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 |
|---|---|---|---|---|
733,280 | <p>I cannot understand why $\log_{49}(\sqrt{ 7})= \frac{1}{4}$. If I take the $4$th root of $49$, I don't get $7$.</p>
<p>What I am not comprehending? </p>
| Sujaan Kunalan | 77,862 | <p>$\log_{49}{\sqrt{7}}=\dfrac{1}{4}$ means that $49^{\frac{1}{4}}={\sqrt{7}}$</p>
<p>In other words, when you take the fourth root of $49$, you should get $\sqrt{7}$, not $7$.</p>
<p>Note that $\Large49^{\frac{1}{4}}=(7^2)^{\frac{1}{4}}=7^\frac{2}{4}=7^{\frac{1}{2}}=\sqrt{7}$</p>
|
2,314,327 | <p>I have a quick question here.</p>
<p>For an exercise, I was asked to factor:</p>
<p>$$11x^2 + 14x - 2685 = 0$$</p>
<p>How do I figure this out quickly without staring at it forever? Is there a quicker mathematical way than guessing number combinations, or do I have to guess until I find the right combination of n... | Janitha357 | 393,345 | <p>Here's one suggestion: Look at $11$ and $-2685$. Factorize $2685$ into primes to get $3.5.179$. Then check the following.</p>
<p>$11.(-3)+(895), 11.(-15)+179, 11.(3)+(-895), 11.(15)+(-179)$ and check which yields $14$. The one that yields $14$ is $11.(-15)+179$. So you get $(11x+179)(x-15)$. </p>
<p>I don't know i... |
18,686 | <p>Let us define the following "dimension" of a Borel subet $B \subset \mathbb{R}^k$:</p>
<p>$\dim(B) = \min\{n \in \mathbb{N}: \exists K \subset \mathbb{R}^n, ~{\rm s.t.} ~ B \sim K\}$,</p>
<p>where $\sim$ denotes "homeomorphic to". Obviously, $0 \leq \dim(B) \leq k$.</p>
<p>I have three questions: Given a $B \sub... | Johannes Hahn | 3,041 | <p>As for 1.) $"dim"(\mathbb{Z}^k)=1$ for all $k\in\mathbb{N}$, because all $\mathbb{Z}^k$ are discrete countable and therefore homeomorphic to each other.</p>
|
1,305,257 | <p>I do not understand how to use the following information: If $f$ is entire, then </p>
<p>$$\lim _{|z| \rightarrow \infty} \frac{f(z)}{z^2}=2i.$$</p>
<p>So if $f$ is entire, it has a power series around $z_0=0$, so $f(z)=\Sigma_{n=0}^\infty a_nz^n$, and then we get </p>
<p>$$\lim _{|z| \rightarrow \infty} \frac{\S... | Hagen von Eitzen | 39,174 | <p>The claim itself is true more generally for any polynomial of even degree with positive leading coefficient.</p>
<p>We do not need that $g(x)<0$ for some $x$ and in fact such $x$ may not exist. What we need is that $\lim_{x\to\pm\infty}g(x)=1$ implies that there exists $M\in \mathbb R$ such that $|x|>M$ impli... |
115,081 | <p>I hope here is the best place to ask this, I will begin my master degree very soon, I've already attended the regular undergraduate courses included Real Analysis, Analysis on manifolds, Abstract Algebra, Field Theory, point-set topology, Algebraic Topology, etc...
I like very much algebraic topology and I found it ... | Alexandre Eremenko | 25,510 | <p>For your first research problem, I recommend that you find an adviser in your department.
If there is no algebraic topologist in your department, find some other adviser, and
ask to suggest an interesting problem.</p>
<p>It is very unlikely that, as a master student, you will be able to find and solve
a reasonable ... |
3,287,424 | <p>I have a function <span class="math-container">$$f(z)=\begin{cases}
e^{-z^{-4}} & z\neq0 \\
0 & z=0
\end{cases}$$</span></p>
<p>I have to show cauchy riemann equation is satisfied everywhere. I have shown that it isn't differentiable at <span class="math-container">$z=0$</span>. </p>
<p>Usually I will hav... | md2perpe | 168,433 | <p>A function <span class="math-container">$f$</span> is holomorphic at <span class="math-container">$z$</span> if <span class="math-container">$\lim_{|h| \to 0} \frac{f(z+h)-f(z)}{h}$</span> exists. This equation is more fundamental than the Cauchy-Riemann equations, which are derived from this.</p>
<p>In the actual ... |
307,701 | <p>Show that if $G$ is a finite group with identity $e$ and with an even number of elements, then there is an $a \neq e$ in $G$, such that $a \cdot a = e$.</p>
<p>I read the solutions here <a href="http://noether.uoregon.edu/~tingey/fall02/444/hw2.pdf" rel="nofollow">http://noether.uoregon.edu/~tingey/fall02/444/hw2.p... | Barbara Osofsky | 59,437 | <p>Consider the relation on $G$ given by $g\equiv h\iff g\in\{\ h\ , h^{-1}\ \}$. It is easy to see that this is symmetric, reflexive, and transitive, and so an equivalence relation with equivalence classes $\{\ h\ ,\ h^{-1}\ \}$. The equivalence class of the identity $e$ of $G$ is $\{\ e\ \}$ containing only one eleme... |
28,568 | <p>Recently, I answered to this problem:</p>
<blockquote>
<p>Given <span class="math-container">$a<b\in \mathbb{R}$</span>, find explicitly a bijection <span class="math-container">$f(x)$</span> from
<span class="math-container">$]a,b[$</span> to <span class="math-container">$[a,b]$</span>.</p>
</blockquote>
<... | Asaf Karagila | 622 | <p>It seems that your construction is fine, however coarse and crude. We usually give this question in the introductory course of set theory, the solution is quite elegant too.</p>
<p>Firstly, it is very clear that this function cannot be continuous. Consider a sequence approaching the ends of the interval, the functi... |
28,568 | <p>Recently, I answered to this problem:</p>
<blockquote>
<p>Given <span class="math-container">$a<b\in \mathbb{R}$</span>, find explicitly a bijection <span class="math-container">$f(x)$</span> from
<span class="math-container">$]a,b[$</span> to <span class="math-container">$[a,b]$</span>.</p>
</blockquote>
<... | xen | 8,229 | <p>Here you have simpler construction.</p>
<p>Let $(a_n)$ be the sequence in $(a,b)$ defined by $a_n = a + \frac{b-a}{2^n}$. Then let $f\colon [a,b] \to (a,b)$ be given by
$$
f(x) = \begin{cases}
a_1, & x = a,\\
a_2, & x = b,\\
a_{n+2}, & x = a_n, n = 1,2,\dots\\
x, & x \in [a,b] \setminus \{a,b,a... |
143,274 | <p>I am trying to find the derivative of $\sqrt{9-x}$ using the definition of a derivative </p>
<p>$$\lim_{h\to 0} \frac {f(a+h)-f(a)}{h} $$</p>
<p>$$\lim_{h\to 0} \frac {\sqrt{9-(a+h)}-\sqrt{9-a}}{h} $$</p>
<p>So to simplify I multiply by the conjugate</p>
<p>$$\lim_{h\to0} \frac {\sqrt{9-(a+h)}-\sqrt{9-a}}{h}\cdo... | Argon | 27,624 | <p>Everything you have done is right except for the last step.</p>
<p>$$\begin{align}
&\lim_{h\to0} \frac {\sqrt{9-(a+h)}-\sqrt{9-a}}{h}\cdot \frac{ \sqrt{9-(a+h)}+ \sqrt{9-a}}{\sqrt{9-(a+h)}+\sqrt{9-a}}=\\
&\lim_{h\to0} \frac{9-(a+h)-(9-a)}{h(\sqrt{9-(a+h)}+\sqrt{9-a})}=\\
&\lim_{h\to0} \frac{9-a-h-9+a}{h... |
4,373,055 | <p>We define a bound vector to be a quantity with a defined starting point, magnitude and direction. A free vector has no defined starting point, just magnitude and direction.</p>
<p>So what is a position vector (of a point)? It is defined relative to something else (Origin), so it has a starting point, size and direct... | ryang | 21,813 | <p>I've not heard of a bound versus free vector, but based on your given definition,</p>
<p>since a position vector <em>does</em> have a <em>defined</em> starting point, magnitude and direction,</p>
<p>then it <em>is</em> a bound vector.</p>
<p>P.S. While its starting point is defined and fixed, its magnitude and direc... |
288,051 | <p>In enumerative combinatorics, a <i>bijective proof</i> that $|A_n| = |B_n|$ (where $A_n$ and $B_n$ are finite sets of combinatorial objects of size $n$) is a proof that constructs an explicit bijection between $A_n$ and $B_n$.
Bijective proofs are often prized because of their beauty and because of the insight that ... | JimN | 62,043 | <p>Does the OEIS count?
