qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
2,196,463 | <blockquote>
<p>Prove that if <span class="math-container">$C \subset B$</span> where <span class="math-container">$B$</span> is a bounded subset of a metric space <span class="math-container">$(X, d)$</span>, then <span class="math-container">$C$</span> is bounded and <span class="math-container">$\operatorname{diam} ... | Henno Brandsma | 4,280 | <p>It can be simplified: the diameter proof need not be done by contradiction. </p>
<p>$B$ is bounded so $B \subseteq B(x, R)$ for some $X \in X, R>0$. (no need for $(X,d)$ in the subscript, as this is clear by context).
Then also $C \subset B$ so the same $x$ and $R$ work to show $C$ is bounded.</p>
<p>Let $c_1, ... |
2,196,463 | <blockquote>
<p>Prove that if <span class="math-container">$C \subset B$</span> where <span class="math-container">$B$</span> is a bounded subset of a metric space <span class="math-container">$(X, d)$</span>, then <span class="math-container">$C$</span> is bounded and <span class="math-container">$\operatorname{diam} ... | Michelle Osorio | 605,830 | <p>You can try to show that </p>
<ol>
<li>if <span class="math-container">$A\subset B$</span> so <span class="math-container">$\sup A\leq \sup B$</span></li>
<li>Show that <span class="math-container">$\{d(x,y)|x,y\in A\}\subset \{d(x',y')|x',y'\in B\}$</span></li>
<li>Let <span class="math-container">$\operatorname{d... |
898,543 | <p>I have the random vector $(X,Y)$ with density function $8x^{2}y$ for $0 < x < 1$, $0 < y < \sqrt{x}$ I am trying to find the marginal distributions of $X$ and $Y$. For $X$ this seems to be simply the integral $\int_{0}^{\sqrt{x}}8x^{2}y = 4x^{3}$, which is also the given solution, and follows the general... | drhab | 75,923 | <p><strong>Hint:</strong></p>
<p>$P\left[L\mid B\right]=\frac{1}{7}$ i.e. $P\left[L\cap B\right]=\frac{1}{7}P\left[B\right]$</p>
<p>$P\left[B\mid L\right]=\frac{1}{3}$ i.e. $P\left[L\cap B\right]=\frac{1}{3}P\left[L\right]$</p>
<p>$1-P\left[L\cup B\right]=\frac{4}{5}$ </p>
<p>These equations are enough to find $P\l... |
72,537 | <blockquote>
<p>Let $A\in M_{n}$ have Jordan canonical form $J_{n_1}(\lambda_{1})\oplus\cdots\oplus J_{n_k}(\lambda_{k})$. If $A$ is non-singular ($\lambda_i\neq 0$), what is the Jordan canonical form of $A^{2}$?</p>
</blockquote>
<p>I can prove that if the eigenvalues of $A$ are $\sigma(A)=\{\lambda_{1},\dots, \lam... | Mariano Suárez-Álvarez | 274 | <p>Everything works blockwise, so you can simply assume that $A$ is one Jordan block...</p>
<p>So let $A=J_n(\lambda)$, which we can write as $\lambda I+N$ with $N=J_n(0)$. Then $A^2=\lambda^2I+2\lambda N+N^2$. The matrix $N'=2\lambda N+N^2$ is nilpotent and (because $\lambda\neq0$) has rank $n-1$, so it is conjug... |
82,254 | <p>Consider the standard form polyhedron, and assume that the rows of the matrix A are linearly independent.</p>
<p>$$ \left \{ x | Ax = b, x \geq 0 \right \} $$</p>
<p>(a) Suppose that two different bases lead to the same basic solution. Show that the basic solution is degenerate (has less than m non-zero entries).<... | Apurv | 240,799 | <p>I have a slightly different proof for part (a).
If the bases, $B$ and $B'$ are distinct, but correspond to the same basic feasible solution $x_b$ ($x_b$ corresponds to the vector of basic variables), then, by definition $Bx_b=b$ and $B'x_b=b$. Hence, $(B-B')x_b=0$. Since $B,B'$ are distinct, $dim(B-B') \geq 1$. Ther... |
4,357,484 | <p>Suppose the following series:
<span class="math-container">\begin{eqnarray}
\sum_{k'}k'f_{k'}
\end{eqnarray}</span>
where <span class="math-container">$f_{k'}$</span> are some Fourier coefficients that result from a periodic function <span class="math-container">$f(t+T)=f(t)$</span>:
<span class="math-container">\be... | tomasz | 30,222 | <p>It's not correct. <span class="math-container">$A$</span> is only isomorphic to <span class="math-container">$\bigoplus_{i=1}^r(K)_i$</span> <em>as a <span class="math-container">$K$</span>-vector space</em>, not as a ring. It is also easy to find counterexamples: take <span class="math-container">$A=\mathbf Q[\sqrt... |
3,394,378 | <p>I am stuck with this Precalculus problem about polynomial functions. The problem:</p>
<blockquote>
<p>Consider <span class="math-container">$f(x)=x^2+ax+b$</span> with <span class="math-container">$a^2-4b>0$</span>. Let <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span> b... | Steven Alexis Gregory | 75,410 | <p>If the roots are <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span>, then the minimum occurs at
<span class="math-container">$x_{min} = \dfrac{\alpha + \beta}{2}$</span>.</p>
<p><span class="math-container">$$\begin{align}
f(x_{min})
&= (x_{min} - \alpha)(x_{m... |
1,482,776 | <blockquote>
<p>Let $(X_t)$ be a continuous nonnegative supermartingale and $T = \inf\{t\geq 0 \colon X_t = 0 \}$ then $X_t = 0$ for every $t\geq T$.</p>
</blockquote>
<p>Idea of solution:</p>
<p>Since $T$ is stopping time, by Doob theorem:
$$E(X_{T+q} 1_{T < \infty} | F_T) \leq X_T 1_{T < \infty} =0 $$
for e... | Math-fun | 195,344 | <p>We have $cos x= \sum_{j=0}^{\infty}\frac{(-1)^j}{(2j)!}x^{2j}$ hence \begin{align}
\frac{x^n}{\cos \sin x -\cos x}&= \frac{x^n}{\sum_{j=0}^{\infty}\frac{(-1)^j}{(2j)!}(\sin^{2j}x-x^{2j})}\\
&= \frac{x^n}{\sum_{j=1}^{\infty}\frac{(-1)^j}{(2j)!}(\sin^{2j}x-x^{2j})}\\
&=\frac{1}{\color{blue}{\frac{(-1)^1}{2... |
3,154,316 | <p>With regard to this curve:
<span class="math-container">$$3xy=x^3+y^3$$</span>
I understand that <span class="math-container">$\frac{dy}{dx}$</span> is not defined at <span class="math-container">$(0,0)$</span>, but, there must be some more information right as there are <span class="math-container">$2$</span> tange... | Community | -1 | <p>If I understand your question correctly, you understand that the Folium of Descarte has certain 'tangent-like' directions at the origin <span class="math-container">$(0,0)$</span>. However (since there are two directions), we cannot identify them by computing the derivative <span class="math-container">$\frac{dy}{d... |
3,154,316 | <p>With regard to this curve:
<span class="math-container">$$3xy=x^3+y^3$$</span>
I understand that <span class="math-container">$\frac{dy}{dx}$</span> is not defined at <span class="math-container">$(0,0)$</span>, but, there must be some more information right as there are <span class="math-container">$2$</span> tange... | Ted Shifrin | 71,348 | <p>Here's an alternative approach, which usually shows up in differential and algebraic geometry. It's called "blowing up" the origin. </p>
<p>Introduce the <em>slope</em> coordinate <span class="math-container">$m$</span> by <span class="math-container">$y=mx$</span> and rewrite the equation. You have
<span class="m... |
1,003,379 | <p>I've been working problems all day so maybe I'm just confusing myself but in order to do this. I have to the take the integral along each contour $C_1-C_4$. My issue is how to convert to parametric functions in order to this so that I can integrate</p>
<p><img src="https://i.stack.imgur.com/HWRoM.jpg" alt="enter im... | dustin | 78,317 | <p>Parametric equations for the square going counter clockwise:
\begin{alignat}{2}
\gamma_1 &= 2 + 2i(2t-1)&&{}\quad 0\leq t\leq 1\\
\gamma_2 &= 2i + 2(3-2t)&&{}\quad 1\leq t\leq 2\\
\gamma_3 &= -2 + 2i(5-2t)&&{}\quad 2\leq t\leq 3\\
\gamma_4 &= -2i + 2(2t - 7)&&{}\quad 3... |
3,581,724 | <p>I suspect a simple wooden toy "lead screw" was made by advancing a cylindrical rotary cutting tool ( <em>Cylindrical End Mill Cutter</em>) along the surface of the rotating wooden dowel (base cylinder), resulting in a helical cut (the axes of the cylinders are orthogonal (<em>skew</em>).</p>
<p><a href="https://i.s... | Narasimham | 95,860 | <p>Trying to understand the motions. At first I imagined that you were referring to a simple twisted tube of <span class="math-container">$(x,y,z)$</span> parametrization:</p>
<p><a href="https://i.stack.imgur.com/Zwx4Y.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Zwx4Y.png" alt="Tube here"></a><... |
690,621 | <p>Consider the Quotient ring $\mathbb{Z}[x]/(x^2+3,5)$. </p>
