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 |
|---|---|---|---|---|
843,763 | <blockquote>
<p>Let $x\in \mathbb{R}$ an irrational number. Define $X=\{nx-\lfloor nx\rfloor: n\in \mathbb{N}\}$. Prove that $X$ is dense on $[0,1)$. </p>
</blockquote>
<p>Can anyone give some hint to solve this problem? I tried contradiction but could not reach a proof.</p>
<p>I spend part of the day studying this... | mm-aops | 81,587 | <p>Ok, since you've asked and it doesn't fit into a comment, there you go. I'll do it on a circle since it's slightly easier to explain and I'll leave it to you to complete it in the case of an interval. let's say you have a circle of length $1$. you take 'steps' along the circle of an irrational length, let's say coun... |
2,213,528 | <p>Combination formula is defined as:</p>
<p>$$\binom{n}{r}=\frac{n!}{(n-r)!\cdot r!}$$</p>
<p>We do $(n-r)!$ because we want the combinations of only $r$ objects given $n$. We do $r!$ again because in combinations, order does not matter so doing this gets rid of repeated results. But why? Why does $r!$ get rid of an... | spaceisdarkgreen | 397,125 | <p>If you have continuous RVs $X_1,X_2$ with joint density function $ f_{X_1X_2}(x_1,x_2)$ then you can define the conditional joint density given $X_2 = x_2^0$ as $$ f_{X_1|X_2}(x_1|x_2^0) = \frac{f_{X_1X_2}(x_1,x_2^0)}{f_{X_2}(x_2^0)} $$ where $f_{X_2}(x_2)$ is the marginal density of $X_2,$ given by $\int_{-\infty}^... |
3,220,135 | <p>How many integer numbers, <span class="math-container">$x$</span>, verify that the following</p>
<p><span class="math-container">\begin{equation*}
\frac{x^3+2x^2+9}{x^2+4x+5}
\end{equation*}</span></p>
<p>is an integer?</p>
<p>I managed to do:</p>
<p><span class="math-container">\begin{equation*}
\frac{x^3+2x^2+... | Servaes | 30,382 | <p>You're off to a good start. Now note that the denominator <span class="math-container">$x^2+4x+5$</span> is quickly larger than the numerator <span class="math-container">$3x+19$</span>; you can quickly reduce the problem to only finitely many values for <span class="math-container">$x$</span> to check.</p>
<p><str... |
3,343,550 | <p>Show that if <span class="math-container">$(a,15)=1$</span>, then <span class="math-container">$a^4\equiv1 \mod 15$</span>, so that we do not have primitive roots of <span class="math-container">$15$</span>
Please help me with this problem.</p>
| J. W. Tanner | 615,567 | <p>If <span class="math-container">$\gcd(a,15)=1,$</span> then <span class="math-container">$a\equiv\pm1, \pm2, \pm4,$</span> or <span class="math-container">$\pm7\mod15$</span>, so <span class="math-container">$a^2\equiv1$</span> or <span class="math-container">$ 4$</span>, so <span class="math-container">$a^4=(a^2)^2... |
4,350,781 | <p><a href="https://i.stack.imgur.com/57qXm.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/57qXm.jpg" alt="enter image description here" /></a></p>
<p>I was able to follow most of the proof but I don’t understand how the author concludes that <span class="math-container">$r=0$</span> at the final pa... | Toby Bartels | 63,003 | <p>For the sloping surface, this is the plane through the points <span class="math-container">$ ( 0 , 0 , 1 ) $</span>, <span class="math-container">$ ( 0 , 1 , 0 ) $</span>, and <span class="math-container">$ ( 1 , 0 , 0 ) $</span>. Since none of these is the origin, this plane has an equation of the form <span class... |
2,043,690 | <p>I'm in the process of studying for an exam and this just popped into my head. Sorry if it's a dumb question</p>
| Noah Schweber | 28,111 | <p>Yes, they are the same. Note that $(-1)+13=12$, so their difference is exactly the thing that is ignored by mod $13$!</p>
|
1,507,519 | <p>How do I solve this question?</p>
<p><a href="https://i.stack.imgur.com/drfiA.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/drfiA.jpg" alt="Question"></a></p>
<p>I tried using the quadratic formula on the question equation and got</p>
<p>$x_1 = 0.25 +1.089724..i = \ln r$</p>
<p>$x_2 = 0.25 -... | Aditya Agarwal | 217,555 | <p>For the $a)$ part, use Vieta's Formula: Sum of roots of a quadratic equation is $\frac{-b}a$.
So answer would be: $\frac12$. <br>
For $b)$ $\frac54$, using Vieta's Formulas for product of roots: $\frac ca$. <br>
For $c)$ the expression becomes $$\frac{\ln r+\ln s}{\ln r\ln s}=\frac{\frac12}{\frac54}=\frac25$$
For ... |
1,250,459 | <p>Let $A\in M_{m \times n}(\mathbb{R})$, $x\in \mathbb{R}^n$ and $b,y\in \mathbb{R}^m$. Show that if $Ax=b$ and $A^ty=0_{\mathbb{R}^m}$, then $\langle b,y\rangle=0$. Also make a geometric interpretation.</p>
<p>I think I may have something to do with overdetermined / underdetermined system, but do not know how to pro... | Ian | 83,396 | <p>A set is open if, from any point in the set, you can wiggle in any direction a little bit and stay inside the set. What "wiggle" means depends on the context at hand. </p>
<p>In metric spaces, "wiggle" means what you might expect: "move a small distance". That is, for each point in an open subset of a metric space,... |
2,985,256 | <blockquote>
<p>Let there are three points <span class="math-container">$(2,5,-3),(5,3,-3),(-2,-3,5)$</span> through which a plane passes. What is the equation of the plane in Cartesian form?</p>
</blockquote>
<p>I know how to find it in using vector form by computing the cross product to get the normal vector and p... | ilovebulbasaur | 586,948 | <p>Arrange the two equations into an augmented matrix to get:</p>
<p>M=
<span class="math-container">\begin{bmatrix}
3 & -2 & 0 & 0 \\
-4 & -8 & 8 & 0
\end{bmatrix}</span></p>
<p>The general solution of this linear system is any scalar multiple of</p>
<p>x =
<span class="math-c... |
1,108,918 | <p>Given two functions $f$ and $g$ whose derivatives $f'$ and $g'$ satisfy : $f'(x)=g(x), g'(x)=-f(x),f(0)=0, g(0)=1$ for all $x$ in an interval $J$. $~~~\cdots(A)$</p>
<p>(a) Prove that $f^2(x)+g^2(x)=1 ~\forall~x \in J $</p>
<p>(b) Let $F$ and $G$ be another pair of functions in $J$ which satisfy the given conditio... | math110 | 58,742 | <p>Hint</p>
<p>Let</p>
<p>$$H(x)=f(x)F(x)+g(x)G(x)$$
since
$$F'=G,G'=-F,f'=g,g'=-f$$
so</p>
<p>\begin{align*}H'(x)&=f'(x)F(x)+f(x)F'(x)+g'(x)G(x)+g(x)G'(x)\\
&=g(x)F(x)+f(x)G(x)+g'(x)G(x)+g(x)G'(x)\\
&=g(x)[F(x)+G'(x)]+G(x)[f(x)+g'(x)]\\
&=0
\end{align*}
so
$$H(x)=H(0)=f(0)F(0)+g(0)G(0)=1$$</p>
|
3,913,005 | <p>So say there is 26 possible characters and you've got 1800 long random string. Now how would you go about finding the chance of there being a specific 5 letter word within?</p>
<p>I already found a related question to this but the given formula in the answer doesn't seem to make sense: <a href="https://math.stackexc... | user2661923 | 464,411 | <p>My previous answer involved Inclusion-Exclusion. <br><br />
My intuition of the proper way of using Inclusion-Exclusion has changed. <br>
Given a set <span class="math-container">$S$</span> with a finite number of elements, let <span class="math-container">$|S|$</span> denote the
number of elements of set <span clas... |
3,390,407 | <p>If <span class="math-container">$t>0,t^2, t+\frac{1}{t},t+t^2,\frac{1}{t}+\frac{1}{t^2}$</span> are all irrational number,
<span class="math-container">$$a_n=n+\left \lfloor \frac{n}{t} \right \rfloor+\left \lfloor \frac{n}{t^2} \right \rfloor,\\
b_n=n+\left \lfloor \frac{n}{t} \right \rfloor +\left \lfloor nt \r... | Hw Chu | 507,264 | <p><strong>Edited 10/24.</strong> Now this is largely expanded, probably longer than what it needs to be. TL;DR: Construct a function <span class="math-container">$f$</span>, record where it ticks. The range of <span class="math-container">$f$</span> is exactly <span class="math-container">$\mathbb N$</span> and covers... |
