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 |
|---|---|---|---|---|
109,928 | <p>I've read lots of examples here on how to set one matrix <code>Equal</code> to another, but how do you nest <code>And</code> and <code>Or</code> around several different equalities for values a matrix of variables can take on? Using what I've found so far, I've been able to cobble something together that works, but ... | Edmund | 19,542 | <p>2D array solution for multiple target arrays for OP's ideal answer (<code>And[Or[V == {A, B, C}]]</code>).</p>
<p>This is demonstrated with <code>Indexed</code> as it make it easier to see the effect of the solution.</p>
<p>First set up the 2D arrays.</p>
<pre><code>index = Table[{r, c}, {r, 2}, {c, 2}];
\[Script... |
173,508 | <p>I am having an issue with the following complex analysis problem. I am suppose to
find the coefficients of $z^{-1}$, $z^{-2}$ and $z^{-3}$ in the Laurent series for
$\displaystyle \frac{1}{\sin z}$ around $z_0 = 0$ which is valid for $2\pi < |z| < 3\pi$.</p>
<p>One way I thought of doing it was to say
$$ ... | pathfinder | 23,431 | <p>$$\sum_{i=0}^2\sum_{j=0}^3(2i+3j)=2\sum_{i=0}^2\sum_{j=0}^3 i+3\sum_{i=0}^2\sum_{j=0}^3 j\\~~~~~~~~~~~~~~=8\sum_{i=0}^2i+9\sum_{j=0}^3 j\\~~~~~~~~~~~~=8\times 3+9\times 6\\\!\!\!\!\!=78.$$</p>
|
3,645 | <p>Being motivated by <a href="https://math.stackexchange.com/questions/3618/solving-an-equation-with-irrational-exponents">this post</a>, I was wondering if there is a proof (analogous to the case of Diophantine equations) that there is no general method for solving <a href="http://en.wikipedia.org/wiki/Transcendental... | Qiaochu Yuan | 232 | <p>Yes. The problem is that the term "transcendental function" is absurdly general. For example, pick a computable bijection between the integers and the set of <a href="http://en.wikipedia.org/wiki/Turing_machine" rel="nofollow">Turing machines</a> and consider the function $f : \mathbb{R} \to \mathbb{R}$ which is $... |
28,878 | <p>I defined a function in Mathematica. It is a function of n, so f[n_]:=, but in the definition, I used a sum, $\sum_{k=0}^n$. So, the $k$ is just an index variable for the sum and no $k$ shows up in the final answer. As I was using this function I tried evaluating f[k-1] and got a weird answer, 0. I finally figur... | Fabian | 7,266 | <p>The function you are looking for is called <code>Module</code>. You can define it as</p>
<p><code>f[n_] := Module[{k}, Sum[a[k], {k,0,n}]]</code></p>
<p>so that the evaluation <code>f[k-1]</code> is possible.</p>
|
2,897,070 | <p>In my analysis class, we had the following function:
$$f: \mathbb{R} \to \mathbb{R}: x \mapsto \begin{cases}
x^2 & \text{ if } x \in \mathbb{Q}\\ -x^2 & \text{elsewhere}.\end{cases}$$</p>
<p>At zero, I can see that this function is differentiable: indeed, we have that
$$\lim_{h \to 0}\Big|{\frac{f(h) -f (... | José Carlos Santos | 446,262 | <p>Consider the point $1$, for instance. In order to prove that $f$ is not differentiable there, you have to prove that the limit $\lim_{x\to1}\frac{f(x)-1}{x-1}$ doesn't exist. Note that there are numbers arbitrarily close to $1$ whose square is irrational. For such a point $x$, $f(x)$ is close to $-1$, and therefore ... |
3,272,824 | <p>If <span class="math-container">$V$</span> is an inner product space, <span class="math-container">$T:V\to V$</span> linear transformation,
show that</p>
<p><span class="math-container">$(\ker(T))^{\bot}=\operatorname{Im}(T^*)$</span> </p>
<p>when <span class="math-container">$T^*=\overline{(T)^T}. $</span></p>
... | paf | 333,517 | <p>The key property you need here is:</p>
<blockquote>
<p><span class="math-container">$$\forall x,y \in V,\; \langle T^*(x), y\rangle = \langle x, T(y)\rangle$$</span></p>
</blockquote>
<p>Thus, if <span class="math-container">$z\in \text{Im}(T^*)$</span>, then <span class="math-container">$\exists x\in V,\;z = T^... |
702,481 | <p>I am trying to solve the following problem: Let $a$ and $b$ be positive integers and let $p$ be a prime number. Prove that if $a^p \equiv$ $b^p$ (mod $p$), then $a \equiv b$ (mod $p$).</p>
<p>My attempt to solve this starts off by using the idea that $b^p-a^p$ is a multiple of $p$, i.e. $pm=b^p-a^p$, for some $m \i... | William Chang | 133,204 | <p>Fermat's Little Theorem is exactly what you use! If $p$ is prime, then $a^p \equiv a \pmod{p}$ and $b^p\equiv b \pmod{p}$. So simply, $a^p \equiv a \equiv b^p\equiv b \pmod{p} \implies a\equiv b\pmod{p}$ and you are done. </p>
<p>If you'd like a proof of Fermat's Little Theorem because you say you're not familiar w... |
3,860,441 | <p>I'm taking up a probability course and my teacher, when explaining the fondamental properties of probability, listed this one:
<span class="math-container">$$\text{If } A \text{ and }B \text{ are mutually exclusive events, then }P(A \cup B) = P(A) + P(B)$$</span></p>
<p>Then, later, under an "other properties&... | Jay | 9,814 | <p>We have <span class="math-container">$A = (A \smallsetminus B) \cup (A \cap B)$</span> and <span class="math-container">$B = (B \smallsetminus A) \cup (A \cap B)$</span>. Notice that <span class="math-container">$A \smallsetminus B$</span> and <span class="math-container">$A \cap B$</span> are disjoint. Similarly <s... |
2,496,658 | <p>Let <span class="math-container">$\mathcal{H}$</span> be a complex finite dimensional Hilbert space (<span class="math-container">$\dim\mathcal{H}=d$</span>). Let <span class="math-container">$A\in M_d(\mathbb{C})$</span>. </p>
<blockquote>
<p>How can I show without using spectral considerations that
<span clas... | Schüler | 496,104 | <p>In <a href="https://perso.univ-rennes1.fr/michel.crouzeix/publis/Final4.pdf" rel="nofollow noreferrer">Michel Crouzeix - Numerical range and functional calculus in Hilbert space</a>, we see that
<a href="https://i.stack.imgur.com/nrNSp.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/nrNSp.png" al... |
85,024 | <p>There are two results famously associated with Cantor's celebrated diagonal argument. The first is the proof that the reals are uncountable.</p>
<p><img src="https://i.stack.imgur.com/v9G8r.png" alt="Cantor's Diagonal Argument"></p>
<p>This clearly illustrates the namesake of the diagonal argument in this case... | Bill Dubuque | 242 | <p><strong>HINT</strong> $\ $ To see the analogy, represent subsets by bit-vectors, so the i'th element of the vector is $1$ if the i'th element is in the subset, and 0 if not. Then the rows of the matrix are the bit-vectors of the subsets, and said powerset diagonalization corresponds simply to complementing the diago... |
2,627,170 | <p>I have a homework question, the subject matter of which the class I'm in has covered terribly:</p>
<p>"Given a group $A$, which has an order of $4$ and has an element of order $4$, prove that $A$ is isomorphic to $\Bbb{Z}_{4}$. "</p>
<p>I feel like this question should be easy, so I am also wondering if there are... | amWhy | 9,003 | <p>The key idea is that if a group $A$ of order $4$ has an element $a$ of order 4, then we know that $a$ <em>generates</em> $A$, i.e., then $$A= \langle a \rangle = \{a^0, a^1, a^2, a^3\}.$$</p>
<p>In other words, $A$ is cyclic, such that $a^4 = a^0, a^5 = a^1,$ etc.</p>
<p>And all cyclic groups of order $4$ are iso... |
3,232,582 | <p>I was wondering if I could get some help understanding the following fact in an academic paper. The setup is as follows: </p>
<p>An open subset of <span class="math-container">$\Omega \subset \mathbb R^k$</span> is an <em>open convex cone</em> if it is closed with respect to taking linear combinations of its elemen... | Moishe Kohan | 84,907 | <p>Let <span class="math-container">$A\subset R^k$</span> be a compact convex subset with nonempty interior invariant under a closed subgroup <span class="math-container">$H< GL(k,R)$</span>. Since <span class="math-container">$A$</span> is <span class="math-container">$H$</span>-invariant then so is the symmetriza... |
186,621 | <p>Let $f$ be a monic univariate polynomial with real coefficients:</p>
<p>$$f_A(x) = x^n + a_{n-1}x^{n-1} + ... + a_{0}$$</p>
<p>The values of $A=(a_{n-1},...,a_0)$ are unknown, but are estimated as $B=(b_{n-1},...,b_0)$ with error $\epsilon$. Therefore, $b_i - \epsilon \leq a_i \leq b_i + \epsilon$. Furthermore, ... | Alex Degtyarev | 44,953 | <p>Polynomials with a multiple root form a Zariski closed set (vanishing of the discriminant, which is a certain polynomial in the coefficients); hence, this set is nowhere dense, whereas its complement is dense. In other words, in any neighborhood of any point, "most" polynomials are nonsingular, and it's very unlikel... |
1,607,670 | <p>I'm getting stuck at solving the following ODE:</p>
