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 |
|---|---|---|---|---|
103,675 | <p>I have defined a recursive sequence</p>
<pre><code>a[0] := 1
a[n_] := Sqrt[3] + 1/2 a[n - 1]
</code></pre>
<p>because I want to calculate the <code>Limit</code> for this sequence when n tends towards infinity.</p>
<p>Unfortunately I get a <code>recursion exceeded</code> error when doing:</p>
<pre><code>Limit[a[n... | Carl Woll | 45,431 | <p>Another possibility is to use <code>SequenceLimit</code>:</p>
<pre><code>SequenceLimit[a /@ Range[10]]
</code></pre>
<blockquote>
<p>2 Sqrt[3]</p>
</blockquote>
|
3,497,420 | <p>Consider the function <span class="math-container">$$f(x,y)=x^6-2x^2y-x^4y+2y^2.$$</span> The point <span class="math-container">$(0,0)$</span> is a critical point. Observe,
<span class="math-container">\begin{align*}
f_x & = 6x^5-4xy-4x^3y, f_x(0,0)=0\\
f_y & = 2x^2-x^4+4y. f_y(0,0)=0\\
f_{xx} & = 30x... | B. Goddard | 362,009 | <p>You might note that your function factors as</p>
<p><span class="math-container">$$(x^2-y)(x^4-2y).$$</span></p>
<p>So there are easy to find regions in the <span class="math-container">$xy$</span>-plane where the function is positive an negative. Close to the origin and between the curves <span class="math-conta... |
4,495,950 | <blockquote>
<p>Why does <span class="math-container">$-\frac{1}{17-x}$</span> equal <span class="math-container">$\frac{1}{x-17}$</span>?</p>
</blockquote>
<p>Is there any simple computation to make this seem a little bit more intuitive? Right now, I cannot wrap my head around the fact that I can just switch signs of ... | Guillermo García Sáez | 696,501 | <p><span class="math-container">$17-x=-1(x-17) $</span> so <span class="math-container">$(-1) (17-x) =(-1) ^2(x-17) =x-17$</span></p>
|
38,659 | <p>I know how to use Matrix Exponentiation to solve problems having linear Recurrence relations (for example Fibonacci sequence). I would like to know, can we use it for linear recurrence in more than one variable too? For example can we use matrix exponentiation for calculating ${}_n C_r$ which follows the recurrence ... | Phira | 9,325 | <p>The likeliest interpretation of your confusion is that you have learned a very constrictive version of the product rule that it is not appropriate for counting non-trivial things.</p>
<p>You multiply things not only when the choices are independent, but also when the <em>number</em> of the second choices are indepe... |
38,659 | <p>I know how to use Matrix Exponentiation to solve problems having linear Recurrence relations (for example Fibonacci sequence). I would like to know, can we use it for linear recurrence in more than one variable too? For example can we use matrix exponentiation for calculating ${}_n C_r$ which follows the recurrence ... | Qiaochu Yuan | 232 | <p>If you're asking what I think you're asking, here is an argument that only uses the "product rule": </p>
<p>First, establish that the number of ways to order the numbers $\{ 1, 2, ... n \}$ is $n!$. Next, let ${n \choose k}$ denote the number of ways to choose $k$ numbers (not in any particular order) out of $\{ 1,... |
1,307,085 | <p>How does one solve this equation?</p>
<blockquote>
<p>$$\cos {x}+\sin {x}-1=0$$</p>
</blockquote>
<p>I have no idea how to start it.</p>
<p>Can anyone give me some hints? Is there an identity for $\cos{x}+\sin{x}$?</p>
<p>Thanks in advance!</p>
| Harish Chandra Rajpoot | 210,295 | <p>Given <span class="math-container">$$\color{blue}{\cos x+\sin x-1=0} $$</span><span class="math-container">$$\cos x+\sin x=1 $$</span> Divide both sides by <span class="math-container">$\color{blue}{\sqrt{2}}$</span> we get <span class="math-container">$$\frac{1}{\sqrt{2}}\cos x+ \frac{1}{\sqrt{2}}\sin x=\frac{1}{\s... |
1,307,085 | <p>How does one solve this equation?</p>
<blockquote>
<p>$$\cos {x}+\sin {x}-1=0$$</p>
</blockquote>
<p>I have no idea how to start it.</p>
<p>Can anyone give me some hints? Is there an identity for $\cos{x}+\sin{x}$?</p>
<p>Thanks in advance!</p>
| Jack Tiger Lam | 186,030 | <p>By inspection, it is obvious, that: $1-\sin{x} \equiv (\cos{\frac{x}{2}} - \sin{\frac{x}{2}})^2$.</p>
<p>From the half angle expansions, $\cos{x} \equiv (\cos{\frac{x}{2}} - \sin{\frac{x}{2}})(\cos{\frac{x}{2}} + \sin{\frac{x}{2}})$.</p>
<p>The equation is thus equivalent to:</p>
<p>$(\cos{\frac{x}{2}} - \sin{\fr... |
713,521 | <p>There are so many notations for differentiation. Some of them are:
$$
f^\prime(x) \qquad \frac{d}{dx}(f(x))\qquad \frac{dy}{dx}\qquad \frac{df}{dx}\qquad
D f(x)\qquad y^\prime\qquad D_x f(x)
$$
Why are there so many ways to say "the derivative of $f(x)$"? Is there a specific use for each notation? What is the differ... | Jacob Wakem | 117,290 | <p>A short answer is that in calculus you do lots of symbolic manipulation, so different notations are worth the bother to minimize eye sore and give you what power you need. For instance, the fraction notation helps if you are doing cancellations or partial derivatives.</p>
|
11,178 | <p>As far as I know, one way to take a homotopy colimit in a model category is to replace (up to acyclic fibration) all arrows in the diagram with cofibrations, and take the strict colimit of the resulting diagram.</p>
<p>In Top with the model structure given by Serre fibrations, cofibrations, and weak equivalences, if... | Reid Barton | 126,667 | <p>Question 1: The model category $\mathcal{C}$ should be <em>left proper</em>, i.e. the pushout of a weak equivalence along a cofibration is again a weak equivalence. (Dually, there is a notion of right proper.) Top is left proper, as is any model category in which every object is cofibrant, such as SSet. There is ... |
3,490,329 | <blockquote>
<p>Show that a 2-dimensional subspace of the space of <span class="math-container">$2\times2$</span> matrices contains a non-zero symmetric matrix. </p>
</blockquote>
<p>I don't know if it should be written like the addition of two symmetric and skew-symmetric matrix or there is another way to show it. ... | Mozhgan Farahani | 736,783 | <p>In general, I think I can write like the following
If A,B are two 2×2 matrices , then
Transpose of (A+B) = Transpose of A + Transpose of B , so it us closed under addition. Then, for any scalar k , Transpose of (kA) = k . Transpose of A = k.A , showing that it is closed under scalar multiplication.
