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 |
|---|---|---|---|---|
108,277 | <p>If Mochizuki's proof of abc is correct, why would this provide a new proof of FLT?</p>
<p>Edit: In proof of asymptotic FLT, does Mochizuki claim a specific value of n and if so what is this value?</p>
| Joël | 9,317 | <p>The two answers of J.H.S and Igor Rivin don't really answer the question : they prove FLT
only for an exposant $n$ large enough.
Since Mochizuki's version of ABC is effective, one could certainly find an effective bound for $n$, and maybe prove by hand the remaining $n$. But as they stand, they are incomplete.</p>
... |
803,608 | <p>I believe that the statement is True, and this is my argument:</p>
<p>Since there exists an element $((1-\sqrt{2})/(1-(\sqrt{2})^2)\in\mathbb{Q}[\sqrt{2}]$ such that their products gives $1$ (multiplicative identity), therefore ( $1+\sqrt{2}$ ) is a unit. </p>
<p>A $unit$ is an element $a\in R|ab=1$ where as $b\in... | Thomas | 26,188 | <p>As noted in the comments, you have a field, so all non-zero elements are units. </p>
<p>But since you asked: to show that $1+\sqrt{2}$ is a unit, you just have to find an element $b$ such that $(1+\sqrt{2})b = 1$. And you can do
$$
(1 + \sqrt{2})(-1+\sqrt{2}) = -1 - \sqrt{2} + \sqrt{2} + 2 = 1.
$$
And that is it. Y... |
3,355,215 | <blockquote>
<p>For <span class="math-container">$x$</span> and <span class="math-container">$k$</span> real numbers, for what values of <span class="math-container">$k$</span> will the graphs of <span class="math-container">$f(x)=-2\sqrt{x+1}$</span> and <span class="math-container">$g(x)=\sqrt{x-2}+k$</span> inters... | Dr. Sonnhard Graubner | 175,066 | <p>Hint: You must solve the equation
<span class="math-container">$$\sqrt{x-2}+k=-2\sqrt{x+1}$$</span> for <span class="math-container">$$x\geq 2$$</span> Writing this equation in the form
<span class="math-container">$$-k-\sqrt{x-2}=2\sqrt{x+1}$$</span> then it must be
<span class="math-container">$$k^2+2\geq x$$</spa... |
438,818 | <p>I am looking for all two consecutive integers $A$ and $A+1$, which can be represented as sums of two squares
$A=a^2+b^2$ and $A+1=c^2+d^2$, $a,b,c,d>0$.</p>
| Tamref Relue | 447,449 | <p>Take a member of the modular group, i.e. a matrix with integer entries and determinant one. Let's call it M={{a,b},{c,d}}. There are plenty of such matrices, in fact factorization of any two consecutive integers is a good source of these beasts <em>(it is a kind of poetic justice that any two consecutive integers wi... |
911,584 | <p><em>Disclaimer: This thread is a record of thoughts.</em></p>
<p><strong>Discussion</strong>
Given a compact set.</p>
<blockquote>
<p>Do mere neighborhood covers admit finite subcovers?
$$C\subseteq\bigcup_{i\in I}N_i\implies C\subseteq N_1\cup\ldots N_n$$
<em>(The idea is that neighborhoods are in some sens... | almagest | 172,006 | <p>@Stones Yes, but the table is not quite right. If there are three or more 1s or 2s, then it does not matter how many 6s there are. So N5 the number of ways of getting five 1s or 2s is 32. Similarly, N4 the number of ways of getting four is $5\times 2^4\times 4=320$ and N4 the number of ways of getting three is $10\t... |
20,802 | <p>Look at the following example:</p>
<p>Which picture has four apples?</p>
<p>A<a href="https://i.stack.imgur.com/Tpm46.png" rel="noreferrer"><img src="https://i.stack.imgur.com/Tpm46.png" alt="enter image description here" /></a></p>
<hr />
<p>B <a href="https://i.stack.imgur.com/AOv29.png" rel="noreferrer"><img src=... | Graham | 6,456 | <h3>If you allowed this, how do you grade answers?</h3>
<p>Suppose the same maths test said "John has 2 apples and Lucy has 3 apples. How many apples do they have in total?"</p>
<p>By your logic, the child could say "1" and be entirely correct. If you have 5 apples and someone asks you "do yo... |
20,802 | <p>Look at the following example:</p>
<p>Which picture has four apples?</p>
<p>A<a href="https://i.stack.imgur.com/Tpm46.png" rel="noreferrer"><img src="https://i.stack.imgur.com/Tpm46.png" alt="enter image description here" /></a></p>
<hr />
<p>B <a href="https://i.stack.imgur.com/AOv29.png" rel="noreferrer"><img src=... | Nardya | 15,952 | <p>I showed this question to my three-year old son. His response - because he counted the apples one by one in each picture, passing "4" each time - was B, C and D. Hence, we need to take into account how children arrive at their conclusion, since they do not apply formal logic. The thought process is very di... |
129,709 | <p>Say we have a continuous function $u(x,y) : \mathbb{R}^2 \rightarrow \mathbb{R}$.</p>
<p>I have seen several textbooks that make the following assertion:</p>
<blockquote>
<p>The length/area element of the zero level set of $u$ is given by
$\lvert\nabla H\left(u\right)\rvert = \delta(u)\lvert\lvert\nabla u\rve... | Charlie | 31,893 | <p>I wanted to remark that the answer above is only true when $\phi$ is a distance function, that is to say $||\nabla \phi|| = 1$ almost everywhere. In the heuristic argument in the accepted answer, the following equality
$$1/2N = |\mathbf{n}(s)| h = h = \textrm{"the distance to $\partial \Omega$"}$$ </p>
<p>is pre... |
546,239 | <p>Myself and my Math teacher are at a disagreement in to what the proper method of solving the question <em>In how many ways can four students be chosen from a group of 12 students?</em> is.</p>
<p>The question comes straight out of a Math revision sheet from a Math book distributed under the national curriculum. The... | ncmathsadist | 4,154 | <p>As asked, the question is seeking to know the number of subsets of size 4 in a set of size 12. A subsequent accounting for the order in which they are chosen is a change in the nature of the question. So there are
$${12\choose 4}$$
ways to do this. Your interpretation is correct. </p>
<p>Were order important, ... |
2,194,762 | <p>I have to find all possible values of <span class="math-container">$a$</span>, <span class="math-container">$b$</span>, <span class="math-container">$c$</span>, and <span class="math-container">$d$</span> for which the <span class="math-container">$2 \times 2$</span> matrix
<span class="math-container">$\begin{pmatr... | Travis Willse | 155,629 | <p>In the setting that the underlying field is algebraically closed (which we henceforth assume), a matrix is semisimple iff it is diagonalizable.</p>
<p>If the eigenvalues of a matrix are (pairwise) distinct then the matrix is diagonalizable. If the eigenvalues of a <span class="math-container">$2 \times 2$</span> mat... |
79,782 | <p>Calculate $17^{14} \pmod{71}$</p>
<p>By Fermat's little theorem:<br>
$17^{70} \equiv 1 \pmod{71}$<br>
$17^{14} \equiv 17^{(70\cdot\frac{14}{70})}\pmod{71}$</p>
<p>And then I don't really know what to do from this point on. In another example, the terms were small enough that I could just simplify down to an answer... | Brian M. Scott | 12,042 | <p>$17$ isn’t particularly close to a multiple of $71$, but as Ragib Zaman pointed out, $17^2=289$ is: $289=4\cdot71+5$. Thus, $17^{14}=(17^2)^7=289^7\equiv 5^7\pmod {71}$. At that point you can use brute force, or you might notice that $5^4=625$ is pretty close to $9\cdot71=639$. In fact $625=639-14$, so $5^4\equiv -1... |
4,135,449 | <p>A few years ago, I got interested in an apparently hard integration problem which had me fascinated. This was the integral of a Sophomore Dream like integral except with the bounds over the real positive numbers denoted by <span class="math-container">$\Bbb R^+$</span> and not from 0 to 1 over the unit square, or un... | JJacquelin | 108,514 | <p>From : <a href="https://fr.scribd.com/doc/34977341/Sophomore-s-Dream-Function" rel="nofollow noreferrer">https://fr.scribd.com/doc/34977341/Sophomore-s-Dream-Function</a>
<span class="math-container">$$\text{Closed form : }\quad\int_0^\infty x^{-x}dx=\text{Sphd}(-1;\infty)$$</span>
<span class="math-container">$$\te... |
675,695 | <p>[<b>Added by PLC</b>: This question is a followup to <a href="https://math.stackexchange.com/questions/298870/inner-product-change-of-axioms/298891">this already answered question</a>.]</p>
<p>Keep the axioms for a real inner product (symmetry, linearity, and homogeneity).
