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 |
|---|---|---|---|---|
496,488 | <p>I have few questions to the proof on Universality of linearly ordered sets. Could anyone advise please? Thank you.</p>
<p>Lemma: Suppose $(A,<_{1})$ is a linearly ordered set and $(B,<_{2})$ is a dense linearly ordered set without end points. Assume $F\subseteq A$ and $E\subseteq B$ are finite and $h:F \to E... | user43208 | 43,208 | <p>There is no particular role of $h(f(0)) = g(0)$ except to start the inductive argument, which traditionally begins with a base case $n=1$. (Although one <em>could</em> start with the case $n=0$, where the base case starts with the function on the empty set. Possibly the author and/or instructor thought that starting... |
4,604,730 | <p>Given a function <span class="math-container">$f$</span> from vectors to scalars, and a vector <span class="math-container">$\vec v$</span>, the directional derivative of <span class="math-container">$f$</span> with respect to <span class="math-container">$\vec v$</span> is defined as <span class="math-container">$\... | Toby Bartels | 63,003 | <p>You're right that the approximation
<span class="math-container">$$ f ( x + h , y + h ) \approx f ( x + h , y ) + f ( x , y + h ) - f ( x , y ) $$</span>
is not always valid. On the other hand, if <span class="math-container">$ f $</span> is linear (even in the affine sense, that is of the form <span class="math-co... |
467,574 | <p>Using permutation or otherwise, prove that $\displaystyle \frac{(n^2)!}{(n!)^n}$ is an integer,where $n$ is a positive integer.</p>
<p>I have no idea how to prove this..!!I am not able to even start this Can u give some hints or the solution.!cheers.!!</p>
| André Nicolas | 6,312 | <p>One can do a little better. We have $n^2$ people, to be divided into $n$ teams, each of which has $n$ people. Let $w$ be the number of ways to do the job.</p>
<p>Do the division into teams this way. Line up the $n^2$ people, put the first $n$ into a team, then the next $n$, and the next, and so on.</p>
<p>Permuti... |
1,290,111 | <p>How one can prove the following statement:</p>
<p>$k(n-1)<n^2-2n$ for all odd $n$ and $k<n$</p>
<p><em>Tried so far</em>: induction on $n$, graphing, and rewriting $n^2−2n$ as $(n−1)^2−1$.</p>
| MCT | 92,774 | <p>Following the hint that $n^2 - 2n = (n-1)^2 - 1$</p>
<p>$k(n-1) < (n-1)^2 - 1$</p>
<p>Since $k < n$, we can take $k = n-1$ to derive a counter example.</p>
<p>$(n-1)^2 < (n-1)^2 - 1$</p>
<p>$1 < 0$.</p>
<p>So, taking $k=n-1$ always gives a counter example.</p>
|
1,776,726 | <p>I'm trying to determine whether or not </p>
<blockquote>
<p>$$\sum_{k=1}^\infty \frac{2+\cos k}{\sqrt{k+1}}$$ </p>
</blockquote>
<p>converges or not. </p>
<p>I have tried using the ratio test but this isn't getting me very far. Is this a sensible way to go about it or should I be doing something else?</p>
| Stefan4024 | 67,746 | <p>Minimizing $1 - \frac{1}{(t - \frac 4t +1)^2 + 2}$ is equivalent to minimizing $(t - \frac 4t +1)^2 + 2$. But obviously the minimal value for this is $2$, as the square of a number is always bigger than $0$. To find the value which minimizes it just solve $t- \frac4t + 1 = 0$</p>
|
908,309 | <p>I'm finding the principal value of $$ i^{2i} $$</p>
<p>And I know it's solved like this:</p>
<p>$$ (e^{ i\pi /2})^{2i} $$ </p>
<p>$$ e^{ i^{2} \pi} $$</p>
<p>$$ e^{- \pi} $$</p>
<p>I understand the process but I don't understand for example where does the $ i $ in $ 2i $ go?</p>
<p>Is this some kind of a prop... | marty cohen | 13,079 | <p>Also remember that
$i = e^{\pi i/2+2\pi i k}
=e^{\pi i(\frac12+2k)}
$
for any integer $k$,
since $e^{2 \pi i k} = 1$.
$k=0$ gives the usual principal value.</p>
<p>Therefore,
$i^{2i}
=(e^{\pi i(\frac12+2k)})^{2i}
=e^{2\pi i^2(\frac12+2k)}
=e^{-2\pi (\frac12+2k)}
=e^{-\pi (1+4k)}
$.
So,
thanks to the joy of
infini... |
1,947,082 | <p>Let us have sum of sequence (I'm not sure how this properly called in English): $$X(n) = \frac{1}{2} + \frac{3}{4}+\frac{5}{8}+...+\frac{2n-1}{2^n}$$</p>
<p>We need</p>
<p>$$\lim_{n \to\infty }X(n)$$</p>
<p>I have a solution, but was unable to find right answer or solution on the internet.</p>
<p>My idea:</p>
<... | Gordon | 169,372 | <p>Let
\begin{align*}
f(x) &= \sum_{n=1}^{\infty} \frac{x^{2n-1}}{2^n}\\
&=\frac{x}{2}\sum_{n=0}^{\infty}\left(\frac{x^2}{2}\right)^n\\
&=\frac{x}{2-x^2}.
\end{align*}
Then
\begin{align*}
f'(x) = \frac{2+x^2}{(2-x^2)^2}.
\end{align*}
Therefore,
\begin{align*}
\lim_{n\rightarrow \infty}X(n) = f'(1)=3.
\end{a... |
3,562,202 | <p>Show that <span class="math-container">$$\int_0^{2\pi}\arctan\Bigl( \frac {\sin x} {\cos x -2 }\Bigl) \, dx=0.$$</span> I can write this as <span class="math-container">$$\mathrm {Im}\int_0^{2\pi}\log( e^{ix}-2)\,dx.$$</span> Making the substitution <span class="math-container">$z =e^{ix} $</span> leads to <span c... | Eugene | 726,796 | <p><span class="math-container">$$
\begin{aligned}
\int_0^{2\pi}\arctan\left(\frac{\sin x}{\cos x - 2}\right)dx &=
\int_0^{\pi}\arctan\left(\frac{\sin x}{\cos x - 2}\right)dx +
\int_{\pi}^{2\pi}\arctan\left(\frac{\sin x}{\cos x - 2}\right)dx = \\
|\text{in a second integral}: x = 2\pi-y| &\Leftrightarrow \\
... |
2,461,820 | <p>I am a bit confused with the following question, I get that P(T|D) = 0.95 and P(D) = 0.0001 however because i'm unable to work out P(T|~D) i'm struggling to apply the theorem, am i missing something? Also i'm unsure about what to do with the information relating to testing negative when you don't have the disease co... | Siong Thye Goh | 306,553 | <p>$\ln e = 1$.</p>
<p>You just have to evaluate $\ln 1$.</p>
<p>The input of a natural log has to be positive, the output need not be positive.</p>
|
2,461,820 | <p>I am a bit confused with the following question, I get that P(T|D) = 0.95 and P(D) = 0.0001 however because i'm unable to work out P(T|~D) i'm struggling to apply the theorem, am i missing something? Also i'm unsure about what to do with the information relating to testing negative when you don't have the disease co... | XRBtoTheMOON | 478,087 | <p>The natural log of a number is whatever you have to raise $e$ to, in order to get that number. So for example $\ln(10)$ is saying, "what do I need to raise $e$ to, in order to get 10?. So you could plug that in your calculator, and see $\ln(10) \approx 2.3025$ and then you could check it by seeing what $e^{2.3025}$ ... |
2,461,820 | <p>I am a bit confused with the following question, I get that P(T|D) = 0.95 and P(D) = 0.0001 however because i'm unable to work out P(T|~D) i'm struggling to apply the theorem, am i missing something? Also i'm unsure about what to do with the information relating to testing negative when you don't have the disease co... | Guillemus Callelus | 361,108 | <p>$$\ln(\ln e)=\ln 1= 0$$
The logarithm on base $a$ of $a$ is always $1$, since $a^{1} = a$. In addition, it has to be $a> 0$ and $a$ other than $1$, so that there are no problems.</p>
|
3,337,382 | <p>I want to find how many roots of the equation <span class="math-container">$z^4+z^3+1=0$</span> lies in the first quadrant.</p>
<p>Using Rouche's Theorem how to find ?</p>
| Lutz Lehmann | 115,115 | <p>Look at the family of polynomials <span class="math-container">$z^4+tz^3+1$</span>. For <span class="math-container">$t=0$</span> we know the solutions <span class="math-container">$z=\sqrt{\frac12}(\pm1\pm i)$</span> which has one root per quadrant. We additionally know that the set of roots is continuous in the co... |
2,793,077 | <blockquote>
<p>Suppose $R$ is a Boolean ring. Prove that $a+a=0$ for all $a\in R$.
