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 |
|---|---|---|---|---|
974,560 | <p>How do I prove that indefinite integral of $\sec x$ is equal to $\ln(\sec x + \tan x) + C$?</p>
<p>I tried to substitute $t = \cos x$ but that didn't help. I have no idea how to integrate it any other way, and my textbook doesn't offer a derivation.</p>
| hjhjhj57 | 150,361 | <p>Hint: Write $$\sec x = \sec x \left(\frac{\sec x + \tan x}{\sec x + \tan x}\right)$$ and remember what the derivatives of $\sec x$ and $\tan x$ are.</p>
|
2,135,151 | <p>Prove that, If $r$ is a real number such that $r^2 = 2$, $r$ is irrational.</p>
<hr>
<p><strong>Proposition:</strong> If $r$ is a real number such that $r^2 = 2$, then $r$ is irrational.</p>
<p><strong>Hypothesis:</strong> If $r$ is a real number such that $r^2 = 2$.</p>
<p><strong>Conclusion:</strong> $r$ is ir... | Palash gupta | 414,024 | <p>This proof is not valid because you are not proving that $\sqrt{2}$ is an irrational number.
The one given in your book is correct.
For your proof to be valid you must further prove that $\sqrt{2}$ is irrational which is again possible through contradiction method.</p>
|
2,135,151 | <p>Prove that, If $r$ is a real number such that $r^2 = 2$, $r$ is irrational.</p>
<hr>
<p><strong>Proposition:</strong> If $r$ is a real number such that $r^2 = 2$, then $r$ is irrational.</p>
<p><strong>Hypothesis:</strong> If $r$ is a real number such that $r^2 = 2$.</p>
<p><strong>Conclusion:</strong> $r$ is ir... | Arnaldo | 391,612 | <p>That is not valid! You are assuming that $\sqrt{2}$ is irrational.</p>
<p>What you have to do is show that $\sqrt{2}$ is irrational. </p>
<p>The most common way to do that is suppose that it is rational:</p>
<p>$$\sqrt{2}=\frac{p}{q} \Leftrightarrow p^2=2q^2$$</p>
<p>Let's take $\gcd(p,q)=1$. The above equation ... |
2,135,151 | <p>Prove that, If $r$ is a real number such that $r^2 = 2$, $r$ is irrational.</p>
<hr>
<p><strong>Proposition:</strong> If $r$ is a real number such that $r^2 = 2$, then $r$ is irrational.</p>
<p><strong>Hypothesis:</strong> If $r$ is a real number such that $r^2 = 2$.</p>
<p><strong>Conclusion:</strong> $r$ is ir... | Behemoth | 414,337 | <p>Proof by contradiction implies that you assume the opposite of what you propose, then prove that the new assumption is invalid, therefore the original proposition valid.</p>
<p>The only way to prove this is to assume that $\sqrt{2}$ is rational, then face a contradiction in the process.</p>
<p><strong>Assumption:<... |
844,403 | <p><img src="https://i.stack.imgur.com/iYn8g.png" alt=" Circumcircles, Incircles, Medians, and Altitudes" /></p>
<p>I tried to use the angle property by which AD=4 and DB=5,but since F is not given as mid point I don't know how to proceed to find length of DG.I think AED as 90 degree is important but I am unable to fig... | happymath | 129,901 | <p>angle FCE=$C/2$ also FEC=$C/2$ implies FC=EF and also FAE=FEA=$90-C/2$ So EF=AF implies F is the midpoint.</p>
|
387,542 | <p>e.g. The function $e^x$ reflected through $y=x$ is $\ln x$. Is this always true OR just in some cases?</p>
| Harald Hanche-Olsen | 23,290 | <p>Yes, but it is the <em>graph</em> of the function that is reflected, not the function itself.</p>
<p>The graph of a function $f$ is the set of all pairs $(x,y)$ with $y=f(x)$. If $f$ has an inverse function $g$, then $y=f(x)$ is equivalent to $x=g(y)$, so when $(x,y)$ belongs to the graph of $f$, then $(y,x)$ belon... |
2,306,250 | <p>When I'm trying to solve this system of equations:</p>
<p>$$\begin{aligned} k_1 \oplus k_2 = a \end{aligned}$$
$$\begin{aligned} k_2 \oplus k_3 = b \end{aligned}$$
$$\begin{aligned} k_3 \oplus k_1 = c \end{aligned}$$</p>
<p>I don't get any adequate result, except like $\begin{aligned} a \oplus c = b \end{align... | Sharat V Chandrasekhar | 400,967 | <p>In general, Idon't see why it wouldn't. If anything your $B$ matrix is like an initial perturbation that dies out. Now if it is stochastic and involves something like a Wiener process, then things could change. </p>
<p>Another potential issue could be with induced instabilities like those in the Lorenz equations th... |
2,091,761 | <p>If we define a set with $2$ elements in it $S=\{a,b\}$ and a variable "density" $d = 1$ here.</p>
<p>Then if we continue to expand the set with more elements relative to variable $d$ arithmetically, in such a way that:</p>
<p>$$(d=2) \to S= \{a, \frac{a+b}{2},b\}$$
$$(d=3) \to S= \{a, \frac{2a+b}{3}, \frac{a+2b}{3... | 5xum | 112,884 | <p>You ask "How can I disprove it", but you didn't really define a strict mathematical statement. Your statement</p>
<blockquote>
<p>As $d\to\infty$, $S=[a,b]$</p>
</blockquote>
<p>lacks definitions. You seem to imply that for a sequence of sets $A_1,A_2,\dots $, there exists a limit $$\lim_{n\to\infty} A_n$$
but l... |
196,460 | <p>Consider the matrix-valued function $f(A) = \frac{A}{\det(A)}$ on the set of $3\times 3$ positive-definite matrices. Is this function matrix-convex ? (i.e., is $tf(A) + (1-t)f(B) - f(tA+(1-t)B)$ positive semi-definite $\forall \ t \in [0,1]$?)</p>
| Suvrit | 8,430 | <p>Not just <span class="math-container">$3\times 3$</span>, but in general, the map <span class="math-container">$A \mapsto \det(A^{-1})A$</span> is operator convex on positive definite matrices.</p>
<p>Proof sketch.
<span class="math-container">$\newcommand{\pfrac}[2]{\left(\tfrac{#1}{#2}\right)}$</span>
If suffices... |
284,053 | <p>Some people have taken the view that morphisms in category theory ought to simply be called functions.</p>
<p>However, not all morphisms look like functions at first sight. For example, in categories of partial orders, such as the category in which the objects are the natural numbers and a morphism exists between t... | Michal R. Przybylek | 83,788 | <p>Let me offer another point of view.</p>
<p>I do not think that invoking Cayley's theorem may justify calling morphisms functions, because such a representation generally does not give a full subcategory of sets, and non-full subcategories rarely inherit interesting properties from they embedding category. It would ... |
1,074,740 | <p>We know that torsion-free plus finitely generated <span class="math-container">$\rightarrow$</span> free and that <span class="math-container">$\mathbf{Q}$</span> is torsion-free is easy. </p>
<blockquote>
<p>But how to show <span class="math-container">$\mathbf{Q}$</span> is not finitely generated and not free?<... | Pedro | 23,350 | <p>Show any finitely generated subgroup of $\mathbf Q$ is cyclic. Since $\mathbf Q$ is not cyclic, it cannot be finitely generated. </p>
<p>$(1)$ <a href="https://math.stackexchange.com/questions/1074740/how-to-show-mathbfq-is-not-free/1074741#comment2186256_1074741">It cannot be free</a>: it is not cyclic, so any put... |
2,085,664 | <p>Let $1$ be the multiplicative identity, so that $1\cdot a = a$ (where $a\in \mathbb{F})$. Let $0$ be the additive identity, so that $a+0=a$. Prove that $0\ne 1$. (Here we don't yet know that $0$ and $1$ must be unique, nor do we know that $0\cdot a = 0$).</p>
<p>My approach:</p>
<p>Suppose that $1=0$, then $a+1 = ... | fleablood | 280,126 | <p>If $0=1$ and all else about the field axioms hold then the field only has one element. If we assume a field has at least two elements we can prove $0\ne 1$.</p>
<p>We start by proving $0*a=0$ for all $a$:</p>
<p>$0*a +0*a = (0+0)*a=0*a $</p>
<p>$0*a+0*a+(-(0*a))=0*a + (-(0*a)) $</p>
<p>$0*a = 0$</p>
<p>Now if ... |
438,925 | <p>There are many statements in abstract algebra, often asked by beginners, which are just <em>too good to be true</em>. For example, if <span class="math-container">$N$</span> is a normal subgroup of a group <span class="math-container">$G$</span>, is <span class="math-container">$G/N$</span> isomorphic to a subgroup ... | Benjamin Steinberg | 15,934 | <p>Every element of a finite simple non-abelian group is a commutator. This is the positive solution to the Ore conjecture (see <a href="https://dx.doi.org/10.4171/JEMS/220" rel="noreferrer">Liebeck, O’Brien, Shalev, and Tiep - The Ore conjecture</a>) and uses the classification.</p>
