qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
2,814,703 | <p>I am reading <a href="https://en.wikipedia.org/wiki/Lower_limit_topology" rel="nofollow noreferrer">lower limit topology</a> on Wikipedia, which states that the lower limit topology </p>
<blockquote>
<p>[...] is the topology generated by the basis of all half-open intervals $[a,b)$, where a and b are real numbers... | Lee Mosher | 26,501 | <p>$$(a,b) = \bigcup_{n=1}^\infty \, \left[\, \left(1-\frac{1}{n}\right)\, a + \frac{1}{n} b ,\, b\right)
$$</p>
|
2,243,083 | <p>I'm writing an advanced interface, but I don't yet have a concept of derivatives or integrals, and I don't have an easy way to construct infinite many functions (which could effectively delay or tween their frame's contributing distance [difference between B and A] over the next few frames).</p>
<p>I can store valu... | user7530 | 7,530 | <p>How about this: each frame, set A's position to a weighted average of B's new position and A's previous position:</p>
<p>$$A = (1-\alpha)A + \alpha B.$$</p>
<p>Tune alpha to taste.</p>
|
1,362,860 | <p>$$\frac{1}{3}+\frac{1}{13}+\frac{1}{23}+\frac{1}{31}+\frac{1}{37}+\frac{1}{43}\cdots$$
Intuitively, I feel that this sum converges, but I really don't know why, (or if I am correct). Can I have a somewhat rigorous proof of whether or not this sum converges or diverges? Thank you lots for any help.</p>
| Charles | 1,778 | <p>By the Prime Number Theorem in arithmetic progressions (the extension of Dirichlet's Theorem), not only are there infinitely many primes which are 3 mod 10, but these primes are asymptotically 1/phi(10) = 1/4 of all primes. Thus, since the reciprocal sum of the primes diverges, so must the reciprocal sum of the prim... |
3,808,575 | <p>Assuming I have the statement ∀x(∀y¬Q(x,y)∨P(x)), can I pull the universal quantifier ∀y out of the parenthesis? Meaning, is this statement equivalent to ∀x∀y(¬Q(x,y)∨P(x)) ?</p>
<p>An approach I tried so far:</p>
<ol>
<li>∀x((∃y Q(x,y) ) => P(x)). (original eq.)</li>
<li>∀x((∀y¬Q(x,y))∨P(x)) (... | Manx | 483,923 | <p>For universal quantifier. In general, if <span class="math-container">$x$</span> apear in both <span class="math-container">$A$</span> and <span class="math-container">$B$</span> we have
<span class="math-container">$$\exists xA(x)\to \forall xB(x)\Rightarrow\forall x(A(x)\to B(x))\tag{1}$$</span>
<span class="math-... |
909,228 | <p>I'm trying to find a closed form for the following sum
$$\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n},$$
where $H_n=\displaystyle\sum_{k=1}^n\frac{1}{k}$ is a harmonic number.</p>
<p>Could you help me with it?</p>
| Robert Israel | 8,508 | <p>Start with the series $$\sum_{n=1}^\infty H_n z^n = - \dfrac{\ln(1-z)}{1-z} = f_0(z) $$ </p>
<p>Then (according to Maple 18)
$$ \sum_{n=1}^\infty \dfrac{H_n}{n} z^n = \int_0^z \dfrac{f_0(t)}{t}\; dt = \operatorname{Li}_{2}(1-z) + \dfrac{\ln(1-z)^2}{2} = f_1(z)$$ </p>
<p>$$\displaystyle \sum_{n=1}^\infty \dfra... |
909,228 | <p>I'm trying to find a closed form for the following sum
$$\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n},$$
where $H_n=\displaystyle\sum_{k=1}^n\frac{1}{k}$ is a harmonic number.</p>
<p>Could you help me with it?</p>
| Tunk-Fey | 123,277 | <p>In the same spirit as Robert Israel's answer and continuing <a href="https://math.stackexchange.com/a/605602/123277">Raymond Manzoni's answer</a> (both of them deserve the credit because of inspiring my answer) we have
$$
\sum_{n=1}^\infty \frac{H_nx^n}{n^2}=\zeta(3)+\frac{1}{2}\ln x\ln^2(1-x)+\ln(1-x)\operatorname{... |
997,602 | <blockquote>
<p>Prove that the function <span class="math-container">$x \mapsto \dfrac 1{1+ x^2}$</span> is uniformly continuous on <span class="math-container">$\mathbb{R}$</span>.</p>
</blockquote>
<p>Attempt: By definition a function <span class="math-container">$f: E →\Bbb R$</span> is uniformly continuous iff for ... | mookid | 131,738 | <p><strong>Hint:</strong> use the inequality
$$x>0\implies
x < 1 + x^2
$$
(if $x<1$ it is true; otherwise via multiplication by $x$, $x>1\implies x^2>x$)</p>
|
997,602 | <blockquote>
<p>Prove that the function <span class="math-container">$x \mapsto \dfrac 1{1+ x^2}$</span> is uniformly continuous on <span class="math-container">$\mathbb{R}$</span>.</p>
</blockquote>
<p>Attempt: By definition a function <span class="math-container">$f: E →\Bbb R$</span> is uniformly continuous iff for ... | Paul | 17,980 | <p>You are nearly finishing the proof.</p>
<p>$$|x - a| (\frac{|x|}{(1 + x^2)(1 + a^2)} + \frac{|a|}{(1 + x^2)(1 + a^2)})\le |x - a| (\frac{1}{2(1 + a^2)} + \frac{1}{2(1 + x^2)})\le |x-a|$$</p>
<p>Take $\delta=\epsilon$.</p>
|
2,449,443 | <p>Set of numbers $\ x_1, \ldots, x_m , y_1, \ldots, y_n $ where $\ x_i=0 $ for $i = 1,\ldots, m$ and $\ y_i=1 $ for $i = 1,\ldots, n$</p>
<p>Show that mean $M$ of this set is given by $\frac{n}{m+n}$ and the standard deviation $S$ by $\frac{ \sqrt{mn}} {m+n} $</p>
<p>I know the definitions of the mean and standard ... | BruceET | 221,800 | <p>The sample variance of observations denoted $x_i$ is $$S^2 = \frac{\sum_{i=1}^n (x_i - \bar x)^3}{n-1}
= \frac{\sum_{i=1}^n x_i^2\; -\; (\sum_{i=1}^n x_i)^2/n}{n-1}.$$</p>
<p>The middle member of the equation is the definition and the last member is
sometimes called the 'computational formula' (easily deduced from... |
2,030,116 | <p>How can i prove that $\sqrt[12]{2}$ is irrational number? </p>
<p>I'm trying: </p>
<p>$$\sqrt[12]{2} = \frac{p}{q}$$ where $p$, $q$ are integers</p>
<p>it follows that :</p>
<p>$$p^{12} = 2q^{12} $$</p>
<p>What is argument of irrationality in this case?
From what we know that the right-hand side has an even n... | mathcounterexamples.net | 187,663 | <p>A (late) proof not using <strong>unique factorization theorem</strong>, but only <strong>Euclid's lemma</strong>.</p>
<p>Suppose that $p,q$ are coprime integers. As $p^{12} = 2q^{12}$, you get that $2 \ | \ p^{12}$. As $2$ is a prime number $2 \ | \ p$. Therefore $p = 2 p^\prime$ and $$2^{12} \left(p^\prime \right... |
105,857 | <p>Let $\mathcal{O}$ be the ring of integers in an algebraic number field. Is $\text{SL}_2(\mathcal{O})$ generated by elementary matrices? If it isn't, is there any other natural generating set for it?</p>
<p>The usual argument shows that this is true for $\mathcal{O} = \mathbb{Z}$ (or, more generally, a Euclidean d... | Charlie Frohman | 4,304 | <p>No. If you look in Charles Frohman and Benjamin Fine, "Some Amalgam Structures for Bianchi Groups," 1988, Proceedings of the American Mathematical Society, Vol. 102, No. 2, pp. 221-229, we construct a splitting of $PSl_2(\mathcal{O})$ where we are in a $\mathbb{Q}[\sqrt{-d}]$ for d a positive square free integer tha... |
2,199,222 | <p>I have the feeling of being stuck or missing something trying to prove
$$ \lim_{N\to\infty}\sum_{k=1}^{N} \frac{1}{N+k} =\int_{1}^{2} \frac{1}{x} dx = ln(2)$$</p>
<p>Using Riemann-Sums I have shown that $$\int_{1}^{a} \frac{1}{x} dx=\lim_{N\to\infty}\sum_{k=1}^{N} (a^{1/N}-1)=\lim_{N\to\infty}N(a^{1/N}-1)=\lim_{h\t... | Stella Biderman | 123,230 | <p>I would call it the sum of a skew-symmetric and a diagonal matrix.</p>
|
1,618,753 | <p>Trying to expand $f(x)=\cot(x)$ to Taylor series (Maclaurin, actually).
