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 |
|---|---|---|---|---|
2,538,521 | <blockquote>
<p>Let $x$ be a function of $C^1(I,R)$ where $I\subset \mathbb{R}$ , such that $$x'(t)\leq a(t) x(t)+b(t),$$ where $a$ and $b$ are continuous functions on $I$ in $R$ then
$$ x(t)\leq x(t_0) \exp\left(\int_{t_0}^{t}a(s)ds\right)+\int_{t_0}^{t}\exp\left(\int_{s}^t a(\sigma)d\sigma\right)b(s)ds$$</p>
... | Diesirae92 | 289,721 | <p>You can solve the differential equation </p>
<p>$x′(t)+d(t)= a(t)x(t)+b(t)$</p>
<p>for some $d\geq 0$. When you get the explicit solution, just recall the sign $d$ had.</p>
|
24,876 | <p>As it is possible to see the last time when you or others visited M.SE, I wonder if one can see a statistics of visits of your own or of a specific user for a period of time (last year, let's say).</p>
| quid | 85,306 | <p>A partial answer:</p>
<p>For your own account this data is available in detail and nicely presented: <a href="https://math.stackexchange.com/users/current?tab=profile">go to your user profile</a>, then clicking on "visited {number} days, {othernumber} consecutive" will give you a calendar marking each and every da... |
450,785 | <p>I want to obtain the formula for binomial coefficients in the following way: elementary ring theory shows that $(X+1)^n\in\mathbb Z[X]$ is a degree $n$ polynomial, for all $n\geq0$, so we can write</p>
<p>$$(X+1)^n=\sum_{k=0}^na_{n,k}X^k\,,\ \style{font-family:inherit;}{\text{with}}\ \ a_{n,k}\in\mathbb Z\,.$$</p>
... | user26872 | 26,872 | <p>A simple-minded approach is to solve the two variable recurrence relation iteratively, that is, knowing $a_{n,0}$ find $a_{n,1}$, then $a_{n,2}$, etc.</p>
<p>We must have<br>
$$\begin{eqnarray*}
a_{n,1} &=& a_{n-1,1}+a_{n-1,0} \\
&=& a_{n-1,1}+1,
\qquad a_{1,1}=1.
\end{eqnarray*}$$
This is a one va... |
2,634,791 | <blockquote>
<p>How can I show that the map $f: GL_n(\mathbb R)\to GL_n(\mathbb R)$ defined by $f(A)=A^{-1}$ is continuous?</p>
</blockquote>
<p>The space $GL_n(\mathbb R)$ is given the operator norm and so I want to show for all $\epsilon$ there exists $\delta$ such that $\|A-B\|<\delta \implies \|A^{-1}-B^{-1}\... | Martín-Blas Pérez Pinilla | 98,199 | <p>In fact is true in any unital Banach algebra. See Lemma 5.1 of <a href="https://www.math.ksu.edu/~nagy/real-an/2-05-b-alg.pdf" rel="nofollow noreferrer">https://www.math.ksu.edu/~nagy/real-an/2-05-b-alg.pdf</a> or <a href="https://math.stackexchange.com/questions/924341/banach-algebras-continuity-of-inversion">Banac... |
1,527,891 | <p>Determine (up to a constant multiplier) the polynomial with a maximum at $(-1,1)$, a minimum $(1,-1)$ and no other critical points.</p>
<p>The only thing I can think of is coming up with an equation with roots $1$ and $-1$ and then integrating it but I don't think that will work.</p>
| molarmass | 119,376 | <p>A polynomial of degree $n=3$ in general has exactly $2$ extreme points so let's use a cubic polynomial $f(x) = ax^3 + bx^2 + cx + d$ with $a \ne 0$. Obviously the derivative of this function is $f'(x) = 3ax^2 + 2bx + c$.</p>
<p>We now need to find values of $(a,b,c,d)$ such that \begin{align}f(-1) &= 1,&f(1... |
3,695,868 | <p>In right triangle <span class="math-container">$ABC,$</span> <span class="math-container">$\angle C = 90^\circ.$</span> Let <span class="math-container">$P$</span> and <span class="math-container">$Q$</span> be points on <span class="math-container">$\overline{AC}$</span> so that <span class="math-container">$AP = P... | Narasimham | 95,860 | <p>Let <span class="math-container">$BA= y, BC=x. $</span> We can solve numerically using Pythagoras thm twice.</p>
<p><span class="math-container">$$ 2 (67.^2 - x^2)^{0.5} = (76^2 - x^2)^{0.5}$$</span> </p>
<p><span class="math-container">$$ 3 (67.^2 - x^2)^{0.5} = (y^2 - x^2)^{0.5}$$</span></p>
<p>Since there are ... |
3,695,868 | <p>In right triangle <span class="math-container">$ABC,$</span> <span class="math-container">$\angle C = 90^\circ.$</span> Let <span class="math-container">$P$</span> and <span class="math-container">$Q$</span> be points on <span class="math-container">$\overline{AC}$</span> so that <span class="math-container">$AP = P... | Harish Chandra Rajpoot | 210,295 | <p>For easy understanding, assume <span class="math-container">$AP=PQ=QC=x$</span> & <span class="math-container">$BC=y$</span> then using Pythagoras theorem in respective right triangles, we get
<span class="math-container">$$QB^2=QC^2+BC^2\iff 67^2=x^2+y^2\tag 1$$</span>
<span class="math-container">$$PB^2=PC^2+... |
277,217 | <p>I am stuck on the following problem, which I do not believe to be so difficult.</p>
<p>Let $X$ and $Y$ be Banach spaces. Let $f:X\times X\rightarrow Y$ be a function such that for any fixed $x_0$, $f(x,x_0)$ and $f(x_0,x)$ are continuous in $x$. Then is $f(x,x)$ continuous in $x$?</p>
<p>I tried taking an arbitrar... | Davide Giraudo | 9,849 | <p>Let $X=Y:=\Bbb R$ and
$$f(x,y):=\begin{cases}\frac{xy}{x^2+y^2},&\mbox{if }(x,y)\neq (0,0);\\
0&\mbox{ if }(x,y)=(0,0).
\end{cases}$$
This function is continuous once a variable is fixed, but is not globally continuous. </p>
|
656,701 | <p>Suppose we have:</p>
<p>$A = \{(x,v,w):x+v=w\}$</p>
<p>$B = \{(x,v):x=v\}$</p>
<p>$C = \{(w,u):\exists x 2x=w\}$</p>
<p>Can we say that $C = A \cup B$?</p>
| Unwisdom | 124,220 | <p>Oh, I see what you're trying to do:
\begin{eqnarray}
A&=&\{\langle x,v,w\rangle :x+v=w\}\\
B&=&\{\langle x,v,w\rangle :x=v\}\\
C&=&\{\langle x,v,w\rangle :2x=w\}
\end{eqnarray}
These are three planes in $\mathbb{R}^{3}$. Planes $A$ and $B$ intersect in a line. Since every solution to the con... |
1,993,217 | <p>Let $\left\{f_{n}\right\}$ be a sequence of equicontinuous functions where $f_n: [0,1] \rightarrow \mathbf{R}$. If $\{f_n(0)\}$ is bounded, why is $\left\{f_{n}\right\}$ uniformly bounded?</p>
| Manoel | 20,988 | <p>Hint: </p>
<p><span class="math-container">$|f_{n}(x)|=|f_{n}(x) +f_{n}(0) -f_{n}(0)|\leq|f_{n}(x) -f_{n}(0)|+|f_{n}(0)|$</span> </p>
<p>now use the Equi-continuity and <span class="math-container">$\{f_{n}(0)\}$</span> bounded</p>
|
1,993,217 | <p>Let $\left\{f_{n}\right\}$ be a sequence of equicontinuous functions where $f_n: [0,1] \rightarrow \mathbf{R}$. If $\{f_n(0)\}$ is bounded, why is $\left\{f_{n}\right\}$ uniformly bounded?</p>
| Martin Sleziak | 8,297 | <p>Let me try to prove this using real induction. You can find some basic description of this proof technique together with some references <a href="https://math.stackexchange.com/questions/4202/induction-on-real-numbers/4204#4204">in this answer</a>. I have tried to give some informal description of real induction <a ... |
907,893 | <p>I wanted to know about this convention :</p>
<p>By rate of growth of R, we normally mean : (change in R) / (change in Time)</p>
<p>But Rate of growth of a geometric sequence "a(1+r)^n" is r, which is strange i feel</p>
<p>I am kind of confused, can anyone clear it </p>
| Calculon | 163,648 | <p>That is not strange at all. The geometric sequence in your question is given by $a_{n+1} = (1+r)a_n$ with $a_0 = a$. In every single "time step" going from $n$ to $n+1$ your $a_n$ becomes $(1+r)a_n$. So your growth rate per time step is $r$. You cannot break up this time step into smaller units of time since $n$ in ... |
290,050 | <p>Are there good lower/upper bounds for
$ \sum\limits_{i = 0}^k {\left( \begin{array}{l} n \\ i \\ \end{array} \right)x^i } $ where $0<x<1$, $k \ll n$?</p>
| zeraoulia rafik | 51,189 | <p><strong>Hint</strong> :for lower Bound we have for $k> 1$:$ (1+\frac{1}{k})^k \leq (e^{1/k})^k =e ,0<x=1/k<1$ , and $1/k << k $</p>
|
290,050 | <p>Are there good lower/upper bounds for
$ \sum\limits_{i = 0}^k {\left( \begin{array}{l} n \\ i \\ \end{array} \right)x^i } $ where $0<x<1$, $k \ll n$?</p>
| Max Alekseyev | 7,076 | <p>Let $p=\frac{x}{1+x}$ and $q=\frac{1}{1+x}$, and thus
$$\sum_{i=0}^k \binom{n}{i} x^i=(1+x)^n\sum_{i=n-k}^n \binom{n}{i} p^{n-i} q^i.$$
Then for $k<np$ <a href="https://en.wikipedia.org/wiki/Chernoff_bound" rel="nofollow noreferrer">Chernoff bound</a> gives
$$\sum_{i=n-k}^n \binom{n}{i} p^{n-i} q^i \le \left( \... |
1,290,516 | <p>Find the values of $m$ if the line $y=mx+2$ is a tangent to the curve $x^2-2y^2=1$.</p>
<p>My working:</p>
<p>First we differentiate $x^2-2y^2=1$ with respect to $y$ to get the gradient. We get $y^2=\frac{1}{2}x^2-\frac{1}{2}\implies y=\pm\sqrt{\frac{1}{2}x^2-\frac{1}{2}}$.</p>
<p>We take the positive one for dem... | Empty | 174,970 | <p><strong>One more simplest way:</strong></p>
<p>Put $y=mx+2$ in the equation $x^2-2y^2=1$. Then it comes to a quadratic equation of $x$. From which we get two values of $x$. Since the line is tangent to the given hyperbola so, it can not intersect at two different points. So, the quadratic equation must give two ide... |
439,620 | <p>As we know, the QR-factorization <span class="math-container">$Q\cdot R=A$</span> of any real symmetric <span class="math-container">$n \times n$</span> matrix <span class="math-container">$A$</span> with full rank is <em><strong>unconditionally</strong></em> <em>numerically stable</em>. Further, when A is rank-1-up... | Daniel Shapero | 49,417 | <p>It may be difficult to meet all your criteria but here's an attempt.