Three of your papers are listed in <a href="https://oeis.org/wiki/Works_Citing_OEIS" rel="nofollow noreferrer">https://oeis.org/wiki/Works_Citing_OEIS</a> as having used the help of either OEIS or Superseeker , I presume to help find a (potential?) bijection.</p>
|
1,599,467 | <p>Here $f$ is a non-zero linear functional on a vector space $X$. I can show this true for one direction, </p>
<blockquote>
<p>Let $x_1, x_2 \in x + N(f)$</p>
<p>$\implies x_1 = x + y_1, \quad x_2 = x + y_2$, where $y_1, y_2 \in N(f).$</p>
<p>Then $f(x_1) = f(x) + f(y_1) = f(x) = f(x) + f(y_2) = f(x_2)$.... | Jendrik Stelzner | 300,783 | <p>As symplectomorphic already pointed out this is just
\begin{align*}
f(x) = f(y)
&\iff f(x) - f(y) = 0
\iff f(x-y) = 0 \\
&\iff x-y \in N(f)
\iff x + N(f) = y + N(f).
\end{align*}</p>
<p>Notice however that this is really just a special case of the more general fact that for every linear map $f \colon V... |
73,629 | <p>I want to use <code>Listplot</code> with <code>Tooltip</code>that displays <code>Position</code>of the element I'm hovering over.</p>
<pre><code>data={{0,1},{1,3},{2,2}};
ListPlot[Tooltip[data]]
</code></pre>
<p>This is displaying the value of the element. Can I use the <code>Position</code>function in the tooltip... | akater | 1,859 | <p><code>Map</code> and <code>MapIndexed</code> solutions are formally valid but <code>Thread</code> is often cleaner. (Poor guy is also underused, I believe.)</p>
<pre><code>data = RandomInteger[10, {10, 2}];
ListPlot@Thread@Tooltip[#, Range@Length@#]&@data
</code></pre>
|
4,358 | <p>I've been reading a bit about how the set of bounds changes for a set depending on what superset one works with. I considered the sets $S\subseteq T\subseteq\mathbb{Q}$ and worked out a few contrived examples:</p>
<p>If $S=T=$ {$x\in\mathbb{Q}\ | \ x^2\lt 2$}, so here $S$ is not bounded above in $T$, but it is boun... | Arturo Magidin | 742 | <p>Yes, it is possible for $\sup_T(S)$ to exist, but $\sup_\mathbb{Q}(S)$ not to exist. Yes, it's possible for both to exist and be distinct.</p>
<p>Here is an example of the latter. Take $S=\{x\in\mathbb{Q}|0\leq x\lt 1\}$. Let $T=S\cup{{2}}$. Then $S$ is bounded above in $T$, and has a supremum, namely $2$. Indeed, ... |
95,314 | <p>To evaluate this type of limits, how can I do, considering $f$ differentiable, and $ f (x_0)> 0 $</p>
<p>$$\lim_{x\to x_0} \biggl(\frac{f(x)}{f(x_0)}\biggr)^{\frac{1}{\ln x -\ln x_0 }},\quad\quad x_0>0,$$</p>
<p>$$\lim_{x\to x_0} \frac{x_0^n f(x)-x^n f(x_0)}{x-x_0},\quad\quad n\in\mathbb{N}.$$</p>
| Did | 6,179 | <p>Hint: write $x=x_0+h$ with $h\to0$ and expand each term up to order $h$.</p>
<p>For example, $\log(x)-\log(x_0)=\log(1+h/x_0)=h/x_0$ and $f(x)=f(x_0)+hf'(x_0)$ hence $f(x)/f(x_0)=1+hf'(x_0)/f(x_0)$, hence...</p>
|
1,336,344 | <p>Given a matrix A of a strongly $k$ regular graph G(srg($n,k,\lambda,\mu$);$\lambda ,\mu >0;k>3$). The matrix A can be divided into 4 sub matrices based on adjacency of vertex $x \in G$.
$A_x$ is the symmetric matrix of the graph $(G-x)$, where $C$ is the symmetric matrix of the graph created by vertices of ... | vzn | 42,153 | <p>this will not be a <em>direct</em> answer at what does not really seem a <em>direct</em> question, which in its apparently big scope/ ambition pushes outside the bounds of SE questions but is nevertheless aimed to be a scientific and guiding response. you seem to be studying <a href="https://en.wikipedia.org/wiki/Gr... |
688,742 | <p>Given $P\colon\mathbb{R} \to \mathbb{R}$ , $P$ is injective (one to one) polynomial function i need to formally prove that $P$ is onto $\mathbb{R}$</p>
<p>my strategy so far .......
polynomial function is continuous and since it one-to-one function it must be strictly monotonic and now i have no idea what to do ..... | Federico Poloni | 65,548 | <p>I can guess that your doubt comes from a first step that you do in your mind that goes like this: "$\lim_{x\rightarrow \infty}{e^{-x}} = 0$, so I can replace $e^{-x}$ with $0$ in the text and I remain with
$$
\lim_{x\to ∞} \frac{11-0}{7},
$$
then what happens to the limit in this last equation?"</p>
<p>This reflect... |
69,590 | <p>Consider the following code.</p>
<pre><code>f[a_,b_]:=x
x=a+b;
f[1,2]
(* a + b *)
</code></pre>
<p>From a certain viewpoint, one might expect it to return <code>3</code> instead of <code>a + b</code>: the symbols <code>a</code> and <code>b</code> are defined during the evaluation of <code>f</code> and <code>a+b</c... | Mr.Wizard | 121 | <h2>Analysis</h2>
<p>In <em>Mathematica</em> when a definition is <em>applied</em> the expressions (arguments) that match pattern objects on the left-hand-side (LHS) are substituted into the matching right-hand-side (RHS) names before it is evaluated. This is separate from the evaluation that does or does not take pl... |
1,487,966 | <p>I have been looking at stereographic projections in books, online but they all seem...I don't know how else to put this, but very pedantic yet skipping the details of calculations.</p>
<p>Say, I have a problem here which asks;</p>
<blockquote>
<p>Let <span class="math-container">$n \geq 1$</span> and put <span class... | Bernard | 202,857 | <p>For part i) you have to find the point of the line (Px) that lies in the hyperplane $x_0=0$, i. e. you have to solve for
$$\lambda +(1-\lambda)x_0=0,\tag{1}$$
since the straight line $(Px)$ has a parametic representation:
$$\lambda(1 ,0,\dots,0)+(1-\lambda)(x_0,x_1,\dots,x_n)=\bigl(\lambda +(1-\lambda)x_0, (1-\lambd... |
3,349,630 | <p>If a,b,c are positive real numbers,prove that
<span class="math-container">$$ \frac{a}{b+2c} + \frac{b}{c+2a} + \frac{c}{a+2b} \ge 1 $$</span>
I tried solving and i have no idea how to proceed I mechanically simplified it it looks promising but im still stuck. This is from the excersice on Cauchy Schwartz Inequality... | Dr. Sonnhard Graubner | 175,066 | <p>Using Cauchy-Schwarz in Engel form we get
<span class="math-container">$$\frac{a^2}{ab+2ac}+\frac{b^2}{bc+ab}+\frac{c^2}{ac+2ac}\geq \frac{(a+b+c)^2}{3ab+3ac+3bc}\geq 1$$</span> if <span class="math-container">$$(a+b+c)^2\geq 3ab+3ac+3bc$$</span>
and this is <span class="math-container">$$a^2+b^2+c^2\geq ab+bc+ca$$<... |
190,948 | <p>So I have </p>
<pre><code>Emin[T_, d_] := If[T == 0, 0, 1/(-1 + E^(1/T)) - d/(-1 + E^(d/T))];
</code></pre>
<p>which I can solve wonderfully for </p>
<pre><code>Solve[Emin[T, 10] == 0.5, T, Reals]
{{T -> 0.910427}}
</code></pre>
<p>Now as you see <code>Emin[T_, d_]</code> is also dependent on d which I set t... | Ulrich Neumann | 53,677 | <p>With the additional information <code>d01,2,3...10</code> try</p>
<pre><code>dT = Table[{d, T /. NSolve[Emin[T, d] == 0.5, T, Reals][[1]]}, {d, 1,10}]
(*{{1, T /. {}[[1]]}, {2, T /. {}[[1]]}, {3, 1.19888}, {4,0.988141},
{5, 0.936864}, {6, 0.919994}, {7, 0.913886}, {8,0.911604}, {9, 0.910747}, {10, 0.910427}}*)
Sho... |
3,869,237 | <p>I know this is quite weird or it does not make much sense, but I was wondering, does <span class="math-container">$\int e^{dx}$</span> has any meaning or whether it makes sense at all? If it does means something, can it be integrated and what is the result?</p>
| zkutch | 775,801 | <p>I agree, that formally it make no sense, but as lunch exercise in mathematical fantasy , if we can give some sense to <span class="math-container">$(dx)^n$</span> as some measure, then we can imagine <span class="math-container">$\int e^{dx}=\int \sum \frac{(dx)^n}{n!}$</span>. Now the point is what is <span class="... |
3,421,858 | <p><span class="math-container">$\sqrt{2}$</span> is irrational using proof by contradiction.</p>
<p>say <span class="math-container">$\sqrt{2}$</span> = <span class="math-container">$\frac{a}{b}$</span> where <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are positive integers. </... | Matt Samuel | 187,867 | <p>Simply calculate <span class="math-container">$$b(\sqrt2-1)\sqrt2=b(2-\sqrt2)=2b-b\sqrt2$$</span>
which is an integer. This is actually the heart of the proof. We assumed <span class="math-container">$b$</span> was the smallest integer satisfying <span class="math-container">$b\sqrt2$</span> being an integer, but we... |
4,383,800 | <p>I can already see that the <span class="math-container">$\lim_\limits{n\to\infty}\frac{n^{n-1}}{n!e^n}$</span> converges by graphing it on Desmos, but I have no idea how to algebraically prove that with L’Hopital’s rule or induction. Where could I even start with something like this?</p>
<p>Edit: For context, I came... | Community | -1 | <p><strong>Hint:</strong></p>
<p>Taking the logarithm, you need the limit of</p>
<p><span class="math-container">$$(n-1)\log n-\sum_{k=1}^n\log n-n.$$</span></p>
<p>Then the summation can be estimated by an integral and the second term is</p>
<p><span class="math-container">$$\sim n(\log n-1).$$</span></p>
<p>A more pr... |
2,529,682 | <p>Right now I'm stuck on the following problem, since I feel like I should be using total probability, but I dont know what numbers to use as what.</p>
<p>Let's say there's a population of students. In this population:</p>
<p>30% have a bike</p>
<p>10% have a motorcycle</p>
<p>12% have a car.</p>
<p>8% have a bik... | Arthur | 15,500 | <p>The derivative is restricted to that domain precisely because the original function is. How would you evaluate the derivative of $\ln x$ at negative $x$? You can't, and therefore the derivative is only defined for positive $x$.</p>
<p>The derivative of $\ln x$ does have a very natural extension to the negative numb... |
1,921,114 | <p><img src="https://i.stack.imgur.com/D8IcM.jpg" alt="enter image description here"></p>
<p>So I solved this system without using matrices, just by (sort of) reverting to high school math instincts.