|
438,925 | <p>There are many statements in abstract algebra, often asked by beginners, which are just <em>too good to be true</em>. For example, if <span class="math-container">$N$</span> is a normal subgroup of a group <span class="math-container">$G$</span>, is <span class="math-container">$G/N$</span> isomorphic to a subgroup ... | spin | 38,068 | <p>Let <span class="math-container">$G$</span> be a finite group and <span class="math-container">$n \mid |G|$</span>.</p>
<p>If <span class="math-container">$S = \{x \in G : x^n = 1\}$</span> contains exactly <span class="math-container">$n$</span> elements, then <span class="math-container">$S$</span> is a subgroup o... |
438,925 | <p>There are many statements in abstract algebra, often asked by beginners, which are just <em>too good to be true</em>. For example, if <span class="math-container">$N$</span> is a normal subgroup of a group <span class="math-container">$G$</span>, is <span class="math-container">$G/N$</span> isomorphic to a subgroup ... | HJRW | 1,463 | <p>Let <span class="math-container">$F$</span> be a non-abelian free group and let <span class="math-container">$G=\prod_\omega F$</span> be the direct product of infinitely many copies of <span class="math-container">$F$</span>. Then the abelianisation of <span class="math-container">$G$</span> has torsion (of order <... |
4,518,734 | <p>Is there a simple formula or distribution curve to answer a question like this?</p>
<p>Assume there are K buckets and we want to randomly assign N balls to them. Each ball has an equal chance of being assigned to any of the buckets. There are more balls than buckets. When all balls have been assigned, what is the li... | StupidDroid | 1,089,132 | <p>Let's start by looking at all the number of possible end states. I'm going to assume from the statement of your problem that the buckets themselves can be distinguished from one another, but that the balls cannot. So our final state can be represented simply by writing, in order, how many balls end up in each of the... |
2,626,597 | <p>In both my textbook (Hungerford's Algebra), and in class, it is claimed that Monoid Homomorphisms are not required to preserve the identity. Interestingly enough, the Wikipedia page for Monoids requires Monoid Homomorphisms to preserve the identity element: <a href="https://en.wikipedia.org/wiki/Monoid#Monoid_homomo... | 57Jimmy | 356,190 | <p>The problem is that you have only proved that $f(e_M)$ is an identity for the elements in the image of $f$, not for all the elements in $N$. This is also specified on Wikipedia. So in general, if you do not require it, it is not true that the identity is preserved. Here is a counterexample:</p>
<p>$$(\mathbb{R},*,1... |
20,540 | <p>How can I combine or separate sums in <em>Mathematica</em> in the way that <code>Together</code> or <code>Expand</code> work for rational expressions？</p>
<p>For example, how does one transform from </p>
<p>$$\text{Sum}\left[\frac{a}{\sqrt{n!}},\{n,0,\infty \}\right]+\text{Sum}\left[\frac{b}{\sqrt{n!}},\{n,0,\inft... | Xerxes | 5,406 | <p>If we don't mind using rules to merge the sums, here's one that can handle prefactors, different variables and limits (assuming the upper limits are always infinite):</p>
<pre><code>mergesums = (ca_. Sum[a_, an_] + cb_. Sum[b_, bn_]) :>
With[{sum = Simplify[
ca a + (cb b /. bn[[1]] -> an[[1]] + bn[[... |
2,719,542 | <blockquote>
<p>Suppose that
$$\int_{-1}^1 f(x)dx=5$$
$$\int_{1}^4 f(x)dx=-2$$
$$\int_{-1}^4 h(x)dx=7$$
Find the value of
$$\int_{-1}^4 (2f(x)+3h(x))dx$$</p>
</blockquote>
<p>I understand how to find definite and indefinite integrals, but I'm not entirely sure how to even begin this problem.</p>
| user | 505,767 | <p>Alternative way by Lagrange's multipliers</p>
<ul>
<li>$4y=8\lambda x$</li>
<li>$4x=18\lambda y$</li>
</ul>
<p>since $\lambda=0$, $y=0$, $x=0$ don't lead to any solution we can divide and obtain</p>
<ul>
<li>$\frac y x = \frac 49\frac x y \implies 9y^2=4x^2 \\\implies 8x^2=3600 \implies x^2=450\implies x=\pm15\sq... |
2,719,542 | <blockquote>
<p>Suppose that
$$\int_{-1}^1 f(x)dx=5$$
$$\int_{1}^4 f(x)dx=-2$$
$$\int_{-1}^4 h(x)dx=7$$
Find the value of
$$\int_{-1}^4 (2f(x)+3h(x))dx$$</p>
</blockquote>
<p>I understand how to find definite and indefinite integrals, but I'm not entirely sure how to even begin this problem.</p>
| mzp | 287,326 | <p>You seem to be going in the right direction. There is a mistake somewhere, but your algebra is a bit convoluted, so I cannot find it exactly. Here is what I got following your procedure:</p>
<p>Solve $4x^2+9y^2=3600$ for $y$ to get $$y=\frac23\sqrt{(30-x)(x+30)}.$$ Then, we want to maximize $$A=4xy=\frac{8x}{3}\sqr... |
836,841 | <p>Calculation of $\displaystyle \lim_{x\rightarrow 1}\frac{(1-x)\cdot(1-x^2)\cdot(1-x^3)\cdots (1-x^{2n})}{\{(1-x)\cdot(1-x^2)\cdot (1-x^3)\cdots(1-x^n)\}^2} = $</p>
<p><b>My Trial</b> After simplification, we get $$\displaystyle \lim_{x\rightarrow 1}\frac{(1-x^{n+1})\cdot(1-x^{n+2})\cdot(1-x^{n+3})\cdots(1-x^{2n})}{... | Community | -1 | <p>By L'Hospital's rule, </p>
<p>\begin{align*}
\lim_{x \to 1} \frac{1 - x^{n + k}}{1 - x^k} &= \lim_{x \to 1} \frac{-(n + k) x^{n + k}}{-k x^k} = \frac{n + k}{k}
\end{align*}</p>
<p>Hence the desired limit is</p>
<p>$$\frac{n + 1}{1} \cdot \frac{n + 2}{2} \cdot \frac{n + 3}{3} \cdots \frac{n + n}{n} = \frac{(2n... |
1,251,457 | <p>I am struggling with the concept of parameterizing curves. I am not even sure if I know what it means so I tried to look some things up.</p>
<p>On Wikipedia it says:</p>
<blockquote>
<p>Parametrization is... the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a va... | Christian Blatter | 1,303 | <p>Often a curve $\gamma$ in the plane is defined as the set of points $(x,y)$ satisfying a certain geometric or algebraic condition. An example is
$$\gamma:=\bigl\{(x,y)\>\bigm|\>{x^2\over a^2}+{y^2\over b^2}=1\bigr\}\ ,\tag{1}$$
whereby the values of $a>0$ and $b>0$ are given. Such a description is <em>im... |
208,485 | <p>Two years ago, I made a conjecture <a href="https://math.stackexchange.com/questions/334448/1-a21-b21-c2-8abc-a-b-c-in-mathbbq-has-infinitely-many-sol/336803#336803">on stackexchange</a>:</p>
<p>Today, I tried to find all solutions in integers $a,b,c$ to
$$(1-a^2)(1-b^2)(1-c^2)=8abc,\quad a,b,c\in \mathbb{Q}^{+}.$$... | Tito Piezas III | 12,905 | <p>I'm late for this party, but using math110's method employing <em>Euler bricks</em>, couldn't resist giving some simple rational solutions to,</p>
<p>$$(1-a^2)(1-b^2)(1-c^2) = 8abc$$</p>
<p><strong>Solution 1:</strong></p>
<p>$$a,\,b,\,c = \frac{-(x-z)(2x+z)}{(2x-z)y},\;\frac{z}{2x},\;\frac{-2y+z}{2y+z}\tag1$$</p... |
2,810,410 | <p>Define $B: \mathbb{R^3} × \mathbb{R^3} \to \mathbb{R}$ with $B((x_1,x_2,x_3),(y_1,y_2,y_3)) := -2x_1y_1-x_2y_3-x_3y_2$.</p>
<p>How to check if vectors $v \in \mathbb{R^3}, v \neq 0$ exist such that $B(v,v) = 0$?</p>
<p>I don't know how to start here.</p>
<p>Also, how to find a basis of $\mathbb{R^3}$ such that th... | drhab | 75,923 | <p>I do not guarantee that this hint will lead to results.</p>
<p>If $F_{n}$ denotes the CDF and $f_{n}$ the PDF of $X_{n}$ then
for $x\in\left[0,1\right]$ we find:</p>
<p>$\begin{aligned}F_{n+1}\left(x\right) & =\int_{0}^{x}P\left(X_{n+1}\leq x\mid X_{n}=y\right)f_{n}\left(y\right)dy+\int_{x}^{1}P\left(X_{n+1}\l... |
23,224 | <p>How to integrate $x^{1/2}e^{-x}$ using integration by parts?</p>
<p>Answer should be $\left(-\sqrt{x} e^{-x}+(1/2)\sqrt{\pi} \mbox{erf}(\sqrt{x})\right)+c$</p>
| Community | -1 | <p>I am not familiar with using <strong>e.r.f</strong> but i shall tell you the procedure. Take $u= \sqrt{x}$ and $dv = e^{-x}$. So you have $du = \frac{1}{2 \cdot \sqrt{x}} \ \rm{dx}$ and $v = -e^{-x}$. Now using the integration formula we have,</p>
<p>\begin{align*}
\int \sqrt{x} e^{-x} \ \textrm{dx} &= \Bigl[ ... |
2,604,377 | <p>Given an orientation-preserving homeomorphism of the circle $$f : S^1 \rightarrow S^1,$$ I want to define a homeomorphism of the cylinder $$F : S^1 \times \mathbb{I} \rightarrow S^1 \times \mathbb{I}$$ such that for all $x \in S^1$, we have: $F(x,0) = (f(x),0)$ and $F(x,1) = (x,1)$.</p>
