qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
2,439,340 | <p>How would one proceed to prove this statement?</p>
<blockquote>
<p>The set of the strictly increasing sequences of natural numbers is not enumerable.</p>
</blockquote>
<p>I've been trying to solve this for quite a while, however I don't even know where to start.</p>
| Misha Lavrov | 383,078 | <p>We can define a very simple injection from the real numbers in the interval $[1,10)$ to your set by mapping $x \in [1,10)$ to the sequence $$\lfloor x \rfloor, \lfloor 10x \rfloor, \lfloor 100x \rfloor, \lfloor 1000x \rfloor, \dots.$$
For example, $\pi$ would map to the sequence $$3, 31, 314, 3141, 31415, 314159, \d... |
2,439,340 | <p>How would one proceed to prove this statement?</p>
<blockquote>
<p>The set of the strictly increasing sequences of natural numbers is not enumerable.</p>
</blockquote>
<p>I've been trying to solve this for quite a while, however I don't even know where to start.</p>
| zhw. | 228,045 | <p>The map from $\{0,1\}^{\mathbb N}$ into the set of strictly increasing sequences of natural numbers given by</p>
<p>$$(a_n) \to (1+a_1,3+a_2, 5+a_3,7+a_4, \dots)$$</p>
<p>is injective. Since $\{0,1\}^{\mathbb N}$ is uncountable, we're done.</p>
|
2,439,340 | <p>How would one proceed to prove this statement?</p>
<blockquote>
<p>The set of the strictly increasing sequences of natural numbers is not enumerable.</p>
</blockquote>
<p>I've been trying to solve this for quite a while, however I don't even know where to start.</p>
| Samuel G. Silva | 229,701 | <p>The family of all strictly increasing sequences of natural numbers, if regarded as a family of functions from naturals into naturals, is a dominating family under both pointwise and mod finite orders, so it size has to be larger than or equal to the dominating number <span class="math-container">$\mathfrak{d}$</spa... |
3,185,317 | <p><span class="math-container">$$\lim_{n\rightarrow 0}\frac{1}{n}\int_{0}^{1}\ln(1+e^{nx})dx$$</span></p>
<p>My try:</p>
<p><span class="math-container">$$\frac{b-a}b\leq \ln b-\ln a\leq \frac{b-a}a \implies \frac{1}{1+e^{nx}}\leq \ln(1+e^{nx})-\ln e^{nx}\leq \frac1{e^{nx}}$$</span></p>
<p>Then I integrated and mul... | DMcMor | 155,622 | <p>Intuitively, what the limit is doing is finding the behavior of the fraction <span class="math-container">$$\frac{-2(x-1)(x+1)}{(x-1)}$$</span> as <span class="math-container">$x$</span> approaches <span class="math-container">$1$</span>. The actual value of the fraction at <span class="math-container">$1$</span> i... |
3,425,373 | <p>Consider:</p>
<p><span class="math-container">$$ 1+1/2^2+2/3^2+1/4^2+2/5^2+1/6^2+...$$</span></p>
<p>Does this sum have a closed form?</p>
<p>If all the numerators are <span class="math-container">$1$</span> then it does have a closed form. </p>
| amir bahadory | 204,172 | <p>We have <span class="math-container">$$\frac{\pi^2}{6}=\sum_{n=1}^\infty\frac{1}{n^2}=\sum_{n=1}^\infty \frac{1}{(2n-1)^2}+\sum_{n=1}^\infty\frac{1}{(2n)^2}$$</span></p>
<p>But <span class="math-container">$\sum_{n=1}^\infty\frac{1}{(2n)^2}=\frac{1}{4}\sum_{n=1}^\infty\frac{1}{n^2}
$</span> so <span class="math-con... |
872,493 | <p>$$\lim_{x\to 0^+} \sin(x)^\frac{1}{\ln(x)} = ... =
\exp \left(\lim_{x\to 0^+} \frac{\ln(\frac{\sin x}{x}) + \ln(x)}{\ln(x)}\right)$$</p>
<p>Now, from continuity we can evaluate each term separately.</p>
<p>$\lim_{x\to 0^+} \ln(\frac{\sin x}{x}) = 0$</p>
<p>Therefore, we have:<br>
$$\exp \left(\lim_{x\to 0^+} \fr... | Hans Lundmark | 1,242 | <p>You can simplify ${\ln x} / {\ln x}$ to $1$ <em>before</em> taking the limit, and therefore there's no problem. But writing ${-\infty}/{-\infty}$ is nonsense.</p>
|
10,873 | <p>We have two tags with identical names on main and meta:</p>
<p><a href="https://math.stackexchange.com/questions/tagged/computer-science">(main:computer-science)</a>
and
<a href="https://math.meta.stackexchange.com/questions/tagged/computer-science">(meta:computer-science)</a></p>
<p>For main tags one can use <co... | Martin Sleziak | 8,297 | <p>Perhaps it is also useful to mention that the syntax <code>[meta-tag:feature-request]</code> is included in the <a href="http://meta.math.stackexchange.com/editing-help#tags">editing help</a> on meta. <a href="https://math.stackexchange.com/editing-help#tags">Editing help</a> on main only mentions the syntax <code>[... |
231,036 | <p>I wonder if there is an example of rational homology sphere that is not a Seifert manifold. If there is, how can one construct such a rational homology sphere from a surgery of a knot in $S^3$?</p>
| Sebastian Goette | 70,808 | <p>If you glue opposite faces of a dedecahedron with a twist of $\frac\pi5$, you obtain the classical Poincaré sphere. It is an integral homology sphere, and a Seifert manifold, see BS' comment below.</p>
<p>But if you do the gluing with a twist of $\frac{3\pi}5$, then you obtain the <a href="https://en.wikipedia.org/... |
3,950,098 | <p>I can evaluate the limit with L'Hospital's rule:</p>
<p><span class="math-container">$\lim_{n\to\infty}n(\sqrt[n]{4}-1)=\lim_{n\to\infty}\cfrac{(4^{\frac1n}-1)}{\dfrac1n}=\lim_{n\to\infty}\cfrac{\dfrac{-1}{n^2}\times 4^{\frac1n}\times\ln4}{\dfrac{-1}{n^2}}=\ln4$</span></p>
<p>But is there any way to do it without us... | 2'5 9'2 | 11,123 | <p>You could use a Maclaurin expansion for <span class="math-container">$e^x$</span>:</p>
<p><span class="math-container">$$\begin{align}
n\left(\sqrt[n]{4}-1\right)
&=n\left(e^{\ln(4)/n}-1\right)\\
&=n\left(\frac{\ln(4)}{n}+\frac{1}{2}\left(\frac{\ln(4)}{n}\right)^2+\cdots\right)\\
&=\ln(4)+\frac{1}{2}\fra... |
1,502,676 | <p>I'm aware of this similiar question:</p>
<p><a href="https://math.stackexchange.com/questions/1249308/what-is-the-difference-between-an-elliptical-and-circular-paraboloid-3d">what is the difference between an elliptical and circular paraboloid? (3D)</a></p>
<p>But I need help in a different way. In my calculus exa... | Narasimham | 95,860 | <p>$$(x-2)^2+(y-4)^2=a^2$$</p>
<p>is a circle with radius $a$ whose center is at $2,4$ </p>
<p>$$(x-2)^2+(y-4)^2+1=z$$</p>
<p>is a circular paraboloid with whose center is at $2,4$</p>
<p>Instead of looking at coefficients of $x^2, y^2$ ( which must be equal for circularity ) you are looking at other displacement c... |
418,258 | <p>Two friends of mine are trying to convince me that they are good at betting.
They infact have a certain profit after n bets. They provided me with all the bets they have made so far: odds, outcome and amount bet.</p>
<ol>
<li>How can I check if their wins are statistically significant or it's just variance that bro... | Hagen von Eitzen | 39,174 | <p>Say you have 32 friends and after betting on coin tosses five times (always betting all their money), one of them approaches you and says: "See? I have prophetical powers. A predicted the outcome of a fair coin five times in a row and made 320\$ from my original 10\$ - you can't explain that with sheer luck. And as ... |
418,258 | <p>Two friends of mine are trying to convince me that they are good at betting.
They infact have a certain profit after n bets. They provided me with all the bets they have made so far: odds, outcome and amount bet.</p>
<ol>
<li>How can I check if their wins are statistically significant or it's just variance that bro... | Nameless | 68,482 | <ol>
<li><p>A very naive way would be to simply compute the mean return per bet, compute the standard error of that mean, and compute confidence intervals with the appropriate t-distribution thresholds. If zero is outside the interval, their return is "significant". That is, with your confidence level you would say the... |
418,258 | <p>Two friends of mine are trying to convince me that they are good at betting.
