qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
3,192,068 | <p><span class="math-container">\begin{matrix}
1 & 2 & 0 & 1 \\
2 & 4 & 1 & 4 \\
3 & 6 & 3 & 9 \\
\end{matrix}</span>
I have tried to transpose it and then reduce it by row echelon form and i get zeros on the last two rows. But i can't grasp if i should be doing that ... | Mathbeginner | 506,526 | <p>In general, the length of this curve is given by
<span class="math-container">$\int_{-\pi}^{\pi} \sqrt{(x')^{2}+(y')^{2}}dt$</span>, which in this case leads to
<span class="math-container">$$\int_{-\pi}^{\pi} \sqrt{(x')^{2}+(y')^{2}}dt = \int_{-\pi}^{\pi} \sqrt{(-3\cos(3t))^{2}+(2)^{2}}dt = \int_{-\pi}^{\pi} \sqrt{... |
1,700,608 | <p>In <a href="https://math.stackexchange.com/a/1700505/132192">this</a> answer the value of $1 - \cos(x)$ has to be evaluated in order to find its upper limit, if it exists.</p>
<p>In particular, $x = 2 \pi / n$. The answer is related to the length of a side of a regular $n$-gon inscribed into a unit-radius circumfer... | ncmathsadist | 4,154 | <p>Let $x\gt 0$. You know that $\sin(x) < x$ for such $x$. Integrating
$$ 1 - \cos(x) =\int_0^x \sin(t)\, dt < \int_0^x t\,dt = {t^2\over 2}$$</p>
|
194,664 | <p>How to generate a list of fixpoint free permutations of n elements in mathematica?</p>
| Roman | 26,598 | <p>As your equation is <code>Sinc[a] == x</code> the formal solution is</p>
<pre><code>A = InverseFunction[Sinc];
</code></pre>
<p>You can plot it with</p>
<pre><code>Plot[A[x], {x, -0.21723362821122166`, 1}]
</code></pre>
<p>but as you see in the result you get a random branch of the solution:</p>
<p><a href="htt... |
194,664 | <p>How to generate a list of fixpoint free permutations of n elements in mathematica?</p>
| cphys | 63,840 | <p>In order to find all of the roots for <span class="math-container">$x$</span> in range <span class="math-container">$\{0,1\}$</span>, (without placing a limit on the range of <span class="math-container">$a$</span>), you should use <a href="https://reference.wolfram.com/language/ref/Reduce.html" rel="nofollow norefe... |
2,061,363 | <p>I have the complex power series $ \sum_{k=1}^{\infty}(\frac{z^4}{4} - \frac{\pi}{7})^k$. </p>
<p>Through algebraic manipulation I obtain $ \sum_{k=1}^{\infty}(\frac{1}{4})^k(z^4 - \frac{4}{7}\pi)^k$. I now argue that this is a power series around $\frac{4}{7}\pi$ with radius of convergence R = 4, using the euler ro... | Learnmore | 294,365 | <p>Put $t=z^4$ so the series is $\sum_{k=0}^\infty \dfrac{1}{4^k}(t-t_0)^k$ where $t_0=\dfrac{4\pi}{7}$</p>
<p>Then the radius of convergence $R=\lim \sup \dfrac{a_k}{a_{k+1}}=4$ where $a_k=\dfrac{1}{4^k}$</p>
<p>Hence the series converges $\forall t$ such that $|t|<4\implies |z|^4<4\implies |z|<\sqrt 2$</p>... |
3,102,336 | <p>I have been looking for fixed points of <a href="https://simple.wikipedia.org/wiki/Riemann_zeta_function" rel="nofollow noreferrer">Riemann Zeta function</a> and find something very interesting, it has two fixed points in <span class="math-container">$\mathbb{C}\setminus\{1\}$</span>.</p>
<p>The first fixed point i... | George Lamprou | 689,811 | <p>Hmm...i did a run in my computer cause i found you question of fixed points interesting so..</p>
<p>the only result i got is this for <span class="math-container">$a=1.8337719154395\cdots$</span> and for <span class="math-container">$b=0$</span></p>
<p><span class="math-container">$\zeta(1.8337719154395\cdots)=1.83... |
1,241,695 | <p>I was reading A First Course in Probability by Sheldon Ross. I think I quite understood the below problem but I still feel fuzzy.</p>
<blockquote>
<p><strong>Problem</strong>: In answering on a multiple choice test, a student either know the answer or guesses. Let p be the probability that the students knows the ans... | paw88789 | 147,810 | <p>Imagine that the test consists of $N$ questions, each with the same parameter $p$ of the student knowing the right answer; and assume that knowing the right answer on any question is independent of knowing the right answer on any other question.</p>
<p>In this scenario, each question will fall into one of three cat... |
1,241,695 | <p>I was reading A First Course in Probability by Sheldon Ross. I think I quite understood the below problem but I still feel fuzzy.</p>
<blockquote>
<p><strong>Problem</strong>: In answering on a multiple choice test, a student either know the answer or guesses. Let p be the probability that the students knows the ans... | wltrup | 232,040 | <p>Apologies for the long post but this material can be confusing so I tried to be clear by explaining every step.</p>
<ol>
<li>Probability that the student knows the answer $= P(KC) = p$ but not $P(C|K)$.</li>
</ol>
<p>It's not. The probability that the student knows the answer is $P(K)$, not $P(KC)$. The probabilit... |
1,265,026 | <p>Suppose I have some vector field
\begin{align}
\vec{F}\left(x\left(t\right),y\left(t\right),z\left(t\right)\right)&=G\textbf{i}+H\textbf{j}+T\textbf{k}.\tag{1}
\end{align}
Would it be correct for me to say
\begin{align}
\mathbb{R}^3\overset{\vec{F}}{\longrightarrow}\mathbb{R}^3\;?\tag{2}
\end{align}</p>
| Thomas | 26,188 | <p>Writing
$$
\mathbb{R}^3\overset{\vec{F}}{\longrightarrow}\mathbb{R}^3
$$
I would think that $\vec{F}$ is a function with domain $\mathbb{R}^3$ and it looks like you have a function with domain $\mathbb{R}$. So the notation isn't good. I also don't think it is a good idea to write $\vec{F}_t: \mathbb{R}^3 \to \mathbb... |
1,271,942 | <p>I am a little bit confused with the definition of finitely presented modules. In Lang's <em>Algebra</em> he defines a module <span class="math-container">$M$</span> to be finitely presented if and only if there is a exact sequence <span class="math-container">$F'\to F\to M \to 0$</span> such that both <span class="m... | Martin Brandenburg | 1,650 | <p>If $F' \to F \to M \to 0$ is exact and $F'$ is finitely generated, choose some finitely generated free module $F''$ which maps <em>onto</em> $F'$. Then $F'' \to F \to M \to 0$ is exact.</p>
<p>This shows: A finitely generated module is finitely related iff it is finitely presented.</p>
<p>Of course, this fails for... |
2,207,572 | <p>Imagine an undirected graph $G = (V,E)$ with $|V| = n$ nodes. Its unweighted edges $E$ are the union of $h$ random Hamiltonian cycles through all nodes, each generated iid uniformly at random from the set of all Hamiltonian cycles.</p>
<p>What is the expected diameter $D$ of $G$?</p>
<p>The case $h=1$ is trivial a... | D.W. | 14,578 | <p>Purely <em>heuristically</em>, I expect the answer to be $O(\log(n)/\log(h))$.</p>
<p>Why? We can imagine that each vertex has an edge to $2h$ randomly chosen other vertices. Then heuristically we can imagine that there are about $(2h)^d$ vertices at distance $\le d$ from a fixed vertex $v$ (as long as $(2h)^d$ i... |
2,207,572 | <p>Imagine an undirected graph $G = (V,E)$ with $|V| = n$ nodes. Its unweighted edges $E$ are the union of $h$ random Hamiltonian cycles through all nodes, each generated iid uniformly at random from the set of all Hamiltonian cycles.</p>
<p>What is the expected diameter $D$ of $G$?</p>
<p>The case $h=1$ is trivial a... | Misha Lavrov | 383,078 | <p>At least for <span class="math-container">$h$</span> constant, taking the union of <span class="math-container">$h$</span> uniformly random Hamiltonian cycles is maybe kind of equivalent to taking a uniformly random <span class="math-container">$2h$</span>-regular graph, whose properties as <span class="math-contain... |
2,634,791 | <blockquote>
<p>How can I show that the map $f: GL_n(\mathbb R)\to GL_n(\mathbb R)$ defined by $f(A)=A^{-1}$ is continuous?</p>
