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 |
|---|---|---|---|---|
434,290 | <p>According to the <a href="http://arxiv.org/abs/0910.5922" rel="nofollow">equation 4</a>,
$$\phi(0,t)= \frac{A_0}{(1+\frac{2t^2}{R^4})^{3/4}}\cos \left(\sqrt2 t+ \frac{3}{2}\tan^{-1}\left[\frac{\sqrt2 t}{R^2}\right]\right)\tag{1}$$
what conditions makes, $$\cos \left(\sqrt2 t+ \frac{3}{2}\tan^{-1}\left[\frac{\sqrt2 ... | Tunk-Fey | 123,277 | <p>Another approach, we can split the denominator part as follows
$$
\frac{1}{x^4+1}=\frac{1}{2i}\left(\frac{1}{x^2-i}-\frac{1}{x^2+i}\right).
$$
Consequently, the integral becomes
$$
\int_{0}^{\infty }\frac {\ln x}{x^4+1}\ dx =\frac{1}{2i}\int_{0}^{\infty }\left(\frac{\ln x}{x^2-i}-\frac{\ln x}{x^2+i}\right)\ dx.
$$
U... |
434,290 | <p>According to the <a href="http://arxiv.org/abs/0910.5922" rel="nofollow">equation 4</a>,
$$\phi(0,t)= \frac{A_0}{(1+\frac{2t^2}{R^4})^{3/4}}\cos \left(\sqrt2 t+ \frac{3}{2}\tan^{-1}\left[\frac{\sqrt2 t}{R^2}\right]\right)\tag{1}$$
what conditions makes, $$\cos \left(\sqrt2 t+ \frac{3}{2}\tan^{-1}\left[\frac{\sqrt2 ... | Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}
\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\down}{\... |
1,265,074 | <p>I simply don't know how to go about answering this question. I've done a good few other questions about point estimation, but I really don't know where I'm going with this one:</p>
<p><img src="https://i.stack.imgur.com/coWTD.png" alt="Unbiased estimator of 1 over lambda"></p>
<p>Thanks for the help!</p>
<p>EDIT:... | Math1000 | 38,584 | <p>Recall that if $X,Y\sim\mathrm{Pois}(\lambda)$ are independent, then $X+Y\sim\mathrm{Pois}(2\lambda)$. So
$$\sum_{i=1}^n X_i\sim\mathrm{Pois}(n\lambda).$$
We can use the law of the unconscious statistician to compute the expectation of the given estimator:
$$
\begin{align*}
\mathbb E\left[\frac n{\sum_{i=1}^n X_i + ... |
1,989,253 | <p>I am trying to evaluate: </p>
<p>$$\lim_{x \to 4}\frac{\sqrt{5-x} - 1}{2-\sqrt{x}}.$$</p>
<p>Even though I tried rationalizing both denominator and numerator, I still end up with the functioning being undefined.</p>
<p>How can I solve this without rationalizing?</p>
| user382540 | 382,540 | <p>If anyone was wondering, after rationalizing I had gotten </p>
<p>$$\lim_{x \to 4}\frac{4 - x}{(2-\sqrt{x})(\sqrt{5-x}+x)}.$$</p>
<p>Of which you need to factor $(4-x)$ to $(2-\sqrt{x})(2+\sqrt{x})$.</p>
<p>From there you can guess where to go.</p>
|
357,557 | <p>I have a function: $f(x)=-\frac{4x^{3}+4x^{2}+ax-18}{2x+3}$ which has only one point of intersection with the $x$-axis.</p>
<p>How can i find the value of $a$?</p>
<p>I tried polynomial division and discriminant, but it didn't help me.</p>
| Peter Smith | 35,151 | <p>Obviously, $x > 1$ is a power of 2 iff (A): every $y > 1$ which divides $x$ is itself divisible by 2. </p>
<p>Use the fact that factors of a number are less than it to bound the quantifiers in formalizing statement (A), and you'll get a $\Delta_0$ wff.</p>
|
1,358,002 | <p>My son did something quite impressive the other day. It was shear luck but I don't think I'll ever see it duplicated again in my lifetime. </p>
<p>I brought my kids to the boardwalk and my son wanted to play an amusement game. It was the arrow spin wheel game. It had 90 different names or possibilities to win. You ... | quid | 85,306 | <p>If I understand the game correctly, he chose $3$ out of $90$, and so had a $3/90=1/30$ chance in winning a game. </p>
<p>Doing this three times in a row has a probability of $(1/30)^3=1/27000$, so it should happen once in $27000$ tries on average. So, it is pretty unlikely but not excessively so. </p>
|
718,166 | <p><strong>Question:</strong></p>
<blockquote>
<p>let $a\in(0,1)$, and such $f(x)\geq0$, $x\in R$ is continuous on $R$,</p>
<p>if
$$f(x)-a\int_x^{x+1}f(t)dt,\forall x\in R $$ is constant,</p>
<p>show that</p>
<p>$f(x)$ is constant;</p>
<p>or $$f(x)=Ae^{bx}+B$$ where $A\ge 0,|B|\le A$ and $A,B$... | Bob Pego | 2,947 | <p>A very pretty problem! Under the hypotheses stated, we will conclude that
$f(x) = A e^{bx}+B$ with $A, B\ge0$. The amplitude of $B$ cannot be restricted. </p>
<p>Since $f$ is continuous, the integral is differentiable and by the OP's calculation,
$$
\frac{d}{dx}(e^{ax} f(x)) = e^{ax}(f'(x)+a f(x)) = e^{ax} a f(x+1... |
1,392,858 | <p>Is is known that the space of symmetric matrices $\mathbb{R}_{sym}^{n \times n}$ has $\binom{n}{2}$ dimensions.</p>
<p>And according to the spectral theorem every symmetric matrix $A \in \mathbb{R}_{sym}^{n \times n}$ has a spectral decomposition in terms of 1-rank matrices.</p>
<p>A = $\sum_{i=1}^n \lambda_i v_i... | Ben Grossmann | 81,360 | <p>You have correctly stated that for any symmetric $A$, there exist rank $1$ matrices $v_iv_i^T$ such that
$$
A = \sum \lambda _i v_i v_i^T
$$
However, it is impossible to select a <strong>fixed</strong> set $\{v_1v_1^T,\dots,v_nv_n^T\}$ such that <strong>every</strong> $A$, there exists a choice of $\lambda_i$ such t... |
1,178,080 | <p>How to calculate the number of solutions of the equation $x_1 + x_2 + x_3 = 9$ when $x_1$, $x_2$ and $x_3$ are integers which can only range from <code>1</code> to <code>6</code>.</p>
| Stefan4024 | 67,746 | <p>Write $x_i = y_i + 1$, where $0\le y_1 \le 5$</p>
<p>Then you have:</p>
<p>$$y_1 + y_2 + y_3 = 6$$</p>
<p>And according to stars and bars we have: $$\binom{6+3-1}{6} = 28 \text{ combinations}$$</p>
<p>Now just exclude the $(6,0,0), (0,6,0)$ and $(0,0,6)$ and you have $25$ solutions </p>
<hr>
<p><strong>UPDATE:... |
139,817 | <p>Studying stability of certain non-autonomous dynamical systems on Lie groups I have come across the following question: Exactly which finite-dimensional, real Lie groups have adjoint representations that are bounded away from zero?</p>
<p>Edit: by "bounded away from zero" I mean that the image of the adjoint repres... | David E Speyer | 297 | <p>The adjoint rep is always bounded away from $0$. Let $\mathfrak{g}_0$ be a simple quotient of $\mathfrak{g}$. (I consider the $1$-dimensional Lie algebra to be simple, so there is always a simple quotient.) Let $\mathfrak{h}$ be the kernel of $\mathfrak{g} \to \mathfrak{g}_0$ and let $H = \exp(\mathfrak{h})$. </p>
... |
3,156,643 | <blockquote>
<p>Prove that <span class="math-container">$\sin(x) < x$</span> when <span class="math-container">$0<x<2\pi.$</span></p>
</blockquote>
<p>I have been struggling on this problem for quite some time and I do not understand some parts of the problem. I am supposed to use rolles theorem and Mean v... | Community | -1 | <p>The inequality obviously holds for <span class="math-container">$x>1$</span>.</p>
<p>Then for <span class="math-container">$0<x\le1$</span>,</p>
<p><span class="math-container">$$\cos x<1$$</span> and by integration from <span class="math-container">$0$</span></p>
<p><span class="math-container">$$\sin x... |
555,239 | <p>Since the polynomial has three irrational roots, I don't know how to solve the equation with familiar ways to solve the similar question. Could anyone answer the question?</p>
| lab bhattacharjee | 33,337 | <p>Rearranging we get $$x^2-x(y+1)+y^2+y=0$$ which is a Quadratic Equation in $x$</p>
<p>As $x$ must be real, the discriminant must be $\ge0$ i.e., </p>
<p>$(y+1)^2-4(y^2+y)=-3y^2-2y+1\ge0$</p>
<p>$\iff 3y^2+2y-1\le0$</p>
<p>$\iff \{y-(-1)\}(y-\frac13)\le0$</p>
<p>$\iff -1\le y\le \frac13$</p>
<p>Now, use the fa... |
555,239 | <p>Since the polynomial has three irrational roots, I don't know how to solve the equation with familiar ways to solve the similar question. Could anyone answer the question?</p>
| ssharma | 108,216 | <p>For second equation:
$$x + y = x^2 + y^2 − xy$$</p>
<p>By dividing $xy$ on both sides</p>
<p>$$\frac{1}{x} + \frac{1}{y} = \frac{x}{y} + \frac{y}{x} -1 = y \text{ (say)}$$</p>
<p>Here for any real no. $a$
$$a + \frac{1}{a}\ge 2$$</p>
<p>So RHS will be $\ge1$.