This is a bit lower-level than Federico Poloni's suggestion to use the eigenvalue factorization.</p>
<p>The <a href="https://en.wikipedia.org/wiki/Lanczos_algorithm" rel="nofollow noreferrer">Lanczos algorithm</a> computes a unitary matrix <span cl... |
1,131,622 | <p>The question itself is a very easy one:<br/></p>
<blockquote>
<p>Somebody has got two kids, one of whom is a girl. Then what's the probability that he's got <strong>at least</strong> one boy?</p>
</blockquote>
<p>My answer is that, since he's already got a girl, then "he's got at least one boy" amounts to "the o... | Timbuc | 118,527 | <p>I agree with your friend, and the reason follows using conditional probability. Define B=the event of having a boy, G= the event of having a girl, and we're in the space defined by "having two kids". Then we want the probability $\;P(B\backslash G)=$ the probability of having a boy <em>knowing</em> that there's alr... |
1,131,622 | <p>The question itself is a very easy one:<br/></p>
<blockquote>
<p>Somebody has got two kids, one of whom is a girl. Then what's the probability that he's got <strong>at least</strong> one boy?</p>
</blockquote>
<p>My answer is that, since he's already got a girl, then "he's got at least one boy" amounts to "the o... | Christoph | 86,801 | <p>I'll try to explain without mentioning conditional probabilities or binomial distributions explicitly:</p>
<p>Having two kids, you can have two boys, two girls, or one girl and a boy. However those 3 possibilities don't have equal probabilities. Two get two boys, both your first and second child have to be boys, ch... |
130,028 | <p>I often want to have the same code at the beginning of every new notebook. Is it possible to configure Mathematica, such that whenever you create a new notebook some user-defined code will always be created with the new document.</p>
<p>E.g. commonly used plot configurations, packages, directory setting etc.</p>
<... | Szabolcs | 12 | <p>I recommend creating a palette with a button that can insert the code for you. Then save the palette and make it easy to access through the palettes menu.</p>
<h3>Create palette</h3>
<p>Suppose your code is (for sake of simplicity),</p>
<pre><code>code1 = HoldComplete[1+1];
</code></pre>
<p>The create the palet... |
394,517 | <p>How can I evaluate $\lim_{x \to \infty}\left(\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}\right)$?</p>
| Zarrax | 3,035 | <p>I guess someone should mention the Taylor approximation approach:
$$\left(\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}\right) = \sqrt{x}\left(\sqrt{1+ {1 \over \sqrt{x}}}-\sqrt{1- {1 \over \sqrt{x}}}\right) $$
$$= \sqrt{x}\bigg(\big(1 + {1 \over 2\sqrt{x}} + O({1 \over x})\big) - \big(1 - {1 \over 2\sqrt{x}} + O({1 \over x})... |
4,196,109 | <p>While studying about inequalities, I came across the following definition (<span class="math-container">$\forall a > 0)$</span>:</p>
<p><span class="math-container">$$
\begin{alignat}{1}
& |x| > a \iff \{ x \mid x < -a \text{ or } x > a \} \\
& |x| < a \iff \{ x \mid -a < x < a \}
\e... | user97357329 | 630,243 | <p>Another framework proposed by Cornel (<strong>answer to the second integral</strong>, <span class="math-container">$\displaystyle \int_0^1\frac{\ln^2(1+x^2)\ln x}{1+x^2}\textrm{d}x$</span>)</p>
<p>Observe that <span class="math-container">$$\int_0^1 \frac{1}{1+x^2}\log^3\left(\frac{2x}{1+x^2}\right)\textrm{d}x$$</s... |
4,336,706 | <p>Let <span class="math-container">$\mathbb {K}$</span> be a field. Let <span class="math-container">$f: \mathbb {K}^2 \rightarrow \mathbb {K}^2; x \mapsto Ax+b$</span> be an affine transformation. Suppose <span class="math-container">$f$</span> has a fixed point line (i.e. a line such that every point on that line is... | Gribouillis | 398,505 | <p>We can assume without loss of generality that <span class="math-container">$f(1)=0$</span>. Now let <span class="math-container">$g(x) = x f(x)$</span>. We have <span class="math-container">$g(0)=g(1)=0$</span>. Hence there is a point <span class="math-container">$c$</span> in <span class="math-container">$(0,1)$</s... |
4,336,706 | <p>Let <span class="math-container">$\mathbb {K}$</span> be a field. Let <span class="math-container">$f: \mathbb {K}^2 \rightarrow \mathbb {K}^2; x \mapsto Ax+b$</span> be an affine transformation. Suppose <span class="math-container">$f$</span> has a fixed point line (i.e. a line such that every point on that line is... | Mr.Gandalf Sauron | 683,801 | <p>To extend on @Gribouillis's solution.</p>
<p>Take <span class="math-container">$g(x)=xf(x)-xf(1)$</span> . Then <span class="math-container">$g(0)=g(1)=0$</span>.</p>
<p>Then there exist <span class="math-container">$c\in(0,1)$</span> such that <span class="math-container">$g'(c)=0$</span>.</p>
<p><span class="math-... |
1,320,874 | <p>I am trying to answer the following: Does the congruence $x^2 \equiv -1$ (mod $p$) have any solutions if $p \equiv 3$ (mod $4$)? If so, how many incongruent solutions does it have? If not, why not?</p>
<p>I know from the previous part of the question that if $p$ is a prime and $p \equiv 1$ (mod $4$), then the congr... | André Nicolas | 6,312 | <p>The following is a proof that uses Wilson's Theorem. There are "easier" (and group-theoretically more natural) proofs that do not use Wilson's Theorem. The idea is due to Dirichlet. </p>
<p>Let $p$ be a prime of the form $4k+3$. We will assume that the congruence $x^2\equiv -1\pmod{p}$ has a solution, and use Wils... |
1,320,874 | <p>I am trying to answer the following: Does the congruence $x^2 \equiv -1$ (mod $p$) have any solutions if $p \equiv 3$ (mod $4$)? If so, how many incongruent solutions does it have? If not, why not?</p>
<p>I know from the previous part of the question that if $p$ is a prime and $p \equiv 1$ (mod $4$), then the congr... | user26486 | 107,671 | <p>We assume $p$ is an odd prime. You know that $$\,p\equiv 1\pmod{\! 4}\,\Rightarrow\, (x^2\equiv -1\pmod{\! p}\,\text{ is solvable})$$ by your constructive proof, namely $x\equiv\pm\left(\frac{p-1}{2}\right)!\pmod{\! p}$ works as a solution. Proofs of this have been discussed <a href="https://math.stackexchange.com/q... |
73,238 | <p>How can I calculate the solid angle that a sphere of radius R subtends at a point P? I would expect the result to be a function of the radius and the distance (which I'll call d) between the center of the sphere and P. I would also expect this angle to be 4π when d < R, and 2π when d = R, and less than 2π when d ... | Ross Millikan | 1,827 | <p>Is your "slope of the axis in the x,y and z axes" the <a href="http://mathworld.wolfram.com/DirectionCosine.html" rel="nofollow">direction cosines</a>? You need a center $c$ as well, presumably a point on the axis. Given the unit vector $\vec{v}$ along the axis one way is to find two perpendicular unit vectors. A... |
2,402,429 | <p>Let $P(x) = x^3 + 2x^2+3x+4$ and $a$ be the root of equation $x^4+x^3+x^2+x+1=0$.</p>
<p>Find the value of $P(a)P(a^2)P(a^3)P(a^4)$</p>
<p>Is my answer correct ?</p>
<p>Since root of equation $x^4+x^3+x^2+x+1=0$ is the $5^{th}$ primitive root of 1,</p>
<p>so $a, a^2, a^3, a^4$ are roots of $x^4+x^3+x^2+x+1=0$ </... | Batominovski | 72,152 | <p>Suppose $P(x)=x^3+2x^2+3x+4=(x-p)(x-q)(x-r)$ for some $p,q,r\in\mathbb{C}$. Then, $$\prod_{j=1}^4\,P\left(a^j\right)=Q(p)\,Q(q)\,Q(r)\,,$$
where $Q(x):=x^4+x^3+x^2+x+1$. Now, $$Q(x)=(x-1)\,P(x)+5\,.$$
Thus, $Q(p)=Q(q)=Q(r)=5$. </p>
<hr>
<p>This is actually quite a nice technique. Let $P(x)$ and $Q(x)$ be two n... |