<p>The same goes with 'homeo... | Moishe Kohan | 84,907 | <p>This is an addendum to Eric Wofsey's answer, regarding what happens in higher dimensions: Given an orientation-preserving homeomorphism (diffeomorphims) $f: S^n\to S^n$, is there a homeomorphism (diffeomorphism) $$F: S^n\times I\to S^n\times I$$
such that $F(x,0)=(f(x),0), F(x,1)=(x,1)$?</p>
<p>The answer in the s... |
88,034 | <p>V10.1:</p>
<pre><code><|a -> 1|>[Key @ a]
(* Missing["KeyAbsent", Key[a]]*)
</code></pre>
<p>V10.0:</p>
<pre><code><|a -> 1|>[Key @ a]
(* 1 *)
</code></pre>
<p>Bug or Design Change?</p>
<p><strong>Update:</strong>
This bug/design change remains in 10.2.</p>
<p>Even though my conclusions in a ... | Mr.Wizard | 121 | <p>I am going to merely <em>guess</em> that it is an intentional change as:</p>
<ol>
<li><p>there exist other ways to extract the same element, e.g.:</p>
<pre><code><|a -> 1|>[[Key[a]]]
Key[a] @ <|a -> 1|>
</code></pre></li>
<li><p>it is more general to allow verbatim <code>Key</code> within keys</... |
88,034 | <p>V10.1:</p>
<pre><code><|a -> 1|>[Key @ a]
(* Missing["KeyAbsent", Key[a]]*)
</code></pre>
<p>V10.0:</p>
<pre><code><|a -> 1|>[Key @ a]
(* 1 *)
</code></pre>
<p>Bug or Design Change?</p>
<p><strong>Update:</strong>
This bug/design change remains in 10.2.</p>
<p>Even though my conclusions in a ... | Szabolcs | 12 | <p>Only the designer could say if this is a bug or a bugfix. But, like others, I also consider this change an improvement. What follows is just personal opinion on why, backed up by observations.</p>
<p>How would you design associations if your goal was to allow any expression to be a key (useful as shown by <code>Gr... |
3,209,740 | <p>In an article the author consider the square matrix <span class="math-container">$(A^{ij}(x))_{i,j=1}^d$</span> defined by
<span class="math-container">$$
A^{ij}(x):= \frac{x_ix_j}{|x|^2} \qquad (x\in R^d).
$$</span>
He writes that the matrix is elliptic.
I don't know to prove that there exists <span class="math-co... | Hans Engler | 9,787 | <p>For any <span class="math-container">$x$</span>, the matrix is positive semidefinite, but not positive definite. This is because
<span class="math-container">$$
\sum_{i,j} A_{ij}(x)\xi_i \xi_j = \frac{|\langle x, \xi \rangle|^2}{|x|^2} \ge 0
$$</span>
with equality if <span class="math-container">$\langle x, \xi \... |
1,511,477 | <p>So I tried and found that $$7^{100} \equiv 1 \pmod{100}$$ but I got stuck with $8^{100}$. Help me out please. </p>
| Barry Cipra | 86,747 | <p>Consider the binomial expansion</p>
<p>$$(2+5)^{100}=2^{100}+100\cdot2^{99}\cdot5+{100\choose2}2^{98}\cdot5^2+\cdots+{100\choose98}2^2\cdot5^{98}+100\cdot2\cdot5^{99}+5^{100}$$</p>
<p>It's easy to see that all the interior terms are divisible by $2^2\cdot5^2=100$, hence</p>
<p>$$7^{100}=(2+5)^{100}\equiv2^{100}+5... |
107,882 | <p>Can someone recommend a good basic book on Geometry? Let me be more specific on what I am looking for. I'd like a book that starts with Euclid's definitions and postulates and goes on from there to prove thereoms about triangles, circles and other plane shapes. I'm not interested (at this time) in a book that tie... | Isaac | 72 | <p>"Geometry for Enjoyment and Challenge" by Rhoad, Milauskas, and Whipple (published by McDougal-Littell) is a fairly common proof-based high school honors geometry textbook, at least in the greater Chicago metropolitan area.</p>
|
600,874 | <p>The question is, find the limit of $\ a_n = (e^n+3^n)/5^n$ as $\ n→∞ $
I tried using L'hopital's rule, but didn't seem to get anything useful, so I figured I may be able to use the squeeze theorem. Would this be an appropriate use of the squeeze theorem, and would there be a better method to proving that the limit a... | Kode Charlie | 114,772 | <p>A double integral is a like a summation over two variables.
In your example, the first sum is taken over a variable x,
with y fixed. Let's call the resulting sum a function of y -- g(y).
The 2nd sum is over a variable y, with the summands or terms equal
to different values of g(y).</p>
|
600,874 | <p>The question is, find the limit of $\ a_n = (e^n+3^n)/5^n$ as $\ n→∞ $
I tried using L'hopital's rule, but didn't seem to get anything useful, so I figured I may be able to use the squeeze theorem. Would this be an appropriate use of the squeeze theorem, and would there be a better method to proving that the limit a... | Brian M. Scott | 12,042 | <p>The righthand expression means the same as $$\int_0^1\int_{\sqrt{y}}^1\frac1{2+x^3}dxdy\;,$$ a form that may be more familiar to you.</p>
|
3,786,127 | <p>Let <span class="math-container">$p$</span> be a prime. I am interested in the set of elements <span class="math-container">$x\in\mathbb{Z}/p\mathbb{Z}$</span> such that <span class="math-container">$x$</span> and <span class="math-container">$x+1$</span> are both quadratic non-residues. Let <span class="math-contai... | egreg | 62,967 | <p>There is no way to predict the value of <span class="math-container">$y(-\sqrt{3})$</span> by knowing the solution over <span class="math-container">$(1,\infty)$</span>.</p>
<p>The problem is ill posed. It is like asking for the value of <span class="math-container">$y(-1)$</span> if <span class="math-container">$y'... |
3,505,675 | <p>Proposition: Let <span class="math-container">$(X, d)$</span> be a metric space, and <span class="math-container">$S$</span> a subset of <span class="math-container">$X$</span>. If <span class="math-container">$B_d(x, r) \cap S \neq \emptyset$</span> for all <span class="math-container">$r >0$</span>, then there ... | Community | -1 | <p>For people who avoided clicking through to the article (which is well written and mostly uncontroversial and very appropriate for newbie math majors), this is what Dr. Lee suggested as a preferred style for the above proof (even though it has a mistake in the factored line):</p>
<blockquote>
<p><a href="https://i... |
3,505,675 | <p>Proposition: Let <span class="math-container">$(X, d)$</span> be a metric space, and <span class="math-container">$S$</span> a subset of <span class="math-container">$X$</span>. If <span class="math-container">$B_d(x, r) \cap S \neq \emptyset$</span> for all <span class="math-container">$r >0$</span>, then there ... | Ted Shifrin | 71,348 | <p>In my experience teaching university mathematics (admittedly in the US) for close to 40 years, the "two-column proof" style that grew out of high school geometry classes seems to have been responsible for making 99% of mathematics students hate proofs (and, in some cases, mathematics). Trying to belabor minutiae, in... |
47,011 | <p>This question was posed originally on <a href="https://math.stackexchange.com/questions/10990/uses-of-divergent-series-and-their-summation-values-in-mathematics">MSE</a>, I put it here because I didn't receive the answer(s) I wished to see.</p>
<p>Dear MO-Community,</p>
<p>When I was trying to find closed-form rep... | Hjalmar Rosengren | 10,846 | <p>It seems likely to me that your derivation can be modified to just involve convergent series and products, but I have not tried it.</p>
<p>There is however a well-known easy derivation of your identity
$\zeta(3) = - (1/2) \psi^{(2)} (1)$ from the product formula, which is quite close to what you are doing. Note tha... |
849,619 | <p>When we draw a Venn diagram, we use circle to represent a Set. We can use any closed plane figure but most of the time it is a circle. Why? are there any specialty about that?</p>
| Emily | 31,475 | <p>Circles are easy to draw and conceptually very nice. There is no requirement for anything other than a circle for Venn diagrams with $n \le 3$ classes. Also, the boundaries of two circles intersect each other at not more than two places, which is an important aspect that has connections to some deeper results in mat... |
1,245,499 | <p>In the proof we say $\left\{\left(\frac1n,1\right):n\geq 1\right\}$ is an infinite cover with no finite subcover.</p>
<p>But, $(0,1)$ set also belongs to cover mentioned above.