But because only integer solutions are required:
LHS ... |
1,878,573 | <p><a href="https://i.stack.imgur.com/3iZQ8.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/3iZQ8.png" alt="enter image description here"></a></p>
<p>I cannot get the $f'(0)$ by using L'Hôpital's rule, because it appears recurrence item. Can you help me?</p>
| Zau | 307,565 | <p>The derivative of f at $0$ can be calculated:</p>
<p>$$ f'(0) = \lim_{ x \to 0 } \frac{f(x) - f(0)}{x} = \frac{({e}^{{x}^{2}} - {e}^{{-x}^{2}} )\sin (\frac{1}{x^3})}{x}$$</p>
<p>Then try to use L'Hôpital's rule or Taylor series to see the derivative.</p>
|
3,227,215 | <p><a href="https://i.stack.imgur.com/7pJ4t.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/7pJ4t.png" alt="enter image description here" /></a></p>
<blockquote>
<p><span class="math-container">$(O, R)$</span> is the circumscribed circle of <span class="math-container">$\triangle ABC$</span>. <span c... | Bartek | 671,751 | <p>As pointed out in the comments the inequality:
<span class="math-container">$$\large \frac{1}{AM \cdot BN} + \frac{1}{BN \cdot CP} + \frac{1}{CP \cdot AM} \le \frac{4}{3(R - OI)}$$</span>
Is not homogeneous and therefore cannot be correct. Take any triangle and any point and even if the given inequality is satisfied... |
821,875 | <p>A school director must randomly select 6 teachers to participate in a training session. There are 30 teachers at the school. In how many different ways can these teachers be selected, if the order of selection does not matter?</p>
| DSinghvi | 148,018 | <p>you must read permutation and combination textbook or section first.
answer is 30C6=(30*29*28*27*26*25)/(6*5*4*3*2*1)=593775
I don't think it can be explained here.</p>
|
2,616,847 | <p>By definition, a function $f:\Bbb R^n \to \Bbb R^m$ is linear if</p>
<ol>
<li>$f(x+y)=f(x)+f(y) \forall x,y\in \Bbb R^n$</li>
<li>$f(ax)=af(x) \forall x\in \Bbb R^n$</li>
</ol>
<p>I want to prove that $f$ is linear iff $f(x)=Ax,A\in\Bbb R^{m\times n}$ and A is unique for any x. </p>
<p>I try to prove it by showin... | Community | -1 | <p>To prove f is linear:
Two parts in your definition are equivalent to say $f(ax+by)=af(x)+bf(y)$.</p>
<p>Now, $f(ax+by)=A(ax+by)=aAx+bAy=af(x)+bf(y)$. So, it is linear.</p>
<p>I think your proof on uniqueness is acceptable. Just make sure to check the case of $x=0$.</p>
|
2,088,346 | <p>I've got the domain of function and I've attempted to find the first derivative at zero but it results in a quartic equation that is too difficult for me to solve. </p>
<p>$f'(x) = \frac{4x-3}{\sqrt{2x^2-3x+4}} + \frac{2x-2}{\sqrt{x^2-2x}}$</p>
<p>For $f'(x) = 0$:</p>
<p>$(4x-3)^2(x^2-2x) = (2-2x)^2(2x^2-3x+4)$</... | xxyshz | 377,743 | <p>We know that $x\in(-\infty,0)or(2,+\infty)$
We can draw the image of the $(x^{2}-2x)^{\frac{1}{2}}$and $(2x^{2}-3x+4)^{\frac{1}{2}}$,you will find the min of this function is 2 when x=0;</p>
|
2,088,346 | <p>I've got the domain of function and I've attempted to find the first derivative at zero but it results in a quartic equation that is too difficult for me to solve. </p>
<p>$f'(x) = \frac{4x-3}{\sqrt{2x^2-3x+4}} + \frac{2x-2}{\sqrt{x^2-2x}}$</p>
<p>For $f'(x) = 0$:</p>
<p>$(4x-3)^2(x^2-2x) = (2-2x)^2(2x^2-3x+4)$</... | zipirovich | 127,842 | <p>First of all, you have a slight error in your derivative: you're missing a factor of $\frac{1}{2}$ in both terms of $f'$. Fortunately, it doesn't affect solving the equation $f'(x)=0$.</p>
<p>You said that you found the domain of this function, so you know that the domain is $(-\infty,0]\cup[2,+\infty)$.</p>
<p>So... |
2,088,346 | <p>I've got the domain of function and I've attempted to find the first derivative at zero but it results in a quartic equation that is too difficult for me to solve. </p>
<p>$f'(x) = \frac{4x-3}{\sqrt{2x^2-3x+4}} + \frac{2x-2}{\sqrt{x^2-2x}}$</p>
<p>For $f'(x) = 0$:</p>
<p>$(4x-3)^2(x^2-2x) = (2-2x)^2(2x^2-3x+4)$</... | Michael Rozenberg | 190,319 | <p>Let $f(x)=\sqrt{2x^2-3x+4}$, $g(x)=\sqrt{x^2-2x}$ and $h(x)=\sqrt{x}$.</p>
<p>Since $h$ is an increasing function, </p>
<p>we see that $f$ and $g$ are decreasing functions on $(-\infty,0]$</p>
<p>and $f$ and $g$ are increasing functions on $[2,+\infty)$.</p>
<p>Thus, $$\min\limits_{(-\infty,0]\cup[2,+\infty)}\le... |
1,848,150 | <blockquote>
<p>Let $m$ and $c$ be non-zero real numbers and $X$ the subspace of $\mathbb R^2$ given by $X =\{ (x,y): y = mx + c \}$. Prove that $X$ is homeomorphic to $\mathbb R$.</p>
</blockquote>
<p>I am struggling to figure out how to define a homeomorphic function between these two sets, can anyone please help?... | Alex Ortiz | 305,215 | <p>Define the bijection $f : \mathbb R \to X$ by $fx = (x, mx + c)$. Let $(p_n)$ be a convergent sequence in $\mathbb R$ with $p_n \to p$. Then, the sequence $(fp_n) = (p_n, mp_n + c)$ also converges to $(p, mp + c)$. The inverse bijection $f^{-1} : X \to \mathbb R$ is defined by $f(x, mx + c) = x$. So, consider a conv... |
1,848,150 | <blockquote>
<p>Let $m$ and $c$ be non-zero real numbers and $X$ the subspace of $\mathbb R^2$ given by $X =\{ (x,y): y = mx + c \}$. Prove that $X$ is homeomorphic to $\mathbb R$.</p>
</blockquote>
<p>I am struggling to figure out how to define a homeomorphic function between these two sets, can anyone please help?... | snulty | 128,967 | <p>Here's a relatively straightforward way to see it.</p>
<p>First note that $\iota :\Bbb R \to \Bbb R\times \{0\}\subset \Bbb R^2$ with the subspace topology is a homeomorphism.</p>
<p>Then, let $m=\tan\theta$ and $A:\Bbb R^2\to \Bbb R^2$ by $(x,y)\mapsto \pmatrix{\cos\theta & -\sin \theta \\ \sin\theta & \c... |
1,848,150 | <blockquote>
<p>Let $m$ and $c$ be non-zero real numbers and $X$ the subspace of $\mathbb R^2$ given by $X =\{ (x,y): y = mx + c \}$. Prove that $X$ is homeomorphic to $\mathbb R$.</p>
</blockquote>
<p>I am struggling to figure out how to define a homeomorphic function between these two sets, can anyone please help?... | tomasz | 30,222 | <p><strong>Hint</strong>: It might be easier to prove a more general fact. Let $f\colon X\to Y$ be any continuous function (between arbitary topological spaces). Show that $x\mapsto (x,f(x))$ defines a homeomorphic embedding of $X$ into $X\times Y$.</p>