500,579 | <p>Is there a formula for the coefficients of $x^n$ for </p>
<p>$$
\prod_{i=1}^N(x+x_i)
$$</p>
<p>in terms of $x_i$?</p>
| Community | -1 | <p>The coefficient of $x$ is
$$\sum_{k=1}^N\prod_{\substack{i=1\\i\ne k}}^N x_i$$</p>
<p>and to see how we find this formula just expand by choosing one $x$ from one factor and the $x_i$ from the other
$$(\color{red}{x}+x_1)(x+\color{red}{x_2})\cdots(x+\color{red}{x_n})$$
and repeat the same thing for all the factors... |
500,579 | <p>Is there a formula for the coefficients of $x^n$ for </p>
<p>$$
\prod_{i=1}^N(x+x_i)
$$</p>
<p>in terms of $x_i$?</p>
| Michael Hardy | 11,667 | <p>$$
\begin{align}
& \phantom{{}={}} (x+x_1)(x+x_2)(x+x_3)(x+x_4) \\[12pt]
& = x^4 & \text{(all size-0 subsets)} \\
& \phantom{=} {} + (x_1+x_2+x_3+x_4) x^3 & \text{(all size-1 subsets)} \\
& \phantom{=} {} + (x_1 x_2 + x_1 x_3 + x_1 x_4 + x_2 x_3 + x_2 x_4 + x_3 x_4) x^2
& \text{(all size-... |
2,834,219 | <p>I'd like to numerically evaluate the following integral using computer software but it has a singularity at $x=1$:</p>
<p>\begin{equation}
\int_1^{\infty} \frac{x}{1-x^4} dx
\end{equation}</p>
<p>I was thinking of a variable transformation, rewriting the expression, or something of a kind. One of my attempts was t... | Mark Viola | 218,419 | <p>Using partial fraction expansion reveals</p>
<p>$$\frac{x}{1-x^4}=\frac{x}{2(x^2+1)}-\frac{1}{4(x-1)}-\frac{1}{4(x+1)}$$</p>
<p>Hence, we see that for $t>1$</p>
<p>$$\begin{align}
\int_t^\infty \frac{x}{1-x^4}\,dx&=\frac14 \log\left(\frac{1-t^2}{1+t^2}\right)\tag1
\end{align}$$</p>
<p>Inasmuch as $\displa... |
112,147 | <p>I have a long vector and some of the values (19 out of 64) are complex. I got them using the Mathematica Rationalize function, so the complex ones are written in the a+bi form. Is there a function I can apply to the entire vector, that would change my complex numbers to the form A<em>Exp[I</em>phi]? </p>
| Alexei Boulbitch | 788 | <p>Try this:</p>
<pre><code>{1, 1 + 2 I, 3 - 5 I, 7, 9 + I} /.x_ /; Head[x] == Complex ->
Sqrt[Re[x]^2 + Im[x]^2]*Exp[ArcTan[Re[x], Im[x]]]
</code></pre>
<p>yielding</p>
<pre><code>(* {1, Sqrt[5] E^ArcTan[2], Sqrt[34] E^-ArcTan[5/3], 7,
Sqrt[82] E^ArcTan[1/9]} *)
</code></pre>
<p><strong>Edit</strong>: ... |
112,147 | <p>I have a long vector and some of the values (19 out of 64) are complex. I got them using the Mathematica Rationalize function, so the complex ones are written in the a+bi form. Is there a function I can apply to the entire vector, that would change my complex numbers to the form A<em>Exp[I</em>phi]? </p>
| Jason B. | 9,490 | <p>The trouble with many methods is that they only work on integer inputs. Trying Alexei's answer with approximate numbers</p>
<pre><code>{1.0, 1.0 + 2 I, 3.0 - 5 I, 7, 9.0 + I} /.
x_ /; Head[x] == Complex ->
Sqrt[Re[x]^2 + Im[x]^2]*Exp[I ArcTan[Re[x], Im[x]]]
(* {1., 1. + 2. I, 3. - 5. I, 7, 9. + 1. I} *)
</... |
482,596 | <p>How do I evaluate $ sin(20)$ exactly? [in degrees]</p>
<p>I derived the relationship between $sin(x) $ and $sin(3x)$ where $x = sin(x)$ and $ y = sin(3x)$</p>
<p><a href="http://www.wolframalpha.com/input/?i=-4x%5E3+%2B+3x+%3D+y%2C+solve+for+x" rel="nofollow">http://www.wolframalpha.com/input/?i=-4x%5E3+%2B+3x+%3... | Nick Peterson | 81,839 | <p><strong>Hint:</strong> To show that it is not unique, all you need to do is find two distinct pairs $(s_1,t_1)$ and $(s_2,t_2)$ such that $1=7s_1+11t_1=7s_2+11t_2$.</p>
|
3,774,400 | <p>Just like the title says, I don't know how to write an example matrix here to look like matrix. If this makes sense, <span class="math-container">$A$</span> can be <span class="math-container">$[1 0 0;0 1 0;0 1 0]$</span> (like in MATLAB syntax) then if we find determinants of <span class="math-container">$A-\lambda... | Fred | 380,717 | <p>If <span class="math-container">$A$</span> is your matrix above, then</p>
<p><span class="math-container">$$ \det(A- \lambda E)=- \lambda(\lambda-1)^2.$$</span></p>
|
324,219 | <p>Given an urn with $M$ unique balls, how many times do I need to draw with replacement before the probability that I have seen each ball at least once is greater than $\epsilon$?</p>
| wece | 65,630 | <p>You can see this as a combinatorial problem.</p>
<p>There are $M^k$ possible run of $k$ draw each with the same probability (the draw are independent).</p>
<p>For a run we have seen each balls at least once if we can choose $M$ indexes such that for those indexes the balls are different. There are ${k\choose M}$ s... |
4,389,997 | <p>In Enderton's <em>A Mathematical Introduction to Logic</em>, he defines <span class="math-container">$n$</span>-tuples recursively using ordered pairs, i.e. <span class="math-container">$\langle x_1,\dots,x_{n+1}\rangle=\langle\langle x_1,\dots,x_n\rangle, x_{n+1}\rangle$</span>. But he also notes,</p>
<blockquote>
... | Tankut Beygu | 754,923 | <p>The author suggests the definition of <em>n</em>-tuples as nested ordered pairs,
<span class="math-container">$$\langle x_1,\dots,x_{n+1}\rangle=\langle\langle x_1,\dots,x_n\rangle, x_{n+1}\rangle ,$$</span></p>
<p>has several cognitional and operational features, which might not be so significant for others. We rea... |
2,906,797 | <p>I want to express this polynomial as a product of linear factors:</p>
<p>$x^5 + x^3 + 8x^2 + 8$</p>
<p>I noticed that $\pm$i were roots just looking at it, so two factors must be $(x- i)$ and $(x + i)$, but I'm not sure how I would know what the remaining polynomial would be. For real roots, I would usually just d... | Mark Bennet | 2,906 | <p>If you spotted this by looking at it you have good intuition. When roots come in complex pairs you can always combine them to find a quadratic factor with real coefficients. Here $(x+i)(x-i)=x^2+1$</p>
<p>If you could do the first bit with intuition, I am sure you can do that division and complete the factorisation... |
662,744 | <h2>Question Statement</h2>
<blockquote>
<p>Let $K$ be a finite field of $q$ elements. Let $U$, $V$ be vector spaces over $K$ with
$\dim(U) = k$, $\dim(V) = l$. How many linear maps $U \rightarrow V$ are there?</p>
</blockquote>
<p>I'm struggling to answer this question. Given that a linear map is uniquely determ... | voldemort | 118,052 | <p>All such linear maps are of the form $c_1y_1+..c_ky_k$ where $y_1,..y_k$ is the image of the basis vectors of $U$, and $c_i$ are scalars. Note that you can have $q$ different choices for each $y_i$</p>
<p>Hope you can finish the problem now.</p>
|
662,744 | <h2>Question Statement</h2>
<blockquote>
<p>Let $K$ be a finite field of $q$ elements. Let $U$, $V$ be vector spaces over $K$ with
$\dim(U) = k$, $\dim(V) = l$. How many linear maps $U \rightarrow V$ are there?</p>
</blockquote>
<p>I'm struggling to answer this question. Given that a linear map is uniquely determ... | Asa Cremin | 77,802 | <p>The answer (if I actually got it right), for anyone still wondering:</p>
<p>Let $\mathbf{e}_1,\mathbf{e}_2,\ldots,\mathbf{e}_k$ be a basis of $U$, $\mathbf{f}_1,\ldots,\mathbf{f}_l$ be a basis of $V$ and $T:U\rightarrow V$ be a linear map.</p>
<p>Then
$$T(\mathbf{e}_1) = \alpha_{11}\mathbf{f}_1 + \alpha_{12}\mathb... |
411,875 | <p>This is an exam question I encountered while studying for my exam for our topology course:</p>
<blockquote>
<p>Give two continuous maps from $S^1$ to $S^1$ which are not homotopic. (Of course, provide a proof as well.)</p>