Thus, both con... |
4,109,827 | <p><span class="math-container">$$f(x,y)=\begin{cases}\dfrac{y^3}{x^2+y^2} &(x,y) \neq \ \mathbb{(0,0)}\\ 0 & (x,y)=(0,0) \\ \end{cases}$$</span></p>
<p>Evaluate <span class="math-container">$f_x(0,0)$</span> and <span class="math-container">$f_y(0,0)$</span> and <span class="math-container">$D_\overrightarrow... | fwd | 897,162 | <p><span class="math-container">$$f_x(0,0) = \lim_{t\rightarrow 0} \frac{f(t,0) - f(0,0)}{t} = 0, \lim_{t\rightarrow 0} \frac{f(0,t) - f(0,0)}{t} = 1\ \ $$</span></p>
<p>Next, I will show that <span class="math-container">$f$</span> is not differentiable at <span class="math-container">$(0,0)$</span>.</p>
<p>Let <span ... |
1,270,042 | <p>$$(a+5)(b-1)=ab-a+5b-5=20-5=15.$$</p>
<p>So, both $a + 5$ and $b-1$ divide $15$. </p>
<p>Then, $a + 5$ is one of $15, -15, 3, -3, 5, -5, 1, -1$, so $a$ is one of $10, -20, -2, -8, 0, -10, -4, -6$ and $b – 1$ is one of $15, -15, 3, -3, 5, -5, 1, -1$, so $b = 14, -14, 4, -2, 6, -4, 2, 0$.</p>
<p>Could all possibili... | Dr. Sonnhard Graubner | 175,066 | <p>Hint: rewrite your equation into $$a=-5+\frac{15}{b-1}$$</p>
|
2,024,997 | <blockquote>
<p>$$\lim_{x \rightarrow +\infty}\frac{\log_{1.1}x}{x}$$</p>
</blockquote>
<p>I can solve this easily by generating the graph with my calculator, but is there is a way to do this analytically?</p>
| DeepSea | 101,504 | <p><strong>hint</strong>: L'hospitale's rule ! . Have you learned this formula yet?</p>
|
92,382 | <p>I was working on a little problem and came up with a nice little equality which I am not sure if it is well-known (or) easy to prove (It might end up to be a very trivial one!). I am curious about other ways to prove the equality and hence I thought I would ask here to see if anybody knows any or can think of any. I... | Community | -1 | <p>For whatever it is worth, below is an explanation on why I was interested in this equality. Consider a rectangle of size $x \times 1$, where $x < 1$. I was interested in covering this rectangle with squares of maximum size whenever possible (i.e. in a greedy sense).</p>
<p>To start off, we can have $\displaystyl... |
635,077 | <p>$$\sin(a+b) = \sin(a) \cos(b) + \cos(a) \sin(b)$$</p>
<p>How can I prove this statement?</p>
| Fly by Night | 38,495 | <p>I typed "<em>Proof of Trigonometric formulae</em>" in to Google and the second hit was an extensive Wikipedia article which supplies proofs to many, many trigonometric identities.</p>
<p><a href="https://en.wikipedia.org/wiki/Proofs_of_trigonometric_identities#Angle_sum_identities" rel="nofollow">Click here</a> for... |
2,694,706 | <p>A rational numbers cannot have irrational as$\exists n \in \mathbb{Z}, \, \sqrt[n]{\frac xy}$, but the two equalities give: $b^2=\frac{(a^2+c^2)}{2} \implies b = \sqrt[2]{\frac {(a^2+c^2)}{2}}$.<br>
To avoid this, need $4\mid a$, & $a=c$; so that if $\exists t=a/2, \, b = \sqrt[2]{\frac {(a^2+c^2)}{2}} => 2t$... | Angina Seng | 436,618 | <p>The general solution of $b^2-a^2=5$ has
$$b=\frac12\left(t+\frac 5t\right)$$
for $t\in\Bbb Q^*$. Then
$$c^2=b^2+5=\frac{t^4+10t^2+25}{4t^2}+5=\frac{t^4+30t^2+25}{4t^2}.$$
The problem boils down to whether the genus-one curve
$$y^2=x^4+30x^2+25$$
has rational points.</p>
|
2,694,706 | <p>A rational numbers cannot have irrational as$\exists n \in \mathbb{Z}, \, \sqrt[n]{\frac xy}$, but the two equalities give: $b^2=\frac{(a^2+c^2)}{2} \implies b = \sqrt[2]{\frac {(a^2+c^2)}{2}}$.<br>
To avoid this, need $4\mid a$, & $a=c$; so that if $\exists t=a/2, \, b = \sqrt[2]{\frac {(a^2+c^2)}{2}} => 2t$... | Oleg567 | 47,993 | <p>We can rewrite these numbers with common denominator $D$:
$a=\dfrac{A}{D}$, $\;b=\dfrac{B}{D}$, $\;c=\dfrac{C}{D}$. Then we'll get diophantine equation (system of diophantine equations):
$$
B^2-A^2=C^2-B^2=5D^2.\tag{1}
$$
The smallest solution of $(1)$ is $(A,B,C,D)=(31,41,49,12)$.
<br>So, example of such three rati... |
2,694,706 | <p>A rational numbers cannot have irrational as$\exists n \in \mathbb{Z}, \, \sqrt[n]{\frac xy}$, but the two equalities give: $b^2=\frac{(a^2+c^2)}{2} \implies b = \sqrt[2]{\frac {(a^2+c^2)}{2}}$.<br>
To avoid this, need $4\mid a$, & $a=c$; so that if $\exists t=a/2, \, b = \sqrt[2]{\frac {(a^2+c^2)}{2}} => 2t$... | Oleg567 | 47,993 | <p><strong>Algebraic approach.</strong> </p>
<p>First, note that $$a^2+c^2=2b^2,$$ and if denote
$$p=\dfrac{a}{b}, \quad q=\dfrac{c}{b}, \tag{1}$$
then we will search rational points on the circle
$$p^2+q^2=2.\tag{2}$$</p>
<p>According to the article <a href="http://www.math.uconn.edu/~kconrad/ross2007/3squarearithpr... |
671,407 | <p>I have problem with equation: $4^x-3^x=1$. </p>
<p>So at once we can notice that $x=1$ is a solution to our equation. But is it the only solution to this problem? How to show that there aren't any other solutions? </p>
| Mark Bennet | 2,906 | <p>Hint: one way of showing that a function takes a value only once is to show that it is increasing.</p>
<p>Hint: For negative values of $x$ you need a different observation.</p>
|
1,689,523 | <p>I need help with this Laplace question.
<span class="math-container">$$f(t) = e^{-t} \sin(t) $$</span></p>
<hr />
<p>Answer should be <span class="math-container">$\dfrac{1}{s^2 + 2s + 2}$</span></p>
<hr />
<p>What I'm currently doing is as follows:</p>
<p><span class="math-container">$u = \sin(t)\qquad$</span> ... | user321205 | 321,205 | <p>Calculate Laplace transform for $f(t)$ and $e^{-t}\cos(t)$ then you'll get a equation system where the unknowns will be the integrals
$$\int\dfrac{ e^{-(s+1)t}\cos(t)}{ -(s+1)} dt \text{ and } \int\dfrac{ e^{-(s+1)t}\sin(t)}{ -(s+1)} dt $$
with this you can conclude.</p>
|
1,999,834 | <p>Let $\varphi : G \rightarrow H$ be a group homomorphism with kernel $K$ and let $a,b \in \varphi(G)$. Let $X = \varphi^{-1}(a)$ and $Y = \varphi^{-1}(b)$. Fix $u \in X$. Let $Z=XY$. Prove that for every $w \in Z$ that there exists $v \in Y$ such that $uv=w$. This is Dummit and Foote exercise 3.1.2.</p>
<p>My attemp... | egreg | 62,967 | <p>Since $w\in Z$, you know that $w=xy$, for some $x\in X$ and $y\in Y$.</p>
<p>By definition, $\varphi(x)=a$ and $\varphi(y)=b$.</p>
<p>Also $\varphi(u)=a$, which implies $u^{-1}x\in\ker\varphi$.</p>
<p>Then
$$
w=xy=u(u^{-1}xy)
$$
Can you finish?</p>
|
4,651,596 | <p>I know the proof of the "<a href="https://en.wikipedia.org/wiki/Doubling_the_cube" rel="nofollow noreferrer">Doubling the cube problem</a>". What is used there is the fact that if a number <span class="math-container">$a$</span> is constructible then <span class="math-container">$[\mathbb{Q}(a):\mathbb{Q}]... | Arthur | 15,500 | <p>The set of constructible numbers is the smallest extension of <span class="math-container">$\Bbb Q$</span> where each positive number has a square root. (That's essentially what straightedge and compass constructions are able to do: field operations and square roots.) Every constructible number can be described by u... |
106,126 | <blockquote>
<p><strong>Problem</strong> Prove that $n! > \sqrt{n^n}, n \geq 3$. </p>
</blockquote>
<p>I'm currently have two ideas in mind, one is to use induction on $n$, two is to find $\displaystyle\lim_{n\to\infty}\dfrac{n!}{\sqrt{n^n}}$. However, both methods don't seem to get close to the answer. I wonder ... | Amihai Zivan | 22,409 | <p>You can prove by induction (a-la Gauss) that $n! = 1 \cdots n = (1 \cdot n)(2 \cdot (n-1))(3 \cdot (n-2))\cdots(n/2(n/2+1)) \geq n^{(n/2)}$ and that finishes the proof.</p>
|
106,126 | <blockquote>
<p><strong>Problem</strong> Prove that $n! > \sqrt{n^n}, n \geq 3$. </p>
</blockquote>
<p>I'm currently have two ideas in mind, one is to use induction on $n$, two is to find $\displaystyle\lim_{n\to\infty}\dfrac{n!}{\sqrt{n^n}}$. However, both methods don't seem to get close to the answer. I wonder ... | Jonas Meyer | 1,424 | <p>To show that $(n!)^2>n^n$ for all $n\geq 3$ by induction, you first check that $(3!)^2>3^3$. Then to get the inductive step, it suffices to show that when $n\geq 3$, $(n+1)^2\geq\frac{(n+1)^{n+1}}{n^n}=(n+1)\left(1+\frac{1}{n}\right)^n$. This is true, and in fact $\left(1+\frac{1}{n}\right)^n<3$ for all $... |
2,578,444 | <blockquote>
<p><span class="math-container">$\tan x> -\sqrt 3$</span></p>
</blockquote>
<p>How do I solve this inequality?</p>
<p>From the <a href="https://www.desmos.com/calculator/qb8bg1vbsf" rel="nofollow noreferrer">graph</a> it is evident that <span class="math-container">$\tan x>-\sqrt 3$</span> for <span ... | Eric Fisher | 476,420 | <p>Tangent has period $\pi$, not $2\pi$. Also, $tan(x) > -\sqrt{3}$ when $x>-\pi/3$. The function is not defined at $\pi/2$, but it’s positive for $x \in [0, \pi/2)$. </p>
|
1,185,108 | <p>empty set is an subset of any sets maybe any collection of sets.</p>
<p>I wonder what about the case of the empty set being a member,not subset, of any collection (family) of sets.</p>
| Alberto Takase | 146,817 | <p>$\varnothing\in\mathcal{P}(\varnothing)$ but $\varnothing\notin \varnothing$.</p>
|
