qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
1,543,542 | <p><strong>Proposition</strong> If $x \in \mathrm{int}(S \cap T)$, then $x \in \mathrm{int}(S) \cap \mathrm{int}(T)$, where $\mathrm{int}(S)$ is the set of interior points of the set $S$.</p>
<p>I have proven this proposition by demonstrating that $\mathrm{int}(S \cap T) \subseteq \mathrm{int}(S) \cap \mathrm{int}(T)$... | Graham Kemp | 135,106 | <p>Any arbitrary point which is interior to the intersection of $S\cap T$ must also a point interior to a set which contains $S\cap T$ (a supraset). (Why?--<em>refer to what "interior point" means here</em>--) Both $S$ and $T$ are suprasets to their intersection $(S\supseteq S\cap T), (T\supseteq S\cap T)... |
534,670 | <p>I am a bit unclear about underflowing in terms of binary representation.</p>
<p>Let's say that an unsigned 8-bit variable gets overflown from the addition of $150+150$. </p>
<p>A signed 8-bit variable gets underflown after the subtraction of $-120-60$.</p>
<p>Now my point is let's think of 8-bit variable, we are ... | zain | 458,919 | <p>The term arithmetic underflow (or "floating point underflow", or just "underflow") is a condition in a computer program where the result of a calculation is a number of smaller absolute value than the computer can actually store in memory.</p>
|
3,304,441 | <p>Let <span class="math-container">$G$</span> be any locally compact group and <span class="math-container">$H$</span> be a compact group. </p>
<p>We know that a map <span class="math-container">$F: G \rightarrow G$</span> is called affine if there exists some <span class="math-container">$\alpha \in G$</span> and an... | José Carlos Santos | 446,262 | <p>If <span class="math-container">$p(X)$</span> has a double root <span class="math-container">$\alpha$</span>, then <span class="math-container">$\alpha$</span> will also be a root of <span class="math-container">$p'(X)$</span>. Therefore, <span class="math-container">$p(X)$</span> and <span class="math-container">$p... |
2,050,385 | <p>Solve $15x$ "congruent to" $20\mod 88$</p>
<p>So far I think I know $15\mod 88$ is $-41$ or if positive $47$`</p>
| Bill Dubuque | 242 | <p>${\rm mod}\ 88\!:\,\ x \equiv \dfrac{20}{15}\equiv \dfrac{4}3\equiv \dfrac{-84}3\equiv -28\equiv 60$</p>
<p><strong>Beware</strong> $\ $ Modular fraction arithmetic is well-defined only for fractions with denominator <em>coprime</em> to the modulus, and we can only cancel factors <em>coprime</em> to the modulus (as... |
142,481 | <ol>
<li><p>Which term is used for model categories whose homotopy categories are triangulated? Stable proper model categories?</p></li>
<li><p>I want $Ho(Pro-M)$ to be triangulated ($Pro-M$ is the category of pro-objects of M) and the functor $Ho(M)\to Ho(Pro-M)$ to be an exact full embedding. Which restrictions on M ... | Tony Huynh | 2,233 | <p>The equivalence relation that HJRW defines in the comments (that is twin vertices may be adjacent) is related to the <a href="http://en.wikipedia.org/wiki/Cocoloring" rel="nofollow">cochromatic number</a> $z(G)$ of a graph $G$. The cochromatic number of $G$ is the minimum number of colours needed to colour $V(G)$ s... |
142,481 | <ol>
<li><p>Which term is used for model categories whose homotopy categories are triangulated? Stable proper model categories?</p></li>
<li><p>I want $Ho(Pro-M)$ to be triangulated ($Pro-M$ is the category of pro-objects of M) and the functor $Ho(M)\to Ho(Pro-M)$ to be an exact full embedding. Which restrictions on M ... | Flo Pfender | 12,487 | <p>In <a href="http://arxiv.org/abs/1108.5699">this paper</a>, they call these objects "blow-up graphs", since the operation of adding twins to a vertex is commonly called a blow-up. Putting integer weights on the vertices of a graph to signify by how much to blow up a vertex like you do above is fairly common. </p>
|
175,803 | <p>This is one of my old unsolved questions when I reading Novikov's book on homology theory. I do not know how to prove it because standard triangulation, fundamental diagram, etc does not help and it should be easy to prove. </p>
| mland | 29,116 | <p>I would agree with Leonid. The claim should definitely be true for open surfaces which are built from closed ones by removing a top handle (i.e. a disc), (this should follow from a handle decomposition and the classification of closed surfaces).
For example (as also stated in wiki if you search for surfaces) you cou... |
967,198 | <p>Given a string consisting of lower-case characters from English alphabets, we want to reverse a substring from the string such that the string becomes a palindrome.</p>
<p><strong>Note :</strong> A Palindrome is a string which equals its reverse.</p>
<p>We need to tell if some substring exists which could be rever... | Henry | 6,460 | <ol>
<li>You can ignore the outer characters which match </li>
<li>If there is a solution, any successful switch will involve one end apart from the ignored outer characters </li>
<li>If there is a solution, you know which characters are at each end of a string you must switch: the outer characters apart from the igno... |
42,326 | <p>Does there exist a nowhere monotonic continuous function from some open subset of $\mathbb{R}$ to $\mathbb{R}$?
Some nowhere differentiable function sort of object?</p>
| N. S. | 9,176 | <p>Check my answer to this question:</p>
<p><a href="https://math.stackexchange.com/questions/41902/is-this-condition-sufficient-to-ensure-monotonicity-of-a-function/41907#41907">Is this condition sufficient to ensure monotonicity of a function?</a></p>
<p>For that function, there are enough details so you can prove ... |
744,309 | <p>Prove or disprove the following assertion. </p>
<blockquote>
<p>The set of all nonzero scalars matrices is a normal subgroup of $GL_2(\mathbb{R})$.</p>
</blockquote>
<p>Proof: </p>
<p>Let $I$ be the identity matrix. Consider the scalar matrix $sI$ where $s$ is some scalar.<br>
Then let $A$ be any other matrix i... | Hayden | 27,496 | <p>Hint: show that the product of any two scalar matrices is a scalar matrix, and that the inverse of a scalar matrix is another scalar matrix.</p>
|
3,695,173 | <p><span class="math-container">$M$</span> is a <span class="math-container">$2\times2$</span> matrix. <span class="math-container">$M$</span> is diagonalizable over <span class="math-container">$\mathbb{R}$</span>. <span class="math-container">$M$</span> has the values <span class="math-container">$1$</span> and <span... | Tsemo Aristide | 280,301 | <p><span class="math-container">$M=\pmatrix{1&a\cr b&2}$</span> the caracteristic polynomial of <span class="math-container">$M$</span> is <span class="math-container">$(1-X)(2-X)-ab=X^2-3X-ab+2$</span> we must have <span class="math-container">$9-4(2-ab)>0$</span> and <span class="math-container">$det(M)=2-... |
3,695,173 | <p><span class="math-container">$M$</span> is a <span class="math-container">$2\times2$</span> matrix. <span class="math-container">$M$</span> is diagonalizable over <span class="math-container">$\mathbb{R}$</span>. <span class="math-container">$M$</span> has the values <span class="math-container">$1$</span> and <span... | Aryaman Maithani | 427,810 | <p>Let <span class="math-container">$M = \begin{pmatrix}1 & a \\ b & 2\end{pmatrix}$</span>. </p>
<p>Note that if <span class="math-container">$M$</span> is diagonalisable over <span class="math-container">$\Bbb R$</span>, then it must have all its eigenvalues real. (This is necessary.)<br>
The eigenvalues of... |
1,984,849 | <p>Can you please clarify whether, for the following question, I need to use the definition of linear transformation, or something else?</p>
<blockquote>
<p>Compute the inverse of the function
$f: \mathbb{R}^3 \rightarrow \mathbb{R}^3$, where $f(x_1,x_2,x_3) := (x_2+x_3, x_1+x_3, x_1+x_2)$.</p>
</blockquote>
| Logician6 | 306,688 | <p>The main thing to realize is that
\begin{align*}
f \left(\left[ \begin{matrix}x_1 \\ x_2 \\ x_3 \end{matrix} \right] \right)=\left[ \begin{matrix}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{matrix} \right]\left[ \begin{matrix}x_1 \\ x_2 \\ x_3 \end{matrix} \right],
\end{align*}
for all $\left[ ... |
3,205,317 | <p>(a) <span class="math-container">$x(v)= 3, y(v)= 4, z(v)= v$</span> for <span class="math-container">$−\infty < v < \infty$</span>,</p>
<p>(b) <span class="math-container">$x(t)= 3\cos(t), y(t)= 2\sin(t), z(t)= 3t−1$</span> for <span class="math-container">$0 \leq t < 2\pi$</span>.</p>
<p>I have no idea w... | DINEDINE | 506,164 | <p>You’re right when writing <span class="math-container">$|\cos{\frac{x}{2}}|=\cos{\frac{x}{2}}$</span> the teacher consider only the cas where <span class="math-container">$\cos\frac{x}{2}\ge 0$</span>. This does not anyway change the structure of the proof in all cases. By the way you can use the fact that <span cla... |
142,220 | <p>Fermat proved that <span class="math-container">$x^3-y^2=2$</span> has only one solution <span class="math-container">$(x,y)=(3,5)$</span>.</p>
<p>After some search, I only found proofs using factorization over the ring <span class="math-container">$Z[\sqrt{-2}]$</span>.</p>
<p>My question is:</p>
<p>Is this Fermat'... | Bob | 76,558 | <p>Here is how Fermat probably did it (it is how I did it - not all of the steps were needed but I have to believe this was close to Fermat's thought process).</p>
<p>Any prime of the form $8n+1$ or $8n+3$ can be written in the form $a^2 +2b^2$. This is proved with descent techniques once realizes that $-2$ and $1$ a... |
29,619 | <p>From the book,<br>
Suppose $p \equiv 1 \pmod{4}$, then by law of quadratic reciprocity, we have:
$$\left(\frac{3}{p}\right) = \left(\frac{p}{3}\right) $$
Next, if $p \equiv 2 \pmod{3}$, then $p \equiv 5 \pmod{12}$
Hence, $$\left(\frac{3}{p}\right) = \left(\frac{p}{3}\right) = -1$$</p>
<p>How do they get those Leg... | yunone | 1,583 | <p>I believe you can find the value of $(3|p)$ without finding $p\equiv 5\pmod{12}$, by use of the second supplement to quadratic reciprocity. Recall that
$$
\left(\frac{2}{p}\right)=(-1)^{(p^2-1)/8}.
