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 |
|---|---|---|---|---|
1,864,604 | <p>What's the difference between $f(x)=f(a-x)$ and $f(x)=f(x-a)$ ?</p>
<p>It's a pretty simple question maybe, but I'm unable to understand this one. </p>
| JonesY | 265,330 | <p>The sign "$-$" here refers to a binary operation, lets call it $O(x,y)=x-y$. Writing it like this shows the difference between $O(x,y)$ and $O(y,x)$</p>
<p>So the difference is $O(x,y)\neq O(y,x)$ since a binary operation does not have to be commutative, the result could be different. </p>
<p>There are a lot of ex... |
117,500 | <p>How would you go about finding the conjugacy classes of the nonabelian group of order 21, $G:=\left\langle x,y | x^7=e=y^3, y^{-1}xy=x^2\right\rangle$?</p>
| Mariano Suárez-Álvarez | 274 | <p>The group has a normal Sylow $7$-subgroup, generated by $x$, and it is clear from the way $y$ acts on $x$ that the conjugacy relation is generated by $x^i\sim x^{2i}$: this gives two conjugacy classes of elements of order $7$.</p>
<p>Using the Sylow theorems for $p=3$ and the fact that a Sylow $3$-subgroup cannot b... |
1,041,731 | <p>I want to prove that if $A$ in an infinite set, then the cartesian product of $A$ with 2 (the set whose only elements are 0 and 1) is equipotent to $A$.</p>
<p>I'm allowed to use Zorn's Lemma, but I can't use anything about cardinal numbers or cardinal arithmetic (since we haven't sotten to that topic in the course... | Git Gud | 55,235 | <p>The sum of the entries on each row is always $1$, so $\left(1,\begin{bmatrix} 1\\1\\1\end{bmatrix}\right)$ is an eigenpair. </p>
<p>Since $\det(A)=2$ (easily seen by Laplace expansion on the first row) and $\text{tr}(A)=4$, the other 'two' eigenvalues follow easily.</p>
<hr>
<p>Actually answering your question, s... |
2,798,206 | <p>How do you prove this using the epsilon-delta definition? I'm unsure of using the min = { } function.</p>
<p>$\lim \limits_{x \to \infty}\frac{2x+1}{1-x}$</p>
<p>These are my steps: </p>
<p>$ |f(x) - L| < \epsilon => |\frac{2x+1}{1-x} +2|< \epsilon $</p>
<p>$ \qquad \qquad \; \; \; \; \; =>|\frac{3}{... | Theo Bendit | 248,286 | <p>A few pointers:</p>
<ul>
<li>The $\varepsilon$-$\delta$ definition of a limit pertains to when you have a limit as $x \to a$, where $a$ is a finite quantity. For a limit as $x \to \infty$, you need an $\varepsilon$-$M$ definition, along the lines of, for all $\varepsilon > 0$, there exists some $M$ such that
$$x... |
1,083,277 | <p>$a,b,c \in \mathbb{R}$ and $a+b+c=0$.
Prove that: $8^{a}+8^{b}+8^{c}\geqslant 2^{a}+2^{b}+2^{c}$</p>
<p>I think that $2^{a}.2^{b}.2^{c}=1$, but i don't know what to do next</p>
| Redundant Aunt | 109,899 | <p>As it was said, you can take $x=2^a$, $y=2^b$ and $z=2^c$ to obtain:
$$
x^3+y^3+z^3\ge x+y+z
$$
With $x,y,z>0$ and $xyz=1$. It is equivalent to:
$$
x^2(x-1)+y^2(y-1)+z^2(z-1)\ge 0 \iff \frac{x^2(x-1)+y^2(y-1)+z^2(z-1)}{3}\ge 0
$$
Since $x^2$ and $x-1$ ar equally ordered, we might apply Chebychevs inequality to ob... |
134,205 | <blockquote>
<p>Find the expectation of a Geometric distribution using $\mathbb{E}(X)= \sum_{k=1}^\infty P(X \ge k)$. </p>
</blockquote>
<p>Okay I know how to find the expectation using the definition of the geometric distribution $$P(X=k)= p \cdot(1-p)^{k-1}$$ and I figured that $P(X \ge k)=(1-p)^{k-1}$ but I don't... | thepiercingarrow | 274,737 | <p>A simpler way would be to plug in $q=1-p$ and solve it that way using formula for geometric sequences:</p>
<p>\begin{align*}
E(X) &= \sum\limits_{k=1}^\infty kpq^{k-1}\\
&= \frac{p}{q} \sum\limits_{k=1}^\infty kq^{k}\\
&= \frac{p}{q} \frac{q}{(1-q)^2}\\
&= \frac{p}{q} \frac{q}{p^2}\\
&= \frac{p}... |
650,395 | <p>I am given generating functions $f(x)= \frac{x}{1-x}$ or $f(x)=\frac{1}{1+x^{2}}$ or $f(x)=\frac{1}{x^2-5x+6}$ and I am obliged to write sequence which are generated by this functions. What is the fastest algorithm to solve these problems? I have problem with even starting. I will be glad if anyone would be so nice ... | Alex | 38,873 | <p>Hint: $\frac{1}{1-x}=(1-x)^{-1}=\sum_{k=0}^{\infty} 1 \cdot x^k$, so the sequence this GF generates is $<1,1, \ldots>$</p>
<p>EDIT: 'in general' you need to represent the GF in the form $\sum_{k=0}^{\infty} a_k x^k$, and $\{ a_k \}$ is the sequence that this GF generates. </p>
|
481,952 | <p>Why is a union of infinitely many bounded sets not necessarily bounded, please? In addition, what condition can we add to make this union bounded, please?</p>
| Amitesh Datta | 10,467 | <p>$\mathbb{R}^n=\bigcup_{n=1}^{\infty} \{x\in\mathbb{R}^n:\left\|x\right|\leq n\}$</p>
<p>A finite union of bounded sets is bounded. </p>
<p>I hope this helps!</p>
|
3,203,607 | <p>"Each cell of a 100 × 100 table is painted either black or white and all
the cells adjacent to the border of the table are black. It is known that in every
2 × 2 square there are cells of both colours. Prove that in the table there is 2 × 2
square that is coloured in the chessboard manner."</p>
<p><a href="https://... | Chathura Gunasekera | 567,103 | <p>Step 1:First reduce the problem to a simpler,but proportional version of the problem
Let's consider a 10 x 10 board where all the border lining boxes are black.
now imagine painting inward as shown in the following figure.
<a href="https://i.stack.imgur.com/QVCRf.jpg" rel="nofollow noreferrer">Click to see image</a... |
1,079,493 | <blockquote>
<p>Prove that <span class="math-container">$f(x) = x^3 + 3x - 1$</span> is irreducible in <span class="math-container">$\mathbb Q[X]$</span>.<br />
Let <span class="math-container">$\theta$</span> be a root of <span class="math-container">$f(x)$</span>. Compute <span class="math-container">$\frac{1}{\thet... | Peter Woolfitt | 145,826 | <p>Let $$g(x)=f(x+1)=x^3+3x^2+3x+1+3x+3-1=x^3+3x^2+6x+3$$</p>
<p>Now we can apply <a href="http://en.wikipedia.org/wiki/Eisenstein%27s_criterion" rel="noreferrer">Eisenstein's Criterion</a> (with $p=3$) to find that $g(x)$ is irreducible in $\mathbb{Q}[x]$, so $f(x)$ is also irreducible in $\mathbb{Q}[x]$.</p>
|
1,079,493 | <blockquote>
<p>Prove that <span class="math-container">$f(x) = x^3 + 3x - 1$</span> is irreducible in <span class="math-container">$\mathbb Q[X]$</span>.<br />
Let <span class="math-container">$\theta$</span> be a root of <span class="math-container">$f(x)$</span>. Compute <span class="math-container">$\frac{1}{\thet... | Mr.Fry | 68,477 | <p>For polynomials of deg $\leq 3$ it is enough to check that there are no rational roots (i.e a linear factor). Applying the Rational Roots Theorem the set of possible roots are $\pm 1$, which clearly $g(\pm 1) \not = 0$.</p>
|
3,403,255 | <p>I am trying to follow wikipedia's page about matrix rotation and having a hard time understanding where the formula comes from.</p>
<p><a href="https://en.wikipedia.org/wiki/Rotation_matrix" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Rotation_matrix</a> Wiki page about it.</p>
<p>what i have so far:</... | DanLewis3264 | 480,329 | <p>Think: where are the unit vectors <span class="math-container">$(1,0)$</span> and <span class="math-container">$(1,1)$</span> sent to under a rotation by angle <span class="math-container">$\theta$</span>? It should not take too long to convince yourself that the answers are <span class="math-container">$(\cos \thet... |
2,360,268 | <p>Draw a triangle given $A-B=90$(degree) and length of $AC,BC$.</p>
<p><strong>My attempt</strong>:I thought It would be a good idea to draw a right angle so I made the picture below:</p>
<p><a href="https://i.stack.imgur.com/yikB7.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/yikB7.png" alt="en... | g.kov | 122,782 | <p>From
\begin{align}
\frac{b}{\sin\beta}
&=
\frac{a}{\sin(90^\circ+\beta)}
=
\frac{a}{\cos\beta}
\end{align}</p>
<p>we have </p>
<p>\begin{align}
\tan\beta&=\frac{b}{a}
,\quad
\sin\beta=\frac{b}{\sqrt{a^2+b^2}}
,\\
2\,R&=\frac{b}{\sin\beta}=\sqrt{a^2+b^2}
.
