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 |
|---|---|---|---|---|
3,397,834 | <blockquote>
<h2><span class="math-container">$$49y″−98y′+48y= 0 \quad\quad\quad \,\, y(2)=3,y′(2)=9.$$</span></h2>
</blockquote>
<p>When I solved, I got that my <span class="math-container">$r_1= \frac67$</span> and <span class="math-container">$r_2= \frac87.$</span> Then I got that <span class="math-container">$y... | b00n heT | 119,285 | <p>Since I'm not completely satisfied with the other answers, rewrite the equation as
<span class="math-container">$$y(x)^5=8-x^5$$</span>
and now partially differentiate with respect to
<span class="math-container">$x$</span> on both sides (using the chain rule):
<span class="math-container">$$5y^4(x)y'(x)=-5x^4$$</s... |
140,459 | <p>There is a long tradition of mathematicians remarking that FLT in itself is a rather isolated claim, attractive only because of its simplicity. And people often note a great thing about current proofs of FLT is their use of the modularity thesis which is just the opposite: arcane, and richly connected to a lot of r... | Alex B. | 35,416 | <p>Corollary 3.17 in <a href="http://arxiv.org/abs/1206.1822">this paper of Stefan Keil</a> uses FLT for exponent 7 to show that if $E/\mathbb{Q}$ is an elliptic curve with a rational 7-torsion point $P$, and $E\rightarrow E'$ is the 7-isogeny with kernel $\langle P\rangle$, then $E'(\mathbb{Q})[7]=0$. There are of cou... |
140,459 | <p>There is a long tradition of mathematicians remarking that FLT in itself is a rather isolated claim, attractive only because of its simplicity. And people often note a great thing about current proofs of FLT is their use of the modularity thesis which is just the opposite: arcane, and richly connected to a lot of r... | Wojowu | 30,186 | <p>Recall that around 1977 Mazur has completely classified the possible torsion groups of elliptic curves over <span class="math-container">$\mathbb Q$</span>. A few years prior, Kubert has worked on this problem and has established a number of partial results, including, in the paper <a href="https://doi.org/10.1112/p... |
1,142,530 | <p>Find an equation of the plane.
The plane that passes through the point
(−3, 2, 1) and contains the line of intersection of the planes
x + y − z = 4
4x − y + 5z = 2</p>
<p>I know the normal to plane 1 is <1,1,-1> and the normal to plane 2 is <4,-1,5>. The cross product of these 2 would give a vector that is ... | kobe | 190,421 | <p>Let $\Pi$ be the plane that we seek. Setting $z = 0$ in the two equations and solving simultaneously for $x$ and $y$, we find that $(6/5,14/5,0)$ is a point in the line of intersection of the two planes. So $(6/5,14/5,0)$ lies on $\Pi$. Since $(-3,2,1)$ also lies on $\Pi$, the vector from $(-3,2,1)$ to $(6/5,14,5,0)... |
3,414,009 | <p>Given functions
<span class="math-container">$$
f(x)=\sum_{i=0}^\infty a_ix^i\,\,\,\text{ and }\,\,\,g(x)=\sum_{j=0}^\infty b_jx^j
$$</span>
the following simplification
<span class="math-container">$$
f(x)g(x)=\left(\sum_{i=0}^\infty a_ix^i\right)\left(\sum_{j=0}^\infty b_jx^j \right)=\sum_{i=0}^\infty\sum_{j=0}^\i... | Hagen von Eitzen | 39,174 | <p>There's no intuition, it's just
<span class="math-container">$$\left(\sum_{i=0}^\infty a_ix^i\right)\cdot c=\sum_{i=0}^\infty a_ix^ic $$</span>
with <span class="math-container">$c:=\sum_{j=0}^\infty b_jx^j$</span>, i.e.,
<span class="math-container">$$\left(\sum_{i=0}^\infty a_ix^i\right)\cdot \left(\sum_{j=0}^\in... |
1,577,838 | <p>Let $p$ and $q$ be primes such that $p=4q+1$. Then $2$ is a primitive root modulo $p$.</p>
<p>Proof. </p>
<p>Note that $q\not=2$ since $4\cdot2+1=9$ is not prime. $\mathrm{ord}_p(2)\vert p-1=4q$, so $\mathrm{ord}_p(2)=1,\;2,\;4,\;q,\;2q,\;\mathrm{or}\;4q$.</p>
<p>Clearly $\mathrm{ord}_p(2) \not= 1$, and $\mathrm{... | Dylan Weil | 499,493 | <p>I'm not certain that this proof is actually valid. Saying that $g^{2iq} \equiv 1 \Rightarrow (p-1)|2iq$ seems to assume that $O_p(2) = p - 1$ which is what you're trying to prove. If this wasn't being implicitly assumed, there'd be no guarantee even that $(p-1) \leq 2iq$.</p>
|
1,331,850 | <p>I haven't done something like this in a long time. How do I set something like this up? Can someone help me with the beginning or give me some direction?</p>
<p><img src="https://i.stack.imgur.com/jluer.png" alt="enter image description here">
<img src="https://i.stack.imgur.com/dv5yu.png" alt="enter image descript... | quazi5 | 249,373 | <p>Notice that your vector field is conservative. You can then calculate the the potential function by finding the integral of F(x,y), in this case V = (1/2)x^2 + (1/2)y^2 + 2xy. Then by plugging in the end point and the beginning point,t = 3 and t = 0 respectively. This leaves us with V(c(3)) - V(c(0)) = V(4,6) - V(1... |
607,044 | <p>I'm looking at some work with Combinatorial Game Theory and I have currently got:
(P-Position is previous player win, N-Position is next player win)</p>
<p>Every Terminal Position is a P-Position,</p>
<p>For every P-Position, any move will result in a N-Position,</p>
<p>For every N-Position, there exists a move t... | sapta | 173,460 | <p>the sequence <span class="math-container">$U_n$</span>is strictly decreasing sequence.
as, <span class="math-container">$U_{n+1}-U_n=\frac{1}{n+1}-\log (1+\frac{1}{n})<0$</span>;which can be checked by Riemann's integral on the function <span class="math-container">$f(x)=\frac{1}{x}$</span>.
now again using R... |
702,879 | <p>I am studying exponentials from MacLane-Moerdijk's book, "Sheaves in geometry and Logic". I do not understand the following: Induced by the product-exponent adjunction, consider the bijection $$\hom(Y\times X,Z)\to\hom(Y,Z^X)\;\;\;\;\;\;(\star)$$
They say: The existence of the above adjunction can be stated in eleme... | user21929 | 133,385 | <p>The naturality of $\alpha_Y$ in $Y$ means that: for any $Y$, for any $W$, for any $g : Y \rightarrow W$, for any $h: W \times X \rightarrow Z$, we have $\alpha_{W}(h) \circ g = \alpha_Y(h \circ (g \times id_X))$, hence ${\alpha_{Y}}^{-1}(\alpha_{W}(h) \circ g) = h \circ (g \times id_X)$. With $W = Z^X$ and $h = {\al... |
3,344,017 | <p>Solve in positive integers:
<span class="math-container">$$
y^3 - x^3 = z^2,$$</span>
where <span class="math-container">$\gcd (x, y, z)=1$</span>.</p>
| Richard Jensen | 658,583 | <p>Let me stress that the way you show set equality is almost always by the following method:</p>
<p>Let <span class="math-container">$A$</span> and <span class="math-container">$B$</span> be sets. Let <span class="math-container">$a \in A$</span>, and <span class="math-container">$b \in B$</span>. If you can show tha... |
3,344,017 | <p>Solve in positive integers:
<span class="math-container">$$
y^3 - x^3 = z^2,$$</span>
where <span class="math-container">$\gcd (x, y, z)=1$</span>.</p>
| J.G. | 56,861 | <p>To prove <span class="math-container">$(A\setminus B)\times C=(A\times C)\setminus(B\times C)$</span>, note both sides have only ordered pairs as elements, so we just need the following: <span class="math-container">$$(x,\,y)\in(A\setminus B)\times C\iff x\in A\setminus B\land y\in C\iff x\in A\land x\notin B\land y... |
69,472 | <blockquote>
<p><strong>Theorem 1</strong><br>
If $g \in [a,b]$ and $g(x) \in [a,b] \forall x \in [a,b]$, then $g$ has a fixed point in $[a,b].$<br>
If in addition, $g'(x)$ exists on $(a,b)$ and a positive constant $k < 1$ exists with
$$|g'(x)| \leq k, \text{ for all } \in (a, b)$$
then the fixed... | Chris Taylor | 4,873 | <p>The reason that the iteration <span class="math-container">$x\leftarrow \tfrac{1}{2}(x+3/x)$</span> converges so rapidly to <span class="math-container">$\sqrt{3}$</span> is because it is derived from Newton's method. Newton's method says that to find a root of a function <span class="math-container">$f(x)$</span>, ... |
69,472 | <blockquote>
<p><strong>Theorem 1</strong><br>
If $g \in [a,b]$ and $g(x) \in [a,b] \forall x \in [a,b]$, then $g$ has a fixed point in $[a,b].$<br>
If in addition, $g'(x)$ exists on $(a,b)$ and a positive constant $k < 1$ exists with
$$|g'(x)| \leq k, \text{ for all } \in (a, b)$$
then the fixed... | Ragib Zaman | 14,657 | <p>As others have already said, your first equation is derived from Newton's method. The idea of it is very simple, and it answers your last question. Basically, to approximate the location of a root of a function, we approximate the function locally by its tangent, find the tangents root instead, and that value is a d... |
1,008,610 | <blockquote>
<p>If <span class="math-container">$a$</span> is a group element, prove that <span class="math-container">$a$</span> and <span class="math-container">$a^{-1}$</span> have the same order.</p>
</blockquote>
<p>I tried doing this by contradiction.</p>
<p>Assume <span class="math-container">$|a|\neq|a^{-1}|$</... | André Nicolas | 6,312 | <p>Suppose that $a$ has infinite order. We show that $a^{-1}$ cannot have finite order. Suppose to the contrary that $(a^{-1})^m=e$ for some positive integer $m$. We have by repeated application of associativity that