19,842 | <p>Seeing <a href="https://stackoverflow.com/help/self-answer">this</a> I though this thing was promoted, and for avoiding for the question becoming boring, I didn't answer it suddenly and waited and I did mentioned that I knew the answer, maybe it's just misunderstanding that I don't know the answer. Anyways, what's t... | GEdgar | 442 | <p><a href="http://meta.math.stackexchange.com/a/4233/442">Puzzle Questions</a> are allowed in math.se ... But include the information in the original post that you know the answer!</p>
<p><a href="https://math.stackexchange.com/questions/351333/evaluation-of-a-continued-fraction">HERE</a> is an example of mine.</p>
|
408,590 | <p>I'm looking for references (books/lecture notes) for :</p>
<ul>
<li>Cardinality without choice, Scott's trick;</li>
<li>Cardinal arithmetic without choice.</li>
</ul>
<p>Any suggestions? Thanks in advance.</p>
| Asaf Karagila | 622 | <ol>
<li>Jech, <strong>The Axiom of Choice</strong>.</li>
<li>Herrlich, <strong>The Axiom of Choice</strong>.</li>
<li>Halbeisen, <strong>Combinatorial Set Theory</strong>.</li>
<li>Jech, <strong>Set Theory, 3rd Millennium Edition</strong>.</li>
</ol>
<p>Jech's (first) book is kinda old, and some progress has been mad... |
3,066,020 | <p>I’m reading Hans Kurzweil ‘s “The Theory of Finite Groups”, where it says</p>
<blockquote>
<p>1.6.4 Let <span class="math-container">$N_1, . . . , N_n$</span> be normal subgroups of <span class="math-container">$G$</span>. Then the mapping <span class="math-container">$$α: G→G/N_1\times ··· \times G/N_n$$</span> ... | cqfd | 588,038 | <blockquote>
<p>I’m confused here: can we write <span class="math-container">$$G/N_1\times ··· \times G/N_n$$</span> ? To
write a product of groups as this, it’s required that each <span class="math-container">$G/N_i$</span> has
only <span class="math-container">$e$</span> as common element.</p>
</blockquote>
<p... |
1,239,211 | <p>I have been allowed to attend some preparatory lectures for a seminar on the Goodwillie Calculus of Functors. I found in my notes from one of the lectures two statements which I would like to ask about.</p>
<p>The first one is probably straightforward and I'm guessing is related to Whitehead-type theorems. Still, I... | Kevin Arlin | 31,228 | <p>For your first claim: every weak homotopy type can be represented by some CW complex $X$. This is one of Whitehead's most famous theorems. But $X$ is given as the union of its finite-dimensional skeleta $X^n$, and such a nested union is a particular example of a filtered colimit.</p>
<p>The reason to restrict to fi... |
1,627,357 | <p>Is there a simple way to prove $$\frac{1}{\sqrt{1-x}} \le e^x$$ on $x \in [0,1/2]$?</p>
<p>Some of my observations from plots, etc.:</p>
<ul>
<li>Equality is attained at $x=0$ and near $x=0.8$.</li>
<li>The derivative is positive at $x=0$, and zero just after $x=0.5$. [I don't know how to find this zero analytical... | André Nicolas | 6,312 | <p>For our interval, the inequality is equivalent to $1-x\ge e^{-2x}$. (We squared and flipped.)</p>
<p>This inequality can be proved using differential calculus. Let $f(x)=1-x-e^{-2x}$. Then $f'(x)=2e^{-2x}-1$. So $f(x)$ is increasing until $x=\frac{\ln 2}{2}\approx 0.34$ and then decreasing. Thus all we need to do ... |
1,627,357 | <p>Is there a simple way to prove $$\frac{1}{\sqrt{1-x}} \le e^x$$ on $x \in [0,1/2]$?</p>
<p>Some of my observations from plots, etc.:</p>
<ul>
<li>Equality is attained at $x=0$ and near $x=0.8$.</li>
<li>The derivative is positive at $x=0$, and zero just after $x=0.5$. [I don't know how to find this zero analytical... | πr8 | 302,863 | <p>If $f(x)=(1-x)e^{2x}$, then $f'(x)=(1-2x)e^{2x}=0$ when $x=\frac{1}{2}$. Drawing a graph/checking the second derivative shows it to be a maximum, whence $1=f(0)\le f(x)\le f(1/2)=\frac{e}{2}$ on $[0,\frac{1}{2}]$. We thus have:</p>
<p>$$1\le(1-x)e^{2x}\le\frac{e}{2}$$</p>
<p>$$\implies \frac{1}{1-x}\le e^{2x}\le\f... |
917,302 | <p>If $p(x)$ is a polynomial of degree 4 such that $p(2)=p(-2)=p(-3)=-1$ and $p(1)=p(-1)=1$, then find $p(0)$.</p>
| user84413 | 84,413 | <p>Using a difference table, with $p(0)=c$, gives</p>
<p>$-1\hspace{.5 in}-1\hspace{.5 in}1\hspace{.5 in}c\hspace{.6 in}1\hspace{.5 in}-1$</p>
<p>$\hspace{.4 in}0\hspace{.64 in}2\hspace{.43 in}c-1\hspace{.35 in}1-c\hspace{.4 in}-2$</p>
<p>$\hspace{.7 in}2\hspace{.47 in}c-3\hspace{.2 in}-2c+2\hspace{.2 in}-3+c$</p>
... |
3,884,581 | <p>Please don't just throw an answer at me, please explain how you arrived at it cause I've been fiddling with this for the past 30min...</p>
| ym94 | 630,901 | <p>Using the binomial theorem, we see that</p>
<p><span class="math-container">$u:=(a+b)^2=a^2+2ab+b^2=(a^2+b^2)+2(ab)=6+8=14$</span>. Therefore, <span class="math-container">$a+b=\pm \sqrt{14}$</span>.</p>
<p>Analogously,</p>
<p><span class="math-container">$v:=(a-b)^2=a^2-2ab+b^2=(a^2+b^2)-2(ab)=6-8=-2$</span>. There... |
622,090 | <p>We are asked to solve the following linear system</p>
<p>$$x_1-3x_2+x_3=1$$
$$2x_1-x_2-2x_3=2$$
$$x_1+2x_2-3x_3=-1$$</p>
<p>by using gauss-jordan elimination method. The augmented matrix of the linear system is $$\left(\begin{array}{ccc|c}1 & -3 & 1 & 1 \\2 & -1 & -2 & 2 \\1 & 2 & -... | Matheman | 117,904 | <p>$$ rank \left(\begin{array}{ccc|c}1 & -3 & 1 & 1 \\2 & -1 & -2 & 2 \\1 & 2 & -3 & -1\end{array}\right)=3$$ because $$det\left(\begin{array}{ccc}-3 & 1 & 1 \\-1 & -2 & 2 \\2 & -3 & -1\end{array}\right)\neq 0$$ and the rank of the coefficient matrix $$\left(\... |
1,989,182 | <p>Why does only one particular solution allow enough degrees of freedom for the general solution?</p>
| lisyarus | 135,314 | <p>This only works if the differential equation is linear, so it can be expressed as $Lx=y$, where $L$ is a <em>linear</em> differential operator. Then it is a basic theorem of linear algebra that if $x$ is some solution, then any solution is of the form $x+a$, where $La=0$.</p>
<ul>
<li><p>First, applying $L$ to $x+a... |
4,564,882 | <p>Suppose there are two types of weathers. Sunny and Rainy. <br />
The probability that a sunny day is followed by a sunny day is 70% and followed by a rainy day is 30%. <br />
The probability that a rainy day is followed by a rainy day is 60% and followed by a sunny day is 40%. <br />
In a year (365 days), how many... | Vercingetorix | 848,746 | <p>Just expanding on what the comments said. Let <span class="math-container">$P$</span> be the ptm. What you want to find is <span class="math-container">$(P^T)^{365}\pmatrix{1\\0}$</span></p>
<p>The idea is that overall long enough timespans (ex:365 days) we reach a stationary probability distribution <span class="ma... |
3,984,230 | <blockquote>
<p><span class="math-container">$2^x=4x$</span></p>
</blockquote>
<p>I cant seem to solve this equation. The furthest I have been able to come is
<span class="math-container">$x-\log_2(x)=2$</span>, but I can't figure how to solve. When I graph <span class="math-container">$2^x$</span> and <span class="mat... | David G. Stork | 210,401 | <p>Classic problem:</p>
<p><span class="math-container">$$ x = -\frac{W\left(-\frac{\log (2)}{4}\right)}{\log (2)}, {\rm or}\ -\frac{W_{-1}\left(-\frac{\log
(2)}{4}\right)}{\log (2)}$$</span></p>
<p>where <span class="math-container">$W$</span> is the <a href="http://wiki.analytica.com/ProductLog#:%7E:text=ProductLo... |
2,221,897 | <p>Show that </p>
<p>$$\lim_{n \to \infty} \sum_{k=3}^n \frac{2k}{k^2+n^2+1} = \ln(2)$$</p>
<p>How many ways are there to prove it ?</p>
<p>Is there a standard way ?</p>
<p>I was thinking about making it a Riemann sum.