They infact have a certain profit after n bets. They provided me with all the bets they have made so far: odds, outcome and amount bet.</p>
<ol>
<li>How can I check if their wins are statistically significant or it's just variance that bro... | Zackkenyon | 79,927 | <p>Without knowing more specifics about the game itself, it is very difficult to establish a correlation between data and confidence. but I will try to give you an upper bound for a broad class of games. That is games with the condition that there is some factor of your bankroll that you cannot make more than in any gi... |
1,850,258 | <p>From where can I learn mathematics from the basic blocks up? I feel like I have a lot of holes in the mathematics that I know and I would like to see where all those concepts come from. I would like to see what are the ideas that are took from granted, as foundation, and which ideas are made from this foundation.</p... | Mr. Brooks | 162,538 | <p>At the time I posted my comment, I couldn't give a fleshed out answer. In the interest of brevity, I assumed a lot of prior knowledge, some of which I learned long ago, some of which I learned just in the past year or so. At the other extreme, an entire book could be written to answer your $4$-part question. I will ... |
3,057,278 | <blockquote>
<p><strong>Question:</strong> Can we show that <span class="math-container">$$\sum_{n=0}^\infty(-1)^{n+1}\frac{(2n-3)!!}{(2n+3)!!}=\frac{\pi}{8} $$</span> ?</p>
</blockquote>
<hr>
<p>According to <a href="https://www.wolframalpha.com/input/?i=sum+(-1)%5E(n%2B1)+%5B(2n-3)!!%2F(2n%2B3)!!%5D+from+n%3D+0+t... | Micah | 30,836 | <p>Note that after cancellation we have</p>
<p><span class="math-container">$$\frac{(2n-3)!!}{(2n+3)!!}=\frac{1}{(2n-1)(2n+1)(2n+3)}$$</span></p>
<p>Taking partial fractions gives
<span class="math-container">$$
\frac{1}{(2n-1)(2n+1)(2n+3)}=\frac{1}{8}\left(-\frac{1}{2n-1}+\frac{2}{2n+1}-\frac{1}{2n+3}\right)
$$</spa... |
1,182,684 | <p>Let $\mathbf{F}(x,y,z) = y \hat{i} + x \hat{j} + z^2 \hat{k}$ be a vector field. Determine if its conservative, and find a potential if it is.</p>
<p><strong>Attempt at solution:</strong></p>
<p>We have that $\frac{\partial F_1}{\partial y} = 1 = \frac{\partial F_2}{\partial x} $, $\frac{\partial F_1}{\partial z} ... | kryomaxim | 212,743 | <p>Take the derivative of $f(x,y,z)$ by y; this must be equal to $-x$. Then solve for $C(y,z)$ by Integration. You will obtain another Integration constant $C(z)$ that you can obtain by derivative by variable $z$ (must be equal to $z^2$).</p>
|
165,853 | <blockquote>
<p>Schauder's conjecture: "<em>Every continuous function, from a nonempty
compact and convex set in a (Hausdorff) topological vector space into
itself, has a fixed point.</em>" [Problem 54 in The Scottish Book]</p>
</blockquote>
<p>I wonder whether this conjecture is resolved. I know R. Cauty [So... | Doug Liu | 153,360 | <p>An old proof (by M. R. Taskovic) is <a href="http://elib.mi.sanu.ac.rs/files/journals/mm/2/Math.%20Moravican2p121-132.pdf" rel="nofollow noreferrer" title="On Schauder's 54th problem in Scottish book">here</a>: Mathematica Moravica, volume 2, (1998).</p>
<p>Summary: " The most famous of many problems in nonline... |
1,984,076 | <p>How to prove that
$$
\sum_{k=1}^\infty\frac{k^k}{k!}x^k=\frac{1}{2}, ~\text{where}~~ x=\frac{1}{3}e^{-1/3}~?
$$
I found this sum in my notes, but I don't remember where I got it. Any hints or references would be nice.</p>
| Martín-Blas Pérez Pinilla | 98,199 | <p>Idea, too long for a comment: use the <a href="https://en.wikipedia.org/wiki/Lambert_W_function" rel="nofollow">Lambert function</a>.
$$W(x) = \sum_{n=1}^\infty\frac{(-n)^{n-1}}{n!}x^n,$$
$$W'(x) = \sum_{n=1}^\infty\frac{(-1)^{n-1}n^n}{n!}x^{n-1} = \sum_{n=1}^\infty\frac{n^n}{n!}(-x)^{n-1}$$
$$\cdots$$</p>
|
1,762,001 | <p>I recently watched a <a href="https://www.youtube.com/watch?v=SrU9YDoXE88" rel="noreferrer">video about different infinities</a>. That there is $\aleph_0$, then $\omega, \omega+1, \ldots 2\omega, \ldots, \omega^2, \ldots, \omega^\omega, \varepsilon_0, \aleph_1, \omega_1, \ldots, \omega_\omega$, etc..</p>
<p>I can't... | galois | 75,830 | <p>As an insight, think of the size of the usual sets of numbers.</p>
<p>Think of the set of all positive integers, and the set of all positive multiples of 5. It's a bit strange to the "uninitiated", but it's ultimately not hard to wrap your head around the fact that there's the same amount of each, because we can li... |
1,165,207 | <p>From browsing the internet so far I've came to the conclusion that an ordered tuple is something in which there is no repettion of the elments
Eg: An ordered 4-tuple is (1,2,3,4) or (5,3,1,7) i.e. no elements are repeated</p>
<p>But an unordered 4-tuple is something like this (1,2,2,3) or (7,1,4,1)</p>
<p>Is this ... | Daniel W. Farlow | 191,378 | <p>Consider the Cartesian product to clarify things a little bit. </p>
<p>The Cartesian product of two sets $A$ and $B$ is defined to be the set of all <strong>ordered pairs</strong> $(a,b)$ where $a\in A$ and $b\in B$. The Cartesian product is often written as
$$
A\times B = \{(a,b)\mid a\in A\space\text{and}\space b... |
3,701,582 | <p>I still struggle mighty with basic conceptions of truth and proof. </p>
<p>For example: The Continuum Hypothesis (CH) is either true or false, i.e. either CH or ~CH holds. Now, Goedel and Cohen proved that CH/~CH are independent from ZFC, so ZFC + CH and ZFC + ~CH are consistent (in case ZFC is consistent but mathe... | Lee Mosher | 26,501 | <p>Your "Example" statement misses an important point: CH is a sentence of set theory, and <em>within any model</em> of set theory CH is either true or false. There are some models of set theory in which CH is true, there are other models of set theory in which it is false, and in every model of set theory it is either... |
2,544,755 | <p>How would I start to prove this set is bounded above and below? A hint would be appreciated!</p>
<p>$A:=[0,1)\cup\{2-\frac{1}{n^5}:n\in \mathbb{N}\}$</p>
| Ltoll | 301,381 | <p>Hint:</p>
<p>A set is bounded above and below if and only if there exists an $M\in\mathbb{R}$ such that $|x|\leq M$ for all $x\in A$.</p>
<p>Now your set is $A=[0,1)\cup\{2-\frac1{n^5}:n\in\mathbb{N}\}$. What can we say about these two sets? Can you find an $M$ that satisfies each of them?</p>
|
1,943,840 | <p>I'm trying to find out for square matrices with $n \geq 2$ :
$$ \det(A-B) = \det(A)-\det(B).$$ </p>
<p>I know that $\det(AB) = \det(A)\det(B)$, but I'm unable to find proof on why a subtraction (or addition) is not equal. Thanks.</p>
| Eman Yalpsid | 94,959 | <p>Consider $A = 5I, B = 3I$, where $I$ is an $n\times n$ identity matrix. In this case $\det A = 5^n, \det B = 3^n$ and $\det (A-B) = \det(2I) = 2^n.$ So, for example if $n=2$, $16 = 25 - 9 = \det A - \det B \neq \det(A-B) = 4$.</p>
|
760,195 | <p>I've seen this proof in a text. I have an issue with it and wanted to check its validity. </p>
<p>Let $X\sim B(n,p)$, we seek the expectation. We let $q=1-p$
\begin{equation}
E(X)=\sum_{j=0}^{n} j {n\choose j} p^{j}q^{n-j}=p\partial_{p}\sum_{j=0}^{n} {n\choose j} p^{j}q^{n-j}=\underline{p\partial_{p} (p+q)^{n}} \q... | mookid | 131,738 | <p>You define a function
$F(p,q)$. You eventually want to compute
$$
F(p,1-p)
$$
and in order to get to that, you compute $F(p,q)$ for every $(p,q)$ and
<em>then</em> plug $q=1-p$.</p>
<p>As you use differentiation technique, plug first and then differentiate yields a different result.</p>
|
149,790 | <p>I know that if $x$ is a rational multiple of $\pi$, then $tan(x)$ is <a href="http://divisbyzero.com/2010/10/28/trigonometric-functions-and-rational-multiples-of-pi/">algebraic</a>.</p>
