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 |
|---|---|---|---|---|
3,151,662 | <p>Consider <span class="math-container">$a_1,\dots,a_n\in\mathbb{R}^n$</span> and identify <span class="math-container">$a_j\in\mathcal{L}(\mathbb{R},\mathbb{R}^n)$</span> via <span class="math-container">$\varphi\mapsto \varphi1$</span>.</p>
<p>Also, consider <span class="math-container">$A\in\mathcal{L}(\mathbb{R}^... | Mason | 464,960 | <p>There are plenty of applications of determinants, but I will just mention one that applies to optimization. A totally unimodular matrix is a matrix (doesn’t have to be square) that every square submatrix has a determinant of 0, 1 or -1. It turns out that (by Cramer’s rule) that if a constraint matrix <span class="... |
3,151,662 | <p>Consider <span class="math-container">$a_1,\dots,a_n\in\mathbb{R}^n$</span> and identify <span class="math-container">$a_j\in\mathcal{L}(\mathbb{R},\mathbb{R}^n)$</span> via <span class="math-container">$\varphi\mapsto \varphi1$</span>.</p>
<p>Also, consider <span class="math-container">$A\in\mathcal{L}(\mathbb{R}^... | J. Wang | 545,729 | <p>My first brief understanding of matrices is that they offer an elegant way to deal with data (combinatorially, sort of). A classical and really concrete example would be a discrete Markov chain (don't be frightened by its name). Say you are given the following information: if today is rainy, then tomorrow has a 0.9 ... |
3,318,993 | <p>Consider the functions <span class="math-container">$x$</span> and <span class="math-container">$x^2$</span> on <span class="math-container">$\mathbb{R}$</span>. Clearly, they are linearly independent.<br>
But consider the following argument.</p>
<p>Consider the matrix
<span class="math-container">$$A = \begin{bma... | Zbigniew | 11,995 | <p><span class="math-container">$x$</span> and <span class="math-container">$x^2$</span> are considered to be elements of the vectorial space <span class="math-container">$\mathbb{R}[X]$</span> or <span class="math-container">$\mathbb{R}[X]_2 $</span> endowed with the basis <span class="math-container">$1,x,x^2$</span... |
120,067 | <p>The <em>theta function</em> is the analytic function $\theta:U\to\mathbb{C}$ defined on the (open) right half-plane $U\subset\mathbb{C}$ by $\theta(\tau)=\sum_{n\in\mathbb{Z}}e^{-\pi n^2 \tau}$. It has the following important transformation property.</p>
<blockquote>
<p><strong>Theta reciprocity</strong>: $\theta... | paul garrett | 15,629 | <p>One way a person could stumble on quadratic reciprocity while looking at theta functions is by trying to prove that Weil's construction of an adelic "Segal-Shale-Weil/oscillator representation" really is a representation, and really produces automorphic forms. This would lead a person to see that certain products of... |
1,841,958 | <p>This is a claim on Wikipedia <a href="https://en.wikipedia.org/wiki/Partially_ordered_set">https://en.wikipedia.org/wiki/Partially_ordered_set</a></p>
<p>I am not sure how to make sense of the claim</p>
<p>What does it mean by ordered by inclusion? Inclusion as in $\subseteq$? </p>
<p>Can someone provide a small ... | Ashwin Ganesan | 157,927 | <p>Let $A$ be the set of all subspaces of $\mathbb{R}^3$. Let $R$ be a binary relation on $A$ defined by $R: = \{(U,V): U \mbox{ is a subspace of } V \}$. So the relation $R$ is a subset of $A \times A$. This relation is reflexive (because every subspace $V$ is a subspace of itself), antisymmetric (if $U$ is a subspa... |
15,159 | <p>Specifically, is it possible for a non-Noetherian ring $R$ to have $R[x]$ Noetherian? Every reference I've seen for the Hilbert basis theorem only states the direction "$R$ Noetherian $\Rightarrow$ $R[x]$ Noetherian", which would certainly seem to imply that the converse is false. Unfortunately, it's tough to think ... | Wanderer | 1,107 | <p>If $A$ is an ideal of $R$, then $A[X]$ is an ideal of $R[X]$, right? So an ascending chain of ideals in $R$ which does not stabilize gives you an ascending chain of ideals in $R[X]$ which doesn't stabilize either?</p>
|
15,159 | <p>Specifically, is it possible for a non-Noetherian ring $R$ to have $R[x]$ Noetherian? Every reference I've seen for the Hilbert basis theorem only states the direction "$R$ Noetherian $\Rightarrow$ $R[x]$ Noetherian", which would certainly seem to imply that the converse is false. Unfortunately, it's tough to think ... | Pete L. Clark | 1,149 | <p>Dear Zev,</p>
<p>There are some sources which give the converse. See e.g. pp. 64-65 of</p>
<p><a href="http://alpha.math.uga.edu/%7Epete/integral.pdf" rel="nofollow noreferrer">http://alpha.math.uga.edu/~pete/integral.pdf</a></p>
<p>For that matter, see also pp. 32-33 of <em>loc. cit.</em> for the Chinese Remainder... |
2,099,828 | <p>Can anyone show me that both $\cos t$ and $\sin t$ are eigen signals. Here is a little bit background of eigen-function. </p>
<blockquote>
<p>The output of a continuous-time, linear time-invariant system is
denoted by $T\{z(t)\}$ where $x(t)$ is the input signal. A signal
$z(t)$ is called eigen-signal of the ... | jnez71 | 295,791 | <p>The output of the LTI system will be the convolution of its impulse response with the input (<a href="https://en.wikipedia.org/wiki/Linear_time-invariant_theory#Impulse_response_and_convolution" rel="nofollow noreferrer">see</a>). Since the impulse response of a Lyapunov stable LTI system is a finite sum of complex ... |
1,039,141 | <blockquote>
<p>Let <span class="math-container">$X = \mathbb{R}$</span> and <span class="math-container">$Y = \{x \in \mathbb{R} :x ≥ 1\}$</span>, and define <span class="math-container">$G : X → Y$</span> by <span class="math-container">$$G(x) = e^{x^2}.$$</span>
Prove that <span class="math-container">$G$</span> is ... | Paul | 17,980 | <p>For any $y\in Y$, there is $x= \sqrt{\ln y} \in X$ such that $G(x)=y$.</p>
|
1,039,141 | <blockquote>
<p>Let <span class="math-container">$X = \mathbb{R}$</span> and <span class="math-container">$Y = \{x \in \mathbb{R} :x ≥ 1\}$</span>, and define <span class="math-container">$G : X → Y$</span> by <span class="math-container">$$G(x) = e^{x^2}.$$</span>
Prove that <span class="math-container">$G$</span> is ... | Graham Kemp | 135,106 | <p>A function is surjective (onto) iff every element in the codomain is mapped to by at least one element in the domain.</p>
<p>So to determine if: $\forall y\in [1, \infty), \exists x\in \mathbb R: y=e^{x^2}$, we ask, do the roots $x=\pm \sqrt[2]{\ln y}$ have a real value for all $y\in [1,\infty)$?</p>
<p>Alternativ... |
2,512,461 | <blockquote>
<p>For any non-zero vector $x$,
$$
\lVert x\rVert_0 \geq \frac{\lVert x\rVert_1^2}{\lVert x\rVert_2^2}
$$</p>
</blockquote>
<p>I am trying to prove this inequality using the definitions of the $\ell_0$ "norm" (the number of none zero elements in the vector) and the definitions of the $\ell_1$ and $\e... | Clement C. | 75,808 | <p>This is a direct consequence of Cauchy—Schwarz: writing $\mathbf{1}_{\{a > 0\}}$ for the indicator of $\{a>0\}$,</p>
<p>$$
\mathbf{1}_{\{a > 0\}} = \begin{cases} 1 &\text{ if } a>0\\0&\text{ otherwise}\end{cases}
$$
we have
$$\begin{align}
\lVert x\rVert_1 = \sum_{i} \lvert x_i\rvert = \sum_{i} ... |
167,326 | <p>Let $ S = \sum_{i=1}^n X_i$ where:</p>
<ul>
<li>Each $X_i$ is independently 3 or 9 (with equal probability), and</li>
<li>The sample size $n$ is itself an independent random variable where $N \sim \text{NegativeBinomial}(r,p)$ e.g. $r = 5$ and $p = \frac34$</li>
</ul>
<p>Let $W = \begin{cases}S-10 & S > 1... | Henrik Schumacher | 38,178 | <p>Simulating random samples is straight-forward:</p>
<pre><code>r = 5;
p = 0.75;
m = 10000000;
Nlist = RandomVariate[NegativeBinomialDistribution[r, p], m];
W = Ramp[Subtract[Total[RandomChoice[{3, 9}, #] & /@ Nlist, {2}], 10]];
</code></pre>
<p>You <em>could</em> obtain the empiric probability density function ... |
112,651 | <p>What is known about the set of well orderings of $\aleph_0$ in set theory without choice? I do not mean the set of countable well-order types, but the set of all subsets of $\aleph_0$ which (relative to a pairing function) code well orderings. And I would be interested in an answer in, say, ZF without choice. My ... | François G. Dorais | 2,000 | <p>This is an aside that I mentioned elsewhere long ago but deserves mention here since it homes in on the counterintuition that probably led Colin to doubt the answer.</p>
