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 |
|---|---|---|---|---|
674,310 | <p>I am having trouble with a proof for linear algebra. Could somebody explain to me how to prove that if $A$ and $B$ are both $n\times n$ non singular matrices, that their product $AB$ is also non singular. </p>
<p>A place to start would be helpful. Thank you for your time. </p>
| Ben Grossmann | 81,360 | <p>Note that a matrix is non-singular if and only if it has an inverse.</p>
<p>Suppose $A$ and $B$ have inverses $A^{-1} B^{-1}$. What do you get when you multiply
$$
(AB)(B^{-1}A^{-1})
$$
and why can we now conclude that $AB$ is non-singular?</p>
|
2,310,109 | <p>I'm an undergraduate with little to no background in functional analysis and topology. The whole concept of function spaces is quite fuzzy to me, and I'm having a difficult time conceptualizing it. (Things like there being different notions of compactness in general topological spaces is one of many things confusing... | Prahlad Vaidyanathan | 89,789 | <p>What you have shown is that $G$ is <em>sequentially closed</em>, which may not imply that $G$ is <em>closed</em>. The two concepts coincide for metric spaces, but not in general. </p>
<p>Here is a standard counterexample which fits your situation as well: if $A = \mathbb{R}, a=0,b=1$, and $G$ is the set of all func... |
2,798,207 | <p>This problem needs also to be extended to $n*m$ chessboard. I tried to think like this:</p>
<p>First I choose a place for the first king in $64$ ways. Then I have a choice $64-5 = 59$ squares for the second king . But this solution is not right because this is not the case if I place the first king in the sidemost... | nonuser | 463,553 | <p>You have ${64\choose 2} = 32\cdot 63=2016$ ways to put $2$ kings on board without restriction (since they are the same). Now we have $7\cdot 8\cdot 2+7\cdot 7\cdot 2 =210$ adjacent pairs of squares (I count that pairs directly, in one row you have $7$ adjacent pairs and we have 8 rows, the same holds for columns, an... |
2,798,207 | <p>This problem needs also to be extended to $n*m$ chessboard. I tried to think like this:</p>
<p>First I choose a place for the first king in $64$ ways. Then I have a choice $64-5 = 59$ squares for the second king . But this solution is not right because this is not the case if I place the first king in the sidemost... | Юрій Ярош | 517,661 | <p>We will solve the problem by cases.<br>
1) We put the first king on the corner square. Then there are 4 ways to do it, and for the second king we have 64-3=61. Then there are $4\cdot61=244$ ways to do it.<br>
2) We put the first king on the edge square. Then for the first king $6+6+6+6=24$ ways to do it, then for th... |
2,798,207 | <p>This problem needs also to be extended to $n*m$ chessboard. I tried to think like this:</p>
<p>First I choose a place for the first king in $64$ ways. Then I have a choice $64-5 = 59$ squares for the second king . But this solution is not right because this is not the case if I place the first king in the sidemost... | Barry Cipra | 86,747 | <p>If you want "non-attacking" placements, in which "adjacent" includes diagonally adjacent, then the answer is</p>
<p>$$4(64-4)+4\cdot6(64-6)+6\cdot6(64-9)\over2$$</p>
<p>That is, the "first" king goes either in one of $4$ corners and the "second" king avoids it and the $3$ adjacent squares, or it goes in one of the... |
322,302 | <p>Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch with what happens in the other fields.</p>
<p><strong>Question</strong> What are the conjectures in your field prove... | Hailong Dao | 2,083 | <p>The <a href="https://en.wikipedia.org/wiki/Homological_conjectures_in_commutative_algebra" rel="noreferrer">homological conjectures in commutative algebra</a> using perfectoid methods. A survey on many recent developments written by André can be found <a href="https://arxiv.org/pdf/1811.09843.pdf" rel="noreferrer">h... |
322,302 | <p>Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch with what happens in the other fields.</p>
<p><strong>Question</strong> What are the conjectures in your field prove... | kodlu | 17,773 | <p>Konstantin Tikhomirov recently <a href="https://arxiv.org/pdf/1812.09016.pdf" rel="noreferrer">proved</a> that the probability that a random <span class="math-container">$n\times n$</span> Bernoulli matrix <span class="math-container">$M_n$</span> with independent <span class="math-container">$\pm 1$</span> entries,... |
322,302 | <p>Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch with what happens in the other fields.</p>
<p><strong>Question</strong> What are the conjectures in your field prove... | Michael | 38,448 | <p>Manolescu refuted the Triangulation Conjecture. The paper is</p>
<blockquote>
<p><em>Pin(2)-equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture</em>, J. Amer. Math. Soc. <strong>29</strong> (2016), 147-176, doi:<a href="https://doi.org/10.1090/jams829" rel="noreferrer">10.1090/jams829</a>, ... |
322,302 | <p>Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch with what happens in the other fields.</p>
<p><strong>Question</strong> What are the conjectures in your field prove... | kodlu | 17,773 | <p>Turan's book <em>On a new method of analysis and its applications</em> focuses on bounds on power sums. The quantity
<span class="math-container">$$
T(m,n)=\inf_{|z_k|=1} \max_{\nu=1,\ldots,m} \left| \sum_{k=1}^n z_k^\nu\right|,
$$</span>
for various choices of <span class="math-container">$m,n$</span> has been of i... |
322,302 | <p>Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch with what happens in the other fields.</p>
<p><strong>Question</strong> What are the conjectures in your field prove... | Louis D | 17,798 | <p>In 2016, Andrew Suk (nearly) solved the <a href="http://www.quantamagazine.org/a-puzzle-of-clever-connections-nears-a-happy-end-20170530" rel="nofollow noreferrer">"happy ending" problem</a>; that is, he proved (<em>On the Erdős-Szekeres convex polygon problem</em>, J. Amer. Math. Soc. <strong>30</strong> ... |
322,302 | <p>Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch with what happens in the other fields.</p>
<p><strong>Question</strong> What are the conjectures in your field prove... | ARG | 18,974 | <p>In 2019 <a href="https://arxiv.org/abs/1802.09077" rel="nofollow noreferrer">Anna Erschler and Tianyi Zheng</a> gave a very sharp estimate of the growth of Grigorchuk's first group. Although it was one of the first example of a group of intermediate growth (finitely generated group whose growth is neither polynomial... |
3,882,566 | <p>We have <span class="math-container">$0<b≤ a$</span>, and:</p>
<p><span class="math-container">$$\underbrace{\dfrac{1+⋯+a^7+a^8}{1+⋯+a^8+a^9}}_{A} \quad \text{and} \quad \underbrace{\dfrac{1+⋯+b^7+b^8}{1+⋯+b^8+b^9}}_{B}$$</span></p>
<p>Source: Lumbreras Editors</p>
<hr />
<p>It was my strategy:</p>
<p><span clas... | Calvin Lin | 54,563 | <p>Step 1: Take the inverses, so we want to compare</p>
<p><span class="math-container">$$ \frac { 1+a+\ldots a^9} { 1+a+\ldots a^8 } \text{ vs } \frac{1+b+\ldots b^9 }{1+b+\ldots b^8 } . $$</span></p>
<p>Step 2: Subtract 1 from both sides, we want to compare</p>
<p><span class="math-container">$$ \frac { a^9} { 1+a+\l... |
3,479,883 | <p>I know that (I might be wrong):</p>
<ul>
<li>Symbol for empty or null set : {Ø} or {}</li>
<li>Null or empty set is 'subset of all sets' as well as 'empty or null set' set</li>
<li>So, { {} } is same as { Ø }</li>
</ul>
<p>I just want to know { {} } or { Ø } is an empty set or not ? And if yes then we can conclud... | nonuser | 463,553 | <p>Yes they are the same and it is a set witch contains an empty set as an element.</p>
<p>But symbols {Ø} and {} are not the same. Second one is an empty set or <span class="math-container">$\emptyset$</span> </p>
|
3,479,883 | <p>I know that (I might be wrong):</p>