\end{align}</p>
<p>$\triangle ABC$ and $\tria... |
3,397,548 | <p>For a sequence <span class="math-container">$\{x_n\}_{n=1}^{\infty}$</span>, define <span class="math-container">$$\Delta x_n:=x_{n+1}-x_n,~\Delta^2 x_n:=\Delta x_{n+1}-\Delta x_n,~(n=1,2,\ldots)$$</span> which are named <strong>1-order</strong> and <strong>2-order difference</strong>, respectively. </p>
<p>The pro... | Daniel Fischer | 83,702 | <p>Suppose that <span class="math-container">$\Delta x_n \not\to 0$</span>. Then there is - without loss of generality, replace <span class="math-container">$x_n$</span> with <span class="math-container">$-x_n$</span> if necessary - a <span class="math-container">$c > 0$</span> such that <span class="math-container"... |
3,811,753 | <p>Show that the equation:</p>
<p><span class="math-container">$$
y’ = \frac{2-xy^3}{3x^2y^2}
$$</span></p>
<p>Has an integration factor that depends on <span class="math-container">$x$</span> And solve it that way.</p>
<hr />
<p>Already we got to:</p>
<p><span class="math-container">$$
y’ + \frac{xy^3}{3x^2y^2} = \fra... | Peanut | 144,455 | <p>You can manipulate the expression and obtain <span class="math-container">$(xy^3)' = 2/x$</span></p>
|
230,504 | <p>Again, this question is related (**) to a <a href="https://mathoverflow.net/questions/101700/large-cardinals-without-the-ambient-set-theory?rq=1">previous one</a>:</p>
<p>in standard books on basic set theory, after stating the axioms of ZFC, ordinal numbers are introduced early on. Afterwards cardinals appear: t... | Thomas Benjamin | 20,597 | <p>You might consider taking a look at a paper by Athanassios Tzouvaras titled "Cardinality without enumeration" (look under title on the Web), especially at Definition 3.1. Since it is short, I will quote it verbatim:</p>
<p>"Definition 3.1 Let $M$ be a model of $ZF$. A <em>notion of cardinality</em> for $M$ is a ... |
231,187 | <p>I'm wondering if there's an efficient way of checking to see if two context free grammars are equivalent, besides working out "test cases" by hand (ie, just trying to see if both grammars can generate the same things, and only the same things, by trial and error).</p>
<p>Thanks!</p>
| Micah | 30,836 | <p>There is not. In fact, there isn't even an inefficient way!</p>
<p>That is, the problem of determining whether two given CFGs represent the same language is undecidable. In fact, an even stronger statement is true: the problem of determining whether a given CFG accepts all strings on its alphabet is undecidable.</p... |
487,123 | <p>How to evaluate the following limit?
$$\lim_{n\to\infty}\dfrac{1!+2!+\cdots+n!}{n!}$$</p>
<p>For this problem I have two methods. But I'd like to know if there are better methods.</p>
<p><strong>My solution 1:</strong></p>
<p>Using Stolz-Cesaro Theorem, we have
$$\lim_{n\to\infty}\dfrac{1!+2!+\cdots+n!}{n!}=\lim_... | Chris | 164,598 | <p>Perhaps you might like the following argument:</p>
<p>Notice that <span class="math-container">$$\frac{\sum_{k=1}^{n} i!}{n!} = 1 + \frac{1}{n}\frac{\sum_{k=1}^{n-1} i!}{n-1!};$$</span>
from this, we get the recurrence</p>
<p><span class="math-container">$$\frac{\sum_{k=1}^{2} i!}{2!} = 1 + \frac{1}{2},$$</span>
<sp... |
1,302,932 | <p>Given $$A=\begin{pmatrix}
2 & 0 & 0\\
a & 2& 0\\
a+3 & a &-1
\end{pmatrix}$$<br>
For which values of $a$ can $A$ be diagonal?<br>
I found that $p_A(x)=(x-2)^2(x+1)$ and tried to find the eigen subspace of 2, to see if the geomtric multiplicity of the eigenvalue $2$ is $2$.<br>
I got a set... | Anurag A | 68,092 | <p>Assuming you got the correct set of equations, from the first equation we get $x$ can be anything, the second equation gives, $y$ can be anything but $ax=0$. From here either we have $a=0$ or $x=0$. </p>
<p>If $a=0$, then the third equation gives $z=0$. But since $x$ and $y$ have no restrictions, therefore you can ... |
1,302,932 | <p>Given $$A=\begin{pmatrix}
2 & 0 & 0\\
a & 2& 0\\
a+3 & a &-1
\end{pmatrix}$$<br>
For which values of $a$ can $A$ be diagonal?<br>
I found that $p_A(x)=(x-2)^2(x+1)$ and tried to find the eigen subspace of 2, to see if the geomtric multiplicity of the eigenvalue $2$ is $2$.<br>
I got a set... | MathNewbie | 24,672 | <p>You'll get for $a \neq 0$, that the nullspace of $A-2I$ is less than $2$. To see why, you have that </p>
<p>\begin{align*} A - 2I = \left(\begin{matrix} 0 & 0 & 0 \\ a & 0 & 0 \\ a+3 & a & - 3 \end{matrix}\right) \end{align*}<br>
is row equivalent to </p>
<p>\begin{align*}\left(\begin{matr... |
3,350,021 | <blockquote>
<p>We have the following quadratic equation:</p>
<p><span class="math-container">$2x^2-\sqrt{3}x-1=0$</span> with roots <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span>.</p>
<p>I have to find <span class="math-container">$x_1^2+x_2^2$</span> and <span clas... | Saketh Malyala | 250,220 | <p>Well, you know that, by the Quadratic Formula, <span class="math-container">$\displaystyle x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$</span>, so the difference between the two roots is <span class="math-container">$\frac{1}{a}\sqrt{b^2-4ac}=\frac{1}{a}\sqrt{3+4(2)(1)}=\frac{1}{2}\sqrt{11}$</span></p>
|
3,350,021 | <blockquote>
<p>We have the following quadratic equation:</p>
<p><span class="math-container">$2x^2-\sqrt{3}x-1=0$</span> with roots <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span>.</p>
<p>I have to find <span class="math-container">$x_1^2+x_2^2$</span> and <span clas... | MPW | 113,214 | <p><strong>Hint:</strong> What is <span class="math-container">$\tfrac{-b+\sqrt{b^2-4ac}}{2a} - \frac{-b-\sqrt{b^2-4ac}}{2a}$</span> ?</p>
|
1,560,411 | <p>If $B_1$ and $B_2$ are the bases of two integer lattices $L_1$ and $L_2$, i.e.</p>
<p>$L_1=\{B_1n:n\in\mathbb Z^d\}$ and $L_2=\{B_2n:n\in\mathbb Z^d\}$,</p>
<p>is there an easy way to determine a basis for $L_1\cap L_2$? Answers of the form "Plug the matrices into a computer and ask for Hermite Normal Form, etc" a... | Steven Stadnicki | 785 | <p>A set of lecture notes up at <a href="http://cseweb.ucsd.edu/classes/wi10/cse206a/lec2.pdf" rel="nofollow">http://cseweb.ucsd.edu/classes/wi10/cse206a/lec2.pdf</a> suggests computing the dual bases $D_1$ and $D_2$ of your lattices, then getting the HNF/orthogonalization of the concatenated matrix $[D_1\mid D_2]$ and... |
227,109 | <p>I keep mixing them up, because they are very similar.</p>
<p>Some contrapositives resemble some contradictions.</p>
| Robert Mastragostino | 28,869 | <p>A contrapositive has truth value equivalent to the original statement:</p>
<p>$$\text{It is raining}\implies\text{I have an umbrella}$$
has a contrapositive (and is equivalent to)
$$\text{I do not have an umbrella}\implies\text{it is not raining}$$</p>
<p>Proving the contrapositive is equivalent to proving the ori... |
892,114 | <p>i have three number
1 2 3 which will always be in this order {123}, i want to find out number of cases can be made,
like {1},{2},{23},{13},{12},{123}{3},{}. but each number has two states like "a" "b", i.e, each one will become different entity,like 2a,2b,3a,3b,1a,
with only exception i.e. 1 will have only one stat... | Maman | 167,819 | <p>$x^{2}-4x-8$=$(x-2)^{2}-4-8$=$(x-2)^{2}-12$</p>
|
1,474,867 | <p>I was trying to prove </p>
<p>$$\left|\int_{0}^{a}{\frac{1-\cos{x}}{x^2}}dx-\frac{\pi}{2}\right|\leq \frac{3}{a}$$ or $\leq \frac{2}{a}$. My work: I would like to use Fubini's theorem to prove it. </p>
<p>I notice that $\frac{1}{x^2}=\int^{\infty}_{0}{ue^{-xu}}du$. </p>
<p>Then, I got $\int_{0}^{a}{\frac{1-\cos{x... | Jack D'Aurizio | 44,121 | <p>Since we have:
$$ \frac{\pi}{2}-\int_{0}^{a}\frac{1-\cos x}{x^2}\,dx = \int_{a}^{+\infty}\frac{1-\cos x}{x^2}\,dx \tag{1}$$
it trivially follows that:
$$\left|\frac{\pi}{2}-\int_{0}^{a}\frac{1-\cos x}{x^2}\,dx\right|\leq \int_{a}^{+\infty}\frac{2}{x^2}\,dx = \frac{2}{a}.\tag{2}$$
If we use integration by parts, from... |
1,108,832 | <p>Q: A team of $11$ is to be chosen out of $15$ cricketers of whom $5$ are bowlers and $2$ others are wicket keepers. In how many ways can this be done so that the team contains at least $4$ bowlers and at least $1$ wicket keeper?</p>
| DeepSea | 101,504 | <p><strong>Hint:</strong> Bowlers: $5$</p>
<p>Keepers: $2$</p>