<p>$\ddot{f}(x)-\dfrac{\dot{f}(x)^2}{f(x)}+C\left(\dfrac{\dot{f}(x)}{f(x)}\right)^2=0$, where $C\neq 0$ constant, and let $f(0)=P$, $\dot{f}(0)=Q$.</p>
<p>Is it possible to find a closed-form solution to this problem? Thanks for help!</p>
| Robert Israel | 8,508 | <p>Maple finds the implicit solution
$$ {\it Ei} \left( 1,{\frac {C}{f \left( x \right) }} \right) -{\frac {Qx
}{P}{{\rm e}^{-{ {C}/{P}}}}}-{\it Ei} \left( 1,{\frac {C}{P}}
\right) =0
$$</p>
|
1,630,335 | <p>Show that the list $(1,2), (3,5)$ is a basis for $\mathbb R^2$.</p>
<p>In order to show it is a basis, I have show that the list is linearly independent and that the list spans $\mathbb R^2$. </p>
<p>So I understand how to show linear independence (you simply set up a system of equations). However, I am uncertain ... | egreg | 62,967 | <p>A linearly independent list with the same cardinality as a basis is a spanning set, so a basis. (Assuming finite dimensional spaces, of course.)</p>
<p>This is because every linearly independent list can be extended to a basis; so if yours wasn't a basis, you'd end up with a basis of $R^2$ with more than two elemen... |
3,393,345 | <p><a href="https://i.stack.imgur.com/CzEtw.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/CzEtw.png" alt="enter image description here"></a></p>
<p>I don’t understand why it is a symbol before every set. I know that we can intersect or reunite two sets, A and B, but these examples make me confused... | StackTD | 159,845 | <p>If you understand how <strong>intersection</strong> and <strong>union</strong> work for two sets, you can probably see how that would work for more than two sets, for example:</p>
<ul>
<li>the union of three sets <span class="math-container">$A \cup B \cup C$</span> (<em>all elements belonging to at least one of th... |
3,393,345 | <p><a href="https://i.stack.imgur.com/CzEtw.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/CzEtw.png" alt="enter image description here"></a></p>
<p>I don’t understand why it is a symbol before every set. I know that we can intersect or reunite two sets, A and B, but these examples make me confused... | Mark Kamsma | 661,457 | <p>It means you take an intersection or union of a lot of sets (in this case infinitely many) at once. Let's take the set <span class="math-container">$B$</span> in your picture as an example. It says:
<span class="math-container">$$
B = \bigcup_{n \in \mathbb{N}} \left[ -1 + \frac{1}{n}, 1 - \frac{1}{n} \right].
$$</s... |
440,466 | <p>This probably has a very quick answer but it's been bugging me for a while.</p>
<p>Take two copies of the Cartesian plane. On each plane cut out the square $S=(0,1)\times(0,1)$. Throw away the rest. Now make a cut in each square from the middle of the top edge $(1/2,1)$ to the center $(1/2,1/2)$. Identify the left ... | Ilya | 5,887 | <p>If you would like to talk about integrals, are you talking about the <em>measurable</em> space, perhaps? A <em>measure</em> space is a <em>measurable</em> space on which a measure is fixed. A <em>measurable</em> space can admit many measures. </p>
<p>Now, your initial space is $X' :=[0,1]^2\times\{0,1\}$ which come... |
440,466 | <p>This probably has a very quick answer but it's been bugging me for a while.</p>
<p>Take two copies of the Cartesian plane. On each plane cut out the square $S=(0,1)\times(0,1)$. Throw away the rest. Now make a cut in each square from the middle of the top edge $(1/2,1)$ to the center $(1/2,1/2)$. Identify the left ... | John C | 20,013 | <p>Let $M'$ be the space you define, then $M := M' \setminus \{0\}$ is a differentiable (actually it's $C^\omega$) manifold as it is an open subset of the riemann surface associated to $x\mapsto \sqrt[2]{x}$ (minus the branching point at $0$).</p>
<p>Let then $p$ be the projection $M\to \Bbb R^2\setminus\{0\}$, and l... |
2,384,939 | <p>There are two boxes, one white, one black. The white box contains 3 white balls and 2 black balls. The black box contains four white balls and six black balls. This particular example 60%/40% and 40%/60% distribution split but in other versions of this problem the distribution could be uneven. I'm hoping for a gener... | Arthur | 15,500 | <p>Intuitive and approximate answer: At any stage, you have $40\%$ probability of switching boxes. After $50$ stages, you will (probably) have swapped back and forth so many times that the probability distribution hardly remembers what you started at, so you're (very close to) equally probable to be standing in front o... |
738,940 | <p>Asked a question to evaluate the Gaussian Integral,</p>
<p>$$\dfrac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty x^2 \exp(-x^2/2) dx $$</p>
<p>using the the following approximation,</p>
<p>$J=\Bbb E[X^2] \sim J_N = 1/N \sum_1^NX_n^2 $</p>
<p>where $x_n \sim N(0, 1)$</p>
<p>Examin $J_N$ for different values of $N$ and... | JPi | 120,310 | <p>The object that you have there is essentially the definition of the variance of a standard normal, which equals one.</p>
|
4,243,754 | <p><span class="math-container">$f(x)=x^3-x+1$</span></p>
<p><span class="math-container">$f'(x)=3x^2-1$</span>, It will be zero at <span class="math-container">$x=1/3,-1/3$</span></p>
<p><span class="math-container">$f"(x)=6x,$</span> so <span class="math-container">$1/3$</span> is a local minima and <span class="math... | Doug M | 317,176 | <p><span class="math-container">$f(x) = x^3 - x + 1\\
f'(x) = 3x^2 - 1\\
f''(x) = 6x\\
\lim_\limits{x\to \infty} = \infty\\
\lim_\limits{x\to -\infty} = -\infty\\
f(0) = 1$</span></p>
<p><span class="math-container">$f'(x) = 0$</span> when <span class="math-container">$x = \pm \sqrt {\frac 13}$</span></p>
<p>The second... |
1,302,459 | <p>I think I understand the proof for square matrices, such that $(AB)^{-1}=B^{-1}A^{-1}$, but I'm not sure if I can just say the same for the bounded linear operators A and B.</p>
<p>Does the existence of $(AB)^{-1}:X\rightarrow X$ imply the existence of $B^{-1}$ and $A^{-1}$?</p>
| Amitai Yuval | 166,201 | <p>The claim is false in the general infinite dimensional setting. For example, let $X$ be the Banach space $\ell^p$, and define $A,B$ by$$A(x_1,x_2,\ldots)=(x_2,x_3,\ldots),\qquad B(x_1,x_2,\ldots)=(0,x_1,x_2,\ldots).$$Then $AB$ is the identity, but $A$ and $B$ are not invertible.</p>
|
4,332,748 | <p>This is slightly different question. First I need to mention that I am neither a mathematician nor a researcher. As an ordinary student the separation " with and without Axiom of choice " trouble me a lot. But I am really interested to know further.</p>
<ol>
<li><p>Every ideal in a ring contained in some ... | HallaSurvivor | 655,547 | <p>The question "why worry about the axiom of choice" (or similar) has been asked many times (see <a href="https://mathoverflow.net/q/22927/145247">here</a>, <a href="https://math.stackexchange.com/q/1253650/655547">here</a>, <a href="https://math.stackexchange.com/q/120925/655547">here</a>, or <a href="https... |
4,354,126 | <p>I know that, for a complex square matrix <span class="math-container">$A$</span>, there exists some non-null vector <span class="math-container">$v$</span> such that <span class="math-container">$Mv=0$</span> if and only if the determinant, <span class="math-container">$\det(M)=0$</span>. That is, <span class="math-... | Sourav Ghosh | 977,780 | <p><span class="math-container">$M\cdot v$</span> : linear combination of the columns of the matrix where the coefficient comes from corresponding components of the vectors.</p>
<p><span class="math-container">\begin{align}M\cdot v&=\pmatrix{col_1 &col_2&...&col_n\\}\cdot \pmatrix{v_1\\v_2\\...\\v_n}\\&... |
118,811 | <p>As everyone knows, the standard middle-thirds Cantor set is constructed by dividing the interval into three equal parts, removing the middle one, then applying the same procedure to the remaining two intervals, etc. </p>
<p>The resulting set has Hausdorff dimension $s=\log 2/\log 3$, in view of Hutchinson's formula... | R W | 8,588 | <p>Since you remove "more", it should be more or less clear that $\text{HD}\; C_\delta \le \text{HD}\; C$ (where $C$ is the standard Cantor set and $C_\delta$ is the "perturbation" you describe). For proving the opposite inequality it is enough to exhibit a measure on $C_\delta$ whose Hausdorff dimension is equal to $\... |
934,343 | <p>How can I show the matrix
$$A = \left(
\begin{array}{c c}
-2 & 0 \\
0 & -1 \\
\end{array}
\right)$$</p>
<p>can't be written as $A = exp(a)$?</p>
<p>I've tried to write A like</p>
<p>$$A = \left(
\begin{array}{c c}
2 & 0 \\
0 & 1 \\
\end{array}
\right)
\left(
\begin{array}{c c}
-1 & 0 ... | Geoff Robinson | 13,147 | <p>Note that if $A = \exp(B),$ then $B$ must commute with $A$, hence must also be diagonal.
