qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
2,481,767 | <p>Let A={$3m-1|m\in Z$} and B={$4m+2|m\in Z$} and let $f:A\rightarrow B$ is defined by </p>
<p>$f(x)=\frac{4(x+1)}{3}-2$ . Is f surjective?</p>
<p>I'm not really sure how to prove this. By trying out certain values it seems it's surjective. This is my work so far:</p>
<p>$f(x)=y \iff \frac{4(x+1)}{3}-2 = y \iff x=\... | Arnaldo | 391,612 | <p>Just see that </p>
<p>$$3m+2=3(m+1)-1=3M-1$$</p>
<p>and $M\in \Bbb Z$</p>
|
4,037,697 | <p><span class="math-container">$a_n $</span> is a sequence defined this way:
<a href="https://i.stack.imgur.com/nHdiD.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/nHdiD.png" alt="enter image description here" /></a></p>
<p>and we define: <a href="https://i.stack.imgur.com/b01hX.png" rel="nofollow... | Nij | 226,212 | <p>Orthogonality is forced by requiring a scalar multiple of the identity; if any two columns <span class="math-container">$a_i, a_j$</span> were not orthogonal, there would be a nonzero entry in the corresponding entries <span class="math-container">$b_{ij}, b_{ji}$</span> of the product.</p>
<p>Orthogonality results ... |
4,492,566 | <blockquote>
<p>To which degree must I rotate a parabola for it to be no longer the graph of a function?</p>
</blockquote>
<p>I have no problem with narrowing the question down by only concerning the standard parabola: <span class="math-container">$$f(x)=x^2.$$</span></p>
<p>I am looking for a specific angle measure. O... | orangeskid | 168,051 | <p>HINT:</p>
<p>The convex region <span class="math-container">$\{(x,y)\ | \ y \ge x^2\}$</span> has <span class="math-container">$\{0\} \times [0, \infty)$</span> as <a href="https://en.wikipedia.org/wiki/Recession_cone" rel="noreferrer">recession cone</a>.</p>
<p><span class="math-container">$\bf{Added:}$</span> We c... |
48,864 | <p>I can't resist asking this companion question to the <a href="https://mathoverflow.net/questions/48771/proofs-that-require-fundamentally-new-ways-of-thinking"> one of Gowers</a>. There, Tim Dokchitser suggested the idea of Grothendieck topologies as a fundamentally new insight. But Gowers' original motivation is to ... | Kimball | 6,518 | <p>I suppose asymptotics for certain functions (e.g., Prime Number Theorem), or any sort of conjecture based on large empirical evidence, would count, but that's probably not what you mean.</p>
<p>Perhaps more interesting is the following. In high school/college, I was briefly interested in automated theorem proving ... |
48,864 | <p>I can't resist asking this companion question to the <a href="https://mathoverflow.net/questions/48771/proofs-that-require-fundamentally-new-ways-of-thinking"> one of Gowers</a>. There, Tim Dokchitser suggested the idea of Grothendieck topologies as a fundamentally new insight. But Gowers' original motivation is to ... | John Pardon | 35,353 | <p>I disagree with the premise of this question.</p>
<p>Conventional computers follow a program written by a human. I think, for example, Daniel Moskovich's answer about simplicial sets is something that a <em>human programming a computer</em> (or a <em>computer scientist</em>) would think of when trying to program a... |
2,014,366 | <p>first of all it's not an exam sheet or some kinda stuff. I'm just preparing myself about quantifiers.</p>
<p>i couldn't find similar task to this one so had to ask here. </p>
<hr>
<p>Let '$x \mathrel{\heartsuit} y$' stand for 'x loves y'. Rewrite the sentence 'Someone loves everyone' using quantifiers in two dif... | Bram28 | 256,001 | <p>Maybe the second interpretation would be something along the lines of: there is someone for eveyone' ... So for evryone there is someone who loves them. So switch the quatifiers for the second reading?</p>
|
3,076,504 | <p>The problem states: </p>
<p>Right Triangle- perimeter of <span class="math-container">$84$</span>, and the hypotenuse is <span class="math-container">$2$</span> greater than the other leg. Find the area of this triangle. </p>
<p>I have tried different methods of solving this problem using Pythagorean Theorem and... | Creep Anonymous | 564,710 | <p>Let the sides of the right triangle be <span class="math-container">$x,y,x+2$</span>.</p>
<p>Given, </p>
<p><span class="math-container">$2x+y=82 \tag{1}$</span></p>
<p><span class="math-container">$x^2 + y^2 = (x+2)^2 \tag{2}$</span>
<span class="math-container">$$\implies x^2 + y^2 = x^2 +4x+4 $$</span>
<sp... |
3,538,305 | <blockquote>
<p>Given that the differential equation</p>
<p><span class="math-container">$f(x,y) \frac {dy}{dx} + x^2 +y = 0$</span> is exact and <span class="math-container">$f(0,y) =y^2$</span> , then <span class="math-container">$f(1,2)$</span> is</p>
</blockquote>
<p>choose the correct option</p>
<p><span cla... | Math_Is_Fun | 590,763 | <p><a href="https://i.stack.imgur.com/DBpj5.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/DBpj5.jpg" alt="enter image description here"></a></p>
<p>This is the solution, your answer should be option (A). </p>
<p>Let me know if you have questions. </p>
|
3,538,305 | <blockquote>
<p>Given that the differential equation</p>
<p><span class="math-container">$f(x,y) \frac {dy}{dx} + x^2 +y = 0$</span> is exact and <span class="math-container">$f(0,y) =y^2$</span> , then <span class="math-container">$f(1,2)$</span> is</p>
</blockquote>
<p>choose the correct option</p>
<p><span cla... | user577215664 | 475,762 | <p><span class="math-container">$$f(x,y) \frac {dy}{dx} + x^2 +y = 0$$</span>
<span class="math-container">$$(x^2+y)dx+fdy=0$$</span>
<span class="math-container">$$Mdx+Ndy=0$$</span>
<span class="math-container">$$ {\partial_y} M=\partial_x N $$</span>
<span class="math-container">$$ \implies 1=\partial_x f$$</span>
A... |
2,947,829 | <p>Indeed this challenges my intuition of how different functions (that are not necessarily differentiable) interact to become differentiable which is nice.</p>
<p>I wonder if my proof suffices to show that it is indeed differentiable at <span class="math-container">$x = 0$</span>.</p>
<p><span class="math-container"... | James | 549,970 | <p>Notice </p>
<p><span class="math-container">$$ \lim_{x \to 0^-} \frac{ f(x)-f(0)}{x-0} = \lim_{x \to 0^-} \frac{ x \cdot (-x)}{x} = \lim_{x \to 0^-} - x = 0 $$</span></p>
<p>and </p>
<p><span class="math-container">$$ \lim_{x \to 0^+} \frac{ f(x)-f(0)}{x-0} = \lim_{x \to 0^+} \frac{ x \cdot (+x)}{x} = \lim_{x \to... |
2,947,829 | <p>Indeed this challenges my intuition of how different functions (that are not necessarily differentiable) interact to become differentiable which is nice.</p>
<p>I wonder if my proof suffices to show that it is indeed differentiable at <span class="math-container">$x = 0$</span>.</p>
<p><span class="math-container"... | Mohammad Riazi-Kermani | 514,496 | <p>Your proof is correct and very elegant indeed. </p>
<p>You have said that it is anti-intuitive for this function to be differentiabe at <span class="math-container">$x=0$</span>, but if you look at the graph of this function you notice that it is very smooth at <span class="math-container">$x=0$</span>, so it is n... |
2,418,448 | <p>Let $\mathbb{N}^{\mathbb{N}}$ be the set of all sequences of positive integers. For $a=(n_1,n_2,\cdots),b=(m_1,m_2,\cdots)$ define $d(a,b)=\frac{1}{\min(i:n_i\neq m_i)}, a\neq b$, $d(a,a)=0$</p>
<hr>
<p>It can be easily shown that $d(a,b)\leq \max\{d(a,c),d(b,c)\}$, I want to prove that if two balls in this metric... | orangeskid | 168,051 | <p>HINT: </p>
<p>Yours is an ultrametric space ( the stronger inequality quoted is satisfied).