<p>Solution: I first tried to take care of $(5)$ in the above ring. Therefor we can consider $\mathbb{Z_5}[x]/(x^2+3)$. Now and interesting point to note here is $(5) \subset (x^2+3)$. So, we can consider $\mathbb{Z_5}[x]/(5)$. But this is just $\mathbb{Z... | janmarqz | 74,166 | <p>The equivalent classes are pieces (chunks) of the set where is the equivalence relation, these pieces include the elements which are interrelated. The set of all these pieces is an example of a partition of the set where the equivalence relation is defined. </p>
<p>For example:</p>
<p>In $\Bbb{Z}$ we define $a\si... |
2,381,406 | <p>Somewhere I saw that </p>
<blockquote>
<p>To show that $x^2-y^3$ is irreducible in $k[x,y]$ it suffices to show that $x^2-y^3$ is irreducible in $k(y)[x]$.</p>
</blockquote>
<p>My question is what is the relation between $k[x,y]$ and $k(y)[x]$ ?</p>
<p>Also there is a confusion that if $k(y)$ is the smallest fi... | Henno Brandsma | 4,280 | <p>No, you cannot make a sequence with all the neighbourhoods of a point $x$, e.g. there are as many neighbourhoods of $0$ in the real line as there are real numbers, for every $r > 0$, we have $O_r = (-r,r)$ which is a neighbourhood of $0$. And Cantor's diagonal argument shows that we cannot put the real numbers in... |
3,816,041 | <blockquote>
<p>How many ways <span class="math-container">$5$</span> identical green balls and <span class="math-container">$6$</span> identical red balls can be arranged into <span class="math-container">$3$</span> distinct boxes such that no box is empty?</p>
</blockquote>
<p>My attempt :</p>
<p>Finding coefficient... | Qiaochu Yuan | 232 | <p>Recall that if <span class="math-container">$f(x) \in \mathbb{F}_p[x]$</span> is any polynomial, the Frobenius map <span class="math-container">$F : x \mapsto x^p$</span> generates the Galois group of its splitting field, and hence to compute its Galois group it suffices to compute the cycle structure of Frobenius a... |
1,074,177 | <p>Suppose a problem
$$\min_{x \in \mathbb{R}^{n}} f(x)$$</p>
<p>subject to $x \in \Omega$ which is a closed and convex set. If $\nabla f(x)$ is Lipschitz continuous in $\Omega$, then prove that</p>
<p>$$e(x) = x - P_{\Omega}(x- \nabla f(x))$$</p>
<p>is also Lipschitz continuous in $\Omega$.</p>
<p>Thanks in advanc... | megas | 191,170 | <p>The key is that projection onto a convex set is non-expansive, that is, for any two points $x, y$,
$$
\| P_{\Omega}(x) - P_{\Omega}(y)\| \le \|x-y\|.
$$
Now, we assume that $\nabla f(y)$ is Lipschitz continuous on $\Omega$, <em>i.e.</em>, there exists some constant $L$ such that
$$
\| \nabla f(x) - \nabla f(y)\| \... |
291,729 | <p>How to show that $\large 3^{3^{3^3}}$ is larger than a googol ($\large 10^{100}$) but smaller than googoplex ($\large 10^{10^{100}}$).</p>
<p>Thanks much in advance!!!</p>
| user1551 | 1,551 | <p>\begin{align}
&\color{red}{100 \log_3 10} < 100\times3 < 729 = 3^6 < \color{red}{3^{3^3}} = 3^{27} < 10^{100} < \color{red}{10^{100} \log_3 10}\\
\Rightarrow&10^{100} < 3^{3^{3^3}} < 10^{10^{100}}.
\end{align}</p>
|
291,729 | <p>How to show that $\large 3^{3^{3^3}}$ is larger than a googol ($\large 10^{100}$) but smaller than googoplex ($\large 10^{10^{100}}$).</p>
<p>Thanks much in advance!!!</p>
| Dave L. Renfro | 13,130 | <p>I know others have beat me by nearly a day, but here's something I came up with just now that seems more straightforward. Each of the inequalities makes frequent use of monotonicity (increasing), either in the base or in the exponent, of an exponentiated expression. (After writing this up, I noticed that my estimate... |
206,227 | <p>I was given the following problem:</p>
<p>Let $V_1, V_2, \dots$ be an infinite sequence of Boolean variables. For each natural number $n$, define a proposition $F_n$ according to the following rules: </p>
<p>$$\begin{align*}
F_0 &= \text{False}\\
F_n &= (F_{n-1} \ne V_n)\;.
\end{align*}$$</p>
<p>Use induc... | hmakholm left over Monica | 14,366 | <p>A quicker way to see intuitively that this works is to notice that if we represent "False" by the number $0$ and "True" by the number $1$, then $\neq$ corresponds exactly to addition modulo $2$.</p>
<p>Therefore $F_n$ is represented by the sum of $V_1$ upto $V_n$ modulo 2, which is $1$ exactly if an <em>odd</em> nu... |
34,959 | <p>$F(x) = \int_{x-1}^{x+1}f(t)dt$ for x an element of the reals.</p>
<p>Show that $F$ is differentiable on Reals, and compute $F^\prime$.</p>
<p>I am unsure about how to showing $F$ is differentiable. I know that I need to use the fundamental theorem of calculus, but can someone please explain how to do so?</p>
| Martin Sleziak | 8,297 | <p>You can simply use definition of the derivative.</p>
<p>You have
$$F(x)=\int_{x-1}^{x+1} f(t) dt.$$</p>
<p>$$F(x+h)-F(x)=\int_{x+h-1}^{x+h+1} f(t) dt-\int_{x-1}^{x+1} f(t) dt=
\int_{x+1}^{x+h+1} f(t) dt - \int_{x-1}^{x+h-1} f(t) dt$$</p>
<p>$$\frac{F(x+h)-F(x)}{h}=\frac{\int_{x+1}^{x+h+1} f(t) dt}{h} - \frac{\in... |
3,805,745 | <p>I am working my way through a linear algebra book and would appreciate some help verifying my proof.</p>
<p><strong>Prove that <span class="math-container">$|u \cdot v| = |u | |v |$</span> if and only if one vector is a scalar multiple of the
other.</strong></p>
<p><strong>PROOF:</strong></p>
<p>Let <span class="mat... | J.G. | 56,861 | <p>Let's make the proof of CS explicit to show @CSquared's answer doesn't require circularity. In fact, it's simpler to run through the proof of CS rather than invoking it, as we don't need to check two directions separately.</p>
<p>Write <span class="math-container">$f(k):=u-kv$</span> so<span class="math-container">$... |
637,061 | <p>I have a problem:</p>
<blockquote>
<p>For a system of linear equations:
$$x_i=\sum_{j=1}^{n}a_{ij}x_j+b_i,\ \ i=1,2, \ldots , n \tag 1$$
Prove that, if<br>
$$\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}^2 \le q<1$$
then $(1)$ has a unique solution.</p>
</blockquote>
<p>==================================</p>
<p... | Nox | 121,022 | <p>The inequality you have derived holds in a given norm.<br>
Now you do not have to show explicitly, that $|| A|| \leq \sum_{i,j} a_{ij}^2$ holds. The double sum which you are given is a specific norm(Hilbert-Schmid or Frobenius) and what you essentially have yet to show to be done with the task is that it is consist... |
209,420 | <p>For instance, when trying to compute <span class="math-container">$\mathbb{E}[\sum_{i=1}^{10}X_i]$</span> where <span class="math-container">$X_i \sim N(0,1)$</span>, I input into Mathematica:</p>
<pre><code>Expectation[Sum[x[i],{i, 1, 10}], x[i] \[Distributed] NormalDistribution[]]
</code></pre>
<p>but, instead o... | Bob Hanlon | 9,362 | <pre><code>Clear["Global`*"]
</code></pre>
<p>Assuming that the variates are i.i.d., the distribution of the sum is normal</p>
<pre><code>distSum[μ_, σ_, n_Integer?Positive] := Assuming[σ > 0,
TransformedDistribution[
Total[Array[x, n]],
Array[x[#] \[Distributed] NormalDistribution[μ, σ] &, n]] //
... |
3,873,138 | <p>Since we have variable coefficients we will use the cauchy-euler method to solve this DE. First we substitute <span class="math-container">$y=x^m$</span> into our given DE. This then gives
"</p>
<p><span class="math-container">$9x(m(m-1)x^{m-2}) + 9mx^{m-1} = 0$</span></p>
<p>Note that:</p>
<p><span class="math... | user577215664 | 475,762 | <p>Since you have that <span class="math-container">$m^2=0 \implies m=0$</span> is a double root you have to multiply by <span class="math-container">$\ln |x|$</span> in order to find the second solution:
<span class="math-container">$$y_1=x^0=1$$</span>
<span class="math-container">$$\implies y_2= 1 \times \ln |x|$$</... |
2,871,655 | <p>I was trying some Cambridge past papers and it said to first separate into partial fractions and then find the sum of the sequence, however after splitting inot partial fractions I'm not getting the terms to cancel out like I normally do with these questions. Is there something I'm missing
.been trying to manipulate... | Robert Israel | 8,508 | <p>You can write this as
$$ -\frac{i}2 \left( {\it polylog} \left( s,i \right) -{\it polylog} \left( s,-
i \right) \right)
$$</p>
<p>EDIT: It does seem that for odd $s$, $f(s)$ is a rational multiple of $\pi^s$.
