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 |
|---|---|---|---|---|
386,018 | <p>I've been working with Spivak's Differential Geometry exercises and I found myself confused with this one: "Let $C\subset \mathbb{R} \subset \mathbb{R}^2$ be the Cantor set. Show that $\mathbb{R}^2 - C$ is homeomorphic to the surface shown at the top of the next page."</p>
<p><img src="https://i.stack.imgur.com/rAR... | Matt E | 221 | <p>Remember that $C$ is obtained via the "middle thirds process": One takes the decreasing chain of closed sets $[0,1] = C_0 \supset C_1 \supset C_2 \supset \cdots,$ where $C_{n+1}$ is obtained from $C_n$ by removing the remaining "middle third" open intervals, and sets $C = \bigcap_{n=0}^{\infty} C_n$.</p>
<p>Thus $\... |
1,488,501 | <p>Let $\varphi:\mathbb{R}\backslash\{3\}\to \mathbb{R}$ a periodic function so that forall $x\in \mathbb{R}$ $$\varphi(x+4)=\frac{\varphi(x)-5}{\varphi(x)-3}$$ Find the period the $\varphi$.</p>
| Winther | 147,873 | <p>With $f(x) = \frac{x-5}{x-3}$ the functional equation can be written</p>
<p>$$\varphi(x+4) = f\circ \varphi(x)$$</p>
<p>A direct calculation shows that
$$f\circ f\circ f\circ f = \text{Id} \implies \varphi(x+16) = \varphi(x)$$
which implies that the period satisfy $T = \frac{16}{m}$ for some $m\in\mathbb{N}$. We c... |
1,765,530 | <p>How many $5$-digit numbers (including leading $0$'s) are there with no digit appearing exactly $2$ times? The solution is supposed to be derived using Inclusion-Exclusion.</p>
<p>Here is my attempt at a solution:</p>
<p>Let $A_0$= sequences where there are two $0$'s that appear in the sequence.</p>
<p>...</p>
<p... | André Nicolas | 6,312 | <p>The <em>structure</em> of the analysis is the same as yours. We count the <em>bad</em> strings, where some digit appears exactly twice. </p>
<p>We first count the strings where say the digit $0$ appears exactly twice. <strong>Where</strong> the $0$'s are can be chosen in $\binom{5}{2}$ ways. For each of these ways,... |
2,865,122 | <p><a href="http://math.sfsu.edu/beck/complex.html" rel="nofollow noreferrer">A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka</a> Exer 3.8</p>
<blockquote>
<p>Suppose <span class="math-container">$f$</span> is holomorphic in region <span class="math-container">$G$<... | BCLC | 140,308 | <p>After discovering the term anti-holomorphic, I have an answer similar to zhw.'s: It starts the same but then doesn't reinvent the wheel or something:</p>
<blockquote>
<p>If <span class="math-container">$f: G \to \mathbb C$</span> is holomorphic on <span class="math-container">$G$</span> and anti-holomorphic on <span... |
223,176 | <p>I made this problem: </p>
<p>$f(x)=e^{f^{\prime \prime}}$ </p>
<p>I have just been taught the first derivative, and was thinking about what if the derivative depended upon it own derivative. I understand that $e^x$ is its "own" derivative, but the problem I made I was thinking that the first derviative is not lo... | Henrique Tyrrell | 17,153 | <p>$\frac{d}{dx} [f(x)] = \frac{d}{dx}\exp (f''(x))$ and by the chain rule its derivative is
$\frac{d}{dx}f''(x) \times \frac{d}{d(f''(x))} \exp(f''(x)) = f'''(x) \times \exp(f''(x))$</p>
|
873,439 | <p>This is a follow-up <a href="https://math.stackexchange.com/questions/872921/prove-that-if-the-square-of-a-number-m-is-a-multiple-of-3-then-the-number-m/872927">question</a>.</p>
<p>The problem is:</p>
<blockquote>
<p>Given two natural numbers, $m$ and $n$, and $n \vert m^2$.</p>
<p>Find necessary and suffi... | hola | 154,508 | <p>I think I found the solution. Do correct me if I am wrong.</p>
<blockquote>
<p>$n|m$ if and only if there is no such prime $p$ such that $p^2|n$.</p>
</blockquote>
<p><code>If</code> part then it is easy, for example $4|14^2$ but $4\nmid 14$.</p>
<p>For the <code>only if</code> part, we can show a contradiction... |
2,994,296 | <p>I'm trying to figure out how to prove, that <span class="math-container">$$\lim_{n\to \infty} \frac{n^{4n}}{(4n)!} = 0$$</span>
The problem is, that <span class="math-container">$$\lim_{n\to \infty} \frac{n^{n}}{n!} = \infty$$</span>
and I have no idea how to prove the first limit equals <span class="math-container"... | Community | -1 | <p>The ratio of successive terms is</p>
<p><span class="math-container">$$\frac{t_{n+1}}{t_n}=\frac{\dfrac{(n+1)^{4n+4}}{(4n+4)!}}{\dfrac{n^{4n}}{(4n)!}}=\left(\left(\frac{n+1}n\right)^n\right)^4\frac{(n+1)^4}{(4n+1)(4n+2)(4n+3)(4n+4)}.$$</span></p>
<p>The first factor is known to describe an increasing sequence that... |
2,880,830 | <p><a href="https://i.stack.imgur.com/a4Wh2.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/a4Wh2.png" alt="enter image description here"></a></p>
<p>Not sure how to do this one. If <span class="math-container">$S$</span> is a field, then I was considering that <span class="math-container">$\exists ... | Bernard | 202,857 | <p><strong>Hints</strong>:</p>
<p>$\Rightarrow$: If $R$ is a field, let $s\in S$, and consider multiplication by $s$ in $S$. Check this is an injective $R$-linear map. What can you conclude, knowing $S$ is a finite dimensional $R$-vector space?</p>
<p>$\Leftarrow$: If $S$ is a field, consider $r\in R$; you know $r^{-... |
3,844,448 | <p>Find all values of <span class="math-container">$h$</span> such that rank(<span class="math-container">$A$</span>) = <span class="math-container">$2$</span>.</p>
<p><span class="math-container">$A$</span> = <span class="math-container">$\begin{bmatrix}
1 & h & -1\\
3 & -1 & 0\\
-4 & 1 & 3
\en... | Andrei | 331,661 | <p>You know that the first row is independent, but the second and third rows must be dependent. In other words, you can write <span class="math-container">$$-1-3h=C(1+4h)\\3=C(-1)$$</span>
Therefore <span class="math-container">$C=-3$</span>. Plug it into the first equation, to find <span class="math-container">$h$</sp... |
79,084 | <p>Let $X$ be a topological space (say a manifold). A result of R. Thom states that the pushforwards of fundamental classes of closed, smooth manifolds generate the rational homology of $X$. This work of Thom predates the development of bordism. Is there now a more elementary proof of this result that does not rely ... | Lost | 18,632 | <p>Thank you everyone for helping with this question. I would like to attempt to provide my own answer (which came to me <em>after</em> reading all the comments): Let <span class="math-container">$B_* (M)$</span> be rational, oriented bordism and <span class="math-container">$H_* (M)$</span> be the rational homology ... |
95,819 | <p>I have a set of parametric equations in spherical coordinates that supposedly form circle trajectories. See below:</p>
<pre><code>r=C1
theta=C2*Sin[beta]*Sin[phi[t]]
phi=(C2*Sin[beta]*(Cos[theta[t]]/Sin[theta[t]])*Cos[phi[t]])+(C2*Cos[beta])
</code></pre>
<p>C1 and C2 are constants and beta is some angle, say 15... | Jack LaVigne | 10,917 | <p>I am quite certain that I don't understand your function but may be able to help with the format for plotting with <code>MeshFunction</code>.</p>
<pre><code>C2 = 1.;
beta = 15. Degree;
StartingTheta = 45. Degree;
StartingPhi = 180. Degree;
tMin = 0;
tMax = 1000.;
</code></pre>
<p>Using these parameters</p>
<pre><... |
95,819 | <p>I have a set of parametric equations in spherical coordinates that supposedly form circle trajectories. See below:</p>
<pre><code>r=C1
theta=C2*Sin[beta]*Sin[phi[t]]
phi=(C2*Sin[beta]*(Cos[theta[t]]/Sin[theta[t]])*Cos[phi[t]])+(C2*Cos[beta])
</code></pre>
<p>C1 and C2 are constants and beta is some angle, say 15... | Maxwell | 34,479 | <p>Thanx for the answers I received concerning this question. @gpap - your answer was/is superb! Below is another take on it:</p>
<p>I used all the coding in the original question above, but this time I inserted the standard Cartesian-Spherical transformations where I had all the question marks. </p>
<pre><code>Traje... |
1,967,847 | <blockquote>
<p>A vector space $V$ is called <strong>finite-dimensional</strong> if there is a finite subset of $V$ that is a basis for $V$. If there is no such finite subset of $V$, then $V$ is called <strong>infinite-dimensional</strong>.</p>
<hr>
<p>We now establish some results about finite-dimensional ... | Junkai Dong | 366,631 | <p>I suppose you are mixing the terms "finite subset" and "finite number of subset".</p>
<p>The first one corresponds to the fact that in the subset there are only finitely many ${\it elements}$; the second one corresponds to the fact that there are finitely many ${\it subsets}$.</p>
<p>Hope I helped. I also mixed th... |
201,688 | <p>There is a right angle corner with width 1 in both directions. One wants to find the largest area shape which can pass through this corner.