$$a^m (a^{-1})^m=e.$$
It follows that $a^m=e$.</p>
|
1,008,610 | <blockquote>
<p>If <span class="math-container">$a$</span> is a group element, prove that <span class="math-container">$a$</span> and <span class="math-container">$a^{-1}$</span> have the same order.</p>
</blockquote>
<p>I tried doing this by contradiction.</p>
<p>Assume <span class="math-container">$|a|\neq|a^{-1}|$</... | coreyman317 | 525,188 | <p>It seems a more straightforward solution exists?</p>
<p>If <span class="math-container">$g$</span> has infinite order then so does <span class="math-container">$g^{-1}$</span> since otherwise, for some <span class="math-container">$m\in\mathbb{Z}^+$</span>, we have <span class="math-container">$(g^{-1})^m=e=(g^m)^{-... |
1,633,722 | <p>Consider the family of $n \times n$ real matrices $A$, for which there is a $n \times n$ real matrix $B$ with $AB-BA=A$. How large can the rank of a matrix in this family be?</p>
<h2>Motivation</h2>
<p>Prasolov's book contains an exercise about proving that if $A,B$ are matrices with $AB-BA=A$ then $A$ cannot be i... | Balloon | 280,308 | <p>You can compute the determinant, see that
$$\begin{vmatrix}1&5&0\\2&1&-2\\0&1&3\end{vmatrix}=5-2\cdot 15=-25\neq 0,$$
and so $(x,y,c)$ is a basis of $\mathbb{R}^3,$ and thus $c\not\in\mathrm{span}(x,y).$ </p>
<p>The error would be that you have written $(2,0,1)$ instead of $(1,2,0),$ as Aure... |
1,633,722 | <p>Consider the family of $n \times n$ real matrices $A$, for which there is a $n \times n$ real matrix $B$ with $AB-BA=A$. How large can the rank of a matrix in this family be?</p>
<h2>Motivation</h2>
<p>Prasolov's book contains an exercise about proving that if $A,B$ are matrices with $AB-BA=A$ then $A$ cannot be i... | Domenico Vuono | 227,073 | <p>If $c$ is in the span of $x$ $y$ then $$c=a \begin{bmatrix}
2 \\
0 \\
1 \\
\end{bmatrix}+b \begin{bmatrix}
5 \\
1 \\
1 \\
\end{bmatrix}$$ you have to solve a linear system to determine $a$ and $b$. If the linear system hasn't solutions then $c$ isn't in... |
221,667 | <p>I'm taking a second course in linear algebra. Duality was discussed in the early part of the course. But I don't see any significance of it. It seems to be an isolated topic, and it hasn't been mentioned anymore. So what's exactly the point of duality?</p>
| Suresh Venkat | 865 | <p>This will probably not be apparent in a linear algebra course, but duality is the workhorse of optimization. Roughly speaking, you can often frame an optimization problem as trying to minimize some quantity subject to linear constraints (that form a matrix). Then in order to solve this problem you usually need to un... |
1,640,373 | <p>The definition of a topological space is a set with a collection of subsets (the topology) satisfying various conditions. A metric topology is given as the set of open subsets with respect to the metric. But if I take an arbitrary topology for a metric space, will this set coincide with the metric topology? </p>
<p... | Jack's wasted life | 117,135 | <p>You don't even need the matrix representation. T will map a vector to $(0,0,0)$ iff its $y$-component is zero. So your basis is the correct one.</p>
|
2,479,305 | <p>I'm studying first-order logic and I saw the textbook put equality symbol $=$ ads a logical symbol.</p>
<p>It's not a logical connective symbol, so $=xy$ is an atomic formula.
<a href="https://en.wikipedia.org/wiki/Atomic_formula" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Atomic_formula</a></p>
<p>Ho... | Bram28 | 256,001 | <p>First, $=$ is a predicate symbol since it says something about two objects, namely that they are identical. Out of all predicate symbols, though, it is the only 'logical' one, so it is indeed not like any other predicate, nor is it like any other logical symbol.</p>
<p>OK, but why is it a logical symbol? One answer... |
929,243 | <p>Hey guys I just need help solving this solution here. Sorry if I didn't type the symbols correctly.</p>
<p>My solution so far:
$$
(¬p \vee ¬(p\wedge¬q)) \wedge (¬p \vee ¬q)≡
(¬p \vee (¬p \vee q)) \wedge (¬p \vee ¬q)≡
$$
at this point I'm stuck. Is there any way I can take care of the not-$p$ $\vee$ not-$p$?</p>
<p... | Adriano | 76,987 | <p>Yes, that would be Idempotent Law:
\begin{align*}
(\neg p \lor \neg p \lor q) \land (\neg p \lor \neg q)
&\equiv (\neg p \lor q) \land (\neg p \lor \neg q) \\
&\equiv \neg p \lor (q \land \neg q) \\
&\equiv \neg p \lor \bot \\
&\equiv \neg p \\
\end{align*}</p>
|
2,982,978 | <p><strong>Exercise :</strong></p>
<blockquote>
<p>Let <span class="math-container">$S: l^1 \to l^1$</span> be the right-shift operator :
<span class="math-container">$$S(x_1,x_2,\dots) = (0,x_1,x_2,\dots)$$</span>
Prove that <span class="math-container">$S$</span> is bounded and find its norm.</p>
</blockquote>... | Matematleta | 138,929 | <p>From scratch: </p>
<p><span class="math-container">$\vec x=(x_1,x_2,\cdots )$</span> and <span class="math-container">$S(\vec x)=(0,x_1,x_2,\cdots)$</span> so </p>
<p><span class="math-container">$\|\vec x\|=\sum^{\infty}_{i=1}|x_i|$</span> and <span class="math-container">$\|S(\vec x)\|=\sum^{\infty}_{i=1}|x_i|=... |
339,090 | <p>I would like to pose a question on a variation on the classical <a href="http://en.wikipedia.org/wiki/Coupon_collector%27s_problem#Extensions_and_generalizations" rel="nofollow">coupon collector's problem</a>: coupon type $i$ is to be collected $k_i$ times. What is the expected stopping time or the expected number o... | Rus May | 17,853 | <p>I have wondered about this exact question and wrote up some results <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v15i1n31/pdf" rel="nofollow">here</a>. This paper gives an explicit answer to your question (and allows for non-uniform probabilities among the coupons). It does not use, as you s... |
2,294,969 | <p>I can't seem to find a path to show that:</p>
<p>$$\lim_{(x,y)\to(0,0)} \frac{x^2}{x^2 + y^2 -x}$$</p>
<p>does not exist.</p>
<p>I've already tried with $\alpha(t) = (t,0)$, $\beta(t) = (0,t)$, $\gamma(t) = (t,mt)$ and with some parabolas... they all led me to the limit being $0$ but this exercise says that there... | Michael Seifert | 248,639 | <p>Other commenters have noted that the function value can be made to approach 1 along the parabolic path $x = y^2$. In fact, the function's value along a path going to the origin can be made to approach any non-zero real number by choosing an elliptical or hyperbolic path instead. If we look at the curve
$$
y^2 - x ... |
2,486,095 | <p>Given an acute-angled triangle $\Delta ABC$ having it's Orthocentre at $H$ and Circumcentre at $O$. Prove that $\vec{HA} + \vec{HB} + \vec{HC} = 2\vec{HO}$</p>
<p>I realise that $\vec{HO} = \vec{BO} + \vec{HB} = \vec{AO} + \vec{HA} =\vec{CO} + \vec{HC}$ which leads to $3\vec{HO} = (\vec{HA} + \vec{HB} + \vec{HC}) +... | marty cohen | 13,079 | <p>5 is a pretty small number of iterations. Try a larger number and see what happens. Maybe 100.</p>
<p>Also, check two residuals at each iteration.</p>
|
832,715 | <blockquote>
<p>Suppose that the distribution of a random variable
$X$ is symmetric with respect to the point $x = 0$. If $\mathbb{E}(X^4)>0$ then $Var(X)$ and $Var(X^2)$ are both positive.</p>
</blockquote>
<p>How is that true? I am getting $Var(X)=\mathbb{E}(X^2)$ and $Var(X^2)=\mathbb{E}(X^4)-(\mathbb{E}(X... | Avraham | 91,378 | <p>I think we can prove everything <strong>but</strong> the fact that $Var(X^2) > 0.$ We are given that $E(X) = 0$ and $E(X^4) > 0$, the question is how to show both Var$(X)$ and Var$(X^2)$ are strictly positive.</p>
<p>Knowing $E(X^4) > 0$ allows us to say that
$$
\begin{align}
\frac{\sum_{i=1}^nx_i^4}{n} &a... |
194,373 | <p>Let $\Omega$ be a bounded smooth domain and define $\mathcal{C} = \Omega \times (0,\infty)$. Below, $x$ refers to the variable in $\Omega$ and $y$ to the variable in $(0,\infty)$. The map $\operatorname{tr}_\Omega:H^1(\mathcal C) \to L^2(\Omega)$ refers to the trace operator ($\operatorname{tr}_\Omega u = u(\cdot,0)... | Jean Van Schaftingen | 42,047 | <p>I think that the trace is <em>defined by the completion</em>. Indeed for every $u \in H^1 (\Omega)$, you have $\Vert\operatorname{tr} u\Vert_{L^2 (\Omega)} \le \Vert u \Vert_{\varepsilon}$. Since $H^1 (\mathcal{C})$ is dense in $H^\varepsilon (\mathcal{C})$, the trace operator $\operatorname{tr}$ is a well-defined c... |
234,466 | <p>This is the second problem in Neukirch's Algebraic Number Theory. I did the proof but it feels a bit too slick and I feel I may be missing some subtlety, can someone check it over real quick?</p>
<p>Show that, in the ring $\mathbb{Z}[i]$, the relation $\alpha\beta =\varepsilon\gamma ^n$, for $\alpha,\beta$ relativ... | Berci | 41,488 | <p>Yes, it seems to hold in every UFD.</p>
|
145,046 | <p>I'm a first year graduate student of mathematics and I have an important question.