$$
Since $p\equiv 2\pmod{3}$, you then have $(p|3)=(2|3)=-1$. So altogether,
$$
\left(\frac{3}{p}\right)\left(\fra... |
1,211,751 | <p>I have an induction homework question that I got stuck in the middle.</p>
<p>Prove by induction that if $a + a^{-1} \in \Bbb{Z}$ then for each $n \in \Bbb{N}$ the following is true:
$$a^{n} + a^{-n} \in \Bbb{Z}$$</p>
<p>If possible, I would like to understand the method for solving those kind of questions (I know ... | user26486 | 107,671 | <p>So we have $a^1+a^{-1}\in\mathbb Z$ and $a^2+a^{-2}=(a^1+a^{-1})^2-2\in\mathbb Z$. </p>
<p>Suppose $a^k+a^{-k}\in\mathbb Z$ for all $k\in\mathbb Z^+, k\le n$, where $a^n+a^{-n}\in\mathbb Z$. </p>
<p>We will prove that then $a^{n+1}+a^{-(n+1)}\in\mathbb Z$ (so your statement follows by strong induction).</p>
<p>... |
470,427 | <p>I am trying to understand the topology on $\{0,1\}^X$, where $X$ is uncountable. The topology on $\{0,1\}$ is the discrete and I am using the product topology on $\{0,1\}^X$.
My question is, who are the basic open sets? From my understanding of the definition of product topology, basic sets should either contain fin... | Asaf Karagila | 622 | <p>It will be easier, perhaps, to think about it as a topology on the power set of $X$. The basic open sets are the sets of the form: $$U_{A,B}=\{Y\subseteq X\mid A\subseteq Y\land B\cap Y=\varnothing\}$$
Where $A,B$ are finite subsets of $X$. </p>
|
2,168,125 | <p>From an exercise list:</p>
<blockquote>
<p>Let $V$ be a inner product space over $\mathbb{C}$ and $T\in \mathcal{L}(V)$ a normal operator such that $T^2=-I$. Prove that T preserves the inner product, i.e. $\langle Tu,Tv\rangle = \langle u,v \rangle, \forall u,v\in V$.</p>
</blockquote>
<p>I found a bunch of equa... | Jonathaniui | 420,908 | <p>Well if it has n <strong>distinct</strong> eigenvalues then yes, each eigenspace must have dimension one. This is because each one has at least dimension one, there is n of them and sum of dimensions is n, if your matrix is of order n it means that the linear transformation it determines goes from and to vector spac... |
3,345,704 | <p>Let <span class="math-container">$X \sim N_p(\mu_1,\Sigma_1)$</span> and <span class="math-container">$Y\sim N_p(\mu_2, \Sigma_2)$</span>, <em>where <span class="math-container">$\Sigma$</span> denotes the covariance matrix</em>, and assume <span class="math-container">$X$</span> and <span class="math-container">$Y$... | MPW | 113,214 | <p>You have NOT shown it's <span class="math-container">$\infty$</span> using rectangular coordinates.</p>
<p>You've only shown that's the case for your specific path of approach (<span class="math-container">$y=mx$</span>). It's possible you could get a different result for a different path of approach.</p>
<p>To ma... |
1,673,967 | <blockquote>
<p>Show algebraically that the Joukowski transformation maps the unit circle, $|z| = 1$, to the straight line segment, $-2 \le u \le 2$ and $v = 0$. </p>
</blockquote>
<p>Other information given is that $u+iv = f(x+iy)$ where $u=u(x,y)$ and $v=v(x,y)$. </p>
<p>For example if $w = z^2$, then $u+iv = (x+... | Qiaochu Yuan | 232 | <p>For commutative rings there is a geometric way to think about these things. Every module $M$ over a commutative ring $R$ has a <a href="https://en.wikipedia.org/wiki/Support_of_a_module">support</a>, which is the set of <a href="https://en.wikipedia.org/wiki/Prime_ideal">prime ideals</a> $P$ such that the <a href="h... |
2,773,457 | <p>I was doing a sample question and came across this question.</p>
<p>A given surface is defined by the equation:
$3x^2+2y^2-z=0$. Describe the normal vector at a point (x, y, z) on the surface. Calculate the normal vector at the point $(1,-1,5)$ on the surface. </p>
<p>The normal vector is
$(6x, 4y, -1)$</p>
<p>Ho... | Mohammad Riazi-Kermani | 514,496 | <p>It is the gradient vector which is normal to the level surface.</p>
<p>The gradient vector is defined to be the vector of partial derivatives which in your case is $(6x, 4y, -1)$</p>
|
3,152,144 | <p>Find the value of <span class="math-container">$x$</span> where <span class="math-container">$f(x)$</span> attains its minimum. (Hint: you will need the Chain Rule.)</p>
<p><span class="math-container">$$f(x) = \int_{-10}^{x^2+2x} e^{t^2}\,dt. $$</span></p>
<p>I'm a little confused by this. I thought this would be... | Robert Z | 299,698 | <p>Yes, here we have to use the Fundamental Theorem of Calculus AND the Chain Rule: it follows that
<span class="math-container">$$f'(x)=e^{(x^2+2x)^2}\cdot (x^2+2x)'=e^{(x^2+2x)^2}\cdot 2(x+1).$$</span>
What is <span class="math-container">$f'(-1)$</span>? Find where <span class="math-container">$f$</span> is increas... |
1,379,849 | <blockquote>
<p>Find the number of seven digit whole numbers in which only $2$ and $3$ are present as digits if no two $2$'s are consecutive in any number?</p>
</blockquote>
<p><strong>My Approach</strong>:
We can make numbers and see like: $2323232$, $2333333$, $2332332$, etc. Please suggest alternate solution of t... | abiessu | 86,846 | <p>One approach to this problem is to reduce the base (as already attempted) as $128\equiv 2\pmod 7$. Next we should reduce the exponent as much as possible to a "simpler" form:</p>
<p>$$128^{128}=(2^7)^{128}=2^{896}$$</p>
<p>which brings us to</p>
<p>$$2^{2^{896}}\pmod 7$$</p>
<p>Note that if we consider $2^{2^n}... |
11,684 | <p>I think it happened dozens of times that I corrected expressions like
$$\huge\mathbb{F_5[x]}$$
to
$$\huge\mathbb F_5[x].$$ </p>
<p>I wonder why this particular misspelling — putting the whole expression as an argument into <code>\mathbb</code> — happens so often.</p>
<p><strong>EDIT</strong>
For an ordinary typo, ... | MvG | 35,416 | <p>Just guessing, but it <em>might</em> be that people consider the whole expression the name of a field, and they are used to writing the names of sets and similar objects in blackboard bold. After all, $\mathbb F$ all by itself usually does not denote a set, but the whole expression does. This does not mean that I ag... |
162,725 | <p>I am working with a very large sparse matrix (for example) given in what follows:</p>
<pre><code>m = 50; n = 40; o = 30; size = m*n*o;
B = SparseArray[{
{i_, i_} -> RandomReal[], {size, size - 1} ->
2., {i_, j_} /; Abs[i - j] == 5 ->
1., {i_, j_} /; Abs[3 i - j] == 2 -> 2.