I know that this is a famous problem, but what is it called?</p>
| Bob Spaghetti | 60,810 | <p><a href="http://en.wikipedia.org/wiki/Moving_sofa_problem">The Moving sofa problem</a>, I believe.</p>
|
201,688 | <p>There is a right angle corner with width 1 in both directions. One wants to find the largest area shape which can pass through this corner.
I know that this is a famous problem, but what is it called?</p>
| Joseph O'Rourke | 6,094 | <p>A supplement to Ian's answer: Here is the largest-area sofa known,
due to Gerver:
<hr />
<img src="https://i.stack.imgur.com/FBlms.png" alt="GerverSofa"></p>
<hr />
<blockquote>
<p>Gerver, Joseph L. (1992). "On Moving a Sofa Around a Corner". <em>Geometriae Dedicata</em> 42 (3): 267–283. (<a href="http://link.sp... |
2,963,886 | <p>What do these statements mean in discrete mathematics?</p>
<p><strong>Example 1:</strong> Let <span class="math-container">$P:\mathbb{Z}\times \mathbb{Z}\to \{T,F\}$</span>, where <span class="math-container">$P(x,y)$</span> denotes "<span class="math-container">$x+y=5$</span>".</p>
<p><strong>Example 2:</... | fleablood | 280,126 | <p>Example 1: is a very abstract and rather obscure but necessary way to say:</p>
<p>"<span class="math-container">$x + y =5$</span> is either true or false"</p>
<p>"Let <span class="math-container">$P...:$</span>" means let <span class="math-container">$P$</span> be a way of writing a statement.</p>
<p>"Let <span c... |
2,419,485 | <blockquote>
<p>In a certain family four girls take turns at washing dishes. Out of a total of four breakages, three were caused by the youngest girl, and she was thereafter called clumsy. Was she justified in attributing the frequency of her breakages to chance?</p>
</blockquote>
<p>I'm not sure how to solve the fo... | Graham Kemp | 135,106 | <ul>
<li>count ways to select a girl and three dishes, multiply by the probability that each selection might happen.</li>
</ul>
<p>The probability that three from four dishes were randomly broken by the same girl is $\tbinom 41\tbinom 43{(\tfrac 14)}^3{(\tfrac 34)}^1$, that is $\tfrac {12}{64}$ or $3/16$.</p>
<p>The ... |
479,594 | <p>I was wandering which is the best way to generate various combinations of $x_i$ such that $$\sum\limits_{i=1}^7 x_i = 1.0$$</p>
<p>where $ x_i \in \{0.0, 0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0\}$</p>
<p>I can generate these using brute-force, i.e checking through all $ 11^7$ combinations and only taking those whi... | robjohn | 13,854 | <p>Subtract the formula for $a_{n-1}$ from that for $a_n$
$$
a_n-a_{n-1}=\frac14-\frac12a_{n-1}
$$
Multiply by $2^n$ and bring $a_{n-1}$ from the left to the right
$$
2^na_n=2^{n-2}+2^{n-1}a_{n-1}
$$
Using the formula for the sum of a geometric series, we get
$$
2^na_n=2^{n-1}+C
$$
Plug in $n=1$ to find that $C=-\frac1... |
3,089,326 | <p>9 person randomly enter 3 different rooms. What is the probability that</p>
<p>a)the first room has 3 person?</p>
<p>b)every room has 3 person?</p>
<p>c)the first room has n person, second room has 3 persons, third room 2 persons?</p>
<p>What i want to know is that which probability techniques i need to use when... | William Elliot | 426,203 | <p><span class="math-container">$\lim_{x\to a} f(x) = \lim_{x\to 0} f(x + a)
= \lim_{x\to 0} (f(x) + f(a))
= \lim_{x\to 0} f(x) + \lim_{x\to 0} f(a)
= f(0) + f(a) = f(0 + a) = f(a)$</span></p>
|
1,012,344 | <p>My knowledge of $C^*$-algebras is very little.</p>
<p>We call an element positive if $a=b^*b$ for some $b$ and make a relation on all positive elements by saying
$$
b \geqslant a \iff b-a \text{ is positive}.
$$
I can't figure out why this gives us a poset.</p>
| Tomasz Kania | 17,929 | <p>Only anti-symmetry seems to be non-trivial here. Let us apply the spectral theorem. </p>
<p>Suppose that $b\leqslant a$ and $a\leqslant b$. Since $a-b$ is positive, $C^*(a-b)$ is commutative and of course $b-a\in C^*(a-b)$. But now you can think of $a-b$ and $b-a$ as continuous functions on $\sigma(a-b)$ which are ... |
3,493,519 | <p>Can I get a verification if this is the right way to approach this problem?</p>
<blockquote>
<p>Give an example of a linear map <span class="math-container">$T$</span> such that <span class="math-container">$\dim(\operatorname{null}T) = 3$</span> and <span class="math-container">$\dim(\operatorname{range}T) = 2$<... | Community | -1 | <p>Let's do an easy variation on your example, where <span class="math-container">$V=\Bbb R^5$</span> and <span class="math-container">$W=\Bbb R^2$</span>. </p>
<p>Define <span class="math-container">$T$</span> by <span class="math-container">$T(x_1,x_2, x_3, x_4, x_5)=(x_1,x_3)$</span>..</p>
<p>Then clearly <span ... |
3,493,519 | <p>Can I get a verification if this is the right way to approach this problem?</p>
<blockquote>
<p>Give an example of a linear map <span class="math-container">$T$</span> such that <span class="math-container">$\dim(\operatorname{null}T) = 3$</span> and <span class="math-container">$\dim(\operatorname{range}T) = 2$<... | tf3 | 805,267 | <p>I was also attempting to solve this question but your approach seemed abstract to me, and upon little reflection, I was able to come up with the following more concrete example which helped increase my understanding of the concept.</p>
<p>The key understanding I got, which prompted me to write this answer, is as fol... |
2,278,431 | <p>"Apply Green's Theorem to evaluate the line integral of F around positively oriented boundary"</p>
<p>$$F(x,y)=x^2yi+xyj$$</p>
<p>C: The region bounded by y=$x^2$ and y=4x+5</p>
| Robert Israel | 8,508 | <p>If $a(n) = n^n/(n! e^n)$, then</p>
<p>$$ \eqalign{\ln \left( \frac{a(n+1)}{a(n)} \right) &= n \log(1+1/n) - 1\cr
&= -\frac{1}{2n} + \frac{1}{3n^2} - \frac{1}{4n^3} + \ldots}$$
I claim this is negative. Indeed,
for $n \ge 1$, $$-\frac{1}{2m n^{2m}} + \frac{1}{(2m+1)n^{2m+1}} < 0 $$
Thus $a(n+1)/a(n) <... |
2,284,208 | <p>Hi I need help with the completion of this proof, I believe I am nearly at the end but I do not know how to end it</p>
<p>Proof: If v1 and v2 are a basis of v, then a<strong>v1</strong>+b<strong>v2</strong>=<strong>v</strong> for all v element of v.
Then if <strong>v1</strong>+<strong>v2</strong>, <strong>v1</stron... | Mark H | 81,870 | <p>You've shown that the new vectors have the same span as the originals (almost--you should find explicit expressions for $c$ and $d$). Now, you need to show that $\vec{v}_1 + \vec{v}_2$ and $\vec{v}_1-\vec{v}_2$ are linearly independent.</p>
|
2,284,208 | <p>Hi I need help with the completion of this proof, I believe I am nearly at the end but I do not know how to end it</p>
<p>Proof: If v1 and v2 are a basis of v, then a<strong>v1</strong>+b<strong>v2</strong>=<strong>v</strong> for all v element of v.