<p>As Colin pointed out, every $R \subset \omega$ can be interpreted as a binary relation on $\omega$ through a pairing function. This leads to a ... |
3,125,263 | <p>I can't solve the last exercises in a worksheet of Pre-Calculus problems. It says:</p>
<p>Quadratic function <span class="math-container">$f(x)=ax^2+bx+c$</span> determines a parabola that passes through points <span class="math-container">$(0, 2)$</span> and <span class="math-container">$(4, 2)$</span>, and its ve... | J. W. Tanner | 615,567 | <p>Since <span class="math-container">$f(0)=c$</span> and we are given <span class="math-container">$f(0)=2$</span>, we see immediately that <span class="math-container">$c=2.$</span></p>
<p>Furthermore, the equation in vertex form is <span class="math-container">$f(x)=a(x-x_v)^2+k$</span>, </p>
<p>and since we are g... |
3,608,441 | <p>It is well known that we can define <span class="math-container">$e^x$</span> by the following limit</p>
<p><span class="math-container">$$e^{x}=\lim_{n\to\infty}\left(1+{x\over n}\right)^n$$</span></p>
<p>I would like to show that the RHS sequence is always less than or equal to <span class="math-container">$e^x$... | Paramanand Singh | 72,031 | <p>Let <span class="math-container">$x=-y$</span> so that <span class="math-container">$0\leq y\leq n$</span>. If <span class="math-container">$a_n$</span> is the sequence in question then consider <span class="math-container">$$b_n=\frac{1}{a_n}=\left(1-\frac{y}{n}\right)^{-n}=1+y+\frac{1/n+1}{2!}y^2+\frac {(1/n+1)(1/... |
3,717,932 | <p>How can this identity convolution be shown?</p>
<p><span class="math-container">$$\int^\infty_{-\infty} f(\tau)\delta(t-\tau)d\tau=f(t)$$</span></p>
<p>I keep getting stuck in traps when trying to show this and need a bit of assistance</p>
| md2perpe | 168,433 | <p><span class="math-container">$$
\int_{-\infty}^{\infty} f(\tau) \, \delta(t-\tau) \, d\tau
= \{ \tau = t-\sigma \}
= \int_{\infty}^{-\infty} f(t-\sigma) \, \delta(\sigma) \, (-d\sigma) \\
= \int_{-\infty}^{\infty} f(t-\sigma) \, \delta(\sigma) \, d\sigma
= f(t-0)
= f(t).
$$</span></p>
|
40,348 | <p>I'm trying to prove the following statement (an exercise in Bourbaki's <em>Set Theory</em>): </p>
<p><em>If $E$ is an infinite set, the set of subsets of $E$ which are equipotent to $E$ is equipotent to $\mathfrak{P}(E)$.</em> </p>
<p>As a hint, there is a reference to a proposition of the book, which reads: </p>
... | JDH | 413 | <p>Using the axiom of choice, every infinite set $X$ can be divided into two disjoint sets $X_0\sqcup X_1$, both of which are equinumerous with $X$. (Just well-order $X$, and take every other point in the enumeration.) </p>
<p>Now, consider all sets of the form $X_0\cup A$ for any $A\subset X_1$. There are $2^X$ many ... |
875,644 | <p>I have a parabolic basin which i am trying to find the equation for so I can reproduce it. I have taken $3$ points along one line of it to find the equation of the parabola, and I'm wondering if there is a way I can go from this to the equation of the parabolic basin.
The equation I have for the parabola is:</p>
<... | Did | 6,179 | <p>$$m=y_1-y_2,\ t=y_1\implies g(y_1-y_2)\mathbf 1_{0\lt y_2\lt y_1\lt1}=g(m)\mathbf 1_{0\lt m\lt t\lt1}$$</p>
|
3,123,857 | <p>I have to find the integral of
<span class="math-container">$$\int_{M_0}^{\infty} q(m, \mu, \sigma) \beta e^{-\beta(m-M_0)}\,\mathrm{d}m,$$</span>
where <span class="math-container">$q(m, \mu, \sigma)$</span> is the normal cumulative distribution function, <span class="math-container">$M_0$</span> is a constant, <s... | Stan Tendijck | 526,717 | <p>I will present to you the solution in a sort of general step-by-step method. I hope you find it useful.</p>
<p>First of all, you have to use Partial Integration to get rid of the normal CDF function in the integral. Can you give it a go? If you have some difficulties, just ask me.</p>
<p>If you have done this corr... |
650,710 | <p>How would I go about simplifying $4(a-2(b-c)-(a-(b-2)))$. Show working out and steps please.</p>
<p>I'd show my working out but I'm not really sure where to start. Firstly, I would want to get rid of the 4 so I'd times everything else by 4 right? No idea. </p>
| Vibhs | 123,765 | <p>4(a−2(b−c)−(a−(b−2)))
=4(a-2b+2c-(a-b+2))
=4(a-2b+2c-a+b-2)
=4(2c-b-2)
=8c-4b-8</p>
|
2,238,734 | <p>Let G be a group of rationals under addition, if $G_1$ and $G_2$ are two non empty subgroups of G, then prove that $G_1 \cap G_2 \neq${0}</p>
| Salvatore Baldino | 374,244 | <p>Well, the second definition is not so useful as $\mathbb R^n$ is a vector space, so if you take any two vectors in a subset of $\mathbb R^n$ it is guaranteed that their sum will belong to $\mathbb R^n$ (analogously for any product by a scalar).</p>
<p>It would be useful to see what does the symbol $<\cdot>$ m... |
1,983,745 | <p>I need to choose weather this is a product notation or a summation. I can figure out which one it is.</p>
<p>I have this expression:</p>
<p>$$2 \times 4 \times 6 \times 8 \times 10 \ldots \times 40$$</p>
<p>The answer is either:</p>
<blockquote>
<p>$$\sum_{m=2}^{40} m$$</p>
</blockquote>
<p><strong>or</strong... | user2825632 | 250,232 | <p>You are multiplying values, so you should probably use the product notation:</p>
<p>$$\prod_{m=1}^{20}2m = 2 \times 4 \times 6 ... \times 40$$</p>
<p>When you have something like $\prod_{m=2}^{40}m$ as in your example, this actually represents the product $2 \times 3 \times 4 ... \times 39 \times 40$ - it includes... |
1,983,745 | <p>I need to choose weather this is a product notation or a summation. I can figure out which one it is.</p>
<p>I have this expression:</p>
<p>$$2 \times 4 \times 6 \times 8 \times 10 \ldots \times 40$$</p>
<p>The answer is either:</p>
<blockquote>
<p>$$\sum_{m=2}^{40} m$$</p>
</blockquote>
<p><strong>or</strong... | Hypergeometricx | 168,053 | <p>Can also be expressed as
$$2^{20} 20!$$</p>
|
3,197,540 | <p>Let a function be defined as:</p>
<p><span class="math-container">$ f(x)=x^2\sin{\left(\frac 1x\right)}$</span> for <span class="math-container">$x \neq 0$</span> and
<span class="math-container">$ f(x)=0$</span> for <span class="math-container">$x=0$</span></p>
<p>I'm trying to prove that f is differentiable at ... | DINEDINE | 506,164 | <p>Hint:</p>
<p><span class="math-container">$$\left|\frac{x^2\sin\left(\frac1x\right)-0}{x-0}\right|\le x$$</span></p>
|
3,027,528 | <p>I am trying to resolve an exercise and there are 2 point that are missing in order to finalize:</p>
<p>Suppose <span class="math-container">$A$</span>, <span class="math-container">$B$</span>, <span class="math-container">$C$</span>, and <span class="math-container">$P$</span> are <span class="math-container">$R$</... | zipirovich | 127,842 | <p>1) No, not necessarily. Here's a counterexample. Let <span class="math-container">$B=\mathbb{Z}$</span> and <span class="math-container">$C=P=\mathbb{Q}$</span> as <span class="math-container">$R=\mathbb{Z}$</span>-modules. Further, let <span class="math-container">$g:B\to C$</span>, i.e. <span class="math-container... |
111,425 | <p>If $R$ is a unital integral ring, then its characteristic is either $0$ or prime. If $R$ is a ring without unit, then the char of $R$ is defined to be the smallest positive integer $p$ s.t. $ pa = 0 $ for some nonzero element $a \in R$. I am not sure how to prove that the characteristic of an integral domain without... | Patrick Da Silva | 10,704 | <p>You don't need to invoke units. As your proof stated, if we assume $(nm)a = 0$ for some $a \in R$ non-zero, then $n(ma) = 0$, and since $nm$ is the <em>least</em> integer with the property that $m(na) = 0 = n(ma)$, then $na \neq 0 \neq ma$. Since
$$
0 = 0a = ((nm)a)a = (nm)a^2 = (na)(ma) \neq 0,
$$
we have a cont... |
656,531 | <p>The definition of a partial derivative is the "derivative of a multi-variable function relative to a single variable when all other variables are held constant".</p>
<p>But isn't the regular derivative (for one-variable functions) just a trivial case of this, where there are no other variables to hold constant? Why... | Lost | 100,183 | <p>Because there exists a notion of "total derivative" that is the multivariate analogue of the 1-D derivative you're familiar.