Let $B = [b_{ij}]$. Then $\exp(b_{11}) =-2$ and $\exp(b_{22}) = -1.$ Therefore $b_{11}$ and $b_{22}$ are not real. </p>
|
550,030 | <p>Hi I would like to prove this statement.</p>
<p>Show that there is no one-to-one correspondence from
the set of positive integers to the power set of the set of
positive integers.</p>
<p>[Hint: Assume that there is such a one-
to-one correspondence. Represent a subset of the set of
positive integers as an infinite b... | user35603 | 84,245 | <p>Hint: Let $(b_1,b_2,...,b_i,...)$ denotes a new bit string. And let $(a_{i1},a_{i2},...,a_{ii},...)$ denotes the $i$th string that corresponds to the $i$th integer. Take $b_1 \neq a_{11}$, $b_2 \neq a_{22}$, ..., $b_i \neq a_{ii}$, and so on. It's called Cantor's diagonal argument.</p>
|
1,209,586 | <p>Find the smallest positive integer $x$ that solves the following simultaneously.</p>
<p>Note: I haven't been taught the Chinese Remainder Theorem, and have had trouble trying to apply it.</p>
<p>$$
\begin{cases}
2x \equiv 11 \pmod{15}\\
3x \equiv 6 \pmod{8}
\end{cases}
$$</p>
<p>I tried solving each congruence in... | Bernard | 202,857 | <p>You must start from Bézout's identity between the moduli $15$ and $8$: $\,2\cdot 8-1\cdot 15=1$. From the solutions to the individual congruences $\color{red} {13} \pmod{15}$ and $\color{red}2 \pmod{8}$, you deduce the solutions to the system:
$$2\cdot 8\cdot\color{red} {13}-1\cdot 15\cdot\color{red}2=178\equiv 58\... |
1,005,660 | <p>I have stuck with ODE problems. </p>
<p>\begin{align}
\frac{f''}{f'}=-2\frac{g'}{g}
\end{align}
Here i know the solution is $ g^2 f' =const$. but i don't know how to obtain those relation.
It seems simple. </p>
<p>Is my approach is correct?</p>
<p>\begin{align}
\int \frac{f''}{f'}dx= -2\int \frac{g'}{g} dx =cons... | GeorgeAlun | 189,803 | <p>In short,</p>
<p>The eigenvector (0,1) is a unit vector in the direction of increasing y, i.e a line x=c.</p>
<p>Your solutions tend asymptotically to the lines x=0 and 4x=3y as can be seen from your phase portrait.</p>
|
40,348 | <p>I discovered experimentally that a certain finite poset (sorry, I cannot give its definition here) seems to be in fact a (non-distributive, non-graded) lattice. The covering relations are reasonably simple, but it seems not so easy to find out whether one element is smaller than the other, let alone find the meet (... | David E Speyer | 297 | <p>I have often found the following lemma of Björner, Eidelman and Ziegler to be useful:</p>
<blockquote>
<p>Let <span class="math-container">$P$</span> be a bounded poset of finite rank such that, for any <span class="math-container">$x$</span> and <span class="math-container">$y$</span> in <span class="math-container... |
2,348,504 | <p>2x/5 - x/4
Can anyone find me the answer?
Question is Express this as asingle fraction </p>
| Josh | 460,396 | <p>For addition and subtraction, you must find a <strong><em>common denominator</em></strong> before doing anything else. Don't be distracted by the '$x$' term for the time being.</p>
<p>$$\frac{2x}{5} - \frac{x}{4} = (\frac{4}{4}*\frac{2x}{5}) - (\frac{5}{5}*\frac{x}{4}) = \frac{8x}{20} - \frac{5x}{20} = \frac{(8x - ... |
51,895 | <p>I generated a spline function called <code>f</code> using the <code>BSplineFunction</code> with domain <code>{t, 0, 1}</code>.</p>
<pre><code>f = BSplineFunction[{{0., 5.81152}, {-0.909122, 5.73997}, {-1.86805, 5.5031},
{-1.79586, 5.52709}, {-2.40811, 5.28912}, {-3.4109, 4.70527}, {-4.68131, 3.501},
{-4.931... | J. M.'s persistent exhaustion | 50 | <p>This is yet another problem that can be solved by explicitly representing the B-spline curve with <code>BSplineBasis[]</code>, as was done <a href="https://mathematica.stackexchange.com/a/25796">here</a>:</p>
<pre><code>pts = {{1, 1}, {2, 3}, {3, -1}, {4, 1}, {5, 0}};
(* piecewise linear interpolant *)
lin = Inter... |
154,726 | <p>The code is supposed to move the center of a circle of radius "d" incrementally <em>on</em> a function (the one in the first line), then solve for the two intercepts and repeat. I noticed that it gave two complex coordinates for every point as well (not really sure why, Id be glad if someone could explain), but I ig... | Thies Heidecke | 47 | <h1>Solution</h1>
<p>I tried to take your code and simplify it a bit to make things easier to understand and faster:</p>
<pre><code>y[x_] = (x Tan[theta]) - ((10*(x^2))/(2*(v^2)*((Cos[theta])^2)))
initialconditions = {x0 -> i, v -> 80, theta -> Pi/4, d -> Rationalize[0.8255], y0 -> y[x0]}
eqs = y1 == y... |
3,557,252 | <p>By " same base, same exponent " rule I mean : </p>
<p><span class="math-container">$$b^x = b^n \iff x=n$$</span></p>
<p>Can the equation:
<span class="math-container">$$ 2^x+4^x = 6$$</span>
be solved by saying
<span class="math-container">$$ 2^x+4^x = 6 = 2+4= 2^1 + 4^1$$</span>
and consequently, that
<span c... | Eric Towers | 123,905 | <p>A perhaps more worrisome example is that
<span class="math-container">$$ (-1)^a = (-1)^b $$</span>
has infinitely many solutions in integers. The equation holds if <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are both even and it holds if <span class="math-container">$a$</sp... |
5,751 | <p>I've just been asked for a good example of a situation in maths where using infinity can greatly shorten an argument. The person who wants the example wants it as part of a presentation to the general public. The best I could think of was Goodstein sequences: if you take a particular instance of Goodstein's theorem,... | Qiaochu Yuan | 290 | <p>The precise <a href="http://en.wikipedia.org/wiki/Partition_%28number_theory%29#Rademacher.27s_series">asymptotics of the partition function</a> $p(n)$ are, as far as I know, totally inaccessible by finitary tools. More generally the methods of complex analysis are a great way to obtain incredibly precise asymptoti... |
5,751 | <p>I've just been asked for a good example of a situation in maths where using infinity can greatly shorten an argument. The person who wants the example wants it as part of a presentation to the general public. The best I could think of was Goodstein sequences: if you take a particular instance of Goodstein's theorem,... | Jonathan Wise | 32 | <p>This is probably neither the sort of infinity nor the sort of simplification you're looking for, but Bézout's theorem is a challenge to prove, let alone state, without a line at infinity. I don't know if it would be suitable for a general audience either, but in its favor, it has a very nice statement using ... |
5,751 | <p>I've just been asked for a good example of a situation in maths where using infinity can greatly shorten an argument. The person who wants the example wants it as part of a presentation to the general public. The best I could think of was Goodstein sequences: if you take a particular instance of Goodstein's theorem,... | Jose Brox | 1,234 | <p>While browsing Wikipedia, I came across this statement:</p>
<p>"The introduction of projective techniques made many theorems in algebraic geometry simpler and sharper: For example, Bézout's theorem on the number of intersection points between two varieties can be stated in its sharpest form only in projective space... |
5,751 | <p>I've just been asked for a good example of a situation in maths where using infinity can greatly shorten an argument. The person who wants the example wants it as part of a presentation to the general public. The best I could think of was Goodstein sequences: if you take a particular instance of Goodstein's theorem,... | Igor Khavkine | 2,622 | <p>A down-to-earth example: the <a href="http://en.wikipedia.org/wiki/Stirling_formula">Stirling approximation</a> to the factorial.</p>
