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 |
|---|---|---|---|---|
803,687 | <p>How to prove : </p>
<blockquote>
<p>$A/m^n$ is Artinian for all $n\geq 0$ if $A$ is a Noetherian ring and $m$ maximal ideal.</p>
</blockquote>
<p>Any suggestions ?</p>
| Keenan Kidwell | 628 | <p>You need to prove that $A/\mathfrak{m}^n$ is of finite length as a module over itself, or equivalently, as an $A$-module (since ideals of $A/\mathfrak{m}^n$ are the same as $A$-submodules). Consider the chain $A/\mathfrak{m}^n\supseteq \mathfrak{m}/\mathfrak{m}^n\supseteq\cdots\supseteq\mathfrak{m}^{n-1}/\mathfrak{m... |
4,404,751 | <p>I want to prove the following: Every symmetric matrix whose entries are calculated as <span class="math-container">$ 1/(n -1) $</span> with <span class="math-container">$n$</span> as the size of the matrix, except for the diagonal which is 0, has a characteristic polynomial with a root at <span class="math-container... | Koro | 266,435 | <p>Note the following lemma:</p>
<p>If the sum of every row of an <span class="math-container">$ m$</span> by <span class="math-container">$m$</span> matrix <span class="math-container">$A$</span> is <span class="math-container">$k$</span>, then <span class="math-container">$k$</span> is an eigenvalue of <span class="m... |
2,243,900 | <p>I've researched this topic a lot, but couldn't find a proper answer to this, and I can't wait a year to learn it at school, so my question is:</p>
<blockquote>
<p>What exactly is calculus? </p>
</blockquote>
<p>I know who invented it, the Leibniz controversy, etc., but I'm not exactly sure what it is. I think I ... | richard1941 | 133,895 | <p>There are several topics of calculus: limits, differentiation, integration, power series, approximation of functions... all are bound together by the concept of INFINITY. That is, they are the result if an infinite process. So I would say that calculus is the study of a particular set of infinite processes that ... |
31,782 | <p>This question is edited following the comment of Joseph. He pointed out that the main object of the first version of this question is the cut locus.</p>
<p>Recall that the cut locus of a set <span class="math-container">$S$</span> in a geodesic space <span class="math-container">$X$</span>
is the closure of the set... | John Sullivan | 46,735 | <p>The 'simple lemma' certainly isn't always true. Let's consider just open disks in the Euclidean plane with a flat metric. Fremlin (Proc. LMS, 1997) calls the set of points without a unique nearest boundary point the "skeleton". He gives an example where the skeleton is somewhere dense (that is, its closure, which ... |
292,639 | <p>Traditional Fourier analysis picks a period and then describes a function as: $$f(x) = \frac{1}{2} a_0 + \sum_{k=1}^\infty\, (a_k \cos{(\omega \cdot kx)} + b_n \sin{(\omega \cdot kx)})$$</p>
<p>I am wondering whether there is a way to Fourier-analyze a function in a way that the period is dependent on $x$. Let $g$ ... | Community | -1 | <p>Note that the function is an odd function and you integrating it from $-4$ to $4$. In general, if $f(x)$ is odd and integrable, we have $$\int_{-a}^a f(x) dx = 0$$where $a \in \mathbb{R}$.</p>
|
162,147 | <p>I have</p>
<pre><code>J = Table[{x10, y10, x10*y10}, {x10, 0, 1, 0.5}, {y10, 0, 1, 0.5}]
L = Table[{x10, y10, 2.0*x10*y10}, {x10, 0, 1, 0.5}, {y10, 0, 1, 0.5}]
</code></pre>
<p>I want the third elements of J and L to be added and the first and second elements are as they are (as they are the same in both cases) fo... | Henrik Schumacher | 38,178 | <pre><code>result = J;
result[[All, All, 3]] += L[[All, All, 3]]
</code></pre>
|
3,033,202 | <p>I have been banging my head against this proof for a few days now, as I can visualize why it is true in my head, but don't know how to prove it in words:</p>
<p>Let <span class="math-container">$A$</span> and <span class="math-container">$B$</span> be nonempty subsets of <span class="math-container">$\mathbb{R}$</s... | José Carlos Santos | 446,262 | <p>You can use the fact that<span class="math-container">$$\arcsin'(x)=\frac1{\sin'(\arcsin x)}=\frac1{\cos(\arcsin x)}=\frac1{\sqrt{1-x^2}}.$$</span>So, and since <span class="math-container">$\arcsin(0)=0$</span>, you get that<span class="math-container">$$\arcsin s=\int_0^s\frac{\mathrm dx}{\sqrt{1-x^2}}.$$</span></... |
177,091 | <p>What would $\int\limits_{-\infty}^\infty e^{ikx}dx$ be equal to where $i$ refers to imaginary unit? What steps should I go over to solve this integral? </p>
<p>I saw this in the Fourier transform, and am unsure how to solve this.</p>
| draks ... | 19,341 | <p>$\int\limits_{-\infty}^\infty e^{ikx}dx$ is a (<em>EDIT: scaled by $\frac1{2\pi}$</em>) representation of <a href="http://en.wikipedia.org/wiki/Dirac_delta_function" rel="nofollow">Dirac's $\delta(k)$ function</a>. For the antiderivative see the other answers...</p>
|
18,090 | <p>I am wondering if anyone could prove the following equivalent definition of recurrent/persistent state for Markov chains:</p>
<p>1) $P(X_n=i,n\ge1|X_0=i)=1$<p>
2) Let $T_{ij}=min\{n: X_n=j|X_0=i\}$, a state $i$ is recurrent if for $\forall j$ such that $P(T_{ij}<\infty)>0$, one has $P(T_{ij}<\infty)=1$</p>... | Did | 6,179 | <p>See <a href="https://mathoverflow.net/questions/53011/proof-of-equivalent-definitions-of-recurrent-states-for-markov-chains">this page</a>.</p>
<p><strong>Edit:</strong> Since incorrect statements continue to pop up on this page, let me explain once again the situation.</p>
<p>First, two notations (standard in the... |
466,873 | <p>I'm trying to prove that if $K$ is a finite field, then every subset of $\mathbb A^n(K)$ is algebraic. I know that if $K$ is finite, then every element of $K$ is algebraic, i.e., for every $a\in K$ there is a polynomial $f\in K$ such that $f(a)=0$, but this didn't help me to solve the question. I almost sure that we... | Alex Youcis | 16,497 | <p>Here is a another approach, I'll give you a hint. Show that every map $K^n\to K$ is a polynomial. Begin by showing that you can write the indicator function at any point as a polynomial (remember you can find a polynomial that has, as roots, any subset of $K$). Then, just add up appropriate multiples of these indica... |
4,159,451 | <p>Kindly see the boldened sentence below. What probability concept does "the most likely value" refer to? How do I symbolize it?</p>
<p>I don't think the author's referring to Expected Value, because it's additive.</p>
<blockquote>
<p> The principle of additivity is so intuitively appealing that it’s ea... | Thomas Andrews | 7,933 | <p>You can only really define the “most likely” value for some random variable. If <span class="math-container">$f$</span> is a pdf for a random <span class="math-container">$X,$</span> then the most like value of <span class="math-container">$X$</span> is the “mode” of <span class="math-container">$X,$</span> a value ... |
3,912,722 | <blockquote>
<p>Circle of radius <span class="math-container">$r$</span> touches the parabola <span class="math-container">$y^2+12x=0$</span> at its vertex. Centre of circle lies left of the vertex and circle lies entirely within the parabola. What is the largest possible value of <span class="math-container">$r$</span... | g.kov | 122,782 | <p>Under given constraints, the maximal value of the radius
<span class="math-container">$r_{\max}=\tfrac1\kappa$</span>,
where <span class="math-container">$\kappa$</span>
is the curvature of the parabola at its vertex.</p>
<p>For the given parabola <span class="math-container">$y^2=-12x$</span>
we can use a convenien... |
3,912,722 | <blockquote>
<p>Circle of radius <span class="math-container">$r$</span> touches the parabola <span class="math-container">$y^2+12x=0$</span> at its vertex. Centre of circle lies left of the vertex and circle lies entirely within the parabola. What is the largest possible value of <span class="math-container">$r$</span... | fleablood | 280,126 | <p>For the circle to be entirely <em>within</em> the parabola means the parabola and the circle intersect at exactly one point.</p>