I like studying math and when I attend, a course I try to study in the best way possible, with different textbooks and moreover I try to understand the concepts rather than worry about the exams. Despite this, months after such an in... | Asaf Karagila | 622 | <p>Usually what remains in your mind is the general idea: a vague outline of the terminology and theorems. It may sound bad, but it's fine. I can honestly say that I hardly remember anything from most courses I took, except the things I had to teach, or directly relate to my work. This includes things from a course I t... |
1,232,036 | <p>I have a midterm tomorrow, and while studying for that I saw this question, however don't have any idea how to solve it. (I could not come up with a legitimate proof. All I could do was, by putting some functions, approving what the problem claims.) I will appreciate if you can help.</p>
<p>Suppose that $ h(t)$ is ... | Chappers | 221,811 | <p>Multiplying by $e^{t}$, we have
$$ (e^{t}y(t))' = e^t h(t). $$
Integrating, we have
$$ y(t) = y(0)e^{-t} + e^{-t}\int_0^{t} e^{s}h(s) \, ds $$
Now we have to find a bounded solution. Since $h(s)\geqslant-M$, we can define $H(s)=h(s)+M$ positive, and hence we can also write the solution as
$$ y(t) = y(0)e^{-t} + e^{-... |
774,434 | <p><a href="https://www.wolframalpha.com/input/?i=Sum%5BBinomial%5B3n,n%5Dx%5En,%20%7Bn,%200,%20Infinity%7D%5D" rel="noreferrer">Wolfram alpha tells me</a> the ordinary generating function of the sequence $\{\binom{3n}{n}\}$ is given by
$$\sum_{n} \binom{3n}{n} x^n = \frac{2\cos[\frac{1}{3}\sin^{-1}(\frac{3\sqrt{3}\sqr... | epi163sqrt | 132,007 | <p>As was already mentioned in the comment section the <em>Lagrange Inversion Formula</em> is a proper method to prove this identity. In the following I use the notation from R. Sprugnolis (etal) paper <a href="http://www.researchgate.net/publication/226195157_Lagrange_Inversion_When_and_How/file/e0b495230d098ec3f7.pdf... |
1,834,756 | <p>The Taylor expansion of the function $f(x,y)$ is:</p>
<p>\begin{equation}
f(x+u,y+v) \approx f(x,y) + u \frac{\partial f (x,y)}{\partial x}+v \frac{\partial f (x,y)}{\partial y} + uv \frac{\partial^2 f (x,y)}{\partial x \partial y}
\end{equation}</p>
<p>When $f=(x,y,z)$ is the following true?</p>
<p>$$\begin{alig... | Hosein Rahnama | 267,844 | <p>The general formula for the Taylor expansion of a sufficiently smooth real valued function <span class="math-container">$f:\mathbb{R}^n \to \mathbb{R}$</span> at <span class="math-container">$\mathbf{x}_0$</span> is</p>
<p><span class="math-container">$$f({\bf{x}})=f({\bf{x}}_0)+\nabla f({\bf{x}}_0) \cdot ({\bf{x}}-... |
639,665 | <p>How can I calculate the inverse of $M$ such that:</p>
<p>$M \in M_{2n}(\mathbb{C})$ and $M = \begin{pmatrix} I_n&iI_n \\iI_n&I_n \end{pmatrix}$, and I find that $\det M = 2^n$. I tried to find the $comM$ and apply $M^{-1} = \frac{1}{2^n} (comM)^T$ but I think it's too complicated.</p>
| Christoph | 86,801 | <p>You can easily just do gaussian elimination to compute this inverse as you would do for any other explicitly given matrix.</p>
<p>We will use row operations and track what we did on the right hand side, so we start with
$$
\pmatrix{I_n & iI_n \\ iI_n & I_n} \,\Bigg|\,
\pmatrix{I_n & 0 \\ 0 & I_n}.