<p>At least the derivation given on the Wikipedia page compares the explicit sum representation of $\ln(n!)$ with a Riemann sum approximation of the integral $\int \ln(x) \mathrm{d}x... |
5,751 | <p>I've just been asked for a good example of a situation in maths where using infinity can greatly shorten an argument. The person who wants the example wants it as part of a presentation to the general public. The best I could think of was Goodstein sequences: if you take a particular instance of Goodstein's theorem,... | Terry Tao | 766 | <p>The Poincare conjecture might qualify as an example. One can phrase this conjecture in a purely combinatorial fashion, involving finite simplicial complexes, but the only known proof of this conjecture uses Ricci flow, and in particular the various monotonicity properties of this flow which are proven by integratio... |
5,751 | <p>I've just been asked for a good example of a situation in maths where using infinity can greatly shorten an argument. The person who wants the example wants it as part of a presentation to the general public. The best I could think of was Goodstein sequences: if you take a particular instance of Goodstein's theorem,... | Mohammad Golshani | 11,115 | <p>Consider the following question of Erdos and Hajnal:</p>
<p><strong>Question (Erdos-Hajnal)</strong> Is there a finite $K_4$-free graph which, when the edges colored by $2$ colors, always contains a monocolored triangle.</p>
<p>The question is answered by Shelah, and his argument uses infinity. I think this is the... |
3,762,322 | <p>I'm looking for the name of the following expression:</p>
<p><span class="math-container">$$
\frac{\sum_{i} x_i^2}{\sum_i x_i}
$$</span></p>
<p>This can be understood as a type of average in which each value is weighted by itself – in other words, a type of average in which near-zero values are ignored. Here are som... | Z Ahmed | 671,540 | <p>Weighted aM is given as <span class="math-container">$$\bar x=\frac{ \sum_{i=1}^n f_i x_i}{\sum_{i=1}^n f_i}$$</span>
In your case it a particular weighted AM, where the frequency of the variables <span class="math-container">$x_i$</span> is <span class="math-container">$x_i$</span> itself: <span class="math-contai... |
810,032 | <p>A consequence of the Lagrange theorem:</p>
<blockquote>
<p>Let $G$ a finite group and $H$ a subgroup of G. Then $|H| \mid |G|$.</p>
</blockquote>
<p>is that each subgroup $\neq <i_d>$ of $D_4$, which has $8$ elements , has either $2$ or $4$ elements..</p>
<p>But.... $1 \text{ divides also }8$..Isn't it p... | leonbloy | 312 | <p>I would make the substitution $y=1/x$ to get $\epsilon \,(y-\epsilon ) -2 \, y^2(y^2-1)=0$ .</p>
<p>Then write $y=y_0 + a \epsilon + b \epsilon^2 +O(\epsilon ^3)$ and replace for each of the limit roots ($y_0=1$, $y_0=-1$, $y_0=0$) to get the respective coefficients. For example, for the first root I get $a=1/4$</... |
810,032 | <p>A consequence of the Lagrange theorem:</p>
<blockquote>
<p>Let $G$ a finite group and $H$ a subgroup of G. Then $|H| \mid |G|$.</p>
</blockquote>
<p>is that each subgroup $\neq <i_d>$ of $D_4$, which has $8$ elements , has either $2$ or $4$ elements..</p>
<p>But.... $1 \text{ divides also }8$..Isn't it p... | etothepitimesi | 30,132 | <p>Your analysis is correct.</p>
<p>Let's look at the original equation
$${\epsilon ^2}{x^4} - \epsilon {x^3} - 2{x^2} + 2 = 0$$
and plot the Newton-Kruskal diagram:</p>
<p><img src="https://i.stack.imgur.com/FpO0k.png" alt="Newton-Kruskal diagram"></p>
<p>There are only two possible placements of straight lines pas... |
1,171,142 | <p>I know this question has been asked before, but I think I'm very close to a new solution and wanted to know if it is a viable approach.</p>
<p>Let $C(X)$ be the ring of continuous functions $X \rightarrow \mathbb{C}$ where $X$ is compact Hausdorff, and let $\mathfrak M$ be a maximal ideal. We want to show that $\m... | Olórin | 187,521 | <p>It's even easier. Take a strict ideal $I$ and show, as you intended to, that $I\subseteq I_x$ for some $x\in X$. For this, suppose the contrary : then for all $x\in X$ you can find an $f_x \in I$ such that $f_x \not\in I_x$ <em>i.e.</em> $f_x (x) \not= 0$. Since $f_x$ is continuous there is a open neighbour hood of ... |
73,454 | <p>Can anyone help me with a proof of the following claim (see for example the book Higher algebraic geometry of Olivier Debarre, proof of Proposition 1.43, page 31):</p>
<p>Let X be a complex manifold, and let W be a complex submanifold of X, with codimension $\geq 2$. Let $\pi :Y \rightarrow X$ be a bimeromorphic mo... | anonymous | 17,326 | <p>Thank you Roy Smith for your excellent answer. </p>
<p>So as I understand, in the smooth case, the universal property goes as follows:</p>
<p>Let $f:Y\rightarrow X$ be a surjective holomorphic map between complex manifold, let W be a submanifold of X so that its inverse image is a submanifold E of Y. Assume moreov... |
2,662,509 | <p>The differential equation is:</p>
<p>${dP\over dT}=kP\Bigl(1-{P\over M}\Bigr)-\alpha P$</p>
<p>where $k > 0, \space M > 0$ and $\alpha > 0$ are constants.</p>
<p>How would I start an approach to solving this differential equation?</p>
| Ross Millikan | 1,827 | <p>The problem with your answer is all the terms that go to the second round. The first, $0.52 \cdot 0.5^2,$ is supposed to be the chance that Bo wins the first round and the first game with Al, then Al wins the second game of the second round. Instead of $0.52$ it should be $0.36$, the chance Bo won the first round.... |
414,642 | <p><strong>Background: Evaluation of</strong> <span class="math-container">$\int_0^\infty \frac{\log(1+x^2)}{(1+x^2)^2}dx$</span></p>
<p>We can prove using the Beta-Function identity that</p>
<p><span class="math-container">$$\int_0^\infty \frac{1}{(1+x^2)^\lambda}dx=\sqrt{\pi}\frac{\Gamma \left(\lambda-\frac{1}{2} \ri... | Ron Gordon | 53,268 | <p>We can attack the other integral</p>
<p><span class="math-container">$$I = \int_0^{\infty} dx \frac{\log{(1+x^3)}}{(1+x^2)^2}$$</span></p>
<p>in a manner similar to what @O.L. outlined in his answer for the other case, but with a different contour. To wit, consider</p>
<p><span class="math-container">$$\oint_C dz \... |
1,225,277 | <p>I tried to prove that any subfield of $\mathbb{C}$ must contain every rational number by contradiction. </p>
<p>Proof:</p>
<p>Let $\mathbb{F}$ be any subfield of $\mathbb{C}$. Thus, $\mathbb{F}$ is itself a field under the usual operations of addition and multiplication of complex numbers. For $\mathb... | J126 | 2,838 | <p>I agree with the first part, if $-q,\dfrac{1}{q}\in\mathbb{F}$, then there is a contradiction. I don't see why if $-q\notin\mathbb{F}$ or $\dfrac{1}{q}\notin\mathbb{F}$, closure implies a contradiction.</p>
<p>You may wish to show that $\mathbb{Z}\subseteq\mathbb{F}$, then use closure to show that $\mathbb{Q}\subs... |
510,750 | <p>I have a molecule with one coordinate system ( denote as x,y,z ) where the origin is center of mass of the molecule. I have to define another coordinate system (p,q,r) for a local motion. (shown in the figure.) I know the unit vectors of p,q,r axises respective to x,y,z coordinates and origin of the p,q,r coordinat... | awwalker | 12,084 | <p>If you plan on talking about matrix diagonalization when you discuss eigenvalues, you could show how eigenvalues can be used to give closed forms for linear recurrences. For example, if $F_n$ denotes the $n$th Fibonacci, then