<p>However solving <span class="math-container">$(x+r)^2 + y^2 = r^2$</span> and <span class="math-container">$y^2 + 12x = 0$</span> we actually will find <em>three</em> (potential) points... |
4,305,593 | <blockquote>
<p>Prove that
<span class="math-container">$$ \int_{0}^{\infty} \frac{\sin^2 x-x\sin x}{x^3} \, dx = \frac{1}{2} - \ln 2 .$$</span></p>
</blockquote>
<p>Integration by parts gives</p>
<p><span class="math-container">\begin{align*}
&\lim_{R\to \infty} \int_{0}^{R} \frac{\sin^2 x-x\sin x}{x^3} \, dx \\
&... | Sangchul Lee | 9,340 | <p>Here is a correct solution that you might want to compare with:</p>
<p><span class="math-container">\begin{align*}
&\int_{0}^{\infty}\frac{\sin^2 x - x \sin x}{x^3} \, \mathrm{d}x \\
&= \lim_{\substack{R\to\infty \\ \varepsilon \to 0^+}} \int_{\varepsilon}^{R}\frac{\sin^2 x - x \sin x}{x^3} \, \mathrm{d}x \\... |
4,305,593 | <blockquote>
<p>Prove that
<span class="math-container">$$ \int_{0}^{\infty} \frac{\sin^2 x-x\sin x}{x^3} \, dx = \frac{1}{2} - \ln 2 .$$</span></p>
</blockquote>
<p>Integration by parts gives</p>
<p><span class="math-container">\begin{align*}
&\lim_{R\to \infty} \int_{0}^{R} \frac{\sin^2 x-x\sin x}{x^3} \, dx \\
&... | Claude Leibovici | 82,404 | <p><span class="math-container">$$I=\int \frac{\sin^2 (x)-x\sin (x)}{x^3}\, dx=\int \frac{\sin^2 (x)}{x^3}\, dx-\int \frac{\sin (x)}{x^2}\, dx$$</span>
<span class="math-container">$$\int \frac{\sin^2 (x)}{x^3}\, dx=\frac 1 2\int \frac{1-\cos(2x)}{x^3}\,dx$$</span>
<span class="math-container">$$\int \frac{1-\cos(2x)}{... |
619,607 | <p>$\mathbf{(1)}$ Find $y^{\prime}$ of $y=8^{\sqrt x}$ </p>
<p>My try: </p>
<p>$\ln y=\ln(8)^{\sqrt x}$<br>
$\dfrac{1}{y}y^{\prime}=\sqrt{x}\ln8$<br>
I don't know how to proceed with right side. </p>
<p>$\mathbf{(2)}$ Find $y^{\prime}$ of $y=(t+4)(t+6)(t+7).$</p>
<p>This one I have no idea what to do so I don't ha... | Michael Albanese | 39,599 | <p>In the first problem, you got down to $\ln y = \ln(8)^{\sqrt{x}}$ which is correct, but you made a mistake in the next line. On the left you differentiated, but on the right you rewrote $\ln(8)^{\sqrt{x}}$ as $\sqrt{x}\ln(8)$. While it is true that $\ln(8)^{\sqrt{x}} = \sqrt{x}\ln(8)$, the equation you wrote is fals... |
3,746,986 | <p>Let <span class="math-container">$\phi:\mathbb (0,\infty) \to [0,\infty)$</span> be a continuous function, and let <span class="math-container">$c \in (0,\infty)$</span> be fixed.</p>
<p>Suppose that "<span class="math-container">$\phi$</span> is convex at <span class="math-container">$c$</span>". i.e. for... | Asaf Shachar | 104,576 | <p>This answer is merely an attempt to fill in the details in the example described <a href="https://math.stackexchange.com/a/3747376/104576">here</a>. Convexity of <span class="math-container">$\phi$</span> at <span class="math-container">$0$</span> means that</p>
<p><span class="math-container">$$
0=\phi(0) \leq \alp... |
4,218,513 | <p>This is an old Schwarz lemma problem from the August 2020 UMD qualifying exam for analysis, which is posted <a href="https://www-math.umd.edu/images/pdfs/quals/Analysis/Analysis-August-2020.pdf" rel="nofollow noreferrer">here</a>. The precise wording from the test is:</p>
<p>Suppose <span class="math-container">$f(z... | leoli1 | 649,658 | <p>Another, perhaps more direct approach, but which also doesn't really use the hint: Let <span class="math-container">$|a|<\frac{1}{8}$</span>. Consider <span class="math-container">$g_a = f\circ \phi_a$</span> where <span class="math-container">$\phi_a(z)=\frac{z-a}{1-\overline{a}z}$</span> as in your answer. If <... |
3,464,383 | <p>I couldn't find any substantial list of 'strange infinite convergent series' so I wanted to ask the MSE community for some. By <em>strange</em>, I mean infinite series/limits that <strong>converge when you would not expect them to and/or converge to something you would not expect</strong>.</p>
<p>My favorite conver... | Ben | 144,928 | <p>Let <span class="math-container">$x_n$</span> be the nth positive solution of <span class="math-container">$\csc(x)=x$</span>, i.e. <span class="math-container">$x_1\approx 1.1141$</span>, <span class="math-container">$x_2\approx 2.7726$</span>, etc. Then,</p>
<p><span class="math-container">$$\sum_{n=1}^{\infty}\f... |
35,579 | <p>Consider the kernel of the homomorphism from two copies of the free group
$F_2 \times F_2$ onto the integers sending every generator to 1. How to see that this subgroup is not finitely presented?</p>
| IJL | 556,441 | <p>Euler characteristics of classifying spaces give a way to see this. The Euler characteristic of the free group $F_2$ is -1, because the figure of 8
has two 1-cells and one 0-cell. Euler characteristics multiply for direct products, so the Euler characteristic of $F_2 \times F_2$ is 1. The Euler characteristic of... |
2,122,955 | <p><em>My question is from T.Y.Lam- A First Course in Noncommutative Rings.</em></p>
<blockquote>
<p>For a ring $R$, if all left $R$-modules are semisimple, then all short exact sequences of left $R$-modules split.</p>
</blockquote>
<p>A module $M$ is a semisimple module if every submodule $N$ of $M$ is a direct su... | Mustafa | 400,050 | <blockquote>
<p><strong>Theorem:</strong> Let be <span class="math-container">$R$</span> is a ring and <span class="math-container">$M,M_1,M_2$</span> are <span class="math-container">$R-$</span>module. If <span class="math-container">$f:M_1\to M$</span>, <span class="math-container">$g:M\to M_2$</span> are <span cla... |
818,850 | <p>I want to generate numbers $1$ to $10$ with uniform probability distribution. So I write the numbers $1$ to $10$ in the natural order. I keep writing the next $10$ number block by permuting the first $10$ numbers. Will this long string can be considered random with uniform distribution? Is there a problem?</p>
| Phil H | 11,736 | <p>The first thing to note is that a decimal representation is just a bunch of fractions; how many whole tenths, how many whole hundredths etc.</p>
<p>Now an irrational number is one that is never a whole number of these fractions, however small you make the fractions. </p>
<p>As someone pointed out, you can demonstr... |
818,850 | <p>I want to generate numbers $1$ to $10$ with uniform probability distribution. So I write the numbers $1$ to $10$ in the natural order. I keep writing the next $10$ number block by permuting the first $10$ numbers. Will this long string can be considered random with uniform distribution? Is there a problem?</p>
| Wilf | 123,393 | <p>As usual with visualizing things, a graph does help - here is a graph of
y=$\sqrt x$ <br><em>(y=$x^\frac {1}{2}$ if you prefer )</em>:</p>
<p><img src="https://i.stack.imgur.com/92fAh.png" alt="a graph of y=x^(1/2) should be here...."></p>
<p>As $x$ gets bigger, $\sqrt x$ gets bigger, but by smaller and smaller a... |
174,676 | <p>When I am faced with a simple linear congruence such as
$$9x \equiv 7 \pmod{13}$$
and I am working without any calculating aid handy, I tend to do something like the following:</p>
<p>"Notice" that adding $13$ on the right and subtracting $13x$ on the left gives:
$$-4x \equiv 20 \pmod{13}$$</p>
<p>so that $$x \eq... | CopyPasteIt | 432,081 | <p>When presented with</p>
<p><span class="math-container">$\tag 1 ax \equiv b \pmod{n}$</span></p>
<p>if <span class="math-container">$a \mid b$</span> the solution is right in front of you.</p>
<p>But there is also a 'plug in' solution if <span class="math-container">$a \mid n-1$</span> or <span class="math-containe... |
1,056,397 | <p>I saw the following problem:
$$\lim_{x\to \infty} \sqrt{9x^2+x}-3x$$
My first thought was to say that the $x$ term is overpowered when $x$ becomes large enough, so the square root becomes just $\sqrt{9x^2} = 3x$, and the value of the limit is zero.</p>
<p>However, the solution given is $1\over6$, and starts with:
$... | Mark Bennet | 2,906 | <p>You should note that $9x^2+x = (3x+\frac 16)^2-\frac 1{36}$ so that $\sqrt {9x^2+x}\approx 3x+\frac 16$ for large $x$</p>
<p>When you subtract $3x$ in the numerator, the dominant term in $x$ vanishes and the highest order term is $\frac 16$.</p>
<p>In the denominator, you add $3x$ and the highest order term is $6... |
248,167 | <p>I want to know why, when I look at the Julia sets of the quadratic family, I see only a finite number of repeating patterns, rather than a countable infinity of them.</p>
<p>My question is specifically about the interaction of these three theorems:</p>
<p><strong>Theorem 1</strong>: Let $z_0\in\mathbb{C}$ be an re... | Per Alexandersson | 1,056 | <p>To extend my comment and emphasize the self-similarity of Julia sets and the Dragon curve, here is an interpolation between the two.</p>
<p><a href="https://i.stack.imgur.com/7ANvL.gif" rel="noreferrer"><img src="https://i.stack.imgur.com/7ANvL.gif" alt="enter image description here"></a></p>
<p>Each frame is gene... |
248,167 | <p>I want to know why, when I look at the Julia sets of the quadratic family, I see only a finite number of repeating patterns, rather than a countable infinity of them.</p>
<p>My question is specifically about the interaction of these three theorems:</p>
<p><strong>Theorem 1</strong>: Let $z_0\in\mathbb{C}$ be an re... | Lasse Rempe | 3,651 | <p>Given that I have started making pictures, I thought it might be worthwhile adding another, shorter, direct answer to your questions, in addition to my longer, more detailed one.</p>
<p><strong>Question 1.</strong> Are the limits in your pictures the same (up to a linear map)?</p>
<p><strong>Answer.</strong> Yes. ... |
190,497 | <p>Q: Is there a reference for a detailed proof of <a href="http://en.wikipedia.org/wiki/Explicit_formula" rel="nofollow">Riemann's explicit formula</a> ?</p>
<p>I am reading the <a href="http://www.claymath.org/millennium/Riemann_Hypothesis/1859_manuscript/EZeta.pdf" rel="nofollow">TeXified version</a> of Riemann's m... | Raymond Manzoni | 21,783 | <p>I would recommend reading Edwards' excellent <a href="http://books.google.com/books?id=ruVmGFPwNhQC&pg=PA1" rel="nofollow">'Riemann's Zeta Function'</a> (it's <a href="http://rads.stackoverflow.com/amzn/click/0486417409" rel="nofollow">cheap</a> !) since it details every part of Riemann's article (with more rigo... |
2,026,406 | <p>As written in the title, does there exists a $\mathbb C$-Banach space isometric to a Hilbert Space but whose norm is not induced by an inner product?</p>
<p>Since an inner product in the Hilbert space has to fulfill the Parallelogram Identity, how could it be that such a $\mathbb C$-Banach space exists?</p>
| marco2013 | 79,890 | <p>Let $f:E \rightarrow F$ an isometry from $E$, a Banach space, to $F$, a Hilbert space with inner product $(x,y) \mapsto L(x,y)$. Then $\forall x \in E, \|f(x)\|_F=\|x\|_E$.</p>
<p>$\forall x,y \in F, \Re L(x,y)=\frac{1}{2}(\|x+y\|_F^2-\|x\|_F^2-\|y\|_F^2)$</p>
<p>because $\|x+y\|_F^2=L(x+y,x+y)=L(x,x)+L(x,y)+L(y,x... |
76,747 | <pre><code>Plot[D[Abs[x], x], {x, -10, 10}, Exclusions -> {0}]
</code></pre>
<p>gives out several lines of error messages and an empty plot.</p>
<pre><code>Plot[Derivative[1][Abs[#1] & ][x], {x, -10, 10}, Exclusions -> {0}]
</code></pre>
<p>just gives out an empty plot.</p>
<p>How do I plot $\left(\left|x... | Michael E2 | 4,999 | <p>To extend the other answers, if you're going to be using the derivative of <code>Abs</code> often in your computations and do not need the complex absolute value, then you can define the <a href="http://reference.wolfram.com/mathematica/ref/Derivative.html" rel="nofollow noreferrer"><code>Derivative</code></a> of <c... |
76,747 | <pre><code>Plot[D[Abs[x], x], {x, -10, 10}, Exclusions -> {0}]
</code></pre>
<p>gives out several lines of error messages and an empty plot.</p>
<pre><code>Plot[Derivative[1][Abs[#1] & ][x], {x, -10, 10}, Exclusions -> {0}]
</code></pre>
<p>just gives out an empty plot.</p>
<p>How do I plot $\left(\left|x... | m_goldberg | 3,066 | <p>You could rewrite it as</p>
<pre><code>Plot[Evaluate @ ComplexExpand @ D[Abs[x], x], {x, -10, 10}, Exclusions -> {0}]
</code></pre>
<p>But I would rewrite it as</p>
<pre><code>dAbs[x_] = ComplexExpand @ D[Abs[x], x];
Plot[dAbs[x], {x, -10, 10}, Exclusions -> {0}]
</code></pre>
<p>Both produce</p>
<p><img ... |
3,851,399 | <p>I am attempting to prove the following statement</p>
<blockquote>
<p>Prove that if n is any integer then 4 either divides n^2 or n^2 − 1</p>
</blockquote>
<p>I have started with the case of n = 2k</p>
<pre><code>Consider the case n = 2k
n = 2k
n^2 = 4k^2
⇒ n = 4k
∴ 4 divides n^2 as there is some integer k in w... | Anonymous | 772,237 | <p>Here's another simple proof using parity. <span class="math-container">$[$</span>Actually the same thing what @Ares did, but not taking the form of <span class="math-container">$2k$</span> or <span class="math-container">$(2k + 1)$$]$</span></p>
<p>Suppose <span class="math-container">$n$</span> is even, then <span ... |
413,982 | <p>How can I divide this using long division?</p>
<p>$$\frac{ax^3-a^2x^2-bx^2+b^2}{ax-b}$$</p>
<p><strong>Edit</strong></p>
<p>Sorry guys I wrote it wrong... Fixed it now.</p>
| Community | -1 | <p>Hint: See if $(ax-b)$ is a factor of $(ax^3-a^2x^2-bx^2+b^2)$ by writing:</p>
<p>$$(ax^3-a^2x^2-bx^2+b^2)=(ax-b)(\square x^2+\square x+\square)$$</p>
<p>and see if you can fill in the $\square$s.</p>
|
1,789,414 | <blockquote>
<p><strong>Question :</strong> Prove that the equation $x + \cos(x) + e^{x} = 0$ has
<em>exactly</em> one root</p>
</blockquote>
<hr>
<p><em>This is what I thought of doing:</em></p>
<p>$$\text{Let} \ \ \ f(x) = x + \cos(x) + e^{x}$$
By using the Intermediate Value Theorem on the open interval $(-\... | Michael Burr | 86,421 | <p>Hint:
Consider the derivative:
$$
1-\sin(x)+e^x.
$$
Since $1\geq \sin(x)$, $1-\sin(x)\geq0$. Therefore, the derivative is always positive. If there were two roots, then there would be a place where the derivative is zero. (why?)</p>
|
1,789,414 | <blockquote>
<p><strong>Question :</strong> Prove that the equation $x + \cos(x) + e^{x} = 0$ has
<em>exactly</em> one root</p>
</blockquote>
<hr>
<p><em>This is what I thought of doing:</em></p>
<p>$$\text{Let} \ \ \ f(x) = x + \cos(x) + e^{x}$$
By using the Intermediate Value Theorem on the open interval $(-\... | Z Ahmed | 671,540 | <p>Let <span class="math-container">$f(x)=x+\sin x + e^x$</span>, then <span class="math-container">$f'(x)=1-\cos x+ e^x>0$</span> implies that <span class="math-container">$f(x)$</span> is monotonicaly increasing so <span class="math-container">$f(x)=0$</span> has almost one real root. Since <span class="math-conta... |
1,792,422 | <p>I need help with an optimization problem. I have a rectangle space being fenced. Three sides are fenced with a material costing 4 dollars and the last side costs 16 dollars. I was given that the area is 250 and asked to minimize the cost.</p>
<p>I got what I think is the correct formula for the cost where $$C(y)=\... | Martin Argerami | 22,857 | <p>$$
\frac1{2\sqrt c}=\frac{\sqrt 4-\sqrt 0}{4-0}=\frac24=\frac12.
$$
Then
$
\sqrt c=1
$, so $c=1$. </p>
|
618,763 | <p>Let $\mathbb{F}$ be a field, $ \mathbb{R} = \mathbb{F}[X,X^{-1}]$ the ring of Laurent polynomials over $\mathbb{F}$</p>
<blockquote>
<blockquote>
<p>a) find the group of units in R.</p>
<p>b) find an Euclidean norm in R.</p>
</blockquote>
</blockquote>
<p>so for a, I understand the units are $uX^{... | 8k14 | 115,868 | <p><strong>Hint</strong> Prove that any $f\in R$ can be represented as $f=x^{-n}g$ for $g\in F[x]$. Then rewrite the condition that $f$ is a unit as a condition for polynomials. </p>
<p>As for b), as you know, $\deg$ is an Euclidean norm for $F[x]$. Try to find its analogue for $R$. For $f=x^{-n}g$ with $g\in F[x], g(... |
1,039,326 | <p>What is the fastest way to solve for $z^3 = -2 (1+i \sqrt 3) \bar z$?</p>
<p>I know how to do this using complex algebra. but that takes a long time.