The total derivative of a function $f: \mathbb{R}^n \to \mathbb{R}^m$ is known as the Jacobian of f. You may have worked with this while doing change of variables with multiple integrals. See... |
656,531 | <p>The definition of a partial derivative is the "derivative of a multi-variable function relative to a single variable when all other variables are held constant".</p>
<p>But isn't the regular derivative (for one-variable functions) just a trivial case of this, where there are no other variables to hold constant? Why... | Martín-Blas Pérez Pinilla | 98,199 | <p>The total derivative/differential of Frechet (see the Lost answer) is the good generalization for several reasons. Two rather big:</p>
<p>(1) existence of partial derivatives does not imply continuity;</p>
<p>(2) the failure of the chain rule for partial derivatives.</p>
|
2,425,157 | <p>How do I show that
$$ \frac 12 \left(\frac 1 {3^2}+\frac 1{4^2}+ \frac 1{5^2}+\dots\right) < \frac 1 {3^2} + \frac 1{5^2} + \frac1{7^2} +\dots \quad ?$$</p>
| Robert Z | 299,698 | <p>After moving the odd terms from the LHS to the RHS, we obtain the following equivalent inequality,
$$\frac 12 \left(\frac 1{4^2}+ \frac 1{6^2}+ \frac 1{8^2}+\dots\right) < \left(1-\frac 12\right)\left( \frac 1 {3^2} + \frac 1{5^2} + \frac1{7^2} +\dots\right).$$
Then note that for all positive integer $n$, each te... |
2,439,340 | <p>How would one proceed to prove this statement?</p>
<blockquote>
<p>The set of the strictly increasing sequences of natural numbers is not enumerable.</p>
</blockquote>
<p>I've been trying to solve this for quite a while, however I don't even know where to start.</p>
| CiaPan | 152,299 | <p>Map any strictly increasing sequence $(a_n)$ to a sequence $(b_n)$ of its increments modulo $2$:
$$\{0,1\}\ni b_n \equiv a_{n+1}-a_n \pmod 2$$
This is, of course, not bijective or even injective, but it is surjective mapping, hence the cardinality of the set of $a$ sequences is not less than that of $b$ sequences.</... |
2,439,340 | <p>How would one proceed to prove this statement?</p>
<blockquote>
<p>The set of the strictly increasing sequences of natural numbers is not enumerable.</p>
</blockquote>
<p>I've been trying to solve this for quite a while, however I don't even know where to start.</p>
| orangeskid | 168,051 | <p>For $\alpha > 1$ consider the strictly increasing sequence $f_{\alpha}(n)= [n \alpha]$</p>
<p>The map $\alpha \mapsto f_{\alpha}(\cdot)$ is injective, since $\lim_{n\to \infty} \frac{[n\alpha]}{n} = \alpha$.</p>
|
3,231,387 | <p>I have been given the following quadratic equation and is asked to find the range of its roots <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span>, where <span class="math-container">$\alpha>\beta$</span>
<span class="math-container">$$(k+1)x^2 - (20k+14)x + 91k +40 =0,$$<... | Maverick | 171,392 | <p>Your equation can be re-written as <span class="math-container">$$f(x)=(x-4)(x-10)+k(x-7)(x-13)$$</span>
So it can be seen that</p>
<p><span class="math-container">$f(4)=27k$</span></p>
<p><span class="math-container">$f(7)=-9$</span></p>
<p><span class="math-container">$f(10)=-9k$</span></p>
<p><span class="mat... |
3,655,545 | <p>What is the asymptotic behaviour of the difference
<span class="math-container">$$
c_j - c_{j+1}
$$</span>
for <span class="math-container">$j\rightarrow \infty$</span> if <span class="math-container">$(c_j)_{j\in\mathbb{N}}$</span> is a null sequence?</p>
| Aladin | 707,258 | <p>Yes, because the limits exist for <span class="math-container">$\lim_{n \to \infty}c_n = 0$</span> and so <span class="math-container">$\lim_{n \to \infty}c_{n+1} = 0$</span> then <span class="math-container">$\lim_{n \to \infty}c_n - c_{n+1} = \lim_{n \to \infty}c_n - \lim_{n \to \infty}c_{n+1}= 0-0=0$</span>. </p>... |
10,873 | <p>We have two tags with identical names on main and meta:</p>
<p><a href="https://math.stackexchange.com/questions/tagged/computer-science">(main:computer-science)</a>
and
<a href="https://math.meta.stackexchange.com/questions/tagged/computer-science">(meta:computer-science)</a></p>
<p>For main tags one can use <co... | Community | -1 | <p>You can address meta tags like <code>[meta-tag:computer-science]</code>: <a href="/questions/tagged/computer-science" class="post-tag" title="show questions tagged 'computer-science'" rel="tag">computer-science</a></p>
|
3,950,098 | <p>I can evaluate the limit with L'Hospital's rule:</p>
<p><span class="math-container">$\lim_{n\to\infty}n(\sqrt[n]{4}-1)=\lim_{n\to\infty}\cfrac{(4^{\frac1n}-1)}{\dfrac1n}=\lim_{n\to\infty}\cfrac{\dfrac{-1}{n^2}\times 4^{\frac1n}\times\ln4}{\dfrac{-1}{n^2}}=\ln4$</span></p>
<p>But is there any way to do it without us... | Dark Malthorp | 532,432 | <p>Here's a trick to prove convergence of the continuous limit<span class="math-container">$$
\lim\limits_{x\rightarrow\infty} x \left(\sqrt[x]4 - 1\right)
$$</span>
if you know also know how to integrate <span class="math-container">$2^x$</span>.
Observe:<span class="math-container">\begin{eqnarray}
x \left(\sqrt[x]4 ... |
2,995,643 | <p>Here is a thought experiment I have. </p>
<p>Say we flip a unique coin where we have a 99.99999999999% chance of it landing on heads, and a .000000000001% chance of it landing on tails (the two possibilities equal to 100%).</p>
<p>And say we have an <em>infinite</em> number of coins flipped all at once (and only o... | Théophile | 26,091 | <p>The probability of getting only heads on <span class="math-container">$n$</span> flips is <span class="math-container">$0.9999999999999^n$</span>. While this is close to <span class="math-container">$1$</span> for small <span class="math-container">$n$</span>, eventually it decreases to <span class="math-container">... |
3,182,532 | <p>I am confused about converting a <strong>Probability Density Function</strong> from <strong>Polar coordinates</strong> to <strong>Cartesian coordinates</strong>. </p>
<p>Here is an example:</p>
<p>In Polar coordinates, we can have a <strong>Gaussian probability function</strong>:</p>
<p><strong><span class="math-... | mjw | 655,367 | <p><strong>UPDATE</strong></p>
<p>If your function in polar coordinates is a circularly symmetric Gaussian centered at the origin,
then it could be written
<span class="math-container">$P_{r\, \theta}(r,\theta)=A\,r\,e^{-r^2/2\sigma^2}$</span> and you can obtain <span class="math-container">$A$</span> from </p>
<p><... |
366,687 | <p>I am interested in the status of the conjecture about the minimum number of edge crossings <span class="math-container">$cr(K_{m,n})$</span> in a drawing of the complete bipartite graph <span class="math-container">$K_{m,n}$</span>.</p>
<p>The Wikipedia article <a href="https://en.wikipedia.org/wiki/Tur%C3%A1n%27s_b... | John Machacek | 51,668 | <p>The Electronic Journal of Combinatorics has many <a href="https://www.combinatorics.org/ojs/index.php/eljc/issue/view/Surveys" rel="noreferrer">Dynamic Surveys</a> one of which is <a href="https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS21/pdf" rel="noreferrer">The Graph Crossing Number and its Varia... |
3,715,522 | <p>I am trying to understand fully how drug half-life works. So I derived this relationship: </p>
<p><span class="math-container">$$\ U_{r} = \frac{1+\ U_{r-1}}{2}$$</span>
Where <span class="math-container">$\ U_{0}=0$</span> and r is a set of natural numbers.</p>
<p>My issue to how to deduce a relationship for the... | quasi | 400,434 | <p>Suppose <span class="math-container">$u_0,u_1,u_2,...$</span> are defined by
<span class="math-container">$$
\left\lbrace
\begin{align*}
u_0\!&=\,0\\[4pt]
u_{n+1}\!&=\frac{a+u_n}{2}\\[4pt]
\end{align*}
\right.
$$</span>
for some constant <span class="math-container">$a > 0$</span>.