$$\left( \begin{matrix} F_{n+1} \\ F_n \end{matrix} \right)=\left(\begin{matrix}1 & 1 ... |
47,259 | <p>Comrades,</p>
<p>Let $S$ be a non-ruled, minimal (smooth projective complex algebraic) surface. Let $K$ be a canonical divisor of $S$ and $H$ a hyperplane section of $S$ (for your favorite embedding).</p>
<p>Suppose I know that $K^2 > 0$. Then supposedly $S$ being non-ruled forces $H.K > 0$. Is this corre... | Sándor Kovács | 10,076 | <p>This actually holds in general, that is, if $d=\dim X$ is arbitrary. At least in characteristic $0$. $X$ being minimal is defined by $K_X$ being nef, for the classical notion of minimal surfaces this is equivalent to being a non-birationally-ruled surface (strictly speaking $\mathbb P^2$ is not ruled, but you would ... |
1,133,066 | <p>Is it possible to create uniformly distributed real pseudo random numbers
<br>
$x_1,x_2$, and $y_1,y_2,y_3\in$ $[0,1]$, subject to the following constraints:</p>
<p>$$x_1^2+x_2^2=1$$</p>
<p>$$y_1^2+y_2^2+y_3^2=1$$</p>
<p>I tried to use sines and cosines but that does not work;
the conventional approach in creatin... | Litho | 197,288 | <p>As @drhab wrote, it's impossible for two variables. It <em>is</em> possible for three variables, though: an interesting fact is that if you choose a random point on a unit sphere in 3 dimensions (in such a way that it's uniformely distributed over the area), then its $x,y$ and $z$ coordinates are all uniformely dist... |
1,657,176 | <p>"Let $\ f: A\to B$ and $\ g:B\to C$ be functions. Prove that if $g\circ f$ is surjective and $g$ is injective, then $f$ is surjective."</p>
| Michael Hardy | 11,667 | <p>Very often the quickest way to <b>find</b> a proof is by contradiction, but after you've found it, you can rearrange it into a simpler and easier-to-understand proof that's not by contradiction.</p>
<p>So by contradiction:</p>
<p>Suppose $f$ is not surjective.</p>
<p>Then for some $b\in B$, there is no $a\in A$ f... |
2,201,391 | <p>$p$ is the probability of heads. Note if $p \le 0.5$, the answer is infinity, so assume $p > 0.5$. </p>
<p>What is the expected number of flips of the coin where we have 2 more heads than tails?</p>
<p>Note you would stop flipping the coin when you first encounter the situation where you have 2 more heads than ... | Chill2Macht | 327,486 | <p>This corresponds to calculating the expectation of the first-passage time of an asymmetric one-dimensional random walk. I believe that even the asymmetric one-dimensional random walk satisfies the (strong) Markov property, so therefore we can assume that the random walk "starts over" at every step. (See <a href="htt... |
267,507 | <p>Figure shows a rectangle $ABCD$ and an isosceles triangle $\triangle DEC$.</p>
<p>$AD=BC=z$;$AB=DC=y$;$DE=CE=x$</p>
<p>One solution is as follows.</p>
<p>We know that the pythagoras theorem holds for a right triangle $\triangle EBC$.</p>
<p>In the right triangle, $\triangle EBC$.</p>
<p>$z^2+(\frac{y}{2})^{2}=x... | Community | -1 | <p>So, let c be the hypotenuse, and a the common leg-length.</p>
<p>First, note that given a length for the hypotenuse, any two isosceles right triangles having that hypotenuse will be congruent.</p>
<p>Then, use that fact to put the special case of Bhaskara’s proof to work, by drawing a square with side-length c, and ... |
1,623,003 | <p>Say you take a dice and you roll it twice so that you have a pair (X,Y) where X represents the first roll, Y represents the second. </p>
<p>When you have a distribution like max(X,Y), what type of distribution is that? It's not uniform, obviously, because the outcomes of max(X,Y) are clearly more favorable towards ... | Ian | 83,396 | <p>Most likely, they want you to give the probability mass function (PMF) of $Z=\max \{ X,Y \}$. You can get that by writing down the values for each outcome $(x,y)$, then count up the number of $1$s, $2$s, etc. and divide by the number of outcomes. So $p(1)=1/36,p(2)=2/36$, and so forth.</p>
|
886 | <p>Okay so <a href="https://math.stackexchange.com/questions/835/optimal-strategy-for-deal-or-no-deal">this question</a> reminded me of one my brother asked me a while back about the hit day-time novelty-worn-off-now snoozathon <a href="http://en.wikipedia.org/wiki/Deal_or_no_deal" rel="noreferrer">Deal or no deal</a>.... | steve | 14,866 | <p>I don't think it matters how you get to the final 2 figures, the question is out of a pool of 20 boxes, or however many it is, what are the chances that I picked the 250k box. The odds are 1 in 20, and that doesn't change. If it's 1 in 20 that I hold the main prize, and it's certainly in 1 of the boxes, then the oth... |
1,634,265 | <p>I have some trouble in how to evaluate this integral:
$$
\int_{0}^{\pi}\theta\ln\left(\tan\left(\theta \over 2\right)\right)
\,\mathrm{d}\theta
$$
I think it maybe has another form
$$
\int_{0}^{\pi}\theta\ln\left(\tan\left(\theta \over 2\right)\right)
\,\mathrm{d}\theta
=
\sum_{n=1}^{\infty}{1 \over n^{2}}
\left[\ps... | user170231 | 170,231 | <p>As an alternative to kishlaya's answer, we have the Fourier series</p>
<p><span class="math-container">$$\ln(\cos(x)) = -\ln(2) - \sum_{k=1}^\infty (-1)^k\frac{\cos(2kx)}k \\ \ln(\sin(x)) = -\ln(2) - \sum_{k=1}^\infty \frac{\cos(2kx)}k$$</span></p>
<p>and hence</p>
<p><span class="math-container">$$\int_0^{\frac\pi2... |
989,043 | <p>I'm trying to prove that, for a finite set $A$, if the map $f: A\rightarrow A$ is a surjection, then it's an injection. I've looked at this post: <a href="https://math.stackexchange.com/questions/63072/surjectivity-implies-injectivity">Surjectivity implies injectivity</a> but the arguments there seem very round-abo... | Asaf Karagila | 622 | <p>The trick, when you do this by induction, is that you replace $f$ by a function $g$ which maps the $n+1$-th element to itself. Without loss of generality, we can assume $A=\{0,\ldots,n\}$. And $f\colon A\to A$ is surjective.</p>
<p>First we can assume that $f(n)\neq n$, for if it were then by simply taking $k$ to b... |
762,476 | <p>Where is the series $\displaystyle\sum_{n=1}^{\infty}\dfrac{\cos(nx)}{n}$ pointwise convergent?
I tried to apply the Dirichlet's test but I couldn't.</p>
| Claude Leibovici | 82,404 | <p><strong>Hint</strong></p>
<p>Following Pedro Tamaroff, may be, you could consider two summations $$I=\displaystyle\sum_{n=1}^{\infty}\dfrac{\cos(nx)}{n}$$ $$J=\displaystyle\sum_{n=1}^{\infty}\dfrac{\sin (nx)}{n}$$ Then $$I+iJ=\displaystyle\sum_{n=1}^{\infty}\dfrac{e^{inx}}{n}=\displaystyle\sum_{n=1}^{\infty}\dfrac{... |
1,036,377 | <p>According to <a href="http://en.wikipedia.org/wiki/Pr%C3%BCfer_group" rel="nofollow">wikipedia</a>, the Pontryagin dual of a Prüfer group is isomorphic to a group of p-adic integers.</p>
<p>Where can I find a proof for it on the internet?</p>
| Hanno | 81,567 | <p>The Pruefer group ${\mathbb Z}_{p^{\infty}}$ is the direct limit of the sequence $${\mathbb Z}/p{\mathbb Z}\hookrightarrow {\mathbb Z}/p^2{\mathbb Z}\hookrightarrow ...$$
Applying $(-)^{\wedge} = \text{Hom}_{\text{cnt}}(-,{\mathbb S}^1)$ shows that $\left({\mathbb Z}_{p^\infty}\right)^{\wedge}$ is the inverse limit ... |
103,728 | <p>The <a href="https://reference.wolfram.com/language/ref/Inactivate.html" rel="nofollow">documentation for <code>Inactivate</code></a> makes the function sound really powerful because it gives granular control over which kind of expressions specifically <strong>not</strong> to evaluate. </p>
<blockquote>
<p><code>... | jkuczm | 14,303 | <p>This statement from documentation might be misleading. It should be read in conjunction with first statement:</p>