Can someone show me a faster way?</p>
| AlexR | 86,940 | <p>Multiply by $z$ to obtain $z^4 = -2 (1+i\sqrt 3)|z|^2$. Looking at absolute values we see that $|z|^2 = \sqrt{4 + 4 \cdot 3} = \sqrt{16} = 4$, finally looking at angles we have
$$4 \arg z = \arg (-1-i\sqrt 3) = -\frac{2\pi}3$$
Combined we get $z = |z| e^{i\arg z} = 2 e^{-i\frac{\pi}6 + k \frac\pi2} = (1-\sqrt 3i) \c... |
1,460,704 | <p>Intuitively we know that $n^2$ grows faster than $n$, thus the difference tends to negative infinity. But I have trouble proving it symbolically because of the indeterminate form $\infty - \infty$. Is there anyway to do this without resorting to the Epsilon-Delta definition ?</p>
| Yes | 155,328 | <p>Let $M < 0$; then $n-n^{2} = n(1-n) < n(1-5) = -4n < M$ if $n > \max \{ 5, |M|/4 \}$.</p>
|
1,778,982 | <p>I can prove that $G$ is cyclic, but I am not sure how to prove the orders. I know I need to use the Fundamental Theorem of Cyclic Groups but I'm not sure how to apply it. Is there something obvious I am missing?</p>
| Domenico Vuono | 227,073 | <p>If $|G|=m$ for a theorem due to a Sylow for each integer $p^i$ with $i$ integer and $p$ prime exist a subgroup of order $p^i$. For this motive you have two possibilities :$pq$ or $p^3$</p>
|
382,295 | <p>Please help. I haven't found any text on how to prove by induction this sort of problem: </p>
<p>$$
\lim_{n\to +\infty}1 + \frac{1}{4} + \frac{1}{4^2} + \cdots+ \frac{1}{4^n} = \frac{4}{3}
$$</p>
<p>I can't quite get how one can prove such. I can prove basic divisible "inductions" but not this. Thanks.</p>
| Rayhunter | 76,236 | <ol>
<li><p>Induction is used to prove statements are true for all natural numbers.</p></li>
<li><p>You want to prove that this is true only for infinitely large n.</p></li>
</ol>
<p>That said, it should be clear that a mathematical induction is inapplicable here. </p>
<p>Trying to use induction to solve this problem... |
2,160,671 | <p>I got stuck with this problem.</p>
<p>\begin{matrix}
1 & 1 & 1 & 1\\
0 & 1 & 1 & 0\\
0 & 0 & 1 & 1\\
\end{matrix}</p>
<p>Consider the $3\times 4$ matrix $\bf A$ (above). Do the columns of $\bf A$ span $\mathbb R^3$? </p>
<p>Prove your answer. Als... | egreg | 62,967 | <p>You <em>can</em> use the augmented matrix method!
\begin{align}
\left[\begin{array}{cccc|ccc}
1 & 1 & 1 & 1 & 1 & 0 & 0 \\
0 & 1 & 1 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 & 1 & 0 & 0 & 1
\end{array}\right]
&\to
\left[\begin{array}{cccc|ccc}
1 & 1... |
2,160,671 | <p>I got stuck with this problem.</p>
<p>\begin{matrix}
1 & 1 & 1 & 1\\
0 & 1 & 1 & 0\\
0 & 0 & 1 & 1\\
\end{matrix}</p>
<p>Consider the $3\times 4$ matrix $\bf A$ (above). Do the columns of $\bf A$ span $\mathbb R^3$? </p>
<p>Prove your answer. Als... | krishna | 417,603 | <p>B is a 4*3 matrix. we will find the columns of B using the fact that the columns of AB are linear combinations of columns of A with column elements of B.
$\\1\begin{bmatrix}
1\\
0\\
0
\end{bmatrix}+0\begin{bmatrix}
1\\
1\\
0
\end{bmatrix}+0\begin{bmatrix}
1\\
1\\
1
\end{bmatrix}+0\begin{bmatrix}
1\\
0\\
1
\e... |
1,856,071 | <p>If I would extend two lines $l_1$ and $l_2$ they would intersect with an angle of 90 degrees. How should I write with math terms that there would be a 90 degree angle. I assume $l_1 \perp l_2$ is wrong if they do not intersect (when not extending them).</p>
<p>Is there a possibility to express a 90 degree angle bet... | Mike Pierce | 167,197 | <p>Generally <em>lines</em> are thought to go on indefinitely (they don't have endpoints), so the $l_1$ and $l_2$ you are describing should really be called <em>line segments</em>. With that in mind you could say something like:</p>
<blockquote>
<p>Let $\ell_1$ and $\ell_2$ be the lines that contain the line segment... |
2,314,369 | <p>My objective is to represent a unit square using a single matrix.<br>The four corners of the unit square are represented by column vectors as follows,</p>
<p>$\left[\begin{matrix} 0 \\ 0 \end{matrix}\right],\left[\begin{matrix} 1 \\ 0 \end{matrix}\right],\left[\begin{matrix} 1 \\ 1 \end{matrix}\right],\left[\begin{... | Christian Sykes | 322,386 | <p>You can't multiply
$$X = \begin{bmatrix}0&1&1&0\\
0&0&1&1
\end{bmatrix}$$
by a $2\times2$ matrix $A$ on the right, but you can on the <em>left</em>. In fact,
if $x_1, \dots, x_k$ are $n\times 1$ vectors, $A$ is any $n\times n$ matrix, and
$$X = \begin{bmatrix}x_1&\dots&x_k\end{bmatrix... |
4,094,333 | <blockquote>
<p>Suppose <span class="math-container">$a_1,a_2>0$</span> and
<span class="math-container">$a_{n+2}=2+\dfrac{1}{a_{n+1}^2}+\dfrac{1}{a_n^2}(n\ge 1)$</span>. Prove
<span class="math-container">$\{a_n\}$</span> converges.</p>
</blockquote>
<p>First, we may show <span class="math-container">$\{a_n\}$</spa... | xpaul | 66,420 | <p>Note that <span class="math-container">$2<a_n<3$</span> for <span class="math-container">$n\ge5$</span>. Since, for <span class="math-container">$n\ge5$</span>,
<span class="math-container">\begin{eqnarray}
&&|a_{n+3}-a_{n+2}|\\
&=&\bigg|\frac{1}{a_{n+2}^2}-\frac{1}{a_{n}^2}\bigg|=\frac{(a_{n+... |
2,572,564 | <p>So I have a straight line, in the classic <span class="math-container">$y = mx + b$</span>, and I'm just trying to translate the formula for the line a certain distance along its normal.</p>
<p>For example, with <a href="https://i.stack.imgur.com/r8vct.jpg" rel="nofollow noreferrer">this graph</a>, how would I tran... | user | 505,767 | <p>By <strong>first order expansion</strong>
$$\frac{log(1+\frac{x^4}{3})}{sen^6(x)} =\frac{\frac{x^4}{3}+o(x^6)}{x^6+o(x^6)} \to +\infty$$</p>
<p>By <strong>standard limits</strong>
$$\frac{log(1+\frac{x^4}{3})}{\frac{x^4}{3}} \frac{\frac{x^4}{3}}{x^6}\frac{x^6}{sen^6(x)}\to +\infty$$</p>
|
581,599 | <p>Is it generally true that
$$\frac{\mathrm d}{\mathrm da} \int_{-\infty}^{a-y} f(x)\, \mathrm dx = f(a-y),$$
where $a$ and $y$ in the expression are constants?</p>
<p>To give context to the question, I am reading about the function convolution of two independent random variables $X, Y$. </p>
| Eastsun | 3,599 | <p>Here is a proof using the definition of derivative:</p>
<p>$$\begin{eqnarray*}\frac{d}{d a}\int_{\infty}^{a-y}f(x)dx &=& \lim_{t\to0}\frac{\int_{-\infty}^{a-y}f(x)dx-\int_{-\infty}^{a-y-t}f(x)dx}{t}\\
&=& \lim_{t\to0}\frac{\int_{a-y-t}^{a-y}f(x)dx}{t} \quad \textbf{mean value theorem for integration... |
1,656,294 | <p>Apart from the Dirchet function ( equals 1 for rational x and 0 for irrational x), and similar constructs (equals some constant a for rational x and b for irrational , where a!=b) , what are the other functions which are defined on all real numbers and discontinuous at all points?</p>
<p>Are there infinite such fun... | jjagmath | 571,433 | <p>Set <span class="math-container">$y = \tan x$</span>. Then <span class="math-container">$y \in (0,1)$</span> and</p>
<ul>
<li><span class="math-container">$t_1 = y^y$</span></li>
<li><span class="math-container">$t_2 = y^{1/y}$</span></li>
<li><span class="math-container">$t_3 = y^{-y}$</span></li>
<li><span class="... |
1,427,992 | <blockquote>
<p>Let $k$ be a divisor of $n^4+n^3+n^2+n+1$ , then either $k$ or $k-1$ is divisible by $5$.</p>
</blockquote>
<p>I solved this problem by considering the prime factorization of $k$ and showing it to be either $5$ or $1$ modulo $5$ with the help of Fermat's Little Theorem and an elementary divisibility ... | f10w | 31,498 | <p>We will show that <span class="math-container">$$f(x)+f^*(u) = \langle x,u \rangle \Longleftrightarrow u \in \partial f(x).$$</span></p>
<p>Indeed, we have
<span class="math-container">\begin{align*}
u \in \partial f(x) &\Longleftrightarrow f(z) \geq f(x) + \langle u, z-x\rangle \quad\forall z \\
&\Longleftr... |
1,427,992 | <blockquote>
<p>Let $k$ be a divisor of $n^4+n^3+n^2+n+1$ , then either $k$ or $k-1$ is divisible by $5$.</p>
</blockquote>
<p>I solved this problem by considering the prime factorization of $k$ and showing it to be either $5$ or $1$ modulo $5$ with the help of Fermat's Little Theorem and an elementary divisibility ... | Svetoslav | 254,733 | <p>Let $f:X\rightarrow \mathbb R$ and $X^*$ is the dual of $X$.
