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 |
|---|---|---|---|---|
4,546,415 | <p>Say <span class="math-container">$I = \mathbb{N} \setminus \{0, 1\}$</span> and</p>
<p><span class="math-container">$$A(n) = \left\{x \in \mathbb{R}\,\middle|\, −1−
\frac{1}{n}
< x \leq
\frac{1}{n}
\text{ or } 1−
\frac{1}{n} \leq x < 2−
\frac{1}{n}
\right\}$$</span>
with <span class="math-container">$n \in I$<... | Nicholas Todoroff | 1,068,683 | <p>To try to reduce clutter, I will use the following notation for the derivative:
<span class="math-container">$$
D^h[f(x)]
$$</span>
where <span class="math-container">$x$</span> is implicitly being differentiated, <span class="math-container">$h$</span> is the point at which the linear map <span class="math-contai... |
3,118 | <p>Can anyone help me out here? Can't seem to find the right rules of divisibility to show this:</p>
<p>If $a \mid m$ and $(a + 1) \mid m$, then $a(a + 1) \mid m$.</p>
| Pete L. Clark | 299 | <p>It is not surprising that you are finding this difficult, because it goes beyond basic divisibility rules -- it rather requires something which is essentially equivalent to the uniqueness of prime factorization. [<b>Edit</b>: Actually this is comment is incorrect -- as Robin Chapman's answer shows, it <em>is</em> p... |
2,448,074 | <p>A red die, a blue die, and a yellow die (all six-sided) are rolled. We are interested in the probability that the number appearing on the blue die is less than that appearing on the yellow die which is less than that appearing on the red die $P(B < Y < R)$.</p>
<p>I know one way to calculate this is:</p>
<p>... | N. F. Taussig | 173,070 | <p>There are $\binom{6}{3}$ ways to obtain three different numbers when three dice are thrown. Given such as outcome, there is only one way to arrange these numbers such that $B < Y < R$. Hence, the probability that the number appearing on the blue die is less than the number appearing on the yellow die and the... |
2,448,074 | <p>A red die, a blue die, and a yellow die (all six-sided) are rolled. We are interested in the probability that the number appearing on the blue die is less than that appearing on the yellow die which is less than that appearing on the red die $P(B < Y < R)$.</p>
<p>I know one way to calculate this is:</p>
<p>... | Donald Splutterwit | 404,247 | <p>Your configuration is of size $216$. You can get the same value ($aaa$) on all $3$ dice in $6$ ways. You can get two equal value and the third different ($aab$) in $6 \times 5 \times 3=90$ ways (the dice taking the value $b$ can be chosen in $3$ different ways). You can get all $3$ different ($abc$) in $6 \times 5 \... |
3,385,921 | <p>What does this mean? </p>
<blockquote>
<p>A matrix is diagonizable if and only if its eigenvectors are invertible.</p>
</blockquote>
| user | 505,767 | <p>It is meaningless! What is true is that a <span class="math-container">$n\times n$</span> matrix is diagonalizable if and only if a <strong>basis of eigenvectors</strong> exists, that is a set of <strong>n linearly independent eigenvectors</strong> exists.</p>
|
3,310,027 | <p>Is the expression <span class="math-container">$\forall a,b,c \in M : \varphi(a,b,c)$</span> equivalent to <span class="math-container">$\forall a \forall b \forall c : (a \in M \land b \in M \land c \in M) \rightarrow \varphi(a,b,c)$</span> ?</p>
| Mark Kamsma | 661,457 | <p>Yes, the former notation is an abbreviation of the latter. We often even leave out the commas, so you would see <span class="math-container">$\forall abc \in M : \varphi(a,b,c)$</span>.</p>
<p>A similar abbreviation is used for the existential quantifier:
<span class="math-container">$$
\exists abc \in M : \varphi(... |
2,043,429 | <p>In my textbook, it states that the general formula for the partial sum </p>
<p>$$\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}$$</p>
<p>My question is, if I have the following sum instead:</p>
<p>$$\sum_{i=1}^n \frac{1}{i^2}$$ </p>
<p>Can I just flip the general formula to get this:</p>
<p>$$\sum_{i=1}^n \frac{1}{i... | Thomas Andrews | 7,933 | <p>As $n$ increases, both sums increase. So it is not possible for the sums to be inverses.</p>
|
4,573,004 | <p>I want to solve <span class="math-container">$y'' +y^3 = 0$</span> with the boundary conditions <span class="math-container">$y(0) = a$</span> and <span class="math-container">$y(k) = b$</span>. My goal is to reduce this problem to <span class="math-container">$y' +y^2 = 0$</span> while solving but I'm not sure it c... | Átila Correia | 953,679 | <p>Since <span class="math-container">$B^{c}\cap A^{c}\subseteq B^{c}$</span>, we may claim the desired result:</p>
<p><span class="math-container">\begin{align*}
(B^{c}\cup(B^{c} - A))^{c} & = (B^{c}\cup(B^{c}\cap A^{c}))^{c} = (B^{c})^{c} = B
\end{align*}</span></p>
<p>Hopefully this helps!</p>
|
4,317 | <p>For computing the present worth of an infinite sequence of equally spaced payments $(n^{2})$ I had the need to evaluate</p>
<p>$$\displaystyle\sum_{n=1}^{\infty}\frac{n^{2}}{x^{n}}=\dfrac{x(x+1)}{(x-1)^{3}}\qquad x>1.$$</p>
<p>The method I used was based on the geometric series $\displaystyle\sum_{n=1}^{\infty}x^{... | Aryabhata | 1,102 | <p>Here is a different look:</p>
<p>The differentiating and multiplying by $x$ gives rise to <a href="http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind" rel="nofollow">Stirling Numbers of the Second Kind</a>.</p>
<p>Say you denote the operator of differentiating and multiplying by $x$ as $D_{x}$</p>
<... |
41,888 | <p>So I've got this line that contains the solution of a partial, non-extraordinary differential equation (because Mathematica doesn't handle extraordinary partial differential equations):</p>
<pre><code>phi6m = NDSolveValue[{D[u[t, x], t, t] -
D[u[t, x], x, x] == -6 u[t, x]^5 + 10.5 u[t, x]^3 - 4.5 u[t, x], u[0,... | Ymareth | 880 | <p>Something along the lines of ...</p>
<pre><code>a /: Set[a,x_]:=((OwnValues[a]={HoldPattern[a]:>x}; Print[{"a was set to:",x}]); a)
a = 1 prints {a was set to:,1}
a on its own is now 1.
