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 |
|---|---|---|---|---|
1,267,395 | <p>Julie is required to pay a 2 percent tax on all income over 3,000. She also has to pay 2.5 percent on all income over 20,000. She earned more than 20,000 and paid 992.50 what was her total income</p>
| maths525 | 235,883 | <p>I think I follow what you're asking. Your question states that for income over \$3,000, the tax rate is 2%, and that for income over \$20,000, the tax rate is 2.5%. Is that correct?</p>
<p>(I doubt that the question is asking you to calculate taxes on her first \$17,000 and her remaining income minus the \$17,000 s... |
1,631,589 | <p>Consider the sequence $\{\frac{x^n}{n!}\}_n$ for any number $x$.</p>
<p>By choosing $m>x$ and letting $n>m$ , show that:</p>
<p>$\frac{x^n}{n!} < \frac{x^n}{m^n} < \frac{m^m}{(m-1)!}$</p>
<p>Am using the squeeze theorem , but unable to start third inequality.</p>
| Archis Welankar | 275,884 | <p>Rhs using simple combinatorics formula simplifies to $\frac{(n)(n-1)^2(n-2)}{8}$ now lhs ${n\choose 2}=\frac{(n)(n-1)}{2}$ so we can write lhs as $\frac{(n^2-n)!}{(2!).(\frac{n^2-n-4}{2})}$ now treat $n^2-n=y$ so it becomes $\frac{(n^2-n)(n^2-n-2)}{8}=\frac{(n)(n-1)^2(n-2)}{8}=Rhs$ </p>
|
1,457,063 | <p>I am utterly confused on how to solve this problem. I found a lemma that says $|A\cup B|=|A|+|B|$ is true if the two sets are disjoint which makes sense, but how do I prove the entire statement. </p>
| Bernard | 202,857 | <p>If the sets are not disjoint, in the right-hand side $\lvert A\rvert+\lvert B\rvert$, formula, the elements of $A\cap B$ are counted twice.</p>
|
1,115,645 | <p>I understand that a primitive polynomial is a polynomial that generates all elements of an extension field from a base field. However I am not sure how to apply this definition to answer my question. Can someone explain to me how I need to start please?</p>
| Bernard | 202,857 | <p>A primitive polynomial root is also the minimal polynomial of a <code>primitive root of unity</code> in $\mathbf F_7$. Let $\xi$ be a root of $f$. The field $\mathbf F_7(\xi)$ is the field $\mathbf F_{49}$ and its nonzero elements form a group of order $48$.
It suffices to show $xi$ has order $48$. Anyway its order... |
1,765,538 | <p>If $N$ is the set of all natural numbers, $R$ is a relation on $N \times N$, defined by $(a,b) \simeq (c,d)$ iff $ad=bc$, how can I prove that $R$ is an equivalence relation ?</p>
| marwalix | 441 | <p>Let's do it step by step:</p>
<p>$$\forall (a,b)\in \Bbb{N}\times \Bbb{N}^{*},\, ab=ab$$</p>
<p>So $(a,b)\mathcal{R}(a,b)$ and $\mathcal{R}$ is <strong>reflexive</strong>.</p>
<p>Assume now that $(a,b)\mathcal{R}(c,d)$ i.e $ad=bc$. Multiplication is commutative so we can write $cb=da$ and this gives $(c,d)\mathca... |
2,426,892 | <blockquote>
<p>Between which two integers does <span class="math-container">$\sqrt{2017}$</span> fall? </p>
</blockquote>
<p>Since <span class="math-container">$2017$</span> is a prime, there's not much I can do with it. However, <span class="math-container">$2016$</span> (the number before it) and <span class="mat... | Duncan Ramage | 405,912 | <p>$44^2 = 1936 < 2017 < 2025 = 45^2$.</p>
<p>Really, I don't think there's much to this one except for "try squaring small integers until you find the right ones".</p>
|
2,426,892 | <blockquote>
<p>Between which two integers does <span class="math-container">$\sqrt{2017}$</span> fall? </p>
</blockquote>
<p>Since <span class="math-container">$2017$</span> is a prime, there's not much I can do with it. However, <span class="math-container">$2016$</span> (the number before it) and <span class="mat... | kingW3 | 130,953 | <p>Knowing the answer lies between $40$ and $50$ the rational thing to do would be to try the number in the middle of $40$ and $50$ i.e $45$. </p>
<p>If you don't feel comfortable with multiplying $45\cdot 45$ you can use the
formula for $(40+5)^2=40^2+2\cdot 40\cdot 5+5^2=1600+400+25=2025$</p>
<p>Now you can use th... |
2,426,892 | <blockquote>
<p>Between which two integers does <span class="math-container">$\sqrt{2017}$</span> fall? </p>
</blockquote>
<p>Since <span class="math-container">$2017$</span> is a prime, there's not much I can do with it. However, <span class="math-container">$2016$</span> (the number before it) and <span class="mat... | Mr. Brooks | 162,538 | <p>You can also use the <em>least</em> significant digits to get your bearings. Since $2016 \equiv 16 \pmod{100}$, if $2016$ is a perfect square, then $n$ in $n^2 = 2016$ is an integer satisfying $n \equiv 4, 6 \pmod{10}$. Clearly $n = 4$ or $6$ is too small.</p>
<p>Then, working our way up, we get $(196, 256), (576, ... |
39,790 | <p>I'm teaching a programming class in Python, and I'd like to start with the mathematical definition of an array before discussing how arrays/lists work in Python.</p>
<p>Can someone give me a definition?</p>
| GEL | 10,825 | <p>I'd go with comparing an array to a matrix. That way when you introduce arrays of arrays, the mental leap will be easier for your students to make.</p>
|
930,949 | <p>Given that the circle C has center $(a,b)$ where $a$ and $b$ are positive constants and that C touches the $x$-axis and that the line $y=x$ is a tangent to C show that $a = (1 + \sqrt{2})b$</p>
| James | 81,163 | <p>When you are confused in this way you need to take a step back. In mathematics many words are used in different ways in different contexts. For instance, in one topology class I took, a "map" was defined to be a continuous function. Then later in a differential geometry class, "map" was the broad term and a "functio... |
4,038,392 | <p>This question is inspired by the problem <a href="https://projecteuler.net/problem=748" rel="nofollow noreferrer">https://projecteuler.net/problem=748</a></p>
<p>Consider the Diophantine equation
<span class="math-container">$$\frac{1}{x^2}+\frac{1}{y^2}=\frac{k}{z^2}$$</span>
<span class="math-container">$k$</span>... | individ | 128,505 | <p>To solve the Diophantine equation.</p>
<p><span class="math-container">$$ \frac{ 1 }{ x^2 } +\frac{ 1 }{ y^2 } = \frac{ q }{ z^2 } $$</span></p>
<p>It is necessary to use the solution of the following equation.</p>
<p><span class="math-container">$$a^2+b^2=qc^2$$</span></p>
<p>There are solutions when the coefficien... |
2,805,975 | <p>For matrices $A, B$, I would like to show and understand the intuition behind the following identity
$$
(A+B)^{-1} = A^{-1} - (A+B)^{-1} B A^{-1}
$$
assuming the inverses exist.</p>
| Jakobian | 476,484 | <p>$$ A^{-1}-(A+B)^{-1}BA^{-1} = (I-(A+B)^{-1}B)A^{-1} = (A+B)^{-1}(A+B-B)A^{-1} = (A+B)^{-1}AA^{-1} = (A+B)^{-1} $$</p>
|
633,985 | <p>How can I find the values of $a$ for which the following function $f:\{0,1,\dots,m-1\} \rightarrow \mathbb{Z}_m$ is bijective for a fixed $m$?