See <a href="https://oeis.org/A053005" rel="nofollow noreferrer">OEIS sequence A053005</a> and <a href=... |
646,779 | <p>Prove that if $p$ and $q$ are polynomials over the field $F$, then the degree of their sum is less than or equal to whichever polynomial's degree is larger</p>
<p>$$\deg(p+q)\leq \max \left\{\deg(p),\deg(q) \right\}$$</p>
<p>Currently, I am taking it case by case, but I was curious if there was a way to do a proof... | Denis | 66,241 | <p>Hint: write polynomials in their general forms, and look at what happens when you sum them. It is impossible to create a nonzero coefficient where the coefficient was zero before, so in particular you cannot augment the maximal degree.</p>
|
304,259 | <p>I am stuck on this problem and I'm not sure how to approach it. Can anyone help me out with figuring how to approach the proof?</p>
<p>My task is to:</p>
<blockquote>
<p>Prove that it is impossible to find integers $\,x,\, y\,$ such that $\;2^x = 4y + 3$. </p>
</blockquote>
<p>I assumed a proof by cases would b... | amWhy | 9,003 | <p><strong>Proof-By-Cases - Sketch:</strong> </p>
<p>We consider $x \in \mathbb{Z}$. For all $x \in \mathbb{Z}$:</p>
<ol>
<li>$x > 0$</li>
<li>$x = 0,\;$ or</li>
<li>$x < 0$</li>
</ol>
<p>$(1)$ For non-negative integer $x (x >0)$: Show the left hand side will always be even, except when $x = 0$, and the rig... |
3,068,534 | <p>Let <span class="math-container">$R$</span> be the ring of algebraic integers of a quadratic imaginary number field <span class="math-container">$\mathbb Q[\sqrt{d}]$</span> for a negative square-free integer <span class="math-container">$d$</span>. For a prime integer <span class="math-container">$p$</span>, <span ... | nowhere dense | 124,875 | <p>I will focus on the case you still don't solve.</p>
<p>Suppose that <span class="math-container">$I$</span> is a proper ideal such that <span class="math-container">$P\overline P\subsetneq I$</span> (notice that the inclusion should be strict, otherwise <span class="math-container">$P\overline P=I$</span> is a coun... |
3,068,534 | <p>Let <span class="math-container">$R$</span> be the ring of algebraic integers of a quadratic imaginary number field <span class="math-container">$\mathbb Q[\sqrt{d}]$</span> for a negative square-free integer <span class="math-container">$d$</span>. For a prime integer <span class="math-container">$p$</span>, <span ... | Wojowu | 127,263 | <p>Here is a straightforward proof. Since we are in a quadratic field, it's not hard to see that <span class="math-container">$R/(p)$</span> has <span class="math-container">$p^2$</span> elements (since, as a group, <span class="math-container">$R$</span> is free abelian on two generators). If <span class="math-contain... |
3,209,722 | <p>I saw in another post on the website a simple proof that <span class="math-container">$$\lim_{n\to\infty} \left( 1+\frac{x}{n} \right)^n = \lim_{m\to\infty} \left( 1+\frac{1}{m} \right)^{mx}$$</span></p>
<p>which consists of substituting <span class="math-container">$n$</span> by <span class="math-container">$mx$</... | Community | -1 | <p><strong>Hint:</strong></p>
<p><span class="math-container">$$ \lim_{n\to\infty} \left( 1+\frac{x}{n} \right)^n \lim_{n\to\infty} \left( 1-\frac{x}{n} \right)^n = \lim_{n\to\infty} \left( 1-\frac{x^2}{n^2} \right)^n = \lim_{n\to\infty} \left( 1-\frac{x^2}{n^2} \right)^{n^2/n }=1.$$</span></p>
|
145,612 | <p>Why are isosceles triangles called that — or called anything? Why is their class given a name? Why did they find their way into the <em>Elements</em> and every single elementary geometry text and course ever since? Did no one ever ask himself, "What use is this, or why is it interesting?"?</p>
<p>Here are som... | J. David Taylor | 30,850 | <p>I believe that one of the reasons why isosceles triangles are discussed in the elements is because Euclid's construction of the regular pentagon hinges on the construction of an isosceles triangle with a nice (will edit with more specifics later) relationship between the length of its sides. </p>
<p>The Greeks were... |
4,196,868 | <p><span class="math-container">$(f_n)$</span> is a sequence of continuous, real valued functions on a metric space <span class="math-container">$M$</span>.</p>
<p>It converges pointwise to a <strong>continuous</strong> function <span class="math-container">$f$</span>.</p>
<p>Suppose that <span class="math-container">$... | IV_ | 292,527 | <p>First check <span class="math-container">$0$</span> and <span class="math-container">$1$</span>: <span class="math-container">$0$</span> is a solution of your first equation.</p>
<p>Neither of your two equations can be solved any further by <a href="https://en.wikipedia.org/wiki/Elementary_function" rel="nofollow no... |
390,532 | <p>I'm trying to solve (for $x$) some problems such as $\arctan(0)=x$, $\arcsin(-\frac{\sqrt{3}}{{2}})=x$, etc.</p>
<p>What is the best way to go about this? So far, I have been trying to solve the problems intuitively (e.g. I ask myself <em>what value of sine will give me $-\frac{\sqrt{3}}{{2}}$?</em>), maybe drawing... | orion | 137,195 | <p>In other words, you know the values of $\arcsin x$/$\arctan x$/$\arccos x$ for some specific values of $x$. That's just fine. Inverse trigonometric functions are transcendental functions and with exceptions of a few well-known values, the result is not nicely expressible with elementary functions (you can use a calc... |
11,916 | <p>In <a href="https://mathoverflow.net/questions/11845/theory-mainly-concerned-with-lambda-calculus/11861#11861">Theory mainly concerned with lambda-calculus?</a>, F. G. Dorais wrote, of the idea that the lambda-calulus defines a domain of mathematics:</p>
<blockquote>
<p>That would never stick unless there's anoth... | Adam | 2,361 | <p>I think most people here would agree that Category Theory is part of mathematics.</p>
<p>The study of strongly-typed functional programming languages is really just the study of cartesian closed categories, so I think that this particular part of functional programming is legitimate mathematics. And Domain Theory ... |
1,397,576 | <p>To me there is a hierarchy where vectors $\subset$ sequences $\subset$ functions $\subset$ operators</p>
<ul>
<li><p>All vectors are sequences, but not all sequences are vectors because
sequences are infinite dimensional</p></li>
<li><p>All sequences are functions, but not all functions are sequences
because functi... | Paul Sinclair | 258,282 | <p>Vectors are not sequences. They can be represented in some cases by finite sequences (as Omnomnomnom has pointed out). But in general a vector is any elements of a vector space, and a vector space is any set where you can add and multiply by scalars. All sequences are vectors, because they form a vector space. All f... |
1,397,576 | <p>To me there is a hierarchy where vectors $\subset$ sequences $\subset$ functions $\subset$ operators</p>
<ul>
<li><p>All vectors are sequences, but not all sequences are vectors because
sequences are infinite dimensional</p></li>
<li><p>All sequences are functions, but not all functions are sequences
because functi... | ASCII Advocate | 260,903 | <p>It's better to think of each as its own type of object, but where some types can be naturally converted into others (similar to type conversion in computer programming). Some of the conversions faithfully translate all of the information, some are reversible, and others lose part of the information. </p>
|
450,410 | <p>I'm trying to teach myself how to do $\epsilon$-$\delta$ proofs and would like to know if I solved this proof correctly. The answer given (Spivak, but in the solutions book) was very different.</p>
<hr>