So why don't you prove this fact: if $B = B(x_0, r)$ is a ball in this space and $x$ a point in $B$ then $B = B(x,r)$. Then your statement is clear, the ball of larger radius contains the other. </p>
|
2,532,280 | <p>If a N×N (N≥3) Hermitian matrix <strong>A</strong> meets the following conditions: </p>
<ol>
<li><strong>A</strong> is positive semi-definite (not positive definite, i.e. <strong>A</strong> has at least M zero eigenvalue, where M is a given paremeter with 1≤M≤N-1).</li>
<li>The sum of each off diagonal results in 0... | Maadhav | 416,874 | <p>$4x^2+(5k+3)x+(2k^2-1)=4x^2-\alpha^2$</p>
<p>$k=\frac{-3}5$</p>
<p>${ \alpha} ^2=1-2\times \frac{9}{25}=\frac{7}{25}$</p>
<p>$ \alpha =\pm \frac{ \sqrt {7}}{5}$</p>
<p>$\alpha \in \mathbb{R}$ so value of $k$ is valid.</p>
|
2,713,391 | <p>I can find the answers to similar questions online, but what I'm trying to do is develop my own intuition so I can find the answers. I am quite sure I am wrong, so could you look over my reasoning?</p>
<p>If $X = (1,2,3,4,5,6,7,8)$,</p>
<blockquote>
<ol>
<li>How many strings over X of length 5?</li>
</ol>
</... | leonbloy | 312 | <p>The problem with the third argument is that it counts some valid strings twice (or more).</p>
<p>For example the string <code>12341</code> is counted as <code>[1][2341]</code> and <code>[1234][1]</code>.</p>
<p>The other way (subtracting the number of strings without '1's from the number of total strings) is corr... |
4,482,707 | <p><a href="https://math.stackexchange.com/questions/942263/really-advanced-techniques-of-integration-definite-or-indefinite/1885401#1885401">Here</a>, I saw the following formula:</p>
<p><span class="math-container">$$\int_{0}^{\infty }\frac{f(t)}{t}dt=\int_{0}^{\infty }\mathcal{L}\left \{ f(t) \right \}ds$$</span></p... | Rebecca J. Stones | 91,818 | <p>You're quite right in that
<span class="math-container">$$
R_1 = \{(x,y):y=x+2,\ x \in X,\ y \in Y\} = \{(1,3),(3,5),(5,7)\}
$$</span>
which is a <a href="https://en.wikipedia.org/wiki/Relation_(mathematics)" rel="nofollow noreferrer">relation</a> over <span class="math-container">$X$</span> and <span class="math-co... |
293,341 | <p>My apologies if this question is more appropriate for mathisfun.com, but I can only get so far reading about combinatrics and set theory before the interlocking logic becomes totally blurred. If this is a totally fundamental concept, feel free just to name it so I can read and understand the math myself.</p>
<p>So ... | Student | 137,056 | <p>I am not sure if this is still relevant to you but I will make the following suggestions that have helped me have that click moment when it comes to proofs. </p>
<p>1) You need to realize how important definitions are. How can you prove a statement involving a concept that you do not fully grasp? It is essential to... |
2,414,011 | <p>In my recent works in PDEs, I'm interested in finding a family of cut-off functions satisfying following properties:</p>
<p>For each $\varepsilon >0$, find a function ${\psi _\varepsilon } \in {C^\infty }\left( \mathbb{R} \right)$ which is a non-decreasing function on $\mathbb{R}$ such that:</p>
<ol>
<li>${\psi... | md2perpe | 168,433 | <p>First let $\Psi(x) = e^{-1/(1-x^2)}$ for $|x|<1$ and $=0$ otherwise (as <a href="https://en.wikipedia.org/wiki/Bump_function#Examples" rel="nofollow noreferrer">here</a>). This satisfies $\Psi \in C_c^\infty(\mathbb R)$ and $0 \leq \Psi(x) \leq e^{-1}$. </p>
<p>Then let $A = \int_{-\infty}^{\infty} \Psi(x) \, dx... |
1,268,431 | <p>$$\lim_{x\to 2} \frac {\sin(x^2 -4)}{x^2 - x -2} $$</p>
<p>Attempt at solution:</p>
<p>So I know I can rewrite denominator:</p>
<p>$$\frac {\sin(x^2 -4)}{(x-1)(x+2)} $$</p>
<p>So what's next? I feel like I'm supposed to multiply by conjugate of either num or denom.... but by what value...?</p>
<p>Don't tell me ... | Martin Argerami | 22,857 | <p>$$
\frac{\sin(x^2-4)}{(x+1)(x-2)}=\frac{\sin(x^2-4)}{x^2-4}\,\frac{x^2-4}{(x+1)(x-2)}
$$</p>
|
271,844 | <p>I've installed the KnotTheory package, following the instructions <a href="http://katlas.org/wiki/Setup" rel="nofollow noreferrer">here</a>. But when I try to use it I get this error:</p>
<p><code>$CharacterEncoding: The byte sequence {139} could not be interpreted as a character in the UTF-8 character encoding.</co... | Nasser | 70 | <p>I just tried it. It works for me, it loads and I can call its functions, but I keep getting warning <code>Get::path: ParentDirectory[File] in $Path is not a string.</code> even though it does give the same output as on the web site.</p>
<p>I think you need to edit <code>init.m</code> to fix these. It seems like old ... |
1,721,565 | <p>I'm having trouble with what I have done wrong with the chain rule below. I have tried to show my working as much as possible for you to better understand my issue here.</p>
<p>So:</p>
<p>Find $dy/dx$ for $y=(x^2-x)^3$
<br> So power to the front will equal = $3(x^2-x)^2 * (2x-1)$</p>
<p>Where did the $-1$ come f... | Aditya Agarwal | 217,555 | <p>So you are evidently confused in how the derivative of $x^2-x$ is computed.
We know that $\frac{d}{dx}k f(x)=k\frac d{dx}f(x)$ and $\frac d{dx}(f(x)+g(x))=\frac d{dx}f(x)+\frac d{dx}g(x)$. <br> So
$$\frac d{dx}(x^2-x)=\frac d{dx}x^2+\frac d{dx}(-x)=2x+(-1)\frac d{dx}(x)=2x+(-1)1=2x-1$$</p>
|
715,361 | <p>Let $\Omega$ be a bounded domain and $f_n\in L^2(\Omega)$ be a sequence such that
$$\int_\Omega f_nq\operatorname{dx}\leq C<\infty\qquad \text{for all}\quad q\in H^1(\Omega),\ \|q\|_{H^1(\Omega)}\leq1,\ n\in\mathbb{N}.\quad (1) $$
Is it then possible to conclude that
$$ \sup_{n\in\mathbb{N}}\|f_n\|_{L^2(\Omega)}... | naslundx | 130,817 | <p>We may never divide by 0, so we require for the solution that the denominator $(x-1) \not =0$, meaning $x \not= 1$.</p>
<p>However, since the only possible value for $x$ (given by both you and the trainer) is $x=1$. Hence we are forced to conclude that no solutions exist. </p>
|
2,026,486 | <p>Let $h,g$ be <em>not</em> injective functions, can function $f:\mathbb{R}\rightarrow \mathbb{R}^2$ such that $f(x) = (h(x), g(x))$ be injective?</p>
<p>I know that, if I pick polynomials for $h$ and $g$, then may be not injective, if picked carefully. For example, I can check zeros of the polynomials, whether they ... | Arthur | 15,500 | <p>$f$ can be injective. For instance, $g(x) = x^2, g(x) = (x + 1)^2$. Then $f(x) = (x^2, (x+1)^2)$ is injective, since $g(a) = g(b)$ and $a \neq b$ means $a = -b$, but $h(a) \neq h(-a)$. Thus, if $a$ and $b$ are such that the first component of $f(a)$ is equal to the first component of $f(b)$, then the second componen... |
1,548,771 | <p>I have come up with the following constrained minimization problem:
\begin{eqnarray}
\min\ \sum_{i=1}^\infty x_i^2\\
\sum_{i=1}^\infty a_ix_i=1
\end{eqnarray}
If it were a finite-dimensional case it would be easily solved via Lagrange multipliers; in this case I ask your help since I don't know where to begin.</p>
| Amr | 29,267 | <p>Assume wlog that there are no zero terms in the sequence by 'ommiting' them.</p>
<p>Assuming that the sequence $a_i $ is square summable, then one can easily use cauchy schwartz inequality to show that </p>
<p>$$x_i := \frac {a_i} {\sum_{i=1}^{\infty}a_i^2} $$</p>
<p>minimizes the quantity $\sum_{i=1}^{\infty}... |
3,910,013 | <p>I'm preparing for a high school math exam and I came across this question in an old exam.</p>
<p>Let <span class="math-container">$f(x) = \dfrac{1}{2(1+x^3)}$</span>.</p>
<p><span class="math-container">$\alpha \in (0, \frac{1}{2})$</span> is the only real number such that <span class="math-container">$f(\alpha) = \... | Michelle | 718,613 | <p>We can write
<span class="math-container">$$
|u_{n+1} - \alpha| \le \frac{1}{2} |u_n - \alpha| \iff |f(u_{n}) - f(\alpha)| \le \frac{1}{2} |u_n - \alpha|
$$</span>
which reminds us of a famous theorem in analysis (mean value theorem). Since <span class="math-container">$f \in \mathcal{C^\infty}$</span>, we only hav... |
3,910,013 | <p>I'm preparing for a high school math exam and I came across this question in an old exam.</p>
<p>Let <span class="math-container">$f(x) = \dfrac{1}{2(1+x^3)}$</span>.</p>
<p><span class="math-container">$\alpha \in (0, \frac{1}{2})$</span> is the only real number such that <span class="math-container">$f(\alpha) = \... | Calvin Lin | 54,563 | <p>As mentioned by others, the key is to show that</p>
<blockquote>
<p>Prove that for <span class="math-container">$x,y \in \left(0,\frac12\right)$</span> we have
<span class="math-container">$$|f(x)-f(y)| \le \frac12 |x-y|.$$</span>
and then apply this to <span class="math-container">$x = u_n$</span> and <span class="... |
52,841 | <p>In classical Mechanics, momentum and position can be paired together to form a symplectic manifold. If you have the simple harmonic oscillator with energy $H = (k/2)x^2 + (m/2)\dot{x}^2$. In this case, the orbits are ellipses. How is the vector field determined by the (symplectic) gradient, then? </p>
<p>Also, ... | Patrick I-Z | 11,885 | <p>The symplectic area contained in a closed curved, that is the boundary of map of a disc, is the "action along the curve".