<p>
Then we have
<span ... |
2,202,529 | <p>I have difficulties thinking the relationship between inverse of a number and gcd.</p>
<p>If I want to know if a specific <code>number module n</code> has an inverse I check if gcd between the number and the module is 1, why?</p>
<pre><code>a≡b(n) has inverse only if gcd(a,n)=1
</code></pre>
<p>I know that the r... | edgar alonso | 329,621 | <p>It does not implies induction. One of your hypotheses is that $f(X)\subseteq X$, and when proceeding by induction you want to prove exactly this for $f(n)=n+1$, so you can deduce $X=\mathbb{N}$.</p>
|
2,202,529 | <p>I have difficulties thinking the relationship between inverse of a number and gcd.</p>
<p>If I want to know if a specific <code>number module n</code> has an inverse I check if gcd between the number and the module is 1, why?</p>
<pre><code>a≡b(n) has inverse only if gcd(a,n)=1
</code></pre>
<p>I know that the r... | EuYu | 9,246 | <p>Like Hayden pointed out in the comments, your theorem doesn't really have much to do with mathematical induction. Notice that the first hypothesis of your theorem says that $Y\backslash X \subseteq f(X)$. So the two hypotheses of your theorem combined says that
$$Y\backslash X \subseteq f(X) \subseteq X.$$
Your the... |
1,700,246 | <p>Let $F=\mathbb{F}_{q}$, where $q$ is an odd prime power. Let $e,f,d$ be a standard basis for the $3$-dimensional orthogonal space $V$, i.e. $(e,e)=(f,f)=(e,d)=(f,d)$ and $(e,f)=(d,d)=1$. I have an element $g\in SO_{3}(q)$ defined by: $g: e\mapsto -e$, $f\mapsto \frac{1}{2}e -f +d$, $d\mapsto e+d$. I would like to de... | mvw | 86,776 | <p>This would mean
$$
X = \sqrt{x} < r < Y = \sqrt{y}
$$
and $X, Y \in \mathbb{R}$ with $X \ne Y$. If I remember right, there is always at least one rational number between two different real numbers. (<a href="https://proofwiki.org/wiki/Between_two_Real_Numbers_exists_Rational_Number" rel="nofollow">Link</a>)</p... |
3,380,081 | <p>Question: Suppose <span class="math-container">$n(S)$</span> is the number of subset of <span class="math-container">$S$</span> and <span class="math-container">$|S|$</span> be the number of elements of <span class="math-container">$S$</span>. If <span class="math-container">$n(A)+n(B)+n(C)=n(A\cup B\cup C)$</span> ... | Martund | 609,343 | <p>You can actually <strong>exactly</strong> evaluate the area, without integration. Observe that required area is
<span class="math-container">$$A=A_1-A_2-A_3$$</span>
where <span class="math-container">$A_1$</span> is the area of the rectangle having vertices <span class="math-container">$(0,0), (0,2), (4,0), (4,2)$... |
165,853 | <blockquote>
<p>Schauder's conjecture: "<em>Every continuous function, from a nonempty
compact and convex set in a (Hausdorff) topological vector space into
itself, has a fixed point.</em>" [Problem 54 in The Scottish Book]</p>
</blockquote>
<p>I wonder whether this conjecture is resolved. I know R. Cauty [So... | Jochen Wengenroth | 21,051 | <p>In <em>Points fixes des applications compactes dans les espaces ULC</em> published in in the <a href="https://arxiv.org/abs/1010.2401" rel="nofollow noreferrer">arXiv</a> in 2010 Robert Cauty wrote</p>
<p><em>il y a d’ailleurs une erreur dans la demonstration du
lemme 3 de [2], qu’il n’y a plus de raison de corriger... |
3,936,187 | <blockquote>
<p>Consider the differential equation <span class="math-container">$$(1+t)y''+2y=0$$</span>
with the variabel coefficient <span class="math-container">$(1+t)$</span>, with <span class="math-container">$t\in \mathbb{R}$</span>.</p>
<p>Set <span class="math-container">$y(t)=\sum_{n=0}^{\infty}a_nt^n$</span>.... | Alann Rosas | 743,337 | <p>If</p>
<p><span class="math-container">$$y(t)=\sum_{n=0}^{\infty}c_n t^n$$</span></p>
<p>then</p>
<p><span class="math-container">$$y''(t)=\sum_{n=2}^{\infty}n(n-1)c_n t^{n-2}$$</span></p>
<p>which is equivalent to</p>
<p><span class="math-container">$$y''(t)=\sum_{n=0}^{\infty}(n+1)(n+2)c_{n+2}t^n$$</span></p>
<p>s... |
1,641,137 | <p>Let $(X,d)$ be a metric space, $a \in X$, and $\delta$ be a positive real number. Then the open ball $B(a;\delta)$ is defined as
$$B(a;\delta) \colon= \left\{ \ x \in X \ \colon \ d(x,a) < \delta \ \right\},$$
whereas the sphere $S(a; \delta)$ is defined as
$$S(a;\delta) \colon= \left\{ \ x \in X \ \colon \ d(x... | Thomas Andrews | 7,933 | <p>This is a global condition - that is, it is both necessary and sufficient to have your condition be true for all $x,\delta$.</p>
<p>You need:</p>
<blockquote>
<p>(Condition 1): Given any $x\neq y$ and any $\epsilon>0$ that there is some $z$ so that $d(y,z)<\epsilon$ and$d(x,z)<d(x,y)$.</p>
</blockquote... |
670,292 | <p>Could someone assist with the following three surface integrals? </p>
<p><strong>Q1</strong> The portion of the cone $z=\sqrt{x^2+y^2}$ that lies inside the cylinder $x^2+y^2 =2x$. </p>
<p><strong>Q2</strong> The portion of the paraboloid $z=1-x^2-y^2$ that lies above the $xy$-plane.</p>
<p><strong>Q3</strong... | Robert Israel | 8,508 | <p>Hint: if $b_n = a_n/(1 + a_n)$, then $a_n = b_n/(1-b_n)$.</p>
<p>By the way, the condition that $a_n$ is bounded is not needed here (it would be needed if you weren't told $\ell \ne 1$).</p>
|
1,762,001 | <p>I recently watched a <a href="https://www.youtube.com/watch?v=SrU9YDoXE88" rel="noreferrer">video about different infinities</a>. That there is $\aleph_0$, then $\omega, \omega+1, \ldots 2\omega, \ldots, \omega^2, \ldots, \omega^\omega, \varepsilon_0, \aleph_1, \omega_1, \ldots, \omega_\omega$, etc..</p>
<p>I can't... | Christian Gaetz | 75,296 | <p>I won't comment on your more philosophical questions, but I will give what I think is one of the more important applications of different sizes of infinity.</p>
<p>There is a rigorous mathematical way of thinking about a computer program, called a Turing machine. One can show that the cardinality of the set of Tur... |
3,568,230 | <p>My question is: why, in general we cannot write down an formula for the <span class="math-container">$n-$</span>th term, <span class="math-container">$S_{n}$</span>, of the sequence of partial sums?</p>
<p>I will explain better in the following but the question is basically that one above.</p>
<p>Suppose then you ... | José Carlos Santos | 446,262 | <p>Yes, you are right: in general we cannot write down a formula for the <span class="math-container">$n$</span><sup>th</sup> partial sum of a sequence. As there is no simple closed expression for most primitives, such as <span class="math-container">$\int\frac1{\log(x)}\,\mathrm dx$</span>, <span class="math-container... |
3,568,230 | <p>My question is: why, in general we cannot write down an formula for the <span class="math-container">$n-$</span>th term, <span class="math-container">$S_{n}$</span>, of the sequence of partial sums?</p>
<p>I will explain better in the following but the question is basically that one above.</p>
<p>Suppose then you ... | johnnyb | 298,360 | <p>So, the answer depends on what you are willing to consider as a solution. If I understand your question, you are asking why, if it is rational to think of a particular series, then it is also rational to think of its partial sums, and there should be no reason we cannot deduce a general formula for the partial sum ... |
180,169 | <p>Can anyone give me suggestions for new books about Besicovitch's almost periodic functions? Thanks a lot. </p>
| Vladimir S Matveev | 14,515 | <p>I add a small $\varepsilon$ to Robert's answer, which is a simple explanation and a simple example to
what he said concerning the 2 dim case. Conformal structure of signature (1,1) on the
surface is essentially the same as a pair of everywhere transversal<br>
foliations. Well, up to a double cover, to be precise... |
3,711,744 | <p>Let <span class="math-container">$C_1\geq C_2\geq\dots\geq C_n$</span> be a fixed set of positive numbers. Maximize the linear function <span class="math-container">$L(x_1, x_2, \dots, x_n)=\sum^n_1C_jx_j$</span> in the closed set described by the inequalities <span class="math-container">$0\leq x_j\leq 1, \sum^n_1 ... | twosigma | 780,083 | <p>Here is a way to get two such maps if we are given a basis of a vector space <span class="math-container">$V$</span>.</p>