}, {size, size... | Ulrich Neumann | 53,677 | <p>Perhaps ReplacePart[] is the answer you are looking for:</p>
<pre><code>index0 = Position[bounindex, 0];
Bnew=ReplacePart[B, index0 -> vector1];
</code></pre>
<p>It is as fast as the Table-version...</p>
|
162,725 | <p>I am working with a very large sparse matrix (for example) given in what follows:</p>
<pre><code>m = 50; n = 40; o = 30; size = m*n*o;
B = SparseArray[{
{i_, i_} -> RandomReal[], {size, size - 1} ->
2., {i_, j_} /; Abs[i - j] == 5 ->
1., {i_, j_} /; Abs[3 i - j] == 2 -> 2.
}, {size, size... | Carl Woll | 45,431 | <p><em>(updated)</em></p>
<p>For later comparisons, I will store the value of <code>B</code>:</p>
<pre><code>B0 = B;
</code></pre>
<p><strong>Original post</strong></p>
<p>Set all the rows to zero at once. First, use <a href="http://reference.wolfram.com/language/ref/Pick" rel="nofollow noreferrer"><code>Pick</code... |
1,302,732 | <p>This is an example from Stoll's <em>Introduction to Analysis</em>. I'm struggling to understand why there's a contradiction here, though I think I'm on the verge of understanding it, but I'd like to understand it more formally. </p>
<p><img src="https://i.stack.imgur.com/vWRwP.jpg" alt="enter image description here... | RRL | 148,510 | <p>Hint: Consider $|f_n(x_n)-0|$ with $x_n = 1 - 1/n.$ What is $\displaystyle\lim_{n \to \infty}\left(1-1/n\right)^n?$</p>
|
1,302,732 | <p>This is an example from Stoll's <em>Introduction to Analysis</em>. I'm struggling to understand why there's a contradiction here, though I think I'm on the verge of understanding it, but I'd like to understand it more formally. </p>
<p><img src="https://i.stack.imgur.com/vWRwP.jpg" alt="enter image description here... | Alberto Debernardi | 140,199 | <p>The rough idea behind uniform convergence is that for any $\varepsilon >0$, "there exists a tube of radius $\varepsilon$" around the limit function in which the $f_n$'s are contained for sufficiently large $n$. In your case, this "tube" does not exist, since your sequence of functions consists only of continuous ... |
100,551 | <p>I need to get a matrix $\{a(x_i-x_j)\}$, where $x_i$ form a partition of an interval, $a(x)$ is a given function. I use </p>
<pre><code>In[67]:= a[x_?NumericQ] := N[Exp[-Abs[x]]];
x = Table[-10 + 0.02 (j - 1), {j, 1, 1001}];
A = Outer[a[#1 - #2] &, x, x]; // AbsoluteTiming
Out[69]= {2.99032, ... | ybeltukov | 4,678 | <p><code>Outer</code> is highly optimized for several built-in functions (<code>Plus</code>, <code>Times</code>, <code>List</code>). Therefore</p>
<pre><code>Exp@-Abs@Outer[Plus, #, -#] &@Range[-10, 10, 0.02]; // RepeatedTiming
(* {0.025, Null} *)
</code></pre>
<p>gives ~50x speedup over <code>Outer[#1 - #2&,... |
3,799,157 | <p>If we consider a sequence of functions <span class="math-container">$f_n$</span> there are theorems relating the properties of the functions <span class="math-container">$f_n$</span> with the properties of the function <span class="math-container">$\displaystyle F(x) = \sum_{n=0}^\infty f_n$</span> (when the series ... | Danny Pak-Keung Chan | 374,270 | <p>Suppose that <span class="math-container">$F(x)=\int_{a}^{b}f(t,x)dt$</span>. Informally, we expect that
<span class="math-container">$F'(x)=\int_{a}^{b}f_{x}(t,x)dt$</span>, where <span class="math-container">$f_{x}$</span> denotes the partial
derivative <span class="math-container">$\frac{\partial f}{\partial x}$<... |
1,575,676 | <p><a href="https://i.stack.imgur.com/xYGNz.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xYGNz.png" alt="enter image description here"></a></p>
<p>This is from Spivak Calculus on Manifolds, section 5.3</p>
<p>I have done part a, but I am stuck on part (b) and have been for a day now:</p>
<p>let... | MPW | 113,214 | <p>The symbol "$\frac d{dx}$" is used to indicate a single derivative (with respect to $x$).</p>
<p>We treat repeated application of this operator symbolically as "powers" of the operator (as if it were ordinary multiplication by an ordinary fraction), writing "$\frac{d^n}{dx^n}$" to indicate $n$ successive applicatio... |
13,460 | <p>there're some students, who belive that <span class="math-container">$$\frac10 = \infty $$</span></p>
<p>I need to teach them that this is not true and <span class="math-container">$\frac10 $</span> is undefined, mathematically and give a good picture (for their minds)</p>
<p>what is the proper way to teach them wit... | Timothy | 11,718 | <p>It is possible for an expression to be undefined. If you work in <span class="math-container">$\mathbb{N}$</span>, you can consider <span class="math-container">$2 - 3$</span> to be undefined. We just have meaningless expressions. Similarly when working in <span class="math-container">$\mathbb{R}$</span>, we can als... |
21,801 | <p>I seem to have severe difficulty finding the right tags to add to a post, which is infuriating when I can't post unless I have added some.</p>
<p>For a simple brackets expansion problem I had, I can't use *expansion *brackets or *multiplying.</p>
<p>Why are these Not available?</p>
<p>What should I do to satisfy ... | Eric Wofsey | 86,856 | <p>If you have difficulty choosing the right tags for a specific question, don't worry too much--others can edit in appropriate tags after you post the question. If you want to increase the likelihood of this happening, you could end your question with a remark inviting them to do so, such as the following:</p>
<bloc... |
21,801 | <p>I seem to have severe difficulty finding the right tags to add to a post, which is infuriating when I can't post unless I have added some.</p>
<p>For a simple brackets expansion problem I had, I can't use *expansion *brackets or *multiplying.</p>
<p>Why are these Not available?</p>
<p>What should I do to satisfy ... | Martin Sleziak | 8,297 | <p>Some additional suggestions to the good answers that were already given:</p>
<p>When you are adding a tag, a short information about the tag call <a href="https://meta.stackexchange.com/questions/74645/display-tag-excerpt-when-asking-a-question">tag-excerpt is shown</a>. If you are not familiar with the tags yet, y... |
2,112,802 | <p>If $A \subseteq \mathbb{R} $
$$
\exists p \in A, \forall q \in A , q \leq p $$</p>
<hr>
<p>Can I just use a specific value for $p$ and arbritary value for $q$ to disprove this?</p>
<p>$p = 3$ and $q = p + 1$, hence $q > p$</p>
<hr>
<p>Also, how would should one go about this one: </p>
<p>If .. $\exists p \... | jsvb | 355,263 | <p>By the Pythagorean theorem, the distance between two points $(x,y)$ and $(a,b)$ is given by $\sqrt{(x-a)^2+(x-b)^2}$. The condition given in a) then translates to
$$\sqrt{(x-4)^2+y^2}=\sqrt{(x-1)^2+y^2}.$$
We can solve this for $y$. Take the square and simplify to obtain
$$y=\pm\sqrt{4-x^2}.$
This equation character... |
2,430,729 | <p>How do you calculate the zeros of $f(x) = x^4-x^2-6$ in best way?
Here are my attempts:</p>
<p>Factorize, but the problem is the $6$:</p>
<p>$$0=x^4-x^2-6 \iff 6 = x^2(x^2-1)$$</p>
<p>This doesn't lead to any good solution...</p>
<p>Here is another attempt, but I don't know if this is allowed:</p>
<p>$$0= x^4-x... | José Carlos Santos | 446,262 | <p>Solve the equation $(x^2)^2-x^2-6=0$. You will get$$x^2=\frac{1\pm\sqrt{25}}2\iff x^2=3\vee x^2=-2.$$But $x^2$ cannot be $-2$. Therefore, $x^2=3$, and this means that the solutions are $\pm\sqrt3$.</p>
|
2,430,729 | <p>How do you calculate the zeros of $f(x) = x^4-x^2-6$ in best way?