$$
\int_\sigma \omega = \int_\sigma d\lambda = \int_{\partial \sigma} \lambda = \int_0^{2\pi} \lambda_{\gamma(t)}(\dot \gamma(t)) dt,
$$
where $\sigma$ is a smooth map from the disc to $M$, and ... |
4,577,651 | <p>From Gallian's "Contemporary Abstract Algebra", Part 2 Chapter 5</p>
<p>It looks like using Lagrange's theorem would work, since <span class="math-container">$|S_n| = n!$</span> and <span class="math-container">$\langle\alpha\rangle$</span> is a subgroup of <span class="math-container">$S_n$</span>. Howeve... | Mike F | 6,608 | <p>You've almost solved it yourself! The numbers <span class="math-container">$|\alpha_i|$</span> are all between <span class="math-container">$1$</span> and <span class="math-container">$n$</span> so, from what you wrote, <span class="math-container">$|\alpha|$</span> is the least common multiple of a set of numbers b... |
133,604 | <p>"A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line." - Wikipedia</p>
<p><img src="https://i.stack.imgur.com/Rl1WS.gif" alt="cycloid animation"></p>
<p>In many calculus books I have, the cycloid, in parametric form, is used in examples to find arc lengt... | Robert Israel | 8,508 | <p>$t$ measures the angle through which the wheel has rotated, starting with your point in the "down" position. Since the wheel is rolling, the distance it has rolled is the distance along the circumference of the wheel from your point to the "down" position, which (since the wheel has radius $r$) is $rt$. So the cen... |
1,879,129 | <p>If $0 < y < 1$ and $-1 < x<1$, then prove that $$\left|\frac{x(1-y)}{1+yx}\right| < 1$$</p>
| Asinomás | 33,907 | <p>to prove $|\frac{x(1-y)}{1+yx}|<1$ it suffices to show $|1+yx|>|x(1-y)|$.</p>
<p>If $x\geq 0$ then $|1+yx|=1+yx>x>x|1-y|=|x(1-y)|$</p>
<p>if $x<0$ then $|1+yx|=1+yx>1-y>|x||1-y|=|x(1-y)|$</p>
|
4,367,893 | <p>I need help finding the multiplicative inverse for a = 123, m = 256</p>
<p>I ran through it python script and it gave me 179</p>
<p>I want to do it by pen paper and see what the algorithm is doing.</p>
<p>so here is what I got so far</p>
<p>the euclidean algorithm</p>
<pre><code>256 = 123 * 2 + 10
123 = 10 * 12 + 3
... | Community | -1 | <p>Being isolated is a property defined only in terms of open sets, so if <span class="math-container">$f$</span> is the homeomorphism, then <span class="math-container">$x$</span> is isolated if and only if <span class="math-container">$f(x)$</span> is. Thus, <span class="math-container">$f$</span> induces a bijection... |
4,367,893 | <p>I need help finding the multiplicative inverse for a = 123, m = 256</p>
<p>I ran through it python script and it gave me 179</p>
<p>I want to do it by pen paper and see what the algorithm is doing.</p>
<p>so here is what I got so far</p>
<p>the euclidean algorithm</p>
<pre><code>256 = 123 * 2 + 10
123 = 10 * 12 + 3
... | Matematleta | 138,929 | <p>If <span class="math-container">$f:X\to Y$</span> is a homeomorphism, then it is an open map. Let <span class="math-container">$x\in U\in \tau_X$</span> such that <span class="math-container">$U\cap \{x\}=\{x\}.$</span> As <span class="math-container">$f\upharpoonleft U$</span> is bijective and open <span class="mat... |
1,985,905 | <p>I was wondering if the cardinality of a set is a well defined function, more specifically, does it have a well defined domain and range?</p>
<p>One would say you could assign a number to every finite set, and a cardinality for an infinite set. So the range would be clear, the set of cardinal numbers. But what about... | Vladimir Kanovei | 304,155 | <p>Being equinumerous or bijective is an equivalence relation on sets: $x\equiv y$ iff there is a bijection $f:x\text{ onto }y$. The problem is to define a total set function (a proper class of course) $F$ satisfying $x\equiv y$ iff $F(x)=F(y)$. </p>
<p>In ZFC this is done by the cardinality function $F(x)=\text{card}... |
108,953 | <p>Given a variety $X$ over $\mathbb{Q}$ with good reduction at $p$, proper smooth base change tells us that its $l$-adic cohomology groups are unramified at $p$ (and I'd guess some $p$-adic Hodge theory tells us its p-adic cohomology is crystalline).</p>
<p>My question is to what extent it's possible to find a conver... | Tzanko Matev | 421 | <p>Unfortunately I don't know much about motives in genereal, but this might be relevant to your question. One result of my thesis, that I am currently writing, is to prove Neron-Ogg-Shafarevich for 1-motives. The proof is not particularly difficult and it ultimately reduces to the corresponding results for the compone... |
2,574,221 | <p>Does divergence of $\sum a_k$ imply divergence of $\sum \frac{a_k}{1+a_k}$?</p>
<p>Note: $a_k > 0 $</p>
<p>I understand that looking at the contrapositive statement, we can say that the convergence of the latter sum implies $\frac{a_k}{1+a_k}\rightarrow 0$ but from here is it possible to deduce that $a_k\right... | Gribouillis | 398,505 | <p>In the case where the $a_n$ can change sign, let
$$b_n = \frac{(-1)^n}{\sqrt{n}}\quad \text{and}\quad a_n=\frac{b_n}{1-b_n} = \frac{(-1)^n}{\sqrt{n}}\frac{1}{1-\frac{(-1)^n}{\sqrt{n}}} =
\frac{(-1)^n}{\sqrt{n}} + \frac{1}{n} + O\left(\frac{1}{n^{3/2}}\right)$$
then one has $b_n = \frac{a_n}{1+a_n}$, the series $\sum... |
1,201,900 | <p>This is a rather soft question to I will tag it as such.</p>
<p>Basically what I am asking, is if anyone has a good explanation of what a homomorphism is and what an isomorphism is, and if possible specifically pertaining to beginner linear algebra.</p>
<p>This is because, in my courses we have talked about vector... | James S. Cook | 36,530 | <p>Two vector spaces are isomorphic if they have the same dimension. This is equivalent to the existence of a bijective linear mapping between the spaces since to say $dim(V)=n$ is to say $\beta = \{ v_1, \dots ,v_n \}$ is a basis for $V$ and to say $dim(W)=n$ is to say $\gamma = \{ w_1, \dots ,w_n \}$ is a basis for $... |
2,985,917 | <p>Would it be possible to calculate which function in the Schwarz class of infinitely differentiable functions with compact support is closest to triangle wave?</p>
<p>Let us measure closeness as <span class="math-container">$$<f-g,f-g>_{L_2}^2 = \int_{-\infty}^{\infty}(f(x)-g(x))^2dx$$</span></p>
<p>I don't e... | Federico | 180,428 | <p>The class of Schwartz class is not closed with respect to the <span class="math-container">$L^2$</span> norm, so there isn't necessarily a closest element. Indeed, in this case there isn't, as you can find arbitrarily close functions (take for instance convolutions with an <a href="https://en.wikipedia.org/wiki/Dira... |
2,410,517 | <p>I feel like I'm missing something very simple here, but I'm confused at how Rudin proved Theorem 2.27 c:</p>
<p>If <span class="math-container">$X$</span> is a metric space and <span class="math-container">$E\subset X$</span>, then <span class="math-container">$\overline{E}\subset F$</span> for every closed set <spa... | K.K.McDonald | 302,349 | <p>Let's go by contradiction. We know from assumption that <span class="math-container">$E \subset F$</span>, thus we just need to prove the limit points of <span class="math-container">$E$</span> are also in <span class="math-container">$F$</span>, i.e. <span class="math-container">$E' \subset F. $</span> Assume <span... |
1,611,390 | <p>How to show that the following function is an injective function?</p>
<p>$ \varphi : \mathbb{N}\times \mathbb{N} \rightarrow \mathbb{N} \\