</blockquote>
<p>The only continuous maps from $S^1$ to $S^1$ I can think of are rotations... | nonlinearism | 59,567 | <p>$F_1:S^1\to S^1 $</p>
<p>$F_1(s)=(\cos(2\pi s),\sin(2\pi s))$ and</p>
<p>$F_2:S^1\to S^1$ </p>
<p>$F_2(s)=(\cos(4\pi s),\sin(4\pi s))$ where $s$ goes from $0$ to $1$, are not homotopic.</p>
|
4,594,043 | <p>For <span class="math-container">$|x|<1,$</span> we have
<span class="math-container">$$
\begin{aligned}
& \frac{1}{1-x}=\sum_{k=0}^{\infty} x^k \quad \Rightarrow \quad \ln (1-x)=-\sum_{k=0}^{\infty} \frac{x^{k+1}}{k+1}
\end{aligned}
$$</span></p>
<hr />
<p><span class="math-container">$$
\begin{aligned}
\int... | user170231 | 170,231 | <p>Without expanding using series at all (taking for granted any properties of special functions whose derivations require series manipulation):</p>
<p><span class="math-container">$$\begin{align*}
I &= \int_0^1 \frac{\log(1-x) \log^n(x)}x \, dx \\[1ex]
&= \int_0^1 \frac{\log(x) \log^n(1-x)}{1-x} \, dx \tag{1} ... |
378,228 | <p>This is probably known, but I have not located a reference.</p>
<p>Let <span class="math-container">$P$</span> be the convex hull of <span class="math-container">$k$</span> points in <span class="math-container">$\mathbb R^n$</span> with rational coordinates. Consider the Euclidean square norm function <span class="... | Claudio Gorodski | 15,155 | <p>I think I may have answered my own question under the assumption that the points are linearly independent as vectors in <span class="math-container">$\mathbb R^n$</span>.</p>
<p>Denote the <span class="math-container">$k$</span> points by <span class="math-container">$v_1,\ldots, v_k\in\mathbb R^n$</span>. Let <span... |
757,702 | <p>I am in my pre-academic year. We recently studied the Remainder sentence (at least that's what I think it translates) which states that any polynomial can be written as <span class="math-container">$P = Q\cdot L + R$</span></p>
<p>I am unable to solve the following:</p>
<blockquote>
<p>Show that <span class="mat... | Andrei Kulunchakov | 140,670 | <p>Hint: try to check if complex roots of $x^2+x+1$ are also roots for $(x+1)^{2n+1}+x^{n+2}$.</p>
<p>Also there is another way to prove it. Let see that $((x+1)^2-x)\cdot (x+1)^{2n-1}$ also divided by $x^2+x+1$. So we just need to prove that $x\cdot (x+1)^{2n-1}+x^{n+2}$ is divided by $x^2+x+1$. In this way we can co... |
757,702 | <p>I am in my pre-academic year. We recently studied the Remainder sentence (at least that's what I think it translates) which states that any polynomial can be written as <span class="math-container">$P = Q\cdot L + R$</span></p>
<p>I am unable to solve the following:</p>
<blockquote>
<p>Show that <span class="mat... | Mark Bennet | 2,906 | <p>Suppose $a$ is a root of $x^2+x+1=0$, then we have both $$a+1=-a^2$$ and $$a^3=1$$</p>
<p>Let $f(x)=(x+1)^{2n+1}+x^{n+2}$ then $$f(a)=(-a^2)^{2n+1}+a^{n+2}=-a^{4n+2}+a^{n+2}=-a^{n+2}+a^{n+2}=0$$</p>
<p>Since the two distinct roots of the quadratic are also roots of $f(x)$ we can use the remainder theorem to conclu... |
2,187,571 | <p>Consider $e^{it}$ with $t$ real. I can rewrite $t$ as $t=2\pi \tau$:</p>
<p>$ e^{i2\pi\tau} = (e^{i2\pi})^\tau = (1)^\tau = 1 $ for arbitrary $\tau$ or $t$. Where is the mistake in this calculation ? My guess that $(e^x)^y = e^{x\cdot y}$ only applies for real $x$ and $y$, but I don't see why this should be the cas... | skyking | 265,767 | <p>The problem lies in the meaning of $a^z$, it's only that well defined in general for complex numbers.</p>
<p>For other $a$ we will fall into the "rabbit hole" of multi-functions, but we can do it nevertheless. You define $a^z$ otherwise as:</p>
<p>$$a^z = \exp(z \ln a)$$</p>
<p>For clarity I use $\exp$ for the ex... |
2,495,176 | <p>For how many positive values of $n$ are both $\frac n3$ and $3n$ four-digit integers?</p>
<p>Any help is greatly appreciated. I think the smallest n value is 3000 and the largest n value is 3333. Does this make sense?</p>
| Lisa | 160,525 | <p>To be absolutely clear: $n$ is a positive integer, right?</p>
<p>Then for $\frac{n}{3}$ to be a $4$-digit number, you need $2999 < n < 29998$. But you also have to consider that not all those numbers are multiples of $3$.</p>
<p>But for $3n$ to be a $4$-digit number, you need $333 < n < 3334$.</p>
<p>... |
101,078 | <p>While reading through several articles concerned with mathematical constants, I kept on finding things like this:</p>
<blockquote>
<p>The continued fraction for $\mu$(Soldner's Constant) is given by $\left[1, 2, 4, 1, 1, 1, 3, 1, 1, 1, 2, 47, 2, ...\right]$. </p>
<p>The <strong>high-water marks</strong> are ... | Michael Lugo | 173 | <p>Your definition seems correct to me -- at least, it agrees with the data you've provided and with my intuition, as a native speaker of English, of how this phrase is used. I don't know of a specific significance of these high-water marks, but it's well-known that cutting off a continued fraction expansion just befo... |
1,431,969 | <p>Just like in the title:</p>
<blockquote>
<p>How to compute the CDF of $X\cdot Y$ if $X,Y$ are independent random variables, uniformly distributed over $(-1,1)$?</p>
</blockquote>
<p>I tried using the next formula:
the density of $X\cdot Y$ is the integral of
$f(u/v)\cdot g(v)$ where $f$ is the density of $X$ a... | Jack D'Aurizio | 44,121 | <p>Let $Z=X\cdot Y$ and $f_Z,f_Y,f_X$ be the pdfs of $X,Y,Z$. </p>
<p>$f_Z$ is quite trivially an even function, supported on $[-1,1]$. </p>
<p>So, let we compute, for any $p\in(0,1)$:</p>
<p>$$\mathbb{P}[0\leq Z\leq p]=2\int_{0}^{1}\int_{0}^{\min\left(1,\frac{p}{x}\right)}\frac{1}{2}\cdot\frac{1}{2}\,dy\,dx=\frac{1... |
1,898,810 | <p>How do I integrate $\frac{1}{1-x^2}$ without using trigonometric identities or partial fractions? Thanks!</p>
| Community | -1 | <p>Lookup a table of derivatives and spot</p>
<p>$$(\text{artanh } x)'=\frac1{1-x^2}.$$</p>
|
3,552,695 | <p>The question is to find all real numbers solutions to the system of equations: </p>
<ul>
<li><span class="math-container">$y=\Large\frac{4x^2}{4x^2+1}$</span>,</li>
<li><span class="math-container">$z=\Large\frac{4y^2}{4y^2+1}$</span>,</li>
<li><span class="math-container">$x=\Large\frac{4z^2}{4z^2+1}$</span>, </... | Dietrich Burde | 83,966 | <p>I had no difficulties with the equations. Substituting I obtain two linear equations with solutions
<span class="math-container">$$
(x,y,z)=(0,0,0), \quad (x,y,z)=\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)
$$</span>
and a polynomial of degree <span class="math-container">$6$</span>, namely
<span class="math-co... |
622,883 | <p>I'm finding maximum and minimum of a function $f(x,y,z)=x^2+y^2+z^2$ subject to $g(x,y,z)=x^3+y^3-z^3=3$.</p>
<p>By the method of Lagrange multiplier, $\bigtriangledown f=\lambda \bigtriangledown g$ and $g=3$ give critical points. So I tried to solve these equalities, i.e.</p>
<p>$\quad 2x=3\lambda x^2,\quad 2y=3\... | mathlove | 78,967 | <p>Let us consider a set $\{a,a+1,\cdots,b-1,b\}$ where $1\lt a\lt b\in\mathbb N$.</p>
<p>Case 1 : If $a,b$ are even, the answer is $1+(b-a)/2$.</p>
<p>Case 2 : If $a,b$ are odd, the answer is $(b-a)/2$.</p>
<p>Case 3 : If $a$ is even and $b$ is odd, then the answer is $1+(b-a-1)/2.$</p>
<p>Case 4 : If $a$ is odd a... |
1,248,329 | <p>Let $R$ be an integral domain, and let $r \in R$ be a non-zero non-unit. Prove that $r$ is irreducible if and only if every divisor of $r$ is either a unit or an associate of $r$.</p>
