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Calculate area of Ellipse without calculus? I like the way integration works, but the final formula $\pi ab$ is too simple.
I know there is a more deeper way to derive it. I just don't like to use calculus here, too many equations.
I'd like to use simple math, which does offer deeper insight into it.
| You may use an affine map $\varphi$ to send an ellipse into a circle. Since affine maps preserve the ratios between areas, the area of the ellipse is $\frac{\text{Area}(\text{circle})}{\left|\det\varphi\right|}=\pi a b$.
|
$x$ is a solution of $0<|x-c|<\delta$
$x$ is a solution of $0<|x-c|<\delta$ iff $c-\delta<x<c+\delta$.
The book says that statement is false, but I can't understand why.
$0<|x-c|$ by definition of the absolute value and it's not 0 unless $x=c$.
$$x-c>0:x-c<\delta\iff x<c+\delta$$
$$x-c<0:c-x<\delta\iff x>c-\delta$$
D... | If $x=c$ then $c-\delta<x<c+\delta$ is true, but $|x-c|=0$ and $0<|x-c|<\delta$ is false. If we changed the first line to $0\le|x-c|<\delta$ then it would be true: $0\le|x-c|$ and not $0<|x-c|$ by the absolute value definition.
|
Why is the energy of Korteweg-deVries equation constant in time? The Korteweg-deVries is $u_t+6uu_x+u_{xxx}=0$ I know that its energy is
$$E(t)=\int_{-\infty}^\infty (\frac12(u_x)^2-u^3)dx$$.
I know $u(x,t)$ and $u'(x,t)$ decays to 0 as $x\rightarrow \pm\infty$. If $E(t)$ is constant w.r.t. $t$ is it sufficient to onl... | This isn't a complete answer but it gives some insight into why the energy is constant for wave solutions, $u(x,t) = f(x - ct)$.
Plugging this into $u_t+6uu_x+u_{xxx}=0$ reduces to a third-order ODE, $-cf' +6ff' + f'''=0$, which can be integrated to $-cf + 3f^2 + f''= C_1$.
If you multiply by $f'$ and integrate a seco... |
Prove that the gcd is 1. I need to show that for $x$ odd the $\gcd\bigg(x,\dfrac{x^2-1}{2},\dfrac{x^2+1}{2}\bigg)=1$. Im trying it doing it pairwise. I already show that $\gcd\bigg(\dfrac{x^2-1}{2},\dfrac{x^2+1}{2}\bigg)=1$, but I dont have an idea how to show that $\gcd\bigg(x,\dfrac{x^2-1}{2}\bigg)=1$ and $\gcd\bigg(... | If $p|x$ then $p|x^2$ and $p\not |x^2 \pm 1$ so $p\not|\frac{x^2 \pm 1}k$.
So $x$ and $\frac{x^2 \pm 1}2$ have no prime factors in common. So $\gcd(x, \frac{x^2 -1}2) = \gcd(x,\frac{x^2 +1}2) = 1$.
And $\gcd(\frac{x^2-1}2,\frac{x^2 + 1}2) = \gcd(\frac{x^2 -1}2, \frac{x^2 +1}2 - \frac{x^2-1}2) = \gcd(\frac{x^2-1}2, 1) =... |
When does $\frac{\partial f}{\partial y} =0$ imply $f(x,y)=g(x)$? Let $U\subseteq\mathbb R^2$ be an open subset and $f:U\to\mathbb R$ a continuously differentiable function such that $$\forall (x,y)\in U:\frac{\partial f}{\partial y} =0.$$
Which conditions must $U$ and $f$ fulfill for $f$ to be a function in $x$ only, ... | Here I make a proof of the theorem with the hypothesis of convexity.
If you use the "directional mean value theorem", you obtain that $\forall \ z_1=(x,y), \ z_2=(x,y') \in U,$ the line segment $\gamma$ between $z_1$ and $z_2$ is totally contained in $U.$ So, you have that $$f(x,y)-f(x,y')=(y'-y) \ \partial_yf(x,c_{y,y... |
Determining which group of order 30 $G$ is given a relation of its elements. Let's say we have a group $G$ where $|G|=30$. I am familiar with the traditional argument using Sylow's theorems that gives rise to the fact that $G$ is isomorphic to one of the following groups:
$$\mathbb{Z}_{30},\mathbb{Z}_3\times D_5, \math... | The order of $b^4$ is the same as that of $b$ without reference to $a,$ since $4$ is relatively prime to $15.$ However, notice that the fact that
$$b^4 = a b a^{-1}$$ implies (by raising both sides to fifth power) that $$b^5 = a b^5 a^{-1}.$$ Which means that $b^5$ is central. Further, $b^5$ generates the Sylow $3$-su... |
Why is $[\mathbb{Q}(\sqrt2,\sqrt3):\mathbb{Q}]=4$? I need to use the so called Ring Tower Theorem to show that $[\mathbb{Q}(\sqrt2,\sqrt3):\mathbb{Q}]=4$, but I'm quite confused with the notation and some concepts. First of all, my book says that if $a$ in some extension $E$ of $\mathbb{Q}$ is algebraic over $\mathbb{Q... | Let $x=\sqrt 2+\sqrt 3$. Then $x-\sqrt 3=\sqrt 2$. So $x^2-2\sqrt 3x+1=0$. But this is not the minimum degree polynomial over $\mathbb{Q}$ since $-2\sqrt 3 \notin\mathbb{Q}$. Thus with $x^2+1=2\sqrt 3x$ squaring both sides gives $x^4-10x^2+1=0$. Hence $[\mathbb{Q}(\sqrt2,\sqrt3):\mathbb{Q}]=4$.
|
Find the number of members of chess club The members of a chess club took part in Round Robin competition in which each plays everyone else once. All members scored the same number of points, except 4 juniors whose total score were 17.5. How many members were there in the club? Assume that for each win a player scores... | Building on the hints and terminology of @Michael we can write$$\frac{(s+4)(s+3)}{2}=\frac{sm}{2}+17.5$$
or $$s^2+(7-m)s-23=0$$
or $$s=\frac 1 2 ((m-7)\pm \sqrt{m^2-14m+141})$$
The problem then becomes finding a whole number $m$ which gives a whole number solution for $s$. I don't know what method @Michael envisioned f... |
How find this maximum of the value $\sum_{i=1}^{6}x_{i}x_{i+1}x_{i+2}x_{i+3}$? Let
$$x_{1},x_{2},x_{3},x_{5},x_{6}\ge 0$$ such that
$$x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}=1$$
Find the maximum of the value of
$$\sum_{i=1}^{6}x_{i}\;x_{i+1}\;x_{i+2}\;x_{i+3}$$
where
$$x_{7}=x_{1},\quad x_{8}=x_{2},\quad x_{9}=x_{3}\,.$... | Let $x_1=a,\ x_2=b,\ x_3=c,\ x_4=d,\ x_5=e,\ x_6=f.$
Objective function is
$$Z(a,b,c,d,e,f) = abcd + bcde + cdef + defa + efab + fabc.$$
Let maximize $Z(a,b,c,d,e,f)$ using Lagrange mulptiplyers method for the function
$$F(a,b,c,d,e,f,\lambda) = abcd + bcde + cdef + defa + efab + fabc + \lambda(1-a-b-c-d-e-f).$$
Equa... |
Find 2 basis $M=\left\{v_1,v_2,v_3\right\}$ and $N=\left\{w_1,w_2,w_3\right\}$ such that $T_{MN(f)}$
$A=\begin{pmatrix} 1 & 3 & -1\\
-1 & 0 & -2\\ 1 & 1 & 1 \end{pmatrix}$ is a real matrix and $f: \mathbb{R}^3 \rightarrow \mathbb{R}^3, \text{ } f(x)= A \cdot x$ is a
linear mapping.
Find two basis $M= \left\{v_1,v_... | If $V$ is a vector space with finite dimension, then, there exists a number $d$ (and only one) such that every basis of $V$ has exactly $d$ vector; said $n$ is the dimension of $V.$
Assume $B = (v_1, \ldots, v_d)$ is one such basis of $V.$ I write a $d$-tuple instead of the more (erroneously used) common usage of a set... |
Why is $P(X \geq a)$ the same as $E[\mathbf{1}_{X \geq a}]$ in a proof of the markov's inequality I didn't understand the last step
the markov's inequality : $$P(X \geq a) \leq \frac{E[X]}{a}$$
let $X : \Omega \to \Bbb{R}$ be a random variable
let $$\mathbf{1}_{X \geq a} :\mathbb{R} \to \{0,1\}$$
$$\mathbf{1}_{X \geq ... | Taking expectation is to integrate a random variable over the outcome space w.r.t the probability measure. Recall that $\int_{\Omega} 1_{A} dP = \int_{A} 1 dP = P(A)$ by basic properties of Lebesgue integration.
Also note that the set $\{ X \geq a \}$ is the shorthand for $\{ \omega \in \Omega \mid X(\omega) \geq a \}$... |
Exercise in Weibel's book Lie cohomology, Homology Can you please help me out to prove the following (rather easy (but not for me obviously)) exercise which I didn't manage to solve (Exercise 7.4.2, page 229)?
