qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
3,510,156 | <p>This is a duplicate question of <a href="https://math.stackexchange.com/questions/2088815/find-integers-solutions-of-x27-y5#">Find integers solutions of $x^2+7=y^5$</a>, however there was no full answer. The solutions <span class="math-container">$(\pm5, 2)$</span> and <span class="math-container">$(\pm 181, 8)$</sp... | Yong Hao Ng | 31,788 | <p>A possible factorization argument leading up till @Kevin's final form, for reference.</p>
<hr>
<p>Since
<span class="math-container">$$
(x+\sqrt{-7}) - (x-\sqrt{-7}) = 2\sqrt{-7}
$$</span>
The possible common factors are <span class="math-container">$\sqrt{-7}$</span> and the prime elements of norm <span class="m... |
1,557,688 | <p>I want to show that the two metrics are equivalent. </p>
<p>Suppose we have a metric space $X \times Y$. Two metrics are defined as:</p>
<p>$d_{X \times Y}((x, y), (x', y')) := \max\{d_X(x, x'), d_Y(y, y')\}$</p>
<p>$d'_{X \times Y}((x, y), (x', y')) := d_X(x, x')+d_Y(y, y')$</p>
<p>Here is my attempt at proof:<... | Em. | 290,196 | <p>Generally, I think, this is taught as or thought of as a box full of "good" items and "bad" items. I will use a slightly different notation. Suppose we have a box full of $N$ balls, and there are $G$ good balls, with $G\leq N$. Let $X$ represent the number of "good" balls I draw when I grab $n$ balls from the box. T... |
2,837,281 | <p>I know the splitting field is generated by $2^{1/4}$ and $i$, I could show $\mathbb{Q} [ 2^{1/4}, i] = \mathbb{Q}[i+2^{1/4}]$ using some algebra. </p>
<p>For the non trivial direction $\mathbb{Q} [ 2^{1/4}, i] \subset \mathbb{Q}[i+2^{1/4}]$. Let us call $\alpha = i+2^{1/4}$, then we know
$$(\alpha-i)^4 - 2 = 0$$
e... | The way of life | 373,966 | <p>To elaborate on @Eric Wofsey 's answer, or perhaps show a more direct view of the injection, define a $G=Gal(K/\mathbb{Q})$-action on $K=\mathbb{Q}[2^{1/4},i]$ by $g.x=g(x)$.</p>
<p>Let us look at the orbit $o(x)$ of $x=2^{1/4}+i$ and the stabilizer $stab_G(x)$.</p>
<p>By orbit-stabilizer theorem $|o(x)|=\frac{|G|... |
912,176 | <p>If $y,z$ are elements of an archimedean field $F$ and if $y<z$, then there is a rational element $r$ of $F$ such that $y<r<z$</p>
<p>The proof begins with saying that it is no loss of generality that we assume that $0<y<z$</p>
<p>I don't understand well why this the case. Please guide me .</p>
<p>... | vociferous_rutabaga | 164,345 | <p>It seems to me that by "without loss of generality," the author really means "since the other case is much easier:" if $z>0$, $y<0$, it's clear that $0$ is the desired rational element. If both are negative, multiply by $-1$.</p>
|
3,756,436 | <p>Recently I was doing a physics problem and I ended up with this quadratic in the middle of the steps:</p>
<p><span class="math-container">$$ 0= X \tan \theta - \frac{g}{2} \frac{ X^2 \sec^2 \theta }{ (110)^2 } - 105$$</span></p>
<p>I want to find <span class="math-container">$0 < \theta < \frac{\pi}2$</span> ... | Narasimham | 95,860 | <p>We can proceed directly.. straightforward way. We have the given relation <span class="math-container">$ (x=X)$</span>:</p>
<p><span class="math-container">$$105 = x\tan\theta-\frac{g}{2} \dfrac{x^2\sec^2\theta}{(110)^2} \tag1 $$</span></p>
<p>To facilitate differentiation let us use symbols for the time being</p>
<... |
493,102 | <p>I have a concern with nested quantifiers.</p>
<p>I have: $$ \forall x \exists y \forall z(x^2-y+z=0) $$ such that $$ x,y,z \in \Bbb Z^+$$ </p>
<p>My first question, can it be read like this:</p>
<p>$$ \forall x \forall z \exists y(x^2-y+z=0) $$</p>
<p>The way I did it, is I started off with $x=1, z=1 $ </p>
... | André Nicolas | 6,312 | <p>The original sentence says that for any $x$, there is a $y$, such that <strong>whatever</strong> $z$ we pick, we have $x^2-y+z=0$.</p>
<p>So the $y$ has to work for all $z$. But that's impossible. If it works for $z=1000$, it fails for all other $z$. The sentence is (very) false. </p>
<p>We <strong>cannot</strong>... |
3,178,860 | <p>I am trying to solve the question 27 of Section 10.4 of Dummit and Foote but I am stuck in the first problem: let me state the question and then I will attach the picture of the page of the corresponding book as well:</p>
<p><em>I am stuck at part c and d</em></p>
<p><strong>Prove that the map <span class="math-co... | fGDu94 | 658,818 | <p>it's <span class="math-container">$-\Delta u$</span> (as a functional...). Why, you may ask...</p>
<p>We require that <span class="math-container">$E(u+h)-E(u) = \langle \nabla E(u) , h \rangle$</span> for any <span class="math-container">$h \in H_0^1 (\Omega)$</span>.</p>
<p><span class="math-container">$E(u+h)-... |
3,178,860 | <p>I am trying to solve the question 27 of Section 10.4 of Dummit and Foote but I am stuck in the first problem: let me state the question and then I will attach the picture of the page of the corresponding book as well:</p>
<p><em>I am stuck at part c and d</em></p>
<p><strong>Prove that the map <span class="math-co... | Matematleta | 138,929 | <p>The Frechet derivative <span class="math-container">$DE$</span>, if it exists, is unique and satisfies</p>
<p><span class="math-container">$$E(u+h)=E(u)+DE(h)+r(h),\ $$</span> where <span class="math-container">$r(h)$</span> is <span class="math-container">$o(h).$</span> So, if we can find a candidate that satisfie... |
978,384 | <p>The following picture is constructed by connecting each corner of a square with the midpoint of a side from the square that is not adjacent to the corner. These lines create the following red octagon:</p>
<p><img src="https://i.stack.imgur.com/PZyGa.jpg" alt="enter image description here"></p>
<p>The question is, ... | flawr | 109,451 | <p>I think the following image will say more than any text. You can divide the image into smaller squares that will allow you immediately to calculate the ratio.</p>
<p>The ratio of the red area within the whole square is the same as the red area in the big green square to the area of the whole green square. And this ... |
978,384 | <p>The following picture is constructed by connecting each corner of a square with the midpoint of a side from the square that is not adjacent to the corner. These lines create the following red octagon:</p>
<p><img src="https://i.stack.imgur.com/PZyGa.jpg" alt="enter image description here"></p>
<p>The question is, ... | user514455 | 514,455 | <p>A calculated answer. Maybe not the simplest.</p>
<p><img src="https://i.stack.imgur.com/12crH.gif" alt="enter image description here"></p>
|
3,773,133 | <p>I have been thinking about this problem for a couple of months, and eventually failed. Could someone help me?</p>
<blockquote>
<p>Let <span class="math-container">$M$</span> and <span class="math-container">$X$</span> be two symmetric matrices with <span class="math-container">$M\succeq 0$</span> and <span class="ma... | greg | 357,854 | <p>Use a colon to denote the trace/Frobenius product
<span class="math-container">$$\eqalign{
A:B = {\rm Tr}(A^TB) = {\rm Tr}(AB^T)
}$$</span>
From this definition and the cyclic property
one can deduce the rearrangement rules
<span class="math-container">$$\eqalign{
A:B &= B:A = B^T:A^T \\
A:BC &= B^TA:C = AC^... |
2,797,902 | <p>AFAIK, every mathematical theory (by which I mean e.g. the theory of groups, topologies, or vector spaces), started out (historically speaking) by formulating a set of axioms that generalize a specific structure, or a specific set of structures. </p>
<p>For example, when people think of a “field” they AFAIK usually... | Gödel | 467,121 | <p>The Peano's arithmetic was thought to axiomatize the structure of natural numbers. However, exists the structure of <a href="https://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic" rel="nofollow noreferrer">non standar naturals</a> where exists a biggest natural number and satisfies that axioms. </p>
|
165,069 | <p>I have a list of the following kind:</p>