Also prove that $R$ is commutative. Give an example (with explanation)
of a Boolean ring.</p>
</blockquote>
<p>From what I know, a Boolean ring is a ring for which $a^2=a$ for all $a\in R$.</p>
<p>Under addition a ring is a comm... | Travis Willse | 155,629 | <p><strong>Hint</strong> Applying the definition of <em>Boolean</em> gives $$a + a = (a + a)^2 = a^2 + a^2 + a^2 + a^2 .$$ What does applying it again give?</p>
|
683,149 | <p>So I am just learning about bijections and I am having difficulty figuring out if these three problems are bijections and how to prove them. </p>
<ol>
<li>$f(x)=x/2$</li>
<li>$f(x)=2x^2$</li>
<li>$f(x)=\lfloor x\rfloor$</li>
</ol>
<p>Sorry I forgot to add the entire question. It is "Determine whether each of these... | Raven | 130,125 | <p>One way to determine whether f is bijective or not by drawing a figure (if possible). If a line parallel to x cuts f at more than one point, then f is not bijective.</p>
|
406,437 | <p>Calculus the extreme value of the $f(x,y)=x^{2}+y^{2}+xy+\dfrac{1}{x}+\dfrac{1}{y}$</p>
<p>pleasee help me.</p>
| Ink | 34,881 | <p>Given an abelian group $M$, let $\operatorname{End}(M)$ denote the set of all homomorphisms $M \to M$ (i.e endomorphisms). This set becomes a ring under pointwise addition and composition. </p>
<p>To see that $\operatorname{End}(M)$ may not be a commutative ring, choose another noncommutative ring $R$ (you alread... |
406,437 | <p>Calculus the extreme value of the $f(x,y)=x^{2}+y^{2}+xy+\dfrac{1}{x}+\dfrac{1}{y}$</p>
<p>pleasee help me.</p>
| Key Ideas | 78,535 | <p>Fairly concrete examples of noncommutative rings arise in calculus when studying differential equations using operator algebra. For example, consider the ring of linear operators generated by the derivative $\, D = d/dx\,$ and the operation of multiplication by $\,x,\,$ i.e. $\, f\mapsto xf,\,$ where $\,(L+M)f = Lf... |
2,825,237 | <p>\begin{cases}
4x \equiv 14 \pmod m \\
3x \equiv 2 \pmod 5
\end{cases}</p>
<p>I want to prove that for $m \in 4\mathbb{Z}$ there are no solutions(1). Moreover, I want to determine all m for which I have solutions(2). </p>
<p>First of all, the second equation is equivalent to $ x \equiv 4$ (mod 5).</p>
<p>If $m... | Jonathan Dunay | 538,622 | <p>The solutions to the second equation are the integers of the form $4+5k$. So, for a given $m$, having a solution to both equations is equivalent to having a $k$ and a $k_1$ such that $4(5k+4)=14+mk_1$ which is true iff $20k-mk_1=-2$. This has a solution iff $gcd(20,m)|2$.</p>
<p>The Chinese Remainder theorem does n... |
2,825,237 | <p>\begin{cases}
4x \equiv 14 \pmod m \\
3x \equiv 2 \pmod 5
\end{cases}</p>
<p>I want to prove that for $m \in 4\mathbb{Z}$ there are no solutions(1). Moreover, I want to determine all m for which I have solutions(2). </p>
<p>First of all, the second equation is equivalent to $ x \equiv 4$ (mod 5).</p>
<p>If $m... | Steven Alexis Gregory | 75,410 | <p>If $m=4n$, then</p>
<p>\begin{align}
4x \equiv 14 \pmod m
&\implies 4x \equiv 14 \pmod{4n} \\
&\implies 4n \mid 14-4x \\
&\implies 2 \mid 7-2x \\
&\implies 2 \mid 7
\end{align}</p>
<p>which is false.</p>
<hr>
<p>$3x \equiv 2 \pmod 5 \implies x \equiv 4 \pmod 5 \implies x = 4 + 5u$ ... |
619,564 | <p>I have to prove if this function is differentiable.</p>
<p>$$f(x,y)= \begin{cases} \frac{\cos x-\cos y}{x-y} \iff x \neq y \\-\sin x \iff x=y \end{cases}$$</p>
<p>if $x \neq y$ it is continuous, but i want to see if it is continuous in x=y too.</p>
<p>i can rewrite f as
$$ f(x,y)= \begin{cases} \frac{g(x)-g(y)}{... | John Hughes | 114,036 | <p>This seems like a perfectly OK way to do your work. </p>
<p>ALternatively, you might look at the function </p>
<p>$$
h(u, y) = f(u+y, y).
$$</p>
<p>Writing that out, you get
$$
h(u, y) = f(u+y,y)= \begin{cases} \frac{\cos(u+y)-\cos(y)}{u} \iff u+y \neq y \\
-\sin(y) \iff u = 0 \end{cases}
$$</p>
<p>Which you ca... |
619,564 | <p>I have to prove if this function is differentiable.</p>
<p>$$f(x,y)= \begin{cases} \frac{\cos x-\cos y}{x-y} \iff x \neq y \\-\sin x \iff x=y \end{cases}$$</p>
<p>if $x \neq y$ it is continuous, but i want to see if it is continuous in x=y too.</p>
<p>i can rewrite f as
$$ f(x,y)= \begin{cases} \frac{g(x)-g(y)}{... | Lutz Lehmann | 115,115 | <p>$\cos x-\cos y=-2\sin\frac{x+y}2\,\sin\frac{x-y}2$ and $\operatorname {sinc}u=\frac{\sin u}{u}$ is a known analytical function. So $f(x,y)=-\sin\frac{x+y}2\operatorname {sinc}\frac{x-y}2$.</p>
|
4,105,854 | <blockquote>
<p>Solve for integers <span class="math-container">$x, y$</span> and <span class="math-container">$z$</span>:</p>
<p><span class="math-container">$x^2 + y^2 = z^3.$</span></p>
</blockquote>
<p>I tried manipulating by adding and subtracting <span class="math-container">$2xy$</span> , but it didn't give me a... | 2'5 9'2 | 11,123 | <p>Factoring over Gaussian integers <span class="math-container">$$(x+iy)(x-iy)=z^3$$</span> so it is sufficient (but not necessary) for <span class="math-container">$x+iy$</span> to be a cube. That is,
<span class="math-container">$$\begin{align}x+iy&=(a+bi)^3\\
&=(a^3-3ab^2)+\left(3a^2b-b^3\right)i
\end{align... |
4,105,854 | <blockquote>
<p>Solve for integers <span class="math-container">$x, y$</span> and <span class="math-container">$z$</span>:</p>
<p><span class="math-container">$x^2 + y^2 = z^3.$</span></p>
</blockquote>
<p>I tried manipulating by adding and subtracting <span class="math-container">$2xy$</span> , but it didn't give me a... | Servaes | 30,382 | <p>Let <span class="math-container">$C$</span>, <span class="math-container">$D$</span>, <span class="math-container">$S$</span> and <span class="math-container">$T$</span> be integers, and define
<span class="math-container">\begin{eqnarray*}
x&=&ab^3X=(C^2+D^2)(CS^3-3DS^2T-3CST^2+DT^3),\\
y&=&ab^3Y=(C... |
217,244 | <p>I would like to know some of the most important definitions and theorems of definite and semidefinite matrices and their importance in linear algebra.
