{2 + \sqrt{3 + ...+\sqrt{n}}}} < \sqrt{1 + \sqrt{2 + \sqrt{3 + ...+\sqrt{n+1}}}}$ , because $\sqrt{n} <\sqrt{n} + \sqrt{n+1}$ which is trivial.<br>
My point here is to give some opinion about b) and c), for me it's better to do the ... |
69,476 | <p>Hello everybody !</p>
<p>I was reading a book on geometry which taught me that one could compute the volume of a simplex through the determinant of a matrix, and I thought (I'm becoming a worse computer scientist each day) that if the result is exact this may not be the computationally fastest way possible to do it... | Gerhard Paseman | 3,206 | <p>The cheapest way of finding the value of a polynomial, given unlimited preprocessing resources, is to look up the precalculated value in the table. However, if you know you are going to need several more values evaluated at successive intervals, you might try a method similar to that desired by Charles Babbage: dif... |
3,226,028 | <h2>Problem</h2>
<p>I want to know how to solve the differential equation
<span class="math-container">$$ \dot{x} + a\cdot x - b\cdot \sqrt{x} = 0 $$</span> for <span class="math-container">$a>0$</span> and both situations: for <span class="math-container">$b > 0$</span> and <span class="math-container">$b < ... | eyeballfrog | 395,748 | <p>Some caution has to be taken here because of the presence of the <span class="math-container">$\sqrt{}$</span> function--we need to keep track of signs carefully. We know that <span class="math-container">$x(t) \ge 0$</span> everywhere, so there is a function <span class="math-container">$u$</span> such that <span c... |
31,308 | <p>Apologies if my question is poorly phrased. I'm a computer scientist trying to teach myself about generalized functions. (Simple explanations are preferred. -- Thanks.)</p>
<p>One of the references I'm studying states that the space of Schwartz test functions of rapid decrease is the set of infinitely differentiabl... | Helge | 3,983 | <p>The Schwartz space $\mathcal{F}$ is just one space, one could use to define distributions. Two other common examples are smooth functions $C^{\infty}$ and smooth functions with compact support $C^{\infty} _c$. Then one has the inclusions
$$
C^{\infty} _{c} \subseteq \mathcal{S} \subseteq C^{\infty}
$$
Now distribut... |
31,308 | <p>Apologies if my question is poorly phrased. I'm a computer scientist trying to teach myself about generalized functions. (Simple explanations are preferred. -- Thanks.)</p>
<p>One of the references I'm studying states that the space of Schwartz test functions of rapid decrease is the set of infinitely differentiabl... | Zen Harper | 6,651 | <p>If you want to extend <strong>differentiation</strong> to all continuous functions, then (provided you have some convenient mathematical properties of the extension) you are FORCED to use distributions or roughly equivalent things; you have no choice! Similarly, to extend <strong>the Fourier transform</strong> you a... |
31,308 | <p>Apologies if my question is poorly phrased. I'm a computer scientist trying to teach myself about generalized functions. (Simple explanations are preferred. -- Thanks.)</p>
<p>One of the references I'm studying states that the space of Schwartz test functions of rapid decrease is the set of infinitely differentiabl... | Peter Luthy | 5,751 | <p>While I'm not saying anything new, I feel the responses thus far either miss the point or are not very complete. Generalized functions (aka distributions) are defined as linear functionals on some class of functions, typically referred to as test functions. To begin with, one usually wants any locally integrable f... |
856,334 | <p>the problem is based on this picture. <img src="https://i.stack.imgur.com/51fmQ.jpg" alt="enter image description here"></p>
<p>at beginning or we say $t=0$, $P$ is a circle of which the center is at the point $(0,r)$, $r_0=1$ is the initial radius of this circle. $AB$ is a vector which has an angle $\theta$ from t... | JJacquelin | 108,514 | <p>Starting from the beautifull result of MvG, it is possible to integrate $$
y'=\frac{\mathrm dy}{\mathrm dx}=
-\frac{3x+\sqrt3\left(4y-\sqrt{3x^2 + 4y^2}\right)}
{-3\sqrt{3} x + 4y - \sqrt{3x^2 + 4y^2}}
$$
and finally obtain an equivalent $-2\sqrt(3)\frac{x}{\ln(x)}$ for the function $y(x)$. This proves that $y(x)$ ... |
191,175 | <p>How to calculate the limit of $(n+1)^{\frac{1}{n}}$ as $n\to\infty$?</p>
<p>I know how to prove that $n^{\frac{1}{n}}\to 1$ and $n^{\frac{1}{n}}<(n+1)^{\frac{1}{n}}$. What is the other inequality that might solve the problem?</p>
| André Nicolas | 6,312 | <p>What about $n^{1/n}\lt (n+1)^{1/n}\le (2n)^{1/n}=2^{1/n}n^{1/n}$, then squeezing.</p>
<p>Or else, for $n \ge 2$,
$$n^{1/n}\lt (n+1)^{1/n}\lt (n^2)^{1/n}=(n^{1/n})(n^{1/n}).$$
Then we don't have to worry about $2^{1/n}$.</p>
|
3,296,122 | <p>I was given a problem in which a matrix <span class="math-container">$A$</span> was specified along with its determinant value, now the determinant of another matrix <span class="math-container">$B$</span> was asked to be found out whose indices were scalar multiplies of <span class="math-container">$A$</span>. What... | Kraigolas | 655,232 | <p>As a brief example, consider </p>
<p><span class="math-container">$$\begin{align}A&=\begin{pmatrix}a & b\\ c& d\end{pmatrix}\\[10 pt]
\det(A)&=ad-bc \\[10pt]
B&=nA \\[10 pt]
&=\begin{pmatrix}na & nb\\ nc& nd\end{pmatrix} \\[10 pt]
\det(B)&=nand-nbnc \\[10pt]
&=n^2ad-n^2bc\\[1... |
1,092,091 | <p>I wonder how I can calculate the distance between two coordinates in a $3D$ coordinate-system.
