qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
3,395,044 | <p>Wikipedia states:</p>
<blockquote>
<p>In mathematics, a <strong>formal power series</strong> is a generalization of a <strong>polynomial</strong>, where the number of terms is allowed to be infinite; this implies giving up the possibility of replacing the variable in the <strong>polynomial</strong> with an arbitr... | Jean-Claude Arbaut | 43,608 | <p>The best way to see this: forget about the <span class="math-container">$X$</span> in a polynomial or a <strong>formal</strong> power series, they are really sequences of coefficients, with no constraint on their values, with some specific rules of computation for addition and multiplication.</p>
<p>A power series,... |
1,983,129 | <p>In the tripple integral to calculate the volume of a sphere
why does setting the limits as follows not work?</p>
<p>$$ \int_{0}^{2\pi} \int_{0}^{\pi}
\int_{0}^{R} p^2 \sin{\phi}
\, dp\,d\theta\,d\phi $$</p>
| Ben Grossmann | 81,360 | <p>Yes. In particular, there's no reason to go through the calculation $\left(\cos{2 \pi k \over n} + i \sin{2 \pi k \over n}\right)^n = 1$. From the problem statement, we <em>already</em> know that $\epsilon^n = 1$.</p>
|
1,983,129 | <p>In the tripple integral to calculate the volume of a sphere
why does setting the limits as follows not work?</p>
<p>$$ \int_{0}^{2\pi} \int_{0}^{\pi}
\int_{0}^{R} p^2 \sin{\phi}
\, dp\,d\theta\,d\phi $$</p>
| hamam_Abdallah | 369,188 | <p>For each root $ (\epsilon \neq1 )$ of the equation $x^n=1, \;$ let
$$S_\epsilon=\sum_{k=0}^{n-1}\epsilon^k.$$</p>
<p>then, for each one of these $\epsilon$, </p>
<p>$$\epsilon S_\epsilon=S_\epsilon.$$</p>
<p>thus</p>
<p>$$S_\epsilon=0,$$</p>
<p>since $\epsilon\neq 0$.</p>
|
2,046,521 | <p>Of course, faster calculations help solve problems quickly. But does that also mean that faster calculations open more opportunities for a career in mathematics (like a researcher)? I like mathematics and can spend weeks trying to solve any problem or understanding any concept. But nowadays, there are many contests ... | Mathily | 375,170 | <p>I don't think being slow at computations will make pursuing research math impossible, but it might make it harder. Getting an undergraduate degree in math (at an American university) still requires a fair amount of test taking and problem sets which involve computation. If you're really slow it may depress your gr... |
793,693 | <p>Since I was interested in maths, I have a question. Is infinity a real or complex quantity? Or it isn't real or complex?</p>
| Michael Hardy | 11,667 | <p>There are many different things that could be called "the infinite" in mathematics. <b>None</b> of them is a real number or a complex number, but some are used in discussing functions or real or complex numbers.</p>
<ul>
<li><p>There are things called $+\infty$ and $-\infty$. Those appear in such expressions as
$... |
2,220,738 | <p>How do I go about finding the dimension of the subspace:
$$$$S:={p($x$) ∈ $P_4$: p($x$)= 2p($x$) for all $x\in\mathbb{R}$} of $P_4$</p>
<p>My textbook says $dim(P_n)=n+1$, but this does not give me the correct answer. All help is appreciated.</p>
| Eman Yalpsid | 94,959 | <p>Let $n=4$, then $p \in S \leq P_n$ has the form of $\sum_{j=0}^{n}a_jx^j$ for some $a_j \in R$. <br> Therefore $p = 2p$ implies that for all $x \in R$,
$$\sum_{j=0}^{n}a_jx^j = 2\sum_{j=0}^{n}a_jx^j \iff 0 = \sum_{j=0}^{n}(2-1)a_jx^j = p(x).$$
What's the dimension of a space where every element has this property? </... |
182,785 | <p>I haved plot a graph from two functions:</p>
<pre><code>η = 52;
h = 0.5682;
dpdx = -4.092*10^(-2);
Fg = dpdx;
Fl = dpdx/η;
Bl = ((Fg - Fl) h^2 - Fg)/(2 h - 2 η*h + 2 η);
Cg = -Fg/2 - η*Bl;
Bg = η*Bl;
Ut1[y_] := Fg*y^2/2 + Bg*y + Cg;
Ut2[y_] := Fl*y^2/2 + Bl*y;
Plot1 = Plot[Ut1[y]*1000, {y, h, 1}];
Plot2 = Plot[U... | Bob Hanlon | 9,362 | <p>Use <a href="https://reference.wolfram.com/language/ref/ParametricPlot.html" rel="nofollow noreferrer"><code>ParametricPlot</code></a> to flip the axes.</p>
<pre><code>η = 52;
h = 0.5682;
dpdx = -4.092*10^(-2);
Fg = dpdx;
Fl = dpdx/η;
Bl = ((Fg - Fl) h^2 - Fg)/(2 h - 2 η*h + 2 η);
Cg = -Fg/2 - η*Bl;
Bg = η*Bl;
Ut... |
2,801,433 | <p>I have made the following conjecture, and I do not know if this is true.</p>
<blockquote>
<blockquote>
<p><strong>Conjecture:</strong></p>
</blockquote>
<p><span class="math-container">\begin{equation*}\sum_{n=1}^k\frac{1}{\pi^{1/n}p_n}\stackrel{k\to\infty}{\longrightarrow}2\verb| such that we denote by | p_n\verb| ... | Claude Leibovici | 82,404 | <p>Numerically, this does not seem to be true.</p>
<p>Considering $$S_m=\sum_{n=1}^{10^m}\frac{1}{\pi^{1/n}p_n}$$ and, using illimited precision, I obtained the following numbers
$$\left(
\begin{array}{cc}
m & S_m \\
1 & 0.891549393 \\
2 & 1.437754209 \\
3 & 1.787152452 \\
4 & 2.038881140 \\
... |
4,394,983 | <p>I am tasked with proving that Th((<span class="math-container">$\mathbb{Z}, <))$</span> has continuum many models.
For this we are given the following construction.</p>
<blockquote>
<p>Let <span class="math-container">$\alpha \in \mathcal{C} = \{0,1\}^{\mathbb{N}}$</span>. We define for each <span class="math-con... | Olivier Roche | 649,615 | <p>Point 1. : let <span class="math-container">$\alpha \neq \beta \in \mathcal{C}$</span>. Then there is <span class="math-container">$n \in \mathbb{N}$</span> such that <span class="math-container">$\alpha(n) \neq \beta(n)$</span>. WLOG, <span class="math-container">$\alpha(n) = 0$</span> and <span class="math-contain... |
3,839,878 | <p>Am currently doing a question that asks about the relationship between a quadratic and its discriminant.</p>
<p>If we know that the quadratic <span class="math-container">$ax^2+bx+c$</span> is a perfect square, then can we say anything about the discriminant?</p>
<p>Specifically, can we be sure that the discriminant... | poetasis | 546,655 | <p>Let us assume that <span class="math-container">$ax^2+bx+c=d=(jx+k)^2 $</span></p>
<p><span class="math-container">\begin{equation}
d=(jx+k)^2\implies j^2x^2+2jkx+k^2-d=0\implies a=j^2\quad b=2jk\quad c=k^2-d\\
x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}
=\frac{-2jk\pm\sqrt{2^2j^2k^2-4j^2(k^2-d)}}{2j}
=\frac{-2jk\pm 2j\sqrt{d}... |
333,467 | <p>I was reading in my analysis textbook that the map $ f: {\mathbf{GL}_{n}}(\mathbb{R}) \to {\mathbf{GL}_{n}}(\mathbb{R}) $ defined by $ f(A) := A^{-1} $ is a continuous map. I also saw that $ {\mathbf{GL}_{n}}(\mathbb{R}) $ is dense in $ {\mathbf{M}_{n}}(\mathbb{R}) $. My question is:</p>
<blockquote>
<p>What is t... | Haskell Curry | 39,362 | <p>Just to add to the two answers above. If you refer to my solution in the thread <a href="https://math.stackexchange.com/questions/290971/left-topological-zero-divisors-in-banach-algebras/291196#291196">Left topological zero-divisors in Banach algebras.</a>, you will see that if $ X \in \partial({\text{GL}_{n}}(\math... |
1,677,359 | <p>$\sum_{i=0}^n 2^i = 2^{n+1} - 1$</p>
<p>I can't seem to find the proof of this. I think it has something to do with combinations and Pascal's triangle. Could someone show me the proof? Thanks</p>
| André Nicolas | 6,312 | <p>We give a combinatorial interpretation.</p>
<p>We are counting the "words" of length $n+1$, over the alphabet $\{0,1\}$, that are not all $0$'s. There are $2^{n+1}-1$ such words. </p>
<p>We count these words another way. Maybe the first $1$ is at the beginning. There are $2^n$ such words.</p>
<p>Maybe the word be... |
