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 |
|---|---|---|---|---|
3,878,723 | <blockquote>
<p>Find the value of <span class="math-container">$k$</span> if the curve <span class="math-container">$y = x^2 - 2x$</span> is tangent to the line <span class="math-container">$y = 4x + k$</span></p>
</blockquote>
<p>I have looked at the solution to this question and the first step is the "equate the... | heropup | 118,193 | <p>If the line <span class="math-container">$y = 4x + k$</span> is tangent to <span class="math-container">$y = x^2 - 2x$</span>, then there exists some value of <span class="math-container">$x$</span> for which the two curves intersect; i.e., <span class="math-container">$$4x + k = x^2 - 2x.$$</span> This results in ... |
3,438,048 | <p>I've recently obtained my University entrance papers from 1967 (yes,52 years ago!) and I found the question below difficult. I presume the answer is a symmetric expression in the differences between alpha,beta and gamma.Am I missing some obvious trick? Any help would be appreciated.</p>
<p>Simplify and evaluate the... | Vijayakumar Muni | 400,662 | <p>First of all in your definition of a topology <span class="math-container">$\tau$</span> on <span class="math-container">$X,$</span> there is a typo in condition (c).</p>
<p>It is supposed to be <span class="math-container">$``$</span>for <span class="math-container">$\alpha\in J,$</span> where <span class="math-co... |
1,772,562 | <p>Let $f:\mathbb{C}\rightarrow\mathbb{C}$ be holomorphic. If we have $|f(z)|\leq|z|^n$ for some $n\in\mathbb{N}$ and all $z\in\mathbb{C}$, then $f$ is a polynomial.</p>
<p>I tried to apply Liouville's theorem but it does not help.</p>
<p>Thanks for your help.</p>
| Martín-Blas Pérez Pinilla | 98,199 | <p>Even true if the the condition is $|f(z)|\le C|z|^n$ for $|z|\ge R>0$. Let be $P_n$ the $n$th degree Taylor polynomial of $f$ at zero and consider $f-P_n$.</p>
|
88,363 | <p>It is easy to truncate Series upto some order, say $n$. My question is how do I remove low orders? Let us say my series is a power series in $x$. I want to remove the terms with negative powers because they diverge at $x = 0$. I can simply write</p>
<p>s1-s2, where</p>
<p>s1=Normal[Series[blah, {x, 0, n}]</p>
<p>... | Acus | 18,792 | <p>Why not to subtract two expansions like in </p>
<pre><code>t1 = Series[1/Sin[x], {x, 0, 10}]
t2 = Series[1/Sin[x], {x, 0, 0}]
</code></pre>
<p>Then </p>
<pre><code>Normal[t1] - Normal[t2]
</code></pre>
<p>Out[3]:=
x/6 + (7 x^3)/360 + (31 x^5)/15120 + (127 x^7)/604800 + (
73 x^9)/3421440</p>
|
3,583,117 | <p>I would like to understand clearly why the following equality is true</p>
<p><span class="math-container">$P[X+Y \leq z] = E_Y[P[X+Y] \leq z | Y]]$</span></p>
<p>I wrote the left part of the equation as follows:</p>
<p><span class="math-container">$E_Y[P[X+Y] \leq z | Y]] = \sum_y y P[X+y \leq z]P(y)$</span></p>
... | roundsquare | 706,295 | <p><span class="math-container">$P[X+Y \le z] = \sum_{y \in Y}P[Y=y] \times P[X + y \le z] = \sum_{y \in Y}P[Y=y] \times P[X + Y \le z | Y=y] = E_Y[P(X + Y \le z | Y=y)]= E_Y[P(X + Y \le z | Y)]$</span></p>
<p>where the last equality is just a bit of notation.</p>
|
3,413,480 | <p>The <a href="https://de.wikipedia.org/wiki/Asymmetrisches_Kryptosystem#Sicherheit" rel="nofollow noreferrer">German Wikipedia article on asymmetric cryptography</a> states that asymmetric cryptography is <em>always</em> based on assumptions which <strong>can not</strong> be proven:</p>
<p><em>Die Sicherheit aller a... | quarague | 169,704 | <p>This seems to be an open research problem. If you look at <a href="https://en.wikipedia.org/wiki/One-way_function" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/One-way_function</a> it says that the existence of one-way functions is currently unproven. So for cryptography applications there are two pieces ... |
3,413,480 | <p>The <a href="https://de.wikipedia.org/wiki/Asymmetrisches_Kryptosystem#Sicherheit" rel="nofollow noreferrer">German Wikipedia article on asymmetric cryptography</a> states that asymmetric cryptography is <em>always</em> based on assumptions which <strong>can not</strong> be proven:</p>
<p><em>Die Sicherheit aller a... | kelalaka | 338,051 | <p>This is due to the fact that the existence of One-Way Functions (OWF) implies that <span class="math-container">$P \neq NP$</span>. In other words, with contrapositive, if <span class="math-container">$P = NP$</span> then OWF doesn't exit. So if we have one you would know it in the news. Therefore, the security of t... |
1,175,632 | <p>Determine whether the following integral converges or diverges: \begin{align*} \iint_Q e^{-xy} \ dA, \end{align*} where $Q$ is the first quadrant of the $xy$-plane.</p>
<p>How should I go about this problem? Should I compare it with another known integral?</p>
| sakas | 521,275 | <p>Let <span class="math-container">$X_i=\begin{cases}
S, & \text{with prob 0.3 } \\
F, & \text{with prob 0.7 } \\
\end{cases}$</span>
for <span class="math-container">$i=1,2,3,4,5$</span> be IID for the outcome of the i-th day.
The probability you are asked is the following (if we are talking fo... |
24,186 | <p>Consider the code below:</p>
<pre><code>s = Solve[(3 - Cos[4*x])*(Sin[x] - Cos[x]) == 2, x, InverseFunctions -> True];
Select[s[[All, 1, 2]], Element[#, Reals] &]
</code></pre>
<p>In MMA 8.0, I get </p>
<pre><code>{-\[Pi], \[Pi]/2, \[Pi]}
</code></pre>
<p>but in MMA 9.0, I get an empty set { }</p>
<p>Ass... | Thies Heidecke | 47 | <p>In those more complicated cases consisting of multiple steps, using Composition clears things up for me while still retaining a pure functional style. In your example of calculating the distance between two points in 2D i would write:</p>
<pre><code>u = {-3, 3}; v = {1, 5};
Composition[Sqrt, #.# &, Subtract][u,... |
3,043,039 | <p>Let <span class="math-container">$f:(0,1) \to \mathbb{R}$</span> be a given function. Explain how the following definition is not equivalent to the definition of the limit</p>
<p><span class="math-container">$\lim\limits_{x \to x_0} f(x) = L$</span></p>
<p>of <span class="math-container">$f$</span> at <span class=... | user | 505,767 | <p>Yes the correct definition requires</p>
<p><span class="math-container">$$\forall \epsilon >0 \quad \exists \delta >0 \quad \ldots$$</span></p>
<p>and the other part of the definition is correct.</p>
<p>Indeed let consider for example <span class="math-container">$f(x)=x$</span> with <span class="math-conta... |
2,848,891 | <p>Find the number of solutions of $$\left\{x\right\}+\left\{\frac{1}{x}\right\}=1,$$ where $\left\{\cdot\right\}$ denotes Fractional part of real number $x$.</p>
<h2>My try:</h2>
<p>When $x \gt 1$ we get</p>