Thanks for your help</p>
| Aamir M. Khan | 623,862 | <p>Positive definite matrices have applications in various domains like physics, chemistry etc. In CS, optimization problems are often treated as quadratic equations of the form <span class="math-container">$Ax=b$</span> where <span class="math-container">$x$</span> can be any higher degree polynomial. To solve such eq... |
1,781,227 | <p>For a simple x and y plane (2 dimensional), to find the distance between two points we would use the formula </p>
<p>$$
a^2 +b^2 = c^2
$$</p>
<p>For a slightly more complicated plane; x,y and z (3 dimensional), to find the distance between two points we would use the formula</p>
<p>$$
d^2 = a^2 + b^2 + c^2
$$</p>... | Ethan Bolker | 72,858 | <p>Good question.</p>
<p>In the theory of relativity the fourth dimension is time, and the distance formula is weird, as @Arthur comments.</p>
<p>But it's quite possible and (for mathematicians) very natural to study spaces with four (or even $n$) geometric dimensions. You just think of a point as a list of its $n$ c... |
2,681,451 | <p>For the following question I'm getting stuck on a proof. Below I've just written out all the things/steps I've tried (even if they might be wrong). Could someone steer me in the right direction?</p>
<blockquote>
<p>Suppose $f$ is a real-valued function $f:\mathbb{R}\to\mathbb{R}$,
which is continuous at $0$, wi... | fleablood | 280,126 | <p>Outline:</p>
<p>Let $\lambda = f(1)$</p>
<p>1) Since $f(x + y) = f(x) + f(y)$ then prove for any $r \in \mathbb Q$ then $f(r) = r*f(1) = r*\lambda$. (This follows inductively.)</p>
<p>2) Since $f$ is continuous prove that $f(x) = x*f(1) = x*\lambda$ by considering a sequence of rations $q_i \to x$. As $f$ is co... |
3,339,647 | <p>I had a class in algebraic topology, our main book is Allen Hatcher, our professor defined a term called "Exponential Law" as the following:</p>
<p><span class="math-container">$Hom (X \times Y, Z) \cong Hom (X, Hom (Y, Z))$</span> </p>
<p><span class="math-container">$\alpha : X \times Y \rightarrow Z $</span></p... | Wlod AA | 490,755 | <p><strong><em>Why is it called</em> exponential law?</strong></p>
<p>Let <span class="math-container">$|X|$</span> be the cardinality of set <span class="math-container">$X$</span> if <span class="math-container">$X$</span> is considered to be a set.</p>
<p>We have, in the <strong>category of sets</strong>,</p>
<p>... |
2,787,227 | <p>I'm trying to compute two integrals involving the Dirac delta, namely
\begin{align}
I_1&=\int^{1}_0\!\!\!\! dx_1\!\cdots\!\int^{1}_0\!\!\!\! dx_{8}\,\delta(x_1-x_2+x_3-x_4+x_5-x_6+x_7-x_8)\,,\\
I_2&=\int^{1}_0\!\!\!\! dx_1\!\cdots\!\int^{1}_0\!\!\!\! dx_{8}\,\delta(x_1-x_2+x_4-x_5)\delta(x_3-x_4+x_6-x_7)\,\d... | greg | 357,854 | <p>Define the matrix
$$\eqalign{
A &= \sqrt{I+\alpha xx^T} \cr
A^2 &= I+\alpha xx^T \cr
}$$
and define separate variables for the length and direction of the $x$ vector
$$\eqalign{
\lambda &= \|x\| \cr
n &= \lambda^{-1}x \cr
}$$</p>
<p>We can create an ortho-projector $P$ with $n$ in its nullspace... |
850,046 | <p>When we calculate sine/cos/tan etc. of a number what exactly are we doing in terms of elementary mathematical concept, please try to explain in an intuitive and theoretical manner and as much as possible explain in the most basic mathematical way.</p>
| 2'5 9'2 | 11,123 | <p>If you have a unit-radius circle centered at the origin, place yourself at $(1,0)$. Now to calculate $\sin(x)$ for the given number $x$, move counter-clockwise around the circle until you have traveled a distance $x$. Wherever you land, the $y$-coordinate is $\sin(x)$. And the $x$-coordinate is $\cos(x)$. The slope ... |
850,046 | <p>When we calculate sine/cos/tan etc. of a number what exactly are we doing in terms of elementary mathematical concept, please try to explain in an intuitive and theoretical manner and as much as possible explain in the most basic mathematical way.</p>
| Tony Piccolo | 71,180 | <p>It is not the whole story, anyhow you could have a "aha!" moment.</p>
<p>I refer to G. A. Jennings $\,$ <em>Modern Geometry with Applications</em> $\,$ (1994), pp. 25-26, paraphrasing the text.</p>
<p>To solve practical problems, initially they drew and measured scale models of triangles; then someone realized tha... |
3,162,056 | <p>In Kleene's "Mathematical Logic" and "Introduction to Metamathematics" for a classical predicate calculus the following two rules of inference are chosen.</p>
<p>If <span class="math-container">$A(x) \Rightarrow C$</span> then <span class="math-container">$(\exists xA(x)) \Rightarrow (C)$</span> and
if <span class=... | Alex Kruckman | 7,062 | <p>The additional rules you give are indeed valid (on non-empty domains). The reason for this is that the implication <span class="math-container">$\forall x\, A(x) \Rightarrow \exists x\, A(x)$</span> is valid (on non-empty domains) - if I recall correctly, this is sometimes called "existential import". </p>
<p>So if... |
2,445,855 | <p>1)How is this letter or symbol pronounced mathematically?</p>
<p>$$\overline k$$</p>
<p>2) $'$ is this sign just a symbol of derivative? For example: </p>
<p>$$k'$$ Do we only understand this as a derivative?</p>
| gen-ℤ ready to perish | 347,062 | <ol>
<li>“Kay bar” (or maybe “bar kay” if $k$ is replaced with something long)</li>
<li>“Kay prime” could hypothetically be used to notate anything, depending on the author and field. However, $k’$ would generally be accepted as any of the following:
<ul>
<li>the derivative of a function $k$</li>
<li>the complement of... |
4,535,612 | <blockquote>
<p>Is a group of order <span class="math-container">$2^kp$</span> not simple, where <span class="math-container">$p$</span> is a prime and <span class="math-container">$k$</span> is an positive integer?</p>
</blockquote>
<p>I did this for the groups of order <span class="math-container">$2^k 3$</span>. Her... | orangeskid | 168,051 | <p>Here is a standard way to get an equation for the image of a plane parametrized curve:</p>
<p><span class="math-container">$$x = \phi(t) \\
y = \psi(t)$$</span></p>
<p>From the first equality obtain a polynomial equation for <span class="math-container">$t$</span> with coefficients dependent on <span class="math-con... |
626,095 | <blockquote>
<p>$R = \mathbb{F}_3[x]/\langle X^3+X^2+1\rangle$ and $\alpha=[X]$ in $R$. How do you prove that the group $R^*$ is not cyclic?</p>
</blockquote>
<p>We have shown that $\alpha$ is a unit in $R$ with order $8$ and that $\alpha^4$ and $-\alpha^4$ are two different elements in $R^*$ both with order 2.</p>
| Josué Tonelli-Cueto | 15,330 | <p>Hint: How many elements of order two have a cyclic group? If $R^*$ has two or more, is it compatible with it being cyclic?</p>
|
1,435,603 | <blockquote>
<p>Let $S$ be a subset of $\mathbb{R}$ with strictly positive Lebesgue measure. Prove that almost every (with respect to Lebesgue measure) real number can be written as the sum of an element of $S$ and an element of $\mathbb{Q}$. </p>
</blockquote>
<p>So, I remember proving a long time ago that $S-S$ c... | Eugene Zhang | 215,082 | <p>This can be formulated and proved as the following theorem.</p>
<p><strong>Theorem:</strong> Let <span class="math-container">$S$</span> be a Lebesgue measurable set on <span class="math-container">$[a,b]$</span> with <span class="math-container">$m(S)>0$</span>. Then<br />
<span class="math-container">$$
\bigcup... |
151,076 | <p>If A and B are partially-ordered-sets, such that there are injective order-preserving maps from A to B and from B to A, is there necessarily an order-preserving bijection between A and B ?</p>
| Brian M. Scott | 12,042 | <p>No. Let $P$ be the disjoint union of chains of every odd length, and let $Q$ be the disjoint union of chains of every positive even length. Clearly each can be embedded in the other, but the two are not isomorphic.</p>
<p>More formally, let $$P=\{\langle 2n+1,k\rangle:n\in\Bbb N\text{ and }1\le k\le 2n+1\}$$ and $$... |
4,467,763 | <p>I have the equation</p>
<p><span class="math-container">$A\vec{x} = \vec{b} \tag{1}.$</span></p>
<p>where <span class="math-container">$A$</span> is an <span class="math-container">$m\times n$</span> matrix of rank <span class="math-container">$m$</span>, so that <span class="math-container">$m<n$</span> and the ... | Charlie Vanaret | 741,916 | <p>A more geometrical interpretation is to project <span class="math-container">$b$</span> onto the column space of <span class="math-container">$A$</span>. This projection is a linear combination of the columns of <span class="math-container">$A$</span>, given by <span class="math-container">$Ax$</span>. Then the &quo... |
3,741,222 | <p>I'm reading the Thompson's book about lattices and sphere packing and got stuck by a sentence of a kind of <span class="math-container">$Z_8$</span> he introduced to reach 2 pages later the full <span class="math-container">$E_8$</span> lattice.
You can find this lattice defined at pages 73-74 and it's basically. To... | riccardoventrella | 264,042 | <p>I think to have found the explanation of the binary trick. First of all, let's consider this lattice has all coordinates even or all odd, by definition.
The idea was suggested to me in the way in which Leech enriched the extended Golay code <span class="math-container">$C_{24}$</span>.
