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Start with $i=\sqrt{-1}$.
This will be $a_1$.
$a_2$ will be $i^i$.
$a_3$ will be $i^{i^{i}}$.
$\vdots$
etc.
In Knuth up-arrow notation:
$$a_n=i\uparrow\uparrow n$$
And, amazingly, you can evaluate $\lim_{n\to\infty}a_n=\lim_{n\to\infty}i\uparrow\uparrow n=e^{-W(-\ln(i))}\approx0.4383+0.3606i$.
You can check this, it do... |
${A * B ^X = C * D ^ X}$
The idea is to find in how much time (X) a small (A) investment with a good tax (B) beats a big investment (C) with a bad tax (D). All values are nonzero and positive.
|
Suppose $f(a)=0$ for some $|a|<1$. If $f$ is analytic in $D$ containing the unit disk and $|f(z)|<M$ finite for all $z:|z|=1$, show for all $z:|z|<1$, that
$$
|f(z)| \leq M \left|\frac{z-a}{\bar a z - 1}\right|.
$$
Attempt
Since $f(a)=0$,
$$
\frac{1}{2\pi i} \int_C \frac{f(z) dz}{z} = 0.
$$
So
$$
\left| \int_C \fra... |
let $T: V \rightarrow V$, if V is finite dimensional, and V = null(T) + range(T) then V is the direct sum of null(T) and range(T).
My attempt at the proof:
let $x_1 ... x_n$ be the basis for V. Then we can write out, by the rank nullity theorem: $x_1 ... x_m$ as a basis for null(T) and $T(x_{x+1}) ... T(x_n)$ as a basi... |
I need to prove the following:
$V(x) = x + G(x)/g(x)$ is an increasing function where $G(x)$ is a CDF and $g(x)$ is the corresponding pdf.
When I take the derivative, I get
$$1 + g(x)^2/g(x)^2 - G(x)g'(x)/g(x)^2=2 - G(x)g'(x)/g(x)^2$$
How can I show that $G(x)g'(x)/g(x)^2$ is less than $2$?
Thanks
|
$$\int\frac{dx}{(x^{2}-36)^{3/2}}$$
My attempt:
the factor in the denominator implies
$$x^{2}-36=x^{2}-6^{2}$$
substituting $x=6\sec\theta$, noting that $dx=6\tan\theta \sec\theta$
$$x^{2}-6^{2}=6^{2}\sec^{2}\theta-6^{2}=6^{2}\tan^{2}\theta$$
$$\int\frac{dx}{(x^{2}-36)^{3/2}}=\int\frac{6\tan\theta \sec\theta}{36\tan^{... |
Please, help me with this. Let $(X,<,>)$, $u,v, w \in X-\{0\} $. Find the solution of the equation:
$$x+<x,u>w=v$$.
Would it be helpful to note by $\lambda=<x,u>$?the equation would become: $x+\lambda w=v$, $\lambda w=v-x$.And I dont know to continue from here..some suggestions, please?
|
As I specified in the other question I asked, analysis is not my field, so I'm sorry if my question is trivial.
Riesz representation theorem states the following:
Given $X$ a locally compact hausdorff space and $\lambda$ a positive, linear and continuous functional on $C_c(X)$, there exists a unique Borel measure $\sig... |
How can I prove that $2p+1$ cannot be a square number if $p$ is prime? Is a contradiction proof enough where I assume true then show it as false eventually?
|
I have this random sum, $Z=\sum_{k=1}^{N}Y_{k}$ where $(Y_1,Y_2,\dots,Y_k,\dots)$ are independent and exponentially distributed random variables (mean is $\mu$). We've defined $N=M+1$ where $M$ is poisson distributed (intensity is $\lambda$).
I need to find the first and second moments of this random sum.
I have found ... |
Find the 233rd subset of [12] with five elements in reverse lexicographic
order.
I am a little confused with the difference between normal and reverse lexicographic order. I think I understand that in lexicographic order. The first few subsets would look like this: {1,2,3,4,5},{1,2,3,4,6},...,{1,2,3,4,12},{1,2,3,5,6}, ... |
If $g(x):= \sqrt x $ for $x \in [0,1]$, show that there does not exist a constant $K$ such that $|g(x)| \leq K|x|$ $ \forall x \in [0,1]$
Conclude that the uniformly continuous function $g$ is not a Lipschitz function on interval $[0,1]$.