Or telescoping.</p>
<p>What is the easiest way ?
What is the shortest way ?</p>
| Jacky Chong | 369,395 | <p>Observe
\begin{align}
\sum^n_{k=3}\frac{2k}{n^2+k^2+1} = \frac{1}{n}\sum^n_{k=3} \frac{2(k/n)}{1+n^{-2}+(k/n)^2}
\end{align}
then we have
\begin{align}
\frac{1}{1+n}\sum^n_{k=3} \frac{2k/(1+n)}{1+k^2(1+n)^{-2}} \leq \frac{1}{n}\sum^n_{k=3} \frac{2(k/n)}{1+n^{-2}+k^2n^{-2}} \leq \frac{1}{n}\sum^n_{k=3} \frac{2(k/n)}{... |
1,265,531 | <p>I understand the question but I am not sure how to solve it. For example, if we flip HHHTTTTT then the next three must be heads because of the question. This however seems counterintuitive. I believe that there are $2^{10}$ possible strings, but I am unsure of how to count all possible strings that begin with HHH.</... | André Nicolas | 6,312 | <p>We do a formal conditional probability calculation.</p>
<p>Let $A$ be the event the first $3$ tosses are heads, and let $B$ be the event we have an equal number of heads and tails in the $10$ tosses. We want $\Pr(A|B)$. By the definition of conditional probability, we have
$$\Pr(A|B)=\frac{\Pr(A\cap B)}{\Pr(B)}.$$
... |
1,265,531 | <p>I understand the question but I am not sure how to solve it. For example, if we flip HHHTTTTT then the next three must be heads because of the question. This however seems counterintuitive. I believe that there are $2^{10}$ possible strings, but I am unsure of how to count all possible strings that begin with HHH.</... | Graham Kemp | 135,106 | <blockquote>
<p>I understand the question but I am not sure how to solve it. For example, if we flip HHHTTTTT then the next three must be heads because of the question. This however seems counterintuitive. I believe that there are $2^{10}$ possible strings, but I am unsure of how to count all possible strings that be... |
959,525 | <p>Could someone tell me what i've done wrong?</p>
<p>I tried to find out the derivative of $3^(2x)-2x+1$ but I got it wrong.
What I did was derivate $3^a-2x+1$ where a = 2x then multiply those two.</p>
<p>$(ln3*3^a - 2)*2$ = $2ln3*3^(2x)-4$</p>
<p>Ps. x = 2 so the answer is supposed to be 176.</p>
| mathlove | 78,967 | <p>It's not true. In general, we have
$$m\lt\frac{a+b}{2}.$$</p>
<p><strong>Proof</strong> : By <a href="http://en.wikipedia.org/wiki/Parallelogram_law" rel="nofollow">parallelogram law</a>, we have
$$a^2+b^2=2\left(m^2+\left(\frac c2\right)^2\right)\Rightarrow 2m=\sqrt{2a^2+2b^2-c^2}.$$
Hence, we have
$$\begin{align}... |
269,242 | <p>The number of primes in each of the $\phi(n)$ residue classes relatively prime to $n$ are known to occur with asymptotically equal frequency (following from the proof of the Prime Number Theorem). Does the same result hold on pairs of consecutive primes on the $\phi(n)^2$ pairs of congruence classes?</p>
<p>To wit:... | Charles | 1,778 | <p>I suspect that the statement is false. The effect of prime gaps making some sizes more likely than others disappears after about $e^{e^n},$ but the effect of (small) primes dividing the modulus seems to be a problem except when $n$ is a prime power.</p>
<p>So the conjecture could either be weakened to the case wher... |
31,502 | <p>This is probably a trivial question, but I don't see the answer, and I haven't found it on <a href="http://en.wikipedia.org/wiki/Cartesian_closed_category" rel="nofollow noreferrer">Wikipedia</a>, <a href="http://ncatlab.org/nlab/show/cartesian+closed+category" rel="nofollow noreferrer">nLab</a>, nor <a href="https:... | thel | 373 | <p>The existence of such an adjunction implies that $B \otimes -$ preserves limits, which doesn't seem very likely.</p>
<p>Here is a counterexample, though probably not the simplest one. Set $B = k[y]$ and consider the inverse limit of $k[x]/(x^{n+1})$. If we take the tensor products first, then we get $k[y][[x]]$ whi... |
2,706,776 | <p>In solving the wave equation
$$u_{tt} - c^2 u_{xx} = 0$$
it is commonly 'factored'</p>
<p>$$u_{tt} - c^2 u_{xx} =
\bigg( \frac{\partial }{\partial t} - c \frac{\partial }{\partial x} \bigg)
\bigg( \frac{\partial }{\partial t} + c \frac{\partial }{\partial x} \bigg)
u = 0$$</p>
<p>to get
$$u(x,t) = f(x+ct) + g(x-c... | akhmeteli | 162,569 | <p>Yes, this is appropriate. You can apply the operators in the brackets one after another and get the same result as with the second-order derivatives. This "factoring" would be wrong, however, if, for example, $c$ were a function of $x$ or $t$.</p>
|
3,613,854 | <p>Let <span class="math-container">$$A=\begin{bmatrix}
3 & 2 \\
2 & 3
\end{bmatrix}.$$</span>
Find the spectral decomposition of <span class="math-container">$A$</span>. This is <span class="math-container">$$A=VDV^{-1}=\begin{bmatrix}
-1 & 1 \\
1 & 1
\end{bmatrix}\begin{bmatrix}
1 & 0 \\
0 & 5... | Mostafa Ayaz | 518,023 | <p><strong>Hint</strong></p>
<p>Note that using that decomposition<span class="math-container">$$A^k=VD^kV^{-1}$$</span>and <span class="math-container">$$2^A=\sum {A^n(\ln 2)^n\over n!}$$</span></p>
|
163,640 | <p>Early in a course in Algebra the result that every group can be embedded as a subgroup<br>
of a symmetric group is introduced. One can further work on it to embed it as a subgroup of a suitable (higher degree) alternating group.</p>
<p>Inverting the view point we can say that the family of simple groups $A_n, n\... | DavidLHarden | 12,610 | <p>Another use of regarding a group (there called $H$) as a subgroup of the symmetric group $S_{|H|}$ is given by Marty Isaacs in <a href="https://mathoverflow.net/questions/173148/subgroup-property-stronger-than-being-characteristic">Subgroup property stronger than being characteristic</a></p>
|
3,910,739 | <p>I am trying to find a pdf for a random variable <span class="math-container">$X$</span> where <span class="math-container">$X=-2Y+1$</span> and <span class="math-container">$Y$</span> is given by <span class="math-container">$N(4,9)$</span></p>
<p>Here is my attempt:</p>
<p>we know <span class="math-container">$\mu=... | Kolmogorov | 551,240 | <p>Let us denote distribution functions by <span class="math-container">$F$</span>, and density functions by <span class="math-container">$f$</span>. Then,
<span class="math-container">\begin{align*}
F_X(x) &= P(X \leqslant x)\\
&= P\left(Y \geqslant \frac{1-x}{2}\right)\\
&= 1 - F_Y\left(\frac{1-x}{2}\righ... |
4,032,983 | <p>I would like to know math websites that are useful for students, PhD students and researchers (useful in the sense most of the students or researchers—of a particular area—are using it). Maybe you can share which math websites you sometime use and why you use it.</p>
<p>Let me give my websites and why I use them:</p... | Qi Zhu | 470,938 | <p>Great question. It would probably also be interesting to think about what could be useful but is not yet out there. Here are a few picks that I can think of:</p>
<ul>
<li>A useful website for ring theorists is the <a href="https://ringtheory.herokuapp.com/" rel="noreferrer">Database of Ring Theory</a>. It is actuall... |
4,032,983 | <p>I would like to know math websites that are useful for students, PhD students and researchers (useful in the sense most of the students or researchers—of a particular area—are using it). Maybe you can share which math websites you sometime use and why you use it.</p>