<ul>
<li>Symbol for empty or null set : {Ø} or {}</li>
<li>Null or empty set is 'subset of all sets' as well as 'empty or null set' set</li>
<li>So, { {} } is same as { Ø }</li>
</ul>
<p>I just want to know { {} } or { Ø } is an empty set or not ? And if yes then we can conclud... | Ethan Bolker | 72,858 | <p>The sets you have written down are the same. Neither is empty since each has just one element: the empty set.</p>
<p>Related: <a href="https://math.stackexchange.com/questions/2620616/what-is-the-difference-between-x-and-x-when-x-itself-is-a-set/2620621#2620621">What is the difference between $x$ and $\{x\}$ when $... |
1,015,826 | <p>For $r>1$, prove the sequence $$X_n=\left(1+r^n\right)^{1/n}$$ is decreasing. I understand the limit is decreasing and that the limit of this sequence is $r$. I am just not sure on the algebra. My thought is to show $X_n>X_{n+1}$ by showing $X_n-X_{n+1}>0$ for all $n$. I could also use induction; however, I... | user84413 | 84,413 | <p>Since $\displaystyle(1+r^n)^\frac{1}{n}=\bigg(r^n\bigg[\left(\frac{1}{r}\right)^n+1\bigg]\bigg)^\frac{1}{n}=r\big(1+s^n\big)^\frac{1}{n}$ where $s=\frac{1}{r}$ satisfies $0<s<1$,</p>
<p>$\hspace{.3 in}$it suffices to show that $\big(1+s^n\big)^{1/n}>(1+s^{n+1})^{1/(n+1)}$ for $0<s<1$:</p>
<p>Since $... |
1,015,826 | <p>For $r>1$, prove the sequence $$X_n=\left(1+r^n\right)^{1/n}$$ is decreasing. I understand the limit is decreasing and that the limit of this sequence is $r$. I am just not sure on the algebra. My thought is to show $X_n>X_{n+1}$ by showing $X_n-X_{n+1}>0$ for all $n$. I could also use induction; however, I... | user84413 | 84,413 | <p>It is enough to show that $(1+r^n)^{n+1}>(1+r^{n+1})^n,\;\;\;$ since then $(1+r^n)^\frac{1}{n}>(1+r^{n+1})^\frac{1}{n+1}$:</p>
<p>$\displaystyle\big(1+r^n\big)^{n+1}-\big(1+r^{n+1}\big)^n=\sum_{k=0}^{n+1}\binom{n+1}{k}(r^n)^k-\sum_{j=0}^n\binom{n}{j}(r^{n+1})^j$</p>
<p>$\displaystyle=\sum_{k=1}^n\binom{n+1}{... |
4,182,153 | <p>Let A be a nonempty compact subset of <span class="math-container">$R$</span> (real numbers) and let B be a nonempty closed subset of R. Recall
that <span class="math-container">$\operatorname{dist}(A, B) = \inf{|x − y| : x ∈ A, y ∈ B}$</span>. Show that there exist <span class="math-container">$a ∈ A$</span> and <s... | Yao Zhao | 889,365 | <p>Proof: From the basic properties of the infimum, we know that we can find elements in a set that are arbitrarily close to its infimum. Then there exists a sequence <span class="math-container">$(|x_n-y_n|)$</span> such that <span class="math-container">$\lim_{n\to \infty} |x_n-y_n|= dis(A,B)$</span> where <span clas... |
4,062,242 | <p>Is <span class="math-container">$$\int_1^\infty \frac{\log x}{x^2}dx$$</span>
finite? How to solve this?</p>
| Community | -1 | <p>If you don't like the <span class="math-container">$\log$</span>, then you can turn it back to exponent by setting <span class="math-container">$u = \log x \implies x = e^u\implies dx = e^udu\implies I = \displaystyle \int_{0}^\infty ue^{-2u}(e^{u}du)= \displaystyle \int_{0}^\infty ue^{-u}du= -\displaystyle \int_{0}... |
542,454 | <p>There are two tangent lines on $f(x) = \sqrt{x}$ each with the $x$-value $a$ and $b$ respectively. </p>
<p>I need to prove that $c$, the $x$ value of the point at which the two lines intersect each other, is equal to $\sqrt{ab}$, the geometric mean of $a$ and $b$. </p>
<p>I have been trying many different ways of ... | André Nicolas | 6,312 | <p>The tangent line at $(a,\sqrt{a})$ has equation $y=\frac{1}{2\sqrt{a}}x+\frac{\sqrt{a}}{2}$. </p>
<p>The tangent line at $(b,\sqrt{b})$ has equation $y=\frac{1}{2\sqrt{b}}x+\frac{\sqrt{b}}{2}$. </p>
<p>Set $\frac{1}{2\sqrt{a}}x+\frac{\sqrt{a}}{2}=\frac{1}{2\sqrt{b}}x+\frac{\sqrt{b}}{2}$ and solve for $x$.</p>
|
2,855,411 | <p>Find all real number(s) $x$ satisfying the equation $\{(x +1)^3\}$ = $x^3$ , where $\{y\}$ denotes the fractional part of $y$ , for example $\{3.1416\ldots\}=0.1416\ldots$.</p>
<p>I am trying all positive real numbers from $1,2,\dots$ but I didn't get any decimals.</p>
<p>Is there a smarter way to solve this pr... | Karn Watcharasupat | 501,685 | <p>Fractional part is always in $[0,1)$. So the domain of $g(x)=x^3$ you are looking for is such that $R_g\in[0,1)$, which happens to be $[0,1)$.</p>
<p>$y=\{(x+1)^3\}$ is basically $y=(x+1)^3$ chopped into appropriate pieces and translated down by some $k\in\mathbb{Z}$ where $k=\lfloor(x+1)^3\rfloor$</p>
<p>So we so... |
3,726,382 | <p>Prove that the series converge using direct comparison or limit comparison
<span class="math-container">$$\sum \limits_{n=1}^{+\infty} \frac{2^n}{n!}.$$</span></p>
<p>I really don't know how to proceed with the comparison tests though I know how to prove its convergence using ratio test.</p>
| Weierstraß Ramirez | 174,035 | <p>Perhaps it is useful to notice that for an <span class="math-container">$n$</span> large enough:</p>
<p><span class="math-container">$$\left(\frac{2}{n}\right)^{n}=\frac{2^n}{n^n}>\frac{2^n}{n!}>\frac{1}{n!}$$</span></p>
<p>After that, one might proceed by observing that:</p>
<p><span class="math-container">$$... |
3,726,382 | <p>Prove that the series converge using direct comparison or limit comparison
<span class="math-container">$$\sum \limits_{n=1}^{+\infty} \frac{2^n}{n!}.$$</span></p>
<p>I really don't know how to proceed with the comparison tests though I know how to prove its convergence using ratio test.</p>
| CNiD | 788,800 | <p>So basically this series is taylor expansion of <span class="math-container">$e^2$</span>.</p>
<p>We can prove it's convergence using Ratio test.</p>
<p><span class="math-container">$\lim_{n\rightarrow\infty}\frac{a_{n+1}}{a_n}=\lim_{n\rightarrow\infty}\frac{2^{n+1}n!}{(n+1)!2^n}=\lim_{n\rightarrow\infty}\frac{2}{n+... |
131,294 | <p>How do I show that $ f(t) = t^2 + t +1 $ is irreducible in $K[t]$, where $K = \{0,1\}$?</p>
<p>I know how to tackle this over $\mathbb{Z}$ or $\mathbb{Q}$ using Guass or Eisenstein say...but I'm a little unsure how to proceed in this case.</p>
<p>Any help is much appreciated.</p>
| Prasad G | 25,314 | <p>Suppose $f(t)$ is reducible.(then we have to show that it is contradiction)</p>
<p>$f(t) = (t+a)(t+b)$ where a and b are in $K$</p>
<p><code>Case 1:</code> $a =0,b=0$</p>
<p>$f(t)= t^2$. This is contradiction.</p>
<p>Similarly we can prove remaining cases.</p>
<p><code>Case 2:</code> $a=1,b=1$</p>
<p><code>Cas... |
925,541 | <p>The exercise states: prove that the limit of the sequence $$a_{n+2}=(a_na_{n+1})^{1/2} \ where \ a_1 \ge 0, a_2 \ge 0 $$</p>
<p>is $L = (a_1a_2^2)^{1/3}$</p>
<p>The solutin says: $$Let \ b_n = \frac{a_{n+1}}{a_n},$$ then $$b_{n+1}= 1/\sqrt{b_n} \ for \ all \ n$$ wich implies that
$$b_{n+1}= b_1^{(-1/2)^n} \rightar... | Dave | 174,047 | <p>Multiply both sides of</p>
<p>$$\prod_{j=2}^{n+1}b_j = \prod_{j=1}^{n}b_j^{-1/2} $$</p>
<p>by $\prod_{j=1}^{n}b_j^{1/2}$ to get</p>
<p>$$b_{n+1}b_1^{1/2} \left(\prod_{j=2}^{n}b_j\right)^{3/2} = b_{n+1}b_1^{1/2} \prod_{j=2}^{n}b_j^{3/2} = 1.$$</p>
<p>This can be rewritten as </p>
<p>$$b_{n+1} (a_2/a_1)^{1/2} (a_... |
2,094,123 | <p>A plane curve is printed on a piece of paper with the directions of both axes specified. How can I (roughly) verify if the curve is of the form $y=a e^{bx}+c$ without fitting or doing any quantitative calculation?</p>
<p>For example, for linear curves, I can choose two points on the curve and check if the midpoint ... | user541686 | 4,890 | <h3>You can't. No, not just "in theory", but also in practice.</h3>
<p>I tried this when doing regression before and I gave up on it once I realized how impossible it is:</p>
<p><a href="https://i.stack.imgur.com/FNYE2.png" rel="noreferrer"><img src="https://i.stack.imgur.com/FNYE2.png" alt="Curve" /></a></p>... |