</code></pre>
<p>Extends what set does with an additional action which here is Print but ,in principle, could be anything.</... |
785,314 | <p>let $$I_n = \int_{\pi/2}^{x} \frac{\cos^{2n+1}t}{\sin(t)} \ dt, n \geq 0$$</p>
<p>show $$2(n+1)I_{n+1} = 2(n+1)I_n +\cos^{2n+2}x$$</p>
<p>I showed the result by considering $I_{n+1} - I_n$ but I'm wondering how could I do it using integration by parts?</p>
<p>Similarly for $J_n = \int_0^x \frac{\sinh^{2n+1}t}{\co... | user144464 | 144,464 | <p>for $I_n$ write $$ I_n = \int_{\pi /2}^x \frac{\cos^{2n+1}t \sin(t)}{\sin^2t} \ dt$$ and let $ u = \cos^{2n+1}t \sin(t)$ $v' = cosec^2(t)$ similarly for $J_n$</p>
|
2,762,040 | <p>I started to think about this problem and then factored $n^5 - n$ to $(n^2 - 1)(n^2 + 1)(n)$, and later to $(n-1)(n)(n+1)(n^2 + 1)$. I know that $(n-1)(n)(n+1)$ is divisible by $6$, but it is not that case $5$ divides $n^2 + 1$ for any integer $n$, so i can´t use the multiplication property. Can anyone help me finis... | Asinomás | 33,907 | <p>This is a particular case of fermats little theorem</p>
|
2,762,040 | <p>I started to think about this problem and then factored $n^5 - n$ to $(n^2 - 1)(n^2 + 1)(n)$, and later to $(n-1)(n)(n+1)(n^2 + 1)$. I know that $(n-1)(n)(n+1)$ is divisible by $6$, but it is not that case $5$ divides $n^2 + 1$ for any integer $n$, so i can´t use the multiplication property. Can anyone help me finis... | lab bhattacharjee | 33,337 | <p>Hint:</p>
<p>$$(n^2-1)(n^2+1)n=(n^2-1)(n^2-4+5)n=\underbrace{(n-2)(n-1)n(n+1)(n+2)}_{\text{product of } 5\text{ consecutive integers}}+5n(n^2-1)$$</p>
|
59,046 | <p>Let $A \in M_n(\mathbb R)$ and suppose its minimal polynomial is:
$$M_A(t)=\prod_{i=1}^{k}(t-\lambda_i)^{\textstyle s_i}.$$</p>
<p>When $\lambda _1,\lambda_2,\lambda _3,......,\lambda _k$ are distinct eigenvalues.</p>
<p>We define a new matrix: $B\in M_{2n}(\mathbb R)$ by:
$$\left(\begin{matrix}
A &I_n \\
0... | Robert Israel | 8,508 | <p>For the matrix $B = \pmatrix{A & I \cr 0 & A\cr}$ and any polynomial $p$, note that $p(B) = \pmatrix{ p(A) & p'(A)\cr 0 & p(A)\cr}$. If the polynomials $p(t)$ and $p'(t)$
are both divisible by $(t - \lambda)^j$, then $p(t)$ must be divisible by $(t - \lambda)^{j+1}$.</p>
|
17,975 | <p>How to systematically classify Mathematica expressions? I can think of using <code>Head[]</code>, <code>Depth[]</code>, <code>Length[]</code>, and some special pattern based on the problems at hand. What other key words, or functions should I consider?</p>
<h2>Update</h2>
<p>I mostly want to group symbols by how... | image_doctor | 776 | <p>The function <code>Depth</code> will separate the examples of the expressions you give into three distinct classes.</p>
<pre><code>exps = {{_String, _Symbol}, {{_Integer}, _String}, _String}
Depth /@ exps
</code></pre>
<blockquote>
<p>{3, 4, 2}</p>
</blockquote>
|
925,746 | <p>I don't really understand Tautologies or how to prove them, so if someone could help, that would be great! </p>
| Michael Hardy | 11,667 | <p>If $Q$ is true, then $P\to Q$ is true, and we're done.</p>
<p>If $Q$ is false, the $Q\to R$ is true, and we're done.</p>
<p>But to go back to basic definitions, a tautology is something that is true in <b>every</b> row of the truth table. In this case there are eight rows. Look at each row and ascertain whether ... |
925,746 | <p>I don't really understand Tautologies or how to prove them, so if someone could help, that would be great! </p>
| David | 119,775 | <p><strong>Truth table method</strong>:
$$\matrix{
p&q&r&p\to q&q\to r&\hbox{answer}\cr
T&T&T&T&T&T\cr
T&T&F&T&F&T\cr}$$
and so on. You can provide six more rows yourself and check that the final answer is always true.</p>
<p><strong>Logical equivalenc... |
1,868,263 | <p>A Relation R on the set N of Natural numbers be defined as (x,y) $\in$R if and only if $x^2-4xy+3y^2=0$ for allx,y $\in$N then show that the relation is reflexive,transitive but not SYMMETRIC.</p>
<p>i got how this relation is reflexive or transitive but i am not able to think of any reason of why this relation is... | Tony Tarng | 308,438 | <p>If the relation is symmetric, then for all $(x,y) \in R$ that satisfy $x^2 - 4xy + 3y^2 = 0 \rightarrow (y,x) \in R$, so $y^2 - 4yx + 3x^2 = 0$. </p>
<p>For $x^2 - 4xy + 3y^2 = (x - 3y)(x - y) = 0$, x must be equal to $3y$ or $y$, but does this ALWAYS imply that $y^2 - 4yx + 3x^2 = (y - 3x)(y - x) = 0$?</p>
|
3,058,019 | <blockquote>
<p>Two numbers <span class="math-container">$297_B$</span> and <span class="math-container">$792_B$</span>, belong to base <span class="math-container">$B$</span> number system. If the first number is a factor of the second number, then what is the value of <span class="math-container">$B$</span>?</p>
</... | Bill Dubuque | 242 | <p>Going <span class="math-container">$1$</span> step more with Euclid's algorithm reveals a common factor <span class="math-container">$\,b\!+\!1.\,$</span> Cancelling it</p>
<p><span class="math-container">$$\dfrac{7b^2\!+\!9b\!+\!2}{2b^2\!+\!9b\!+\!7} = \color{#c00}{\dfrac{7b\!+\!2}{2b\!+\!7}}\in\Bbb Z\ \, \Rightar... |
3,058,019 | <blockquote>
<p>Two numbers <span class="math-container">$297_B$</span> and <span class="math-container">$792_B$</span>, belong to base <span class="math-container">$B$</span> number system. If the first number is a factor of the second number, then what is the value of <span class="math-container">$B$</span>?</p>
</... | Steven Alexis Gregory | 75,410 | <p><span class="math-container">$$2B^2+9B+7\mid 7B^2+9B+2$$</span></p>
<p>Let's write <span class="math-container">$aB^2+bB + c$</span> as <span class="math-container">$[a,b,c]_B$</span> to emphasis that <span class="math-container">$a,b,c$</span> are digits base <span class="math-container">$B$</span>.</p>
<p>Then <... |
1,604,573 | <p>Consider the random graph $G(n,\frac{1}{n})$. I'm trying to estimate the size of the maximum matching in $G$. </p>
<p>If we look at one vertex, the expected value of its degree is $\frac{n-1}{n}$ so it seems like with high prob it should be 1.</p>
<p>So if I can show that with high probability half of the vertices... | D Poole | 83,727 | <p>First, just because the expected value of a fixed vertex is $\frac{n-1}{n}$ does not mean w.h.p. its degree is 1. In fact,
$$
\text{Pr(deg(}v\text{)=1)} = {n-1 \choose 1} \frac{1}{n} \left(1-\frac{1}{n}\right)^{n-2} \approx 1 \cdot e^{-1}.
$$
Now if $X$ is the random number of vertices with degree exactly 1, then $... |
770,544 | <p>It's clear that a system of two quadratic equations can have none, one or two solutions. </p>
<p>For example: $y = x^2 + 2$ and $y = - x^2 + 1$ have none. $y = x^2$, $2x^2 - 8x + 8$ and $y = - x^2 + 8x - 8$ have $4$ as common solution. And $2x^2 - 8x + 8 = x^2 - 4x + 5$ have $1$ and $3$ as solutions.</p>
<p>Is it ... | user88595 | 88,595 | <p>Intuitively you are correct however you have forgotten a trivial case, when both quadratics are the same in which case you have infinitely many solutions.</p>
<p>Apart from this very specific case here's how to prove that indeed they only have none,one or two solutions.
Consider a system where you have :
\begin{cas... |
19,605 | <p>Let <span class="math-container">$P=(x_1,y_1)$</span> be a non torsion point on an elliptic curve <span class="math-container">$y^2=x^3+Ax+B$</span>.
Let <span class="math-container">$(x_n,y_n)=P^{2^n}. x_n,y_n$</span> are rationals with heights growing rapidly. Can <span class="math-container">${x_n} {y_n}$</span... | Kevin Buzzard | 1,384 | <p>EDIT: this answer is <em>wrong</em>. I misread the question as looking at the group generated by P, not the points obtained by repeated doubling. I would be OK if the subset of S^1 generated by taking a non-torsion point and repeatedly doubling came arbitrarily close to the origin---but it may not, as the comments b... |
2,779,379 | <blockquote>
<p>Let $f:\mathbb R\to \mathbb R$ be a continuous function and $\Phi(x)=\int_0^x (x-t)f(t)\,dt$. Justify that $\Phi(x)$ is twice differentiable and calculate $\Phi''(x)$.</p>
</blockquote>
<p>I'm having a hard time finding the first derivative of $\Phi(x)$. Here's what I tried so far:</p>
<p>Since $f$ ... | zhw. | 228,045 | <p>We can write</p>
<p>$$\Phi(x)=x\int_0^xf(t)\,dt- \int_0^xtf(t)\,dt$$</p>
<p>Using the product rule and the FTC, we get</p>
<p>$$\Phi'(x)=\int_0^xf(t)\,dt + xf(x) - xf(x) = \int_0^xf(t)\,dt$$.</p>
<p>Using FTC again, we have $$\Phi''(x) = f(x),$$ and we're done.</p>
|
2,225,650 | <p>Given that $\vec{a}$ and $\vec{b}$ are two non-zero vector. The two vectors form 4 resultant vectors such that $\vec{a} + 3\vec{b}$ and $2\vec{a} - 3\vec{b}$ are perpendicular, $\vec{a} - 4\vec{b}$ and $\vec{a} + 2\vec{b}$ are perpendicular. How can I find the angle between $\vec{a}$ and $\vec{b}$?</p>
<p>The answe... | N.Bach | 433,527 | <p>I don't have the tools to draw my proposal, but I still think I can (hopefully) give you some pointers.</p>
<p>Try a function $r(u,v)$ that satisfies these constraints:</p>
<ul>
<li>$\forall 0\le v\le 1$, $r(0,v) = 1$</li>
<li>$\forall 0\le u\le 1$, $r(u,0) = 1 = r(u,1)$</li>
<li><strong>EDIT</strong>: my first pr... |
2,919,841 | <blockquote>
<p><span class="math-container">$$\Large\bigcup\limits_{k\in\bigcup\limits_{i\in I}J_i}A_k=\bigcup\limits_{i\in I}\bigg(\bigcup\limits_{k\in J_i}A_k\bigg)$$</span></p>
</blockquote>
<hr />
<p><strong>My attempt:</strong></p>
<p><span class="math-container">$\large x\in\bigcup\limits_{k\in\bigcup\limits_{i\... | mengdie1982 | 560,634 | <h1>Another Proof</h1>
<p>First, we can prove that $$\forall k \in \mathbb{N}:\lim_{n \to \infty}\frac{n^{k}}{c^n}=0.$$</p>
<p>For this purpose, we assume that $c=1+h(h>0)$. Then $$\forall n \geq k+1:(1+h)^n\geq \frac{n(n-1)\cdots(n-k)}{(k+1)!}h^{k+1}.$$Thus
\begin{align*}
0 \leq \frac{n^k}{c^n} &\leq \frac{(k... |
19,285 | <p>Is anyone aware of Mathematica use/implementation of <a href="http://en.wikipedia.org/wiki/Random_forest">Random Forest</a> algorithm?</p>
| Andy Ross | 43 | <p>Here I will attempt to provide a basic implementation of the random forest algorithm for classification. This is by no means fast and doesn't scale very well but otherwise is a nice classifier. I recommend reading <a href="http://www.stat.berkeley.edu/~breiman/RandomForests/cc_home.htm">Breiman and Cutler's page</a>... |
3,817,104 | <p>For <span class="math-container">$a,b,c \in \Big[\dfrac{1}{3},3\Big].$</span> Prove<span class="math-container">$:$</span></p>
<p><span class="math-container">$$(a+b+c) \Big(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\Big) \leqslant 25.$$</span></p>
<p>Assume <span class="math-container">$a\equiv \text{mid}\{a,b,c\},$</s... | Mike Daas | 317,530 | <p>By AM-GM we have
<span class="math-container">$$
\frac{(a+b+c) + (\frac{1}{a}+\frac{1}{b}+\frac{1}{c})}{2} \geq \sqrt{(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)}.