$... |
2,916,887 | <p>The formula for Shannon entropy is as follows,</p>
<p>$$\text{Entropy}(S) = - \sum_i p_i \log_2 p_i
$$</p>
<p>Thus, a fair six sided dice should have the entropy,</p>
<p>$$- \sum_{i=1}^6 \dfrac{1}{6} \log_2 \dfrac{1}{6} = \log_2 (6) = 2.5849...$$</p>
<p>However, the entropy should also correspond to the average ... | Ahmad Bazzi | 310,385 | <p>There is nothing wrong with what you did. <strong>In the book "Elements on Information Theory", there is a proof that the average number of questions needed lies between $H(X)$ and $H(X)+1$, which agrees with what you did</strong>. So, in terms of "questions", the entropy gives you an accuracy within $1$ question. T... |
3,985,917 | <p>I am trying to show algebraically that <span class="math-container">$8^3>9^{8/3}$</span>. This came from trying to complete the base case of an induction proof.</p>
<p>I have struggled because <span class="math-container">$8$</span> and <span class="math-container">$9$</span> cannot be manipulated to be the same ... | user2661923 | 464,411 | <p>An approach that is alternative to (but inferior to) David Lui's answer is if you happen to know that</p>
<p><span class="math-container">$$\log_{10} ~2 \approx 0.301 ~~\text{and}~~ \log_{10} ~3 \approx 0.477.$$</span></p>
<p>Then, you simply compare</p>
<p><span class="math-container">$$0.301 \times (3 \times 3) ~~... |
3,403,364 | <p>Here's a little number puzzle question with strange answer:</p>
<blockquote>
<p>In an apartment complex, there is an even number of rooms. Half of the rooms have one occupant, and half have two occupants. How many roommates does the average person in the apartment have? </p>
</blockquote>
<p>My gut instinct was ... | rogerl | 27,542 | <p>If there are <span class="math-container">$2n$</span> rooms, then there are <span class="math-container">$3n$</span> people. Clearly <span class="math-container">$2n$</span> of them have exactly one roommate...</p>
|
1,790,311 | <p>Show the following equalities $5 \mathbb{Z} +8= 5\mathbb{Z} +3= 5\mathbb{Z} +(-2)$.</p>
<p>$5 \mathbb{Z} +8=\{5z_{1}+8: z_{1} \in \mathbb{Z}\}$,</p>
<p>$5 \mathbb{Z} +3=\{5z_{2}+3: z_{2} \in \mathbb{Z}\}$,</p>
<p>$5 \mathbb{Z} +(-2)=\{5z_{3}+(-2): z_{3} \in \mathbb{Z}\}$.</p>
<p>So, how can prove to use these de... | Pipicito | 93,689 | <p>Note that for any $n \in \mathbb{Z}$ you have $\mathbb{Z} = \mathbb{Z} + n$.</p>
<p>Thus, $5 \mathbb{Z} +8 = 5 \mathbb{Z} + 5 + 3 = 5 (\mathbb{Z} + 1)+ 3 = 5\mathbb{Z} + 3$.</p>
<p>Similarly, you have $5 \mathbb{Z} + 8 = 5 \mathbb{Z} + 10 - 2 = 5\mathbb{Z} - 2$.</p>
|
23,312 | <p>What is the importance of eigenvalues/eigenvectors? </p>
| lhf | 589 | <p>The behaviour of a linear transformation can be obscured by the choice of basis. For some transformations, this behaviour can be made clear by choosing a basis of eigenvectors: the linear transformation is then a (non-uniform in general) scaling along the directions of the eigenvectors. The eigenvalues are the scale... |
23,312 | <p>What is the importance of eigenvalues/eigenvectors? </p>
| daven11 | 10,607 | <p>This made it clear for me <a href="https://www.youtube.com/watch?v=PFDu9oVAE-g" rel="nofollow noreferrer">https://www.youtube.com/watch?v=PFDu9oVAE-g</a></p>
<p>There's a lot of other linear algebra videos as well from 3 blue 1 brown</p>
|
142,677 | <p>Consider the following list of equations:</p>
<p>$$\begin{align*}
x \bmod 2 &= 1\\
x \bmod 3 &= 1\\
x \bmod 5 &= 3
\end{align*}$$</p>
<p>How many equations like this do you need to write in order to uniquely determine $x$?</p>
<p>Once you have the necessary number of equations, how would you actually ... | Community | -1 | <p>This is a classic example of <a href="http://en.wikipedia.org/wiki/Chinese_remainder_theorem" rel="nofollow noreferrer">Chinese remainder theorem</a>. To solve it, one typically proceeds as follows. We have <span class="math-container">$$x = 2k_2 + 1 = 3k_3 + 1 = 5k_5 + 3.$$</span>
Since <span class="math-container"... |
1,693,630 | <p><a href="https://i.stack.imgur.com/TVeGv.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/TVeGv.jpg" alt="enter image description here"></a></p>
<p>This is my attempt at finding $\frac{d^2y}{dx^2}$. Can some one point out where I'm going wrong here?</p>
| Emilio Novati | 187,568 | <p>You are wrong because you have not computed
$$
\frac{d}{dx} \frac{dy}{dx}
$$
but</p>
<p>$$
\frac{d}{dt} \frac{dy}{dx}
$$</p>
<p>if you set
$$
\frac{dy}{dt}=\dot y \qquad \frac{dx}{dt}=\dot x
$$
than the formula for the second derivative is:
$$
\frac{d^2y}{dx^2}=\frac{d}{dt} \frac{dy}{dx}\frac{dt}{dx}=\frac{\do... |
885,129 | <p>I want to speed up the convergence of a series involving rational expressions the expression is $$\sum _{x=1}^{\infty }\left( -1\right) ^{x}\dfrac {-x^{2}-2x+1} {x^{4}+2x^{2}+1}$$
If I have not misunderstood anything the error in the infinite sum is at most the absolute value of the last neglected term. The formula ... | WimC | 25,313 | <p>There are several methods to speed up the summation of series. For example <a href="https://en.wikipedia.org/wiki/Euler_summation" rel="nofollow noreferrer">Euler summation</a> or the <a href="https://en.wikipedia.org/wiki/Shanks_transformation" rel="nofollow noreferrer">Shanks transformation</a>. Here is a simple... |
507,454 | <p>I had a geometry class which was proctored using the Moore method, where the questions were given but not the answers, and the students were the source of all answers in the class. One of the early questions which we never solved is listed in the title.</p>
<p>In this case, use any reasonable definition of "betwee... | Stefan4024 | 67,746 | <p>It's well-known fact that every prime number, except for $2$ and $3$ can be written as $6k \pm 1$. SO we have:</p>
<p>$$(6k \pm 1)^2 - (6l \pm 1)^2 = 36k^2 \pm 12k + 1 - 36l^2 \mp 12l - 1$$ </p>
<p>$$= 36(k^2-l^2) \pm 12(k - l) = 12(3k^2 - 3l^2 \pm k \mp l)$$</p>
<p>The sum in the parenthesis is always even. Why?... |
1,918,144 | <p>Suppose $\sum\limits_{k=0}^{\infty} b_k $ converges absolutely and has the sum $b$. Suppose $a \in\mathbb R$ with $|a|<1$. What is the sum of the series $\sum\limits_{k=0}^{\infty}(a^kb_0 +a^{k-1}b_1 +a^{k-2}b_2 +...+b_k)$?</p>
| Bernard | 202,857 | <p>This is the *Cauchy product of the series $\sum_{k=0}^\infty b_k$ and $\sum_{k=0}^\infty a_k$. As both series converge absolutely, their Cauchy product converge to the product of each sum, $b\cdot\dfrac1{1-a}$.</p>
|
625,975 | <p>I'm just starting to learn computability. Some treatments of the subject use a relation they call $T$, which I <em>think</em> is called the universal recursive relation. It's defined something like this (<a href="http://www.its.caltech.edu/~jclemens/courses/02ma117a/handouts/handout6.pdf" rel="nofollow">http://www.i... | tomasz | 30,222 | <p>Just because we can tell if a given computation is the halting computation doesn't mean we can tell if one exists at all.</p>
<p>The halting relation is is $\exists c T_n(m,c,x_1,\ldots,x_n)$.</p>
|
265 | <p>I'm a mod at Computational Science SE. Sometimes, users ask questions on Computational Science about doing something in Wolfram Alpha. As far as I can tell, Wolfram Alpha uses Mathematica for its math engine. Are those questions on topic here?</p>
<p>Note: I was made aware of <a href="https://mathematica.meta.stack... | Szabolcs | 12 | <p>We haven't had questions like that so far, so there is no precedent.</p>
<p>But I'd say questions about how to do something with Wolfram|Alpha <em>without</em> using Mathematica should be explicitly off topic. Based on the discussion you linked to I think most will agree.</p>
|
1,823,487 | <p>There are $m$ different people and a <strong>circle</strong> that has $m+r$ identical seats. How many ways can we put those people in the circle?</p>
<p>If the seats were not identical then the solution was: $ \frac{1*(m+r-1)!}{r!} $</p>
<p>I can't understand how the fact that the seats are identical affects my s... | Brian M. Scott | 12,042 | <p>First imagine that one seat is marked. Then we can think of the seats as actually being arranged in a line, with the marked seat at the beginning of the line. There are $\binom{m+r}m$ ways to pick $m$ of the seats to be occupied by people, and those $m$ people can be arranged in $m!$ ways in the chosen seats, so the... |