1,666,396 | <p>I can show the convergence of the following infinite product and some bounds for it:</p>
<p>$$\prod_{k \geq 2}\sqrt[k]{1+\frac{1}{k}}=\sqrt{1+\frac{1}{2}} \sqrt[3]{1+\frac{1}{3}} \sqrt[4]{1+\frac{1}{4}} \cdots<$$</p>
<p>$$<\left(1+\frac{1}{4} \right)\left(1+\frac{1}{9} \right)\left(1+\frac{1}{16} \right)\cdo... | Rob | 274,944 | <p>As already discussed above by Yuriy S and others, the product is intimately linked with the series $\sum_{n=1}^{\infty} { \ln(n+1) \over n(n+1) } \approx 1.2577\dots$. The connection is derived as follows:</p>
<p>$$ \prod_{k=2}^{\infty} \left( 1+ {1\over k} \right)^{1/k} \\
=\exp \left( \ln \left( \prod_{k=2}^{\in... |
1,752,021 | <blockquote>
<p>Let $G=S_3\times S_3$ where $S_3$ is the symmetric group. Let $p=
\begin{pmatrix}
1 & 2 & 3 \\
2 & 3 & 1 \\
\end{pmatrix}
$, let $L=(p)$, $K=L\times L$ and $H=\{(I_3,I_3),(p,p),(p^2,p^2)\}$. Show that $K\triangleleft G$, $H\triangleleft K$ but $H$ no is ... | JKim | 290,442 | <p>$K \trianglelefteq G$ is easy to be seen because each $L \trianglelefteq S_3$ as $L$ has index 2 in $S_3$. For $H \trianglelefteq K$, we can use the fact:</p>
<p>If $H$ has a prime index $p$ in $G$ and there is no prime divisor of $|G|$ less than $p$, then $H \trianglelefteq G$. </p>
<p>$|K:H| = 3$ and there is no... |
14,007 | <p>A colleague of mine will be teaching 3 classes (pre-calculus and two sections of calculus, at the university level) next semester with an additional grader in only one of those classes (pre-calculus). With an upper bound of 35 students a class, there is potential for a large amount of grading that needs to happen wi... | Daniel R. Collins | 5,563 | <p>One I recently learned -- for the order of the signs in factoring a sum or difference of cubes, remember <strong>SOAP</strong>: Same sign, Opposite sign, Always a Plus.</p>
<blockquote>
<p>Sum of Cubes: <span class="math-container">$x^3 + a^3 = (x + a)(x^2 - ax + a^2)$</span></p>
<p>Difference of Cubes: <span class=... |
14,007 | <p>A colleague of mine will be teaching 3 classes (pre-calculus and two sections of calculus, at the university level) next semester with an additional grader in only one of those classes (pre-calculus). With an upper bound of 35 students a class, there is potential for a large amount of grading that needs to happen wi... | JRN | 77 | <p>Here is my way of memorizing the three main trigonometric functions. An angle $\theta$ is in standard position locating a point $(x,y)$ on a circle with a radius $r$ centered at the origin. There is a convertible (a car with the roof removed) being driven down a road during the daytime.</p>
<p>The sun (sine) is a... |
14,007 | <p>A colleague of mine will be teaching 3 classes (pre-calculus and two sections of calculus, at the university level) next semester with an additional grader in only one of those classes (pre-calculus). With an upper bound of 35 students a class, there is potential for a large amount of grading that needs to happen wi... | Steven Gubkin | 117 | <p>I tell students to visualize $<$ and $>$ as mouths. They always want to eat the bigger number.</p>
|
1,657,694 | <p>The Algorithms course I am taking on Coursera does not require discrete math to find discrete sums. Dr. Sedgewick recommends replacing sums with integrals in order to get basic estimates.</p>
<p>For example: $$\sum _{ i=1 }^{ N }{ i } \sim \int _{ x=1 }^{ N }{ x } dx \sim \frac { 1 }{ 2 } N^2$$</p>
<p>How would I ... | DanielWainfleet | 254,665 | <p>Let $D$ be the set of open intervals removed from $[0,1]$ in the construction of the Cantor set.</p>
<p>For $d=(a,b)\in D,$ let $(x_{n,a})_{n\in N}$ be strictly decreasing , converging to $a$; and let $(y_{n,d})_{n\in N}$ be strictly increasing, converging to $b$; with $x_{1,d}=y_{1,d}=(a+b)/2.$</p>
<p>Let $C$ be ... |
3,429,489 | <blockquote>
<p>Let <span class="math-container">$\mathcal F = \{f \mid f : \mathbb R \rightarrow \mathbb R\}$</span> and
define relationship <span class="math-container">$R$</span> on <span class="math-container">$\mathcal F$</span> as follows:</p>
<p><span class="math-container">$$R = \{(f,g) \in \mathcal F... | Community | -1 | <p>[ EDITED ] I'm not competent enough to assess the proof you have proposed ( though it seems right to me). . Let me try an indirect proof of (-->) using contraposition. </p>
<p><a href="https://i.stack.imgur.com/25CyC.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/25CyC.png" alt="enter image desc... |
3,301,115 | <p>I'm currently taking an introductory course in graph theory, and this problem is giving me a bit of a hard time. Where would I even start? Thanks a bunch?</p>
| chenbai | 59,487 | <p>hint:</p>
<p><span class="math-container">$\dfrac{a}{\sqrt{2-2bc}} \le \dfrac{a}{\sqrt{1+a^2}}$</span></p>
<p>if you can prove <span class="math-container">$f(x)=\sqrt{\dfrac{x}{1+x}}$</span> is concave function, then the problem is solved.</p>
|
2,147,458 | <p>Solve the following integral:
$$
\frac{2}{\pi}\int_{-\pi}^\pi\frac{\sin\frac{9x}{2}}{\sin\frac{x}{2}}dx
$$</p>
| xpaul | 66,420 | <p>By using
$$ \sin A-\sin B=2\cos(\frac{A+B}{2})\sin(\frac{A-B}{2})$$
one has
\begin{eqnarray}
&&\frac{2}{\pi}\int_{-\pi}^\pi\frac{\sin\frac{9x}{2}}{\sin\frac{x}{2}}dx\\
&=&\frac{2}{\pi}\int_{-\pi}^\pi\frac{\sum_{n=1}^4\bigg[\sin\frac{(2n+1)x}{2}-\sin\frac{(2n-1)x}{2}\bigg]+\sin\frac{x}{2}}{\sin\frac{x... |
14,712 | <p>I have matrix <code>in</code> as shown, consisting of real numbers and 0. How can I sort it to become <code>out</code> as shown?</p>
<pre><code>in ={
{0, 0, 3.411, 0, 1.343},
{0, 0, 4.655, 2.555, 3.676},
{0, 3.888, 0, 3.867, 1.666}
};
out ={
{1.343, 3.411, 0, 0, 0},
{2.555, 3.676, 4.655, 0, 0},
... | Andy Ross | 43 | <p>You can map <code>Sort</code> over the rows using a custom ordering function which treats 0 as infinity.</p>
<pre><code>data = RandomChoice[{0, 1}, {5, 5}]*RandomReal[{1, 10}, {5, 5}];
f[0|0.]= \[Infinity];
f[x_] := x
Sort[#, f[#1] <= f[#2] &] & /@ data
(*{{6.07883, 7.33113, 0., 0., 0.}, {2.74761, 0., ... |
14,712 | <p>I have matrix <code>in</code> as shown, consisting of real numbers and 0. How can I sort it to become <code>out</code> as shown?</p>
<pre><code>in ={
{0, 0, 3.411, 0, 1.343},
{0, 0, 4.655, 2.555, 3.676},
{0, 3.888, 0, 3.867, 1.666}
};
out ={
{1.343, 3.411, 0, 0, 0},
{2.555, 3.676, 4.655, 0, 0},
... | ssch | 1,517 | <p>Since what you are doing is basically sorting each row, but 0 is treated as highest value. One way is to replace all zeros with <code>Infinity</code> before sorting and changing back after </p>
<pre><code>r = RandomChoice[{0, Random[]}, {3, 5}];
r // MatrixForm
(Sort[#] & /@ (r /. {(0 | 0.) -> Infinity})) /... |
14,712 | <p>I have matrix <code>in</code> as shown, consisting of real numbers and 0. How can I sort it to become <code>out</code> as shown?</p>
<pre><code>in ={
{0, 0, 3.411, 0, 1.343},
{0, 0, 4.655, 2.555, 3.676},
{0, 3.888, 0, 3.867, 1.666}
};
out ={
{1.343, 3.411, 0, 0, 0},
{2.555, 3.676, 4.655, 0, 0},
... | user1066 | 106 | <pre><code>PadRight[#, Length@in[[1]], 0] & /@ Sort /@ DeleteCases[in, 0 | 0., 2]
</code></pre>
<p>=></p>
<blockquote>
<p>{{1.343, 3.411, 0, 0, 0}, {2.555, 3.676, 4.655, 0, 0}, {1.666, 3.867,
3.888, 0, 0}}</p>
</blockquote>
|
2,936,028 | <p>The question is:</p>
<p>Prove that If the sum of the elements of each row of a square matrix is k, then the sum of the elements in each row of the inverse matrix is 1/k ?</p>
<p>In the text book the answer is:</p>
<p>Let A be <span class="math-container">${m\times m}$</span>, non-singular, with the stated propert... | peterwhy | 89,922 | <p>Let <span class="math-container">$A$</span> be the invertible square matrix.</p>
