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 |
|---|---|---|---|---|
2,685,424 | <p>What I don't understand is that why can't we find the general solution of non homogeneous differential equation from the non homogeneous one itself. Currently we use the homogeneous equation also. </p>
<p>Why isn't it that general solution is not available from the non-homogeneous equation itself?</p>
| Christian Blatter | 1,303 | <p>What you are observing here is a fundamental principle valid in the "linear world". You are given an equation or system of equations
$$Ax=b\ ,\tag{1}$$
whereby $A$ operates linearly on the input vector $x$, and $b$ is a given constant vector. Such an equation may have no solutions. If it has solutions then they are ... |
454,426 | <blockquote>
<p>In set theory and combinatorics, the cardinal number $n^m$ is the size of the set of functions from a set of size m into a set of size $n$.</p>
</blockquote>
<p>I read this from this <a href="http://en.wikipedia.org/wiki/Empty_product#0_raised_to_the_0th_power" rel="nofollow noreferrer">Wikipedia pag... | Chris Culter | 87,023 | <p>Consider a small example: the number of functions from a 2-element set $\{a,b\}$ to a 3-element set $\{1,2,3\}$. They are:</p>
<p>$$\begin{align}
a,b\mapsto1,1\\
a,b\mapsto1,2\\
a,b\mapsto1,3\\
a,b\mapsto2,1\\
a,b\mapsto2,2\\
a,b\mapsto2,3\\
a,b\mapsto3,1\\
a,b\mapsto3,2\\
a,b\mapsto3,3 \end{align}$$</p>
<p>See? T... |
3,218,662 | <p>Let <span class="math-container">$T: X \to Y$</span> be a linear operator between normed Banach spaces <span class="math-container">$X$</span> and <span class="math-container">$Y$</span>.
The definition of the operator norm
<span class="math-container">$$
\| T \|
:= \sup_{x \neq 0} \frac{\| Tx \|}{\| x \|}
$$</span>... | Aweygan | 234,668 | <p>First of all, you need to assume that <span class="math-container">$T^{-1}$</span> is bounded. If <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are Banach spaces, this holds automatically, but not for normed spaces. For example, consider <span class="math-container">$X=Y=c_{00}(... |
3,211,264 | <p>I can use the exponents laws only <span class="math-container">$m,n \in \mathbb{N}$</span>, and need to prove them for <span class="math-container">$m,n \in \mathbb{Z}$</span>.</p>
<p>note that <span class="math-container">$0 \neq a \in \mathbb{R}$</span></p>
<p>I proved some cases (mainly the trivial ones) and I... | J.G. | 56,861 | <p>Defining <span class="math-container">$a^{-k}:=\frac{1}{a^k}$</span> for <span class="math-container">$k>0$</span>, we first prove <span class="math-container">$a^{m+1}=a^m a$</span> for all <span class="math-container">$m\in\Bbb Z$</span>: the desired result is the definition of <span class="math-container">$a^{... |
3,211,264 | <p>I can use the exponents laws only <span class="math-container">$m,n \in \mathbb{N}$</span>, and need to prove them for <span class="math-container">$m,n \in \mathbb{Z}$</span>.</p>
<p>note that <span class="math-container">$0 \neq a \in \mathbb{R}$</span></p>
<p>I proved some cases (mainly the trivial ones) and I... | zwim | 399,263 | <p><strong>Important note:</strong> I consider always <span class="math-container">$m,n,k\in\mathbb N$</span>, and use <span class="math-container">$-n,-m,-k$</span> for negative exponents.</p>
<p>If both negative exponents then <span class="math-container">$$a^{-m-n}=(\frac 1a)^{m+n}\color{red}=(\frac 1a)^n(\frac 1a)... |
1,105,126 | <p><img src="https://i.stack.imgur.com/XtrB7.png" alt="enter image description here"></p>
<p>My attempt at the solution is to let P(n) be $10^{3n} + 13^{n+1}$</p>
<p>P(1)= $10^3 + 13^2 = 1169$</p>
<p>Thus P(1) is true.</p>
<p>Suppose P(k) is true for all $k \in N$
$\Rightarrow P(k) = 10^{3k} + 13^{k+1} = 10^{3k} +... | drhab | 75,923 | <p>In $200$ minutes working together they deliver $5+4=9$ papers. So how many minutes they need for delivering $1$ paper?</p>
|
453,212 | <p>Consider a number Q in a made up base system:</p>
<p>The base system is as follows:</p>
<p>It encodes a number as a sum of odd numbers:</p>
<p>1 3 5 7 9 ...</p>
<p>If the number can be expressed as a sum of unique odds. For example, the number 16 is expressed as:</p>
<p>1110 = 7 + 5 + 3 + 1</p>
<p>The system i... | qaphla | 85,568 | <p>If $u$ is even and greater than two: represent $u$ as $110\dots0$, where the number of $0$s is equal to $\frac{u}{2} - 2$.</p>
<p>If $u$ is odd: represent $u$ as $10\dots0$, where the number of $0$s is equal to $\frac{u - 1}{2}$.</p>
<p>$2$ cannot be represented in this system, but is the only nonnegative number f... |
1,570,754 | <p>Let $I$ be an interval and $f\colon I \to \mathbb{R}$ a differentiable function. Suppose the following definitions:</p>
<p>For $x_0 \in I$ the point $(x_0,f(x_0))$ is called <em>saddle point</em> if $f'(x_0) = 0$ but $x_0$ is not a local extremum of $f$.</p>
<p>For $x_W \in I$ the point $(x_W,f(x_W))$ is called <... | Altrouge | 298,678 | <p>OK for the first part.</p>
<p>Let's take a look at the second part (the $\Leftarrow$ implication). It is indeed true.</p>
<p>Let us suppose $(x,f(x))$ is a point of inflexion and $f'(x) = 0$. Then there exists an interval $J = [a,b]$, $x \in J$ and $J \subset I$, where we can suppose by symmetry, that $f'$ is incr... |
1,570,754 | <p>Let $I$ be an interval and $f\colon I \to \mathbb{R}$ a differentiable function. Suppose the following definitions:</p>
<p>For $x_0 \in I$ the point $(x_0,f(x_0))$ is called <em>saddle point</em> if $f'(x_0) = 0$ but $x_0$ is not a local extremum of $f$.</p>
<p>For $x_W \in I$ the point $(x_W,f(x_W))$ is called <... | Narasimham | 95,860 | <p>Can only be brief, sorry you can fill in the gaps. Information available in Wiki.. For z = f(x,y) ; second derivative test. <0 for max, >0 for min, test fails but at saddle points the both signs prevail. E.g., monkey saddle.$ f(x,y) = x^3 - 3 x y^2 $. when considering inflection points along certain directions (... |
677,859 | <p>$f(x)= f(x+1)+3$ and $f(2)= 5$, determine the value of $f(8)$.</p>
<p>I don't understand how $f(x)$ can equal $f(x+1)+3$</p>
| WhizKid | 87,019 | <p>Essentially re arrange the equation to: $f(x+1)=f(x)-3$</p>
<p>so $f(8)=f(7)-3=f(6)-3-3=.....=f(2)-18=5-18=-13$ using $f(2)=5$</p>
<p>so $f(8)=-13$</p>
|
1,914,752 | <p>dividing by a whole number i can describe by simply saying split this "cookie" into two pieces, then you now have half a cookie. </p>
<p>does anyone have an easy way to describe dividing by a fraction? 1/2 divided by 1/2 is 1</p>
| fleablood | 280,126 | <p>$a \div b $ means "how many $b $s does it take to get $a $"</p>
<p>So "$2 \frac 12 \div \frac 12$" is "how many $\frac 12$s does it take to get $2\frac 12$?" The answer is $5$.</p>
<p>So how many half cookies does it take to make half a cookie? The answer is one.</p>
|
203,827 | <p>Suppose I have the following lists: </p>
<pre><code>prod = {{"x1", {"a", "b", "c", "d"}}, {"x2", {"e", "f",
"g"}}, {"x3", {"h", "i", "j", "k", "l"}}, {"x4", {"m",
"n"}}, {"x5", {"o", "p", "q", "r"}}}
</code></pre>
<p>and </p>
<pre><code>sub = {{"m", "n"}, {"o", "p", "r", "q"}, {"g", "f", "e"}};
</code><... | MelaGo | 63,360 | <pre><code>sortedsub = Sort /@ sub;
Select[prod, MemberQ[sortedsub, Sort[#[[2]]]] &]
</code></pre>
<blockquote>
<p>{{"x2", {"e", "f", "g"}}, {"x4", {"m", "n"}}, {"x5", {"o", "p", "q", "r"}}}</p>
</blockquote>
|
90,712 | <p>How many <em>unique</em> pairs of integers between $1$ and $100$ (inclusive) have a sum that is even? The solution I got was</p>
<p>$${100 \choose 1}{99 \choose 49}$$</p>
<p>I don't have a way to verify it, but I figured you pick one card from the 100, then you can pick 49 of the other cards (if the first card is ... | TurlocTheRed | 397,318 | <p>An equivalent variation: To get an even sum, both numbers have to be even or both numbers have to be odd. For either case, there are C(50,2) possible combinations. So the final answer is 2*C(50,2). </p>
|
4,263,631 | <p>so i have question about existence of function <span class="math-container">$f:\mathbb{R} \to \mathbb{R}$</span> such <span class="math-container">$f$</span> is not the pointwise limit of a sequence of continuous functions <span class="math-container">$\mathbb{R} \to \mathbb{R}$</span>.</p>
<p>i'm created a family o... | principal-ideal-domain | 131,887 | <p>If you want to cover it simply use rectangles of width <span class="math-container">$1$</span> and height of the maximum of the function for that inverval of length <span class="math-container">$1$</span>. So you have
<span class="math-container">$$m_2(A) \le \sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}.$$</spa... |
13,989 | <p>Suppose $E_1$ and $E_2$ are elliptic curves defined over $\mathbb{Q}$.