129,875 | <p>The Fourier transform of the Heaviside step function $u(t)$ <a href="http://fourier.eng.hmc.edu/e101/lectures/handout3/node3.html" rel="nofollow noreferrer">is</a> $\dfrac{1}{iω} + π δ(ω)$.<br>
The Laplace transform of the same function <a href="http://leevaraiya.org/releases/LeeVaraiya_DigitalV2_02.pdf#page=569" re... | Tom Copeland | 27,786 | <p>Actually they do match in the sense that the Laplace transform provides an analytic continuation of the Fourier transform result to the complex plane. Look at the limits of the real and imaginary parts of</p>
<p>$\frac{1}{s}=\frac{s^{*}}{|s|^2}=\frac{\sigma-i\omega}{\sigma^2+\omega^2}$</p>
<p>as the real part of $... |
129,875 | <p>The Fourier transform of the Heaviside step function $u(t)$ <a href="http://fourier.eng.hmc.edu/e101/lectures/handout3/node3.html" rel="nofollow noreferrer">is</a> $\dfrac{1}{iω} + π δ(ω)$.<br>
The Laplace transform of the same function <a href="http://leevaraiya.org/releases/LeeVaraiya_DigitalV2_02.pdf#page=569" re... | user541686 | 4,890 | <p><em>(Realizing 6 years later that I never accepted an answer...)</em></p>
<p>The answer was that <strong>the premise of my question was simply false</strong>.</p>
<p>The (bilateral) Laplace transform of the unit-step function is <strong><em>not</em></strong> $\dfrac{1}{s}$ everywhere.<br>
Rather, the statement <a ... |
354,250 | <p><strong>Remark:</strong> All the answers so far have been very insightful and on point but after receiving public and private feedback from other mathematicians on the MathOverflow I decided to clarify a few notions and add contextual information. 08/03/2020.</p>
<h2>Motivation:</h2>
<p>I recently had an interesting... | Community | -1 | <blockquote>
<p>Is it possible to accurately simulate any non-trivial physics without computing partial derivatives?</p>
</blockquote>
<p>Yes. An example is the nuclear shell model as formulated by Maria Goeppert Mayer in the 1950's. (The same would also apply to, for example, the <a href="https://en.wikipedia.org/wiki... |
354,250 | <p><strong>Remark:</strong> All the answers so far have been very insightful and on point but after receiving public and private feedback from other mathematicians on the MathOverflow I decided to clarify a few notions and add contextual information. 08/03/2020.</p>
<h2>Motivation:</h2>
<p>I recently had an interesting... | Mozibur Ullah | 35,706 | <p>I'd query the contention that organisms or even inorganic matter compute in the sense described.</p>
<p>For example, if I drop a stone on the surface of the earth, it falls in a straight line. To call this as 'computing' a straight line seems rather a stretch of the word computation; to my thinking, to compute, mea... |
229,161 | <p>A sequence of positive integer is defined as follows</p>
<blockquote>
<ul>
<li>The first term is $1$.</li>
<li>The next two terms are the next two even numbers $2$, $4$.</li>
<li>The next three terms are the next three odd numbers $5$, $7$, $9$.</li>
<li>The next $n$ terms are the next $n$ even numbers if... | Hagen von Eitzen | 39,174 | <p>The general term is
$$\tag1a_n=2n-\left\lceil \sqrt{2 n}-\frac12\right\rceil.$$
Why?
We have $a_{n+1}=a_{n}+2$ unless there is an integer $\ge\frac12 +\sqrt{2n}$ and $<\frac12+\sqrt{2(n+1)}$.
We need $a_{n+1}=a_n+1$ iff $n$ is one of the numbers $1, 3, 6, 10, \ldots$, i.e. a number of the form $k \choose 2$. This... |
229,161 | <p>A sequence of positive integer is defined as follows</p>
<blockquote>
<ul>
<li>The first term is $1$.</li>
<li>The next two terms are the next two even numbers $2$, $4$.</li>
<li>The next three terms are the next three odd numbers $5$, $7$, $9$.</li>
<li>The next $n$ terms are the next $n$ even numbers if... | Arthur | 15,500 | <p><a href="http://oeis.org/A001614" rel="nofollow">http://oeis.org/A001614</a> has a few formulas:
$$
a_n = 2n - \left \lfloor \frac{1+ \sqrt{8n-7}}{2} \right\rfloor
$$</p>
|
3,605,636 | <p>Let <span class="math-container">$m_{a},m_{b},m_{c}$</span> be the lengths of the medians and <span class="math-container">$a,b,c$</span> be the lengths of the sides of a given triangle , Prove the inequality : </p>
<p><span class="math-container">$$m_{a}m_{b}m_{c}\leq\frac{Rs^{2}}{2}$$</span></p>
<p>Where : </p>
... | Michael Rozenberg | 190,319 | <p>In the standard notation we need to prove that:
<span class="math-container">$$\frac{1}{8}\sqrt{\prod_{cyc}(2a^2+2b^2-c^2)}\leq\frac{1}{2}\cdot\frac{abc}{4S}\cdot\frac{(a+b+c)^2}{4}$$</span> or
<span class="math-container">$$a^2b^2c^2(a+b+c)^3\geq\prod_{cyc}(2a^2+2b^2-c^2)\prod_{cyc}(a+b-c).$$</span>
Now, let <span ... |
138,243 | <p><a href="https://i.stack.imgur.com/kRmeb.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/kRmeb.png" alt="Area and Perimeter"></a></p>
<p>How can I draw the figure shown above in rectangular coordinates, calculate the area and perimeter of the shaded region as a function of radius <code>r</code> o... | yode | 21,532 | <p>Show it:</p>
<pre><code>RegionPlot[
region = RegionUnion[
Sequence @@
RegionIntersection @@@
Subsets[{Disk[{-1, 0}], Disk[{0, -1}], Disk[{1, 0}],
Disk[{0, 1}]}, {2}],
Fold[RegionDifference, {Disk[{0, 0}, 2], Disk[{-1, 0}],
Disk[{0, -1}], Disk[{1, 0}], Disk[{0, 1}]}]], Frame -> Fa... |
138,243 | <p><a href="https://i.stack.imgur.com/kRmeb.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/kRmeb.png" alt="Area and Perimeter"></a></p>
<p>How can I draw the figure shown above in rectangular coordinates, calculate the area and perimeter of the shaded region as a function of radius <code>r</code> o... | bobbym | 8,585 | <p>Your question wants a relationship for r which I assume is the radius of the larger circle. You can get it like this:</p>
<pre><code>c1 = ImplicitRegion[(x - r)^2 + y^2 <= r^2, {x, y}];
c2 = ImplicitRegion[x^2 + (y - r)^2 <= r^2, {x, y}];
Assuming[r > 0, Area[RegionIntersection[c1, c2]]]
</code></pre>
<p>... |
138,243 | <p><a href="https://i.stack.imgur.com/kRmeb.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/kRmeb.png" alt="Area and Perimeter"></a></p>
<p>How can I draw the figure shown above in rectangular coordinates, calculate the area and perimeter of the shaded region as a function of radius <code>r</code> o... | ubpdqn | 1,997 | <p>Just for fun. By symmetry we need only consider the first quadrant.</p>
<pre><code>Graphics[{EdgeForm[Black], FaceForm[None], Disk[{0, 0}, 2, {0, Pi/2}],
Disk[{1, 0}, 1, {0, Pi}], Disk[{0, 1}, 1, {-Pi/2, Pi/2}],
Line[{{1, 0}, {1, 1}}], Line[{{0, 0}, {1, 1}}],
Line[{{0, 1}, {1, 1}}],
Text[Style["A", 20], ... |
344,725 | <p>in $\Delta ABC$,and </p>
<p>$$\dfrac{\sin{(\dfrac{B}{2}+C)}}{\sin^2{B}}=\dfrac{\sin{(\dfrac{C}{2}+B)}}{\sin^2{C}}$$</p>
<p>prove that $B=C$</p>
<p>I think $\sin{(\dfrac{B}{2}+C)}\sin^2{C}=\sin{(\dfrac{C}{2}+B)}\sin^2{B}$</p>
<p>then
$$\sin{(\dfrac{B}{2})}\cos{C}\sin^2{C}+\cos{\dfrac{B}{2}}\sin^3C=\sin{\dfrac{C}{... | Christian Blatter | 1,303 | <p>The following picture suggests that the statement might be wrong. It shows that near $\beta=\gamma=1.417079$ there are angles $\beta\ne\gamma$ satisfying the stated equality.</p>
<p><img src="https://i.stack.imgur.com/NVxai.jpg" alt="enter image description here"></p>
|
842,271 | <p>Evaluation of $\displaystyle \int \frac{\sqrt[3]{x+\sqrt[4]{x}}}{\sqrt{x}}dx$</p>
<p>$\bf{My\; Try::}$ Let $x=t^4\;,$ Then $dx = 4t^3dt$</p>
<p>So Integral is $\displaystyle \int\frac{\sqrt[3]{t^4+t}}{t^2} \cdot 4t^3dt$</p>
<p>So Integral is $\displaystyle 4\int t^{\frac{7}{3}}\cdot (1+t^{-3})^{\frac{1}{3}}$</p>
... | achille hui | 59,379 | <p>Let $\mathcal{I}$ be the integral. You can actually evaluate it using the substitution $x = t^4$.</p>
<p>$$\mathcal{I} =
\int\frac{\sqrt[3]{x + \sqrt[4]{x}}}{\sqrt{x}}dx
= \int\frac{\sqrt[3]{(t^3 + 1)t}}{t^2}4t^3dt
= 4\int\sqrt[3]{1+t^3}t^{4/3}dt
$$
For $|x| < 1$, we can expand the integrand at RHS using follow... |
2,471,680 | <p>I am working with a theorem and i need the reference of the above limit.