<p>Others: $8$</p>
<p>$A = A_{41} + A_{51} + A_{42} + A_{52}$</p>
<p>$A_{41} = \binom{5}{4}\binom{2}{1}\binom{8}{6}$</p>
<p>$A_{51} = \binom{5}{5}\binom{2}{1}\binom{8}{5}$, etc..</p>
|
2,510,322 | <p>$\left( f(x) \right ) =\min_{t<x}\left(t^2\right)$</p>
<p>How do I sketch this function for all real x? I don't get what minimum means in this context how do I sketch such a function when t is in the function but x isn't the square term? </p>
| Siong Thye Goh | 306,553 | <p>Guide:</p>
<p>Suppose $f(x) = \min_{t\le x} t^2$.</p>
<p>Let's illustrate how to evaluate this function value at $-1$ and $1$.</p>
<p>$f(-1) =\min_{t \leq {-1}}t^2$, since $t^2$ is a decreasing function for non-positive value, the minimimum value occur at $-1$, hence $$f(1) =\min_{t \leq {-1}}t^2=(-1)^2=1$$</p>
... |
439,302 | <p>@HansEngler Left the following response to <a href="https://math.stackexchange.com/questions/260656/cant-argue-with-success-looking-for-bad-math-that-gets-away-with-it">this question</a> regarding "bad math" that works,</p>
<blockquote>
<p>Here's another classical freshman calculus example: </p>
<p><strong>F... | nbubis | 28,743 | <p>Look at $y = f(u(x),v(x))$:
$$\frac{dy}{dx} = \frac{\partial f}{\partial u}\cdot\frac{du}{dx}+\frac{\partial f}{\partial v}\cdot\frac{dv}{dx}$$
Now, note that $x^{\sin x}$ can be written as:
$$y=x^{\sin x} = f(x,\sin x), \ f(u,v) = u^v$$
So that:
$$\frac{dy}{dx} = \frac{\partial f}{\partial u}\cdot\frac{du}{dx}+\fra... |
596,374 | <p>I solved this , but I am not sure if I did in the right way.</p>
<p>$$2^{2x + 1} - 2^{x + 2} + 8 = 0$$</p>
<p>$$2^{x + 2} - 2^{2x + 2} = 8$$</p>
<p>$$\log_22^{x + 2} - \log_22^{2x + 2} = \log_28$$</p>
<p>$$x + 2- 2x - 2 = 3$$</p>
<p>solving for $x$:</p>
<p>$$x = -2$$</p>
<p>any feedback would be appreciated.<... | abkds | 112,225 | <p>You can do it without including logarithms . Just take $t = 2^x$. It will become a quadratic in $t$ solve for $t$ . If $t$ has any negative value , neglect it beacuse $t> 0$ . </p>
<p>$t^2-2t+4 = 0 $
which has complex roots and hence there is no solution for $x$ in the real domain .</p>
|
3,850,422 | <p>For a few days now I've been trying to find a closed form expression for the determinant of the following <span class="math-container">$n\times n$</span> tridiagonal matrix</p>
<p><span class="math-container">$$\begin{pmatrix}c_1+b_1+a_1 & b_1 & 0 & \ddots & 0 \\ c_2 & c_2+b_2+a_2 & b_2 &... | Servaes | 30,382 | <p>Your matrix is a general tridiagonal matrix, with <span class="math-container">$d_i:=a_i+b_i+c_i$</span> along the diagonal. If we denote the determinant of the <span class="math-container">$n\times n$</span>-matrix by <span class="math-container">$f_n$</span>, then we have the recurrence relation
<span class="math-... |
2,637,983 | <p>I was working on a program to carry out some computations, and ran into an issue of needing to compare some algebraic numbers, but not having enough precision to do it without exact arithmetic, and not knowing how to do it with exact arithmetic.</p>
<p>A little algebra shows that the statement
$$a+b\sqrt{n}>0$$
... | Dap | 467,147 | <p>Yes.</p>
<p>For the sledgehammer approach, note that the set of $(c_0,\dots,c_{\deg(\alpha)-1})$ such that $\sum_{n=0}^{\deg(\alpha)-1} c_n\alpha^n>0$ can be defined in the language of real-closed fields, so by the Tarski-Seidenberg theorem can be expressed by polynomial inequalities.</p>
<p>This is sort of cir... |
1,138,789 | <p><a href="http://en.wikipedia.org/wiki/Free_object" rel="nofollow noreferrer">Wikipedia</a> defines free objects as follows:</p>
<blockquote>
<p>Let <span class="math-container">$(\mathcal{C},F)$</span> be a concrete category (i.e. <span class="math-container">$F : \mathcal{C} \to {\rm \bf{Set}}$</span> is a faithful... | individ | 128,505 | <p>For the equation:</p>
<p>$$3x^2+xy+5y^2=z^3$$</p>
<p>write the formula so that it was easier to go through. To facilitate calculations will make the replacement.</p>
<p>$$p=15k^2+3q^2-21n^2-11qk$$</p>
<p>$$s=25n^2+15k^2+3q^2+10qn-11qk-38kn$$</p>
<p>$$a=42n^2-30k^2-46qn+60qk-16q^2$$</p>
<p>$$t=42n^2-6q^2+8k^2-4... |
18 | <p>Some teachers make memorizing formulas, definitions and others things obligatory, and forbid "aids" in any form during tests and exams. Other allow for writing down more complicated expressions, sometimes anything on paper (books, tables, solutions to previously solved problems) and in yet another setting students a... | Sue VanHattum | 60 | <p>I allow notes on tests, because math is less about memory than about understanding, and I don't want students to focus on the memory part. I don't allow notes on quizzes, because they are on just one problem type, and I want students to be ready to think it through. </p>
<p>You may find this blog post helpful: <a h... |
18 | <p>Some teachers make memorizing formulas, definitions and others things obligatory, and forbid "aids" in any form during tests and exams. Other allow for writing down more complicated expressions, sometimes anything on paper (books, tables, solutions to previously solved problems) and in yet another setting students a... | Geoff | 685 | <p>I’m a high school maths teacher and I use a way to test where crib sheets are allowed; yet pupils are rewarded for not using it. </p>
<p>At the start of the test, the pupils use black ink and are not allowed to use the test aids (such as crib sheet and calculator). When they want to, they indicate to me that they w... |
18 | <p>Some teachers make memorizing formulas, definitions and others things obligatory, and forbid "aids" in any form during tests and exams. Other allow for writing down more complicated expressions, sometimes anything on paper (books, tables, solutions to previously solved problems) and in yet another setting students a... | ncr | 1,537 | <p>In many of the classes I've taught, I've allowed students to have a sheet of notes (e.g., a course on differential equations where some of the recipes they're asked to apply can be easily mixed up) but with the following two features:</p>
<ol>
<li>I talk a lot about how, like others mention above, if they're going ... |
2,257,365 | <p>In a semicircle of diameter $CD$ there's a chord $AB$ of length 7, and it's parallel to the diameter. There's also a small semicircle that is tangent to $AB$ and its diameter is a segment in $CD$ . Find the area of the semicircle without the small semicircle.</p>
<p>I'm pretty curious about this problem, i've tried... | egreg | 62,967 | <p>You have
$$
\tan\frac{\alpha}{2}=\frac{1/2}{1}=\frac{1}{2}
$$
Then use
$$
\tan2\beta=\frac{2\tan\beta}{1-\tan^2\beta}
$$
With $\beta=\alpha/2$, we get
$$
\tan\alpha=\frac{1}{1-1/4}=\frac{4}{3}
$$</p>
<hr>
<p>Using your tools it can be done as well; set $r=AM$, for simplicity. Then
$$
\frac{1}{2}=\frac{1}{2}r^2\sin... |
2,257,365 | <p>In a semicircle of diameter $CD$ there's a chord $AB$ of length 7, and it's parallel to the diameter. There's also a small semicircle that is tangent to $AB$ and its diameter is a segment in $CD$ . Find the area of the semicircle without the small semicircle.</p>
<p>I'm pretty curious about this problem, i've tried... | Steven Alexis Gregory | 75,410 | <p>You've already found the lengths shown below.</p>
<p><a href="https://i.stack.imgur.com/pZykI.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/pZykI.jpg" alt="enter image description here" /></a></p>
<p>Hence
<span class="math-container">\begin{align}
u^2 &= \left(\frac{\sqrt 5}{2}\right)^2 ... |
743,227 | <p>I've this question:</p>
<blockquote>
<p>Find the area of the intersection between the sphere $x^2 + y^2 + z^2 = 1$ and the cylinder $x^2 + y^2 - y = 0$.</p>
</blockquote>
<p>Is this second equation even a closed shape? If one were to plot points satisfying that equation, one gets things like $(2, \sqrt{-2})$, $(... | Guy | 127,574 | <p>Yes it is. </p>
<p>Consider this equation only in the $xy$ ,i.e, $(z=0)$ plane.</p>
<p>Clearly it is a circle(why? Prove)</p>
<p>Now since it is independent of $z$, this equation will form a circle for any plane $z\in \Bbb R$</p>
<p>Do you see why that is a cylinder?</p>
|
2,926,270 | <p>The base step is pretty obvious: <span class="math-container">$1 \geq \frac{2}{3}$</span>.</p>