\varphi(\langle n, k\rangle) = \frac{1}{2}(n+k+1)(n+k)+n$</p>
<p>I'm starting with $ \frac{1}{2}(a+b+1)(a+b)+a = \frac{1}{2}(c+d+1)(c+d)+c$, but how am I supposed to show from ... | Pedro | 23,350 | <p>You should <em>really</em> make a drawing. The function enumerates the pairs $(n,k)$ in a diagonal manner. Note that $$(0,0)\mapsto 0,(0,1)\mapsto 1,(1,0)\mapsto 2, (0,2)\mapsto 3,(1,1)\mapsto 4,(2,0)\mapsto 5,\ldots$$</p>
<p>Thus, at least empirically, the function is enumerating $\Bbb N\times\Bbb N$ by travesing ... |
512,590 | <p>According to the definition my professor gave us its okay for a matrix in echelon form to have a zero row, but a system of equations in echelon form cannot have an equation with no leading variable.</p>
<p>Why is this? Aren't they supposed to represent the same thing?</p>
| preferred_anon | 27,150 | <p>Assuming you are familiar with the formula $z^{n}=(\cos(\alpha)+i\sin(\alpha))^{n}=\cos(n\alpha)+i\sin(n\alpha)$ at least for integers $n$, then you can see that$$z^{n}+z^{-n}=\cos(n\alpha)+i\sin(n\alpha)+\cos(-n\alpha)+i\sin(n\alpha)$$
Since $\cos$ is even and $\sin$ is odd, this simplifies to your result.</p>
|
325,765 | <p>Is there any method which allows us to describe all continuous functions (maps to $\mathbb{R}$) on the quotient space?</p>
<p>For examle, how could I classify all continuous functions on $\mathbb{R}/[x\sim2x]$?</p>
| Julien | 38,053 | <p>You can classify them easily: these are the constant functions, whatever Hausdorff topological space $Z$ they land in.</p>
<p>If $f:\mathbb{R}/[x\sim 2x]\longrightarrow Z$ is continuous, then
$$
g(x):=f(\bar{x})
$$
is continuous on $\mathbb{R}$ by composition with the canonical surjection $x\longmapsto \bar{x}$ fro... |
325,765 | <p>Is there any method which allows us to describe all continuous functions (maps to $\mathbb{R}$) on the quotient space?</p>
<p>For examle, how could I classify all continuous functions on $\mathbb{R}/[x\sim2x]$?</p>
| Damien L | 59,825 | <p>Let's say that your quotient is describe by a relation $\rm R$. Then the <strong>Universal Property</strong> of the quotient topology tells you that there is a bijection $$ \text{ continuous functions on } \mathbb R/\mathrm{R} \longleftrightarrow \mathrm{R}-\text{invariant continuous functions on } \mathbb R.$$</p>
... |
112,226 | <p>Prove that there are exactly</p>
<p>$$\displaystyle{\frac{(a-1)(b-1)}{2}}$$ </p>
<p>positive integers that <em>cannot</em> be expressed in the form </p>
<p>$$ax\hspace{2pt}+\hspace{2pt}by$$</p>
<p>where $x$ and $y$ are non-negative integers, and $a, b$ are positive integers such that $\gcd(a,b) =1$.</p>
| Gerry Myerson | 8,269 | <p>Hints: Prove </p>
<p>If $ax+by=c$, and $ax'+by'=c$, then $b$ divides $x-x'$, and $a$ divides $y-y'$, and $(x-x')/b=(y'-y)/a$. </p>
<p>$n$ can be expressed if and only if $((a-1)(b-1)/2)-1-n$ can't. </p>
|
2,909,480 | <p>Please notice the following before reading: the following text is translated from Swedish and it may contain wrong wording. Also note that I am a first year student at an university - in the sense that my knowledge in mathematics is limited.</p>
<p>Translated text:</p>
<p><strong>Example 4.4</strong> Show that it ... | ArsenBerk | 505,611 | <p>Since the $n$-th statement is assumed to be true, first it writes $n^3 - n = 3b$ because it should be divisible by $3$.</p>
<p>Then, for $(n+1)$-th statement, it rearranges the expression $(n+1)^3-(n+1)$ as $(n^3+3n^2+3n+1) - (n+1) = (n^3-n) + (3n^2+3n)$, then puts $3b$ in the place of $(n^3-n)$. Then it concludes ... |
409,220 | <p>$$f(x,y)=6x^3y^2-x^4y^2-x^3y^3$$
$$\frac{\delta f}{\delta x}=18x^2y^2-4x^3y^2-3x^2y^3$$
$$\frac{\delta f}{\delta y}=12x^3y-2x^4y-3x^3y^2$$
Points, in which partial derivatives ar equal to 0 are: (3,2), (x,0), (0,y), x,y are any real numbers. Now I find second derivatives
$$\Delta_1=\frac{\delta f}{\delta x^2}=36xy^2... | john | 79,781 | <p>You could try looking at further derivatives but generally in this case it's better to think of the function itself. Imagine you're at a point (0,y) for instance. How does f change when you move a little in the y-direction? How does f change when you move a little in the x-direction? </p>
|
2,619,131 | <p>How one can prove the following inequality?</p>
<p>$$58x^{10}-42x^9+11x^8+42x^7+53x^6-160x^5+118x^4+22x^3-56x^2-20x+74\geq 0$$ </p>
<p>I plotted the graph on Wolfram Alpha and found that the inequality seems to hold. I was unable to represent the polynomial as a sum of squares. </p>
<p>It looks quite boring to ap... | Johan Löfberg | 37,404 | <p>A sum-of-squares decomposition is given by $z^TQz$ where $z = (1,x,x^2,x^3,x^4,x^5)$ and the positive definite matrix $Q$ is</p>
<p>$Q = \begin{bmatrix}
74 & -10 & -38 & 9 & 8 & -30\\
-10 & 20 & 2 & -8 & -22 & 9\\
-38 & 2 & 118 ... |
760,767 | <p>I don't understand the last part of this proof:</p>
<p><a href="http://www.proofwiki.org/wiki/Intersection_of_Normal_Subgroup_with_Sylow_P-Subgroup" rel="nofollow">http://www.proofwiki.org/wiki/Intersection_of_Normal_Subgroup_with_Sylow_P-Subgroup</a></p>
<p>where they say: $p \nmid \left[{N : P \cap N}\right]$, t... | Mahkoe | 124,475 | <p>I just realized where I went wrong. The length of an arc is the <em>radius of curvature</em> times the angle, not the radial distance between the arc and the origin times the angle. Thanks to all the other answers</p>
|
1,372,376 | <p>For what values of $a$ and $b$, the two functions $f_a(x)=ax^2+3x+1$ and $g_b(x)=\frac{b}{x}$ are tangent to each other at a point where the $x\text{-coordinate}=1$.</p>
<p>The points of intersection are where:
$f_a(1)=g_b(1)$</p>
<p>which gives
$$a+4=b\text{ and } b-4=a$$</p>
<p>Now what to do with this informa... | Harish Chandra Rajpoot | 210,295 | <p>Notice, the slope of tangent of $f_{a}(x)=ax^2+3x+1$ at a general point is given as $$\frac{d}{dx}(f_{a}(x))=\frac{d}{dx}(ax^2+3x+1)$$$$\color{red}{f'_{a}(x)=2ax+3}$$ Similarly, the slope of tangent of $g_{b}(x)=\frac{b}{x}$ at a general point is given as $$\frac{d}{dx}(g_{b}(x))=\frac{d}{dx}\left(\frac{b}{x}\right)... |
1,372,376 | <p>For what values of $a$ and $b$, the two functions $f_a(x)=ax^2+3x+1$ and $g_b(x)=\frac{b}{x}$ are tangent to each other at a point where the $x\text{-coordinate}=1$.</p>
<p>The points of intersection are where:
$f_a(1)=g_b(1)$</p>
<p>which gives
$$a+4=b\text{ and } b-4=a$$</p>
<p>Now what to do with this informa... | Michael Hoppe | 93,935 | <p>Isn't “to be tangent to each other in $x_0$” defined by $f(x_0)=g(x_0)$ and $g'(x_0)=f'(x_0)$? know that $f(1)=g(1)$ and $f'(1)=g'(1)$. In this case we derive two linear equations with solutions $a=-7/3$ and $b=5/3$.</p>
|
1,274,816 | <p>It seems known that there are infinitely many numbers that can be expressed as a sum of two positive cubes in at least two different ways (per the answer to this post: <a href="https://math.stackexchange.com/questions/1192338/number-theory-taxicab-number">Number Theory Taxicab Number</a>).</p>
<p>We know that</p>
... | SHOUNAK GUPTA | 857,041 | <p>I found a number recently which can be expressed as a sum of two cubes in exactly two different ways.</p>
<pre><code> 65673928=164³+394³=103³+401³
</code></pre>
|
2,626,920 | <p>I am working on a homework problem and I am given this situation:</p>
<p>Let $A$ be the event that the $r$ numbers we
obtain are all different from each other. So, for example, if $n = 3$ and $r = 2$ the sample space is