Then if <strong>v1</strong>+<strong>v2</strong>, <strong>v1</stron... | Jean Marie | 305,862 | <p>Here is another - more direct - proof: </p>
<p>Basis $\{v_1,v_2\}$ is sent onto system $\{v_1+v_2,v_1-v_2\}$ by a linear application $L$ with matrix </p>
<p>$$\begin{pmatrix}1&\ \ 1\\1&-1\end{pmatrix}$$</p>
<p>which is invertible. Thus $L$ is bijective ; therefore $\{v_1+v_2,v_1-v_2\}$ is a basis also.</... |
1,390,423 | <p>An acquaintance of mine proposed a scenario. Imagine parents who ground their child. Initially, the grounding is for 5 days, but for every day the child misbehaves while they're grounded, the parents will tack on an extra 2 days. The child is very predictable and has a 30% chance of misbehaving on any given day, ... | true blue anil | 22,388 | <p>Expected # of days misbehaved = $0.3\times5 = 1.5$</p>
<p>Expected # of additional days tacked on = $2\times1.5 = 3$</p>
<p>so it is a G.P. with <em>a</em> = 5, <em>r</em> = $\dfrac35$ =0.6</p>
<p>$S_\infty = \dfrac{a}{(1-r)} = \dfrac{5}{0.4} = 12.5$</p>
|
1,390,423 | <p>An acquaintance of mine proposed a scenario. Imagine parents who ground their child. Initially, the grounding is for 5 days, but for every day the child misbehaves while they're grounded, the parents will tack on an extra 2 days. The child is very predictable and has a 30% chance of misbehaving on any given day, ... | Community | -1 | <p>This answer comes in two parts:</p>
<ul>
<li>Show that the expectation is finite</li>
<li>Calculate the expectation.</li>
</ul>
<h2>Showing the expectation is finite</h2>
<p>Let's ignore the punishment for a moment and just consider the kids actions for all eternity. Each day, starting with day 1, the child is ei... |
3,408,846 | <p>This is an example in Serge Lang "Introduction to Linear Algebra", page 48. I try to multiply these two <span class="math-container">$2$</span>x<span class="math-container">$3$</span> and <span class="math-container">$3$</span>x<span class="math-container">$2$</span> matrices but fail to obtain the result as mention... | user | 505,767 | <p>We have</p>
<p><span class="math-container">$$\left( \begin{array}{cc}
\color{red}{2} & \color{red}{1} & \color{red}{5}\\
1 & 3 & 2
\end{array} \right)
%
\left( \begin{array}{c}
3 & \color{red}{4} \\
-1 & \color{red}{2} \\
2 & \color{red}{1}
\end{array} \right)
=\left( \begin{array}{c}
1... |
3,454,725 | <p>I am reading real analysis book and encountered this symbol <span class="math-container">$\wedge$</span> and <span class="math-container">$\vee.$</span></p>
<p>The author says following:</p>
<ol>
<li><span class="math-container">$f\wedge g=\frac{1}{2}(f+g-
|f-g|)$</span>,</li>
<li><span class="math-container">$f\v... | azif00 | 680,927 | <p>Define <span class="math-container">$I = \{1,\dots,|\phi_M|\}$</span> and put
<span class="math-container">$$A = \{\{x_i\}:\, i\in I\} \cup \{\{x_i,x_j\}:\, (i,j)\in I^2\} \cup \cdots \cup \{\{\{x_{i_1}\},\dots,\{x_{i_k}\}\} :\, (i_1,\dots,i_k)\in I^k, \, k=|\phi_M|-1\} \cup \phi_M$$</span></p>
|
324,594 | <p>You have three buckets, two big buckets holding <code>8 litres</code> of water each and one small empty bucket that can hold <code>3 litres</code> of water. How will you split the <code>16 litres</code> of water to <code>four people</code> evenly? Each person has a container but once water is distributed to someone ... | joriki | 6,622 | <pre><code>8 8 0 [0 0 0 0]
8 5 3 [0 0 0 0]
8 5 0 [3 0 0 0]
8 2 3 [3 0 0 0]
8 0 3 [3 2 0 0]
8 3 0 [3 2 0 0]
5 3 3 [3 2 0 0]
5 6 0 [3 2 0 0]
2 6 3 [3 2 0 0]
2 8 1 [3 2 0 0]
2 8 0 [3 2 1 0]
0 8 2 [3 2 1 0]
0 7 3 [3 2 1 0]
3 7 0 [3 2 1 0]
3 4 3 [3 2 1 0]
6 4 0 [3 2 1 0]
6 1 3 [3 2 1 0]
6 0 3 [3 2 1 1]
8 0 1 [3 2 1 1]
8 0 0... |
324,594 | <p>You have three buckets, two big buckets holding <code>8 litres</code> of water each and one small empty bucket that can hold <code>3 litres</code> of water. How will you split the <code>16 litres</code> of water to <code>four people</code> evenly? Each person has a container but once water is distributed to someone ... | Théophile | 26,091 | <p>Here is a solution using 24 steps (one fewer than joriki's solution). I believe this is minimal, since I wrote a shortest path algorithm to find it:</p>
<pre><code>8 8 0 [0 0 0 0]
8 5 3 [0 0 0 0]
8 5 0 [3 0 0 0]
8 2 3 [3 0 0 0]
8 0 3 [3 2 0 0]
8 3 0 [3 2 0 0]
5 3 3 [3 2 0 0]
5 6 0 [3 2 0 0]
2 6 3 [3 2 0 0]
2 8 1 [3... |
4,480,905 | <p>When <span class="math-container">$T$</span> is any linear operator acting on a vector space <span class="math-container">$V$</span>, and <span class="math-container">$n$</span> is a natural number, <span class="math-container">$T^n$</span> means <span class="math-container">$T$</span> applied <span class="math-cont... | Randall | 464,495 | <p>We define <span class="math-container">$T^0=I$</span> for <strong>any</strong> linear operator <span class="math-container">$T:V \to V$</span> so that the usual laws of exponents hold for composition. So, you should take this as a definition which is convenient and not anything extraordinarly deep. Here's why.</p>... |
3,836,542 | <p>In last lines in the image from Lawrece. C. Evans, Partial Differentail Equations, he states that in Hilbert space every bounded sequence contains a weakly convergent subsequence.</p>
<p>What is wrong in my counter example ?</p>
<p>Let <span class="math-container">$\mathcal H$</span> be an infinite dimensional Hilbe... | Will Jagy | 10,400 | <p>I get <span class="math-container">$$ \left(x^2 + n^2 \right)^2 - \left(2nx+1 \right)^2 = (x^2 - n^2)^2 - 4nx-1 $$</span></p>
<p>Once written as a difference of squares, note that <span class="math-container">$A^2 - B^2 = 0$</span> means <span class="math-container">$(A+B)(A-B) = 0,$</span> so that <span class="ma... |
3,836,542 | <p>In last lines in the image from Lawrece. C. Evans, Partial Differentail Equations, he states that in Hilbert space every bounded sequence contains a weakly convergent subsequence.</p>
<p>What is wrong in my counter example ?</p>
<p>Let <span class="math-container">$\mathcal H$</span> be an infinite dimensional Hilbe... | sirous | 346,566 | <p>Another approach:</p>
<p><span class="math-container">$$(x^2-2018^2)^2=8072 x+1$$</span></p>
<p><span class="math-container">$$(x-2018)^2(x+2018)^2=4\times 2018 x+1$$</span></p>
<p>A:</p>
<p><span class="math-container">$ \begin{cases}(x-2018)^2=4\times 2018 x+1 \\ (x+2018)^2=1\end {cases}$</span></p>
<p><span class... |
916,963 | <p>I am reading an introductory book on mathematical proofs and I don't seem to understand the mechanics of proof by contradiction. Consider the following example. </p>
<p>$\textbf{Theorem:}$ If $P \rightarrow Q$ and $R \rightarrow \neg Q$, then $P \rightarrow \neg R$.</p>
<p>$\textbf{Proof:}$ (by contradiction)
Ass... | Jean-Claude Arbaut | 43,608 | <p>To answer <em>It also could have been that our first assumption, namely, P, was false. Or both of them could be false</em>: yes, it could, but if you assume P, then R must be false, what you write $P\rightarrow\neg R$. You could have concluded $R\rightarrow \neg P$, of course (and it's the <a href="https://en.wikipe... |
14,508 | <p>Suppose that $f$ is a weight $k$ newform for $\Gamma_1(N)$ with attached $p$-adic Galois representation $\rho_f$. Denote by $\rho_{f,p}$ the restriction of $\rho_f$ to a decomposition group at $p$. When is $\rho_{f,p}$ semistable (as a representation of
$\mathrm{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$?</p>
<... | Kevin Buzzard | 1,384 | <p>The right way to do this sort of question is to apply Saito's local-global theorem, which says that the (semisimplification of the) Weil-Deligne representation built from $D_{pst}(\rho_{f,p})$ by forgetting the filtration is precisely the one attached to $\pi_p$, the representation of $GL_2(\mathbf{Q}_p)$ attached t... |
3,478,098 | <p><a href="https://i.stack.imgur.com/dcxhi.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/dcxhi.jpg" alt="enter image description here" /></a></p>