<p>$$\left\{x\right\}+\frac{1}{x}=1$$ $\implies$</p>
<p>$$\left\{x\right\}=1-\frac{1}{x}.$$</p>
<p>Letting... | Community | -1 | <p>Let $x:=n+f$. The equation is</p>
<p>$$f+\frac1{n+f}=1,$$ </p>
<p>giving the solutions in $f$</p>
<p>$$f=\frac{\pm\sqrt{(n+1)^2-4}-n+1}2.$$</p>
<p>The negative sign cannot work, nor the negative $n$. Then $n\ge1$ is required, but $n=1$ yields $x=1$, which is wrong. Finally,</p>
<p>$$f=\frac{\sqrt{(n+1)^2-4}-n+1... |
2,848,891 | <p>Find the number of solutions of $$\left\{x\right\}+\left\{\frac{1}{x}\right\}=1,$$ where $\left\{\cdot\right\}$ denotes Fractional part of real number $x$.</p>
<h2>My try:</h2>
<p>When $x \gt 1$ we get</p>
<p>$$\left\{x\right\}+\frac{1}{x}=1$$ $\implies$</p>
<p>$$\left\{x\right\}=1-\frac{1}{x}.$$</p>
<p>Letting... | fleablood | 280,126 | <p>It's simpler if you realize that:</p>
<p>EDIT (new text): </p>
<p>$\{a\} + \{\frac b\} = (a + b) - ([a]+ [b])$ and $([a] + [b])$ is always an integer. So $\{a\} + \{frac b\}$ is an integer if and only if $a + b$ is an integer.</p>
<p>Now $0\le \{a \} < 1$ so $0 \le \{a\} + \{b\} < 2$.</p>
<p>So $\{a \} +... |
75,795 | <p><strong>The problem:</strong></p>
<p>If we have</p>
<blockquote>
<p>$P(H_\eta|E_1,E_2,...,E_e)(1 \leq \eta \leq \mathbb{H})$</p>
</blockquote>
<p>and</p>
<blockquote>
<p>$P(E_1,E_2,...,E_e)$</p>
</blockquote>
<p>for all <strong>True</strong> and <strong>False</strong> values of $E_\epsilon(1 \leq \epsilon \... | karmic_mishap | 17,529 | <p>Yes, the theorem which allows you to calculate $P(H_{h})$ from the given probabilities is called [Bayes' Theorem]. The extended form listed on the Wikipedia entry linked should cover this situation nicely.</p>
|
23,566 | <p>I love math, and I used to be very good at it. The correct answers came fast and intuitively. I never studied, and redid the demonstration live for the tests (sometimes inventing new ones). I was the one who answered the tricky questions in class (8 hours of math/week in high school)... You get the idea.</p>
<p>As ... | Carl Offner | 2,579 | <p>One thing I remember realizing in high school was that I would often see something, or read something, and think to myself, "yes, that makes sense". But then a day later I wouldn't have a clue about it. That was my first introduction to the difference between what I later learned was "active understanding" as oppo... |
1,022,950 | <p>I was reading linear dependence between vectors, where I come across below explanation:</p>
<hr>
<p>In a rectangular xy-coordinate system every vector in the plane can be expressed in
exactly one way as a linear combination of the standard unit vectors. For example, the
only way to express the vector (3, 2) as a l... | peterwhy | 89,922 | <p>Assuming $n, \epsilon > 0$,
$$\begin{align*}
\left|\sqrt{\frac{n+1}n}-1\right| &< \epsilon\\
\left|\sqrt{1+\frac1n}-1\right| &< \epsilon\\
\sqrt{1+\frac1n}-1 &< \epsilon\\
\sqrt{1+\frac1n} &< 1 + \epsilon\\
1+\frac1n &< (1+\epsilon)^2\\
\frac1n &< (1+\epsilon)^2-1\\
n &am... |
1,184,961 | <p>I need to prove/show that $n^3 \leq 3^n$ for all natural numbers $n$ by strong induction. I have no clue where to begin!!!! :( I know how to do the beginning steps of showing that it's true for $k = 0$ and $k = 1$, etc but get suck on how to start the strong inductive step.</p>
| DanielV | 97,045 | <p>For the strong inductive step, you want to assume:</p>
<p>$$k^3 \le 3^n$$</p>
<p>and use that to prove</p>
<p>$$(k+1)^3 \le 3^{n+1}$$</p>
<p>Use the fact that $a < b$ and $b < c$ implies $a < c$.</p>
|
1,184,961 | <p>I need to prove/show that $n^3 \leq 3^n$ for all natural numbers $n$ by strong induction. I have no clue where to begin!!!! :( I know how to do the beginning steps of showing that it's true for $k = 0$ and $k = 1$, etc but get suck on how to start the strong inductive step.</p>
| Daniel W. Farlow | 191,378 | <p>We can prove that $n^3 < 3^n$ for all $n\geq 4$ (which is basically the same thing as proving that $n^3 \leq 3^n$ for $n\geq 0$), and we can prove this using <strong>weak</strong> induction (there's no need to use strong induction here).</p>
<p>Start by noting that
$$
3n^2+3n+1<2(3^n)\tag{1}
$$
is true for $n... |
99,572 | <p>One of the most useful tools in the study of convex polytopes is to move from polytopes (through their fans) to toric varieties and see how properties of the associated toric variety reflects back on the combinatorics of the polytopes. This construction requires that the polytope is rational which is a real restrict... | Dan Zaffran | 2,109 | <p>Dear Gil, we have a nonrational construction with F. Battaglia in the case of simplicial fans here: <a href="http://arxiv.org/abs/1108.1637%20%22D.%20Zaffran,%20F.%20Battaglia%3A%20Foliations%20modelling%20nonrational%20simplicial%20toric%20varieties%22" rel="nofollow">arXiv:1108.1637</a>, where we use foliated ... |
120,260 | <p>Let $X$ be a simply connected smooth projective variety, whose Picard group is generated by the classes of the irreducible codimension 1 loci $D_1, \ldots, D_k$. Let $E_1, \ldots, E_r$ be other irreducible codimension 1 loci, and suppose that $X^0$ is the complement in $X$ of the divisors $D_i$ and $E_j$.</p>
<p>Su... | alvarezpaiva | 21,123 | <p>I can't say that what I'll relate is fundamental, but it does fit into the new ideas category. Since I and (my collaborator) Florent Balacheff have given talks on the subject and the paper will be in the ArXiv in a few days I feel free to comment on it. <strong>This post is an annoucement of joint work with Florent ... |
463,650 | <p>Consider the sequence $\left \{ x_{n} \right \}$ that satisfies the condition:
$$\left | x_{n+1}-x_{n} \right |< \frac{1}{2^{n}}
\ \ \ for\ all\ n=1,2,3,...$$
Part (1): Prove that the sequence $\left \{ x_{n} \right \}$ is convergent.</p>
<p>Part (2): Does the result in part (1) hold if we only assume that $\l... | Brian M. Scott | 12,042 | <p>Let $x_1=1$, say, and let $x_{n+1}=x_n+\frac1{2n}$ for each $n\in\Bbb Z^+$.</p>
|
1,714,654 | <p>Show that a box (rectangular parallelopiped) of maximum volume V with prescribed surface area is a cube.
Let $$V=xyz$$
$$S=2xy + 2yz + 2zx$$
$S$ is constant.</p>
<p>Using Lagrange method, I am stuck at $V_x$$_x$=$0$=$V_y$$_y$=$V_z$$_z$ at the (only) critical point. How to approach this. </p>
| RRL | 148,510 | <p>We have the inequality $z - z^3/6 \leqslant \sin z \leqslant z$ for $z > 0$.</p>
<p>Hence, for $x > y$</p>
<p>$$1 - \frac{(x^2 - y^2)^2}{6} \leqslant \frac{\sin (x^2 - y^2)}{x^2 - y^2} \leqslant 1,$$</p>
<p>and by the squeeze theorem the limit is $1$ as $(x,y) \to (0,0)$ with $x > y.$</p>
<p>Similarly,... |
438,231 | <p>How should I state the general solution for the equation $\sin(4\phi)=\cos(2\phi)$.