<p>Let <span class="math-container">$v_1, …, v_k, v$</span> be a basis of <span class="math-container">$V$</span>. Then observe that <span class="math-container">$v_1, …, v_k, v + v_1$</span> is... |
1,102,709 | <p>I'd like to know if there is an explicit atlas for the manifold $\mathbb{R}P^3$ which is defined as the quotient of the three-sphere by the antipodal mapping.</p>
<p>Thanks.</p>
| Per Erik Manne | 33,572 | <p>There is a version of Weierstrass' approximation theorem, due to Torsten Carleman (1927), which is valid for the interval $(-\infty,\infty)$, but it requires you to replace the approximating polynomial by a convergent power series:</p>
<p>If $\epsilon : \bf R\rm \to (0,\infty)$ is a positive, continuous function (f... |
2,973,314 | <p>If we have to find the sum of n terms of a G.P. then we have two formulas for it (1) <span class="math-container">$a(1-r^n)/(1-r)$</span> and (2) <span class="math-container">$a(r^n-1)/(r-1)$</span>. Now I know how the (1) has been derived but dont know about the (2)(is it obtained by multiplying denominator and num... | Sam Streeter | 487,113 | <p>As you have mentioned in your post and as Dr. Sonnhard Graubner has mentioned in his answer, you can get one expression from the other by multiplying by <span class="math-container">$-1$</span> on the numerator and denominator. Which one you use is just a matter of preference. In particular, when working with geomet... |
360,608 | <p>In physics I came across these kind of equations when I am trying to find the asymptotic behaviour of some function.</p>
<p>Can anyone explain if there is any sense in talking about $\sin(x)$ or $\cos(x)$ as $x$ tends to infinity?</p>
<p>$$\lim_{x\rightarrow\infty}\;\sin(x)?$$</p>
| Community | -1 | <p>If we take $x_n=2\pi n$ and $x'_n=2\pi n+\frac{\pi}{2}$ then we have
$$\lim_{n\to\infty}x_n=\lim_{n\to\infty}x'_n=+\infty$$
but
$$\lim_{n\to\infty}\sin(x_n)=0\neq1=\lim_{n\to\infty}\sin(x'_n)$$
hence $\displaystyle\lim_{x\to\infty}\sin x$ does not exist.</p>
|
149,790 | <p>I know that if $x$ is a rational multiple of $\pi$, then $tan(x)$ is <a href="http://divisbyzero.com/2010/10/28/trigonometric-functions-and-rational-multiples-of-pi/">algebraic</a>.</p>
<p>Is there a fairly simple way to express $x$ as $\pi\ m/n$, if $tan(x)$ is given as a square root of a rational?</p>
| ile | 43,192 | <p>If $x \in \mathbb{Q}$ and $tan^2(x\pi) \in \mathbb{Q}$, then $tan(x\pi) \in \{0, \pm\sqrt{3}, \pm\frac{1}{\sqrt{3}}, \pm 1 \}$.</p>
<p>Chapter 11 of the <em><a href="http://books.google.com/books?id=emE6SLT3CLsC&dq=Angles+Whose+Squared+Trigonometric+Functions+Are+Rational&source=gbs_navlinks_s" rel="nofollo... |
282,050 | <p>I have equation $y = -x^2 + 2x + 7$.
How can I change it to canonical form, which looks like $y^2 = 2px$ ?
($p$ will be parameter)</p>
<p>What i ve tried so far:
$$\begin{align}
y &= -x^2 + 2x + 7\\
y &= -(x^2 - 2x + 1) + 8\\
(y-8) &= -(x-1)^2 \\
(y-8)^2 &= 2*(0.5)*(x-1)^4
\end{align}
$$</p>
<p>... | Adi Dani | 12,848 | <p>$$y = -x^2 + 2x + 7 $$
$$y = -(x^2 - 2x +1)+8 $$
$$y = -(x- 1)^2+8 $$
$$(x- 1)^2=-(y-8) $$
$$(x- 1)^2=2(-\frac{1}{2})(y-8)\Rightarrow p=-\frac{1}{2},x-1=Y,y-8=X $$
$$Y^2=2pX$$</p>
|
1,278,848 | <p>Based on <a href="https://math.stackexchange.com/questions/1267021/let-m-subseteq-mathbbrk-manifold-topology-vs-trace-topology/1267760?noredirect=1#comment2573732_1267760">this</a> question I'd like to know: Are there compact (sub)manifolds without boundary in $\mathbb{R}^n$ ? Because, as that question shows, the to... | RobertCRH | 395,336 | <p>A compact manifold without a boundary is called a <a href="https://en.wikipedia.org/wiki/Closed_manifold" rel="nofollow noreferrer">closed manifold</a> so it is certainly an important class of manifolds.</p>
<p>Example: A $n-1$sphere ${\cal S}^{n-1}$ is a closed manifold, but the unit ball enclosed $\{\mathbf{x}\in... |
3,746,630 | <p>So I am solving some probability/finance books and I've gone through two similar problems that conflict in their answers.</p>
<h2>Paul Wilmott</h2>
<p>The first book is Paul Wilmott's <a href="https://smile.amazon.com/Frequently-Asked-Questions-Quantitative-Finance/dp/0470748753" rel="nofollow noreferrer">Frequently... | T_M | 562,248 | <p>Joshi's problem is a much easier problem and he is correct. Wilmott's problem is a little bit more subtle, and I think he is misleading about what he is computing. The main point is that returns are not additive, so the trap is to compute expectation of the return on a given day and then "add it up" to con... |
1,884,852 | <p>Suppose there are <em>k</em> dice thrown. Let <em>M</em> denote the minimum of the <em>k</em> numbers rolled. </p>
<p>I've learned that finding the individual probability is:</p>
<p>$$P(M = m) = P(M \ge m) - P(M \ge m + 1) $$</p>
<p>Can someone please explain this to me? I've tried plugging in values for $m = 1, ... | André Nicolas | 6,312 | <p>How can the minimum be $\ge m$? There are two possibilities: (i) the minimum is exactly $m$ or (ii) the minimum is greater than $m$. </p>
<p>The minimum is greater than $m$ precisely if the minimum is $\ge m+1$.</p>
<p>The possibilities (i) and (ii) are disjoint, so $$\Pr(M\ge m)=\Pr(M=m)+\Pr(M\ge m+1).$$
From thi... |
4,283,707 | <p>When solving this problem I arrive to a cubic equation not very friendly, is there any algebraic shortcut?</p>
<p><span class="math-container">$$W=\frac{3+\left [ \sqrt[3]{4+\sqrt[3]{4+...}} \right ]^{2}}{1+\left [ \sqrt[3]{4+\sqrt[3]{4+...}} \right ]^{-1}}$$</span></p>
<p>to do this
<span class="math-container">$$P... | march | 852,914 | <p>Assuming that the nested cube-root converges, which I believe it does, let <span class="math-container">$x$</span> be the unique real root of <span class="math-container">$P^3-P-4$</span>. Then <span class="math-container">$x^3 = x+4$</span>, and therefore
<span class="math-container">$$
W=\frac{3+x^2}{1+x^{-1}}= \f... |
4,036,896 | <p>Let <span class="math-container">$\boldsymbol{A}$</span> is a real symmetric matrix, <span class="math-container">$\boldsymbol{B}$</span> is a real antisymmetric matrix, <span class="math-container">$\boldsymbol{A}^2 = \boldsymbol{B}^2$</span>, prove <span class="math-container">$\boldsymbol{A} = \boldsymbol{B} = \b... | TheSilverDoe | 594,484 | <p>Let <span class="math-container">$\mu$</span> be a complex eigenvalue of <span class="math-container">$B$</span>. Because <span class="math-container">$B$</span> is antisymmetric, then <span class="math-container">$\mu \in i\mathbb{R}$</span>, so <span class="math-container">$\mu^2$</span> is a nonpositive real numb... |
4,036,896 | <p>Let <span class="math-container">$\boldsymbol{A}$</span> is a real symmetric matrix, <span class="math-container">$\boldsymbol{B}$</span> is a real antisymmetric matrix, <span class="math-container">$\boldsymbol{A}^2 = \boldsymbol{B}^2$</span>, prove <span class="math-container">$\boldsymbol{A} = \boldsymbol{B} = \b... | Fred | 380,717 | <p>Let us denote the usual inner product on <span class="math-container">$ \mathbb R^n$</span> by <span class="math-container">$(\cdot|\cdot).$</span> Then</p>
<p><span class="math-container">$$||Ax||^2=(Ax|Ax)=(A^TAx|x)=(A^2x|x)=(B^2x|x)=(Bx|B^Tx)=(Bx|-Bx)=-(Bx|Bx)=-||Bx||^2$$</span></p>
<p>for all <span class="math-c... |
297,812 | <p>If $a-b=b-c$ .How to find the value of $a^2-2b^2+c^2$</p>
| André Nicolas | 6,312 | <p>Note that $a-b=b-c$ is equivalent to $c=2b-a$.</p>
<p>Substitute for $c$ in $a^2-2b^2+c^2$. We get $a^2-2b^2+(2b-a)^2$.</p>
<p>Expand the square. We get $a^2-2b^2+(4b^2-4ab+a^2)$.</p>
<p>This simplifies to $2a^2-4ab+2b^2$, which simplifies to $2(a-b)^2$.</p>
<p>Further simplification is not possible, since we ha... |
1,029,868 | <p>Let $$
A=\begin{bmatrix}
1 & 1 & 2\\
1 & 2 & 1\\
2 & 1 & 1
\end{bmatrix}$$</p>
<p>Show that $ A^-=\dfrac{1}{4}(-A^2+4A+I)$</p>
<p>I have absolutely no clue how to do this. Could someone be kind enough to explain and provide and answer? I believe it has something to do with the Cayley-Hamili... | Martin Argerami | 22,857 | <p>The characteristic polynomial is $(\lambda-4)(\lambda-1)(\lambda+1)=\lambda^3-4\lambda^2-\lambda+4$. So, by Cayley-Hamilton,
$$
A^3-4A^2-A+4I=0.