<p>The product <span class="math-container">$A \pmatrix{1\\1\\\vdots\\1} $</span> gives a column matrix, with elements equal to sum of elements in a row of <span class="math-container">$A$</span>.</p>
<p><span class="math-container">$... |
2,905,022 | <p>I recently stumbled upon the problem $3\sqrt{x-1}+\sqrt{3x+1}=2$, where I am supposed to solve the equation for x. My problem with this equation though, is that I do not know where to start in order to be able to solve it. Could you please give me a hint (or two) on what I should try first in order to solve this equ... | Batominovski | 72,152 | <p>Let $A:=\sqrt{3x+1}$ and $B:=\sqrt{3x-3}$. Then, $A+\sqrt{3}B=2$ and $A^2-B^2=4$. That is,
$$(A+\sqrt{3}B)^2=A^2-B^2\,.$$</p>
<blockquote class="spoiler">
<p> Suppose that we are solving over the reals. Thus, $\left(2\sqrt{3}A+4B\right)\,B=0$. Since $A$ and $B$ are nonnegative and cannot simultaneously be $0... |
2,905,022 | <p>I recently stumbled upon the problem $3\sqrt{x-1}+\sqrt{3x+1}=2$, where I am supposed to solve the equation for x. My problem with this equation though, is that I do not know where to start in order to be able to solve it. Could you please give me a hint (or two) on what I should try first in order to solve this equ... | user247327 | 247,327 | <p>Similar to what others have said but I think a little simpler: write $3\sqrt{x- 1}+ \sqrt{3x+ 1}= 2$ as $3\sqrt{x- 1}= 2- \sqrt{3x+ 1}$ and square both sides: $9(x- 1)= 4- 4\sqrt{3x+ 1}+ 3x+ 1$. Now write that as $6x- 14= -4\sqrt{3x+ 1}$ and square again: $36x^2- 168x+ 196= 16(3x+ 1)= 48x+ 16$.</p>
<p>$36x^2- 120... |
206,305 | <p>Prove: $s_n \to s \implies \sqrt{s_n} \to \sqrt{s}$ by the definition of the limit. $s \geq 0$ and $s_n$ is a sequence of non-negative real numbers.</p>
<p>This is my preliminary computation:</p>
<p>$|\sqrt{s_n} - \sqrt{s}| < \epsilon$</p>
<p>multiply by the conjugate:</p>
<p>$|\dfrac{s_n - s}{\sqrt{s_n}+\sqr... | Mathematics | 22,687 | <p>well, you can actually take $\epsilon={\sqrt{s}}\epsilon'$ for any $\epsilon'>0$ given that for all $n>N, \epsilon'>|s_n-s|$</p>
|
206,305 | <p>Prove: $s_n \to s \implies \sqrt{s_n} \to \sqrt{s}$ by the definition of the limit. $s \geq 0$ and $s_n$ is a sequence of non-negative real numbers.</p>
<p>This is my preliminary computation:</p>
<p>$|\sqrt{s_n} - \sqrt{s}| < \epsilon$</p>
<p>multiply by the conjugate:</p>
<p>$|\dfrac{s_n - s}{\sqrt{s_n}+\sqr... | Pedro | 23,350 | <p><strong>ADD</strong> You got to</p>
<p>$$\left| {\frac{{{s_n} - s}}{{\sqrt {{s_n}} + \sqrt s }}} \right| < \frac{{\left| {{s_n} - s} \right|}}{{\sqrt s }}$$</p>
<p>Since $s_n\to s$, for every $\epsilon >0$ there is an $n_0$ for wich $$\left| {{s_n} - s} \right| < \varepsilon \sqrt s $$
whenever $n\geq n_... |
99,506 | <p>I am trying to show that how the binary expansion of a given positive integer is unique.</p>
<p>According to this link, <a href="http://www.math.fsu.edu/~pkirby/mad2104/SlideShow/s5_3.pdf" rel="nofollow">http://www.math.fsu.edu/~pkirby/mad2104/SlideShow/s5_3.pdf</a>, All I see is that I can recopy theorem 3-1's pro... | Amit Kumar Gupta | 8,953 | <p>Assume for contradiction that $n$ is the smallest positive integer with two different binary expansions.</p>
<p>Then $n=a_m\dots a_0=b_m\dots b_0$, allowing leading zeroes in at most one of the expansions.</p>
<p>Let $l$ be the smallest index so that $a_l\neq b_l$. It follows that $a_m\dots a_l = b_m\dots b_l$ are... |
1,852,889 | <blockquote>
<p>A fair coin is tossed independently four times . The probability of event "the no. Of times head show up is more than the no. Of times tails shows up" is</p>
</blockquote>
<p>The answer is $5/16$.</p>
<p>I did</p>
<p>$${_4\mathsf C}_4 (1/2)^4 (1/2)^0 + {_4\mathsf C}_3(1/2)^3 (1/2)$$</p>
<p>Is it ... | samerivertwice | 334,732 | <p>There are two combinations of tails only coming up once or not at all:</p>
<p>All heads: This can only happen 1 successful way.</p>
<p>Or once tails and three heads: The tail can come 1st, 2nd, 3rd, or 4th: So there are 4 succesful ways.</p>
<p>There are $2^4=16$ possible outcomes in total as 4 tosses each have... |
1,852,889 | <blockquote>
<p>A fair coin is tossed independently four times . The probability of event "the no. Of times head show up is more than the no. Of times tails shows up" is</p>
</blockquote>
<p>The answer is $5/16$.</p>
<p>I did</p>
<p>$${_4\mathsf C}_4 (1/2)^4 (1/2)^0 + {_4\mathsf C}_3(1/2)^3 (1/2)$$</p>
<p>Is it ... | Em. | 290,196 | <p>You're correct. You claimed that you calculated it, but it simply looks like you meant ${_4 \mathsf C}_4$ <strong>not</strong> ${_5\mathsf C }_4$. If you fix that, then it is correct. In other words, the number of heads is 3 or 4 are the only two cases where the number of heads is greater than the number of tails, t... |
633,858 | <p>If G is cyclic group of 24 order, then how many element of 4 order in G?
I can't understand how to find it, step by step. </p>
| user44441 | 44,441 | <p>Any element of order 4 will generate a cyclic group of order 4. Inside any finite cyclic group, there is a unique cyclic subgroup of any order dividing the order of the group. Therefore, there is a unique group of order 4 inside this group of order 24 and every element of order 4 is inside this subgroup. In a group ... |
3,298,445 | <p>A random variable is defined by it distribution function. The density function is the derivative of the distribution function. Thus the density function exisst iff the distribution function is absolutely continuous. However, can we construct a distribution function without a density function, except for the finite d... | forgottenarrow | 531,585 | <p>Adding to Masacroso's answer, there are examples of continuous random variables that do not have densities. The standard construction involves the <em>Cantor function</em>.</p>
<p>If you're not familiar with the Cantor set, this set is constructed via the following iteration:</p>
<ul>
<li>Start with <span class="m... |
2,706,141 | <p>I've been working on a math problem recently whose small subpart part is this. I don't want to post the whole problem and be spoon fed it, but I've been struggling with this sub part of it and since my math skills are still trivial the solution may require maths which I have to learn so,</p>
<p>Can the product $\ma... | José Carlos Santos | 446,262 | <p><strong>Hint:</strong> Just express $\dfrac1{3x^2-7}$ as$$\frac13\left(\frac a{x-\sqrt{\frac73}}+\frac b{x+\sqrt{\frac73}}\right).$$</p>
|
2,706,141 | <p>I've been working on a math problem recently whose small subpart part is this. I don't want to post the whole problem and be spoon fed it, but I've been struggling with this sub part of it and since my math skills are still trivial the solution may require maths which I have to learn so,</p>
<p>Can the product $\ma... | shere | 524,467 | <p>you can use this substitution: $x=\dfrac{\sqrt{7}}{\sqrt{3}}\sin u$ so $\mathrm dx= \dfrac{\sqrt{7}}{\sqrt{3}}\cos u$ and we have:</p>
<p>$$\int \dfrac{\frac{\sqrt{7}}{\sqrt{3}}\cos u}{7\cos^{2}u}\,\mathrm du=\frac{\sqrt{7}}{7\sqrt{3}}\int \sec u\,\mathrm du$$</p>
<p>which is easy to find.</p>
|
380,177 | <p>In mathematics, I want to know what is indeed the difference between a <strong>ring</strong> and an <strong>algebra</strong>?</p>
| Stephen | 146,439 | <p>For a commutative ring $k$, a $k$-<em>algebra</em> is a ring $A$ together with an extra datum: a homomorphism from $k$ into the center of $A$.</p>