Like this. I've read about <em><a href="http://www.purplemath.com/modules/distform.htm" rel="nofollow">the distance formula</a></em>:</p>
<p>$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$</p>
<p>(How) Can I use that it $3D... | amWhy | 9,003 | <p>In three dimensions, the distance between two points (which are each triples of the form $(x, y, z))$ is given by $$d= \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$$</p>
|
2,280,243 | <blockquote>
<p>A tribonacci sequence is a sequence of numbers such that each term from the fourth onward is the sum of the previous three terms. The first three terms in a tribonacci sequence are called its <em>seeds</em> For example, if the three seeds of a tribonacci sequence are $1,2$,and $3$, it's 4th terms is $... | Claude Leibovici | 82,404 | <p>In <a href="http://www.kurims.kyoto-u.ac.jp/EMIS/journals/GMN/yahoo_site_admin/assets/docs/8_GMN-3532-V17N1.243113959.pdf" rel="nofollow noreferrer">this paper</a>, the authors show that, if $S_1,S_2,S_3$ are the seeds, then </p>
<p>$$S_n=T_{n-2}\,S_1+(T_{n-2}+T_{n-3})\,S_2+T_{n-1}\,S_3$$ where $T_k$ is the "usual"... |
2,927,079 | <p><a href="https://i.stack.imgur.com/ih7X2.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ih7X2.png" alt="enter image description here"></a></p>
<p>In the second paragraph, Munkres assumes that there exists a separation of <span class="math-container">$Y$</span> (in the sense he defined in Lemma 2... | yushang | 764,253 | <p>In my understanding, <span class="math-container">$Y$</span> can be viewed as a subset then the first paragraph is the proving of <span class="math-container">$\overline{A}\cap B=\emptyset$</span> and <span class="math-container">$A\cap\overline{B}=\emptyset$</span>. This means <span class="math-container">$A,B$</sp... |
1,651,427 | <blockquote>
<p>Let $f$ be a bounded function on $[0,1]$. Assume that for any $x\in[0,1)$, $f(x+)$ exists. Define $g(x)=f(x+)$, $x\in [0,1)$, and $g(1)=f(1)$. Is $g(x)$ right continuous? </p>
</blockquote>
<p>Prove it or give me a counterexample.</p>
<p>My ideas:</p>
<p>$(1)$If $f$ is of bounded variation, then $g... | Andres Mejia | 297,998 | <p>Define $f:[0,1] \to \mathbb{R}$ by</p>
<p>$f(x) = \begin{cases}
\ 1 \textrm{ if $x \in [0,1)$} \\
\ -1 \textrm{ if $x \in [1/2,1]$} \\
\end{cases}$</p>
<p>then $g(x)$ is well defined for all $x \in [0,1]$. It is not continuous, despite the fact that $g(1)=f(1)$.</p>
|
52,299 | <p>Hello everybody.</p>
<p>I'm looking for an "easy" example of a (non-zero) holomorphic function $f$ with almost everywhere vanishing radial boundary limits: $\lim\limits_{r \rightarrow 1-} f(re^{i\phi})=0$.</p>
<p>Does anyone know such an example.</p>
<p>Best
CJ</p>
| GH from MO | 11,919 | <p>To complement Andrey Rekalo's response, Lusin's construction was generalized by Bagemihl and Seidel (Math. Zeitschrift 61 (1954), online <a href="http://www.digizeitschriften.de/main/dms/img/?PPN=GDZPPN002384698" rel="nofollow noreferrer">here</a>). See their Corollary 4 whose proof takes about 2 pages, much less th... |
939,868 | <p>There are a lot math journals with title "acta" includes, for instance, Acta Mathematica, acta arithmetica, etc. Would you explain what "acta" means?</p>
| Claude Leibovici | 82,404 | <p>In latin, $actum$ means $fact$ and $acta$ is, in latin, the plural form of $actum$.</p>
|
2,354,004 | <p>I'm struggling with the following sum:</p>
<p>$$\sum_{n=0}^\infty \frac{n!}{(2n)!}$$</p>
<p>I know that the final result will use the error function, but will not use any other non-elementary functions. I'm fairly sure that it doesn't telescope, and I'm not even sure how to get $\operatorname {erf}$ out of that.</... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\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{\exp... |
150,180 | <p>I try to read Gross's paper on Heegner points and it seems ambiguous for me on some points:</p>
<p>Gross (page 87) said that $Y=Y_{0}(N)$ is the open modular curve over $\mathbb{Q}$ which classifies ordered pairs $(E,E^{'})$ of elliptic curves together with cyclic isogeny $E\rightarrow E^{'}$ of degree $N$. Gross u... | ya-tayr | 29,980 | <p>Mimicking the theory of projective resolutions, try this:</p>
<p>Start with the category whose objects are pairs $(V_1,V_0,d:V_1 \to V_0)$ where the $V_i$ are vector bundles, and whose morphisms are pairs $(f_i:V_i \to W_i)_{i \in \{0,1\}}$ intertwining with $d$. </p>
<p>Now divide each Hom group by the subgroup ... |
150,180 | <p>I try to read Gross's paper on Heegner points and it seems ambiguous for me on some points:</p>
<p>Gross (page 87) said that $Y=Y_{0}(N)$ is the open modular curve over $\mathbb{Q}$ which classifies ordered pairs $(E,E^{'})$ of elliptic curves together with cyclic isogeny $E\rightarrow E^{'}$ of degree $N$. Gross u... | Daniel Schäppi | 1,649 | <p>Here are a few comments that might be useful. I don't think there is a chance that this can work unless the scheme in question has the resolution property (meaning every coherent sheaf is a quotient of a locally free sheaf of finite rank). Otherwise the category of locally free sheaves does not even form a generator... |
18,659 | <p>This is more of a philosophy/foundation question.</p>
<p>I usually come across things like "the set of all men", or for example sets of symbols, i.e. sets of non-mathematical objects.</p>
<p>This confuses me, because as I understand it, the only kind of objects that exists in set theory are sets. It doesn't make s... | Pete L. Clark | 1,149 | <p>If you are being, say, at least semiformal in your approach to set theory, whether or not objects which are not sets exist depends upon the particular brand of set theory you choose. The most common contemporary set theory, ZFC, is a "pure set theory", in which every object is itself a set, so the men indeed do not... |
2,468,067 | <p>Can we say that that if $f(x)$ and $f^{-1}(x)$ intersect, then at least one point of intersection will lie on $y=x$? </p>
<p>Also there are many function e.g. $f(x)=1-x^3$ where point of intersection exists outside $y=x$ There will be $5$(odd) point of intersection of $f(x)=1-x^3$ and $f^{-1}(x)=(1-x)^{1/3}$ ou... | Fly by Night | 38,495 | <p>It is perfectly possible for the graphs of $f$ and $f^{-1}$ to cross away from the line $y=x$. </p>
<p>We need is $f(p)=q$ and $f^{-1}(p)=q$. </p>