$$</span>
Note that by the assumption, we have
<span class="math-container">$$
3 + \frac{1}{3} \geq a + \frac{1}{a}
$$</span>
and similarly f... |
519,764 | <p>Question: show that the following three points in 3D space A = <-2,4,0>, B = <1,2,-1> C = <-1,1,2> form the vertices of an equilateral triangle.</p>
<p>How do i approach this problem?</p>
| Novice | 97,093 | <p>Apart for showing their length are the same, you will have to show they are not collinear. Construct a line equation with any two given points and check that the remaining point does not lie on the line. The other way is to check the cross product of AB and BC to make sure that it is not equal to 0. :)</p>
|
37,804 | <p>I'm trying to gain some intuition for the usefullness of the spectral theory for bounded self adjoint operators. I work in PDE and any interesting applications/examples I've ever encountered are concerning <em>compact operators</em> and <em>unbounded operators</em>. Here I have the examples of $-\Delta$, the laplaci... | user36539 | 36,539 | <p>The original answer was deleted this is a refined answer :</p>
<p>The Fourier transform $F$ is a bounded operator and non compact see <a href="http://en.wikipedia.org/wiki/Fourier_transform" rel="nofollow">http://en.wikipedia.org/wiki/Fourier_transform</a> where some properties are given. $F$ is defined on $S$ the ... |
198,555 | <p>I am having difficulties understanding how do I perform set operation like union or intersection on Relations. </p>
<p>In a question, I am asked to prove/disprove: </p>
<ul>
<li>If R & S are symmetric, is $R \cap S$ symmetric? </li>
<li>If R & S are transitive, is $R \cup S$ transitive?</li>
</ul>
<p>How ... | Aang | 33,989 | <p>For $R\cap S$ being symmetric,</p>
<p>Let $(x,y)\in R\cap S\implies (x,y)\in R $ and $(x,y)\in S\implies (y,x)\in R$ and $(y,x)\in S$ (as $R,S$are symmetric) $\implies (y,x)\in R\cap S$. </p>
<p>Thus, $R\cap S$ is symmetric. </p>
<p>For transitive part($R\cup S$),</p>
<p>Let $(x,y)$ and $(y,z)\in R\cup S\implies... |
198,555 | <p>I am having difficulties understanding how do I perform set operation like union or intersection on Relations. </p>
<p>In a question, I am asked to prove/disprove: </p>
<ul>
<li>If R & S are symmetric, is $R \cap S$ symmetric? </li>
<li>If R & S are transitive, is $R \cup S$ transitive?</li>
</ul>
<p>How ... | martini | 15,379 | <p>A binary relation is nothing but a set of ordered pairs, where instead of $(x,y) \in R$ we usually write $x\, R\,y$. Now you have to use the definition of 'symmetic' and $\cap$ (resp. transitive and $\cup$) to show these properties hold or give an example where it doesn't hold. I will show you what I mean using ano... |
1,348,127 | <p>I'm struggling with this problem, because I'm not sure how to integrate $1/\ln(x)$</p>
<blockquote>
<p>Suppose that you have the following information about a function
$F(x)$:</p>
<p>$$F(0)=1, F(1)=2, F(2)=5$$ $$F'(x)=\frac1{\ln(x)}$$</p>
<p>Using the Fundamental Theorem of Calculus evaluate $$\int_0... | Mark Viola | 218,419 | <p>In <a href="https://math.stackexchange.com/questions/1206631/proving-the-series-of-partial-sums-of-sin-in-is-bounded/1206665#1206665">this</a> answer, I showed that </p>
<p>$$\begin{align}
\left|\sum_{n=1}^N \sin(nx)\right| \le \frac12\left(1+\left|\cos (\frac{x}{2})\right|\right)\left|\csc\left(\frac{x}{2}\right)\... |
14,448 | <p>Here's the most common way that I've seen letter grades assigned in undergrad math courses. At the end of the semester, the professor: 1) computes the raw score (based on homework, quizzes, and tests) for each student; 2) writes down all the raw scores in order; 3) somewhat arbitrarily clusters the scores into group... | guest | 10,126 | <p>60 D
70 C
80 B
90 A</p>
<p>IF you are familiar with the course, you should be able to set appropriate difficulty tests. If you are less familiar, than sneak in easy test (or hard one) if you think you need to correct a little for scores being out of whack halfway through. but even then try to have some personal g... |
1,343,995 | <p>We know that $i^i$ is real. But how to explain it geometrically maybe in terms of rotation. like we can explain geometrically multiplication of two complex numbers and so on. Can someone show me a little bit about geometric interpretation of $i^i$ and tell me if what I think below is true?</p>
<p>Note :(Some info... | Job Bouwman | 274,003 | <p>I don't have one, but maybe this helps. </p>
<p>I'd like to call $z^i$ the 'semiprocal' of $z$, because the $i^{\text{th}}$ power brings us 'halfway' to the reciprocal of $z$, being $\left(z^i\right)^i = z^{-1} = \frac1z$ </p>
<p>Likewise, $i^i$ brings us halfway to the reciprocal of $i$, being $\left(i^i\right)^i... |
2,394,716 | <p>Let $A = \{\frac{x}{2} - \lfloor\frac{x+1}{2}\rfloor : x \in \mathbb{R} \}$ </p>
<p>Does <strong>supremum</strong> and <strong>infimum</strong> of $A$ exist ? If the answer is yes then find them .</p>
<p>My try : I rewrite the expression $\frac{x}{2} - (\lfloor 2x \rfloor - \lfloor x \rfloor)$ but it doesn't h... | hmakholm left over Monica | 14,366 | <p>Usually such a requirement is not implied by saying that the equation is in standard/canonical/whatever form.</p>
<p>If, for some particular application, it is easier for you to handle equations with non-negative $C$, then you're of course free to <em>set up</em> such a requirement yourself. (If you get an equation... |
132,591 | <p>Let $f(x)$ be a positive function on $[0,\infty)$ such that $f(x) \leq 100 x^2$. I want to bound $f(x) - f(x-1)$ from above. Of course, we have $$f(x) - f(x-1) \leq f(x) \leq 100 x^2.$$ This is not good for me though. I need a bound which is linear (or at worst linear-times-root) in $x$.</p>
<p>Is there an inequali... | akkkk | 28,826 | <p>There is no such bound. Let $f(2)=0$, and $f(x)=100x^2$ for other $x$. Surely $f(3)-f(2)=900\le100x^2$, but equality holds (!).</p>
|
308,452 | <p>"Use direction field and Isocline make a qualitative sketch of the solution , determine equilibrium values and classify them ?"</p>
<p>$y'=y-\sqrt y $
<img src="https://i.stack.imgur.com/fOVG6.jpg" alt="enter image description here"></p>
<p>It is clear that $y=0 , y=1 $ are equilibrium points but <strong>for these... | Emanuele Paolini | 59,304 | <p>You have an <em>autonomous</em> equation:
$$
y' = f(y).
$$
Every zero of $f$ identifies a constant solution: if $f(y_0)=0$ then $y(x)=y_0$ is a solution.