<p>Let me give my websites and why I use them:</p... | storluffarn | 891,289 | <p>I really enjoy the following schematic overview of various statistical, distributions, their relationships and properties. It's quite handy for giving students (and others!) a quick way of relating new distributions to distributions that they already know about.</p>
<p><a href="http://www.math.wm.edu/%7Eleemis/chart... |
870,240 | <p>Which number is larger? $\underbrace{888\cdots8}_\text{19 digits}\times\underbrace{333\cdots3}_\text{68 digits}$ or $\underbrace{444\cdots4}_\text{19 digits}\times\underbrace{666\cdots67}_\text{68 digits}$? Why? How much is it larger?</p>
| please delete me | 164,934 | <p>Let those four numbers be $a,b,c,d$ respectively. Then $a=2c$ and $d=2b+1$. So $cd-ab=c$.</p>
|
1,665,443 | <p>How do we show the ring homomorphism for </p>
<p>$\phi :\mathbb F_p(\alpha) \rightarrow\mathbb F_p(\alpha)$ which is defined as $ \phi(\alpha)=\alpha +1$.</p>
<p>This is a very basic fact but I am unable to prove it by the definition of ring homomorphism. Same thing happens for ring homomorphism over $\phi :K[x] \... | Andreas Caranti | 58,401 | <p>Let us start with your second question.</p>
<p>Let $K$ be a field, and $K[x]$ be the polynomial rings. Let $B$ be a commutative ring with unity containing $K$ as a subring, and let $\beta \in B$. Then there is a unique ring homomorphism
$$
v_{\beta} : K[x] \to B
$$
which satisfies
$$\begin{cases}
v_{\beta}: &a ... |
1,448,416 | <p>It states that nth difference of a polynomial of n degree is constant thus (n+1)th difference will be zero.</p>
<ul>
<li>how can i show that the nth difference is constant? </li>
<li>forward difference of a constant is zero but how can i prove it?</li>
</ul>
| lisyarus | 135,314 | <p>Let $p(x) = c$, where $c$ is a constant. Then $p(x+h)-p(x) = c-c=0$, thus forward difference of a constant is zero.</p>
<p>Let $p(x)=\sum\limits_{k=0}^{n}a_k x^k$. Then $p(x+h)-p(x) = \sum\limits_{k=0}^{n}a_k ((x+h)^k - x^k)$.</p>
<p>$(x+h)^k-x^k = {0 \choose k} h^0 x^k + {1 \choose k} h^1 x^{k-1} + \dots + {k \ch... |
1,300,273 | <p>I have a question about evaluating the limit:</p>
<p>$$\lim_{x \to\infty }\left(x^{f(x)}-x \right)$$</p>
<p>where:</p>
<p>$f(x)$ is a continuous map from the positive reals to the positive reals , and</p>
<p>$\lim_{x\rightarrow \infty }f(x)= 1$.</p>
<p>I attempted to apply L'Hôpital's rule by writing:</p>
<p>... | gjh | 37,021 | <p>I now think the problem is simpler than it originally seemed to me when I posted the question.</p>
<p>Is the following how the form of $f(x)$ determines $\lim_{x \to\infty }\left(x^{f(x)}-x \right)$?</p>
<p><strong>CASE ONE</strong> : $\lim_{x \to\infty }\left(x^{f(x)}-x \right) = L$ (finite)</p>
<p>This limit ... |
1,474,123 | <p>I have tried to use u-substitution but for some reason am not doing it right and thus not getting the correct answer. I want to know the most obvious/ intuitive way to solve this integral.</p>
| Vamsi Spidy | 279,085 | <p>$x=z\tan(k)$</p>
<p>$\mathrm{d}x=z\sec^{2}(k)\mathrm{d}k$</p>
<p>$(a^2 + x^2)^{\frac32} = z^4 (\sec k)^3 (sec(k))^2 \mathrm{d}k
=z^4 (1+(\tan k)(\tan k))^{\frac32}(\sec k)^2\mathrm{d}k$ </p>
<p>now put $\tan(k)=t$, $\mathrm{d}k=(\sec k)(\sec k)\mathrm{d}k$ .
And then integrate it very easily</p>
|
3,531,693 | <p>Let <span class="math-container">$A \subset \mathbb{R}^n$</span> be a compact set with positive Lebesgue measure on <span class="math-container">$\mathbb{R}^n$</span>. Can we find an open set <span class="math-container">$B \subset \mathbb{R}^n$</span> such that <span class="math-container">$B \subset A$</span>?</p>... | Milo Brandt | 174,927 | <p>Although the other answer is really a good answer, it's worth noting that you can cook up a lot of examples in a similar manner: choose your favorite compact set <span class="math-container">$C$</span> (in <span class="math-container">$\mathbb R^n$</span> or some other nicely behaved space) with positive measure. No... |
147,095 | <p>I was wondering if there is any stationary distribution for bipartite graph? Can we apply random walks on bipartite graph? since we know the stationary distribution can be found from Markov chain, but we have two different islands in bipartite graph and connections occur between nodes from different groups. </p>
| R W | 8,588 | <p>There is no problem with dealing with random walks and stationary distributions on bipartite graphs. Actually, integer lattices $\mathbb Z^d$ or finite cyclic groups of even order $Z_{2p}$ all give rise to bipartite graphs with respect to natural generating sets. The simple random walk on a finite connected graph al... |
240,699 | <p>I have the following equation which I want to solve:</p>
<p><span class="math-container">$$
I_D = [Li_2(-e^{V_D-I_D})-Li_{2}(e^{I_D})]
$$</span></p>
<p>Here <span class="math-container">$Li_2(x)$</span> is the PolyLog function of order <span class="math-container">$2$</span>. Is there a way to solve this equation it... | bbgodfrey | 1,063 | <p>A typical solution of the equation</p>
<pre><code>id - PolyLog[2, -Exp[vd - id]] - PolyLog[2, Exp[id]] == 0
</code></pre>
<p>can be obtained by plotting this expression.</p>
<pre><code>ReImPlot[(id - PolyLog[2, -Exp[vd - id]] - PolyLog[2, Exp[id]]) /. vd -> .5, id, -1, 1},
ImageSize -> Large, AxesLabel -&... |
672,707 | <p><img src="https://i.stack.imgur.com/Tr5Jy.gif" alt="enter image description here" /></p>
<p>2 How do I solve this equation involving a logarithm? 3</p>
| voligno | 67,047 | <p>Use the properties of the log function: </p>
<p>$\log_a {\frac{b}{c}} = \log_a b -\log_a c$ </p>
<p>and $\log_a x = \log_a b \times \log_b x$. </p>
|
481,313 | <p>Show that in an abelian group the product of two elements of finite order is itself an element of finite order.</p>
<p>I need some hint to start with, I am familiar with the basic</p>
| Prahlad Vaidyanathan | 89,789 | <p>$(ab)^n = a^nb^n$, so take $n = lcm(|a|,|b|)$</p>
|
285,227 | <p>I am trying to prove $\exp(x+y) = \exp(x) \exp(y)$.</p>
<p>I may use that $$\exp(x) = \sum_{n=0}^\infty \frac {x^n}{n!}$$
I further know how to multiply two power series in one point, i.e. if $f(x) = \sum_{n=0}^\infty c_n(x-a)^n$ and $g(x) = \sum_{k=0}^\infty d_n(x-a)^n$ then
$$
f(x)g(x) = \sum_{n=0}^\infty e_n(x-a... | Bumblebee | 156,886 | <p>Euler formula says that (exponential form of a complex number) $$e^{i\theta}=\cos\theta+i\sin\theta.$$ Therefor $$e^{x+y}=\cos(-i(x+y))+i\sin(-i(x+y))\\=\cos(xi+yi)-i\sin(xi+yi)\\=(\cos ix\cos iy-\sin ix\sin iy)+i(\sin ix\cos iy+\cos ix\sin iy)\\=(\cos ix+i\sin iy)(\cos iy+i\sin iy)\\=e^xe^y.$$</p>
|
1,850,069 | <p>Let the incircle (with center $I$) of $\triangle{ABC}$ touch the side $BC$ at $X$, and let $A'$ be the midpoint of this side. Then prove that line $A'I$ (extended) bisects $AX$.<a href="https://i.stack.imgur.com/pd7Di.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/pd7Di.png" alt="enter image desc... | Stefan4024 | 67,746 | <p>First denote the intersection of $A'I$ and $AX$ with $M$. Now let $IX$ intersect the incenter for a second time at $Y$. Then let $AY$ intersect $BC$ at $W$. It's well-known that $W$ is the tangent point of the excircle and $BC$ (you can check the proof of this lemma <a href="http://yufeizhao.com/olympiad/geolemmas.p... |