<p>Then we assume that <span class="math-container">$P(k)$</span> is true for some <span class="math-container">$k \in \mathbb{Z}^{+}$</span> and try to prove <span class="math-container">$P(k+1)$</span>. So I have</p>
<... | Hagen von Eitzen | 39,174 | <p>You hope to have
<span class="math-container">$$\frac23k\sqrt k+\sqrt{k+1}\stackrel?\ge \frac23(k+1)\sqrt{k+1} $$</span>
or equivalently after simple transformations,
<span class="math-container">$$\frac23k\sqrt k+\sqrt{k+1}\stackrel?\ge \frac23k\sqrt{k+1} +\frac23\sqrt{k+1},$$</span>
<span class="math-container">$$... |
1,433,456 | <p>Given matrics $A = \begin{bmatrix}1 && 5 && 2 \\ -1 && 0 && 1 \\ 3 && 2 && 4\end{bmatrix}$ and $B = \begin{bmatrix}6 && 1 && 3 \\ -1 && 1 && 2 \\ 4 && 1 && 3\end{bmatrix}$, find $-3\mathop{Tr}(A-3B)$.</p>
<p>I am not exa... | Hirshy | 247,843 | <p>Hint: for a matrix $A\in \mathbb R^{n\times n}$ the trace of $A$ is a real number. Thus $3\operatorname{trace}(A-3B)$ is just...can you take it from here?</p>
|
1,433,456 | <p>Given matrics $A = \begin{bmatrix}1 && 5 && 2 \\ -1 && 0 && 1 \\ 3 && 2 && 4\end{bmatrix}$ and $B = \begin{bmatrix}6 && 1 && 3 \\ -1 && 1 && 2 \\ 4 && 1 && 3\end{bmatrix}$, find $-3\mathop{Tr}(A-3B)$.</p>
<p>I am not exa... | Emilio Novati | 187,568 | <p>I suppose that you know that the <a href="https://en.wikipedia.org/wiki/Trace_(linear_algebra)" rel="nofollow">trace</a> of a matrix is the sum of the elements on the principal diagonal. You have correctly found $A-3B$, so: trace$(A-3B)=-25$ and $-3 \mbox{trace} (A-3B)= -3\times (-25)$ .</p>
|
663,736 | <p>For a very large number n, how many divisibility tests are required to establish if its prime?</p>
<p>I know this has something to do with the Golden Number, but I can't figure out what. I did try searching for an answer but not much luck.</p>
<hr>
<p>!!EDIT!!
(It wont let me answer my own question for upto 8hour... | Newb | 98,587 | <p>To test if some $x$ is prime, we generally have to do divisibility tests only up to and including $\sqrt{x}$. </p>
<p>That's because if some $y > \sqrt{x}$ were a factor of $x$, then there would have to be some $z$ such that $zy = x$. And $z < \sqrt{x}$ because if $z > \sqrt{x}$, then clearly $zy > x$ (... |
2,654,538 | <p>If $2\tan^2x - 5\sec x = 1$ has exactly $7$ distinct solutions for $x\in[0,\frac{n\pi}{2}]$, $n\in N$, then the greatest value of $n$ is?</p>
<p>My attempt:</p>
<p>Solving the above quadratic equation, we get $\cos x = \frac{1}{3}$</p>
<p>The general solution of the equation is given by $\cos x = 2n\pi \pm \cos^{... | Dr. Sonnhard Graubner | 175,066 | <p>simplifying the given equation we get
$$-5\cos(x)^2-5\cos(x)+4=0$$
with $$\cos(x)=t$$ we get the quadratic equation
$$-5t^2-5t+4=0$$
solving this we get
$$t_{1,2}=-\frac{1}{2}\pm\frac{\sqrt{105}}{10}$$
can you finish?</p>
|
2,205,950 | <p>If $f:\mathbb{R} \rightarrow \mathbb{R}$ satisfies $f'(a) \neq 0$ for all $a \in \mathbb{R}$, show that $f$ is one-to-one for all $a\in \mathbb{R}$.</p>
<h2>My attempt</h2>
<p>We know that $f(a)$ is not a constant because $f'(a)\neq 0$.Define $f$ by $f(a)=bx$. $f'(a)=x\neq 0$</p>
<p>If $f(x)=f(v)$ then $$bx=bv$$<... | szw1710 | 130,298 | <p>The intermediate value property of a derivative is not needed to get our conclusion. Simply apply the Mean Value Theorem: if $x\ne y$ then there exists an intermmediate point $a$ between $x$ and $y$ s.t. $f(y)-f(x)=f'(a)(y-x)\ne 0$ which means that $f(x)\ne f(y)$.</p>
|
334,701 | <p>The question is really simple, its just terminology.</p>
<p>For simplicity we work on smooth algebraic surfaces and we consider the intersection form on curves on the surface.</p>
<p>So let $S$ be a surface and $D \in \operatorname{Pic}(S)$ a divisor. Then $D$ is said to be nef if
$$
D.C \geq 0
$$
for all curves $... | Community | -1 | <p>Yes, this is confusing. Let me try to clarify.</p>
<p>The term "nef" was indeed coined by Reid. But it was <strong>not</strong> supposed to be an abbreviation of "numerically effective"! Rather, it was meant as an acronym standing for "numerically eventually free". (This is why sometimes, in older references, you m... |
4,590,677 | <p>This is perhaps a silly question related to calculating with surds. I was working out the area of a regular pentagon ABCDE of side length 1 today and I ended up with the following expression :</p>
<p><span class="math-container">$$\frac{\sqrt{5+2\sqrt5}+\sqrt{10+2\sqrt{5}}}{4}$$</span></p>
<p>obtained by summing the... | Hirofumi Ryo | 300,531 | <p>Yes. There is a way to formalize this particular type of sum of square roots, similar to the way a determinant is developed for quadratic equations.</p>
<p>We can write the generic form of the expression in the first place as follows.</p>
<p><span class="math-container">$\sqrt{a+b\sqrt{s}}+\sqrt{c+d\sqrt{s}}=\sqrt{x... |
2,006,118 | <p>When defining the notion of a measurable cardinal using the ultrafilter definition, why do we require the ultrafilter to be $\kappa$-complete? I know this makes our definition of a measurable cardinal more restrictive (by requiring more sets to be in the ultrafilter) but intuitively speaking, what effect does this c... | Stefan Mesken | 217,623 | <p>Your question leaves room for interpretations and may therefore not have a definitive answer. Thus, instead of trying to come up with one, let me highlight two factors such an answer would have to take into account.</p>
<p>First, there is a historical reason. The definition of a measurable cardinal is ultimately du... |
2,006,118 | <p>When defining the notion of a measurable cardinal using the ultrafilter definition, why do we require the ultrafilter to be $\kappa$-complete? I know this makes our definition of a measurable cardinal more restrictive (by requiring more sets to be in the ultrafilter) but intuitively speaking, what effect does this c... | Mitchell Spector | 350,214 | <blockquote>
<p>"When defining the notion of a measurable cardinal using the ultrafilter definition, why do we require the ultrafilter to be $\kappa$-complete? I know this makes our definition of a measurable cardinal more restrictive (by requiring more sets to be in the ultrafilter)..."</p>
</blockquote>
<p>It's no... |
681,543 | <p>So I have a function </p>
<p>$$r= ( x^2 + y^2)^{1/2}$$</p>
<p>and I want to show that </p>
<p>$$\operatorname{grad} f(r) = f'(r)(\operatorname{grad} r).$$</p>
<p>I don't really know where to begin do you say that $f(r) = (f \circ r)(x,y)$ and then use the definition of gradient to work it out. Please give a rel... | Nitish | 61,574 | <p>Let's break it down:</p>
<p><span class="math-container">\begin{align}
f &: \Re^+\to \Re\\f(z) &=\sqrt z\\f'(z)&=\frac{1}{2\sqrt{z}}\\
\\g&:\Re^2\to\Re\\ g(x,y)&=x^2+y^2\\\nabla g(x,y)&=\binom{2x}{2y}
\\\mbox{So,}
\\f'(g(x,y))&=f'(g(x,y))\cdot\nabla g(x,y)\\
&=\frac{1}{2\sqrt{x^2+y^2}... |
3,043,846 | <p>I want to rewrite a question not so well written on this site and clarified by Mr. Lahtonen (thank you again).</p>
<p>So here the question:</p>
<blockquote>
<p>Let the extention <span class="math-container">$GF(p^m) \supset GF(p)$</span> that contains roots of
<span class="math-container">$p(x)=x^{p^{m}}-1$</s... | user1729 | 10,513 | <p>As another example, both the trivial group <span class="math-container">$Id$</span> and the cyclic group of order two <span class="math-container">$C_2$</span> have trivial automorphism group:
<span class="math-container">$$\operatorname{Aut}(Id)\cong Id\cong\operatorname{Aut(C_2)}$$</span></p>
<p>This is the small... |
895,759 | <p>Is there some sorts of Krull's theorem (that every ring has maximal ideal) for rings that do not have multiplicative identity (unit)? So I know that non-unital rings do not satisfy Krull's theorem, but for some types of non-unital rings, theorem does get satisfied. So what is it?</p>
<p>Edit: Wikipedia seems to men... | Xam | 133,781 | <p>I found a theorem given in David Burton's book: "A first course in rings and ideals" where he proves a "non-unitary" version of Krull theorem. It says that every nonzero finitely generated ring $R$ has a maximal ideal. The proof is essentially the same than for the usual version of Krull theorem.