<p><strong>Exercise:</strong> Prove $\lim_{x \to 1} \sqrt{x} = 1$ using $\epsilon$-$\delta$.</p>
<p><strong>My ... | Emanuele Paolini | 59,304 | <p>The proof is correct but can be simplified. You don't need the part "Now let $\delta=1$...". In fact it is always true that
$$
\frac{1}{\sqrt x + 1} \le 1
$$
since $\sqrt x \ge 0$.</p>
<p>Also, a matter of style. In the first line you don't <em>have</em> $0 < |x-1|<\delta$ but you <em>suppose</em> it (this ... |
920,050 | <p>The answer is $\frac1{500}$ but I don't understand why that is so. </p>
<p>I am given the fact that the summation of $x^{n}$ from $n=0$ to infinity is $\frac1{1-x}$. So if that's the case then I have that $x=\frac15$ and plugging in the values I have $\frac1{1-(\frac15)}= \frac54$.</p>
| Aldo | 171,035 | <p>$$S_n = (1/5)^4+...+(1/5)^n\ \ \ \ \ (i)$$ </p>
<p>$$-(1/5)S_n = -(1/5)^5-...-(1/5)^n-(1/5)^{n+1}\ \ \ \ (ii)$$</p>
<p>$(i)+(ii)$ $$S_n(1-1/5) = (1/5)^4 - (1/5)^{n+1} \Rightarrow (4/5)S_n = 1/625 - (1/5)^{n+1}$$</p>
<p>$\Rightarrow S_n = 1/500 - (5/4)(1/5)^{n+1}$</p>
<p>but</p>
<p>$(1/5)^n \rightarrow 0$</p>
|
2,871,105 | <p>I am trying to prove the following statement, but starting to doubt its correctness.</p>
<p>Suppose that $H$ is a Hausdorf topological space (I am formulating generally, though my specific case is $H=S'(R)$ - a space of tempered distributions). </p>
<p>Suppose I have a set of nested subsets $\Omega_i \subseteq H$ ... | Daniel Schepler | 337,888 | <p>For a counterexample using sequential closures, consider the topological space whose underlying set is $\mathbb{N}^2 \sqcup \{ x_0 \}$, and with the topology such that $U$ is open if and only if $x_0 \notin U$ or for some function $f : \mathbb{N} \to \mathbb{N}$, $\{ (x, y) \in \mathbb{N}^2 \mid y > f(x) \} \subs... |
2,535,933 | <p>let assume i have a position function in 1 dimension with constant acceleration.</p>
<p>$$ x(t) = x_0 + v_0t + \frac{1}{2}at^2 $$</p>
<p>then it's first derivative is a velocity function:
$$ \frac{dx}{dt} = v(t) = v_0 + at $$</p>
<p>then it's second derivative is an acceleration function:</p>
<p>$$ \frac{dv}{dt}... | Falrach | 506,310 | <p>Shorter:</p>
<p><span class="math-container">$(X-\Bbb{E}[X])^2 \geq 0$</span> by definition. So it follows directly from <span class="math-container">$\Bbb{E}[(X-\Bbb{E}[X])^2] = \operatorname{Var}(X) = 0$</span>, that <span class="math-container">$(X-\Bbb{E}[X])^2 = 0$</span> almost surely. We conclude <span class=... |
2,477,137 | <p>$\left(1+3+5...+(2n+1)\right ) + \left(3.5+5+6.5+...+(\frac{7+3n}{2})\right)=105$ </p>
<p>It is the equation that I did not understand how to find $n.$</p>
| Disintegrating By Parts | 112,478 | <p>If $\mu$ is a Borel measure on $\mathbb{R}$ with no atoms, then $m(\lambda)=\mu(-\infty,\lambda]$ is continuous, and
$$
\int_{-\infty}^{\infty}\lambda^2 dm(\lambda)=\int_{0}^{\infty}\lambda d(m(\sqrt{\lambda})-m(-\sqrt{\lambda})).
$$
However, there are problems in the case that $m$ has atoms, if you want a ... |
107,915 | <p>I randomly place $k$ rooks on an (arbitrarily sized) $N$ by $M$ chessboard. Until only one rook remains, for each of $P$ time intervals we move the pieces as follows:</p>
<p>(1) We choose one of the $k$ rooks on the board with uniform probability. </p>
<p>(2) We choose a direction for the rook, $(N, W, E, S)$, w... | Per Alexandersson | 1,056 | <p>So, I did some numerical experiments on 4 rooks on a k times k board.
Each data point is the mean of 500 runs.</p>
<p><a href="https://i.stack.imgur.com/k00FG.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/k00FG.png" alt=""></a>
</p>
<p>The x axis is the width/height of the board, the y axis is... |
3,239,185 | <p>Let <span class="math-container">$f,g$</span> be two analytic functions on the domain <span class="math-container">$\Omega$</span> such that <span class="math-container">$|f(z)|=|g(z)|$</span> throughout <span class="math-container">$\Omega$</span>.</p>
<p>I believe <span class="math-container">$h(z)=f/g$</span> on... | Martin R | 42,969 | <p>The problem with your approach is that <span class="math-container">$\arg f(z_0) = \arg 0$</span> and <span class="math-container">$\arg g(z_0) = \arg 0$</span> are not defined, and actually <span class="math-container">$w \mapsto \arg w$</span> <em>cannot</em> be defined as a continuous function in the neighborhood... |
4,528,489 | <p>I hope I could get clarification on a minor detail in the proof to theorem 3.11 (b) in Rudin's Principles of Mathematical Analysis. The theorem and proof are as follows.
<br>Theorem:</p>
<blockquote>
<p>If X is a compact metric space and if {<span class="math-container">$p_n$</span>} is a Cauchy sequence in X, then ... | Anne Bauval | 386,889 | <p>It is relatively elementary to prove that <span class="math-container">$f$</span> is bijective on the four following (invariant) subsets, which form a partition of <span class="math-container">$\mathbb Z/(pq\mathbb Z):$</span>
<span class="math-container">$$\{0\},\{(pk)\bmod{pq}:q\nmid k\},\{(qk)\bmod{pq}:p\nmid k\}... |
976,617 | <p>Find supremum and infimum of the set:
$B={ \frac{x}{1+ \mid x \mid }} \ for \ x\in \mathbb{R}$
For me it is visible that it will be 1 and -1 respectively but how to prove it properly?</p>
| Ruben | 153,329 | <p>For all $x\in \mathbb R$ we have $\frac{x}{1+|x|} < 1$, hence the supremum is less than or equal to 1. Suppose the supremum is smaller than 1, then we can write it as $1-\epsilon$ for some $\epsilon > 0$. Can you find an $x\in \mathbb R$ such that $\frac{x}{1+|x|} \in (1-\epsilon, 1)$?</p>
|
75,862 | <blockquote>
<p>In quadilateral $ABCD$ (usual clockwise or anticlockwise naming), $AB=16\sqrt{2}$ cm, $CD=10$ cm, $DA=8.5$ cm, $\angle D = 120^\circ $ and $\angle ACB = 45^\circ$. How to find $\angle ABC$?</p>
</blockquote>
<p><a href="http://testfunda.com/examprep/learningresources/smsqod/cat-sms-question-of-the-... | robjohn | 13,854 | <p>Using the Law of Cosines, I get that $|AC|^2=8.5^2+10^2+85=257.25$ since $\cos(ADC)=-\frac{1}{2}$. Next, $\sin^2(ACB)=\frac{1}{2}$ and $|AB|^2=512$. Law of Sines says that
$$
\frac{\sin^2(ACB)}{|AB|^2}=\frac{\sin^2(ABC)}{|AC|^2}
$$
Therefore,
$$
\sin^2(ABC)=\frac{1}{2}\frac{257.25}{512}\approx\frac{1}{4}
$$
Thus... |
1,707,675 | <p>How can I find the indefinite integral which is $$\int \frac{\ln(1-x)}{x}\text{d}x$$</p>
<p>I tried to use substitution by assigning $$\ln(1-x)\text{d}x = \text{d}v $$ and $$\frac{1}{x}=u$$ but, it is meaningless but true, the only thing I came up from integration by part is that $$\int \frac{\ln(1-x)}{x^2}\text{d... | Enrico M. | 266,764 | <p>This integral has no primitive. Indeed the result is a so well known Special function called Logarithm Integral:</p>
<p>$$\int\frac{\ln(1-x)}{x}\ \text{d}x = -\text{Li}_2(x)$$</p>
<p>More here</p>
<p><a href="https://en.wikipedia.org/wiki/Logarithmic_integral_function" rel="nofollow">https://en.wikipedia.org/wiki... |
4,253,564 | <p>I am having trouble with the following integral</p>
<blockquote>
<p>Prove that <span class="math-container">$$ \int_0^1\frac{x\ln(x)}{1+x^2+x^4}dx=\frac{1}{36}\Big(\psi^{(1)}(2/3)-\psi^{(1)}(1/3)\Big)$$</span></p>
</blockquote>
<p><span class="math-container">$$I=\int_0^1\frac{x\ln(x)}{1+x^2+x^4}dx=\int_0^1\frac{\ln... | projectilemotion | 323,432 | <p>Firstly, note that after your substitution it should be
<span class="math-container">$$I=\int_0^1\frac{x\ln(x)}{1+x^2+x^4}~dx=\color{red}{\frac{1}{4}}\int_0^1\frac{\ln(u)}{1+u+u^2}~du.$$</span>
To evaluate the latter integral, the geometric series shows that
<span class="math-container">$$\begin{align*} \int_0^1\fra... |
401,898 | <p>the function $f_n(x)=x^n-x^{2n}$ converge to $f(x)=0$ in $(-1,1]$. Intuativly the function does not converge uniformally in (-1,1]. How can I prove it?