<p>Guys, Why is this weird statement true? It seems counterintuitive to me I cannot understand or lack creativity understanding it can you help me expla... | Kavi Rama Murthy | 142,385 | <p>If there is a rational number <span class="math-container">$x$</span> in this interval with <span class="math-container">$q \leq N$</span> then <span class="math-container">$x \in Q_N$</span> and the definition of <span class="math-container">$\delta$</span> shows that <span class="math-container">$\delta \leq |x-... |
1,575,764 | <p>$Q$ is a linear operator from $V \to V$ with $V$ being a finite dimensional complex inner-product-space. </p>
<p>Given: $Q^*=5Q$, $Q^*$ being the adjoint.</p>
<p>Show that $0$ is the only eigenvalue of $Q$. </p>
<p>I've been staring at this problem for quite a while now. I think the answer lies in the eigenvalues... | Justpassingby | 293,332 | <p>If you have one nonzero eigenvalue, then its multiples by all powers of $5$ must also be eigenvalues.</p>
|
131,283 | <p>I came across a question which required us to find $\displaystyle\sum_{n=3}^{\infty}\frac{1}{n^5-5n^3+4n}$. I simplified it to $\displaystyle\sum_{n=3}^{\infty}\frac{1}{(n-2)(n-1)n(n+1)(n+2)}$ which simplifies to $\displaystyle\sum_{n=3}^{\infty}\frac{(n-3)!}{(n+2)!}$. I thought it might have something to do with pa... | robjohn | 13,854 | <p>The <a href="http://en.wikipedia.org/wiki/Heaviside_cover-up_method">Heaviside Method</a> gives the partial fraction decomposition
$$
\begin{align}
&\frac{1}{(n-2)(n-1)n(n+1)(n+2)}\\[6pt]
&=\frac{1}{24(n-2)}-\frac{1}{6(n-1)}+\frac{1}{4n}-\frac{1}{6(n+1)}+\frac{1}{24(n+2)}\tag{1}
\end{align}
$$
Notice that
$$... |
3,185,226 | <p>I have a simple question that confuses me for a while:</p>
<blockquote>
<p><span class="math-container">$$f(X) = \text{tr} \left( [ \log(X) ]^2 \right)$$</span></p>
<p>where <span class="math-container">$X$</span> is an <span class="math-container">$m \times m$</span> symmetric positive definite (SPD) matrix and <sp... | Community | -1 | <p>Let <span class="math-container">$\phi:X\in U\mapsto \log(X)$</span>, where <span class="math-container">$U$</span> is the set of <span class="math-container">$n\times n$</span> complex matrices that have no eigenvalues in <span class="math-container">$\mathbb{R}^-=(-\infty,0]$</span> (we use the principal <span cla... |
4,281,654 | <p>My professor gave me an exercise where I had to show that the special linear group <span class="math-container">$SL(2,\mathbb{R})$</span> is a lie subgroup of <span class="math-container">$GL(2,\mathbb{R})$</span>. I was able to do this part. However, I was then asked to do the following:</p>
<p>All real <span class... | orangeskid | 168,051 | <p>Let's first solve the problem using Lagrange multiplier. We consider the system
<span class="math-container">$$\operatorname{grad} ((a^2 + b^2 + c^2 + d^2) - 2\lambda (a d - b c - 1)) = 0$$</span>
that is
<span class="math-container">$$ (a, b, c, d) = \lambda(d,-c,-b,a)$$</span>
Since <span class="math-container"... |
290,910 | <p>Which sequences of adjacent edges of a polyhedron could be considered to be a geodesic? The edges of a face most surely will not, but the "equator" of the octahedron eventually will. But for what reasons? How do the defining property of a geodesic - having zero geodesic curvature - apply to a sequence of edges?</p>
... | Gerry Myerson | 8,269 | <p>Some thoughts are given by Konrad Polthier and Markus Schmies, <a href="http://page.mi.fu-berlin.de/polthier/articles/straightest/straightest_preprint.pdf" rel="nofollow">Straightest geodesics on polyhedral surfaces</a>. Form the abstract: </p>
<p>Geodesic curves are the fundamental concept in geometry to generaliz... |
1,016,682 | <p>is my proof correct?</p>
<p>Definition:</p>
<p>Let $X\subset\mathbb R$ and let $x'\in\mathbb R$, we say that $x'$ is an adherent point of $X$ iff $\forall\epsilon>0\exists x\in X \text{ s.t. }d(x′,x)≤ε$. the closure of X is denoted as $\overline X$ and is defined to be the set of all the adherent points of $X$.... | 5xum | 112,884 | <p>From $z\notin \overline{X}$, you cannot conclude that there exists such an $x'$ that $|z-x'|\leq \epsilon.$</p>
<p>Since $z\in \overline X$ is equivalent to $$\forall \epsilon \exists x \in X: d(z, x)\leq \epsilon$$</p>
<p>the statement $z\notin \overline X$ is equivalent to $$\exists \epsilon:\forall x\in X: d(z,... |
1,016,682 | <p>is my proof correct?</p>
<p>Definition:</p>
<p>Let $X\subset\mathbb R$ and let $x'\in\mathbb R$, we say that $x'$ is an adherent point of $X$ iff $\forall\epsilon>0\exists x\in X \text{ s.t. }d(x′,x)≤ε$. the closure of X is denoted as $\overline X$ and is defined to be the set of all the adherent points of $X$.... | user141240 | 141,240 | <p>The closure of a set is by definition the intersection of all closed sets containing the set.
But any intersection of closed sets is closed, so the closure is closed, and closure of a closed set is itself.</p>
|
3,963,661 | <p>I am trying to find that <span class="math-container">$L ={\{w\text{ | } w ∈ {\{a, b\}} * \text{is not a palindrome}\}}$</span></p>
<p>This is related to <a href="https://math.stackexchange.com/questions/1034989/w-w-%E2%88%88-a-b-is-not-a-palindrome-prove-this-language-is-not-regular">this previous question</a>, tho... | Brian M. Scott | 12,042 | <p>There are several problems here. First, you can’t use just any <span class="math-container">$N$</span>: you need to specify that <span class="math-container">$N$</span> is at least as big as the pumping length. Next, the pumping lemma says that there is <strong>at least one</strong> partition <span class="math-conta... |
3,963,661 | <p>I am trying to find that <span class="math-container">$L ={\{w\text{ | } w ∈ {\{a, b\}} * \text{is not a palindrome}\}}$</span></p>
<p>This is related to <a href="https://math.stackexchange.com/questions/1034989/w-w-%E2%88%88-a-b-is-not-a-palindrome-prove-this-language-is-not-regular">this previous question</a>, tho... | Humberto Longo | 925,330 | <p>Consider the string <span class="math-container">$w = xyz = a^pba^{p+p!} \in L$</span>, where <span class="math-container">$p$</span> is the pumping length provided by the Pumping Lemma for regular languages. Since <span class="math-container">$|xy| \leqslant p$</span>, <span class="math-container">$z = a^{p-i-j}ba^... |
3,865,388 | <p>Let <span class="math-container">$A$</span> be the set of all <span class="math-container">$2\times2$</span> boolean matrices and <span class="math-container">$R$</span> be a relation defined on <span class="math-container">$A$</span> as <span class="math-container">$M \mathrel{R} N$</span> if and only if <span clas... | amrsa | 303,170 | <p>If you have two matrices <span class="math-container">$A=\{a_{ij}:0\leq i,j\leq1\}$</span> and <span class="math-container">$B=\{b_{ij}:0\leq i,j\leq1\}$</span>, then
<span class="math-container">$$A \wedge B = C = \{c_{ij}:0\leq i,j\leq1\}$$</span>
and
<span class="math-container">$$A\vee B = D =\{d_{ij}:0\leq i,j\... |
1,199,304 | <p>Let $M\neq \{0\}$ be a semi-simple left $R$ module .Prove that it contains a simple sub-module.</p>
<p>An $R-$ module $M$ is said to be semi-simple if every submodule of $M$ is a direct summand of M
<strong>My solution</strong></p>
<p>Since $M\neq \{0\}$; $\exists m\in M$ such that $m\neq 0$.Then I can consider th... | 1123581321 | 482,390 | <blockquote>
<p>If <span class="math-container">$M\not=0$</span> is a semisimple <span class="math-container">$R-$</span>module then there exists a simple <span class="math-container">$R-$</span>module <span class="math-container">$N\leq M$</span></p>
</blockquote>
<p>Let <span class="math-container">$x\in M-\{0\}$</sp... |
3,203,100 | <p>This equation just came to my mind, I tried solving it but can't find any solution to this problem. Can anyone please tell what is the process to approach this problem? </p>
| Dr. Sonnhard Graubner | 175,066 | <p>Considering the equation <span class="math-container">$$x=n^x$$</span> and taking the logarithm on both sides we get
<span class="math-container">$$\frac{\ln(x)}{x}=\ln(n)$$</span> so <span class="math-container">$$n=e^{\frac{\ln(x)}{x}}$$</span> now you can use calculus for <span class="math-container">$$e^{\frac{\... |
2,840,192 | <p>I have this problem and I was thinking to use the mean value theorem, but in the hypothesis i don't have the conditions (explicitly at least). Any ideas of how to start?</p>