But make the fourth be
$$\langle x,x \ran... | Pete L. Clark | 299 | <p>Suppose that $\langle x, x \rangle > 0$ and $\langle y, y \rangle < 0$. I claim that there are $a,b \in \mathbb{R}$ such that $ax+by \neq 0$ and $\langle ax+by, ax+by \rangle = 0$. This shows that if
$q(v) = \langle v, v \rangle$ assumes both positive and negative value, then there is some nonzero $v$ with... |
1,329,539 | <p>I'm trying to figure out how to solve this word problem. I'm pretty sure it involves calculus or something even harder, but I don't know how to solve <strong><em>the general form</em></strong>.</p>
<p>Let me start with the concrete form, however:</p>
<p><strong>Concrete Form:</strong></p>
<p>You start with 5 scie... | jose_castro_arnaud | 292,886 | <p>Here is a solution, although probably it's not optimal.</p>
<p>To complete all projects at once in one year, one needs (30 * 200) + (50 * 150) + (25 * 120) = 6000 + 7500 + 3000 = 16500 scientists. </p>
<p>Training almost all of them can be done in 10 years: </p>
<ul>
<li>Year 0: 5 scientists</li>
<li>Year 5: 5 + ... |
1,329,539 | <p>I'm trying to figure out how to solve this word problem. I'm pretty sure it involves calculus or something even harder, but I don't know how to solve <strong><em>the general form</em></strong>.</p>
<p>Let me start with the concrete form, however:</p>
<p><strong>Concrete Form:</strong></p>
<p>You start with 5 scie... | Ross Millikan | 1,827 | <p>There are some features to this problem that you might not think are present in the general form. All the projects require a fixed number of scientist-years of research and all the scientists are interchangeable. We just need to assign $22500$ scientist-years to research. The minimum number per project does not m... |
1,232,023 | <p>For me, intuitively, integral $2\pi y~dx$ make more sense. I know intuition can not be proof, but by far, most part of math I've learned does match with my intuition. So, I think this one should 'make sense' as well. Probably I didn't understand the way surface area is measured. It will be great if any one could tel... | Simon S | 21,495 | <p>Here's some intuition:</p>
<p>Draw a diagram of some element of a graph of $y = f(x)$ you are rotating. If the line element described by $dx$ or $ds$ is not parallel to the $x$ axis, then notice that $ds > dx$; that is, the length of the line element is longer than just $dx$ because there is a component in the $... |
1,232,023 | <p>For me, intuitively, integral $2\pi y~dx$ make more sense. I know intuition can not be proof, but by far, most part of math I've learned does match with my intuition. So, I think this one should 'make sense' as well. Probably I didn't understand the way surface area is measured. It will be great if any one could tel... | user492893 | 492,893 | <p>A bit late but either way... </p>
<p>The surface area depends on $ds$ because $ds$ runs on the surface, unlike $dx$ which runs through the shape. </p>
<p>Consider two points A and B (not mathematical points, just locations), which are separated by a mountain. Person X simply goes through the mountain tunnel to get... |
250,314 | <p>I have a problem about proving the following. Can you please help me?</p>
<p>First of all, boundary of A is the set of points that for every r>0 we can find a ball B(x,r) such that B contains points from both A and outside of A.</p>
<p>Secondly, definition of closure of A is the intersection of all closed sets con... | Brian M. Scott | 12,042 | <p>$\newcommand{\cl}{\operatorname{cl}}\newcommand{\bdry}{\operatorname{bdry}}$One way to show that $\cl A\subseteq A\cup\bdry A$ is to show that if $x\in X\setminus(A\cup\bdry A)$, then $x\notin\cl A$. To show that $x\notin\cl A$, find a closed set $F$ such that $A\subseteq F$ and $x\notin F$.</p>
<p>So suppose that ... |
250,314 | <p>I have a problem about proving the following. Can you please help me?</p>
<p>First of all, boundary of A is the set of points that for every r>0 we can find a ball B(x,r) such that B contains points from both A and outside of A.</p>
<p>Secondly, definition of closure of A is the intersection of all closed sets con... | Grifulkin | 17,398 | <p>So let $a\in\bar{A}$ the closure. Then every open set $U$ such that $a\in U$ we have the following: $$U\cap A \neq \emptyset.$$ And if $a \not\in A \Rightarrow U \cap (X-A) \neq \emptyset$ for all open sets $U$. So $a \in \bar{A} \cap \overline{X-A} = \partial(A)=bdry \ A.$ So either $a \in A \mbox{ or } a\in \p... |
1,351,881 | <p>This argument comes up once every while on Lambda the Ultimate. I want to know where the flaw is.</p>
<p><em>Take a countable number of TMs all generating different bitstreams. Construct a Cantor TM which runs every nth TM upto the nth bit and outputs the reversed bit.</em></p>
<p><em>Now I have established there ... | Qiaochu Yuan | 232 | <p>There are uncountably many countable sets of Turing machines, and only countably many of these can be enumerated by a Turing machine (since there are, in fact, only countably many Turing machines). So for most such subsets, your Cantor TM can't exist. </p>
|
1,351,881 | <p>This argument comes up once every while on Lambda the Ultimate. I want to know where the flaw is.</p>
<p><em>Take a countable number of TMs all generating different bitstreams. Construct a Cantor TM which runs every nth TM upto the nth bit and outputs the reversed bit.</em></p>
<p><em>Now I have established there ... | Taumatawhakatangihangakoauauot | 144,375 | <p>There are a couple of issues here. To begin, I'll assume that by a Turing Machine we mean a Turing Machine with unary input alphabet $\{ \mathsf{1} \}$, and tape alphabet $\{ \mathsf{1} , \mathsf{-} \}$, where $\mathsf{-}$ is the blank symbol. The natural number $k$ is inputted into such a Turing Machine in the usua... |
3,019,882 | <p>Í already have found a proof About the statment which I did not understand, I will write it down, so maybe someone can explain me the part that I did not understand. But if there are easier ways to prove the statement I would be delighted to know. </p>
<p><span class="math-container">$\alpha:=\sqrt{2}+\sqrt{3}\noti... | Wuestenfux | 417,848 | <p>Well, <span class="math-container">$\alpha\not\in {\Bbb Q}(\sqrt 2)$</span> is equivalent to <span class="math-container">$\sqrt 3\not\in {\Bbb Q}(\sqrt 2)$</span>.
On the contrary, <span class="math-container">$\sqrt 3 = a+b\sqrt 2$</span> for some rational numbers <span class="math-container">$a,b$</span>. Squarin... |
3,019,882 | <p>Í already have found a proof About the statment which I did not understand, I will write it down, so maybe someone can explain me the part that I did not understand. But if there are easier ways to prove the statement I would be delighted to know. </p>
<p><span class="math-container">$\alpha:=\sqrt{2}+\sqrt{3}\noti... | Stockfish | 362,664 | <p>Me neither - but if <span class="math-container">$r(s-1) \neq 0$</span> you can rewrite the last equation to obtain <span class="math-container">$$\mathbb{Q} \ni \frac{3-r^2-2(s-1)^2}{2r(s-1)} = \sqrt{2} \notin \mathbb{Q},$$</span> a contradiction. If <span class="math-container">$s=1$</span> we have <span class="ma... |
4,489,898 | <p>After 18 months of studying an advanced junior high school mathematics course, I'm doing a review of the previous 6 months, starting with solving difficult quadratics that are not easily factored, for example:
<span class="math-container">$$x^2+6x+2=0$$</span>
This could be processed via the quadratic equation but t... | Oscar Lanzi | 248,217 | <p>This may not be an example of what the OP wants, but there is an inside-out version of completing the square. Given the expression</p>
<p><span class="math-container">$a^4+4b^4,$</span></p>
<p>we recognize that adding the middle term <span class="math-container">$4a^2b^2$</span> forms a square and at the same time, ... |
718,850 | <p>I see similar questions asked on here and obviously I did some research and read my book, but it seems like every explanation contradicts another in some way. There are basically infinite scenarios using these and every example problem/scenario I seem to convince myself it could be both!</p>
<p><strong>Here are som... | StephanCasey | 157,220 | <p>a) this one depends on the question at hand.</p>
<p>Say you have 3 people (A, B, C) and you want to put them in a row. Finding all the arrangements would include</p>
<p>A B C</p>
<p>A C B</p>
<p>B C A</p>
<p>B A C</p>
<p>C A B</p>
<p>C B A</p>
<p>3P3 = 3! = 6 arrangements</p>
<p>but the same three people we... |
1,515,417 | <p>I understand the idea that some infinities are "bigger" than other infinities. The example I understand is that all real numbers between 0 and 1 would not be able to "fit" on an infinite list.</p>
<p>I have to show whether these sets are countable or uncountable. If countable, how would you enumerate the set? If un... | Ittay Weiss | 30,953 | <p>Hints: 1) You just listed these numbers, so.... 2) Use the exact same technique of diagnolization you say you understand.</p>
|
335,258 | <p>Find the domain of the function:
$$f(x)= \sqrt{x^2 - 4x - 45}$$</p>
<p>I'm just guessing here; how about if I square everything and then put it in the graphing calculator?