Assume $\mathfrak{g}$ is the free $\mathbb{k}$-module on basis $\{ e_1, e_2,..., e_n \}$, made it into abelia... | The $a$ be a finite dimensional abelian Lie algebra, let $\{x_1,\dots,x_n\}$ be a basis for $a$ and let $U$ be its enveloping algebra. It is easy to see that $U$ is a polinomial ring with generators $x_1,\dots,x_n$. You can check that $x_1,\dots,x_n$ is a regular sequence in $U$, so using the result of Chapter 4 in the... |
Prove that $d(x,y) = \sum_{i\in\mathbb{N}} a_i\frac{d_i(x_i,y_i)}{1+d_i(x_i,y_i)}$ satisfies the triangle inequality Let $(X_i
, d_i), i ∈ \Bbb N$, be a collection of metric spaces.
Define the metric \begin{align}d(x,y) = \sum_{i\in\mathbb{N}} a_i\frac{d_i(x_i,y_i)}{1+d_i(x_i,y_i)} \end{align} on the infinite product... | Let $\rho_k(x_k,y_k) = \frac{d_k(x_k,y_k)}{1+d_k(x_k,y_k)}$ and note that each $\rho_k$ are a metric, so for all $k$
$$
\rho_k(x_k,y_k)\le \rho_k(x_k,z_k) + \rho_k(z_k,y_k).
$$
Then multiplying by $a_k$ and summing from $k=1$ up to $k=n$ we have
$$
\sum_{k=1}^na_k\rho_k(x_k,y_k)\le \sum_{k=1}^na_k\rho_k(x_k,z_k) + \sum... |
A general way to solve $p(x)\cdot T(x)=1$ distributional equation?
DISCLAMER : I first should apologize, it could be that my question does not make much sense or could be imprecise. Just take this question as a naive question from a physics guy who is trying to understand what he is doing...
Is there a generic way to... | If the roots of $p$ are real and distinct, then the intuition suggests that $$T" =" \frac{1}{(x-x_1)(x-x_2)}.$$ We can rewrite it as $$T(x) "=" \frac{1}{x_2-x_1}\left(\frac{1}{x-x_1} - \frac{1}{x-x_2}\right).$$ Now we apply a standard workaround to avoid non-local-integrability of terms $\frac{1}{x-x_i}$, we consider p... |
The dimension of the kernel of $X \to AX-XA$ is the sum of the squares of the multiplicities of the eigenvalues of $A$ Let $A$ be a diagonalizable matrix and let $F$ be this application
$$F:M(n,\mathbb{R})\to M(n,\mathbb{R})$$
$$F(X)=AX-XA$$
Prove that the dimension of the kernel of F is the sum of the multiplicities o... | Let $E_{ij}$ be the elementary matrix whose only nonzero entry is in the $i$th row and $j$th column.
Lemma: Let $D \in Mat_n (\mathbb{R})$ be a diagonal matrix. Then $[ D, E_{ij} ] = 0 \iff D_{ii} = D_{jj}$.
Proof: $D E_{ij} = D_{ii} E_{ij}$ and $D E_{ij} = D_{jj} E_{ij}$.
Lemma: for a diagonal matrix $D \in Mat_n (\... |
Show such a function has a maximum Let $f:[0, \infty)$ be a continuous function.
$f(0) = 1 $ and $\forall x \in [0, \infty)$ $f(x)\leq \frac{x+2}{x+1}$
Show that $f$ gets a maximal value in $[0, \infty)$.
My intuition:
if $f(0)$ is the maximum i'm done if not the function you showed me converges to $1$ I want to show ... | Assume that $\sup_{x\geq 0}f(x)>1$ and that the maximum does not exist. Then there is a sequence $(x_n)_{n\in\Bbb N}$ in $[0,\infty)$ (wlog monotonically increasing) such that $f(x_n)\to\sup_{x\geq 0}f(x)$ for $n\to\infty$. Furthermore, the sequence $(x_n)_{n\in\Bbb N}$ can be chosen as $x_n\to\infty$ (otherwise the se... |
Prove $\lim_{(x,y) \to (-1,8)} xy = -8$ using only the definition. I'm having problems trying to prove this limit using only the definition with delta and epsilon.
I need to see that:
$$ \forall\epsilon \ \exists \delta \ : \sqrt{(x+1)^2+(y-8)^2} < \delta \Rightarrow |xy+8|<\epsilon$$
I want to make $|xy+8|$ look like... | For a slightly different approach consider the following inequality. \begin{align} |xy + 8| &= |(x+1)(y-8) \, \, + 8x - y + 16 |
\\ &= |(x+1)(y-8)\, \, + 8(x+1) - (y-8)|
\\&\leq |(x+1)(y-8)|+ 8|x+1| + |y-8|
\\&= |x+1|\cdot|y-8|\, + 8|x+1| + |y-8|\end{align}
If you want to use the euclidean norm, then note that usi... |
Tan function and isosceles triangles I have a non-right-angled isosceles triangle with two longer sides, X, and a short base Y.
I know the length of the long sides, X.
I also know the acute, vertex angle opposite the base Y, let's call it angle 'a'
I have been told I can calculate the length of the base Y by:
Y = tan(a... | Cut the iscoles triangle in half to get a two right triangles with opposite side $\frac 12 y$ and hypotenuse $x$.
$\frac 12 Y = \sin (\frac 12 a) x$
So apparently this is claiming $ 2\sin(\frac 12 a) = \tan a$ which isn't true but is apparently an approximation. $\tan a = \frac{\sin (\frac 12 a + \frac 12 a)}{\cos(\... |
Taking a sum of exponentials and turning it into a fraction I'm curious as to how they made this jump in logic:
$$A e^{i \omega t} \left[1 + e^{i \phi} + e^{2i\phi} + \dots + e^{(N-1)i\phi}\right] = A e^{i\omega t} \frac{e^{i N\phi} - 1}{e^{i\phi} - 1}$$
How did they convert the sum within the brackets into the express... | They used:
1. Power laws: $(e^x)^a=e^{xa}$
2. The sum of a geometric series
|
Prove $X^tX$, where $X$ is a matrix of full column rank, is positive definite? Let $X$ be a matrix of dimension $n\times k$ where $n>k$, $\text{rk}(X)=k$ so $X$ is of full column rank. Then how do I prove $X^tX$ is always positive definite, where $X^t$ is transpose of $X$? This is given sortta like a lemma in our lect... | For any real invertible matrix $X$ we can show that the product $X^tX$ is a positive defined matrix. In fact, let's just take a vector $v$ non-zero. So we can easily see that: $$v^tX^tXv=||Xv||^2>0$$
because since the matrix $X$ is invertible, $Xv\neq 0.$
|
Why does $\epsilon$ come first in the $\epsilon-\delta$ definition of limit? As we know, $\underset{x\rightarrow c}{\lim}f(x)=L\Leftrightarrow$ for every $\epsilon>0$ there exists $\delta>0$ such that if $0<|x-c|<\delta$, then $|f(x)-L|<\epsilon$.
My question is: why do we say that for every $\epsilon>0$ there exists $... | Good question!
The answer is that if we put $\delta$ first, then our definition would no longer correspond to what we mean by the limit of a function.
Proposition Let $f$ be the constant function $0$. Let $c,L$ be any real numbers. Then:
For all $\delta>0$ there exists $\epsilon>0$ such that if $0<|x-c|<\delta$, the... |
problem involving inertia
Update my lagrangian is
$$L=\frac{ml^2}2\left(\dot \theta_1^2+\frac 13\dot \theta_2^2+\dot \theta_1\dot \theta_2\cos\delta\right)+mgl(cos\theta_1+\frac 12\cos\theta_2)$$
and for my linearised equations of motion question (5) i got the following
$$(\ddot \theta_1) +\frac{2}{3}\ddot\theta_2=\... | *
*For an arbitrary point on the rod, let $p$ be its distance from the point of connection to the wire. Since the mass is uniformly distributed, then
$$dm=\frac ml dp$$
and the distance of that point from the reference of rotation can be obtained from the law of cosines:
$$r^2=l^2+p^2+2lp\cos\delta$$
Here, $\delta=\th... |
Equality of limit and a sequence and its terms Two numbers $a$ and $b$ are equal if and only if for every $\epsilon > 0 $,
$|a-b| < \epsilon $
A sequence is said to converge to a limit $L$ if for any $\epsilon >0,\
\exists m \in $ N such that $ |x_n - L| <\epsilon\ for\ all\ n\ge m $
Does the second statement imply... | No, here is a counterexample. Let $x_n = \frac{1}{n}$. We want to show that it converges to $L = 0$. Let $\epsilon > 0$. Then pick some $m$ such that $m > \frac{1}{\epsilon}$. For all $n \geq m - \frac{1}{\epsilon}$ we have
$$ |x_n - 0 | = | \frac{1}{n} | \leq |\frac{1}{m} | < \epsilon. $$
But, obviously, no $x_n$ is e... |
Looking for help in understanding a solution to a Calc III problem about surfaces Studying for a Calc III midterm and I'm trying to shore up my intuitions. The question I'm looking at asks:
Show that the curve with parametric equations $x = sin (t)$, $y = cos
(t)$, $z = sin^2 (t)$ is the curve of intersection of the s... | It's helpful to visualize what the surfaces you are to trying to intersect. In this case, you have two "cylinders". One is perpedicular to the xy plane, while the other is perpendicular to xz plane. So when you tried intersecting these surfaces, and obtain $$2x^2+y^2-z=1,$$ you have to keep in mind that this intersecti... |
Differentiating $ x^{a}y^{b} = c $, in its simplest form. $$ x^{a}y^{b} = c, $$
where a, b and c are constants. My attempts so far
$$ \frac{dy}{dx} = ax^{a - 1}by^{b - 1}$$
$$ \frac{d^2y}{dx^2} = (a^2 - a)x^{a-2}(b^2 - b)y^{b - 2} $$
I think that these first and second derivatives are correct, however my issue is, are... | Assuming this is implicit then,
$$\frac{d}{dx}y^n=ny^{n-1}\frac{dy}{dx}$$
Use together with the product rule for correct solution
|
Dense transfer of a set with positive lebesgue measure: is it conull? I'm facing a problem in measure theory and I need to prove the following conjecture to move on.