</blockquote>
<p>The space $GL_n(\mathbb R)$ is given the operator norm and so I want to show for all $\epsilon$ there exists $\delta$ such that $\|A-B\|<\delta \implies \|A^{-1}-B^{-1}\... | C. Falcon | 285,416 | <p>One has a formula for the inverse, namely:
$$A^{-1}=\frac{1}{\det(A)}\operatorname{adj}(A),$$
where $\operatorname{adj}(A)$ is the <a href="https://en.wikipedia.org/wiki/Adjugate_matrix" rel="nofollow noreferrer">adjugate</a> of $A$. Whence, $A^{-1}$ is a rational fraction in the coefficients of $A$.</p>
<hr>
<p>A... |
1,993,217 | <p>Let $\left\{f_{n}\right\}$ be a sequence of equicontinuous functions where $f_n: [0,1] \rightarrow \mathbf{R}$. If $\{f_n(0)\}$ is bounded, why is $\left\{f_{n}\right\}$ uniformly bounded?</p>
| Eugene Zhang | 215,082 | <p>Since $[0,1]$ is compact, $f_n$ is uniform continuous. So given $\epsilon>0$, there is a $\delta'>0$ such that for any $x,y\in [0,1], \:|x-y|<\delta'$
$$
|f_n(x)-f_n(y)|<\epsilon\tag1
$$
Since $f_n$ is equicontinuous, $(1)$ holds for all $n$. </p>
<p>Take $\delta=\delta'/2$. For any open cover on $\bigc... |
130,028 | <p>I often want to have the same code at the beginning of every new notebook. Is it possible to configure Mathematica, such that whenever you create a new notebook some user-defined code will always be created with the new document.</p>
<p>E.g. commonly used plot configurations, packages, directory setting etc.</p>
<... | c186282 | 4,515 | <p>Place the code in your init.m file. It will then be run each time the kernel is started. On Linux the init.m file is in ~/.Mathematica/Kernel/. I forget where it is on Windows but just look at the output of <code>$Path</code> and you will be able to find the "Kernel" directory within your account directories. </p>
|
3,883,164 | <p>I evaluated following limit with taylor series but for a practice I am trying to evaluate it using L'Hopital's Rule:</p>
<p><span class="math-container">$$\lim_{x\to 0}\frac{\sinh x-x\cosh x+\frac{x^3}3}{x^2\tan^3x}=\lim_{x\to0}\cfrac{f(x)}{g(x)}$$</span>
<span class="math-container">$f(x)=\sinh x-x\cosh x+\frac{x^3... | Community | -1 | <p>The denominator is of the fifth degree (after linearizing the tangent), so if there is a finite answer you will need five successive applications of L'Hospital.</p>
<p>The numerator is easy:</p>
<p><span class="math-container">$$\sinh x-x\cosh x+\frac{x^3}3,$$</span></p>
<p><span class="math-container">$$-x\sinh x+x... |
1,392,576 | <p>How can the followin question be solved algebraically?</p>
<p>A certain dealership has a total of 100 vehicles consisting of cars and trucks. 1/2 of the cars are used and 1/3 of the trucks are used. If there are 42 used vehicles used altogether, how many trucks are there?</p>
| haqnatural | 247,767 | <p>$$\lim _{ x\rightarrow 0 }{ \frac { { x }^{ 2 }+2\sqrt { { x }^{ 2 } } }{ x } } =\lim _{ x\rightarrow 0 }{ \frac { { x }^{ 2 }+2\left| x \right| }{ x } } =\lim _{ x\rightarrow 0 }{ \left( x+2\frac { \left| x \right| }{ x } \right) } \\ \lim _{ x\rightarrow 0- }{ \left( x+2\frac { \left| x \right| }{ x } \ri... |
1,392,576 | <p>How can the followin question be solved algebraically?</p>
<p>A certain dealership has a total of 100 vehicles consisting of cars and trucks. 1/2 of the cars are used and 1/3 of the trucks are used. If there are 42 used vehicles used altogether, how many trucks are there?</p>
| LeastSquaresWonderer | 233,263 | <p>$$ \lim_{ x \to 0} \frac{x^2 + 2\sqrt{x^2}}{x}. $$</p>
<p>$$ \lim_{ x \to 0} \frac{x^2 + 2|x|}{x}. $$</p>
<p>We will look at the limits aproaching 0 from both sides</p>
<p>from the left -> 0^-
$$ \lim_{ x \to 0} \frac{x^2 + 2x}{-x}. $$</p>
<p>$$ \lim_{ x \to 0} -(x + 2) = -2 $$</p>
<p>from the right -> 0^2
$$ ... |
73,238 | <p>How can I calculate the solid angle that a sphere of radius R subtends at a point P? I would expect the result to be a function of the radius and the distance (which I'll call d) between the center of the sphere and P. I would also expect this angle to be 4π when d < R, and 2π when d = R, and less than 2π when d ... | J. M. ain't a mathematician | 498 | <p>For this answer, I make a few (not too drastic) assumptions:</p>
<ol>
<li><p>The circle you are interested in is centered at the origin (thus, the plane your circle lies in has to pass through the origin).</p></li>
<li><p>The plane your circle lies in is already in <a href="http://mathworld.wolfram.com/HessianNorma... |
73,238 | <p>How can I calculate the solid angle that a sphere of radius R subtends at a point P? I would expect the result to be a function of the radius and the distance (which I'll call d) between the center of the sphere and P. I would also expect this angle to be 4π when d < R, and 2π when d = R, and less than 2π when d ... | SadMakesMeCalcface | 84,846 | <p>If your circle has a unit normal vector < cos(a), cos(b), cos(c)> then depending on c, you have:</p>
<p>< cos(t-arcsin(cos(b)/sin(c)))/sqrt(sin(t)^2 +cos(t)^2*sec(c)^2), sin(t-arcsin(cos(b)/sin(c)))/sqrt(sin(t)^2 +cos(t)^2*sec(c)^2), cos(t)*sin(c)/sqrt(cos(t)^2+sin(t)^2*cos(c)^2> 0
<p>< sin(t)*sin(a), -s... |
858,494 | <p>Where does the definition of the $L_\infty$ norm come from?</p>
<p>$$\|x\|_\infty=\max \{|x_1|,\dots,|x_k|\}$$</p>
| Oria Gruber | 76,802 | <p>The $p$ norm of a vector is defined as such:</p>
<p>$\|x\|_p = (\sum_{i=1}^{n}|x_i|^p)^\frac{1}{p}$.</p>
<p>Notice that when $p=2$ this is the simple euclidean norm.</p>
<p>You asked about the infinity norm.</p>
<p>When $p$ tends to infinity, we can see that:</p>
<p>$$\lim_ {p \to \infty} \|x\|_p = \lim_ {p \to... |
10,974 | <p>Is the following true: If two chain complexes of free abelian groups have isomorphic homology modules then they are chain homotopy equivalent.</p>
| Tyler Lawson | 360 | <p>Yes, this is true. Suppose $C_*$ is such a chain complex of free abelian groups.</p>
<p>For each $n$, choose a splitting of the boundary map $C_n \to B_{n-1}$, so that $C_n \cong Z_n \oplus B_{n-1}$. (You can do this because $B_{n-1}$, as a subgroup of a free group, is free.) For all $n$, you then have a sub-cha... |
3,329,363 | <p>Sorry for the long text; this is a nebulous question that has always been in the back of my mind, and I've had trouble putting into a short form.</p>
<hr>
<p><strong>"Natural" Definition</strong></p>
<p>If someone on the street hears the word "permutation," I think they will naturally assume that a permutation:</... | robjohn | 13,854 | <p>"A bijective map from a set to itself" does not require the set to be ordered, but when applied to an ordered set, this map acts to reorder the set.</p>
<p>This definition is therefore a generalization of the idea of "reordering an ordered set" to a more general setting.</p>
<p>Often, in mathematics, a name lifts ... |
1,627,357 | <p>Is there a simple way to prove $$\frac{1}{\sqrt{1-x}} \le e^x$$ on $x \in [0,1/2]$?</p>
<p>Some of my observations from plots, etc.:</p>
<ul>
<li>Equality is attained at $x=0$ and near $x=0.8$.</li>
<li>The derivative is positive at $x=0$, and zero just after $x=0.5$. [I don't know how to find this zero analytical... | RRL | 148,510 | <p>We have</p>
<p>$$-2 \ln \sqrt{1-x}=-\ln(1-x)= \int_{1-x}^1\frac{dt}{t} \leqslant \frac{x}{1-x}.$$</p>
<p>For $0 \leqslant x \leqslant 1/2$, we have $2(1-x) \geqslant 1$ and </p>
<p>$$-\ln \sqrt{1-x} < \frac{x}{2(1-x)} \leqslant x.$$</p>
<p>Hence,</p>
<p>$$\frac{1}{\sqrt{1-x}} = \exp[-\ln(\sqrt{1-x})]\leqslan... |
3,026,097 | <p>I was studying an article where I encountered <span class="math-container">$\mathbb{R}^E_{\gt 0}$</span>. I couldn't find out what does this notation mean exactly.