But I keep "adding up" infinities when using the formula. (Because of $\cot(0)=\infty$) Could you perhaps give me a hint on how to proceed?</p>
| Travis Willse | 155,629 | <p>Since $f$ is not defined at $0$, its Maclaurin series is undefined.</p>
<p>On the other hand, the pole of $f$ at $0$ is simple, and it's not hard to compute that the residue of $f$ there is $1$, so one <em>can</em> compute a Maclaurin series for $\cot x - \frac{1}{x}$ there, namely,
$$\cot x - \frac{1}{x} \sim -\fr... |
644,163 | <p>The question asks:
Find the line through $(3,1,-2)$ that intersects and is perpendicular to</p>
<p>$$x = -1 + t, y = -2 + t, z = -1 + t.$$</p>
<p>My thoughts:
Say the point of intersection is $(x_0,y_0,z_0)$, then my line can be of the form</p>
<p>$$L(s) = (3,1,-2) + (x_0- 3,y_0- 1,z_0+ 2)s$$</p>
<p>Then I tried... | Mhenni Benghorbal | 35,472 | <p>Here is an approach. Find a normal vector to the directional vector $(-1,-2,-1)$ of the given line, then use it (together with finding a point on the line $L$) to find the equation of the perpendicular line. </p>
|
898,683 | <p>Given a pool of 30 balls (5 of each color). When drawing 8 balls without replacement, what is the probability of getting at least one of each color?</p>
<p>Related: <a href="https://math.stackexchange.com/questions/897730/probability-of-drawing-at-least-one-red-and-at-least-one-green-ball">Probability of drawing at... | bobbym | 77,276 | <p>Similar problems appear in</p>
<p>S. Ghahramani, Fundamentals of Probability with Stochastic Processes, 3rd ed. 2005. p73</p>
<p>and </p>
<p>P. J. Nahin, Digital Dice: Computational Solutions to Practical Probability Problems 2008., p. 237</p>
<p>using their methods and Mathematica which is not as succinct as us... |
2,475,757 | <p>I want to determine if the following integrals converge or diverge.</p>
<ol>
<li>$\int_{0}^\infty \frac{\sqrt{x}}{\sqrt[3]{x^5+1}}dx.$</li>
<li>$\int_{0}^\infty \sin\frac{1}{x^2+1}dx$.</li>
<li>$\int_{\sqrt{2}}^2 \frac{dx}{\sqrt{x^2-2}}dx.$</li>
<li>$\int_{0}^1 \frac{\ln{x}}{x}dx.$</li>
</ol>
<hr>
<p><strong>(1):... | Amanda R. | 335,482 | <p>$S={(1,1),(1,2), (1,3)
(2,1), (2,2), (2,3),
(3,1), (3,2), (3,3)}$</p>
<p>Therefore the sample space contains 9 items/events.</p>
<p>We assume that each combinations has an equal chance of being chosen, so each combination has a $1/9$ chance of being chosen.</p>
<p>$Pr(X=2)=Pr((1,2), (2,1) or (2,2))=1/... |
2,941,106 | <p>I have tried 29.2/8.44 and tried multiplying this to get whole numbers but doesn't seem like it's working </p>
| Phil H | 554,494 | <p><span class="math-container">$844:2920$</span> divided by <span class="math-container">$4 = 211:730$</span>. </p>
<p><span class="math-container">$211$</span> is a prime number so it won't reduce further.</p>
|
2,516,123 | <p>Problem 11985, by Donald Knuth, <em>American Mathematical Monthly</em>, June-July, 2017:</p>
<blockquote>
<p>For fixed $s,t \in \mathbb{N}$. with $s\leq t$. let $a_{n}=\sum\limits_{k=s}^{t}$ $ {n}\choose{k}$. Prove that this sequence is log-concave, namely that $a_{n}^{2}\geq a_{n-1}a_{n+1} \ \forall n\geq 1$. </... | epi163sqrt | 132,007 | <p>This answer is based upon a result stated as example 1.3 in the paper <em><a href="https://arxiv.org/abs/math/0504164" rel="nofollow noreferrer">Log-concavity and LC-positivity</a></em> by Yi Wang and Yeong-Nan Yeh.</p>
<blockquote>
<p>In the following we consider natural numbers <span class="math-container">$0\leq ... |
1,022,098 | <p>Determine whether the series </p>
<p>$\sum _{n=1}^{\infty }\:\frac{n^2-5n}{\sqrt{n^7+2n+1}}$</p>
<p>is convergent or divergent.</p>
<p>So far in class i've learned a lot of different test to use, but i'm having trouble finding out what test would be 'most pratical' for this problem. The ratio test would be too co... | Stella Biderman | 123,230 | <p>Sets are defined by their elements. If you list out the elements, it immediately forms a set from the rules of ZFC (which is the axiomatic definition of how one can construct sets). This also works for infinite cases, for example, $\{1,2,\ldots\}$ defines a set as well, with the minor counterexample of exceptionally... |
1,022,098 | <p>Determine whether the series </p>
<p>$\sum _{n=1}^{\infty }\:\frac{n^2-5n}{\sqrt{n^7+2n+1}}$</p>
<p>is convergent or divergent.</p>
<p>So far in class i've learned a lot of different test to use, but i'm having trouble finding out what test would be 'most pratical' for this problem. The ratio test would be too co... | hmakholm left over Monica | 14,366 | <p>I assume that $\langle ~,~\rangle$ means an ordered pair.</p>
<p><strong>No</strong>, in standard set theory there is no set which has everything of the form $\langle A,\varnothing\rangle$ as elements.</p>
<p>Namely, if such a thing existed -- let's call it $X$ -- we could make a variant of Russell's paradox, by c... |
2,710,681 | <p>If I have a function of three variables and I want to create a new function in which it equals the other function squared, could I literally just square the other function or does this violate any rules? Would this also mean its gradient vector is just squared at a certain point?</p>
| Community | -1 | <blockquote>
<p>could I literally just square the other function </p>
</blockquote>
<p>Yes. The square of $f(x,y,z)$ is $[f(x,y,z)]^2$. No violations. For example, if $f(x,y,z) = xyz$, then the square of this function is $g(x,y,z) = (xyz)^2 = x^2y^2z^2$.</p>
<blockquote>
<p>Would this also mean its gradient ve... |
76,683 | <p>How do I force mathematica to display the below expression as a sum <code>a+b</code> with a scaling factor of <code>1/r</code>.</p>
<p>(a+b)/r</p>
<p>I would like Mathematica to display (1/r) (a+b), ie. I want it to show 1/r as a scaling factor. </p>
<p>currently, it shows (a+b)/r , with r as a common denomi... | Themis | 7,863 | <p>Here is something close to what you want</p>
<pre><code>expr = (a + b)/r;
expr Denominator[expr] HoldForm[1/Denominator[expr] // Evaluate]
</code></pre>
<p><a href="https://i.stack.imgur.com/O2EHD.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/O2EHD.png" alt="output"></a></p>
|
476,899 | <p>Does someone know a proof that $\{1,e,e^2,e^3\}$ is linearly independent over $\mathbb{Q}$?</p>
<p>The proof should not use that $e$ is transcendental.</p>
<p>$e:$ Euler's number.</p>
<p><a href="http://paramanands.blogspot.com/2013/03/proof-that-e-is-not-a-quadratic-irrationality.html#.Uhv87tJFUnl">$\{1,e,e^2\}... | Git Gud | 55,235 | <p>Below is my attempt which is too long for a comment and may be saveable, (doubt it).</p>
<p>Consider the differential equation $y^{(4)}-6y^{(3)}+11y''-6y'=\textbf 0$, where $\bf 0$ is the null function over some non-trivial interval $I$ containing $1$.</p>
<p>The theory of ODE tells us that a basis of <a href="htt... |
3,016,252 | <p>A function like <span class="math-container">$f(x) = 2x$</span> can be defined over the reals so its “type signature” or in set theory domain and codomain is <span class="math-container">$f: \mathbb{R} \rightarrow \mathbb{R}$</span>. </p>
<p>I want to define a function <span class="math-container">$f(x) = 5$</span>... | Hans Hüttel | 289,137 | <p>Define the type <span class="math-container">$\mathsf{Five}$</span> as the singleton type populated by the type axiom <span class="math-container">$E \vdash 5 : \mathsf{Five}$</span>. Then the type of <span class="math-container">$f$</span> is <span class="math-container">$f : \mathbb{R} \rightarrow \mathsf{Five}$</... |
3,016,252 | <p>A function like <span class="math-container">$f(x) = 2x$</span> can be defined over the reals so its “type signature” or in set theory domain and codomain is <span class="math-container">$f: \mathbb{R} \rightarrow \mathbb{R}$</span>. </p>
<p>I want to define a function <span class="math-container">$f(x) = 5$</span>... | Rob Arthan | 23,171 | <p>The answer to this really depends on your type theory. The type theories of proof assistants like Coq or Agda will let you express any of the following classes as types:</p>
<p><span class="math-container">$$
\{f : \Bbb{R} \to \Bbb{R} \mid \forall x:\Bbb{R}\cdot f(x) = 5\} \\
\{f : \Bbb{R} \to \Bbb{R} \mid \exists ... |
4,051,403 | <p>I'm not a math major, but a philosophy major that likes to know that he knows what he's talking about. This may seem like a super stupid question, but here I go.</p>
<p>So Euclid made a lot of sense when he gave the example of the nature of multiplication. For example.
