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 |
|---|---|---|---|---|
1,016,302 | <p>I'm having a bit of trouble with a the proof in Ross' Elementary Analysis. The theorem is the $\epsilon-\delta$ one. Theorem: Let $f$ be continuous at $x_0$ in $dom(f)$ if and only if for each $\epsilon>o$ there exists $\delta>0$ such that $x \in dom(f)$ and $|x-x_0|<\delta$ implies $|f(x)-f(x_0)|$ $(1)$. <... | wes.stueve | 189,075 | <p>It appears he is assuming $|f(x) - f(x_0)| \lt \epsilon$ fails for each natural. Or to put it another way $|f(x_0) - f(x_n)| \ge \epsilon$ to arrive at a contradiction.</p>
|
1,782,423 | <p>Gödel's completeness theorem says that for any first order theory $F$, the statements derivable from $F$ are precisely those that hold in all models of $F$. Thus, it is not possible to have a theorem that is "true" (in the sense that it holds in the intersection of all models of $F$) but unprovable in $F$.</p>
<p>H... | Noah Schweber | 28,111 | <p><em>I should clarify that I believe all the math here is well-known folklore, but I haven't ever actually seen it written up or even stated explicitly. So this is not new, but it is original in the sense that I can't point you towards a reference.</em></p>
<hr>
<p>There is a very interesting <a href="https://en.wi... |
1,740,032 | <p>In layman's terms, why would anyone ever want to change basis? Do eigenvalues have to do with changing basis? </p>
| Carsten S | 90,962 | <p>\begin{equation}
\sum_{k=0}^{n-1}k^3 = \sum_{k=0}^{n-1}6\binom k3+6\binom k2 + \binom k1
= 6\binom n4+6\binom n3 + \binom n2=\frac{n^4-2n^3+n^2}4.
\end{equation}
Hey, that change of basis in the space of polynomials was useful.</p>
|
1,740,032 | <p>In layman's terms, why would anyone ever want to change basis? Do eigenvalues have to do with changing basis? </p>
| Bart | 355,805 | <p>Solid mechanics. Computing the principal stresses/strains in a material involves a change of basis. With the principal stresses, we can compute the maximum normal stress and maximum shear stress the material experiences. And these maxima tell us whether the material will become damaged or not.</p>
|
232,825 | <p>I have to read and process a file that looks like this:</p>
<pre><code> MASSA TMASS1
uscita da elettrodi:
0 1 0.56705 -19.98160 2.80000 -0.87939 0.66823 -0.63034 0.39513
ingresso tmass1:
0 1 0.56705 -19.13351 2.00000 -0.37791 0.66823 -0.63034 0.39513
MAS... | Bob Hanlon | 9,362 | <p>Using your formula</p>
<pre><code>Clear["Global`*"]
ages = {14, 15, 16, 22, 24, 25};
counts = {1, 1, 3, 2, 2, 5};
Evaluate[n /@ ages] = counts;
?n
</code></pre>
<p><a href="https://i.stack.imgur.com/UOYEW.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/UOYEW.png" alt="enter image des... |
232,825 | <p>I have to read and process a file that looks like this:</p>
<pre><code> MASSA TMASS1
uscita da elettrodi:
0 1 0.56705 -19.98160 2.80000 -0.87939 0.66823 -0.63034 0.39513
ingresso tmass1:
0 1 0.56705 -19.13351 2.00000 -0.37791 0.66823 -0.63034 0.39513
MAS... | kglr | 125 | <pre><code>values = {14, 15, 16, 22, 24, 25};
weights = {1, 1, 3, 2, 2, 5};
</code></pre>
<p>You can also use <a href="https://reference.wolfram.com/language/ref/WeightedData.html" rel="noreferrer"><code>WeightedData</code></a>:</p>
<pre><code>Mean @ WeightedData[values, weights]
</code></pre>
<blockquote>
<pre><code>2... |
232,825 | <p>I have to read and process a file that looks like this:</p>
<pre><code> MASSA TMASS1
uscita da elettrodi:
0 1 0.56705 -19.98160 2.80000 -0.87939 0.66823 -0.63034 0.39513
ingresso tmass1:
0 1 0.56705 -19.13351 2.00000 -0.37791 0.66823 -0.63034 0.39513
MAS... | Henrik Schumacher | 38,178 | <p>These are all quite fancy solutions so far. For beginners' sake, I'd like to add a somewhat more elementary one that uses somewhat more universal tools:</p>
<pre><code>ages = {14, 15, 16, 22, 24, 25};
counts = {1, 1, 3, 2, 2, 5};
ages.counts/Total[counts]
</code></pre>
<blockquote>
<p>21</p>
</blockquote>
<p>The <c... |
86,277 | <p>I have a question: does the Heine-Borel theorem hold for the space $\mathbb{R}^\omega$ (where $\mathbb{R}^\omega$ is the space of countable sequences of real numbers with the product topology). That is, prove that a subspace of $\mathbb{R}^\omega$ is compact if and only if it is the product of closed and bounded su... | Jonas Meyer | 1,424 | <p>[<strong>Edit</strong>: This answer does not answer the question. I tried an answer before the question was clarified, and it turns out to be an answer to the wrong question.]</p>
<p>A metric space with the Heine-Borel property (that every closed and bounded subspace is compact) must be locally compact and $\sigma... |
424,853 | <p>As an interested outsider, I have been intrigued by the number of times that homotopy theory seems to have revamped its foundations over the past fifty years or so. Sometimes there seems to have been a narrowing of focus, via a choice to ignore certain "pathological"—or at least intractably complicated—ph... | David White | 11,540 | <p>I think Dmitri Pavlov does a nice job laying out a timeline. Instead of writing a competing answer, let me just try to add in a couple of things I think he left out, addressing a few points from the OP.</p>
<p>1940s <strong>Leray</strong> invents <a href="https://en.wikipedia.org/wiki/Spectral_sequence" rel="nofollo... |
1,410,889 | <p>Consider the function with domain $A = \{ (x,y) \in \, \mathbb{R}^2: (x,y) \neq (0,0)\}$ given by</p>
<p>$$\frac{2x^2y}{x^4+y^2}$$</p>
<blockquote>
<p>Letting $(x,y)$ approach $(0,0)$ along the straight line $y=ax$ , where $a$ is a real constant, we find that the limit is zero. This is not enough to conclude tha... | Patrick | 263,218 | <p>$$\frac{2x^2y}{x^4+y^2}$$</p>
<p>I order for a limit to exist it must so follow that at (0,0) we get the same limit among all paths. The limit 0 for f(x,y) along y=ax approaching (0,0) is only one direction. It must follow through for other directions.</p>
<p><strong>A long the path $y=ax$</strong></p>
<p>$$\lim... |
1,787,744 | <p>There are two measurable spaces with Borel sigma-algebras on them $(X, \mathcal B (X)) $ and $(Y, \mathcal B (Y)) $. There is also a continuous function $f:(X, \mathcal B (X)) \to (Y, \mathcal B (Y)) $. Prove that this function is measurable with respect to Borel sigma-algebra. </p>
| mariob6 | 326,028 | <p>Use definition of continuity: </p>
<p>$f:(X, \mathcal M) \to (Y, \mathcal N) $ is continuous iff $\ \ \forall A \subset Y \ \ A \text{ open} $ we have $f^{-1}(A)\text{ is open} $</p>
<p>By definition of Borel sigma algebra you get the thesis</p>
|
2,344,931 | <p>It is given that $a,b$ are roots of $3x^2+2x+1$ then find the value of:
$$\left(\dfrac{1-a}{1+a}\right)^3+\left(\dfrac{1-b}{1+b}\right)^3$$</p>
<p>I thought to proceed in this manner:</p>
<p>We know $a+b=\frac{-2}{3}$ and $ab=\frac{1}{3}$. Using this I tried to convert everything to <em>sum and product of roots</e... | Martin Hansen | 646,413 | <p>Although three years old, this is a good question with some terrific answers. I thought I'd add mine to the collection...</p>
<p>Let<span class="math-container">$$x=\left(\frac{1-a}{1+a}\right), y=\left(\frac{1-b}{1+b}\right)$$</span>
Then,
<span class="math-container">$$x+y=\left(\frac{1-a}{1+a}\right)+\left(\frac{... |
303,283 | <p>A basic PDE I would like to understand much better is the viscous Hamilton-Jacobi equation, such as:
\begin{equation*}
u - \epsilon \Delta u + H(Du) = f(x)
\end{equation*}
or
\begin{equation*}
u_{t} - \epsilon \Delta u + H(Du) = f(x)
\end{equation*}
with or without boundary conditions in the stationary case, or the... | Toni Mhax | 121,643 | <p>Some way to do this is to say <span class="math-container">$S=\langle a,b,c\rangle=\langle 3,3k+1,3j+2\rangle$</span> then it is easy to see that <span class="math-container">$g(S)=k+j$</span>. Now take a generator of <span class="math-container">$S$</span> it can not be less than <span class="math-container">$F(S)... |
701,681 | <p>It seems intuitive that, if a function differentiable on [a,b] is such that f'(a) < 0 < f'(b) then there exists some c in the open interval (a,b) such that f'(c)=0, but I can't prove it rigourously...</p>
<p>I expect to need to use Fermat's Theorem on stationary points but I would then need to prove that c is... | Yiorgos S. Smyrlis | 57,021 | <p>This is the Famous <a href="http://en.wikipedia.org/wiki/Darboux%27s_theorem_%28analysis%29" rel="nofollow">Darboux's Theorem</a>.</p>
<p>The idea of the proof is that if $f'(a)<0$, then there is a $x_1>a$, where $f(x_1)<f(a)$.