The angles are $15$, $45$, $75$ and $135$ if I restrict myself within the range $[0,360]$</p>
| lab bhattacharjee | 33,337 | <p>As $\cos2\phi=\sin4\phi=\cos(90^\circ-4\phi)$</p>
<p>$\implies 2\phi=n360^\circ\pm(90^\circ-4\phi)$ where $n$ is any integer</p>
<p>Taking '+' sign, $2\phi=n360^\circ+90^\circ-4\phi$</p>
<p>$\implies 6\phi=n360^\circ+90^\circ \implies \phi=n60^\circ+15^\circ$</p>
<p>As $0\le \phi<360^\circ, 0\le n60^\circ+15^... |
106,464 | <p>I'd like to prove the following:</p>
<blockquote>
<p>If $\mathfrak{a} \subseteq k[x_0, \ldots, x_n]$ is a homogeneous ideal, and if $f \in k[x_0,\ldots,x_n]$ is a homogeneous polynomial with $\mathrm{deg} \ f > 0$, such that $f(P) = 0 $ for all $P \in Z(\mathfrak{a})$ in $\mathbb P^n$, then $f^q \in \mathfrak{... | Georges Elencwajg | 3,217 | <p>Consider $V(\mathfrak a)\subset \mathbb A^{n+1}(k)=k^{n+1}$, the cone in <em>affine</em> $n+1$ space defined by the ideal $\mathfrak a$ .<br>
Your polynomial $f$ will vanish on $V(\mathfrak a)$ because that's <em>exactly</em> what it means that it vanishes on $Z(\mathfrak a)$ (see below)<br>
The usual Nullstelle... |
322,302 | <p>Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch with what happens in the other fields.</p>
<p><strong>Question</strong> What are the conjectures in your field prove... | Eran | 121,912 | <p>Ladner's theorem states that there exist <span class="math-container">$\mathsf{NP}$</span>-intermediate problems when <span class="math-container">$\mathsf{P}\neq\mathsf{NP}$</span>. However, the problem constructed in Ladner's proof is rather 'unnatural'. The question arises of whether any 'natural' examples of pro... |
322,302 | <p>Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch with what happens in the other fields.</p>
<p><strong>Question</strong> What are the conjectures in your field prove... | Rodrigo A. Pérez | 13,923 | <p>Graph theory / Discrete dynamics: In 2007, A. Trahtman <a href="https://arxiv.org/abs/0709.0099" rel="noreferrer">proved</a> the <a href="https://en.m.wikipedia.org/wiki/Road_coloring_theorem" rel="noreferrer">Road Coloring Conjecture</a>, which had been posited 37 years earlier by R. Adler and B. Weiss.</p>
|
322,302 | <p>Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch with what happens in the other fields.</p>
<p><strong>Question</strong> What are the conjectures in your field prove... | zhoraster | 8,146 | <p>A remarkable example is the <a href="https://en.wikipedia.org/wiki/Gaussian_correlation_inequality" rel="noreferrer">Gaussian correlation conjecture</a> (which only recently became the Gaussian correlation inequality). The formulation is very simple:</p>
<blockquote>
<p>For arbitrary centered Gaussian measure, an... |
322,302 | <p>Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch with what happens in the other fields.</p>
<p><strong>Question</strong> What are the conjectures in your field prove... | Orr Shalit | 1,193 | <p>Here are two examples from operator theory/operator algebras. Both were open problems for more than forty years. The first example is remarkable because it is was quite well known, it is so simple to state and was elusive for so much time. The second example is notable because of its importance and because it was op... |
322,302 | <p>Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch with what happens in the other fields.</p>
<p><strong>Question</strong> What are the conjectures in your field prove... | Mozibur Ullah | 35,706 | <p>The Baez-Dolan corbordism hypothesis or conjecture which states that the higher corbordism category is the free symmetric higher monoidal category on a single object was formalised by Lurie and proven in his paper classifying topological field theiries in 2008.</p>
|
3,686,921 | <blockquote>
<p>Prove that for all triangles with angles <span class="math-container">$\alpha, \beta, \gamma$</span>, <span class="math-container">$$\frac{\sin\alpha}{\cos\alpha + 1} + \frac{\sin\beta}{\cos\beta + 1} + \frac{\sin\gamma}{\cos\gamma + 1} = \frac{\cos\alpha + \cos\beta + \cos\gamma + 3}{\sin\alpha + \si... | lab bhattacharjee | 33,337 | <p>For <span class="math-container">$1+\cos\alpha\ne0,$</span>
<span class="math-container">$$\dfrac{\sin\alpha}{1+\cos\alpha}=\cdots=\tan\dfrac\alpha2$$</span></p>
<p>Now,
<span class="math-container">$$\tan\dfrac\alpha2+\tan\dfrac\beta2+\tan\dfrac\gamma2$$</span></p>
<p><span class="math-container">$$=\dfrac{\sin\l... |
3,882,566 | <p>We have <span class="math-container">$0<b≤ a$</span>, and:</p>
<p><span class="math-container">$$\underbrace{\dfrac{1+⋯+a^7+a^8}{1+⋯+a^8+a^9}}_{A} \quad \text{and} \quad \underbrace{\dfrac{1+⋯+b^7+b^8}{1+⋯+b^8+b^9}}_{B}$$</span></p>
<p>Source: Lumbreras Editors</p>
<hr />
<p>It was my strategy:</p>
<p><span clas... | Michael Rozenberg | 190,319 | <p><span class="math-container">$$1-A=1-\frac{1+a+...+a^8}{1+a+...+a^9}=\frac{a^9}{1+a+...+a^9}=$$</span>
<span class="math-container">$$=\frac{1}{\frac{1}{a^9}+\frac{1}{a^8}+...+1}\geq\frac{1}{\frac{1}{b^9}+\frac{1}{b^8}+...+1}=1-B,$$</span> which gives <span class="math-container">$$A\leq B.$$</span></p>
|
3,536,822 | <p>A man has three bags filled with balls. One bag contains balls weighing <span class="math-container">$9$</span> grams, the second bag contains balls weighing <span class="math-container">$10$</span> grams and the third bag contains balls weighing <span class="math-container">$11$</span> grams. The man got confused a... | lemontree | 344,246 | <p>Your basic idea is correct, but you are missing an important bit: There is an implicit universal quantification in the definitions:</p>
<blockquote>
<p>Soundness: <em>For all</em> formulas <span class="math-container">$A, B$</span>, if <span class="math-container">$A \vdash B$</span>, then <span class="math-conta... |
2,855,411 | <p>Find all real number(s) $x$ satisfying the equation $\{(x +1)^3\}$ = $x^3$ , where $\{y\}$ denotes the fractional part of $y$ , for example $\{3.1416\ldots\}=0.1416\ldots$.</p>
<p>I am trying all positive real numbers from $1,2,\dots$ but I didn't get any decimals.</p>
<p>Is there a smarter way to solve this pr... | Henry | 6,460 | <p>Hints:</p>
<ul>
<li>$0 \le \{y\} \lt 1$</li>
<li>so any solution to $\{(x +1)^3\} = x^3$ has $0 \le x^3 \lt 1$ and thus $0 \le x \lt 1$</li>
<li>so $1 \le x+1 \lt 2$ and $1 \le (x+1)^3 \lt 8$</li>
<li>any solution has $(x+1)^3 = x^3 +n$ for $n \in \{1,2,3,4,5,6,7\}$, which gives you seven quadratic equations to che... |
131,294 | <p>How do I show that $ f(t) = t^2 + t +1 $ is irreducible in $K[t]$, where $K = \{0,1\}$?</p>
<p>I know how to tackle this over $\mathbb{Z}$ or $\mathbb{Q}$ using Guass or Eisenstein say...but I'm a little unsure how to proceed in this case.</p>
<p>Any help is much appreciated.</p>
| Belgi | 21,335 | <p>As said by a couple of users, this polynomial does not have a root
in $\mathbb{F}_{2}$ and since it is of degree $2$, it is irreducible.</p>
<p>You ask if $f$ can have a quadratic factor or a factor of higher
degree, assume that such factor $g$ exist, i.e. $f=gh$ for some
polynomial $h\neq0$. </p>
<p>Then $\deg(f)... |
1,321,233 | <p>German Wikipedia states that the Ramsey`s theorem is a generalization of the Pigeonhole principle <a href="http://de.wikipedia.org/wiki/Satz_von_Ramsey" rel="noreferrer" title="source">source</a></p>
<p>But does not say why this is true. I am doing a presentation about the Ramsey theory and also wanna explain why t... | Hagen von Eitzen | 39,174 | <p>The pigeonhole principle states that for a given number of pigeons, if only there are enough holes then at least one empty hole is guaranteed to exist.</p>
<p>Ramsey' s theorem states that if only there are enough vertices then at least one thingy (e.g., red or blue triangle) is guaranteed to exist.</p>
|
3,299,661 | <p>I am familiar with all 3 of the entities I have listed in my question. I know the definitions of "reflexive", "symmetric", and "transitive". However, I am afraid I do not mechanistically understand the "flow" of how we ultimately generate equivalence classes from a particular relation that exhibits the 3 properties ... | Ruben | 386,073 | <p>If you'd extend your set just a bit you'd see transitivity happen. Consider the set <span class="math-container">$\{1,2,3,4,5,6,7 \}$</span> for example. Now your <span class="math-container">$R_1$</span> contains <span class="math-container">$(1,4), (4,7)$</span> and <span class="math-container">$(1,7)$</span>.</p>... |
2,094,123 | <p>A plane curve is printed on a piece of paper with the directions of both axes specified. How can I (roughly) verify if the curve is of the form $y=a e^{bx}+c$ without fitting or doing any quantitative calculation?</p>
<p>For example, for linear curves, I can choose two points on the curve and check if the midpoint ... | Community | -1 | <p>Assuming $c=0$ (it's not the graph of a exponential function otherwise):</p>
<p>Pick a point on the curve. Draw its tangent and extend it until it meets the $x$-axis. Also drop a vertical from the point to the $x$-axis. Now you have a right triangle.</p>
<p>Do this for lots of points. The bases of all the triangle... |
1,802,515 | <blockquote>
<p>Say you have a bank account in which your invested money yields 3% every year, continuously compounded. Also, you have estimated that you spend $1000 every month to pay your bills, that are withdrawn from this account.</p>
<p>Create a differential model for that, find its equilibriums and determine its ... | Doug M | 317,162 | <p>The ballance must be sufficient to generate $1,000 / month in cash flow</p>
<p>$B(e^{0.0025}-1) = 1000\\
B = \dfrac{1000}{e^{0.0025}-1} = \$399,500$</p>
|
1,615,177 | <p>The primes $p$ are, of course, in one-to-one correspondence with the squares of primes $p^2$. But is there any interval $a < x < b$ possible where the primes thin out so much, that it contains more squares of primes than primes?</p>
| Tito Piezas III | 4,781 | <p>I think the intent of your question was that the number of primes in the interval, call it $\pi(n)'$, is <strong><em>non-zero</em></strong>. If so, then the simplest case is,</p>
<p>$$p_n^2\leq x \leq p_{n+1}^2$$</p>
<p>for prime $p_i$. Your question then assumes the count as $\pi(n)'<2.$ However,</p>
<ol>
<li... |
129,875 | <p>The Fourier transform of the Heaviside step function $u(t)$ <a href="http://fourier.eng.hmc.edu/e101/lectures/handout3/node3.html" rel="nofollow noreferrer">is</a> $\dfrac{1}{iω} + π δ(ω)$.<br>
The Laplace transform of the same function <a href="http://leevaraiya.org/releases/LeeVaraiya_DigitalV2_02.pdf#page=569" re... | Julián Aguirre | 4,791 | <p>The integral defining $\mathcal{L}(x)$ converges for all $s>0$ (or, more generally, for all $s\in\mathbb{C}$ with $\operatorname{Re}(s)>0$.) However, the integral defining $\mathcal{F}(x)$ does not converge for any $\omega\in\mathbb{R}$. As noted n the comments, $\mathcal{F}(x)$ is not a function, but a distri... |
4,215 | <p>I suspect it is impossible to split a (any) 3d solid into two, such that each of the pieces is identical in shape (but not volume) to the original. How can I prove this?</p>
| David E Speyer | 448 | <p>You can certainly take a rectangular box, $2^{1/3} \times 2^{2/3} \times 2$ and slice it into two boxes of size $1 \times 2^{1/3} \times 2^{2/3}$.</p>
|
433,403 | <ol>
<li>Let F(x,y) be the statement, “x can fool y,” where the domain consists of all of the people in the world. Translate this statement into symbolic logic.