"2 x 3" is really 2 added to itself 3... | JJacquelin | 108,514 | <p>There was two mistakes in the equation :
<span class="math-container">$$y^2=\lambda x^\color{red}{2}-\frac{x^4}{\color{red}{2} }+C$$</span>
( Corrected in red ).</p>
<p>Then :
<span class="math-container">$$y=x'=\pm \sqrt{\lambda x-\frac{x^4}{2}+C}$$</span>
<span class="math-container">$$\frac{dx}{dt}=\pm \sqrt{\la... |
88,565 | <p>Today I had an argument with my math teacher at school. We were answering some simple True/False questions and one of the questions was the following:</p>
<p><span class="math-container">$$x^2\ne x\implies x\ne 1$$</span></p>
<p>I immediately answered true, but for some reason, everyone (including my classmates and ... | Colton Fitzjarrald | 1,086,061 | <p>If you'd like a more informal understanding of why the statement given is indeed true, consider the argument of the proposition. It's purely stating that if <span class="math-container">$x^2 \neq x$</span>, then <span class="math-container">$x \neq 1.$</span><br />
This statement is true. It's not arguing that 1 is ... |
2,757,870 | <p>I've come to this:
$$f: \mathbb{N} \to \{\ldots, -6,-4,-2,0,2,4,6, \ldots\},\qquad f(n) =
\begin{cases}
2n & \text{ if } n \text{ is odd} \\
-n & \text{ if } n \text{ is even}
\end{cases}$$</p>
<p>I don't know what to do with this though. I never know how to format a proof correctly.</p>
| feynhat | 359,886 | <p>As I pointed out in the comments, the function that you described is not surjective because $4$ (or any multiple of $4$) has no inverse image.</p>
<p>Assuming you define the set of even integers as, $\mathbb{E} = 2\mathbb{Z} = \{\dots, -6, -4, -2, 0, 2, 4, 6, \dots\}$</p>
<p>and the set of naturals as, $\mathbb{N}... |
3,415,624 | <p>I want to show that the sequence given in the title is convergent and find its limit. I'm not sure if I should use the monotone convergence theorem, because when I try using induction, I don't seem to get anywhere. And I also don't know how to find a suitable candidate for a limit. I do know what the definition of c... | Marios Gretsas | 359,315 | <p><span class="math-container">$$0 \leq a_n \leq \frac{\sqrt[n]{n^3}}{n} \to 0 $$</span></p>
<p>since <span class="math-container">$\sqrt[n]{n^3}\to 1$</span></p>
|
3,935,494 | <p>Usually the inverse of a square <span class="math-container">$n \times n$</span> matrix <span class="math-container">$A$</span> is defined as a matrix <span class="math-container">$A'$</span> such that:</p>
<p><span class="math-container">$A \cdot A' = A' \cdot A = E$</span></p>
<p>where <span class="math-container... | TheSimpliFire | 471,884 | <p>Let <span class="math-container">$X\in\{0,1,2,3\}$</span> be the number of boys. Let <span class="math-container">$A$</span> be the event that <span class="math-container">$X=3$</span> and <span class="math-container">$B$</span> be the event that <span class="math-container">$X\ge1$</span>. By definition, we have <... |
80,918 | <p>Could anyone please tell me what could be the math function to get the number of zeros in given decimal representation of numbers? I scratched my head on Combination and Permutation but couldn't come up with generic answer. The number length can be up to 1000 digits, so you can represent a number as a String.</p>
<... | Adam Boddington | 103,086 | <p>There's a nice algorithm for doing this calculation which is explained <a href="http://blog.codility.com/2011/12/mu-2011-certificate-solution.html" rel="nofollow">here</a>.</p>
<p>If $x$ is the number we're given, $f(x)$ is the number of zeros that appear in the range $1..x$. Using a simple program we can calculate... |
137,136 | <p><a href="https://i.stack.imgur.com/tk4kk.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/tk4kk.png" alt="Mathcad example of solving coil impedance"></a></p>
<p>I am new to Mathematica.</p>
<p>I am trying to figure out how to do the same thing in Mathematica as the image depicts done in Mathcad.<... | WReach | 142 | <p>The problem is due to two factors:</p>
<ol>
<li>The type system cannot infer the data type that results from a call to <code>MinMax</code>.</li>
<li>A type-checked application of the <code>Transpose</code> query operator will fail when applied to an argument with an unknown type (i.e. from <code>MinMax</code>).</li... |
564,378 | <p>Suppose we start with a rational number $a_0$, and define $a_{n+1}=2a_n^2-1$ for $n\geq 0$. For what $a_0$ will it be the case that $a_i=a_j$ for some $i\neq j$?</p>
<p>We can start with something like $a_0=1$, then $a_1=1$ so $a_0=a_1$.</p>
<p>If $a_0=0$, we get $0, -1, 1, 1, \ldots$</p>
<p>Likewise if $a_0=-1$,... | Sangchul Lee | 9,340 | <p>Let $a_{0} = \cos \theta$. Then it is easy to check that $a_{n} = \cos (2^{n}\theta)$. So if $a_{i} = a_{j}$ for some $i \neq j$, then we must have</p>
<p>\begin{align*}
\cos(2^{i}\theta) = \cos(2^{j}\theta)
&\quad \Longleftrightarrow \quad 2^{i}\theta = 2n\pi \pm 2^{j}\theta, \quad n \in \Bbb{Z} \\
&\quad... |
3,527,919 | <p>I've tried to prove this property of Bessel function but I don't seem to be going anywhere</p>
<p><span class="math-container">$$\sqrt{\frac 12 \pi x} J_\frac 32 (x) = \cfrac{\sin x}{x} - \cos x$$</span></p>
<p>I have tried substituting <span class="math-container">$\frac 32$</span> for <span class="math-container... | Kemono Chen | 521,015 | <p>Using one of the integral representations of Bessel function:
<span class="math-container">$$J_n(x)=\frac{2^{1-n} x^n \int_0^{\frac{\pi }{2}} \sin ^{2 n}t \cos (x \cos t) dt}{\sqrt{\pi } \Gamma \left(n+\frac{1}{2}\right)},$$</span>
We get
<span class="math-container">$$J_{3/2}(x)=\frac{x^{3/2}}{\sqrt{2\pi}}\int_0^{... |
1,931,754 | <p>I am trying to show that the interval $[0,1)$ is a closed subset of $(-1,1)$ by using the definition that a closed subset contains all of its limit points.
So for a convergent sequence $\{x_n\}$ in $[0,1)$ we have that $0 \leq x_{n} < 1$ for all $n \in \mathbf{N}$. How can I show that $\lim_{n \rightarrow \infty... | John Doe | 341,321 | <p>By applying an exponential of a logarithm to the expression we get
$$\lim_{x\to\infty} \left( 1+\log \left( \frac{x}{x-1} \right)\right)^x=\lim_{x\to\infty} \mathrm{exp}\left( x\log \left( 1+\log \left( \frac{x}{x-1}\right)\right)\right)$$</p>
<p>We will show that $$L:=\lim_{x\to\infty} \left( x\log \left( 1 + \log... |
1,355,133 | <p>A while ago I asked a question about probability here <a href="https://math.stackexchange.com/questions/1353044/why-is-binomial-probability-used-here/">Why is binomial probability used here?</a></p>
<p>I get that you can find how many ways of choosing the $6$ correct out of $10$ questions.</p>
<p>But why do we <st... | danimal | 202,026 | <p>$\binom{n}{k}$ is the number of ways of choosing $k$ things from $n$ things (without caring about the order), for example to get a $x^2$ term from (the binomial) $(1+x)^3=(1+x)(1+x)(1+x)$, I have to choose two $x$s but am free to choose which two, so I have $\binom{3}{2} = 3$ choices for this - I can discard any of ... |
2,361,516 | <p>Let $X$ be a real Hilbert space.Let $x,y \in X$ such that $\langle x,y\rangle >0$. If $\alpha \geq 1$.</p>
<p>I want to prove that $\Vert \alpha x-y \Vert \leq \Vert x-y \Vert$</p>
| stity | 285,341 | <p>Inequality is wrong : for $x=y \ne 0$ and $\alpha = 0$ you get $||\alpha x-y|| = ||y|| > 0 = ||x-y||$</p>
|
2,361,516 | <p>Let $X$ be a real Hilbert space.Let $x,y \in X$ such that $\langle x,y\rangle >0$. If $\alpha \geq 1$.</p>
<p>I want to prove that $\Vert \alpha x-y \Vert \leq \Vert x-y \Vert$</p>
| tattwamasi amrutam | 90,328 | <p>The inequality doesn't hold true in general. What you can show is this:
$$\|\alpha x+y \|^2=\langle \alpha x+y, \alpha x+y\rangle=|\alpha|^2\|x\|^2+2\alpha\langle x,y\rangle+\|y\|^2 \ge ||x+y||^2$$</p>
|
475,863 | <p>Let $f,P,Q$ three analytic functions. Here $P$ is a polynomial.</p>
<p>I want to solve this equation: $$f(s)=P(s)\exp(Q(s)).$$</p>
<p>The unknown here are $P, Q$ and $f$ is known. </p>
| Hagen von Eitzen | 39,174 | <p>Assume $f$ is entire and not the zero function.