Let's write a generic point st... |
1,513,056 | <p>Why is a limit of an <strong>integer</strong> function $f(x)$ also integer? For example, a function that's defined on interval $[a, \infty)$ and the limit is $L$</p>
| Lutz Lehmann | 115,115 | <p>Do you mean $f:[a,\infty)\to\Bbb Z$ as integer function? In $\Bbb R$, the set $\Bbb Z$ consists only of isolated points.</p>
|
3,987,357 | <p>Let
<span class="math-container">$$
\psi(x)=
\left\{
\begin{array}{cll}
x \sin\Big(\dfrac{1}{x}\Big) & \text{if} & x\in (0,1],\, \\
0 & \text{if} & x=0,
\end{array}
\right.
$$</span>
and let <span class="math-container">$f:[-1,1]\rightarrow \mathbb{R}$</span> be Riemann integrable.</p>
<p>How can I s... | zhw. | 228,045 | <p>i) If <span class="math-container">$\psi:[\alpha,\beta]\to [a,b]$</span> is <span class="math-container">$C^1$</span> and <span class="math-container">$\psi'$</span> has only finitely many zeros, then <span class="math-container">$f\circ \psi$</span> is RI on <span class="math-container">$[\alpha,\beta]$</span> for ... |
791,020 | <p>From my textbook. </p>
<p>$$\sum\limits_{k=0}^\infty (-\frac{1}{5})^k$$</p>
<p>My work:</p>
<p>So a constant greater than or equal to $1$ raised to ∞ is ∞.</p>
<p>A number $n$ for $0<n<1$ is $0$. So when taking the limit of this series you get 0 but when formatting the problem a different way $(-1)^k/(5^k)... | Pavelshu | 82,360 | <p>If what you meant to ask about was a binomial expansion, then the approach is as follows: </p>
<p>First, note that the binomial theorem states that
\begin{align*}
(a+b)^n = \sum_{k=0}^{\infty}\binom{n}{k}a^kb^{n-k}
\end{align*}
Thus, by substituting for the above values, we have
\begin{align*}
(x-2)^\frac{1}{2} = ... |
1,844,894 | <p>To explain my question, here is an example.</p>
<p>Below is an AP:</p>
<p>2, 6, 10, 14....n</p>
<p>Calculating the nth term in this sequence is easy because we have a formula. The common difference (d = 4) in AP is constant and that's why the formula is applicable, I think.</p>
<p>But what about this sequence:</... | parsiad | 64,601 | <p>I will change your notation a bit and use $x_{-1}\equiv n$ instead.</p>
<p><strong>Hint</strong>: for $m>0$,
$$
\sum_{x_{0}=0}^{x_{-1}-1}\cdots\sum_{x_{m}=0}^{x_{m-1}-1}1=\sum_{x_{0}=0}^{x_{-1}-1}\left[\sum_{x_{1}=0}^{x_{0}-1}\cdots\sum_{x_{m}=0}^{x_{m-1}-1}1\right]=\cdots
$$
You should use an induction hypothes... |
983,200 | <p>Find the derivative of the following: $$f(x)=(x^3-4x+6)\ln(x^4-6x^2+9)$$ Would I use the chain rule and product rule?
So far I have:</p>
<p>$$\begin{align}g(x)=x^3-4x+6
\\g'(x)=2x^2-4\end{align}$$</p>
<p>would $h(x)$ be $\ln(x^4-6x^2+9)$?
If so, how would I find $h'(x)$?</p>
| Dr. Sonnhard Graubner | 175,066 | <p>given $f(x)=(x^3-4*x+6)\ln(x^4-6x^2+9)$ we find by the product and chaine rule
$f'(x)=(3x^2-4)\ln(x^4-6x^2+9)+(x^3-4x+6)\frac{4x^3-12x}{x^4-6x^2+9}$
simplifying this in a few minutes
simplifying this i got
$f'(x)=\frac{4 x^4-16 x^2+3 x^4 \ln \left(x^4-6 x^2+9\right)-13 x^2 \ln \left(x^4-6
x^2+9\right)+12 \ln \lef... |
1,523,427 | <p>Is it possible to cover all of $\mathbb{R}^2$ using balls $\{ B(x_n,n^{-1/2})\}_{n=1}^\infty$ of decreasing radius $n^{-1/2}$? I know that if we chose e.g. radius $n^{-1}$ it could never work because $\sum \pi (n^{-1})^2 < \infty$. But in this case the balls cover an infinite amount of area, so it seems that it m... | detnvvp | 85,818 | <p>Yes, this can be done.</p>
<p>Let $n\in\mathbb N$, consider $m\in\{1,\dots n\}$, and set $$a(n,m)=n^2+(m-1)n.$$ Note then that $$\sum_{k=a(n,m)+1}^{a(n,m+1)}\frac{1}{\sqrt{k}}\geq\sum_{k=a(n,m)+1}^{a(n,m+1)}\frac{1}{\sqrt{n^2+mn}}=\frac{n}{\sqrt{n^2+mn}}\geq\frac{n}{\sqrt{n^2+n^2}}=\frac{1}{\sqrt{2}}.$$ Therefore, ... |
354,986 | <p>I have to show that $h$ is measurable as well as $\int h d(\mu \times \nu) < \infty$ .</p>
<p>I tried showing by contradiction that $\int h$ had to be finite but I'm stuck with showing how it is measurable.</p>
| icurays1 | 49,070 | <p>Hint: show that $\int f(x)g(y) d(\mu\times \nu)=\int f(x)(\int g(y)d\nu)d\mu$.</p>
|
1,985,653 | <p>I want to show that {$\sqrt 2$} $\cup$ {$\sqrt 2$ + $1/n$ : $n$ $∈$ $\mathbb{N}$} is closed. I'm having trouble. I've been trying to show that the complement is open, but the presence of {$\sqrt 2$} is confusing me.</p>
| 5xum | 112,884 | <p>Let's call your set $A$.</p>
<hr>
<p>The easiest way to show that $A$ is closed is by seeing that the complement of $A$ is the set</p>
<p>$$(-\infty, \sqrt{2})\cup (\sqrt 2+1,\infty)\cup \bigcup_{n=1}^\infty \left(\sqrt 2+ \frac{1}{n+1}, \sqrt 2 + \frac1n\right)$$</p>
<p>which is pretty clearly open. Of course, ... |
1,511,733 | <p>B = matrix given below. I is identity matrix.</p>
<pre><code> [1 2 3 4]
[3 2 4 3]
[1 3 2 4]
[5 4 3 7]
</code></pre>
<p>So What will be the relation between the matrices A and C if AB = I and BC = I?</p>
<p>I think that A = C because both AB and BC have B in common and both of their product is an identity matri... | egreg | 62,967 | <p>Suppose $AB=I$ and $BC=I$; then
$$
A=AI=A(BC)=(AB)C=IC=C
$$</p>
|
4,274,085 | <p>A linear equation in one variable <span class="math-container">$x$</span>, <span class="math-container">$$ax+b=k$$</span> has only one non-negative integer solution. For example, <span class="math-container">$2x+3=85$</span> has a solution 41.</p>
<p>How to find the number of non-negative integer solutions of a line... | Chris Langfield | 978,915 | <p>Given <span class="math-container">$a, b, k$</span>, your linear equation defines a line on the plane: <span class="math-container">$y = k - \frac{a}{b} x$</span></p>
<p>The solutions will be limited to <span class="math-container">$x$</span> less than the <span class="math-container">$x$</span>-intercept, and <span... |
335,483 | <p>Let $N$ be a set of non-negative integers. Of course we know that $a+b=0$ implies that $a=b=0$ for $a, b \in N$.</p>
<p>How do (or can) we prove this fact if we don't know the subtraction or order?</p>
<p>In other words, we can only use the addition and multiplication.</p>
<p>Please give me advise.</p>
<p>EDIT</... | marty cohen | 13,079 | <p>What is the addition law?</p>
<p>If it the one from Peano arithmetic,
it is
$x+0=x$ and $x+S(y) = S(x+y)$,
where $S(x)$ is the successor of $x$.</p>
<p>If $x+y=0$,
suppose $y$ was not zero.