<p>Proof. ($\leftarrow$) Suppose $r$ is reducible then $r$ can be expressed as $r = ab$ where $a$, $b$ are not units. This contradic... | Ronnie Brown | 28,586 | <p>Since the question asks only for hints I refer to the following paper (available <a href="http://pages.bangor.ac.uk/~mas010/pdffiles/bhk-gpds-mv.pdf" rel="nofollow">here</a>) </p>
<p>R. Brown, P/R. Heath, and K.H. Kamps, ``Groupoids and the
Mayer-Vietoris sequence'', <em>J. Pure Appl. Alg</em>. 30 (1983)
109-... |
2,727,237 | <p>$$
\begin{matrix}
1 & 0 & -2 \\
0 & 1 & 1 \\
0 & 0 & 0 \\
\end{matrix}
$$</p>
<p>I am told that the span of vectors equal $R^m$ where $m$ is the rows which has a pivot in it. So when describing the span of the above vectors, is it correct it saying that they don't span $... | Theo Bendit | 248,286 | <p>A basis being orthonormal is dependent on the inner product used. Have a think: why are the coordinate vectors $(1, 0, 0, \ldots, 0)$ and $(0, 1, 0 ,\ldots, 0)$ orthogonal? Traditionally, if they were just considered vectors in $\mathbb{R}^n$, then <strong>under the dot product</strong>, they are orthogonal because ... |
1,299,032 | <p>The following problem is from Taylor's PDE I. I do not get why $D \det(I)B=\frac{d}{d t}\det(I+t B)|_{t=0}$ hold. Since $\det(I)=1$ and $D$ is $n\times n$ matrix, the left side seems to be a matrix, while the right side is clearly a scalar. Could anyone tell me how I should correctly interpret $\det(I)$ in this case... | abel | 9,252 | <p>suppose the eigenvalues of $B$ are $\lambda_1, \lambda_2, \cdots, \lambda_n.$ then the eigenvalues of $I + tB$ are $1 + t\lambda_1, 1 + t\lambda_2, \cdots, 1 + t\lambda_n.$ we will use the fact that the determinant of a matrix is the product of its eigenvalues.</p>
<p>now we have $$det(I + tB) = (1 + t\lambda_1)... |
4,361,626 | <p><span class="math-container">$4$</span> balls are randomly distributed into <span class="math-container">$3$</span> cells (<span class="math-container">$3^4=81$</span> possibilities of equal probability).</p>
<p>What is the probability that there is a cell that contains exactly <span class="math-container">$2$</span... | Wuestenfux | 417,848 | <p>Here <span class="math-container">$J=\langle p(x)\rangle$</span> is the ideal generated by the underlying irreducible polynomial.</p>
<p>The result basically says that <span class="math-container">$\bar x = x+J$</span> is a zero of <span class="math-container">$p(x)$</span> which lies in the extension field <span cl... |
510,488 | <p>Using the fact that $x^2 + 2xy + y^2 = (x + y)^2 \ge 0$, show that the assumption $x^2 + xy + y^2 < 0$ leads to a contradiction...
So do I start off with...</p>
<p>"Assume that $x^2 + xy + y^2 <0$, then blah blah blah"?</p>
<p>It seems true...because then I go $(x^2 + 2xy + y^2) - (x^2 + xy + y^2) \ge 0$.
It... | Ted Shifrin | 71,348 | <p>Assuming everything in the picture is a real number, you've arrived at $xy\ge 0$, and so $x^2+xy+y^2\ge 0$. This contradicts our assumption.</p>
|
3,861,820 | <p>Given the following <span class="math-container">$\frac{1}{2^2}+
\frac{1}{2^3}+\frac{1}{2^4}+...+\frac{1}{3^2}+\frac{1}{3^3}+...+$</span></p>
<p>Can this be symbolized as:
<span class="math-container">$$\sum_{n=2}^{\infty}2^{-n}+(n+1)^{-n}$$</span></p>
<p>and if so, are the following values for <span class="math-con... | marwalix | 441 | <p>Let me follow up on my comment.</p>
<p>We start with the inner sum, a geometric series</p>
<p><span class="math-container">$$\sum_{n=2}^\infty{1\over m^n}={1\over m^2}{1\over 1-{1\over m}}={1\over m^2-m}$$</span></p>
<p>Now we décompose the rational fraction</p>
<p><span class="math-container">$${1\over m(m-1)}={1\o... |
3,815,891 | <p>Let <span class="math-container">$x_{n}$</span> be the positive real root of equation
<span class="math-container">$$(x^{n-1}+2^n)^{n+1} = (x^{n} + 2^{n+1})^{n}$$</span></p>
<p>How to prove that <span class="math-container">$x_{n} > x_{n + 1}$</span>?</p>
<p>Actually, <span class="math-container">$x_{n} > 2$</... | Claude Leibovici | 82,404 | <p><em>This is not a proof.</em></p>
<p>Consider that you look for the non-trivial positive zero of function
<span class="math-container">$$f_n(x)=(x^{n-1}+2^n)^{n+1} - (x^{n} + 2^{n+1})^{n}$$</span> If you compute the series of <span class="math-container">$\frac{f_n(x)}{x^n}$</span> to <span class="math-container">$O... |
3,815,891 | <p>Let <span class="math-container">$x_{n}$</span> be the positive real root of equation
<span class="math-container">$$(x^{n-1}+2^n)^{n+1} = (x^{n} + 2^{n+1})^{n}$$</span></p>
<p>How to prove that <span class="math-container">$x_{n} > x_{n + 1}$</span>?</p>
<p>Actually, <span class="math-container">$x_{n} > 2$</... | yisishoujo | 339,918 | <p>replace <span class="math-container">$x$</span> by <span class="math-container">$2x$</span>, we have
<span class="math-container">\begin{equation}
1+\frac{x^n}{2} = \Big(1+\frac{x^{n-1}}{2}\Big)^{1+\frac{1}{n}} > 1 + \big(1+\frac{1}{n}\big)\frac{x^{n-1}}{2}
\end{equation}</span>
hence <span class="math-container"... |
1,518,258 | <p>I haven't been able to come up with a counterexample so far. </p>
| Dylan | 146,990 | <p>In general, this does not hold for any $n$ except for $n \leq 2$.</p>
<p>You can check that the result holds for $n \leq 2$ by just considering all of the possible values of $n$, $a$ and $b$. (There aren't that many)</p>
<p>If $n \geq 3$, then by Bertrand's Postulate, there is a prime $p$ strictly between $\frac{n... |
2,928,806 | <p><span class="math-container">$m(t)=\frac{t^2}{1-t^2},t<1$</span></p>
<p>The suggested answer provided by our teacher states that m(0)=0, so there's no real value with this function as mgf; hence mgf doesn't exist.</p>
<p>Intuitively I understand that <span class="math-container">$e^{xt}>0$</span> should alwa... | drhab | 75,923 | <p><span class="math-container">$$m_X(t)=\mathbb Ee^{tX}\text{ so that }m_X(0)=\mathbb Ee^{0\cdot X}=\mathbb E1=1$$</span></p>
<p>showing that for <em>every</em> moment generating function we have <span class="math-container">$m(0)=1\neq0$</span>.</p>
|
2,168,554 | <p>Assuming you are playing roulette.</p>
<p>The probabilities to win or to lose are:</p>
<p>\begin{align}
P(X=\mathrm{win})&=\frac{18}{37}\\
P(X=\mathrm{lose})&=\frac{19}{37}
\end{align}</p>
<p>Initially 1$ is used. Everytime you lose, you double up the stake. If you win once, you stop playing. If required ... | spaceisdarkgreen | 397,125 | <p>It is absolutely the case that you will win eventually with probability one. However this means your expected winnings is one, since when you inevitably win and walk away you win a single bet. I don't understand your calculation of expected winnings but rest assured it's wrong.</p>
<p>The usual caveat to mention wi... |
33,743 | <p>I have a lot of sum questions right now ... could someone give me the convergence of, and/or formula for, $\sum_{n=2}^{\infty} \frac{1}{n^k}$ when $k$ is a fixed integer greater than or equal to 2? Thanks!!</p>
<p>P.S. If there's a good way to google or look up answers to these kinds of simple questions ... I'd lo... | Arturo Magidin | 742 | <p>The fact that these series converge follows from the integral test; their values are somewhat more complicated, as indicated by the comments: they correspond to evaluating Riemann's zeta function at $k$, and this is highly nontrivial, even for positive integral $k$ (called the <a href="http://en.wikipedia.org/wiki/Z... |
854,671 | <p>So I'm a bit confused with calculating a double integral when a circle isn't centered on $(0,0)$. </p>
<p>For example: Calculating $\iint(x+4y)\,dx\,dy$ of the area $D: x^2-6x+y^2-4y\le12$.