I'm sorry if my question is basic, I searched this community but I didn't find the answer to my question.</p>
<p>Here is a part of that article:</p>
<b... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
323,128 | <p>Show that in every (not necessarily connected) graph there is a path from every vertex $u$ of odd degree to some other vertex $v$ ($u \neq v$), also of odd degree.</p>
| joriki | 6,622 | <p>If there isn't then $u$ is in a connected component consisting of itself and vertices of even degree. But then the sum of degrees in that connected component is odd, which it can't be, since it counts every edge twice.</p>
|
323,128 | <p>Show that in every (not necessarily connected) graph there is a path from every vertex $u$ of odd degree to some other vertex $v$ ($u \neq v$), also of odd degree.</p>
| hardmath | 3,111 | <p>The conclusion holds in finite graphs, but no finiteness assumption was stated. For an infinite counterexample consider the natural numbers with edges connecting each number and its successor. Only one vertex has odd degree.</p>
|
3,757,763 | <p>Let <span class="math-container">$T: V\rightarrow V$</span> be a linear operator of the vector space <span class="math-container">$V$</span>.</p>
<p>We write <span class="math-container">$V=U\oplus W$</span>, for subspaces <span class="math-container">$U,W$</span> of <span class="math-container">$V$</span>, if <span... | Batominovski | 72,152 | <p>Let <span class="math-container">$\mathbb{K}$</span> be the base field. If <span class="math-container">$T:V\to V$</span> is such that <span class="math-container">$\ker(T)\cap\text{im}(T)=0$</span> and there exists <span class="math-container">$p(X)\in\mathbb{K}[X]$</span> such that <span class="math-container">$p... |
917,302 | <p>If $p(x)$ is a polynomial of degree 4 such that $p(2)=p(-2)=p(-3)=-1$ and $p(1)=p(-1)=1$, then find $p(0)$.</p>
| lab bhattacharjee | 33,337 | <p>Let $\displaystyle p(x)=(Ax+B)(x-2)(x+2)(x+3)-1,$ where $A,B$ are arbitrary finite constants</p>
<p>$$p(0)=(+B)(-2)(+2)(+3)+1$$</p>
<p>Set $x=1,-1$ one by one to find $B$</p>
|
917,302 | <p>If $p(x)$ is a polynomial of degree 4 such that $p(2)=p(-2)=p(-3)=-1$ and $p(1)=p(-1)=1$, then find $p(0)$.</p>
| Kelenner | 159,886 | <p>Put $q(x)=p(x)-p(-x)$. Then $p$ is an odd polynomial, of degree $\leq 3$. Hence $p(x)-p(-x)=xq(x^2)$ with degree of $q\leq 1$, say $q(y)=ay+b$. We have $q(1)=q(4)=0$, hence as degree of $q$ $\leq 1$, $q=0$ and $p(x)=p(-x)$. We get $p(-3)=p(3)=-1$, and so $p(x)=c(x^2-4)(x^2-9)-1$ for a constant $c$, and we finish eas... |
917,302 | <p>If $p(x)$ is a polynomial of degree 4 such that $p(2)=p(-2)=p(-3)=-1$ and $p(1)=p(-1)=1$, then find $p(0)$.</p>
| Community | -1 | <p>For later simplification, consider $p_1(x)=p(x)+1$, that has three known roots. We see that $p_1(x)$ is an even function, so that $p_1(x)=ax^4+bx^2+c=a(x^2)^2+bx^2+c=q(x^2)$.</p>
<p>$q(x^2)$ is of the <strong>second degree</strong> in $x^2$, and much easier to interpolate.
We have $q(1)=2$, $q(4)=q(9)=0$, hence by... |
2,531,714 | <p>Given is: </p>
<p>$(1-x^2)\dfrac{d^2y(x)}{dx^2} + 2x\dfrac{dy(x)}{dx} - 2y(x) = 0 $</p>
<p>The Solution is: </p>
<p>$y(x) =C_1x + C_2(x^2+1)$ </p>
<p>How do I factor the $x$ out in order to get it into a normal "linear" form that contains only coefficients to show that the solution is valid? </p>
<p>Edit: The e... | Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\... |
3,884,581 | <p>Please don't just throw an answer at me, please explain how you arrived at it cause I've been fiddling with this for the past 30min...</p>
| Deepak | 151,732 | <p>Hints:</p>
<p><span class="math-container">$a^2 + b^2 + 2ab = (a+b)^2$</span></p>
<p>and <span class="math-container">$a^2 + b^2 - 2ab = (a-b)^2$</span></p>
<p>You can easily find both <span class="math-container">$a+b$</span> and <span class="math-container">$a-b$</span> and you're left with only linear simultaneo... |
2,861,949 | <blockquote>
<p>Use the method of least squares in order to find the best approximation
to a solution for the system
$$3x + y = 1\\
x − y = 2\\
x + 3y = −1$$</p>
</blockquote>
<p><strong>My Try:</strong>
$$Ax=B$$
$$\begin{bmatrix} 1 & 1 \\ 1 & -1 \\ 1 & 3 \end{bmatrix}\begin{bmatrix} x \\ y \end{bmat... | John Bentin | 875 | <p>Analysis of the posted text is complicated by it being a faulty translation from the original Italian. However, some features appear to be a fair representation of the author's writing, and some of those are nonstandard. The notation $f : X ⊆ S → T$ is nonstandard. We would normally write $f : X → T$, with $X ⊆ S$ ... |
2,861,949 | <blockquote>
<p>Use the method of least squares in order to find the best approximation
to a solution for the system
$$3x + y = 1\\
x − y = 2\\
x + 3y = −1$$</p>
</blockquote>
<p><strong>My Try:</strong>
$$Ax=B$$
$$\begin{bmatrix} 1 & 1 \\ 1 & -1 \\ 1 & 3 \end{bmatrix}\begin{bmatrix} x \\ y \end{bmat... | trying | 309,917 | <p>You need not be scared by terms used unconventionally as long as those terms are unambiguously defined by the Author. Yes, today <em>codomain</em> is a term that all agree to define in the same ways, but still some 30 years ago in non-English speaking countries <em>codomain</em> could be defined as the <em>image</em... |
2,231,092 | <p>I am reading <a href="http://people.ucalgary.ca/~rzach/static/open-logic/open-logic-complete.pdf" rel="nofollow noreferrer">Open Logic TextBook</a>. In which there is a proposition about Extensionality of first order sentences (6.12) It goes like this, </p>
<p>Let $\phi$ be a sentence, and $M$ and $M'$
be structure... | user12345 | 394,065 | <p><strong>Hint</strong>: First, find the equation of the line in Cartesian ($y=mx+b$ (shouldn't be very difficult)). You know the slope is $\frac{1}{7}=m$. They gave you another point that it goes through/satisfies. Use that to find $b$ and you'll have the equation of your line in Cartesian Coordinates. Then, use the... |
2,576,344 | <p>This problem is about expected value, and it's a real world problem.</p>
<p>I know so far that $f$ is strictly increasing, if that makes the proof more concise (but if you can also prove it without this assumption, that would be awesome). Find all solutions for $f$ when $f(P_a \cdot a+P_b \cdot b)=P_a \cdot f(a)+P_... | Przemysław Scherwentke | 72,361 | <p>HINT: For all $P_a$, $P_b\in\mathbb{R}$ and all $\lambda\in[0,1]$ your condition gives that the fuction is convex and concave, so its graph is an interval with $x$-ends $P_a$ and $P_b$.</p>
|
1,301,522 | <p>Many texts will define a manifold as "a second-countable Hausdorff space that is locally homeomorphic to Euclidean space".
