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 |
|---|---|---|---|---|
787,322 | <p>For a group $[S,*]$ where $S=\{a,b\}$, how come there are $2^4$ binary operations that can be defined on $S$ instead of $2^2$? I can only see $a*a$, $a*b$, $b*a$, and $b*b$, which is $4=2^2$. What other operations can possibly exist? What am I not seeing here?</p>
| Marcel Besixdouze | 29,892 | <p>By definition a binary operation on $S$ is a function from $S^2$ to $S$.</p>
<p>You are correct that $S^2$ has cardinality $2^2$, that is to say $a\ast b$, $a\ast a$, $b\ast a$ and $b\ast b$ are the only <em>inputs</em> for a binary operation on $S$, but there are $2$ choices for each <em>output</em> corresponding ... |
2,258,147 | <p>I'm trying to see how to go about this problem for my revision. </p>
<blockquote>
<p><strong>Question 16</strong>
<p>$(a)$ If $g(x)=1-3x$, find $g(x-1)$ in terms of $x$
<p>$(b)$ On the axes below sketch the graph of $g(x)$ and $g(x-1)$</p>
</blockquote>
<p>I have a test coming up but I can't remember how yo... | scarface | 173,820 | <p>Hint: If $a,b \geq 0$ then</p>
<p>$$\left(\dfrac{a^3+b^3}{2}\right)^{1/3}\geq \left(\dfrac{a^2+b^2}{2}\right)^{1/2}$$</p>
|
2,258,147 | <p>I'm trying to see how to go about this problem for my revision. </p>
<blockquote>
<p><strong>Question 16</strong>
<p>$(a)$ If $g(x)=1-3x$, find $g(x-1)$ in terms of $x$
<p>$(b)$ On the axes below sketch the graph of $g(x)$ and $g(x-1)$</p>
</blockquote>
<p>I have a test coming up but I can't remember how yo... | Bernard | 202,857 | <p><strong>Hint:</strong></p>
<p>You can suppose $0\le x\le\pi/2$. The problem comes down to finding the minimum of
$$\cos^3x+\sin^3x=(\cos x+\sin x)(\cos^2x-\sin x\cos x+\sin^2x)=\sqrt2\sin\bigl(x+\tfrac\pi4\bigr)\bigl(1-\tfrac12\sin2x\bigr).$$
On the interval $(0,\pi/2)$, $\sin\bigl(x+\tfrac\pi4\bigr)\ge\frac{\sqrt2... |
2,786,648 | <p>Let $f:[0,1]\rightarrow[0,\infty)$ be a continuous function such that $\int_0^1f(x)dx=1$, and let $M=\max f(x)$. Show that:
$$\frac{1}{2M}\le\int_0^1xf(x)dx\le1-\frac{1}{2M}$$</p>
<p>We have $1=\int_0^1f(x)dx\le\int_0^1Mdx=M$, so $M\ge1$. I tried several inequalities, but none of them seem to be working in this cas... | Rigel | 11,776 | <p>Let $F(x) := \int_0^x f(s)\, ds$, $x\in [0,1]$.
Then $F$ is continuously differentiable, monotone non-decreasing function, satisfying
$$
F(0) = 0, \quad
F(1) = 1, \quad
0 \leq F'(x) \leq M, \ \forall x\in [0,1].
$$
As a consequence,
$$
\max\{0, 1- M(1-x)\} \leq F(x) \leq \min\{1, Mx\},
\qquad \forall x\in [0,1],
$$
... |
2,130,062 | <p>I have 4 decks, each with five unique cards. If I select 3 from the first, 4 from the second, 2 from the third and 2 from the fourth. In case order matters and I do not put cards back, in how many ways can I arrange the 11 cards if the order matters for each selected card? </p>
<p>I think that I use the following f... | Hagen von Eitzen | 39,174 | <p>If $A$ is that matrix, and we let $v=\begin{pmatrix}a\\b\\c\end{pmatrix}$, then we notice that $Ax = (v\cdot x)v+x$ for all vectors $x$. In particular, $Av=(|v|^2+1)v$ whereas $Aw=w$ for $w\perp v$. Thus, we can express $A$ with respect to a suitable basis $v,w_1,w_2$ as
$$\begin{pmatrix}|v|^2+1&0&0\\0&1... |
4,489,437 | <p>Is it possible to find the value of n (n belongs to Real number) such that it satisfies the equation: <span class="math-container">$2^n = n^8$</span> <strong>without any help of computer or graph generator</strong> (i.e. only manually). If possible, please explain and if it requires explanation of some large number ... | PC1 | 960,197 | <p>If you're happy to work with some approximation, you can find the root using something like <a href="https://en.wikipedia.org/wiki/Newton%27s_method" rel="nofollow noreferrer">Newton-Raphson's method</a> for example. It is tedious but it works "by hand".</p>
<p>In any case, there seems to exist 3 roots to ... |
152,626 | <p>Is there any simple way of computing the following sum?</p>
<p>$$\sum_{k=1}^\infty \frac1{k\space k!}$$</p>
| user26872 | 26,872 | <p>$\def\d{\delta}
\def\e{\epsilon}
\def\g{\gamma}
\def\pv{\mathrm{PV}}
\def\pv{\mathcal{P}}
\def\pv{\mathrm{P}}$We show another way to get the integral representation of the sum and explain its relation to the exponential integral.</p>
<p>Let
$$S(x) = \sum_{k=1}^\infty \frac{x^k}{k k!}.$$
The sum we are interested in... |
3,670,384 | <p>I’ve read just the basics of some introductory analysis books and sometimes they show that we can characterize things like limits, continuity, compactness, etc. in terms of sequences. </p>
<p>I’ve heard that these sequential criteria hold for general metric spaces, but that in topology for example one encounters si... | Will Jagy | 10,400 | <p>James R. Munkres, in <em>Topology: A First Course</em> section 3-7, defines two reasonable weakenings of compactness. On page 178, limit point compactness is when every infinite subset of the space has a limit point. On page 179, when every sequence has a convergent subsequence, he calls it sequential compactness. ... |
2,644,610 | <p>How is it possible to prove that:
$$
|e^{ia}-e^{ib}|=2\sin\frac{|a-b|}{2}\leq|a-b|?
$$</p>
<p>Specifically, I'm looking for an analytic technique to show that the equality $|e^{ia}-e^{ib}|=2\sin\frac{|a-b|}{2}$ is correct.</p>
| egreg | 62,967 | <p>A useful trick when you have expressions such as $e^{ia}-e^{ib}$ is to collect $e^{ib}$, so
$$
e^{ia}-e^{ib}=e^{ib}(e^{i(a-b)}-1)
$$
Now set $a-b=2c$:
$$
e^{i(a-b)}-1=e^{ic}(e^{ic}-e^{-ic})=2ie^{ic}\sin c
$$
Thus
$$
|e^{ia}-e^{ib}|=|e^{ib}|\,|e^{ic}|\,|2ie^{ic}\sin c|=2\lvert\sin c\rvert=
2\sin\lvert c\rvert=2\sin\f... |
2,860,195 | <p>The operation $@$ is defined on the real numbers as $a @ b= ab + b + a$</p>
<p>a) Show that $0$ is an identity for the operation.</p>
<p>b) Show that some real numbers have inverses under the operation.</p>
<p>c) Find a counter-example to show that, for this operation inverses do not exist for all the real number... | Cornman | 439,383 | <blockquote>
<p>a) Show that 0 is an identity for the operation.</p>
</blockquote>
<p>(I write $\circ$ instead of @)</p>
<p>We have $a\circ b:=ab+b+a$. Let $a\in\mathbb{R}$ be arbitrary, then:</p>
<p>$0\circ a=0a+a+0=a$</p>
<p>$a\circ 0=a0+0+a=a$</p>
<blockquote>
<p>b) Show that some real numbers have inverses... |
1,320,365 | <p>I am trying to prove that $\mathbb Z[i]/ \langle 1+2i \rangle$ is isomorphic to $\mathbb Z_5$. </p>
<p>The only thing that came to my mind was trying to apply the first isomorphism theorem using an appropiate function. If I consider the euclidean function $N: Z[i] \setminus \{0 \} \to \mathbb N$ defined as $N(a... | jgon | 90,543 | <p>One more proof. It turns out that in this case, it is easier to obtain the isomorphism by going the other direction.</p>
<p>Let $f: \Bbb{Z}\to \Bbb{Z}[i]/(1+2i)$ be the natural mapping, i.e. let $n$ map to $\overline{n}$ where $\overline{n}$ is the residue of $n$ as a gaussian integer mod $1+2i$. Now what is the im... |
2,964,512 | <p>A metric <span class="math-container">$d(x,y)$</span> takes two points from some domain <span class="math-container">$X$</span> and returns a non-negative real number. It is the distance between two points.</p>
<p>A norm <span class="math-container">$n(x)$</span> takes only one point from <span class="math-containe... | Henno Brandsma | 4,280 | <p>A norm <em>induces</em> a metric, using <span class="math-container">$d(x,y) = \|x-y\|$</span>. But a norm is a function <span class="math-container">$X \to \mathbb{R}$</span> not <span class="math-container">$X \times X \to \mathbb{R}$</span>, so in that sense it's simpler.</p>
<p>It only makes sense to talk about... |
2,964,512 | <p>A metric <span class="math-container">$d(x,y)$</span> takes two points from some domain <span class="math-container">$X$</span> and returns a non-negative real number. It is the distance between two points.</p>
<p>A norm <span class="math-container">$n(x)$</span> takes only one point from <span class="math-containe... | Xander Henderson | 468,350 | <p>A metric is, in some sense, more general than a norm. It might be helpful here to look at somewhat precise definitions:</p>
<blockquote>
<p><em>Definition:</em> Let <span class="math-container">$V$</span> be a vector space over <span class="math-container">$\mathbb{C}$</span> (or, perhaps more generally, over a val... |
27,126 | <p>$$e^{\pi i} + 1 = 0$$</p>
<p>I have been searching for a convincing interpretation of this. I understand how it comes about but what is it that it is telling us? </p>