We can say $\{(0,1)\}$ is a subcover of mentioned above cover.</p>
<p>I am not able to understand what I am doing wrong.</p>
| Aman Pandey | 469,000 | <p>Consider the open cover <span class="math-container">$C=\Big\{(\frac{1}{n},1):n\geq 2\Big\}$</span>,. Since every <span class="math-container">$x\in (0,1)$</span> has <span class="math-container">$x>\frac{1}{n}$</span> for some <span class="math-container">$n$</span>. But there is no finite subcover; if <span cl... |
287,154 | <p>Note: I asked the question below last week on MathSE but received no answer. </p>
<p>Background:</p>
<p>I have read the claim that perverse sheaves behave more like sheaves than like complexes of sheaves. This refers to the fact that they can be glued. </p>
<p>For instance, suppose that $X$ is a complex analyti... | Zhaoting Wei | 24,965 | <p>I'm not sure if you are still interested in this question. Actually for an open cover <span class="math-container">$\{U_i\}$</span> and complexes of sheaves on each <span class="math-container">$U_i$</span>, we could give the "higher" descent data and "higher" cocycle conditions in terms of <em>twisted complexes</em... |
1,676,427 | <p>I was asked to use Romberg integration to evaluate the integral$$\int_0^1x^{-x}dx=\sum_{n=1}^\infty n^{-n}$$ and compare the result with the result I get from the sum. And I also need to estimate how many function evaluation Romberg integration will require to achieve 12 digit accuracy. I looked it up on Wikipedia b... | marty cohen | 13,079 | <p>Romberg integration involves
successive subdivisions
of the region of integration
combined with extrapolation
(from step $h$ to step $h/2$)
to estimate the integral
and get an approximation
to the error.</p>
<p>It is usually done using
a canned routine,
but the formulas are simple enough
that you should be able to
... |
28,562 | <p>In the course of my research I have come across the following integral:</p>
<p>$\int_{0}^{\infty} e^{- \Lambda \sqrt{(z^2+a)^2+b^2}}\mathrm{d}z$.</p>
<p>This initially looks like it should be solvable by some suitable change of variable which will allow you to get it into a gaussian form. Unfortunately after tryi... | Harry Peter | 83,346 | <p>When <span class="math-container">$a=0$</span> , <span class="math-container">$b$</span> is a real number,</p>
<p>Then <span class="math-container">$\int_0^\infty e^{-\Lambda\sqrt{(z^2+a)^2+b^2}}~dz$</span></p>
<p><span class="math-container">$=\int_0^\infty e^{-\Lambda\sqrt{z^4+b^2}}~dz$</span></p>
<p><span clas... |
4,421,862 | <p>I am trying to prove that <span class="math-container">$\frac{d}{dx}x^e=ex^{e-1}$</span>.</p>
<p><span class="math-container">$\displaystyle \frac{d}{dx}x^e=\lim_{h\to 0}\dfrac{(x+h)^e-x^e}{h}=\lim_{h\to 0}x^e \cdot \dfrac {e^{e \ln \left({1 + \frac h x} \right)} - 1} {e \ln {\left(1 + \dfrac h x\right)} } \cdot \... | Obinna Nwakwue | 307,490 | <p>This is a simple application of the power rule, i.e. <span class="math-container">$$\frac{d}{dx} x^n = nx^{n-1}$$</span>
However, if the question assumes that one doesn't know the power rule, then the limit definition is the only other way to prove it.</p>
|
2,132,994 | <p>I need to prove that absolute value of any real number is greater than or equal to that real number, where $|a| = a ; a\ge0 , |a| = -a ; a<0 $</p>
<p>I came across this on real analysis.
I need this proven Filed and Order Axioms and basic definitions.</p>
| JMP | 210,189 | <p>$$|a|-a=
\begin{cases}
0 & a\ge0\\
2|a| & a\le0
\end{cases}
$$</p>
<p>So $2|a|\ge|a|-a\ge 0$ and $|a|\ge-a$ from the left inequality, and $|a|\ge a$ from the right inequality.</p>
|
1,740,981 | <blockquote>
<p>Let $B$ be a group of all the continuous functions in the interval $[a,b]$ such that $f(a)=0$. Prove that $A$ is close group in the metric space $C[a,b]$</p>
</blockquote>
<p><strong>My attempt:</strong></p>
<p>Metric space $C[a,b]$ defined by $d(x(t),y(t))=\max\limits_{a\leqslant t\leqslant b} \mid... | Balloon | 280,308 | <p>Let $E_a:C([a,b])\to\mathbb{R}$ defined as $E_a(f)=f(a).$ You can see that $E_a$ is linear :
$$E_a(f+g)=(f+g)(a)=f(a)+g(a)=E_a(f)+E_a(g),E_a(\lambda f)=(\lambda f)(a)=\lambda f(a)=\lambda E_a(f)$$ and that $E_a$ is continuous :
$$|E(f)|=|f(a)|\leq ||f||_{\infty},$$
where $||f||_\infty=\max\limits_{x\in[a,b]}|f(x)|$... |
3,452,030 | <p>Let <span class="math-container">$m, n \in N$</span> and <span class="math-container">$a, b \in Z$</span> so that <span class="math-container">$ m\mid n$</span> and <span class="math-container">$a \equiv b \mod n.$</span> Then
<span class="math-container">$a \equiv b \mod m$</span></p>
| EDS | 513,888 | <p>To say <span class="math-container">$m|n$</span> means there exists some natural number <span class="math-container">$x$</span> such that <span class="math-container">$mx=n$</span>.</p>
<p>To say <span class="math-container">$a \equiv b$</span> <span class="math-container">$mod n$</span> means that <span class="mat... |
399,135 | <p>I need to compute the line integral for the vector $\vec{F} = \langle x^2,xy\rangle$, for the curve specified: part of circle $x^2+y^2=9$ with $x \le0,y \ge 0$,oriented clockwise.</p>
<p>Once again, I'm stuck at the setup (this happens a lot with me). I know that I need to parameterize F, but how would I go about ... | Mikasa | 8,581 | <p>We have $$F(x,y)=x^2\textbf{i}+xy\textbf{j}$$ and $$C: x^2+y^2=1, x\le0, y\ge0$$ and want to evaluate $$\oint_CF\cdot dr=\int_{\pi}^{\pi/2} F(\cos t,\sin t)\cdot(-\sin t,\cos t)dt=\int_{\pi}^{\pi/2}(\cos^2 t,\sin t\cos t)\cdot(-\sin t,\cos t)dt\\\\\\ =\int_{\pi}^{\pi/2}(-\sin t\cos^2 t+\sin t\cos^2 t)dt=0$$</p>
<p>... |
1,480,010 | <p>Express $A \cup B$ in set buildier notation when $A=\{2n+1 \mid n \in \mathbb{Z}\}$ and $B=\{3n+2 |n \in \mathbb{Z}\}$</p>
<p>We know that the rooster notation version of the union set is $A \cup B = \{1,2,3,5,7,8,9,11,13,14,15,17,19,20,21,\ldots\}$</p>
<p>It is a combination of every odd integer and every multipl... | Vim | 191,404 | <p>Assuming you are talking about differentiation operator, consider
$$f_n=x^n\in C^1[0,1],\,n=1,2,3\cdots$$
Using the usual sup norm, then $|f_n|\equiv 1$ but
$$|Tf_n|=|nx^{n-1}|=n$$
which is obviously unbounded. </p>
|
2,242,634 | <ol>
<li>Let $\mathcal{P}$ be the plane containing the points $(-3,4,-2)$, $(1,4,0)$, and $(3,2,-1)$.
Find the point in this plane that is closest to $(0,3,-1)$.</li>
</ol>
<hr>
<ol start="2">
<li>Let $\mathcal{P}$ be the plane containing the points $(-3,4,-2)$, $(1,4,0)$, and $(3,2,-1)$.