4,402,839 | <p>I need to prove that</p>
<blockquote>
<p>Let <span class="math-container">$f:[a,b] \to \mathbf{R}$</span> be a bounded map and let <span class="math-container">$f$</span> be an integrable map on the interval <span class="math-container">$[c,b]$</span> for all <span class="math-container">$c \in (a,b).$</span> Then, ... | Alan | 175,602 | <p>I find it easier to talk about the equivalent in Darboux integral terms, that a funct ion is integrable if the upper sum minus the lower sum for any partition is less than any <span class="math-container">$\epsilon$</span> you want. So to show intgrability over <span class="math-container">$[a,b]$</span> we start w... |
4,402,839 | <p>I need to prove that</p>
<blockquote>
<p>Let <span class="math-container">$f:[a,b] \to \mathbf{R}$</span> be a bounded map and let <span class="math-container">$f$</span> be an integrable map on the interval <span class="math-container">$[c,b]$</span> for all <span class="math-container">$c \in (a,b).$</span> Then, ... | Mathematician | 971,859 | <p>So I guess I've found out:</p>
<p>By hypotesis, <span class="math-container">$f:[a,b] \to \mathbf{R}$</span> is bounded, so there exists <span class="math-container">$0 < K \in \mathbf{R}$</span> such that <span class="math-container">$|f(x)| \leq K$</span>, for all <span class="math-container">$x \in [a,b].$</sp... |
4,177,829 | <p>Given angles <span class="math-container">$0<\theta_{ij}<\pi$</span> for <span class="math-container">$1\leq i<j\leq k$</span>, what conditions are there on the angles to ensure that there exists <span class="math-container">$k$</span> unit vector <span class="math-container">$v_i\in \mathbb R^k$</span> so ... | NN2 | 195,378 | <p><strong>Note</strong>: It's not the final answer. It provides a necessary condition (but not sufficient condition).</p>
<p>Let us represent the unit vectors <span class="math-container">$\{\boldsymbol{v}_i \}_{1 \le i \le k} \in \Bbb R^k$</span>, begining from the origine <span class="math-container">$\boldsymbol{O}... |
4,407,210 | <p><span class="math-container">$$y''-\frac{1}{x}y'=2x\cdot cos(x)$$</span></p>
<p>For the homogeneous part I multiplied through with <span class="math-container">$x^2$</span> and got a second order Cauchy Euler equation with the general solution: <span class="math-container">$$y_h (x)=A x^2 +B$$</span></p>
<p>Then for... | Doug M | 317,176 | <p>This reduces the problem to a first order Diff Eq.
<span class="math-container">$u= y'\\
xu' - u = 2x^2\cos x$</span></p>
<p>Choose a candidate for the particular solution that could work.</p>
<p><span class="math-container">$u_p = x\sin x$</span><br />
"Generalize" this by adding terms that might come up ... |
2,067,097 | <p>Given are two points on a line with coordinates. How do we calculate the third forming a perfect 60 degree triangle? So we have X,Y, but need Z...</p>
<p>X: 0,0    ( 0,0 i.e. horizontal, vertical )<br>
Y: 50, 0 <br>
Z: 25, ??</p>
<p>How to calculate the missing horizontal coordinate for Z? Formi... | Fred | 380,717 | <p>I gave your question to Pythagoras (a friend of mine): He said: the missing horizontal coordinate is given by</p>
<p>$$ \sqrt{50^2+25^2}.$$</p>
<p>OOps !</p>
<p>Edit: coordinate is given by</p>
<p>$$ \sqrt{50^2-25^2}.$$</p>
|
427,564 | <p>I'm supposed to give a 30 minutes math lecture tomorrow at my 3-grade daughter's class. Can you give me some ideas of mathemathical puzzles, riddles, facts etc. that would interest kids at this age?</p>
<p>I'll go first - Gauss' formula for the sum of an arithmetic sequence.</p>
| ralu | 20,840 | <p>Probability, throwing dice and gambling. How likley is to get #6 throwing dice and what implications this have.</p>
|
427,564 | <p>I'm supposed to give a 30 minutes math lecture tomorrow at my 3-grade daughter's class. Can you give me some ideas of mathemathical puzzles, riddles, facts etc. that would interest kids at this age?</p>
<p>I'll go first - Gauss' formula for the sum of an arithmetic sequence.</p>
| Alexander Gruber | 12,952 | <p>Teach 'em how to play <a href="http://en.wikipedia.org/wiki/Sprouts_%28game%29">sprouts.</a></p>
|
427,564 | <p>I'm supposed to give a 30 minutes math lecture tomorrow at my 3-grade daughter's class. Can you give me some ideas of mathemathical puzzles, riddles, facts etc. that would interest kids at this age?</p>
<p>I'll go first - Gauss' formula for the sum of an arithmetic sequence.</p>
| Robert Mastragostino | 28,869 | <p>Eeny-meeny-miney-mo is essentially counting up to $16$. I'm not sure if they'd be comfortable enough with multiplication and remainders to totally grasp it, but I think kids would like to be able to "cheat the system" and intentionally predict who to land on (by, say, realizing that 16 is one more than a multiple of... |
427,564 | <p>I'm supposed to give a 30 minutes math lecture tomorrow at my 3-grade daughter's class. Can you give me some ideas of mathemathical puzzles, riddles, facts etc. that would interest kids at this age?</p>
<p>I'll go first - Gauss' formula for the sum of an arithmetic sequence.</p>
| Shivendra | 83,703 | <p>may be a nice multiplication speed improvement class will be helpful.
tell the students to use multiplication in day to day life using the distributivity property of multiplication</p>
<ul>
<li><p>multiply 19x15 :
it can be changed to 15x(20-1) in mind and then 300-15 =285
such tricks will help speed up their mathe... |
1,722,995 | <blockquote>
<blockquote>
<p>Question: Given the circle $x^2+y^2=25$ is inscribed in triangle $\triangle ABC$, where vertex $B$ lies on the first quadrant. Slope of $AB$ is $\sqrt 3$ and has a positive y-coordinate, and $|AB|=|AC|$. Find the equations for $AC$ and $BC$</p>
</blockquote>
</blockquote>
<p>I foun... | manshu | 287,678 | <p>Hint: Use the formula of incentre. It is given by $$(x,y)=(\dfrac{ax_1+bx_2+cx_3}{a+b+c},\dfrac{ay_1+by_2+cy_3}{a+b+c})$$</p>
<p>Here $a,b,c $ are the lengths of the sides of the triangle. Length a is opposite to the point A. Length b is opposite to the point B. Length c is opposite to the point C.<br>
Here you can... |
3,041,907 | <p>I am unable to isolate the variable <span class="math-container">$x$</span> of this inequality <span class="math-container">$y \leq \sqrt{2x-x^2}$</span> ( where <span class="math-container">$0 \leq y \leq 1 $</span>)</p>
<p>Is it correct doing this: <span class="math-container">$y^2 \leq 2x-x^2$</span>?
I found ... | Joel Pereira | 590,578 | <p>When you get to <span class="math-container">$x^2-2x+y^2$</span> = 0, you can complete the square for the x-terms and get
<span class="math-container">$$ (x^2-2x+1) + y^2 = 1$$</span>
<span class="math-container">$$ (x-1)^2 + y^2 = 1$$</span></p>
<p>This is a circle of radius 1 centered at (1,0). So now you just ne... |
366,311 | <blockquote>
<p>Show that the sequence $\displaystyle (x_n)=\left( \sum_{i=1}^n\frac 1 i\right)$ diverge by epsilon delta definition.</p>
</blockquote>
<p>I'm not familiar with proving divergent sequence. Do anyone have any des? Thank you.</p>
| Alex Ravsky | 71,850 | <p>Hint: Estimate from below the sums $A_n=\sum_{i=2^n}^{2^{n+1}-1}\frac 1i$.</p>
|
278 | <p>If you take a look at our status in the <a href="http://area51.stackexchange.com/proposals/64216/mathematics-educators">area51</a>, all criteria seem to be satisfied (soon) but not the number of questions asked (which seems to be decreasing, actually). Do you think this is a problem for us? Is it something we should... | Benjamin Dickman | 262 | <p>I think the fundamental piece is to accumulate <strong>users</strong> who ask high quality questions and/or provide high quality answers. The "we"/"us" you refer to is not static, of course, and the hope must be that the user-base on MESE continues to increase. How this accretion will manifest is still unclear (to m... |
34,215 | <p>How do professional mathematicians learn new things? How do they expand their comfort zone? By talking to colleagues? </p>
| Deane Yang | 613 | <p>It seems to me that the most important thing to learn when you're a graduate student is how to learn more mathematics. Everything else is detail. So you do what you learned to do as a graduate student (in order of increasing effectiveness, at least for me):</p>
<ul>
<li>Read papers and books (I'm actually unable to... |
1,858,529 | <p>I know, there are some threads dealing with this sum but I want to solve it with the integral test for convergence(<a href="https://en.wikipedia.org/wiki/Integral_test_for_convergence" rel="nofollow">more</a>)</p>
<blockquote>
<p>$$\sum\limits_{n=3}^{\infty} \frac{1}{n\log(n)\log(\log(n))}$$</p>
</blockquote>
<p... | Olivier Oloa | 118,798 | <p><strong>Hint</strong>. One has, for $x\ge3$,
$$
(\log(\log x))'=\frac{\frac1x}{\log x}=\frac1{x\log x}
$$ giving
$$
\int\limits_3^{\infty} \frac{1}{x\log(x)\log(\log x)}dx=\int\limits_3^{\infty} \frac{\frac1{x\log x}}{\log(\log x)}dx=\int\limits_3^{\infty} \frac{(\log(\log x))'}{\log(\log x)}dx.