<p>Given that $f^{-1}$ exists, and so $f$ and $f^{-1}$ are one-to-one, the condition that $f^{-1}(p)=q$ becomes $f(f^{-1}(p)) = f(q)$, i.e. $p=f(q)$. Hence:</p>
<p>The... |
1,032,650 | <p><img src="https://i.stack.imgur.com/GVk1i.png" alt="enter image description here"></p>
<p>Here, ABCD is a rectangle, and BC = 3 cm. An Equilateral triangle XYZ is inscribed inside the rectangle as shown in the figure where YE = 2 cm. YE is perpendicular to DC. Calculate the length of the side of the equilateral tri... | Ross Millikan | 1,827 | <p>Hint: Extend $EY$ to meet $AB$ at $F$. Drop a vertical line from $X$, hitting $CD$ at $G$. You now have three right triangles with the hypotenuse being the side of the equilateral triangle.</p>
|
1,032,650 | <p><img src="https://i.stack.imgur.com/GVk1i.png" alt="enter image description here"></p>
<p>Here, ABCD is a rectangle, and BC = 3 cm. An Equilateral triangle XYZ is inscribed inside the rectangle as shown in the figure where YE = 2 cm. YE is perpendicular to DC. Calculate the length of the side of the equilateral tri... | g.kov | 122,782 | <p><a href="https://i.stack.imgur.com/d1dCS.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/d1dCS.png" alt="enter image description here"></a></p>
<p>Let <span class="math-container">$|XY|=|YZ|=|ZX|=a$</span>,
<span class="math-container">$\angle EYZ=\theta$</span>, <span class="math-container">$|FY... |
302,061 | <p>Can you say how to find number of non-abelian groups of order n?</p>
<p>Suppose n is 24 ,then from structure theorem of finite abelian group we know that there are 3 abelian groups.But what can you say about the number of non-abelian groups of order 24?</p>
<p>The following link is a list of number of groups of or... | Chris Godsil | 16,143 | <p>The short answer is that there is no formula for the number of non-abelian groups of order $n$, nor is there an algorithm for computing the number of such groups of order $n$.</p>
<p>Note that we do not really have a formula for the number of abelian groups.
We can express the number in terms of the number of parti... |
2,487,638 | <p>$X = C([0, 1], \mathbb{R})$, $T = T(d_\infty)$. where $(X,T)$ is a topological space could some one please explain what $X=C([0,1],\mathbb{R})$ means, my teacher never explained ? Is it some kind of cover ?</p>
| Ben P. | 422,515 | <p>$C([0,1],\mathbb{R})$ is the set of all continuous functions $f:[0,1]\rightarrow\mathbb{R}$</p>
|
2,487,638 | <p>$X = C([0, 1], \mathbb{R})$, $T = T(d_\infty)$. where $(X,T)$ is a topological space could some one please explain what $X=C([0,1],\mathbb{R})$ means, my teacher never explained ? Is it some kind of cover ?</p>
| cronos2 | 148,305 | <p>It's just the collection of all continuous mappings from $[0, 1]$ to $\Bbb R$ (with respect to the usual topology of $[0, 1]$ inherited from $\Bbb R$), endowed with the topology given by the uniform norm $||•||_{\infty}$.</p>
<p>Let me know if you don't understand anything. </p>
|
1,509,340 | <p>I'm just wondering, what are the advantages of using either the Newton form of polynomial interpolation or the Lagrange form over the other?
It seems to me, that the computational cost of the two are equal, and seeing as the interpolated polynomial is unique, why ever use one over the other?</p>
<p>I get that they ... | lhf | 589 | <p>Here are two differences:</p>
<ul>
<li><p>Lagrange's form is more efficient when you have to interpolate several data sets on the same data points.</p></li>
<li><p>Newton's form is more efficient when you have to interpolate data incrementally.</p></li>
</ul>
|
554,025 | <p>I have a question as such:</p>
<blockquote>
<p>Class A has 45 students in it, and class B has 30 students in it. In class A, every student attends any particular lecture with probability 0.7 independent of the other students. For class B, two thirds of lectures are attended by everyone, with probability 1/3 that a s... | bof | 97,206 | <p>A <em>crossing</em> in a drawing of $K_n$ is an unordered pair $\{e,f\}$ of edges which cross each other but have no endpoint in common, i.e., they have $4$ distinct endpoints between them. If $x=\{e,f\}$ is a crossing, let $V(X)$ be the set consisting of the $4$ endpoints of $e$ and $f$.</p>
<p>Let $f(n)$ be the <... |
719,681 | <p>There are 2 similar questions on <span class="math-container">$\log$</span> that I'm unable to solve. </p>
<ol>
<li><p>Given that <span class="math-container">$\log_a xy^2 = p$</span> and <span class="math-container">$\log_a x^2/y^3 = q $</span>. Express <span class="math-container">$\log_a 1/\sqrt{xy}$</span> or ... | Ali | 136,967 | <p>log(xy^2) = log(x) + 2 log(y) = p Eq-1
log(x^2/y^3) = 2 log(x) -3 log(y) = q Eq-2</p>
<p>solving Eq-1 and 2 by multiply Eq-1 by -2 both sides</p>
<p>-2 log(x) -4 log(y) = -2p
2 log(x) -3 log(y) = q
---------------------------- by adding
-7 log(y) = q-2p
log(y) =... |
2,012,318 | <p>Find the volume of:</p>
<p>$V=[(x,y,z): 0 \leqslant z \leqslant 4 - \sqrt{x^2+y^2}, 2x \leqslant x^2+ y^2 \leqslant 4x] $</p>
<p>I should somehow construct triple integral here in order to solve this, which means that i have to find limits of integration for three variables, but i am just not quite sure how, i ass... | Tyler | 383,143 | <p>To show two sets $X$ and $Y$ are equal, you should show $X\subset Y$ and $Y\subset X$.</p>
<p>Let $x\in A\times (B-C)$. Then $x=(a,y)$, where $a\in A$ and $y\in B-C$. What does that tell us about the element $x$? Can you deduce that it must be an element of $(A\times B)-(A\times C)$? If so, that shows $A\times(B-C)... |
4,575,771 | <p>I need to show that <span class="math-container">$\int_0^1 (1+t^2)^{\frac 7 2} dt < \frac 7 2 $</span>. I've checked numerically that this is true, but I haven't been able to prove it.</p>
<p>I've tried trigonometric substitutions. Let <span class="math-container">$\tan u= t:$</span></p>
<p><span class="math-cont... | Gary | 83,800 | <p>By the Cauchy–Schwarz inequality
<span class="math-container">\begin{align*}
\int_0^1 {(1 + t^2 )^{7/2} {\rm d}t} & \le \sqrt {\int_0^1 {(1 + t^2 )^3 {\rm d}t} \int_0^1 {(1 + t^2 )^4 {\rm d}t} } = \sqrt {\frac{{42496}}{{3675}}} \\ & = \sqrt {\frac{{49}}{4}\frac{{169984}}{{180075}}} < \sqrt {\frac{{49}}{... |
402,427 | <p><em>Sorry if I don't use the words properly, I haven't learnt these things in English, only some of the words.