Let $\ell$ be the line conta... | PTDS | 277,299 | <p>Problem 1.</p>
<p>The equation of the plane containing the points <span class="math-container">$(−3,4,−2)$</span>, <span class="math-container">$(1,4,0)$</span> and <span class="math-container">$(3,2,−1)$</span> is given by
<span class="math-container">\begin{align*}
\begin{vmatrix}
x & y & z & 1 \\
-... |
2,654,606 | <p>In Chapter 7 ("Inverse limits and direct limits"), subchapter 4 ("Conditions for an inverse limit to be non-empty"), Bourbaki lets $(E_\alpha)_{\alpha \in I}$ be a projective system of sets with connecting maps $(f_{\alpha \beta}) _{\alpha, \beta \in I}$, and for each $\alpha \in I$ he lets $\mathfrak S_\alpha$ be a... | Community | -1 | <p>This is one of those statements which are true vacuously:</p>
<ul>
<li>The empty intersection of subsets of a set $A$ is the whole $A$.</li>
<li>The empty union of subsets of $A$ is an empty set.</li>
</ul>
<p>In other words: if $\emptyset=\mathcal F\subseteq\mathcal P(A)$ is the empty family of subsets of $A$, th... |
2,870,910 | <blockquote>
<p>I wish to show that if $z$ is real, then
$$\left|\frac{e^{iz}}{z^2+1}\right|\leq\frac{1}{|z|^2+1}$$</p>
</blockquote>
<p>I have shown this result, although my inequality is the wrong way around.</p>
<p>I considered
\begin{align}
|z^2+1|&\leq |z^2|+|1| \ \ \ \ \ \ \ \text{(triangle inequality)... | Angina Seng | 436,618 | <p>If $z$ is real: $|e^{iz}|=1$, $|z^2+1|=z^2+1=|z|^2+1$, so that
$$\left|\frac{e^{iz}}{z^2+1}\right|=\frac{1}{|z|^2+1}.$$</p>
<p>If $z=x+iy$ is complex, with $y$ a large negative number, then $|e^{iz}|
=e^{-y}$ is huge, and so
$$\left|\frac{e^{iz}}{z^2+1}\right|\gg\frac{1}{|z|^2+1}.$$</p>
|
1,722,628 | <p>I assume that I need to use the theorem that states that the sum of the degrees of the vertices is equal to twice the number of edges. Then, because $k$ must be greater than or equal to 3, the there must be 1.5 edges, but that is impossible. How else can I continue this proof, using the stated theorem?</p>
| Alex Kruckman | 7,062 | <p>There <em>is</em> a simple necessary and sufficient condition for the existence of a prime model: "isolated types are dense". If $T$ is a complete theory in a countable language, then </p>
<ol>
<li>If $M\models T$, then $M$ is prime if and only if $M$ is countable and atomic (meaning that $M$ realizes only isolated... |
2,964,503 | <p>Let <span class="math-container">$U_n = 1 + p^n \mathbb{Z}_p = \{1+p^n x \mid x \in \mathbb{Z}_p\}$</span> for <span class="math-container">$n \in \mathbb{Z}_{\geq 1}$</span>, where <span class="math-container">$U_0 = \mathbb{Z}_p^{*}$</span>. I have two questions: I managed to show that <span class="math-container"... | nguyen quang do | 300,700 | <p>@reuns </p>
<p>Here is a proof of <span class="math-container">$\mathbf Z^*_p = <\omega> \times U_1$</span> without Hensel's lemma (of course it's academic, because why shouldn't we use Hensel's lemma) ?</p>
<p><strong><em>Algebraic lemma</em></strong>: Let <span class="math-container">$0\to A\to B\to C\to 0... |
11,070 | <p>I am a game developer currently making a game based on basic arithmetic problems for kids and teens. </p>
<p>Are there any research papers on the average time to solve basic arithmetic problems or on how to evaluate success and improvement rate. </p>
<p>That would help us a lot in creation of this game! </p>
<p>E... | JTP - Apologise to Monica | 64 | <p>At the risk of posting an answer that doesn't directly answer the question, I'm going to suggest that as a game or computer based software, you consider levels. Start with, say, 10 seconds per problem, and at each level, retire the time a bit, .5 seconds or so, to produce the number of levels you wish. </p>
<p>Bett... |
11,070 | <p>I am a game developer currently making a game based on basic arithmetic problems for kids and teens. </p>
<p>Are there any research papers on the average time to solve basic arithmetic problems or on how to evaluate success and improvement rate. </p>
<p>That would help us a lot in creation of this game! </p>
<p>E... | Marian Minar | 6,845 | <p>If you would like to create a game that adapts to its user's skill level, then the best way to go is to research <strong>Computerized Adaptive Assessment</strong> and <strong>Item Response Theory</strong>. </p>
<p>As a starting point for this, you would need to create a database of questions that are graded (i.e. i... |
13,649 | <p>I've been exposed to various problems involving infinite circuits but never seen an extensive treatment on the subject. The main problem I am referring to is</p>
<blockquote>
<p>Given a lattice L, we turn it into a circuit by placing a unit resistance in each edge. We would like to calculate the effective resista... | Steve Huntsman | 1,847 | <p><a href="http://www.google.com/search?hl=en&q=%22random+walks+and+electrical+networks%22" rel="nofollow">A Google Scholar search for "random walks and electrical networks"</a> will bring up <a href="http://arxiv.org/abs/math.PR/0001057" rel="nofollow">a text by Doyle and Snell that is now available online</a>; f... |
13,649 | <p>I've been exposed to various problems involving infinite circuits but never seen an extensive treatment on the subject. The main problem I am referring to is</p>
<blockquote>
<p>Given a lattice L, we turn it into a circuit by placing a unit resistance in each edge. We would like to calculate the effective resista... | Agelos | 26,286 | <p>If you are still interested in this, you may want to have a look in Section 6 of <a href="http://www.sciencedirect.com/science/article/pii/0095895690900658" rel="nofollow">http://www.sciencedirect.com/science/article/pii/0095895690900658</a> by Thomassen. He proves for example that the effective resistance between a... |
1,710,469 | <p>If $p_1 = 2$ and $p_{n+1} = \frac{p_n}{2}+ \frac{1}{p_n}$, determine $p_n$ is decreasing or increasing.</p>
<p>Here are the first few terms:
$$p_2 = \frac{3}{2}, p_3 = \frac{3}{4} + \frac{2}{3} = \frac{17}{12}, p_4 = \frac{17}{24} + \frac{12}{17} = \frac{577}{408}$$</p>
<p>The sequence seems decreasing to me so I ... | almagest | 172,006 | <p>AM/GM. $\frac{x}{2}+\frac{1}{x}<x$ iff $x^2>2$. So it is enough to show that if $x^2>2$ then $(\frac{x}{2}+\frac{1}{x})^2>2$ or $\frac{x^2}{4}+\frac{1}{x^2}>1$. But by AM/GM $(\frac{x^2}{4}+\frac{1}{x^2})/2>\sqrt{\frac{1}{4}}$.</p>
|
1,710,469 | <p>If $p_1 = 2$ and $p_{n+1} = \frac{p_n}{2}+ \frac{1}{p_n}$, determine $p_n$ is decreasing or increasing.</p>
<p>Here are the first few terms:
$$p_2 = \frac{3}{2}, p_3 = \frac{3}{4} + \frac{2}{3} = \frac{17}{12}, p_4 = \frac{17}{24} + \frac{12}{17} = \frac{577}{408}$$</p>
<p>The sequence seems decreasing to me so I ... | Tryss | 216,059 | <p>You have that $p_{n+1} - p_{n} = \frac{1}{p_n}-\frac{p_n}{2}$</p>
<p>So $p_n$ is decreasing if $ \forall n, \frac{1}{p_n} -\frac{p_n}{2} \leq 0$</p>
<p>And this is true if $\forall n, p_n^2 -2 \geq 0$, ie. $p_n > \sqrt{2}$ (as $p_n > 0$ )</p>
<p>You can prove this by induction. First, let's study the functi... |
618,068 | <p>I was wondering if it's erroneous to use the quadratic formula on a quadratic equation where there is no constant term. What I figured I'd try was to just assome the constant term is +0.</p>
<p>I was doing a trigonometric equation, which looks like this:</p>
<ul>
<li>$2\cos^2 x - 3\sqrt 3 \cos x = 0$</li>
</ul>
<... | amWhy | 9,003 | <p>That is correct, you can use the quadratic formula for $c = 0$. And your work is all fine.</p>
<p>But note, you can save yourself time by simply factoring your equaton, and noting that when once has factors $a, b$, then $$a b = 0 \;\text{ if and only if } \;a= 0 \;\text{ or }\;b = 0$$</p>
<hr>
<p>$$\begin{align} ... |
618,068 | <p>I was wondering if it's erroneous to use the quadratic formula on a quadratic equation where there is no constant term. What I figured I'd try was to just assome the constant term is +0.</p>
<p>I was doing a trigonometric equation, which looks like this:</p>
<ul>
<li>$2\cos^2 x - 3\sqrt 3 \cos x = 0$</li>
</ul>
<... | Bill Dubuque | 242 | <p>Yes, you can <em>mechanically</em> apply the quadratic formula when $\,c=0,\,$ so the trinomial $\,ax^2+bx+c\,$ degenerates to the binomial $\,ax^2+bx.\,$ But that is not a very inefficient way to proceed because binomials are easily solved by <em>factoring</em>, viz. $\,ax^2+bx = (ax+b)x$.</p>
<p>Why does the quad... |
4,203,431 | <p>Can I infer <span class="math-container">$AB=C$</span> from <span class="math-container">$AB \vec r = C \vec r$</span>? where <span class="math-container">$\vec r$</span> is <span class="math-container">$n \times 1$</span> vector(<span class="math-container">$n$</span> rows and <span class="math-container">$1$</span... | Michael Seifert | 248,639 | <p>Strictly speaking, all that the statement <span class="math-container">$(AB)\vec{r} = C \vec{r}$</span> allows you to say is that <span class="math-container">$\vec{r}$</span> is in the null space of <span class="math-container">$AB - C$</span> (since the above equation implies that <span class="math-container">$(AB... |
222,849 | <p>If $|G|=p^rm$ with $(p,m)=1$, suppose that $x\in G$ is an element such that $o(x)=p^{r_1}m_1$ with $r_1>0$ and $(m_1,p)=1$. I dont understand why exist $a,b\in G$ such that: </p>
<p>1) $a$ has order a power of $p$</p>
<p>2) $b$ has order coprime with $p$</p>
<p>3) $x=ab$ and $[a,b]=1$</p>
<p>This fact is of... | someone | 47,724 | <p>Write the cyclic group $\langle x \rangle$ as direct product of a $p$-subgroup and a $p'$-subgroup, and take the two projections of $x$ into the two factors.</p>
|
1,841,644 | <p>I have a square matrix called A. How can I find $A ^ {-1/2}$. Should I compute $a_{ij} ^ {-1/2}$ for all of its elements?</p>
<p>Thanks</p>
| Junning Li | 350,065 | <p>Eigen-decomposition: $A = U * \Lambda * U^t$.</p>
<p>$A^{-1/2} = U * \Lambda^{-1/2} * U^t$.</p>
<p>This method is not a very good for numerical computation.</p>
|
4,149,478 | <p><span class="math-container">$$y'' - y' = \frac{2-x}{x^3}e^x$$</span></p>
<p>The solution of the homogenous equation is <span class="math-container">$C_1 + C_2e^x$</span>.</p>
<p>Now, onto the variation of parameters:</p>
<p>In this case, the Wronskian would simply be <span class="math-container">$e^x$</span>. There... | user577215664 | 475,762 | <p><span class="math-container">$$y'' - y' = \frac{2-x}{x^3}e^x$$</span>
Rewrite the DE as:
<span class="math-container">$$(y'e^{-x} )' = \dfrac{2}{x^3}-\dfrac 1 {x^2}$$</span>
Then integrate both sides.