a. Everyone can be fooled by somebody.</li>
</ol>
<p>Would it be: For every x.y in W, F(x,y) is in W?</p>
<p>I am not getting the gist of this...</p>
| Austin Mohr | 11,245 | <p>I find it helpful to write an intermediate step that is halfway between English and symbolic logic.</p>
<p>"Everyone can be fooled by somebody" is the same as "For every person $x$, there exists a person $y$ such that $y$ can fool $x$". If we replace all the English parts with symbols, this becomes "$\forall x \exi... |
354,250 | <p><strong>Remark:</strong> All the answers so far have been very insightful and on point but after receiving public and private feedback from other mathematicians on the MathOverflow I decided to clarify a few notions and add contextual information. 08/03/2020.</p>
<h2>Motivation:</h2>
<p>I recently had an interesting... | Abdelmalek Abdesselam | 7,410 | <p>Well if you take out partial derivatives, at least quantum field theory and in particular conformal field theory will survive the massacre. The reason is explained in my MO answer:
<a href="https://mathoverflow.net/questions/259155/p-adic-numbers-in-physics/259160#259160">$p$-adic numbers in physics</a></p>
<p>One ... |
354,250 | <p><strong>Remark:</strong> All the answers so far have been very insightful and on point but after receiving public and private feedback from other mathematicians on the MathOverflow I decided to clarify a few notions and add contextual information. 08/03/2020.</p>
<h2>Motivation:</h2>
<p>I recently had an interesting... | rimu | 175,280 | <p>For one example of non-trivial physics without partial derivatives, one can look into Volume 1 of the Feynman lectures. In Chapter 28, Feynman starts to develop electrodynamics without partial derivatives — they only appear in Volume 2.</p>
<p>Instead of the Maxwell equation, Feynman uses a somewhat complex formula ... |
2,059,192 | <p>I was reading about Sobolev spaces and came across the notation $\dot{H}^1, \dot{H}^{-1}, \dot{H}^t$. I'm familiar with $H^1, H^{-1}, H^t$, but not the dot, and I can't find these spaces defined anywhere. Is this notation common, and could you explain it to me or point me to a reference?</p>
<p>I have more or less ... | Matt | 206,546 | <p>I have seen the dot notation to mean trace-free: $\dot{H}^1 \equiv H^1_0$. The space $L^2_t,H_x^2$ should mean that the function $u=u(x,t)$ is $L^2$ in time and $H^2$ in space. The norm would be</p>
<p>$$\|u\|^2_{L^2_t,H^2_x} = \int_0^T \|u(\cdot,t)\|^2_{H^2(\Omega)} dt.$$</p>
<p>In general, a common notation is $... |
842,271 | <p>Evaluation of $\displaystyle \int \frac{\sqrt[3]{x+\sqrt[4]{x}}}{\sqrt{x}}dx$</p>
<p>$\bf{My\; Try::}$ Let $x=t^4\;,$ Then $dx = 4t^3dt$</p>
<p>So Integral is $\displaystyle \int\frac{\sqrt[3]{t^4+t}}{t^2} \cdot 4t^3dt$</p>
<p>So Integral is $\displaystyle 4\int t^{\frac{7}{3}}\cdot (1+t^{-3})^{\frac{1}{3}}$</p>
... | Claude Leibovici | 82,404 | <p>As Pranav Arora showed, there is no nice answer to this antiderivative. Personally, the only way I can think about it is a Taylor expansion of the integrand followed by a term by term integration.</p>
<p>As, you did, starting with $x=t^4$,we have $$\frac{\sqrt[3]{x+\sqrt[4]{x}}}{\sqrt{x}}=t^{-\frac{5}{3}} \sqrt[3] ... |
2,471,680 | <p>I am working with a theorem and i need the reference of the above limit.
Kindly guide.</p>
| Community | -1 | <p>Not too hard to prove it from first principles.</p>
<p>We have to prove $$\forall \epsilon > 0: \exists N: \forall n > N: |k^n - 0|< \epsilon$$</p>
<p>which is the same as saying </p>
<p>$$\forall \epsilon > 0: \exists N: \forall n > N: k^n< \epsilon$$</p>
<p><strong>Proof:</strong> </p>
<p>Pi... |
2,802,156 | <p>I have a function:</p>
<p>$${{\mathop{\rm F}\nolimits} _i}\left( {\bf{\xi }} \right) = \sum\limits_k^N {{\mathop{\rm D}\nolimits} \left( {\frac{1}{N}\sum\limits_j^N {{\mathop{\rm G}\nolimits} \left( {j,{\mathop{\rm I}\nolimits} \left( {j,{\bf{\xi }}} \right)} \right)} - {\mathop{\rm G}\nolimits} \left( {k,{\mathop... | Fimpellizzeri | 173,410 | <p>Personally, I think the total derivative chain rule is easiest to remember:</p>
<p>$$D_{f\circ g}(x) = D_f(g(x)) \cdot D_g(x)$$</p>
<p>What the $D$'s look like $($as $m\times n$ matrices$)$ of course depends on the domain and codomain of each function.</p>
|
3,917,601 | <p>Let <span class="math-container">$(X,\mathcal{M},\mu)$</span> be a measure space. Suppose <span class="math-container">$E_n\in \mathcal{M}$</span> such that <span class="math-container">$$\sum_{n=1}^\infty \mu(E_n) < \infty$$</span> show <span class="math-container">$\mu(\lim\sup_{n\to\infty} E_n) = 0.$</span></p... | varpi | 677,270 | <p>Proof of the first part:</p>
<p>The result you mention is the first Borel-Cantelli Lemma. First recall that
<span class="math-container">$$\limsup_n E_n = \bigcap_{n=1}^\infty \bigcup_{k=n}^\infty E_k.$$</span>
By continuity from above (check the conditions as an exercise) and sub-additivity, we have
<span class="ma... |
2,752,511 | <p>Prove that if $X$ is Hausdorff, $\Delta=\{(x, x)\mid x\in X\}$ is closed in $X\times X$ (with the product topology).</p>
<p><strong>My attempt:</strong></p>
<p>Let $x_1, x_2\in X$ s.t. $x_1\ne x_2$.</p>
<p>There exist neighborhoods $U_1$ and $U_2$ of $x_1$ and $x_2$ that are disjoint.</p>
<p>$U_1\times U_2$ is a... | Siddhartha | 257,185 | <p>Let $(x, y)\in X\times X-\Delta$</p>
<p>$\implies(x, y)\in X\times X$ and $(x, y)\notin\Delta$</p>
<p>$\implies x, y\in X\text{ and }x\ne y$</p>
<p>There exist neighborhoods $U_x$ and $U_y$ of $x$ and $y$ respectively that are disjoint.</p>
<p>$U_x\times U_y$ is a basis element in the product topology on $X\time... |
187,545 | <p><span class="math-container">$\DeclareMathOperator\GL{GL}\DeclareMathOperator\L{\mathfrak{L}}$</span>The free Lie algebra <span class="math-container">$\L(V)$</span> generated by an <span class="math-container">$r$</span>-dimensional vector space <span class="math-container">$V$</span> is, in the
language of <a href... | Tom Church | 250 | <p>Fix a primitive <span class="math-container">$k$</span>-th root of unity <span class="math-container">$\zeta_k$</span>, and let <span class="math-container">$\rho\in S_k$</span> be a <span class="math-container">$k$</span>-cycle. (I am working over <span class="math-container">$\mathbb{C}$</span> here, obviously.) K... |
3,451,374 | <p>Given that I have a random variable <span class="math-container">$\max\{K-X, 0\}$</span> where <span class="math-container">$k>0$</span> is a constant and <span class="math-container">$x$</span> is uniformly distributed on <span class="math-container">$[-K, K]$</span> or I guess more generally with any distributi... | Kavi Rama Murthy | 142,385 | <p>In general <span class="math-container">$E\max \{K-X,0\}=E(K-X) I_{X \leq K}=KP(X \leq K)-EXI_{X \leq K}$</span>. In your special case this gives <span class="math-container">$E\max \{K-X,0\}=K-EX=K$</span></p>
|
2,941,456 | <blockquote>
<p>Given <span class="math-container">$K$</span> elements between <span class="math-container">$1$</span> and <span class="math-container">$7$</span> (inclusive), how many ways can you arrange the elements s.t. their sum adds to <span class="math-container">$N$</span>? </p>
</blockquote>
<p>I can brute-... | Rushabh Mehta | 537,349 | <p>The best way to solve this is via a <a href="https://en.wikipedia.org/wiki/Generating_function" rel="nofollow noreferrer">generating function</a>. We treat the powers of <span class="math-container">$x$</span> as the values of <span class="math-container">$k$</span></p>
<p>So, to represent the fact that each elemen... |
14,391 | <p>I am currently tutoring a student who is really lacking in math. At first I thought she was just resistant; perhaps she thought that her teacher had no choice but to pass her. What truly stunned me is that she is going to tenth-grade and has very little common sense with numbers.</p>
<p>I once asked her what is 3... | JRN | 77 | <p>I do not specialize in learning disabilities, but it's possible that your student has <em>dyscalculia</em>. From <a href="https://en.wikipedia.org/wiki/Dyscalculia" rel="noreferrer">Wikipedia</a>:</p>
<blockquote>
<p>Dyscalculia is difficulty in learning or comprehending arithmetic, such as difficulty in understand... |
3,638,028 | <p>find <span class="math-container">$f\circ f$</span> for the function <span class="math-container">$f\colon \mathbb R^2\to \mathbb R^2$</span> (,)=(−,)
I know that if (,)=(−,), then () is its inverse reflected about the -axis. If this is the case then <span class="math-container">$f\circ f$</span> = f^-1(−f^-1(−)). I... | Jeppe Stig Nielsen | 70,134 | <p>With colors:
<span class="math-container">$$\frac{\color{blue}{k}\color{red}{(k+1)}+\color{green}{2}\color{red}{(k+1)}}{2}
=\frac{(\color{blue}{k}+\color{green}{2})\color{red}{(k+1)}}{2}$$</span>
The red part <span class="math-container">$\color{red}{(k+1)}$</span> is seen in both terms of the left-hand-side, so can... |
1,241,970 | <p>A fair coin is tossed three times. Let $X$ be the number of heads that turn up on the first two tosses and $Y$ the number of heads that turn up on the third toss. Give the distribution of $X$, $Y$, $X + Y$, $X − Y$ and $XY$.</p>
| Karolina Sz | 232,662 | <p>Is it good answer?