By definition of homeomorphism, shouldn't this really and officially read as "locally homeomorphic to a <em>subset</em> of Euclidean space"?</p>
| nullUser | 17,459 | <p>Note that $B(0,1) \simeq \mathbb{R}^n$. Can you answer your own question now?</p>
|
279,707 | <p>$p \land \lnot q \lor q \land \lnot r \lor \lnot p \lor r $
$\equiv$$(p \lor \lnot p) \land (\lnot q \lor q) \land (\lnot r \lor r)$</p>
<p>Is this move "legal"? Or can you only apply the associative property on like operators? </p>
| mjqxxxx | 5,546 | <p>No, mixed expressions like this are not associative; instead, they obey distributive laws:
$$
(a\wedge b)\vee c \equiv (a\vee c) \wedge (b \vee c)
$$
and
$$
a \wedge (b\vee c) \equiv(a\wedge b) \vee (a \wedge c).
$$</p>
|
279,707 | <p>$p \land \lnot q \lor q \land \lnot r \lor \lnot p \lor r $
$\equiv$$(p \lor \lnot p) \land (\lnot q \lor q) \land (\lnot r \lor r)$</p>
<p>Is this move "legal"? Or can you only apply the associative property on like operators? </p>
| amWhy | 9,003 | <p><strong>Associativity</strong> applies only when the connectives involved are exclusively $\land$ or exclusively $\lor$:</p>
<p>$$p \land q \land r \equiv (p \land q)\land r \equiv p \land (q\land r)$$</p>
<p>$$p \lor q \lor r \equiv (p \lor q)\lor r \equiv p \lor (q\lor r)$$</p>
<p>Because of associativity of $\... |
1,919,159 | <p>I don't get this step in proof of <a href="https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_theorem_(convex_hull)" rel="nofollow">Carathéodory's theorem (convex hull)</a>
Why:</p>
<blockquote>
<p>Suppose k > d + 1 (otherwise, there is nothing to prove). Then, the points $x_2 − x_1, ..., x_k − x_1$ are linearly... | Surb | 154,545 | <p>What is the cardinality of $\{x_2 − x_1, ..., x_k − x_1\}$? Now remember that there aren't any linearly independent set of cardinality greater than $d$ in $\Bbb R^d$.</p>
|
1,969,903 | <blockquote>
<p>a) Evaluate the one-dimensional Gaussian integral</p>
<p><span class="math-container">$I(a)$</span> = <span class="math-container">$\int_R exp(-ax^2)dx$</span>, <span class="math-container">$a>0$</span></p>
<p>b) evaluate the two-dimensional Gaussian integral using a)</p>
<p><span class="math-contai... | Karl | 880,798 | <p>Note the suggested answer in the original question is wrong. Should be:
<span class="math-container">$$I = \sqrt{\frac{\pi}{a}}$$</span></p>
|
40,572 | <p>Dummit and Foote, p. 204</p>
<p>They suppose that $G$ is simple with a subgroup of index $k = p$ or $p+1$ (for a prime $p$), and embed $G$ into $S_k$ by the action on the cosets of the subgroup. Then they say</p>
<p>"Since now Sylow $p$-subgroups of $S_k$ are precisely the groups generated by a $p$-cycle, and dist... | JavaMan | 6,491 | <p>First and foremost, your notation $c(9,39)$ should read $c(39,9)$. That is presumably a typo. Now:</p>
<p>The inclusion-exclusion principle helps you find the cardinality of the sets $A \cup B$, $A \cup B \cup C$, $A \cup B \cup C \cup D$, etc. In your case, you have to find the cardinality of the set $A \cup B$... |
2,221,897 | <p>Show that </p>
<p>$$\lim_{n \to \infty} \sum_{k=3}^n \frac{2k}{k^2+n^2+1} = \ln(2)$$</p>
<p>How many ways are there to prove it ?</p>
<p>Is there a standard way ?</p>
<p>I was thinking about making it a Riemann sum.
Or telescoping.</p>
<p>What is the easiest way ?
What is the shortest way ?</p>
| RRL | 148,510 | <p>Note that</p>
<p>$$\sum_{k=3}^n \frac{2k}{k^2+n^2+1} = \sum_{k=1}^n \frac{2k}{k^2+n^2+1} - \frac{2}{2+n^2} - \frac{4}{5+n^2}.$$</p>
<p>We can ignore the last two terms since they converge to $0$.</p>
<p>Consider</p>
<p>$$\sum_{k=1}^n \frac{2k}{k^2+n^2+1} = \frac{1}{n}\sum_{k=1}^n \frac{2(k/n)}{1+(k/n)^2 + (1/n^2... |
995,775 | <p>So we have the regular $\delta$-$\epsilon$ definition of continuity as: </p>
<p>(1) For all $\epsilon>0$, there exists a $\delta>0$ such that, if $|x-a|<\delta$, then $|f(x)-f(a)|<\epsilon$.</p>
<p>My question is why is the following definition incorrect?</p>
<p>(2) There exists a $\delta>0$ such t... | hmakholm left over Monica | 14,366 | <p>The English sentence</p>
<blockquote>
<p>There exists $A$ such that $P(A,B)$ for all $B$.</p>
</blockquote>
<p>is <strong>ambiguous</strong> -- it can either mean $\exists A\forall B\,P(A,B)$ or $\forall B\exists A\,P(A,B)$, and there is no generally observed convention about which of them it ought to be underst... |
943,048 | <p><strong>Question:</strong></p>
<blockquote>
<p>let $x_{i}=1$ or $-1$,$i=1,2,\cdots,1990$, show that
$$x_{1}+2x_{2}+\cdots+1990x_{1990}\neq 0$$</p>
</blockquote>
<p>this problem it seem is easy,But I think is not easy. </p>
<p>I think note
$$1+2+3+\cdots+1990\equiv \pmod { 1990}?$$</p>
| Petite Etincelle | 100,564 | <p>suppose all the $x_i$ are $1$, then we have $$1+2+3+\cdots+1990 = \frac{1991\times 1990}{2}$$ is odd.</p>
<p>Then each time you change one of $x_i$ from $1$ to $-1$, you change the sum by an even number, so the sum is always odd</p>
|
33,153 | <p>Here is one definition of a differential equation:</p>
<blockquote>
<p>"An equation containing the derivatives of one or more dependent variables, with respect to one of more independent variables, is said to be a differential equation (DE)" <em>(Zill - A First Course in Differential Equations)</em></p>
</... | Qiaochu Yuan | 232 | <p>Arnold simply means that most books are not being precise. A slightly more precise version of the first few definitions is that a differential equation (in one variable) is an equation of the form $f(t, x, x', x'', ...) = 0$. This rules out Arnold's example. </p>
|
31,502 | <p>This is probably a trivial question, but I don't see the answer, and I haven't found it on <a href="http://en.wikipedia.org/wiki/Cartesian_closed_category" rel="nofollow noreferrer">Wikipedia</a>, <a href="http://ncatlab.org/nlab/show/cartesian+closed+category" rel="nofollow noreferrer">nLab</a>, nor <a href="https:... | BCnrd | 3,927 | <p>Set $A = B = k[x]$ and figure out for yourself what that is a counterexample. (Hint: rigorously prove that there's no "universal polynomial" over $k$-algebras.)</p>
|
2,706,776 | <p>In solving the wave equation
$$u_{tt} - c^2 u_{xx} = 0$$
it is commonly 'factored'</p>
<p>$$u_{tt} - c^2 u_{xx} =
\bigg( \frac{\partial }{\partial t} - c \frac{\partial }{\partial x} \bigg)
\bigg( \frac{\partial }{\partial t} + c \frac{\partial }{\partial x} \bigg)
u = 0$$</p>
<p>to get
$$u(x,t) = f(x+ct) + g(x-c... | Community | -1 | <p>Let us define $$ f(u)=\frac{\partial u}{\partial t}-c\frac{\partial u}{\partial x},\ g(u)=\frac{\partial u}{\partial t}-c\frac{\partial u}{\partial x} $$</p>
<p>Then, wave equation can be expressed by composition of two functions.</p>
<p>$$ f(g(u))=g(f(u))=\frac{\partial^2 u}{\partial t^2}-c^2\frac{\partial^2 u}{\... |
1,071,433 | <p>consider the region bounded by $ \displaystyle y=4{{x}^{2}}$ and $ \displaystyle 2x+y=6$. What is the volume of solid of revolution about $\displaystyle x$-axis.</p>
<p>What is thought about setting the integral:</p>
<p>I split the region into two parts</p>
<p>$\displaystyle V=4\pi \int\limits_{0}^{4}{y\left( 1-\... | MathMajor | 113,330 | <p>Below is the plot for $y = 4x^2$, and $y=6-2x:$</p>
<p><img src="https://i.stack.imgur.com/1JuF6.png" alt="enter image description here"></p>
<p>There is absolutely no need to separate $\mathfrak{R}$ into two regions. Find the intersection of the two curves and use $r = r_{out} - r_{in}$. Recall that </p>
<p>$$
\... |
1,071,433 | <p>consider the region bounded by $ \displaystyle y=4{{x}^{2}}$ and $ \displaystyle 2x+y=6$. What is the volume of solid of revolution about $\displaystyle x$-axis.</p>
<p>What is thought about setting the integral:</p>
<p>I split the region into two parts</p>