On the other hand when $f$ is positive the solutions are increasing, when $f$ is negative the solutions are decreasing. Hence if $f'(y_0) > 0$ the solution is i... |
308,452 | <p>"Use direction field and Isocline make a qualitative sketch of the solution , determine equilibrium values and classify them ?"</p>
<p>$y'=y-\sqrt y $
<img src="https://i.stack.imgur.com/fOVG6.jpg" alt="enter image description here"></p>
<p>It is clear that $y=0 , y=1 $ are equilibrium points but <strong>for these... | Amzoti | 38,839 | <p>This is a powerful qualitative tool that helps us get a general understanding of the behavior. When the DEQ is not solvable (other than using numerical methods or otherwise), this is the next best thing - we do qualitative analyses. Just look at how much information you were able to garner just from the direction fi... |
548,470 | <p>Prove $$(x+1)e^x = \sum_{k=0}^{\infty}\frac{(k+1)x^k}{k!}$$ using Taylor Series.</p>
<p>I can see how the $$\sum_{k=0}^{\infty}\frac{x^k}{k!}$$ plops out, but I don't understand how $(x+1)$ can become $(k+1)$.</p>
| Haha | 94,689 | <p>$(k+1)x^{k}/k!=x \frac {x^{k-1}}{k-1!} + \frac {x^k}{k!}$</p>
|
3,794,507 | <p>On <span class="math-container">$(5),$</span> <span class="math-container">$(6),$</span> and <span class="math-container">$(7),$</span> what's the difference between <span class="math-container">$S^2$</span> and <span class="math-container">$\sigma_x^2$</span>?</p>
<p>Also, why does:</p>
<p><span class="math-contain... | Antoni Parellada | 152,225 | <p>You may find your answers both in the <em>sample variance</em> subheading of the <a href="https://en.wikipedia.org/wiki/Variance#Sample_variance" rel="nofollow noreferrer">variance entry in Wikipedia</a>, or possibly in this <a href="https://stats.stackexchange.com/a/100050/67822">answer in Stack Exchange CV</a>.</p... |
1,504,433 | <p>I was wondering if you could help me with this question, in discrete math.</p>
<p>Prove that gcd(m, n) | lcm(n, m) for any non-zero integers m, n</p>
<p>any help is appreciated!</p>
| lagrange103 | 174,961 | <p>If there are only two inputs, I suggest you label the amount of the 0.6 input $x$ and the amount of the 0.2 input $1-x$. Or if you want to find the percentage, $100-x$. You then set up your equation $0.60x + 0.2(1-x)=0.5$. </p>
<p>Solving for $x$, you get $x=0.75$. This means that you need 75% of your input to be ... |
1,717,149 | <p>Is it true or false that if $V$ is a vector space and $T:V \to W$ is a linear transformation such that $T^2 = 0$, then $Im(T) \subseteq Ker(T)$ ?<br>
I don't understand it that much. It doesn't seem related... I can have a vector $v$ from $V$ that its power by 2 equals zero but $T(v) \neq 0_{v}$ </p>
| Dave L. Renfro | 13,130 | <p>Two books not yet mentioned that the OP might want to consider are:</p>
<p><a href="http://rads.stackoverflow.com/amzn/click/B000JBZYV0" rel="nofollow"><strong>Groups in the New Mathematics</strong></a> by Irving Adler (1967)</p>
<p><a href="http://rads.stackoverflow.com/amzn/click/088385614X" rel="nofollow"><stro... |
2,410,243 | <p>Suppose we want to
$ min_i$ median$(a_i)$</p>
<p>$a_i$ are real numbers</p>
<p>Does someone know how to pose this as an integer programming problem or point me in the direction of a resource? </p>
| Johan Löfberg | 37,404 | <p>EDIT: Removing a simpler model which was incorrect.</p>
<p>The way median </p>
<pre><code>y = median(a)
</code></pre>
<p>is implemented in the optimization modelling toolbox YALMIP is roughly by (writing in MATLAB pseudo code)</p>
<pre><code>y = s(length(a)/2); s = sort(a);
</code></pre>
<p>Hence, to model medi... |
468 | <p>Textbook writers are blessed with only solving problems with neat answers. Numerical coefficients are small integers, many terms cancel, polynomials split into simple factors, angles have trigonometric functions with known values. Pure bliss.</p>
<p>The "real life" is different (as any of us knows).</p>
<p>Giving ... | Neil Strickland | 76 | <p>When teaching linear algebra, I rely heavily on carefully constructed examples where everything can be done with rational numbers of small denominator. (It is faintly amusing that the construction of such examples often requires mathematics that is far harder than the actual content of the course.) However, I also... |
1,650,881 | <p>I found this problem in a book on undergraduate maths in the Soviet Union (<a href="http://www.ftpi.umn.edu/shifman/ComradeEinstein.pdf" rel="nofollow">http://www.ftpi.umn.edu/shifman/ComradeEinstein.pdf</a>):</p>
<blockquote>
<p>A circle is inscribed in a face of a cube of side a. Another circle is circumscribed... | David K | 139,123 | <p>I'll consider just the two-spheres method.
For simplicity, I'll assume $a = 2$; the general solution will be
proportional to the solution for this special case.</p>
<p>Both spheres have their centers at the body center of the cube;
call that point $O$.
The sphere containing the smaller circle is tangent to all twel... |
201,807 | <p>I heard this problem, so I might be missing pieces. Imagine there are two cities separated by a very long road. The road has only one lane, so cars cannot overtake each other. $N$ cars are released from one of the cities, the cars travel at constant speeds $V$ chosen at random and independently from a probability di... | André Nicolas | 6,312 | <p>As Hagen von Eitzen has pointed out, the number of groups is the number of record low speeds as we scan the cars from the first car to the last. We calculate the expected number in a much simpler way. Label the cars, in order they start out, $1,2,\dots,n$.</p>
<p>For $i=1$ to $n$, let $X_1=1$ if car $i$ is slower t... |
808,389 | <p>I roll a dice $3$ times. What is the probability that only $2$ of the sides show up, or put equivalently, what is the probability that 4 of the sides don't show up at all?</p>
<p>More generally lets say I have a $20$ numbered balls in a bag. I pull one out, write down its number and put it back, I then pull another... | Will Orrick | 3,736 | <p>If $8$ balls are drawn in order, then an example of an outcome in which exactly $6$ different balls appear is
$$
8,\ 6,\ 8,\ 9,\ 19,\ 3,\ 6,\ 2.