<p>The definition allows non-associative algebras if you allow non-associative rings. The most well-understood case occurs when $k$ is a field, and the right way to thi... |
1,212,425 | <p>This is a homework problem that I cannot figure out. I have figured out that if $n^2 + 1$ is a perfect square it can be written as such:</p>
<p>$n^2 + 1 = k^2$.</p>
<p>and if $n$ is even it can be written as such:</p>
<p>$n = 2m$</p>
<p>I believe I'm supposed to use the fact that if $n \pmod{4} \equiv 0$ or $1$ ... | Dan Brumleve | 1,284 | <p>If $n^2+1$ is a perfect square then $n=0$, and $0$ is even. You can prove this by factoring the equation as $(k-n) \cdot (k+n) = 1$.</p>
|
1,212,425 | <p>This is a homework problem that I cannot figure out. I have figured out that if $n^2 + 1$ is a perfect square it can be written as such:</p>
<p>$n^2 + 1 = k^2$.</p>
<p>and if $n$ is even it can be written as such:</p>
<p>$n = 2m$</p>
<p>I believe I'm supposed to use the fact that if $n \pmod{4} \equiv 0$ or $1$ ... | Divide1918 | 706,588 | <p>Suppose n is odd, write <span class="math-container">$n=2m+1$</span>. Then <span class="math-container">$n^2+1=(2m+1)^2+1=4m^2+4m+2$</span>, which is even. Therefore if this is a perfect square, it must be divisible by <span class="math-container">$4$</span>. But clearly <span class="math-container">$n^2+1\equiv 2\... |
576,519 | <p>Assume that $x+\frac{1}{x} \in \mathbb{N}$. Prove by induction that $$x^2+\frac1{x^2}, x^3+\frac1{x^3}, \dots , x^n+\frac1{x^n}$$ is also a member of $\mathbb{N}$.</p>
<p>I have my <em>base</em>, it is indeed true for $n=1$..</p>
<p>I can assume it is true for $x^k+x^{-k}$ and then proove it is true for $x^{k+1}+x... | Community | -1 | <p><strong>Hint</strong></p>
<p>By induction using</p>
<p>$$(x^n+x^{-n})(x+x^{-1})=x^{n+1}+x^{-(n+1)}+x^{n-1}+x^{1-n}\in\mathbb N$$</p>
|
884,362 | <blockquote>
<p>Compute the integral
$$\int_{0}^{2\pi}\frac{x\cos(x)}{5+2\cos^2(x)}dx$$</p>
</blockquote>
<p>My Try: I substitute $$\cos(x)=u$$</p>
<p>but it did not help. Please help me to solve this.Thanks </p>
| Claude Leibovici | 82,404 | <p><strong>This is not an answer to the post but a reply to David's comment</strong></p>
<p>The antiderivative does not express in terms of elementary functions. For your curiosity, I write it down, but, as said, it looks like a nightmare.</p>
<p>$$4 \sqrt{14}\int\frac{x\cos(x)}{5+2\cos^2(x)}dx=-2 i \text{Li}_2\left(... |
2,136,937 | <p>Find $$\lim_{z \to \exp(i \pi/3)} \dfrac{z^3+8}{z^4+4z+16}$$</p>
<p>Note that
$$z=\exp(\pi i/3)=\cos(\pi/3)+i\sin(\pi/3)=\dfrac{1}{2}+i\dfrac{\sqrt{3}}{2}$$
$$z^2=\exp(2\pi i/3)=\cos(2\pi/3)+i\sin(2\pi/3)=-\dfrac{1}{2}+i\dfrac{\sqrt{3}}{2}$$
$$z^3=\exp(3\pi i/3)=\cos(\pi)+i\sin(\pi)=1$$
$$z^4=\exp(4\pi i/3)=\cos(4\... | Dan Velleman | 414,884 | <p>If you like "How To Prove It," you could try:</p>
<p>Velleman, Calculus: A Rigorous First Course, Dover Publications, 2016.</p>
|
4,264,558 | <p>I calculated homogenous already, I'm just struggling a bit with the right side. Would <span class="math-container">$y_p$</span> be <span class="math-container">$= ++e^x$</span> or <span class="math-container">$= ++e^{2x}$</span>?</p>
<p>Would the power in front of the root be the roots found from the homogenous part... | Paradox | 969,042 | <p>In general, if you are to calculate
<span class="math-container">$$
\lim_{x \to a} [f(x) + g(x)]
$$</span>
where <span class="math-container">$a$</span> can be a real number or <span class="math-container">$\pm \infty$</span>, you cannot blindly move the limit inside the parantheses like this:
<span class="math-cont... |
1,740,151 | <p>$$\lim_\limits {x \to \pi} \frac{(e^{\sin x} -1)}{(x-\pi)}$$</p>
<p>I found $-1$ as the answer and what I did was: </p>
<p>$\lim_\limits {x \to \pi} \frac{(e^{\sin x} -1)}{(x-\pi)}$ $\Rightarrow$ $\lim \frac{(f(x) - f(a))}{(x-a)}$ $\Rightarrow$ $f(x)=(e^{\sin x})$ </p>
<p>$f(a)=1$ </p>
<p>$x=x$ </p>
<p>and $a=... | Asaf Karagila | 622 | <p>This is false without the axiom of choice.</p>
<p>Mostowski constructed a model of $\sf ZFA$ (set theory with atoms), and in that model for every $n\in\Bbb N$ there is some $A$ such that: $$|A|<|A|^2<\ldots<|A|^n=|A|^{n+1}$$
So taking a large enough $n$ (e.g. $n=2$) we can take $X=A^{n-1}$ and $Y=A^n$. The... |
2,208,943 | <p>I am about to finish my first year of studying mathematics at university and have completed the basic linear algebra/calculus sequence. I have started to look at some real analysis and have really enjoyed it so far.</p>
<p>One thing I feel I am lacking in is motivation. That is, the difference in rigour between the... | lhf | 589 | <p>You may enjoy these books. The first one is a classic.</p>
<ul>
<li><p><a href="http://store.doverpublications.com/0486605094.html" rel="noreferrer">The History of the Calculus and Its Conceptual Development</a>,
by Carl B. Boyer</p></li>
<li><p><a href="http://bookstore.ams.org/hmath-24" rel="noreferrer">A History... |
2,208,943 | <p>I am about to finish my first year of studying mathematics at university and have completed the basic linear algebra/calculus sequence. I have started to look at some real analysis and have really enjoyed it so far.</p>
<p>One thing I feel I am lacking in is motivation. That is, the difference in rigour between the... | Count Iblis | 155,436 | <p>Rigor is essential in mathematics, there is just no other way to do math than to proceed on the basis of rigorously proven theorems. This does not mean that calculus necessarily needs to be set up in the same way as it currently is. You may rail against the rigorous definition of limits, but you need to come up with... |
275,371 | <p>I was wondering if it is possible to decompose any symmetric matrix into a positive definite and a negative definite component. I can't seem to think of a counterexample if the statement is false.</p>
| adam W | 43,193 | <p>Yes, see one of my <a href="https://math.stackexchange.com/q/209834/43193">question</a>s with the details. I will type up some more:</p>
<p>Given $A$ such that $A = A^\top$, $A$ with both positive and negative eigenvalues, the LDU factorization will have $U=L^\top$ (follows directly from symmetry) and $D$ diagonal ... |
1,407,797 | <p>P is the middle of the median line from vertex A, of ABC triangle. Q is the point of intersection between lines AC and BP.</p>
<p><a href="https://i.stack.imgur.com/ka8E8.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ka8E8.png" alt="enter image description here"></a></p>
| Macavity | 58,320 | <p>Starting you off in more detail....</p>
<p>With $\mathbb{a, b}$ denoting the vertices $A, B$ and $C$ being the Origin, one gets $$\mathbb m = \tfrac12\mathbb b, \quad \mathbb p = \tfrac12(\mathbb{m+a})=\tfrac12\mathbb a+\tfrac14 \mathbb b$$</p>
<p>Now $Q$ is located on the intersection of $\vec{BP} = t\mathbb b+(1... |
1,407,797 | <p>P is the middle of the median line from vertex A, of ABC triangle. Q is the point of intersection between lines AC and BP.</p>
<p><a href="https://i.stack.imgur.com/ka8E8.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ka8E8.png" alt="enter image description here"></a></p>
| mathlove | 78,967 | <p>Let $\vec a=\vec{CA},\vec b=\vec{CB}$. Then, we have
$$\vec{CM}=\frac 12\vec b$$$$\vec{CP}=\frac 12\vec{CA}+\frac 12\vec{CM}=\frac 12\vec a+\frac 14\vec b\tag 1$$</p>
<p>Also, setting $QC:AQ=s:1-s,BP:PQ=t:1-t$ gives</p>
<p>$$\vec{CP}=t\vec{CQ}+(1-t)\vec{CB}=ts\vec a+(1-t)\vec b\tag2$$</p>
<p>Now comparing $(2)$ w... |
329,513 | <p>$$
\int \frac{\sqrt{\frac{x+1}{x-2}}}{x-2}dx
$$</p>
<p>I tried:
$$
t =x-2
$$
$$
dt = dx
$$
but it didn't work.