<p>Is there a fairly simple way to express $x$ as $\pi\ m/n$, if $tan(x)$ is given as a square root of a rational?</p>
| Carlo Beenakker | 11,260 | <p>An angle $x$ with $\tan^2 x$ rational has been called "geodetic" by Conway, Radin, and Sadun, <A HREF="http://arxiv.org/abs/math--ph/9812019">On Angles Whose Squared Trigonometric Functions are Rational</A>. Geodetic angles have a simple representation, see their Theorem 2, but they are not in general rational multi... |
149,790 | <p>I know that if $x$ is a rational multiple of $\pi$, then $tan(x)$ is <a href="http://divisbyzero.com/2010/10/28/trigonometric-functions-and-rational-multiples-of-pi/">algebraic</a>.</p>
<p>Is there a fairly simple way to express $x$ as $\pi\ m/n$, if $tan(x)$ is given as a square root of a rational?</p>
| joro | 12,481 | <p>You can numerically compute $\arctan(\tan(x))=\pi \frac{m}{n}$. Then divide by $\pi$ and numerically search for linear relation between $(K,1)$ to recover $m,n$. I am not sure there is always a solution.</p>
|
3,602,323 | <p>Let <span class="math-container">$ m $</span>, <span class="math-container">$ m+1 $</span>, <span class="math-container">$ m+2 $</span>, <span class="math-container">$ \dots $</span>, <span class="math-container">$ m+p-1 $</span> be an integers and let <span class="math-container">$ p $</span> be an odd prime. I wan... | Servaes | 30,382 | <p>Here's a nice alternative to the approach suggested in the comments by user rtybase:</p>
<p>If you are familiar with a bit of group theory, you may know that for every odd prime <span class="math-container">$p$</span>, the group <span class="math-container">$(\Bbb{Z}/p\Bbb{Z})^{\times}$</span> of units modulo <spa... |
348,324 | <p>In an interview the interviewer asked me the following but I failed to give the answer. </p>
<p>$\{0,1\}^\mathbb{N}$ with product topology is homeomorphic to which subset of $\mathbb{R}$?</p>
<p>Can anyone give me the answer and explain me please? thanks for your kind help.</p>
| Asaf Karagila | 622 | <p>I am going to assume that the topology on $\{0,1\}$ is the discrete topology. </p>
<p>The space $\{0,1\}^\Bbb N$ is called the Cantor space. It is a zero-dimensional compact metric space which has no isolated points. From these properties also follow that it is totally disconnected as well.</p>
<p>But there is a n... |
1,412,869 | <p>Does it make sense to talk about Christoffel symbols in flat space time? Do they have non-zero values? I understand that the Christoffel symbols appear as an indication of curvature in space. So, are they non-existent in flat space-time?</p>
| frakbak | 91,962 | <p>Yes, it makes sense to talk about Christoffel symbols in flat spacetime. Every coordinate system has associated Christoffel symbols. On Minkowski spacetime in the standard coordinates, the Christoffel symbols are all zero. But in different coordinates (e.g., spherical coordinates), they will not be zero. The Christo... |
2,225 | <p>If $f: \mathbb{R} \to \mathbb{R}$ is a continuous function and satisfies $f(x)=f(2x+1)$, then its not to hard to show that $f$ is a constant.</p>
<p>My question is suppose $f$ is continuous and it satisfies $f(x)=f(2x+1)$, then can the domain of $f$ be restricted so that $f$ doesn't remain a constant. If yes, then ... | JDH | 413 | <p>Let $f$ have value $1$ on $[0,\infty)$ and value $0$ on $(-\infty,-1]$. This function is not constant (although it is locally constant), and satisfies $f(x)=f(2x+1)$ whenever $x$ is in its domain. </p>
|
4,283,707 | <p>When solving this problem I arrive to a cubic equation not very friendly, is there any algebraic shortcut?</p>
<p><span class="math-container">$$W=\frac{3+\left [ \sqrt[3]{4+\sqrt[3]{4+...}} \right ]^{2}}{1+\left [ \sqrt[3]{4+\sqrt[3]{4+...}} \right ]^{-1}}$$</span></p>
<p>to do this
<span class="math-container">$$P... | Rene Schipperus | 149,912 | <p><span class="math-container">$$W=\frac{3+P^2}{1+\frac{1}{P}}$$</span></p>
<p>multiply top and bottom by <span class="math-container">$P$</span>,</p>
<p><span class="math-container">$$W=\frac{3P+P^3}{1+P}=\frac{4P+4}{1+P}=4$$</span></p>
|
104,818 | <p>Can anyone help me with this? I want to know how to solve it.</p>
<blockquote>
<p>Let $f:\mathbb R \longrightarrow \mathbb R$ be a continuous function with period $P$. Also suppose that $$\frac{1}{P}\int_0^Pf(x)dx=N.$$ Show that $$\lim_{x\to 0^+}\frac 1x\int_0^x f\left(\frac{1}{t}\right)dt=N.$$</p>
</blockquote>... | robjohn | 13,854 | <p>Note that if $f\in L^1([0,P])$, then
$$
F(x)=\int_0^x(f(t)-N)\;\mathrm{d}t\tag{1}
$$
is a continuous function of period P.</p>
<p>Then, with $M=\max\limits_{[0,P]}|F|$, we have</p>
<p>$$
\begin{align}
\lim_{x\to0^+}\frac 1x\int_0^x f\left(\frac{1}{t}\right)dt
&=\lim_{x\to0^+}\frac1x\int_\frac1x^\infty\fra... |
623,819 | <p>I do not understand a remark in Adams' Calculus (page 628 <span class="math-container">$7^{th}$</span> edition). This remark is about the derivative of a determinant whose entries are functions as quoted below.</p>
<blockquote>
<p>Since every term in the expansion of a determinant of any order is a product involving... | John Gowers | 26,267 | <p>The author is probably referring to the fact that the determinant is given by: </p>
<p>$$
\sum_{i,j,k=1}^n\varepsilon_{ijk}a_{1i}a_{2j}a_{3k}
$$</p>
<p>where $\varepsilon_{ijk}$ is $1$ if $(ijk)$ is an even permutation of $(123)$, $-1$ if $(ijk)$ is an odd permutation of $(123)$ and $0$ if two or more of $i,j,k$ a... |
2,916,099 | <p>Find a Mobius transformation $T$ from the unit disk to the right half plane with condition $T(0)=3$.</p>
<p>First, the transformation from the unit circle to the upper half plane is $T_1(z)=(1-i)\frac{z-i}{z-1}$.</p>
<p>So from the unit circle to the right half plane, $T_2(z)=-i(1-i)\frac{z-i}{z-1}$</p>
<p>How ca... | zhw. | 228,045 | <p>Let <span class="math-container">$\mathbb D$</span> be the open unit disc, <span class="math-container">$U$</span> the open right half plane. Suppose <span class="math-container">$f:\mathbb D\to U$</span> is bilholomorphic, with <span class="math-container">$f(0)= 1-i$</span> (just as your map <span class="math-cont... |
2,916,099 | <p>Find a Mobius transformation $T$ from the unit disk to the right half plane with condition $T(0)=3$.</p>
<p>First, the transformation from the unit circle to the upper half plane is $T_1(z)=(1-i)\frac{z-i}{z-1}$.</p>
<p>So from the unit circle to the right half plane, $T_2(z)=-i(1-i)\frac{z-i}{z-1}$</p>
<p>How ca... | Maxim | 491,644 | <p>You took only one possible transformation that maps the unit disk to the right half-plane. Generally, since the conjugation wrt the unit circle becomes the conjugation wrt the imaginary axis, <span class="math-container">$T(0) = 3$</span> implies <span class="math-container">$T(\infty) = -3$</span>. Therefore, <span... |
2,624,669 | <blockquote>
<p>Find global maxima and global minima of
$$f(x)=3(x-2)^{\frac{2}{3}}-(x-2)$$
over the interval $[0,20]$.</p>
</blockquote>
<p><strong>My input:</strong> Derivative vanishes at $x=10$ and left neighborhood gives positive derivative and right neighborhood gives negative derivative . Therefore $x=1... | Robert Z | 299,698 | <p>You are wrong, here $x=10$ is not a global maximum point.</p>
<p>Notice that the function $f$ is continuous in $I=[0,20]$, but it is not differentiable at $2$. So, by <a href="https://en.wikipedia.org/wiki/Fermat%27s_theorem_(stationary_points)" rel="nofollow noreferrer">Fermat's theorem</a>, if $x_0$ is a global... |
3,270,856 | <p>So i have to calculate this triple integral:</p>
<p><span class="math-container">$$\iiint_GzdV$$</span>
Where G is defined as: <span class="math-container">$$x^2+y^2-z^2 \geq 6R^2, x^2+y^2+z^2\leq12R^2, z\geq0$$</span></p>
<p>So with drawing it it gives this:</p>