$$</p>
|
1,858,529 | <p>I know, there are some threads dealing with this sum but I want to solve it with the integral test for convergence(<a href="https://en.wikipedia.org/wiki/Integral_test_for_convergence" rel="nofollow">more</a>)</p>
<blockquote>
<p>$$\sum\limits_{n=3}^{\infty} \frac{1}{n\log(n)\log(\log(n))}$$</p>
</blockquote>
<p... | Ben Grossmann | 81,360 | <p>Make the substitution $t = \ln(n)$. We find that
$$
\int\limits_3^{\infty} \frac{1}{n\log(n)\log(\log(n))} =
\int_3^\infty \frac{1}{t(n)\ln(t(n))} \frac{1}n dn = \\
\int_{\ln 3}^\infty \frac{1}{t\ln(t)}\,dt
$$
Integrate this using the further substitution $u = \ln(t)$.</p>
<p>Alternatively, start with the substit... |
696,285 | <p>Let $X$ and $Y$ be some infinite dimensional Banach spaces. Let $T:X\longrightarrow Y$ be some compact linear operator. It is easy to understand that $T$ cannot be surjective: the Open Mapping Theorem due to Banach, states that surjective operators should be open, and it follows that the image of the unit ball $\mat... | Daniel Fischer | 83,702 | <p>Let $F\subset T(X)$ be a closed (in $Y$) subspace. Then $E = T^{-1}(F)$ is a closed subspace of $X$, and $T\lvert_E \colon E \to F$ is a compact surjective operator. Since $F$ is closed in $Y$, the open mapping theorem implies that $F$ is finite-dimensional.</p>
<p>So: the image of a compact operator cannot contain... |
696,285 | <p>Let $X$ and $Y$ be some infinite dimensional Banach spaces. Let $T:X\longrightarrow Y$ be some compact linear operator. It is easy to understand that $T$ cannot be surjective: the Open Mapping Theorem due to Banach, states that surjective operators should be open, and it follows that the image of the unit ball $\mat... | user133339 | 133,339 | <p>A compact operator has a closed range iff it has a finite dimensional range.
Without loss of generality,we can assume that the range of A is closed,otherwise we can consider the restriction of A to E where E=T−1(F),In all cases the theorem assures that F is finite dimensional.
to prove the theorem,consider the canon... |
2,445,655 | <p>Challenge: A Good Deal</p>
<p>You are currently learning some important aspects of collusion and cartels. This challenge puts you in the position of a bad guy, namely a price-fixing sales manager. Suppose that you find yourself in a so-called “smoke-filled room” to fix prices for the upcoming year with the sales ma... | Eric M. Schmidt | 48,235 | <p>Making the substitution $u = 1/x$, we aim to find
$$\lim_{u\to 0^{\large{-}}} \left(2/u + \sqrt{4/u^2 + 1/u}\ \right).$$
When $u < 0$, we have $\sqrt{u^2} = -u$, so $\sqrt{4/u^2 + 1/u} = \sqrt{(4+u)/u^2}=-\sqrt{4 + u}/u$. Hence, we obtain
$$\lim_{u\to 0^{\large{-}}} \frac{2-\sqrt{4 + u}}{u}.$$
The numerator is $0... |
3,917,255 | <p>Why does Chebyshev's inequality demand that <span class="math-container">$\mathbb{E(}X^2) < \infty$</span>?</p>
| José Carlos Santos | 446,262 | <p>Every closed subset of a compact metric space is compact. And a continuous map maps compact sets onto compact sets. And, finally, every compact subset of a metric space is closed. So, yes, your map is compact.</p>
|
477,477 | <p>Prove that $e^x=-x^2+2x+5 $ have exactly two solutions.</p>
<p>Is it enoguht that Vertex of the parabola is over $y=e^x$ and arms of it looks down</p>
| bubba | 31,744 | <blockquote>
<p>is there any trivial quad tessellation that minimizes distortion in this case</p>
</blockquote>
<p>As indicated in a comment, a cube is a tesselation of a sphere using squares as the tiles. Not a very interesting/useful tesselation, from a graphics point of view, though, unless the spherical object i... |
3,154,032 | <p>Suppose we have a 4 dimension positive signature clifford algebra. In <a href="https://math.stackexchange.com/questions/443555/calculating-the-inverse-of-a-multivector">Calculating the inverse of a multivector</a> and <a href="https://math.stackexchange.com/questions/556247/inverse-of-a-general-nonfactorizable-multi... | Nicholas Todoroff | 1,068,683 | <p><a href="https://doi.org/10.1007/s40314-021-01536-0" rel="nofollow noreferrer">This</a> paper by D. S. Shirokov (2021) claims to give a basis-free formula using only basic operations and various involutions on multivectors. <a href="https://arxiv.org/abs/2005.04015" rel="nofollow noreferrer">Preprint</a>.</p>
|
2,479,918 | <p>Every vector space $V$ could be embedded into $V^{\ast}$ (see <a href="https://en.wikipedia.org/wiki/Dual_basis" rel="noreferrer">here</a>) after choosing a basis, for a given vector $v \in V$ denote this embedding by $v^{\ast}\in V^{\ast}$. Now for given vector spaces $V_1, \ldots, V_k$ over some field $F$, let $V ... | Eric Wofsey | 86,856 | <p>Let me mention something which has not been explicitly stated in any of the other answers: your construction is emphatically wrong for infinite-dimensional vector spaces (not just, you have to tweak it or do more work prove it works, but it just doesn't give the right answer at all).</p>
<p>This is easiest to see b... |
760,926 | <p>Show that $\binom{n}{0} - \binom{n}{1} + \binom{n}{2} - ...+(-1)^k * \binom{n}{k} = (-1)^k * \binom{n-1}{k}$.</p>
<p>I know this has to do with permutations and combination problems, but I'm not sure how would I start with this problem. </p>
| ml0105 | 135,298 | <p>We have the identity $\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$. So we see the series telescope:</p>
<p>$$\sum_{i=0}^{k} (-1)^{i} \binom{n}{i} = \binom{n}{0} + \sum_{i=1}^{k} \binom{n-1}{i-1} + \sum_{i=0}^{k} \binom{n-1}{i}$$ </p>
<p>So we see $\binom{n}{0} = 1$. Then $\binom{n}{1} = \binom{n-1}{0} + \bino... |
4,520,485 | <p>Suppose there are 78 heroes. Only one of them is considered to be 'Tier 1'. At the beginning of some game you are given a choice between either 2 heroes or 4 heroes. The question is: how advantegeous is it to choose out of 4 heroes to choosing out of 2, if by advantageous we mean to have a higher probability of gett... | Robert Israel | 8,508 | <p>It goes up to <span class="math-container">$n$</span>, not <span class="math-container">$k+n$</span>.
In this case the product has only one factor namely <span class="math-container">$k+1 = 3$</span>.</p>
|
1,568,233 | <p>$$\sum_{n=1}^{\infty}n^210^{-n} = \frac{110}{3^6}$$
I noticed this while playing around on my calculator. Is it true and how come?</p>
| Ron Gordon | 53,268 | <p>$$\sum_{n=1}^{\infty} n^2 x^n = x \frac{d}{dx} \left [x \frac{d}{dx} \left (\frac1{1-x} \right ) \right ] $$</p>
<p>because </p>
<p>$$\frac1{1-x} = \sum_{n=0}^{\infty} x^n$$</p>
<p>Evaluating the derivatives, we get</p>
<blockquote>
<p>$$\sum_{n=1}^{\infty} n^2 x^n = \frac{x(1+ x)}{(1-x)^3}$$</p>
</blockquote>... |
1,898,207 | <blockquote>
<p>A man speaks the truth $8$ out of $10$ times. A fair die is thrown. The man says that the number on the upper face is $5$. Find the probability that the original number on the upper face is $5$.</p>
</blockquote>
<p>While solving I find two ways (shown in the image). I think one of them is correct an... | lulu | 252,071 | <p>Surely the answer is $\frac 8{10}$ as all you are asking is: "is the fellow telling the truth?"</p>
<p>Let's see this following the approach you took.</p>
<p>There are two ways in which one might hear the answer "$5$" from the fellow. Either it is $5$, and he tells the truth about it, or it is not $5$ and he lie... |
2,210,893 | <p>A lot of times when proving for example inequalities like $$x \leq y$$
for real numbers $x,y$ the argument looks like
$$x \leq y + \varepsilon$$
for all $\varepsilon > 0$, hence $x \leq y$. </p>
<p>Now this is obviously very intuitive, but is there a "proof" that this conclusion is correct? And is it always suf... | joeb | 362,915 | <p>Suppose $x \leqslant y + \varepsilon$ for every positive $\varepsilon$, and for the sake of contradiction, suppose $x > y$. </p>
<p>For the specific error $\varepsilon := \frac{1}{2}(x - y) > 0$ we have that $$x \leqslant y + \varepsilon < y + 2\varepsilon = y + (x - y) = x,$$ which is a desired contradic... |
1,942,578 | <p>Consider the following wedge</p>
<p><a href="https://i.stack.imgur.com/xiaPX.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xiaPX.png" alt=""></a> cut from a cylinder of radius r.