I tried using the definition $\lim \limits_{n\to\infty}\sup \limits_{ x\in (-1,1]}|f_n(x)-f(x)|$ function is continial fractional on $[-1,1]$ and $x=0,(\frac 1 2 )^{... | DonAntonio | 31,254 | <p>An idea: say for $\,n>2\,$</p>
<p>$$f_n(x)=x^n-x^{2n}\implies f'_n(x)=nx^{n-1}-2nx^{2n-1}=nx^{n-1}\left(1-2x^{n}\right)=0\iff$$</p>
<p>$$x=0\,,\,\frac1{\sqrt[n]2}$$</p>
<p>$$f_n''(x)=n(n-1)x^{n-2}-2n(2n-1)x^{2n-2}\implies\begin{cases}f''(0)=0\\{}\\f''\left(\frac1{\sqrt[n]2}\right)=\frac{n\sqrt[n]4}4\left(-2n\r... |
401,898 | <p>the function $f_n(x)=x^n-x^{2n}$ converge to $f(x)=0$ in $(-1,1]$. Intuativly the function does not converge uniformally in (-1,1]. How can I prove it?
I tried using the definition $\lim \limits_{n\to\infty}\sup \limits_{ x\in (-1,1]}|f_n(x)-f(x)|$ function is continial fractional on $[-1,1]$ and $x=0,(\frac 1 2 )^{... | Tim | 74,128 | <p>Choose an arbitrarily large <em>odd</em> value of $n$. There exists some $0<x<1$ such that $x^n>\dfrac 12$.</p>
<p>Then $$\begin{array}{rl}f_n(-x) &= (-x)^n - (-x)^{2n}
\\ &= -\left(x^n + x^2n\right) \\ &\leq -\frac 34 \end{array}$$</p>
<p>So $f_n$ does not converge uniformly on $(-1,1]$.</p>... |
3,163,580 | <p>I'm having troubles to show that if <span class="math-container">$0<|\alpha|<1$</span> then the elements <span class="math-container">$f_k=\lbrace 1, \alpha^k, \alpha^{2k}, \alpha^{3k}, \cdots \rbrace$</span> span <span class="math-container">$\ell^2$</span> for <span class="math-container">$k \geq 1$</span>. ... | anomaly | 156,999 | <p>To clarify, not every element of <span class="math-container">$\ell^2$</span> is the sum of finitely many <span class="math-container">$f_k$</span>; take an element of <span class="math-container">$\ell^2$</span> that does not decay as <span class="math-container">$e^{-tn}$</span> for some <span class="math-containe... |
633,799 | <p>I am a little confused about the basic definition of inclusion.</p>
<p>I understand that, for example, $\{4\}\subset\{4\}$.</p>
<p>I also understand that $4\in\{4\}$, and that it is false to say that $\{4\}\in\{4\}$.</p>
<p>However, is it possible to say that $4\subset\{4\}$?</p>
| dani_s | 119,524 | <p>Technically it depends on the definition of 4 and the axioms of set theory you are using. With the standard definitions it is false. (Note though that $4 \subset \{4\}$ is a valid statement)</p>
|
633,799 | <p>I am a little confused about the basic definition of inclusion.</p>
<p>I understand that, for example, $\{4\}\subset\{4\}$.</p>
<p>I also understand that $4\in\{4\}$, and that it is false to say that $\{4\}\in\{4\}$.</p>
<p>However, is it possible to say that $4\subset\{4\}$?</p>
| Asaf Karagila | 622 | <p>First of all, sets <em>can</em> be elements of other sets too. For example if $X$ is a set then $\mathcal P(X)$ is the power set of $X$, and it is a set whose elements are all sets. But now that's out of the way, let us focus on the question whether or not $4\subseteq\{4\}$ makes sense.</p>
<p>It is possible if you... |
3,665,879 | <p>We all are familiar with the sum and difference formulas for <span class="math-container">$\sin$</span> and <span class="math-container">$\cos$</span>, but is there an analogue for the sum and difference formulas for secant and cosecant? That is, </p>
<p><span class="math-container">$$\csc (A\pm B) = ?$$</span> an... | Community | -1 | <p>Let <span class="math-container">$x \in X$</span> and <span class="math-container">$B_1{x}$</span> an open ball around <span class="math-container">$x$</span> with radius <span class="math-container">$1$</span> with respect the metric <span class="math-container">$d$</span>. Obviously, <span class="math-container">... |
97,131 | <p>I have the following problem:</p>
<p>I have a convex hull $\Omega$ defined by a set of n-dimensional hyperplanes $S = [(n_1,d_1), (n_2,d_2),...,(n_k,d_k)]$ such that a point $p \in \Omega$ if $n_i^T p \geq d_i \quad \forall (n_i,d_i) \in S $. Now I have a "joining" hyperplane $(n_{k+1},d_{k+1})$ and I want to know ... | Joseph O'Rourke | 6,094 | <p>If you search for <em>detection of redundant constraints in linear programming</em> you will find many hits, including one to an MO question, "<a href="https://mathoverflow.net/questions/69662/">Detection of Redundant Constraints</a>."
One source paper is</p>
<blockquote>
<p>J. Gondzio. Presolve analysis of linea... |
2,604,206 | <p>Can anyone provide links to a concrete proof? Intuitively, the two-dimensional real space is infinite. so there should be infinitely many subspaces. But how do I go about a proof?</p>
| asdq | 466,346 | <p>Take $v_\epsilon=(\epsilon,1)$, for $\epsilon \in [0,1]$. Then $\lambda v_\epsilon + \mu v_\nu=0$ implies $\lambda \epsilon + \nu \nu =0$ and $\lambda + \nu =0$, hence $\lambda (\epsilon -\nu)=0$. This shows that for $\epsilon \neq \nu$, $v_\epsilon$ and $v_\nu$ are linearly independent and therefore span different ... |
1,811,109 | <p>How can we cause this relation to be true?</p>
<blockquote>
<p>$$x \sin\theta + y \cos\theta = \sqrt{ x^2 + y^2 } \tag{$\star$}$$</p>
</blockquote>
<p>I know the identity</p>
<p>$$x \sin\theta + y \cos\theta = \sqrt{x^2+y^2}\; \sin\left(\theta + \operatorname{atan}\frac{y}{x}\right)$$
What can make the sine pa... | Prasanna Venkatesan | 320,861 | <p>First of all you should've tried this out yourself and should give your details about your effort made.
Anyway it is not an identity. Here's workout:-</p>
<p>set: </p>
<p>$$ x=r \cos(\alpha)~~ \text{and}~~ y=r \sin(\alpha)$$</p>
<p>Substituting back in he equation we get</p>
<p>$$ r \cos(\alpha) \sin(\theta) + ... |
2,600,679 | <p>Provided two real number sequences: $a_1,a_2,...,a_n$;$b_1,b_2,...,b_n$, define their means respectively:
$$\bar a=\frac{1}{n}\sum_{i=1}^n a_i,\bar b=\frac{1}{n}\sum_{i=1}^n b_i$$
and define their variances and covariance respectively:
$$var(a)=\frac{1}{n}\sum_{i=1}^n (a_i-\bar a)^2,var(b)=\frac{1}{n}\sum_{i=1}^n (b... | Enrico M. | 266,764 | <p>Yes.</p>
<p>$$\sum_{k = 0}^{+\infty} \frac{x^k}{(k!)^2} = I_0\left(2 \sqrt{x}\right)$$</p>
<p>Where $I_0$ is the modified Bessel Function of the first kind.</p>
<p><a href="http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html" rel="nofollow noreferrer">http://mathworld.wolfram.com/ModifiedBessel... |
619,040 | <p>An exponential object $B^{A}$ is defined to be the representing object of the functor $$\mathcal{C}\left(- \times A,B\right): \mathcal{C} \rightarrow Set$$
or equivalently, as the terminal object of $\left(-\times A \downarrow B\right)$. The dual concept is of the co-exponential object which is the initial object o... | Henry Story | 253,728 | <p>I have given <a href="https://math.stackexchange.com/questions/3621660/examples-of-co-implication-a-k-a-co-exponential/3624965#3624965">an explanation with example</a> and a lot of references about how to think of co-exponentials.