Now we know that both curves are isomorphic over $\mathbb{C}$ iff
they have the same $j$-invariant.</p>
<p>But $E_1$ and $E_2$ could also be isomorphic over a subfield of $\mathbb{C}$.
As is the case for $E$ and its quadratic twist $E_d$. Now th... | Sam Derbyshire | 362 | <p>A concrete explanation: for elliptic curves defined using short Weierstrass equations $E_i : y^2 = x^3 + a_ix + b_i$ over $K$ (not of characteristic $2$ or $3$), all isomorphisms over $L$ (an extension of $K$) are just given by $f(x,y) = (\lambda^2 x,\lambda^3 y)$ for some $\lambda \in L^\times$, so we then need $\l... |
3,130,059 | <p>I have faced this differential problem: <span class="math-container">$(y'(x))^3 = 1/x^4$</span>. </p>
<p>From the fundamental theorem of algebra i know there exist 3 solutions <span class="math-container">$y_1$</span>, <span class="math-container">$y_2$</span>, <span class="math-container">$y_3$</span>, but formall... | Fred | 380,717 | <p>If <span class="math-container">$y'(x)^3=\frac{1}{x^4}$</span>, then <span class="math-container">$y(x)=-3 x^{-1/3}+c$</span> ........</p>
|
1,788,435 | <p>My math book says, a Linear equation has exactly one solution. Because $ax + b = 0$; $x =-\frac{b}{a}$. But I've solved many linear equations with multiple solutions before.
(I'm not very good in math. Need help...)</p>
| Rodrigo de Azevedo | 339,790 | <p>If we have $2$ unknowns, then the linear system</p>
<p>$$a_1 x_1 + a_2 x_2 = b$$</p>
<p>has, in general, infinitely many solutions. Why is that? Assuming that $a_1 \neq 0$, we write</p>
<p>$$x_1 = \frac{b}{a_1} - \left(\frac{a_2}{a_1}\right) x_2$$</p>
<p>Let $x_2 = \gamma$, where $\gamma \in \mathbb{R}$. Then, t... |
3,878,380 | <p>Do the columns of a matrix always represent different vectors? If so, I don't understand how if I have a <span class="math-container">$3\times3$</span> matrix where the rows represent the dimensions and I multiply it by a <span class="math-container">$3\times1$</span> column vector with the same dimensions, it will ... | Watercrystal | 571,790 | <p>Well, on the first level your answer is "because that is how we defined things", but this is hardly a satisfying explanation.
The real reason is that matrices are just a convenient way to represent linear functions between (finite dimensional) vector spaces, i.e. functions of the form <span class="math-con... |
3,656,978 | <p>I understand to say that a bounded linear operator <span class="math-container">$T$</span> is called "polynomially compact" if there is a nonzero polynomial <span class="math-container">$p$</span> such that <span class="math-container">$p(T)$</span> is compact. </p>
<p>Can anyone help me with examples of polynomial... | RedLapm | 779,251 | <p>This should be the correct way to solve:</p>
<p><span class="math-container">$\mathbf I$$\vec x$</span> - <span class="math-container">$\mathbf A$$\vec x$</span> = <span class="math-container">$\vec d$</span></p>
<p>(<span class="math-container">$\mathbf I$</span>-<span class="math-container">$\mathbf A$</span>)<s... |
175,971 | <p>Let's $F$ be a field. What is $\operatorname{Spec}(F)$? I know that $\operatorname{Spec}(R)$ for ring $R$ is the set of prime ideals of $R$. But field doesn't have any non-trivial ideals.</p>
<p>Thanks a lot!</p>
| Rene Schoof | 36,713 | <p>$Spec(Z)$ does not have the cofinite topology. The non-empty
open sets are precisely the cofinite sets that contain the zero ideal.</p>
|
2,041,484 | <p>Solve the system of equations for all real values of $x$ and $y$
$$5x(1 + {\frac {1}{x^2 +y^2}})=12$$
$$5y(1 - {\frac {1}{x^2 +y^2}})=4$$</p>
<p>I know that $0<x<{\frac {12}{5}}$ which is quite obvious from the first equation.<br>
I also know that $y \in \mathbb R$ $\sim${$y:{\frac {-4}{5}}\le y \le {\frac 4... | Dr. Sonnhard Graubner | 175,066 | <p>for $$x,y\ne 0$$ we obtain
$$1+\frac{1}{x^2+y^2}=\frac{12}{5x}$$
$$1-\frac{1}{x^2+y^2}=\frac{4}{5y}$$
adding both we get
$$5=\frac{6}{x}+\frac{2}{y}$$
from here we obtain
$$y=\frac{2x}{5x-6}$$
can you proceed?
after substitution and factorizing we get this equation:
$$- \left( x-2 \right) \left( 5\,x-2 \right) \le... |
4,019,748 | <p>Let <span class="math-container">$H_1=(H_1, (\cdot, \cdot )_1)$</span> and <span class="math-container">$H_2=(H_2, (\cdot, \cdot )_2)$</span> be Hilbert spaces. Suppose that <span class="math-container">$H_1$</span> is continuously and densely embedded in <span class="math-container">$H_2$</span>. Simbolically, <spa... | alepopoulo110 | 351,240 | <p>Of course not, the question fails trivially in this generality: take any Hilbert space <span class="math-container">$H$</span> and let <span class="math-container">$H_1=H_2=H$</span>. Let <span class="math-container">$X,Y$</span> be closed subspaces with <span class="math-container">$X\subset Y$</span> and <span cl... |
4,363,327 | <p>Consider the one-layer neural network <span class="math-container">$y=\mathbf{w}^T\mathbf{x} +b$</span> and the optimization objective <span class="math-container">$J(\mathbf{w}) = \mathbb{E}\left[ \frac12 (1-y\cdot t) \right]$</span> where <span class="math-container">$t\in\{-1,1\}$</span> is the label of our data ... | Trevor Gunn | 437,127 | <p>Neither is "the correct way of doing it." Nor are either wrong. There are two conventions, and as long as you follow the same convention consistently, you will get sensible answers.</p>
<p>If you treat the derivative as a linear operator approximating your function, then the derivative of a function <span ... |
204,043 | <p>I am looking at the following optimization problem</p>
<p><span class="math-container">$$
\begin{align*}
\max\ & 1000 r_1 + \frac{1}{2}r_2 + \frac{1}{3}r_3\\
\text{s.t. }& 1000^2 r_1 + \frac{1}{4}r_2 + \frac{1}{9}r_3 = \frac{1}{9},\\
& 1000^2 p_1 + \frac{1}{4}p_2 + \frac{1}{9}p_3 = \frac{1}{9},\\
& ... | user64494 | 7,152 | <p>I don't find it a bug. The feasible set consists of only one element</p>
<pre><code>Reduce[{1000^2 r1 + 1/4 r2 + 1/9 r3 == 1/9, r1 + r2 + r3 + r4 == 2,
r1^2 <= p1, r2^2 <= p2, r3^2 <= p3, r4^2 <= p4, 1000^2 p1 + 1/4 p2 + 1/9 p3 == 1/9,
p1 + p2 + p3 + p4 == 2, 0 <= p1 <= 1, 0 <= p2 <= 1, 0 ... |
81,588 | <p>A certain function contains points $(-3,5)$ and $(5,2)$. We are asked to find this function,of course this will be simplest if we consider slope form equation </p>
<p>$$y-y_1=m(x-x_1)$$</p>
<p>but could we find for general form of equation? for example quadratic? cubic?</p>
| rogerl | 27,542 | <p>One more proof, similar to Greg Martin's: Suppose $\alpha$ is a root of $f(x)=x^p-x+a$ in some splitting field; then
\begin{equation*}
(\alpha+1)^p - (\alpha+1) + a
= \alpha^p + 1 - \alpha - 1 + a
= \alpha^p - \alpha + a = 0,
\end{equation*}
so that $\alpha+1$ is also a root. It follows that the roots ... |
467 | <p>Moderators have started incorporating the old faq material in the new <a href="https://mathoverflow.net/help">help system</a>. It wasn't a perfect fit, a lot of stuff is no longer relevant, redundant, missing or broken. You can help by going through the <a href="https://mathoverflow.net/help">help center</a> and pos... | Neil Strickland | 10,366 | <p>I don't know what things you can or can't edit, but ideally there should be a couple of lines about the scope of the site and the relationship with MSE right near the top of <a href="https://mathoverflow.net/help">https://mathoverflow.net/help</a>. I suggest:</p>
<blockquote>
<p>This site is for questions about ... |