<blockquote>
<p><code>Inactivate[expr]</code></p>
<p>replaces all instances of <code>f</code> with <code>Inactive[f]</code> for symbols <code>f</code> used as heads in <code>expr</code>.</p>
</bl... |
1,685,714 | <p>Let $f(z)=\sum_{n} a_n z^n$ has radius of convergence $R>0$ and $0<r<R$. Show:</p>
<p>$$\frac{1}{2\pi} \int_0^{2\pi} |f(re^{it})|^2 dt= \sum_{n}|a_n|^2r^{2n}$$</p>
<p>Use this equality to prove Liouvelle's Theorem. </p>
<p>Can anyone give me a hint to how to proceed with this?</p>
| Mann | 126,204 | <p>We have $f(z)=\sum_n c_nz^n$.</p>
<p>Substituting $z=re^{it}$ and evaluating, </p>
<p>$\left|f(re^{it})\right|^2=\sum_n\left|c_n\right|^2r^{2n}+\sum_n\sum_mg(n,m,r)\times e^{i(n-m)t}$</p>
<p>Integrating both sides from $t=0$ to $t=2\pi$ should do it.</p>
|
1,397,400 | <p>Are there any books/papers talking about inner product on vectors over finite fields? In particular, I'd like to learn things on $F_p^n$, or simply $F_2^n$. I read some proofs using the inner product on $F_2^n$ in some papers, but I can only understand the general ideas. However, since it's possible that $v\cdot v =... | Booldy | 261,261 | <p>$lim_{x \rightarrow 0} (\sqrt{ax+b}-2)=0 \Rightarrow b=4$ so we have </p>
<p>$lim_{x \rightarrow 0} \dfrac {(\sqrt{ax+4}-2)} x=lim_{x \rightarrow 0} \dfrac {(ax+4-4)} {x(\sqrt{ax+4}+2)}=lim_{x \rightarrow 0} \dfrac {a} {\sqrt{ax+4}+2}=\dfrac a 4 $ </p>
<p>so we have
$a=4$</p>
|
1,535,323 | <p>Let $u$ and $v$ be an $n \times 1$ and $m \times 1$ unit norm (L-2 norm) vectors, respectively.<br>
Let us define matrix $M$ (of dimension $n \times m$) as the kronecker product of $u$ and $v^*$ $$M= u \otimes v^*,$$ where $v^*$ denotes the conjugate transpose of $v$. </p>
<p>My question: can we claim that the re... | K. Miller | 264,375 | <p>The matrix $M$ is given by </p>
<p>$$
M = u\otimes v^* = uv^* = \left[
\begin{array}{c}
u_1v^*\\
\vdots\\
u_n v^*
\end{array}
\right]
$$</p>
<p>The goal is to find the maximum eigenvalue of $M^*M$. Since $\|u\|_2 = 1$, we have</p>
<p>$$
M^*M = \sum_{i=1}^n |u_i|^2 vv^* = vv^*
$$</p>
<p>Observing that $(vv^*)v = ... |
187,938 | <p>let $ABC$ be an acute triangle with all angles greater than $45^o$
Prove that
$$\frac{2}{1+\tan A}+\frac{2}{1+\tan B}+\frac{2}{1+\tan C} \le 3(\sqrt{3}-1)$$</p>
<p>I let $\tan A=a$, $\tan B=b$, $\tan C=c$ with $a+b+c=abc$ then the inequality equivalent to $\frac{2}{1+a}+\frac{2}{1+b}+\frac{2}{1+c} \le 3(\sqrt{3}-1... | N. S. | 9,176 | <p>What about Calculus:</p>
<p>$f(x)=\frac{1}{1+\tan(x)}$ has a positive second derivative on $(\frac{\pi}{4}, \frac{\pi}{2})$. This solves the problem.</p>
|
967,084 | <p>I am asked to find all equilibrium solutions to this system of differential equations:</p>
<p>$$\begin{cases} x ' = x^2 + y^2 - 1 \\ y'= x^2 - y^2 \end{cases} $$</p>
<p>and to determine if they are stable, asymptotically stable or unstable.</p>
<p>I do not know how to proceed, I tried switching to polar coordinat... | Monolite | 157,701 | <p>Continuing from where Amzoti left (by the way thanks).</p>
<p>Using the dot notation for derivatives.</p>
<p>The Jacobian of the system is: $$J(x, y) = \begin{bmatrix} \dfrac{\partial {\dot x}}{\partial x} & \dfrac{\partial {\dot x}}{\partial y} \\ \dfrac{\partial {\dot y}}{\partial x} & \dfrac{\partial {\... |
51,247 | <p>I have a matrix of 0's and 1's forming a number of disjoint paths:
<hr />
<img src="https://i.stack.imgur.com/pUlT7.jpg" alt="RandomPaths">
<hr />
I would like to find the lengths of the paths, and from that "spectrum,"
the longest length (in the above example: 27, starting at {1,14}).
(The shortest length po... | rm -rf | 5 | <p>As I expressed in my comment above, it is possible (and easy) to use the image processing functions for this. Taking <code>m</code> to be the matrix above the following steps illustrate the idea:</p>
<pre><code>img = Image@m;
ComponentMeasurements[img, "PerimeterCount"]
(* {1 -> 3, 2 -> 27, 3 -> 9, 4 ->... |
51,247 | <p>I have a matrix of 0's and 1's forming a number of disjoint paths:
<hr />
<img src="https://i.stack.imgur.com/pUlT7.jpg" alt="RandomPaths">
<hr />
I would like to find the lengths of the paths, and from that "spectrum,"
the longest length (in the above example: 27, starting at {1,14}).
(The shortest length po... | george2079 | 2,079 | <p>Here is a direct approach w/o the graphics/graph functions.</p>
<p>Algorithm:</p>
<ul>
<li>create an array with every '1' changed to a unique number (pad zeros around array to avoid dealing with edge issues )</li>
<li>loop over nonzero entries. If any of the four neighbors (on further thought you may only need to ... |
410,071 | <p>Let the function $f$ be increasing on the closed internal $[a,b]$. If $a \le f(a)$ and $f(b)\le b$, prove that:</p>
<blockquote>
<p>$\exists x_0\in [a,b]$, such that $f(x_0)=x_0$.</p>
</blockquote>
<p>Thanks for your help.</p>
<p><strong>Note that $f$ need not be continuous.</strong></p>
| Clement C. | 75,808 | <p><strong>Hint:</strong> consider $g:x\mapsto f(x) -x$.</p>
<p>PS: are you sure there is no assumption on $f$, e.g. continuity?
PPS: after the question has been edited — without continuity, see comments below.</p>
|
2,987,055 | <blockquote>
<p>Let <span class="math-container">$\ V $</span> be a linear space over <span class="math-container">$\ F $</span>, <span class="math-container">$\ \dim V = n $</span> and <span class="math-container">$\ T_1,T_2 : V \rightarrow F $</span> are linear transofrmations. <span class="math-container">$\ T_1 ,... | user1551 | 1,551 | <p>Let <span class="math-container">$A=\pmatrix{P&q\\ q^T&r}$</span> be the matrix representation of a quadratic form, where <span class="math-container">$P$</span> is symmetric and non-singular, <span class="math-container">$q$</span> is a vector and <span class="math-container">$r$</span> is a scalar. Then th... |
263,397 | <p>I'm having a trouble with this integral expression:</p>
<p>$$\int_0^{2\pi} \frac{d\theta}{A+B \cos\theta}$$</p>
<p>I've done this substitution: $t= \tan(\theta/2)$</p>
<p>and get: $\displaystyle \cos\theta= \frac{1-t^2}{1+t^2}$ and $\displaystyle d\theta=\frac{2}{1+t^2}dt$ where $\displaystyle \cos^2\theta/2=\fra... | Community | -1 | <p>To avoid confusion with limits, note that
$$\int_0^{\pi} \dfrac{dx}{a+b \cos(x)} = \int_{\pi}^{2\pi} \dfrac{dx}{a+b \cos(x)}$$
Hence, we have
$$I = \int_0^{2\pi} \dfrac{dx}{a+b \cos(x)} = 2\int_0^{\pi} \dfrac{dx}{a+b \cos(x)}$$
Now use your substitution $t = \tan(x/2)$ and note that $t$ goes from $0$ to $\infty$ as ... |
48,118 | <p>I'm reading an article by Ricardo Mañé, "The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces" (<a href="https://doi.org/10.1007/BF02585431" rel="nofollow noreferrer">https://doi.org/10.1007/BF02585431</a>). I'm having a technical problem. Sorry for my ignorance, but I would like a referenc... | Ian Morris | 1,840 | <p>I found it surprisingly difficult to find a reference for this when I was studying Mane's papers on multiplicative ergodic theorems. My hypothesis was that people working with the Grassmannian in other areas are happy with the fact that the Grassmannian is metrisable for abstract topological reasons, and don't actua... |
2,222,116 | <p>Regards.. i would like to ask about some high quality books about the history of mathematics. The ideal books should provide detailed and thorough history of mathematics, with good <strong>images</strong> and <strong>manuscripts</strong>.</p>
<p>Books available at libraries are preferred, information about the libr... | NSZ | 432,374 | <p>Be $N_k$ your succes count, i.e. the number of successes at $k$th trial.