<p>$\displaystyle V=4\pi \int\limits_{0}^{4}{y\left( 1-\... | Ivo Terek | 118,056 | <p>I don't see why you need to split the region. Notice that $y = 4x^2$ and $y = 6 - 2x$ both intercept at: $$4x^2 = 6-2x \implies2x^2 + x - 3 =0 \implies x = \frac{-1\pm\sqrt{1+24}}{4} \implies x = \frac{-1\pm 5}{4},$$ so $x = -3/2$ and $x = 1$. Since the area of revolution of $y = f(x)$ around the $x$ axis is $$A = \... |
4,032,983 | <p>I would like to know math websites that are useful for students, PhD students and researchers (useful in the sense most of the students or researchers—of a particular area—are using it). Maybe you can share which math websites you sometime use and why you use it.</p>
<p>Let me give my websites and why I use them:</p... | Amarjeet Jayanthi | 250,743 | <p><a href="http://www.algebra.com" rel="nofollow noreferrer">http://www.algebra.com</a> is a very good site for algebra problems and solutions.</p>
|
870,240 | <p>Which number is larger? $\underbrace{888\cdots8}_\text{19 digits}\times\underbrace{333\cdots3}_\text{68 digits}$ or $\underbrace{444\cdots4}_\text{19 digits}\times\underbrace{666\cdots67}_\text{68 digits}$? Why? How much is it larger?</p>
| cirpis | 152,276 | <p>note that $$\underbrace{888\cdots8}_\text{19 digits}=2*\underbrace{444\cdots4}_\text{19 digits}$$
and $$2*\underbrace{333\cdots3}_\text{68 digits}=\underbrace{666\cdots6}_\text{68 digits}$$
further
$$\underbrace{666\cdots6}_\text{68 digits}+1=\underbrace{666\cdots7}_\text{68 digits}$$
Thus
$$\underbrace{888\cdots8}_... |
4,115,069 | <p>I understand 'functionals' as functions of functions, for example:</p>
<p><span class="math-container">$$ S[y(x)]= \int_{t_1}^{t_2} \sqrt{1+(y')^2} dx$$</span></p>
<p>Which is the famous arc length integral</p>
<p>Now, in a similar way, a limit we can write as:</p>
<p><span class="math-container">$$L(a, [y(x)] ) = \... | Son Gohan | 865,323 | <p>Why not? There are lots of examples in which old concepts can be seen as functionals or operators from some abstract space to another abstract space or the real numbers. And, indeed, now this seems obvious to us but this is the case just because we are used to functional analysis point of view, where it is natural t... |
16,627 | <p>Yesterday, I wrote <a href="https://math.stackexchange.com/a/904777">this answer</a>, but then realized that the OP had considered breaking things into specific cases, so I deleted my answer. Right after I deleted my answer, I saw that the OP had accepted my answer. I undeleted my answer and commented to the OP, ask... | Willie Wong | 1,543 | <p>As a side note:</p>
<blockquote>
<p>... and hope that it can be fixed so that others won't miss reputation they've earned.</p>
</blockquote>
<p>Deleting an <strong>accepted</strong> answer is a mod-only power. So that this potential bug only affects moderators. (Remember, except in very compelling circumstances ... |
1,448,416 | <p>It states that nth difference of a polynomial of n degree is constant thus (n+1)th difference will be zero.</p>
<ul>
<li>how can i show that the nth difference is constant? </li>
<li>forward difference of a constant is zero but how can i prove it?</li>
</ul>
| R.N | 253,742 | <p>Hint: i) Use induction and $f^{(n)}(x)=n!$ where $f(x)=x^n$</p>
<p>ii) For $f(x)=c$ you have $f'(x_0)=\lim_{h\to 0}\frac{f(x_0+h)-f(x_0)}{x-x_0}=\lim_{h\to 0}\frac{c-c}{x-x_0}=0$</p>
|
3,098,587 | <p>I have no answers to refer to, hence, would be great if someone could check up if my procedure to solve the following problem is correct. Also, I am struggling to solve for event B from part b) - any tips would be much much appreciated! I'm preparing for an exam, hence, it is of vital importance. Thank you! </p>
<p... | Kulisty | 170,765 | <p>I'm quite positive rotation does not involve Euclid's fifth postulate. I'll try to present strict definition of rotation in synthetic setting.</p>
<p><strong>Definition 1</strong>. An ordered pair <span class="math-container">$(A,B)$</span> of halflines having the same origin will be called a directed angle. Also d... |
240,699 | <p>I have the following equation which I want to solve:</p>
<p><span class="math-container">$$
I_D = [Li_2(-e^{V_D-I_D})-Li_{2}(e^{I_D})]
$$</span></p>
<p>Here <span class="math-container">$Li_2(x)$</span> is the PolyLog function of order <span class="math-container">$2$</span>. Is there a way to solve this equation it... | Stephen Luttrell | 1,393 | <p>You can investigate what your function does when you iterate it by plotting how it updates points in the complex <span class="math-container">$I_D$</span> plane, and I find that using <code>VectorPlot</code> to plot a vector field is a useful way of visualising this.</p>
<p>Define your function <span class="math-con... |
14,340 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://mathematica.stackexchange.com/questions/3247/consistent-plot-styles-across-multiple-mma-files-and-data-sets">Consistent Plot Styles across multiple MMA files and data sets</a> </p>
</blockquote>
<p>So, here's my problem; I have a lot of d... | b.gates.you.know.what | 134 | <p>Another possibility is to add your settings to the options :</p>
<pre><code>SetOptions[Plot, PlotStyle -> {RGBColor[1, 0, 0], Frame -> True, BaseStyle -> {FontSize -> 20}}]
Plot[Sin[x], {x, -1, 1}]
</code></pre>
<p><img src="https://i.stack.imgur.com/QrLNY.png" alt="plot1"></p>
<p>but</p>
<pre><code... |
3,389,659 | <p>if <span class="math-container">$S_n={a_1,a_2,a_3,...,a_{2n}}$</span>} where <span class="math-container">$a_1,a_2,a_3,...,a_{2n}$</span> are all distinct integers.Denote by <span class="math-container">$T$</span> the product
<span class="math-container">$$T=\prod_{i,j\epsilon S_n,i<j}{(a_i-a_j})$$</span> Prove t... | richrow | 633,714 | <p>Let me give a sketch of the solution.</p>
<p>Well, there is a following fact:</p>
<blockquote>
<p><strong>Proposition.</strong> Given <span class="math-container">$n$</span> distinct positive integers <span class="math-container">$a_1, a_2, \ldots, a_n$</span>. Then,
<span class="math-container">$$
\prod_{i>... |
3,915,771 | <p>I'm given the series:</p>
<p><span class="math-container">$$\sum_{n=2}^{\infty} \frac{n^2}{n^4-n-3}$$</span></p>
<p>I know it converges, however I'm meant to show that by the comparison test. What would be a good choice here? <span class="math-container">$\frac{1}{k^2}$</span> and <span class="math-container">$\frac... | hamam_Abdallah | 369,188 | <p>For <span class="math-container">$ n\ge 3$</span>, we have</p>
<p><span class="math-container">$$n\le \frac{n^4}{3}$$</span>
and</p>
<p><span class="math-container">$$3\le \frac{n^4}{3}$$</span></p>
<p>thus</p>
<p><span class="math-container">$$n^4-n-3\ge \frac{n^4}{3}$$</span></p>
<p>and</p>
<p><span class="math-co... |
413,108 | <p>Given a commutative ring <span class="math-container">$ R $</span> and a multiplicatively closed subset <span class="math-container">$ S $</span> of <span class="math-container">$ R $</span>, there are two ways to consturct <span class="math-container">$ S^{-1}R $</span>:</p>
<ol>
<li><p>define an equivalence relati... | Johannes Hahn | 3,041 | <p>Wlog assume <span class="math-container">$n=m$</span>. Set <span class="math-container">$S:=\{a_1^{k_1}\cdots a_n^{k_n} \mid k_i\in\mathbb{N}\}$</span>. Then <span class="math-container">$R[x_1,\ldots,x_n]/I$</span> is precisely the localisation <span class="math-container">$S^{-1} R$</span> that inverts <span class... |
118,763 | <p>Hello,</p>
<p>Let $X'X$ be a positive definite matrix and let $\mathbf{1}$ denote the vector of ones. </p>
<p>I'm hoping to construct a positive, diagonal matrix $W$ such that
$$(W X'X W) \mathbf{1} = \mathbf{1}$$</p>
<p>$X$ and $W$ are all assumed to have real-valued entries, and $X'$ denotes the transpose of $X... | David Bryant | 30,601 | <p>The relevant reference is </p>
<p>Marshall, A. and Olkin, I. Scaling of Matrices to Achieve Specified Row and Column Sums. Numerische Mathematik 12, 83-90 (1968)</p>