4,055,850 | <p>I was thinking it could be plugging in <span class="math-container">$x^2+x$</span> to the <span class="math-container">$f'(x)$</span>, then using the Chain Rule to solve it, but I'm not sure if it is right. Please help!</p>
| Ikhtesad Mannan | 1,019,660 | <p>That's taken from the IB Oxford AA HL textbook. In IB, the real scale is the x-axis and the imaginary scale is the y-axis. According to that, the book is wrong. It has gotten the conjugate and the conjugate of the opposite the wrong way around.</p>
|
2,780,216 | <p>How many integer solutions are there to the equation $a + b + c = 21$ if $a \geq3$,
$b \geq 1$, and $c \geq 2b$?</p>
| Abhishek Choudhary | 452,208 | <p>$a$ can vary from $3$ to $18$ $b$ can vary from $1$ to $16$ $c$ can vary from $2$ to $17$ Hence according to me number of solutions should be $16*16*16=4096$</p>
|
1,722,692 | <p>I am asked to find</p>
<p>$$\lim_{x \to 0} \frac{\sqrt{1+x \sin(5x)}-\cos(x)}{\sin^2(x)}$$</p>
<p>and I tried not to use L'Hôpital but it didn't seem to work. After using it, same thing: the fractions just gets bigger and bigger.</p>
<p>Am I missing something here?</p>
<p>The answer is $3$</p>
| Claude Leibovici | 82,404 | <p>Let us play with Taylor series to get even more than just the limit considering $$\sin(5x)=5 x-\frac{125 x^3}{6}+O\left(x^5\right)$$ $$1+x \sin(5x)=1+5 x^2-\frac{125 x^4}{6}+O\left(x^6\right)$$ $$\sqrt{1+x \sin(5x)}=1+\frac{5 x^2}{2}-\frac{325 x^4}{24}+O\left(x^6\right)$$ $$\sqrt{1+x \sin(5x)}-\cos(x)=3 x^2-\frac{1... |
30,305 | <p>I want to call <code>Range[]</code> with its arguments depending on a condition. Say we have </p>
<pre><code>checklength = {5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6}
</code></pre>
<p>I then want to call <code>Range[]</code> 13 times (the length of <code>checklength</code>) and do <code>Range[5]</code> when <code>... | jVincent | 1,194 | <p>The solution you are asking for is to pass the sequence of 2,6 which is nicely defined as <code>Sequence[2,6]</code> However if we simply put that as the last argument in <code>If</code>, it will be misinterpreted, for instance <code>If[False,Sequence[1,2]]</code> would return 2. So we need to use <code>Unevaluated<... |
4,000,935 | <p>How do you show that <span class="math-container">$7a^2 - 12b^2 = 8c^2$</span> has no integer solutions</p>
<hr />
<p>When <span class="math-container">$x = a/c$</span> and <span class="math-container">$y = b/c$</span> then <span class="math-container">$\gcd(a,b,c) = 1$</span> I believe.</p>
<p>If we use mod5, and n... | lab bhattacharjee | 33,337 | <p>Hint</p>
<p>WLOG <span class="math-container">$(a,c)=1$</span></p>
<p>Take modulo <span class="math-container">$\pmod3$</span></p>
<p><span class="math-container">$$a^2\equiv2c^2\pmod3$$</span></p>
<p>But for any integer <span class="math-container">$d,$</span></p>
<p><span class="math-container">$$d^2\equiv0,1\pmod... |
4,000,935 | <p>How do you show that <span class="math-container">$7a^2 - 12b^2 = 8c^2$</span> has no integer solutions</p>
<hr />
<p>When <span class="math-container">$x = a/c$</span> and <span class="math-container">$y = b/c$</span> then <span class="math-container">$\gcd(a,b,c) = 1$</span> I believe.</p>
<p>If we use mod5, and n... | Community | -1 | <p>Using mod <span class="math-container">$4$</span>, a is divisible by <span class="math-container">$2$</span><br>
Let <span class="math-container">$a = 2k$</span></p>
<p><span class="math-container">$7k^2 - 3b^2 = 2c^2$</span></p>
<p>HINT: GO mod 3 and use that <span class="math-container">$t^2 \equiv 0,1 \pmod 3$</s... |
733,101 | <p>I've been stuck for a while on this question and haven't found applicable resources.</p>
<p>I have 10 choices and can select 3 at a time. I am allowed to repeat choices (combination), but the challenge is that ABA and AAB are not unique.</p>
<p>10 choose 3 is the question.</p>
<p>I have been working on a smaller ... | user2566092 | 87,313 | <p>Hint: First consider the case that all 3 chosen elements are distinct, and count the possibilities (taking into account that order doesn't matter). Then consider the case when exactly 2 chosen elements are equal, and count the possibilities. Finally count the possibilities when all 3 elements are equal.</p>
|
394,294 | <p>I would like to know the asymptotics of the following sequences of integrals:
<span class="math-container">$$ I_n = \int _0
^{+ \infty}
e^{-t} \left ( \dfrac{t}{1 + t} \right )^n
\ dt
$$</span></p>
<p>I have tried using Laplace method ou saddle node method, but I hav... | MathTolliob | 171,738 | <p>Let us assume the following conjecture:</p>
<blockquote>
<p>Let <span class="math-container">$h: \mathbb{R}^+ \longrightarrow \mathbb{R}$</span> be a continuous function.
Let also <span class="math-container">$g_n: \mathbb{R}^+ \longrightarrow \mathbb{R}$</span> be a <span class="math-container">$\mathcal{C}^2$</spa... |
3,319,122 | <p>This is from Tao's Analysis I: </p>
<p><a href="https://i.stack.imgur.com/DYQxE.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/DYQxE.png" alt="enter image description here"></a></p>
<p>So far I managed to show (inductively) that these sets do exist for for every <span class="math-container">$\m... | José Carlos Santos | 446,262 | <p>Formally, your question makes no sense, because <span class="math-container">$f$</span> is undefined at <span class="math-container">$0$</span>. But suppose that you extend the domain of <span class="math-container">$f$</span> to <span class="math-container">$\mathbb R$</span>, defining<span class="math-container">$... |
2,701,582 | <p>I thought I was doing this right until I checked my answer online and got a different one. I worked through the problem again and got my original answer a second time so this one is bothering me since the other similar ones I have done checked out okay. Please let me know if I'm doing something wrong, thanks!</p>
<... | Dirk | 379,594 | <p>One famous generating set of $S_n$ is
$$A = \{ (1,2), (1,2,3,4,\ldots, n)\}.$$</p>
<p>The proof that this sets generates the whole group is basically the proof of correctness for the bubble sort algorithm.
Now you can only generate four subgroups from subsets of this set, the trivial one, the whole group and two c... |
2,701,582 | <p>I thought I was doing this right until I checked my answer online and got a different one. I worked through the problem again and got my original answer a second time so this one is bothering me since the other similar ones I have done checked out okay. Please let me know if I'm doing something wrong, thanks!</p>
<... | Ivan Di Liberti | 36,248 | <p>Given a prime number $p$, $S_p$ is always generated by a transposition and a $p$ cycle. There is no way to find a subst of this set that generates $A_n$.</p>
|
3,370,521 | <p>I have to prove that the following series converges:
<span class="math-container">$$
\sum_{n=1}^{\infty}\frac{\sin n}{n(\sqrt{n}+\sin n)}
$$</span>
I tried to use Dirichlet's test but I was not really sure whether the denominator is a monotonically decreasing function. If it is than the problem becomes easy.</p>
| José Carlos Santos | 446,262 | <p>No. Take<span class="math-container">$$\begin{array}{rccc}f\colon&D(0,1)&\longrightarrow&\mathbb C\\&z&\mapsto&\frac1{1-z}\end{array}$$</span>and <span class="math-container">$a_n=-1+\frac1n$</span> for each <span class="math-container">$n\in\mathbb N$</span>.</p>
|
2,317,166 | <p>I just pondered this question and have tried out several methods to solve it(mainly using trigonometry). However, I am not satisfied with my trigonometrical proof and is looking for better proofs.</p>
<p>Give an <em>elegant proof</em> that the diagonals are the longest lines in a square.
It would rather be nice if ... | Andrew D. Hwang | 86,418 | <p>Consider the axis-oriented rectangle $R$ consisting of all points whose Cartesian coordinates $(x, y)$ satisfy
$$
x_{0} \leq x \leq x_{1},\qquad
y_{0} \leq y \leq y_{1}.