So I kind of understand how to center the circle and solve this with polar coordinates. Since the circle equation is $(x-3)^2... | Martin Sleziak | 8,297 | <p>Let us ignore leap years for a moment.</p>
<p>And let us assume that some date (e.g., 13th) falls on Monday in January. </p>
<p>What happens in February? January has $31=4\cdot 7+3$ days, so the days move by 3 to Thursday.</p>
<p>So if we denote the days in week by numbers $\{0,1,2,3,4,5,6\}$, we only have to con... |
611,198 | <p>A corollary to the Intermediate Value Theorem is that if $f(x)$ is a continuous real-valued function on an interval $I$, then the set $f(I)$ is also an interval or a single point.</p>
<p>Is the converse true? Suppose $f(x)$ is defined on an interval $I$ and that $f(I)$ is an interval. Is $f(x)$ continuous on $I$? <... | Andrés E. Caicedo | 462 | <p>Here is a bad counterexample: <a href="http://en.wikipedia.org/wiki/Conway_base_13_function" rel="nofollow">Conway's base-$13$ function</a> takes <em>all values</em> on any (non-degenerate) interval. (This is stronger than just asking that $f(I)$ is an interval: You get that $f(J)$ is an interval for all intervals $... |
2,990,642 | <p><span class="math-container">$\lim_{n\to \infty}(0.9999+\frac{1}{n})^n$</span></p>
<p>Using Binomial theorem:</p>
<p><span class="math-container">$(0.9999+\frac{1}{n})^n={n \choose 0}*0.9999^n+{n \choose 1}*0.9999^{n-1}*\frac{1}{n}+{n \choose 2}*0.9999^{n-2}*(\frac{1}{n})^2+...+{n \choose n-1}*0.9999*(\frac{1}{n})... | Travis Willse | 155,629 | <p><strong>Hint</strong> Instead of using the binomial expansion, observe that for <span class="math-container">$n > 10^5$</span> the quantity in parentheses is at most <span class="math-container">$1 - \frac{1}{10^5 (10^5 + 1)}$</span>. Now apply the Squeeze Theorem.</p>
|
1,651,922 | <p>I'm solving this equation:</p>
<p>$$\sin(3x) = 0$$</p>
<p>The angle is equal to 0, therefore:</p>
<p>$$3x=0+2k\pi \space\vee\space3x= (\pi-0)+2k\pi$$ </p>
<p>$$x = \frac {2}{3}k\pi \space \vee \space x = \frac {\pi + 2k\pi}{3}$$</p>
<p>Though, the answer is </p>
<p>$$x = k\frac {\pi}{3}$$</p>
<p>It looks like... | mathmandan | 198,422 | <p>Your solutions are:
$$
x = \frac{\pi}{3}(2k),\phantom{NNNNNNNN}
x = \frac{\pi}{3}(2k + 1).
$$
So, you've shown that $x$ is either $\frac{\pi}{3}$ times an even integer, or else $\frac{\pi}{3}$ times an odd integer. In other words, $x$ is $\frac{\pi}{3}$ times an integer.</p>
|
932,430 | <p>I am studying about how a real number is defined by its properties. The three type of properties that make the real numbers what they are.</p>
<ol>
<li><p>Algebraic properties i.e, the axioms of addition, subtraction multiplication and division.</p></li>
<li><p>Order properties i.e., the inequality properties</p></... | MJD | 25,554 | <p>The crucial problem with rational numbers is that they are <em>incomplete</em>. It was discovered about 2300 years ago that there is no rational number whose square is 2. But why should there be a square root of 2? One reason is that it's easy to find a sequence of rational numbers that appears to be getting close... |
29,443 | <p>I'm looking at a <a href="http://en.wikipedia.org/wiki/Partition_function_%28statistical_mechanics%29#Relation_to_thermodynamic_variables" rel="nofollow noreferrer">specific</a> derivation on wikipedia relevant to statistical mechanics and I don't understand a step.</p>
<p>$$ Z = \sum_s{e^{-\beta E_s}} $$</p>
<p>$... | Fabian | 7,266 | <p>The answer is valid for the partition sum $Z$ (which is closely related to the <a href="http://en.wikipedia.org/wiki/Moment-generating_function" rel="nofollow">moment generating function</a>). The reason is the special structure of the partition sum $$Z = \sum_s e^{-\beta E_s}.$$
The system is characterized
with <em... |
851,959 | <p>In Halmos's Naive Set Theory about well-ordering set, it states that if a collecton <span class="math-container">$\mathbb{C}$</span> of well-ordered set is a chain w.r.t continuation, then the union of these sets is a well-ordered set. However, I cannot see why it is well-ordered. That is, I cannot see why each of i... | Asaf Karagila | 622 | <p>Note that in general, the union of a $\subseteq$-chain of well-ordered sets is not necessarily a well-ordered set. For example, write $\Bbb Q$ as $\{q_n\mid n\in\Bbb N\}$, then it is the increasing union of $Q_n=\{q_i\mid i<n\}$, with the linear order induced by $\Bbb Q$. Then $\{Q_n\mid n\in\Bbb N\}$ is a chain ... |
159,510 | <p>I have data points of the form <code>{x,y,z}</code>:</p>
<pre><code>s = {{0, 1, 2}, {3, 4, 5}, {6, 7, 8}}
</code></pre>
<p>And I need to remove <em>entire data points</em> based on a condition. For example, let's say that I want to remove any data points where <code>z > 6</code>. The result should be this:</p>
... | Nasser | 70 | <p>This works, but may be there is easier way</p>
<pre><code>SetDirectory[NotebookDirectory[]]
data = Flatten@Import["input.txt","CSV"];
data2 = If[StringQ[#],ToExpression[StringDelete[#,{"[","]"}]],#]&/@data;
data2 = ArrayReshape[data2,{Length@data2/3,3}]
</code></pre>
<p>now</p>
<pre><code>MatrixForm[data2]
<... |
2,467,095 | <p>I'm considering the original coupon collector problem with a small modification. For the sake of completeness I shall state the original problem again first, where <strong>my question is at the end</strong>. </p>
<p>say there is a coupon inside every packet of wafers, for the moment let's assume there are only tw... | Deep | 386,936 | <p>I am not sure I correctly understand your question. Are you looking for sequences: $C_2C_1,C_2C_2C_1,C_2C_2C_2C_1,...$. If $p$ is the probability of obtaining coupon $C_2$ then probability of a sequence of the sort mentioned which contains $n~C_2$ coupons and $1~C_1$ coupon is $P(n)=p^n(1-p)$. Expected number of waf... |
1,129,052 | <p>How many 3-digits numbers possess the following property: </p>
<blockquote>
<p>After subtracting $297$ from such a number, we get a $3$-digit number consisting of the same digits in the reverse order.</p>
</blockquote>
| Mufasa | 49,003 | <p>You have:$$\text{ }abc$$$$-297$$$$cba$$The 10's column has $b-9=b$. This implies there first must have been a borrow from the 10's column to the units column, and then a borrow from the 100's column into the 10's column. This information tells us that:$$a=c+3$$and $b$ can be any digit from $0$ to $9$.</p>
<p>Now yo... |
1,129,052 | <p>How many 3-digits numbers possess the following property: </p>
<blockquote>
<p>After subtracting $297$ from such a number, we get a $3$-digit number consisting of the same digits in the reverse order.</p>
</blockquote>
| Arian | 172,588 | <p>Let one of these three digit number be $\overline{abc}$ where $a,b$ and $c$ are the digits. Then the problem can be expressed as
$$\overline{abc}-297=\overline{cba}$$
or writing it in powers of $10$ as follows
$$100a+10b+c-297=100c+10b+a\Rightarrow 99(a-c)=297\Rightarrow a-c=3$$
So we have any triple $(a,b,c)=(c+3,... |
3,097,816 | <p>Show that the product of four consecutive odd integers is 16 less than a square.</p>
<p>For the first part I first did
<span class="math-container">$n=p(p+2)(p+4)(p+6)
=(p^2+6p)(p^2+6p+8).$</span>
I know that you are supposed to rearrange this to give an equation in the form <span class="math-container">$(ap^2+bp+c... | Wolfgang Kais | 640,973 | <p>The middle number is <span class="math-container">$p+3 =: m_1$</span>.</p>
<p><span class="math-container">$$n = p(p+2)(p+4)(p+6) = p(p+6)(p+2)(p+4)
= (m_1-3)(m_1+3)(m_1-1)(m_1+1)
= (m_1^2-3^2)(m_1^2-1^2)
= (m_1^2-9)(m_1^2-1)$$</span></p>
<p>Now, with a new "middle number" <span class="math-container">$m_2 :... |
1,536,904 | <blockquote>
<p>A father send his $3$ sons to sell watermelons.The first son took $10$
watermelons,the second $20$ and the third $30$ watermelons.The father
gave order to sell all in the same price and collect and the same
number of money.How is this possible?</p>
</blockquote>
<p>Any ideas for this puzzle?</p... | Empy2 | 81,790 | <p>Son 3 sells 30 watermelons, then buys 20 watermelons from Son 2.