467 | <p>Moderators have started incorporating the old faq material in the new <a href="https://mathoverflow.net/help">help system</a>. It wasn't a perfect fit, a lot of stuff is no longer relevant, redundant, missing or broken. You can help by going through the <a href="https://mathoverflow.net/help">help center</a> and pos... | Neil Strickland | 10,366 | <p>I suggest that the topics in the 'Asking' menu should be reordered as follows. Roughly speaking, stuff that all new posters should read comes at the top, and stuff that only becomes relevant when there is a problem goes further down.</p>
<ul>
<li>What topics can I ask about here?</li>
<li>How do I ask a good quest... |
2,506,182 | <p>The question speaks for itself, since the question comes from a contest environment, one where the use of calculators is obviously not allowed, can anyone perhaps supply me with an easy way to calculate the first of last digit of such situations?</p>
<p>My intuition said that I can look at the cases of $3$ with an ... | Aaron Montgomery | 485,314 | <p>Hint: Consider the first few powers of $3$ (say, the first five) and look for a pattern.</p>
|
1,056,041 | <p>Look at problem 8 :</p>
<blockquote>
<p>Let $n\geq 1$ be a fixed integer. Calculate the distance:
$$\inf_{p,f}\max_{x\in[0,1]}|f(x)-p(x)|$$ where $p$ runs over
polynomials with degree less than $n$ with real coefficients and $f$
runs over functions $$ f(x)=\sum_{k=n}^{+\infty}c_k\, x^k$$ defined on
the c... | d125q | 112,944 | <p>You can think of the number of favorable arrangements in the following way: choose the empty box in $\binom{n}{1}$ ways. For each such choice, choose the box that will have at least $2$ balls (there has to be one such box) in $\binom{n - 1}{1}$ ways. And for this box, choose the balls that will go inside in $\binom{... |
2,663,130 | <p>Let $f:\mathbb{R}^2\to \mathbb{R}^2$ be function $f(x,y)=(\frac{1}{2}x+y,x-2y)$. Find a image of set $A\subset\mathbb{R}^2$ bounded with lines $x-2y=0, x-2y+2=0, x+2y-2=0, x+2y-3=0.$</p>
<p>Set $A$ is parallelogram with vertices $(1,\frac{1}{2}), (\frac{3}{2},\frac{3}{4}), (\frac{1}{2},\frac{5}{4}), (0,1)$.</p>
<p... | Peter Szilas | 408,605 | <p>Hint: $\dfrac{n}{n+1}= 1- \dfrac{1}{n+1}.$</p>
|
2,557,520 | <p>PS: Before posting it, one tried to grasp <a href="https://math.stackexchange.com/questions/1996141/if-ffx-x2-x1-what-is-f0">this</a> (although didn't understand mfl's answer completely either).</p>
<p>I was shown the way to solve it: if I set $$\frac{3x-2}{2}=0 \Rightarrow x=\frac{2}{3} \Rightarrow x^2-x-1=-\frac{... | John | 7,163 | <p>Hint: To calculate $f(0)$ given an expression for $f((3x-2)/2)$, you could find the value of $x$ that makes the argument zero, and then use that value in the expression.</p>
|
72,478 | <p>I am trying to find the intervals on which f is increasing or decreasing, local min and max, and concavity and inflextion points for $f(x)=\sin x+\cos x$ on the interval $[0,\pi]$.</p>
<p>I know at $\pi/4$ the derivative will equal zero. So that gives me my critical numbers, positive and negative $\pi/4$ so now I n... | NoChance | 15,180 | <p>$f(x)=\sin(x)+\cos(x)$</p>
<p>$f'(x)=\cos(x)-\sin(x)$</p>
<p>critical points are when $f'(x)=0$: </p>
<p>i.e, at:</p>
<p>$\cos(x)=\sin(x)$ which can be satisfied by the values of x such as:</p>
<p>...,$-7{\pi}/4$ , $-3{\pi}/4$, ${\pi}/4$, $3{\pi}/4$,...</p>
<p>now, you need to examine the second derivative's s... |
2,949,224 | <p>How do I show that if <span class="math-container">$\sqrt{n}(X_n - \theta)$</span> converges in distribution, then <span class="math-container">$X_n$</span> converges in probability to <span class="math-container">$\theta$</span>? </p>
<p>Setting <span class="math-container">$Y_n = \sqrt{n}(X_n - \theta)$</span> , ... | yurnero | 178,464 | <p>We have:
<span class="math-container">$$
\frac{1}{\sqrt{n}}\to 0\implies\frac{1}{\sqrt{n}}\overset{L}\to 0\implies X_n-\theta=\frac{1}{\sqrt{n}}[\sqrt{n}(X_n-\theta)]\overset{L}{\to} 0\implies X_n-\theta\overset{P}{\to} 0.
$$</span>
Here, <span class="math-container">$\overset{L}{\to}$</span> indicates convergence i... |
3,657,428 | <p>My textbook says that</p>
<blockquote>
<p>If <span class="math-container">$f(x)$</span> is piecewise continuous on <span class="math-container">$(a,b)$</span> and satisfies <span class="math-container">$f(x) = \frac{1}{2} [f(x_{-})+f(x_{+})]$</span> for all <span class="math-container">$x\in(a,b)$</span>, and if ... | Wlod AA | 490,755 | <p>When <span class="math-container">$\ f(x_0)>0\ $</span> then <span class="math-container">$\ f(x_-)> 0\ $</span> or <span class="math-container">$\ f(x_+) > 0. $</span>
Thus, either <span class="math-container">$\ f\ $</span> is positive in an interval
<span class="math-container">$\ (x_0-h;x_0)\ $</span> o... |
2,075,485 | <p>Let $[a,b]$ be a finite closed interval on $\mathbb{R}$, $f$ be a continuous differentible function on $[a,b]$. Prove that
$$\max_{x\in [a,b]} |f(x)|\le \Bigg|\frac{1}{b-a} \int_a^b f(x)dx\Bigg|+\int_a^b |f'(x)|dx$$</p>
<p>I think this is similar to Sobolev embedding theorem but have no idea about how to use it. I... | na1201 | 397,984 | <p>although Kobe has explained well. I had already started writing the answer.
So using triangle inequality we have $|f(c)| \leq |f(\xi)|+|f(c) - f(\xi)|$. By fundamental theorem of calculus $f(c) - f(\xi) = \int_{\xi}^cf'(x)dx$. Taking absolute values we have $$|f(c) - f(\xi)| = \int_{\xi}^cf'(x)dx \leq \int_{\xi}^c|f... |
1,440,106 | <p>I am currently studying how to prove the Fibonacci Identity by Simple Induction, shown <a href="http://mathforum.org/library/drmath/view/52718.html">here</a>, however I do not understand how $-(-1)^n$ becomes $(-1)^{n+1}$. Can anybody explain to me the logic behind this?</p>
| mweiss | 124,095 | <p>The negative sign <em>outside</em> the parentheses can be re-written as $-1$:
$$-(-1)^n = (-1)(-1)^n$$
That first factor of $(-1)$ can be written as $(-1)^1$, so we have
$$(-1)^1(-1)^n$$
and finally the two factors can be combined by adding the exponents:
$$(-1)^{n+1}$$</p>
|
1,440,106 | <p>I am currently studying how to prove the Fibonacci Identity by Simple Induction, shown <a href="http://mathforum.org/library/drmath/view/52718.html">here</a>, however I do not understand how $-(-1)^n$ becomes $(-1)^{n+1}$. Can anybody explain to me the logic behind this?</p>
| Kushal Bhuyan | 259,670 | <p>Simple there is a $-$ sign in front so you can write $-(-1)^n=(-1)(-1)^n=(-1)^{n+1}$.</p>
|
804,532 | <p>I a working my way through some old exam papers but have come up with a problem. One question on sequences and induction goes:</p>
<p>A sequence of integers $x_1, x_2,\cdots, x_k,\cdots$ is defind recursively as follows: $x_1 = 2$ and $x_{k+1} = 5x_k,$ for $k \geq 1.$</p>
<p>i) calculate $x_2, x_3, x_4$</p>
<p>ii... | mfl | 148,513 | <p>Your answer is correct.</p>
<p>To write down the proof by induction you have:</p>
<p>Your formula is correct for $n=1$ because $x_1=2\cdot 5^{1-1}=2\cdot 5^0=2.$</p>
<p>Now you suppose that it is correct for $n,$ that is, $x_n=2\cdot 5^{n-1},$ and you need to prove that it hols for $n+1.$ We have:</p>
<p>$x_{n+1... |
1,593,282 | <p>Say we have the function $f:A \rightarrow B$ which is pictured below.<a href="https://i.stack.imgur.com/WfCxs.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/WfCxs.jpg" alt="enter image description here"></a></p>
<p>This function is not bijective, so the inverse function $f^{-1}: B \rightarrow A$... | AnotherPerson | 185,237 | <p>If it is not injective then no, because we don't know what preimage you are referring to when you write $f^{-1}(x)$ if $x$ has multiple preimages. But what you can write is $f^{pre}(x)$ indicating the preimage (or set of preimages) of $x$. </p>
|
3,600,528 | <p>Is there a general formula for determining multiplicity of <span class="math-container">$2$</span> in <span class="math-container">$n!\;?$</span>
I was working on a Sequence containing subsequences of 0,1. 0 is meant for even quotient, 1 for odd quotient.