Thanks,
Lauri</p>
| André Nicolas | 6,312 | <p>Note that $x^2-4x-45=(x-2)^2-49$. We completed the square. This is an almost universally useful move when we are dealing with quadratics.</p>
<p>Now when is the thing under the square root sign bad (negative)?</p>
|
2,550,655 | <p>When we're integrating a one variable function, there is only one path to follow between points $a$ and $b$.</p>
<p>$F(b)=F(a)+h(f(a)+f(a+h)+f(a+2h)+........)$ where $h$ is very small. So, we approach from $a$ to $b$ in steps of $h$ along the line joining unique path joining $a$ and $b$.</p>
<p>I was wondering wha... | davidlowryduda | 9,754 | <p>Generically, yes. The integral of $f$ along a path depends on the choice of the path.</p>
<p>For perhaps the simplest example, suppose $f(x,y) \equiv 1$, and choose a path $\gamma(t)$. When I say choose a path $\gamma(t)$, I think of $t$ as indicating 'time' and $\gamma(t)$ as indicating the location of a particle ... |
1,822,179 | <p>Pointwise limit of the sequence of functions </p>
<p>$$h_n(x)=\begin{cases}1,&\text{if }x\ge \frac1n\\nx,&\text{if }x\in[0,\frac1n)\end{cases}$$</p>
<p>The trouble with this question is that I think that $h(0)$ is not defined but Im not completely sure. We can see that for any $n\in\Bbb N$ we have that $h_... | Michael Hardy | 11,667 | <p>You have $h_1(0)=0$, $h_2(0)=0$, $h_3(0)=0$, $h_4(0)= 0$, etc. So the sequence $h_1(0), h_2(0), h_3(0), h_4(0), h_5(0),\ldots$ is just the sequence $0,0,0,0,0,\ldots$ in which every term is $0$. That sequence does have a limit and the limit is $0$. In other words $\lim_{n\to\infty} h_n(0) = 0$. The limit does ex... |
5,363 | <p>There is something in the definition of the <a href="http://en.wikipedia.org/wiki/Free_product" rel="nofollow noreferrer">free product</a> of two groups that annoys me, and it's this "word" thing:</p>
<blockquote>
<p>If <span class="math-container">$G$</span> and <span class="math-container">$H$</span> are groups... | Jack Lee | 1,421 | <p>In answer to your question about whether this is written somewhere: the construction of free products in my book <em>Introduction to Topological Manifolds</em> proceeds very much along these lines. (I'm not the only one who's written it down, but I don't have other references handy.) I don't go through the set-theor... |
4,570,396 | <p>Is the set <span class="math-container">$\{\frac 1n :n\in\mathbb{N}\}$</span> countable?</p>
<p>This was part of the Exercises in a book on Real Analysis. The answer given is that this set is uncountable.</p>
<p><strong>My work:</strong></p>
<p>I'm just starting on Real Analysis and , to my knowledge, this set is co... | Adam Rubinson | 29,156 | <blockquote>
<p>...To my knowledge, this set is countable, since we can map every element of this set to natural numbers as
<span class="math-container">\begin{eqnarray}1&\to&1\\
2&\to&\frac12\\
3&\to&\frac13
\end{eqnarray}</span>
and so on.</p>
</blockquote>
<p>This is sort of the correct reaso... |
719,056 | <p>I'm writing a Maple procedure and I have a line that is "if ... then ..." and I would like it to be if k is an integer (or if k^2 is a square) - how would onw say that in Maple? Thanks!</p>
| Tony | 39,296 | <p>The notation $\frac{dy}{dx} = \frac{dy}{du} * \frac{du}{dx}$ is valid. However, you cannot prove the chain rule just by "cancelling" the two $du$'s; it doesn't work that way.</p>
|
400,715 | <p>Consider the metric space $(\mathbb{Q},d)$ where $\mathbb{Q}$ denotes the rational numbers and $d(x,y)=|x-y|$. Let $$E:=\{x \in\mathbb{Q}:x>0, 2<x^2<3\}$$</p>
<p>Is $E$ closed and bounded in $\mathbb{Q}?$ Is it compact? Justify your answers.</p>
| ncmathsadist | 4,154 | <p>It is a metric space. A compact metric space is necessarily complete. This will tell you about compactness.</p>
|
26,192 | <p>I have a list of rules that represents a list of parameters to be applied to a circuit model (Wolfram SystemModeler model): </p>
<pre><code>sk = {
{R1 -> 10080., R2 -> 10080., C1 -> 1.*10^-7, C2 -> 9.8419*10^-8},
{R1 -> 10820., R2 -> 4984.51, R3 -> 10000., R4 -> 10000., C1 -> 1.*10^-7,
C... | andre314 | 5,467 | <p>Here is a solution :</p>
<pre><code>myFormat[ruleList_, {type_, counter_}] :=
MapAt[type <> ToString[counter] <> "." <> ToString[#] &, #, 1] & /@ ruleList
(* example : myFormat[{R1->1,R2->2}, {"sallenKey",5}] gives : *)
(* {"sallenKey5.R1"->1, "sallenKey5.R2"->2} ... |
186,477 | <p>The question is.</p>
<p>Is the converse true: In a simply connected domain every harmonic function has its conjugate?</p>
<p>I am not able to get an example to disprove the statement.</p>
| timur | 2,473 | <p>No, to the question as stated by OP in the comments. Take the real part of a holomorphic function on an annulus.</p>
<p>On the other hand, look at the comment by Jonas.</p>
|
186,477 | <p>The question is.</p>
<p>Is the converse true: In a simply connected domain every harmonic function has its conjugate?</p>
<p>I am not able to get an example to disprove the statement.</p>
| Federico | 128,293 | <p>The answer is yes. If a domain is not simply connected, you can always construct a harmonic function without a harmonic conjugate there. For example, for an annulus centred at the origin, take $f(x,y)=\log\sqrt{x^2+y^2}$ in . This cannot have a harmonic conjugate, as if it did you would get a branch of the logarithm... |
2,404,258 | <p>I'm asked to prove that </p>
<h2>$|z-i||z+i|=2$</h2>
<p>defines an ellipse in the plane.</p>
<p>I have tried replacing $z = x+iy $ in the previous equation and brute forcing the result to no avail. Considering that $ |z-i||z+i| = |z^2+1| $ eases the algebra a bit but didn't help me that much.</p>
<p>Edit: I kno... | user1337 | 62,839 | <p>The locus of the points $P$ which have the product of their distances from two given points (foci) constant is known as a the <a href="https://en.wikipedia.org/wiki/Cassini_oval" rel="nofollow noreferrer">Cassini oval</a>. Thus I also think that there should be a + sign rather than multiplication in your expression.... |
1,468,097 | <p>Below is the problem:</p>
<p>Choose a point uniformly at random from the triangle with vertices (0,0), (0, 30), and (20, 30). Let (X, Y ) be the coordinates of the chosen point. (a) Find the cumulative distribution function of X. (b) Use part (a) to find the density of X.</p>
<p>First, for the first part of the qu... | Empiricist | 189,188 | <p>Consider $V = \mathbb{C}^2$ and $u = (1,0), v = (0,1), w = (1,1)$. Then $u,v,w$ are pairwise linearly independent and but the whole set it not.</p>
|
141,112 | <p>I'm trying to put the following equation in determinant form:
$12h^3 - 6ah^2 + ha^2 - V = 0$, where $h, a, V$ are variables (this is a volume for a pyramid frustum with $1:3$ slope, $h$ is the height and $a$ is the side of the base, $V$ is the volume). </p>
<p>The purpose of identifying the determinant is to constr... | Jason Walker | 53,357 | <p>It's a genus 1 type as described <a href="http://www.projectrho.com/nomogram/standardForms.html" rel="nofollow">here</a> with determinantal form:</p>
<p>$$\left| \begin{array}{ccc}
0 & V & 1 \\
1 & 6a-a^2 & 1 \\
\frac{h}{h+1} & \frac{12h^3}{h+1} & 1 \end{array} \right|=0$$</p>
|
141,112 | <p>I'm trying to put the following equation in determinant form:
$12h^3 - 6ah^2 + ha^2 - V = 0$, where $h, a, V$ are variables (this is a volume for a pyramid frustum with $1:3$ slope, $h$ is the height and $a$ is the side of the base, $V$ is the volume). </p>