I have not yet worked out how useful they are for myself under that name.</p>
<p>But... |
304,209 | <p>I am trying to learn weak derivatives. In that, we call <span class="math-container">$\mathbb{C}^{\infty}_{c}$</span> functions as test functions and we use these functions in weak derivatives. I want to understand why these are called <em>test functions</em> and why the functions with these properties are needed. I... | nicomezi | 316,579 | <p>To understand why we are calling them test functions, we have to understand what distributions are and where they come from.</p>
<p>Usually, to evaluate a function, we compute its value at the point where we want to know it. But remember that there are spaces of functions (or equivalence classes of functions) such ... |
987,054 | <p>Prove that the sequence
$$b_n=\left(1+\frac{1}{n}\right)^{n+1}$$
Is decreasing.</p>
<p>I have calculated $b_n/b_{n-1}$ but it is obtain:
$$\left(1-\frac{1}{n^2}\right)^n \left(1+\frac{1}{n}\right)^n$$
But I can't go on.</p>
<p>Any suggestions please?</p>
| Community | -1 | <p>$$y=\left(1+\frac{1}{x}\right)^{x+1}$$</p>
<p>$$\ln y=({x+1})\cdot\ln\left(1+\frac{1}{x}\right)$$</p>
<p>$$y'\frac{1}{y}=\ln\left(1+\frac{1}{x}\right)+(x+1)\cdot \frac{1}{1+\frac{1}{x}}\cdot\left(-\frac{1}{x^2}\right)$$</p>
<p>$$y'=\left(\ln\left(1+\frac{1}{x}\right)-\frac{1}{x}\right)\cdot\left(1+\frac{1}{x}\rig... |
372,211 | <p>I'm trying to write an <a href="http://developer.android.com/reference/android/view/animation/Interpolator.html" rel="nofollow noreferrer">interpolator</a> for a translate animation, and I'm stuck. The animation passes a single value to the function. This value maps a value representing the elapsed fraction of an an... | bubba | 31,744 | <p>Let $s(t)$ denote the distance moved at time $t$. Choose some value $h$ with $0 \le h \le 1$. Then the required function is:</p>
<p>$$s(t) = \frac{2t}{1+h} \quad \text{for } 0 \le t \le h $$</p>
<p>$$s(t) = \frac{t^2 - 2t + h^2}{h^2 - 1} \quad \text{for } h \le t \le 1 $$</p>
<p>The first part of the curve (where... |
1,990,670 | <blockquote>
<p>Assume that $0 < \theta < \pi$. Solve the following equation for $\theta$. $$\frac{1}{(\cos \theta)^2} = 2\sqrt{3}\tan\theta - 2$$ </p>
</blockquote>
<p><a href="https://i.stack.imgur.com/SoU8A.png" rel="nofollow noreferrer">Question and Answer</a></p>
<p>Regarding to the attached image, that... | EnlightenedFunky | 372,659 | <p>HINT: Write $\frac{1}{\cos^2(x)}$ as $\sec^2(x)$ then look up Pythagorean identities. And possibly solutions to a quadratic equations.</p>
|
3,807,708 | <p>I was asked to prove the following identity (starting from the left-hand side):
<span class="math-container">$$(a+b)³(a⁵+b⁵)+5ab(a+b)²(a⁴+b⁴)+15a²b²(a+b)(a³+b³)+35a³b³(a²+b²)+70a⁴b⁴=(a+b)^8.$$</span>
I'm trying to solve it by a sort of "inspection", but I haven't made it yet. Of course I could try to expan... | user | 505,767 | <p>Since the expression is symmetric and in decreasing order, it suffices to consider only the expansion for <span class="math-container">$a$</span> that is</p>
<p><span class="math-container">$$a^8+3a^7+3a^6+a^5+5a^7+10a^6+5a^5+15a^6+15a^5+35a^5+70a^4$$</span></p>
<p><span class="math-container">$$a^8+8a^7+28a^6+56a^5... |
4,565,584 | <blockquote>
<p>Let <span class="math-container">$X = (-1,1)^{\Bbb N}$</span> have the product topology. Is the subset <span class="math-container">$(0,1)^{\Bbb N}$</span> open?</p>
</blockquote>
<p>To consider whether <span class="math-container">$(0,1)^{\Bbb N}$</span> is open I know that it is if I can find a basic ... | Theo Bendit | 248,286 | <p>Neither, but the second is closest.</p>
<p>Simply finding a basic open subset is insufficient to show that a set is open (e.g. on the real line, <span class="math-container">$(0, 1]$</span> contains the basic open set <span class="math-container">$(0, 1)$</span>, but <span class="math-container">$(0, 1]$</span> is n... |
4,565,584 | <blockquote>
<p>Let <span class="math-container">$X = (-1,1)^{\Bbb N}$</span> have the product topology. Is the subset <span class="math-container">$(0,1)^{\Bbb N}$</span> open?</p>
</blockquote>
<p>To consider whether <span class="math-container">$(0,1)^{\Bbb N}$</span> is open I know that it is if I can find a basic ... | Jakobian | 476,484 | <p>For any product of topological spaces <span class="math-container">$X_i$</span>, if <span class="math-container">$U\subseteq \prod_{i\in I} X_i$</span> is open and non-empty, and <span class="math-container">$\pi_j:\prod_{i\in I} X_i\to X_j$</span> is the projection onto the <span class="math-container">$j$</span>th... |
508,059 | <p>How it is possible to considerably shorten the list of properties that define a vector space by using definitions from abstract algebra?</p>
| DonAntonio | 31,254 | <p>A vector space is a module over a division ring.</p>
<p>Less short? The pair $\;(V,\Bbb F)\;$ is a vector space if $\;V\;$ is an additive (abelian) group and $\;V\;$ is a module over $\;\Bbb F\;$.</p>
|
508,059 | <p>How it is possible to considerably shorten the list of properties that define a vector space by using definitions from abstract algebra?</p>
| Matemáticos Chibchas | 52,816 | <p>A $R$-module ($R$ a commutative ring with unity) is an abelian group $M$ endowed with a ring (with unity) homomorphism $R\to\mathrm{End}(M,M)$.</p>
|
89,845 | <p>first,I think we can avoid set theory to bulid the first order logic , by the operation of the finite string.but I have The following questions:</p>
<p>How does "meta-logic" work. I don't really know this stuff yet, but from what I can see right now, meta-logic proves things about formal languages and logics in g... | Community | -1 | <p>There are two roles for metalanguage: First, to avoid contradictions like the liar paradox, because in the liar paradox we have a statement that speaks about itself so it does not respect the hierarchy language-metalanguage.
The other role is to allow us to speak freely and to use theorems of the language as meta-th... |
293,921 | <p>The problem I am working on is:</p>
<p>An ATM personal identification number (PIN) consists of four digits, each a 0, 1, 2, . . . 8, or 9, in succession.</p>
<p>a.How many different possible PINs are there if there are no restrictions on the choice of digits?</p>
<p>b.According to a representative at the author’s... | joriki | 6,622 | <p>For b): Which is the descending sequence starting with $1$ that you counted?</p>
<p>For c): Good question; the problem is badly worded in that regard. Taking it literally, I'd tend to interpret it as referring to unordered pairs, but since it makes little sense to couple two different PINs in this manner, I suspect... |
386,799 | <blockquote>
<p>P1086: For a closed surface, the positive orientation is the one for which the normal vectors point outward from the surface, and inward-pointing normals give the negative orientation.</p>
<p>P1087: If <span class="math-container">$S$</span> is a smooth orientable surface given in parametric form by a v... | Christian Blatter | 1,303 | <p>A surface $S$ as such is a "two-dimensional smooth set of points" embedded in three-space. At each point $p\in S$ we have a tangent plane $T_p(\sim{\mathbb R}^2)$ with origin at $p$, and this tangent plane has a well defined orthogonal complement $N:=T_p^\perp$, a line through $p$ with origin at
$p$. On this line o... |
4,285,426 | <p>Intuitively it is quite easy to see why <span class="math-container">$$a \equiv (a \bmod m) \pmod m.$$</span></p>
<p>When you divide a by m you get a remainder in the range <span class="math-container">$0, \dots, m-1.$</span> When you divide the remainder by m again, you get the same number again as the remainder, ... | Thomas Andrews | 7,933 | <p>At heart, you need this version of the division algorithm to even define <span class="math-container">$(a\bmod m).$</span></p>
<blockquote>
<p>Let <span class="math-container">$a,m\in\mathbb Z, m\neq 0.$</span> Then there is a unique pair <span class="math-container">$q,r\in \mathbb Z$</span> such that <span class="... |
4,114,034 | <p>In Linear Algebra Done Right by Axler, there are two sentences he uses to describe the uniqueness of Linear Maps (3.5) which I cannot reconcile. Namely, whether the uniqueness of Linear Maps is determined by the choice of 1) <em>basis</em> or 2) <em>subspace</em>. These two seem like very different statements to me ... | DonAntonio | 31,254 | <p>Simple, direct meaning of definition: for any function <span class="math-container">$\;f\;$</span> , a point <span class="math-container">$\;a\;$</span> in its domain of definion is a fixed point if <span class="math-container">$\;f(a)=a\;$</span> . For your function, this means that we must have</p>
<p><span class=... |
2,392,114 | <p>It is possible to rewrite the equation $x^3+ax^2+bx+c=0$ as $y^3+3hy+k=0$ by setting $y=x+a/3$</p>
<p>How do you find the coefficient h in the equation $y^3+3hy+k=0$?</p>
| Mark Bennet | 2,906 | <p>You put $$x^3+ax^2+bx+c=\left(y-\frac a3\right)^3+a\left(y-\frac a3\right)^2+b\left(y-\frac a3\right)+c=$$$$=y^3-ay^2+\frac {a^2}3y-\frac {a^3}{27}+ay^2-\frac {2a^2}3y+\frac {a^3}9+by-\frac {ab}3+c=$$and collect terms$$=y^3+3\left(\frac {3b-a^2}9\right)y+\frac{-2a^3-9ab+27c}{27}$$</p>
|
150,472 | <p>Let $h\in C_0([a,b])$ arbitrary, that is $h$ is continuous and vanishes on the boundary.