$$
Then $$I=\frac14\,(A+4A^2-A^3),$$ and since $A$ is invertible (its determinant is $4\ne0$) we have, multiplying by $A^{-1}$,
$$
A^{-1}=\frac14\,(I+4A-A^2).
$$</p>
|
1,846,592 | <p>I know that a discrete topological space is where all singletons are open.</p>
<p>For example, $\mathbb{N}$ with the subspace topology inherited from $(\mathbb{R}, \mathfrak{T}_{usual})$. This is the case because we can find $\{n\} = (a,b) \cap \mathbb{N}$ which is open. Hence all singletons are open.</p>
<p>But a... | J.-E. Pin | 89,374 | <p><strong>Hint</strong>. Use the fact that open sets are closed under arbitrary union.</p>
|
2,010,255 | <p>While finding the Taylor Series of a function, <strong>when</strong> are you allowed to substitute? And <strong>why</strong>?</p>
<p>For example:</p>
<p>Around $x=0$ for $e^{2x}$ I apparently am allowed to substitute $u=2x$ and then use the known series for $e^u$. But for $e^{x+1}$ I am not allowed to substitute $... | hamam_Abdallah | 369,188 | <p>If $ f(x)=P_n(x)+x^n\epsilon(x)$ then</p>
<p>$f(u(x))=P(u(x))+(u(x))^n\epsilon(u(x))$</p>
<p>with $\lim_{x\to 0}\epsilon(x)=0$.</p>
<p>thus we need that</p>
<p>$\lim_{x\to 0}\epsilon(u(x))=0$</p>
<p>so, we must have</p>
<p>$$\lim_{x\to 0}u(x)=0$$</p>
|
85,052 | <p>A housemate of mine and I disagree on the following question: </p>
<p>Let's say that we play a game of yahtzee. Of the five dice you throw, two dice obtain the value 1, two other dice obtain the value 2, and one die shows you six dots on the top side. Given the fact that you haven't thrown a "full house" yet, you s... | JavaMan | 6,491 | <p>You are right. The easiest way to see this is to recognize the probability that you roll a $1$ or a $2$ as </p>
<p>$$
1 - Pr(\text{not rolling a }1 \text{ or } 2) = 1 - (4/6)^2 = 5/9.
$$</p>
|
121,909 | <p>I came across this question while studying primitive roots. I know it has something to do with the fact that if the order of $a$ is $m$ then for every $k \in \mathbb{Z}$, the order of $a^k$ is $m/(m,k)$. The question is as follows: </p>
<blockquote>
<p>Let $p$ be an odd prime. Prove that $a^2$ is never a primi... | Bill Dubuque | 242 | <p>Your hunch is correct: $\rm\:(m,k)>1\:\Rightarrow\:ord(a^k) = m/(m,k) < m = ord(a).\:$ Since $\rm\:a^k\:$ has smaller order than $\rm\:a,\:$ it doesn't have maximal order in $\rm\left<a\right>.\:$ Your problem is the special case $\rm\:2 = k\ |\ m$</p>
|
3,530,492 | <blockquote>
<p>Evaluate</p>
<p><span class="math-container">$$ \int_0^{e^{\pi}} |\cos\ (\ln x)|dx$$</span></p>
</blockquote>
<p><em>My ideas:</em> I substituted <span class="math-container">$u = \ln x$</span> and tried to evaluate</p>
<p><span class="math-container">$$\int_{-\infty}^\pi |\cos u|\ e^u du$$</sp... | Z Ahmed | 671,540 | <p><span class="math-container">$$I=\int_{0}^{e^{\pi}}|\cos (\ln x)| dx= -\int_{-\infty} ^{\pi} e^t~ |\cos t| dt$$</span>
<span class="math-container">$$\implies I=-\int_{-\infty}^{0} e^{t} |\cos t|~ dt-\int_{0}^{\pi/2} e^{t} \cos t ~dt+\int_{\pi/2}^{\pi} e^{t} \cos t~ dt$$</span>
<span class="math-container">$$\implie... |
1,736,376 | <p>Let $R$ be a ring and $I$ the set of non-invertible elements of $R$. </p>
<p>If $(I,+)$ is an additive subgroup of $(R,+)$, then show that $I$ is an ideal of $R$ and so $R$ is local. </p>
<p>$$$$ </p>
<p>I have done the following: </p>
<p>Since $(I,+)$ is an additive subgroup of $(R,+)$, we have that $\forall a,... | Alex M. | 164,025 | <p>I believe that somewhere in your textbook $R$ is assumed to be commutative. Otherwise, the product of a non-invertible element with an arbitrary element of the ring <a href="https://math.stackexchange.com/questions/627562/can-the-product-of-two-non-invertible-elements-in-a-ring-be-invertible">may turn out to be inve... |
1,237,528 | <p>$$ \displaystyle {\int_{0}^{z}} \sqrt {1 + \tan^2(\dfrac{\pi}{4} \dfrac{z}{H} )} dz $$</p>
<p>_</p>
<p>$$ gives $$ </p>
<p>_</p>
<p>$$ \dfrac{4H}{\pi} {\sinh^{-1}} ( {\tan \dfrac{\pi}{4} \dfrac{z}{H} } ) $$</p>
<p>Please advise solution</p>
<p>edit:- </p>
<p>I can get to </p>
<p>$$\dfrac{4H}{\pi} \displaysty... | tired | 101,233 | <p>1.) Define $\frac{\pi/4 }{H}=a $</p>
<p>2.) Substitute $a z'=\arctan(r),\quad dz'=\frac{dr}{a(1+r^2)}$. This gives</p>
<p>$$
I(a)=\frac{1}{a}\int_0^{\tan(a z)}\frac{\sqrt{1+r^2}}{1+r^2}=\frac{1}{a}\int_0^{\tan(a z)}\frac{1}{\sqrt{1+r^2}}
$$</p>
<p>Furthermore $\int\frac{1}{\sqrt{1+r^2}}=\text{arcsinh}(r)+C$</p>
... |
2,624,669 | <blockquote>
<p>Find global maxima and global minima of
$$f(x)=3(x-2)^{\frac{2}{3}}-(x-2)$$
over the interval $[0,20]$.</p>
</blockquote>
<p><strong>My input:</strong> Derivative vanishes at $x=10$ and left neighborhood gives positive derivative and right neighborhood gives negative derivative . Therefore $x=1... | Dr. Sonnhard Graubner | 175,066 | <p>you are correct, the Maximum will be attained for $x=10$ gives $$f(10)$$
the Minimum for $x=2$ this gives $$f(2)=0$$</p>
|
3,263,076 | <p>Let <span class="math-container">$\Gamma\subset PSL_2(\mathbb{R})$</span> be a cofinite Fuchsian group (e.g. a Fuchsian group with finite fundamental domain). Does <span class="math-container">$\Gamma$</span> necessarily contain a hyperbolic element? </p>
<p>At first, I tried to use the fact that <span class="math-... | Severin Schraven | 331,816 | <p>We show that a pretty ring has exactly one unit. Indeed, if <span class="math-container">$0 \neq u$</span> is not a unit and <span class="math-container">$e$</span> is a unit, then
<span class="math-container">$$ (u+e) + 0 = e + u $$</span>
tells us that <span class="math-container">$u+e$</span> is not a unit (other... |
3,130,877 | <p>Normal geometry concepts, such as parallel, angle, area, triangle, do they still apply in Mobius band?</p>
<p>If not, in which case will they fail to do so?</p>
<p>For example, what would three lines on a Mobius band form? A triangle if not parallel? or it might be totally something else?</p>
| Mark Fischler | 150,362 | <p>Away from the edges, the geometry of a Moebius band is locally Euclidean, so the usual concepts and theorems of geometry all apply. The issues involving edges are qualitatively no different than if you were trying to do geometry on a disk instead of a plane: non-parallel lines may still never meet because they fal... |
70,801 | <p>I am asked to find how many there are $k$-dimensional subspaces in vector space $V$ over $\mathbb F_p$, $\dim V = n$.</p>
<p>My attempt:
1) Let's find a total number of elements in $V$: assume that $\{v_1, v_2, \cdots, v_n\}$ is a basis in $V$. Then, for every $v \in V$ we can write down
$$ v = a_1 v_1 + a_2 v_2 + ... | Marc van Leeuwen | 18,880 | <p>Just for the record, the number asked for here is the <a href="http://en.wikipedia.org/wiki/Gaussian_binomial_coefficient">Gaussian binomial coefficient</a> $\binom nk_q$ evaluated at $q=p$. The indeterminate of the Gaussian binomial coefficient is traditionally called $q$, and I guess this is in particular because ... |
70,801 | <p>I am asked to find how many there are $k$-dimensional subspaces in vector space $V$ over $\mathbb F_p$, $\dim V = n$.</p>
<p>My attempt:
1) Let's find a total number of elements in $V$: assume that $\{v_1, v_2, \cdots, v_n\}$ is a basis in $V$. Then, for every $v \in V$ we can write down
$$ v = a_1 v_1 + a_2 v_2 + ... | Abdolzadeh.H | 330,189 | <p>Let $n=\dim V$, where $V$ is a vector space over any finite field $\mathbb{F}$ with $|\mathbb{F}|=r$ and $W$ is a $k$-dimensional subspace of $V$. First of all we count the number of basis of $W$. There are $r^k-1$ non zero vectors in $W$, so to select a base for $W$, first member could be selected in $r^k-1$ way. S... |
2,178,318 | <p>On a test I wrote an implication arrow "$\implies$" to show that I deduced one statement from the previous one, but I didn't get full score since it was more accurate to use an equivalence arrow "$\iff$".