$$
In this example, ball $8$ came up on draws $1$ and $3,$ ball $6$ came up on draws $2$ and $7,$ and balls $9,$ $19,$ $3,$ and $2$ came up on draws $4,$ $5,$ $6,$ and $8,$... |
2,236,862 | <p>$$\frac{1^2}{1!}+ \frac{2^2}{2!}+ \frac{3^2}{3!} + \frac{4^2}{4!} + \dotsb$$</p>
<p>I wrote it as:
$$\lim_{n\to \infty}\sum_{r=1}^n \frac{(r^2)}{r!}.$$</p>
<p>Then I thought of sandwich theorem, it didn't work. Now I am trying to convert it into difference of two consecutive terms but can't. Need hints. </p>
| k.Vijay | 428,609 | <p>The $n^{th}$ term of this series, suppose:</p>
<p>$t_n=\dfrac{n^2}{n!}=\dfrac{n(n-1)+n}{n!}=\dfrac{n(n-1)}{n!}+\dfrac{n}{n!}=\dfrac{1}{(n-2)!}+\dfrac{1}{(n-1)!}$</p>
<p>Sum of terms:
\begin{align*}
S_n&=\dfrac{1^2}{1!}+\dfrac{2^2}{2!}+\dfrac{3^2}{3!} +\cdots\infty\\
&=\sum\limits_{n=1}^{\infty}t_n\\
&=... |
238,627 | <p>Kindly consider it soft question as I am a software engineer.and I know only software but I have a doubt in my mind that there would be something like null in mathmatics as well. </p>
<p>If Mathematics <code>NULL</code> IS Equivalent to <code>ZERO</code>?</p>
| Berci | 41,488 | <p>In 'mathematics' everything is possible, and in theory everything is renamable. So, we can have a theory where 'Zero' and 'Null' have different meanings, however you want to mean it..</p>
<p>For example, we can just consider the set of natural numbers $\Bbb N$ equipped with one more element, which we can call '<em>... |
19,373 | <p>I posted this question earlier today on the Mathematics site (<a href="https://math.stackexchange.com/q/3988907/96384">https://math.stackexchange.com/q/3988907/96384</a>), but was advised it would be better here.</p>
<p>I had a heated argument with someone online who claimed to be a school mathematics teacher of man... | Flydog57 | 13,120 | <p><strong>It depends</strong></p>
<p>I agree with just about everyone that the answer is 35, or perhaps 35.4 (a number I like better, see below). An answer of 35.405598 km is precise to the millimeter. I've ridden horses; they don't work in millimeters.</p>
<p><em>Update:</em> For what it's worth, after all this disc... |
19,373 | <p>I posted this question earlier today on the Mathematics site (<a href="https://math.stackexchange.com/q/3988907/96384">https://math.stackexchange.com/q/3988907/96384</a>), but was advised it would be better here.</p>
<p>I had a heated argument with someone online who claimed to be a school mathematics teacher of man... | Alan | 15,317 | <p>I would say that based on the words of the question, the answer is 35.</p>
<p>It is not a distance or measurement of 22 miles between points A & B. It is a journey between locations A & B.</p>
<p>Next time you see the guy who scolded you, ask him how one should answer if asked, “What is the numerical value ... |
613,105 | <p>I was observing some nice examples of equalities containing the numbers $1,2,3$ like $\tan^{-1}1+\tan^{-1}2+\tan^{-1}3=\pi$ and $\log 1+\log 2+ \log 3=\log (1+2+3)$. I found out this only happens because $1+2+3=1*2*3=6$.<br> I wanted to find other examples in small numbers, but I failed. How can we find all of the s... | David Holden | 79,543 | <p>leaving aside the solution in which $a=b=c=0$ order the numbers so $a \le b \le c$</p>
<p>as OP shows, $c|(a+b)$ so we have:</p>
<p>$$
\frac{a+b}{c} \le 2
$$</p>
<p>thus only the values $1$ and $2$ are possible for $\frac{a+b}c$. these give $a+b=c$ and $a+b=2c$ respectively. however the latter is only possible if... |
3,703,495 | <p><a href="https://en.m.wikipedia.org/wiki/Doomsday_argument" rel="nofollow noreferrer">https://en.m.wikipedia.org/wiki/Doomsday_argument</a></p>
<p>Suppose each new human born has the knowledge of the total number of humans born so far. So in their life, each human multiplies that number by 20 to get the upper bound... | user3257842 | 365,433 | <p>For each possible world, 95% of all humans <strong>in that world</strong> will get the correct answer. But it isn't the "same" 95% in every world. Because there are infinite potential humans to choose from. You're assuming a consistent identity across all worlds, instead of picking at random from the infinite born ... |
3,538,786 | <p>Given
<span class="math-container">$$a_n=\sqrt{2-a_{n-1}},a_1=\sqrt{2}$$</span></p>
<p>I calculated <span class="math-container">$a_1$</span> to <span class="math-container">$a_5$</span>
<span class="math-container">$$\sqrt{2},
\sqrt{2-\sqrt{2}},
\sqrt{2-\sqrt{2-\sqrt{2}}}, \\
\sqrt{2-\sqrt{2-\sqrt{2-\sqrt{2}}}}, \... | rtybase | 22,583 | <p>If the question is about finding the limit, let's consider <span class="math-container">$a_{n+1}=f(a_n)$</span>, where <span class="math-container">$f(x)=\sqrt{2-x}$</span>. Then we have</p>
<blockquote>
<p>If <span class="math-container">$0\leq x \leq \sqrt{2}$</span> then <span class="math-container">$0\leq f(x... |
3,538,786 | <p>Given
<span class="math-container">$$a_n=\sqrt{2-a_{n-1}},a_1=\sqrt{2}$$</span></p>
<p>I calculated <span class="math-container">$a_1$</span> to <span class="math-container">$a_5$</span>
<span class="math-container">$$\sqrt{2},
\sqrt{2-\sqrt{2}},
\sqrt{2-\sqrt{2-\sqrt{2}}}, \\
\sqrt{2-\sqrt{2-\sqrt{2-\sqrt{2}}}}, \... | Fabio Lucchini | 54,738 | <p>Let <span class="math-container">$x_n$</span> for <span class="math-container">$n\in\Bbb N$</span> be the sequence defined by
<span class="math-container">\begin{align}
x_0&=\frac\pi 2&
x_n&=\frac{\pi-x_{n-1}}2
\end{align}</span>
and <span class="math-container">$a_n=2\cos(x_n)$</span>.
Then <span class=... |
951 | <p>I think that complex analysis is hard because graphs of even basic functions are 4 dimensional. Does anyone have any good visual representations of basic complex functions or know of any tools for generating them?</p>
| Jackson Walters | 13,181 | <p>Oh yes, there's a way to do this. Here is my exploration into the topic about a month ago using Mathematica. The easiest thing to do is to plot the vector field and let the direction of the arrows represent the phase and let the color represent the magnitude. This is a great way to get all four dimensions on a plane... |
3,443,082 | <p>Find all the positive integral solutions of, <span class="math-container">$\tan^{-1}x+\cos^{-1}\dfrac{y}{\sqrt{y^2+1}}=\sin^{-1}\dfrac{3}{\sqrt{10}}$</span></p>
<p>Assuming <span class="math-container">$x\ge1,y\ge1$</span> as we have to find positive integral solutions of <span class="math-container">$(x,y)$</span>... | lab bhattacharjee | 33,337 | <p>We can actually utilize <span class="math-container">$x,y$</span> are positive integers</p>
<p><span class="math-container">$$\tan^{-1}x=\tan^{-1}3-\tan^{-1}\dfrac1y<\tan^{-1}3$$</span></p>
<p>As <span class="math-container">$\tan^{-1}x$</span> is an increasing function
<span class="math-container">$$\implies x... |
65,059 | <p>I have two points ($P_1$ & $P_2$) with their coordinates given in two different frames of reference ($A$ & $B$). Given these, what I'd like to do is derive the transformation to be able to transform any point $P$ ssfrom one to the other.</p>
<p>There is no third point, but there <em>is</em> an extra constra... | Andrew Szymczak | 202,838 | <p>I have a simplified way of thinking of the problem (imo). Suppose you have a reference frame </p>
<p>$$ \mathcal{R} = (r_1,r_2,r_3,o)$$ </p>
<p>where $r_1,r_2,r_3$ are the normalized basis vectors and $o$ is the origin. We assume that these are all given in Euclidean space, which I will call world space $\mathca... |
1,199,746 | <p>Let $f : (−\infty,0) → \mathbb{R}$ be the function given by $f(x) = \frac{x}{|x|}$. Use the $\epsilon -\delta$ definition of a $\lim\limits_{x \to 0^-} f(x) = -1.$</p>
<p>Workings:</p>
<p>Informal Thinking:
We want $|f(x) - L| < \epsilon$</p>
<p>$\left|\frac{x}{|x|} - -1\right| < \epsilon$</p>
<p>$\left|\f... | Cameron Buie | 28,900 | <p>You went astray when you dropped the absolute value bars. Instead, remember that we are only considering $x<0,$ so what is $|x|$?</p>
<p>Also, as graydad points out, your $\delta$ is only allowed to depend on the choice of $\epsilon,$ not on $x$.</p>
|
3,700,938 | <p>I know that the equations are equivalent by doing the math with the same value for x, but I don't understand the rules for changing orders or operations.<br>
When it is not the first addition or subtraction happening in the equation, parentheses make the addition subtraction and vice versa? Are there any other rules... | Plushkin Neponemaet | 794,422 | <p>That's a simple general rule for bracket notation: <span class="math-container">$-(a+b)=-a-b$</span>. It's more or less a definition</p>
|
2,561,968 | <p>Is there a sequence of integers such that for ∀ k it contains an arithmetic subsequence of length k but it does have no infinitely long arithmetic subsequence?</p>