56,162 | <p>I'm trying to understand the Cartan decomposition of a semisimple Lie algebra, $\mathfrak g=\mathfrak k \oplus \mathfrak p$, where $[\mathfrak k,\mathfrak p] \subseteq \mathfrak p$, cf. the wikipedia article on <a href="http://en.wikipedia.org/wiki/Cartan_decomposition" rel="noreferrer">Cartan decomposition</a>.</p>... | Kelly Davis | 1,400 | <p>First consider the case $M = S^3$. Generalizing, consider the connected sum of a generic M with a sphere $M = M\# S^3$</p>
<p><strong>Edit</strong> Here's what I was thinking (Still not sure if it's all correct, but it seems closer to the spirit of Witten's paper than the obstruction arguments.) </p>
<p>Consider a... |
56,162 | <p>I'm trying to understand the Cartan decomposition of a semisimple Lie algebra, $\mathfrak g=\mathfrak k \oplus \mathfrak p$, where $[\mathfrak k,\mathfrak p] \subseteq \mathfrak p$, cf. the wikipedia article on <a href="http://en.wikipedia.org/wiki/Cartan_decomposition" rel="noreferrer">Cartan decomposition</a>.</p>... | Paul | 3,874 | <p>@kwl1026. Gauge transformations are sections of the Ad bundle $P\times_{Ad} g$ where $P\to M$ is the principal $G$ bundle; $g$ the lie algebra. When $G$ is abelian the adjoint action is trivial so, e.g. the $U(1)$ gauge group is always $Map(M,U(1))$ whether or not $P$ is trivial. Its homotopy classes are then $[M, ... |
2,127,679 | <p>I need to find $\frac{a}{b} \mod c$.<br>
This is equal to $(a\cdot b^{\phi(c)-1}\mod c)$, when $b,c$ are co-prime. But what if that's not the case?<br>
To be more clear, I need $$\frac{10^{a\cdot b}-1}{10^b-1}\mod P$$ </p>
| Ben Grossmann | 81,360 | <p>What' you're looking for is a solution to
$$
bx = a \pmod c
$$
If $b$ and $c$ are not coprime, write $d = \gcd(b,c)$, and write $c = (md)n$ in such a way that $md$ and $n$ are relatively prime. With the Chinese remainder theorem, it suffices to solve the system of equations
$$
bx = a \pmod {md}\\
bx = a \pmod n
$$... |
235,430 | <p>Suppose that a bounded sequence of real numbers $s_i$ ($i\in\omega$) has a limit $\alpha$ along some ultrafilter $\mu_1\in \beta{\Bbb N}\setminus{\Bbb N}$. Then given another ultrafilter $\mu_2\in \beta{\Bbb N}\setminus{\Bbb N}$, surely there exists some rearrangement $s_{r(i)}$ of $s_i$ that has the same limit $\al... | Paata Ivanishvili | 50,901 | <p>If diagonal entries of $Y$ are zero then there is an open question of Pełczyński which asks whether we have the lower bound<br>
$$
\mathbb{E} |x^{T}Yx| \geq \frac{1}{2} \sqrt{\mathbb{E} |x^{T}Yx|^{2}} ?
$$</p>
<p>K. Oleszkiewicz writes in his slides (see slides 170)
<a href="https://simons.berkeley.edu/sites/def... |
235,430 | <p>Suppose that a bounded sequence of real numbers $s_i$ ($i\in\omega$) has a limit $\alpha$ along some ultrafilter $\mu_1\in \beta{\Bbb N}\setminus{\Bbb N}$. Then given another ultrafilter $\mu_2\in \beta{\Bbb N}\setminus{\Bbb N}$, surely there exists some rearrangement $s_{r(i)}$ of $s_i$ that has the same limit $\al... | Henry.L | 25,437 | <p>The expectation can be computed in closed form, and I think that without further assumptions on entries of the matrix $Y$, the Jensen bound is sharp according to following calculation:
$\begin{align}\mathbb{E}\left[\boldsymbol{X^{t}YX}\right] & =\mathbb{E}\left[\sum_{\substack{1\le i,j\le n}
}X_{i}X_{j}Y_{ij}\ri... |
3,033,344 | <p>Question: Tom only have 2 type of coins: coins: 4 cents and 5 cents. Write a proof by induction that every amount n ≥ a can indeed be paid with Tom coins</p>
<p>1) Base Case: Tom can pay <span class="math-container">$12, $</span>13, <span class="math-container">$14, $</span>15, <span class="math-container">$16 and ... | Melody | 598,521 | <p>You can show that <span class="math-container">$\phi:\mathbb{N}\to\mathbb{N}$</span> defined by <span class="math-container">$$\phi(n)=\#\{m\in\mathbb{N}:\text{gcd}(m,n)=1,1\leq m\leq n\}$$</span>
is multiplicative. That is, if <span class="math-container">$m,n\in\mathbb{N}$</span> are relatively prime, then <span ... |
3,752,770 | <p>I tested this in python using:</p>
<pre><code>import numpy as np
import matplotlib.pyplot as plt
x = np.linspace(0, 10*2*np.pi, 10000)
y = np.sin(x)
plt.plot(y/y)
plt.plot(y)
</code></pre>
<p>Which produces:</p>
<p><a href="https://i.stack.imgur.com/pCwoV.png" rel="nofollow noreferrer"><img src="https://i.stack.img... | Knight wants Loong back | 569,595 | <p>There are few misconceptions regarding the rational functions. For example, if
<span class="math-container">$$
f(x) = \frac{x^2 -1}{x-1}
$$</span>
Then, we usually find that people write it out as
<span class="math-container">$$
f(x) = x+1
$$</span>
But they are not equivalent, they differ from each other at <span... |
1,765,222 | <p>I have proven this by the induction method but would like to know if it can be proven using an alternative method.</p>
| Roman83 | 309,360 | <p>$$\frac{n(n^4-1)}{5}=\frac{n(n^2-1)(n^2+1)}{5}=\frac{(n-1)n(n+1)(n^2+1)}{5}$$
If $n=5k$, then $5|n$</p>
<p>If $n=5k+1$, then $5|n-1$</p>
<p>If $n=5k-1$, then $5|n+1$</p>
<p>If $n=5k\pm2$, then $n^2+1=(5k\pm2)^2+1=25k^2\pm10k+4+1=25k^2\pm10k+5=5(5k^2\pm2k+1)$, then $5|(n^2+1)$</p>
|
1,765,222 | <p>I have proven this by the induction method but would like to know if it can be proven using an alternative method.</p>
| S.C.B. | 310,930 | <p>By <a href="https://en.wikipedia.org/wiki/Fermat%27s_little_theorem" rel="nofollow">Fermat's Little Theorem</a>, we have that $$n^5 \equiv n \pmod 5 \Leftrightarrow \frac{n^5-n}{5} \in \mathbb{Z}$$</p>
|
769,504 | <p>It is mentioned in <a href="http://ac.els-cdn.com/0166864182900657/1-s2.0-0166864182900657-main.pdf?_tid=2ecebd88-ccce-11e3-ae74-00000aab0f6c&acdnat=1398467347_2b1c578dc3ae8c1a9107e7444203edb6" rel="nofollow noreferrer">this</a> article, that, the one point compactification of an uncountable discrete space, is A... | user642796 | 8,348 | <p>Suppose that $\mathscr{B} \subseteq \mathscr{A}$, and $B \subseteq \omega$. It is not too difficult to show that $\mathscr{B} \cup B$ is a compact subset of $\Psi$ iff $\mathscr{B}$ is finite, and $B \setminus \bigcup \mathscr{B}$ is finite. From this it follows that if we make the very mild assumption that $\omega ... |
62,177 | <p>One of the most mind boggling results in my opinion is, with the axiom of choice/well-ordering principle, there exist such things as uncountable well-ordered sets $(A,\leq)$. </p>
<p>With this is mind, does there exist some well ordered set $(B,\leq)$ with some special element $b$ such that the set of all elements ... | Craig | 15,279 | <p>The best explanation I've seen of this for the layman has to be "The Pancake at the Bottom", by Scott Aaronson: <a href="http://www.scottaaronson.com/writings/pancake.html" rel="nofollow">http://www.scottaaronson.com/writings/pancake.html</a></p>
|
2,762,230 | <blockquote>
<p>Let $I:=[a,b]$ a perfect interval and $\gamma\in C(I,\Bbb R^n)$ an injective path such that $\Gamma:=\gamma(I)$ is rectifiable. Show that $\dim_H(\Gamma)=1$.</p>
</blockquote>
<p>Here $\dim_H$ is the Hausdorff dimension. My work so far: </p>
<p>Note that the canonical projections $\pi_k$ are Lipschi... | Masacroso | 173,262 | <p>After some mistakes I think I found a valid answer.</p>