For example:
$$ 2x = 4 \implies x = 2 $$
but it's also true the other way around:
$$ 2x = 4 \impliedby x = 2$$
s... | Gordon Geringas | 571,600 | <blockquote>
<p>"Given this i would assume that if Q⟹P is true, then Q⟸P is false.
Is this correct?"</p>
</blockquote>
<p>No that is not implied. However you may be thinking of Modus Tollens which goes:</p>
<blockquote>
<p>P⇒Q, notQ hence notP</p>
</blockquote>
<p>For example, if it rains (P) then it is wet o... |
1,505,076 | <p>This might sound a stupid question but it is indeed a real one.</p>
<p>I'm trying to figure a Confidence interval for the average age of my population.</p>
<p>Given i have a population of 100 individual, and i sample 3 of them. From CLT, i can say that $Var[\bar{x}_3] = \frac{s^2}{3} $. Alright. </p>
<p>I want to... | A.S. | 274,197 | <p>You are sampling from a finite population so you samples are not in fact independent - once you sampled a person, they are out and you are sampling from a smaller pool of people. Once your sample size gets compatible (order of magnitude) with population size, these effects start to matter more and more.</p>
|
2,358,838 | <p>I can see the answer to this in my textbook; however, I am not quite sure how to solve this for myself . . . the book has the following:</p>
<blockquote>
<p>To take advantage of the inductive hypothesis, we use these steps:</p>
<p>$ 7^{(k+1)+2} + 8^{2(k+1)+1} = 7^{k+3} + 8^{2k+3} $</p>
<p>$$
= 7\cdot7^... | G Cab | 317,234 | <p>"..for every non-negative integer $n$" means that it shall be valid starting from $n=0$.<br>
In fact $F(0)= 7^2+8=57$.</p>
<p>You already found that $F(n+1)=7 \,F(n)+57\,8^{2n+1}$ which is clearly divisible by 57.<br>
So $$57\backslash F(n)\quad \left| {\;0 \le n} \right.$$</p>
|
198,995 | <p>From Barbeau's <em>Polynomials</em>:</p>
<blockquote>
<ul>
<li>(a) Is it possible to find a polynomial, apart from the constant $0$ itself, which is identically equal to $0$ (i.e. a polynomial
$P(t)$ with some nonzero coefficient such that $P(c)=0$ for each
number $c$)?</li>
</ul>
</blockquote>
<p>And t... | Marc van Leeuwen | 18,880 | <p>I guess there is no way to forbid people to present contorted arguments to prove a simple result, but they do the reader a disservice by masking the essential point. Here the essential point is that we want a polynomial that vanishes <em>in more points than its degree</em>, and this can only be achieved by the zero ... |
65,304 | <p>I have a plane curve $C$ described by parametric equations $x(t)$ and $y(t)$ and a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$. The line integral of $f$ along $C$ is the area of the "fence" whose path is governed by $C$ and height is governed by $f$.</p>
<p><img src="https://i.stack.imgur.com/4rmZy.png" alt="... | ubpdqn | 1,997 | <p>I have not done the labeling but this is a start:</p>
<pre><code>axes[n_] :=
With[{uv = n IdentityMatrix[3]},
Graphics3D[{Arrow[{{0, 0, 0}, #}] & /@ uv,
MapThread[
Text[#1, 1.1 #2] &, {Style[#, 20] & /@ {"x", "y", "z"}, uv}]},
Boxed -> False]];
p = ParametricPlot3D[{Cos[t], Si... |
1,889,957 | <p>I'm a bit rusty on my math notations and I'd like to write that:</p>
<blockquote>
<p>It exists a unique element $z$ such that $z$ belongs to the collection of values returned by $f(x,y)$</p>
</blockquote>
<p>Honestly I'm not just rusty I'm also mostly ignorant of math except from basic functions and basic matrix... | Michael Rozenberg | 190,319 | <p>$\sum\limits_{cyc}\frac{a}{b+c}=\sum\limits_{cyc}\frac{a^2}{ab+ac}\geq\frac{(a+b+c+s)^2}{\sum\limits_{cyc}(ab+ac)}\geq2$</p>
<p>Because the last inequality it's $(a-c)^2+(b-d)^2\geq0$. Done!</p>
|
2,723,585 | <p>If $\textbf{A}$ is a square matrix, how can I prove that, by using the power series of matrices that the above equality holds?