Kindly guide.</p>
| Peter Szilas | 408,605 | <p>$0<k<1$, then $1/k >1.$</p>
<p>$1/k= :(1+x)$, with $x \gt 0.$</p>
<p>$(1/k)^n = (1+x)^n \ge 1 +nx.$
(Bernouilli's inequality)</p>
<p>Let $M \gt 1/\epsilon.$</p>
<p>Choose $n_0$ such that $1+n_0x \gt M.$
(Archimedes)</p>
<p>For $n \ge n_0 :$</p>
<p>$(1/k)^n = (1+x)^n \ge 1+nx \gt M \gt 1/\epsilon$,<... |
2,293,600 | <p>How to calculate X $\cap$ $\{X\}$ for finite sets to develop an intuition for intersections?</p>
<p>If $X$ = $\{$1,2,3$\}$, then what is $X$ $\cap$ $\{X\}$? </p>
| Dragonite | 395,130 | <p>As far as developing intuition for intersection, the idea of $A \cap B$ are the elements that $A$ and $B$ both have in common. So if we're looking at $X \cap \left\{ X \right\}$ where $X = \left\{ 1,2,3 \right\}$ then it is a matter of
$$X \cap \left\{ X \right\} = \left\{ 1,2,3 \right\} \cap \left\{ \left\{ 1,2,3 ... |
2,293,600 | <p>How to calculate X $\cap$ $\{X\}$ for finite sets to develop an intuition for intersections?</p>
<p>If $X$ = $\{$1,2,3$\}$, then what is $X$ $\cap$ $\{X\}$? </p>
| fleablood | 280,126 | <p>$\{X\}$ contains one element and one element only. So as $E \cap F \subset F$ we know $E \cap \{X\} \subset \{X\}$. So either $E \cap \{X\} = \{X\}$ if $X \in E$ or $E \cap \{X\} = \emptyset$ if $X \not \in E$. </p>
<p>It violates the axioms of set theory to have a set such that $X \in X$ (a set can't be an elem... |
3,613,120 | <p>In elementary algebra and beyond, we are taught to use a sequence of equations to derive a relationship. For instance, to show that <span class="math-container">$a \le 2b - 1$</span> follows from <span class="math-container">$\frac{a+1}{2} = b$</span>, one would use the following sequence of equations, where each eq... | rae306 | 168,956 | <p>This holds in general:</p>
<blockquote>
<p>If <span class="math-container">$f\in C[0,1]$</span>, then the sequence of Bernstein polynomials <span class="math-container">$B_nf$</span> converges uniformly to <span class="math-container">$f$</span> on <span class="math-container">$[0,1]$</span>.</p>
</blockquote>
<... |
3,613,120 | <p>In elementary algebra and beyond, we are taught to use a sequence of equations to derive a relationship. For instance, to show that <span class="math-container">$a \le 2b - 1$</span> follows from <span class="math-container">$\frac{a+1}{2} = b$</span>, one would use the following sequence of equations, where each eq... | s.harp | 152,424 | <p>The first important thing to see is that <span class="math-container">$n(1-e^{1/n})\to1$</span>, as <span class="math-container">$n\to\infty$</span>, which you can show in any way you like. Now the following lemma instantly gives you the result you are looking for:</p>
<blockquote>
<p><strong>Lemma.</strong> Let ... |
354,124 | <p>I was stumbled with a basic calculus question by a friend.</p>
<p>The question first asks to find unit vectors $v,w$ s.t $|u+v|$ is
maximal and $|u-w|$ is minimal where $u=(-2,5,3)$.</p>
<p>Then the question asks to find unit vectors $v,w$ s.t $u\cdot v$
is maximal and $|u\cdot w|$ is minimal.</p>
<p>It's easy to... | Eckhard | 53,115 | <p>The right direction is to consider vectors $v$ and $w$ which are collinear with $u$.</p>
<p>For the second part of the question, observe that $u\cdot v=|u||v|\cos\left(\angle(u,v)\right)$. In your case, the absolute values of $u$ and $v$ are fixed, so the values of $u\cdot v$ and $|u\cdot w|$ depend only on the ang... |
2,752,511 | <p>Prove that if $X$ is Hausdorff, $\Delta=\{(x, x)\mid x\in X\}$ is closed in $X\times X$ (with the product topology).</p>
<p><strong>My attempt:</strong></p>
<p>Let $x_1, x_2\in X$ s.t. $x_1\ne x_2$.</p>
<p>There exist neighborhoods $U_1$ and $U_2$ of $x_1$ and $x_2$ that are disjoint.</p>
<p>$U_1\times U_2$ is a... | TheMagicSnoot | 555,837 | <p>You're basically there, you just need to interpret your result. You found that for any point $(x_1,x_2)\in X\times X-\Delta$, there exists a neighborhood of $(x_1,x_2)$ contained in $X\times X-\Delta$. That is, $X\times X-\Delta$ is open. Therefore $\Delta$ is...</p>
|
1,497,898 | <p>Consider the polynomial $$f(x)=x^4-x^3+14x^2+5x+16$$ and $\mathbb{F}_p$ be the field with $p$ elements, where $p$ is prime. Then</p>
<ol>
<li>Considering $f$ as a polynomial with coefficients in $\mathbb{F_3}$, it has no roots in $\mathbb{F_3}$.</li>
<li><p>Considering $f$ as a polynomial with coefficients in $\mat... | 2'5 9'2 | 11,123 | <p>To see if the 4th degree polynomial over $\mathbb{Z}_7$ has an irreducible 3rd degree factor, simply check to see if it has a root in $\mathbb{Z}_7$ first by trying all $7$ options. If so, divide by the corresponding factor to leave you with a cubic. Now either the cubic is irreducible, or it has a root in $\mathbb{... |
1,557,165 | <p>Prove that
$$\int_1^\infty\frac{e^x}{x (e^x+1)}dx$$
does not converge.</p>
<p>How can I do that? I thought about turning it into the form of $\int_b^\infty\frac{dx}{x^a}$, but I find no easy way to get rid of the $e^x$.</p>
| Martin Argerami | 22,857 | <p>For $x\geq1$, you have $e^x>1$, so $2e^x>e^x+1$, or
$$
\frac {e^x}{e^x+1}>\frac12.
$$
Then
$$
\int_1^\infty \frac {e^x}{x (e^x+1)}\,dx\geq\int_1^\infty\frac1x\,dx=\infty .
$$</p>
|
898,151 | <p>I have encountered an statement several times while proving determinant of a block matrix. </p>
<blockquote>
<p>$$\det\pmatrix{A&0\\0&D}\; = \det(A)\det(D)$$</p>
</blockquote>
<p>where $A$ is $k\times k$ and $D$ is $n\times n$ matrix. How to prove this?</p>
<p>Thanks in advance.</p>
| Community | -1 | <p>This is more general result
$$\det\pmatrix{A&B\\0&D} = \det A\det D$$
and to prove it notice that</p>
<p>$$\pmatrix{A&B\\0&D}=\pmatrix{I_k&0\\0&D}\pmatrix{A&B\\0&I_n}$$
and we develop along the $k$ first rows we find
$$\det\pmatrix{I_k&0\\0&D}=\det D$$
and along the last $n$ ... |
1,029,485 | <p>I wish to show the following statement:</p>
<p>$
\forall x,y \in \mathbb{R}
$</p>
<p>$$
(x+y)^4 \leq 8(x^4 + y^4)
$$</p>
<p>What is the scope for generalisaion?</p>
<p><strong>Edit:</strong></p>
<p>Apparently the above inequality can be shown using the Cauchy-Schwarz inequality. Could someone please elaborate,... | Joel | 85,072 | <p>If you instead consider $$\left( \frac{x}{2} + \frac{y}{2} \right)^4$$ we know that the function $(\cdot)^4$ is convex. This leads to: $$\left( \frac{x}{2} + \frac{y}{2} \right)^4 \le \frac12 x^4 + \frac12 y^4$$</p>
<p>Multiply both sides by $16$ and we have: $$(x+y)^4 \le 8x^4 + 8y^4.$$</p>
<p>This process works ... |
187,545 | <p><span class="math-container">$\DeclareMathOperator\GL{GL}\DeclareMathOperator\L{\mathfrak{L}}$</span>The free Lie algebra <span class="math-container">$\L(V)$</span> generated by an <span class="math-container">$r$</span>-dimensional vector space <span class="math-container">$V$</span> is, in the
language of <a href... | Vladimir Dotsenko | 1,306 | <p>The Whitehouse module referred to in one of the other answers is not necessary, since it is related to the <em>cyclic</em> operad Lie, that is to the representation of <span class="math-container">$S_{n+1}$</span> in <span class="math-container">$Lie(n)$</span>.</p>
<p>The decomposition in terms of Young diagrams is... |
3,275,423 | <p>How do I see that for a <span class="math-container">$K$</span>-vector space <span class="math-container">$V$</span> the map</p>
<blockquote>
<p><span class="math-container">$\bigwedge^d(V^*) \times \bigwedge^d(V) \rightarrow K, (f_1 \wedge ... \wedge f_d, x_1 \wedge ... \wedge x_d) \mapsto det(f_i(x_i)_{i,j})$</... | Berci | 41,488 | <p>The given formula is certainly giving a well defined mapping
<span class="math-container">$$\varphi:V^*\times\dots\times V^*\ \times\ V\times\dots\times V \longrightarrow K$$</span>
Fixing all but one arguments makes the matrix of the determinant varying linearly in one row or column. <br>
This shows that <span clas... |
716,121 | <p>Construct the matrix corresponding to a rotation of 90 degrees about the y-axis together with a reflection about the (x,z) plane. </p>
<p>Reviewing Linear Algebra and seem to have forgotten some stuff. Not sure what to do with this problem</p>
| Klaas van Aarsen | 134,550 | <p>Figure out what the images are of each of the 3 standard unit vectors.