<blockquote>
<p>Suppose that $\lim_{x\to a} f(x) = 0$, and that exist $\lim_{x\to a} \frac{1}{f'(x)}$. Show that exist an open interval with... | Paramanand Singh | 72,031 | <p>Since the limit of $1/f'$ exists the derivative $f'$ is non-zero in some deleted neighborhood of $a$ and therefore by intermediate value property it maintains a constant sign on each side of $a$ in this deleted neighborhood. Thus $f$ is strictly monotone on each side of $a$ in the deleted neighborhood. And since $f(... |
2,840,192 | <p>I have this problem and I was thinking to use the mean value theorem, but in the hypothesis i don't have the conditions (explicitly at least). Any ideas of how to start?</p>
<blockquote>
<p>Suppose that $\lim_{x\to a} f(x) = 0$, and that exist $\lim_{x\to a} \frac{1}{f'(x)}$. Show that exist an open interval with... | Torsten Schoeneberg | 96,384 | <p>Here's a proof idea which uses the MVT, or rather just its special case known as Rolle's Theorem.</p>
<p>Assume that for every $\epsilon >0$, there is $x\in (a, a+\epsilon)$ with $f(x)=0$.</p>
<p>Then for every $\epsilon$, there are actually infinitely many such $x$ (why?). Choose two of them, $x_{1,\epsilon}$ ... |
1,796,792 | <p>Is $\log_27$ a rational number?</p>
| Rob | 274,944 | <p>Suppose $\log_2 7 = {a\over b}$ for two positive integers $a$ and $b$. </p>
<p>$\log_2 7 = { \ln 7 \over \ln 2} = {a \over b}$</p>
<p>Cross multiply,</p>
<p>$b \ln 7 = a \ln 2 \implies \ln ( 7^b ) = \ln (2^a)$</p>
<p>Take the $e^{( \ \ )}$ of both sides, </p>
<p>$7^b = 2^a$</p>
<p>This is impossible for intege... |
376,796 | <p>This is more of a pedagogical question rather than a strictly mathematical one, but I would like to find good ways to visually depict the notion of curvature. It would be preferable to have pictures which have a reasonably simple mathematical formalization and even better if there is a related diagram that explains ... | Joseph O'Rourke | 6,094 | <p>With advances in discrete differential geometry, it is now nearly
routine to compute curvature on meshed surfaces. Here are two
of many possible color-coded examples.</p>
<hr />
<img src="https://i.stack.imgur.com/motmV.png" width="300" />
<img src="https://i.stack.imgur.com/xPIQj.png" ... |
376,796 | <p>This is more of a pedagogical question rather than a strictly mathematical one, but I would like to find good ways to visually depict the notion of curvature. It would be preferable to have pictures which have a reasonably simple mathematical formalization and even better if there is a related diagram that explains ... | Sebastian | 4,572 | <p>This is a very similar picture to that in the answer by Gabe, but concerning the sectional curvature of a Riemannian metric. Consider a point <span class="math-container">$p\in M$</span>, and a plane <span class="math-container">$V\subset T_pM.$</span> For small radius <span class="math-container">$r$</span> conside... |
4,338,382 | <blockquote>
<p>Let <span class="math-container">$a,b>0$</span>. Prove that: <span class="math-container">$$\frac{1}{a^2}+b^2\ge\sqrt{2\left(\frac{1}{a^2}+a^2\right)}(b-a+1)$$</span></p>
</blockquote>
<p>Anyone can help me get a nice solution for this tough question?
My approach works for 2 cases:</p>
<p>Case 1: <sp... | obscurans | 619,038 | <p>Alternate way, since who the knight is is important:</p>
<p><strong>Suppose A is a knight</strong>: then B is a spy, and C is a knave.</p>
<p><strong>Suppose C is a knight</strong>: then A is a knave, so B is <em>not</em> a spy, which is a contradiction.</p>
<p><strong>Suppose B is a knight</strong>: A is lying, but... |
3,501,332 | <blockquote>
<p>A cryptoanalist, while trying to decipher a message, found that the
most frequent blocks were RH and NI, which must correspond to TH and
HE, which are the most common in the english language. Supposing the
text was codified using a 2x2 block cipher, what was the used matrix?</p>
</blockquote>
<... | lonza leggiera | 632,373 | <p>The equation
<span class="math-container">$$
\begin{bmatrix}
a&b\\c&d
\end{bmatrix}
\begin{bmatrix}
19&7\\7&4
\end{bmatrix}
\equiv
\begin{bmatrix}
17&13\\7&8
\end{bmatrix}
\pmod{26}
$$</span>
gives you four equations in the four unknowns <span class="math-container">$\ a,b,c,d\ $</span>. Two... |
275,539 | <p>Kind of leading on from my other question, how would I solve for $i$? Or how would I check that it is possible to have such an $i$?</p>
<p>First I had to check for all $2^i$ and clearly this doesn't happen as all $2^i$ are even and so I will just get even $x's$ such that $2^i \equiv x \mod 28$. So the next one I go... | Matt E | 221 | <p>If you know the Chinese Remainder Theorem, then you see that you can check this separately mod $4$ and mod $7$. Already $3^2 \equiv 1 \bmod 4,$ and so you
are left to finding the smallest power of $3$ that is $1 \bmod 7$. By Euler it is a factor of $7- 1,$ i.e. of $6$, and one easily rules out $1,2,$ and $3$. Thu... |
1,368,988 | <p>I was thinking about different ways of finding $\pi$ and stumbled upon what I'm sure is a very old method: dividing a circle of radius $r$ up into $n$ isosceles triangles each with radial side length $r$ and central angle $$\theta=\frac{360^\circ}{n}$$ Use $s$ for the side opposite to $\theta$.</p>
<p><img src="htt... | Mark Viola | 218,419 | <p>Since there were already two solid answers that provided efficient approaches, I thought that it would be instructive to see a different way forward. </p>
<p>Here, we will use the expansion of the cosine as </p>
<p>$$\cos x=1-\frac12 x^2+O(x^4) \tag 1$$</p>
<p>Letting $x=\frac{2\pi}{n}$ in $(1)$ yields </p>
<... |
2,934,028 | <blockquote>
<p>A particle moves along the top of the
parabola <span class="math-container">$y^2 = 2x$</span> from left to right at a constant speed of 5 units
per second. Find the velocity of the particle as it moves through
the point <span class="math-container">$(2, 2)$</span>. </p>
</blockquote>
<p>So I is... | PabloG. | 597,199 | <p>Consider the curve in a parametric form <span class="math-container">$\vec{z}=(x(t),y(t))$</span>, in that case we know that </p>
<p><span class="math-container">$$
\vec{v}=\frac{\mathrm{d}\vec{z}}{\mathrm{d}t}=(x'(t),y'(t))
$$</span></p>
<p>Additionally, we know that the speed is <span class="math-container">$\|\... |
200,903 | <p>My teacher was explaining quadratics in my class and it was a little bit unclear to me. The problem was <br> <br>
Suppose $at^2 + 5t + 4 > 0$, show that $a > 25/16$ . <br> <br></p>
<p>My teacher said that there are no solutions for this function when it is greater than $0$ and used $b^2-4ac \lt 0$, and this ... | Community | -1 | <p>The question really means that $at^2+5t+4>0$ for all real values of $t$. This is only possible if $a>0$ and the quadratic curve does not intersect the horizontal axis. In this case there are no real solutions to the equation $at^2+5t+4=0$ so that the discriminant is negative.</p>
|
103,675 | <p>I have defined a recursive sequence</p>
<pre><code>a[0] := 1
a[n_] := Sqrt[3] + 1/2 a[n - 1]
</code></pre>
<p>because I want to calculate the <code>Limit</code> for this sequence when n tends towards infinity.</p>
<p>Unfortunately I get a <code>recursion exceeded</code> error when doing:</p>
<pre><code>Limit[a[n... | IPoiler | 30,913 | <p>You can attempt to convert your sequence in terms of a function which is not recursive, then take the <code>Limit</code> of that function</p>
<pre><code>a[0] = 1;
a[n_] := a[n] = Sqrt[3] + 1/2 a[n - 1]
seq30 = Table[a[ic], {ic, 0, 30}];
func = FindSequenceFunction[seq30, n]
Limit[func, n -> Infinity]
</code></... |
1,903,473 | <p>Given three permutations $p_1,p_2,p_3$ of $\{1,2,\ldots,n^3+1\}$, prove that two of them have a common subsequence of length $n+1$.</p>
<p>I have tried to solve this using the pigenhole principle but I didnt progress too much, any help would be appreciated</p>
<p>edit: when I say subsequence I mean that there are ... | Kaligule | 182,303 | <p>I don't think that statement is true.</p>
<p>I am not shure what you think a permutation of a set is, so lets just assume we talk about sequences.