<span class="math-container">$$y'=e^{x} \left ( -\dfrac{1}{x^2}+\dfrac 1 {x}+C \right)$$</span></p>
<p><span class="... |
64,716 | <p>We know that by using Stirling approximation:
$\log n! \approx n \log n$</p>
<p>So how to approximate $\log {m \choose n}$?</p>
| Dan Brumleve | 1,284 | <p>Using $\log(n!) ≈ n \log(n)$ and the <a href="http://en.wikipedia.org/wiki/Binomial_coefficient#Factorial_formula">definition of the binomial coefficient</a>,
$\log{m \choose n} ≈ m \log{m} - (m-n) \log{(m-n)} - n \log{n}$. The same should work for any of the more precise statements of <a href="http://en.wikipedia.... |
6,712 | <p>I had received the "warmth" of an angry user which decided to downvote no less than eight of my questions within the span of a minute.</p>
<p>I know who the user is and I can prove their identity beyond reasonable doubt.</p>
<p>Surely the software will catch the serial voting by tomorrow and reverse it, but I was ... | CodeMed | 39,744 | <p>Feel compassion for the idiot who has nothing better to do than try to take a few points away from you? </p>
<p>My understanding is that down-voting only takes away a measly two points each time, which is much less than the number of points awarded for an accepted answer, or even for a "+1". So, despite this pers... |
1,180,743 | <p>I need help finding out the basis in the following question :</p>
<blockquote>
<p>Let $~~W=\big<[1~~2~~1~~0~~1]^t~,[1~~0~~1~~1~~1]^t~,[1~~2~~1~~3~~1]^t\big >~$ be a subspace of $\mathbb R^5$ . Find a basis of $\mathbb R^5/W.$ </p>
</blockquote>
<p>I can't figure out the basis , kindly help with some hint... | Bernard | 202,857 | <p>Row reduction of the matrix:
$$\begin{bmatrix}
1&1&1&x\\
2&0&2&y\\
1&1&1&z\\
0&1&3&t\\
1&1&1&u\end{bmatrix}
\rightsquigarrow
\begin{bmatrix}
1&1&1&x\\
0&1&3&t\\
0&0&6&y-2x+2t\\
0&0&0&z-x\\
0&0&0&u-... |
1,690,210 | <p>What is $$\int \frac{4t}{1-t^4}dt$$ is there some kind of substitution which might help .Note that here $t=\tan(\theta)$</p>
| Enrico M. | 266,764 | <p>Well aside the $4$ constant, you have</p>
<p>$$\int\frac{t}{1 - t^4}\ \text{d}t$$</p>
<p>Use</p>
<p>$y = t^2$ so $\text{d}y = 2t\ \text{d}t$ </p>
<p>so</p>
<p>$$2\int \frac{1}{1 - y^2}\ \text{d}y = 2\ \text{arctanh}(y) ~~~ \to ~~~ 2\ \text{arctanh}(t^2)$$</p>
|
1,690,210 | <p>What is $$\int \frac{4t}{1-t^4}dt$$ is there some kind of substitution which might help .Note that here $t=\tan(\theta)$</p>
| GoodDeeds | 307,825 | <p>$$I=\int \frac{4t}{1-t^4}dt$$
Let $t=\sqrt{\sin\theta}$. Then, $dt=\frac{1}{2\sqrt{\sin\theta}}\cos\theta d\theta$.
Then,
$$I=\int \frac{4\sqrt{\sin\theta}\cos\theta d\theta}{2\sqrt{\sin\theta}\cos^2\theta}=2\int\sec\theta d\theta=2\ln|\sec\theta+\tan\theta|+c=2\ln\left|\frac1{\cos\theta}+\frac{\sin\theta}{\cos\thet... |
1,690,210 | <p>What is $$\int \frac{4t}{1-t^4}dt$$ is there some kind of substitution which might help .Note that here $t=\tan(\theta)$</p>
| Mark Fischler | 150,362 | <p>The useful technique to find a good substitution is to let $t = f(u;a)$ where the function $f$ is of some simple form but contains a parameter $a$. In this case, the natural thing to try is to let
$$
t = u^a.
$$
Then
$$
\int \frac{4t}{1-t^4}dt = \int \frac{4u^a}{1-u^{4a}} au^{a-1} du = \int\frac{4au^{2a-1}}{1-u^{... |
23,454 | <p>In an exercise asking to mark true or false, it shows:</p>
<p>$$\frac{1}{a/x-b/x}=\frac{1}{a-b}$$</p>
<p>It really look like <strong>false</strong> to me. But the answer is <strong>true</strong>! How can it be?</p>
| Ross Millikan | 1,827 | <p>Are you sure it is not $\frac{1}{a/x-b/x}=\frac{x}{a-b}$? Otherwise I agree with you.</p>
|
125,630 | <p>Let $L$ be a line bundle on an (algebraic) K3 surface over a field $k$. The Riemann-Roch theorem specializes to </p>
<p>$$
\chi(X, L)=\frac{1}{2}(L\cdot L)+2
$$ </p>
<p>which can be rewritten as
$$
h^0(X, L)+h^0(X, L^\ast)=\frac{1}{2}(L\cdot L)+2+h^1(X, L)
$$</p>
<p>(I use Serre's duality to identify $H^2(X, L)... | Francesco Polizzi | 7,460 | <p>Since the canonical class of $X$ is trivial, by adjunction one has $$2p_a(L)-2 = L^2,$$ where $p_a$ denotes the arithmetic genus.</p>
<p>Now assume that $L=\mathcal{O}_X(C)$, where $C$ is an effective curve. If $C$ is connected and reduced then $p_a(C) \geq 0$, so $L^2 < -2$ implies that $C$ is either non-reduce... |
2,330,438 | <blockquote>
<p>$$x\frac{dy}{dx}=x^2 +y$$ </p>
</blockquote>
<p>given that $\\ y\left( 1 \right) =0$</p>
<p>When i got partial derivatives of both sides, found it's not an exact equation..please can anybody can give a clue to solve this..</p>
| Teh Rod | 389,818 | <p>$x\frac{dy}{dx}-y=x^2$. Divide both sides by $x$ to get $y'-\frac{y}{x}=x$, the integrating factor is $e^{\int-\frac{1}{x}\,\mathrm{d}x}=\frac{1}{x}$. Our differential equation then becomes $\left(\frac{y}{x}\right)'=1\implies\frac{y}{x}=x+C\implies\boxed{y=x^2+Cx}$ solving for the constant gives us $C=-1$</p>
|
740,154 | <p>I'm trying to prove this sequence: $a_n = \sqrt{n}-\sqrt{n^2-1}$ to be divergent. How would I do this? I'm thinking of proving that it's not bounded below, but I'm not sure how to do that with induction, as I've only done that to prove it's bounded.</p>
| Clement C. | 75,808 | <p>$$
a_n = \sqrt{n} - \sqrt{n^2-1} = \sqrt{n} - n\sqrt{1-\frac{1}{n^2}} = \sqrt{n}\underbrace{\left( 1-\sqrt{n}\sqrt{1-\frac{1}{n^2}}\right)}_{b_n}
$$
Now, $\sqrt{n}\xrightarrow[n\to\infty]{}\infty$; and $\sqrt{1-\frac{1}{n^2}}\xrightarrow[n\to\infty]{}1$, so $\sqrt{n}\sqrt{1-\frac{1}{n^2}}\xrightarrow[n\to\infty]{}+\... |
1,683,375 | <blockquote>
<p>If Salvatore has achieved the test and Carmela has achieved the
test,then Benedetto also has achieved the test.But Salvatore didn't
achieve the test.So:</p>
<p>A)Benedetto didn't achieve the test.</p>
<p>B)Benedetto could have achieved the test.</p>
<p>C)Benedetto or Carmela didn't ... | Ross Millikan | 1,827 | <p>The correct answer is B, not E. Your hypotheses are $(S \wedge C) \to B, \lnot S$ Having $S$ be false makes the antecedent of the implication false, so the implication is true regardless of the truth of $C$ or $B$.</p>
|
3,816,808 | <p>I was solving some functions problems and those exercises asked for stating the domain and range of the functions. In this process, I had my doubts about the function notation. I would like something to relate the Domain and Range. Considering the function <span class="math-container">$f$</span> I've seen notations ... | Mark S. | 26,369 | <p>It depends on context.</p>
<p>When you are first learning to find the range of an expression like <span class="math-container">$(4-t^2)/(2-t)$</span>, then it is likely that no one is focused on the codomain. In that case, then you could write something like "We may consider <span class="math-container">$f$</sp... |
3,012,042 | <p>We learned expression of deduce, i.e. =>.</p>
<p>But now I dont have capable reason for I agree I represent True if assumption is False.</p>
<p>Anybady there having to explain reason for its deduce?</p>
<p>Best regards,</p>
| Mark Bennet | 2,906 | <p>Note that <span class="math-container">$3^2=11-2$</span></p>
<p>Then <span class="math-container">$$3^{10}=(11-2)^5= 11^5- \dots +\binom 51\times 11\times 2^4-2^5\equiv 880-32 \bmod 121$$</span></p>
<p>using the binomial expansion, and simply <span class="math-container">$848=7\times 121 +1$</span></p>
<p>Other s... |
2,469,841 | <p>I wonder why this is true</p>
<p>$$ \sum_{m,n = - \infty}^{\infty} \frac{(-1)^m}{m^2 + 58 n^2} = - \frac{\pi \ln( 27 + 5 \sqrt {29})}{\sqrt {58}} $$</p>
<p>Where the sum omits the case $n = m = 0$ ofcourse.</p>