Put them next to each other in a matrix... and presto! :)</p>
|
716,121 | <p>Construct the matrix corresponding to a rotation of 90 degrees about the y-axis together with a reflection about the (x,z) plane. </p>
<p>Reviewing Linear Algebra and seem to have forgotten some stuff. Not sure what to do with this problem</p>
| jamisans | 102,913 | <p>I'm assuming this is in 3-space and your rotation is CCW. We can figure out the matrix for this transformation by seeing where it sends the standard basis vectors. $[1, 0, 0]^T$ gets rotated to $[0, 1, 0]^T$ then reflected to $[0, -1, 0]^T$. $[0, 1, 0]^T$ gets rotated to $[-1, 0, 0]^T$ and it is fixed by the reflect... |
397,274 | <p>Suppose you have a group isomorphism given by the first isomorphism theorem:</p>
<p><span class="math-container">$$G/\ker(\phi) \simeq \operatorname{im}(\phi)$$</span></p>
<p>What can we say about the group <span class="math-container">$\ker(\phi)\times \operatorname{im}(\phi)$</span>? In particular, when does the f... | Andrea Marino | 177,070 | <p>In your special case you actually have a morphism <span class="math-container">$GL_n ^+ \to SL_n \times \mathbb{R}^+ $</span> given by</p>
<p><span class="math-container">$$ M \mapsto (M/(\det M)^{1/n}, \det M) $$</span></p>
<p>the inverse being given by <span class="math-container">$(N, t) \mapsto t^{1/n} N$</span>... |
2,281,510 | <p>Why do we replace y by x and then calculate y for calculating the inverse of a function?</p>
<p>So, my teacher said that in order to find the inverse of any function, we need to replace y by x and x by y and then calculate y. The reason being inverse takes y as input and produces x as output.</p>
<p>My question is... | Ahmed S. Attaalla | 229,023 | <p>Let $f(x)=y$, we would like to find $f^{-1}(x)$, to do this note by definition:</p>
<p>$$f(f^{-1}(x))=x$$</p>
<p>If we for the moment call $f^{-1}(x)$ as $y$, then by solving,</p>
<p>$$f(y)=x$$</p>
<p>For $y$ we have found $f^{-1}(x)$. Note in the above we have switched $x$ with $y$ and vice versa.</p>
<p>Actua... |
1,840,778 | <p>In rectangle $ABCD$, we have $AD = 3$ and $AB = 4$. Let $M$ be the midpoint of $\overline{AB}$, and let $X$ be the point such that $MD = MX$, $\angle MDX = 77^\circ$, and $A$ and $X$ lie on opposite sides of $\overline{DM}$. Find $\angle XCD$, in degrees. </p>
<p><img src="https://i.stack.imgur.com/3TsZm.png" alt="... | Jack D'Aurizio | 44,121 | <p>Since $MC=MD=MX$, the points $C,X,D$ lie on a circle centered at $M$ and
$$\widehat{XCD}=\frac{1}{2}\widehat{XMD} = \color{red}{13^\circ}.$$</p>
|
440,791 | <p>I am trying to figure out if the infinite product <span class="math-container">$$\omega=\frac{5\sqrt{3}}{12}\prod\limits_{\substack{p\equiv 1\pmod3 \\
p\ge 13}}\left(\frac{p-2}{p-1}\right)\prod\limits_{\substack{p\equiv 2\pmod3 \\
p\ge 13}}\left(\frac{p}{p-1}\right)$$</span> is asymptotically equal to the infinite p... | KConrad | 3,272 | <p>I don't know why you are restricting the products to <span class="math-container">$p \geq 13$</span> or where the factor <span class="math-container">$5\sqrt{3}/12$</span> is coming from. I am going to ignore that and discuss the following product over all primes <span class="math-container">$p$</span>:
<span class... |
215,474 | <p>Suppose I have a function <code>cyclePart</code> which has a definition for the case</p>
<pre><code>cyclePart[list_->{},n_,Δn_,cycle_:True]:=...
</code></pre>
<p>But for example in the algorithm, if it encounters a case like</p>
<pre><code>cyclePart[{a,b,c,d,e,f,g,h,i,j}->{b,d,i,j},5,2,True]
</code></pre>
... | bbgodfrey | 1,063 | <p>Progress can be made by integrating over <code>y</code> only (with the correction that <code>{…}</code> be replaced by <code>(…)</code>).</p>
<pre><code>Integrate[x DiracDelta[r x - y] Exp[1/g^2 (Cos[x2 - x] + Cos[x2] + Cos[y + x2])],
{y, 0, 2 Pi}, Assumptions -> r > 0 && 0 < x < 2 Pi]
(* E... |
215,474 | <p>Suppose I have a function <code>cyclePart</code> which has a definition for the case</p>
<pre><code>cyclePart[list_->{},n_,Δn_,cycle_:True]:=...
</code></pre>
<p>But for example in the algorithm, if it encounters a case like</p>
<pre><code>cyclePart[{a,b,c,d,e,f,g,h,i,j}->{b,d,i,j},5,2,True]
</code></pre>
... | Andreas | 69,887 | <p>if the range of the x2 integration may be shifted to {0,2π} then it can be done and plotted like:</p>
<pre><code>g = 2.; Plot[{NIntegrate[
E^((Cos[x - x2] + Cos[x2] + Cos[r x + x2])/g^2) x HeavisideTheta[2 Pi - r x], {x, 0, 2 Pi}, {x2, 0, 2 Pi}],
2 Pi NIntegrate [x BesselI[0, 1/g^2 \[Sqrt](3 + 2 Cos[x] + 2 C... |
231,887 | <p>I'm learning to do proofs, and I'm a bit stuck on this one.