Son 2 buys 10 watermelons from Son 1.</p>
|
1,536,904 | <blockquote>
<p>A father send his $3$ sons to sell watermelons.The first son took $10$
watermelons,the second $20$ and the third $30$ watermelons.The father
gave order to sell all in the same price and collect and the same
number of money.How is this possible?</p>
</blockquote>
<p>Any ideas for this puzzle?</p... | mvw | 86,776 | <p>Son 3 sells 20 watermelons then gives 10 watermelons to son 1. Son 2 and son 1 then sell 20 watermelons each.</p>
|
1,874,159 | <blockquote>
<p>Given real numbers $a_0, a_1, ..., a_n$ such that $\dfrac {a_0}{1} + \dfrac {a_1}{2} + \cdots + \dfrac {a_n}{n+1}=0,$ prove that $a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n=0$ has at least one real solution.</p>
</blockquote>
<p>My solution:</p>
<hr>
<p>Let $$f(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a... | Hagen von Eitzen | 39,174 | <p>Why not write it the other way round?</p>
<p>The polynomial function
$$F(x)=\sum_{k=0}^n\frac{a_k}{k+1}x^{k+1} $$
is a differentiable function $\Bbb R\to\Bbb R$ with derivative $$F'(x)=\sum_{k=0}^na_kx^k.$$
We are given that $F(1)=0$, and clearly $F(0)=0$. Hence by <a href="https://en.wikipedia.org/wiki/Rolle%27s_... |
2,465,705 | <p>In differential geometry, you are often asked to reparameterize a curve using arc-length. I understand the process of how to do this, but I don't understand what we are reparameterizing from.</p>
<p>What is the curve originally parameterized by (before we REparameterize it by arc-length)?</p>
| user7530 | 7,530 | <p>It's completely arbitrary.</p>
<p>For example the "standard" parameterization of the unit circle is $\gamma(t) = [\cos(t),\sin(t)]$. But you could also sweep out a circle using the curves</p>
<ul>
<li>$\gamma(t) = [\cos(t^2), \sin(t^2)]$, which "speeds up" as you draw it;</li>
<li>$\gamma(t) = [\cos(\sqrt{t}), \si... |
402,970 | <p>I had an exercise in my text book:</p>
<blockquote>
<p>Find $\frac{\mathrm d}{\mathrm dx}(6)$ by first principles.</p>
</blockquote>
<p>The answer that they gave was as follows:</p>
<p>$$\lim_{h\to 0} \frac{6-6}{h} = 0$$</p>
<p>However surely that answer would be undefined if we let $h$ tend towards $0$, as fo... | Michael Hardy | 11,667 | <p>In all cases in which the $\lim\limits_{h\to0}\dfrac{f(x+h)-f(x)}{h}$ exists, <b>both</b> the numerator and the denominator approach $0$. And even if the limit does not exist the numerator and denominator both become $0$ when $h=0$.</p>
<p>But $\lim\limits_{h\to0}$ of <em>anything</em> depends only on the behavior... |
1,859,652 | <p>I've just been studying cyclic quads in geometry at school and I'm thinking see seems pretty interesting, but where would I actually find these in the real world? They seem pretty useless to me...</p>
| Michael Hardy | 11,667 | <p>Ptolemy's theorem says that if $a,b,c,d$ are the lengths of the sides of a cyclic quadrilateral, with $a$ opposite $c$ and $b$ opposite $d$, and $e,f$ are the diagonals, then $ac+ bd = ef$. In the second century AD, Ptolemy used that to prove identities that today we would express as
\begin{align}
\sin(a+b) & =... |
4,502,150 | <p>There is a "direct" way to prove this equality.</p>
<p><span class="math-container">$\displaystyle \binom{1/2}{n}=\frac{2(-1)^{n-1}}{n4^n}\binom{2n-2}{n-1}$</span></p>
<p>I am trying to skip the induction. Maybe there is a rule or formula that will help me.</p>
<p>Thank you</p>
| runway44 | 681,431 | <p>Just move the numbers around in the formula:</p>
<p><span class="math-container">$$\begin{array}{cl}
\displaystyle\binom{1/2}{n} & \displaystyle = \frac{(\frac{1}{2})(\frac{1}{2}-1)\cdots(\frac{1}{2}-(n-1))}{n!} \\[5pt]
& \displaystyle = \frac{(1)(1-2)(1-4)\cdots(1-2(n-1))}{2^nn!} \\[5pt]
& \displaystyl... |
1,290,316 | <p>Let $F_i$ be a family of closed sets, then we know that $\bigcup_{i=1}^nF_i$ is closed.</p>
<p>Proving that statement is equivalent to proving:</p>
<blockquote>
<p>If $p$ is a limit point of $\bigcup_{i=1}^nF_i$ then $p\in\bigcup_{i=1}^nF_i$</p>
</blockquote>
<p>It is easy to prove the contrapositive: if $p\not... | KyleW | 252,356 | <p>I know from your other questions you like to see rigorous answers; so I thought I'd do the same on this one too. I did this a slightly longer way since I do not know which definition of closed you are using (so I used a standard definition of open)</p>
<p>def. $U$ is open $\Leftrightarrow \forall x \in U\; \exists ... |
388,950 | <p>I've been trying to get my head around this for days. I understand what is going on with the calculation of a linear recurrence and I also understand how the characteristic is obtained.</p>
<p>What is confusion me is the general solution.</p>
<p>The general solution to the recurrence $at(n) + bt(n-1) ct(n-2) = 0$<... | amWhy | 9,003 | <p>Your steps in using the two point formula to find a relation between $F$ and $C$ is correct, and done well.</p>
<p>The resulting relation (equation) is your solution: $$F = \frac{9}{5}C+32$$</p>
<p>Note that given this equation, you can also represent $C$ as a function of $F$:</p>
<p>$$F = \frac{9}{5}C+32 \iff F ... |
2,867,718 | <p>How do I integrate $\sin\theta + \sin\theta \tan^2\theta$ ?</p>
<p>First thing,
I have been studying maths for business for approximately 3 months now. Since then, I studied algebra and then I started studying calculus. Yet, my friend stopped me there, and asked me to study Fourier series as we'll need it for our ... | Bernard | 202,857 | <p><em>Bioche's rules</em> say we can make the substitution $u=\cos\theta$, $\mathrm d u=-\sin\theta\,\mathrm d\theta$. Indeed
$$ \int(1+\tan^2\theta)\sin\theta\,\mathrm d\theta=\int \frac1{\cos^2\theta}\,\sin\theta\,\mathrm d\theta=\int -\frac{\mathrm du}{u^2}=\frac1u=\frac1{\cos\theta}. $$</p>
|
1,485,463 | <p>At a school function, the ratio of teachers to students is $5:18$. The ratio of female students to male students is $7:2$. If the ratio of the female teachers to female students is $1:7$, find the ratio of the male teachers to male students. </p>
| tony0858 | 281,122 | <p>Let A = # of Female Teachers</p>
<p>Let B = # of Male Teachers</p>
<p>Let X = # of Female Students</p>
<p>Let Y = # of Male Students</p>
<p>Then the statement "the ratio of Teachers to students is 5:18" can be represented by the following equation, EQ1:</p>
<p>EQ1: (A+B)/(X+Y) = 5/18 _________(A+B = number of t... |
141,101 | <p>I have given a high dimensional input $x \in \mathbb{R}^m$ where $m$ is a big number. Linear regression can be applied, but in generel it is expected, that a lot of these dimensions are actually irrelevant.</p>
<p>I ought to find a method to model the function $y = f(x)$ and at the same time uncover which dimension... | Xiaorui Zhu | 372,354 | <p>The subgradient should be like this: </p>
<p>$$\frac{\partial}{\partial \beta} L^{\text lasso}(\beta) = - \sum_{i = 1}^n (2\phi(x_i)(y_i - \phi(x_i)^T \beta) + \lambda \;sign(\beta))$$</p>
|
1,412,091 | <p>The typewriter sequence is an example of a sequence which converges to zero in measure but does not converge to zero a.e.</p>
<p>Could someone explain why it does not converge to zero a.e.?</p>
<blockquote>
<p><span class="math-container">$f_n(x) = \mathbb 1_{\left[\frac{n-2^k}{2^k}, \frac{n-2^k+1}{2^k}\right]} \tex... | grand_chat | 215,011 | <p>Draw a picture of the generic function $f_n$ in the typewriter sequence. It's a rectangle of height 1 over an interval of width $1/2^k$, with value zero elsewhere. As the sequence progresses, the rectangles slide across the unit interval, the way a typewriter moves across the page. At each 'carriage return' of the t... |
3,839,244 | <p>Is <span class="math-container">$\{3\}$</span> a subset of <span class="math-container">$\{\{1\},\{1,2\},\{1,2,3\}\}$</span>?</p>