So this result work... |
825,318 | <p>Can someone please help me with these True and False questions? I've tried them myself, but I'm not very good at discrete math... Thank you in advance!</p>
<ol>
<li><p>Any set $A$ and $B$ with $B\subseteq A$ and $f: B \to A$ be $1$-$1$ and onto, then $B = A$</p>
<p>False?</p></li>
<li><p>Let $A$ and $B$ be nonempt... | André Nicolas | 6,312 | <p>It is an unfortunate fact that when one sees, for example, $N(20.6,0.8)$, one does not immediately know whether $0.8$ is the variance or the standard deviation. Each convention is used. I suspect that in "applied" settings, the second parameter is more frequently the standard deviation. </p>
<p>There is a similar i... |
50,227 | <p>The problem I'm having is mapping a 3D triangle into 2 dimensions. I have three points in $(x,y,z)$ form, and want to map them onto the plane described by the normal of the triangle, such that I end up with three points in $(x,y)$ form.</p>
<p>My guess would be it'd assign an arbitrary up vector and then doing some... | Ross Millikan | 1,827 | <p>You have not specified the problem well enough. Do you have three points $(x_1,y_1,z_1), (x_2,y_2,z_2), (x_3,y_3,z_3)$ to map to two dimensional points? The simplest is to ignore the third coordinate. This is not as stupid as it sounds-you are projecting the triangle on the $xy$ plane. If you want to project ont... |
50,227 | <p>The problem I'm having is mapping a 3D triangle into 2 dimensions. I have three points in $(x,y,z)$ form, and want to map them onto the plane described by the normal of the triangle, such that I end up with three points in $(x,y)$ form.</p>
<p>My guess would be it'd assign an arbitrary up vector and then doing some... | Glorious Nathalie | 948,761 | <p>Suppose you have three points A(x_1, y_1, z_1) , B(x_2, y_2, z_2), C(x_3, y_3, z_3). You can have the axes of the coordinate system attached to the plane containing <span class="math-container">$A,B,C$</span> anywhere on this plane, and the orientation of the its <span class="math-container">$x', y'$</span> axes ca... |
2,157,914 | <p>I am struggling with the next exercise of my HW:</p>
<p>How many conjugacy classes are in $GL_3(\mathbb{F}_p)$? And how many in $SL_2(\mathbb{F}_p)$?</p>
<p>It's on the topic of Frobenius normal form of finitely generated modules over $\mathbb{F}_p$.</p>
<p>I'd appreciate any idea.</p>
| N. S. | 9,176 | <p><strong>Hint</strong> $$k-\frac{1}{2} \leq \sqrt{n} < k+\frac{1}{2} \Leftrightarrow k^2-k+\frac{1}{4} \leq n <k^2+k+\frac{1}{4}$$</p>
<p>There are $2k$ integers for which $f(n)=\frac{1}{k}$.</p>
|
1,814,823 | <p>In this question , multiple concepts of graphical transformations are involved. I am facing problems in applying all of them in a single question.</p>
| Spenser | 39,285 | <p>The <a href="https://en.wikipedia.org/wiki/Graph_of_a_function" rel="nofollow">graph</a> of your function is by definition the set
$$\left\{\left(x,\frac{x}{1+|x|}\right):x\in\Bbb R\right\}\subseteq\Bbb R^2.$$</p>
|
353,947 | <p>Problem. Let $f:[a,b]\to\mathbb{R}$ be a function such that $ f\in C^3([a,b])$ and $f(a)=f(b)$. Prove that $$ \left|\int\limits_{a}^{\frac{a+b}{2}}f(x)dx-\int\limits_{\frac{a+b}{2}}^{b}f(x)dx\right|\leq\frac{(b-a)^4}{192}\max_{x\in [a,b]}|f'''(x)|.$$
Any idea are welcome.</p>
| r9m | 129,017 | <p>Idea : Making use of successive Integration by parts ,</p>
<p>$\int P(x)f^{(3)}(x)\,dx = P(x)f^{(2)}(x)-P^{(1)}(x)f^{(1)}(x)+P^{(2)}(x)f(x)-\int P^{(3)}(x)f(x)\,dx$</p>
<p>Consider the two third degree monic polynomials, $P_1(x)$ and $P_2(x)$.</p>
<p>Now, we compute the difference of the definite integrals: </p>
... |
4,158,431 | <p>Say we have the set of real numbers. Can we construct a different ring than the usual ring of real numbers?</p>
<p>I am trying to wrap my head around the idea of rings and I couldn't find two other operations that will hold distributive property other than usual addition and multiplication.</p>
<p>If we can find ano... | Kavi Rama Murthy | 142,385 | <p>(31) shows that <span class="math-container">$\phi_n(t)\to \phi (t)$</span> for each <span class="math-container">$t\neq x$</span>. Now let <span class="math-container">$m \to \infty$</span> in the inequality <span class="math-container">$$|\phi_n(t)-\phi_m (t)| \leq \frac {\epsilon} {b-a}$$</span> to see that <spa... |
4,158,431 | <p>Say we have the set of real numbers. Can we construct a different ring than the usual ring of real numbers?</p>
<p>I am trying to wrap my head around the idea of rings and I couldn't find two other operations that will hold distributive property other than usual addition and multiplication.</p>
<p>If we can find ano... | user912011 | 912,011 | <p>Since <span class="math-container">$f_n(t) \rightarrow f(t)$</span> for any <span class="math-container">$a \leq t \leq b$</span>, by theorem 3.3,
<span class="math-container">$$ f_n(t)-f_n(x) \rightarrow f(t)-f(x) $$</span>
and hence
<span class="math-container">$$ \frac{f_n(x)-f_n(t)}{t-x} \rightarrow \frac{f(t)-f... |
2,261,927 | <p>How to get alternative form from equation 1)</p>
<p>$$ 1) -a^2 + a + b^2 -b $$</p>
<p>to equation 2)</p>
<p>$$ 2) (a-b)(a+b-1)$$</p>
| StackTD | 159,845 | <p>Group terms and factor, e.g. by first using $\color{blue}{a^2-b^2=(a-b)(a+b)}$ and then putting the common term $\color{red}{a-b}$ up front:
$$\begin{align}-a^2 + a + b^2 -b
& = -\left(\color{blue}{a^2-b^2}\right)+a-b \\[4 pt]
& = -\color{blue}{\left(a-b\right)\left(a+b\right)}+a-b \\[4 pt]
& = -\color{r... |
1,407,714 | <p>Why is:
$$\bigcap_{n=0}^\infty\,\mathopen{]}0,e^{-n}\mathclose{[}\,=\emptyset\quad?$$
Indeed,
$$\forall n\in\mathbb N, 0\in\mathopen{]}0,e^{-n}\mathclose{[},$$
and thus $0\in\bigcap_{n=0}^\infty\,\mathopen{]}0,e^{-n}\mathclose{[}$. So, what's wrong here ? </p>
| Augustin | 241,520 | <blockquote>
<p>$$\forall n\in\mathbb N, 0\in ]0,e^{-n}[$$</p>
</blockquote>
<p>This is wrong. $0\notin ]0,e^{-n}[$.</p>
|
1,407,714 | <p>Why is:
$$\bigcap_{n=0}^\infty\,\mathopen{]}0,e^{-n}\mathclose{[}\,=\emptyset\quad?$$
Indeed,
$$\forall n\in\mathbb N, 0\in\mathopen{]}0,e^{-n}\mathclose{[},$$
and thus $0\in\bigcap_{n=0}^\infty\,\mathopen{]}0,e^{-n}\mathclose{[}$. So, what's wrong here ? </p>
| Scientifica | 164,983 | <p><strong>Hint</strong>: Try to show that if $\bigcap_{n=0}^\infty\neq \emptyset$ which means $\exists x\in\mathbb{R},\,x\in\bigcap_{n=0}^\infty]0,\,e^{-n}[$ then $\exists m\in\mathbb{N},\,e^{-m}<x$ and so $x\notin]0,\,e^{-m}[$ which leads to a contradiction. Note that since $x\in\bigcap_{n=0}^\infty]0,\,e^{-n}[$ t... |
1,581,545 | <p>I am looking for a proof of Euclid's Lemma, i.e if a prime number divides a product of two numbers then it must at least divide one of them.</p>
<p>I am coding this proof in Coq, and i'm doing it over <em>natural numbers</em>. I aim to prove the uniqueness of prime factorization (So I cannot use this lemma!). Howev... | CopyPasteIt | 432,081 | <p>The OP's stated goal is to prove the uniqueness of prime factorization using elementary/foundational techniques, perhaps by proving Euclid's lemma in a different way.</p>
<p>Since the end goal is to prove uniqueness in the FTA, of interest here are two similar elementary proofs.</p>
<p>(1) The wikipedia article: <... |
987,895 | <blockquote>
<p>Let $(G,\cdot)$ be a group, $g \in G$.</p>
<p>For $a,b \in G$ define $a * b := a \cdot g^{-1} b$. Show that $(G,*)$ is a group with the neutral element $g$ and $f : (G,*) \rightarrow (G,\cdot), a \mapsto a \cdot g^{-1}$ is a group isomorphism.</p>
</blockquote>
<p>In order to show that $(G,*)$ ... | Hagen von Eitzen | 39,174 | <p>It is best to start from the end, i.e. first to show that $f$ has the isomorphism property (even if we don't know yet that $(G,*)$ is a group), that is
$$\tag1 f(a*b)=f(a)\cdot f(b)\qquad\text{for all $a,b\in G$}$$
and
$$\tag2 f\text{ is bijective}.$$