The question asks to prove for any positive integer $k \ne 0$, $\gcd(k, k+1) = 1$.</p>
<p>First I tried: $\gcd(k,k+1) = 1 = kx + (k+1)y$ : But I couldn't get anywhere.</p>
<p>Then I tried assuming that $\gcd(k,k+1) \ne 1$ , therefore $k$ and $k+1$ are no... | apnorton | 23,353 | <p>Old John's answer in the comments is better than this, but I'm hoping to provide some intuition...</p>
<p>Look at $k!$ instead of $k$:</p>
<ul>
<li>2 divides $k!$, thus it cannot divide $k!+1$. (2 divides every other integer)</li>
<li>3 divides $k!$, thus it cannot divide $k!+1$. (3 divides every third integer)<... |
373,906 | <p>(This question is <a href="https://math.stackexchange.com/questions/3859476">originally from Math.SE</a> where it was suggested that I ask the question here)</p>
<p>Let <span class="math-container">$G$</span> be a finite group with fewer than <span class="math-container">$p^2$</span> Sylow <span class="math-containe... | Richard Lyons | 99,221 | <p>The conjecture follows quickly from <strong>Brodkey's Theorem</strong>: Let <span class="math-container">$G$</span> be a finite group and <span class="math-container">$p$</span> a prime. Suppose that Sylow <span class="math-container">$p$</span>-subgroups of <span class="math-container">$G$</span> are abelian. If <s... |
1,953,517 | <p>Let $X,Y$ be two independet Poisson variables with parameters $\mu,\lambda>0$. Let $N:=Y+X$
what is $\mathbb{E}(X\vert N=n)$?</p>
<p>I already computed $P(X=k\vert N=n)$ for $k,n\in \mathbb{Z}_{+}$ which is $$P(X=k\vert N=n)=\binom{n}{k}\frac{\mu^{n-k}\lambda^k}{(\mu+\lambda)^n}$$ if $n>k$ else $0$.</p>
<p>I... | Andreas Caranti | 58,401 | <p>As a variant on the solution of Benson Lin, and basically as suggested in comments by Doug M and csts, start with only the constraint of at least 10 bicycles in warehouse 1 and 2. This gives you
$$
\binom{100-10-10+3}{3} = \binom{83}{3}
$$
possibilities. Now count the possibilities when warehouse 1 gets at least 10,... |
819,704 | <p>Here is the problem I have </p>
<p>$\lim \limits_{x \to -1} (x + 1)^2 sin (\frac{1}{x + 1})$</p>
<p>I approached it like this:</p>
<p>\begin{align}
-1 \le sin(\displaystyle \frac{1}{x + 1}) \le 1 \\
-(x + 1) \le sin(1) \le (x + 1)
\end{align}</p>
<p>I then go on to solve the limit by replacing $sin (\frac{1}{x ... | mm-aops | 81,587 | <p>it's not, you should've just written that $|\sin (\frac{1}{x+1})|$ is bounded by $1$ (which you did) and then say that $(x+1)^2 \rightarrow 0$, hence the product goes to zero as well, kinda like $$ - (x+1)^2 \leq (x+1)^2 \sin (\frac{1}{x+1}) \leq (x+1)^2$$ it isn't true that $\sin(\frac{1}{x+1}) = (x+1)\sin(1)$ whic... |
288,974 | <p>Alright this maybe really funny but I want to know why is this wrong. We often come across identities which we prove by multiplying both the sides of the identity by a certain entity but why don't we multiply it by $0$. That way every identity will be proved in one single line. That is so stupid. I mean, by that way... | guest196883 | 43,798 | <p>Because to prove $a=b$ you don't have to prove that $$a=b \implies c=c$$You have to prove that some trivial statement, such as $$c=c$$or another axiom, or logical tautology, or proved statement, and derive from that that $a=b$. So in your example, </p>
<p>$$\begin{array}{lcl}&& \sin^2 \theta & = & t... |
1,579,170 | <p>This problem is dependent because it matters which one you choose, So i don't think we can do the multiplication thing in this one. </p>
<ul>
<li>Probability of ( non defective ) = 6/10 </li>
</ul>
<p>What does the question mean when it says all will be non-defective? is "all" the 2 randomly chosen telephone? How ... | André Nicolas | 6,312 | <p>We need to assume that $p$ and $q$ are <em>distinct</em>.</p>
<p>Let $L$ be our lcm. Since $p-1$ divides $L$, we have $a^L\equiv 1\pmod{p}$. Similarly, $a^L\equiv 1\pmod{q}$. Thus $p$ divides $a^L-1$ and $q$ divides $a^L-1$. Since $p$ and $q$ are relatively prime, it follows that $pq$ divides $a^L-1$.</p>
|
1,303,577 | <p>I have started to learn about the properties of the <a href="http://en.wikipedia.org/wiki/Quadratic_residue" rel="nofollow">quadratic residues modulo n (link)</a> and reviewing the list of quadratic residues modulo $n$ $\in [1,n-1]$ I found the following possible property:</p>
<blockquote>
<p>(1) $\forall\ p \gt ... | 111 | 100,311 | <p>Say $N=2p$ then
$$\#\{y:x^2\equiv y(\ {\rm{mod}} N)\}$$ is equal (from CRT)
$\#\{y:x^2\equiv y(\ {\rm{mod}} p)\}\times 2-1=p$ (we subtract $-1$ since we add the solution $x=0$ twice. We have $\frac{p+1}{2}\times 2$ equations but two of them are the same.).
More general if $N=p_1p_2\cdots p_k$ and $p_j$ are distin... |
292,122 | <p>This question actually came out of a question. In some other post, I saw a reference and going through, found this, $n>0$.</p>
<p>Solve for n explicitly without calculator:
$$\frac{3^n}{n!}\le10^{-6}$$</p>
<p>And I appreciate hint rather than explicit solution.</p>
<p>Thank You.</p>
| A Ricko Maulidar | 60,472 | <p>how about make a function?$f(n)=\frac{3^n}{n!}-10^{-6}$ or maybe $f(n)=\frac{3^n}{n!10^{-6}}$</p>
|
1,897,849 | <p>I'm reading <strong>Awodey's</strong>: Category Theory. I harbor a little confusion: When we speak about a category, say: $\mathbf{Set}$. In the book, usually he talks about this category but there is no notion of quantity of sets. How many sets are actually there?</p>
<p>Another thing that is getting me confused i... | SixWingedSeraph | 318 | <p>First, "Set" is one specific category. Its objects are all the sets and its arrows are all the functions between sets. </p>
<p>In the second paragraph, you suppose you have a category with n sets. To define the category you have to say specifically which n sets the objects are. You then have to say what the arrows... |
3,142,339 | <p>Let <span class="math-container">$p$</span> be a real number. I am looking for all <span class="math-container">$(x,y)$</span> such that <span class="math-container">$\ln[e^{x}+e^{y}]=px+(1-p)y$</span>. My effort:</p>
<p>Take exponent of both sides to obtain <span class="math-container">$e^{x}+e^{y}=e^{px}e^{(1-p)y... | Vasily Mitch | 398,967 | <p>The problem with this expression is that function <span class="math-container">$f(x)=(i+x)^{-1}$</span> is a tricky one:</p>
<p><span class="math-container">$$
f(f(x)) = \frac{1}{i+\frac{1}{i+x}}=\frac{i+x}{-1+ix+1}=-i+\frac1x,\\
f(f(f(x))) = \frac{1}{i+(-i+1/x)}=x.
$$</span></p>
<p>So every number generates the o... |
1,319,288 | <p>There is a <a href="https://math.stackexchange.com/questions/1103723">similar question</a> in this site but I am not satisfied with the answer, which is basically the same as the proof in the mentioned textbook.</p>
<p>The book(Karel Hrbacek&Thomas Jech, <em>Introduction to Set Theory 3e</em>, p165) states a le... | Noah Schweber | 28,111 | <p>As bof says, the statement "$2^{\aleph_\omega}=\lim 2^{\aleph_n}$" is false - to see why, consider the following analogous claim: $$2^\omega=\lim 2^n.$$ They are each wrong for precisely the same reason.</p>
<p>EDIT: Specifically, if we try to "cover" $2^\omega$ by $\bigcup 2^n$ in the obvious way, we miss all the ... |
661,182 | <p>I'm taking a discrete structures class and I would appreciate some help with a homework problem. The problem is</p>
<blockquote>
<p>Attempt to find a closed form for the sum $\displaystyle \sum_{k=1}^n k^3$
by perturbation, only to find a closed form for the following sum $\displaystyle \sum_{k=1}^n k^2$.</p>
<... | drhab | 75,923 | <p>I am not familiar with perturbation so cannot really help you.</p>
<p>However, maybe this can be useful anyhow:</p>
<p>$$\sum_{k=r}^{n}\binom{k}{r}=\binom{n+1}{r+1}$$
is a nice closed form that can be proved easily by induction. This for any nonnegative $r$.</p>
<p>Closed forms for: $$\sum_{k=1}^{n}k^{r}$$
can be... |
3,055,324 | <p>I need some help with constructing a proof for the following statement,<span class="math-container">$ \frac{P_1 P_2}{hcf(m,n)} = lcm(P_1,P_2)$</span> where <span class="math-container">$P_1$</span> and <span class="math-container">$P_2$</span> are polynomials with real coefficients.</p>
<p>I know how to do the sam... | Quang Hoang | 91,708 | <p>It works pretty much the same for integers if you modify the argument a little. Let <span class="math-container">$L = lcm(P_1, P_2)$</span> and <span class="math-container">$G=gcd(P_1, P_2)$</span>. Then
<span class="math-container">$$P_1 = Gh_1, P_2 = Gh_2,$$</span>
with <span class="math-container">$gcd(h_1, h_2) ... |
293,047 | <p>When I am reading through higher Set Theory books I am frequently met with statements such as '$V$ is a model of ZFC' or '$L$ is a model of ZFC' where $V$ is the Von Neumann Universe, and $L$ the Constructible Universe. For instance, in Jech's 'Set Theory' pg 176, in order to prove the consistency of the Axiom of Ch... | Asaf Karagila | 7,206 | <p>Yes, that is true. But note that in its nature statements like $\operatorname{Con}(T)$ are <em>meta-theoretic</em> statements. So when we say that $V$ is a model of $\sf ZF$, we mean that in the meta-theory it is a model of $\sf ZF$.</p>
<p>This is often something which is not stressed enough in introductions to $V... |
2,964,359 | <blockquote>
<p>Let <span class="math-container">$(X, d)$</span> be a metric space with no isolated points, and let <span class="math-container">$A$</span>
be a relatively discrete subset of <span class="math-container">$X$</span>. Prove that <span class="math-container">$A$</span> is nowhere
dense in <span class... | Hagen von Eitzen | 39,174 | <p>With your <span class="math-container">$x$</span> and <span class="math-container">$\epsilon$</span>, <span class="math-container">$$\overline A\setminus B_d(x,\epsilon/2)$$</span> is a strictly smaller closed set than <span class="math-container">$\overline A$</span>, hence cannot contain all of <span class="math-c... |
1,375,365 | <p>Find all polynomials for which </p>
<p>What I have done so far:
for $x=8$ we get $p(8)=0$
for $x=1$ we get $p(2)=0$</p>
<p>So there exists a polynomial $p(x) = (x-2)(x-8)q(x)$</p>
<p>This is where I get stuck. How do I continue?</p>
<p><strong>UPDATE</strong></p>
<p>After substituting and simplifying I get
$(x-... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>Let the highest of power of $x$ be $n$</p>
<p>So, $(x-8)[a(2x)^n+\cdots]=8(x-1)[ax^n+\cdots]$</p>
<p>Comparing the coefficients of $x^{n+1},$
$$a2^n=8a\implies n=3$$</p>
<p>Let $p(x)=(x-2)(x-8)(ax+b)$ where $a,b$ are arbitrary constants to be determined</p>
<p>Hope you take it from here?</p>
|
2,946,384 | <p>How to prove that any integer n which is not divisible by 2 or 3 is not divisible by 6?</p>
<p>The point was to prove separately inverse, converse and contrapositive statements of the given statement: "for all integers n, if n is divisible by 6, then n is divisible by 3 and n is divisible by 2".