$$
If $(x, y)$ and $(x', y')$ are points of $R$, then $|x' - x| \leq |x_{1} - x_{0}|$ and $|y' - y| \leq |y_{1} - y_{0}|$, so their distance satisf... |
493,600 | <p>I am presented with the following task:</p>
<p>Can you use the chain rule to find the derivatives of $|x|^4$ and $|x^4|$ in $x = 0$? Do the derivatives exist in $x = 0$? I solved the task in a rather straight-forward way, but I am worried that there's more to the task:</p>
<p>First of all, both functions is a vari... | user71352 | 71,352 | <p>You can apply the chain rule to $(x^{2})^{2}$, which happens to equal $|x^{4}|=|x|^{4}$, since it is the composition of two functions differentiable at $0$ (i.e take $f(x)=x^{2}$ then this is $f\circ f$). You could actually proceed quicker since $|x^{4}|=|x|^{4}=x^{4}$ so we don't need chain rule. What you are using... |
246,384 | <p>How can I plot a hexagon inside this cylinder so that it is always in the same direction as the cylinder, no matter what the orientation is? In addition, the hexagon must always be the same size as the cylinder.</p>
<p>For example, start at the origin (10,9,8) and go to (1,2,3)?</p>
<pre><code>Graphics3D [{Cylinder ... | Rohit Namjoshi | 58,370 | <p>The nested lists are saved as strings so <code>ToExpression</code> has to be mapped.</p>
<pre><code>data = {{1, {{2, 3, 4}, {5, 6, 7, {8, 9}}}, 10, {11, 12}}};
export = ExportString[data, "CSV"];
import = ImportString[export, "CSV"];
importData = import // Map[ToExpression]
importData == data
(... |
1,272,647 | <p>I guess it's a simple question, but it really escaped my memory.</p>
<p>If $a + b =1$, then how can I call those $a$ and $b$ numbers? </p>
<p>$a$ is not an inversion of $b$, and it's not reciprocal of $b$.. but I'm sure that they do have a 'name'.</p>
| k1.M | 132,351 | <p>This pair of numbers has no special name in mathematics as I know. Because usually mathematicians use $\lambda$ and $1-\lambda$ instead of $a$ and $b$ and there is no need to use a special name...</p>
|
1,272,647 | <p>I guess it's a simple question, but it really escaped my memory.</p>
<p>If $a + b =1$, then how can I call those $a$ and $b$ numbers? </p>
<p>$a$ is not an inversion of $b$, and it's not reciprocal of $b$.. but I'm sure that they do have a 'name'.</p>
| quapka | 112,628 | <p>I've considered posting this as just a comment, because I am quite sure, this does not really answer the question (but is a bit related). But since I tend to write more, I've chosen to post an answer.</p>
<p><strong>Background</strong></p>
<p>Consider $V$ a vector space over the field $\mathbb{K}$ (e.g. $\mathbb{K... |
128,122 | <p>Original Question: Suppose that $X$ and $Y$ are metric spaces and that $f:X \rightarrow Y$. If $X$ is compact and connected, and if to every $x\in X$ there corresponds an open ball $B_{x}$ such that $x\in B_{x}$ and $f(y)=f(x)$ for all $y\in B_{x}$, prove that f is constant on $X$. </p>
<p>Here's my attempt:
Cover ... | Bill Cook | 16,423 | <p>Notice that since $f(X)$ is a finite set and $Y$ is Hausdorff you can cover each $a_i$ with an open ball disjoint from all others. For example: let $B_i = B_{a_i}(\epsilon_i)$ where $\epsilon_i$ is $1/4$ the minimal distance between $a_i$ and the rest of the $a$'s (this exists because you have a finite set). Then yo... |
85,814 | <p>how to solve $\pm y \equiv 2x+1 \pmod {13}$ with Chinese remainder theorem or iterative method?</p>
<p>It comes from solving $x^2+x+1 \equiv 0 \pmod {13}$ (* ) and background is following:</p>
<blockquote>
<p>13 is prime. (* ) holds under Euclidean lemma if and only if $4(x^2+x+1) \equiv \pmod {13}$
or if and ... | Arturo Magidin | 742 | <p>To solve $x^2 + x + 1 \equiv 0 \pmod{13}$, you can use the usual quadratic formula, interpreted appropriately. The solutions are given by
$$x = \frac{-1 \pm\sqrt{-3}}{2},$$
where "$\sqrt{-3}$" means an integer $y$ such that $y^2\equiv -3\pmod{13}$, and "$\frac{1}{2}$" means an integer $z$ such that $2z\equiv 1\pmod{... |
3,263,795 | <p>Let <span class="math-container">$A = (a_{ij})$</span> be an invertible <span class="math-container">$n\times n$</span> matrix. I wonder how to prove that <span class="math-container">$A$</span> is a product of elementary matrices. I suspect that we need to transform it into the identity matrix by using elementary r... | Toby Mak | 285,313 | <p>Adding to John Omielan's answer, <a href="https://math.stackexchange.com/questions/2295360/prove-the-perpendicular-bisector-of-chord-passes-through-the-centre-of-the-circle">according to the perpendicular bisector theorem</a>, if a point is on the perpendicular bisector, then it is equidistant from the segment's end... |
275,308 | <p>Problems with calculating </p>
<p>$$\lim_{x\rightarrow0}\frac{\ln(\cos(2x))}{x\sin x}$$</p>
<p>$$\lim_{x\rightarrow0}\frac{\ln(\cos(2x))}{x\sin x}=\lim_{x\rightarrow0}\frac{\ln(2\cos^{2}(x)-1)}{(2\cos^{2}(x)-1)}\cdot \left(\frac{\sin x}{x}\right)^{-1}\cdot\frac{(2\cos^{2}(x)-1)}{x^{2}}=0$$</p>
<p>Correct answer i... | user 1591719 | 32,016 | <p>$$\lim_{x\to 0}\frac{\ln(\cos(2x))}{x\sin x}=\lim_{x\to 0}\frac{\ln(\cos(2x))}{1-\cos 2x}\times\lim_{x\to 0}\frac{x}{\sin x}\times\lim_{x\to 0}\frac{1-\cos 2x }{(2x)^2}\times4=-1\times1\times2=-2$$</p>
|
1,025,117 | <p>Let $V$ be finite dim $K-$vector space. If w.r.t. any basis of $V$, the matrix of $f$ is a diagonal matrix, then I need to show that $f=\lambda Id$ for some $\lambda\in K$. </p>
<p>I am trying a simple approach: to show that $(f-\lambda Id)(e_i)=0$ where $(e_1,...,e_2)$ is a basis of $V$. Let the diagonal matrix b... | mfl | 148,513 | <p>Using L'Hospital:</p>
<p>$$\lim_{x\to \infty}x\ \log\left(\frac{x+c}{x-c}\right)=\lim_{x\to \infty}\frac{\log\left(\frac{x+c}{x-c}\right)}{\frac 1x}\underbrace{=}_{\mathrm{L'Hospital}} \lim_{x\to \infty} \frac{\frac{x-c}{x+c}\cdot \frac{-2c}{(x-c)^2}}{-\frac {1}{x^2}}=\lim_{x\to \infty} \frac{x-c}{x+c}\cdot \frac{2... |
1,025,117 | <p>Let $V$ be finite dim $K-$vector space. If w.r.t. any basis of $V$, the matrix of $f$ is a diagonal matrix, then I need to show that $f=\lambda Id$ for some $\lambda\in K$. </p>
<p>I am trying a simple approach: to show that $(f-\lambda Id)(e_i)=0$ where $(e_1,...,e_2)$ is a basis of $V$. Let the diagonal matrix b... | Ivo Terek | 118,056 | <p>We have:$$\lim_{x\to \infty}x\ \log\left(\frac{x+c}{x-c}\right) = \lim_{x\to \infty}\ \log\left(\frac{x+c}{x-c}\right)^x = \log \lim_{x \to +\infty}\left(\frac{1+\frac{c}{x}}{1-\frac{c}{x}}\right)^x = \log \lim_{x \to +\infty} \frac{\left(1+\frac{c}{x}\right)^x}{\left(1-\frac{c}{x}\right)^x}$$</p>
<p>Using the fund... |
1,071,321 | <p>The problem is quite simple to formulate. If you have a large group of people (n > 365), and their birthdays are uniformly distributed over the year (365 days), what's the probability that every day of the year is someone's birthday?</p>
<p>I am thinking that the problem should be equivalent to finding the number o... | Graham Kemp | 135,106 | <p>Birthday Coverage is basically a <a href="http://en.wikipedia.org/wiki/Coupon_collector%27s_problem" rel="nofollow">Coupon Collector's</a> problem.</p>
<p>You have $n$ people who drew birthdays with repetition, and wish to find the probability that all $365$ different days were drawn among all $n$ people. ($n\geq ... |
1,470,819 | <p>Let $f$ be defined (and real-valued) on $[a,b]$. For any $x\in [a,b]$ form a quotient $$\phi(t)=\dfrac{f(t)-f(x)}{t-x} \quad (a<t<b, t\neq x),$$ and define $$f'(x)=\lim \limits_{t\to x}\phi(t),$$ provided this limit exists in accordance with Defintion 4.1. </p>
<p>I have one question. Why Rudin considers $t\i... | user133281 | 133,281 | <p>Let the endpoints be $a$ and $b$. In the semilog plot, we plot $xf(x) = e^y f(e^y)$ as a function of $y = \log(x)$. The endpoints then are $\log(a)$ and $\log(b)$ and the area under the curve is
$$
\int_{\log(a)}^{\log(b)} e^y f(e^y) \, dy = \int_{a}^{b} f(x) \, dx = Q
$$
which follows from the <a href="https://en.w... |
320,704 | <p>Given an irrational <span class="math-container">$a$</span>, the sequence <span class="math-container">$b_n := na$</span> is dense and equidistributed in <span class="math-container">$\mathbb S^1$</span> where we view <span class="math-container">$\mathbb S^1$</span> as <span class="math-container">$[0, 1]$</span> ... | Aaron Meyerowitz | 8,008 | <p>I think that the answer from @burtonpeterj is in fact pretty much best possible. Note that it does not utilize <span class="math-container">$p.$</span> I don't think there is a way to work general values of <span class="math-container">$p$</span> into the estimate.</p>