Necessary definitions:
Let $A \subseteq \Bbb R$. A function $f: A \to \Bbb R$ is... |
In one of the papers I was reading for my masters thesis I came across a theorem with no references.
Theorem:
If $(X_t)$ is an $(\mathcal{F}_t)$ martingale then there exists a $(\mathcal{F}^L_t)$ martingale $\tilde{X}_t$ such that
$$X_{t \wedge L}=\tilde{X}_t+\int_0^{t \wedge L} \frac{d \langle X, M^L \rangle_s}{Z^L_{... |
It is well known that a lower bound on Ricci curvature gives an upper bound on the volume of balls. What are conditions that gives a lower bound on the volume of balls? It is reasonable to think that an upper bound on curvature would give some kind of lower bound on the volume of balls, is this not true? If not, what a... |
Let $M$ and $N$ be finite rank free modules over a ring with invariant basis number such that $N \subseteq M$ is a submodule of $M$. Suppose $M$ and $N$ have the same rank $n$. Is it possible for
$$M=N \oplus P$$
for some nonzero module $P$? Would this contradict the assumption that the rank of $M$ is $n$?
|
How can I find the integer solutions of a polynomial $f(x,y)$ such that
$a<|x|,|y|<A$
using Mathematica?
|
Take a random variable $Y$. You're given $f_Y(x)=xe^{-x}$ when $x>0$. Given $Y$, a new random variable, $M$, is uniform over the interval $(0,Y)$.
I need to calculate the marginal density for $Y$, the conditional density for $M$ given $Y$, and I must show that $M$ and $(M-Y)$ are independent, then proceed to find their... |
In yesterday's Iowa Caucus, Hillary Clinton beat Bernie Sanders in six out of six tied counties by a coin-toss*. I believe we would have heard the uproar about it by now if this was somehow rigged in her favor, but I wanted to calculate the odds of this happening, assuming she really was that lucky, and assuming she ri... |
Find
$ \int_C \frac{dz}{z} $ from $1 - 5i$ to $5 + 6i$
I want to use
$$\int_C f(Z) dz = \int_a^b f[z(t)]z'(t)dt$$
My guess is that I need to find $z(t)$ such that
$$z(a) = 1 - 5i$$
$$z(b) = 5 + 6i$$
But how?
|
Hi I am having problem with proving $x_{n-1}$ converges to $x$ when $x_n$ converges to $x$. Intuitively, it seems obvious. But when it comes to proving it by using rigorous mathematical definition of limit, I get
lost... Need your help!!
|
I have started a middle/highschool level geometry text. The text is warming me up for proofs by going through basic logic, which was introduced alongside Euler diagrams.
I am given the following problem:
"Each of the lettered statements below is followed by some other statements. Identify the relation of each of them... |
Can you help me to find pints of intersection of given system of equesions? Can't do it by myself.
$xy + x - 2y + 3 = 0$
$x^2 + 4y^2 - 9 = 0$
|
Let $x,y$ be two variables. Consider the following system:
$$ \begin{cases} x=a_0+b (1+y^2/x^2)^{-1/2} \\ y=a_1+b(1+x^2/y^2)^{-1/2}, \end{cases}$$
where $a_0,a_1,b$ are parameters.
I can solve this using numerical solvers, but it's very inefficient. The two equations appear very symmetrical and somewhat 'simple' in for... |
Let $\mathbb{R}/{\sim}$ be the quotient space given by the equivalence relation $a \sim b$ if $a$ and $b$ are rational. I am trying to understand general properties of the quotient topology and this example seems worth fleshing out in full. It's also a very strange example to me so I'd appreciate feedback on what I've ... |
I am currently computing the row sums of a reduced rank factored matrix by reconstructing a row subset of the original (approximated) matrix.
The matrix was factored using SVD: A -> U, S, V -> U, SxV = W So the reconstruction is SxW ~ A (using the desired rows in S).