similarly, as $f'(b)>0$, there is a $x_2<b$, with $f(x_2)<f(b)$. Hence... |
1,080,746 | <p>I want to calculate the limit of this sum :</p>
<p>$$\lim\limits_{x \to 1} {\left(x - x^2 + x^4 - x^8 + {x^{16}}-\dotsb\right)}$$</p>
<p>My efforts to solve the problem are described in the <a href="https://math.stackexchange.com/a/1308281/202081">self-answer below</a>.</p>
| Gottfried Helms | 1,714 | <p><em>This is a copy of <a href="http://mathoverflow.net/questions/198665/on-an-example-of-an-eventually-oscillating-function/198871#198871">my answer in mathoverflow</a> which I've linked to in my OP's comment but it might really be overlooked</em><br>
<hr>
<em>This is more an extended comment than an answer.</em> ... |
1,640,471 | <p>Good day.
I am trying to solve the following equation:
$$\ddot{y}(x)-\frac{A}{x}\dot{y}(x)+\frac{Bx^2}{2}y(x)=0.$$
WolframAlpha says it is an Emden‐Fowler equation, but I have no idea how to solve this. Can you give me some tips?
In case of $$A=B/2=1$$ WolframALpha gives an analytical solution $$y(x)=c_1 sin\frac{x^... | Chinny84 | 92,628 | <p>For the example you give with the solution you have
$$
y''-\frac{1}{x}y'+x^2y = 0 \implies \frac{1}{x^2}y''-\frac{1}{x^3}y' +y = 0
$$
multiply by $y'$ we find
$$
\frac{1}{x^2}\dfrac{d}{dx}\frac{y'^2}{2}-\frac{1}{x^3}y'^2+yy'= \frac{1}{2x^2}\dfrac{d}{dx}y'^2-\frac{1}{x^3}y'^2+\dfrac{d}{dx}\frac{y^2}{2} = 0
$$
thus
$$... |
1,640,471 | <p>Good day.
I am trying to solve the following equation:
$$\ddot{y}(x)-\frac{A}{x}\dot{y}(x)+\frac{Bx^2}{2}y(x)=0.$$
WolframAlpha says it is an Emden‐Fowler equation, but I have no idea how to solve this. Can you give me some tips?
In case of $$A=B/2=1$$ WolframALpha gives an analytical solution $$y(x)=c_1 sin\frac{x^... | JJacquelin | 108,514 | <p>$$\frac{d^2y}{dx^2}-\frac{A}{x}\frac{dy}{dx}+\frac{Bx^2}{2}y(x)=0.$$
Changing $x$ into $-x$ doesn't change the equation. This draw us to a change of variable </p>
<p>$t=x^2 \quad;\quad dt=2x\:dx \quad;\quad \frac{dy}{dx}=\frac{dy}{dt}\frac{dt}{dx}=2x\frac{dy}{dt}\quad;\quad \frac{d^2y}{dx^2}=2\frac{dy}{dt}+2x\frac{... |
2,649,557 | <p>I've been studying topology recently, and I've gotten to the part of the book that deals with quotient spaces. For the most part, it's fairly clear, but one thing that has been confusing me a bit is how the unit circle is represented.</p>
<p>Sometimes $\mathbf S^1$ is denoted as $\{(x,y)\in\mathbb R^2|(x-a)^2+(y-b)... | Mohammad Riazi-Kermani | 514,496 | <p>$$\{(x,y)\in\mathbb R^2|(x-a)^2+(y-b)^2=1\}$$ is the same set as$$ \{z\in\mathbb C| \; |z-w|=1\}$$ with $w=a+bi.$</p>
<p>For the subject of metric topology it is probably easier to talk about d(z,w) = |z-w| than using the square root notation.</p>
|
4,293,199 | <p>let me preface by saying this <em>is</em> from a homework question, but the question is not to plot the decision boundary, just to train the model and do some predictions. I have already done that and my predictions <em>seem</em> to be correct.</p>
<p>But I would like to verify my results by plotting the decision bo... | Albus Dumbledore | 769,226 | <p>Hint: <span class="math-container">$$\cos^5\theta =\frac{ 10 \cos(θ) + 5 \cos(3 θ) + \cos(5 θ)}{16}$$</span>because <span class="math-container">$$\cos^5 x=1/32{(e^{ix}+e^{-ix})}^5=\frac{e^{5ix}+e^{-5ix}+5(e^{3ix}+e^{-3ix})+10(e^{ix}+e^{-ix})}{32}$$</span></p>
<p>Note that there will be infinitely many solutions as... |
1,191,176 | <p>I know how to prove the result for $n=2$ by contradiction, but does anyone know a proof for general integers $n$ ?</p>
<p>Thank you for your answers.</p>
<p>Marcus</p>
| Surb | 154,545 | <p>Suppose that $\sqrt[n]2$ is rational. Then, for some $p,q\in\mathbb Q$,</p>
<p>$$\sqrt[n]2=\frac{p}{q}\implies 2=\frac{p^n}{q^n}\implies p^n=2q^n=q^n+q^n.$$</p>
<p>Contradiction with Fermat last theorem.</p>
|
4,383,098 | <p>This was a question I had ever since I started studying Formal mathematics. Take ZFC for example, in it the axioms tell us 'tests' to check if something is a set or not and how the object, if they are set, behave with some other operations defined on the set.</p>
<p>My question is how exactly do we find objects do f... | Tita | 791,224 | <p>Given that <span class="math-container">$(A\cap B)\cup(B\cap C)\cup(C\cap A)$</span> and from the distributive property <span class="math-container">$X\cap(Y\cup Z)=(X\cap Y)\cup (X\cap Z)$</span> we get that:</p>
<p><span class="math-container">$$ (A\cap B)\cup(B\cap C) = B\cap(A\cup C) $$</span></p>
<p>so our init... |
2,998,811 | <p><strong>Exercise :</strong></p>
<blockquote>
<p>We consider a market of one period <span class="math-container">$(\Omega, \mathcal{F}, \mathbb P, S^0, S^1)$</span>, where the sample space <span class="math-container">$\Omega$</span> has a finite number of elements and the <span class="math-container">$\sigma-$</span... | Rebellos | 335,894 | <p><strong>Note :</strong> The following answer was provided by Daneel Olivaw from a post regarding a similar question of mine for finance mathematics at Quantitive Finace Stack exchange : <a href="https://quant.stackexchange.com/questions/42637/equivalent-martingale-measure-exists-if-and-only-if-a-s-011r-b">https://qu... |
1,958,616 | <p>Question: Let $z=x+iy $. Using complex notation, find an equation of a circle with radius $5$ and center $(3,-6) $.</p>
<p>The answer seems simple to derive, but I'm curious as to whether or not there is more that must be done to establish the equation than what I have accomplished.</p>
<p>The circle with the abov... | DonAntonio | 31,254 | <p>Develop the idea (the proof): it is</p>
<p>$$|z-3+6i|=5$$</p>
|
6,661 | <p>Is there a simple explanation of what the Laplace transformations do exactly and how they work? Reading my math book has left me in a foggy haze of proofs that I don't completely understand. I'm looking for an explanation in layman's terms so that I understand what it is doing as I make these seemingly magical tran... | rcollyer | 2,283 | <p>I'm going to come at this one from left-field. In quantum mechanics, we deal with infinite dimensional vector spaces (Hilbert spaces), so I tend to think of integral transforms in those terms. For instance, </p>
<p>$$\int_{-\infty}^{\infty} K(x,y) f(y) dy = F(x)$$</p>
<p>can be thought of as </p>
<p>$$ \mathbf{... |
6,661 | <p>Is there a simple explanation of what the Laplace transformations do exactly and how they work? Reading my math book has left me in a foggy haze of proofs that I don't completely understand. I'm looking for an explanation in layman's terms so that I understand what it is doing as I make these seemingly magical tran... | vonjd | 346 | <p>Have also a look here - many great resources for the Laplace transform:<br>
<a href="https://mathoverflow.net/questions/383/motivating-the-laplace-transform-definition/2141#2141">https://mathoverflow.net/questions/383/motivating-the-laplace-transform-definition/2141#2141</a></p>
|
384,991 | <p>If $f$ is defined as a function of real variables to real values, and $c \in cl(Domain)$ as its limit value (i.e. $\lim_{x \to c} {f(x)} = 0 $) how to prove that this implies: $\lim_{x \to c} {1 \over f(x)} = \infty$.</p>
<p>It seems logical that the values will be always bigger, but when tried to construct a cont... | Cameron Buie | 28,900 | <p>This is problematic, even if you consider $1/|f(x)|$ instead of $1/f(x)$. For example, let $$f(x)=\begin{cases}x\sin(1/x) & x\ne0\\ 0 & \text{otherwise.}\end{cases}$$ This is everywhere defined and continuous on $\Bbb R$, and $$\lim_{x\to 0}f(x)=0,$$ but since there is no $x$-interval around $0$ on which $1/... |
3,338,597 | <p>I am struggling to find the cases for which <span class="math-container">$I(X;Y|Z)>I(X;Y)$</span>. The only mathematical example I could find for such a case is the following:
<span class="math-container">$$
I(X;Y) + I(X;Z|Y) = I(X;Z) + I(X;Y|Z).