$$w+x=-5$$
$$x+y=4$$
$$y+z=1$$
$$w+z=8$$
From this, I got $w=-9,x=4,y=0$ and $z=1$. How would I convert this back to wh... | Sam Waggoner | 1,145,242 | <p>In the equations, you have four variables, <em>x1</em>, <em>x2</em>, <em>x3</em>, and <em>x4</em>. The challenge is to 1) solve for each, and 2) state them in terms of some single constant s, instead of the other variables.
The first step is to solve for each of the variables, which will result in this:</p>
<p><span... |
2,489,988 | <p>A sequence of numbers is formed from the numbers $1, 2, 3, 4, 5, 6, 7$ where all $7!$ permutations are equally likely. What is the probability that anywhere in the sequence there will be, at least, five consecutive positions in which the numbers are in increasing order?</p>
<p>I approached this problem in the follo... | CogitoErgoCogitoSum | 52,938 | <p>${}_7 C_2 = 21$, are the number of ways of choosing two out of the seven numbers. We pull them out. The remaining five retain their increasing order. So now find the number of ways you can replace those two values at either the beginning and/or end of the sequence, and multiply this result by 21. If they happen to b... |
237,142 | <p>I am having a problem with the final question of this exercise.</p>
<p>Show that $e$ is irrational (I did that). Then find the first $5$ digits in a decimal expansion of $e$ ($2.71828$).</p>
<p>Can you approximate $e$ by a rational number with error $< 10^{-1000}$ ? </p>
<p>Thank you in advance</p>
| Belgi | 21,335 | <p>I don't think we have to really know anything about $e$ to say that
we can approximate it with a rational number with an error less then
$10^{-1000}$.</p>
<p>Say $e=a_0.a_1 a_2 \ldots$</p>
<p>There is clearly a rational number $q=a_0.a_1 \ldots a_{1001}$ and
$|q-e|\leq10^{-1000}$ (note that the difference is bound... |
1,566,111 | <p>prove $(n)$ prime ideal of $\mathbb{Z}$ iff $n$ is prime or zero</p>
<hr>
<p><strong>Defintions</strong></p>
<p>Def of prime Ideal (n)
$$ ab\in (n) \implies a\in(n) \vee b\in(n) $$
Def 1] integer n is prime if $n \neq 0,\pm 1 $ and only divisors are $\pm n,\pm 1$ </p>
<p>Def 2 of n is prime] If $n\neq0,\pm1$ ... | Robert Soupe | 149,436 | <p>Another way would be to show that $\mathbb{Z}$ is a principal ideal domain or that it has unique factorization. Don't you also need the definition that a prime ideal has to be properly contained within the whole ring?</p>
<p>If $n = \pm 1$, then $\langle n \rangle = \mathbb{Z}$ and thus it can't be a prime ideal. I... |
2,130,658 | <p>How would I go about proving this mathematically? Having looked at a proof for a similar question I think it requires proof by induction. </p>
<p>It seems obvious that it would be even by thinking about the first few cases. As for $n=0$ there will be no horizontal dominoes which is even, and for $n=1$ there can onl... | Brian Tung | 224,454 | <p>We assume, given the description of the problem in the OP, that the rectangle is a horizontal strip of height $2$ and width $n$.</p>
<p><strong>Proof by Contradiction.</strong> Suppose we had a covering of the $2 \times n$ rectangle containing an odd number of horizontal dominoes. Then either the upper row or the ... |
1,088,338 | <p>There are at least a few things a person can do to contribute to the mathematics community without necessarily obtaining novel results, for example:</p>
<ul>
<li>Organizing known results into a coherent narrative in the form of lecture notes or a textbook</li>
<li>Contributing code to open-source mathematical softw... | Community | -1 | <p>Even people with no mathematical background at all can contribute to mathematics.</p>
<p>One obvious way is by running software such as the <a href="http://www.mersenne.org/">GIMPS</a> client for finding Mersenne primes, though the value of such primes to theoretical mathematics is debatable.</p>
<p>Another, vastl... |
1,088,338 | <p>There are at least a few things a person can do to contribute to the mathematics community without necessarily obtaining novel results, for example:</p>
<ul>
<li>Organizing known results into a coherent narrative in the form of lecture notes or a textbook</li>
<li>Contributing code to open-source mathematical softw... | Jair Taylor | 28,545 | <p>It is useful to contribute to databases of mathematical objects such as <a href="http://oeis.org/">Sloane's Encyclopedia of Integer Sequences</a>, <a href="http://findstat.org/">FindStat</a> and others. These resources have become invaluable for mathematicians searching for references to objects that they don't kno... |
2,666,568 | <p>I have a dynamical system: $\dot{\mathbf x}$= A$\mathbf x$ with $\mathbf x$=
$\bigl( \begin{smallmatrix} x \\ y\end{smallmatrix} \bigr)$ and A =
$\bigl( \begin{smallmatrix} 3 & 0 \\ \beta & 3 \end{smallmatrix} \bigr). \beta$ real, time-independent.</p>
<p>I calculated the eigenvalue $\lambda$ = 3 with the ... | gt6989b | 16,192 | <p>When $\beta = 0$, you have $A = 3I$ so the eigenvectors are the standard basis and the eigenvalues are both $1$.</p>
<p>When $\beta \ne 0$, the matrix is not diagonalizable, it only has a Jordan form.</p>
|
4,231,509 | <p>I'm trying to prove that the group <span class="math-container">$(\mathbb{R}^*, \cdot)$</span> is not cyclic (similar to [1]). My efforts until now culminated into the following sentence:</p>
<blockquote>
<p>If <span class="math-container">$(\mathbb{R}^*,\cdot)$</span> is cyclic, then <span class="math-container">$\... | Infinity_hunter | 826,797 | <p>Suppose <span class="math-container">$(\mathbb{R}^*, \cdot)$</span> is cyclic then there exist <span class="math-container">$x\in \mathbb{R}^*$</span> such that <span class="math-container">$\mathbb{R}^* = \langle x \rangle =\{x^n \vert \, n \in \mathbb{Z}\}$</span>. However we see that the set <span class="math-co... |
1,722,287 | <p>So far I know that when matrices A and B are multiplied, with B on the right, the result, AB, is a linear combination of the columns of A, but I'm not sure what to do with this. </p>
| Tryss | 216,059 | <p>If you know the rank-nullity theorem, it's easy :</p>
<p>$\forall x \in \text{Ker} B, ABx = A(Bx) = A(0) = 0$</p>
<p>So $x\in \text{Ker} AB$, hence $\text{Ker} B \subset \text{Ker} AB$, that imply that </p>
<p>$\text{dim Ker} B \leq \text{dim Ker} AB$ , and by the rank nullity theorem, you get that $\text{rank} B... |
1,007,399 | <p>I came across following problem</p>
<blockquote>
<p>Evaluate $$\int\frac{1}{1+x^6} \,dx$$</p>
</blockquote>
<p>When I asked my teacher for hint he said first evaluate</p>
<blockquote>
<p>$$\int\frac{1}{1+x^4} \,dx$$</p>
</blockquote>
<p>I've tried to factorize $1+x^6$ as</p>
<p>$$1+x^6=(x^2 + 1)(x^4 - x^2 +... | John_dydx | 82,134 | <p>As suggested, you can transform your denominator $1+x^6$ into $(1+x^2)(1-x^2+x^4)$. You can then use partial fractions to split the following into something that can be integrated:</p>
<p>$\Large\int \frac{1}{(1+x^2)(1-x^2+x^4)}dx$ </p>
<p>Hope that helps. It looks like a very nasty and messy integration to do. Be... |
1,007,399 | <p>I came across following problem</p>
<blockquote>
<p>Evaluate $$\int\frac{1}{1+x^6} \,dx$$</p>
</blockquote>
<p>When I asked my teacher for hint he said first evaluate</p>
<blockquote>
<p>$$\int\frac{1}{1+x^4} \,dx$$</p>
</blockquote>
<p>I've tried to factorize $1+x^6$ as</p>
<p>$$1+x^6=(x^2 + 1)(x^4 - x^2 +... | Quanto | 686,284 | <p>With <span class="math-container">$1+x^6= (1+x^2)(x^4-x^2+1)$</span>, decompose the integrand</p>
<p><span class="math-container">\begin{align}
& \int \frac{dx}{1+x^6}
=\frac13\int \left( \frac{1}{1+x^2}+\frac12\frac{x^2+1}{x^4-x^2+1}- \frac32\frac{x^2-1}{x^4-x^2+1}\right)dx \\
&\hspace{15mm}=\frac13\int \fr... |
637,819 | <p>$$x\in(\cap F)\cap(\cap G)=[\forall A\in F(x\in A)]\land[\forall A\in G(x\in A)]$$</p>
<p>Since the variable $A$ is bounded by universal quantifier, it is regarded as bounded variable, according to the rules, the variable is free to change to other letters while the meaning statement remains unchanged. But,the abov... | Mauro ALLEGRANZA | 108,274 | <p>In a quantified expression like $\forall x A(x)$ the occurrences of the variable $x$ into the formuala $A(x)$ are <em>bounded</em> by the quantifier (i.e. $\forall x$) becuase they occur into the <strong>scope</strong> of the quantifier.</p>
<p>So, in the example :</p>
<blockquote>
<p>$\forall x (A(x) \land B(x)... |
592,560 | <p>Let G be an abelian group. Show that, if G is not cyclic, then for all $x\in G$, there is a divisor $d$ of $n = |G|$ which is strictly smaller than n satisfying $x^d=1$. </p>
<p>I'm guessing that this is a consequence of Lagrange's Theorem. We can have that G is a disjoint union of left cosets that all have the sam... | amWhy | 9,003 | <p>You've shown that $d$ divides $n$, but you haven't shown that $d \neq n$ (i.e., you haven't shown that $d$ is strictly smaller than $n$). </p>
<p>This is where the "non-cyclic" condition on $G$ comes in. Suppose for the sake of contradiction that $H = G$, and that $H$ is the only non-trivial subgroup of $G$, and he... |
1,798,261 | <p>what is multilinear coefficient? I heard it a couple of times and I tried to google it, all I am getting is multiple linear regression.