<p>Best that I can figure out is that it just emphasizes that the various definitions mathematicians have provided for non-intuitive operations (com... | Aj Brown | 85,509 | <p>Though I would contribute with this animated version of vonjd's illustration:</p>
<p><img src="https://orig06.deviantart.net/5004/f/2012/108/5/c/euler__s_formula_3d_visualization_by_woodmath-d4g48y9.gif" alt="An animation."></p>
<p>This animation is from <a href="http://woodmath.deviantart.com/art/Euler-s-formula-... |
27,126 | <p>$$e^{\pi i} + 1 = 0$$</p>
<p>I have been searching for a convincing interpretation of this. I understand how it comes about but what is it that it is telling us? </p>
<p>Best that I can figure out is that it just emphasizes that the various definitions mathematicians have provided for non-intuitive operations (com... | D. Thomine | 75,670 | <p>A way to introduce the exponential is by the ODE:</p>
<p>$$f' = \alpha f, \ \ f(0) = 1,$$</p>
<p>whose unique solution is $t \mapsto exp(\alpha t)$. This is common when $\alpha$ is natural, but what if $\alpha$ is complex? And, in particular, what if $\alpha = i$? Of course, $f$ must be complex-valued instead of r... |
26,651 | <p>Hi, everybody. I'm recently reading W.Bruns and J.Herzog's famous book-Cohen-Macaulay Rings. I personally believe that it would be perfect if the authors provide for readers more concrete examples. After reading the first two sections of this book, I have two questions.</p>
<ol>
<li><p>Given a non-negative integer ... | Emerton | 2,874 | <p>(1) A commutative Noetherian ring is reduced if and only if it is generically reduced
(i.e. $R_0$, i.e. regular after localization at all height zero primes) and $S_1$
(i.e. every prime of height at least one has depth at least one).</p>
<p>Since a ring is Cohen--Macaulay iff it is $S_i$ for all $i$, to construct ... |
69,839 | <p>Is it possible to get all points on a Polyhedron surface using two surface parameters, say </p>
<p>$ \phi,\theta $ spherical co-ordinates?</p>
<p>Just like in <code>ParametricPlot3D</code>, can we start with <code>PolyhedronData["Tetrahedron"]</code> to obtain spatial point positions?. The tip of position vector s... | chyanog | 2,090 | <pre><code>regularPolyhedraParametricEquation[poly_String]:=
With[{sp={Sin[θ] Cos[ϕ], Sin[θ] Sin[ϕ], Cos[θ]}},
sp/Max[(Cross[#2-#1,#3-#1]&@@@PolyhedronData[poly,"Polygons"][[All,1]]).sp]];
regularPolyhedraList = PolyhedronData["Platonic"]
paraEqn = regularPolyhedraParametricEquation /@ regular... |
1,541,623 | <p>So I'm just getting the grasp of set theory and I have this question.</p>
<blockquote>
<p>Let $|A| = m$ and $|B| = n$. What is the cardinality of the set $A \times B
$?</p>
</blockquote>
<p>I put $\{1,1\}$ as the answer however I wasn't totally sure what the two vertical bars between set $A$ and set $B$ mean. If... | mweiss | 124,095 | <p>I suspect you are interpreting the question to be saying that set $A$ contains a single element $m$ and set $B$ contains a single element $n$. That would explain where the $1$s in your answer are coming from, at least -- although I'm not sure why you think $\{1,1\}$ is the answer. </p>
<p>But in any case, that is... |
786,596 | <p>So I'm trying to solve this practice exam question, </p>
<blockquote>
<p>Let $G$ be a planar graph with at least two edges and does not contain $K_{3}$ as a subgraph. Prove that $|E|\leq 2|V|-4$.</p>
</blockquote>
<p>Now I started doing this by induction, but it seems to me like the base-case is a counter-examp... | Roger Burt | 130,875 | <p><strong>Hint:</strong> Can you factor $x^n-y^n$?</p>
|
786,596 | <p>So I'm trying to solve this practice exam question, </p>
<blockquote>
<p>Let $G$ be a planar graph with at least two edges and does not contain $K_{3}$ as a subgraph. Prove that $|E|\leq 2|V|-4$.</p>
</blockquote>
<p>Now I started doing this by induction, but it seems to me like the base-case is a counter-examp... | Andrew D | 55,458 | <p>A quicker/non-inductive method is as $11 \equiv 2 \mod 9$, for all integers $n \geq 1$, $11^n \equiv 2^n \mod 9$, so $11^n - 2^n \equiv 0 \mod 9$ and hence $11^n - 2^n$ is divisible by $9$.</p>
|
4,134,365 | <p>This is a question in a book that I am studying, and I have attempted to answer it but got it wrong.<br />
There were two parts that I wrong.<br />
The question is about five digit numbers where the digits are 1, 2, 3, or 4.</p>
<p>The first part asked how many numbers were such that the sum of the digits was even. ... | Math Lover | 801,574 | <p>Your approach to the first part is correct and so is the answer. Please note that -</p>
<p><span class="math-container">$\displaystyle \small \binom{5}{0} \cdot 2^5 +\binom{5}{2} \cdot 2^5 + \binom{5}{4} \cdot 2^5 = 512$</span></p>
<p>and so is the book answer,</p>
<p><span class="math-container">$\displaystyle \sma... |
267,552 | <p>While using <span class="math-container">$\LaTeX$</span>, there are two characters, \Re and \Im, that represent the real and imaginary parts of a complex number. These look like this:</p>
<p><a href="https://i.stack.imgur.com/0MKrr.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/0MKrr.png" alt="en... | Syed | 81,355 | <p>Not sure if this is an answer or what particular use you have in mind, but a font that resembles such <span class="math-container">$\LaTeX$</span> typography closely is the "Euclid Fraktur". Not sure if there are better matches in the "Math Fraktur" family of fonts.</p>
<p><a href="https://i.sta... |
267,552 | <p>While using <span class="math-container">$\LaTeX$</span>, there are two characters, \Re and \Im, that represent the real and imaginary parts of a complex number. These look like this:</p>
<p><a href="https://i.stack.imgur.com/0MKrr.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/0MKrr.png" alt="en... | gwr | 764 | <p>Just take a look at the fonts that you have installed, as <code>"Euklid Fraktur"</code> is not available for everyone.</p>
<pre><code>Dataset @* Association @@ ({# -> Style["R, I", FontFamily -> #]} & /@ $FontFamilies)
</code></pre>
<p><a href="https://i.stack.imgur.com/GzBIo.png" rel="... |
121,645 | <p>I have a (presumably simple) Laplace Transform problem which I'm having trouble with:</p>
<p>$$\mathcal L\big\{t \sinh(4t)\big\} = ?$$</p>
<p>How would I go about solving this? Would you please show working if possible, or alternatively point me in the right direction regarding how to go about solving this?</p>
<... | MattyZ | 7,571 | <p>$\mathcal{L}\{f(t)\}' = F'(s)= \int_0^\infty\frac{d}{ds}[e^{-st} f(t)] dt = \int_0^\infty e^{-st}[-t f(t)]dt$, </p>
<p>so $\mathcal{L}\{tf(t)\} = -F'(s).$</p>
<p>Since $\mathcal{L}\{\sinh(4t)\} = \frac{4}{s^2 - 8}$, (I "cheated" and checked a table!)</p>
<p>Then $\mathcal{L}\{t*\sinh(4t)\} = -\mathcal... |
981,949 | <blockquote>
<p>Find the derivative and evaluate at $f\;'(2):$ $$\log_4(2x^2+1)$$ </p>
</blockquote>
<p>$\log_4(2x^2+1)=y$<br>
$4^y=2x^2+1$ </p>
<p>$4^y\ln4 \times y\;'=4x$<br>
$y\;'=\dfrac{4x}{4^y\ln4}\implies \dfrac{4x}{(2x^2+1)\ln4}$ </p>
<p>What am I doing wrong? I evaluated at $2$ and got $1.154$</p>
| Sherlock Holmes | 173,514 | <p>There seem to be no issues with your derivation; when I plug in x=2 I get 0.641 to three decimal places...perhaps you substituted in the wrong value? </p>
|
1,305,151 | <p>I want to prove this without using any of the properties about the field of algebraic numbers (specifically that it is one). Essentially I just want to find a polynomial for which $\cos\frac{2\pi}{n}$ is a root.</p>
<p>I know roots of unity and De Moivre's theorem is clearly going to be important here but I just ca... | hmakholm left over Monica | 14,366 | <p>Let $\theta=\frac{2\pi}{n}$. By De Moivre's formula we have
$$ (\cos \theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta = 1 $$</p>
<p>Expand the left-hand side using the binomial theorem!</p>
<p>Every other term of the expansion contains a $\pm i$ and an <em>odd</em> power of $\sin\theta$; we know these terms ... |
2,018,703 | <p>Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function with the property that $\lim_{x \rightarrow \infty} f(x)$ and $\lim_{x \rightarrow -\infty} f(x)$ exist and are equal. Prove that $\forall d > 0$ there exists $x_1, x_2 \in \mathbb{R}$ such that $x_1 - x_2 = d$ and $f(x_1) = f(x_2)$.</p>
<p>I a... | Community | -1 | <p>Fix $d>0$ and put $g(x) = f(x+d)-f(x)$. If for all $x \in \Bbb R$, $g(x) \neq 0$, then $g(\Bbb R) \subset (-\infty,0)\cup(0,\infty)$. </p>