<p>The purpose of identifying the determinant is to constr... | Glen_b | 47,748 | <p>As it stands there are 6 linearly independent terms to deal with ($a$, $a^2$, $h$, $h^2$, $h^3$ and $V$). Without being able to spot a common factor somewhere it's not clear one can accommodate so many terms (the three terms in $h$ in particular make it hard). </p>
<p>Saint-Robert's criterion indicates that it can'... |
3,207,453 | <p>studying the series <span class="math-container">$\sum_\limits{n=2}^\infty \frac{1}{n(\log n)^ {2}}$</span>.</p>
<p>I've tried with the root criterion</p>
<p><span class="math-container">$\lim_{n \to \infty} \sqrt[n]{\frac{1}{n(\log n)^ {2}}}>1$</span>
and the series should diverge.</p>
<p>But I'm not sure
Ca... | José Carlos Santos | 446,262 | <p>It turns out that that limit is equal to <span class="math-container">$1$</span>, not bigger.</p>
<p>On the other hand, you can use the integral test: in order to determine whether the integral<span class="math-container">$$\int_2^\infty\frac1{x\log^2x}\,\mathrm dx$$</span>converges or not, use the fact that iy is ... |
4,215,824 | <p>Is there a formal definition of "almost always less than" or "almost always greater than"? I think one could define it using probabilities but not sure how to go about it. If one could show the following, then I think you could say <span class="math-container">$X$</span> is almost always less tha... | Ethan Bolker | 72,858 | <p>The precise meaning of "almost always" depends on context. In measure theory it means "except on a set of measure <span class="math-container">$0$</span>".</p>
<p>Probability is often discussed in a measure theoretic context. In that context your probability assertion makes sense. For example, on... |
3,784,889 | <p><span class="math-container">$$\begin{array}{ll} \text{minimize} & f(x,y,z) := x^2 + y^2 + z^2\\ \text{subject to} & g(x,y,z) := xy - z + 1 = 0\end{array}$$</span></p>
<hr />
<p>I tried the Lagrange multipliers method and the system resulted from has no solution. So I posted it to see if the question is wron... | Michael Rozenberg | 190,319 | <p><span class="math-container">$$x^2+y^2+z^2=x^2+y^2+z^2+2(xy-z+1)=(x+y)^2+(z-1)^2+1\geq1.$$</span>
The equality occurs for <span class="math-container">$x=y=0$</span> and <span class="math-container">$z=1$</span>, which says that we got a minimal value.</p>
|
1,932,599 | <p>Find the solution of the differential equation that satisfies the given initial condition <span class="math-container">$$~y'=\frac{x~y~\sin x}{y+1}, ~~~~~~y(0)=1~$$</span></p>
<p>When I integrate this function I get
<span class="math-container">$$y+\ln(y)= -x\cos x + \sin x + C.$$</span></p>
<p>Have I integrated ... | Dr. Sonnhard Graubner | 175,066 | <p>we write the equation in the form
$$\left(1+\frac{1}{y}\right)dy=x\sin(x)dx$$
integrating we obtain
$$y+\ln|y|=\sin(x)-x\cos(x)+C$$</p>
|
3,102,218 | <p>Given a fraction:</p>
<p><span class="math-container">$$\frac{a}{b}$$</span></p>
<p>I now add a number <span class="math-container">$n$</span> to both numerator and denominator in the following fashion:</p>
<p><span class="math-container">$$\frac{a+n}{b+n}$$</span></p>
<p>The basic property is that the second fr... | Rhys Hughes | 487,658 | <p>Let <span class="math-container">$a=kb$</span>. (<span class="math-container">$k$</span> doesnt necessarily have to be an integer). Then:</p>
<p><span class="math-container">$$\frac ab = k$$</span></p>
<p><span class="math-container">$$\frac{a+n}{b+n}=\frac{k(b+n)-(k-1)n}{b+n}$$</span>
<span class="math-container"... |
299,795 | <p>I seem to have completely lost my bearing with implicit differentiation. Just a quick question:</p>
<p>Given $y = y(x)$ what is $$\frac{d}{dx} (e^x(x^2 + y^2))$$</p>
<p>I think its the $\frac d{dx}$ confusing me, I don't what effect it has compared to $\frac{dy}{dx}$. Any help will be greatly appreciated.</p>
| Librecoin | 53,846 | <p>I just want to add that Wolfram Alpha's solutions $\left\{\text{$\color{blue}{x_\mathrm{I} = 0}$, $\color{green}{x_\mathrm{II} = (-1)^{2/3}}$, $\color{red}{x_\mathrm{III} = -\sqrt[3]{-1}}$} \right\}$ are indeed correct.</p>
<p>As others already have noted, the equation can be factorized into $x^7(x^2+x+1)=0$. Clear... |
3,194,249 | <p>If we are given <span class="math-container">$d$</span> an integer, then I've seen it written that the floor of <span class="math-container">$\sqrt{d^2-1}=d-1$</span>, but how is this found?</p>
<p>It's not immediately obvious, at least not to me.It makes me feel there must be an algorithm that can be used . </p>
... | Martin Argerami | 22,857 | <p>Saying that <span class="math-container">$\lfloor\sqrt{ d^2-1}\rfloor=d-1$</span> amounts to say
<span class="math-container">$$
d-1\leq\sqrt{d^2-1}<d.
$$</span>
The inequality on the right is trivial since <span class="math-container">$d^2-1<d^2$</span>. For the inequality on the left,
<span class="math-co... |
2,259,099 | <blockquote>
<p><em><strong>Theorem 29.2.</strong></em> Let <span class="math-container">$X$</span> be a Hausdorff space. Then <span class="math-container">$X$</span> is locally compact if and only if given <span class="math-container">$x \in X$</span>, and given a neighborhood <span class="math-container">$U$</span> o... | Tob Ernack | 275,602 | <p>The set $\overline{V}$ is closed, and since $Y$ is compact, closed subsets of $Y$ are compact by Theorem 26.2 in Munkres.</p>
|
2,081,612 | <p>I'd like to evaluate the following integral:</p>
<p>$$\int \frac{\cos^2 x}{1+\tan x}dx$$</p>
<p>I tried integration by substitution, but I was not able to proceed. </p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>$$\frac{\cos^2x}{1+\tan x}=\frac{\cos^3x}{\cos x+\sin x}$$</p>
<p>Now $\cos x+\sin x=\sqrt2\cos\left(x-\dfrac\pi4\right)$</p>
<p>Set $x-\dfrac\pi4=u\implies\cos x=\cos\left(u+\dfrac\pi4\right)=\sqrt2(\cos u-\sin u)$</p>
|
2,081,612 | <p>I'd like to evaluate the following integral:</p>
<p>$$\int \frac{\cos^2 x}{1+\tan x}dx$$</p>
<p>I tried integration by substitution, but I was not able to proceed. </p>
| StackTD | 159,845 | <p>Here's another way; rewrite:
$$\int \frac{\cos^2 x}{1+\tan x} \,\mbox{d}x
= \int \frac{\sec^2 x}{\left(1+\tan x\right) \sec^4 x} \,\mbox{d}x
= \int \frac{\sec^2 x}{\left(1+\tan x\right)\left( 1+\tan^2x\right)^2} \,\mbox{d}x $$</p>
<p>Now set $u = \tan x$ to get:
$$\int \frac{1}{\left(1+u\right)\left( 1+u^2\right)^... |
2,081,612 | <p>I'd like to evaluate the following integral:</p>
<p>$$\int \frac{\cos^2 x}{1+\tan x}dx$$</p>
<p>I tried integration by substitution, but I was not able to proceed. </p>
| Alexandre C. | 61,517 | <p>This one always works for rational functions of $\sin x$ and $\cos x$ but can be a bit tedious. Set:
$$ z = \tan x / 2$$
so that
$$ \mathrm{d}x = \frac{2\,\mathrm{d} z}{1 + z^2}$$
$$ \cos x = \frac{1 - z^2}{1 + z^2}$$
$$\sin x = \frac{2z}{1 + z^2}$$
Now, you have a rational fraction in $z$ that you can integrate by ... |
5,927 | <p>I have a problem with the binomial coefficient $\binom{5}{7}$. I know that the solution is zero, but I have problems to reproduce that:</p>
<p>${\displaystyle \binom{5}{7}=\frac{5!}{7!\times(5-7)!}=\frac{5!}{7!\times(-2)!}=\frac{120}{5040\times-2}=\frac{120}{-10080}=-\frac{1}{84}}$</p>
<p>Where is my mistake?</p>
| Sy123 | 61,808 | <p>Although not as formal, one by relying on only a combinatoric definition of the binomial coefficient we can find that it is zero straight away.