$S = \{(1, 1),(1, 2),(1, 3),(2, 1),(2, 2),(2, 3),(3, 1),(3, 2),(3, 3)\}$
and the event $A$ is
$A... | patentfox | 17,495 | <p>Size of sample space can be calculated by <code>n^r</code>, as we are selecting r elements out of n choices, with repetition.</p>
<p>No of allowed outcomes = <code>nCr</code></p>
<p>So, the formula comes out to be</p>
<pre><code>nCr/n^r
</code></pre>
|
476,095 | <p>I am attempting to learn about mathematical proofs on my own and this is where I've started. I think I can prove this by induction. Something like:</p>
<p>$n = 2k+1$ is odd by definition</p>
<p>$n = 2k+1 + 2$ (this is where I'm stuck, how do I show that this is odd?)</p>
<p>$n = 2(k+1) + 1$ (if I can show that it... | Argon | 27,624 | <p>Odd numbers have a remainder of $1$ when divided by $2$, thus</p>
<p>$$n+2 = 2k+3 \equiv 3 \equiv 1 \pmod 2 $$</p>
|
3,144,813 | <blockquote>
<p>Let <span class="math-container">$X : \mathbb{R} \to \mathbb{R}^n$</span> be a <span class="math-container">$C^1$</span> function. Let <span class="math-container">$\| .\|$</span> be the norm : <span class="math-container">$\| v \| = \max_{1 \leq i \leq N} \mid v_i \mid$</span>. Then is it true that :... | dmtri | 482,116 | <p>The pde alone has infinite number of solutions. We can visualize them as surfases, but when we want it that goes through a specific curve like the one you have here, there is only one solution surface. </p>
|
44,771 | <p>A capital delta ($\Delta$) is commonly used to indicate a difference (especially an incremental difference). For example, $\Delta x = x_1 - x_0$</p>
<p><strong>My question is: is there an analogue of this notation for ratios?</strong></p>
<p>In other words, what's the best symbol to use for $[?]$ in $[?]x = \dfrac... | J. M. ain't a mathematician | 498 | <p>Not entirely standard, but in Peter Henrici's discussion of the (justly famous) quotient-difference (QD) algorithm in the books <a href="http://rads.stackoverflow.com/amzn/click/0471372412" rel="noreferrer"><em>Elements of Numerical Analysis</em></a> (see p. 163) and <a href="http://rads.stackoverflow.com/amzn/click... |
392,835 | <p>In a concrete category (i.e., where the morphisms are functions between sets), I define a <strong>base</strong> of an object <span class="math-container">$A$</span> to be a set of elements <span class="math-container">$M$</span> of <span class="math-container">$A$</span> such that for any morphisms <span class="math... | Martin Brandenburg | 2,841 | <p>The term "base" should not be used, since, as you say, you are actually generalizing the notion of a generating set.</p>
<p>It is an <strong>epi-sink</strong>, also known as <strong>jointly epimorphic family</strong>. See <a href="http://katmat.math.uni-bremen.de/acc/acc.pdf" rel="nofollow noreferrer">Joy ... |
2,096,711 | <p>I need help finding a closed form of this finite sum. I'm not sure how to deal with sums that include division in it.</p>
<p>$$\sum_{i=1}^n \frac{2^i}{2^n}$$</p>
<p>Here's one of the attempts I made and it turned out to be wrong:</p>
<p>$$\frac{1}{2^n}\sum_{i=1}^n {2^i} = \frac{1}{2^n} (2^{n +1} - 1) = \frac{2^{n... | haqnatural | 247,767 | <p>$$\sum _{ i=1 }^{ n }{ 2^{ i } } =\frac { 2\left( 1-{ 2 }^{ n } \right) }{ 1-2 } ={ 2 }^{ n+1 }-2$$</p>
|
2,353,272 | <p>Suppose that we are given the function $f(x)$ in the following product form:
$$f(x) = \prod_{k = -K}^K (1-a^k x)\,,$$
Where $a$ is some real number. </p>
<p>I would like to find the expansion coefficients $c_n$, such that:
$$f(x) = \sum_{n = 0}^{2K+1} c_n x^n\,.$$</p>
<p>A closed form solution for $c_n$, or at le... | Ronald Blaak | 458,842 | <p>Let the function $f_n(x)$ be given by
$$
f_n(x) = \prod_{k=-n}^{n} \left( 1 - a^k x \right)
$$
Since it is clear that it is a polynomial of degree $2 n +1$, it can be expressed as:
$$
f_n(x) = \sum_{k=0}^{2n+1} c_{n,k} x^k
$$
in some yet unknown coefficients $c_{n,k}$. For these coefficients it is easy to see that $... |
4,013,559 | <p>In the video <a href="https://youtu.be/eI4an8aSsgw?t=16354" rel="nofollow noreferrer">https://youtu.be/eI4an8aSsgw?t=16354</a> the professor says that the following equation
<span class="math-container">$$\sqrt{(x+c)^2 + y^2} + \sqrt{(x-c)^2+y^2}=2a$$</span>
simplifies to
<span class="math-container">$$
(a^2-c^2)x^2... | zwim | 399,263 | <p>Let's call <span class="math-container">$\begin{cases}U=\sqrt{(x+c)^2+y^2}\\V=\sqrt{(x-c)^2+y^2}\end{cases}\quad$</span> and we start from <span class="math-container">$U+V=2a$</span></p>
<p>Squaring we get <span class="math-container">$\quad U^2+2UV+V^2=4a^2$</span></p>
<p><span class="math-container">$U^2+V^2=(x+c... |
101,098 | <p>I apologize in advance because I don't know how to enter code to format equations, and I apologize for how elementary this question is. I am trying to teach myself some differential geometry, and it is helpful to apply it to a simple case, but that is where I am running into a wall.</p>
<p>Consider $M=\mathbb{R}^2$... | mathNotebook | 23,648 | <p>This isomorphism is usually established in elemantary textbooks (see Schutz. Geometrical Methods or Wheeler.Gravitation) via the directional derivative. I could go through the argument, but you can find it here:</p>
<p><a href="http://en.wikipedia.org/wiki/Vector_(geometric)" rel="nofollow">http://en.wikipedia.org/... |
681,737 | <p>What is the simplest way we can find which one of $\cos(\cos(1))$ and $\cos(\cos(\cos(1)))$ [in radians] is greater without using a calculator [pen and paper approach]? I thought of using some inequality relating $\cos(x)$ and $x$, but do not know anything helpful.
We can use basic calculus. Please help. </p>
| Henno Brandsma | 4,280 | <p>The example given in the other answer shows why it is false. Note that the diameters of these sets are all infinite. If indeed we have countably many such closed sets <strong>and</strong> their diameters tend to $0$ as well, then completeness will give a point in the intersection (as picking points in the intersecti... |
2,524,809 | <p>I am trying to solve the following problem:</p>
<blockquote>
<p>Show that $\frac{dy}{dt}=f(y/t)$ is equal to $t\frac{dv}{dt}+v=f(v)$, (which is a separable differential equation) by using substitution of $y = t \cdot v$ or $v =\frac{y}{t}$. </p>
</blockquote>
<p>I did the following:</p>
<p>By using the chain-ru... | Dr. Sonnhard Graubner | 175,066 | <p>if we assume, that $y=y(x)$ we get by the chain rule
$$2x+2yy'=0$$</p>
|
63,525 | <p>I asked this question in math.stackexchange but I didn't have much luck. It might be more appropiate for this forum. Let $z_1,z_2,…,z_n$ be i.i.d random points on the unit circle ($|z_i|=1$) with uniform distribution on the unit circle. Consider the random polynomial $P(z)$ given by
$$
P(z)=\prod_{i=1}^{n}(z−z_i).
$... | Seva | 9,924 | <p>I believe you can obtain very reasonable bounds for your problem using the following approach. (I myself was too lazy to carry out the computations.) Split the unit circle into the union of an interval $I$ of length $4\pi/(n\log n)$ and $N\sim \pi n/\log n$ intervals $J_k$ of length about $2\log n/n$ each. (You may ... |
63,525 | <p>I asked this question in math.stackexchange but I didn't have much luck. It might be more appropiate for this forum. Let $z_1,z_2,…,z_n$ be i.i.d random points on the unit circle ($|z_i|=1$) with uniform distribution on the unit circle. Consider the random polynomial $P(z)$ given by
$$
P(z)=\prod_{i=1}^{n}(z−z_i).