Here are my attempts:</p>
<p>Factorize, but the problem is the $6$:</p>
<p>$$0=x^4-x^2-6 \iff 6 = x^2(x^2-1)$$</p>
<p>This doesn't lead to any good solution...</p>
<p>Here is another attempt, but I don't know if this is allowed:</p>
<p>$$0= x^4-x... | An aedonist | 143,679 | <p>Once you got to $x^2 (x^2-1) = 6$ (in plain English, "a number times (itself - $1$) $= 6$), you could have thought that $6 = 3 \cdot 2$ , and conclude in one go, finding the two real roots $\pm \sqrt{3}$.</p>
|
3,200,995 | <p>I would like to approximate <span class="math-container">$$\frac{\cos^2{x}}{\sin(x) \tan(x)}$$</span> using the small angle approximations. </p>
<p>Throughout I will use <span class="math-container">$\sin(x) \approx x$</span>, <span class="math-container">$\tan(x) \approx x$</span>, <span class="math-container">$\c... | vegardlarsen85 | 682,295 | <p>Constant Hamiltonian in Optimal Control Theory are related to the Beltrami Identity appearing in Calculus of Variations.</p>
<p>In Calculus of Variations, if the Lagrangian <span class="math-container">$ \mathcal{L} $</span> don't explicetly depend on time such that <span class="math-container">$ J = \int \mathcal{... |
4,621,925 | <p>I'm reading Lang's Complex Analysis (GTM 103) as an introduction to complex analysis. I came across the theorem which states that a set of complex numbers is compact if and only if it's closed and bounded.</p>
<p>The definitions used in the book is presented below:</p>
<p><strong>Accumulation points</strong>
Let <sp... | Paul Frost | 349,785 | <p>From Lang's book:</p>
<blockquote>
<p>Let <span class="math-container">$S$</span> be a subset of the plane. A <strong>boundary point</strong> of <span class="math-container">$S$</span> is a point <span class="math-container">$\alpha$</span> such that <strong>every</strong> disc <span class="math-container">$D(\alpha... |
1,557,097 | <p>Show that a finite domain $F$ is a field.</p>
<p>Let $I$ a proper ideal of $F$ and let $a\in I$. In particular, $a$ is not invertible, otherwise $I$ wouldn't be proper. </p>
<p>I would like to show that $I=(a)=(0)$, but without success. </p>
| egreg | 62,967 | <p>Since $F$ is finite, it has a minimal non zero ideal $I$. If $a\in I$, $a\ne0$, then $\{0\}\ne aF\subseteq I$; by minimality, $I=aF$. Now consider $a^2$; since $F$ is a domain, then $a^2\ne0$, so $\{0\}\ne a^2F\subseteq aF$ and, by minimality, $a^2F=aF$. In particular $a=a^2x$, for some $x\in F$. Therefore $1=ax$ an... |
2,714,190 | <p>Find the last three digits of $6^{2002}$. I did some work and figured out that the last two digits is 36. Can anyone help me with the hundredth digit? By the way, I used modular arithmetic and the recursion method for the tens digit, but it fell short when I attempted to do the hundreds digit. Thank you in advance!<... | Ross Millikan | 1,827 | <p>I did the spreadsheet approach. Put $6$ in a cell, =mod(6*up,1000) in the cell below, copy down, and look for a repeat. I found a repeat of $25$, so $6^{2002} \equiv 6^{27}\equiv 536 \pmod {1000}$. You can't use $6^2$ because you need the value to be a multiple of $8$.</p>
|
2,714,190 | <p>Find the last three digits of $6^{2002}$. I did some work and figured out that the last two digits is 36. Can anyone help me with the hundredth digit? By the way, I used modular arithmetic and the recursion method for the tens digit, but it fell short when I attempted to do the hundreds digit. Thank you in advance!<... | fleablood | 280,126 | <p>$3^{400}\equiv 1\mod 1000$ so $3^{2002}\equiv 9\mod 1000$.</p>
<p>$2^{100}\equiv 1\mod 125$ so $2^{2002}\equiv 4\mod 125$.</p>
<p>And $2^{2002}\equiv 0\mod 8$. So by Chinese Remainder theorem $2^{2002}\equiv 504 \mod 1000$.</p>
<p>So $6^{2002}\equiv 9*504\equiv 536\mod 1000$.</p>
|
2,714,190 | <p>Find the last three digits of $6^{2002}$. I did some work and figured out that the last two digits is 36. Can anyone help me with the hundredth digit? By the way, I used modular arithmetic and the recursion method for the tens digit, but it fell short when I attempted to do the hundreds digit. Thank you in advance!<... | András Salamon | 3,362 | <p>It is possible to do this without the Chinese Remainder Theorem, just using the binomial expansion. Note that
$$\begin{align}2^{500k} &= (30+2)^{100k} = 2^{100k} \pmod{1000},\\
2^{10k} &= (1000+24)^k \equiv 24^k \pmod{1000},\\
24^{5k} &= 2^{15k}3^{5k} = (32000+768)^k 243^k \equiv (768\cdot 243)^k \pmod{... |
2,326,259 | <p>I tried the following $$I = \langle X^2,X+1\rangle =\langle X^2,X+1,X^2+2(X+1)\rangle =\langle X^2,X+1,(X+1)^2+1 \rangle$$</p>
<p>Yet no matter how I arrange it, I cannot obtain $1$. Can someone help me out?</p>
| Hagen von Eitzen | 39,174 | <p>Note that $I$ contains
$$X^2+(X+1)(-X+1) =1$$</p>
|
1,702,064 | <p>While solving Linear algebra and Its application by Gilbert Strang, I am not getting any idea how to solve the problem 6.4.14, which says </p>
<p>From the zero submatrix decide the signs of the $n$ eigenvalues:
$$\pmatrix{0&.&0&1 \\ .&.&0&2 \\0&0&0&.\\1&2&.&n}$$ </p>
... | Jean Marie | 305,862 | <p>What I am going to write is not fully rigorous, but this site is also a place where people practising mathematics show sometimes how they have an intuitive grasp on certain situations.</p>
<p>I have thought a certain time asking myself what do they mean when they say "from the zero submatrix decide the signs of the... |
1,427,970 | <p>Let $G$ act on $\Omega$ transitively, and let $|G| = |\Omega| + 1$ (both sets are assumed to be finite). I want to show from first principles (using maybe arguments like the pigeonhole principle, but not Burnside's lemma) that there exists a non-trivial element having a fixed point. For example let $\Omega = \{\alph... | M.U. | 261,888 | <p>$G$ acts transitively on $\Omega$ (both finite). If the action were free then $|G| = |\Omega|$. Thus by your assumption the action is <em>not</em> free, i.e. for some $\omega \in \Omega$ the set $Stab(\omega) = \{g \in G \mid g \cdot \omega = \omega \}$ contains at least two elements, thus some non-trivial element $... |
593,438 | <p>I was reading Mathematics for Economists by Simon and Blume.</p>
<blockquote>
<p>The level set $x^2+y^2+z^2=1$ is a two-dimensional sphere of radius $1$.</p>
</blockquote>
<p>How to actually know that it is two dimensional?</p>
| Community | -1 | <p>It is probably easier if we look at in terms of spherical coordinates:
$$x = \sin(\theta) \cos(\phi), y = \sin(\theta) \sin(\phi) \text{ and } z = \cos(\theta)$$
Note that $x^2 + y^2 + z^2 = 1$.
We only have two degrees of freedom, namely, $\theta$ and $\phi$. Hence, it is a two dimensional surface.</p>
|
954,147 | <p>I want to solve the congruence for $k$ such that $k^2\equiv 5k\pmod {15}, 2\leq k\leq 30$.</p>
<p>For this, if $\gcd(15,k)=1$, then $k\equiv 5\pmod{15}$. Is my approach correct?