<pre><code>{{1,0.5},{2,0.6},{3,0.8},{-4,0.9},{-3,0.95}}
</code></pre>
<p>The important property is, that somewhere in the list, the first element of the sublists changes sign (above is from + to -, but could be from - to +). How can I most efficiently split this into two li... | m_goldberg | 3,066 | <p>This works on your example data, but it might not be general enough to satisfy you, but it is the best I can do with only one example.</p>
<pre><code>data = {{1, 0.5}, {2, 0.6}, {3, 0.8}, {-4, 0.9}, {-3, 0.95}};
Column[{d1, d2} = SplitBy[data, Sign @* First]]
</code></pre>
<p><a href="https://i.stack.imgur.com/Vpd... |
1,227,419 | <p>$$\int\limits_6^{16}\left(\frac{1}{\sqrt{x^3+7x^2+8x-16}}\right)\,\mathrm dx=\frac{\pi }{k}$$</p>
<p><strong>Note:</strong> $k$ is a constant.</p>
| Prasun Biswas | 215,900 | <p>$$x^3+7x^2+8x-16=(x-1)(x+4)^2\implies \sqrt{x^3+7x^2+8x-16}=\sqrt{x-1}(x+4)$$</p>
<p>Denote the indefinite integral as $I$.</p>
<p>$$I=\int\frac{\mathrm dx}{(x+4)\sqrt{x-1}}$$</p>
<p>Make the substitution $u^2=x-1$ and $2u\,\mathrm du=\mathrm dx$ to get,</p>
<p>$$I=2\int\frac{\mathrm du}{u^2+5}$$</p>
<p>This is... |
180,647 | <p>Two persons have 2 uniform sticks with equal length which can be cut at any point. Each person will cut the stick into $n$ parts ($n$ is an odd number). And each person's $n$ parts will be permuted randomly, and be compared with the other person's sticks one by one. When one's stick is longer than the other person's... | Jonathan Gleason | 10,109 | <p>You can't actually add a scalar and a matrix. In general, you can't add two matrices unless they are of the same dimension. However, it is often the case that we denote a scalar matrix (a diagonal matrix all of whose entries are the same) by a scalar. For example, you might write $4$ to denote the matrix $\begin{... |
120,177 | <p>here's a question I had in an exam today:</p>
<blockquote>
<p>Four people are checking 230 exams. In how many ways can you split the
papers between the four of them if you want each one to check at least
15?</p>
</blockquote>
<p>So, after checking $4 \cdot 15$ papers we are left with 170 and so the result is... | Alex Becker | 8,173 | <p>One definition of continuity at a point $p$ is that the value of a function at $p$ is equal to the limit of the function as it approaches $p$. Thus what you want to do is set $$L=f(0)=\lim\limits_{x\to 0}\frac{\sin(.19x)}{x}$$
and I will let you compute that limit yourself.</p>
|
120,177 | <p>here's a question I had in an exam today:</p>
<blockquote>
<p>Four people are checking 230 exams. In how many ways can you split the
papers between the four of them if you want each one to check at least
15?</p>
</blockquote>
<p>So, after checking $4 \cdot 15$ papers we are left with 170 and so the result is... | Arturo Magidin | 742 | <p>By definition, a function $f(x)$ is continuous at $a$ if and only if:</p>
<ol>
<li>$f(x)$ is <em>defined</em> at $x=a$;</li>
<li>$\lim\limits_{x\to a}f(x)$ exists; and</li>
<li>$\lim\limits_{x\to a}f(x) = f(a)$.</li>
</ol>
<p>So for your $f(x)$ to be continuous at $0$ you need it to be defined at $0$ (which it is)... |
3,936,676 | <p>Well this format of a limit <span class="math-container">$0^0$</span> is an indeterminate form.</p>
<p>I claim that whatever this limit is (which depends on the exact question) should always be in between <span class="math-container">$[0,1]$</span>.</p>
<p>Is my claim correct?</p>
<p>I have no mathematical proof for... | Théophile | 26,091 | <p>Your claim is wrong. You can choose any nonnegative number as the limit:
<span class="math-container">$$\lim_{x\to0^+}\left(e^{-1/x}\right)^{ax} = e^{-a}.$$</span></p>
<p>This example is from the <a href="https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero" rel="noreferrer">Wikipedia article</a> on <span class="... |
2,867,479 | <p>From <a href="https://math.stackexchange.com/questions/2867457">ETS Major Field Test in Mathematics</a></p>
<blockquote>
<p>A student is given an exam consisting of
8 essay questions divided into 4 groups of
2 questions each. The student is required to
select a set of 6 questions to answer,
including at l... | Bram28 | 256,001 | <blockquote>
<p>I'm thinking <span class="math-container">$$\binom{2}{1}^4 \binom{4}{2}$$</span></p>
<p>because we have to pick 1 from each of the 4 groups of 2 and then from the remaining 4 questions we pick 2.</p>
</blockquote>
<p>You are overcounting here: there will be two groups from which you end up picking both ... |
3,074,035 | <p>I am trying to find a simplified form for this summation:</p>
<p><span class="math-container">$$B(k,j) \equiv \sum_{i=1}^k (-1)^{k+i} {i \choose j} (k-1)_{i-1} \quad \quad \quad \text{for } 1 \leqslant j \leqslant k,$$</span></p>
<p>where the terms <span class="math-container">$(k-1)_{i-1} = (k-1) \cdots (k-i+1)$<... | robjohn | 13,854 | <p><strong>Some Asymptotic Approximations</strong></p>
<p>The AM-GM inequality says
<span class="math-container">$$
\frac{k!}{(k-i)!}\le\left(k-\frac{i-1}{2}\right)^i\tag1
$$</span>
The fact that
<span class="math-container">$$
\frac{k(k-i+1)}{\left(k-\frac{i-1}2\right)^2}=1-\left(\frac{\frac{i-1}2}{k-\frac{i-1}2}\rig... |
458,088 | <p>I would like to find an approximation when $ n \rightarrow\infty$ of $ \frac{n!}{(n-2x)!}(n-1)^{-2x} $. Using Stirling formula, I obtain $$e^{\frac{-4x^2+x}{n}}. $$ The result doesn't seem right!</p>
<p>Below is how I derive my approximation. I use mainly Stirling Approximation and $e^x =(1+\frac{x}{n})^n $.</p>
... | Emily | 31,475 | <p><em>This is a long comment; hence, community wiki status.</em></p>
<p>Gamification is not the embedding of educational content into a game. Gamification is the construction of elements typically found in games around a traditional paradigm, whether it be education, training, marketing, etc. This mistaken understand... |
458,088 | <p>I would like to find an approximation when $ n \rightarrow\infty$ of $ \frac{n!}{(n-2x)!}(n-1)^{-2x} $. Using Stirling formula, I obtain $$e^{\frac{-4x^2+x}{n}}. $$ The result doesn't seem right!</p>
<p>Below is how I derive my approximation. I use mainly Stirling Approximation and $e^x =(1+\frac{x}{n})^n $.</p>
... | Ilmari Karonen | 9,602 | <p>Well, let me list some ideas from math games (using both words fairly loosely) that <em>I've</em> found interesting. Hopefully, I'm not a complete outlier here, so others may find them fun too.</p>
<h3><a href="http://en.wikipedia.org/wiki/Fractal" rel="noreferrer">Fractals</a></h3>
<p>There are a number of const... |
1,329,398 | <p>So, I've posted a question regarding Wikipedia's quartic page. This was from the first question.</p>
<blockquote>
<p>I'm trying to implement the general quartic solution for use in a ray tracer, but I'm having some trouble. The solvers I've found do cause some strange false negatives leaving holes in the tori I'm t... | Rolf Hoyer | 228,612 | <p>If $\alpha$ is a root of $x^3 = A$, then so too is $\gamma \alpha$, where $\gamma$ is a complex cube root of unity. This follows since $(\gamma\alpha)^3 = \gamma^3\alpha^3 = \alpha^3 = A$.</p>
<p>The nontrivial values of these are given by $\gamma = -\frac{1}{2} + \frac{i\sqrt{3}}{2}$ or $\gamma = -\frac{1}{2} - ... |
1,329,398 | <p>So, I've posted a question regarding Wikipedia's quartic page. This was from the first question.</p>
<blockquote>
<p>I'm trying to implement the general quartic solution for use in a ray tracer, but I'm having some trouble. The solvers I've found do cause some strange false negatives leaving holes in the tori I'm t... | Tito Piezas III | 4,781 | <p>Try this version. Given,</p>
<p>$$x^4+ax^3+bx^2+cx+d=0$$</p>
<p>then,</p>
<p>$$x_{1,2} = -\tfrac{1}{4}a+\tfrac{1}{2}\sqrt{u}\pm\tfrac{1}{4}\sqrt{3a^2-8b-4u+\frac{-a^3+4ab-8c}{\sqrt{u}}}\tag1$$</p>
<p>$$x_{3,4} = -\tfrac{1}{4}a-\tfrac{1}{2}\sqrt{u}\pm\tfrac{1}{4}\sqrt{3a^2-8b-4u-\frac{-a^3+4ab-8c}{\sqrt{u}}}\tag2... |
2,359,372 | <blockquote>