| Bart Michels | 43,288 | <p>Note that a sequence with arbitrarily large gaps cannot contain an infinite arithmetic sequence.</p>
<p>This observations allows to construct many counterexamples, e.g. $$[1,10]\cup[100,1000]\cup\ldots\cup[10^{2k},10^{2k+1}]\cup\ldots$$</p>
<p>Any sequence with arbitrarily large gaps and whose complement has arbit... |
2,115,484 | <p>I am not too sure how to prove that a hyperplane in $\mathbb{R}^{n}$ is convex? So far I know the definition of what convex is, but how do we add that hyperplane in $\mathbb{R}^{n}$ is convex?</p>
<p>Thanks in advance!</p>
| Gokulakrishnan CANDASSAMY | 917,973 | <p>Here is another elegant way of proving that a hyperplance <span class="math-container">$\mathcal{H}_n \subset \mathbb{R}^n $</span> is a convex set (while going through some important results in the theory of convex sets).</p>
<p><strong>Claim 1:</strong> Every hyperplane given by <span class="math-container">$a^Tx ... |
337,655 | <p>I am seeking for some intuition why norm (for any reasonable norm on functions) of a function is smaller if the function is smoother.</p>
| spitespike | 67,974 | <p>Norms may not necessarily be related smoothness in any way. </p>
<p>The uniform norm $\|f\|_u=\sup_{x \in [0,1]} |f(x)|$ on the space of continuous functions $C[0,1]$ is unrelated to smoothness. There are nowhere differential functions of arbitrarily small uniform norm.</p>
<p>But on the space of $L_2[0,1]$ absolu... |
643,851 | <p>Let the ODE
$$\frac{dy}{dx}=\frac{y+x-2}{y+x-4}$$</p>
<p>I got the general (implicit) solution:
$$y=\ln|x+y-3|+x+A$$ A is arbitrary constant.</p>
<p>My question is:
is $3=y+x$ a solution of this ODE? I know it's not contained in the general solution.</p>
| Bill Dubuque | 242 | <p>$n\ge 40\,\Rightarrow\, 3$ divides one of $\,n\!-\!a,\ a \in\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \overbrace{\{9,25,35\}}^{\large\qquad\ \equiv\ \{0,\ \ 1,\ \ 2 \}\,\pmod{\! 3}}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!$ so $\ n = (n-a)+ a\,$ works, by $\,3\mid n-a > 3.\,$
Else $\,39 \ge n = a + b,\,$ wlog $a \le b\,$ ... |
422,948 | <p>How could I/is it possible to take a fourier transform of text? i.e. What domain would/does text exist in? Any help would be great.</p>
<p>NOTE: I do not mean text as an image. I understand it's value, but I'm wondering if it is possible to map text to some domain and transform text on the basis of letters. This is... | Matt L. | 70,664 | <p>You could take the text as a 2-D image and use a 2-D Fourier transform. This could be useful e.g. to find the orientation of the text and subsequently - if necessary - apply an appropriate rotation, which makes it easier for text recognition methods to give satisfactory results.</p>
|
4,106,273 | <p>In how many ways can a committee of four be formed from 10 men (including Richard) <br>
and 12 women (including Isabel and Kathleen) if it is to have two men and two women <br></p>
<p>a) Isabel refuses to serve with Richard,</p>
<p>b) Isabel will serve only if Kathleen does, too</p>
<p>My Thoughts : <br>
a) Total nu... | N. F. Taussig | 173,070 | <blockquote>
<p>In how many ways can a committee of four be formed from <span class="math-container">$10$</span> men (including Richard) and <span class="math-container">$12$</span> women (including Isabel and Kathleen) if it is to have two men and two women and Isabel refuses to serve with Richard?</p>
</blockquote>
<... |
3,105,546 | <p>Suppose we have two functions <span class="math-container">$g:\mathbb{R}\rightarrow [0,1]$</span> and <span class="math-container">$f:\mathbb{R}\rightarrow [0,1]$</span> such that
<span class="math-container">$$\hspace{1cm}\lim\limits_{y\rightarrow -\infty}g(y)=\lim\limits_{y\rightarrow -\infty}f(y)=0 \tag{1}$$</spa... | Servaes | 30,382 | <p>Yes, it is not finite; if it were finite, say <span class="math-container">$\lim_{y\rightarrow -\infty}\frac{f(y)}{g(y)}=L$</span>, then this would imply that
<span class="math-container">$$0\cdot L=\left(\lim_{y\rightarrow -\infty}\frac{g(y)}{f(y)}\right)\left(\lim_{y\rightarrow -\infty}\frac{f(y)}{g(y)}\right)=\li... |
2,716,081 | <p>A stone of mass 50kg starts from rest and is dragged 35m up a slope inclined at 7 degrees to the horizontal by a rope inclined at 25 degrees to the slope. the tension in the rope is 120N and the resistance to motion of the stone is 20N. calculate the speed of the stone after it has moved 35m up the slope.</p>
<p>An... | Jack D'Aurizio | 44,121 | <p>You may also consider $f_n(x) = \sqrt{n}\, e^{-n^2 x^2}$. In explicit terms, $\int_{-\infty}^{+\infty} f_n(x)^2\,dx = \sqrt{\frac{\pi}{2}}$ while
$$ \int_{-\infty}^{+\infty}f_n'(x)^2\,dx = \color{red}{n^2} \sqrt{\frac{\pi}{2}}.$$</p>
|
4,253,598 | <p>My textbook states that if <span class="math-container">$f(x) \to 0$</span> as <span class="math-container">$x \to 0$</span> <span class="math-container">$$\lim_{x \to 0} (1+f(x))^\frac{1}{g(x)} = e^l$$</span> where <span class="math-container">$$l=\lim_{x \to 0} \frac{f(x)}{g(x)}$$</span><br />
I try to do this as... | Alann Rosas | 743,337 | <p><strong>Note</strong>: we must have <span class="math-container">$f(x)>-1$</span> for every <span class="math-container">$x\in\text{dom}[f]$</span> for <span class="math-container">$(1+f(x))^\frac{1}{g(x)}$</span> to be well defined.</p>
<p>If <span class="math-container">$f$</span> is nonzero in a sufficiently s... |
4,487,494 | <blockquote>
<p><strong>Problem:</strong> Let <span class="math-container">$x$</span> and <span class="math-container">$y$</span> be non-zero vectors in <span class="math-container">$\mathbb{R}^n$</span>.<br>
(a) Suppose that <span class="math-container">$\|x+y\|=\|x−y\|$</span>. Show that <span class="math-container">... | emacs drives me nuts | 746,312 | <p>It's unclear from where you got the starting relations. That <span class="math-container">$x+y$</span> and <span class="math-container">$x-y$</span> are perpendicular means that
<span class="math-container">$$(x+y)\cdot(x-y) = 0 \tag 1$$</span>
and you can start reasoning from there:</p>
<p><span class="math-contai... |
358,184 | <p>We have a lot of probabilities lower bounding as (e.g. chernoff bound, reverse markov inequality, Paley–Zygmund inequality)
<span class="math-container">$$
P( X-E(X) > a) \geq c, a > 0 \quad and \quad P(X > (1-\theta)E[X]) \geq c, 0<\theta < 1
$$</span></p>
<p>However, It would be great to know if t... | coudy | 6,129 | <p>The <a href="https://en.wikipedia.org/wiki/Cantelli%27s_inequality" rel="nofollow noreferrer">Cantelli</a> inequality asserts that</p>
<p><span class="math-container">$$
\Pr(X-\mathbb{E}[X]\ge\lambda)\quad\begin{cases}
\le \frac{\sigma^2}{\sigma^2 + \lambda^2} & \text{if } \lambda > 0, \\[8pt]
\ge 1 - \frac{\... |
4,226,030 | <blockquote>
<p>I want to solve
<span class="math-container">$$C\cos(\sqrt\lambda \theta) + D\sin(\sqrt\lambda \theta) = C\cos(\sqrt\lambda (\theta + 2m\pi)) + D\sin(\sqrt\lambda (\theta + 2m\pi))$$</span>
The solution must be valid for all <span class="math-container">$\theta$</span> in <span class="math-container">$\... | Akil | 875,365 | <p>Use the formula <span class="math-container">$\cos(x) = \frac{e^{ix} + e^{-ix}}{2}$</span> and <span class="math-container">$\sin(x) = \frac{e^{ix} - e^{-ix}}{2i}$</span>. When you substitute this into your expression, on the left-hand side, you would get <span class="math-container">$ C(\frac{e^{i\sqrt{\lambda}\th... |
1,842,365 | <p>I recently got acquainted with a theorem:</p>
<p>If $f(x)$ is a periodic function with period $P$, then $f(ax+b)$ is periodic with period $\dfrac{P}{a}$ , $a>0$.</p>
<p>I am having a difficulty in understanding this theorem. Does this theorem mean that $f(ax+b)=f(ax+b+ \dfrac{P}{a})$? </p>
<p>If the above mean... | joriki | 6,622 | <p>Let $(Z_1,Z_2)$ be uniformly distributed on $\{(-1,0),(0,1),(1,-1)\}$. Then $Z_1$ and $Z_2$ are identically distributed but dependent. The value of $Z_1+Z_2$ fixes the (different) values of $Z_1$ and $Z_2$, so $\mathbb E[Z_1\mid Z_1+Z_2]\ne\mathbb E[Z_2\mid Z_1+Z_2]$.</p>
|
24,550 | <p>Let <span class="math-container">$H$</span> and <span class="math-container">$K$</span> be affine group schemes over a field <span class="math-container">$k$</span> of characteristic zero. Let <span class="math-container">$\varphi:H\to Aut(K)$</span> be a group action. Then we can form the semi-direct product <span ... | Victor Protsak | 5,740 | <p>Even when viewed as an additive category, Rep(G) is not semisimple, so it's not really clear to me what such a description would entail... But simple objects in it are (in a different language) irreducible representations of a semidirect product of groups and Mackey theory was invented precisely with the goal of det... |
1,504,214 | <p>When Mr. and Mrs. Smith took the airplane, they had together 94 pounds of baggage.