Attention: I'm not sure the following statement is true.
Let $A \subset \mathbb{R}$ be a measurable set such that $m(A)>0$ and $H$ be a countable, dense s... | Use Lebesgue density theorem (LDT) which has an elementary proof.
Towards a contradiction, suppose $B = \mathbb{R} \setminus (A + H)$ has positive measure. Using LDT, choose open intervals $I, J$ of same length such that $B \cap I$ has $\geq 99$ percent measure of $I$ and $A \cap J$ has $\geq 99$ percent measure of $J$... |
Square classes p-adic numbers isomorphism I was reading about the Hilbert symbol and the Hasse-Minkowski theorem and found this statement in the book I was reading :
$ |\frac{Q_p^{*}}{(Q_p^{*})^{2}} |=2^r$; with $r=2$ for $p \neq 2$ and $r=3$ for $p=2$
I was thinking about proving it directly by showing an isomorpishm,... | It may be possible to prove this via Hensel's lemma, but I think a more persuasive argument (perhaps the most standard one) is by p-adic versions of exponential and logarithm (yes, taking p-adic inputs and producing p-adic outputs). The $n!$ in the denominator of the exponential is no longer large, p-adically, indeed, ... |
How do you construct a function that is continuous over $(0,1)$ whose image is the entire real line? How do you construct a continuous function over the interval $(0,1)$ whose image is the entire real line?
When I first saw this problem, I thought $\frac{1}{x(x-1)}$ might work since it is continuous on $(0,1)$, but whe... | Here's another example. Take the continuous function $f:(0,1)\to\mathbb R$ defined by $$f(x)=\frac{1}{x}\cos\frac{\pi}x.$$ For $n\in\mathbb N$ we have $f(\frac1{2n})=2n$ and $f(\frac1{2n+1})=-2n-1$, so $[-2n-1,2n]$ lies in the image of $f$ for each $n\in\mathbb N$. Therefore, the image of $f$ is $\mathbb R$.
This examp... |
Galois Group of any polynomial over $\mathbb{R}$ This problem came up on a homework and I immediately came up with the following: The Galois Group is $Z/2Z$ if the polynomial has a complex root ($\mathbb{C} = \mathbb{R}[i]$ is algebraically closed).
Am I correct or have I made a trivial error?
| This is correct.
By the fundamental theorem of algebra, any polynomial $f(x) \in \mathbb R[x]$ splits completely into linear factors over $\mathbb C$. So the splitting field of $f(x)$ is certainly some intermediate field between $\mathbb R$ and $\mathbb C$.
But $[\mathbb C: \mathbb R] = 2$. So the only such intermediat... |
If I roll D copies of an S-sided dice, which is the probability that I will get at least M matches? If I roll $D$ copies of an $S$-sided dice, which is the probability that I will get at least $M$ matches?
For example, consider $D=3$, $S=4$, $M=2$: rolling three $4$-sided dice, looking for two or more of a kind.
There ... | Edit: this was an answer given for a version with the question with a typo on it. It's an approximation assuming that $D$ and $S$ are both large, and $D$ is much smaller than $S$.
If we let $X_k$ count the number of dice that come up $k$, then $X_k$ counts lots and lots of independent events that are each very very un... |
Prove that the real projective line cannot be embedded into Euclidean space Can the real projective line $RP^1$ be embedded into $\mathbb{R}^n$ for any $n$?
At first, I thought it could because $RP^2$ can be embedded in four-dimensional space, and $RP^1$ seemed like a simpler object to deal with than $RP^2$. But after... | $\mathbb{R}P^1$ is just the circle $S^1$ and this cannot be embedded in the reals: an infinite connected subset of $\mathbb{R}$ (is an interval so) always has a cut-point (a point we can remove to leave a disconnected subset), and the circle remains connected if we remove any point.
|
Integral and limit: $\lim_{n \to \infty}\int_{-1}^{1}\frac{t{e}^{nt}}{{\left( {{e}^{2nt}}+1 \right)}^{2}}dt$ Show that, $$\underset{n\to \infty }{\mathop{\lim }}\,{{n}^{2}}\int_{-1}^{1}{\frac{t{{e}^{nt}}}{{{\left( {{e}^{2nt}}+1 \right)}^{2}}}dt=-\frac{\pi }{4}}$$
i reached this result after using two steps of subsitut... | Hint. Let's consider
$$
\int _0^{\infty} \frac {\ln y} {y^2+a^2} \:dy,\qquad a>0,
$$ then, by the change of variable $y=ax$, $\ln y= \ln a +\ln x$, $dy=adx$, one has
$$
\int _0^{\infty} \frac {\ln y} {y^2+a^2} \:dy=\frac{\ln a}{a}\int _0^{\infty} \frac {1} {x^2+1} \:dx+\frac{1}{a}\int _0^{\infty} \frac {\ln x} {x^2+1}... |
Morphism of varieties from $\mathbb A^1 \to \mathbb A^1 \{0\}$?
Suppose $f$ is a morphism of varieties from $\mathbb A^1 \to \mathbb A^1 \backslash \{0\}$. What can we say about $f$ ?
I am unable to deduce much. I know that $f$ will induce a $k$ algebra map $k[x] \to k[x]$ by pulling back the regular functions. What ... | Are you aware that $\mathbb A^1\backslash \{ 0 \}$ is also affine? It is isomorphic to $V(xy - 1) \subset \mathbb A^2$ via the mapping $x \mapsto (x, 1/x)$. The coordinate ring of $V(xy - 1) \subset \mathbb A^2$ is
$$ k[x,y]/(xy - 1) \cong k[x,x^{-1}].$$
Therefore, following your line of reasoning, your geometric probl... |
Preference relation Let $X= \{(a,b)|a,b\in \mathbb R\}$. Suppose that we have a weak preference relation $R$ (its strict part is denoted by $P$ defined on $X$. Assume that $P$ is coordinate-wise strictly monotonic increasing, that is,
if $a>c$, then $(a,b)P(c,b)$ for all $b$, and
if $b>d$, then $(a,b)P(a,d)$ for all $a... | For a positive example, define $(a,b) R (c,d)$ if and only if $a+b>c+d$. The representing function is $v(x,y) = x + y$ which is of course component-wise strictly monotonic.
EDIT
For a counterexample, use the lexicographic preference relation, defined by $(a,b) R (c,d)$ if and only if $a > c$ or [$a=c$ and $b>d$]. The p... |
A multiple choice question related to singularities , poles. 1)Let $f(z)=\frac{1}{e^z-1}$ for all z$\in C$ such that $e^z\ne1$ then
a) f is meromorphic
b) the only singularities of f are poles
c) f has infinitely many poles on the imaginary axis
d) each pole of f is simple.
2)For z $\in C$, define $f(z)=\frac{e^z}{e^... | Hint: Using L'Hospital Calulate lim$_{z\to 2n\pi i}\frac{z-2n\pi i}{e^z-1}$,hence deduce that the given function has simple poles at $z= 2n \pi i$, $\forall n \in \mathbb N$
|
Definition of the gamma function for non-integer negative values The gamma function is defined as
$$\Gamma(x)=\int_0^\infty t^{x-1}e^{-t}dt$$
for $x>0$.
Through integration by parts, it can be shown that for $x>0$,
$$\Gamma(x)=\frac{1}{x}\Gamma(x+1).$$
Now, my textbook says we can use this definition to define $\Gamma... | The integral $\int_0^\infty t^{x-1}e^{-t}\,dt$ is a "representation" of the Gamma function for $x>0$. That is, for $x>0$
$$\int_0^\infty t^{x-1}e^{-t}\,dt=\Gamma(x)$$
But the Gamma Function exists for all complex values of $x$ provided $x$ is not $0$ or a negative integer.
The idea of "representing" a function on a ... |
Graph Theory - Application of Kirchoff's Matrix Tree Theorem Calculate the number of spanning trees of the graph that you obtain by removing one edge from $K_n$.
(Hint: How many of the spanning trees of $K_n$ contain the edge?)