Take $n=2$, so $n^3+1 = 9$, so our sequence is $l=(1,2,3,4,5,6,7,8,9)$. Now I have these three permutations:</p>
<ul>
<li>$p_1(l) = (1,2,3,4,5,6,7,8,9) = l$</li>
<li>$p... |
4,495,950 | <blockquote>
<p>Why does <span class="math-container">$-\frac{1}{17-x}$</span> equal <span class="math-container">$\frac{1}{x-17}$</span>?</p>
</blockquote>
<p>Is there any simple computation to make this seem a little bit more intuitive? Right now, I cannot wrap my head around the fact that I can just switch signs of ... | G Tony Jacobs | 92,129 | <p>You can break this down into a couple of basic facts. First of all, we have <span class="math-container">$\frac{1}{-1}=-1$</span>. Thus, if you want to see the <em>opposite</em>, or negative, of a fraction, we can multiply either the numerator or denominator by <span class="math-container">$-1$</span>.</p>
<p>Second... |
2,512,294 | <p>So I've been given the following problem:</p>
<p>How many positive integers are there that can not be written as a sum of 5's and 7's? For example, 4 is one of those integers, but 19 is not because 19 = 5 + 7 + 7. How to solve this? </p>
| Piquito | 219,998 | <p>It is easy enough to conclude that a such positive integer $n$ must be of the form $$n=5a+7b+c$$ where $a,b$ are non-negative integers and $c$ are positive integers with $c\lt 35$, coprime with $35$ and such that $c$ is not solution of $5a+7b=c$. </p>
<p>There are $\phi(35)=(5-1)(7-1)=24$ positive integers coprimes... |
1,307,085 | <p>How does one solve this equation?</p>
<blockquote>
<p>$$\cos {x}+\sin {x}-1=0$$</p>
</blockquote>
<p>I have no idea how to start it.</p>
<p>Can anyone give me some hints? Is there an identity for $\cos{x}+\sin{x}$?</p>
<p>Thanks in advance!</p>
| Community | -1 | <p>Rewrite as:</p>
<p>$$\cos(x)+\sin(x)=1$$
Then square:
$$\cos^2(x)+2\cos(x)\sin(x)+\sin^2(x)=1$$
Note an important identity:
$$1+2\cos(x)\sin(x)=1$$
Then simplify and note another identity:
$$\sin(2x)=0$$</p>
<p>Can you take it from here?</p>
|
1,307,085 | <p>How does one solve this equation?</p>
<blockquote>
<p>$$\cos {x}+\sin {x}-1=0$$</p>
</blockquote>
<p>I have no idea how to start it.</p>
<p>Can anyone give me some hints? Is there an identity for $\cos{x}+\sin{x}$?</p>
<p>Thanks in advance!</p>
| Bhaskara-III | 246,676 | <p>starting from, $$\cos x+\sin x-1=0$$ $$(\sin x+\cos x)^2=1^2$$ completing square, i get $$\sin^2 x+\cos^2 x+2\sin \cos x=1$$ $$1+\sin 2x=1$$$$\sin 2x=0$$
$$2x=k\pi$$
$$x=\frac{k\pi}{2}$$ where, $k=0, \pm 1, \pm2, \ldots$</p>
|
1,044,507 | <p>Sample:
$$∀x ∈ R+,∃y ∈ R+, x < y ⇒ x > y$$</p>
<p>Say I tried <code>y = 5</code>. Do I need to check if the consequent is true for just the x values less than 5?</p>
<p>Secondly, Since there is no value y that makes the antecedent true, is this statement true since there are no counter examples? The implicat... | hmakholm left over Monica | 14,366 | <p>Note that $\forall x$ is <em>before</em> $\exists y$. This means that the $y$ you use is allowed to depend on $x$ -- you don't have to select a single $y$ that must work with all possible $x$.</p>
<p>What the formula says is that <em>once</em> someone chooses an $x$, <em>then</em> it is possible to find a particula... |
713,521 | <p>There are so many notations for differentiation. Some of them are:
$$
f^\prime(x) \qquad \frac{d}{dx}(f(x))\qquad \frac{dy}{dx}\qquad \frac{df}{dx}\qquad
D f(x)\qquad y^\prime\qquad D_x f(x)
$$
Why are there so many ways to say "the derivative of $f(x)$"? Is there a specific use for each notation? What is the differ... | Guy | 127,574 | <p>$f'(x)$ is equivalent to $\frac{d}{dx}(f(x))$. The difference is that in the first you aren't making explicit that you are differentiating <em>with respect to</em> $x$, while in the second that distinction is made clear. Although when we write $f'(x)$ is usually implied that the differentiation is with respect to $x... |
846,108 | <p>How do you solve this equation: $2x+8=6x-12$ by using the guess and check method?</p>
<p>I divide $2x+8$ and I get $4$ then I divide $6x-12$ and I get $-2$ but I don't know what to do next or is it wrong?</p>
| afedder | 29,604 | <p>The answer is $x = 5$. You can get this solution using normal analytical methods (algebraic manipulation), i.e.,
$$\begin{align} 2x + 8 = 6x - 12 &\iff -4x + 8 = -12 \\&\iff -4x = -20 \\&\iff x = 5 \,\,. \end{align}$$
In terms of a "guess and check method", here's my strategy: factor $2$ out of the LHS a... |
1,354,490 | <p>Prove this function is injective $f(x)=x+\mod(x,7)$.</p>
<p><strong>Attempt:</strong></p>
<p>I tried separating in two cases: $x \equiv y \pmod 7$ and $x \not \equiv y \pmod 7 $:</p>
<p>First case:
$$f(x)=f(y) \iff x+\mod(x,7)=y+ \mod (y,7)\implies x= y
$$</p>
<p>But I couldn't prove the second case.</p>
| Batominovski | 72,152 | <p>Note: $-7<\text{mod}(x,7)-\text{mod}(y,7)<7$ and $x-y\equiv \text{mod}(x,7)-\text{mod}(y,7)\pmod{7}$.</p>
<p>If $f(x)=f(y)$, then $-(x-y)=\text{mod}(x,7)-\text{mod}(y,7)$. What happens then?</p>
<p><strong>P.S.</strong> </p>
<p>(1) It's "injective," and not "inyective."</p>
<p>(2) We can use any other odd... |
1,354,490 | <p>Prove this function is injective $f(x)=x+\mod(x,7)$.</p>
<p><strong>Attempt:</strong></p>
<p>I tried separating in two cases: $x \equiv y \pmod 7$ and $x \not \equiv y \pmod 7 $:</p>
<p>First case:
$$f(x)=f(y) \iff x+\mod(x,7)=y+ \mod (y,7)\implies x= y
$$</p>
<p>But I couldn't prove the second case.</p>
| Asinomás | 33,907 | <p>Take two different numbers $7a+b$ and $7c+d$ with $0\leq b,d<7$. notice $f(7a+b)=7a+2b$ and $f(7c+d)=7c+2d$. The difference is therefore $7(a-b)+2(d-c)$, if this number was zero then $2(d-c)$ would have to be a multiple of $7$, the only possibility is $0$, of course this means $b=d$, but if $b=d$ then $f(7a+b)=7a... |
3,490,329 | <blockquote>
<p>Show that a 2-dimensional subspace of the space of <span class="math-container">$2\times2$</span> matrices contains a non-zero symmetric matrix. </p>
</blockquote>
<p>I don't know if it should be written like the addition of two symmetric and skew-symmetric matrix or there is another way to show it. ... | JMoravitz | 179,297 | <p>Note that the space of <span class="math-container">$2\times 2$</span> matrices is <span class="math-container">$4$</span>-dimensional.</p>
<p>Note further that the subspace of symmetric <span class="math-container">$2\times 2$</span> matrices is <span class="math-container">$3$</span>-dimensional</p>
<hr>
<p>Now... |
4,109,827 | <p><span class="math-container">$$f(x,y)=\begin{cases}\dfrac{y^3}{x^2+y^2} &(x,y) \neq \ \mathbb{(0,0)}\\ 0 & (x,y)=(0,0) \\ \end{cases}$$</span></p>
<p>Evaluate <span class="math-container">$f_x(0,0)$</span> and <span class="math-container">$f_y(0,0)$</span> and <span class="math-container">$D_\overrightarrow... | M. Strochyk | 40,362 | <p>Using polar coordinates
<span class="math-container">$$\begin{gather}x = \rho\cos{\varphi}, \\ y= \rho\sin{\varphi}.\end{gather}$$</span>
we have
<span class="math-container">$${\frac{f(x,\,y) - f(0,\,0)}{\sqrt{x^2+y^2}}} ={\frac{f(\rho\cos{\varphi},\,\rho\sin{\varphi}) - 0}{\rho}} = {\frac{\rho^3\sin^3{\varphi}}{\... |
1,829,030 | <p>The limit isn't too bad using l'hospital's rule, but I was wondering if there was a way to do it without l'hospital's. </p>
<p>Looking around the section limits without lhopital's, it seems usually evaluating without requires some clever factoring, while here the $\arctan$ seems to muck things up. </p>
<p>Here is ... | Pierpaolo Vivo | 302,446 | <p>Rewrite as
$$
\lim_{x\rightarrow 0}\exp\left[\frac{2}{x}\ln\left(1+\arctan(\frac{x}{2})\right)\right]
$$
and then use $\arctan(x/2)\sim x/2$ for $x\to 0$, and $\ln (1+x/2)\sim x/2$ for $x\to 0$. Therefore the limit is $=\exp[(2/x)(x/2)]=\mathrm{e}$.</p>
|
1,829,030 | <p>The limit isn't too bad using l'hospital's rule, but I was wondering if there was a way to do it without l'hospital's. </p>
<p>Looking around the section limits without lhopital's, it seems usually evaluating without requires some clever factoring, while here the $\arctan$ seems to muck things up. </p>