Start with k=3, k should be odd at start, if odd find (k-1)... | J. W. Tanner | 615,567 | <p><strong>Hint:</strong></p>
<p><span class="math-container">$n!$</span> is the product of the numbers from <span class="math-container">$1$</span> to <span class="math-container">$n$</span>.</p>
<p>How many multiples of <span class="math-container">$2$</span> are there in <span class="math-container">$\{1,2,...,n\}... |
259,119 | <p>I have a string like this:</p>
<p><code>string="there is a humble-bee in Hanna's garden";</code></p>
<p>Now I want to exclude those words that contain "-" and "'". My own solution would be:</p>
<p><code>StringDelete[string,Cases[StringSplit[string," "], _?(StringContainsQ[#, {... | Daniel Huber | 46,318 | <p>You can give a string pattern to "StringDelete" like:</p>
<pre><code>string = "there is a humble-bee in Hanna's garden";
pat = WordCharacter ... ~~ ("-" | "'") ~~ WordCharacter ...;
StringDelete[string, pat]
(*"there is a in garden"*)
</code></pre>
|
182,101 | <p>With respect to assignments/definitions, when is it appropriate to use $\equiv$ as in </p>
<blockquote>
<p>$$M \equiv \max\{b_1, b_2, \dots, b_n\}$$</p>
</blockquote>
<p>which I encountered in my analysis textbook as opposed to the "colon equals" sign, where this example is taken from Terence Tao's <a href="http... | Michael Hardy | 11,667 | <p>The $\equiv$ symbol has different standard meanings in different contexts:</p>
<ul>
<li>Congruence in number theory, and various generalizations;</li>
<li>Geometric congruence;</li>
<li>Equality for <b>all</b> values of the variables, as opposed to an equation in which one seeks the values that make the equation tr... |
165,328 | <p>What is the difference between $\cap$ and $\setminus$ symbols for operations on sets?</p>
| Asaf Karagila | 622 | <p>Their definition is different:</p>
<ul>
<li><p>$A\cap B=\{x\mid x\in A\text{ and } x\in B\}$, we take all the elements which appear both in $A$ and in $B$, but not just in one of them. </p></li>
<li><p>$A\setminus B=\{x\mid x\in A\text{ and } x\notin B\}$, we take only the part of $A$ which is not a part of $B$. </... |
214,475 | <p>Function:
<a href="https://i.stack.imgur.com/sH7mh.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/sH7mh.png" alt="enter image description here"></a></p>
<p>I am to solve for <span class="math-container">$T_{12}(4.8), T_{24}(1.2)$</span>, using <strong>If</strong> and <strong>Which</strong> funct... | AsukaMinato | 68,689 | <pre><code>T[x_, n_] := Block[{temp = n}, Which[
temp == 0, Return[1], temp == 1, Return[x], temp > 1,
1/x T[x, temp - 2] - 2/7 T[x, temp - 1]]]
</code></pre>
<p>T[1, 4] gives</p>
<blockquote>
<p>167/343</p>
</blockquote>
<p>Or as what you want.</p>
<pre><code>T[n_] := Block[{temp = n},
Which[temp == 0, Retu... |
3,247,176 | <p>I have this statement:</p>
<blockquote>
<p>It can be assured that | p | ≤ 2.4, if it is known that:</p>
<p>(1) -2.7 ≤ p <2.3</p>
<p>(2) -2.2 < p ≤ 2.6</p>
</blockquote>
<p>My development was:</p>
<p>First, <span class="math-container">$ -2.4 \leq p \leq 2.4$</span></p>
<p>With <span class="math... | Vineet | 196,541 | <p><span class="math-container">$ -2.4 \leq p \leq 2.4$</span></p>
<p><span class="math-container">$ -2.7 \leq p < 2.3$</span></p>
<p><span class="math-container">$-2.2 < p \leq 2.6 $</span></p>
<p>Your answer is common intersection of these inequalities. </p>
<p><span class="math-container">$ p \in (-2.2, 2.... |
361,740 | <p>Spivak's <em>Calculus on Manifolds</em> asks the reader to prove this (problem 1-8, pp.4-5):</p>
<blockquote>
<p>If there is a basis $x_1, x_2, ..., x_n$ of $\mathbb{R}^n$ and numbers $\lambda_1, \lambda_2, ..., \lambda_n$ such that $T(x_i) = \lambda_i x_i$, $1 \leq i \leq n$, prove that $T$ is angle-preserving i... | user1551 | 1,551 | <p>You are correct. This is one of the few errors in Spivak's <em>Calculus on Manifolds</em>. For this particular exercise, see the following questions:</p>
<ul>
<li><a href="https://math.stackexchange.com/questions/177005/question-about-angle-preserving-operators">Question about Angle-Preserving Operators</a></li>
<l... |
361,740 | <p>Spivak's <em>Calculus on Manifolds</em> asks the reader to prove this (problem 1-8, pp.4-5):</p>
<blockquote>
<p>If there is a basis $x_1, x_2, ..., x_n$ of $\mathbb{R}^n$ and numbers $\lambda_1, \lambda_2, ..., \lambda_n$ such that $T(x_i) = \lambda_i x_i$, $1 \leq i \leq n$, prove that $T$ is angle-preserving i... | Matt S | 109,082 | <p>A possible "true 'version' of this statement the author had in mind" is</p>
<blockquote>
<p>Suppose <span class="math-container">$T(x_i)=\lambda_ix_i$</span> for some basis <span class="math-container">$x_1,\dots,x_n$</span> of <span class="math-container">$\mathbb R^n$</span> and numbers <span class="math... |
1,994,021 | <p>In one of the research article it is written that the following limit is equal to zero $$\lim_{x \to 0 }\frac{d}{2^{b+c/x}-1}\left[a2^{b+c/x}-a-a\frac{c\ln{(2)}2^{b+c/x}}{2x}-\frac{c\ln{(2)}}{2x^2}\frac{2^{b+c/x}}{\sqrt{2^{b+c/x}-1}}\right]\left(e^{-ax\sqrt{2^{b+c/x}-1}}\right)=0$$ where $a,b,c,d$ are all positive c... | zhw. | 228,045 | <p>Try something simpler. Get rid of most of the constants. Instead of $x\to 0^+,$ replace $x$ by $1/x$ and let $x\to \infty.$ (For me it's easier to think this way.) Throw away the $1$'s you keep subtracting, they're nothing compared with $2^x.$ So here's what I looked at:</p>
<p>$$\tag 1 \frac{1}{2^x}\left [ 2^x + x... |
223,008 | <p>Ok so my teacher said we can use this sentence:
<strong>If $a$ is not a multiple of $5$, then $a^2$ is not a multiple of $5$ neither.</strong></p>
<p>to prove this sentence:
<strong>If $a^2$ is a multiple of $5$, then $a$ itself is a multiple of $5$</strong></p>
<p>I don't understand the logic behind it, I mean wh... | Mark Bennet | 2,906 | <p>There are two possibilities:</p>
<p>$a$ is a multiple of 5 - in which case we prove that $a^2$ is a multiple of 5</p>
<p>$a$ is not a multiple of 5 - in which case we prove that $a^2$ is not a multiple of 5</p>
<p>So we have proved those facts.</p>
<p>Now suppose we are given a square number, and it is a multipl... |
1,597,247 | <p>Give the continued fraction expansion of two real numbers $a,b \in \mathbb R$, is there an "easy" way to get the continued fraction expansion of $a+b$ or $a\cdot b$?</p>
<p>If $a,b$ are rational it is easy as you can easily conver the back to the 'rational' form, add or multiply and then conver them back to continu... | djechlin | 79,767 | <p>You need a representation of the real number to start with. Real numbers such as $e = \sum\frac1{n!}$ are easy to work with (in particular, using that representation). Real numbers such as $\gamma = \lim_{n\rightarrow\infty}\left(\sum_{k=1}^n\frac1k - \ln n\right)$ are going to be a bit "harder" to work with. And tr... |
290,527 | <p>What would be a good metric on $C^k(0,1)$, space of $k$ times continuously differentiable real valued functions on $(0,1)$ and $C^\infty(0,1)$, space of infinitely differentiable real valued functions on $(0,1)$? </p>
<p>It is of course open to interpretation what good would mean, I want it to bring a good notion o... | Sh4pe | 14,497 | <p>The space $C^k([a,b])$ is a normed space for each $k$ and for each pair $a < b$ of real numbers with the norm</p>
<p>$\|f\|_{C^0} = \sup_{x\in[a,b]} |f(x)|$ for $k=0$</p>
<p>and</p>
<p>$\|f\|_{C^k} = \sum_{|s|\le k} \|\partial^sf\|_{C^0}$ for $k>1$</p>