<p>Let $A=g^{-1}((-\infty,0))$ and $B=g^{-1}((0,\infty))$. Assume that neither is empty. Note that $A\cup B = \Bbb R$, and $A$ and $B$ are both open (by continuity of $g$) an... |
58,306 | <p>Let $X$ be a topological space, and let $\mathscr{F}, \mathscr{G}$ be sheaves of sets on $X$. It is well-known that a morphism $\varphi : \mathscr{F} \to \mathscr{G}$ is epic (in the category of sheaves on $X$) if and only if the induced map of stalks $\varphi_P : \mathscr{F}_P \to \mathscr{G}_P$ is surjective for e... | Gregory Grant | 217,398 | <p>Alex B. gave an example above that was supposed to be surjective on all stalks but not surjective. According to Hartshorn exercise II.1.2(b) a morphism is surjective $\Leftrightarrow$ it is surjective on all stalks. So that pursuit is in vain.</p>
<p>But the question was to find a morphism that is surjective but ... |
58,306 | <p>Let $X$ be a topological space, and let $\mathscr{F}, \mathscr{G}$ be sheaves of sets on $X$. It is well-known that a morphism $\varphi : \mathscr{F} \to \mathscr{G}$ is epic (in the category of sheaves on $X$) if and only if the induced map of stalks $\varphi_P : \mathscr{F}_P \to \mathscr{G}_P$ is surjective for e... | user2747939 | 402,814 | <p>We seek two surjective local homeomorphisms
<span class="math-container">$\pi_{\mathcal E}\colon\mathcal E\rightarrow\mathcal X$</span> and
<span class="math-container">$\pi_{\mathcal H}\colon\mathcal H\rightarrow\mathcal X$</span> and a surjective
sheaf morphism <span class="math-container">$f\colon\mathcal E\right... |
1,234,500 | <p>I am a student in 12th grade and am fond of mathematics. I enjoy reading mathematics but when it comes to problems I just get completely stuck. Its not that I don't understand the problem but often don't know how to go about tackling it. When I see the solution, often I understand it perfectly but arriving at that s... | abel | 9,252 | <p>the value of $m$ you are looking for is the slope of the tangent line to the graph $y = -x^2 - 4x _ 3, -3 < x < 1.$ it has a tangent at the $x$-coordinate $2 - \sqrt {15}$ with a slope of $-2a - 4 = -8 + 2\sqrt{15}.$ any line with a bigger slope cuts at two points and any with a positive and smaller slope will... |
3,132,009 | <p>This question is a <strong>cross post</strong> from <a href="https://mathoverflow.net/questions/333204/reference-request-introduction-to-finsler-manifolds-from-the-metric-geometry-po">MathOverflow</a>. I have requested the migration of the question, but unfortunately it is not possible after two months of posting.</... | HK Lee | 37,116 | <p>I suggest the paper "On intrinsic geometry of surfaces in normed
spaces - Burago and Ivanov" which deals second variation on Finsler
surface, isometric embedding of Finsler surface and a geodesic line
on Finsler surface.</p>
|
3,463,970 | <p>I am trying to see if someone can help me understand the isomorphism between <span class="math-container">$V$</span> and <span class="math-container">$V''$</span> a bit more <strong>intuitively</strong>.</p>
<p>I understand that the dual space of <span class="math-container">$V$</span> is the set of linear maps fro... | Calum Gilhooley | 213,690 | <p>The intuitive difficulty you are having seems to be that you wish to
write <span class="math-container">$\varphi(v) = g,$</span> or <span class="math-container">$v \mapsto g$</span>, where <span class="math-container">$g$</span> is an expression
that denotes a function in the same way in which <span class="math-cont... |
617,598 | <p>Does anyone know any examples of $f$'s for which $-\triangle u(x) = k f(u(x))$ has an explicit solution (i.e. a formula for the solution, not a numerical approximation scheme) in terms of $k$?</p>
<p>I am interested in examples where $f\geq 0$ is neither constant nor linear. Optimally I would be interested in a smo... | Adi Dani | 12,848 | <p>The cancellation make sense if instead of $x^2+16y^2$ we use $x^2-16y^2$
$$\frac{x^2-16y^2}{x}\frac{x^2+4xy}{x-4y}=\frac{x^2-(4y)^2}{x}\frac{x(x+4y)}{x-4y}=$$
$$=\frac{(x-4y)(x+4y)}{x}\frac{x(x+4y)}{x-4y}=(x+4y)^2$$</p>
|
2,728,317 | <p>As I know when you move to "bigger" number systems (such as from complex to quaternions) you lose some properties (e.g. moving from complex to quaternions requires loss of commutativity), but does it hold when you move for example from naturals to integers or from reals to complex and what properties do you lose?</p... | Jordan Hardy | 202,814 | <p>The biggest thing you lose when you move from the reals to the complex numbers is the ordering. You can, of course, find some ordering on the complex numbers, but the ordering will have nothing to do with the algebraic structure.</p>
<p>On the real numbers, if $a < b$, and $c$ is positive, then $ac < bc$. And... |
217,483 | <p><strong>Prove that if $I \subset \mathbb{R}$ is an open interval and $f: I \to \mathbb{R}$ differentiable, and $f$ has only one critical point $x_0$ and this critical point is a local minimum, then $x_0$ is also the absolute min of $f$, using Rolle's and IVT.</strong></p>
<p>To me this seems "obvious", because if $... | Hagen von Eitzen | 39,174 | <p>I assume that $0\notin \mathbb N$ and $x<y:\Leftrightarrow \exists z\colon x+z=y$.</p>
<p>You may already have proved
$$\tag1\forall x,y\colon \exists z\colon x+y=S(z)$$
and
$$\tag2 \forall x,y,z\colon (x+y=x+z\rightarrow y=z).$$</p>
<p>Now the assumption $\alpha<\beta \land \beta<\alpha+1$ implies that ... |
233,846 | <p>This may be related to <a href="https://mathematica.stackexchange.com/q/98147">How to discretize a BezierCurve?</a>, but this question deals with <code>BSplineCurve</code>s with specific <code>SplineWeights</code>, so I don't think the answers there will help here.</p>
<hr />
<p><strong>Background</strong></p>
<p>I ... | robjohn | 232 | <p><strong>Additional Problem</strong></p>
<p>In addition to the two problems I mentioned above, there was a third problem that</p>
<pre><code>segments = {
BSplineCurve[{{1,0},{1,1},{0,1}},SplineWeights->{1,1/Sqrt[2],1}],
BSplineCurve[{{0,1},{-1,1},{-1,0}},SplineWeights->{1,1/Sqrt[2],1}]
};
Graphics[{Arrow[... |
2,040,678 | <p>I have a confusion regarding the symmetry of the volume in the following question. </p>
<p>Find the volume common to the sphere $x^2+y^2+z^2=16$ and cylinder $x^2+y^2=4y$.</p>
<p>The author used polar coordinates $x=rcos\theta$ snd $y=rsin\theta$ and does something like this:</p>
<p>Required volume $V=4\int_0^{π/... | Narasimham | 95,860 | <p>Before that what happened to the latitude $\phi$ ?</p>
<p>The picture shows a situation <a href="https://www.google.co.in/search?hl=en&site=imghp&tbm=isch&source=hp&biw=1028&bih=521&q=viviani&oq=viviani&gs_l=img.1.1.0l10.17336.288445.0.294391.7.6.0.1.1.0.301.884.1j2j1j1.5.0....0...1... |
261,361 | <ol>
<li>In a group of 200 people, number of people having at least primary education (assuming - <em>Category I</em>): number of people having at least middle school education (<em>Category II</em>): number of people having at least high school education (<em>Category III</em>) are in the ratio 7 : 3 : 1</li>
<li>Out ... | lamb_da_calculus | 54,044 | <p>First of all the idea here is that you give some indication that you've tried the problem before posting it, i.e. point out places where you get stuck or areas of confusion. That makes it easier to write a helpful answer. Not doing that may also be why you've been downvoted.</p>
<p>Having said that, here's how I ap... |
356,497 | <p>I usually solve a quadratic equation:</p>
<p>$$ax^2+bx+c=0$$</p>
<p>Through a method I learned in school: For a monic quadratic, you make $x=y-\frac{b}{2}$.</p>
<p>The method is intended for a monic equation but in this case (non-monic equation), I divide all the equation by $a$ to transform it in a monic equatio... | justt | 71,809 | <p>1/x is not a polynomial in x, so there is no such coefficient. However, 1/x is a rational function in x, with 1/x being formed from the 1/x "monom" with coefficient 1, and that's what wolfram|alpha's Coefficient[] function was thinking about. But it won't help you in your problem.</p>
<p>You have a very simple fact... |
749,090 | <p>Prove $\ a_{n}<2^{n} $ for every natural number n, where $\ a_{n} $ is defined recursively by $$ a_{1}=1, a_{2}=2, a_{3}=3, a_{n}=a_{n-3}+a_{n-2}+a_{n-1},\ for\ n>=4$$
Once I get the explicit equation, proving this would be easy with induction, however I'm having trouble finding it. I can't find the connecti... | MCT | 92,774 | <p>Hint: Imagine a $3 \times 3$ square with coordinates $(0,0), (0,3), (3,0), (3,3)$. When yo pick $x$ and $y$, you are choosing any point on the interior of this square, each being equally likely. Find the intersection of this square with the inequality graph $y \geq x + 1$.</p>