You want to find $P(N_7=1,N_8=2,N_9=3)$ because you want exactely one success between trial $1$ and $7$.
$$P(N_7=1,N_8=2,N_9=3)=P(N_7=1)P(N_8=2|N_7=1)P(N_9=3|N_8=2,N_7=1)$$
but $P(N_8=2|N_7=1)$ is equal to $P(X_8=1)=p$ and $P(N_9=3|N_8=2,N_7=1)... |
3,254,239 | <p>Given <span class="math-container">$f(x) = | \sin( | x | ) | $</span>, I am told to found the local and global minimum and maximum of <span class="math-container">$f$</span> (If they exist). </p>
<p>Simply from sketching the graph of <span class="math-container">$f$</span> I get that the function maximizes periodi... | PierreCarre | 639,238 | <p>Your general reasoning is correct. The global minimum is <span class="math-container">$0$</span>, and it is attained in points of the form <span class="math-container">$x = k \pi$</span>. The global maximum is 1, and it is attained in points of the form <span class="math-container">$x= \frac{\pi}{2}+ k \pi$</span>. ... |
14,757 | <p>So I think the following image already shows the problem, similar to <a href="https://mathematica.stackexchange.com/questions/2629/avoiding-white-lines-inside-filled-area-in-regionplot-exported-as-pdf-or-ps">Avoiding white lines inside filled area in RegionPlot exported as PDF or PS</a> the pdf export of the chart c... | Thomas Auzinger | 4,710 | <h2>Solution</h2>
<p>The problem seems to be that the polygonal regions in the exported PDF file do not have edge strokes. It can be partially solved by reimporting the figure and correct this.</p>
<pre><code>(* FILENAME.pdf contains an exported BarChart3D figure *)
(* 't' gives the desired edge thickness *)
chart = ... |
77,527 | <p>I am interested in some geometrical aspects of spaces $L(E)$, of bounded operators on a given Banach space $E$. I am unable to estimate if my problem deserves to be asked at MO, but let me try.</p>
<p>Is there an infinite-dimensional Banach space (non-separable preferably) $E$ such that for some non-zero </p>
<p>$... | Philip Brooker | 848 | <p>In what follows I show that such an operator exists if $E$ can be written (isometrically) as the $\ell_\infty$-direct sum of two (nonzero) subspaces (I have not tried the Hilbert space case, but I started writing my answer before the edits were made to the question.)</p>
<p>Let $E = X\oplus_\infty Y$, where $X$ and... |
1,195,337 | <p>If i suppose that $I$ is an interval but it is not connected,i.e there existe two open sets on $I$: $U,V$ such that $I=U\cup V$ et $U\cap V=\emptyset$ </p>
<p>Let $x\in U$ and $y\in V$ (for example $x<y$) and let $B_x=\{z\in U, [x,z]\subset U\}$ </p>
<p>Then $B_x$ is bounded by y .</p>
<p>My question is why $... | 5xum | 112,884 | <p>Because if $z\geq y$, then $y\in [x,z]$, this means that $[x,z]$ cannot be a subset of $U$ </p>
<p>(because it contains at least one element, $y$, from $V$, and since $y\in V$, we know that $y\notin U$).</p>
<hr>
<p>For part two, you show that there exists such an $\epsilon$ that $(M-\epsilon, M+\epsilon)$ is a s... |
1,801,599 | <p>Hey guys for this parametric equations its giving me negative Question is:</p>
<blockquote>
<p>A dart is thrown from a point 5 feet above the ground with an inital velocity of 58 ft/sec and angle of elvation of 41∘. Assume the onnly force acting on the dart is gravity. What is the maximum height reached by the da... | Triatticus | 75,861 | <p>I will begin where you write your two equations of interest now including the initial height information in y:</p>
<p>\begin{array}{l}
x_t = 58 (\text{ ft/s})\cos{(41^{\circ})} t \\
y_t = -16 (\text{ ft}/\text{s}^2) t^2 + 58 (\text{ ft/s})\sin{(41^{\circ})} t
+5 \text{ ft}
\end{array}</p>
<p>To find the time to h... |
3,338,312 | <p>In order to compute the derivative of the exponential function in an "early transcendentals" approach (where exponentials are defined not via power series or differential equations, but instead arithmetically via repeated multiplication, extended to reals), you first need to prove some fact like convexity of the exp... | Community | -1 | <p>Suppose the function <span class="math-container">$b^x$</span>, where <span class="math-container">$b>1$</span>, is <strong>not</strong> strictly convex. Then there are <span class="math-container">$U=b^{x_1}$</span> and <span class="math-container">$V=b^{x_2}$</span>, with <span class="math-container">$x_1<x_... |
1,049,711 | <p>I'm having trouble finding a function to compare it to on [-1,0]...</p>
| 5xum | 112,884 | <p><strong>Hint:</strong>
$e^x$ is a bounded function which has no roots.</p>
|
7,090 | <p>Let $C$ be a category. I'd like to say that a property $P$ of objects of $C$ (or rather isomorphism
classes of objects) is a "Yoneda property" or a "maps-in property" if there is a property $P'$ of
contravariant functors $h:C\to\mathrm{Set}$ such that the functor $\mathrm{Hom}(-,X)$ has $P'$ if and
only if $X$ has $... | Charles Siegel | 622 | <p>Well, I'll take a stab at it. I think you're going to want properties $P$ of functors $F$ of the form</p>
<p>$F$ has property $P$ if and only if $\forall X$, $F(X)$ has property $P'$</p>
<p>as you'd suggested above. I think we can avoid $P'$ being a natural isomorphism with $h_X$ by making sure it's a property o... |
248,385 | <p>For example, dividing by 2, means we cut something in two.</p>
<p>But dividing by 0.5, can only be explained with multiplying something by 2.</p>
<p>So, is there a "physical" explanation of dividing by 0.5? Is it "I divide by an entity that internally multiplies' or something as so bizarre?</p>
| amWhy | 9,003 | <p>$$\frac{1}{0.5} = \frac{2(\not{.5})}{\not{.5}} = 2.$$</p>
<p>$$\frac{1}{\frac12} = \frac{2\cdot\not{\frac{1}{2}}}{\not{\frac12}} = 2.$$</p>
<hr>
<p>In words, "how many times does one-half fit into the whole?"</p>
<p>And more generally, "how many times does $\;\dfrac1n\;$ fit into $\;1\;$?"</p>
|
1,031,315 | <p>I don't know why $\sin x$ and $\cos x$ are lineary independent since if we take linear combination $a\cdot \sin x + b \cdot \cos x=0$ and for $a=\sqrt{3}$ and $b=1$ and $\displaystyle x=\frac{\pi}{6}$ we have that</p>
<p>$a\cdot \sin x + b \cdot \cos x=0$ and $a \neq 0 $ and $b \neq 0$ so it suggest they aren't ind... | paw88789 | 147,810 | <p>For two functions $f$, $g$ to be linearly independent, you need a nontrivial linear combination to be the zero-function. That is, you'd need $a,b\in\mathbb{R}$, not both $0$, so that $a\sin(x)+b\cos(x)=0$ for all $x$.</p>
<p>This can't happen, because checking the equation at $x=0$ and $x=\frac{\pi}{2}$ shows $a$ ... |
405,368 | <p>I'm doing some work in linear algebra and these came up and I realized I don't know how to solve them as they have quadratics in them. I'm sure I've done this before but if someone could give me a crash course on how to find the values of the variables, it would be most appreciated. For the following questions, let ... | Martin | 49,437 | <p>To answer the question in the title, there is the following general theorem:</p>
<blockquote>
<p>The natural map $I \colon L^\infty(X) \to (L^1(X))^\ast$ is</p>
<ol>
<li>an isometric injection if and only if $(X,\Sigma,\mu)$ is semifinite.</li>
<li>an isometric isomorphism if and only if $(X,\Sigma,\mu)$... |
1,088 | <p>OK, so <a href="http://chat.stackexchange.com">chat</a> is now available... but; it has been suggested that for Mathematics we should have TeX support.</p>
<p>The current TeX processing has some non-trivial client impact. Before I even attempt trying to hack this in, is this something that the community would want ... | robjohn | 13,854 | <p>There are four bookmarks which are located on <a href="http://www.math.ucla.edu/~robjohn/math/mathjax.html" rel="nofollow noreferrer">this installation page</a>:</p>
<ol>
<li><p><code>start ChatJax</code> installs <a href="http://docs.mathjax.org/en/latest/index.html" rel="nofollow noreferrer">MathJax</a> and start... |
3,125,081 | <p>We know that a line can be written as <span class="math-container">$ ⟨x,y,z⟩ = ⟨x_0,y_0,z_0⟩ + t⟨a,b,c⟩ $</span> where <span class="math-container">$ x_0, y_0, z_0 $</span> is a point on the line and <span class="math-container">$ ⟨a, b, c⟩$</span> is a direction vector of the line. In a similar way, a plane can be ... | Haris Gušić | 450,231 | <p>You can use the following approach.</p>
<p>Find a vector that is normal to your plane of interest, let's call it <span class="math-container">$\vec n$</span>.