<p>If you take the fractional parts of <span c... |
320,619 | <p>Consider $\ell^\infty $ the vector space of real bounded sequences endowed with the sup norm, that is
$||x|| = \sup_n |x_n|$ where $x = (x_n)_{n \in \Bbb N}$. </p>
<p>Prove that $B'(0,1) = \{x \in l^\infty : ||x|| \le 1\} $ is not compact.</p>
<p>Now, we are given a hint that we can use the equivalence of sequent... | saz | 36,150 | <p>You have to find a sequence in $B'(0,1)$ which does not contain a convergent subsequence. Because the existenc of such a sequence implies (right from the definition) that $B'(0,1)$ isn't sequentially compact.</p>
<p><strong>Hint</strong> Consider the sequence $(y^n)_n$ defined by $$y^n_k := \begin{cases} 1 & n=... |
3,420,459 | <p>I have this <a href="https://i.stack.imgur.com/ew8Id.png" rel="nofollow noreferrer">question:</a></p>
<blockquote>
<p>If <span class="math-container">$a\otimes b=a^b-b^a$</span>, what is <span class="math-container">$(3\otimes 2)\otimes (4\otimes 1)$</span>?</p>
</blockquote>
<p>The answer in the solution set I ... | K B Dave | 534,616 | <p>If the problem had been posed this way:</p>
<blockquote>
<p>Let <span class="math-container">$f$</span> be the function with</p>
<ul>
<li>domain pairs of positive numbers, </li>
<li>range all real numbers, and</li>
<li>value at <span class="math-container">$(a,b)$</span> given by <span class="math-cont... |
536,805 | <p>Three cards are drawn sequentially from a deck that contains 16 cards numbered 1 to 16 in an arbitrary order. Suppose the first card drawn is a 6.</p>
<p>Define the event of interest, A, as the set of all increasing 3-card sequences, i.e. A={(x1,x2,x3)|x1 < x2 < x3}, where x1,x2,x3∈{1,⋯,16}. Define event B as... | Henry | 6,460 | <p>Hint: </p>
<ul>
<li>$S_{x_3=8} = \{(6,7,8)\}$</li>
<li>$S_{x_3=9} = \{(6,7,9),(6,8,9)\}$</li>
<li>$S_{x_3=t} = \{(6,7,t),\ldots,(6,t-1,t)\}$</li>
<li>Count each</li>
<li>Add up the counts</li>
</ul>
|
891,137 | <p>Count all $n$-length strings of digits $0, 1,\dots, m$ that have an equal number of $0$'s and $1$'s. Is there a closed form expression?</p>
| Jack D'Aurizio | 44,121 | <p>$f(x)=\log(1+x)$ is a concave function over $(-1,+\infty)$, since its second derivative equals:
$$f''(x) = -\frac{1}{(1+x)^2}<0.$$
Concavity implies that the graphics of $f(x)$ always lies under the graphics of any tangent line. </p>
<p>So, consider the equation of the tangent line in $x=0$ in order to have:
$$\... |
891,137 | <p>Count all $n$-length strings of digits $0, 1,\dots, m$ that have an equal number of $0$'s and $1$'s. Is there a closed form expression?</p>
| Varun Iyer | 118,690 | <p>So the inequality is:</p>
<p>$$\log(1+x) \le x$$</p>
<p>First, notice that $x > -1$ for all $x$ in the function $f(x) = \log(1+x)$</p>
<p>Since, start plugging in values, and see that for any $x$, where $x > -1$, $x > \log(1+x)$, except when $x=0$, which in that case $x = \log(1+x)$</p>
<p>So the soluti... |
891,137 | <p>Count all $n$-length strings of digits $0, 1,\dots, m$ that have an equal number of $0$'s and $1$'s. Is there a closed form expression?</p>
| André Nicolas | 6,312 | <p><strong>A start:</strong> Let $f(x)=x-\ln(1+x)$. Note that $f(0)=0$. Use the <em>derivative</em> of $f(x)$ to conclude that $f(x)$ reaches a minimum at $x=0$.</p>
|
3,369,438 | <p><span class="math-container">$\begin{array}{|l} \forall xP(x) \vee \forall x \neg P(x) \quad premise
\\ \exists xQ(x) \rightarrow \neg P(x) \quad premise \\ \forall xQ(x) \quad premise
\\\hline \begin{array}{|l} \forall xP(x) \quad assumption
\\\hline \vdots \quad
\\ \forall x\neg P(x) \quad \end{array}
\\\begin{a... | Graham Kemp | 135,106 | <p>Hint: you have an existential and a universal in the premises. Assume a witness for the existance and see what happens under the assumption of <span class="math-container">$\forall x~P(x)$</span>.</p>
<p><span class="math-container">$\begin{array}{|l} \forall xP(x) \vee \forall x \neg P(x) \quad premise
\\ \exist... |
2,056,499 | <p>I am trying to prove that if
$$
\lim_{x \to c} (f(x)) = L_1
\\ \lim_{x \to c} (g(x)) = L_2
\\ L_1, L_2 \geq 0
$$
Then
$$
\lim_{x \to c} f(x)^{g(x)} = (L_1)^{L_2}
$$</p>
<p>I am doing this for fun, and my prof said that it shouldn't be too hard, but all I got so far is
$$
\forall \epsilon >0 \ \exists \delta >... | RyRy the Fly Guy | 412,727 | <p>Given
<span class="math-container">$$
\lim_{x \to c} (f(x)) = L_1
\\ \lim_{x \to c} (g(x)) = L_2
$$</span></p>
<p>as long as <span class="math-container">$L_1 > 0$</span>, then your statement is true as follows:</p>
<p><span class="math-container">$$\lim_{x \rightarrow c} f(x)^{g(x)} = \lim_{x \rightarrow c} e^... |
35,964 | <p>This is kind of an odd question, but can somebody please tell me that I am crazy with the following question, I did the math, and what I am told to prove is simply wrong:</p>
<p>Question:
Show that a ball dropped from height of <em>h</em> feet and bounces in such a way that each bounce is $\frac34$ of the height of... | Yuval Filmus | 1,277 | <p>Note that when the ball bounces it goes both up and down. So from the second term onwards, you need to count each term twice. Therefore the answer is $2 \cdot 4h - h = 7h$ ($h$ is the first term, which is only counted once).</p>
|
35,964 | <p>This is kind of an odd question, but can somebody please tell me that I am crazy with the following question, I did the math, and what I am told to prove is simply wrong:</p>
<p>Question:
Show that a ball dropped from height of <em>h</em> feet and bounces in such a way that each bounce is $\frac34$ of the height of... | Arturo Magidin | 742 | <p>Your computation does not give the <em>total distance traveled</em>, it only gives the distance it traveled <em>downward.</em></p>
<p>The ball first falls $h$. Then it <em>rises</em> $\frac{3}{4}h$, and falls $\frac{3}{4}h$ again; then it <em>rises</em> $(\frac{3}{4})^2h$, and falls that much again. Etc.</p>
<p>So... |
40,500 | <blockquote>
<p>What are the most
fundamental/useful/interesting ways in
which the concepts of Brownian motion,
martingales and markov chains are
related?</p>
</blockquote>
<p>I'm a graduate student doing a crash course in probability and stochastic analysis. At the moment, the world of probability is a conf... | Steven Heston | 6,769 | <p>Girsanov's Theorem.</p>
|
1,893,609 | <p>I am trying to show that $A=\{(x,y) \in \Bbb{R} \mid -1 < x < 1, -1< y < 1 \}$ is an open set algebraically. </p>
<p>Let $a_0 = (x_o,y_o) \in A$. Suppose that $r = \min\{1-|x_o|, 1-|y_o|\}$ then choose $a = (x,y) \in D_r(a_0)$. Then</p>
<p>Edit: I am looking for the proof of the algebraic implication t... | Parcly Taxel | 357,390 | <p>The question states that all the points are on the circumference. Three non-collinear points already determine a unique circle passing through them, so any three of the four given points may be chosen and the fourth will automatically lie on the circle.</p>
<p>The problem of finding the centre of the circle through... |
11,629 | <p>The basic logic course in school gives the impression that logic has both the syntax and the semantics aspects. Recently, I wonder whether the syntax part still plays an essential role in the current studies. Below are some of my observations, I hope the idea from the community can make them more complete.</p>
<p>M... | Adam | 2,361 | <p>I would posit that proof theory is exactly the syntactical part of logic, and the other three branches (set theory, model theory, and recursion theory) are what's left when you get rid of the syntax. Which is probably why those three branches get more attention nowadays.</p>
|
1,952,562 | <p>Find an equation of the tangent line to the curve $y = sin(3x) + sin^2 (3x)$ given the point (0,0). Answer is $y = 3x$, but please explain solution steps. </p>
| dantopa | 206,581 | <p><strong>Goal</strong></p>
<p>Find the slope $m$, and intercept $b$, for the line
$$
y = mx + b,
$$
tangent at the origin to the curve
$$
f(x) = \sin ^2(3 x)+\sin (3 x).