|
3,845,520 | <blockquote>
<p><strong>The length approximately equals width. The length is three times the height. The volume of the building is about <span class="math-container">$0.009 km^3$</span>.</strong></p>
</blockquote>
<hr />
<h2><em>The answer is 100 m by 300 m by 300 m.</em></h2>
<p>This question is supposed to be solved ... | hamam_Abdallah | 369,188 | <p>Is this allowed ?</p>
<p><span class="math-container">$$(1(13-13))^3=(10)^3$$</span></p>
|
1,158,601 | <p><img src="https://i.stack.imgur.com/P3crI.jpg" alt="enter image description here"></p>
<p>The solution is $x=50^{\circ}$.</p>
<p>How to prove $x=50^{\circ}$ without trigonometry?</p>
| Karl | 203,893 | <p><strong>This is incorrect and just left for reference</strong></p>
<p>The strategy to take appears to be to find all the angles in the centre at $D$ in terms of $x$ and then equate to $360$ but I don't get $x=50$</p>
<p>$\angle BDC=180-20-x=160-x$</p>
<p>$\angle ADB=180-30-10=140$</p>
<p>Use the largest triangl... |
1,031,038 | <p>"The price of a train-ticket is 110 dollars for grown-ups and 90 dollars for children. To a train, 120 tickets were sold for a total of 11640 dollars. How many grown-ups bought a ticket to the train?"</p>
<p>So here's my thought process:</p>
<p>$11640/120 = 97$</p>
<p>So now I have the mean value, but I have no i... | peterwhy | 89,922 | <p>A grown-up ticket costs $\$20$ more than a child ticket. Every ticket costs at least $\$90$, so $120$ tickets cost at least $\$10800$. The difference between the total price, $\$11640$, and this minimum cost, $\$10800$, represents how much more is paid to "upgrade" to grown-up tickets.</p>
<p>The difference $\$1164... |
581,497 | <p>Case $1$: $4$ games: Team A wins first $4$ games, team B wins none = $\binom{4}{4}\binom{4}{0}$</p>
<p>Case $2$: $5$ games: Team A wins $4$ games, team B wins one = $\binom{5}{4}\binom{5}{1}-1$...minus $1$ for the possibility of team A winning the first four.</p>
<p>Case $3$: $6$ games: Team A wins $4$ games, team... | Emily | 31,475 | <p>There are $\begin{pmatrix} 7 \\ 4\end{pmatrix}$ ways of arranging $4$ W's and $3$ L's. This is the answer.</p>
<p>Note that the following sequences are essentially identical:</p>
<p>$$WWLWWLL$$
$$WWLWW$$</p>
<p>Why?</p>
<p>Because once the winning team has amassed four wins, it doesn't matter if we count the rem... |
731,764 | <p>I understand what the epsilon-delta definition is saying in regards to the distance from a point c and the distance from your limit, but I don't understand how this defines a limit. Any help is appreciated.
Thanks!</p>
| Will Orrick | 3,736 | <p>In expressing the idea that
$$\displaystyle\lim_{x\to a}f(x)=L,$$
we often use intuitive phrases such as "$f(x)$ approaches $L$ as $x$ approaches $a$" or "$f(x)$ gets closer and closer to $L$ as $x$ closer and closer to $a$". While these phrases do capture some aspects of limits, the problem with them is that they ... |
3,109,300 | <p>For the case that <span class="math-container">$m\geq0$</span> I don't need to apply L'Hospital.</p>
<p>Let <span class="math-container">$m<0$</span></p>
<p>We have <span class="math-container">$x^m=\frac{1}{x^{-m}}$</span></p>
<p>We also know that <span class="math-container">$x^{-m}\rightarrow 0$</span> as <... | J.G. | 56,861 | <p>With <span class="math-container">$m:=-k<0,\,y:=1/x$</span> we want <span class="math-container">$\lim_{y\to\infty}y^k\exp -y$</span>. This is <span class="math-container">$0$</span> because <span class="math-container">$\int_0^\infty y^k\exp -y dy=k!$</span> is finite.</p>
|
194,123 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/150482/probability-of-a-random-binary-string-containing-a-long-run-of-1s">Probability of a random binary string containing a long run of 1s?</a> </p>
</blockquote>
<p><strong><em>EDIT</strong>: Cocopuffs b... | leonbloy | 312 | <p>To add to mjqxxxx's answer: </p>
<p>The probability of failure $P_f(L)=P(X_1<L ; X_2<L+X_1;X_3<L+X_1+X_2; \dots)$ can be bounded by the other side by</p>
<p>$$P_f(L) \le P(X_1<L) \, P(X_2<L + X_1 ) \, P(X_3<L + X_1+X_2 ) \cdots $$</p>
<p>(the inequality sign comes from the observation that if w... |
1,090,418 | <p>I just started reading D. Eisenbud <em>Commutative algebra with a view towards algebraic geometry</em> and I wonder about a theorem on page 42:</p>
<p>If $M$ is a finitely generated graded module over $k[x_1,...,x_r]$ then $H_M(s)$ agrees, for large $s$, with a polynomial of degree $\leq r-1$, where $H_M(s):=\dim_k... | Pp.. | 203,995 | <p>It means that there is a polynomial $p(n)=a_{r-1}n^r+...+a_0$, and a number $N$ such that $H_M(s)=p(s)$ for $s>N$. </p>
<p>Maybe more important to understand than that statement is the following equivalent one.</p>
<p>Define $\Delta f(n):=f(n+1)-f(n)$ and $\Delta^{m}f(n)=\Delta\Delta^{m-1}f(n)$.</p>
<p>Then th... |
15,784 | <p>When I edit a cell in the notebook and re-evaluate it, it "overwrites" the input and the output of the previous edit. I want it to copy my edited version to the bottom or something and give me a separate output, not overwriting the old results or old input.</p>
| Mike Honeychurch | 77 | <p>I'm not sure how your input would get overwritten -- can you add a screen grab showing this?</p>
<p>As for the output there is a default option for cells called <code>CellAutoOverwrite</code> and also an option for notebooks called <code>OutputAutoOverwrite</code> which is <code>True</code> by default. If you want ... |
1,478,103 | <p>I'm looking for examples of subtle errors in reasoning in a mathematical proof. An example of a 'false' proof would be</p>
<blockquote>
<p>Let $a=b>0$. Then $a^2 - b^2 = ab - b^2$. Factoring, we have $(a-b)(a+b) = b(a-b)$, which after cancellation yields $b = a = 2b$ and thus $1=2$. </p>
</blockquote>
<p>Howe... | Cheerful Parsnip | 2,941 | <p>This is a logical proof. </p>
<p><strong>Theorem:</strong> Either a statement or its converse must be true.</p>
<p><strong>Proof:</strong> Consider the proposition $$(P\to Q)\vee (Q\to P)$$ and check that it's a tautology, say using truth tables. The point is that if $P$ is false, then $P\to Q$ is true, and simila... |
1,478,103 | <p>I'm looking for examples of subtle errors in reasoning in a mathematical proof. An example of a 'false' proof would be</p>
<blockquote>
<p>Let $a=b>0$. Then $a^2 - b^2 = ab - b^2$. Factoring, we have $(a-b)(a+b) = b(a-b)$, which after cancellation yields $b = a = 2b$ and thus $1=2$. </p>
</blockquote>
<p>Howe... | Barry Cipra | 86,747 | <p>There is a famous old "proof" that all triangles are equilateral. I first came across it in Harold Stark's <em>An Introduction to Number Theory</em> (pp. 223-224). Carlo Sèquin gives a nice, five-minute <a href="https://www.youtube.com/watch?v=Yajonhixy4g" rel="nofollow">youtube demonstration</a> of it.</p>
|
299,304 | <p>I had posted the following problem on <a href="https://math.stackexchange.com/questions/2722893/the-modulus-of-a-polynomial-are-the-same-is-1">stack exchange</a> before.</p>
<blockquote>
<p>Suppose $\lambda$ is a real number in $\left( 0,1\right)$, and let $n$ be a positive integer. Prove that all the roots of t... | Lucia | 38,624 | <p>This is a special case of the Lee-Yang theorem. Let $G$ be a finite graph with vertex set $V$ and edge set $E$. Let $\beta \in (0,1)$ be a real number. Let $\sigma$ denote a ``spin function" $\sigma: V \to \{ -1, 1\}$. Put $m(\sigma)$ to be the number of vertices with positive spin, and $d(\sigma)$ to be the num... |
299,304 | <p>I had posted the following problem on <a href="https://math.stackexchange.com/questions/2722893/the-modulus-of-a-polynomial-are-the-same-is-1">stack exchange</a> before.</p>
<blockquote>
<p>Suppose $\lambda$ is a real number in $\left( 0,1\right)$, and let $n$ be a positive integer. Prove that all the roots of t... | David Hughes | 101,646 | <p>This follows from Julienne's answer. Fix $\lambda \in (0,1)$ and let
$$ f_n(x) = \sum_{k=0}^n {n \choose k} \lambda^{k(n-k)}x
^k. $$
Each $f_n$ satisfies condition (a) for $\mu = 1$. As for (b), we can compute the derivative of $f_n$ to get
$$f_n ' (x) = n \lambda^{n-1} f_{n-1} \left(\frac{x}{\lambda}\right).$$
Now ... |
48,359 | <p>I'm trying to numerically integrate a function which has a vector-valued slow part and a much faster component which is shared by all the components, i.e. an integral of the form
$$
\int_a^b\begin{pmatrix}f(x)\\ g(x) \\ h(x)\end{pmatrix}w(x)\,\text dx.