I know the number is $(n-2)n^{n-3}$ and that Kirchoff's matrix tree theorem applies but how... | All edges of $K_n$ are identical. So if there are $n^{n-2}$ spanning trees of $K_n$, and each includes $n-1$ edges out of $\binom n2$, then each edge is included in $$\frac{n-1}{\binom n2} \cdot n^{n-2} = \frac2n \cdot n^{n-2} = 2n^{n-3}$$ spanning trees.
(Normally, this would be "included in an average of $2n^{n-3}$ s... |
Sequence and subsequence problem Let $(x_n)$ be a bounded sequence and for each $n ∈ ℕ$ let $s_n:=\sup\{x_k \mid k\geq n\}$ and $S:=\inf\{s_n\}.$ Show that there exist a subsequence of $(x_n)$ that converges to $S$.
I have that $(s_n)$ is a bounded decreasing sequence (by intuition, idk how to prove that).
Because $(s_... | Sn >=Sn+1 hence Sn is contracting sequence.
Sn has a bound
As. Inf(Xn)<=Sn<=Sup(Xn)
now inf of Sn is S. So for €>0 there exist Sn such that S <=Sn < S+€ for n >= k. So S-€ < S <= Sn < S+€
Hence |S-Sn| < € for n>=k
Where one may assume Sn as a subsequence of the given sequence
|
Proving that $T:V \rightarrow W$ has a unique adjoint transformation Let $(V, \langle . , \rangle_{V})$,$(W, \langle . , \rangle_{W})$ be inner product spaces, and let $T: V \rightarrow W$ be a linear transformation. A function $T^{*}: W \rightarrow V$ is called an adjoint of $T$ if $\langle T(x),y \rangle_{W} = \langl... | Hint/sketch: choose orthonormal bases $\{v_i\}$ and $\{w_j\}$ of $V$ and $W$, and represent $T$ by a matrix $A$ with respect to these bases. Then, the conjugate transpose $A^*$ will define a function $T^*:W\to V$, and you can use the orthogonality + the definition of the action of a matrix on a basis element to show th... |
Let ${a_n}$ and ${b_n}$ be sequences. Prove that if $\lim_{n\rightarrow \infty}(a_n^2 + b_n^2) = 0$ then $\lim_{n\rightarrow \infty}a_n=0$ Let ${a_n}$ and ${b_n}$ be sequences. Prove that if $\lim_{n\rightarrow \infty}(a_n^2 + b_n^2) = 0$ then $\lim_{n\rightarrow \infty}a_n=0$
This was a question on my Real Analysis u... | Hint Let $\epsilon >0$. If $N$ is such that ${a_n}^2+{b_n}^2 < \epsilon^2$ for all $n >N$, show that
$$\left| a_n -0 \right| < \epsilon \qquad \forall n >N$$
|
Proof C(n,r) = C(n, n-r) Hello just want to see if my proof is right, and if not could someone please guide me because I am not clearly seeing the steps to this proof. I don't know if I correctly solve the proof in the second to last step. If I did any mistake it would be great if someone could point at it.
$$
C(n, n-r... | $ C(n, k)$ denotes the number of ways to select $k$ out $n$ objects without regard for the order in which they are selected. To prove $C(n,r) = C(n, n-r)$ one needs to observe that whenever $k$ items are selected, $n-k$ items are left over, (un)selected of sorts.
|
Why is $5^{n+1}+2\cdot 3^n+1$ a multiple of $8$ for every natural number $n$? I have to show by induction that this function is a multiple of 8. I have tried everything but I can only show that is multiple of 4, some hints? The function is
$$5^{n+1}+2\cdot 3^n+1 \hspace{1cm}\forall n\ge 0$$, because it is a multiple of... | $\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newc... |
Are regular languages closed by a full-shuffle operation?
Let be two languages $L_1,L_2$ we define the operation full-shuffle as $S(L_1,L_2)=\{w \mid w=w_1w_2...w_k\}$ such that $w_1,w_3,...\in L_1$ and $w_2,w_4...\in L_2$. In other words, the language $L$ only contains words that we can build from a word $L_1$ follow... | Just use regular expressions instead of automata. Indeed,
$$S(L_1,L_2) = (L_1L_2)^* \cup (L_1L_2)^*L_1$$
|
Sketching for isotherms $T(x, y) = \text{ const}$ Hi I came across this question
Let the temperature $$T(x,y) = 4x^2+16y^2$$ in a body be independent of $z$, identify the isotherms $T(x, y) = \text{ const}$. Sketch it.
I do not understand the question. What type of equation(s) do I have to sketch?
| For several different values of $C$, you should draw lines containing all the points $(x, y)$ that solve $4x^2 + 16y^2 = C$, and you should also describe what the isotherms look like for a general value of $C$. For example, if $C = 16$, then the graph you would draw is $\frac{x^2}{4} + y^2 = 1$ (this is an ellipse with... |
Doubt in Category Theory about dualization. I have to prove that the Category of Abelian Groups has pushouts.
I want to use a theorem that says that if a category has equalizers and products, then it has pulbacks. Then I want to dualize the statement and say that if a category has coequalizer and coproducts, then it h... | Your strategy is correct; knowing that products and equalizers gives you pullbacks tells you that coproducts and coequalizers give you pushouts.
The reason for this is that for any category $\mathcal C$, you get $\mathcal C^{op}$ simply by having the new domain function of $\mathcal C^{op}$ be the codomain function of... |
Finding general solution $\frac{d^2f}{dx^2}$ for an implicit function Let's say I have a function $F:\mathbb R \times \mathbb R \rightarrow \mathbb R$, which meets the requirements of the implicit function theorem. Now I would like to find a general expression for:
$$\frac{d^2f}{dx^2}$$
From the theorem I know that $$\... | HINT:
Let $\displaystyle F_x(x,y)=\frac{\partial F(x,y)}{\partial x}$ and $\displaystyle F_y(x,y)=\frac{\partial F(x,y)}{\partial y}$.
$$\begin{align}
\frac{d}{dx}F_x(x,f(x))&=F_{xx}(x,f(x))+F_{xy}(x,f(x))\frac{df(x)}{dx}\\\\
&=F_{xx}(x,f(x))-F_{xy}(x,f(x))\frac{F_x(x,f(x))}{F_y(x,f(x))}
\end{align}$$
|
Solving $\lim_{ x \to 0^+} \sqrt{\tan x}^{\sqrt{x}}$ The limit to be calculated is:
$$\lim_{x \to 0^+}\sqrt{\tan x}^{\sqrt{x}}$$
I tried:
$$L =\lim_{x \to 0^+}\sqrt{\tan x}^{\sqrt{x}}$$
$$\log L = \lim _{x\to 0^+} \ \ \dfrac{1}{2}\cdot\dfrac{\sqrt{x}}{\frac{1}{\log(\tan x)}}$$
To apply the L'hospital theorem but failed... | Well, we have
$$
\begin{align}\lim_{x\to 0^+}\sqrt{x}\ln\sqrt{\tan x}&=\lim_{x\to 0^+}\frac{\sqrt{x}}{2}\ln(\tan x)\\
&=\lim_{x\to 0^+}\frac{\ln(\tan x)}{\frac{2}{\sqrt{x}}}\qquad\text{which takes the form } \frac{-\infty}{+\infty}\text{and apply LHR to get}\\
&=\lim_{x\to 0^+}\frac{\frac{\sec^2x}{\tan x}}{-\frac{1}{\... |
A function with convex level sets but is not a convex function I have the function $$f(x)=\sqrt{|x|},$$ with $x\in\mathbb{R}$. I understand that for some values $x,y\in$ dom$f$ and $t\in[0,1]$ that the function is not convex. However, I am having trouble proving that this function has convex level sets. We can define t... | You can attain your goal by proving that $\sqrt{|X|}$ is a quasi-convex function; that is, $f(tx+(1-t)y) \le \max \{f(x),f(y)\}$.
Assume wlog $x \le y$.
For $x \ge 0$, $f(x)$ is increasing; so, if $0<x<y$, $f(tx+(1-t)y) \le f(y)$.
Similarly, for $x \le 0$, $f(x)$ is decreasing; so, if $y \le 0$, $f(tx+(1-t)y) \le f(... |
Surface area of a sphere with integration of disks Why it is not correct to say that the surface area of a sphere is:
$$
2 \int_{0}^{R} 2\pi r \text{ } dr
$$
In my mind we are summing up the perimeters of disks from $r=0$ to $r=R$, so by 1 integration, we would have $\frac{1}{2}$ of the surface area of the sphere.
I kn... | $$\text{Area will be}~~: 2 \int_{0}^{R} 2\pi x ~ ds$$
Where $ds$ is width of strip bounded by circles of radius $x$ and $x+dx$ situated at height $y$. Also $ds \neq dr$ it's tilted in $y$ direction too. Only horizontal projection of $ds$ is $dr$.What you have done is valid for Disk See image below.
$$(ds)^2=(dx)^2+(dy... |
Finding an integral curve (applications of differential equations) Find a curve whose distance of every tangent from the origin $ON$ is equal to the $x$ axis coordinate of the point of intersection between the curve and that tangent $OU$.
How to set up the graph for this kind of problems in general?