I have the proof f... | J.G. | 56,861 | <p>Since <span class="math-container">$dx=dt/t$</span>, you need to divide the whole integrand by <span class="math-container">$t$</span>.</p>
|
444,486 | <p>I am teaching myself real analysis, and in this particular set of lecture notes, the <a href="http://www.math.louisville.edu/~lee/RealAnalysis/IntroRealAnal-ch01.pdf" rel="nofollow">introductory chapter on set theory</a> when explaining that not all sets are countable, states as follows:</p>
<blockquote>
<p>If $S... | Muphrid | 45,296 | <p>The relationship between $\mathbb C$ and $\mathbb R^2$ becomes clearer using Clifford algebra.</p>
<p>Clifford algebra admits a "geometric product" of vectors (and more than just two vectors). The so-called complex plane can instead be seen as the algebra of geometric products of two vectors.</p>
<p>These objects... |
444,486 | <p>I am teaching myself real analysis, and in this particular set of lecture notes, the <a href="http://www.math.louisville.edu/~lee/RealAnalysis/IntroRealAnal-ch01.pdf" rel="nofollow">introductory chapter on set theory</a> when explaining that not all sets are countable, states as follows:</p>
<blockquote>
<p>If $S... | Wlod AA | 490,755 | <p>For the sake of easy communication, it is common to identify <span class="math-container">$\ \mathbb C\ $</span> and
<span class="math-container">$\ \mathbb R^2\ $</span> via the algebraic connecting <span class="math-container">$\ \mathbb C\ $</span> with field
<span class="math-container">$\mathbb R[i]/(i^2+1).\ $... |
444,486 | <p>I am teaching myself real analysis, and in this particular set of lecture notes, the <a href="http://www.math.louisville.edu/~lee/RealAnalysis/IntroRealAnal-ch01.pdf" rel="nofollow">introductory chapter on set theory</a> when explaining that not all sets are countable, states as follows:</p>
<blockquote>
<p>If $S... | Allawonder | 145,126 | <p>The basic difference between <span class="math-container">$\mathrm C$</span> and <span class="math-container">$\mathrm R^2$</span> which makes electrical engineers prefer working with complex quantities is that <span class="math-container">$\mathrm C$</span> is not usually thought of as just a set (yes, it's an abus... |
1,671,357 | <p>I'm trying to solve a minimization problem whose purpose is to optimize a matrix whose square is close to another given matrix. But I can't find an effective tool to solve it.</p>
<p>Here is my problem:</p>
<blockquote>
<p>Assume we have an unknown Q with parameter $q11, q12,q14,q21,q22,q23,q32,q33,q34,q41,q43... | Samrat Mukhopadhyay | 83,973 | <p>For the modified question, let me try to give an answer that can address situations with matrices having the structure that you have. </p>
<p>Basically, your matrix $G$ has the following structure $$G=uv^T$$ where I have taken $u$ as the all $1$'s vector and $v$ a vector of positive coordinates such that $v^Tu=1$, ... |
3,334,031 | <p>I was doing some practice problems that my professor had sent us and I have not been able to figure out one of them. The given equation is:</p>
<p><span class="math-container">$-y^2dx +x^2dy = 0$</span></p>
<p>He then asks us to verify that:</p>
<p><span class="math-container">$ u(x, y) = \frac{1}{(x-y)^2}$</span... | N. S. | 9,176 | <p>It has (kinda) exponential growth.</p>
<p>Let <span class="math-container">$f_n$</span> be the number of solutions. Then, it is easy to see that
<span class="math-container">$$f_{m+n-1}\geq f_n\cdot f_m$$</span></p>
<p>Indeed, if <span class="math-container">$x_1,.., x_n$</span> is a solution for <span class="mat... |
3,334,031 | <p>I was doing some practice problems that my professor had sent us and I have not been able to figure out one of them. The given equation is:</p>
<p><span class="math-container">$-y^2dx +x^2dy = 0$</span></p>
<p>He then asks us to verify that:</p>
<p><span class="math-container">$ u(x, y) = \frac{1}{(x-y)^2}$</span... | David E Speyer | 448 | <p>If we restrict ourselves to sequences which stay between <span class="math-container">$A$</span> and <span class="math-container">$B$</span>, we can get an exact count, and thus a lower bound for the original problem. Let <span class="math-container">$M$</span> be the matrix with rows and columns indexed by <span cl... |
654,408 | <p>I know that the volume form on $S^1$ is $\omega= ydx-xdy$. But how I can derive that? The only things that I know are the definition of differential q-form, and the fact that the vector field $v= y \frac{\partial}{\partial x}-x\frac{\partial}{\partial y}$ never vanishes on $S^1$.</p>
| Willie Wong | 1,543 | <ol>
<li><p>You speak of "the" volume form. There are <em>many</em> volume forms on any orientable smooth manifold. Given $\omega$ a volume form, and a non-vanishing smooth function $f$, then $f\omega$ is again a volume form. </p></li>
<li><p>The task of finding "a" volume form is to find a non-vanishing top degree for... |
189,069 | <p>The Survival Probability for a walker starting at the origin is defined as the probability that the walker stays positive through n steps. Thanks to the Sparre-Andersen Theorem I know this PDF is given by</p>
<pre><code>Plot[Binomial[2 n, n]*2^(-2 n), {n, 0, 100}]
</code></pre>
<p>However, I want to validate this ... | Carl Lange | 57,593 | <p>We can do this using an implementation of <code>FoldWhileList</code>.</p>
<p>First, implement <code>FoldWhileList</code> using <a href="https://mathematica.stackexchange.com/a/19105/57593">this great answer</a>.</p>
<pre><code>FoldWhileList[f_, test_, start_, secargs_List] :=
Module[{tag},
If[# === {}, {start... |
2,792,061 | <p>I would like to understand how to apply <em>well-founded induction</em>. I have found two definitions which I list now, followed by the question.</p>
<blockquote>
<p>(1) A binary relation $\prec$ is <a href="http://www.cs.cornell.edu/courses/cs6110/2013sp/lectures/lec07-sp13.pdf" rel="noreferrer"><em>well-founded... | qualcuno | 362,866 | <p>Note that since you have $m+1$ polynomial, lineal independence would in particular imply that $<p_0,...,p_m>$ generates $\mathbb{F}_{\leq m}$. However, this is not so: clearly</p>
<p>$$
<p_0, \dots p_m> \subseteq \{p \in \mathbb{F}_{\leq m} : p(2) = 0\}
$$</p>
<p>which is a proper subspace of $\mathbb{... |
203,505 | <p>Let <span class="math-container">$P(x)$</span> be a non-constant polynomial with real coefficients.</p>
<p>Can <a href="http://en.wikipedia.org/wiki/Natural_density" rel="noreferrer">natural density</a> of</p>
<p><span class="math-container">$$\{n\ |\ \lfloor P(n)\rfloor \ \text{is prime.}\}$$</span></p>
<p>be posit... | Igor Rivin | 11,142 | <p>The Bateman-Horn conjecture says no. See <a href="http://projecteuclid.org/download/pdf_1/euclid.bams/1183551839">this paper of Kevin McCurley</a> for related results.</p>
|
536,362 | <p>Let $\Sigma = \sigma(\mathcal C)$ be the $\sigma$-algebra generated by the countable collection of sets $\mathcal C \subset \mathcal{P}(X)$. How can I prove that if $\mu$ is a $\sigma$-finite measure on $(X,\Sigma)$ then $L^p(X)$ is separable for $1 \le p < \infty$?</p>
<p>I know that simple functions are dense ... | Davide Giraudo | 9,849 | <p>Before going into a formal proof, here is the idea. The space is $\sigma$-finite, so we can "break" it into countably many spaces of finite measure. Up to some technical considerations, we are reduced to the case $X$ of finite measure. An algebra generated by a countable class is countable and we can approximate ele... |
1,631,589 | <p>Consider the sequence $\{\frac{x^n}{n!}\}_n$ for any number $x$.</p>
<p>By choosing $m>x$ and letting $n>m$ , show that:</p>
<p>$\frac{x^n}{n!} < \frac{x^n}{m^n} < \frac{m^m}{(m-1)!}$</p>
<p>Am using the squeeze theorem , but unable to start third inequality.</p>
| Maffred | 279,068 | <p>LHS can be interpreted as counting the ways in which you can create two different pairs of two different elements, taken from a set of $n$ toys; while the pairs must be different, they need not be disjoint: they might have a toy in common.</p>