<p><a href="https://i.stack.imgur.com/60hok.jpg" r... | TeM | 247,735 | <p>Wanting to calculate the following <em>triple integral</em>:
<span class="math-container">$$I := \iiint\limits_{\Omega} z\,\text{d}x\,\text{d}y\,\text{d}z$$</span>
with:
<span class="math-container">$$\Omega := \left\{ (x,\,y,\,z) \in \mathbb{R}^3 : x^2 + y^2 \ge 6R^2 + z^2, \; x^2 +y^2 + z^2 \le 12R^2, \; z \ge 0 \... |
1,505,076 | <p>This might sound a stupid question but it is indeed a real one.</p>
<p>I'm trying to figure a Confidence interval for the average age of my population.</p>
<p>Given i have a population of 100 individual, and i sample 3 of them. From CLT, i can say that $Var[\bar{x}_3] = \frac{s^2}{3} $. Alright. </p>
<p>I want to... | Clement C. | 75,808 | <p>You are sampling in an i.i.d. fashion, i.e. with replacement. A sample of size 100 does not necessarily include all 100 individuals; most likely, it will not (you have both people sampled several times and people that are not sampled). So you still have uncertainty, and variance in your estimator.</p>
<p>In short: ... |
2,358,838 | <p>I can see the answer to this in my textbook; however, I am not quite sure how to solve this for myself . . . the book has the following:</p>
<blockquote>
<p>To take advantage of the inductive hypothesis, we use these steps:</p>
<p>$ 7^{(k+1)+2} + 8^{2(k+1)+1} = 7^{k+3} + 8^{2k+3} $</p>
<p>$$
= 7\cdot7^... | lhf | 589 | <p>I don't think there is an algorithm for that, but I'd start by simplifying
$$
7^{n+2} + 8^{2n+1} = 49\cdot 7^n + 8 \cdot 64^n
$$
Now note that $49+8=57$ and $64-7=57$, which at least suggests where $57$ comes from.</p>
<p>This observation is probably useful if you write $64^n=(57+7)^n$ and use the binomial theorem.... |
2,358,838 | <p>I can see the answer to this in my textbook; however, I am not quite sure how to solve this for myself . . . the book has the following:</p>
<blockquote>
<p>To take advantage of the inductive hypothesis, we use these steps:</p>
<p>$ 7^{(k+1)+2} + 8^{2(k+1)+1} = 7^{k+3} + 8^{2k+3} $</p>
<p>$$
= 7\cdot7^... | Jack D'Aurizio | 44,121 | <p>By the Chinese remainder theorem something is $\equiv 0\pmod{57}$ iff it is $\equiv 0\pmod{3}$ and $\equiv 0\pmod{19}$. On the other hand
$$ a_n = 7^{n+2}+8^{2n+1} = 49\cdot 7^n + 8\cdot 64^n $$
fulfills
$$ a_n \equiv 49\cdot 1^n+8\cdot 1^n \equiv 57 \equiv 0\pmod{3} $$
$$ a_n \equiv 49\cdot 7^n+8\cdot 7^n \equiv (... |
2,358,838 | <p>I can see the answer to this in my textbook; however, I am not quite sure how to solve this for myself . . . the book has the following:</p>
<blockquote>
<p>To take advantage of the inductive hypothesis, we use these steps:</p>
<p>$ 7^{(k+1)+2} + 8^{2(k+1)+1} = 7^{k+3} + 8^{2k+3} $</p>
<p>$$
= 7\cdot7^... | Bill Dubuque | 242 | <p><strong>Hint</strong> $ $ To induct simply <em>multiply</em> the first two congruences below (by the <a href="https://math.stackexchange.com/a/879262/242">Congruence Product Rule)</a></p>
<p>$$\begin{align} \bmod 57\!:\qquad\ \ 7&\equiv 8^{\large 2}\\[.3em]
{-}7^{\large n+2}&\equiv 8^{\large 2n+1}\ \ \ {\... |
3,452,707 | <p>It is well known that <span class="math-container">$\sum_{k=0}^n{n\choose k} =2^n$</span>.</p>
<p><strong>My question:</strong> If <span class="math-container">$z$</span> is the limit point of an infinite sequence of real numbers <span class="math-container">$\{ a_n \}$</span>, then does <span class="math-container"... | Community | -1 | <p>Below is not entirely rigorous.</p>
<p>Since <span class="math-container">$\{a_n\}$</span> is convergent, we can pick <span class="math-container">$N$</span> such that <span class="math-container">$a_m$</span> is close to <span class="math-container">$z$</span> for any <span class="math-container">$m\ge N$</span>.<... |
2,529,262 | <p>I have five real numbers $a,b,c,d,e$ and their arithmetic mean is $2$. I also know that the arithmetic mean of $a^2, b^2,c^2,d^2$, and $e^2$ is $4$. Is there a way by which I can prove that the range of $e$ (or any ONE of the numbers) is $[0,16/5]$. I ran across this problem in a book and am stuck on it. Any help w... | Arthur | 15,500 | <p>If we have
$$
\frac{a+b+c+d+e}{5} = 2 = \sqrt{\frac{a^2 + b^2 + c^2 + d^2 + e^2}{5}}
$$
then all the five numbers are necessarily equal to $2$, as dictated by the <a href="https://brilliant.org/wiki/power-mean-qagh/" rel="nofollow noreferrer">AM-QM inequality</a>.</p>
<p>PS. Technically, the power mean inequality i... |
3,087,933 | <p>I read in the book <em>A First Course in Probability</em> by Sheldon Ross the following statement:</p>
<blockquote>
<p><strong>Technical Remark.</strong> We have supposed that <span class="math-container">$P(E)$</span> is defined for all the events <span class="math-container">$E$</span> of the sample space. Actu... | Erik Parkinson | 630,372 | <p>This gets into a branch of math called measure theory. The idea is, given a set of real numbers, can we give certain measure to it? Probably the most common measure (as far as I know), is Lebesgue measure. It extends the idea of lengths of intervals. For example, any interval <span class="math-container">$[a,b]$</sp... |
65,270 | <p>On <a href="https://crypto.stanford.edu/pbc/notes/elliptic/divisor.html" rel="nofollow noreferrer">this page</a>, the author states:</p>
<blockquote>
<p>It turns out this definition can be extended to points of order 2, and also the point O (when we homogenize the functions and work over the projective plane). Moreo... | Ted | 15,012 | <p>Homogenization refers to the process of going from the "affine" to the "projective" plane, by adding "points at infinity". It's easier to work over the projective plane because there aren't any "missing points"; for example, in the projective plane, <em>every</em> 2 distinct lines intersect at one point (instead of... |
2,420,727 | <p>I'm trying to evaluate </p>
<blockquote>
<p>$$\lim _{ x\to -\infty } \frac { 2x-3 }{ \sqrt { x^{ 2 }+7x-2 } } $$</p>
</blockquote>
<p>by rationalizing the denominator, but I am not getting anywhere. Can someone please help me with this?</p>
<p>Thanks</p>
| hamam_Abdallah | 369,188 | <p><strong>hint</strong></p>
<p>near $-\infty $,</p>
<p>$$\sqrt {x^2+7x-2}=\sqrt {x^2 (1+\frac {7}{x}-\frac {2}{x^2})} $$</p>
<p>$$=\color {red}{-}x\sqrt {1+\frac {7}{x}-\frac {2}{x^2}} $$</p>
|
2,807,611 | <p>I know the answer is $n=6$, but can't figure out how to solve.
I tried dividing by $n!$, but didn't work because there isn't one in RHS to simplify... also tried using Gamma function properties, but didn't work either... </p>
<p>Any help would be appreciated.</p>
<p>Thanks.</p>
| Lord KK | 446,929 | <p>Write $330 = 2\times3\times5\times11 $ & write $(n+2)!-n! = n!\times{(n^2+3n+1)}$ </p>
<p>Clearly the RHS doesn't have 7 as its factor, so you must have $n\le6$</p>
<p>The other thing which we can see is that after simplification, we will be getting a quadratic equation of form $n^2 +3n-m =0$ where $m =\frac... |
311,380 | <p>Prove that the relation $x \sim y$ iff $y$ is an element of the connected component of $x$ is an equivalence relation.</p>
<p>This question is confusing me, do I simply go about showing the relation is reflexive, symmetric, and transitive? I don't really see how to do this for this question. Any suggestions or hint... | Asaf Karagila | 622 | <p>Yes. Whenever you are asked to verify that a certain mathematical object (in this case a relation) has a specific property (in this case, an equivalence relation), you really just need to verify that the definition of this property hold for this object.</p>
<p>In this case, you simply need to show that the relation... |
5,231 | <p>I have coordinates for 4 vertices/points that define a plane and the normal/perpendicular.