|
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 ... | WalterJ | 344,100 | <p>As a sort of hint:</p>
<p>I suppose you have heard of convolution
\begin{equation}
y(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)d\tau=\int_{-\infty}^{\infty}h(\tau)x(t-\tau)d\tau
\end{equation}
Then what happens if our input is of the form $x(t)=e^{st}$? In that case $y(t)=H(s)e^{st}$ with $H(s)=\int_{-\infty}^{\inf... |
1,039,141 | <blockquote>
<p>Let <span class="math-container">$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 ... | jchun | 187,232 | <p>Pick an arbitrary element from within the range of the function, and show that the preimage of the element is non-empty.</p>
|
2,337,332 | <p>Tried a lot. Though unable to find starting point.</p>
| Siong Thye Goh | 306,553 | <p>Hint:</p>
<p>Since</p>
<p>$$9 \equiv -5 \pmod{14}$$</p>
<p>$$9^{16} - 5^{16} \equiv (-5)^{16} - 5^{16} \pmod{14}$$</p>
|
2,298,665 | <p>*prior to the body, note that title might be insufficient or inappropriate. Please edit it if it's needed. </p>
<p>I am proving the claim below: </p>
<p>Let $f: [a,b] \to \Bbb R$ be of bounded variation. </p>
<p>$f(x) \ge c \gt 0$ for all $x \in [a, b]$ where $c$ is a constant</p>
<p>$\Rightarrow$ $h(x)$ = $1 \o... | Community | -1 | <p>This is a pure transcendental extension. $\mathbb{Q}(\pi)$ is the field of rational functions of one variable; any additional 'arithmetic' meaning of $\pi$ is completely irrelevant.</p>
<p>It's <a href="https://math.stackexchange.com/questions/13129/automorphism-of-the-field-of-rational-functions">well-known</a> th... |
1,930,901 | <p>I need to prove that:</p>
<p>$$f(z) = \frac{Re(z)}{|z|}$$</p>
<p>and </p>
<p>$$g(z) = \frac{Im(z^2)}{|z^2|}$$</p>
<p>both have limit at $z=0$</p>
<p>If I see $z$ as $z = x+iy$ I have:</p>
<p>$$\lim_{(x,y)\to (0,0)}\frac{x}{\sqrt{x^2+y^2}}$$</p>
<p>but if I take this limit at $y = x$ we have:</p>
<p>$$\lim_{(... | mathworker21 | 366,088 | <p>First of all, the result should be intuitive: if $a_j \to a$, then $a_j$ becomes closer and closer to $a$, so as $j \to \infty$, the averages $\frac{a_1+a_2+\dots+a_n}{n}$ should tends toward $a$. We can make this intuition rigorous and thus prove the result by using epsilons and deltas; we don't need any convergenc... |
1,930,901 | <p>I need to prove that:</p>
<p>$$f(z) = \frac{Re(z)}{|z|}$$</p>
<p>and </p>
<p>$$g(z) = \frac{Im(z^2)}{|z^2|}$$</p>
<p>both have limit at $z=0$</p>
<p>If I see $z$ as $z = x+iy$ I have:</p>
<p>$$\lim_{(x,y)\to (0,0)}\frac{x}{\sqrt{x^2+y^2}}$$</p>
<p>but if I take this limit at $y = x$ we have:</p>
<p>$$\lim_{(... | Beni Bogosel | 7,327 | <p>A nice (and immediate) proof can be given using <a href="https://en.wikipedia.org/wiki/Stolz%E2%80%93Ces%C3%A0ro_theorem" rel="nofollow noreferrer">Stolz-Cesaro Theorem</a>. This result is quite useful and helps solve rapidly many such problems. It is almost like l'Hopital's rule, which allows to "differentiate... |
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... | Brian Borchers | 6,310 | <p>rearrange the elements of $x$ so that the nonzeros are in entries $1$, $2$, $\ldots$, $n$, where $n=\| x \|_{0}$. .You can rewrite the inequality as</p>
<p>$n(x_{1}^{2}+\ldots+x_{n}^{2}) \geq (|x_{1}|+\ldots +|x_{n}|)^{2}$. </p>
<p>After taking the square root of both sides, this is equivalent to the well-known i... |
207,515 | <p>Suppose I have the following list, </p>
<pre><code>l = {{"b", "c", "d"}, {"e", "b"}, {"a", "b", "d", "e"}}
</code></pre>
<p>and further suppose I have the following association, </p>
<pre><code>l1=<|1 -> "a", 2 -> "b", 3 -> "c", 4 -> "d", 5 -> "e"|>
</code></pre>
<p>I wonder how can I replac... | Carl Woll | 45,431 | <p>For this particular mapping, you could also use <a href="http://reference.wolfram.com/language/ref/ToCharacterCode" rel="nofollow noreferrer"><code>ToCharacterCode</code></a>:</p>
<pre><code>ToCharacterCode[StringJoin /@ l] - First@ToCharacterCode@"a" + 1
</code></pre>
<blockquote>
<p>{{2, 3, 4}, {5, 2}, {1, 2, ... |
2,354,467 | <p>I am trying to evaluate the following
\begin{equation}
I(a,b) = \int_{a}^{\frac{a+b}{2}} (x-a)^{\alpha-1} \, x^n \, dx + \int_{\frac{a+b}{2}}^{b} (b-x)^{\alpha-1} \, x^n \, dx,
\end{equation}
where $0<\alpha<1$. Wolfram alpha gives no solution. I tried integration by parts without success. My problem is that ... | tempx | 357,017 | <p>Why did you separate the integration into two pieces? If I did not read anything wrong we state the integral as </p>
<p>$$
I(a,b) = \int_{a}^{b} (x-a)^{\alpha-1}x^ndx
$$</p>
<p>Please correct me if this is wrong. Then, by change of variables (it may not be necessary but for the ease of calculation) and assuming th... |
62,539 | <p>I am using two books for my calculus refresher.</p>
<ol>
<li>Thomas' Calculus </li>
<li>Higher Math for Beginners by Ya. B. Zeldovich</li>
</ol>
<p><strong>My question is :</strong> When applying Integral Calculus for calculation of volumes of solids, generated by curves revolved around an axis, we use slices of '... | TonyK | 1,508 | <p>You could use "sliced portions of cones" as your infinitesimal volumes, but the answer would the same as if you used cylinders -- the difference between the two tends to zero faster than the volume itself, so it disappears in the limit. This is not the case with the surface area of a sliced portion of a cone -- its ... |
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 ... | Andrés E. Caicedo | 6,085 | <p>Colin, there are continuum many, as you suspect. </p>
<p>In fact, there are continuum many well-orderings of type $\omega$. The set of infinite binary sequences has size continuum. Given such a sequence $x=(x_0,x_1,\dots)$, let $i\in\{0,1\}$ be least such that $x_n=i$ infinitely often. Consider the enumeration of t... |
4,192,687 | <p>Let <span class="math-container">$f: [0,1] \rightarrow \mathbb{R}$</span> be a continuous function. <br />
How can I show that <span class="math-container">$ \lim_{s\to\infty} \int_0^1 f(x^s) \, dx$</span> exists?<br />
It is difficult for me to calculate a limit, as no concrete function or function values are given... | Matt E. | 948,077 | <p>To calculate the limit, use the fact that <span class="math-container">$f$</span> is continuous.