$$f(n) = \sum_{k=0}^n a^k$$</p>
| Nate | 91,364 | <p>First choose 3 points arbitrarily this gives $\dbinom{25}{3}=2300$ possibly degenerate triangles.</p>
<p>Now we need to subtract off triples that all lie in a line:</p>
<p>There are 12 lines (5 vertical, 5 horizontal, and the main diagonals) containing 5 points these contribute $12\cdot\dbinom{5}{3}=120$ degenerat... |
3,653,148 | <p>Let <span class="math-container">$w$</span> be a primitive 5th root of unity. Then the difference equation <span class="math-container">$$x_nx_{n+2}=x_n-(w^2+w^3)x_{n+1}+x_{n+2}$$</span> generates a cycle of period 5 for general initial values:
<span class="math-container">$$u,v,\frac{u-(w^2+w^3)v}{u-1},\frac{uv-(w... | Pavel Kozlov | 143,912 | <p>Another simple case periodic difference equation is the next one, with zero coefficient at <span class="math-container">$x_{n+1}$</span>:
<span class="math-container">$$x_nx_{n+2}=ax_n+x_{n+2}.$$</span></p>
<p>Simplicity of this case expressed in existence of nice closed form description for sequence members. Of co... |
1,949,966 | <h2>Q 1a</h2>
<p>Is it possible to define a number $x$ such that $|x|=-1$, where $|\cdot|$ means absolute value, in the same manner that we define $i^2=-1$?</p>
<p>I have no idea if it makes sense, but then again, $\sqrt{-1}$ used to not be a thing either.</p>
<p>To be more explicit, I want as many properties to hol... | Patrick Sheehan | 599,007 | <p>What about extending the real number absolute value to complex numbers in a different way?</p>
<p>For <span class="math-container">$z = re^{iθ}$</span>, let <span class="math-container">$|z| = re^{2iθ}$</span></p>
<p>You still get that the "new" absolute value for all reals matches the regular one.</p>
<p>... |
1,706,939 | <p>Can anyone share an easy way to approximate $\log_2(x)$, given $x$ is between $0$ and 1?</p>
<p>I'm trying to solve this using an old fashioned calculator (i.e. no logs)</p>
<p>Thanks!</p>
<p>EDIT: I realize that I stepped a bit ahead. The x comes in the form of a fraction, e.g. 3/8, which is indeed between 0 and... | seoanes | 322,225 | <p>Using that $\log_2(x)=\mbox{ln}(x)/\mbox{ln}(2)$, now you can use the Taylor expansion:
\begin{equation}
\mbox{ln}(x)=\mbox{ln}\left((x-1)+1\right)=\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k}(x-1)^k
\end{equation}
The same expansion can be used for determining the $\mbox{ln}(2)$. The first terms of the expansion are
\be... |
288,340 | <p>I am having difficulty understanding the recursive definition of a language. The problem asked how to write this non recursively. But I want to understand just how a recursive definition of a language works.</p>
<p>Recursive definition of a subset of L of $\{a,b\}^*$.</p>
<p>Basis : $a\in L$ </p>
<p>Recursive D... | Brian M. Scott | 12,042 | <p>The recursive definition says that once you have a word of $L$, you can prefix an $a$ or suffix a $b$ to get another word of $L$. You can repeat either of these operations any number of times, so you can prefix $a^n$ for any $n\ge 0$ or suffix $b^n$ for any $n\ge 0$, or both (not necessarily with the same $n$). If y... |
2,041,251 | <p>Suppose $f$ is a twice-differentiable function with $f(0) = 0$, $f\left(\frac12\right) = \frac12$ and $f'(0) = 0$. Prove that $|f''(x)| \ge 4$ for some $x \in \left[0,\frac12\right]$.</p>
<p>I've been stuck on this question for a while now without any idea on how to get started. Is it possible for someone to help ... | Community | -1 | <p>[Paramanand's answer is definitely the way to go. I just want to share another elementary approach in the $f''$-is-continuous case.]</p>
<p>If $f$ is twice <em>continuously</em> differentiable then by the (second) <em>Fundamental theorem of Calculus</em> we can write
$$
f(x)=0+\int_0^xf'(y)\, dy=\int_0^x\left(0+\in... |
1,180,854 | <p>I was solving some probability and combination problems and I came across this one,
that I couldn't solve. Any tips appreciated.</p>
<blockquote>
<p>Chance to win with one lottery ticket is <span class="math-container">$\frac13$</span>. How many tickets must be purchased for the probability of winning at least with ... | Stefan4024 | 67,746 | <p>Assume that $z=a+bi$ then we have for the equation.</p>
<p>$$a + bi + 1 -ai +b=0$$
$$(a+b+1) + (b-a)i=0$$</p>
<p>Now set $a+b+1=0$ and $a-b=0$ to get $a=b=-\frac 12$, hence $z = -\frac 12 - \frac i2$</p>
|
2,961,796 | <p>I was trying to solve the inequality <span class="math-container">$$a-\sqrt[3]{a^3-c\cdot a^2}<b-\sqrt[3]{b^3-c\cdot b^2}$$</span> where <span class="math-container">$a>b>0$</span> and <span class="math-container">$c>0$</span>. I managed to pack the part inside the cube root: <span class="math-container"... | Michael Rozenberg | 190,319 | <p>We need to solve
<span class="math-container">$$\sqrt[3]{a^2(a-c)}-\sqrt[3]{b^2(b-c)}-(a-b)>0$$</span> or since <span class="math-container">$$a^2(a-c)=-b^2(b-c)=-(a-b)^3$$</span> gives
<span class="math-container">$$\frac{a^2b^2}{a^2+b^2}+(a-b)^2=0,$$</span> which is impossible, we need to solve
<span class="mat... |
3,789,494 | <p>I'm stuck in solving this strange and beautiful formula : <span class="math-container">$3= 3^{z}$</span> since it says 'Solve' and not 'Prove'</p>
<p>Also i really don't understand what does it means by saying <span class="math-container">$3^{z}$</span>? Will <span class="math-container">$3^z$</span> form a set ?</p... | mjw | 655,367 | <p>Well, if the question is asking to find all <span class="math-container">$z$</span> such that</p>
<p><span class="math-container">$$3^z = 3$$</span></p>
<p>Then</p>
<p><span class="math-container">$$\begin{aligned}
3^z &= 3e^{2 k \pi i } \\
z \log 3 &=\log 3 + 2 k \pi i \\
\end{aligned}$$</span>
<span cla... |
3,245,796 | <p>Let <span class="math-container">$f$</span> have a continuous second derivative. Prove that</p>
<p><span class="math-container">$$f(x) = f(a) + (x - a)f'(a) + \int_a^x(x - t)f''(t) dt.$$</span></p>
<p>This is a modification of exercise 6.6.4 from Advanced Calculus by Fitzpatrick. I have seen that this question has... | uniquesolution | 265,735 | <p>Integration by parts is not necessary.</p>
<p>Put
<span class="math-container">$$g(x)=f(x)-f(a)-(x-a)f'(a)$$</span>
and
<span class="math-container">$$h(x)=\int_a^x(x-t)f''(t)dt$$</span>
and
<span class="math-container">$$k(x)=g(x)-h(x)$$</span>
Then <span class="math-container">$k'(x)=0$</span> for all <span class... |
1,685 | <p>Are there books or article that develop (or sketch the main points) of Euclidean geometry without fudging the hard parts such as angle measure, but might at times use coordinates, calculus or other means so as to maintain rigor or avoid the detail involved in Hilbert-type axiomatizations?</p>
<p>I am aware of Hilbe... | Julien Narboux | 239,646 | <p>There are some axioms systems such as Birkoff axioms which assume the existence of a field from the beginning.</p>
<p>For the synthetic approach the main axiom systems are those of Hilbert and Tarski.</p>
<p>You can also use Tarski's axiom as described in W. Schwabhäuser,
W Szmielew, A. Tarski, Metamathematische M... |
322,140 | <p>$$\int \left ( r\sqrt{R^2-r^2} \right )dr$$</p>
<p>It looks simple. I know that the derivative of </p>
<p>$$\left (R^2-r^2 \right )^\frac{3}{2}$$</p>
<p>Is the stuff in the integral.</p>
<p>However, what about if I don't know?</p>
<p>How in general do we solve integral of</p>
<p>$$G(r)^n$$</p>
| marty cohen | 13,079 | <p>Note that $(r^2)' = 2r$, so that
$(R^2-r^2)' = -2r$
(with respect to $r$).</p>
<p>Therefore, if $u = R^2 - r^2$,
$du = -2r\ dr$,
so, for any function $f$,
$\int r f(R^2-r^2) dr
= -\frac1{2}\int (-2r) f(R^2-r^2) dr
= -\frac1{2}\int f(R^2-r^2) (-2r\ dr)
= -\frac1{2} \int f(u) du
$.</p>
<p>Putting $f(u) = \sqrt{u}$ ... |
1,595,118 | <p>What is value of $a+b+c+d+e$? If given :</p>
<p>$$abcde=45$$</p>
<p>And $a,b, c, d, e$ all are distinct integer.</p>
<hr>
<p>My attempt :</p>
<p>I calculated, $45 = 3^2 \times 5$.</p>
<blockquote>
<p>Can you explain, how do I find the distinct values of $a,b, c, d, e$ ?</p>
</blockquote>
| Jimmy R. | 128,037 | <p>So $$45=5\cdot(\pm 3)^2\cdot(\pm1)^2$$ which implies that the only possibility (up to renaming) is $a=1, b=-1, c=3, d=-3, e=5$. </p>
|
2,749,539 | <p>I have tried for some time to solve this problem and I'm stuck, so any help would be greatly appreciated. I'm not a math guy, so I apologize if I am missing something basic.</p>
<p>I have a function $f(x)$ $$f(x) = \sum_{i=1}^x dr^{i-1}$$ where<br>
$x$ is a positive integer<br>
$d$ is an initial delta and<br>
$r$ ... | mzp | 287,326 | <p>Notice that using the geometric progression formula we have that</p>
<p>$$f(n) = \sum_{i=1}^n dr^{i-1} = \begin{cases}\frac{d(1-r^n)}{1-r},& \quad\text{if}\;r\neq 1 \\ dn, &\quad\text{if}\;r= 1\end{cases}$$</p>
<p><em>1) How can I solve for $r$?</em></p>
<p>You would need to solve the polynomial</p>
<p>$... |
2,645,611 | <blockquote>
<p>Prove that:
<span class="math-container">$$\frac{(n+0)!}{0!}+\frac{(n+1)!}{1!}+...+\frac{(n+n)!}{n!}=\frac{(2n+1)!}{(n+1)!}$$</span></p>
</blockquote>
<h3>My work so far:</h3>
<p><span class="math-container">$$\frac{(n+0)!}{0!}+\frac{(n+1)!}{1!}+...+\frac{(n+n)!}{n!}=\frac{(2n+1)!}{(n+1)!}$$</span>
<spa... | Seewoo Lee | 350,772 | <p>Try to use the following identity:
$$
\binom{n}{k} = \binom{n-1}{k}+\binom{n-1}{k-1}.