I want to show that
$\int\limits_a^b h(x)\sin(nx)dx \rightarrow 0$.</p>
<p>If $h\in C^1$, integration by parts immediately yields the claim, since $h'$ is continuous and thence bounded on the compact interval, using also the ze... | Pedro | 23,350 | <p>I think this is from Apostol. It is an informal approach to the following Lemma, if I'm not recalling wrongly:</p>
<p>Let $f$ be integrable in $[a,b]$. Then</p>
<p>$$\lim \limits_{\lambda \to \infty } \int\limits_a^b f\left( x \right)\sin \lambda xdx = 0$$</p>
<p>$(1)$ Let $f$ be constant. Then </p>
<p>$$\lim ... |
162,836 | <p>I would like to find the surface normal for a point on a 3D filled shape in Mathematica. </p>
<p>I know how to calculate the normal of a parametric surface using the cross product but this method will not work for a shape like <code>Cone[]</code> or <code>Ball[]</code>.</p>
<ol>
<li>Is there some sort of <code>Reg... | Michael E2 | 4,999 | <pre><code>(* put inequality into u ≤ 0 form, return just u *)
standardize[a_ <= b_] := a - b;
standardize[a_ >= b_] := b - a;
regnormal[reg_, {x_, y_, z_}] := Module[{impl},
impl = LogicalExpand@ Simplify[RegionMember[reg, {x, y, z}], {x, y, z} ∈ Reals];
If[Head@impl === Or,
impl = List @@ impl,
i... |
499,044 | <p>I "know" that $\mathbb{C} \otimes_\mathbb{R} \mathbb{C} \cong \mathbb{C} \oplus \mathbb{C}$ as rings, but I don't really know it, what I mean with this is that I don't know any explicit isomorphism $f: \mathbb{C} \otimes_\mathbb{R} \mathbb{C} \rightarrow \mathbb{C} \oplus \mathbb{C}$. I suspect that such an isomorph... | Martin Brandenburg | 1,650 | <p>You don't mean the direct sum of rings (or algebras), you mean the direct product of rings. See also <a href="https://math.stackexchange.com/questions/345501">here</a>.</p>
<p>In general, if $K/k$ is a field extension and $f \in k[x]$ is a polynomial which splits over $K$ into $n$ distinct linear factors $x-\alpha_... |
2,658,563 | <p><strong>(Brazil National Olympiad)</strong></p>
<p><em>Let $n$ be a positive integer. In how many ways can we distribute $n+1$ toys to $n$ kids, such that each kid gets at least one toy?</em></p>
<p><strong><em>My approach</em></strong>:</p>
<p>For each child we can assign a number $k$ to it, representing the to... | Badam Baplan | 164,860 | <p>A standard combinatorial way to think about it would be to recognize the role of the <a href="https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind" rel="nofollow noreferrer">Stirling Numbers of the Second Kind</a>. We proceed as follows:</p>
<p>(1) partition the set of $n+1$ toys into $n$ nonempty subs... |
123,018 | <p>I'm doing some homework for a computer science class. It's been so long since I've done math, I have a question that assumes math knowledge that confuses me.</p>
<p>Given:
<em>Whether a diophantine polynomial in a single variable has integer roots.</em></p>
<p>With the given question I need to determine if that qu... | Ross Millikan | 1,827 | <p><a href="http://en.wikipedia.org/wiki/Diophantine_equation" rel="nofollow">A Diophantine equation</a> is one where the variables are required to take integer values. A root of a <a href="http://en.wikipedia.org/wiki/Polynomial" rel="nofollow">polynomial</a> is a value of the variable(s) that gives the polynomial th... |
1,420,277 | <p>I have to solve this:</p>
<p>$$[(\nabla \times \nabla)\cdot \nabla](x^2 + y^2 + z^2)$$</p>
<p>But I am really drowning in the sand..</p>
<p>Can anybody help me please?</p>
| MathAdam | 266,049 | <p>Let <strong><em>E</em></strong> = number of Estrada supporters<br>
Let <strong><em>A</em></strong> = number of Arrayo supporters </p>
<p>Then <strong><em>E</em></strong> + <strong><em>A</em></strong> = <strong>8600</strong> </p>
<p>Estrada's majority -- we don't know what it is yet -- is <strong><em>E</em></stro... |
3,368,655 | <p>I came across a problem that asked if it is posible for a function to be Riemann integrable function in <span class="math-container">$[0,+\infty)$</span> but also <span class="math-container">$|f(x)|\geq 1$</span> for all <span class="math-container">$x\geq 0$</span>. </p>
<p>At first I thought it was imposible, bu... | MathematicsStudent1122 | 238,417 | <blockquote>
<p>I came across a problem that asked if it is posible for a function to be Riemann integrable function in <span class="math-container">$[0,+\infty)$</span> but also <span class="math-container">$|f(x)|\geq 1$</span> for all <span class="math-container">$x\geq 0$</span>. </p>
</blockquote>
<p>Yes, altho... |
292,331 | <p>Suppose that $(n_k)_{k\in \mathbb{N}}$ is a given increasing sequence of positive integers. </p>
<p>Does there exist an (irrational) number $a$ such that
$\{an_k\}:=(a n_k)\text{mod }1 \rightarrow 1/2$ as $k \rightarrow \infty$? </p>
| GH from MO | 11,919 | <p>The answer is no in general. For many increasing sequences $(n_k)_{k\in \mathbb{N}}$ of positive integers, it happens for every irrational number $a$ that $\{an_k\}$ is dense or even equidistributed in the unit circle. See <a href="https://en.wikipedia.org/wiki/Equidistribution_theorem" rel="nofollow noreferrer">thi... |
2,168,906 | <blockquote>
<p>The task is to find necessary and sufficient condition on <span class="math-container">$b$</span> and <span class="math-container">$c$</span> for the equation <span class="math-container">$x^3-3b^2x+c=0$</span> to have three distinct real roots.</p>
</blockquote>
<p>Are there any formulas (such as <spa... | Jonathaniui | 420,908 | <p>Note that $ln (y) = x^x$ and not equal to $x ln (e^x) $</p>
<p>Thsi is because $ln (e^(x^x))$ cancels the $e $ and the exponent $x^x $ is tue result, from there derivate as normal and obtain the result.</p>
|
300,105 | <p>I want to find the proof of the spectrum of the hypercube</p>
| Mariano Suárez-Álvarez | 274 | <p>This is just a data point:</p>
<p>Computing characteristic polynomials of their adjacency matrices, one finds the roots for the $d$-dimensional hypercube are $d$, $d-2$, $d-4$, $\dots$, $-d$, with multiplicities $\binom{d}{0}$, $\binom{d}{1}$, $\binom{d}{2}$, $\dots$, $\binom{d}{d}$.</p>
<p>Since this is extraordi... |
1,101,371 | <p>Any book that I find on abstract algebra is somehow advanced and not OK for self-learning. I am high-school student with high-school math knowledge. Please someone tell me a book can be fine on abstract algebra? Thanks a lot. </p>
| Community | -1 | <p>Abstract Algebra
Theory and Applications
Thomas W. Judson </p>
<p>You can also find it online in <a href="http://abstract.ups.edu/download/aata-20130816.pdf" rel="nofollow">here.</a> </p>
|
1,101,371 | <p>Any book that I find on abstract algebra is somehow advanced and not OK for self-learning. I am high-school student with high-school math knowledge. Please someone tell me a book can be fine on abstract algebra? Thanks a lot. </p>
| Dave L. Renfro | 13,130 | <p>Three books I know of that <strong>really are high school level</strong> are listed below. Although the books thus far listed (Gallian, Herstein, Fraleigh, Pinter, etc.) are fine texts, these are standard upper undergraduate college level textbooks, not books <em>specifically written</em> for good (but not necessari... |
701,122 | <p>Part 1
Let $f(x) = ax^n$, where $a$ is any real number. Prove that $f$ is even if $n$ is an even integer. (Integers can be negative too)</p>
<p>Part 2
Prove that if you add any two even functions, you get an even function</p>
<p>I'm confused as to how you would prove adding two even functions would get you an eve... | Ishfaaq | 109,161 | <p>The first part is simple. Suppose $f(x) = ax^n$for an even integer $n$. $f(-x) = a(-x)^n = ax^n = f(x)$ since $n$ is even. </p>
<p>As for the second part, suppose $h(x) = f(x) + g(x)$ for each $x$ in the domain of $h$, where $f$ and $g$ are even functions. Then, $h(-x) = f(-x) + g(-x) = f(x) + g(x) = h(x) $</p>
<p... |
1,007,309 | <p><img src="https://i.stack.imgur.com/xJKtU.png" alt="enter image description here"></p>
<p>I cannot understand part ii) in this solution. I cannot see the significance of arbitrarily close to 0 points for which $|sin(\frac{1}{x_n})|=1$</p>
| Ben Grossmann | 81,360 | <p>Our goal is to show that $g^{-1}(B)$ fails to be open because, although $0 \in g^{-1}(B)$, no open neighborhood of $0$ lies inside the set $g^{-1}(B)$.</p>
<p>So, we want to show that for any $\delta > 0$, the set $(-\delta,\delta)$ (the neighborhood of $0$ of radius $\delta$) is not a subset of $g^{-1}(B)$. Th... |
962,287 | <p>I am trying to isolate x in the equation $$(x-20)^{2} = -(y-40)^{2} - 525.$$ How can I do it?</p>
| Jasser | 170,011 | <p>$x=(\sqrt {(y-40)^ 2+525})i+20$ where $i=\sqrt {-1}$.</p>
<p>If domain of x is real number set than there is no solution.</p>
|
87,583 | <p>Next task to complete:</p>
<ul>
<li><p>Count <code>*</code>-symbol in such expression as <code>a + s^2*b - c/y + o^3 + n*m*u</code> (in this case count of <code>*</code> should be 6)</p></li>
<li><p>Powers such $o^3$ should be expand to $o*o*o$</p></li>
</ul>
<p>I try, but my code is pretty ugly.</p>
<p><img src=... | m_goldberg | 3,066 | <p>I think this is easier to do by working with strings.</p>
<p>First write a function that will expand strings of the form "Power(x,k)" where k is an integer in "x<em>x</em>...<em>x" with k - 1 "</em>"s.</p>
<pre><code>f[x_, k_] :=
Module[{i = Abs[ToExpression[k]] - 1},
Nest[StringJoin[#, "*" <> x] &... |
2,666,772 | <blockquote>
<p>$W$ = $\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & -1 & 1 & -1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & -1 & 1 \end{bmatrix}$. Use W to build an 8x8 matrix encoding an orthonormal basis in $R^8$ by scaling A = $\begin{bmatrix} W & W \\ W & -W \end{bmatrix}$ in t... | Pietro Paparella | 414,530 | <p>The matrix $W$ above is a <a href="https://en.wikipedia.org/wiki/Hadamard_matrix" rel="nofollow noreferrer">Hadamard matrix</a> of order four and therefore satisfies $WW^\top = W^\top W = 4I_4$.</p>
<p>If
$$
A = \frac{1}{\sqrt{8}}\begin{bmatrix}
W & W \\
W & -W
\end{bmatrix},
$$
then $A$ is an orthogonal m... |
34,724 | <h3>Overview</h3>
<p>For integers n ≥ 1, let T(n) = {0,1,...,n}<sup>n</sup> and B(n)= {0,1}<sup>n</sup>. Note that |T(n)|=(n+1)<sup>n</sup> and |B(n)| = 2<sup>n</sup>.