$$</span>
This makes sense since they are both definitions of <sp... | Stelios | 153,212 | <p>Actually, the case you consider, that is, with <span class="math-container">$X$</span> and <span class="math-container">$Z$</span> being independent, is a well-known case where conditioning increases the mutual information. </p>
<p>To provide some intuition/interpretation of this result, consider a communication ch... |
142,734 | <p>What are some books that discuss elementary mathematical topics ('school mathematics'), like arithmetic, basic non-abstract algebra, plane & solid geometry, trigonometry, etc, in an insightful way? I'm thinking of books like Klein's <em>Elementary Mathematics from an Advanced Standpoint</em> and the books of the... | Shuchang | 39,246 | <p>I recommmend <em>How to prove it</em> by Daniel J. Velleman. The book introduces the basic logic and proof method to beginners and have many good examples and exercises to make students better understanding on what is a proof in the very elementary mathematics.</p>
|
142,734 | <p>What are some books that discuss elementary mathematical topics ('school mathematics'), like arithmetic, basic non-abstract algebra, plane & solid geometry, trigonometry, etc, in an insightful way? I'm thinking of books like Klein's <em>Elementary Mathematics from an Advanced Standpoint</em> and the books of the... | Tom Copeland | 12,178 | <p>In "On teaching mathematics", V. Arnold mentions Numbers and Figures by Rademacher and Töplitz, Geometry and the Imagination by Hilbert and Cohn-Vossen, What is Mathematics? by Courant and Robbins, How to Solve It and Mathematics and Plausible Reasoning by Polya, and Development of Mathematics in the 19th Century by... |
142,734 | <p>What are some books that discuss elementary mathematical topics ('school mathematics'), like arithmetic, basic non-abstract algebra, plane & solid geometry, trigonometry, etc, in an insightful way? I'm thinking of books like Klein's <em>Elementary Mathematics from an Advanced Standpoint</em> and the books of the... | Johann Cigler | 5,585 | <p>I would suggest "Numbers and functions from a classical-experimental mathematicians point of view" by Victor H.Moll. It contains very elementary but also some more sophisticated themes. </p>
<p>I once also wrote an elementary book "Grundideen der Mathematik", B.I. Wissenschaftsverlag 1992. But it is out of print an... |
2,378,042 | <p>I have been learning that the sum of the squares of the first <span class="math-container">$n $</span> natural numbers is given by <span class="math-container">$$\Sigma= \frac {1}{6} n (n+1)(2n+1) $$</span> while the sum of the cubes of the first <span class="math-container">$n $</span> natural numbers is given by:... | Doug M | 317,162 | <p>$\sum_\limits{i=1}^n i^k$</p>
<p>One way to do this is to assume that there exists a polynomial in terms of $n$ that equals the series.</p>
<p>And then $p(n+1) - p(n) = (n+1)^k$ and $p(1)= 1$ allows you to solve for the coefficients.</p>
<p>This is basically the same algebra of the induction proof.</p>
<p>An alt... |
375,016 | <p>I just started reading the ABC of category theory using the appendix of a text, the first chapter of a text that I have never read, and above all (I found out now that they handle well the theory) the wikipedia pages. I want to know only this: in all three sources are given the definition of the category, and in all... | Giorgio Mossa | 11,888 | <p>The relationship between category theory and set theory is much complex.
I suppose that by set theory you mean ZFC.
In such context you can define what are categories, functors and natural transformations and develop most of the basic theory.
The problems arise when you want to deal with <em>huge categories</em> su... |
1,275,810 | <p>Let <span class="math-container">$ c_0 = \{ x = (x_n)_{n \in \mathbb N} \in l^\infty : \lim_{n \to \infty} x_n = 0\}$</span>. Show that <span class="math-container">$c_0$</span> is a Banach space with the norm <span class="math-container">$\rVert \cdot \lVert_\infty$</span></p>
<p>I am capable of showing the space ... | copper.hat | 27,978 | <p>Suppose $x^k \in c_0$ and $x^k \to x$. Let $\epsilon>0$ and pick $N$ such that
$\|x^k-x\|_\infty < {1 \over 2 } \epsilon$ for $k \ge N$. Since $x^N \in c_0$,
there is some $N'$ such that $|x_i^N| < {1 \over 2 } \epsilon$ for $i \ge N'$.
Then $|x_i| \le |x_i^N| +|x_i-x_i^N| \le |x_i^N| +\|x-x^N\|_\infty <... |
4,038,802 | <p>Say I have <span class="math-container">$2^n +1 < n! -n$</span> for all <span class="math-container">$n \ge 4$</span>, and <span class="math-container">$n$</span> is an integer.</p>
<p>My inductive steps says, consider a <span class="math-container">$k$</span> that is a arbitrary integer, assuming <span class="ma... | HallaSurvivor | 655,547 | <p>Welcome to MSE!</p>
<p>Here is a hint:</p>
<p>We know that every permutation can be written as a product of transpositions
(see <a href="https://math.stackexchange.com/questions/1953960/permutation-written-as-product-of-transpositions">here</a>, for instance). So if you can get every transposition, then you can get ... |
1,025,079 | <p>Consider two related problems:</p>
<ol>
<li>You have $n$ cannisters that must go into $m$ trucks that can each carry $k$ cannisters. You require that no truck becomes overloaded, and for each cannister, there is a specified subset of trucks in which it may be safely carried. Is there a way to load all $n$ cannister... | Irvan | 172,851 | <p>For question (1), yes, there is, using maximum matching / flow. Consider the bipartite graph $(L, R)$ -- each of the nodes $l_i$ in $L$ correspond to a canister, and each of the nodes $r_i$ in $R$ correspond to a truck. For every pair of allowed canister $(l_i, r_i)$, such that it is allowed to put canister $l_i$ in... |
3,387,049 | <p>Say a deck of cards is dealt out equally to four players (each player receives 13 cards).</p>
<p>A friend of mine said he believed that if one player is dealt four-of-a-kind (for instance), then the likelihood of another player having four-of-a-kind is increased - compared to if no other players had received a four... | Reese Johnston | 351,805 | <p>First off: there are</p>
<p><span class="math-container">$$\frac{1}{4!} \cdot \left(\begin{array}{c}52\\13\end{array}\right) \cdot \left(\begin{array}{c}39\\13\end{array}\right) \cdot \left(\begin{array}{c}26\\13\end{array}\right) = 22,351,974,068,953,663,683,015,600,000$$</span></p>
<p>distinct combinations of fo... |
176,386 | <p>I stumbled across this "dictionary for noncommutative topology" <a href="http://planetmath.org/noncommutativetopology" rel="nofollow">http://planetmath.org/noncommutativetopology</a>
and I would be very interested in learning more on the subject, particularly I'd like to see why these results are true. </p>
<p>Wha... | ABIM | 36,886 | <p><em>Les C$^*$-algèbres et leurs représentations</em> (answers most of these). </p>