How can I get the values of $k$.</p>
| mathlove | 78,967 | <p>Well, no because if $\gcd(15,k)=1$, then $k\not\equiv 5\pmod{15}$.</p>
<p>Since there is an integer $s$ such that
$$k^2-5k=15s\iff k^2=5(k+3s),$$
there is an integer $t$ such that $k=5t$. So, we have
$$(5t)^2=5(5t+3s)\iff 5t(t-1)=3s.$$</p>
<p>Hence, we have
$$t=3u\ \ \text{or}\ \ t-1=3v$$
where $u,v$ are integer... |
265,494 | <p>I have two lists given by:</p>
<pre><code>t1 = {{1, 2}, {3, 4}, {5, 6}};
t2 = {a, b, c};
</code></pre>
<p>and want to replace the second parts of <code>t1</code> with <code>t2</code> to get</p>
<pre><code>{{1,a},(3,b},{5,c}}
</code></pre>
<p>I tried</p>
<pre><code>t1 /. {u_, v_} -> {u, #} & /@ t2
</code></pre... | bmf | 85,558 | <p>An alternative way</p>
<pre><code>t1 = {{1, 2}, {3, 4}, {5, 6}};
t2 = {a, b, c};
list = Flatten[MapThread[List, {t1, #}] &@t2];
Partition[Drop[list, {2, Length@list, 3}], 2]
</code></pre>
<p>Edit: thanks to @lericr for the relevant comment about timings.</p>
<p>I think it should be sufficient to say that all of ... |
2,743,266 | <p>I am able to find the sixth derivative of $\cos(x^2)$ by simply replacing the $x$ in the Taylor series for $\cos(x)$ with $x^2$ but beyond simple substitutions, I am struggling... </p>
<p>Thanks for any help!</p>
| Sri-Amirthan Theivendran | 302,692 | <p>Since you need to differentiate $6$ times, you can just brute force it. Indeed
$$
\begin{align}
f(x)&=\cos^2x\\
f'(x)&=-2\cos x\sin x=-\sin(2x)\\
f''(x)&=-2\cos(2x)\\
&\vdots
\end{align}
$$
and so on. The rest is just remembering the trig derivatives and using the chain rule.</p>
|
385,887 | <p>I'm looking for an example of a mathematical relation that is symmetric but not reflexive. A standard non-mathematical example is siblinghood. </p>
| rschwieb | 29,335 | <p>For any equivalence relation $\sim$, $x\nsim y$.</p>
<p>This captures the example of "equality" that people came up with earlier, and grabs other similar things like "is isomorphic to", etc.</p>
<p>Strictly speaking, you are not using transitivity at all, so any reflexive symmetric relation would do.</p>
<p>There... |
385,887 | <p>I'm looking for an example of a mathematical relation that is symmetric but not reflexive. A standard non-mathematical example is siblinghood. </p>
| Austin Mohr | 11,245 | <p>Here is an example that isn't really in the spirit of the question, but I'll provide it anyway.</p>
<p>Let $R$ be any symmetric relation (possibly even reflexive) on a set $X$. Define a new relation $R^\prime$ by
$$
R^\prime = R \setminus \{(x,x) \mid x \in X\}.
$$</p>
<p>What we've done is take a symmetric relati... |
1,967,928 | <p>I came across a question that interested me recently. It asked the following:</p>
<blockquote>
<p>Prove that if <span class="math-container">$\mathbb R$</span> is homeomorphic to <span class="math-container">$X \times Y$</span>, then <span class="math-container">$X$</span> or <span class="math-container">$Y$</span> ... | Orest Bucicovschi | 378,410 | <p>Let $X$, $Y$ be topological spaces so that $\mathbb{R}$ is homeomorphic to $X \times Y$. Since $\mathbb{R}$ is connected, clearly so is $X \times Y$. Now $X$ is a surjetive image of $X\times Y$, and so is $Y$. Hence both $X$ and $Y$ are also connected. </p>
<p>Note now that $\mathbb{R}$- and so $X\times Y$- has the... |
1,967,928 | <p>I came across a question that interested me recently. It asked the following:</p>
<blockquote>
<p>Prove that if <span class="math-container">$\mathbb R$</span> is homeomorphic to <span class="math-container">$X \times Y$</span>, then <span class="math-container">$X$</span> or <span class="math-container">$Y$</span> ... | bof | 111,012 | <p>If $X$ and $Y$ are connected topological spaces, each containing at least two points, then the product space $X\times Y$ has no cut point.</p>
<p>Proof. Consider any point $(a,b)\in X\times Y;$ I have to show that $X\times Y\setminus\{(a,b)\}$ is connected.</p>
<p>Choose $x_0\in X\setminus\{a\}$ and $y_0\in Y\setm... |
386 | <p>The Reshetikhin-Turaev construction take as input a Modular Tensor Category (MTC) and spits out a 3D TQFT. I've been told that the other main construction of 3D TQFTs, the Turaev-Viro State sum construction, factors through the RT construction in the sense that for each such TQFT Z there exists a MTC M such that the... | Kevin Walker | 284 | <p>If you have a 3d TQFT, with no anomaly, and which goes down to points, and where things are sufficiently finite and semisimple, then I think you can show that it comes from a Turaev-Viro type construction on the 2-category Z(pt).</p>
<p>If you have a 3d TQFT, possibly with anomaly, which goes down to circles, and w... |
386 | <p>The Reshetikhin-Turaev construction take as input a Modular Tensor Category (MTC) and spits out a 3D TQFT. I've been told that the other main construction of 3D TQFTs, the Turaev-Viro State sum construction, factors through the RT construction in the sense that for each such TQFT Z there exists a MTC M such that the... | Noah Snyder | 22 | <p>Kevin's parenthetical about needing things to be sufficiently finite and semisimple suggests that thought of another way the answer is "no." In particular, there are known non-semisimple TQFTs. I know very little about these, but Alexis Virelizier was very into them. His papers (for example, <a href="http://arxiv... |
2,544,705 | <p>I got introduced to the idea of fractals, and the idea that fractals can have dimensions that are non-integers. </p>
<p>This got me thinking, the space of real-valued functions has a dimensionality as well, and it seems likely to me that the space of continuous functions has a dimensionality strictly lower than tha... | Cardinal | 254,200 | <p>We want to show</p>
<p>$$\int_{\mu}^{\infty} \frac{1}{\Gamma(k)} z^{k-1} e^{-z}dz = \sum_{x = 0}^{k-1} \frac{\mu^x}{x!e^\mu}$$</p>
<p>So, the easiest way is to use the series representation of the incomplete Gamma function:</p>
<p>$$ \int_{\mu}^{\infty} \frac{1}{\Gamma(k)} z^{k-1} e^{-z}dz = \frac{1}{\Gamma(k)} \... |
1,558,530 | <p>Let $\alpha, \beta$ be random variables, $P(\alpha = i) = P(\beta = i) = \frac{1}{N}$, $i \in \{1, \ldots, N\}$.</p>
<p>What is the probability that $\alpha^3 + \beta^3 = 3 t, t \in \mathbb{N}$? </p>
| djechlin | 79,767 | <p>Use the fact that $a^3 \cong a \mod 3$.</p>
|
1,533 | <p>What is up with this site: <a href="https://mathoverflow.net/">https://mathoverflow.net/</a> ? Is it a clone or something? I wasn't paying attention and went to login, and it says that my name is unknown... What's up? I revoked access from my Google account just to be sure.</p>
| InterestedGuest | 3,731 | <p>-overflow and -stackexchange are sister sites, mathoverflow specifically is research-level math-oriented, while math.stackexchange is open to questions of all levels (thanks Arturo).</p>
|
1,365,882 | <p>The theorem statement is "if $f$ is continuous on $[a,b]$, $f$ is bounded on $[a,b]$". This is proven in the textbook Calculus by the author Apostol by the "method of successive bisection", which I'm sure many are familiar with. The proof is done by contradiction. </p>
<p>Here is my concern with this proof: we take... | David C. Ullrich | 248,223 | <p>No, you don't have to "answer" the question of in which subinterval the function is unbounded! If a function is unbounded in $A\cup B$ then it is unbounded in $A$ or unbounded in $B$ (or both). So you let $C=A$ or $C=B$, in such a way that $f$ is unbounded in $C$. Which is it, $A$ or $B$? We don't know and we don't ... |
1,018,292 | <p>A Markov chain with discrete time dependence and stationary transition probabilities is defined as follows. Let $S$ be a countable set, $p_{ij}$ be a nonnegative number for each $i,j\in S$ and assume that these numbers satisfy $\sum_{j\in S}p_{ij}=1$ for each $i$. A sequence $\{X_j\}$ of random variables with ranges... | Did | 6,179 | <p>One can consider <em>primitives</em> of stochastic processes, in the following sense. Assume that $(U_n)$ is i.i.d. and define $(X_n)$ and $(S_n)$ recursively by $X_0=S_0=0$ and, for every $n\geqslant0$, $$X_{n+1}=X_n+U_{n+1},\qquad S_{n+1}=S_n+X_{n+1}.