Do you have any other ideas?</p>
| Pantelis Sopasakis | 8,357 | <p>$\renewcommand{\Re}{\mathbb{R}}\newcommand{\<}{\langle}\newcommand{\>}{\rangle}\newcommand{\barre}{\bar{\Re}}$Let us first introduce the <em>convex conjugate</em> of an extended-real-valued convex proper function $f:\Re^n\to\barre$ which is a function $f^*:\Re^n\to\barre$ defined as</p>
<p>$$
f^*(y) = \sup_x ... |
4,021,994 | <p>I was taught in high school algebra to translate word problems into algebraic expressions. So when I encountered <a href="https://artofproblemsolving.com/wiki/index.php/2016_AMC_10A_Problems/Problem_3" rel="nofollow noreferrer">this</a> problem I tried to reason out an algebra formula for it</p>
<blockquote>
<p>For ... | Robert Israel | 8,508 | <p>Another way, with no <span class="math-container">$x$</span>'s needed:</p>
<p>The first condition says the difference between their spending is <span class="math-container">$1/4$</span> of what Ben spends. That difference is <span class="math-container">$\$12.50$</span>, so Ben's amount is <span class="math-contain... |
319,262 | <p>If the first 10 positive integer is placed around a circle, in any order, there exists 3 integer in consecutive locations around the circle that have a sum greater than or equal to 17? </p>
<p>This was from a textbook called "Discrete math and its application", however it does not provide solution for this question... | Brandon | 212,011 | <p>Solution: For any given the first 10 positive integers placed around a circle, in any order, there are exactly
10 choices of 3 consecutive numbers around the circle. And each number appears exactly 3 times among the
10 choices. Hence, the sum of all numbers in 10 choices of 3 adjacent number is
(1 + 2 + · · · + 10) ... |
1,092,665 | <p>My question is really simple, how can I write symbolically this phrase: </p>
<blockquote>
<p>$x=\sum a_mx^m$ where $m$ range over
$\{1,\ldots,g\}\setminus\{t_1,\ldots,t_u\}$</p>
</blockquote>
<p>Being more specific, I would like to know how to write with mathematical symbols this part: "range over $\{1,\ldots,... | Ross Millikan | 1,827 | <p>Often we see something like$$x=\sum_{m\in\{1,\ldots,g\}\setminus\{t_1,\ldots,t_u\}} a_mx^m$$</p>
|
177,519 | <p>Let $\mathfrak{g}$ be a simple lie algebra over $\mathbb{C}$ and let $\hat{\mathfrak{g}}$ be the Kac-Moody algebra obtained as the canonical central extension of the algebraic loop algebra $\mathfrak{g} \otimes \mathbb{C}[t,t^{-1}]$. In a sequence of papers, Kazhdan and Lusztig constructed a braided monoidal structu... | dhy | 51,424 | <p>(Written on my phone - apologies for any typos.) </p>
<p>A few comments:</p>
<p>a) First, as to the source of the braided monoidal structure on the Kazhdan-Lusztig category. The category of integrable affine Lie algebra reps is naturally a factorization category, which is close morally to an E2/braided monoidal ca... |
3,386,530 | <p>Let <span class="math-container">$(\Omega,\mathcal{F},\mathbb{P})$</span> be a probability space and <span class="math-container">$(\mathcal{X},d)$</span> be a complete, separable, locally compact metric space. Suppose that <span class="math-container">$X,X_1,X_2,X_3,... : \Omega\to\mathcal{X}$</span> are <span clas... | egreg | 62,967 | <p>The proof is correct. Here's a different way to present the same idea.</p>
<p>First a useful lemma: <em>if <span class="math-container">$A\subseteq B$</span> and <span class="math-container">$x$</span> is a limit point of <span class="math-container">$A$</span>, then <span class="math-container">$x$</span> is a lim... |
395,791 | <p>I am searching for examples of manifolds which are not symmetric spaces but where Jacobi fields can be computed in closed form. For now, I am aware of</p>
<ul>
<li>Gaussian distribution with the Wasserstein metric: <a href="https://arxiv.org/pdf/2012.07106.pdf" rel="noreferrer">https://arxiv.org/pdf/2012.07106.pdf</... | R W | 8,588 | <p><strong>Damek-Ricci spaces</strong> are obtained by equipping certain solvable Lie groups with appropriate invariant Riemannian metrics. This class is larger than the class of symmetric spaces (this was precisely the point of their construction as counterexamples to the <a href="https://en.wikipedia.org/wiki/Lichner... |
370,058 | <p>How can I take this integral?</p>
<p>$$\int_{0}^{x} (z- u)_+^2 du $$</p>
<p>which <code>+</code> means If $z$ is bigger than u its equal $z - u$ and else it's equal zero.</p>
| Community | -1 | <p>If $t\leq\tau$, we have
$$\int_0^t (\tau-u)_+^2 du = \int_0^t (\tau-u)^2 du = \dfrac{(t-\tau)^3+\tau^3}3$$
If $t\geq\tau$, we have
$$\int_0^t (\tau-u)_+^2 du = \int_0^{\tau} (\tau-u)^2 du = \dfrac{\tau^3}3$$
Hence, we get that
$$\int_0^t (\tau-u)_+^2 du = \begin{cases} \dfrac{(t-\tau)^3+\tau^3}3 & t \leq \tau\\ ... |
4,447,522 | <p>Show that <span class="math-container">$$\cot\left(\dfrac{\pi}{4}+\beta\right)+\dfrac{1+\cot\beta}{1-\cot\beta}=-2\tan2\beta$$</span>
I'm supposed to solve this problem only with sum and difference formulas (identities).</p>
<p>So the LHS is <span class="math-container">$$\dfrac{\cot\dfrac{\pi}{4}\cot\beta-1}{\cot\d... | user2661923 | 464,411 | <p>The alternative approach is to stay with your original approach and complete it.</p>
<p><span class="math-container">$\displaystyle \frac{4\cot\beta}{1-\cot^2\beta}
= \frac{\frac{4}{\tan(\beta)}}{1 - \frac{1}{\tan^2(\beta)}}
= \frac{\frac{4}{\tan(\beta)}}{\frac{\tan^2(\beta) - 1}{\tan^2(\beta)}}
$</span></p>
<p><spa... |
3,999,488 | <p><strong>Question:</strong> How is the differentiation of <span class="math-container">$xy=constant$</span> equal to <span class="math-container">$x\text{d}y+y\text{d}x$</span>?</p>
<p><strong>My Approach:</strong> I first tried using partial differentiation, which I know very little of. Basically, it's the different... | Community | -1 | <p>In general, the <em>differential</em> of the function <span class="math-container">$f(x,y)$</span> is given by
<span class="math-container">$$
df=f_x(x,y)dx+f_y(x,y)dy
$$</span>
where <span class="math-container">$f_x$</span> and <span class="math-container">$f_y$</span> denote the partial derivatives.</p>
<p>In you... |
3,476,022 | <p>I was watching this Mathologer video (<a href="https://youtu.be/YuIIjLr6vUA?t=1652" rel="noreferrer">https://youtu.be/YuIIjLr6vUA?t=1652</a>) and he says at 27:32</p>
<blockquote>
<p>First, suppose that our initial <em>chunk</em> is part of a parabola, or if you like a cubic, or any polynomial. If I then tell you... | Alberto Saracco | 715,058 | <p>“There is one and only one polynomial” means two things:</p>
<p>1) There is at most one polynomial.</p>
<p>2) There is at least one polynomial.</p>
<p>Only the first affirmation is true.</p>
<hr>
<p>1) There is at most one polynomial:</p>
<p>Proof by contradiction.</p>
<p>Assume <span class="math-container">$... |
1,175,993 | <p>I want to show $T=d/dx$ is unbounded on $C^1[a,b]$ with $b>1$. Take a sequence $f(x)=x^n$, and $\|T\|=\sup_{x\in[a,b]}\frac{\|Tx\|}{\|x\|}=\frac{\|n\cdot b^{n-1}\|}{\|b\|}$. I want to claim as $n$ goes to infinity, the operator norm goes to infinity, and hence it's unbounded. But the definition of operator norm o... | Pedro M. | 21,628 | <p>Your definition of operator norm does not compile: $Tx$ makes no sense for $x \in [a,b]$, as $T$ operates on functions. It should have been $$\sup_{f \in C^1([a,b])} \frac{\|Tf\|}{\|f\|} \geq \frac{\|Tf_n\|}{\|f_n\|} = nb^{n-2},$$
and thus $T$ is unbounded ($f_n(x) = x^n$).</p>
|
1,787,985 | <p>I have the following differential equation: $$\Big(\frac{dy}{dx}\Big)^2=\frac{y^2-A^2}{A^2}.$$ I am looking to obtain a solution $$y(x)=A\cosh\Big({\frac{x+B}{A}}\Big),$$ where B and A are constants. </p>