<p>Given that
$$\log_a(3x-4a)+\log_a(3x)=\frac2{\log_2a}+\log_a(1-2a)$$
where $0<a<\frac12$, find $x$.</p>
</blockquote>
<p>My question is how do we find the value of $x$ but we don't know the exact value of $a$? </p>
| Dhruv Kohli | 97,188 | <p>$x$ might come out as a function of $a$.</p>
<p>$$(3x-4a)(3x) = 4(1-2a) \implies 9x^2 - 12ax - 4(1-2a) = 0$$</p>
<p>Solve this quadratic equation.</p>
|
2,359,372 | <blockquote>
<p>Given that
$$\log_a(3x-4a)+\log_a(3x)=\frac2{\log_2a}+\log_a(1-2a)$$
where $0<a<\frac12$, find $x$.</p>
</blockquote>
<p>My question is how do we find the value of $x$ but we don't know the exact value of $a$? </p>
| Parcly Taxel | 357,390 | <p>$$\log_a(3x-4a)+\log_a(3x)=\frac2{\log_2a}+\log_a(1-2a)$$
$$\log_a(3x(3x-4a))=\frac2{\log_aa/\log_a2}+\log_a(1-2a)$$
$$\log_a(3x(3x-4a))=2\log_a2+\log_a(1-2a)$$
$$\log_a(3x(3x-4a))=\log_a4(1-2a)$$
$$3x(3x-4a)=4(1-2a)$$
$$9x^2-12ax+8a-4=0$$
$$x=\frac{12a\pm\sqrt{144a^2-4\cdot9(8a-4)}}{18}$$
$$=\frac{12a\pm12(a-1)}{18... |
114,754 | <p>I have several questions concerning the proof. I don't think I quite understand the details and motivation of the proof. Here is the proof given by our professor.</p>
<p>The space of polynomials $F[x]$ is not finite-dimensional.</p>
<p><em>Proof</em>. Suppose
$$F[x] = \operatorname{span}\{f_1,f_2,\dots,f_n\}$$</p>... | Yuval Filmus | 1,277 | <p>The idea of the proof is this. A basis for the set of all polynomials is
$$1,x,x^2,x^3,\ldots,x^n,\ldots$$
If you have a finite basis, there will be some polynomials that you will be missing, in particular polynomials with high degree. For example, if you take the span of
$$x^3+105x^4+x^7, 1+x^8$$
then for sure you'... |
49,074 | <p>It might sound silly, but I am always curious whether Hölder's inequality $$\sum_{k=1}^n |x_k\,y_k| \le \biggl( \sum_{k=1}^n |x_k|^p \biggr)^{\!1/p\;} \biggl( \sum_{k=1}^n |y_k|^q \biggr)^{\!1/q}
\text{ for all }(x_1,\ldots,x_n),(y_1,\ldots,y_n)\in\mathbb{R}^n\text{ or }\mathbb{C}^n.$$
can be derived from the ... | Peter Patzt | 10,856 | <p>I suggest the following consideration. We will prove the above inequality for rational $p,q\in(1,\infty)$ with $\frac1p+\frac1q= 1$, and the irrational cases follow by continuity.</p>
<p>If $p$ and $q$ are rational, let $p=\frac ab$ and $q=\frac ac$ with $b+c=a$ and $2^m\ge a$. Now by induction
$$ \sum |x^{(1)}\dot... |
161,029 | <p>I have not seen a problem like this so I have no idea what to do.</p>
<p>Find an equation of the tangent to the curve at the given point by two methods, without elimiating parameter and with.</p>
<p>$$x = 1 + \ln t,\;\; y = t^2 + 2;\;\; (1, 3)$$</p>
<p>I know that $$\dfrac{dy}{dx} = \dfrac{\; 2t\; }{\dfrac{1}{t}}... | talmid | 19,603 | <p>One way to do this is by considering the parametric form of the curve:
$(x,y)(t) = (1 + \log t, t^2 + 2)$, so $(x,y)'(t) = (\frac{1}{t}, 2t)$
We need to find the value of $t$ when $(x,y)(t) = (1 + \log t, t^2 + 2) = (1,3)$, from where we deduce $t=1$. The tangent line at $(1,3)$ has direction vector $(x,y)'(1) = (1,... |
3,554,393 | <p>Namely, I need to prove <span class="math-container">${\max\limits_i} |\lambda_i| \leq {\max\limits_i}{\sum\limits_j} |M_{ij}|\mid$</span>, where <span class="math-container">$M$</span> is the matrix and <span class="math-container">$\lambda_i$</span> are its eigenvalues.</p>
<p>I'm not sure if there is any helpful... | Math1000 | 38,584 | <p>In fact a stronger statement is true. Let <span class="math-container">$\rho(M) = \max_i|\lambda_i|$</span> be the spectral radius of <span class="math-container">$M$</span>. Then for each positive integer <span class="math-container">$k$</span>, we have <span class="math-container">$\rho(M)\leqslant \|M^k\|^{\frac1... |
440,242 | <p>I'm pretty sure almost all mathematicians have been in a situation where they found an interesting problem; they thought of many different ideas to tackle the problem, but in all of these ideas, there was something missing- either the "middle" part of the argument or the "end" part of the argumen... | Dirk | 9,652 | <p>Here is an answer which may be math-specific: If you are stuck in some proof of some claim that you believe is true:</p>
<h3>Add the missing piece as assumption and continue as planned.</h3>
|
1,097,134 | <p>this is something that came up when working with one of my students today and it has been bothering me since. It is more of a maths question than a pedagogical question so i figured i would ask here instead of MESE.</p>
<p>Why is $\sqrt{-1} = i$ and not $\sqrt{-1}=\pm i$?</p>
<p>With positive numbers the square r... | egreg | 62,967 | <p>The square root function <em>doesn't</em> return two values for positive numbers, or it wouldn't be a function.</p>
<p>It's a fact that, if $x$ is a positive real number, there are two real numbers whose square is $x$. The positive one is denoted by $\sqrt{x}$, so the negative one is $-\sqrt{x}$.</p>
<p>In this wa... |
3,306,341 | <p>Let <span class="math-container">$f\in Hom(R,R')$</span> be a surjective map and let <span class="math-container">$I$</span> be an ideal of <span class="math-container">$R$</span></p>
<p>Assume that <span class="math-container">$Ker(f)\subseteq I$</span> , prove that <span class="math-container">$f^{-1}(f(I))=I$</s... | egreg | 62,967 | <blockquote>
<p>Let <span class="math-container">$f\colon R\to R'$</span> be a surjective ring homomorphism and <span class="math-container">$I$</span> an ideal of <span class="math-container">$R$</span>. Then <span class="math-container">$f^{-1}(f(I))=I$</span> if and only if <span class="math-container">$\ker f\sub... |
668,959 | <p>I don't know if this is an already existing conjecture, or has been proven: There is at least one prime number between <span class="math-container">$N$</span> and <span class="math-container">$N-\sqrt{N}$</span>.</p>
<p>Some examples:
<span class="math-container">$N=100$</span></p>
<p><span class="math-container">$... | Kevin Arlin | 31,228 | <p>I'm afraid this is false whether we consider the strong form in which endpoints are allowed, or not. $\sqrt{126}<12$ and $113$ is the next prime below $126$. </p>
|
1,680,269 | <p>Here $\mathbb{Z}_{n}^{*}$ means $\mathbb{Z}_{n}-{[0]_{n}}$</p>
<p>My attempt:</p>
<p>$(\leftarrow )$</p>
<p>$p$ is a prime, then, for every $[x]_{n},[y]_{n},[z]_{n}$ $\in (\mathbb{Z}_{n}^{*},.)$ are verified the following:</p>
<p>1) $[x]_{n}.([y]_{n}.[z]_{n}) = ([x]_{n}.[y]_{n}).[z]_{n}$, since from the operatio... | T. Eskin | 22,446 | <p><strong>Hint:</strong> Assume that $d=0$ would be true, and take sequences $(x_{n})\subset K$ and $(y_{n})\subset L$ so that $d(x_n,y_n)\to 0$. Use compactness to conclude a contradiction.</p>
|
4,549,300 | <p>Matrix C of size n<span class="math-container">$\times$</span>n is symmetric . Zero is a simple eigenvalue of C. The associated eigenvector is q. For <span class="math-container">$\epsilon$</span>>0, the equation <span class="math-container">$Cx+\epsilon x=d$</span> in x, where x and d are n-dimensional Column v... | snowman | 1,057,706 | <p>Thanks @<a href="https://math.stackexchange.com/users/71348/ted-shifrin">Ted Shifrin</a>! Here is my answer:</p>
<p>First,
<span class="math-container">$(C+\epsilon I)x = d$</span>, so <span class="math-container">$x=(C+\epsilon I)^{-1}d$</span> and then <span class="math-container">$\epsilon x=\epsilon (C+\epsilon ... |
2,067,003 | <p>(Mathematics olympiad Netherlands) Let $A,B$ and $C$ denote chess players in a tournament. The winner of each match plays the next match against the oponent that did not play the current. At the end of the tournament $A$, $B$ and $C$ played $10$, $15$ and $17$ times respectively. Each match only ended up in a win. <... | Yiorgos S. Smyrlis | 57,021 | <p>$$
\sum_{k=1}^{n}\frac{(-1)^{k+1}}{k(k+1)}=\sum_{k=1}^{n}(-1)^{k+1}\left(\frac1k-\frac1{k+1}\right)
=\sum_{k=1}^n\frac{(-1)^{k+1}}{k}+\sum_{k=1}^n\frac{(-1)^{k+2}}{k+1}
\\=2\sum_{k=1}^n\frac{(-1)^{k+1}}{k}-1-\frac{(-1)^{n+1}}{n+1}.