| Paramanand Singh | 72,031 | <p>Finally I managed to sum this series using <a href="http://paramanands.blogspot.com/2012/03/ramanujans-class-invariants.html" rel="nofollow noreferrer">Ramanujan's class invariants</a>. We have the definition $$g(q) = 2^{-1/4}q^{-1/24}\prod_{n = 1}^{\infty}(1 - q^{2n - 1}), \, g_{p} = g(e^{-\pi\sqrt{p}})\tag{1}$$ Ra... |
1,372,558 | <p>$y=\sqrt{x^x}$</p>
<p>How do I convert this into a form that is workable and what indicates that I should do so? </p>
<p>Anyway, I tried this method of logging both sides of the equation but I don't know if I am right.</p>
<p>$\ln\ y=\sqrt{x} \ln\ x$</p>
<p>$\frac{dy}{dx}\cdot \frac{1}{y}=\sqrt{x}\ \frac{1}{x} +... | sbares | 198,998 | <p>You made a few simple mistakes, most notably, you got the log wrong:</p>
<p>$$\ln y = \frac12 x \ln x$$</p>
<p>Differentiating then gives:</p>
<p>$$\frac{y'}{y}=\frac12\ln x+\frac12$$</p>
<p>And hence:</p>
<p>$$y'=y\left(\frac12\ln x +\frac12\right)=\frac12\sqrt{x^x}\left(\ln x+1\right)$$</p>
|
3,471,147 | <p>I was in need of a method to compute the intersection of two lines given two points along each line. While searching for such a method, I came across one on <a href="https://en.wikipedia.org/wiki/Line%E2%80%93line_intersection#Given_two_points_on_each_line" rel="nofollow noreferrer">Wikipedia</a> that listed the res... | Ben Grossmann | 81,360 | <p>Here's an attempt at the (usually inadvisable) approach of reverse-engineering the systems of equations from the solutions. We have
<span class="math-container">$$
P_x =
\frac{\begin{vmatrix} \begin{vmatrix} x_1 & y_1\\x_2 & y_2\end{vmatrix} & \begin{vmatrix} x_1 & 1\\x_2 & 1\end{vmatrix} \\\... |
519,571 | <p>I would appreciate help, please, as to how to verify this relation from Kato's "Fermat's Dream" p.96.</p>
<p>He say: By the definition of $B_n(x)$, the Bernoulli polynomial, we have</p>
<p>$$\sum_{n=0}^{\infty}\frac{B_n(x)}{n!}u^n = \frac{u e^{xu}}{e^u - 1}$$</p>
<p>The definition for Bernoulli polynomials is, fo... | gammatester | 61,216 | <p>Remember that the method of generating functions works with formal power series. Using the generating series for the Bernoulli numbers and the exponential series we have:
$$\frac{u e^{xu}}{e^u - 1} = \frac{u}{e^u - 1} \cdot e^{xu}
= \left(\sum_{k=0}^{\infty}\frac{B_k}{k!} u^k\right)
\left(\sum_{k=0}^{\infty}\frac{... |
519,571 | <p>I would appreciate help, please, as to how to verify this relation from Kato's "Fermat's Dream" p.96.</p>
<p>He say: By the definition of $B_n(x)$, the Bernoulli polynomial, we have</p>
<p>$$\sum_{n=0}^{\infty}\frac{B_n(x)}{n!}u^n = \frac{u e^{xu}}{e^u - 1}$$</p>
<p>The definition for Bernoulli polynomials is, fo... | Slade | 33,433 | <p>I am going to basically steal gammatester's answer and say it backwards, which I think is a much more intuitive way to look at things.</p>
<p>You are trying to compute $G(u) = \sum_{n=0}^\infty \frac{B_n(x)}{n!}u^n$, which is exactly the exponential generating function of the sequence $\{B_0 (x), B_1 (x), B_2(x), \... |
2,732,220 | <p>A friend of mine asked me to help him evaluate the series</p>
<p>$$\mathcal{S} = \sum_{n=1}^{\infty} \frac{\sin (n \pi y) \sin \left ( n \pi x \right )}{n^2 \pi^2} \quad , \quad x , y \in (0, 1)$$</p>
<p>It does not ring any bells as to what it could be behind. The only thing I see is Fourier series and probably a... | Ron Gordon | 53,268 | <p>This sum can be evaluated explicitly using Parseval's theorem: given</p>
<p>$$A(w) = \sum_{n=-\infty}^{\infty} a_n \, e^{i n w} $$
$$B(w) = \sum_{n=-\infty}^{\infty} b_n \, e^{i n w} $$</p>
<p>Then</p>
<p>$$\sum_{n=-\infty}^{\infty} a_n \bar{b}_n = \frac1{2 \pi} \int_{-\pi}^{\pi} dw \, A(w) \bar{B}(w) $$</p>
<p>... |
53,001 | <p>Consider one of the standard methods used for defining the <a href="http://en.wikipedia.org/wiki/Riemann_integral" rel="noreferrer">Riemann integrals</a>: </p>
<blockquote>
<p>Suppose $\sigma$ denotes any subdivision $a=x_0<x_1<x_2\cdots<x_{n-1}<x_n=b$, and let $x_{i-1}\leq \xi_i\leq x_i$. Then if
... | Qiaochu Yuan | 232 | <p>It is the limit of a <a href="http://en.wikipedia.org/wiki/Net_(mathematics)">net</a>. Nets are a generalization of sequences which make all the familiar statements about sequences true for spaces that are not first-countable (for example a point lies in the closure of a subspace if and only if there is a net conver... |
53,001 | <p>Consider one of the standard methods used for defining the <a href="http://en.wikipedia.org/wiki/Riemann_integral" rel="noreferrer">Riemann integrals</a>: </p>
<blockquote>
<p>Suppose $\sigma$ denotes any subdivision $a=x_0<x_1<x_2\cdots<x_{n-1}<x_n=b$, and let $x_{i-1}\leq \xi_i\leq x_i$. Then if
... | Robert Israel | 8,508 | <p>It can be stated in terms of the ordinary definition of limit. Let $A(\sigma)$ and $B(\sigma)$ respecively be the supremum and infimum of $\sum_i f(\xi_i) (x_i - x_{i-1})$ over all subdivisions of "norm" $\sigma$ and all choices of the $\xi_i$. Then if
$\lim_{\sigma \to 0} A(\sigma) = \lim_{\sigma \to 0} B(\sigma... |
171,521 | <p>I have nested associations like this:</p>
<pre><code><|{"Country1","YEAR1"} -> {<|{"VARIABLE1", "MEASURE1"} ->
"Value1"|>, <|{"VARIABLE2", "MEASURE2"} -> "Value2"|>}, {"Country1",
"YEAR2"} -> {<|{"VARIABLE1", "MEASURE1"} ->
"Value1"|>, <|{"VARIABLE2", "MEASURE2"} -> "Va... | Carl Woll | 45,431 | <p>Fixing syntax errors and adding assumptions on your variables:</p>
<pre><code>Assuming[Y > 0 && Y1 > 0,
X = 2 π Y^2 Y1 Integrate[1/((Y^2+s)^2*Sqrt[(Y1^2+s)]), {s,0,∞}]
];
X //TeXForm
</code></pre>
<blockquote>
<p>$\frac{2 \pi Y \operatorname{Y1} \left(Y \cos ^{-1}\left(\frac{\operatorname{Y1}}... |
1,329,112 | <p>Given a measurable $E\subset \Bbb R^d $ and a measurable function $f:E\rightarrow \Bbb R^d $, prove that :</p>
<p>$$
\int (\left\lvert f \right\rvert)^r d\mu = r\int_{0}^\infty t^{r-1} \mu(\{x \in E \mid \left\lvert f(x) \right\rvert>t\})\,dt
$$<br>
where $r \ge 1$</p>
<p>The $r=1$ case is simple; howe... | copper.hat | 27,978 | <p>The basic idea is to use $t^r = r \int_0^t s^{r-1} ds$ combined with Fubini Tonelli.</p>
<p>Suppose $g$ is a non negative measurable function.</p>
<p>Then
\begin{eqnarray}
\int g(x)^r d \mu(x) &=& \int r \int_0^{g(x)}s^{r-1}ds d \mu(x) \\
&=& \int r \int_0^\infty 1_{[0,g(x))}(s)s^{r-1}ds d \mu(x) \... |
3,285,697 | <p>I have a matrix <span class="math-container">$B$</span> which it's dimension is <span class="math-container">$nm$</span> (with <span class="math-container">$n>m$</span>). During an iterative process I'll change it to get a desired state of matrix <span class="math-container">$B$</span>, but in each step, I should... | dcolazin | 654,562 | <p>You calculate <span class="math-container">$x$</span> as above, call <span class="math-container">$a = \sqrt{\frac{M}{\text{Trace}(x)}}$</span>. The new <span class="math-container">$B$</span> is <span class="math-container">$aB$</span>.</p>
|
4,371 | <p>I would like to learn Graph Theory from the beginning. It seems to me that one does not need to be familiar with many abstract type subjects to be able to understand the more basic concepts of graphs.</p>
<ol>
<li><p>Which subjects should one know prior to learn Graph Theory at the introductory level?</p></li>
<li>... | blaklaybul | 1,838 | <p>I am learning some graph theory myself as an independent study in college.</p>
<p>I started with a very simple, but informative text, Introductory Graph Theory by Chatrand. It is a Dover book, and can be bought for very cheap on Amazon.