$$X=\{0,1,2\}, Y=\{0,1\}$$
$$P(Y=1)=1/2, P(Y=0)=1/2, P(X=0)=1/4, P(X=1)=1/2, P(X=2)=1/4$$
$$P(X+Y=0)=1/8, P(X+Y=1)=3/8, P(X+Y=2)=3/8, P(X+Y=3)=1/8$$
$$P(X-Y=-1)=1/8, P(X-Y=0)=3/8, P(X-Y=1)=3/8, P(X-Y=2)=1/8$$
$P(XY=0)=5/8, P(XY=1)=2/8, P(XY=2)=1/8$ </p>
|
397,274 | <p>Suppose you have a group isomorphism given by the first isomorphism theorem:</p>
<p><span class="math-container">$$G/\ker(\phi) \simeq \operatorname{im}(\phi)$$</span></p>
<p>What can we say about the group <span class="math-container">$\ker(\phi)\times \operatorname{im}(\phi)$</span>? In particular, when does the f... | Najib Idrissi | 10,014 | <p>Let $N = ker(\phi)$ and $K = im(\phi)$, then you're asking when, given an exact sequence $1 \to N \to G \to K \to 1$ is trivial.</p>
<ul>
<li>First you need the extension to be split, that is, there must exist a morphism $s : K \to G$ such that the composition $\phi \circ s$ is the identity. In this case $G \simeq ... |
1,840,778 | <p>In rectangle $ABCD$, we have $AD = 3$ and $AB = 4$. Let $M$ be the midpoint of $\overline{AB}$, and let $X$ be the point such that $MD = MX$, $\angle MDX = 77^\circ$, and $A$ and $X$ lie on opposite sides of $\overline{DM}$. Find $\angle XCD$, in degrees. </p>
<p><img src="https://i.stack.imgur.com/3TsZm.png" alt="... | Huang | 18,931 | <p>Hint:
It's easy to see MC=MD, hence points D X and C are located on a circle, with M being the center. Now, we know angle XCD is half of angle DMX (why?)</p>
|
440,791 | <p>I am trying to figure out if the infinite product <span class="math-container">$$\omega=\frac{5\sqrt{3}}{12}\prod\limits_{\substack{p\equiv 1\pmod3 \\
p\ge 13}}\left(\frac{p-2}{p-1}\right)\prod\limits_{\substack{p\equiv 2\pmod3 \\
p\ge 13}}\left(\frac{p}{p-1}\right)$$</span> is asymptotically equal to the infinite p... | Henri Cohen | 81,776 | <p>You have almost obtained the answer yourself:
<span class="math-container">$$\prod_{p\equiv1\pmod{3}}(1-1/p)\prod_{p\equiv2\pmod{3}}(1+1/p)=1/L(\chi,1)$$</span>
where <span class="math-container">$\chi(p)=(-3/p)$</span> the Legendre symbol (as in KConrad's answer), and
<span class="math-container">$L(\chi,1)=\pi/(3\... |
43,505 | <p>I am looking to make a physics based Mathematica project. Ideally the project would take around 12 hours, gathering any experimental data and analyse the findings.</p>
<p>I'd have full access to university physics labs. The project would be for 2nd year physics students in the end and would aim to introduce using M... | Chris Degnen | 363 | <p>You could try making an intuitive explanation of Planck's constant. There are a several formulae and plots to interrelate and explain, notably Wein's law, the Raleigh-Jeans curve, and then Planck's law. Some plotting challenges.</p>
<p><a href="http://en.wikipedia.org/wiki/Planck_constant#Black-body_radiation" re... |
231,887 | <p>I'm learning to do proofs, and I'm a bit stuck on this one.
The question asks to prove for any positive integer $k \ne 0$, $\gcd(k, k+1) = 1$.</p>
<p>First I tried: $\gcd(k,k+1) = 1 = kx + (k+1)y$ : But I couldn't get anywhere.</p>
<p>Then I tried assuming that $\gcd(k,k+1) \ne 1$ , therefore $k$ and $k+1$ are no... | Bill Dubuque | 242 | <p><strong>Hint</strong> <span class="math-container">$\ $</span> Both <span class="math-container">$\rm\:1+k\:$</span> and <span class="math-container">$\rm\:k\:$</span> are multiples of <span class="math-container">$\rm\:c = gcd(k,1+k),\:$</span> and the set of multiples of any integer <span class="math-container">$\... |
231,887 | <p>I'm learning to do proofs, and I'm a bit stuck on this one.
The question asks to prove for any positive integer $k \ne 0$, $\gcd(k, k+1) = 1$.</p>
<p>First I tried: $\gcd(k,k+1) = 1 = kx + (k+1)y$ : But I couldn't get anywhere.</p>
<p>Then I tried assuming that $\gcd(k,k+1) \ne 1$ , therefore $k$ and $k+1$ are no... | cansomeonehelpmeout | 413,677 | <p>Let <span class="math-container">$d=\gcd(k.k+1)$</span>, then <span class="math-container">$d\mid k+1-k=1$</span>. So <span class="math-container">$d=\pm 1$</span>.</p>
|
299,824 | <p>I've posted this <a href="https://math.stackexchange.com/q/2733614/214353">to Math.SE</a> about a month ago:</p>
<p>Seems like
$$
\Delta(a_0+a_1t^d+a_2t^{2d}+...+a_nt^{nd})=(-1)^{n\frac{d(d-1)}2}d^{nd}(a_0a_n)^{d-1}[\Delta(a_0+a_1t+a_2t^2+...+a_nt^n)]^d,
$$
where $\Delta$ is the discriminant.</p>
<p>Presumably thi... | Igor Rivin | 11,142 | <p>I don't know of a standard reference, but there is <a href="http://faculty.bard.edu/cullinan/disccomp.pdf" rel="nofollow noreferrer">a note by John Cullinan at Bard College</a> where he computes the discriminant of a <em>composition</em> of two polynomials, so a nontrivially more general result than the one you ment... |
299,824 | <p>I've posted this <a href="https://math.stackexchange.com/q/2733614/214353">to Math.SE</a> about a month ago:</p>
<p>Seems like
$$
\Delta(a_0+a_1t^d+a_2t^{2d}+...+a_nt^{nd})=(-1)^{n\frac{d(d-1)}2}d^{nd}(a_0a_n)^{d-1}[\Delta(a_0+a_1t+a_2t^2+...+a_nt^n)]^d,
$$
where $\Delta$ is the discriminant.</p>
<p>Presumably thi... | Joe Silverman | 11,926 | <p>You say "presumably not difficult to prove." Did you try? It seems like a pretty easy exercise, using
$$ \Delta(F(t)) = \prod_{F(a)=0} F'(a). $$
Taking $F(t)=f(t^d)$, the roots of $F$ are the $d$'th roots of the roots of $f$, while $F'(t)=dt^{d-1}f'(t^d)$. So I doubt you'll find this formula in a reference, but if y... |
75,900 | <p>I have got a following equation: </p>
<pre><code>-(c - x)/Sqrt[b^2 + (c - x)^2] + x/Sqrt[a^2 + x^2] == 0.