The plane has an arbitrary rotation applied to it.</p>
<p>How can I 'un-rotate'/translate the points so that the plane has rotation 0 on x,y,z ?</p>
<p>I've tried to get the plane rotation from the plane's normal:</p>
<pre>... | George Profenza | 304 | <p>At the moment, I went with a somewhat simple solution that allows me to draw a plane/face with 4 vertices with arbitrary rotations, in 2D:</p>
<p>Here's how it works:</p>
<pre><code>PVector[] unRotateVerts(PVector[] verts,PVector n){
//get the angle between the face4 normal and Y
angle = PVector.angleBetween(n... |
872,017 | <p>$$\int_0^1 xe^{\sqrt{x}} dx = ? $$</p>
<p>All I can think of is the integration by parts rule, where
$ u = x $ and $ dv= e^{\sqrt(x)} $ $ \Rightarrow du = 1$ and $ v= e^{\sqrt(x)} $ </p>
<p>The answer I get is $e^{\sqrt(x)}(x-1)$ , which is wrong.</p>
<p>Can anyone please explain in detail?</p>
| lemon | 89,931 | <p>Let $u=\sqrt{x}$ so $du=\frac{1}{2\sqrt{x}}dx=\frac{1}{2u}dx$ so the integral becomes</p>
<p>$$ \int_0^1 2u^3e^u du$$</p>
<p>then evaluate this by integrating by parts.</p>
|
1,397,160 | <p>How do I prove that in a finite group G, for each element in G there is natural power (say $k$) which depends on g,such that $g^k=e$ ?
I need to show the existence and the dependence on which $g$ I choose.</p>
<p>I tried write it that way, but I don't have any direction in the proof: </p>
<p>$$G\:=\:\left|n\right|... | wltrup | 232,040 | <p>The idea here is the following. Suppose you have a thin sheet of a metal, in the shape of a square of side $s$, and suppose also that the metal has a coefficient of <strong>linear</strong> expansion $\alpha$.</p>
<p>What that means is that if you increase the temperature of the metal from its current temperature $T... |
863,167 | <p>How can the signed area be 0? For example if you have 3 on positive x side and 3 on the negative x side then you get the signed area of 0? How can area be 0?</p>
| Idris Addou | 192,045 | <p>I think the deep question by the OP is the following. </p>
<p>If one takes a set and compute its signed area and finds $0$, so ok, it is $0$ for this particular set. </p>
<p>But if one takes a set such that its signed area is $0$ so how it is? how can one imagine it? </p>
<p>It is this second part which he ask a... |
43,282 | <p>This question is somewhat related to Tilmans notorious problem in <a href="https://mathoverflow.net/questions/17532/does-linearization-of-categories-reflect-isomorphism">this post</a>. Let $(M,\cdot)$ be a monoid with unit $1$ and set
$$(M,\cdot)^{\times} := \lbrace x \in M \mid \exists y \in M : xy=yx=1 \rbrace.$$
... | Benjamin Steinberg | 15,934 | <p>If your monoid $M$ is finite the non-invertible elements form an ideal of the monoid and hence they span an ideal of kM. So any invertible element of the algebra must have an invertible element of the monoid in its support. So the answer to your second question is no.</p>
|
791,719 | <p>I have this inequation:
$$5-3|x-6|\leq 3x -7$$</p>
<p>i solved this this way: </p>
<p>i said, for $x\geq6$ is the modulus positive, so I made 2 cases in which the modulus gives + or - : </p>
<p>1) for $x\geq6$ (positive): </p>
<p>$5-3x+6\leq 3x -7\\
6x\geq30\\
x\geq5$</p>
<p>2) for $x<6$ (negative): </... | lab bhattacharjee | 33,337 | <p>As $\displaystyle\tan\left(\frac\pi2-z\right)=\cot z,\arctan u+\text{arccot}u=\frac\pi2$</p>
<p>and use <a href="https://math.stackexchange.com/questions/304399/are-arccotx-and-arctan1-x-the-same-function">Are $\mathrm{arccot}(x)$ and $\arctan(1/x)$ the same function?</a> or <a href="http://www.sosmath.com/CBB/vie... |
791,719 | <p>I have this inequation:
$$5-3|x-6|\leq 3x -7$$</p>
<p>i solved this this way: </p>
<p>i said, for $x\geq6$ is the modulus positive, so I made 2 cases in which the modulus gives + or - : </p>
<p>1) for $x\geq6$ (positive): </p>
<p>$5-3x+6\leq 3x -7\\
6x\geq30\\
x\geq5$</p>
<p>2) for $x<6$ (negative): </... | DonAntonio | 31,254 | <p>hints:</p>
<p>Proof that</p>
<p>$$\arctan x=\frac\pi2+\arctan\frac1x\;,\;\;0<x\neq\begin{cases}\frac\pi2\\{}\\\frac2\pi\end{cases}+n\pi\;,\;\;n\in\Bbb Z\;\;:\;$$</p>
<p>$$f(x):=\arctan x+\arctan\frac1x\implies f'(x)=\frac1{1+x^2}-\frac1{x^2}\frac1{1+\frac1{x^2}}=0$$</p>
<p>and thus $\;f(x)=k=$ a constant. Now... |
1,242,570 | <p>I want to use the standard definition $x_n \rightarrow x$ if for all $\epsilon>0$ there exists $N$ such that if $n>N$ then $|x_N-x|<\epsilon$. </p>
<p>So my claim is $x_n\rightarrow 0$ If I set $N=\epsilon^2,$ then the following expression $|\sqrt{n^2+1}-n-0|<\epsilon$ will hold true. I solved for $N$... | E.H.E | 187,799 | <p>multiply by $$\frac{\sqrt{n^2+1}+n}{\sqrt{n^2+1}+n}$$</p>
|
2,749,624 | <p>Prove: $k^3 - k( b c + c a + a b ) + 2 a b c = 0$ always has a negative root with all positive parameters $a, b, c$</p>
<p>I tried: Write $f(x)=x^3-x(ab+ac+bc)+2abc$ then $f(-\infty)=-\infty,f(0)>0$. Now use the Intermediate Value Theorem. I can' t continue. Help me! Thanks!</p>
| Lutz Lehmann | 115,115 | <p>Additionally to the already observed $f(0)=2abc>0$ one also has
$$
f(\sqrt[3]{abc})=-\sqrt[3]{abc}(ab+bc+ac)+3abc
$$
As per mean inequalities
$$
\frac{ab+bc+ac}3\ge\sqrt[3]{a^2b^3c^2}
$$
this function value is non-positive, which implies that there are 2 positive real roots or a double root at $\sqrt[3]{abc}$ in... |
3,971,025 | <p>I need to find the number of conjugated to the permutation (12)(34) in the symmetric group <span class="math-container">$S_6$</span> of rank 6</p>
<p>My answer is 6! = 720</p>
<p>Is this correct?</p>
<p>I concluded that (12)(34)=(12)(34)(5)(6) and the number of combinations for <span class="math-container">$S_6$</sp... | Alex Ravsky | 71,850 | <p>The statement essentially says that each natural number is a unique product of prime numbers.</p>
|
324,557 | <p>Map the common part of the disks $|z|<1$ and $|z-1|<1$ on the inside of the unit circle. Choose the mapping sot hat the two symmetries are preserved.</p>
<p>I don't really know how to approach this??</p>
<p>Any suggestions on how to start constructing such a linear transformation??</p>
<p>Thanks in advance!... | Zev Chonoles | 264 | <p><strong>Hint:</strong> Try making the condition of transitivity degenerate; what if $A$ only had two elements?</p>
|
383,194 | <p>What are the free objects in the category of $G$-sets for a group $G$? </p>
<p>After considerable deliberation (I'm not very bright), I'm pretty sure they are the $G$-sets $X$ on which $G$ acts freely, that is in such a way that only $e$ fixes any elements in $X$. I can prove it -- almost.</p>
<p>Suppose $X$ is a ... | Bartek | 23,371 | <p>I want to show that if a $G$-set $X$ is a free object in the category of $G$-sets, then its basis $B$ has to generate $X$. In other words, </p>
<p>$$\langle B\rangle=\bigcup_{b\in B}\mathrm{orb}(b)=X.$$</p>
<p>Let's say that an equivalence relation $\sim$ on $X$ is a congruence whenever for any $x,x'\in X$ and $g\... |
455,259 | <p>This person has been on all seven continents. But this same person has never been to Brazil.</p>
<p>Contrary/Consistent: I would say it's consistent because Brazil is not a continent.</p>
<p>am i right?</p>
| Eric Tressler | 26,785 | <p>Yes. You just have to reason that a person who has never been to Brazil may nevertheless have been to all seven continents.</p>
|
2,972,938 | <p>When is it possible to make a change of variables in the limit?</p>
<p>For example <span class="math-container">$\lim_{x \to \infty}(\ln x/x)$</span>, can I change <span class="math-container">$x=e^{y}$</span>?</p>
<p>Then <span class="math-container">$\lim_{x \to \infty}(\ln x/x)= \lim_{y \to \infty}(y/e^{y})$</s... | user1551 | 1,551 | <p>Here is another approach. It is not as nice as the other answer, but it is shorter. Perform the elementary row operations (in C programming language's notation):