<p>Find it <a href="http://www.google.ca/url?sa=t&rct=j&q=&esrc=s&source=web&cd=6&ved=0CDgQFjAF&url=http%3A%2F%2Fiecl.univ-lorraine.fr%2F~Nicolas.Prudhon%2Fgnc%2Fbibliographie%2FOperator%2520Algebras%2FDi... |
3,489,898 | <p>In order to show that a metric space <span class="math-container">$(X, d)$</span> is not complete one may apply the definition and look for a Cauchy sequence <span class="math-container">$\{x_n\}\subset X$</span> which does not converge with respect to the metric <span class="math-container">$d$</span>. Now I have o... | QuantumSpace | 661,543 | <p>The point here is that there are two metric spaces involved.</p>
<p>Basically, the following situation happens. Let <span class="math-container">$X$</span> be a metric space and <span class="math-container">$Y$</span> be a metric subspace of <span class="math-container">$X$</span> (thus <span class="math-container"... |
2,648,273 | <p>I am reading "Linear Algebra" by Takeshi SAITO.</p>
<p>Why $n \geq 0$ instead of $n \geq 1$?<br>
Why $K^0 = \{0\}$?<br>
Is $K^0 = \{0\}$ a definition or not?</p>
<p>He wrote as follows in his book:</p>
<p>Let $K$ be a field, and $n \geq 0$ be a natural number. </p>
<p>$$K^n = \left\{\begin{pmatrix}
... | Thomas Andrews | 7,933 | <p>It can be thought of as a "useful" definition. Any subspace of $K^n$ is isomorphic to $K^m$ for some $m\leq n$. If you don't define $K^0=\{0\},$ then this isn't true for the $0$-subspace.</p>
<p>Another approach is to define $K^n$ as the set of functions from a set of $n$ elements to $K$. When $n=0$, the set of fun... |
1,752,468 | <p>I've tried some values of $m$, and found that the equality $C^{m}_{2m}=\Sigma_{i=1}^{m}C^{i}_{m+1}C^{i-1}_{m-1}$ holds. But I can't give it a proof. Can anybody give some suggestions?</p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>Compare the coefficients of $x^m$ in $$(1+x)^{2m}=(1+x)^{m-1}(x+1)^{m+1}$$</p>
|
3,186,192 | <p>Is the function below always positive for <span class="math-container">$0< x <1$</span>? (I am determining if the function requires the modulus sign or not.)</p>
<p><span class="math-container">$$\frac{1}{2}\log\left|\frac{1+\log(x)}{1-\log(x)}\right|$$</span></p>
<p>My first instinct is that it cannot be, b... | Peter Foreman | 631,494 | <p>This function is equivalent to
<span class="math-container">$$\mathrm{artanh}\,(\ln{(x)})$$</span>
which is only defined for <span class="math-container">$\frac1e\lt x\lt e$</span> and is negative for <span class="math-container">$\frac1e\lt x\lt1$</span>.</p>
|
2,727,598 | <p>Given p is a prime number greater than 2, and</p>
<p>$ 1 + \frac{1}{2} + \frac{1}{3} + ... \frac{1}{p-1} = \frac{N}{p-1}$ </p>
<p>how do I show, $ p | N $ ???</p>
<p>The previous part of this question had me factor $ x^{p-1} -1$ mod $p$. Which I think is just plainly $(x-1) ... (x-(p-1))$ </p>
| TheSimpliFire | 471,884 | <p>By Wolstenholme's Theorem, we have that $$\sum_{k=1}^{p-1}\frac{(p-1)!}k\equiv0\pmod{p^2}$$ so $$(p-1)!\left(1+\frac12+\cdots+\frac1{p-1}\right)=np^2$$ for some integer $n$. </p>
<p>From your condition (assuming typo of $(p-1)!$ instead of $p-1$) we get $$\frac{np^2}{(p-1)!}=\frac N{(p-1)!}\implies p^2=\frac Nn$$ a... |
1,356,524 | <p>EDIT: the OP has since edited the question fixing all the issues mentioned here. Yay!</p>
<p>There was a <a href="https://puzzling.stackexchange.com/questions/17627/knights-knaves-and-normals-smullyans-error/17628">question asked on Puzzling recently</a>, titled <code>Knights, Knaves and Normals "(Smullyan's Er... | psmears | 134,915 | <p>Short answer: you're both wrong! (Or, perhaps more optimistically, you're both right.) But on balance you're probably less wrong :-)</p>
<p>The thing is, to answer questions like this definitively, you have to be very precise about the rules of the “game” – and there are many sets of rules that co... |
476,955 | <p>Let $X$ be a set with 4 elements. Is it possible to have a topology on $X$ with 14 open sets?</p>
| André Nicolas | 6,312 | <p>We do most of the problem, leaving the last step to you.</p>
<p>If singletons are open, everything is open.</p>
<p>Suppose that $\{1\}$ is not open. Then $\{1,2,3\}$, $\{1,2,4\}$, and $\{1,3,4\}$ cannot all be open, else their intersection would be.</p>
<p>We may assume that $\{1,2,3\}$ is not open. Show this is ... |
373,397 | <p>the question just like in the title: </p>
<p>How to solve $x^3-bx^2+c=0$ for x analitically, when b and c are constants, and all (coefficients and variables) are reals.</p>
| Somaye | 83,530 | <p>$x^3-bx^2+c=0$ is continue and we know that a polynomial from n degree cant have more than n root.so you must find that where that $x^3-bx^2+c$ have change at it's sign then you can find roots are in which interval </p>
|
1,678,687 | <p>Find the point $(x_0, y_0)$ on the line $ax + by = c$ that is closest to the origin. </p>
<p>According to this <a href="http://www.intmath.com/plane-analytic-geometry/perpendicular-distance-point-line.php" rel="nofollow noreferrer">source</a>, I thought that $\left( -\frac{ac}{a^{2}+b^{2}}, -\frac{bc}{a^{2}+b^{... | Community | -1 | <p>The requested point lies on the perpendicular ($(a,b)\to(b,-a)$) to the line drawn from the origin. Solve</p>
<p>$$\begin{align}ax+by&=c\\bx-ay&=0.\end{align}$$</p>
|
1,586,607 | <p>This is something I've been thinking about lately;</p>
<p>$$\int_0^1 \int_0^1 \frac{1}{1-(xy)^2} dydx$$
Solutions I've read involve making the substitutions: $x= \frac{sin(u)}{cos(v)}$ and $y= \frac{sin(v)}{cos(u)}$. This reduces the integral to the area of a right triangle with both legs of length $\frac{\pi}{2}$... | Winther | 147,873 | <p>Here is an alternative derivation. The form of the integrand suggest expanding it a geometrical series
$$\frac{1}{1-(xy)^2} = \sum_{n=0}^\infty x^{2n}y^{2n}$$
Now integrating term by term we get
$$\int_0^1\int_0^1\frac{{\rm d}x{\rm d}y}{1-(xy)^2} = \sum_{n=0}^\infty \frac{1}{(2n+1)^2}$$
and from the <a href="https:/... |
1,549,079 | <p>According to my textbook (Grinstead and Snell's Introduction to Probability), $\int_{0}^{\infty} \lambda e^{(t-\lambda)x} dx = \frac{\lambda}{\lambda-t}$.</p>
<p>But it seems to me like there are two solutions. If $t \gt \lambda$, then $\int_{}^{}\ldots = \frac{\lambda}{t - \lambda}$, and if $t < \lambda$, then ... | Jason | 195,308 | <p>If $t\ge\lambda$, the integral diverges. It is implicit that $t<\lambda$.</p>
|
1,549,079 | <p>According to my textbook (Grinstead and Snell's Introduction to Probability), $\int_{0}^{\infty} \lambda e^{(t-\lambda)x} dx = \frac{\lambda}{\lambda-t}$.</p>
<p>But it seems to me like there are two solutions. If $t \gt \lambda$, then $\int_{}^{}\ldots = \frac{\lambda}{t - \lambda}$, and if $t < \lambda$, then ... | Enrico M. | 266,764 | <p>$$\lambda\int_0^{+\infty} e^{ax} \text{d}x$$</p>
<p>where $a = t-\lambda$</p>
<p>Solving that ad you will get</p>
<p>$$\lambda\left(\frac{e^{ax}}{a}\bigg|_{0}^{+\infty}\right)$$</p>
<p>$a$ has to be negative, otherwise the integral diverges. So you have to have $t-\lambda < 0$</p>
|
52,361 | <p>I have seen the following one. Please give the proof of the observation.