Anyway, I'm practicing to one of my exams and sadly this task seemed more challanging for me than it should be. Some kind of explain would help a lot!</em></p>
<p>10 meters of clothes have 6 holes in it.</... | Community | -1 | <p>If it's not obvious that $\mathbb{Z}[X]/(2,X) \cong \mathbb{F}_2$, then quotient out by one element at a time:</p>
<p>$$ \mathbb{Z}[X]/(2,X) = \left( \mathbb{Z}[X] / (2) \right) / (X) $$</p>
<p>or</p>
<p>$$ \mathbb{Z}[X]/(2,X) = \left( \mathbb{Z}[X] / (X) \right) / (2) $$</p>
<p>and maybe it will be easier.</p>
|
4,236,077 | <p>Suppose that <span class="math-container">$A\subset B$</span> and <span class="math-container">$A\subset C$</span>. Why does this imply <span class="math-container">$A\subset B\cup C?$</span></p>
<p>If <span class="math-container">$x\in A$</span>, then since <span class="math-container">$A\subset B$</span> and <span... | user0102 | 322,814 | <p>The following result is useful to have at hand:</p>
<p><strong>Proposition</strong></p>
<p>Given some set <span class="math-container">$U$</span> as well as <span class="math-container">$X\subseteq U$</span> and <span class="math-container">$Y\subseteq U$</span>, <span class="math-container">$X\subseteq Y$</span> if... |
4,237,342 | <p>I am a researcher and encountered the following challenging function in my work:</p>
<p><span class="math-container">$$f(S)=\sum_{k=1}^{S-1}(\ln (S)-\ln (k))^2 \bigg [ \frac{1}{(S-k)^2}+\frac{1}{(S+k)^2} \bigg ]$$</span></p>
<p>And I am only interested in the first term of the Taylor expansion of this function when ... | Jack D'Aurizio | 44,121 | <p><span class="math-container">$$f(S)=\frac{1}{S}\cdot\underbrace{\frac{1}{S}\sum_{k=1}^{S-1}\log^2\left(\frac{k}{S}\right)\left(\frac{1}{\left(1-\frac{k}{S}\right)^2}+\frac{1}{\left(1+\frac{k}{S}\right)^2}\right)}_{\text{Riemann sum}}$$</span></p>
<p>And the Riemann sum converges to</p>
<p><span class="math-container... |
4,062,987 | <p>I was reading this question here: <a href="https://math.stackexchange.com/questions/1230688/what-are-the-semisimple-mathbbz-modules">What are the semisimple $\mathbb{Z}$-modules?</a> and I understood everything except why we need <span class="math-container">$\alpha_p$</span> copies here <span class="math-container... | Community | -1 | <p>Without looking at the dup solution, you can find a recursive relation for <span class="math-container">$a_{n+1}$</span> and <span class="math-container">$a_n$</span>. One has: <span class="math-container">$a_{n+1} = \dfrac{1\cdot 3\cdot 5\cdots (2n+1)}{2\cdot 4\cdot 6\cdots (2n+2)}= \dfrac{2n+1}{2n+2}\cdot a_n< ... |
1,936,043 | <p>I would like to prove that the sequence $n^{(-1)^{n}}$ is divergent. </p>
<p>My thoughts: I know $(-1)^n$ is divergent, so $n$ to the power of a divergent sequence is still divergent? I am not sure how to give a proper proof, pls help!</p>
| Darío A. Gutiérrez | 353,218 | <p>If $n$ even then $-1^n = 1 $
$$n^1 = n\Rightarrow \lim\limits_{n \rightarrow \infty}{({n})} = \infty$$ </p>
<p>If $n$ odd then $-1^n = -1 $
$$n^{-1} = \frac{1}{n}\Rightarrow \lim\limits_{n \rightarrow \infty}{(\frac{1}{n})} = 0$$</p>
|
1,936,043 | <p>I would like to prove that the sequence $n^{(-1)^{n}}$ is divergent. </p>
<p>My thoughts: I know $(-1)^n$ is divergent, so $n$ to the power of a divergent sequence is still divergent? I am not sure how to give a proper proof, pls help!</p>
| Olivier Oloa | 118,798 | <p><strong>Hint</strong>. By setting $$u_n:=n^{(-1)^n}$$ one gets that
$$
\lim_{n \to \infty}u_{2n}=\infty \neq 0 =\lim_{n \to \infty}u_{2n+1}
$$ thus the sequence $\left\{ u_n \right\}$ is <em>divergent</em>.</p>
|
2,868,047 | <p>My question is in relation to a problem I am trying to solve <a href="https://math.stackexchange.com/questions/2867002/finding-mathbbpygx">here</a>. If $g(.)$ is a monotonically increasing function and $a <b$, is it always true that $a<g(a)<g(b)<b$? Why or why not?</p>
| MJD | 25,554 | <p>In what follows I will refer to “pattern” in the dice rolls. For example the pattern <code>AAABC</code> means that three of the dice show the same number and the other two dice are different from the first three and also different from each other. The roll $1 2 2 4 2$ has this pattern, but $1 2 2 1 2$ does not; th... |
1,913,835 | <p>I'm having a difficult time explaining/understanding a (seemingly) simple argument of an algorithm that I know I can use to determine if a directed graph <strong>G</strong> is strongly connected.</p>
<p>The algorithm that I know (does this have a name?) goes like this:</p>
<pre><code>Use BFS (breadth-first-search)... | Ashwin Ganesan | 157,927 | <p>When you do a BFS starting at a node $S$ of a directed graph, the neighbors of $S$ in the BFS tree would be the out-neighbors of $S$ (not the in-neighbors). The resulting BFS tree tells you the shortest directed path from $S$ to all other nodes. The important observation here is that BFS on a digraph considers onl... |
4,219,303 | <p>I'm trying to solve <span class="math-container">$y''-3y^2 =0$</span>, i use the substitution <span class="math-container">$w=\frac{dy}{dx}$</span>.</p>
<p>Using the chain rule i have:
<span class="math-container">$$\frac{d^2y}{dx^2} = \frac{dw}{dx} = \frac{dw}{dy}\cdot \frac{dy}{dx} = w \cdot \frac{dw}{dy}$$</span>... | Alfredo Maussa | 954,357 | <p>Assuming that what you did is right, you could integrate as separable variables:</p>
<p><span class="math-container">$\int wdw = \int 3y^2dy$</span> , so <span class="math-container">$\frac{w^2}2 = y^3 +C_1 => w=\pm\sqrt{2y^3+2C_1}$</span></p>
<p>then</p>
<p><span class="math-container">$\int \frac1{w(y)}dy = \in... |
1,603,323 | <p>If <span class="math-container">$A$</span> is a positive definite matrix can it be concluded that the kernel of <span class="math-container">$A$</span> is <span class="math-container">$\{0\}$</span>? </p>
<p>pf: R.T.P <span class="math-container">$\ker A = 0$</span>.