<p>I have tried square-rooting and Taylor expanding, substitution of an integral but am getting nowhere. </p>
... | Omar Antolín-Camarena | 1,070 | <p>Let $V$ have basis $e_1, \ldots, e_n$. There is a basis $\delta_, \ldots, \delta_n$ of $V^\vee$ called the dual basis characterized by the property $\delta_i(e_j) = \begin{cases}1 & \text{if }i=j \\ 0 & \text{otherwise}\end{cases}$.</p>
<p>The element $"\mathrm{id}" \in T \otimes T^\vee$ corresponding to th... |
1,787,985 | <p>I have the following differential equation: $$\Big(\frac{dy}{dx}\Big)^2=\frac{y^2-A^2}{A^2}.$$ I am looking to obtain a solution $$y(x)=A\cosh\Big({\frac{x+B}{A}}\Big),$$ where B and A are constants. </p>
<p>I have tried square-rooting and Taylor expanding, substitution of an integral but am getting nowhere. </p>
... | Alex Saad | 184,547 | <p>Not really an answer, but I wanted to post this here in case anyone else ends up thinking about this thing. It might help set you on the right track.</p>
<p>Anyway, after browsing through <a href="http://www.math.washington.edu/~julia/AMS_SFSU_2014/Serganova.pdf" rel="nofollow noreferrer">this</a> collection of sli... |
168,053 | <p>If g is a positive, twice differentiable function that is decreasing and has limit zero at infinity, does g have to be convex? I am sure, from drawing a graph of a function which starts off as being concave and then becomes convex from a point on, that g does not have to be convex, but can someone show me an example... | copper.hat | 27,978 | <p>Let $f(x) = \begin{cases} e^{1\over x}, & x < 0 \\
0, & \text{otherwise} \end{cases}$.</p>
|
139,021 | <p>Can you, please, recommend a good text about algebraic operads?</p>
<p>I know the main one, namely, <a href="http://www-irma.u-strasbg.fr/~loday/PAPERS/LodayVallette.pdf" rel="nofollow noreferrer">Loday, Vallette "Algebraic operads"</a>. But it is very big and there is no way you can read it fast. Also there are no... | Al-Amrani | 34,304 | <p>A recent good book is <a href="http://www-irma.u-strasbg.fr/~loday/PAPERS/LodayVallette.pdf" rel="nofollow noreferrer">Algebraic Operads by Jean-Louis Loday and Bruno Vallette</a>.</p>
|
139,021 | <p>Can you, please, recommend a good text about algebraic operads?</p>
<p>I know the main one, namely, <a href="http://www-irma.u-strasbg.fr/~loday/PAPERS/LodayVallette.pdf" rel="nofollow noreferrer">Loday, Vallette "Algebraic operads"</a>. But it is very big and there is no way you can read it fast. Also there are no... | Pedro | 21,326 | <p>The book by M. Bremner and V. Dotsenko titled <em>Algebraic Operads: an algorithmic companion</em> (published in 2016) is (in my perhaps biased opinion) a must-have for those wishing to complement their reading of Loday--Vallette. As the authors explain :</p>
<blockquote>
<p>It is fairly accurate to say that the aim... |
2,473,220 | <p>From how I understood the question and judging from solutions I've been provided with (see graph below),</p>
<p><a href="https://i.stack.imgur.com/73RU3.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/73RU3.png" alt="enter image description here"></a></p>
<p>$f(x)$ starts from an x-position, whi... | Rory Daulton | 161,807 | <p>Your first graph is the correct one for $f(x)$, and $f$ is well-defined by the definition in your title--although I might rather say "...to a nearest integer" since more than one could be "nearest."</p>
<p>You are correct that all integers are critical points of $f$. Those are the bottom corners in your graph. Howe... |
2,780,403 | <p>I've tried to solve this but I don't seem to get anywhere.</p>
<p>The question states:</p>
<blockquote>
<p>Tom's home is $1800$ m from his school. One morning he walked part of the way to school and then ran the rest. If it took him $20$ mins or less to get to school, and he walks at $70$ m/min and runs at $210$... | trancelocation | 467,003 | <p>The gradient at $P(1,1,1)$ is $\nabla T_P= (2,4,4)$.</p>
<p>So, the <strong>magnitude</strong> of the rate of change is
$$|\nabla T_P| = 2\cdot |(1,2,2)| = 6$$</p>
<p>Nevertheless, the direction of maximum decrease is in the direction of $-(2,4,4)$, which has the same direction as $-(1,2,2)$ which give the same u... |
2,496,817 | <p>My task is to prove the above, with $m,n \in \mathbb{N}$</p>
<p>Here is what I have:</p>
<p>$7 | (100m + n) \iff (100m +n) \mod 7 = 0$</p>
<p>$\iff (100m \mod 7 + n \mod 7) \mod 7 = 0 $</p>
<p>$\iff (2m +n) \mod 7 = 0$ </p>
<p>That is where I am stuck.</p>
| lhf | 589 | <p>More generally,</p>
<blockquote>
<p>$7 \mid (100m + n) \iff 7 \mid (m + 4n)$</p>
</blockquote>
<p>Indeed, let $a=100m + n$ and $b=m + 4n$. Then
$$
a-2b = 98m -7n \equiv 0 \bmod 7
$$
Therefore, $a \equiv 2b \bmod 7$. The result follows because $b \equiv 0 \bmod 7$ iff $2b \equiv 0 \bmod 7$.</p>
|
942,470 | <p>I am trying to count how many functions there are from a set $A$ to a set $B$. The answer to this (and many textbook explanations) are readily available and accessible; I am <strong>not</strong> looking for the answer to that question and <strong>please do not post it</strong>. Instead I want to know what fundamen... | Daniel McLaury | 3,296 | <p>A function $f : A \to B$ sends each element of $A$ to exactly one element of $B$.</p>
|
3,096,572 | <p>I am trying to find whether the following is stable absolutely using the improved Euler and the Adams-Bashforth 2 scheme,
<span class="math-container">$u'=\begin{bmatrix} -20&0&0\\ 20&-1&0\\0&1&0\end{bmatrix}u=Au$</span>, where the timestep is <span class="math-container">$\frac{1}{2}$</span... | Lutz Lehmann | 115,115 | <p>In the improved Euler method, your ODE gives the step
<span class="math-container">\begin{align}
k_1&=Au_n\\
k_2&=A(u_n+hk_1)=(A+hA^2)u_n\\
u_{n+1}&=u_n+\frac h2 (k_1+k_2)=(I+hA+\frac{h^2}2A^2)u_n
\end{align}</span>
What you want for stability is that the matrix factor is contracting if all eigenvalues o... |
4,507,155 | <p>In <a href="https://math.stackexchange.com/questions/4454551/are-fracp212-and-fracp5np5-12-are-coprime-to-each-other">previous post</a>, I got the answer that <span class="math-container">$\gcd \left(\frac{p^2+1}{2}, \frac{p^5-1}{2} \right)=1$</span>, where <span class="math-container">$p$</span> is prime number.</p... | Joseph Camacho | 731,433 | <p>Suppose that
<span class="math-container">$$\gcd\left(\frac{p^m + 1}{2}, \frac{p^n - 1}{2}\right) \neq 1.$$</span>
There are two cases:</p>
<ol>
<li>Both <span class="math-container">$p^m + 1$</span> and <span class="math-container">$p^n - 1$</span> are multiples of <span class="math-container">$4$</span>.</li>
<li>... |
3,112,043 | <p>The first part of the problem is:</p>
<p>Prove that for all integers <span class="math-container">$n \ge 1$</span> and real numbers <span class="math-container">$t>1$</span>,
<span class="math-container">$$ (n+1)t^n(t-1)>t^{n+1}-1>(n+1)(t-1)$$</span></p>
<p>I have done the first part by induction on <span... | didgogns | 392,996 | <p>We only need to prove the second part of question.</p>
<p>For the left inequality, let's apply induction on <span class="math-container">$m$</span>.</p>
<p>Base case: <span class="math-container">$\frac{1}{n+1}<1$</span>, trivial.</p>
<p>Induction case: Note that <span class="math-container">$1^n+\cdots+m^n>... |
637,199 | <p>If $K^T=K$, $K^3=K$, $K1=0$ and $K\left[\begin{matrix}1\\2 \\-3\end{matrix}\right]=\left[\begin{matrix}1\\2 \\-3\end{matrix}\right]$,</p>
<p>how can I find the trace of $K$ and the determinant of $K$?</p>
<p>I think for determinant of $K$, since $K^3-K=(K^2-I)K=0$, then $K^2=I$ since $K$ is nonzero. Then this impl... | copper.hat | 27,978 | <p>There are three matrices that satisfy the given conditions.</p>