$$</p>
<p><strong>Intercept $b$</strong></p>
<p>Because the function goes through the origin, the tangent line will also go through the origin.... |
4,480,570 | <p>I am reading over the proof of <strong>Lemma 10.32 (Local Frame Criterion for Subbundles)</strong> in Lee's <em>Introduction to Smooth Manifolds</em>.</p>
<p>The lemma says</p>
<blockquote>
<p>Let <span class="math-container">$\pi: E \rightarrow M$</span> be a smooth vector bundle and suppose that for each <span cla... | subrosar | 602,170 | <p>Suppose that there is an open cover <span class="math-container">$\{U_i\}$</span> of <span class="math-container">$E$</span> such that <span class="math-container">$D\cap U_i\hookrightarrow E$</span> is a smooth embedding and <span class="math-container">$D\cap U_i$</span> is open in <span class="math-container">$D$... |
2,362,942 | <p>How could I notate a matrix rotation?</p>
<p>Example:
$ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix},\:\:\: A_{\text{rotated}} = \begin{pmatrix} c & a \\ d & b\end{pmatrix}$. </p>
<p>Notice the whole matrix is "rotated" clockwise. Is there any notation for this, and anyway to compute it genera... | Jim Ferry | 458,592 | <p>A bit more generally, how does one express the actions of $D_4$ on a square matrix? The transpose expresses a flip about the main diagonal. Multiplying on the left by the matrix
$$F = \left(\begin{matrix} 0 & 0 & \cdots & 1 \\ \vdots & \vdots & \ddots & \vdots \\0 & 1 & \cdots &... |
46 | <p>I have solved a couple of questions myself in the past, and I think some of them are interesting to the public and will most likely appear in the future. One example for this is the question how to enable antialiasing in the Linux frontend, for which there is no native support right now. My question would now be whe... | Robert Cartaino | 80 | <p>If your questions are genuinely interesting or intriguing problems, then posting self-answered questions is generally considered okay. </p>
<p>My only caution this early in a beta is with regard to asking questions simply to
"seed" the site. Stack Exchange sites don't generally need to be seeded for the purpose of... |
810,190 | <blockquote>
<p>Page 29 of Source 1: Denote the complex conjugate by * :
$\mathbf{u \cdot v} = \sum_{1 \le i \le n} u_i^*v_i = (\mathbf{v \cdot u})^*$</p>
<p><a href="http://www.math.sunysb.edu/~eitan/la13.pdf" rel="nofollow">Page 1 of Source 2:</a> $\mathbf{u \cdot v} = \mathbf{u}^T\mathbf{ \bar{v} }$.</p>
... | Peter Franek | 62,009 | <p>Source 1. and 3. are identical. It depends on the convention: there is not really a big difference, just in one case it is linear in the first entry and antilinear ($(u, \alpha v)=\bar\alpha (u,v)$) in the second entry (Source 2) and in the other convention, it is antilinear in the first and linear in the second ent... |
133,936 | <p>I am trying to understand a part of the following theorem:</p>
<blockquote>
<p><strong>Theorem.</strong> Assume that $f:[a,b]\to\mathbb{R}$ is bounded, and let $c\in(a,b)$. Then, $f$ is integrable on $[a,b]$ if and only if $f$ is integrable on $[a,c]$ and $[c,b]$. In this case, we have
$$\int_a^bf=\int_a^cf+\in... | nullUser | 17,459 | <p>Think about how $U(f,P_1)-L(f,P_1)$ relates to $U(f,P)-L(f,P)$. Suppose $P_1=\{ t_0,t_1,\ldots,t_n \}$ and $P = \{ t_0, t_1, \ldots, t_n,\ldots,t_{n+k}\}$. Then we have</p>
<p>$$
U(f,P_1)-L(f,P_1) = \sum_{i=1}^n (\sup\limits_{[t_{i-1},t_i]}(f)-\inf\limits_{[t_{i-1},t_i]}(f))|t_i-t_{i-1}|
\leq \sum_{i=1}^{n+k} (\su... |
2,931,762 | <blockquote>
<p>Given a function <span class="math-container">$f$</span> is defined for integers <span class="math-container">$m$</span> and <span class="math-container">$n$</span> as given:
<span class="math-container">$$f(mn) = f(m)\,f(n) - f(m+n) + 1001$$</span>
where either <span class="math-container">$m$</s... | ℋolo | 471,959 | <p>You got $$f(x)=2f(x)-f(x+1)+1001$$Now, let's find $f(x+1)$ using $f(x)$:$$f(x)=2f(x)-f(x+1)+1001\iff f(x+1)=f(x)+1001$$Now, let $y=x+1, x=y-1$ you get $$f(y)=f(y-1)+1001$$</p>
<p>(Note, both $x$ and $y$ are just names of the variables, so if it is more convenient to you, you can rewrite the final line using $x$ ins... |
2,931,762 | <blockquote>
<p>Given a function <span class="math-container">$f$</span> is defined for integers <span class="math-container">$m$</span> and <span class="math-container">$n$</span> as given:
<span class="math-container">$$f(mn) = f(m)\,f(n) - f(m+n) + 1001$$</span>
where either <span class="math-container">$m$</s... | fleablood | 280,126 | <blockquote>
<p>Given a function f is defined for integers m and n as given:
f(mn)=f(m)f(n)−f(m+n)+1001
where either m or n is equal to 1</p>
</blockquote>
<p>As multiplication and addition is commutative we can assume, wolog <span class="math-container">$n = 1$</span> and that is just a convoluted way of simply... |
2,304,332 | <p>This is related to <a href="https://math.stackexchange.com/questions/2162294/prove-that-there-exists-f-g-mathbbr-to-mathbbr-such-that-fgx-i/2294324#2294324">Prove that there exists $f,g : \mathbb{R}$ to $\mathbb{R}$ such that $f(g(x))$ is strictly increasing and $g(f(x))$ is strictly decreasing.</a></p>
<p>But acco... | José Carlos Santos | 446,262 | <p>If $f(g(x))=x$, then $f$ is surjective and $g$ is injective. And if $g(f(x))=-x$, then $g$ is surjective and $f$ is injective. Therefore, $f$ and $g$ are bijections and then it follows from $f(g(x))=x$ that $g=f^{-1}$. But this contradicts the fact that $g(f(x))=-x$.</p>
|
3,513,581 | <p>I'm referring to Sakasegawa's forumla for calculating average line length in a queuing system.</p>
<p><a href="https://i.stack.imgur.com/Ydg3Y.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Ydg3Y.png" alt="enter image description here"></a></p>
<p>I don't understand the result intuitively. For... | Mark Spearman | 1,078,333 | <p>This formula is for an M/M/m queue which has exponential interarrival times and exponential service times and is an approximation for <span class="math-container">$m>1$</span>.</p>
|
853,878 | <p>I'm learning calc and after learning about how to differentiate using product rule and chain rule etc. I came across marginal cost and marginal revenue. I'm pretty familiar with cost, profit and revenue. Why would I want to find the marginal revenue (aka rate of change for revenue or cost)? What does this rate of ch... | Puzzle_riddle | 161,144 | <p>Great question and there are many potential answers to this question. Let me give you one that from my experience is used extensively.</p>
<p>Suppose you have a firm that produces widgets. A widget sells for \$300, and it costs \$100 per unit, if you only produce 5; the next 5 will cost \$200 per unit to produce, t... |
540,217 | <p><strong>Question:</strong></p>
<p>$\int ^1_0 \frac {\ln x}{1-x^2}dx$ - converges or diverges?</p>
<p><strong>What we did:</strong></p>
<p>We tried to compare with $-\frac 1x$ and $-\frac 1{x-1}$ but ended up finding that these convergence tests fail. Our book says this integral diverges, but Wolfram on the other ... | Ron Gordon | 53,268 | <p>There are two potential sources of divergence if the integral were to diverge (it doesn't): at $x=0$ and $x=1$. At $x=0$, the integral behaves as $\ln{x}$, which has antiderivative $x \ln{x}-x$. You may show that the limit of this expression as $x \to 0$ is $0$ using e.g.,L'Hopital. So the integrand is integrable... |
588,488 | <p>I know that for the harmonic series
$\lim_{n \to \infty} \frac1n = 0$ and
$\sum_{n=1}^{\infty} \frac1n = \infty$.</p>
<p>I was just wondering, is there a sequence ($a_n =\dots$) that converges "faster" (I am not entirely sure what's the exact definition here, but I think you know what I mean...) than $\frac1n$ to ... | Fixed Point | 30,261 | <p>I had wondered about this too a long time ago and then came across this. The series</p>
<p>$$\sum_{n=3}^{\infty}\frac{1}{n\ln n (\ln\ln n)}=\infty$$</p>
<p>diverges and it can be very easily proven by the integral test. But here is the kicker. This series actually requires googolplex number of terms before the p... |
249,332 | <p>Consider the following list</p>
<pre><code>list={{{0,0,0},0},{{0,0,1},a},{{0,0,-1},-a},{{1,0,1},b},{{1,0,0},-b},{{1,1,1},a+b},{{1,1,1},a-b},{{-1,0,-1},{-a-b}},{{-1,0,-1},{-a+b}}};
</code></pre>
<p>how can this list be sorted such that a>b>0, therefore the expected result would be</p>
<pre><code>list={{{1,1,1},... | kglr | 125 | <pre><code>ordering = Reverse @
Ordering[list[[All, 2]] /. FindInstance[a > b > 0 && 2 b > a, {a, b}][[1]]];
list[[ordering]]
</code></pre>
<blockquote>
<pre><code>{{{1, 1, 1}, a + b}, {{0, 0, 1}, a}, {{1, 0, 1}, b}, {{1, 1, 1}, a - b},
{{0, 0, 0}, 0}, {{-1, 0, -1}, -a + b}, {{1, 0, 0}, -b}, {{0... |
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