Suppose not, i.e., there exists some <span class... | YiFan | 496,634 | <p>Your proof is correct. As an alternative, we have a nonzero <span class="math-container">$x$</span> so that <span class="math-container">$Ax=0=0x$</span>, which means <span class="math-container">$0$</span> is an eigenvalue of <span class="math-container">$A$</span>. But the eigenvalues of a positive definite matrix... |
19,356 | <p>So I was wondering: are there any general differences in the nature of "what every mathematician should know" over the last 50-60 years? I'm not just talking of small changes where new results are added on to old ones, but fundamental shifts in the nature of the knowledge and skills that people are expected to acqui... | James Weigandt | 4,872 | <p>The general question of what a professional mathematician should know was asked by Phil Davis at the end of <a href="http://www.siam.org/news/news.php?id=1642">this article</a>. Barry Mazur <a href="http://www.math.harvard.edu/~mazur/preprints/math_ed_2.pdf">posted a brief response</a> about a year ago.</p>
<p>I'm ... |
1,995,663 | <p>My brother in law and I were discussing the four color theorem; neither of us are huge math geeks, but we both like a challenge, and tonight we were discussing the four color theorem and if there were a way to disprove it.</p>
<p>After some time scribbling on the back of an envelope and about an hour of trial-and-e... | Arkya | 276,417 | <p>This works, as you can check..</p>
<p><a href="https://i.stack.imgur.com/wJgmS.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/wJgmS.png" alt="enter image description here"></a></p>
|
1,995,663 | <p>My brother in law and I were discussing the four color theorem; neither of us are huge math geeks, but we both like a challenge, and tonight we were discussing the four color theorem and if there were a way to disprove it.</p>
<p>After some time scribbling on the back of an envelope and about an hour of trial-and-e... | Especially Lime | 341,019 | <p>To answer the "algorithmically" question, this map has some regions that only border four others. There is a relatively short, algorithmic proof that if you can 4-colour all but one of the regions of a map, and the last region, R, only borders four others (call them R_1, R_2, R_3, R_4 in clockwise ordering about R),... |
2,835,474 | <p>What is linear about a linear combination of things?. In linear algebra, the "things" we are dealing with are usually vectors and the linear combination gives the span of the vectors. Or it could be a linear combination of variables and functions. But why not just call it combination. Why is the term "linear" includ... | OnceUponACrinoid | 246,291 | <p>It is linear because such a combination has the form</p>
<p>$$
(const_1) (quantity_1)+(const_2)(quantity_2)+\ldots
$$</p>
<p>as opposed to expressions such as
$$
(const)(quantity_1)(quantity_2)
$$
or</p>
<h2>$$
(const)(quantity)^3
$$</h2>
<p>More generally, such forms preserve "linearity", in the sense that sca... |
2,520,044 | <p>$$\lim_{x\to2}{\frac{\sqrt{3x-2}-\sqrt{5x-6}}{\sqrt{2x-1}-\sqrt{x+1}}}$$</p>
<p>Evaluate the limit.</p>
<p>Thanks for any help</p>
| lab bhattacharjee | 33,337 | <p>Hint:</p>
<p>Set $x-2=h$</p>
<p>and rationalize the denominator & the numerator.</p>
<p>Setting limit to $0$ often eases of the calculation</p>
|
3,449,826 | <p><span class="math-container">$C$</span> is any closed curve encompassing the whole branch cut.
The approach to this problem would involve using the residue theorem:</p>
<p>1) We first want to find the residues at infinity so we change the form to where we are able to perform a series expansion.</p>
<p>2) Obtain th... | GEdgar | 442 | <p>More complete version of Pedrpan's answer... As <span class="math-container">$z \to \infty$</span>, we have
<span class="math-container">$$
\frac{z-a}{z-b} = \frac{1-a/z}{1-b/z} =
\left(1-\frac{a}{z}\right)\left(1+\frac{b}{z}+O(z^{-2})\right)
= 1 +\frac{b-a}{z}+O(z^{-2})
$$</span>
then one branch of the square roo... |
2,965,821 | <p>Let <span class="math-container">$(x_k)_{k\in N}$</span> <span class="math-container">$\subset \mathbb{R^4} $</span>.</p>
<p>Then there's this series, which I have to check for convergence and its limit.</p>
<p><a href="https://i.stack.imgur.com/o6zpZ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.... | Bertrand Wittgenstein's Ghost | 606,249 | <p>Chain rule states: <span class="math-container">$\frac {d}{dx}f(g(x))=f'(g(x))\cdot g'(x)$</span>, then <span class="math-container">$$(e^{x^3+2})'=e^{x^3+2}\cdot 3x^2$$</span></p>
|
14,140 | <p>One of the most annoying "features" of <em>Mathematica</em> is that the <code>Plot</code> family does extrapolation on <code>InterpolatingFunction</code>s without any warning. I'm sure it was discussed to hell previously, but I cannot seem to find any reference. While I know how to simply overcome the problem by def... | Mr.Wizard | 121 | <p>It seems that indeed messages are suppressed somewhere in the evaluation of <code>Plot</code>. Interestingly this appears to happen some time after the first numeric evaluation. If we issue this message for each (numeric) point plotted we see that it is printed only once:</p>
<pre><code>f[x_?NumericQ] := (Message... |
704,073 | <p>I encountered something interesting when trying to differentiate $F(x) = c$.</p>
<p>Consider: $\lim_{x→0}\frac0x$. </p>
<p>I understand that for any $x$, no matter how incredibly small, we will have $0$ as the quotient. But don't things change when one takes matters to infinitesimals?