You can find it as the cross product of any two vectors that are parallel to your plane. Keep in mind that these two vectors <strong>must not be parallel to ... |
3,291,503 | <p>Say I have a circle D of area/probability 1 (radius <span class="math-container">$\frac{1}{\sqrt{\pi}}$</span>) centered at (0,0). I also have a circle A of area <span class="math-container">$\frac{8 \pi}{27 \sqrt{3}} \approx 0.537422$</span> and two circles B and C, each of area <span class="math-container">$\frac{... | Paul B. Slater | 217,460 | <p>This is by no means a direct answer to the question posed. It is a "work-around"--introducing new details--of the main presentational difficulty, which seemed to be the depiction of the void intersection of the circles B and C within the circle A. The values ("bound-entangled probabilities") <span class="math-contai... |
315,051 | <p>Unlike the collection <span class="math-container">$L$</span> of Lebesgue measurable sets, the collection <span class="math-container">$J$</span> of Jordan measurable sets do not form a Sigma algebra. (A set is Jordan measurable if and only if its characteristic function is Riemann integrable, and the characteristi... | Dave L Renfro | 15,780 | <p>Regarding the collections of analytic sets and Jordan measurable sets, <strong>a simple cardinality argument</strong> shows there exist Jordan measurable sets that are not analytic. Any subset of the Cantor middle thirds set is Jordan measurable (indeed, any such subset has Jordan measure zero), and thus there are <... |
2,269 | <p>Manin <a href="http://www.ihes.fr/document?id=1704&id_attribute=48" rel="noreferrer" title="Manin talk">stressed</a> that every projective scheme should have a quantum-cohomology structure. I'd like to know more about that. And since the varieties considered in texts about monodromy resp. vanishing cycles which ... | Arend Bayer | 7,437 | <p>About monodromy and quantum cohomology: while everybody knows that Gromov-Witten invariants are invariant under deformations, the immediate consequence that they are invariant under the monodromy action for any smooth family of varieties is sometimes forgotten. In other words, they are invariant under the mapping cl... |
2,867,489 | <p>I want to rigorously verify that
$$\overline{\left(\frac{z_1}{z_2}\right)}={\left(\frac{\overline{z_1}}{\overline{z_2}}\right)}$$</p>
<p>Here's my attempt at thinking:</p>
<p>$$\overline{\left(\frac{z_1}{z_2}\right)}=\overline{\left({z_1}\cdot \frac{1}{z_2}\right)}={\left(\overline{z_1}\cdot \overline{\left(\frac{... | grentank | 325,199 | <p>I suppose, that you should just prove
$$
\overline{\left(\dfrac1{z_2}\right)} = \dfrac1{\overline{z_2}}
$$
Simply multiply both sides by $\overline{z_2}$ and get
$$
\overline{\left(\dfrac1{z_2}\right)}\cdot\overline{z_2} = \overline{\left(\dfrac1{z_2}\right)\cdot z_2} = \overline{1} = 1 = \dfrac1{\overline{z_2}}\cdo... |
1,213,594 | <p>Define the <em>differentiation operator</em> $D$ to be the transformation
$D : C_\infty(\mathbb R)\to C_\infty(\mathbb R)$ given by $D( f ) = f'$.</p>
<p>How can I show that every real number $\lambda$ is an eigenvalue of $D$ by finding a corresponding eigenfunction and what will be the dimension of its eigenspace ... | Muneeb | 225,020 | <p>$D(f)$ = $f^{'}$ = $\lambda$.${e^{\lambda.x}}$ = $\lambda. f$ hence $f(x)$ = ${e^{\lambda.x}}$ is in the eigenspace {${f | D(f)= \lambda.x}$}, the eigenspace = Span {${e^{\lambda.x}}$} and $dim$ ( such eigenspace) = 1</p>
|
18,753 | <p>Consider a compact manifold <em>M</em>. For a vector field <em>X</em> on <em>M</em>, let $\phi_X$ denote the diffeomorphism of <em>M</em> given by the time 1 flow of <em>X</em>.</p>
<p>If <em>X</em> and <em>Y</em> are two vector fields, is $\phi_X \circ \phi_Y$ necessarily of the form $\phi_Z$ for some vector field... | Martin M. W. | 1,227 | <p>To answer your first question, the composition of two time-1 flows won't necessarily be another time-1 flow.</p>
<p>One way to see this is to note that when a time-1 flow $\phi_X$ has a periodic point $P$ (period > 1), then $P$ can't be hyperbolic since it lies on a closed orbit of the flow for $X$. (The eigenvecto... |
2,675,882 | <p>A lattice $(L, \le)$ is a <a href="https://en.wikipedia.org/wiki/Modular_lattice#Introduction" rel="nofollow noreferrer">modular lattice @ wiki</a> if the following modular law holds:
$$\forall x \in L: a \le b \implies a \lor (x \land b) = (a \lor x) \land b.$$</p>
<p>The <a href="https://en.wikipedia.org/wiki/Mod... | ancient mathematician | 414,424 | <p>I think you are confused about two different things. (I sympathise, I shared the confusion for many years!)</p>
<p>Firstly, addition of natural numbers is not defined by the algorithm we use to calculate $(a+b)$ in base 10, or any other base. We can add two numbers, irrespective of how they are represented.</p>
<p... |
2,889,830 | <p>There is a given curve $ α(t) = (3 \cos t − \cos 3t, 3 \sin t − \sin 3t)$ and I have to determine if it's a closed one. I tried to find the answer and failed. What should I do? Thanks.</p>
| Sarvesh Ravichandran Iyer | 316,409 | <p>The issue is the following : $f$ continuous, means that if $x_n$ is a <em>convergent</em> sequence to $x$, then so is $f(x_n)$ , converging to $f(x)$.</p>
<p>Note that every convergent sequence is Cauchy. However, the converse is true(in $\mathbb R$), only in a closed set. Note that $(0,1)$ is not a closed set. The... |
3,197,274 | <p>Given the bottom left coordinates of a square and the side length of the square, how do you find out the top right coordinates of the square?</p>
<p>Example: <span class="math-container">$(-10,-2)$</span> is the bottom left coordinates of the square. And the side length of the square is <span class="math-container">... | Jamāl | 506,409 | <p>Add <span class="math-container">$7$</span> to each coordinate,
<span class="math-container">$(-10+7,-2+7)=(-3,5)$</span>.</p>
|
1,118,816 | <p>Let $\{a_n\}$ be a sequence with limit $\alpha$, and define $b_n=a_{n+1}$ where $n\in \mathbb{N}$. Show that $\{b_n\}\rightarrow \alpha$.</p>
<p>What I have:</p>
<p>Since $\{a_n\}\rightarrow \alpha$ we know that for all $\epsilon>0$ there exists $N\in \mathbb{N}$ such that $|a_n-\alpha|<\epsilon$ for all $n\... | Trajan | 119,537 | <p>Suppose that $a_n \rightarrow a$ as $n \to \infty$ we have that,</p>
<p>Let $\epsilon >0, \exists N_1 \in \mathbb{N}$ such that $\forall n \geq N_1, |a_n -a|<\epsilon/2$.</p>
<p>Also $\exists N_2 \in \mathbb{N}$ , such that for all $n,m \geq N_2$ $|a_n-a_m|=|a_n-a+a-a_m| \leq |a_n-a|+|a_m-a|<\epsilon/4+\e... |
181,167 | <p>I would like to find a good polynomial fit of the square root.
I tried the following:</p>
<pre><code>m = Fit[Table[Sqrt[x], {x, 0, Pi}], {x, x^2}, x]
Plot[{Sqrt[x], m}, {x, 0, Pi}]
</code></pre>
<p>But the results are not convincing:</p>
<p><a href="https://i.stack.imgur.com/UwF72.jpg" rel="nofollow noreferrer"><... | Anton Antonov | 34,008 | <p>You can use a Quantile Regression fit of Chebyshev polynomials through the package "MonadicQuantileRegression.m". See <a href="https://mathematicaforprediction.wordpress.com/2018/08/01/a-monad-for-quantile-regression-workflows/" rel="nofollow noreferrer">this blog post</a> for details.</p>
<pre><code>sqPoints = Tab... |
898,310 | <p>$$\Large\frac{\left(\frac{\tan \frac \pi 4+\tan x}{1-\tan\frac \pi 4\tan x}\right)}{\left(\frac{\tan \frac \pi 4-\tan x}{1+\tan\frac \pi 4\tan x}\right)}=\frac{\left(\frac{1+\tan x}{1-\tan x}\right)}{\left(\frac{1-\tan x}{1+\tan x}\right)}$$</p>
<p>I am having a little trouble figuring out how we got from the first... | MathGems | 168,836 | <p>Hint: $$\tan\frac\pi4=1\hspace{5mm} ;-){}{}{}{}$$</p>
|
1,350,245 | <p>How can I show this? I tried with the definiton of the exponential function but it didn't work. </p>
<p>$$\displaystyle \lim\limits_{n\to\infty} \frac{(4n^7+3^n)^n}{3^{n^2}+(-9)^n\log(n)}=1$$</p>
<p>I hope somebody could help. </p>
| Jamal Farokhi | 244,152 | <p>For term $(4n^{7}+3^{n})^{n}$ since $n\longrightarrow \infty$ is equal to $3^{n^{2}}$. The denominator is the same and result is routine.</p>
|
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