Then there is a $z$ such that
$S(z) = y$.</p>
<p>Then $0 = x+y = x+S(z) = S(x+z)$
which is a contradiction, since
$0$ is ... |
3,156,962 | <blockquote>
<p>Find general term of <span class="math-container">$1+\frac{2!}{3}+\frac{3!}{11}+\frac{4!}{43}+\frac{5!}{171}+....$</span></p>
</blockquote>
<p>However it has been ask to check convergence but how can i do that before knowing the general term. I can't see any pattern,comment quickly!</p>
| John Omielan | 602,049 | <p>As you've already shown, factoring out the cube of the GCD of <span class="math-container">$a$</span> and <span class="math-container">$b$</span> gives a new equation</p>
<p><span class="math-container">$$e = \gcd\left(\alpha^3-3\alpha \beta^2, \beta^3-3\beta \alpha^2 \right) \tag{1}\label{eq1}$$</span></p>
<p>whe... |
3,155,463 | <blockquote>
<p><span class="math-container">$$
\lim _{n \rightarrow \infty}\left[n-\frac{n}{e}\left(1+\frac{1}{n}\right)^{n}\right] \text { equals }\_\_\_\_
$$</span></p>
</blockquote>
<p>I tried to expand the term in power using binomial theorem but still could not obtain the limit. </p>
| MarcM | 655,795 | <p>The problem is that you cannot use Binomial Theorem because you cannot control the behaviour of <span class="math-container">$\binom{n}{k}\frac{1}{n^k}$</span> when <span class="math-container">$n$</span> and <span class="math-container">$k$</span> increase towards infinity. One must expand <span class="math-contain... |
98,196 | <p>Ok, so I am running a classic linear regression where betahat = (X'X)^-1X'y</p>
<p>Due to performance issues, I would like to estimate betahat with an additional data point (x1,x2,x3,x4,...,y) without recalculating based on the whole history. </p>
<p>Could I do some sort of multiplication based on Xmu or std devia... | Brian Borchers | 9,022 | <p>There's a very large literature on updating solutions to least squares problems as new data are added. The naive formula $\hat{\beta}=(X^{T}X)^{-1}X^{T}y$ can be problematic in practice because of numerical issues with ill-conditioning. The QR factorization is typically a much better choice in practice. There are... |
98,196 | <p>Ok, so I am running a classic linear regression where betahat = (X'X)^-1X'y</p>
<p>Due to performance issues, I would like to estimate betahat with an additional data point (x1,x2,x3,x4,...,y) without recalculating based on the whole history. </p>
<p>Could I do some sort of multiplication based on Xmu or std devia... | user44099 | 44,099 | <p>You might look into the Kalman filter <a href="http://www.lce.hut.fi/~ssarkka/course_k2012/handout2.pdf" rel="nofollow">http://www.lce.hut.fi/~ssarkka/course_k2012/handout2.pdf</a>
to do the update.</p>
|
877,687 | <p>So my textbook's explanation of the derivative of e is very sketchy. They used lots of approximations and plugging things into the calculator. Basically I want to know how you can work out as h approaches 0</p>
<p>$$
\lim_{h\to0}\frac {10^{x+h}-10^x }h
$$</p>
| mvw | 86,776 | <p>We try to model the family of lines and then try to infer the envelope.</p>
<p>The guiding lines (left and right arms of the V shape) are
$$
g(t) = u_g \, (1-t) + v_g \, t \quad h(t) = u_h \, (1-t) + v_h \, t
$$
for $t \in [0, 1]$, where $u$ is the start point and $v$ the end point of that line.</p>
<p>A line $f_r... |
1,610,616 | <blockquote>
<p>$6$ letters are to be posted in three letter boxes.The number of ways
of posting the letters when no letter box remains empty is?</p>
</blockquote>
<p>I solved the sum like dividing into possibilities $(4,1,1),(3,2,1)$ and $(2,2,2)$ and calculated the three cases separately getting $90,360$ and $90... | Henry | 6,460 | <p><a href="https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind" rel="nofollow">Stirling numbers of the second kind</a>: your particular example of six letters and 3 letterboxes has a solution of $3! \,S_2(6,3) = 6 \times 90 =540$ possible surjections, the same as the result you found</p>
<p>Using expone... |
1,518,697 | <blockquote>
<p>For this homework exercise, we are asked to show that the ideal $I=(3,1+\sqrt{-5})$ is a flat $\mathbb{Z}[\sqrt{-5}]$-module. The hint is to show that $I$ becomes principal (and thus free as a module) when we invert $2$ <strong>or</strong> $3$, so that $I$ is locally flat.</p>
</blockquote>
<p>I'm ha... | B. Pasternak | 283,676 | <p><em>So, a couple days later, a little wiser. Here it goes.</em></p>
<p>Indeed, as Gerry suggested, inverting $(2)$ makes $I$ equal to $(\tfrac{1}{2}(1+\sqrt{-5}))$ and inverting $(3)$ makes $I$ equal to the whole ring since $I$ contains a unit, so in both cases $I$ is principal and thus free as a module.</p>
<p><s... |
3,256,881 | <p>How to prove that?
I try to use the comparasion test, but i don't know with that function compare.</p>
| azif00 | 680,927 | <p><span class="math-container">$$-1\le \sin\left(x+\frac{1}{x}\right)\le1$$</span>
And, therefore
<span class="math-container">$$-1\le \int_0^1\sin\left(x+\frac{1}{x}\right)\mathrm{d}x\le1$$</span></p>
|
3,501,052 | <p>I want to find the number of real roots of the polynomial <span class="math-container">$x^3+7x^2+6x+5$</span>.
Using Descartes rule, this polynomial has either only one real root or 3 real roots (all are negetive). How will we conclude one answer without doing some long process?</p>
| Community | -1 | <h2>Use the Discriminant formula for a 3rd ordered polynomial.Click <a href="https://en.wikipedia.org/wiki/Discriminant" rel="nofollow noreferrer" title="this text appears when you mouse over">here</a>!</h2>
<p><strong>N.B</strong> <em>The discriminant is zero if and only if at least two roots are equal. If the coeffic... |
2,869,031 | <p>Let $A$ a domain, i.e. $ab\in A\implies a=0$ or $b=0$. It's written that all domains are commutative. Is it by definition, or can we prove that domains are commutative? I mean, do we only consider domaind for commutative rings, or is a ring that is a domain then commutative?</p>
| quid | 85,306 | <p>It is "by definition" though, as the other answer indicates, some might disagree with the terminology. </p>
<p>It is perfectly possible to have a non-commutative ring that has no zero-divisors (except for $0$). Examples would be, for example, non-commutative <a href="https://en.wikipedia.org/wiki/Division_ring" re... |
941,632 | <p>Is the Set $$S=\{e^{2x},e^{3x}\}$$ linearly independent?? And answer says Linearly independent over any interval $(a,b)$,only when $0$ doesnot belong to $(a,b)$</p>
<p>How do I proceed??</p>
<p>Thanks for the help!!</p>
| tattwamasi amrutam | 90,328 | <p>Consider $$c_1e^{2x}+c_2e^{3x}=0$$. Then Differentiating once you will get $$2c_1e^{2x}+3c_2e^{3x}=0$$. Again differentiate it to get $$4c_1e^{2x}+9c_2e^{3x}=0$$. Plug in a value $x_0$ such that $x_0$ belongs to $(a,b)$. Then the equation reduces to </p>
<p>$$2c_1e^{2x_0}+3c_2e^{3x_0}=0$$ and $$4c_1e^{2x_0}+9c_2e^{... |
941,632 | <p>Is the Set $$S=\{e^{2x},e^{3x}\}$$ linearly independent?? And answer says Linearly independent over any interval $(a,b)$,only when $0$ doesnot belong to $(a,b)$</p>
<p>How do I proceed??</p>
<p>Thanks for the help!!</p>
| Belgi | 21,335 | <p>Hint: Consider the Wronskian.
For more details see <a href="http://en.wikipedia.org/wiki/Wronskian" rel="nofollow">http://en.wikipedia.org/wiki/Wronskian</a></p>
|
4,518,908 | <p>For sufficiently large integer <span class="math-container">$m$</span>, in order to prove</p>
<p><span class="math-container">$\frac{(m+1)}{m}<\log(m)$</span></p>
<p>is it sufficient to point out that</p>
<p><span class="math-container">$ \displaystyle\lim_{m \to \infty} \frac{(m+1)}{m}=1 $</span></p>
<p>while</p... | ryang | 21,813 | <blockquote>
<p>Suppose that <span class="math-container">$$\forall x {\in} \mathbb{R} \begin{bmatrix} P(x) \implies \begin{bmatrix} \forall y {\in}\mathbb{R} \;P(x + y) \end{bmatrix} \end{bmatrix}.$$</span> Is this sufficient to conclude that <span class="math-container">$$\begin{bmatrix}\forall x {\in}\mathbb{R}\; P(... |
273,308 | <p>Find the standard form of the conic section $x^2-3x+4xy+y^2+21y-15=0$.</p>
<p>I understand the approach in trying to solve these problems. But the $4xy$ is confusing me. I am not sure of where to start on this one.</p>
<p><strong>UPDATE:</strong></p>
<p>i have found a way of doing this with reference to <a href="... | lab bhattacharjee | 33,337 | <p>So, $$m=\frac{-4\pm\sqrt{4^2-4\cdot1\cdot5}}2=-2\pm i$$</p>
<p>Then as the roots are unequal, $$y=Ae^{(-2+i)x}+Be^{(-2-i)x}$$ where $A,B$ are arbitrary constants.</p>
<p>$$y=e^{-2x}(Ae^{ix}+Be^{-ix})$$
$$=e^{-2x}\{(A+B)\cos x+i(A-B)\sin x\}$$ using <a href="http://mathworld.wolfram.com/EulerFormula.html" rel="nofo... |
273,308 | <p>Find the standard form of the conic section $x^2-3x+4xy+y^2+21y-15=0$.</p>
<p>I understand the approach in trying to solve these problems. But the $4xy$ is confusing me. I am not sure of where to start on this one.</p>
<p><strong>UPDATE:</strong></p>
<p>i have found a way of doing this with reference to <a href="... | Community | -1 | <p>If you want all solution real: </p>
<p>$m^2+4m+5=0$ $\implies$ $m=-2\pm i$.</p>
<p>Then $e^{(-2+i)t}=e^{-2t}(\cos{t}+i\sin{t}) $ and $e^{(-2-i)t}=e^{-2t}(\cos{t}-i\sin{t})$ are two solutions.</p>
<p>Clearly linear combinations these solutions still are solutions. Then</p>
<p>$\phi(t)=\dfrac{e^{(-2+i)t}+e^{(-2-i)... |
3,777,042 | <p>What is the particular solution to <span class="math-container">$ \frac{dy}{dx} =\cos(x^2)$</span> with the initial condition <span class="math-container">$y\left( \sqrt{\frac{\pi}{2}}\right)=3$</span>?</p>
<p>A.) <span class="math-container">$y = 3 + \int_{0}^{x}\cos(t^2)dt$</span></p>
<p>B.) <span class="math-cont... | Community | -1 | <p>I'll give a outline about it problem.