Consider that:</p>
<p>$\binom{m}{k}$ is the amount of combinations of k elements from m numbered set.</p>
<p>It is obvious that we cannot select more than what we have, so... |
2,356,593 | <blockquote>
<p>Quoting:" Prove: if $f$ and $g$ are continuous on $(a,b)$ and $f(x)=g(x)$ for every $x$ in a dense subset of $(a,b)$, then $f(x)=g(x)$ for all $x$ in $(a,b)$."</p>
</blockquote>
<p>Let $S \subset (a,b)$ be a dense subset such that every point $x \in (a,b)$ either belongs to S or is a limit point of S... | gary | 6,595 | <p>Assuming these are Real-valued functions of a Real variable. Consider the $0$ function on a dense subset. Since it is uniformly -continuous, we can extended it into the complement using sequential continuity $( x_n \rightarrow x ) \rightarrow (f(x_n) \rightarrow f(x))$, which implies $f(x)=0$. Assume $f(x)= a>0 ... |
978,399 | <p>A friend of mine posed this question to me a couple days ago and it's been bugging me ever since. He told me to take the square root of 5 twenty times, subtract 1 from it, and then multiply it by 2^20, giving me a rough approximation of ln (5). I can't for the life of my figure out why this is.</p>
<p>Also: How can... | Mats Granvik | 8,530 | <p>I checked this in Mathematica. It really does seem to converge to Log(5) accurately after many iterations:</p>
<pre><code>a = N[Sqrt[5], 30];
Do[
a = Sqrt[a], {n, 1, 10000 - 1}]
(a - 1)*2^10000
</code></pre>
<p>Output:</p>
<p>1.60943791243410037460075933323</p>
<p>compared to:</p>
<pre><code>N[Log[5], 30]
</co... |
978,399 | <p>A friend of mine posed this question to me a couple days ago and it's been bugging me ever since. He told me to take the square root of 5 twenty times, subtract 1 from it, and then multiply it by 2^20, giving me a rough approximation of ln (5). I can't for the life of my figure out why this is.</p>
<p>Also: How can... | paw88789 | 147,810 | <p>Let $a$ be a positive real number.</p>
<p>Then
$$\lim_{n\to\infty}(a^\frac1n-1)\cdot n=\lim_{n\to\infty}\frac{a^{\frac1n}-1}{\frac1n}=\lim_{n\to\infty} \frac{a^\frac1n\cdot\ln a\cdot\frac{-1}{n^2}}{\frac{-1}{n^2}}=\ln a$$</p>
<p>Your result is a special case of this with $a=5$, and $n=2^{20}$</p>
|
2,338,321 | <p>Under given standardized random variable $Z = (X-\mu)/\sigma$ Show that </p>
<p>$$C_3(Z) = M_3(Z)\;\;\text{and}\;\; C_4(Z) = M_4(Z) -3$$</p>
<p>where $C_r(Z)$ refer to $r$-th cumulant, $M_r(Z)$ refer to $r$-th moment.</p>
<hr>
<p><strong>Question</strong></p>
<p>I've been required to show above relationship by ... | Michael Hardy | 11,667 | <p>The validity of an answer might depend on which characterizations of cumulants you're looking for.</p>
<p>The $r$th cumulant is a polynomial function of the first $r$ moments. The $r$th cumulant is shift-invariant if $r>1.$ "Shift invariant" means that for random variables $X$ and constants $c$ ("constant" = not... |
2,736,286 | <p>Just wondering if I am anywhere near the correct answer.</p>
<p>We have the set {z $\in \mathbb{C} : |z - 1| < |z + i|$}</p>
<p>So we let $0 < \delta < |z + i| - |z - 1|$ then use the triangle inequality to prove $\delta + |z - 1| < |z + i|$.</p>
<p>Thanks!</p>
| S. D. ZHU | 156,697 | <p>You are as far as $\delta$ to $0$. </p>
<p>Consider its geometric meaning instead: this set depicts the points closer to $1$ than to $-i$. Can you draw it? </p>
|
1,594,968 | <p>How much knowledge of group theory is needed in order to begin Galois Theory? Which topics are most relevant?</p>
| p Groups | 301,282 | <p>Some elementary: definitions and examples of groups (of small orders and especially $S_n,A_n$), normal subgroups, cyclic and abelian groups, quotient groups, isomorphism theorem. </p>
<p>[These of these terminologies enter in the (two) main theorems of Galois; examples are needed to know just to understand what is ... |
1,578,378 | <p>Why does $$\int_{-L}^{L} \sum_{n=1}^{\infty}a_n\cos \frac{n\pi x}{L}=\sum_{n=1}^{\infty}a_n\int_{-L}^{L}\cos \frac{n\pi x}{L}$$</p>
<p>This is used in a derivation of the Fourier coefficients. I see why they flip them but I don't understand why this is true and when we are allowed to do such a thing.</p>
<p>Thanks... | Julián Aguirre | 4,791 | <p>These are conditions allowing the interchange of integration and summation, from stronger to weaker:</p>
<ol>
<li>$\sum_{n=1}^\infty|a_n|<\infty$.</li>
<li>$\sum_{n=1}^\infty a_n\cos\dfrac{n\,\pi\,x}{L}$ converges uniformly.</li>
<li>$\sum_{n=1}^\infty a_n\cos\dfrac{n\,\pi\,x}{L}$ is the Fourier series of a Riem... |
2,839,554 | <p>I would like to formalise some operations I am doing, however it is unclear how I should deal with categorical variables. </p>
<p>Imagine a dataset with 15 distinct couples (<code>ID</code>). Each couple was observed 3 times (<code>time</code>). </p>
<p>Each partner has responded to two questions: <code>p</code> a... | jnez71 | 295,791 | <p>Such a choice is arbitrary. Do you view $A$ and $B$ as time varying parameters, or as inputs like $u$? (Probably the former, since you described (1) and (2) as a "linear system"). It's all about context. In control theory, equations (3) and (4) are the most common for a nonlinear ODE system in explicit state-space f... |
4,074,031 | <p>You have been teaching Dennis and Inez math using the Moore method for their entire lives, and you're currently deep into topology class.</p>
<p>You've convinced them that topological manifolds aren't quite the right object of study because you "can't do calculus on them," and you've managed to motivate th... | Moishe Kohan | 84,907 | <p>Regarding the sentence:</p>
<p>"Alternatively, answer this question in the negative by proving that Dennis and Inez will always invent integration if you make them think about higher exterior derivatives enough."</p>
<p>The answer is "yes" (but "long enough" might take longer than their... |
1,381,545 | <p>I have the following problem:</p>
<blockquote>
<p>Show by induction that $F_n \geq 2^{0.5 \cdot n}$, for $n \geq 6$</p>
</blockquote>
<p>Where $F_n$ is the $nth$ Fibonacci number. </p>
<hr>
<h3>Proof</h3>
<p><strong>Basis</strong></p>
<p>$n = 6$. </p>
<p>$F_6 = 8 \geq 2^{0.5 \cdot 6} = 2^{\frac{6}{2}} = 2^3... | Kelechi Nze | 164,374 | <p>Suppose we knew for 2 values of n $F_n \ge 2^{\frac n2}$ i.e for n = 6 and n = 7.</p>
<p>We know this holds for n=6 and n=7.</p>
<p>We also know that $F_{n+1} = F_{n} + F_{n-1}$</p>
<p>So we assume for some k and k-1 (7 and 6) $F_{k-1} \ge 2^{\frac {k-1}{2}}$ and $F_{k+1} \ge 2^{\frac {k+1}{2}}$</p>
<p>We know $... |
2,005,365 | <p>For example, p(getting two heads from tossing a coin twice) = 0.5 * 0.5...</p>
<p>I passed my probability course in college, but I am still having trouble getting the intuition for this.</p>
| Mark Viola | 218,419 | <p><strong>HINT:</strong></p>
<p>Rationalize the numerator to get</p>
<p>$$\frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt x}=-\frac{1}{\sqrt{x+h}\sqrt{x}(\sqrt{x+h}+\sqrt{x})}$$</p>
|
3,807,504 | <p>Suppose that <span class="math-container">$z=f(x,y)$</span> is given. where <span class="math-container">$f(x,y)=x^3+y^3$</span>.