$$</p>
|
2,645,611 | <blockquote>
<p>Prove that:
<span class="math-container">$$\frac{(n+0)!}{0!}+\frac{(n+1)!}{1!}+...+\frac{(n+n)!}{n!}=\frac{(2n+1)!}{(n+1)!}$$</span></p>
</blockquote>
<h3>My work so far:</h3>
<p><span class="math-container">$$\frac{(n+0)!}{0!}+\frac{(n+1)!}{1!}+...+\frac{(n+n)!}{n!}=\frac{(2n+1)!}{(n+1)!}$$</span>
<spa... | Gerry Myerson | 8,269 | <p>First, note ${n\choose0}={n+1\choose0}$, then repeatedly use ${k\choose r}+{k\choose r+1}={k+1\choose r+1}$</p>
|
2,645,611 | <blockquote>
<p>Prove that:
<span class="math-container">$$\frac{(n+0)!}{0!}+\frac{(n+1)!}{1!}+...+\frac{(n+n)!}{n!}=\frac{(2n+1)!}{(n+1)!}$$</span></p>
</blockquote>
<h3>My work so far:</h3>
<p><span class="math-container">$$\frac{(n+0)!}{0!}+\frac{(n+1)!}{1!}+...+\frac{(n+n)!}{n!}=\frac{(2n+1)!}{(n+1)!}$$</span>
<spa... | lab bhattacharjee | 33,337 | <p>$$\sum_{r=0}^n\binom{n+r}r$$ is the coefficient of $x^n$ in $$\sum_{r=0}^n(1+x)^{n+r}$$ which is a Finite Geometric Series and is $$=(1+x)^n\cdot\dfrac{(1+x)^{n+1}-1}{1+x-1}=\dfrac{(1+x)^{2n+1}-(1+x)^n}x$$</p>
<p>Now the coefficient of $x^{n+1}$ in $$(1+x)^{2n+1}-(1+x)^n$$</p>
<p>$$=\binom{2n+1}{n+1}-0$$</p>
|
1,808,441 | <p>I want to know for two circles why the ratio of arc length is equal to the ratio of the two central angles in geometry. It must have something to do with the concept of similarity in geometry. I have scoured the Internet looking answers but only found the ratio of circumferences to the ratio of their diameter.</p>
... | Christian Blatter | 1,303 | <p>It is a basic feature of euclidean geometry, not present in spherical or hyperbolical geometry, that we have available a large family of "special" maps $T_{O,\rho}\,$, called <em>homotheties</em>. These maps are stretching the base space ${\mathbb E}^2$ linearly from a given center $O$ with a given factor $\rho>0... |
1,528,235 | <p>Recall that <a href="http://en.wikipedia.org/wiki/Tetration" rel="noreferrer">tetration</a> ${^n}x$ for $n\in\mathbb N$ is defined recursively: ${^1}x=x,\,{^{n+1}}x=x^{({^n}x)}$. </p>
<p>Its inverse function with respect to $x$ is called <a href="http://en.wikipedia.org/wiki/Tetration#Super-root" rel="noreferrer">s... | mick | 39,261 | <p>$$\mathcal L=\lim\limits_{n\to\infty}\frac{\sqrt[n]2_s-\sqrt2}{(\ln2)^n}\tag1$$</p>
<p>Notice the resemblance with the Koenigs function</p>
<p><a href="https://en.m.wikipedia.org/wiki/Koenigs_function" rel="nofollow">https://en.m.wikipedia.org/wiki/Koenigs_function</a></p>
<p>In fact it is a Koenigs function with... |
4,180,750 | <p>Let's consider inner product space with vectors <span class="math-container">$x, y, z$</span> which satisfies:</p>
<p><span class="math-container">$$\|x+y+z\|^2 = 14$$</span></p>
<p><span class="math-container">$$\|x+y-z\|^2 = 2$$</span></p>
<p><span class="math-container">$$\|x-y+z\|^2 = 6$$</span></p>
<p><span cla... | alepopoulo110 | 351,240 | <p>I assume you work with a real and not a complex vector space. Is that true?</p>
<p>Add the first and second one: by the <a href="https://en.wikipedia.org/wiki/Parallelogram_law#The_parallelogram_law_in_inner_product_spaces" rel="nofollow noreferrer">parallelogram identity</a> you get
<span class="math-container">$$2... |
1,838,002 | <p>There is famous <a href="https://en.wikipedia.org/wiki/Quillen-Suslin_theorem" rel="nofollow">Quillen-Suslin theorem</a> which states that every finitely generated projective module over a ring of polynomials $k[x_1,...,x_n]$, where $k$ is a field, is free.</p>
<p>I have never carefully read a proof of this theorem... | Mohan | 245,104 | <p>Ok Slup, here goes.<br/></p>
<p>Let $R$ be any commutative ring and let $A$ be a polynomial ring over $R$. Let $P$ be any projective module over $R$. Then Quillen (and Suslin a bit later in this generality) proved that if for every maximal ideal $\mathfrak{m}$ of $R$, $P_{\mathfrak{m}}$ is of the form $Q\otimes_{R_... |
2,097,584 | <blockquote>
<p>How many ways can the number $2160$ be written as a product of factors which are relatively prime to each other?</p>
</blockquote>
<p>I was confused by this question because couldn't we just add $1$ into the factorization every time? For example, $2160$ and $2160 \cdot 1$ would count as distinct fact... | hmakholm left over Monica | 14,366 | <p>You're right that the problem only makes sense if factors of $1$ are not allowed.</p>
<p>Under this interpretation, your analysis seems to make sense: The number of ways to write $n$ as a product of coprime factors $>1$ (not counting the order of these factors) is exactly the number of partitions of the set of p... |
64,406 | <p>It's often said that there are only two nonabelian groups of order 8 up to isomorphism, one is the quaternion group, the other given by the relations $a^4=1$, $b^2=1$ and $bab^{-1}=a^3$. </p>
<p>I've never understood why these are the only two. Is there a reference or proof walkthrough on how to show any nonabelian... | Beginner | 15,847 | <p>Let $G$ be non-abelian group of order 8.</p>
<p>By Sylow theorem, $G$ will have subgroups of order 2.</p>
<ul>
<li><p>If $G$ has unique subgroup of order 2, then $G$ is quaternion (<em>Ref. Theorem 12.5.2, Theory of groups- Marshall Hall</em>). ($G$ can be cyclic also, but we have assumed it is non-abelian.)</p></... |
64,406 | <p>It's often said that there are only two nonabelian groups of order 8 up to isomorphism, one is the quaternion group, the other given by the relations $a^4=1$, $b^2=1$ and $bab^{-1}=a^3$. </p>
<p>I've never understood why these are the only two. Is there a reference or proof walkthrough on how to show any nonabelian... | Marc Olschok | 15,825 | <p>You probably meant $bab = a^3$ in the above.</p>
<p>The following steps should also work. In order not to spoil the fun for you,
I omit all the details.</p>
<p>Let $G$ be a nonabelian group of order $8$ and let $Z$ be its centre.</p>
<p>(1) From the class equation we know that $|Z|$ is divisible by $2$,
and hence... |
2,328,505 | <p>Let $X$ be an exponential random variable with $\lambda =5$ and $Y$ a uniformly distributed random variable on $(-3,X)$. Find $\mathbb E(Y)$.</p>
<p>My attempt:</p>
<p>$$\mathbb E(Y)= \mathbb E(\mathbb E(Y|X))$$ </p>
<p>$$\mathbb E(Y|X) = \int^{x}_{-3} y \frac{1}{x+3} dy = \frac{x^2+9}{2(x+3)}$$</p>
<p>$$ \mathb... | callculus42 | 144,421 | <p><strong>Hint:</strong> $$\int^{x}_{-3} y \frac{1}{x+3} dy=\frac{\frac12\cdot y^2}{x+3}\bigg|_{-3}^x=\frac12\cdot \frac{x^2-(-3)^2}{x+3}=\frac12\cdot \frac{x^2-9}{x+3}$$</p>
<p>At the numerator you can use the second binomial formula. </p>
|
1,439,429 | <p>Is it possible to calculate the Sagitta, knowing the Segment Area and Radius?