$$
Hence,
$$
\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n(n+1)}=2\sum_{n=1}^\infty\frac{(-... |
2,239,240 | <p>I'm looking to do some independent reading and I haven't been able to find rough prerequisites for Differential Topology at the level of Milnor or Guillemin and Pollack.</p>
<p>Is a semester of analysis (Pugh) and a semester of topology (Munkres) enough to make sense of most of it or should I take a second semester ... | mrnovice | 416,020 | <p>If I'm understanding your question correctly, then note:</p>
<p>$(2+3)^2 = (5)^2=25 $</p>
<p>$(2+3)^2 = 2^2+2\cdot 2\cdot 3+3^2=4+12+9 = 25$</p>
<p>You can expand the brackets, or simplify the contents of the brackets and then square it and obtain the same answer via both methods, if that's what you mean. </p>
|
2,239,240 | <p>I'm looking to do some independent reading and I haven't been able to find rough prerequisites for Differential Topology at the level of Milnor or Guillemin and Pollack.</p>
<p>Is a semester of analysis (Pugh) and a semester of topology (Munkres) enough to make sense of most of it or should I take a second semester ... | celtschk | 34,930 | <p>It seems you confuse the <em>precedence</em> of operations, which describe how to interpret an expression (and thus which transformations are actually valid), with the order in which you <em>apply valid transformations</em> to an expression.</p>
<p>For your specific formula, $(2+3)^2$, the precedence rules say that... |
1,613,185 | <p>There are five red balls and five green balls in a bag. Two balls are taken out at random. What is the probability that both the balls are of the same colour</p>
| barak manos | 131,263 | <p>The number of ways to choose $2$ balls is $\binom{5+5}{2}=45$.</p>
<p>The number of ways to choose $2$ red balls is $\binom{5}{2}=10$.</p>
<p>The number of ways to choose $2$ green balls is $\binom{5}{2}=10$.</p>
<p>Hence the probability of choosing $2$ balls of the same color is $\frac{10+10}{45}=\frac49$.</p>
|
3,172,485 | <p>Consider the familiar trigonometric identity: <span class="math-container">$\cos^3(x) = \frac{3}{4} \cos(x) + \frac{1}{4} \cos(3x)$</span></p>
<p>Show that the identity above can be interpreted as Fourier series expansion.</p>
<p>so we know that cos is periodic between <span class="math-container">$\pi$</span> and... | John Doe | 399,334 | <p>Using the result given in your question,<span class="math-container">$$\begin{align}\int_{-\pi}^\pi\cos^3 x &= \int_{-\pi}^\pi\left(\frac34 \cos x + \frac 14 \cos 3x\right)\,dx\\
&=\left[\frac34\sin x+\frac1{12}\sin 3x\right]_{-\pi}^\pi\\
&=0\end{align}$$</span> since <span class="math-container">$\sin$<... |
1,776,850 | <p>Given a square $ABCD$ such that the vertex $A$ is on the $x$-axis and the vertex $B$ is on the $y$-axis. The coordinates of vertex $C$ are $(u,v)$. Find the area of square in terms of $u$ and $v$ only.</p>
<p><strong>What I have done</strong></p>
<p>Let the coordinate of $A$ be $(x,0)$ and $B$ be $(0,y)$. Also let... | Roman83 | 309,360 | <p>Let $A(x;0), B(0,y), C(u,v)$, and $AB=a$. Then $$S=a^2$$.
$$\begin{cases}
AB=a=\sqrt{x^2+y^2}
\\
BC=a=\sqrt{u^2+(v-y)^2}{}
\\
AC=\sqrt2a=\sqrt{(u-x)^2+v^2}
\end{cases}$$</p>
<p>$$\begin{cases}
a^2=x^2+y^2 (1)
\\
a^2=u^2+(v-y)^2 (2)
\\
2a^2=(u-x)^2+v^2 (3)
\end{cases}$$</p>
<p>(3)+(2) $3a^2=u^2-2ux+x^2+v^2+u^2+v^2-... |
1,581,161 | <p>Let the triangle $ABC$ and the angle $\widehat{ BAC}<90^\circ$ </p>
<p>Let the perpendicular to $AB$ passing by the point $C$ and the perpendicular to $AC$ passing by $B$ intersect the circumscribed circle of $ABC$ on $D$ and $E$ respectively .
We suppose that $DE=BC$</p>
<p>What is the angle $\widehat{BAC}$ ... | Narasimham | 95,860 | <p>Let $AECFBD $ (labeled anti-clockwise) be a <em>regular hexagon</em> inscribed in a circle center $O$. All the given conditions are satisfied by this assumption ( perpendicularities) and $(DE=BC).$ Simply by symmetry, angle BAC = $60^0.$ </p>
|
1,530,148 | <p>In $(C[0,1],d_\infty)$, consider $U=\{f\in C[0,1]: f(x)\neq 0, \forall x \in [0,1]\}$. Prove that $U$ is open and find his connected components.
I know that for proof the first thing, i have to show the existence of an $\epsilon\gt0$ so for all $f\in U$, $B(f,\epsilon)\subseteq U$. First of all, is that correct? Sec... | Simon S | 21,495 | <p><em>Hint:</em></p>
<p>There is no such $\epsilon$ for <strong>all</strong> $f \in U$.</p>
<p>You want to show that for any $f \in U$ there exists an $\epsilon$ ball $B(f,\epsilon) \subset U$. </p>
<p>Note that as $f$ is continuous, $f$ is either strictly positive or strictly negative (candidate connected componen... |
1,530,148 | <p>In $(C[0,1],d_\infty)$, consider $U=\{f\in C[0,1]: f(x)\neq 0, \forall x \in [0,1]\}$. Prove that $U$ is open and find his connected components.
I know that for proof the first thing, i have to show the existence of an $\epsilon\gt0$ so for all $f\in U$, $B(f,\epsilon)\subseteq U$. First of all, is that correct? Sec... | Georges Elencwajg | 3,217 | <p>The connected components of $U$ are the open subsets $$U_+=\{f\in C[0,1]: f(x)\gt 0, \forall x \in [0,1]\}$$ and $$U_-=\{f\in C[0,1]: f(x)\lt 0, \forall x \in [0,1]\}$$ Indeed if $f,g\in U_+$ (resp $f,g\in U_-$), then $tf+(1-t)g\in U_+$ (resp. $tf+(1-t)g\in U_-$) for all values of $t\in [0,1]$.<br>
This proves that... |
1,876,708 | <p>Let $m<n$. Why $\mathbb R^m$ is closed in $\mathbb R^n$ ? For example, let us take $\mathbb R^3$ and the subspace $\mathbb R^2$. It looks weird to me that $\mathbb R^2$ is closed in $\mathbb R^3$. To me it looks impossible. It may be open, but not closed. Any explanation is welcome.</p>
| Christian Blatter | 1,303 | <p>Let $V$ be any plane in ${\mathbb R}^3$, e.g., the $(x_1,x_2)$-plane $x_3=0$. If ${\bf p}\notin V$ then we can drop a normal $n$ from ${\bf p}$ to $V$, which then will hit $V$ at some point ${\bf p}'$. Note that all points ${\bf x}\in V$ satisfy $$|{\bf x}-{\bf p}|\geq|{\bf p}'-{\bf p}|=:d>0\ .$$ It follows that ... |
1,876,708 | <p>Let $m<n$. Why $\mathbb R^m$ is closed in $\mathbb R^n$ ? For example, let us take $\mathbb R^3$ and the subspace $\mathbb R^2$. It looks weird to me that $\mathbb R^2$ is closed in $\mathbb R^3$. To me it looks impossible. It may be open, but not closed. Any explanation is welcome.</p>
| Bernard | 202,857 | <p>$\mathbf R^m$ is a finite intersection of hyperplanes in $\mathbf R^n$. Each of these is closed, as it's the kernel of a (continuous) linear form. </p>
|
8,816 | <p>What is the result of multiplying several (or perhaps an infinite number) of binomial distributions together?</p>
<p>To clarify, an example.</p>
<p>Suppose that a bunch of people are playing a game with k (to start) weighted coins, such that heads appears with probability p < 1. When the players play a round, t... | David E Speyer | 297 | <p>It is not always true that the automorphism group of an algebraic variety has a natural algebraic group structure. For example, the automorphism group of $\mathbb{A}^2$ includes all the maps of the form $(x,y) \mapsto (x, y+f(x))$ where $f$ is any polynomial. I haven't thought through how to say this precisely in te... |
3,632,576 | <p>Considering that input <span class="math-container">$x$</span> is a scalar, the data generation process works as follows:</p>
<ul>
<li>First, a target t is sampled from {0, 1} with equal probability.</li>
<li>If t = 0, x is sampled from a uniform distribution over the interval
[0, 1]. </li>
<li>If t = 1, x is sampl... | Duchamp Gérard H. E. | 177,447 | <p>In fact, you must combine your idea (cut the integral) and that of Reveillark. Indeed, you begin, as it was suggested by writing
<span class="math-container">$$
\left | \frac{1}{x}\int_0^x f(t)\,dt-a\right|=\left | \frac{1}{x}\int_0^x (f(t)-a) \,dt\right|
$$</span>
now, let us exploit the fact that <span class="ma... |
1,282,486 | <p>Given the function $f(x) = |8x^3 − 1|$ in the set $A = [0, 1].$
Prove that the function is not differentiable at $x = \frac12.$ </p>
<p>The answer in my book is as follows:</p>
<p>$$\lim_{x \to \frac12-} \dfrac{f(x)-f(1/2)}{x-1/2} = -6$$
$$\lim_{x \to \frac12+} \dfrac{f(x)-f(1/2)}{x-1/2} = 6$$ </p>
<p>Can anyone... | Rob Arthan | 23,171 | <p><strong>Hint</strong>: $f(x) = |g(x)|$ where $g(x) = 8x^3 - 1$. What is the derivative of $g(x)$ when $x = \frac{1}{2}$?</p>
|
2,431,375 | <p>A continuous function $f$ on $[a,b]$, differentiable in $(a,b)$, has only 1 point where its derivative vanishes. What is true about this function?</p>
<p>A. $f$ cannot have an even number of extrema.</p>
<p>B. $f$ cannot have a maximum at one endpoint and minimum at the other.</p>
<p>C. $f$ might be monotonically... | Mathemagical | 446,771 | <p>To see that A is false, consider the function $f(x) = (x-0.5)^2$ on [-1,1]. To see that B is false and also that C is true, consider the function $f(x) = x^3$ on the same domain.</p>
|
3,620,612 | <p>I had a question in the exercises of a complex analysis course I couldn't solve, It asked me to evaluate this integral <span class="math-container">$$\int_{-\pi}^{\pi}\frac{dx}{\cos^2(x) + 1}$$</span></p>
<p>I tried to evaluate it without using residues, the antiderivative of this function contains tan, which is no... | Elsa | 278,945 | <p>Here's an answer using residues.</p>
<p>Using
<span class="math-container">$cos(x) = \frac{1}{2}\left(e^{ix} + e^{-ix}\right)$</span>
and defining <span class="math-container">$z=e^{ix}$</span> which implies <span class="math-container">$dz = izdx$</span>, we write the integral as</p>
<p><span class="math-containe... |
666,297 | <p>Find the value of $x$, what is the value of $x$ in this equation, step by step solution will be great.