How to form an ODE?... | The equation of the tangent at abscissa $x$ is
$$Y-y(x)-y'(x)(X-x)=0$$ so that the distance to the origin is
$$\frac{|0-y-y'\cdot(0-x)|}{\sqrt{1+y'^2}}$$ and is known to equal $x$.
Squaring and rearranging,
$$y^2-2xyy'+x^2y'^2=x^2+x^2y'^2,$$
$$y^2-2xyy'=x^2,$$
$$\left(\frac{y^2}x\right)'=-1.$$
Then after integration
$$... |
For what values of λ does Rank(A) = 1? Rank(A)= 2? Rank(A) = 3? $$ A = \begin{pmatrix} λ & 1 & 1 \\ 1 & λ & 1 \\ 1 & 1 & λ \end{pmatrix} $$
My attempt:
For Rank(A) = 1, the rows of matrix A must be scalar multiples of each other. Thus, the only value that works in this case would be λ = 1.
For Rank(A) = 2, the determ... | If the rank is not three, then it is not invertible and hence, its determinant must be $0$ (iff). Finding the characteristi polynomial, we can see when this happens. $$\left\vert\begin{pmatrix} λ & 1 & 1 \\ 1 & λ & 1 \\ 1 & 1 & λ \end{pmatrix}\right\vert=\lambda\cdot\left\vert\begin{pmatrix} \lambda & 1 \\ 1 & \lambda\... |
Find sum binomial coefficients $$
\sum_{k>=1}^{\infty} {2N \choose N-k}k
$$
How to find this sum?
I know that the answer is $ \frac{1}{2}N{2N \choose N}$
But it is very interesting to know the solution :)
| Our first goal in dealing with this sum is to get rid of the $k$. The standard approach is to rewrite something like $\binom{n}{k} \cdot k$ as $\frac nk \binom{n-1}{k-1} \cdot k$, or $n \binom{n-1}{k-1}$. Here, the bottom index doesn't match the extra factor, but we can make it so with a little extra work:
\begin{align... |
Linear transformation and its matrix I have two bases:
$A = \{v_1, v_2, v_3\}$ and $B = \{2v_1, v_2+v_3, -v_1+2v_2-v_3\}$
There is also a linear transformation: $T: \mathbb R^3 \rightarrow \mathbb R^3$
Matrix in base $A$:
$M_{T}^{A} = \begin{bmatrix}1 & 2 &3\\4 & 5 & 6\\1 & 1 & 0\end{bmatrix}$
Now I am to find matrix o... | Some notation: for a vector $v \in \Bbb R^3$, let $[v]_A$ denote the coordinate vector of $v$ with respect to the basis $A$, and let $[v]_B$ denote the coordinate vector of $v$ with respect to the basis $B$. both of these are column vectors. To put this another way,
$$
[v]_A = \pmatrix{a_1\\a_2\\a_3} \iff v = a_1v_1 +... |
Zero is least element of ordinal Definition. An ordinal is a well-ordered set $X$ such that for all $x\in X$, $(−∞, x) = x$.
Lemma. Zero is least element of ordinal.
Proof. Let $\alpha$ be an ordinal. Let $x$ be least element of $\alpha$. So, $x=x\cap\alpha=\emptyset$. Thus $\emptyset$ is least element of $\alpha$, tha... | If $\alpha$ is an ordinal and $x\in\alpha$ then, according to the definition you just gave, $x=(-\infty,x)$ and so $x\cap\alpha=(-\infty,x)\cap\alpha.$ If $x$ is the least element of $\alpha,$ then $(-\infty,x)
\cap\alpha=\emptyset.$
|
Show that $\| f \|_{Lip(\mathbb{R^n})} = \| \nabla f \|_{L^{\infty}{\mathbb{(R^n)}}}.$ For a Lipschitz function $f : \mathbb{R}^n \rightarrow \mathbb{R}$, define the Lipschitz norm acting on the set of Lipschitz functions as $\| f \|_{Lip(\mathbb{R^n})} = \sup_{x \neq y}\frac{|f(x) - f(y)|}{\| x - y \|}.$
In this paper... | By definition $\Vert \nabla f\Vert_{\infty}$ is the $L^\infty$ norm of the Euclidean norm of $\nabla f$. Note that when $f$ is Lipschitz, $f$ is differentiable (in the multivariable sense) almost everywhere, this is Rademacher's theorem.
Call the Lipschitz norm $L$.
Consider then that $\left|\frac{f(x+h)-f(x)}{\Vert h ... |
5x5 board Bingo Question There is a game which I play, it is like bingo. It starts with a 5x5.
Lets say horizontally it goes ABCDE from left to right and vertically it goes 12345 from bottom to top.
I have 2 random generators which will generate a letter and a number giving me a box to cross. So for example A2.
Suppos... | Another simple way to evaluate is probability is to use Monte Carlo Simulation,
See my python code:
from random import randint
from functools import reduce
def isBingo(board):
for i in range(5):
horizontal = reduce(lambda x,y: x and y, map(lambda j: board[i][j],range(5)), True)
vertical = reduce(lambda x,y: ... |
Why have the Mathematicians defined the boundedness in Functional Analysis so much different from that in Calculus? Let us consider $R$ with the norm $||x||=|x| $ for the whole discussion. Now the identity mapping $I$ from $R$ to $R$ is unbounded in Calculus but in Functional Analysis treating the same identity mapping... | Let $X$ be a Banach space. One way to connect the two definitions of boundedness is to say that a linear map $T : X \to X$ is bounded in the sense of functional analysis, if its restriction to the unit ball $B = \{ x \,:\, |x| < 1 \}$ is bounded in the sense of calculus.
Why the restriction to the unit ball? The only l... |
Summation of Binomial Coefficient Proof Prove that $$\sum_{k=0}^n(-1)^k {{n}\choose{k}} = 0$$
I have no idea about how to approach this problem.
| By the binomial theorem, we know that $$\sum_{k=0}^n(-1)^k {{n}\choose{k}} =\sum_{k=0}^n(-1)^k \times 1^{n-k} {{n}\choose{k}} =(1+(-1))^{n}= 0$$
We have the desired result. Also note that elementary sums involving binomial coefficients usually involves the binomial theorem.
|
Order of $f(n) = \sqrt[\log n]{n} \cdot n^{\sqrt[\log n]{n}}$ This is as far as i've gone:
$$f(n) = \sqrt[\log n]{n} \cdot n^{\sqrt[\log n]{n}} \iff f(n)^{1 / \sqrt[\log n]{n}} = (\sqrt[\log n]{n})^{1 / \sqrt[\log n]{n} } \cdot n \iff f(n) = n^{\sqrt[\log n]{n}} $$
since $ \lim \limits_{x\to \infty} \sqrt[x]{x} = 1$
N... | $$
\sqrt[\log n]{n} = n^{1/\log(n)} = \left(e^{\log(n)}\right)^{1/\log(n)} = e
$$
So overall you have $f(n) = e \cdot n^e$
|
Showing that the hyperintegers are uncountable In class, we constructed the hyperintegers as follows:
Let $N$ be a normal model of the natural number with domain $\mathbb{N}$ in the language $\{0, 1, +, \cdot, <, =\} $. Also let $F$ be a fixed nonprincipal ultrafilter on $\omega$. Then we have $N^*$ as the ultrapower $... | A further variation on this theme is to choose a fixed infinite hyperinteger $H$ and note that the partial map $f\colon{}^\ast\mathbb{N}\to\mathbb{R}$ given by $f(n)=\text{st}(\frac{n}{H})$ whenever this is defined, is surjective.
|
Showing that $\sum_{k=0}^{n}(-1)^k{n\choose k}{1\over k+1}\sum_{j=0}^{k}{H_{j+1}\over j+1}={1\over (n+1)^3}$ Consider this double sums $(1)$
$$\sum_{k=0}^{n}(-1)^k{n\choose k}{1\over k+1}\sum_{j=0}^{k}{H_{j+1}\over j+1}={1\over (n+1)^3}\tag1$$
Where $H_n$ is the n-th harmonic
An attempt:
Rewrite $(1)$ as
$$\sum_{k=... | We seek to show that
$$\sum_{k=0}^n (-1)^k {n\choose k} \frac{1}{k+1}
\sum_{j=0}^k \frac{H_{j+1}}{j+1} = \frac{1}{(1+n)^3}.$$
This is
$$\sum_{k=0}^n (-1)^k {n+1\choose k+1} \frac{k+1}{n+1} \frac{1}{k+1}
\sum_{j=0}^k \frac{H_{j+1}}{j+1} = \frac{1}{(1+n)^3}$$
or
$$\sum_{k=0}^n (-1)^k {n+1\choose k+1}
\sum_{j=0}^k \frac{... |
Lesson in an induction problem I'm trying to do this problem but I'm having a basic misunderstanding that just needs some clarification.
Consider the proposition that $P(n) = n^2 + 5n + 1$ is even.
Prove $P(k) \to P(k+1)$ $\forall k \in \mathbb N$.
For which values is this actually true?
What is the moral here?
This pr... | While you can prove the step that $P(k) \rightarrow P(k+1)$ for all $k \in \mathbb{N}$, it does not follow that "$n^2+5n+1$ is even" for all $n \in \mathbb{N}$.