A certain set S(n) ⊂ T(n), defined below, contains B(n). The question is about the growth rate of |S(n)|. Does it grow exponentially, like |B(n)... | David E Speyer | 297 | <p>Your sequence is bounded by $(125+\epsilon)^n$. Obviously, this isn't close to a good bound, but it answers the question.</p>
<p>We start by bounding a different question: Let $\Gamma_n$ be the convex hull of $(0,0)$, $(0,n)$ and $(n,n)$. (So $\Gamma$ is rotated $180^{\circ}$ with respect to your $\Delta$.) Let $q_... |
441,962 | <p>I looking for a proof for the theorem but I have not find yet.</p>
<p>A link or even sketch for How it goes will be very appreciate.</p>
<p>A linear map is self adjoint </p>
<p>iff </p>
<p>the matrix representation according to orthonormal basis is self adjoint.</p>
<p>by the way is not that true for all self a... | al-Hwarizmi | 68,686 | <p>I try to understand what you really asking for. If I correctly understand you are looking for a theorem in literature that reflects your case. This is according to my understanding the spectral theorem <a href="https://en.wikipedia.org/wiki/Spectral_theorem" rel="nofollow">>>> here</a>. From either the Cauchy or von... |
438,070 | <p>I stumbled across this question and I cannot figure out how to use the value of $\cos(\sin 60^\circ)$ which would be $\sin 0.5$ and $\cos 0.5$ seems to be a value that you can only calculate using a calculator or estimate at the very best.</p>
| Community | -1 | <p>To show that $F_k$ is a subspace of $\mathbb R^\mathbb N$ you should verify that $F_k$ is a <strong>non empty set</strong> and <strong>any linear combination of two elements of $F_k$ remains in $F_k$</strong>.</p>
<p>Let's show an example:</p>
<p>Clearly $F_1$ is a non empty set since the zero sequence is bounded.... |
2,771,059 | <p>This question is similar to a question I posted earlier.<br/>
<span class="math-container">$$z=\cos\frac{\pi}{3}+j\sin\frac{\pi}{3}$$</span>
<br/>
This time I have to do the sum <span class="math-container">$z^4+z$</span><br/>
<br/>
I have used the approach I was shown in my previous question. Here is what I've done... | Rhys Hughes | 487,658 | <p>Via de Moivre's Theorem, <span class="math-container">$z^4=\cos\big(\frac{4\pi}{3}\big)+j\sin\big(\frac{4\pi}{3}
\big)=-\frac{1}{2} -\frac{\sqrt{3}}{2}j$</span>
<span class="math-container">$$\cos{\frac{\pi}{3}}+j\sin{\frac{\pi}{3}}=\frac{1}{2}+\frac{\sqrt{3}}{2}j$$</span></p>
<p>Adding those together yields <span ... |
148,032 | <p>What is the larger of the two numbers?</p>
<p>$$\sqrt{2}^{\sqrt{3}} \mbox{ or } \sqrt{3}^{\sqrt{2}}\, \, \; ?$$
I solved this, and I think that is an interesting elementary problem. I want different points of view and solutions. Thanks!</p>
| Robert Israel | 8,508 | <p>Hint: If $a$ and $b$ are positive numbers, $a^b < b^a$ if and only if $\dfrac{\ln a}{a} < \dfrac{\ln b}{b}$. Find intervals on which $\dfrac{\ln x}{x}$ is increasing or decreasing.</p>
|
148,032 | <p>What is the larger of the two numbers?</p>
<p>$$\sqrt{2}^{\sqrt{3}} \mbox{ or } \sqrt{3}^{\sqrt{2}}\, \, \; ?$$
I solved this, and I think that is an interesting elementary problem. I want different points of view and solutions. Thanks!</p>
| checkmath | 25,077 | <p>Hint: Use the Logarithm function.</p>
|
148,032 | <p>What is the larger of the two numbers?</p>
<p>$$\sqrt{2}^{\sqrt{3}} \mbox{ or } \sqrt{3}^{\sqrt{2}}\, \, \; ?$$
I solved this, and I think that is an interesting elementary problem. I want different points of view and solutions. Thanks!</p>
| Jeff | 10,832 | <p>We have $\sqrt{2}>1$ and $\sqrt{3}>1$, so raising either of these to powers $>1$ makes them larger.</p>
<p>Call $x=\sqrt{2}^\sqrt{3}$ and $y=\sqrt{3}^\sqrt{2}$.</p>
<p>We have $x^{2\sqrt{3}}$=8 and $y^{2\sqrt{2}}=9.$</p>
<p>Since $2\sqrt{2} < 2\sqrt{3}$, we conclude $y>x$.</p>
|
88,469 | <p>These vectors form a basis on $\mathbb R^3$: $$\begin{bmatrix}1\\0\\-1\\\end{bmatrix},\begin{bmatrix}2\\-1\\0\\\end{bmatrix} ,\begin{bmatrix}1\\2\\1\\\end{bmatrix}$$</p>
<p>Can someone show how to use the Gram-Schmidt process to generate an orthonormal basis of $\mathbb R^3$?</p>
| Paul | 16,158 | <p>Let $u_1=\begin{bmatrix}1\\0\\-1\\\end{bmatrix} ,u_2=\begin{bmatrix}2\\-1\\0\\\end{bmatrix} ,u_3=\begin{bmatrix}1\\2\\1\\\end{bmatrix}$. To find the required orthonormal basis $\{w_1,w_1,w_3\}$, first we have
$$w_1=\frac{u_1}{\|u_1\|}=\begin{bmatrix}\frac{1}{\sqrt{2}}\\0\\-\frac{1}{\sqrt{2}}\\\end{bmatrix}.$$</p>
<... |
1,794,221 | <p>I am asked to show that the tangent space of $M$={ $(x,y,z)\in \mathbb{R}^3 : x^{2}+y^{2}=z^{2}$} at the point p=(0,0,0) is equal to $M$ itself.</p>
<p>I have that $f(x,y,z)=x^{2}+y^{2}-z^{2}$ but as i calculate $<gradf_p,v>$ i get zero for any vector.Where am i making a disastrous error?</p>
| Mathematician 42 | 155,917 | <p>Suppose $\phi$ is injective, then $\phi(g)=e_H$ implies that $g=e_G$ since $\phi$ is injective and $\phi(e_G)=e_H$. Hence $\ker(\phi)$ is trivial. Conversely, suppose that the kernel is trivial, then $\phi(g)=\phi(h)$ implies that $\phi(gh^{-1})=e_H$, hence $gh^{-1}\in \ker(\phi)$. Since this kernel is trivial, it f... |
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