$... | Johan Wästlund | 14,302 | <p>I think $z$ should be chosen so that the deviations tend to go in the positive direction on all scales. The following approach seems to work: Suppose $n$ is a power of 2 (some fix is needed if it isn't). Suppose the points $z_i$ are sorted, say in counter-clockwise direction, and assume without loss of generality th... |
4,321,675 | <p>I'm struggling to derive the Finsler geodesic equations. The books I know either skip the computation or use the length functional directly. I want to use the energy. Let <span class="math-container">$(M,F)$</span> be a Finsler manifold and consider the energy functional <span class="math-container">$$E[\gamma] = \f... | Ivo Terek | 118,056 | <p>Someone (not on this website) also pointed me to the Bao, Chern, Shen book, but namely, to Exercise 1.2.1 on page 11 and to relation (1.4.5) on page 23. Using the suggestive coordinate notation on the book, the exercise says that</p>
<p>(a) <span class="math-container">$y^iF_{y^i} = F$</span> (I already knew that)</... |
2,217,454 | <p>Let $\{ x_i : i \in I \}$ be a family of numbers $x_i \in \mathbb R$ with $I$ an arbitrary index set. We say that this family is summable with value $s$ (and write $s = \sum_{i \in I} x_i$ then) if for every $\varepsilon > 0$ there exists some finite set $I_{\varepsilon}$ such that for every finite superset $J \s... | Gio67 | 355,873 | <p>You cannot find such a sequence. Take a look at my answer in
<a href="https://math.stackexchange.com/questions/2126816/on-a-necessary-and-sufficient-condition-for-sum-k-in-mathbbza-k-l-a-k-i">on-a-necessary-and-sufficient-condition-for-sum-k-in-mathbbza-k-l-a-k-i</a></p>
<p>If $\sum_{i\in I}x_i=1$, then $\sum_{i\i... |
2,217,454 | <p>Let $\{ x_i : i \in I \}$ be a family of numbers $x_i \in \mathbb R$ with $I$ an arbitrary index set. We say that this family is summable with value $s$ (and write $s = \sum_{i \in I} x_i$ then) if for every $\varepsilon > 0$ there exists some finite set $I_{\varepsilon}$ such that for every finite superset $J \s... | s.harp | 152,424 | <p>If $\sum_{i\in I} x_i$ converges as a sum in $\Bbb R$ then $I(x_i{\neq0}):=\{i\in I\mid x_i\neq0\}$ has to be countable. This is because
$$I_{\neq0}=\bigcup_{n\in\Bbb N} I(x_i{>\frac1n})\cup I(x_i{<\frac{-1}n})$$
and if this were uncountable, there would have to be one term in the union that is uncountable (si... |
1,478,314 | <p>In this particular case, I am trying to <strong>find all points $(x,y)$ on the graph of $f(x)=x^2$ with tangent lines passing through the point $(3,8)$</strong>. </p>
<p>Now then, I know the <a href="http://www.meta-calculator.com/online/?panel-102-graph&data-bounds-xMin=-10&data-bounds-xMax=10&data-bo... | Michael Burr | 86,421 | <p>Step 1: For a point $(x,y)$ on the graph of $f(x)=x^2$, find the slope of the line between $(x,y)$ and $(3,8)$.</p>
<p>Step 2: Compute the slope of the tangent line to $f(x)=x^2$ at the point $(x,y)$. </p>
<p>Step 3: Set these two slopes equal to each other and find candidate $x$ values.</p>
<p>Step 4: Check you... |
2,114,276 | <p>How to show that $(x^{1/4}-y^{1/4})(x^{3/4}+x^{1/2}y^{1/4}+x^{1/4}y^{1/2}+y^{3/4})=x-y$</p>
<p>Can anyone explain how to solve this question for me? Thanks in advance. </p>
| BranchedOut | 364,830 | <p>Start with $$(x^{1/4}-y^{1/4})(x^{3/4}+x^{1/2}y^{1/4}+x^{1/4}y^{1/2}+y^{3/4}).$$
Then distribute the terms:
$$=(x+x^{3/4}y^{1/4}+x^{1/2}y^{1/2}+x^{1/4}y^{3/4})-(x^{3/4}y^{1/4}+x^{1/2}y^{1/2}+x^{1/4}y^{3/4}+y).$$
Regrouping them gives
$$(x-y)+(x^{3/4}y^{1/4}-x^{3/4}y^{1/4})+(x^{1/2}y^{1/2}-x^{1/2}y^{1/2})+(x^{1/4}y^{... |
4,216,602 | <p>In this book - <a href="https://www.oreilly.com/library/view/machine-learning-with/9781491989371/" rel="nofollow noreferrer">https://www.oreilly.com/library/view/machine-learning-with/9781491989371/</a> - I came to the differentiation of these to terms like this:</p>
<p>Train - Applying a learning algorithm to data ... | 1 2 100 | 47,907 | <p>In simple words, Train is referring to making a choice of algorithm which you want to train your ML model. Meaning, which algorithm you want to use to train your model to accomplish your task. Here you use your expertise and/or intuition which algorithm might work for you ML model. Example which Regression algorithm... |
24,318 | <p>I have an expression as below:</p>
<pre><code>Equations = 2.0799361919940695` x[1] + 3.3534325557330327` x[1]^2 -
4.335179297091139` x[1] x[2] + 1.1989715511881491` x[2]^2 -
3.766597877399148` x[1] x[3] - 0.33254815073371535` x[2] x[3] +
1.9050048836042945` x[3]^2 + 1.1386715715291826` x[1] x[4] +
2... | Dr. belisarius | 193 | <pre><code>f = {# &, 3*# - 5 &, 0.1*#^2 &};
xvalues = Range[0, 500, 2.5];
t1 = Through[f[xvalues]] /. x_ /; x < 0 -> 0;
ListPlot[t1, DataRange -> {0, 500}]
</code></pre>
<p><img src="https://i.stack.imgur.com/kh38S.png" alt="enter image description here"></p>
|
2,119,178 | <p>I have this question:</p>
<blockquote>
<blockquote>
<p>Known that:
$$3pq-5p+4q=22$$
Find the value of $p + q$</p>
</blockquote>
</blockquote>
<p>I have solved 2 variables with 2 equations or more, but I have never encountered 1 equation with 2 variables. The answer is a positive integer. Can I have... | Anurag A | 68,092 | <p>The given equation can be written as
$$(3p+4)(3q-5)=46$$
Since $p$ and $q$ are both integers (as what OP has mentioned in the comments). Therefore, we want factors of $46=ab$ such that
\begin{align*}
3p+4 & =a \\
3q-5 & =b
\end{align*}
Thus
$$3(p+q)=a+b+1 \implies a+b+1 \equiv 0 \pmod{3}.$$
But the only pos... |
2,119,178 | <p>I have this question:</p>
<blockquote>
<blockquote>
<p>Known that:
$$3pq-5p+4q=22$$
Find the value of $p + q$</p>
</blockquote>
</blockquote>
<p>I have solved 2 variables with 2 equations or more, but I have never encountered 1 equation with 2 variables. The answer is a positive integer. Can I have... | Lutz Lehmann | 115,115 | <p>Multiply with $3$, add a constant term and factorize to get
$$
(3p+4)(3q-5)=3(3pq−5p+4q)-20=46
$$</p>
<p>If the quadratic equation $$0=z^2-az+46$$ has the solutions $z_1=(3p+4)$, $z_2=(3q−5)$ then by the Viete rules $$a=z_1+z_2=(3p+4)+(3q−5)=3(p+q)-1$$ which is independent of the order of the roots.</p>
<p>As $46=... |
1,927,394 | <blockquote>
<p>Number of all positive continuous function <span class="math-container">$f(x)$</span> in <span class="math-container">$\left[0,1\right]$</span> which satisfy <span class="math-container">$\displaystyle \int^{1}_{0}f(x)dx=1$</span> and <span class="math-container">$\displaystyle \int^{1}_{0}xf(x)dx=\alp... | Joel Cohen | 10,553 | <p>I believe there are no such functions. Indeed, by combining the equations, we get that for every polynomial $P$ of degree $\le2$, we have </p>
<p>$$\int_0^1 P(x) f(x) dx = P(\alpha) $$</p>
<p>In particular, for $P=(x-\alpha)^2 $, we get </p>
<p>$$\int_0^1 (x-\alpha)^2 f(x) dx = 0$$</p>
<p>The integrand $x \m... |
2,222,514 | <p>A straight line OL rotates around the point O with a constant angular velocity !. A point M moves along the line OL with a speed proportional to the distance OM. Find the equation of the curve described by the point M</p>
<p>As it says angular velocity is constant which i think means
$$... | Narasimham | 95,860 | <p>There a constant ratio between any two velocities. There are several ways to say the same thing.</p>
<p>$$\frac { r \,d \theta}{ dr} = \frac { r \,d \theta/dt}{ dr/dt} =\frac { V_{\theta}}{ V_{r}}= \tan \psi = const. \tag{1} $$</p>
<p>Log variation ( instead of $ \cot \psi$ say $k$ )</p>
<p>$$\frac{dr}{d\theta}... |
122,503 | <p>It seems that the current state of quantum Brownian motion is ill-defined. The best survey I can find is <a href="http://arxiv.org/pdf/1009.0843v1.pdf">this one</a> by László Erdös, but the closest the quantum Brownian motion comes to appearing is in this conjecture (p. 30):</p>
<blockquote>
<p>[<b>Quantum Browni... | Uwe Franz | 30,364 | <p>No answer, just another question:</p>
<p>Is quantum Brownian motion related to Quantum Noise or the quantum Wiener process? I think these notions have a well-established mathematical theory, e.g. there are quantum stochastic integrals defined for them.</p>
<p>For a more physical approach see</p>
<p>Gardiner, Zoll... |
122,503 | <p>It seems that the current state of quantum Brownian motion is ill-defined. The best survey I can find is <a href="http://arxiv.org/pdf/1009.0843v1.pdf">this one</a> by László Erdös, but the closest the quantum Brownian motion comes to appearing is in this conjecture (p. 30):</p>
<blockquote>
<p>[<b>Quantum Browni... | Carlo Beenakker | 11,260 | <p><strong>A.</strong> To the extent that you think of Brownian motion as a random walk, the natural quantum extension is the <em>quantum random walk</em>. For a physics perspective, see <A HREF="http://arxiv.org/abs/quant-ph/0303081">Quantum random walks - an introductory overview</A>, but you might prefer the more ma... |
36,568 | <p>To do Algebraic K-theory, we need a technical condition that a ring $R$ satisfies $R^m=R^n$ if and only if $m=n$. I know some counterexamples for a ring $R$ satisfies $R=R^2$. </p>
<p>Are there any some example that $R\neq R^3$ but $R^2 = R^4$ or something like that?</p>
<p>(c.f. if $R^2=R^4$, then we need that $R... | Pace Nielsen | 3,199 | <p>There are examples of exotic behavior like that which you propose. The specific objects which you should look for are the Leavitt algebras of type (2,2). A very good source on how to create many such examples is George Bergman's paper "Coproducts and some universal ring constructions" although there are easier met... |
2,909,022 | <p>I don't understand how to get from the first to the second step here and get $1/3$ in front.</p>
<p>In the second step $g(x)$ substitutes $x^3 + 1$.</p>
<p>\begin{align*}
\int_0^2 \frac{x^2}{x^3 + 1} \,\mathrm{d}x
&= \frac{1}{3} \int_{0}^{2} \frac{1}{g(x)} g'(x) \,\mathrm{d}x
= \frac{1}{3} \int_{1}^{... | Jan | 529,121 | <p>Let $u:=x^3+1$. Then it follows that </p>
<p>$$\frac{\operatorname{d}u}{\operatorname{d}x}=3x^2 \Longleftrightarrow \operatorname{d}x=\frac{\operatorname{d}u}{3x^2}.$$</p>
<p>Putting this into the integral leads to</p>
<p>$$\int_{0}^2 \frac{x^2}{x^3+1}\operatorname{d}x= \int_{1}^9 \frac{x^2}{u} \cdot \frac{\opera... |
2,989,494 | <p>I am trying to derive properties of natural log and exponential just from the derivative properties.</p>
<p>Let <span class="math-container">$f : (0,\infty) \to \mathbb{R}$</span> and <span class="math-container">$g : \mathbb{R} \to \mathbb{R}$</span>.