<hr>
<p>First note that the function $\tilde\gamma: I\to\Gamma,\, t\mapsto\gamma(t)$ is bijective and note that for every closed set $C\subset I$ then $\gamma(C)$ is compact, and by the injectivity of $\gamma$ we find that
$$
\gamma(I\setminus C)=\gamma(I\cap... |
1,291,511 | <p>This may seem like a silly question, but I just wanted to check. I know there are proofs that if $f(x)=f'(x)$ then $f(x)=Ae^x$. But can we 'invent' another function that obeys $f(x)=f'(x)$ which is <strong>non-trivial</strong>?</p>
| Paramanand Singh | 72,031 | <p>I think other answers given here assume the existence of a nice function $e^{x}$ and this makes the proof considerably simpler. However I believe that it is better to approach the problem of solving $f'(x) = f(x)$ without knowing anything about $e^{x}$.</p>
<p>When we go down this path our final result is the follo... |
1,822,008 | <p>Here are two functions:
$f\left(u,v\right)=u^{2}+3v^{2}$</p>
<p>$g\left(x,y\right)=\begin{pmatrix} e^{x}\cos y \\ e^{x}\sin y \end{pmatrix} $</p>
<p>I need to make Jacobian matrix of $f\circ g$. I found derivative of their composition:</p>
<p>$\frac{d\left(f\circ g\right) }{d\left(x,y\right) }=2e^{2x}\cos^{2}{y... | Community | -1 | <p>$$(f\circ g)(x,y) = h(x,y) = e^{2x}\cos^2(y)+3e^{2x}\sin^2(y)$$ Now just build the Jacobian matrix (AKA gradient because $h$ is a scalar-valued function) like normal: $$\pmatrix{\frac{\partial h}{\partial x} & \frac{\partial h}{\partial y}}$$</p>
|
3,374,248 | <p>I haven't worked out all the details yet, but it seems to be true for the following functions:</p>
<ul>
<li><span class="math-container">$f(k) = 1$</span></li>
<li><span class="math-container">$f(k) = 1/k!$</span></li>
<li><span class="math-container">$f(k) = a^k$</span></li>
<li><span class="math-container">$f(k) ... | Vincent Granville | 574,948 | <p>This is not an answer, but rather an upper bound. Using the Cauchy-Schwartz inequality, it is easy to obtain
<span class="math-container">$$\sum_{k=1}^n f(k)f(n-k)\leq \sum_{k=0}^n f^2(k)\sim \int_0^nf^2(x) dx.$$</span></p>
<p>For a full solution, consider two independently and identically distributed random variab... |
154,757 | <p>I have this data:</p>
<ul>
<li><p>$a=6$</p></li>
<li><p>$b=3\sqrt2 -\sqrt6$ </p></li>
<li><p>$\alpha = 120°$</p></li>
</ul>
<p><strong>How to calculate the area of this triangle?</strong></p>
<p>there is picture:</p>
<p><img src="https://i.stack.imgur.com/hr2Cp.jpg" alt=""></p>
| Peter | 152,834 | <p>Area: S = 3.80384750844</p>
<p>Triangle calculation with its picture:</p>
<p><a href="http://www.triangle-calculator.com/?what=ssa&a=1.7931509&b=6&b1=120&submit=Solve" rel="nofollow">http://www.triangle-calculator.com/?what=ssa&a=1.7931509&b=6&b1=120&submit=Solve</a></p>
<p>Only on... |
3,895,314 | <p>How do I prove <span class="math-container">$x ^ {1 - x}(1 - x) ^ {x} \le \frac{1}{2}$</span>, for every <span class="math-container">$x \in (0, 1)$</span>.</p>
<hr />
<p>For <span class="math-container">$x = \frac {1}{2}$</span> the LHS is equal to one half. I tried studying what happens when <span class="math-cont... | Albus Dumbledore | 769,226 | <p>let <span class="math-container">$a=x,b=1-x$</span>,</p>
<p><span class="math-container">$a+b=1$</span>,</p>
<p>By AM-GM<span class="math-container">$$\frac{1}{2}=\frac{{(a+b)}^2}{2}\ge 2ab=\frac{ab+ba}{a+b}\ge \sqrt[a+b]{a^b b^a}=a^bb^a=x^{1-x}{(1-x)}^{x}$$</span></p>
|
3,895,314 | <p>How do I prove <span class="math-container">$x ^ {1 - x}(1 - x) ^ {x} \le \frac{1}{2}$</span>, for every <span class="math-container">$x \in (0, 1)$</span>.</p>
<hr />
<p>For <span class="math-container">$x = \frac {1}{2}$</span> the LHS is equal to one half. I tried studying what happens when <span class="math-cont... | xpaul | 66,420 | <p>Let
<span class="math-container">$$ f(x)=\ln[x ^ {1 - x}(1 - x) ^ {x}]=(1-x)\ln x+x\ln(1-x) $$</span>
and then
<span class="math-container">$$ f'(x)=-\ln x+\frac{1-x}{x}+\ln(1-x)-\frac{x}{1-x}, f''(x)=-\frac{1-x+x^2}{x^2(1-x)^2} .$$</span>
Clearly <span class="math-container">$x=\frac12$</span> is the only point in ... |
90,876 | <p>$$2x-\dfrac{x+1}{2} + \dfrac{1}{3}(x+3)= \dfrac{7}{3}$$</p>
<p>When I solve this I always end up with 11x = 5, which is wrong, no matter which way I solve it. Does anyone know how to solve it? Steps? (Because I know the answer should be x=1)</p>
| Jesko Hüttenhain | 11,653 | <p>$$\begin{align*}
&& 2x-\frac{x+1}{2}+\frac{x+3}{3} &= \frac{7}{3} & \cdot 6 \\
&\Leftrightarrow& 12x - 3x - 3 + 2x +6 &= 14 & \text{rearrange} \\
&\Leftrightarrow& 11x&=11
\end{align*}$$</p>
|
3,231,271 | <blockquote>
<p>Suppose <span class="math-container">$X$</span> is Banach and <span class="math-container">$T\in B(X)$</span> (i.e. <span class="math-container">$T$</span> is a linear and continuous map and <span class="math-container">$T:X \to X$</span>). Also, suppose <span class="math-container">$\exists c > 0$... | Robert Israel | 8,508 | <p>Hint: if not, the image of the unit ball of <span class="math-container">$X$</span> contains a ball in an infinite-dimensional space.</p>
|
2,966,871 | <blockquote>
<p>Define the unit sphere as <span class="math-container">$S^1=\{x\in \mathbb{R}^2: \|x\|=1\}$</span></p>
<p>Also define the real projective line as <span class="math-container">$\mathbb{R}P^1=S^1/(x\sim-x)$</span></p>
</blockquote>
<p>We can consider the mapping <span class="math-container">$f:S^1\rightar... | Henno Brandsma | 4,280 | <p>The reason Ashvin gave for surjectivity, using the polar coordinates representation is perfectly fine:</p>
<blockquote>
<p>The map <span class="math-container">$f$</span> is surjective because in polar coordinates, it is given by <span class="math-container">$e^{i\theta} \mapsto e^{2i\theta}$</span>, and every an... |
81,982 | <p>I am beggining to do some work with cubical sets and thought that I should have an understanding of various extra structures that one may put on cubical sets (for purposes of this question, connections). I know that cubical sets behave more nicely when one has an extra set of degeneracies called connections. The que... | Tim Porter | 3,502 | <p>A list of precise references for connections on cubical sets has to start with :</p>
<p>R. Brown, P. J. Higgins and R. Sivera, 2011, Nonabelian Algebraic Topology: Filtered spaces, crossed complexes, cubical homotopy groupoids , volume 15 of EMS Monographs in Mathematics , European Mathematical Society.</p>
<p>as ... |
1,190,345 | <p>If $f$ is Riemann integrable on $[a,b]$ , is $|f|$ Riemann integrable on $[a,b]$ ?
(The metric is $\mathbb R$ usual)</p>
<p>The other is question is $f$ is Riemann integrable on $[a,b]$ , can I claim $f$ is bounded on $[a,b]$ ? (I think the answer can be either yes or no that depend on considerating generalized fun... | Learnmore | 294,365 | <p>Let $\epsilon >0$ be arbitrary</p>
<p>Let $P=\{a=x_0<x_1<x_2<....<x_n=b\}$ be a partition of $[a,b]$ with $||P||<\delta$</p>
<p>let $M_i=\sup _{({x_{i-1},x_i})}f$ and $m_i=\inf _{({x_{i-1},x_i})}f$</p>
<p>let $M_i^{'}=\sup _{({x_{i-1},x_i})}|f|$ and $m_i^{'}=\inf _{({x_{i-1},x_i})}|f|$</p>
<p>T... |
1,190,345 | <p>If $f$ is Riemann integrable on $[a,b]$ , is $|f|$ Riemann integrable on $[a,b]$ ?