Since the exponential is never zero, $P$ should capture all zeroes of $f$.
That is: $(s-s_0)$ is a factor of $P$ if and only if $f(s_0)=0$. More precisely, for $s\in\mathbb C$ let $$\nu(s)=\min\{\,n\in\mathbb N_0\mid f^{(n)}(s)\ne0\,\}.$$
Then we can (and up to a con... |
1,406,878 | <p>Given is following sequence:</p>
<p>$a_{n+1} = a_n - \frac{a_n - v}{s}$</p>
<p>I found out that</p>
<p>$\forall a_0, v, s \in \mathbb{R}, s>0: \lim\limits_{n \to \infty}a_n=v$</p>
<p>But I do not know why. I tried to write down $a_2$ , $a_3$, but the term becomes very long and complex, and it doesn't help me ... | Augustin | 241,520 | <p>You can write $a_{n+1}=\left(1-\frac{1}{s}\right)a_n+\frac{v}{s}$.</p>
<p>The form is $a_{n+1}=Aa_n+B$, with $A\neq 1$. Now let $r=\frac{B}{1-A}$. You can show that $b_n=a_n-r$ is a geometric sequence. So $b_n=b_0A^n$ and $a_n=r+(a_0-r)A^n$. I let you substitue with the values of $A$ and $B$, so you can find the ge... |
1,159,860 | <p>If $$f:[a,b]\times [c,d] \to \mathbb{R}$$ is continuous and $f_{y}$ is continuous, let $$F(x,y)=\int_{a}^{x} f(t,y)dt.$$ </p>
<ol>
<li>Find $F_x$ and $F_y$.</li>
<li>If $G(x)=\int_{a}^{g(x)}f(t,x)dt$, find $G'(x)$</li>
</ol>
<p>My try: </p>
<p>For (1) $$F(x+h,y)-F(x,y)=\int_{a}^{x+h} f(t,y)dt-\int_{a}^{x}f(t,y)dt... | RE60K | 67,609 | <p>$$\newcommand{\d}[1]{{\rm #1}}
\d P(\d A-\d B)=\d P(\d A)-\d P(\d A\d B)=\d P(\d A)-\d P(\d A)\d P(\d B)=a-ab$$
Since for independant events: $\d P(\d A\d B)=\d P(\d A)\d P(\d B)$</p>
|
1,535,731 | <p>I am currently self-learning about matrix exponential and found that determining the matrix of a diagonalizable matrix is pretty straight forward :).</p>
<p>I do not, however, know how to find the exponential matrix of a non-diagonalizable matrix.</p>
<p>For example, consider the matrix
$$\begin{bmatrix}1 & 0... | amd | 265,466 | <p>In the $2\times2$ case, you can find the exponential of a matrix $A$ without having to decompose it into $BMB^{-1}$ form. In particular, you only need the eigenvalues—you don’t need to find any eigenvectors. There are three cases, as follows.</p>
<hr>
<p><strong>Distinct Real Eigenvalues:</strong> Let $P_1 = (A-\l... |
2,346,489 | <p>I am studying the Proposition: Let $D$ be a Dedekind domain, $F$ its field of fractions, $E$ a finite dimensional extension field of $F$ and $D'$ the subring of $E$ of $D$ integral elements. Assume that $E/F$ is a finite separable field extension. Then $D'$ is a finitely generated $D$-module.</p>
<p>I have to show ... | D_S | 28,556 | <p>Every ideal of $D'$ is a $D$-submodule of $D'$, so if you have an ascending chain </p>
<p>$$I_1 \subseteq I_2 \subseteq \cdots$$</p>
<p>of ideals of $D'$, then it must terminate, because $D'$ is a Noetherian $D$-module.</p>
|
446,966 | <p>If a, b, c and d are real numbers, I would probably consider the following expressions equivalent.</p>
<p>$$a = b \cdot c \cdot d$$
$$a = (b \cdot c) \cdot d$$
$$a = b\cdot(c\cdot d)$$</p>
<p>If a, b, c, and d are four matrices, then the order is most defiantly right to left, like so:</p>
<p>$$a = b \cdot c \cdot... | amWhy | 9,003 | <p>Actually, matrix multiplication <strong>is associative</strong>, just as is scalar multiplication. </p>
<p>(Recall the properties of <a href="https://en.wikipedia.org/wiki/Matrix_multiplication#Properties_of_matrix_multiplication" rel="nofollow">matrix multiplication</a>.)</p>
<p>So for $A$, expressed as the pro... |
446,966 | <p>If a, b, c and d are real numbers, I would probably consider the following expressions equivalent.</p>
<p>$$a = b \cdot c \cdot d$$
$$a = (b \cdot c) \cdot d$$
$$a = b\cdot(c\cdot d)$$</p>
<p>If a, b, c, and d are four matrices, then the order is most defiantly right to left, like so:</p>
<p>$$a = b \cdot c \cdot... | Ben Grossmann | 81,360 | <p>Matrix multiplication is associative, so $(AB)C=A(BC)$ for compatible matrices $A,B,C$. It is not, on the other hand, commutative. That is, $AB\neq BA$ (assuming the latter multiplication can even be carried out).</p>
|
1,672,847 | <p>A stick of total length $1$ is split at a randomly selected point $X$, i.e. $X$ is uniformly distributed in the interval $[0, 1]$.</p>
<p>Determine the expected length of the piece that contains the point $1/3$.</p>
<p>I've figured out so far that I need to determine a function $f(x)$ so that the length of the pie... | deanavery | 418,076 | <p>Because of uniform, $X$ falls to the left of $1/3$ with probability $1/3$. The length of the piece that includes $1/3$ in this case is of length $1 - X$, which can range from $2/3$ to $1$. In expectation this length is $5/6$.</p>
<p>With probability $2/3$, $X$ falls to the right of $1/3$. The length of the piece th... |
3,258,642 | <blockquote>
<p>If the roots of quadratic equation <span class="math-container">$$x^2 − 2ax + a^2 + a – 3 = 0$$</span>
are real and less than <span class="math-container">$3$</span>, find the range of <span class="math-container">$a$</span>.</p>
</blockquote>
<p>The roots are <span class="math-container">$a... | user10354138 | 592,552 | <p>You can square both sides of an inequality <span class="math-container">$x<y$</span> to get <span class="math-container">$x^2<y^2$</span>, provided <span class="math-container">$x\geq 0$</span>.</p>
|
3,258,642 | <blockquote>
<p>If the roots of quadratic equation <span class="math-container">$$x^2 − 2ax + a^2 + a – 3 = 0$$</span>
are real and less than <span class="math-container">$3$</span>, find the range of <span class="math-container">$a$</span>.</p>
</blockquote>
<p>The roots are <span class="math-container">$a... | auscrypt | 675,509 | <p>It's not always allowable; take <span class="math-container">$a = 2.99$</span>, then <span class="math-container">$-\sqrt{3-a} = -0.1 < 0.01 = 3-a$</span>. But squaring both sides gives <span class="math-container">$0.01<0.0001$</span>, which is false. The issue is that negative signs cause issues when you squ... |
3,258,642 | <blockquote>
<p>If the roots of quadratic equation <span class="math-container">$$x^2 − 2ax + a^2 + a – 3 = 0$$</span>
are real and less than <span class="math-container">$3$</span>, find the range of <span class="math-container">$a$</span>.</p>
</blockquote>
<p>The roots are <span class="math-container">$a... | Peter Szilas | 408,605 | <p><span class="math-container">$y=(x-a)^2+(a-3)$</span>;</p>
<p><span class="math-container">$a-3 >0:$</span> This parabola does not cut the <span class="math-container">$x-$</span>axis (<span class="math-container">$(x-a)^\ge 0)$</span>.</p>
<p>Hence <span class="math-container">$a-3\le 0$</span>.</p>
<p>Roots:... |
3,275,732 | <p>How can I solve it without using matrix? I tried it to solve by using systems. But I have no idea how deal with "<span class="math-container">$0$</span>"</p>
| miky | 685,127 | <p>You can substitute every point in the equation and solve a system in 4 unknown (a,b,c,d) with four equation, one for every point.