I.e. why is the function $\fr... | wroobell | 130,926 | <p>You are in fact considering function $f: \mathbb R \setminus \{0\} \mapsto \mathbb R$ that is defined $f(x) = \frac 0x$ so it i equal $0$ on all its domain. Lets look at limit $\lim \limits_{x \rightarrow 0^+} f(x)$. Function is identically equal $0$ on every open interval $(0,\varepsilon)$ for $\varepsilon>0$. H... |
704,073 | <p>I encountered something interesting when trying to differentiate $F(x) = c$.</p>
<p>Consider: $\lim_{x→0}\frac0x$. </p>
<p>I understand that for any $x$, no matter how incredibly small, we will have $0$ as the quotient. But don't things change when one takes matters to infinitesimals?
I.e. why is the function $\fr... | user133943 | 133,943 | <p>A function isn't just an expression, but you can think whether a single expression can be applied to an argument. The expression $0^{-1}$ is rather meaningless, so you don't know how to get the behavior of the function $f(x)=0\cdot x^{-1}$ at $x=0$ from the expression.</p>
<p>Limits are just a way to describe the b... |
3,046,083 | <p>Is it true that the intersection of the closures of sets <span class="math-container">$A$</span> and <span class="math-container">$B$</span> is equal to the closure of their intersection?
<span class="math-container">$ cl(A)\cap{cl(B)}=cl(A\cap{B})$</span> ?</p>
| Kavi Rama Murthy | 142,385 | <p>No. Take <span class="math-container">$A=(-1,0), B=(0,1)$</span>. Note that <span class="math-container">$0$</span> is in the closure of both of these sets.</p>
|
2,534,999 | <p>I tried to solve $z^3=(iz+1)^3$. I noticed that $(iz+1)^3=i(z-1)^3$ so $(\frac{z-1}{z})^3=i$. How to finish it?</p>
| Fred | 380,717 | <p>If $z^3=(iz+1)^3$, then $|z|=|iz+1|$, hence, if $z=x+iy$, we have $y=1/2$.</p>
<p>Therefore $z=x+\frac{i}{2}$.</p>
<p>Can you proceed ?</p>
|
4,527,429 | <p>I am confused as to how we open the abs value, do we get <span class="math-container">$e=0$</span> and <span class="math-container">$e=2x$</span>, or does the identity not exist?</p>
<p>Thanks.</p>
| Shaun | 104,041 | <p>Note that, if the identity <span class="math-container">$e$</span> did exist, then for negative <span class="math-container">$n$</span> we would have</p>
<p><span class="math-container">$$n=n*e=|n-e|\ge 0.$$</span></p>
|
1,767,682 | <p>I was thinking about sequences, and my mind came to one defined like this:</p>
<p>-1, 1, -1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, 1, ...</p>
<p>Where the first term is -1, and after the nth occurrence of -1 in the sequence, the next n terms of the sequence are 1, followed by -1, and so on. Which led me to perhaps a str... | marty cohen | 13,079 | <p>As André Nicolas wrote,
the Cesaro mean,
for which the $n$-th term
is the average of the
first $n$ terms
will do what you want.</p>
<p>In both your cases,
for large $n$,
if $(a_n)$ is your sequence,
if
$b_n = \frac1{n}\sum_{k=1}^n a_k
$,
then
$b_n \to 1$
since the number of $-1$'s
gets arbitrarily small
compared ... |
3,287,016 | <blockquote>
<p>Show <span class="math-container">$\lvert\sin z\rvert \leq C \lvert z\rvert$</span> <span class="math-container">$\forall \lvert z\rvert\leq 1$</span></p>
</blockquote>
<p>The question asks me to show the constant <span class="math-container">$C$</span> exists and to estimate it.</p>
<p>I only know ... | Robert Israel | 8,508 | <p>Hint: <span class="math-container">$|e^{iz}| \le e^{|z|}$</span> so <span class="math-container">$|\sin(z)| \le e$</span> for <span class="math-container">$|z|=1$</span>.</p>
|
3,287,016 | <blockquote>
<p>Show <span class="math-container">$\lvert\sin z\rvert \leq C \lvert z\rvert$</span> <span class="math-container">$\forall \lvert z\rvert\leq 1$</span></p>
</blockquote>
<p>The question asks me to show the constant <span class="math-container">$C$</span> exists and to estimate it.</p>
<p>I only know ... | copper.hat | 27,978 | <p>Note that <span class="math-container">$\sin' = \cos$</span> and <span class="math-container">$|\cos z| \le \cosh (\operatorname{im} z) \le \cosh 1$</span> for <span class="math-container">$|z| \le 1$</span>. Hence <span class="math-container">$|\sin z| \le (\cosh 1) |z|$</span>.</p>
|
3,356,951 | <p>On SAT,scores range from 2000 to 2400, with two thirds of the scores falling in the range of 2200 to 2300. If we further assume that test scores are normally distributed in this range from 2000 to 2400, determine the mean and standard deviation.</p>
| Hassan A | 446,166 | <p>No matter how you fit a normal distribution to this data, it will always put positive probability outside of 2000 to 2400 range. So it is not possible to fix the range to a bounded set and assume that the underlying distribution is normal.