<code>row 3 += row 2; col 2 -= col 3</code>. We see that <span class="math-container">$A$</span> is similar to
<span class="math-container">$$
B=\pmatrix{
... |
45,973 | <p>Let $B,C,D \geq 1$ be positive integers and $(b_n)_{n\geq 0}$ be a sequence with $b_0 = 1, b_n = B b_{n-1} + C B^n + D$ for $n \geq 1$.</p>
<p>Prove that </p>
<p>(a) $\sum_{n\geq 0}^\infty b_n t^n$ ist a rational function</p>
<p>(b) identify a formula for $b_n$</p>
<hr>
<p>Hi!</p>
<p>(a)</p>
<p>As I know I ne... | SteveH | 12,251 | <p>Use the recursion $b_n=B b_{n-1}+CB^n+D$ in $\sum_{n=1}^{\infty} b_n t^n$ and equate the two sides. You will have the sum showing up on both sides after a little manipulation. Hope this helps. </p>
|
1,510 | <p>It's hard to prove a number is transcendental (non-algebraic) yet there are some wonderful examples amongst them like π,e and Liouville's number. What's so special about them? </p>
<p>Are most numbers transcendental?</p>
| Peter Arndt | 733 | <p>So, yes, most numbers are transcendental. But you have to be more inventive to define one; you cannot just say take the solution of this or that polynomial ...</p>
|
1,832,757 | <p>Consider the following premises.</p>
<ol>
<li>If A = B then B = C.</li>
<li>B != C.</li>
<li>If C > D then D < E.</li>
<li>F != G and A = B.</li>
<li>A = B or C > D.</li>
</ol>
<p>Alternatives:</p>
<p>a) F != G</p>
<p>b) F != G and D < E</p>
<p>c) A = B</p>
<p>d) B = C or D < E</p>
<p>e) D < E</p>... | user21820 | 21,820 | <p>You are right; the question is wrong. From (4) we get A = B, and then by (1) we get B = C, contradicting (2).</p>
|
1,832,757 | <p>Consider the following premises.</p>
<ol>
<li>If A = B then B = C.</li>
<li>B != C.</li>
<li>If C > D then D < E.</li>
<li>F != G and A = B.</li>
<li>A = B or C > D.</li>
</ol>
<p>Alternatives:</p>
<p>a) F != G</p>
<p>b) F != G and D < E</p>
<p>c) A = B</p>
<p>d) B = C or D < E</p>
<p>e) D < E</p>... | QTHalfTau | 88,718 | <p>I suspect that there is a typo on your source's part, namely being that (4) should be a disjunction. </p>
<p>Making this correction, we have</p>
<p>(6) $A\neq B$. Take the contrapositive of (1) and applying modus ponens with (2).</p>
<p>(7) $C>D$. Use disjunctive syllogism on (5) with (6).</p>
<p>(8) $D<E$... |
4,652,260 | <p>I am self-studying Functional Analysis from Kreyszig's <em>Introductory Functional Analysis with Applications</em>. In section-1.3, he proves that the sequence space <span class="math-container">$\ell^\infty$</span> with the metric <span class="math-container">$d_\infty(x,y)=\sup_i\{|x_i-y_i|\}$</span> is not separa... | Martin Argerami | 22,857 | <p>What you are doing is what I would call the standard proof that <span class="math-container">$\ell^\infty$</span> is not separable. There's no real need to use any theorems or anything. It is trivial to check that <span class="math-container">$\|y_1-y_2\|_\infty=1$</span> for all <span class="math-container">$y_1,y_... |
2,180,091 | <p>I'm working through some exercises in Sedgewick's Analysis of Algorithms, but I'm stuck on 7.45:</p>
<blockquote>
<p>Find the CGF for the total number of inversions in all involutions of length <span class="math-container">$N$</span>. Use this to find the average number of inversions in an involution.</p>
</block... | arctic tern | 296,782 | <p>The proof is valid if you're allowed to use the fact that $\phi$ is multiplicative. Personally, the Chinese Remainder Theorem is what I consider to be <em>the reason</em> that $\phi$ is multiplicative, so this would be circular for me.</p>
<p>Yes, we do have $(\prod_i R_i)^\times=\prod_i R_i^\times$. It is actually... |
2,180,091 | <p>I'm working through some exercises in Sedgewick's Analysis of Algorithms, but I'm stuck on 7.45:</p>
<blockquote>
<p>Find the CGF for the total number of inversions in all involutions of length <span class="math-container">$N$</span>. Use this to find the average number of inversions in an involution.</p>
</block... | Adren | 405,819 | <p><strong>Remark</strong></p>
<p>I think that the multiplicativeness of $\phi$ should rather be considered as a <em>consequence</em> of the fact that the groups of units of $\mathbb{Z}/n\mathbb{Z}$ and $\displaystyle{\prod_{i=1}^j\left(\mathbb{Z}/p_i^{k_i}\mathbb{Z}\right)}$ are isomorphic and hence equipotent.</p>
... |
4,214,329 | <p>For convenience, let <span class="math-container">$(f(x), g(x))$</span> be a solution to the problem. Now,
<span class="math-container">\begin{align*}
f(x) + g(x) &= f(x)g(x) \\
f(x)g(x) - f(x) - g(x) &= 0 \\
f(x)g(x) - f(x) - g(x) + 1 &= 1 \\
(f(x) - 1)(g(x) - 1) &= 1
\end{align*}</s... | José Carlos Santos | 446,262 | <p>No, because then<span class="math-container">\begin{align}\deg\bigl(f(x)g(x)\bigr)&=\deg f(x)+\deg g(x)\\&>\max\bigl\{\deg f(x),\deg g(x)\bigr\}\\&\geqslant\deg\bigl(f(x)+g(x)\bigr).\end{align}</span></p>
|
1,043,734 | <p>I found the following on Wikipedia, <a href="http://en.wikipedia.org/w/index.php?title=Elementary_algebra&oldid=633621020#Properties_of_inequality" rel="nofollow">on the page for Inequalities</a>:</p>
<blockquote>
<p>If $a<b$ and $c<d$ then $a+c < b+d$.</p>
</blockquote>
<p>It references <a href="ht... | Simon S | 21,495 | <p>I don't know about the name. </p>
<p>But proof: yes. If $p$ and $q$ are positive numbers, then $p + q$ is a positive number; this follows from a commonly used postulate of the real number system. We can take $a < b$ to mean $b - a$ is positive.</p>
<p>Now let $p = b - a$ and $q = d - c$. Both are positive hence... |
1,904,354 | <p>Yeah the title says everything I will explain this quick if someone is so smart and nice than he has my ammiration! Here you are :: if we take an irrational number like π or e or whatever and we write this π+π-π… (ecc) at infinity of this series what could possibly come out?? I hope somebody can explain this thanks ... | Med | 261,160 | <p>$I_n+5I_{n-1}=\int_{0}^{1}\frac{x^n}{x+5}dx+5\int_{0}^{1}\frac{x^{n-1}}{x+5}dx=\int_{0}^{1}(\frac{x^n}{x+5}+5\frac{x^{n-1}}{x+5})dx$</p>
<p>$I_n+5I_{n-1}=\int_{0}^{1}x^{n-1}dx=\frac{x^n}{n}|_{0}^{1}=\frac{1}{n}$</p>
|
587,275 | <p>I was trying to understand why $e^{x}$ is special by finding the derivatives of other exponential functions and comparing the results. So I tried ${\rm f}\left(x\right) = 2^{x}$, but now I'm stuck.</p>
<p>Here's my final step:
<strong>$\displaystyle{{\rm f}'\left(x\right)
= \lim_{h \to 0}{2^{x}\left(2^{h} - 1\right... | littleO | 40,119 | <p>You discovered that
\begin{equation*}
\frac{d}{dx} 2^x = c 2^x
\end{equation*}
where $c = \lim_{h \to 0} (2^h - 1)/h$.</p>
<p>But note that $c \neq 1$, which is kind of annoying.</p>
<p>If you had used $e$ instead of $2$, you would have had $c = \lim_{h \to 0} (e^h - 1)/h$, which actually is equal to $1$. In fact... |
15,235 | <p>More precisely, is there a map of schemes $X$ --> $Y$ such that $f$ gives a homeomorphism between $X$ and a closed subset of $Y$, but the corresponding map on sheaves is not surjective?</p>
| Rebecca Bellovin | 88 | <p>Here's an example: Consider the morphism $f:\text{Spec} k[\varepsilon]/(\varepsilon^2)\rightarrow \text{Spec} k$ corresponding to the inclusion $k\hookrightarrow k[\varepsilon]/(\varepsilon^2)$. More generally, you can get lots of examples from non-reducedness.</p>
|
2,050,698 | <p>So I'm studying a few special families of square matrices, the diagonal matrices, upper triangular matrices, lower triangular matrices and symmetric matrices and I just had a few questions. </p>
<p>I know...</p>
<p>a diagonal matrix is if every nondiagonal entry is zero, $a_{ij}$=0 whenever $i$ doesn't equal $j
$.... | Fimpellizzeri | 173,410 | <p>What have you tried?</p>