How can I do this without having to reconstruct SxW,... |
I am trying to show that the following function is injective in some neighborhood of $(0, 0)$: $f:\mathbb R^2 \rightarrow \mathbb R^2$ given by
$$f(x, y)=(\sin(x^3)\cosh(y), \cos(x^3)\sinh(y))$$
I tried to use the inverse function theorem but $df(0, 0)$ is not invertible.
|
If Legendre's conjecture is true, couldn't we easily obtain $\sqrt{p_{n+1}}-\sqrt{p_{n}}<1$ where $p_{n}$ is the $n$th prime?
$$p_{n+1}<(\lfloor \sqrt{p_{n}} \rfloor + 1)^{2}<( \sqrt{p_{n}}+ 1)^{2}$$
$$\sqrt{p_{n+1}}<1+\sqrt{p_{n}}$$
$$\sqrt{p_{n+1}}-\sqrt{p_{n}}<1$$
which is Andrica's conjecture.
|
The vertex reconstruction conjecture states that a graph on n>2 vertices can be discovered from only knowing its proper induced subgraphs.
Kelly proved this for trees in 1961. I saw his proof and I was asking the question: is his original proof still the only or simplest out there?
|
Let's assume we the average income for a journalist is $10000$ (in the whole country).
In a certain state X the average income for this profession is $10700$, therefore 7% higher. Also, the average income (in the whole country) is $11500$ (15% higher), when working for a big company.
Can I say something about the aver... |
So the question says: consider the subset $W$ of $\mathbb{R}^3$ consisting of all vectors
$$
\begin{bmatrix}
x \\ y \\ z
\end{bmatrix}
\qquad \text{such that } x+y+z \geq -1
$$
Select all statements that are correct:
A. There exists a $3 \times 3$ matrix $B$ whose image is $W$.
B. $W$ is closed under addition.
C. $W$ c... |
“From data on global average temperatures over the last $100$ years, it turns out that of the $10$ years with the highest temperatures, $9$ occurred in the last $11$ years.” Compute the probability for this to happen, under the assumption that these highest temperatures are distributed randomly over the $100$ years.
No... |
I was thinking and found the following question :
Let $f:\mathbb{R} \to \mathbb{R}$ a differentiable function and consider the restrictions $f|_\mathbb{N}$ and $f'_\mathbb{N}$
i) Which functions satisfies $f(\mathbb{N}) \subseteq \mathbb{N}$?
ii) Which functions satisfies $f'(\mathbb{N}) \subseteq \mathbb{N}$?
Example... |
We denote the composite of relation R and relation S by $S \circ R$.
Since the order matters, meaning composite of R and S is not composite of S and R. I am trying to understand why the denotation of composite of R and S puts S before R, which is unintuitive for me.
Can anyone help me understand this? Thanks
|
The problem is to evaluate the improper integral $I = \int_0^{\infty} \frac{\sin xt \sin yt \cos zt}{t^2} dt$.
This can be written as $\int_0^{\infty} dt \int_0^y \frac{\sin xt \cos st \cos zt}{t} ds$, noting that $\int_0^y \cos(st) ds = \frac{\sin yt}{t}$.
I want to interchange the order of the two integrals for $I$ s... |
Let $p: \mathbb{R} \rightarrow \mathbb{R^n}$ be a differentiable function. Define $h:\mathbb{R^2} \rightarrow \mathbb{R^n}$ by:
$h(s,t) :=$ $\left\{\begin{matrix}
\frac{p(s)-p(t)}{s-t}& \textrm{if } s \neq t\\
p'(s) & \textrm{if } s=t
\end{matrix}\right.$
Prove that if $p'$ is continuous at $t \in \mathbb{R}$, then $... |
Prove that splitting field of a polynomial $f \in \mathbb{k}[x]$ ($f = 0$ solvable by radicals) has a solvable group of automorphisms $Aut_\mathbb{k}(\mathbb{F})$.
I have just started learning Galois theory, any advice will be highly appreciated. Thanks in advance.
|
I am curious to know how to recover the differential equation that goes with a phase portrait. I have seen the following posts but the first one was a $y'$ (and easy enough to "do in my head") and the second one lacks the figure.
As an example, I solved (in Mathematica) the non-linear pendulum equation and plotted the... |
I would like to resolve this exercise:
Let $W$ be a Brownian motion with $T_1=1 \text{ year}$ and $T_2=2 \text{ years}$.
I want to compute the probability that $W_{T_1}$ be positive and $W_{T_2}$ negative.