</code></pre>
<p>Trying to solve it for x, so I evaluate</p>
<pre><code>Solve[-(c-x)/Sqrt[b^2+(c-x)^2]+x/Sqrt[a^2+x^2] == 0,x].
</code></pre>
<p>This produces</p>
<blockquote>
<pre><code>{{x -> (a*c)/(a ... | m_goldberg | 3,066 | <p>Your argument concluding that <code>(a c)/(a + b)</code> is a root but <code>(a c)/(a - b)</code> is not a root is not sound. </p>
<p>You accept that <code>(a c)/(a + b)</code> is a root because your equation is satisfied by the triple <code>{a, b, c} = {2, 3, 4}</code>. However, it immediately follows that <code>(... |
3,132,380 | <p>To compute this I used the fact that <span class="math-container">$S(n,2) = 2^{n-1}-1$</span> and used the recurrence relation <span class="math-container">$S(n,k) = kS(n-1,k) + S(n-1,k-1)$</span>, and used induction to get that <span class="math-container">$S(n,3)=\dfrac{3^{n-1}+1}{2}-2^{n-1}$</span>.</p>
<p>But i... | Marko Riedel | 44,883 | <p>With Stirling numbers of the second kind we have the combinatorial
class</p>
<p><span class="math-container">$$\def\textsc#1{\dosc#1\csod}
\def\dosc#1#2\csod{{\rm #1{\small #2}}}
\textsc{SET}_{=k}(\textsc{SET}_{\ge 1}(\mathcal{Z})).$$</span></p>
<p>This yields the EGF</p>
<p><span class="math-container">$$\frac{(... |
1,953,517 | <p>Let $X,Y$ be two independet Poisson variables with parameters $\mu,\lambda>0$. Let $N:=Y+X$
what is $\mathbb{E}(X\vert N=n)$?</p>
<p>I already computed $P(X=k\vert N=n)$ for $k,n\in \mathbb{Z}_{+}$ which is $$P(X=k\vert N=n)=\binom{n}{k}\frac{\mu^{n-k}\lambda^k}{(\mu+\lambda)^n}$$ if $n>k$ else $0$.</p>
<p>I... | Benson Lin | 371,844 | <p>First, remove 20 bikes at the beginning. This is for the "at least" restraints in the question. Now consider for each possible value of the number of bikes in warehouse 2, which are $10,11,\cdots,20$.</p>
<p>For each case, we remove another $0$ to $10$ bikes.</p>
<p>Now we simply have to divide the remaining bikes... |
327,990 | <p>So i was working on this:</p>
<p>$$
\lim\limits_{x\to1} \frac{x + \sqrt{x} - 2}{x - 1}
$$</p>
<p>and I thought to simpify my top by multiplying by a conjugate, taking everything other than the $x$ to be the $b$ from $a+b$ so that my conjugate looked like $x - \sqrt{x} + 2$.</p>
<p>The multiplication, if correct, ... | André Nicolas | 6,312 | <p><strong>Hint:</strong> I think you will find things <strong>much</strong> easier if you let $x=t^2$. This makes no real mathematical difference, but will send you in the right direction. The $\sqrt{x}$ was causing unnecessary confusion. After the substitution, the work will take seconds only.</p>
|
1,700 | <p>Is there an algorithm in literature to compute an efficient (pareto optimal) and envy-free cake cutting when there are only $n=2$ players and a mediator?</p>
| Joseph Malkevitch | 1,618 | <p>Take a look at:</p>
<p><a href="http://ideas.repec.org/p/pad/wpaper/0022.html" rel="nofollow">http://ideas.repec.org/p/pad/wpaper/0022.html</a></p>
<p>or the description of Crawford Divide and Choose as described in the book Equity: In Theory and Practice, by H. Peyton Young, Princeton U. Press.</p>
|
819,704 | <p>Here is the problem I have </p>
<p>$\lim \limits_{x \to -1} (x + 1)^2 sin (\frac{1}{x + 1})$</p>
<p>I approached it like this:</p>
<p>\begin{align}
-1 \le sin(\displaystyle \frac{1}{x + 1}) \le 1 \\
-(x + 1) \le sin(1) \le (x + 1)
\end{align}</p>
<p>I then go on to solve the limit by replacing $sin (\frac{1}{x ... | DanZimm | 37,872 | <p>There appears to be a silly mistake which @mm-aops touched on, but I want to point out more directly.</p>
<p>If we have an inequality like so:</p>
<p>$$
a \le f\left( \frac{b}{c} \right) \le d
$$
then we cannot conclude that
$$
a \cdot c \le f(b) \le d \cdot c
$$
If you want a direct counter example consider $f(x)... |
2,712,631 | <p>The problem is to use a power series to evaluate the integral to six decimal places. The upper limit of integration is one and the lower limit of integration is zero.</p>
<p>To start the problem I factored $x$ out and focused on $\arctan(3x)$.
I knew that by taking the derivative I could get this equation
in the f... | Claude Leibovici | 82,404 | <p><em>Just added for your curiosity.</em></p>
<p>After geometryfan's answer, writing
$$\arctan(3x)=\sum_{n=0}^{p-1}(-1)^n\frac{(3x)^{2n+1}}{2n+1}$$ you search $p$ such that, for a given value of $x$
$$\frac{(3x)^{2p+1}}{2p+1} \leq \epsilon$$ For simplicity, let $3x=a$ and $2p+1=k$.</p>
<p>The solution of $a^k=\epsi... |
238,577 | <p>The following working program uses
<a href="https://mathematica.stackexchange.com/questions/58560/graph-and-markov-chain">Graph and Markov Chain</a></p>
<pre><code>P = {{1/2, 1/2, 0, 0}, {1/2, 1/2, 0, 0}, {1/4, 1/4, 1/4, 1/4}, {0, 0,
0, 1}}; proc = DiscreteMarkovProcess[3, P];
Graph[proc, GraphStyle -> "... | florin | 54,979 | <p>@kglr Thanks! Your code can be shortened a bit, since tm2 is precisely P, to</p>
<pre><code> P = {{0, 1/4, 1/2, 1/4, 0, 0}, {0, 1, 0, 0, 0, 0}, {0, 0, 1/3, 0, 2/3, 0}, {0, 0, 0, 0, 0, 1}, {0, 0, 1/4, 0, 3/4, 0}, {1/4, 0, 0, 0, 3/4, 0}};
proc = DiscreteMarkovProcess[1, P];
Graph[proc, EdgeLabels -> {DirectedE... |
288,974 | <p>Alright this maybe really funny but I want to know why is this wrong. We often come across identities which we prove by multiplying both the sides of the identity by a certain entity but why don't we multiply it by $0$. That way every identity will be proved in one single line. That is so stupid. I mean, by that way... | Kaz | 28,530 | <p>You are confusing derivation with proving.</p>
<p>If you want to prove that some $X = Y$ statement is true, you have to show that that statement can be derived from some other statement which is already known to be true. You're doing it backwards: you're deriving from $X = Y$ some statement which is true, namely $0... |
288,974 | <p>Alright this maybe really funny but I want to know why is this wrong. We often come across identities which we prove by multiplying both the sides of the identity by a certain entity but why don't we multiply it by $0$. That way every identity will be proved in one single line. That is so stupid. I mean, by that way... | notes | 39,595 | <p>The "usual way" is often used with inequalities, but we take care to make sure that we can reverse our steps. For instance, suppose we want to prove
$$a+b \geq 2\sqrt{ab}.$$
Since both sides are positive, this statement is true iff
$$(a+b)^2 \geq 4ab,$$
which is true iff
$$(a+b)^2-4ab \geq 0 \Leftrightarrow (a-b)^2 ... |
3,110,660 | <p>Let <span class="math-container">$f:\mathbb{R}^n\to \mathbb{R}^n$</span> a function of class <span class="math-container">$C^1$</span> such that <span class="math-container">$Df(x)$</span> is invertible for all <span class="math-container">$x\in \mathbb{R}^n$</span>. Show that <span class="math-container">$\{x\in \m... | zhw. | 228,045 | <p>Let <span class="math-container">$Z$</span> be the zero set of <span class="math-container">$f.$</span> If <span class="math-container">$Z$</span> were uncountable, it would have a limit point <span class="math-container">$a\in \mathbb R^n.$</span> Thus there would exist a sequence of distinct points <span class="ma... |
292,122 | <p>This question actually came out of a question. In some other post, I saw a reference and going through, found this, $n>0$.</p>
<p>Solve for n explicitly without calculator:
$$\frac{3^n}{n!}\le10^{-6}$$</p>
<p>And I appreciate hint rather than explicit solution.</p>
<p>Thank You.</p>
| Ross Millikan | 1,827 | <p>I would use <a href="http://en.wikipedia.org/wiki/Stirling%27s_approximation" rel="nofollow">Stirling's approximation</a> $n!\approx \frac {n^n}{e^n}\sqrt{2 \pi n}$ to get $\left( \frac {3e}n\right)^n \sqrt{2 \pi n} \lt 10^{-6}$. Then for a first cut, ignore the square root part an set $3e \approx 8$ so we have $\l... |