In other words, the moral is to never forget to prove the base case for induction, for otherwise you might be proving things that are just not true.
Here is a ... |
Can I apply L'Hôpital to $\lim_{x \to \infty} \frac{x+\ln x}{x-\ln x}$? Can I apply L'Hôpital to this limit:
$$\lim_{x \to \infty} \frac{x+\ln x}{x-\ln x}?$$
I am not sure if I can because I learnt that I use L'Hôpital only if we have $\frac{0}{0}$ or $\frac{\infty}{\infty}$ and here $x-\ln x$ is $\infty-\infty$ and x... | $x -\ln x$ goes to $+\infty$ if and only if $e^{x-\ln x}$ does. And this is the case, since
$$e^{x-\ln x} = \frac{e^x}{x} \ge \frac{1+x+\dfrac{x^2}{2}}{x} \to +\infty $$
as $x \to +\infty$. So yes, you can apply de l'Hopital from the beginning.
|
How to explain why Integration by parts apparently "fails" in the case of $\int \frac{f'(x)}{f(x)}dx$ without resorting to definite integrals? Integrating by parts the following integral $$I=\int \frac{f'(x)}{f(x)}dx$$
gives us
$$\begin{align*}
I&=\int \frac{f'(x)}{f(x)}\,dx\\
&=\int\frac1{f(x)}f'(x)\,dx\\
&=\frac1{f(x... |
Integrating by parts the following integral $$I=\int \frac{f'(x)}{f(x)}dx$$
gives us
$$I=\int \frac{f'(x)}{f(x)}dx=\int\frac1{f(x)}f'(x)dx=\\\frac1{f(x)}f(x)-\int\Big(\frac1{f(x)}\Big)'f(x)dx=\\1+\int \frac{f'(x)}{f(x)}dx\Rightarrow\\
I=1+I$$
Be careful here. The $I$ on the left and the $I$ on the right are not exac... |
Question about determining whether vector field is conservative and about determining a potential function for said vector field. So, I've received this question to solve. Can anyone help me? I do not understand how to show that the vector field is conservative in this case. A detailed solution would help to understand... | If we manage to find a potential function $U$ for $\mathbf{F}$, we are done so let's try to do that. We need:
$$ \frac{\partial U}{\partial x} = 2x + y \implies U(x,y,z) = \int (2x + y) \, dx + G(y,z) = x^2 + yx + G(y,z) $$
for some function $G = G(y,z)$ which depends only on $y,z$. Next,
$$ \frac{\partial U}{\partial ... |
Equation translation: $V=\{f\in C^0([0,1]);\ f(0)=f(1)=0\}$ What does this equation say in English? More specifically, what does $C^0([0,1])$ mean?
$V=\{f\in C^0([0,1]);\ f(0)=f(1)=0\}$
If it helps, this is a set that I need to prove whether or not is a vector space.
EDIT: Okay, I just read something that says,
Here w... | $C^0([0,1])$ represents the set of continuous functions on the interval $[0,1]$. Therefore, $V$ is the set of continuous functions on the interval $[0,1]$ vanishing at the endpoints. For example, the functions $|x-\frac{1}{2}|-\frac{1}{2}$ and $\sin(\pi x)$ both belong to $V$.
More generally, the notation $C^n(A)$ repr... |
Context free grammar for language { {a,b}*: where the number of a's = the number of b's}
My professor said that converting a language to a CFG is more of an art than anything else. I looked at this problem and didn't even know how to get started, or how I would reason my way to the solution (which I understand how it ... | For the language given, you need the number of $a$'s to match up with the number of $b$'s. Notice that the strategy used to find a CFG for the language is to make sure that whenever we introduce an $a$, that we also introduce a $b$ at the same time. By doing this, we make sure that the number of $a$'s and $b$'s are equ... |
Summing sines of different frequencies Is there a general formula for solving the following equation:
$$A \sin(Bt+C) + D \sin(Et+F) = G \sin(Ht+I)$$
All constants on the left side of the equation are known (t is a variable). Is there a formula for calculating G, H and I? Is this even solvable in general?
I searched th... | If you mean $G$, $H$, $I$ to be constant (i.e. independent of $t$), then the answer is No. The sum of sinusoids of different frequencies is never a sinusoid.
|
sphere bundles isomorphic to vector bundle. Bredon claims that sphere bundles in certain cases are isomorphic to vector bundles. For example he says just replace $S^{n-1}$ with $R^n$. But for example the circle and the plane are not even isomorphic. How can we talk about bundle isomorphism when even the fibers are not ... | Apparently, in this context you have a sphere bundle say $M$ with the orthogonal group as structure group. This is a very special case of a sphere bundle. This means that you have a family of trivialization $\phi _i$ of your bundel $F$ over open sets $U_i$, say $\phi _i : U_i\times S \to M$ such that the change of char... |
Find the values of x for which the series $\sum_{n=0}^{\infty} \frac{(x-1)^n}{(-3)^n}$ Find the values of x for which the series
$$\sum_{n=0}^{\infty} \frac{(x-1)^n}{(-3)^n}$$
converge?
Not really sure how to properly answer this question considering its edge terms. Here goes my attempt: $$\sum_{n=0}^{\infty} \left(\f... | You close by saying you don't know if it converges when $x = 4$ or $x = -2$.
At these values, note that $x-1 = 3$ or $-3$, respectively, so that in the former case you would be considering the infinite series $1-1+1-1+\cdots$ and in the latter you have $1+1+1+\cdots$
Each of these series diverges, as the $n^{\text{th}}... |
Find the particular solution that satisfies the initial condition This is how I am going about it
$yy'-e^{x}=0$ ; $y(0)= 4$
I put it in standard form
$\frac{dy}{dx}-\frac{e^{x}}y$=0
$P(x)=e^{x}$
$Q(x)=0$
$I(x) = e^{\int e^{x}}$= ?
I'm not sure if I am doing it correctly but if so, what would I(x) come out to be?
| $$\frac{ydy}{dx} - \exp^x =0$$
$$\frac{dy}{dx} = \frac{\exp^x}{y}$$
$$\int(y)dy = \int(\exp^x)dx$$
$$\frac{y^{2}}{2} = \exp^x + c $$
$$y^2 = 2 \exp^x + c_{}1$$ where c_{1} = 2c
$$y = \surd (2(\exp^x) + c_{1})$$
$$ now y(0) = 4 $$
$$ c_{1} = 14 $$
$$so the answer is y = \surd(2(\exp^x) + 14) $$
|
Prove there exists $t>0$ such that $\cos(t) < 0$. I want to prove that there exists a positive real number $t$ such that $\cos(t)$ is negative.
Here's what I know
$$\cos(x) := \sum_{n=0}^\infty{x^{2n}(-1)^n\over(2n)!}, \;\;(x\in\mathbb R)$$ $${d\over dx}\cos(x) = -\sin(x)$$ $$\cos\left({\pi\over2}\right) = 0, \;\; \co... | Perhaps a more formal way is to invoke the mean value theorem which says that
$$\cos \left( \frac{\pi}{2}+\epsilon \right)-\cos \left( \frac{\pi}{2} \right)=-\sin \left( c \right) \epsilon $$
for some $c \in \left( \frac{\pi}{2},\frac{\pi}{2}+\epsilon \right)$. Using the continuity of $\sin$, you can take $\epsilon$ sm... |
How do we get the final formula of the Bernoulli number? I was trying to understand Bernoulli numbers. When I googled, I found this link.
It starts by saying that, The Bernoulli numbers are defined via the coefficients of the power series expansion of
$\frac{t}{e^{t}-1}$,
then they write the expansion,for $m \geq 0$.... | With recurrences of this type I like the representation in a matrix format. Note, that the binomial-coefficients in your recursive formula occur in the manner of the lower triangular Pascal-matrix ( = "P") and that we need a modified/trimmed version (= "Q" ) of it.
The following is the multiplication-scheme fo... |
Properties of gamma function What is the simplest way to prove the two following properties of $
\mathit{\Gamma}{\mathrm{(}}{z}{\mathrm{)}}
$
a)
$
\mathit{\Gamma}{\mathrm{(}}\frac{3}{2}{\mathrm{)}}\mathrm{{=}}\frac{1}{2}\mathit{\Gamma}{\mathrm{(}}\frac{1}{2}{\mathrm{)}}\mathrm{{=}}\frac{\sqrt{\mathit{\pi}}}{2}
$
b) $
... | For the first property we are looking for :
$\Gamma (z+1)=z\Gamma(z)$ which was introduced by Gauss and is sometimes called $\Pi (z)$.
This can be derived from the integral representation of Gamma.
The second property is called Euler' reflection formula. The proof is available here :
https://proofwiki.org/wiki/Euler's... |
When two unbiased dice are rolled one by one, what is the probability that either the first one is $2$ or the sum of the two is less than $5$?
When two unbiased dice are rolled one by one, what is the probability that either the first one is $2$ or the sum of the two is less than $5$?
a) $\dfrac 16$
b) $\dfrac 29$
c) ... | You need to calculate the probability that either the first die is $2$, OR the sum is less than $5$. In math, "OR" always includes the possibility that both things occur (i.e. "either A or B" includes when A and B both happen).