What I did:
$X=W_{T_1}$ and $ Y=W_{T_2}-W_{T_1} $ independent $\sim$ Gaussian
So our probability is:
$$P(X>0,X... |
Let $\gamma$ be a curve in $\mathbb{R}^2$ and $d:\mathbb{R^2}\times\mathbb{R}^2\to\mathbb{R}$ be the distance function between two points of $\gamma$. I'm trying to show that if the distance between two points $p_1,p_2$ is maximal, then the line spanned by connecting these points is perpendicular to the $\gamma'$ at $p... |
i have a problem in the partial derivatives of this function :
so consider this integral :
$u(x,t)=\int_0^t(\int_{\mathbb{R}^n}\phi_n(y,s)f(x-y,t-s)dy)ds$
can somone tell me why we have this equality :
$\frac{\partial u}{\partial t}(x,t)=\int_0^t(\int_{\mathbb{R}^n}\phi_n(y,s)\frac{\partial f}{\partial t}(x-y,t-s)dy)ds... |
Someone claimed that the following two expressions are equal. But after much manipulation, I cannot establish their equality. Is there in fact a typo somewhere?
$C(x_1^2+x_2^2)^{-(a+p)/2} x_1^p$
$C(x_1)^{-a}(1+x_2/x_1)^{-(a+p)/2}$
I did find that the first expression is equal to $C(x_1^2)^{-(a+p)/2}(1+x_2^2/x_1^2)^{-... |
Hi I dont have much experiance with spherical coordinates, but to me it seems as if the following problem requires it. The problem I am having is finding the limits of integration.
I want to solve $$\iiint_{R} (x^2+y^2)dV$$, where R is the region that lies above the cone $z=c\sqrt{x^2+y^2}$ , and inside the sphere $x^2... |
If $a_1, a_2, \dots \in \mathbb{R}$ and $f$ is a $C^1$ function in an open set that contains $[1, \infty)$, prove that
$\sum_{1\le k \le x}a_kf(k)=f(x)\sum_{1\le k \le x}a_k-\int_1^xf'(t)\left( \sum_{1\le k \le t} a_k\right)\,dt,\,\,\,\,\,x\ge 1$.
Does my proof work?
Let $a_1, a_2, \dots \in \mathbb{R}$ and $f$ be a $... |
Here is my work so far, I am missing something quite obvious but I can't seem to link it together:
Proof.
Let $n$ be an integer.
Suppose $n$ is odd.
This means that there is an integer $k$ such that $n = 2k + 1$.
So $-n = -2k-1$ ...
Thus, there is an integer $m$ such that $-n = 2m + 1$; namely, $m = ?$
Therefore, $-n$ ... |
Let $\phi : F \rightarrow F_1$ be an isomophism of fields and $f(x)
> \in F[x]$.
Let $\Phi : F[x] \rightarrow F_1[x]$ be the unique ring isomorphism
which extends $\phi$ and maps $x$ to $x$.
Let $f_1(x) = \Phi(f(x))$ and $E/F$, $E_1/F_1$ be the splitting fields
of $f(x)$ and $f_1(x)$ respectively.
Then there exi... |
I am supposed to find the radius of convergence for the complex power series $$\sum_{n=0}^{\infty}(-1)^n2^nz^{2n+2}$$
I know that the radius of convergence is calculated by $\frac{1}{R}=\limsup_{n\to\infty}|a_n|^{1/n}$, which in this case means
$$\frac{1}{R}=\limsup_{n\to\infty}|(-2)^n|^{1/n}=2$$
$$\Rightarrow R=\frac{... |
I was skimming some of Lurie's work and he mentioned "sheaves of spaces."
Is this a functor from Top^op -> Top?
Or is it meant to just mean a sheaf on a space?
|
How come $4x(1-x)$ is onto in $[0,1]$ but $2x(1-x)$ is not? Isn't it true that for any $y$ in the range interval, there exist two $x$ such that $f(x)=y$?
|
The title pretty much says it, $p$ is some real $>0$, $a$ and $b$ both real, $f$ bounded and $\int d\alpha$ is the Riemann-Stieltjes integral.
It is trivially true if $f>0$ since then $f^p = e^{p \log f}$ over the whole interval but I'm not sure how to start if we allow points with $f(x)=0$. In this case $f^p$ is not ... |
Find $\sqrt{(1 + \sqrt{3i})}$. I am trying to use the fact that $\sqrt{(1 + \sqrt{3i})} = re^{i \theta} = r(cos \theta + i sin\theta)$ but I am having trouble figuring out where to go from here. Any help would be appreciated.
|
Hi I'm quite new to induction so I don't really know how I should tackle this problem.