$$
Because <code>NIntegrate</code> is nicely Listable on its fir... | Simon Woods | 862 | <p>Your earlier approach would have worked if you had actually tested the argument to the function (<code>a</code>) rather than the undefined symbol <code>x</code>...</p>
<pre><code>AlgebraicQ[a_] := True /; Element[a, Algebraics]
AlgebraicQ[a_] := False /; ! Element[a, Algebraics]
AlgebraicQ /@ {7, Pi, Pi + E}
(* {T... |
2,010,158 | <p>Let $b >0$ , let $B= \{ f \in C^r([-b,b]) : f(x) = f(-x) for \ \ 0\leq x\leq b\}$, and let $A$ be the set of all polynomials that contain only terms of even degree (with domains restricted to $[-b,b]$). Then the uniform closure of $A$ is $B$.</p>
<p>I am not getting any clue how to solve the problem. Help Neede... | user64066 | 64,066 | <p>I think the following observation would be enough to show this result</p>
<blockquote>
<p>If <span class="math-container">$P$</span> is a polynomial, <span class="math-container">$Q$</span> is the part of <span class="math-container">$P$</span> which contains only even degree terms and <span class="math-containe... |
392,580 | <p>How to evaluate the following
$$\int_0^{\infty} \frac{\sin (ax)}{e^{\pi x} \sinh(\pi x)} dx $$
Given hints says to construct a rectangle $0\to R\to R+i\to i \to 0$ and consider $\displaystyle f(z):=\frac{e^{iaz}}{e^{2\pi z}-1} $ and evaluate around it but that does not help.</p>
<p><strong>ADDED::</strong> I need ... | 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}{\... |
1,563,686 | <p>Let $\sim$ be define so that $a\sim b$ exactly when $a \times b$ is divisible by $3$. Is this an equivalence relation? If not, which of the three properties (reflexive, symmetric, transitive) does not hold?</p>
<p>Solution:</p>
<p>We need to test each of the following cases to see if they hold.</p>
<p>Here are my... | Noah Schweber | 28,111 | <p>I think you're making things more complicated by using notation, rather than thinking about what the question means.</p>
<p><strong>Reflexivity</strong>. Is it the case that, for every number $a$, $a^2$ is divisible by $3$? If yes, then the relation is reflexive. If no, then the relation is not reflexive.</p>
<p><... |
1,563,686 | <p>Let $\sim$ be define so that $a\sim b$ exactly when $a \times b$ is divisible by $3$. Is this an equivalence relation? If not, which of the three properties (reflexive, symmetric, transitive) does not hold?</p>
<p>Solution:</p>
<p>We need to test each of the following cases to see if they hold.</p>
<p>Here are my... | Mikasa | 8,581 | <p>It is not transitive! Take $3|(2\times 3),~~3|(3\times 4)$ but $3$ does not divide $2\times 4$.</p>
|
662,590 | <p>It is mentioned that using the interpolation inequality</p>
<p><span class="math-container">$$\Vert f \Vert_{p} \leq \Vert f \Vert^{1/p}_{1} \Vert f \Vert_{\infty}^{1-1/p}$$</span></p>
<p>one can deduce that the space <span class="math-container">$L^{1} \cap L^{\infty}$</span> is dense in <span class="math-contain... | Yiorgos S. Smyrlis | 57,021 | <p>Let <span class="math-container">$f\in L^p(X)$</span>, where <span class="math-container">$X$</span> is a measure space, and set
<span class="math-container">$$
f_n(x)=\left\{\begin{array}{lll}
f(x) & \text{if} & \frac{1}{n}\le |f(x)|\le n, \\
0 & \text{if} & |f(x)|> n \,\,\text{or}\,\,|f(x)|<\... |
3,117,260 | <p>So according to the commutative property for multiplication:</p>
<p><span class="math-container">$a \times b = b \times a$</span> </p>
<p>However this does not hold for division</p>
<p><span class="math-container">$a \div b \neq b \div a$</span> </p>
<p>Why is it that in the following case:</p>
<p><span class="... | Angela Pretorius | 15,624 | <p>The non-zero real numbers form an abelian (commutative) group under multiplication. The notation <span class="math-container">$a\div b$</span> is shorthand for <span class="math-container">$ab^{-1}$</span>. </p>
<p>So <span class="math-container">$ab^{-1}\ne ba^{-1}$</span> but <span class="math-container">$56\time... |
3,117,260 | <p>So according to the commutative property for multiplication:</p>
<p><span class="math-container">$a \times b = b \times a$</span> </p>
<p>However this does not hold for division</p>
<p><span class="math-container">$a \div b \neq b \div a$</span> </p>
<p>Why is it that in the following case:</p>
<p><span class="... | Andrei | 331,661 | <p>Notice that you have always <span class="math-container">$\div 8$</span>, no matter the order of the other terms. You don't divide by a different number. It might help to think <span class="math-container">$\div c$</span> as a multiplication with <span class="math-container">$d=1/c$</span>. Then everything would loo... |
50,209 | <p>Over the past few days I have been pondering about this: I enjoy technical things (like programming and stuff) and try to find the patterns and algorithms in everything. My life is number oriented. I'll spend all day working on a programmatic problem. I'll spend however much time is needed to think of an elegant/eff... | Michael Chen | 541 | <p>High school math classes generally reward rote memorization and a fairly straightforward application of formulas to problems. There is not a lot of <em>why</em> or <em>how</em> or <em>what if</em> in the curriculum.</p>
<p>The mathematics that you speak of is the proof-based, abstract thinking classes that one sees... |
1,025,642 | <p>Let $X$ be a non-empty set. Suppose that $d_1$ and $d_2$ are two possibly different metrics on $X$. Let $\tau_i$ denote the topology generated by the metric $d_i$ ($i\in\{1,2\}$).</p>
<p>The following are known:</p>
<ul>
<li>$\tau_1=\tau_2\equiv\tau$;</li>
<li>$(X,d_1)$ is a totally bounded metric space;</li>
<li>... | zaira | 562,589 | <p><span class="math-container">$\forall k\in\mathbb{N}$</span><br/>
<span class="math-container">$k^2+2k<k^2+2k+1\Rightarrow k(k+2)<(k+1)^2\Rightarrow\frac{k}{k+1}<\frac{k+1}{k+2}$</span>
<br/><br/>
<span class="math-container">$\therefore a_n=\frac{1\cdot3\cdot5...(2n-1)}{2\cdot4\cdot6...2n}=\frac{1}{2}\cdot... |
979,299 | <p>Assuming that I have $\{x_1,\ldots, x_N\}$ - an iid (independent identically distributed) sample size $N$ of observations of random variable $\xi$ with unknown mean $m_1$, variance (second central moment) $m_{c_2}$ and second raw moment $m_2$. I try to use sample mean $\overline{x}=\frac{1}{N}\sum_{i=1}^Nx_i$ as an ... | ajotatxe | 132,456 | <p>The area that you have "computed by symmetry" is</p>