The system is
<span class="math-container">$$d=1$$</span>
<span class="math-container">$$-a+b-c+d=-2$$</span>
<span class="math-container">$$a+b+c+d=2$$</span>
<span class="math-containe... |
3,275,732 | <p>How can I solve it without using matrix? I tried it to solve by using systems. But I have no idea how deal with "<span class="math-container">$0$</span>"</p>
| xrfxlp | 678,937 | <p>Given equation, <span class="math-container">$$y = ax^3 + bx^2 + cx + d$$</span> ,On putting the points <span class="math-container">$(0,1),(-1,-2),(1,2),(2,9)$</span> following system of equation are obtained <span class="math-container">$$\begin{align} d &=1\\ -a+b-c+d &=-2\\a+b+c+d &=2\\8a+4b+2c+d &am... |
2,801,936 | <p>To me, it seems obvious that the binary quadratic form $x^2+8y^2$ does not properly represent 3. However, I have managed to prove that it does so I think I must be doing something stupid.
I have used the following:</p>
<p><strong>Let f be a a binary quadratic form and n an integer. We say that f <em>properly repres... | Robert Z | 299,698 | <p>Yes, you are correct so far. Now by the multiplicative property of the <a href="https://en.wikipedia.org/wiki/M%C3%B6bius_function" rel="nofollow noreferrer">Mobius function</a> and its definition, it follows that
$$\begin{align*}\sum_{0\leq a_i\leq e_i, \forall i}\mu (p_1^{a_1}\cdots p_k^{a_k})\cdot p_1^{a_1}\cdot... |
2,558,870 | <p>Suppose $f:[0,1]\to \mathbb{R}$ is uniformly continuous, and $(p_n)_{n\in\mathbb{N}}$ is a sequence of polynomial functions converging uniformly to $f$.</p>
<p>Does it follow that $\mathcal{F}=\{p_n\mid n\in\mathbb{N}\}\cup \{f\}$ is equicontinuous?</p>
<p>Also, if $C_n$ are the Lipschitz constants of the polynomi... | GNUSupporter 8964民主女神 地下教會 | 290,189 | <p>$$Av=\lambda v \\
\iff (A-pI)v=(\lambda-p)v \\
\iff (A-pI)^{-1}v=(\lambda-p)^{-1}v
$$
Since $p$ is not an eigenvalue of $A$, $A-pI$ is invertible. We assume $(\lambda,v)$ an eigenpair of $A$ on the top, and $v$ to be an eigenvector at the bottom.</p>
|
3,997,968 | <p>I'm trying to figure out how to get the point x = 3 :
What's given here are the points S and G .
(Assuming the 2 angles are equal)
<a href="https://i.stack.imgur.com/COFMn.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/COFMn.png" alt="enter image description here" /></a></p>
<p>Apparently, we can... | Mick | 42,351 | <p>If <span class="math-container">$G = (G_x, G_y)$</span>, then <span class="math-container">$G' = (G_x, ??)$</span>.</p>
<p>Next, can you use two-point form to write the equation of SG'?</p>
<p><span class="math-container">$ P(P_x, 0)$</span> is a point on SG'.</p>
<p>Putting the above together with S = (1, 1) and G=... |
4,381,145 | <blockquote>
<p>Show that the three vector fields <span class="math-container">$X = y\frac{\partial}{\partial z}-z\frac{\partial}{\partial y}, Y = z\frac{\partial}{\partial x}-x\frac{\partial}{\partial z}$</span> and <span class="math-container">$Z=x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x}$</span> on <s... | Aaron | 9,863 | <p>Here is a hint: The sphere is the collection of points <span class="math-container">$(x,y,z)$</span> such that <span class="math-container">$f(x,y,z)=0$</span> where <span class="math-container">$f(x,y,z)=x^2+y^2+z^2-1$</span>. Following Lee's suggestion, a necessary condition to be tangent is for <span class="math... |
503,358 | <p>I remember I saw this question somewhere in Lang's undergraduate real analysis.</p>
<blockquote>
<p>Given any real number $\ge0$, show that it has a square root.</p>
</blockquote>
| AlexR | 86,940 | <p>This means to show that for $a\geq 0$, the Polynomial $x^2-a$ has at least one real root.
Chose $x_0 := 1$ and use <a href="http://en.wikipedia.org/wiki/Newton%27s_method">Newton's method</a> with
$$x_{n+1} = x_n - \frac{x_n^2 - a}{2x_n}$$
Then since $(x_n)$ is cauchy and $a \geq 0$ is a fixpoint of the iteration
$$... |
982,780 | <p>I have the following system of <span class="math-container">$M$</span> linear equations in <span class="math-container">$N$</span> unknowns.</p>
<p><span class="math-container">$$
\begin{bmatrix}
3 & 0 & 1 & 0 & -1 & -3 & 2\\
1 & 2 & 0 & 4 & 0 & 0 & -1\\
1 & 1 &a... | Asaf Karagila | 622 | <p>This is quite similar to Andreas Blass' answer, but as a positive proof instead of a negative proof. I will use standard ordinals notation, so $0$ is the least element, and a successor ordinal means a point that has a predecessor in the order. I will also use $(x,y)$ to denote the open interval from $x$ to $y$.</p>
... |
3,078,097 | <blockquote>
<p>Why it is impossible to split the natural numbers into two sets <span class="math-container">$A$</span> and <span class="math-container">$B$</span> such that for distinct elements <span class="math-container">$m, n \in A$</span> we have <span class="math-container">$m + n \in B$</span> and vice-versa?... | Stockfish | 362,664 | <p>Vice versa means that for distinct <span class="math-container">$m,n \in B$</span>, <span class="math-container">$m+n \in A$</span>.</p>
<p>I guess you have to work through some cases.</p>
<p>For instance, suppose that <span class="math-container">$1 \in A$</span> and <span class="math-container">$2 \in A$</span>.... |
1,274,317 | <blockquote>
<p>Let $f:R \longrightarrow S$ a surjective ring homomorphism. Is the inverse image $f^{-1}(M)$ a maximal left ideal of $R$ for any maximal
left ideal $M$ of $S$?</p>
</blockquote>
<p><strong>Comments:</strong> I tied something like this: if $M$ is maximal then</p>
<p>$M \neq S$ and if $J$ is a le... | user26857 | 121,097 | <blockquote>
<p>If $I$ is a left ideal such that $f^{-1}(M) \varsubsetneq I \subseteq R \Rightarrow I = R$.</p>
</blockquote>
<p>Let $a\in I$, $a\notin f^{-1}(M)$. It follows $f(a)\notin M$. Since $f^{-1}(M)\subset I$ we get $f(f^{-1}(M))\subset f(I)$. $f$ surjective gives you $M=f(f^{-1}(M))$, so $M\subset f(I)$. B... |
25,853 | <p>With regard to an undergraduate statistics course, I am developing a standardized list of point deductions with the TAs (doctoral students) so that graders are consistent in what they are taking off intermediate points for. For example, most problems are 10 points total, and my proposed point deductions for interme... | Nick C | 470 | <p>I favor an additive grading scheme, where points are <em>earned</em> toward a possible maximum (say 10) instead of deducting points for the myriad possible mistakes one could make. Here, I would try to adopt a set of markers I am looking for and awarding points if they appear in the written work. This could help in ... |
564,360 | <p>Lets take the example, if we take the expression $\frac{X!}{y_1!\cdot y_2!\cdots y_n!} $as long as summation $S=y_1+y_2+...y_n$ is less than or equals $X$, the remainder is always $0$. Thats How the permutation of $X$ things where there is $y_1$ things same , $y_2$ things same works. My question is, why does this ha... | Thanos Darkadakis | 105,049 | <p>This is not exactly a proof.
What I know from combinations is that:</p>
<p>$\binom{N}{m}=\frac{N!}{m!(N-m)!}$ is always integer.</p>
<p>That means that if N=a+b, then a!b! divides N!</p>
<p>You can use it more and more. Eg if also a=c+d then c!d! divides a!.</p>
<p>So b!c!d! divides N! where N=b+c+d. And you con... |
8,658 | <p>$f(x) = \frac{1}{\cos x}$</p>
<p>$f'(x) = \frac{\sin(x)}{\cos^2(x)}$</p>
<p>$f''(x) = \frac{2\sin^2(x)+\cos^2(x)}{\cos^3(x)}$</p>
<p>$f^{(3)}(x) = \frac{6\sin^3(x)+5\cos^2(x)\sin(x)}{cos^4(x)}$</p>
<p>$\vdots$</p>
<p>$f^{(n)}(x) = \frac{ ?}{cos^{n+1}(x)}$</p>
<p>Some of these are easy: <a href="http://darkwing... | Robin Chapman | 226 | <p>This is asking for the $n$-th derivative of the secant function.