<p>who prove the result in the affirmative for positive definite matrices (and some generalizations). The proof is elegant and construction: the dia... |
15,237 | <p><a href="https://matheducators.stackexchange.com/questions/176/knowing-mathematics-does-not-translate-to-knowing-to-teach-mathematics-why">A question</a> has been asked about why great mathematicians are not necessarily great teachers. On the other hand, I am wondering if knowing more mathematics actually helps with... | Michael Joyce | 1,397 | <p>I'll give an answer by analogy. When you are a kid, you should be exposed to playing with other kids or participating in activities such as team sports. It's natural to ask, "Does playing soccer at seven years old help a child become a better adult?" And the answer seems to be that, in general, yes it does help. Not... |
285,227 | <p>I am trying to prove $\exp(x+y) = \exp(x) \exp(y)$.</p>
<p>I may use that $$\exp(x) = \sum_{n=0}^\infty \frac {x^n}{n!}$$
I further know how to multiply two power series in one point, i.e. if $f(x) = \sum_{n=0}^\infty c_n(x-a)^n$ and $g(x) = \sum_{k=0}^\infty d_n(x-a)^n$ then
$$
f(x)g(x) = \sum_{n=0}^\infty e_n(x-a... | David Zhang | 80,762 | <p>This can actually be done without writing a single sum. Consider the function $$ f(x, y) = \frac{e^x e^y}{e^{x+y}}. $$ Observe that $$ \frac{\partial f}{\partial x} = \frac{e^x e^y e^{x+y} - e^x e^y e^{x+y}}{(e^{x+y})^2} = 0. $$
Similarly, $$ \frac{\partial f}{\partial y} = \frac{e^x e^y e^{x+y} - e^x e^y e^{x+y}}... |
1,860,134 | <p>In <em>The logic of provability</em>, by G. Boolos, there is a remark in chapter 7 saying that $\diamond^{m} \top\implies \diamond^{n} \top$ is false if $m<n$ <strong>(unless $PA$ is 1-inconsistent)</strong>.</p>
<p>Now, it seems to me that the parenthetical expression is not necessary, since earlier in the chap... | Jsevillamol | 240,304 | <p>Nevermind, I just realized that in the proof of arithmetical completeness the author also needs to assume 1-consistency (To show that Bew(False) is not provable).</p>
|
7,268 | <p>I'm a private tutor working with a 7th grader who is struggling with solving equations. Given a simple equation, he is able to solve it using a formulaic procedure, but it is very obvious that he has no idea what the solution really means. Hence, if he gets a problem that's slightly different from ones he's solved b... | Karl | 4,668 | <p>I would backtrack and revisit substition, solving equations should fall out as a bi-product or the student hasn't grasped that topic fully. With regards to solving equations I believe choosing the numbers to provide a rich example is extremely important. Consider for example $$\frac{x}{4}=12$$</p>
<p>This is far m... |
3,692,435 | <p>prove the following identity:</p>
<p><span class="math-container">$\displaystyle\sum_{k=0}^{n}\frac{1}{k+1}\binom{2k}{k}\binom{2n-2k}{n-k} = \binom{2n+1}{n}$</span></p>
<p>what I tried:</p>
<p>I figured that: <span class="math-container">$\displaystyle\binom{2n+1}{n} = (2n+1) C_n$</span>
and <span class="math-con... | Brian M. Scott | 12,042 | <p>Here’s a combinatorial argument. <span class="math-container">$\binom{2n+1}n$</span> is the number of lattice paths from <span class="math-container">$\langle 0,0\rangle$</span> to <span class="math-container">$\langle n,n+1\rangle$</span> using <span class="math-container">$n$</span> right steps and <span class="ma... |
1,521,518 | <p>Determine if the given vectors span $\mathbb{R}^4$</p>
<p>${(1, 1, 1, 1), (0, 1, 1, 1), (0, 0, 1, 1), (0, 0, 0, 1)}$.</p>
<p>I'm completely confused on this question. My textbook gives a different problem but in $\mathbb{R}^3$. How would i go about this?</p>
| uniquesolution | 265,735 | <ol>
<li>Arrange the vectors as rows of a matrix.</li>
<li>Compute the determinant. It is particularly easy to compute as the matrix is upper triangular, so the determinant is just the product of the diagonal entries. It is equal to $1$.</li>
<li>Conclude that the rows of the matrix are linearly independent.</li>
<li>A... |
1,559,906 | <p>Is there an equivalent in mathematical language to the modulo (or <code>mod</code>) function in computing? </p>
| Mike Pierce | 167,197 | <p>I have seen the percent sign (%) represent <em>mod</em> in computing. So when you are thinking of as <code>x % n</code>, a mathematician would equivalently write this as $x \bmod n$. </p>
<p>There is a subtlety here, though. When you are using <code>mod</code> in computing, you are thinking of it as a function that... |
235,430 | <p>Suppose that a bounded sequence of real numbers $s_i$ ($i\in\omega$) has a limit $\alpha$ along some ultrafilter $\mu_1\in \beta{\Bbb N}\setminus{\Bbb N}$. Then given another ultrafilter $\mu_2\in \beta{\Bbb N}\setminus{\Bbb N}$, surely there exists some rearrangement $s_{r(i)}$ of $s_i$ that has the same limit $\al... | MJ73550 | 89,993 | <p>Since $Y$ is diagonal you get $M$ such that $M^TM=Y$ so $x^TYx=|Mx|^2$</p>
<p>let's say $X_1$ is centered</p>
<p>then $$E(X^T Y X)=\sum_{i=1}^n\text{Var}(\sum_{j=1}^n M_{ij}X_j)$$</p>
<p>then since $X_j$ are independent you get :
$$\text{Var}(\sum_{j=1}^n M_{ij}X_j)=\sum_{j=1} M_{ij}^2\text{Var}(X_j)$$
because th... |
3,033,344 | <p>Question: Tom only have 2 type of coins: coins: 4 cents and 5 cents. Write a proof by induction that every amount n ≥ a can indeed be paid with Tom coins</p>
<p>1) Base Case: Tom can pay <span class="math-container">$12, $</span>13, <span class="math-container">$14, $</span>15, <span class="math-container">$16 and ... | davidlowryduda | 9,754 | <p>What primes do you need to worry about? The primes that divide <span class="math-container">$360$</span> are <span class="math-container">$2,3,5$</span>, so you really want to find those numbers which are not divisible by <span class="math-container">$2, 3$</span>, or <span class="math-container">$5$</span>.</p>
<... |
1,309,578 | <blockquote>
<p>Prove\Disprove:<br>
$A$ is bounded from above $\iff$ $A\cap \mathbb{Z}$ is bounded from above.</p>
</blockquote>
<p>Let $A=\{a\in \mathbb{Q} \setminus \mathbb{Z}: a<0\}$ is bounded from above, $A\cap \mathbb{Z}=\emptyset $ and $\emptyset$ is not bounded from above</p>
<p>Is it a valid contradic... | acradis | 229,012 | <p>The empty set is trivially bounded </p>
|
1,309,578 | <blockquote>
<p>Prove\Disprove:<br>
$A$ is bounded from above $\iff$ $A\cap \mathbb{Z}$ is bounded from above.</p>
</blockquote>
<p>Let $A=\{a\in \mathbb{Q} \setminus \mathbb{Z}: a<0\}$ is bounded from above, $A\cap \mathbb{Z}=\emptyset $ and $\emptyset$ is not bounded from above</p>
<p>Is it a valid contradic... | drhab | 75,923 | <p>I presume you are working in $\mathbb R$ or $\mathbb Q$.</p>
<p>If $s$ serves as upper bound of $A$ then it also serves as upper bound of any subset of $A$. So if $A$ is bounded from above then any subset of $A$, including $A\cap\mathbb Z$, is bounded from above as well. </p>
<p>If e.g. $A=\mathbb Q-\mathbb Z$ the... |
1,879,076 | <p>How to show that $(1+x/n)^n\geq (1+x/10)^{10}$? (for $n\geq 10$)</p>
<p>I see that if I consider $n\to\infty$, the LHS approaches $e^x$, but that's all I could really see.</p>
<p>Please assume that $x\in[0,\infty)$</p>
| parsiad | 64,601 | <p>Since the two expressions are equal at $n=10$, it is sufficient to
show that $$\frac{\partial}{\partial n}(1+x/n)^{n}\geq0 \text{ for } n\geq1 \text{ and } x\geq 0.$$</p>
<p>You can check that this is equivalent to showing $$\log((n+x)/n)\geq x/(n+x)\text{ for } n\geq1 \text{ and } x\geq 0.$$</p>
<p>Moreover, this... |
1,765,222 | <p>I have proven this by the induction method but would like to know if it can be proven using an alternative method.</p>
| Michael Burr | 86,421 | <p>Using a little linear algebra, we can write:
$$
\frac{n(n^4-1)}{5}=24\binom{n}{5}+48\binom{n}{4}+30\binom{n}{3}+6\binom{n}{2}.