I am confused at this point. </p>
| Zelos Malum | 197,853 | <p>By what you write we have that $c=0$ which means we've already lost 1 dimension, then we have $a+b=0$ which means it's a line only, a line is definitionally a subspace of the 3 dimensional space.</p>
|
912,176 | <p>If $y,z$ are elements of an archimedean field $F$ and if $y<z$, then there is a rational element $r$ of $F$ such that $y<r<z$</p>
<p>The proof begins with saying that it is no loss of generality that we assume that $0<y<z$</p>
<p>I don't understand well why this the case. Please guide me .</p>
<p>... | Adam Hughes | 58,831 | <p>It's not because in an ordered field if you have arbitrary elements $a,b,c\in F$ then</p>
<p>$$a<b\iff a+c<b+c$$</p>
<p>Hence if $0<y$, for example, then you can just add $1-y$ to both sides to get</p>
<p>$$0<1<z+1-y$$</p>
<p>and you have $y'=1, z'=z+1-y$ so that your new question is only about po... |
3,756,436 | <p>Recently I was doing a physics problem and I ended up with this quadratic in the middle of the steps:</p>
<p><span class="math-container">$$ 0= X \tan \theta - \frac{g}{2} \frac{ X^2 \sec^2 \theta }{ (110)^2 } - 105$$</span></p>
<p>I want to find <span class="math-container">$0 < \theta < \frac{\pi}2$</span> ... | Anatoly | 90,997 | <p>I reworded the problem setting <span class="math-container">$X=y$</span> and <span class="math-container">$\theta=x$</span>. Considering <span class="math-container">$g=9.81$</span>, solving the quadratic equation for <span class="math-container">$y$</span> gives the two solutions</p>
<p><span class="math-container... |
493,102 | <p>I have a concern with nested quantifiers.</p>
<p>I have: $$ \forall x \exists y \forall z(x^2-y+z=0) $$ such that $$ x,y,z \in \Bbb Z^+$$ </p>
<p>My first question, can it be read like this:</p>
<p>$$ \forall x \forall z \exists y(x^2-y+z=0) $$</p>
<p>The way I did it, is I started off with $x=1, z=1 $ </p>
... | MJD | 25,554 | <p>Consider $$\forall y. \exists x. x\text{ is the mother of }y$$ means that for every person $y$, there is some person $x$ who is $y$'s mother, which is true; every person has a mother. But $$\exists x.\forall y. x\text{ is the mother of }y$$ says that there is some person $x$, so that for every person $y$, $x$ is $... |
244,679 | <p>I have a two variable function <code>z[x,y] = f[x,y] + g[x,y]</code>, such that I know the functional form of <code>f[x,y]</code> but not of <code>g[x,y]</code>. I have to do some symbolic calculations with the function <code>z[x,y]</code>, but I would like to keep only the first order in <code>g[x,y]</code> (treati... | Henrik Schumacher | 38,178 | <p>First OP's implementation with timing on my machine:</p>
<pre><code>First@AbsoluteTiming[
u1 = input;
Do[f = SparseArray[{{i_} :>
If[MemberQ[lll, i],
0, -2.0*u1[[i - 1]] - 2.0*u1[[i + 1]] - 2.0*u1[[i + dos]] -
2.0*u1[[i - dos]] + (1. + I)*u1[[i]]]}, {dos*dos}];
u1 = LinearSolve[s, f];,... |
3,773,133 | <p>I have been thinking about this problem for a couple of months, and eventually failed. Could someone help me?</p>
<blockquote>
<p>Let <span class="math-container">$M$</span> and <span class="math-container">$X$</span> be two symmetric matrices with <span class="math-container">$M\succeq 0$</span> and <span class="ma... | rych | 73,934 | <p>Let <span class="math-container">$f(A) = \operatorname{tr}(A)$</span> then</p>
<p><span class="math-container">$df_A[H]=\operatorname{tr}H=(\operatorname{tr}\circ\operatorname{id})[H]=(\operatorname{tr}\circ\, dA_A)[H]$</span></p>
<p>for <span class="math-container">$A=M^pX$</span>, and e.g., <span class="math-conta... |
2,797,902 | <p>AFAIK, every mathematical theory (by which I mean e.g. the theory of groups, topologies, or vector spaces), started out (historically speaking) by formulating a set of axioms that generalize a specific structure, or a specific set of structures. </p>
<p>For example, when people think of a “field” they AFAIK usually... | Ethan Bolker | 72,858 | <p>The useful <a href="https://en.wikipedia.org/wiki/Zariski_topology" rel="nofollow noreferrer">Zariski topology</a> in algebraic geometry satisfies the usual axioms for a topology in a context that doesn't really match "the structure of which those axioms were originally intended as a generalization".</p>
|
1,227,419 | <p>$$\int\limits_6^{16}\left(\frac{1}{\sqrt{x^3+7x^2+8x-16}}\right)\,\mathrm dx=\frac{\pi }{k}$$</p>
<p><strong>Note:</strong> $k$ is a constant.</p>
| mickep | 97,236 | <p>Hint: Since $x^3+7x^2+8x-16=(x-1)(x+4)^2$, I suggest you to do the substitution $u=\sqrt{x-1}$. In the end, you will find that a primitive function is given by
$$
\frac{2}{\sqrt{5}}\arctan(\sqrt{x-1}/\sqrt{5}).
$$
The value of $k$? Well, I leave that fun to you!</p>
|
2,623,324 | <p>Assume that the measure space is finite for this to make sense. Also, we know that $L^p$ spaces satisfy log convexity, that is -
$$\|f\|_r \leq \|f\|_p^\theta \|f\|_q^{1-\theta}$$
where $\frac{1}{r}=\frac{\theta}{p} +\frac{1-\theta}{q}$.