As the derivative of $\sec$ is $\sec\tan$ and that of $\tan$ is
$\sec^2=1+\tan^2$ then the $n$-th derivative of $\sec$ is
$\sec f_n(\tan)$ where $f_0(t)=1$ and $f_{n+1}(t)=tf_n(t)+(t^2+1)f_n'(t)$.</p>
<p>It's probably too much to hope to find a nice f... |
2,612,134 | <p>One of the exercise in Artin's algebra gives an eigenvector of an element of $SO(3)$, in one possible case. Namely, it is asked to show that </p>
<blockquote>
<p>If $A=[a_{ij}]$ is a rotation in $SO(3)$, then the vector
$$v=\begin{bmatrix} (a_{23}+a_{32})^{-1}\\ (a_{13}+a_{31})^{-1} \\ (a_{12}+a_{21})^{-1}\end... | Community | -1 | <p>If $A\not= A^T$ (that is, if $A$ is not $I_3$ or a U-turn), then $x\in \ker(A-A^T)$ iff $Ax=x$.</p>
<p>It suffices to show that the first component of $(A-A^T)v$ is zero</p>
<p>$\dfrac{a_{1,2}-a_{2,1}}{a_{1,3}+a_{3,1}}+\dfrac{a_{1,3}-a_{3,1}}{a_{1,2}+a_{2,1}}=0$ iff ${a_{1,2}}^2+{a_{1,3}^2}-{a_{2,1}^2}-{a_{3,1}}^2... |
889,111 | <p>This is a problem from Apostol's Real Analysis book.
$$\text{Find if }\sum_{n=1}^{\infty}\dfrac{1}{n^{1+\frac{1}{n}}}\text{ converges or diverges. }$$
I tried to compare with $\displaystyle \sum_{n=1}^{\infty}\dfrac{1}{n^p}$ for suitable $p$, but $p>1$ fails always. I tried to show $\displaystyle \sum_{k=1}^{\in... | André Nicolas | 6,312 | <p><strong>Outline:</strong> One can prove, say by induction, that $2^n\gt n$ for
every positive integer $n$. </p>
<p>It follows that $n^{1/n}\lt 2$. </p>
<p>From this we can conclude by Comparison that our series diverges. </p>
|
1,410,586 | <blockquote>
<p>For certain pairs $ (m,n)$ of positive integers with $ m\ge n$ there are exactly $ 50$ distinct positive integers $ k$ such that $ |\log m - \log k| < \log n$. Find the sum of all possible values of the product $ mn$.</p>
</blockquote>
<p><strong>HINTS ONLY!</strong></p>
<p>Obviously, converting ... | mathlove | 78,967 | <p>HINT : </p>
<p>We have $$\frac 1n\lt\frac mk\lt n\iff \frac mn\lt k\lt mn\tag 1$$</p>
<p>Setting $\lfloor\frac mn\rfloor=s$ gives
$$(1)\iff s+1\le k\le mn-1$$with $s\le\frac mn\lt s+1$.</p>
<p>Hence, one has $50=(mn-1)-(s+1)+1$.</p>
|
2,206,247 | <p><strong>Question:</strong> Consider the following non linear recurrence relation defined for $n \in \mathbb{N}$:</p>
<p>$$a_1=1, \ \ \ a_{n}=na_0+(n-1)a_1+(n-2)a_2+\cdots+2a_{n-2}+a_{n-1}$$</p>
<p>a) Calculate $a_1,a_2,a_3,a_4.$</p>
<p>b) Use induction to prove for all positive integers that:</p>
<p>$$a_n=\dfra... | Community | -1 | <p><strong>HINT:</strong> $$2\sqrt{ab}\ge 0$$</p>
<p>Add $a+b$ to both sides.</p>
|
64,643 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/4467/a1-2-is-either-an-integer-or-an-irrational-number">$a^{1/2}$ is either an integer or an irrational number</a> </p>
</blockquote>
<p>I know how to prove $\sqrt 2$ is an irrational number. Who can tell ... | kuch nahi | 8,365 | <p>The proof is very similar to the irrationality of square root of two. </p>
<p>Let $\sqrt{3} = \frac a b$, where a and b have no common factors besides $1$</p>
<p>As $3b^2 = a^2$ so $a^2$ is a multiple of $3$, and hence $a$ should be a multiple of $3$. Let $a = 3k$, then $b^2 = 3k^2$, and $b$ must also be a multi... |
64,643 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/4467/a1-2-is-either-an-integer-or-an-irrational-number">$a^{1/2}$ is either an integer or an irrational number</a> </p>
</blockquote>
<p>I know how to prove $\sqrt 2$ is an irrational number. Who can tell ... | Community | -1 | <p>Alternatively, you can use the <a href="http://en.wikipedia.org/wiki/Rational_root_theorem" rel="nofollow">rational root test</a> on the polynomial equation $x^2-3=0$ (whose solutions are $\pm \sqrt{3}$). If $\frac{a}{b}$ is a solution to the equation (with $a,b\in \mathbb{Z}$ and $b\not=0$), then $b \vert 1$ and $... |
39,423 | <ul>
<li><p>case1</p>
<pre><code>Options[f] = {"t" -> "0"};
f[___, OptionsPattern[]] := StringReplace["content", "t" :> OptionValue["t"]]
f[]
(*
con0en0
*)
</code></pre></li>
<li><p>case2</p>
<pre><code>rule = {"t" -> OptionValue["t1"]};
Options[gg] = {"t1" -> "T1", "t2" -> "1"};
gg[___, OptionsPat... | ubpdqn | 1,997 | <p>Perhaps not what you are after:</p>
<pre><code>op[] := Function[x,
x /. MapThread[
Rule[#1, (#2)] &, {{"t", "t1", "t2"},
Thread["t" -> {"0", "T1", "1"}]}]];
srf[opts_: "t"] := StringReplace["content", op[][opts]]
</code></pre>
<p>Note:</p>
<pre><code>srf[]
</code></pre>
<p>yields the defau... |
2,464,890 | <p>Here is link to some limit questions:</p>
<p><a href="https://i.stack.imgur.com/2rM9f.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/2rM9f.png" alt="Example" /></a>
Can anyone explain how has answers were derived? In (a), how can we cancel out <span class="math-container">$(x-2)$</span>? And how ... | fleablood | 280,126 | <p>Big Picture.</p>
<p>1) If $f(x) = \frac {(x-2)^2}{(x-2)(x+2)}$ then $f(0) = \frac 00$ which is undefined (not $\infty$ by the way; simply undefined and meaningless). </p>
<p>However and values very very close to $x =2$ but not actually precisely $x=2$ all the $f(x)$ will have values that are very close to a value... |
2,647,194 | <p>show that $p(x)=x^3-x^2-4x+5$ is irreducible in $\mathbb{Q}[x]$ </p>
<p>How do we decide if a polynomial $p (x)$ in $\mathbb{Q}[x]$ is irreducible?</p>
| Arthur | 15,500 | <p>There isn't one single all-encompassing way to show that a polynomial is irreducible. There are a lot of theorems that apply in specific cases, though, and those we try to use whenever possible to see whether we get any results.</p>
<p>In this case, we see that the polynomial is of degree three, which means that if... |
1,987,317 | <blockquote>
<p>Prove that the system $$A^T A x = A^T b$$ always has a solution. The matrices and vectors are all real. The matrix $A$ is $m \times n$. </p>
</blockquote>
<p>I think it makes sense intuitively but I can't prove it formally.</p>
| H. H. Rugh | 355,946 | <p>The matrix $A^TA$ need not be invertible. So what you need to prove is that $A^T b$ lies in the image $V= {\rm im\;} A^T A$. </p>
<p>Now, $A^T b\in V$ is equivalent to $V^\perp \subset (A^T b)^\perp$ so it suffices to show that the orthogonal complement of ${\rm im} A^T A$ is also orthogonal to $A^T b$.</p>
<p>So... |
1,987,317 | <blockquote>
<p>Prove that the system $$A^T A x = A^T b$$ always has a solution. The matrices and vectors are all real. The matrix $A$ is $m \times n$. </p>
</blockquote>
<p>I think it makes sense intuitively but I can't prove it formally.</p>
| Rodrigo de Azevedo | 339,790 | <p>Let the SVD of $\mathrm A \in \mathrm R^{m \times n}$ be</p>
<p>$$\mathrm A = \mathrm U \Sigma \mathrm V^{\top} = \begin{bmatrix} \mathrm U_1 & \mathrm U_2\end{bmatrix} \begin{bmatrix} \hat\Sigma & \mathrm O\\ \mathrm O & \mathrm O\end{bmatrix} \begin{bmatrix} \mathrm V_1^{\top}\\ \mathrm V_2^{\top}\end... |
687,179 | <p>This is a question that asks the reader for a $strategy$ to solve a particular problem. I cannot solve this problem myself so I am looking around for general methods one might use to confront it with. Suppose
$$f(x)=a_0+a_1x+..., g(x)=b_0+b_1x_...$$
and given
$$\lim\limits_{x\to 1^-}\frac{f^{(n)}(x)}{g^{(n)}(x)}=1$... | Daniel Fischer | 83,702 | <p>Under the given assumptions,</p>
<p>$$\lim_{n\to\infty} \frac{a_n}{b_n} = 1$$</p>
<p>need not hold. Counterexample:</p>
<p>$$\begin{align}
f(x) &= \frac{1}{1-x^2} = \sum_{k=0}^\infty x^{2k},\\
g(x) &= f(x) + e^x.