Once you have shown $(1)$ and $(2)$, the group properties are s... |
215,835 | <p>According to Willard,</p>
<p>If $(X,\tau)$ is a topological space, a base for $\tau$ is a collection $\mathscr{B} \subset \tau$ such that $\tau=\{ \bigcup_{B \in \mathscr C} : \mathscr C \subset \mathscr B\}$. Evidently, $\mathscr B$ is a base for $X$ iff whenever $G$ is an open set in $X$ and $p \in G$ there is so... | Rudy the Reindeer | 5,798 | <p>Question 1: Yes. It says so in the first sentence. Unless I am misunderstanding your question.</p>
<p>Question 2: Every base is a subset of the topology it generates. By definition, all sets in a topology are open. Hence all sets in a base also have to be open. Hence "no" to your last question in the comment to que... |
1,859,719 | <blockquote>
<p>Let be $U (x,y) = x^\alpha y^\beta$. Find the maximum of the function $U(x,y)$ subject to the equality constraint $I = px + qy$.</p>
</blockquote>
<p>I have tried to use the Lagrangian function to find the solution for the problem, with the equation</p>
<p>$$\nabla\mathscr{L}=\vec{0}$$</p>
<p>where... | Jashaszun | 258,167 | <p>You can certainly simplify that system!</p>
<p>$$\lambda = \frac{\alpha x^{\alpha - 1}y^\beta} p$$
$$\lambda = \frac{\beta y^{\beta - 1}x^\alpha} q$$</p>
<p>Thus</p>
<p>$$\frac{\alpha x^{\alpha - 1}y^\beta} p = \frac{\beta y^{\beta - 1}x^\alpha} q$$</p>
<p>and</p>
<p>$$q\left(\alpha x^{\alpha - 1}y^\beta\right)... |
2,781,801 | <p>When asked to evaluate $g$ at the point specified above we would get $\dfrac{1}{e} \cdot \log_e(\frac{1}{\sqrt e})$ and that evaluates to some -0.18393... but the correct answer is -1/2e. How does it get simplified to that?</p>
| Claude Leibovici | 82,404 | <p>$$g(x)=x^2 \log(x) \implies g\left(\frac{1}{\sqrt{e}}\right)=\left(\frac 1 {{\sqrt{e}}}\right)^2 \log\left(\frac{1}{\sqrt{e}}\right)=-\frac 1 e\times \log(\sqrt{e})=-\frac 1 e\times\frac 12 \log(e)$$ and, by definition $\log(e)=1$.</p>
|
108,253 | <p>I would like to assign 'x' individuals to 'y' groups, randomly. For example, I would like to divide 50 individuals into 100 groups randomly. Of course, with more groups than individuals many of the groups will have zero individuals, while some groups will have multiple individuals. That is fine. With random assignme... | Bob Hanlon | 9,362 | <pre><code>randChoice[
individuals_Integer?NonNegative,
groups_Integer?Positive] :=
RandomChoice[Range[groups], individuals];
choices = randChoice[50, 100]
(* {58, 83, 34, 28, 25, 97, 6, 73, 28, 91, 6, 42, 93, 48, 56, 64, 20, \
88, 73, 11, 79, 65, 34, 3, 16, 18, 4, 18, 53, 30, 20, 97, 79, 30, 91, \
35, 35, 4... |
3,306,571 | <p>I know that the function <span class="math-container">$f(x) = \frac{x}{x}$</span> is not differentiable at <span class="math-container">$x = 0$</span>, but according to the definition of differentiable functions:</p>
<blockquote>
<p>A differentiable function of one real variable is a function whose derivative exi... | fleablood | 280,126 | <blockquote>
<p>"I know that the function <span class="math-container">$f(x)=\frac xx$</span> is not differentiable at <span class="math-container">$x=0$</span>"</p>
</blockquote>
<p>The function isn't <em>ANYTHING</em> at <span class="math-container">$x=0$</span>. <span class="math-container">$0$</span> is not in ... |
336,196 | <p>Can anyone help me with the following SDE?</p>
<p>Solve the following stochastic differential equation:
$$dY_t=aY_tdt+(b(t)+cY_t)dB_t$$
with $Y_0=0$.</p>
<p>Hint: Try a solution of the form $Z_tH_t$ where $Z_t = exp(cB_t+(a-\frac{1}{2}c^2t))$ and $dH_t=F(t)dt+G(t)dB_t$ for some adapted process F and G which need ... | gt6989b | 16,192 | <p><strong>Here is an idea to get you started.</strong></p>
<p>Note that
$$
dZ_t = cZ_t dB_t + \frac{c^2}{2}Z_t dt = Z_t \left(cdB_t + \frac{c^2}{2}dt\right)
$$</p>
<p>and let $dH_t = P dt + Q dB_t$.</p>
<p>Consider $A_t = Z_t H_t$. Then, by Ito's Lemma, and expansion for $dZ_t$,</p>
<p>$$\begin{split}
dA_t &= ... |
2,567,332 | <p>A Greek urn contains a red, blue, yellow, and orange ball. A ball is drawn from the urn at random and then replaced. If one does this $4$ times, what is the probability that all $4$ colors were selected?</p>
<p>I approached this questions by doing $(1/4)^4$ because there's always a $1/4$ chance of selected a specif... | Austin Weaver | 480,825 | <p>The probability of drawing $4$ different balls is the product of the probabilities of drawing a new ball on all $4$ draws.</p>
<p>The first draw yields a new ball, guaranteed:
$$P(\text{ball 1 new})=1$$</p>
<p>For the second draw, there are $3$ possible new balls and $4$ total balls, so:
$$P(\text{ball 2 new})=\fr... |
690,331 | <p>Does it make a different when you parametrize a counterclockwise full circle and a clockwise circle in the complex plane? </p>
<p>For example, I am looking at computing an integral $\int_\gamma {1\over{z+4}}dz$ where $\gamma$ is the circle of radius $1$, centered at $-4$, oriented <strong>counterclockwise.</strong>... | Bill Cook | 16,423 | <p>No. This doesn't work. You have to check that $E_i \cap \left(E_1+E_2+\cdots+E_{i-1}+E_{i+1}+\cdots+E_p\right) = \{0\}$ for each $i$. </p>
<p>Just checking that the subspaces don't pairwise intersect is not enough.</p>
<p>Consider the example: $U = \{ (0,y) \;|\; y \in \mathbb{R} \}$, $V =\{ (x,0)\;|\; x \in \math... |
1,672,131 | <p>A card game is played with a deck whose cards can be one of 6 suits, one of the suits being hearts, and one of 11 ranks. A hand is a subset of 3 cards. What is the probability that a hand has exactly two hearts given that it has the 2 of hearts? Please explain.</p>
| Hagen von Eitzen | 39,174 | <p><strong>Hint:</strong> $10^n$ has $n+1$ digits.</p>
|
2,619,185 | <p>Let $$P=(X+2)^m+(X+3)^{2m+3}$$ and $$Q=X^2+5X+7.$$ I need to show that $Q$ divides $P$ for any $m$ natural. </p>
<p>I said like this: let $a$ be a root of $X^2+5X+7=0$. Then $a^2+5a+7=0$. </p>
<p>Now, I know I need to show that $P(a)=0$, but I do not know if it is the right path since I have not found any way to d... | Yash Jain | 522,158 | <p>Let us restate the value of $P$ first.</p>
<p>$P = (X+2)^m+(X+2)^{2m+3}+1^{2m+3} = (X+2)^m+(X+2)^{2m+3}+1 =$</p>
<p>$(X+2)^m+(X+2)^{2m+3}+(X+2)^0 = (X+2)^{3m+3} = (X+2)^{(m+1)^{3}}$</p>
<p>And, Let us restate the value of $Q$.</p>
<p>$Q = X^2+4X+4+X+3 = X^2+4X+4+X+2+1 = (X+2)^2+(X+2)^1+(X+2)^0$</p>
<p>Do you no... |
4,265,001 | <p>I'm a bit confused about expanding out the notation of product of matrices, in the context of quadratic forms.</p>
<p>If <span class="math-container">$x \in \mathbb{R}^n, \, \, A \in \mathbb{R}^{n \times n}$</span> then</p>
<p><span class="math-container">$$x^TAx = \sum_{i,j=1}^na_{ij}x_ix_j$$</span></p>
<p>But then... | Mas | 973,301 | <p>The <span class="math-container">$(i,j)$</span>-coefficient of the matrix <span class="math-container">$X^TAX$</span> is given by
<span class="math-container">\begin{equation}
\sum_{k,l=1}^nx_{ki}a_{kl}x_{lj},
\end{equation}</span>
where <span class="math-container">$x_{ij}$</span> (resp. <span class="math-container... |
2,060,156 | <p>First thing I want to mention is that this is not a topic about why $1+2+3+... = -1/12$ but rather the connection between this summation and $\zeta$.</p>
<p>I perfectly understand that the definition using the summation $\sum_{k=1}^\infty k^{-s}$ of the zeta function is only valid for $Re(s) > 1$ and that the fu... | snulty | 128,967 | <p>$$\frac{7k-5}{5k-3}=\frac{6l-1}{4l-3}$$</p>
<p>$$28kl-20l-21k+15=30kl-18l-5k+3$$</p>
<p>$$2kl+2l+16k-12=0$$</p>
<p>$$kl+l+8k-6=0$$</p>
<p>either:</p>
<p>$$l(1+k)=2(3-4k)$$
so
$$l=2\frac{3-4k}{1+k}$$</p>
<p>or:</p>
<p>$$k(l+8)=6-l$$</p>
<p>so</p>
<p>$$k=\frac{6-l}{l+8}$$</p>
<p>Let's go with this second one... |
64,544 | <blockquote>
<p>Please let me know what is the standard notation for group action.</p>
</blockquote>
<p>I saw the following three notations for group action.