<p>Here is ... | haqnatural | 247,767 | <p>$$\lim _{ x\rightarrow 0 } \left( 1+\arctan \left( \frac { x }{ 2 } \right) \right) ^{ \frac { 2 }{ x } }=\lim _{ x\rightarrow 0 }{ \left[ \left( 1+\arctan \left( \frac { x }{ 2 } \right) \right) ^{ \frac { 1 }{ \arctan \left( \frac { x }{ 2 } \right) } } \right] } ^{ \frac { 2 }{ x } \arctan \left( \fr... |
1,829,030 | <p>The limit isn't too bad using l'hospital's rule, but I was wondering if there was a way to do it without l'hospital's. </p>
<p>Looking around the section limits without lhopital's, it seems usually evaluating without requires some clever factoring, while here the $\arctan$ seems to muck things up. </p>
<p>Here is ... | Claude Leibovici | 82,404 | <p>From Taylor series, you can get more than just the limit. Consider $$y=\left(1+\arctan\left(\frac{x}{2}\right)\right)^{\frac{2}{x}}$$ and, as usual for this kind of problem, take logarithms $$\log(y)=\frac{2}{x}\,\log\left(1+\arctan(\frac{x}{2}) \right)$$ Now, use the classical $$\arctan(y)=y-\frac{y^3}{3}+O\left(y^... |
2,135,191 | <p>In the sequence $a_{1}, a_{2}, a_{3}, ..., a_{100}$, the $k$th term is defined by $$a_{k} = \frac{1}{k} - \frac{1}{k+1}$$ for all integers $k$ from $1$ through $100$. What is the sum of $100$ terms of this sequence? </p>
<p>The answer given is $\frac{100}{101}$, but I am not sure how.</p>
<p>So far I am have plugg... | Community | -1 | <p>Such a sum is called <strong>telescop sum</strong>. We have
$$\sum_{k=1}^{100} \left( \frac{1}{k}- \frac{1}{k+1} \right) \stackrel{\star}{=}1 - \frac{1}{101}.$$
In $(\star)$ we split the sum into two and make an index shift.</p>
|
2,849,643 | <p>Consider the following recurrence problem:
\begin{align}
d_{i-1} &= 2\varphi_{i+1}+4\varphi_i + 8d_i-7d_{i+1} - F \left( \delta_{i,N} + \delta_{i,N+1} \right) \, , \\
\varphi_{i-1} &= -7\varphi_{i+1}-16\varphi_{i} + 24 \left( d_{i+1}-d_{i} \right) + F \left( \delta_{i,N} + \delta_{i,N+1} \right) \, ,
\end{a... | Empy2 | 81,790 | <p>Except for i=N and i=N+1, you can write the recursion for a 4-vector
$$\vec{v}_i=(d_i,d_{i-1},\phi_i,\phi_{i-1})^t$$
and a 4x4 matrix $A$. Write $d_i$ and $\phi_i$ in terms of the eigenvalues and eigenvectors of $A$, then deal with the boundary conditions at $0,N$ and $N+1$</p>
|
2,849,643 | <p>Consider the following recurrence problem:
\begin{align}
d_{i-1} &= 2\varphi_{i+1}+4\varphi_i + 8d_i-7d_{i+1} - F \left( \delta_{i,N} + \delta_{i,N+1} \right) \, , \\
\varphi_{i-1} &= -7\varphi_{i+1}-16\varphi_{i} + 24 \left( d_{i+1}-d_{i} \right) + F \left( \delta_{i,N} + \delta_{i,N+1} \right) \, ,
\end{a... | Staufenberg | 140,351 | <p>Thanks again to Yuri for the helpful insights. I can say that I have now found the solution to my problem.</p>
<p>First of all, let us define $D_i = d_i-d_{i-1}$, in the same way as Yuri did, and let us get a recurrence equation that involves $\varphi_i$ only. </p>
<p>Actually, original problem (which is equivalen... |
92,382 | <p>I was working on a little problem and came up with a nice little equality which I am not sure if it is well-known (or) easy to prove (It might end up to be a very trivial one!). I am curious about other ways to prove the equality and hence I thought I would ask here to see if anybody knows any or can think of any. I... | Srivatsan | 13,425 | <p>I'll let you decide if it's trivial :-): </p>
<p>$$
\begin{align*}
\sum_{n=0}^{\infty} \left \lfloor \frac{g_n}{g_{n+1}} \right \rfloor g_{n+1}^2
&= \sum_{n=0}^{\infty} \left( \left \lfloor \frac{g_n}{g_{n+1}} \right \rfloor g_{n+1} \right) \cdot g_{n+1}
\\&= \sum_{n=0}^{\infty} \left( g_{n} - g_{n+2} \... |
3,373,528 | <p>Supposing <span class="math-container">$b > 0$</span> and <span class="math-container">$a < b$</span>, how could I prove:</p>
<p><span class="math-container">$$ \frac{a}{b} < \frac{a+1}{b+1} $$</span></p>
| Bman72 | 119,527 | <p>It's equivalent to
<span class="math-container">$$ab + {\color{red}a} < ab +{\color{red}b}$$</span></p>
|
3,373,528 | <p>Supposing <span class="math-container">$b > 0$</span> and <span class="math-container">$a < b$</span>, how could I prove:</p>
<p><span class="math-container">$$ \frac{a}{b} < \frac{a+1}{b+1} $$</span></p>
| Mike | 544,150 | <p>Hint: </p>
<p><span class="math-container">$\dfrac{a}{b} = \dfrac{a(b+1)}{b(b+1)} = \dfrac{ab+a}{b(b+1)}$</span>. However, <span class="math-container">$\dfrac{a+1}{b+1} = \dfrac{(a+1)(b)}{(b)(b+1)} = \dfrac{ab+b}{b(b+1)}$</span>. </p>
<p>If <span class="math-container">$a<b$</span> then what about <span class=... |
708,604 | <blockquote>
<p>Let $f: \mathbb{R^4} \to \mathbb{R}$ be a linear transformation defined by $f(a,b,c,d)=a+b+c+d$. Find a basis for the $Im(f)$.</p>
</blockquote>
<p>So, $Im(f)=\{f(a,b,c,d) \in \mathbb{R}: (a,b,c,d) \in \mathbb{R^4} \}$.</p>
<p>Then $Im(f)=\{a+b+c+d \in \mathbb{R}: a,b,c,d \in \mathbb{R} \}=\mathbb{R... | Martin Argerami | 22,857 | <p>It is simpler than that. $\mathbb R$ is a one-dimensional vector space. So any nonzero "vector" will be a basis. For example, $\{1\}$ is a basis for the image of $f$. So are $\{-2\}$, $\{\pi/7\}$ and $\{102.35\}$.</p>
|
129,293 | <p>I'm writing a survey that involves Levy processes and wanted to mention the different forms of the Levy-Khintchine formula found in literature.</p>
<p>The most common version seems to give the Levy symbol as</p>
<p>$$\Psi(u) = i\langle b,u \rangle - \frac{1}{2} \langle u,\Sigma u\rangle + \int_{\mathbb{R}^d} {(} e... | Tarasenya | 27,003 | <p>I consider this book: V. V. Petrov, Sums of independent random variables might be useful.</p>
|
4,177,639 | <p>I have an object with known coordinates in in 3D but on the ground (<code>z=0</code>). The object has a direction vector. My goal is to move this object on the ground (so <code>z</code> stays <code>0</code>) using its direction vector and via randomly-generated velocity vectors with one condition: I want to ensure t... | Community | -1 | <p>In general, there is no analytical solution.</p>
<p>If <span class="math-container">$$b=a^{m/n}$$</span> where <span class="math-container">$m,n$</span> are naturals, the equation can be written as</p>
<p><span class="math-container">$$z^n+z^m=1$$</span> where <span class="math-container">$z:=a^{x/m}$</span>. There ... |
510,080 | <p>This is not too obvious to me - what is the size of alternating group?</p>
<p>Following the hint in the comment, should it be $A_n = S_n/2$?</p>
<p>So I don't feel right up to here.....</p>
| Michael Joyce | 17,673 | <p>The key is to use the existence of the sign homomorphism $\text{sgn} : S_n \rightarrow \{ \pm 1 \}$. By definition $A_n$ is the kernel of $\text{sgn}$. Since $\text{sgn}$ is surjective, it follows immediately that $[S_n : A_n ] = 2$, so $|A_n| = |S_n| / 2 = n! / 2$.</p>
<p>Edit: As noted by Jared, one must assume... |
3,371,922 | <p>The definition of the limit states that limit of <span class="math-container">$f(x)$</span> when <span class="math-container">$x$</span> approaches <span class="math-container">$c$</span> is <span class="math-container">$L$</span> iff for every <span class="math-container">$\epsilon > 0$</span> there exists <spa... | user | 505,767 | <p>Indeed by the definition of limit </p>
<p><span class="math-container">$$\lim_{x\rightarrow c} f(x) = L \iff (\forall \varepsilon >0\, \exists \delta > 0: \forall x\in D\quad \color{red}{0<\vert x-c\vert <\delta} \implies \vert f(x)-L\vert <\varepsilon $$</span></p>
<p>it suffices that <span class="... |
2,578,444 | <blockquote>
<p><span class="math-container">$\tan x> -\sqrt 3$</span></p>
</blockquote>
<p>How do I solve this inequality?</p>
<p>From the <a href="https://www.desmos.com/calculator/qb8bg1vbsf" rel="nofollow noreferrer">graph</a> it is evident that <span class="math-container">$\tan x>-\sqrt 3$</span> for <span ... | ℋolo | 471,959 | <p>First a side note: when you are working with graphs like this you should always work with the closest points to $0$, it will make your life easier</p>
<hr>
<p>So $\tan x>-\sqrt 3$, to find the values of $x$ first let's find where those 2 are equal, you will get $-\frac{\pi}3$(this is the closest point to $0$, o... |
87,437 | <p>Let <span class="math-container">$R$</span> be a rectangular region of the integer lattice <span class="math-container">$\mathbb{Z}^2$</span>,
each of whose unit squares is labeled with a number
in <span class="math-container">$\lbrace 1, 2, 3, 4, 5, 6 \rbrace$</span>.