Alternatively, is there a way to calculate the Chord Length, knowing the Segment Area and Radius?</p>
| Rubi Shnol | 90,296 | <p>In convex functions, all chords lie above the function values.</p>
<p>You can apply gradient descent to non-convex problems provided that they are smooth, but the solutions you get may be only local. Use global optimization techniques in that case such simulated annealing, genetic algorithms etc.</p>
|
1,439,429 | <p>Is it possible to calculate the Sagitta, knowing the Segment Area and Radius?
Alternatively, is there a way to calculate the Chord Length, knowing the Segment Area and Radius?</p>
| Shiv Tavker | 687,825 | <p>Yes. The function you have is non-convex. Nevertheless, gradient descent will still take you to the global optimum as you have correctly pointed out "the function still has a single minimum". This function is quasi-convex. Gradient descent almost always work for quasi convex functions but we do not have co... |
4,398,864 | <p>How many ways are there to select a three digit number <span class="math-container">$\underline{A}\ \underline{B}\ \underline{C}$</span> so that <span class="math-container">$A \neq B$</span>, <span class="math-container">$B \leq C$</span>, and <span class="math-container">$A < C$</span>?</p>
<hr />
<p>I found th... | Keryann Massin | 450,912 | <p>If I understood your question, you want to find how many choices of <span class="math-container">$A,B,C\in\{0,1,2,3,\dots, 9\}$</span> such that <span class="math-container">$A\ne B$</span>, <span class="math-container">$B\leq C$</span> and <span class="math-container">$A<C$</span>.</p>
<p>First, <span class="mat... |
3,487,105 | <p>I'm kinda a newbie in calculus, but what are the conditions for the power rule to happen?
For example, if we have the number <span class="math-container">$e$</span>, with the property <span class="math-container">$\frac{d}{dt}{ {e^t} } = {e^t}$</span> we can get, using the power rule, that <span class="math-containe... | Andrew Chin | 693,161 | <p>This is a mistake common to many calculus students, and it is evidence of a lack of fundamentals.</p>
<p>The power rule is used to differentiate <em>powers</em> of functions. These are functions that have some constant in the exponent (e.g. <span class="math-container">$x^2$</span>, <span class="math-container">$\... |
3,487,105 | <p>I'm kinda a newbie in calculus, but what are the conditions for the power rule to happen?
For example, if we have the number <span class="math-container">$e$</span>, with the property <span class="math-container">$\frac{d}{dt}{ {e^t} } = {e^t}$</span> we can get, using the power rule, that <span class="math-containe... | James Turbett | 721,003 | <p>The power rule only works for functions raised to a power, like x^3, x^4, (x+2)^5, or sqrt(x), etc. The power isn't a variable, it's a constant. When the power is a variable, like e^x, 2^x, we call that an exponential function, and you can't use the power rule to differentiate it. Think about the definition of the d... |
2,523,660 | <p>I'm trying to solve the next question:</p>
<p>For all $m\in I=(0,1)$ there is a subset $A_m \subseteq \mathbb{R}$ that $A_{m} = \{ a\in \mathbb{R} : a-\lfloor a \rfloor < m \} $. Find $\bigcap\limits_{m\in I} A_{m}$</p>
<p>So I think that the solution is $\bigcap\limits_{m\in I} A_m=\mathbb{Z}$, and I tried to ... | Robert Dixon | 322,951 | <p>The inverse projection takes x to the set {x} × Y which would only have a meaningful definition of open and closed in the power set of the product of X and Y. Topologies for power sets do exist, but I don't think you are looking for that avenue of thought.</p>
|
4,164,820 | <p>CGM = Continuous Geometric Mean. Heuristics and mathematics are described in:</p>
<ul><li><a href="https://math.stackexchange.com/questions/4162869/subset-as-arithmetic-mean-of-geometric-means-not-really">Subset as arithmetic mean of geometric means. Not really?</a></li></ul>
A shortcut to the question as presented ... | Han de Bruijn | 96,057 | <p>Here comes a replay of <a href="https://www.cantorsparadise.com/richard-feynmans-integral-trick-e7afae85e25c" rel="nofollow noreferrer">Richard Feynman’s Integral Trick</a>,
adapted ( if not <i>improved</i> ) for our purpose.
Define:
<span class="math-container">$$
I(a)=\int_{0}^{2\pi}\ln\left(1-2a\cos x+a^2\right)\... |
3,328,737 | <p>For any rational number, <span class="math-container">$\frac{p}{q}$</span> , <span class="math-container">$p$</span> and <span class="math-container">$q$</span> should be integers, <span class="math-container">$q\neq0$</span> and <span class="math-container">$p,q$</span> should not have any common factors.
Now, if w... | Fred | 380,717 | <p>If <span class="math-container">$m$</span> and <span class="math-container">$n$</span> are integers and <span class="math-container">$n \ne 0$</span>, then <span class="math-container">$\frac{m}{n}$</span> is rational.</p>
|
1,673,854 | <p>Taylor Series of $f(x) = \sqrt{x}$ about $c = 1$</p>
<p>I've tried doing this problem but stuck at finding a pattern..</p>
<p>Work:</p>
<p>$$T_n = \sum^\infty_{n=0}\frac{f^n(c)}{n!}(x-c)^n = f(a) + \frac{f'(c)}{1!}(x-a)^1 + \frac{f''(c)}{2!}(x-a)^2+... $$</p>
<p>So $f(x) = \sqrt{x}$</p>
<p>$$ f'(x)=\frac12x^{-\... | Em. | 290,196 | <p>Let's take a smaller example. Consider
$$5! = 5\cdot4\cdot3\cdot2\cdot1.$$
Notice that this can be expressed as
$$5! = 5\cdot(4\cdot3\cdot2\cdot1\cdot) = 5\cdot4!.$$
Similarly
$$5! = 5\cdot4\cdot(3\cdot2\cdot1) = 5\cdot4\cdot3!$$
We can generalize to $k!$,
$$k! = k(k-1)(k-2)\dotsm 2\cdot1$$
and in particular
$$k! ... |
198,945 | <p>Why and how publishing a paper in proceedings?<br>
What are the difference with a "classical" journal?<br>
What's the list of the main proceedings in which one can publish?<br>
Do proceedings papers (never, sometimes, often or always) appear on mathscinet?</p>
| Andreas Blass | 6,794 | <p>Proceedings of conferences are often published as special issues of "classical" journals. But even those that are not are usually included in MathSciNet if they include a statement (often a footnote on the first page of each paper) to the effect that the papers are in final form and will not be published elsewhere. ... |
35,321 | <p>I need to do some simplification of an expression involving averages over a stochastic variable (in order to verify a long analytical calculation).
The easiest way to do that, I figured, were if I could implement an operator which would basically be short-hand for the averaging procedure, with all the appropriate p... | Sooner | 10,361 | <p>I think I got it.</p>
<pre><code>av /: D[av[f___], x_] := av[D[f, x]]
av[y_ + z_] := av[y] + av[z]
av[c_ y_] := c av[y] /; FreeQ[c, x]
av[c_] := c /; FreeQ[c, x]
D[av[x y], x]
(* y *)
D[av[Exp[-x y]], x]
(* -y av[E^(-x y)] *)
</code></pre>
<p>I wasn't using <code>UpValues</code> correctly before.</p>
<p>EDIT: ... |
1,646,135 | <p>I have the subset $\left[0,1\right] \backslash \mathbb{Q}$ in $\mathbb{R} \backslash \mathbb{Q}$. </p>
<p>Am I right in thinking that this set is open and not closed in the space given?</p>
<p>Also, how do I go about finding the interior, closure and boundary?</p>
| noctusraid | 185,359 | <p>I don't think such a function exists.
You can construct an example where all the inputs are non-zero but the function gives back 0:</p>
<p>$$A=\{1,2,3,4,5\}, B=\{1,6\},C=\{5,6\} \implies f(_\cdots)=0$$</p>
<p>You can easily construct an example where you have the same cardinalities but the intersection is non triv... |
1,646,135 | <p>I have the subset $\left[0,1\right] \backslash \mathbb{Q}$ in $\mathbb{R} \backslash \mathbb{Q}$. </p>
<p>Am I right in thinking that this set is open and not closed in the space given?</p>
<p>Also, how do I go about finding the interior, closure and boundary?</p>
| Nick Graham | 547,268 | <p>You can estimate $|A \cup B \cup C|$ by computing its mean over a population where every element is three sets that meet the known conditions. This assumes that the presence of an element in a set is independent from and identically distributed to the presence of every other element in that set or other sets.</p>
<... |
3,621,223 | <p>I use a software called Substance Designer which has a Pixel Processor where I can assign to every pixel of a image a gray-scale value defined by a series of operations.</p>
<p>I am basically trying to generate a <a href="https://i.stack.imgur.com/6jhYa.jpg" rel="nofollow noreferrer">"normal gradient"</a> generated... | Anonymous Coward | 770,101 | <p>This is an extended comment, to solve the following problem that came up in the comments:</p>
<p>We have an axis-aligned ellipse, centered at origin, width <span class="math-container">$2$</span>, height <span class="math-container">$2 h$</span> — so semi-major axis is <span class="math-container">$1$</span> ... |
1,661,439 | <p>I'm trying to take the Lagrange polynomial $P_3(x)$ that passes through four points- $(x_1,y_1), (x_2,y_2), (x_3,y_3)$, and $(x_4,y_4)$, and integrate it (maybe deriving Simpson's rule in the process!).