\end{align}$$</p>
<p>We have $\dfrac{a_n}{b_n} = 0$ for all odd $n$, but nevertheless</p>
<... |
285,034 | <p>I currently don't see how to solve the following integral:</p>
<p>$$\int_{-1/2}^{1/2} \cos(x)\ln\left(\frac{1+x}{1-x}\right) \,dx$$</p>
<p>I tried to solve it with integration by parts and with a Taylor series, but nothing did help me so far.</p>
| ՃՃՃ | 48,751 | <p>Hint: The integrand is an odd function.</p>
|
1,640,653 | <p>I have this matrix below and I'm trying to find it's inverse, I know I augment it with I<sub>2</sub> but I don't know where to go from that.</p>
<p>\begin{bmatrix}
2&1\\
a&a
\end{bmatrix}</p>
| MPW | 113,214 | <p>Perhaps quicker to use
$$\begin{bmatrix}
a&b\\
c&d
\end{bmatrix}^{-1} = \frac1{ad-bc}\begin{bmatrix}
d&-b\\
-c&a
\end{bmatrix}$$</p>
<p>so that</p>
<p>$$\begin{bmatrix}
2&1\\
a&a
\end{bmatrix}^{-1} = \frac1a\begin{bmatrix}
a&-1\\
-a&2
\end{bmatrix}=\begin{bmatrix}
1&-\frac1a\\
-... |
2,130,141 | <p>I am having troubles with the following excercise:</p>
<p>$P(A\times B) = Q$ and $Q = \lbrace V\times W \ \vert \ V\in P(A), W\in P(B)\rbrace$ </p>
<p>So I have to prove or disprove. I know that $P(A\times B) \neq Q$ and being specific $P(A\times B) \not\subset Q$ and $Q \subset P(A\times B)$. In addition; </p>
<... | Rob Arthan | 23,171 | <p>Assume $d$ is a metric on the set $M = \{a, b\}$ where $a \neq b$ and let $\delta = d(a, b)$. Then $\delta \neq 0$ because otherwise $a = b$. The set $U = \{x \in M \mid d(x, a) < \delta\}$ is then open by definition of the topology on a metric space. But $U = \{a\}$ is not open in the topology described in the q... |
59,567 | <p>I am looking for a way to add a legend showing the identity of various atoms (with different colours) to this picture. Any Clues?</p>
<pre><code>Import["ExampleData/1PPT.pdb", "Rendering" -> "BallAndStick"]
</code></pre>
<p><img src="https://i.stack.imgur.com/FSFoH.png" alt="enter image description here"></p>
| Bob Hanlon | 9,362 | <pre><code>bas = Import["ExampleData/1PPT.pdb", "Rendering" -> "BallAndStick",
ImageSize -> 500];
elements =
Import["ExampleData/1PPT.pdb", "ResidueAtoms"] // Flatten // Union;
legend = GraphicsColumn[{
{Graphics[{#[[1]], Disk[{0, 0}, 1]}, ImageSize -> 10], #[[2]]} & /@
Thread[{
El... |
3,878,723 | <blockquote>
<p>Find the value of <span class="math-container">$k$</span> if the curve <span class="math-container">$y = x^2 - 2x$</span> is tangent to the line <span class="math-container">$y = 4x + k$</span></p>
</blockquote>
<p>I have looked at the solution to this question and the first step is the "equate the... | PL Wang | 803,716 | <p>Additionally, you can also solve with calculus.
Take the derivative.
<span class="math-container">$(x^2-2x)' = 2x-2$</span>. We want the derivative to be <span class="math-container">$4$</span>, which is the slope of the line. This happens when <span class="math-container">$2x-2 = 4$</span>, at <span class="math-con... |
3,438,048 | <p>I've recently obtained my University entrance papers from 1967 (yes,52 years ago!) and I found the question below difficult. I presume the answer is a symmetric expression in the differences between alpha,beta and gamma.Am I missing some obvious trick? Any help would be appreciated.</p>
<p>Simplify and evaluate the... | José Carlos Santos | 446,262 | <p>The empty set belongs to <span class="math-container">$\tau$</span> trivially.</p>
<p>The set <span class="math-container">$\mathbb R$</span> belongs to <span class="math-container">$\tau$</span> because is <span class="math-container">$x\in U$</span> and you take <span class="math-container">$\varepsilon=1$</span>... |
47,926 | <p>Is there any known two-dimensional Conway's game of life variation where each cell can not be just on/off but able to hold more states, maybe 4 or 5?</p>
| El'endia Starman | 10,537 | <p>Adding on to anon's comment, here's a cellular automaton called <a href="http://en.wikipedia.org/wiki/Wireworld" rel="nofollow">Wireworld</a>.</p>
|
47,926 | <p>Is there any known two-dimensional Conway's game of life variation where each cell can not be just on/off but able to hold more states, maybe 4 or 5?</p>
| alan2here | 6,982 | <p>I can recomend the software Golly, it's so far ahead of anything else and easy to use. Generations rules are similar to Lifelike rules such as Conways Life but have more states. Alternitivly WireWorld as mentioned above is an example in Golly of a Rule Table, and their are many other for more human engineered cellul... |
1,343,909 | <p>I was reading some examples about linear functionals from the book Introductory functional analysis with applications of Kreysig and one of the examples states the following </p>
<p>Let <span class="math-container">$L:C[0,1]\rightarrow C[0,1]$</span></p>
<p><span class="math-container">$$L[f(x)]=\int_{0}^{x}f(s)ds... | egreg | 62,967 | <p>Hint: fundamental theorem of calculus.</p>
<p>Hint 2: if $f\in C^1([0,1])$ and $f(0)=0$, then we can consider $f'\in C([0,1])$ and also
$$
g(x)=\int_0^x f'(s)\,ds
$$
By the fundamental theorem of calculus, $f'=g'$ and $g=L(f')$, so…</p>
|
3,075,869 | <p>The following is quoted from <a href="https://en.wikipedia.org/wiki/Quotient_space_(topology)" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Quotient_space_(topology)</a></p>
<p>Quotient maps q : X → Y are characterized among surjective maps by the following property: if Z is any topological space and f :... | Henno Brandsma | 4,280 | <p>Suppose <span class="math-container">$q: X \to Y$</span> obeys the "composition property". Suppose <span class="math-container">$U$</span> is a subset of <span class="math-container">$Y$</span> such that <span class="math-container">$q^{-1}[U]$</span> is open. Define <span class="math-container">$g: Y \to \{0,1\}$</... |
176,260 | <blockquote>
<p>Let $\left\{ f_{n}\right\} $ denote the set of functions on
$[0,\infty) $ given by $f_{1}\left(x\right)=\sqrt{x} $ and
$f_{n+1}\left(x\right)=\sqrt{x+f_{n}\left(x\right)} $ for $n\ge1 $.