Without knowing or stating that <span class="math-container">$f... | Martund | 609,343 | <p>Assume the contrary, then x<50 and y<50, adding both inequalities, we get x+y<100, a contradiction.
Hence proved.</p>
|
2,348,131 | <p>In our class, we encountered a problem that is something like this: "A ball is thrown vertically upward with ...". Since the motion of the object is rectilinear and is a free fall, we all convene with the idea that the acceleration $a(t)$ is 32 feet per second square. However, we are confused about the sign of $a(t)... | Jr Antalan | 207,778 | <p>I will try to answer my own question but correct me if I am wrong. This answer was due to @Yves's answer and with the help of Serway's book which states that the negative in $a$ simply means that the acceleration is on the negative direction. </p>
<p>Clearly, if we set the upward direction to be positive, the gravi... |
634,344 | <p>Im trying to go alone through Fultons, Introduction to algebraic topology. He asks whether there is a function $g$ on a region such that $dg$ is the form:
$$\omega =\dfrac{-ydx+xdy}{x^2+y^2}$$
in some regions. I know you can do it on the upper half plane by considering $-arctan(x/y)$. But Im a bit confused. I know t... | Eric Auld | 76,333 | <p>By the way, this form is notable in that it is closed but not exact on $\mathbb{R}^2 \setminus \{0\}$. (You can tell it is not exact by integrating around the unit circle and getting a nonzero result.)</p>
<p>Be careful of the meaning he intends for the word "region"...I suppose the general meaning is "open connect... |
634,344 | <p>Im trying to go alone through Fultons, Introduction to algebraic topology. He asks whether there is a function $g$ on a region such that $dg$ is the form:
$$\omega =\dfrac{-ydx+xdy}{x^2+y^2}$$
in some regions. I know you can do it on the upper half plane by considering $-arctan(x/y)$. But Im a bit confused. I know t... | Brian Rushton | 51,970 | <p>Yes, you can let $g$ be the function that takes a polar representation with $r>0$ and $-\pi<\theta<\pi$ and returns just $\theta$. This is equal to shifted versions of your arctan function pieced together.</p>
|
2,325,968 | <p>I was trying to calculate : $e^{i\pi /3}$.
So here is what I did : $e^{i\pi /3} = (e^{i\pi})^{1/3} = (-1)^{1/3} = -1$</p>
<p>Yet when I plug : $e^{i\pi /3}$ in my calculator it just prints : $0.5 + 0.866i$</p>
<p>Where am I wrong ? </p>
| MaudPieTheRocktorate | 256,345 | <p>$(e^{i\pi /3})^3=-1$, but that doesn't mean $e^{i\pi /3}=(-1)^{1/3}$. Similarly, $(-1)^2=1$, but $-1\neq1^{1/2}=1$</p>
<p>There are three different cubic roots of $-1$, and $-1$ is just one of them. $e^{i\pi /3}$ is another, and $e^{2i\pi /3}$ is the third one.</p>
<p>The problem is essentially that taking the cub... |
2,325,968 | <p>I was trying to calculate : $e^{i\pi /3}$.
So here is what I did : $e^{i\pi /3} = (e^{i\pi})^{1/3} = (-1)^{1/3} = -1$</p>
<p>Yet when I plug : $e^{i\pi /3}$ in my calculator it just prints : $0.5 + 0.866i$</p>
<p>Where am I wrong ? </p>
| Saketh Malyala | 250,220 | <p>There exist three numbers that all have the property that $z^3=- 1$.</p>
<p>Your calculator uses $e^{ix}=\cos(x)+i\sin(x)$, which is why it spit out the value $\displaystyle \frac{1}{2}+\frac{\sqrt{3}}{2}i$.</p>
<p>The other two numbers are $-1$ and $\displaystyle \frac{1}{2}-\frac{\sqrt{3}}{2}i$.</p>
|
2,433,174 | <p>I'm struggling with the following (<em>is it true?</em>):</p>
<blockquote>
<p>Let <span class="math-container">$X$</span> be a set and denote <span class="math-container">$\aleph(X)$</span> the <em><strong>cardinality</strong></em> of <span class="math-container">$X$</span>. Suppose that <span class="math-container"... | Deepesh Meena | 470,829 | <p>assume $h=4x-y$ and $k=3x-2y$ now take the image of the point $(h,k)$ with respect to the line $4x+2y=6$ you will get a point now you can get the slope easily</p>
|
1,603,272 | <p>I'm trying to figure out if the sequence $e^{(-n)^n}$ where n is a natural number has a convergent subsequence? It's in a past exam paper. I know that obviously I can't apply the Bolzano-Weirstrass theorem because its not a convergence sequence but im not sure how to test for a convergent subsequence if the original... | adjan | 219,722 | <p>The subsequence of odd indices $(a_{2n+1})$ should converge to zero, since you only have negative values in the exponent.</p>
|
1,552 | <p>Closely related: what is the smallest known composite which has not been factored? If these numbers cannot be specified, knowing their approximate size would be interesting. E.g. can current methods factor an arbitrary 200 digit number in a few hours (days? months? or what?).
Can current methods certify that an a... | Michael Lugo | 143 | <p>I don't think this is the right question to be asking. People aren't going to store long lists of primes. You might be able to store the first 10^12 or so primes (in some compressed form) on your hard drive; but testing the first number not on your table for primality, or factoring it, would be trivial.</p>
<p>Th... |
1,552 | <p>Closely related: what is the smallest known composite which has not been factored? If these numbers cannot be specified, knowing their approximate size would be interesting. E.g. can current methods factor an arbitrary 200 digit number in a few hours (days? months? or what?).