We know that, The difference between the sum of the odd numbered digits (1st, 3rd, 5th...) and the sum of the even numbered digits (2nd, 4th...) is divisible by 11. I have checked the same for other numbers in different base system. For example... | Bill Dubuque | 242 | <p>Yes, the test for divisibility by $11$ generalizes to arbitrary radix. Using modular arithmetic this amounts to simply evaluating a polynomial. Notice that radix notation has polynomial form. Namely in radix $\rm\:b\:$ the digit string $\rm\ d_n\ \cdots\ d_1\ d_0\ $ denotes $\rm\ d_n\ b^n +\:\cdots\: + d_1\ b + d_0... |
904,074 | <p>Is this Convergent or Divergent</p>
<p>$$\int_0^1 \frac{1}{\sin(x)}\mathrm dx $$</p>
<p>So little background to see if I am solid on this topic otherwise correct me please :) </p>
<p>To check for convergence I can look for a "bigger" function and compare if that is convergent, If yes then for sure the one in ques... | Cao | 168,664 | <p>We have
$$\frac{1}{{\sin x}}:\frac{1}{x} = \frac{x}{{\sin x}} = \frac{1}{{\sin x/x}} \to 1$$
as $x\to 0^+$. In other word, $\frac{1}{\sin x}\sim \frac{1}{x}$ as $x\to 0^+$. But, it is well known that $\int_0^{1}\frac{1}{x}dx$ is divergent. So, we conclude that $\int_0^1 {\frac{1}{{\sin x}}dx}$ is divergent.</p>
|
3,306,209 | <p>I was given this problem and told to find it's derivative: <span class="math-container">$$f(x)=\int^{x^2}_1 t^2+t+1\ dt$$</span>I thought the derivative was simply the inside function in terms of x - <span class="math-container">$x^2+x+1$</span>. But, this is not correct. Where am I going wrong? What piece am I miss... | Chinnapparaj R | 378,881 | <p>Hint:</p>
<p>If <span class="math-container">$h$</span> is continuous , <span class="math-container">$f$</span> and <span class="math-container">$g$</span> are differentiable and <span class="math-container">$$F(x)=\int_{f(x)}^{g(x)}h(t)\;dt$$</span> then <span class="math-container">$F'(x)=h(g(x)) \cdot g'(x)- h(f... |
3,306,209 | <p>I was given this problem and told to find it's derivative: <span class="math-container">$$f(x)=\int^{x^2}_1 t^2+t+1\ dt$$</span>I thought the derivative was simply the inside function in terms of x - <span class="math-container">$x^2+x+1$</span>. But, this is not correct. Where am I going wrong? What piece am I miss... | Community | -1 | <p>You're missing the chain rule. Plus you need to evaluate the integrand at <span class="math-container">$x^2$</span>.</p>
<p>So <span class="math-container">$F'(x)=((x^2)^2+x^2+1)2x=2x^5+2x^3+2x$</span>. </p>
|
239,045 | <p>I'm trying to write a short program that does the following over and over again.</p>
<ol>
<li>Import all files in a folder</li>
<li>Automatically generate names for the imported files based on source file name.</li>
<li>Perform a simple operation on the file.</li>
<li>Re-Export the modified files with a new name.</l... | TumbiSapichu | 53,229 | <p>Suppose you have many .csv files in your current directory, named "File1.csv", "File2.csv", etc.</p>
<p>You can first get the names of all .csv files:</p>
<pre><code>myCurrentCSVs=FileNames["*.csv"];
</code></pre>
<p>Then, you can make some operation, opening each at a time, for instanc... |
2,439,069 | <p>$$\int_0^{\infty}\frac{dx}{2+\cosh (x)}$$</p>
<p>I'm not sure how to approach this problem. I tried substituting $\cosh (x) = \frac{e^x + e^{-x}}{2}$, and following the substitution $y = e^x$, I ended up with the integral </p>
<p>$$2 \cdot \int_1^{\infty} \frac{1}{(y+2)^2-3} dy$$</p>
<p>Which I wasn't able to ev... | Dr. Sonnhard Graubner | 175,066 | <p>write your Integrand in the form
$$\frac{2e^x}{e^{2x}+4e^x+1}$$ and set $$e^x=t$$</p>
|
2,439,069 | <p>$$\int_0^{\infty}\frac{dx}{2+\cosh (x)}$$</p>
<p>I'm not sure how to approach this problem. I tried substituting $\cosh (x) = \frac{e^x + e^{-x}}{2}$, and following the substitution $y = e^x$, I ended up with the integral </p>
<p>$$2 \cdot \int_1^{\infty} \frac{1}{(y+2)^2-3} dy$$</p>
<p>Which I wasn't able to ev... | Raffaele | 83,382 | <p>$$\int_0^{\infty}\frac{dx}{2+\cosh (x)}=\int_0^{\infty}\frac{dx}{2+\frac{e^x+e^{-x}}{2}}$$
substitute $e^x=y\to x =\log y\to dx =\dfrac{dy}{y}$</p>
<p>limits become $x=0\to y=1;\;x\to\infty,\;y\to\infty$ so the integral becomes</p>
<p>$$\int_1^{\infty}\frac{dy}{2+\frac{y+\frac{1}{y}}{2}}\,\dfrac{1}{y}=\int_1^{\inf... |
1,968,267 | <p>When doing induction should you always try to put your final answer as the "<em>desired</em> " form? For example if: $$\sum^{n}_{k=1}(k+2)(k+4) = \frac{2n^{3} + 21n^{2} + 67n}{6}$$ we ought to give the final answer as $$\frac{2(k+1)^{3} + 21(k+1)^{2} + 67(k+1)}{6}?$$</p>
<p>I just expanded both the $\text{LHS}_{k+1... | barak manos | 131,263 | <p><strong>First, show that this is true for $n=1$:</strong></p>
<p>$\sum\limits_{k=1}^{1}(k+2)(k+4)=\frac{2\cdot1^3+21\cdot1^2+67\cdot1}{6}$</p>
<p><strong>Second, assume that this is true for $n$:</strong></p>
<p>$\sum\limits_{k=1}^{n}(k+2)(k+4)=\frac{2n^3+21n^2+67n}{6}$</p>
<p><strong>Third, prove that this is t... |
1,968,267 | <p>When doing induction should you always try to put your final answer as the "<em>desired</em> " form? For example if: $$\sum^{n}_{k=1}(k+2)(k+4) = \frac{2n^{3} + 21n^{2} + 67n}{6}$$ we ought to give the final answer as $$\frac{2(k+1)^{3} + 21(k+1)^{2} + 67(k+1)}{6}?$$</p>
<p>I just expanded both the $\text{LHS}_{k+1... | Community | -1 | <p>As both members are cubic polynomials, it suffices to show equality for $4$ distinct values of $n$.</p>
<p>$$0=0=\frac{0+0+0}6,\\
0+3\cdot5=15=\frac{2+21+67}6,\\
0+3\cdot5+4\cdot6=39=\frac{2\cdot8+21\cdot4+67\cdot2}6,\\
0+3\cdot5+4\cdot6+5\cdot7=74=\frac{2\cdot27+21\cdot9+67\cdot3}6.\\
$$</p>
<p>QED.</p>
|
282,139 | <p>Assume that $\gamma$ is an analytic simple closed curve in $\mathbb{R}^2$ which surrounds origin.</p>
<blockquote>
<p>Is there a polynomial vector field on the plane which is tangent to $\gamma$? In the other word, can an arbitrary analytic simple closed curve be realized as a closed orbit or a limit cycle of a p... | john mangual | 1,358 | <p>Far as I can tell... <a href="https://en.wikipedia.org/wiki/Hex_(board_game)" rel="nofollow noreferrer">Hex</a> is a "Princeton" phenomenon. It originated in their Math department as invented by John Nash / Piet Hein. The game is not widely available (but these days with Amazon, relatively easy to find). And... t... |
2,381,496 | <p>I am trying to write the explicit formula of all solutions of a linear system in the form :</p>
<p>$Ax=b$ where $A$ is an $m \times n$ matrix ($n$ different from $m$), $x$ is $n$-dimensional vector and $b$ is $m$-dimensional vector.</p>
| poetasis | 546,655 | <p>With Euclid's formula <span class="math-container">$F(m,k)$</span>, we can solve <span class="math-container">$R=$</span>area/perimeter for <span class="math-container">$k$</span> and test a defined range of <span class="math-container">$m$</span>-values to see which yield integers.