However, you added the probability that the first die was $2$ ($6/36$) and the probability ... |
Calculate the volume of the solid bounded laterally. How do i find the volume bounded below by the plane $xy$ and bounded above by $x^2+y^2+4z^2=16$ and laterally by the cylinder $x^2+y^2-4y=0$.
Since when i change to polar coordinates $x^2+y^2-4y=0$. is equal to $4sin(\theta)$.
And for the limits. $z=\frac{\sqrt{16-x... | You have some mistake in the limits of integration. Using the symmetry of the solid around the $y-z$ plane ( see the figure), we can take for $\theta$ the values between $0$ and $\frac{\pi}{2}$ and duplicate the integral, so the limits becomes:
$$
0<\theta<\frac{\pi}{2} \qquad 0<r<4\sin \theta \qquad 0<z< \frac{1}{... |
Duality between universal enveloping algebra and algebras of functions There is a well known duality (of Hopf algebras) between universal enveloping algebra $U(\mathfrak{g})$ of a complex Lie algebra $\mathfrak{g}$ of a compact group $G$ and the algebra of continuous functions $C(G)$.
My question is, is there in the... | I would suggest Hochschild's "Basic Theory of Algebraic Groups and Lie Algebras". It is about affine algebraic groups in general and not just compact Lie groups, but I think it is worthy to have a look there.
|
Entire function bounded on every horizontal and vertical line , then is it bounded on every horizontal and vertical strip? Let $f:\mathbb C \to \mathbb C$ be an entire function such that $f$ is bounded on every horizontal and every vertical line , then is it true that $f$ is bounded on any set of the form $V_{[a,b]}:=\... | This is not true. Let $G\subset \mathbb{C}$ be the set $\{x+iy:|x|<\pi/2, y> -1, |y-\tan x|<1\}$. This is a connected open set that does not contain any line, or even a half-line. Let $E=\mathbb{C}\setminus G$. The function $f(z)=1/z$ is holomorphic on $E$. By Arakelyan's approximation theorem there exists an entire fu... |
$n^{n-1}-1$ is a multiple of $k$
Find the number of integers $k$ with $2 \leq k \leq 1000$ satisfying the following property:
*
*For every positive integer $n$ relatively prime to $k$, $n^{n-1}-1$ is a multiple of $k$.
Let $k = 2^{\alpha_1}3^{\alpha_2} \cdots p_n^{\alpha_n}$ be the prime decomposition of $k$. Then... | If $\gcd(n,k)=1$ then $\gcd(n+k,k)=1$. So:
$$1\equiv (n+k)^{n+k-1}\equiv n^{n+k-1}=n^{n-1}n^k\pmod{k}$$
and hence $n^k\equiv 1\pmod{k}$ for all $n$ relatively prime to $k$.
Now if $0<n<k$ with $\gcd(n,k)=1$, then:
$$1\equiv (k-n)^{k-n-1} \equiv (-1)^{k-n-1} n^kn^{-(n-1)}n^{-2}\pmod{k}$$
But $n^k\equiv n^{-(n-1)}\equiv ... |
Variation of Weierstrass Approximation Theorem Let $f:[-1,1] \to \mathbb R$ be a continuous even function. Show that for any $\epsilon >0$ there exists an even polynomial $p(x)= \sum _{k=0} ^{n} a_k x^{2k}$ such that $|f(x)-p(x)|<\epsilon$ for any $x \in [-1,1]$. Show a similar result for a continuous odd function.
I ... | The polynomial furnished by Weierstrass Approximation Theorem doesn't need to be even/odd in this case. But you can decompose $p$ in its even and odd parts $p=p_e+p_o$ where $p_e(x)=\frac{p(x)+p(-x)}{2}$ and $p_o(x)=\frac{p(x)-p(-x)}{2}$. Then $p_e$ will still be a good enough approximation to $f$ since $p_o$ will be s... |
Proof by Induction: $n! > 2^{n+1}$ for all integers $n \geq 5.$ I have to answer this question for my math class and am having a little trouble with it.
Use mathematical induction to prove that $n! > 2^{n+1}$ for all integers $n \geq 5.$
For the basis step: $(n = 5)$
$5! = 120$
$2^{5+1} = 2^6 = 64$
So $120 > 64$, whi... | $(n+1)!=(n+1)\times n!>(n+1)2^{n+1}>2\times2^{n+1}=2^{n+2}$
Make sure you understand all the steps and ask if you got trouble.
|
Euclid 1999 Question 4(a) - Circle Tangent Intersection Below is a question and the intended solution to a math contest problem.
I understand that if for both circles, if you assume that a circle's centre, the two points on the circumference that touch a tangent line each, and the intersection of the tangent lines; th... | It is given that the two tangents form a 90 degree angle with each other. it is also known that the angle formed by the tangent to a circle and the radius to that point is 90 degrees as well. Since the figure emerging is a quadrilateral, the sum of the interior angles is 360 degrees, so the last remaining angle formd b... |
Roots of Complex Polynomials
Find all $x \in \mathbb{C}$ satisfying $(x - \sqrt{3} + 2i)^3 - 8i = 0$
I was able to find one value, $x = \sqrt{3} - 4i$.
I can also see that $x = -i$ also works although I am not able to devise a formal way for ending up with this value.
I am also unsure of how to find other values of $... | Hint: Divide your polynomial by $(x-(\sqrt{3} - 4i))$. You will get a quadratic polynomial. You found that $x = -i$ is also a root (great work!) so you should be to divide again by $(x+i)$.
|
If $A$ & $B$ are $4\times 4$ matrices with $\det(A)=-5$ & $\det(B)=10$ then evaluate... If $A$ & $B$ are $4\times 4$ matrices with $\det(A)=-5 $ & $\det(B)=10$ then evaluate...
a) $\det\left(A+\operatorname{adj}\left(A^{-1}\right)\right)$
b) $\det(A+B)$
Yes, those are meant to be addition signs. I wouldn't be asking if... | Answer to a):
It holds that:
$$ A \cdot \operatorname{adj(A)} = \det (A) \cdot I .$$
Thus, we have:
$$ A^{-1}\cdot \operatorname{adj}\left(A^{-1}\right) = \det \left(A^{-1}\right) \cdot I.$$
However, $\det\left(A^{-1}\right) = -\frac{1}{5}$ and you need to apply the identity $\det(\lambda A) = \lambda^n \det A,$ where ... |
Show $\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$ I know there are various methods showing that $\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$, but I want to know how to derive it from letting $t\rightarrow 0^{+}$ for the following identity:
$$\sum_{n=-\infty}^{\infty}\frac{1}{t^2+n^2}=\frac{\pi}{t}\frac{1+e^... | Note that we have
$$\sum_{n=-\infty}^\infty \frac{1}{t^2+n^2}=\frac1{t^2}+2\sum_{n=1}^\infty\frac{1}{t^2+n^2}$$
Therefore, using $\sum_{n=-\infty}^\infty \frac{1}{t^2+n^2}=\frac\pi t \frac{1+e^{-2\pi t}}{1-e^{-2\pi t}}$, we find that
$$\sum_{n=1}^\infty\frac{1}{t^2+n^2}=\frac12\left(\frac\pi t \frac{1+e^{-2\pi t}}{1-e... |
Determine the mode of the gamma distribution with parameters $\alpha$ and $\beta$ How do you determine the mode of a gamma distribution with parameters $\alpha$ and $\beta$ ? Without looking on Wikipedia.
| Hint: you want to maximize $x^{\alpha-1} e^{-\beta x}$ over $x \in (0,\infty)$.
The derivative is $e^{-\beta x}[(\alpha-1)x^{\alpha-2} - \beta x^{\alpha-1}] = x^{\alpha-2} e^{-\beta x} (\alpha-1-\beta x)$, which is zero when $x= \frac{\alpha-1}{\beta}$ or $x=0$.
*
*If $\alpha \ge 1$, direct inspection shows that $x=... |
Expected absolute difference between two iid variables Suppose $X$ and $Y$ are iid random variables taking values in $[0,1]$, and let $\alpha > 0$. What is the maximum possible value of $\mathbb{E}|X-Y|^\alpha$?
I have already asked this question for $\alpha = 1$ here: one can show that $\mathbb{E}|X-Y| \leq 1/2$ by i... | This isn't a full solution, but it's too long for a comment.
For fixed $0<\alpha<1$ we can get an approximate solution by considering the problem discretized to distributions that only take on values of the form $\frac{k}{n}$ for some reasonably large $n$. Then the problem becomes equivalent to
$$\max_x x^T A x$$
wh... |
Plotkin bound for the minimal distance of linear code over $\mathbb{F_q}$ I want to understand the proof that the minimal distance $d_C$ of a linear $[n,k]$ code $C$ over $\mathbb{F_q}$ is less or equal to $\frac{nq^{k-1}(q-1)}{q^k-1}$. The proof I'm reading says that the sum of the weights of all the words in $C$ is n... | In a linear code over $\mathbb F_q$, in each coordinate position, either all the codewords have a $0$, or each element of $\mathbb F_q$ appears equally often. That is, if we form a $q^k\times n$ array in which the rows are the codewords, then in each column, we either have all zeroes, or each element of $\mathbb F_q$ a... |
Expansion of this expression Let $x$ be a real number in $\left[0,\frac{1}{2}\right].$ It is well known that
$$\frac{1}{1-x}=\sum_{n=0}^{+\infty} x^n.$$
What is the expansion or the series of the expression $(\frac{1}{1-x})^2$?