He paid 1.50 and she paid 2.00 for excess weight. If Mr. Smith made the trip by himself with the combined baggage of both of them, he would have to pay $13.50. How many pounds of baggage can one person take along without being charged... | josh | 11,815 | <p>They want you to find how many pounds of baggage each person can take on the plane for free. Let</p>
<p>$x = $ the number of pounds someone can bring for free.</p>
<p>We also need to introduce a variable that represents the price per pound for excess baggage, which isn't mentioned in the word problem. This is call... |
948,076 | <p>of course the problem is how to prove if <code>a</code> and <code>b</code> are both algebraic <code>real</code> numbers then <code>a+b</code> and <code>ab</code>
is also an algebraic number .</p>
<p>would you explain it without using vector spaces or extensions or etc.
things we know are :</p>
<ol>
<li><p>there ex... | Marc Bogaerts | 118,955 | <p>I would say because it can. The subgroup of $SL(2,11)$ generated by $\begin{pmatrix} 0&-1\\ 1 & 0 \\\end{pmatrix} $ and $\begin{pmatrix} 3 &1\\ 0 & 4 \\\end{pmatrix}$ is isomorphic to $SL(2,5)$</p>
|
948,076 | <p>of course the problem is how to prove if <code>a</code> and <code>b</code> are both algebraic <code>real</code> numbers then <code>a+b</code> and <code>ab</code>
is also an algebraic number .</p>
<p>would you explain it without using vector spaces or extensions or etc.
things we know are :</p>
<ol>
<li><p>there ex... | C Monsour | 552,399 | <p>Let's give a little more explanation here. First of all, $SL(2,5)$ can be presented as $<x,y,z|o(x)=5,o(y)=3,o(z)=2,xz=zx,yz=zy,(xy)^2=z>$. (See Passman, <em>Permutation Groups</em>, Prop 13.7). This can be realized over any field of characteristic $0$ or greater than $5$ that has square roots of -1 and of ... |
1,189,814 | <p>Can a finitely generated module $M$ over a commutative ring have $\operatorname{Ann}(x) \neq 0$ for all $x \in M$ while $\operatorname{Ann}(M) = 0$?</p>
<p>It's not difficult to show that there is no such module if the ring is a integral domain. For general, I guess the answer is yes. But I failed to find a desired... | Gregory Grant | 217,398 | <p>What if you take $C=\{i^r | r\in \mathbb Q\}$, where $i\in\mathbb C$. $C$ is closed under multiplication. Let $\mathbb Q$ act on it by $r\cdot z = z^r$. Then it's finitely generated as a $\mathbb Q$ module (generated by $i$). Then $C$ is a $\mathbb Q$ module (I think) and every element has non-zero annihilator (... |
1,488,752 | <p><s>I would like to know if the following question has an intelligent solution:</p>
<p>Determine the largest bet that cannot be made using chips of $7$ and $9$ dollars.</p>
<p>After not being able to solve it I found a solution online which writes out all combinations of $7$ and $9$ up to $90$ and then notes that w... | vonbrand | 43,946 | <p>A very nice explanation is given at <a href="http://www.cut-the-knot.org/blue/Sylvester2.shtml" rel="nofollow">Cut the knot</a>.</p>
<p>You want the number for $p, q$ with $\gcd(p, q) = 1$ (otherwise it makes no sense). Consider the $q$ sequences:</p>
<p>$\begin{align}
&0 + 0, 0 + q, 0 + 2 q, \dotsc \\
&... |
584,171 | <blockquote>
<p>Show that every graph can be embedded in $\mathbb{R}^3$ with all
edges straight. </p>
</blockquote>
<p>(Hint: Embed the vertices inductively, where should you
not put the new vertex?)</p>
<p>I've also received a tip about using the curve ${(t, t^2 , t^3 : t \in \mathbb{R} )}$ but I'm not sure ho... | user99680 | 99,680 | <p>A graph can be seen as a manifold; there is a result that the graph of a smooth function is a manifold; in this case, it is a 1-manifold. Then use Whitney's theorem that guarantees that a smooth m-manifold can be smoothly-embedded in $\mathbb R^{2m}$.</p>
|
315,386 | <p>I think Wikipedia's polar coordinate elliptical equation isn't correct. Here is my explanation: Imagine constants <span class="math-container">$a$</span> and <span class="math-container">$b$</span> in this format -
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/6/65/Ellipse_Properties_of_Directrix_a... | robjohn | 13,854 | <p><strong>Polar Equation from the Center of the Ellipse</strong></p>
<p>The equation of an ellipse is
<span class="math-container">$$
\left(\frac{x}{a}\right)^2+\left(\frac{y}{a\sqrt{1-e^2}}\right)^2=1\tag1
$$</span>
Using <span class="math-container">$x=r\cos(\theta)$</span> and <span class="math-container">$y=r\sin... |
120,992 | <p>An algorithm book <a href="http://rads.stackoverflow.com/amzn/click/1849967202" rel="nofollow">Algorithm Design Manual</a> has given an description:</p>
<blockquote>
<p>Consider a graph that represents the street map of Manhattan in New York City. Every junction of two streets will be a vertex of the graph. Neigh... | Desiato | 26,955 | <p>If you have 15 vertical and 200 horizontal lines, parallel for each direction, they'll have 15*200 crossings. An edge is a line segment between each crossing. Each crossing connects 4 line segments (ignoring the outer bounds) and each line segment connects two crossings, so it's double the number of crossings.</p>
|
2,724,686 | <blockquote>
<p>Set $B = \{1,2,3,4,5\}$, $S$ - equivalence relation. It is given that for all $x,y \in B$ if $(x,y)\in S$ and if $x+y$ is an even number then $x = y$. In such case is it true that:</p>
<ol>
<li>the number of elements in each equivalence class of $S$ is at most $2$</li>
<li>any relation $S$ wo... | Jimmy R. | 128,037 | <p>Assume that $S$ has an equivalence class with $2$ even numbers, namely $2$ and $4$. Then, $(x,y)\in S$ and $x+y=6$ which is even, but this contradicts $x=y$. Hence, two even numbers cannot be in the same equivalence class.</p>
<p>Now, assume that there exists an equivalence class with two odd numbers $x\neq y$. The... |
1,639,232 | <p>A really simple question, but I thought I'd ask anyway. Does $n<x^n<(n+1)$ imply $\sqrt[n] n < x < \sqrt[n] {n+1}$?</p>
<p>Thank you very much.</p>
| Hagen von Eitzen | 39,174 | <p>$\{ceps_q\}_{q=0}^Q$ is the finite sequence (or <code>array</code> or <code>vector</code> in programmese) $$ceps_0,ceps_1,\ldots, ceps_Q.$$</p>
<p>Likewise, $\{a_q\}_{q=1}^p$ denotes $$a_1,a_2,\ldots, a_p.$$</p>
|
2,417,542 | <p>$$\sum_{i,j}{n\brack i+j}\binom{i+j}i$$
Does this have a combinatorial interpretation? I don't see how to use Stirling numbers of the first kind in interpretations. I know that the answer is $(n+1)!$ , but the original question didn't provide it.</p>
| John Alexiou | 3,301 | <p>You can look at this graphically. The columns of the rotation matrix are the components of the local $\hat{i}$ and $\hat{j}$ vectors.</p>
<p>$$\begin{align} \hat{i} & = \pmatrix{0 \\ -1} \\ \hat{j} &= \pmatrix{1 \\ 0} \end{align}$$</p>
<p>Graphically you are asked to find the angle $\theta$ the produces th... |
1,218,354 | <p>I've read in some textbooks that $\vdash$ and $\vDash$ are symbols present only in metalanguage. From this, I infer that their use in object language is unacceptable.</p>
<p>I would like to know why. Can't we define them as relation symbols in a structure? Or introduce them in statements for the sake of formal proo... | Mauro ALLEGRANZA | 108,274 | <p>Consider for example <em>propositional logic</em></p>
<p>The <em>syntax</em> specifications of the <strong>language</strong> allows us to build formulae from propositional variables : $p, q, \ldots$ and constants : $\bot, \top$ with the <em>connectives</em>; usually :</p>
<blockquote>
<p>$\lnot, \lor, \land, \to... |
125,661 | <p>When typing the name of a built-in function like <code>Integrate</code>, the button (<em>ℹ︎</em>) appears next to that name in the autocomplete:</p>
<p><img src="https://i.stack.imgur.com/ISWj1.png" alt="enter image description here"></p>