<span class="math-container">$$ \... |
1,252,326 | <p>"A student sits 6 examination papers, each worth 100 marks. In how many possible ways can he score 40% of the total possible marks?" I could not think of a way to attack this from first principles and thought there must be some theoretical generality that would make it quite easy. Haven't done any serious mathematic... | Christian Blatter | 1,303 | <p>Assume that there are just two papers worth $100$ points each, and we want to know the number of ways to score a total of $137$ points, say. It is a well known "trick" to consider the product
$$\bigl(1+x+x^2+\ldots +x^{99}+x^{100}\bigr)\cdot \bigl(1+x+x^2+\ldots +x^{99}+x^{100}\bigr)\ .$$
Each possible pair $(p,q)$... |
1,372,997 | <p>This question originates from the 1996 Canada National Olympiad.</p>
<blockquote>
<p>Let $r_1, r_2, \dots, r_m$ be a given set of $m$ positive rational numbers such that</p>
<p>$\sum\limits^{m}_{k=1}{r_k} = 1 \tag{1}$</p>
<p>Define the function $f$ by </p>
<p>$f(n) = n − \sum\limits^{m}_{k=1}{\lflo... | Community | -1 | <p>The minimum is $0$ when $n=$ common multiplier of $q_i$ in OP's notation.</p>
<p>The maximum is $m-1$, to achieve the result, we need to prove </p>
<ol>
<li><p>$f(n)<m$</p></li>
<li><p>For any set of $r_i$ exist $n$ such that $f(n)=m-1$</p></li>
<li><p>$f(n)$ is always integer</p></li>
</ol>
<p>Prove of 1. is ... |
202,530 | <p>I'd like to check if all elements of a list are numbers. I've tried </p>
<pre><code>t = {5/4, 12}
MatrixQ[t, NumberQ]
MemberQ[t, NumberQ]
And @@ Table[NumberQ[t[[i]]], {i, 1, Length[t]}]
</code></pre>
<p>but only the last one yields the desired result. Is there a better way to check?</p>
| kglr | 125 | <p>You can use <a href="https://reference.wolfram.com/language/ref/Element.html" rel="nofollow noreferrer"><code>Element</code></a>:</p>
<blockquote>
<p>Element[{<span class="math-container">$x_1 , x_2 , \ldots$</span>}, dom]
<em>asserts that all the</em> <span class="math-container">$x_i$</span> <em>are elements ... |
1,321,704 | <p>For example: $3^\sqrt5$ versus $5^\sqrt3$</p>
<p>I tried to write numbers as this:</p>
<p>$3^{5^{\frac{1}{2}}}$ and then as
$3^{\frac{1}{2}^5}$</p>
<p>But this method gives the wrong answer because $a^{(b^c)} \ne a^{bc}$</p>
| RowanS | 246,774 | <p>Consider </p>
<p>$f(x)=x^{\sqrt{5}}-5^{\sqrt{x}}$</p>
<p>$f(\sqrt{5})=0 $</p>
<p>$f'(x)=(\sqrt{5}-1)x-\frac {log(5)}{2 \sqrt{x}}e^{\sqrt{x}log5}$</p>
<p>so $ f(x) < 0$ for $x$ less than $\sqrt{5}$ so we have $$ 5^{\sqrt{3}} > 3^{\sqrt{5}}$$</p>
|
1,321,704 | <p>For example: $3^\sqrt5$ versus $5^\sqrt3$</p>
<p>I tried to write numbers as this:</p>
<p>$3^{5^{\frac{1}{2}}}$ and then as
$3^{\frac{1}{2}^5}$</p>
<p>But this method gives the wrong answer because $a^{(b^c)} \ne a^{bc}$</p>
| Zach466920 | 219,489 | <p>$$3^{\sqrt 5} \lt 5^{\sqrt3}$$
$$\sqrt 5 \cdot \ln(3) \lt \sqrt3 \cdot \ln(5)$$
$$\sqrt{5 \over 3} \cdot \log_5 (3) \lt \sqrt{5 \over 3} \cdot {3 \over 4} \lt 1$$
because $\log_a(b) \lt 1$ for $a \gt b$, $\ln(3) \lt {3 \over 2}$, and $\ln(5) \lt 2$</p>
|
972,385 | <p>Could someone please explain this question to me? I know that such integers do NOT exist, but I could not prove it. "Either solve it or give a brief explanation as to why it is impossible." Thank you!</p>
<p>Find integers $s$ and $t$ such that $15s + 11t = 1$.</p>
| curious | 184,154 | <p>Try $s=3$ and $t=-4$. (More generally, $s=3+11k$ and $t=-4-15k$ for any integer $k$ will work.)</p>
<p>You can know that such integers exist in advance from <a href="http://en.wikipedia.org/wiki/B%C3%A9zout's_identity" rel="nofollow">Bezout's identity</a>, and find them using the Euclidean algorithm.</p>
|
3,188,966 | <p>Find the number of real solutions of <span class="math-container">$x^7 + 2x^5 + 3x^3 + 4x = 2018$</span>?</p>
<p>What is the general approach to solving this kind of questions? I am interested in the thought process. </p>
<hr>
<p>Few of my thoughts after seeing this question:
since <span class="math-container">$x... | DINEDINE | 506,164 | <p>If <span class="math-container">$x\le 0$</span> the left hand side is negative therefore no solution. We suppose <span class="math-container">$x>0$</span> and we consider <span class="math-container">$f(x)=x^7+2x^5+3x^3+4x$</span>, then <span class="math-container">$f$</span> is the sum of increasing functions t... |
540 | <p>A user has been using multiple unregistered accounts (with the same name and the same avatar). For reference, the accounts I've been able to find are 8280, 8281, 8284, 8289, 8294, 8308, 8309, 8310, 8311, 8312, 8313, 8314, 8315, 8316, 8317, 8318, and 8319, but I expect more to be added soon.</p>
<p>If these were re... | quid | 143 | <p>Using (multiple) unregistered accounts is not in itself something that is against the SE platform's norms. If SE would not want to allow contributions of this form, it would be easy to enforce sign-up to contribute. Instead, it is considered a feature that one can contribute in passing. </p>
<p>What makes this part... |
540 | <p>A user has been using multiple unregistered accounts (with the same name and the same avatar). For reference, the accounts I've been able to find are 8280, 8281, 8284, 8289, 8294, 8308, 8309, 8310, 8311, 8312, 8313, 8314, 8315, 8316, 8317, 8318, and 8319, but I expect more to be added soon.</p>
<p>If these were re... | James S. Cook | 128 | <p>It may not be against the sitewide policy, but, I for one will downvote such posts. Even I can figure out how to get and use a username. It's a small thing to expect. I find it far more offensive than questions that are "off-topic" or "too-localized" (thankfully, we have not attracted the gaming legalists the other ... |
746,751 | <p>I'm studying the theory of computation, and I know there are pumping lemmas for regular and context-free languages, but why not for recursively enumerable languages? Is there something about a Turing machine that would make a pumping lemma impossible? My guess is that no matter how you pump a string, a Turing machin... | user2566092 | 87,313 | <p>A Turing machine can decide languages like the set of all strings that consist of $N$ substrings that are each consecutive 0's and consecutive 1's that alternate where the substrings have length $1,2,3,..,N$, for some natural number $N$ (the language is the union over all possible $N$). It's hard to imagine any kind... |
746,751 | <p>I'm studying the theory of computation, and I know there are pumping lemmas for regular and context-free languages, but why not for recursively enumerable languages? Is there something about a Turing machine that would make a pumping lemma impossible? My guess is that no matter how you pump a string, a Turing machin... | Carl Mummert | 630 | <p>The reason is that context free and regular languages are recognized by much simpler kinds of automata that have limitations on how they use their memory, while a Turing machine does not have those limitations.</p>
<p>For example, a regular language is recognizable by a finite state machine. The finite state machin... |
746,751 | <p>I'm studying the theory of computation, and I know there are pumping lemmas for regular and context-free languages, but why not for recursively enumerable languages? Is there something about a Turing machine that would make a pumping lemma impossible? My guess is that no matter how you pump a string, a Turing machin... | user642796 | 8,348 | <p>Part of the problem is that TMs are allowed to go <em>backwards</em> through their tapes. This makes concocting a Pumping Lemma difficult for a basic reason.</p>
<p>Recall that the basic proof of the Pumping Lemma for regular languages was to take a DFA $D$ for $L$, and then count the number of local configuration... |
2,088,564 | <blockquote>
<p>Sum of binomial product $\displaystyle \sum^{n}_{r=0}\binom{p}{r}\binom{q}{r}\binom{n+r}{p+q}$</p>
</blockquote>
<p>Simplifying $\displaystyle \frac{p!}{r!\cdot (p-r)!} \cdot \frac{q!}{r!\cdot (q-r)!}\cdot \frac{(n+r)!}{(p+q)! \cdot (n+r-p-q)!}$.</p>
<p>Could some help me with this, thanks</p>
| Robert Z | 299,698 | <p>This is Bizley’s identity (6.42 in Gould's book "Combinatorial identities"):
$$\sum^{n}_{r=0}\binom{p}{r}\binom{q}{r}\binom{n+r}{p+q}=\binom{n}{p}\binom{n}{q}.$$
A nice combinatorial proof can be found in the following paper by Székely as a special case of a more general identity (see Corollary 3):</p>
<p><a href="... |
1,876,590 | <p>In "<a href="https://web.archive.org/web/20151219180208/http://apollonius.math.nthu.edu.tw/d1/ne01/jyt/linkjstor/regular/7.pdf" rel="nofollow">Angle Trisection, the Heptagon, and the Triskaidecagon</a>", published in the <em>American Mathematical Monthly</em> in March 1988, Andrew Gleason discusses what regular poly... | Tito Piezas III | 4,781 | <p>The question essentially asks about transforming <em>solvable</em> equations from one form to another. </p>