I took out a calculator and checked the results
2,16,167,1667 I sort of see a pattern but how would I start the induction?
|
In the title, $v$ is an integer such that $v>1$.
|
It is the problem 1.2.2 of Karlin's book Introduction to stochastic modeling:
Let $N$ cards carry distinct numbers $x_1, x_2, ..., x_n$. If two cards are drawn at random without replacement show that the correlation coefficient $\rho$ between the numbers appearing in the two cards is $-1/(N-1)$
What I did was define ... |
On a symmetrical random walk, I am trying to deduce
P[$M_{n}$ = k] = $(\frac{1}2)^n$ ${n \choose \frac{n+k}2}$
where n is the total number of steps and ${n \choose \frac{n+k}2}$ is the number of paths possible knowing the integers n,k.
we know that for each n and k, the # of steps up can be calculated using $\frac{n... |
I'm getting lost in notation here.
We have $f_X(x)=xe^{-x}$ and we have a new random variable, $Y$, which is uniform from $(0,X)$.
We want three things:
Marginal density for $Y$
Conditional density for $X$ given $Y$
Joint density function.
From the given information, the conditional density of $Y$ is $f_{Y|X=x}(y)=\f... |
When evaluating whether an improper integral is convergent or divergent, I'm sometimes unsure whether I simplified enough to be sure. For example, given
$$
\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \sec \theta \, \textrm{d}\theta = \lim_{b \to \frac{\pi}{2}^-} [\ln \left|\frac{1 + \sin b}{\cos b}\right|] + \ln |1 + \sqrt{2}... |
A famous paper by Leon Henkin ("Completeness in the theory of types") begins as follows:
"The first order functional calculus was proved complete by Gödel in 1930. Roughly speaking, this proof demonstrates that each formula of the calculus is a formal theorem which becomes a true sentence under every one of a certain ... |
Let $V$ be a vector space over a field $k$ and let $U,W$ be finite-dimensional subspaces of $V$. Prove that $U+W = \{u+w\mid u\in U, w\in W\}$ is a finite-dimensional subspace of $V$.
I know how to prove that $U\cap W$ is a subspace of $V$ but I'm having a hard time grasping how to prove $U+W$
This is what I was doing ... |
Let $\{a_n\}$ be a sequence of polynomials in $\mathbb{Q}[x,y]$ with $a_0=0,a_1=1$, and
$$a_n=xa_{n-1}+ya_{n-2}$$
The first few look like
$$a_3:y+x^2$$
$$a_4:2xy+x^3=x(2y+x^2)$$
$$a_5:y^2+3x^2y+x^4$$
$$a_6:3xy^2+4x^3y+x^5=x(y+x^2)(3y+x^2)$$
Note that $a_6=0$ has many obvious rational points, for example $(3,-9)$.
I co... |
Take $2^{1/n}$. Since $1/n$ can be simplified as $n^{-1}$, the original term can become $2^{n^{-1}}$. The exponents can then be multiplied to result in $2^{-n}$ which is $1/(2^n)$. However it is obvious that
$2^{1/n} \ne 1/(2^n)$
because the first item approaches 1 as $n \rightarrow \infty$ but the second item approach... |
I am trying to proof the theorem that states that:
"A map to a smaller dimensional space is not injective"
So I first suppose that $V$ and $W$ are finite-dimensional vector spaces such that $\dim V > \dim W$ and I let $T$ be the linear transformation from $V$ to $W$.
Given the above, you have that:
$\dim(\operatorname{... |
I already read this, and so wish to intuit 3 without relying on (only rearranging) the definition of Conditional Probability.
I modified the following's source for concision.
$1.$ Now look at $\Pr(A \cap B)$. We know that if $A$ has happened, then $A \cap B$ happens with probability $\Pr(B\mid A)$.