$$
Then, the process $(U_n)$ is i.i.d. by definition (hence a ... |
3,664,433 | <p>I'm reading Peter Scott's <em><a href="https://homepages.warwick.ac.uk/~masgar/Teach/2012_MA4J2/geometry.pdf" rel="nofollow noreferrer">The Geometry of 3-Manifolds</a></em> and am trying to understand the argument behind this statement, which arises in the proof of Corollary 3.3:</p>
<blockquote>
<p>If <span clas... | Kyle Miller | 172,988 | <p>Here's an argument that doesn't involve covering spaces that illustrates some important facts about Seifert fibered spaces. I'm basically following Hatcher's <a href="http://pi.math.cornell.edu/~hatcher/3M/3Mdownloads.html" rel="nofollow noreferrer">"Notes on basic 3-manifold topology"</a>, Propositions 1.11 and 1.1... |
3,528,227 | <blockquote>
<p>How to transform the <span class="math-container">$\tanh$</span> sigmoid function so that it starts from <span class="math-container">$f(0)=0$</span>, goes asymptotically to <span class="math-container">$1$</span>, and has <span class="math-container">$f(0.1)=a$</span> and <span class="math-container"... | Mostafa Ayaz | 518,023 | <p>Starting from <span class="math-container">$$f(x)=p+q\tanh (rx+s)$$</span>with <span class="math-container">$$\tanh(u)={e^{2u}-1\over e^{2u}+1}$$</span>we obtain <span class="math-container">$$q,r>0$$</span>
<span class="math-container">$${p+q=1\\p+q\tanh(s)=0\\
p+q\tanh(0.1r+s)=a\\
p+q\tanh(0.9r+s)=b\\
}$$</span... |
2,682,599 | <p>Several teams played a baseball tournament (as a reminder, there are no ties in baseball); each team played every other team exactly once. We say that team $A$ is dominating team $B$ if either $A$ beat $B$ heads up or if there exists a team $C$ such that $A$ beat $C$ and $C$ beat $B$. (Notice that it is entirely pos... | Community | -1 | <p>(note: the end of this post contains a precise statement of what is what is meant by the statement quoted in the OP)</p>
<p>There are lots of strange functions; many operations you'd like to do to functions in analysis aren't really well-behaved (or even well-<em>defined</em>) for <em>all</em> of them.</p>
<p>For ... |
1,715,945 | <p>I need help solving/understanding this question:</p>
<p>L (x,y) : "x loves y".
Translate "there are exactly two people whom Lynn loves".
Its answer includes a variable "z". I do not get that part with the variable "z". How did it come here when it was not introduced in the question? Detailed solution is appreciated... | DonAntonio | 31,254 | <p>I think the following definition (the most easy to grab and/or standard I know) of a free group will solve the problem: </p>
<p>The group $\;F_n\;$ is free on $\;S:=\{s_1,...,s_n\}\;$ iff for <em>any function</em> $\;f:S\to G\;,\;\;G\;$ any group,</p>
<p>there exists a unique homomorphism $\;\phi:F_n\to G\;$ exten... |
3,628,919 | <blockquote>
<p>Give an example of a <span class="math-container">$T\in\mathcal L\left(\mathbb R^2\right)$</span> s. t. <span class="math-container">$Ker(T) = Im(T)$</span>.</p>
</blockquote>
<p><strong>MY APPROACH</strong></p>
<p>According to the rank-nullity theorem, <span class="math-container">$\dim Ker(T) = \o... | Physical Mathematics | 592,278 | <p>The rank-nullity theorem tells you that the sum of the dimensions of the range and nullspace (also called the kernel) must equal the dimension of the domain, i.e. 2. Thus in your example, you are looking for 1D subspaces of <span class="math-container">$\mathbb{R}$</span>. You might as well just take that be the <sp... |
3,365,569 | <p>I want to calculate the value of √2 but using any common probability distribution, preferably based on Bernoulli Trials.</p>
<p>I will perform a test in real and observe the output of the test and the output of this test should lead me to the value of √2 <strong>like as we can compute the value of π using <a href="... | Joitandr | 704,255 | <p>I think, the easiest way to handle this problem is to represent Dirichlet distribution in terms of exponential family: <a href="https://en.wikipedia.org/wiki/Exponential_family" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Exponential_family</a></p>
|
2,963,964 | <p>I'm usually good at determining divergence and using the comparison test, but I can't figure out what function I can use to determine if
<span class="math-container">$$ \int_{0}^{1} \frac{e^{x^2}}{x^2} \, dx$$</span>
is divergent. If anyone can help me, that would be greatly appreciated. </p>
| Nosrati | 108,128 | <p>For <span class="math-container">$x>0$</span>, <span class="math-container">$e^{x^2}>1$</span> then
<span class="math-container">$$\dfrac{e^{x^2}}{x^2}>\dfrac{1}{x^2}$$</span></p>
|
2,963,964 | <p>I'm usually good at determining divergence and using the comparison test, but I can't figure out what function I can use to determine if
<span class="math-container">$$ \int_{0}^{1} \frac{e^{x^2}}{x^2} \, dx$$</span>
is divergent. If anyone can help me, that would be greatly appreciated. </p>
| farruhota | 425,072 | <p>Using the Taylor series:
<span class="math-container">$$\begin{align}\int_{0}^{1} \frac{e^{x^2}}{x^2} \, dx&=\int_{0}^{1} \frac{1+x^2+\frac{x^4}{2!}+O(x^6)}{x^2} \, dx=\\
&=\int_{0}^{1} \frac{1}{x^2}+1+\frac{x^2}{2!}+O(x^4) \, dx>\\
&>\int_{0}^{1} \frac{1}{x^2} \, dx=\\
&=\lim_{x\to 1^-}\left(-... |
1,184,963 | <p>Toss two fair dice. There are $36$ outcomes in the sample space $\Omega$, each with probability $\frac{1}{36}$. Let:</p>
<ul>
<li>$A$ be the event '$4$ on first die'.</li>
<li>$B$ be the event 'sum of numbers is $7$'.</li>
<li>$C$ be the event 'sum of numbers is $8$'.</li>
</ul>
<p>It says here $A$ and $B$ are ind... | Graham Kemp | 135,106 | <p>Since we are using <em>fair</em> dice, the atomic outcomes have equal probability measure. </p>
<p>$\begin{array}{l}
A = \{(4,1), (4,2), (4,3), (4,4), (4,5), (4, 6)\}
\\
B = \{(1,6), (2,5), (3,4), \color{blue}{(4,3)}, (5,2), (6,1)\}, & A\cap B=\{(4,3)\}
\\
C = \{(2,6), (3,5), \color{blue}{(4,4)}, (5,3), (6,2)\}... |
2,246,777 | <p>I was curious as to whether $$\lim_{x \to 0}\frac{1}{x^2}=\infty $$
Or the limit does not exist? Because doesn't a limit exist if and only if the limit tends to a finite number? </p>
| Mark Viola | 218,419 | <p>Note that for any $B>0$, $\frac{1}{x^2}>B$ whenever $0<|x|<\frac{1}{\sqrt{B}}$. By definition the limit is $\infty$.</p>
|
2,538,390 | <p>I don't understand the concept of circle inversion.</p>
<p>$OP \cdot OP' = k^2$</p>
<p>For example, in a circle $x^2+y^2=k^2$.
If I set a general point $P(x,y)$, why is its image $P'(\frac{xk^2}{x^2+y^2}, \frac{yk^2}{x^2+y^2})$?</p>
<p>Also, why does a line become a circle through O?</p>
<p>Sorry for my English,... | Stefan4024 | 67,746 | <p>Note that $O, P, P'$ must be collinear. Therefore if we take $P = (x,y)$ we must have $P' = (ax,ay)$ for some positive integer $a$. Now from the condition that $OP \cdot OP' = k^2$ we must have:</p>
<p>$$k^2 = \sqrt{x^2 + y^2} \sqrt{a^2x^2 + a^2y^2} = a(x^2+y^2) \implies a = \frac{k^2}{x^2+y^2}$$</p>
<p>For the se... |
2,619,478 | <p>I am working on a problem connected to shallow water waves.
I have a vector:</p>
<p>$U = \begin{bmatrix} h \\
h \cdot v_1\\
h \cdot v_2\end{bmatrix}$</p>
<p>and a function</p>
<p>$f(U) = \begin{bmatrix} h \cdot v_1 \\
h \cdot v_1^2 + 0.5\cdot gh^2\\
h \cdot v_1 \cdot v_2\end{bmatrix}$</p>
<p>I now want to calcu... | Lee Mosher | 26,501 | <p>To do this, simply get rid of the "products of variables" by a substitution.</p>
<p>First, set $w_1 = h v_1$ and $w_2 = h v_2$.</p>
<p>Next, solve for $v_1 = w_1 / h$ and $v_2 = w_2 / h$.</p>
<p>Next, substitute for $v_1,v_2$ and simplify: </p>
<p>$U = \begin{bmatrix} h \\
w_1\\
w_2\end{bmatrix}$</p>
<p>$f(U) =... |
219,965 | <p>I am currently reading up on nuclear spaces in Jarchow, "Locally Convex Spaces", but I got confused and don't seem to find my mistake. In said book, theorem 21.5.9 states:</p>
<p>Let $F$ be a nuclear Frechet space. Then, $F'_\beta \otimes_{\pi} F = L_\beta (F,F)$.
i.e. the projective tensor product of $F$ with its ... | J.L.R. | 81,148 | <p>Your answer is wrong (unfortunately, I can't comment, so here is another answer instead).