<blockquote>
<p><strong>I. Cubic</strong></p>
</blockquote>
<p>Using just a linear transformation, the general cubic $P(x)=0$ can be transformed to the form,</p>
<p>$$y^3+3ay+b = 0\tag1$$</p>
<p>with so... |
637,699 | <p>We have a triangle $ABC$. Whats the value of angle $C$?</p>
<p>$$\sin^2(A)+\sin^2(B)-\sin^2(C)=1$$</p>
<p>I made a small java program and it gave me an answer. I want to know how to make it through other ways.</p>
| lab bhattacharjee | 33,337 | <p>From my comment,</p>
<p>$$\cos C=\frac{-\cos(A-B)\pm\sqrt{\cos^2(A-B)+4}}2$$</p>
<p>For a non-trivial triangle, $\displaystyle 0<C<\pi\implies -1<\cos C<1$</p>
<p>Now as $\displaystyle |A-B|<\pi,\cos(A-B)>0 \implies \cos(A-B)+\sqrt{\cos^2(A-B)+4}>2$</p>
<p>Now if $\cos C_1,\cos C_2$ are the ... |
159,495 | <p>I have a text file where each line is a data point in the form:</p>
<pre><code>[ -495.01172, -158.35966, 2705.0 ]
[ -489.15576, -127.229675, 2673.0 ]
[ -487.6918, -97.679855, 2665.0 ]
[ -487.32578, -68.4594, 2663.0 ]
[ -485.86182, -39.19415, 2655.0 ]
[ -485.3128, -10.12311, 2652.0 ]
[ -484.03183, 18.853745, 2645.0 ... | george2079 | 2,079 | <p>another approach.</p>
<pre><code>ImportString[
StringReplace[
Import["test.txt", "Text"], {"[", "]"} -> ""], "CSV"]
</code></pre>
|
572,429 | <p>If A is invertible, prove that $\lambda \neq 0$, and $\vec{v}$ is also an eigenvector for $A^{-1}$, what is the corresponding eigenvalue?</p>
<p>I don't really know where to start with this one. I know that $p(0)=det(0*I_{n}-A)=det(-A)=(-1)^{n}*det(A)$, thus if both $p(0)$ and $det(0) = 0$ then $0$ is an eigenvalue... | LASV | 89,736 | <p>Hint: Look at $I_nv=A^{-1}(Av)$.</p>
|
2,475,054 | <p>I'm wondering if someone could give me some hints as to how to approach this question, or some theory to understand what it's actually asking:</p>
<p><a href="https://i.stack.imgur.com/FXvsz.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/FXvsz.png" alt="enter image description here"></a></p>
<p... | Peter Szilas | 408,605 | <p>$F(x,y,z) =0$ is the surface, let $P := (x_0,y_0z_0) $ be a point on the surface.</p>
<p>Normal vector $\vec n$ to the surface at $P$ is:</p>
<p>$\vec n = (F_x(P),F_y(P),F_z(P)).$</p>
<p>This normal vector specifies the tangent plane to the surface.</p>
<p>If the normal points in the $z$ direction, </p>
<p>t... |
1,589,577 | <p>Try to find the solution of the functional equation $$f(x+y)=\frac{f(x)+f(y)}{1-4f(x)f(y)}$$ with $f'(1)=1/2$.</p>
| J.G. | 56,861 | <p>Define $g(x)=\arctan (2f(x))$ so $f(x)=\frac{1}{2}\tan g(x)$ and $$\tan g(x+y)=\frac{\tan g(x) + \tan g(y)}{1-\tan g(x) tan g(y)}=\tan (g(x)+g(y)).\quad(*)$$ Thus $$g(x+y)=g(x)+g(y)+\Pi(x,\,y),\quad(**)$$ where $\Pi(x,\,y)$ is a function with range $\subseteq \pi\mathbb{Z}$. Thus $g'(x+y)=g'(x)+g'(y)+\frac{\partial ... |
1,589,577 | <p>Try to find the solution of the functional equation $$f(x+y)=\frac{f(x)+f(y)}{1-4f(x)f(y)}$$ with $f'(1)=1/2$.</p>
| orangeskid | 168,051 | <p>HINT: Notice that the equation is equivalent to
$$2 f(x+y) = \frac{2 f(x) + 2 f(y)} { 1 - (2 f(x)) \cdot (2 f(y))}$$</p>
<p>while we know that
$$\tan(a(x+y)) = \frac{\tan(ax) + \tan(ay)}{1 - \tan(a x) \cdot \tan(a y)}$$</p>
<p>so we might just have $2 f(x) \equiv \tan a x$ for some convenient $a$. </p>
|
2,903,110 | <p>How do I prove that the function $$f:(0,1)\rightarrow \mathbb R$$ defined by:</p>
<p>$$f(x) = \frac{-2x+1}{(2x-1)^2-1}$$</p>
<p>is onto?</p>
| José Carlos Santos | 446,262 | <p>Just check that$$\lim_{x\to0^+}f(x)=-\infty\text{ and that }\lim_{x\to1^-}f(x)=+\infty.$$Now, the fact that the restricton of $f$ to $(0,1)$ is onto is a consequenc of the intermediate value theorem.</p>
|
2,531,523 | <p>Knowing that $S_3=\{\text{id},\sigma,\sigma^2,\tau,\tau\circ\sigma,\tau\circ\sigma^2\}$ ($\tau=(1 2), \sigma=(123)$), why does a group homomorphism $f:S_3\to\mathbb{Z}/2\mathbb{Z}$ satisfy $f(a)=0$ for all $a\in S_3$ such that $a^2\neq e$?</p>
| Hector Blandin | 170,571 | <p>Write
$$ L=\lim_{x \to 0^+}x^{a}\ln(x) = \lim_{x \to 0^+}\frac{\ln(x)}{x^{-a}}$$</p>
<p>and change the variable $\displaystyle{x=\frac{1}{u}}$, so $u\to \infty$</p>
<p>$$\underset{u\to\infty}{\lim}{\ \frac{\ln\left(\frac{1}{u}\right)}{u^{a}}}=\underset{u\to\infty}{\lim}{\ \frac{-\ln(u)}{u^{a}}}$$
then $u=e^{t}$, ... |
2,075 | <p>I've come across so many posts here and on the "main" math-SE site that voice complaints, frustrations, pet-peeves, grievances, or else are critical of another post/question, user, OP, etc. It is really an energy sapper! Certainly not a boost for morale.</p>
<p>Since I'm pretty new here, and feeling a bit ambival... | Mitch | 1,919 | <p>Despite my <a href="http://meta.math.stackexchange.com/questions/2069/question-in-imperative-homework">annoyance</a> with imperatives and homeworks, I like answering the questions here. There is very little noise here; even what you seem to notice as discord here at meta is at least -substantive- disagreement. The q... |
2,075 | <p>I've come across so many posts here and on the "main" math-SE site that voice complaints, frustrations, pet-peeves, grievances, or else are critical of another post/question, user, OP, etc. It is really an energy sapper! Certainly not a boost for morale.</p>
<p>Since I'm pretty new here, and feeling a bit ambival... | Qiaochu Yuan | 232 | <p>The signal-to-noise ratio here is great. The Stack Exchange team has worked very hard to provide a platform with tools to keep it great: the voting system, community moderation, etc. Reputation may seem silly, but it is "an important form of silliness": getting upvotes is psychologically rewarding and incentivizes p... |
18,421 | <p>I am teaching 4th-grade kids. The topic is Fraction. Basic understanding of a fraction as a part of the whole and as part of the collection is clear to the kids. Several concrete ways exist to teach this basic concept. But when it comes to fraction addition/subtraction I could not find a way that teaches it concrete... | Nick C | 470 | <p>I like the cutting paper approach the best because it is cheap and easy, and it forces students to confront basic fractions by folding/cutting them out. But here’s another thought...</p>
<p>For you to demonstrate: have several clear-plastic, cylindrical cups, each marked off in different fractions of a whole (maybe... |
3,947,632 | <p>It is almost obvious that we have to use derivative in order to find the answer. However, I couldn't see this one to be derivative of something.</p>
<p>It would be amazing if you share your thoughts on this problem. Also, I would be thankful if you would explain what was intuition behind your solution. Thanks!</p>
| saulspatz | 235,128 | <p>Let <span class="math-container">$$f(x)=\sum_{k=0}^nx^k,$$</span> so that <span class="math-container">$$
\begin{align}
f'(x)&=\sum_{k=1}^n kx^{k-1}\\
xf'(x)&=\sum_{k=1}^n kx^k\\
f'(x)+xf''(x)&=\sum_{k=1}^n k^2x^k\\
\end{align}$$</span></p>
<p>Now just evaluate <span class="math-container">$f(x)$</span> ... |
358,328 | <p>How do I go about doing this? I'm clueless..
Thank you.</p>
<p>My attempt:</p>
<p>Using the product rule and making:</p>
<p>$\eqalign{
& u = u \cr
& v = {v^{ - 1}} \cr} $</p>
<p>so:</p>
<p>${{du} \over {dx}} = 1$
and ${{dv} \over {dx}} = - {v^{ - 2}}$
so:</p>
<p>$\eqalign{
& {{dy} \over {d... | Xiaolang | 71,857 | <p>$$\frac{dy}{dx}=u\frac{d(\frac1v)}{dx}+\frac{du}{vdx}$$
and
$$\frac{d(\frac1v)}{dx}=-\frac{v'}{v^2}$$</p>
<p>then</p>
<p>$$\frac{dy}{dx}=\frac{-uv'}{v^2}+\frac{u'}{v}=\frac{u'v-uv'}{v^2}$$</p>
|
342,491 | <p>How to prove the following:</p>
<p>$a_n = \left\{\left(1+\frac{1}{n}\right)^n\right\}$ is bounded sequence, $ n\in\mathbb{N}$</p>
| Siméon | 51,594 | <p>This can be shown very quickly with the inequality $\log (1+u) \leq u$. Indeed,
$$
\left(1+\frac{1}{n}\right)^n = \exp\left(n\log(1+\frac{1}{n})\right) \leq\exp\left(n\times\frac{1}{n}\right)
$$
This gives you $e$ as an upper bound. Actually, this is the best possible bound since it is also the limit of the sequence... |
3,932,279 | <p>Here, I am taking Lebesgue measure. There is a notion that a set <span class="math-container">$E$</span> is measruable iff <span class="math-container">$m_*(E) = m^*(E)$</span>. Let <span class="math-container">$E' = [0,1] \cap \mathbb I$</span>, where <span class="math-container">$\mathbb I$</span> denote the set o... | Community | -1 | <p>It's easy to construct such an <span class="math-container">$F$</span>.</p>
<p>Let <span class="math-container">$(q_k)_{k=1}^{\infty}$</span> be an enumeration of the rationals in <span class="math-container">$[0,1]$</span>, let <span class="math-container">$0 < c < 1$</span> be a constant, and let <span class... |
2,321,850 | <p>The base is the semicircle $$y=\sqrt{16−x^2},$$
where -4 $\le$ $x$ $\le$ 4. The cross-sections perpendicular to the $x$-axis are squares.