Can current methods certify that an a... | Ian H | 151 | <p>What is the smallest integer whose even/odd status is not known?</p>
|
624,715 | <p>Find this follow ODe solution
$$y''-y'+y=x^2e^x\cos{x}$$</p>
<p>I konw solve this follow three case
$$y''-y'+y=x^2\cos{x}$$
$$y''-y'+y=e^x\cos{x}$$
$$y''-y'+y=x^2e^x$$</p>
<p>But for $f(x)=x^2e^x\cos{x}$, I can't.</p>
<p>Thank you very much!</p>
| Mikasa | 8,581 | <p>Honestly, I don't know how the three cases you made is related to the original ODE. I point you a criteria:</p>
<blockquote>
<p>Let $a_ny^{(n)}+a_{n-1}y^{(n-1)}+\cdots+a_1y'+a_0y=Q(x)$ where $a_n\neq0$ and $Q(x)$ is not zero in an interval $I$. If <em>No term</em> of $Q(x)$ is the same as a term of $y_c(x)$, the... |
624,715 | <p>Find this follow ODe solution
$$y''-y'+y=x^2e^x\cos{x}$$</p>
<p>I konw solve this follow three case
$$y''-y'+y=x^2\cos{x}$$
$$y''-y'+y=e^x\cos{x}$$
$$y''-y'+y=x^2e^x$$</p>
<p>But for $f(x)=x^2e^x\cos{x}$, I can't.</p>
<p>Thank you very much!</p>
| Leox | 97,339 | <p>One more way to solve it is using the Laplace transformation $\mathcal{L}.$</p>
<p>Let $\mathcal{L}(y)=Y(p).$ Then
$$
\mathcal{L}(y')=p Y(p)-y(0),\\
\mathcal{L}(y'')=p^2 Y(p)-y'(0)-p y(0),\\ \text{and, using the standard rules for L.t.}\\
\mathcal{L}({x}^{2}{{\rm e}^{x}}\cos \left( x \right))=\frac{1}{\left( p-1-... |
142,127 | <p>A simplex is regular if its all edges have the same length.</p>
<p>How to test in Mathematica whether a <code>Simplex</code> is regular or not, without checking all the edges manually? I'm not really familiar with loops in Mathematica. I also can't find in the documentation how to access the vertices of a <code>Sim... | J. M.'s persistent exhaustion | 50 | <p>As I mentioned in my comment, you can use <code>Subsets[]</code> to enumerate the edges of your simplex:</p>
<pre><code>regularSimplexQ[Simplex[vertices_]] :=
MatrixQ[vertices] && Subtract @@ Dimensions[vertices] == 1 &&
Equal @@ EuclideanDistance @@@ Subsets[vertices, {2}];
regularS... |
3,125,958 | <p>I've been asked to consider this parabolic equation. </p>
<p><span class="math-container">$ 3\frac{∂^2u}{∂x^2} + 6\frac{∂^2u}{∂x∂y} +3\frac{∂^2u}{∂y^2} - \frac{∂u}{∂x} - 4\frac{∂u}{∂y} + u = 0$</span></p>
<p>I calculated the characteristic coordinates to be <span class="math-container">$ξ = y - x, η = x$</span>. T... | Dylan | 135,643 | <p>I'm using subscripts as partials to save time</p>
<p><span class="math-container">\begin{align}
u_x &= u_\xi\xi_x + u_\eta\eta_x = - u_\xi + u_\eta \\
u_y &= u_\xi\xi_y + u_\eta\eta_y = u_\xi
\end{align}</span></p>
<p>Using the chain rule again</p>
<p><span class="math-container">\begin{align}
u_{xx} &... |
126,901 | <p>How to evaluate this determinant $$\det\begin{bmatrix}
a& b&b &\cdots&b\\ c &d &0&\cdots&0\\c&0&d&\ddots&\vdots\\\vdots &\vdots&\ddots&\ddots& 0\\c&0&\cdots&0&d
\end{bmatrix}?$$</p>
<p>I am looking for the different approaches.</p>
| bgins | 20,321 | <p>Let $A_n=(a_{ij})$ be the $n\times n$ matrix with
$$
a_{ij}=\left\{\matrix{a&i=j=1\\b&i=1\ne j\\c&i\ne1=j\\d&i=j\ne1\\0&\text{otherwise}}\right.
$$
and $\Delta_n=\det A_n$. Then ($\Delta_1=a$ according to my definition above, which may differ from your implicit definition), $\Delta_2=ad-bc$ and... |
1,579,349 | <p>Need help setting this thing up don't really get how to get the derivative is it $0$? If you just plug everything in since there will be no variable.</p>
| user299317 | 299,317 | <p>I think I got it I was confused whether you plug everything in at first which I was stuck at you have to find derivative first, then plug the x and get your answer
Y= 6*2(2^2-1)^2= 108</p>
<p>Thanks for the help</p>
|
2,309,721 | <p>The problem is: Prove that $7|x^2+y^2$ only if $7|x$ and $7|y$ for $x,y∈Z$.</p>
<p>I found a theorem in my book that allows to do the following transformation:
if $a|b$ and $a|c$ -> $a|(b+c)$</p>
<p>So, can I prove it like this: $7|x^2+y^2 =>7|x^2, 7|y^2 => 7|x*x, 7|y*y => 7|x, 7|y$ ?</p>
<p>I am no... | Qwerty | 290,058 | <p>No you cant!</p>
<p>The book says if $a|b$ and $a|c\implies a|b+c$ and <strong>not</strong> the reverse.. The case you have is a simple contradiction.</p>
<p>As far the problem goes , I can give you a proceeding.</p>
<p>Take $x=7\alpha +\beta; y=7\gamma +\delta$ Find out what $x^2+y^2$ evaluates to. Check what mu... |
67,516 | <p>The book by Durrett "Essentials on Stochastic Processes" states on page 55 that:</p>
<blockquote>
<p>If the state space S is finite then there is at least on stationary
distribution.</p>
</blockquote>
<ol>
<li><p>How can I find the stationary distribution for example for the square 2x2 matrix $[[a,b],[1-a, 1-b... | Robert Israel | 8,508 | <p>Draw tangent lines at three points on the circle, and see where they intersect.</p>
|
67,516 | <p>The book by Durrett "Essentials on Stochastic Processes" states on page 55 that:</p>
<blockquote>
<p>If the state space S is finite then there is at least on stationary
distribution.</p>
</blockquote>
<ol>
<li><p>How can I find the stationary distribution for example for the square 2x2 matrix $[[a,b],[1-a, 1-b... | davidlowryduda | 9,754 | <p>It is. But it is not unique, i.e. infinitely many triangles can be drawn from a single circle. (To see this, draw many non-similar triangles, find their incircles, and then scale them so that the circles are all the same size. Then the triangles have the same incircle, though they're different).</p>
<p>I think the ... |
2,176,081 | <p>I am trying to compute </p>
<blockquote>
<p>$$ \int_0^\infty \frac{\ln x}{x^2 +4}\,dx,$$</p>
</blockquote>
<p>which I find <a href="https://math.stackexchange.com/questions/2173289/integrating-int-0-infty-frac-ln-xx24-dx-with-residue-theorem/2173342">here</a>, without complex analysis. I am consistently getting ... | Jack D'Aurizio | 44,121 | <p>It is probably easier to perform the following manipulations:
$$ \int_{0}^{+\infty}\frac{\log x}{x^2+4}\,dx\stackrel{x\mapsto 2z}{=} 2\int_{0}^{+\infty}\frac{\log(2)+\log(z)}{4+4z^2}\,dz \\= \color{red}{\frac{\log(2)}{2}\int_{0}^{+\infty}\frac{dz}{1+z^2}}+\color{blue}{\frac{1}{2}\int_{0}^{+\infty}\frac{\log z}{1+z^2... |
155,429 | <p>Consider a square skew-symmetric $n\times n$ matrix $A$. We know that $\det(A)=\det(A^T)=(-1)^n\det(A)$, so if $n$ is odd, the determinant vanishes.</p>
<p>If $n$ is even, my book claims that the determinant is the square of a polynomial function of the entries, and Wikipedia confirms this. The polynomial in questi... | Matt E | 221 | <p>Here is an elaboration of Qiaochu's comment above: </p>
<p>A $2n\times 2n$ matrix $A$ induces a pairing (say on column vectors), namely
$$\langle v,w \rangle := v^T A w.$$
Thus we can think of $A$ as being an element of $(V\otimes V)^*$ (which is
the space of all bilinear pairings on $V$), where $V$ is the space of... |
155,429 | <p>Consider a square skew-symmetric $n\times n$ matrix $A$. We know that $\det(A)=\det(A^T)=(-1)^n\det(A)$, so if $n$ is odd, the determinant vanishes.</p>
<p>If $n$ is even, my book claims that the determinant is the square of a polynomial function of the entries, and Wikipedia confirms this. The polynomial in questi... | Qmechanic | 11,127 | <p>Here is an approach using (possibly complex) <a href="https://en.wikipedia.org/wiki/Grassmann_number" rel="nofollow noreferrer">Grassmann variables</a> and Berezin integration<span class="math-container">$^1$</span> to prove the required relation <span class="math-container">$${\rm Det}(A)~=~{\rm Pf}(A)^2. \tag{1}$$... |
162,655 | <p>Does there exist a Ricci flat Riemannian or Lorentzian manifold which is geodesic complete but not flat? And is there any theorm about Ricci-flat but not flat? </p>
<p>I am especially interset in the case of Lorentzian Manifold whose sign signature is (- ,+ ,+ , + ). Of course, the example is not constricted in L... | Otis Chodosh | 1,540 | <p>Your claim that "all solutions from general relativity are geodesically incomplete" is not true. The classical Schwarzschild/Kerr black hole solutions are geodesically incomplete, and along with Minkowski space (which is flat), these are the most well known explicit metrics. </p>
<p>However, many solutions to the (... |
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