<p>(For the $C^0$-case, it is important that the inter... |
2,725,839 | <p>The question is below.<a href="https://i.stack.imgur.com/k3UMf.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/k3UMf.png" alt="enter image description here"></a></p>
<p>I was able to solve part (a) because the $x$-coordinate would just be the circumference of the circle, which is $2\pi$. Therefor... | CY Aries | 268,334 | <p>The $x$-coordinate is just the length of the "rotated arc", i.e., $\displaystyle \frac{\pi}{2}$.</p>
|
131,842 | <p>Let $X,Y$ be normed linear spaces. Let $T: X\to Y$ be linear. If $X$ is finite dimensional, show
that $T$ is continuous. If $Y$ is finite dimensional, show that $T$ is continuous if and only if $\ker T$ is closed. </p>
<p>I am able to show that $X$, finite dimensional $\implies$ $T$ is bounded, hence continuous.... | Balbichi | 24,690 | <p>I am doing for $Y=\mathbb{R}$; Clearly if $f$ is continuous then its kernel is closed set. for the converse, assume that $f\neq0$ and that $f^{-1}(\{0\})$ is a closed set. Pick some $e$ in $X$ with $f(e)=1$. Suppose by way of contradiction $||f||=\infty$. Then there exists a sequence $\{x_n\}$ in $X$ with $||x_n||=1... |
4,623,022 | <p>I have a question that I have been curious about for years.</p>
<p>In differential geometry, since the exterior derivative satisfies property <span class="math-container">$d^2=0$</span>, we can make a de Rham cohomology from it.</p>
<p>Then if we write <span class="math-container">$\iota_X:\Omega^n\rightarrow\Omega^... | Mariano Suárez-Álvarez | 274 | <p>You can do this calculation purely algebraically. I'd suggest doing first the following variation.</p>
<p>Consider a vector space <span class="math-container">$V$</span>, a vector <span class="math-container">$v$</span> in <span class="math-container">$V$</span>, the exterior algebra <span class="math-container">$\L... |
804,871 | <p>Prove that if x and y are odd natural numbers, then $x^2+y^2$ is never a perfect square.</p>
<p>Let $x=2m+1$ and $y=2l+1$ where m,l are integers.</p>
<p>$x^2+y^2=(2m+1)^2+(2l+1)^2=4(m^2+m+l^2+l)+2$</p>
<p>Where do I go from here?</p>
| Deepak | 151,732 | <p>The square of an integer is congruent to 0 or 1 (mod 4). In fact an even number (remember the sum is even) will always be congruent to 0 (mod 4). What you end up with is congruent to 2 (mod 4), which means it's not a perfect square.</p>
|
804,871 | <p>Prove that if x and y are odd natural numbers, then $x^2+y^2$ is never a perfect square.</p>
<p>Let $x=2m+1$ and $y=2l+1$ where m,l are integers.</p>
<p>$x^2+y^2=(2m+1)^2+(2l+1)^2=4(m^2+m+l^2+l)+2$</p>
<p>Where do I go from here?</p>
| BlackAdder | 74,362 | <p>You can now look at all the natural numbers modulo $4$. We know that numbers must either be even or odd, hence they have the form
$$2n\text{ or } 2n+1.$$
In modulo $4$, they are just $2n\text{ or } 2n+1\mod4$. Now, look at the squares of these numbers, we have that
$$(2n)^2\equiv 4n^2\equiv0\mod4.$$
Also, the odds ... |
141,522 | <p>First a summary of the general problem I'm trying to solve:
I want to get <strong>a</strong> set of inequalities for a very complex function (If you are interested is the no-arbitrage conditions Black-Scholes equation with a volatility given by an SVI function)</p>
<p>So basically I'm trying to find the parameters ... | mikado | 36,788 | <p>I have found <code>CylindricalDecomposition</code> very useful for analysing inequalities. The result you get will depend on the order in which you list the variables in the second argument.</p>
<p>I think the result you are looking for is given by</p>
<pre><code>f = a*b*(1 + x^2);
CylindricalDecomposition[f >... |
1,583,747 | <p>I just started a course on queue theory, yet equations are given for granted without any demonstrations, which is very frustrating... Thus</p>
<ol>
<li>Why is the mean number of people in a queue system following an $M/M/1$ system</li>
</ol>
<p>$$E(L)=\frac{\rho}{1-\rho}$$</p>
<p>with $\rho=\frac{\lambda}{\mu}$ w... | JKnecht | 298,619 | <p>For every common queue system you can follow this route:</p>
<ol>
<li><p>Set up the balance equations: Inflow equal outflow in steady state.</p></li>
<li><p>Solve the balance equations. Always straight forward.</p></li>
<li><p>Calculate $p_0$ and $E[L]$ </p></li>
</ol>
<p>And its not hard to see that this gives $E... |
1,583,747 | <p>I just started a course on queue theory, yet equations are given for granted without any demonstrations, which is very frustrating... Thus</p>
<ol>
<li>Why is the mean number of people in a queue system following an $M/M/1$ system</li>
</ol>
<p>$$E(L)=\frac{\rho}{1-\rho}$$</p>
<p>with $\rho=\frac{\lambda}{\mu}$ w... | JKnecht | 298,619 | <p>There is a pretty good <a href="https://www.youtube.com/watch?v=AsTuNP0N7DU" rel="nofollow">series of video lectures on youtube</a> that might help you out.</p>
|
3,237,242 | <p>I have the following problem:</p>
<p>I need to prove that given the following integral</p>
<p><span class="math-container">$\int_{0}^{1}{c(k,l)x^k(1-x)^l}dx = 1$</span>,</p>
<p>we the constant <span class="math-container">$c(k,l) = (k+l+1) {{k+l}\choose{k}} = \frac{(k+l+1)!}{k!l!}$</span>,</p>
<p>with the use of... | Community | -1 | <p>Integration by parts relates <span class="math-container">$I_{k,l}$</span> to <span class="math-container">$I_{k+1,l-1}$</span>. So by setting <span class="math-container">$l=n-k$</span>, you relate <span class="math-container">$I_{k,n-k}$</span> to <span class="math-container">$I_{k+1,n-(k+1)}$</span>, which forms ... |
80,783 | <p>How do you convert $(12.0251)_6$ (in base 6) into fractions?</p>
<p>I know how to convert a fraction into base $x$ by constantly multiplying the fraction by $x$ and simplifying, but I'm not sure how to go the other way?</p>
| wendy.krieger | 78,024 | <p>You could use continued fractions. </p>
<pre><code> Cf A / B
12.0251 1 / 0 In the left, we produce a
1.0000 -12.0000 8 8 / 1 continued fraction from
-0.5420 0.0251 12 97 / 12 12.0521 and 1.0000, in base 6
0.0140 ... |
3,999,325 | <p>Let me start with some objects. Consider the <span class="math-container">$\mathrm{C}^*$</span>-algebra <span class="math-container">$A$</span> defined by:
<span class="math-container">$$A=M_1(\mathbb{C})\oplus M_2(\mathbb{C})\subset B(\mathbb{C}^3).$$</span>
Let <span class="math-container">$x=\mathbb{C}^3$</span> ... | JP McCarthy | 19,352 | <p>The confusion is that the (introductory) quantum mechanical texts that I am reading are using the full <span class="math-container">$B(\mathsf{H})$</span> rather than closed self-adjoint subalgebras.</p>
<p>For example in full <span class="math-container">$B(\mathbb{C}^3)$</span>, the state associated to the same ve... |
4,011,864 | <p><span class="math-container">$$\lim_{n \to \infty}(3^n+1)^{\frac{1}{n}}$$</span></p>
<p>I'm fairly sure I can't bring the limit inside the 1/n and I don't think I can use l'Hôpital's rule. I'm pretty sure I'm meant to use the sandwich theorem but I'm not quite sure how to do that in this circumstance.</p>
| Aryan | 866,404 | <p>Since <span class="math-container">$\ln\big(3^n+1\big)^{\frac{1}{n}}=\frac{\ln(3^n+1)}{n}$</span> , one can observe <span class="math-container">$\displaystyle\lim_{n\to\infty}{(3^n+1)^{\frac{1}{n}}}=\displaystyle\lim_{n\to\infty}e^{\frac{\ln(3^n+1)}{n}}=e^{\tiny{{\displaystyle\lim_{n\to\infty}{\frac{\ln(3^n+1)}{n}}... |
4,011,864 | <p><span class="math-container">$$\lim_{n \to \infty}(3^n+1)^{\frac{1}{n}}$$</span></p>