\begin{equation}
0.4x+15=x
\end{equation}</p>
| Warren Hill | 86,986 | <p>As your asking for help on such a simple question. I'll explain each step</p>
<p>Starting with the original equation </p>
<p>$$0.4 x + 15 = x$$</p>
<p>First we want to get the 15 on its own which we can do by subracting $0.4 x$ from the left had side (LHS) of the equation because we do that on the LHS we need to... |
481,086 | <blockquote>
<p>Find a formula (provide your answer in terms of $f$ and its derivatives) for the curvature of a curve in $\mathbb{R}^3$ given by
$\{(x,y,z)\ | \ x=y, f(x)=z\}$.</p>
</blockquote>
<p>How will I be able to do this problem? </p>
<p>I know that a regular parametrization of a curve then the curvature a... | nbubis | 28,743 | <p><a href="http://en.wikipedia.org/wiki/Curvature" rel="nofollow">Wikipedia</a> gives the following for a curve $(x(t),y(t),z(t))$:
$$\kappa=\frac{\sqrt{(z''y'-y''z')^2+(x''z'-z''x')^2+(y''x'-x''y')^2}}{(x'^2+y'^2+z'^2)^{3/2}}$$</p>
|
1,988,420 | <p>An Ant is on a vertex of a triangle. Each second, it moves randomly to an adjacent vertex. What is the expected number of seconds before it arrives back at the original vertex?</p>
<p>My solution: I dont know how to use markov chains yet, but Im guessing that could be a way to do this. I was wondering if there was ... | Dennis | 381,631 | <p>Let the 3 vertices be A, B, and C. Without loss of generality, assume that the ant starts at A.</p>
<p>Let <span class="math-container">$E_{A}$</span>, <span class="math-container">$E_{B}$</span>, and <span class="math-container">$E_{C}$</span> be the expected number of seconds to get back to vertex A when it's at A... |
512,037 | <p>This is a question from our reviewer for our exam for linear algebra. I just want to have some ideas how to tackle the problem.</p>
<p>If $A$ is an $n\times n$ matrix with integer coefficients, such that the sum of each row's elements is equal to $m$, show that $m$ divides the determinant.</p>
| Casteels | 92,730 | <p>Here's a guide: Why is the "all 1's" vector an eigenvector of $A$? What is its eigenvalue? How does $\det(A)$ relate to the eigenvalues of $A$?</p>
|
3,280,095 | <p>Given the function
<span class="math-container">$$\int \frac{\sqrt{x}}{\sqrt{x}-3}dx $$</span>
You would need to have <span class="math-container">$u=\sqrt{x}-3$</span> and <span class="math-container">$du=\frac{1}{2 \sqrt{x}}$</span>, when I use a online calculator it suggests to rewrite the numerator as
<span clas... | azif00 | 680,927 | <p>If <span class="math-container">$u=\sqrt{x}-3$</span>, then <span class="math-container">$x=(u+3)^2$</span> and <span class="math-container">$dx=2(u+3)du$</span>. Therefore
<span class="math-container">$$\int \frac{\sqrt x}{\sqrt{x}-3}dx=\int \frac{u+3}{u}\cdot 2(u+3)du=\int \frac{2(u+3)^2}{u}du$$</span>
My recommen... |
1,652,165 | <p>On a empty shelf you have to arrange $3$ cans of soup, $4$ cans of beans, and $5$ cans of tomato sauce. What is the probability that none of the cans of soup are next to each other?</p>
<p>I tried working this out but get very stuck because I'm not sure that I'm including all the possible outcomes. </p>
| Marco Disce | 17,010 | <p>You have 9 objets different from soup, put them in a row and consider all the spaces adiacent and in between (the spaces where you could put the soups avoiding to have 2 adjacent soups). There are 10 such spaces therefore:</p>
<ul>
<li>For the first soup you have $10$ choices.</li>
<li>For the second soup you have ... |
1,684,741 | <p>I'm able to show it isn't absolutely convergent as the sequence $\{1^n\}$ clearly doesn't converge to $0$ as it is just an infinite sequence of $1$'s. How do I prove the series isn't conditionally convergent to prove divergence!</p>
| N. F. Taussig | 173,070 | <p>First, factor the equivalence as a difference of squares.<br>
\begin{align*}
y^4 & \equiv 4 \pmod{7}\\
y^4 - 4 & \equiv 0 \pmod{7}\\
(y^2 + 2)(y^2 - 2) & \equiv 0 \pmod{7}
\end{align*}
Hence, $$y^2 + 2 \equiv 0 \pmod{7} \implies y^2 \equiv -2 \equiv 5 \pmod{7}$$ or $$y^2 - 2 \equiv 0 \pmod{7} \implies y^... |
4,050,893 | <p>Given a linear transformation <span class="math-container">$T: V \rightarrow W$</span> where <span class="math-container">$V$</span> and <span class="math-container">$W$</span> are finite dimensional, then is it true that nullity(<span class="math-container">$T$</span>) = nullity(<span class="math-container">$[T]_\b... | Petrus1904 | 808,320 | <p>I have made an algebraic attempt that might help you somewhere. Lets start with the second and third equation:
<span class="math-container">$$AB(I+D) = -BCB \rightarrow AB = -BCB(I+D)^{-1}$$</span>
<span class="math-container">$$(I+D)CA = -CBC \rightarrow CA = -(I+D)^{-1}CBC$$</span>
Now plug these results in the fi... |
51,246 | <p>In undergraduate courses we compute the sum $S$ of some series
of the form $\frac{1}{P(n)}$ where $P(x)$ is some simple
polynomial of degree $2$ with integer coefficients, by the following procedure:</p>
<p>(sketch)</p>
<p>(a) Choose an appropriate periodic function $f(x)$ defined over a domain $D.$</p>
<p>(b) C... | Michael Renardy | 12,120 | <p>There is a systematic method for evaluating series of this type by residue calculus. It is explained in many texts on complex analysis. Using this method, certain sums over all integers can be evaluated. This makes use of functions like cotangent or cosecant, which have poles at all integers. An even function summed... |
43,611 | <p>I posted this on Stack Exchange and got a lot of interest, but no answer.</p>
<p>A recent <a href="http://people.missouristate.edu/lesreid/POW12_0910.html" rel="nofollow">Missouri State problem</a> stated that it is easy to decompose the plane into half-open intervals and asked us to do so with intervals pointing i... | S. Carnahan | 121 | <p>Here is a solution to the half-open interval problem:</p>
<ol>
<li>Start with the interval from $(0,0)$ to $(1,0)$ that contains the endpoint $(0,0)$.</li>
<li>Fill in the closed unit disc minus the point $(1,0)$ using half-open intervals that point inward.</li>
<li>Add the interval from $(1,0)$ to $(1,1)$ that con... |