Say that such a labeled <span class="math-conta... | domotorp | 955 | <p>UPDATE: I played around and came up with a construction (chance of containing a mistake is high!), below it I leave my original answer for explanation.</p>
<p>$\begin{array}{ccccccccccccccccccccccccc}
3&-&2& &2&-&1&-&5&-&6& &5&-&1&-&2&-&6&-... |
4,278,505 | <p>I would like to clear up a confusion which might be trivial. In a proof the author proved <span class="math-container">$T = T'$</span> as following:</p>
<p>The author showed if <span class="math-container">$x \in T$</span> then <span class="math-container">$x \in T'$</span>, the next line is -</p>
<blockquote>
<p>.... | Michael | 179,940 | <p>When we prove all the elements of <span class="math-container">$T$</span> is also elements of <span class="math-container">$T'$</span>, it has two meanings:</p>
<ol>
<li><p><span class="math-container">$T=T'$</span> (note it does not contradict <span class="math-container">$T\subset T'$</span>),</p>
</li>
<li><p><sp... |
1,185,108 | <p>empty set is an subset of any sets maybe any collection of sets.</p>
<p>I wonder what about the case of the empty set being a member,not subset, of any collection (family) of sets.</p>
| Gal Porat | 221,699 | <p>This not true in general. For example, the empty set is a collection of sets that does not contain the empty set (because it does not contain any members).</p>
|
661,771 | <p>I am stuck on the following problem that says : </p>
<blockquote>
<p>Which of the following is a solution to the differential equation $y'=|y|^{\frac12},y(0)=0\,$ where square root means the
positive square root ? </p>
<ol>
<li><p>$y(t)=\frac{t^2}{4}$ </p></li>
<li><p>$y(t)=-\frac{t^2}{4}$ </p></li>... | Community | -1 | <p>Your proof consists of some correct steps done in the wrong order, which makes it something other than a valid proof. It looks more like scratchwork done in preparation for a proof. I rewrite it below, with some of the more important additions in bold. I will also change <span class="math-container">$t$</span> to ... |
787,926 | <p>I need some help to solve this integral:</p>
<p>$$\int_0^1 dy\int_0^{1-y} \cos \left(\frac{x-y}{x+y} \right) \mathrm dx$$</p>
<p>Thank you.</p>
| Julián Aguirre | 4,791 | <p>Let $u_n\ne1$ be the $n$-th 5-smooth number. At least one of
$$
\frac32\,u_n, \frac43\,u_n, \frac65\,u_n
$$
is a 5-smooth number greater than $u_n$. Let $U_n$ be the smallest of them. Then
$$
u_n<u_{n+1}\le U_n.
$$
Check the numbers $u_n+1,u_n+2,\dots$ until you find a 5-smooth number. The algorithm can be made f... |
427,564 | <p>I'm supposed to give a 30 minutes math lecture tomorrow at my 3-grade daughter's class. Can you give me some ideas of mathemathical puzzles, riddles, facts etc. that would interest kids at this age?</p>
<p>I'll go first - Gauss' formula for the sum of an arithmetic sequence.</p>
| john | 79,781 | <p>The <a href="http://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg">Seven Bridges of Königsberg</a> is a nice one. From memory, third-grade is around the time when kids like to try those problems of "can you draw a house without raising your pen and only drawing each line once", etc. which is essentially wha... |
427,564 | <p>I'm supposed to give a 30 minutes math lecture tomorrow at my 3-grade daughter's class. Can you give me some ideas of mathemathical puzzles, riddles, facts etc. that would interest kids at this age?</p>
<p>I'll go first - Gauss' formula for the sum of an arithmetic sequence.</p>
| Little Endian | 67,892 | <p>When I was in second grade, we did an experiment where we tossed a pin n times and counted the number of times the pin fell across some line, given that it fell between two other lines. Years later I learned to compute this - but the idea was there. My vote is for doing something related to probability and statistic... |
427,564 | <p>I'm supposed to give a 30 minutes math lecture tomorrow at my 3-grade daughter's class. Can you give me some ideas of mathemathical puzzles, riddles, facts etc. that would interest kids at this age?</p>
<p>I'll go first - Gauss' formula for the sum of an arithmetic sequence.</p>
| Jacob Raccuia | 67,366 | <p>I don't know if it's too advanced but...</p>
<p>I'm 21 years old and I just recently learned about 'using your hands to find the multiples of 9'.</p>
<p>For those who don't know what I'm talking about, hold your two hands out in front of you.</p>
<p>Say we want to find 9 x 8. Starting from your left hand pinky,... |
427,564 | <p>I'm supposed to give a 30 minutes math lecture tomorrow at my 3-grade daughter's class. Can you give me some ideas of mathemathical puzzles, riddles, facts etc. that would interest kids at this age?</p>
<p>I'll go first - Gauss' formula for the sum of an arithmetic sequence.</p>
| Brian Rushton | 51,970 | <p>Knot theory can be fun for third graders; the trefoil knit can be formed with one person and a stick, and some knits can be formed with 2 people, so you could show a picture of a knot, and have them try and make it.</p>
<p>Or better yet, a lot of people play a game where you have a lot of people randomly grab hands... |
1,722,995 | <blockquote>
<blockquote>
<p>Question: Given the circle $x^2+y^2=25$ is inscribed in triangle $\triangle ABC$, where vertex $B$ lies on the first quadrant. Slope of $AB$ is $\sqrt 3$ and has a positive y-coordinate, and $|AB|=|AC|$. Find the equations for $AC$ and $BC$</p>
</blockquote>
</blockquote>
<p>I foun... | amd | 265,466 | <p>I found it fairly straightforward to describe these lines with vector equations. </p>
<p>The line $AB$ is given as having a slope of $\sqrt3$, so it can immediately be parametrized as $\vec P_{AB}=\vec A+t\,(1,\sqrt3)$. </p>
<p>The line $AC$ is the reflection of this line in the line $AO$. A parametrization for ... |
2,469,798 | <p>Let $S = \left\{x \in \mathbb{Q} \mid 1 \leqslant {x}^2 \leqslant 29 \right\}$</p>
<p>What can we say about the supremum and infimum of this set? Would it be non-existent?</p>
<p>Would it be correct to say the following?</p>
<p>Suppose $ \sup S < \sqrt{29} $ then $ \exists x \in S $ such that $ x > \sup S$... | fleablood | 280,126 | <p>The supremum, or least upper bound, of a set need not be elements of the set (and for open sets they <em>never</em> are).</p>
<p>Example. Let $T = (0,1)$. Then $\sup T = 1 \not \in S$. That's it.</p>
<p>So in your set, $\sup S = \sqrt{29} \not \in S$. That's it.</p>
<p>....</p>
<p>Okay, but listen up, this i... |
3,041,907 | <p>I am unable to isolate the variable <span class="math-container">$x$</span> of this inequality <span class="math-container">$y \leq \sqrt{2x-x^2}$</span> ( where <span class="math-container">$0 \leq y \leq 1 $</span>)</p>
<p>Is it correct doing this: <span class="math-container">$y^2 \leq 2x-x^2$</span>?
I found ... | Bernard | 202,857 | <p>First, you have to determine the domain of validity of this inequation:
<span class="math-container">$$x^2-2x\ge 0\iff x(2-x)\ge 0\iff x\in[0,2].$$</span>
Next</p>
<ul>
<li>if <span class="math-container">$y\le 0$</span>, the inequation is satisfied for all <span class="math-container">$x\in[0,2]$</span>,</li>
<li... |
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