First recall the <strong>definition of $f^*: X^*\rightarrow \mathbb R$</strong></p>
<p>$f^*(u^*)=\sup\limits_{x\in X}{\{\langle u^*,x\rangle - f(x)\}}$, where $u^*\in X^*$ and $\langle .,.\rangle$ is the duality pairing in $X^*\times X$.</p>
<p>Then the ... |
833,353 | <p>Let $a,b \geq 0$ and $0<p,q < 1$ s.t. $p + q = 1$.</p>
<p>Is it true that $a^p b^q \leq a+b$?</p>
| PhoemueX | 151,552 | <p>Regarding the edited question, assume w.l.o.g. That $a \leq b$.</p>
<p>Then</p>
<p>$$a^p b^q \leq b^p b^q = b \leq a+b.$$</p>
|
833,353 | <p>Let $a,b \geq 0$ and $0<p,q < 1$ s.t. $p + q = 1$.</p>
<p>Is it true that $a^p b^q \leq a+b$?</p>
| Alexander Vigodner | 92,276 | <p>Since for $1/q+1/q=1$
$$
\frac{a^p}{p}+\frac{b^q}{q}\ge ab
$$
We can redefine $\tilde p=1/p$, $\tilde q=1/q$, $\tilde a=a^p$, $\tilde b =b^q$
Then the above inequality implies
$$
\tilde a+\tilde b\ge p \tilde a +q \tilde b \ge \tilde a ^{\tilde p} b^{\tilde q}
$$
REMARK: I believe $p,q>0$ and not $>1$ as wr... |
2,675,758 | <p>For example:</p>
<p>There are two green marbles and two red marbles in a bag. Two marbles are chosen at random, what is the probability that the two marbles which have been chosen are of the same colour?</p>
<p>Ordered + distinct: <br>
Marbles = $G1, G2, R1, R2$ <br>
Event space = $\{|G1~G2|, |G2~G1|, |R1~R2|, |R2... | Community | -1 | <p>$$x^2+4x+4=100$$</p>
<p>$$x^2+4x-96=0$$</p>
<p>$$(x-8)(x+12)=0$$</p>
<p>So, $x=8$ and $x=-12$.</p>
<p>$8>0$, but $-12 \ngtr 0$, so applying the restriction, $x=8$</p>
|
2,614,776 | <p><strong>Exercise :</strong></p>
<blockquote>
<p><strong>(A.1)</strong> Let G be a group. The subgroup <span class="math-container">$Z(G) = \{z \in G | zg =gz \space \forall \space g \in G\}$</span> is called the center of <span class="math-container">$G$</span>. Show that <span class="math-container">$Z(G)$</span> i... | Nick A. | 412,202 | <p>A2:Use the relations $σ^3=e,τ^2=ε,στσ=τ^{-1}$.</p>
<p>A4: Do you know what conjugate ellements is and cycle representation? If yes then this is a very easy excersise sincethe center $Z(G)$ consists of precisly the ellements only conjugate to themselves. So if there is an element $σ$ in $S_n$ then this means that it... |
270,875 | <p>When I run</p>
<pre><code>FullSimplify[Sqrt[(1 - Sqrt[1 - 2 x^2 y^2])^2], {1 > 2 x^2 y^2 > 0}]
</code></pre>
<p>I obtain <code>1 - Sqrt[1 - 2 x^2 y^2]</code>. However, when I run</p>
<pre><code>FullSimplify[ExpandAll[Sqrt[(1 - Sqrt[1 - 2 x^2 y^2])^2]], {1 > 2 x^2 y^2 > 0}]
</code></pre>
<p>I am never abl... | Akku14 | 34,287 | <p>Eliminate the term you want to ged rid of with Solve</p>
<pre><code>f = Sqrt[(1 - Sqrt[1 - 2 x^2 y^2])^2];
fe = ExpandAll[f];
fsol /. First@
Solve[{fe == fsol, g == Sqrt[1 - 2 x^2 y^2], g >= 0}, fsol, {g},
Reals] // FullSimplify[#, 1 > 2 x^2 y^2 > 0] &
(* 1 - Sqrt[1 - 2 x^2 y^2] *)
</code... |
3,489,896 | <p>It is clear to me that <span class="math-container">$\liminf X_n \le \limsup X_n$</span> but every time that I see that <span class="math-container">$\liminf X_n \le \sup X_n$</span> it sound to me not so obvious awkward. Could someone help me to understand this inequality?<span class="math-container">$ $</span> <sp... | mathcounterexamples.net | 187,663 | <p><strong>Hint</strong></p>
<p>As <span class="math-container">$\limsup X_n \le \sup X_n$</span>, this second inequality <span class="math-container">$\liminf X_n \le \sup X_n$</span> should be obvious to you if <span class="math-container">$\liminf X_n \le \limsup X_n$</span> is.</p>
<p>So you're left to prove and ... |
115,630 | <p>Let $V$ and $W$ be two algebraic structures, $v\in V$, $w\in W$ be two arbitrary elements.</p>
<p>Then, what is the geometric intuition of $v\otimes w$, and more complex $V\otimes W$ ? Please explain for me in the most concrete way (for example, $v, w$ are two vectors in 2 dimensional vector spaces $V, W$)</p>
<p>... | Michael Hardy | 11,667 | <p>The difference between the ordered pair $(v,w)$ of vectors and the tensor product $v\otimes w$ of vectors is that for a scalar $c\not\in\{0,1\}$, the pair $(cv,\;w/c)$ is different from the pair $(v,w)$, but the tensor product $(cv)\otimes(w/c)$ is the same as the tensor product $v\otimes w$.</p>
|
115,630 | <p>Let $V$ and $W$ be two algebraic structures, $v\in V$, $w\in W$ be two arbitrary elements.</p>
<p>Then, what is the geometric intuition of $v\otimes w$, and more complex $V\otimes W$ ? Please explain for me in the most concrete way (for example, $v, w$ are two vectors in 2 dimensional vector spaces $V, W$)</p>
<p>... | Mnifldz | 210,719 | <p>The way I like to think about this is that if we have two vector spaces $V$ and $W$, their tensor product $V \otimes W$ intuitively is the idea "for each vector $v\in V$, attach to it the entire vector space $W$." Notice however that since we have $V\otimes W \cong W \otimes V$ we can rephrase this as the idea "for... |
115,630 | <p>Let $V$ and $W$ be two algebraic structures, $v\in V$, $w\in W$ be two arbitrary elements.</p>
<p>Then, what is the geometric intuition of $v\otimes w$, and more complex $V\otimes W$ ? Please explain for me in the most concrete way (for example, $v, w$ are two vectors in 2 dimensional vector spaces $V, W$)</p>
<p>... | rehctawrats | 407,385 | <blockquote>
<p>geometric intuition of <span class="math-container">$v\otimes w$</span> [...] in the most concrete way</p>
</blockquote>
<p>Let me try to throw in an answer on <strong>freshmen level</strong>. I deliberately try to leave out any information that is not essential to the problem, e.g. don't bother abou... |
3,649,125 | <p>Bridge is a game of four players in which each player is dealt 13 cards from a standard 52 card deck. Bridge players (such as myself) are interested in the number of possible deals, where each player is distinct. This can be counted by</p>
<p><span class="math-container">$$\binom{52}{13}\binom{39}{13}\binom{26}{13}... | Ross Millikan | 1,827 | <p>There are <span class="math-container">$16$</span> cards of interest, so distributing them would give you <span class="math-container">$4^{16}$</span> possibilities. The problem is that no hand can contain more than <span class="math-container">$13$</span> cards, so we subtract off the ones that have a hand of more... |
1,200,878 | <p>Integrate the following integral:</p>
<p>$$\int \frac{\sin x (2 \cos x - \sin x)}{2\sin x + \cos x} dx$$</p>
<p>I have tried it by using by parts by considering the $\sin x$ as first function. Again in the following step i got stuck. Please help me.</p>
| Elaqqad | 204,937 | <p><strong>Hints</strong> First you can notice that:</p>
<p>$$\frac{\sin x (2 \cos x - \sin x)}{2\sin x + \cos x}=\cos x-\frac{1}{2\sin x + \cos x} $$</p>
<p>and remember that it's very useful to make use of $t=tan(x/2)$ for the second integral, and you can find the following primitive :</p>
<p>$$\sin(x)-\frac{1}{\s... |
3,081,918 | <p>It was claimed <a href="https://math.stackexchange.com/a/2387782/446262">here</a> that the convergence of the series<span class="math-container">$$\sum_{n=2}^\infty \frac{\Lambda(n)-1}{n^{1/2}\log^3 n}\tag1$$</span>(where <span class="math-container">$\Lambda$</span> is the <a href="https://en.wikipedia.org/wiki/Von... | Conrad | 298,272 | <p>A good place to start is by reading Terry Tao's post <a href="https://terrytao.wordpress.com/2013/07/19/the-riemann-hypothesis-in-various-settings/" rel="nofollow noreferrer">"The Riemann hypothesis in various settings"</a></p>
|
509,487 | <p>Prove $1+{n \choose 1}2+{n \choose 2}4+...+{n \choose n-1}2^{n-1}+{n \choose n}2^n=3^n$ using combinatorial arguments. I have no idea how to begin solving this, a nudge in the right direction would be appreciated. </p>