<p>If the set contained <span class="math-container">$\{3\}$</span> plain and simply I would know but does the element <span class="math-container">$\{1,2,3\}$</span> include <span class=... | the_candyman | 51,370 | <p>As you can see, there are plenty of good answers here.</p>
<p>I think that the most important thing to understand is:</p>
<p><span class="math-container">$$3 \neq \{ 3 \}.$$</span></p>
<p>These are two different objects.</p>
<p>Moreover the followings hold:</p>
<p><span class="math-container">$$ 3 \in \{3\}$$</span>... |
159,438 | <p>Can be easily proved that the following series onverges/diverges?</p>
<p>$$\sum_{k=1}^{\infty} \frac{\tan(k)}{k}$$</p>
<p>I'd really appreciate your support on this problem. I'm looking for some easy proof here. Thanks.</p>
| leonbloy | 312 | <p>A proof that the sequence $\frac{\tan(n)}{n}$ does not have a limit for $n\to \infty$ is given in <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.112.5431&rep=rep1&type=pdf" rel="nofollow">this article</a> <sub>(Sequential tangents, Sam Coskey)</sub>. This, of course, implies that the serie... |
765,738 | <p>I am trying to prove the following inequality:</p>
<p>$$(\sqrt{a} - \sqrt{b})^2 \leq \frac{1}{4}(a-b)(\ln(a)-\ln(b))$$</p>
<p>for all $a>0, b>0$.</p>
<p>Does anyone know how to prove it?</p>
<p>Thanks a lot in advance!</p>
| user136725 | 136,725 | <p>Wlog a>b, then by Cauchy-Swartz $$\left(\int_b^a \ 1\cdot\frac{1}{\sqrt{x}}\ dx\right)^2\leq\int_b^a \ 1\ dx \cdot\int^a_b \frac{1}{x}\ dx$$</p>
|
7,072 | <p>I want to replace $x^{i+1}$ with $z_i$.</p>
<p>EDIT:
I have some latex equations, and wish to import them to MMa.</p>
<p>How to replace x_i (NOT $x_i$) with $x[i]$</p>
<p>But it seems the underscore "_" has a special meaning (function), can we do such substitution in MMa?</p>
<p><strong>EDIT2:</strong></p>
<p>F... | Jens | 245 | <p>To put in the offset by <code>1</code> that was desired in the question, one could do this:</p>
<pre><code>x^(i+1) /. x^exponent_ :> Subscript[z, exponent - 1]
</code></pre>
<p>The rule works independently of whether <code>i</code> is an integer or not. I didn't take care of the special case <code>i=0</code>, b... |
870,174 | <p>If I have $6$ children and $4$ bedrooms, how many ways can I arrange the children if I want a maximum of $2$ kids per room?</p>
<p>The problem is that there are two empty slots, and these empty slots are not unique.</p>
<p>So, I assumed there are $8$ objects, $6$ kids and $2$ empties.</p>
<p>$$C_2^8 \cdot C_2^6 \... | Juanito | 153,015 | <p>Only possible distributions are (2,2,2,0) and (2,2,1,1).
So, total ways=$^4C_1*^6C_2*^4C_2+^4C_2*^6C_2*^4C_2*2=1440$
$\\$
If you do not want to leave any room empty as an added requirement, the only pattern possible is (2,2,1,1), so the answer is 1080. hence, OP is right!</p>
|
599,656 | <p>If the objects of a category are algebraic structures in their own right, this often places additional structure on the homsets. Is there somewhere I can learn more about this general idea?</p>
<hr>
<p><strong>Example 1.</strong> Let $\cal M$ denote the category of magmas and <strong>functions</strong>, not necess... | zrbecker | 19,536 | <p>The quadratic formula is $$x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}.$$
Thus in your problem, $$x = \frac{-104\pm\sqrt{104^2 - 4(-896)}}{2}
=\frac{-104 \pm\sqrt{14400}}{2} = \frac{-104 \pm 120}{2}.$$
This gives $x = 8$ or $-112$.</p>
<p>It seems the problem in your computation was you forgot the negative on $896$ when you... |
1,035,091 | <p>I try to get the limit of the following expression (should be valid for $a\geq 0$):</p>
<p>$\lim_{n\to\infty}\sum_{k=0}^n\frac{a}{ak+n}$</p>
<p>Unfortunately nothing worked. I tried to rewrite the expressiona and to use l'hospitals rule, but it didn't work.</p>
<p>Thank you for any help:)</p>
<p>regards</p>
<p>... | Community | -1 | <p>Rewrite the given sum on the Riemann sum's form:</p>
<p>$$\frac1n\sum_{k=0}^n\frac{a}{a\frac kn+1}\xrightarrow{n\to\infty}\int_0^1\frac{a}{ax+1}dx=\ln(ax+1)\Bigg|_0^1=\ln(a+1)$$</p>
|
548,563 | <p>Calculate $$\sum \limits_{k=0}^{\infty}\frac{1}{{2k \choose k}}$$</p>
<p>I use software to complete the series is $\frac{2}{27} \left(18+\sqrt{3} \pi \right)$</p>
<p>I have no idea about it. :|</p>
| Pedro | 23,350 | <p><a href="http://www.emis.de/journals/INTEGERS/papers/g27/g27.pdf" rel="noreferrer">This</a> paper is very relevant to your question. In particular, $\bf Theorems \;\;3.4-5$ and $\bf Theorem \;\;3.7$</p>
|
1,865,868 | <p>the problem goes like that "in urn $A$ white balls, $B$ black balls. we take out without returning 5 balls. (we assume $A,B\gt4$) what would be the probability that at the 5th ball removal, there was a white ball while we know that at the 3rd was a black ball".</p>
<p>What I did is I build a conditional probability... | Felix Marin | 85,343 | <p>$\newcommand{\angles}[1]{\left\langle\,{#1}\,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\half}{{1 ... |
191,307 | <p>How can I calculate the perimeter of an ellipse? What is the general method of finding out the perimeter of any closed curve?</p>
| Aang | 33,989 | <p>For general closed curve(preferably loop), perimeter=$\int_0^{2\pi}rd\theta$ where (r,$\theta$) represents polar coordinates.</p>
<p>In ellipse, $r=\sqrt {a^2\cos^2\theta+b^2\sin^2\theta}$</p>
<p>So, perimeter of ellipse = $\int_0^{2\pi}\sqrt {a^2\cos^2\theta+b^2\sin^2\theta}d\theta$ </p>
<p>I don't know if close... |
191,307 | <p>How can I calculate the perimeter of an ellipse? What is the general method of finding out the perimeter of any closed curve?</p>
| NinjaDarth | 181,639 | <p>If the semi-major axis has length <span class="math-container">$a$</span> and the eccentricity is <span class="math-container">$e$</span>, then the perimeter is
<span class="math-container">$$
2πa \left(\frac{1}{1} \left(\frac{(-1)!!}{0!!} e^0\right)^2 + \frac{1}{-1} \left(\frac{1!!}{2!!} e^1\right)^2 + \frac{1}{-3}... |
1,993,034 | <p>if I have say $x_4$ = free, what value goes in the $x_4$ position of the parametric form, is it $1$ or $0$ or can it be any value since it's free?</p>
| Matthew Leingang | 2,785 | <p>"Free" is not a value and "x4=free" is an abuse of the equal sign. You mean $x_4$ <em>is</em> free. <em>Is</em> can denote equality ($=$), but here it denotes membership ($\in$). I had originally put this as a postscript, but your comments on my first drafts indicate this might be a stumbling block for you.</p>
... |
3,450,692 | <p>How to solve this equation?</p>
<p><span class="math-container">$$
\frac{dy}{dx}=\frac{x^{2}+y^{2}}{2xy}
$$</span></p>
| IrbidMath | 255,977 | <p>Try the following let <span class="math-container">$u = \frac{y}{x} $</span> </p>
|
3,450,692 | <p>How to solve this equation?</p>
<p><span class="math-container">$$
\frac{dy}{dx}=\frac{x^{2}+y^{2}}{2xy}
$$</span></p>
| Z Ahmed | 671,540 | <p><span class="math-container">$$\frac{dy}{dx}=\frac{x^2+y^2}{2xy} ~~~~(1)$$</span>
Let <span class="math-container">$y=vx \implies \frac{dy}{dx}=v+x \frac{dv}{dx}.$</span>
Then
<span class="math-container">$$v+xv'=\frac{1+v^2}{2v} \implies \int \frac{2v}{1-v^2}= \frac{dx}{x}
\implies -\ln (1-v^2)= \ln Cx \implies C... |
102,402 | <p>I wanted to make a test bank of graphs of linear equations for my algebra classes. I want the $y$-intercept of each graph to be an integer no less than $-10$ and no greater than $10$. Generally, you want these graphs to be small, so i've decided on a $20 \times 20$ grid (10 units from the origin). Additionally, i ... | André Nicolas | 6,312 | <p>As a start, we compute exactly the number of such lines with $y$-intercept $0$. </p>