The plane that cuts the wedge goes through the very bottom of the cylinder leading to an ellipse as the cross sectio... | Mark Viola | 218,419 | <blockquote>
<p>Recall that the definition of the negation of uniform convergence states that a sequence of functions $f_n(x)$ , which converges to $f(x)$, fails to converge uniformly to $f(x)$ for $x\in A$ if there exists an $\epsilon>0$, such that for all $N\in \mathbb{N}$, there exists an $x\in A$ and a number ... |
137,414 | <p>I am trying to evaluate the following expression numerically
$$\frac{d^2}{dt^2}e^{-2t^2}\int_0^\infty\frac{\xi/\sqrt{2}}{\xi^{3/2}}e^{(-\xi^2/2-2\xi t))}$$</p>
<p>My code is as follows</p>
<pre><code>f[t_]:=Exp[-2*t^2]*NIntegrate[Erf[\[Xi]/Sqrt[2]]/\[Xi]^(3/2)*Exp[-(\[Xi]^2/2)-2*\[Xi]*t],{\[Xi],0,Infinity}]
Der[t... | Bob Hanlon | 9,362 | <pre><code>Needs["NumericalCalculus`"]
f[t_?NumericQ] :=
Exp[-2*t^2]*
NIntegrate[
Erf[ξ/Sqrt[2]]/ξ^(3/2)*
Exp[-(ξ^2/2) - 2*ξ*t], {ξ, 0, Infinity}]
</code></pre>
<p>Use <a href="http://reference.wolfram.com/language/NumericalCalculus/ref/ND.html" rel="nofollow noreferrer"><code>ND</code></a></p>
<pre><code... |
3,430,305 | <blockquote>
<p>Let <span class="math-container">$P$</span> be a partition of <span class="math-container">$[0, b]$</span> defined as <span class="math-container">$P = \{ 0 = x_0 < x_1 <
> \ldots < x_n = b\}$</span>, and let <span class="math-container">$c_i \in [x_{i-1}, x_i]$</span> for every <span clas... | Pebeto | 605,486 | <p>The approach described above assumed <span class="math-container">$0 \notin P$</span>. Here is the general case.</p>
<p>A hyperplane is such <span class="math-container">$\{ x \in \mathbb{R}^n\ s.t. \ c^t x = c_0 \},$</span> where <span class="math-container">$c \in \mathbb{R}^n, \ c \neq 0$</span> and <span class=... |
2,485,261 | <blockquote>
<p>$\displaystyle \sum_{k=0}^n k {n \choose k} p^k (1-p)^{n-k}$ with $0<p<1$</p>
</blockquote>
<p>I know of one way to evaluate it (from statistics) but I was wondering if there are any other ways. </p>
<p>This is the way I know:</p>
<p>Let </p>
<p>$$M(t)=\displaystyle \sum_{k=0}^n e^{kt} {n \c... | Clement C. | 75,808 | <p>Yes.</p>
<p>$$\begin{align}
\sum_{k=0}^n k \binom{n}{k} p^k (1-p)^{n-k}
&= \sum_{k=1}^n k \binom{n}{k} p^k (1-p)^{n-k}
= \sum_{k=1}^n k\frac{n!}{k!(n-k)!} p^k (1-p)^{n-k}\\
&=\sum_{k=1}^n \frac{n!}{(k-1)!(n-1-(k-1))!} p^k (1-p)^{n-k}\\
&=np\sum_{k=1}^n \frac{(n-1)!}{(k-1)!(n-1-(k-1))!} p^{k-1} (1-p)^{n-... |
4,284,803 | <p>I am solving a question and I can't get over this step: proving <span class="math-container">$$\sin \frac{1}{k} > \frac{1}{k} - \frac{1}{k^2}$$</span>
where <span class="math-container">$k$</span> is a positive integer.</p>
<p>I tried using induction, but I failed. One of my friends managed to prove it using deri... | Piquancy | 979,182 | <p>We just have to think about:<span class="math-container">$$f(x)=\sin x-x+x^2,x\in[0,1].\\$$</span>
<span class="math-container">$$f'(x)=\cos x-1+2x.\\$$</span>Let<span class="math-container">$\ g(x)=f'(x),$</span>than<span class="math-container">$$g'(x)=-\sin x+2.\\$$</span>
When <span class="math-container">$x\in[0... |
1,772,650 | <p><strong>Note: in relation to the answer of the duplicate question, I see that the second picture below refers to the triangulation when we consider <em>simplicial complexes</em>. I do not understand why the triangles are used as they are, however, so would like some help trying to understand this</strong></p>
<p>I ... | Eric Towers | 123,905 | <p>In a triangulation (specifically, a simplicial complex), the three vertices of a triangle are <em>distinct</em>. (Technically, the two 0-cells at the boundary of each 1-cell are distinct, the three 1-cells at the boundary of each 2-cell are distinct, et c. This leads to: the vertex set of a $k$-cell contains $k-1$... |
3,419,550 | <p>Let <span class="math-container">$D ⊂ \mathbb{R}$</span> </p>
<p>Let <span class="math-container">$D_A$</span> be the set of all accumulation points of <span class="math-container">$D$</span>. The set <span class="math-container">$\bar{D} := D
\cup
D_A$</span> is called
the closure of <span class="math-container">$... | José Carlos Santos | 446,262 | <p>If <span class="math-container">$D$</span> is bounded, then <span class="math-container">$D\subset[a,b]$</span> for some interval <span class="math-container">$[a,b]$</span>. If <span class="math-container">$x\in\mathbb R\setminus[a,b]$</span>, then <span class="math-container">$x\notin D$</span> and <span class="ma... |
1,451,301 | <p>We're given a function $P_n(x)$ for $-1\leq x\leq1$ as follows :</p>
<p>$$P_n(x) = \int \limits_0^\pi \dfrac{1}{\pi}(x+i\sqrt{1-x^2} \cos\theta)^n \, d\theta$$</p>
<p>for $n=(0,1,2,3,\ldots)$, we need to prove that $|P_n(x)| \leq 1$.</p>
<p>I tried the following :</p>
<p>Let $z=x+i\sqrt{1-x^2}\cos\theta$ </p>
<... | Przemysław Scherwentke | 72,361 | <p>HINT: $(x^2+(1-x^2) \cos^2\theta) \leq (x^2+(1-x^2))=1$, because $\theta$ is real.</p>
|
1,021,599 | <p>Let $X$ be a metric space and $q \in X$. I want to show that the distance function $d(q,p)$ is a uniformly continuous function of $p$. </p>
<p>I know how to show that $d$ is continuous, but I am stuck on how to show UC. </p>
<p>Given $\epsilon >0$ let $\delta =?$. Then if $d(x,y) <\delta$, then $|d(q,x)-d(q,... | Vera | 169,789 | <p>$d(q,x)\leq d(q,y)+d(y,x)$ so that $d(q,x)-d(q,y)\leq d(y,x)=d(x,y)$</p>
<p>By symmetry: $d(q,y)-d(q,x)\leq d(x,y)$</p>
<p>This together allows the conclusion that $|d(q,x)-d(q,y)|\leq d(x,y)$</p>
|
2,008,263 | <p>Solve $$(1+y^2\sin2x) \;dx - 2y\cos^2x \;dy = 0$$</p>
<p>Well, first of all I've written $M = 1+y^2\sin2x$ , $N = 2y\cos^2x$.</p>
<p>Then, I noticed that $M'_y$ <strong>does not</strong> equal to $N'_x$.</p>
<p>I'm trying to find something to multiply the equation with, but my math skills sucks. So I'm going for ... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>$$y^2\sin2x\ dx-2y\cos^2x\ dy=y^2d(-\cos^2x)+d(y^2)(-\cos^2x)=d(-\cos^2x\cdot y^2)$$</p>
|
954,933 | <p>Let $\phi\in\ell^\infty$. For $p\in[1,\infty]$, define $M_\phi:\ell^p\to\ell^p$ by</p>
<p>$$M_\phi(f)=\phi f.$$</p>
<p>Show that $\Vert M_\phi\Vert=\Vert\phi\Vert_\infty$, and $M_\phi$ is compact if and only if $\phi\in c_0$, i.e. $\phi$ is a sequence that converges to $0$.</p>
<p>I only have problem with the par... | Yiorgos S. Smyrlis | 57,021 | <p>If $\boldsymbol\varphi\in\ell^\infty(\mathbb N)\smallsetminus c_0(\mathbb N)$, then there is an $\varepsilon>0$, and a subsequence $\{\varphi(k_n)\}_{n\in\mathbb N}$, such that
$$
\lvert \varphi(k_n)\rvert\ge \varepsilon.