| robjohn | 13,854 | <p>Using red, green, and blue, how many ways can you paint a fence with $n$ pickets?</p>
<p><strong>Hint:</strong> Count the number of ways to paint the fence if $k$ pickets are to be red or green. There are $\binom{n}{k}$ ways to select $k$ pickets to be painted red or green and for each of those choices, there are $... |
1,457,797 | <p>I'm told that a circle intersects the y-axis at $(0,0)$ and $(0, -4)$.</p>
<p>I have a tangent that starts at $(0, -6)$. I want to find the point of intersection.</p>
<p>I am taking the midpoint of the circle to be $(0, -2)$ and the radius to be $2$ from the points above.</p>
<p>The equation of the circle then is... | BLAZE | 144,533 | <blockquote>
<p>I am taking the equation of the line to be ${y = -6}$.</p>
<p>If I substitute ${y = -6}$ into ${x^2 + y^2 + 4y = 0}$ I get</p>
<p>${x^2 + 36 - 24 = 0}$</p>
<p>${x^2 = -12}$</p>
<p>I think I have gone wrong somewhere.</p>
</blockquote>
<p>You have not done anything wrong, the reaso... |
4,135,473 | <p>Can you explain how the lower integral is the supremum of a sum? Isn't <span class="math-container">$L(f,P)$</span> just a sum and therefore a single number? Definition 5.13 says from <span class="math-container">$a$</span> to <span class="math-container">$b$</span>, not from each <span class="math-container">$x_j$<... | WhatsUp | 256,378 | <p>For <span class="math-container">$x > 0$</span>:</p>
<p><span class="math-container">$[2x]$</span> is the number of positive integers <span class="math-container">$\leq 2x$</span>. Count them by separating the even and odd numbers.</p>
<p>For general <span class="math-container">$x$</span>: add a sufficiently lar... |
3,629,379 | <p>I'm trying to understand the following proof that a space <span class="math-container">$X$</span> is compact if and only if every net has a cluster point. I have a specific confusion with how cluster points relate to closure which is expanded on after the proof. </p>
<p>A cluster point is defined as </p>
<p><a hre... | Jacob Manaker | 330,413 | <p>Yes, there are counterexamples. Apart from the trivial group, there is also the two element group <span class="math-container">$C_2=\{e,a\}$</span>, where <span class="math-container">$a^2=e$</span>. In this group, <span class="math-container">$a$</span> is the unique non-identity element and <span class="math-conta... |
2,596,347 | <p>Suppose $X$ is compact and Hausdorff and that $f:X \to Y$ is continuous, closed, and surjective. How can I show that $Y$ is Hausdorff?</p>
| Henno Brandsma | 4,280 | <p>Let $y_1 \neq y_2$ be distinct points of $Y$. By surjectivity we have $x_1 ,x_2 \in X$ such that $f(x_1) = y_1 , f(x_2) = y_2$. In particular, $f[\{x_i\}] = y_i$ for both $i$ and as in $X$ singletons are closed (Hausdorff implies $T_1$) and $f$ is a closed map (first time we use it), we have that both $\{y_i\}$ are ... |
1,580,736 | <p>I know that $\arcsin(x) + \arccos(x) = \frac{\pi}2$,</p>
<p>but how to use that to solve the following question?</p>
<p>$$2\arcsin(x)-3\arccos(x)=\frac{\pi}6 $$</p>
| Narasimham | 95,860 | <p>Eliminate either arcsin or arcos.</p>
<p>If arcsin is eliminated, using both equations and simplifying $ x= \cos (\pi/6)= \sqrt3 /2. $</p>
<p>(The first one is an identity. The second one is an equation to be solved for $x$. An identity can be used to <em>simplify</em> part of the equation but not <em>solve</em> i... |
116,220 | <p>Say two graphs are not isomorphic but are both strongly regular with the same set of parameters. Are there any parameters (other than the usual such as order, degrees, eigenvalues and multiplicities, etc.) that are determined, e.g., independence number, chromatic number, etc.?</p>
<p>Thanks for any help</p>
| GeoffDS | 13,886 | <p>Okay, well, I checked Brouwer's website and combined that with the <a href="https://mathoverflow.net/questions/39312/smallest-non-isomorphic-strongly-regular-graphs">comment</a> to the accepted answer of a question on this site. I checked the complement of the Shrikhande graph versus the complement of the line grap... |
116,220 | <p>Say two graphs are not isomorphic but are both strongly regular with the same set of parameters. Are there any parameters (other than the usual such as order, degrees, eigenvalues and multiplicities, etc.) that are determined, e.g., independence number, chromatic number, etc.?</p>
<p>Thanks for any help</p>
| Dima Pasechnik | 11,100 | <p>It's a classic result that a graph parameter called <i>Lovasz theta-function</i> $\theta(\Gamma)$ of a strongly regular graph $\Gamma$ is determined by its parameters. And the significance of $\theta(\Gamma)$ is that it is <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v1i1a1" rel="nofollow">"... |
4,412,371 | <p>Would appreciate some help with proving the following statement:</p>
<p>Let <span class="math-container">$a_{n}$</span> be a sequence, and <span class="math-container">$\lim _{n\rightarrow \infty }a_{n}=1 $</span>.</p>
<p>Prove that <span class="math-container">$\lim _{n\rightarrow \infty }a_{1}+a_{2}+\ldots +a_{n}=... | Falcon | 766,785 | <p>If <span class="math-container">$a_n \to 1$</span> then there is <span class="math-container">$N \in \mathbb N$</span> such that <span class="math-container">$a_n \ge 1/2$</span> for all <span class="math-container">$n \ge N$</span>. Therefore, for <span class="math-container">$n \ge N$</span>,
<span class="math-con... |
290,507 | <p>Every metric space is associated with a topological space in a canonical way. According to <a href="http://topospaces.subwiki.org/wiki/Induced_topology_from_metric_is_functorial" rel="nofollow">this</a> source, this amounts to a <a href="http://en.wikipedia.org/wiki/Full_and_faithful_functors" rel="nofollow">full fu... | Martin Brandenburg | 1,650 | <p>Let $(X,d)$ be a metric space. Then the canonical topology $\tau_{can}$ is the coarsest topology on $X$ which makes (with the product topology) $d : X \times X \to \mathbb{R}$ continuous.</p>
<p>Proof: It follows from the triangle inequality that $d$ is continuous in the canonical topology (in fact short when we en... |
290,507 | <p>Every metric space is associated with a topological space in a canonical way. According to <a href="http://topospaces.subwiki.org/wiki/Induced_topology_from_metric_is_functorial" rel="nofollow">this</a> source, this amounts to a <a href="http://en.wikipedia.org/wiki/Full_and_faithful_functors" rel="nofollow">full fu... | Ittay Weiss | 30,953 | <p>The open ball topology is unique in the following sense. Firstly, following on ideas of Flagg one can consider a generalization of metric spaces where instead of taking values in <span class="math-container">$[0,\infty ]$</span> the metric takes values in a value quantale. If <span class="math-container">$Met$</span... |
4,502,175 | <p>I want to create a simple demo of moving an eye (black circle) inside a bigger circle with a black stroke when moving a mouse. I have cursor positions mouseX and mouseY on a canvas and I need to map the value of mouse position into a circle so the eye is moving inside the circle.</p>
<p>This should be trivial but I ... | peterwhy | 89,922 | <p>By your use of the <a href="https://p5js.org/reference/#/p5/map" rel="nofollow noreferrer">map</a> function on both <span class="math-container">$x$</span> and <span class="math-container">$y$</span>, the centre of the black circle (with diameter <span class="math-container">$20$</span>) will be limited to a rectang... |
1,241,586 | <p>I want to prove that for all positive integers
$n$, $2^1+2^2+2^3+...+2^n=2^{n+1}-2$.