I then went to my university library and took out Modern Graph Theory by Bollob... |
1,406,219 | <p>Hi I am reviewing partial derivatives.
For the question below, I am not sure why $(x-1)$ appears. Could anyone give me a explanation on this?</p>
<p>$y = x\sin(z)e^{-x}$ </p>
<p>$\partial y/\partial x = -e^{-x}(x-1)\sin(z)$</p>
| Mark Viola | 218,419 | <p>Let $f=u+iv$, where $u$ and $v$ satisfy the Cauchy-Riemann Equations and are harmonic. Then, straightforward use of the chain rule exposes that</p>
<p>$$\nabla^2 |f|^p=\frac p2\left(\frac p2 -1\right)|f|^{p-4}\left(\left(\frac{\partial |f|^2}{\partial x}\right)^2+\left(\frac{\partial |f|^2}{\partial y}\right)^2\ri... |
3,553,697 | <p>We all have seen and know of Euclid’s proof by contradiction, but I’m wondering if there is such thing as a direct proof? Also, the theorem isn’t posed as an “if-then” statement, so I can not even imagine how an alternative proof would be structured. </p>
| Henno Brandsma | 4,280 | <p>What Euclid's proof doesn't show is "there is a bijection between the set of primes and the set of integers", or some such more modern idea. But "infinite" (apeiron in Greek) means "without end" for him, so he shows that for every <em>finite</em> list of primes there is some prime <em>not</em> in that list. Any proo... |
2,863,265 | <p>Let a and b be elements of a group, with $a^2 = e, b^6 = e $ and $ ab =b^4 a $ . Find the order of ab, and
express its inverse in each of the forms $a^mb^n$ and $b^ma^n?$</p>
<p>Though it seems very simple I'm unable to find a power such that $(ab)^x =e$. Please help.</p>
<p>Im all confused with this problem.</p>... | 1ENİGMA1 | 255,913 | <p>Let be <span class="math-container">$|ab|=m$</span>. Then <span class="math-container">$(ab)^m=e$</span>.</p>
<p>Observe that <span class="math-container">$(ab)^3=(ab)(ab)(ab)=ab(b^4a)(ab)=ab^5a^2b=a$</span>. So</p>
<p><span class="math-container">$((ab)^{3})^{2}=a^2=e$</span>. So <span class="math-container">$|ab|=... |
20,634 | <p>From the <a href="https://en.wikipedia.org/wiki/Peter%E2%80%93Weyl_theorem" rel="nofollow noreferrer">Peter–Weyl theorem in Wikipedia</a>, this theorem applies for compact group. I wonder whether there is a non-compact version for this theorem.</p>
<p>I suspect it because the proof of the Peter–Weyl theorem heavily ... | Mariano Suárez-Álvarez | 1,409 | <p>The conclusion of the part of the Peter-Weyl theorem stating that unitary representations of the group split as an orthogonal direct sum of finite dimensional ones is false if the group is not compact. There are even non-compact groups which simply do not admit <em>any</em> unitary finite dimensional representations... |
2,982,942 | <p>Our professor gave us definitions for closed and open intervals. </p>
<p>A set <span class="math-container">$U$</span> is open if <span class="math-container">$\forall x \in U$</span>, <span class="math-container">$\exists \epsilon \gt 0$</span> such that <span class="math-container">$(x- \epsilon,x+ \epsilon)\subs... | user | 505,767 | <p>Yes we have that</p>
<p><span class="math-container">$$\frac{xy\sin{y}}{3x^2+y^2}=\frac{\sin y}y\frac{xy^2}{3x^2+y^2} \to 1 \cdot 0=0$$</span></p>
<p>indeed by <span class="math-container">$x=\frac r {\sqrt 3} \cos \theta$</span> and <span class="math-container">$y=r \sin \theta$</span> we can easily show that <sp... |
2,499,778 | <p>Let $ABC$ a triangle with $AB <AC$. The angle bisector of $\angle BAC$ intersects $(BC)$ in $D$. The perpendicular from $B$ on $AD$ intersects the circumscribed circle of the triangle $ABD$ in $E$ for the second time. </p>
<p>Show that the center of the circumscribed circle $S$ of the triangle $ABC$ is on the l... | Especially Lime | 341,019 | <p>There must be a turning point between every two roots. The derivative is $15x^4-30x^2-120=15(x^2-4)(x^2+2)$. So there are only two turning points, at $x=\pm2$, and so at most three real roots. If both turning points are the same side of the $x$-axis there will be one root, and if they are on different sides there wi... |
768,911 | <p>Now, I have an idea how to attempt this question with modulo arithmetic, but I was thinking if there was a solution that did not involve modular arithmetic.</p>
<p>If $7 |(b^2+c^2)$ iff $7|b$ and $7|c$.</p>
<p>I can prove it in the reverse direction $(\Leftarrow )$. </p>
<p>We simply use the fact that if $a|b$ an... | lab bhattacharjee | 33,337 | <p>As requested for "<strong>Without modular arithmetic</strong>"</p>
<p>Any $n=7k,7k\pm1,7k\pm2,7k\pm3$ where $k$ is any integer</p>
<p>$\implies n^2=49k^2,49k^2\pm14k+1,49k^2\pm28k+4,49k^2\pm42k+9$</p>
<p>Observe that if $7\nmid bc, b^2,c^2$ will have to be of the form $\in \{49k^2\pm14k+1,49k^2\pm28k+4,49k^2\pm4... |
768,911 | <p>Now, I have an idea how to attempt this question with modulo arithmetic, but I was thinking if there was a solution that did not involve modular arithmetic.</p>
<p>If $7 |(b^2+c^2)$ iff $7|b$ and $7|c$.</p>
<p>I can prove it in the reverse direction $(\Leftarrow )$. </p>
<p>We simply use the fact that if $a|b$ an... | ziang chen | 38,195 | <p>$p\equiv3\pmod 4, p|(a^2+b^2)$, then $p|a,p|b$</p>
<p>in fact, if $p|(a^2+b^2)$, we have
$$a^2\equiv-b^2 \pmod p$$</p>
<p>so</p>
<p>$$ (\frac ap)^2=(\frac {-1}p)(\frac bp)^2$$
that is
$$ (\frac ap)^2=-(\frac bp)^2$$</p>
<p>we must have</p>
<p>$$ (\frac ap)=(\frac bp)=0$$</p>
<p>$p|a,p|b$</p>
|
216,268 | <p>Suppose $k>0$ is some fixed constant, and $n$ is a positive integer tending to infinity. Find $j\equiv j(n,k)$ such that
$$ \frac{\binom{n}{j}}{j!} \sim k. $$</p>
<p>The asymptotic expression for $(n!)^{-1}$ (<a href="https://stackoverflow.com/questions/3084937/how-to-calculate-the-inverse-factorial-of-a-real-... | Brendan McKay | 9,025 | <p>(CORRECTED EDITION)
By mucking around with expansions like Igor suggested, I found
$$ j \approx J(n,k)=
en^{1/2} - \tfrac14\ln(n) -\tfrac12\ln(2\pi k)-\tfrac14 e^2-\tfrac12.
$$</p>
<p>It seems good when $k$ is small but not extremally small. For example, $\binom{100}{22}/22!\approx 6.523187$ and $J(100,6.523187... |
216,268 | <p>Suppose $k>0$ is some fixed constant, and $n$ is a positive integer tending to infinity. Find $j\equiv j(n,k)$ such that
$$ \frac{\binom{n}{j}}{j!} \sim k. $$</p>
<p>The asymptotic expression for $(n!)^{-1}$ (<a href="https://stackoverflow.com/questions/3084937/how-to-calculate-the-inverse-factorial-of-a-real-... | Dan Romik | 78,525 | <p>This is not quite an answer to the question (Brendan McKay already gave quite a satisfactory one), but it's worth adding that </p>
<ol>
<li><p>$\binom{n}{j}/j!$ has an interesting probabilistic interpretation, namely it is the expected number of increasing subsequences of length $j$ in a (uniformly) random permutat... |
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