<p>$$\int_0^1\frac{dx}{x^2}-1=-1-1+\lim_{x\to 0^+}\frac1x$$</p>
<p>that is, $1/x^2$ is not even integrable in $(0,\infty)$.</p>
|
1,447,089 | <p>I am trying to prove that (0.1) is uncountable given that R is uncountable.</p>
<p>I start by assuming that (0.1) is countable. </p>
<p>Then there exists a bijective map between (0.1) and N.</p>
<p>I guess then we can construct bijective map for (1.2) also.</p>
<p>this shows that each (i-1,i) for i=intergers is ... | Thomas Pouget | 273,168 | <p>Hint:</p>
<p>study the variations of the function:</p>
<p>f(x) = (left hand side of your inequality)</p>
<p>this should then yield the answer very quickly</p>
|
1,447,089 | <p>I am trying to prove that (0.1) is uncountable given that R is uncountable.</p>
<p>I start by assuming that (0.1) is countable. </p>
<p>Then there exists a bijective map between (0.1) and N.</p>
<p>I guess then we can construct bijective map for (1.2) also.</p>
<p>this shows that each (i-1,i) for i=intergers is ... | Euler88 ... | 252,332 | <p>Note that $\sqrt{c+1}\geq \sqrt{c-1}$ and so $\frac{1}{\sqrt{c}+\sqrt{c-1}}\geq \frac{1}{\sqrt{c}+\sqrt{c+1}}$. Write $\sqrt{c}-\sqrt{c-1}=\frac{1}{\sqrt{c}+\sqrt{c-1}}$ and $\sqrt{c+1}-\sqrt{c}=\frac{1}{\sqrt{c}+\sqrt{c+1}}.$ Thus you have the result.</p>
|
443,099 | <p>I remember hearing someone say "almost infinite" on one of the science-esque youtube channels. I can't remember which video exactly, but if I do, I'll include it for reference.</p>
<p>As someone who hasn't studied very much math, "almost infinite" sounds like nonsense. Either something ends or it doesn't, there rea... | dfeuer | 17,596 | <p>In standard mathematics, this is indeed a meaningless concept. Some people have attempted, apparently unsuccessfully as yet, to develop a framework of <a href="http://en.m.wikipedia.org/wiki/Ultrafinitism" rel="nofollow">ultrafinitism</a>, which would give this concept some meaning.</p>
<p>The notion has more poten... |
16,080 | <p>I'm having trouble solving problem 12 from Section 1.2 in Hatcher's "Algebraic Topology".</p>
<p>Here's the relevant image for the problem:
<img src="https://i.stack.imgur.com/QNb5W.png" alt="enter image description here"></p>
<p>I'm trying to find $\pi_1(R^3-Z)$, where $Z$ is the graph shown in the first figure. ... | Sean Tilson | 627 | <p>This is admittedly not an answer, but it seems to me like you have already done a lot of the work. So here are my suggestions:</p>
<p>1) Name the generators of each copy of the integers, think about $\pi_1(A \cup B)$ in terms of generators and relations by using van Kampens' theorem as you are doing.</p>
<p>2) Rem... |
1,871,565 | <p>I am reading about deformation theory. I am treating mostly the algebraic case, but I would like to know a bit about all facets of this field of mathematics, so the geometric case is also of great interest to me. What are good references for the theory of deformations of complex analytic structures on a manifold?</p... | Libertron | 8,767 | <p>Using polar coordinates, consider $x= r \cos \theta, y= r \sin \theta, dxdy= r dr d\theta$. By the mean value theorem for harmonic functions and what is given, we have that $$f(0,0)= \frac{1}{2 \pi} \int_{0}^{2 \pi} f(r \cos \theta, r \sin \theta) d\theta=1 \implies \int_{0}^{2 \pi} f(r \cos \theta, r \sin \theta) d... |
2,364,690 | <p>The problem states:</p>
<blockquote>
<p>Let $mn$ be integers. Show that $mn$ is even if and only if $m$ is even or $n$ is even.</p>
</blockquote>
<p>They are asking to prove an iff statement. So it can be said that $P→Q$ and $Q→P$. I can prove that $¬P→¬Q$ and $¬Q→¬P$ as far is I understood. </p>
<p>I tried it ... | Especially Lime | 341,019 | <p>Choose a strictly increasing sequence of integers $0=a_0<a_1<\cdots$. Now define $f$ as follows. If $a_k\leq x\leq a_{k+1}-1$ for some $k$ then $f(x)=2^{-k}$; on the interval $[a_{k+1}-1,a_{k+1}]$ the function is linear.</p>
<p>Now on the interval $(a_{k+1}-1,a_{k+1})$ we have $g(x)$ is differentiable with de... |
2,364,690 | <p>The problem states:</p>
<blockquote>
<p>Let $mn$ be integers. Show that $mn$ is even if and only if $m$ is even or $n$ is even.</p>
</blockquote>
<p>They are asking to prove an iff statement. So it can be said that $P→Q$ and $Q→P$. I can prove that $¬P→¬Q$ and $¬Q→¬P$ as far is I understood. </p>
<p>I tried it ... | Kavi Rama Murthy | 142,385 | <p>Let $a_n=\frac{\ln(1)}{n^{1/2}}$ and $b_n=\sum_{k=2}^n (\ln(k))^{-1}$ for $n>3$. Let $f(b_n)=a_n$ and $f$ have a straight line graph between $b_n$ and $b_{n+1}$. Then f is defined on $[c,\infty)$ for some $c>0$ and $f(x+c)$ has the desired properties. </p>
|
137,691 | <p>What is the eigenvalue/eigenvector relationship between matrix A,B and AB?</p>
| Peter Michor | 26,935 | <p>See the following paper. The review describes only results for $A+B$, but this readily transforms to $AB$, as described in the paper.</p>
<ul>
<li>MR1957068 (2004b:14093) Reviewed
Klyachko, Alexander(TR-BILK)
Vector bundles, linear representations, and spectral problems. (English summary) Proceedings of the Intern... |
410,411 | <p>Let $f(x) = x^4 + 1 \in \mathbb{Q}[x]$. We can show that if $\alpha$ is a zero of $f(x)$, then the full set of zeros is given by $\{\alpha, -\alpha, i\alpha, -i\alpha\}$. Since $\alpha^2 = \pm i$ we can easily see that $L = \mathbb{Q}(\alpha)$ is the splitting field of $f$ over $\mathbb{Q}$. Since $L$ is the splitti... | Lubin | 17,760 | <p>Expanding on my comment, here’s my way of looking at things. Let $\zeta$ be a root of your polynomial $f(x)=x^4+1$. You know, of course, by looking at $f(x+1)$ and applying Eissenstein, that $f$ is irreducible. (In fact, it’s one of the cyclotomic polynomials, all of which are irreducible over $\mathbb Q$.) Now look... |
157,301 | <p>Here is the limit I'm trying to find out:</p>
<p><span class="math-container">$$\lim_{x\rightarrow 0} \frac{x^3}{\tan^3(2x)}$$</span></p>
<p>Since it is an indeterminate form, I simply applied l'Hopital's Rule and I ended up with:</p>
<p><span class="math-container">$$\lim_{x\rightarrow 0} \frac{x^3}{\tan^3(2x)} = \... | Gigili | 181,853 | <p>$$\lim_{x \to 0} \frac{x^3}{\tan (2x)^3}=\lim_{x \to 0} \frac{x^3}{(2x)^3}=\frac18$$</p>
|
157,301 | <p>Here is the limit I'm trying to find out:</p>