By mathematical induction:</p>
<p>1) it holds for $n=1$, since $2^1=2^2-2=4-2=2$</p>
<p>2) if $2+2^2+2^3+...+2^n=2^{n+1}-2$, then prove that $2+2^2+2^3+...+2^n+2^{n+1}=2^{n+2}-2$ holds.</p>
<p>I have no idea how to proceed wit... | Michael Hoppe | 93,935 | <p>Define $S=2^1+\cdots+2^n$ and realize that $2S=2^2+\cdots+2^{n+1}=S+2^{n+1}-2$.</p>
|
81,016 | <p>I'm trying to learn about derived categories of algebraic stacks. To be honest, as of now, I don't need anything fancy nor deep. In my setup I have a scheme $X$ (well, a smooth and projective variety over $\mathbb{C}$) and a finite group $G$ acting on it. I would like to understand simple things like: what does a co... | Yosemite Sam | 16,857 | <p>In case anyone is interested, I've been finding this thesis useful
<a href="http://tobias-lib.uni-tuebingen.de/volltexte/2007/2941/pdf/diss.pdf" rel="nofollow">http://tobias-lib.uni-tuebingen.de/volltexte/2007/2941/pdf/diss.pdf</a></p>
|
2,437,466 | <p>A point, $p$, is defined as a cluster point of a set $S$ if $\forall \epsilon > 0,$ there exists an open ball of radius $\epsilon$ centered at $p$ that contains infinitely many points in $S$.</p>
<p>We want to prove that a subset $S$ of a metric space $(E,d)$ is closed iff $S$ contains all of its cluster points.... | Jaroslaw Matlak | 389,592 | <p>As your first steps of mathematical induction are right, I'll just perform a proof of inductive thesis for $n=k+1$.</p>
<p>First notice, that for $k>1$
$$\frac{4(k+1)}{(k+2)} < \frac{(2k+2)(2k+1)}{(k+1)^2}$$
proof:
$$(k-1)k(k+1)^2(k+2)>0\\
\frac{k(k-1)}{(k+1)^2(k+2)}>0\\
\frac{4k^2-4k}{(k+1)^2(k+2)}>... |
128,009 | <p>Hi,
Is there any known sequence such that the sum of a combination of one subsequence never equals another subsequence sum. The subsequences should have elements only from the parent sequence.</p>
<p>Thanks
Sundi</p>
| Pedro Lauridsen Ribeiro | 11,211 | <p>I have never seen such a characterization (and I'm inclined to agree with André that there may not be one at the same level of simplicity as in the case of invariant differential operators), but my best bet towards such one would be to employ a Beals-Cordes-type characterization of pseudodifferential operators: a bo... |
128,009 | <p>Hi,
Is there any known sequence such that the sum of a combination of one subsequence never equals another subsequence sum. The subsequences should have elements only from the parent sequence.</p>
<p>Thanks
Sundi</p>
| Dick Palais | 7,311 | <p>It seems to me that an interesting "first step" or sub-problem---and one that should be fairly straightforward, is to characterize the space of invariant symbols of pseudo differential operators on a Lie group (or more generally on a homogeneous space of a Lie group).</p>
|
1,504,888 | <p>Let $R$ be an integral domain. The following is a well known fact:</p>
<blockquote>
<p>Let $a,b \in R$. Then $(a)=(b)$ if and only if there exist a unit $u \in R$ such that $a=ub$.</p>
</blockquote>
<p>I would like to generalize this result for ideals which are generated by two elements. So the question is:</p>
... | Olivier Bégassat | 11,258 | <p>Concerning domains, there is a nice partial result covering the case where the ideal is principal.</p>
<blockquote>
<p><strong>Lemma.</strong> Let <span class="math-container">$R$</span> be a domain, and let <span class="math-container">$a,b,x\in R$</span> be elements of <span class="math-container">$R$</span>. The ... |
81,261 | <p>We have $Z_1,Z_2,Z_3$ are independent standard normal random variables.
Find</p>
<p>a) $\mathbb P(Z_1<Z_2+Z_3)$</p>
<p>b) $\mathrm{Var}(Z_1Z_2^2)$</p>
<p>c) $\mathbb P(Z_1/Z_2>1)$</p>
<p>d) $\mathbb P(Z_1^2>Z_2^2+Z_3^2)$</p>
| Did | 6,179 | <p>Re d), $(Z_2,Z_3)$ is a random point of the complex plane whose module and argument are independent, the argument being uniformly distributed and the module $R=\sqrt{Z_2^2+Z_3^2}$ having density
$$
u(r)=r\mathrm e^{-r^2/2}\cdot[r\gt0].
$$
<strong>First application:</strong> For every $r\gt0$,
$$
\mathrm P(R^2\... |
392,297 | <p>For which sets of primes <span class="math-container">$P$</span> is there a finite type <span class="math-container">$\mathbb{Z}$</span>-algebra <span class="math-container">$A$</span> such that<span class="math-container">$$p\in P\iff\mathrm{Hom}(A, \mathbb{F}_p)=\emptyset?$$</span>Do all the finite <span class="ma... | R.P. | 17,907 | <p>Here's a sketch of an answer. I think the answer is that you can get three types of sets: (i) finite sets, (ii) co-finite sets, and (iii) sets of the form
<span class="math-container">$$
S_f = \{ p : f(x) ~ \textrm{has a root in $\mathbb{F}_p$}\} - S_0
$$</span>
for some polynomial <span class="math-container">$f \i... |
1,628,616 | <p>I need to prove that the set of functions of the form $\{g\colon [-1,1]\to \mathbb{R}\, |$ there exists a constant $c$ such that $g(x) = c\cdot x\}$ is complete. How shall I proceed? I am taking a Cauchy sequence and want to prove that it converges to a function in this set, but how can I guarantee the limit will ha... | HK Lee | 37,116 | <p>$$ B= \{ (x_1,\cdots,x_{n+1} )\in {\bf R}^{n+1} | \sum_i x_i^2 =1,\
x_i\geq 0 \} $$</p>
<p>So we have $$ f: B\rightarrow B',\ f(x_1,\cdots,x_{n+1} ) =
(x_1,\cdots, x_n,0) $$</p>
<p>Fix $p=(\epsilon,\cdots,\epsilon,0)\in B'$ Define $$ C:=
\{(x_1,\cdots, x_n,0) \in B'| \sum_i x_i^2=1 \} $$</p>
<p>Consider $$ c(t)= ... |
1,507,859 | <blockquote>
<p>$$\int \tan \sqrt {x} \,dx$$</p>
</blockquote>
<p>I was trying to solve this. But it took very long time and three pages. Could someone please tell me how to solve this quickly. </p>
| Zhanxiong | 192,408 | <p>Since $a > 1$, we can write $a = 1 + \delta$ for some $\delta > 0$, by binomial formula, for each $n$,
$$a^n = (1 + \delta)^n = 1 + n\delta + \cdots + \delta^n > 1 + n\delta.$$
Since $\{n\delta\}$ is unbounded (for fixed $\delta$) by Archimedean property of real numbers, the result follows. </p>
|
1,507,859 | <blockquote>
<p>$$\int \tan \sqrt {x} \,dx$$</p>
</blockquote>
<p>I was trying to solve this. But it took very long time and three pages. Could someone please tell me how to solve this quickly. </p>
| Paramanand Singh | 72,031 | <p>This is another approach which uses proof by contradiction. Suppose that the sequence $s_{n} = a^{n}$ is bounded above. Since $a > 1$ we can see that $$s_{n + 1} = a^{n + 1} = a\cdot a^{n} = as_{n} > s_{n}$$ so that the sequence $s_{n}$ is strictly increasing. Since it is also bounded above it follows that the... |
1,391,014 | <p>Let $f,g:[0,1] \rightarrow [0,1]$ be continuous functions such that $f\circ g =g\circ f$. Prove that there exists $x \in [0,1]$ such that $f(x)=g(x)$</p>
| PhoemueX | 151,552 | <p>This answer was inspired by the one of @PaulSinclair.</p>
<p>Let us assume that the claim is false, so that $f\left(x\right)\neq g\left(x\right)$
for all $x\in I:=\left[0,1\right]$. By swapping $f,g$, we can assume
$f\left(0\right)<g\left(0\right)$. If we had $f\left(x\right)\geq g\left(x\right)$
for some $x\in ... |
73,767 | <p>I have the following expressions $g^{ab+cd+...}$ or in the full form <code>Power[g,Plus[Times[a,b],Times[c,d],...]]</code></p>
<p>How to convert this expression into</p>
<p><code>Power[g,Times[a,b]]Power[g,Times[c,d]]...</code>?</p>
<p>Here $...$ could be either nothing or one more, but of course it would be nice... | ybeltukov | 4,678 | <p>These forms are equivalent from mathematics point of view. So if you want to <em>display</em> exponents separately you can use</p>
<pre><code>MakeBoxes[Power[z_, Plus@x__], StandardForm] :=
RowBox[SuperscriptBox[ToBoxes@z, ToBoxes@#] & /@ {x}]
g^(a b + c d)
</code></pre>
<p><img src="https://i.stack.imgur.c... |
2,636,929 | <p>I know that it can be obtained by simply differentiating the equation and finding the roots of the derivative, but it is a lengthy and tricky process. I am looking for a faster and more straightforward way.</p>