I am a bit confused about the notation of <span class="math-container">$z_x$</span> and <span class="math-container">$f_x$</span>.</p>
<p>A lot of people use the below notation for <span ... | Claude Leibovici | 82,404 | <p>There is an antiderivative in terms of the gaussian hypergeometric function
<span class="math-container">$$\int \frac{b\,x}{x^a-b}\,dx=-\frac{1}{2} x^2 \, _2F_1\left(1,\frac{2}{a};\frac{a+2}{a};\frac{x^a}{b}\right)$$</span> If <span class="math-container">$a >0$</span>
<span class="math-container">$$I_a=\int_0^R ... |
16,740 | <p>Excuse me for my unclear question title, math terms have never been my forté. However being new here and knowing I shall be staying here for a while, I thought it's best to do things right with the community and site.</p>
<p>I just asked my first <a href="https://math.stackexchange.com/questions/920192/differentiat... | GEdgar | 442 | <p>Generally, I would be opposed to "check my proof" questions even if the link is to a permanent place like the ArXiV.</p>
|
182,024 | <p>Following my previous question <a href="https://math.stackexchange.com/q/182000/21813">Relationship between cross product and outer product</a> where I learnt that the Exterior Product generalises the Cross Product whereas the Inner Product generalises the Dot Product, I was wondering if the simple map that I have d... | rschwieb | 29,335 | <p>I'm not sure what you're reading, but it certainly contains a lot of terms talked about in <a href="http://en.wikipedia.org/wiki/Clifford_algebra" rel="nofollow">Clifford algebra</a> (or "<a href="http://en.wikipedia.org/wiki/Geometric_algebra" rel="nofollow">geometric algebra</a>").</p>
<p>Clifford algebras were b... |
2,051,785 | <p>$PQRS$ is a convex quadrilateral. The intersection of the two diagonals is $O$. If the areas of triangles $PQS$, $QRP$, and $SPR$ are $1, 2,$ and $3$, respectively. What is the area of triangle $PQO$?</p>
| Triskele | 397,755 | <p>If you want a hint, read on. </p>
<p>First determine $\mathrm{Area}(\triangle QRS)$. </p>
<p>Now, let $x = \mathrm{Area}(\triangle SPO)$, $y = \mathrm{Area}(\triangle PQO)$, and note that $\mathrm{Area}(\triangle RSO) : \mathrm{Area}(\triangle QRO) = x : y$. (Why?) </p>
|
3,480,890 | <p>I do understand pure mathematical concepts of probability space and random variables as a (measurable) functions. </p>
<p>The question is: what is the real-world meaning of probability and how can we apply the machinery of probability to the real situations?</p>
<p>Ex1: probability of heads for fair coin is 1/2. W... | kludg | 42,926 | <p>General interpretation of probabilities is Bayesian: probabilities are subjective opinions or beliefs; of course probabilities must satisfy probability axioms, but this is the only requirement; any probabilistic model satisfying probability axioms is good. I believe it is helpful to think about probabilities as subj... |
855,088 | <p>Consider two vectors $V_1$ and $V_2$ in $\mathbb{R}^3$.
When we take their dot product we get a real number.
How is that number related to the vectors?
Is there any way we can visualize it? </p>
| Eric Towers | 123,905 | <p>Temporarily imagine that $V_2$ is of unit length. Then, $V_1 \cdot V_2$ is the projection of the vector $V_1$ onto the vector $V_2$. Picture <a href="http://en.wikipedia.org/wiki/File%3aDot_Product.svg">here</a>. Now we let $V_2$ have its original length and to do so we multiply the result of the dot product by t... |
889,728 | <p>Evaluate $$\int_0^{1/2}x^3\arctan(x)\,dx$$</p>
<p>My work so far:
$x^3\arctan(x) = \sum_{n=0}^\infty(-1)^n \dfrac{x^{2n+4}}{2n+1}$</p>
<p>$$\int_0^{1/2}x^3\arctan(x)\,dx = \sum_{n=0}^\infty \frac{(-1)^{n+1}(1/2)^{2n+4}}{2n+1}$$</p>
<p>The desired accuracy needed is to four decimal places, that is were I am stuck.... | Claude Leibovici | 82,404 | <p>You wrote correctly $$x^3\arctan(x) = \sum_{n=0}^\infty(-1)^n \dfrac{x^{2n+4}}{2n+1}$$ but you forgot to integrate with respect to $x$. If you do it $$\int_0^{1/2}x^3\arctan(x)\,dx = \sum_{n=0}^\infty \frac{(-1)^{n}(1/2)^{2n+5}}{(2n+1)(2n+5)}$$</p>
<p>As said in the comments, you could start searching for which val... |
1,326,816 | <p>How to find the Maclaurin series of the function $$f(x)=\frac{1}{(9-x^2)^2}$$
I guess we are gonna use derivatives but i have no idea how the final answer should be formed.</p>
| abel | 9,252 | <p>$\bf hint:$ </p>
<p>(a) use the binomial series $$(1-u)^{-2} = 1+\frac21 u+\frac{2 \cdot 3}{1 \cdot 2}u^2+ \frac{2 \cdot 3 \cdot 4}{1 \cdot 2 \cdot 3}u^3+ \cdots \tag 1$$ </p>
<p>(b) $$\frac{1}{(9-x^2)^2} =\frac{1}{9^2(1-x^2/9)^2} = \frac1{81}\left(1-\frac{x^2}{9}\right)^{-2} \tag 2$$</p>
|
96,957 | <p>Stacks, of varying kinds, appear in algebraic geometry whenever we have moduli problems, most famously the stacks of (marked) curves. But these seem to be to be very geometric in motivation, so I was wondering if there are natural examples of stacks that arise in arithmetic geometry or number theory.</p>
<p>To me, ... | David Zureick-Brown | 2 | <p>Here are two applications of stacks to number theory.</p>
<p>1) Section 3 of <a href="http://math.mit.edu/~poonen/papers/pss.pdf" rel="nofollow noreferrer">this</a> paper, which solves the diophantine equation $x^2 + y^3 = z^7$, explains the connection between stacks and generalized Fermat equations.</p>
<p>2) <a ... |
390,644 | <p>I'm trying to solve this recurrence relation:</p>
<p>$$
a_n = \begin{cases}
0 & \mbox{for } n = 0 \\
5 & \mbox{for } n = 1 \\
6a_{n-1} - 5a_{n-2} + 1 & \mbox{for } n > 1
\end{cases}
$$</p>
<p>I calculated generator function as:
$$
A = \frac{31x - 24x^2}{1 - 6x + 5x^2} + \frac{x^3}{(... | Brian M. Scott | 12,042 | <p>I did not check your work, so I’ll outline what you need to do to finish. You have something of the form</p>
<p>$$A(x)=\frac{a}{1-x}+\frac{b}{5-x}+\frac{c}{(1-x)^2}=\frac{a}{1-x}+\frac{b/5}{1-\frac{x}5}+\frac{c}{(1-x)^2}\;.$$</p>
<p>Expand these three terms into power series:</p>
<p>$$\begin{align*}
A(x)&=a\s... |
2,239,058 | <blockquote>
<p>Find the spherically symmetric solution to $$\nabla^2u=1$$ in the
region $r=|\mathbf{r}|\le a$ for $a>0$ that satisfies the following
boundary condition at $r=a$:</p>
<p>$\frac{\partial u}{\partial n}=0$</p>
</blockquote>
<p>The solution I have looked at states to begin with $\frac{\parti... | user1551 | 1,551 | <p>This is just a change of basis. Let $A$ and $B$ be the matrices of the same bilinear form, but with respect to two ordered bases $\{x_1,\ldots,x_n\}$ and $\{y_1,\ldots,y_n\}$ respectively. Let $P$ be the matrix such that $y_j=\sum_ip_{ij}x_i$, i.e. the matrix such that $\sum_jc_jy_j=\sum_i(Pc)_ix_i$ for any $c\in\ma... |
1,541,800 | <p>I happened to stumble upon the following matrix:
$$ A = \begin{bmatrix}
a & 1 \\
0 & a
\end{bmatrix}
$$</p>
<p>And after trying a bunch of different examples, I noticed the following remarkable pattern. If $P$ is a polynomial, then:
$$ P(A)=\begin{bmatrix}
P(a) & P'(a) \... | Vectornaut | 16,063 | <p><em>Here's one for the algebra lovers. It can probably be improved substantially, since I'm not an algebra lover myself. For simplicity, I'll use rational functions with rational coefficients; sophisticated readers may substitute their favorite coefficient fields.</em></p>
<p>A rational function in the variable $x$... |
1,344,993 | <p>Let $g_1, g_2, h_1, h_2 : \mathbb{R} \rightarrow \mathbb{R}$ be non-decreasing and right-continuous. Define
$$
\begin{align}
f_1 & := g_1 - h_1 \\
f_2 & := g_2 - h_2
\end{align}
$$
and suppose $f_1 = f_2$. In other words, for every $a, b \in \mathbb{R}$ with $a < b$, the restriction of $g_1 - h_1$ and $g_... | David C. Ullrich | 248,223 | <p>Actually no, a function has at most <em>one</em> Jordan decomposition.