4,031,476 | <p>I recently completed a variation of a problem I found from a mathematical olympiad which is as follows:</p>
<p>Prove that, for all <span class="math-container">$n \in \mathbb{Z}^+$</span>, <span class="math-container">$n \geq 1$</span>, <span class="math-container">$$\sum_{k=1}^n \frac{k}{2^k} < 2 $$</span></p>
<... | heropup | 118,193 | <p>There are some major issues. First, when <span class="math-container">$n = 1$</span>, the integral is trivially <span class="math-container">$0$</span>, so you cannot make the claim that <span class="math-container">$$S(n) = \sum_{k=1}^n \frac{k}{2^k} < \int_{x=1}^n \frac{x}{2^x} \, dx$$</span> for all <span cla... |
4,031,476 | <p>I recently completed a variation of a problem I found from a mathematical olympiad which is as follows:</p>
<p>Prove that, for all <span class="math-container">$n \in \mathbb{Z}^+$</span>, <span class="math-container">$n \geq 1$</span>, <span class="math-container">$$\sum_{k=1}^n \frac{k}{2^k} < 2 $$</span></p>
<... | Community | -1 | <p><span class="math-container">$$2S_n-S_n=\sum_{k=1}^nk2^{1-k}-\sum_{k=1}^nk2^{-k}=\sum_{k=0}^{n-1}(k+1)2^{-k}-\sum_{k=1}^nk2^{-k}
\\=1-n2^{-n}+\sum_{k=0}^{n-1}2^{-k}<1+1.$$</span></p>
|
4,008,420 | <p>Suppose we had a differentiable curve <span class="math-container">$C$</span> in <span class="math-container">$\mathbb{R}^2$</span> that serves as our "light container". Light is shining in from all directions, so the space of incoming light-beams is <span class="math-container">$\mathbb{R} \times S^1$</sp... | Intelligenti pauca | 255,730 | <p>A possible answer is given by S. Tabachnikov in his book "Geometry and billiards" and originally proposed by R. Peirone (Reflections can be trapped. Amer. Math. Monthly 101 (1994), 259–260).</p>
<p>First of all, one can construct a trap for light rays parallel to a given direction, see figure below (taken ... |
3,860,330 | <p>I am interested in proving what family of functions have the property
<span class="math-container">$$f'(x)=f^{-1}(x)$$</span>
I've never dealt with a differential equation of this form, hence I could only go as far as to gather a little data:</p>
<p><span class="math-container">$$f'(x)=f^{-1}(x)\implies f(f'(x))=x$$... | Yiorgos S. Smyrlis | 57,021 | <p>This is a partial answer.</p>
<p>In what follows we seek for a solution of the form <span class="math-container">$f(x)=ax^b$</span>.</p>
<p>Then <span class="math-container">$f'(x)=abx^{b-1}$</span> while <span class="math-container">$f^{-1}(x)=(x/a)^{1/b}=a^{-1/b}x^{1/b}$</span>.</p>
<p>So, setting, <span class="ma... |
10,977 | <p>When I taught calculus, I posted my notes after the lecture. Then I had the students fill out a mid-quarter evaluation, and a lot of them wanted me to post my notes before class.</p>
<p>What I started doing was printing and handing out the notes to them, leaving the examples blank so they can fill those in. Many ... | Kenneth Laeremans | 6,516 | <p>Well, what I've got during my classes, like GeraldEdgar already said, is that you don't have one solution to solve it all. What you could do is posting your notes the night (or maybe two nights) before the lecture. That way the students who want to use your notes, can print it and take it with them. They're will alw... |
1,579,528 | <p>You decide to play a holiday drinking game. You start with 100 containers of eggnog in a row. The 1st container contains 1 liter of eggnog, the 2nd contains 2 liters, all the way until the 100th, which contains 100 liters. You select a container uniformly at random and take a one liter sip from it. If the container ... | fredq | 297,080 | <p><strong>EDIT</strong> This answer is valid only in the assumption that the probability of taking a sip from a bottle is proportional to the number of sips remaining. </p>
<p>Let $\alpha$ be the number of ways we can arrange <em>all</em> the sips (even counting the one in the end that are not taken)</p>
<p>$$\alpha... |
438,263 | <p>Is there a concrete example of a <span class="math-container">$4$</span> tensor <span class="math-container">$R_{ijkl}$</span> with the same symmetries as the Riemannian curvature tensor, i.e.
<span class="math-container">\begin{gather*}
R_{ijkl} = - R_{ijlk},\quad R_{ijkl} = R_{jikl},\quad R_{ijkl} = R_{klij}, \\
R... | Thomas Kojar | 99,863 | <p>The website <a href="https://www.scilag.net/" rel="nofollow noreferrer">https://www.scilag.net/</a> is also meant as a database.</p>
|
438,263 | <p>Is there a concrete example of a <span class="math-container">$4$</span> tensor <span class="math-container">$R_{ijkl}$</span> with the same symmetries as the Riemannian curvature tensor, i.e.
<span class="math-container">\begin{gather*}
R_{ijkl} = - R_{ijlk},\quad R_{ijkl} = R_{jikl},\quad R_{ijkl} = R_{klij}, \\
R... | Alex W | 72,323 | <p>For open problems in group theory: <a href="https://arxiv.org/abs/1401.0300" rel="nofollow noreferrer">https://arxiv.org/abs/1401.0300</a></p>
|
754,583 | <p>Write <span class="math-container">$$\phi_n\stackrel{(1)}{=}n+\cfrac{n}{n+\cfrac{n}{\ddots}}$$</span> so that <span class="math-container">$\phi_n=n+\frac{n}{\phi_n},$</span> which gives <span class="math-container">$\phi_n=\frac{n\pm\sqrt{n^2+4n}}{2}.$</span> We know <span class="math-container">$\phi_1=\phi$</span... | user88595 | 88,595 | <p>Found why it's $-1$.<br>
Rewrite the equation as :$$n\bigg(\frac{1+\frac1n\sqrt{n^2 + 4n}}{2}\bigg)$$
The square root can be written as $\sqrt{n^2(1+4/n)} = |n|\sqrt{1+4/n} = -n\sqrt{1+4/n}$ because $n<0$.
Then you obtain :
\begin{eqnarray*}
\phi(n) &=& \frac{n}{2}\big(1-\sqrt{1+4/n}\big)\\
&=& \f... |
680,364 | <p>I want some verification for my proof to a homework problem. (Is it correct? Is there a simpler way to do this?)</p>
<p>Let $G$ be a finite group of odd order and suppose there is an element $g$ that is conjugate to its own inverse. In other words, there is $h \neq e$ such that $h^{-1}gh = g^{-1}$. We will show $g=... | Ted Shifrin | 71,348 | <p><strong>HINT:</strong> Note that $gh=hg^{-1}$. What is $(gh)^2$?</p>
|
150,809 | <p>An Iwasawa manifold is a compact quotient of a 3-dimensional complex Heisenberg group by a cocompact, discrete subgroup.