However, we can approximate a normal distribution assuming that the probabil... |
1,671,111 | <p>I'm looking for an elegant way to show that, among <em>non-negative</em> numbers,
$$
\max \{a_1 + b_1, \dots, a_n + b_n\} \leq \max \{a_1, \dots, a_n\} + \max \{b_1, \dots, b_n\}
$$</p>
<p>I can show that $\max \{a+b, c+d\} \leq \max \{a,c\} + \max \{b,d\}$ by exhaustively checking all possibilities of orderings am... | user247608 | 247,608 | <p>How about proof by induction?</p>
<p>You've proved the following by slogging it out.
max(a+b,c+d) <= max(a,c) + max(b,d)</p>
<p>To show the induction step, consider the case for three terms:</p>
<p>max(a+b,c+d,e+f) = max(max(a+b,c+d), e+f ) ( max is associative).</p>
<pre><code> = max( max(a,c... |
4,631,618 | <p>Consider this absolute value quadratic inequality</p>
<p><span class="math-container">$$ |x^2-4| < |x^2+2| $$</span></p>
<p>Right side is always positive for all real numbers,so the absolute value is not needed.</p>
<p>Now consider the cases for the left absolute value</p>
<ol>
<li><span class="math-container">$$... | Tuvasbien | 702,179 | <p><strong>Little introduction to hyperbolic geometry :</strong></p>
<p>Let <span class="math-container">$\operatorname{ch}(x)=\frac{e^x+e^{-x}}{2}$</span> and <span class="math-container">$\operatorname{sh}(x)=\frac{e^x-e^{-x}}{2}$</span>. Using the Taylor expansion of <span class="math-container">$\exp$</span>, namel... |
4,451,894 | <p><strong>Problem</strong><br />
There is a knight on an infinite chessboard. After moving one step, there are <span class="math-container">$8$</span> possible positions, and after moving two steps, there are <span class="math-container">$33$</span> possible positions. The possible position after moving n steps is <sp... | Community | -1 | <h2>An intuitive setup</h2>
<p>The growth of the number of knight moves for <span class="math-container">$n \ge 3$</span> can be modelled by an octagon increasing linearly in size, simulated below</p>
<p><a href="https://i.stack.imgur.com/TRVOA.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/TRVOA.gi... |
172,366 | <blockquote>
<p>What will be the value of $P(12)+P(-8)$ if $P(x)=x^{4}+ax^{3}+bx^{2}+cx+d$
provided that $P(1)=10$, $P(2)=20$, $P(3)=30$?</p>
</blockquote>
<p>I put these values and got three simultaneous equations in $a, b, c, d$. What is the smarter way to approach these problems?</p>
| Did | 6,179 | <p>Two remarks, to avoid almost every computation:</p>
<ul>
<li>The polynomial $P(x)-10x$ has roots $1$, $2$ and $3$, hence there exists a polynomial $Q$ such that $P(x)-10x=(x-1)(x-2)(x-3)Q(x)$. </li>
<li>The polynomial $P(x)-10x$ has degree $4$ and leading coefficient $1$, hence $Q(x)=x+z$ for some unknown constant ... |
172,366 | <blockquote>
<p>What will be the value of $P(12)+P(-8)$ if $P(x)=x^{4}+ax^{3}+bx^{2}+cx+d$
provided that $P(1)=10$, $P(2)=20$, $P(3)=30$?</p>
</blockquote>
<p>I put these values and got three simultaneous equations in $a, b, c, d$. What is the smarter way to approach these problems?</p>
| Yuki | 31,172 | <p>Other way doing this:</p>
<p>We try to find reals $e,f,g$ such that $P(12)+P(-8)=eP(1)+fP(2)+gP(3)$. So, if we try to equal "$x^k$ evaluated", we gain a system of equations:
$$
\left\{\begin{array}{ccc}
1^ke+2^kf+3^kg&=&12^k+(-8)^k
\end{array}\right.,\quad k=0,\cdots,4
$$
In particular,
$$
\left\{\begin{arr... |
3,454,095 | <p>Minimize <span class="math-container">$\;\;\displaystyle \frac{(x^2+1)(y^2+1)(z^2+1)}{ (x+y+z)^2}$</span>, if <span class="math-container">$x,y,z>0$</span>.
By setting gradient to zero I found <span class="math-container">$x=y=z=\frac{1}{\displaystyle\sqrt{2}}$</span>, which could minimize the function.</p>
<bl... | Xiaohai Zhang | 628,555 | <p>You first fix <span class="math-container">$y, z$</span> and let <span class="math-container">$x > 0$</span> vary. Taking derivative with respect to <span class="math-container">$x$</span>, dropping all those nonnegative terms such as <span class="math-container">$y^2+1$</span> to simplify notation, leads to
<spa... |
2,247,522 | <p>Suppose $X$ and $Y$ are discrete random variables. Show that $$E(X \mid Y)=E(X \mid Y^3).$$</p>
<p>The conditional expected value of a discrete random variable is expressed as
$$E(X \mid Y)=\sum xp_{X \mid Y}(x \mid y),$$
where$$p_{X \mid Y}(x \mid y)=\frac{p_{X,Y}(x,y)}{p_Y(y)}.$$ </p>
<p>Similarly, you can say t... | Jay Zha | 379,853 | <p>Provide a generic approach. Strictly speaking $\mathbb E[X|Y]$ is just a notation, what it really means is $\mathbb E[X|\sigma(Y)]$, where $\sigma(Y)$ is the $\sigma$-field generated by $Y$ (see its definition below).</p>
<p><em>Proof.</em> We need to show that $\sigma(Y)=\sigma(Y^3)$.</p>
<p>Let $f(x)=x^3$, $x \i... |
2,041,441 | <p>$\binom{74}{37}-2$ is divisible by :</p>
<p>a) $1369$</p>
<p>b) $38$</p>
<p>c) $36$ </p>
<p>d) $none$ $of$ $ these$</p>
<p>I have no idea how to solve this...I tried writing $\binom{74}{37}$ in some useful form but its not helping...any clues?? Thanks in advance!!</p>
| 2'5 9'2 | 11,123 | <p>The first line of the following is true by a combinatorial argument (among other arguments) where you count how many ways to choose $p$ marbles from a collection of $2p$ marbles, where half are red and half are blue.