<p>$K^3=K$ shows that all eigenvalues are 0 or satisfy $\lambda^2 = 1$. Since $K$ is symmetric (and presumably real?) we have $\lambda \in \{-1,0,1\}$.</p>
<p>Since $K e = 0$, we see that $K$ is singular, hence $\det K = 0$. We are given that one eige... |
431,236 | <p>I have a cylinder of radius 4 and height 10 that is at a 30 degree angle. I need to find the volume.</p>
<p>I have no clue how to do this, I have spent quite a while on it and went through many ideas but I think my best idea was this.</p>
<p>I know that the radius is 4 so if I cut the cylinder in half from corner ... | Lucozade | 83,956 | <p>You should use the standard formula for volume of a cylinder, except that the height is now the projected height of the axis length onto the perpendicular to the base plane. Thus, if the length of the axis of your cylinder is 10, then the projected height is 10*sin(30deg) = 5 (to see this, consider the projection of... |
431,236 | <p>I have a cylinder of radius 4 and height 10 that is at a 30 degree angle. I need to find the volume.</p>
<p>I have no clue how to do this, I have spent quite a while on it and went through many ideas but I think my best idea was this.</p>
<p>I know that the radius is 4 so if I cut the cylinder in half from corner ... | DJohnM | 58,220 | <p>To expand on the answer above, consider an ordinary cylinder as a stack of thin discs:coins, poker chips, whatever. The stack has a height, given by the total thickness of all the coins, and a certain volume, given by the total volume of all the coins.</p>
<p>Now, sort of srounch the stack over at an angle, so tha... |
4,019,956 | <p><strong>Preface</strong></p>
<p>We will use the following facts</p>
<p>i) The sequence <span class="math-container">$ \left\lbrace a_n \right\rbrace $</span> is convergent to <span class="math-container">$a$</span> if for each <span class="math-container">$ \varepsilon >0$</span> there exists <span class="math-co... | trancelocation | 467,003 | <p>Just do some square completion:</p>
<p><span class="math-container">\begin{eqnarray*}a^2+ab+b^2-a-2b
& = & \left(a+\frac b2\right)^2 + 3\left(\frac b2\right)^2 - a - 2b \\
& = & \left(a+\frac b2\right)^2 + 3\left(\frac b2\right)^2 - \left(a+\frac b2\right) - 3\frac b2 \\
& \stackrel{u=a+\frac b2,... |
1,998,244 | <p>Given the equation of a damped pendulum:</p>
<p>$$\frac{d^2\theta}{dt^2}+\frac{1}{2}\left(\frac{d\theta}{dt}\right)^2+\sin\theta=0$$</p>
<p>with the pendulum starting with $0$ velocity, apparently we can derive:</p>
<p>$$\frac{dt}{d\theta}=\frac{1}{\sqrt{\sqrt2\left[\cos\left(\frac{\pi}{4}+\theta\right)-e^{-(\the... | Dion Leijnse | 385,638 | <p>It says that there are 18 with at least one sister, and 5 without brothers/sisters. So there have to be at least 23. Now we have to look if something implies that this has to be higher. Because these eighteen said that they had at least one sister, those 17 with at least one brother can all be combined with this, so... |
3,745,273 | <p>I am looking for a way to solve :</p>
<p><span class="math-container">$$\int_{-\infty}^{\infty} \frac{x\sin(3x)}{x^4+1}\,dx $$</span></p>
<p>without making use of complex integration.</p>
<p>What I tried was making use of integration by parts, but that didn't reach any conclusive result. (i.e. I integrated <span cla... | Nanayajitzuki | 611,558 | <p>I only write the key step for central issue, let for a>0 (this makes problem easy to deal without abs function)
<span class="math-container">$$
f(a) = \int_{0}^{\infty} \frac{\sin(ax)}{x(x^4+1)} \,\mathrm{d}x
$$</span>
then you have ODE
<span class="math-container">$$
f^{(4)}(a) + f(a) = \int_{0}^{\infty} \frac{\... |
2,421,145 | <p>I am practicing problems around NFA and DFA.</p>
<p>I have seen many questions on how to convert NFA to DFA and DFA to Regular expression etc.</p>
<p>But I have seen very different question and I am stuck on how to proceed with the following question? </p>
<p>Given DFA. Convert this DFA to NFA with 5 states.
<a h... | SSequence | 469,108 | <p>I believe at least the intention of question seems to understand the language of the DFA and then use that to build the NFA. </p>
<p>I think I understand the language of DFA. Hopefully you can use that description to find a corresponding NFA. It seems to me that it shouldn't be too difficult, but if you have troubl... |
339,142 | <p>I'm trying to understand the difference between the sense, orientation, and direction of a vector. According to
<a href="http://www.eng.auburn.edu/users/marghdb/MECH2110/c1_2110.pdf">this</a>,
sense is specified by two points on a line parallel to a vector. Orientation is specified by the relationship between the ve... | Mehdi | 67,913 | <p>I think(I'm not sure) that direction of a vector is an intrinsic property of that vector, so one can define direction of a vector without any reference to the outside world, but orientation is an extrinsic property, it depends on the relation between the vector and outside world(how it is placed w.r.t other vectors ... |
1,955,591 | <p>I have to prove that ' (p ⊃ q) ∨ ( q ⊃ p) ' is a tautology.I have to start by giving assumptions like a1 ⇒ p ⊃ q and then proceed by eliminating my assumptions and at the end i should have something like ⇒(p ⊃ q) ∨ ( q ⊃ p) but could not figure out how to start.</p>
| marty cohen | 13,079 | <p>(p ⊃ q) ∨ ( q ⊃ p) </p>
<p>I assume that
"⊃" means "implies".
Actually,
since your statement
is symmetric in p and q,
it doesn't matter if
"a ⊃ b" means
"a implies b"
or
"b implies a".</p>
<p>Since
"a ⊃ b"
is equivalent to
"~a ∨ b",
your statement is
equivalent to
"(~p ∨ q) ∨ ( ~q ∨ p)".</p>
<p>Since
"∨" is commu... |
4,137,362 | <p>If we refer to the <code>Minimum</code> of a set of numbers, we mean the lowest number. <code>Min(12, 7, 18) = 7</code> and <code>Min(5, -8) = -8</code>.</p>
<p>Is there a technical term for the 'number closest to zero'? e.g. where <code>fn(5, -8) = 5</code> and <code>fn(-5, 8) = -5</code></p>
<p>Similarly, what w... | SolubleFish | 918,393 | <p>If <span class="math-container">$c_2$</span> or <span class="math-container">$c_3$</span> is non zero, then the exponentials in <span class="math-container">$x(t)$</span> will mean that <span class="math-container">$\|x(t)\| \to \infty$</span>.</p>
<p>The solution with <span class="math-container">$c_1= 1$</span>, <... |
4,137,362 | <p>If we refer to the <code>Minimum</code> of a set of numbers, we mean the lowest number. <code>Min(12, 7, 18) = 7</code> and <code>Min(5, -8) = -8</code>.</p>
<p>Is there a technical term for the 'number closest to zero'? e.g. where <code>fn(5, -8) = 5</code> and <code>fn(-5, 8) = -5</code></p>
<p>Similarly, what w... | the_candyman | 51,370 | <p>The solutions with <span class="math-container">$|c1| \leq 1$</span> and <span class="math-container">$c_2 = c_3 = 0$</span> are constant and of norm less or equal to <span class="math-container">$1$</span>, i.e.:</p>
<p><span class="math-container">$$x(t) = \begin{bmatrix}c_1 \\ 0 \\ 0 \end{bmatrix} ~\forall t \geq... |
140,500 | <p>The diagonals of a rectangle are both 10 and intersect at (0,0). Calculate the area of this rectangle, knowing that all of its vertices belong to the curve $y=\frac{12}{x}$.</p>
<p>At first I thought it would be easy - a rectanlge with vertices of (-a, b), (a, b), (-a, -b) and (a, -b). However, as I spotted no ment... | bgins | 20,321 | <p>To follow J.M.'s hint, you need to solve the two equations
$$
\eqalign{
x^2+y^2&=25\\
xy&=12
}
$$
and a nice way to do this would be to notice that then
$$
\left(x+y\right)^2=x^2+2xy+y^2=25+2\cdot12=49
$$
so that $x+y=\pm7$. Next, if you note that
$$
\left(x-y\right)^2=x^2-2xy+y^2=25-2\cdot12=1
$$
you will s... |
816,088 | <blockquote>
<p>The sum of two variable positive numbers is $200$.