$$
Let now
$$
\boldsymbol{u}_n=\boldsymbol{e}_{k_n},
$$
where $\boldsymbol{e}_{n}$ is the s... |
2,114,446 | <p>But, just to get across the idea of a generating function, here is how a generatingfunctionologist might answer the question: the nth Fibonacci number, $F_{n}$, is the coefficient of $x^{n}$ in the expansion of the function $\frac{x}{(1 − x − x^2)}$ as a power series about the origin.</p>
<p>I am reading a book abo... | lab bhattacharjee | 33,337 | <p>$$(1-ax)^{-1}-(1-bx)^{-1}=\dfrac{x(a-b)}{1-(a+b)x+abx^2}$$</p>
<p>Now the $r$th term of $(1-ax)^{-1}$ is $$(ax)^r$$</p>
<p>So, the coefficient of $x^n$ will be $$a^n-b^n$$</p>
<p>Here $a+b=1, ab=-1$</p>
|
267,236 | <blockquote>
<p><strong>Possible Duplicate:</strong><br />
<a href="https://math.stackexchange.com/questions/30156/why-is-this-entangled-circle-not-a-retract-of-the-solid-torus">Why is this entangled circle not a retract of the solid torus?</a></p>
</blockquote>
<p>I am stuck with exercise 16 (c), pag.39 of Hatcher's ... | Community | -1 | <p>This is indeed a duplicate. Just note that the map $S^1 \cong A \to S^1 \times D^2$ induces a map</p>
<p>$\pi_1 S^1 \to \pi_1 (S^1 \times D^2)$</p>
<p>which is the zero map, $0: \mathbb{Z} \to \mathbb{Z}$. This is because $A$ can be shrunk to a point in $S^1 \times D^2$. (You're allowed to homotope $A$ through its... |
1,102,758 | <p><strong>Problem</strong></p>
<p>Given a pre-Hilbert space $\mathcal{H}$.</p>
<p>Consider unbounded operators:
$$S,T:\mathcal{H}\to\mathcal{H}$$</p>
<p>Suppose they're formal adjoints:
$$\langle S\varphi,\psi\rangle=\langle\varphi,T\psi\rangle$$</p>
<p>Regard the completion $\hat{\mathcal{H}}$.</p>
<p>Here they'... | m_gnacik | 182,603 | <p>Let $S_0$, $T_0$ be densely defined linear operators (this is required for adjoints to exists) on a Hilbert space such that
$$(1) \ \qquad S_0 \subseteq T_0^* \quad \& \quad T_0 \subseteq S_0^*. $$
Hence, we can conclude that $S_0$, $T_0$ are closable ($S_0^*$, $T_0^*$ are closed) and since the closure is the s... |
1,746,748 | <p>My calculus teacher gave us this problem in class:</p>
<p>Which is easier to integrate?</p>
<p>$$\int \sin^{100}x\cos x dx$$</p>
<p>or</p>
<p>$$\int \sin^{50}xdx$$</p>
<p>By easier, I assume the teacher means which integral would take less work. I'm unsure of how to approach this problem because of the relative... | zz20s | 213,842 | <p>Which would you prefer to do?</p>
<p>Let $u=\sin x$ and $du=\cos x$, transforming the integral into $\displaystyle \int u^{100} du$.</p>
<p>Or use the reduction formula: $\displaystyle\int \sin^n x dx=\frac{-\sin^{n-1}x \cos x}{n}+\frac{n-1}{n}\int \sin^{n-2}x dx$ for $n=50$? I found the formula <a href="http://... |
2,127,494 | <p>Given two $3$D vectors $\mathbf{u}$ and $\mathbf{v}$ their cross-product $\mathbf{u} \times \mathbf{v}$ can be defined by the property that, for any vector $\mathbf{x}$ one has $\langle \mathbf{x} ; \mathbf{u} \times \mathbf{v} \rangle = {\rm det}(\mathbf{x}, \mathbf{u},\mathbf{v})$.
From this a number of properties... | mechanodroid | 144,766 | <p>I believe I've found an elegant proof.</p>
<ol>
<li><p>Assume that <span class="math-container">$ \mathbf{u}\times \mathbf{v} \ne 0$</span>. For <span class="math-container">$\mathbf{x} = \mathbf{u}\times \mathbf{v}$</span> we have
<span class="math-container">\begin{align}
\|\mathbf{u}\times \mathbf{v}\|^4 &= \... |
398,388 | <p>The classification of finite simple groups has been called one of the great intellectual achievements of humanity, but I don't even know one single application of it. Even worse, I know a lot of applications of simple <em>modules</em> over some ring/algebra <span class="math-container">$A$</span>, but I can barely k... | arsmath | 3,711 | <p>There's an entire book on this subject, "Applying the Classification of Finite Simple Groups: A User’s Guide" by Stephen D. Smith, published through the AMS, though you can find a draft version <a href="http://homepages.math.uic.edu/%7Esmiths/book.pdf" rel="noreferrer">here</a>.</p>
<p>The applications ar... |
333,807 | <p><strong>Notations</strong>: For a scalar $a\in\mathbb{R}$, denote
$$\mathrm{sgn}(a)=\left\{
\begin{array}{l l}
1 & \mbox{if } a>0\\
0 & \mbox{if } a=0\\
-1 & \mbox{if } a<0
\end{array}.\right.$$
For a vector $r\in\mathbb{R}^n$, $\mathrm{sgn... | adam W | 43,193 | <p>The following uses $p=n$ thus strictly speaking is incorrect. It may serve as an example as to why that restriction is necessary for the question though.</p>
<p>Using row vectors, look at
$$\{r_1,\dots,r_p\} = \pmatrix{1 & 7 & 1 \\ -1 & 1 & 1 \\ -2 & -6 & 0}$$
$$ x = \pmatrix{3 & -1 &... |
1,621,019 | <blockquote>
<p>Let $a,b,$ and $c$ be the lengths of the sides of a triangle, prove that $$a(b^2+c^2-a^2)+b(a^2+c^2-b^2)+c(a^2+b^2-c^2) \le 3abc.$$</p>
</blockquote>
<p>I can't really factor this into something nice. Also using AM-GM or Cauchy-Schwarz doesn't look like it will help. I am thinking we need to bound th... | Nikunj | 287,774 | <p>Note: Using the cosine rule, this inequality can be written as:</p>
<p>$2abc(cosA+cosB+cosC)\le3abc$</p>
<p>Now, we have to prove that $cosA+cosB+cosC\le3/2$</p>
<p>Proof:
$f(A,B) = cos(A) + cos(B) + cos(π-A-B)$</p>
<p>$f(A,B) = cos(A) + cos(B) - cos(A+B)$</p>
<p>within the region R:</p>
<p>$$A+B<π,$$</p>
... |
773,880 | <p>What approach would be ideal in finding the integral $\int4^{-x}dx$?</p>
| Felix Marin | 85,343 | <p>$$
\int 4^{-x}\,{\rm d}x =
-\,{1 \over \ln(4)}\int\left[-\ln(4)\,4^{-x}\right]\,{\rm d}x
=-\,{1 \over \ln(4)}\int{{\rm d 4^{-x}} \over {\rm d}x}\,{\rm d}x
=-\,{4^{-x} \over \ln(4)} + \mbox{a constant}
$$</p>
|
202,040 | <p>I'd like to get separate plots for the functions in a list, and I'm trying the following, which doesn't work. What is the correct way to do that?</p>
<pre><code>Table[ContourPlot3D[f, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}], {f, {x + y + z + x y z == 0, x + y + z^2 + x y z^2 == 0, x + y^2 + z + x y^2 z == 0}}]
</code><... | NonDairyNeutrino | 46,490 | <p>Instead of <code>Table</code> you could use <code>Map</code></p>
<pre><code>ContourPlot3D[#, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}] & /@ {x + y + z + x y z == 0, x + y + z^2 + x y z^2 == 0, x + y^2 + z + x y^2 z == 0}
</code></pre>
|
1,342,570 | <p>So, this was my initial proof:</p>
<hr>
<p>Assume $R$ is a ring, and $a,b\in R$</p>
<p>Let $x_1$ and $x_2$ be solutions of $ax=b$</p>
<p>Hence, $ax_1=b=ax_2 \Rightarrow ax_1-ax_2=0_R \Rightarrow a(x_1-x_2)=0_R$</p>
<p>Thus, we have $x_1-x_2=0_R \Rightarrow x_1=x_2$, and only one solution exists.</p>
<hr>
<p>O... | Robert Lewis | 67,071 | <p>If</p>
<p>$ax_1 = ax_2 = b, \tag{1}$</p>
<p>with $a \in R$ a unit, then since we have $c \in R$ with $ac = ca = 1_R$, </p>
<p>$x_1 = 1_R x_1 = (ca)x_1 = c(ax_1)$
$= c(ax_2) = (ca)x_2 = 1_R x_2 = x_2, \tag{2}$</p>
<p>so the solution is unique. We further note that</p>
<p>$ax = b \tag{3}$</p>
<p>yields</p>
<p>... |
1,416,998 | <p>In the definition of martingales, one finds in Stroock and Varadhan (Multidimensional Diffusion processes - page 20) the strange request that it be right-continuous process.</p>
<p><a href="https://i.stack.imgur.com/0Nni7.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/0Nni7.png" alt="enter image... | Inuyaki | 244,122 | <p>If $(a, b) = d$, then we have