Many thanks.
| Or do by brute force multiplication and gathering terms:
$$\frac{1}{(1-x)^2}=$$
$$
(1+x+x^2+x^3+x^4+\ldots)(1+x+x^2+x^3+x^4+\ldots)\\
=(1+x+x^2+x^3+x^4+\ldots)\\
+x+x^2+x^3+x^4+\ldots\\
\quad +x^2+x^3+x^4+\ldots\\
\quad \quad +x^3+x^4+\ldots\\
=1+2x+3x^2+4x^3+\ldots
$$
|
Tangent to a fiber bundle I am trying to prove that the kernel of a push-forward is the fiber.
Let $π : E → M $ be a fiber be bundle with a fiber $F$ . What is the meaning of a tangent space to a bundle? Does it means that if we have a vector, $X$ tangent to curve $\lambda$, that curve must pass to all points of the fi... | The definition of the tangent space of the bundle $E$ is the same as the definition of the tangent space to any manifold. A tangent vector at a point $p \in E$ is just an equivalence class of curves $[\alpha]$ with $\alpha(0) = p$. Since $E$ is a fiber bundle, instead of considering all the curves you can consider curv... |
Improper integral $\int_a^bf(x)dx$ is convergent and $g(x)\ge0$. If $\int_a^bg(x)dx=0$, show that $\int_a^bf(x)g(x)dx=0.$ Assume $f(x)\in\mathrm{C}[a,b),\lim_{x\to b}f(x)=+\infty$, $\int_a^bf(x)dx$ is convergent. And the non-negative function $g(x)$ on the interval $[a,b]$ is Riemann-integrable. If $\int_a^bg(x)dx=0$, ... | Hint: $g(x)$ is non-negative on $[a,b]$ and it's integral is zero.
|
Is this normed linear space a Banach space? Let $E$ be a measurable set of finite measure and $1 < p_1 < p_2 < \infty$. Consider the linear space $L^{p_2} (E)$ normed by $||.||_{p_1}$ . Is this normed linear space a Banach space?
| Okay lets suppose that $(L^{p_2},\|.\|_{p_1})$ is a Banach space then the map
\begin{align*}
\Phi:(L^{p_2},\|.\|_{p_2}) &\to (L^{p_2},\|.\|_{p_1}), \\f &\mapsto f
\end{align*}
is not only continuous but also bicontinuous. This follows from the fact that $\|.\|_{p_1} \leq C \|.\|_{p_2}$ and the open map theorem. Now thi... |
Is exponentiation open? Already for $2\times 2$ matrices the exponential map is not open. However, the diagonalization trick does not work for algebras of functions. Hence the question
Is the map $f\mapsto \exp(f)$ open on the complex space $C[0,1]$?
| Yes. Suppose $f\in C[0,1]$ and let $$C=\inf\{|\exp(f(x))|:x\in[0,1]\}.$$ Given $g\in C[0,1]$, let $H:[0,1]\times[0,1]\to\mathbb{C}$ be the linear homotopy from $\exp(f)$ to $g$ (that is, $H(x,t)=t\exp(f(x))+(1-t)g(x)$). Note that if $\|g-\exp(f)\|<C$, then the image of $H$ is contained in $\mathbb{C}\setminus\{0\}$.... |
How to find the limit of this sequence $u_n$? (defined by recurrence) $\left(u_n\right)$ is a sequence defined by recurrence as follows:
$
\begin{cases}
u_1=\displaystyle\frac{8}{3}\\
u_{n+1}=\displaystyle\frac{16}{8-u_n}, \forall n\in \mathbb{N}
\end{cases}
$
The first part of this question is to show that $u_n<4, \fo... | At $n \rightarrow \infty$ , $u_n \approx u_{n+1}$
$$\lim_{n \to \infty} u_{n}=\lim_{n \to \infty}\displaystyle\frac{16}{8-u_n} \implies \lim_{n \to \infty}u_n(8-u_n)=16 \implies \lim_{n \to \infty}u_n=4$$
|
What is wrong with this way of solving trig equations? Let's suppose I have to find the values of $\theta$ and $\alpha$ that satisfy these equations:
*
*$\cos^3 \theta$ = $\cos \theta $
*$3\tan^3 \alpha = \tan \alpha$
on the interval $[0; 2 \pi]$.
If I try to solve, for instance, the first equation like this:
$$\co... | The equation $x^3=x$ has three solutions: you can write it as
$$
x^3-x=0
$$
so
$$
x(x-1)(x+1)=0
$$
and the roots are $0$, $1$ and $-1$.
You cannot “divide by $x$”, which is the mistake you make when you “divide by $\cos\theta$”.
Thus your equation becomes
$$
\cos\theta=0
\quad\text{or}\quad
\cos\theta=1
\quad\text{or}\... |
If symmetric matrix $A\geq0$, $P>0$, does $APA\leq \lambda_{max}^2(A) P$ always hold? If symmetric matrix $A\geq0$, $P>0$, can $APA\leq \lambda_{max}^2(A) P$ always hold?
Notation:
$\lambda_{max}(A)$ means matrix $A$'s largest eigenvalue.
$A\geq0$ means matrix A is a positive semi-definite matrix.
$P>0$ means matri... | No, it is not always, the case. For example
$$A=\left(\begin{array}{cc}1&1\\1&1\end{array}\right), P=\left(\begin{array}{cc}100&0\\0&1\end{array}\right)$$
Then
$$APA = \left(\begin{array}{cc}101&101\\101&101\end{array}\right)\not\le 4P$$
|
Sum of Gaussian Curves I have two Gaussian curves, and I would like to sum them to have a curve with two bells, to fit some bimodal histogram.
Is doing N(m1 + m2; sig1 + sig2) the good way or should I do something else ?
For instance, I would like to obtain something like the green curve :
Gaussian curves
Thanks for t... | $N(\mu_1+\mu_2,\sigma_1^2+\sigma_2^2)$ corresponds to the curve $\displaystyle x \mapsto \frac 1 {\sqrt{2\pi}\sqrt{\sigma_1^2+\sigma_2^2}} e^{-(x-(\mu_1+\mu_2))^2/(2(\sigma_1^2+\sigma_2^2))},$ and that has just one "bell", centered at $\mu_1+\mu_2.$
What you need is
$$
w_1 \frac 1 {\sigma_1\sqrt{2\pi}} e^{-(x-\mu_1)^2/... |
Two-sided ideal $I$ in exterior algebra $T(V)/I$. I have a confusion regarding two definitions of the two-sided ideal in exterior algebra.
Def 1)
In one definition, the exterior algebra $\Lambda(V)$ is defined as $T(V)/I$, where $I$ is the two-sided ideal generated by the graded commutators $$[a,b]=ab-(-1)^{|a||b|}ba$$... | Since for all $a,b\in V$ we have $$(a+b)\otimes(a+b)=a\otimes a+a\otimes b+b\otimes a+b\otimes b\in J$$ and since also $(a+b)\otimes(a+b),a\otimes a,b\otimes b\in J$, we deduce that all $a\otimes b+b\otimes a\in J$, so that $I\subset J$ and we have a surjective algebra map $f:T(V)/I\to T(V)/J=\Lambda V $.
Over a... |
Range of gradient map for coercive function I am given a differentiable (and therefore continuous) function $f: \mathbf{E} \to \mathbb{R}$ which satisfies the following growth condition:
$$
\lim_{||x|| \to \infty} \frac{f(x)}{||x||} \to +\infty
$$
Just for the sake of notation, $\mathbf{E}$ is some arbitrary Euclidean ... | The assumption
$$\lim_{\|x\| \to \infty} \frac{f(x)}{\|x\|} \to +\infty$$
implies that the function $g(x) = f(x) - a^T x$ satisfies
$$\lim_{\|x\| \to \infty} \frac{g(x)}{\|x\|} \to +\infty$$
as the contribution of $a^T x/\|x\|$ is bounded.
Hence $g(x)\to +\infty$ as $\|x\|\to \infty$. A continuous function with this... |
Zeros of complex function Consider the function
$f(z)=e^{z}+\varepsilon_1e^{\varepsilon_1 z}+\varepsilon_2e^{\varepsilon_2 z}$ of a complex variable $z=x+i y$, where
$\varepsilon_1=-\frac{1}{2}+i\frac{\sqrt{3}}{2}$, $\varepsilon_2=-\frac{1}{2}-i\frac{\sqrt{3}}{2}$.
Numerical calculations show that all zeros of the func... | Since $f(\varepsilon_1 z) = \varepsilon_2f(z)$, it is enough to study $f$ on a "third" of the complex plane, for example the infinite cone centered on the negative real axis with an angle of $2\pi/3$.
There, $|\exp(z)|$ will quickly get negligible compared to the other two exponentials :
$f(z) = \exp(z) + \exp(2i\pi/3+... |
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