All the x-values are a distance h apart from each other.
Once I did some simplifications using h, here's what I ... | Carl Christian | 307,944 | <p>While Acheille Hui has outlined a good strategy I would like to continue with your approach.</p>
<p>First of all, I can confirm that your polynomial passes through the four given values, hence you have the correct expression. </p>
<p>You can always change the interval of integration from, say, $[a,b]$ to $[-1,1]$ ... |
1,926,839 | <p>I have to find a function $f\in C^\infty(\mathbb{R})$ which, for fixed $a,b\in \mathbb{R}$, $a<b$, is identically 1 for $x\le a$, identically $0$ for $x\ge b$ and decreases in $a\le x\le b$. I've tried many times to write $f$ in $a\le x\le b$ using an exponential, but it didn't work. Can you help me?</p>
| Daniel Robert-Nicoud | 60,713 | <p><strong>Hint:</strong> Use the function
$$\varphi(x) = \cases{0 & if $x\le0$,\\e^{-\frac{1}{x^2}} & if $x\ge0$}$$
as a building block. You can easily prove that $\varphi\in C^\infty(\mathbb{R})$.</p>
|
3,640,307 | <p>The extended reals are taken to be the set <span class="math-container">$\mathbb{R}\cup\{+\infty,-\infty\}$</span>. But is there a <em>natural</em> way to define <span class="math-container">$+\infty$</span> and <span class="math-container">$-\infty$</span> as sets, in a pure set-theoretic theme, following the const... | J.G. | 56,861 | <p>Roughly speaking, the Cauchy definition identifies each real with the Cauchy sequences of rationals that converge to it, whereas the Dedekine definition identifies each real with the set of rational numbers less than it (or an ordered pair of that set, together with the set of greater rationals). Since you asked abo... |
4,329,888 | <p>I have a problem of understanding how to find shaded regions in Complex Plane.</p>
<p><span class="math-container">\begin{array}{l}
|z-2i|\ \geqslant \ |z+6+4i|\\
\\
\sqrt{x^{2} +( y-2)^{2}} =\sqrt{( x+6)^{2} +( y+4)^{2}}\\
x^{2} +( y-2)^{2} =( x+6)^{2} +( y+4)^{2}\\
x^{2} +y^{2} -4y+4=x^{2} +12x+36+y^{2} +8y+16\\
1... | Gary | 83,800 | <p>You can continue by changing the order of summation and then splitting the sum into two parts:
<span class="math-container">\begin{align*}
& \sum\limits_{n = 0}^\infty {\frac{{( - 1)^{n + 1} \pi ^{2n + 1} }}{{(2n + 1)!}}\left( {\sum\limits_{k = 0}^{2n + 1} {( - 1)^k \binom{2n + 1}{k}(z + 1)^k } } \right)}
\\ &... |
23,502 | <p><em>Edit: I wrote the following question and then immediately realized an answer to it, and moonface gave the same answer in the comments. Namely, $\mathbb C(t)$, the field of rational functions of $\mathbb C$, gives a nice counterexample. Note that it is of dimension $2^{\mathbb N}$.</em></p>
<p>The following is... | Manny Reyes | 778 | <p>The idea behind Schur's Lemma is the following. The endomorphism ring of any simple $R$-module is a division ring. On the other hand, a finite dimensional division algebra over an algebraically closed field $k$ must be equal to $k$ (this is because any element generates a finite dimensional subfield over $k$, whic... |
23,502 | <p><em>Edit: I wrote the following question and then immediately realized an answer to it, and moonface gave the same answer in the comments. Namely, $\mathbb C(t)$, the field of rational functions of $\mathbb C$, gives a nice counterexample. Note that it is of dimension $2^{\mathbb N}$.</em></p>
<p>The following is... | Victor Protsak | 5,740 | <p>It all depends on what you call "Schur's Lemma". If M is a simple module over a ring R then D=End<sub>R</sub>M is always a division ring (think of it as a weak Schur's lemma). The question is, can the endomorphism ring be pinned down more concretely, for example, if R is an algebra over a field k? In the usual Schur... |
2,662,033 | <p>My question is how to prove that $(X,d)$ is complete if and only if $(X,d')$ is complete.</p>
<p>I have that $d$ and $d'$ are strongly equivalent metrics and I have used this to show that a sequence $x_{n}$ is Cauchy in $(X,d)$ if and only if it is Cauchy in $(X,d')$.</p>
<p>I have the definition of complete as:
"... | José Carlos Santos | 446,262 | <p><strong>Hint:</strong> You are assuming that $d$ and $d'$ are strongly equivalent, right?! Now, given a sequence $(x_n)_{n\in\mathbb N}$ of elements of $X$, prove that</p>
<ol>
<li>if $x\in X$, then $(x_n)_{n\in\mathbb N}$ converges to $x$ in $(X,d)$ if and only if it converges to $x$ in $(X,d')$;</li>
<li>$(x_n)_{... |
3,318,130 | <p>Let <span class="math-container">$(a_j)_{j \in \mathbb{N}}$</span> and <span class="math-container">$(b_j)_{j \in \mathbb{N}}$</span> two real valued sequences such that <span class="math-container">$a_j \nearrow +\infty$</span> and <span class="math-container">$b_j \nearrow +\infty$</span>.
Is it possible to extra... | José Carlos Santos | 446,262 | <p>Not necessarily. Suppose that <span class="math-container">$a_j=j^2$</span> and that <span class="math-container">$b_j=j$</span>. Then <span class="math-container">$\frac{a_j}{b_j}=j$</span> for each <span class="math-container">$j\in\mathbb N$</span>.</p>
|
365,808 | <p>Sorry if something like this has already been asked, I searched but I couldn't find anything similar to my question.</p>
<p>I'm a senior undergraduate and currently doing my senior thesis. My senior thesis is not original work, however it's quite demanding and I'm learning a lot of high level topics. I have been lur... | David White | 11,540 | <p>I'm adding this answer in response to Annie Lee's question, because it's too long to fit in a comment.</p>
<p>Publishing as a grad student should definitely be done in consultation with an advisor. Unlike publishing as an undergraduate, a grad student's first papers serve as their introduction to the experts in thei... |
4,049,293 | <p>I am learning about the cross entropy, defined by Wikipedia as
<span class="math-container">$$H(P,Q)=-\text{E}_P[\log Q]$$</span>
for distributions <span class="math-container">$P,Q$</span>.</p>
<p>I'm not happy with that notation, because it implies symmetry, <span class="math-container">$H(X,Y)$</span> is often us... | robjohn | 13,854 | <p><strong>Fermat's Little Theorem</strong></p>
<p>In particular, this polynomial admits a lot of simplification.