<p>Notice that if $M$ is the targt mean, then for each $x\in \mathbb{R}$ a pair of data points $(M+x, M-x)$ have mean $M$ and contribute $2x^2$ to the variance. Can you think of how to use this observation to craft however many datasets with mean $M$ and distinct standard deviations you wis... |
1,812,914 | <p>What is the correct approach to solving a log equation with more than one non log value? Please demonstrate using the following equation:
$$\log(2x-1)=-x+3$$</p>
| 2'5 9'2 | 11,123 | <p>For an equation that is similar to this one, you will typically need <a href="https://en.wikipedia.org/wiki/Lambert_W_function" rel="nofollow">the Lambert $W$ function</a>. I assume that your $\log$ is $\ln$.</p>
<p>$$\begin{align}
\ln(2x-1)&=-x+3\\
2x-1&=e^{-x+3}\\
(2x-1)e^{x-3}&=1\\
\left(x-\frac12\ri... |
147,363 | <blockquote>
<p>If $\alpha$ is an algebraic element of $\mathbb{C}$, then there is a unique non-zero polynomial $f \in \mathbb{Q}[x]$ with leading coefficient $1$ such that $f(\alpha) = 0$, and $f$ is irreducible. </p>
</blockquote>
<p>The first part of this proof would be proving that $f$ is not a unit, but what do... | DonAntonio | 31,254 | <p>A unit in the ring of polynomials over some structure (say, a field) is the same as in any other ring: an element that has a multiplicative inverse within that ring. </p>
<p>If you want to focus on polynomials over fields then the units in that ring are precisely the constant polynomials $\,f(x)=k\,,\,0\neq k\,$ a ... |
3,129,248 | <p>I am solving ordinary differential equation in <span class="math-container">$S'$</span> (dual to Schwartz space) given as:</p>
<p><span class="math-container">$y' + ay = \delta$</span>, where <span class="math-container">$\delta$</span> is a Dirac delta function.</p>
<p>The general solution of homogenous equation ... | md2perpe | 168,433 | <p>Here is a solution fully inside the theory of distributions:</p>
<p>First we multiply on both sides with the smooth and everywhere nonzero integrating factor <span class="math-container">$e^{ax}$</span>:
<span class="math-container">$$y' e^{ax} + y \, a e^{ax} = e^{ax} \delta.$$</span></p>
<p>Now, the left hand si... |
3,079,023 | <p>A vector space with norm <span class="math-container">$\parallel\cdot\parallel$</span> Satisfy for two vectors the following</p>
<p><span class="math-container">$\parallel x+y\parallel=\parallel x\parallel +\parallel y\parallel$</span></p>
<p>i need to proof the fallowing statement</p>
<p><span class="math-contai... | Kavi Rama Murthy | 142,385 | <p>I have changed <span class="math-container">$\alpha, \beta$</span> to <span class="math-container">$a,b$</span>. Assume without loss of generality that <span class="math-container">$b\leq a$</span>. Then <span class="math-container">$\left\Vert ax+by\right\Vert =\left\Vert
a(x+y)+(b-a)y\right\Vert \geq a\left\Vert x... |
2,526,695 | <p>I've got following sequence formula:
$ a_{n}=2a_{n-1}-a_{n-2}+2^{n}+4$</p>
<p>where $ a_{0}=a_{1}=0$</p>
<p>I know what to do when I deal with sequence in form like this:</p>
<p>$ a_{n}=2a_{n-1}-a_{n-2}$
- when there's no other terms but previous terms of the sequence.
Can You tell me how to deal with this typ... | Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\... |
1,127,551 | <p>I have an equation, where I need to find <em>n</em>, that I need help solving.</p>
<p>I already cheated a little bit by using a CAS (<em>Maple</em>) to solve the equation, so i know what the result should be, but I need to know how to get the to result without using a CAS.</p>
<p>The equation:</p>
<p>$$\frac{2400... | KittyL | 206,286 | <p>So the CAS tells you how to transform it into a quadratic equation. Now to solve $40n^2 -200n-12000=0$:</p>
<p>Divide by $40$: $n^2 -5n-300=0$</p>
<p>Factorize: $(n-20)(n+15)=0$ (or use quadratic formula $\frac{5\pm \sqrt{(-5)^2-4(-300)}}{2}$)</p>
<p>So $n=20$ or $-15$.</p>
|
2,967,615 | <p>Given for example 2 functions,<span class="math-container">$\ n^{100} $</span> and<span class="math-container">$\ 2^n$</span>. I know that<span class="math-container">$\ 2^n$</span> grows faster and that therefore there is some<span class="math-container">$\ n$</span> where it will eventually overtake <span class="m... | Henry Lee | 541,220 | <p><span class="math-container">$$n^{100}=2^n$$</span>
<span class="math-container">$$100\ln(n)=n\ln(2)$$</span>
<span class="math-container">$$\frac{\ln(n)}{n}=\frac{\ln(2)}{100}$$</span>
let <span class="math-container">$n=e^{-x}$</span>
<span class="math-container">$$xe^{x}=\frac{-\ln(2)}{100}$$</span>
then you can ... |
2,967,615 | <p>Given for example 2 functions,<span class="math-container">$\ n^{100} $</span> and<span class="math-container">$\ 2^n$</span>. I know that<span class="math-container">$\ 2^n$</span> grows faster and that therefore there is some<span class="math-container">$\ n$</span> where it will eventually overtake <span class="m... | Paramanand Singh | 72,031 | <p>If you know that <span class="math-container">$2^n$</span> grows faster than <span class="math-container">$n^{100}$</span> then you understand that it is a special case of the more general result that exponential functions grow faster than polynomials.</p>
<p>Moreover knowing this fact may mean one has heard or rea... |
2,681,621 | <p>I'm trying to calculate the following limit:</p>
<p>$$\lim_{x\to\pi} \dfrac{1}{x-\pi}\left(\sqrt{\dfrac{4\cos²x}{2+\cos x}}-2\right)$$</p>
<p>I thought of calculating this:</p>
<p>$$\lim_{t\to0} \dfrac{1}{t}\left(\sqrt{\dfrac{4\cos²(t+\pi)}{2+\cos(t+\pi)}}-2\right)$$</p>
<p>Which is the same as:</p>
<p>$$\lim_{... | Amarildo | 307,377 | <p>$$
\begin{aligned}
\lim _{x\to \pi }\:\frac{1}{x-\pi }\left(\sqrt{\frac{4\cos^²x}{2+\cos x}}-2\right) &= \lim _{y\to 0}\:\frac{1}{\left(y+\pi \right)-\pi }\left(\sqrt{\frac{4\cos^²(y+\pi)}{2+\cos(y+\pi)}}-2\right)
\\ &= \lim _{y\to 0} \frac{2\cos y-2\sqrt{-\cos y+2}}{y\sqrt{-\cos y+2}}
\\ &= \lim _{y\to ... |
4,502,916 | <p>I know solving inequalities using table method ,i.e , finding each values of variable that makes it zero , and writing the root a line and change the sign when you have odd root. It is classical high school method. However , i encounter with different question style such that it contains exponential function. Howev... | W.Döblin | 1,082,497 | <p>it's equivalent to <span class="math-container">$x(x-2) \leq 0,$</span> therefore the set of solutions is <span class="math-container">$[0,2].$</span></p>
|
4,502,916 | <p>I know solving inequalities using table method ,i.e , finding each values of variable that makes it zero , and writing the root a line and change the sign when you have odd root. It is classical high school method. However , i encounter with different question style such that it contains exponential function. Howev... | Cathedral | 933,575 | <p>Simply note that <span class="math-container">$2^{x-2}>0 \space\forall \space x \in \mathbb R$</span>. The inequality now reduces to <span class="math-container">$$\frac{3x(x^2-4)}{8x+16}\le0$$</span>
which is something you mentioned as being comfortable with.</p>
|
544,008 | <p>We know since $\mathbb{Q}$ is countable that there exist a bijection $f : \mathbb{Z} \to \mathbb{Q} $. If we view $\mathbb{Q}$ and $\mathbb{Z}$ are topological subspaces of $\mathbb{R}$, are theo homeomorphic??</p>
| Seirios | 36,434 | <p><strong>Solution 1:</strong> Is $\mathbb{Q}$ a discrete topological space?</p>
<p><strong>Solution 2:</strong> When does a sequence in $\mathbb{Z}$ converge?</p>
<p><strong>Solution 3:</strong> Is $\mathbb{Q}$ locally compact?</p>
|
2,885,918 | <p>We consider the following random variable $X$: We have a uniform distribution of the numbers of the unit interval $[0,1]$. After a number $x$ from $[0,1]$ is chosen, numbers from $[0,1]$ are chosen until a number $y$ with $x\leq y$ pops up.</p>