<p>But I don't get that (<em>ℹ︎</em>) button for my package functions, even ... | b3m2a1 | 38,205 | <p>So I mentioned this in a comment but I'll get it out here for the bounty poster too. Searching for the <code>"JLink`"</code> in the autocomplete directory gave me this:</p>
<pre><code>FileNameJoin@{$InstallationDirectory, "SystemFiles", "Components", "AutoCompletionData", "Main", "documentedContexts.m"}
</code></pr... |
1,972,927 | <p>What's the function determined by the series $1+\sin(x)\cos(x) + \sin(x)^2 \cos(x)^2 + \cdot \cdot \cdot$?</p>
<hr>
<p>Note, the series converges uniformly.</p>
| E.H.E | 187,799 | <p>$$\frac1{1-x}=\sum_{n=0}^\infty x^n\quad{\text{ for }}|x|<1$$</p>
<hr>
<p><em>Let $x\to\sin x\cos x$.</em></p>
|
2,812,314 | <p>For integers $n\geq 1$ we denote the square-free kernel as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,$$ that is the product of distinct primes dividing an integer $n>1$ with the definition $\operatorname{rad}(1)=1$ (the Wikipedia's article dedicated to this multiplicative function is <a ... | mechanodroid | 144,766 | <p>It is not in $\ell^2$:</p>
<p>\begin{align}
\sum_{n=1}^\infty\frac{n-2n\sqrt n+n^2}{n^3}&=\sum_{n=1}^\infty\frac{1-2\sqrt n+n}{n^2} \\
&= \sum_{n=1}^\infty\frac{(\sqrt{n}-1)^2}{n^2} \\
&\ge \sum_{n=1}^3\frac{(\sqrt{n}-1)^2}{n^2} + \sum_{n=4}^\infty\frac{\left(\frac12\sqrt{n}\right)^2}{n^2} \\
&= \s... |
2,812,314 | <p>For integers $n\geq 1$ we denote the square-free kernel as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,$$ that is the product of distinct primes dividing an integer $n>1$ with the definition $\operatorname{rad}(1)=1$ (the Wikipedia's article dedicated to this multiplicative function is <a ... | Fred | 380,717 | <p>Put $a_n=\frac{1}{n}-\frac{1}{\sqrt{n}}, b_n =\frac{1}{n}$ and $c_n=\frac{1}{\sqrt{n}}$.</p>
<p>Suppose that $(a_n) \in \ell^2$. Since $(b_n) \in \ell^2$ and since $\ell^2$ is a vector space, we get that $(c_n)=(b_n)-(a_n) \in \ell^2$, a contradiction. Hence $(a_n) \notin \ell^2$.</p>
|
720,282 | <p>I was trying to come up with reasons, why we naturally consider the topology of uniform convergence on compact sets as the appropriate framework for spaces of holomorphic functions such as e.g. $H(\mathbb{C}^n)$ (which is the space of entire functions on $\mathbb{C}^n$).</p>
<p>I understand that e.g. by Weierstrass... | OR. | 26,489 | <p>Another reason could steam from <a href="http://en.wikipedia.org/wiki/Montel%27s_theorem" rel="nofollow">Montel-type's results</a>. </p>
<p>Assume you have a family of holomorphic functions that omit two values. Then the family will be <a href="http://en.wikipedia.org/wiki/Normal_family" rel="nofollow">normal</a>.<... |
171,364 | <p>So I'm looking for a function that takes in the degree of the polynomial and the range of coefficients from -c to c, and outputs a list of all the monic polynomials of that degree and with coefficients in that range.</p>
<p>I already have code to numerically compute the roots and plot in the complex plane, I just n... | kglr | 125 | <p>Note that we can use <code>FromDigits[{1, a, b, c}, x]</code> to get a polynomial of degree <code>Length[{a,b,c}]</code> in <code>x</code> with leading coefficient <code>1</code>:</p>
<pre><code> Expand @ FromDigits[{1, a, b, c}, x]
</code></pre>
<blockquote>
<p>c + b x + a x^2 + x^3</p>
</blockquote>
<p>Using ... |
1,182,432 | <p>Is it possible, that everyone is a pseudo-winner in a tournament with 25 people?
(pseudo-winner means that either he won against everyone, or if he lost against someone, then he beated someone else, who beated the one who he lost to).</p>
<p>In the "language" of graph theory: Is it possible in the directed K(25) (a... | Rebecca J. Stones | 91,818 | <p>There are <a href="http://mathworld.wolfram.com/SteinerTripleSystem.html" rel="nofollow">Steiner triple systems</a> of order $25 \equiv 1 \pmod 6$. If we take one, and replace each undirected $3$-cycle with a (coherent) directed $3$-cycle, we get a everyone's a pseudo-winner tournament.</p>
|
2,935,693 | <p>I am trying to prove that </p>
<p><span class="math-container">$(a\to(b\to c))\to((a\to b)\to(a\to c))$</span></p>
<p>holds in natural deduction, in particular when I work backwards from a Fitch style proof I can only get so far:</p>
<p><a href="https://i.stack.imgur.com/w2BYf.png" rel="nofollow noreferrer"><img ... | fleablood | 280,126 | <p>In line 7. you Hypothesize B. </p>
<p>Don't Hypothesize when you can prove.</p>
<p>You have 4: <span class="math-container">$A\to B$</span>. and 5: <span class="math-container">$A$</span> so you <em>know</em> <span class="math-container">$B$</span> via "elim 4, 5".</p>
<p>Replace "7. B Hyp" with "7. B elim 4,5"</... |
28,104 | <p>It occured to me that the Sieve of Eratosthenes eventually generates the same prime numbers, independently of the ones chosen at the beginning. (We generally start with 2, but we could chose any finite set of integers >= 2, and it would still end up generating the same primes, after a "recalibrating" phase).</p>
<p... | Bruno Joyal | 6,779 | <p>This is not exacly what you are asking for, but it's relevant enough to mention : <a href="http://en.wikipedia.org/wiki/Lucky_number" rel="nofollow">lucky numbers</a>.</p>
|
2,875 | <p>I've heard that irreducible unitary representations of noncompact forms of simple Lie groups, the first example of such a group <code>G</code> being <code>SL(2, R)</code>, can be completely described and that there is a discrete and continuous part of the spectrum of <code>L^2(G)</code>.</p>
<ol>
<li>How are those ... | LSpice | 2,383 | <p>I think that <a href="https://mathoverflow.net/questions/2875/unitary-representations-of-sl2-r/2940#2940">Rob H.</a>'s answer is probably best; but, for (1) and (2), if you are interested in small general and special linear groups particularly, you could do worse than to consult Lang's $\operatorname{SL}_2(\mathbb R... |
1,607,190 | <p>Prove by induction that $8^{n} − 1$ for any positive integer $n$ is divisible by $7$. </p>
<p>Hint: It is easy to represent divisibility by $7$ in the following way: $8^{n} − 1 = 7 \cdot k$ where k is a positive integer.</p>
<p>This question confused me because I think the hint isn't true. If $n = 1$ and $k = 2$ f... | Community | -1 | <p><span class="math-container">$$8^{n+1}-1=7\cdot8^n+8^n-1\equiv 8^n-1\mod 7$$</span> and <span class="math-container">$$8^0-1=0.$$</span></p>
|
1,016,884 | <p>Four friends, Andrew, Bob, Chris and David, all have different heights. The sum of their heights is 670 cm.
Andrew is 8cm taller than Chris and Bob is 4cm shorter than David.
The sum of the heights of the tallest and shortest of the friends is 2cm more than the sum of the heights of the other two.
Find the height of... | Ross Millikan | 1,827 | <p>Hint: let the heights be $a,b,c,d$. The second and third sentences give you three equations-can you write them? From the fourth sentence, we know the tallest is either Andrew or David and the shortest is either Bob or Chris. That gives you four possibilities for the fourth equation. All the equations are linear... |
1,016,884 | <p>Four friends, Andrew, Bob, Chris and David, all have different heights. The sum of their heights is 670 cm.
Andrew is 8cm taller than Chris and Bob is 4cm shorter than David.
The sum of the heights of the tallest and shortest of the friends is 2cm more than the sum of the heights of the other two.
Find the height of... | Bhaskar Vashishth | 101,661 | <p>Let us dnote there heights by a,b,c,d resp. then equation are
$$a+b+c+d=670$$ $$c+8=a$$ $$b+4=d$$ they together gives $$2c+2b+12=670 \implies c+b=329$$ </p>
<p>Now Andrew is 8cm taller than Chris and Bob is 4cm shorter than David says that chris is not tallest and neither is bob, which says andrew or david is tal... |
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