<span class="math-container">$$ \displaystyle\lim_{s\to\infty} f(x^s) = f\left( \lim_{s\to\infty} x^s\right) = f( \chi_{\{1\}}(x)).$$</span>
In other words, <span class="math-container">$\displaystyle\lim_{s\to\infty} f(... |
24,873 | <p>It is very elementary to show that $\mathbb{R}$ isn't homeomorphic to $\mathbb{R}^m$ for $m>1$: subtract a point and use the fact that connectedness is a homeomorphism invariant.</p>
<p>Along similar lines, you can show that $\mathbb{R^2}$ isn't homeomorphic to $\mathbb{R}^m$ for $m>2$ by subtracting a point ... | Pete L. Clark | 299 | <p>Well, I might recast your proofs of the first two cases as follows:</p>
<p>Suppose that $\mathbb{R}^n$ and $\mathbb{R}^m$ are homeomorphic. Then for any $P \in \mathbb{R}^n$, there must exist a point $Q \in \mathbb{R}^m$ such that $\mathbb{R}^n \setminus \{P\}$ and $\mathbb{R}^m \setminus \{Q\}$ are homeomorphic. ... |
24,873 | <p>It is very elementary to show that $\mathbb{R}$ isn't homeomorphic to $\mathbb{R}^m$ for $m>1$: subtract a point and use the fact that connectedness is a homeomorphism invariant.</p>
<p>Along similar lines, you can show that $\mathbb{R^2}$ isn't homeomorphic to $\mathbb{R}^m$ for $m>2$ by subtracting a point ... | Jesse Railo | 223,052 | <p>One possible approach is the following:</p>
<ol>
<li>Show <a href="https://en.wikipedia.org/wiki/Borsuk%E2%80%93Ulam_theorem" rel="noreferrer">Borsuk–Ulam theorem</a>.</li>
<li>Deduce that $S^n$ cannot be embedded to $\mathbb{R}^n$.</li>
<li>Let us consider $\mathbb{R}^n$ and $\mathbb{R}^m$, where $m > n$. Now $... |
1,858,297 | <p>Suppose the diameter of a nonempty set $A$ is defined as </p>
<p>$$\sigma(A) := \sup_{x,y \in A} d(x,y)$$</p>
<p>where $d(x,y)$ is a metric.</p>
<p>Is $\sigma(.)$ a 'measurement'? I.e., how do I prove the countable additivity for this particular case?</p>
| fleablood | 280,126 | <p>... not to mention</p>
<p>$\sigma( \text{rational numbers between A and B}) + \sigma( \text {irrational numbers between A and B}) \ne \sigma( \text{ real numbers between A and B})$. </p>
<p>This is pretty much the perfect example of something that absolutely can not be a measure and illustrates why we need a conc... |
806,532 | <p>This question takes place in a general metric space $X$. </p>
<p>Let $x$ be an interior* point of $E \subset X$ iff there exists a deleted neighborhood of $x$ that is contained in $E$. </p>
<p>This is like the normal definition of "interior point", except it uses "deleted neighborhood" instead of "neighborhood",... | echinodermata | 122,187 | <p>The notions of "interior point" and "limit point", as you point out, are not dual. And if we use interior point to define interior and limit point to define closure, then we've just used two non-dual notions to define two dual notions. This is highly dissatisfying, as you know. Which one should we keep and which one... |
3,450,598 | <blockquote>
<p>Prove that <span class="math-container">$\sum_{i = m}^n a_i + \sum_{i = n + 1}^p a_i = \sum_{i = m}^p a_i$</span>, where <span class="math-container">$m ≤ n<p$</span> are integers, and <span class="math-container">$a_i$</span> is a real number assigned to each integer <span class="math-container">$... | user | 505,767 | <p>I suppose it is convenient apply induction in a different way, that is</p>
<ul>
<li>base case, <span class="math-container">$p=n+1 \implies \sum_{i = m}^n a_i + \sum_{i = n + 1}^{n+1} a_i = \sum_{i = m}^n a_i + a_{n+1}=\sum_{i = m}^{n+1} a_i$</span></li>
</ul>
<p>and for the induction step assuming that <span cla... |
4,124,324 | <p>I am trying to find the complex function <span class="math-container">$f(z)$</span> who's derivative equals the complex conjugate of its reciprocal</p>
<p><span class="math-container">$$\dfrac{\mathrm{d} f(z)}{\mathrm{d} z} = \dfrac{1}{f(z)^*}$$</span></p>
<p>which is equivalent to</p>
<p><span class="math-container... | Aryaman Maithani | 427,810 | <p>(I shall use the notation <span class="math-container">$\overline{z}$</span> to denote the conjugate of <span class="math-container">$z$</span>. I will assume that your equation is defined on a nonempty open subset <span class="math-container">$\Omega \subset \Bbb C$</span>.)</p>
<p>There is no such <span class="mat... |
765,404 | <p>Can anyone explain the partial derivative below:</p>
<p>$\frac{\partial a^tX^{-1}b}{\partial X} = -X^{-t}ab^tX^{-t}$</p>
<p>I was trying to derive this equation using the below formula, but failed.</p>
<p><img src="https://i.stack.imgur.com/apR2q.png" alt="enter image description here"></p>
| Set | 26,920 | <p>Here's another way you might consider computing the derivative of <span class="math-container">$f(X)=a^TX^{-1}b$</span>,</p>
<p><span class="math-container">\begin{align}
f(X+H)=a^T(X+H)^{-1}b&=a^T((I+HX^{-1})X)^{-1}b\\[10pt]
&=a^TX^{-1}(I+HX^{-1})^{-1}b\\[1pt]
&=a^{T}X^{-1}\sum_{n=0}^\infty(-1)^n(HX^{-1... |
178,319 | <p>I asked this initially in <a href="https://math.stackexchange.com/questions/894399/identities-that-connect-antipode-with-multiplication-and-comultiplication">math.stackexchange</a>:</p>
<p>The group algebra $k(G)$ of any group $G$ satisfies as a Hopf algebra the following identities:
$$
S\otimes S\circ \Delta=\sigm... | Tilman | 4,183 | <p>That's true in any Hopf algebra. See Sweedler, <em>Hopf algebras</em>, Prop. 4.0.1.</p>
|
1,613,645 | <p>Let's get started:</p>
<p>$$\hat f(n) = \frac{1}{2\pi}\int_0^{2\pi} |x|e^{-inx} dx$$</p>
<p>since $|x|$ is an even function:</p>
<p>$$= \frac{1}{\pi}\int_0^{\pi} xe^{-inx} dx$$</p>
<p>Integration by parts yields:</p>
<p>$$e^{-inx}\Big|_0^{\pi} + \frac{1}{in} \int_0^\pi e^{-inx} dx = (-1)^n - 1 + \frac{1}{in} \l... | Akshat VIjoy | 546,552 | <p>The question is referring to the MINUTE hand, NOT hour hand. Read it properly.