<p>I'm fairly sure I can't bring the limit inside the 1/n and I don't think I can use l'Hôpital's rule. I'm pretty sure I'm meant to use the sandwich theorem but I'm not quite sure how to do that in this circumstance.</p>
| Kyky | 423,726 | <p>Consider the natural logarithm of the limit, <span class="math-container">$$\lim_{n\to\infty}\frac1n\ln\left(3^n+1\right)$$</span></p>
<p>Note that <span class="math-container">$\lim_{n\to\infty}\ln\left(3^n\right)-\ln\left(3^n+1\right)=\ln\left(\lim_{n\to\infty}\frac{3^n}{3^n+1}\right)=\ln1=0$</span>. Hence we have... |
134,523 | <p>Is there any recursively axiomized system with infinitely many proofs for some propositions or a proposition? So we will have at least one proposition which is deduced from the recursively axiomatic system in infinite ways.Could any one give an example or proof?</p>
<p><strong>EDIT</strong>:We define a proof or a... | Noah Schweber | 8,133 | <p>How about the theorem "There exists at least one prime number?" There are infinitely many distinct proofs of this result, none of which includes another as a subproof. Certainly this example is trivial in some sense, but I think it is not obvious how to pin down why this shouldn't be counted.</p>
<p>Perhaps a bette... |
449,296 | <p>I am trying to deduce how mathematicians decide on what axioms to use and why not other axioms, I mean surely there is an infinite amount of available axioms. What I am trying to get at is surely if mathematicians choose the axioms then we are inventing our own maths, surely that is what it is but as it has been pra... | Ronnie Brown | 28,586 | <p>Of course the (presumably) first axiom system was that of Euclid's Geometry. This book though was a systematisation of knowledge at the time, and it seems reasonable to suppose it started as a teaching course. If you are giving a course, you have to decide where you are going to start, and it seems reasonable to s... |
449,296 | <p>I am trying to deduce how mathematicians decide on what axioms to use and why not other axioms, I mean surely there is an infinite amount of available axioms. What I am trying to get at is surely if mathematicians choose the axioms then we are inventing our own maths, surely that is what it is but as it has been pra... | Andreas Blass | 48,510 | <p>There are (at least) four types of sources for axiomatic systems. Here are the scenarios that I have in mind:</p>
<p>(1) Some mathematical structure, like the plane in geometry or the system of natural numbers, has been recognized as useful for applications and has therefore been studied extensively. So many facts... |
4,122,419 | <blockquote>
<p>From the triangle <span class="math-container">$\triangle ABC$</span> we have <span class="math-container">$AB=3$</span>, <span class="math-container">$BC=5$</span>, <span class="math-container">$AC=7$</span>. If
the point <span class="math-container">$O$</span> placed inside the triangle <span class="m... | Andrei | 331,661 | <p>Let's call the length of the race <span class="math-container">$L$</span>, and the time required by A to finish it <span class="math-container">$t$</span>. Then <span class="math-container">$$v_A=\frac Lt\\v_B=\frac{L-10}t\\v_C=\frac{L-20}t$$</span>
When they race for the second time, we want to calculate the time r... |
4,013,796 | <p>If we make a regular polygon with n vertices (n edges) and triangulate on the inside with n-3 edges, then triangulate on the outside with (n-3) edges (or draw dotted lines inside again), a Maximal Planar Graph is formed. Edges shouldn't be repeated and there's no loops or directions.</p>
<p>How many distinct graphs... | John Hunter | 721,154 | <p>Here is the work so far. In what follows the number of distinct Hamiltonian Maximal Planar Graphs with n vertices is <span class="math-container">$X(n)$</span>. It should follow A000109 <a href="https://oeis.org/A000109/list" rel="nofollow noreferrer">https://oeis.org/A000109/list</a> up to <span class="math-conta... |
2,311,848 | <p>$X$ and $Y$ are independent r.v.'s and we know $F_X(x)$ and $F_Y(y)$. Let $Z=max(X,Y)$. Find $F_Z(z)$.</p>
<p>Here's my reasoning: </p>
<p>$F_Z(z)=P(Z\leq z)=P(max(X,Y)\leq z)$. </p>
<p>I claim that we have 2 cases here: </p>
<p>1) $max(X,Y)=X$. If $X<z$, we are guaranteed that $Y<z$, so $F_Z(z)=P(Z\leq z)... | Community | -1 | <p>$\max(X,Y)\le z$ means that <em>both</em> $X$ and $Y$ are $\le z$.</p>
|
2,311,848 | <p>$X$ and $Y$ are independent r.v.'s and we know $F_X(x)$ and $F_Y(y)$. Let $Z=max(X,Y)$. Find $F_Z(z)$.</p>
<p>Here's my reasoning: </p>
<p>$F_Z(z)=P(Z\leq z)=P(max(X,Y)\leq z)$. </p>
<p>I claim that we have 2 cases here: </p>
<p>1) $max(X,Y)=X$. If $X<z$, we are guaranteed that $Y<z$, so $F_Z(z)=P(Z\leq z)... | Math-fun | 195,344 | <p>I think you are fine with separating the cases, but then do not take care of them correctly. Since when you say in your case 1 that the maximum is $X$, you are "conditioning on" $X>Y$ and that changes the space over which you calculate the probabilities. </p>
<p>We have two cases that either of which happens:</p... |
536,073 | <p>I came across several questions like this in the problem section of a book on coding theory & cryptography and I have no idea how to tackle them. There must be a certain trick that allows for efficiently solving such problems by hand.</p>
| André Nicolas | 6,312 | <p>There are a lot of tricks. A useful one for your problems is <strong>Fermat's Theorem</strong>, which says that if $p$ is prime and $a$ is not divisible by $p$, then $a^{p-1}\equiv 1\pmod{p}$.</p>
<p>We look for example at $2^{170}$ modulo $19$. Note that $170=9\cdot 18+8$. Thus
$$2^{170}=(2^{18})^9 2^8\equiv 1^9\c... |
536,073 | <p>I came across several questions like this in the problem section of a book on coding theory & cryptography and I have no idea how to tackle them. There must be a certain trick that allows for efficiently solving such problems by hand.</p>
| Dennis Meng | 35,665 | <p>For those specific examples, <a href="http://en.wikipedia.org/wiki/Fermat%27s_little_theorem" rel="nofollow">Fermat's Little Theorem</a> is the way to go. It states that if $p$ is prime and $gcd(a,p) = 1$, then $$a^{p-1} \equiv 1 \bmod{p}$$.</p>
<p>For the more general case where the modulus is not prime, then you ... |
34,775 | <p><strong>What is the goal of MSE? Is it to get a repository of interesting questions and well-written answers. Or are we instead an online math tutoring site where we help anyone as long as they seem to be trying. These two goals are often in contradiction with each other!</strong></p>
<p>I am afraid that we are head... | discipulus | 1,060,368 | <p>(writing this from new user, but I have some experience on this in other SE's and MSE in the recent years)</p>
<ul>
<li>I believe that any change that is imposed from "above" will ultimately be difficult to manage and cannot be successful without drastic change in site that would (I bet on that) make this ... |
61,316 | <p>Hi all,</p>
<p>I heard a claim that if I have a matrix $A\in\mathbb R^{n\times n}$ such that $A^n \to 0 \ (\text{for }n\to\infty )$
(that is, every entry of $A^n$ converges to $0$ where $n\to \infty$)
then $I-A$ is invertible.</p>
<p>anyone knows if there is a name for such a matrix or how (for general knowledge)... | Community | -1 | <p>The matrices you are looking for are exactly those that have spectral radius (the max. of the absolute value of the eigenvalues) strictly less than one.