$$\\$$
So far this is what I have:</p>
<p>$\int (Area)\,dx$</p>
<p>$\implies$ $\int \frac{π}{2} ($$r^2$$)\ dx$.</p>
<p>$r$ = $\frac{\sqrt{16−x^2}}{2}$</p>
<p>$\implies$ $ \fra... | Anurag A | 68,092 | <p>You should be considering the slice which is square in shape. Imagine a vertical line drawn at some point $x$ that meets the semi-circle. This will be the side of your square slice. Note the length of this line is simply the $y-$value of the point where the line meets the semicircle.</p>
<p>Now compute the area of... |
1,341,771 | <p>I'm trying to show that given three distinct points $z_1,z_2,z_3\in\mathbb C$, the rational function
$$
f(z) = \frac{(z-z_1)(z_2 - z_3)}{(z - z_3)(z_2 - z_1)} = \frac{(z_2 - z_3)z + (z_1z_3 - z_1z_2)}{(z_2 - z_1)z + (z_1z_3 - z_2z_3)} = \frac{az + b}{cz + d}
$$
is a Möbius transformation. That is, I must show that $... | Michael Albanese | 39,599 | <p>To show that your final expression is non-zero, note that</p>
<p>\begin{align*}
(z_3 - z_2)(z_3 - z_1)(z_2 - z_1) &= (z_3^2 - z_1z_3 - z_2z_3 + z_1z_2)(z_2 - z_1)\\
&= z_2z_3^2 - z_1z_2z_3 - z_2^2z_3 + z_1z_2^2 - z_1z_3^2 + z_1^2z_3 + z_1z_2z_3 - z_1^2z_2\\
&= z_2z_3^2 - z_2^2z_3 + z_1z_2^2 - z_1z_3^2 +... |
3,497,554 | <blockquote>
<p>Show that <span class="math-container">$\frac{(2n-1)!}{(n)!(n-1)!}$</span> is odd or even according as <span class="math-container">$n$</span> is or is not a power of <span class="math-container">$2$</span>.</p>
</blockquote>
<p>I know that the index of the highest power of <span class="math-containe... | Community | -1 | <p>For any integer <span class="math-container">$k$</span> let <span class="math-container">$v(k)$</span> be the highest power of <span class="math-container">$2$</span> dividing <span class="math-container">$k$</span>. </p>
<p>Then <span class="math-container">$v\left((2n-1)!\right)=v\left(2\times 4 \times 6 ...\time... |
830,755 | <p>I would like to compute the following,
$$
\int_0^{\infty}\int_0^{\infty}e^{-x^2-2xy-y^2}\ dx\,dy
$$
It is obvious that we can rewrite the integral above to,
$$
\int_0^{\infty}\int_0^{\infty}e^{-(x+y)^2}\ dx\,dy
$$
so we are ending up with something looking like a gaussian integral. I think that a smart substitution ... | Tunk-Fey | 123,277 | <blockquote>
<p><strong><a href="http://tutorial.math.lamar.edu/Classes/CalcII/ParametricIntro.aspx">Theorem</a> :</strong> $$ \iint_A f(x,y)\ dx\,dy=\iint_B g(u,v) |J|\ du\,dv, $$
where $J$ is <a href="http://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant">Jacobian</a>.</p>
</blockquote>
<p>Now, using para... |
2,710,900 | <blockquote>
<p>Let $S=\{1,2t,-2+4t^2,-12t+8t^3\}$ be a set of polynomials in $P_3$.
Show that these polynomials make up a basis for $P_3$ and determine
the coordinates for $p=7-12t-8t^2+12t^3$ in this basis.</p>
</blockquote>
<p>The first part of the problem was easy, showing that they make up a basis by showin... | MPW | 113,214 | <p>In the absence of grouping symbols (parentheses), the expression $a^{b^c}$ means $a^{(b^c)}$, not $(a^b)^c$. Calculators may or may not perform the implicit grouping correctly.</p>
<p>So on a calculator, you should either use parentheses, or store the intermediate result (the exponent $r=b^c$), then compute the fin... |
3,382,119 | <p><span class="math-container">$x\cdot\begin{bmatrix}0\\-3\\1\end{bmatrix}$</span> +
<span class="math-container">$y\cdot\begin{bmatrix}-5\\2\\-4\end{bmatrix}$</span> +
<span class="math-container">$z\cdot\begin{bmatrix}-20\\-1\\-13\end{bmatrix}$</span>
=<span class="math-container">$\begin{bmatrix}0\\0\\0\end{bmatrix... | wgb22 | 711,553 | <p>Your matrix is full rank which can be verified with numerical maths software (such as MATLAB).</p>
<p>That means your vectors are in fact <strong>linearly independent</strong>.</p>
<p>Therefore, only the trivial solution exists:
<span class="math-container">$\begin{bmatrix}x\\y\\z\end{bmatrix} = \bf{0}$</span></p>... |
1,741,229 | <p>As the title.
Or rather, for any integer $m$ which is not the characteristic, does such an 'integer division' exist?</p>
| Ross Millikan | 1,827 | <p>As long as $m \neq 0$ it must have an inverse. The fact that $m$ is integer does not matter.</p>
|
882,540 | <p>Does anyone know of a non-trivial (i.e. cardinality $\geq 2)$ algebraic structure $(X,+,-)$ satisfying the following identities?</p>
<ol>
<li><p>$(x+a)-a=x$</p></li>
<li><p>$(x-a)+a=x$</p></li>
<li><p>$(x+y)+a = (x+a)+(y+a)$</p></li>
<li><p>$(x-y)+a = (x+a)-(y+a)$</p></li>
</ol>
<p><em>Remark.</em> The Abelian gro... | achille hui | 59,379 | <p>Instead of $+$ and $-$, let us use $\tilde{+}$ and $\tilde{-}$ as the algebraic operations
we wish to invent.</p>
<p>Let $\mathcal{D} = \{\; a + b \epsilon : a, b \in \mathbb{R} \;\}$ be the the set of
<a href="http://en.wikipedia.org/wiki/Dual_number" rel="nofollow">dual numbers</a> over $\mathbb{R}$. i.e.
the a... |
56,804 | <p>I know the Galois group is $S_3$. And obviously we can swap the imaginary cube roots. I just can't figure out a convincing, "constructive" argument to show that I can swap the "real" cube root with one of the imaginary cube roots. </p>
<p>I know that if you have a 3-cycle and a 2-cycle operating on three elements, ... | Qiaochu Yuan | 232 | <p>I don't know what kind of argument you'd like to see, but the standard way to compute the Galois group of an irreducible cubic (characteristic not $2$) is to look at the sign of its <a href="http://en.wikipedia.org/wiki/Discriminant">discriminant</a> $\Delta = \prod_{i \neq j} (r_i - r_j)$. It has a square root $\pr... |
674,598 | <p>(a) $f(x) = |x|; -\infty <x < \infty $</p>
<p>The way i approach is to take the derivative for x from 0 to infinity which gives me 1 and take the derivative for -x from 0 to negative infinity which gives me negative 1...therefore, since there is not constant can be 1 and -1 at the same time ....my answer is t... | Community | -1 | <p>No. Recall the definition of Lipschitz:</p>
<blockquote>
<p>A function $f$ is Lipschitz if there exists a constant $C$ for which $|f(x) - f(y)| \le C |x - y|$ for all $x, y$.</p>
</blockquote>
<p>So in order to tell if $f(x) = |x|$ is Lipschitz, you need to see whether there is a constant $C$ such that
$$\Big||x... |
2,666,421 | <p>Consider a linear regression model, i.e., $Y = \beta_0 + \beta_1 x_i + \epsilon_i$, where $\epsilon_i$ satisfies the classical assumptions. The estimation method of the coefficients $(\beta_0 , \beta)$ is the least-squared method. What would be an intuitive explanation of why the sum of residuals is $0$? I know t... | Nicolas Hemelsoet | 491,630 | <p>$Z$ has no reason to be a principal divisor, e.g if $Z \subset \Bbb P^2$ is a line. In any case you have a map $\Bbb Z \to Pic(X), 1 \mapsto [Z]$.</p>
<p>The map $Pic(X) \to Pic(U)$ is given by restriction :$D \mapsto D \cap U$ where $D$ is a prime divisor and extend by linearity. In particular the composition is ... |
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