$$a+b = d\cdot\left(\frac{a}{d} + \frac{b}{d}\right)\text{ and }a-b = d\cdot\left(\frac{a}{d} - \frac{b}{d}\right)$$
and obviously then
$$((a+b), (a-b)) = \left(d\cdot\left(\frac{a}{d} + \frac{b}{d}\right), d\cdot\left(\frac{a}{d} - \frac{b}{d}\right)\right) = d\cdot\left(\left(\frac{a}... |
3,250,061 | <blockquote>
<p>Prove that if <span class="math-container">$p\equiv 5\pmod{8}$</span>, <span class="math-container">$p>5$</span> then <span class="math-container">$\zeta_p$</span> not constructible </p>
</blockquote>
<p>How to do this? There is a theorem in my book that says that the regular <span class="math-con... | lhf | 589 | <p><span class="math-container">$\zeta_p$</span> has degree <span class="math-container">$\phi(p)=p-1 \equiv 4 \bmod 8$</span>.</p>
<p>Now, <span class="math-container">$p-1 > 4$</span> because <span class="math-container">$p>5$</span>. If <span class="math-container">$p-1$</span> were a power of <span class="ma... |
1,409,545 | <p>I'm not sure if there is an actual solution to this problem or not, but thought I would give it a shot here to see if anyone has any ideas. So here goes:</p>
<p>I basically have three vertices of a rigid triangle with known 3D coordinates. The vertices are projected onto a 2D plane (by projection, I mean that each ... | haqnatural | 247,767 | <p>Your work is correct
$$\frac { d\left( 7+5^{ x^{ 2 }+2x-1 } \right) }{ dx } =\frac { d\left( 7 \right) }{ dx } +\frac { d\left( 5^{ x^{ 2 }+2x-1 } \right) }{ dx } =0+\left( 5^{ x^{ 2 }+2x-1 } \right) \ln { 5 } \frac { d\left( x^{ 2 }+2x-1 \right) }{ dx } =\\ =\left( 5^{ x^{ 2 }+2x-1 } \right) \ln { 5 } \left( 2x... |
2,091,766 | <p>Suppose $h:R \longrightarrow R$ is differentiable everywhere and $h'$ is continuous on $[0,1]$, $h(0) = -2$ and $h(1) = 1$. Show that:
<p> $|h(x)|\leq max(|h'(t)| , t\in[0,1])$ for all $x\in[0,1]$</p>
<p>I attempted the problem the following way:
Since $h(x)$ is differentiable everywhere then it is also continuous ... | quid | 85,306 | <p>First, your definition of $S$ is not quite precise. Let us sidestep this and take for the final set the union of all the sets you write down. That is, </p>
<p>$$(d=2) \to S_2= \{a, \frac{a+b}{2},b\}$$
$$(d=3) \to S_3= \{a, \frac{2a+b}{3}, \frac{a+2b}{3}, b\}$$
$$(d=4)\to S_4= \{a, \frac{3a+b}{4}, \frac{a+b}{2}, \fr... |
1,666,642 | <p><a href="https://mathematica.stackexchange.com/questions/107859/puzzle-with-mathematica">This puzzle</a> was found from the <em>Hot Network Questions</em> on the right. A repost of a question in a way (from Mathematica SE), but I was wondering if the following puzzle could be done:</p>
<p><a href="https://i.stack.im... | joriki | 6,622 | <p>You can reduce the effort to a level that can be handled without a computer.</p>
<p>Let's write the calculation as</p>
<p>\begin{array}{cc}
&A&B\\
\times&&C\\\hline
&D&E\\
+&F&G\\\hline
&H&I
\end{array}</p>
<p>There are strong restrictions on $A$ and $C$, since their produc... |
1,666,642 | <p><a href="https://mathematica.stackexchange.com/questions/107859/puzzle-with-mathematica">This puzzle</a> was found from the <em>Hot Network Questions</em> on the right. A repost of a question in a way (from Mathematica SE), but I was wondering if the following puzzle could be done:</p>
<p><a href="https://i.stack.im... | Ravi Yadav | 773,069 | <p><span class="math-container">$1 7* 4$</span> </p>
<p><span class="math-container">$=6 8
+2 5$</span> </p>
<p><span class="math-container">$9 3$</span></p>
<p>is the right answer! </p>
<blockquote>
<p><span class="math-container">$$17*4 = 68 + 25 = 93$$</span> </p>
</blockquote>
<p>There may be more combinatio... |
99,378 | <p>The following equation in $\mathbb{C}$:</p>
<p>$4z^2+8|z|^2-3=0$</p>
<p>is not algebraic and has 4 solutions : $\pm\frac{1}{2}$ and $\pm i\frac{\sqrt{3}}{2}$.
The Solve function in Mathematica only returns the 2 real values :</p>
<pre><code>Solve[4 z^2 + 8 Abs[z]^2 - 3 == 0, Complexes]
(* {{z -> -(1/2)}, {z -... | Georges Perros | 35,555 | <p>Actually, Reduce did it :</p>
<pre><code>Reduce[4 z^2 + 8 Abs[z]^2 - 3 == 0,z, Complexes]
</code></pre>
<blockquote>
<p>z == -(1/2) || z == 1/2 || z == -((I Sqrt[3])/2) || z == (I Sqrt[3])/2</p>
</blockquote>
<p>Or using the option Method-> Reduce in Solve :</p>
<pre><code>Solve[ 4 z^2 + 8 Abs[z]^2 - 3 == 0, z... |
103,397 | <p>Is there functionality in <em>Mathematica</em> to expand a function into a series with Chebyshev polynomials? </p>
<p>The <code>Series</code> function only approximates with Taylor series.</p>
| Jason B. | 9,490 | <p>You can just take <a href="https://mathematica.stackexchange.com/users/9362/bob-hanlon">Bob Hanlon</a>'s <a href="http://forums.wolfram.com/mathgroup/archive/2006/Aug/msg00165.html" rel="noreferrer">answer</a> from 2006 directly, and modify the plot just a bit to update it.</p>
<pre><code>ChebyshevApprox[n_Integer?... |
103,397 | <p>Is there functionality in <em>Mathematica</em> to expand a function into a series with Chebyshev polynomials? </p>
<p>The <code>Series</code> function only approximates with Taylor series.</p>
| Michael E2 | 4,999 | <p>Here's a way to leverage the Clenshaw-Curtis rule of <code>NIntegrate</code> and Anton Antonov's answer, <a href="https://mathematica.stackexchange.com/questions/26401/determining-which-rule-nintegrate-selects-automatically/96663#96663">Determining which rule NIntegrate selects automatically</a>, to construct a piec... |
3,516,921 | <p>Let <span class="math-container">$f : [−1, 0] → \mathbb{R}, x → x − x^2, n ∈ \mathbb{N}$</span> and let <span class="math-container">$P_n : x_0, . . . , x_n$</span> be an equal partition of <span class="math-container">$[−1, 0]$</span>.</p>
<ul>
<li>Compute the Riemann sum <span class="math-container">$S_{P_n} (f, ... | Community | -1 | <p><span class="math-container">$x_j=-1+j/n\implies z_j=(x_j-x_{j-1})/2+x_{j-1}=x_{j-1}+1/(2n)=-1+(j-1)/n+1/(2n)=-1+(2j-1)/(2n)$</span>. </p>
<p>So <span class="math-container">$f(z_j)=-1+(2j-1)/(2n)-(-1+(2j-1)/(2n))^2=-2+3(2j-1)/(2n)-(2j-1)^2/(4n^2)$</span>.</p>
<p>So <span class="math-container">$R=\sum_{j=1}^n f(... |
258,332 | <blockquote>
<p>Prove that if $a^x=b^y=(ab)^{xy}$, then $x+y=1$.</p>
</blockquote>
<p>How do I use logarithms to approach this problem?</p>
| Brian M. Scott | 12,042 | <p>No logarithms are needed:</p>
<p>$$a^x=(ab)^{xy}=a^{xy}b^{xy}=\left(a^x\right)^y\left(b^y\right)^x=\left(a^x\right)^y\left(a^x\right)^x=\left(a^x\right)^{x+y}$$</p>
|
258,332 | <blockquote>
<p>Prove that if $a^x=b^y=(ab)^{xy}$, then $x+y=1$.</p>
</blockquote>
<p>How do I use logarithms to approach this problem?</p>
| P.K. | 34,397 | <p>How about this?$$\begin{align} &(ab)^{xy} \\ =& a^{xy}\cdot b^{xy} \\ = & (a^x)^y \cdot (b^y)^x \\ = & (a^x)^y \cdot (a^x)^x \\ = & (a^x)^{x + y} \end{align}$$It suffices to say that $x + y = 1.$</p>
|
712,736 | <p>For any continuous function $f(z)$ of period $1$. Show that $\varphi'=2\pi \varphi+f(t)$ has a unique solution of period $1$.</p>
<p>Is this problem wrong with the counter example $\varphi(t)=e^{2\pi t}$. Shall we change it into $\varphi'=2\pi i\varphi+f(t)$</p>
| mookid | 131,738 | <p>Look for a solution $\varphi(t) = z(t)\exp(2\pi t)$, then
$$
z'(t) = f(t) \exp(-2\pi t)\\
z(t) = z(0) + \int_0^t f(s) \exp(-2\pi s) ds\\
\varphi(t) = \varphi(0)\exp(2\pi t)+ \int_0^t f(s) \exp(2\pi (t-s))ds
$$</p>
<p>Now if the solution is 1 periodic:
$$