The text which I am following says 'Indeed this is trivial when $q=\infty$, and... | Dr. Sonnhard Graubner | 175,066 | <p>write your Limit in the form $$e^{\lim_{n\to \infty}\frac{\ln(n+1)}{\sqrt{n}}}$$</p>
|
2,651,054 | <p>I have this expression:
$$(x + y + z’)(x’ + y’ + z)$$ which I am trying to simplify. I decide to multiply it out in order to get, $${\color{red}{(xx')}}+(xy')+(xz)+(yx')+{\color{red}{(yy')}}+(yz)+(z'x')+(z'y')+{\color{red}{(z'z)}}.$$
I know that the $xx', yy'$ and $zz'$ would just be $0$, however, now I am stuck. ... | user284331 | 284,331 | <p>For $f(z)=z^{3}+2$, then $f(z)=f(1)+f'(1)(z-1)+\dfrac{f''(1)}{2!}(z-1)^{2}+\dfrac{f^{(3)}(1)}{3!}(z-1)^{3}$.</p>
|
2,651,054 | <p>I have this expression:
$$(x + y + z’)(x’ + y’ + z)$$ which I am trying to simplify. I decide to multiply it out in order to get, $${\color{red}{(xx')}}+(xy')+(xz)+(yx')+{\color{red}{(yy')}}+(yz)+(z'x')+(z'y')+{\color{red}{(z'z)}}.$$
I know that the $xx', yy'$ and $zz'$ would just be $0$, however, now I am stuck. ... | David | 119,775 | <p>As suggested in comments, other methods are probably easier, but you certainly can do this by division if you wish. We want
$$z^3+2=a_0+a_1(z-1)+a_2(z-1)^2+a_3(z-1)^3\ .\tag{$*$}$$
Dividing the LHS by $z-1$ gives
$$z^3+2=(z-1)(z^2+z+1)+3\ .$$
Looking at $(*)$, when we divide the RHS by $z-1$ the remainder is obviou... |
269,178 | <p>i would like to know where i could find a plot of</p>
<p>$$ J_{ia}(2\pi i)$$ (1)</p>
<p>using Quantum mechanics i have conjectured that if $ a= \frac{x}{2} $ and $ i= \sqrt{-1} $ then </p>
<p>$$ J_{it}(2\pi i)\approx0=\zeta (1/2+2it)$$ at least for big $ t \to \infty $ (2)</p>
<p>however i do not know how to che... | Ron Gordon | 53,268 | <p>Here is a plot over some sample values in Wolfram Alpha:</p>
<p><a href="http://www.wolframalpha.com/share/clip?f=d41d8cd98f00b204e9800998ecf8427emmfihklhut" rel="nofollow">http://www.wolframalpha.com/share/clip?f=d41d8cd98f00b204e9800998ecf8427emmfihklhut</a></p>
|
324,119 | <p>I've been reading about the Artin Spin operation. It's defined as taking the classical <span class="math-container">$n$</span>-knot (<span class="math-container">$S^n\hookrightarrow S^{n+2}$</span>) to an <span class="math-container">$(n+1)$</span>-knot. For the <span class="math-container">$1$</span>-knot case (in ... | Friedrich Knop | 89,948 | <p>There are various generalizations of the Jordan-Hölder theorem. Beyond groups and groups with operators it holds for any equational theory which contains a Mal'cev operation. This means that from the given operations one can form a ternary operation <span class="math-container">$m(x,y,z)$</span> satisfying <span cla... |
4,218,729 | <p>You have a database of <span class="math-container">$25,000$</span> potential criminals. The probability that this database includes the art thief is <span class="math-container">$0.1$</span>. In a stroke of luck, you found a DNA sample of this thief from the crime scene. You compare this sample with a database of <... | true blue anil | 22,388 | <p>** Recast answer(12 Aug) **</p>
<p><em>Answer recast to include possibility that</em> a <strong>wrong match</strong> <em>may be found even if thief is in the database, and to present with greater clarity and simplicity through a contingency table</em></p>
<p>If the probability of a match being wrong is <span class... |
3,936,676 | <p>Well this format of a limit <span class="math-container">$0^0$</span> is an indeterminate form.</p>
<p>I claim that whatever this limit is (which depends on the exact question) should always be in between <span class="math-container">$[0,1]$</span>.</p>
<p>Is my claim correct?</p>
<p>I have no mathematical proof for... | Mike | 544,150 | <p>That's not true in general. Take <span class="math-container">$$y(t)=e^{-1/t}$$</span> and <span class="math-container">$$x(t)=-5t$$</span> say.
Then <span class="math-container">$$\lim_{t \rightarrow 0^+} y(t) = \lim_{t \rightarrow 0^+} x(t)=0,$$</span> but <span class="math-container">$$\lim_{t \rightarrow 0^+} (... |
957,940 | <p>I'm "walking" through the book "A walk through combinatorics" and stumbled on an example I don't understand. </p>
<blockquote>
<p><strong>Example 3.19.</strong> A medical student has to work in a hospital for five
days in January. However, he is not allowed to work two consecutive
days in the hospital. In how... | Trold | 180,885 | <p>Henry's answer for the first part is already very good, so this only tries to tackle the sentence ***. </p>
<p>Understanding the correspondence between five element subsets of $[27]$ and the non-consecutive days problem might be easier by going from the five element subsets of 27 to the days problem asked, rather t... |
1,002,719 | <p>If we have</p>
<p>$f: \{1, 2, 3\} \to \{1, 2, 3\}$</p>
<p>and</p>
<p>$f \circ f = id_{\{1,2,3\}}$</p>
<p>is the following then always true for every function?</p>
<p>$f = id_{\{1,2,3\}}$</p>
| Dustan Levenstein | 18,966 | <p>If $f: S \to S$ is a function satisfying $f(f(x)) = x$ for all $x \in S$, consider what the implications are of having a fixed $x \in S$ so that $f(x) \neq x$; setting $y := f(x)$, we must have:</p>
<p>$$f(x) = y$$ by definition, and $$f(y) = f(f(x)) = x.$$</p>
<p>Given this observation, can you then describe exac... |
19,253 | <p><strong>Bug introduced in 7.0.1 or earlier and fixed in 10.0.2 or earlier</strong></p>
<hr>
<p>I have access to two versions of Mathematica, version 7.0.1 on Linux and version 8 on Windows. When I try the following two lines on version 7, the kernel quits when it tries to plot. In version 8 it plots just fine. ... | Community | -1 | <p>It turns out to indeed be a problem with the triangulation algorithm. I figured it out by plotting the list incrimentally as in <code>ListContourPlot[temp[[;;n]]]</code> and found it would plot for <code>n<103</code> but then gave an error "The data generates an inconsistent triangulation. You can perturb the da... |
85,957 | <p>Assume the annually gross salary is $100,000.</p>
<p>Tax brackets:</p>
<ol>
<li>0 - 50K - 10% = 5K in taxes</li>
<li>50K - 70K - 20% = 4K in taxes</li>
<li>70K - $90K - 30% = 6K in taxes</li>
<li>90K and up - 40% = 4K in taxes</li>
</ol>
<p>The income tax on 100K would be 19K.</p>
<p>So the net salary would be 8... | Arturo Magidin | 742 | <p>First, determine the maximum net for each bracket:</p>
<ol>
<li><p>If your gross is 0-50,000, then your net is 0-45,000.</p></li>
<li><p>If your gross is 50,001-70,000, then your net is 45,001-61,000 (the most you will pay in taxes is 5,000 for the first 50,000, and 4,000 for the next 20,000).</p></li>
<li><p>If yo... |
85,957 | <p>Assume the annually gross salary is $100,000.</p>
<p>Tax brackets:</p>
<ol>
<li>0 - 50K - 10% = 5K in taxes</li>
<li>50K - 70K - 20% = 4K in taxes</li>
<li>70K - $90K - 30% = 6K in taxes</li>
<li>90K and up - 40% = 4K in taxes</li>
</ol>
<p>The income tax on 100K would be 19K.</p>
<p>So the net salary would be 8... | dku.rajkumar | 20,185 | <pre><code>consider net salary is x, now we will find out gross salary y
if x < 45000(0.9 * 50000)
y = x/0.9
end
if 45000 < x < 61000 (45000 + 0.8 * 20000)
y = 50000 + (x- 45000)/0.8
end
if 61000 < x < 75000 (61000 + 0.7 * 20000)
y = 70000 + (x - 61000)/0.7
end
if 75000 < x
y = 90000 + (x- 75000)/... |
114,754 | <p>I have several questions concerning the proof. I don't think I quite understand the details and motivation of the proof. Here is the proof given by our professor.</p>
<p>The space of polynomials $F[x]$ is not finite-dimensional.</p>
<p><em>Proof</em>. Suppose
$$F[x] = \operatorname{span}\{f_1,f_2,\dots,f_n\}$$</p>... | Pierre-Yves Gaillard | 660 | <p>I think the proof in your question is incorrect. Let's assume that $F$ is a field and $x$ an indeterminate. A correct proof might (I believe) go as follows:</p>
<p>Suppose
$$
F[x]=\operatorname{span}\{f_1,f_2,\dots,f_n\}.
$$</p>
<p>Let us choose a positive integer $N$ such that $N > \deg (f_i)$ for all $i=1,\... |
2,881,673 | <p>I've searched all over the internet and cannot seem to factorise this polynomial.</p>
<p>$x^4 - 2x^3 + 8x^2 - 14x + 7$</p>
<p>The result should be $(x − 1)(x^3 − x^2 + 7x − 7)$</p>
<p>What are the steps to get to that result?