(All the images obtained as <code>G\acts X</code> for different deinitions of <code>\acts</code>.) </p>
<p>(1) <img src="https://lh5.googleusercontent.com/_7... | Stefan Waldmann | 12,482 | <p>IN addition to what has been said, I would like to mention the following aspect: a group action usually does not come alone. When $G$ acts on a set $X$ you will almost for sure be interested also in the induced action of $G$ on certain kind of functions on $X$: think of a smooth manifold $M$ and of smooth functions,... |
3,115,168 | <p>I've converted <span class="math-container">$\cos^3(x)$</span> into <span class="math-container">$\cos^2(x)\cos(x)$</span> but still have not gotten the answer. </p>
<p>The answer is <span class="math-container">$\dfrac{\sin(x)(3\cos^2x + 2\sin^2x)}{3}$</span></p>
<p>My answer was the same except I did not have a ... | José Carlos Santos | 446,262 | <p>Since <span class="math-container">$\cos^3x=(1-\sin^2 x)\cos x$</span>, you can do <span class="math-container">$\sin x=t$</span> and <span class="math-container">$\cos x\,\mathrm dx=\mathrm dt$</span>, thereby getting<span class="math-container">$$\int1-t^2\,\mathrm dt.$$</span></p>
|
3,115,168 | <p>I've converted <span class="math-container">$\cos^3(x)$</span> into <span class="math-container">$\cos^2(x)\cos(x)$</span> but still have not gotten the answer. </p>
<p>The answer is <span class="math-container">$\dfrac{\sin(x)(3\cos^2x + 2\sin^2x)}{3}$</span></p>
<p>My answer was the same except I did not have a ... | Sri-Amirthan Theivendran | 302,692 | <p>Note that
<span class="math-container">$$
\int\cos^3x\, dx=\int\cos^2x\, d(\sin x)=\cos^2x\sin x+\int2\sin^2 x\cos x\, dx
$$</span>
To compute the last integral make the substitution <span class="math-container">$u=\sin x$</span>.</p>
|
777,186 | <p>My equation is the following, and I would like to find which $k$ can make it a circle.</p>
<p>$$x^2+y^2+4x-6y+k=0$$</p>
<p>My naive approach is to have $k$ to be $-4x+6y+c$ where c is any number, so that I can have any circle that is in 0. However k is a parameter and I can't really figure that out if I am missing... | Fly by Night | 38,495 | <p><strong>HINT</strong></p>
<p>Complete the square on $x^2+4x$ and $y^2-6y$ separately.</p>
<p>Take all of the numbers, and the $k$, over to the right hand side.</p>
<p>You will have something like $(x-a)^2 + (y-b)^2 = t$, where $a$ and $b$ are numbers and $t$ is a mixture of numbers and the letter $k$.</p>
<p>You... |
1,594,722 | <p>The ODE is</p>
<p>($xy^{3} + x^{2}y^{7}) \frac{dy}{dx} = 1$</p>
<p>I have tried everything like integrating factor,it is not homogenous and not linear differential equation..What should be done now?</p>
| SchrodingersCat | 278,967 | <p>Re-arranging your differential equation, we have
$$\frac{dx}{dy}=xy^3+x^2y^7$$
$$\frac{dx}{dy}-xy^3=x^2y^7$$
$$\frac{1}{x^2}\cdot \frac{dx}{dy}-\frac{1}{x}\cdot y^3=y^7$$
$$-\frac{d}{dy}\left(\frac{1}{x}\right)-y^3\cdot \left(\frac{1}{x}\right) =y^7$$</p>
<p>Put $u=\frac{1}{x}$</p>
<p>You get $$\frac{du}{dy}+uy^3... |
1,969,169 | <p>We have to do the following integral.
$$\int_1^{\frac{1+\sqrt{5}}{2}}\frac{x^2+1}{x^4-x^2+1}\ln\left(1+x-\frac{1}{x}\right)dx$$
I tried it a lot.
I substitute $t=1+x-(1/x)$, $dt=1+(1/x^2)$</p>
<p>But then I stuck at
$$\int\limits_{1}^{2} \frac{\ln(t)}{(t-1)^{2} + 1} \mathrm{d}t$$</p>
<p>But now how to proceed.</p>... | Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\ic... |
3,981,809 | <p>Imagine that we have two pairs of integers <span class="math-container">$(a_1,b_1)$</span> and <span class="math-container">$(a_2, b_2)$</span> where</p>
<p><span class="math-container">$$ a_1b_1\equiv 0,\,\ a_2b_2\equiv 0,\,\ a_1b_2+a_2b_1\equiv 0\pmod n$$</span></p>
<p>Does this imply that
<span class="math-contai... | Bill Dubuque | 242 | <p>Hypotheses <span class="math-container">$\Rightarrow \dfrac{a_1b_2}n$</span> & <span class="math-container">$\dfrac{a_2b_1}n\,$</span> have sum & product <span class="math-container">$\in\Bbb Z\,$</span> <a href="https://math.stackexchange.com/a/3900209/242">thus</a> both are <span class="math-container">$\i... |
3,470,208 | <p><span class="math-container">$$f(x)=\begin{cases}
\dfrac{x}{\sin x}, & x>0\\
2-x, & x\le0
\end{cases}$$</span></p>
<p><span class="math-container">$$g(x)=\begin{cases}
x+3, &x<1\\
x^2-2x-2, &1\le x<2\\
x-5, & x\ge2
\end{cases}$$</span></p>
<p>Find left hand limi... | user | 505,767 | <p>We have that since <span class="math-container">$x>\sin x$</span> for <span class="math-container">$x\neq 0$</span></p>
<ul>
<li>as <span class="math-container">$x\to0^+, y=f(x)\to1^+ \implies \lim_{x\to0^+}
g(f(x))=\lim_{y\to1^+} g(y)=-3$</span></li>
</ul>
<p>and</p>
<ul>
<li>as <span class="math-container... |
149,558 | <p>I always use <code>InputForm</code> to check the result object,such as <code>Dataset</code> or <code>Graphics</code> or other objects.But if you are in the result of <code>InputForm</code>,you cannot use the Front-End function of balance the bracket. Note this gif</p>
<p><a href="https://i.stack.imgur.com/51OYd.gif"... | Carl Woll | 45,431 | <p>Per my comment, I like using <code>expr //InputForm //SequenceForm</code>, but another similar possibility is to use a custom head with a custom <a href="http://reference.wolfram.com/language/ref/MakeBoxes" rel="noreferrer"><code>MakeBoxes</code></a> rule. For instance, let's call the custom form <code>myInputForm</... |
2,755,143 | <p>Find Number of integers satisfying $$\left[\frac{x}{100}\left[\frac{x}{100}\right]\right]=5$$ where $[.]$ is Floor function.</p>
<p>I assumed $$x=100q+r$$ where $0 \le r \le 99$</p>
<p>Then we have </p>
<p>$$\left[\left(q+\frac{r}{100}\right)q\right]=5$$ $\implies$</p>
<p>$$q^2+\left[\frac{rq}{100}\right]=5$$</... | xpaul | 66,420 | <p>Using
$$ [x]\le x<[x]+1 $$
one has
$$ 5\le\frac{x}{100}\left[\frac{x}{100}\right]<6. \tag{1}$$
Let $x=100q+r$, where $q$ and $r$ are an integer and $r\in[0,99]$. Then (1) becomes
$$ 500\le (100q+r)q<600. \tag{2}$$
Noting that
$$ 100q^2 \le (100q+r)q<600$$
implies $q^2<6$, one has $q=\pm1,\pm2$. If $q=... |
3,519,515 | <p>Here, I wonder what is a good way to use the epsilon delta definition or converging sequences to show that the set S containing quotients on [0,1] have/does not have volume 0, (i.e. whether there exist a <strong>finite</strong> number of intervals which union contain all of S such that the <strong>sum</strong> of le... | AZM | 440,612 | <p>In order to provide an effective answer, just think of the volume of the complementary set, which is the whole interval but a countable union of zero-measure sets, hence the complementary set has measure one (i.e., the set you are looking for has measure zero).</p>
|
237,031 | <p>The question is: if I assert in ZF that there exists a Reinhardt cardinal, do I really get a theory of higher consistency strength than when I assert in ZFC that there exists an I0 cardinal (the strongest large cardinal not known to be inconsistent with choice, as I understand)? This is implicit in the ordering of t... | Mohammad Golshani | 11,115 | <p>The answer to your question is (almost) yes (almost is because of the addition of DC to the statement).</p>
<p>Recently Gabriel Goldberg has proved </p>