We can also refer to Griffiths and Harris's Principles of Algebraic Geometry p. 444 for simpler description.</p>
<p>I want to compute the automorphism of Iwasawa manifold,i.e.the group of biholo... | Robert Bryant | 13,972 | <p>I'm sure this can be found in the literature, though I don't know exactly where to look. On the other hand, it is easy to calculate the automorphism group directly from the following observations: As in Griffiths--Harris, let $M = G/\Gamma$ where $G$ is the $3$-dimensional complex Heisenberg group and $\Gamma\subs... |
2,292,713 | <blockquote>
<p><strong>Definition.</strong> Let <span class="math-container">$E$</span> be a nonempty subset of <span class="math-container">$X$</span>, and let <span class="math-container">$S$</span> be the set of all real numbers of the form <span class="math-container">$d(p, q)$</span>, with <span class="math-conta... | Ted Shifrin | 71,348 | <p>You're missing an important point. If $x<c$ for all $x\in S$, then $\sup S\le c$. (For example, take $S = (0,1)$. For every $x\in S$, we have $x<1$, but $\sup S = 1$.)</p>
|
418,724 | <p>This question arises in STEP 2011 Paper III, question 2. The paper can be found <a href="http://www.admissionstestingservice.org/our-services/subject-specific/step/preparing-for-step/" rel="nofollow">here</a>. </p>
<p>The first part of the question requires us to prove the result that if the polynomial
$$x^{n}+a_{... | Ross Millikan | 1,827 | <p>You have identified the possible rational roots correctly and shown that none of them work. It is a fine application of the rational root theorem.</p>
|
418,724 | <p>This question arises in STEP 2011 Paper III, question 2. The paper can be found <a href="http://www.admissionstestingservice.org/our-services/subject-specific/step/preparing-for-step/" rel="nofollow">here</a>. </p>
<p>The first part of the question requires us to prove the result that if the polynomial
$$x^{n}+a_{... | Key Ideas | 78,535 | <p>Yes, it's correct. Simpler: Rational Root Test $\,\Rightarrow\,x\in\Bbb Z\,\Rightarrow\, 7 = 5x\!-\!x^n\,$ is even, contradiction.</p>
|
445,816 | <p>I have to show that</p>
<blockquote>
<blockquote>
<p>$\mathbb{C}=\overline{\mathbb{C}\setminus\left\{0\right\}}$,</p>
</blockquote>
</blockquote>
<p>what is very probably an easy task; nevertheless I have some problems.</p>
<p>In words this means: $\mathbb{C}$ is the smallest closed superset of $\mathbb{C... | Jay | 9,814 | <p>The elements of the sequence of complex numbers $1$, $\frac{1}{2}$, $\frac{1}{3}$,$\cdots$ are all contained in $\overline{\mathbb{C} \setminus \{ 0 \} }$ but the limit of the sequence is $0$; hence $0$ is in the closure.</p>
|
2,423,569 | <p>I am asked to show that if $T(z) = \dfrac{az+b}{cz+d}$ is a mobius transformation such that $T(\mathbb{R})=\mathbb{R}$ and that $ad-bc=1$ then $a,b,c,d$ are all real numbers or they all are purely imaginary numbers. </p>
<p>So far I've tried multiplying by the conjugate of $cz+d$ numerator and denominator and see i... | Henno Brandsma | 4,280 | <p>Suppose $T$ is as asked for. </p>
<ul>
<li>Suppose $c\neq 0$. Then if $\frac{-d}{c} \in \mathbb{R}$ and we know that $T(\frac{-d}{c}) = \infty \notin \mathbb{R}$ unless the numerator is $0$ as well, in which $a(\frac{-d}{c}) = b = 0$ which means $\frac{-ad}{c} = -b$ or $-ad = -bc$ or $ad=bc$ and we have a contradic... |
161,616 | <p>There is a well known result that every one dimensional topological manifold without boundary is homeomorphic either to the circle or to the whole real line. However there is one detail hidden: manifold is understood to be second countable (or paracompact). If we drop this assumption it becomes possible to construct... | Mirko | 48,481 | <p>The long ray and the long line are the only non-metrizable 1-manifolds, see e.g. a paper by Peter Nyikos (who also discusses larger dimensions)
<a href="http://www.math.sc.edu/~nyikos/Manifolds.pdf"><code>here</code></a>
(p.2, just after Main Theorem). No proof is given in the above paper (just saying it is easy).... |
234,851 | <p>Find the length of the curve $x=0.5y\sqrt{y^2-1}-0.5\ln(y+\sqrt{y^2-1})$ from y=1 to y=2.</p>
<p>My attempt involves finding $\frac {dy}{dx}$ of that function first, which leaves me with a massive equation.</p>
<p>Next, I used this formula, </p>
<p>$$\int_1^2\sqrt{1+(\frac{dy}{dx})^2}$$</p>
<p>this attempt leave... | Golob | 48,784 | <p>$$\frac{\mathrm{d}}{\mathrm{d}y} \Big[\frac{1}{2}y \sqrt{y^2-1}- \frac{1}{2} \ln(\sqrt{y^2-1}+y) \Big]=$$</p>
<p>$$=\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}y} \Big[y \sqrt{y^2-1} \Big] - \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}y} \Big[\ln(\sqrt{y^2-1}+y) \Big]=$$</p>
<p>$$=\frac{1}{2}\Big[\frac{2y^2-1}{\sqrt{y^2-1}}... |
165,489 | <p>I have problem solving this equation, smallest n such that $1355297$ divides $10^{6n+5}-54n-46$. I tried everything using my scientific calculator, but I never got the correct results(!).and finally I gave up!. Could you help me find the first 2 solutions for this equation ? (thanks.)</p>
| Henrik Schumacher | 38,178 | <p>$n_1 = 2331259$, $n_2 = 3776127$, </p>
<p>Obtained from this Mathematica code:</p>
<pre><code>cf = Compile[{{m, _Integer}},
Block[{n, a, b, p, counter = 0,result},
result = ConstantArray[0, m];
p = 1355297;
n = 0;
a = Mod[10^(5), p];
b = 0;
While[counter < m,
n++;
a = Mod[a 1000000, ... |
1,476,847 | <p>I am a little puzzled by some notations in optimization community. Is there anyone who can explain why $f_1:\mathbb{R}^n\rightarrow\mathbb{R}$ is a finite valueed but $f_2:\mathbb{R}^n\rightarrow\mathbb{R}\cup\{\infty\}$ is not?? I have never have this kind of notations. For function $f_1$ I always calculated limit ... | PhoemueX | 151,552 | <p>This is false even for $2\times2$ diagonal matrices. For these, what
you want reduces to
\begin{eqnarray*}
\left(a_{1}+b_{1}\right)\left(a_{2}+b_{2}\right) & = & \det\left(\left(\begin{array}{cc}
a_{1}\\
& a_{2}
\end{array}\right)+\left(\begin{array}{cc}
b_{1}\\
& b_{2}
\end{array}\right)\right)\\
... |
1,558,256 | <p>The standard Normal distribution probability density function is $$p(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2},\int_{-\infty}^{\infty}p(t)\,dt = 1$$ i.e., mean 0 and variance 1. The cumulative distribution function is given by the improper integral $$P(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-t^2/2}\,dt$$ Describe... | m_gnacik | 182,603 | <p>First note that cumulative distribution of Normal distribution
$$ P(x) = \frac{1}{2}+\frac{1}{2}\mathrm{erf}\left(\frac{x}{\sqrt{2}}\right)$$
where $\mathrm{erf}$ is the error function. </p>
<p>Abramowitz and Stegun gave several approximations of the error function,
<a href="https://en.wikipedia.org/wiki/Error_fun... |
1,558,256 | <p>The standard Normal distribution probability density function is $$p(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2},\int_{-\infty}^{\infty}p(t)\,dt = 1$$ i.e., mean 0 and variance 1. The cumulative distribution function is given by the improper integral $$P(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-t^2/2}\,dt$$ Describe... | Math-fun | 195,344 | <p>Say we look at $x<0$ only. Then consider a change of variable of the form $u=\frac{t}{1-t}$ to have$$P(x)=\int_{-1}^{\frac{x}{1-x}}\frac{e^{-\frac{u^2}{2 (u+1)^2}}}{\sqrt{2 \pi } (u+1)^2}du$$ which is manageable stuff since $$\lim_{u\to -1}\frac{e^{-\frac{u^2}{2 (u+1)^2}}}{\sqrt{2 \pi } (u+1)^2}=0.$$</p>
|
3,886,523 | <p>Each of <span class="math-container">$3$</span> urns contains twenty balls. First urn contains ten white balls, second urn contains six white balls and third urn contains two white balls. All other balls are black.
One ball is drawn from the random urn with return in the same urn. The ball's color is white.