<span class="math-container">$$\frac{dy}{dx}=\cos(x^2) \implies y=\int \cos(x^2)dx +K$$</span>
But we know that <span class="math-container">$$\int \cos(x^2)dx=\sqrt{\frac{\pi}{2}}C\left( \sqrt{\frac{2}{\pi}}x\right)+K$$</span> where <span class="math-container">$C(x)$</span> is... |
3,777,042 | <p>What is the particular solution to <span class="math-container">$ \frac{dy}{dx} =\cos(x^2)$</span> with the initial condition <span class="math-container">$y\left( \sqrt{\frac{\pi}{2}}\right)=3$</span>?</p>
<p>A.) <span class="math-container">$y = 3 + \int_{0}^{x}\cos(t^2)dt$</span></p>
<p>B.) <span class="math-cont... | Digitallis | 741,526 | <p>For each proposed solution we need to check if <span class="math-container">$\frac{dy}{dx} = \cos(x^2)$</span> and <span class="math-container">$y\left(\sqrt \frac \pi 2 \right) = 3$</span>. The fundamental theorem of calculus will come in handy !</p>
<p>Let <span class="math-container">$x_0 = \sqrt \frac \pi 2$</sp... |
2,069,001 | <p>A team of seven netballers is to be chosen from a squad of twelve players A,B,D, E, F, G, H, I, J, K, L. In how many ways can they be chosen:<br>
a) with no restriction
This is fairly easy. 12C7 = 792</p>
<p>b) if the captain C is to be included
11C6 = 462</p>
<p>c) If J and K are both to be excluded
10C7 = 120</p... | Robert Z | 299,698 | <p>As you noted in d) if F is included but L is not then the number of teams is $\binom{10}{6}$.</p>
<p>Similarly if L is included but F is not then the number is $\binom{10}{6}$.</p>
<p>These two sets of teams are disjoint therefore the total number is $$\binom{10}{6}+\binom{10}{6}=420.$$.</p>
|
1,204,864 | <blockquote>
<p>$$\text{Find }\,\dfrac{d}{dx}\Big(\cos^2(5x+1)\Big).$$</p>
</blockquote>
<p>I have tried using the rules outlined in my standard derivatives notes but I've failed to find the point of application.</p>
| Jordan Glen | 225,803 | <p>$$y = \cos^2(5x+1)=\Big(\cos(5x+1)\Big)^2$$</p>
<p>$$\frac{dy}{dx} = 2(\cos(5x+1))\cdot (-\sin(5x+1)) \cdot \frac{d}{dx}(5x+1)$$</p>
<p>$$=-10\cos(5x+1) \sin (5x+1)$$
We applied the power-rule first, then used the chain-rule, twice.</p>
|
3,224,102 | <p>For a given curve: <span class="math-container">$$C: \frac {ax^2+bx+c}{dx+e} $$</span> where <span class="math-container">$a,b,c,d,e$</span> are integers. Let <strong><span class="math-container">$f(x)=ax^2+bx+c$</span></strong> .</p>
<hr>
<p>Oblique asymptote can be found by long division of numerator by denomina... | Claude Leibovici | 82,404 | <p>For large values of <span class="math-container">$x$</span>, write <span class="math-container">$$y=\frac {ax^2+bx+c}{dx+e}=m+n x+\epsilon$$</span> and cross multiply
<span class="math-container">$${ax^2+bx+c}=(m+n x+\epsilon)(d x+e)=d n x^2+x (d m+e n +d \epsilon) +e( m+ \epsilon)$$</span> </p>
<p>Compare the coef... |
2,485,276 | <p>Is the statement true? if it is, how to prove it?</p>
<p>If $\binom{p}{k} \mod p=0 $ for $k=1,2,..,p−1$ then $p$ is prime.</p>
| Ricardo Largaespada | 305,474 | <p>Hint.$${p \choose k} = \frac{p \cdot (p-1) \cdot (p-2) \cdots (p-k+1)}{1 \cdot 2 \cdot 3 \cdots k}$$</p>
|
29,155 | <p>Do we have a pullback operation on singular simplicial chains,that is if f:X-->Y is a continuous map between topological space X and Y,and C is a singular simplicial chain on Y,then do we have a singular simplicial chain on X which is the pullback of C along f?</p>
| Daniel Litt | 6,950 | <p>No, there is a pullback on singular cochains, given by composition.</p>
|
2,210,531 | <blockquote>
<p>Prove that the limit is zero:
$$ F(x, y)= \frac{x^2y^3}{3x^2+ 2y^3}$$</p>
</blockquote>
<p>Definition1. Let $ U ⊂ R^n$ be an open set and letf : $U→R^m$ beafunction with domain U. Let x0 be a vector in U or on the boundary of U. Let b∈Rm. We say that the limit of f as x approaches x0 is b, written... | Mark Viola | 218,419 | <p><strong>HINT:</strong></p>
<p>The limit, $\lim_{(x,y)\to(0,0)}\left(\frac{x^2y^3}{3x^2+2y^3} \right)$,fails to exist.</p>
<p>Examine the limit along the curve described parametrically by $x=t$ and $2y^3=-3t^2+t^a$, $a\ge4$. Then, note that </p>
<p>$$\left|\frac{x^2y^3}{3x^2+2y^3} \right|=\left|\frac{t^{a-2}+3}{2... |
178,342 | <p>This is an exercise from Kunen's book.</p>
<p>Write a formula expressing $z=\langle \langle x,y\rangle, \langle u,v\rangle \rangle$ using just $\in$ and $=$.</p>
<p>What I've tried: because $\langle x,y\rangle= \{\{x\},\{x,y\}\}$ and $\langle u,v\rangle= \{\{u\},\{u,v\}\}$, and hence $z=\{\{\langle x,y\rangle\},\... | Asaf Karagila | 622 | <p>Write a formula:</p>
<p>$$\varphi(u,v,z)=
\forall x\bigg(x\in z\leftrightarrow
\underbrace{\forall y(y\in x\leftrightarrow y=u\lor y=v)}_{\Large x=\{u,v\}}\lor\underbrace{\forall y(y\in x\leftrightarrow y=u)}_{\Large x=\{u\}}\bigg)$$</p>
<p>Note that $\varphi(u,v,z)$ holds if and only if $z=\langle u,v\rangle$.</... |
483,442 | <p>I am trying to learn about velocity vectors but this word problem is confusing me.</p>
<p>A boat is going 20 mph north east, the velocity u of the boat is the durection of the boats motion, and length is 20, the boat's speed. If the positive y axis represents north and x is east the boats direction makes an angle o... | Caleb Stanford | 68,107 | <p>Hints:</p>
<ul>
<li>In how many ways can you pull 6 socks out, one at a time, so that there are <strong>no</strong> matching pairs among them? (How many choices do you have for the first sock? The second? The third?)</li>
<li>In how many ways total can you pull 6 socks out, one at a time?</li>
<li>What does this... |
1,279,328 | <p>Can someone please explain to me if a 2/3D Poisson's equation is separable or non separable?
Thank you</p>
| ApproximatelyTrue | 240,066 | <p>For a homogenous linear PDE to be separable, you must be able to write the differential operator as the sum of two or more differential operators involving non-empty, non-overlapping subsets of the variables in the original PDE. Then, if you solve the eigenvalue problem for these differential operators (which shou... |
2,206,938 | <p>Context: <a href="http://www.hairer.org/notes/Regularity.pdf" rel="nofollow noreferrer">http://www.hairer.org/notes/Regularity.pdf</a>, section 4.1 (pages 15-16)</p>
<blockquote>
<p>Define
$$(\Pi_x\Xi^0)(y)=1 \qquad (\Pi_x\Xi)(y)=0 \qquad (\Pi_x\Xi^2)(y)=c$$
and
$$(\Pi^{(n)}_x\Xi^0)(y)=1 \qquad (\Pi^{(n)}_x... | Mohsen Shahriari | 229,831 | <p>As it's noted by @5xum, the concepts of Lipschitz continuity and Hölder continuity are related to global behaviour of the function, but I suppose you have an intuition about local properties of <span class="math-container">$ f $</span>.