<p>A more effective and quick way to find the answer via simple differentiation will also be appreciated.... | J.G. | 56,861 | <p>Let $y=\arcsin x$, so you're trying to extremise $y^4+(\frac{\pi}{2}-y)^4$. The first two derivatives with respect to $y$ are proportional to $y^3-(\frac{\pi}{2}-y)^3,\,y^2+(\frac{\pi}{2}-y)^2$. The former vanishes when $y=\frac{\pi}{4}$ i.e. $x=\frac{1}{\sqrt{2}}$, while the latter is non-negative so we have obtain... |
3,113,243 | <p>Let <span class="math-container">${\mathcal D} = C^\infty_c$</span>, the space of <em>test functions</em> (smooth functions <span class="math-container">$\Bbb R \to \Bbb C$</span> with compact support). Let <span class="math-container">$\mathcal D^*$</span> be the space of <em>distributions</em> (continuous linear ... | Aphelli | 556,825 | <p><span class="math-container">$$|\rho(x)-\rho_n(x)|=\left|\sum_m{\int_{m/n}^{(m+1)/n}{(\varphi(v)\psi(x-v)-\varphi(m/n)\psi(x-m/n))\,dv}}\right| \leq \sum_m{\int_{m/n}^{(m+1)/n}{|\varphi(v)\psi(x-v)-\varphi(v)\psi(x-m/n)+\varphi(v)\psi(x-m/n)-\varphi(m/n)\psi(m/n)|\, dv}}$$</span></p>
<p>Thus </p>
<p><span class="m... |
3,113,243 | <p>Let <span class="math-container">${\mathcal D} = C^\infty_c$</span>, the space of <em>test functions</em> (smooth functions <span class="math-container">$\Bbb R \to \Bbb C$</span> with compact support). Let <span class="math-container">$\mathcal D^*$</span> be the space of <em>distributions</em> (continuous linear ... | Greg Grunberg | 129,384 | <p>I didn't quite understand Mindlack's answer, but it did suggest some critical steps which I've been able to exploit to form my own answer:</p>
<p>Write
<span class="math-container">$$\rho(x) := \int_{-M}^M {\varphi(v) \psi(x-v) \, dv} = \sum_{m=-Mn+1}^{Mn} \int_{(m-1)/n}^{m/n} {\varphi(v) \psi(x-v) \, dv},$$</span>... |
40,473 | <p>I found this « Worm on the rubber band » problem in Concrete Mathematics book.</p>
<p>A slow worm $W$ starts at one end of a meter-long rubber band and crawls one centimetre per minute toward the other end.</p>
<p>At the end of each minute, a keeper of the band $K$ stretches it one meter.</p>
<p>Does the worm eve... | Jean-Pierre | 6,240 | <p>Thanks to Lubos, I get the following results:</p>
<p>With the data of the question: rubber band of length L, u(t)=a*t and v(t)=b*(t) with a et b constants, the insect will ALWAYS reach the end of a rubber band in a time:</p>
<p>T= [(2.L)/b * ( Exp(b/a) -1) ] ^(1/2)</p>
<p>If u is constant and v= b(t) with b con... |
55,853 | <p>Given a function $f:X\to X$, let $\text{Fix}(f)=\{x\in X\mid x=f(x)\}$.</p>
<p>In a recent <a href="https://math.stackexchange.com/questions/55846/is-the-set-of-fixed-points-in-a-non-hausdorff-space-always-closed/55847#55847">comment</a>, I wondered whether
$X$ is Hausdorff $\iff$ $\text{Fix}(f)\subseteq X$ is clos... | Sam | 3,208 | <p>Let me propose the following counterexample:</p>
<p>Take $X = \overline{\mathbb Q}$ the one-point compactification of $\mathbb Q$. This space is not Hausdorff, since $\mathbb Q$ is not locally compact (the problem is $\infty$).</p>
<p>Now let $f: \overline{\mathbb Q} \to \overline{\mathbb Q}$ be a continuous funct... |
2,124,560 | <p>A pentagon ABCDE contains 5 triangles whose areas are each one. The triangles are ABC, BCD, CDE, DEA, and EAB. Find the area of ABCDE?</p>
<p>Is there a theorem for overlapping triangle areas?</p>
| Jack D'Aurizio | 44,121 | <p>If $ABC$ and $BCD$ have the same area, then $BC\parallel AD$ and so on. It follows that $ABCDE$ is not necessarily a regular pentagon, but it is for sure the image of a regular pentagon through an affine map. Affine maps preserve the ratios between areas, hence it is enough to understand what is the area of a regula... |
1,001,913 | <p>If we want to graph a horizontal line, we will do the following:</p>
<pre><code>y = 0x + 3
</code></pre>
<p>No matter the domain for x, the range for y will always be 3. Therefore, we have a horizontal line.</p>
<pre><code>y = 0(0) + 3 = (0,3)
y = 0(1) + 3 = (1,3)
y = 0(2) + 3 = (2,3)
</code></pre>
<p>Now the fo... | Brian M. Scott | 12,042 | <p>There is a fairly easy combinatorial proof. We count the number of subsets of $\{0,1,\ldots,n\}$ of size $m+1$ in two ways. First, of course, there are $\binom{n+1}{m+1}$ of them. Now suppose that $S=\{a_0,\ldots,a_m\}$ is such a subset, where $a_0<\ldots<a_m$. Clearly $a_k\ge k$; let $t=a_k-k\ge 0$. Moreover,... |
1,816,783 | <p>When can one conclude that a character table has non-real entries?</p>
<p>In particular, by constructing the character table for $\mathbb{Z}/3\mathbb{Z}$ or $A_4$ how does one determine that some of the entries will be nonreal?
Is the reason that the same complex values in the table for $\mathbb{Z}/3\mathbb{Z}$ als... | Gerry Myerson | 8,269 | <p>Characters of degree 1 are homomorphisms from the group to the multiplicative group of nonzero complex numbers. It follows that if, say, $g$ is a group element of order 3, then $(\chi(g))^3=1$, so $\chi(g)$ is 1, $\omega=e^{2\pi i/3}$, or $\omega^2$. Once you have found all the linear characters with $\chi(g)=1$, if... |
1,653,569 | <p>Consider, in the first-order NGB theory of sets, the following axioms:
$$\vdash\exists x\forall y(y\notin x)$$
and
$$\vdash\forall y(y\notin\varnothing)$$</p>
<p>The second one defines the constant $\varnothing$.
Moreover, from the second axiom we get the first by $\exists$I inference rule.</p>
<p>So my question i... | Mauro ALLEGRANZA | 108,274 | <p>Yes, but we have to assume the individual constant $\emptyset$ as primitive in the language.</p>
<p>The way used e.g. into Mendelson's textbook [see 6th ed., 2015, page 231] is to formalize $\mathsf {NBG}$ in a f-o language with a single predicate letter $A^2_2$ (i.e. $\in$) but no function letter or individual con... |
1,356,585 | <p>I am reading Hajime Sato's: <em>Algebraic Topology, an Intuitive Approach</em>. His Sample Problem 1.3 is: <em>Show that the topological spaces X and I are not homeomorphic.</em> (Note that this requires a font where I appears as a simple straight vertical line.) </p>
<p>Sato argues by contradiction. Brief sketc... | Ivo Terek | 118,056 | <p>The issue is that connectedness is preserved by homeomorphisms. More exactly, two homeomorphic spaces must have the same number of connected components. If $X \cong I$, then $X \setminus \{\rm crossing~point\} \cong I \setminus \{\rm one ~point\}$. But $X \setminus \{\rm crossing~point\}$ has four connected componen... |
748,083 | <p>I'm trying to find the Jordan Normal Form of the matrix $A = \begin{bmatrix} 1 & 0 & 1 & -1 \\0 & 1 & 1 & 0\\ 0 & 0 & 1 & 1\\ 0 & 0 & 0 & 1\end{bmatrix}$ </p>
<p>(That is, a matrix J and a matrix S such that $A = S J S^{-1})$.</p>
<p>Finding the eigenvalues and eigen... | Scrooge McDuck | 83,481 | <p>The value of the function in 0 is 0 for every a. The function is continuous, so you could do the derivative and use binomial theorem to prove that for every x > 0 is positive and for every x < 0 is negative.</p>
<p>EDIT: <a href="https://i.imgur.com/CZcJutJ.jpg" rel="nofollow noreferrer">http://i.imgur.com/CZcJu... |
2,354,094 | <p>I've been stumped by this problem:</p>
<blockquote>
<p>Find three non-constant, pairwise unequal functions $f,g,h:\mathbb R\to \mathbb R$ such that
$$f\circ g=h$$
$$g\circ h=f$$
$$h\circ f=g$$
or prove that no three such functions exist.</p>
</blockquote>
<p>I highly suspect, by now, that no non-trivial ... | lulu | 252,071 | <p>An example:</p>
<p>$$f(x) =
\begin{cases}
x, & \text{if $x\in \mathbb Q$} \\
-x, & \text{if $x\notin \mathbb Q$}
\end{cases}$$</p>
<p>$$g(x) =
\begin{cases}
-x, & \text{if $x\in \mathbb Q$} \\
x, & \text{if $x\notin \mathbb Q$}
\end{cases}$$</p>
<p>$$h(x)=-x$$</p>
|
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