I do not know whether there is a more specific name.
(A matrix such that a finite power would be exactly the zero-matrix would be called nilpotent; but this is a ... |
238,970 | <p>Recently I've come across 'tetration' in my studies of math, and I've become intrigued how they can be evaluated when the "tetration number" is not whole. For those who do not know, tetrations are the next in sequence of iteration functions. (The first three being addition, multiplication, and exponentiation, while ... | The_Sympathizer | 11,172 | <p>Ah yes, a fave topic of mine. Basically, there is no universally-agreed on way to do this. The problem is, that, in general, there isn't a unique way to interpolate the values of tetration at integer "height" (which is what the "number of exponents in the 'tower'" may be called). So in theory, you could define it to... |
238,970 | <p>Recently I've come across 'tetration' in my studies of math, and I've become intrigued how they can be evaluated when the "tetration number" is not whole. For those who do not know, tetrations are the next in sequence of iteration functions. (The first three being addition, multiplication, and exponentiation, while ... | Gottfried Helms | 1,714 | <p>Here is a q&d - implementation in Pari/GP to get some intuition about what is going on at all. The "Kneser-Method" is much more involved, but it seems there is a good possibility, that the simple method below (I call it the "polynomial method") is asymptotic to/approximates the Kneser-method when the size of the... |
238,970 | <p>Recently I've come across 'tetration' in my studies of math, and I've become intrigued how they can be evaluated when the "tetration number" is not whole. For those who do not know, tetrations are the next in sequence of iteration functions. (The first three being addition, multiplication, and exponentiation, while ... | Mark Hunter | 700,582 | <p>Some attempts at defining <span class="math-container">$e$</span> to itself <span class="math-container">$s$</span> times when <span class="math-container">$s$</span> is a real number besides just a whole number involve adding a sort of socket variable to turn the problem into finding continuous iterates of <span cl... |
550,441 | <p>Say I roll a 6-sided die until its sum exceeds $X$. What is E(rolls)?</p>
| Community | -1 | <p>Let $h(s)$ be the expected number of rolls to exceed $X$, starting with a sum of $s$.
Then "first step analysis" gives the recursive formula $h(s)=1+{1\over 6}\sum_{j=1}^6 h(s+j)$ for $0\leq s\leq X$, while $h(s)=0$ for $s>X$. You use this equation to calculate $h(s)$ for $s=X,X-1,X-2,\dots$ and eventually work... |
2,995,495 | <p>I'm trying to prove that, for every <span class="math-container">$x \geq 1$</span>:</p>
<p><span class="math-container">$$\left|\arctan (x)-\frac{π}{4}-\frac{(x-1)}{2}\right| \leq \frac{(x-1)^2}{2}.$$</span> </p>
<p>I could do it graphically on <span class="math-container">$\Bbb R$</span>, but how to make a formal... | hamza boulahia | 406,464 | <h2> Hint: </h2>
<p>Use the Taylor expansion of <span class="math-container">$\arctan(x)$</span> near <span class="math-container">$1$</span> to the second order. And the fact that <span class="math-container">$\arctan(x)$</span> is concave when <span class="math-container">$x\geq1$</span></p>
<hr>
<h2> Answer: </h2... |
2,995,495 | <p>I'm trying to prove that, for every <span class="math-container">$x \geq 1$</span>:</p>
<p><span class="math-container">$$\left|\arctan (x)-\frac{π}{4}-\frac{(x-1)}{2}\right| \leq \frac{(x-1)^2}{2}.$$</span> </p>
<p>I could do it graphically on <span class="math-container">$\Bbb R$</span>, but how to make a formal... | zhw. | 228,045 | <p>Hint: For each <span class="math-container">$x >1,$</span> Taylor gives</p>
<p><span class="math-container">$$\arctan (x)=\frac{\pi}{4}+\frac{(x-1)}{2} + \frac{\arctan'' (c_x)}{2}(x-1)^2,$$</span></p>
<p>where <span class="math-container">$1<c_x<x.$</span> Thus all you need to show is that <span class="ma... |
112,137 | <p>I'm guessing the answer to this question is well-known:</p>
<p>Suppose that $Y:C \to P$ and $F:C \to D$ are functors with $D$ cocomplete. Then one can define the point-wise Kan extension $\mathbf{Lan}_Y\left(F\right).$ Under what conditions does $\mathbf{Lan}_Y\left(F\right)$ preserve colimits? Notice that if $C=P$... | Tom Leinster | 586 | <p>$F$ preserving colimits doesn't imply that $\text{Lan}_Y(F)$ preserves colimits, even if all the categories are cocomplete. </p>
<p>Consider, for example, the case $C = D$ and $F = 1_C$. Then the left Kan extension $\text{Lan}_Y(1_C)$ exists if and only if $Y$ has a right adjoint, and if it does exist, it <em>is... |
112,137 | <p>I'm guessing the answer to this question is well-known:</p>
<p>Suppose that $Y:C \to P$ and $F:C \to D$ are functors with $D$ cocomplete. Then one can define the point-wise Kan extension $\mathbf{Lan}_Y\left(F\right).$ Under what conditions does $\mathbf{Lan}_Y\left(F\right)$ preserve colimits? Notice that if $C=P$... | John Bourke | 17,696 | <p>The pointwise left Kan extension of F along Y is a coend of functors $Lan_{Y}(F) = \int^{x}P(Yx,-).Fx$ where each functor $P(Yx,-).Fx$ is the composite of the representable $P(Yx,-):P \to Set$ and the copower functor $(-.Fx):Set \to D$. As a coend (colimit) of the $P(Yx,-).Fx$ the left Kan extension preserves any c... |
480,195 | <p>Three friends brought 3 pens together each 10 dollars. Next day they got 5 dollars cash back so they shared each 1 dollar and donated 2 dollars. Now the pen cost for each guy will be 9 dollars (\$10 -\$1).</p>
<p>But if you add all 9+9+9 = 27 dollars and donated amount is 2 dollars so total 29 dollars. </p>
<p>Whe... | Tomas | 83,498 | <p>The last conclusion is simply wrong. You are right, they paid $27$ dollars altogether. The pens however cost $25$ dollars ($30$ dollars initially, then $5$ discount), so that's the two dollar donation difference. </p>
<p>There is no sense in adding the $2$ dollars, since the nine dollars each friend spent includes ... |
2,601,412 | <p>"A game is played by tossing an unfair coin ($P(head) = p$) until $A$ heads or $A$ tails (not necessarily consecutive) are observed. What is the expected number of tosses in one game?"</p>
<p>My approach is the following:</p>
<p>Let's represent represent a head by $H$ and a tail by $T$, and call $H_n$ the event "t... | BallBoy | 512,865 | <p>The approach seems correct. There are a couple of different ways to get to the terms in the sum, and rearrange the sum, but no significantly more elegant method that I see.</p>
|
317,753 | <p>I am taking real analysis in university. I find that it is difficult to prove some certain questions. What I want to ask is:</p>
<ul>
<li>How do we come out with a proof? Do we use some intuitive idea first and then write it down formally?</li>
<li>What books do you recommended for an undergraduate who is studying ... | amWhy | 9,003 | <p>While this doesn't speak, directly, to Real Analysis, it is a recommendation that will help you there, and in other courses you're encounter, or will encounter soon:</p>
<p>In terms of both reading and writing proofs, in general, an excellent book to work through and/or have as a reference is Velleman's great text ... |
1,005,576 | <p>How can I write this term in a compact form where $a$ only appears once on the RHS (in particular without cases)?</p>
<p>$T(a) =
\begin{cases}
a^2 &,\text{ if $a \leq 0$}\\
2a^2 &,\text{ if $a > 0$}\\
\end{cases}$</p>
<p>I have already thought about $T(a) = \max\{\sqrt{2}a,|a|\}^2$ or $T(a)... | matheburg | 155,537 | <p>Another class of solutions to this problem is the "trivial substitution class".</p>
<p><strong>Examples</strong></p>
<p>$$T(a) = 2\int_0^a x\cdot(\max\{x,0\}+1)dx$$</p>
<p>or even more trivial</p>
<p>$$T(a) = \max\{\sqrt{2}x,|x|\}^2\bigg|_{x=a}$$</p>
|
22,101 | <p>The general rule used in LaTeX doesn't work: for example, typing <code>M\"{o}bius</code> and <code>Cram\'{e}r</code> doesn't give the desired outputs.</p>
| Community | -1 | <p>I also wanted to type Möbius, etc, just as in LaTeX, by typing <kbd>M</kbd><kbd>\</kbd><kbd>"</kbd><kbd>o</kbd>... so I've made a <a href="https://github.com/normalhuman/MathShortcuts2" rel="nofollow">userscript</a> for that. To use it,</p>
<ol>
<li>Install a userscript manager (e.g., Tampermonkey extension for Chr... |
22,101 | <p>The general rule used in LaTeX doesn't work: for example, typing <code>M\"{o}bius</code> and <code>Cram\'{e}r</code> doesn't give the desired outputs.</p>
| Rob | 510,296 | <p>Slightly less ugly than one answer offered is: \ddot{\mathsf a} <span class="math-container">$\ddot{\mathsf a}$</span> - but it's slightly bold and for all the extra typing it's better to find a webpage with the characters and copy/paste them (assuming you're using a cellphone or a keyboard without the characters).<... |
1,845,663 | <p>$\left\{a,b,c\right\}\in \mathbb{R}^3$ are linearly independent vectors.</p>
<p>Find the value of $\lambda $, so the dimension of the subspace generated by the vectors:</p>
<p>$2a-3b,\:\:\left(\lambda -1\right)b-2c,\:\:3c-a,\:\:\lambda c-b$ is 2.</p>
<p>So, if I understand this correctly the span of the given vec... | Zau | 307,565 | <p>As Michael said, by using row reduction:</p>
<p>$$\begin{pmatrix}2&0&-1&0\\ -3&\lambda -1&0&-1\\ 0&-2&3&\lambda \end{pmatrix}$$</p>
<p>$$ \iff \begin{pmatrix}2&0&-1&0\\ 0&\lambda -1&-\frac{3}{2}&-1\\ 0&-2&3&\lambda \end{pmatrix} $$</p>
<p>$$ ... |
1,845,663 | <p>$\left\{a,b,c\right\}\in \mathbb{R}^3$ are linearly independent vectors.</p>
<p>Find the value of $\lambda $, so the dimension of the subspace generated by the vectors:</p>
<p>$2a-3b,\:\:\left(\lambda -1\right)b-2c,\:\:3c-a,\:\:\lambda c-b$ is 2.</p>
<p>So, if I understand this correctly the span of the given vec... | Marc van Leeuwen | 18,880 | <p>I'd say column reduction is easier here, since there are already two linearly independent columns that do not involve $\lambda$ at all; the remaining columns must be linear combinations of them. Column-reduction of $A$ (starting with moving the second column to the end) gives
$$
A'=\begin{pmatrix}2&0&0&... |
878,115 | <p>Question1:
I found 30 boxes. In 10 boxes i found 15 balls. In 20 boxes i found 0 balls.