I've tried grouping but doesn't seem to work...</p>
| yakobyd | 569,127 | <p>Considering $(a-b)^2 = a^2 - 2ab + b^2$, notice that you may manipulate the polynomial as follows:</p>
<p>\begin{align}
x^4 - 2x^3 + 8x^2 -14x + 7 &= (x^4 -2x^3 + x^2) + (7x^2 - 14x + 7) \\
&= x^2(x^2 - 2x + 1) + 7(x^2 - 2x + 1) \\
&= (x-1)^2(x^2 + 7)
\end{align}</p>
<p>which yields a complete factor... |
375,372 | <p>Using the $\epsilon-M $ definition of the limit, calculate
$$\lim_{x\to\infty}\frac{3x^2+7}{x^2+x+8}.$$</p>
<p>Working so far: </p>
<p>$$\lim_{x\to\infty}\frac{3x^2+7}{x^2+x+8}=3$$</p>
<p>Given $\epsilon>0$, I want M s.t. $x>M \implies \left|\frac{3x^2+7}{x^2+x+8}-3 \right|<\epsilon$</p>
<p>$$\left|\fr... | mathemagician | 49,176 | <p>Let $x>\max\{\sqrt{17},\frac{3}{\epsilon-1}\}$. We have $x>0$. This gives us</p>
<p>$$\left|\frac{-3x-17}{x^2+x+8} \right|= \left|\frac{3x+17}{x^2+x+8} \right|=\frac{3x+17}{x^2+x+8}<\frac{3x+17}{x^2}<\frac{3}{x}+1<\epsilon$$</p>
<p>This is incomplete as it assumes $\epsilon\neq 1$. When $\epsilon=1$... |
681,608 | <blockquote>
<p>Prop. 6.9: Let $X \to Y$ be a finite morphism of non-singular curves, then for any divisor $D$ on $Y$ we have $\deg f^*D=\deg f\deg D$.</p>
</blockquote>
<p>I can not understand two points in the proof:</p>
<p>(1) (Line 9) Now $A'$ is torsion free, and has rank equal to $r=[K(X):K(Y)]$.</p>
<p>Sinc... | Tomo | 62,940 | <p>I like @SomeEE's answer for Line 15. I want to give a different justification for Line 9, because I wasn't able to justify to myself the statement 'Passing to quotient fields.' Unfortunately in my notation below, B = A' and A is the local ring at Q; noting that normalization commutes with localization we see that A ... |
161,029 | <p>I have not seen a problem like this so I have no idea what to do.</p>
<p>Find an equation of the tangent to the curve at the given point by two methods, without elimiating parameter and with.</p>
<p>$$x = 1 + \ln t,\;\; y = t^2 + 2;\;\; (1, 3)$$</p>
<p>I know that $$\dfrac{dy}{dx} = \dfrac{\; 2t\; }{\dfrac{1}{t}}... | Lemon | 26,728 | <p><strong>Method 1</strong> Eliminating. I think they want to write everything in terms of x first. </p>
<p>$x = 1 + ln(t) \iff e^{x - 1} = t$</p>
<p>$y = t^2 + 2 = e^{2x -2} + 2 \implies y' = 2e^{2x -2}$</p>
<p>At (1,3) $y' = 2$. So the tangent line is $y = 2(x- 1) + 3$ or parametrically let $x - 1 = t
\iff x = 1... |
192,883 | <p>Can anyone please give an example of why the following definition of $\displaystyle{\lim_{x \to a} f(x) =L}$ is NOT correct?:</p>
<p>$\forall$ $\delta >0$ $\exists$ $\epsilon>0$ such that if $0<|x-a|<\delta$ then $|f(x)-L|<\epsilon$</p>
<p>I've been trying to solve this for a while, and I think it w... | Pedro | 23,350 | <p>I'm a little confused when you ask wether it is correct or not. Do you ask as opposed to the well known definition? That is, as opposed to </p>
<blockquote>
<p>We say that $\lim\limits_{x\to a}f(x)=\mathscr L$ if for every $\epsilon >0$ there exists a $\delta >0$ such that, for all $x$, if $0<|x-a|<\d... |
3,554,393 | <p>Namely, I need to prove <span class="math-container">${\max\limits_i} |\lambda_i| \leq {\max\limits_i}{\sum\limits_j} |M_{ij}|\mid$</span>, where <span class="math-container">$M$</span> is the matrix and <span class="math-container">$\lambda_i$</span> are its eigenvalues.</p>
<p>I'm not sure if there is any helpful... | Arthur | 15,500 | <p>Hint: Take an eigenvector <span class="math-container">$v$</span> corresponding to the maximum eigenvalue, and scale it so that the largest entry (in absolute value) is <span class="math-container">$1$</span>, at the <span class="math-container">$j$</span>th row. Look at the <span class="math-container">$j$</span>th... |
440,242 | <p>I'm pretty sure almost all mathematicians have been in a situation where they found an interesting problem; they thought of many different ideas to tackle the problem, but in all of these ideas, there was something missing- either the "middle" part of the argument or the "end" part of the argumen... | Jon Bannon | 6,269 | <p>A little buddhism goes a long way here, it seems.</p>
<p>There is a slight difference between being stuck and being obsessed.</p>
<p>As the saying goes, doing the same thing over and over again and expecting a different result is the definition of insanity.</p>
<p>So, if by stuck you mean obsessively trying to remov... |
440,242 | <p>I'm pretty sure almost all mathematicians have been in a situation where they found an interesting problem; they thought of many different ideas to tackle the problem, but in all of these ideas, there was something missing- either the "middle" part of the argument or the "end" part of the argumen... | Boris Bukh | 806 | <p>Not much changes with tenure -- we do not suddenly change our habits after a decade of working.</p>
<p>Our job is and has always been to advance the science. Since the most important unsolved problems tend to be also the most well-known, it usually means that we are almost (but not always!) perpetually stuck on the ... |
3,424,259 | <p>The system in question is <span class="math-container">$$\begin{cases} x_1 -x_2 + x_3 = -1 \\ -3x_1 +5x_2 + 3x_3 = 7 \\ 2x_1 -x_2 + 5x_3 = 4 \end{cases}$$</span></p>
<p>After writing this in matrix-form and performing row-operations we can show that</p>
<p><span class="math-container">$$
\begin{matrix}
-1 ... | G. Gare | 568,973 | <p>You can do one step more and transform the last line to <span class="math-container">$(0, 0, 0 | 4)$</span> so that it's clear that the system is not compatible and hence admits no solution (contradiction <span class="math-container">$0=4$</span> for any value of the variables <span class="math-container">$x_1, x_2,... |
126,168 | <p>$$\zeta(it)=2it\pi it−1\sin(i\pi t/2)\Gamma(1−it)\zeta(1−it).$$
Everything on the RHS is never zero,</p>
<p>Does that means LHS has no zeros, since $\sin(s)$ has a simple zero at $s=0$ while $\zeta(1−s)$ has a simple pole at $s=0$ (the Laurent expansion), so the product $\sin(i\pi t/2)\zeta(1−it)$ is finite and no... | Aryabhata | 1,102 | <p>Do you mean the functional equation</p>
<p>$$\zeta(s) = 2(2\pi)^{s-1} \Gamma(1-s) \zeta(1-s) \sin (\frac{1}{2} \pi s)$$</p>
<p>?</p>
<p>which is valid for all $s \neq 1$ and $s \neq 0$.</p>
<p>Even though you cannot directly substitute $s =0$ or $s=1$, what you can do is multiply by $s-1$ to get</p>
<p>$$[\zeta... |
126,168 | <p>$$\zeta(it)=2it\pi it−1\sin(i\pi t/2)\Gamma(1−it)\zeta(1−it).$$
Everything on the RHS is never zero,</p>
<p>Does that means LHS has no zeros, since $\sin(s)$ has a simple zero at $s=0$ while $\zeta(1−s)$ has a simple pole at $s=0$ (the Laurent expansion), so the product $\sin(i\pi t/2)\zeta(1−it)$ is finite and no... | Max Clifford | 71,661 | <p>The LHS has no zeroes and therein lies the proof of the Prime Number Theorem.</p>
<p>You can get $\zeta(0)=-1/2$ from the functional equation; though you must be comfortable with limit manipulation as the poster above rightly states that the functional equation is not valid for $s=0,1$. The derivation at the end of... |
1,680,269 | <p>Here $\mathbb{Z}_{n}^{*}$ means $\mathbb{Z}_{n}-{[0]_{n}}$</p>
<p>My attempt:</p>
<p>$(\leftarrow )$</p>
<p>$p$ is a prime, then, for every $[x]_{n},[y]_{n},[z]_{n}$ $\in (\mathbb{Z}_{n}^{*},.)$ are verified the following:</p>
<p>1) $[x]_{n}.([y]_{n}.[z]_{n}) = ([x]_{n}.[y]_{n}).[z]_{n}$, since from the operatio... | Aloizio Macedo | 59,234 | <p>Consider the function</p>
<p>$$f: K \times L \rightarrow \mathbb{R}$$
$$(x,y) \mapsto |x-y|.$$</p>
<p>$f=\vert\cdot\vert \circ -|_{K \times L},$ where $-: \mathbb{R} \times \mathbb{R}$ is the subtraction and $| \cdot|$ is the module function. Hence, $f$ is the composition of continuous functions, therefore continu... |
1,298,971 | <p>Any help on this problem is greatly appreciated! I'm completely stuck</p>
<p>School board officials are debating whether to require all high school seniors to take a proficiency exam before graduating. A student passing all three parts (mathematics, language skills, and general knowledge) would be awarded a diploma... | Barry Cipra | 86,747 | <p>The length of a side, $L$, is analogous to the <em>diameter</em> of a circle, not its radius. So the appropriate variable to use is $\ell=L/2$, in which case the perimeter is $8\ell$, which integrates to $4\ell^2$, and that <em>is</em> correct, since $4\ell^2=(2\ell)^2=L^2$.</p>
<p>(Remark: I posted this before r... |
2,166,917 | <p>$20$ questions in a test. The probability of getting correct first $10$ questions is $1$. The probability of getting correct next $5$ questions is $\frac 13$. The probability of getting correct last $5$ questions is $\frac 15$. What is the probability of getting exactly $11$ questions correctly?</p>
<p>This is the ... | Jaroslaw Matlak | 389,592 | <p>For $k<0$:</p>
<p>$P(k)=0.$</p>
<p>For $k\geq10$:</p>
<p>$$P(k)=\sum_{i=\max(k-15,0)}^{\min(k-10,5)}p(i,k-10-i)$$
Where
$$p(i,j)={5 \choose i}\left(\frac{1}{3}\right)^i\left(\frac{2}{3}\right)^{5-i} {5 \choose j}\left(\frac{1}{5}\right)^j\left(\frac{4}{5}\right)^{5-j}$$</p>
<p>Finally: for $k\in \{0,1,2,...,... |
2,231,949 | <p>To find the minimal polynomial of $i\sqrt{-1+2\sqrt{3}}$, I need to prove that
$x^4-2x^2-11$ is irreducible over $\Bbb Q$. And I am stuck. Could someone please help? Thanks so much!</p>
| dxiv | 291,201 | <p>As noted already, the rational root theorem excludes rational roots, which only leaves a product of rational quadratics as a potential factorization.</p>
<p>By the way the polynomial was constructed, it is known that its roots are $\,\pm i \sqrt{2 \sqrt{3}-1}\,$ and $\,\pm \sqrt{2 \sqrt{3}+1}\,$. Any quadratic fact... |
3,328,387 | <p>Suppose I have two positive semi-definite <span class="math-container">$n$</span>-by-<span class="math-container">$n$</span> matrices <span class="math-container">$A$</span>, <span class="math-container">$B$</span> and an <span class="math-container">$n$</span>-by-<span class="math-container">$n$</span> identity mat... | Kwin van der Veen | 76,466 | <p>Your equation is equivalent to solving the following equation for <span class="math-container">$P$</span></p>
<p><span class="math-container">$$
A\,P + P\,A^\top = B. \tag{1}
$$</span></p>
<p>Such equation also solves for the <a href="https://en.wikipedia.org/wiki/Controllability_Gramian#Controllability_Gramian" r... |
92,660 | <p>Let $X$ be a nonsingular projective variety over $\mathbb{C}$, and let $\widetilde{X}$ be the blow-up of X at a point $p\in X$.