Prove that this sequence is convergent and find the limit function.</p>
</blockquote>
<p>We can easily show ... | David Mitra | 18,986 | <p>In a sense, the answer is "yes". You can apply the Monotone Convergence Theorem pointwise. That is, for a fixed value of $x$, show that the sequence of nonnegative numbers $\bigl(f_n(x)\bigr)$ is bounded above and increasing. Then of course for each $x$, the sequence $\bigl(f_n(x)\bigr)$ converges to some $f(x)$. (... |
1,175,632 | <p>Determine whether the following integral converges or diverges: \begin{align*} \iint_Q e^{-xy} \ dA, \end{align*} where $Q$ is the first quadrant of the $xy$-plane.</p>
<p>How should I go about this problem? Should I compare it with another known integral?</p>
| awkward | 76,172 | <p>I think your answer of 0.0648 is correct.</p>
<p>For a more general approach, consider the probability of <em>not</em> having 3 successes in a row out of $n$ trials; call this probability $P(n)$, and let's say the probability of a single success is $p$, with $q=1-p$. If we have $n$ trials, condition the probability... |
2,360,819 | <p>Probability question that I don't understand? It is on our assignment for probability and I can't seem to figure out how it has to do with probability or how to solve it.</p>
| Michael Burr | 86,421 | <p>Consider the following chart
$$
\begin{matrix}
\text{Sides}&\text{Diagonals}\\\hline
3&0\\
4&2\\
5&5\\
6&9
\end{matrix}
$$
It appears that the number of diagonals is increasing by $n-2$ each time (where $n$ is the number of sides of the new shape).</p>
<p>If you poke around a bit (writing down t... |
2,360,819 | <p>Probability question that I don't understand? It is on our assignment for probability and I can't seem to figure out how it has to do with probability or how to solve it.</p>
| DanielWainfleet | 254,665 | <p>There are $\binom {8}{2}=28$ pairs of vertices and $\binom {8}{1}=8$ of those are pairs of adjacent vertices, so there are $28-8=20$ pairs of non-adjacent vertices and therefore $20$ diagonals.</p>
<p>Probability often involves counting cases, which often involves combinatorics, which often uses various methods of ... |
24,186 | <p>Consider the code below:</p>
<pre><code>s = Solve[(3 - Cos[4*x])*(Sin[x] - Cos[x]) == 2, x, InverseFunctions -> True];
Select[s[[All, 1, 2]], Element[#, Reals] &]
</code></pre>
<p>In MMA 8.0, I get </p>
<pre><code>{-\[Pi], \[Pi]/2, \[Pi]}
</code></pre>
<p>but in MMA 9.0, I get an empty set { }</p>
<p>Ass... | amr | 950 | <p>In my continuing mission to provide smartass answers, you can do, for example:</p>
<pre><code>u = {-3, 3}; v = {1, 5};
d = Function[{u, v},
((Abs[u[[1]]] - Abs[v[[1]]])^2 + (Abs[u[[2]]] - Abs[v[[2]]])^2)^(1/2)][u, v]
</code></pre>
<p>I think the reason why it's suggested to convert to pure functions is for perf... |
545,003 | <p>I have a proof that I am trying to prove and I am getting stuck at the inductive hypothesis. This is my theorem:</p>
<blockquote>
<p>For all real numbers $n>3$, the following is true: $n + 3 < n!$.</p>
</blockquote>
<p>I have proven true for $n = 4$, and will assume true for some arbitrary value $k$, i.e.,... | Community | -1 | <p>Suppose $n+3< n!$ and $(n+1)!\leq n+1+3$ , we see the following:</p>
<p>$$(n+1)!\leq n+1+3 \Rightarrow (n+1)n!\leq n+1+3 $$</p>
<p>$$\Rightarrow (n+1)(n+3)\leq(n+1)n!\leq n+1+3 $$</p>
<p>$$(n+1)(n+3)\leq (n+1)+3 \Rightarrow (n+1)(n+3)-(n+1) \leq 3$$</p>
<p>$$ \Rightarrow (n+1)(n+3-1)\leq 3$$</p>
<p>$$(n+1)(n... |
1,213,344 | <p>A solid half-ball $H$ of radius $a$ with density given by $k(2a-\rho)$, where $k$ is a constant. Find its mass.</p>
<p>You of course use spherical coordinates so $dV=\rho ^2 \sin\phi d\rho d\phi d\theta$. It is clear to see that the limits are $\rho \in [0,a]$ and $\theta \in [0,2\pi]$. The limits for $\phi$ are no... | Scientifica | 164,983 | <p><strong>Hints</strong>:</p>
<p>1)We have $x\in [0,1]\Rightarrow x^{n+1}\le x^n$. Then prove that $\frac{x^n}{x^n+1}\ge\frac{x^{n+1}}{x^{n+1}+1}$. You can use the fact that for any real number $a\neq -1$ we have $\frac{a}{a+1}=1-\frac{1}{a+1}$.</p>
<p>2)For any $x>0$ we have $2x-1<2x$</p>
|
874,697 | <p>I'm trying to understand the supremem of a sequence of functions so I came up with a trivial case as follows -</p>
<p>Let $(f_n(x))$ be a sequence of functions with $n$ having a value of either $1$ or $2$. Ie. A sequence with only two elements.</p>
<p>Now if we define $f_1(x)=1$ and $f_2(x)=x$ what is the sup $(f(... | Clive Newstead | 19,542 | <p><strong>Hint:</strong> The supremum is a function of $x$, which you can define piecewise. Consider what the supremum is in the two cases $x \ge 1$ and $x < 1$.</p>
|
533,534 | <p>Let X = {x(i)} be a group of n data with mean = μ(x) and variance $= σ(x)^2$.</p>
<p>We use the symbol S(x(i)) to represent the sum of all the x's.</p>
<p>Similar notations will be used for the group Y.</p>
<p>Supposed that Y is formed by adding an extra element x(n+1) to X and the value of that element is greate... | Cameron Buie | 28,900 | <p><strong>Hint</strong>: Find the two limits $\lim\limits_{x\to 0^+}f(x)$ and $\lim\limits_{x\to 0^-}f(x).$ Keep in mind that $f(x)$ has a different formula when $x>0$ than it does when $x<0$.</p>
|
2,848,891 | <p>Find the number of solutions of $$\left\{x\right\}+\left\{\frac{1}{x}\right\}=1,$$ where $\left\{\cdot\right\}$ denotes Fractional part of real number $x$.</p>
<h2>My try:</h2>
<p>When $x \gt 1$ we get</p>
<p>$$\left\{x\right\}+\frac{1}{x}=1$$ $\implies$</p>
<p>$$\left\{x\right\}=1-\frac{1}{x}.$$</p>
<p>Letting... | Ng Chung Tak | 299,599 | <p>Using <a href="https://en.wikipedia.org/wiki/Continued_fraction" rel="nofollow noreferrer"><strong>continued fraction</strong></a>:</p>
<p>\begin{align}
f+\frac{1}{n+f} &= 1 \tag{$n>1$, $0<f<1$} \\[5pt]
n+f &= n+1-\frac{1}{n+f} \\[5pt]
x &= n+\frac{n+f-1}{n+f} \\[5pt]
&= n+\frac{1}{... |
2,068,906 | <p>Recall, with the birthday problem, with 23 people, the odds of a shared birthday is APPROXIMATELY .5 (correct?)</p>
<p>P(no sharing of dates with 23 people) = $$\frac{365}{365}*\frac{364}{365}*\frac{363}{365}*...*\frac{343}{365} $$</p>
<p>$$= \frac{365!}{342!}*\frac{1}{365^{23}} $$</p>
<p>I want to do this multip... | Aloizio Macedo | 59,234 | <p>With respect to the question in the title, by doing the second line, you are making your calculator attempt to compute a number greater than $100^{200}$. It won't.</p>
<p>By doing the first line, you are making a multiplication of about $20$ numbers close to $1$. It will handle this just fine.</p>
|
2,068,906 | <p>Recall, with the birthday problem, with 23 people, the odds of a shared birthday is APPROXIMATELY .5 (correct?)</p>
<p>P(no sharing of dates with 23 people) = $$\frac{365}{365}*\frac{364}{365}*\frac{363}{365}*...*\frac{343}{365} $$</p>
<p>$$= \frac{365!}{342!}*\frac{1}{365^{23}} $$</p>
<p>I want to do this multip... | vadim123 | 73,324 | <p>The following command works in LibreOffice, and would probably work in Excel as well:</p>
<p>=COMBIN(365,342)*FACT(23)/(365^23)</p>
<p>The key is that $365!$ and $342!$ are both enormous, but all the other numbers are manageable, so we need to find a built-in function to cancel these two monsters. <a href="https:... |
2,068,906 | <p>Recall, with the birthday problem, with 23 people, the odds of a shared birthday is APPROXIMATELY .5 (correct?)</p>
<p>P(no sharing of dates with 23 people) = $$\frac{365}{365}*\frac{364}{365}*\frac{363}{365}*...*\frac{343}{365} $$</p>
<p>$$= \frac{365!}{342!}*\frac{1}{365^{23}} $$</p>
<p>I want to do this multip... | Rolazaro Azeveires | 39,752 | <p>You can make the calculation with just about any calculator out there. You only need the basics (* /). Stay clear of any large or small numbers, to avoid overflowing (or underflowing). So go divide-multiply-divide-multiply-and-so-on:</p>
<p>$$P \approx 364 / 365 * 363 / 365 * 362 / 365 \ldots $$</p>
|
2,152,872 | <p>I am working on a problem in Artin's Algebra related to the algebraic geometry talked in Chapter 11. The problem number is 9.2., F.Y.I.</p>
<p>Here goes the problem:</p>
<blockquote>
<p>Let $f_1, \dots, f_r$ be complex polynomials in the variables $x_1, \dots, x_n$, let $V$ be the variety of their common zeros, ... | M. Winter | 415,941 | <p>Mostly try and error. What I learned so far is that there are no <em>good</em> characterizations of degree sequences.</p>
<p><a href="https://i.stack.imgur.com/GpCpv.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/GpCpv.png" alt="enter image description here"></a></p>
|
2,122,389 | <p>The problem goes so : you have a parking lot with 8 parking spaces and 8 cars, of which 4 are red and 4 are white. What is the probability of :</p>
<p>a) 4 white cars being parked next to each other ?</p>
<p>b) 4 white cars and 4 red cars being parked next to each other ?</p>
<p>c) red and white cars being parked... | DXT | 372,201 | <p>$$\int \ln{\left(1+2m\cos{x}+2m^2\right)}dx=x\ln{(1+2m^2)}+\int\ln{\left(1+\frac{2m}{1+2m^2}\cos{x}\right)}dx=$$
where $\displaystyle \left|\frac{2m\cos{x}}{1+2m^2}\right|<1$</p>
<p>$\displaystyle =x\ln{(1+2m^2)}+\sum\limits_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}\left(\frac{2m}{1+2m^2}\right)^n\int\cos^n{x}dx$</p>
... |
23,566 | <p>I love math, and I used to be very good at it. The correct answers came fast and intuitively. I never studied, and redid the demonstration live for the tests (sometimes inventing new ones). I was the one who answered the tricky questions in class (8 hours of math/week in high school)... You get the idea.</p>
<p>As ... | Bill Dubuque | 242 | <p>I'd like to emphasize a remark that Eric made in a comment to his answer. Introspection is essential when learning mathematics - not only to analyze problem solving techniques - but also in many other ways. The web of mathematics is connected in many mysterious and marvelous ways. Spending a little effort attempting... |
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