<span class="math-container">$$
\begin{align}
3n^5+5n^3+7n
&\equiv2n^3+n&\pmod3\tag{1a}\\
&\equiv2n+n&\pmod3\tag{1b}\\
&\equiv0&\pmod3\tag{1c}
\end{align}
$$</span>
Explanation:<br /... |
3,841,806 | <p>Using spherical coordinates I have to find the volume of a cone <span class="math-container">$z=\sqrt{x^2+y^2}$</span> inscribed in a sphere <span class="math-container">$(x-1)^2+y^2+z^2=4.$</span></p>
<p>I can`t find <span class="math-container">$\rho$</span> because the center of sphere is displaced from the origi... | David G. Stork | 210,401 | <p>Not an answer (so please don't downvote), but a figure that should help you visualize the problem:</p>
<p><a href="https://i.stack.imgur.com/IWExy.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/IWExy.png" alt="enter image description here" /></a></p>
<p>Clearly you want to integrate over <span cl... |
3,841,806 | <p>Using spherical coordinates I have to find the volume of a cone <span class="math-container">$z=\sqrt{x^2+y^2}$</span> inscribed in a sphere <span class="math-container">$(x-1)^2+y^2+z^2=4.$</span></p>
<p>I can`t find <span class="math-container">$\rho$</span> because the center of sphere is displaced from the origi... | Doug M | 317,162 | <p><span class="math-container">$x=\rho\cos\theta\sin\phi\\
y=\rho\sin\theta\sin\phi\\
z=\rho\cos\phi$</span></p>
<p>Plug these substitutions into the given equations.</p>
<p><span class="math-container">$x^2 - 2x + 1 + y^2 + z^2 = 4$</span></p>
<p><span class="math-container">$\rho^2 - 2\rho\cos\theta\sin\phi - 3 = 0$... |
1,534,694 | <p>I tried to solve for the following limit: </p>
<p>$$\lim_{x\rightarrow \infty} (e^{2x}+x)^{1/x}$$
and I reached to the indeterminate form:
$${4e^{2x}}\over {4e^{2x}}$$
if I plug in, I will get another indeterminate form! </p>
| Claude Leibovici | 82,404 | <p>Another way, considering $$A=(e^{2x}+x)^{1/x}$$ Taking logarithms $$\log(A)=\frac 1x \log(e^{2x}+x)=\frac 1x \left(\log(e^{2x})+\log(1+\frac x {e^{2x}})\right)=\frac 1x \left(2x+\log(1+\frac x {e^{2x}})\right)$$ $$\log(A)=2+\frac 1x \log(1+\frac x {e^{2x}})\approx 2+\frac 1x \frac x {e^{2x}}=2+\frac 1 {e^{2x}}$$</p>... |
1,138,212 | <p>I am given $f(x) = 1 + x - \frac{sin(x)}{(x e^x)} $ and am asked to solve this for when x ≃ 0.</p>
<p>I'm doing the following steps but am getting stuck halfway through:</p>
<p>$$f(x) = 1 + x - \frac {x - \frac{x^3}{6} + \frac{x^5}{120}}{xe^x} $$</p>
<p>$$= 1 + x - \frac{e^{-x} (x - \frac{x^3}{6} + \frac{x^5}{120... | abel | 9,252 | <p>i will take it from your last step.<br>
$\begin{align} 1 + x - \frac{e^{-x} \sin x}{x} &= 1 + x -
\left(1 - \frac{x}{1!} + \frac{x^2}{2!} - \frac{x^3}{3!} + \cdots \right)
\left( 1 - \frac{x^2}{3!} + \frac{x^4}{5!} + \cdots\right)\\
&=1 + x -
\left(1 - x + x^2(\frac{1}{2!} - \frac{1}{3!}) + \cdots \right)... |
1,138,212 | <p>I am given $f(x) = 1 + x - \frac{sin(x)}{(x e^x)} $ and am asked to solve this for when x ≃ 0.</p>
<p>I'm doing the following steps but am getting stuck halfway through:</p>
<p>$$f(x) = 1 + x - \frac {x - \frac{x^3}{6} + \frac{x^5}{120}}{xe^x} $$</p>
<p>$$= 1 + x - \frac{e^{-x} (x - \frac{x^3}{6} + \frac{x^5}{120... | Jack D'Aurizio | 44,121 | <p>Since in a neighbourhood of the origin we have:
$$\frac{\sin x}{x}=1-\frac{x^2}{6}+o(x^2),\qquad e^{-x}=1-x+\frac{x^2}{2}+o(x^2)$$
it happens that:
$$\frac{\sin x}{x e^x}=1-x+\frac{x^2}{3}+o(x^2) $$
so:
$$ f(x) = 1+x-\frac{\sin x}{x e^x}\approx 2x-\frac{x^2}{3}$$
and provided that $g(x)$ is the inverse function of $... |
1,290,176 | <p>Can anybody help me with this limit?
I think the answer should be $0$ as $0$ to the power $1$ should be $0$ but it doesn't match with the book's answer.</p>
<p>$$ \lim_{x\to 0} |x|^{\lfloor\cos{x}\rfloor}$$</p>
| marwalix | 441 | <p>So the edit is right and $\lfloor\cos{x}\rfloor=0$ for $x\neq 0$ in a small neighborhood of $0$ and $\lfloor\cos{0}\rfloor=1$.</p>
<p>One has $|x|^{\lfloor\cos{x}\rfloor}=1$ is a constant function for $x\neq 0$ and the limit you're looking for is $1$</p>
|
2,878,777 | <p>Usually mathematicians consider isomorphic fields as equal fields. That is, if the $(A,+,\cdot)$ is isomorphic to $(B,\oplus,\odot)$, then I can consider those fields as equals. Thinking about it, I thought about the following interpretation:</p>
<p>Let $A$ and $B$ be two sets. I think we can interpret that $A$ and... | Mohammad Riazi-Kermani | 514,496 | <p>Two sets are considered equal iff every element of the first set is an element of the second set and every element of the second set is an element of the first set.</p>
<p>Sets of the same Cardinality are not the same sets but they enjoy having a one to one correspondence between them.</p>
<p>You have answered you... |
3,174,982 | <p>I tried for few primes but they do not satisfy Eisenstein criterion.Also is there any approach other than brute force with the help of which we can find that prime.</p>
| Community | -1 | <p>You can use the reducite criterium.</p>
<p>I.e., consider the polynomial over the field <span class="math-container">$\mathbb{F}_2$</span>.I.e., we get <span class="math-container">$X^3 + X^2 +1$</span>. If this reduced polynomial is irreducible over <span class="math-container">$\mathbb{F}_2$</span>, it is also ir... |
178,302 | <p>Assume that $H$ is a separable Hilbert space.
Is there a polynomial $p(z)\in \mathbb{C}[x]$ with $deg(p)>1$ with the following property?:</p>
<p>Every densely defined operator $A:D(A)\to D(A),\;D(A)\subset H$ with $p(A)=0$ is necessarily a bounded operator on $H$.</p>
<p>That is the polynomial-operator e... | Robert Israel | 13,650 | <p>You mean an infinite-dimensional separable Hilbert space. The answer is no.</p>
<p>Suppose $p(z)$ has distinct roots $\alpha_1, \alpha_2$. Define a sequence $x_1, x_2, \ldots$ in the unit sphere of $H$ such that </p>
<ol>
<li>$x_1,\ldots, x_n$ are linearly independent for all $n$.</li>
<li>$\|x_i - x_{i+1}\| \to ... |
124,280 | <p>Show that the sequence ($x_n$) defined by $$x_1=1\quad \text{and}\quad x_{n+1}=\frac{1}{x_n+3} \quad (n=1,2,\ldots)$$ converges and determine its limit ? </p>
<p>I try to show ($x_n$) is a Cauchy sequence or ($x_n$) is decreasing (or increasing) and bounded sequence but I fail every step of all.</p>
| David Mitra | 18,986 | <p>You can easily show that $(x_n)$ is bounded below by 0 and bounded above by 1.</p>
<p>You can then show (by induction e.g.) that $(x_{2n})$ is decreasing and that $(x_{2n+1})$ is increasing. Then you can argue that the sequence $(x_{2n})$ converges to some number $L$ and that the sequence $(x_{2n+1})$ converge... |
912,426 | <p>A bag contains six chips, numbered 1 through 6. If two chips are chosen at random without replacement and the values on those two chips are multiplied, what is the probability that this product will be greater than 20?</p>
<p>I tried to solve by counting the total possibilities (36) and solving for 6 choices that w... | Harald Hanche-Olsen | 23,290 | <p>Take logarithms:
$$\ln\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)}=\sum_{k=1}^n\ln\frac{2k-1}{2k}=\sum_{k=1}^n\ln\Bigl(1-\frac1{2k}\Bigr)$$
and use $$\ln(1-x)=-x+O(x^2)$$ together with the divergence of the harmonic series.</p>
|
3,684,799 | <p>When the theory of groups is built up from its axioms, it is often necessary to establish very simple results such as</p>
<p><span class="math-container">$ax = xa \Longrightarrow a^{-1}x = xa^{-1}. \tag 1$</span></p>
<p>Thus we ask how the title question might be proved.</p>
| Disintegrating By Parts | 112,478 | <p>If <span class="math-container">$T$</span> is self-adjoint and <span class="math-container">$T^2x=0$</span>, then <span class="math-container">$Tx=0$</span> because
<span class="math-container">$$
\|Tx\|^2=\langle T^2 x,x\rangle = 0.