Note that the $x \in \mathbb{N}$ and $\textbf{A}$ is a square matrix.</p>
| David C. Ullrich | 248,223 | <p>If $AB=BA$ it follows by induction on $n$ that the binomial theorem holds for $(A+B)^n$ ($n\in\Bbb N$). Now if you simply mulitply the two power series and collect terms this shows that$$e^{A+B}=e^Ae^B\quad(AB=BA).$$</p>
<p>By induction on $n$ this shows that $$\left(e^A\right)^n=e^{nA}\quad(n\in\Bbb N).$$</p>
<p>... |
2,416,510 | <p>I have a matrix $A \in R^{n×n}$. I would like to choose two diagonal matrices $D_1,D_2 \in R^{n×n}$ such that $\text{cond}(D_1AD_2)$ should be minimal. How to provide such diagonal matrices? </p>
| Jaroslaw Matlak | 389,592 | <p><img src="https://i.imgur.com/ZUa8pHw.png" alt=""></p>
<p>What you need to compute is $\alpha+\beta+\gamma+\delta=?$</p>
|
48,726 | <p>I'm trying to plot a 3d revolution plot from a set of 2d points. These data points form a 2d curve, then we rotate that curve around y axis and get a 3d surface. @<a href="https://mathematica.stackexchange.com/users/50/50">J. M.</a> has a well explained and very helpful post at <a href="https://mathematica.stackexch... | Community | -1 | <p>Sounds like what you actually need after your edit is a way to smooth a list of data while keeping the endpoints fixed. Here's a dumb approach that will work with any "symmetrical" smoothing filter, including <code>GaussianFilter</code>, <code>MeanFilter</code>, even <code>MedianFilter</code>. It won't work with <co... |
4,092,994 | <p>The question is</p>
<blockquote>
<p>Find the solutions to the equation <span class="math-container">$$2\tan(2x)=3\cot(x) , \space 0<x<180$$</span></p>
</blockquote>
<p>I started by applying the tan double angle formula and recipricoal identity for cot</p>
<p><span class="math-container">$$2* \frac{2\tan(x)}{1-... | Quanto | 686,284 | <p>Factorize the equation as follows</p>
<p><span class="math-container">\begin{align}
2\tan(2x)-3\cot(x)
=& \frac{2\sin2x}{\cos 2x} - \frac{3\cos x}{\sin x}\\
=& \frac{2\sin2x\sin x-3 \cos x\cos2x }{ \sin x\cos 2x}\\
=& \frac{\cos x(10\sin^2x-3 )}{ \sin x\cos 2x}\\
\end{align}</span>
where the factor <sp... |
1,478,142 | <p>Evaluate these limits by relating them to a derivative. </p>
<p>$\lim\limits_{x \to 0} \frac{\sqrt{\cos{x}}-1}{x}$</p>
| MPW | 113,214 | <p><strong>Hint:</strong> Put $f(t) = \sqrt{\cos t}$, note that $f(0)=1$ and $f(0+x) = f(x) = \sqrt{\cos x}$, and $$\frac{f(0+x)-f(0)}{x}=\frac{\sqrt{\cos x} - 1}{x}$$</p>
<p>Can you take it from here?</p>
<p><strong>Note:</strong> You may recognize the form on the left side of the last line better if you write $h$ (... |
1,784,912 | <p>In this question, I know that $\text{C},\text{R},\text{T},\text{A}\in\mathbb{R}^+$</p>
<p>I've this circuit (the bottom of the resitor is connected to earth ($0$)):
<a href="https://i.stack.imgur.com/hfKGJ.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/hfKGJ.jpg" alt="enter image description her... | Rorschach 0007 | 334,308 | <p>Your circuit is a passive differentiator I think. There is one zero and one pole and the operation of the differentiation on the square wave (hence the spikes to be expected as your picture shows) occurs at frequencies below the pole frequency. You should be getting a Dirac delta function in the output for this, I t... |
71,117 | <p>I have this assertion: if $p$ is a prime such that $p\equiv 11 \pmod{56}$, then $p$ splits in $\mathbb{Z}[\sqrt{14}]$ (the discriminant of $\mathbb{Z}[\sqrt{14}]$ is $56$.)</p>
<p>Why? Does $p\equiv 11\pmod{56}$ imply $14$ is a quadratic residue mod $p$?</p>
| Brandon Carter | 1,016 | <p>If we have a quadratic field $K = \mathbb{Q}(\sqrt{d})$ with $d$ squarefree, then an odd prime $p$ splits if and only if $\left(\frac{d}{p}\right) = 1$. </p>
<p><em>Claim</em>: If $q$ is an odd prime with $q \equiv 3 \pmod 4$, then $q$ is a quadratic residue mod $p$ if and only if $p \equiv \pm b^2 \pmod {4q}$, whe... |
386,921 | <p>I am 16 years old at the time of writing (so I have no supervisors to seek advice from) and I have written a mathematics research paper, which I plan on submitting to a journal for publication. I asked an <a href="https://academia.stackexchange.com/questions/164114/how-to-structure-a-proof-by-induction-in-a-maths-re... | 2734364041 | 111,215 | <p>Writing a proof for school is very different from writing a proof for a research paper. Perhaps the most important distinction is that the audiences are completely different. In school, your audience is your instructor, whose job is to assess your ability to learn and apply a principle. The audience of a research... |
386,921 | <p>I am 16 years old at the time of writing (so I have no supervisors to seek advice from) and I have written a mathematics research paper, which I plan on submitting to a journal for publication. I asked an <a href="https://academia.stackexchange.com/questions/164114/how-to-structure-a-proof-by-induction-in-a-maths-re... | AfterMath | 153,908 | <p>Welcome to math overflow!</p>
<p>There is no need to list up the names of the steps of the induction (Base case, Inductive step etc.) if it clear from context and/or obvious what you're doing. This will likely only make the argument take up more physical space on the page than strictly necessary. It should however b... |
258,132 | <p>Consider the following simple example as motivation for my question. If it were the case that, say, the Riemann hypothesis turned out to be independent of ZFC, I have no doubt it would be accepted by many as a new axiom (or some stronger principle which implied it). This is because we intuitively think that if we ... | Andrej Bauer | 1,176 | <p>It cannot be done in a consistent way.</p>
<p>Consider a <em>closed</em> statement $\psi$ which is independent of a theory $T$, and take $\forall x . \psi$ and $\forall x . \lnot\psi$. (I made the closed statement have a dummy free variable to satisfy your condition.) Both statements are of the kind you are asking ... |
258,132 | <p>Consider the following simple example as motivation for my question. If it were the case that, say, the Riemann hypothesis turned out to be independent of ZFC, I have no doubt it would be accepted by many as a new axiom (or some stronger principle which implied it). This is because we intuitively think that if we ... | Erfan Khaniki | 83,598 | <p>This post is not an answer to your question, but it explains the reason that if <span class="math-container">$\bf GC$</span> (or <span class="math-container">$\bf RH$</span>) is independent of <span class="math-container">$\bf ZFC$</span>, <span class="math-container">$\bf GC$</span> (or <span class="math-container"... |
258,132 | <p>Consider the following simple example as motivation for my question. If it were the case that, say, the Riemann hypothesis turned out to be independent of ZFC, I have no doubt it would be accepted by many as a new axiom (or some stronger principle which implied it). This is because we intuitively think that if we ... | Joel David Hamkins | 1,946 | <p>The phenomenon accords more strongly with your philosophical explanation if you ask also that the sentences have complexity $\Pi^0_1$. That is, the universal statement $\forall x\ \varphi(x)$ should have $\varphi(x)$ involving only bounded quantifiers, so that we can check $\varphi(x)$ for any particular $x$ in fini... |
4,122,425 | <p>Let’s say a corona test is correct with <code>p=0.8</code>. If I now take two tests. What’s the probability that I get a correct result?</p>
<p>I think thought of <code>0.8*0.8</code>, but that makes now sense, since it should not decrease and <code>0.8+0.8</code> gives a probability over 1, which makes no sense eit... | MathR | 876,250 | <p>0.2<span class="math-container">$\cdot$</span>0.8+0.2<span class="math-container">$\cdot$</span>0.8+0.8<span class="math-container">$\cdot$</span>0.8=0.96=P(“at least one positive test”)</p>
|
2,420,727 | <p>I'm trying to evaluate </p>
<blockquote>
<p>$$\lim _{ x\to -\infty } \frac { 2x-3 }{ \sqrt { x^{ 2 }+7x-2 } } $$</p>
</blockquote>
<p>by rationalizing the denominator, but I am not getting anywhere. Can someone please help me with this?</p>
<p>Thanks</p>
| Siong Thye Goh | 306,553 | <p>$$\lim_{x\to -\infty}\left(\frac{2x-3}{\sqrt{x^2+7x-2}}\right) = \lim_{x\to -\infty}\left(\frac{2\frac{x}{|x|}-\frac{3}{|x|}}{\sqrt{1+\frac{7}{x}-\frac{2}{x^2}}}\right)=-2$$</p>
|
2,420,727 | <p>I'm trying to evaluate </p>
<blockquote>
<p>$$\lim _{ x\to -\infty } \frac { 2x-3 }{ \sqrt { x^{ 2 }+7x-2 } } $$</p>
</blockquote>
<p>by rationalizing the denominator, but I am not getting anywhere. Can someone please help me with this?</p>
<p>Thanks</p>
| Nosrati | 108,128 | <p>\begin{align}
\lim_{x\to\infty}\frac{2x-3}{\sqrt{x^2+7x-2}}
&= \lim_{x\to\infty}\frac{x(2-\dfrac3x)}{\sqrt{x^2(1+\dfrac7x-\dfrac{2}{x^2})}} \\
&= \lim_{x\to\infty}\frac{x}{\sqrt{x^2}} \times \lim_{x\to\infty}\frac{2-\dfrac3x}{\sqrt{1+\dfrac7x-\dfrac{2}{x^2}}} \\
&= \lim_{x\to\infty}\frac{x}{\sqrt{x^2}} \... |
3,954,865 | <p>I am trying to solve a question but stuck with the steps. I can not find any similar questions. With help of some online resources to calculate some parts of the question but I can see that is not enough. I know my approach has lack of information but, this is the only thing I have reached, I was covid ill at the cl... | BruceET | 221,800 | <p>Notice that <em>all nine</em> of the observations are \$460 and below? Just from common sense, what does that tell you about the claim that average cost is \$500.</p>
<p>You already have a thoughtful Answer from @tommik (+1), but because you ask I will show some additional detail.</p>
<hr />
<p>Here is a relevant t ... |
311,380 | <p>Prove that the relation $x \sim y$ iff $y$ is an element of the connected component of $x$ is an equivalence relation.</p>
<p>This question is confusing me, do I simply go about showing the relation is reflexive, symmetric, and transitive? I don't really see how to do this for this question. Any suggestions or hint... | Cameron Buie | 28,900 | <p>Let's call our space $S$. For any $x\in S,$ we can define $C_x$, the connected component of $x$, to be the $\subseteq$-greatest connected subset $A$ of $S$ such that $x\in A.$ Put another way, $C_x$ is the union of all connected subsets of $S$ containing $x$ as an element--we can show that such a union is connected ... |
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