The MINUTE hand does not only move 360 degrees in 12 hours. That's the HOUR hand. The MINUTE hand moves 360 degrees in an hour. This is because the MINUTE cycle restarts after an hour. It moves the full clock around after ONE hour, NOT TW... |
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... | gultu | 383,558 | <p>the integral is: </p>
<p><span class="math-container">$I= \int_{M_0}^{\infty} q(m,\mu,\sigma). \beta e^{-\beta (m-M_0)} dm \\
=\frac{1}{2}+\frac{1}{2} \beta e^{\beta M_0} * I_{11}$</span></p>
<p>where,
<span class="math-container">$I_{11}= \int_{M_0}^{\infty} erf(\frac{(m-\mu)}{\sigma \sqrt{2}}) e^{- \beta m} dm\... |
4,296,967 | <p>Is there any subset of the real numbers that is not a Unique Factorization Domain? (i.e. where within that subset, a "prime" is a number that cannot be written as a product of any numbers in that set except itself and 1, and where there is at least one number that can be written as the product of two diff... | Davide Trono | 494,745 | <p>Consider <span class="math-container">$\mathbb{Z}[\sqrt{10}]$</span>, then <span class="math-container">$9=3\cdot3=(1+\sqrt{10})(\sqrt{10}-1)$</span> and <span class="math-container">$3, 1+\sqrt{10}, \sqrt{10} - 1$</span> are all prime numbers in this ring (although the proof of them being prime is far from obvious)... |
4,296,967 | <p>Is there any subset of the real numbers that is not a Unique Factorization Domain? (i.e. where within that subset, a "prime" is a number that cannot be written as a product of any numbers in that set except itself and 1, and where there is at least one number that can be written as the product of two diff... | MH.Lee | 980,971 | <p>You asked the subset of real numbers, so I'll give an example <span class="math-container">$\mathbb Z[\sqrt5]$</span>.</p>
<p>In that case, <span class="math-container">$2, -2, 1\pm\sqrt5$</span> is all prime, but <span class="math-container">$-2\times2=(1-\sqrt5)(1+\sqrt5)$</span>.</p>
<p>So, <span class="math-cont... |
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>
| bryan.blackbee | 45,767 | <p>Consider re-writing the equation in different brackets. Mathematics has three different type of parentheses for a reason - to distinguish between each pair of brackets.
$$
\begin{align}
4(a-2(b-c)-(a-(b-2)))&=4\left\{a-2[b-c]-[a-(b-2)]\right\}\\
&=4\left\{a-2[b-c]-[a-b+2]\right\}\\
&=4\left\{a-2[b-c]-a+b... |
185,766 | <p>After studying general a linear algebra course, how would an advanced linear algebra course differ from the general course? </p>
<p>And would an advanced linear algebra course be taught in graduate schools?</p>
| Alexander Gruber | 12,952 | <p>At my <em>[undergraduate]</em> university <em>[which was University of Cincinnati, at the time of this post]</em>, the first linear algebra sequence is taught to sophomores. It is mostly computational. Everything takes place in the reals and complex numbers. The class begins with row reducing and culminates with ... |
3,197,540 | <p>Let a function be defined as:</p>
<p><span class="math-container">$ f(x)=x^2\sin{\left(\frac 1x\right)}$</span> for <span class="math-container">$x \neq 0$</span> and
<span class="math-container">$ f(x)=0$</span> for <span class="math-container">$x=0$</span></p>
<p>I'm trying to prove that f is differentiable at ... | Sri-Amirthan Theivendran | 302,692 | <p>You don't have to compute that limit. Indeed
<span class="math-container">$$
f'(0)=\lim_{h\to0}\frac{f(0+h)-f(0)}{h}=\lim_{h\to 0}\frac{f(h)}{h}=\lim_{h\to 0}h\sin(1/h)
$$</span>
which you can compute using the squeeze theorem since
<span class="math-container">$$
0\leq |h\sin(1/h)|\leq |h|.
$$</span></p>
|
1,322,016 | <p>Find the vertical asymptotes (if any) of the graph of the function. (Use $n$ as an arbitrary integer if necessary.)</p>
<p>$$s(t)= \frac{8t}{\sin{t}}$$</p>
<p>$t= ?$, where n cannot $=?$</p>
<p>I need a general rule for the asymptotes with where the exception of $n$ is. </p>
| OnceUponACrinoid | 246,291 | <p>Hint: A good way to find vertical asymptotes for a function defined as a fraction is to think of where the denominator vanishes, assuming you have the fraction in a factored from. Of course, those aren't the only possibilities for where vertical asymtotes may occur.</p>
<p>Also -- Have you tried graphing the given ... |
376,120 | <p>Let <span class="math-container">$X$</span> be a proper and smooth scheme over <span class="math-container">$\mathbf{C}$</span> and let <span class="math-container">$\mathbb{L}$</span> be a local system of finite dimensional <span class="math-container">$\mathbf{C}$</span>-vector spaces. By the Riemann Hilbert corre... | Chris | 116,075 | <p>This is true for any local system underlying a polarized <span class="math-container">$\mathbb{C}$</span>-variation of Hodge structure, e.g. it is true for <span class="math-container">$R^nf_*\mathbb{C}$</span> where <span class="math-container">$f:X\to Y$</span> is smooth proper and <span class="math-container">$n\... |
376,120 | <p>Let <span class="math-container">$X$</span> be a proper and smooth scheme over <span class="math-container">$\mathbf{C}$</span> and let <span class="math-container">$\mathbb{L}$</span> be a local system of finite dimensional <span class="math-container">$\mathbf{C}$</span>-vector spaces. By the Riemann Hilbert corre... | Donu Arapura | 4,144 | <p>Let me supplement Chris' answer with a few additional remarks. When <span class="math-container">$\mathcal{F}$</span> underlies a polarizable complex variation of Hodge structure, then it carries a filtration <span class="math-container">$F^\bullet\mathcal{F}$</span> which induces one on the de Rham complex
<span cl... |
300,460 | <p>How would we go about proving that $$\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \frac{1}{3 \cdot 4} +\ldots +\frac{1}{n(n+1)} = \frac{n}{n+1}$$</p>
| preferred_anon | 27,150 | <p><strong>Hint</strong>: $\frac{1}{n}-\frac{1}{n+1}=\frac{1}{n(n+1)}$</p>
|
300,460 | <p>How would we go about proving that $$\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \frac{1}{3 \cdot 4} +\ldots +\frac{1}{n(n+1)} = \frac{n}{n+1}$$</p>
| Adi Dani | 12,848 | <p>$$\sum_{i=1}^{n}\frac{1}{i(i+1)}=\sum_{i=1}^{n}\Big(\frac{1}{i}-\frac{1}{i+1}\Big)=$$
$$=1+\sum_{i=2}^{n}\frac{1}{i}-\Big(\sum_{j=2}^{n}\frac{1}{j}+\frac{1}{n+1}\Big)=1-\frac{1}{n+1}$$</p>
|
2,439,340 | <p>How would one proceed to prove this statement?</p>
<blockquote>
<p>The set of the strictly increasing sequences of natural numbers is not enumerable.</p>
</blockquote>
<p>I've been trying to solve this for quite a while, however I don't even know where to start.</p>
| DanielWainfleet | 254,665 | <p>For a countably infinite set $F$ of strictly increasing functions from $\Bbb N$ to $\Bbb N$ let $F=\{f_n:n\in \Bbb N\}.$ Define $g:\Bbb N \to \Bbb N$ by $g(n)=1+\sum_{j=1}^n f_j(n).$ </p>
<p>Then $g(n+1)-g(n)=f_{n+1}(n+1)+\sum_{j=1}^n (f_j(n+1)-f_j(n))>0$ so $g$ is strictly increasing. </p>
<p>And $g\not \in F$... |
2,439,340 | <p>How would one proceed to prove this statement?</p>
<blockquote>
<p>The set of the strictly increasing sequences of natural numbers is not enumerable.</p>
</blockquote>
<p>I've been trying to solve this for quite a while, however I don't even know where to start.</p>
| Eric Lippert | 21,264 | <p>As other answers note, there are lots of fancy ways to prove this. But we can always go back to the basics. A straightforward diagonalization proof-by-contradiction suffices. Suppose there is such an enumeration. Maybe this is it:</p>
<pre><code>1 --> 1, 2, 3, 5, ...
2 --> 4, 5, 7, 100, ...
3 --> 1, 2, 3, ... |
2,439,340 | <p>How would one proceed to prove this statement?</p>
<blockquote>
<p>The set of the strictly increasing sequences of natural numbers is not enumerable.</p>
</blockquote>
<p>I've been trying to solve this for quite a while, however I don't even know where to start.</p>
| Hagen von Eitzen | 39,174 | <p>There are uncountably many subsets of $\Bbb N$, but only countably many finite subsets, hence uncountably many infinite subsets.