$$</span>
Therefore, if <span class="math-container">$T^k=0$</span> for s... |
38,193 | <p>For simplicity, let me pick a particular instance of Gödel's Second Incompleteness
Theorem:</p>
<p>ZFC (Zermelo-Fraenkel Set Theory plus the Axiom of Choice, the usual foundation of mathematics) does not prove Con(ZFC), where Con(ZFC) is a formula that expresses that
ZFC is consistent.</p>
<p>(Here ZFC can be replac... | Henry Towsner | 8,991 | <p>While it's not directly a philosophical benefit, the Second Incompleteness Theorem is quite useful for giving concrete unprovability results: if we want to prove that theory T does not prove theorem X, it suffices to show that X implies the consistency of T. For instance, Harvey Friedman has a number of results sho... |
38,193 | <p>For simplicity, let me pick a particular instance of Gödel's Second Incompleteness
Theorem:</p>
<p>ZFC (Zermelo-Fraenkel Set Theory plus the Axiom of Choice, the usual foundation of mathematics) does not prove Con(ZFC), where Con(ZFC) is a formula that expresses that
ZFC is consistent.</p>
<p>(Here ZFC can be replac... | abo | 20,716 | <p>It is an open question whether what I have called Hilbert's ultrafinitist program is possible, that is whether a natural base theory can prove the consistency of natural stronger theories. Please see </p>
<p><a href="https://mathoverflow.net/questions/120258/is-an-ultrafinitist-hilberts-program-doomed">Is an ultra... |
3,768,333 | <blockquote>
<p>Let <span class="math-container">$f: [a,b] \to R$</span> be a differentiable function of one variable such that <span class="math-container">$|f'(x)| \le 1$</span> for all <span class="math-container">$x\in [a,b]$</span>. Prove that <span class="math-container">$f$</span> is a contraction. (Hint: use MV... | M Kupperman | 593,668 | <p>You're almost there! A (weak) contraction is defined as a function <span class="math-container">$f: A \to \mathbb{R}$</span> such that <span class="math-container">$$|f(x) - f(y)| \leq k | x - y| ~\forall x,y \in A,\:\:\: 0 \leq k \leq 1$$</span> (in your case <span class="math-container">$A = [a,b]$</span>.)</p>
<... |
4,117,409 | <blockquote>
<p>Prove or disprove: if for every <span class="math-container">$n\in\Bbb{N}, |a_{n+1}-a_n|<\frac{1}{n^2}$</span> then <span class="math-container">$a_n$</span> converges.</p>
</blockquote>
<p>I think this is true, and tried using Cauchy's theorem - I take some <span class="math-container">$\varepsilon ... | José Carlos Santos | 446,262 | <p>Take <span class="math-container">$\varepsilon>0$</span>. Since the series <span class="math-container">$\sum_{n=1}^\infty\frac1{n^2}$</span> converges, there is some <span class="math-container">$N\in\Bbb N$</span> such that <span class="math-container">$\sum_{n=N}^\infty\frac1{n^2}<\varepsilon$</span> . So, ... |
210,735 | <p>The Cantor set is closed, so its complement is open. So the complement can be written as a countable union of disjoint open intervals. Why can we not just enumerate all endpoints of the countably many intervals, and conclude the Cantor set is countable?</p>
| Carl Mummert | 630 | <p>The reason that does not work is that there are some points in the Cantor set which are not the endpoint of any interval that is removed during the construction of the Cantor set. In the proof that is suggested in the question, it would be necessary to show that every point in the set is one of these endpoints, but ... |
210,735 | <p>The Cantor set is closed, so its complement is open. So the complement can be written as a countable union of disjoint open intervals. Why can we not just enumerate all endpoints of the countably many intervals, and conclude the Cantor set is countable?</p>
| Adam Rubinson | 29,156 | <p>Just because the open intervals approach a number, doesn't mean the number is the endpoint of one of the intervals. Consider:</p>
<p><span class="math-container">$$D_1= \bigcup\limits_{\substack{n~\in~\mathbb{N},~n~\text{odd} \\}} \left(\frac{1}{n+1}, \frac{1}{n}\right). $$</span></p>
<p>This looks like:</p>
<p><a ... |
4,249,794 | <p>i have to determine whether this graph is bipartite or not:</p>
<p><img src="https://i.stack.imgur.com/AjxQzl.png" alt="" /></p>
<p>I have found an answer but i am not sure about it. If we divide the vertices set into <span class="math-container">$\{a,d,c,h\}$</span> and <span class="math-container">$\{b,f,e,g\}$</s... | AKSHAY V | 963,523 | <p>"The given graph is bipartite by the fact that there are no odd cycles in it". If you just want to prove it theoretically, this statement should be enough.</p>
<p>If you want to find out how it is bipartite then your sets of {a,d,c,h} and {b,f,e,g} are correct as an example.</p>
|
1,612,808 | <p>Suppose that $X$ is a finite $G$-set. A group $G$ is of prime power if $|G|=p^n$ for $p$ prime.</p>
<p>The fixed point set $X_G=\{x\in X : gx=x$ $\forall g\in G\}$.</p>
<p>I'm asked to prove that $|X|=|X_G|$ (mod $p$), but I'm unsure of how I should start.</p>
| Dor Marciano | 78,839 | <p>Remember for each $x \in X$, denoting $G_x = \{g\in G : gx=x\}$ and $O_x = \{y\in X | \exists g\in G : gx = y\}$, a well known theorem in Group Theory claims that $|O_x| = \frac{|G|}{|G_x|}$.</p>
<p>Notice that $|O_x|=1$ iff $x\in X_G$. If $|O_x|>1$ then, since $|O_x| \mid |G|$, we have that $p \mid |O_x|$. Now,... |
30,220 | <p>Jeremy Avigad and Erich Reck claim that one factor leading to abstract mathematics in the late 19th century (as opposed to concrete mathematics or hard analysis) was <em>the use of more abstract notions to obtain the same results with fewer calculations.</em></p>
<p>Let me quote them from their remarkable historical... | O.R. | 5,506 | <p>All I see here are calculations. It only changed the nature of the object which you calculate with and its relation to the the final goal. For this reason I still can not make clear sense of the question.</p>
|
1,818,976 | <p>Let there be many numbers $a_1,a_2,a_3,\dots,a_n$.</p>
<p>I want to find the first digit of their product, i.e. of $A=a_1\times a_2\times a_3\times a_4\times \dots\times a_n$.</p>
<p>These numbers are huge and multiplying all of them exceeds the time limit.</p>
<p>Is there any shortcut to find the most significan... | quid | 85,306 | <p>Your approach works fine, just apply it twice.</p>
<p>The real part of the sequence is the sequence $(\cos (2\pi r n))_n$ this is clearly bounded and thus has a convergent subsequence $(\cos (2\pi r n_k))_k$. </p>
<p>The imaginary part of the respective subsequence of the original sequence is $\sin (2\pi r n_k)... |
4,275,780 | <blockquote>
<p>If <span class="math-container">$0<a<b$</span> and <span class="math-container">$0<c<d$</span> then <span class="math-container">$\frac{c+a}{d+a} <\frac{c+b}{d+b}.$</span></p>
</blockquote>
<p>I get to <span class="math-container">$$d+a<d+b \Longrightarrow \frac{1}{d+b} < \frac{1}{d... | Ankit Saha | 876,128 | <p><span class="math-container">$$ \dfrac{c+a}{d+a} =\dfrac{d+a+c-d}{d+a} = 1 + \dfrac{c-d}{d+a}$$</span>
<span class="math-container">$$ \dfrac{c+b}{d+b} =\dfrac{d+b+c-d}{d+b} = 1 + \dfrac{c-d}{d+b}$$</span>
You have already proved that
<span class="math-container">$$ \frac{1}{d+b} < \frac{1}{d+a}$$</span>
As <sp... |
4,549,340 | <p>I have heard people say that the flight time from Fort Lauderdale to Seattle is the longest possible flight time within the continental United States. However, upon further consideration, I realized that the curvature of the Earth may cause the visible distance on a map to decrease when traveling north (the circumfe... | Bruno B | 1,104,384 | <p>(Important edit: Damian is right, the induction itself is off in your proof. However, I still deem it informative to let my answer stay, for clarity about the symmetry argument that would've let you finish, were your statement true!)</p>
<p>Let's do as if you'd proven that, for <span class="math-container">$m > n... |
1,653,106 | <p>I was following a calculus tutorial that factored the equation $x^4-16$ into $(x^2 +4) (x+2)(x-2)$.</p>
<p>Why is the factorization of $x^4-16 = (x^2 + 4)(x+2)(x-2)$ rather than $(x^2 - 4)(x^2 +4)$? </p>
| 3SAT | 203,577 | <p>Notice that $x^4-16=(x^2)^2-4^2\\
=(x^2-4)(x^2+4)\\
=(x+2)(x-2)(x^2+4)\\
=(x-2)(x+2)(x+2\color{red}i)(x-2\color{red}i)$</p>
|
4,205,906 | <p>Since X and Y are independent and uniform I have the joint density function f(x,y)=1 but Im not sure where to go from there. I keep getting that the answer is f(z)=1 but this doesnt make sense since the range of z is from 0 to infinity.</p>