Now, I'm quoting from "Notes on locally convex vector spaces" by J.L. Taylor. </p>
<p>Since $F$ is nuclear, the (completed, you are right, I meant that) injective and projective tensor products agree, so "is smaller than" doe... |
313,388 | <p>Computing without Taylor series or l'Hôpital's rule </p>
<p>$$\lim_{n\to\infty}\prod_{k=1}^{n}\cos \frac{k}{n\sqrt{n}}$$</p>
<p>What options would I have here? Thanks!</p>
| marty cohen | 13,079 | <p>I am just going to prove that the product converges.
To get the exact value,
you need more advanced stuff.</p>
<p>Start with $|\cos(x)-1| \le x^2/2$.
This can be proved from
$\cos(2x) = \cos^2(x)-\sin^2(x) = 1 - 2 \sin^2(x)$,
so $|\cos(2x)-1| = 2|\sin^2(x)|$.
Since $|\sin(x)| \le x$ for
all $x$,
$|\cos(2x)-1| \le ... |
507,109 | <p>If we have two homeomorphisms $f:A\to X$ and $g:B\to Y$, then is it true that $f\times g:A\times B\to X\times Y$ defined by $(f\times g)(a,b)=(f(a),g(b))$ is again a homeomorphism?</p>
<p>I think the answer is yes; </p>
<p>It's clearly a bijection. Intuitively it seems to be continuous but I don't know how to show... | Stefan Hamcke | 41,672 | <p>To check that it is continuous, you just have to check the individual coordinates, since $f\times g$ is continuous iff $p_X\circ (f\times g)$ and $p_Y\circ (f\times g)$ are. But these are just the mappings $(a,b)\mapsto f(a)$ and $(a,b)\mapsto g(b)$. Can you express them as compositions of continuous functions?</p>
... |
3,988,837 | <p>Let <span class="math-container">$A, B$</span> two rings and <span class="math-container">$I_A: {}_A\text{Mod} \to{}_A\text{Mod}$</span> the identity functor. I am trying to show that if <span class="math-container">$A, B$</span> are Morita equivalent, then <span class="math-container">$\underline{\text{Nat}}(I_A, I... | Berci | 41,488 | <p>Note that the given data should already ensure such a canonical isomorphism <span class="math-container">$\varphi:N\to P\otimes_A M$</span>, namely using the specific <span class="math-container">$M=Q\otimes_B N$</span> and the natural isomorphism <span class="math-container">$P\otimes_A(Q\otimes_B N)\cong N$</span>... |
1,739,052 | <p>After years of mathematics, I am struggling with this simple question. </p>
<p>If we have 3 r.v. $X,Y,Z$ and we have $X$ independent to $Y$ and to $Z$, then do we have that $X$ is also independent to $YZ$ ?</p>
<p>At first sight, I thought that if $X$ is independent to $Y$ and $Z$, it is also independent to the si... | wece | 65,630 | <p>You can see it as having information on $Y$ is not enough to determine $X$ an identically with information on $Z$ however if you have combined informations it could determined $X$.</p>
<p>An example I came up with is (may be not a good one but it helps me understand whats going on).</p>
<p>You have a set of people... |
13,635 | <p>I want to define</p>
<pre><code>isGood[___] = False;
isGood[#] = True & /@ list
</code></pre>
<p>where <code>list</code> is a list of several million integers. What's the fastest way of doing this?</p>
| Ajasja | 745 | <p>Summary: undocumented <code>HashTable</code> is a bit faster (at least in version 9) both in storage and in retrieval than <code>DownValues</code>.</p>
<h2>DownValues</h2>
<pre><code>list = RandomInteger[{-10^9, 10^9}, 10^6];
ret = RandomInteger[{-10^9, 10^9}, 10^6];
isGood[___] = False;
Scan[(isGood[#] = True) ... |
13,635 | <p>I want to define</p>
<pre><code>isGood[___] = False;
isGood[#] = True & /@ list
</code></pre>
<p>where <code>list</code> is a list of several million integers. What's the fastest way of doing this?</p>
| Henrik Schumacher | 38,178 | <p>What you aim at can be done quicker now with <code>Association</code>. I compare only to the <code>Do</code> method, which is also reasonably fast:</p>
<pre><code>list = RandomInteger[{-10^9, 10^9}, 10^6];
ret = RandomInteger[{-10^9, 10^9}, 10^6];
ClearAll[isGood];
isGood[___] = False;
Do[isGood@i = True, {i, list}... |
2,967,366 | <p>I need to integrate function <span class="math-container">$\int_0^1 pur\mathrm{d}r$</span>, where I only have discrete values for <span class="math-container">$p$</span>,<span class="math-container">$u$</span> and <span class="math-container">$r$</span>. So, if I multiply these values, would it be correct to integra... | Community | -1 | <p>If you're saying that <span class="math-container">$p$</span> and <span class="math-container">$u$</span> are scalar values and <span class="math-container">$r$</span> is a function when you numerically integrate just take the <span class="math-container">$p \cdot u$</span> outside </p>
<p><span class="math-contain... |
1,691,963 | <p>Given the differential equation $dy/dx = f(x,y)$ with initial condition $y(x_0)=y_0$. Let $f$ be a continuous function in $x$ and $y$ and Lipschitz-continuous in $y$ with Lipschitz constant L. Let F be a mapping on the space $C^0(I)$ of continuous functions $u:I\rightarrow \mathbb{R}$. $I = [x_0,M]$.
$$Fu(x) = y_0 +... | KonKan | 195,021 | <p>since $x$ is a positive integer and $x-1<x$, then:
$$x! \leq x(x-1)^{x-1}\leq x x^{x-1}=x^x$$</p>
|
313,470 | <p>Consider the set <span class="math-container">$\mathbf{N}:=\left\{1,2,....,N \right\}$</span> and let <span class="math-container">$$\mathbf M:=\left\{ M_i; M_i \subset \mathbf N \text{ such that } \left\lvert M_i \right\rvert=2 \text{ or }\left\lvert M_i \right\rvert=1 \right\}$$</span>
be the set of all subsets of... | Christian Remling | 48,839 | <p>The matrices are of the form
<span class="math-container">$$
A=\begin{pmatrix} 1 & C \\ C^* & D \end{pmatrix}, \quad\quad
B= \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix} A \begin{pmatrix} 1 & 0\\ 0&2\end{pmatrix} ,
$$</span>
with the blocks corresponding to the sizes of the sets <span class="... |
2,072,729 | <p>Given that $n\in \mathbb{N}$.</p>
<p>I know that it converges to $1$ if $ \alpha=3$ and to $0$ if $\left | \alpha \right |< 3$ intuitively but I am not able to convince myself algebraically. </p>
<p>I tried writing it as $e^{2^{n}ln\left ( \frac{\alpha }{3} \right )}$ which tells me that my exponent needs to c... | Community | -1 | <p>You are making things complicated by writing the exponential function. For $|q|<1$ where $q=\alpha/3$, you want to show that
$$
\lim_{n\to\infty}q^{2^n}=0\tag{1}
$$</p>
<hr>
<p>Note that
$$
q^{2^n}=(q^2)^{2^{n-1}}.