Every strictly increasing sequence of naturals corresponds to an infinite subset of $\Bbb N$.</p>
|
3,185,317 | <p><span class="math-container">$$\lim_{n\rightarrow 0}\frac{1}{n}\int_{0}^{1}\ln(1+e^{nx})dx$$</span></p>
<p>My try:</p>
<p><span class="math-container">$$\frac{b-a}b\leq \ln b-\ln a\leq \frac{b-a}a \implies \frac{1}{1+e^{nx}}\leq \ln(1+e^{nx})-\ln e^{nx}\leq \frac1{e^{nx}}$$</span></p>
<p>Then I integrated and mul... | D.B. | 530,972 | <p>You are actually canceling the factor <span class="math-container">$x-1$</span> from numerator and denominator. This works as long as <span class="math-container">$x \ne 1$</span>. Keep in mind that in the limit, <span class="math-container">$x$</span> is approaching <span class="math-container">$1$</span>; never ... |
3,231,387 | <p>I have been given the following quadratic equation and is asked to find the range of its roots <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span>, where <span class="math-container">$\alpha>\beta$</span>
<span class="math-container">$$(k+1)x^2 - (20k+14)x + 91k +40 =0,$$<... | Mick | 42,351 | <p>First, we need to find for what values of k such that the equation has real roots (<span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span>).</p>
<p>To this end, we set <span class="math-container">$(10k + 7)^2 \ge (k+1)((91k + 40)$</span></p>
<p>This is reduced to <span class="... |
3,950,098 | <p>I can evaluate the limit with L'Hospital's rule:</p>
<p><span class="math-container">$\lim_{n\to\infty}n(\sqrt[n]{4}-1)=\lim_{n\to\infty}\cfrac{(4^{\frac1n}-1)}{\dfrac1n}=\lim_{n\to\infty}\cfrac{\dfrac{-1}{n^2}\times 4^{\frac1n}\times\ln4}{\dfrac{-1}{n^2}}=\ln4$</span></p>
<p>But is there any way to do it without us... | GEdgar | 442 | <p>You could try this. As <span class="math-container">$n \to \infty$</span>,
<span class="math-container">$$
4^{1/n} = \exp\left(\frac{\log 4}{n}\right)
= 1 + \frac{\log 4}{n} + O(1/n^2)
\\
4^{1/n}-1 = \frac{\log 4}{n} + O(1/n^2)
\\
n\left(4^{1/n}-1\right) = \log 4 + O(1/n)
\\
\lim_{n\to\infty} n\left(4^{1/n}-1\right... |
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... | Vishu | 751,311 | <p>Let <span class="math-container">$t=\frac 1n \to 0$</span>: <span class="math-container">$$\lim_{t\to 0} \frac{4^t-1}{t} $$</span> which is of the well-known form <span class="math-container">$\lim_{x\to 0} \frac{a^x-1}{x} =\ln a $</span>.</p>
|
663,563 | <p>it seems obvious that this integral is zero and so is the limit but what theorem we are using here?</p>
<p>I see it's connected to Riemann sums with an interval=zero Right ?</p>
<p>The function $\mathrm{f}$ is continuous.</p>
<p>$$\lim_{x \to 0}\int_0^x\mathrm{f}(x)\ \mathrm{d}x= \ ?$$</p>
| copper.hat | 27,978 | <p>Assuming $f$ is Riemann integrable, it is bounded by some $B$, so $0 \le |\int_0^x f(x) dx | \le \int_0^x |f(x)| dx \le Bx$.</p>
|
663,563 | <p>it seems obvious that this integral is zero and so is the limit but what theorem we are using here?</p>
<p>I see it's connected to Riemann sums with an interval=zero Right ?</p>
<p>The function $\mathrm{f}$ is continuous.</p>
<p>$$\lim_{x \to 0}\int_0^x\mathrm{f}(x)\ \mathrm{d}x= \ ?$$</p>
| Thomas | 26,188 | <p>We are not using any theorem. The definition of the definite integral is
$$
\int_a^b f(x) \; dx = \lim_{n\to \infty} \sum_{i=1}^n f(x_i)\Delta x.
$$
where $x_i = a + i\Delta x$ and $\Delta x = \frac{b - a}{n}$. If $a=b=0$, then $\Delta x = 0$ and so the integral is zero:
$$
\int_0^0 f(x)\; dx = \lim_{n\to \infty}\su... |
4,315,572 | <p>exercise:</p>
<p>Let us assume that the function f has derivatives of all orders.</p>
<p>Suppose that all zeros of <span class="math-container">$f$</span> have finite multiplicity. Let <span class="math-container">$a$</span> and <span class="math-container">$b$</span> be points of <span class="math-container">$A$</s... | TonyK | 1,508 | <p>I think it's just to avoid a proliferation of cases in the proof. If you allow <span class="math-container">$f(a)=0$</span>, for instance, you have to say "<span class="math-container">$n$</span> times right-differentiable" every time, instead of just "<span class="math-container">$n$</span> times dif... |
3,380,081 | <p>Question: Suppose <span class="math-container">$n(S)$</span> is the number of subset of <span class="math-container">$S$</span> and <span class="math-container">$|S|$</span> be the number of elements of <span class="math-container">$S$</span>. If <span class="math-container">$n(A)+n(B)+n(C)=n(A\cup B\cup C)$</span> ... | Paramanand Singh | 72,031 | <p>It is not necessary to choose equispaced division points. You can choose the division points as <span class="math-container">$$x_k=\left(1+\frac{k}{n}\right)^2$$</span> and use the Riemann sum <span class="math-container">$$S_n=\sum_{k=1}^{n}f(x_k)(x_k-x_{k-1})$$</span> where <span class="math-container">$f(x) =\sqr... |
173,286 | <p>I have these two functions <code>fun</code> and <code>microstep</code>.Fun makes use of a Module construct within which I define the <code>Array</code> I need to store the values of magnetization for different temperatures (each case stored in a different row).
<code>microstep</code> is the function that store the... | Henrik Schumacher | 38,178 | <p>Does this work out for you?</p>
<p>Here I added <code>magnetization</code> as additional argument and gave <code>microstep</code> the attibute <code>HoldAll</code> to allow for call by reference.</p>
<pre><code>SetAttributes[microstep, HoldAll];
microstep[tindex_, temp_, matrix_, mcindex_, magnetization_] :=
Mod... |
1,641,137 | <p>Let $(X,d)$ be a metric space, $a \in X$, and $\delta$ be a positive real number. Then the open ball $B(a;\delta)$ is defined as
$$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... | tomasz | 30,222 | <p>An example of a meaningful, sufficient condition for this is that $X$ is a <a href="https://en.wikipedia.org/wiki/Intrinsic_metric" rel="nofollow">length space</a>. This is certainly not a necessary condition: for example, a dense subspace of a length space also has this property (more generally, the property is inh... |
91,645 | <p>I asked a similar question previously, though this is more specific and directed. In the writing of mathematics research papers, when is information cited, such as definitions? I have read that if it is fairly recent, then cite it. But what is "fairly recent?" Also, should books from whence a definition came be ... | Potato | 18,240 | <p>The first rule of academic honesty is: if in doubt, always cite. There is simply no downside, and it may help readers unfamiliar with the literature.</p>
|
3,701,582 | <p>I still struggle mighty with basic conceptions of truth and proof. </p>
<p>For example: The Continuum Hypothesis (CH) is either true or false, i.e. either CH or ~CH holds. Now, Goedel and Cohen proved that CH/~CH are independent from ZFC, so ZFC + CH and ZFC + ~CH are consistent (in case ZFC is consistent but mathe... | Reveillark | 122,262 | <p>Before getting to the mathematics, here is some philosophy: </p>
<p>To a first approximation, there are three ways of approaching mathematics:</p>