Let $x$ be one of the numbers and let the product of these two numbers be $y$. Find the maximum value of $y$.</p>
</blockquote>
<p><em>NB</em>: I'm currently on the stationary points of the calculus section of a text book. I can work this out in my... | Samrat Mukhopadhyay | 83,973 | <p>The problem is the following: $$\mbox{Find}\\ y=\max_{x\ge 0}\left(x(200-x)\right)$$ So differentiate the function $f(x)=x(200-x)$ to get $x=100$ as the one that maximizes it (since $f''(100)=-2$. So, $y=100^2$.</p>
|
17,713 | <p>I am a bit perplexed in trying to find values <span class="math-container">$a,b,c$</span> so that the approximation is as precise as possible:</p>
<p><span class="math-container">$$\sum_{k=n}^{\infty}\frac{(\ln(k))^{2}}{k^{3}} \approx \frac{1}{n^{2}}[a(\ln (n))^{2}+b \ln(n) + c]$$</span></p>
<p>I can see from Wolf... | Nabyl Bod | 5,838 | <p>Invoke an <a href="http://en.wikipedia.org/wiki/Integral_test_for_convergence" rel="nofollow">Integral test for convergence</a>
then two Integration by parts give you (up to a mistake in my calculus)</p>
<p>$a=1/2$, $b=1/2$, and $c=1/4$.</p>
|
3,075,979 | <p>Prove that <span class="math-container">$$\frac{k^7}{7}+\frac{k^5}{5}+\frac{2k^3}{3}-\frac{k}{105}$$</span> is an integer using mathematical induction.</p>
<p>I tried using mathematical induction but using binomial formula also it becomes little bit complicated.</p>
<p>Please show me your proof.</p>
<p>Sorry if t... | Klaas van Aarsen | 134,550 | <p>We have:
<span class="math-container">$$\frac{k^7}{7}+\frac{k^5}{5}+\frac{2k^3}{3}-\frac{k}{105}
=\frac{15k^7+21k^5+70k^3-k}{3\cdot 5\cdot 7}
$$</span>
To prove this is an integer we need that:
<span class="math-container">$$15k^7+21k^5+70k^3-k\equiv 0 \pmod{3\cdot 5\cdot 7}$$</span>
According to the <em>Chinese Rem... |
100,955 | <p>I'm trying to find the most general harmonic polynomial of form $ax^3+bx^2y+cxy^2+dy^3$. I write this polynomial as $u(x,y)$. </p>
<p>I calculate
$$
\frac{\partial^2 u}{\partial x^2}=6ax+2by,\qquad \frac{\partial^2 u}{\partial y^2}=2cx+6dy
$$
and conclude $3a+c=0=b+3d$. Does this just mean the most general harmon... | David E Speyer | 448 | <p>This is a community wiki answer to note that the question was answered in the comments, and thus remove this question from the unanswered list -- the answer is yes, the OP is correct.</p>
|
2,759,827 | <blockquote>
<p>Let $\{x_n\}$ be a bounded sequence and $s=\sup\{x_n|n\in\mathbb N\}.$
Show that if $s\notin \{x_n|n\in\mathbb N\}, $then there exists a
subsequence convereges to $s$.</p>
</blockquote>
<p>$s-1$ cannot be the upperbound and $s\notin \{x_n|n\in\mathbb N\}, \exists n_1\in \mathbb N:n_1\ge1:s-1<x... | Alex Ortiz | 305,215 | <p>Using the definition of supremum and the assumption that $s\notin \{x_n\}$, show that for any $\epsilon > 0$, there are infinitely many $n$ such that $s-\epsilon < x_n < s$. Having done so, find an increasing sequence $n_1 < n_2 < n_3 < \dots,$ such that $s-1/n_i < x_{n_i} <s$.</p>
|
7,110 | <p>In 1974, Aharoni proved that every separable metric space (X, d) is Lipschitz isomorphic to a subset of the Banach space c_0.
Thus, for some constant L, there is a map K: X --> c_0 that satisfies the inequality d(u,v) <= || Ku - Kv || <= Ld(u,v) for all u and v in X.
Now, suppose X = l_1 (in this case, L = 2 ... | fedja | 1,131 | <p>To answer Bill Johnson's question, a monotone linear bi-Lipschitz embedding (actually, an isometric one) $\ell^1\to\ell^\infty$ is very easy to construct. Just take any antisymmetric matrix $A$ of $\pm 1$s with the property that for each $n$ every combination of signs in the first $n$ positions appears in some row o... |
4,301,632 | <blockquote>
<p><span class="math-container">$$X^2 = \begin{bmatrix}1&a\\0&1\\\end{bmatrix}$$</span> where <span class="math-container">$a \in \Bbb R \setminus \{0\}$</span>. Solve for matrix <span class="math-container">$X$</span>.</p>
</blockquote>
<hr />
<p>I was practicing for matrix equations and this is t... | Theo Bendit | 248,286 | <p>Let
<span class="math-container">$$M(a) = \begin{bmatrix} 1 & a \\ 0 & 1 \end{bmatrix}.$$</span>
Note <span class="math-container">$M(a)$</span> has one eigenvalue: <span class="math-container">$1$</span>, and is not diagonalisable. This means that, if <span class="math-container">$X^2 = M(a)$</span>, then <... |
2,554,448 | <p>Beside using l'Hospital 10 times to get
$$\lim_{x\to 0} \frac{x(\cosh x - \cos x)}{\sinh x - \sin x} = 3$$ and lots of headaches, what are some elegant ways to calculate the limit?</p>
<p>I've tried to write the functions as powers of $e$ or as power series, but I don't see anything which could lead me to the righ... | egreg | 62,967 | <p>It's not difficult to show that
$$
\lim_{x\to0}\frac{\cosh x-1}{x^2}=
\lim_{x\to0}\frac{\cosh^2x-1}{x^2(\cosh x+1)}=
\lim_{x\to0}\frac{\sinh^2x}{x^2}\frac{1}{\cosh x+1}=\frac{1}{2}
$$
Similarly,
$$
\lim_{x\to0}\frac{1-\cos x}{x^2}=\frac{1}{2}
$$
hence
$$
\lim_{x\to0}\frac{\cosh x-\cos x}{x^2}=1
$$
Therefore your lim... |
1,425,935 | <p>How would I solve this trigonometric equation?</p>
<p>$$3\cos x \cot x + \sin 2x = 0$$</p>
<p>I got to this stage: $$3 \cos x = -2 \sin^2x$$</p>
<p>Is is a dead end or is there a easier way solve this equation? </p>
| Aaron Maroja | 143,413 | <p><strong>Hint:</strong> Use $\sin^2 x = 1 - \cos^2 x $ then $$-2 \cos^2 x + 3 \cos x + 2 = 0$$</p>
<p>take $y = \cos x$. </p>
|
1,955,212 | <p>I couldn't find this on the whole internet. My life depends on solving this. Please help.
I must write a formula for this sequence
<span class="math-container">$-8, -14, 8, 14, -8, -14, 8, 14$</span>.</p>
| coffeemath | 30,316 | <p>$f(x)=14 \cos (\pi x/2)-8 \sin(\pi x/2)$ does it, for $x=1,2,3,4,\cdots .$</p>
|
201,173 | <p>I have problem solving this equation:
$$
\left(\frac{1+iz}{1-iz}\right)^4 = \frac12 + i {\sqrt{3}\over 2}
$$
I know how to solve equations that are like:
$$
w^4 = \frac12 + i {\sqrt{3}\over 2}
$$
And I have solved it to:
$$
w = \cos(-\frac{\pi}{12} + \frac{\pi k}{2})) + i\sin(-\frac{\pi}{12} + \frac{\pi k}{2})... | Did | 6,179 | <p>$$w=\frac{1+\mathrm iz}{1-\mathrm iz}\iff z=\mathrm i\cdot\frac{1-w}{1+w}$$
<strong>Edit:</strong> On the road are the identities $(1-\mathrm iz)\cdot w=1+\mathrm iz$ and $1-w=-\mathrm i\cdot(1+w)\cdot z$.</p>
|
119,584 | <p>It is known that there are multiplicative version concentration inequalities for
sums of independent random variables. For example, the following
multiplicative version <strong>Chernoff</strong> bound.</p>
<hr>
<p><strong>Chernoff bound:</strong></p>
<p>Let $X_1,\ldots,X_n$ be independent random variables and $X_... | Neal | 27,773 | <p>Yes, such bounds are possible. You can adapt the proof of Azuma's inequality to the multiplicative-error case, if you set it up correctly.
For example:</p>
<p><strong>Lemma 10 [<a href="http://www.cs.ucr.edu/~neal/Koufogiannakis13Nearly.pdf" rel="nofollow noreferrer">this paper</a>]</strong>.
<em>Let $Y=\sum_{t=1}... |
18,048 | <p>When taking a MOOC in calculus the exercises contain 5 options to select from. I then solve the question and select the option that matches my answer. Obviously only one of the options is correct. But there are (quite a few) times where my solution is wrong even though it is one of the available options. </p>
<p>My... | Amy B | 5,321 | <p>I freelance as an item writer, someone who writes questions for standardized tests. When making up alternate choices, I always have to justify my reasons for the "wrong answers" or distractors.
Here are some strategies I use.</p>
<ol>
<li>I focus on common misconceptions for students at that grade level.This is ea... |
18,048 | <p>When taking a MOOC in calculus the exercises contain 5 options to select from. I then solve the question and select the option that matches my answer. Obviously only one of the options is correct. But there are (quite a few) times where my solution is wrong even though it is one of the available options. </p>
<p>My... | JRN | 77 | <p>If a teacher has taught the course before, and has asked questions that are free-response (not multiple-choice), then the teacher can look at the incorrect answers previously given by the students.</p>
<p>If not, then the teacher can ask other teachers who have had this experience.</p>
<p>Errors that "appear to st... |
18,048 | <p>When taking a MOOC in calculus the exercises contain 5 options to select from. I then solve the question and select the option that matches my answer. Obviously only one of the options is correct. But there are (quite a few) times where my solution is wrong even though it is one of the available options. </p>
<p>My... | niicole16 | 13,676 | <p>So. I started working as a teaching assistant for a course, and the professor showed me "test generating" software. It comes down to the fact that the editor of a textbook creates various open, multiple choice, and true/false questions and answers, and you - using the software - can just click "generate X number of ... |
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