$$
Thus you don't need to worry about the case when $-1<q<0$. To show (1) it suffices to s... |
64,265 | <p>I've been using a DateList plot to visualise property information but I don't think it's the best way display my data. My data is formatted as {time (hours), property} where property is an integer between 1 and 20</p>
<pre><code>data = {{0, 0}, {0.2187, 3}, {0.25, 1}, {0.3715, 15}, {0.868,
1}, {1.261, 15}, {1.4595... | Verbeia | 8 | <p>Completely rewritten following the clarification. If those are timestamps, then the height of the bars should be the <code>Differences</code> of the first column. Then style these heights according to the value of the second element of the pair, like this.</p>
<pre><code>transformeddata =
With[{max = Max[data[[Al... |
1,710,929 | <p>If $3$ people are dealt $3$ cards from a standard deck, determine the probability that none of them is dealt three of a kind?</p>
<p>Here is my attempt:</p>
<p>The total number of hands is
$${_{52}\mathsf C}_3\times{_{49}\mathsf C}_3\times{_{46}\mathsf C}_3=7.75262759\times 10^{12}.$$
The number of ways we can de... | Mark Fischler | 150,362 | <p>You have incorrectly written $+ 109824$ instead if $-109824$. But evidently you actually did subtract, because your answer is correct.</p>
|
3,880,630 | <blockquote>
<p>In right <span class="math-container">$\Delta ABC$</span>, <span class="math-container">$\angle C = 90^\circ$</span>. <span class="math-container">$E$</span> is on <span class="math-container">$BC$</span> such that <span class="math-container">$AC = BE$</span>. <span class="math-container">$D$</span> is... | Michael Rozenberg | 190,319 | <p>Also, <span class="math-container">$uvw$</span> helps.</p>
<p>Indeed, in my previous solution it's enough to prove that:
<span class="math-container">$$64(2a+b)^3(2a+c)^3(b+c)^2\geq729(2a+b+c)^4a^2bc.$$</span>
Now, let <span class="math-container">$b+c=2u$</span> and <span class="math-container">$bc=v^2$</span>, whe... |
1,025,880 | <p>Why </p>
<blockquote>
<p>$(cf)' = c(f)'$</p>
</blockquote>
<p>but not </p>
<blockquote>
<p>$(cf)' = (c)' (f)' = 0 f = 0$</p>
</blockquote>
<p>?</p>
| user2345215 | 131,872 | <p>Because the formula for the derivative of a product is
$$g'\!f+gf'$$
Which gives you $c'\!f+cf'=0+cf'=cf'.$</p>
|
250,426 | <p>I want to factorize any quadratic expressions into two complex-valued linear expressions.</p>
<p>My effort below</p>
<pre><code>a := 1;(*needed*)
p := 2;(*needed*)
q := 3;(*needed*)
f[x_] := a (x - p)^2 + q;(*needed*)
AA := Coefficient[f[x], x^2];
BB := Coefficient[f[x], x];
CC := f[0];
DD = BB^2 - 4 AA CC;
EE = Tim... | Michael Seifert | 27,813 | <p>This is inelegant, but it'll work. Basically we look at the coefficient of <code>x</code> in each factor, divide each factor by that coefficient, multiply all the <code>x</code> coefficients together as a leading coefficient for the factorization, and then multiply it all back together.</p>
<p>Use your code as abov... |
250,426 | <p>I want to factorize any quadratic expressions into two complex-valued linear expressions.</p>
<p>My effort below</p>
<pre><code>a := 1;(*needed*)
p := 2;(*needed*)
q := 3;(*needed*)
f[x_] := a (x - p)^2 + q;(*needed*)
AA := Coefficient[f[x], x^2];
BB := Coefficient[f[x], x];
CC := f[0];
DD = BB^2 - 4 AA CC;
EE = Tim... | JimB | 19,758 | <p>Look at the coefficients of the expanded quadratics and solve:</p>
<pre><code>cl1 = CoefficientList[(I*x + a) (I*x + b), x]
(* {a b,I a + I b,-1} *)
cl2 = CoefficientList[e (x + I*c) (x + I*d), x]
(* {-c d e, I c e + I d e, e} *)
sol = Solve[MapThread[Equal, {cl1, cl2}], {c, d, e}]
(* {{c -> -a, d -> -b, e-&g... |
12,544 | <blockquote>
<p>Can <span class="math-container">$n!$</span> be a perfect square when <span class="math-container">$n$</span> is an integer greater than <span class="math-container">$1$</span>?</p>
</blockquote>
<p>Clearly, when <span class="math-container">$n$</span> is prime, <span class="math-container">$n!$</span> ... | curious | 223 | <p>There is a prime between n/2 and n, if I am not mistaken. </p>
|
49,630 | <p>Most programmers (including me) are painfully aware of quadratic behavior resulting from a loop that internally performs 1, 2, 3, 4, 5 and so on operations per iteration,</p>
<p>$$\sum_{i=1}^n i = \frac{n \left(n+1\right)}{2} $$</p>
<p>It’s very easy to derive, e.g. considering the double of the sum like $(1... | leonbloy | 312 | <p>Your derivation is actually quite nice. I doubt you'll find some very conceptually simple way of computing the general formula (so that you could look at it and say: <em>aha! that was why...!</em>). There are some "visual proofs" for power <a href="http://www.takayaiwamoto.com/Sums_and_Series/sumint_1.html" rel="nof... |
926,581 | <p>I find the <a href="https://en.wikipedia.org/wiki/Surreal_number" rel="nofollow noreferrer">surreal numbers</a> very interesting. I have tried my best to work through John Conway's <em>On Numbers and Games</em> and teach myself from some excellent <a href="http://www.tondering.dk/claus/sur16.pdf" rel="nofollow noref... | SK19 | 509,159 | <p>The answer to 3. is affirmative. We can prove it with a straight calculation.</p>
<h2>Fun with surreals</h2>
<p>Now let <span class="math-container">$(a_n)$</span> be a real sequence with <span class="math-container">$0< a_n$</span> for all <span class="math-container">$n\in\mathbb{N}$</span> and <span class="mat... |
3,540,956 | <p>Let G be a finite group, H a maximal proper subgroup of G and K a subgroup of H. Is the normalizer of K in G, <span class="math-container">$N_GK$</span>, a subgroup of H.
Now <span class="math-container">$N_GK$</span> is certainly contained in some maximal subgroup, maybe more than one, but why is it contained in H... | David K | 139,123 | <p>A sketch of a proof that <span class="math-container">$T_N,$</span> the total age of the <span class="math-container">$N$</span> oldest students is at least <span class="math-container">$20N.$</span></p>
<p>Consider the age of the <span class="math-container">$N$</span>th oldest student, <span class="math-container... |
2,605,626 | <p>$\sin(t)$ is continuous on $[0,x]$ and $\frac{1}{1+t}$ is continuous on $[0,x]$ so $\frac{\sin(t)}{1+t}$ is continuous on $[0,x]$ so the function is integrable.</p>
<p>How do I proceed? What partition should I consider ? </p>
<p>Edit : We haven't done any properties of the integral so far except the basic definiti... | Community | -1 | <p>Here is an outline that you can follow. The key idea is to draw the graph and recognize that the area contained within each arc of the function is decreasing, and the first area is positive.</p>
<hr>
<ol>
<li><p>Conclude that for all $x \in [0, \pi]$, $\int_0^x \frac{\sin t}{1 + t} \, dt > 0$. </p></li>
<li><p>... |
484,273 | <p>$$\int_0^1\frac{ \arcsin x}{x}\,\mathrm dx$$</p>
<p>I was looking in my calculus text by chance when I saw this example , the solution is written also but it uses very tricky methods for me ! I wonder If there is a nice way to find this integral.</p>
<p>The idea of the solution in the text is in brief , Assume $y... | Mhenni Benghorbal | 35,472 | <p>Using integration by parts with $u=\arcsin(x)$ yields</p>
<blockquote>
<p>$$ \int_0^1 \frac{\arcsin x}{x}\,dx = -\int_0^1 \frac{\ln(x)}{\sqrt{1-x^2}}\,dx=I .$$</p>
</blockquote>
<p>Now, consider the integral</p>
<blockquote>
<p>$$ F = \int_0^1 \frac{x^\alpha}{\sqrt{1-x^2}}\,dx = \frac{\sqrt{\pi}}{2}{\frac {\G... |
1,394,490 | <p>I stumbled upon "the God proof" which goes:</p>
<p>$0 = 0 + 0 + 0...$</p>
<p>$ = (1-1) + (1-1) + (1-1) + ...$</p>
<p>$= 1 - 1 + 1 - 1 + 1 - 1 + ...$ </p>
<p>$= 1 + (-1+1) + (-1+1) + (-1+1) + ...$ </p>
<p>$= 1$</p>
<p>Even though this result is obviously wrong, I can't quite pinpoint exactly what the 'illegal'... | joe1984 | 261,363 | <p>See <a href="https://en.wikipedia.org/wiki/Grandi's_series" rel="nofollow">Grandi's series</a>. Answer comes out to be $0.5$</p>
|
2,745,623 | <blockquote>
<p>Maximize the generic bivariate quadratic form constrained to the unit circle.</p>
<p>$$\begin{array}{ll} \text{maximize} & f(x_1, x_2) := ax_1^2 + 2bx_1 x_2 + cx_2^2\\ \text{subject to} & g(x_1, x_2) := x_1^2 + x_2^2 - 1 = 0\end{array}$$</p>
</blockquote>
<p>Using the standard Lagrange M... | Cesareo | 397,348 | <p>This problem can be easily solved without the Lagrange multipliers </p>
<p>Make the transformation</p>
<p>$$
x_1 = \cos\theta\\
x_2 = \sin\theta
$$</p>
<p>then the objective function will read</p>
<p>$$
f(\theta) = a \cos^2\theta + 2 b \sin \theta\cos\theta + c \sin^2\theta
$$</p>
<p>and now the stationary poin... |
8,193 | <p><strong>NB. Some answers appear to be for a question I did not ask, namely, "Why is standardized testing bad?" Indeed, these answers tend to underscore the premise of my actual question, which can be found above.</strong></p>
<p>As a foreigner who has spent some time in the US, it seems to me that in the U... | supercat | 1,740 | <p>In many cases where people are distrustful about certain forms of testing, such distrust stems from a belief that the results of such testing will be put to uses that those harboring the distrust would oppose. In the case of standardized testing of educational performance, some people are worried that test results ... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.