$$\begin{align}
\binom{p+p}{p}
&=\sum_{k=0}^p\binom{p}{k}\binom{p}{p-k}\\
&=2+\sum_{k=1}^{p-... |
96,369 | <p>Is there a mathematical term which would include both liminf and limsup? (In a similar way we talk about extrema to describe both maxima and minima?)</p>
<p>The only thing I was able to find was that some authors use the name extreme limits; see google books: <a href="http://www.google.com/search?tbm=bks&tbo=1&... | Will Jagy | 10,400 | <p>I would go with cluster point, as the target <span class="math-container">$\mathbb R$</span> is a metric space. See <a href="https://www.planetmath.org/typesoflimitpoints" rel="nofollow noreferrer">this PlanetMath link</a>.</p>
|
3,998,098 | <p>I was asked to determine the locus of the equation
<span class="math-container">$$b^2-2x^2=2xy+y^2$$</span></p>
<p>This is my work:</p>
<blockquote>
<p>Add <span class="math-container">$x^2$</span> to both sides:
<span class="math-container">$$\begin{align}
b^2-x^2 &=2xy+y^2+x^2\\
b^2-x^2 &=\left(x+y\right... | Piquito | 219,998 | <p>HINT.-A way to study the curve is to consider it as a union of two graphs of functions (similar to how you would see two functions <span class="math-container">$y=\pm\sqrt{r^2-x^2}$</span> for a circle) so you get two explicits functions:
<span class="math-container">$$y=-x+\sqrt{b^2-x^2}\\y=-x-\sqrt{b^2-x^2}$$</spa... |
2,900,372 | <p>Let $R=\mathbb{Z}[\sqrt{-2}]$</p>
<blockquote>
<p>1) Is it true that for every free $R-$module $M$ of rank=$n$ that $M$ is a free $\mathbb{Z}-$module of rank=$2n$?</p>
<p>2) Find two non-isomorphic $R$-modules with $19$ elements each.</p>
</blockquote>
<p>For (1) I tried to begin with a basis $\{m_1,...,m_n... | Angina Seng | 436,618 | <p>For $2$ look for modules $R/I$ where $I$ is an ideal of $R$. As $R$
is a principal ideal domain, $I=\left<\alpha\right>$ for some $\alpha=a+b\sqrt{-2}$. Then $|R/\left<\alpha\right>|=|\alpha|^2=a^2+2b^2$. If we
have $\alpha=\eta\beta$ where $\eta$ is a unit in $R$. Then $\left<\alpha\right>
=\left&... |
1,284,388 | <p>Solving for variable $d$:</p>
<p>$v = \frac{1}{2}hd^2 + 9.9$</p>
<p>$-2(v - 9.9) = hd^2$</p>
<p>$-2v + 19.8 = hd^2$</p>
<p>$d = \sqrt{\frac{-2v + 19.8}{h}}$</p>
<p>The correct answer is:</p>
<p>$d = \pm\sqrt{\frac{2v - 19.8}{h}}$</p>
| user4640007 | 230,456 | <p>$$V = \frac{1}{2}hd^2 + 9.9$$</p>
<p>Subtract 9.9 to the LHS and multiply everything by 2</p>
<p>$$2(V-9.9) = hd^2$$</p>
<p>1) Divide by $h$ (likely the height in the problem)</p>
<p>2) Solve for $d$ by taking the square root of both sides</p>
|
963,503 | <p>Vectors $a$, $b$ and $c$ all have length one. $a + b + c = 0$. Show that
$$
|a-c| = |a-b| = |b-c|
$$
I am not sure how to get started, as writing out the norms didn't help and there is no way to manipulate
$$
|a-c| \le |a-b| + |b-c|
$$
to get an equality. I just need an idea of where to start.</p>
| Paul | 17,980 | <p><strong>Proof:</strong> If there is <span class="math-container">$x \in Z\setminus X$</span>, then</p>
<p><span class="math-container">$x\notin X\cap (Y \cup Z)$</span>, however, <span class="math-container">$x\in (X\cap Y) \cup Z$</span>. A tradiction!</p>
|
204,842 | <p>A probability measure defined on a sample space $\Omega$ has the following properties:</p>
<ol>
<li>For each $E \subset \Omega$, $0 \le P(E) \le 1$</li>
<li>$P(\Omega) = 1$</li>
<li>If $E_1$ and $E_2$ are disjoint subsets $P(E_1 \cup E_2) = P(E_1) + P(E_2)$</li>
</ol>
<p>The above definition defines a measure that... | hot_queen | 72,316 | <p>In this interesting <a href="http://www.math.wisc.edu/~akumar/INDUCED_IDEALS.pdf" rel="nofollow">note</a>, the author proves the following: It is consistent to have a finitely additive total extension of Lebesgue measure on $[0, 1]$ such that, although, the measure zero sets form a sigma ideal, there is no real valu... |
3,965,834 | <p>Does this sum converge or diverge?</p>
<p><span class="math-container">$$ \sum_{n=0}^{\infty}\frac{\sin(n)\cdot(n^2+3)}{2^n} $$</span></p>
<p>To solve this I would use <span class="math-container">$$ \sin(z) = \sum \limits_{n=0}^{\infty}(-1)^n\frac{z^{2n+1}}{(2n+1)!} $$</span></p>
<p>and make it to <span class="math... | Uri Toti | 834,026 | <p>Since you don't know how the <span class="math-container">$sin(x)$</span> as <span class="math-container">$x\to \infty$</span> behaves, you can use the fact that, if a series converges absolutely, then the original series converges. So you can see that</p>
<p><span class="math-container">$\sum^{\infty}_{n=0} \frac{s... |
2,825,522 | <p>I have this problem:</p>
<p><a href="https://i.stack.imgur.com/blD6N.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/blD6N.png" alt="enter image description here"></a></p>
<p>I have not managed to solve the exercise, but this is my breakthrough:</p>
<p><a href="https://i.stack.imgur.com/0dTdO.j... | soktinpk | 188,945 | <p>An easy, general way to solve these problems is to draw a circle around them. Then by symmetry, each of the 6 arcs has a measure of $360^\circ/6=60^\circ$. The angle intercepts two arcs on one side and on arc on the other, resulting in $\frac{120^\circ+60^\circ}{2}=90^\circ$.</p>
<p>In the circle below arc $DCB$ me... |
2,268,345 | <p>Find the value of $$S=\sum_{n=1}^{\infty}\left(\frac{2}{n}-\frac{4}{2n+1}\right)$$ </p>
<p>My Try:we have</p>
<p>$$S=2\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{2}{2n+1}\right)$$ </p>
<p>$$S=2\left(1-\frac{2}{3}+\frac{1}{2}-\frac{2}{5}+\frac{1}{3}-\frac{2}{7}+\cdots\right)$$ so</p>
<p>$$S=2\left(1+\frac{1}{2}-... | Στέλιος | 403,502 | <p>We can use the digamma function $\psi$. It is known that</p>
<p>$$\psi(z+1)=-\gamma+\sum_{n=1}^{\infty}\dfrac{z}{n(n+z)},\ \ \forall z \in \mathbb{C} \setminus \{-1,-2,...\}.$$</p>
<p>Thus for $z=\frac{1}{2}$ we obtain</p>
<p>$$\psi\left(\dfrac{3}{2}\right)=-\gamma+\sum_{n=1}^{\infty}\dfrac{1}{n(2n+1)}=-\gamma+\s... |
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