What is ... | Community | -1 | <p>For the most general case in which the base doesn't even need to be a polygon, it is an application of <a href="https://en.wikipedia.org/wiki/Cavalieri%27s_principle" rel="nofollow noreferrer">Cavalieri's Principle</a>. Imagine a pyramid with a blob-shaped base and a square pyramid next to it with the same height a... |
1,318,462 | <p>I am struggling with the following problem.Any help will be appreciated.</p>
<p>If the following statement true then please give a proof otherwise give a counterexample.</p>
<ol>
<li><p>If $a^{27} \equiv 1 \pmod{37}$, then $a^9 \equiv 1 \pmod{37}$ </p></li>
<li><p>$a^{9} \equiv 1 \pmod{37}$, then $a^3 \equiv 1 \pm... | tong_nor | 213,691 | <p>The first one is true: if $a^{27}\equiv 1\pmod{37}$, then $a\bmod{37}$ is one of numbers $1,7,9,10,12,16,26,33,34$. For each of them you have $a^{9}\equiv 1\pmod{37}$.</p>
<p>The second one is false: $7^{9}\equiv 1\pmod{37}$, but $7^3\equiv 10\pmod{37}$.</p>
<p>The third one is also false, try to find a counterexa... |
1,119,634 | <p>Find the point on the curve $y=x^2+2$ where the tangent is parallel to the line $2x+y-1=0$</p>
<p>I understand the answer is $(-1,3)$ but I can't find a way to get there... Thanks </p>
| WW1 | 88,679 | <p>The line $2x+y-1=0$ has a slope of $-2$</p>
<p>the derivative of $x^2+2$ is $2x$</p>
<p>for the $x$-coordinate solve $2x=-2$</p>
<p>for the $y$-coordinate plug the value of $x$ into $y=x^2+2$</p>
|
11,973 | <p>I have a list of strings called <code>mylist</code>:</p>
<pre><code>mylist = {"[a]", "a", "a", "[b]", "b", "b", "[ c ]", "c", "c"};
</code></pre>
<p>I would like to split <code>mylist</code> by "section headers." Strings that begin with the character <code>[</code> are section headers in my application. Thus, I ... | rm -rf | 5 | <p>Here's one approach using <code>FixedPoint</code> and <code>Replace</code>:</p>
<pre><code>sectionQ := ! StringFreeQ[#, "["] &;
FixedPoint[
Replace[#, {h___, sec_?sectionQ, Longest[x___?(! sectionQ@# &)], t___} :> {h, t, {sec, x}}] &,
mylist]
(* {{"[a]", "a", "a"}, {"[b]", "b", "b"}, {"[ c ... |
11,973 | <p>I have a list of strings called <code>mylist</code>:</p>
<pre><code>mylist = {"[a]", "a", "a", "[b]", "b", "b", "[ c ]", "c", "c"};
</code></pre>
<p>I would like to split <code>mylist</code> by "section headers." Strings that begin with the character <code>[</code> are section headers in my application. Thus, I ... | kglr | 125 | <pre><code>Split[mylist, StringFreeQ["["] @ #2 &]
</code></pre>
<blockquote>
<p>{{"[a]", "a", "a"}, {"[b]", "b", "b"}, {"[ c ]", "c", "c"}}</p>
</blockquote>
<pre><code>SequenceCases[mylist, a:{_?(!StringFreeQ[ "["]@#&),__?(StringFreeQ[ "["])}:> {a}]
</code></pre>
<blockquote>
<p>{{"[a]", "a", "a"}, {... |
233,238 | <p>I am just practicing making some new designs with Mathematica and I thought of this recently. I want to make a tear drop shape (doesn't matter the orientation) constructed of mini cubes. I am familiar with the preliminary material, I am just having some difficulty getting it to work.</p>
| AsukaMinato | 68,689 | <pre><code>list // Map[Flatten]
</code></pre>
<blockquote>
<p>{{Position, Code}, {1, 0, 1}, {2, 100, 11}, {3, 110, 111}, {4, 1000,
1001}, {5, 1100, 1011}, {6, 1110, 1111}}</p>
</blockquote>
|
233,238 | <p>I am just practicing making some new designs with Mathematica and I thought of this recently. I want to make a tear drop shape (doesn't matter the orientation) constructed of mini cubes. I am familiar with the preliminary material, I am just having some difficulty getting it to work.</p>
| m_goldberg | 3,066 | <p><code>ReplacePart</code> can do it.</p>
<pre><code>data =
{{Position, {Code}},
{1, {0000, 0001}}, {2, {0100, 0011}}, {3, {0110, 0111}},
{4, {1000, 1001}}, {5, {1100, 1011}}, {6, {1110, 1111}}};
ReplacePart[data, {i_, 2} :> Sequence @@ data[[i, 2]]]
</code></pre>
<blockquote>
<pre><code>{{Position, Code... |
14,847 | <p>I thought a simple Mathematica kerning machine (for adjusting the space between characters) would be interesting, but I'm having trouble with the locators. (There are a number of other questions related to this, and I've read the answers, but as yet without finding a solution, or understanding them that well.)</p>
... | m_goldberg | 3,066 | <p>This is just an addendum to jVincent's answer. In order to constrain the letters/locators to a horizontal line, you need to use the second argument of <code>Dynamic</code>. The following modified version of jVincent's answer adds the needed constraint: </p>
<pre><code>With[{text = "Wolfram", fontsize = 96, font = ... |
674,448 | <p>Prove $F: \mathbb{R}\to\mathbb{R}$ where $F(x) = \int_a^x f(t)\, dt$ ($a<x$) is surjective. </p>
<p>$f$ is continuous and bounded below by $m>0$. Also $a$ belongs to $\mathbb{R}$ (reals).</p>
| John Hughes | 114,036 | <p>OK. The $a < x$ condition is wrong, and makes the theorem false, as others have pointed out. So let's get rid of it. </p>
<p>Let $u \in \mathbb R$ be nonnegative. Let $x = a + u/m$. Now estimate $F(x)$:
\begin{align}
F(x) &= \int_a^x f(t) ~dt \\
&\ge \int_a^x m ~dt \\
&= mx - ma \\
&= m(a + u... |
424,209 | <p>I am a Computer Science student. While going through some random maths topics I came across Chaos Theory. I wanted to know if there are any applications of it in CS.
I tried searching on the internet about this but ended up only with <a href="https://security.stackexchange.com/questions/31000/does-chaos-theory-have-... | Angela Pretorius | 15,624 | <ol>
<li><p>A small world network introduced with time delay generates chaos. Databases that are linked in a small world network are faster to extract information from. Most neural networks are small world networks.</p></li>
<li><p>Computer graphics? Generating realistic-looking animations of flames, flowing water etc.... |
13,166 | <p>I taught IT in an engineering school during three years in <a href="https://en.wikipedia.org/wiki/Problem-based_learning" rel="nofollow noreferrer">problem based learning</a> (PBL) only. Now I teach maths to pupils between 10 and 15 years old who have a lot of educational difficulties.</p>
<p>I'm thinking to use PB... | James S. | 1,798 | <p>PBL, especially if you have a lot of students that struggle with mathematics, is definitely something that you will have to design carefully; ordinary project based approaches can sometimes exacerbate already existing difficulties. At the same time, it is important not to lower the rigor or expectations for those w... |
3,525,814 | <p>One reasonably well-known property of the Thue-Morse sequence is that it can be used to provide solutions to the <a href="https://en.wikipedia.org/wiki/Prouhet%E2%80%93Tarry%E2%80%93Escott_problem" rel="nofollow noreferrer">Prouhet–Tarry–Escott problem</a> - for example, splitting the first eight nonnegative integer... | Sam | 746,289 | <p>Mathematician, "Chen Shuwen" has givn solution </p>
<p>for, <span class="math-container">$m=4$</span> & is shown below.</p>
<p><span class="math-container">$m=1,2,3,4$</span></p>
<p><span class="math-container">$(401,521,641,881,911)^m=(431,461,701,821,941)^m$</span></p>
<p>The link to his web page's is give... |
3,858,517 | <p>Is it possible to count exactly the number of binary strings of length <span class="math-container">$n$</span> that contain no two adjacent blocks of 1s of the same length? More precisely, if we represent the string as <span class="math-container">$0^{x_1}1^{y_1}0^{x_2}1^{y_2}\cdots 0^{x_{k-1}}1^{y_{k-1}}0^{x_k}$</s... | RobPratt | 683,666 | <p>I confirm your results for <span class="math-container">$n \le 16$</span>. It might be useful to compute the values by conditioning on <span class="math-container">$k\in\{1,\dots,\lfloor(n+3)/2\rfloor\}$</span>:
<span class="math-container">\begin{matrix}
n\backslash k & 1 & 2 & 3 & 4 & 5 & ... |
2,416,671 | <p>Here is the problem: Let $K$ be a compact subset of $ \mathbb{R}^{m} $ ($m>1$) with empty interior and such that $\mathbb{R}^{m}\setminus K $ has no bounded component. For $n=1,2,...$, we define
$$K_{n}=\lbrace x\in \mathbb{R^{m}}: distance (x,K)=1/n\rbrace.$$ Prove that for all $x\in K$, there is a sequence $(y... | Daniel Fischer | 83,702 | <p>Since $K$ has empty interior, for every $r > 0$ there is an $y \in B(x,r/2) \setminus K$. Let $\delta = \operatorname{dist}(y,K)$, and choose $z \in K$ with $\lVert y-z\rVert = \delta$. For $t \in (0,1]$, let $p(t) = ty + (1-t)z$. Then $$\operatorname{dist}(p(t),K) = \lVert p(t) - z\rVert = t\delta$$ since $B(p(t... |
2,637,812 | <p>Here is dice game question about probability.</p>
<p>Play a game with $2$ die. What is the probability of getting a sum greater than $7$?</p>
<p>I know how the probability for this one is easy, $\cfrac{1+2+3+4+5}{36}=\cfrac 5{12}$.</p>
<p>I don't know how to solve the follow-up question:</p>
<p>Play a game with ... | user | 505,767 | <p>Note that by Taylor's series</p>
<ul>
<li>$\log(1+x^2)=x^2-\frac{x^4}{2}+o(x^4)$</li>
<li>$\sin x^2=x^2+o(x^2)$</li>
</ul>
<p>thus
$$\frac{x^2-\log(1+x^2)}{x^2\sin^2x}=\frac{x^2-x^2+\frac{x^4}{2}+o(x^4)}{x^4+o(x^4)}=\frac{\frac12+o(1)}{1+o(1)}\to \frac12$$</p>
|
207,243 | <p>I am using FindFit function in order to fit my data and get two parameters: c and m.</p>
<p>The function that I am using has the following form:</p>
<pre><code>function = (m/x*(x/c)^m)*Exp[-1*(x/c)^m];
</code></pre>
<p>The answer should be c = 64.68 and m = 2.47, but I am constantly getting the error message Over... | kglr | 125 | <p>Using <code>pts</code> from mikado's answer:</p>
<pre><code>DelaunayMesh[pts]["Faces"]
</code></pre>
<blockquote>
<p>{{3, 4, 1}, {1, 4, 5}, {1, 2, 3}, {1, 5, 6}} </p>
</blockquote>
<pre><code>Grid @ %
</code></pre>
<p><a href="https://i.stack.imgur.com/7Nk5E.png" rel="noreferrer"><img src="https://i.stack.imgu... |
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