<p><span class="math-container">$$\lim_{x\rightarrow 0} \frac{x^3}{\tan^3(2x)}$$</span></p>
<p>Since it is an indeterminate form, I simply applied l'Hopital's Rule and I ended up with:</p>
<p><span class="math-container">$$\lim_{x\rightarrow 0} \frac{x^3}{\tan^3(2x)} = \... | user 1591719 | 32,016 | <p>We may resort to $\sin(x)<x<\tan(x),\space 0< x <\frac{\pi}{2}$ and solve it elementarily. By Squeeze's theorem we get that:</p>
<p>$$\lim_{x\rightarrow0}\frac{x^3 \cos^3(2x)}{{(2x)}^3}\leq \lim_{x\rightarrow0}\frac{x^3}{\tan^3(2x)}\leq \lim_{x\rightarrow0}\frac{x^3}{(2x)^3}$$</p>
<p>Therefore, taking... |
4,364,370 | <p>For <span class="math-container">$$\Phi(x) = (2\pi)^{-\frac 12}\int_{-\infty}^x \mathrm{e}^{-t^2/2}\,\mathrm{d}t,$$</span> it is claimed in the proof of Lemma 8.12 of <a href="https://www.hse.ru/data/2016/11/24/1113029206/Concentration%20inequalities.pdf" rel="nofollow noreferrer">this book</a> that we have the asym... | JetfiRex | 1,014,890 | <p>I try to turn the way around: for <span class="math-container">$x\to +\infty$</span>, <span class="math-container">$\log ( (2\pi)^{-\frac 12}\int_{x}^\infty \mathrm{e}^{-t^2/2}\,\mathrm{d}t )\sim -x^2/2$</span>. Since this is asymptotic, we omit the <span class="math-container">$(2\pi)^{-\frac 12}$</span> and only f... |
542,639 | <blockquote>
<h3>Problem:</h3>
<p>The collection $\mathcal{B}$ of subsets of the form $V = \{ x + yk : k
> \in \mathbb{Z} \} $ for $x,y \in \mathbb{Z}$ is a basis for some
topology of $\mathbb{Z}$</p>
</blockquote>
<h3>Solution attempt:</h3>
<p>Pick $n \in \mathbb{Z}$. and Put $N = \{ nk : k \in \mathbb{Z... | egreg | 62,967 | <p>Let $\mathscr{F}$ be a filter of subgroups in an abelian group $G$ (written additively). Consider the family
$$\mathscr{B}=\{\,x+H:x\in G,H\in\mathscr{F}\,\}.$$
Then $\mathscr{B}$ is a base for a topology on $G$. We need to see that the intersection of two elements of the family is a union of elements of the family... |
407,253 | <p>An edge will have the same vertices as another edge that it is parallel to, so how can it be uniquely described? </p>
| Chris Godsil | 16,143 | <p>I would prefer to use a variant of the second definition offered by Zev. A multigraph has a set of vertices $V$, a set of edges $E$. The connection between vertices and edges can be described by a relation on $V\times E$, such that
each edge is incident with one or two vertices (just one if you're not inclined to al... |
5,119 | <p>If $\mathcal{F}_1 \subset \mathcal{F}_2 \subset \dotsb$ are sigma algebras, what is wrong with claiming that $\cup_i\mathcal{F}_i$ is a sigma algebra?</p>
<p>It seems closed under complement since for all $x$ in the union, $x$ has to belong to some $\mathcal{F}_i$, and so must its complement.</p>
<p>It seems close... | hot_queen | 72,316 | <p>Something more drastic is true: If $\langle \mathcal{F}_n: n \geq 1\rangle$ is a <em>strictly</em> increasing sequence of sigma algebras over some set $X$ then $ \bigcup_{n \geq 1} \mathcal{F_n}$ is not a sigma algebra. As a corollary, there is no countably infinite sigma algebra. See, for example, "A comment on uni... |
3,429,020 | <p>This is the complex function: </p>
<p><span class="math-container">$$f(z) = 6\bar z^2-2\bar z - 4i |z|^2$$</span></p>
<p>which is also problem number 7.4, b on page 46 of this book: <a href="https://nnquan.files.wordpress.com/2013/01/giao-trinh-ham-phuc.pdf" rel="nofollow noreferrer">https://nnquan.files.wordpress... | José Carlos Santos | 446,262 | <p>If <span class="math-container">$x=0$</span>, then <span class="math-container">$f_n(x)=0$</span> and therefore <span class="math-container">$\lim_{n\to\infty}f_n(x)=0$</span>.</p>
<p>And if <span class="math-container">$x\in(0,1]$</span>, then<span class="math-container">$$\lim_{n\to\infty}ne^{-nx^2}=\lim_{n\to\in... |
4,448,865 | <p>Let the continuous spectrum of a densely defined linear operator <span class="math-container">$L$</span> over a Separable Hilbert space, be defined as the set of all <span class="math-container">$\lambda \in \mathbb C$</span> such that:</p>
<p>(i) <span class="math-container">$L-\lambda$</span> is injective,</p>
<p>... | Kavi Rama Murthy | 142,385 | <p>The statement is true.</p>
<p><span class="math-container">$$\int_{0}^{1}|f_n(x)g_n(x)-f(x)g(x)|dx\leq \int_{0}^{1}|g_n(x)||f_n(x)-f(x)|dx\\~~~~~~~~ ~~~~~~~ ~~~~~~~ ~~~~~~~ ~~~~~~~ ~~~~~~~ ~~~~~+\int_{0}^{1}|f(x)||g_n(x)-g(x)|dx.$$</span>
The first term clearly tends to <span class="math-container">$0$</span>. For t... |
2,902,058 | <p>I am solving the following problem:
$$\lim_{R\rightarrow \infty} \int_{C_R} \frac{6z^6 + 5z^5}{z^6 + z^5 + 10}dz,$$
where $C_R=\{z \in \mathbb{C} : |z|=R \}$ for $R>0$.</p>
<p>The only one idea I have is to use the Residue theorem. </p>
<p>But I couldn't apply the theorem to the above problem.</p>
| qualcuno | 362,866 | <p>Here is a possible argument: since $x$ is not an integer, it is contained in some interval $(k, k+1)$ with $k \in \mathbb{Z}$. This is not hard to prove, try making a drawing and play with the floor function.</p>
<p>Now, since $x_n \to x$, there exists $n_0 \in \mathbb{N}$ such that $x_n \in (k,k+1)$ for $n \geq n_... |
634,510 | <p>Consider a uniformly selected random vector $V= (v_1,v_2,\dots, v_n)$ where $v_i \in \{0,1\}$. Let us define $V_1 = (v_1,v_2,\dots, v_n)$, $V_2 = (v_n, v_1,v_2,\dots, v_{n-1})$, $V_3 = (v_{n-1}, v_n, v_1,v_2,\dots,v_{n-2})$ and so on.</p>
<p>I am trying to work out the probability that there exists $i \ne j$ such ... | Daniel Pietrobon | 17,824 | <p>You are overcounting in the following way. </p>
<p>If the men were elements of $M = \{A,B,C,D,E,F,G\}$ then you might choose $A,B,C$ from $M$ and then $D,E$ from the remaining four or you might choose $A,B,D$ and $C,E$ from the remaining four but you get the same five men in both cases. </p>
<p>For any given five ... |
634,510 | <p>Consider a uniformly selected random vector $V= (v_1,v_2,\dots, v_n)$ where $v_i \in \{0,1\}$. Let us define $V_1 = (v_1,v_2,\dots, v_n)$, $V_2 = (v_n, v_1,v_2,\dots, v_{n-1})$, $V_3 = (v_{n-1}, v_n, v_1,v_2,\dots,v_{n-2})$ and so on.</p>
<p>I am trying to work out the probability that there exists $i \ne j$ such ... | arkadeep | 120,499 | <p>Hi lets us concider a small real life example!
See....
let us concider that we have 5 members such as A B C D E ok.