You have two decompositions, but at most one is the <em>Jordan</em> decomposition.</p>
<p>Anyway, finiteness for one decomposition does not imply finiteness for the other. Consider $$0=0-0=x^+-x^+.$$</p>
<p><em>If</em> $f$ has bounded variation... |
1,344,993 | <p>Let $g_1, g_2, h_1, h_2 : \mathbb{R} \rightarrow \mathbb{R}$ be non-decreasing and right-continuous. Define
$$
\begin{align}
f_1 & := g_1 - h_1 \\
f_2 & := g_2 - h_2
\end{align}
$$
and suppose $f_1 = f_2$. In other words, for every $a, b \in \mathbb{R}$ with $a < b$, the restriction of $g_1 - h_1$ and $g_... | Evan Aad | 37,058 | <p>I'd like to expand upon <a href="https://math.stackexchange.com/users/248223/david-c-ullrich">David C. Ullrich</a>'s answer to question #2. The answer, as he wrote, is: yes, $\phi_1 = \phi_2$. Indeed, let $B$ be a Borel set on the real line. Then for $i \in \{1, 2\}$,
$$
\phi_i(B) = \phi_i\left(\cup_{n \in \mathbb{Z... |
4,206,994 | <p>Let <span class="math-container">$F(X, Y)$</span> be an irreducible polynomial of <span class="math-container">$\mathbb{C}[X, Y]$</span> with <span class="math-container">$F(0, 0) = 0$</span> and non-singular there, and let <span class="math-container">$(X(t), Y(t))$</span> be a local analytical parametrization of t... | Aphelli | 556,825 | <p>The result that decides your problem is the following: let <span class="math-container">$f,g \in \mathbb{C}[x,y]$</span> be two non-associated irreducible polynomials. Then the set of common zeros of <span class="math-container">$f$</span> and <span class="math-container">$g$</span> is finite.</p>
<p>Proof: we can s... |
2,185,434 | <p>I recently read this problem</p>
<blockquote>
<p>A bag contains $500$ beads, each of the same size, but in $5$ different colors. Suppose there are $100$ beads of each color and I am blindfolded. What is the least number of beads I must pick before I can be sure there are $5$ beads of the same color among the bead... | hunter | 108,129 | <p>Your answer is the number of beads you would have to pick to be sure there was at least one of each color. But that's not the question. You just want there to be one color such that there are at least 5 beads in that color. So 21 beads will do, since it's greater than (4 beads)*(5 colors) = 20.</p>
|
90,548 | <p>Suppose we are handed an algebra $A$ over a field $k$. What should we look at if we want to determine whether $A$ can or cannot be equipped with structure maps to make it a Hopf algebra?</p>
<p>I guess in order to narrow it down a bit, I'll phrase it like this: what are some necessary conditions on an algebra for ... | Peter May | 14,447 | <p>Let $I$ be the kernel of $\epsilon\colon A\to k$. Filter $A$ by the powers of $I$.
If $A$ is a Hopf algebra, then the associated graded algebra $E^0A$ is
a primitively generated Hopf algebra. If the characteristic of $k$ is
zero, $E^0A$ is isomorphic to the universal enveloping algebra of its
Lie algebra of primi... |
60,081 | <p>I have a stochastic matrix $A \in R^{n \times n}$ whose sum of the entries in each row is $1$. When I found out the eigenvalues and eigenvectors for this stochastic matrix, it always happens that one of the eigenvalues is $1$. </p>
<p>Is it true that for any square <a href="https://en.wikipedia.org/wiki/Stochastic_... | Geoff Robinson | 13,147 | <p>The column vector with every entry $1$ is an eigenvector with eigenvalue $1$ for your matrix. It is not necessarily true that the eigenvalue $1$ occurs with <em>algebraic multiplicity</em> $1$ as an eigenvalue for your matrix $A$. By the Frobenius Perron-Theorem, that is the case if the entries of $A$ (or even some ... |
782,929 | <p>prove or disprove following matrix is
positive definite matrix ?</p>
<p>$$\begin{bmatrix}
\dfrac{\sin(a_1-a_1)}{a_1-a_1}&\cdots&\dfrac{\sin(a_1-a_n)}{a_1-a_n}\\
\vdots& &\vdots\\
\dfrac{\sin(a_n-a_1)}{a_n-a_1}&\cdots&\dfrac{\sin(a_n-a_n)}{a_n-a_n}
\end{bmatrix}_{n\times n}$$
and where we ... | Omran Kouba | 140,450 | <blockquote>
<p>Let $b_{kl}=\frac{\sin{(a_k-a_l)}}{a_k-a_l}$.
The symmetric $n\times n$ matrix $M=(b_{kl})$ is positive definite, if and only if $(a_1,\ldots,a_n)$ are distinct. </p>
</blockquote>
<p>Note that
$$\frac{\sin a}{a}=\frac{1}{2}\int_{-1}^1e^{iat}dt$$
Thus, for ${\bf x}=(x_1,\ldots,x_n)\in\Bbb{C}^n$ we... |
1,760,148 | <p>If I have a connected metric space $X$, is any ball around a point $x\in X$ also connected?</p>
| hmakholm left over Monica | 14,366 | <p>There are even <em>complete path-connected</em> metric spaces that contain a point $x$ such that <em>no</em> ball around $x$ is connected, for example</p>
<p>$$ \{\langle x,0\rangle \mid x\ge 1\} \cup
\{\langle 1,y\rangle \mid 0\le y\le 1 \} \cup
\{\langle x,\tfrac1x \rangle \mid x\ge 1 \} \cup
\bigcup_{n=3}^\infty... |
1,760,148 | <p>If I have a connected metric space $X$, is any ball around a point $x\in X$ also connected?</p>
| Giulio Bresciani | 118,028 | <p>Just take a plane without a segment, and consider a small ball near the center of the segment. There's no need to get complicated stuff.</p>
|
2,941,896 | <p>I want to ask you for following statement</p>
<blockquote>
<p><span class="math-container">$f$</span> is continuous from <span class="math-container">$X$</span> to <span class="math-container">$Y$</span>, where <span class="math-container">$X$</span> is compact Hausdorff, connected and <span class="math-container... | 5xum | 112,884 | <p><span class="math-container">$3\log x - 2\log x = \log x$</span>, just like <span class="math-container">$3y-2y=y$</span> no matter what <span class="math-container">$y$</span> is equal to.</p>
<p>Alternatively, you can get
<span class="math-container">$$3\log x - 2\log x = \log(x^3)-\log(x^2) = \log\left(\frac{x^3... |
2,941,896 | <p>I want to ask you for following statement</p>
<blockquote>
<p><span class="math-container">$f$</span> is continuous from <span class="math-container">$X$</span> to <span class="math-container">$Y$</span>, where <span class="math-container">$X$</span> is compact Hausdorff, connected and <span class="math-container... | KM101 | 596,598 | <p><span class="math-container">$$3\log x-2\log x$$</span>
Here, we use two identities.