(The metric is $\mathbb R$ usual)</p>
<p>The other is question is $f$ is Riemann integrable on $[a,b]$ , can I claim $f$ is bounded on $[a,b]$ ? (I think the answer can be either yes or no that depend on considerating generalized fun... | TomGrubb | 223,701 | <p>I will use the fact that $f(x)$ is Riemann integrable on $[a,b]$ if and only if it is bounded and continuous almost everywhere on $[a,b]$.</p>
<p>Let $f(x)$ be Riemann integrable on [a,b]. Then $f$ is bounded and continuous almost everywhere. Define $f_+(x)$ by $f_+(x)=f(x)$ if $f(x)>0$, and $f_+(x)=0$ otherwise... |
2,403,201 | <p>How do I solve for $x$:</p>
<p>$$\log\left(\frac{1.07^x}{1050-2.5x}\right)=\log\left(\frac{1.2}{828}\right)$$</p>
<p>If I raise both sides to the power of $10$, I get:
$\dfrac{1.07^x}{1050-2.5x}=\frac{1}{690}$</p>
<p>Then I'm stuck. How do I solve this ?</p>
<p>As suggest by @Kevin, I have decided to add my take... | Claude Leibovici | 82,404 | <p>As said in comments, the solution is given in terms of Lambert function.</p>
<p>If you plot the function $$f(x)=\frac{1.07^x}{1050-2.5x}-\frac{1}{690}$$ you should notice that the solution is very close to $x=6$; this means that you could start Newton method and converge quite fast as shown below
$$\left(
\begin{ar... |
1,983,614 | <p>Consider a measurable space $(\Omega, \mathcal{F})$ and let $I$ be an arbitrary index set. </p>
<p>Is the following true?</p>
<blockquote>
<p>If $\left( A_i \right)_{i \in I}$ is a chain in $\mathcal{F}$ – that is, $\forall i \in I$, $A_i \in \mathcal{F}$ and for all $i, j \in I$, we have $A_i \subseteq A_... | bof | 111,012 | <p>No. Consider Lebesgue measure on the real line. Let $\kappa$ be the minimum cardinality of a non-measurable set, and let $A$ be a non-measurable set of cardinality $\kappa.$ Then $A$ is the union of a chain of sets of cardinality less than $\kappa,$ which are of course measurable sets.</p>
|
3,917,912 | <p>I am reading an article where the author seems to use a known relationship between the sum of a finite sequence of real positive numbers <span class="math-container">$a_1 +a_2 +... +a_n = m$</span> and the sum of their reciprocals. In particular, I suspect that
<span class="math-container">\begin{equation}
\sum_{i=1... | Bart Michels | 43,288 | <p>The author is using the Arithmetic Mean - Harmonic Mean ("AM-HM") inequality:
<a href="https://en.wikipedia.org/wiki/Harmonic_mean#Relationship_with_other_means" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Harmonic_mean#Relationship_with_other_means</a></p>
<p>This is a popular inequality in th... |
668,291 | <p>If $h$ and $k$ are any two distinct integers, then $h^n-k^n$ is divisible by $h-k$.</p>
<p>Let's start with the basis. Let $n=1$, then
$h^1-k^1 = h-k$</p>
<p>Now for the induction, I can't use $k$ because I don't want to be confused. So let $P(r)$ for $h^n-k^n$ and that's $h^r-k^r$</p>
<p>$h^r-k^r = h-k$</p>
<... | robjohn | 13,854 | <p>For $n=0$: $h-k\mid h^0-k^0$.</p>
<p>Suppose $h-k\mid h^n-k^n$, then
$$
\begin{align}
h^{n+1}-k^{n+1}
&=h\cdot h^n-k\cdot k^n\\
&=(\color{#C00000}{h-k})h^n+k(\color{#C00000}{h^n-k^n})
\end{align}
$$
Since $h-k$ and $h^n-k^n$ are divisible by $h-k$, so is $h^{n+1}-k^{n+1}$.</p>
|
2,960,734 | <p>So basically, I am given the following to prove:</p>
<blockquote>
<p>Let <span class="math-container">$+\gamma$</span> be a positively oriented smooth Jordan arc, and let <span class="math-container">$\omega$</span> denote the interior of <span class="math-container">$+\gamma$</span>. Recall that if <span class="mat... | Community | -1 | <p>I assume that <span class="math-container">$z$</span> is <span class="math-container">$\gamma$</span> in the definition of <span class="math-container">$\operatorname{diameter}(+\gamma)$</span>. Without loss of generality, we can assume that <span class="math-container">$\gamma(a)=(0,0)$</span> (otherwise, we can t... |
19,305 | <p>If I compute the eigenvalues and eigenvectors using <code>numpy.linalg.eig</code> (from Python), the eigenvalues returned seem to be all over the place. Using, for example, <a href="http://archive.ics.uci.edu/ml/machine-learning-databases/iris/iris.data" rel="nofollow">the Iris dataset</a>, the normalized Eigenvalue... | leonbloy | 312 | <p>The sum of the eigenvalues equals the trace of the matriz. For a $N \times N$ covariance matriz, this would amount to $N VAR$ - where VAR is the variance of each variable (assuming they are equal - otherwise it would be the mean variance).
Put in other way, the mean value of the eigenvalues is equal to the mean v... |
3,763,744 | <p>The helix is a curve <span class="math-container">$x(t) \in \mathbb{R}^3$</span> defined by:</p>
<p><span class="math-container">$$
x(t) = \begin{bmatrix}
\sin(t) \\
\cos(t) \\
t
\end{bmatrix}
$$</span></p>
<p>and it takes the classic shape:</p>
<p><a href="https://en.wikipedia.org/wiki/File:Rising_circular.gif" rel... | kdbanman | 426,612 | <p>After a few hours of digging around and thinking, I've found a way to more naturally express the spherical spiral idea in my question.</p>
<p><strong>I'm still not sure if my construction or properties make sense though, so I won't mark my own answer as correct here.</strong> Someone else with broader geometry know... |
3,441,346 | <p>I was asked to prove that a set <span class="math-container">$X$</span> is closed if and only if it contains all its limit points. I proceeded like so:</p>
<p>Let <span class="math-container">$X^\dagger=\partial X \cap X´$</span> and <span class="math-container">$X^\ast=\partial X \backslash X´$</span> with <span c... | Will Cai | 470,180 | <p>Assume <span class="math-container">$a_n \in X$</span>, and <span class="math-container">$a_n \to a$</span>, if <span class="math-container">$a \notin X$</span>, <span class="math-container">$a\in X'$</span>, then <span class="math-container">$\exists \epsilon, B(a,\epsilon)\in X'$</span>. So <span class="math-conta... |
4,496,736 | <p>Question: Use the variation of parameter method to find the general solution of the following differential equation
<span class="math-container">$$(\cos x) y''+(2\sin x) y'-(\cos x) y =0\;\;\;\;,\;\;\;\;0<x<1$$</span></p>
<p><strong>My Try:</strong></p>
<p>I think the question is wrong, since the right hand si... | Z Ahmed | 671,540 | <p>Note that <span class="math-container">$y_1(x)=\sin x$</span> is one solution of the second order linear ODE
<span class="math-container">$$y''+2\tan x y'-y=0$$</span>
If <span class="math-container">$y_1$</span> is one solution of ODE
<span class="math-container">$$y''+P(x)y'+Q(x)y=0.$$</span>
Then the other soluti... |
2,853,668 | <blockquote>
<p>Show that $$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{x^n}$$ converges for every $x>1$.</p>
</blockquote>
<p>let $a(x)$ be the sum of the series. does $a$ continious at $x=2$? differentiable?</p>
<p>I guess the first part is with leibniz but I am not sure about it.</p>
| mechanodroid | 144,766 | <p><strong>Hint:</strong></p>
<p>Use the geometric series:</p>
<p>$$a(x) = \frac1{x}\sum_{n=0}^\infty \frac{(-1)^n}{x^n} = \frac{1}{x\left(1+\frac1x\right)} = \frac{1}{x+1}$$</p>
|
2,969,004 | <p>I have seen several references to "order" of an element in the Symmetric Group. Specifically, that the order of a cycle is the least common multiple of the lengths of the cycles in its decomposition.</p>
<p>But the Symmetric Group is not cyclic, and I'm only familiar with the concept of "order" for cyclic groups. S... | Shweta Aggrawal | 581,242 | <p>Since you know about cyclic groups. </p>
<p>Think of order of an element in the symmetric group <span class="math-container">$S_n$</span> as the size of the cyclic group generated by it.</p>
<p>Example 1: Let us take <span class="math-container">$S_3$</span>. Take <span class="math-container">$g=(123)$</span>. Con... |
3,433,249 | <p>I need to find the number of <span class="math-container">$7$</span>s if we write all the numbers from <span class="math-container">$1$</span> to <span class="math-container">$1000000$</span>(so <span class="math-container">$77$</span>, for example, counts as two <span class="math-container">$7$</span>s and not one)... | fleablood | 280,126 | <p>One. the numbers <span class="math-container">$0$</span> to <span class="math-container">$999999$</span> (as <span class="math-container">$1000000$</span> doesn't have any <span class="math-container">$7$</span>) have <span class="math-container">$6$</span> digits. Not seven.</p>
<p>Three somehow you went from th... |
3,134,991 | <p>If nine coins are tossed, what is the probability that the number of heads is even?</p>
<p>So there can either be 0 heads, 2 heads, 4 heads, 6 heads, or 8 heads.</p>
<p>We have <span class="math-container">$n = 9$</span> trials, find the probability of each <span class="math-container">$k$</span> for <span class="... | Ethan Bolker | 72,858 | <p>If there are an even number of heads then there must be an odd number of tails. But heads and tails are symmetrical, so the probability must be <span class="math-container">$1/2$</span>.</p>
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.