292,122 | <p>This question actually came out of a question. In some other post, I saw a reference and going through, found this, $n>0$.</p>
<p>Solve for n explicitly without calculator:
$$\frac{3^n}{n!}\le10^{-6}$$</p>
<p>And I appreciate hint rather than explicit solution.</p>
<p>Thank You.</p>
| half-integer fan | 54,125 | <p>How about we <strong>overestimate</strong> $3^n$ as $\sqrt{10}^n$ and <strong>underestimate</strong> every contribution to the factorial beyond $10$ as only $10$? Then $$\frac{3^n}{n!} \le \frac{10^5 \cdot 10^{\frac{n-10}2}}{10! \cdot 10^{n-10}} $$ Since $10!$ is about $10^{6.5}$ we have $$ 10^{-1.5} \cdot 10^{-\fr... |
801,562 | <p>We consider that $R$ is a commutative ring with $1_R$.</p>
<p>Each $c \in R^*$(if we see it as a constant polynomial), divides each polynomial of $R[X]$.</p>
<p>($c \in R^*$ means that $c$ is invertible.)</p>
<p>I haven't undersotod it..Could you explain it to me?</p>
<p>Does it mean that if we have a polynomial... | rschwieb | 29,335 | <p>If "reversible" mean "invertible," then this is true because invertible elements divide everything. (This is true in any ring with units and doesn't depend on the polynomial ring.)</p>
<p>If $u$ is invertible, then $a=(au^{-1})u$, and this says that $u|a$.</p>
|
3,142,339 | <p>Let <span class="math-container">$p$</span> be a real number. I am looking for all <span class="math-container">$(x,y)$</span> such that <span class="math-container">$\ln[e^{x}+e^{y}]=px+(1-p)y$</span>. My effort:</p>
<p>Take exponent of both sides to obtain <span class="math-container">$e^{x}+e^{y}=e^{px}e^{(1-p)y... | Lubin | 17,760 | <p>Well, Vasily has put his finger on one problem, but I would like to point out a much more serious one.</p>
<p>We write a continued fraction to get a number that is the limit of the convergents, that is, of the expressions that you get when you cut your continued fraction off, to be a finite c.f.</p>
<p>The first c... |
1,507,181 | <p><a href="https://i.stack.imgur.com/nuhUB.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/nuhUB.png" alt="enter image description here"></a></p>
<p><em>We know $G(0) = 0$</em></p>
<p>Okay, so I have the above graph but I'm having a difficult time translating it into the graph of $G(x)$.</p>
<p>W... | tomi | 215,986 | <p>Another way to attack this problem is to draw a tangent field diagram. This is usually done when the gradient is a function of both $x$ and $y$, but there is no reason why not to go ahead in this instance.</p>
<p>The basic idea is to draw a short line with gradient $G'(x)$ at every point $(x,y)$:</p>
<p><a href="h... |
934,660 | <p>Prove that for $ n \geq 2$, n has at least one prime factor.</p>
<p>I'm trying to use induction. For n = 2, 2 = 1 x 2. For n > 2, n = n x 1, where 1 is a prime factor. Is this sufficient to prove the result? I feel like I may be mistaken here.</p>
| Francesco Alem. | 175,276 | <p>let $n \in \mathbb{N}, n \geq 2$ and let's consider cases:</p>
<p>$n\mbox{ is prime}$: done. </p>
<p>$n\mbox{ is not prime} \iff n \mbox{ is mixed} \implies n=\prod_{i=1}^j P_i^{q_i}$, where $P_k \mbox { is prime}$, $q_k \mbox { is the exponent}$ $\implies$ $n$ has prime factors: done.</p>
|
3,604,745 | <p>(I Prefer to open new question because those are my homework and i want to understand my way)</p>
<p>In my homework i need to solve the integral: </p>
<p><span class="math-container">$$
\int \frac{e^x}{2e^x + \sqrt{e^x}}dx
$$</span></p>
<p>I tried the substitution method: </p>
<p><span class="math-container">$$
... | jeea | 550,450 | <p>Following your work:</p>
<p><span class="math-container">$$\int \frac{dt}{2t + \sqrt{t}} = \int\frac{dt}{\sqrt{t}(2\sqrt{t}+1)} = \ln(2\sqrt{t}+1)+c$$</span></p>
<p>This is because the derivative of <span class="math-container">$2\sqrt{t}+1 = \frac{1}{\sqrt{t}}$</span> is present so essentially we have <span class... |
747,400 | <p>I've run into something confusing.</p>
<p>The problem is that I have to find the power series representation of $g(x)$ using the given $f(x)$, specifically $g(x) = \ln(1 - 3x)$ using $f(x) = \frac{1}{1-x}$.</p>
<p>Now, I did it, but my result is incorrect:
$$\ln(1-3x) = \int \frac{dx}{1-3x} = \int \sum 3^{k}x^{k} ... | anon | 11,763 | <p>If you interpret $\underbrace{A\oplus\cdots\oplus A}_n$ as $\bigoplus_{i\in I}A$ for infinite $n$ then you're right back at the definition.</p>
|
747,400 | <p>I've run into something confusing.</p>
<p>The problem is that I have to find the power series representation of $g(x)$ using the given $f(x)$, specifically $g(x) = \ln(1 - 3x)$ using $f(x) = \frac{1}{1-x}$.</p>
<p>Now, I did it, but my result is incorrect:
$$\ln(1-3x) = \int \frac{dx}{1-3x} = \int \sum 3^{k}x^{k} ... | rschwieb | 29,335 | <p>Sure, it is always true that a free module (under this definition) will be isomorphic to $A^{(I)}$ for some index set $I$, no matter if the set $I$ is finite or infinite.</p>
<p>Suppose you are given $M=\bigoplus_{i\in I} M_i$ with isomorphisms $\phi_i:M_i\to A$. This can be composed with the map that injects $A$ i... |
173,387 | <p>How can I indent properly long code in <em>Mathematica</em>?
Are there some best practices?</p>
| GenericAccountName | 38,159 | <p>I have a feeling many are going to find things they don't like slightly about my indentation, and this is totally a subjective question based on preference. I don't think there's a standardized best practice for coding style.</p>
<p>For me, similarly to other languages, have the brackets line up vertically with the... |
3,055,324 | <p>I need some help with constructing a proof for the following statement,<span class="math-container">$ \frac{P_1 P_2}{hcf(m,n)} = lcm(P_1,P_2)$</span> where <span class="math-container">$P_1$</span> and <span class="math-container">$P_2$</span> are polynomials with real coefficients.</p>
<p>I know how to do the sam... | Joel Pereira | 590,578 | <p>Think of the irreducible factors of P<span class="math-container">$_1$</span> and P<span class="math-container">$_2$</span> as your prime factors. Suppose P<span class="math-container">$_1$</span> = gcd(P<span class="math-container">$_1$</span>,P<span class="math-container">$_2$</span>)(<span class="math-container">... |
3,055,324 | <p>I need some help with constructing a proof for the following statement,<span class="math-container">$ \frac{P_1 P_2}{hcf(m,n)} = lcm(P_1,P_2)$</span> where <span class="math-container">$P_1$</span> and <span class="math-container">$P_2$</span> are polynomials with real coefficients.</p>
<p>I know how to do the sam... | William Grannis | 332,311 | <p>Do it the exact same way. Suppose that the hcf/gcd of <span class="math-container">$P_1$</span> and <span class="math-container">$P_2$</span> is <span class="math-container">$G$</span>. Because <span class="math-container">$G$</span> is a factor of <span class="math-container">$P_1$</span>, there exists an <span c... |
549,347 | <p>How would I solve the following question. And determine if its true or false.</p>
<p>1.$\forall x \in R , \exists y\in R, x^2+y^2=-1$</p>
<p>2: $\exists x\in R,\forall y \in R, x^2+y^2=-1$</p>
<p>For the first one I think I can justify it is false.</p>
<p>As for any arbitrary x must y must be </p>
<p>$y=\sqrt{-... | meh | 70,191 | <p>Another simple way to think about this questions is that for any
$$
x ∈ \mathbb{R}, x^2 \geq 0
$$</p>
<p>Then $$x^2 + y^2 \geq 0 $$</p>
|
2,566,193 | <p>Let's say we draw $49$ numbers from $1,\ldots,100$ without returning them back. Then we use the arithmetic mean from the sample.