<p>RHS counts the same amount of pairs in a different way: you can creat... |
3,042,149 | <p>We can't exactly draw a line of length square root of 2 but in an isosceles right angle triangle of sides 1 unit each, the length of hypotenuse will be the square root of 2. Now does it mean we can get the line of exact such length?</p>
<p>How is it possible? How can we get a line of exact length square root of 2 w... | PrincessEev | 597,568 | <p>A lot of numbers are called "constructible", in that, if <span class="math-container">$n$</span> is constructible, then we can find a construction in which a line of <span class="math-container">$n$</span> units can be made. Granted, this is not a trivial task depending on the construction and number involved.</p>
... |
1,457,063 | <p>I am utterly confused on how to solve this problem. I found a lemma that says $|A\cup B|=|A|+|B|$ is true if the two sets are disjoint which makes sense, but how do I prove the entire statement. </p>
| R.N | 253,742 | <p>suppose $x\in A\cup B$ then in left hand you have considered it 1 time. what a bout in right hand 2 event may happen, if $x\notin A\cap B$ then in right hand you count it one time, just in A or B, but if $x\in A\cap B$ then $\color{red}{1=1+1-1}$</p>
|
1,765,538 | <p>If $N$ is the set of all natural numbers, $R$ is a relation on $N \times N$, defined by $(a,b) \simeq (c,d)$ iff $ad=bc$, how can I prove that $R$ is an equivalence relation ?</p>
| Michael Burr | 86,421 | <p>Hint: To prove that this is an equivalence relation, you must prove reflexivity, symmetry, and transitivity. Each of these is a different statement that you must prove.</p>
<ul>
<li><p>Reflexivity: Suppose that $(a,b)\in\mathbb{N}\times\mathbb{N}$. Then, you want to prove that $(a,b)R(a,b)$. This can get a littl... |
602,286 | <p>I'm reading a paper which uses the following fact; it appears to be standard but I am not sure where to look for a proof.</p>
<blockquote>
<p><strong>Claim.</strong> Let $M$ be a complete Riemannian manifold (assumed to be second countable, so no long lines). There is an increasing sequence of open sets $U_n$ wi... | Lame-Ov2.0 | 114,476 | <p>You can factor out the $x^2$ so your equation looks like,</p>
<p>$\dfrac{dy}{dx} = x^2(y+1)-y-1$</p>
<p>The rest shouldn't be too bad.</p>
|
602,286 | <p>I'm reading a paper which uses the following fact; it appears to be standard but I am not sure where to look for a proof.</p>
<blockquote>
<p><strong>Claim.</strong> Let $M$ be a complete Riemannian manifold (assumed to be second countable, so no long lines). There is an increasing sequence of open sets $U_n$ wi... | neela | 365,061 | <p>After separating the variables, integrate on both sides.
$\log(y+1)=(x^3)-x+c$
Is the solution of given differential equation</p>
|
667,293 | <p>We are given a $n \times m$ rectangular matrix in which every cell there is a light bulb, together with the information whether the bulb is ON or OFF.</p>
<p>Now i am required to switch OFF all the bulbs but i can perform only one operation that is as follows:</p>
<ul>
<li>I can simultaneously flip all the bulbs f... | Hagen von Eitzen | 39,174 | <p>There are $n={N+1\choose 2}{M+1\choose2}$ ways to pick a submatrix, each corresponding to a vector $v_i\in\mathbb F_2^{NM}$, $1\le i\le n$. You are looking for a solution of $\sum c_iv_i=A$ that minimizes the weight $w(c)$ (i.e. the number of $i$ with $c_i=1$)</p>
<p><strong>Proposition:</strong> Among all minimum... |
2,426,892 | <blockquote>
<p>Between which two integers does <span class="math-container">$\sqrt{2017}$</span> fall? </p>
</blockquote>
<p>Since <span class="math-container">$2017$</span> is a prime, there's not much I can do with it. However, <span class="math-container">$2016$</span> (the number before it) and <span class="mat... | Will Jagy | 10,400 | <p>As to the next question, Pell -1 and Pell +1..Hmm, seems to be showing the exponents incorrectly.</p>
<p>$$ {106515299132603184503844444}^2 - 2017 \cdot 2371696115380807559791481^2 = -1 $$</p>
<p>$$ 22691017898615873418283839489716246568157231499338273^2 - 2017 \cdot 5052438423628393473350846837568851798192790... |
2,426,892 | <blockquote>
<p>Between which two integers does <span class="math-container">$\sqrt{2017}$</span> fall? </p>
</blockquote>
<p>Since <span class="math-container">$2017$</span> is a prime, there's not much I can do with it. However, <span class="math-container">$2016$</span> (the number before it) and <span class="mat... | Robert Soupe | 149,436 | <p>Try graphing $\sqrt x$ for $x \geq 0$. You should see a fairly smooth curve that goes upwards, which means that if $a < b$ and they're both positive, then $\sqrt a < \sqrt b$.</p>
<p>From this, it's clear that the integers you want are $\lfloor \sqrt{2017} \rfloor$ and $\lceil \sqrt{2017} \rceil$. A calculato... |
2,426,892 | <blockquote>
<p>Between which two integers does <span class="math-container">$\sqrt{2017}$</span> fall? </p>
</blockquote>
<p>Since <span class="math-container">$2017$</span> is a prime, there's not much I can do with it. However, <span class="math-container">$2016$</span> (the number before it) and <span class="mat... | mathreadler | 213,607 | <p>You can use the Newton square root method <a href="https://en.wikipedia.org/wiki/Methods_of_computing_square_roots" rel="nofollow noreferrer">wikipedia</a> for integers:</p>
<p>$$x_{n+1} = \frac 1 2 \left(x_n+ \frac S {x_n}\right)$$</p>
<p>Let us start with a crappy guess </p>
<ol>
<li>$x_1 = 500$:</li>
<li>$x_2 ... |
2,040,293 | <p>I am trying to follow this tutorial: <a href="http://ctms.engin.umich.edu/CTMS/index.php?example=InvertedPendulum&section=SystemModeling" rel="nofollow noreferrer">http://ctms.engin.umich.edu/CTMS/index.php?example=InvertedPendulum&section=SystemModeling</a></p>
<p>I am stuck to understand how to make a sta... | Community | -1 | <p>We fix $n$ and prove induction on $m$. When $m=0$, we have to show that $x^{n+0}=x^{n}\times x^{0}$. But the left hand side is $x^{n+0}=x^{n}$ while the right hand side is $x^{n}\times 1= x^{n}$, so we are done with $m=0$.</p>
<hr>
<p>Now suppose inductively we have already proven that $x^{n+m}=x^{n}\times x^{m}$;... |
39,790 | <p>I'm teaching a programming class in Python, and I'd like to start with the mathematical definition of an array before discussing how arrays/lists work in Python.</p>
<p>Can someone give me a definition?</p>
| Tim van Beek | 7,556 | <p>An array is a tuple with elements taken from a specific set $S$. When the array can contain variables of a specific type, then the set is the set $S$ consists of all possible values of this type. </p>
<p>This is the most general definition, as a mathematician I don't talk about implementation details like the memor... |
930,949 | <p>Given that the circle C has center $(a,b)$ where $a$ and $b$ are positive constants and that C touches the $x$-axis and that the line $y=x$ is a tangent to C show that $a = (1 + \sqrt{2})b$</p>
| Mazdak | 161,745 | <p>As we know <strong>any indicative sentence that is True or False but not both ,is a statement</strong> , so in this case $D$ is a set and we don't know about its content and we say for every $x$ in the set of $D$, $P(x)$ and this point that "we didn't know what set $D$ is" may be cause that our statement be False or... |
1,017,411 | <blockquote>
<p>Let <span class="math-container">$R$</span> be a commutative Ring with <span class="math-container">$1$</span> and <span class="math-container">$M$</span> a <span class="math-container">$R$</span>-Module. <span class="math-container">$$\varphi: \begin{cases}R & \longrightarrow \text{end}_R(M) \\ a &... | Avitus | 80,800 | <p>Extending the comments:</p>
<ul>
<li><strong>Ring structure on</strong> $\operatorname{End}_R(M)$ (sketch)</li>
</ul>
<p>The composition $\circ$ is nothing but $(\phi\circ\psi)(m):=\phi(\psi(m))\in M$, for all $\phi, \psi\in \operatorname{End}_R(M)$ and $m\in M$. It is clearly associative with unit $1_{\operatorna... |
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