<p>The random variable $X$ counts the number of trials to obtain $y$. H... | drhab | 75,923 | <p>Let $U$ denote the number chosen uniformly from $[0,1]$.</p>
<p>Then $P(X=n\mid U=u)=u^{n-1}(1-u)$ and $f_U(u)=1_{[0,1]}$ so that:$$P(X=n)=\int_0^1P(X=n\mid U=u)du=\left[\frac{u^{n}}{n}-\frac{u^{n+1}}{n+1}\right]_{0}^{1}=\frac{1}{n}-\frac{1}{n+1}$$</p>
<p>Here $X$ counts the number of trials needed to get a number... |
2,607,090 | <p>I have a function for which I know:</p>
<p>$f(2) = 2x -3y \\
f(3) = 5x - 6y \\
f(4) = 9x - 10 y \\
f(5) = 14x - 15y$</p>
<p>Assuming that $f$ is a polynomial, how do I find the general expression for $f$? After many minutes of fiddling I eventually found that this general expression works:</p>
<p>$f(N) = \frac{N(... | Community | -1 | <p>First of all, a <strong>disclaimer</strong>: I am not answering how to extend a number sequence here. The answer in that case would not be unique, because despite anything that we know about the values of a function at given points - the function may have <em>any</em> values at any other point. However, we are restr... |
2,607,090 | <p>I have a function for which I know:</p>
<p>$f(2) = 2x -3y \\
f(3) = 5x - 6y \\
f(4) = 9x - 10 y \\
f(5) = 14x - 15y$</p>
<p>Assuming that $f$ is a polynomial, how do I find the general expression for $f$? After many minutes of fiddling I eventually found that this general expression works:</p>
<p>$f(N) = \frac{N(... | zwim | 399,263 | <p>We notice $\begin{cases}
f(3)-f(2)=3x-3y=3(x-y)\\
f(4)-f(3)=4x-4y=4(x-y)\\
f(5)-f(4)=5x-5y=5(x-y)\\
\end{cases}$</p>
<p>So let assume the pattern is verified for all integers we get $f(n)-f(n-1)=n(x-y)$</p>
<p>This is a telescopic sum $f(n)-f(0)=(x-y)\sum\limits_{k=1}^{n} k\iff f(n)=f(0)+\dfrac{n(n+1)}2(x-y)$</p>
... |
635,195 | <p>I'm trying to calculate the following limit: </p>
<p>$$\mathop {\lim }\limits_{x \to {0^ + }} {\left( {\frac{{\sin x}}{x}} \right)^{\frac{1}{x}}}$$</p>
<p>What I did is writing it as: </p>
<p>$${e^{\frac{1}{x}\ln \left( {\frac{{\sin x}}{x}} \right)}}$$</p>
<p>Therefore, we need to calculate: </p>
<p>$$\matho... | Fabio Lucchini | 54,738 | <p>Note that as $x\to 0^+$ you have
$$\frac{\ln\frac{\sin x}x}x\sim \frac{\frac{\sin x}x-1}x\sim \frac{\sin x-x}{x^2}\sim \frac{-x^3}{6x^2}\to 0$$.</p>
|
1,683,238 | <p>Let $[x]$ denote the fractional part of x. I'm quite lost about how to solve this problem. I suspect the solution is elementary, but all I can determine is that $x\notin\Bbb{Q}$.</p>
| Calum Gilhooley | 213,690 | <p>Let any $\alpha$ be given in the interval $(0, 1),$ or indeed in $[0,1).$</p>
<p>For any $x$ in $[0, 1),$ and any positive integer $n,$ we say that $x$ <strong>fails for</strong> $n$ if
$$
[nx] \leqslant \alpha^n.
$$</p>
<p>For any $x$ in $[0, 1),$ we say simply that $x$ <strong>fails</strong> if $x$ fails for som... |
2,016,995 | <p>Is is possible to find a metric $d$ on $\mathbb{R}$ so that $(\mathbb{R}, d)$ would have a <strong>finite</strong> number of open sets?</p>
| Taku | 390,458 | <p>I cannot write English very well ,but I do my best from Japan.</p>
<p>The metric $d$ is defined by
$d(x,y)=\left\{ \begin{matrix} 1 & (x \not= y ) \\
0 & (x=y)\end{matrix} \right. $ ,we call it discrete metric.
The metric $d$ yields discrete topology.
So, there will be open finite sets.</p>
<p>Let me know... |
1,505,336 | <p>Please can someone explain if this identity is correct:</p>
<p>|a| = $\sqrt{a^2} \ $ <p>
I thought it should be:<p> |a| = $(\sqrt{a})^2\ $</p>
<p>being that the former would produce an answer that is either positive or negative.</p>
<p>Thank you for your help.</p>
<p>PS: The full question was comparing say:</p>
... | Emilio Novati | 187,568 | <p>Since $\sqrt{x}$ is defined in $\mathbb{R}$ only for $x\ge 0$ and it's always positive:</p>
<p>the first is correct and the absolute value is necessary , e.g. $\sqrt{(-2)^2}=|-2|=2$</p>
<p>the second is redundant since the square root exists only if $a>0$</p>
<p>An answer to the PS. require a discussion of the... |
1,221,056 | <p>Have assigment and will use it as example, found solution computationaly, want to understand idea.</p>
<p>It is about <em>SubBytes</em> procedure in AES, particulary about finding inverse of polynomial.</p>
<p>Suppose we have element $A=x^5+1$ in finite field $F=\mathbb{Z}_2[x]/x^8+x^4+x^3+x+1$ and it is required ... | Jordan Glen | 225,803 | <p>What you want to prove: $$\sec (x)=\tan (x)+\frac{\cos (x)}{1+\sin (x)}$$ is the same as proving
$$\begin{align}
\frac{1}{\cos(x)} & = \frac{\sin(x)}{\cos(x) } +\frac{\cos (x)}{1+\sin (x)}\tag{1}\\ \\
&= \dfrac{\sin(x)(1+\sin x) +\cos^2 x}{\cos(x)(1+\sin(x))}\tag{2}\\ \\
&= \frac{\sin x + \overbrace{\s... |
1,326,626 | <p><a href="https://books.google.com/books?id=e-YlBQAAQBAJ&pg=PA144&lpg=PA144&dq=hungerford%20ring%20of%20fractions&source=bl&ots=ztyrAtNHmr&sig=l9Xc098vOOKXs0VP58_lzMkrdBc&hl=en&sa=X&ved=0CCQQ6AEwAWoVChMIi-Xn7fmRxgIVQdCACh3sigQm#v=onepage&q=hungerford%20ring%20of%20fractions&... | wlad | 228,274 | <p>Let's use an example. $\Bbb Z$ is an integral domain; we can build the field of fractions $\Bbb Q$ out of it. If an integral domain $D$ is already a field, then its fields of fractions is isomorphic to $D$.</p>
<p><strong>Exercise</strong>: What is the general element of the field of fractions of a polynomial ring ... |
635,351 | <p>It is well known that if a series $\sum\limits_{k= 0}^\infty a_k$ converges, then $a_k \to 0$. </p>
<p>However, this is not true for integrals. What makes them different? Is it simply that they are "smoother?" Is there a rigorous way to explain this difference?</p>
| Phira | 9,325 | <p>Here are three typical situations where you should try to use integration by parts.</p>
<p>You want to apply integration by parts if</p>
<ul>
<li>your integrand has one factor which gets simpler by differentiation and a second factor which can be easily integrated.</li>
</ul>
<blockquote>
<p>Typical examples: $... |
3,234,420 | <p>This question is from my homework and I don't have any idea how to solve it. Find the number of 2019-length walks between two vertices of a triangle.</p>
| Dirk | 379,594 | <p>From each vertex of the triangle, you always have two choices, go left or go right (clockwise/anticlockwise).<br>
Now say that you go <span class="math-container">$x$</span> times left and <span class="math-container">$y$</span> times right. Then <span class="math-container">$x + y = 2019$</span>. Furthermore, you s... |
3,234,420 | <p>This question is from my homework and I don't have any idea how to solve it. Find the number of 2019-length walks between two vertices of a triangle.</p>
| Ross Millikan | 1,827 | <p>Pick one vertex as the starting vertex. Let <span class="math-container">$A(n)$</span> be the number of walks of length <span class="math-container">$n$</span> that end at that vertex. Let <span class="math-container">$B(n)$</span> be the number of walks of length <span class="math-container">$n$</span> that end a... |
3,120,187 | <p>Suppose tall matrix <span class="math-container">$A$</span> is <span class="math-container">$n \times k$</span> and that its columns are orthogonal, i.e., <span class="math-container">$A' A = I_k$</span>. Suppose further that diagonal <span class="math-container">$M$</span> is <span class="math-container">$n \times ... | Ben Dyer | 164,207 | <p>No, consider <span class="math-container">$n=k$</span> then the rank of <span class="math-container">$A’MA$</span> equals the rank of <span class="math-container">$M$</span>, which only equals <span class="math-container">$k$</span> if <span class="math-container">$M$</span> is the identity.</p>
<p>Edit: Still no. ... |
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