Now you are willing to choose any 3 members from them to form a team.So then you will use the combination concept among them.right!fine...this will be simply 5C3 = ((120)/((2!)*(3!)))= 10.</p>
<p><str... |
3,077,882 | <p>Simplify the expression
<span class="math-container">$$\sin\left(\tan^{-1}(x)\right)$$</span>
Using a triangle with an angle <span class="math-container">$\theta$</span>, opposite is x and adjacent is 1 meaning the hypo. is <span class="math-container">${\sqrt {x^2+1}}$</span> </p>
<p>Now because the problem has s... | lab bhattacharjee | 33,337 | <p>Let <span class="math-container">$\tan^{-1}x=y\implies x=\tan y$</span> and <span class="math-container">$-\dfrac\pi2<y<\dfrac\pi2$</span> using <a href="https://en.wikipedia.org/wiki/Inverse_trigonometric_functions#Principal_values" rel="nofollow noreferrer">Principal values</a></p>
<p><span class="math-cont... |
818,015 | <p>I try to get the variables for this equation:</p>
<p>$$\begin{cases}
6x_1 + 4x_2 + 8x_3 + 17x_4 &= -20\\
3x_1 + 2x_2 + 5x_3 + 8x_4 &= -8\\
3x_1 + 2x_2 + 7x_3 + 7x_4 &= -4\\
0x_1 + 0x_2 + 2x_3 -1x_4 &= 4
\end{cases}$$</p>
<p>So i started with:</p>
<p>$$ \begin{pmatrix}
6 & 4 &am... | 5xum | 112,884 | <p>Using $\cos x = 1-2\sin^2\frac x2$ gives you $$\frac{1-\cos x}{x^2} = \frac{2\sin^2\frac x2}{x^2} = \frac12\left(\frac{\sin\frac x2}{\frac x2}\right)^2.$$</p>
<p>Using similar identities will help with the second limit as well.</p>
|
2,959,026 | <blockquote>
<p>I have the matrix</p>
<p><span class="math-container">$$Q=\begin{bmatrix}-1&2\\0&1\\\end{bmatrix}$$</span></p>
<p>and want to find the invariant points.</p>
</blockquote>
<p>To do this, I solve the equation:</p>
<p><span class="math-container">$$\begin{bmatrix}-1&2\\0&1\\\end{bmatrix}\be... | gandalf61 | 424,513 | <p><span class="math-container">$Q$</span> has rank <span class="math-container">$2$</span> and eigenvalues <span class="math-container">$+1$</span> and <span class="math-container">$-1$</span>. The invariant points correspond to the eigenvectors with eigenvalue <span class="math-container">$+1$</span>. As you have fou... |
1,353,892 | <p>Just a small thought popped up in my mind; and now I'm stuck on it. Any idea on how to find the value of $\tan 20^\circ$? I tried doing it by using the multiple angle formulas, but I didn't get an answer... How do I proceed? </p>
| John Hughes | 114,036 | <p>$\sin(20^\circ)$ and $\cos(20^\circ)$ can both be found using the triple-angle formulas. (I'm assuming your "20" was in degrees, not radians.)</p>
<p>Unfortunately, both of those will end up with you having to solve a cubic; you can use Cardano's formula to do that. </p>
<p>You can't solve it with just quadratics ... |
1,353,892 | <p>Just a small thought popped up in my mind; and now I'm stuck on it. Any idea on how to find the value of $\tan 20^\circ$? I tried doing it by using the multiple angle formulas, but I didn't get an answer... How do I proceed? </p>
| user2662833 | 89,483 | <p>I'm assuming you mean via multiple angle formulas, and I'm assuming you want an exact answer. The simple answer is, it would require a ridiculous amount of work. You can write:</p>
<p>$$ \tan \left(45-30+5\right)$$</p>
<p>You could perform the double angle formula twice, and reduce the problem to finding tan(5), w... |
1,353,892 | <p>Just a small thought popped up in my mind; and now I'm stuck on it. Any idea on how to find the value of $\tan 20^\circ$? I tried doing it by using the multiple angle formulas, but I didn't get an answer... How do I proceed? </p>
| Community | -1 | <p>$20°$ is a third of $60°$, for which the value of the tangent is well known to be $\sqrt3$. Let us denote $x:=\tan(20°)$.</p>
<p>Then, by the addition formula, </p>
<p>$$\tan(40°)=\frac{2x}{1-x^2},$$
and
$$\tan(60°)=\frac{x+\dfrac{2x}{1-x^2}}{1-x\dfrac{2x}{1-x^2}}=\frac{3x-x^3}{1-3x^2}=\sqrt3,$$
or
$$x^3-3\sqrt3x... |
1,353,892 | <p>Just a small thought popped up in my mind; and now I'm stuck on it. Any idea on how to find the value of $\tan 20^\circ$? I tried doing it by using the multiple angle formulas, but I didn't get an answer... How do I proceed? </p>
| Claude Leibovici | 82,404 | <p>Since, obviously, numerical calculations would be required, I cannot resist the pleasure of reusing a 1400 years old approximation proposed by Mahabhaskariya of Bhaskara I, a seventh-century Indian mathematician $$\sin(x) \simeq \frac{16 (\pi -x) x}{5 \pi ^2-4 (\pi -x) x}$$ A similar one $$\cos(x)\simeq \frac{\pi^2-... |
258,071 | <p>Let $P(x),Q(x),R(x)$ be the statements $x$ is a clear explanation,$x$ is satisfactory,$x$ is an excuse,respectively. Suppose that the domain for $x$ consists of all the English text. Express each of these statements using quantifiers, logical connectives and $P(x),Q(x),R(x)$.</p>
<p>a. All clear explanations are sa... | Ross Millikan | 1,827 | <p>Assuming that by Vx you mean $\forall x$ (for all x) you are correct on the first one. The reason you use implication is that you are transcribing "all A are B" into "A(x) implies B(x)" which is an implication.</p>
|
129,294 | <p>I found the answer in this <a href="https://mathematica.stackexchange.com/questions/67306/insert-at-specific-resulting-positions/67309#67309">post</a> very interesting to do what I need, but I would like something where I could provide a <code>list to be modified</code>, a <code>list with values</code> that will be ... | george2079 | 2,079 | <pre><code>listaInicial = {0, 1, 2, 3, 4, 5, 6, 7};
listaModificadora = {a, b, c, d, e};
listaPos = {2, 6, 7, 8, 13};
listIpos =
Complement[Range@Length@Join[listaInicial, listaModificadora],
listaPos]
Normal@SparseArray[listIpos~Join~listaPos ->
listaInicial~Join~listaModificadora]
</code></pr... |
3,630,967 | <p>More precisely, define <span class="math-container">$\phi(x) = \frac1{\sqrt{2\pi}}e^{-\frac{x^2}{2}}$</span>. </p>
<p>Does there exists a constant <span class="math-container">$L$</span> such that <span class="math-container">$$|\phi(x)-\phi(y)|\le L|x-y|,$$</span> for all <span class="math-container">$x,y\in \math... | Tuvasbien | 702,179 | <p><span class="math-container">$$\forall x\in\mathbb{R},\phi'(x)=-\frac{x}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} $$</span>
Thus <span class="math-container">$\lim\limits_{x\rightarrow\pm\infty}\phi'(x)=0$</span> and since <span class="math-container">$\phi'$</span> is continuous, it is bounded on <span class="math-container"... |
2,448,082 | <p>Trying to solve $$\int 27x^3(9x^2+1)^{12} dx$$
I know the process and formula of integration by parts. When I set $u = 9x^2 + 1$, $du = 18x dx$. I am stuck on the next step as 18x does not line up with the $27x^3$. </p>
| operatorerror | 210,391 | <p>As you saw, substitution is not the way to go, because the powers do not work out nicely. Try reducing your problem to an easier one to use substitution on by taking
$$
u=27x^2\\
\mathrm dv=x(9x^2+1)^{12}
$$
in the formula
$$
\int u\mathrm dv=uv-\int v\mathrm du
$$</p>
|
318,753 | <p>I tried to show that the following group is abelian by manipulation the relations but they didn't work. Please show me the right way. The group is $$G:=\left<x,y \mid xyxy^2=yxyx^2=1\right>$$</p>
| Seirios | 36,434 | <p><strong>Hint:</strong> You can identify $xyxy^2$ as a subword of $yxyx^2$. In details:</p>
<blockquote class="spoiler">
<p> $$\begin{array}{ll} yxyx^2=1 & \Rightarrow xyxyx^2=x \\ & \Rightarrow (xyxy^2)y^{-1}x^2=x \\ & \Rightarrow y^{-1}x^2=x \\ & \Rightarrow y^{-1}x=1 \\ & \Rightarrow x=y \en... |
203,995 | <p>Let $\mathbb{C}[x,y]$ be the polynomial ring with variables $x,y$ and coefficient in $\mathbb{C}$.</p>
<p>Let $f,g\in \mathbb{C}[x,y]$. </p>
<p>Let $(f,g)$ be the ideal of $\mathbb{C}[x,y]$ generated by $f,g$. </p>
<p>Given $h\in \mathbb{C}[x,y]$, how to determine whether $h\in (f,g)$ or not? </p>
<p>I have trie... | Igor Rivin | 11,142 | <p>The algorithm is described in <a href="http://people.csail.mit.edu/madhu/ST12/scribe/lect15.pdf" rel="nofollow">these notes by Madhu Sudan.</a> The sage implementation is <a href="http://www.sagemath.org/doc/reference/polynomial_rings/sage/rings/polynomial/multi_polynomial_ideal.html" rel="nofollow">described in the... |
2,992,411 | <p><span class="math-container">\begin{cases}
\frac {dP}{dt} = rP(t)(1-\frac {P(t)}{K}) ,t \geq 0 \\
P(0) = P_o
\end{cases}</span></p>
<p><span class="math-container">$r, K$</span> and <span class="math-container">$P_o$</span> are positive constants.</p>
<p>We say that <span class="math-container">$P(t), t \geq0... | David Lui | 445,002 | <p>Consider the bijection <span class="math-container">$\{-1, 1\}^{\mathbb{Z}^2}$</span> to <span class="math-container">$P(\mathbb{Z}^2)$</span> by sending a function <span class="math-container">$f$</span> to the set <span class="math-container">$\{x \in \mathbb{Z}^2 : f(x) = 1\}$</span>. The inverse is <span class="... |
3,314,414 | <p>Need </p>
<ul>
<li><p>an example of a matrix with a square root but it does not admit any cubic roots </p></li>
<li><p>an example of a matrix with a cubic root but it does not admit any square root</p></li>
</ul>
| Ben Grossmann | 81,360 | <p><strong>Hint:</strong> You can find suitable examples among the powers of <a href="https://en.wikipedia.org/wiki/Jordan_matrix" rel="nofollow noreferrer">Jordan blocks</a> for <span class="math-container">$\lambda = 0$</span>.</p>
|
3,314,414 | <p>Need </p>
<ul>
<li><p>an example of a matrix with a square root but it does not admit any cubic roots </p></li>
<li><p>an example of a matrix with a cubic root but it does not admit any square root</p></li>
</ul>
| Piquito | 219,998 | <p>COMMENT.- The cube of <span class="math-container">$A=\begin{pmatrix}1&2&3\\4&5&6\\7&8&9\end{pmatrix}$</span> is equal to
<span class="math-container">$B=\begin{pmatrix}468&576&684\\1062&1305&1548\\1656&2034&2412\end{pmatrix}$</span> so it is evident that <span class="... |
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