So I give a hint to make a counterexample for a local version of your question.<... |
3,916,130 | <p>Assume the weight of a person follows a normal distribution N(71,7). What is the probability of 4 people weighing more than 300kg?</p>
<p>I tried solving this by multiplying the values by 4, so it'd be N(284,28). I converted that into <span class="math-container">$x=284+28z$</span> which lead to <span class="math-c... | J.G. | 56,861 | <p>Your mistake is that variances add, not standard deviations. If <span class="math-container">$7$</span> is the one-person variance, work with <span class="math-container">$300=284+\sqrt{28}z$</span>; if <span class="math-container">$7$</span> (<span class="math-container">$49$</span>) is the one-person srandard devi... |
167,848 | <p>I am trying to solve numerically an equation and generate some results. I use the following code </p>
<pre><code>u[c_] := (c^(1 - σ) - 1)/(1 - σ)
f[s_] := g s (1 - s/sbar1)
h[s_] := (2 hbar)/(1 + Exp[η (s/sbar - 1)])
co[a_] := ϕ (a^2)/2
ψ[k_] := wbar (ω + (1 - ω) Exp[-γ k])
</code></pre>
<p>The equation I try to s... | george2079 | 2,079 | <p>one approach here is to use <code>ContourPlot</code> </p>
<pre><code>p = ContourPlot[(adap[k, s] /. paramFinal2) == 0, {s, 0, 10},
{k, 0, 30}, PlotPoints -> 100]
</code></pre>
<p><a href="https://i.stack.imgur.com/TlfZA.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/TlfZA.png" alt="enter ... |
45,211 | <p>Instances of SAT induce a bipartite graph between clauses vertices and variable vertices, and for planar 3SAT, the resulting bipartite graph is planar. </p>
<p>It would be very convenient if there was a planar layout that had all the variable vertices in one line and all the clause vertices in a straight line. This... | Louigi Addario-Berry | 3,401 | <p><b>Edit:</b> When I posted this I was assuming you also wanted a straight-line drawing, which I now realize you did not say. The below relates only to straight-line drawings. </p>
<hr>
<p>This is not possible. The $3$-cube is already a counterexample. Viewing the cube as the Hamming cube, up to symmetries there is... |
248,182 | <p>Some textbooks I've seen declare inequalities such as $-2>x>2$ to have no solution, or to be ill-defined, which I disagree with. I'm curious to know if anyone else thinks the same.</p>
<p>Inequalities can always be written two ways. For example, $x>2$ is the same as $2<x$. So far as I understand, the sa... | Michael Hardy | 11,667 | <p>I wouldn't say the system of inequalities $2<x<-2$ is "ill defined", but certainly it has no solutions: there is no number that is both bigger than $2$ and less than $-2$.</p>
|
3,346,676 | <blockquote>
<p><strong>Question.</strong> Find a divergent sequence <span class="math-container">$\{X_n\}$</span> in <span class="math-container">$\mathbb{R}$</span> such that for any <span class="math-container">$m\in\mathbb{N}$</span>,
<span class="math-container">$$\lim_{n\to\infty}|X_{n+m}-X_n|=0$$</span></p>
... | Peter Foreman | 631,494 | <p>Let <span class="math-container">$X_n=\sqrt{n}$</span> then we have
<span class="math-container">$$\begin{align}
\lim_{n\to\infty}(\sqrt{n+m}-\sqrt{n})
&=\lim_{n\to\infty}\left(\sqrt{n}\left(\sqrt{1+\frac{m}n}-1\right)\right)\\
&=\lim_{n\to\infty}\left(\sqrt{n}\left(1+\frac{m}{2n}+o\left(\frac1n\right)-1\rig... |
1,530,057 | <p>At my multivariable calculus class we gave this definition for the limit of a function:</p>
<blockquote>
<p><em>Definition:</em></p>
<p>Let <span class="math-container">$ \mathbb{R}^n \supset A $</span> be a open set , let <span class="math-container">$f:A \to\mathbb{R}^m $</span> be a function, let <span class="mat... | BCLC | 140,308 | <p>$Y|X=x$ is a uniformly distributed random variable on the interval $(0,x)$</p>
<p>Hence, $$E[Y|X=x] = \int_{0}^{x} y\frac{1}{x} dy$$</p>
<p>$$= y^2\frac{1}{2x} |_{0}^{x}$$</p>
<p>$$= x^2\frac{1}{2x}$$</p>
<p>$$= \frac{x}{2}$$</p>
|
767,304 | <p>Prove that there are no real numbers $x$ such that</p>
<p>$$\sum_{n\,=\,0}^\infty \frac {(-1)^{n + 1}} {n^x} = 0$$</p>
<p>Can I have a hint please?</p>
| Daniel Fischer | 83,702 | <p>For $x \leqslant 0$, the terms of the series don't converge to $0$, hence the series diverges then. Therefore, we need only consider $x > 0$.</p>
<p>For $x > 0$, the sequence $\left(\frac{1}{n^x}\right)_{n\in\mathbb{Z}^+}$ is strictly decreasing and converges to $0$, thus by Leibniz' criterion</p>
<p>$$\eta(... |
3,006,046 | <p>How to find the Newton polygon of the polynomial product <span class="math-container">$ \ \prod_{i=1}^{p^2} (1-iX)$</span> ?</p>
<p><strong>Answer:</strong></p>
<p>Let <span class="math-container">$ \ f(X)=\prod_{i=1}^{p^2} (1-iX)=(1-X)(1-2X) \cdots (1-pX) \cdots (1-p^2X).$</span></p>
<p>If I multiply , then we... | Lubin | 17,760 | <p>It’s really quite simple. There are <span class="math-container">$p^2-p$</span> roots <span class="math-container">$\rho$</span> with <span class="math-container">$v(\rho)=0$</span>, <span class="math-container">$p-1$</span> roots with <span class="math-container">$v(\rho)=-1$</span>, and one root with <span class="... |
4,069,185 | <p>I got stuck here:</p>
<p>The probability that it will rain today is that it did not rain in the previous two days is <span class="math-container">$0.3$</span>, but if it rained in one of the last two days then the probability of rain today is <span class="math-container">$0.6$</span>.</p>
<p><span class="math-contai... | Kavi Rama Murthy | 142,385 | <p>If <span class="math-container">$0$</span> is not an eigenvalue then <span class="math-container">$T$</span> is invertible and so is any power of <span class="math-container">$T$</span>. Hence, <span class="math-container">$ker (T^{n-2})=ker(T^{n-1})=\{0\}$</span>.</p>
|
4,044,708 | <p>[3,4] is closed in R <-- R-[3,4] is open</p>
<p>[5,6] is closed in R <-- R-[5,6] is open</p>
<p>Show that [3,4] x [5,6] is closed in R x R by writing it as the complement of the intersection of two open sets in R x R.</p>
<p>(R - [3,4]) x (R - [5,6]) not equal R x R - [3,4] x [5,6]</p>
| Peter Morfe | 711,689 | <p>Martin gives a very reasonable answer extending the question's second argument; here's how to extend the first one.</p>
<p>If <span class="math-container">$u: \mathbb{R}^{d} \to \mathbb{R}$</span> is continuous and convex, then mollification of <span class="math-container">$u$</span> gives a family of smooth functio... |
2,131,679 | <p>Let $f: \mathbb{R} \to \mathbb{R}$ be continuous and $D \subset \mathbb{R}$ be a dense subset of $\mathbb{R}$. Furthermore, $\forall y_1,y_2 \in D \ f(y_1)=f(y_2)$. Should $f$ be a constant function?</p>
<p>My attempt:
Since $f$ is continuous
$$\forall x_0 \ \forall \varepsilon >0 \ \exists \delta>0 \ \forall... | Michael Rozenberg | 190,319 | <p>Since $BD=CE=AF=k$, we get $OP=OQ=PQ=\frac{3}{7}k$ and since $AP=PQ$, </p>
<p>we get$AP=PQ=PO$ and we are done!</p>
|
726,574 | <p>Ant stands at the end of a rubber string which has 1km of length. Ant starts going to the other end at speed 1cm/s. Every second the string becomes 1km longer. </p>
<p>For readers from countries where people use imperial system: 1km = 1000m = 100 000cm</p>
<p><strong>Will the ant ever reach the end of the string? ... | Community | -1 | <p>Here's an another method using point slope form of straight line.</p>
<ul>
<li><span class="math-container">$m = \dfrac12$</span></li>
<li><span class="math-container">$(x_1,y_1) = (2,-2)$</span></li>
</ul>
<p>Equation of straight line is given by,</p>
<p><span class="math-container">$(y-y_1) = m(x-x_1)$</span></p>
... |