$$
Since the binomial coefficients are always integers, this exhibits that your fraction is always an integer.</p>
<p>This method is kinda silly, I prefer the Fermat's little theorem approa... |
62,177 | <p>One of the most mind boggling results in my opinion is, with the axiom of choice/well-ordering principle, there exist such things as uncountable well-ordered sets $(A,\leq)$. </p>
<p>With this is mind, does there exist some well ordered set $(B,\leq)$ with some special element $b$ such that the set of all elements ... | William | 13,579 | <p>Let $\aleph_1$ be the first uncountable ordinal. It can be defined to be the smallest ordinal which is not in bijection with $\omega$. This set exists using the ZF axioms. The ordinal $\aleph_1 + 1 $ is defined as $\aleph_1 \cup \{\aleph_1\}$. Hence $\aleph_1 + 1$ is the desired set which contains an element particu... |
1,291,511 | <p>This may seem like a silly question, but I just wanted to check. I know there are proofs that if $f(x)=f'(x)$ then $f(x)=Ae^x$. But can we 'invent' another function that obeys $f(x)=f'(x)$ which is <strong>non-trivial</strong>?</p>
| Ivo Terek | 118,056 | <p>No. Suppose that $f \not\equiv 0$. We <strong>solve</strong> the ODE: $$f(x) = f'(x) \implies \frac{f'(x)}{f(x)} = 1 \implies \int \frac{f'(x)}{f(x)}\,{\rm d}x = \int \,{\rm d}x \implies \ln |f(x)| = x+c, \quad c \in \Bbb R$$With this, $|f(x)| = e^{x+c} = e^ce^x$. Call $e^c = A > 0$. Eliminating the absolute valu... |
1,291,511 | <p>This may seem like a silly question, but I just wanted to check. I know there are proofs that if $f(x)=f'(x)$ then $f(x)=Ae^x$. But can we 'invent' another function that obeys $f(x)=f'(x)$ which is <strong>non-trivial</strong>?</p>
| Michael Hardy | 11,667 | <p>OK, let's try a plodding pedestrian approach:
$$
f'(x)-f(x) = 0
$$
The idea of an exponential multiplier $m(x)$ is that we write
$$
0 = m(x)f'(x) + (-m(x))f(x) = m(x)f'(x)+m'(x)f(x) = \Big( m(x)f(x)\Big)'.
$$
For this to work we would need $-m(x)=m'(x)$. We <b>do not need <em>all</em></b> functions $m$ satisfying t... |
863,364 | <p><img src="https://i.stack.imgur.com/w6R2g.jpg" alt="enter image description here"></p>
<p>I am missing the 3D graph for the equation $x^2+2z^2=1$.</p>
| Andrew D | 55,458 | <p>What happens depends on what variable(s) you are applying the Fourier transform to; if we suppose we are making the Fourier transform with regards to $x \rightsquigarrow \xi$, then if we are using the one-dimensional convention that</p>
<p>$$ \hat{f}(\xi) = \int^{\infty}_{-\infty}f(x)e^{-i\xi x} dx$$</p>
<p>so in... |
1,175,297 | <p>Note: The following definitions from my book, Discrete Mathematics and Its Applications [7th ed, 598].</p>
<p>This is my book's definition for a reflexive relation
<img src="https://i.stack.imgur.com/og5wE.png" alt="enter image description here"></p>
<p>This is my book's definition for a anti symmetric relation
<i... | N. S. | 9,176 | <p>They are not the same thing. For example, on $\mathbb R$ the strict inequality is anti-symmetric but not reflexive.... If you want one which is not vacuous, change it to be $\leq $ if at least one of the numbers is negative.</p>
<p>On $\mathbb Z$, if we define $(x,y) \in R$ if and only if $x-y$ is even, then this i... |
1,175,297 | <p>Note: The following definitions from my book, Discrete Mathematics and Its Applications [7th ed, 598].</p>
<p>This is my book's definition for a reflexive relation
<img src="https://i.stack.imgur.com/og5wE.png" alt="enter image description here"></p>
<p>This is my book's definition for a anti symmetric relation
<i... | ajotatxe | 132,456 | <p>No.</p>
<p>First, you can have a reflexive relation which is not antisymmetric. For example: an integer number $a$ is related to other integer $b$ if and only if $a$ and $b$ have the same parity. This is not antysymmetric, because $2R4$ and $4R2$, but $2\neq 4$. But it is reflexive, since every integer has the same... |
154,757 | <p>I have this data:</p>
<ul>
<li><p>$a=6$</p></li>
<li><p>$b=3\sqrt2 -\sqrt6$ </p></li>
<li><p>$\alpha = 120°$</p></li>
</ul>
<p><strong>How to calculate the area of this triangle?</strong></p>
<p>there is picture:</p>
<p><img src="https://i.stack.imgur.com/hr2Cp.jpg" alt=""></p>
| Isaac | 72 | <p>Because the angle at $A$ is obtuse, the given information uniquely determines a triangle. To find the area of a triangle, we might want:</p>
<ul>
<li>the length of a side and the length of the altitude to that side (we don't have altitudes)</li>
<li>all three side lengths (we're short 1)</li>
<li>two side lengths ... |
3,222,871 | <p>Let <span class="math-container">$P(x, y, 1)$</span> and <span class="math-container">$Q(x, y, z)$</span> lie on the curves <span class="math-container">$$\frac{x^2}{9}+\frac{y^2}{4}=4$$</span> and <span class="math-container">$$\frac{x+2}{1}=\frac{y-\sqrt{3}}{\sqrt{3}}=\frac{z-1}{2}$$</span> respectively. Then find... | Cesareo | 397,348 | <p>We can solve this problem proposing a lagrangian. So calling</p>
<p><span class="math-container">$$
d^2 = (x_1-x_2)^2+(y_1-y_2)^2+(1-z_2)^2\\
g_1 = \frac{x_1^2}{9}+\frac{y_1^2}{4}-4\\
g_3 = x_2+2-\lambda\\
g_4 = y_2-\sqrt 3-\sqrt 3\lambda\\
g_5 = z_2-1-2\lambda
$$</span></p>
<p>and forming</p>
<p><span class="ma... |
1,639,081 | <p>I have been unable to solve the following question, </p>
<p>If $$\sin(2x) - \tan(x) = 0$$</p>
<p>Find $x$ , $-\pi\le x\le \pi$</p>
<p>So far my workings have been
Use following identity: </p>
<p>$$\sin(2x) = 2\sin(x)\cos(x)\\2\sin(x)\cos(x) - \tan(x) = 0\\2\sin(x)\cos(x) - \frac{\sin(x)}{\cos(x)} = 0\\
2\frac{\s... | 100001 | 310,245 | <p>Once you got to
$$2\sin(x)\cos(x) - \frac{\sin(x)}{\cos(x)} = 0$$
you can pull out a factor of $\sin (x)$ to get
$$\sin(x)\left[2\cos(x)- \frac{1}{\cos(x)}\right]$$
Now, either $\sin(x) = 0$ or $2\cos(x)- \frac{1}{\cos(x)} = 0$. For $\sin(x) = 0$, we have $-180$, $0$, and $180$. For $2\cos(x)- \frac{1}{\cos(x)} ... |
1,639,081 | <p>I have been unable to solve the following question, </p>
<p>If $$\sin(2x) - \tan(x) = 0$$</p>
<p>Find $x$ , $-\pi\le x\le \pi$</p>
<p>So far my workings have been
Use following identity: </p>
<p>$$\sin(2x) = 2\sin(x)\cos(x)\\2\sin(x)\cos(x) - \tan(x) = 0\\2\sin(x)\cos(x) - \frac{\sin(x)}{\cos(x)} = 0\\
2\frac{\s... | vrugtehagel | 304,329 | <p>The equation we want to solve is $$\sin(2x)-\tan(x)$$
You deduced correctly that we now have to solve $$2\sin(x)\cos(x)-\frac{\sin(x)}{\cos(x)}=0$$ which we can rewrite to $$2\sin(x)\cos(x)=\frac{\sin(x)}{\cos(x)}$$
or $$2\sin(x)\cos(x)^2=\sin(x)$$
Now, either $\sin(x)=0$ (in which case $x\in\{-180,0,180\}$, given t... |
204,218 | <p>I can't bear an expression containing radicals of imaginary numbers,
in case it can be expressed as in terms of radicals of real numbers only.</p>
<p>For example, I can't bear the expression</p>
<pre><code>Sqrt[2 + I]
</code></pre>
<p>because
it can be expressed as </p>
<pre><code>Sqrt[1/2 (2 + Sqrt[5])] + Sqrt[... | Bill Watts | 53,121 | <p>Try</p>
<pre><code>Sqrt[2 + I] // ComplexExpand // FunctionExpand
</code></pre>
<p>It may not create a result simplified in the exact form you want, but it will be closer.</p>
|
204,218 | <p>I can't bear an expression containing radicals of imaginary numbers,
in case it can be expressed as in terms of radicals of real numbers only.</p>
<p>For example, I can't bear the expression</p>
<pre><code>Sqrt[2 + I]
</code></pre>
<p>because
it can be expressed as </p>
<pre><code>Sqrt[1/2 (2 + Sqrt[5])] + Sqrt[... | Bob Hanlon | 9,362 | <pre><code>{1, I}.(Sqrt[2 + I] // ReIm // ComplexExpand // FunctionExpand //
FullSimplify)
(* I Sqrt[1/2 (-2 + Sqrt[5])] + Sqrt[1/2 (2 + Sqrt[5])] *)
</code></pre>
<p>or</p>
<pre><code>% // Simplify
(* (I Sqrt[-2 + Sqrt[5]] + Sqrt[2 + Sqrt[5]])/Sqrt[2] *)
</code></pre>
<p>Verifying that these are equivalent to... |
1,400,394 | <p>Given that $u(x,y)$ can someone please explain to me how the result as asked in the question is achieved? Steps would be really appreciated, thanks.</p>
| Ian Taylor | 236,192 | <p>It's analogous to $$\frac{d^2y}{dx^2}=0$$. The solution to this is obtained by saying $\frac{dy}{dx}=k_1$ (a constant, not a function of $x$).