<p>So if I make the substitution Z=Y/X and W=X, I get Y=ZW and X=W. The 4 ... | MSIS | 678,294 | <p>The <span class="math-container">$p$</span>-value is the probability of obtaining the value of the Statistic, or, as pointed out below, a value as extreme as the one you obtained that you obtained while assuming your <span class="math-container">$H_0$</span>; Null Hypothesis is correct.</p>
<p>In Inferential Statist... |
330,775 | <p>Let $G$ be a Lie Group, how to prove that the tangent Space of $\operatorname{Aut}(T_e G)$ at identity is $\operatorname{End}(T_e G)$? Thanks.</p>
| Martin Brandenburg | 1,650 | <p>If $V$ is a finite-dimensional vector space, considered as a manifold, then the tangent space is $V$ at any point. And $\mathrm{Aut}$ is an open subspace of the vector space $\mathrm{End}$. This doesn't have anything to do with Lie groups.</p>
|
330,775 | <p>Let $G$ be a Lie Group, how to prove that the tangent Space of $\operatorname{Aut}(T_e G)$ at identity is $\operatorname{End}(T_e G)$? Thanks.</p>
| Julien | 38,053 | <p>Assume $E$ is a Banach space. Let $B(E)$ be the Banach algebra of bounded linear operators on $E$, equipped with the induced operator norm. You can call it the algebra of endomorphisms if you prefer. Then denote $GL(E)$ the group of invertible elements in $B(E)$, that is the group of automorphisms.</p>
<p>Fix $T_0$... |
322,858 | <p>Let <span class="math-container">$G$</span> be a split semisimple real Lie group in characteristic zero, and let <span class="math-container">$B=TU$</span> be a Borel subgroup with unipotent radical <span class="math-container">$U$</span> and Levi <span class="math-container">$T$</span>. Fix an ordering on the root... | Andrei Smolensky | 5,018 | <p>"Simple groups of Lie type" by R. W. Carter, a table after Section 12.4. But there only the values of <span class="math-container">$C_{\alpha\beta11}$</span> are listed. An explicit form of commutator formulas inside <span class="math-container">$U^+$</span> is given in Table IV of "Chevalley groups over commutative... |
2,579,156 | <p>I found the solution of series on Wolfram Alpha
<a href="http://www.wolframalpha.com/input/?i=sum+1%2F(2k%2B1)%2F(2k%2B2)+from+1+to+n" rel="nofollow noreferrer">http://www.wolframalpha.com/input/?i=sum+1%2F(2k%2B1)%2F(2k%2B2)+from+1+to+n</a></p>
<p>$ \sum\limits_{k=1}^{n} \left(\frac{1}{2k+1} - \frac{1}{2k+2}\righ... | Claude Leibovici | 82,404 | <p>If you use polygamma functions
$$S_1=\sum\limits_{k=1}^{n} \frac{1}{2k+1}=\frac{1}{2} \left(\psi ^{(0)}\left(n+\frac{3}{2}\right)-\psi
^{(0)}\left(\frac{3}{2}\right)\right)$$
$$S_2=\sum\limits_{k=1}^{n} \frac{1}{2k+2}=\frac{1}2\sum\limits_{k=1}^{n} \frac{1}{k+1}=\frac{1}{2} (\psi ^{(0)}(n+2)+\gamma -1)$$ making
... |
424,404 | <p>I want to create an algorithm/calculation that helps me figure out if the price on a used vehicle is high or low.</p>
<p>My thoughts are that to calculate this i need a critical mass of previous vehicle sales with the same characteristics; manufacturer, model, year and trim. Having this data I must figure out each ... | Nameless | 68,482 | <p>I would also suggest running a regression. With ordinary least squares regression, you explain the observed price (of past sales) as linear combination of car characteristics. For example, an equation you estimate could be (for a particular model)
$$price_i=\beta_0+\beta_1 LowMiles_i+\beta_2 MidMiles_i+\beta_3 HighM... |
113,854 | <p>I'm importing a *.pdb file containing a single protein. <em>Mathematica</em> automatically produces a plot of the protein.</p>
<p>I want to specify the color of each residue independently, in this plot. Is this possible?</p>
<p>Additionally, I would like to change the type of plot to "cartoon". How can I do this?<... | Jason B. | 9,490 | <p>Okay, so this is what it looks like with standard coloring of the residues:</p>
<pre><code>Import["ExampleData/1PPT.pdb", ColorFunction -> "Residue"]
</code></pre>
<p><a href="https://i.stack.imgur.com/upk6k.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/upk6k.png" alt="enter image descripti... |
3,527,350 | <p>I need to calculate the spectrum of the operator <span class="math-container">$T$</span> for <span class="math-container">$f \in L^2([0,1])$</span> defined by: </p>
<p><span class="math-container">\begin{equation}
(Tf)(x) = \int_0^1 (x+y)f(y)dy.
\end{equation}</span></p>
<p>I know that <span class="math-container"... | operatorerror | 210,391 | <p>To continue on your path, note that <em>once you have justified</em> that <span class="math-container">$f$</span> must in fact be smooth to be an eigenvector for nonzero eigenvalue <span class="math-container">$\lambda$</span>, you can differentiate twice as you did and note that
<span class="math-container">$$
\la... |
2,807,356 | <blockquote>
<p><strong>If $z_1,z_2$ are two complex numbers such that $\vert
z_1+z_2\vert=\vert z_1\vert+\vert z_2\vert$,then it is necessary that</strong> </p>
<p>$1)$$z_1=z_2$</p>
<p>$2)$$z_2=0$</p>
<p>$3)$$z_1=\lambda z_2$for some real number $\lambda.$</p>
<p>$4)$$z_1z_2=0$ or $z_1=\lambda z... | trancelocation | 467,003 | <p>$$\vert z_1+z_2\vert=\vert z_1\vert+\vert z_2\vert \Rightarrow \vert z_1+z_2\vert^2= (\vert z_1\vert+\vert z_2\vert)^2 \Rightarrow Re(z_1\bar z_2) = |z_1||z_2|$$
Now, note that $Re(z_1\bar z_2)$ is a real 2-dimensional scalar product
$$Re(z_1\bar z_2) = x_1x_2+y_1y_2 \mbox{ where } z_1 = x_1+iy_1, \, z_2 = x_2+iy_2$... |
2,476,717 | <p>Let $f: \mathbb R^n \rightarrow \mathbb R^n$ with arbitrary norm $\|\cdot\|$. It exists a $x_0 \in \mathbb R^n$ and a number $r \gt 0 $ with </p>
<p>$(1)$ on $B_r(x_0)=$ {$x\in \mathbb R^n: \|x-x_0\| \leq r$} $f$ is a contraction with Lipschitz constant L</p>
<p>$(2)$ it applies $\|f(x_0)-x_0\| \le (1-L)r$</p>
<... | Guy Fsone | 385,707 | <p>Hint: Prove by induction
Assume that $x_k\in B_r(x_0)$ then you have </p>
<p>$$ \|x_{k+1} -x_0\| = \|f(x_k) -f(x_{0}) +f(x_{0}) -x_0\| \\\le \|f(x_k) -f(x_{0})\|+\|f(x_{0}) -x_0\| \\ \le L
\|x_k -x_{0}\|+\|f(x_{0}) -x_0\| $$</p>
<p>That is </p>
<p>$$ \|x_{k+1} -x_0\| \le L \|x_k -x_{0}\|+\|f(x_{0}) -x_0\| ... |
2,662,554 | <p>I have to use Proof by contradiction to show what if $n^2 - 2n + 7$ is even then $n + 1$ is even. </p>
<p>Assume $n^2 - 2n + 7$ is even then $n + 1$ is odd. By definition of odd integers, we have $n = 2k+1$. </p>
<p>What I have done so far:</p>
<p>\begin{align}
& n + 1 = (2k+1)^2 - 2(2k+1) + 7 \\
\implies &am... | Bram28 | 256,001 | <p>Since you are doing a proof by contradiction, your start is: Assume $n+1$ is odd. But if $n+1$ is odd, then $n$ is even, and so yo should plug in $n=2k$, rather than what you did, which was plugging in $2k+1$</p>
|
2,662,554 | <p>I have to use Proof by contradiction to show what if $n^2 - 2n + 7$ is even then $n + 1$ is even. </p>
<p>Assume $n^2 - 2n + 7$ is even then $n + 1$ is odd. By definition of odd integers, we have $n = 2k+1$. </p>
<p>What I have done so far:</p>
<p>\begin{align}
& n + 1 = (2k+1)^2 - 2(2k+1) + 7 \\
\implies &am... | marty cohen | 13,079 | <p><span class="math-container">$n^2-2n+7
=n^2+2n+1-(4n+6)
=(n+1)^2-2(2n+3)
$</span>.</p>
<p>If this is even then,
since <span class="math-container">$2(2n+3)$</span> is even,
their sum is even,
so <span class="math-container">$(n+1)^2$</span> is even
so <span class="math-container">$n+1$</span> is even.</p>
|
3,806,122 | <p>I tried using Chinese remainder theorem but I kept getting 19 instead of 9.</p>
<p>Here are my steps</p>
<p><span class="math-container">$$
\begin{split}
M &= 88 = 8 \times 11 \\
x_1 &= 123^{456}\equiv 2^{456} \equiv 2^{6} \equiv 64 \equiv 9 \pmod{11} \\
y_1 &= 9^{-1} \equiv 9^9 \equiv (-2)^9 \equiv -512... | Bill Dubuque | 242 | <p>You used an incorrect CRT formula. It should be: for coprime <span class="math-container">$\,m,n,\,$</span> and <span class="math-container">$\,c^{-1}_{\ n}:= c^{-1}\bmod n$</span></p>
<p><span class="math-container">$\qquad\begin{align} &x\equiv a\!\!\pmod{\!m}\\ &x\equiv b\!\!\pmod{\!n}\end{align}\iff x\,\... |
713,104 | <p>Are there any combinatorial games whose order (in the usual addition of combinatorial games) is finite but neither $1$ nor $2$?</p>
<p>Finding examples of games of order $2$ is easy (for example any impartial game), but I have not been able to think up an example with finite order where the order did not come from ... | Adola | 406,858 | <p>The hockey stick identity states that
<span class="math-container">$$\sum_{k=r}^{n} \binom{k}{r}=\binom{n+1}{r+1}$$</span></p>
<p>Note that <span class="math-container">$$1+2+...+n = \frac{n(n+1)}{2}=\binom{n+1}{2}.$$</span></p>
<p>Hence</p>
<p><span class="math-container">$$f(n)=\binom{2}{2}+\binom{3}{2}+...+\binom... |
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