3,088,620 | <p>Let <span class="math-container">$M$</span> be a second countable smooth manifold. When I learned about differential geometry, a side note was made about how if <span class="math-container">$E$</span> is a vector bundle, <span class="math-container">$\Gamma(E)$</span> is a <span class="math-container">$C^\infty(M)$<... | anomaly | 156,999 | <p>Let <span class="math-container">$X$</span> be a closed, smooth manifold, and let <span class="math-container">$E\to X$</span> be a vector bundle. Put <span class="math-container">$R = C^\infty(X)$</span>. By induction on <span class="math-container">$\dim E$</span> (and splitting off a trivial subbundle), it's suff... |
3,384,280 | <p>I'm trying to solve the limit of this sequence without the use an upper bound o asymptotic methods:</p>
<p><span class="math-container">$$\lim_{n\longrightarrow\infty}\frac{\sqrt{4n^2+1}-2n}{\sqrt{n^2-1}-n}=\left(\frac{\infty-\infty}{\infty-\infty}\right)$$</span></p>
<p>Here there are my differents methods:</p>
... | Barry Cipra | 86,747 | <p><strong>Hint</strong>:</p>
<p><span class="math-container">$$\begin{align}
{\sqrt{4n^2+1}-2n\over\sqrt{n^2-1}-n}
&={\sqrt{4n^2+1}-2n\over\sqrt{n^2-1}-n}\cdot{\sqrt{4n^2+1}+2n\over\sqrt{4n^2+1}+2n}\cdot{\sqrt{n^2-1}+n\over\sqrt{n^2-1}+n}\\
&={(4n^2+1)-4n^2\over(n^2-1)-n^2}{\sqrt{n^2-1}+n\over\sqrt{4n^2+1}+2n... |
3,384,280 | <p>I'm trying to solve the limit of this sequence without the use an upper bound o asymptotic methods:</p>
<p><span class="math-container">$$\lim_{n\longrightarrow\infty}\frac{\sqrt{4n^2+1}-2n}{\sqrt{n^2-1}-n}=\left(\frac{\infty-\infty}{\infty-\infty}\right)$$</span></p>
<p>Here there are my differents methods:</p>
... | xpaul | 66,420 | <p>Note
<span class="math-container">\begin{eqnarray}
\lim_{n\to\infty}\frac{\sqrt{4n^2+1}-2n}{\sqrt{n^2-1}-n}&=&\lim_{n\to\infty}\frac{(\sqrt{4n^2+1}-2n)(\sqrt{4n^2+1}+2n)}{(\sqrt{n^2-1}-n)(\sqrt{n^2-1}+n)}\cdot\frac{\sqrt{n^2-1}+n}{\sqrt{4n^2+1}+2n}\\
&=&-\lim_{n\to\infty}\frac{\sqrt{n^2-1}+n}{\sqrt{4... |
2,764,141 | <p>This is what I have tried so far: </p>
<p>Since $g(z)$ is bounded, then $\lim\limits_{z\rightarrow 0} zg(z)=0$ and hence $z=0$ is a removable singularity of $g(z)$. We can define $g(0) = \lim\limits_{z\rightarrow 0} f(z)f(\frac{1}{z})$ and make $g$ entire.</p>
<p>Then $g(z)$ is a bounded entire function and hence ... | Chappers | 221,811 | <p>Suppose that $c \neq 0$, or the equation $f(z)f(1/z)=c$ has only zero solutions. $f$ is analytic, so it has a zero of order $n$ at $z=0$, where $n$ can be zero. Since $c \neq 0$, $f \neq 0$ since $f(1)^2=c$, for example.</p>
<p>Then $h(z) = f(z)/z^n$ can be extended to an entire function by adding the value $h(0) =... |
4,241 | <p>I was preparing for an area exam in analysis and came across a problem in the book <em>Real Analysis</em> by Haaser & Sullivan. From p.34 Q 2.4.3, If the field <em>F</em> is isomorphic to the subset <em>S'</em> of <em>F'</em>, show that <em>S'</em> is a subfield of <em>F'</em>. I would appreciate any hints on ho... | User3568 | 1,767 | <p>Yes, you are right in saying that field axioms are restrictive. Some other examples of the restriction [for finite fields] are:</p>
<p>1) You can't have field of arbitrary order. Only of the order $p^n$ are possible. </p>
<p>2) Non-zero elements of a field form a multiplicative group. When the field is finite, thi... |
3,520,269 | <p><img src="https://i.stack.imgur.com/mdM8B.png" alt="enter image description here"></p>
<p>I also know that given the length of 2 sides in a kite and the angle of one of the other angles (which aren't included angles), you can find the area by multiplying these two sides and the sine of the angle. I am unable to fin... | Quanto | 686,284 | <p>The quadrilateral consists of two triangles of the same base <span class="math-container">$d_1$</span>. Therefore the area is </p>
<p><span class="math-container">$$A=\frac12h_1b_1+ \frac12h_2b_1=\frac12(h_1+h_2)d_1$$</span></p>
<p>Let the two segments of <span class="math-container">$d_2=x+y$</span> and recognize... |
3,858,962 | <p>given a rectangle <span class="math-container">$ABCD$</span> how to construct a triangle such that <span class="math-container">$\triangle X, \triangle Y$</span> and <span class="math-container">$\triangle Z$</span> have equal areas.i dont know where to start. .i tried some algebra with the area of the triangles an... | Shreyas Reddy | 834,702 | <p><img src="https://i.stack.imgur.com/lLSyI.png" alt="enter image description here" /></p>
<p>well this is my first answer so i cant upload a picture, so i am using the picture in the above answer.
Consider, AB=CD=a...
BD=AC=b...
CF=c..
CE=d.'' so, AE=b-d, FD=a-c..
as the areas of 3 triangles(right angled) equal,
a(b-... |
36,756 | <p>In a class, 18 students like to play chess, 23 like to play soccer, 21 like biking, and 17 like jogging. The number of those who like to play both chess and soccer is 9. We also know
that 7 students like chess and biking, 6 students like chess and jogging, 12 like soccer and
biking, 9 like soccer and jogging, and fi... | Arturo Magidin | 742 | <p>Take $S$ to be the set of kids who play soccer; $C$ of kids who play chess; $B$ for biking; and $J$ for jogging.</p>
<p>If you just add up $|S|+|C|+|J|+|B|$, then you are overcounting: any kid who likes more than one sport is getting counted as many times as there are sports he likes. So you need to compensate for ... |
1,041,212 | <p>I want to show that the function $f(X) = -log \ det(X)$ is convex on the space $S$ of positive definite matrices. </p>
<p>What I have done:</p>
<p>It seems like this problem could be tackled by considering the restriction of $f(X)$ to a line through a given point $X \in S$ so that $g(\alpha) = f(X + \alpha V)$ for... | Michael Grant | 52,878 | <p>Since $X$ is positive definite, it admits a symmetric, invertible square root $X^{1/2}$. Then
$$\begin{aligned}
f(X)&=-\log\det(X+tV)\\
&=-\log\det X^{1/2}(I+tX^{-1/2}VX^{-1/2})X^{1/2}\\
&=-\log(\det X^{1/2})^2\det(I+tX^{-1/2}VX^{-1/2}) \\
&=-\log\det X-\log\det(I+t\tilde{V})
\end{aligned... |
1,569,411 | <p>Express $log_3(a^2 + \sqrt{b})$ in terms of m and k where
$m = log_{3}a$</p>
<p>$k = log_{3}b$</p>
<p>Given this information I made
$a = 3^m$</p>
<p>$b = 3^k$</p>
<p>Therefore
= $log_{3} ((3^m)^2 + (3^k))^{\frac{1}{2}}$</p>
<p>= $log_{3} (3^{2m} + 3^{\frac{k}{2}})$</p>
<p>I don't know if I'm done or there is ... | Community | -1 | <p>You are correct. If $P$ is a Markov transition matrix, then $P^n$ converges
as $n\to\infty$ if and only if $P$'s only eigenvalue of modulus one is $\lambda=1.$ </p>
|
125,705 | <p>Find an open cover $\mathcal{G}$ of the set $E = \{ 1/n : n \in \mathbb{Z^{+}}\}$ such that any proper subset of $\mathcal{G}$ is not an open cover of $E$.</p>
| David Mitra | 18,986 | <p>Hint: for each positive integer $n$, find an open interval $O_n$ containing $1\over n$ that contains no other point of $E$. (I'm assuming you're working in $\Bbb R$ with the usual topology.) </p>
|
125,705 | <p>Find an open cover $\mathcal{G}$ of the set $E = \{ 1/n : n \in \mathbb{Z^{+}}\}$ such that any proper subset of $\mathcal{G}$ is not an open cover of $E$.</p>
| Brett Frankel | 22,405 | <p>You can cover each point of $E$ with an interval so small that it contains no other points in $E$. Since this looks like it might be homework, I'll leave you to work out the cover explicitly.</p>
|
125,705 | <p>Find an open cover $\mathcal{G}$ of the set $E = \{ 1/n : n \in \mathbb{Z^{+}}\}$ such that any proper subset of $\mathcal{G}$ is not an open cover of $E$.</p>
| Brian M. Scott | 12,042 | <p>The easiest way is to make sure that every member of $\mathcal{G}$ is needed in order to cover $E$. That means that for each $G\in\mathcal{G}$ you want a point $e_G\in E$ that is covered by $G$ and by no other member of $\mathcal{G}$. One easy way to do this is to arrange matters so that each member of $\mathcal{G}$... |
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