What relationships exist between the degrees of the Chern classes of $X$ (i.e. of the tangent bundle of $X$) and the degrees of the Chern classes of $\widetilde{X}$?</p>
<p>Thanks.</p>
| Johannes Nordström | 13,061 | <p>Like Georges says, 15.4 of Fulton's Intersection Theory deals with the general theory. For this special case it's not too hard to work out the Chern classes by hand though.</p>
<p>Let $f : \widetilde X \to X$ be the projection and $E \cong \mathbb{C}P^{n-1}$ the exceptional divisor.
$H^*(\widetilde X) \cong f^*H^*(... |
424,694 | <p>Let <span class="math-container">$p$</span> be a prime, and consider <span class="math-container">$$S_p(a)=\sum_{\substack{1\le j\le a-1\\(p-1)\mid j}}\binom{a}{j}\;.$$</span>
I have a rather complicated (15 lines) proof that <span class="math-container">$S_p(a)\equiv0\pmod{p}$</span>. This must be
extremely classic... | Zhi-Wei Sun | 124,654 | <p>Actually, a further extension was given in my paper <a href="http://maths.nju.edu.cn/%7Ezwsun/92a.pdf" rel="nofollow noreferrer">Combinatorial congruences and Stirling numbers</a> [Acta Arith. 126 (2007), 387-398]. Now I state Corollary 1.3 (a consequence of Theorem 1.1) in the 2007 paper of mine.</p>
<p>Let <span c... |
143,655 | <p>According to <a href="http://en.wikipedia.org/wiki/Lipschitz_continuity#Properties" rel="nofollow noreferrer">wikipedia</a> a function <span class="math-container">$f\colon \mathbb{R}^n\to\mathbb{R}^n$</span> that is continuously differentiable, is also locally Lipschitz.</p>
<p>I there someone who knows a good refe... | Shuhao Cao | 7,200 | <p>The exact quote on wiki was:</p>
<blockquote>
<p>In particular, any $C^1$ function is locally Lipschitz, as continuous functions on a locally compact space are locally bounded so its gradient is.</p>
</blockquote>
<p>The logic here is we would like to show the gradient of a $C^1$-function is locally bounded on a... |
3,906,920 | <p>A string in <span class="math-container">$\{0, 1\}*$</span> has even parity if the symbol <span class="math-container">$1$</span> occurs in the word an even number of times; otherwise, it has odd parity.</p>
<p>(a) How many words of length <span class="math-container">$n$</span> have even parity?</p>
<p>(b) How many... | ho boon suan | 436,996 | <p>Let <span class="math-container">$X\in\mathfrak{X}(M)$</span> be a smooth vector field on <span class="math-container">$M$</span>. If all you have is a smooth bijection <span class="math-container">$F\colon M\to N$</span> with an inverse <span class="math-container">$F^{-1}$</span> that isn’t smooth, there is no way... |
2,067,003 | <p>(Mathematics olympiad Netherlands) Let $A,B$ and $C$ denote chess players in a tournament. The winner of each match plays the next match against the oponent that did not play the current. At the end of the tournament $A$, $B$ and $C$ played $10$, $15$ and $17$ times respectively. Each match only ended up in a win. <... | MatheMagic | 397,530 | <p>I got my answer in this way:
$\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n(n+1)}=\sum_{n=1}^{\infty}(-1)^{n+1}\left(\frac1n-\frac1{n+1}\right)=\sum_{n=1}^{\infty}\frac {(-1)^{n+1}}{n}-\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n+1}=\\log2+\log2-1=2\log2-1$</p>
|
2,239,240 | <p>I'm looking to do some independent reading and I haven't been able to find rough prerequisites for Differential Topology at the level of Milnor or Guillemin and Pollack.</p>
<p>Is a semester of analysis (Pugh) and a semester of topology (Munkres) enough to make sense of most of it or should I take a second semester ... | Zev Chonoles | 264 | <p><strong>As a general rule, you can do anything you want in any order as long as each step is a valid operation.</strong></p>
<p>Replacing $2+3$ by $5$ is certainly fine. </p>
<p>Expanding an expression $(a+b)^2$ as $a^2+2ab+b^2$ is also correct. </p>
<p>Do whichever you want.</p>
|
2,239,240 | <p>I'm looking to do some independent reading and I haven't been able to find rough prerequisites for Differential Topology at the level of Milnor or Guillemin and Pollack.</p>
<p>Is a semester of analysis (Pugh) and a semester of topology (Munkres) enough to make sense of most of it or should I take a second semester ... | Ethan Bolker | 72,858 | <p>This is a very good question.</p>
<p>There are some standard <em>conventions</em> for order of operations - for example, when you see
$2 + 3 \times 4$ you do the multiplication first. This convention is not really a part of arithmetic, it's just an agreed upon way to write the calculation that avoids having to w... |
107,399 | <p>Let's say we have a set a\of associations:</p>
<pre><code>dataset = {
<|"type" -> "a", "subtype" -> "I", "value" -> 1|>,
<|"type" -> "a", "subtype" -> "II", "value" -> 2|>,
<|"type" -> "b", "subtype" -> "I", "value" -> 1|>,
<|"type" -> "b", "subtype" -> ... | gwr | 764 | <p>Here is the approach I would take to transform your dataset to a nested <code>Association</code>:</p>
<pre><code>Clear[ makeNested ];
makeNested[ assoc_, keylist_] := GroupBy[
assoc,
keylist
] // Apply[
Association,
#,
{ Length @ keylist }
] &
</code></pre>
<p>Now <code>makeNested[ dataset... |
1,581,161 | <p>Let the triangle $ABC$ and the angle $\widehat{ BAC}<90^\circ$ </p>
<p>Let the perpendicular to $AB$ passing by the point $C$ and the perpendicular to $AC$ passing by $B$ intersect the circumscribed circle of $ABC$ on $D$ and $E$ respectively .