<ul>
<li><p><em>Platonism</em> is the belief that mathematical statements have an intrinsic truth value, and that mathematical objects really exist in some ideal univer... |
3,387,458 | <p>Show that a bounded sequence having one limit point is convergent. </p>
<p>The converse holds true. The fact that a convergent seq is bounded has been shown in Baby Rudin. The fact that it will have only one limit point can be found <a href="https://math.stackexchange.com/questions/3386703/prove-that-a-convergent-s... | user284331 | 284,331 | <p>Consider <span class="math-container">$\limsup x_{n}=\lim_{k}x_{n_{k}}$</span> and <span class="math-container">$\liminf x_{n}=\lim_{l}y_{n_{l}}$</span> for some subsequences of <span class="math-container">$(x_{n_{k}})$</span> and <span class="math-container">$(y_{n_{l}})$</span> of <span class="math-container">$(x... |
2,973,314 | <p>If we have to find the sum of n terms of a G.P. then we have two formulas for it (1) <span class="math-container">$a(1-r^n)/(1-r)$</span> and (2) <span class="math-container">$a(r^n-1)/(r-1)$</span>. Now I know how the (1) has been derived but dont know about the (2)(is it obtained by multiplying denominator and num... | Dr. Sonnhard Graubner | 175,066 | <p>Multiply numerator and denominator by <span class="math-container">$(-1)$</span>
<span class="math-container">$$\frac{a(r^n-1)(-1)}{(r-1)(-1)}$$</span></p>
|
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>
| preferred_anon | 27,150 | <p>I think the easiest way to express what a limit really means, is to say that you get arbitrarily close to the limit as you get closer and closer to your desired input. </p>
<p>As $x$ goes to infinity, $\sin(x)$ and $\cos(x)$ take the values $-1$ and $1$ infinitely often, and therefore do not get as close as we mi... |
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>
| Thomas | 26,188 | <p>The limit
$$
\lim_{x\to \infty} \sin(x)
$$
does not exist. What we mean by saying that it doesn't exist is that there is not an $L$ such that $\sin(x)$ can be made arbitrarily close to $L$ for $x$ "large enough". You could try to prove that such an $L$ doesn't exist by assuming that such did exist and then getting a... |
869,506 | <p>In a paper I am reading, there is a step that seems to come from the following inequality:
$$(1+x)^\alpha \le 1+2^\alpha x,$$
where $0<x<1$. (Also, $3\le \alpha \le 9/2$ in the context of the paper, but the above probably holds for more general $\alpha$, say, $\alpha\ge 1$.) It is stated with no explanation, a... | robjohn | 13,854 | <p>Consider
$$
f(x)=1+(2^\alpha-1)x-(1+x)^\alpha\tag{1}
$$
Note that $f(0)=f(1)=0$. The Mean Value Theorem says that for some $0\lt x_\alpha\lt1$, we have $f'(x_\alpha)=0$.</p>
<p>Furthermore, since $\alpha-1\ge0$, $f'(x)=(2^\alpha-1)-\alpha(1+x)^{\alpha-1}$ is non-increasing.</p>
<p>Thus, $f'(x_\alpha)\ge0$ for $0\l... |
3,602,323 | <p>Let <span class="math-container">$ m $</span>, <span class="math-container">$ m+1 $</span>, <span class="math-container">$ m+2 $</span>, <span class="math-container">$ \dots $</span>, <span class="math-container">$ m+p-1 $</span> be an integers and let <span class="math-container">$ p $</span> be an odd prime. I wan... | metamorphy | 543,769 | <p>Yet another way of computing the sum, or (following @Servaes) even more general <span class="math-container">$$S_k=\sum_{a\in\mathbb{F}_q^{\times}}a^k$$</span> where <span class="math-container">$\mathbb{F}_q$</span> is a finite field with <span class="math-container">$q$</span> elements (<span class="math-container... |
614,962 | <blockquote>
<p>We have a continuous function <span class="math-container">$f:(a,b)\to \mathbb R$</span></p>
<p>Prove that: <span class="math-container">$\forall n: x_1...x_n\in(a,b):\exists x\in(a,b)$</span> such that:</p>
<p><span class="math-container">$$f(x)=\frac1n ( f(x_1)+...+f(x_n) ) $$</span></p>
</blockquote>... | DonAntonio | 31,254 | <p>$$\frac{x\csc x}{\cos 2x}=\frac{2x}{\sin 2x}\frac1{2\cos 5x}\xrightarrow[x\to 0]{}1\cdot\frac12$$</p>
|
614,962 | <blockquote>
<p>We have a continuous function <span class="math-container">$f:(a,b)\to \mathbb R$</span></p>
<p>Prove that: <span class="math-container">$\forall n: x_1...x_n\in(a,b):\exists x\in(a,b)$</span> such that:</p>
<p><span class="math-container">$$f(x)=\frac1n ( f(x_1)+...+f(x_n) ) $$</span></p>
</blockquote>... | Adi Dani | 12,848 | <p>$$\lim_{x \to 0} \frac{x \csc(2x)}{\cos(5x)}=\lim_{x \to 0} \frac{x }{\sin2x\cos(5x)}$$
$$=\lim_{x\to0}\frac{2x}{\sin(2x)}\frac{1}{2\cos(5x)}=\frac{1}{2}$$</p>
|
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... | Robert Shore | 640,080 | <p>The difference is that a <span class="math-container">$50$</span>% loss and a <span class="math-container">$50$</span>% gain (in either sequence) result in a net loss (AM-GM inequality), whereas halving and doubling (in either sequence) do not result in a net loss. Joshi is presenting (and solving) a different prob... |
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... | Ross Millikan | 1,827 | <p>The crucial thing is that Wilmott asks about the chance of making a profit, regardless of how large the profit or loss is. Joshi is asking about expected value of the portfolio. Those are very different questions. If I pay <span class="math-container">$1$</span> to bet on something and win <span class="math-conta... |
297,812 | <p>If $a-b=b-c$ .How to find the value of $a^2-2b^2+c^2$</p>
| Peder | 59,704 | <p>$a-b=b-c$ means $b=(a+c)/2$ therefore $a^2-2b^2+c^2=a^2+c^2-2(\frac{a+c}{2})^2=\frac{2a^2+2c^2-(a+c)^2}{2}=\frac{a^2+c^2-2ac}{2}=\frac{(a-c)^2}{2}$</p>
|
297,812 | <p>If $a-b=b-c$ .How to find the value of $a^2-2b^2+c^2$</p>
| Aang | 33,989 | <p>$a-b=b-c\implies a,b,c$ are in A.P.</p>
<p>$a^2-2b^2+c^2=(a^2-b^2)-(b^2-c^2)=(a-b)(a+b)-(b-c)(c+b)=(a-b)(a+b)-(a-b)(c+b)=(a-b)(a+b-c-b)=(a-b)(a-c)$</p>
|
2,246,025 | <p>How do I solve this?
$$y^\prime = y^2 -4$$
I think I am supposed to use the separable equations method and then use partial fractions.</p>
| Harambe | 357,206 | <p>Disclaimer: technically I'm not answering your question, because I have used the word "structure" in a more general way than I believe you intended. Also I feel like talking about some maths I love.</p>
<p>I can't say much about "recently discovered", but there are two main approaches to adding "structure" to sets:... |
2,246,025 | <p>How do I solve this?
$$y^\prime = y^2 -4$$
I think I am supposed to use the separable equations method and then use partial fractions.</p>
| epi163sqrt | 132,007 | <p>With respect to quaternions you mention, there were some <em>exciting</em> new numbers, called <strong>periods</strong> discovered just a few years ago.</p>
<blockquote>
<p>The definition of periods below is from the fascinating introductory <a href="http://www.maths.ed.ac.uk/%7Eaar/papers/kontzagi.pdf" rel="nofollo... |
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 $... | epi163sqrt | 132,007 | <blockquote>
<p>A function $f(x)$ analytic at $x=0$ can be represented as power series within an open disc with radius of convergence $R$.