Afer i collected all 15 balls i put them randomly inside the boxes.</p>
<p>How much is the chance that all balls are in only 10 boxes or less?</p>
<p>Question2:
I found 30 boxes. In 10 boxes i found 15 balls. In 20 boxes i foun... | hardmath | 3,111 | <p>Random trials/Monte Carlo simulations are notoriously slow to converge, with an expected error inversely proportional to the square root of the number of trials.</p>
<p>In this case it is not hard (given a programming language that provides big integers) to do an exact count of cases. Effectively the outcomes are ... |
927,261 | <p>I was doing a presentation on Limits and I was using this $$f(x)=\frac{x^2+2x-8}{x^2-4}$$ to explain different types of limits. </p>
<p>I know that the function is not defined at $x=-2$ or $x=2$. I showed the graph and everyone was ok with the graph at $x=-2$ but one member of the audience didn't like how the grap... | acer | 12,448 | <p>As mentioned, you are seeing artefacts of floating-point computation. These can be alleviated by increasing the working precision, by adjusting the <code>Digits</code> environment variable.</p>
<p>The default value of <code>Digits</code> is 10. Also, for an expression containing only arithmetic operations (and elem... |
1,766,264 | <p>A store sells 8 kinds of candy. How many ways can you pick out 15 candies total to throw unordered into a bag and take home.</p>
<p>here 15 candies..
so we choose 8 from out of 15 is ..=$^{15}C_8$ is i am right</p>
| André Nicolas | 6,312 | <p>Call the various types of candy Type 1, Type 2, and so on up to Type 8. Let $x_1$ be the number of Type 1 candies we get, $x_2$ the number of Type 2 candies we get, and so on up to $x_8$.</p>
<p>Then the $x_i$ are non-negative integers, and $x_1+x_2+\cdots +x_8=15$.</p>
<p>Conversely, if $x_1,x_2,\dots, x_8$ are n... |
718,266 | <p>Is there a simple intuitive graphical explanation of Clifford Algebra for the layman? Since Clifford Algebra is a Geometric Algebra, surely the best way to present those concepts is with graphical figures.</p>
| Paul Siegel | 1,509 | <p>I'm not sure I agree with the premise of the question; I would say that the point of introducing Clifford algebras is to work with certain geometric data that cannot be easily visualized.</p>
<p>But Clifford algebras do make contact with conventional geometry via the twisted adjoint representation. Given a unit ve... |
4,272,964 | <p>I want to solve the equation following in a set of complex numbers:</p>
<p><span class="math-container">$$z^2 + \bar z = \frac 1 2$$</span></p>
<p><strong>My work so far</strong></p>
<p>Apparently I have a problem with transforming equation above into form that will be easy to solve. I tried to multiply sides by <sp... | Reveillark | 122,262 | <p>Write <span class="math-container">$z=x+iy$</span>, so <span class="math-container">$z^2=x^2-y^2+i2xy$</span>. So, equating real and imaginary parts,
<span class="math-container">$$
x^2-y^2+x=\frac{1}{2}
$$</span>
and
<span class="math-container">$$
2xy-y=0
$$</span>
So this means that <span class="math-container">$... |
1,793,182 | <p>My task was to find the directional derivative of function:<br>
$$z = y^2 - \sin(xy)$$ at the point $(0, -1)$ in direction of vector $u = (-1, 10) $. </p>
<p>The result I found was $-21/\sqrt{101}$. But I can't figure out what is the interpretation of this result. </p>
<p>Does it mean that the function grows fa... | Venkata Karthik Bandaru | 303,300 | <p>[This is from Prop 2.3.4 of "A Course in Metric Geometry" by Burago-Burago-Ivanov. As in the book, <span class="math-container">${ \vert p q \vert }$</span> stands for <span class="math-container">${ d(p,q) }$</span>]</p>
<p><strong>Def</strong>: Let <span class="math-container">${ (X, d) }$</span> be a me... |
1,865,364 | <p>After having seen a lengthy and painful calculation showing
$\operatorname{Gal}(\mathbb Q[e^\frac{2\pi i}3, \sqrt[\leftroot{-2}\uproot{2}3]{2}]/\mathbb Q)\cong S_3$, I'm wondering whether there's a slick proof $\operatorname{Gal}(\mathbb Q[e^\frac{2\pi i}p, \sqrt[\leftroot{-2}\uproot{2}p]{2}]/\mathbb Q)\cong S_p$ fo... | M. Van | 337,283 | <p>Here is an 'easy' group it is isomorphic to:</p>
<p>$$
\left\{\begin{pmatrix}
a & b \\ 0 & 1
\end{pmatrix} : a, b \in \mathbb{F}_p, a \neq 0 \right\}
$$
with the following isomorphism. If $\sigma \in \text{Gal}(\mathbb{Q}(\zeta,\sqrt[p]{2}))$ with $\sigma(\zeta)= \zeta^a$ and $ \sigma ( \sqrt{2} ) = \zeta^... |
2,943,461 | <p>I'm stumped on a math puzzle and I can't find an answer to it anywhere!
A man is filling a pool from 3 hoses. Hose A could fill it in 2 hours, hose B could fill it in 3 hours and hose C can fill it in 6 hours. However, there is a blockage in hose A, so the guy starts by using hoses B and C. When the blockage in hose... | farruhota | 425,072 | <blockquote>
<p>But the blockage in hose A is still bothering me, does it make a difference?</p>
</blockquote>
<p>Finetuning MRobinson's solution. </p>
<p>Let the pool can fit <span class="math-container">$x$</span> units of water. </p>
<p>Let the rates of hoses be: <span class="math-container">$r_A=\frac{x}{2}; r... |
4,351,504 | <p>A question from Herstein's Abstract Algebra book goes-</p>
<blockquote>
<p>Let <span class="math-container">$(R,+,\cdot)$</span> be a ring with unit element. Using its elements we define a ring <span class="math-container">$(\tilde R,\oplus,\odot)$</span> by defining <span class="math-container">$a\oplus b = a + b +... | Jan Eerland | 226,665 | <p>Well, we are trying to find:</p>
<p><span class="math-container">$$\text{y}_\text{k}\left(\text{n}\space;x\right):=\mathscr{L}_\text{s}^{-1}\left[-\sqrt{\frac{\text{k}}{\text{s}}}\cdot\exp\left(-\text{n}\cdot\sqrt{\frac{\text{s}}{\text{k}}}\right)\right]_{\left(x\right)}\tag1$$</span></p>
<p>Using the linearity of t... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.