So $$y=k_1x+k_2$$ where $k_2$ is another constant, not a function of $x$.
In your example with partial derivatives $f(y)$ and $g(y)$ are like the constants $k_1$ and $k_2$, ... |
2,431,861 | <p>Let $P(z)=\displaystyle \sum_{0\le k\le n}a_kz^k$ a complex polynomial. What conditions must satisfy the coefficients $a_k$ to have $$P(z)=-\overline{P(\overline z)}\space \space ?$$</p>
| Gabriel Romon | 66,096 | <p>Note that $\overline{P(\overline z)}=\overline{\sum_{k=0}^na_k\overline z^k}=\sum_{k=0}^n\overline{a_k}z^k$, hence $P(z)=-\overline{P(\overline z)}$ if and only if $\sum_{k=0}^n(a_k+\overline{a_k})z^k=0$.</p>
<p>A polynomial is $0$ if and only if its coefficients are $0$, hence $P(z)=-\overline{P(\overline z)}$ if ... |
4,492,930 | <p>Let <span class="math-container">$\mathfrak{c}$</span> denote the cardinality of the continuum. I sketch an intuitive but non-rigorous argument that <span class="math-container">$|\mathbb{R}^\mathbb{N}| = \mathfrak{c}$</span>, with the question:</p>
<p><strong>Question</strong>: can this argument be made rigorous?</... | sadman-ncc | 942,091 | <p>Inner products, by definition, are linear in each position. By linearity in the first position, we have:
<span class="math-container">$$ \langle Ax ,x \rangle + \langle Bx, x \rangle=\langle Ax+Bx,x \rangle\\
= \langle (A+B)x,x \rangle = \langle \lambda x,x \rangle .$$</span> So, your statement is always right.</p>
|
1,579,781 | <blockquote>
<p>If $x+y+z=6$ and $xyz=2$, then find the value of $$\cfrac{1}{xy}
+\cfrac{1}{yz}+\cfrac{1}{zx}$$</p>
</blockquote>
<p>I've started by simply looking for a form which involves the given known quantities ,so:</p>
<p>$$\cfrac{1}{xy} +\cfrac{1}{yz} +\cfrac{1}{zx}=\cfrac{yz\cdot zx +xy \cdot zx +xy \cdot... | Kushal Bhuyan | 259,670 | <p>$$\cfrac{1}{xy} +\cfrac{1}{yz} +\cfrac{1}{zx}=\cfrac{yz\cdot zx +xy \cdot zx +xy \cdot yz}{(xyz)^2}=\frac{x+y+z}{xyz}$$</p>
|
928,644 | <blockquote>
<p>Let $f,g$ be $\mathcal E$-$\mathcal B(\mathbb R)$-measurable functions. I want to show piecewise function $h$ of $f$ and $g$ is also measurable.</p>
</blockquote>
<p>Suppose $(X, \mathcal E)$ is a measure space, let $f,g$ be $\mathcal E$-$\mathcal B(\mathbb R)$-measurable functions and let $A \in \ma... | Stefan Hansen | 25,632 | <p>For any Borel set $B$ one has
$$
h^{-1}(B)=\big(f^{-1}(B)\cap A\big)\cup \big(g^{-1}(B)\cap A^c\big).
$$</p>
|
4,186,743 | <p>Is there a trigonometric function explaining <span class="math-container">$\cos(x)+\sin(x)$</span>? if not, what is <span class="math-container">$\cos(x)+\sin(x)$</span> as a function of <span class="math-container">$\cos(x)$</span>?</p>
| Sayan Dutta | 943,723 | <p>We know that <span class="math-container">$\cos{x}+\sin{x}=\cos{x}+\cos{(\frac\pi2-x)}$</span>.</p>
<p>You may keep it as it is, or you may apply the formula for <span class="math-container">$\cos{C}+\cos{D}$</span> according to your needs.</p>
<p>Another representation may also be <span class="math-container">$\cos... |
3,946,591 | <blockquote>
<p>The sides of a triangle are on the lines <span class="math-container">$2x+3y+4=0$</span>, <span class="math-container">$ \ \ x-y+3=0$</span>, and <span class="math-container">$5x+4y-20=0$</span>. Find the equations of the altitudes of the triangle.</p>
</blockquote>
<p>Should I find the vertices first? ... | Robert Shore | 640,080 | <p>The answer is</p>
<p><span class="math-container">$$3^n+ \sum_{k=1}^n 3^{k-1}\cdot 3^{n-k}=n3^{n-1}+3^n.$$</span></p>
<p>There are <span class="math-container">$3^n$</span> possible words with no occurrence of <span class="math-container">$b$</span>. Assume the first <span class="math-container">$b$</span> occurs a... |
3,231,271 | <blockquote>
<p>Suppose <span class="math-container">$X$</span> is Banach and <span class="math-container">$T\in B(X)$</span> (i.e. <span class="math-container">$T$</span> is a linear and continuous map and <span class="math-container">$T:X \to X$</span>). Also, suppose <span class="math-container">$\exists c > 0$... | Matematleta | 138,929 | <p>Here is an idea: let <span class="math-container">$T(X) \ni y_n=T(x_n)\to y\in \overline{T(X)}$</span>. Then, since <span class="math-container">$\|Tx_n\| \ge c\|x_n\|,\ x_n\to x\in X$</span> and continuity of <span class="math-container">$T$</span> implies that <span class="math-container">$y_n=T(x_n)\to T(x)$</spa... |
81,982 | <p>I am beggining to do some work with cubical sets and thought that I should have an understanding of various extra structures that one may put on cubical sets (for purposes of this question, connections). I know that cubical sets behave more nicely when one has an extra set of degeneracies called connections. The que... | Ronnie Brown | 19,949 | <p>The point I want to make was that the notion of connection on cubical set was forced on us in the following way. </p>
<p>Go back to</p>
<p><a href="http://groupoids.org.uk/pdffiles/brown-spencerCTDC_1977_18_4_450_0.pdf" rel="nofollow">[21]</a>. (with C.B. SPENCER), ``Double groupoids and crossed
modules'', Cah. ... |
3,023,726 | <p>I'm trying to solve a problem that I can't seem to work out.</p>
<blockquote>
<p><span class="math-container">$f$</span> is an entire function. Prove that <span class="math-container">$|f^{(n)}(0)|< n!n^n$</span> for at least 1 <span class="math-container">$n$</span>. </p>
</blockquote>
<p>I've been thinking ... | achille hui | 59,379 | <p>Let <span class="math-container">$M = \sup \{ |f(z)| : |z| = 1\}$</span>. Since <span class="math-container">$f$</span> is entire, for all <span class="math-container">$n \ge 0$</span>, we have</p>
<p><span class="math-container">$$f^{(n)}(0) = \frac{n!}{2\pi i}\int_{|z|=1} \frac{f(z)}{z^{n+1}}dz$$</span>
Whenever... |
198,739 | <p>I'm actually doing an exercise where I have to draw graphs of functions.
I understand r=|s| but not |r|=|s|.
Are they the same?</p>
| Emily | 31,475 | <p>Let $r = 4$ and $s = -4$. Then, $r = |s|$ and $|r| = |s|$.</p>
<p>Then, switch it around. Let $r = -4$ and $s = 4$. Then, $|r| = |s|$ but $r \neq |s|$.</p>
<p>So they are not always the same.</p>
|
258,226 | <p>As an algebraist, I have some strong intuitions about what it means for an algebraic result to be true. In particular, my intuition would lead me to believe that if I cannot construct a counter-example to a claim, then the claim must be true. This is what motivated <a href="https://mathoverflow.net/questions/25813... | Joel David Hamkins | 1,946 | <p>Yes, your theory is the same as the true arithmetic. In
particular, yes, it is consistent.</p>
<p>I claim that at stage $n$, your theory $T_n$ consists of PA plus
the collection of true $\Pi^0_n$ sentences (that is, true in the
standard model). This starts out true with $T_0$. If $T_n$ is like
that, then consider $... |
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