$2.$ If we do NOT k... |
Let $A$= $\begin{bmatrix}3 & 1 & -1\\0 & 4 &0\\6 &3&-2\end{bmatrix}$ and $x= \begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}$ and $b=\begin{bmatrix}0\\4\\1 \end{bmatrix}$
Does $Ax = b$ have a solution for every vector $b$ in $\mathbb{R}^3$
So I went about this first by doing the matrix multiplication and got
$A$= $\begin{b... |
$AB' + B'A + CDE + C'DE + EC'D$
Since $AB' = B'A$:
$AB' + CDE + C'DE + EC'D$
Since $C'DE = EC'D$:
$AB' + CDE + C'DE$Is this as far as it can be simplified according to commutative law?
|
A committee of 11 members is voting on a proposal. Each member casts a YES or NO vote. On a random voting basis, what is the probability that the proposal wins by a vote of 8 to 3?
|
I'm trying to work out an algorithm analysis problem, and I'm having some difficulty determining how a jump is made between two steps in the answer.
$$
\begin{align} &\frac{1}{2}\sum_{i=1}^{n-1}n^2+n-i^2-i \\ = {} &\frac{1}{2}\left((n-1)n^2 + (n-1)n - \left(\frac{n(n+1)(2n+1)}{6} - n^2\right) - \left(\frac{n(n+1)}... |
First of all, I am not a mathematician. I'm mainly asking the question to see if what I want to do is even possible via math -- and whether I then could computerize this math.
So you may throw this question out. If so, I understand.
But here is the question:
100 musicians participate in a chamber music workshop.
A mus... |
$1/x$ is an odd function, so it makes sense to me intuitively that the area would be $0$, and similarly I would expect that $\int_{-1}^{2}\frac{dx}{x} = \ln(2)$.
Proof Wiki seems to confirm my intuition, but with the exception of functions that don't have a primitive (i.e. integral?), which I guess this one doesn't, be... |
I am trying to pick two points from $[0, 1]$ at random and with uniform probability.
Let the result be the pair $(X, Y)$ in the square $[0, 1] \times [0, 1]$. Suppose that the distribution of $(X, Y)$ in the square is uniform, that is, the probability of $(X, Y)$ being in a particular region in the square is the area
o... |
It is easily shown that the forgetful functor $F: \mathbf{Man} \to \mathbf{Set}$ preserves limits ($F$ is representable), but does it preserve colimits?
It certainly preserves all examples of colimits that know of which exist in $\mathbf{Man}$, namely
coproducts (disjoint unions)
the glueing of open manifolds alon... |
Can you please let me know if my proof is reasonable?
Prove: If $A$ and $B$ are conneted in $\mathbb{R}^n$ and $A\cap B\neq \emptyset$, then $A\cup B$ is connected
Proof: Suppose that $A\cap B$ is not connected. Then there exist sets $U$ and $V$ such that $U\cup V = A\cap B$, $U=C\cap(A\cap B)$, $V=D\cap(A\cap B)$ for ... |
I'm completely stuck on this one. I only know that it converges thanks to Wolfram, but I don't know how to evaluate it.
$$\int_1^{\infty}\frac{33e^{-\sqrt{x}}}{\sqrt{x}}$$
Thank you for the help.
|
Find all incongruent solutions to $21x \equiv 14 \pmod{91}$.
I am able to work out the solution using Euclidean algorithm techniques, but the signs on the expression do not match up with the initial expression when I check my work. So by the linear congruence theorem, my solution has to satisfy: $$21x - 91y = 14$$ b... |
Can someone please point out where I am (If I am) going wrong during the solution process of the following question:
I have been presented with the following :
$$4sinh(2ln(2))-cosh(ln2)$$
and told by my tutor the solution is 10. however I cannot obtain this value, the steps I take are as follows :
$$(4*(e^{ln(2)^2} - e... |
How to use Rolle's theorem to verify the location of roots ?
$f(x)=x^3+4/x^2+7$ has exactly one zero in ($-\infty$,$0$)
I can do it without Rolle's theorem by finding the stationary point which is at about $1.2$ (taking derivative).
|
I already read this, and so wish to intuit 3 without relying on (only rearranging) the definition of Conditional Probability.
I pursue only intuition; do not answer with formal proofs.
Which region in my 2D Venn Diagram below matches $Pr(A|B)$?