<p>There are $2$ (the axes) plus twice the number with positive slope. How positive slopes are possible? There is the special case slope $1$. Then there are the proper (reduced) fractions such as $7/9$ and $1/3$, <em>plus</em> th... |
1,910,085 | <p>For all integers $n \ge 0$, prove that the value $4^n + 1$ is not divisible by 3.</p>
<p>I need to use Proof by Induction to solve this problem. The base case is obviously 0, so I solved $4^0 + 1 = 2$. 2 is not divisible by 3.</p>
<p>I just need help proving the inductive step. I was trying to use proof by contrad... | Soham | 242,402 | <p>$$4\equiv1\pmod3$$</p>
<p>$$\implies4^n\equiv1^n\pmod3$$</p>
<p>Also,$1\equiv1\pmod3$</p>
<p>Adding,$$4^n+1\equiv2\pmod3$$</p>
|
245,623 | <p><em>For the following vectors $v_1 = (3,2,0)$ and $v_2 = (3,2,1)$, find a third vector $v_3 = (x,y,z)$ which together build a base for $\mathbb{R}^3$.</em></p>
<p>My thoughts:</p>
<p>So the following must hold:</p>
<p>$$\left(\begin{matrix}
3 & 3 & x \\
2 & 2 & y \\
0 & 1 & z
\end{matrix}\... | Lubin | 17,760 | <p>But we’re talking about vector spaces over $\mathbb R$ here. If the dimension of the vector space is $n$, then any set of fewer than $n$ vectors spans a lower-dimensional subspace, whose complement is <em>open and dense</em> in the whole. You should think of this as telling you that one more vector has almost no cha... |
1,791,990 | <p>I have to prove that integral</p>
<p>$I = \int_{0}^{+\infty}\sin(t^2)dt$ is convergent. Could you tell me if it's ok?</p>
<p>Let $t^2=u$ then $dt=\frac{du}{2\sqrt{u}}$</p>
<p>Now $$I = \int_{0}^{+\infty}\frac{\sin(u)du}{2\sqrt{u}}$$</p>
<p>Which is equal to $$\int_{0}^{1}\frac{\sin(u)du}{2\sqrt{u}} + \int_{1}^{+... | Aritra Das | 229,480 | <p>Putting $\sqrt{u}=t$ and $\dfrac{du}{2\sqrt{u}}=dt$, </p>
<p>$$I = \int_{0}^{+\infty}\frac{\sin(u)du}{2\sqrt{u}}\\=\int_0^{+\infty} \sin(t^2) dt\\=\int_0^{+\infty}\Im(e^{it^2})dt\\=\Im\left(\int_0^{+\infty}e^{it^2}dt\right)$$</p>
<p>Putting $it^2=-z$ and $t=\sqrt{-\frac1iz}=\sqrt{i}\sqrt{z}$, $$dt = \sqrt{i}\frac{... |
4,465,150 | <p>Let <span class="math-container">$A_1,A_2,…,A_n$</span> be events in a probability space <span class="math-container">$(\Omega,\Sigma,P)$</span>.</p>
<p>If <span class="math-container">$A_1,A_2,…,A_n$</span> are independent then <span class="math-container">$A_1^c,A_2^c,…,A_n^c$</span> are also independent, (where ... | mark leeds | 49,818 | <p>Hi Inquirer: My apologies for the delay. I've had a lot going on so I just got back to this today.</p>
<p>I figured I'd put it in an answer since you get more space that way.</p>
<p>So, you agree that you proved the statement for <span class="math-container">$i = 2$</span> case, right. You showed that if <span class... |
185,020 | <p>In literature, there are many proofs of the well-known result $$\zeta(2) = \frac{\pi^2}{6}.$$ </p>
<p>However, as far as I know, they do not offer an <em>intuitive</em> explanation of why this result <em>should</em> be true. </p>
<p>So my question is: </p>
<blockquote>
<p>What is the key intuition -- that is, t... | Alex R. | 934 | <p>The word intuitive by definition means that something "feels correct", and is very myopic. The fact that there are so many different proofs is in itself a gift because you can pick your favorite and intuit all you want. </p>
<p>I like Euler's proof because it shows how it's related to the taylor series of $\sin(x)$... |
185,020 | <p>In literature, there are many proofs of the well-known result $$\zeta(2) = \frac{\pi^2}{6}.$$ </p>
<p>However, as far as I know, they do not offer an <em>intuitive</em> explanation of why this result <em>should</em> be true. </p>
<p>So my question is: </p>
<blockquote>
<p>What is the key intuition -- that is, t... | user60745 | 60,745 | <p>Here is a short note by Robert E. Greene (UCLA):</p>
<p><a href="http://www.math.ucla.edu/~greene/How%20Geometry.pdf" rel="nofollow">How Geometry implies $\sum \frac{1}{k^2} = \pi^2/6$</a></p>
|
577,163 | <p>Let $A=(a_{ij})_{n\times n}$ such
$a_{ij}>0$ and $\det(A)>0$.</p>
<p>Defining the matrix $B:=(a_{ij}^{\frac{1}{n}})$, show that $\det(B)>0?$.</p>
<p>This problem is from my friend, and I have considered sometimes, but I can't. Thank you </p>
| Marc van Leeuwen | 18,880 | <p>This is wrong.
$$
\begin{pmatrix}27&8&1\\1&1&\epsilon\\8&\epsilon&1\end{pmatrix}
$$
where $\epsilon>0$ is very small.</p>
|
577,163 | <p>Let $A=(a_{ij})_{n\times n}$ such
$a_{ij}>0$ and $\det(A)>0$.</p>
<p>Defining the matrix $B:=(a_{ij}^{\frac{1}{n}})$, show that $\det(B)>0?$.</p>
<p>This problem is from my friend, and I have considered sometimes, but I can't. Thank you </p>
| Hu Zhengtang | 53,845 | <p>It is <strong>false</strong> when $n\ge 3$. Counter-example: for $x>0$ small, let</p>
<p>$$A(x)=\begin{pmatrix}8 & (1+3x)^3 & 1\\ 1 & (1+x)^3 & 1 \\ x^3 & 1& 1\end{pmatrix}.$$
Then</p>
<p>$$B(x)=\begin{pmatrix}2 & 1+3x & 1\\ 1 & 1+x & 1 \\ x & 1& 1\end{pmatrix}.$$... |
3,614,875 | <p><strong>Prop:</strong> Every sequence has a monotone subsequence.</p>
<p><strong>Pf:</strong> Suppose <span class="math-container">$\{a_n\}_{n\in \mathbb{N}}$</span> is a sequence. Choose <span class="math-container">$a_{n_1} \in \{a_1,a_2,...\}$</span>. Further choose smallest possible <span class="math-container"... | DonAntonio | 31,254 | <p>Your proof will work, with the natural changes, if you argue as follows: let <span class="math-container">$\;a_n\;$</span> be an infinite sequence. If for some <span class="math-container">$\;N\in\Bbb N\;$</span> we have that the sequence is monotonic (whatever: ascending or descending) for <span class="math-contain... |
1,668,487 | <p>$$2^{-1} \equiv 6\mod{11}$$</p>
<p>Sorry for very strange question. I want to understand on which algorithm there is a computation of this expression. Similarly interested in why this expression is equal to two?</p>
<p>$$6^{-1} \equiv 2\mod11$$</p>
| vrugtehagel | 304,329 | <p>First, we need to understand what $2^{-1}\mod 11$ stands for. Raising to the power $-1$ is usually used to denote the <em>multiplicative inverse</em>. That is, $b$ is a multiplicative inverse of $a$ if and only if $a\cdot b=1$. Thus, $2^{-1}\mod 11$ denotes the number such that $$2^{-1}\cdot2\equiv1\mod11$$ Since $2... |
847,672 | <p>I repeat the Peano Axioms:</p>
<ol>
<li><p>Zero is a number.</p></li>
<li><p>If a is a number, the successor of a is a number.</p></li>
<li><p>zero is not the successor of a number.</p></li>
<li><p>Two numbers of which the successors are equal are themselves equal.</p></li>
<li><p>If a set S of numbers contains zer... | David Z | 1,190 | <p>I'm not fully sure I understand your question, but perhaps you will find this argument enlightening: consider the set</p>
<p>$$\mathbb{N}' = \{0', s(0'), s(s(0')), \ldots\}$$</p>
<p>which contains $0'$ and all its successors. (This is the same $\mathbb{N}'$ you defined in the question.) Hopefully it is clear that ... |
847,672 | <p>I repeat the Peano Axioms:</p>
<ol>
<li><p>Zero is a number.</p></li>
<li><p>If a is a number, the successor of a is a number.</p></li>
<li><p>zero is not the successor of a number.</p></li>
<li><p>Two numbers of which the successors are equal are themselves equal.</p></li>
<li><p>If a set S of numbers contains zer... | Dan Christensen | 3,515 | <p><strong>Theorem</strong></p>
<p>Suppose we have zeroes $0$ and $0'$ in a <em>Peano set</em> such that:</p>
<ol>
<li><p>$0$ is a number.</p></li>
<li><p>$0'$ is a number</p></li>
<li><p>If <em>x</em> is a number, the successor of <em>x</em> is a number.</p></li>
<li><p>$0$ is not the successor of any number.</p></l... |
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