\begin{align*}
f(x)=\sum_{n=0}^\infty a_nx^n\qquad\qquad \qquad |x|<R
\end{align*}</p>
<p><em>Any</em> substitution $x=g(u)$ is admissible as long as we respect the <em>... |
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... | 2'5 9'2 | 11,123 | <p>You are correct. And you can tell your housemate that you are using conditional probability for the second part of your calculation. <em>Given</em> that you did not complete a full house on roll 1, the probability of rolling it on the second try is $\frac{1}{3}$, just as he says. However, that scenario is only on... |
4,453,784 | <p>For a time homogeneous Markov chain <span class="math-container">$(X_n)_{n\ge 0}$</span> with state space <span class="math-container">$I$</span> with no self loop . Given <span class="math-container">$X_0 = i \in I$</span> , define
the first return time <span class="math-container">$T_i = \inf\{n\ge 1 : X_n = i\} $... | Mason | 752,243 | <p>I'd write like this:
<span class="math-container">\begin{align}
P_i(T_i < \infty) &= \sum_{j \in I}P_i(T_i < \infty \mid X_1 = j)P_i(X_1 = j)
\end{align}</span>
Now, in order to use the Markov property, we write <span class="math-container">$T_i$</span> as <span class="math-container">$T_i(X_{0 + \cdot})$<... |
4,453,784 | <p>For a time homogeneous Markov chain <span class="math-container">$(X_n)_{n\ge 0}$</span> with state space <span class="math-container">$I$</span> with no self loop . Given <span class="math-container">$X_0 = i \in I$</span> , define
the first return time <span class="math-container">$T_i = \inf\{n\ge 1 : X_n = i\} $... | Calvinfwc | 897,319 | <p>I think @Mason suggests to more explicitly write the stoping times in terms of random variable , so I try the following approach . Consider the expression on then RHS of the second equality in the original post , I first try to show ,</p>
<blockquote>
<p><span class="math-container">$$
\sum_{j\in I} P(T_i < \inft... |
623,819 | <p>I do not understand a remark in Adams' Calculus (page 628 <span class="math-container">$7^{th}$</span> edition). This remark is about the derivative of a determinant whose entries are functions as quoted below.</p>
<blockquote>
<p>Since every term in the expansion of a determinant of any order is a product involving... | Toan Nguyen Dinh | 118,406 | <p>That remarks has said most of what it needs to explain.However, I think a more precise explaination for the example is necessary.Hence, I'll cite one.
<br>
i) $a_{11}(t).a_{23}(t).a_{32}(t) $ is abitrary term in expansion of left determinant<br>
ii) $ (a_{11}(t).a_{23}(t).a_{32}(t))' = a'_{11}(t)a_{23}(t).a_{32}(t)... |
623,819 | <p>I do not understand a remark in Adams' Calculus (page 628 <span class="math-container">$7^{th}$</span> edition). This remark is about the derivative of a determinant whose entries are functions as quoted below.</p>
<blockquote>
<p>Since every term in the expansion of a determinant of any order is a product involving... | pikunsia | 178,025 | <p>If a $n$x$n$ matrix $A(t)=A_{ij}(t)$ is differentiable, then $d(det(A(t))/dt$ can be performed as follows (for brevity, we shall take the particular case $n=3$). The determinant of a matrix $A(t)=A_{ij}(t)$ is given by
$$det(A_{ij}(t))=e_{ijk}A_{i1}(t)A_{j2}(t)A_{k3}(t),$$
where $e_{ijk}$ is the Cartesian alternator... |
2,090,512 | <p>You can calculate the <strong>volume of a parallelepiped</strong> by $|(A \times B) \cdot C|$, where $A$, $B$ and $C$ are vectors. I wonder does the order matter? If it does how, is it determined. I know I can just put it in a matrix and calculate the determinant but I would like to know how it is in this case. </p... | TheGeekGreek | 359,887 | <p>We have $a \times b = -(b \times a)$. So you get also a change of sign since the (real) standard inner product is symmetric, so there the order does not matter. The reason why the cross product changes its sign if you permute the arguments, is simply that it is like the determinant an alternating mapping (this is im... |
4,298,184 | <p><a href="https://i.stack.imgur.com/f5ny0.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/f5ny0.png" alt="enter image description here" /></a>
as we can see, we are supposed to use stars and bars where n = 10 and r = 3. but what i dont understand is why we use stars and bars when stars and bars is ... | Mina | 777,827 | <p>You can assume that you have 10 balls and 2 sticks. The different ways you put these sticks between these balls divide your balls into three groups. The number of the balls in the first group is <span class="math-container">$x_1$</span> the number of the balls in the second group is <span class="math-container">$x_2... |
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-... | Sunny | 304,150 | <p>Only a partial answer. Suppose <span class="math-container">$n \geq 3$</span> be the characteristic of ring. Then <span class="math-container">$n.1 = 0$</span> implies that <span class="math-container">$ 1 = -(n-1).1 = -(n-2).1 + (-1).1$</span>, since <span class="math-container">$-1$</span> is unit. If <span class=... |
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 + ... | Florian | 1,609 | <p>Here is a hint: Find a formula for the number of possibilities to choose $k$ linearly indepenent vectors in $\mathbb{F}_p ^n$ (where order matters). Each of these choices serves as a basis for a $k$-dimensional subspace, but for each subspace there are several bases, so you have to divide by the number of bases for ... |
1,765,022 | <p>The problem is:</p>
<blockquote>
<p>$Prove$ $that$ $|\sin^2 (x)-\sin^2 (y)|\le |x-y|$ $ for $ $ all $ $ x,y>0$.</p>
</blockquote>
<p><em>$My$ $work$ :</em> $$\sin^2 (x)\le|\sin x|\le|x|\le|x-y|+|y|$$ and so is $$|\sin^ 2 (x)-\sin^2 (y)|\le |x-y|+|y|$$ But this is not the actual result I want. I think I have ... | Michael Hardy | 11,667 | <p>The integral $\displaystyle\int_0^R\int_0^R\cdots \,dx\,dy$ is over a <b>square</b>, $[0,R]^2$.</p>
<p>But the integral $\displaystyle \int_0^\text{something} \int_0^R \cdots \, r \, dr \, d\theta$ is over a <b>sector of a circle</b>, with an arc of a circle as a part of its boundary.</p>
|
3,033,812 | <p>My problem: If there are 5 different candies in a jar and a child wants to take out one or more candies, how many ways can this be done? </p>
<p>I said it is <span class="math-container">$^5C_1 -\; ^5C_0 = 5-1 = 4$</span> ways. The <span class="math-container">$-1$</span> for the unwanted case using this trick:</p>... | Michael Burr | 86,421 | <p>Adding to the other answers:</p>
<p><span class="math-container">$^5C_1-\, ^5C_0$</span> doesn't make sense for the following reasons:</p>
<ul>
<li><p>The problem states that the child takes one <em>or more</em> candies. None of these quantities in the expression includes data about the case where more than one c... |
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="... | Sander | 14,625 | <pre><code> ListPointPlot3D[
Table[{Cos[t], Sin[t], 2 + Sin[t] Cos[t]^2} ,{t, 0, π, 0.01}] ,
Filling -> 0]
</code></pre>
<p><img src="https://i.stack.imgur.com/Ey0fO.png" alt="Mathematica graphics"></p>
|
1,167,880 | <p>Given $a = [-5, 8, 1]$ and $b = [2, -7, -3]$, find a vector $c$ such that $a \cdot (b × c) = 0$</p>
<p>I don't know how to get it, I've been looking for examples, but I don't know..</p>
| Jason Knapp | 8,454 | <p>What you said is exactly right, that $\delta$ can be selected independently on bounded intervals but not on unbounded ones. You have the picture from your link you consider, but if you want more what you should think about is this - for $x^2$ the $\delta$ you need to use becomes smaller (to $0$) based on the size o... |
3,452,707 | <p>It is well known that <span class="math-container">$\sum_{k=0}^n{n\choose k} =2^n$</span>.</p>
<p><strong>My question:</strong> If <span class="math-container">$z$</span> is the limit point of an infinite sequence of real numbers <span class="math-container">$\{ a_n \}$</span>, then does <span class="math-container"... | QC_QAOA | 364,346 | <p>Let <span class="math-container">$\epsilon>0$</span> and define <span class="math-container">$b_n=a_n-z$</span>. Since <span class="math-container">$a_n\to z$</span> there exists <span class="math-container">$N$</span> such that for all <span class="math-container">$n\geq N$</span>, <span class="math-container">$... |
148,972 | <p>I am working with solving a linear system that becomes a tridiagonal matrix. In order to speed up the process for large matrices, I want to use sparse matrices. My problem is that the values are not constant along the bands, but change based on their horizontal position in the matrix (x position, if you will). For i... | kglr | 125 | <pre><code>func1[a_, b_] := a + b;
func2[a_, b_] := 1 + a + b;
func3[a_, b_] := a + 2 b;
n = 10;
sa = SparseArray[{Band[{1, 1}] -> (func1[0, #] & /@ Range[n]),
Band[{1, 2}] -> (func2[0, #] & /@ Range[n - 1]),
Band[{2, 1}] -> (func3[0, #] & /@ Range[n - 1])}, {n, n}];
sa // MatrixForm
</... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.