On the left, the green = $\Pr(A\cap B)$. The right is herefrom, but fa... |
I'm trying to find the complexity of a program and reduced the question to the following one:
Let $g$ be a function from natural numbers (including $0$) to natural numbers. Assume that for every $n \geq 1 $,
$$\sum_{k=0}^{2n} g(k) \leq a\cdot g(2n-1)$$ and
$$\sum_{k=0}^{2n-1} g(k) \leq a\cdot g(2n-2)$$
and $0 < g(k... |
Given two-sided ideals $B$ and $C$ of a ring $A$,
(a) show that $BC \subseteq B \cap C$.
(b) If the ring $A$ is commutative and $B + C = A$, show that $BC = B \cap C$.
Here's what i have but I am unsure if it is correct.
(a) Since $A,B$ are both two-sided ideals, $aba' \in B$ and $aca' \in C$.
So, $BC = aba'(aca') = ... |
How can one make $66$ with only $1,1,1,1,1$? You cannot combine these two numbers to make a new number, such as this: $66=11 \times (1+1+1)!$.
This was inspired a game of dice that I used to play, where you were given $5$ numbers between $1$ and $6$ to make a two digit number between $11$ and $66$. My mother had once ... |
Can anybody please give a relationship among these objects.
Varieties, schemes, moduli spaces, stacks, algebraic spaces, groupoids.
I mean here, we can define an abstract Variety as a scheme with certain properties and so on, and a stack as a algebraic space.
I just wanna see how these related in general in a way simil... |
How do I formally show that the sequence $a_n = {{2^n}\over n}$ diverges using a $\delta$-$\epsilon$ argument?
|
Denoting the Cauchy-Schwarz inequality as Wikipedia does,
$$\vert\langle x,y \rangle \vert \leq \Vert x \Vert \cdot \Vert y \Vert$$
and noting that
$$\vert\langle x,y \rangle \vert = \Vert x\cdot y \Vert$$
Which gives the version of Cauchy-Schwarz that I'm used to seeing, my question is why does the Wikipedia page re... |
I'm confused by what Wolfram Alpha is doing with my function:
$$-4\sin{(2t - (\pi/2))}$$
on why the it gets replaced by
$$4\cos{(2t)}$$.
Is it equal? Link: See behavior here
|
If $X$ is a noncompact LCH space (locally compact, Hausdorff) then its one point compactification is $X^*=X\cup \{\infty\}$ with topology $\mathcal{T^*}$ given by $U \in \mathcal{T^*}$ iff either
a) $U \subset X$ is open, or
b) if $\infty \in U$ then $U^c \subset X$ is compact
What whould be the definition and the topo... |
Let $\mathcal{B}$ be the braid category, as in Categories for the Working Mathematician §XI.4 p.262 (objects are natural numbers and morphisms are the braids $n\to n$).
Then this can be given a natural braided monoidal category structure $(\mathcal{B},\otimes,\varnothing)$, where $\otimes$ corresponds to addition (on o... |
$X$ and $Y$ are iid r.v.s where $f(x)=\lambda e^{-\lambda x}$, $x>0$. We are given $Z=X/Y$ and asked to find the CDF and density for $Z$.
I tried doing this using the multivariate transformation formula, where
$f_{W_1,W_2}(w_1,w_2)=f_{X_1,X_2}(x_1,x_2)|J|$, but I ended up getting that $f_Z(z)=\frac{1}{(1+z)^2}$, which... |
I am a Calc 2 student and am having trouble seeing this limit, the assignment is to use the definition of the hyperbolic functions to find the limit.
$\lim_{x\to 0+} Coth(x)$
the answer is $+\infty$
If I work it out I get to this point: $lim_{x\ to 0+}$ $(e^(2x)+1)/(e^(2x) -1)$
Please explain it using the definition ... |
If $(u,v)$ is a point on $4x^2+a^2y^2=4a^2$,where $4<a^2<8$,that is farthest from $(0,-2)$ then $u+v$ is equal to?
My Approach:
I took a parametric point $(t,4-4t^2/a^2)$.And then tried to find the minima of the distance.But that is too lengthy method.Any other suggestions?
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Let $P(x,y)$ be a polynomial with integer coefficients that is constant neither in the horizontal nor vertical direction. Prove that $\sup_{-2\leq x,y\leq 2}|P(x,y)|\geq 4$. I suspect we might be able to show a stronger statement that the $\sup$ over pairs of integers in $[-2,2]$ is at least $4$, but I'm not sure.
|
I am attempting to prove that
$$\frac{1}{E'} - \frac{1}{E} = \frac{1}{m_e c^2} \cdot (1-\cos\theta)$$
can be derived from
$$E + m_ec^2 - E' = c^2(p^2 - 2pp'\cos\theta + p'^2) + m_e^2c^4 $$
where $p\cdot c = E$ and $p'\cdot c = E'$.