$$M=\frac{1}{49}\sum_{i=1}^{49}X_i$$</p>
<p>They gave me the hint that $M$ is roughly normal distributed despite the dependencies in the draws. Now I have to determine an symmetric interv... | Dietrich Burde | 83,966 | <p>In addition to Jose's answer, we can give explicit $3\times 3$ matrices $A$ satisfying $A^2+4A+3I=0$, with different traces, e.g., $A=-I$, or $A=-3I$, or
$$
A=\begin{pmatrix} 0 & 1/r & -4 \cr r & 0 & -4r \cr 1 & 1/r & -5\end{pmatrix}
$$</p>
<p>$$
A=\begin{pmatrix} 0 & 9/r & -12 \cr r... |
1,842,340 | <p>A polynomial with integer coefficients is called
primitive if its coefficients are relatively prime. For
example, $$3{x^2} + 7x + 9$$ is primitive while $$10{x^2} + 5x + 15$$
is not.</p>
<p>(a) Prove that the product of two primitive polynomials is primitive.</p>
<p>(b) Use this to prove Gauss's Lemma: If a polyno... | Jyrki Lahtonen | 11,619 | <p>Supplementing Benjamin Lindqvist's answer with the details of how this can be shown <em>in this case</em> from first principles. All this assuming that the codes are to be binary for larger field alphabets the claim is false.</p>
<p>Assume that a binary linear code $C$ of length $n=k+2$, dimension $k$ and minimum d... |
444,486 | <p>I am teaching myself real analysis, and in this particular set of lecture notes, the <a href="http://www.math.louisville.edu/~lee/RealAnalysis/IntroRealAnal-ch01.pdf" rel="nofollow">introductory chapter on set theory</a> when explaining that not all sets are countable, states as follows:</p>
<blockquote>
<p>If $S... | celtschk | 34,930 | <p>The difference is that in the complex plane, you've got a multiplication $\mathbb C\times\mathbb C\to\mathbb C$ defined, which makes $\mathbb C$ into a field (which basically means that all the usual rules of arithmetics hold.)</p>
|
444,486 | <p>I am teaching myself real analysis, and in this particular set of lecture notes, the <a href="http://www.math.louisville.edu/~lee/RealAnalysis/IntroRealAnal-ch01.pdf" rel="nofollow">introductory chapter on set theory</a> when explaining that not all sets are countable, states as follows:</p>
<blockquote>
<p>If $S... | Emily | 31,475 | <p>The big difference between $\mathbb{R}^2$ and $\mathbb{C}$: differentiability.</p>
<p>In general, a function from $\mathbb{R}^n$ to itself is differentiable if there is a linear transformation $J$ such that the limit exists:</p>
<p>$$\lim_{h \to 0} \frac{\mathbf{f}(\mathbf{x}+\mathbf{h})-\mathbf{f}(\mathbf{x})-\ma... |
444,486 | <p>I am teaching myself real analysis, and in this particular set of lecture notes, the <a href="http://www.math.louisville.edu/~lee/RealAnalysis/IntroRealAnal-ch01.pdf" rel="nofollow">introductory chapter on set theory</a> when explaining that not all sets are countable, states as follows:</p>
<blockquote>
<p>If $S... | Kendra Lynne | 83,385 | <p>Since everyone is defining the space, I figured I could give an example of why we use it (relating to your "Electrical Engineering" reference). The <span class="math-container">$i$</span> itself is what makes using complex numbers/variables ideal for numerous applications. For one, note that:</p>
<p><span class="ma... |
12,359 | <p>There are, IMO, quite a lot of badly tagged questions and... not very good tags. Some of them were discussed on meta recently; some of these discussions show, IMO, that users who created these tags don't always understand tagging system of Math.SE well enough.</p>
<p>On Meta.SO one needs 500 rep to create a new tag... | Grace Note | 14,141 | <p>After a quick check on this, we've increased the threshold to the suggested 1000 reputation. That seems a saner number than 1500 for here. Only a handful of tags are created in the 1000-1500 reputation range, many of which were things like <a href="https://math.stackexchange.com/questions/tagged/conditional-probabil... |
189,069 | <p>The Survival Probability for a walker starting at the origin is defined as the probability that the walker stays positive through n steps. Thanks to the Sparre-Andersen Theorem I know this PDF is given by</p>
<pre><code>Plot[Binomial[2 n, n]*2^(-2 n), {n, 0, 100}]
</code></pre>
<p>However, I want to validate this ... | kirma | 3,056 | <p>Count number of steps before random walk value either goes negative or over <span class="math-container">$m$</span> steps are already taken, for <span class="math-container">$n$</span> walks. Then count amount of last successful steps on each integer bin, reverse it, accumulate these values (essentially extend last ... |
1,488,388 | <p><strong>The Statement of the Problem:</strong></p>
<p>Let $G$ be a finite abelian group. Let $w$ be the product of all the elements in $G$. Prove that $w^2 = 1$.</p>
<p><strong>Where I Am:</strong></p>
<p>Well, I know that the commutator subgroup of $G$, call it $G'$, is simply the identity element, i.e. $1$. But... | fleablood | 280,126 | <p>List the elements of G as {$g_1, ...., g_n$} for each $g_i$ there is precisely one element $g_i^{-1}$ so that $g_i^{-1}g_i = 1$. It is possible that $g_i = g_i^{-1}$ or it is possible $g_i^{-1} = g_j$ for some other j. It doesn't matter.</p>
<p>The set of all inverses {$g_1^{-1}.... g_n^{-1}$} is the same set as ... |
2,792,061 | <p>I would like to understand how to apply <em>well-founded induction</em>. I have found two definitions which I list now, followed by the question.</p>
<blockquote>
<p>(1) A binary relation $\prec$ is <a href="http://www.cs.cornell.edu/courses/cs6110/2013sp/lectures/lec07-sp13.pdf" rel="noreferrer"><em>well-founded... | tchappy ha | 384,082 | <p>I am reading "Linear Algebra Done Right 3rd Edition" by Sheldon Axler.</p>
<p>This book contains the following problem. (Exercise 2.A 16 on p.38)</p>
<blockquote>
<p>Suppose <span class="math-container">$p_0, p_1, \dots, p_m$</span> are polynomials in <span class="math-container">$\mathcal{P}_m(\mathbb{F})... |
2,503,306 | <p>Suppose $g{^n}$=e. Show the order of $g$ divides $n$.</p>
<p>Would I use Eulers Theorem???;</p>
<p>$a{^{\phi p}}$ $\equiv1 \pmod p$</p>
<p>$a{^{p-1}}\equiv1 \pmod p$</p>
<p>$a{^p}\equiv a\pmod p$</p>
<p>So then I would have </p>
<p>$g{^n}\equiv g\pmod n$</p>
<p>then I think you use the $\gcd$, which states $\... | Asinomás | 33,907 | <p>You don't need Euler's theorem. You just need division with remainder.</p>
<p>Suppose the order of $g$ is $d$ and $g^n=e$.</p>
<p>Suppose $n$ is not a multiple of $d$, then $n=kd+r$ with $0<r<d$.</p>
<p>It follows that $e=g^n=g^{kd}g^r=eg^r=g^r\implies g^r=e,$ contradicting the fact that $d$ is the order of... |
203,505 | <p>Let <span class="math-container">$P(x)$</span> be a non-constant polynomial with real coefficients.</p>
<p>Can <a href="http://en.wikipedia.org/wiki/Natural_density" rel="noreferrer">natural density</a> of</p>
<p><span class="math-container">$$\{n\ |\ \lfloor P(n)\rfloor \ \text{is prime.}\}$$</span></p>
<p>be posit... | David E Speyer | 297 | <p>No. Let $\omega(p)$ be the number of roots of $f$ modulo $p$. Clearly, for any finite set $S$, the upper asymptotic density of your set is bounded by $\prod_{p \in S} (1-\omega(p)/p)$. (Because the probability that $p \nmid f(n)$ is $1-\omega(p)/p$, these probabilities are independent for distinct primes, and $f(n)$... |
138,708 | <p>I'm dealing with derivatives of scalar functions of matrices and wondering if Mathematica can help me here.</p>
<p>The standard approach of expanding it in terms of components is cumbersome. As an motivating example, I want to minimize the following function, where $X$ is a matrix</p>
<p>$$f(X) = \text{tr}(X'X)$$... | mikado | 36,788 | <p>It is certainly fairly easy to check these relations for specific sizes of array:</p>
<pre><code>X = Array[x, {7, 11}];
Map[D[Tr[Transpose[X].X], #] &, X, {2}] == 2 X
(* True *)
Y = Array[y, {17, 17}];
Map[D[Tr[Y.Y], #] &, Y, {2}] == 2 Transpose[Y]
(* True *)
</code></pre>
<p>Generalisation to the determi... |
2,917,535 | <p>I have found this problem in a 10th grade textbook and it's given me headaches trying to solve it. It says, determine the set:</p>
<p>$$ A = \left \{ x \in \mathbb Z| \root3\of{\frac{7x+2}{x+5}} \in \mathbb Z\right \} $$</p>
<p>So I have to find a condition for x so that the expression under the radical is a perfe... | fleablood | 280,126 | <p>Let $ \root3\of{\frac{7x+2}{x+5}} = N$</p>
<p>$\frac {7x+2}{x+5} = N^3$</p>
<p>$\frac {7x + 35 - 33}{x + 5} = N^3$</p>
<p>$\frac {7x + 35}{x+5} - \frac {33}{x+5} = N^3$</p>
<p>$7 - \frac {33}{x+5} = N^3 \in \mathbb Z$.</p>
<p>So $x+5$ divides $33$. So $x+5 =\pm 1, \pm 3, \pm 11$ or $\pm 33$.</p>
<p>That means... |
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