\varphi(0) =\varphi(1)\\
= \varphi(0)\exp(2\pi )+ \int_0^1 f... |
3,279,878 | <p>I got this equation while I was trying to solve a certain math Olympiad problem. I tried modulus and whatnot, but I haven't got anywhere. Is there a way to prove this?</p>
| Will Jagy | 10,400 | <p>Just for culture:</p>
<p><a href="https://i.stack.imgur.com/7kBhg.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/7kBhg.jpg" alt="enter image description here"></a></p>
<p>Showing what numbers are NOT represented is the easy part. These 102 ternary forms are special (among positive forms) in tha... |
3,840,699 | <p>I need to calculate something of the form</p>
<p><span class="math-container">\begin{equation}
\int_{D} f(\mathbf{x}) d\mathbf{x}
\end{equation}</span></p>
<p>with <span class="math-container">$D \subseteq \mathbb{R^2}$</span>, but I only have available <span class="math-container">$f(\mathbf{x})$</span> at given sa... | nicomezi | 316,579 | <p>There is no "best" estimation method in general. For an extreme example, if <span class="math-container">$f$</span> is linear over <span class="math-container">$\mathbb{R}^2$</span>, then knowing <span class="math-container">$f$</span> at three non-aligned points is enough to compute the integral exactly o... |
4,618,433 | <p>Just a heads up: "<span class="math-container">$a$</span>" and "<span class="math-container">$α$</span>" are different</p>
<p>Let <span class="math-container">$a,b \in \Bbb R$</span> and suppose <span class="math-container">$a^2 − 4b \neq 0$</span>. Let <span class="math-container">$\alpha$</span... | Ethan Bolker | 72,858 | <p>Here's an answer that needs very little algebra.</p>
<p>If the roots are real their difference <span class="math-container">$c$</span> is real.</p>
<p>If the roots are complex they are conjugates, so their difference is a real multiple of <span class="math-container">$i$</span>.</p>
|
3,391,280 | <p>Prove by Induction on n that <span class="math-container">$\exists x,y,z \in Z$</span> s.t. <span class="math-container">$x\ge 2, y\ge 2, z\ge 2$</span> satisfies <span class="math-container">$x^2+y^2=z^{2n+1}$</span> </p>
<p>I'm a lot more comfortable with proving induction with <span class="math-container">$\for... | Batominovski | 72,152 | <p><strong>Remark.</strong> This problem is much easier to prove without induction. But, well, since it is required, I will oblige. However, if you look carefully, this is exactly the same as what I wrote in my comment under the OP's question.</p>
<p>For each integer <span class="math-container">$n\geq 0$</span>, w... |
3,905,331 | <p>I need to prove that <span class="math-container">$\lim_\limits{n\to \infty}$$\sqrt{\frac{n^2+3}{2n+1}} = \infty$</span> (series) by using the definition:</p>
<p>"A sequence <span class="math-container">$a_n$</span> converges to <span class="math-container">$\infty$</span> if, for every number <span class="math... | Ameet Sharma | 122,510 | <p>The proof isn't right. What you proved is that</p>
<p><span class="math-container">$\frac{n^2+3}{2n+1}>N \implies n>M^2$</span> for <span class="math-container">$N=M^2$</span></p>
<p>This strategy would have worked if you had double implication type deductions... so then your reasoning would work backwards as ... |
2,264,021 | <p>Can you help me explain the basic difference between Interior Point Methods, Active Set Methods, Cutting Plane Methods and Proximal Methods.</p>
<p>What is the best method and why?
What are the pros and cons of each method?
What is the geometric intuition for each algorithm type?</p>
<p>I am not sure I understand ... | littleO | 40,119 | <p>This is only a partial answer, but it's too long for a comment:</p>
<p>Interior point methods are similar in spirit to Newton's method. Like Newton's method, they require solving a large linear system of equations at each iteration, and they converge to high accuracy in a small number of iterations (typically 30 o... |
700,004 | <p>I have been working on this proof for a few hours and I can not make it work out.</p>
<p>$$\sum_{i=1}^{n}\frac{1}{i(i+1)}=1-\frac{1}{(n+1)}$$</p>
<p>i need to get to $1-\frac{1}{k+2}$</p>
<p>I get as far as $$1-\frac{1}{k+1}+\frac{1}{(k+1)(k+2)}$$
then I have tried $1-\frac{(k+2)+1}{(k+1)(k+2)}$ which got me exac... | MCT | 92,774 | <p>To finish your method:</p>
<p>We have $1 - \frac{1}{k+1} + \frac{1}{(k+1)(k+2)}$. You made a slight algebraic mistake -- you didn't distribute the negative when you combined fractions. If you did, you would have gotten $1 - \frac{(k+2) - 1}{(k+1)(k+2)} = 1 - \frac{1}{k+2}$, thus completing the proof.</p>
|
308,856 | <p>A set $E\subseteq \mathbb{R}^d$ is said to be Jordan measurable if its inner measure $m_{*}(E)$ and outer measure $m^{*}(E)$ are equal.However, Lebesgue mesure theory is developed with only outer measure. </p>
<p>A function is Riemann integrable iff its upper integral and lower integral are equal.However, in Lebesg... | Fedor Petrov | 4,312 | <p>I think, the reason is that if the ground space has infinite measure, you can not define the measurable sets as those for which inner measure equals the outer measure: it may happen that both are infinite, while the set is still not measurable.</p>
<p>Note also (this may be related) that outer and inner regularity b... |
308,856 | <p>A set $E\subseteq \mathbb{R}^d$ is said to be Jordan measurable if its inner measure $m_{*}(E)$ and outer measure $m^{*}(E)$ are equal.However, Lebesgue mesure theory is developed with only outer measure. </p>
<p>A function is Riemann integrable iff its upper integral and lower integral are equal.However, in Lebesg... | Martin Väth | 165,275 | <p>The asymmetry has only historical reasons. It is possible to develop Lebesgue theory (moreover, all extension theorems and thus the theory of product measures) from the “inner approach”. This was done by Heinz König in a couple of papers and monographs.</p>
<p>Although this “inner approach” is for the Lebesgue measu... |
2,446,282 | <p>The maximum value of the function $f(x)= ax^2+bx+c$ is 10. Given that $f(3)=f(-1)=2$, find $f(2)$</p>
<p>The answer is $f(2)=8$</p>
<p>I thought that by maximum value it meant that c=10, but the equation I got gave as a result $f(2)=10$</p>
<p>Any hint on how to solve it?</p>
| Carl Christian | 307,944 | <p>Machine epsilon $\epsilon$ is the distance between 1 and the next floating point number. </p>
<p>Machine precision $u$ is the accuracy of the basic arithmetic operations. This number is also know as the unit roundoff.</p>
<p>When the precision is $p$ and the radix is $\beta$ we have
$$ \epsilon = \beta^{1-p}.$$
To... |
2,705,794 | <p>I ran across this problem on a practice Putnam worksheet. Completely stumped.</p>
<p>Is $$\large \frac{m^{6} + 3m^{4} + 12m^{3} + 8m^{2}}{24}$$ an integer for all $m \in \mathbb{N}$?</p>
<p>I suspect it is an integer for any $m$. It checks out for small cases.</p>
<p>Any hints for proving the general case?</p>
| VyshnavPT | 546,126 | <p>This can also be tackled using Mathematical induction.
It is clear that m=1 holds.
Assume it holds for a number k.Substitue k+1in place of k and cancel the divisors of 24 and then we get another polynomial.Then we must start the whole process for this polynomial(ie prove that 24 divides the new polynomial).Eventuall... |
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