And have become found that
$$E - E' = \frac{1}{m_ec^2} \cdot (1-\cos\theta)$$
This l... |
What it says on the tin. Using the Borel $\sigma$-algebra on the reals instead of the Lebesgue $\sigma$-algebra has the advantage that it allows a broader class of measures, many of which are quite natural: For example the "uniform" measure on the Cantor set is defined on the Borel $\sigma$-algebra, but cannot be defin... |
Let $K$ a field and $L$ a subfield of $K$. Let the set $\overline{L}:= \{k
\in K: k$ is algebraic over $L$ $\}$ is another subfield of $K$. Show that
$\overline{\overline{L}}=\overline{L}$. [Hint : If $c_0+c_1\alpha+ \dots +
c_n\alpha^n=0$ where $\alpha \in K$ and $c_0,c_1, \dots, c_n \in
\overline{L}$, then con... |
If a subset $W$ of a vector space $V$ is a subspace of $V$, I want to show that $\operatorname{span}(W)=W$.
But is it possible to define $\operatorname{span}(W)$? $W$ can either be a finite or infinite set and according to what I know $\operatorname{span}(W)$ can only be defined on a finite set $W$.
Is there any proble... |
$$u(x,y)=\frac{x}{x^2+y^2}$$
I'm trying to figure out if this function is harmonic. I worked out all the algebra and my conclusion is no because the numerator of the final fraction is $2x^3+2xy^2$ which doesn't cancel to zero. I did this by finding the second derivatives of both $x$ and $y$ and adding them together. Do... |
Does the following limit exists
\begin{align}
\lim_{a \to \infty} \int \frac{e^{-(x-a)^2}}{\frac{1}{a^2}e^{-\frac{(x-a)^2}{2}}+e^{-\frac{x^2}{2}}} dx
\end{align}
Observe that if we do
\begin{align}
\lim_{a \to \infty} \int \frac{e^{-(x-a)^2}}{\frac{1}{a^2}e^{-\frac{(x-a)^2}{2}}+e^{-\frac{x^2}{2}}} dx \le \lim_{a \to ... |
To define the Cohomology (with values in $\mathbb{C}$) on a lie algebra $L$, we define a coboundary map $\delta:\Lambda^n(L)\to \Lambda^{n+1}(L)$. There is a general formula for the coboundary map but I'm only interested in one and two-forms. Say $\eta\in \Lambda^1(L)$ and $\omega\in \Lambda^2(L)$ and $X,Y,Z\in L$ are ... |
Wikipedia gives the equation for a sawtooth waveform composed as a sum of sines as:
$$
x_\mathrm{sawtooth}(t) = \frac{A}{2}-\frac {A}{\pi}\sum_{k=1}^{\infty}\frac {\sin (2\pi kft)}{k}
$$
Where $A$ is amplitude.
Is that correct? My reading of that (and graphs thereof) suggest that the initial term ($A/2-$) biases the ... |
Let V be the Volterra operator on $\mathscr{L^2(0,1)}$.$V(f)(x)=\int_{0}^{x}{f(y)dy}$.
Show that $C^*(V)$, the smallest C* algebra generated with V, is $\mathbb{C}+\mathscr{B_0(L^2(0,1))}$ where
$\mathbb{C}$ denotes the complex field and $ \mathscr{B_0(L^2(0,1))}$ denotes the algebra of compact operators on $\mathscr{L... |
Source: Example 1.11, p 26, *Introduction to Probability (1 Ed, 2002) by Bertsekas, Tsitsiklis.
Hereafter abbreviate graduate students to GS and undergraduate students to UG.
Example 1.11. A class consisting of 4 graduate and 12 undergraduate students
is randomly divided into 4 groups of 4. What is the probability ... |
I know that bases vectors must span and be linearly independent.
The (i) is not bases because the last vector contains $\pi$.
The (iii) is not bases because they are not linearly independent.
The (iv) is not bases because there are only 3 vectors here and bases for $\mathbb{R}^4$ must contain at least 4 vectors.
The (... |
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