We suppose that $DE=BC$</p>
<p>What is the angle $\widehat{BAC}$ ... | James Pak | 187,056 | <p><a href="https://i.stack.imgur.com/YElQu.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/YElQu.png" alt="enter image description here"></a></p>
<p>Note that $\angle BAC=\angle BEC$, and that $\angle BAC=180^\circ-\angle DFE=\angle CFE$. As $DE=BC$, $\angle BEC=\angle DCE$. Therefore, $\angle BAC ... |
273,499 | <blockquote>
<p>Show that every group $G$ of order 175 is abelian and list all isomorphism types of these groups. [HINT: Look at Sylow $p$-subgroups and use the fact that every group of order $p^2$ for a prime number $p$ is abelian.]</p>
</blockquote>
<p>What I did was this. $|G| = 175$. Splitting 175 gives us $175 ... | DonAntonio | 31,254 | <p>What you did looks fine, albeit slightly messy and overkill: if you already know there's one unique Sylow $\,5-$subgroup $\,P\,$ of order $\,25\,$ and one single Sylow $\,7-$ subgroup $\,Q\,$ of order $\,7\,$ , both of them abelian, and then you already know:</p>
<p>(1) $\,P,Q\triangleleft G\,$</p>
<p>(2) $\,P\cap... |
2,484 | <p>It's been quite a while since I was tutoring a high school student and even longer since not a gifted one.</p>
<p>However, this time, something was amiss. I have asked him to show me how he does some exercise, and then another and the only thing I wanted to do was to shout:</p>
<blockquote>
<p><strong>You are do... | Community | -1 | <p><em>What does he want?</em> The question says more about what he doesn't want.</p>
<p>Maybe he wants to pass an exam to avoid math for a few years, and wants to improve his chance of passing. Maybe he wants to run a business, and would be interested in business algebra.</p>
<p>Starting from his goals can make th... |
2,805,192 | <p><a href="https://i.stack.imgur.com/OhGi4.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/OhGi4.png" alt="enter image description here"></a></p>
<p>Definition:</p>
<p><a href="https://i.stack.imgur.com/mV4NU.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/mV4NU.png" alt="enter ... | Igor Sikora | 464,503 | <p>Your argument is nearly valid. It doesn't show that every $x \in X$ belongs to $E$, but rather that every $x$ is in the closure of $E$. And that's ok, because this is the definition of a dense subset: $E$ is dense in X if $\bar{E}=X$.</p>
<p>Also, having said that, you cannot fix $\delta$. That is because of the de... |
2,805,192 | <p><a href="https://i.stack.imgur.com/OhGi4.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/OhGi4.png" alt="enter image description here"></a></p>
<p>Definition:</p>
<p><a href="https://i.stack.imgur.com/mV4NU.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/mV4NU.png" alt="enter ... | Berci | 41,488 | <p>Let indeed $x_1$ be arbitrary, and set $\delta_1=1$. Denote $X_1$ the finite set $\{x_1,\dots, x_{n_1}\}$ obtained as in the hint for $\delta_1$. <br>
Let $x_{n_1+1}, \dots$ be points satisfying $d(x_k, x_j) \ge\delta_2:=\frac12$. Denote their set $X_2$, it's finite again. <br>
Continue with $\delta_n:=\frac1n$. </p... |
754,012 | <p>Is it possible to show that the harmonic series is divergent by showing that the sequence of partial sums is a monotone increasing sequence that is unbounded?</p>
| marty cohen | 13,079 | <p>Another way,
which is moderately equivalent.</p>
<p>Let
$S_n = \sum_{k=1}^n \frac1{k}$.
$S_n$ is obviously an increasing sequence.</p>
<p>$S_{2n}-S_n
=\sum_{k=1}^{2n} \frac1{k}-\sum_{k=1}^n \frac1{k}
=\sum_{k=n+1}^{2n} \frac1{k}
>\sum_{k=n+1}^{2n} \frac1{2n}
=\frac{n}{2n}
=\frac12
$</p>
<p>so, by induction
$S_... |
3,632,576 | <p>Considering that input <span class="math-container">$x$</span> is a scalar, the data generation process works as follows:</p>
<ul>
<li>First, a target t is sampled from {0, 1} with equal probability.</li>
<li>If t = 0, x is sampled from a uniform distribution over the interval
[0, 1]. </li>
<li>If t = 1, x is sampl... | S.C. | 544,640 | <p>Here is a more detailed/adapted version of Spivak's proposed solution:</p>
<p>By assumption, we know that there exists an <span class="math-container">$N \gt 0: \forall t \geq N: \left|f(t)-a\right|\lt \frac{\varepsilon}{3}$</span>. Consider <span class="math-container">$f$</span> restricted to the domain <span clas... |
2,528,716 | <p>How can I prove </p>
<blockquote>
<p>$$x^2+y^2-x-y-xy+1≥0$$</p>
</blockquote>
<p>I tried $(x+y)^2-3xy-(x+y)+1≥0 \rightarrow(x+y-1)(x-y)-3xy+1≥0$ I can not continue</p>
| mathlove | 78,967 | <p>Another way :$$\begin{align}x^2+y^2-x-y-xy+1&=x^2+(-1-y)x+y^2-y+1\\\\&=\left(x+\frac{-1-y}{2}\right)^2-\left(\frac{-1-y}{2}\right)^2+y^2-y+1\\\\&=\left(x-\frac{1+y}{2}\right)^2+\frac 34(y-1)^2\ge 0\end{align}$$</p>
|
2,528,716 | <p>How can I prove </p>
<blockquote>
<p>$$x^2+y^2-x-y-xy+1≥0$$</p>
</blockquote>
<p>I tried $(x+y)^2-3xy-(x+y)+1≥0 \rightarrow(x+y-1)(x-y)-3xy+1≥0$ I can not continue</p>
| Michael Rozenberg | 190,319 | <p>We need to prove that
$$x^2-(y+1)x+y^2-y+1\geq0,$$ which is a quadratic inequality of $x$.</p>
<p>Thus, it's enough to prove that
$$(y+1)^2-4(y^2-y+1)\leq0$$ or
$$(y-1)^2\geq0,$$
which is obvious.</p>
|
1,282,486 | <p>Given the function $f(x) = |8x^3 − 1|$ in the set $A = [0, 1].$
Prove that the function is not differentiable at $x = \frac12.$ </p>
<p>The answer in my book is as follows:</p>
<p>$$\lim_{x \to \frac12-} \dfrac{f(x)-f(1/2)}{x-1/2} = -6$$
$$\lim_{x \to \frac12+} \dfrac{f(x)-f(1/2)}{x-1/2} = 6$$ </p>
<p>Can anyone... | Przemysław Scherwentke | 72,361 | <p>$(8x^3-1)/(x-1/2)=8(x^3-1/8)/(x-1/2)=8(x^2+(1/2)x+1/4)$, which is equal to 6 at $x=1/2$.</p>
|
1,282,486 | <p>Given the function $f(x) = |8x^3 − 1|$ in the set $A = [0, 1].$
Prove that the function is not differentiable at $x = \frac12.$ </p>
<p>The answer in my book is as follows:</p>
<p>$$\lim_{x \to \frac12-} \dfrac{f(x)-f(1/2)}{x-1/2} = -6$$
$$\lim_{x \to \frac12+} \dfrac{f(x)-f(1/2)}{x-1/2} = 6$$ </p>
<p>Can anyone... | zhw. | 228,045 | <p>It's almost easier to prove a more general result: If $f(a) = 0$ and $f'(a) \ne 0,$ then $|f(x)|$ is not differentiable at $a.$</p>
|
232,562 | <p>Ax-Grothendieck Theorem states that if $\mathbf K$ is an algebraically closed field, then any injective polynomial map $P:\mathbf K^n\longrightarrow \mathbf K^n$ is bijective.</p>
<blockquote><b>Question 1.</b> What does the inverse map of $P$ look like ? What kind of map is that ?</blockquote>
<p>$P^{-1}$ need no... | R. van Dobben de Bruyn | 82,179 | <p>In light of abx's comment, I came up with the following argument. I'm sure some version of this must be in the literature somewhere.</p>
<p>Recall that a dominant morphism $Y \to X$ of varieties is <em>separable</em> if $K(X) \to K(Y)$ is a separable field extension. In characteristic $0$, this is automatic.</p>
<... |
2,471,633 | <p>Let $B$ an open ball in $\mathbb{R}^{n}$, and $(K_{j})_{j}$ be an increasing sequence of compact subsets of $B$ whose union equals $B$. For each $j$, let $\rho_{j}$ be a cut-off function in $C_{c}^{\infty}(B)$ that equals 1 on a neighborhood of $K_{j}$ and whose support is in $K_{j+1}$. Finally, let $\theta$ be a s... | Dap | 467,147 | <p>Using the divergence theorem,</p>
<p>$$\int_{K_{j+1}\setminus K_j} \Delta \phi_j + \int_{K_j} \Delta \phi_j=\int_B \Delta \phi_j = \int_{\partial B} \nabla \phi_j \cdot dS=0.$$</p>
<p>But $\int_{K_j} \Delta \phi_j = \operatorname{vol}(K_j),$ so</p>
<p>$$\sup_{K_{j+1}\setminus K_j}(-\Delta\phi_j)\geq \frac{1}{\ope... |
1,611,506 | <blockquote>
<p>$$\int (2x^2+1)e^{x^2} \, dx$$</p>
</blockquote>
<p>It's part of my homework, and I have tried a few things but it seems to lead to more difficult integrals. I'd appreciate a hint more than an answer but all help is valued.</p>
| Ben Longo | 137,131 | <p>Start by expanding the integrand.</p>
<p>$$\begin{align}
I&=\int \left(2x^2e^{x^2}+e^{x^2}\right)\,dx\\
&=\int 2x^2e^{x^2}\,dx+\int {e^{x^2}}\,dx\tag{a}\\
&=x e^{x^2}-\int 2x^2 e^{x^2}\,dx+\int 2x^2 e^{x^2}\,dx\\
&=x e^{x^2}+C
\end{align}$$</p>
<p>$(\text{a})$: Use integration by parts on the secon... |
58,024 | <p><img src="https://i.stack.imgur.com/cTpA2.jpg" alt="Show that..."></p>
<p>The picture says it all. "Vis at" means "show that". My first thought was that h is 2x, which is not correct. Maybe the formulas for area size is useful? </p>
<p>EDIT: (To make the question less dependent from the <a href="https://math.meta.... | robjohn | 13,854 | <p>Consider the area of the whole triangle and the areas of the constituent triangles and square.</p>
|
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