<blockquote>
<p>''Con(NBG+DC+Reinhardt)<span class="math-container">$ \implies$</span> Con(ZFC+I0)''. </p>
</blockquote>
<p>See the abstract of the talk by Gab... |
533,855 | <p>I need to show that $\{x_{n}\}$ is Cauchy given that there exists $0<C<1$ s.t. $|x_{n+1}-x_{n}|\leq C|x_{n}-x_{n-1}|$. Intuitively, that statement clearly implies $\{x_{n}\}$ is Cauchy, since it implies the sequence terms become arbitrarily close. But how to make it precise? </p>
<p>Couldn't it also be said ... | André Nicolas | 6,312 | <p>We change the question, to what is more likely to appear first, TT or HT. If We get TT at the beginning (probability $\frac{1}{4}$, then TT wins. In all other cases, HT wins.</p>
<p>(For the original question, TT and TH are equally likely to be first.) </p>
|
2,316,159 | <p>I'm interested in the differences in the groups but also in the Lie algebra associated. I know that two groups can have the same lie algebra if they differ from discrete elements, for instance: $SO(n)$ and $O(n)$ should have the same algebra. But then if I have a group $O(2,2)$, what is the associated Lie algebra? D... | Travis Willse | 155,629 | <p>No, the groups $O(2, 2)$ and $O(2) \times O(2)$ (and the corresponding algebras) are different---in fact, they have different dimensions as Lie groups.</p>
<p>The group $O(2, 2)$ is the group preserving an inner product of signature $(2, 2)$ on a $4$-dimensional real vector space $\Bbb V$. Concretely, if $g$ is the... |
470,739 | <p>Assume $S$ and $T$ are diagonalizable maps on $\mathbb{R}^n$ such that $S\circ T$=$T \circ S$. Then $S$ and $T$ have a common eigenvector.</p>
<p>I already have proof, but I just need validation in one part.
My proof:
Let $F$ be an eigenvector of $T$. This means $\exists \; \lambda \in R$ such that $T(v)=\lambda v$... | Community | -1 | <p>Since $T$ is diagonalizable then its minimal polynomial factors into distinct linear factors and the restriction of $T$ to any invariant subspace annihilates the minimal polynomial so this restriction is also diagonalizable since its minimal polynomial divides the minimal polynomial of $T$ and then also it factors i... |
2,684,805 | <p>This question is asked by my 12 yr old cousin and I seem to be failing to give him a convincing explanation. Here is the summary of our discussion so far - </p>
<p>Case1 : $a>0, b>0$<br>
<a href="https://i.stack.imgur.com/fuoZS.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/fuoZS.png" alt=... | Christian Blatter | 1,303 | <p>Let ${\mathbb Z}_3$ be the set of colors. There are six ways of assigning $\pm1$ to the edges of the square such that the sum is $=0$ mod $3$, namely four of cyclic type $(1,1,-1,-1)$ and two of type $(1,-1,1,-1)$. Given such an assignment $s$ there are three colorings $f$ of the vertices such that (in the obvious ... |
3,775,749 | <p>So far I know that it’s possible to draw angles which are multiples of <strong>15°</strong> (ex. <em><strong>15°</strong></em>, <em><strong>30°</strong></em>, <em><strong>45°</strong></em> etc.).</p>
<p>Could anybody please tell me if it's possible to draw other angles which are not multiples of 15° using only a com... | Ethan Bolker | 72,858 | <p>You can construct a regular <span class="math-container">$n$</span>-gon with straightedge and compass if and only if <span class="math-container">$n$</span> is a power of <span class="math-container">$2$</span> times a product of <a href="https://en.wikipedia.org/wiki/Fermat_number" rel="nofollow noreferrer">Fermat ... |
158,896 | <p>Being interested in the very foundations of mathematics, I'm trying to build a rigorous proof on my own that $a + b = b + a$ for all $\left[a, b\in\mathbb{R}\right] $. Inspired by interesting properties of the complex plane and some researches, I realized that defining multiplication as repeated addition will lead m... | mboratko | 17,349 | <p>Please don't let the comments discourage you - it is good that you are trying to understand the fundamentals of Mathematics, and your effort is to be applauded.</p>
<p>In order to prove a theorem, one must have in place a set of axioms and definitions (as well as acceptable rules of logic) from which the theorem ca... |
158,896 | <p>Being interested in the very foundations of mathematics, I'm trying to build a rigorous proof on my own that $a + b = b + a$ for all $\left[a, b\in\mathbb{R}\right] $. Inspired by interesting properties of the complex plane and some researches, I realized that defining multiplication as repeated addition will lead m... | André Nicolas | 6,312 | <p>Something of this kind was done by Hilbert a bit over a century ago, in his <a href="http://en.wikipedia.org/wiki/Hilbert%27s_axioms" rel="nofollow">Foundations of Geometry.</a> The axioms of the geometric substrate were laid out in <em>great detail</em>. Then an arithmetic was defined on the points of a particula... |
623,190 | <p>What would be the formula, to determine a rectagles edges, when given the perimeter and space? for example, the rectagles space is 80, and the perimeter is 36, and the edge would be 8 and 10, but how do I find them.</p>
<p>I know that the formula for the perimeter would be
2x+2y=per, or 2(space/y)+2y=per
However I'... | lsp | 64,509 | <p>length = $l$, breadth = $b$</p>
<p>space, $lb = 80$</p>
<p>perimeter, $2(l+b) = 36$</p>
<p>$$l + \frac{80}{l} - 18 = 0$$
$$l^2-18.l+80=0$$
$$(l-10)(l-8)$$
So, $l=10$ or $l=8$.</p>
<p>But since length > breadth:
$$length=10$$
$$breadth=8$$</p>
|
3,267,883 | <p>I apologize in advanced as my literacy in this subject is not too great and this question may either be trivial or impossible as of yet. </p>
<p>I have seen many questions on stack exchange utilizing the Chinese Remainder Theorem to find solutions of <span class="math-container">$a^2\equiv 1\mod (p*q)$</span>, wher... | J. W. Tanner | 615,567 | <p><strong>Hint:</strong> You're asking for <span class="math-container">$2^k|(a+1)(a-1).$</span> <span class="math-container">$\gcd(a+1,a-1)$</span> is at most <span class="math-container">$2$</span>.</p>
|
1,182,523 | <p>I have stumbled across this question:
Let $a,b,c$ be integers, not all $0$ such that $\max(|a|,|b|,|c|)<10^6$. Prove that $|a+b \sqrt{2} + c \sqrt{3}| > 10^{-21}$. </p>
<p>Could anybody help by solving this? Elementary solution is preferred.</p>
| math110 | 58,742 | <p>Let $$\begin{cases}f_{1}=a+b\sqrt{2}+c\sqrt{3}\\
f_{2}=a-b\sqrt{2}+c\sqrt{3}\\
f_{3}=a-b\sqrt{2}-c\sqrt{3}\\
f_{4}=a+b\sqrt{2}-c\sqrt{3}\end{cases}$$
It is clear $$f_{1}f_{2}f_{3}f_{4}\in Z,a,b,c\in Z$$
since $a,b,c$ are integer,and not all 0 ,so $f_{k}\neq 0,k=1,2,3,4$.and Note
$$\max\{|a|,|b|,|c|\}<10^6\Longrig... |
2,646,890 | <blockquote>
<p>If <span class="math-container">$p(x)$</span> is a polynomial of degree <span class="math-container">$n$</span> such that <span class="math-container">$$p(-2)=-15,\ p(-1)=1,\ p(0)=7,\ p(1)=9,\ p(2)=13,\ p(3)=25.$$</span>
Then smalest possible value of <span class="math-container">$n$</span> is</p>
<p>Op... | user5713492 | 316,404 | <p>Construct a difference table.
$$\begin{array}{rrrrr}-15&&&&\\
&16&&&\\
1&&-10&&\\
&6&&6&\\
7&&-4&&0\\
&2&&6&\\
9&&2&&0\\
&4&&6&\\
13&&8&&\\
&12&&&\\
25&&... |
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