<span class="math-container">$$\log a^b = b\cdot\log a$$</span>
<span class="math-container">$$\log_a b-\log_a c = \log_a \big(\frac{b}{c}\big)$$</span>
Use the first identity to reach the following expression.<br>
<span class="math... |
782,427 | <p><strong>I - First doubt</strong> : free variables on open formulas . </p>
<p>I'm having a hard time discovering what different kinds of variables in an open formula of fol are refering to., For example, lets take some open formulas : </p>
<p>1: $"x+2=5"$<br>
2: $"x=x"$<br>
3: $"x+y = y+x"$<br>
4: $"x+2 = ... | Dan Christensen | 3,515 | <p>Quite often, quantifiers are are left out of statements in informal discussions. When we talk about the commutativity of addition on the natural numbers, for example, some authors may simply write $x+y=y+x$. To be more precise, they should write $\forall x,y\in N: x+y=y+x.$ </p>
<p>In a statement in a <em>formal</e... |
414,432 | <p>We define $f_n:\mathbb{R}\to\mathbb{R}$ by $f_n(x)=\dfrac{x}{1+nx^2}$ for each $n\ge 1$.</p>
<p>I compute that $f(x):= \displaystyle\lim_{n\to \infty}f_n(x) = 0$ for each $x\in\mathbb{R}$.</p>
<p>Now, I want to know in which intervals $I\subseteq \mathbb{R}$ the convergence is uniform.</p>
<p>Any hint? Thanks.</p... | Euler....IS_ALIVE | 38,265 | <p>The sequence converges uniformly on $I$ if for every $\epsilon >0, \exists N$ such that $\forall x \in I $ and $\forall n \geq N$, we have $|f_n(x) - f(x)| < \epsilon$</p>
<p>For the interval $|x| \leq 1$, we see that $x(1+nx) = x +nx^2 \leq 1 + nx^2$, so that $\frac{x}{1+nx^2} \leq \frac{x}{x(1+nx)} = \frac{... |
221,762 | <p>Let $f$ be a newform of level $\Gamma_1(N)$ and character $\chi$ which is not induced by a character mod $N/p$. I learned from <a href="http://wstein.org/books/ribet-stein/main.pdf" rel="nofollow">these notes</a> by Ribet and Stein that $|a_p|=p^{(k-1)/2}$ where $k$ is the weight of $f$. So I wonder</p>
<p>1, the ... | Myshkin | 43,108 | <p>I think you can find the proof you want in:</p>
<ul>
<li>Andrew Ogg, <a href="http://wwwuser.gwdg.de/~subtypo3/gdz/pdf/PPN235181684_0179/PPN235181684_0179___LOG_0039.pdf" rel="nofollow">On the Eigenvalues of Hecke Operators</a> (1969)</li>
</ul>
<p>Perhaps you can find it more thoroughly explained in Shimura's boo... |
221,762 | <p>Let $f$ be a newform of level $\Gamma_1(N)$ and character $\chi$ which is not induced by a character mod $N/p$. I learned from <a href="http://wstein.org/books/ribet-stein/main.pdf" rel="nofollow">these notes</a> by Ribet and Stein that $|a_p|=p^{(k-1)/2}$ where $k$ is the weight of $f$. So I wonder</p>
<p>1, the ... | paul Monsky | 6,214 | <p>This is really a comment, but the system won't let me post one. I think there's a confusion and that the absolute value is $p^((k-2)/2)$ where $k$ is the weight. For example there's a level $\Gamma_0 (2)$ newform of weight 8, $f=(\eta(z)\eta(2z))^8$, with $f^3 =\delta(z)\delta(2z)=q^3-24q^4+...$ Then $f=q-8q^2+...$,... |
1,319,761 | <p>I'm stuck trying to figure out how to solve the following integral:</p>
<p>$\int_{C(0,1)^+}\sin(z)dz$</p>
<p>I've tried parameterizing z(t) but then I get</p>
<p>$\int_0^{2\pi}\sin(e^{it})ie^{it}$ which I don't know how to integrate.</p>
<p>So then I'm looking to use Cauchy's Integral formula but I'm not sure if... | Jack D'Aurizio | 44,121 | <p>The integral of any entire function over any closed curve is just zero. That follows from:
$$ \forall n\in\mathbb{Z},\qquad \int_{0}^{2\pi} e^{nit}\,dt = 2\pi\cdot\delta(n). $$
In your case:
$$ \sin z = \sum_{n\geq 0}\frac{(-1)^n}{(2n+1)!}z^{2n+1} $$
can be integrated termwise. Can you see what happens, then?</p>
|
659,988 | <p>I understood the definition of a $\sigma$-algebra, that its elements are closed under complementation and countable union, but as I am not very good at maths, I could not visualize or understand the intuition behind the meaning of "closed under complementation and countable union". </p>
<p>If we consider the set X... | Shivam Singh | 377,513 | <p>steps to form the smallest sigma field
step 1: break the whole space into sets that are mutually exclusive and exhaustive .
step 2: then form a new set using these sets(sets obtained in step 1) in this way{ take single sets ,take sum of two at a time ,take sum of three at a time ,..........,take sum of all } . the ... |
1,045,941 | <p>Usually this is just given as a straight up definition in a calculus course. I am wondering how you prove it?
I tried using the limit definition, $$\lim\limits_{h\rightarrow 0} \dfrac{\log(x+h)-\log(x)}{h}$$
but this led to no developments.</p>
| Darrin | 117,053 | <p>Usually, this statement is defined in the following way: $$\int_0^x \frac{1}{t}dt = \ln(x), x \in (0, \infty)$$. </p>
<p>Nonetheless, suppose you were not working with this as a definition - how could we verify this from, say, what we know about $f(x)=e^x$ and then proceeding to define $\ln(x)$ as that function $f^... |
2,583,232 | <p>Prove that $\lim_{n \to \infty} \dfrac{2n^2}{5n^2+1}=\dfrac{2}{5}$</p>
<p>$$\forall \epsilon>0 \ \ :\exists N(\epsilon)\in \mathbb{N} \ \ \text{such that} \ \ \forall n >N(\epsilon) \ \ \text{we have} |\dfrac{-2}{25n^2+5}|<\epsilon$$
$$ |\dfrac{-2}{25n^2+5}|<\epsilon \\ |\dfrac{2}{25n^2+5}|<\epsi... | Higurashi | 204,434 | <p>Note that $\frac{2n^2}{5n^2+1} = \frac{2}{5} \frac{n^2}{n^2+\frac{1}{5}}$. Subtract $\frac{2}{5}$, and you get:
$\frac{2}{5} \left|\frac{n^2}{n^2+\frac{1}{5}}-1\right| = \frac{2}{5}\left|\frac{\frac{1}{5}}{n^2+\frac{1}{5}}\right|$.
Now we can always take $N>0$ such that for all $n>N$, this is smaller than $\ep... |
3,984,778 | <p>Ackermann's function and all the up-arrow notation is based on exponentiating. We know for a fact that the factorial function grows faster than any power law so why not build an iterative sequence based on factorials? I am thinking of the simple function defined by
<span class="math-container">$$
a(n)=a(n-1)!
$$</sp... | Wuestenfux | 417,848 | <p>So you mean the function <span class="math-container">$a(n)$</span> defined by <span class="math-container">$a(1)=3$</span> and <span class="math-container">$a(n) = a(n-1)!$</span>.</p>
<p>This function is primitive recursive.
Multiplication can be defined by a primitive recursive scheme. Based on this, the factoria... |
1,051,335 | <p>Purely out of interest, I wanted to try and construct a sequence of differentiable functions converging to a non-differentiable function. I began with the first non-differentiable function that sprung to my mind, namely
\begin{align}
&f:\mathbb{R}\to\mathbb{R}\\
&f(x)=|x|.
\end{align}
After some testing I co... | Lutz Lehmann | 115,115 | <p>Why do you think your function is differentiable? Did you calculate the derivatives? What does the picture look like for $ε=1$ or $ε=10$?</p>
<hr>
<p>An easier differentiable approximation of the abs function is
$$\sqrt{ε^2+x^2}$$ or $$\sqrt{ε^2+x^2}-ε.$$</p>
<p>Differentiability here is obvious by the chain rule... |
722,663 | <p>I am totally stack in how to do this exercise :
How can I find the irreducible components of $ V(X^2 - XY - X^2 Y + X^3) $ in $A^2$(R)?</p>
<p>Given an algebraic set, what is the consideration I have to make to find them? I know the definitions of components and irreducible components but these don't help in to so... | BoZenKhaa | 60,953 | <p>$V(f)$, the zero set of a polynomial $f \in R[x,y]$, is given as set of points such that $f(x,y) = 0$. If this polynomial is reducible, i.e. $f=gh$, the points $(x_0, y_0)$ in $V(f)$ are such that $g(x_0, y_0) = 0$ or $h(x_0, y_0) = 0$. This means that $V(f)$ is a union of $V(g)$ and $V(h)$. </p>
<p>In your case,... |
3,760,140 | <p>My professor asked us to find the boundary points and isolated points of <span class="math-container">$\Bbb{Q}$</span>; my answer that there is no boundary point or either isolated point for rational numbers set. However the question for confusing me, which asks "to find", means that there exist boundary p... | Siddharth Bhat | 261,373 | <ul>
<li><p><a href="https://www.desmos.com/calculator/cpecxbejut" rel="nofollow noreferrer">Here's a desmos link to the function <code>peek-a-boo</code> showed me</a></p>
</li>
<li><p>Here's a drawing. The bump function is in purple, the function <span class="math-container">$h(x)$</span> is in red.</p>
</li>
</ul>
<i... |
3,428,546 | <p>Consider whether the following series converges or diverges <span class="math-container">$$\sum_{n=1}^\infty\frac{(-1)^n\sin\frac{n\pi}{3n+1}}{\sqrt{n+3}}$$</span></p>
<p>I have tried Leibniz's test but failed to show that <span class="math-container">$\sin\frac{n\pi}{3n+1} / \sqrt{n+3}$</span> is monotone, which I... | WoolierThanThou | 686,397 | <p>The point is that often-times, your question only depends on the <em>distributions</em> of the involved random variables, not about their behaviour as a map.</p>
<p>For instance, if <span class="math-container">$X:\Omega\to \mathbb{R}$</span> has the property that <span class="math-container">$X$</span> only ever a... |
1,410,048 | <p>I am unsure of how to proceed about finding the solution to this problem. $$\sum_{i=6}^8(\sum_{j=i}^8 (j+1)^2)$$</p>
<p>Obviously the last step is not to difficult, but the fact that the lower limit for the summation in brackets is i I am not sure how to solve this. In classes so far we have only really dealt with ... | Christopher Carl Heckman | 261,187 | <p>Since there are only three values of $i$, I'd rewrite this as
$$\sum\limits_{j=6}^8 (j+1)^2 + \sum\limits_{j=7}^8 (j+1)^2 + \sum\limits_{j=8}^8 (j+1)^2$$</p>
|
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